LMFDB - Embedded newform 8004.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

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:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.44612$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.44612 q^{5} +4.67774 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.44612 q^{5} +4.67774 q^{7} +1.00000 q^{9} -3.77983 q^{11} -3.51025 q^{13} -3.44612 q^{15} -8.12984 q^{17} +5.33577 q^{19} +4.67774 q^{21} -1.00000 q^{23} +6.87575 q^{25} +1.00000 q^{27} +1.00000 q^{29} -9.72230 q^{31} -3.77983 q^{33} -16.1200 q^{35} +7.45986 q^{37} -3.51025 q^{39} +3.95562 q^{41} -2.47386 q^{43} -3.44612 q^{45} -3.53158 q^{47} +14.8812 q^{49} -8.12984 q^{51} +9.38979 q^{53} +13.0258 q^{55} +5.33577 q^{57} +0.971003 q^{59} -12.8969 q^{61} +4.67774 q^{63} +12.0967 q^{65} +14.4797 q^{67} -1.00000 q^{69} +9.25213 q^{71} +10.1166 q^{73} +6.87575 q^{75} -17.6811 q^{77} +14.1749 q^{79} +1.00000 q^{81} +10.7640 q^{83} +28.0164 q^{85} +1.00000 q^{87} -15.4033 q^{89} -16.4200 q^{91} -9.72230 q^{93} -18.3877 q^{95} -14.6078 q^{97} -3.77983 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.44612	−1.54115	−0.770576	−	0.637348i	$-0.780031\pi$
$5$	−3.44612	−1.54115	−0.770576	+	0.637348i	$0.780031\pi$
$6$	0	0
$7$	4.67774	1.76802	0.884009	−	0.467470i	$-0.154834\pi$
$7$	4.67774	1.76802	0.884009	+	0.467470i	$0.154834\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.77983	−1.13966	−0.569831	−	0.821762i	$-0.692991\pi$
$11$	−3.77983	−1.13966	−0.569831	+	0.821762i	$0.692991\pi$
$12$	0	0
$13$	−3.51025	−0.973568	−0.486784	−	0.873522i	$-0.661830\pi$
$13$	−3.51025	−0.973568	−0.486784	+	0.873522i	$0.661830\pi$
$14$	0	0
$15$	−3.44612	−0.889785
$16$	0	0
$17$	−8.12984	−1.97178	−0.985888	−	0.167407i	$-0.946461\pi$
$17$	−8.12984	−1.97178	−0.985888	+	0.167407i	$0.946461\pi$
$18$	0	0
$19$	5.33577	1.22411	0.612055	−	0.790815i	$-0.290343\pi$
$19$	5.33577	1.22411	0.612055	+	0.790815i	$0.290343\pi$
$20$	0	0
$21$	4.67774	1.02077
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	6.87575	1.37515
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−9.72230	−1.74618	−0.873088	−	0.487562i	$-0.837886\pi$
$31$	−9.72230	−1.74618	−0.873088	+	0.487562i	$0.837886\pi$
$32$	0	0
$33$	−3.77983	−0.657984
$34$	0	0
$35$	−16.1200	−2.72479
$36$	0	0
$37$	7.45986	1.22639	0.613197	−	0.789930i	$-0.289883\pi$
$37$	7.45986	1.22639	0.613197	+	0.789930i	$0.289883\pi$
$38$	0	0
$39$	−3.51025	−0.562090
$40$	0	0
$41$	3.95562	0.617764	0.308882	−	0.951100i	$-0.400045\pi$
$41$	3.95562	0.617764	0.308882	+	0.951100i	$0.400045\pi$
$42$	0	0
$43$	−2.47386	−0.377259	−0.188630	−	0.982048i	$-0.560405\pi$
$43$	−2.47386	−0.377259	−0.188630	+	0.982048i	$0.560405\pi$
$44$	0	0
$45$	−3.44612	−0.513717
$46$	0	0
$47$	−3.53158	−0.515134	−0.257567	−	0.966260i	$-0.582921\pi$
$47$	−3.53158	−0.515134	−0.257567	+	0.966260i	$0.582921\pi$
$48$	0	0
$49$	14.8812	2.12589
$50$	0	0
$51$	−8.12984	−1.13841
$52$	0	0
$53$	9.38979	1.28979	0.644893	−	0.764272i	$-0.276902\pi$
$53$	9.38979	1.28979	0.644893	+	0.764272i	$0.276902\pi$
$54$	0	0
$55$	13.0258	1.75639
$56$	0	0
$57$	5.33577	0.706740
$58$	0	0
$59$	0.971003	0.126414	0.0632069	−	0.998000i	$-0.479867\pi$
$59$	0.971003	0.126414	0.0632069	+	0.998000i	$0.479867\pi$
$60$	0	0
$61$	−12.8969	−1.65128	−0.825638	−	0.564201i	$-0.809184\pi$
$61$	−12.8969	−1.65128	−0.825638	+	0.564201i	$0.809184\pi$
$62$	0	0
$63$	4.67774	0.589339
$64$	0	0
$65$	12.0967	1.50042
$66$	0	0
$67$	14.4797	1.76898	0.884491	−	0.466557i	$-0.154506\pi$
$67$	14.4797	1.76898	0.884491	+	0.466557i	$0.154506\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	9.25213	1.09803	0.549013	−	0.835814i	$-0.315004\pi$
$71$	9.25213	1.09803	0.549013	+	0.835814i	$0.315004\pi$
$72$	0	0
$73$	10.1166	1.18406	0.592030	−	0.805916i	$-0.298327\pi$
$73$	10.1166	1.18406	0.592030	+	0.805916i	$0.298327\pi$
$74$	0	0
$75$	6.87575	0.793944
$76$	0	0
$77$	−17.6811	−2.01494
$78$	0	0
$79$	14.1749	1.59480	0.797399	−	0.603453i	$-0.206209\pi$
$79$	14.1749	1.59480	0.797399	+	0.603453i	$0.206209\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.7640	1.18151	0.590753	−	0.806852i	$-0.298831\pi$
$83$	10.7640	1.18151	0.590753	+	0.806852i	$0.298831\pi$
$84$	0	0
$85$	28.0164	3.03881
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−15.4033	−1.63275	−0.816373	−	0.577525i	$-0.804019\pi$
$89$	−15.4033	−1.63275	−0.816373	+	0.577525i	$0.804019\pi$
$90$	0	0
$91$	−16.4200	−1.72129
$92$	0	0
$93$	−9.72230	−1.00816
$94$	0	0
$95$	−18.3877	−1.88654
$96$	0	0
$97$	−14.6078	−1.48319	−0.741597	−	0.670846i	$-0.765931\pi$
$97$	−14.6078	−1.48319	−0.741597	+	0.670846i	$0.765931\pi$
$98$	0	0
$99$	−3.77983	−0.379887
$100$	0	0
$101$	1.45378	0.144657	0.0723284	−	0.997381i	$-0.476957\pi$
$101$	1.45378	0.144657	0.0723284	+	0.997381i	$0.476957\pi$
$102$	0	0
$103$	7.66441	0.755197	0.377598	−	0.925970i	$-0.376750\pi$
$103$	7.66441	0.755197	0.377598	+	0.925970i	$0.376750\pi$
$104$	0	0
$105$	−16.1200	−1.57316
$106$	0	0
$107$	20.2562	1.95824	0.979122	−	0.203274i	$-0.0651583\pi$
$107$	20.2562	1.95824	0.979122	+	0.203274i	$0.0651583\pi$
$108$	0	0
$109$	−6.12731	−0.586890	−0.293445	−	0.955976i	$-0.594802\pi$
$109$	−6.12731	−0.586890	−0.293445	+	0.955976i	$0.594802\pi$
$110$	0	0
$111$	7.45986	0.708059
$112$	0	0
$113$	4.43980	0.417661	0.208831	−	0.977952i	$-0.433034\pi$
$113$	4.43980	0.417661	0.208831	+	0.977952i	$0.433034\pi$
$114$	0	0
$115$	3.44612	0.321352
$116$	0	0
$117$	−3.51025	−0.324523
$118$	0	0
$119$	−38.0292	−3.48614
$120$	0	0
$121$	3.28712	0.298829
$122$	0	0
$123$	3.95562	0.356666
$124$	0	0
$125$	−6.46407	−0.578164
$126$	0	0
$127$	4.12568	0.366095	0.183047	−	0.983104i	$-0.441404\pi$
$127$	4.12568	0.366095	0.183047	+	0.983104i	$0.441404\pi$
$128$	0	0
$129$	−2.47386	−0.217811
$130$	0	0
$131$	11.1176	0.971352	0.485676	−	0.874139i	$-0.338573\pi$
$131$	11.1176	0.971352	0.485676	+	0.874139i	$0.338573\pi$
$132$	0	0
$133$	24.9593	2.16425
$134$	0	0
$135$	−3.44612	−0.296595
$136$	0	0
$137$	−18.5291	−1.58305	−0.791524	−	0.611137i	$-0.790712\pi$
$137$	−18.5291	−1.58305	−0.791524	+	0.611137i	$0.790712\pi$
$138$	0	0
$139$	4.48110	0.380082	0.190041	−	0.981776i	$-0.439138\pi$
$139$	4.48110	0.380082	0.190041	+	0.981776i	$0.439138\pi$
$140$	0	0
$141$	−3.53158	−0.297413
$142$	0	0
$143$	13.2681	1.10954
$144$	0	0
$145$	−3.44612	−0.286185
$146$	0	0
$147$	14.8812	1.22738
$148$	0	0
$149$	6.33782	0.519214	0.259607	−	0.965714i	$-0.416407\pi$
$149$	6.33782	0.519214	0.259607	+	0.965714i	$0.416407\pi$
$150$	0	0
$151$	5.55727	0.452244	0.226122	−	0.974099i	$-0.427395\pi$
$151$	5.55727	0.452244	0.226122	+	0.974099i	$0.427395\pi$
$152$	0	0
$153$	−8.12984	−0.657259
$154$	0	0
$155$	33.5042	2.69112
$156$	0	0
$157$	−2.14436	−0.171138	−0.0855692	−	0.996332i	$-0.527271\pi$
$157$	−2.14436	−0.171138	−0.0855692	+	0.996332i	$0.527271\pi$
$158$	0	0
$159$	9.38979	0.744659
$160$	0	0
$161$	−4.67774	−0.368657
$162$	0	0
$163$	−8.26347	−0.647245	−0.323623	−	0.946186i	$-0.604901\pi$
$163$	−8.26347	−0.647245	−0.323623	+	0.946186i	$0.604901\pi$
$164$	0	0
$165$	13.0258	1.01405
$166$	0	0
$167$	9.48777	0.734186	0.367093	−	0.930184i	$-0.380353\pi$
$167$	9.48777	0.734186	0.367093	+	0.930184i	$0.380353\pi$
$168$	0	0
$169$	−0.678155	−0.0521658
$170$	0	0
$171$	5.33577	0.408037
$172$	0	0
$173$	−1.73038	−0.131558	−0.0657790	−	0.997834i	$-0.520953\pi$
$173$	−1.73038	−0.131558	−0.0657790	+	0.997834i	$0.520953\pi$
$174$	0	0
$175$	32.1630	2.43129
$176$	0	0
$177$	0.971003	0.0729850
$178$	0	0
$179$	14.0890	1.05306	0.526531	−	0.850156i	$-0.323493\pi$
$179$	14.0890	1.05306	0.526531	+	0.850156i	$0.323493\pi$
$180$	0	0
$181$	8.89981	0.661517	0.330759	−	0.943715i	$-0.392695\pi$
$181$	8.89981	0.661517	0.330759	+	0.943715i	$0.392695\pi$
$182$	0	0
$183$	−12.8969	−0.953364
$184$	0	0
$185$	−25.7076	−1.89006
$186$	0	0
$187$	30.7294	2.24716
$188$	0	0
$189$	4.67774	0.340255
$190$	0	0
$191$	−22.4148	−1.62188	−0.810940	−	0.585129i	$-0.801044\pi$
$191$	−22.4148	−1.62188	−0.810940	+	0.585129i	$0.801044\pi$
$192$	0	0
$193$	22.4446	1.61559	0.807797	−	0.589461i	$-0.200660\pi$
$193$	22.4446	1.61559	0.807797	+	0.589461i	$0.200660\pi$
$194$	0	0
$195$	12.0967	0.866266
$196$	0	0
$197$	2.86856	0.204377	0.102188	−	0.994765i	$-0.467416\pi$
$197$	2.86856	0.204377	0.102188	+	0.994765i	$0.467416\pi$
$198$	0	0
$199$	11.8844	0.842459	0.421230	−	0.906954i	$-0.361599\pi$
$199$	11.8844	0.842459	0.421230	+	0.906954i	$0.361599\pi$
$200$	0	0
$201$	14.4797	1.02132
$202$	0	0
$203$	4.67774	0.328313
$204$	0	0
$205$	−13.6315	−0.952068
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−20.1683	−1.39507
$210$	0	0
$211$	−6.40524	−0.440955	−0.220477	−	0.975392i	$-0.570762\pi$
$211$	−6.40524	−0.440955	−0.220477	+	0.975392i	$0.570762\pi$
$212$	0	0
$213$	9.25213	0.633945
$214$	0	0
$215$	8.52521	0.581414
$216$	0	0
$217$	−45.4784	−3.08727
$218$	0	0
$219$	10.1166	0.683618
$220$	0	0
$221$	28.5378	1.91966
$222$	0	0
$223$	2.73456	0.183120	0.0915600	−	0.995800i	$-0.470815\pi$
$223$	2.73456	0.183120	0.0915600	+	0.995800i	$0.470815\pi$
$224$	0	0
$225$	6.87575	0.458384
$226$	0	0
$227$	27.5888	1.83113	0.915566	−	0.402168i	$-0.131743\pi$
$227$	27.5888	1.83113	0.915566	+	0.402168i	$0.131743\pi$
$228$	0	0
$229$	−20.8421	−1.37728	−0.688641	−	0.725102i	$-0.741793\pi$
$229$	−20.8421	−1.37728	−0.688641	+	0.725102i	$0.741793\pi$
$230$	0	0
$231$	−17.6811	−1.16333
$232$	0	0
$233$	−10.1311	−0.663712	−0.331856	−	0.943330i	$-0.607675\pi$
$233$	−10.1311	−0.663712	−0.331856	+	0.943330i	$0.607675\pi$
$234$	0	0
$235$	12.1703	0.793900
$236$	0	0
$237$	14.1749	0.920757
$238$	0	0
$239$	10.7910	0.698015	0.349007	−	0.937120i	$-0.386519\pi$
$239$	10.7910	0.698015	0.349007	+	0.937120i	$0.386519\pi$
$240$	0	0
$241$	8.64672	0.556984	0.278492	−	0.960438i	$-0.410165\pi$
$241$	8.64672	0.556984	0.278492	+	0.960438i	$0.410165\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−51.2825	−3.27632
$246$	0	0
$247$	−18.7299	−1.19175
$248$	0	0
$249$	10.7640	0.682143
$250$	0	0
$251$	−12.1724	−0.768313	−0.384156	−	0.923268i	$-0.625508\pi$
$251$	−12.1724	−0.768313	−0.384156	+	0.923268i	$0.625508\pi$
$252$	0	0
$253$	3.77983	0.237636
$254$	0	0
$255$	28.0164	1.75446
$256$	0	0
$257$	−31.6720	−1.97564	−0.987822	−	0.155591i	$-0.950272\pi$
$257$	−31.6720	−1.97564	−0.987822	+	0.155591i	$0.950272\pi$
$258$	0	0
$259$	34.8953	2.16829
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	18.9941	1.17123	0.585613	−	0.810591i	$-0.300854\pi$
$263$	18.9941	1.17123	0.585613	+	0.810591i	$0.300854\pi$
$264$	0	0
$265$	−32.3584	−1.98776
$266$	0	0
$267$	−15.4033	−0.942667
$268$	0	0
$269$	−0.758985	−0.0462761	−0.0231381	−	0.999732i	$-0.507366\pi$
$269$	−0.758985	−0.0462761	−0.0231381	+	0.999732i	$0.507366\pi$
$270$	0	0
$271$	−19.8284	−1.20449	−0.602244	−	0.798312i	$-0.705727\pi$
$271$	−19.8284	−1.20449	−0.602244	+	0.798312i	$0.705727\pi$
$272$	0	0
$273$	−16.4200	−0.993785
$274$	0	0
$275$	−25.9892	−1.56721
$276$	0	0
$277$	−1.27693	−0.0767232	−0.0383616	−	0.999264i	$-0.512214\pi$
$277$	−1.27693	−0.0767232	−0.0383616	+	0.999264i	$0.512214\pi$
$278$	0	0
$279$	−9.72230	−0.582059
$280$	0	0
$281$	8.58038	0.511863	0.255931	−	0.966695i	$-0.417618\pi$
$281$	8.58038	0.511863	0.255931	+	0.966695i	$0.417618\pi$
$282$	0	0
$283$	−6.23788	−0.370804	−0.185402	−	0.982663i	$-0.559359\pi$
$283$	−6.23788	−0.370804	−0.185402	+	0.982663i	$0.559359\pi$
$284$	0	0
$285$	−18.3877	−1.08919
$286$	0	0
$287$	18.5033	1.09222
$288$	0	0
$289$	49.0943	2.88790
$290$	0	0
$291$	−14.6078	−0.856322
$292$	0	0
$293$	10.5887	0.618596	0.309298	−	0.950965i	$-0.399906\pi$
$293$	10.5887	0.618596	0.309298	+	0.950965i	$0.399906\pi$
$294$	0	0
$295$	−3.34619	−0.194823
$296$	0	0
$297$	−3.77983	−0.219328
$298$	0	0
$299$	3.51025	0.203003
$300$	0	0
$301$	−11.5720	−0.667002
$302$	0	0
$303$	1.45378	0.0835176
$304$	0	0
$305$	44.4442	2.54487
$306$	0	0
$307$	12.7072	0.725237	0.362618	−	0.931938i	$-0.381883\pi$
$307$	12.7072	0.725237	0.362618	+	0.931938i	$0.381883\pi$
$308$	0	0
$309$	7.66441	0.436013
$310$	0	0
$311$	−0.170695	−0.00967922	−0.00483961	−	0.999988i	$-0.501541\pi$
$311$	−0.170695	−0.00967922	−0.00483961	+	0.999988i	$0.501541\pi$
$312$	0	0
$313$	−0.414275	−0.0234162	−0.0117081	−	0.999931i	$-0.503727\pi$
$313$	−0.414275	−0.0234162	−0.0117081	+	0.999931i	$0.503727\pi$
$314$	0	0
$315$	−16.1200	−0.908262
$316$	0	0
$317$	−17.3331	−0.973526	−0.486763	−	0.873534i	$-0.661822\pi$
$317$	−17.3331	−0.973526	−0.486763	+	0.873534i	$0.661822\pi$
$318$	0	0
$319$	−3.77983	−0.211630
$320$	0	0
$321$	20.2562	1.13059
$322$	0	0
$323$	−43.3790	−2.41367
$324$	0	0
$325$	−24.1356	−1.33880
$326$	0	0
$327$	−6.12731	−0.338841
$328$	0	0
$329$	−16.5198	−0.910766
$330$	0	0
$331$	−0.522129	−0.0286988	−0.0143494	−	0.999897i	$-0.504568\pi$
$331$	−0.522129	−0.0286988	−0.0143494	+	0.999897i	$0.504568\pi$
$332$	0	0
$333$	7.45986	0.408798
$334$	0	0
$335$	−49.8990	−2.72627
$336$	0	0
$337$	−30.0909	−1.63916	−0.819578	−	0.572968i	$-0.805792\pi$
$337$	−30.0909	−1.63916	−0.819578	+	0.572968i	$0.805792\pi$
$338$	0	0
$339$	4.43980	0.241137
$340$	0	0
$341$	36.7486	1.99005
$342$	0	0
$343$	36.8662	1.99059
$344$	0	0
$345$	3.44612	0.185533
$346$	0	0
$347$	−18.1601	−0.974884	−0.487442	−	0.873155i	$-0.662070\pi$
$347$	−18.1601	−0.974884	−0.487442	+	0.873155i	$0.662070\pi$
$348$	0	0
$349$	8.88270	0.475480	0.237740	−	0.971329i	$-0.423593\pi$
$349$	8.88270	0.475480	0.237740	+	0.971329i	$0.423593\pi$
$350$	0	0
$351$	−3.51025	−0.187363
$352$	0	0
$353$	32.0169	1.70409	0.852043	−	0.523472i	$-0.175364\pi$
$353$	32.0169	1.70409	0.852043	+	0.523472i	$0.175364\pi$
$354$	0	0
$355$	−31.8840	−1.69222
$356$	0	0
$357$	−38.0292	−2.01272
$358$	0	0
$359$	11.0964	0.585645	0.292823	−	0.956167i	$-0.405405\pi$
$359$	11.0964	0.585645	0.292823	+	0.956167i	$0.405405\pi$
$360$	0	0
$361$	9.47048	0.498446
$362$	0	0
$363$	3.28712	0.172529
$364$	0	0
$365$	−34.8631	−1.82482
$366$	0	0
$367$	15.4290	0.805388	0.402694	−	0.915335i	$-0.368074\pi$
$367$	15.4290	0.805388	0.402694	+	0.915335i	$0.368074\pi$
$368$	0	0
$369$	3.95562	0.205921
$370$	0	0
$371$	43.9230	2.28037
$372$	0	0
$373$	−11.8381	−0.612955	−0.306478	−	0.951878i	$-0.599150\pi$
$373$	−11.8381	−0.612955	−0.306478	+	0.951878i	$0.599150\pi$
$374$	0	0
$375$	−6.46407	−0.333803
$376$	0	0
$377$	−3.51025	−0.180787
$378$	0	0
$379$	19.6109	1.00734	0.503672	−	0.863895i	$-0.331982\pi$
$379$	19.6109	1.00734	0.503672	+	0.863895i	$0.331982\pi$
$380$	0	0
$381$	4.12568	0.211365
$382$	0	0
$383$	10.2295	0.522703	0.261352	−	0.965244i	$-0.415832\pi$
$383$	10.2295	0.522703	0.261352	+	0.965244i	$0.415832\pi$
$384$	0	0
$385$	60.9310	3.10533
$386$	0	0
$387$	−2.47386	−0.125753
$388$	0	0
$389$	23.7177	1.20253	0.601267	−	0.799048i	$-0.294663\pi$
$389$	23.7177	1.20253	0.601267	+	0.799048i	$0.294663\pi$
$390$	0	0
$391$	8.12984	0.411144
$392$	0	0
$393$	11.1176	0.560811
$394$	0	0
$395$	−48.8483	−2.45783
$396$	0	0
$397$	−28.6869	−1.43975	−0.719877	−	0.694102i	$-0.755802\pi$
$397$	−28.6869	−1.43975	−0.719877	+	0.694102i	$0.755802\pi$
$398$	0	0
$399$	24.9593	1.24953
$400$	0	0
$401$	4.38220	0.218837	0.109418	−	0.993996i	$-0.465101\pi$
$401$	4.38220	0.218837	0.109418	+	0.993996i	$0.465101\pi$
$402$	0	0
$403$	34.1277	1.70002
$404$	0	0
$405$	−3.44612	−0.171239
$406$	0	0
$407$	−28.1970	−1.39767
$408$	0	0
$409$	−13.8465	−0.684664	−0.342332	−	0.939579i	$-0.611217\pi$
$409$	−13.8465	−0.684664	−0.342332	+	0.939579i	$0.611217\pi$
$410$	0	0
$411$	−18.5291	−0.913974
$412$	0	0
$413$	4.54209	0.223502
$414$	0	0
$415$	−37.0942	−1.82088
$416$	0	0
$417$	4.48110	0.219440
$418$	0	0
$419$	−30.9630	−1.51264	−0.756320	−	0.654202i	$-0.773005\pi$
$419$	−30.9630	−1.51264	−0.756320	+	0.654202i	$0.773005\pi$
$420$	0	0
$421$	30.7315	1.49776	0.748880	−	0.662706i	$-0.230592\pi$
$421$	30.7315	1.49776	0.748880	+	0.662706i	$0.230592\pi$
$422$	0	0
$423$	−3.53158	−0.171711
$424$	0	0
$425$	−55.8988	−2.71149
$426$	0	0
$427$	−60.3282	−2.91948
$428$	0	0
$429$	13.2681	0.640592
$430$	0	0
$431$	25.9916	1.25197	0.625985	−	0.779835i	$-0.284697\pi$
$431$	25.9916	1.25197	0.625985	+	0.779835i	$0.284697\pi$
$432$	0	0
$433$	−0.170347	−0.00818634	−0.00409317	−	0.999992i	$-0.501303\pi$
$433$	−0.170347	−0.00818634	−0.00409317	+	0.999992i	$0.501303\pi$
$434$	0	0
$435$	−3.44612	−0.165229
$436$	0	0
$437$	−5.33577	−0.255245
$438$	0	0
$439$	7.83101	0.373754	0.186877	−	0.982383i	$-0.440163\pi$
$439$	7.83101	0.373754	0.186877	+	0.982383i	$0.440163\pi$
$440$	0	0
$441$	14.8812	0.708629
$442$	0	0
$443$	28.0164	1.33110	0.665550	−	0.746354i	$-0.268197\pi$
$443$	28.0164	1.33110	0.665550	+	0.746354i	$0.268197\pi$
$444$	0	0
$445$	53.0816	2.51631
$446$	0	0
$447$	6.33782	0.299769
$448$	0	0
$449$	23.3399	1.10148	0.550738	−	0.834678i	$-0.314346\pi$
$449$	23.3399	1.10148	0.550738	+	0.834678i	$0.314346\pi$
$450$	0	0
$451$	−14.9516	−0.704042
$452$	0	0
$453$	5.55727	0.261103
$454$	0	0
$455$	56.5854	2.65276
$456$	0	0
$457$	30.1467	1.41021	0.705103	−	0.709105i	$-0.250901\pi$
$457$	30.1467	1.41021	0.705103	+	0.709105i	$0.250901\pi$
$458$	0	0
$459$	−8.12984	−0.379468
$460$	0	0
$461$	−7.02110	−0.327005	−0.163503	−	0.986543i	$-0.552279\pi$
$461$	−7.02110	−0.327005	−0.163503	+	0.986543i	$0.552279\pi$
$462$	0	0
$463$	2.05569	0.0955358	0.0477679	−	0.998858i	$-0.484789\pi$
$463$	2.05569	0.0955358	0.0477679	+	0.998858i	$0.484789\pi$
$464$	0	0
$465$	33.5042	1.55372
$466$	0	0
$467$	6.46555	0.299190	0.149595	−	0.988747i	$-0.452203\pi$
$467$	6.46555	0.299190	0.149595	+	0.988747i	$0.452203\pi$
$468$	0	0
$469$	67.7324	3.12759
$470$	0	0
$471$	−2.14436	−0.0988068
$472$	0	0
$473$	9.35076	0.429948
$474$	0	0
$475$	36.6875	1.68334
$476$	0	0
$477$	9.38979	0.429929
$478$	0	0
$479$	−38.1412	−1.74272	−0.871358	−	0.490648i	$-0.836760\pi$
$479$	−38.1412	−1.74272	−0.871358	+	0.490648i	$0.836760\pi$
$480$	0	0
$481$	−26.1860	−1.19398
$482$	0	0
$483$	−4.67774	−0.212844
$484$	0	0
$485$	50.3401	2.28583
$486$	0	0
$487$	−14.3846	−0.651827	−0.325914	−	0.945400i	$-0.605672\pi$
$487$	−14.3846	−0.651827	−0.325914	+	0.945400i	$0.605672\pi$
$488$	0	0
$489$	−8.26347	−0.373687
$490$	0	0
$491$	28.7557	1.29773	0.648863	−	0.760905i	$-0.275245\pi$
$491$	28.7557	1.29773	0.648863	+	0.760905i	$0.275245\pi$
$492$	0	0
$493$	−8.12984	−0.366150
$494$	0	0
$495$	13.0258	0.585464
$496$	0	0
$497$	43.2790	1.94133
$498$	0	0
$499$	20.6705	0.925339	0.462669	−	0.886531i	$-0.346892\pi$
$499$	20.6705	0.925339	0.462669	+	0.886531i	$0.346892\pi$
$500$	0	0
$501$	9.48777	0.423882
$502$	0	0
$503$	31.0955	1.38648	0.693240	−	0.720706i	$-0.256182\pi$
$503$	31.0955	1.38648	0.693240	+	0.720706i	$0.256182\pi$
$504$	0	0
$505$	−5.00991	−0.222938
$506$	0	0
$507$	−0.678155	−0.0301179
$508$	0	0
$509$	−12.5883	−0.557966	−0.278983	−	0.960296i	$-0.589997\pi$
$509$	−12.5883	−0.557966	−0.278983	+	0.960296i	$0.589997\pi$
$510$	0	0
$511$	47.3229	2.09344
$512$	0	0
$513$	5.33577	0.235580
$514$	0	0
$515$	−26.4125	−1.16387
$516$	0	0
$517$	13.3488	0.587079
$518$	0	0
$519$	−1.73038	−0.0759551
$520$	0	0
$521$	−13.1795	−0.577405	−0.288702	−	0.957419i	$-0.593224\pi$
$521$	−13.1795	−0.577405	−0.288702	+	0.957419i	$0.593224\pi$
$522$	0	0
$523$	−9.75682	−0.426636	−0.213318	−	0.976983i	$-0.568427\pi$
$523$	−9.75682	−0.426636	−0.213318	+	0.976983i	$0.568427\pi$
$524$	0	0
$525$	32.1630	1.40371
$526$	0	0
$527$	79.0407	3.44307
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.971003	0.0421379
$532$	0	0
$533$	−13.8852	−0.601435
$534$	0	0
$535$	−69.8054	−3.01795
$536$	0	0
$537$	14.0890	0.607985
$538$	0	0
$539$	−56.2485	−2.42279
$540$	0	0
$541$	4.48118	0.192661	0.0963304	−	0.995349i	$-0.469289\pi$
$541$	4.48118	0.192661	0.0963304	+	0.995349i	$0.469289\pi$
$542$	0	0
$543$	8.89981	0.381927
$544$	0	0
$545$	21.1155	0.904487
$546$	0	0
$547$	20.8881	0.893109	0.446554	−	0.894757i	$-0.352651\pi$
$547$	20.8881	0.893109	0.446554	+	0.894757i	$0.352651\pi$
$548$	0	0
$549$	−12.8969	−0.550425
$550$	0	0
$551$	5.33577	0.227312
$552$	0	0
$553$	66.3063	2.81963
$554$	0	0
$555$	−25.7076	−1.09123
$556$	0	0
$557$	40.4571	1.71422	0.857112	−	0.515130i	$-0.172257\pi$
$557$	40.4571	1.71422	0.857112	+	0.515130i	$0.172257\pi$
$558$	0	0
$559$	8.68385	0.367288
$560$	0	0
$561$	30.7294	1.29740
$562$	0	0
$563$	24.3385	1.02575	0.512873	−	0.858464i	$-0.328581\pi$
$563$	24.3385	1.02575	0.512873	+	0.858464i	$0.328581\pi$
$564$	0	0
$565$	−15.3001	−0.643679
$566$	0	0
$567$	4.67774	0.196446
$568$	0	0
$569$	−20.7044	−0.867974	−0.433987	−	0.900919i	$-0.642894\pi$
$569$	−20.7044	−0.867974	−0.433987	+	0.900919i	$0.642894\pi$
$570$	0	0
$571$	23.9870	1.00383	0.501913	−	0.864918i	$-0.332630\pi$
$571$	23.9870	1.00383	0.501913	+	0.864918i	$0.332630\pi$
$572$	0	0
$573$	−22.4148	−0.936393
$574$	0	0
$575$	−6.87575	−0.286739
$576$	0	0
$577$	−8.22207	−0.342289	−0.171145	−	0.985246i	$-0.554747\pi$
$577$	−8.22207	−0.342289	−0.171145	+	0.985246i	$0.554747\pi$
$578$	0	0
$579$	22.4446	0.932764
$580$	0	0
$581$	50.3513	2.08893
$582$	0	0
$583$	−35.4918	−1.46992
$584$	0	0
$585$	12.0967	0.500139
$586$	0	0
$587$	−7.75665	−0.320151	−0.160076	−	0.987105i	$-0.551174\pi$
$587$	−7.75665	−0.320151	−0.160076	+	0.987105i	$0.551174\pi$
$588$	0	0
$589$	−51.8760	−2.13751
$590$	0	0
$591$	2.86856	0.117997
$592$	0	0
$593$	24.4988	1.00605	0.503023	−	0.864273i	$-0.332221\pi$
$593$	24.4988	1.00605	0.503023	+	0.864273i	$0.332221\pi$
$594$	0	0
$595$	131.053	5.37267
$596$	0	0
$597$	11.8844	0.486394
$598$	0	0
$599$	−3.81077	−0.155704	−0.0778519	−	0.996965i	$-0.524806\pi$
$599$	−3.81077	−0.155704	−0.0778519	+	0.996965i	$0.524806\pi$
$600$	0	0
$601$	−30.3072	−1.23626	−0.618128	−	0.786077i	$-0.712109\pi$
$601$	−30.3072	−1.23626	−0.618128	+	0.786077i	$0.712109\pi$
$602$	0	0
$603$	14.4797	0.589661
$604$	0	0
$605$	−11.3278	−0.460541
$606$	0	0
$607$	−9.05353	−0.367471	−0.183736	−	0.982976i	$-0.558819\pi$
$607$	−9.05353	−0.367471	−0.183736	+	0.982976i	$0.558819\pi$
$608$	0	0
$609$	4.67774	0.189551
$610$	0	0
$611$	12.3967	0.501518
$612$	0	0
$613$	−16.2807	−0.657573	−0.328786	−	0.944404i	$-0.606640\pi$
$613$	−16.2807	−0.657573	−0.328786	+	0.944404i	$0.606640\pi$
$614$	0	0
$615$	−13.6315	−0.549677
$616$	0	0
$617$	−11.6136	−0.467546	−0.233773	−	0.972291i	$-0.575107\pi$
$617$	−11.6136	−0.467546	−0.233773	+	0.972291i	$0.575107\pi$
$618$	0	0
$619$	35.3699	1.42164	0.710818	−	0.703376i	$-0.248325\pi$
$619$	35.3699	1.42164	0.710818	+	0.703376i	$0.248325\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−72.0526	−2.88673
$624$	0	0
$625$	−12.1028	−0.484111
$626$	0	0
$627$	−20.1683	−0.805445
$628$	0	0
$629$	−60.6475	−2.41817
$630$	0	0
$631$	−11.2301	−0.447063	−0.223532	−	0.974697i	$-0.571759\pi$
$631$	−11.2301	−0.447063	−0.223532	+	0.974697i	$0.571759\pi$
$632$	0	0
$633$	−6.40524	−0.254585
$634$	0	0
$635$	−14.2176	−0.564208
$636$	0	0
$637$	−52.2368	−2.06970
$638$	0	0
$639$	9.25213	0.366008
$640$	0	0
$641$	23.2220	0.917215	0.458608	−	0.888639i	$-0.348348\pi$
$641$	23.2220	0.917215	0.458608	+	0.888639i	$0.348348\pi$
$642$	0	0
$643$	37.1524	1.46515	0.732573	−	0.680688i	$-0.238319\pi$
$643$	37.1524	1.46515	0.732573	+	0.680688i	$0.238319\pi$
$644$	0	0
$645$	8.52521	0.335680
$646$	0	0
$647$	−3.30473	−0.129922	−0.0649611	−	0.997888i	$-0.520692\pi$
$647$	−3.30473	−0.129922	−0.0649611	+	0.997888i	$0.520692\pi$
$648$	0	0
$649$	−3.67023	−0.144069
$650$	0	0
$651$	−45.4784	−1.78244
$652$	0	0
$653$	−19.4065	−0.759436	−0.379718	−	0.925102i	$-0.623979\pi$
$653$	−19.4065	−0.759436	−0.379718	+	0.925102i	$0.623979\pi$
$654$	0	0
$655$	−38.3127	−1.49700
$656$	0	0
$657$	10.1166	0.394687
$658$	0	0
$659$	23.7540	0.925323	0.462662	−	0.886535i	$-0.346894\pi$
$659$	23.7540	0.925323	0.462662	+	0.886535i	$0.346894\pi$
$660$	0	0
$661$	−41.2657	−1.60505	−0.802525	−	0.596618i	$-0.796511\pi$
$661$	−41.2657	−1.60505	−0.802525	+	0.596618i	$0.796511\pi$
$662$	0	0
$663$	28.5378	1.10831
$664$	0	0
$665$	−86.0129	−3.33544
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	2.73456	0.105724
$670$	0	0
$671$	48.7480	1.88190
$672$	0	0
$673$	−23.0875	−0.889959	−0.444980	−	0.895541i	$-0.646789\pi$
$673$	−23.0875	−0.889959	−0.444980	+	0.895541i	$0.646789\pi$
$674$	0	0
$675$	6.87575	0.264648
$676$	0	0
$677$	5.46122	0.209892	0.104946	−	0.994478i	$-0.466533\pi$
$677$	5.46122	0.209892	0.104946	+	0.994478i	$0.466533\pi$
$678$	0	0
$679$	−68.3312	−2.62231
$680$	0	0
$681$	27.5888	1.05720
$682$	0	0
$683$	4.06714	0.155625	0.0778124	−	0.996968i	$-0.475206\pi$
$683$	4.06714	0.155625	0.0778124	+	0.996968i	$0.475206\pi$
$684$	0	0
$685$	63.8536	2.43972
$686$	0	0
$687$	−20.8421	−0.795175
$688$	0	0
$689$	−32.9605	−1.25570
$690$	0	0
$691$	24.1398	0.918322	0.459161	−	0.888353i	$-0.348150\pi$
$691$	24.1398	0.918322	0.459161	+	0.888353i	$0.348150\pi$
$692$	0	0
$693$	−17.6811	−0.671648
$694$	0	0
$695$	−15.4424	−0.585764
$696$	0	0
$697$	−32.1585	−1.21809
$698$	0	0
$699$	−10.1311	−0.383194
$700$	0	0
$701$	−12.0106	−0.453632	−0.226816	−	0.973938i	$-0.572832\pi$
$701$	−12.0106	−0.453632	−0.226816	+	0.973938i	$0.572832\pi$
$702$	0	0
$703$	39.8041	1.50124
$704$	0	0
$705$	12.1703	0.458358
$706$	0	0
$707$	6.80041	0.255756
$708$	0	0
$709$	−2.50719	−0.0941594	−0.0470797	−	0.998891i	$-0.514991\pi$
$709$	−2.50719	−0.0941594	−0.0470797	+	0.998891i	$0.514991\pi$
$710$	0	0
$711$	14.1749	0.531599
$712$	0	0
$713$	9.72230	0.364103
$714$	0	0
$715$	−45.7236	−1.70997
$716$	0	0
$717$	10.7910	0.402999
$718$	0	0
$719$	15.2810	0.569886	0.284943	−	0.958544i	$-0.408025\pi$
$719$	15.2810	0.569886	0.284943	+	0.958544i	$0.408025\pi$
$720$	0	0
$721$	35.8521	1.33520
$722$	0	0
$723$	8.64672	0.321575
$724$	0	0
$725$	6.87575	0.255359
$726$	0	0
$727$	32.3691	1.20050	0.600251	−	0.799812i	$-0.295067\pi$
$727$	32.3691	1.20050	0.600251	+	0.799812i	$0.295067\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.1121	0.743871
$732$	0	0
$733$	−33.3689	−1.23251	−0.616254	−	0.787548i	$-0.711350\pi$
$733$	−33.3689	−1.23251	−0.616254	+	0.787548i	$0.711350\pi$
$734$	0	0
$735$	−51.2825	−1.89158
$736$	0	0
$737$	−54.7310	−2.01604
$738$	0	0
$739$	−12.4338	−0.457385	−0.228693	−	0.973499i	$-0.573445\pi$
$739$	−12.4338	−0.457385	−0.228693	+	0.973499i	$0.573445\pi$
$740$	0	0
$741$	−18.7299	−0.688060
$742$	0	0
$743$	−2.90265	−0.106488	−0.0532439	−	0.998582i	$-0.516956\pi$
$743$	−2.90265	−0.106488	−0.0532439	+	0.998582i	$0.516956\pi$
$744$	0	0
$745$	−21.8409	−0.800189
$746$	0	0
$747$	10.7640	0.393836
$748$	0	0
$749$	94.7533	3.46221
$750$	0	0
$751$	−29.1303	−1.06298	−0.531491	−	0.847064i	$-0.678368\pi$
$751$	−29.1303	−1.06298	−0.531491	+	0.847064i	$0.678368\pi$
$752$	0	0
$753$	−12.1724	−0.443586
$754$	0	0
$755$	−19.1510	−0.696978
$756$	0	0
$757$	0.171557	0.00623535	0.00311767	−	0.999995i	$-0.499008\pi$
$757$	0.171557	0.00623535	0.00311767	+	0.999995i	$0.499008\pi$
$758$	0	0
$759$	3.77983	0.137199
$760$	0	0
$761$	−19.7066	−0.714365	−0.357182	−	0.934035i	$-0.616263\pi$
$761$	−19.7066	−0.714365	−0.357182	+	0.934035i	$0.616263\pi$
$762$	0	0
$763$	−28.6619	−1.03763
$764$	0	0
$765$	28.0164	1.01294
$766$	0	0
$767$	−3.40846	−0.123072
$768$	0	0
$769$	20.9009	0.753705	0.376853	−	0.926273i	$-0.377006\pi$
$769$	20.9009	0.753705	0.376853	+	0.926273i	$0.377006\pi$
$770$	0	0
$771$	−31.6720	−1.14064
$772$	0	0
$773$	2.23704	0.0804609	0.0402304	−	0.999190i	$-0.487191\pi$
$773$	2.23704	0.0804609	0.0402304	+	0.999190i	$0.487191\pi$
$774$	0	0
$775$	−66.8481	−2.40126
$776$	0	0
$777$	34.8953	1.25186
$778$	0	0
$779$	21.1063	0.756211
$780$	0	0
$781$	−34.9715	−1.25138
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	7.38972	0.263750
$786$	0	0
$787$	−4.75870	−0.169629	−0.0848146	−	0.996397i	$-0.527030\pi$
$787$	−4.75870	−0.169629	−0.0848146	+	0.996397i	$0.527030\pi$
$788$	0	0
$789$	18.9941	0.676207
$790$	0	0
$791$	20.7682	0.738432
$792$	0	0
$793$	45.2712	1.60763
$794$	0	0
$795$	−32.3584	−1.14763
$796$	0	0
$797$	32.0330	1.13467	0.567334	−	0.823488i	$-0.307975\pi$
$797$	32.0330	1.13467	0.567334	+	0.823488i	$0.307975\pi$
$798$	0	0
$799$	28.7112	1.01573
$800$	0	0
$801$	−15.4033	−0.544249
$802$	0	0
$803$	−38.2391	−1.34943
$804$	0	0
$805$	16.1200	0.568157
$806$	0	0
$807$	−0.758985	−0.0267175
$808$	0	0
$809$	−13.4503	−0.472887	−0.236444	−	0.971645i	$-0.575982\pi$
$809$	−13.4503	−0.472887	−0.236444	+	0.971645i	$0.575982\pi$
$810$	0	0
$811$	−19.3121	−0.678139	−0.339069	−	0.940761i	$-0.610112\pi$
$811$	−19.3121	−0.678139	−0.339069	+	0.940761i	$0.610112\pi$
$812$	0	0
$813$	−19.8284	−0.695411
$814$	0	0
$815$	28.4769	0.997503
$816$	0	0
$817$	−13.1999	−0.461807
$818$	0	0
$819$	−16.4200	−0.573762
$820$	0	0
$821$	13.6697	0.477074	0.238537	−	0.971133i	$-0.423332\pi$
$821$	13.6697	0.477074	0.238537	+	0.971133i	$0.423332\pi$
$822$	0	0
$823$	−44.2534	−1.54258	−0.771289	−	0.636485i	$-0.780388\pi$
$823$	−44.2534	−1.54258	−0.771289	+	0.636485i	$0.780388\pi$
$824$	0	0
$825$	−25.9892	−0.904827
$826$	0	0
$827$	−3.49234	−0.121441	−0.0607203	−	0.998155i	$-0.519340\pi$
$827$	−3.49234	−0.121441	−0.0607203	+	0.998155i	$0.519340\pi$
$828$	0	0
$829$	45.0956	1.56623	0.783117	−	0.621875i	$-0.213629\pi$
$829$	45.0956	1.56623	0.783117	+	0.621875i	$0.213629\pi$
$830$	0	0
$831$	−1.27693	−0.0442962
$832$	0	0
$833$	−120.982	−4.19177
$834$	0	0
$835$	−32.6960	−1.13149
$836$	0	0
$837$	−9.72230	−0.336052
$838$	0	0
$839$	−11.1486	−0.384893	−0.192446	−	0.981307i	$-0.561642\pi$
$839$	−11.1486	−0.384893	−0.192446	+	0.981307i	$0.561642\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	8.58038	0.295524
$844$	0	0
$845$	2.33701	0.0803954
$846$	0	0
$847$	15.3763	0.528335
$848$	0	0
$849$	−6.23788	−0.214084
$850$	0	0
$851$	−7.45986	−0.255721
$852$	0	0
$853$	−13.2489	−0.453632	−0.226816	−	0.973938i	$-0.572832\pi$
$853$	−13.2489	−0.453632	−0.226816	+	0.973938i	$0.572832\pi$
$854$	0	0
$855$	−18.3877	−0.628847
$856$	0	0
$857$	−56.7504	−1.93856	−0.969278	−	0.245966i	$-0.920895\pi$
$857$	−56.7504	−1.93856	−0.969278	+	0.245966i	$0.920895\pi$
$858$	0	0
$859$	14.4878	0.494318	0.247159	−	0.968975i	$-0.420503\pi$
$859$	14.4878	0.494318	0.247159	+	0.968975i	$0.420503\pi$
$860$	0	0
$861$	18.5033	0.630592
$862$	0	0
$863$	28.5463	0.971729	0.485864	−	0.874034i	$-0.338505\pi$
$863$	28.5463	0.971729	0.485864	+	0.874034i	$0.338505\pi$
$864$	0	0
$865$	5.96309	0.202751
$866$	0	0
$867$	49.0943	1.66733
$868$	0	0
$869$	−53.5786	−1.81753
$870$	0	0
$871$	−50.8275	−1.72222
$872$	0	0
$873$	−14.6078	−0.494398
$874$	0	0
$875$	−30.2372	−1.02220
$876$	0	0
$877$	8.39579	0.283506	0.141753	−	0.989902i	$-0.454726\pi$
$877$	8.39579	0.283506	0.141753	+	0.989902i	$0.454726\pi$
$878$	0	0
$879$	10.5887	0.357147
$880$	0	0
$881$	6.79317	0.228868	0.114434	−	0.993431i	$-0.463495\pi$
$881$	6.79317	0.228868	0.114434	+	0.993431i	$0.463495\pi$
$882$	0	0
$883$	−1.86314	−0.0626997	−0.0313498	−	0.999508i	$-0.509981\pi$
$883$	−1.86314	−0.0626997	−0.0313498	+	0.999508i	$0.509981\pi$
$884$	0	0
$885$	−3.34619	−0.112481
$886$	0	0
$887$	16.9765	0.570015	0.285007	−	0.958525i	$-0.408004\pi$
$887$	16.9765	0.570015	0.285007	+	0.958525i	$0.408004\pi$
$888$	0	0
$889$	19.2988	0.647262
$890$	0	0
$891$	−3.77983	−0.126629
$892$	0	0
$893$	−18.8437	−0.630581
$894$	0	0
$895$	−48.5524	−1.62293
$896$	0	0
$897$	3.51025	0.117204
$898$	0	0
$899$	−9.72230	−0.324257
$900$	0	0
$901$	−76.3375	−2.54317
$902$	0	0
$903$	−11.5720	−0.385094
$904$	0	0
$905$	−30.6698	−1.01950
$906$	0	0
$907$	−4.12320	−0.136909	−0.0684544	−	0.997654i	$-0.521807\pi$
$907$	−4.12320	−0.136909	−0.0684544	+	0.997654i	$0.521807\pi$
$908$	0	0
$909$	1.45378	0.0482189
$910$	0	0
$911$	−19.7395	−0.653998	−0.326999	−	0.945025i	$-0.606037\pi$
$911$	−19.7395	−0.653998	−0.326999	+	0.945025i	$0.606037\pi$
$912$	0	0
$913$	−40.6862	−1.34652
$914$	0	0
$915$	44.4442	1.46928
$916$	0	0
$917$	52.0054	1.71737
$918$	0	0
$919$	15.8701	0.523505	0.261752	−	0.965135i	$-0.415700\pi$
$919$	15.8701	0.523505	0.261752	+	0.965135i	$0.415700\pi$
$920$	0	0
$921$	12.7072	0.418716
$922$	0	0
$923$	−32.4773	−1.06900
$924$	0	0
$925$	51.2922	1.68648
$926$	0	0
$927$	7.66441	0.251732
$928$	0	0
$929$	20.1494	0.661080	0.330540	−	0.943792i	$-0.392769\pi$
$929$	20.1494	0.661080	0.330540	+	0.943792i	$0.392769\pi$
$930$	0	0
$931$	79.4028	2.60232
$932$	0	0
$933$	−0.170695	−0.00558830
$934$	0	0
$935$	−105.897	−3.46321
$936$	0	0
$937$	35.2269	1.15081	0.575406	−	0.817868i	$-0.304844\pi$
$937$	35.2269	1.15081	0.575406	+	0.817868i	$0.304844\pi$
$938$	0	0
$939$	−0.414275	−0.0135194
$940$	0	0
$941$	−2.20126	−0.0717591	−0.0358795	−	0.999356i	$-0.511423\pi$
$941$	−2.20126	−0.0717591	−0.0358795	+	0.999356i	$0.511423\pi$
$942$	0	0
$943$	−3.95562	−0.128813
$944$	0	0
$945$	−16.1200	−0.524385
$946$	0	0
$947$	−0.304367	−0.00989059	−0.00494530	−	0.999988i	$-0.501574\pi$
$947$	−0.304367	−0.00989059	−0.00494530	+	0.999988i	$0.501574\pi$
$948$	0	0
$949$	−35.5118	−1.15276
$950$	0	0
$951$	−17.3331	−0.562065
$952$	0	0
$953$	7.99275	0.258911	0.129455	−	0.991585i	$-0.458677\pi$
$953$	7.99275	0.258911	0.129455	+	0.991585i	$0.458677\pi$
$954$	0	0
$955$	77.2442	2.49957
$956$	0	0
$957$	−3.77983	−0.122185
$958$	0	0
$959$	−86.6743	−2.79886
$960$	0	0
$961$	63.5231	2.04913
$962$	0	0
$963$	20.2562	0.652748
$964$	0	0
$965$	−77.3467	−2.48988
$966$	0	0
$967$	−16.7469	−0.538544	−0.269272	−	0.963064i	$-0.586783\pi$
$967$	−16.7469	−0.538544	−0.269272	+	0.963064i	$0.586783\pi$
$968$	0	0
$969$	−43.3790	−1.39353
$970$	0	0
$971$	12.9854	0.416721	0.208361	−	0.978052i	$-0.433187\pi$
$971$	12.9854	0.416721	0.208361	+	0.978052i	$0.433187\pi$
$972$	0	0
$973$	20.9614	0.671992
$974$	0	0
$975$	−24.1356	−0.772958
$976$	0	0
$977$	9.88074	0.316113	0.158056	−	0.987430i	$-0.449477\pi$
$977$	9.88074	0.316113	0.158056	+	0.987430i	$0.449477\pi$
$978$	0	0
$979$	58.2219	1.86078
$980$	0	0
$981$	−6.12731	−0.195630
$982$	0	0
$983$	16.0024	0.510399	0.255199	−	0.966888i	$-0.417859\pi$
$983$	16.0024	0.510399	0.255199	+	0.966888i	$0.417859\pi$
$984$	0	0
$985$	−9.88541	−0.314975
$986$	0	0
$987$	−16.5198	−0.525831
$988$	0	0
$989$	2.47386	0.0786640
$990$	0	0
$991$	7.97246	0.253253	0.126627	−	0.991950i	$-0.459585\pi$
$991$	7.97246	0.253253	0.126627	+	0.991950i	$0.459585\pi$
$992$	0	0
$993$	−0.522129	−0.0165693
$994$	0	0
$995$	−40.9549	−1.29836
$996$	0	0
$997$	12.2848	0.389062	0.194531	−	0.980896i	$-0.437681\pi$
$997$	12.2848	0.389062	0.194531	+	0.980896i	$0.437681\pi$
$998$	0	0
$999$	7.45986	0.236020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.2	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.2	✓	18	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.44612 q^{5} +4.67774 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.44612 q^{5} +4.67774 q^{7} +1.00000 q^{9} -3.77983 q^{11} -3.51025 q^{13} -3.44612 q^{15} -8.12984 q^{17} +5.33577 q^{19} +4.67774 q^{21} -1.00000 q^{23} +6.87575 q^{25} +1.00000 q^{27} +1.00000 q^{29} -9.72230 q^{31} -3.77983 q^{33} -16.1200 q^{35} +7.45986 q^{37} -3.51025 q^{39} +3.95562 q^{41} -2.47386 q^{43} -3.44612 q^{45} -3.53158 q^{47} +14.8812 q^{49} -8.12984 q^{51} +9.38979 q^{53} +13.0258 q^{55} +5.33577 q^{57} +0.971003 q^{59} -12.8969 q^{61} +4.67774 q^{63} +12.0967 q^{65} +14.4797 q^{67} -1.00000 q^{69} +9.25213 q^{71} +10.1166 q^{73} +6.87575 q^{75} -17.6811 q^{77} +14.1749 q^{79} +1.00000 q^{81} +10.7640 q^{83} +28.0164 q^{85} +1.00000 q^{87} -15.4033 q^{89} -16.4200 q^{91} -9.72230 q^{93} -18.3877 q^{95} -14.6078 q^{97} -3.77983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

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Database

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