LMFDB - Embedded newform 7600.2.a.bw.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7600,2,Mod(1,7600)] 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("7600.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2,0,0,0,0,0,-3,0,-1,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.801938 q^{3} +1.69202 q^{7} -2.35690 q^{9} +0.911854 q^{11} -1.55496 q^{13} -5.29590 q^{17} +1.00000 q^{19} -1.35690 q^{21} +4.24698 q^{23} +4.29590 q^{27} +5.00969 q^{29} -1.82908 q^{31} -0.731250 q^{33} -6.29590 q^{37} +1.24698 q^{39} +4.18060 q^{41} +7.31767 q^{43} -2.04892 q^{47} -4.13706 q^{49} +4.24698 q^{51} +2.70171 q^{53} -0.801938 q^{57} -9.87800 q^{59} +0.542877 q^{61} -3.98792 q^{63} -13.9976 q^{67} -3.40581 q^{69} +12.8780 q^{71} +2.80731 q^{73} +1.54288 q^{77} -1.59419 q^{79} +3.62565 q^{81} +12.2349 q^{83} -4.01746 q^{87} +2.91723 q^{89} -2.63102 q^{91} +1.46681 q^{93} -1.55496 q^{97} -2.14914 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{9} - q^{11} - 5 q^{13} - 2 q^{17} + 3 q^{19} + 8 q^{23} - q^{27} - 7 q^{29} + 5 q^{31} - 10 q^{33} - 5 q^{37} - q^{39} + q^{41} + 5 q^{43} + 3 q^{47} - 7 q^{49} + 8 q^{51} - 19 q^{53}+ \cdots - 20 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^9 - q^11 - 5 * q^13 - 2 * q^17 + 3 * q^19 + 8 * q^23 - q^27 - 7 * q^29 + 5 * q^31 - 10 * q^33 - 5 * q^37 - q^39 + q^41 + 5 * q^43 + 3 * q^47 - 7 * q^49 + 8 * q^51 - 19 * q^53 + 2 * q^57 - 10 * q^59 - 17 * q^61 + 7 * q^63 - q^67 + 3 * q^69 + 19 * q^71 + q^73 - 14 * q^77 - 18 * q^79 - q^81 + 13 * q^83 - 28 * q^87 + 2 * q^89 + 7 * q^91 + q^93 - 5 * q^97 - 20 * 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	0
$2$	0	0
$3$	−0.801938	−0.462999	−0.231499	−	0.972835i	$-0.574363\pi$
$3$	−0.801938	−0.462999	−0.231499	+	0.972835i	$0.574363\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.69202	0.639524	0.319762	−	0.947498i	$-0.396397\pi$
$7$	1.69202	0.639524	0.319762	+	0.947498i	$0.396397\pi$
$8$	0	0
$9$	−2.35690	−0.785632
$10$	0	0
$11$	0.911854	0.274934	0.137467	−	0.990506i	$-0.456104\pi$
$11$	0.911854	0.274934	0.137467	+	0.990506i	$0.456104\pi$
$12$	0	0
$13$	−1.55496	−0.431268	−0.215634	−	0.976474i	$-0.569182\pi$
$13$	−1.55496	−0.431268	−0.215634	+	0.976474i	$0.569182\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.29590	−1.28444	−0.642222	−	0.766519i	$-0.721987\pi$
$17$	−5.29590	−1.28444	−0.642222	+	0.766519i	$0.721987\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.35690	−0.296099
$22$	0	0
$23$	4.24698	0.885556	0.442778	−	0.896631i	$-0.353993\pi$
$23$	4.24698	0.885556	0.442778	+	0.896631i	$0.353993\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.29590	0.826746
$28$	0	0
$29$	5.00969	0.930276	0.465138	−	0.885238i	$-0.346005\pi$
$29$	5.00969	0.930276	0.465138	+	0.885238i	$0.346005\pi$
$30$	0	0
$31$	−1.82908	−0.328513	−0.164257	−	0.986418i	$-0.552523\pi$
$31$	−1.82908	−0.328513	−0.164257	+	0.986418i	$0.552523\pi$
$32$	0	0
$33$	−0.731250	−0.127294
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.29590	−1.03504	−0.517520	−	0.855671i	$-0.673145\pi$
$37$	−6.29590	−1.03504	−0.517520	+	0.855671i	$0.673145\pi$
$38$	0	0
$39$	1.24698	0.199677
$40$	0	0
$41$	4.18060	0.652901	0.326450	−	0.945214i	$-0.394147\pi$
$41$	4.18060	0.652901	0.326450	+	0.945214i	$0.394147\pi$
$42$	0	0
$43$	7.31767	1.11593	0.557967	−	0.829863i	$-0.311582\pi$
$43$	7.31767	1.11593	0.557967	+	0.829863i	$0.311582\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.04892	−0.298865	−0.149433	−	0.988772i	$-0.547745\pi$
$47$	−2.04892	−0.298865	−0.149433	+	0.988772i	$0.547745\pi$
$48$	0	0
$49$	−4.13706	−0.591009
$50$	0	0
$51$	4.24698	0.594696
$52$	0	0
$53$	2.70171	0.371108	0.185554	−	0.982634i	$-0.440592\pi$
$53$	2.70171	0.371108	0.185554	+	0.982634i	$0.440592\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.801938	−0.106219
$58$	0	0
$59$	−9.87800	−1.28601	−0.643003	−	0.765864i	$-0.722312\pi$
$59$	−9.87800	−1.28601	−0.643003	+	0.765864i	$0.722312\pi$
$60$	0	0
$61$	0.542877	0.0695082	0.0347541	−	0.999396i	$-0.488935\pi$
$61$	0.542877	0.0695082	0.0347541	+	0.999396i	$0.488935\pi$
$62$	0	0
$63$	−3.98792	−0.502430
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.9976	−1.71008	−0.855040	−	0.518562i	$-0.826467\pi$
$67$	−13.9976	−1.71008	−0.855040	+	0.518562i	$0.826467\pi$
$68$	0	0
$69$	−3.40581	−0.410012
$70$	0	0
$71$	12.8780	1.52834	0.764169	−	0.645016i	$-0.223149\pi$
$71$	12.8780	1.52834	0.764169	+	0.645016i	$0.223149\pi$
$72$	0	0
$73$	2.80731	0.328571	0.164286	−	0.986413i	$-0.447468\pi$
$73$	2.80731	0.328571	0.164286	+	0.986413i	$0.447468\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.54288	0.175827
$78$	0	0
$79$	−1.59419	−0.179360	−0.0896800	−	0.995971i	$-0.528584\pi$
$79$	−1.59419	−0.179360	−0.0896800	+	0.995971i	$0.528584\pi$
$80$	0	0
$81$	3.62565	0.402850
$82$	0	0
$83$	12.2349	1.34295	0.671477	−	0.741025i	$-0.265660\pi$
$83$	12.2349	1.34295	0.671477	+	0.741025i	$0.265660\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.01746	−0.430717
$88$	0	0
$89$	2.91723	0.309226	0.154613	−	0.987975i	$-0.450587\pi$
$89$	2.91723	0.309226	0.154613	+	0.987975i	$0.450587\pi$
$90$	0	0
$91$	−2.63102	−0.275806
$92$	0	0
$93$	1.46681	0.152101
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.55496	−0.157882	−0.0789410	−	0.996879i	$-0.525154\pi$
$97$	−1.55496	−0.157882	−0.0789410	+	0.996879i	$0.525154\pi$
$98$	0	0
$99$	−2.14914	−0.215997
$100$	0	0
$101$	−16.6015	−1.65191	−0.825955	−	0.563737i	$-0.809363\pi$
$101$	−16.6015	−1.65191	−0.825955	+	0.563737i	$0.809363\pi$
$102$	0	0
$103$	−4.84548	−0.477439	−0.238720	−	0.971089i	$-0.576728\pi$
$103$	−4.84548	−0.477439	−0.238720	+	0.971089i	$0.576728\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.46681	−0.431823	−0.215912	−	0.976413i	$-0.569272\pi$
$107$	−4.46681	−0.431823	−0.215912	+	0.976413i	$0.569272\pi$
$108$	0	0
$109$	18.8267	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$109$	18.8267	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$110$	0	0
$111$	5.04892	0.479222
$112$	0	0
$113$	−20.0368	−1.88491	−0.942453	−	0.334337i	$-0.891488\pi$
$113$	−20.0368	−1.88491	−0.942453	+	0.334337i	$0.891488\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.66487	0.338818
$118$	0	0
$119$	−8.96077	−0.821433
$120$	0	0
$121$	−10.1685	−0.924411
$122$	0	0
$123$	−3.35258	−0.302292
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.8702	−1.58573	−0.792863	−	0.609399i	$-0.791411\pi$
$127$	−17.8702	−1.58573	−0.792863	+	0.609399i	$0.791411\pi$
$128$	0	0
$129$	−5.86831	−0.516676
$130$	0	0
$131$	−7.44265	−0.650267	−0.325134	−	0.945668i	$-0.605409\pi$
$131$	−7.44265	−0.650267	−0.325134	+	0.945668i	$0.605409\pi$
$132$	0	0
$133$	1.69202	0.146717
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.68664	−0.485843	−0.242921	−	0.970046i	$-0.578106\pi$
$137$	−5.68664	−0.485843	−0.242921	+	0.970046i	$0.578106\pi$
$138$	0	0
$139$	−3.61596	−0.306701	−0.153351	−	0.988172i	$-0.549006\pi$
$139$	−3.61596	−0.306701	−0.153351	+	0.988172i	$0.549006\pi$
$140$	0	0
$141$	1.64310	0.138374
$142$	0	0
$143$	−1.41789	−0.118570
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.31767	0.273637
$148$	0	0
$149$	3.29052	0.269570	0.134785	−	0.990875i	$-0.456966\pi$
$149$	3.29052	0.269570	0.134785	+	0.990875i	$0.456966\pi$
$150$	0	0
$151$	10.2131	0.831133	0.415566	−	0.909563i	$-0.363583\pi$
$151$	10.2131	0.831133	0.415566	+	0.909563i	$0.363583\pi$
$152$	0	0
$153$	12.4819	1.00910
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.3448	−1.14484	−0.572420	−	0.819960i	$-0.693995\pi$
$157$	−14.3448	−1.14484	−0.572420	+	0.819960i	$0.693995\pi$
$158$	0	0
$159$	−2.16660	−0.171823
$160$	0	0
$161$	7.18598	0.566335
$162$	0	0
$163$	−19.5308	−1.52977	−0.764885	−	0.644167i	$-0.777204\pi$
$163$	−19.5308	−1.52977	−0.764885	+	0.644167i	$0.777204\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.8823	0.919481	0.459741	−	0.888053i	$-0.347942\pi$
$167$	11.8823	0.919481	0.459741	+	0.888053i	$0.347942\pi$
$168$	0	0
$169$	−10.5821	−0.814008
$170$	0	0
$171$	−2.35690	−0.180236
$172$	0	0
$173$	−15.4722	−1.17633	−0.588164	−	0.808741i	$-0.700149\pi$
$173$	−15.4722	−1.17633	−0.588164	+	0.808741i	$0.700149\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.92154	0.595420
$178$	0	0
$179$	−2.16421	−0.161761	−0.0808803	−	0.996724i	$-0.525773\pi$
$179$	−2.16421	−0.161761	−0.0808803	+	0.996724i	$0.525773\pi$
$180$	0	0
$181$	16.8974	1.25597	0.627986	−	0.778225i	$-0.283879\pi$
$181$	16.8974	1.25597	0.627986	+	0.778225i	$0.283879\pi$
$182$	0	0
$183$	−0.435353	−0.0321822
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.82908	−0.353138
$188$	0	0
$189$	7.26875	0.528724
$190$	0	0
$191$	−5.92394	−0.428641	−0.214320	−	0.976763i	$-0.568754\pi$
$191$	−5.92394	−0.428641	−0.214320	+	0.976763i	$0.568754\pi$
$192$	0	0
$193$	4.43535	0.319264	0.159632	−	0.987177i	$-0.448969\pi$
$193$	4.43535	0.319264	0.159632	+	0.987177i	$0.448969\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.4722	−1.17359	−0.586797	−	0.809734i	$-0.699612\pi$
$197$	−16.4722	−1.17359	−0.586797	+	0.809734i	$0.699612\pi$
$198$	0	0
$199$	24.4131	1.73060	0.865300	−	0.501255i	$-0.167128\pi$
$199$	24.4131	1.73060	0.865300	+	0.501255i	$0.167128\pi$
$200$	0	0
$201$	11.2252	0.791765
$202$	0	0
$203$	8.47650	0.594934
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.0097	−0.695721
$208$	0	0
$209$	0.911854	0.0630743
$210$	0	0
$211$	4.34050	0.298813	0.149406	−	0.988776i	$-0.452264\pi$
$211$	4.34050	0.298813	0.149406	+	0.988776i	$0.452264\pi$
$212$	0	0
$213$	−10.3274	−0.707619
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.09485	−0.210092
$218$	0	0
$219$	−2.25129	−0.152128
$220$	0	0
$221$	8.23490	0.553939
$222$	0	0
$223$	26.2379	1.75702	0.878509	−	0.477725i	$-0.158539\pi$
$223$	26.2379	1.75702	0.878509	+	0.477725i	$0.158539\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.6853	0.974699	0.487349	−	0.873207i	$-0.337964\pi$
$227$	14.6853	0.974699	0.487349	+	0.873207i	$0.337964\pi$
$228$	0	0
$229$	−21.6407	−1.43006	−0.715029	−	0.699095i	$-0.753587\pi$
$229$	−21.6407	−1.43006	−0.715029	+	0.699095i	$0.753587\pi$
$230$	0	0
$231$	−1.23729	−0.0814078
$232$	0	0
$233$	−27.1183	−1.77658	−0.888289	−	0.459286i	$-0.848105\pi$
$233$	−27.1183	−1.77658	−0.888289	+	0.459286i	$0.848105\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.27844	0.0830435
$238$	0	0
$239$	11.5308	0.745865	0.372933	−	0.927858i	$-0.378352\pi$
$239$	11.5308	0.745865	0.372933	+	0.927858i	$0.378352\pi$
$240$	0	0
$241$	11.8194	0.761354	0.380677	−	0.924708i	$-0.375691\pi$
$241$	11.8194	0.761354	0.380677	+	0.924708i	$0.375691\pi$
$242$	0	0
$243$	−15.7952	−1.01326
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.55496	−0.0989396
$248$	0	0
$249$	−9.81163	−0.621787
$250$	0	0
$251$	9.66487	0.610041	0.305021	−	0.952346i	$-0.401337\pi$
$251$	9.66487	0.610041	0.305021	+	0.952346i	$0.401337\pi$
$252$	0	0
$253$	3.87263	0.243470
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.1860	−0.884897	−0.442449	−	0.896794i	$-0.645890\pi$
$257$	−14.1860	−0.884897	−0.442449	+	0.896794i	$0.645890\pi$
$258$	0	0
$259$	−10.6528	−0.661932
$260$	0	0
$261$	−11.8073	−0.730854
$262$	0	0
$263$	−21.7942	−1.34389	−0.671943	−	0.740603i	$-0.734540\pi$
$263$	−21.7942	−1.34389	−0.671943	+	0.740603i	$0.734540\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.33944	−0.143171
$268$	0	0
$269$	−24.7265	−1.50760	−0.753800	−	0.657104i	$-0.771781\pi$
$269$	−24.7265	−1.50760	−0.753800	+	0.657104i	$0.771781\pi$
$270$	0	0
$271$	13.2295	0.803636	0.401818	−	0.915720i	$-0.368378\pi$
$271$	13.2295	0.803636	0.401818	+	0.915720i	$0.368378\pi$
$272$	0	0
$273$	2.10992	0.127698
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.560335	0.0336673	0.0168336	−	0.999858i	$-0.494641\pi$
$277$	0.560335	0.0336673	0.0168336	+	0.999858i	$0.494641\pi$
$278$	0	0
$279$	4.31096	0.258091
$280$	0	0
$281$	27.6039	1.64671	0.823355	−	0.567527i	$-0.192100\pi$
$281$	27.6039	1.64671	0.823355	+	0.567527i	$0.192100\pi$
$282$	0	0
$283$	−15.9608	−0.948769	−0.474385	−	0.880318i	$-0.657329\pi$
$283$	−15.9608	−0.948769	−0.474385	+	0.880318i	$0.657329\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.07367	0.417546
$288$	0	0
$289$	11.0465	0.649796
$290$	0	0
$291$	1.24698	0.0730992
$292$	0	0
$293$	−25.3327	−1.47995	−0.739977	−	0.672632i	$-0.765164\pi$
$293$	−25.3327	−1.47995	−0.739977	+	0.672632i	$0.765164\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.91723	0.227301
$298$	0	0
$299$	−6.60388	−0.381912
$300$	0	0
$301$	12.3817	0.713666
$302$	0	0
$303$	13.3134	0.764832
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−11.5574	−0.659613	−0.329806	−	0.944049i	$-0.606983\pi$
$307$	−11.5574	−0.659613	−0.329806	+	0.944049i	$0.606983\pi$
$308$	0	0
$309$	3.88577	0.221054
$310$	0	0
$311$	18.3424	1.04010	0.520052	−	0.854135i	$-0.325913\pi$
$311$	18.3424	1.04010	0.520052	+	0.854135i	$0.325913\pi$
$312$	0	0
$313$	18.9119	1.06896	0.534481	−	0.845181i	$-0.320507\pi$
$313$	18.9119	1.06896	0.534481	+	0.845181i	$0.320507\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.2784	−1.13895	−0.569475	−	0.822008i	$-0.692854\pi$
$317$	−20.2784	−1.13895	−0.569475	+	0.822008i	$0.692854\pi$
$318$	0	0
$319$	4.56810	0.255765
$320$	0	0
$321$	3.58211	0.199934
$322$	0	0
$323$	−5.29590	−0.294672
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−15.0978	−0.834912
$328$	0	0
$329$	−3.46681	−0.191132
$330$	0	0
$331$	4.77479	0.262446	0.131223	−	0.991353i	$-0.458110\pi$
$331$	4.77479	0.262446	0.131223	+	0.991353i	$0.458110\pi$
$332$	0	0
$333$	14.8388	0.813160
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−24.3967	−1.32897	−0.664487	−	0.747300i	$-0.731350\pi$
$337$	−24.3967	−1.32897	−0.664487	+	0.747300i	$0.731350\pi$
$338$	0	0
$339$	16.0683	0.872710
$340$	0	0
$341$	−1.66786	−0.0903196
$342$	0	0
$343$	−18.8442	−1.01749
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.99761	−0.536700	−0.268350	−	0.963322i	$-0.586478\pi$
$347$	−9.99761	−0.536700	−0.268350	+	0.963322i	$0.586478\pi$
$348$	0	0
$349$	21.9584	1.17541	0.587703	−	0.809077i	$-0.300033\pi$
$349$	21.9584	1.17541	0.587703	+	0.809077i	$0.300033\pi$
$350$	0	0
$351$	−6.67994	−0.356549
$352$	0	0
$353$	36.8786	1.96285	0.981425	−	0.191848i	$-0.0614479\pi$
$353$	36.8786	1.96285	0.981425	+	0.191848i	$0.0614479\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.18598	0.380322
$358$	0	0
$359$	−16.5187	−0.871824	−0.435912	−	0.899989i	$-0.643574\pi$
$359$	−16.5187	−0.871824	−0.435912	+	0.899989i	$0.643574\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.15452	0.428001
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6.83148	−0.356600	−0.178300	−	0.983976i	$-0.557060\pi$
$367$	−6.83148	−0.356600	−0.178300	+	0.983976i	$0.557060\pi$
$368$	0	0
$369$	−9.85325	−0.512940
$370$	0	0
$371$	4.57135	0.237333
$372$	0	0
$373$	4.76271	0.246604	0.123302	−	0.992369i	$-0.460652\pi$
$373$	4.76271	0.246604	0.123302	+	0.992369i	$0.460652\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.78986	−0.401198
$378$	0	0
$379$	−14.3773	−0.738514	−0.369257	−	0.929327i	$-0.620388\pi$
$379$	−14.3773	−0.738514	−0.369257	+	0.929327i	$0.620388\pi$
$380$	0	0
$381$	14.3308	0.734190
$382$	0	0
$383$	−30.6708	−1.56721	−0.783603	−	0.621261i	$-0.786621\pi$
$383$	−30.6708	−1.56721	−0.783603	+	0.621261i	$0.786621\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−17.2470	−0.876713
$388$	0	0
$389$	−12.9215	−0.655148	−0.327574	−	0.944825i	$-0.606231\pi$
$389$	−12.9215	−0.655148	−0.327574	+	0.944825i	$0.606231\pi$
$390$	0	0
$391$	−22.4916	−1.13745
$392$	0	0
$393$	5.96854	0.301073
$394$	0	0
$395$	0	0
$396$	0	0
$397$	29.8471	1.49798	0.748992	−	0.662579i	$-0.230538\pi$
$397$	29.8471	1.49798	0.748992	+	0.662579i	$0.230538\pi$
$398$	0	0
$399$	−1.35690	−0.0679298
$400$	0	0
$401$	−28.7101	−1.43371	−0.716856	−	0.697221i	$-0.754420\pi$
$401$	−28.7101	−1.43371	−0.716856	+	0.697221i	$0.754420\pi$
$402$	0	0
$403$	2.84415	0.141677
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.74094	−0.284568
$408$	0	0
$409$	−13.6203	−0.673479	−0.336739	−	0.941598i	$-0.609324\pi$
$409$	−13.6203	−0.673479	−0.336739	+	0.941598i	$0.609324\pi$
$410$	0	0
$411$	4.56033	0.224945
$412$	0	0
$413$	−16.7138	−0.822432
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.89977	0.142002
$418$	0	0
$419$	2.13946	0.104519	0.0522596	−	0.998634i	$-0.483358\pi$
$419$	2.13946	0.104519	0.0522596	+	0.998634i	$0.483358\pi$
$420$	0	0
$421$	−15.0562	−0.733795	−0.366897	−	0.930261i	$-0.619580\pi$
$421$	−15.0562	−0.733795	−0.366897	+	0.930261i	$0.619580\pi$
$422$	0	0
$423$	4.82908	0.234798
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.918559	0.0444522
$428$	0	0
$429$	1.13706	0.0548979
$430$	0	0
$431$	−4.37435	−0.210705	−0.105353	−	0.994435i	$-0.533597\pi$
$431$	−4.37435	−0.210705	−0.105353	+	0.994435i	$0.533597\pi$
$432$	0	0
$433$	33.1564	1.59340	0.796698	−	0.604377i	$-0.206578\pi$
$433$	33.1564	1.59340	0.796698	+	0.604377i	$0.206578\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.24698	0.203161
$438$	0	0
$439$	−32.2368	−1.53858	−0.769290	−	0.638900i	$-0.779390\pi$
$439$	−32.2368	−1.53858	−0.769290	+	0.638900i	$0.779390\pi$
$440$	0	0
$441$	9.75063	0.464316
$442$	0	0
$443$	21.7362	1.03272	0.516358	−	0.856373i	$-0.327287\pi$
$443$	21.7362	1.03272	0.516358	+	0.856373i	$0.327287\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.63879	−0.124811
$448$	0	0
$449$	−24.9584	−1.17786	−0.588929	−	0.808185i	$-0.700450\pi$
$449$	−24.9584	−1.17786	−0.588929	+	0.808185i	$0.700450\pi$
$450$	0	0
$451$	3.81210	0.179505
$452$	0	0
$453$	−8.19029	−0.384814
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13.1347	−0.614414	−0.307207	−	0.951643i	$-0.599394\pi$
$457$	−13.1347	−0.614414	−0.307207	+	0.951643i	$0.599394\pi$
$458$	0	0
$459$	−22.7506	−1.06191
$460$	0	0
$461$	−35.3726	−1.64746	−0.823732	−	0.566979i	$-0.808112\pi$
$461$	−35.3726	−1.64746	−0.823732	+	0.566979i	$0.808112\pi$
$462$	0	0
$463$	14.6963	0.682997	0.341498	−	0.939882i	$-0.389066\pi$
$463$	14.6963	0.682997	0.341498	+	0.939882i	$0.389066\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.2121	−1.53687	−0.768435	−	0.639927i	$-0.778964\pi$
$467$	−33.2121	−1.53687	−0.768435	+	0.639927i	$0.778964\pi$
$468$	0	0
$469$	−23.6843	−1.09364
$470$	0	0
$471$	11.5036	0.530060
$472$	0	0
$473$	6.67264	0.306809
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.36765	−0.291555
$478$	0	0
$479$	−24.6219	−1.12500	−0.562502	−	0.826796i	$-0.690161\pi$
$479$	−24.6219	−1.12500	−0.562502	+	0.826796i	$0.690161\pi$
$480$	0	0
$481$	9.78986	0.446379
$482$	0	0
$483$	−5.76271	−0.262212
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−29.5646	−1.33970	−0.669851	−	0.742496i	$-0.733642\pi$
$487$	−29.5646	−1.33970	−0.669851	+	0.742496i	$0.733642\pi$
$488$	0	0
$489$	15.6625	0.708282
$490$	0	0
$491$	−36.2978	−1.63810	−0.819049	−	0.573724i	$-0.805498\pi$
$491$	−36.2978	−1.63810	−0.819049	+	0.573724i	$0.805498\pi$
$492$	0	0
$493$	−26.5308	−1.19489
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21.7899	0.977409
$498$	0	0
$499$	−7.84415	−0.351152	−0.175576	−	0.984466i	$-0.556179\pi$
$499$	−7.84415	−0.351152	−0.175576	+	0.984466i	$0.556179\pi$
$500$	0	0
$501$	−9.52888	−0.425719
$502$	0	0
$503$	−20.4166	−0.910330	−0.455165	−	0.890407i	$-0.650420\pi$
$503$	−20.4166	−0.910330	−0.455165	+	0.890407i	$0.650420\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.48619	0.376885
$508$	0	0
$509$	−14.7530	−0.653916	−0.326958	−	0.945039i	$-0.606024\pi$
$509$	−14.7530	−0.653916	−0.326958	+	0.945039i	$0.606024\pi$
$510$	0	0
$511$	4.75004	0.210129
$512$	0	0
$513$	4.29590	0.189668
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.86831	−0.0821683
$518$	0	0
$519$	12.4077	0.544639
$520$	0	0
$521$	37.1487	1.62751	0.813756	−	0.581206i	$-0.197419\pi$
$521$	37.1487	1.62751	0.813756	+	0.581206i	$0.197419\pi$
$522$	0	0
$523$	−16.3623	−0.715472	−0.357736	−	0.933823i	$-0.616451\pi$
$523$	−16.3623	−0.715472	−0.357736	+	0.933823i	$0.616451\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.68664	0.421957
$528$	0	0
$529$	−4.96316	−0.215790
$530$	0	0
$531$	23.2814	1.01033
$532$	0	0
$533$	−6.50066	−0.281575
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.73556	0.0748950
$538$	0	0
$539$	−3.77240	−0.162489
$540$	0	0
$541$	−2.08947	−0.0898335	−0.0449168	−	0.998991i	$-0.514302\pi$
$541$	−2.08947	−0.0898335	−0.0449168	+	0.998991i	$0.514302\pi$
$542$	0	0
$543$	−13.5506	−0.581514
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.86054	−0.0795511	−0.0397756	−	0.999209i	$-0.512664\pi$
$547$	−1.86054	−0.0795511	−0.0397756	+	0.999209i	$0.512664\pi$
$548$	0	0
$549$	−1.27950	−0.0546079
$550$	0	0
$551$	5.00969	0.213420
$552$	0	0
$553$	−2.69740	−0.114705
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.80061	−0.415265	−0.207633	−	0.978207i	$-0.566576\pi$
$557$	−9.80061	−0.415265	−0.207633	+	0.978207i	$0.566576\pi$
$558$	0	0
$559$	−11.3787	−0.481266
$560$	0	0
$561$	3.87263	0.163502
$562$	0	0
$563$	−38.0170	−1.60222	−0.801112	−	0.598514i	$-0.795758\pi$
$563$	−38.0170	−1.60222	−0.801112	+	0.598514i	$0.795758\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.13467	0.257632
$568$	0	0
$569$	−7.32975	−0.307279	−0.153640	−	0.988127i	$-0.549099\pi$
$569$	−7.32975	−0.307279	−0.153640	+	0.988127i	$0.549099\pi$
$570$	0	0
$571$	−37.8775	−1.58513	−0.792563	−	0.609791i	$-0.791254\pi$
$571$	−37.8775	−1.58513	−0.792563	+	0.609791i	$0.791254\pi$
$572$	0	0
$573$	4.75063	0.198460
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.6993	1.19477	0.597384	−	0.801955i	$-0.296207\pi$
$577$	28.6993	1.19477	0.597384	+	0.801955i	$0.296207\pi$
$578$	0	0
$579$	−3.55688	−0.147819
$580$	0	0
$581$	20.7017	0.858852
$582$	0	0
$583$	2.46357	0.102030
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.72348	−0.153684	−0.0768422	−	0.997043i	$-0.524484\pi$
$587$	−3.72348	−0.153684	−0.0768422	+	0.997043i	$0.524484\pi$
$588$	0	0
$589$	−1.82908	−0.0753661
$590$	0	0
$591$	13.2097	0.543373
$592$	0	0
$593$	27.7399	1.13914	0.569570	−	0.821943i	$-0.307110\pi$
$593$	27.7399	1.13914	0.569570	+	0.821943i	$0.307110\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.5778	−0.801266
$598$	0	0
$599$	23.5579	0.962551	0.481276	−	0.876569i	$-0.340174\pi$
$599$	23.5579	0.962551	0.481276	+	0.876569i	$0.340174\pi$
$600$	0	0
$601$	−7.36898	−0.300587	−0.150293	−	0.988641i	$-0.548022\pi$
$601$	−7.36898	−0.300587	−0.150293	+	0.988641i	$0.548022\pi$
$602$	0	0
$603$	32.9909	1.34349
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.4198	1.15352	0.576762	−	0.816912i	$-0.304316\pi$
$607$	28.4198	1.15352	0.576762	+	0.816912i	$0.304316\pi$
$608$	0	0
$609$	−6.79763	−0.275454
$610$	0	0
$611$	3.18598	0.128891
$612$	0	0
$613$	−6.48129	−0.261777	−0.130888	−	0.991397i	$-0.541783\pi$
$613$	−6.48129	−0.261777	−0.130888	+	0.991397i	$0.541783\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.4650	0.984924	0.492462	−	0.870334i	$-0.336097\pi$
$617$	24.4650	0.984924	0.492462	+	0.870334i	$0.336097\pi$
$618$	0	0
$619$	22.2457	0.894128	0.447064	−	0.894502i	$-0.352470\pi$
$619$	22.2457	0.894128	0.447064	+	0.894502i	$0.352470\pi$
$620$	0	0
$621$	18.2446	0.732130
$622$	0	0
$623$	4.93602	0.197757
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.731250	−0.0292033
$628$	0	0
$629$	33.3424	1.32945
$630$	0	0
$631$	15.5888	0.620581	0.310290	−	0.950642i	$-0.399574\pi$
$631$	15.5888	0.620581	0.310290	+	0.950642i	$0.399574\pi$
$632$	0	0
$633$	−3.48081	−0.138350
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.43296	0.254883
$638$	0	0
$639$	−30.3521	−1.20071
$640$	0	0
$641$	8.17496	0.322892	0.161446	−	0.986882i	$-0.448384\pi$
$641$	8.17496	0.322892	0.161446	+	0.986882i	$0.448384\pi$
$642$	0	0
$643$	11.1836	0.441038	0.220519	−	0.975383i	$-0.429225\pi$
$643$	11.1836	0.441038	0.220519	+	0.975383i	$0.429225\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.2121	0.637362	0.318681	−	0.947862i	$-0.396760\pi$
$647$	16.2121	0.637362	0.318681	+	0.947862i	$0.396760\pi$
$648$	0	0
$649$	−9.00730	−0.353567
$650$	0	0
$651$	2.48188	0.0972725
$652$	0	0
$653$	−24.7952	−0.970312	−0.485156	−	0.874427i	$-0.661237\pi$
$653$	−24.7952	−0.970312	−0.485156	+	0.874427i	$0.661237\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.61655	−0.258136
$658$	0	0
$659$	20.2868	0.790262	0.395131	−	0.918625i	$-0.370699\pi$
$659$	20.2868	0.790262	0.395131	+	0.918625i	$0.370699\pi$
$660$	0	0
$661$	20.4222	0.794332	0.397166	−	0.917747i	$-0.369994\pi$
$661$	20.4222	0.794332	0.397166	+	0.917747i	$0.369994\pi$
$662$	0	0
$663$	−6.60388	−0.256473
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.2760	0.823812
$668$	0	0
$669$	−21.0411	−0.813498
$670$	0	0
$671$	0.495024	0.0191102
$672$	0	0
$673$	−28.7254	−1.10728	−0.553641	−	0.832755i	$-0.686762\pi$
$673$	−28.7254	−1.10728	−0.553641	+	0.832755i	$0.686762\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−44.0062	−1.69130	−0.845648	−	0.533740i	$-0.820786\pi$
$677$	−44.0062	−1.69130	−0.845648	+	0.533740i	$0.820786\pi$
$678$	0	0
$679$	−2.63102	−0.100969
$680$	0	0
$681$	−11.7767	−0.451284
$682$	0	0
$683$	8.48427	0.324642	0.162321	−	0.986738i	$-0.448102\pi$
$683$	8.48427	0.324642	0.162321	+	0.986738i	$0.448102\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17.3545	0.662116
$688$	0	0
$689$	−4.20105	−0.160047
$690$	0	0
$691$	−20.2586	−0.770673	−0.385336	−	0.922776i	$-0.625915\pi$
$691$	−20.2586	−0.770673	−0.385336	+	0.922776i	$0.625915\pi$
$692$	0	0
$693$	−3.63640	−0.138135
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.1400	−0.838614
$698$	0	0
$699$	21.7472	0.822553
$700$	0	0
$701$	−23.4101	−0.884188	−0.442094	−	0.896969i	$-0.645764\pi$
$701$	−23.4101	−0.884188	−0.442094	+	0.896969i	$0.645764\pi$
$702$	0	0
$703$	−6.29590	−0.237454
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.0901	−1.05644
$708$	0	0
$709$	30.3472	1.13971	0.569857	−	0.821744i	$-0.306999\pi$
$709$	30.3472	1.13971	0.569857	+	0.821744i	$0.306999\pi$
$710$	0	0
$711$	3.75733	0.140911
$712$	0	0
$713$	−7.76809	−0.290917
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.24698	−0.345335
$718$	0	0
$719$	38.3230	1.42921	0.714604	−	0.699529i	$-0.246607\pi$
$719$	38.3230	1.42921	0.714604	+	0.699529i	$0.246607\pi$
$720$	0	0
$721$	−8.19865	−0.305334
$722$	0	0
$723$	−9.47842	−0.352506
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.69069	0.0997923	0.0498961	−	0.998754i	$-0.484111\pi$
$727$	2.69069	0.0997923	0.0498961	+	0.998754i	$0.484111\pi$
$728$	0	0
$729$	1.78986	0.0662910
$730$	0	0
$731$	−38.7536	−1.43335
$732$	0	0
$733$	−18.9651	−0.700491	−0.350246	−	0.936658i	$-0.613902\pi$
$733$	−18.9651	−0.700491	−0.350246	+	0.936658i	$0.613902\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.7638	−0.470160
$738$	0	0
$739$	−29.8278	−1.09723	−0.548616	−	0.836075i	$-0.684845\pi$
$739$	−29.8278	−1.09723	−0.548616	+	0.836075i	$0.684845\pi$
$740$	0	0
$741$	1.24698	0.0458089
$742$	0	0
$743$	8.78448	0.322271	0.161136	−	0.986932i	$-0.448484\pi$
$743$	8.78448	0.322271	0.161136	+	0.986932i	$0.448484\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−28.8364	−1.05507
$748$	0	0
$749$	−7.55794	−0.276161
$750$	0	0
$751$	18.1142	0.660998	0.330499	−	0.943806i	$-0.392783\pi$
$751$	18.1142	0.660998	0.330499	+	0.943806i	$0.392783\pi$
$752$	0	0
$753$	−7.75063	−0.282449
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.222816	0.00809840	0.00404920	−	0.999992i	$-0.498711\pi$
$757$	0.222816	0.00809840	0.00404920	+	0.999992i	$0.498711\pi$
$758$	0	0
$759$	−3.10560	−0.112726
$760$	0	0
$761$	−6.07798	−0.220327	−0.110163	−	0.993913i	$-0.535137\pi$
$761$	−6.07798	−0.220327	−0.110163	+	0.993913i	$0.535137\pi$
$762$	0	0
$763$	31.8552	1.15323
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.3599	0.554613
$768$	0	0
$769$	−14.6267	−0.527453	−0.263726	−	0.964598i	$-0.584952\pi$
$769$	−14.6267	−0.527453	−0.263726	+	0.964598i	$0.584952\pi$
$770$	0	0
$771$	11.3763	0.409706
$772$	0	0
$773$	4.05728	0.145930	0.0729651	−	0.997334i	$-0.476754\pi$
$773$	4.05728	0.145930	0.0729651	+	0.997334i	$0.476754\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.54288	0.306474
$778$	0	0
$779$	4.18060	0.149786
$780$	0	0
$781$	11.7429	0.420192
$782$	0	0
$783$	21.5211	0.769102
$784$	0	0
$785$	0	0
$786$	0	0
$787$	19.5657	0.697442	0.348721	−	0.937227i	$-0.386616\pi$
$787$	19.5657	0.697442	0.348721	+	0.937227i	$0.386616\pi$
$788$	0	0
$789$	17.4776	0.622218
$790$	0	0
$791$	−33.9028	−1.20544
$792$	0	0
$793$	−0.844150	−0.0299767
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.9051	−1.37809	−0.689046	−	0.724718i	$-0.741970\pi$
$797$	−38.9051	−1.37809	−0.689046	+	0.724718i	$0.741970\pi$
$798$	0	0
$799$	10.8509	0.383876
$800$	0	0
$801$	−6.87561	−0.242938
$802$	0	0
$803$	2.55986	0.0903355
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.8291	0.698017
$808$	0	0
$809$	−22.5730	−0.793625	−0.396812	−	0.917900i	$-0.629884\pi$
$809$	−22.5730	−0.793625	−0.396812	+	0.917900i	$0.629884\pi$
$810$	0	0
$811$	46.3605	1.62794	0.813968	−	0.580909i	$-0.197303\pi$
$811$	46.3605	1.62794	0.813968	+	0.580909i	$0.197303\pi$
$812$	0	0
$813$	−10.6093	−0.372083
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.31767	0.256013
$818$	0	0
$819$	6.20105	0.216682
$820$	0	0
$821$	−17.8194	−0.621901	−0.310951	−	0.950426i	$-0.600647\pi$
$821$	−17.8194	−0.621901	−0.310951	+	0.950426i	$0.600647\pi$
$822$	0	0
$823$	−32.5394	−1.13425	−0.567126	−	0.823631i	$-0.691945\pi$
$823$	−32.5394	−1.13425	−0.567126	+	0.823631i	$0.691945\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.8256	1.38487	0.692436	−	0.721479i	$-0.256537\pi$
$827$	39.8256	1.38487	0.692436	+	0.721479i	$0.256537\pi$
$828$	0	0
$829$	38.7525	1.34593	0.672966	−	0.739674i	$-0.265020\pi$
$829$	38.7525	1.34593	0.672966	+	0.739674i	$0.265020\pi$
$830$	0	0
$831$	−0.449354	−0.0155879
$832$	0	0
$833$	21.9095	0.759118
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.85756	−0.271597
$838$	0	0
$839$	−2.76749	−0.0955445	−0.0477723	−	0.998858i	$-0.515212\pi$
$839$	−2.76749	−0.0955445	−0.0477723	+	0.998858i	$0.515212\pi$
$840$	0	0
$841$	−3.90302	−0.134587
$842$	0	0
$843$	−22.1366	−0.762425
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.2054	−0.591183
$848$	0	0
$849$	12.7995	0.439279
$850$	0	0
$851$	−26.7385	−0.916586
$852$	0	0
$853$	−24.1390	−0.826503	−0.413252	−	0.910617i	$-0.635607\pi$
$853$	−24.1390	−0.826503	−0.413252	+	0.910617i	$0.635607\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.16229	0.108022	0.0540109	−	0.998540i	$-0.482799\pi$
$857$	3.16229	0.108022	0.0540109	+	0.998540i	$0.482799\pi$
$858$	0	0
$859$	−4.86426	−0.165967	−0.0829833	−	0.996551i	$-0.526445\pi$
$859$	−4.86426	−0.165967	−0.0829833	+	0.996551i	$0.526445\pi$
$860$	0	0
$861$	−5.67264	−0.193323
$862$	0	0
$863$	4.54958	0.154870	0.0774348	−	0.996997i	$-0.475327\pi$
$863$	4.54958	0.154870	0.0774348	+	0.996997i	$0.475327\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.85862	−0.300855
$868$	0	0
$869$	−1.45367	−0.0493122
$870$	0	0
$871$	21.7657	0.737502
$872$	0	0
$873$	3.66487	0.124037
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.6595	0.630086	0.315043	−	0.949077i	$-0.397981\pi$
$877$	18.6595	0.630086	0.315043	+	0.949077i	$0.397981\pi$
$878$	0	0
$879$	20.3153	0.685217
$880$	0	0
$881$	19.4306	0.654632	0.327316	−	0.944915i	$-0.393856\pi$
$881$	19.4306	0.654632	0.327316	+	0.944915i	$0.393856\pi$
$882$	0	0
$883$	25.6437	0.862979	0.431490	−	0.902118i	$-0.357988\pi$
$883$	25.6437	0.862979	0.431490	+	0.902118i	$0.357988\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.0062	−1.04109	−0.520544	−	0.853835i	$-0.674271\pi$
$887$	−31.0062	−1.04109	−0.520544	+	0.853835i	$0.674271\pi$
$888$	0	0
$889$	−30.2368	−1.01411
$890$	0	0
$891$	3.30606	0.110757
$892$	0	0
$893$	−2.04892	−0.0685644
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.29590	0.176825
$898$	0	0
$899$	−9.16315	−0.305608
$900$	0	0
$901$	−14.3080	−0.476668
$902$	0	0
$903$	−9.92931	−0.330427
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−17.7676	−0.589964	−0.294982	−	0.955503i	$-0.595314\pi$
$907$	−17.7676	−0.589964	−0.294982	+	0.955503i	$0.595314\pi$
$908$	0	0
$909$	39.1280	1.29779
$910$	0	0
$911$	−41.7313	−1.38262	−0.691309	−	0.722559i	$-0.742966\pi$
$911$	−41.7313	−1.38262	−0.691309	+	0.722559i	$0.742966\pi$
$912$	0	0
$913$	11.1564	0.369224
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.5931	−0.415862
$918$	0	0
$919$	−30.4088	−1.00309	−0.501547	−	0.865130i	$-0.667236\pi$
$919$	−30.4088	−1.00309	−0.501547	+	0.865130i	$0.667236\pi$
$920$	0	0
$921$	9.26828	0.305400
$922$	0	0
$923$	−20.0248	−0.659123
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11.4203	0.375091
$928$	0	0
$929$	−28.8219	−0.945616	−0.472808	−	0.881165i	$-0.656760\pi$
$929$	−28.8219	−0.945616	−0.472808	+	0.881165i	$0.656760\pi$
$930$	0	0
$931$	−4.13706	−0.135587
$932$	0	0
$933$	−14.7095	−0.481567
$934$	0	0
$935$	0	0
$936$	0	0
$937$	50.4601	1.64846	0.824230	−	0.566255i	$-0.191608\pi$
$937$	50.4601	1.64846	0.824230	+	0.566255i	$0.191608\pi$
$938$	0	0
$939$	−15.1661	−0.494928
$940$	0	0
$941$	1.10082	0.0358857	0.0179428	−	0.999839i	$-0.494288\pi$
$941$	1.10082	0.0358857	0.0179428	+	0.999839i	$0.494288\pi$
$942$	0	0
$943$	17.7549	0.578180
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.9353	−0.452836	−0.226418	−	0.974030i	$-0.572701\pi$
$947$	−13.9353	−0.452836	−0.226418	+	0.974030i	$0.572701\pi$
$948$	0	0
$949$	−4.36526	−0.141702
$950$	0	0
$951$	16.2620	0.527333
$952$	0	0
$953$	41.3400	1.33913	0.669567	−	0.742751i	$-0.266480\pi$
$953$	41.3400	1.33913	0.669567	+	0.742751i	$0.266480\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.66334	−0.118419
$958$	0	0
$959$	−9.62192	−0.310708
$960$	0	0
$961$	−27.6544	−0.892079
$962$	0	0
$963$	10.5278	0.339254
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.26875	−0.169432	−0.0847158	−	0.996405i	$-0.526998\pi$
$967$	−5.26875	−0.169432	−0.0847158	+	0.996405i	$0.526998\pi$
$968$	0	0
$969$	4.24698	0.136433
$970$	0	0
$971$	5.15346	0.165382	0.0826911	−	0.996575i	$-0.473649\pi$
$971$	5.15346	0.165382	0.0826911	+	0.996575i	$0.473649\pi$
$972$	0	0
$973$	−6.11828	−0.196143
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.77612	−0.152802	−0.0764008	−	0.997077i	$-0.524343\pi$
$977$	−4.77612	−0.152802	−0.0764008	+	0.997077i	$0.524343\pi$
$978$	0	0
$979$	2.66009	0.0850168
$980$	0	0
$981$	−44.3726	−1.41671
$982$	0	0
$983$	−28.9758	−0.924186	−0.462093	−	0.886832i	$-0.652901\pi$
$983$	−28.9758	−0.924186	−0.462093	+	0.886832i	$0.652901\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.78017	0.0884937
$988$	0	0
$989$	31.0780	0.988222
$990$	0	0
$991$	38.5042	1.22313	0.611564	−	0.791195i	$-0.290541\pi$
$991$	38.5042	1.22313	0.611564	+	0.791195i	$0.290541\pi$
$992$	0	0
$993$	−3.82908	−0.121512
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.1491	−0.321427	−0.160713	−	0.987001i	$-0.551379\pi$
$997$	−10.1491	−0.321427	−0.160713	+	0.987001i	$0.551379\pi$
$998$	0	0
$999$	−27.0465	−0.855714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bw.1.1		3
4.3	odd	2		475.2.a.d.1.3	✓	3
5.4	even	2		7600.2.a.bn.1.3		3
12.11	even	2		4275.2.a.bn.1.1		3
20.3	even	4		475.2.b.c.324.3		6
20.7	even	4		475.2.b.c.324.4		6
20.19	odd	2		475.2.a.h.1.1	yes	3
60.59	even	2		4275.2.a.z.1.3		3
76.75	even	2		9025.2.a.be.1.1		3
380.379	even	2		9025.2.a.w.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.3	✓	3	4.3	odd	2
475.2.a.h.1.1	yes	3	20.19	odd	2
475.2.b.c.324.3		6	20.3	even	4
475.2.b.c.324.4		6	20.7	even	4
4275.2.a.z.1.3		3	60.59	even	2
4275.2.a.bn.1.1		3	12.11	even	2
7600.2.a.bn.1.3		3	5.4	even	2
7600.2.a.bw.1.1		3	1.1	even	1	trivial
9025.2.a.w.1.3		3	380.379	even	2
9025.2.a.be.1.1		3	76.75	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-0.801938 q^{3} +1.69202 q^{7} -2.35690 q^{9} +0.911854 q^{11} -1.55496 q^{13} -5.29590 q^{17} +1.00000 q^{19} -1.35690 q^{21} +4.24698 q^{23} +4.29590 q^{27} +5.00969 q^{29} -1.82908 q^{31} -0.731250 q^{33} -6.29590 q^{37} +1.24698 q^{39} +4.18060 q^{41} +7.31767 q^{43} -2.04892 q^{47} -4.13706 q^{49} +4.24698 q^{51} +2.70171 q^{53} -0.801938 q^{57} -9.87800 q^{59} +0.542877 q^{61} -3.98792 q^{63} -13.9976 q^{67} -3.40581 q^{69} +12.8780 q^{71} +2.80731 q^{73} +1.54288 q^{77} -1.59419 q^{79} +3.62565 q^{81} +12.2349 q^{83} -4.01746 q^{87} +2.91723 q^{89} -2.63102 q^{91} +1.46681 q^{93} -1.55496 q^{97} -2.14914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{9} - q^{11} - 5 q^{13} - 2 q^{17} + 3 q^{19} + 8 q^{23} - q^{27} - 7 q^{29} + 5 q^{31} - 10 q^{33} - 5 q^{37} - q^{39} + q^{41} + 5 q^{43} + 3 q^{47} - 7 q^{49} + 8 q^{51} - 19 q^{53}+ \cdots - 20 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^9 - q^11 - 5 * q^13 - 2 * q^17 + 3 * q^19 + 8 * q^23 - q^27 - 7 * q^29 + 5 * q^31 - 10 * q^33 - 5 * q^37 - q^39 + q^41 + 5 * q^43 + 3 * q^47 - 7 * q^49 + 8 * q^51 - 19 * q^53 + 2 * q^57 - 10 * q^59 - 17 * q^61 + 7 * q^63 - q^67 + 3 * q^69 + 19 * q^71 + q^73 - 14 * q^77 - 18 * q^79 - q^81 + 13 * q^83 - 28 * q^87 + 2 * q^89 + 7 * q^91 + q^93 - 5 * q^97 - 20 * q^99

Introduction

L-functions

Modular forms

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