LMFDB - Embedded newform 95.3.d.c.94.3

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			94.3
Root			$-1.83901$ 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)

$f(q)$	$=$	$q+1.13657 q^{2} +3.67802 q^{3} -2.70820 q^{4} +5.00000 q^{5} +4.18034 q^{6} -7.62436 q^{8} +4.52786 q^{9} +5.68286 q^{10} +13.4164 q^{11} -9.96084 q^{12} -20.1266 q^{13} +18.3901 q^{15} +2.16718 q^{16} +5.14624 q^{18} -19.0000 q^{19} -13.5410 q^{20} +15.2487 q^{22} -28.0426 q^{24} +25.0000 q^{25} -22.8754 q^{26} -16.4486 q^{27} +20.9017 q^{30} +32.9606 q^{32} +49.3459 q^{33} -12.2624 q^{36} -73.3555 q^{37} -21.5949 q^{38} -74.0263 q^{39} -38.1218 q^{40} -36.3344 q^{44} +22.6393 q^{45} +7.97096 q^{48} +49.0000 q^{49} +28.4143 q^{50} +54.5071 q^{52} +69.4725 q^{53} -18.6950 q^{54} +67.0820 q^{55} -69.8825 q^{57} -49.8042 q^{60} +120.748 q^{61} +28.7933 q^{64} -100.633 q^{65} +56.0851 q^{66} +122.701 q^{67} -34.5221 q^{72} -83.3738 q^{74} +91.9506 q^{75} +51.4559 q^{76} -84.1362 q^{78} +10.8359 q^{80} -101.249 q^{81} -102.291 q^{88} +25.7312 q^{90} -95.0000 q^{95} +121.230 q^{96} +185.023 q^{97} +55.6920 q^{98} +60.7477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.13657	0.568286	0.284143	−	0.958782i	$-0.408291\pi$
$2$	1.13657	0.568286	0.284143	+	0.958782i	$0.408291\pi$
$3$	3.67802	1.22601	0.613004	−	0.790080i	$-0.289961\pi$
$3$	3.67802	1.22601	0.613004	+	0.790080i	$0.289961\pi$
$4$	−2.70820	−0.677051
$5$	5.00000	1.00000
$5$	5.00000	1.00000
$6$	4.18034	0.696723
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−7.62436	−0.953045
$9$	4.52786	0.503096
$10$	5.68286	0.568286
$11$	13.4164	1.21967	0.609837	−	0.792527i	$-0.291235\pi$
$11$	13.4164	1.21967	0.609837	+	0.792527i	$0.291235\pi$
$12$	−9.96084	−0.830070
$13$	−20.1266	−1.54820	−0.774102	−	0.633061i	$-0.781798\pi$
$13$	−20.1266	−1.54820	−0.774102	+	0.633061i	$0.781798\pi$
$14$	0	0
$15$	18.3901	1.22601
$16$	2.16718	0.135449
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	5.14624	0.285902
$19$	−19.0000	−1.00000
$19$	−19.0000	−1.00000
$20$	−13.5410	−0.677051
$21$	0	0
$22$	15.2487	0.693123
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−28.0426	−1.16844
$25$	25.0000	1.00000
$26$	−22.8754	−0.879823
$27$	−16.4486	−0.609208
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	20.9017	0.696723
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.9606	1.03002
$33$	49.3459	1.49533
$34$	0	0
$35$	0	0
$36$	−12.2624	−0.340622
$37$	−73.3555	−1.98258	−0.991291	−	0.131691i	$-0.957959\pi$
$37$	−73.3555	−1.98258	−0.991291	+	0.131691i	$0.957959\pi$
$38$	−21.5949	−0.568286
$39$	−74.0263	−1.89811
$40$	−38.1218	−0.953045
$41$	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			$\pi$
$44$	−36.3344	−0.825781
$45$	22.6393	0.503096
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	7.97096	0.166062
$49$	49.0000	1.00000
$50$	28.4143	0.568286
$51$	0	0
$52$	54.5071	1.04821
$53$	69.4725	1.31080	0.655401	−	0.755281i	$-0.272500\pi$
$53$	69.4725	1.31080	0.655401	+	0.755281i	$0.272500\pi$
$54$	−18.6950	−0.346205
$55$	67.0820	1.21967
$56$	0	0
$57$	−69.8825	−1.22601
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−49.8042	−0.830070
$61$	120.748	1.97947	0.989735	−	0.142915i	$-0.0456475\pi$
$61$	120.748	1.97947	0.989735	+	0.142915i	$0.0456475\pi$
$62$	0	0
$63$	0	0
$64$	28.7933	0.449896
$65$	−100.633	−1.54820
$66$	56.0851	0.849775
$67$	122.701	1.83136	0.915682	−	0.401903i	$-0.131651\pi$
$67$	122.701	1.83136	0.915682	+	0.401903i	$0.131651\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−34.5221	−0.479473
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−83.3738	−1.12667
$75$	91.9506	1.22601
$76$	51.4559	0.677051
$77$	0	0
$78$	−84.1362	−1.07867
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	10.8359	0.135449
$81$	−101.249	−1.24999
$82$	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
$88$	−102.291	−1.16240
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	25.7312	0.285902
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−95.0000	−1.00000
$96$	121.230	1.26281
$97$	185.023	1.90745	0.953726	−	0.300677i	$-0.0972126\pi$
$97$	185.023	1.90745	0.953726	+	0.300677i	$0.0972126\pi$
$98$	55.6920	0.568286
$99$	60.7477	0.613613
$100$	−67.7051	−0.677051
$101$	−201.246	−1.99254	−0.996268	−	0.0863148i	$-0.972491\pi$
$101$	−201.246	−1.99254	−0.996268	+	0.0863148i	$0.972491\pi$
$102$	0	0
$103$	−81.1215	−0.787587	−0.393794	−	0.919199i	$-0.628838\pi$
$103$	−81.1215	−0.787587	−0.393794	+	0.919199i	$0.628838\pi$
$104$	153.453	1.47551
$105$	0	0
$106$	78.9605	0.744911
$107$	−74.6821	−0.697964	−0.348982	−	0.937129i	$-0.613473\pi$
$107$	−74.6821	−0.697964	−0.348982	+	0.937129i	$0.613473\pi$
$108$	44.5462	0.412465
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	76.2436	0.693123
$111$	−269.803	−2.43066
$112$	0	0
$113$	40.8682	0.361666	0.180833	−	0.983514i	$-0.442121\pi$
$113$	40.8682	0.361666	0.180833	+	0.983514i	$0.442121\pi$
$114$	−79.4265	−0.696723
$115$	0	0
$116$	0	0
$117$	−91.1307	−0.778895
$118$	0	0
$119$	0	0
$120$	−140.213	−1.16844
$121$	59.0000	0.487603
$122$	137.238	1.12491
$123$	0	0
$124$	0	0
$125$	125.000	1.00000
$126$	0	0
$127$	−134.350	−1.05788	−0.528938	−	0.848660i	$-0.677410\pi$
$127$	−134.350	−1.05788	−0.528938	+	0.848660i	$0.677410\pi$
$128$	−99.1166	−0.774349
$129$	0	0
$130$	−114.377	−0.879823
$131$	−118.000	−0.900763	−0.450382	−	0.892836i	$-0.648712\pi$
$131$	−118.000	−0.900763	−0.450382	+	0.892836i	$0.648712\pi$
$132$	−133.639	−1.01241
$133$	0	0
$134$	139.459	1.04074
$135$	−82.2431	−0.609208
$136$	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	$-0.757656\pi$
$139$	−201.246	−1.44781	−0.723907	+	0.689898i	$0.757656\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−270.027	−1.88830
$144$	9.81272	0.0681439
$145$	0	0
$146$	0	0
$147$	180.223	1.22601
$148$	198.662	1.34231
$149$	228.079	1.53073	0.765366	−	0.643596i	$-0.222558\pi$
$149$	228.079	1.53073	0.765366	+	0.643596i	$0.222558\pi$
$150$	104.508	0.696723
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	144.863	0.953045
$153$	0	0
$154$	0	0
$155$	0	0
$156$	200.478	1.28512
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	255.522	1.60705
$160$	164.803	1.03002
$161$	0	0
$162$	−115.077	−0.710352
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	246.729	1.49533
$166$	0	0
$167$	−331.734	−1.98643	−0.993215	−	0.116291i	$-0.962900\pi$
$167$	−331.734	−1.98643	−0.993215	+	0.116291i	$0.962900\pi$
$168$	0	0
$169$	236.082	1.39694
$170$	0	0
$171$	−86.0294	−0.503096
$172$	0	0
$173$	299.247	1.72975	0.864874	−	0.501988i	$-0.167398\pi$
$173$	299.247	1.72975	0.864874	+	0.501988i	$0.167398\pi$
$174$	0	0
$175$	0	0
$176$	29.0758	0.165204
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−61.3119	−0.340622
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	444.113	2.42685
$184$	0	0
$185$	−366.778	−1.98258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−107.974	−0.568286
$191$	−302.000	−1.58115	−0.790576	−	0.612364i	$-0.790219\pi$
$191$	−302.000	−1.58115	−0.790576	+	0.612364i	$0.790219\pi$
$192$	105.903	0.551576
$193$	−66.9161	−0.346716	−0.173358	−	0.984859i	$-0.555462\pi$
$193$	−66.9161	−0.346716	−0.173358	+	0.984859i	$0.555462\pi$
$194$	210.292	1.08398
$195$	−370.132	−1.89811
$196$	−132.702	−0.677051
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	69.0441	0.348708
$199$	322.000	1.61809	0.809045	−	0.587746i	$-0.199985\pi$
$199$	322.000	1.61809	0.809045	+	0.587746i	$0.199985\pi$
$200$	−190.609	−0.953045
$201$	451.299	2.24527
$202$	−228.731	−1.13233
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−92.2004	−0.447575
$207$	0	0
$208$	−43.6182	−0.209703
$209$	−254.912	−1.21967
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−188.146	−0.887480
$213$	0	0
$214$	−84.8816	−0.396643
$215$	0	0
$216$	125.410	0.580603
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−181.672	−0.825781
$221$	0	0
$222$	−306.651	−1.38131
$223$	−386.289	−1.73224	−0.866120	−	0.499837i	$-0.833393\pi$
$223$	−386.289	−1.73224	−0.866120	+	0.499837i	$0.833393\pi$
$224$	0	0
$225$	113.197	0.503096
$226$	46.4497	0.205529
$227$	442.075	1.94747	0.973733	−	0.227694i	$-0.0731187\pi$
$227$	442.075	1.94747	0.973733	+	0.227694i	$0.0731187\pi$
$228$	189.256	0.830070
$229$	−415.909	−1.81619	−0.908097	−	0.418759i	$-0.862465\pi$
$229$	−415.909	−1.81619	−0.908097	+	0.418759i	$0.862465\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	−103.577	−0.442635
$235$	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	$0.434274\pi$
$240$	39.8548	0.166062
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	67.0578	0.277098
$243$	−224.359	−0.923290
$244$	−327.009	−1.34020
$245$	245.000	1.00000
$246$	0	0
$247$	382.406	1.54820
$248$	0	0
$249$	0	0
$250$	142.072	0.568286
$251$	−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
$254$	−152.699	−0.601177
$255$	0	0
$256$	−227.827	−0.889948
$257$	504.396	1.96263	0.981315	−	0.192407i	$-0.0616292\pi$
$257$	504.396	1.96263	0.981315	+	0.192407i	$0.0616292\pi$
$258$	0	0
$259$	0	0
$260$	272.535	1.04821
$261$	0	0
$262$	−134.116	−0.511891
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−376.231	−1.42512
$265$	347.363	1.31080
$266$	0	0
$267$	0	0
$268$	−332.300	−1.23993
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−93.4752	−0.346205
$271$	−40.2492	−0.148521	−0.0742606	−	0.997239i	$-0.523660\pi$
$271$	−40.2492	−0.148521	−0.0742606	+	0.997239i	$0.523660\pi$
$272$	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			$\pi$
$278$	−228.731	−0.822772
$279$	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
$285$	−349.412	−1.22601
$286$	−306.906	−1.07310
$287$	0	0
$288$	149.241	0.518198
$289$	289.000	1.00000
$290$	0	0
$291$	680.519	2.33855
$292$	0	0
$293$	−5.92126	−0.0202091	−0.0101045	−	0.999949i	$-0.503216\pi$
$293$	−5.92126	−0.0202091	−0.0101045	+	0.999949i	$0.503216\pi$
$294$	204.837	0.696723
$295$	0	0
$296$	559.289	1.88949
$297$	−220.681	−0.743035
$298$	259.228	0.869893
$299$	0	0
$300$	−249.021	−0.830070
$301$	0	0
$302$	0	0
$303$	−740.188	−2.44287
$304$	−41.1765	−0.135449
$305$	603.738	1.97947
$306$	0	0
$307$	47.3076	0.154096	0.0770482	−	0.997027i	$-0.475450\pi$
$307$	47.3076	0.154096	0.0770482	+	0.997027i	$0.475450\pi$
$308$	0	0
$309$	−298.367	−0.965589
$310$	0	0
$311$	−362.243	−1.16477	−0.582384	−	0.812914i	$-0.697880\pi$
$311$	−362.243	−1.16477	−0.582384	+	0.812914i	$0.697880\pi$
$312$	564.403	1.80898
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−516.045	−1.62790	−0.813951	−	0.580933i	$-0.802688\pi$
$317$	−516.045	−1.62790	−0.813951	+	0.580933i	$0.802688\pi$
$318$	290.419	0.913267
$319$	0	0
$320$	143.967	0.449896
$321$	−274.683	−0.855709
$322$	0	0
$323$	0	0
$324$	274.204	0.846307
$325$	−503.166	−1.54820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	280.426	0.849775
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−332.144	−0.997429
$334$	−377.039	−1.12886
$335$	613.507	1.83136
$336$	0	0
$337$	−60.2832	−0.178882	−0.0894409	−	0.995992i	$-0.528508\pi$
$337$	−60.2832	−0.178882	−0.0894409	+	0.995992i	$0.528508\pi$
$338$	268.324	0.793859
$339$	150.314	0.443405
$340$	0	0
$341$	0	0
$342$	−97.7786	−0.285902
$343$	0	0
$344$	0	0
$345$	0	0
$346$	340.115	0.982992
$347$	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
$351$	331.056	0.943179
$352$	442.213	1.25629
$353$	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	$-0.470217\pi$
$359$	67.0820	0.186858	0.0934290	+	0.995626i	$0.470217\pi$
$360$	−172.610	−0.479473
$361$	361.000	1.00000
$362$	0	0
$363$	217.003	0.597806
$364$	0	0
$365$	0	0
$366$	504.766	1.37914
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	−416.869	−1.12667
$371$	0	0
$372$	0	0
$373$	708.219	1.89871	0.949355	−	0.314205i	$-0.101738\pi$
$373$	708.219	1.89871	0.949355	+	0.314205i	$0.101738\pi$
$374$	0	0
$375$	459.753	1.22601
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	257.279	0.677051
$381$	−494.144	−1.29697
$382$	−343.245	−0.898547
$383$	755.008	1.97130	0.985651	−	0.168798i	$-0.0539885\pi$
$383$	755.008	1.97130	0.985651	+	0.168798i	$0.0539885\pi$
$384$	−364.553	−0.949358
$385$	0	0
$386$	−76.0550	−0.197034
$387$	0	0
$388$	−501.080	−1.29144
$389$	−742.000	−1.90746	−0.953728	−	0.300672i	$-0.902789\pi$
$389$	−742.000	−1.90746	−0.953728	+	0.300672i	$0.902789\pi$
$390$	−420.681	−1.07867
$391$	0	0
$392$	−373.594	−0.953045
$393$	−434.007	−1.10434
$394$	0	0
$395$	0	0
$396$	−164.517	−0.415447
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	365.976	0.919538
$399$	0	0
$400$	54.1796	0.135449
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	512.934	1.27595
$403$	0	0
$404$	545.016	1.34905
$405$	−506.246	−1.24999
$406$	0	0
$407$	−984.168	−2.41810
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	219.694	0.533237
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−663.386	−1.59468
$417$	−740.188	−1.77503
$418$	−289.726	−0.693123
$419$	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
$424$	−529.683	−1.24925
$425$	0	0
$426$	0	0
$427$	0	0
$428$	202.254	0.472557
$429$	−993.167	−2.31508
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−35.6472	−0.0825167
$433$	679.615	1.56955	0.784774	−	0.619781i	$-0.212779\pi$
$433$	679.615	1.56955	0.784774	+	0.619781i	$0.212779\pi$
$434$	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			$\pi$
$440$	−511.457	−1.16240
$441$	221.865	0.503096
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	730.683	1.64568
$445$	0	0
$446$	−439.046	−0.984407
$447$	838.880	1.87669
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	128.656	0.285902
$451$	0	0
$452$	−110.679	−0.244866
$453$	0	0
$454$	502.450	1.10672
$455$	0	0
$456$	532.809	1.16844
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−472.710	−1.03212
$459$	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			$\pi$
$468$	246.801	0.527352
$469$	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
$477$	314.562	0.659459
$478$	111.384	0.233021
$479$	−898.899	−1.87662	−0.938308	−	0.345800i	$-0.887608\pi$
$479$	−898.899	−1.87662	−0.938308	+	0.345800i	$0.887608\pi$
$480$	606.149	1.26281
$481$	1476.40	3.06944
$482$	0	0
$483$	0	0
$484$	−159.784	−0.330132
$485$	925.114	1.90745
$486$	−255.001	−0.524693
$487$	−971.807	−1.99550	−0.997748	−	0.0670685i	$-0.978635\pi$
$487$	−971.807	−1.99550	−0.997748	+	0.0670685i	$0.978635\pi$
$488$	−920.623	−1.88652
$489$	0	0
$490$	278.460	0.568286
$491$	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
$494$	434.632	0.879823
$495$	303.738	0.613613
$496$	0	0
$497$	0	0
$498$	0	0
$499$	979.398	1.96272	0.981360	−	0.192176i	$-0.0615544\pi$
$499$	979.398	1.96272	0.981360	+	0.192176i	$0.0615544\pi$
$500$	−338.525	−0.677051
$501$	−1220.13	−2.43538
$502$	−206.856	−0.412064
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−1006.23	−1.99254
$506$	0	0
$507$	868.316	1.71265
$508$	363.848	0.716237
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	137.525	0.268604
$513$	312.524	0.609208
$514$	573.282	1.11534
$515$	−405.608	−0.787587
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1100.64	2.12069
$520$	767.264	1.47551
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	−888.647	−1.69913	−0.849567	−	0.527481i	$-0.823137\pi$
$523$	−888.647	−1.69913	−0.849567	+	0.527481i	$0.823137\pi$
$524$	319.568	0.609863
$525$	0	0
$526$	0	0
$527$	0	0
$528$	106.942	0.202541
$529$	529.000	1.00000
$530$	394.803	0.744911
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−373.411	−0.697964
$536$	−935.519	−1.74537
$537$	0	0
$538$	0	0
$539$	657.404	1.21967
$540$	222.731	0.412465
$541$	335.410	0.619982	0.309991	−	0.950740i	$-0.399674\pi$
$541$	335.410	0.619982	0.309991	+	0.950740i	$0.399674\pi$
$542$	−45.7461	−0.0844025
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−119.951	−0.219290	−0.109645	−	0.993971i	$-0.534971\pi$
$547$	−119.951	−0.219290	−0.109645	+	0.993971i	$0.534971\pi$
$548$	0	0
$549$	546.729	0.995863
$550$	381.218	0.693123
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1349.02	−2.43066
$556$	545.016	0.980244
$557$	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
$563$	−1086.03	−1.92901	−0.964503	−	0.264071i	$-0.914935\pi$
$563$	−1086.03	−1.92901	−0.964503	+	0.264071i	$0.914935\pi$
$564$	0	0
$565$	204.341	0.361666
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	−397.132	−0.696723
$571$	−415.909	−0.728386	−0.364193	−	0.931323i	$-0.618655\pi$
$571$	−415.909	−0.728386	−0.364193	+	0.931323i	$0.618655\pi$
$572$	731.289	1.27848
$573$	−1110.76	−1.93851
$574$	0	0
$575$	0	0
$576$	130.372	0.226341
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	328.469	0.568286
$579$	−246.119	−0.425076
$580$	0	0
$581$	0	0
$582$	773.458	1.32897
$583$	932.072	1.59875
$584$	0	0
$585$	−455.654	−0.778895
$586$	−6.72994	−0.0114845
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−488.081	−0.830070
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−158.975	−0.268539
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−250.820	−0.422257
$595$	0	0
$596$	−617.684	−1.03638
$597$	1184.32	1.98379
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−701.064	−1.16844
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	555.575	0.921352
$604$	0	0
$605$	295.000	0.487603
$606$	−841.277	−1.38825
$607$	653.857	1.07719	0.538597	−	0.842563i	$-0.318954\pi$
$607$	653.857	1.07719	0.538597	+	0.842563i	$0.318954\pi$
$608$	−626.251	−1.03002
$609$	0	0
$610$	686.192	1.12491
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	53.7685	0.0875709
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−339.115	−0.548731
$619$	13.4164	0.0216743	0.0108372	−	0.999941i	$-0.496550\pi$
$619$	13.4164	0.0216743	0.0108372	+	0.999941i	$0.496550\pi$
$620$	0	0
$621$	0	0
$622$	−411.715	−0.661922
$623$	0	0
$624$	−160.429	−0.257097
$625$	625.000	1.00000
$626$	0	0
$627$	−937.572	−1.49533
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1140.39	1.80728	0.903641	−	0.428291i	$-0.140884\pi$
$631$	1140.39	1.80728	0.903641	+	0.428291i	$0.140884\pi$
$632$	0	0
$633$	0	0
$634$	−586.522	−0.925114
$635$	−671.752	−1.05788
$636$	−692.005	−1.08806
$637$	−986.206	−1.54820
$638$	0	0
$639$	0	0
$640$	−495.583	−0.774349
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−312.197	−0.486288
$643$	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			$\pi$
$648$	771.960	1.19130
$649$	0	0
$650$	−571.885	−0.879823
$651$	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			$\pi$
$660$	−668.193	−1.01241
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−377.505	−0.566825
$667$	0	0
$668$	898.403	1.34491
$669$	−1420.78	−2.12374
$670$	697.295	1.04074
$671$	1620.00	2.41431
$672$	0	0
$673$	604.221	0.897802	0.448901	−	0.893581i	$-0.351816\pi$
$673$	604.221	0.897802	0.448901	+	0.893581i	$0.351816\pi$
$674$	−68.5162	−0.101656
$675$	−411.216	−0.609208
$676$	−639.358	−0.945796
$677$	1204.14	1.77864	0.889319	−	0.457288i	$-0.151179\pi$
$677$	1204.14	1.77864	0.889319	+	0.457288i	$0.151179\pi$
$678$	170.843	0.251981
$679$	0	0
$680$	0	0
$681$	1625.96	2.38761
$682$	0	0
$683$	539.633	0.790093	0.395046	−	0.918661i	$-0.370729\pi$
$683$	539.633	0.790093	0.395046	+	0.918661i	$0.370729\pi$
$684$	232.985	0.340622
$685$	0	0
$686$	0	0
$687$	−1529.72	−2.22667
$688$	0	0
$689$	−1398.25	−2.02939
$690$	0	0
$691$	−1381.89	−1.99984	−0.999920	−	0.0126162i	$-0.995984\pi$
$691$	−1381.89	−1.99984	−0.999920	+	0.0126162i	$0.995984\pi$
$692$	−810.421	−1.17113
$693$	0	0
$694$	0	0
$695$	−1006.23	−1.44781
$696$	0	0
$697$	0	0
$698$	−588.744	−0.843473
$699$	0	0
$700$	0	0
$701$	−952.565	−1.35887	−0.679433	−	0.733738i	$-0.737774\pi$
$701$	−952.565	−1.35887	−0.679433	+	0.733738i	$0.737774\pi$
$702$	376.269	0.535995
$703$	1393.75	1.98258
$704$	386.303	0.548726
$705$	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
$715$	−1350.14	−1.88830
$716$	0	0
$717$	360.446	0.502715
$718$	76.2436	0.106189
$719$	1247.73	1.73536	0.867681	−	0.497121i	$-0.165609\pi$
$719$	1247.73	1.73536	0.867681	+	0.497121i	$0.165609\pi$
$720$	49.0636	0.0681439
$721$	0	0
$722$	410.303	0.568286
$723$	0	0
$724$	0	0
$725$	0	0
$726$	246.640	0.339725
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	86.0433	0.118029
$730$	0	0
$731$	0	0
$732$	−1202.75	−1.64310
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	901.116	1.22601
$736$	0	0
$737$	1646.21	2.23367
$738$	0	0
$739$	1402.00	1.89716	0.948579	−	0.316540i	$-0.102521\pi$
$739$	1402.00	1.89716	0.948579	+	0.316540i	$0.102521\pi$
$740$	993.309	1.34231
$741$	1406.50	1.89811
$742$	0	0
$743$	1483.35	1.99644	0.998219	−	0.0596489i	$-0.0189981\pi$
$743$	1483.35	1.99644	0.998219	+	0.0596489i	$0.0189981\pi$
$744$	0	0
$745$	1140.39	1.53073
$746$	804.942	1.07901
$747$	0	0
$748$	0	0
$749$	0	0
$750$	522.542	0.696723
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−669.400	−0.888978
$754$	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
$760$	724.314	0.953045
$761$	−1435.56	−1.88641	−0.943203	−	0.332216i	$-0.892204\pi$
$761$	−1435.56	−1.88641	−0.943203	+	0.332216i	$0.892204\pi$
$762$	−561.630	−0.737048
$763$	0	0
$764$	817.878	1.07052
$765$	0	0
$766$	858.122	1.12026
$767$	0	0
$768$	−837.952	−1.09108
$769$	−684.237	−0.889775	−0.444887	−	0.895587i	$-0.646756\pi$
$769$	−684.237	−0.889775	−0.444887	+	0.895587i	$0.646756\pi$
$770$	0	0
$771$	1855.18	2.40620
$772$	181.223	0.234744
$773$	−705.469	−0.912638	−0.456319	−	0.889816i	$-0.650832\pi$
$773$	−705.469	−0.912638	−0.456319	+	0.889816i	$0.650832\pi$
$774$	0	0
$775$	0	0
$776$	−1410.68	−1.81789
$777$	0	0
$778$	−843.336	−1.08398
$779$	0	0
$780$	1002.39	1.28512
$781$	0	0
$782$	0	0
$783$	0	0
$784$	106.192	0.135449
$785$	0	0
$786$	−493.280	−0.627583
$787$	913.562	1.16082	0.580408	−	0.814326i	$-0.302893\pi$
$787$	913.562	1.16082	0.580408	+	0.814326i	$0.302893\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−463.162	−0.584800
$793$	−2430.25	−3.06462
$794$	0	0
$795$	1277.61	1.60705
$796$	−872.042	−1.09553
$797$	−758.698	−0.951942	−0.475971	−	0.879461i	$-0.657903\pi$
$797$	−758.698	−0.951942	−0.475971	+	0.879461i	$0.657903\pi$
$798$	0	0
$799$	0	0
$800$	824.015	1.03002
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−1222.21	−1.52016
$805$	0	0
$806$	0	0
$807$	0	0
$808$	1534.37	1.89898
$809$	−1118.00	−1.38195	−0.690977	−	0.722877i	$-0.742819\pi$
$809$	−1118.00	−1.38195	−0.690977	+	0.722877i	$0.742819\pi$
$810$	−575.385	−0.710352
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−148.038	−0.182088
$814$	−1118.58	−1.37417
$815$	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			$\pi$
$824$	618.499	0.750606
$825$	1233.65	1.49533
$826$	0	0
$827$	−1501.64	−1.81576	−0.907881	−	0.419227i	$-0.862301\pi$
$827$	−1501.64	−1.81576	−0.907881	+	0.419227i	$0.862301\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−579.514	−0.696531
$833$	0	0
$834$	−841.277	−1.00873
$835$	−1658.67	−1.98643
$836$	690.353	0.825781
$837$	0	0
$838$	520.550	0.621181
$839$	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
$845$	1180.41	1.39694
$846$	0	0
$847$	0	0
$848$	150.560	0.177547
$849$	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
$855$	−430.147	−0.503096
$856$	569.403	0.665191
$857$	−1609.23	−1.87774	−0.938872	−	0.344266i	$-0.888128\pi$
$857$	−1609.23	−1.87774	−0.938872	+	0.344266i	$0.888128\pi$
$858$	−1128.81	−1.31562
$859$	−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
$863$	−217.704	−0.252264	−0.126132	−	0.992013i	$-0.540256\pi$
$863$	−217.704	−0.252264	−0.126132	+	0.992013i	$0.540256\pi$
$864$	−542.156	−0.627496
$865$	1496.23	1.72975
$866$	772.431	0.891953
$867$	1062.95	1.22601
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2469.57	−2.83532
$872$	0	0
$873$	837.758	0.959631
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44.5577	−0.0508069	−0.0254035	−	0.999677i	$-0.508087\pi$
$877$	−44.5577	−0.0508069	−0.0254035	+	0.999677i	$0.508087\pi$
$878$	0	0
$879$	−21.7786	−0.0247765
$880$	145.379	0.165204
$881$	1677.05	1.90358	0.951788	−	0.306756i	$-0.0992434\pi$
$881$	1677.05	1.90358	0.951788	+	0.306756i	$0.0992434\pi$
$882$	252.166	0.285902
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1489.99	1.67980	0.839902	−	0.542737i	$-0.182612\pi$
$887$	1489.99	1.67980	0.839902	+	0.542737i	$0.182612\pi$
$888$	2057.08	2.31653
$889$	0	0
$890$	0	0
$891$	−1358.40	−1.52458
$892$	1046.15	1.17281
$893$	0	0
$894$	953.447	1.06650
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−306.559	−0.340622
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−311.594	−0.344683
$905$	0	0
$906$	0	0
$907$	1522.18	1.67826	0.839131	−	0.543929i	$-0.183064\pi$
$907$	1522.18	1.67826	0.839131	+	0.543929i	$0.183064\pi$
$908$	−1197.23	−1.31853
$909$	−911.215	−1.00244
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−151.448	−0.166062
$913$	0	0
$914$	0	0
$915$	2220.56	2.42685
$916$	1126.37	1.22966
$917$	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
$921$	173.999	0.188923
$922$	−679.670	−0.737169
$923$	0	0
$924$	0	0
$925$	−1833.89	−1.98258
$926$	0	0
$927$	−367.307	−0.396232
$928$	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
$933$	−1332.34	−1.42802
$934$	0	0
$935$	0	0
$936$	694.813	0.742322
$937$	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
$950$	−539.872	−0.568286
$951$	−1898.03	−1.99582
$952$	0	0
$953$	356.649	0.374238	0.187119	−	0.982337i	$-0.440085\pi$
$953$	356.649	0.374238	0.187119	+	0.982337i	$0.440085\pi$
$954$	357.523	0.374762
$955$	−1510.00	−1.58115
$956$	−265.404	−0.277619
$957$	0	0
$958$	−1021.66	−1.06645
$959$	0	0
$960$	529.513	0.551576
$961$	961.000	1.00000
$962$	1678.04	1.74432
$963$	−338.150	−0.351143
$964$	0	0
$965$	−334.581	−0.346716
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−449.837	−0.464708
$969$	0	0
$970$	1051.46	1.08398
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	607.611	0.625114
$973$	0	0
$974$	−1104.53	−1.13401
$975$	−1850.66	−1.89811
$976$	261.682	0.268117
$977$	−1000.60	−1.02416	−0.512080	−	0.858938i	$-0.671125\pi$
$977$	−1000.60	−1.02416	−0.512080	+	0.858938i	$0.671125\pi$
$978$	0	0
$979$	0	0
$980$	−663.510	−0.677051
$981$	0	0
$982$	338.698	0.344907
$983$	12.0703	0.0122791	0.00613954	−	0.999981i	$-0.498046\pi$
$983$	12.0703	0.0122791	0.00613954	+	0.999981i	$0.498046\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−1035.63	−1.04821
$989$	0	0
$990$	345.221	0.348708
$991$	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			$\pi$
$998$	1113.16	1.11539
$999$	1206.60	1.20781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.3.d.c.94.3	yes	4
3.2	odd	2		855.3.g.d.379.2		4
5.2	odd	4		475.3.c.d.151.3		4
5.3	odd	4		475.3.c.d.151.2		4
5.4	even	2	inner	95.3.d.c.94.2	✓	4
15.14	odd	2		855.3.g.d.379.3		4
19.18	odd	2	inner	95.3.d.c.94.2	✓	4
57.56	even	2		855.3.g.d.379.3		4
95.18	even	4		475.3.c.d.151.3		4
95.37	even	4		475.3.c.d.151.2		4
95.94	odd	2	CM	95.3.d.c.94.3	yes	4
285.284	even	2		855.3.g.d.379.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.d.c.94.2	✓	4	5.4	even	2	inner
95.3.d.c.94.2	✓	4	19.18	odd	2	inner
95.3.d.c.94.3	yes	4	1.1	even	1	trivial
95.3.d.c.94.3	yes	4	95.94	odd	2	CM
475.3.c.d.151.2		4	5.3	odd	4
475.3.c.d.151.2		4	95.37	even	4
475.3.c.d.151.3		4	5.2	odd	4
475.3.c.d.151.3		4	95.18	even	4
855.3.g.d.379.2		4	3.2	odd	2
855.3.g.d.379.2		4	285.284	even	2
855.3.g.d.379.3		4	15.14	odd	2
855.3.g.d.379.3		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)

\(f(q)\)	\(=\)	\(q+1.13657 q^{2} +3.67802 q^{3} -2.70820 q^{4} +5.00000 q^{5} +4.18034 q^{6} -7.62436 q^{8} +4.52786 q^{9} +5.68286 q^{10} +13.4164 q^{11} -9.96084 q^{12} -20.1266 q^{13} +18.3901 q^{15} +2.16718 q^{16} +5.14624 q^{18} -19.0000 q^{19} -13.5410 q^{20} +15.2487 q^{22} -28.0426 q^{24} +25.0000 q^{25} -22.8754 q^{26} -16.4486 q^{27} +20.9017 q^{30} +32.9606 q^{32} +49.3459 q^{33} -12.2624 q^{36} -73.3555 q^{37} -21.5949 q^{38} -74.0263 q^{39} -38.1218 q^{40} -36.3344 q^{44} +22.6393 q^{45} +7.97096 q^{48} +49.0000 q^{49} +28.4143 q^{50} +54.5071 q^{52} +69.4725 q^{53} -18.6950 q^{54} +67.0820 q^{55} -69.8825 q^{57} -49.8042 q^{60} +120.748 q^{61} +28.7933 q^{64} -100.633 q^{65} +56.0851 q^{66} +122.701 q^{67} -34.5221 q^{72} -83.3738 q^{74} +91.9506 q^{75} +51.4559 q^{76} -84.1362 q^{78} +10.8359 q^{80} -101.249 q^{81} -102.291 q^{88} +25.7312 q^{90} -95.0000 q^{95} +121.230 q^{96} +185.023 q^{97} +55.6920 q^{98} +60.7477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} + 20 q^{5} - 28 q^{6} + 36 q^{9} + 116 q^{16} - 76 q^{19} + 80 q^{20} - 300 q^{24} + 100 q^{25} - 172 q^{26} - 140 q^{30} + 264 q^{36} + 8 q^{39} - 360 q^{44} + 180 q^{45} + 196 q^{49} - 200 q^{54}+ \cdots - 240 q^{99}+O(q^{100}) \) 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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