LMFDB - Embedded newform 95.3.d.c.94.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [95,3,Mod(94,95)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("95.94"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 95 = 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	95.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,16,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.58856251142$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7600.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 19 $ x^4 - 9*x^2 + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			94.4
Root			$2.37024$ of defining polynomial
Character	$\chi$	$=$	95.94

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.83513 q^{2} -4.74048 q^{3} +10.7082 q^{4} +5.00000 q^{5} -18.1803 q^{6} +25.7268 q^{8} +13.4721 q^{9} +19.1756 q^{10} -13.4164 q^{11} -50.7620 q^{12} -16.4596 q^{13} -23.7024 q^{15} +55.8328 q^{16} +51.6674 q^{18} -19.0000 q^{19} +53.5410 q^{20} -51.4536 q^{22} -121.957 q^{24} +25.0000 q^{25} -63.1246 q^{26} -21.2001 q^{27} -90.9017 q^{30} +111.219 q^{32} +63.6002 q^{33} +144.262 q^{36} +9.74513 q^{37} -72.8674 q^{38} +78.0263 q^{39} +128.634 q^{40} -143.666 q^{44} +67.3607 q^{45} -264.674 q^{48} +49.0000 q^{49} +95.8782 q^{50} -176.253 q^{52} +80.0598 q^{53} -81.3050 q^{54} -67.0820 q^{55} +90.0691 q^{57} -253.810 q^{60} -120.748 q^{61} +203.207 q^{64} -82.2979 q^{65} +243.915 q^{66} +53.8551 q^{67} +346.595 q^{72} +37.3738 q^{74} -118.512 q^{75} -203.456 q^{76} +299.241 q^{78} +279.164 q^{80} -20.7508 q^{81} -345.161 q^{88} +258.337 q^{90} -95.0000 q^{95} -527.230 q^{96} +58.3314 q^{97} +187.921 q^{98} -180.748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4} + 20 q^{5} - 28 q^{6} + 36 q^{9} + 116 q^{16} - 76 q^{19} + 80 q^{20} - 300 q^{24} + 100 q^{25} - 172 q^{26} - 140 q^{30} + 264 q^{36} + 8 q^{39} - 360 q^{44} + 180 q^{45} + 196 q^{49} - 200 q^{54}+ \cdots - 240 q^{99}+O(q^{100}) $ 4 * q + 16 * q^4 + 20 * q^5 - 28 * q^6 + 36 * q^9 + 116 * q^16 - 76 * q^19 + 80 * q^20 - 300 * q^24 + 100 * q^25 - 172 * q^26 - 140 * q^30 + 264 * q^36 + 8 * q^39 - 360 * q^44 + 180 * q^45 + 196 * q^49 - 200 * q^54 + 464 * q^64 + 600 * q^66 - 92 * q^74 - 304 * q^76 + 580 * q^80 - 244 * q^81 - 380 * q^95 - 812 * q^96 - 240 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$.

$n$	$21$	$77$
$\chi(n)$	$-1$	$-1$

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 $

$2$	3.83513	1.91756	0.958782	−	0.284143i	$-0.0917091\pi$
$2$	3.83513	1.91756	0.958782	+	0.284143i	$0.0917091\pi$
$3$	−4.74048	−1.58016	−0.790080	−	0.613004i	$-0.789961\pi$
$3$	−4.74048	−1.58016	−0.790080	+	0.613004i	$0.789961\pi$
$4$	10.7082	2.67705
$5$	5.00000	1.00000
$5$	5.00000	1.00000
$6$	−18.1803	−3.03006
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	25.7268	3.21585
$9$	13.4721	1.49690
$10$	19.1756	1.91756
$11$	−13.4164	−1.21967	−0.609837	−	0.792527i	$-0.708765\pi$
$11$	−13.4164	−1.21967	−0.609837	+	0.792527i	$0.708765\pi$
$12$	−50.7620	−4.23017
$13$	−16.4596	−1.26612	−0.633061	−	0.774102i	$-0.718202\pi$
$13$	−16.4596	−1.26612	−0.633061	+	0.774102i	$0.718202\pi$
$14$	0	0
$15$	−23.7024	−1.58016
$16$	55.8328	3.48955
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	51.6674	2.87041
$19$	−19.0000	−1.00000
$19$	−19.0000	−1.00000
$20$	53.5410	2.67705
$21$	0	0
$22$	−51.4536	−2.33880
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−121.957	−5.08156
$25$	25.0000	1.00000
$26$	−63.1246	−2.42787
$27$	−21.2001	−0.785188
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−90.9017	−3.03006
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	111.219	3.47558
$33$	63.6002	1.92728
$34$	0	0
$35$	0	0
$36$	144.262	4.00729
$37$	9.74513	0.263382	0.131691	−	0.991291i	$-0.457959\pi$
$37$	9.74513	0.263382	0.131691	+	0.991291i	$0.457959\pi$
$38$	−72.8674	−1.91756
$39$	78.0263	2.00067
$40$	128.634	3.21585
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−143.666	−3.26513
$45$	67.3607	1.49690
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−264.674	−5.51405
$49$	49.0000	1.00000
$50$	95.8782	1.91756
$51$	0	0
$52$	−176.253	−3.38947
$53$	80.0598	1.51056	0.755281	−	0.655401i	$-0.227500\pi$
$53$	80.0598	1.51056	0.755281	+	0.655401i	$0.227500\pi$
$54$	−81.3050	−1.50565
$55$	−67.0820	−1.21967
$56$	0	0
$57$	90.0691	1.58016
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−253.810	−4.23017
$61$	−120.748	−1.97947	−0.989735	−	0.142915i	$-0.954353\pi$
$61$	−120.748	−1.97947	−0.989735	+	0.142915i	$0.954353\pi$
$62$	0	0
$63$	0	0
$64$	203.207	3.17510
$65$	−82.2979	−1.26612
$66$	243.915	3.69568
$67$	53.8551	0.803807	0.401903	−	0.915682i	$-0.368349\pi$
$67$	53.8551	0.803807	0.401903	+	0.915682i	$0.368349\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	346.595	4.81382
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	37.3738	0.505052
$75$	−118.512	−1.58016
$76$	−203.456	−2.67705
$77$	0	0
$78$	299.241	3.83642
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	279.164	3.48955
$81$	−20.7508	−0.256182
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−345.161	−3.92229
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	258.337	2.87041
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−95.0000	−1.00000
$96$	−527.230	−5.49198
$97$	58.3314	0.601354	0.300677	−	0.953726i	$-0.402787\pi$
$97$	58.3314	0.601354	0.300677	+	0.953726i	$0.402787\pi$
$98$	187.921	1.91756
$99$	−180.748	−1.82573
$100$	267.705	2.67705
$101$	201.246	1.99254	0.996268	−	0.0863148i	$-0.0275091\pi$
$101$	201.246	1.99254	0.996268	+	0.0863148i	$0.0275091\pi$
$102$	0	0
$103$	189.355	1.83840	0.919199	−	0.393794i	$-0.128838\pi$
$103$	189.355	1.83840	0.919199	+	0.393794i	$0.128838\pi$
$104$	−423.453	−4.07166
$105$	0	0
$106$	307.039	2.89660
$107$	−200.546	−1.87426	−0.937129	−	0.348982i	$-0.886527\pi$
$107$	−200.546	−1.87426	−0.937129	+	0.348982i	$0.886527\pi$
$108$	−227.015	−2.10199
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−257.268	−2.33880
$111$	−46.1966	−0.416186
$112$	0	0
$113$	−222.274	−1.96703	−0.983514	−	0.180833i	$-0.942121\pi$
$113$	−222.274	−1.96703	−0.983514	+	0.180833i	$0.942121\pi$
$114$	345.426	3.03006
$115$	0	0
$116$	0	0
$117$	−221.746	−1.89526
$118$	0	0
$119$	0	0
$120$	−609.787	−5.08156
$121$	59.0000	0.487603
$122$	−463.083	−3.79576
$123$	0	0
$124$	0	0
$125$	125.000	1.00000
$126$	0	0
$127$	215.560	1.69732	0.848660	−	0.528938i	$-0.177410\pi$
$127$	215.560	1.69732	0.848660	+	0.528938i	$0.177410\pi$
$128$	334.449	2.61288
$129$	0	0
$130$	−315.623	−2.42787
$131$	−118.000	−0.900763	−0.450382	−	0.892836i	$-0.648712\pi$
$131$	−118.000	−0.900763	−0.450382	+	0.892836i	$0.648712\pi$
$132$	681.044	5.15942
$133$	0	0
$134$	206.541	1.54135
$135$	−106.000	−0.785188
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	201.246	1.44781	0.723907	−	0.689898i	$-0.242344\pi$
$139$	201.246	1.44781	0.723907	+	0.689898i	$0.242344\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	220.829	1.54426
$144$	752.187	5.22352
$145$	0	0
$146$	0	0
$147$	−232.283	−1.58016
$148$	104.353	0.705087
$149$	−228.079	−1.53073	−0.765366	−	0.643596i	$-0.777442\pi$
$149$	−228.079	−1.53073	−0.765366	+	0.643596i	$0.777442\pi$
$150$	−454.508	−3.03006
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−488.810	−3.21585
$153$	0	0
$154$	0	0
$155$	0	0
$156$	835.522	5.35591
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−379.522	−2.38693
$160$	556.094	3.47558
$161$	0	0
$162$	−79.5819	−0.491246
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	318.001	1.92728
$166$	0	0
$167$	−38.8411	−0.232581	−0.116291	−	0.993215i	$-0.537100\pi$
$167$	−38.8411	−0.232581	−0.116291	+	0.993215i	$0.537100\pi$
$168$	0	0
$169$	101.918	0.603065
$170$	0	0
$171$	−255.971	−1.49690
$172$	0	0
$173$	−173.688	−1.00398	−0.501988	−	0.864874i	$-0.667398\pi$
$173$	−173.688	−1.00398	−0.501988	+	0.864874i	$0.667398\pi$
$174$	0	0
$175$	0	0
$176$	−749.076	−4.25611
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	721.312	4.00729
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	572.402	3.12788
$184$	0	0
$185$	48.7257	0.263382
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−364.337	−1.91756
$191$	−302.000	−1.58115	−0.790576	−	0.612364i	$-0.790219\pi$
$191$	−302.000	−1.58115	−0.790576	+	0.612364i	$0.790219\pi$
$192$	−963.297	−5.01717
$193$	−380.156	−1.96972	−0.984859	−	0.173358i	$-0.944538\pi$
$193$	−380.156	−1.96972	−0.984859	+	0.173358i	$0.944538\pi$
$194$	223.708	1.15314
$195$	390.132	2.00067
$196$	524.702	2.67705
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−693.190	−3.50096
$199$	322.000	1.61809	0.809045	−	0.587746i	$-0.199985\pi$
$199$	322.000	1.61809	0.809045	+	0.587746i	$0.199985\pi$
$200$	643.170	3.21585
$201$	−255.299	−1.27014
$202$	771.805	3.82081
$203$	0	0
$204$	0	0
$205$	0	0
$206$	726.200	3.52524
$207$	0	0
$208$	−918.985	−4.41820
$209$	254.912	1.21967
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	857.296	4.04385
$213$	0	0
$214$	−769.118	−3.59401
$215$	0	0
$216$	−545.410	−2.52505
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−718.328	−3.26513
$221$	0	0
$222$	−177.170	−0.798062
$223$	−222.927	−0.999674	−0.499837	−	0.866120i	$-0.666607\pi$
$223$	−222.927	−0.999674	−0.499837	+	0.866120i	$0.666607\pi$
$224$	0	0
$225$	336.803	1.49690
$226$	−852.450	−3.77190
$227$	−103.373	−0.455389	−0.227694	−	0.973733i	$-0.573119\pi$
$227$	−103.373	−0.455389	−0.227694	+	0.973733i	$0.573119\pi$
$228$	964.478	4.23017
$229$	415.909	1.81619	0.908097	−	0.418759i	$-0.137535\pi$
$229$	415.909	1.81619	0.908097	+	0.418759i	$0.137535\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	−850.423	−3.63429
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	98.0000	0.410042	0.205021	−	0.978758i	$-0.434274\pi$
$239$	98.0000	0.410042	0.205021	+	0.978758i	$0.434274\pi$
$240$	−1323.37	−5.51405
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	226.273	0.935010
$243$	289.169	1.19000
$244$	−1292.99	−5.29914
$245$	245.000	1.00000
$246$	0	0
$247$	312.732	1.26612
$248$	0	0
$249$	0	0
$250$	479.391	1.91756
$251$	−182.000	−0.725100	−0.362550	−	0.931964i	$-0.618094\pi$
$251$	−182.000	−0.725100	−0.362550	+	0.931964i	$0.618094\pi$
$252$	0	0
$253$	0	0
$254$	826.699	3.25472
$255$	0	0
$256$	469.827	1.83526
$257$	−98.8970	−0.384813	−0.192407	−	0.981315i	$-0.561629\pi$
$257$	−98.8970	−0.384813	−0.192407	+	0.981315i	$0.561629\pi$
$258$	0	0
$259$	0	0
$260$	−881.263	−3.38947
$261$	0	0
$262$	−452.545	−1.72727
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	1636.23	6.19784
$265$	400.299	1.51056
$266$	0	0
$267$	0	0
$268$	576.691	2.15183
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−406.525	−1.50565
$271$	40.2492	0.148521	0.0742606	−	0.997239i	$-0.476340\pi$
$271$	40.2492	0.148521	0.0742606	+	0.997239i	$0.476340\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−335.410	−1.21967
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	771.805	2.77628
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	450.345	1.58016
$286$	846.906	2.96121
$287$	0	0
$288$	1498.35	5.20262
$289$	289.000	1.00000
$290$	0	0
$291$	−276.519	−0.950236
$292$	0	0
$293$	−585.970	−1.99990	−0.999949	−	0.0101045i	$-0.996784\pi$
$293$	−585.970	−1.99990	−0.999949	+	0.0101045i	$0.996784\pi$
$294$	−890.837	−3.03006
$295$	0	0
$296$	250.711	0.846998
$297$	284.429	0.957672
$298$	−874.712	−2.93527
$299$	0	0
$300$	−1269.05	−4.23017
$301$	0	0
$302$	0	0
$303$	−954.003	−3.14852
$304$	−1060.82	−3.48955
$305$	−603.738	−1.97947
$306$	0	0
$307$	−612.175	−1.99405	−0.997027	−	0.0770482i	$-0.975450\pi$
$307$	−612.175	−1.99405	−0.997027	+	0.0770482i	$0.975450\pi$
$308$	0	0
$309$	−897.633	−2.90496
$310$	0	0
$311$	362.243	1.16477	0.582384	−	0.812914i	$-0.302120\pi$
$311$	362.243	1.16477	0.582384	+	0.812914i	$0.302120\pi$
$312$	2007.37	6.43387
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	368.312	1.16187	0.580933	−	0.813951i	$-0.302688\pi$
$317$	368.312	1.16187	0.580933	+	0.813951i	$0.302688\pi$
$318$	−1455.51	−4.57709
$319$	0	0
$320$	1016.03	3.17510
$321$	950.683	2.96163
$322$	0	0
$323$	0	0
$324$	−222.204	−0.685813
$325$	−411.490	−1.26612
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	1219.57	3.69568
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	131.288	0.394258
$334$	−148.961	−0.445990
$335$	269.275	0.803807
$336$	0	0
$337$	671.299	1.99198	0.995992	−	0.0894409i	$-0.0285080\pi$
$337$	671.299	1.99198	0.995992	+	0.0894409i	$0.0285080\pi$
$338$	390.868	1.15642
$339$	1053.69	3.10822
$340$	0	0
$341$	0	0
$342$	−981.680	−2.87041
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−666.115	−1.92519
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−518.000	−1.48424	−0.742120	−	0.670267i	$-0.766180\pi$
$349$	−518.000	−1.48424	−0.742120	+	0.670267i	$0.766180\pi$
$350$	0	0
$351$	348.944	0.994143
$352$	−1492.16	−4.23908
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−67.0820	−0.186858	−0.0934290	−	0.995626i	$-0.529783\pi$
$359$	−67.0820	−0.186858	−0.0934290	+	0.995626i	$0.529783\pi$
$360$	1732.98	4.81382
$361$	361.000	1.00000
$362$	0	0
$363$	−279.688	−0.770491
$364$	0	0
$365$	0	0
$366$	2195.23	5.99791
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	186.869	0.505052
$371$	0	0
$372$	0	0
$373$	−234.397	−0.628410	−0.314205	−	0.949355i	$-0.601738\pi$
$373$	−234.397	−0.628410	−0.314205	+	0.949355i	$0.601738\pi$
$374$	0	0
$375$	−592.560	−1.58016
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−1017.28	−2.67705
$381$	−1021.86	−2.68204
$382$	−1158.21	−3.03196
$383$	129.299	0.337596	0.168798	−	0.985651i	$-0.446012\pi$
$383$	129.299	0.337596	0.168798	+	0.985651i	$0.446012\pi$
$384$	−1585.45	−4.12877
$385$	0	0
$386$	−1457.95	−3.77706
$387$	0	0
$388$	624.624	1.60986
$389$	−742.000	−1.90746	−0.953728	−	0.300672i	$-0.902789\pi$
$389$	−742.000	−1.90746	−0.953728	+	0.300672i	$0.902789\pi$
$390$	1496.20	3.83642
$391$	0	0
$392$	1260.61	3.21585
$393$	559.376	1.42335
$394$	0	0
$395$	0	0
$396$	−1935.48	−4.88758
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	1234.91	3.10279
$399$	0	0
$400$	1395.82	3.48955
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−979.103	−2.43558
$403$	0	0
$404$	2154.98	5.33412
$405$	−103.754	−0.256182
$406$	0	0
$407$	−130.745	−0.321240
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	2027.65	4.92148
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1830.61	−4.40051
$417$	−954.003	−2.28778
$418$	977.619	2.33880
$419$	458.000	1.09308	0.546539	−	0.837433i	$-0.315945\pi$
$419$	458.000	1.09308	0.546539	+	0.837433i	$0.315945\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	2059.68	4.85774
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−2147.48	−5.01749
$429$	−1046.83	−2.44017
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1183.66	−2.73995
$433$	−536.731	−1.23956	−0.619781	−	0.784774i	$-0.712779\pi$
$433$	−536.731	−1.23956	−0.619781	+	0.784774i	$0.712779\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−1725.81	−3.92229
$441$	660.135	1.49690
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−494.683	−1.11415
$445$	0	0
$446$	−854.954	−1.91694
$447$	1081.20	2.41880
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	1291.68	2.87041
$451$	0	0
$452$	−2380.16	−5.26583
$453$	0	0
$454$	−396.450	−0.873237
$455$	0	0
$456$	2317.19	5.08156
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	1595.06	3.48267
$459$	0	0
$460$	0	0
$461$	−598.000	−1.29718	−0.648590	−	0.761138i	$-0.724641\pi$
$461$	−598.000	−1.29718	−0.648590	+	0.761138i	$0.724641\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−2374.50	−5.07372
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−475.000	−1.00000
$476$	0	0
$477$	1078.58	2.26117
$478$	375.843	0.786281
$479$	898.899	1.87662	0.938308	−	0.345800i	$-0.112392\pi$
$479$	898.899	1.87662	0.938308	+	0.345800i	$0.112392\pi$
$480$	−2636.15	−5.49198
$481$	−160.401	−0.333474
$482$	0	0
$483$	0	0
$484$	631.784	1.30534
$485$	291.657	0.601354
$486$	1109.00	2.28189
$487$	65.3247	0.134137	0.0670685	−	0.997748i	$-0.478635\pi$
$487$	65.3247	0.134137	0.0670685	+	0.997748i	$0.478635\pi$
$488$	−3106.45	−6.36568
$489$	0	0
$490$	939.606	1.91756
$491$	298.000	0.606925	0.303462	−	0.952843i	$-0.401857\pi$
$491$	298.000	0.606925	0.303462	+	0.952843i	$0.401857\pi$
$492$	0	0
$493$	0	0
$494$	1199.37	2.42787
$495$	−903.738	−1.82573
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−979.398	−1.96272	−0.981360	−	0.192176i	$-0.938446\pi$
$499$	−979.398	−1.96272	−0.981360	+	0.192176i	$0.938446\pi$
$500$	1338.53	2.67705
$501$	184.125	0.367516
$502$	−697.993	−1.39042
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	1006.23	1.99254
$506$	0	0
$507$	−483.140	−0.952939
$508$	2308.26	4.54381
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	464.050	0.906349
$513$	402.801	0.785188
$514$	−379.282	−0.737904
$515$	946.775	1.83840
$516$	0	0
$517$	0	0
$518$	0	0
$519$	823.364	1.58644
$520$	−2117.26	−4.07166
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	551.745	1.05496	0.527481	−	0.849567i	$-0.323137\pi$
$523$	551.745	1.05496	0.527481	+	0.849567i	$0.323137\pi$
$524$	−1263.57	−2.41139
$525$	0	0
$526$	0	0
$527$	0	0
$528$	3550.98	6.72534
$529$	529.000	1.00000
$530$	1535.20	2.89660
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1002.73	−1.87426
$536$	1385.52	2.58492
$537$	0	0
$538$	0	0
$539$	−657.404	−1.21967
$540$	−1135.07	−2.10199
$541$	−335.410	−0.619982	−0.309991	−	0.950740i	$-0.600326\pi$
$541$	−335.410	−0.619982	−0.309991	+	0.950740i	$0.600326\pi$
$542$	154.361	0.284799
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1087.40	1.98794	0.993971	−	0.109645i	$-0.0349713\pi$
$547$	1087.40	1.98794	0.993971	+	0.109645i	$0.0349713\pi$
$548$	0	0
$549$	−1626.73	−2.96308
$550$	−1286.34	−2.33880
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−230.983	−0.416186
$556$	2154.98	3.87587
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	297.344	0.528142	0.264071	−	0.964503i	$-0.414935\pi$
$563$	297.344	0.528142	0.264071	+	0.964503i	$0.414935\pi$
$564$	0	0
$565$	−1111.37	−1.96703
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	1727.13	3.03006
$571$	415.909	0.728386	0.364193	−	0.931323i	$-0.381345\pi$
$571$	415.909	0.728386	0.364193	+	0.931323i	$0.381345\pi$
$572$	2364.68	4.13405
$573$	1431.62	2.49847
$574$	0	0
$575$	0	0
$576$	2737.63	4.75283
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1108.35	1.91756
$579$	1802.12	3.11247
$580$	0	0
$581$	0	0
$582$	−1060.48	−1.82214
$583$	−1074.11	−1.84239
$584$	0	0
$585$	−1108.73	−1.89526
$586$	−2247.27	−3.83493
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−2487.34	−4.23017
$589$	0	0
$590$	0	0
$591$	0	0
$592$	544.098	0.919085
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	1090.82	1.83640
$595$	0	0
$596$	−2442.32	−4.09785
$597$	−1526.43	−2.55684
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−3048.94	−5.08156
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	725.543	1.20322
$604$	0	0
$605$	295.000	0.487603
$606$	−3658.72	−6.03750
$607$	1022.87	1.68513	0.842563	−	0.538597i	$-0.181046\pi$
$607$	1022.87	1.68513	0.842563	+	0.538597i	$0.181046\pi$
$608$	−2113.16	−3.47558
$609$	0	0
$610$	−2315.41	−3.79576
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−2347.77	−3.82373
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−3442.54	−5.57045
$619$	−13.4164	−0.0216743	−0.0108372	−	0.999941i	$-0.503450\pi$
$619$	−13.4164	−0.0216743	−0.0108372	+	0.999941i	$0.503450\pi$
$620$	0	0
$621$	0	0
$622$	1389.25	2.23352
$623$	0	0
$624$	4356.43	6.98146
$625$	625.000	1.00000
$626$	0	0
$627$	−1208.40	−1.92728
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1140.39	−1.80728	−0.903641	−	0.428291i	$-0.859116\pi$
$631$	−1140.39	−1.80728	−0.903641	+	0.428291i	$0.859116\pi$
$632$	0	0
$633$	0	0
$634$	1412.52	2.22795
$635$	1077.80	1.69732
$636$	−4064.00	−6.38993
$637$	−806.520	−1.26612
$638$	0	0
$639$	0	0
$640$	1672.24	2.61288
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	3645.99	5.67911
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−533.851	−0.823845
$649$	0	0
$650$	−1578.12	−2.42787
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−590.000	−0.900763
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	3405.22	5.15942
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	503.505	0.756014
$667$	0	0
$668$	−415.918	−0.622632
$669$	1056.78	1.57964
$670$	1032.71	1.54135
$671$	1620.00	2.41431
$672$	0	0
$673$	−1202.76	−1.78716	−0.893581	−	0.448901i	$-0.851816\pi$
$673$	−1202.76	−1.78716	−0.893581	+	0.448901i	$0.851816\pi$
$674$	2574.52	3.81976
$675$	−530.002	−0.785188
$676$	1091.36	1.61444
$677$	−619.168	−0.914576	−0.457288	−	0.889319i	$-0.651179\pi$
$677$	−619.168	−0.914576	−0.457288	+	0.889319i	$0.651179\pi$
$678$	4041.02	5.96021
$679$	0	0
$680$	0	0
$681$	490.039	0.719587
$682$	0	0
$683$	1254.89	1.83732	0.918661	−	0.395046i	$-0.129271\pi$
$683$	1254.89	1.83732	0.918661	+	0.395046i	$0.129271\pi$
$684$	−2740.99	−4.00729
$685$	0	0
$686$	0	0
$687$	−1971.61	−2.86988
$688$	0	0
$689$	−1317.75	−1.91256
$690$	0	0
$691$	1381.89	1.99984	0.999920	−	0.0126162i	$-0.00401597\pi$
$691$	1381.89	1.99984	0.999920	+	0.0126162i	$0.00401597\pi$
$692$	−1859.89	−2.68770
$693$	0	0
$694$	0	0
$695$	1006.23	1.44781
$696$	0	0
$697$	0	0
$698$	−1986.60	−2.84613
$699$	0	0
$700$	0	0
$701$	952.565	1.35887	0.679433	−	0.733738i	$-0.262226\pi$
$701$	952.565	1.35887	0.679433	+	0.733738i	$0.262226\pi$
$702$	1338.25	1.90633
$703$	−185.158	−0.263382
$704$	−2726.30	−3.87259
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	202.000	0.284908	0.142454	−	0.989801i	$-0.454501\pi$
$709$	202.000	0.284908	0.142454	+	0.989801i	$0.454501\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1104.14	1.54426
$716$	0	0
$717$	−464.567	−0.647932
$718$	−257.268	−0.358312
$719$	−1247.73	−1.73536	−0.867681	−	0.497121i	$-0.834391\pi$
$719$	−1247.73	−1.73536	−0.867681	+	0.497121i	$0.834391\pi$
$720$	3760.94	5.22352
$721$	0	0
$722$	1384.48	1.91756
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−1072.64	−1.47747
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1184.04	−1.62420
$730$	0	0
$731$	0	0
$732$	6129.39	8.37349
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−1161.42	−1.58016
$736$	0	0
$737$	−722.541	−0.980382
$738$	0	0
$739$	1402.00	1.89716	0.948579	−	0.316540i	$-0.102521\pi$
$739$	1402.00	1.89716	0.948579	+	0.316540i	$0.102521\pi$
$740$	521.764	0.705087
$741$	−1482.50	−2.00067
$742$	0	0
$743$	−88.6382	−0.119298	−0.0596489	−	0.998219i	$-0.518998\pi$
$743$	−88.6382	−0.119298	−0.0596489	+	0.998219i	$0.518998\pi$
$744$	0	0
$745$	−1140.39	−1.53073
$746$	−898.942	−1.20502
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−2272.54	−3.03006
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	862.767	1.14577
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	−2444.05	−3.21585
$761$	1435.56	1.88641	0.943203	−	0.332216i	$-0.107796\pi$
$761$	1435.56	1.88641	0.943203	+	0.332216i	$0.107796\pi$
$762$	−3918.95	−5.14298
$763$	0	0
$764$	−3233.88	−4.23282
$765$	0	0
$766$	495.878	0.647361
$767$	0	0
$768$	−2227.20	−2.90000
$769$	684.237	0.889775	0.444887	−	0.895587i	$-0.353244\pi$
$769$	684.237	0.889775	0.444887	+	0.895587i	$0.353244\pi$
$770$	0	0
$771$	468.819	0.608066
$772$	−4070.78	−5.27304
$773$	1375.66	1.77963	0.889816	−	0.456319i	$-0.150832\pi$
$773$	1375.66	1.77963	0.889816	+	0.456319i	$0.150832\pi$
$774$	0	0
$775$	0	0
$776$	1500.68	1.93387
$777$	0	0
$778$	−2845.66	−3.65767
$779$	0	0
$780$	4177.61	5.35591
$781$	0	0
$782$	0	0
$783$	0	0
$784$	2735.81	3.48955
$785$	0	0
$786$	2145.28	2.72936
$787$	1281.75	1.62865	0.814326	−	0.580408i	$-0.197107\pi$
$787$	1281.75	1.62865	0.814326	+	0.580408i	$0.197107\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−4650.06	−5.87129
$793$	1987.46	2.50625
$794$	0	0
$795$	−1897.61	−2.38693
$796$	3448.04	4.33171
$797$	1401.86	1.75892	0.879461	−	0.475971i	$-0.157903\pi$
$797$	1401.86	1.75892	0.879461	+	0.475971i	$0.157903\pi$
$798$	0	0
$799$	0	0
$800$	2780.47	3.47558
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−2733.79	−3.40024
$805$	0	0
$806$	0	0
$807$	0	0
$808$	5177.42	6.40770
$809$	−1118.00	−1.38195	−0.690977	−	0.722877i	$-0.742819\pi$
$809$	−1118.00	−1.38195	−0.690977	+	0.722877i	$0.742819\pi$
$810$	−397.909	−0.491246
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−190.801	−0.234687
$814$	−501.423	−0.615998
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	122.000	0.148599	0.0742996	−	0.997236i	$-0.476328\pi$
$821$	122.000	0.148599	0.0742996	+	0.997236i	$0.476328\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	4871.50	5.91202
$825$	1590.00	1.92728
$826$	0	0
$827$	−693.401	−0.838454	−0.419227	−	0.907881i	$-0.637699\pi$
$827$	−693.401	−0.838454	−0.419227	+	0.907881i	$0.637699\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−3344.70	−4.02007
$833$	0	0
$834$	−3658.72	−4.38696
$835$	−194.205	−0.232581
$836$	2729.65	3.26513
$837$	0	0
$838$	1756.49	2.09605
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	509.590	0.603065
$846$	0	0
$847$	0	0
$848$	4469.96	5.27118
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	−1279.85	−1.49690
$856$	−5159.40	−6.02734
$857$	590.072	0.688532	0.344266	−	0.938872i	$-0.388128\pi$
$857$	590.072	0.688532	0.344266	+	0.938872i	$0.388128\pi$
$858$	−4014.74	−4.67918
$859$	−1702.00	−1.98137	−0.990687	−	0.136160i	$-0.956524\pi$
$859$	−1702.00	−1.98137	−0.990687	+	0.136160i	$0.956524\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1712.22	−1.98403	−0.992013	−	0.126132i	$-0.959744\pi$
$863$	−1712.22	−1.98403	−0.992013	+	0.126132i	$0.959744\pi$
$864$	−2357.84	−2.72899
$865$	−868.440	−1.00398
$866$	−2058.43	−2.37694
$867$	−1370.00	−1.58016
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−886.432	−1.01772
$872$	0	0
$873$	785.848	0.900170
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1753.43	1.99935	0.999677	−	0.0254035i	$-0.00808704\pi$
$877$	1753.43	1.99935	0.999677	+	0.0254035i	$0.00808704\pi$
$878$	0	0
$879$	2777.78	3.16016
$880$	−3745.38	−4.25611
$881$	−1677.05	−1.90358	−0.951788	−	0.306756i	$-0.900757\pi$
$881$	−1677.05	−1.90358	−0.951788	+	0.306756i	$0.900757\pi$
$882$	2531.70	2.87041
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	962.816	1.08547	0.542737	−	0.839902i	$-0.317388\pi$
$887$	962.816	1.08547	0.542737	+	0.839902i	$0.317388\pi$
$888$	−1188.49	−1.33839
$889$	0	0
$890$	0	0
$891$	278.401	0.312459
$892$	−2387.15	−2.67618
$893$	0	0
$894$	4146.55	4.63820
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	3606.56	4.00729
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−5718.41	−6.32567
$905$	0	0
$906$	0	0
$907$	−986.687	−1.08786	−0.543929	−	0.839131i	$-0.683064\pi$
$907$	−986.687	−1.08786	−0.543929	+	0.839131i	$0.683064\pi$
$908$	−1106.94	−1.21910
$909$	2711.22	2.98263
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	5028.81	5.51405
$913$	0	0
$914$	0	0
$915$	2862.01	3.12788
$916$	4453.63	4.86205
$917$	0	0
$918$	0	0
$919$	−1582.00	−1.72144	−0.860718	−	0.509082i	$-0.829985\pi$
$919$	−1582.00	−1.72144	−0.860718	+	0.509082i	$0.829985\pi$
$920$	0	0
$921$	2902.00	3.15092
$922$	−2293.41	−2.48743
$923$	0	0
$924$	0	0
$925$	243.628	0.263382
$926$	0	0
$927$	2551.02	2.75190
$928$	0	0
$929$	−878.000	−0.945102	−0.472551	−	0.881303i	$-0.656667\pi$
$929$	−878.000	−0.945102	−0.472551	+	0.881303i	$0.656667\pi$
$930$	0	0
$931$	−931.000	−1.00000
$932$	0	0
$933$	−1717.21	−1.84052
$934$	0	0
$935$	0	0
$936$	−5704.81	−6.09489
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−1821.69	−1.91756
$951$	−1745.97	−1.83593
$952$	0	0
$953$	1872.33	1.96467	0.982337	−	0.187119i	$-0.0599150\pi$
$953$	1872.33	1.96467	0.982337	+	0.187119i	$0.0599150\pi$
$954$	4136.48	4.33593
$955$	−1510.00	−1.58115
$956$	1049.40	1.09770
$957$	0	0
$958$	3447.39	3.59853
$959$	0	0
$960$	−4816.48	−5.01717
$961$	961.000	1.00000
$962$	−615.158	−0.639457
$963$	−2701.78	−2.80559
$964$	0	0
$965$	−1900.78	−1.96972
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1517.88	1.56806
$969$	0	0
$970$	1118.54	1.15314
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	3096.48	3.18568
$973$	0	0
$974$	250.529	0.257216
$975$	1950.66	2.00067
$976$	−6741.68	−6.90746
$977$	−1678.36	−1.71788	−0.858938	−	0.512080i	$-0.828875\pi$
$977$	−1678.36	−1.71788	−0.858938	+	0.512080i	$0.828875\pi$
$978$	0	0
$979$	0	0
$980$	2623.51	2.67705
$981$	0	0
$982$	1142.87	1.16382
$983$	−1965.96	−1.99996	−0.999981	−	0.00613954i	$-0.998046\pi$
$983$	−1965.96	−1.99996	−0.999981	+	0.00613954i	$0.998046\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	3348.80	3.38947
$989$	0	0
$990$	−3465.95	−3.50096
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1610.00	1.61809
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−3756.12	−3.76364
$999$	−206.597	−0.206804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.3.d.c.94.4	yes	4
3.2	odd	2		855.3.g.d.379.1		4
5.2	odd	4		475.3.c.d.151.4		4
5.3	odd	4		475.3.c.d.151.1		4
5.4	even	2	inner	95.3.d.c.94.1	✓	4
15.14	odd	2		855.3.g.d.379.4		4
19.18	odd	2	inner	95.3.d.c.94.1	✓	4
57.56	even	2		855.3.g.d.379.4		4
95.18	even	4		475.3.c.d.151.4		4
95.37	even	4		475.3.c.d.151.1		4
95.94	odd	2	CM	95.3.d.c.94.4	yes	4
285.284	even	2		855.3.g.d.379.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.d.c.94.1	✓	4	5.4	even	2	inner
95.3.d.c.94.1	✓	4	19.18	odd	2	inner
95.3.d.c.94.4	yes	4	1.1	even	1	trivial
95.3.d.c.94.4	yes	4	95.94	odd	2	CM
475.3.c.d.151.1		4	5.3	odd	4
475.3.c.d.151.1		4	95.37	even	4
475.3.c.d.151.4		4	5.2	odd	4
475.3.c.d.151.4		4	95.18	even	4
855.3.g.d.379.1		4	3.2	odd	2
855.3.g.d.379.1		4	285.284	even	2
855.3.g.d.379.4		4	15.14	odd	2
855.3.g.d.379.4		4	57.56	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:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	95.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.83513 q^{2} -4.74048 q^{3} +10.7082 q^{4} +5.00000 q^{5} -18.1803 q^{6} +25.7268 q^{8} +13.4721 q^{9} +19.1756 q^{10} -13.4164 q^{11} -50.7620 q^{12} -16.4596 q^{13} -23.7024 q^{15} +55.8328 q^{16} +51.6674 q^{18} -19.0000 q^{19} +53.5410 q^{20} -51.4536 q^{22} -121.957 q^{24} +25.0000 q^{25} -63.1246 q^{26} -21.2001 q^{27} -90.9017 q^{30} +111.219 q^{32} +63.6002 q^{33} +144.262 q^{36} +9.74513 q^{37} -72.8674 q^{38} +78.0263 q^{39} +128.634 q^{40} -143.666 q^{44} +67.3607 q^{45} -264.674 q^{48} +49.0000 q^{49} +95.8782 q^{50} -176.253 q^{52} +80.0598 q^{53} -81.3050 q^{54} -67.0820 q^{55} +90.0691 q^{57} -253.810 q^{60} -120.748 q^{61} +203.207 q^{64} -82.2979 q^{65} +243.915 q^{66} +53.8551 q^{67} +346.595 q^{72} +37.3738 q^{74} -118.512 q^{75} -203.456 q^{76} +299.241 q^{78} +279.164 q^{80} -20.7508 q^{81} -345.161 q^{88} +258.337 q^{90} -95.0000 q^{95} -527.230 q^{96} +58.3314 q^{97} +187.921 q^{98} -180.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} + 20 q^{5} - 28 q^{6} + 36 q^{9} + 116 q^{16} - 76 q^{19} + 80 q^{20} - 300 q^{24} + 100 q^{25} - 172 q^{26} - 140 q^{30} + 264 q^{36} + 8 q^{39} - 360 q^{44} + 180 q^{45} + 196 q^{49} - 200 q^{54}+ \cdots - 240 q^{99}+O(q^{100}) \) 4 * q + 16 * q^4 + 20 * q^5 - 28 * q^6 + 36 * q^9 + 116 * q^16 - 76 * q^19 + 80 * q^20 - 300 * q^24 + 100 * q^25 - 172 * q^26 - 140 * q^30 + 264 * q^36 + 8 * q^39 - 360 * q^44 + 180 * q^45 + 196 * q^49 - 200 * q^54 + 464 * q^64 + 600 * q^66 - 92 * q^74 - 304 * q^76 + 580 * q^80 - 244 * q^81 - 380 * q^95 - 812 * q^96 - 240 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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