LMFDB - Embedded newform 76.5.j.a.21.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,5,Mod(13,76)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("76.13");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	76.j (of order $18$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.85611719437$
Analytic rank:	$0$
Dimension:	$42$
Relative dimension:	$7$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			21.3
Character	$\chi$	$=$	76.21
Dual form			76.5.j.a.29.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.504186 + 0.600866i) q^{3} +(-8.75235 + 3.18559i) q^{5} +(-3.87876 - 6.71820i) q^{7} +(13.9587 + 79.1635i) q^{9} +O(q^{10})$	\(q+(-0.504186 + 0.600866i) q^{3} +(-8.75235 + 3.18559i) q^{5} +(-3.87876 - 6.71820i) q^{7} +(13.9587 + 79.1635i) q^{9} +(-94.7123 + 164.047i) q^{11} +(-152.490 - 181.730i) q^{13} +(2.49870 - 6.86512i) q^{15} +(-73.3453 + 415.962i) q^{17} +(171.374 + 317.729i) q^{19} +(5.99236 + 1.05661i) q^{21} +(-669.617 - 243.721i) q^{23} +(-412.322 + 345.979i) q^{25} +(-109.627 - 63.2931i) q^{27} +(1556.26 - 274.410i) q^{29} +(-389.047 + 224.616i) q^{31} +(-50.8173 - 139.619i) q^{33} +(55.3497 + 46.4439i) q^{35} -923.124i q^{37} +186.079 q^{39} +(1592.45 - 1897.81i) q^{41} +(374.881 - 136.446i) q^{43} +(-374.354 - 648.400i) q^{45} +(189.007 + 1071.91i) q^{47} +(1170.41 - 2027.21i) q^{49} +(-212.957 - 253.793i) q^{51} +(-293.848 + 807.340i) q^{53} +(306.370 - 1737.51i) q^{55} +(-277.317 - 57.2217i) q^{57} +(-2307.63 - 406.897i) q^{59} +(5176.31 + 1884.02i) q^{61} +(477.695 - 400.833i) q^{63} +(1913.56 + 1104.80i) q^{65} +(-6476.74 + 1142.02i) q^{67} +(484.055 - 279.469i) q^{69} +(877.159 + 2409.97i) q^{71} +(-3639.17 - 3053.62i) q^{73} -422.188i q^{75} +1469.46 q^{77} +(-7424.75 + 8848.48i) q^{79} +(-6025.19 + 2192.99i) q^{81} +(5155.43 + 8929.47i) q^{83} +(-683.142 - 3874.29i) q^{85} +(-619.759 + 1073.45i) q^{87} +(1402.20 + 1671.08i) q^{89} +(-629.430 + 1729.35i) q^{91} +(61.1878 - 347.013i) q^{93} +(-2512.09 - 2234.95i) q^{95} +(6417.59 + 1131.59i) q^{97} +(-14308.6 - 5207.89i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) $ 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9	$ 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9} - 45 q^{11} + 33 q^{13} - 393 q^{15} + 909 q^{17} + 1242 q^{19} + 1107 q^{21} - 360 q^{23} - 810 q^{25} - 7056 q^{27} - 2889 q^{29} + 2808 q^{31} + 10875 q^{33} + 6741 q^{35} - 3480 q^{39} - 3060 q^{41} - 8079 q^{43} - 4320 q^{45} - 2655 q^{47} - 474 q^{49} - 12222 q^{51} - 6705 q^{53} + 4623 q^{55} - 8022 q^{57} + 24309 q^{59} + 7104 q^{61} + 12063 q^{63} + 25245 q^{65} + 15573 q^{67} - 10881 q^{69} - 25506 q^{71} + 3036 q^{73} + 12924 q^{77} - 16839 q^{79} - 2208 q^{81} - 6363 q^{83} - 37890 q^{85} - 21924 q^{87} - 22644 q^{89} + 17418 q^{91} + 8184 q^{93} - 82413 q^{95} + 13383 q^{97} + 23565 q^{99}+O(q^{100}) $ 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9 - 45 * q^11 + 33 * q^13 - 393 * q^15 + 909 * q^17 + 1242 * q^19 + 1107 * q^21 - 360 * q^23 - 810 * q^25 - 7056 * q^27 - 2889 * q^29 + 2808 * q^31 + 10875 * q^33 + 6741 * q^35 - 3480 * q^39 - 3060 * q^41 - 8079 * q^43 - 4320 * q^45 - 2655 * q^47 - 474 * q^49 - 12222 * q^51 - 6705 * q^53 + 4623 * q^55 - 8022 * q^57 + 24309 * q^59 + 7104 * q^61 + 12063 * q^63 + 25245 * q^65 + 15573 * q^67 - 10881 * q^69 - 25506 * q^71 + 3036 * q^73 + 12924 * q^77 - 16839 * q^79 - 2208 * q^81 - 6363 * q^83 - 37890 * q^85 - 21924 * q^87 - 22644 * q^89 + 17418 * q^91 + 8184 * q^93 - 82413 * q^95 + 13383 * q^97 + 23565 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{1}{18}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.504186	+	0.600866i	−0.0560207	+	0.0667629i	−0.793327	−	0.608796i	$-0.791653\pi$
$3$	−0.504186	+	0.600866i	−0.0560207	+	0.0667629i	0.737306	+	0.675558i	$0.236097\pi$
$4$		0			0
$5$	−8.75235	+	3.18559i	−0.350094	+	0.127424i	−0.511081	−	0.859533i	$-0.670755\pi$
$5$	−8.75235	+	3.18559i	−0.350094	+	0.127424i	0.160987	+	0.986957i	$0.448532\pi$
$6$		0			0
$7$	−3.87876	−	6.71820i	−0.0791583	−	0.137106i	0.823729	−	0.566984i	$-0.191890\pi$
$7$	−3.87876	−	6.71820i	−0.0791583	−	0.137106i	−0.902887	+	0.429878i	$0.858557\pi$
$8$		0			0
$9$	13.9587	+	79.1635i	0.172329	+	0.977328i
$10$		0			0
$11$	−94.7123	+	164.047i	−0.782746	+	1.35576i	0.147590	+	0.989049i	$0.452849\pi$
$11$	−94.7123	+	164.047i	−0.782746	+	1.35576i	−0.930336	+	0.366708i	$0.880485\pi$
$12$		0			0
$13$	−152.490	−	181.730i	−0.902307	−	1.07533i	−0.996811	−	0.0798029i	$-0.974571\pi$
$13$	−152.490	−	181.730i	−0.902307	−	1.07533i	0.0945038	−	0.995524i	$-0.469874\pi$
$14$		0			0
$15$	2.49870	−	6.86512i	0.0111053	−	0.0305117i
$16$		0			0
$17$	−73.3453	+	415.962i	−0.253790	+	1.43931i	0.545370	+	0.838195i	$0.316389\pi$
$17$	−73.3453	+	415.962i	−0.253790	+	1.43931i	−0.799160	+	0.601118i	$0.794722\pi$
$18$		0			0
$19$	171.374	+	317.729i	0.474721	+	0.880136i
$19$	171.374	+	317.729i	0.474721	+	0.880136i
$20$		0			0
$21$	5.99236	+	1.05661i	0.0135881	+	0.00239595i
$22$		0			0
$23$	−669.617	−	243.721i	−1.26582	−	0.460720i	−0.380100	−	0.924945i	$-0.624110\pi$
$23$	−669.617	−	243.721i	−1.26582	−	0.460720i	−0.885717	+	0.464226i	$0.846333\pi$
$24$		0			0
$25$	−412.322	+	345.979i	−0.659715	+	0.553567i
$26$		0			0
$27$	−109.627	−	63.2931i	−0.150380	−	0.0868217i
$28$		0			0
$29$	1556.26	−	274.410i	1.85048	−	0.326290i	0.865765	−	0.500450i	$-0.166832\pi$
$29$	1556.26	−	274.410i	1.85048	−	0.326290i	0.984717	+	0.174160i	$0.0557211\pi$
$30$		0			0
$31$	−389.047	+	224.616i	−0.404835	+	0.233732i	−0.688568	−	0.725172i	$-0.741760\pi$
$31$	−389.047	+	224.616i	−0.404835	+	0.233732i	0.283733	+	0.958903i	$0.408427\pi$
$32$		0			0
$33$	−50.8173	−	139.619i	−0.0466642	−	0.128209i
$34$		0			0
$35$	55.3497	+	46.4439i	0.0451834	+	0.0379134i
$36$		0			0
$37$		−	923.124i		−	0.674305i	−0.941450	−	0.337153i	$-0.890536\pi$
$37$		−	923.124i		−	0.674305i	0.941450	−	0.337153i	$-0.109464\pi$
$38$		0			0
$39$	186.079			0.122340
$40$		0			0
$41$	1592.45	−	1897.81i	0.947322	−	1.12897i	−0.0441984	−	0.999023i	$-0.514073\pi$
$41$	1592.45	−	1897.81i	0.947322	−	1.12897i	0.991520	−	0.129952i	$-0.0414822\pi$
$42$		0			0
$43$	374.881	−	136.446i	0.202748	−	0.0737942i	−0.238650	−	0.971106i	$-0.576705\pi$
$43$	374.881	−	136.446i	0.202748	−	0.0737942i	0.441398	+	0.897311i	$0.354483\pi$
$44$		0			0
$45$	−374.354	−	648.400i	−0.184866	−	0.320198i
$46$		0			0
$47$	189.007	+	1071.91i	0.0855623	+	0.485248i	0.997234	+	0.0743281i	$0.0236812\pi$
$47$	189.007	+	1071.91i	0.0855623	+	0.485248i	−0.911672	+	0.410920i	$0.865208\pi$
$48$		0			0
$49$	1170.41	−	2027.21i	0.487468	−	0.844319i
$50$		0			0
$51$	−212.957	−	253.793i	−0.0818752	−	0.0975751i
$52$		0			0
$53$	−293.848	+	807.340i	−0.104609	+	0.287412i	−0.980944	−	0.194291i	$-0.937759\pi$
$53$	−293.848	+	807.340i	−0.104609	+	0.287412i	0.876335	+	0.481703i	$0.159982\pi$
$54$		0			0
$55$	306.370	−	1737.51i	0.101279	−	0.574383i
$56$		0			0
$57$	−277.317	−	57.2217i	−0.0853546	−	0.0176121i
$58$		0			0
$59$	−2307.63	−	406.897i	−0.662921	−	0.116891i	−0.167944	−	0.985797i	$-0.553713\pi$
$59$	−2307.63	−	406.897i	−0.662921	−	0.116891i	−0.494978	+	0.868906i	$0.664824\pi$
$60$		0			0
$61$	5176.31	+	1884.02i	1.39111	+	0.506321i	0.925526	−	0.378684i	$-0.123623\pi$
$61$	5176.31	+	1884.02i	1.39111	+	0.506321i	0.465580	+	0.885006i	$0.345846\pi$
$62$		0			0
$63$	477.695	−	400.833i	0.120356	−	0.100991i
$64$		0			0
$65$	1913.56	+	1104.80i	0.452915	+	0.261490i
$66$		0			0
$67$	−6476.74	+	1142.02i	−1.44280	+	0.254405i	−0.839609	−	0.543191i	$-0.817216\pi$
$67$	−6476.74	+	1142.02i	−1.44280	+	0.254405i	−0.603193	+	0.797596i	$0.706105\pi$
$68$		0			0
$69$	484.055	−	279.469i	0.101671	−	0.0586997i
$70$		0			0
$71$	877.159	+	2409.97i	0.174005	+	0.478075i	0.995784	−	0.0917320i	$-0.0292403\pi$
$71$	877.159	+	2409.97i	0.174005	+	0.478075i	−0.821779	+	0.569807i	$0.807018\pi$
$72$		0			0
$73$	−3639.17	−	3053.62i	−0.682898	−	0.573020i	0.233953	−	0.972248i	$-0.424834\pi$
$73$	−3639.17	−	3053.62i	−0.682898	−	0.573020i	−0.916852	+	0.399228i	$0.869278\pi$
$74$		0			0
$75$		−	422.188i		−	0.0750557i
$76$		0			0
$77$	1469.46			0.247844
$78$		0			0
$79$	−7424.75	+	8848.48i	−1.18967	+	1.41780i	−0.304516	+	0.952507i	$0.598495\pi$
$79$	−7424.75	+	8848.48i	−1.18967	+	1.41780i	−0.885158	+	0.465291i	$0.845950\pi$
$80$		0			0
$81$	−6025.19	+	2192.99i	−0.918334	+	0.334246i
$82$		0			0
$83$	5155.43	+	8929.47i	0.748357	+	1.29619i	0.948610	+	0.316448i	$0.102490\pi$
$83$	5155.43	+	8929.47i	0.748357	+	1.29619i	−0.200253	+	0.979744i	$0.564176\pi$
$84$		0			0
$85$	−683.142	−	3874.29i	−0.0945525	−	0.536234i
$86$		0			0
$87$	−619.759	+	1073.45i	−0.0818813	+	0.141823i
$88$		0			0
$89$	1402.20	+	1671.08i	0.177023	+	0.210968i	0.847259	−	0.531181i	$-0.178251\pi$
$89$	1402.20	+	1671.08i	0.177023	+	0.210968i	−0.670236	+	0.742148i	$0.733807\pi$
$90$		0			0
$91$	−629.430	+	1729.35i	−0.0760090	+	0.208833i
$92$		0			0
$93$	61.1878	−	347.013i	0.00707455	−	0.0401218i
$94$		0			0
$95$	−2512.09	−	2234.95i	−0.278347	−	0.247640i
$96$		0			0
$97$	6417.59	+	1131.59i	0.682070	+	0.120267i	0.503939	−	0.863739i	$-0.331884\pi$
$97$	6417.59	+	1131.59i	0.682070	+	0.120267i	0.178131	+	0.984007i	$0.442995\pi$
$98$		0			0
$99$	−14308.6	−	5207.89i	−1.45991	−	0.531363i
$100$		0			0
$101$	826.363	−	693.401i	0.0810081	−	0.0679738i	−0.601384	−	0.798960i	$-0.705384\pi$
$101$	826.363	−	693.401i	0.0810081	−	0.0679738i	0.682393	+	0.730986i	$0.260940\pi$
$102$		0			0
$103$	−3188.70	−	1841.00i	−0.300566	−	0.173532i	0.342131	−	0.939652i	$-0.388851\pi$
$103$	−3188.70	−	1841.00i	−0.300566	−	0.173532i	−0.642697	+	0.766120i	$0.722185\pi$
$104$		0			0
$105$	−55.8131	+	9.84136i	−0.00506242	+	0.000892641i
$106$		0			0
$107$	13571.7	−	7835.61i	1.18540	−	0.684393i	0.228145	−	0.973627i	$-0.426734\pi$
$107$	13571.7	−	7835.61i	1.18540	−	0.684393i	0.957258	+	0.289234i	$0.0934007\pi$
$108$		0			0
$109$	4348.57	+	11947.6i	0.366010	+	1.00561i	0.976864	+	0.213862i	$0.0686042\pi$
$109$	4348.57	+	11947.6i	0.366010	+	1.00561i	−0.610853	+	0.791744i	$0.709174\pi$
$110$		0			0
$111$	554.673	+	465.426i	0.0450185	+	0.0377750i
$112$		0			0
$113$			9061.84i			0.709675i	0.934928	+	0.354837i	$0.115464\pi$
$113$			9061.84i			0.709675i	−0.934928	+	0.354837i	$0.884536\pi$
$114$		0			0
$115$	6637.12			0.501862
$116$		0			0
$117$	12257.9	−	14608.3i	0.895453	−	1.06716i
$118$		0			0
$119$	3079.00	−	1120.67i	0.217428	−	0.0791375i
$120$		0			0
$121$	−10620.3	−	18395.0i	−0.725384	−	1.25640i
$122$		0			0
$123$	337.436	+	1913.70i	0.0223039	+	0.126492i
$124$		0			0
$125$	5417.28	−	9383.00i	0.346706	−	0.600512i
$126$		0			0
$127$	−14942.6	−	17807.9i	−0.926444	−	1.10409i	−0.994323	−	0.106400i	$-0.966068\pi$
$127$	−14942.6	−	17807.9i	−0.926444	−	1.10409i	0.0678790	−	0.997694i	$-0.478377\pi$
$128$		0			0
$129$	−107.024	+	294.047i	−0.00643137	+	0.0176700i
$130$		0			0
$131$	−2940.86	+	16678.4i	−0.171369	+	0.971880i	0.770884	+	0.636976i	$0.219815\pi$
$131$	−2940.86	+	16678.4i	−0.171369	+	0.971880i	−0.942252	+	0.334904i	$0.891296\pi$
$132$		0			0
$133$	1469.85	−	2383.72i	0.0830940	−	0.134757i
$134$		0			0
$135$	1161.12	+	204.736i	0.0637102	+	0.0112338i
$136$		0			0
$137$	9299.62	+	3384.78i	0.495478	+	0.180339i	0.577659	−	0.816278i	$-0.303966\pi$
$137$	9299.62	+	3384.78i	0.495478	+	0.180339i	−0.0821812	+	0.996617i	$0.526189\pi$
$138$		0			0
$139$	15521.6	−	13024.2i	0.803356	−	0.674096i	−0.145656	−	0.989335i	$-0.546529\pi$
$139$	15521.6	−	13024.2i	0.803356	−	0.674096i	0.949012	+	0.315240i	$0.102085\pi$
$140$		0			0
$141$	−739.370	−	426.876i	−0.0371898	−	0.0214715i
$142$		0			0
$143$	44254.9	−	7803.33i	2.16416	−	0.381600i
$144$		0			0
$145$	−12746.7	+	7359.33i	−0.606266	+	0.350028i
$146$		0			0
$147$	627.977	+	1725.35i	0.0290609	+	0.0798441i
$148$		0			0
$149$	2762.13	+	2317.70i	0.124415	+	0.104396i	0.702872	−	0.711316i	$-0.251901\pi$
$149$	2762.13	+	2317.70i	0.124415	+	0.104396i	−0.578457	+	0.815713i	$0.696345\pi$
$150$		0			0
$151$		−	17076.6i		−	0.748940i	−0.927239	−	0.374470i	$-0.877825\pi$
$151$		−	17076.6i		−	0.748940i	0.927239	−	0.374470i	$-0.122175\pi$
$152$		0			0
$153$	−33952.8			−1.45042
$154$		0			0
$155$	2689.54	−	3205.26i	0.111947	−	0.133414i
$156$		0			0
$157$	−23631.5	+	8601.16i	−0.958720	+	0.348945i	−0.773532	−	0.633758i	$-0.781512\pi$
$157$	−23631.5	+	8601.16i	−0.958720	+	0.348945i	−0.185188	+	0.982703i	$0.559289\pi$
$158$		0			0
$159$	−336.949	−	583.613i	−0.0133282	−	0.0230850i
$160$		0			0
$161$	959.917	+	5443.96i	0.0370324	+	0.210021i
$162$		0			0
$163$	11840.5	−	20508.4i	0.445651	−	0.771891i	−0.552446	−	0.833549i	$-0.686305\pi$
$163$	11840.5	−	20508.4i	0.445651	−	0.771891i	0.998097	+	0.0616580i	$0.0196388\pi$
$164$		0			0
$165$	889.542	+	1060.11i	0.0326737	+	0.0389390i
$166$		0			0
$167$	−13247.0	+	36395.9i	−0.474991	+	1.30503i	0.438706	+	0.898631i	$0.355437\pi$
$167$	−13247.0	+	36395.9i	−0.474991	+	1.30503i	−0.913697	+	0.406397i	$0.866785\pi$
$168$		0			0
$169$	−4813.19	+	27297.0i	−0.168523	+	0.955743i
$170$		0			0
$171$	−22760.4	+	18001.7i	−0.778373	+	0.615631i
$172$		0			0
$173$	4285.42	+	755.635i	0.143186	+	0.0252476i	0.244782	−	0.969578i	$-0.421284\pi$
$173$	4285.42	+	755.635i	0.143186	+	0.0252476i	−0.101596	+	0.994826i	$0.532395\pi$
$174$		0			0
$175$	3923.66	+	1428.09i	0.128119	+	0.0466317i
$176$		0			0
$177$	1407.97	−	1181.42i	0.0449413	−	0.0377102i
$178$		0			0
$179$	37315.9	+	21544.3i	1.16463	+	0.672399i	0.952409	−	0.304823i	$-0.0985975\pi$
$179$	37315.9	+	21544.3i	1.16463	+	0.672399i	0.212220	+	0.977222i	$0.431931\pi$
$180$		0			0
$181$	−10363.9	+	1827.44i	−0.316349	+	0.0557808i	−0.329568	−	0.944132i	$-0.606903\pi$
$181$	−10363.9	+	1827.44i	−0.316349	+	0.0557808i	0.0132194	+	0.999913i	$0.495792\pi$
$182$		0			0
$183$	−3741.87	+	2160.37i	−0.111734	+	0.0645098i
$184$		0			0
$185$	2940.70	+	8079.50i	0.0859225	+	0.236070i
$186$		0			0
$187$	−61290.4	−	51428.7i	−1.75271	−	1.47070i
$188$		0			0
$189$			981.994i			0.0274907i
$190$		0			0
$191$	10198.0			0.279543			0.139772	−	0.990184i	$-0.455363\pi$
$191$	10198.0			0.279543			0.139772	+	0.990184i	$0.455363\pi$
$192$		0			0
$193$	−32382.4	+	38591.8i	−0.869349	+	1.03605i	0.129660	+	0.991558i	$0.458611\pi$
$193$	−32382.4	+	38591.8i	−0.869349	+	1.03605i	−0.999010	+	0.0444917i	$0.985833\pi$
$194$		0			0
$195$	−1628.63	+	592.772i	−0.0428304	+	0.0155890i
$196$		0			0
$197$	11238.5	+	19465.7i	0.289585	+	0.501576i	0.973711	−	0.227788i	$-0.0731493\pi$
$197$	11238.5	+	19465.7i	0.289585	+	0.501576i	−0.684126	+	0.729364i	$0.739816\pi$
$198$		0			0
$199$	8018.46	+	45475.0i	0.202481	+	1.14833i	0.901354	+	0.433083i	$0.142574\pi$
$199$	8018.46	+	45475.0i	0.202481	+	1.14833i	−0.698873	+	0.715246i	$0.746315\pi$
$200$		0			0
$201$	2579.28	−	4467.44i	0.0638420	−	0.110578i
$202$		0			0
$203$	−7879.88	−	9390.88i	−0.191217	−	0.227884i
$204$		0			0
$205$	−7892.02	+	21683.2i	−0.187794	+	0.515958i
$206$		0			0
$207$	9946.83	−	56411.3i	0.232137	−	1.31651i
$208$		0			0
$209$	−68353.6	−	1979.49i	−1.56484	−	0.0453169i
$210$		0			0
$211$	47835.1	+	8434.62i	1.07444	+	0.189453i	0.682755	−	0.730647i	$-0.260782\pi$
$211$	47835.1	+	8434.62i	1.07444	+	0.189453i	0.391684	+	0.920100i	$0.371893\pi$
$212$		0			0
$213$	−1890.32	−	688.021i	−0.0416655	−	0.0151650i
$214$		0			0
$215$	−2846.43	+	2388.44i	−0.0615777	+	0.0516698i
$216$		0			0
$217$	3018.03	+	1742.46i	0.0640921	+	0.0370036i
$218$		0			0
$219$	3669.63	−	647.056i	0.0765129	−	0.0134913i
$220$		0			0
$221$	86777.3	−	50100.9i	1.77673	−	1.02580i
$222$		0			0
$223$	16111.5	+	44266.0i	0.323986	+	0.890145i	0.989600	+	0.143848i	$0.0459478\pi$
$223$	16111.5	+	44266.0i	0.323986	+	0.890145i	−0.665613	+	0.746297i	$0.731830\pi$
$224$		0			0
$225$	−33144.4	−	27811.5i	−0.654705	−	0.549362i
$226$		0			0
$227$			11619.6i			0.225497i	0.993624	+	0.112748i	$0.0359654\pi$
$227$			11619.6i			0.225497i	−0.993624	+	0.112748i	$0.964035\pi$
$228$		0			0
$229$	38184.6			0.728145			0.364072	−	0.931371i	$-0.381386\pi$
$229$	38184.6			0.728145			0.364072	+	0.931371i	$0.381386\pi$
$230$		0			0
$231$	−740.884	+	882.951i	−0.0138844	+	0.0165467i
$232$		0			0
$233$	−62685.9	+	22815.8i	−1.15467	+	0.420266i	−0.847191	−	0.531289i	$-0.821708\pi$
$233$	−62685.9	+	22815.8i	−1.15467	+	0.420266i	−0.307480	+	0.951554i	$0.599486\pi$
$234$		0			0
$235$	−5068.93	−	8779.65i	−0.0917870	−	0.158980i
$236$		0			0
$237$	−1573.29	−	8922.56i	−0.0280099	−	0.158852i
$238$		0			0
$239$	25095.4	−	43466.4i	0.439337	−	0.760954i	−0.558302	−	0.829638i	$-0.688547\pi$
$239$	25095.4	−	43466.4i	0.439337	−	0.760954i	0.997638	+	0.0686844i	$0.0218801\pi$
$240$		0			0
$241$	−45715.6	−	54481.7i	−0.787100	−	0.938030i	0.212131	−	0.977241i	$-0.431960\pi$
$241$	−45715.6	−	54481.7i	−0.787100	−	0.938030i	−0.999231	+	0.0392116i	$0.987515\pi$
$242$		0			0
$243$	5227.02	−	14361.1i	0.0885201	−	0.243207i
$244$		0			0
$245$	−3785.97	+	21471.3i	−0.0630732	+	0.357706i
$246$		0			0
$247$	31608.2	−	79594.4i	0.518090	−	1.30463i
$248$		0			0
$249$	−7964.71	−	1404.39i	−0.128461	−	0.0226511i
$250$		0			0
$251$	32866.8	+	11962.5i	0.521687	+	0.189878i	0.589423	−	0.807825i	$-0.299355\pi$
$251$	32866.8	+	11962.5i	0.521687	+	0.189878i	−0.0677358	+	0.997703i	$0.521578\pi$
$252$		0			0
$253$	103403.	−	86765.0i	1.61544	−	1.35551i
$254$		0			0
$255$	2672.36	+	1542.89i	0.0410974	+	0.0237276i
$256$		0			0
$257$	89242.7	−	15735.9i	1.35116	−	0.238246i	0.549233	−	0.835669i	$-0.314920\pi$
$257$	89242.7	−	15735.9i	1.35116	−	0.238246i	0.801926	+	0.597423i	$0.203809\pi$
$258$		0			0
$259$	−6201.73	+	3580.57i	−0.0924514	+	0.0533769i
$260$		0			0
$261$	43446.5	+	119368.i	0.637784	+	1.75230i
$262$		0			0
$263$	−15528.7	−	13030.1i	−0.224504	−	0.188381i	0.523597	−	0.851966i	$-0.324590\pi$
$263$	−15528.7	−	13030.1i	−0.224504	−	0.188381i	−0.748101	+	0.663585i	$0.769034\pi$
$264$		0			0
$265$		−	8002.20i		−	0.113951i
$266$		0			0
$267$	−1711.06			−0.0240018
$268$		0			0
$269$	−13981.4	+	16662.4i	−0.193218	+	0.230268i	−0.853952	−	0.520352i	$-0.825801\pi$
$269$	−13981.4	+	16662.4i	−0.193218	+	0.230268i	0.660734	+	0.750620i	$0.270245\pi$
$270$		0			0
$271$	−104267.	+	37950.1i	−1.41974	+	0.516743i	−0.933972	−	0.357346i	$-0.883682\pi$
$271$	−104267.	+	37950.1i	−1.41974	+	0.516743i	−0.485766	+	0.874089i	$0.661459\pi$
$272$		0			0
$273$	−721.755	−	1250.12i	−0.00968421	−	0.0167735i
$274$		0			0
$275$	−17704.7	−	100409.i	−0.234112	−	1.32772i
$276$		0			0
$277$	56652.4	−	98124.8i	0.738344	−	1.27885i	−0.214897	−	0.976637i	$-0.568942\pi$
$277$	56652.4	−	98124.8i	0.738344	−	1.27885i	0.953241	−	0.302212i	$-0.0977250\pi$
$278$		0			0
$279$	−23212.0	−	27663.0i	−0.298197	−	0.355378i
$280$		0			0
$281$	−12444.0	+	34189.7i	−0.157597	+	0.432995i	−0.993212	−	0.116321i	$-0.962890\pi$
$281$	−12444.0	+	34189.7i	−0.157597	+	0.432995i	0.835614	+	0.549317i	$0.185112\pi$
$282$		0			0
$283$	−20132.3	+	114176.i	−0.251374	+	1.42562i	0.553836	+	0.832626i	$0.313164\pi$
$283$	−20132.3	+	114176.i	−0.251374	+	1.42562i	−0.805210	+	0.592990i	$0.797948\pi$
$284$		0			0
$285$	2609.46	−	382.596i	0.0321263	−	0.00471033i
$286$		0			0
$287$	−18926.6	−	3337.26i	−0.229778	−	0.0405160i
$288$		0			0
$289$	−89160.5	−	32451.8i	−1.06752	−	0.388546i
$290$		0			0
$291$	−3915.60	+	3285.58i	−0.0462394	+	0.0387995i
$292$		0			0
$293$	1781.50	+	1028.55i	0.0207515	+	0.0119809i	0.510340	−	0.859973i	$-0.329520\pi$
$293$	1781.50	+	1028.55i	0.0207515	+	0.0119809i	−0.489588	+	0.871954i	$0.662853\pi$
$294$		0			0
$295$	21493.4	−	3789.86i	0.246979	−	0.0435491i
$296$		0			0
$297$	20766.0	−	11989.3i	0.235418	−	0.135919i
$298$		0			0
$299$	57818.4	+	158855.i	0.646731	+	1.77688i
$300$		0			0
$301$	−2370.74	−	1989.29i	−0.0261668	−	0.0219566i
$302$		0			0
$303$			846.137i			0.00921627i
$304$		0			0
$305$	−51306.6			−0.551535
$306$		0			0
$307$	72507.7	−	86411.3i	0.769321	−	0.916841i	−0.229078	−	0.973408i	$-0.573571\pi$
$307$	72507.7	−	86411.3i	0.769321	−	0.916841i	0.998399	+	0.0565673i	$0.0180156\pi$
$308$		0			0
$309$	2713.89	−	987.776i	0.0284234	−	0.0103453i
$310$		0			0
$311$	−8320.72	−	14411.9i	−0.0860281	−	0.149005i	0.819801	−	0.572649i	$-0.194084\pi$
$311$	−8320.72	−	14411.9i	−0.0860281	−	0.149005i	−0.905829	+	0.423644i	$0.860751\pi$
$312$		0			0
$313$	27818.3	+	157766.i	0.283950	+	1.61036i	0.709011	+	0.705198i	$0.249142\pi$
$313$	27818.3	+	157766.i	0.283950	+	1.61036i	−0.425060	+	0.905165i	$0.639747\pi$
$314$		0			0
$315$	−2904.06	+	5029.97i	−0.0292674	+	0.0506926i
$316$		0			0
$317$	−101170.	−	120569.i	−1.00677	−	1.19982i	−0.979757	−	0.200189i	$-0.935844\pi$
$317$	−101170.	−	120569.i	−1.00677	−	1.19982i	−0.0270146	−	0.999635i	$-0.508600\pi$
$318$		0			0
$319$	−102381.	+	281288.i	−1.00609	+	2.76421i
$320$		0			0
$321$	−2134.50	+	12105.4i	−0.0207151	+	0.117481i
$322$		0			0
$323$	−144733.	+	47981.3i	−1.38727	+	0.459903i
$324$		0			0
$325$	125750.	+	22173.1i	1.19053	+	0.209923i
$326$		0			0
$327$	−9371.39	−	3410.91i	−0.0876413	−	0.0318988i
$328$		0			0
$329$	6468.21	−	5427.48i	0.0597575	−	0.0501425i
$330$		0			0
$331$	−95092.2	−	54901.5i	−0.867939	−	0.501105i	−0.00127600	−	0.999999i	$-0.500406\pi$
$331$	−95092.2	−	54901.5i	−0.867939	−	0.501105i	−0.866663	+	0.498895i	$0.833739\pi$
$332$		0			0
$333$	73077.7	−	12885.6i	0.659017	−	0.116202i
$334$		0			0
$335$	53048.7	−	30627.7i	0.472699	−	0.272913i
$336$		0			0
$337$	−12804.4	−	35179.9i	−0.112746	−	0.309767i	0.870468	−	0.492226i	$-0.163817\pi$
$337$	−12804.4	−	35179.9i	−0.112746	−	0.309767i	−0.983213	+	0.182459i	$0.941594\pi$
$338$		0			0
$339$	−5444.95	−	4568.85i	−0.0473799	−	0.0397565i
$340$		0			0
$341$		−	85095.7i		−	0.731811i
$342$		0			0
$343$	−36784.7			−0.312665
$344$		0			0
$345$	−3346.34	+	3988.02i	−0.0281146	+	0.0335057i
$346$		0			0
$347$	−169961.	+	61860.6i	−1.41153	+	0.513754i	−0.931578	−	0.363542i	$-0.881567\pi$
$347$	−169961.	+	61860.6i	−1.41153	+	0.513754i	−0.479950	+	0.877296i	$0.659345\pi$
$348$		0			0
$349$	66787.2	+	115679.i	0.548330	+	0.949736i	0.998389	+	0.0567371i	$0.0180697\pi$
$349$	66787.2	+	115679.i	0.548330	+	0.949736i	−0.450059	+	0.892999i	$0.648597\pi$
$350$		0			0
$351$	5214.70	+	29574.1i	0.0423268	+	0.240047i
$352$		0			0
$353$	19867.3	−	34411.1i	0.159437	−	0.276153i	−0.775229	−	0.631680i	$-0.782366\pi$
$353$	19867.3	−	34411.1i	0.159437	−	0.276153i	0.934666	+	0.355528i	$0.115699\pi$
$354$		0			0
$355$	−15354.4	−	18298.7i	−0.121836	−	0.145199i
$356$		0			0
$357$	−879.022	+	2415.09i	−0.00689705	+	0.0189495i
$358$		0			0
$359$	−12869.3	+	72985.6i	−0.0998543	+	0.566302i	0.893297	+	0.449467i	$0.148386\pi$
$359$	−12869.3	+	72985.6i	−0.0998543	+	0.566302i	−0.993151	+	0.116835i	$0.962725\pi$
$360$		0			0
$361$	−71582.6	+	108901.i	−0.549279	+	0.835639i
$362$		0			0
$363$	16407.6	+	2893.09i	0.124518	+	0.0219558i
$364$		0			0
$365$	41578.9	+	15133.5i	0.312095	+	0.113593i
$366$		0			0
$367$	46243.9	−	38803.3i	0.343339	−	0.288095i	−0.454770	−	0.890609i	$-0.650279\pi$
$367$	46243.9	−	38803.3i	0.343339	−	0.288095i	0.798109	+	0.602514i	$0.205834\pi$
$368$		0			0
$369$	172465.	+	99573.0i	1.26663	+	0.731289i
$370$		0			0
$371$	6563.64	−	1157.35i	0.0476867	−	0.00840845i
$372$		0			0
$373$	122597.	−	70781.6i	0.881178	−	0.508748i	0.0101310	−	0.999949i	$-0.496775\pi$
$373$	122597.	−	70781.6i	0.881178	−	0.508748i	0.871046	+	0.491201i	$0.163442\pi$
$374$		0			0
$375$	2906.61	+	7985.84i	0.0206692	+	0.0567882i
$376$		0			0
$377$	−287182.	−	240974.i	−2.02057	−	1.69546i
$378$		0			0
$379$		−	121623.i		−	0.846715i	−0.905963	−	0.423358i	$-0.860851\pi$
$379$		−	121623.i		−	0.846715i	0.905963	−	0.423358i	$-0.139149\pi$
$380$		0			0
$381$	18234.0			0.125613
$382$		0			0
$383$	−51324.3	+	61165.9i	−0.349885	+	0.416977i	−0.912070	−	0.410035i	$-0.865517\pi$
$383$	−51324.3	+	61165.9i	−0.349885	+	0.416977i	0.562185	+	0.827012i	$0.309961\pi$
$384$		0			0
$385$	−12861.3	+	4681.12i	−0.0867685	+	0.0315812i
$386$		0			0
$387$	16034.3	+	27772.3i	0.107061	+	0.185434i
$388$		0			0
$389$	5919.35	+	33570.3i	0.0391178	+	0.221848i	0.998100	−	0.0616190i	$-0.0196264\pi$
$389$	5919.35	+	33570.3i	0.0391178	+	0.221848i	−0.958982	+	0.283467i	$0.908515\pi$
$390$		0			0
$391$	150492.	−	260659.i	0.984372	−	1.70498i
$392$		0			0
$393$	−8538.76	−	10176.1i	−0.0552853	−	0.0658865i
$394$		0			0
$395$	36796.4	−	101097.i	0.235836	−	0.647955i
$396$		0			0
$397$	16108.0	−	91352.8i	0.102202	−	0.579617i	−0.890099	−	0.455767i	$-0.849365\pi$
$397$	16108.0	−	91352.8i	0.102202	−	0.579617i	0.992301	−	0.123849i	$-0.0395240\pi$
$398$		0			0
$399$	691.219	+	2085.02i	0.00434180	+	0.0130968i
$400$		0			0
$401$	20240.3	+	3568.92i	0.125872	+	0.0221946i	0.236229	−	0.971697i	$-0.424088\pi$
$401$	20240.3	+	3568.92i	0.125872	+	0.0221946i	−0.110357	+	0.993892i	$0.535199\pi$
$402$		0			0
$403$	100145.	+	36449.9i	0.616624	+	0.224433i
$404$		0			0
$405$	45748.6	−	38387.6i	0.278912	−	0.234035i
$406$		0			0
$407$	151435.	+	87431.2i	0.914194	+	0.527810i
$408$		0			0
$409$	192872.	−	34008.5i	1.15298	−	0.203302i	0.435704	−	0.900090i	$-0.356499\pi$
$409$	192872.	−	34008.5i	1.15298	−	0.203302i	0.717277	+	0.696788i	$0.245388\pi$
$410$		0			0
$411$	−6722.54	+	3881.26i	−0.0397970	+	0.0229768i
$412$		0			0
$413$	6217.11	+	17081.4i	0.0364493	+	0.100144i
$414$		0			0
$415$	−73567.8	−	61730.7i	−0.427161	−	0.358431i
$416$		0			0
$417$			15893.0i			0.0913977i
$418$		0			0
$419$	71668.6			0.408226			0.204113	−	0.978947i	$-0.434569\pi$
$419$	71668.6			0.408226			0.204113	+	0.978947i	$0.434569\pi$
$420$		0			0
$421$	−78684.7	+	93772.7i	−0.443942	+	0.529069i	−0.940891	−	0.338711i	$-0.890009\pi$
$421$	−78684.7	+	93772.7i	−0.443942	+	0.529069i	0.496949	+	0.867780i	$0.334454\pi$
$422$		0			0
$423$	−82218.1	+	29924.9i	−0.459501	+	0.167245i
$424$		0			0
$425$	−113672.	−	196886.i	−0.629328	−	1.09003i
$426$		0			0
$427$	−7420.39	−	42083.1i	−0.0406978	−	0.230809i
$428$		0			0
$429$	−17624.0	+	30525.6i	−0.0957611	+	0.165863i
$430$		0			0
$431$	−63488.9	−	75663.1i	−0.341777	−	0.407314i	0.567588	−	0.823313i	$-0.307877\pi$
$431$	−63488.9	−	75663.1i	−0.341777	−	0.407314i	−0.909365	+	0.415998i	$0.863432\pi$
$432$		0			0
$433$	−6334.35	+	17403.5i	−0.0337852	+	0.0928240i	−0.955438	−	0.295191i	$-0.904617\pi$
$433$	−6334.35	+	17403.5i	−0.0337852	+	0.0928240i	0.921653	+	0.388015i	$0.126839\pi$
$434$		0			0
$435$	2004.76	−	11369.6i	0.0105946	−	0.0600848i
$436$		0			0
$437$	−37318.1	−	254524.i	−0.195414	−	1.33280i
$438$		0			0
$439$	47901.5	+	8446.33i	0.248554	+	0.0438267i	0.296537	−	0.955021i	$-0.404168\pi$
$439$	47901.5	+	8446.33i	0.248554	+	0.0438267i	−0.0479832	+	0.998848i	$0.515279\pi$
$440$		0			0
$441$	176819.	+	64356.7i	0.909181	+	0.330915i
$442$		0			0
$443$	133793.	−	112265.i	0.681749	−	0.572055i	−0.234768	−	0.972051i	$-0.575433\pi$
$443$	133793.	−	112265.i	0.681749	−	0.572055i	0.916517	+	0.399996i	$0.130989\pi$
$444$		0			0
$445$	−17595.9	−	10159.0i	−0.0888570	−	0.0513016i
$446$		0			0
$447$	−2785.26	+	491.116i	−0.0139396	+	0.00245793i
$448$		0			0
$449$	−180265.	+	104076.i	−0.894165	+	0.516247i	−0.875303	−	0.483576i	$-0.839338\pi$
$449$	−180265.	+	104076.i	−0.894165	+	0.516247i	−0.0188626	+	0.999822i	$0.506005\pi$
$450$		0			0
$451$	160504.	+	440981.i	0.789102	+	2.16804i
$452$		0			0
$453$	10260.7	+	8609.78i	0.0500014	+	0.0419562i
$454$		0			0
$455$		−	17141.0i		−	0.0827965i
$456$		0			0
$457$	78903.9			0.377804			0.188902	−	0.981996i	$-0.439507\pi$
$457$	78903.9			0.377804			0.188902	+	0.981996i	$0.439507\pi$
$458$		0			0
$459$	34368.1	−	40958.3i	0.163129	−	0.194409i
$460$		0			0
$461$	−74686.5	+	27183.7i	−0.351431	+	0.127910i	−0.511703	−	0.859162i	$-0.670985\pi$
$461$	−74686.5	+	27183.7i	−0.351431	+	0.127910i	0.160272	+	0.987073i	$0.448763\pi$
$462$		0			0
$463$	−70223.0	−	121630.i	−0.327580	−	0.567385i	0.654451	−	0.756104i	$-0.272900\pi$
$463$	−70223.0	−	121630.i	−0.327580	−	0.567385i	−0.982031	+	0.188719i	$0.939566\pi$
$464$		0			0
$465$	569.906	+	3232.10i	0.00263571	+	0.0149479i
$466$		0			0
$467$	−18527.0	+	32089.7i	−0.0849514	+	0.147140i	−0.905371	−	0.424622i	$-0.860407\pi$
$467$	−18527.0	+	32089.7i	−0.0849514	+	0.147140i	0.820419	+	0.571763i	$0.193740\pi$
$468$		0			0
$469$	32794.0	+	39082.4i	0.149090	+	0.177679i
$470$		0			0
$471$	6746.53	−	18535.9i	0.0304116	−	0.0835551i
$472$		0			0
$473$	−13122.4	+	74421.0i	−0.0586532	+	0.332639i
$474$		0			0
$475$	−180589.	−	71714.8i	−0.800395	−	0.317849i
$476$		0			0
$477$	−68013.6	−	11992.6i	−0.298923	−	0.0527082i
$478$		0			0
$479$	−329391.	−	119889.i	−1.43563	−	0.522525i	−0.497087	−	0.867701i	$-0.665597\pi$
$479$	−329391.	−	119889.i	−1.43563	−	0.522525i	−0.938538	+	0.345176i	$0.887819\pi$
$480$		0			0
$481$	−167760.	+	140767.i	−0.725099	+	0.608430i
$482$		0			0
$483$	−3755.07	−	2167.99i	−0.0160962	−	0.00929314i
$484$		0			0
$485$	−59773.8	+	10539.7i	−0.254113	+	0.0448071i
$486$		0			0
$487$	262839.	−	151750.i	1.10823	−	0.639839i	0.169862	−	0.985468i	$-0.445668\pi$
$487$	262839.	−	151750.i	1.10823	−	0.639839i	0.938371	+	0.345629i	$0.112334\pi$
$488$		0			0
$489$	6352.95	+	17454.6i	0.0265679	+	0.0729948i
$490$		0			0
$491$	−37420.5	−	31399.5i	−0.155220	−	0.130245i	0.561870	−	0.827225i	$-0.310082\pi$
$491$	−37420.5	−	31399.5i	−0.155220	−	0.130245i	−0.717090	+	0.696980i	$0.754526\pi$
$492$		0			0
$493$			667470.i			2.74623i
$494$		0			0
$495$	141824.			0.578814
$496$		0			0
$497$	12788.4	−	15240.6i	0.0517731	−	0.0617007i
$498$		0			0
$499$	−32724.7	+	11910.8i	−0.131424	+	0.0478345i	−0.406895	−	0.913475i	$-0.633388\pi$
$499$	−32724.7	+	11910.8i	−0.131424	+	0.0478345i	0.275471	+	0.961309i	$0.411166\pi$
$500$		0			0
$501$	−15190.1	−	26310.0i	−0.0605180	−	0.104820i
$502$		0			0
$503$	46195.2	+	261986.i	0.182583	+	1.03548i	0.929021	+	0.370026i	$0.120651\pi$
$503$	46195.2	+	261986.i	0.182583	+	1.03548i	−0.746438	+	0.665455i	$0.768238\pi$
$504$		0			0
$505$	−5023.73	+	8701.35i	−0.0196990	+	0.0341196i
$506$		0			0
$507$	−13975.1	−	16654.8i	−0.0543673	−	0.0647925i
$508$		0			0
$509$	−127751.	+	350992.i	−0.493091	+	1.35476i	0.404746	+	0.914429i	$0.367360\pi$
$509$	−127751.	+	350992.i	−0.493091	+	1.35476i	−0.897837	+	0.440327i	$0.854862\pi$
$510$		0			0
$511$	−6399.42	+	36292.9i	−0.0245075	+	0.138989i
$512$		0			0
$513$	1322.82	−	45678.4i	0.00502652	−	0.173571i
$514$		0			0
$515$	33773.3	+	5955.15i	0.127338	+	0.0224532i
$516$		0			0
$517$	−193745.	−	70517.4i	−0.724852	−	0.263824i
$518$		0			0
$519$	−2614.68	+	2193.98i	−0.00970699	+	0.00814513i
$520$		0			0
$521$	−164152.	−	94773.2i	−0.604743	−	0.349149i	0.166162	−	0.986098i	$-0.446862\pi$
$521$	−164152.	−	94773.2i	−0.604743	−	0.349149i	−0.770905	+	0.636950i	$0.780196\pi$
$522$		0			0
$523$	83466.4	−	14717.4i	0.305147	−	0.0538056i	−0.0189782	−	0.999820i	$-0.506041\pi$
$523$	83466.4	−	14717.4i	0.305147	−	0.0538056i	0.324125	+	0.946014i	$0.394930\pi$
$524$		0			0
$525$	−2836.35	+	1637.57i	−0.0102906	+	0.00594128i
$526$		0			0
$527$	−64897.0	−	178303.i	−0.233670	−	0.642004i
$528$		0			0
$529$	174617.	+	146521.i	0.623985	+	0.523586i
$530$		0			0
$531$		−	188360.i		−	0.668035i
$532$		0			0
$533$	−587721.			−2.06879
$534$		0			0
$535$	−93823.0	+	111814.i	−0.327795	+	0.390650i
$536$		0			0
$537$	−31759.4	+	11559.5i	−0.110135	+	0.0400857i
$538$		0			0
$539$	221705.	+	384004.i	0.763128	+	1.32178i
$540$		0			0
$541$	−82338.9	−	466967.i	−0.281326	−	1.59548i	−0.718120	−	0.695919i	$-0.754997\pi$
$541$	−82338.9	−	466967.i	−0.281326	−	1.59548i	0.436794	−	0.899562i	$-0.356114\pi$
$542$		0			0
$543$	4127.29	−	7148.68i	0.0139980	−	0.0242452i
$544$		0			0
$545$	−76120.4	−	90716.8i	−0.256276	−	0.305418i
$546$		0			0
$547$	67687.8	−	185971.i	0.226223	−	0.621541i	−0.773705	−	0.633546i	$-0.781599\pi$
$547$	67687.8	−	185971.i	0.226223	−	0.621541i	0.999928	+	0.0120041i	$0.00382113\pi$
$548$		0			0
$549$	−76891.4	+	436073.i	−0.255113	+	1.44682i
$550$		0			0
$551$	353890.	+	447441.i	1.16564	+	1.47378i
$552$		0			0
$553$	88244.7	+	15559.9i	0.288561	+	0.0508812i
$554$		0			0
$555$	−6337.36	−	2306.61i	−0.0205742	−	0.00748838i
$556$		0			0
$557$	414735.	−	348004.i	1.33678	−	1.12169i	0.354341	−	0.935116i	$-0.384705\pi$
$557$	414735.	−	348004.i	1.33678	−	1.12169i	0.982440	−	0.186576i	$-0.0597391\pi$
$558$		0			0
$559$	−81961.8	−	47320.7i	−0.262294	−	0.151435i
$560$		0			0
$561$	61803.5	−	10897.6i	0.196376	−	0.0346263i
$562$		0			0
$563$	69780.4	−	40287.7i	0.220149	−	0.127103i	−0.385870	−	0.922553i	$-0.626099\pi$
$563$	69780.4	−	40287.7i	0.220149	−	0.127103i	0.606019	+	0.795450i	$0.292765\pi$
$564$		0			0
$565$	−28867.3	−	79312.4i	−0.0904295	−	0.248453i
$566$		0			0
$567$	38103.2	+	31972.4i	0.118521	+	0.0994510i
$568$		0			0
$569$		−	219752.i		−	0.678749i	−0.940651	−	0.339374i	$-0.889785\pi$
$569$		−	219752.i		−	0.678749i	0.940651	−	0.339374i	$-0.110215\pi$
$570$		0			0
$571$	−71941.6			−0.220652			−0.110326	−	0.993895i	$-0.535190\pi$
$571$	−71941.6			−0.220652			−0.110326	+	0.993895i	$0.535190\pi$
$572$		0			0
$573$	−5141.70	+	6127.64i	−0.0156602	+	0.0186631i
$574$		0			0
$575$	360420.	−	131182.i	1.09012	−	0.396771i
$576$		0			0
$577$	48217.9	+	83515.8i	0.144829	+	0.250852i	0.929309	−	0.369303i	$-0.120403\pi$
$577$	48217.9	+	83515.8i	0.144829	+	0.250852i	−0.784480	+	0.620154i	$0.787070\pi$
$578$		0			0
$579$	−6861.76	−	38914.9i	−0.0204681	−	0.116081i
$580$		0			0
$581$	39993.3	−	69270.5i	0.118477	−	0.205209i
$582$		0			0
$583$	−104610.	−	124670.i	−0.307778	−	0.366796i
$584$		0			0
$585$	−60748.8	+	166906.i	−0.177511	+	0.487708i
$586$		0			0
$587$	67100.9	−	380548.i	0.194739	−	1.10442i	−0.718052	−	0.695990i	$-0.754966\pi$
$587$	67100.9	−	380548.i	0.194739	−	1.10442i	0.912790	−	0.408428i	$-0.133923\pi$
$588$		0			0
$589$	−138040.	−	85118.0i	−0.397900	−	0.245353i
$590$		0			0
$591$	−17362.6	−	3061.49i	−0.0497094	−	0.00876511i
$592$		0			0
$593$	109933.	+	40012.2i	0.312620	+	0.113784i	0.493565	−	0.869709i	$-0.335693\pi$
$593$	109933.	+	40012.2i	0.312620	+	0.113784i	−0.180945	+	0.983493i	$0.557916\pi$
$594$		0			0
$595$	−23378.5	+	19616.9i	−0.0660364	+	0.0554111i
$596$		0			0
$597$	−31367.1	−	18109.8i	−0.0880089	−	0.0508119i
$598$		0			0
$599$	−127777.	+	22530.6i	−0.356123	+	0.0627940i	−0.348847	−	0.937180i	$-0.613427\pi$
$599$	−127777.	+	22530.6i	−0.356123	+	0.0627940i	−0.00727514	+	0.999974i	$0.502316\pi$
$600$		0			0
$601$	30321.7	−	17506.3i	0.0839470	−	0.0484668i	−0.457439	−	0.889241i	$-0.651233\pi$
$601$	30321.7	−	17506.3i	0.0839470	−	0.0484668i	0.541386	+	0.840774i	$0.317900\pi$
$602$		0			0
$603$	−180813.	−	496780.i	−0.497274	−	1.36625i
$604$		0			0
$605$	151552.	+	127167.i	0.414048	+	0.347428i
$606$		0			0
$607$		−	469725.i		−	1.27487i	−0.770504	−	0.637435i	$-0.779995\pi$
$607$		−	469725.i		−	1.27487i	0.770504	−	0.637435i	$-0.220005\pi$
$608$		0			0
$609$	9615.58			0.0259263
$610$		0			0
$611$	165977.	−	197804.i	0.444597	−	0.529850i
$612$		0			0
$613$	224865.	−	81844.2i	0.598413	−	0.217805i	−0.0250126	−	0.999687i	$-0.507963\pi$
$613$	224865.	−	81844.2i	0.598413	−	0.217805i	0.623426	+	0.781883i	$0.285740\pi$
$614$		0			0
$615$	−9049.62	−	15674.4i	−0.0239265	−	0.0414420i
$616$		0			0
$617$	49410.0	+	280218.i	0.129791	+	0.736081i	0.978346	+	0.206974i	$0.0663617\pi$
$617$	49410.0	+	280218.i	0.129791	+	0.736081i	−0.848555	+	0.529107i	$0.822527\pi$
$618$		0			0
$619$	−176197.	+	305181.i	−0.459850	+	0.796484i	−0.998953	−	0.0457565i	$-0.985430\pi$
$619$	−176197.	+	305181.i	−0.459850	+	0.796484i	0.539103	+	0.842240i	$0.318763\pi$
$620$		0			0
$621$	57982.1	+	69100.4i	0.150353	+	0.179183i
$622$		0			0
$623$	5787.84	−	15902.0i	0.0149122	−	0.0409708i
$624$		0			0
$625$	40892.7	−	231914.i	0.104685	−	0.593699i
$626$		0			0
$627$	35652.4	−	40073.3i	0.0906887	−	0.101934i
$628$		0			0
$629$	383984.	+	67706.8i	0.970537	+	0.171132i
$630$		0			0
$631$	402796.	+	146606.i	1.01164	+	0.368207i	0.794062	−	0.607836i	$-0.207962\pi$
$631$	402796.	+	146606.i	1.01164	+	0.368207i	0.217578	+	0.976043i	$0.430184\pi$
$632$		0			0
$633$	−29185.9	+	24489.8i	−0.0728392	+	0.0611193i
$634$		0			0
$635$	187512.	+	108260.i	0.465030	+	0.268485i
$636$		0			0
$637$	−546881.	+	96429.9i	−1.34777	+	0.237647i
$638$		0			0
$639$	−178538.	+	103079.i	−0.437249	+	0.252446i
$640$		0			0
$641$	11509.8	+	31622.9i	0.0280125	+	0.0769637i	0.952911	−	0.303251i	$-0.0980722\pi$
$641$	11509.8	+	31622.9i	0.0280125	+	0.0769637i	−0.924898	+	0.380215i	$0.875850\pi$
$642$		0			0
$643$	−15196.8	−	12751.6i	−0.0367562	−	0.0308421i	0.624225	−	0.781245i	$-0.285415\pi$
$643$	−15196.8	−	12751.6i	−0.0367562	−	0.0308421i	−0.660981	+	0.750403i	$0.729859\pi$
$644$		0			0
$645$		−	2914.54i		−	0.00700568i
$646$		0			0
$647$	609119.			1.45510			0.727551	−	0.686053i	$-0.240658\pi$
$647$	609119.			1.45510			0.727551	+	0.686053i	$0.240658\pi$
$648$		0			0
$649$	285311.	−	340020.i	0.677375	−	0.807264i
$650$		0			0
$651$	−2568.64	+	934.908i	−0.00606095	+	0.00220601i
$652$		0			0
$653$	−318930.	−	552403.i	−0.747944	−	1.29548i	−0.948807	−	0.315858i	$-0.897708\pi$
$653$	−318930.	−	552403.i	−0.747944	−	1.29548i	0.200863	−	0.979619i	$-0.435626\pi$
$654$		0			0
$655$	−27391.3	−	155344.i	−0.0638455	−	0.362086i
$656$		0			0
$657$	190938.	−	330714.i	0.442345	−	0.766163i
$658$		0			0
$659$	201555.	+	240204.i	0.464111	+	0.553106i	0.946438	−	0.322885i	$-0.104653\pi$
$659$	201555.	+	240204.i	0.464111	+	0.553106i	−0.482327	+	0.875991i	$0.660208\pi$
$660$		0			0
$661$	160266.	−	440328.i	0.366808	−	1.00780i	−0.609759	−	0.792587i	$-0.708734\pi$
$661$	160266.	−	440328.i	0.366808	−	1.00780i	0.976568	−	0.215211i	$-0.0690440\pi$
$662$		0			0
$663$	−13648.0	+	77401.7i	−0.0310486	+	0.176085i
$664$		0			0
$665$	−5271.07	+	25545.5i	−0.0119194	+	0.0577659i
$666$		0			0
$667$	−1.10897e6	−	195542.i	−2.49270	−	0.439530i
$668$		0			0
$669$	−34721.1	−	12637.5i	−0.0775786	−	0.0282363i
$670$		0			0
$671$	−799327.	+	670715.i	−1.77533	+	1.48968i
$672$		0			0
$673$	418973.	+	241894.i	0.925031	+	0.534067i	0.885237	−	0.465141i	$-0.153996\pi$
$673$	418973.	+	241894.i	0.925031	+	0.534067i	0.0397942	+	0.999208i	$0.487330\pi$
$674$		0			0
$675$	67099.6	−	11831.5i	0.147269	−	0.0259676i
$676$		0			0
$677$	−439608.	+	253808.i	−0.959154	+	0.553768i	−0.895913	−	0.444230i	$-0.853477\pi$
$677$	−439608.	+	253808.i	−0.959154	+	0.553768i	−0.0632415	+	0.997998i	$0.520144\pi$
$678$		0			0
$679$	−17290.0	−	47503.9i	−0.0375021	−	0.103036i
$680$		0			0
$681$	−6981.83	−	5858.45i	−0.0150548	−	0.0126325i
$682$		0			0
$683$			560367.i			1.20124i	0.799533	+	0.600622i	$0.205080\pi$
$683$			560367.i			1.20124i	−0.799533	+	0.600622i	$0.794920\pi$
$684$		0			0
$685$	−92176.1			−0.196443
$686$		0			0
$687$	−19252.2	+	22943.8i	−0.0407912	+	0.0486130i
$688$		0			0
$689$	191527.	−	69710.1i	0.403452	−	0.146844i
$690$		0			0
$691$	197689.	+	342407.i	0.414024	+	0.717111i	0.995325	−	0.0965773i	$-0.0307895\pi$
$691$	197689.	+	342407.i	0.414024	+	0.717111i	−0.581301	+	0.813689i	$0.697456\pi$
$692$		0			0
$693$	20511.8	+	116328.i	0.0427107	+	0.242224i
$694$		0			0
$695$	−94361.0	+	163438.i	−0.195354	+	0.338364i
$696$		0			0
$697$	672616.	+	801593.i	1.38453	+	1.65002i
$698$		0			0
$699$	17896.1	−	49169.3i	0.0366273	−	0.100633i
$700$		0			0
$701$	47480.8	−	269277.i	0.0966233	−	0.547978i	−0.897615	−	0.440781i	$-0.854701\pi$
$701$	47480.8	−	269277.i	0.0966233	−	0.547978i	0.994238	−	0.107197i	$-0.0341874\pi$
$702$		0			0
$703$	293303.	−	158200.i	0.593480	−	0.320107i
$704$		0			0
$705$	7831.08	+	1380.83i	0.0157559	+	0.00277819i
$706$		0			0
$707$	−7863.67	−	2862.14i	−0.0157321	−	0.00572602i
$708$		0			0
$709$	−223386.	+	187443.i	−0.444389	+	0.372886i	−0.837349	−	0.546669i	$-0.815895\pi$
$709$	−223386.	+	187443.i	−0.444389	+	0.372886i	0.392960	+	0.919556i	$0.371451\pi$
$710$		0			0
$711$	−804116.	−	464257.i	−1.59067	−	0.918373i
$712$		0			0
$713$	315256.	−	55588.1i	0.620132	−	0.109346i
$714$		0			0
$715$	−362476.	+	209276.i	−0.709034	+	0.409361i
$716$		0			0
$717$	13464.8	+	36994.1i	0.0261915	+	0.0719605i
$718$		0			0
$719$	59480.3	+	49909.9i	0.115058	+	0.0965448i	0.698501	−	0.715609i	$-0.253851\pi$
$719$	59480.3	+	49909.9i	0.115058	+	0.0965448i	−0.583444	+	0.812154i	$0.698295\pi$
$720$		0			0
$721$			28563.1i			0.0549459i
$722$		0			0
$723$	55785.4			0.106719
$724$		0			0
$725$	−546739.	+	651578.i	−1.04017	+	1.23962i
$726$		0			0
$727$	−727130.	+	264654.i	−1.37576	+	0.500737i	−0.920891	−	0.389821i	$-0.872537\pi$
$727$	−727130.	+	264654.i	−1.37576	+	0.500737i	−0.454872	+	0.890557i	$0.650315\pi$
$728$		0			0
$729$	−253687.	−	439399.i	−0.477357	−	0.826807i
$730$		0			0
$731$	29260.4	+	165944.i	0.0547577	+	0.310546i
$732$		0			0
$733$	−521382.	+	903059.i	−0.970393	+	1.68077i	−0.276025	+	0.961150i	$0.589017\pi$
$733$	−521382.	+	903059.i	−0.970393	+	1.68077i	−0.694368	+	0.719620i	$0.744316\pi$
$734$		0			0
$735$	−10992.5	−	13100.4i	−0.0203481	−	0.0242499i
$736$		0			0
$737$	426082.	−	1.17065e6i	0.784437	−	2.15522i
$738$		0			0
$739$	56940.7	−	322926.i	0.104264	−	0.591309i	−0.887248	−	0.461293i	$-0.847386\pi$
$739$	56940.7	−	322926.i	0.104264	−	0.591309i	0.991512	−	0.130017i	$-0.0415031\pi$
$740$		0			0
$741$	31889.1	+	59122.7i	0.0580773	+	0.107676i
$742$		0			0
$743$	336541.	+	59341.2i	0.609621	+	0.107493i	0.469932	−	0.882703i	$-0.344278\pi$
$743$	336541.	+	59341.2i	0.609621	+	0.107493i	0.139689	+	0.990195i	$0.455390\pi$
$744$		0			0
$745$	−31558.4	−	11486.3i	−0.0568594	−	0.0206951i
$746$		0			0
$747$	−634925.	+	532766.i	−1.13784	+	0.954762i
$748$		0			0
$749$	−105283.	−	60784.9i	−0.187669	−	0.108351i
$750$		0			0
$751$	−577391.	+	101810.i	−1.02374	+	0.180513i	−0.660220	−	0.751073i	$-0.729537\pi$
$751$	−577391.	+	101810.i	−1.02374	+	0.180513i	−0.363522	+	0.931586i	$0.618426\pi$
$752$		0			0
$753$	−23758.9	+	13717.2i	−0.0419021	+	0.0241922i
$754$		0			0
$755$	54399.1	+	149460.i	0.0954328	+	0.262200i
$756$		0			0
$757$	342910.	+	287735.i	0.598395	+	0.502113i	0.890929	−	0.454142i	$-0.150054\pi$
$757$	342910.	+	287735.i	0.598395	+	0.502113i	−0.292534	+	0.956255i	$0.594499\pi$
$758$		0			0
$759$			105877.i			0.183788i
$760$		0			0
$761$	741840.			1.28098			0.640488	−	0.767968i	$-0.278732\pi$
$761$	741840.			1.28098			0.640488	+	0.767968i	$0.278732\pi$
$762$		0			0
$763$	63399.4	−	75556.4i	0.108902	−	0.129784i
$764$		0			0
$765$	297167.	−	108160.i	0.507782	−	0.184818i
$766$		0			0
$767$	277944.	+	481414.i	0.472462	+	0.818329i
$768$		0			0
$769$	−75631.6	−	428928.i	−0.127894	−	0.725323i	−0.979547	−	0.201217i	$-0.935510\pi$
$769$	−75631.6	−	428928.i	−0.127894	−	0.725323i	0.851653	−	0.524107i	$-0.175601\pi$
$770$		0			0
$771$	−35539.8	+	61556.7i	−0.0597869	+	0.103554i
$772$		0			0
$773$	260878.	+	310902.i	0.436595	+	0.520313i	0.938813	−	0.344428i	$-0.111927\pi$
$773$	260878.	+	310902.i	0.436595	+	0.520313i	−0.502218	+	0.864741i	$0.667483\pi$
$774$		0			0
$775$	82700.0	−	227216.i	0.137690	−	0.378300i
$776$		0			0
$777$	975.385	−	5531.69i	0.00161560	−	0.00916253i
$778$		0			0
$779$	875893.	+	180732.i	1.44337	+	0.297824i
$780$		0			0
$781$	−478426.	−	84359.4i	−0.784355	−	0.138303i
$782$		0			0
$783$	−187976.	−	68417.5i	−0.306604	−	0.111595i
$784$		0			0
$785$	179431.	−	150561.i	0.291178	−	0.244327i
$786$		0			0
$787$	334739.	+	193261.i	0.540451	+	0.312030i	0.745262	−	0.666772i	$-0.232325\pi$
$787$	334739.	+	193261.i	0.540451	+	0.312030i	−0.204811	+	0.978802i	$0.565658\pi$
$788$		0			0
$789$	15658.7	−	2761.06i	0.0251537	−	0.00443528i
$790$		0			0
$791$	60879.3	−	35148.7i	0.0973008	−	0.0561767i
$792$		0			0
$793$	−446950.	−	1.22799e6i	−0.710743	−	1.95275i
$794$		0			0
$795$	4808.25	+	4034.60i	0.00760769	+	0.00638361i
$796$		0			0
$797$		−	390632.i		−	0.614966i	−0.951554	−	0.307483i	$-0.900513\pi$
$797$		−	390632.i		−	0.614966i	0.951554	−	0.307483i	$-0.0994868\pi$
$798$		0			0
$799$	−459737.			−0.720139
$800$		0			0
$801$	−112715.	+	134329.i	−0.175678	+	0.209365i
$802$		0			0
$803$	845610.	−	307777.i	1.31141	−	0.477315i
$804$		0			0
$805$	−25743.8	−	44589.5i	−0.0397265	−	0.0688083i
$806$		0			0
$807$	−2962.63	−	16801.9i	−0.00454915	−	0.0257995i
$808$		0			0
$809$	−446878.	+	774016.i	−0.682798	+	1.18264i	0.291326	+	0.956624i	$0.405904\pi$
$809$	−446878.	+	774016.i	−0.682798	+	1.18264i	−0.974123	+	0.226017i	$0.927430\pi$
$810$		0			0
$811$	−590057.	−	703202.i	−0.897123	−	1.06915i	−0.997245	−	0.0741749i	$-0.976368\pi$
$811$	−590057.	−	703202.i	−0.897123	−	1.06915i	0.100122	−	0.994975i	$-0.468077\pi$
$812$		0			0
$813$	29767.1	−	81784.4i	0.0450355	−	0.123734i
$814$		0			0
$815$	−38300.9	+	217215.i	−0.0576626	+	0.327021i
$816$		0			0
$817$	107598.	+	95727.3i	0.161198	+	0.143414i
$818$		0			0
$819$	−145687.	−	25688.6i	−0.217197	−	0.0382977i
$820$		0			0
$821$	−218235.	−	79431.0i	−0.323771	−	0.117843i	0.175021	−	0.984565i	$-0.444001\pi$
$821$	−218235.	−	79431.0i	−0.323771	−	0.117843i	−0.498792	+	0.866722i	$0.666223\pi$
$822$		0			0
$823$	712295.	−	597687.i	1.05162	−	0.882417i	0.0583605	−	0.998296i	$-0.481413\pi$
$823$	712295.	−	597687.i	1.05162	−	0.882417i	0.993263	+	0.115878i	$0.0369683\pi$
$824$		0			0
$825$	69258.5	+	39986.4i	0.101757	+	0.0587496i
$826$		0			0
$827$	−1.07287e6	+	189177.i	−1.56869	+	0.276603i	−0.889353	−	0.457221i	$-0.848845\pi$
$827$	−1.07287e6	+	189177.i	−1.56869	+	0.276603i	−0.679340	+	0.733824i	$0.737734\pi$
$828$		0			0
$829$	51152.8	−	29533.1i	0.0744322	−	0.0429734i	−0.462322	−	0.886712i	$-0.652984\pi$
$829$	51152.8	−	29533.1i	0.0744322	−	0.0429734i	0.536754	+	0.843739i	$0.319650\pi$
$830$		0			0
$831$	30396.5	+	83513.6i	0.0440171	+	0.120936i
$832$		0			0
$833$	757398.	+	635532.i	1.09153	+	0.915899i
$834$		0			0
$835$		−	360749.i		−	0.517408i
$836$		0			0
$837$	56866.6			0.0811720
$838$		0			0
$839$	841547.	−	1.00292e6i	1.19551	−	1.42476i	0.316083	−	0.948732i	$-0.397632\pi$
$839$	841547.	−	1.00292e6i	1.19551	−	1.42476i	0.879431	−	0.476027i	$-0.157923\pi$
$840$		0			0
$841$	1.68200e6	−	612200.i	2.37813	−	0.865568i
$842$		0			0
$843$	−14269.3	−	24715.2i	−0.0200793	−	0.0347784i
$844$		0			0
$845$	−44830.3	−	254246.i	−0.0627854	−	0.356074i
$846$		0			0
$847$	−82387.5	+	142699.i	−0.114840	+	0.198909i
$848$		0			0
$849$	−58454.1	−	69662.9i	−0.0810960	−	0.0966465i
$850$		0			0
$851$	−224984.	+	618139.i	−0.310666	+	0.853547i
$852$		0			0
$853$	−11939.6	+	67712.7i	−0.0164093	+	0.0930620i	−0.991913	−	0.126923i	$-0.959490\pi$
$853$	−11939.6	+	67712.7i	−0.0164093	+	0.0930620i	0.975503	+	0.219985i	$0.0706009\pi$
$854$		0			0
$855$	141861.	−	230062.i	0.194058	−	0.314712i
$856$		0			0
$857$	590017.	+	104036.i	0.803347	+	0.141652i	0.560221	−	0.828343i	$-0.310716\pi$
$857$	590017.	+	104036.i	0.803347	+	0.141652i	0.243126	+	0.969995i	$0.421827\pi$
$858$		0			0
$859$	93942.4	+	34192.3i	0.127314	+	0.0463384i	0.404891	−	0.914365i	$-0.367309\pi$
$859$	93942.4	+	34192.3i	0.127314	+	0.0463384i	−0.277578	+	0.960703i	$0.589532\pi$
$860$		0			0
$861$	11547.8	−	9689.72i	0.0155773	−	0.0130709i
$862$		0			0
$863$	−1.10332e6	−	637004.i	−1.48143	−	0.855304i	−0.481652	−	0.876363i	$-0.659963\pi$
$863$	−1.10332e6	−	637004.i	−1.48143	−	0.855304i	−0.999778	+	0.0210585i	$0.993296\pi$
$864$		0			0
$865$	−39914.6	+	7038.02i	−0.0533457	+	0.00940629i
$866$		0			0
$867$	64452.7	−	37211.8i	0.0857438	−	0.0495042i
$868$		0			0
$869$	−748346.	−	2.05607e6i	−0.990976	−	2.72268i
$870$		0			0
$871$	1.19518e6	+	1.00287e6i	1.57542	+	1.32193i
$872$		0			0
$873$			523835.i			0.687331i
$874$		0			0
$875$	−84049.3			−0.109779
$876$		0			0
$877$	57433.7	−	68446.9i	0.0746737	−	0.0889927i	−0.727414	−	0.686199i	$-0.759278\pi$
$877$	57433.7	−	68446.9i	0.0746737	−	0.0889927i	0.802088	+	0.597206i	$0.203723\pi$
$878$		0			0
$879$	−1516.23	+	551.862i	−0.00196240	+	0.000714254i
$880$		0			0
$881$	337422.	+	584431.i	0.434732	+	0.752977i	0.997274	−	0.0737915i	$-0.0235099\pi$
$881$	337422.	+	584431.i	0.434732	+	0.752977i	−0.562542	+	0.826769i	$0.690177\pi$
$882$		0			0
$883$	−133261.	−	755759.i	−0.170915	−	0.969308i	−0.942754	−	0.333489i	$-0.891774\pi$
$883$	−133261.	−	755759.i	−0.170915	−	0.969308i	0.771839	−	0.635818i	$-0.219337\pi$
$884$		0			0
$885$	−8559.47	+	14825.4i	−0.0109285	+	0.0189287i
$886$		0			0
$887$	903796.	+	1.07710e6i	1.14874	+	1.36902i	0.918276	+	0.395940i	$0.129581\pi$
$887$	903796.	+	1.07710e6i	1.14874	+	1.36902i	0.230468	+	0.973080i	$0.425974\pi$
$888$		0			0
$889$	−61678.5	+	169460.i	−0.0780423	+	0.214419i
$890$		0			0
$891$	210907.	−	1.19612e6i	0.265666	−	1.50667i
$892$		0			0
$893$	−308187.	+	243751.i	−0.386466	+	0.305664i
$894$		0			0
$895$	−395233.	−	69690.2i	−0.493409	−	0.0870013i
$896$		0			0
$897$	−124602.	−	45351.3i	−0.154860	−	0.0563644i
$898$		0			0
$899$	−543819.	+	456318.i	−0.672876	+	0.564610i
$900$		0			0
$901$	−314270.	−	181444.i	−0.387127	−	0.223508i
$902$		0			0
$903$	2390.59	−	421.526i	0.00293177	−	0.000516950i
$904$		0			0
$905$	84887.0	−	49009.6i	0.103644	−	0.0598389i
$906$		0			0
$907$	132260.	+	363381.i	0.160773	+	0.441721i	0.993756	−	0.111579i	$-0.0355907\pi$
$907$	132260.	+	363381.i	0.160773	+	0.441721i	−0.832982	+	0.553300i	$0.813368\pi$
$908$		0			0
$909$	66427.0	+	55738.9i	0.0803928	+	0.0674575i
$910$		0			0
$911$		−	260514.i		−	0.313902i	−0.987606	−	0.156951i	$-0.949834\pi$
$911$		−	260514.i		−	0.313902i	0.987606	−	0.156951i	$-0.0501665\pi$
$912$		0			0
$913$	−1.95313e6			−2.34310
$914$		0			0
$915$	25868.1	−	30828.4i	0.0308974	−	0.0368221i
$916$		0			0
$917$	123456.	−	44934.3i	0.146816	−	0.0534367i
$918$		0			0
$919$	24980.7	+	43267.9i	0.0295783	+	0.0512312i	0.880436	−	0.474166i	$-0.157250\pi$
$919$	24980.7	+	43267.9i	0.0295783	+	0.0512312i	−0.850857	+	0.525397i	$0.823917\pi$
$920$		0			0
$921$	15364.2	+	87134.8i	0.0181130	+	0.102724i
$922$		0			0
$923$	304208.	−	526903.i	0.357081	−	0.618482i
$924$		0			0
$925$	319382.	+	380624.i	0.373273	+	0.444850i
$926$		0			0
$927$	101230.	−	278127.i	0.117801	−	0.323656i
$928$		0			0
$929$	−27203.8	+	154280.i	−0.0315208	+	0.178764i	−0.996504	−	0.0835497i	$-0.973374\pi$
$929$	−27203.8	+	154280.i	−0.0315208	+	0.178764i	0.964983	+	0.262313i	$0.0844854\pi$
$930$		0			0
$931$	844682.	+	24461.6i	0.974527	+	0.0282218i
$932$		0			0
$933$	12854.8	+	2266.65i	0.0147674	+	0.00260388i
$934$		0			0
$935$	700266.	+	254876.i	0.801014	+	0.291545i
$936$		0			0
$937$	−128884.	+	108146.i	−0.146798	+	0.123178i	−0.713229	−	0.700931i	$-0.752768\pi$
$937$	−128884.	+	108146.i	−0.146798	+	0.123178i	0.566431	+	0.824109i	$0.308324\pi$
$938$		0			0
$939$	−108822.	−	62828.2i	−0.123420	−	0.0712563i
$940$		0			0
$941$	95914.7	−	16912.3i	0.108319	−	0.0190996i	−0.119226	−	0.992867i	$-0.538041\pi$
$941$	95914.7	−	16912.3i	0.108319	−	0.0190996i	0.227545	+	0.973768i	$0.426930\pi$
$942$		0			0
$943$	−1.52887e6	+	882691.i	−1.71928	+	0.992625i
$944$		0			0
$945$	−3128.23	−	8594.75i	−0.00350296	−	0.00962431i
$946$		0			0
$947$	442250.	+	371092.i	0.493138	+	0.413792i	0.855149	−	0.518382i	$-0.173465\pi$
$947$	442250.	+	371092.i	0.493138	+	0.413792i	−0.362011	+	0.932174i	$0.617910\pi$
$948$		0			0
$949$			1.12699e6i			1.25138i
$950$		0			0
$951$	123454.			0.136504
$952$		0			0
$953$	−776882.	+	925852.i	−0.855400	+	1.01943i	0.144153	+	0.989555i	$0.453954\pi$
$953$	−776882.	+	925852.i	−0.855400	+	1.01943i	−0.999554	+	0.0298712i	$0.990490\pi$
$954$		0			0
$955$	−89256.6	+	32486.7i	−0.0978664	+	0.0356205i
$956$		0			0
$957$	−117398.	−	203339.i	−0.128185	−	0.222022i
$958$		0			0
$959$	−13331.3	−	75605.5i	−0.0144956	−	0.0822084i
$960$		0			0
$961$	−360856.	+	625020.i	−0.390739	+	0.676780i
$962$		0			0
$963$	809738.	+	965008.i	0.873156	+	1.04059i
$964$		0			0
$965$	160484.	−	440926.i	0.172337	−	0.473491i
$966$		0			0
$967$	−45497.4	+	258029.i	−0.0486557	+	0.275940i	−0.999423	−	0.0339655i	$-0.989186\pi$
$967$	−45497.4	+	258029.i	−0.0486557	+	0.275940i	0.950767	+	0.309906i	$0.100297\pi$
$968$		0			0
$969$	44141.9	−	111156.i	0.0470115	−	0.118382i
$970$		0			0
$971$	−1.41989e6	−	250365.i	−1.50597	−	0.265543i	−0.641067	−	0.767485i	$-0.721508\pi$
$971$	−1.41989e6	−	250365.i	−1.50597	−	0.265543i	−0.864901	+	0.501943i	$0.832619\pi$
$972$		0			0
$973$	−147704.	−	53759.8i	−0.156015	−	0.0567848i
$974$		0			0
$975$	−76724.4	+	64379.4i	−0.0807095	+	0.0677233i
$976$		0			0
$977$	169383.	+	97793.1i	0.177451	+	0.102452i	0.586095	−	0.810243i	$-0.300665\pi$
$977$	169383.	+	97793.1i	0.177451	+	0.102452i	−0.408643	+	0.912694i	$0.633998\pi$
$978$		0			0
$979$	−406940.	+	71754.5i	−0.424585	+	0.0748658i
$980$		0			0
$981$	−885114.	+	511021.i	−0.919732	+	0.531007i
$982$		0			0
$983$	−219280.	−	602467.i	−0.226930	−	0.623485i	0.773010	−	0.634393i	$-0.218750\pi$
$983$	−219280.	−	602467.i	−0.226930	−	0.623485i	−0.999941	+	0.0109081i	$0.996528\pi$
$984$		0			0
$985$	−160373.	−	134569.i	−0.165295	−	0.138699i
$986$		0			0
$987$			6622.99i			0.00679860i
$988$		0			0
$989$	−284281.			−0.290640
$990$		0			0
$991$	400726.	−	477567.i	0.408038	−	0.486281i	−0.522415	−	0.852691i	$-0.674969\pi$
$991$	400726.	−	477567.i	0.408038	−	0.486281i	0.930453	+	0.366411i	$0.119413\pi$
$992$		0			0
$993$	80932.6	−	29457.1i	0.0820777	−	0.0298738i
$994$		0			0
$995$	−215045.	−	372469.i	−0.217212	−	0.376222i
$996$		0			0
$997$	178904.	+	1.01461e6i	0.179982	+	1.02073i	0.932235	+	0.361854i	$0.117856\pi$
$997$	178904.	+	1.01461e6i	0.179982	+	1.02073i	−0.752253	+	0.658875i	$0.771033\pi$
$998$		0			0
$999$	−58427.3	+	101199.i	−0.0585443	+	0.101402i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.j.a.21.3	✓	42
19.10	odd	18	inner	76.5.j.a.29.3	yes	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.j.a.21.3	✓	42	1.1	even	1	trivial
76.5.j.a.29.3	yes	42	19.10	odd	18	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.j (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.504186 + 0.600866i) q^{3} +(-8.75235 + 3.18559i) q^{5} +(-3.87876 - 6.71820i) q^{7} +(13.9587 + 79.1635i) q^{9} +O(q^{10})\)	\(q+(-0.504186 + 0.600866i) q^{3} +(-8.75235 + 3.18559i) q^{5} +(-3.87876 - 6.71820i) q^{7} +(13.9587 + 79.1635i) q^{9} +(-94.7123 + 164.047i) q^{11} +(-152.490 - 181.730i) q^{13} +(2.49870 - 6.86512i) q^{15} +(-73.3453 + 415.962i) q^{17} +(171.374 + 317.729i) q^{19} +(5.99236 + 1.05661i) q^{21} +(-669.617 - 243.721i) q^{23} +(-412.322 + 345.979i) q^{25} +(-109.627 - 63.2931i) q^{27} +(1556.26 - 274.410i) q^{29} +(-389.047 + 224.616i) q^{31} +(-50.8173 - 139.619i) q^{33} +(55.3497 + 46.4439i) q^{35} -923.124i q^{37} +186.079 q^{39} +(1592.45 - 1897.81i) q^{41} +(374.881 - 136.446i) q^{43} +(-374.354 - 648.400i) q^{45} +(189.007 + 1071.91i) q^{47} +(1170.41 - 2027.21i) q^{49} +(-212.957 - 253.793i) q^{51} +(-293.848 + 807.340i) q^{53} +(306.370 - 1737.51i) q^{55} +(-277.317 - 57.2217i) q^{57} +(-2307.63 - 406.897i) q^{59} +(5176.31 + 1884.02i) q^{61} +(477.695 - 400.833i) q^{63} +(1913.56 + 1104.80i) q^{65} +(-6476.74 + 1142.02i) q^{67} +(484.055 - 279.469i) q^{69} +(877.159 + 2409.97i) q^{71} +(-3639.17 - 3053.62i) q^{73} -422.188i q^{75} +1469.46 q^{77} +(-7424.75 + 8848.48i) q^{79} +(-6025.19 + 2192.99i) q^{81} +(5155.43 + 8929.47i) q^{83} +(-683.142 - 3874.29i) q^{85} +(-619.759 + 1073.45i) q^{87} +(1402.20 + 1671.08i) q^{89} +(-629.430 + 1729.35i) q^{91} +(61.1878 - 347.013i) q^{93} +(-2512.09 - 2234.95i) q^{95} +(6417.59 + 1131.59i) q^{97} +(-14308.6 - 5207.89i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) \) 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9} - 45 q^{11} + 33 q^{13} - 393 q^{15} + 909 q^{17} + 1242 q^{19} + 1107 q^{21} - 360 q^{23} - 810 q^{25} - 7056 q^{27} - 2889 q^{29} + 2808 q^{31} + 10875 q^{33} + 6741 q^{35} - 3480 q^{39} - 3060 q^{41} - 8079 q^{43} - 4320 q^{45} - 2655 q^{47} - 474 q^{49} - 12222 q^{51} - 6705 q^{53} + 4623 q^{55} - 8022 q^{57} + 24309 q^{59} + 7104 q^{61} + 12063 q^{63} + 25245 q^{65} + 15573 q^{67} - 10881 q^{69} - 25506 q^{71} + 3036 q^{73} + 12924 q^{77} - 16839 q^{79} - 2208 q^{81} - 6363 q^{83} - 37890 q^{85} - 21924 q^{87} - 22644 q^{89} + 17418 q^{91} + 8184 q^{93} - 82413 q^{95} + 13383 q^{97} + 23565 q^{99}+O(q^{100}) \) 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9 - 45 * q^11 + 33 * q^13 - 393 * q^15 + 909 * q^17 + 1242 * q^19 + 1107 * q^21 - 360 * q^23 - 810 * q^25 - 7056 * q^27 - 2889 * q^29 + 2808 * q^31 + 10875 * q^33 + 6741 * q^35 - 3480 * q^39 - 3060 * q^41 - 8079 * q^43 - 4320 * q^45 - 2655 * q^47 - 474 * q^49 - 12222 * q^51 - 6705 * q^53 + 4623 * q^55 - 8022 * q^57 + 24309 * q^59 + 7104 * q^61 + 12063 * q^63 + 25245 * q^65 + 15573 * q^67 - 10881 * q^69 - 25506 * q^71 + 3036 * q^73 + 12924 * q^77 - 16839 * q^79 - 2208 * q^81 - 6363 * q^83 - 37890 * q^85 - 21924 * q^87 - 22644 * q^89 + 17418 * q^91 + 8184 * q^93 - 82413 * q^95 + 13383 * q^97 + 23565 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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