LMFDB - Embedded newform 475.3.c.d.151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(151,475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.7600.1
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:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			151.1
Root			$2.37024i$ of defining polynomial
Character	$\chi$	$=$	475.151
Dual form			475.3.c.d.151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.83513i q^{2} -4.74048i q^{3} -10.7082 q^{4} -18.1803 q^{6} +25.7268i q^{8} -13.4721 q^{9} +O(q^{10})$	$q-3.83513i q^{2} -4.74048i q^{3} -10.7082 q^{4} -18.1803 q^{6} +25.7268i q^{8} -13.4721 q^{9} -13.4164 q^{11} +50.7620i q^{12} -16.4596i q^{13} +55.8328 q^{16} +51.6674i q^{18} +19.0000 q^{19} +51.4536i q^{22} +121.957 q^{24} -63.1246 q^{26} +21.2001i q^{27} -111.219i q^{32} +63.6002i q^{33} +144.262 q^{36} -9.74513i q^{37} -72.8674i q^{38} -78.0263 q^{39} +143.666 q^{44} -264.674i q^{48} -49.0000 q^{49} +176.253i q^{52} +80.0598i q^{53} +81.3050 q^{54} -90.0691i q^{57} -120.748 q^{61} -203.207 q^{64} +243.915 q^{66} -53.8551i q^{67} -346.595i q^{72} -37.3738 q^{74} -203.456 q^{76} +299.241i q^{78} -20.7508 q^{81} -345.161i q^{88} -527.230 q^{96} -58.3314i q^{97} +187.921i q^{98} +180.748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 28 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 16 * q^4 - 28 * q^6 - 36 * q^9	$ 4 q - 16 q^{4} - 28 q^{6} - 36 q^{9} + 116 q^{16} + 76 q^{19} + 300 q^{24} - 172 q^{26} + 264 q^{36} - 8 q^{39} + 360 q^{44} - 196 q^{49} + 200 q^{54} - 464 q^{64} + 600 q^{66} + 92 q^{74} - 304 q^{76} - 244 q^{81} - 812 q^{96} + 240 q^{99}+O(q^{100}) $ 4 * q - 16 * q^4 - 28 * q^6 - 36 * q^9 + 116 * q^16 + 76 * q^19 + 300 * q^24 - 172 * q^26 + 264 * q^36 - 8 * q^39 + 360 * q^44 - 196 * q^49 + 200 * q^54 - 464 * q^64 + 600 * q^66 + 92 * q^74 - 304 * q^76 - 244 * q^81 - 812 * q^96 + 240 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 3.83513i	− 1.91756i	−0.284143	−	0.958782i	$-0.591709\pi$
$2$	− 3.83513i	− 1.91756i	0.284143	−	0.958782i	$-0.408291\pi$
$3$	− 4.74048i	− 1.58016i	−0.613004	−	0.790080i	$-0.710039\pi$
$3$	− 4.74048i	− 1.58016i	0.613004	−	0.790080i	$-0.289961\pi$
$4$	−10.7082	−2.67705
$5$	0	0
$5$	0	0
$6$	−18.1803	−3.03006
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	25.7268i	3.21585i
$9$	−13.4721	−1.49690
$10$	0	0
$11$	−13.4164	−1.21967	−0.609837	−	0.792527i	$-0.708765\pi$
$11$	−13.4164	−1.21967	−0.609837	+	0.792527i	$0.708765\pi$
$12$	50.7620i	4.23017i
$13$	− 16.4596i	− 1.26612i	−0.774102	−	0.633061i	$-0.781798\pi$
$13$	− 16.4596i	− 1.26612i	0.774102	−	0.633061i	$-0.218202\pi$
$14$	0	0
$15$	0	0
$16$	55.8328	3.48955
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	51.6674i	2.87041i
$19$	19.0000	1.00000
$19$	19.0000	1.00000
$20$	0	0
$21$	0	0
$22$	51.4536i	2.33880i
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	121.957	5.08156
$25$	0	0
$26$	−63.1246	−2.42787
$27$	21.2001i	0.785188i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 111.219i	− 3.47558i
$33$	63.6002i	1.92728i
$34$	0	0
$35$	0	0
$36$	144.262	4.00729
$37$	− 9.74513i	− 0.263382i	−0.991291	−	0.131691i	$-0.957959\pi$
$37$	− 9.74513i	− 0.263382i	0.991291	−	0.131691i	$-0.0420407\pi$
$38$	− 72.8674i	− 1.91756i
$39$	−78.0263	−2.00067
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	143.666	3.26513
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	− 264.674i	− 5.51405i
$49$	−49.0000	−1.00000
$50$	0	0
$51$	0	0
$52$	176.253i	3.38947i
$53$	80.0598i	1.51056i	0.655401	+	0.755281i	$0.272500\pi$
$53$	80.0598i	1.51056i	−0.655401	+	0.755281i	$0.727500\pi$
$54$	81.3050	1.50565
$55$	0	0
$56$	0	0
$57$	− 90.0691i	− 1.58016i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−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
$65$	0	0
$66$	243.915	3.69568
$67$	− 53.8551i	− 0.803807i	−0.915682	−	0.401903i	$-0.868349\pi$
$67$	− 53.8551i	− 0.803807i	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			$\pi$
$72$	− 346.595i	− 4.81382i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−37.3738	−0.505052
$75$	0	0
$76$	−203.456	−2.67705
$77$	0	0
$78$	299.241i	3.83642i
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−20.7508	−0.256182
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	− 345.161i	− 3.92229i
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−527.230	−5.49198
$97$	− 58.3314i	− 0.601354i	−0.953726	−	0.300677i	$-0.902787\pi$
$97$	− 58.3314i	− 0.601354i	0.953726	−	0.300677i	$-0.0972126\pi$
$98$	187.921i	1.91756i
$99$	180.748	1.82573
$100$	0	0
$101$	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
$103$	189.355i	1.83840i	0.393794	+	0.919199i	$0.371162\pi$
$103$	189.355i	1.83840i	−0.393794	+	0.919199i	$0.628838\pi$
$104$	423.453	4.07166
$105$	0	0
$106$	307.039	2.89660
$107$	200.546i	1.87426i	0.348982	+	0.937129i	$0.386527\pi$
$107$	200.546i	1.87426i	−0.348982	+	0.937129i	$0.613473\pi$
$108$	− 227.015i	− 2.10199i
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−46.1966	−0.416186
$112$	0	0
$113$	− 222.274i	− 1.96703i	−0.180833	−	0.983514i	$-0.557879\pi$
$113$	− 222.274i	− 1.96703i	0.180833	−	0.983514i	$-0.442121\pi$
$114$	−345.426	−3.03006
$115$	0	0
$116$	0	0
$117$	221.746i	1.89526i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	59.0000	0.487603
$122$	463.083i	3.79576i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 215.560i	− 1.69732i	−0.528938	−	0.848660i	$-0.677410\pi$
$127$	− 215.560i	− 1.69732i	0.528938	−	0.848660i	$-0.322590\pi$
$128$	334.449i	2.61288i
$129$	0	0
$130$	0	0
$131$	−118.000	−0.900763	−0.450382	−	0.892836i	$-0.648712\pi$
$131$	−118.000	−0.900763	−0.450382	+	0.892836i	$0.648712\pi$
$132$	− 681.044i	− 5.15942i
$133$	0	0
$134$	−206.541	−1.54135
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	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
$143$	220.829i	1.54426i
$144$	−752.187	−5.22352
$145$	0	0
$146$	0	0
$147$	232.283i	1.58016i
$148$	104.353i	0.705087i
$149$	228.079	1.53073	0.765366	−	0.643596i	$-0.222558\pi$
$149$	228.079	1.53073	0.765366	+	0.643596i	$0.222558\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	488.810i	3.21585i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	835.522	5.35591
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	379.522	2.38693
$160$	0	0
$161$	0	0
$162$	79.5819i	0.491246i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	38.8411i	0.232581i	0.993215	+	0.116291i	$0.0371004\pi$
$167$	38.8411i	0.232581i	−0.993215	+	0.116291i	$0.962900\pi$
$168$	0	0
$169$	−101.918	−0.603065
$170$	0	0
$171$	−255.971	−1.49690
$172$	0	0
$173$	− 173.688i	− 1.00398i	−0.864874	−	0.501988i	$-0.832602\pi$
$173$	− 173.688i	− 1.00398i	0.864874	−	0.501988i	$-0.167398\pi$
$174$	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			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	572.402i	3.12788i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−302.000	−1.58115	−0.790576	−	0.612364i	$-0.790219\pi$
$191$	−302.000	−1.58115	−0.790576	+	0.612364i	$0.790219\pi$
$192$	963.297i	5.01717i
$193$	− 380.156i	− 1.96972i	−0.173358	−	0.984859i	$-0.555462\pi$
$193$	− 380.156i	− 1.96972i	0.173358	−	0.984859i	$-0.444538\pi$
$194$	−223.708	−1.15314
$195$	0	0
$196$	524.702	2.67705
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	− 693.190i	− 3.50096i
$199$	−322.000	−1.61809	−0.809045	−	0.587746i	$-0.800015\pi$
$199$	−322.000	−1.61809	−0.809045	+	0.587746i	$0.800015\pi$
$200$	0	0
$201$	−255.299	−1.27014
$202$	− 771.805i	− 3.82081i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	726.200	3.52524
$207$	0	0
$208$	− 918.985i	− 4.41820i
$209$	−254.912	−1.21967
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	− 857.296i	− 4.04385i
$213$	0	0
$214$	769.118	3.59401
$215$	0	0
$216$	−545.410	−2.52505
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	177.170i	0.798062i
$223$	− 222.927i	− 0.999674i	−0.866120	−	0.499837i	$-0.833393\pi$
$223$	− 222.927i	− 0.999674i	0.866120	−	0.499837i	$-0.166607\pi$
$224$	0	0
$225$	0	0
$226$	−852.450	−3.77190
$227$	103.373i	0.455389i	0.973733	+	0.227694i	$0.0731187\pi$
$227$	103.373i	0.455389i	−0.973733	+	0.227694i	$0.926881\pi$
$228$	964.478i	4.23017i
$229$	−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		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	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	$-0.565726\pi$
$239$	−98.0000	−0.410042	−0.205021	+	0.978758i	$0.565726\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	− 226.273i	− 0.935010i
$243$	289.169i	1.19000i
$244$	1292.99	5.29914
$245$	0	0
$246$	0	0
$247$	− 312.732i	− 1.26612i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−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
$257$	98.8970i	0.384813i	0.981315	+	0.192407i	$0.0616292\pi$
$257$	98.8970i	0.384813i	−0.981315	+	0.192407i	$0.938371\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	452.545i	1.72727i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−1636.23	−6.19784
$265$	0	0
$266$	0	0
$267$	0	0
$268$	576.691i	2.15183i
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	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
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	771.805i	2.77628i
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	846.906	2.96121
$287$	0	0
$288$	1498.35i	5.20262i
$289$	−289.000	−1.00000
$290$	0	0
$291$	−276.519	−0.950236
$292$	0	0
$293$	− 585.970i	− 1.99990i	−0.0101045	−	0.999949i	$-0.503216\pi$
$293$	− 585.970i	− 1.99990i	0.0101045	−	0.999949i	$-0.496784\pi$
$294$	890.837	3.03006
$295$	0	0
$296$	250.711	0.846998
$297$	− 284.429i	− 0.957672i
$298$	− 874.712i	− 2.93527i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 954.003i	− 3.14852i
$304$	1060.82	3.48955
$305$	0	0
$306$	0	0
$307$	612.175i	1.99405i	0.0770482	+	0.997027i	$0.475450\pi$
$307$	612.175i	1.99405i	−0.0770482	+	0.997027i	$0.524550\pi$
$308$	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	$0.302120\pi$
$312$	− 2007.37i	− 6.43387i
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 368.312i	− 1.16187i	−0.813951	−	0.580933i	$-0.802688\pi$
$317$	− 368.312i	− 1.16187i	0.813951	−	0.580933i	$-0.197312\pi$
$318$	− 1455.51i	− 4.57709i
$319$	0	0
$320$	0	0
$321$	950.683	2.96163
$322$	0	0
$323$	0	0
$324$	222.204	0.685813
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	131.288i	0.394258i
$334$	148.961	0.445990
$335$	0	0
$336$	0	0
$337$	− 671.299i	− 1.99198i	−0.0894409	−	0.995992i	$-0.528508\pi$
$337$	− 671.299i	− 1.99198i	0.0894409	−	0.995992i	$-0.471492\pi$
$338$	390.868i	1.15642i
$339$	−1053.69	−3.10822
$340$	0	0
$341$	0	0
$342$	981.680i	2.87041i
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−666.115	−1.92519
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	518.000	1.48424	0.742120	−	0.670267i	$-0.233820\pi$
$349$	518.000	1.48424	0.742120	+	0.670267i	$0.233820\pi$
$350$	0	0
$351$	348.944	0.994143
$352$	1492.16i	4.23908i
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	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	$0.470217\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	− 279.688i	− 0.770491i
$364$	0	0
$365$	0	0
$366$	2195.23	5.99791
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 234.397i	− 0.628410i	−0.949355	−	0.314205i	$-0.898262\pi$
$373$	− 234.397i	− 0.628410i	0.949355	−	0.314205i	$-0.101738\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−1021.86	−2.68204
$382$	1158.21i	3.03196i
$383$	129.299i	0.337596i	0.985651	+	0.168798i	$0.0539885\pi$
$383$	129.299i	0.337596i	−0.985651	+	0.168798i	$0.946012\pi$
$384$	1585.45	4.12877
$385$	0	0
$386$	−1457.95	−3.77706
$387$	0	0
$388$	624.624i	1.60986i
$389$	742.000	1.90746	0.953728	−	0.300672i	$-0.0972110\pi$
$389$	742.000	1.90746	0.953728	+	0.300672i	$0.0972110\pi$
$390$	0	0
$391$	0	0
$392$	− 1260.61i	− 3.21585i
$393$	559.376i	1.42335i
$394$	0	0
$395$	0	0
$396$	−1935.48	−4.88758
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	1234.91i	3.10279i
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	979.103i	2.43558i
$403$	0	0
$404$	−2154.98	−5.33412
$405$	0	0
$406$	0	0
$407$	130.745i	0.321240i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	− 2027.65i	− 4.92148i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1830.61	−4.40051
$417$	954.003i	2.28778i
$418$	977.619i	2.33880i
$419$	−458.000	−1.09308	−0.546539	−	0.837433i	$-0.684055\pi$
$419$	−458.000	−1.09308	−0.546539	+	0.837433i	$0.684055\pi$
$420$	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
$428$	− 2147.48i	− 5.01749i
$429$	1046.83	2.44017
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1183.66i	2.73995i
$433$	− 536.731i	− 1.23956i	−0.784774	−	0.619781i	$-0.787221\pi$
$433$	− 536.731i	− 1.23956i	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			$\pi$
$440$	0	0
$441$	660.135	1.49690
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	494.683	1.11415
$445$	0	0
$446$	−854.954	−1.91694
$447$	− 1081.20i	− 2.41880i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	2380.16i	5.26583i
$453$	0	0
$454$	396.450	0.873237
$455$	0	0
$456$	2317.19	5.08156
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	1595.06i	3.48267i
$459$	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		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	− 2374.50i	− 5.07372i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1078.58i	− 2.26117i
$478$	375.843i	0.786281i
$479$	−898.899	−1.87662	−0.938308	−	0.345800i	$-0.887608\pi$
$479$	−898.899	−1.87662	−0.938308	+	0.345800i	$0.887608\pi$
$480$	0	0
$481$	−160.401	−0.333474
$482$	0	0
$483$	0	0
$484$	−631.784	−1.30534
$485$	0	0
$486$	1109.00	2.28189
$487$	− 65.3247i	− 0.134137i	−0.997748	−	0.0670685i	$-0.978635\pi$
$487$	− 65.3247i	− 0.134137i	0.997748	−	0.0670685i	$-0.0213646\pi$
$488$	− 3106.45i	− 6.36568i
$489$	0	0
$490$	0	0
$491$	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
$495$	0	0
$496$	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	$0.0615544\pi$
$500$	0	0
$501$	184.125	0.367516
$502$	697.993i	1.39042i
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	483.140i	0.952939i
$508$	2308.26i	4.54381i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	− 464.050i	− 0.906349i
$513$	402.801i	0.785188i
$514$	379.282	0.737904
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−823.364	−1.58644
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	551.745i	1.05496i	0.849567	+	0.527481i	$0.176863\pi$
$523$	551.745i	1.05496i	−0.849567	+	0.527481i	$0.823137\pi$
$524$	1263.57	2.41139
$525$	0	0
$526$	0	0
$527$	0	0
$528$	3550.98i	6.72534i
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	1385.52	2.58492
$537$	0	0
$538$	0	0
$539$	657.404	1.21967
$540$	0	0
$541$	−335.410	−0.619982	−0.309991	−	0.950740i	$-0.600326\pi$
$541$	−335.410	−0.619982	−0.309991	+	0.950740i	$0.600326\pi$
$542$	− 154.361i	− 0.284799i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1087.40i	− 1.98794i	−0.109645	−	0.993971i	$-0.534971\pi$
$547$	− 1087.40i	− 1.98794i	0.109645	−	0.993971i	$-0.465029\pi$
$548$	0	0
$549$	1626.73	2.96308
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	2154.98	3.87587
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	297.344i	0.528142i	0.964503	+	0.264071i	$0.0850653\pi$
$563$	297.344i	0.528142i	−0.964503	+	0.264071i	$0.914935\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	415.909	0.728386	0.364193	−	0.931323i	$-0.381345\pi$
$571$	415.909	0.728386	0.364193	+	0.931323i	$0.381345\pi$
$572$	− 2364.68i	− 4.13405i
$573$	1431.62i	2.49847i
$574$	0	0
$575$	0	0
$576$	2737.63	4.75283
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	1108.35i	1.91756i
$579$	−1802.12	−3.11247
$580$	0	0
$581$	0	0
$582$	1060.48i	1.82214i
$583$	− 1074.11i	− 1.84239i
$584$	0	0
$585$	0	0
$586$	−2247.27	−3.83493
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	− 2487.34i	− 4.23017i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 544.098i	− 0.919085i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	−1090.82	−1.83640
$595$	0	0
$596$	−2442.32	−4.09785
$597$	1526.43i	2.55684i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	725.543i	1.20322i
$604$	0	0
$605$	0	0
$606$	−3658.72	−6.03750
$607$	− 1022.87i	− 1.68513i	−0.538597	−	0.842563i	$-0.681046\pi$
$607$	− 1022.87i	− 1.68513i	0.538597	−	0.842563i	$-0.318954\pi$
$608$	− 2113.16i	− 3.47558i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	2347.77	3.82373
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	− 3442.54i	− 5.57045i
$619$	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
$622$	− 1389.25i	− 2.23352i
$623$	0	0
$624$	−4356.43	−6.98146
$625$	0	0
$626$	0	0
$627$	1208.40i	1.92728i
$628$	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
$635$	0	0
$636$	−4064.00	−6.38993
$637$	806.520i	1.26612i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	− 3645.99i	− 5.67911i
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	− 533.851i	− 0.823845i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	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
$668$	− 415.918i	− 0.622632i
$669$	−1056.78	−1.57964
$670$	0	0
$671$	1620.00	2.41431
$672$	0	0
$673$	− 1202.76i	− 1.78716i	−0.448901	−	0.893581i	$-0.648184\pi$
$673$	− 1202.76i	− 1.78716i	0.448901	−	0.893581i	$-0.351816\pi$
$674$	−2574.52	−3.81976
$675$	0	0
$676$	1091.36	1.61444
$677$	619.168i	0.914576i	0.889319	+	0.457288i	$0.151179\pi$
$677$	619.168i	0.914576i	−0.889319	+	0.457288i	$0.848821\pi$
$678$	4041.02i	5.96021i
$679$	0	0
$680$	0	0
$681$	490.039	0.719587
$682$	0	0
$683$	1254.89i	1.83732i	0.395046	+	0.918661i	$0.370729\pi$
$683$	1254.89i	1.83732i	−0.395046	+	0.918661i	$0.629271\pi$
$684$	2740.99	4.00729
$685$	0	0
$686$	0	0
$687$	1971.61i	2.86988i
$688$	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	$0.00401597\pi$
$692$	1859.89i	2.68770i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 1986.60i	− 2.84613i
$699$	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	$0.262226\pi$
$702$	− 1338.25i	− 1.90633i
$703$	− 185.158i	− 0.263382i
$704$	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	$-0.545499\pi$
$709$	−202.000	−0.284908	−0.142454	+	0.989801i	$0.545499\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	464.567i	0.647932i
$718$	− 257.268i	− 0.358312i
$719$	1247.73	1.73536	0.867681	−	0.497121i	$-0.165609\pi$
$719$	1247.73	1.73536	0.867681	+	0.497121i	$0.165609\pi$
$720$	0	0
$721$	0	0
$722$	− 1384.48i	− 1.91756i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−1072.64	−1.47747
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	1184.04	1.62420
$730$	0	0
$731$	0	0
$732$	− 6129.39i	− 8.37349i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	722.541i	0.980382i
$738$	0	0
$739$	−1402.00	−1.89716	−0.948579	−	0.316540i	$-0.897479\pi$
$739$	−1402.00	−1.89716	−0.948579	+	0.316540i	$0.897479\pi$
$740$	0	0
$741$	−1482.50	−2.00067
$742$	0	0
$743$	− 88.6382i	− 0.119298i	−0.998219	−	0.0596489i	$-0.981002\pi$
$743$	− 88.6382i	− 0.119298i	0.998219	−	0.0596489i	$-0.0189981\pi$
$744$	0	0
$745$	0	0
$746$	−898.942	−1.20502
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	862.767i	1.14577i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1435.56	1.88641	0.943203	−	0.332216i	$-0.107796\pi$
$761$	1435.56	1.88641	0.943203	+	0.332216i	$0.107796\pi$
$762$	3918.95i	5.14298i
$763$	0	0
$764$	3233.88	4.23282
$765$	0	0
$766$	495.878	0.647361
$767$	0	0
$768$	− 2227.20i	− 2.90000i
$769$	−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
$771$	468.819	0.608066
$772$	4070.78i	5.27304i
$773$	1375.66i	1.77963i	0.456319	+	0.889816i	$0.349168\pi$
$773$	1375.66i	1.77963i	−0.456319	+	0.889816i	$0.650832\pi$
$774$	0	0
$775$	0	0
$776$	1500.68	1.93387
$777$	0	0
$778$	− 2845.66i	− 3.65767i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−2735.81	−3.48955
$785$	0	0
$786$	2145.28	2.72936
$787$	− 1281.75i	− 1.62865i	−0.580408	−	0.814326i	$-0.697107\pi$
$787$	− 1281.75i	− 1.62865i	0.580408	−	0.814326i	$-0.302893\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	4650.06i	5.87129i
$793$	1987.46i	2.50625i
$794$	0	0
$795$	0	0
$796$	3448.04	4.33171
$797$	− 1401.86i	− 1.75892i	−0.475971	−	0.879461i	$-0.657903\pi$
$797$	− 1401.86i	− 1.75892i	0.475971	−	0.879461i	$-0.342097\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	2733.79	3.40024
$805$	0	0
$806$	0	0
$807$	0	0
$808$	5177.42i	6.40770i
$809$	1118.00	1.38195	0.690977	−	0.722877i	$-0.257181\pi$
$809$	1118.00	1.38195	0.690977	+	0.722877i	$0.257181\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	− 190.801i	− 0.234687i
$814$	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		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	−4871.50	−5.91202
$825$	0	0
$826$	0	0
$827$	693.401i	0.838454i	0.907881	+	0.419227i	$0.137699\pi$
$827$	693.401i	0.838454i	−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
$832$	3344.70i	4.02007i
$833$	0	0
$834$	3658.72	4.38696
$835$	0	0
$836$	2729.65	3.26513
$837$	0	0
$838$	1756.49i	2.09605i
$839$	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
$845$	0	0
$846$	0	0
$847$	0	0
$848$	4469.96i	5.27118i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−5159.40	−6.02734
$857$	− 590.072i	− 0.688532i	−0.938872	−	0.344266i	$-0.888128\pi$
$857$	− 590.072i	− 0.688532i	0.938872	−	0.344266i	$-0.111872\pi$
$858$	− 4014.74i	− 4.67918i
$859$	1702.00	1.98137	0.990687	−	0.136160i	$-0.0434762\pi$
$859$	1702.00	1.98137	0.990687	+	0.136160i	$0.0434762\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1712.22i	− 1.98403i	−0.126132	−	0.992013i	$-0.540256\pi$
$863$	− 1712.22i	− 1.98403i	0.126132	−	0.992013i	$-0.459744\pi$
$864$	2357.84	2.72899
$865$	0	0
$866$	−2058.43	−2.37694
$867$	1370.00i	1.58016i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−886.432	−1.01772
$872$	0	0
$873$	785.848i	0.900170i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1753.43i	− 1.99935i	−0.0254035	−	0.999677i	$-0.508087\pi$
$877$	− 1753.43i	− 1.99935i	0.0254035	−	0.999677i	$-0.491913\pi$
$878$	0	0
$879$	−2777.78	−3.16016
$880$	0	0
$881$	−1677.05	−1.90358	−0.951788	−	0.306756i	$-0.900757\pi$
$881$	−1677.05	−1.90358	−0.951788	+	0.306756i	$0.900757\pi$
$882$	− 2531.70i	− 2.87041i
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 962.816i	− 1.08547i	−0.839902	−	0.542737i	$-0.817388\pi$
$887$	− 962.816i	− 1.08547i	0.839902	−	0.542737i	$-0.182612\pi$
$888$	− 1188.49i	− 1.33839i
$889$	0	0
$890$	0	0
$891$	278.401	0.312459
$892$	2387.15i	2.67618i
$893$	0	0
$894$	−4146.55	−4.63820
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	5718.41	6.32567
$905$	0	0
$906$	0	0
$907$	986.687i	1.08786i	0.839131	+	0.543929i	$0.183064\pi$
$907$	986.687i	1.08786i	−0.839131	+	0.543929i	$0.816936\pi$
$908$	− 1106.94i	− 1.21910i
$909$	−2711.22	−2.98263
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	− 5028.81i	− 5.51405i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	4453.63	4.86205
$917$	0	0
$918$	0	0
$919$	1582.00	1.72144	0.860718	−	0.509082i	$-0.170015\pi$
$919$	1582.00	1.72144	0.860718	+	0.509082i	$0.170015\pi$
$920$	0	0
$921$	2902.00	3.15092
$922$	2293.41i	2.48743i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2551.02i	− 2.75190i
$928$	0	0
$929$	878.000	0.945102	0.472551	−	0.881303i	$-0.343333\pi$
$929$	878.000	0.945102	0.472551	+	0.881303i	$0.343333\pi$
$930$	0	0
$931$	−931.000	−1.00000
$932$	0	0
$933$	− 1717.21i	− 1.84052i
$934$	0	0
$935$	0	0
$936$	−5704.81	−6.09489
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	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		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−1745.97	−1.83593
$952$	0	0
$953$	1872.33i	1.96467i	0.187119	+	0.982337i	$0.440085\pi$
$953$	1872.33i	1.96467i	−0.187119	+	0.982337i	$0.559915\pi$
$954$	−4136.48	−4.33593
$955$	0	0
$956$	1049.40	1.09770
$957$	0	0
$958$	3447.39i	3.59853i
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	615.158i	0.639457i
$963$	− 2701.78i	− 2.80559i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	1517.88i	1.56806i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 3096.48i	− 3.18568i
$973$	0	0
$974$	−250.529	−0.257216
$975$	0	0
$976$	−6741.68	−6.90746
$977$	1678.36i	1.71788i	0.512080	+	0.858938i	$0.328875\pi$
$977$	1678.36i	1.71788i	−0.512080	+	0.858938i	$0.671125\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 1142.87i	− 1.16382i
$983$	− 1965.96i	− 1.99996i	−0.00613954	−	0.999981i	$-0.501954\pi$
$983$	− 1965.96i	− 1.99996i	0.00613954	−	0.999981i	$-0.498046\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	3348.80i	3.38947i
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 3756.12i	− 3.76364i
$999$	206.597	0.206804

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.d.c.94.1	✓	4	5.3	odd	4
95.3.d.c.94.1	✓	4	95.37	even	4
95.3.d.c.94.4	yes	4	5.2	odd	4
95.3.d.c.94.4	yes	4	95.18	even	4
475.3.c.d.151.1		4	1.1	even	1	trivial
475.3.c.d.151.1		4	95.94	odd	2	CM
475.3.c.d.151.4		4	5.4	even	2	inner
475.3.c.d.151.4		4	19.18	odd	2	inner
855.3.g.d.379.1		4	15.2	even	4
855.3.g.d.379.1		4	285.113	odd	4
855.3.g.d.379.4		4	15.8	even	4
855.3.g.d.379.4		4	285.227	odd	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.83513i q^{2} -4.74048i q^{3} -10.7082 q^{4} -18.1803 q^{6} +25.7268i q^{8} -13.4721 q^{9} +O(q^{10})\)	\(q-3.83513i q^{2} -4.74048i q^{3} -10.7082 q^{4} -18.1803 q^{6} +25.7268i q^{8} -13.4721 q^{9} -13.4164 q^{11} +50.7620i q^{12} -16.4596i q^{13} +55.8328 q^{16} +51.6674i q^{18} +19.0000 q^{19} +51.4536i q^{22} +121.957 q^{24} -63.1246 q^{26} +21.2001i q^{27} -111.219i q^{32} +63.6002i q^{33} +144.262 q^{36} -9.74513i q^{37} -72.8674i q^{38} -78.0263 q^{39} +143.666 q^{44} -264.674i q^{48} -49.0000 q^{49} +176.253i q^{52} +80.0598i q^{53} +81.3050 q^{54} -90.0691i q^{57} -120.748 q^{61} -203.207 q^{64} +243.915 q^{66} -53.8551i q^{67} -346.595i q^{72} -37.3738 q^{74} -203.456 q^{76} +299.241i q^{78} -20.7508 q^{81} -345.161i q^{88} -527.230 q^{96} -58.3314i q^{97} +187.921i q^{98} +180.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 28 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 16 * q^4 - 28 * q^6 - 36 * q^9	\( 4 q - 16 q^{4} - 28 q^{6} - 36 q^{9} + 116 q^{16} + 76 q^{19} + 300 q^{24} - 172 q^{26} + 264 q^{36} - 8 q^{39} + 360 q^{44} - 196 q^{49} + 200 q^{54} - 464 q^{64} + 600 q^{66} + 92 q^{74} - 304 q^{76} - 244 q^{81} - 812 q^{96} + 240 q^{99}+O(q^{100}) \) 4 * q - 16 * q^4 - 28 * q^6 - 36 * q^9 + 116 * q^16 + 76 * q^19 + 300 * q^24 - 172 * q^26 + 264 * q^36 - 8 * q^39 + 360 * q^44 - 196 * q^49 + 200 * q^54 - 464 * q^64 + 600 * q^66 + 92 * q^74 - 304 * q^76 - 244 * q^81 - 812 * q^96 + 240 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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