LMFDB - Embedded newform 76.5.j.a.13.5

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			13.5
Character	$\chi$	$=$	76.13
Dual form			76.5.j.a.41.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.65366 - 7.29088i) q^{3} +(0.405227 + 2.29816i) q^{5} +(23.2726 - 40.3094i) q^{7} +(15.9346 + 13.3707i) q^{9} +O(q^{10})$	\(q+(2.65366 - 7.29088i) q^{3} +(0.405227 + 2.29816i) q^{5} +(23.2726 - 40.3094i) q^{7} +(15.9346 + 13.3707i) q^{9} +(-88.3294 - 152.991i) q^{11} +(-37.7242 - 103.646i) q^{13} +(17.8309 + 3.14407i) q^{15} +(-57.1483 + 47.9531i) q^{17} +(-192.492 - 305.398i) q^{19} +(-232.133 - 276.646i) q^{21} +(15.6411 - 88.7052i) q^{23} +(582.191 - 211.900i) q^{25} +(684.034 - 394.927i) q^{27} +(-121.887 + 145.259i) q^{29} +(670.369 + 387.038i) q^{31} +(-1349.84 + 238.013i) q^{33} +(102.068 + 37.1497i) q^{35} +1355.70i q^{37} -855.781 q^{39} +(-316.776 + 870.336i) q^{41} +(298.395 + 1692.28i) q^{43} +(-24.2709 + 42.0384i) q^{45} +(1738.43 + 1458.72i) q^{47} +(117.269 + 203.116i) q^{49} +(197.968 + 543.913i) q^{51} +(-1141.83 - 201.336i) q^{53} +(315.804 - 264.991i) q^{55} +(-2737.43 + 593.012i) q^{57} +(2592.52 + 3089.64i) q^{59} +(702.333 - 3983.13i) q^{61} +(909.805 - 331.142i) q^{63} +(222.909 - 128.697i) q^{65} +(1781.28 - 2122.85i) q^{67} +(-605.233 - 349.431i) q^{69} +(5745.13 - 1013.02i) q^{71} +(-4484.03 - 1632.05i) q^{73} -4806.99i q^{75} -8222.63 q^{77} +(1602.75 - 4403.53i) q^{79} +(-771.592 - 4375.92i) q^{81} +(-3711.86 + 6429.14i) q^{83} +(-133.362 - 111.904i) q^{85} +(735.618 + 1274.13i) q^{87} +(-1547.47 - 4251.63i) q^{89} +(-5055.87 - 891.486i) q^{91} +(4600.78 - 3860.51i) q^{93} +(623.850 - 566.132i) q^{95} +(3365.63 + 4011.01i) q^{97} +(638.106 - 3618.88i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{5}{18}\right)$	$1$

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
$3$	2.65366	−	7.29088i	0.294851	−	0.810098i	−0.700488	−	0.713664i	$-0.747034\pi$
$3$	2.65366	−	7.29088i	0.294851	−	0.810098i	0.995339	−	0.0964336i	$-0.0307435\pi$
$4$		0			0
$5$	0.405227	+	2.29816i	0.0162091	+	0.0919263i	0.991839	−	0.127496i	$-0.0406939\pi$
$5$	0.405227	+	2.29816i	0.0162091	+	0.0919263i	−0.975630	+	0.219422i	$0.929583\pi$
$6$		0			0
$7$	23.2726	−	40.3094i	0.474952	−	0.822641i	−0.524637	−	0.851326i	$-0.675799\pi$
$7$	23.2726	−	40.3094i	0.474952	−	0.822641i	0.999588	+	0.0286856i	$0.00913217\pi$
$8$		0			0
$9$	15.9346	+	13.3707i	0.196723	+	0.165070i
$10$		0			0
$11$	−88.3294	−	152.991i	−0.729995	−	1.26439i	−0.956885	−	0.290468i	$-0.906189\pi$
$11$	−88.3294	−	152.991i	−0.729995	−	1.26439i	0.226889	−	0.973921i	$-0.427144\pi$
$12$		0			0
$13$	−37.7242	−	103.646i	−0.223220	−	0.613293i	0.776641	−	0.629943i	$-0.216922\pi$
$13$	−37.7242	−	103.646i	−0.223220	−	0.613293i	−0.999861	+	0.0166508i	$0.994700\pi$
$14$		0			0
$15$	17.8309	+	3.14407i	0.0792486	+	0.0139737i
$16$		0			0
$17$	−57.1483	+	47.9531i	−0.197745	+	0.165928i	−0.736284	−	0.676673i	$-0.763421\pi$
$17$	−57.1483	+	47.9531i	−0.197745	+	0.165928i	0.538539	+	0.842601i	$0.318977\pi$
$18$		0			0
$19$	−192.492	−	305.398i	−0.533218	−	0.845978i
$19$	−192.492	−	305.398i	−0.533218	−	0.845978i
$20$		0			0
$21$	−232.133	−	276.646i	−0.526379	−	0.627314i
$22$		0			0
$23$	15.6411	−	88.7052i	0.0295673	−	0.167685i	−0.966449	−	0.256860i	$-0.917312\pi$
$23$	15.6411	−	88.7052i	0.0295673	−	0.167685i	0.996016	+	0.0891754i	$0.0284232\pi$
$24$		0			0
$25$	582.191	−	211.900i	0.931505	−	0.339040i
$26$		0			0
$27$	684.034	−	394.927i	0.938318	−	0.541738i
$28$		0			0
$29$	−121.887	+	145.259i	−0.144931	+	0.172721i	−0.833626	−	0.552329i	$-0.813739\pi$
$29$	−121.887	+	145.259i	−0.144931	+	0.172721i	0.688696	+	0.725051i	$0.258184\pi$
$30$		0			0
$31$	670.369	+	387.038i	0.697574	+	0.402745i	0.806443	−	0.591311i	$-0.201390\pi$
$31$	670.369	+	387.038i	0.697574	+	0.402745i	−0.108869	+	0.994056i	$0.534723\pi$
$32$		0			0
$33$	−1349.84	+	238.013i	−1.23952	+	0.218561i
$34$		0			0
$35$	102.068	+	37.1497i	0.0833208	+	0.0303263i
$36$		0			0
$37$			1355.70i			0.990283i	0.868812	+	0.495141i	$0.164884\pi$
$37$			1355.70i			0.990283i	−0.868812	+	0.495141i	$0.835116\pi$
$38$		0			0
$39$	−855.781			−0.562644
$40$		0			0
$41$	−316.776	+	870.336i	−0.188445	+	0.517749i	−0.997553	−	0.0699111i	$-0.977728\pi$
$41$	−316.776	+	870.336i	−0.188445	+	0.517749i	0.809108	+	0.587660i	$0.199951\pi$
$42$		0			0
$43$	298.395	+	1692.28i	0.161382	+	0.915243i	0.952717	+	0.303860i	$0.0982754\pi$
$43$	298.395	+	1692.28i	0.161382	+	0.915243i	−0.791335	+	0.611383i	$0.790613\pi$
$44$		0			0
$45$	−24.2709	+	42.0384i	−0.0119856	+	0.0207597i
$46$		0			0
$47$	1738.43	+	1458.72i	0.786978	+	0.660353i	0.944995	−	0.327083i	$-0.106066\pi$
$47$	1738.43	+	1458.72i	0.786978	+	0.660353i	−0.158018	+	0.987436i	$0.550510\pi$
$48$		0			0
$49$	117.269	+	203.116i	0.0488417	+	0.0845962i
$50$		0			0
$51$	197.968	+	543.913i	0.0761123	+	0.209117i
$52$		0			0
$53$	−1141.83	−	201.336i	−0.406491	−	0.0716753i	−0.0333356	−	0.999444i	$-0.510613\pi$
$53$	−1141.83	−	201.336i	−0.406491	−	0.0716753i	−0.373155	+	0.927769i	$0.621724\pi$
$54$		0			0
$55$	315.804	−	264.991i	0.104398	−	0.0876004i
$56$		0			0
$57$	−2737.43	+	593.012i	−0.842545	+	0.182521i
$58$		0			0
$59$	2592.52	+	3089.64i	0.744763	+	0.887574i	0.996783	−	0.0801483i	$-0.0255394\pi$
$59$	2592.52	+	3089.64i	0.744763	+	0.887574i	−0.252020	+	0.967722i	$0.581095\pi$
$60$		0			0
$61$	702.333	−	3983.13i	0.188749	−	1.07045i	−0.732295	−	0.680987i	$-0.761551\pi$
$61$	702.333	−	3983.13i	0.188749	−	1.07045i	0.921044	−	0.389459i	$-0.127338\pi$
$62$		0			0
$63$	909.805	−	331.142i	0.229228	−	0.0834321i
$64$		0			0
$65$	222.909	−	128.697i	0.0527595	−	0.0304607i
$66$		0			0
$67$	1781.28	−	2122.85i	0.396811	−	0.472900i	−0.530234	−	0.847851i	$-0.677896\pi$
$67$	1781.28	−	2122.85i	0.396811	−	0.472900i	0.927045	+	0.374951i	$0.122340\pi$
$68$		0			0
$69$	−605.233	−	349.431i	−0.127123	−	0.0733945i
$70$		0			0
$71$	5745.13	−	1013.02i	1.13968	−	0.200956i	0.428215	−	0.903677i	$-0.359143\pi$
$71$	5745.13	−	1013.02i	1.13968	−	0.200956i	0.711466	+	0.702721i	$0.248032\pi$
$72$		0			0
$73$	−4484.03	−	1632.05i	−0.841440	−	0.306259i	−0.114895	−	0.993378i	$-0.536653\pi$
$73$	−4484.03	−	1632.05i	−0.841440	−	0.306259i	−0.726545	+	0.687119i	$0.758875\pi$
$74$		0			0
$75$		−	4806.99i		−	0.854577i
$76$		0			0
$77$	−8222.63			−1.38685
$78$		0			0
$79$	1602.75	−	4403.53i	0.256810	−	0.705581i	−0.742549	−	0.669792i	$-0.766383\pi$
$79$	1602.75	−	4403.53i	0.256810	−	0.705581i	0.999359	−	0.0357890i	$-0.0113944\pi$
$80$		0			0
$81$	−771.592	−	4375.92i	−0.117603	−	0.666959i
$82$		0			0
$83$	−3711.86	+	6429.14i	−0.538810	+	0.933247i	0.460158	+	0.887837i	$0.347793\pi$
$83$	−3711.86	+	6429.14i	−0.538810	+	0.933247i	−0.998968	+	0.0454098i	$0.985541\pi$
$84$		0			0
$85$	−133.362	−	111.904i	−0.0184584	−	0.0154884i
$86$		0			0
$87$	735.618	+	1274.13i	0.0971883	+	0.168335i
$88$		0			0
$89$	−1547.47	−	4251.63i	−0.195362	−	0.536754i	0.802872	−	0.596152i	$-0.203304\pi$
$89$	−1547.47	−	4251.63i	−0.195362	−	0.536754i	−0.998234	+	0.0593976i	$0.981082\pi$
$90$		0			0
$91$	−5055.87	−	891.486i	−0.610538	−	0.107654i
$92$		0			0
$93$	4600.78	−	3860.51i	0.531943	−	0.446353i
$94$		0			0
$95$	623.850	−	566.132i	0.0691246	−	0.0627293i
$96$		0			0
$97$	3365.63	+	4011.01i	0.357704	+	0.426295i	0.914646	−	0.404257i	$-0.132470\pi$
$97$	3365.63	+	4011.01i	0.357704	+	0.426295i	−0.556942	+	0.830552i	$0.688025\pi$
$98$		0			0
$99$	638.106	−	3618.88i	0.0651062	−	0.369235i
$100$		0			0
$101$	−6086.56	+	2215.33i	−0.596663	+	0.217168i	−0.622658	−	0.782494i	$-0.713947\pi$
$101$	−6086.56	+	2215.33i	−0.596663	+	0.217168i	0.0259944	+	0.999662i	$0.491725\pi$
$102$		0			0
$103$	12871.6	−	7431.44i	1.21327	−	0.700484i	0.249803	−	0.968297i	$-0.419634\pi$
$103$	12871.6	−	7431.44i	1.21327	−	0.700484i	0.963471	+	0.267812i	$0.0863007\pi$
$104$		0			0
$105$	541.708	−	645.583i	0.0491346	−	0.0585563i
$106$		0			0
$107$	−13745.2	−	7935.81i	−1.20056	−	0.693144i	−0.239881	−	0.970802i	$-0.577108\pi$
$107$	−13745.2	−	7935.81i	−1.20056	−	0.693144i	−0.960680	+	0.277658i	$0.910442\pi$
$108$		0			0
$109$	10969.9	−	1934.29i	0.923314	−	0.162805i	0.308271	−	0.951299i	$-0.400250\pi$
$109$	10969.9	−	1934.29i	0.923314	−	0.162805i	0.615043	+	0.788493i	$0.289139\pi$
$110$		0			0
$111$	9884.23	+	3597.56i	0.802226	+	0.291986i
$112$		0			0
$113$			24537.3i			1.92163i	0.277193	+	0.960814i	$0.410596\pi$
$113$			24537.3i			1.92163i	−0.277193	+	0.960814i	$0.589404\pi$
$114$		0			0
$115$	210.197			0.0158939
$116$		0			0
$117$	784.706	−	2155.96i	0.0573239	−	0.157496i
$118$		0			0
$119$	602.969	+	3419.61i	0.0425796	+	0.241481i
$120$		0			0
$121$	−8283.67	+	14347.7i	−0.565786	+	0.979970i
$122$		0			0
$123$	5504.89	+	4619.16i	0.363864	+	0.305318i
$124$		0			0
$125$	1452.15	+	2515.20i	0.0929378	+	0.160973i
$126$		0			0
$127$	−1306.34	−	3589.15i	−0.0809935	−	0.222528i	0.892586	−	0.450878i	$-0.148889\pi$
$127$	−1306.34	−	3589.15i	−0.0809935	−	0.222528i	−0.973579	+	0.228350i	$0.926667\pi$
$128$		0			0
$129$	13130.1	+	2315.19i	0.789020	+	0.139125i
$130$		0			0
$131$	−25945.5	+	21770.9i	−1.51189	+	1.26862i	−0.651905	+	0.758301i	$0.726030\pi$
$131$	−25945.5	+	21770.9i	−1.51189	+	1.26862i	−0.859983	+	0.510323i	$0.829526\pi$
$132$		0			0
$133$	−16790.2	+	651.813i	−0.949189	+	0.0368485i
$134$		0			0
$135$	1184.79	+	1411.98i	0.0650092	+	0.0774750i
$136$		0			0
$137$	−771.241	+	4373.93i	−0.0410912	+	0.233040i	−0.998436	−	0.0559085i	$-0.982194\pi$
$137$	−771.241	+	4373.93i	−0.0410912	+	0.233040i	0.957345	+	0.288948i	$0.0933056\pi$
$138$		0			0
$139$	9764.51	−	3553.99i	0.505383	−	0.183944i	−0.0767305	−	0.997052i	$-0.524448\pi$
$139$	9764.51	−	3553.99i	0.505383	−	0.183944i	0.582114	+	0.813107i	$0.302226\pi$
$140$		0			0
$141$	15248.6	−	8803.77i	0.766992	−	0.442823i
$142$		0			0
$143$	−12524.8	+	14926.5i	−0.612491	+	0.729938i
$144$		0			0
$145$	−383.219	−	221.252i	−0.0182268	−	0.0105233i
$146$		0			0
$147$	1792.08	−	315.993i	0.0829323	−	0.0146232i
$148$		0			0
$149$	17999.4	+	6551.26i	0.810748	+	0.295088i	0.713933	−	0.700214i	$-0.246912\pi$
$149$	17999.4	+	6551.26i	0.810748	+	0.295088i	0.0968155	+	0.995302i	$0.469134\pi$
$150$		0			0
$151$			16772.4i			0.735598i	0.929905	+	0.367799i	$0.119889\pi$
$151$			16772.4i			0.735598i	−0.929905	+	0.367799i	$0.880111\pi$
$152$		0			0
$153$	−1551.80			−0.0662909
$154$		0			0
$155$	−617.822	+	1697.45i	−0.0257158	+	0.0706535i
$156$		0			0
$157$	−6522.77	−	36992.5i	−0.264626	−	1.50077i	−0.770097	−	0.637926i	$-0.779792\pi$
$157$	−6522.77	−	36992.5i	−0.264626	−	1.50077i	0.505471	−	0.862844i	$-0.331319\pi$
$158$		0			0
$159$	−4497.96	+	7790.69i	−0.177918	+	0.308164i
$160$		0			0
$161$	−3211.64	−	2694.89i	−0.123901	−	0.103965i
$162$		0			0
$163$	−11507.2	−	19931.0i	−0.433105	−	0.750160i	0.564034	−	0.825752i	$-0.309249\pi$
$163$	−11507.2	−	19931.0i	−0.433105	−	0.750160i	−0.997139	+	0.0755914i	$0.975916\pi$
$164$		0			0
$165$	−1093.98	−	3005.69i	−0.0401829	−	0.110402i
$166$		0			0
$167$	−7735.07	−	1363.90i	−0.277352	−	0.0489046i	0.0332420	−	0.999447i	$-0.489417\pi$
$167$	−7735.07	−	1363.90i	−0.277352	−	0.0489046i	−0.310594	+	0.950543i	$0.600528\pi$
$168$		0			0
$169$	12559.5	−	10538.7i	0.439744	−	0.368989i
$170$		0			0
$171$	1016.11	−	7440.14i	0.0347494	−	0.254442i
$172$		0			0
$173$	11562.8	+	13780.0i	0.386341	+	0.460424i	0.923805	−	0.382863i	$-0.125062\pi$
$173$	11562.8	+	13780.0i	0.386341	+	0.460424i	−0.537464	+	0.843287i	$0.680617\pi$
$174$		0			0
$175$	5007.55	−	28399.2i	0.163512	−	0.927321i
$176$		0			0
$177$	29405.9	−	10702.9i	0.938616	−	0.341628i
$178$		0			0
$179$	19162.8	−	11063.7i	0.598072	−	0.345297i	−0.170211	−	0.985408i	$-0.554445\pi$
$179$	19162.8	−	11063.7i	0.598072	−	0.345297i	0.768283	+	0.640111i	$0.221111\pi$
$180$		0			0
$181$	20579.6	−	24525.8i	0.628175	−	0.748629i	−0.354278	−	0.935140i	$-0.615273\pi$
$181$	20579.6	−	24525.8i	0.628175	−	0.748629i	0.982453	+	0.186511i	$0.0597179\pi$
$182$		0			0
$183$	−27176.8	−	15690.5i	−0.811513	−	0.468527i
$184$		0			0
$185$	−3115.61	+	549.365i	−0.0910330	+	0.0160516i
$186$		0			0
$187$	12384.3	+	4507.51i	0.354150	+	0.128900i
$188$		0			0
$189$		−	36764.0i		−	1.02920i
$190$		0			0
$191$	−54589.8			−1.49639			−0.748196	−	0.663478i	$-0.769080\pi$
$191$	−54589.8			−1.49639			−0.748196	+	0.663478i	$0.769080\pi$
$192$		0			0
$193$	−18704.7	+	51390.8i	−0.502154	+	1.37966i	0.387013	+	0.922074i	$0.373507\pi$
$193$	−18704.7	+	51390.8i	−0.502154	+	1.37966i	−0.889167	+	0.457582i	$0.848716\pi$
$194$		0			0
$195$	−346.786	−	1966.72i	−0.00911994	−	0.0517218i
$196$		0			0
$197$	3388.48	−	5869.03i	0.0873118	−	0.151229i	−0.819062	−	0.573705i	$-0.805506\pi$
$197$	3388.48	−	5869.03i	0.0873118	−	0.151229i	0.906374	+	0.422476i	$0.138839\pi$
$198$		0			0
$199$	−49441.1	−	41486.0i	−1.24848	−	1.04760i	−0.996812	−	0.0797869i	$-0.974576\pi$
$199$	−49441.1	−	41486.0i	−1.24848	−	1.04760i	−0.251669	−	0.967813i	$-0.580980\pi$
$200$		0			0
$201$	−10750.5	−	18620.4i	−0.266095	−	0.460891i
$202$		0			0
$203$	3018.67	+	8293.73i	0.0732527	+	0.201260i
$204$		0			0
$205$	−2128.53	−	375.318i	−0.0506492	−	0.00893083i
$206$		0			0
$207$	1435.29	−	1204.35i	0.0334964	−	0.0281068i
$208$		0			0
$209$	−29720.5	+	56425.1i	−0.680398	+	1.29175i
$210$		0			0
$211$	43222.2	+	51510.2i	0.970827	+	1.15699i	0.987578	+	0.157127i	$0.0502233\pi$
$211$	43222.2	+	51510.2i	0.970827	+	1.15699i	−0.0167510	+	0.999860i	$0.505332\pi$
$212$		0			0
$213$	7859.82	−	44575.3i	0.173242	−	0.982505i
$214$		0			0
$215$	−3768.22	+	1371.52i	−0.0815190	+	0.0296705i
$216$		0			0
$217$	31202.5	−	18014.8i	0.662628	−	0.382569i
$218$		0			0
$219$	−23798.2	+	28361.6i	−0.496200	+	0.591348i
$220$		0			0
$221$	7126.05	+	4114.23i	0.145903	+	0.0842371i
$222$		0			0
$223$	3287.92	−	579.750i	0.0661168	−	0.0116582i	−0.140492	−	0.990082i	$-0.544868\pi$
$223$	3287.92	−	579.750i	0.0661168	−	0.0116582i	0.206609	+	0.978424i	$0.433757\pi$
$224$		0			0
$225$	12110.2	+	4407.76i	0.239214	+	0.0870669i
$226$		0			0
$227$		−	74854.6i		−	1.45267i	−0.687341	−	0.726335i	$-0.741222\pi$
$227$		−	74854.6i		−	1.45267i	0.687341	−	0.726335i	$-0.258778\pi$
$228$		0			0
$229$	69586.5			1.32695			0.663474	−	0.748199i	$-0.269081\pi$
$229$	69586.5			1.32695			0.663474	+	0.748199i	$0.269081\pi$
$230$		0			0
$231$	−21820.1	+	59950.2i	−0.408915	+	1.12348i
$232$		0			0
$233$	5567.08	+	31572.5i	0.102545	+	0.581563i	0.992172	+	0.124876i	$0.0398532\pi$
$233$	5567.08	+	31572.5i	0.102545	+	0.581563i	−0.889627	+	0.456688i	$0.849036\pi$
$234$		0			0
$235$	−2647.91	+	4586.31i	−0.0479476	+	0.0830477i
$236$		0			0
$237$	−27852.4	−	23371.0i	−0.495869	−	0.416083i
$238$		0			0
$239$	12567.6	+	21767.7i	0.220017	+	0.381081i	0.954813	−	0.297207i	$-0.0960553\pi$
$239$	12567.6	+	21767.7i	0.220017	+	0.381081i	−0.734796	+	0.678289i	$0.762722\pi$
$240$		0			0
$241$	10430.9	+	28658.6i	0.179592	+	0.493425i	0.996524	−	0.0833092i	$-0.0265489\pi$
$241$	10430.9	+	28658.6i	0.179592	+	0.493425i	−0.816932	+	0.576735i	$0.804327\pi$
$242$		0			0
$243$	29054.4	+	5123.07i	0.492038	+	0.0867596i
$244$		0			0
$245$	−419.271	+	351.810i	−0.00698494	+	0.00586106i
$246$		0			0
$247$	−24391.8	+	31472.0i	−0.399807	+	0.515858i
$248$		0			0
$249$	37024.0	+	44123.5i	0.597152	+	0.711658i
$250$		0			0
$251$	7734.70	−	43865.7i	0.122771	−	0.696269i	−0.859836	−	0.510571i	$-0.829434\pi$
$251$	7734.70	−	43865.7i	0.122771	−	0.696269i	0.982607	−	0.185699i	$-0.0594548\pi$
$252$		0			0
$253$	−14952.7	+	5442.33i	−0.233603	+	0.0850244i
$254$		0			0
$255$	−1169.78	+	675.371i	−0.0179896	+	0.0103863i
$256$		0			0
$257$	−65549.4	+	78118.7i	−0.992436	+	1.18274i	−0.00928251	+	0.999957i	$0.502955\pi$
$257$	−65549.4	+	78118.7i	−0.992436	+	1.18274i	−0.983153	+	0.182782i	$0.941490\pi$
$258$		0			0
$259$	54647.3	+	31550.6i	0.814647	+	0.470337i
$260$		0			0
$261$	−3884.43	+	684.929i	−0.0570224	+	0.0100546i
$262$		0			0
$263$	113180.	+	41194.3i	1.63629	+	0.595560i	0.986384	−	0.164456i	$-0.0525870\pi$
$263$	113180.	+	41194.3i	1.63629	+	0.595560i	0.649904	+	0.760017i	$0.274809\pi$
$264$		0			0
$265$		−	2705.70i		−	0.0385290i
$266$		0			0
$267$	−35104.6			−0.492426
$268$		0			0
$269$	−47107.1	+	129426.i	−0.651001	+	1.78861i	−0.0369886	+	0.999316i	$0.511777\pi$
$269$	−47107.1	+	129426.i	−0.651001	+	1.78861i	−0.614013	+	0.789296i	$0.710446\pi$
$270$		0			0
$271$	−15656.1	−	88790.1i	−0.213179	−	1.20900i	−0.884038	−	0.467414i	$-0.845186\pi$
$271$	−15656.1	−	88790.1i	−0.213179	−	1.20900i	0.670859	−	0.741585i	$-0.265925\pi$
$272$		0			0
$273$	−19916.3	+	34496.0i	−0.267229	+	0.462854i
$274$		0			0
$275$	−83843.4	−	70352.9i	−1.10867	−	0.930287i
$276$		0			0
$277$	−62079.9	−	107525.i	−0.809080	−	1.40137i	−0.913502	−	0.406834i	$-0.866633\pi$
$277$	−62079.9	−	107525.i	−0.809080	−	1.40137i	0.104423	−	0.994533i	$-0.466701\pi$
$278$		0			0
$279$	5507.09	+	15130.6i	0.0707479	+	0.194378i
$280$		0			0
$281$	47169.6	+	8317.28i	0.597379	+	0.105334i	0.464158	−	0.885752i	$-0.346357\pi$
$281$	47169.6	+	8317.28i	0.597379	+	0.105334i	0.133221	+	0.991086i	$0.457468\pi$
$282$		0			0
$283$	−22734.0	+	19076.1i	−0.283860	+	0.238187i	−0.773589	−	0.633688i	$-0.781540\pi$
$283$	−22734.0	+	19076.1i	−0.283860	+	0.238187i	0.489729	+	0.871875i	$0.337096\pi$
$284$		0			0
$285$	−2472.11	−	6050.74i	−0.0304354	−	0.0744935i
$286$		0			0
$287$	27710.5	+	33024.1i	0.336419	+	0.400928i
$288$		0			0
$289$	−13536.8	+	76771.2i	−0.162077	+	0.919185i
$290$		0			0
$291$	38175.0	−	13894.6i	0.450810	−	0.164081i
$292$		0			0
$293$	121893.	−	70375.0i	1.41986	−	0.819754i	0.423570	−	0.905864i	$-0.360777\pi$
$293$	121893.	−	70375.0i	1.41986	−	0.819754i	0.996286	+	0.0861098i	$0.0274436\pi$
$294$		0			0
$295$	−6049.93	+	7210.03i	−0.0695194	+	0.0828500i
$296$		0			0
$297$	−120841.	−	69767.3i	−1.36993	−	0.790932i
$298$		0			0
$299$	−9784.03	+	1725.19i	−0.109440	+	0.0192972i
$300$		0			0
$301$	75159.4	+	27355.8i	0.829564	+	0.301937i
$302$		0			0
$303$			50255.1i			0.547388i
$304$		0			0
$305$	9438.46			0.101462
$306$		0			0
$307$	−33386.2	+	91727.9i	−0.354234	+	0.973251i	0.626760	+	0.779213i	$0.284381\pi$
$307$	−33386.2	+	91727.9i	−0.354234	+	0.973251i	−0.980994	+	0.194038i	$0.937841\pi$
$308$		0			0
$309$	−20024.8	−	113566.i	−0.209725	−	1.18941i
$310$		0			0
$311$	72667.7	−	125864.i	0.751313	−	1.30131i	−0.195874	−	0.980629i	$-0.562754\pi$
$311$	72667.7	−	125864.i	0.751313	−	1.30131i	0.947187	−	0.320682i	$-0.103912\pi$
$312$		0			0
$313$	−101844.	−	85457.2i	−1.03955	−	0.872288i	−0.0475958	−	0.998867i	$-0.515156\pi$
$313$	−101844.	−	85457.2i	−1.03955	−	0.872288i	−0.991957	+	0.126579i	$0.959600\pi$
$314$		0			0
$315$	1129.69	+	1956.69i	0.0113852	+	0.0197197i
$316$		0			0
$317$	−54530.5	−	149821.i	−0.542651	−	1.49092i	−0.843436	−	0.537230i	$-0.819471\pi$
$317$	−54530.5	−	149821.i	−0.542651	−	1.49092i	0.300784	−	0.953692i	$-0.402752\pi$
$318$		0			0
$319$	32989.5	+	5816.93i	0.324186	+	0.0571627i
$320$		0			0
$321$	−94334.2	+	79155.8i	−0.915502	+	0.768197i
$322$		0			0
$323$	25645.4	+	8222.39i	0.245813	+	0.0788122i
$324$		0			0
$325$	−43925.4	−	52348.2i	−0.415861	−	0.495604i
$326$		0			0
$327$	15007.7	−	85113.1i	0.140352	−	0.795978i
$328$		0			0
$329$	99258.0	−	36127.0i	0.917010	−	0.333764i
$330$		0			0
$331$	78104.0	−	45093.4i	0.712881	−	0.411582i	−0.0992457	−	0.995063i	$-0.531643\pi$
$331$	78104.0	−	45093.4i	0.712881	−	0.411582i	0.812127	+	0.583481i	$0.198310\pi$
$332$		0			0
$333$	−18126.6	+	21602.5i	−0.163466	+	0.194812i
$334$		0			0
$335$	5600.47	+	3233.43i	0.0499039	+	0.0288120i
$336$		0			0
$337$	−9306.78	+	1641.04i	−0.0819482	+	0.0144497i	−0.214472	−	0.976730i	$-0.568803\pi$
$337$	−9306.78	+	1641.04i	−0.0819482	+	0.0144497i	0.132524	+	0.991180i	$0.457692\pi$
$338$		0			0
$339$	178898.	+	65113.7i	1.55671	+	0.566595i
$340$		0			0
$341$		−	136747.i		−	1.17601i
$342$		0			0
$343$	122672.			1.04269
$344$		0			0
$345$	557.791	−	1532.52i	0.00468634	−	0.0128756i
$346$		0			0
$347$	−30236.7	−	171481.i	−0.251117	−	1.42415i	−0.805847	−	0.592124i	$-0.798289\pi$
$347$	−30236.7	−	171481.i	−0.251117	−	1.42415i	0.554730	−	0.832031i	$-0.312822\pi$
$348$		0			0
$349$	−85890.6	+	148767.i	−0.705172	+	1.22139i	0.261458	+	0.965215i	$0.415797\pi$
$349$	−85890.6	+	148767.i	−0.705172	+	1.22139i	−0.966630	+	0.256178i	$0.917537\pi$
$350$		0			0
$351$	−66737.4	−	55999.3i	−0.541695	−	0.454536i
$352$		0			0
$353$	−52519.6	−	90966.6i	−0.421475	−	0.730016i	0.574609	−	0.818428i	$-0.305154\pi$
$353$	−52519.6	−	90966.6i	−0.421475	−	0.730016i	−0.996084	+	0.0884119i	$0.971821\pi$
$354$		0			0
$355$	4656.17	+	12792.7i	0.0369464	+	0.101509i
$356$		0			0
$357$	26532.0	+	4678.32i	0.208178	+	0.0367074i
$358$		0			0
$359$	−131354.	+	110219.i	−1.01919	+	0.855200i	−0.989525	−	0.144360i	$-0.953888\pi$
$359$	−131354.	+	110219.i	−1.01919	+	0.855200i	−0.0296621	+	0.999560i	$0.509443\pi$
$360$		0			0
$361$	−56214.8	+	117573.i	−0.431356	+	0.902182i
$362$		0			0
$363$	82625.6	+	98469.4i	0.627049	+	0.747288i
$364$		0			0
$365$	1933.67	−	10966.4i	0.0145143	−	0.0823146i
$366$		0			0
$367$	−41854.0	+	15233.6i	−0.310745	+	0.113102i	−0.492685	−	0.870208i	$-0.663985\pi$
$367$	−41854.0	+	15233.6i	−0.310745	+	0.113102i	0.181940	+	0.983310i	$0.441762\pi$
$368$		0			0
$369$	−16684.7	+	9632.92i	−0.122537	+	0.0707465i
$370$		0			0
$371$	−34689.2	+	41341.0i	−0.252027	+	0.300354i
$372$		0			0
$373$	−192034.	−	110871.i	−1.38026	−	0.796894i	−0.388070	−	0.921630i	$-0.626858\pi$
$373$	−192034.	−	110871.i	−1.38026	−	0.796894i	−0.992190	+	0.124736i	$0.960192\pi$
$374$		0			0
$375$	22191.6	−	3912.97i	0.157807	−	0.0278256i
$376$		0			0
$377$	19653.6	+	7153.34i	0.138280	+	0.0503299i
$378$		0			0
$379$			146879.i			1.02254i	0.859419	+	0.511271i	$0.170825\pi$
$379$			146879.i			1.02254i	−0.859419	+	0.511271i	$0.829175\pi$
$380$		0			0
$381$	−29634.7			−0.204150
$382$		0			0
$383$	−946.882	+	2601.54i	−0.00645503	+	0.0177351i	−0.942878	−	0.333137i	$-0.891893\pi$
$383$	−946.882	+	2601.54i	−0.00645503	+	0.0177351i	0.936423	+	0.350872i	$0.114115\pi$
$384$		0			0
$385$	−3332.04	−	18896.9i	−0.0224796	−	0.127488i
$386$		0			0
$387$	−17872.2	+	30955.6i	−0.119332	+	0.206689i
$388$		0			0
$389$	35161.7	+	29504.2i	0.232365	+	0.194977i	0.751534	−	0.659694i	$-0.229314\pi$
$389$	35161.7	+	29504.2i	0.232365	+	0.194977i	−0.519169	+	0.854671i	$0.673759\pi$
$390$		0			0
$391$	3359.83	+	5819.39i	0.0219768	+	0.0380649i
$392$		0			0
$393$	89878.1	+	246938.i	0.581927	+	1.59883i
$394$		0			0
$395$	10769.5	+	1898.95i	0.0690241	+	0.0121708i
$396$		0			0
$397$	91174.4	−	76504.4i	0.578485	−	0.485406i	−0.305964	−	0.952043i	$-0.598979\pi$
$397$	91174.4	−	76504.4i	0.578485	−	0.485406i	0.884449	+	0.466637i	$0.154534\pi$
$398$		0			0
$399$	−39803.2	+	124145.i	−0.250019	+	0.779800i
$400$		0			0
$401$	130832.	+	155920.i	0.813628	+	0.969644i	0.999917	−	0.0128506i	$-0.00409060\pi$
$401$	130832.	+	155920.i	0.813628	+	0.969644i	−0.186289	+	0.982495i	$0.559646\pi$
$402$		0			0
$403$	14825.9	−	84082.0i	0.0912876	−	0.517718i
$404$		0			0
$405$	9743.87	−	3546.48i	0.0594048	−	0.0216216i
$406$		0			0
$407$	207410.	−	119748.i	1.25210	−	0.722902i
$408$		0			0
$409$	−67298.2	+	80202.8i	−0.402306	+	0.479450i	−0.928722	−	0.370778i	$-0.879091\pi$
$409$	−67298.2	+	80202.8i	−0.402306	+	0.479450i	0.526416	+	0.850227i	$0.323536\pi$
$410$		0			0
$411$	29843.2	+	17230.0i	0.176669	+	0.102000i
$412$		0			0
$413$	184876.	−	32598.7i	1.08388	−	0.191117i
$414$		0			0
$415$	−16279.3	−	5925.19i	−0.0945236	−	0.0344038i
$416$		0			0
$417$		−	80622.9i		−	0.463646i
$418$		0			0
$419$	−214521.			−1.22192			−0.610959	−	0.791662i	$-0.709216\pi$
$419$	−214521.			−1.22192			−0.610959	+	0.791662i	$0.709216\pi$
$420$		0			0
$421$	−76569.4	+	210373.i	−0.432007	+	1.18693i	0.512571	+	0.858645i	$0.328693\pi$
$421$	−76569.4	+	210373.i	−0.432007	+	1.18693i	−0.944579	+	0.328286i	$0.893529\pi$
$422$		0			0
$423$	8197.12	+	46488.2i	0.0458122	+	0.259814i
$424$		0			0
$425$	−23109.9	+	40027.6i	−0.127944	+	0.221606i
$426$		0			0
$427$	−144212.	−	121009.i	−0.790946	−	0.663682i
$428$		0			0
$429$	75590.7	+	130927.i	0.410727	+	0.711400i
$430$		0			0
$431$	−37160.5	−	102098.i	−0.200045	−	0.549618i	0.798589	−	0.601877i	$-0.205580\pi$
$431$	−37160.5	−	102098.i	−0.200045	−	0.549618i	−0.998634	+	0.0522588i	$0.983358\pi$
$432$		0			0
$433$	117896.	+	20788.3i	0.628818	+	0.110878i	0.478970	−	0.877832i	$-0.341010\pi$
$433$	117896.	+	20788.3i	0.628818	+	0.110878i	0.149848	+	0.988709i	$0.452122\pi$
$434$		0			0
$435$	−2630.06	+	2206.88i	−0.0138991	+	0.0116627i
$436$		0			0
$437$	−30101.2	+	12298.3i	−0.157623	+	0.0643992i
$438$		0			0
$439$	146363.	+	174429.i	0.759456	+	0.905084i	0.997813	−	0.0660931i	$-0.0210534\pi$
$439$	146363.	+	174429.i	0.759456	+	0.905084i	−0.238357	+	0.971177i	$0.576609\pi$
$440$		0			0
$441$	−847.169	+	4804.53i	−0.00435605	+	0.0247044i
$442$		0			0
$443$	−67184.3	+	24453.1i	−0.342342	+	0.124602i	−0.507469	−	0.861670i	$-0.669419\pi$
$443$	−67184.3	+	24453.1i	−0.342342	+	0.124602i	0.165127	+	0.986272i	$0.447197\pi$
$444$		0			0
$445$	9143.84	−	5279.20i	0.0461752	−	0.0266592i
$446$		0			0
$447$	95528.8	−	113847.i	0.478101	−	0.569778i
$448$		0			0
$449$	−14220.1	−	8210.00i	−0.0705361	−	0.0407240i	0.464317	−	0.885669i	$-0.346300\pi$
$449$	−14220.1	−	8210.00i	−0.0705361	−	0.0407240i	−0.534853	+	0.844945i	$0.679633\pi$
$450$		0			0
$451$	161134.	−	28412.3i	0.792200	−	0.139686i
$452$		0			0
$453$	122285.	+	44508.2i	0.595907	+	0.216892i
$454$		0			0
$455$		−	11980.4i		−	0.0578695i
$456$		0			0
$457$	−236201.			−1.13097			−0.565483	−	0.824760i	$-0.691310\pi$
$457$	−236201.			−1.13097			−0.565483	+	0.824760i	$0.691310\pi$
$458$		0			0
$459$	−20153.4	+	55371.0i	−0.0956583	+	0.262819i
$460$		0			0
$461$	8044.64	+	45623.4i	0.0378534	+	0.214677i	0.997867	−	0.0652750i	$-0.0207925\pi$
$461$	8044.64	+	45623.4i	0.0378534	+	0.214677i	−0.960014	+	0.279952i	$0.909681\pi$
$462$		0			0
$463$	−107047.	+	185410.i	−0.499358	+	0.864913i	−1.00000		0.000741640i	$-0.999764\pi$
$463$	−107047.	+	185410.i	−0.499358	+	0.864913i	0.500642	+	0.865654i	$0.333097\pi$
$464$		0			0
$465$	10736.4	+	9008.93i	0.0496539	+	0.0416646i
$466$		0			0
$467$	−90821.3	−	157307.i	−0.416441	−	0.721298i	0.579137	−	0.815230i	$-0.303390\pi$
$467$	−90821.3	−	157307.i	−0.416441	−	0.721298i	−0.995579	+	0.0939325i	$0.970056\pi$
$468$		0			0
$469$	−44115.6	−	121207.i	−0.200561	−	0.551037i
$470$		0			0
$471$	−287017.	−	50608.8i	−1.29380	−	0.228131i
$472$		0			0
$473$	232547.	−	195130.i	1.03941	−	0.872172i
$474$		0			0
$475$	−176781.	−	137011.i	−0.783516	−	0.607250i
$476$		0			0
$477$	−15502.6	−	18475.3i	−0.0681348	−	0.0811999i
$478$		0			0
$479$	37321.6	−	211661.i	0.162663	−	0.922508i	−0.788778	−	0.614678i	$-0.789286\pi$
$479$	37321.6	−	211661.i	0.162663	−	0.922508i	0.951441	−	0.307830i	$-0.0996028\pi$
$480$		0			0
$481$	140513.	−	51142.6i	0.607333	−	0.221051i
$482$		0			0
$483$	−28170.7	+	16264.4i	−0.120755	+	0.0697177i
$484$		0			0
$485$	−7854.08	+	9360.13i	−0.0333896	+	0.0397922i
$486$		0			0
$487$	127784.	+	73775.9i	0.538787	+	0.311069i	0.744587	−	0.667525i	$-0.232646\pi$
$487$	127784.	+	73775.9i	0.538787	+	0.311069i	−0.205800	+	0.978594i	$0.565980\pi$
$488$		0			0
$489$	−175851.	+	31007.2i	−0.735405	+	0.129672i
$490$		0			0
$491$	−256367.	−	93309.9i	−1.06341	−	0.387048i	−0.249698	−	0.968324i	$-0.580331\pi$
$491$	−256367.	−	93309.9i	−1.06341	−	0.387048i	−0.813707	+	0.581276i	$0.802554\pi$
$492$		0			0
$493$		−	14146.1i		−	0.0582028i
$494$		0			0
$495$	8575.33			0.0349978
$496$		0			0
$497$	92870.0	−	255158.i	0.375978	−	1.03299i
$498$		0			0
$499$	29635.6	+	168072.i	0.119018	+	0.674985i	0.984682	+	0.174360i	$0.0557856\pi$
$499$	29635.6	+	168072.i	0.119018	+	0.674985i	−0.865664	+	0.500626i	$0.833103\pi$
$500$		0			0
$501$	−30470.3	+	52776.1i	−0.121395	+	0.210263i
$502$		0			0
$503$	12725.1	+	10677.7i	0.0502952	+	0.0422027i	0.667589	−	0.744530i	$-0.267326\pi$
$503$	12725.1	+	10677.7i	0.0502952	+	0.0422027i	−0.617294	+	0.786733i	$0.711771\pi$
$504$		0			0
$505$	−7557.61	−	13090.2i	−0.0296348	−	0.0513290i
$506$		0			0
$507$	−43507.6	−	119536.i	−0.169258	−	0.465033i
$508$		0			0
$509$	182544.	+	32187.4i	0.704581	+	0.124237i	0.514448	−	0.857522i	$-0.327997\pi$
$509$	182544.	+	32187.4i	0.704581	+	0.124237i	0.190133	+	0.981758i	$0.439108\pi$
$510$		0			0
$511$	−170142.	+	142766.i	−0.651584	+	0.546744i
$512$		0			0
$513$	−252281.	−	132882.i	−0.958626	−	0.504931i
$514$		0			0
$515$	22294.5	+	26569.6i	0.0840590	+	0.100178i
$516$		0			0
$517$	69616.1	−	394813.i	0.260453	−	1.47710i
$518$		0			0
$519$	131152.	−	47735.5i	0.486902	−	0.177218i
$520$		0			0
$521$	376417.	−	217325.i	1.38674	−	0.800633i	0.393790	−	0.919200i	$-0.371164\pi$
$521$	376417.	−	217325.i	1.38674	−	0.800633i	0.992946	+	0.118568i	$0.0378303\pi$
$522$		0			0
$523$	12503.6	−	14901.2i	0.0457122	−	0.0544776i	−0.742704	−	0.669620i	$-0.766457\pi$
$523$	12503.6	−	14901.2i	0.0457122	−	0.0544776i	0.788416	+	0.615142i	$0.210901\pi$
$524$		0			0
$525$	−193767.	−	111871.i	−0.703009	−	0.405883i
$526$		0			0
$527$	−56870.1	+	10027.7i	−0.204768	+	0.0361062i
$528$		0			0
$529$	255341.	+	92936.4i	0.912449	+	0.332104i
$530$		0			0
$531$			83896.0i			0.297545i
$532$		0			0
$533$	102157.			0.359596
$534$		0			0
$535$	12667.8	−	34804.5i	0.0442582	−	0.121598i
$536$		0			0
$537$	−29812.2	−	169073.i	−0.103382	−	0.586308i
$538$		0			0
$539$	20716.6	−	35882.2i	0.0713084	−	0.123510i
$540$		0			0
$541$	346656.	+	290879.i	1.18442	+	0.993844i	0.999939	+	0.0110070i	$0.00350370\pi$
$541$	346656.	+	290879.i	1.18442	+	0.993844i	0.184477	+	0.982837i	$0.440941\pi$
$542$		0			0
$543$	−124204.	−	215127.i	−0.421245	−	0.729617i
$544$		0			0
$545$	8890.60	+	24426.7i	0.0299322	+	0.0822379i
$546$		0			0
$547$	−312719.	−	55140.9i	−1.04515	−	0.184289i	−0.375393	−	0.926866i	$-0.622492\pi$
$547$	−312719.	−	55140.9i	−1.04515	−	0.184289i	−0.669761	+	0.742577i	$0.733603\pi$
$548$		0			0
$549$	64448.7	−	54078.8i	0.213830	−	0.179425i
$550$		0			0
$551$	67823.9	+	9262.78i	0.223398	+	0.0305097i
$552$		0			0
$553$	−140203.	−	167088.i	−0.458467	−	0.546379i
$554$		0			0
$555$	−4262.41	+	24173.3i	−0.0138379	+	0.0784785i
$556$		0			0
$557$	−63243.6	+	23018.8i	−0.203848	+	0.0741945i	−0.441926	−	0.897051i	$-0.645705\pi$
$557$	−63243.6	+	23018.8i	−0.203848	+	0.0741945i	0.238078	+	0.971246i	$0.423482\pi$
$558$		0			0
$559$	164142.	−	94767.7i	0.525288	−	0.303275i
$560$		0			0
$561$	65727.4	−	78330.9i	0.208843	−	0.248890i
$562$		0			0
$563$	178538.	+	103079.i	0.563268	+	0.325203i	0.754456	−	0.656351i	$-0.227901\pi$
$563$	178538.	+	103079.i	0.563268	+	0.325203i	−0.191188	+	0.981553i	$0.561234\pi$
$564$		0			0
$565$	−56390.5	+	9943.17i	−0.176648	+	0.0311478i
$566$		0			0
$567$	−194347.	−	70736.7i	−0.604523	−	0.220028i
$568$		0			0
$569$			532584.i			1.64499i	0.568771	+	0.822496i	$0.307419\pi$
$569$			532584.i			1.64499i	−0.568771	+	0.822496i	$0.692581\pi$
$570$		0			0
$571$	−197356.			−0.605309			−0.302654	−	0.953100i	$-0.597873\pi$
$571$	−197356.			−0.605309			−0.302654	+	0.953100i	$0.597873\pi$
$572$		0			0
$573$	−144863.	+	398008.i	−0.441213	+	1.21222i
$574$		0			0
$575$	−9690.52	−	54957.7i	−0.0293097	−	0.166224i
$576$		0			0
$577$	50452.7	−	87386.6i	0.151542	−	0.262478i	−0.780253	−	0.625464i	$-0.784910\pi$
$577$	50452.7	−	87386.6i	0.151542	−	0.262478i	0.931794	+	0.362986i	$0.118243\pi$
$578$		0			0
$579$	325048.	+	272748.i	0.969596	+	0.813588i
$580$		0			0
$581$	172770.	+	299246.i	0.511818	+	0.886494i
$582$		0			0
$583$	70054.8	+	192474.i	0.206111	+	0.566285i
$584$		0			0
$585$	5272.73	+	929.724i	0.0154072	+	0.00271670i
$586$		0			0
$587$	−368685.	+	309363.i	−1.06999	+	0.897826i	−0.995051	−	0.0993621i	$-0.968320\pi$
$587$	−368685.	+	309363.i	−1.06999	+	0.897826i	−0.0749363	+	0.997188i	$0.523875\pi$
$588$		0			0
$589$	−10840.0	−	279231.i	−0.0312464	−	0.804883i
$590$		0			0
$591$	−33798.5	−	40279.5i	−0.0967659	−	0.115321i
$592$		0			0
$593$	−65332.1	+	370517.i	−0.185788	+	1.05366i	0.739151	+	0.673540i	$0.235227\pi$
$593$	−65332.1	+	370517.i	−0.185788	+	1.05366i	−0.924939	+	0.380116i	$0.875884\pi$
$594$		0			0
$595$	−7614.46	+	2771.44i	−0.0215083	+	0.00782837i
$596$		0			0
$597$	−433670.	+	250379.i	−1.21678	+	0.702505i
$598$		0			0
$599$	−51498.6	+	61373.7i	−0.143530	+	0.171052i	−0.833020	−	0.553243i	$-0.813390\pi$
$599$	−51498.6	+	61373.7i	−0.143530	+	0.171052i	0.689490	+	0.724295i	$0.257835\pi$
$600$		0			0
$601$	−206392.	−	119160.i	−0.571403	−	0.329900i	0.186306	−	0.982492i	$-0.440348\pi$
$601$	−206392.	−	119160.i	−0.571403	−	0.329900i	−0.757710	+	0.652592i	$0.773682\pi$
$602$		0			0
$603$	56768.0	−	10009.7i	0.156124	−	0.0275288i
$604$		0			0
$605$	−36330.2	−	13223.1i	−0.0992559	−	0.0361262i
$606$		0			0
$607$			46881.4i			0.127240i	0.997974	+	0.0636199i	$0.0202645\pi$
$607$			46881.4i			0.127240i	−0.997974	+	0.0636199i	$0.979735\pi$
$608$		0			0
$609$	68479.1			0.184639
$610$		0			0
$611$	85610.0	−	235212.i	0.229320	−	0.630052i
$612$		0			0
$613$	62444.3	+	354139.i	0.166177	+	0.942438i	0.947842	+	0.318740i	$0.103260\pi$
$613$	62444.3	+	354139.i	0.166177	+	0.942438i	−0.781665	+	0.623698i	$0.785629\pi$
$614$		0			0
$615$	−8384.81	+	14522.9i	−0.0221689	+	0.0383976i
$616$		0			0
$617$	−34916.8	−	29298.6i	−0.0917199	−	0.0769622i	0.595775	−	0.803152i	$-0.296845\pi$
$617$	−34916.8	−	29298.6i	−0.0917199	−	0.0769622i	−0.687495	+	0.726189i	$0.741290\pi$
$618$		0			0
$619$	179541.	+	310975.i	0.468579	+	0.811603i	0.999355	−	0.0359092i	$-0.0114327\pi$
$619$	179541.	+	310975.i	0.468579	+	0.811603i	−0.530776	+	0.847512i	$0.678099\pi$
$620$		0			0
$621$	−24333.0	−	66854.4i	−0.0630976	−	0.173359i
$622$		0			0
$623$	−207394.	−	36569.2i	−0.534343	−	0.0942191i
$624$		0			0
$625$	291437.	−	244545.i	0.746079	−	0.626034i
$626$		0			0
$627$	332521.	+	366422.i	0.845832	+	0.932065i
$628$		0			0
$629$	−65009.9	−	77475.8i	−0.164315	−	0.195824i
$630$		0			0
$631$	−68745.3	+	389874.i	−0.172657	+	0.979187i	0.768156	+	0.640262i	$0.221174\pi$
$631$	−68745.3	+	389874.i	−0.172657	+	0.979187i	−0.940814	+	0.338925i	$0.889937\pi$
$632$		0			0
$633$	490252.	−	178437.i	1.22352	−	0.445326i
$634$		0			0
$635$	7719.07	−	4456.61i	0.0191433	−	0.0110524i
$636$		0			0
$637$	16628.3	−	19816.9i	0.0409798	−	0.0488378i
$638$		0			0
$639$	105091.	+	60674.4i	0.257374	+	0.148595i
$640$		0			0
$641$	−9953.74	+	1755.11i	−0.0242254	+	0.00427158i	−0.185748	−	0.982597i	$-0.559471\pi$
$641$	−9953.74	+	1755.11i	−0.0242254	+	0.00427158i	0.161522	+	0.986869i	$0.448360\pi$
$642$		0			0
$643$	147595.	+	53720.0i	0.356984	+	0.129931i	0.514285	−	0.857619i	$-0.328057\pi$
$643$	147595.	+	53720.0i	0.356984	+	0.129931i	−0.157302	+	0.987551i	$0.550279\pi$
$644$		0			0
$645$			31113.2i			0.0747868i
$646$		0			0
$647$	63979.4			0.152838			0.0764190	−	0.997076i	$-0.475651\pi$
$647$	63979.4			0.152838			0.0764190	+	0.997076i	$0.475651\pi$
$648$		0			0
$649$	243692.	−	669539.i	0.578565	−	1.58959i
$650$		0			0
$651$	−48542.6	−	275299.i	−0.114541	−	0.649595i
$652$		0			0
$653$	−365655.	+	633333.i	−0.857522	+	1.48527i	0.0167642	+	0.999859i	$0.494664\pi$
$653$	−365655.	+	633333.i	−0.857522	+	1.48527i	−0.874286	+	0.485411i	$0.838670\pi$
$654$		0			0
$655$	−60546.7	−	50804.7i	−0.141126	−	0.118419i
$656$		0			0
$657$	−49629.5	−	85960.8i	−0.114977	−	0.199145i
$658$		0			0
$659$	−47157.2	−	129563.i	−0.108587	−	0.298340i	0.873484	−	0.486853i	$-0.161855\pi$
$659$	−47157.2	−	129563.i	−0.108587	−	0.298340i	−0.982071	+	0.188513i	$0.939633\pi$
$660$		0			0
$661$	474797.	+	83719.5i	1.08669	+	0.191612i	0.688171	−	0.725549i	$-0.258414\pi$
$661$	474797.	+	83719.5i	1.08669	+	0.191612i	0.398517	+	0.917161i	$0.369525\pi$
$662$		0			0
$663$	48906.5	−	41037.4i	0.111260	−	0.0933582i
$664$		0			0
$665$	−8301.81	−	38322.4i	−0.0187728	−	0.0866581i
$666$		0			0
$667$	10978.8	+	13084.0i	0.0246775	+	0.0294095i
$668$		0			0
$669$	4498.16	−	25510.3i	0.0100504	−	0.0569985i
$670$		0			0
$671$	−671420.	+	244377.i	−1.49125	+	0.542769i
$672$		0			0
$673$	−680011.	+	392605.i	−1.50136	+	0.866813i	−0.501365	+	0.865236i	$0.667168\pi$
$673$	−680011.	+	392605.i	−1.50136	+	0.866813i	−0.999999	+	0.00157637i	$0.999498\pi$
$674$		0			0
$675$	314553.	−	374869.i	0.690377	−	0.822759i
$676$		0			0
$677$	139771.	+	80696.9i	0.304958	+	0.176068i	0.644668	−	0.764463i	$-0.276996\pi$
$677$	139771.	+	80696.9i	0.304958	+	0.176068i	−0.339710	+	0.940530i	$0.610329\pi$
$678$		0			0
$679$	240008.	−	42320.0i	0.520579	−	0.0917922i
$680$		0			0
$681$	−545756.	−	198639.i	−1.17681	−	0.428322i
$682$		0			0
$683$		−	328136.i		−	0.703417i	−0.936110	−	0.351708i	$-0.885601\pi$
$683$		−	328136.i		−	0.703417i	0.936110	−	0.351708i	$-0.114399\pi$
$684$		0			0
$685$	−10364.5			−0.0220886
$686$		0			0
$687$	184659.	−	507347.i	0.391253	−	1.07496i
$688$		0			0
$689$	22207.0	+	125942.i	0.0467790	+	0.265297i
$690$		0			0
$691$	397198.	−	687967.i	0.831862	−	1.44083i	−0.0646987	−	0.997905i	$-0.520609\pi$
$691$	397198.	−	687967.i	0.831862	−	1.44083i	0.896560	−	0.442922i	$-0.146058\pi$
$692$		0			0
$693$	−131024.	−	109942.i	−0.272826	−	0.228928i
$694$		0			0
$695$	12124.5	+	21000.2i	0.0251011	+	0.0434764i
$696$		0			0
$697$	−23632.1	−	64928.6i	−0.0486448	−	0.133651i
$698$		0			0
$699$	244964.	+	43193.8i	0.501359	+	0.0884031i
$700$		0			0
$701$	−94166.3	+	79014.9i	−0.191628	+	0.160795i	−0.733554	−	0.679631i	$-0.762140\pi$
$701$	−94166.3	+	79014.9i	−0.191628	+	0.160795i	0.541926	+	0.840426i	$0.317695\pi$
$702$		0			0
$703$	414027.	−	260961.i	0.837757	−	0.528037i
$704$		0			0
$705$	26411.6	+	31476.1i	0.0531393	+	0.0633290i
$706$		0			0
$707$	−52351.8	+	296902.i	−0.104735	+	0.593984i
$708$		0			0
$709$	−809937.	+	294793.i	−1.61124	+	0.586442i	−0.981684	−	0.190517i	$-0.938984\pi$
$709$	−809937.	+	294793.i	−1.61124	+	0.586442i	−0.629551	+	0.776959i	$0.716761\pi$
$710$		0			0
$711$	84417.5	−	48738.5i	0.166991	−	0.0964124i
$712$		0			0
$713$	44817.6	−	53411.5i	0.0881595	−	0.105064i
$714$		0			0
$715$	−39378.8	−	22735.4i	−0.0770284	−	0.0444724i
$716$		0			0
$717$	192056.	−	33864.7i	0.373585	−	0.0658732i
$718$		0			0
$719$	−512487.	−	186530.i	−0.991346	−	0.360821i	−0.205105	−	0.978740i	$-0.565754\pi$
$719$	−512487.	−	186530.i	−0.991346	−	0.360821i	−0.786241	+	0.617919i	$0.787976\pi$
$720$		0			0
$721$		−	691797.i		−	1.33078i
$722$		0			0
$723$	236627.			0.452676
$724$		0			0
$725$	−40180.9	+	110396.i	−0.0764440	+	0.210028i
$726$		0			0
$727$	38500.0	+	218344.i	0.0728437	+	0.413117i	0.999324	+	0.0367730i	$0.0117078\pi$
$727$	38500.0	+	218344.i	0.0728437	+	0.413117i	−0.926480	+	0.376344i	$0.877181\pi$
$728$		0			0
$729$	294411.	−	509934.i	0.553986	−	0.959532i
$730$		0			0
$731$	−98203.1	−	82402.2i	−0.183777	−	0.154207i
$732$		0			0
$733$	−77402.2	−	134065.i	−0.144061	−	0.249521i	0.784961	−	0.619545i	$-0.212683\pi$
$733$	−77402.2	−	134065.i	−0.144061	−	0.249521i	−0.929022	+	0.370024i	$0.879349\pi$
$734$		0			0
$735$	1452.40	+	3990.44i	0.00268851	+	0.00738663i
$736$		0			0
$737$	−482117.	−	85010.2i	−0.887600	−	0.156508i
$738$		0			0
$739$	360117.	−	302174.i	0.659409	−	0.553310i	−0.250501	−	0.968116i	$-0.580595\pi$
$739$	360117.	−	302174.i	0.659409	−	0.553310i	0.909910	+	0.414807i	$0.136151\pi$
$740$		0			0
$741$	164731.	+	261354.i	0.300012	+	0.475984i
$742$		0			0
$743$	−514022.	−	612588.i	−0.931117	−	1.10966i	−0.993750	−	0.111626i	$-0.964394\pi$
$743$	−514022.	−	612588.i	−0.931117	−	1.10966i	0.0626329	−	0.998037i	$-0.480050\pi$
$744$		0			0
$745$	−7761.96	+	44020.3i	−0.0139849	+	0.0793122i
$746$		0			0
$747$	−145109.	+	52815.4i	−0.260048	+	0.0946498i
$748$		0			0
$749$	−639775.	+	369374.i	−1.14042	+	0.658420i
$750$		0			0
$751$	−502229.	+	598533.i	−0.890475	+	1.06123i	0.107278	+	0.994229i	$0.465787\pi$
$751$	−502229.	+	598533.i	−0.890475	+	1.06123i	−0.997753	+	0.0669978i	$0.978658\pi$
$752$		0			0
$753$	−299294.	−	172797.i	−0.527847	−	0.304753i
$754$		0			0
$755$	−38545.6	+	6796.62i	−0.0676208	+	0.0119234i
$756$		0			0
$757$	−187799.	−	68353.4i	−0.327719	−	0.119280i	0.172920	−	0.984936i	$-0.444680\pi$
$757$	−187799.	−	68353.4i	−0.327719	−	0.119280i	−0.500640	+	0.865656i	$0.666902\pi$
$758$		0			0
$759$			123460.i			0.214311i
$760$		0			0
$761$	718953.			1.24146			0.620728	−	0.784026i	$-0.286837\pi$
$761$	718953.			1.24146			0.620728	+	0.784026i	$0.286837\pi$
$762$		0			0
$763$	177328.	−	487206.i	0.304600	−	0.836880i
$764$		0			0
$765$	−628.833	−	3566.29i	−0.00107451	−	0.00609387i
$766$		0			0
$767$	222430.	−	385260.i	0.378096	−	0.654882i
$768$		0			0
$769$	−307970.	−	258418.i	−0.520782	−	0.436988i	0.344122	−	0.938925i	$-0.388177\pi$
$769$	−307970.	−	258418.i	−0.520782	−	0.436988i	−0.864904	+	0.501937i	$0.832621\pi$
$770$		0			0
$771$	395608.	+	685214.i	0.665513	+	1.15270i
$772$		0			0
$773$	341555.	+	938415.i	0.571612	+	1.57049i	0.801956	+	0.597383i	$0.203793\pi$
$773$	341555.	+	938415.i	0.571612	+	1.57049i	−0.230344	+	0.973109i	$0.573985\pi$
$774$		0			0
$775$	472296.	+	83278.5i	0.786340	+	0.138653i
$776$		0			0
$777$	375048.	−	314702.i	0.621218	−	0.521264i
$778$		0			0
$779$	326776.	−	70789.7i	0.538486	−	0.116653i
$780$		0			0
$781$	−662447.	−	789474.i	−1.08605	−	1.29430i
$782$		0			0
$783$	−26007.9	+	147498.i	−0.0424211	+	0.240582i
$784$		0			0
$785$	82371.3	−	29980.7i	0.133671	−	0.0486522i
$786$		0			0
$787$	−113438.	+	65493.7i	−0.183152	+	0.105743i	−0.588773	−	0.808299i	$-0.700389\pi$
$787$	−113438.	+	65493.7i	−0.183152	+	0.105743i	0.405621	+	0.914041i	$0.367055\pi$
$788$		0			0
$789$	600685.	−	715869.i	0.964924	−	1.14995i
$790$		0			0
$791$	989082.	+	571047.i	1.58081	+	0.912681i
$792$		0			0
$793$	−439332.	+	77466.1i	−0.698629	+	0.123187i
$794$		0			0
$795$	−19726.9	−	7180.01i	−0.0312123	−	0.0113603i
$796$		0			0
$797$			115451.i			0.181753i	0.995862	+	0.0908763i	$0.0289668\pi$
$797$			115451.i			0.181753i	−0.995862	+	0.0908763i	$0.971033\pi$
$798$		0			0
$799$	−169299.			−0.265192
$800$		0			0
$801$	32189.0	−	88438.7i	0.0501699	−	0.137841i
$802$		0			0
$803$	146382.	+	830175.i	0.227017	+	1.28747i
$804$		0			0
$805$	4891.83	−	8472.90i	0.00754883	−	0.0130750i
$806$		0			0
$807$	818621.	+	686905.i	1.25700	+	1.05475i
$808$		0			0
$809$	−289577.	−	501562.i	−0.442453	−	0.766351i	0.555418	−	0.831571i	$-0.312558\pi$
$809$	−289577.	−	501562.i	−0.442453	−	0.766351i	−0.997871	+	0.0652204i	$0.979225\pi$
$810$		0			0
$811$	−210873.	−	579369.i	−0.320612	−	0.880874i	−0.990389	−	0.138312i	$-0.955832\pi$
$811$	−210873.	−	579369.i	−0.320612	−	0.880874i	0.669777	−	0.742563i	$-0.266390\pi$
$812$		0			0
$813$	−688904.	−	121472.i	−1.04226	−	0.183779i
$814$		0			0
$815$	41141.6	−	34521.9i	0.0619392	−	0.0519732i
$816$		0			0
$817$	459381.	−	416880.i	0.688223	−	0.624550i
$818$		0			0
$819$	−68643.4	−	81806.0i	−0.102337	−	0.121960i
$820$		0			0
$821$	−167837.	+	951854.i	−0.249002	+	1.41216i	0.562010	+	0.827130i	$0.310028\pi$
$821$	−167837.	+	951854.i	−0.249002	+	1.41216i	−0.811012	+	0.585030i	$0.801083\pi$
$822$		0			0
$823$	454734.	−	165510.i	0.671364	−	0.244357i	0.0162291	−	0.999868i	$-0.494834\pi$
$823$	454734.	−	165510.i	0.671364	−	0.244357i	0.655135	+	0.755512i	$0.272612\pi$
$824$		0			0
$825$	−735427.	+	424599.i	−1.08052	+	0.623837i
$826$		0			0
$827$	−415744.	+	495464.i	−0.607876	+	0.724438i	−0.978935	−	0.204171i	$-0.934550\pi$
$827$	−415744.	+	495464.i	−0.607876	+	0.724438i	0.371059	+	0.928609i	$0.378995\pi$
$828$		0			0
$829$	596643.	+	344472.i	0.868171	+	0.501239i	0.866740	−	0.498760i	$-0.166211\pi$
$829$	596643.	+	344472.i	0.868171	+	0.501239i	0.00143079	+	0.999999i	$0.499545\pi$
$830$		0			0
$831$	−948695.	+	167280.i	−1.37380	+	0.242238i
$832$		0			0
$833$	−16441.7	−	5984.31i	−0.0236951	−	0.00862430i
$834$		0			0
$835$		−	18329.1i		−	0.0262886i
$836$		0			0
$837$	611406.			0.872728
$838$		0			0
$839$	−229836.	+	631469.i	−0.326508	+	0.897074i	0.662480	+	0.749080i	$0.269504\pi$
$839$	−229836.	+	631469.i	−0.326508	+	0.897074i	−0.988988	+	0.147994i	$0.952718\pi$
$840$		0			0
$841$	116574.	+	661126.i	0.164820	+	0.934743i
$842$		0			0
$843$	185813.	−	321837.i	0.261469	−	0.452877i
$844$		0			0
$845$	29309.0	+	24593.2i	0.0410477	+	0.0344431i
$846$		0			0
$847$	385566.	+	667820.i	0.537442	+	0.930877i
$848$		0			0
$849$	78753.2	+	216373.i	0.109258	+	0.300184i
$850$		0			0
$851$	120257.	+	21204.6i	0.166055	+	0.0292800i
$852$		0			0
$853$	−404189.	+	339155.i	−0.555502	+	0.466122i	−0.876799	−	0.480857i	$-0.840326\pi$
$853$	−404189.	+	339155.i	−0.555502	+	0.466122i	0.321297	+	0.946979i	$0.395881\pi$
$854$		0			0
$855$	17510.4	−	679.771i	0.0239532	−	0.000929887i
$856$		0			0
$857$	−111406.	−	132768.i	−0.151686	−	0.180773i	0.684850	−	0.728684i	$-0.259868\pi$
$857$	−111406.	−	132768.i	−0.151686	−	0.180773i	−0.836537	+	0.547911i	$0.815423\pi$
$858$		0			0
$859$	−74590.2	+	423022.i	−0.101087	+	0.573293i	0.891624	+	0.452776i	$0.149566\pi$
$859$	−74590.2	+	423022.i	−0.101087	+	0.573293i	−0.992711	+	0.120517i	$0.961545\pi$
$860$		0			0
$861$	314309.	−	114399.i	0.423985	−	0.154318i
$862$		0			0
$863$	−843301.	+	486880.i	−1.13230	+	0.653733i	−0.944512	−	0.328477i	$-0.893465\pi$
$863$	−843301.	+	486880.i	−1.13230	+	0.653733i	−0.187786	+	0.982210i	$0.560131\pi$
$864$		0			0
$865$	−26983.1	+	32157.2i	−0.0360628	+	0.0429780i
$866$		0			0
$867$	523808.	+	302421.i	0.696841	+	0.402321i
$868$		0			0
$869$	−815271.	+	143754.i	−1.07960	+	0.190362i
$870$		0			0
$871$	−287223.	−	104541.i	−0.378602	−	0.137800i
$872$		0			0
$873$			108915.i			0.142908i
$874$		0			0
$875$	135182.			0.176564
$876$		0			0
$877$	−115276.	+	316717.i	−0.149878	+	0.411787i	−0.991798	−	0.127816i	$-0.959203\pi$
$877$	−115276.	+	316717.i	−0.149878	+	0.411787i	0.841920	+	0.539603i	$0.181426\pi$
$878$		0			0
$879$	−189633.	−	1.07546e6i	−0.245434	−	1.39193i
$880$		0			0
$881$	−66029.0	+	114366.i	−0.0850713	+	0.147348i	−0.905422	−	0.424514i	$-0.860445\pi$
$881$	−66029.0	+	114366.i	−0.0850713	+	0.147348i	0.820350	+	0.571861i	$0.193778\pi$
$882$		0			0
$883$	576917.	+	484091.i	0.739932	+	0.620877i	0.932819	−	0.360344i	$-0.117341\pi$
$883$	576917.	+	484091.i	0.739932	+	0.620877i	−0.192887	+	0.981221i	$0.561785\pi$
$884$		0			0
$885$	36513.0	+	63242.3i	0.0466187	+	0.0807460i
$886$		0			0
$887$	−201026.	−	552313.i	−0.255508	−	0.702002i	−0.999431	−	0.0337365i	$-0.989259\pi$
$887$	−201026.	−	552313.i	−0.255508	−	0.702002i	0.743923	−	0.668265i	$-0.232963\pi$
$888$		0			0
$889$	−175079.	−	30871.1i	−0.221528	−	0.0390614i
$890$		0			0
$891$	−601322.	+	504569.i	−0.757446	+	0.635572i
$892$		0			0
$893$	110856.	−	811706.i	0.139013	−	1.01788i
$894$		0			0
$895$	33191.4	+	39555.9i	0.0414361	+	0.0493816i
$896$		0			0
$897$	−13385.4	+	75912.2i	−0.0166359	+	0.0943467i
$898$		0			0
$899$	−137930.	+	50202.3i	−0.170662	+	0.0621160i
$900$		0			0
$901$	74908.5	−	43248.5i	0.0922745	−	0.0532747i
$902$		0			0
$903$	398895.	−	475385.i	0.489197	−	0.583002i
$904$		0			0
$905$	64703.7	+	37356.7i	0.0790009	+	0.0456112i
$906$		0			0
$907$	−774708.	+	136602.i	−0.941724	+	0.166051i	−0.623375	−	0.781923i	$-0.714239\pi$
$907$	−774708.	+	136602.i	−0.941724	+	0.166051i	−0.318348	+	0.947974i	$0.603128\pi$
$908$		0			0
$909$	−126607.	−	46081.3i	−0.153226	−	0.0557695i
$910$		0			0
$911$		−	651620.i		−	0.785159i	−0.919718	−	0.392580i	$-0.871583\pi$
$911$		−	651620.i		−	0.785159i	0.919718	−	0.392580i	$-0.128417\pi$
$912$		0			0
$913$	1.31147e6			1.57332
$914$		0			0
$915$	25046.5	−	68814.7i	0.0299161	−	0.0821938i
$916$		0			0
$917$	273750.	+	1.55251e6i	0.325548	+	1.84628i
$918$		0			0
$919$	836911.	−	1.44957e6i	0.990942	−	1.71636i	0.379175	−	0.925325i	$-0.376208\pi$
$919$	836911.	−	1.44957e6i	0.990942	−	1.71636i	0.611768	−	0.791037i	$-0.290459\pi$
$920$		0			0
$921$	580181.	+	486830.i	0.683982	+	0.573929i
$922$		0			0
$923$	−321727.	−	557247.i	−0.377645	−	0.654100i
$924$		0			0
$925$	287272.	+	789274.i	0.335746	+	0.922453i
$926$		0			0
$927$	304468.	+	53685.9i	0.354309	+	0.0624742i
$928$		0			0
$929$	1.13934e6	−	956016.i	1.32014	−	1.10773i	0.333865	−	0.942621i	$-0.391647\pi$
$929$	1.13934e6	−	956016.i	1.32014	−	1.10773i	0.986275	−	0.165108i	$-0.0527973\pi$
$930$		0			0
$931$	39457.8	−	74911.8i	0.0455233	−	0.0864272i
$932$		0			0
$933$	−724825.	−	863813.i	−0.832664	−	0.992330i
$934$		0			0
$935$	−5340.52	+	30287.6i	−0.00610886	+	0.0346451i
$936$		0			0
$937$	−142188.	+	51752.2i	−0.161951	+	0.0589454i	−0.421723	−	0.906725i	$-0.638575\pi$
$937$	−142188.	+	51752.2i	−0.161951	+	0.0589454i	0.259772	+	0.965670i	$0.416352\pi$
$938$		0			0
$939$	−893318.	+	515757.i	−1.01315	+	0.584944i
$940$		0			0
$941$	875851.	−	1.04380e6i	0.989124	−	1.17879i	0.00523986	−	0.999986i	$-0.498332\pi$
$941$	875851.	−	1.04380e6i	0.989124	−	1.17879i	0.983884	−	0.178806i	$-0.0572235\pi$
$942$		0			0
$943$	72248.6	+	41712.7i	0.0812467	+	0.0469078i
$944$		0			0
$945$	84489.4	−	14897.8i	0.0946103	−	0.0166824i
$946$		0			0
$947$	−988738.	−	359871.i	−1.10251	−	0.401280i	−0.274267	−	0.961654i	$-0.588435\pi$
$947$	−988738.	−	359871.i	−1.10251	−	0.401280i	−0.828240	+	0.560374i	$0.810657\pi$
$948$		0			0
$949$			526322.i			0.584412i
$950$		0			0
$951$	−1.23703e6			−1.36779
$952$		0			0
$953$	260609.	−	716017.i	0.286948	−	0.788384i	−0.709541	−	0.704664i	$-0.751098\pi$
$953$	260609.	−	716017.i	0.286948	−	0.788384i	0.996489	−	0.0837199i	$-0.0266801\pi$
$954$		0			0
$955$	−22121.3	−	125456.i	−0.0242551	−	0.137558i
$956$		0			0
$957$	129953.	−	225086.i	0.141894	−	0.245768i
$958$		0			0
$959$	158361.	+	132881.i	0.172192	+	0.144486i
$960$		0			0
$961$	−162164.	−	280877.i	−0.175594	−	0.304137i
$962$		0			0
$963$	−112917.	−	310237.i	−0.121761	−	0.334535i
$964$		0			0
$965$	−125684.	−	22161.5i	−0.134966	−	0.0237982i
$966$		0			0
$967$	408657.	−	342904.i	0.437025	−	0.366708i	−0.397570	−	0.917572i	$-0.630146\pi$
$967$	408657.	−	342904.i	0.437025	−	0.366708i	0.834595	+	0.550864i	$0.185702\pi$
$968$		0			0
$969$	128003.	−	165158.i	0.136324	−	0.175894i
$970$		0			0
$971$	−710180.	−	846359.i	−0.753234	−	0.897669i	0.244166	−	0.969733i	$-0.421486\pi$
$971$	−710180.	−	846359.i	−0.753234	−	0.897669i	−0.997400	+	0.0720641i	$0.977041\pi$
$972$		0			0
$973$	83986.7	−	476312.i	0.0887125	−	0.503113i
$974$		0			0
$975$	−498228.	+	181340.i	−0.524105	+	0.190759i
$976$		0			0
$977$	−193555.	+	111749.i	−0.202775	+	0.117072i	−0.597949	−	0.801534i	$-0.704018\pi$
$977$	−193555.	+	111749.i	−0.202775	+	0.117072i	0.395174	+	0.918606i	$0.370684\pi$
$978$		0			0
$979$	−513774.	+	612292.i	−0.536052	+	0.638842i
$980$		0			0
$981$	200664.	+	115853.i	0.208512	+	0.120384i
$982$		0			0
$983$	763466.	−	134620.i	0.790101	−	0.139316i	0.235986	−	0.971756i	$-0.424168\pi$
$983$	763466.	−	134620.i	0.790101	−	0.139316i	0.554115	+	0.832440i	$0.313057\pi$
$984$		0			0
$985$	14861.1	+	5408.98i	0.0153171	+	0.00557498i
$986$		0			0
$987$		−	819547.i		−	0.841278i
$988$		0			0
$989$	154782.			0.158244
$990$		0			0
$991$	399466.	−	1.09753e6i	0.406755	−	1.11755i	−0.552130	−	0.833758i	$-0.686185\pi$
$991$	399466.	−	1.09753e6i	0.406755	−	1.11755i	0.958886	−	0.283793i	$-0.0915928\pi$
$992$		0			0
$993$	−121509.	−	689109.i	−0.123228	−	0.698859i
$994$		0			0
$995$	75306.5	−	130435.i	0.0760653	−	0.131749i
$996$		0			0
$997$	−542528.	−	455235.i	−0.545798	−	0.457979i	0.327717	−	0.944776i	$-0.393721\pi$
$997$	−542528.	−	455235.i	−0.545798	−	0.457979i	−0.873515	+	0.486797i	$0.838165\pi$
$998$		0			0
$999$	535401.	+	927342.i	0.536474	+	0.929200i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.j.a.13.5	✓	42
19.3	odd	18	inner	76.5.j.a.41.5	yes	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.j.a.13.5	✓	42	1.1	even	1	trivial
76.5.j.a.41.5	yes	42	19.3	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)

\(f(q)\)	\(=\)	\(q+(2.65366 - 7.29088i) q^{3} +(0.405227 + 2.29816i) q^{5} +(23.2726 - 40.3094i) q^{7} +(15.9346 + 13.3707i) q^{9} +O(q^{10})\)	\(q+(2.65366 - 7.29088i) q^{3} +(0.405227 + 2.29816i) q^{5} +(23.2726 - 40.3094i) q^{7} +(15.9346 + 13.3707i) q^{9} +(-88.3294 - 152.991i) q^{11} +(-37.7242 - 103.646i) q^{13} +(17.8309 + 3.14407i) q^{15} +(-57.1483 + 47.9531i) q^{17} +(-192.492 - 305.398i) q^{19} +(-232.133 - 276.646i) q^{21} +(15.6411 - 88.7052i) q^{23} +(582.191 - 211.900i) q^{25} +(684.034 - 394.927i) q^{27} +(-121.887 + 145.259i) q^{29} +(670.369 + 387.038i) q^{31} +(-1349.84 + 238.013i) q^{33} +(102.068 + 37.1497i) q^{35} +1355.70i q^{37} -855.781 q^{39} +(-316.776 + 870.336i) q^{41} +(298.395 + 1692.28i) q^{43} +(-24.2709 + 42.0384i) q^{45} +(1738.43 + 1458.72i) q^{47} +(117.269 + 203.116i) q^{49} +(197.968 + 543.913i) q^{51} +(-1141.83 - 201.336i) q^{53} +(315.804 - 264.991i) q^{55} +(-2737.43 + 593.012i) q^{57} +(2592.52 + 3089.64i) q^{59} +(702.333 - 3983.13i) q^{61} +(909.805 - 331.142i) q^{63} +(222.909 - 128.697i) q^{65} +(1781.28 - 2122.85i) q^{67} +(-605.233 - 349.431i) q^{69} +(5745.13 - 1013.02i) q^{71} +(-4484.03 - 1632.05i) q^{73} -4806.99i q^{75} -8222.63 q^{77} +(1602.75 - 4403.53i) q^{79} +(-771.592 - 4375.92i) q^{81} +(-3711.86 + 6429.14i) q^{83} +(-133.362 - 111.904i) q^{85} +(735.618 + 1274.13i) q^{87} +(-1547.47 - 4251.63i) q^{89} +(-5055.87 - 891.486i) q^{91} +(4600.78 - 3860.51i) q^{93} +(623.850 - 566.132i) q^{95} +(3365.63 + 4011.01i) q^{97} +(638.106 - 3618.88i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) \) 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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