LMFDB - Embedded newform 925.2.a.l.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(1,925)] 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("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,-5,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.38616218697$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 $ x^9 - 4x^8 - 6x^7 + 30x^6 + 15x^5 - 70x^4 - 22x^3 + 44x^2 + 4x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.296889$ of defining polynomial
Character	$\chi$	$=$	925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.703111 q^{2} +1.64070 q^{3} -1.50564 q^{4} -1.15359 q^{6} -0.501390 q^{7} +2.46485 q^{8} -0.308108 q^{9} -1.80401 q^{11} -2.47029 q^{12} -1.63218 q^{13} +0.352533 q^{14} +1.27821 q^{16} -1.92844 q^{17} +0.216634 q^{18} -0.869173 q^{19} -0.822630 q^{21} +1.26842 q^{22} -4.86522 q^{23} +4.04408 q^{24} +1.14761 q^{26} -5.42761 q^{27} +0.754910 q^{28} -3.15000 q^{29} +2.39134 q^{31} -5.82842 q^{32} -2.95984 q^{33} +1.35591 q^{34} +0.463898 q^{36} +1.00000 q^{37} +0.611125 q^{38} -2.67792 q^{39} +8.24040 q^{41} +0.578400 q^{42} -11.6931 q^{43} +2.71619 q^{44} +3.42079 q^{46} +1.43311 q^{47} +2.09715 q^{48} -6.74861 q^{49} -3.16400 q^{51} +2.45747 q^{52} -10.7692 q^{53} +3.81621 q^{54} -1.23585 q^{56} -1.42605 q^{57} +2.21480 q^{58} +8.71701 q^{59} -7.84555 q^{61} -1.68138 q^{62} +0.154482 q^{63} +1.54161 q^{64} +2.08110 q^{66} -11.8599 q^{67} +2.90353 q^{68} -7.98236 q^{69} -6.06069 q^{71} -0.759439 q^{72} +5.56086 q^{73} -0.703111 q^{74} +1.30866 q^{76} +0.904514 q^{77} +1.88288 q^{78} +7.44924 q^{79} -7.98075 q^{81} -5.79391 q^{82} +4.70795 q^{83} +1.23858 q^{84} +8.22154 q^{86} -5.16820 q^{87} -4.44662 q^{88} +5.16332 q^{89} +0.818361 q^{91} +7.32524 q^{92} +3.92347 q^{93} -1.00764 q^{94} -9.56268 q^{96} +6.48639 q^{97} +4.74502 q^{98} +0.555830 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} - 8 q^{3} + 11 q^{4} + 2 q^{6} - 8 q^{7} - 15 q^{8} + 13 q^{9} - 16 q^{12} - 6 q^{13} - 4 q^{14} + 11 q^{16} - 18 q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{21} - 6 q^{22} - 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) $ 9 * q - 5 * q^2 - 8 * q^3 + 11 * q^4 + 2 * q^6 - 8 * q^7 - 15 * q^8 + 13 * q^9 - 16 * q^12 - 6 * q^13 - 4 * q^14 + 11 * q^16 - 18 * q^17 + 3 * q^18 - 4 * q^19 + 4 * q^21 - 6 * q^22 - 16 * q^23 + 6 * q^24 - 6 * q^26 - 32 * q^27 + 20 * q^28 - 2 * q^29 - 6 * q^31 - 35 * q^32 - 6 * q^33 + 6 * q^34 + 11 * q^36 + 9 * q^37 - 4 * q^38 - 16 * q^39 + 2 * q^41 - 4 * q^42 - 14 * q^43 + 4 * q^44 + 16 * q^46 - 34 * q^47 - 6 * q^48 - 7 * q^49 + 8 * q^51 - 16 * q^52 + 4 * q^53 - 18 * q^54 - 24 * q^56 + 4 * q^57 + 22 * q^58 + 10 * q^59 + 8 * q^61 - 26 * q^62 - 18 * q^63 + 15 * q^64 - 34 * q^66 - 16 * q^67 - 36 * q^68 - 10 * q^69 + 8 * q^71 - 9 * q^72 - 8 * q^73 - 5 * q^74 - 48 * q^76 - 14 * q^77 + 76 * q^78 - 26 * q^79 + 13 * q^81 + 4 * q^82 - 70 * q^83 - 24 * q^84 - 16 * q^86 + 12 * q^87 + 36 * q^88 - 8 * q^89 - 18 * q^91 - 16 * q^92 - 2 * q^93 - 26 * q^94 + 52 * q^96 + 2 * q^97 - 21 * q^98 + 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.703111	−0.497174	−0.248587	−	0.968610i	$-0.579966\pi$
$2$	−0.703111	−0.497174	−0.248587	+	0.968610i	$0.579966\pi$
$3$	1.64070	0.947258	0.473629	−	0.880724i	$-0.342944\pi$
$3$	1.64070	0.947258	0.473629	+	0.880724i	$0.342944\pi$
$4$	−1.50564	−0.752818
$5$	0	0
$5$	0	0
$6$	−1.15359	−0.470952
$7$	−0.501390	−0.189508	−0.0947538	−	0.995501i	$-0.530206\pi$
$7$	−0.501390	−0.189508	−0.0947538	+	0.995501i	$0.530206\pi$
$8$	2.46485	0.871456
$9$	−0.308108	−0.102703
$10$	0	0
$11$	−1.80401	−0.543931	−0.271965	−	0.962307i	$-0.587674\pi$
$11$	−1.80401	−0.543931	−0.271965	+	0.962307i	$0.587674\pi$
$12$	−2.47029	−0.713112
$13$	−1.63218	−0.452686	−0.226343	−	0.974048i	$-0.572677\pi$
$13$	−1.63218	−0.452686	−0.226343	+	0.974048i	$0.572677\pi$
$14$	0.352533	0.0942183
$15$	0	0
$16$	1.27821	0.319552
$17$	−1.92844	−0.467717	−0.233858	−	0.972271i	$-0.575135\pi$
$17$	−1.92844	−0.467717	−0.233858	+	0.972271i	$0.575135\pi$
$18$	0.216634	0.0510611
$19$	−0.869173	−0.199402	−0.0997010	−	0.995017i	$-0.531789\pi$
$19$	−0.869173	−0.199402	−0.0997010	+	0.995017i	$0.531789\pi$
$20$	0	0
$21$	−0.822630	−0.179513
$22$	1.26842	0.270428
$23$	−4.86522	−1.01447	−0.507234	−	0.861808i	$-0.669332\pi$
$23$	−4.86522	−1.01447	−0.507234	+	0.861808i	$0.669332\pi$
$24$	4.04408	0.825494
$25$	0	0
$26$	1.14761	0.225064
$27$	−5.42761	−1.04454
$28$	0.754910	0.142665
$29$	−3.15000	−0.584940	−0.292470	−	0.956275i	$-0.594477\pi$
$29$	−3.15000	−0.584940	−0.292470	+	0.956275i	$0.594477\pi$
$30$	0	0
$31$	2.39134	0.429498	0.214749	−	0.976669i	$-0.431107\pi$
$31$	2.39134	0.429498	0.214749	+	0.976669i	$0.431107\pi$
$32$	−5.82842	−1.03033
$33$	−2.95984	−0.515243
$34$	1.35591	0.232537
$35$	0	0
$36$	0.463898	0.0773163
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0.611125	0.0991375
$39$	−2.67792	−0.428811
$40$	0	0
$41$	8.24040	1.28693	0.643467	−	0.765474i	$-0.277495\pi$
$41$	8.24040	1.28693	0.643467	+	0.765474i	$0.277495\pi$
$42$	0.578400	0.0892490
$43$	−11.6931	−1.78318	−0.891590	−	0.452844i	$-0.850410\pi$
$43$	−11.6931	−1.78318	−0.891590	+	0.452844i	$0.850410\pi$
$44$	2.71619	0.409481
$45$	0	0
$46$	3.42079	0.504368
$47$	1.43311	0.209041	0.104521	−	0.994523i	$-0.466669\pi$
$47$	1.43311	0.209041	0.104521	+	0.994523i	$0.466669\pi$
$48$	2.09715	0.302698
$49$	−6.74861	−0.964087
$50$	0	0
$51$	−3.16400	−0.443048
$52$	2.45747	0.340790
$53$	−10.7692	−1.47927	−0.739633	−	0.673010i	$-0.765001\pi$
$53$	−10.7692	−1.47927	−0.739633	+	0.673010i	$0.765001\pi$
$54$	3.81621	0.519320
$55$	0	0
$56$	−1.23585	−0.165148
$57$	−1.42605	−0.188885
$58$	2.21480	0.290817
$59$	8.71701	1.13486	0.567429	−	0.823422i	$-0.307938\pi$
$59$	8.71701	1.13486	0.567429	+	0.823422i	$0.307938\pi$
$60$	0	0
$61$	−7.84555	−1.00452	−0.502260	−	0.864717i	$-0.667498\pi$
$61$	−7.84555	−1.00452	−0.502260	+	0.864717i	$0.667498\pi$
$62$	−1.68138	−0.213535
$63$	0.154482	0.0194629
$64$	1.54161	0.192701
$65$	0	0
$66$	2.08110	0.256165
$67$	−11.8599	−1.44891	−0.724456	−	0.689321i	$-0.757909\pi$
$67$	−11.8599	−1.44891	−0.724456	+	0.689321i	$0.757909\pi$
$68$	2.90353	0.352105
$69$	−7.98236	−0.960963
$70$	0	0
$71$	−6.06069	−0.719272	−0.359636	−	0.933093i	$-0.617099\pi$
$71$	−6.06069	−0.719272	−0.359636	+	0.933093i	$0.617099\pi$
$72$	−0.759439	−0.0895007
$73$	5.56086	0.650849	0.325424	−	0.945568i	$-0.394493\pi$
$73$	5.56086	0.650849	0.325424	+	0.945568i	$0.394493\pi$
$74$	−0.703111	−0.0817350
$75$	0	0
$76$	1.30866	0.150113
$77$	0.904514	0.103079
$78$	1.88288	0.213194
$79$	7.44924	0.838105	0.419052	−	0.907962i	$-0.362362\pi$
$79$	7.44924	0.838105	0.419052	+	0.907962i	$0.362362\pi$
$80$	0	0
$81$	−7.98075	−0.886750
$82$	−5.79391	−0.639830
$83$	4.70795	0.516765	0.258382	−	0.966043i	$-0.416811\pi$
$83$	4.70795	0.516765	0.258382	+	0.966043i	$0.416811\pi$
$84$	1.23858	0.135140
$85$	0	0
$86$	8.22154	0.886551
$87$	−5.16820	−0.554089
$88$	−4.44662	−0.474012
$89$	5.16332	0.547311	0.273656	−	0.961828i	$-0.411767\pi$
$89$	5.16332	0.547311	0.273656	+	0.961828i	$0.411767\pi$
$90$	0	0
$91$	0.818361	0.0857875
$92$	7.32524	0.763709
$93$	3.92347	0.406845
$94$	−1.00764	−0.103930
$95$	0	0
$96$	−9.56268	−0.975987
$97$	6.48639	0.658593	0.329297	−	0.944227i	$-0.393188\pi$
$97$	6.48639	0.658593	0.329297	+	0.944227i	$0.393188\pi$
$98$	4.74502	0.479319
$99$	0.555830	0.0558631
$100$	0	0
$101$	2.50611	0.249367	0.124684	−	0.992197i	$-0.460208\pi$
$101$	2.50611	0.249367	0.124684	+	0.992197i	$0.460208\pi$
$102$	2.22464	0.220272
$103$	6.40079	0.630688	0.315344	−	0.948977i	$-0.397880\pi$
$103$	6.40079	0.630688	0.315344	+	0.948977i	$0.397880\pi$
$104$	−4.02309	−0.394496
$105$	0	0
$106$	7.57195	0.735453
$107$	−11.6711	−1.12829	−0.564146	−	0.825675i	$-0.690794\pi$
$107$	−11.6711	−1.12829	−0.564146	+	0.825675i	$0.690794\pi$
$108$	8.17200	0.786351
$109$	7.93573	0.760105	0.380053	−	0.924965i	$-0.375906\pi$
$109$	7.93573	0.760105	0.380053	+	0.924965i	$0.375906\pi$
$110$	0	0
$111$	1.64070	0.155728
$112$	−0.640881	−0.0605575
$113$	−16.2261	−1.52643	−0.763213	−	0.646146i	$-0.776380\pi$
$113$	−16.2261	−1.52643	−0.763213	+	0.646146i	$0.776380\pi$
$114$	1.00267	0.0939088
$115$	0	0
$116$	4.74275	0.440353
$117$	0.502888	0.0464920
$118$	−6.12902	−0.564222
$119$	0.966903	0.0886358
$120$	0	0
$121$	−7.74553	−0.704139
$122$	5.51629	0.499421
$123$	13.5200	1.21906
$124$	−3.60049	−0.323333
$125$	0	0
$126$	−0.108618	−0.00967646
$127$	9.56408	0.848675	0.424337	−	0.905504i	$-0.360507\pi$
$127$	9.56408	0.848675	0.424337	+	0.905504i	$0.360507\pi$
$128$	10.5729	0.934523
$129$	−19.1848	−1.68913
$130$	0	0
$131$	−20.2974	−1.77340	−0.886698	−	0.462350i	$-0.847006\pi$
$131$	−20.2974	−1.77340	−0.886698	+	0.462350i	$0.847006\pi$
$132$	4.45644	0.387884
$133$	0.435795	0.0377882
$134$	8.33879	0.720362
$135$	0	0
$136$	−4.75333	−0.407594
$137$	11.0879	0.947300	0.473650	−	0.880713i	$-0.342936\pi$
$137$	11.0879	0.947300	0.473650	+	0.880713i	$0.342936\pi$
$138$	5.61248	0.477766
$139$	14.1716	1.20202	0.601009	−	0.799242i	$-0.294765\pi$
$139$	14.1716	1.20202	0.601009	+	0.799242i	$0.294765\pi$
$140$	0	0
$141$	2.35131	0.198016
$142$	4.26134	0.357603
$143$	2.94448	0.246230
$144$	−0.393826	−0.0328188
$145$	0	0
$146$	−3.90990	−0.323585
$147$	−11.0724	−0.913239
$148$	−1.50564	−0.123762
$149$	−17.5785	−1.44008	−0.720042	−	0.693930i	$-0.755878\pi$
$149$	−17.5785	−1.44008	−0.720042	+	0.693930i	$0.755878\pi$
$150$	0	0
$151$	10.4870	0.853422	0.426711	−	0.904388i	$-0.359672\pi$
$151$	10.4870	0.853422	0.426711	+	0.904388i	$0.359672\pi$
$152$	−2.14238	−0.173770
$153$	0.594168	0.0480357
$154$	−0.635974	−0.0512482
$155$	0	0
$156$	4.03197	0.322816
$157$	14.1135	1.12638	0.563189	−	0.826328i	$-0.309574\pi$
$157$	14.1135	1.12638	0.563189	+	0.826328i	$0.309574\pi$
$158$	−5.23764	−0.416684
$159$	−17.6690	−1.40125
$160$	0	0
$161$	2.43937	0.192249
$162$	5.61135	0.440869
$163$	−0.261155	−0.0204553	−0.0102276	−	0.999948i	$-0.503256\pi$
$163$	−0.261155	−0.0204553	−0.0102276	+	0.999948i	$0.503256\pi$
$164$	−12.4070	−0.968826
$165$	0	0
$166$	−3.31021	−0.256922
$167$	−11.7929	−0.912560	−0.456280	−	0.889836i	$-0.650818\pi$
$167$	−11.7929	−0.912560	−0.456280	+	0.889836i	$0.650818\pi$
$168$	−2.02766	−0.156437
$169$	−10.3360	−0.795075
$170$	0	0
$171$	0.267799	0.0204791
$172$	17.6055	1.34241
$173$	−13.5257	−1.02834	−0.514170	−	0.857688i	$-0.671900\pi$
$173$	−13.5257	−1.02834	−0.514170	+	0.857688i	$0.671900\pi$
$174$	3.63382	0.275479
$175$	0	0
$176$	−2.30591	−0.173814
$177$	14.3020	1.07500
$178$	−3.63039	−0.272109
$179$	0.443280	0.0331323	0.0165661	−	0.999863i	$-0.494727\pi$
$179$	0.443280	0.0331323	0.0165661	+	0.999863i	$0.494727\pi$
$180$	0	0
$181$	11.4380	0.850182	0.425091	−	0.905151i	$-0.360242\pi$
$181$	11.4380	0.850182	0.425091	+	0.905151i	$0.360242\pi$
$182$	−0.575398	−0.0426513
$183$	−12.8722	−0.951539
$184$	−11.9920	−0.884064
$185$	0	0
$186$	−2.75863	−0.202273
$187$	3.47894	0.254405
$188$	−2.15775	−0.157370
$189$	2.72135	0.197949
$190$	0	0
$191$	10.1652	0.735531	0.367765	−	0.929919i	$-0.380123\pi$
$191$	10.1652	0.735531	0.367765	+	0.929919i	$0.380123\pi$
$192$	2.52932	0.182538
$193$	8.94479	0.643860	0.321930	−	0.946763i	$-0.395668\pi$
$193$	8.94479	0.643860	0.321930	+	0.946763i	$0.395668\pi$
$194$	−4.56065	−0.327436
$195$	0	0
$196$	10.1609	0.725782
$197$	8.65157	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$197$	8.65157	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$198$	−0.390810	−0.0277737
$199$	10.2145	0.724084	0.362042	−	0.932162i	$-0.382080\pi$
$199$	10.2145	0.724084	0.362042	+	0.932162i	$0.382080\pi$
$200$	0	0
$201$	−19.4584	−1.37249
$202$	−1.76207	−0.123979
$203$	1.57938	0.110851
$204$	4.76383	0.333534
$205$	0	0
$206$	−4.50046	−0.313562
$207$	1.49901	0.104188
$208$	−2.08627	−0.144657
$209$	1.56800	0.108461
$210$	0	0
$211$	13.6414	0.939113	0.469557	−	0.882902i	$-0.344414\pi$
$211$	13.6414	0.939113	0.469557	+	0.882902i	$0.344414\pi$
$212$	16.2145	1.11362
$213$	−9.94377	−0.681336
$214$	8.20611	0.560958
$215$	0	0
$216$	−13.3782	−0.910274
$217$	−1.19899	−0.0813930
$218$	−5.57970	−0.377905
$219$	9.12369	0.616522
$220$	0	0
$221$	3.14758	0.211729
$222$	−1.15359	−0.0774241
$223$	6.66288	0.446180	0.223090	−	0.974798i	$-0.428386\pi$
$223$	6.66288	0.446180	0.223090	+	0.974798i	$0.428386\pi$
$224$	2.92231	0.195255
$225$	0	0
$226$	11.4088	0.758900
$227$	−5.71859	−0.379556	−0.189778	−	0.981827i	$-0.560777\pi$
$227$	−5.71859	−0.379556	−0.189778	+	0.981827i	$0.560777\pi$
$228$	2.14711	0.142196
$229$	29.9549	1.97948	0.989738	−	0.142895i	$-0.0456412\pi$
$229$	29.9549	1.97948	0.989738	+	0.142895i	$0.0456412\pi$
$230$	0	0
$231$	1.48404	0.0976424
$232$	−7.76427	−0.509750
$233$	25.4277	1.66583	0.832913	−	0.553404i	$-0.186671\pi$
$233$	25.4277	1.66583	0.832913	+	0.553404i	$0.186671\pi$
$234$	−0.353586	−0.0231147
$235$	0	0
$236$	−13.1246	−0.854341
$237$	12.2220	0.793902
$238$	−0.679840	−0.0440675
$239$	−10.9704	−0.709615	−0.354808	−	0.934939i	$-0.615454\pi$
$239$	−10.9704	−0.709615	−0.354808	+	0.934939i	$0.615454\pi$
$240$	0	0
$241$	23.8516	1.53642	0.768208	−	0.640200i	$-0.221149\pi$
$241$	23.8516	1.53642	0.768208	+	0.640200i	$0.221149\pi$
$242$	5.44597	0.350080
$243$	3.18882	0.204563
$244$	11.8125	0.756220
$245$	0	0
$246$	−9.50606	−0.606084
$247$	1.41865	0.0902666
$248$	5.89430	0.374288
$249$	7.72433	0.489509
$250$	0	0
$251$	18.9886	1.19855	0.599275	−	0.800543i	$-0.295455\pi$
$251$	18.9886	1.19855	0.599275	+	0.800543i	$0.295455\pi$
$252$	−0.232594	−0.0146520
$253$	8.77692	0.551800
$254$	−6.72461	−0.421939
$255$	0	0
$256$	−10.5172	−0.657322
$257$	−5.12928	−0.319956	−0.159978	−	0.987121i	$-0.551142\pi$
$257$	−5.12928	−0.319956	−0.159978	+	0.987121i	$0.551142\pi$
$258$	13.4891	0.839793
$259$	−0.501390	−0.0311549
$260$	0	0
$261$	0.970538	0.0600748
$262$	14.2713	0.881687
$263$	−27.1591	−1.67470	−0.837351	−	0.546666i	$-0.815897\pi$
$263$	−27.1591	−1.67470	−0.837351	+	0.546666i	$0.815897\pi$
$264$	−7.29557	−0.449011
$265$	0	0
$266$	−0.306412	−0.0187873
$267$	8.47146	0.518445
$268$	17.8566	1.09077
$269$	19.0743	1.16298	0.581492	−	0.813552i	$-0.302469\pi$
$269$	19.0743	1.16298	0.581492	+	0.813552i	$0.302469\pi$
$270$	0	0
$271$	−18.0630	−1.09725	−0.548626	−	0.836068i	$-0.684849\pi$
$271$	−18.0630	−1.09725	−0.548626	+	0.836068i	$0.684849\pi$
$272$	−2.46495	−0.149460
$273$	1.34268	0.0812629
$274$	−7.79600	−0.470973
$275$	0	0
$276$	12.0185	0.723430
$277$	−22.9367	−1.37813	−0.689067	−	0.724698i	$-0.741979\pi$
$277$	−22.9367	−1.37813	−0.689067	+	0.724698i	$0.741979\pi$
$278$	−9.96420	−0.597613
$279$	−0.736790	−0.0441105
$280$	0	0
$281$	14.1793	0.845869	0.422934	−	0.906160i	$-0.361000\pi$
$281$	14.1793	0.845869	0.422934	+	0.906160i	$0.361000\pi$
$282$	−1.65323	−0.0984484
$283$	8.55472	0.508525	0.254263	−	0.967135i	$-0.418167\pi$
$283$	8.55472	0.508525	0.254263	+	0.967135i	$0.418167\pi$
$284$	9.12519	0.541480
$285$	0	0
$286$	−2.07030	−0.122419
$287$	−4.13165	−0.243884
$288$	1.79578	0.105817
$289$	−13.2811	−0.781241
$290$	0	0
$291$	10.6422	0.623858
$292$	−8.37262	−0.489971
$293$	−26.2220	−1.53191	−0.765953	−	0.642896i	$-0.777733\pi$
$293$	−26.2220	−1.53191	−0.765953	+	0.642896i	$0.777733\pi$
$294$	7.78515	0.454039
$295$	0	0
$296$	2.46485	0.143266
$297$	9.79148	0.568159
$298$	12.3596	0.715973
$299$	7.94093	0.459236
$300$	0	0
$301$	5.86280	0.337926
$302$	−7.37354	−0.424300
$303$	4.11177	0.236215
$304$	−1.11098	−0.0637193
$305$	0	0
$306$	−0.417766	−0.0238821
$307$	−20.8170	−1.18809	−0.594046	−	0.804431i	$-0.702470\pi$
$307$	−20.8170	−1.18809	−0.594046	+	0.804431i	$0.702470\pi$
$308$	−1.36187	−0.0775997
$309$	10.5018	0.597425
$310$	0	0
$311$	15.9789	0.906078	0.453039	−	0.891491i	$-0.350340\pi$
$311$	15.9789	0.906078	0.453039	+	0.891491i	$0.350340\pi$
$312$	−6.60068	−0.373690
$313$	4.27429	0.241597	0.120799	−	0.992677i	$-0.461454\pi$
$313$	4.27429	0.241597	0.120799	+	0.992677i	$0.461454\pi$
$314$	−9.92333	−0.560006
$315$	0	0
$316$	−11.2158	−0.630940
$317$	6.19424	0.347903	0.173952	−	0.984754i	$-0.444346\pi$
$317$	6.19424	0.347903	0.173952	+	0.984754i	$0.444346\pi$
$318$	12.4233	0.696664
$319$	5.68264	0.318167
$320$	0	0
$321$	−19.1488	−1.06878
$322$	−1.71515	−0.0955815
$323$	1.67615	0.0932636
$324$	12.0161	0.667561
$325$	0	0
$326$	0.183621	0.0101698
$327$	13.0201	0.720015
$328$	20.3113	1.12151
$329$	−0.718549	−0.0396149
$330$	0	0
$331$	−32.0400	−1.76108	−0.880540	−	0.473972i	$-0.842820\pi$
$331$	−32.0400	−1.76108	−0.880540	+	0.473972i	$0.842820\pi$
$332$	−7.08846	−0.389030
$333$	−0.308108	−0.0168842
$334$	8.29169	0.453701
$335$	0	0
$336$	−1.05149	−0.0573636
$337$	−12.5183	−0.681917	−0.340959	−	0.940078i	$-0.610752\pi$
$337$	−12.5183	−0.681917	−0.340959	+	0.940078i	$0.610752\pi$
$338$	7.26733	0.395291
$339$	−26.6222	−1.44592
$340$	0	0
$341$	−4.31401	−0.233617
$342$	−0.188292	−0.0101817
$343$	6.89341	0.372209
$344$	−28.8217	−1.55396
$345$	0	0
$346$	9.51007	0.511264
$347$	−3.30472	−0.177407	−0.0887033	−	0.996058i	$-0.528272\pi$
$347$	−3.30472	−0.177407	−0.0887033	+	0.996058i	$0.528272\pi$
$348$	7.78142	0.417128
$349$	−4.45130	−0.238273	−0.119136	−	0.992878i	$-0.538013\pi$
$349$	−4.45130	−0.238273	−0.119136	+	0.992878i	$0.538013\pi$
$350$	0	0
$351$	8.85886	0.472851
$352$	10.5146	0.560428
$353$	−31.9145	−1.69864	−0.849319	−	0.527879i	$-0.822987\pi$
$353$	−31.9145	−1.69864	−0.849319	+	0.527879i	$0.822987\pi$
$354$	−10.0559	−0.534464
$355$	0	0
$356$	−7.77408	−0.412026
$357$	1.58640	0.0839610
$358$	−0.311675	−0.0164725
$359$	31.8986	1.68354	0.841771	−	0.539835i	$-0.181513\pi$
$359$	31.8986	1.68354	0.841771	+	0.539835i	$0.181513\pi$
$360$	0	0
$361$	−18.2445	−0.960239
$362$	−8.04220	−0.422689
$363$	−12.7081	−0.667002
$364$	−1.23215	−0.0645823
$365$	0	0
$366$	9.05057	0.473081
$367$	22.7089	1.18540	0.592698	−	0.805425i	$-0.298063\pi$
$367$	22.7089	1.18540	0.592698	+	0.805425i	$0.298063\pi$
$368$	−6.21876	−0.324175
$369$	−2.53893	−0.132171
$370$	0	0
$371$	5.39958	0.280332
$372$	−5.90732	−0.306280
$373$	28.7105	1.48657	0.743287	−	0.668973i	$-0.233266\pi$
$373$	28.7105	1.48657	0.743287	+	0.668973i	$0.233266\pi$
$374$	−2.44608	−0.126484
$375$	0	0
$376$	3.53241	0.182170
$377$	5.14138	0.264794
$378$	−1.91341	−0.0984151
$379$	30.3515	1.55905	0.779526	−	0.626370i	$-0.215460\pi$
$379$	30.3515	1.55905	0.779526	+	0.626370i	$0.215460\pi$
$380$	0	0
$381$	15.6918	0.803914
$382$	−7.14729	−0.365687
$383$	−35.4366	−1.81073	−0.905363	−	0.424639i	$-0.860401\pi$
$383$	−35.4366	−1.81073	−0.905363	+	0.424639i	$0.860401\pi$
$384$	17.3470	0.885234
$385$	0	0
$386$	−6.28918	−0.320111
$387$	3.60273	0.183137
$388$	−9.76614	−0.495801
$389$	−17.2794	−0.876102	−0.438051	−	0.898950i	$-0.644331\pi$
$389$	−17.2794	−0.876102	−0.438051	+	0.898950i	$0.644331\pi$
$390$	0	0
$391$	9.38230	0.474483
$392$	−16.6343	−0.840159
$393$	−33.3020	−1.67986
$394$	−6.08301	−0.306458
$395$	0	0
$396$	−0.836878	−0.0420547
$397$	4.68683	0.235225	0.117613	−	0.993060i	$-0.462476\pi$
$397$	4.68683	0.235225	0.117613	+	0.993060i	$0.462476\pi$
$398$	−7.18190	−0.359996
$399$	0.715008	0.0357951
$400$	0	0
$401$	−22.3427	−1.11574	−0.557871	−	0.829928i	$-0.688382\pi$
$401$	−22.3427	−1.11574	−0.557871	+	0.829928i	$0.688382\pi$
$402$	13.6814	0.682368
$403$	−3.90311	−0.194428
$404$	−3.77329	−0.187728
$405$	0	0
$406$	−1.11048	−0.0551121
$407$	−1.80401	−0.0894216
$408$	−7.79878	−0.386097
$409$	−5.17691	−0.255982	−0.127991	−	0.991775i	$-0.540853\pi$
$409$	−5.17691	−0.255982	−0.127991	+	0.991775i	$0.540853\pi$
$410$	0	0
$411$	18.1918	0.897338
$412$	−9.63725	−0.474793
$413$	−4.37062	−0.215064
$414$	−1.05397	−0.0517998
$415$	0	0
$416$	9.51306	0.466416
$417$	23.2513	1.13862
$418$	−1.10248	−0.0539239
$419$	−15.8651	−0.775063	−0.387531	−	0.921857i	$-0.626672\pi$
$419$	−15.8651	−0.775063	−0.387531	+	0.921857i	$0.626672\pi$
$420$	0	0
$421$	6.46992	0.315325	0.157662	−	0.987493i	$-0.449604\pi$
$421$	6.46992	0.315325	0.157662	+	0.987493i	$0.449604\pi$
$422$	−9.59142	−0.466903
$423$	−0.441553	−0.0214691
$424$	−26.5445	−1.28912
$425$	0	0
$426$	6.99157	0.338743
$427$	3.93368	0.190364
$428$	17.5725	0.849398
$429$	4.83101	0.233243
$430$	0	0
$431$	−24.9727	−1.20289	−0.601445	−	0.798914i	$-0.705408\pi$
$431$	−24.9727	−1.20289	−0.601445	+	0.798914i	$0.705408\pi$
$432$	−6.93761	−0.333786
$433$	38.7308	1.86128	0.930641	−	0.365934i	$-0.119250\pi$
$433$	38.7308	1.86128	0.930641	+	0.365934i	$0.119250\pi$
$434$	0.843026	0.0404665
$435$	0	0
$436$	−11.9483	−0.572220
$437$	4.22872	0.202287
$438$	−6.41496	−0.306519
$439$	−32.8425	−1.56749	−0.783743	−	0.621085i	$-0.786692\pi$
$439$	−32.8425	−1.56749	−0.783743	+	0.621085i	$0.786692\pi$
$440$	0	0
$441$	2.07930	0.0990142
$442$	−2.21310	−0.105266
$443$	−24.4262	−1.16052	−0.580262	−	0.814430i	$-0.697050\pi$
$443$	−24.4262	−1.16052	−0.580262	+	0.814430i	$0.697050\pi$
$444$	−2.47029	−0.117235
$445$	0	0
$446$	−4.68475	−0.221829
$447$	−28.8410	−1.36413
$448$	−0.772947	−0.0365183
$449$	−23.8188	−1.12408	−0.562040	−	0.827110i	$-0.689983\pi$
$449$	−23.8188	−1.12408	−0.562040	+	0.827110i	$0.689983\pi$
$450$	0	0
$451$	−14.8658	−0.700003
$452$	24.4307	1.14912
$453$	17.2061	0.808411
$454$	4.02080	0.188706
$455$	0	0
$456$	−3.51500	−0.164605
$457$	−35.0439	−1.63928	−0.819642	−	0.572876i	$-0.805828\pi$
$457$	−35.0439	−1.63928	−0.819642	+	0.572876i	$0.805828\pi$
$458$	−21.0616	−0.984145
$459$	10.4668	0.488550
$460$	0	0
$461$	7.94572	0.370069	0.185035	−	0.982732i	$-0.440760\pi$
$461$	7.94572	0.370069	0.185035	+	0.982732i	$0.440760\pi$
$462$	−1.04344	−0.0485453
$463$	−23.9463	−1.11288	−0.556440	−	0.830888i	$-0.687833\pi$
$463$	−23.9463	−1.11288	−0.556440	+	0.830888i	$0.687833\pi$
$464$	−4.02635	−0.186919
$465$	0	0
$466$	−17.8785	−0.828206
$467$	−42.2147	−1.95346	−0.976732	−	0.214464i	$-0.931200\pi$
$467$	−42.2147	−1.95346	−0.976732	+	0.214464i	$0.931200\pi$
$468$	−0.757166	−0.0350000
$469$	5.94641	0.274580
$470$	0	0
$471$	23.1559	1.06697
$472$	21.4861	0.988979
$473$	21.0945	0.969926
$474$	−8.59339	−0.394707
$475$	0	0
$476$	−1.45580	−0.0667266
$477$	3.31808	0.151924
$478$	7.71340	0.352803
$479$	34.3187	1.56806	0.784031	−	0.620721i	$-0.213160\pi$
$479$	34.3187	1.56806	0.784031	+	0.620721i	$0.213160\pi$
$480$	0	0
$481$	−1.63218	−0.0744212
$482$	−16.7703	−0.763867
$483$	4.00227	0.182110
$484$	11.6619	0.530089
$485$	0	0
$486$	−2.24210	−0.101704
$487$	3.03995	0.137753	0.0688766	−	0.997625i	$-0.478059\pi$
$487$	3.03995	0.137753	0.0688766	+	0.997625i	$0.478059\pi$
$488$	−19.3381	−0.875394
$489$	−0.428477	−0.0193764
$490$	0	0
$491$	−11.4565	−0.517023	−0.258511	−	0.966008i	$-0.583232\pi$
$491$	−11.4565	−0.517023	−0.258511	+	0.966008i	$0.583232\pi$
$492$	−20.3562	−0.917728
$493$	6.07460	0.273586
$494$	−0.997468	−0.0448782
$495$	0	0
$496$	3.05663	0.137247
$497$	3.03877	0.136307
$498$	−5.43106	−0.243371
$499$	−39.9114	−1.78668	−0.893340	−	0.449381i	$-0.851645\pi$
$499$	−39.9114	−1.78668	−0.893340	+	0.449381i	$0.851645\pi$
$500$	0	0
$501$	−19.3485	−0.864429
$502$	−13.3511	−0.595889
$503$	14.8201	0.660794	0.330397	−	0.943842i	$-0.392817\pi$
$503$	14.8201	0.660794	0.330397	+	0.943842i	$0.392817\pi$
$504$	0.380775	0.0169611
$505$	0	0
$506$	−6.17115	−0.274341
$507$	−16.9582	−0.753141
$508$	−14.4000	−0.638897
$509$	−22.3612	−0.991145	−0.495572	−	0.868567i	$-0.665042\pi$
$509$	−22.3612	−0.991145	−0.495572	+	0.868567i	$0.665042\pi$
$510$	0	0
$511$	−2.78816	−0.123341
$512$	−13.7511	−0.607719
$513$	4.71753	0.208284
$514$	3.60645	0.159074
$515$	0	0
$516$	28.8854	1.27161
$517$	−2.58536	−0.113704
$518$	0.352533	0.0154894
$519$	−22.1916	−0.974103
$520$	0	0
$521$	−2.48042	−0.108669	−0.0543346	−	0.998523i	$-0.517304\pi$
$521$	−2.48042	−0.108669	−0.0543346	+	0.998523i	$0.517304\pi$
$522$	−0.682396	−0.0298677
$523$	0.452507	0.0197867	0.00989336	−	0.999951i	$-0.496851\pi$
$523$	0.452507	0.0197867	0.00989336	+	0.999951i	$0.496851\pi$
$524$	30.5605	1.33504
$525$	0	0
$526$	19.0958	0.832618
$527$	−4.61157	−0.200883
$528$	−3.78330	−0.164647
$529$	0.670348	0.0291455
$530$	0	0
$531$	−2.68578	−0.116553
$532$	−0.656148	−0.0284476
$533$	−13.4498	−0.582577
$534$	−5.95637	−0.257758
$535$	0	0
$536$	−29.2328	−1.26266
$537$	0.727289	0.0313848
$538$	−13.4114	−0.578205
$539$	12.1746	0.524396
$540$	0	0
$541$	−16.4102	−0.705531	−0.352765	−	0.935712i	$-0.614759\pi$
$541$	−16.4102	−0.705531	−0.352765	+	0.935712i	$0.614759\pi$
$542$	12.7003	0.545525
$543$	18.7664	0.805341
$544$	11.2398	0.481902
$545$	0	0
$546$	−0.944055	−0.0404018
$547$	−8.18537	−0.349981	−0.174991	−	0.984570i	$-0.555990\pi$
$547$	−8.18537	−0.349981	−0.174991	+	0.984570i	$0.555990\pi$
$548$	−16.6943	−0.713144
$549$	2.41727	0.103167
$550$	0	0
$551$	2.73789	0.116638
$552$	−19.6753	−0.837437
$553$	−3.73497	−0.158827
$554$	16.1271	0.685173
$555$	0	0
$556$	−21.3372	−0.904901
$557$	−25.8788	−1.09652	−0.548259	−	0.836309i	$-0.684709\pi$
$557$	−25.8788	−1.09652	−0.548259	+	0.836309i	$0.684709\pi$
$558$	0.518045	0.0219306
$559$	19.0853	0.807221
$560$	0	0
$561$	5.70789	0.240987
$562$	−9.96965	−0.420544
$563$	−24.4404	−1.03004	−0.515021	−	0.857178i	$-0.672216\pi$
$563$	−24.4404	−1.03004	−0.515021	+	0.857178i	$0.672216\pi$
$564$	−3.54021	−0.149070
$565$	0	0
$566$	−6.01491	−0.252826
$567$	4.00147	0.168046
$568$	−14.9387	−0.626814
$569$	5.31751	0.222922	0.111461	−	0.993769i	$-0.464447\pi$
$569$	5.31751	0.222922	0.111461	+	0.993769i	$0.464447\pi$
$570$	0	0
$571$	−13.2022	−0.552496	−0.276248	−	0.961086i	$-0.589091\pi$
$571$	−13.2022	−0.552496	−0.276248	+	0.961086i	$0.589091\pi$
$572$	−4.43332	−0.185366
$573$	16.6781	0.696737
$574$	2.90501	0.121253
$575$	0	0
$576$	−0.474982	−0.0197909
$577$	41.4001	1.72351	0.861755	−	0.507325i	$-0.169365\pi$
$577$	41.4001	1.72351	0.861755	+	0.507325i	$0.169365\pi$
$578$	9.33808	0.388413
$579$	14.6757	0.609901
$580$	0	0
$581$	−2.36052	−0.0979308
$582$	−7.48265	−0.310166
$583$	19.4278	0.804618
$584$	13.7067	0.567186
$585$	0	0
$586$	18.4370	0.761625
$587$	−11.4771	−0.473712	−0.236856	−	0.971545i	$-0.576117\pi$
$587$	−11.4771	−0.473712	−0.236856	+	0.971545i	$0.576117\pi$
$588$	16.6710	0.687502
$589$	−2.07849	−0.0856426
$590$	0	0
$591$	14.1946	0.583889
$592$	1.27821	0.0525340
$593$	11.1191	0.456608	0.228304	−	0.973590i	$-0.426682\pi$
$593$	11.1191	0.456608	0.228304	+	0.973590i	$0.426682\pi$
$594$	−6.88449	−0.282474
$595$	0	0
$596$	26.4668	1.08412
$597$	16.7589	0.685894
$598$	−5.58335	−0.228320
$599$	7.25813	0.296559	0.148280	−	0.988945i	$-0.452626\pi$
$599$	7.25813	0.296559	0.148280	+	0.988945i	$0.452626\pi$
$600$	0	0
$601$	−14.6121	−0.596040	−0.298020	−	0.954560i	$-0.596326\pi$
$601$	−14.6121	−0.596040	−0.298020	+	0.954560i	$0.596326\pi$
$602$	−4.12220	−0.168008
$603$	3.65411	0.148807
$604$	−15.7896	−0.642471
$605$	0	0
$606$	−2.89103	−0.117440
$607$	−26.3898	−1.07113	−0.535564	−	0.844495i	$-0.679901\pi$
$607$	−26.3898	−1.07113	−0.535564	+	0.844495i	$0.679901\pi$
$608$	5.06591	0.205450
$609$	2.59128	0.105004
$610$	0	0
$611$	−2.33911	−0.0946301
$612$	−0.894601	−0.0361621
$613$	40.8554	1.65013	0.825067	−	0.565035i	$-0.191137\pi$
$613$	40.8554	1.65013	0.825067	+	0.565035i	$0.191137\pi$
$614$	14.6367	0.590689
$615$	0	0
$616$	2.22949	0.0898288
$617$	−28.2802	−1.13852	−0.569260	−	0.822158i	$-0.692770\pi$
$617$	−28.2802	−1.13852	−0.569260	+	0.822158i	$0.692770\pi$
$618$	−7.38390	−0.297024
$619$	−7.17028	−0.288198	−0.144099	−	0.989563i	$-0.546028\pi$
$619$	−7.17028	−0.288198	−0.144099	+	0.989563i	$0.546028\pi$
$620$	0	0
$621$	26.4065	1.05966
$622$	−11.2349	−0.450479
$623$	−2.58884	−0.103720
$624$	−3.42294	−0.137027
$625$	0	0
$626$	−3.00530	−0.120116
$627$	2.57262	0.102740
$628$	−21.2497	−0.847957
$629$	−1.92844	−0.0768921
$630$	0	0
$631$	−7.70444	−0.306709	−0.153354	−	0.988171i	$-0.549008\pi$
$631$	−7.70444	−0.306709	−0.153354	+	0.988171i	$0.549008\pi$
$632$	18.3613	0.730372
$633$	22.3814	0.889582
$634$	−4.35524	−0.172969
$635$	0	0
$636$	26.6031	1.05488
$637$	11.0150	0.436429
$638$	−3.99553	−0.158184
$639$	1.86735	0.0738710
$640$	0	0
$641$	24.3375	0.961274	0.480637	−	0.876920i	$-0.340405\pi$
$641$	24.3375	0.961274	0.480637	+	0.876920i	$0.340405\pi$
$642$	13.4637	0.531372
$643$	−27.8512	−1.09834	−0.549171	−	0.835710i	$-0.685057\pi$
$643$	−27.8512	−1.09834	−0.549171	+	0.835710i	$0.685057\pi$
$644$	−3.67280	−0.144729
$645$	0	0
$646$	−1.17852	−0.0463683
$647$	−24.6280	−0.968228	−0.484114	−	0.875005i	$-0.660858\pi$
$647$	−24.6280	−0.968228	−0.484114	+	0.875005i	$0.660858\pi$
$648$	−19.6713	−0.772763
$649$	−15.7256	−0.617284
$650$	0	0
$651$	−1.96719	−0.0771002
$652$	0.393205	0.0153991
$653$	27.8960	1.09165	0.545827	−	0.837898i	$-0.316216\pi$
$653$	27.8960	1.09165	0.545827	+	0.837898i	$0.316216\pi$
$654$	−9.15460	−0.357973
$655$	0	0
$656$	10.5329	0.411242
$657$	−1.71334	−0.0668438
$658$	0.505219	0.0196955
$659$	−18.3903	−0.716384	−0.358192	−	0.933648i	$-0.616607\pi$
$659$	−18.3903	−0.716384	−0.358192	+	0.933648i	$0.616607\pi$
$660$	0	0
$661$	2.05323	0.0798614	0.0399307	−	0.999202i	$-0.487286\pi$
$661$	2.05323	0.0798614	0.0399307	+	0.999202i	$0.487286\pi$
$662$	22.5277	0.875564
$663$	5.16423	0.200562
$664$	11.6044	0.450338
$665$	0	0
$666$	0.216634	0.00839439
$667$	15.3254	0.593403
$668$	17.7558	0.686991
$669$	10.9318	0.422647
$670$	0	0
$671$	14.1535	0.546389
$672$	4.79463	0.184957
$673$	−16.8287	−0.648698	−0.324349	−	0.945938i	$-0.605145\pi$
$673$	−16.8287	−0.648698	−0.324349	+	0.945938i	$0.605145\pi$
$674$	8.80178	0.339032
$675$	0	0
$676$	15.5622	0.598546
$677$	20.8182	0.800108	0.400054	−	0.916491i	$-0.368991\pi$
$677$	20.8182	0.800108	0.400054	+	0.916491i	$0.368991\pi$
$678$	18.7184	0.718874
$679$	−3.25221	−0.124808
$680$	0	0
$681$	−9.38248	−0.359538
$682$	3.03323	0.116148
$683$	−6.40527	−0.245091	−0.122545	−	0.992463i	$-0.539106\pi$
$683$	−6.40527	−0.245091	−0.122545	+	0.992463i	$0.539106\pi$
$684$	−0.403207	−0.0154170
$685$	0	0
$686$	−4.84683	−0.185053
$687$	49.1470	1.87507
$688$	−14.9462	−0.569819
$689$	17.5774	0.669644
$690$	0	0
$691$	−0.406659	−0.0154700	−0.00773502	−	0.999970i	$-0.502462\pi$
$691$	−0.406659	−0.0154700	−0.00773502	+	0.999970i	$0.502462\pi$
$692$	20.3648	0.774153
$693$	−0.278688	−0.0105865
$694$	2.32358	0.0882020
$695$	0	0
$696$	−12.7388	−0.482864
$697$	−15.8911	−0.601920
$698$	3.12976	0.118463
$699$	41.7193	1.57797
$700$	0	0
$701$	2.39744	0.0905501	0.0452750	−	0.998975i	$-0.485584\pi$
$701$	2.39744	0.0905501	0.0452750	+	0.998975i	$0.485584\pi$
$702$	−6.22876	−0.235089
$703$	−0.869173	−0.0327815
$704$	−2.78108	−0.104816
$705$	0	0
$706$	22.4394	0.844520
$707$	−1.25654	−0.0472570
$708$	−21.5336	−0.809281
$709$	22.0154	0.826804	0.413402	−	0.910549i	$-0.364340\pi$
$709$	22.0154	0.826804	0.413402	+	0.910549i	$0.364340\pi$
$710$	0	0
$711$	−2.29517	−0.0860755
$712$	12.7268	0.476958
$713$	−11.6344	−0.435712
$714$	−1.11541	−0.0417432
$715$	0	0
$716$	−0.667418	−0.0249426
$717$	−17.9991	−0.672189
$718$	−22.4282	−0.837014
$719$	−9.29180	−0.346526	−0.173263	−	0.984876i	$-0.555431\pi$
$719$	−9.29180	−0.346526	−0.173263	+	0.984876i	$0.555431\pi$
$720$	0	0
$721$	−3.20929	−0.119520
$722$	12.8279	0.477406
$723$	39.1333	1.45538
$724$	−17.2215	−0.640032
$725$	0	0
$726$	8.93519	0.331616
$727$	29.4352	1.09169	0.545846	−	0.837886i	$-0.316208\pi$
$727$	29.4352	1.09169	0.545846	+	0.837886i	$0.316208\pi$
$728$	2.01714	0.0747600
$729$	29.1741	1.08052
$730$	0	0
$731$	22.5495	0.834023
$732$	19.3808	0.716335
$733$	3.71935	0.137377	0.0686887	−	0.997638i	$-0.478118\pi$
$733$	3.71935	0.137377	0.0686887	+	0.997638i	$0.478118\pi$
$734$	−15.9669	−0.589349
$735$	0	0
$736$	28.3565	1.04524
$737$	21.3953	0.788107
$738$	1.78515	0.0657122
$739$	20.2734	0.745768	0.372884	−	0.927878i	$-0.378369\pi$
$739$	20.2734	0.745768	0.372884	+	0.927878i	$0.378369\pi$
$740$	0	0
$741$	2.32758	0.0855057
$742$	−3.79650	−0.139374
$743$	−1.11337	−0.0408457	−0.0204229	−	0.999791i	$-0.506501\pi$
$743$	−1.11337	−0.0408457	−0.0204229	+	0.999791i	$0.506501\pi$
$744$	9.67077	0.354547
$745$	0	0
$746$	−20.1867	−0.739086
$747$	−1.45056	−0.0530730
$748$	−5.23802	−0.191521
$749$	5.85179	0.213820
$750$	0	0
$751$	−37.9213	−1.38377	−0.691885	−	0.722008i	$-0.743219\pi$
$751$	−37.9213	−1.38377	−0.691885	+	0.722008i	$0.743219\pi$
$752$	1.83182	0.0667995
$753$	31.1546	1.13534
$754$	−3.61496	−0.131649
$755$	0	0
$756$	−4.09736	−0.149019
$757$	41.8794	1.52213	0.761066	−	0.648674i	$-0.224676\pi$
$757$	41.8794	1.52213	0.761066	+	0.648674i	$0.224676\pi$
$758$	−21.3405	−0.775121
$759$	14.4003	0.522697
$760$	0	0
$761$	−12.6827	−0.459749	−0.229875	−	0.973220i	$-0.573832\pi$
$761$	−12.6827	−0.459749	−0.229875	+	0.973220i	$0.573832\pi$
$762$	−11.0331	−0.399685
$763$	−3.97889	−0.144046
$764$	−15.3051	−0.553721
$765$	0	0
$766$	24.9159	0.900247
$767$	−14.2278	−0.513735
$768$	−17.2555	−0.622653
$769$	34.0483	1.22781	0.613907	−	0.789378i	$-0.289597\pi$
$769$	34.0483	1.22781	0.613907	+	0.789378i	$0.289597\pi$
$770$	0	0
$771$	−8.41560	−0.303081
$772$	−13.4676	−0.484709
$773$	20.6092	0.741261	0.370630	−	0.928780i	$-0.379142\pi$
$773$	20.6092	0.741261	0.370630	+	0.928780i	$0.379142\pi$
$774$	−2.53312	−0.0910511
$775$	0	0
$776$	15.9880	0.573935
$777$	−0.822630	−0.0295117
$778$	12.1494	0.435576
$779$	−7.16233	−0.256617
$780$	0	0
$781$	10.9336	0.391234
$782$	−6.59680	−0.235901
$783$	17.0970	0.610995
$784$	−8.62613	−0.308076
$785$	0	0
$786$	23.4150	0.835185
$787$	49.1315	1.75135	0.875674	−	0.482903i	$-0.160417\pi$
$787$	49.1315	1.75135	0.875674	+	0.482903i	$0.160417\pi$
$788$	−13.0261	−0.464036
$789$	−44.5599	−1.58637
$790$	0	0
$791$	8.13562	0.289269
$792$	1.37004	0.0486822
$793$	12.8054	0.454732
$794$	−3.29536	−0.116948
$795$	0	0
$796$	−15.3793	−0.545103
$797$	−1.87847	−0.0665389	−0.0332694	−	0.999446i	$-0.510592\pi$
$797$	−1.87847	−0.0665389	−0.0332694	+	0.999446i	$0.510592\pi$
$798$	−0.502729	−0.0177964
$799$	−2.76368	−0.0977720
$800$	0	0
$801$	−1.59086	−0.0562103
$802$	15.7094	0.554718
$803$	−10.0319	−0.354017
$804$	29.2973	1.03324
$805$	0	0
$806$	2.74432	0.0966645
$807$	31.2952	1.10164
$808$	6.17719	0.217313
$809$	−15.4989	−0.544912	−0.272456	−	0.962168i	$-0.587836\pi$
$809$	−15.4989	−0.544912	−0.272456	+	0.962168i	$0.587836\pi$
$810$	0	0
$811$	6.06459	0.212956	0.106478	−	0.994315i	$-0.466043\pi$
$811$	6.06459	0.212956	0.106478	+	0.994315i	$0.466043\pi$
$812$	−2.37797	−0.0834503
$813$	−29.6360	−1.03938
$814$	1.26842	0.0444582
$815$	0	0
$816$	−4.04425	−0.141577
$817$	10.1633	0.355570
$818$	3.63994	0.127267
$819$	−0.252143	−0.00881059
$820$	0	0
$821$	−3.79024	−0.132280	−0.0661402	−	0.997810i	$-0.521068\pi$
$821$	−3.79024	−0.132280	−0.0661402	+	0.997810i	$0.521068\pi$
$822$	−12.7909	−0.446133
$823$	−2.64216	−0.0920999	−0.0460499	−	0.998939i	$-0.514663\pi$
$823$	−2.64216	−0.0920999	−0.0460499	+	0.998939i	$0.514663\pi$
$824$	15.7770	0.549617
$825$	0	0
$826$	3.07303	0.106924
$827$	−21.8953	−0.761376	−0.380688	−	0.924704i	$-0.624313\pi$
$827$	−21.8953	−0.761376	−0.380688	+	0.924704i	$0.624313\pi$
$828$	−2.25696	−0.0784349
$829$	7.03307	0.244269	0.122134	−	0.992514i	$-0.461026\pi$
$829$	7.03307	0.244269	0.122134	+	0.992514i	$0.461026\pi$
$830$	0	0
$831$	−37.6323	−1.30545
$832$	−2.51619	−0.0872332
$833$	13.0143	0.450919
$834$	−16.3482	−0.566093
$835$	0	0
$836$	−2.36084	−0.0816512
$837$	−12.9793	−0.448629
$838$	11.1549	0.385341
$839$	−49.4466	−1.70709	−0.853544	−	0.521021i	$-0.825551\pi$
$839$	−49.4466	−1.70709	−0.853544	+	0.521021i	$0.825551\pi$
$840$	0	0
$841$	−19.0775	−0.657845
$842$	−4.54907	−0.156771
$843$	23.2640	0.801256
$844$	−20.5390	−0.706981
$845$	0	0
$846$	0.310461	0.0106739
$847$	3.88353	0.133440
$848$	−13.7653	−0.472703
$849$	14.0357	0.481704
$850$	0	0
$851$	−4.86522	−0.166778
$852$	14.9717	0.512922
$853$	−2.94665	−0.100891	−0.0504457	−	0.998727i	$-0.516064\pi$
$853$	−2.94665	−0.100891	−0.0504457	+	0.998727i	$0.516064\pi$
$854$	−2.76581	−0.0946441
$855$	0	0
$856$	−28.7676	−0.983257
$857$	27.7795	0.948929	0.474465	−	0.880275i	$-0.342642\pi$
$857$	27.7795	0.948929	0.474465	+	0.880275i	$0.342642\pi$
$858$	−3.39673	−0.115963
$859$	−50.2134	−1.71326	−0.856629	−	0.515933i	$-0.827445\pi$
$859$	−50.2134	−1.71326	−0.856629	+	0.515933i	$0.827445\pi$
$860$	0	0
$861$	−6.77879	−0.231021
$862$	17.5585	0.598046
$863$	1.11827	0.0380663	0.0190331	−	0.999819i	$-0.493941\pi$
$863$	1.11827	0.0380663	0.0190331	+	0.999819i	$0.493941\pi$
$864$	31.6344	1.07622
$865$	0	0
$866$	−27.2320	−0.925382
$867$	−21.7903	−0.740037
$868$	1.80525	0.0612741
$869$	−13.4385	−0.455871
$870$	0	0
$871$	19.3575	0.655903
$872$	19.5604	0.662398
$873$	−1.99851	−0.0676392
$874$	−2.97326	−0.100572
$875$	0	0
$876$	−13.7369	−0.464128
$877$	−23.0843	−0.779502	−0.389751	−	0.920920i	$-0.627439\pi$
$877$	−23.0843	−0.779502	−0.389751	+	0.920920i	$0.627439\pi$
$878$	23.0919	0.779314
$879$	−43.0224	−1.45111
$880$	0	0
$881$	−10.1304	−0.341301	−0.170650	−	0.985332i	$-0.554587\pi$
$881$	−10.1304	−0.341301	−0.170650	+	0.985332i	$0.554587\pi$
$882$	−1.46198	−0.0492273
$883$	−0.620421	−0.0208788	−0.0104394	−	0.999946i	$-0.503323\pi$
$883$	−0.620421	−0.0208788	−0.0104394	+	0.999946i	$0.503323\pi$
$884$	−4.73910	−0.159393
$885$	0	0
$886$	17.1743	0.576983
$887$	−0.762728	−0.0256099	−0.0128049	−	0.999918i	$-0.504076\pi$
$887$	−0.762728	−0.0256099	−0.0128049	+	0.999918i	$0.504076\pi$
$888$	4.04408	0.135710
$889$	−4.79533	−0.160830
$890$	0	0
$891$	14.3974	0.482330
$892$	−10.0319	−0.335892
$893$	−1.24562	−0.0416832
$894$	20.2784	0.678211
$895$	0	0
$896$	−5.30116	−0.177099
$897$	13.0287	0.435015
$898$	16.7473	0.558864
$899$	−7.53272	−0.251230
$900$	0	0
$901$	20.7678	0.691877
$902$	10.4523	0.348023
$903$	9.61909	0.320103
$904$	−39.9950	−1.33021
$905$	0	0
$906$	−12.0978	−0.401921
$907$	−44.6062	−1.48112	−0.740562	−	0.671988i	$-0.765441\pi$
$907$	−44.6062	−1.48112	−0.740562	+	0.671988i	$0.765441\pi$
$908$	8.61011	0.285737
$909$	−0.772152	−0.0256107
$910$	0	0
$911$	21.5382	0.713591	0.356796	−	0.934182i	$-0.383869\pi$
$911$	21.5382	0.713591	0.356796	+	0.934182i	$0.383869\pi$
$912$	−1.82279	−0.0603586
$913$	−8.49321	−0.281084
$914$	24.6397	0.815010
$915$	0	0
$916$	−45.1012	−1.49018
$917$	10.1769	0.336072
$918$	−7.35935	−0.242895
$919$	−32.1570	−1.06076	−0.530380	−	0.847760i	$-0.677951\pi$
$919$	−32.1570	−1.06076	−0.530380	+	0.847760i	$0.677951\pi$
$920$	0	0
$921$	−34.1545	−1.12543
$922$	−5.58672	−0.183989
$923$	9.89217	0.325605
$924$	−2.23442	−0.0735069
$925$	0	0
$926$	16.8369	0.553295
$927$	−1.97213	−0.0647733
$928$	18.3595	0.602681
$929$	53.7540	1.76361	0.881805	−	0.471614i	$-0.156328\pi$
$929$	53.7540	1.76361	0.881805	+	0.471614i	$0.156328\pi$
$930$	0	0
$931$	5.86571	0.192241
$932$	−38.2849	−1.25406
$933$	26.2165	0.858290
$934$	29.6816	0.971212
$935$	0	0
$936$	1.23954	0.0405158
$937$	32.7426	1.06965	0.534827	−	0.844961i	$-0.320377\pi$
$937$	32.7426	1.06965	0.534827	+	0.844961i	$0.320377\pi$
$938$	−4.18098	−0.136514
$939$	7.01283	0.228855
$940$	0	0
$941$	−7.12766	−0.232355	−0.116178	−	0.993228i	$-0.537064\pi$
$941$	−7.12766	−0.232355	−0.116178	+	0.993228i	$0.537064\pi$
$942$	−16.2812	−0.530470
$943$	−40.0913	−1.30555
$944$	11.1422	0.362646
$945$	0	0
$946$	−14.8318	−0.482222
$947$	14.6389	0.475702	0.237851	−	0.971302i	$-0.423557\pi$
$947$	14.6389	0.475702	0.237851	+	0.971302i	$0.423557\pi$
$948$	−18.4018	−0.597663
$949$	−9.07634	−0.294630
$950$	0	0
$951$	10.1629	0.329554
$952$	2.38327	0.0772422
$953$	51.2467	1.66004	0.830021	−	0.557731i	$-0.188328\pi$
$953$	51.2467	1.66004	0.830021	+	0.557731i	$0.188328\pi$
$954$	−2.33298	−0.0755329
$955$	0	0
$956$	16.5174	0.534211
$957$	9.32350	0.301386
$958$	−24.1299	−0.779601
$959$	−5.55934	−0.179521
$960$	0	0
$961$	−25.2815	−0.815532
$962$	1.14761	0.0370003
$963$	3.59597	0.115878
$964$	−35.9118	−1.15664
$965$	0	0
$966$	−2.81404	−0.0905403
$967$	12.5599	0.403899	0.201949	−	0.979396i	$-0.435272\pi$
$967$	12.5599	0.403899	0.201949	+	0.979396i	$0.435272\pi$
$968$	−19.0916	−0.613627
$969$	2.75006	0.0883447
$970$	0	0
$971$	−5.56513	−0.178594	−0.0892968	−	0.996005i	$-0.528462\pi$
$971$	−5.56513	−0.178594	−0.0892968	+	0.996005i	$0.528462\pi$
$972$	−4.80120	−0.153999
$973$	−7.10549	−0.227792
$974$	−2.13742	−0.0684874
$975$	0	0
$976$	−10.0282	−0.320996
$977$	25.6883	0.821843	0.410922	−	0.911671i	$-0.365207\pi$
$977$	25.6883	0.821843	0.410922	+	0.911671i	$0.365207\pi$
$978$	0.301267	0.00963346
$979$	−9.31471	−0.297699
$980$	0	0
$981$	−2.44506	−0.0780647
$982$	8.05516	0.257050
$983$	21.5725	0.688057	0.344028	−	0.938959i	$-0.388208\pi$
$983$	21.5725	0.688057	0.344028	+	0.938959i	$0.388208\pi$
$984$	33.3248	1.06236
$985$	0	0
$986$	−4.27112	−0.136020
$987$	−1.17892	−0.0375255
$988$	−2.13597	−0.0679543
$989$	56.8894	1.80898
$990$	0	0
$991$	9.15527	0.290827	0.145413	−	0.989371i	$-0.453549\pi$
$991$	9.15527	0.290827	0.145413	+	0.989371i	$0.453549\pi$
$992$	−13.9377	−0.442524
$993$	−52.5681	−1.66820
$994$	−2.13659	−0.0677686
$995$	0	0
$996$	−11.6300	−0.368511
$997$	−21.1522	−0.669896	−0.334948	−	0.942237i	$-0.608719\pi$
$997$	−21.1522	−0.669896	−0.334948	+	0.942237i	$0.608719\pi$
$998$	28.0621	0.888292
$999$	−5.42761	−0.171722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.a.l.1.5		9
3.2	odd	2		8325.2.a.cr.1.5		9
5.2	odd	4		185.2.b.a.149.8	✓	18
5.3	odd	4		185.2.b.a.149.11	yes	18
5.4	even	2		925.2.a.m.1.5		9
15.2	even	4		1665.2.c.e.334.11		18
15.8	even	4		1665.2.c.e.334.8		18
15.14	odd	2		8325.2.a.cq.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.b.a.149.8	✓	18	5.2	odd	4
185.2.b.a.149.11	yes	18	5.3	odd	4
925.2.a.l.1.5		9	1.1	even	1	trivial
925.2.a.m.1.5		9	5.4	even	2
1665.2.c.e.334.8		18	15.8	even	4
1665.2.c.e.334.11		18	15.2	even	4
8325.2.a.cq.1.5		9	15.14	odd	2
8325.2.a.cr.1.5		9	3.2	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:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.703111 q^{2} +1.64070 q^{3} -1.50564 q^{4} -1.15359 q^{6} -0.501390 q^{7} +2.46485 q^{8} -0.308108 q^{9} -1.80401 q^{11} -2.47029 q^{12} -1.63218 q^{13} +0.352533 q^{14} +1.27821 q^{16} -1.92844 q^{17} +0.216634 q^{18} -0.869173 q^{19} -0.822630 q^{21} +1.26842 q^{22} -4.86522 q^{23} +4.04408 q^{24} +1.14761 q^{26} -5.42761 q^{27} +0.754910 q^{28} -3.15000 q^{29} +2.39134 q^{31} -5.82842 q^{32} -2.95984 q^{33} +1.35591 q^{34} +0.463898 q^{36} +1.00000 q^{37} +0.611125 q^{38} -2.67792 q^{39} +8.24040 q^{41} +0.578400 q^{42} -11.6931 q^{43} +2.71619 q^{44} +3.42079 q^{46} +1.43311 q^{47} +2.09715 q^{48} -6.74861 q^{49} -3.16400 q^{51} +2.45747 q^{52} -10.7692 q^{53} +3.81621 q^{54} -1.23585 q^{56} -1.42605 q^{57} +2.21480 q^{58} +8.71701 q^{59} -7.84555 q^{61} -1.68138 q^{62} +0.154482 q^{63} +1.54161 q^{64} +2.08110 q^{66} -11.8599 q^{67} +2.90353 q^{68} -7.98236 q^{69} -6.06069 q^{71} -0.759439 q^{72} +5.56086 q^{73} -0.703111 q^{74} +1.30866 q^{76} +0.904514 q^{77} +1.88288 q^{78} +7.44924 q^{79} -7.98075 q^{81} -5.79391 q^{82} +4.70795 q^{83} +1.23858 q^{84} +8.22154 q^{86} -5.16820 q^{87} -4.44662 q^{88} +5.16332 q^{89} +0.818361 q^{91} +7.32524 q^{92} +3.92347 q^{93} -1.00764 q^{94} -9.56268 q^{96} +6.48639 q^{97} +4.74502 q^{98} +0.555830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} - 8 q^{3} + 11 q^{4} + 2 q^{6} - 8 q^{7} - 15 q^{8} + 13 q^{9} - 16 q^{12} - 6 q^{13} - 4 q^{14} + 11 q^{16} - 18 q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{21} - 6 q^{22} - 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 8 * q^3 + 11 * q^4 + 2 * q^6 - 8 * q^7 - 15 * q^8 + 13 * q^9 - 16 * q^12 - 6 * q^13 - 4 * q^14 + 11 * q^16 - 18 * q^17 + 3 * q^18 - 4 * q^19 + 4 * q^21 - 6 * q^22 - 16 * q^23 + 6 * q^24 - 6 * q^26 - 32 * q^27 + 20 * q^28 - 2 * q^29 - 6 * q^31 - 35 * q^32 - 6 * q^33 + 6 * q^34 + 11 * q^36 + 9 * q^37 - 4 * q^38 - 16 * q^39 + 2 * q^41 - 4 * q^42 - 14 * q^43 + 4 * q^44 + 16 * q^46 - 34 * q^47 - 6 * q^48 - 7 * q^49 + 8 * q^51 - 16 * q^52 + 4 * q^53 - 18 * q^54 - 24 * q^56 + 4 * q^57 + 22 * q^58 + 10 * q^59 + 8 * q^61 - 26 * q^62 - 18 * q^63 + 15 * q^64 - 34 * q^66 - 16 * q^67 - 36 * q^68 - 10 * q^69 + 8 * q^71 - 9 * q^72 - 8 * q^73 - 5 * q^74 - 48 * q^76 - 14 * q^77 + 76 * q^78 - 26 * q^79 + 13 * q^81 + 4 * q^82 - 70 * q^83 - 24 * q^84 - 16 * q^86 + 12 * q^87 + 36 * q^88 - 8 * q^89 - 18 * q^91 - 16 * q^92 - 2 * q^93 - 26 * q^94 + 52 * q^96 + 2 * q^97 - 21 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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