LMFDB - Embedded newform 6004.2.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95303 q^{3} +2.03258 q^{5} +1.63097 q^{7} +0.814318 q^{9} +O(q^{10})$	$q-1.95303 q^{3} +2.03258 q^{5} +1.63097 q^{7} +0.814318 q^{9} -2.53238 q^{11} +4.11043 q^{13} -3.96969 q^{15} +5.05831 q^{17} +1.00000 q^{19} -3.18533 q^{21} -3.14690 q^{23} -0.868602 q^{25} +4.26870 q^{27} +3.27745 q^{29} -9.46380 q^{31} +4.94581 q^{33} +3.31508 q^{35} +9.34168 q^{37} -8.02778 q^{39} +6.40846 q^{41} +4.34690 q^{43} +1.65517 q^{45} +3.69870 q^{47} -4.33994 q^{49} -9.87902 q^{51} -0.751212 q^{53} -5.14728 q^{55} -1.95303 q^{57} -8.75518 q^{59} +12.3011 q^{61} +1.32813 q^{63} +8.35479 q^{65} -5.77709 q^{67} +6.14599 q^{69} -6.10040 q^{71} +2.21270 q^{73} +1.69640 q^{75} -4.13024 q^{77} +1.00000 q^{79} -10.7798 q^{81} +13.4930 q^{83} +10.2814 q^{85} -6.40094 q^{87} -4.87005 q^{89} +6.70398 q^{91} +18.4831 q^{93} +2.03258 q^{95} -5.39797 q^{97} -2.06216 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * 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
$3$	−1.95303	−1.12758	−0.563791	−	0.825918i	$-0.690657\pi$
$3$	−1.95303	−1.12758	−0.563791	+	0.825918i	$0.690657\pi$
$4$	0	0
$5$	2.03258	0.908999	0.454500	−	0.890747i	$-0.349818\pi$
$5$	2.03258	0.908999	0.454500	+	0.890747i	$0.349818\pi$
$6$	0	0
$7$	1.63097	0.616448	0.308224	−	0.951314i	$-0.400265\pi$
$7$	1.63097	0.616448	0.308224	+	0.951314i	$0.400265\pi$
$8$	0	0
$9$	0.814318	0.271439
$10$	0	0
$11$	−2.53238	−0.763542	−0.381771	−	0.924257i	$-0.624686\pi$
$11$	−2.53238	−0.763542	−0.381771	+	0.924257i	$0.624686\pi$
$12$	0	0
$13$	4.11043	1.14003	0.570014	−	0.821635i	$-0.306938\pi$
$13$	4.11043	1.14003	0.570014	+	0.821635i	$0.306938\pi$
$14$	0	0
$15$	−3.96969	−1.02497
$16$	0	0
$17$	5.05831	1.22682	0.613410	−	0.789765i	$-0.289797\pi$
$17$	5.05831	1.22682	0.613410	+	0.789765i	$0.289797\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.18533	−0.695095
$22$	0	0
$23$	−3.14690	−0.656175	−0.328087	−	0.944647i	$-0.606404\pi$
$23$	−3.14690	−0.656175	−0.328087	+	0.944647i	$0.606404\pi$
$24$	0	0
$25$	−0.868602	−0.173720
$26$	0	0
$27$	4.26870	0.821511
$28$	0	0
$29$	3.27745	0.608606	0.304303	−	0.952575i	$-0.401576\pi$
$29$	3.27745	0.608606	0.304303	+	0.952575i	$0.401576\pi$
$30$	0	0
$31$	−9.46380	−1.69975	−0.849874	−	0.526985i	$-0.823322\pi$
$31$	−9.46380	−1.69975	−0.849874	+	0.526985i	$0.823322\pi$
$32$	0	0
$33$	4.94581	0.860955
$34$	0	0
$35$	3.31508	0.560351
$36$	0	0
$37$	9.34168	1.53576	0.767881	−	0.640592i	$-0.221311\pi$
$37$	9.34168	1.53576	0.767881	+	0.640592i	$0.221311\pi$
$38$	0	0
$39$	−8.02778	−1.28547
$40$	0	0
$41$	6.40846	1.00083	0.500417	−	0.865785i	$-0.333180\pi$
$41$	6.40846	1.00083	0.500417	+	0.865785i	$0.333180\pi$
$42$	0	0
$43$	4.34690	0.662896	0.331448	−	0.943474i	$-0.392463\pi$
$43$	4.34690	0.662896	0.331448	+	0.943474i	$0.392463\pi$
$44$	0	0
$45$	1.65517	0.246738
$46$	0	0
$47$	3.69870	0.539510	0.269755	−	0.962929i	$-0.413057\pi$
$47$	3.69870	0.539510	0.269755	+	0.962929i	$0.413057\pi$
$48$	0	0
$49$	−4.33994	−0.619992
$50$	0	0
$51$	−9.87902	−1.38334
$52$	0	0
$53$	−0.751212	−0.103187	−0.0515934	−	0.998668i	$-0.516430\pi$
$53$	−0.751212	−0.103187	−0.0515934	+	0.998668i	$0.516430\pi$
$54$	0	0
$55$	−5.14728	−0.694059
$56$	0	0
$57$	−1.95303	−0.258685
$58$	0	0
$59$	−8.75518	−1.13983	−0.569914	−	0.821705i	$-0.693023\pi$
$59$	−8.75518	−1.13983	−0.569914	+	0.821705i	$0.693023\pi$
$60$	0	0
$61$	12.3011	1.57499	0.787496	−	0.616320i	$-0.211377\pi$
$61$	12.3011	1.57499	0.787496	+	0.616320i	$0.211377\pi$
$62$	0	0
$63$	1.32813	0.167328
$64$	0	0
$65$	8.35479	1.03628
$66$	0	0
$67$	−5.77709	−0.705784	−0.352892	−	0.935664i	$-0.614802\pi$
$67$	−5.77709	−0.705784	−0.352892	+	0.935664i	$0.614802\pi$
$68$	0	0
$69$	6.14599	0.739890
$70$	0	0
$71$	−6.10040	−0.723985	−0.361992	−	0.932181i	$-0.617903\pi$
$71$	−6.10040	−0.723985	−0.361992	+	0.932181i	$0.617903\pi$
$72$	0	0
$73$	2.21270	0.258977	0.129489	−	0.991581i	$-0.458666\pi$
$73$	2.21270	0.258977	0.129489	+	0.991581i	$0.458666\pi$
$74$	0	0
$75$	1.69640	0.195884
$76$	0	0
$77$	−4.13024	−0.470684
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−10.7798	−1.19776
$82$	0	0
$83$	13.4930	1.48104	0.740522	−	0.672032i	$-0.234578\pi$
$83$	13.4930	1.48104	0.740522	+	0.672032i	$0.234578\pi$
$84$	0	0
$85$	10.2814	1.11518
$86$	0	0
$87$	−6.40094	−0.686253
$88$	0	0
$89$	−4.87005	−0.516224	−0.258112	−	0.966115i	$-0.583100\pi$
$89$	−4.87005	−0.516224	−0.258112	+	0.966115i	$0.583100\pi$
$90$	0	0
$91$	6.70398	0.702768
$92$	0	0
$93$	18.4831	1.91660
$94$	0	0
$95$	2.03258	0.208539
$96$	0	0
$97$	−5.39797	−0.548081	−0.274041	−	0.961718i	$-0.588360\pi$
$97$	−5.39797	−0.548081	−0.274041	+	0.961718i	$0.588360\pi$
$98$	0	0
$99$	−2.06216	−0.207255
$100$	0	0
$101$	−0.995468	−0.0990528	−0.0495264	−	0.998773i	$-0.515771\pi$
$101$	−0.995468	−0.0990528	−0.0495264	+	0.998773i	$0.515771\pi$
$102$	0	0
$103$	18.0301	1.77655	0.888277	−	0.459308i	$-0.151903\pi$
$103$	18.0301	1.77655	0.888277	+	0.459308i	$0.151903\pi$
$104$	0	0
$105$	−6.47444	−0.631841
$106$	0	0
$107$	11.9399	1.15427	0.577137	−	0.816648i	$-0.304170\pi$
$107$	11.9399	1.15427	0.577137	+	0.816648i	$0.304170\pi$
$108$	0	0
$109$	−12.2840	−1.17660	−0.588299	−	0.808643i	$-0.700202\pi$
$109$	−12.2840	−1.17660	−0.588299	+	0.808643i	$0.700202\pi$
$110$	0	0
$111$	−18.2446	−1.73170
$112$	0	0
$113$	6.97832	0.656465	0.328233	−	0.944597i	$-0.393547\pi$
$113$	6.97832	0.656465	0.328233	+	0.944597i	$0.393547\pi$
$114$	0	0
$115$	−6.39635	−0.596462
$116$	0	0
$117$	3.34720	0.309448
$118$	0	0
$119$	8.24994	0.756271
$120$	0	0
$121$	−4.58704	−0.417004
$122$	0	0
$123$	−12.5159	−1.12852
$124$	0	0
$125$	−11.9284	−1.06691
$126$	0	0
$127$	20.0001	1.77472	0.887361	−	0.461075i	$-0.152536\pi$
$127$	20.0001	1.77472	0.887361	+	0.461075i	$0.152536\pi$
$128$	0	0
$129$	−8.48961	−0.747469
$130$	0	0
$131$	6.26754	0.547597	0.273799	−	0.961787i	$-0.411720\pi$
$131$	6.26754	0.547597	0.273799	+	0.961787i	$0.411720\pi$
$132$	0	0
$133$	1.63097	0.141423
$134$	0	0
$135$	8.67649	0.746753
$136$	0	0
$137$	−7.74214	−0.661455	−0.330728	−	0.943726i	$-0.607294\pi$
$137$	−7.74214	−0.661455	−0.330728	+	0.943726i	$0.607294\pi$
$138$	0	0
$139$	−0.151833	−0.0128783	−0.00643914	−	0.999979i	$-0.502050\pi$
$139$	−0.151833	−0.0128783	−0.00643914	+	0.999979i	$0.502050\pi$
$140$	0	0
$141$	−7.22365	−0.608341
$142$	0	0
$143$	−10.4092	−0.870459
$144$	0	0
$145$	6.66168	0.553223
$146$	0	0
$147$	8.47603	0.699091
$148$	0	0
$149$	−0.920346	−0.0753977	−0.0376988	−	0.999289i	$-0.512003\pi$
$149$	−0.920346	−0.0753977	−0.0376988	+	0.999289i	$0.512003\pi$
$150$	0	0
$151$	−10.5755	−0.860619	−0.430310	−	0.902681i	$-0.641596\pi$
$151$	−10.5755	−0.860619	−0.430310	+	0.902681i	$0.641596\pi$
$152$	0	0
$153$	4.11907	0.333007
$154$	0	0
$155$	−19.2360	−1.54507
$156$	0	0
$157$	−16.3192	−1.30242	−0.651209	−	0.758899i	$-0.725738\pi$
$157$	−16.3192	−1.30242	−0.651209	+	0.758899i	$0.725738\pi$
$158$	0	0
$159$	1.46714	0.116352
$160$	0	0
$161$	−5.13250	−0.404498
$162$	0	0
$163$	−8.48843	−0.664865	−0.332432	−	0.943127i	$-0.607869\pi$
$163$	−8.48843	−0.664865	−0.332432	+	0.943127i	$0.607869\pi$
$164$	0	0
$165$	10.0528	0.782608
$166$	0	0
$167$	0.701665	0.0542965	0.0271482	−	0.999631i	$-0.491357\pi$
$167$	0.701665	0.0542965	0.0271482	+	0.999631i	$0.491357\pi$
$168$	0	0
$169$	3.89563	0.299664
$170$	0	0
$171$	0.814318	0.0622724
$172$	0	0
$173$	8.27354	0.629026	0.314513	−	0.949253i	$-0.398159\pi$
$173$	8.27354	0.629026	0.314513	+	0.949253i	$0.398159\pi$
$174$	0	0
$175$	−1.41666	−0.107090
$176$	0	0
$177$	17.0991	1.28525
$178$	0	0
$179$	4.06086	0.303523	0.151761	−	0.988417i	$-0.451505\pi$
$179$	4.06086	0.303523	0.151761	+	0.988417i	$0.451505\pi$
$180$	0	0
$181$	13.9133	1.03417	0.517083	−	0.855935i	$-0.327018\pi$
$181$	13.9133	1.03417	0.517083	+	0.855935i	$0.327018\pi$
$182$	0	0
$183$	−24.0243	−1.77593
$184$	0	0
$185$	18.9877	1.39601
$186$	0	0
$187$	−12.8096	−0.936729
$188$	0	0
$189$	6.96211	0.506419
$190$	0	0
$191$	−13.8489	−1.00207	−0.501034	−	0.865428i	$-0.667047\pi$
$191$	−13.8489	−1.00207	−0.501034	+	0.865428i	$0.667047\pi$
$192$	0	0
$193$	2.69551	0.194027	0.0970136	−	0.995283i	$-0.469071\pi$
$193$	2.69551	0.194027	0.0970136	+	0.995283i	$0.469071\pi$
$194$	0	0
$195$	−16.3171	−1.16849
$196$	0	0
$197$	13.7111	0.976875	0.488438	−	0.872599i	$-0.337567\pi$
$197$	13.7111	0.976875	0.488438	+	0.872599i	$0.337567\pi$
$198$	0	0
$199$	−5.79421	−0.410741	−0.205370	−	0.978684i	$-0.565840\pi$
$199$	−5.79421	−0.410741	−0.205370	+	0.978684i	$0.565840\pi$
$200$	0	0
$201$	11.2828	0.795829
$202$	0	0
$203$	5.34541	0.375174
$204$	0	0
$205$	13.0257	0.909757
$206$	0	0
$207$	−2.56258	−0.178112
$208$	0	0
$209$	−2.53238	−0.175169
$210$	0	0
$211$	−23.8903	−1.64467	−0.822337	−	0.569001i	$-0.807330\pi$
$211$	−23.8903	−1.64467	−0.822337	+	0.569001i	$0.807330\pi$
$212$	0	0
$213$	11.9143	0.816352
$214$	0	0
$215$	8.83544	0.602572
$216$	0	0
$217$	−15.4352	−1.04781
$218$	0	0
$219$	−4.32147	−0.292018
$220$	0	0
$221$	20.7918	1.39861
$222$	0	0
$223$	22.6081	1.51395	0.756975	−	0.653444i	$-0.226676\pi$
$223$	22.6081	1.51395	0.756975	+	0.653444i	$0.226676\pi$
$224$	0	0
$225$	−0.707318	−0.0471546
$226$	0	0
$227$	10.3559	0.687346	0.343673	−	0.939089i	$-0.388329\pi$
$227$	10.3559	0.687346	0.343673	+	0.939089i	$0.388329\pi$
$228$	0	0
$229$	−3.92852	−0.259604	−0.129802	−	0.991540i	$-0.541434\pi$
$229$	−3.92852	−0.259604	−0.129802	+	0.991540i	$0.541434\pi$
$230$	0	0
$231$	8.06646	0.530734
$232$	0	0
$233$	14.5807	0.955214	0.477607	−	0.878573i	$-0.341504\pi$
$233$	14.5807	0.955214	0.477607	+	0.878573i	$0.341504\pi$
$234$	0	0
$235$	7.51791	0.490414
$236$	0	0
$237$	−1.95303	−0.126863
$238$	0	0
$239$	−12.1439	−0.785522	−0.392761	−	0.919641i	$-0.628480\pi$
$239$	−12.1439	−0.785522	−0.392761	+	0.919641i	$0.628480\pi$
$240$	0	0
$241$	12.9477	0.834032	0.417016	−	0.908899i	$-0.363076\pi$
$241$	12.9477	0.834032	0.417016	+	0.908899i	$0.363076\pi$
$242$	0	0
$243$	8.24723	0.529060
$244$	0	0
$245$	−8.82130	−0.563572
$246$	0	0
$247$	4.11043	0.261540
$248$	0	0
$249$	−26.3521	−1.67000
$250$	0	0
$251$	−0.221753	−0.0139969	−0.00699845	−	0.999976i	$-0.502228\pi$
$251$	−0.221753	−0.0139969	−0.00699845	+	0.999976i	$0.502228\pi$
$252$	0	0
$253$	7.96916	0.501017
$254$	0	0
$255$	−20.0799	−1.25745
$256$	0	0
$257$	4.31424	0.269115	0.134557	−	0.990906i	$-0.457039\pi$
$257$	4.31424	0.269115	0.134557	+	0.990906i	$0.457039\pi$
$258$	0	0
$259$	15.2360	0.946718
$260$	0	0
$261$	2.66888	0.165200
$262$	0	0
$263$	15.2718	0.941702	0.470851	−	0.882213i	$-0.343947\pi$
$263$	15.2718	0.941702	0.470851	+	0.882213i	$0.343947\pi$
$264$	0	0
$265$	−1.52690	−0.0937968
$266$	0	0
$267$	9.51134	0.582085
$268$	0	0
$269$	25.0666	1.52834	0.764170	−	0.645015i	$-0.223149\pi$
$269$	25.0666	1.52834	0.764170	+	0.645015i	$0.223149\pi$
$270$	0	0
$271$	31.9552	1.94114	0.970569	−	0.240821i	$-0.0774168\pi$
$271$	31.9552	1.94114	0.970569	+	0.240821i	$0.0774168\pi$
$272$	0	0
$273$	−13.0931	−0.792428
$274$	0	0
$275$	2.19963	0.132643
$276$	0	0
$277$	−24.8023	−1.49023	−0.745114	−	0.666937i	$-0.767605\pi$
$277$	−24.8023	−1.49023	−0.745114	+	0.666937i	$0.767605\pi$
$278$	0	0
$279$	−7.70654	−0.461379
$280$	0	0
$281$	2.98537	0.178092	0.0890461	−	0.996028i	$-0.471618\pi$
$281$	2.98537	0.178092	0.0890461	+	0.996028i	$0.471618\pi$
$282$	0	0
$283$	−18.4350	−1.09585	−0.547925	−	0.836528i	$-0.684582\pi$
$283$	−18.4350	−1.09585	−0.547925	+	0.836528i	$0.684582\pi$
$284$	0	0
$285$	−3.96969	−0.235144
$286$	0	0
$287$	10.4520	0.616962
$288$	0	0
$289$	8.58649	0.505088
$290$	0	0
$291$	10.5424	0.618006
$292$	0	0
$293$	−6.78900	−0.396618	−0.198309	−	0.980140i	$-0.563545\pi$
$293$	−6.78900	−0.396618	−0.198309	+	0.980140i	$0.563545\pi$
$294$	0	0
$295$	−17.7956	−1.03610
$296$	0	0
$297$	−10.8100	−0.627258
$298$	0	0
$299$	−12.9351	−0.748058
$300$	0	0
$301$	7.08965	0.408641
$302$	0	0
$303$	1.94418	0.111690
$304$	0	0
$305$	25.0030	1.43167
$306$	0	0
$307$	16.4987	0.941628	0.470814	−	0.882232i	$-0.343960\pi$
$307$	16.4987	0.941628	0.470814	+	0.882232i	$0.343960\pi$
$308$	0	0
$309$	−35.2132	−2.00321
$310$	0	0
$311$	27.1349	1.53868	0.769340	−	0.638839i	$-0.220585\pi$
$311$	27.1349	1.53868	0.769340	+	0.638839i	$0.220585\pi$
$312$	0	0
$313$	18.6015	1.05142	0.525708	−	0.850665i	$-0.323800\pi$
$313$	18.6015	1.05142	0.525708	+	0.850665i	$0.323800\pi$
$314$	0	0
$315$	2.69953	0.152101
$316$	0	0
$317$	−3.56601	−0.200287	−0.100144	−	0.994973i	$-0.531930\pi$
$317$	−3.56601	−0.200287	−0.100144	+	0.994973i	$0.531930\pi$
$318$	0	0
$319$	−8.29974	−0.464696
$320$	0	0
$321$	−23.3189	−1.30154
$322$	0	0
$323$	5.05831	0.281452
$324$	0	0
$325$	−3.57033	−0.198046
$326$	0	0
$327$	23.9911	1.32671
$328$	0	0
$329$	6.03245	0.332580
$330$	0	0
$331$	−11.7086	−0.643561	−0.321780	−	0.946814i	$-0.604281\pi$
$331$	−11.7086	−0.643561	−0.321780	+	0.946814i	$0.604281\pi$
$332$	0	0
$333$	7.60710	0.416866
$334$	0	0
$335$	−11.7424	−0.641557
$336$	0	0
$337$	8.64400	0.470869	0.235434	−	0.971890i	$-0.424349\pi$
$337$	8.64400	0.470869	0.235434	+	0.971890i	$0.424349\pi$
$338$	0	0
$339$	−13.6289	−0.740218
$340$	0	0
$341$	23.9660	1.29783
$342$	0	0
$343$	−18.4951	−0.998641
$344$	0	0
$345$	12.4922	0.672560
$346$	0	0
$347$	−36.6808	−1.96913	−0.984565	−	0.175018i	$-0.944002\pi$
$347$	−36.6808	−1.96913	−0.984565	+	0.175018i	$0.944002\pi$
$348$	0	0
$349$	15.2195	0.814680	0.407340	−	0.913277i	$-0.366456\pi$
$349$	15.2195	0.814680	0.407340	+	0.913277i	$0.366456\pi$
$350$	0	0
$351$	17.5462	0.936546
$352$	0	0
$353$	26.4930	1.41008	0.705040	−	0.709168i	$-0.250929\pi$
$353$	26.4930	1.41008	0.705040	+	0.709168i	$0.250929\pi$
$354$	0	0
$355$	−12.3996	−0.658102
$356$	0	0
$357$	−16.1124	−0.852757
$358$	0	0
$359$	−19.7196	−1.04076	−0.520379	−	0.853935i	$-0.674209\pi$
$359$	−19.7196	−1.04076	−0.520379	+	0.853935i	$0.674209\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.95862	0.470206
$364$	0	0
$365$	4.49750	0.235410
$366$	0	0
$367$	−0.568689	−0.0296853	−0.0148427	−	0.999890i	$-0.504725\pi$
$367$	−0.568689	−0.0296853	−0.0148427	+	0.999890i	$0.504725\pi$
$368$	0	0
$369$	5.21853	0.271666
$370$	0	0
$371$	−1.22520	−0.0636094
$372$	0	0
$373$	16.2272	0.840214	0.420107	−	0.907475i	$-0.361993\pi$
$373$	16.2272	0.840214	0.420107	+	0.907475i	$0.361993\pi$
$374$	0	0
$375$	23.2966	1.20303
$376$	0	0
$377$	13.4717	0.693828
$378$	0	0
$379$	−6.85726	−0.352234	−0.176117	−	0.984369i	$-0.556354\pi$
$379$	−6.85726	−0.352234	−0.176117	+	0.984369i	$0.556354\pi$
$380$	0	0
$381$	−39.0608	−2.00114
$382$	0	0
$383$	30.7628	1.57191	0.785953	−	0.618286i	$-0.212173\pi$
$383$	30.7628	1.57191	0.785953	+	0.618286i	$0.212173\pi$
$384$	0	0
$385$	−8.39505	−0.427851
$386$	0	0
$387$	3.53976	0.179936
$388$	0	0
$389$	10.3993	0.527263	0.263632	−	0.964623i	$-0.415080\pi$
$389$	10.3993	0.527263	0.263632	+	0.964623i	$0.415080\pi$
$390$	0	0
$391$	−15.9180	−0.805009
$392$	0	0
$393$	−12.2407	−0.617460
$394$	0	0
$395$	2.03258	0.102270
$396$	0	0
$397$	5.84240	0.293222	0.146611	−	0.989194i	$-0.453164\pi$
$397$	5.84240	0.293222	0.146611	+	0.989194i	$0.453164\pi$
$398$	0	0
$399$	−3.18533	−0.159466
$400$	0	0
$401$	6.76413	0.337784	0.168892	−	0.985635i	$-0.445981\pi$
$401$	6.76413	0.337784	0.168892	+	0.985635i	$0.445981\pi$
$402$	0	0
$403$	−38.9003	−1.93776
$404$	0	0
$405$	−21.9109	−1.08876
$406$	0	0
$407$	−23.6567	−1.17262
$408$	0	0
$409$	26.6769	1.31909	0.659545	−	0.751665i	$-0.270749\pi$
$409$	26.6769	1.31909	0.659545	+	0.751665i	$0.270749\pi$
$410$	0	0
$411$	15.1206	0.745844
$412$	0	0
$413$	−14.2794	−0.702644
$414$	0	0
$415$	27.4256	1.34627
$416$	0	0
$417$	0.296534	0.0145213
$418$	0	0
$419$	22.0644	1.07792	0.538959	−	0.842332i	$-0.318818\pi$
$419$	22.0644	1.07792	0.538959	+	0.842332i	$0.318818\pi$
$420$	0	0
$421$	30.6011	1.49141	0.745703	−	0.666279i	$-0.232114\pi$
$421$	30.6011	1.49141	0.745703	+	0.666279i	$0.232114\pi$
$422$	0	0
$423$	3.01191	0.146444
$424$	0	0
$425$	−4.39366	−0.213124
$426$	0	0
$427$	20.0627	0.970900
$428$	0	0
$429$	20.3294	0.981513
$430$	0	0
$431$	8.21734	0.395815	0.197908	−	0.980221i	$-0.436585\pi$
$431$	8.21734	0.395815	0.197908	+	0.980221i	$0.436585\pi$
$432$	0	0
$433$	20.1716	0.969383	0.484692	−	0.874685i	$-0.338932\pi$
$433$	20.1716	0.969383	0.484692	+	0.874685i	$0.338932\pi$
$434$	0	0
$435$	−13.0105	−0.623803
$436$	0	0
$437$	−3.14690	−0.150537
$438$	0	0
$439$	−29.3866	−1.40255	−0.701273	−	0.712893i	$-0.747385\pi$
$439$	−29.3866	−1.40255	−0.701273	+	0.712893i	$0.747385\pi$
$440$	0	0
$441$	−3.53409	−0.168290
$442$	0	0
$443$	30.8054	1.46361	0.731804	−	0.681515i	$-0.238679\pi$
$443$	30.8054	1.46361	0.731804	+	0.681515i	$0.238679\pi$
$444$	0	0
$445$	−9.89879	−0.469247
$446$	0	0
$447$	1.79746	0.0850170
$448$	0	0
$449$	5.66479	0.267338	0.133669	−	0.991026i	$-0.457324\pi$
$449$	5.66479	0.267338	0.133669	+	0.991026i	$0.457324\pi$
$450$	0	0
$451$	−16.2287	−0.764179
$452$	0	0
$453$	20.6542	0.970418
$454$	0	0
$455$	13.6264	0.638816
$456$	0	0
$457$	11.8184	0.552841	0.276421	−	0.961037i	$-0.410852\pi$
$457$	11.8184	0.552841	0.276421	+	0.961037i	$0.410852\pi$
$458$	0	0
$459$	21.5924	1.00785
$460$	0	0
$461$	4.26847	0.198803	0.0994013	−	0.995047i	$-0.468307\pi$
$461$	4.26847	0.198803	0.0994013	+	0.995047i	$0.468307\pi$
$462$	0	0
$463$	25.8451	1.20112	0.600561	−	0.799579i	$-0.294944\pi$
$463$	25.8451	1.20112	0.600561	+	0.799579i	$0.294944\pi$
$464$	0	0
$465$	37.5684	1.74219
$466$	0	0
$467$	9.25292	0.428174	0.214087	−	0.976815i	$-0.431322\pi$
$467$	9.25292	0.428174	0.214087	+	0.976815i	$0.431322\pi$
$468$	0	0
$469$	−9.42226	−0.435079
$470$	0	0
$471$	31.8719	1.46858
$472$	0	0
$473$	−11.0080	−0.506149
$474$	0	0
$475$	−0.868602	−0.0398542
$476$	0	0
$477$	−0.611725	−0.0280090
$478$	0	0
$479$	−4.35740	−0.199095	−0.0995473	−	0.995033i	$-0.531739\pi$
$479$	−4.35740	−0.199095	−0.0995473	+	0.995033i	$0.531739\pi$
$480$	0	0
$481$	38.3983	1.75081
$482$	0	0
$483$	10.0239	0.456104
$484$	0	0
$485$	−10.9718	−0.498205
$486$	0	0
$487$	−15.7233	−0.712492	−0.356246	−	0.934392i	$-0.615944\pi$
$487$	−15.7233	−0.712492	−0.356246	+	0.934392i	$0.615944\pi$
$488$	0	0
$489$	16.5781	0.749689
$490$	0	0
$491$	31.0168	1.39977	0.699884	−	0.714256i	$-0.253235\pi$
$491$	31.0168	1.39977	0.699884	+	0.714256i	$0.253235\pi$
$492$	0	0
$493$	16.5783	0.746651
$494$	0	0
$495$	−4.19152	−0.188395
$496$	0	0
$497$	−9.94957	−0.446299
$498$	0	0
$499$	7.44206	0.333153	0.166576	−	0.986029i	$-0.446729\pi$
$499$	7.44206	0.333153	0.166576	+	0.986029i	$0.446729\pi$
$500$	0	0
$501$	−1.37037	−0.0612237
$502$	0	0
$503$	26.6145	1.18668	0.593341	−	0.804952i	$-0.297809\pi$
$503$	26.6145	1.18668	0.593341	+	0.804952i	$0.297809\pi$
$504$	0	0
$505$	−2.02337	−0.0900389
$506$	0	0
$507$	−7.60827	−0.337895
$508$	0	0
$509$	−26.8195	−1.18875	−0.594377	−	0.804187i	$-0.702601\pi$
$509$	−26.8195	−1.18875	−0.594377	+	0.804187i	$0.702601\pi$
$510$	0	0
$511$	3.60885	0.159646
$512$	0	0
$513$	4.26870	0.188468
$514$	0	0
$515$	36.6476	1.61489
$516$	0	0
$517$	−9.36651	−0.411939
$518$	0	0
$519$	−16.1585	−0.709278
$520$	0	0
$521$	35.1241	1.53882	0.769408	−	0.638758i	$-0.220551\pi$
$521$	35.1241	1.53882	0.769408	+	0.638758i	$0.220551\pi$
$522$	0	0
$523$	−19.0947	−0.834954	−0.417477	−	0.908687i	$-0.637086\pi$
$523$	−19.0947	−0.834954	−0.417477	+	0.908687i	$0.637086\pi$
$524$	0	0
$525$	2.76678	0.120752
$526$	0	0
$527$	−47.8708	−2.08529
$528$	0	0
$529$	−13.0970	−0.569435
$530$	0	0
$531$	−7.12950	−0.309394
$532$	0	0
$533$	26.3415	1.14098
$534$	0	0
$535$	24.2688	1.04923
$536$	0	0
$537$	−7.93097	−0.342247
$538$	0	0
$539$	10.9904	0.473390
$540$	0	0
$541$	−42.0237	−1.80674	−0.903370	−	0.428863i	$-0.858914\pi$
$541$	−42.0237	−1.80674	−0.903370	+	0.428863i	$0.858914\pi$
$542$	0	0
$543$	−27.1730	−1.16611
$544$	0	0
$545$	−24.9684	−1.06953
$546$	0	0
$547$	−0.804533	−0.0343994	−0.0171997	−	0.999852i	$-0.505475\pi$
$547$	−0.804533	−0.0343994	−0.0171997	+	0.999852i	$0.505475\pi$
$548$	0	0
$549$	10.0170	0.427514
$550$	0	0
$551$	3.27745	0.139624
$552$	0	0
$553$	1.63097	0.0693558
$554$	0	0
$555$	−37.0836	−1.57411
$556$	0	0
$557$	−34.5286	−1.46302	−0.731512	−	0.681828i	$-0.761185\pi$
$557$	−34.5286	−1.46302	−0.731512	+	0.681828i	$0.761185\pi$
$558$	0	0
$559$	17.8676	0.755720
$560$	0	0
$561$	25.0175	1.05624
$562$	0	0
$563$	39.8199	1.67821	0.839105	−	0.543969i	$-0.183079\pi$
$563$	39.8199	1.67821	0.839105	+	0.543969i	$0.183079\pi$
$564$	0	0
$565$	14.1840	0.596726
$566$	0	0
$567$	−17.5816	−0.738357
$568$	0	0
$569$	5.10199	0.213887	0.106943	−	0.994265i	$-0.465894\pi$
$569$	5.10199	0.213887	0.106943	+	0.994265i	$0.465894\pi$
$570$	0	0
$571$	−30.9776	−1.29637	−0.648186	−	0.761482i	$-0.724472\pi$
$571$	−30.9776	−1.29637	−0.648186	+	0.761482i	$0.724472\pi$
$572$	0	0
$573$	27.0472	1.12991
$574$	0	0
$575$	2.73341	0.113991
$576$	0	0
$577$	−1.62092	−0.0674799	−0.0337399	−	0.999431i	$-0.510742\pi$
$577$	−1.62092	−0.0674799	−0.0337399	+	0.999431i	$0.510742\pi$
$578$	0	0
$579$	−5.26441	−0.218781
$580$	0	0
$581$	22.0066	0.912987
$582$	0	0
$583$	1.90236	0.0787875
$584$	0	0
$585$	6.80346	0.281288
$586$	0	0
$587$	−32.0531	−1.32297	−0.661487	−	0.749957i	$-0.730074\pi$
$587$	−32.0531	−1.32297	−0.661487	+	0.749957i	$0.730074\pi$
$588$	0	0
$589$	−9.46380	−0.389949
$590$	0	0
$591$	−26.7782	−1.10151
$592$	0	0
$593$	−8.77629	−0.360399	−0.180200	−	0.983630i	$-0.557674\pi$
$593$	−8.77629	−0.360399	−0.180200	+	0.983630i	$0.557674\pi$
$594$	0	0
$595$	16.7687	0.687450
$596$	0	0
$597$	11.3163	0.463143
$598$	0	0
$599$	−7.55117	−0.308532	−0.154266	−	0.988029i	$-0.549301\pi$
$599$	−7.55117	−0.308532	−0.154266	+	0.988029i	$0.549301\pi$
$600$	0	0
$601$	4.52915	0.184748	0.0923740	−	0.995724i	$-0.470554\pi$
$601$	4.52915	0.184748	0.0923740	+	0.995724i	$0.470554\pi$
$602$	0	0
$603$	−4.70439	−0.191578
$604$	0	0
$605$	−9.32355	−0.379056
$606$	0	0
$607$	25.0505	1.01677	0.508385	−	0.861130i	$-0.330243\pi$
$607$	25.0505	1.01677	0.508385	+	0.861130i	$0.330243\pi$
$608$	0	0
$609$	−10.4397	−0.423039
$610$	0	0
$611$	15.2032	0.615057
$612$	0	0
$613$	31.4040	1.26839	0.634197	−	0.773171i	$-0.281331\pi$
$613$	31.4040	1.26839	0.634197	+	0.773171i	$0.281331\pi$
$614$	0	0
$615$	−25.4396	−1.02583
$616$	0	0
$617$	31.9276	1.28536	0.642678	−	0.766136i	$-0.277823\pi$
$617$	31.9276	1.28536	0.642678	+	0.766136i	$0.277823\pi$
$618$	0	0
$619$	19.1592	0.770072	0.385036	−	0.922902i	$-0.374189\pi$
$619$	19.1592	0.770072	0.385036	+	0.922902i	$0.374189\pi$
$620$	0	0
$621$	−13.4332	−0.539055
$622$	0	0
$623$	−7.94290	−0.318225
$624$	0	0
$625$	−19.9025	−0.796101
$626$	0	0
$627$	4.94581	0.197517
$628$	0	0
$629$	47.2531	1.88410
$630$	0	0
$631$	−22.2579	−0.886071	−0.443036	−	0.896504i	$-0.646099\pi$
$631$	−22.2579	−0.886071	−0.443036	+	0.896504i	$0.646099\pi$
$632$	0	0
$633$	46.6583	1.85450
$634$	0	0
$635$	40.6519	1.61322
$636$	0	0
$637$	−17.8390	−0.706808
$638$	0	0
$639$	−4.96767	−0.196518
$640$	0	0
$641$	23.8553	0.942226	0.471113	−	0.882073i	$-0.343852\pi$
$641$	23.8553	0.942226	0.471113	+	0.882073i	$0.343852\pi$
$642$	0	0
$643$	29.3317	1.15673	0.578366	−	0.815778i	$-0.303691\pi$
$643$	29.3317	1.15673	0.578366	+	0.815778i	$0.303691\pi$
$644$	0	0
$645$	−17.2559	−0.679449
$646$	0	0
$647$	18.5959	0.731081	0.365540	−	0.930795i	$-0.380884\pi$
$647$	18.5959	0.731081	0.365540	+	0.930795i	$0.380884\pi$
$648$	0	0
$649$	22.1715	0.870306
$650$	0	0
$651$	30.1453	1.18149
$652$	0	0
$653$	−45.6960	−1.78822	−0.894112	−	0.447844i	$-0.852192\pi$
$653$	−45.6960	−1.78822	−0.894112	+	0.447844i	$0.852192\pi$
$654$	0	0
$655$	12.7393	0.497765
$656$	0	0
$657$	1.80184	0.0702966
$658$	0	0
$659$	−38.9462	−1.51713	−0.758564	−	0.651598i	$-0.774099\pi$
$659$	−38.9462	−1.51713	−0.758564	+	0.651598i	$0.774099\pi$
$660$	0	0
$661$	14.6800	0.570986	0.285493	−	0.958381i	$-0.407843\pi$
$661$	14.6800	0.570986	0.285493	+	0.958381i	$0.407843\pi$
$662$	0	0
$663$	−40.6070	−1.57705
$664$	0	0
$665$	3.31508	0.128553
$666$	0	0
$667$	−10.3138	−0.399352
$668$	0	0
$669$	−44.1542	−1.70710
$670$	0	0
$671$	−31.1510	−1.20257
$672$	0	0
$673$	−0.343524	−0.0132419	−0.00662094	−	0.999978i	$-0.502108\pi$
$673$	−0.343524	−0.0132419	−0.00662094	+	0.999978i	$0.502108\pi$
$674$	0	0
$675$	−3.70780	−0.142713
$676$	0	0
$677$	8.70202	0.334446	0.167223	−	0.985919i	$-0.446520\pi$
$677$	8.70202	0.334446	0.167223	+	0.985919i	$0.446520\pi$
$678$	0	0
$679$	−8.80392	−0.337864
$680$	0	0
$681$	−20.2254	−0.775039
$682$	0	0
$683$	31.9220	1.22146	0.610730	−	0.791839i	$-0.290876\pi$
$683$	31.9220	1.22146	0.610730	+	0.791839i	$0.290876\pi$
$684$	0	0
$685$	−15.7365	−0.601262
$686$	0	0
$687$	7.67251	0.292724
$688$	0	0
$689$	−3.08780	−0.117636
$690$	0	0
$691$	−3.09027	−0.117560	−0.0587798	−	0.998271i	$-0.518721\pi$
$691$	−3.09027	−0.117560	−0.0587798	+	0.998271i	$0.518721\pi$
$692$	0	0
$693$	−3.36332	−0.127762
$694$	0	0
$695$	−0.308613	−0.0117064
$696$	0	0
$697$	32.4160	1.22784
$698$	0	0
$699$	−28.4765	−1.07708
$700$	0	0
$701$	−0.0318468	−0.00120284	−0.000601418	−	1.00000i	$-0.500191\pi$
$701$	−0.0318468	−0.00120284	−0.000601418		1.00000i	$0.500191\pi$
$702$	0	0
$703$	9.34168	0.352328
$704$	0	0
$705$	−14.6827	−0.552982
$706$	0	0
$707$	−1.62358	−0.0610609
$708$	0	0
$709$	12.2353	0.459507	0.229754	−	0.973249i	$-0.426208\pi$
$709$	12.2353	0.459507	0.229754	+	0.973249i	$0.426208\pi$
$710$	0	0
$711$	0.814318	0.0305393
$712$	0	0
$713$	29.7817	1.11533
$714$	0	0
$715$	−21.1575	−0.791247
$716$	0	0
$717$	23.7173	0.885740
$718$	0	0
$719$	−39.2418	−1.46347	−0.731737	−	0.681587i	$-0.761290\pi$
$719$	−39.2418	−1.46347	−0.731737	+	0.681587i	$0.761290\pi$
$720$	0	0
$721$	29.4064	1.09515
$722$	0	0
$723$	−25.2871	−0.940439
$724$	0	0
$725$	−2.84680	−0.105727
$726$	0	0
$727$	38.4360	1.42551	0.712757	−	0.701411i	$-0.247446\pi$
$727$	38.4360	1.42551	0.712757	+	0.701411i	$0.247446\pi$
$728$	0	0
$729$	16.2324	0.601202
$730$	0	0
$731$	21.9880	0.813254
$732$	0	0
$733$	−30.7243	−1.13483	−0.567414	−	0.823432i	$-0.692056\pi$
$733$	−30.7243	−1.13483	−0.567414	+	0.823432i	$0.692056\pi$
$734$	0	0
$735$	17.2282	0.635473
$736$	0	0
$737$	14.6298	0.538896
$738$	0	0
$739$	18.8361	0.692898	0.346449	−	0.938069i	$-0.387387\pi$
$739$	18.8361	0.692898	0.346449	+	0.938069i	$0.387387\pi$
$740$	0	0
$741$	−8.02778	−0.294908
$742$	0	0
$743$	6.17504	0.226540	0.113270	−	0.993564i	$-0.463867\pi$
$743$	6.17504	0.226540	0.113270	+	0.993564i	$0.463867\pi$
$744$	0	0
$745$	−1.87068	−0.0685364
$746$	0	0
$747$	10.9876	0.402014
$748$	0	0
$749$	19.4736	0.711550
$750$	0	0
$751$	−38.3495	−1.39939	−0.699696	−	0.714440i	$-0.746681\pi$
$751$	−38.3495	−1.39939	−0.699696	+	0.714440i	$0.746681\pi$
$752$	0	0
$753$	0.433089	0.0157826
$754$	0	0
$755$	−21.4955	−0.782302
$756$	0	0
$757$	21.4671	0.780235	0.390117	−	0.920765i	$-0.372434\pi$
$757$	21.4671	0.780235	0.390117	+	0.920765i	$0.372434\pi$
$758$	0	0
$759$	−15.5640	−0.564937
$760$	0	0
$761$	−23.7830	−0.862134	−0.431067	−	0.902320i	$-0.641863\pi$
$761$	−23.7830	−0.862134	−0.431067	+	0.902320i	$0.641863\pi$
$762$	0	0
$763$	−20.0349	−0.725312
$764$	0	0
$765$	8.37236	0.302703
$766$	0	0
$767$	−35.9875	−1.29944
$768$	0	0
$769$	−18.5695	−0.669632	−0.334816	−	0.942284i	$-0.608674\pi$
$769$	−18.5695	−0.669632	−0.334816	+	0.942284i	$0.608674\pi$
$770$	0	0
$771$	−8.42582	−0.303449
$772$	0	0
$773$	−26.2958	−0.945793	−0.472896	−	0.881118i	$-0.656792\pi$
$773$	−26.2958	−0.945793	−0.472896	+	0.881118i	$0.656792\pi$
$774$	0	0
$775$	8.22028	0.295281
$776$	0	0
$777$	−29.7563	−1.06750
$778$	0	0
$779$	6.40846	0.229607
$780$	0	0
$781$	15.4486	0.552793
$782$	0	0
$783$	13.9904	0.499977
$784$	0	0
$785$	−33.1702	−1.18390
$786$	0	0
$787$	7.79556	0.277882	0.138941	−	0.990301i	$-0.455630\pi$
$787$	7.79556	0.277882	0.138941	+	0.990301i	$0.455630\pi$
$788$	0	0
$789$	−29.8263	−1.06185
$790$	0	0
$791$	11.3814	0.404677
$792$	0	0
$793$	50.5627	1.79553
$794$	0	0
$795$	2.98208	0.105763
$796$	0	0
$797$	−46.8762	−1.66044	−0.830221	−	0.557435i	$-0.811786\pi$
$797$	−46.8762	−1.66044	−0.830221	+	0.557435i	$0.811786\pi$
$798$	0	0
$799$	18.7091	0.661882
$800$	0	0
$801$	−3.96577	−0.140124
$802$	0	0
$803$	−5.60341	−0.197740
$804$	0	0
$805$	−10.4322	−0.367688
$806$	0	0
$807$	−48.9558	−1.72333
$808$	0	0
$809$	8.25810	0.290339	0.145170	−	0.989407i	$-0.453627\pi$
$809$	8.25810	0.290339	0.145170	+	0.989407i	$0.453627\pi$
$810$	0	0
$811$	−2.17274	−0.0762952	−0.0381476	−	0.999272i	$-0.512146\pi$
$811$	−2.17274	−0.0762952	−0.0381476	+	0.999272i	$0.512146\pi$
$812$	0	0
$813$	−62.4094	−2.18879
$814$	0	0
$815$	−17.2534	−0.604362
$816$	0	0
$817$	4.34690	0.152079
$818$	0	0
$819$	5.45917	0.190759
$820$	0	0
$821$	−46.9110	−1.63721	−0.818603	−	0.574360i	$-0.805251\pi$
$821$	−46.9110	−1.63721	−0.818603	+	0.574360i	$0.805251\pi$
$822$	0	0
$823$	−42.1513	−1.46930	−0.734651	−	0.678445i	$-0.762654\pi$
$823$	−42.1513	−1.46930	−0.734651	+	0.678445i	$0.762654\pi$
$824$	0	0
$825$	−4.29595	−0.149566
$826$	0	0
$827$	−9.19947	−0.319897	−0.159948	−	0.987125i	$-0.551133\pi$
$827$	−9.19947	−0.319897	−0.159948	+	0.987125i	$0.551133\pi$
$828$	0	0
$829$	16.2464	0.564261	0.282130	−	0.959376i	$-0.408959\pi$
$829$	16.2464	0.564261	0.282130	+	0.959376i	$0.408959\pi$
$830$	0	0
$831$	48.4397	1.68035
$832$	0	0
$833$	−21.9528	−0.760618
$834$	0	0
$835$	1.42619	0.0493554
$836$	0	0
$837$	−40.3981	−1.39636
$838$	0	0
$839$	26.8632	0.927422	0.463711	−	0.885986i	$-0.346518\pi$
$839$	26.8632	0.927422	0.463711	+	0.885986i	$0.346518\pi$
$840$	0	0
$841$	−18.2584	−0.629598
$842$	0	0
$843$	−5.83051	−0.200813
$844$	0	0
$845$	7.91820	0.272394
$846$	0	0
$847$	−7.48132	−0.257061
$848$	0	0
$849$	36.0041	1.23566
$850$	0	0
$851$	−29.3974	−1.00773
$852$	0	0
$853$	−47.8910	−1.63976	−0.819879	−	0.572537i	$-0.805959\pi$
$853$	−47.8910	−1.63976	−0.819879	+	0.572537i	$0.805959\pi$
$854$	0	0
$855$	1.65517	0.0566056
$856$	0	0
$857$	31.8863	1.08922	0.544608	−	0.838690i	$-0.316678\pi$
$857$	31.8863	1.08922	0.544608	+	0.838690i	$0.316678\pi$
$858$	0	0
$859$	−1.48491	−0.0506644	−0.0253322	−	0.999679i	$-0.508064\pi$
$859$	−1.48491	−0.0506644	−0.0253322	+	0.999679i	$0.508064\pi$
$860$	0	0
$861$	−20.4131	−0.695675
$862$	0	0
$863$	−20.7979	−0.707970	−0.353985	−	0.935251i	$-0.615174\pi$
$863$	−20.7979	−0.707970	−0.353985	+	0.935251i	$0.615174\pi$
$864$	0	0
$865$	16.8167	0.571784
$866$	0	0
$867$	−16.7697	−0.569527
$868$	0	0
$869$	−2.53238	−0.0859052
$870$	0	0
$871$	−23.7463	−0.804614
$872$	0	0
$873$	−4.39566	−0.148771
$874$	0	0
$875$	−19.4549	−0.657695
$876$	0	0
$877$	−4.73601	−0.159924	−0.0799619	−	0.996798i	$-0.525480\pi$
$877$	−4.73601	−0.159924	−0.0799619	+	0.996798i	$0.525480\pi$
$878$	0	0
$879$	13.2591	0.447219
$880$	0	0
$881$	−30.0628	−1.01284	−0.506420	−	0.862287i	$-0.669032\pi$
$881$	−30.0628	−1.01284	−0.506420	+	0.862287i	$0.669032\pi$
$882$	0	0
$883$	−11.3030	−0.380375	−0.190188	−	0.981748i	$-0.560910\pi$
$883$	−11.3030	−0.380375	−0.190188	+	0.981748i	$0.560910\pi$
$884$	0	0
$885$	34.7554	1.16829
$886$	0	0
$887$	−9.89556	−0.332260	−0.166130	−	0.986104i	$-0.553127\pi$
$887$	−9.89556	−0.332260	−0.166130	+	0.986104i	$0.553127\pi$
$888$	0	0
$889$	32.6195	1.09402
$890$	0	0
$891$	27.2987	0.914540
$892$	0	0
$893$	3.69870	0.123772
$894$	0	0
$895$	8.25404	0.275902
$896$	0	0
$897$	25.2627	0.843496
$898$	0	0
$899$	−31.0171	−1.03448
$900$	0	0
$901$	−3.79986	−0.126592
$902$	0	0
$903$	−13.8463	−0.460776
$904$	0	0
$905$	28.2799	0.940056
$906$	0	0
$907$	−32.7951	−1.08894	−0.544471	−	0.838779i	$-0.683270\pi$
$907$	−32.7951	−1.08894	−0.544471	+	0.838779i	$0.683270\pi$
$908$	0	0
$909$	−0.810627	−0.0268868
$910$	0	0
$911$	−36.4619	−1.20804	−0.604018	−	0.796970i	$-0.706435\pi$
$911$	−36.4619	−1.20804	−0.604018	+	0.796970i	$0.706435\pi$
$912$	0	0
$913$	−34.1693	−1.13084
$914$	0	0
$915$	−48.8315	−1.61432
$916$	0	0
$917$	10.2222	0.337565
$918$	0	0
$919$	0.851151	0.0280769	0.0140384	−	0.999901i	$-0.495531\pi$
$919$	0.851151	0.0280769	0.0140384	+	0.999901i	$0.495531\pi$
$920$	0	0
$921$	−32.2223	−1.06176
$922$	0	0
$923$	−25.0753	−0.825363
$924$	0	0
$925$	−8.11421	−0.266793
$926$	0	0
$927$	14.6822	0.482226
$928$	0	0
$929$	35.5935	1.16778	0.583892	−	0.811832i	$-0.301529\pi$
$929$	35.5935	1.16778	0.583892	+	0.811832i	$0.301529\pi$
$930$	0	0
$931$	−4.33994	−0.142236
$932$	0	0
$933$	−52.9953	−1.73499
$934$	0	0
$935$	−26.0365	−0.851486
$936$	0	0
$937$	−4.10493	−0.134102	−0.0670511	−	0.997750i	$-0.521359\pi$
$937$	−4.10493	−0.134102	−0.0670511	+	0.997750i	$0.521359\pi$
$938$	0	0
$939$	−36.3292	−1.18556
$940$	0	0
$941$	32.6925	1.06574	0.532872	−	0.846196i	$-0.321113\pi$
$941$	32.6925	1.06574	0.532872	+	0.846196i	$0.321113\pi$
$942$	0	0
$943$	−20.1668	−0.656722
$944$	0	0
$945$	14.1511	0.460335
$946$	0	0
$947$	47.8774	1.55581	0.777903	−	0.628384i	$-0.216284\pi$
$947$	47.8774	1.55581	0.777903	+	0.628384i	$0.216284\pi$
$948$	0	0
$949$	9.09516	0.295241
$950$	0	0
$951$	6.96452	0.225840
$952$	0	0
$953$	9.92047	0.321355	0.160678	−	0.987007i	$-0.448632\pi$
$953$	9.92047	0.321355	0.160678	+	0.987007i	$0.448632\pi$
$954$	0	0
$955$	−28.1490	−0.910879
$956$	0	0
$957$	16.2096	0.523983
$958$	0	0
$959$	−12.6272	−0.407753
$960$	0	0
$961$	58.5635	1.88915
$962$	0	0
$963$	9.72287	0.313315
$964$	0	0
$965$	5.47885	0.176371
$966$	0	0
$967$	15.8645	0.510169	0.255084	−	0.966919i	$-0.417897\pi$
$967$	15.8645	0.510169	0.255084	+	0.966919i	$0.417897\pi$
$968$	0	0
$969$	−9.87902	−0.317360
$970$	0	0
$971$	−50.5232	−1.62137	−0.810684	−	0.585484i	$-0.800904\pi$
$971$	−50.5232	−1.62137	−0.810684	+	0.585484i	$0.800904\pi$
$972$	0	0
$973$	−0.247634	−0.00793880
$974$	0	0
$975$	6.97295	0.223313
$976$	0	0
$977$	−11.7361	−0.375472	−0.187736	−	0.982220i	$-0.560115\pi$
$977$	−11.7361	−0.375472	−0.187736	+	0.982220i	$0.560115\pi$
$978$	0	0
$979$	12.3328	0.394159
$980$	0	0
$981$	−10.0031	−0.319375
$982$	0	0
$983$	32.1233	1.02457	0.512287	−	0.858814i	$-0.328798\pi$
$983$	32.1233	1.02457	0.512287	+	0.858814i	$0.328798\pi$
$984$	0	0
$985$	27.8690	0.887979
$986$	0	0
$987$	−11.7816	−0.375011
$988$	0	0
$989$	−13.6793	−0.434976
$990$	0	0
$991$	−10.8563	−0.344861	−0.172430	−	0.985022i	$-0.555162\pi$
$991$	−10.8563	−0.344861	−0.172430	+	0.985022i	$0.555162\pi$
$992$	0	0
$993$	22.8672	0.725667
$994$	0	0
$995$	−11.7772	−0.373363
$996$	0	0
$997$	−31.5861	−1.00034	−0.500171	−	0.865927i	$-0.666730\pi$
$997$	−31.5861	−1.00034	−0.500171	+	0.865927i	$0.666730\pi$
$998$	0	0
$999$	39.8768	1.26165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.8	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.8	✓	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.95303 q^{3} +2.03258 q^{5} +1.63097 q^{7} +0.814318 q^{9} +O(q^{10})\)	\(q-1.95303 q^{3} +2.03258 q^{5} +1.63097 q^{7} +0.814318 q^{9} -2.53238 q^{11} +4.11043 q^{13} -3.96969 q^{15} +5.05831 q^{17} +1.00000 q^{19} -3.18533 q^{21} -3.14690 q^{23} -0.868602 q^{25} +4.26870 q^{27} +3.27745 q^{29} -9.46380 q^{31} +4.94581 q^{33} +3.31508 q^{35} +9.34168 q^{37} -8.02778 q^{39} +6.40846 q^{41} +4.34690 q^{43} +1.65517 q^{45} +3.69870 q^{47} -4.33994 q^{49} -9.87902 q^{51} -0.751212 q^{53} -5.14728 q^{55} -1.95303 q^{57} -8.75518 q^{59} +12.3011 q^{61} +1.32813 q^{63} +8.35479 q^{65} -5.77709 q^{67} +6.14599 q^{69} -6.10040 q^{71} +2.21270 q^{73} +1.69640 q^{75} -4.13024 q^{77} +1.00000 q^{79} -10.7798 q^{81} +13.4930 q^{83} +10.2814 q^{85} -6.40094 q^{87} -4.87005 q^{89} +6.70398 q^{91} +18.4831 q^{93} +2.03258 q^{95} -5.39797 q^{97} -2.06216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more