LMFDB - Embedded newform 6525.2.a.bu.1.7

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 = [7,-2,0,10,0,0,-1,-3,0,0,-4] 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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 $ x^7 - 2x^6 - 10x^5 + 19x^4 + 24x^3 - 44x^2 - 3x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.47806$ 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)

$f(q)$	$=$	$q+2.47806 q^{2} +4.14079 q^{4} -2.83516 q^{7} +5.30500 q^{8} -6.09835 q^{11} +3.00195 q^{13} -7.02571 q^{14} +4.86455 q^{16} -2.74182 q^{17} +8.03698 q^{19} -15.1121 q^{22} -7.56737 q^{23} +7.43901 q^{26} -11.7398 q^{28} +1.00000 q^{29} -5.32278 q^{31} +1.44464 q^{32} -6.79439 q^{34} -6.28158 q^{37} +19.9161 q^{38} +9.04088 q^{41} -9.06154 q^{43} -25.2520 q^{44} -18.7524 q^{46} -0.0839748 q^{47} +1.03815 q^{49} +12.4304 q^{52} +3.82861 q^{53} -15.0405 q^{56} +2.47806 q^{58} -11.9122 q^{59} -0.275669 q^{61} -13.1902 q^{62} -6.14919 q^{64} -7.36750 q^{67} -11.3533 q^{68} -15.2213 q^{71} +8.77173 q^{73} -15.5661 q^{74} +33.2794 q^{76} +17.2898 q^{77} +1.64999 q^{79} +22.4039 q^{82} -1.38875 q^{83} -22.4551 q^{86} -32.3518 q^{88} +1.05825 q^{89} -8.51101 q^{91} -31.3349 q^{92} -0.208095 q^{94} +9.18751 q^{97} +2.57260 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 10 q^{4} - q^{7} - 3 q^{8} - 4 q^{11} - q^{13} - 15 q^{14} + 12 q^{16} - 8 q^{17} + 15 q^{19} + 3 q^{22} - 14 q^{23} - 6 q^{26} - 24 q^{28} + 7 q^{29} + 5 q^{31} - 18 q^{32} + 7 q^{34}+ \cdots + 59 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 + 10 * q^4 - q^7 - 3 * q^8 - 4 * q^11 - q^13 - 15 * q^14 + 12 * q^16 - 8 * q^17 + 15 * q^19 + 3 * q^22 - 14 * q^23 - 6 * q^26 - 24 * q^28 + 7 * q^29 + 5 * q^31 - 18 * q^32 + 7 * q^34 - 6 * q^37 + 18 * q^38 - 22 * q^41 - 19 * q^43 - 15 * q^44 - 4 * q^46 - 22 * q^47 + 12 * q^49 + 11 * q^52 - 10 * q^53 - 14 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 - 40 * q^62 + 5 * q^64 - 13 * q^67 + 7 * q^68 - 26 * q^71 - 24 * q^73 + 10 * q^74 + 46 * q^76 - 4 * q^77 + 14 * q^79 + 16 * q^82 - 10 * q^83 - 44 * q^86 - 66 * q^88 - 14 * q^89 + 13 * q^91 - 58 * q^92 - 3 * q^94 - 31 * q^97 + 59 * 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 $

$2$	2.47806	1.75225	0.876127	−	0.482080i	$-0.160119\pi$
$2$	2.47806	1.75225	0.876127	+	0.482080i	$0.160119\pi$
$3$	0	0
$3$	0	0
$4$	4.14079	2.07039
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.83516	−1.07159	−0.535795	−	0.844348i	$-0.679988\pi$
$7$	−2.83516	−1.07159	−0.535795	+	0.844348i	$0.679988\pi$
$8$	5.30500	1.87560
$9$	0	0
$10$	0	0
$11$	−6.09835	−1.83872	−0.919361	−	0.393416i	$-0.871293\pi$
$11$	−6.09835	−1.83872	−0.919361	+	0.393416i	$0.871293\pi$
$12$	0	0
$13$	3.00195	0.832591	0.416295	−	0.909229i	$-0.363328\pi$
$13$	3.00195	0.832591	0.416295	+	0.909229i	$0.363328\pi$
$14$	−7.02571	−1.87770
$15$	0	0
$16$	4.86455	1.21614
$17$	−2.74182	−0.664988	−0.332494	−	0.943105i	$-0.607890\pi$
$17$	−2.74182	−0.664988	−0.332494	+	0.943105i	$0.607890\pi$
$18$	0	0
$19$	8.03698	1.84381	0.921905	−	0.387415i	$-0.126632\pi$
$19$	8.03698	1.84381	0.921905	+	0.387415i	$0.126632\pi$
$20$	0	0
$21$	0	0
$22$	−15.1121	−3.22191
$23$	−7.56737	−1.57791	−0.788953	−	0.614453i	$-0.789377\pi$
$23$	−7.56737	−1.57791	−0.788953	+	0.614453i	$0.789377\pi$
$24$	0	0
$25$	0	0
$26$	7.43901	1.45891
$27$	0	0
$28$	−11.7398	−2.21862
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.32278	−0.956000	−0.478000	−	0.878360i	$-0.658638\pi$
$31$	−5.32278	−0.956000	−0.478000	+	0.878360i	$0.658638\pi$
$32$	1.44464	0.255379
$33$	0	0
$34$	−6.79439	−1.16523
$35$	0	0
$36$	0	0
$37$	−6.28158	−1.03268	−0.516342	−	0.856382i	$-0.672707\pi$
$37$	−6.28158	−1.03268	−0.516342	+	0.856382i	$0.672707\pi$
$38$	19.9161	3.23082
$39$	0	0
$40$	0	0
$41$	9.04088	1.41195	0.705974	−	0.708238i	$-0.250509\pi$
$41$	9.04088	1.41195	0.705974	+	0.708238i	$0.250509\pi$
$42$	0	0
$43$	−9.06154	−1.38187	−0.690936	−	0.722916i	$-0.742801\pi$
$43$	−9.06154	−1.38187	−0.690936	+	0.722916i	$0.742801\pi$
$44$	−25.2520	−3.80688
$45$	0	0
$46$	−18.7524	−2.76489
$47$	−0.0839748	−0.0122490	−0.00612449	−	0.999981i	$-0.501949\pi$
$47$	−0.0839748	−0.0122490	−0.00612449	+	0.999981i	$0.501949\pi$
$48$	0	0
$49$	1.03815	0.148307
$50$	0	0
$51$	0	0
$52$	12.4304	1.72379
$53$	3.82861	0.525900	0.262950	−	0.964809i	$-0.415305\pi$
$53$	3.82861	0.525900	0.262950	+	0.964809i	$0.415305\pi$
$54$	0	0
$55$	0	0
$56$	−15.0405	−2.00988
$57$	0	0
$58$	2.47806	0.325385
$59$	−11.9122	−1.55084	−0.775421	−	0.631444i	$-0.782462\pi$
$59$	−11.9122	−1.55084	−0.775421	+	0.631444i	$0.782462\pi$
$60$	0	0
$61$	−0.275669	−0.0352958	−0.0176479	−	0.999844i	$-0.505618\pi$
$61$	−0.275669	−0.0352958	−0.0176479	+	0.999844i	$0.505618\pi$
$62$	−13.1902	−1.67515
$63$	0	0
$64$	−6.14919	−0.768648
$65$	0	0
$66$	0	0
$67$	−7.36750	−0.900083	−0.450042	−	0.893008i	$-0.648591\pi$
$67$	−7.36750	−0.900083	−0.450042	+	0.893008i	$0.648591\pi$
$68$	−11.3533	−1.37679
$69$	0	0
$70$	0	0
$71$	−15.2213	−1.80643	−0.903215	−	0.429188i	$-0.858800\pi$
$71$	−15.2213	−1.80643	−0.903215	+	0.429188i	$0.858800\pi$
$72$	0	0
$73$	8.77173	1.02665	0.513327	−	0.858193i	$-0.328413\pi$
$73$	8.77173	1.02665	0.513327	+	0.858193i	$0.328413\pi$
$74$	−15.5661	−1.80953
$75$	0	0
$76$	33.2794	3.81741
$77$	17.2898	1.97036
$78$	0	0
$79$	1.64999	0.185638	0.0928191	−	0.995683i	$-0.470412\pi$
$79$	1.64999	0.185638	0.0928191	+	0.995683i	$0.470412\pi$
$80$	0	0
$81$	0	0
$82$	22.4039	2.47409
$83$	−1.38875	−0.152435	−0.0762176	−	0.997091i	$-0.524284\pi$
$83$	−1.38875	−0.152435	−0.0762176	+	0.997091i	$0.524284\pi$
$84$	0	0
$85$	0	0
$86$	−22.4551	−2.42139
$87$	0	0
$88$	−32.3518	−3.44871
$89$	1.05825	0.112174	0.0560872	−	0.998426i	$-0.482138\pi$
$89$	1.05825	0.112174	0.0560872	+	0.998426i	$0.482138\pi$
$90$	0	0
$91$	−8.51101	−0.892196
$92$	−31.3349	−3.26689
$93$	0	0
$94$	−0.208095	−0.0214633
$95$	0	0
$96$	0	0
$97$	9.18751	0.932850	0.466425	−	0.884561i	$-0.345542\pi$
$97$	9.18751	0.932850	0.466425	+	0.884561i	$0.345542\pi$
$98$	2.57260	0.259872
$99$	0	0
$100$	0	0
$101$	4.02738	0.400739	0.200369	−	0.979720i	$-0.435786\pi$
$101$	4.02738	0.400739	0.200369	+	0.979720i	$0.435786\pi$
$102$	0	0
$103$	−5.12350	−0.504833	−0.252417	−	0.967619i	$-0.581225\pi$
$103$	−5.12350	−0.504833	−0.252417	+	0.967619i	$0.581225\pi$
$104$	15.9253	1.56161
$105$	0	0
$106$	9.48753	0.921511
$107$	−11.6381	−1.12510	−0.562549	−	0.826764i	$-0.690179\pi$
$107$	−11.6381	−1.12510	−0.562549	+	0.826764i	$0.690179\pi$
$108$	0	0
$109$	−13.7720	−1.31912	−0.659561	−	0.751651i	$-0.729258\pi$
$109$	−13.7720	−1.31912	−0.659561	+	0.751651i	$0.729258\pi$
$110$	0	0
$111$	0	0
$112$	−13.7918	−1.30320
$113$	−16.4395	−1.54650	−0.773250	−	0.634101i	$-0.781370\pi$
$113$	−16.4395	−1.54650	−0.773250	+	0.634101i	$0.781370\pi$
$114$	0	0
$115$	0	0
$116$	4.14079	0.384462
$117$	0	0
$118$	−29.5193	−2.71747
$119$	7.77350	0.712596
$120$	0	0
$121$	26.1899	2.38090
$122$	−0.683125	−0.0618473
$123$	0	0
$124$	−22.0405	−1.97930
$125$	0	0
$126$	0	0
$127$	7.47638	0.663422	0.331711	−	0.943381i	$-0.392374\pi$
$127$	7.47638	0.663422	0.331711	+	0.943381i	$0.392374\pi$
$128$	−18.1273	−1.60225
$129$	0	0
$130$	0	0
$131$	7.63547	0.667114	0.333557	−	0.942730i	$-0.391751\pi$
$131$	7.63547	0.667114	0.333557	+	0.942730i	$0.391751\pi$
$132$	0	0
$133$	−22.7862	−1.97581
$134$	−18.2571	−1.57717
$135$	0	0
$136$	−14.5454	−1.24725
$137$	8.83004	0.754401	0.377201	−	0.926132i	$-0.376887\pi$
$137$	8.83004	0.754401	0.377201	+	0.926132i	$0.376887\pi$
$138$	0	0
$139$	1.29281	0.109655	0.0548274	−	0.998496i	$-0.482539\pi$
$139$	1.29281	0.109655	0.0548274	+	0.998496i	$0.482539\pi$
$140$	0	0
$141$	0	0
$142$	−37.7192	−3.16533
$143$	−18.3069	−1.53090
$144$	0	0
$145$	0	0
$146$	21.7369	1.79896
$147$	0	0
$148$	−26.0107	−2.13806
$149$	10.9561	0.897561	0.448780	−	0.893642i	$-0.351859\pi$
$149$	10.9561	0.897561	0.448780	+	0.893642i	$0.351859\pi$
$150$	0	0
$151$	−15.0875	−1.22781	−0.613903	−	0.789382i	$-0.710401\pi$
$151$	−15.0875	−1.22781	−0.613903	+	0.789382i	$0.710401\pi$
$152$	42.6362	3.45826
$153$	0	0
$154$	42.8452	3.45257
$155$	0	0
$156$	0	0
$157$	17.1333	1.36739	0.683693	−	0.729770i	$-0.260373\pi$
$157$	17.1333	1.36739	0.683693	+	0.729770i	$0.260373\pi$
$158$	4.08877	0.325285
$159$	0	0
$160$	0	0
$161$	21.4547	1.69087
$162$	0	0
$163$	3.12031	0.244402	0.122201	−	0.992505i	$-0.461005\pi$
$163$	3.12031	0.244402	0.122201	+	0.992505i	$0.461005\pi$
$164$	37.4364	2.92329
$165$	0	0
$166$	−3.44141	−0.267105
$167$	7.92935	0.613591	0.306796	−	0.951775i	$-0.400743\pi$
$167$	7.92935	0.613591	0.306796	+	0.951775i	$0.400743\pi$
$168$	0	0
$169$	−3.98831	−0.306793
$170$	0	0
$171$	0	0
$172$	−37.5219	−2.86102
$173$	−12.5868	−0.956956	−0.478478	−	0.878100i	$-0.658811\pi$
$173$	−12.5868	−0.956956	−0.478478	+	0.878100i	$0.658811\pi$
$174$	0	0
$175$	0	0
$176$	−29.6657	−2.23614
$177$	0	0
$178$	2.62241	0.196558
$179$	−12.3291	−0.921520	−0.460760	−	0.887525i	$-0.652423\pi$
$179$	−12.3291	−0.921520	−0.460760	+	0.887525i	$0.652423\pi$
$180$	0	0
$181$	18.2467	1.35626	0.678132	−	0.734940i	$-0.262790\pi$
$181$	18.2467	1.35626	0.678132	+	0.734940i	$0.262790\pi$
$182$	−21.0908	−1.56335
$183$	0	0
$184$	−40.1449	−2.95952
$185$	0	0
$186$	0	0
$187$	16.7206	1.22273
$188$	−0.347722	−0.0253602
$189$	0	0
$190$	0	0
$191$	−1.43334	−0.103713	−0.0518564	−	0.998655i	$-0.516514\pi$
$191$	−1.43334	−0.103713	−0.0518564	+	0.998655i	$0.516514\pi$
$192$	0	0
$193$	−1.01402	−0.0729910	−0.0364955	−	0.999334i	$-0.511619\pi$
$193$	−1.01402	−0.0729910	−0.0364955	+	0.999334i	$0.511619\pi$
$194$	22.7672	1.63459
$195$	0	0
$196$	4.29876	0.307054
$197$	−2.30413	−0.164162	−0.0820812	−	0.996626i	$-0.526157\pi$
$197$	−2.30413	−0.164162	−0.0820812	+	0.996626i	$0.526157\pi$
$198$	0	0
$199$	6.61638	0.469023	0.234511	−	0.972113i	$-0.424651\pi$
$199$	6.61638	0.469023	0.234511	+	0.972113i	$0.424651\pi$
$200$	0	0
$201$	0	0
$202$	9.98008	0.702196
$203$	−2.83516	−0.198989
$204$	0	0
$205$	0	0
$206$	−12.6963	−0.884596
$207$	0	0
$208$	14.6031	1.01254
$209$	−49.0123	−3.39025
$210$	0	0
$211$	15.7936	1.08728	0.543639	−	0.839319i	$-0.317046\pi$
$211$	15.7936	1.08728	0.543639	+	0.839319i	$0.317046\pi$
$212$	15.8535	1.08882
$213$	0	0
$214$	−28.8399	−1.97146
$215$	0	0
$216$	0	0
$217$	15.0910	1.02444
$218$	−34.1280	−2.31144
$219$	0	0
$220$	0	0
$221$	−8.23079	−0.553663
$222$	0	0
$223$	7.24477	0.485146	0.242573	−	0.970133i	$-0.422009\pi$
$223$	7.24477	0.485146	0.242573	+	0.970133i	$0.422009\pi$
$224$	−4.09579	−0.273662
$225$	0	0
$226$	−40.7381	−2.70986
$227$	−4.03611	−0.267886	−0.133943	−	0.990989i	$-0.542764\pi$
$227$	−4.03611	−0.267886	−0.133943	+	0.990989i	$0.542764\pi$
$228$	0	0
$229$	21.8237	1.44215	0.721076	−	0.692856i	$-0.243648\pi$
$229$	21.8237	1.44215	0.721076	+	0.692856i	$0.243648\pi$
$230$	0	0
$231$	0	0
$232$	5.30500	0.348291
$233$	−19.0324	−1.24685	−0.623425	−	0.781883i	$-0.714260\pi$
$233$	−19.0324	−1.24685	−0.623425	+	0.781883i	$0.714260\pi$
$234$	0	0
$235$	0	0
$236$	−49.3261	−3.21085
$237$	0	0
$238$	19.2632	1.24865
$239$	9.13075	0.590619	0.295309	−	0.955402i	$-0.404577\pi$
$239$	9.13075	0.590619	0.295309	+	0.955402i	$0.404577\pi$
$240$	0	0
$241$	−15.1578	−0.976401	−0.488201	−	0.872731i	$-0.662347\pi$
$241$	−15.1578	−0.976401	−0.488201	+	0.872731i	$0.662347\pi$
$242$	64.9001	4.17194
$243$	0	0
$244$	−1.14149	−0.0730763
$245$	0	0
$246$	0	0
$247$	24.1266	1.53514
$248$	−28.2374	−1.79307
$249$	0	0
$250$	0	0
$251$	19.1722	1.21014	0.605070	−	0.796172i	$-0.293145\pi$
$251$	19.1722	1.21014	0.605070	+	0.796172i	$0.293145\pi$
$252$	0	0
$253$	46.1485	2.90133
$254$	18.5269	1.16248
$255$	0	0
$256$	−32.6223	−2.03889
$257$	−0.186589	−0.0116391	−0.00581954	−	0.999983i	$-0.501852\pi$
$257$	−0.186589	−0.0116391	−0.00581954	+	0.999983i	$0.501852\pi$
$258$	0	0
$259$	17.8093	1.10662
$260$	0	0
$261$	0	0
$262$	18.9212	1.16895
$263$	18.6625	1.15078	0.575391	−	0.817879i	$-0.304850\pi$
$263$	18.6625	1.15078	0.575391	+	0.817879i	$0.304850\pi$
$264$	0	0
$265$	0	0
$266$	−56.4655	−3.46212
$267$	0	0
$268$	−30.5072	−1.86353
$269$	6.55984	0.399960	0.199980	−	0.979800i	$-0.435912\pi$
$269$	6.55984	0.399960	0.199980	+	0.979800i	$0.435912\pi$
$270$	0	0
$271$	−6.03232	−0.366437	−0.183219	−	0.983072i	$-0.558652\pi$
$271$	−6.03232	−0.366437	−0.183219	+	0.983072i	$0.558652\pi$
$272$	−13.3377	−0.808717
$273$	0	0
$274$	21.8814	1.32190
$275$	0	0
$276$	0	0
$277$	−4.24630	−0.255136	−0.127568	−	0.991830i	$-0.540717\pi$
$277$	−4.24630	−0.255136	−0.127568	+	0.991830i	$0.540717\pi$
$278$	3.20367	0.192143
$279$	0	0
$280$	0	0
$281$	17.2019	1.02618	0.513091	−	0.858334i	$-0.328500\pi$
$281$	17.2019	1.02618	0.513091	+	0.858334i	$0.328500\pi$
$282$	0	0
$283$	−11.1095	−0.660393	−0.330196	−	0.943912i	$-0.607115\pi$
$283$	−11.1095	−0.660393	−0.330196	+	0.943912i	$0.607115\pi$
$284$	−63.0280	−3.74002
$285$	0	0
$286$	−45.3657	−2.68253
$287$	−25.6324	−1.51303
$288$	0	0
$289$	−9.48244	−0.557790
$290$	0	0
$291$	0	0
$292$	36.3219	2.12558
$293$	−31.4210	−1.83564	−0.917818	−	0.397000i	$-0.870051\pi$
$293$	−31.4210	−1.83564	−0.917818	+	0.397000i	$0.870051\pi$
$294$	0	0
$295$	0	0
$296$	−33.3238	−1.93691
$297$	0	0
$298$	27.1499	1.57275
$299$	−22.7169	−1.31375
$300$	0	0
$301$	25.6910	1.48080
$302$	−37.3878	−2.15143
$303$	0	0
$304$	39.0963	2.24233
$305$	0	0
$306$	0	0
$307$	−16.7120	−0.953804	−0.476902	−	0.878957i	$-0.658240\pi$
$307$	−16.7120	−0.953804	−0.476902	+	0.878957i	$0.658240\pi$
$308$	71.5935	4.07942
$309$	0	0
$310$	0	0
$311$	13.1232	0.744147	0.372073	−	0.928203i	$-0.378647\pi$
$311$	13.1232	0.744147	0.372073	+	0.928203i	$0.378647\pi$
$312$	0	0
$313$	−21.7294	−1.22822	−0.614110	−	0.789221i	$-0.710485\pi$
$313$	−21.7294	−1.22822	−0.614110	+	0.789221i	$0.710485\pi$
$314$	42.4574	2.39601
$315$	0	0
$316$	6.83225	0.384344
$317$	17.8775	1.00410	0.502049	−	0.864839i	$-0.332580\pi$
$317$	17.8775	1.00410	0.502049	+	0.864839i	$0.332580\pi$
$318$	0	0
$319$	−6.09835	−0.341442
$320$	0	0
$321$	0	0
$322$	53.1661	2.96283
$323$	−22.0359	−1.22611
$324$	0	0
$325$	0	0
$326$	7.73233	0.428254
$327$	0	0
$328$	47.9619	2.64825
$329$	0.238082	0.0131259
$330$	0	0
$331$	1.52663	0.0839111	0.0419555	−	0.999119i	$-0.486641\pi$
$331$	1.52663	0.0839111	0.0419555	+	0.999119i	$0.486641\pi$
$332$	−5.75052	−0.315601
$333$	0	0
$334$	19.6494	1.07517
$335$	0	0
$336$	0	0
$337$	−12.7719	−0.695732	−0.347866	−	0.937544i	$-0.613094\pi$
$337$	−12.7719	−0.695732	−0.347866	+	0.937544i	$0.613094\pi$
$338$	−9.88327	−0.537579
$339$	0	0
$340$	0	0
$341$	32.4602	1.75782
$342$	0	0
$343$	16.9028	0.912666
$344$	−48.0715	−2.59184
$345$	0	0
$346$	−31.1908	−1.67683
$347$	17.1670	0.921571	0.460785	−	0.887512i	$-0.347568\pi$
$347$	17.1670	0.921571	0.460785	+	0.887512i	$0.347568\pi$
$348$	0	0
$349$	−25.0051	−1.33849	−0.669247	−	0.743040i	$-0.733383\pi$
$349$	−25.0051	−1.33849	−0.669247	+	0.743040i	$0.733383\pi$
$350$	0	0
$351$	0	0
$352$	−8.80993	−0.469571
$353$	6.33813	0.337345	0.168672	−	0.985672i	$-0.446052\pi$
$353$	6.33813	0.337345	0.168672	+	0.985672i	$0.446052\pi$
$354$	0	0
$355$	0	0
$356$	4.38200	0.232245
$357$	0	0
$358$	−30.5523	−1.61474
$359$	36.2964	1.91565	0.957827	−	0.287347i	$-0.0927731\pi$
$359$	36.2964	1.91565	0.957827	+	0.287347i	$0.0927731\pi$
$360$	0	0
$361$	45.5931	2.39964
$362$	45.2163	2.37652
$363$	0	0
$364$	−35.2423	−1.84720
$365$	0	0
$366$	0	0
$367$	−0.515494	−0.0269086	−0.0134543	−	0.999909i	$-0.504283\pi$
$367$	−0.515494	−0.0269086	−0.0134543	+	0.999909i	$0.504283\pi$
$368$	−36.8118	−1.91895
$369$	0	0
$370$	0	0
$371$	−10.8547	−0.563550
$372$	0	0
$373$	25.6201	1.32656	0.663279	−	0.748372i	$-0.269164\pi$
$373$	25.6201	1.32656	0.663279	+	0.748372i	$0.269164\pi$
$374$	41.4346	2.14253
$375$	0	0
$376$	−0.445487	−0.0229742
$377$	3.00195	0.154608
$378$	0	0
$379$	31.7258	1.62965	0.814824	−	0.579709i	$-0.196834\pi$
$379$	31.7258	1.62965	0.814824	+	0.579709i	$0.196834\pi$
$380$	0	0
$381$	0	0
$382$	−3.55190	−0.181731
$383$	20.8586	1.06582	0.532912	−	0.846170i	$-0.321098\pi$
$383$	20.8586	1.06582	0.532912	+	0.846170i	$0.321098\pi$
$384$	0	0
$385$	0	0
$386$	−2.51281	−0.127899
$387$	0	0
$388$	38.0435	1.93137
$389$	−36.7636	−1.86399	−0.931993	−	0.362476i	$-0.881931\pi$
$389$	−36.7636	−1.86399	−0.931993	+	0.362476i	$0.881931\pi$
$390$	0	0
$391$	20.7484	1.04929
$392$	5.50739	0.278165
$393$	0	0
$394$	−5.70977	−0.287654
$395$	0	0
$396$	0	0
$397$	−8.62743	−0.432999	−0.216499	−	0.976283i	$-0.569464\pi$
$397$	−8.62743	−0.432999	−0.216499	+	0.976283i	$0.569464\pi$
$398$	16.3958	0.821847
$399$	0	0
$400$	0	0
$401$	−17.9078	−0.894274	−0.447137	−	0.894465i	$-0.647556\pi$
$401$	−17.9078	−0.894274	−0.447137	+	0.894465i	$0.647556\pi$
$402$	0	0
$403$	−15.9787	−0.795956
$404$	16.6765	0.829687
$405$	0	0
$406$	−7.02571	−0.348680
$407$	38.3072	1.89882
$408$	0	0
$409$	32.6668	1.61527	0.807633	−	0.589685i	$-0.200748\pi$
$409$	32.6668	1.61527	0.807633	+	0.589685i	$0.200748\pi$
$410$	0	0
$411$	0	0
$412$	−21.2153	−1.04520
$413$	33.7732	1.66187
$414$	0	0
$415$	0	0
$416$	4.33674	0.212626
$417$	0	0
$418$	−121.456	−5.94059
$419$	23.5677	1.15136	0.575679	−	0.817676i	$-0.304738\pi$
$419$	23.5677	1.15136	0.575679	+	0.817676i	$0.304738\pi$
$420$	0	0
$421$	21.7941	1.06218	0.531089	−	0.847316i	$-0.321783\pi$
$421$	21.7941	1.06218	0.531089	+	0.847316i	$0.321783\pi$
$422$	39.1376	1.90519
$423$	0	0
$424$	20.3108	0.986379
$425$	0	0
$426$	0	0
$427$	0.781567	0.0378227
$428$	−48.1909	−2.32940
$429$	0	0
$430$	0	0
$431$	24.6107	1.18545	0.592727	−	0.805403i	$-0.298051\pi$
$431$	24.6107	1.18545	0.592727	+	0.805403i	$0.298051\pi$
$432$	0	0
$433$	16.7103	0.803045	0.401523	−	0.915849i	$-0.368481\pi$
$433$	16.7103	0.803045	0.401523	+	0.915849i	$0.368481\pi$
$434$	37.3963	1.79508
$435$	0	0
$436$	−57.0271	−2.73110
$437$	−60.8189	−2.90936
$438$	0	0
$439$	−30.8206	−1.47099	−0.735494	−	0.677532i	$-0.763050\pi$
$439$	−30.8206	−1.47099	−0.735494	+	0.677532i	$0.763050\pi$
$440$	0	0
$441$	0	0
$442$	−20.3964	−0.970158
$443$	−25.0299	−1.18921	−0.594604	−	0.804019i	$-0.702691\pi$
$443$	−25.0299	−1.18921	−0.594604	+	0.804019i	$0.702691\pi$
$444$	0	0
$445$	0	0
$446$	17.9530	0.850098
$447$	0	0
$448$	17.4339	0.823676
$449$	2.48864	0.117446	0.0587230	−	0.998274i	$-0.481297\pi$
$449$	2.48864	0.117446	0.0587230	+	0.998274i	$0.481297\pi$
$450$	0	0
$451$	−55.1344	−2.59618
$452$	−68.0726	−3.20186
$453$	0	0
$454$	−10.0017	−0.469405
$455$	0	0
$456$	0	0
$457$	−21.5308	−1.00717	−0.503585	−	0.863946i	$-0.667986\pi$
$457$	−21.5308	−1.00717	−0.503585	+	0.863946i	$0.667986\pi$
$458$	54.0805	2.52702
$459$	0	0
$460$	0	0
$461$	14.1345	0.658311	0.329156	−	0.944276i	$-0.393236\pi$
$461$	14.1345	0.658311	0.329156	+	0.944276i	$0.393236\pi$
$462$	0	0
$463$	−11.7525	−0.546184	−0.273092	−	0.961988i	$-0.588046\pi$
$463$	−11.7525	−0.546184	−0.273092	+	0.961988i	$0.588046\pi$
$464$	4.86455	0.225831
$465$	0	0
$466$	−47.1633	−2.18480
$467$	−28.0936	−1.30002	−0.650009	−	0.759926i	$-0.725235\pi$
$467$	−28.0936	−1.30002	−0.650009	+	0.759926i	$0.725235\pi$
$468$	0	0
$469$	20.8881	0.964521
$470$	0	0
$471$	0	0
$472$	−63.1945	−2.90876
$473$	55.2605	2.54088
$474$	0	0
$475$	0	0
$476$	32.1884	1.47535
$477$	0	0
$478$	22.6266	1.03491
$479$	−5.25749	−0.240221	−0.120110	−	0.992761i	$-0.538325\pi$
$479$	−5.25749	−0.240221	−0.120110	+	0.992761i	$0.538325\pi$
$480$	0	0
$481$	−18.8570	−0.859803
$482$	−37.5620	−1.71090
$483$	0	0
$484$	108.447	4.92939
$485$	0	0
$486$	0	0
$487$	−4.48076	−0.203043	−0.101521	−	0.994833i	$-0.532371\pi$
$487$	−4.48076	−0.203043	−0.101521	+	0.994833i	$0.532371\pi$
$488$	−1.46243	−0.0662009
$489$	0	0
$490$	0	0
$491$	10.5211	0.474810	0.237405	−	0.971411i	$-0.423703\pi$
$491$	10.5211	0.474810	0.237405	+	0.971411i	$0.423703\pi$
$492$	0	0
$493$	−2.74182	−0.123485
$494$	59.7872	2.68995
$495$	0	0
$496$	−25.8929	−1.16263
$497$	43.1547	1.93575
$498$	0	0
$499$	29.4564	1.31865	0.659325	−	0.751858i	$-0.270842\pi$
$499$	29.4564	1.31865	0.659325	+	0.751858i	$0.270842\pi$
$500$	0	0
$501$	0	0
$502$	47.5100	2.12047
$503$	−17.5927	−0.784419	−0.392210	−	0.919876i	$-0.628289\pi$
$503$	−17.5927	−0.784419	−0.392210	+	0.919876i	$0.628289\pi$
$504$	0	0
$505$	0	0
$506$	114.359	5.08387
$507$	0	0
$508$	30.9581	1.37354
$509$	−2.87587	−0.127471	−0.0637354	−	0.997967i	$-0.520301\pi$
$509$	−2.87587	−0.127471	−0.0637354	+	0.997967i	$0.520301\pi$
$510$	0	0
$511$	−24.8693	−1.10015
$512$	−44.5854	−1.97041
$513$	0	0
$514$	−0.462378	−0.0203946
$515$	0	0
$516$	0	0
$517$	0.512108	0.0225225
$518$	44.1325	1.93907
$519$	0	0
$520$	0	0
$521$	−37.9466	−1.66247	−0.831235	−	0.555921i	$-0.812366\pi$
$521$	−37.9466	−1.66247	−0.831235	+	0.555921i	$0.812366\pi$
$522$	0	0
$523$	23.0805	1.00924	0.504619	−	0.863342i	$-0.331633\pi$
$523$	23.0805	1.00924	0.504619	+	0.863342i	$0.331633\pi$
$524$	31.6169	1.38119
$525$	0	0
$526$	46.2469	2.01646
$527$	14.5941	0.635729
$528$	0	0
$529$	34.2651	1.48979
$530$	0	0
$531$	0	0
$532$	−94.3527	−4.09071
$533$	27.1403	1.17557
$534$	0	0
$535$	0	0
$536$	−39.0846	−1.68820
$537$	0	0
$538$	16.2557	0.700832
$539$	−6.33100	−0.272695
$540$	0	0
$541$	−14.2524	−0.612758	−0.306379	−	0.951910i	$-0.599117\pi$
$541$	−14.2524	−0.612758	−0.306379	+	0.951910i	$0.599117\pi$
$542$	−14.9485	−0.642091
$543$	0	0
$544$	−3.96094	−0.169824
$545$	0	0
$546$	0	0
$547$	−11.6415	−0.497754	−0.248877	−	0.968535i	$-0.580062\pi$
$547$	−11.6415	−0.497754	−0.248877	+	0.968535i	$0.580062\pi$
$548$	36.5633	1.56191
$549$	0	0
$550$	0	0
$551$	8.03698	0.342387
$552$	0	0
$553$	−4.67799	−0.198928
$554$	−10.5226	−0.447062
$555$	0	0
$556$	5.35326	0.227029
$557$	−15.2321	−0.645406	−0.322703	−	0.946500i	$-0.604592\pi$
$557$	−15.2321	−0.645406	−0.322703	+	0.946500i	$0.604592\pi$
$558$	0	0
$559$	−27.2023	−1.15053
$560$	0	0
$561$	0	0
$562$	42.6275	1.79813
$563$	−35.7787	−1.50789	−0.753946	−	0.656937i	$-0.771852\pi$
$563$	−35.7787	−1.50789	−0.753946	+	0.656937i	$0.771852\pi$
$564$	0	0
$565$	0	0
$566$	−27.5301	−1.15718
$567$	0	0
$568$	−80.7488	−3.38815
$569$	−3.07401	−0.128869	−0.0644347	−	0.997922i	$-0.520524\pi$
$569$	−3.07401	−0.128869	−0.0644347	+	0.997922i	$0.520524\pi$
$570$	0	0
$571$	0.569487	0.0238323	0.0119161	−	0.999929i	$-0.496207\pi$
$571$	0.569487	0.0238323	0.0119161	+	0.999929i	$0.496207\pi$
$572$	−75.8051	−3.16957
$573$	0	0
$574$	−63.5186	−2.65121
$575$	0	0
$576$	0	0
$577$	−34.4259	−1.43317	−0.716584	−	0.697501i	$-0.754295\pi$
$577$	−34.4259	−1.43317	−0.716584	+	0.697501i	$0.754295\pi$
$578$	−23.4981	−0.977390
$579$	0	0
$580$	0	0
$581$	3.93733	0.163348
$582$	0	0
$583$	−23.3482	−0.966984
$584$	46.5341	1.92559
$585$	0	0
$586$	−77.8633	−3.21650
$587$	21.6502	0.893600	0.446800	−	0.894634i	$-0.352563\pi$
$587$	21.6502	0.893600	0.446800	+	0.894634i	$0.352563\pi$
$588$	0	0
$589$	−42.7791	−1.76268
$590$	0	0
$591$	0	0
$592$	−30.5570	−1.25589
$593$	−13.0532	−0.536031	−0.268015	−	0.963415i	$-0.586368\pi$
$593$	−13.0532	−0.536031	−0.268015	+	0.963415i	$0.586368\pi$
$594$	0	0
$595$	0	0
$596$	45.3670	1.85830
$597$	0	0
$598$	−56.2938	−2.30202
$599$	−2.53099	−0.103413	−0.0517066	−	0.998662i	$-0.516466\pi$
$599$	−2.53099	−0.103413	−0.0517066	+	0.998662i	$0.516466\pi$
$600$	0	0
$601$	28.0697	1.14499	0.572493	−	0.819910i	$-0.305977\pi$
$601$	28.0697	1.14499	0.572493	+	0.819910i	$0.305977\pi$
$602$	63.6638	2.59474
$603$	0	0
$604$	−62.4743	−2.54204
$605$	0	0
$606$	0	0
$607$	12.5431	0.509109	0.254554	−	0.967058i	$-0.418071\pi$
$607$	12.5431	0.509109	0.254554	+	0.967058i	$0.418071\pi$
$608$	11.6106	0.470871
$609$	0	0
$610$	0	0
$611$	−0.252088	−0.0101984
$612$	0	0
$613$	−47.7414	−1.92826	−0.964129	−	0.265435i	$-0.914485\pi$
$613$	−47.7414	−1.92826	−0.964129	+	0.265435i	$0.914485\pi$
$614$	−41.4133	−1.67131
$615$	0	0
$616$	91.7225	3.69561
$617$	−34.9338	−1.40638	−0.703191	−	0.711001i	$-0.748242\pi$
$617$	−34.9338	−1.40638	−0.703191	+	0.711001i	$0.748242\pi$
$618$	0	0
$619$	−21.1586	−0.850437	−0.425218	−	0.905091i	$-0.639803\pi$
$619$	−21.1586	−0.850437	−0.425218	+	0.905091i	$0.639803\pi$
$620$	0	0
$621$	0	0
$622$	32.5200	1.30393
$623$	−3.00032	−0.120205
$624$	0	0
$625$	0	0
$626$	−53.8468	−2.15215
$627$	0	0
$628$	70.9453	2.83103
$629$	17.2229	0.686723
$630$	0	0
$631$	−48.5973	−1.93463	−0.967313	−	0.253585i	$-0.918390\pi$
$631$	−48.5973	−1.93463	−0.967313	+	0.253585i	$0.918390\pi$
$632$	8.75320	0.348183
$633$	0	0
$634$	44.3015	1.75944
$635$	0	0
$636$	0	0
$637$	3.11647	0.123479
$638$	−15.1121	−0.598293
$639$	0	0
$640$	0	0
$641$	−9.13307	−0.360735	−0.180367	−	0.983599i	$-0.557729\pi$
$641$	−9.13307	−0.360735	−0.180367	+	0.983599i	$0.557729\pi$
$642$	0	0
$643$	26.2899	1.03677	0.518386	−	0.855146i	$-0.326533\pi$
$643$	26.2899	1.03677	0.518386	+	0.855146i	$0.326533\pi$
$644$	88.8395	3.50077
$645$	0	0
$646$	−54.6064	−2.14846
$647$	28.7440	1.13004	0.565021	−	0.825077i	$-0.308868\pi$
$647$	28.7440	1.13004	0.565021	+	0.825077i	$0.308868\pi$
$648$	0	0
$649$	72.6450	2.85157
$650$	0	0
$651$	0	0
$652$	12.9206	0.506008
$653$	14.7225	0.576135	0.288068	−	0.957610i	$-0.406987\pi$
$653$	14.7225	0.576135	0.288068	+	0.957610i	$0.406987\pi$
$654$	0	0
$655$	0	0
$656$	43.9798	1.71712
$657$	0	0
$658$	0.589982	0.0229999
$659$	−43.6117	−1.69887	−0.849435	−	0.527693i	$-0.823057\pi$
$659$	−43.6117	−1.69887	−0.849435	+	0.527693i	$0.823057\pi$
$660$	0	0
$661$	31.3884	1.22087	0.610433	−	0.792068i	$-0.290995\pi$
$661$	31.3884	1.22087	0.610433	+	0.792068i	$0.290995\pi$
$662$	3.78308	0.147034
$663$	0	0
$664$	−7.36733	−0.285908
$665$	0	0
$666$	0	0
$667$	−7.56737	−0.293010
$668$	32.8337	1.27038
$669$	0	0
$670$	0	0
$671$	1.68113	0.0648992
$672$	0	0
$673$	−27.3226	−1.05321	−0.526604	−	0.850110i	$-0.676535\pi$
$673$	−27.3226	−1.05321	−0.526604	+	0.850110i	$0.676535\pi$
$674$	−31.6496	−1.21910
$675$	0	0
$676$	−16.5147	−0.635182
$677$	0.598834	0.0230151	0.0115075	−	0.999934i	$-0.496337\pi$
$677$	0.598834	0.0230151	0.0115075	+	0.999934i	$0.496337\pi$
$678$	0	0
$679$	−26.0481	−0.999634
$680$	0	0
$681$	0	0
$682$	80.4383	3.08014
$683$	11.9236	0.456243	0.228122	−	0.973633i	$-0.426742\pi$
$683$	11.9236	0.456243	0.228122	+	0.973633i	$0.426742\pi$
$684$	0	0
$685$	0	0
$686$	41.8862	1.59922
$687$	0	0
$688$	−44.0803	−1.68055
$689$	11.4933	0.437859
$690$	0	0
$691$	34.6437	1.31791	0.658955	−	0.752183i	$-0.270999\pi$
$691$	34.6437	1.31791	0.658955	+	0.752183i	$0.270999\pi$
$692$	−52.1192	−1.98128
$693$	0	0
$694$	42.5408	1.61483
$695$	0	0
$696$	0	0
$697$	−24.7884	−0.938929
$698$	−61.9642	−2.34538
$699$	0	0
$700$	0	0
$701$	16.1632	0.610477	0.305239	−	0.952276i	$-0.401264\pi$
$701$	16.1632	0.610477	0.305239	+	0.952276i	$0.401264\pi$
$702$	0	0
$703$	−50.4849	−1.90408
$704$	37.4999	1.41333
$705$	0	0
$706$	15.7063	0.591113
$707$	−11.4183	−0.429428
$708$	0	0
$709$	−26.8631	−1.00887	−0.504433	−	0.863451i	$-0.668298\pi$
$709$	−26.8631	−1.00887	−0.504433	+	0.863451i	$0.668298\pi$
$710$	0	0
$711$	0	0
$712$	5.61403	0.210395
$713$	40.2795	1.50848
$714$	0	0
$715$	0	0
$716$	−51.0522	−1.90791
$717$	0	0
$718$	89.9448	3.35671
$719$	−6.96465	−0.259738	−0.129869	−	0.991531i	$-0.541456\pi$
$719$	−6.96465	−0.259738	−0.129869	+	0.991531i	$0.541456\pi$
$720$	0	0
$721$	14.5259	0.540975
$722$	112.983	4.20478
$723$	0	0
$724$	75.5556	2.80800
$725$	0	0
$726$	0	0
$727$	−36.7261	−1.36210	−0.681048	−	0.732239i	$-0.738475\pi$
$727$	−36.7261	−1.36210	−0.681048	+	0.732239i	$0.738475\pi$
$728$	−45.1509	−1.67341
$729$	0	0
$730$	0	0
$731$	24.8451	0.918929
$732$	0	0
$733$	−39.3167	−1.45219	−0.726097	−	0.687592i	$-0.758668\pi$
$733$	−39.3167	−1.45219	−0.726097	+	0.687592i	$0.758668\pi$
$734$	−1.27743	−0.0471507
$735$	0	0
$736$	−10.9321	−0.402964
$737$	44.9296	1.65500
$738$	0	0
$739$	−31.4897	−1.15837	−0.579184	−	0.815197i	$-0.696629\pi$
$739$	−31.4897	−1.15837	−0.579184	+	0.815197i	$0.696629\pi$
$740$	0	0
$741$	0	0
$742$	−26.8987	−0.987482
$743$	−30.2195	−1.10864	−0.554322	−	0.832302i	$-0.687022\pi$
$743$	−30.2195	−1.10864	−0.554322	+	0.832302i	$0.687022\pi$
$744$	0	0
$745$	0	0
$746$	63.4882	2.32447
$747$	0	0
$748$	69.2363	2.53153
$749$	32.9959	1.20565
$750$	0	0
$751$	14.3956	0.525304	0.262652	−	0.964891i	$-0.415403\pi$
$751$	14.3956	0.525304	0.262652	+	0.964891i	$0.415403\pi$
$752$	−0.408499	−0.0148964
$753$	0	0
$754$	7.43901	0.270913
$755$	0	0
$756$	0	0
$757$	6.44800	0.234356	0.117178	−	0.993111i	$-0.462615\pi$
$757$	6.44800	0.234356	0.117178	+	0.993111i	$0.462615\pi$
$758$	78.6186	2.85556
$759$	0	0
$760$	0	0
$761$	−36.8289	−1.33505	−0.667523	−	0.744589i	$-0.732646\pi$
$761$	−36.8289	−1.33505	−0.667523	+	0.744589i	$0.732646\pi$
$762$	0	0
$763$	39.0460	1.41356
$764$	−5.93516	−0.214726
$765$	0	0
$766$	51.6889	1.86760
$767$	−35.7599	−1.29122
$768$	0	0
$769$	−13.5699	−0.489344	−0.244672	−	0.969606i	$-0.578680\pi$
$769$	−13.5699	−0.489344	−0.244672	+	0.969606i	$0.578680\pi$
$770$	0	0
$771$	0	0
$772$	−4.19885	−0.151120
$773$	3.84475	0.138286	0.0691429	−	0.997607i	$-0.477974\pi$
$773$	3.84475	0.138286	0.0691429	+	0.997607i	$0.477974\pi$
$774$	0	0
$775$	0	0
$776$	48.7398	1.74966
$777$	0	0
$778$	−91.1023	−3.26618
$779$	72.6614	2.60337
$780$	0	0
$781$	92.8245	3.32152
$782$	51.4157	1.83862
$783$	0	0
$784$	5.05013	0.180362
$785$	0	0
$786$	0	0
$787$	−18.1619	−0.647400	−0.323700	−	0.946160i	$-0.604927\pi$
$787$	−18.1619	−0.647400	−0.323700	+	0.946160i	$0.604927\pi$
$788$	−9.54091	−0.339881
$789$	0	0
$790$	0	0
$791$	46.6087	1.65722
$792$	0	0
$793$	−0.827545	−0.0293870
$794$	−21.3793	−0.758723
$795$	0	0
$796$	27.3970	0.971062
$797$	34.3762	1.21767	0.608834	−	0.793298i	$-0.291638\pi$
$797$	34.3762	1.21767	0.608834	+	0.793298i	$0.291638\pi$
$798$	0	0
$799$	0.230244	0.00814543
$800$	0	0
$801$	0	0
$802$	−44.3767	−1.56700
$803$	−53.4931	−1.88773
$804$	0	0
$805$	0	0
$806$	−39.5962	−1.39472
$807$	0	0
$808$	21.3652	0.751627
$809$	−19.8011	−0.696169	−0.348084	−	0.937463i	$-0.613168\pi$
$809$	−19.8011	−0.696169	−0.348084	+	0.937463i	$0.613168\pi$
$810$	0	0
$811$	6.59998	0.231756	0.115878	−	0.993263i	$-0.463032\pi$
$811$	6.59998	0.231756	0.115878	+	0.993263i	$0.463032\pi$
$812$	−11.7398	−0.411987
$813$	0	0
$814$	94.9277	3.32721
$815$	0	0
$816$	0	0
$817$	−72.8275	−2.54791
$818$	80.9502	2.83036
$819$	0	0
$820$	0	0
$821$	18.0216	0.628959	0.314479	−	0.949264i	$-0.398170\pi$
$821$	18.0216	0.628959	0.314479	+	0.949264i	$0.398170\pi$
$822$	0	0
$823$	12.5622	0.437892	0.218946	−	0.975737i	$-0.429738\pi$
$823$	12.5622	0.437892	0.218946	+	0.975737i	$0.429738\pi$
$824$	−27.1802	−0.946866
$825$	0	0
$826$	83.6920	2.91202
$827$	12.1147	0.421270	0.210635	−	0.977565i	$-0.432447\pi$
$827$	12.1147	0.421270	0.210635	+	0.977565i	$0.432447\pi$
$828$	0	0
$829$	−14.0551	−0.488152	−0.244076	−	0.969756i	$-0.578485\pi$
$829$	−14.0551	−0.488152	−0.244076	+	0.969756i	$0.578485\pi$
$830$	0	0
$831$	0	0
$832$	−18.4595	−0.639969
$833$	−2.84642	−0.0986225
$834$	0	0
$835$	0	0
$836$	−202.950	−7.01916
$837$	0	0
$838$	58.4023	2.01747
$839$	−7.95419	−0.274609	−0.137305	−	0.990529i	$-0.543844\pi$
$839$	−7.95419	−0.274609	−0.137305	+	0.990529i	$0.543844\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	54.0070	1.86121
$843$	0	0
$844$	65.3980	2.25109
$845$	0	0
$846$	0	0
$847$	−74.2525	−2.55135
$848$	18.6245	0.639567
$849$	0	0
$850$	0	0
$851$	47.5350	1.62948
$852$	0	0
$853$	43.0370	1.47356	0.736779	−	0.676133i	$-0.236346\pi$
$853$	43.0370	1.47356	0.736779	+	0.676133i	$0.236346\pi$
$854$	1.93677	0.0662750
$855$	0	0
$856$	−61.7402	−2.11024
$857$	−19.2579	−0.657837	−0.328919	−	0.944358i	$-0.606684\pi$
$857$	−19.2579	−0.657837	−0.328919	+	0.944358i	$0.606684\pi$
$858$	0	0
$859$	23.5213	0.802535	0.401268	−	0.915961i	$-0.368570\pi$
$859$	23.5213	0.802535	0.401268	+	0.915961i	$0.368570\pi$
$860$	0	0
$861$	0	0
$862$	60.9868	2.07722
$863$	−16.8210	−0.572593	−0.286296	−	0.958141i	$-0.592424\pi$
$863$	−16.8210	−0.572593	−0.286296	+	0.958141i	$0.592424\pi$
$864$	0	0
$865$	0	0
$866$	41.4091	1.40714
$867$	0	0
$868$	62.4884	2.12100
$869$	−10.0622	−0.341337
$870$	0	0
$871$	−22.1168	−0.749401
$872$	−73.0607	−2.47415
$873$	0	0
$874$	−150.713	−5.09794
$875$	0	0
$876$	0	0
$877$	−28.6157	−0.966283	−0.483142	−	0.875542i	$-0.660504\pi$
$877$	−28.6157	−0.966283	−0.483142	+	0.875542i	$0.660504\pi$
$878$	−76.3753	−2.57754
$879$	0	0
$880$	0	0
$881$	−11.5327	−0.388546	−0.194273	−	0.980948i	$-0.562235\pi$
$881$	−11.5327	−0.388546	−0.194273	+	0.980948i	$0.562235\pi$
$882$	0	0
$883$	0.259436	0.00873072	0.00436536	−	0.999990i	$-0.498610\pi$
$883$	0.259436	0.00873072	0.00436536	+	0.999990i	$0.498610\pi$
$884$	−34.0820	−1.14630
$885$	0	0
$886$	−62.0257	−2.08379
$887$	25.6339	0.860702	0.430351	−	0.902662i	$-0.358390\pi$
$887$	25.6339	0.860702	0.430351	+	0.902662i	$0.358390\pi$
$888$	0	0
$889$	−21.1968	−0.710917
$890$	0	0
$891$	0	0
$892$	29.9991	1.00444
$893$	−0.674904	−0.0225848
$894$	0	0
$895$	0	0
$896$	51.3940	1.71695
$897$	0	0
$898$	6.16699	0.205795
$899$	−5.32278	−0.177525
$900$	0	0
$901$	−10.4974	−0.349718
$902$	−136.627	−4.54917
$903$	0	0
$904$	−87.2117	−2.90062
$905$	0	0
$906$	0	0
$907$	−38.4925	−1.27812	−0.639060	−	0.769156i	$-0.720677\pi$
$907$	−38.4925	−1.27812	−0.639060	+	0.769156i	$0.720677\pi$
$908$	−16.7127	−0.554630
$909$	0	0
$910$	0	0
$911$	−48.7894	−1.61646	−0.808232	−	0.588864i	$-0.799575\pi$
$911$	−48.7894	−1.61646	−0.808232	+	0.588864i	$0.799575\pi$
$912$	0	0
$913$	8.46909	0.280286
$914$	−53.3547	−1.76482
$915$	0	0
$916$	90.3674	2.98582
$917$	−21.6478	−0.714873
$918$	0	0
$919$	1.29772	0.0428079	0.0214040	−	0.999771i	$-0.493186\pi$
$919$	1.29772	0.0428079	0.0214040	+	0.999771i	$0.493186\pi$
$920$	0	0
$921$	0	0
$922$	35.0263	1.15353
$923$	−45.6934	−1.50402
$924$	0	0
$925$	0	0
$926$	−29.1234	−0.957053
$927$	0	0
$928$	1.44464	0.0474227
$929$	6.26915	0.205684	0.102842	−	0.994698i	$-0.467206\pi$
$929$	6.26915	0.205684	0.102842	+	0.994698i	$0.467206\pi$
$930$	0	0
$931$	8.34359	0.273450
$932$	−78.8089	−2.58147
$933$	0	0
$934$	−69.6178	−2.27796
$935$	0	0
$936$	0	0
$937$	−6.64562	−0.217103	−0.108552	−	0.994091i	$-0.534621\pi$
$937$	−6.64562	−0.217103	−0.108552	+	0.994091i	$0.534621\pi$
$938$	51.7619	1.69009
$939$	0	0
$940$	0	0
$941$	6.21026	0.202449	0.101224	−	0.994864i	$-0.467724\pi$
$941$	6.21026	0.202449	0.101224	+	0.994864i	$0.467724\pi$
$942$	0	0
$943$	−68.4157	−2.22792
$944$	−57.9477	−1.88604
$945$	0	0
$946$	136.939	4.45226
$947$	−13.9468	−0.453209	−0.226604	−	0.973987i	$-0.572762\pi$
$947$	−13.9468	−0.453209	−0.226604	+	0.973987i	$0.572762\pi$
$948$	0	0
$949$	26.3323	0.854782
$950$	0	0
$951$	0	0
$952$	41.2384	1.33655
$953$	17.9504	0.581469	0.290735	−	0.956804i	$-0.406100\pi$
$953$	17.9504	0.581469	0.290735	+	0.956804i	$0.406100\pi$
$954$	0	0
$955$	0	0
$956$	37.8085	1.22281
$957$	0	0
$958$	−13.0284	−0.420928
$959$	−25.0346	−0.808409
$960$	0	0
$961$	−2.66800	−0.0860645
$962$	−46.7287	−1.50659
$963$	0	0
$964$	−62.7653	−2.02154
$965$	0	0
$966$	0	0
$967$	−46.8801	−1.50756	−0.753782	−	0.657125i	$-0.771772\pi$
$967$	−46.8801	−1.50756	−0.753782	+	0.657125i	$0.771772\pi$
$968$	138.937	4.46561
$969$	0	0
$970$	0	0
$971$	42.6028	1.36719	0.683595	−	0.729861i	$-0.260415\pi$
$971$	42.6028	1.36719	0.683595	+	0.729861i	$0.260415\pi$
$972$	0	0
$973$	−3.66533	−0.117505
$974$	−11.1036	−0.355782
$975$	0	0
$976$	−1.34101	−0.0429246
$977$	−14.4907	−0.463599	−0.231800	−	0.972764i	$-0.574461\pi$
$977$	−14.4907	−0.463599	−0.231800	+	0.972764i	$0.574461\pi$
$978$	0	0
$979$	−6.45359	−0.206258
$980$	0	0
$981$	0	0
$982$	26.0719	0.831987
$983$	−47.0175	−1.49963	−0.749813	−	0.661649i	$-0.769857\pi$
$983$	−47.0175	−1.49963	−0.749813	+	0.661649i	$0.769857\pi$
$984$	0	0
$985$	0	0
$986$	−6.79439	−0.216378
$987$	0	0
$988$	99.9032	3.17834
$989$	68.5721	2.18046
$990$	0	0
$991$	30.1662	0.958262	0.479131	−	0.877743i	$-0.340952\pi$
$991$	30.1662	0.958262	0.479131	+	0.877743i	$0.340952\pi$
$992$	−7.68951	−0.244142
$993$	0	0
$994$	106.940	3.39193
$995$	0	0
$996$	0	0
$997$	26.4657	0.838176	0.419088	−	0.907946i	$-0.362350\pi$
$997$	26.4657	0.838176	0.419088	+	0.907946i	$0.362350\pi$
$998$	72.9948	2.31061
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bu.1.7		7
3.2	odd	2		2175.2.a.bb.1.1	yes	7
5.4	even	2		6525.2.a.bx.1.1		7
15.2	even	4		2175.2.c.o.349.2		14
15.8	even	4		2175.2.c.o.349.13		14
15.14	odd	2		2175.2.a.ba.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.ba.1.7	✓	7	15.14	odd	2
2175.2.a.bb.1.1	yes	7	3.2	odd	2
2175.2.c.o.349.2		14	15.2	even	4
2175.2.c.o.349.13		14	15.8	even	4
6525.2.a.bu.1.7		7	1.1	even	1	trivial
6525.2.a.bx.1.1		7	5.4	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)

\(f(q)\)	\(=\)	\(q+2.47806 q^{2} +4.14079 q^{4} -2.83516 q^{7} +5.30500 q^{8} -6.09835 q^{11} +3.00195 q^{13} -7.02571 q^{14} +4.86455 q^{16} -2.74182 q^{17} +8.03698 q^{19} -15.1121 q^{22} -7.56737 q^{23} +7.43901 q^{26} -11.7398 q^{28} +1.00000 q^{29} -5.32278 q^{31} +1.44464 q^{32} -6.79439 q^{34} -6.28158 q^{37} +19.9161 q^{38} +9.04088 q^{41} -9.06154 q^{43} -25.2520 q^{44} -18.7524 q^{46} -0.0839748 q^{47} +1.03815 q^{49} +12.4304 q^{52} +3.82861 q^{53} -15.0405 q^{56} +2.47806 q^{58} -11.9122 q^{59} -0.275669 q^{61} -13.1902 q^{62} -6.14919 q^{64} -7.36750 q^{67} -11.3533 q^{68} -15.2213 q^{71} +8.77173 q^{73} -15.5661 q^{74} +33.2794 q^{76} +17.2898 q^{77} +1.64999 q^{79} +22.4039 q^{82} -1.38875 q^{83} -22.4551 q^{86} -32.3518 q^{88} +1.05825 q^{89} -8.51101 q^{91} -31.3349 q^{92} -0.208095 q^{94} +9.18751 q^{97} +2.57260 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 10 q^{4} - q^{7} - 3 q^{8} - 4 q^{11} - q^{13} - 15 q^{14} + 12 q^{16} - 8 q^{17} + 15 q^{19} + 3 q^{22} - 14 q^{23} - 6 q^{26} - 24 q^{28} + 7 q^{29} + 5 q^{31} - 18 q^{32} + 7 q^{34}+ \cdots + 59 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 + 10 * q^4 - q^7 - 3 * q^8 - 4 * q^11 - q^13 - 15 * q^14 + 12 * q^16 - 8 * q^17 + 15 * q^19 + 3 * q^22 - 14 * q^23 - 6 * q^26 - 24 * q^28 + 7 * q^29 + 5 * q^31 - 18 * q^32 + 7 * q^34 - 6 * q^37 + 18 * q^38 - 22 * q^41 - 19 * q^43 - 15 * q^44 - 4 * q^46 - 22 * q^47 + 12 * q^49 + 11 * q^52 - 10 * q^53 - 14 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 - 40 * q^62 + 5 * q^64 - 13 * q^67 + 7 * q^68 - 26 * q^71 - 24 * q^73 + 10 * q^74 + 46 * q^76 - 4 * q^77 + 14 * q^79 + 16 * q^82 - 10 * q^83 - 44 * q^86 - 66 * q^88 - 14 * q^89 + 13 * q^91 - 58 * q^92 - 3 * q^94 - 31 * q^97 + 59 * q^98

Introduction

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