LMFDB - Embedded newform 6525.2.a.bh.1.2

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 = [3,3,0,5,0,0,2,9,0,0,-8] 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:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ 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)

$f(q)$	$=$	$q+1.53919 q^{2} +0.369102 q^{4} -0.630898 q^{7} -2.51026 q^{8} -0.290725 q^{11} +0.921622 q^{13} -0.971071 q^{14} -4.60197 q^{16} +4.97107 q^{17} -6.04945 q^{19} -0.447480 q^{22} +2.29072 q^{23} +1.41855 q^{26} -0.232866 q^{28} -1.00000 q^{29} +10.0494 q^{31} -2.06278 q^{32} +7.65142 q^{34} -1.55252 q^{37} -9.31124 q^{38} -0.340173 q^{41} +5.70928 q^{43} -0.107307 q^{44} +3.52586 q^{46} -1.12783 q^{47} -6.60197 q^{49} +0.340173 q^{52} -0.340173 q^{53} +1.58372 q^{56} -1.53919 q^{58} -9.75872 q^{59} +3.07838 q^{61} +15.4680 q^{62} +6.02893 q^{64} +5.70928 q^{67} +1.83483 q^{68} -9.07838 q^{71} +6.94441 q^{73} -2.38962 q^{74} -2.23287 q^{76} +0.183417 q^{77} +12.3896 q^{79} -0.523590 q^{82} +2.78765 q^{83} +8.78765 q^{86} +0.729794 q^{88} -4.73820 q^{89} -0.581449 q^{91} +0.845512 q^{92} -1.73594 q^{94} +15.8927 q^{97} -10.1617 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8} - 8 q^{11} + 6 q^{13} + 12 q^{14} + 5 q^{16} - 2 q^{22} + 14 q^{23} - 10 q^{26} + 22 q^{28} - 3 q^{29} + 12 q^{31} + 11 q^{32} - 14 q^{34} - 4 q^{37} - 2 q^{38}+ \cdots + 23 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 5 * q^4 + 2 * q^7 + 9 * q^8 - 8 * q^11 + 6 * q^13 + 12 * q^14 + 5 * q^16 - 2 * q^22 + 14 * q^23 - 10 * q^26 + 22 * q^28 - 3 * q^29 + 12 * q^31 + 11 * q^32 - 14 * q^34 - 4 * q^37 - 2 * q^38 + 10 * q^41 + 10 * q^43 - 12 * q^44 + 8 * q^46 + 18 * q^47 - q^49 - 10 * q^52 + 10 * q^53 + 32 * q^56 - 3 * q^58 - 4 * q^59 + 6 * q^61 + 14 * q^62 + 33 * q^64 + 10 * q^67 - 36 * q^68 - 24 * q^71 + 4 * q^73 + 22 * q^74 + 16 * q^76 - 4 * q^77 + 8 * q^79 + 14 * q^82 - 2 * q^83 + 16 * q^86 - 38 * q^88 - 22 * q^89 - 16 * q^91 + 22 * q^92 + 36 * q^97 + 23 * 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 $

$2$	1.53919	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$2$	1.53919	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$3$	0	0
$3$	0	0
$4$	0.369102	0.184551
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.630898	−0.238457	−0.119228	−	0.992867i	$-0.538042\pi$
$7$	−0.630898	−0.238457	−0.119228	+	0.992867i	$0.538042\pi$
$8$	−2.51026	−0.887511
$9$	0	0
$10$	0	0
$11$	−0.290725	−0.0876568	−0.0438284	−	0.999039i	$-0.513955\pi$
$11$	−0.290725	−0.0876568	−0.0438284	+	0.999039i	$0.513955\pi$
$12$	0	0
$13$	0.921622	0.255612	0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622	0.255612	0.127806	+	0.991799i	$0.459207\pi$
$14$	−0.971071	−0.259530
$15$	0	0
$16$	−4.60197	−1.15049
$17$	4.97107	1.20566	0.602831	−	0.797869i	$-0.294039\pi$
$17$	4.97107	1.20566	0.602831	+	0.797869i	$0.294039\pi$
$18$	0	0
$19$	−6.04945	−1.38784	−0.693919	−	0.720053i	$-0.744118\pi$
$19$	−6.04945	−1.38784	−0.693919	+	0.720053i	$0.744118\pi$
$20$	0	0
$21$	0	0
$22$	−0.447480	−0.0954031
$23$	2.29072	0.477649	0.238825	−	0.971063i	$-0.423238\pi$
$23$	2.29072	0.477649	0.238825	+	0.971063i	$0.423238\pi$
$24$	0	0
$25$	0	0
$26$	1.41855	0.278201
$27$	0	0
$28$	−0.232866	−0.0440075
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	10.0494	1.80493	0.902467	−	0.430759i	$-0.141754\pi$
$31$	10.0494	1.80493	0.902467	+	0.430759i	$0.141754\pi$
$32$	−2.06278	−0.364651
$33$	0	0
$34$	7.65142	1.31221
$35$	0	0
$36$	0	0
$37$	−1.55252	−0.255233	−0.127616	−	0.991824i	$-0.540733\pi$
$37$	−1.55252	−0.255233	−0.127616	+	0.991824i	$0.540733\pi$
$38$	−9.31124	−1.51048
$39$	0	0
$40$	0	0
$41$	−0.340173	−0.0531261	−0.0265630	−	0.999647i	$-0.508456\pi$
$41$	−0.340173	−0.0531261	−0.0265630	+	0.999647i	$0.508456\pi$
$42$	0	0
$43$	5.70928	0.870656	0.435328	−	0.900272i	$-0.356632\pi$
$43$	5.70928	0.870656	0.435328	+	0.900272i	$0.356632\pi$
$44$	−0.107307	−0.0161772
$45$	0	0
$46$	3.52586	0.519859
$47$	−1.12783	−0.164510	−0.0822552	−	0.996611i	$-0.526212\pi$
$47$	−1.12783	−0.164510	−0.0822552	+	0.996611i	$0.526212\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	0	0
$52$	0.340173	0.0471735
$53$	−0.340173	−0.0467264	−0.0233632	−	0.999727i	$-0.507437\pi$
$53$	−0.340173	−0.0467264	−0.0233632	+	0.999727i	$0.507437\pi$
$54$	0	0
$55$	0	0
$56$	1.58372	0.211633
$57$	0	0
$58$	−1.53919	−0.202105
$59$	−9.75872	−1.27048	−0.635239	−	0.772316i	$-0.719098\pi$
$59$	−9.75872	−1.27048	−0.635239	+	0.772316i	$0.719098\pi$
$60$	0	0
$61$	3.07838	0.394146	0.197073	−	0.980389i	$-0.436856\pi$
$61$	3.07838	0.394146	0.197073	+	0.980389i	$0.436856\pi$
$62$	15.4680	1.96444
$63$	0	0
$64$	6.02893	0.753616
$65$	0	0
$66$	0	0
$67$	5.70928	0.697499	0.348749	−	0.937216i	$-0.386606\pi$
$67$	5.70928	0.697499	0.348749	+	0.937216i	$0.386606\pi$
$68$	1.83483	0.222506
$69$	0	0
$70$	0	0
$71$	−9.07838	−1.07741	−0.538703	−	0.842496i	$-0.681085\pi$
$71$	−9.07838	−1.07741	−0.538703	+	0.842496i	$0.681085\pi$
$72$	0	0
$73$	6.94441	0.812782	0.406391	−	0.913699i	$-0.366787\pi$
$73$	6.94441	0.812782	0.406391	+	0.913699i	$0.366787\pi$
$74$	−2.38962	−0.277788
$75$	0	0
$76$	−2.23287	−0.256127
$77$	0.183417	0.0209024
$78$	0	0
$79$	12.3896	1.39394	0.696971	−	0.717100i	$-0.254531\pi$
$79$	12.3896	1.39394	0.696971	+	0.717100i	$0.254531\pi$
$80$	0	0
$81$	0	0
$82$	−0.523590	−0.0578209
$83$	2.78765	0.305985	0.152992	−	0.988227i	$-0.451109\pi$
$83$	2.78765	0.305985	0.152992	+	0.988227i	$0.451109\pi$
$84$	0	0
$85$	0	0
$86$	8.78765	0.947597
$87$	0	0
$88$	0.729794	0.0777963
$89$	−4.73820	−0.502249	−0.251124	−	0.967955i	$-0.580800\pi$
$89$	−4.73820	−0.502249	−0.251124	+	0.967955i	$0.580800\pi$
$90$	0	0
$91$	−0.581449	−0.0609524
$92$	0.845512	0.0881507
$93$	0	0
$94$	−1.73594	−0.179048
$95$	0	0
$96$	0	0
$97$	15.8927	1.61366	0.806829	−	0.590785i	$-0.201182\pi$
$97$	15.8927	1.61366	0.806829	+	0.590785i	$0.201182\pi$
$98$	−10.1617	−1.02648
$99$	0	0
$100$	0	0
$101$	12.2557	1.21948	0.609741	−	0.792600i	$-0.291273\pi$
$101$	12.2557	1.21948	0.609741	+	0.792600i	$0.291273\pi$
$102$	0	0
$103$	7.86603	0.775063	0.387532	−	0.921856i	$-0.373328\pi$
$103$	7.86603	0.775063	0.387532	+	0.921856i	$0.373328\pi$
$104$	−2.31351	−0.226858
$105$	0	0
$106$	−0.523590	−0.0508556
$107$	12.7298	1.23064	0.615318	−	0.788279i	$-0.289028\pi$
$107$	12.7298	1.23064	0.615318	+	0.788279i	$0.289028\pi$
$108$	0	0
$109$	12.4391	1.19145	0.595723	−	0.803190i	$-0.296865\pi$
$109$	12.4391	1.19145	0.595723	+	0.803190i	$0.296865\pi$
$110$	0	0
$111$	0	0
$112$	2.90337	0.274343
$113$	12.5730	1.18277	0.591386	−	0.806389i	$-0.298581\pi$
$113$	12.5730	1.18277	0.591386	+	0.806389i	$0.298581\pi$
$114$	0	0
$115$	0	0
$116$	−0.369102	−0.0342703
$117$	0	0
$118$	−15.0205	−1.38275
$119$	−3.13624	−0.287498
$120$	0	0
$121$	−10.9155	−0.992316
$122$	4.73820	0.428977
$123$	0	0
$124$	3.70928	0.333103
$125$	0	0
$126$	0	0
$127$	20.9132	1.85575	0.927874	−	0.372895i	$-0.121635\pi$
$127$	20.9132	1.85575	0.927874	+	0.372895i	$0.121635\pi$
$128$	13.4052	1.18487
$129$	0	0
$130$	0	0
$131$	13.4680	1.17670	0.588352	−	0.808605i	$-0.299777\pi$
$131$	13.4680	1.17670	0.588352	+	0.808605i	$0.299777\pi$
$132$	0	0
$133$	3.81658	0.330940
$134$	8.78765	0.759138
$135$	0	0
$136$	−12.4787	−1.07004
$137$	−13.5525	−1.15787	−0.578935	−	0.815374i	$-0.696531\pi$
$137$	−13.5525	−1.15787	−0.578935	+	0.815374i	$0.696531\pi$
$138$	0	0
$139$	−4.89496	−0.415185	−0.207593	−	0.978215i	$-0.566563\pi$
$139$	−4.89496	−0.415185	−0.207593	+	0.978215i	$0.566563\pi$
$140$	0	0
$141$	0	0
$142$	−13.9733	−1.17262
$143$	−0.267938	−0.0224061
$144$	0	0
$145$	0	0
$146$	10.6888	0.884608
$147$	0	0
$148$	−0.573039	−0.0471035
$149$	12.5236	1.02597	0.512986	−	0.858397i	$-0.328539\pi$
$149$	12.5236	1.02597	0.512986	+	0.858397i	$0.328539\pi$
$150$	0	0
$151$	7.60197	0.618639	0.309320	−	0.950958i	$-0.399899\pi$
$151$	7.60197	0.618639	0.309320	+	0.950958i	$0.399899\pi$
$152$	15.1857	1.23172
$153$	0	0
$154$	0.282314	0.0227495
$155$	0	0
$156$	0	0
$157$	24.8865	1.98616	0.993081	−	0.117428i	$-0.0374648\pi$
$157$	24.8865	1.98616	0.993081	+	0.117428i	$0.0374648\pi$
$158$	19.0700	1.51713
$159$	0	0
$160$	0	0
$161$	−1.44521	−0.113899
$162$	0	0
$163$	−0.447480	−0.0350493	−0.0175247	−	0.999846i	$-0.505579\pi$
$163$	−0.447480	−0.0350493	−0.0175247	+	0.999846i	$0.505579\pi$
$164$	−0.125559	−0.00980448
$165$	0	0
$166$	4.29072	0.333025
$167$	19.8660	1.53728	0.768640	−	0.639682i	$-0.220934\pi$
$167$	19.8660	1.53728	0.768640	+	0.639682i	$0.220934\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	0	0
$171$	0	0
$172$	2.10731	0.160681
$173$	−25.4329	−1.93363	−0.966815	−	0.255478i	$-0.917767\pi$
$173$	−25.4329	−1.93363	−0.966815	+	0.255478i	$0.917767\pi$
$174$	0	0
$175$	0	0
$176$	1.33791	0.100848
$177$	0	0
$178$	−7.29299	−0.546633
$179$	−14.8371	−1.10898	−0.554489	−	0.832191i	$-0.687086\pi$
$179$	−14.8371	−1.10898	−0.554489	+	0.832191i	$0.687086\pi$
$180$	0	0
$181$	−5.91548	−0.439694	−0.219847	−	0.975534i	$-0.570556\pi$
$181$	−5.91548	−0.439694	−0.219847	+	0.975534i	$0.570556\pi$
$182$	−0.894960	−0.0663389
$183$	0	0
$184$	−5.75031	−0.423919
$185$	0	0
$186$	0	0
$187$	−1.44521	−0.105684
$188$	−0.416283	−0.0303606
$189$	0	0
$190$	0	0
$191$	−7.02893	−0.508595	−0.254298	−	0.967126i	$-0.581844\pi$
$191$	−7.02893	−0.508595	−0.254298	+	0.967126i	$0.581844\pi$
$192$	0	0
$193$	−17.8660	−1.28603	−0.643013	−	0.765856i	$-0.722316\pi$
$193$	−17.8660	−1.28603	−0.643013	+	0.765856i	$0.722316\pi$
$194$	24.4619	1.75626
$195$	0	0
$196$	−2.43680	−0.174057
$197$	6.09890	0.434528	0.217264	−	0.976113i	$-0.430287\pi$
$197$	6.09890	0.434528	0.217264	+	0.976113i	$0.430287\pi$
$198$	0	0
$199$	−9.75872	−0.691778	−0.345889	−	0.938276i	$-0.612423\pi$
$199$	−9.75872	−0.691778	−0.345889	+	0.938276i	$0.612423\pi$
$200$	0	0
$201$	0	0
$202$	18.8638	1.32725
$203$	0.630898	0.0442803
$204$	0	0
$205$	0	0
$206$	12.1073	0.843556
$207$	0	0
$208$	−4.24128	−0.294080
$209$	1.75872	0.121653
$210$	0	0
$211$	9.86603	0.679206	0.339603	−	0.940569i	$-0.389707\pi$
$211$	9.86603	0.679206	0.339603	+	0.940569i	$0.389707\pi$
$212$	−0.125559	−0.00862340
$213$	0	0
$214$	19.5936	1.33939
$215$	0	0
$216$	0	0
$217$	−6.34017	−0.430399
$218$	19.1461	1.29674
$219$	0	0
$220$	0	0
$221$	4.58145	0.308182
$222$	0	0
$223$	−10.9711	−0.734677	−0.367339	−	0.930087i	$-0.619731\pi$
$223$	−10.9711	−0.734677	−0.367339	+	0.930087i	$0.619731\pi$
$224$	1.30140	0.0869536
$225$	0	0
$226$	19.3523	1.28729
$227$	−12.5464	−0.832732	−0.416366	−	0.909197i	$-0.636697\pi$
$227$	−12.5464	−0.832732	−0.416366	+	0.909197i	$0.636697\pi$
$228$	0	0
$229$	23.3607	1.54372	0.771859	−	0.635794i	$-0.219327\pi$
$229$	23.3607	1.54372	0.771859	+	0.635794i	$0.219327\pi$
$230$	0	0
$231$	0	0
$232$	2.51026	0.164807
$233$	12.4703	0.816954	0.408477	−	0.912769i	$-0.366060\pi$
$233$	12.4703	0.816954	0.408477	+	0.912769i	$0.366060\pi$
$234$	0	0
$235$	0	0
$236$	−3.60197	−0.234468
$237$	0	0
$238$	−4.82726	−0.312905
$239$	−13.7587	−0.889978	−0.444989	−	0.895536i	$-0.646792\pi$
$239$	−13.7587	−0.889978	−0.444989	+	0.895536i	$0.646792\pi$
$240$	0	0
$241$	−14.6803	−0.945644	−0.472822	−	0.881158i	$-0.656765\pi$
$241$	−14.6803	−0.945644	−0.472822	+	0.881158i	$0.656765\pi$
$242$	−16.8010	−1.08001
$243$	0	0
$244$	1.13624	0.0727401
$245$	0	0
$246$	0	0
$247$	−5.57531	−0.354748
$248$	−25.2267	−1.60190
$249$	0	0
$250$	0	0
$251$	−15.4413	−0.974649	−0.487324	−	0.873221i	$-0.662027\pi$
$251$	−15.4413	−0.974649	−0.487324	+	0.873221i	$0.662027\pi$
$252$	0	0
$253$	−0.665970	−0.0418692
$254$	32.1894	2.01974
$255$	0	0
$256$	8.57531	0.535957
$257$	−6.28231	−0.391880	−0.195940	−	0.980616i	$-0.562776\pi$
$257$	−6.28231	−0.391880	−0.195940	+	0.980616i	$0.562776\pi$
$258$	0	0
$259$	0.979481	0.0608620
$260$	0	0
$261$	0	0
$262$	20.7298	1.28069
$263$	10.0761	0.621320	0.310660	−	0.950521i	$-0.399450\pi$
$263$	10.0761	0.621320	0.310660	+	0.950521i	$0.399450\pi$
$264$	0	0
$265$	0	0
$266$	5.87444	0.360185
$267$	0	0
$268$	2.10731	0.128724
$269$	−28.1711	−1.71762	−0.858812	−	0.512291i	$-0.828797\pi$
$269$	−28.1711	−1.71762	−0.858812	+	0.512291i	$0.828797\pi$
$270$	0	0
$271$	28.8020	1.74960	0.874799	−	0.484485i	$-0.160993\pi$
$271$	28.8020	1.74960	0.874799	+	0.484485i	$0.160993\pi$
$272$	−22.8767	−1.38710
$273$	0	0
$274$	−20.8599	−1.26019
$275$	0	0
$276$	0	0
$277$	−0.0266620	−0.00160196	−0.000800982	−	1.00000i	$-0.500255\pi$
$277$	−0.0266620	−0.00160196	−0.000800982		1.00000i	$0.500255\pi$
$278$	−7.53427	−0.451875
$279$	0	0
$280$	0	0
$281$	28.0722	1.67465	0.837325	−	0.546706i	$-0.184119\pi$
$281$	28.0722	1.67465	0.837325	+	0.546706i	$0.184119\pi$
$282$	0	0
$283$	20.8143	1.23728	0.618641	−	0.785674i	$-0.287683\pi$
$283$	20.8143	1.23728	0.618641	+	0.785674i	$0.287683\pi$
$284$	−3.35085	−0.198836
$285$	0	0
$286$	−0.412408	−0.0243862
$287$	0.214614	0.0126683
$288$	0	0
$289$	7.71154	0.453620
$290$	0	0
$291$	0	0
$292$	2.56320	0.150000
$293$	−15.4101	−0.900270	−0.450135	−	0.892961i	$-0.648624\pi$
$293$	−15.4101	−0.900270	−0.450135	+	0.892961i	$0.648624\pi$
$294$	0	0
$295$	0	0
$296$	3.89723	0.226522
$297$	0	0
$298$	19.2762	1.11664
$299$	2.11118	0.122093
$300$	0	0
$301$	−3.60197	−0.207614
$302$	11.7009	0.673309
$303$	0	0
$304$	27.8394	1.59670
$305$	0	0
$306$	0	0
$307$	28.4307	1.62262	0.811312	−	0.584614i	$-0.198754\pi$
$307$	28.4307	1.62262	0.811312	+	0.584614i	$0.198754\pi$
$308$	0.0676998	0.00385756
$309$	0	0
$310$	0	0
$311$	19.6248	1.11282	0.556409	−	0.830909i	$-0.312179\pi$
$311$	19.6248	1.11282	0.556409	+	0.830909i	$0.312179\pi$
$312$	0	0
$313$	−22.9093	−1.29491	−0.647456	−	0.762103i	$-0.724167\pi$
$313$	−22.9093	−1.29491	−0.647456	+	0.762103i	$0.724167\pi$
$314$	38.3051	2.16168
$315$	0	0
$316$	4.57304	0.257254
$317$	22.8599	1.28394	0.641970	−	0.766730i	$-0.278118\pi$
$317$	22.8599	1.28394	0.641970	+	0.766730i	$0.278118\pi$
$318$	0	0
$319$	0.290725	0.0162775
$320$	0	0
$321$	0	0
$322$	−2.22446	−0.123964
$323$	−30.0722	−1.67326
$324$	0	0
$325$	0	0
$326$	−0.688756	−0.0381467
$327$	0	0
$328$	0.853922	0.0471500
$329$	0.711543	0.0392286
$330$	0	0
$331$	24.0905	1.32413	0.662066	−	0.749445i	$-0.269680\pi$
$331$	24.0905	1.32413	0.662066	+	0.749445i	$0.269680\pi$
$332$	1.02893	0.0564698
$333$	0	0
$334$	30.5776	1.67313
$335$	0	0
$336$	0	0
$337$	−12.7877	−0.696588	−0.348294	−	0.937385i	$-0.613239\pi$
$337$	−12.7877	−0.696588	−0.348294	+	0.937385i	$0.613239\pi$
$338$	−18.7021	−1.01726
$339$	0	0
$340$	0	0
$341$	−2.92162	−0.158215
$342$	0	0
$343$	8.58145	0.463355
$344$	−14.3318	−0.772717
$345$	0	0
$346$	−39.1461	−2.10451
$347$	8.41628	0.451810	0.225905	−	0.974149i	$-0.427466\pi$
$347$	8.41628	0.451810	0.225905	+	0.974149i	$0.427466\pi$
$348$	0	0
$349$	22.1978	1.18822	0.594110	−	0.804384i	$-0.297504\pi$
$349$	22.1978	1.18822	0.594110	+	0.804384i	$0.297504\pi$
$350$	0	0
$351$	0	0
$352$	0.599701	0.0319642
$353$	−6.18342	−0.329110	−0.164555	−	0.986368i	$-0.552619\pi$
$353$	−6.18342	−0.329110	−0.164555	+	0.986368i	$0.552619\pi$
$354$	0	0
$355$	0	0
$356$	−1.74888	−0.0926906
$357$	0	0
$358$	−22.8371	−1.20698
$359$	−5.05559	−0.266824	−0.133412	−	0.991061i	$-0.542593\pi$
$359$	−5.05559	−0.266824	−0.133412	+	0.991061i	$0.542593\pi$
$360$	0	0
$361$	17.5958	0.926096
$362$	−9.10504	−0.478550
$363$	0	0
$364$	−0.214614	−0.0112488
$365$	0	0
$366$	0	0
$367$	−29.5402	−1.54199	−0.770994	−	0.636843i	$-0.780240\pi$
$367$	−29.5402	−1.54199	−0.770994	+	0.636843i	$0.780240\pi$
$368$	−10.5418	−0.549531
$369$	0	0
$370$	0	0
$371$	0.214614	0.0111422
$372$	0	0
$373$	−14.4124	−0.746246	−0.373123	−	0.927782i	$-0.621713\pi$
$373$	−14.4124	−0.746246	−0.373123	+	0.927782i	$0.621713\pi$
$374$	−2.22446	−0.115024
$375$	0	0
$376$	2.83114	0.146005
$377$	−0.921622	−0.0474660
$378$	0	0
$379$	14.1340	0.726013	0.363007	−	0.931787i	$-0.381750\pi$
$379$	14.1340	0.726013	0.363007	+	0.931787i	$0.381750\pi$
$380$	0	0
$381$	0	0
$382$	−10.8188	−0.553541
$383$	−15.7815	−0.806397	−0.403199	−	0.915112i	$-0.632102\pi$
$383$	−15.7815	−0.806397	−0.403199	+	0.915112i	$0.632102\pi$
$384$	0	0
$385$	0	0
$386$	−27.4992	−1.39967
$387$	0	0
$388$	5.86603	0.297803
$389$	13.8166	0.700529	0.350264	−	0.936651i	$-0.386092\pi$
$389$	13.8166	0.700529	0.350264	+	0.936651i	$0.386092\pi$
$390$	0	0
$391$	11.3874	0.575883
$392$	16.5727	0.837045
$393$	0	0
$394$	9.38735	0.472928
$395$	0	0
$396$	0	0
$397$	−9.05172	−0.454293	−0.227146	−	0.973861i	$-0.572940\pi$
$397$	−9.05172	−0.454293	−0.227146	+	0.973861i	$0.572940\pi$
$398$	−15.0205	−0.752911
$399$	0	0
$400$	0	0
$401$	−19.7587	−0.986704	−0.493352	−	0.869830i	$-0.664228\pi$
$401$	−19.7587	−0.986704	−0.493352	+	0.869830i	$0.664228\pi$
$402$	0	0
$403$	9.26180	0.461363
$404$	4.52359	0.225057
$405$	0	0
$406$	0.971071	0.0481934
$407$	0.451356	0.0223729
$408$	0	0
$409$	−1.71769	−0.0849341	−0.0424670	−	0.999098i	$-0.513522\pi$
$409$	−1.71769	−0.0849341	−0.0424670	+	0.999098i	$0.513522\pi$
$410$	0	0
$411$	0	0
$412$	2.90337	0.143039
$413$	6.15676	0.302954
$414$	0	0
$415$	0	0
$416$	−1.90110	−0.0932093
$417$	0	0
$418$	2.70701	0.132404
$419$	35.5318	1.73584	0.867922	−	0.496701i	$-0.165456\pi$
$419$	35.5318	1.73584	0.867922	+	0.496701i	$0.165456\pi$
$420$	0	0
$421$	−12.0722	−0.588365	−0.294182	−	0.955749i	$-0.595047\pi$
$421$	−12.0722	−0.588365	−0.294182	+	0.955749i	$0.595047\pi$
$422$	15.1857	0.739228
$423$	0	0
$424$	0.853922	0.0414701
$425$	0	0
$426$	0	0
$427$	−1.94214	−0.0939868
$428$	4.69860	0.227115
$429$	0	0
$430$	0	0
$431$	−19.8310	−0.955224	−0.477612	−	0.878571i	$-0.658497\pi$
$431$	−19.8310	−0.955224	−0.477612	+	0.878571i	$0.658497\pi$
$432$	0	0
$433$	−14.8143	−0.711931	−0.355965	−	0.934499i	$-0.615848\pi$
$433$	−14.8143	−0.711931	−0.355965	+	0.934499i	$0.615848\pi$
$434$	−9.75872	−0.468434
$435$	0	0
$436$	4.59129	0.219883
$437$	−13.8576	−0.662900
$438$	0	0
$439$	−17.8576	−0.852298	−0.426149	−	0.904653i	$-0.640130\pi$
$439$	−17.8576	−0.852298	−0.426149	+	0.904653i	$0.640130\pi$
$440$	0	0
$441$	0	0
$442$	7.05172	0.335416
$443$	33.5936	1.59608	0.798039	−	0.602606i	$-0.205871\pi$
$443$	33.5936	1.59608	0.798039	+	0.602606i	$0.205871\pi$
$444$	0	0
$445$	0	0
$446$	−16.8865	−0.799601
$447$	0	0
$448$	−3.80364	−0.179705
$449$	−7.07838	−0.334049	−0.167025	−	0.985953i	$-0.553416\pi$
$449$	−7.07838	−0.334049	−0.167025	+	0.985953i	$0.553416\pi$
$450$	0	0
$451$	0.0988967	0.00465686
$452$	4.64074	0.218282
$453$	0	0
$454$	−19.3112	−0.906322
$455$	0	0
$456$	0	0
$457$	5.81658	0.272088	0.136044	−	0.990703i	$-0.456561\pi$
$457$	5.81658	0.272088	0.136044	+	0.990703i	$0.456561\pi$
$458$	35.9565	1.68014
$459$	0	0
$460$	0	0
$461$	32.3090	1.50478	0.752390	−	0.658718i	$-0.228901\pi$
$461$	32.3090	1.50478	0.752390	+	0.658718i	$0.228901\pi$
$462$	0	0
$463$	−1.44134	−0.0669846	−0.0334923	−	0.999439i	$-0.510663\pi$
$463$	−1.44134	−0.0669846	−0.0334923	+	0.999439i	$0.510663\pi$
$464$	4.60197	0.213641
$465$	0	0
$466$	19.1941	0.889149
$467$	−11.7503	−0.543740	−0.271870	−	0.962334i	$-0.587642\pi$
$467$	−11.7503	−0.543740	−0.271870	+	0.962334i	$0.587642\pi$
$468$	0	0
$469$	−3.60197	−0.166323
$470$	0	0
$471$	0	0
$472$	24.4969	1.12756
$473$	−1.65983	−0.0763189
$474$	0	0
$475$	0	0
$476$	−1.15759	−0.0530582
$477$	0	0
$478$	−21.1773	−0.968626
$479$	−17.1689	−0.784465	−0.392233	−	0.919866i	$-0.628297\pi$
$479$	−17.1689	−0.784465	−0.392233	+	0.919866i	$0.628297\pi$
$480$	0	0
$481$	−1.43084	−0.0652405
$482$	−22.5958	−1.02921
$483$	0	0
$484$	−4.02893	−0.183133
$485$	0	0
$486$	0	0
$487$	4.10277	0.185914	0.0929572	−	0.995670i	$-0.470368\pi$
$487$	4.10277	0.185914	0.0929572	+	0.995670i	$0.470368\pi$
$488$	−7.72753	−0.349809
$489$	0	0
$490$	0	0
$491$	−40.7708	−1.83996	−0.919981	−	0.391963i	$-0.871796\pi$
$491$	−40.7708	−1.83996	−0.919981	+	0.391963i	$0.871796\pi$
$492$	0	0
$493$	−4.97107	−0.223886
$494$	−8.58145	−0.386098
$495$	0	0
$496$	−46.2472	−2.07656
$497$	5.72753	0.256915
$498$	0	0
$499$	−18.4703	−0.826843	−0.413421	−	0.910540i	$-0.635666\pi$
$499$	−18.4703	−0.826843	−0.413421	+	0.910540i	$0.635666\pi$
$500$	0	0
$501$	0	0
$502$	−23.7671	−1.06078
$503$	−21.4947	−0.958400	−0.479200	−	0.877706i	$-0.659073\pi$
$503$	−21.4947	−0.958400	−0.479200	+	0.877706i	$0.659073\pi$
$504$	0	0
$505$	0	0
$506$	−1.02505	−0.0455692
$507$	0	0
$508$	7.71912	0.342480
$509$	−3.75872	−0.166602	−0.0833012	−	0.996524i	$-0.526546\pi$
$509$	−3.75872	−0.166602	−0.0833012	+	0.996524i	$0.526546\pi$
$510$	0	0
$511$	−4.38121	−0.193813
$512$	−13.6114	−0.601546
$513$	0	0
$514$	−9.66967	−0.426511
$515$	0	0
$516$	0	0
$517$	0.327887	0.0144204
$518$	1.50761	0.0662404
$519$	0	0
$520$	0	0
$521$	−12.8059	−0.561037	−0.280518	−	0.959849i	$-0.590506\pi$
$521$	−12.8059	−0.561037	−0.280518	+	0.959849i	$0.590506\pi$
$522$	0	0
$523$	21.1278	0.923855	0.461928	−	0.886918i	$-0.347158\pi$
$523$	21.1278	0.923855	0.461928	+	0.886918i	$0.347158\pi$
$524$	4.97107	0.217162
$525$	0	0
$526$	15.5090	0.676226
$527$	49.9565	2.17614
$528$	0	0
$529$	−17.7526	−0.771851
$530$	0	0
$531$	0	0
$532$	1.40871	0.0610753
$533$	−0.313511	−0.0135797
$534$	0	0
$535$	0	0
$536$	−14.3318	−0.619038
$537$	0	0
$538$	−43.3607	−1.86941
$539$	1.91935	0.0826725
$540$	0	0
$541$	32.7382	1.40753	0.703763	−	0.710435i	$-0.251502\pi$
$541$	32.7382	1.40753	0.703763	+	0.710435i	$0.251502\pi$
$542$	44.3318	1.90421
$543$	0	0
$544$	−10.2542	−0.439646
$545$	0	0
$546$	0	0
$547$	22.1073	0.945240	0.472620	−	0.881266i	$-0.343308\pi$
$547$	22.1073	0.945240	0.472620	+	0.881266i	$0.343308\pi$
$548$	−5.00227	−0.213686
$549$	0	0
$550$	0	0
$551$	6.04945	0.257715
$552$	0	0
$553$	−7.81658	−0.332395
$554$	−0.0410378	−0.00174353
$555$	0	0
$556$	−1.80674	−0.0766229
$557$	−39.8720	−1.68943	−0.844715	−	0.535216i	$-0.820230\pi$
$557$	−39.8720	−1.68943	−0.844715	+	0.535216i	$0.820230\pi$
$558$	0	0
$559$	5.26180	0.222550
$560$	0	0
$561$	0	0
$562$	43.2085	1.82264
$563$	10.1217	0.426578	0.213289	−	0.976989i	$-0.431582\pi$
$563$	10.1217	0.426578	0.213289	+	0.976989i	$0.431582\pi$
$564$	0	0
$565$	0	0
$566$	32.0372	1.34662
$567$	0	0
$568$	22.7891	0.956209
$569$	24.4391	1.02454	0.512270	−	0.858825i	$-0.328805\pi$
$569$	24.4391	1.02454	0.512270	+	0.858825i	$0.328805\pi$
$570$	0	0
$571$	−28.2511	−1.18227	−0.591136	−	0.806572i	$-0.701320\pi$
$571$	−28.2511	−1.18227	−0.591136	+	0.806572i	$0.701320\pi$
$572$	−0.0988967	−0.00413508
$573$	0	0
$574$	0.330332	0.0137878
$575$	0	0
$576$	0	0
$577$	−46.1171	−1.91988	−0.959941	−	0.280202i	$-0.909598\pi$
$577$	−46.1171	−1.91988	−0.959941	+	0.280202i	$0.909598\pi$
$578$	11.8695	0.493707
$579$	0	0
$580$	0	0
$581$	−1.75872	−0.0729642
$582$	0	0
$583$	0.0988967	0.00409588
$584$	−17.4323	−0.721352
$585$	0	0
$586$	−23.7191	−0.979828
$587$	−0.715418	−0.0295285	−0.0147642	−	0.999891i	$-0.504700\pi$
$587$	−0.715418	−0.0295285	−0.0147642	+	0.999891i	$0.504700\pi$
$588$	0	0
$589$	−60.7936	−2.50496
$590$	0	0
$591$	0	0
$592$	7.14465	0.293643
$593$	15.5441	0.638320	0.319160	−	0.947701i	$-0.396599\pi$
$593$	15.5441	0.638320	0.319160	+	0.947701i	$0.396599\pi$
$594$	0	0
$595$	0	0
$596$	4.62249	0.189344
$597$	0	0
$598$	3.24951	0.132882
$599$	9.59809	0.392167	0.196084	−	0.980587i	$-0.437178\pi$
$599$	9.59809	0.392167	0.196084	+	0.980587i	$0.437178\pi$
$600$	0	0
$601$	6.81044	0.277804	0.138902	−	0.990306i	$-0.455643\pi$
$601$	6.81044	0.277804	0.138902	+	0.990306i	$0.455643\pi$
$602$	−5.54411	−0.225961
$603$	0	0
$604$	2.80590	0.114171
$605$	0	0
$606$	0	0
$607$	−31.6970	−1.28654	−0.643271	−	0.765639i	$-0.722423\pi$
$607$	−31.6970	−1.28654	−0.643271	+	0.765639i	$0.722423\pi$
$608$	12.4787	0.506077
$609$	0	0
$610$	0	0
$611$	−1.03943	−0.0420508
$612$	0	0
$613$	−1.20394	−0.0486265	−0.0243133	−	0.999704i	$-0.507740\pi$
$613$	−1.20394	−0.0486265	−0.0243133	+	0.999704i	$0.507740\pi$
$614$	43.7602	1.76602
$615$	0	0
$616$	−0.460425	−0.0185511
$617$	−37.9337	−1.52715	−0.763577	−	0.645716i	$-0.776559\pi$
$617$	−37.9337	−1.52715	−0.763577	+	0.645716i	$0.776559\pi$
$618$	0	0
$619$	−4.60424	−0.185060	−0.0925299	−	0.995710i	$-0.529495\pi$
$619$	−4.60424	−0.185060	−0.0925299	+	0.995710i	$0.529495\pi$
$620$	0	0
$621$	0	0
$622$	30.2062	1.21116
$623$	2.98932	0.119765
$624$	0	0
$625$	0	0
$626$	−35.2618	−1.40934
$627$	0	0
$628$	9.18568	0.366549
$629$	−7.71769	−0.307724
$630$	0	0
$631$	−8.41241	−0.334893	−0.167446	−	0.985881i	$-0.553552\pi$
$631$	−8.41241	−0.334893	−0.167446	+	0.985881i	$0.553552\pi$
$632$	−31.1012	−1.23714
$633$	0	0
$634$	35.1857	1.39740
$635$	0	0
$636$	0	0
$637$	−6.08452	−0.241077
$638$	0.447480	0.0177159
$639$	0	0
$640$	0	0
$641$	−32.5380	−1.28517	−0.642586	−	0.766213i	$-0.722139\pi$
$641$	−32.5380	−1.28517	−0.642586	+	0.766213i	$0.722139\pi$
$642$	0	0
$643$	2.09293	0.0825372	0.0412686	−	0.999148i	$-0.486860\pi$
$643$	2.09293	0.0825372	0.0412686	+	0.999148i	$0.486860\pi$
$644$	−0.533431	−0.0210201
$645$	0	0
$646$	−46.2868	−1.82113
$647$	45.1955	1.77682	0.888410	−	0.459051i	$-0.151811\pi$
$647$	45.1955	1.77682	0.888410	+	0.459051i	$0.151811\pi$
$648$	0	0
$649$	2.83710	0.111366
$650$	0	0
$651$	0	0
$652$	−0.165166	−0.00646840
$653$	−2.14834	−0.0840712	−0.0420356	−	0.999116i	$-0.513384\pi$
$653$	−2.14834	−0.0840712	−0.0420356	+	0.999116i	$0.513384\pi$
$654$	0	0
$655$	0	0
$656$	1.56547	0.0611211
$657$	0	0
$658$	1.09520	0.0426953
$659$	45.0843	1.75624	0.878118	−	0.478444i	$-0.158799\pi$
$659$	45.0843	1.75624	0.878118	+	0.478444i	$0.158799\pi$
$660$	0	0
$661$	−36.3234	−1.41281	−0.706407	−	0.707806i	$-0.749685\pi$
$661$	−36.3234	−1.41281	−0.706407	+	0.707806i	$0.749685\pi$
$662$	37.0798	1.44115
$663$	0	0
$664$	−6.99773	−0.271565
$665$	0	0
$666$	0	0
$667$	−2.29072	−0.0886972
$668$	7.33260	0.283707
$669$	0	0
$670$	0	0
$671$	−0.894960	−0.0345496
$672$	0	0
$673$	17.4719	0.673491	0.336746	−	0.941596i	$-0.390674\pi$
$673$	17.4719	0.673491	0.336746	+	0.941596i	$0.390674\pi$
$674$	−19.6826	−0.758146
$675$	0	0
$676$	−4.48482	−0.172493
$677$	−40.0372	−1.53875	−0.769377	−	0.638796i	$-0.779433\pi$
$677$	−40.0372	−1.53875	−0.769377	+	0.638796i	$0.779433\pi$
$678$	0	0
$679$	−10.0267	−0.384788
$680$	0	0
$681$	0	0
$682$	−4.49693	−0.172196
$683$	2.07611	0.0794402	0.0397201	−	0.999211i	$-0.487353\pi$
$683$	2.07611	0.0794402	0.0397201	+	0.999211i	$0.487353\pi$
$684$	0	0
$685$	0	0
$686$	13.2085	0.504302
$687$	0	0
$688$	−26.2739	−1.00168
$689$	−0.313511	−0.0119438
$690$	0	0
$691$	26.7070	1.01598	0.507991	−	0.861362i	$-0.330388\pi$
$691$	26.7070	1.01598	0.507991	+	0.861362i	$0.330388\pi$
$692$	−9.38735	−0.356854
$693$	0	0
$694$	12.9542	0.491737
$695$	0	0
$696$	0	0
$697$	−1.69102	−0.0640521
$698$	34.1666	1.29322
$699$	0	0
$700$	0	0
$701$	21.9155	0.827736	0.413868	−	0.910337i	$-0.364177\pi$
$701$	21.9155	0.827736	0.413868	+	0.910337i	$0.364177\pi$
$702$	0	0
$703$	9.39189	0.354222
$704$	−1.75276	−0.0660596
$705$	0	0
$706$	−9.51745	−0.358194
$707$	−7.73206	−0.290794
$708$	0	0
$709$	−4.60811	−0.173061	−0.0865306	−	0.996249i	$-0.527578\pi$
$709$	−4.60811	−0.173061	−0.0865306	+	0.996249i	$0.527578\pi$
$710$	0	0
$711$	0	0
$712$	11.8941	0.445751
$713$	23.0205	0.862125
$714$	0	0
$715$	0	0
$716$	−5.47641	−0.204663
$717$	0	0
$718$	−7.78151	−0.290403
$719$	−6.80590	−0.253817	−0.126909	−	0.991914i	$-0.540506\pi$
$719$	−6.80590	−0.253817	−0.126909	+	0.991914i	$0.540506\pi$
$720$	0	0
$721$	−4.96266	−0.184819
$722$	27.0833	1.00794
$723$	0	0
$724$	−2.18342	−0.0811461
$725$	0	0
$726$	0	0
$727$	26.9711	1.00030	0.500151	−	0.865938i	$-0.333278\pi$
$727$	26.9711	1.00030	0.500151	+	0.865938i	$0.333278\pi$
$728$	1.45959	0.0540960
$729$	0	0
$730$	0	0
$731$	28.3812	1.04972
$732$	0	0
$733$	30.0638	1.11043	0.555216	−	0.831706i	$-0.312635\pi$
$733$	30.0638	1.11043	0.555216	+	0.831706i	$0.312635\pi$
$734$	−45.4680	−1.67825
$735$	0	0
$736$	−4.72526	−0.174175
$737$	−1.65983	−0.0611405
$738$	0	0
$739$	51.1422	1.88130	0.940648	−	0.339383i	$-0.110218\pi$
$739$	51.1422	1.88130	0.940648	+	0.339383i	$0.110218\pi$
$740$	0	0
$741$	0	0
$742$	0.330332	0.0121269
$743$	11.1857	0.410363	0.205181	−	0.978724i	$-0.434222\pi$
$743$	11.1857	0.410363	0.205181	+	0.978724i	$0.434222\pi$
$744$	0	0
$745$	0	0
$746$	−22.1834	−0.812193
$747$	0	0
$748$	−0.533431	−0.0195042
$749$	−8.03120	−0.293454
$750$	0	0
$751$	−18.3630	−0.670074	−0.335037	−	0.942205i	$-0.608749\pi$
$751$	−18.3630	−0.670074	−0.335037	+	0.942205i	$0.608749\pi$
$752$	5.19022	0.189268
$753$	0	0
$754$	−1.41855	−0.0516606
$755$	0	0
$756$	0	0
$757$	−15.8927	−0.577630	−0.288815	−	0.957385i	$-0.593261\pi$
$757$	−15.8927	−0.577630	−0.288815	+	0.957385i	$0.593261\pi$
$758$	21.7548	0.790172
$759$	0	0
$760$	0	0
$761$	−13.8843	−0.503305	−0.251652	−	0.967818i	$-0.580974\pi$
$761$	−13.8843	−0.503305	−0.251652	+	0.967818i	$0.580974\pi$
$762$	0	0
$763$	−7.84778	−0.284109
$764$	−2.59439	−0.0938619
$765$	0	0
$766$	−24.2907	−0.877660
$767$	−8.99386	−0.324749
$768$	0	0
$769$	−35.4063	−1.27678	−0.638391	−	0.769712i	$-0.720400\pi$
$769$	−35.4063	−1.27678	−0.638391	+	0.769712i	$0.720400\pi$
$770$	0	0
$771$	0	0
$772$	−6.59439	−0.237337
$773$	−0.488518	−0.0175708	−0.00878539	−	0.999961i	$-0.502797\pi$
$773$	−0.488518	−0.0175708	−0.00878539	+	0.999961i	$0.502797\pi$
$774$	0	0
$775$	0	0
$776$	−39.8948	−1.43214
$777$	0	0
$778$	21.2663	0.762435
$779$	2.05786	0.0737304
$780$	0	0
$781$	2.63931	0.0944419
$782$	17.5273	0.626775
$783$	0	0
$784$	30.3820	1.08507
$785$	0	0
$786$	0	0
$787$	1.99159	0.0709925	0.0354962	−	0.999370i	$-0.488699\pi$
$787$	1.99159	0.0709925	0.0354962	+	0.999370i	$0.488699\pi$
$788$	2.25112	0.0801927
$789$	0	0
$790$	0	0
$791$	−7.93230	−0.282040
$792$	0	0
$793$	2.83710	0.100748
$794$	−13.9323	−0.494439
$795$	0	0
$796$	−3.60197	−0.127668
$797$	−17.2702	−0.611742	−0.305871	−	0.952073i	$-0.598948\pi$
$797$	−17.2702	−0.611742	−0.305871	+	0.952073i	$0.598948\pi$
$798$	0	0
$799$	−5.60650	−0.198344
$800$	0	0
$801$	0	0
$802$	−30.4124	−1.07390
$803$	−2.01891	−0.0712458
$804$	0	0
$805$	0	0
$806$	14.2557	0.502134
$807$	0	0
$808$	−30.7649	−1.08230
$809$	56.5068	1.98667	0.993336	−	0.115254i	$-0.0367680\pi$
$809$	56.5068	1.98667	0.993336	+	0.115254i	$0.0367680\pi$
$810$	0	0
$811$	−8.77924	−0.308281	−0.154140	−	0.988049i	$-0.549261\pi$
$811$	−8.77924	−0.308281	−0.154140	+	0.988049i	$0.549261\pi$
$812$	0.232866	0.00817199
$813$	0	0
$814$	0.694722	0.0243500
$815$	0	0
$816$	0	0
$817$	−34.5380	−1.20833
$818$	−2.64384	−0.0924398
$819$	0	0
$820$	0	0
$821$	28.1568	0.982678	0.491339	−	0.870969i	$-0.336508\pi$
$821$	28.1568	0.982678	0.491339	+	0.870969i	$0.336508\pi$
$822$	0	0
$823$	−20.7442	−0.723096	−0.361548	−	0.932353i	$-0.617752\pi$
$823$	−20.7442	−0.723096	−0.361548	+	0.932353i	$0.617752\pi$
$824$	−19.7458	−0.687877
$825$	0	0
$826$	9.47641	0.329726
$827$	−9.12783	−0.317406	−0.158703	−	0.987326i	$-0.550731\pi$
$827$	−9.12783	−0.317406	−0.158703	+	0.987326i	$0.550731\pi$
$828$	0	0
$829$	31.8576	1.10646	0.553230	−	0.833028i	$-0.313395\pi$
$829$	31.8576	1.10646	0.553230	+	0.833028i	$0.313395\pi$
$830$	0	0
$831$	0	0
$832$	5.55640	0.192633
$833$	−32.8188	−1.13711
$834$	0	0
$835$	0	0
$836$	0.649149	0.0224513
$837$	0	0
$838$	54.6902	1.88924
$839$	−27.4413	−0.947380	−0.473690	−	0.880692i	$-0.657078\pi$
$839$	−27.4413	−0.947380	−0.473690	+	0.880692i	$0.657078\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−18.5814	−0.640359
$843$	0	0
$844$	3.64158	0.125348
$845$	0	0
$846$	0	0
$847$	6.88655	0.236625
$848$	1.56547	0.0537583
$849$	0	0
$850$	0	0
$851$	−3.55640	−0.121912
$852$	0	0
$853$	56.0515	1.91917	0.959584	−	0.281422i	$-0.0908061\pi$
$853$	56.0515	1.91917	0.959584	+	0.281422i	$0.0908061\pi$
$854$	−2.98932	−0.102292
$855$	0	0
$856$	−31.9551	−1.09220
$857$	−6.08452	−0.207843	−0.103922	−	0.994585i	$-0.533139\pi$
$857$	−6.08452	−0.207843	−0.103922	+	0.994585i	$0.533139\pi$
$858$	0	0
$859$	35.5936	1.21444	0.607218	−	0.794535i	$-0.292285\pi$
$859$	35.5936	1.21444	0.607218	+	0.794535i	$0.292285\pi$
$860$	0	0
$861$	0	0
$862$	−30.5236	−1.03964
$863$	−12.8287	−0.436694	−0.218347	−	0.975871i	$-0.570066\pi$
$863$	−12.8287	−0.436694	−0.218347	+	0.975871i	$0.570066\pi$
$864$	0	0
$865$	0	0
$866$	−22.8020	−0.774844
$867$	0	0
$868$	−2.34017	−0.0794306
$869$	−3.60197	−0.122188
$870$	0	0
$871$	5.26180	0.178289
$872$	−31.2253	−1.05742
$873$	0	0
$874$	−21.3295	−0.721481
$875$	0	0
$876$	0	0
$877$	1.50307	0.0507551	0.0253776	−	0.999678i	$-0.491921\pi$
$877$	1.50307	0.0507551	0.0253776	+	0.999678i	$0.491921\pi$
$878$	−27.4863	−0.927616
$879$	0	0
$880$	0	0
$881$	−23.4908	−0.791425	−0.395712	−	0.918375i	$-0.629502\pi$
$881$	−23.4908	−0.791425	−0.395712	+	0.918375i	$0.629502\pi$
$882$	0	0
$883$	29.0433	0.977385	0.488693	−	0.872456i	$-0.337474\pi$
$883$	29.0433	0.977385	0.488693	+	0.872456i	$0.337474\pi$
$884$	1.69102	0.0568753
$885$	0	0
$886$	51.7068	1.73712
$887$	−19.0700	−0.640307	−0.320153	−	0.947366i	$-0.603734\pi$
$887$	−19.0700	−0.640307	−0.320153	+	0.947366i	$0.603734\pi$
$888$	0	0
$889$	−13.1941	−0.442516
$890$	0	0
$891$	0	0
$892$	−4.04945	−0.135586
$893$	6.82273	0.228314
$894$	0	0
$895$	0	0
$896$	−8.45732	−0.282539
$897$	0	0
$898$	−10.8950	−0.363570
$899$	−10.0494	−0.335168
$900$	0	0
$901$	−1.69102	−0.0563362
$902$	0.152221	0.00506839
$903$	0	0
$904$	−31.5616	−1.04972
$905$	0	0
$906$	0	0
$907$	−5.54023	−0.183960	−0.0919802	−	0.995761i	$-0.529320\pi$
$907$	−5.54023	−0.183960	−0.0919802	+	0.995761i	$0.529320\pi$
$908$	−4.63090	−0.153682
$909$	0	0
$910$	0	0
$911$	−53.2990	−1.76587	−0.882937	−	0.469492i	$-0.844437\pi$
$911$	−53.2990	−1.76587	−0.882937	+	0.469492i	$0.844437\pi$
$912$	0	0
$913$	−0.810439	−0.0268216
$914$	8.95282	0.296133
$915$	0	0
$916$	8.62249	0.284895
$917$	−8.49693	−0.280593
$918$	0	0
$919$	37.5897	1.23997	0.619985	−	0.784614i	$-0.287139\pi$
$919$	37.5897	1.23997	0.619985	+	0.784614i	$0.287139\pi$
$920$	0	0
$921$	0	0
$922$	49.7296	1.63776
$923$	−8.36683	−0.275398
$924$	0	0
$925$	0	0
$926$	−2.21849	−0.0729041
$927$	0	0
$928$	2.06278	0.0677140
$929$	−37.3197	−1.22442	−0.612209	−	0.790696i	$-0.709719\pi$
$929$	−37.3197	−1.22442	−0.612209	+	0.790696i	$0.709719\pi$
$930$	0	0
$931$	39.9383	1.30892
$932$	4.60281	0.150770
$933$	0	0
$934$	−18.0860	−0.591790
$935$	0	0
$936$	0	0
$937$	22.8638	0.746927	0.373463	−	0.927645i	$-0.378170\pi$
$937$	22.8638	0.746927	0.373463	+	0.927645i	$0.378170\pi$
$938$	−5.54411	−0.181022
$939$	0	0
$940$	0	0
$941$	−0.523590	−0.0170686	−0.00853428	−	0.999964i	$-0.502717\pi$
$941$	−0.523590	−0.0170686	−0.00853428	+	0.999964i	$0.502717\pi$
$942$	0	0
$943$	−0.779243	−0.0253756
$944$	44.9093	1.46167
$945$	0	0
$946$	−2.55479	−0.0830633
$947$	−10.0228	−0.325697	−0.162848	−	0.986651i	$-0.552068\pi$
$947$	−10.0228	−0.325697	−0.162848	+	0.986651i	$0.552068\pi$
$948$	0	0
$949$	6.40012	0.207757
$950$	0	0
$951$	0	0
$952$	7.87277	0.255158
$953$	−8.15676	−0.264223	−0.132112	−	0.991235i	$-0.542176\pi$
$953$	−8.15676	−0.264223	−0.132112	+	0.991235i	$0.542176\pi$
$954$	0	0
$955$	0	0
$956$	−5.07838	−0.164246
$957$	0	0
$958$	−26.4261	−0.853789
$959$	8.55025	0.276102
$960$	0	0
$961$	69.9914	2.25779
$962$	−2.20233	−0.0710059
$963$	0	0
$964$	−5.41855	−0.174520
$965$	0	0
$966$	0	0
$967$	−15.7671	−0.507037	−0.253518	−	0.967331i	$-0.581588\pi$
$967$	−15.7671	−0.507037	−0.253518	+	0.967331i	$0.581588\pi$
$968$	27.4007	0.880691
$969$	0	0
$970$	0	0
$971$	17.8804	0.573810	0.286905	−	0.957959i	$-0.407374\pi$
$971$	17.8804	0.573810	0.286905	+	0.957959i	$0.407374\pi$
$972$	0	0
$973$	3.08822	0.0990037
$974$	6.31494	0.202344
$975$	0	0
$976$	−14.1666	−0.453462
$977$	−55.1071	−1.76303	−0.881517	−	0.472153i	$-0.843477\pi$
$977$	−55.1071	−1.76303	−0.881517	+	0.472153i	$0.843477\pi$
$978$	0	0
$979$	1.37751	0.0440255
$980$	0	0
$981$	0	0
$982$	−62.7540	−2.00256
$983$	−1.29687	−0.0413637	−0.0206818	−	0.999786i	$-0.506584\pi$
$983$	−1.29687	−0.0413637	−0.0206818	+	0.999786i	$0.506584\pi$
$984$	0	0
$985$	0	0
$986$	−7.65142	−0.243671
$987$	0	0
$988$	−2.05786	−0.0654692
$989$	13.0784	0.415868
$990$	0	0
$991$	−3.11942	−0.0990915	−0.0495458	−	0.998772i	$-0.515777\pi$
$991$	−3.11942	−0.0990915	−0.0495458	+	0.998772i	$0.515777\pi$
$992$	−20.7298	−0.658172
$993$	0	0
$994$	8.81575	0.279618
$995$	0	0
$996$	0	0
$997$	−30.2472	−0.957940	−0.478970	−	0.877831i	$-0.658990\pi$
$997$	−30.2472	−0.957940	−0.478970	+	0.877831i	$0.658990\pi$
$998$	−28.4292	−0.899912
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bh.1.2		3
3.2	odd	2		725.2.a.d.1.2		3
5.4	even	2		1305.2.a.o.1.2		3
15.2	even	4		725.2.b.d.349.2		6
15.8	even	4		725.2.b.d.349.5		6
15.14	odd	2		145.2.a.d.1.2	✓	3
60.59	even	2		2320.2.a.s.1.1		3
105.104	even	2		7105.2.a.p.1.2		3
120.29	odd	2		9280.2.a.bu.1.1		3
120.59	even	2		9280.2.a.bm.1.3		3
435.434	odd	2		4205.2.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.d.1.2	✓	3	15.14	odd	2
725.2.a.d.1.2		3	3.2	odd	2
725.2.b.d.349.2		6	15.2	even	4
725.2.b.d.349.5		6	15.8	even	4
1305.2.a.o.1.2		3	5.4	even	2
2320.2.a.s.1.1		3	60.59	even	2
4205.2.a.e.1.2		3	435.434	odd	2
6525.2.a.bh.1.2		3	1.1	even	1	trivial
7105.2.a.p.1.2		3	105.104	even	2
9280.2.a.bm.1.3		3	120.59	even	2
9280.2.a.bu.1.1		3	120.29	odd	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)

\(f(q)\)	\(=\)	\(q+1.53919 q^{2} +0.369102 q^{4} -0.630898 q^{7} -2.51026 q^{8} -0.290725 q^{11} +0.921622 q^{13} -0.971071 q^{14} -4.60197 q^{16} +4.97107 q^{17} -6.04945 q^{19} -0.447480 q^{22} +2.29072 q^{23} +1.41855 q^{26} -0.232866 q^{28} -1.00000 q^{29} +10.0494 q^{31} -2.06278 q^{32} +7.65142 q^{34} -1.55252 q^{37} -9.31124 q^{38} -0.340173 q^{41} +5.70928 q^{43} -0.107307 q^{44} +3.52586 q^{46} -1.12783 q^{47} -6.60197 q^{49} +0.340173 q^{52} -0.340173 q^{53} +1.58372 q^{56} -1.53919 q^{58} -9.75872 q^{59} +3.07838 q^{61} +15.4680 q^{62} +6.02893 q^{64} +5.70928 q^{67} +1.83483 q^{68} -9.07838 q^{71} +6.94441 q^{73} -2.38962 q^{74} -2.23287 q^{76} +0.183417 q^{77} +12.3896 q^{79} -0.523590 q^{82} +2.78765 q^{83} +8.78765 q^{86} +0.729794 q^{88} -4.73820 q^{89} -0.581449 q^{91} +0.845512 q^{92} -1.73594 q^{94} +15.8927 q^{97} -10.1617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8} - 8 q^{11} + 6 q^{13} + 12 q^{14} + 5 q^{16} - 2 q^{22} + 14 q^{23} - 10 q^{26} + 22 q^{28} - 3 q^{29} + 12 q^{31} + 11 q^{32} - 14 q^{34} - 4 q^{37} - 2 q^{38}+ \cdots + 23 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 5 * q^4 + 2 * q^7 + 9 * q^8 - 8 * q^11 + 6 * q^13 + 12 * q^14 + 5 * q^16 - 2 * q^22 + 14 * q^23 - 10 * q^26 + 22 * q^28 - 3 * q^29 + 12 * q^31 + 11 * q^32 - 14 * q^34 - 4 * q^37 - 2 * q^38 + 10 * q^41 + 10 * q^43 - 12 * q^44 + 8 * q^46 + 18 * q^47 - q^49 - 10 * q^52 + 10 * q^53 + 32 * q^56 - 3 * q^58 - 4 * q^59 + 6 * q^61 + 14 * q^62 + 33 * q^64 + 10 * q^67 - 36 * q^68 - 24 * q^71 + 4 * q^73 + 22 * q^74 + 16 * q^76 - 4 * q^77 + 8 * q^79 + 14 * q^82 - 2 * q^83 + 16 * q^86 - 38 * q^88 - 22 * q^89 - 16 * q^91 + 22 * q^92 + 36 * q^97 + 23 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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