LMFDB - Embedded newform 6525.2.a.bf.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,0,5,0,0,4,0,0,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.39138$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39138 q^{2} +3.71871 q^{4} +1.32733 q^{7} +4.11009 q^{8} -3.00000 q^{11} -6.11009 q^{13} +3.17415 q^{14} +2.39138 q^{16} +0.672673 q^{17} -5.43742 q^{19} -7.17415 q^{22} -7.89286 q^{23} -14.6116 q^{26} +4.93594 q^{28} +1.00000 q^{29} -2.50147 q^{32} +1.60862 q^{34} +3.89286 q^{37} -13.0029 q^{38} -4.32733 q^{41} +8.45544 q^{43} -11.1561 q^{44} -18.8748 q^{46} +4.76475 q^{47} -5.23820 q^{49} -22.7217 q^{52} -13.3303 q^{53} +5.45544 q^{56} +2.39138 q^{58} -3.21724 q^{59} +7.43742 q^{61} -10.7647 q^{64} +1.10714 q^{67} +2.50147 q^{68} -12.3483 q^{71} -9.20217 q^{73} +9.30931 q^{74} -20.2202 q^{76} -3.98198 q^{77} +11.6576 q^{79} -10.3483 q^{82} -0.889908 q^{83} +20.2202 q^{86} -12.3303 q^{88} +2.33028 q^{89} -8.11009 q^{91} -29.3512 q^{92} +11.3943 q^{94} +12.6756 q^{97} -12.5265 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} + 4 q^{7} - 9 q^{11} - 6 q^{13} - 9 q^{14} + q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{22} + q^{23} - 13 q^{26} + 21 q^{28} + 3 q^{29} + 11 q^{32} + 11 q^{34} - 13 q^{37} - 2 q^{38} - 13 q^{41}+ \cdots - 50 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 + 4 * q^7 - 9 * q^11 - 6 * q^13 - 9 * q^14 + q^16 + 2 * q^17 - 4 * q^19 - 3 * q^22 + q^23 - 13 * q^26 + 21 * q^28 + 3 * q^29 + 11 * q^32 + 11 * q^34 - 13 * q^37 - 2 * q^38 - 13 * q^41 + 13 * q^43 - 15 * q^44 - 32 * q^46 + 2 * q^47 + 9 * q^49 - 25 * q^52 - 3 * q^53 + 4 * q^56 + q^58 - 22 * q^59 + 10 * q^61 - 20 * q^64 + 28 * q^67 - 11 * q^68 - 3 * q^73 + 28 * q^74 - 36 * q^76 - 12 * q^77 - 2 * q^79 + 6 * q^82 - 15 * q^83 + 36 * q^86 - 30 * q^89 - 12 * q^91 - 14 * q^92 - 9 * q^94 + q^97 - 50 * q^98

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 $

$2$	2.39138	1.69096	0.845481	−	0.534005i	$-0.179314\pi$
$2$	2.39138	1.69096	0.845481	+	0.534005i	$0.179314\pi$
$3$	0	0
$3$	0	0
$4$	3.71871	1.85935
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.32733	0.501683	0.250841	−	0.968028i	$-0.419293\pi$
$7$	1.32733	0.501683	0.250841	+	0.968028i	$0.419293\pi$
$8$	4.11009	1.45314
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−6.11009	−1.69463	−0.847317	−	0.531087i	$-0.821784\pi$
$13$	−6.11009	−1.69463	−0.847317	+	0.531087i	$0.821784\pi$
$14$	3.17415	0.848327
$15$	0	0
$16$	2.39138	0.597846
$17$	0.672673	0.163147	0.0815735	−	0.996667i	$-0.474005\pi$
$17$	0.672673	0.163147	0.0815735	+	0.996667i	$0.474005\pi$
$18$	0	0
$19$	−5.43742	−1.24743	−0.623715	−	0.781652i	$-0.714377\pi$
$19$	−5.43742	−1.24743	−0.623715	+	0.781652i	$0.714377\pi$
$20$	0	0
$21$	0	0
$22$	−7.17415	−1.52953
$23$	−7.89286	−1.64577	−0.822887	−	0.568205i	$-0.807638\pi$
$23$	−7.89286	−1.64577	−0.822887	+	0.568205i	$0.807638\pi$
$24$	0	0
$25$	0	0
$26$	−14.6116	−2.86556
$27$	0	0
$28$	4.93594	0.932806
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−2.50147	−0.442202
$33$	0	0
$34$	1.60862	0.275876
$35$	0	0
$36$	0	0
$37$	3.89286	0.639982	0.319991	−	0.947421i	$-0.396320\pi$
$37$	3.89286	0.639982	0.319991	+	0.947421i	$0.396320\pi$
$38$	−13.0029	−2.10936
$39$	0	0
$40$	0	0
$41$	−4.32733	−0.675815	−0.337907	−	0.941179i	$-0.609719\pi$
$41$	−4.32733	−0.675815	−0.337907	+	0.941179i	$0.609719\pi$
$42$	0	0
$43$	8.45544	1.28944	0.644721	−	0.764418i	$-0.276974\pi$
$43$	8.45544	1.28944	0.644721	+	0.764418i	$0.276974\pi$
$44$	−11.1561	−1.68185
$45$	0	0
$46$	−18.8748	−2.78294
$47$	4.76475	0.695010	0.347505	−	0.937678i	$-0.387029\pi$
$47$	4.76475	0.695010	0.347505	+	0.937678i	$0.387029\pi$
$48$	0	0
$49$	−5.23820	−0.748315
$50$	0	0
$51$	0	0
$52$	−22.7217	−3.15093
$53$	−13.3303	−1.83105	−0.915527	−	0.402256i	$-0.868226\pi$
$53$	−13.3303	−1.83105	−0.915527	+	0.402256i	$0.868226\pi$
$54$	0	0
$55$	0	0
$56$	5.45544	0.729013
$57$	0	0
$58$	2.39138	0.314004
$59$	−3.21724	−0.418848	−0.209424	−	0.977825i	$-0.567159\pi$
$59$	−3.21724	−0.418848	−0.209424	+	0.977825i	$0.567159\pi$
$60$	0	0
$61$	7.43742	0.952264	0.476132	−	0.879374i	$-0.342038\pi$
$61$	7.43742	0.952264	0.476132	+	0.879374i	$0.342038\pi$
$62$	0	0
$63$	0	0
$64$	−10.7647	−1.34559
$65$	0	0
$66$	0	0
$67$	1.10714	0.135259	0.0676295	−	0.997711i	$-0.478456\pi$
$67$	1.10714	0.135259	0.0676295	+	0.997711i	$0.478456\pi$
$68$	2.50147	0.303348
$69$	0	0
$70$	0	0
$71$	−12.3483	−1.46547	−0.732736	−	0.680513i	$-0.761757\pi$
$71$	−12.3483	−1.46547	−0.732736	+	0.680513i	$0.761757\pi$
$72$	0	0
$73$	−9.20217	−1.07703	−0.538516	−	0.842615i	$-0.681015\pi$
$73$	−9.20217	−1.07703	−0.538516	+	0.842615i	$0.681015\pi$
$74$	9.30931	1.08219
$75$	0	0
$76$	−20.2202	−2.31941
$77$	−3.98198	−0.453789
$78$	0	0
$79$	11.6576	1.31158	0.655791	−	0.754942i	$-0.272335\pi$
$79$	11.6576	1.31158	0.655791	+	0.754942i	$0.272335\pi$
$80$	0	0
$81$	0	0
$82$	−10.3483	−1.14278
$83$	−0.889908	−0.0976801	−0.0488400	−	0.998807i	$-0.515552\pi$
$83$	−0.889908	−0.0976801	−0.0488400	+	0.998807i	$0.515552\pi$
$84$	0	0
$85$	0	0
$86$	20.2202	2.18040
$87$	0	0
$88$	−12.3303	−1.31441
$89$	2.33028	0.247009	0.123504	−	0.992344i	$-0.460587\pi$
$89$	2.33028	0.247009	0.123504	+	0.992344i	$0.460587\pi$
$90$	0	0
$91$	−8.11009	−0.850169
$92$	−29.3512	−3.06008
$93$	0	0
$94$	11.3943	1.17524
$95$	0	0
$96$	0	0
$97$	12.6756	1.28701	0.643507	−	0.765440i	$-0.277479\pi$
$97$	12.6756	1.28701	0.643507	+	0.765440i	$0.277479\pi$
$98$	−12.5265	−1.26537
$99$	0	0
$100$	0	0
$101$	−3.56258	−0.354490	−0.177245	−	0.984167i	$-0.556719\pi$
$101$	−3.56258	−0.354490	−0.177245	+	0.984167i	$0.556719\pi$
$102$	0	0
$103$	12.9109	1.27215	0.636073	−	0.771629i	$-0.280558\pi$
$103$	12.9109	1.27215	0.636073	+	0.771629i	$0.280558\pi$
$104$	−25.1130	−2.46254
$105$	0	0
$106$	−31.8778	−3.09624
$107$	4.22018	0.407981	0.203990	−	0.978973i	$-0.434609\pi$
$107$	4.22018	0.407981	0.203990	+	0.978973i	$0.434609\pi$
$108$	0	0
$109$	−11.8748	−1.13740	−0.568702	−	0.822544i	$-0.692554\pi$
$109$	−11.8748	−1.13740	−0.568702	+	0.822544i	$0.692554\pi$
$110$	0	0
$111$	0	0
$112$	3.17415	0.299929
$113$	14.2382	1.33942	0.669709	−	0.742624i	$-0.266419\pi$
$113$	14.2382	1.33942	0.669709	+	0.742624i	$0.266419\pi$
$114$	0	0
$115$	0	0
$116$	3.71871	0.345274
$117$	0	0
$118$	−7.69364	−0.708257
$119$	0.892857	0.0818481
$120$	0	0
$121$	−2.00000	−0.181818
$122$	17.7857	1.61024
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.20217	0.639089	0.319544	−	0.947571i	$-0.396470\pi$
$127$	7.20217	0.639089	0.319544	+	0.947571i	$0.396470\pi$
$128$	−20.7397	−1.83315
$129$	0	0
$130$	0	0
$131$	−14.2382	−1.24400	−0.621999	−	0.783018i	$-0.713679\pi$
$131$	−14.2382	−1.24400	−0.621999	+	0.783018i	$0.713679\pi$
$132$	0	0
$133$	−7.21724	−0.625814
$134$	2.64760	0.228718
$135$	0	0
$136$	2.76475	0.237075
$137$	15.0950	1.28965	0.644827	−	0.764328i	$-0.276929\pi$
$137$	15.0950	1.28965	0.644827	+	0.764328i	$0.276929\pi$
$138$	0	0
$139$	−6.43742	−0.546015	−0.273007	−	0.962012i	$-0.588018\pi$
$139$	−6.43742	−0.546015	−0.273007	+	0.962012i	$0.588018\pi$
$140$	0	0
$141$	0	0
$142$	−29.5295	−2.47806
$143$	18.3303	1.53285
$144$	0	0
$145$	0	0
$146$	−22.0059	−1.82122
$147$	0	0
$148$	14.4764	1.18995
$149$	−1.21724	−0.0997198	−0.0498599	−	0.998756i	$-0.515877\pi$
$149$	−1.21724	−0.0997198	−0.0498599	+	0.998756i	$0.515877\pi$
$150$	0	0
$151$	−2.32733	−0.189395	−0.0946976	−	0.995506i	$-0.530188\pi$
$151$	−2.32733	−0.189395	−0.0946976	+	0.995506i	$0.530188\pi$
$152$	−22.3483	−1.81269
$153$	0	0
$154$	−9.52244	−0.767340
$155$	0	0
$156$	0	0
$157$	−9.43742	−0.753188	−0.376594	−	0.926378i	$-0.622905\pi$
$157$	−9.43742	−0.753188	−0.376594	+	0.926378i	$0.622905\pi$
$158$	27.8778	2.21784
$159$	0	0
$160$	0	0
$161$	−10.4764	−0.825656
$162$	0	0
$163$	21.3303	1.67072	0.835358	−	0.549706i	$-0.185260\pi$
$163$	21.3303	1.67072	0.835358	+	0.549706i	$0.185260\pi$
$164$	−16.0921	−1.25658
$165$	0	0
$166$	−2.12811	−0.165173
$167$	5.30931	0.410847	0.205423	−	0.978673i	$-0.434143\pi$
$167$	5.30931	0.410847	0.205423	+	0.978673i	$0.434143\pi$
$168$	0	0
$169$	24.3332	1.87179
$170$	0	0
$171$	0	0
$172$	31.4433	2.39753
$173$	22.7677	1.73100	0.865498	−	0.500913i	$-0.167002\pi$
$173$	22.7677	1.73100	0.865498	+	0.500913i	$0.167002\pi$
$174$	0	0
$175$	0	0
$176$	−7.17415	−0.540772
$177$	0	0
$178$	5.57258	0.417683
$179$	−17.6576	−1.31979	−0.659896	−	0.751357i	$-0.729400\pi$
$179$	−17.6576	−1.31979	−0.659896	+	0.751357i	$0.729400\pi$
$180$	0	0
$181$	10.5655	0.785330	0.392665	−	0.919682i	$-0.371553\pi$
$181$	10.5655	0.785330	0.392665	+	0.919682i	$0.371553\pi$
$182$	−19.3943	−1.43760
$183$	0	0
$184$	−32.4404	−2.39154
$185$	0	0
$186$	0	0
$187$	−2.01802	−0.147572
$188$	17.7187	1.29227
$189$	0	0
$190$	0	0
$191$	3.23820	0.234308	0.117154	−	0.993114i	$-0.462623\pi$
$191$	3.23820	0.234308	0.117154	+	0.993114i	$0.462623\pi$
$192$	0	0
$193$	3.30931	0.238209	0.119105	−	0.992882i	$-0.461998\pi$
$193$	3.30931	0.238209	0.119105	+	0.992882i	$0.461998\pi$
$194$	30.3123	2.17629
$195$	0	0
$196$	−19.4794	−1.39138
$197$	−1.89286	−0.134860	−0.0674302	−	0.997724i	$-0.521480\pi$
$197$	−1.89286	−0.134860	−0.0674302	+	0.997724i	$0.521480\pi$
$198$	0	0
$199$	−5.56258	−0.394321	−0.197160	−	0.980371i	$-0.563172\pi$
$199$	−5.56258	−0.394321	−0.197160	+	0.980371i	$0.563172\pi$
$200$	0	0
$201$	0	0
$202$	−8.51949	−0.599429
$203$	1.32733	0.0931601
$204$	0	0
$205$	0	0
$206$	30.8748	2.15115
$207$	0	0
$208$	−14.6116	−1.01313
$209$	16.3123	1.12834
$210$	0	0
$211$	15.7497	1.08425	0.542126	−	0.840297i	$-0.317619\pi$
$211$	15.7497	1.08425	0.542126	+	0.840297i	$0.317619\pi$
$212$	−49.5714	−3.40458
$213$	0	0
$214$	10.0921	0.689880
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−28.3973	−1.92331
$219$	0	0
$220$	0	0
$221$	−4.11009	−0.276475
$222$	0	0
$223$	2.01802	0.135136	0.0675682	−	0.997715i	$-0.478476\pi$
$223$	2.01802	0.135136	0.0675682	+	0.997715i	$0.478476\pi$
$224$	−3.32028	−0.221845
$225$	0	0
$226$	34.0490	2.26490
$227$	−20.6756	−1.37229	−0.686145	−	0.727465i	$-0.740698\pi$
$227$	−20.6756	−1.37229	−0.686145	+	0.727465i	$0.740698\pi$
$228$	0	0
$229$	14.2562	0.942078	0.471039	−	0.882113i	$-0.343879\pi$
$229$	14.2562	0.942078	0.471039	+	0.882113i	$0.343879\pi$
$230$	0	0
$231$	0	0
$232$	4.11009	0.269841
$233$	20.7677	1.36054	0.680268	−	0.732963i	$-0.261863\pi$
$233$	20.7677	1.36054	0.680268	+	0.732963i	$0.261863\pi$
$234$	0	0
$235$	0	0
$236$	−11.9640	−0.778788
$237$	0	0
$238$	2.13516	0.138402
$239$	−3.30931	−0.214061	−0.107031	−	0.994256i	$-0.534134\pi$
$239$	−3.30931	−0.214061	−0.107031	+	0.994256i	$0.534134\pi$
$240$	0	0
$241$	−8.78571	−0.565938	−0.282969	−	0.959129i	$-0.591319\pi$
$241$	−8.78571	−0.565938	−0.282969	+	0.959129i	$0.591319\pi$
$242$	−4.78276	−0.307448
$243$	0	0
$244$	27.6576	1.77060
$245$	0	0
$246$	0	0
$247$	33.2231	2.11394
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.8037	−1.50248	−0.751239	−	0.660030i	$-0.770543\pi$
$251$	−23.8037	−1.50248	−0.751239	+	0.660030i	$0.770543\pi$
$252$	0	0
$253$	23.6786	1.48866
$254$	17.2231	1.08068
$255$	0	0
$256$	−28.0670	−1.75419
$257$	−27.3303	−1.70482	−0.852408	−	0.522877i	$-0.824859\pi$
$257$	−27.3303	−1.70482	−0.852408	+	0.522877i	$0.824859\pi$
$258$	0	0
$259$	5.16710	0.321068
$260$	0	0
$261$	0	0
$262$	−34.0490	−2.10355
$263$	−21.7497	−1.34114	−0.670571	−	0.741845i	$-0.733951\pi$
$263$	−21.7497	−1.34114	−0.670571	+	0.741845i	$0.733951\pi$
$264$	0	0
$265$	0	0
$266$	−17.2592	−1.05823
$267$	0	0
$268$	4.11714	0.251495
$269$	−10.3303	−0.629848	−0.314924	−	0.949117i	$-0.601979\pi$
$269$	−10.3303	−0.629848	−0.314924	+	0.949117i	$0.601979\pi$
$270$	0	0
$271$	−6.25622	−0.380038	−0.190019	−	0.981780i	$-0.560855\pi$
$271$	−6.25622	−0.380038	−0.190019	+	0.981780i	$0.560855\pi$
$272$	1.60862	0.0975368
$273$	0	0
$274$	36.0980	2.18076
$275$	0	0
$276$	0	0
$277$	−31.6815	−1.90356	−0.951779	−	0.306784i	$-0.900747\pi$
$277$	−31.6815	−1.90356	−0.951779	+	0.306784i	$0.900747\pi$
$278$	−15.3943	−0.923291
$279$	0	0
$280$	0	0
$281$	−27.9138	−1.66520	−0.832600	−	0.553875i	$-0.813148\pi$
$281$	−27.9138	−1.66520	−0.832600	+	0.553875i	$0.813148\pi$
$282$	0	0
$283$	−23.5295	−1.39868	−0.699342	−	0.714788i	$-0.746523\pi$
$283$	−23.5295	−1.39868	−0.699342	+	0.714788i	$0.746523\pi$
$284$	−45.9197	−2.72483
$285$	0	0
$286$	43.8347	2.59200
$287$	−5.74378	−0.339045
$288$	0	0
$289$	−16.5475	−0.973383
$290$	0	0
$291$	0	0
$292$	−34.2202	−2.00258
$293$	−6.23820	−0.364440	−0.182220	−	0.983258i	$-0.558328\pi$
$293$	−6.23820	−0.364440	−0.182220	+	0.983258i	$0.558328\pi$
$294$	0	0
$295$	0	0
$296$	16.0000	0.929981
$297$	0	0
$298$	−2.91087	−0.168622
$299$	48.2261	2.78899
$300$	0	0
$301$	11.2231	0.646891
$302$	−5.56553	−0.320260
$303$	0	0
$304$	−13.0029	−0.745770
$305$	0	0
$306$	0	0
$307$	−3.45249	−0.197044	−0.0985220	−	0.995135i	$-0.531411\pi$
$307$	−3.45249	−0.197044	−0.0985220	+	0.995135i	$0.531411\pi$
$308$	−14.8078	−0.843755
$309$	0	0
$310$	0	0
$311$	−11.0000	−0.623753	−0.311876	−	0.950123i	$-0.600957\pi$
$311$	−11.0000	−0.623753	−0.311876	+	0.950123i	$0.600957\pi$
$312$	0	0
$313$	32.9908	1.86475	0.932376	−	0.361490i	$-0.117732\pi$
$313$	32.9908	1.86475	0.932376	+	0.361490i	$0.117732\pi$
$314$	−22.5685	−1.27361
$315$	0	0
$316$	43.3512	2.43870
$317$	4.63664	0.260419	0.130210	−	0.991486i	$-0.458435\pi$
$317$	4.63664	0.260419	0.130210	+	0.991486i	$0.458435\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	0	0
$322$	−25.0531	−1.39615
$323$	−3.65760	−0.203515
$324$	0	0
$325$	0	0
$326$	51.0088	2.82512
$327$	0	0
$328$	−17.7857	−0.982052
$329$	6.32438	0.348674
$330$	0	0
$331$	−22.2261	−1.22166	−0.610828	−	0.791763i	$-0.709163\pi$
$331$	−22.2261	−1.22166	−0.610828	+	0.791763i	$0.709163\pi$
$332$	−3.30931	−0.181622
$333$	0	0
$334$	12.6966	0.694726
$335$	0	0
$336$	0	0
$337$	27.5655	1.50159	0.750795	−	0.660535i	$-0.229671\pi$
$337$	27.5655	1.50159	0.750795	+	0.660535i	$0.229671\pi$
$338$	58.1900	3.16512
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.2441	−0.877099
$344$	34.7526	1.87374
$345$	0	0
$346$	54.4463	2.92705
$347$	−22.3273	−1.19859	−0.599297	−	0.800527i	$-0.704553\pi$
$347$	−22.3273	−1.19859	−0.599297	+	0.800527i	$0.704553\pi$
$348$	0	0
$349$	−20.5115	−1.09795	−0.548977	−	0.835837i	$-0.684983\pi$
$349$	−20.5115	−1.09795	−0.548977	+	0.835837i	$0.684983\pi$
$350$	0	0
$351$	0	0
$352$	7.50442	0.399987
$353$	−28.8748	−1.53685	−0.768426	−	0.639938i	$-0.778960\pi$
$353$	−28.8748	−1.53685	−0.768426	+	0.639938i	$0.778960\pi$
$354$	0	0
$355$	0	0
$356$	8.66562	0.459277
$357$	0	0
$358$	−42.2261	−2.23172
$359$	−2.76770	−0.146073	−0.0730367	−	0.997329i	$-0.523269\pi$
$359$	−2.76770	−0.146073	−0.0730367	+	0.997329i	$0.523269\pi$
$360$	0	0
$361$	10.5655	0.556081
$362$	25.2662	1.32796
$363$	0	0
$364$	−30.1591	−1.58077
$365$	0	0
$366$	0	0
$367$	−25.4584	−1.32892	−0.664458	−	0.747325i	$-0.731338\pi$
$367$	−25.4584	−1.32892	−0.664458	+	0.747325i	$0.731338\pi$
$368$	−18.8748	−0.983919
$369$	0	0
$370$	0	0
$371$	−17.6936	−0.918608
$372$	0	0
$373$	4.03604	0.208978	0.104489	−	0.994526i	$-0.466679\pi$
$373$	4.03604	0.208978	0.104489	+	0.994526i	$0.466679\pi$
$374$	−4.82585	−0.249539
$375$	0	0
$376$	19.5835	1.00994
$377$	−6.11009	−0.314686
$378$	0	0
$379$	20.4043	1.04810	0.524050	−	0.851687i	$-0.324420\pi$
$379$	20.4043	1.04810	0.524050	+	0.851687i	$0.324420\pi$
$380$	0	0
$381$	0	0
$382$	7.74378	0.396206
$383$	−5.80078	−0.296406	−0.148203	−	0.988957i	$-0.547349\pi$
$383$	−5.80078	−0.296406	−0.148203	+	0.988957i	$0.547349\pi$
$384$	0	0
$385$	0	0
$386$	7.91382	0.402803
$387$	0	0
$388$	47.1370	2.39302
$389$	32.2592	1.63560	0.817802	−	0.575499i	$-0.195192\pi$
$389$	32.2592	1.63560	0.817802	+	0.575499i	$0.195192\pi$
$390$	0	0
$391$	−5.30931	−0.268503
$392$	−21.5295	−1.08740
$393$	0	0
$394$	−4.52654	−0.228044
$395$	0	0
$396$	0	0
$397$	−22.1841	−1.11339	−0.556695	−	0.830717i	$-0.687931\pi$
$397$	−22.1841	−1.11339	−0.556695	+	0.830717i	$0.687931\pi$
$398$	−13.3023	−0.666782
$399$	0	0
$400$	0	0
$401$	38.6045	1.92782	0.963909	−	0.266233i	$-0.0857790\pi$
$401$	38.6045	1.92782	0.963909	+	0.266233i	$0.0857790\pi$
$402$	0	0
$403$	0	0
$404$	−13.2482	−0.659123
$405$	0	0
$406$	3.17415	0.157530
$407$	−11.6786	−0.578885
$408$	0	0
$409$	28.0980	1.38936	0.694678	−	0.719321i	$-0.255547\pi$
$409$	28.0980	1.38936	0.694678	+	0.719321i	$0.255547\pi$
$410$	0	0
$411$	0	0
$412$	48.0118	2.36537
$413$	−4.27032	−0.210129
$414$	0	0
$415$	0	0
$416$	15.2842	0.749371
$417$	0	0
$418$	39.0088	1.90799
$419$	−3.56553	−0.174188	−0.0870938	−	0.996200i	$-0.527758\pi$
$419$	−3.56553	−0.174188	−0.0870938	+	0.996200i	$0.527758\pi$
$420$	0	0
$421$	−17.6216	−0.858823	−0.429411	−	0.903109i	$-0.641279\pi$
$421$	−17.6216	−0.858823	−0.429411	+	0.903109i	$0.641279\pi$
$422$	37.6635	1.83343
$423$	0	0
$424$	−54.7887	−2.66077
$425$	0	0
$426$	0	0
$427$	9.87189	0.477734
$428$	15.6936	0.758581
$429$	0	0
$430$	0	0
$431$	33.7916	1.62768	0.813842	−	0.581086i	$-0.197372\pi$
$431$	33.7916	1.62768	0.813842	+	0.581086i	$0.197372\pi$
$432$	0	0
$433$	−23.3663	−1.12291	−0.561457	−	0.827506i	$-0.689759\pi$
$433$	−23.3663	−1.12291	−0.561457	+	0.827506i	$0.689759\pi$
$434$	0	0
$435$	0	0
$436$	−44.1591	−2.11484
$437$	42.9168	2.05299
$438$	0	0
$439$	−16.1461	−0.770613	−0.385306	−	0.922789i	$-0.625904\pi$
$439$	−16.1461	−0.770613	−0.385306	+	0.922789i	$0.625904\pi$
$440$	0	0
$441$	0	0
$442$	−9.82880	−0.467508
$443$	9.38337	0.445817	0.222909	−	0.974839i	$-0.428445\pi$
$443$	9.38337	0.445817	0.222909	+	0.974839i	$0.428445\pi$
$444$	0	0
$445$	0	0
$446$	4.82585	0.228511
$447$	0	0
$448$	−14.2883	−0.675061
$449$	−17.5685	−0.829108	−0.414554	−	0.910025i	$-0.636062\pi$
$449$	−17.5685	−0.829108	−0.414554	+	0.910025i	$0.636062\pi$
$450$	0	0
$451$	12.9820	0.611298
$452$	52.9477	2.49045
$453$	0	0
$454$	−49.4433	−2.32049
$455$	0	0
$456$	0	0
$457$	17.4194	0.814845	0.407423	−	0.913240i	$-0.366428\pi$
$457$	17.4194	0.814845	0.407423	+	0.913240i	$0.366428\pi$
$458$	34.0921	1.59302
$459$	0	0
$460$	0	0
$461$	0.113041	0.00526484	0.00263242	−	0.999997i	$-0.499162\pi$
$461$	0.113041	0.00526484	0.00263242	+	0.999997i	$0.499162\pi$
$462$	0	0
$463$	26.8568	1.24814	0.624071	−	0.781368i	$-0.285478\pi$
$463$	26.8568	1.24814	0.624071	+	0.781368i	$0.285478\pi$
$464$	2.39138	0.111017
$465$	0	0
$466$	49.6635	2.30062
$467$	0.726727	0.0336289	0.0168145	−	0.999859i	$-0.494648\pi$
$467$	0.726727	0.0336289	0.0168145	+	0.999859i	$0.494648\pi$
$468$	0	0
$469$	1.46954	0.0678571
$470$	0	0
$471$	0	0
$472$	−13.2231	−0.608644
$473$	−25.3663	−1.16634
$474$	0	0
$475$	0	0
$476$	3.32028	0.152185
$477$	0	0
$478$	−7.91382	−0.361970
$479$	9.12516	0.416939	0.208470	−	0.978029i	$-0.433152\pi$
$479$	9.12516	0.416939	0.208470	+	0.978029i	$0.433152\pi$
$480$	0	0
$481$	−23.7857	−1.08454
$482$	−21.0100	−0.956979
$483$	0	0
$484$	−7.43742	−0.338065
$485$	0	0
$486$	0	0
$487$	−35.5714	−1.61190	−0.805948	−	0.591987i	$-0.798344\pi$
$487$	−35.5714	−1.61190	−0.805948	+	0.591987i	$0.798344\pi$
$488$	30.5685	1.38377
$489$	0	0
$490$	0	0
$491$	4.44037	0.200391	0.100196	−	0.994968i	$-0.468053\pi$
$491$	4.44037	0.200391	0.100196	+	0.994968i	$0.468053\pi$
$492$	0	0
$493$	0.672673	0.0302957
$494$	79.4492	3.57459
$495$	0	0
$496$	0	0
$497$	−16.3902	−0.735202
$498$	0	0
$499$	−40.3663	−1.80704	−0.903522	−	0.428541i	$-0.859028\pi$
$499$	−40.3663	−1.80704	−0.903522	+	0.428541i	$0.859028\pi$
$500$	0	0
$501$	0	0
$502$	−56.9238	−2.54063
$503$	23.7117	1.05725	0.528625	−	0.848855i	$-0.322708\pi$
$503$	23.7117	1.05725	0.528625	+	0.848855i	$0.322708\pi$
$504$	0	0
$505$	0	0
$506$	56.6245	2.51727
$507$	0	0
$508$	26.7828	1.18829
$509$	0.568478	0.0251974	0.0125987	−	0.999921i	$-0.495990\pi$
$509$	0.568478	0.0251974	0.0125987	+	0.999921i	$0.495990\pi$
$510$	0	0
$511$	−12.2143	−0.540328
$512$	−25.6396	−1.13312
$513$	0	0
$514$	−65.3571	−2.88278
$515$	0	0
$516$	0	0
$517$	−14.2942	−0.628660
$518$	12.3565	0.542913
$519$	0	0
$520$	0	0
$521$	−21.6576	−0.948837	−0.474418	−	0.880299i	$-0.657342\pi$
$521$	−21.6576	−0.948837	−0.474418	+	0.880299i	$0.657342\pi$
$522$	0	0
$523$	−14.0180	−0.612965	−0.306483	−	0.951876i	$-0.599152\pi$
$523$	−14.0180	−0.612965	−0.306483	+	0.951876i	$0.599152\pi$
$524$	−52.9477	−2.31303
$525$	0	0
$526$	−52.0118	−2.26782
$527$	0	0
$528$	0	0
$529$	39.2972	1.70857
$530$	0	0
$531$	0	0
$532$	−26.8388	−1.16361
$533$	26.4404	1.14526
$534$	0	0
$535$	0	0
$536$	4.55046	0.196550
$537$	0	0
$538$	−24.7036	−1.06505
$539$	15.7146	0.676876
$540$	0	0
$541$	20.8748	0.897479	0.448740	−	0.893663i	$-0.351873\pi$
$541$	20.8748	0.897479	0.448740	+	0.893663i	$0.351873\pi$
$542$	−14.9610	−0.642631
$543$	0	0
$544$	−1.68267	−0.0721440
$545$	0	0
$546$	0	0
$547$	−4.45249	−0.190375	−0.0951873	−	0.995459i	$-0.530345\pi$
$547$	−4.45249	−0.190375	−0.0951873	+	0.995459i	$0.530345\pi$
$548$	56.1340	2.39793
$549$	0	0
$550$	0	0
$551$	−5.43742	−0.231642
$552$	0	0
$553$	15.4735	0.657998
$554$	−75.7626	−3.21885
$555$	0	0
$556$	−23.9389	−1.01524
$557$	34.4253	1.45865	0.729323	−	0.684169i	$-0.239835\pi$
$557$	34.4253	1.45865	0.729323	+	0.684169i	$0.239835\pi$
$558$	0	0
$559$	−51.6635	−2.18513
$560$	0	0
$561$	0	0
$562$	−66.7526	−2.81579
$563$	19.6756	0.829229	0.414614	−	0.909997i	$-0.363917\pi$
$563$	19.6756	0.829229	0.414614	+	0.909997i	$0.363917\pi$
$564$	0	0
$565$	0	0
$566$	−56.2680	−2.36512
$567$	0	0
$568$	−50.7526	−2.12953
$569$	−26.2942	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$569$	−26.2942	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$570$	0	0
$571$	37.0600	1.55091	0.775455	−	0.631402i	$-0.217520\pi$
$571$	37.0600	1.55091	0.775455	+	0.631402i	$0.217520\pi$
$572$	68.1650	2.85012
$573$	0	0
$574$	−13.7356	−0.573312
$575$	0	0
$576$	0	0
$577$	14.2202	0.591994	0.295997	−	0.955189i	$-0.404348\pi$
$577$	14.2202	0.591994	0.295997	+	0.955189i	$0.404348\pi$
$578$	−39.5714	−1.64595
$579$	0	0
$580$	0	0
$581$	−1.18120	−0.0490044
$582$	0	0
$583$	39.9908	1.65625
$584$	−37.8217	−1.56508
$585$	0	0
$586$	−14.9179	−0.616254
$587$	32.0980	1.32483	0.662413	−	0.749139i	$-0.269533\pi$
$587$	32.0980	1.32483	0.662413	+	0.749139i	$0.269533\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	9.30931	0.382610
$593$	−27.1871	−1.11644	−0.558220	−	0.829693i	$-0.688516\pi$
$593$	−27.1871	−1.11644	−0.558220	+	0.829693i	$0.688516\pi$
$594$	0	0
$595$	0	0
$596$	−4.52654	−0.185414
$597$	0	0
$598$	115.327	4.71607
$599$	−31.8037	−1.29947	−0.649733	−	0.760163i	$-0.725119\pi$
$599$	−31.8037	−1.29947	−0.649733	+	0.760163i	$0.725119\pi$
$600$	0	0
$601$	−13.3453	−0.544368	−0.272184	−	0.962245i	$-0.587746\pi$
$601$	−13.3453	−0.544368	−0.272184	+	0.962245i	$0.587746\pi$
$602$	26.8388	1.09387
$603$	0	0
$604$	−8.65465	−0.352153
$605$	0	0
$606$	0	0
$607$	37.6635	1.52871	0.764357	−	0.644793i	$-0.223056\pi$
$607$	37.6635	1.52871	0.764357	+	0.644793i	$0.223056\pi$
$608$	13.6016	0.551616
$609$	0	0
$610$	0	0
$611$	−29.1130	−1.17779
$612$	0	0
$613$	−38.3663	−1.54960	−0.774800	−	0.632206i	$-0.782150\pi$
$613$	−38.3663	−1.54960	−0.774800	+	0.632206i	$0.782150\pi$
$614$	−8.25622	−0.333194
$615$	0	0
$616$	−16.3663	−0.659418
$617$	−34.6245	−1.39393	−0.696965	−	0.717105i	$-0.745467\pi$
$617$	−34.6245	−1.39393	−0.696965	+	0.717105i	$0.745467\pi$
$618$	0	0
$619$	0.00589781	0.000237053 0	0.000118526	−	1.00000i	$-0.499962\pi$
$619$	0.00589781	0.000237053 0	0.000118526		1.00000i	$0.499962\pi$
$620$	0	0
$621$	0	0
$622$	−26.3052	−1.05474
$623$	3.09304	0.123920
$624$	0	0
$625$	0	0
$626$	78.8937	3.15323
$627$	0	0
$628$	−35.0950	−1.40044
$629$	2.61862	0.104411
$630$	0	0
$631$	42.3001	1.68394	0.841971	−	0.539523i	$-0.181395\pi$
$631$	42.3001	1.68394	0.841971	+	0.539523i	$0.181395\pi$
$632$	47.9138	1.90591
$633$	0	0
$634$	11.0880	0.440360
$635$	0	0
$636$	0	0
$637$	32.0059	1.26812
$638$	−7.17415	−0.284027
$639$	0	0
$640$	0	0
$641$	−14.7297	−0.581787	−0.290894	−	0.956755i	$-0.593953\pi$
$641$	−14.7297	−0.581787	−0.290894	+	0.956755i	$0.593953\pi$
$642$	0	0
$643$	−37.9820	−1.49786	−0.748932	−	0.662647i	$-0.769433\pi$
$643$	−37.9820	−1.49786	−0.748932	+	0.662647i	$0.769433\pi$
$644$	−38.9587	−1.53519
$645$	0	0
$646$	−8.74673	−0.344135
$647$	31.5144	1.23896	0.619480	−	0.785012i	$-0.287344\pi$
$647$	31.5144	1.23896	0.619480	+	0.785012i	$0.287344\pi$
$648$	0	0
$649$	9.65171	0.378863
$650$	0	0
$651$	0	0
$652$	79.3211	3.10645
$653$	−38.8929	−1.52200	−0.760998	−	0.648755i	$-0.775290\pi$
$653$	−38.8929	−1.52200	−0.760998	+	0.648755i	$0.775290\pi$
$654$	0	0
$655$	0	0
$656$	−10.3483	−0.404033
$657$	0	0
$658$	15.1240	0.589595
$659$	32.2792	1.25742	0.628709	−	0.777641i	$-0.283584\pi$
$659$	32.2792	1.25742	0.628709	+	0.777641i	$0.283584\pi$
$660$	0	0
$661$	−29.8447	−1.16082	−0.580412	−	0.814323i	$-0.697109\pi$
$661$	−29.8447	−1.16082	−0.580412	+	0.814323i	$0.697109\pi$
$662$	−53.1511	−2.06577
$663$	0	0
$664$	−3.65760	−0.141943
$665$	0	0
$666$	0	0
$667$	−7.89286	−0.305613
$668$	19.7438	0.763910
$669$	0	0
$670$	0	0
$671$	−22.3123	−0.861355
$672$	0	0
$673$	−10.5505	−0.406690	−0.203345	−	0.979107i	$-0.565181\pi$
$673$	−10.5505	−0.406690	−0.203345	+	0.979107i	$0.565181\pi$
$674$	65.9197	2.53913
$675$	0	0
$676$	90.4882	3.48032
$677$	40.0600	1.53963	0.769815	−	0.638268i	$-0.220349\pi$
$677$	40.0600	1.53963	0.769815	+	0.638268i	$0.220349\pi$
$678$	0	0
$679$	16.8247	0.645673
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.2913	0.929480	0.464740	−	0.885447i	$-0.346148\pi$
$683$	24.2913	0.929480	0.464740	+	0.885447i	$0.346148\pi$
$684$	0	0
$685$	0	0
$686$	−38.8459	−1.48314
$687$	0	0
$688$	20.2202	0.770887
$689$	81.4492	3.10297
$690$	0	0
$691$	−43.9017	−1.67010	−0.835050	−	0.550174i	$-0.814561\pi$
$691$	−43.9017	−1.67010	−0.835050	+	0.550174i	$0.814561\pi$
$692$	84.6665	3.21854
$693$	0	0
$694$	−53.3932	−2.02678
$695$	0	0
$696$	0	0
$697$	−2.91087	−0.110257
$698$	−49.0508	−1.85660
$699$	0	0
$700$	0	0
$701$	39.9197	1.50775	0.753874	−	0.657020i	$-0.228183\pi$
$701$	39.9197	1.50775	0.753874	+	0.657020i	$0.228183\pi$
$702$	0	0
$703$	−21.1671	−0.798332
$704$	32.2942	1.21713
$705$	0	0
$706$	−69.0508	−2.59876
$707$	−4.72871	−0.177841
$708$	0	0
$709$	11.4885	0.431461	0.215730	−	0.976453i	$-0.430787\pi$
$709$	11.4885	0.431461	0.215730	+	0.976453i	$0.430787\pi$
$710$	0	0
$711$	0	0
$712$	9.57765	0.358938
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−65.6635	−2.45396
$717$	0	0
$718$	−6.61862	−0.247005
$719$	1.00295	0.0374037	0.0187018	−	0.999825i	$-0.494047\pi$
$719$	1.00295	0.0374037	0.0187018	+	0.999825i	$0.494047\pi$
$720$	0	0
$721$	17.1370	0.638214
$722$	25.2662	0.940311
$723$	0	0
$724$	39.2901	1.46021
$725$	0	0
$726$	0	0
$727$	20.4404	0.758091	0.379046	−	0.925378i	$-0.376252\pi$
$727$	20.4404	0.758091	0.379046	+	0.925378i	$0.376252\pi$
$728$	−33.3332	−1.23541
$729$	0	0
$730$	0	0
$731$	5.68774	0.210369
$732$	0	0
$733$	−30.9669	−1.14379	−0.571895	−	0.820327i	$-0.693791\pi$
$733$	−30.9669	−1.14379	−0.571895	+	0.820327i	$0.693791\pi$
$734$	−60.8807	−2.24715
$735$	0	0
$736$	19.7438	0.727765
$737$	−3.32143	−0.122346
$738$	0	0
$739$	3.62157	0.133222	0.0666108	−	0.997779i	$-0.478781\pi$
$739$	3.62157	0.133222	0.0666108	+	0.997779i	$0.478781\pi$
$740$	0	0
$741$	0	0
$742$	−42.3123	−1.55333
$743$	−25.1691	−0.923364	−0.461682	−	0.887046i	$-0.652754\pi$
$743$	−25.1691	−0.923364	−0.461682	+	0.887046i	$0.652754\pi$
$744$	0	0
$745$	0	0
$746$	9.65171	0.353374
$747$	0	0
$748$	−7.50442	−0.274389
$749$	5.60157	0.204677
$750$	0	0
$751$	−9.63760	−0.351681	−0.175841	−	0.984419i	$-0.556264\pi$
$751$	−9.63760	−0.351681	−0.175841	+	0.984419i	$0.556264\pi$
$752$	11.3943	0.415509
$753$	0	0
$754$	−14.6116	−0.532122
$755$	0	0
$756$	0	0
$757$	17.4282	0.633440	0.316720	−	0.948519i	$-0.397418\pi$
$757$	17.4282	0.633440	0.316720	+	0.948519i	$0.397418\pi$
$758$	48.7946	1.77230
$759$	0	0
$760$	0	0
$761$	−2.27622	−0.0825130	−0.0412565	−	0.999149i	$-0.513136\pi$
$761$	−2.27622	−0.0825130	−0.0412565	+	0.999149i	$0.513136\pi$
$762$	0	0
$763$	−15.7618	−0.570615
$764$	12.0419	0.435662
$765$	0	0
$766$	−13.8719	−0.501212
$767$	19.6576	0.709795
$768$	0	0
$769$	29.4433	1.06175	0.530877	−	0.847449i	$-0.321863\pi$
$769$	29.4433	1.06175	0.530877	+	0.847449i	$0.321863\pi$
$770$	0	0
$771$	0	0
$772$	12.3064	0.442916
$773$	14.2562	0.512761	0.256380	−	0.966576i	$-0.417470\pi$
$773$	14.2562	0.512761	0.256380	+	0.966576i	$0.417470\pi$
$774$	0	0
$775$	0	0
$776$	52.0980	1.87021
$777$	0	0
$778$	77.1440	2.76575
$779$	23.5295	0.843032
$780$	0	0
$781$	37.0449	1.32557
$782$	−12.6966	−0.454029
$783$	0	0
$784$	−12.5265	−0.447377
$785$	0	0
$786$	0	0
$787$	27.1311	0.967118	0.483559	−	0.875312i	$-0.339344\pi$
$787$	27.1311	0.967118	0.483559	+	0.875312i	$0.339344\pi$
$788$	−7.03899	−0.250753
$789$	0	0
$790$	0	0
$791$	18.8988	0.671962
$792$	0	0
$793$	−45.4433	−1.61374
$794$	−53.0508	−1.88270
$795$	0	0
$796$	−20.6856	−0.733182
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	3.20511	0.113389
$800$	0	0
$801$	0	0
$802$	92.3182	3.25987
$803$	27.6065	0.974212
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−14.6425	−0.515123
$809$	13.5324	0.475775	0.237888	−	0.971293i	$-0.423545\pi$
$809$	13.5324	0.475775	0.237888	+	0.971293i	$0.423545\pi$
$810$	0	0
$811$	−40.4492	−1.42036	−0.710182	−	0.704018i	$-0.751387\pi$
$811$	−40.4492	−1.42036	−0.710182	+	0.704018i	$0.751387\pi$
$812$	4.93594	0.173218
$813$	0	0
$814$	−27.9279	−0.978873
$815$	0	0
$816$	0	0
$817$	−45.9758	−1.60849
$818$	67.1930	2.34935
$819$	0	0
$820$	0	0
$821$	21.6016	0.753900	0.376950	−	0.926234i	$-0.376973\pi$
$821$	21.6016	0.753900	0.376950	+	0.926234i	$0.376973\pi$
$822$	0	0
$823$	−25.6995	−0.895830	−0.447915	−	0.894076i	$-0.647833\pi$
$823$	−25.6995	−0.895830	−0.447915	+	0.894076i	$0.647833\pi$
$824$	53.0649	1.84860
$825$	0	0
$826$	−10.2120	−0.355320
$827$	−19.6016	−0.681613	−0.340807	−	0.940133i	$-0.610700\pi$
$827$	−19.6016	−0.681613	−0.340807	+	0.940133i	$0.610700\pi$
$828$	0	0
$829$	−4.22608	−0.146778	−0.0733889	−	0.997303i	$-0.523381\pi$
$829$	−4.22608	−0.146778	−0.0733889	+	0.997303i	$0.523381\pi$
$830$	0	0
$831$	0	0
$832$	65.7736	2.28029
$833$	−3.52360	−0.122085
$834$	0	0
$835$	0	0
$836$	60.6606	2.09799
$837$	0	0
$838$	−8.52654	−0.294545
$839$	2.45839	0.0848729	0.0424365	−	0.999099i	$-0.486488\pi$
$839$	2.45839	0.0848729	0.0424365	+	0.999099i	$0.486488\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−42.1399	−1.45224
$843$	0	0
$844$	58.5685	2.01601
$845$	0	0
$846$	0	0
$847$	−2.65465	−0.0912150
$848$	−31.8778	−1.09469
$849$	0	0
$850$	0	0
$851$	−30.7258	−1.05327
$852$	0	0
$853$	−39.8427	−1.36419	−0.682094	−	0.731264i	$-0.738931\pi$
$853$	−39.8427	−1.36419	−0.682094	+	0.731264i	$0.738931\pi$
$854$	23.6075	0.807831
$855$	0	0
$856$	17.3453	0.592852
$857$	−16.9518	−0.579064	−0.289532	−	0.957168i	$-0.593500\pi$
$857$	−16.9518	−0.579064	−0.289532	+	0.957168i	$0.593500\pi$
$858$	0	0
$859$	38.3902	1.30986	0.654929	−	0.755691i	$-0.272699\pi$
$859$	38.3902	1.30986	0.654929	+	0.755691i	$0.272699\pi$
$860$	0	0
$861$	0	0
$862$	80.8087	2.75235
$863$	49.6995	1.69179	0.845896	−	0.533348i	$-0.179066\pi$
$863$	49.6995	1.69179	0.845896	+	0.533348i	$0.179066\pi$
$864$	0	0
$865$	0	0
$866$	−55.8778	−1.89880
$867$	0	0
$868$	0	0
$869$	−34.9728	−1.18637
$870$	0	0
$871$	−6.76475	−0.229215
$872$	−48.8067	−1.65280
$873$	0	0
$874$	102.630	3.47153
$875$	0	0
$876$	0	0
$877$	40.9168	1.38166	0.690831	−	0.723017i	$-0.257245\pi$
$877$	40.9168	1.38166	0.690831	+	0.723017i	$0.257245\pi$
$878$	−38.6116	−1.30308
$879$	0	0
$880$	0	0
$881$	26.9079	0.906551	0.453276	−	0.891370i	$-0.350255\pi$
$881$	26.9079	0.906551	0.453276	+	0.891370i	$0.350255\pi$
$882$	0	0
$883$	6.69069	0.225160	0.112580	−	0.993643i	$-0.464089\pi$
$883$	6.69069	0.225160	0.112580	+	0.993643i	$0.464089\pi$
$884$	−15.2842	−0.514065
$885$	0	0
$886$	22.4392	0.753860
$887$	−46.0859	−1.54741	−0.773706	−	0.633545i	$-0.781599\pi$
$887$	−46.0859	−1.54741	−0.773706	+	0.633545i	$0.781599\pi$
$888$	0	0
$889$	9.55963	0.320620
$890$	0	0
$891$	0	0
$892$	7.50442	0.251267
$893$	−25.9079	−0.866976
$894$	0	0
$895$	0	0
$896$	−27.5283	−0.919657
$897$	0	0
$898$	−42.0129	−1.40199
$899$	0	0
$900$	0	0
$901$	−8.96691	−0.298731
$902$	31.0449	1.03368
$903$	0	0
$904$	58.5203	1.94636
$905$	0	0
$906$	0	0
$907$	18.7117	0.621310	0.310655	−	0.950523i	$-0.399452\pi$
$907$	18.7117	0.621310	0.310655	+	0.950523i	$0.399452\pi$
$908$	−76.8866	−2.55157
$909$	0	0
$910$	0	0
$911$	−25.2923	−0.837970	−0.418985	−	0.907993i	$-0.637614\pi$
$911$	−25.2923	−0.837970	−0.418985	+	0.907993i	$0.637614\pi$
$912$	0	0
$913$	2.66972	0.0883550
$914$	41.6564	1.37787
$915$	0	0
$916$	53.0147	1.75166
$917$	−18.8988	−0.624092
$918$	0	0
$919$	−45.6396	−1.50551	−0.752756	−	0.658300i	$-0.771276\pi$
$919$	−45.6396	−1.50551	−0.752756	+	0.658300i	$0.771276\pi$
$920$	0	0
$921$	0	0
$922$	0.270324	0.00890265
$923$	75.4492	2.48344
$924$	0	0
$925$	0	0
$926$	64.2249	2.11056
$927$	0	0
$928$	−2.50147	−0.0821149
$929$	21.2733	0.697953	0.348977	−	0.937131i	$-0.386529\pi$
$929$	21.2733	0.697953	0.348977	+	0.937131i	$0.386529\pi$
$930$	0	0
$931$	28.4823	0.933470
$932$	77.2290	2.52972
$933$	0	0
$934$	1.73788	0.0568652
$935$	0	0
$936$	0	0
$937$	46.9489	1.53375	0.766877	−	0.641794i	$-0.221810\pi$
$937$	46.9489	1.53375	0.766877	+	0.641794i	$0.221810\pi$
$938$	3.51424	0.114744
$939$	0	0
$940$	0	0
$941$	18.5986	0.606298	0.303149	−	0.952943i	$-0.401962\pi$
$941$	18.5986	0.606298	0.303149	+	0.952943i	$0.401962\pi$
$942$	0	0
$943$	34.1550	1.11224
$944$	−7.69364	−0.250407
$945$	0	0
$946$	−60.6606	−1.97224
$947$	40.0800	1.30242	0.651212	−	0.758896i	$-0.274261\pi$
$947$	40.0800	1.30242	0.651212	+	0.758896i	$0.274261\pi$
$948$	0	0
$949$	56.2261	1.82518
$950$	0	0
$951$	0	0
$952$	3.66972	0.118936
$953$	−5.03309	−0.163038	−0.0815188	−	0.996672i	$-0.525977\pi$
$953$	−5.03309	−0.163038	−0.0815188	+	0.996672i	$0.525977\pi$
$954$	0	0
$955$	0	0
$956$	−12.3064	−0.398016
$957$	0	0
$958$	21.8217	0.705029
$959$	20.0360	0.646997
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−56.8807	−1.83391
$963$	0	0
$964$	−32.6715	−1.05228
$965$	0	0
$966$	0	0
$967$	−1.72871	−0.0555916	−0.0277958	−	0.999614i	$-0.508849\pi$
$967$	−1.72871	−0.0555916	−0.0277958	+	0.999614i	$0.508849\pi$
$968$	−8.22018	−0.264207
$969$	0	0
$970$	0	0
$971$	16.1071	0.516903	0.258451	−	0.966024i	$-0.416788\pi$
$971$	16.1071	0.516903	0.258451	+	0.966024i	$0.416788\pi$
$972$	0	0
$973$	−8.54456	−0.273926
$974$	−85.0649	−2.72565
$975$	0	0
$976$	17.7857	0.569307
$977$	−13.2323	−0.423339	−0.211669	−	0.977341i	$-0.567890\pi$
$977$	−13.2323	−0.423339	−0.211669	+	0.977341i	$0.567890\pi$
$978$	0	0
$979$	−6.99083	−0.223428
$980$	0	0
$981$	0	0
$982$	10.6186	0.338854
$983$	3.08913	0.0985278	0.0492639	−	0.998786i	$-0.484312\pi$
$983$	3.08913	0.0985278	0.0492639	+	0.998786i	$0.484312\pi$
$984$	0	0
$985$	0	0
$986$	1.60862	0.0512288
$987$	0	0
$988$	123.547	3.93056
$989$	−66.7376	−2.12213
$990$	0	0
$991$	30.9138	0.982010	0.491005	−	0.871157i	$-0.336630\pi$
$991$	30.9138	0.982010	0.491005	+	0.871157i	$0.336630\pi$
$992$	0	0
$993$	0	0
$994$	−39.1953	−1.24320
$995$	0	0
$996$	0	0
$997$	25.3863	0.803993	0.401996	−	0.915641i	$-0.368316\pi$
$997$	25.3863	0.803993	0.401996	+	0.915641i	$0.368316\pi$
$998$	−96.5313	−3.05564
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bf.1.3		3
3.2	odd	2		2175.2.a.u.1.1		3
5.4	even	2		1305.2.a.q.1.1		3
15.2	even	4		2175.2.c.m.349.1		6
15.8	even	4		2175.2.c.m.349.6		6
15.14	odd	2		435.2.a.i.1.3	✓	3
60.59	even	2		6960.2.a.cl.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.i.1.3	✓	3	15.14	odd	2
1305.2.a.q.1.1		3	5.4	even	2
2175.2.a.u.1.1		3	3.2	odd	2
2175.2.c.m.349.1		6	15.2	even	4
2175.2.c.m.349.6		6	15.8	even	4
6525.2.a.bf.1.3		3	1.1	even	1	trivial
6960.2.a.cl.1.2		3	60.59	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39138 q^{2} +3.71871 q^{4} +1.32733 q^{7} +4.11009 q^{8} -3.00000 q^{11} -6.11009 q^{13} +3.17415 q^{14} +2.39138 q^{16} +0.672673 q^{17} -5.43742 q^{19} -7.17415 q^{22} -7.89286 q^{23} -14.6116 q^{26} +4.93594 q^{28} +1.00000 q^{29} -2.50147 q^{32} +1.60862 q^{34} +3.89286 q^{37} -13.0029 q^{38} -4.32733 q^{41} +8.45544 q^{43} -11.1561 q^{44} -18.8748 q^{46} +4.76475 q^{47} -5.23820 q^{49} -22.7217 q^{52} -13.3303 q^{53} +5.45544 q^{56} +2.39138 q^{58} -3.21724 q^{59} +7.43742 q^{61} -10.7647 q^{64} +1.10714 q^{67} +2.50147 q^{68} -12.3483 q^{71} -9.20217 q^{73} +9.30931 q^{74} -20.2202 q^{76} -3.98198 q^{77} +11.6576 q^{79} -10.3483 q^{82} -0.889908 q^{83} +20.2202 q^{86} -12.3303 q^{88} +2.33028 q^{89} -8.11009 q^{91} -29.3512 q^{92} +11.3943 q^{94} +12.6756 q^{97} -12.5265 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} + 4 q^{7} - 9 q^{11} - 6 q^{13} - 9 q^{14} + q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{22} + q^{23} - 13 q^{26} + 21 q^{28} + 3 q^{29} + 11 q^{32} + 11 q^{34} - 13 q^{37} - 2 q^{38} - 13 q^{41}+ \cdots - 50 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 + 4 * q^7 - 9 * q^11 - 6 * q^13 - 9 * q^14 + q^16 + 2 * q^17 - 4 * q^19 - 3 * q^22 + q^23 - 13 * q^26 + 21 * q^28 + 3 * q^29 + 11 * q^32 + 11 * q^34 - 13 * q^37 - 2 * q^38 - 13 * q^41 + 13 * q^43 - 15 * q^44 - 32 * q^46 + 2 * q^47 + 9 * q^49 - 25 * q^52 - 3 * q^53 + 4 * q^56 + q^58 - 22 * q^59 + 10 * q^61 - 20 * q^64 + 28 * q^67 - 11 * q^68 - 3 * q^73 + 28 * q^74 - 36 * q^76 - 12 * q^77 - 2 * q^79 + 6 * q^82 - 15 * q^83 + 36 * q^86 - 30 * q^89 - 12 * q^91 - 14 * q^92 - 9 * q^94 + q^97 - 50 * q^98

Introduction

L-functions

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