LMFDB - Embedded newform 475.3.c.e.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:	4.0.462080.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 16x^{2} + 45 $ x^4 + 16*x^2 + 45
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			151.1
Root			$-3.51552i$ of defining polynomial
Character	$\chi$	$=$	475.151
Dual form			475.3.c.e.151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.51552i q^{2} +0.751268i q^{3} -8.35890 q^{4} +2.64110 q^{6} +15.3238i q^{8} +8.43560 q^{9} +O(q^{10})$	$q-3.51552i q^{2} +0.751268i q^{3} -8.35890 q^{4} +2.64110 q^{6} +15.3238i q^{8} +8.43560 q^{9} +17.4356 q^{11} -6.27978i q^{12} -25.8704i q^{13} +20.4356 q^{16} -29.6555i q^{18} -19.0000 q^{19} -61.2952i q^{22} -11.5123 q^{24} -90.9479 q^{26} +13.0988i q^{27} -10.5466i q^{32} +13.0988i q^{33} -70.5123 q^{36} -44.9793i q^{37} +66.7949i q^{38} +19.4356 q^{39} -145.742 q^{44} +15.3526i q^{48} -49.0000 q^{49} +216.248i q^{52} -105.735i q^{53} +46.0492 q^{54} -14.2741i q^{57} +17.4356 q^{61} +44.6657 q^{64} +46.0492 q^{66} -124.844i q^{67} +129.265i q^{72} -158.126 q^{74} +158.819 q^{76} -68.3263i q^{78} +66.0796 q^{81} +267.180i q^{88} +7.92330 q^{96} +172.077i q^{97} +172.261i q^{98} +147.080 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{16} - 76 q^{19} + 76 q^{24} - 172 q^{26} - 160 q^{36} + 8 q^{39} - 304 q^{44} - 196 q^{49} - 304 q^{54} - 48 q^{64} - 304 q^{66} - 92 q^{74} + 304 q^{76} + 892 q^{81} + 84 q^{96} + 1216 q^{99}+O(q^{100}) $ 4 * q - 16 * q^4 + 28 * q^6 - 36 * q^9 + 12 * q^16 - 76 * q^19 + 76 * q^24 - 172 * q^26 - 160 * q^36 + 8 * q^39 - 304 * q^44 - 196 * q^49 - 304 * q^54 - 48 * q^64 - 304 * q^66 - 92 * q^74 + 304 * q^76 + 892 * q^81 + 84 * q^96 + 1216 * 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 $

$2$	− 3.51552i	− 1.75776i	−0.477042	−	0.878881i	$-0.658291\pi$
$2$	− 3.51552i	− 1.75776i	0.477042	−	0.878881i	$-0.341709\pi$
$3$	0.751268i	0.250423i	0.992130	+	0.125211i	$0.0399609\pi$
$3$	0.751268i	0.250423i	−0.992130	+	0.125211i	$0.960039\pi$
$4$	−8.35890	−2.08972
$5$	0	0
$5$	0	0
$6$	2.64110	0.440184
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	15.3238i	1.91548i
$9$	8.43560	0.937288
$10$	0	0
$11$	17.4356	1.58505	0.792527	−	0.609837i	$-0.208765\pi$
$11$	17.4356	1.58505	0.792527	+	0.609837i	$0.208765\pi$
$12$	− 6.27978i	− 0.523315i
$13$	− 25.8704i	− 1.99003i	−0.0997310	−	0.995014i	$-0.531798\pi$
$13$	− 25.8704i	− 1.99003i	0.0997310	−	0.995014i	$-0.468202\pi$
$14$	0	0
$15$	0	0
$16$	20.4356	1.27722
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 29.6555i	− 1.64753i
$19$	−19.0000	−1.00000
$19$	−19.0000	−1.00000
$20$	0	0
$21$	0	0
$22$	− 61.2952i	− 2.78615i
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−11.5123	−0.479679
$25$	0	0
$26$	−90.9479	−3.49800
$27$	13.0988i	0.485141i
$28$	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			$\pi$
$32$	− 10.5466i	− 0.329580i
$33$	13.0988i	0.396934i
$34$	0	0
$35$	0	0
$36$	−70.5123	−1.95867
$37$	− 44.9793i	− 1.21566i	−0.794068	−	0.607829i	$-0.792041\pi$
$37$	− 44.9793i	− 1.21566i	0.794068	−	0.607829i	$-0.207959\pi$
$38$	66.7949i	1.75776i
$39$	19.4356	0.498349
$40$	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	$0.5\pi$
$44$	−145.742	−3.31233
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	15.3526i	0.319846i
$49$	−49.0000	−1.00000
$50$	0	0
$51$	0	0
$52$	216.248i	4.15861i
$53$	− 105.735i	− 1.99501i	−0.0706256	−	0.997503i	$-0.522500\pi$
$53$	− 105.735i	− 1.99501i	0.0706256	−	0.997503i	$-0.477500\pi$
$54$	46.0492	0.852762
$55$	0	0
$56$	0	0
$57$	− 14.2741i	− 0.250423i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	17.4356	0.285829	0.142915	−	0.989735i	$-0.454353\pi$
$61$	17.4356	0.285829	0.142915	+	0.989735i	$0.454353\pi$
$62$	0	0
$63$	0	0
$64$	44.6657	0.697901
$65$	0	0
$66$	46.0492	0.697715
$67$	− 124.844i	− 1.86335i	−0.363296	−	0.931674i	$-0.618349\pi$
$67$	− 124.844i	− 1.86335i	0.363296	−	0.931674i	$-0.381651\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	129.265i	1.79535i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−158.126	−2.13684
$75$	0	0
$76$	158.819	2.08972
$77$	0	0
$78$	− 68.3263i	− 0.875978i
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	66.0796	0.815798
$82$	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
$88$	267.180i	3.03613i
$89$	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
$96$	7.92330	0.0825344
$97$	172.077i	1.77399i	0.461775	+	0.886997i	$0.347213\pi$
$97$	172.077i	1.77399i	−0.461775	+	0.886997i	$0.652787\pi$
$98$	172.261i	1.75776i
$99$	147.080	1.48565
$100$	0	0
$101$	17.4356	0.172630	0.0863148	−	0.996268i	$-0.472491\pi$
$101$	17.4356	0.172630	0.0863148	+	0.996268i	$0.472491\pi$
$102$	0	0
$103$	76.5326i	0.743035i	0.928426	+	0.371518i	$0.121162\pi$
$103$	76.5326i	0.743035i	−0.928426	+	0.371518i	$0.878838\pi$
$104$	396.433	3.81185
$105$	0	0
$106$	−371.715	−3.50674
$107$	194.615i	1.81884i	0.415883	+	0.909418i	$0.363473\pi$
$107$	194.615i	1.81884i	−0.415883	+	0.909418i	$0.636527\pi$
$108$	− 109.492i	− 1.01381i
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	33.7916	0.304428
$112$	0	0
$113$	− 128.273i	− 1.13516i	−0.823317	−	0.567581i	$-0.807879\pi$
$113$	− 128.273i	− 1.13516i	0.823317	−	0.567581i	$-0.192121\pi$
$114$	−50.1809	−0.440184
$115$	0	0
$116$	0	0
$117$	− 218.232i	− 1.86523i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	183.000	1.51240
$122$	− 61.2952i	− 0.502420i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	57.4236i	0.452155i	0.974109	+	0.226077i	$0.0725902\pi$
$127$	57.4236i	0.452155i	−0.974109	+	0.226077i	$0.927410\pi$
$128$	− 199.209i	− 1.55632i
$129$	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	$0.648712\pi$
$132$	− 109.492i	− 0.829482i
$133$	0	0
$134$	−438.893	−3.27532
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	191.792	1.37980	0.689898	−	0.723907i	$-0.257656\pi$
$139$	191.792	1.37980	0.689898	+	0.723907i	$0.257656\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 451.065i	− 3.15430i
$144$	172.386	1.19713
$145$	0	0
$146$	0	0
$147$	− 36.8122i	− 0.250423i
$148$	375.978i	2.54039i
$149$	191.792	1.28719	0.643596	−	0.765366i	$-0.277442\pi$
$149$	191.792	1.28719	0.643596	+	0.765366i	$0.277442\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	− 291.152i	− 1.91548i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−162.460	−1.04141
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	79.4356	0.499595
$160$	0	0
$161$	0	0
$162$	− 232.304i	− 1.43398i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 262.036i	− 1.56908i	−0.620079	−	0.784539i	$-0.712900\pi$
$167$	− 262.036i	− 1.56908i	0.620079	−	0.784539i	$-0.287100\pi$
$168$	0	0
$169$	−500.276	−2.96021
$170$	0	0
$171$	−160.276	−0.937288
$172$	0	0
$173$	88.7834i	0.513199i	0.966518	+	0.256599i	$0.0826021\pi$
$173$	88.7834i	0.513199i	−0.966518	+	0.256599i	$0.917398\pi$
$174$	0	0
$175$	0	0
$176$	356.307	2.02447
$177$	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
$183$	13.0988i	0.0715782i
$184$	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	$-0.209781\pi$
$191$	302.000	1.58115	0.790576	+	0.612364i	$0.209781\pi$
$192$	33.5559i	0.174770i
$193$	316.127i	1.63797i	0.573818	+	0.818983i	$0.305462\pi$
$193$	316.127i	1.63797i	−0.573818	+	0.818983i	$0.694538\pi$
$194$	604.942	3.11826
$195$	0	0
$196$	409.586	2.08972
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	− 517.062i	− 2.61142i
$199$	322.000	1.61809	0.809045	−	0.587746i	$-0.199985\pi$
$199$	322.000	1.61809	0.809045	+	0.587746i	$0.199985\pi$
$200$	0	0
$201$	93.7916	0.466625
$202$	− 61.2952i	− 0.303442i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	269.052	1.30608
$207$	0	0
$208$	− 528.677i	− 2.54171i
$209$	−331.276	−1.58505
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	883.831i	4.16901i
$213$	0	0
$214$	684.175	3.19708
$215$	0	0
$216$	−200.724	−0.929276
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	− 118.795i	− 0.535112i
$223$	430.781i	1.93175i	0.259001	+	0.965877i	$0.416607\pi$
$223$	430.781i	1.93175i	−0.259001	+	0.965877i	$0.583393\pi$
$224$	0	0
$225$	0	0
$226$	−450.948	−1.99534
$227$	− 239.498i	− 1.05506i	−0.849537	−	0.527529i	$-0.823119\pi$
$227$	− 239.498i	− 1.05506i	0.849537	−	0.527529i	$-0.176881\pi$
$228$	119.316i	0.523315i
$229$	191.792	0.837518	0.418759	−	0.908097i	$-0.362465\pi$
$229$	191.792	0.837518	0.418759	+	0.908097i	$0.362465\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−767.200	−3.27863
$235$	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			$\pi$
$242$	− 643.341i	− 2.65843i
$243$	167.533i	0.689436i
$244$	−145.742	−0.597305
$245$	0	0
$246$	0	0
$247$	491.537i	1.99003i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	182.000	0.725100	0.362550	−	0.931964i	$-0.381906\pi$
$251$	182.000	0.725100	0.362550	+	0.931964i	$0.381906\pi$
$252$	0	0
$253$	0	0
$254$	201.874	0.794780
$255$	0	0
$256$	−521.663	−2.03775
$257$	286.731i	1.11569i	0.829947	+	0.557843i	$0.188371\pi$
$257$	286.731i	1.11569i	−0.829947	+	0.557843i	$0.811629\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	414.832i	1.58333i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−200.724	−0.760317
$265$	0	0
$266$	0	0
$267$	0	0
$268$	1043.56i	3.89388i
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	540.503	1.99448	0.997239	−	0.0742606i	$-0.0236597\pi$
$271$	540.503	1.99448	0.997239	+	0.0742606i	$0.0236597\pi$
$272$	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	$0.5\pi$
$278$	− 674.248i	− 2.42535i
$279$	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
$286$	−1585.73	−5.54451
$287$	0	0
$288$	− 88.9666i	− 0.308912i
$289$	−289.000	−1.00000
$290$	0	0
$291$	−129.276	−0.444249
$292$	0	0
$293$	418.530i	1.42843i	0.699926	+	0.714216i	$0.253216\pi$
$293$	418.530i	1.42843i	−0.699926	+	0.714216i	$0.746784\pi$
$294$	−129.414	−0.440184
$295$	0	0
$296$	689.255	2.32856
$297$	228.386i	0.768975i
$298$	− 674.248i	− 2.26258i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.0988i	0.0432304i
$304$	−388.276	−1.27722
$305$	0	0
$306$	0	0
$307$	399.421i	1.30105i	0.759486	+	0.650523i	$0.225450\pi$
$307$	399.421i	1.30105i	−0.759486	+	0.650523i	$0.774550\pi$
$308$	0	0
$309$	−57.4965	−0.186073
$310$	0	0
$311$	−505.632	−1.62583	−0.812914	−	0.582384i	$-0.802120\pi$
$311$	−505.632	−1.62583	−0.812914	+	0.582384i	$0.802120\pi$
$312$	297.827i	0.954575i
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	104.463i	0.329537i	0.986332	+	0.164769i	$0.0526877\pi$
$317$	104.463i	0.329537i	−0.986332	+	0.164769i	$0.947312\pi$
$318$	− 279.258i	− 0.878169i
$319$	0	0
$320$	0	0
$321$	−146.208	−0.455478
$322$	0	0
$323$	0	0
$324$	−552.353	−1.70479
$325$	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
$333$	− 379.427i	− 1.13942i
$334$	−921.194	−2.75807
$335$	0	0
$336$	0	0
$337$	− 432.053i	− 1.28206i	−0.767517	−	0.641029i	$-0.778508\pi$
$337$	− 432.053i	− 1.28206i	0.767517	−	0.641029i	$-0.221492\pi$
$338$	1758.73i	5.20335i
$339$	96.3677	0.284271
$340$	0	0
$341$	0	0
$342$	563.455i	1.64753i
$343$	0	0
$344$	0	0
$345$	0	0
$346$	312.120	0.902081
$347$	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	$-0.766180\pi$
$349$	−518.000	−1.48424	−0.742120	+	0.670267i	$0.766180\pi$
$350$	0	0
$351$	338.871	0.965445
$352$	− 183.886i	− 0.522403i
$353$	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
$359$	714.859	1.99125	0.995626	−	0.0934290i	$-0.0297828\pi$
$359$	714.859	1.99125	0.995626	+	0.0934290i	$0.0297828\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	137.482i	0.378739i
$364$	0	0
$365$	0	0
$366$	46.0492	0.125817
$367$	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
$373$	− 335.043i	− 0.898238i	−0.893472	−	0.449119i	$-0.851738\pi$
$373$	− 335.043i	− 0.898238i	0.893472	−	0.449119i	$-0.148262\pi$
$374$	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
$381$	−43.1406	−0.113230
$382$	− 1061.69i	− 2.77929i
$383$	625.300i	1.63264i	0.577602	+	0.816318i	$0.303988\pi$
$383$	625.300i	1.63264i	−0.577602	+	0.816318i	$0.696012\pi$
$384$	149.660	0.389739
$385$	0	0
$386$	1111.35	2.87915
$387$	0	0
$388$	− 1438.38i	− 3.70716i
$389$	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
$392$	− 750.867i	− 1.91548i
$393$	− 88.6497i	− 0.225572i
$394$	0	0
$395$	0	0
$396$	−1229.42	−3.10461
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	− 1132.00i	− 2.84422i
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	− 329.726i	− 0.820215i
$403$	0	0
$404$	−145.742	−0.360748
$405$	0	0
$406$	0	0
$407$	− 784.242i	− 1.92688i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	− 639.728i	− 1.55274i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−272.844	−0.655874
$417$	144.087i	0.345532i
$418$	1164.61i	2.78615i
$419$	−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
$424$	1620.27	3.82139
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 1626.77i	− 3.80087i
$429$	338.871	0.789910
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	267.682i	0.619634i
$433$	101.034i	0.233335i	0.993171	+	0.116668i	$0.0372212\pi$
$433$	101.034i	0.233335i	−0.993171	+	0.116668i	$0.962779\pi$
$434$	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
$441$	−413.344	−0.937288
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−282.460	−0.636172
$445$	0	0
$446$	1514.42	3.39556
$447$	144.087i	0.322342i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	1072.22i	2.37218i
$453$	0	0
$454$	−841.961	−1.85454
$455$	0	0
$456$	218.734	0.479679
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	− 674.248i	− 1.47216i
$459$	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	$0.5\pi$
$468$	1824.18i	3.89782i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 891.940i	− 1.86990i
$478$	344.521i	0.720756i
$479$	−331.276	−0.691600	−0.345800	−	0.938308i	$-0.612392\pi$
$479$	−331.276	−0.691600	−0.345800	+	0.938308i	$0.612392\pi$
$480$	0	0
$481$	−1163.63	−2.41919
$482$	0	0
$483$	0	0
$484$	−1529.68	−3.16049
$485$	0	0
$486$	588.966	1.21186
$487$	640.980i	1.31618i	0.752939	+	0.658090i	$0.228635\pi$
$487$	640.980i	1.31618i	−0.752939	+	0.658090i	$0.771365\pi$
$488$	267.180i	0.547499i
$489$	0	0
$490$	0	0
$491$	−298.000	−0.606925	−0.303462	−	0.952843i	$-0.598143\pi$
$491$	−298.000	−0.606925	−0.303462	+	0.952843i	$0.598143\pi$
$492$	0	0
$493$	0	0
$494$	1728.01	3.49800
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	191.792	0.384352	0.192176	−	0.981360i	$-0.438446\pi$
$499$	191.792	0.384352	0.192176	+	0.981360i	$0.438446\pi$
$500$	0	0
$501$	196.859	0.392933
$502$	− 639.825i	− 1.27455i
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 375.842i	− 0.741305i
$508$	− 479.998i	− 0.944879i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1037.08i	2.02555i
$513$	− 248.877i	− 0.485141i
$514$	1008.01	1.96111
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−66.7002	−0.128517
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	238.226i	0.455499i	0.973720	+	0.227749i	$0.0731367\pi$
$523$	238.226i	0.455499i	−0.973720	+	0.227749i	$0.926863\pi$
$524$	986.350	1.88235
$525$	0	0
$526$	0	0
$527$	0	0
$528$	267.682i	0.506974i
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	1913.09	3.56920
$537$	0	0
$538$	0	0
$539$	−854.344	−1.58505
$540$	0	0
$541$	−1028.70	−1.90148	−0.950740	−	0.309991i	$-0.899674\pi$
$541$	−1028.70	−1.90148	−0.950740	+	0.309991i	$0.899674\pi$
$542$	− 1900.15i	− 3.50582i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	684.092i	1.25063i	0.780374	+	0.625313i	$0.215029\pi$
$547$	684.092i	1.25063i	−0.780374	+	0.625313i	$0.784971\pi$
$548$	0	0
$549$	147.080	0.267905
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−1603.17	−2.88339
$557$	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
$563$	557.686i	0.990561i	0.868733	+	0.495280i	$0.164935\pi$
$563$	557.686i	0.990561i	−0.868733	+	0.495280i	$0.835065\pi$
$564$	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
$571$	1063.57	1.86265	0.931323	−	0.364193i	$-0.118655\pi$
$571$	1063.57	1.86265	0.931323	+	0.364193i	$0.118655\pi$
$572$	3770.41i	6.59163i
$573$	226.883i	0.395956i
$574$	0	0
$575$	0	0
$576$	376.782	0.654135
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	1015.99i	1.75776i
$579$	−237.497	−0.410184
$580$	0	0
$581$	0	0
$582$	454.474i	0.780883i
$583$	− 1843.56i	− 3.16219i
$584$	0	0
$585$	0	0
$586$	1471.35	2.51084
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	307.709i	0.523315i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 919.180i	− 1.55267i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	802.895	1.35167
$595$	0	0
$596$	−1603.17	−2.68988
$597$	241.908i	0.405207i
$598$	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
$603$	− 1053.14i	− 1.74649i
$604$	0	0
$605$	0	0
$606$	46.0492	0.0759887
$607$	− 1185.63i	− 1.95326i	−0.214936	−	0.976628i	$-0.568954\pi$
$607$	− 1185.63i	− 1.95326i	0.214936	−	0.976628i	$-0.431046\pi$
$608$	200.385i	0.329580i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	1404.18	2.28693
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	202.130i	0.327072i
$619$	1237.93	1.99988	0.999941	−	0.0108372i	$-0.00344964\pi$
$619$	1237.93	1.99988	0.999941	+	0.0108372i	$0.00344964\pi$
$620$	0	0
$621$	0	0
$622$	1777.56i	2.85782i
$623$	0	0
$624$	397.178	0.636503
$625$	0	0
$626$	0	0
$627$	− 248.877i	− 0.396934i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	540.503	0.856582	0.428291	−	0.903641i	$-0.359116\pi$
$631$	540.503	0.856582	0.428291	+	0.903641i	$0.359116\pi$
$632$	0	0
$633$	0	0
$634$	367.243	0.579247
$635$	0	0
$636$	−663.994	−1.04402
$637$	1267.65i	1.99003i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	513.999i	0.800622i
$643$	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	$0.5\pi$
$648$	1012.59i	1.56264i
$649$	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
$666$	−1333.89	−2.00283
$667$	0	0
$668$	2190.33i	3.27894i
$669$	−323.632	−0.483755
$670$	0	0
$671$	304.000	0.453055
$672$	0	0
$673$	− 423.232i	− 0.628873i	−0.949279	−	0.314437i	$-0.898184\pi$
$673$	− 423.232i	− 0.628873i	0.949279	−	0.314437i	$-0.101816\pi$
$674$	−1518.89	−2.25355
$675$	0	0
$676$	4181.76	6.18603
$677$	413.636i	0.610983i	0.952195	+	0.305492i	$0.0988208\pi$
$677$	413.636i	0.610983i	−0.952195	+	0.305492i	$0.901179\pi$
$678$	− 338.783i	− 0.499680i
$679$	0	0
$680$	0	0
$681$	179.927	0.264210
$682$	0	0
$683$	− 1268.92i	− 1.85786i	−0.370252	−	0.928932i	$-0.620729\pi$
$683$	− 1268.92i	− 1.85786i	0.370252	−	0.928932i	$-0.379271\pi$
$684$	1339.73	1.95867
$685$	0	0
$686$	0	0
$687$	144.087i	0.209734i
$688$	0	0
$689$	−2735.41	−3.97012
$690$	0	0
$691$	17.4356	0.0252324	0.0126162	−	0.999920i	$-0.495984\pi$
$691$	17.4356	0.0252324	0.0126162	+	0.999920i	$0.495984\pi$
$692$	− 742.131i	− 1.07244i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1821.04i	2.60894i
$699$	0	0
$700$	0	0
$701$	−1028.70	−1.46748	−0.733738	−	0.679433i	$-0.762226\pi$
$701$	−1028.70	−1.46748	−0.733738	+	0.679433i	$0.762226\pi$
$702$	− 1191.31i	− 1.69702i
$703$	854.607i	1.21566i
$704$	778.773	1.10621
$705$	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
$715$	0	0
$716$	0	0
$717$	− 73.6243i	− 0.102684i
$718$	− 2513.10i	− 3.50015i
$719$	714.859	0.994241	0.497121	−	0.867681i	$-0.334391\pi$
$719$	714.859	0.994241	0.497121	+	0.867681i	$0.334391\pi$
$720$	0	0
$721$	0	0
$722$	− 1269.10i	− 1.75776i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	483.321	0.665732
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	468.855	0.643148
$730$	0	0
$731$	0	0
$732$	− 109.492i	− 0.149579i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2176.73i	− 2.95351i
$738$	0	0
$739$	1402.00	1.89716	0.948579	−	0.316540i	$-0.102521\pi$
$739$	1402.00	1.89716	0.948579	+	0.316540i	$0.102521\pi$
$740$	0	0
$741$	−369.276	−0.498349
$742$	0	0
$743$	− 986.213i	− 1.32734i	−0.748026	−	0.663670i	$-0.768998\pi$
$743$	− 986.213i	− 1.32734i	0.748026	−	0.663670i	$-0.231002\pi$
$744$	0	0
$745$	0	0
$746$	−1177.85	−1.57889
$747$	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
$753$	136.731i	0.181581i
$754$	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
$761$	−505.632	−0.664431	−0.332216	−	0.943203i	$-0.607796\pi$
$761$	−505.632	−0.664431	−0.332216	+	0.943203i	$0.607796\pi$
$762$	151.662i	0.199031i
$763$	0	0
$764$	−2524.39	−3.30417
$765$	0	0
$766$	2198.26	2.86979
$767$	0	0
$768$	− 391.909i	− 0.510298i
$769$	−1377.41	−1.79117	−0.895587	−	0.444887i	$-0.853244\pi$
$769$	−1377.41	−1.79117	−0.895587	+	0.444887i	$0.853244\pi$
$770$	0	0
$771$	−215.412	−0.279393
$772$	− 2642.48i	− 3.42290i
$773$	473.894i	0.613058i	0.951861	+	0.306529i	$0.0991676\pi$
$773$	473.894i	0.613058i	−0.951861	+	0.306529i	$0.900832\pi$
$774$	0	0
$775$	0	0
$776$	−2636.88	−3.39804
$777$	0	0
$778$	− 2608.52i	− 3.35285i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1001.34	−1.27722
$785$	0	0
$786$	−311.650	−0.396501
$787$	1552.32i	1.97245i	0.165405	+	0.986226i	$0.447107\pi$
$787$	1552.32i	1.97245i	−0.165405	+	0.986226i	$0.552893\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	2253.82i	2.84573i
$793$	− 451.065i	− 0.568809i
$794$	0	0
$795$	0	0
$796$	−2691.57	−3.38136
$797$	454.785i	0.570621i	0.958435	+	0.285310i	$0.0920967\pi$
$797$	454.785i	0.570621i	−0.958435	+	0.285310i	$0.907903\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−783.994	−0.975117
$805$	0	0
$806$	0	0
$807$	0	0
$808$	267.180i	0.330668i
$809$	−1118.00	−1.38195	−0.690977	−	0.722877i	$-0.742819\pi$
$809$	−1118.00	−1.38195	−0.690977	+	0.722877i	$0.742819\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	406.063i	0.499463i
$814$	−2757.02	−3.38700
$815$	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	$0.5\pi$
$824$	−1172.77	−1.42327
$825$	0	0
$826$	0	0
$827$	− 1552.13i	− 1.87681i	−0.345531	−	0.938407i	$-0.612301\pi$
$827$	− 1552.13i	− 1.87681i	0.345531	−	0.938407i	$-0.387699\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	− 1155.52i	− 1.38884i
$833$	0	0
$834$	506.541	0.607363
$835$	0	0
$836$	2769.11	3.31233
$837$	0	0
$838$	1610.11i	1.92137i
$839$	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
$848$	− 2160.76i	− 2.54807i
$849$	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
$856$	−2982.25	−3.48394
$857$	− 720.651i	− 0.840900i	−0.907316	−	0.420450i	$-0.861872\pi$
$857$	− 720.651i	− 0.840900i	0.907316	−	0.420450i	$-0.138128\pi$
$858$	− 1191.31i	− 1.38847i
$859$	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
$863$	1364.66i	1.58130i	0.612271	+	0.790648i	$0.290256\pi$
$863$	1364.66i	1.58130i	−0.612271	+	0.790648i	$0.709744\pi$
$864$	138.148	0.159893
$865$	0	0
$866$	355.188	0.410148
$867$	− 217.117i	− 0.250423i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−3229.77	−3.70811
$872$	0	0
$873$	1451.58i	1.66274i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1208.36i	1.37783i	0.724842	+	0.688916i	$0.241913\pi$
$877$	1208.36i	1.37783i	−0.724842	+	0.688916i	$0.758087\pi$
$878$	0	0
$879$	−314.429	−0.357712
$880$	0	0
$881$	540.503	0.613511	0.306756	−	0.951788i	$-0.400757\pi$
$881$	540.503	0.613511	0.306756	+	0.951788i	$0.400757\pi$
$882$	1453.12i	1.64753i
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1734.39i	− 1.95535i	−0.210127	−	0.977674i	$-0.567388\pi$
$887$	− 1734.39i	− 1.95535i	0.210127	−	0.977674i	$-0.432612\pi$
$888$	517.815i	0.583125i
$889$	0	0
$890$	0	0
$891$	1152.14	1.29308
$892$	− 3600.86i	− 4.03683i
$893$	0	0
$894$	506.541	0.566601
$895$	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
$904$	1965.64	2.17438
$905$	0	0
$906$	0	0
$907$	− 378.653i	− 0.417479i	−0.977971	−	0.208739i	$-0.933064\pi$
$907$	− 378.653i	− 0.417479i	0.977971	−	0.208739i	$-0.0669361\pi$
$908$	2001.94i	2.20478i
$909$	147.080	0.161804
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	− 291.700i	− 0.319846i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−1603.17	−1.75018
$917$	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
$921$	−300.073	−0.325812
$922$	2102.28i	2.28013i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	645.598i	0.696438i
$928$	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
$933$	− 379.866i	− 0.407144i
$934$	0	0
$935$	0	0
$936$	3344.15	3.57281
$937$	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
$951$	−78.4799	−0.0825236
$952$	0	0
$953$	− 1576.13i	− 1.65386i	−0.562304	−	0.826930i	$-0.690085\pi$
$953$	− 1576.13i	− 1.65386i	0.562304	−	0.826930i	$-0.309915\pi$
$954$	−3135.64	−3.28683
$955$	0	0
$956$	819.172	0.856875
$957$	0	0
$958$	1164.61i	1.21567i
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	4090.78i	4.25237i
$963$	1641.70i	1.70477i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	2804.26i	2.89696i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 1400.39i	− 1.44073i
$973$	0	0
$974$	2253.38	2.31353
$975$	0	0
$976$	356.307	0.365068
$977$	1894.32i	1.93891i	0.245265	+	0.969456i	$0.421125\pi$
$977$	1894.32i	1.93891i	−0.245265	+	0.969456i	$0.578875\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	1047.63i	1.06683i
$983$	− 1381.61i	− 1.40550i	−0.711435	−	0.702752i	$-0.751954\pi$
$983$	− 1381.61i	− 1.40550i	0.711435	−	0.702752i	$-0.248046\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 4108.71i	− 4.15861i
$989$	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	$0.5\pi$
$998$	− 674.248i	− 0.675599i
$999$	589.176	0.589766

Twists

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

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

\(f(q)\)	\(=\)	\(q-3.51552i q^{2} +0.751268i q^{3} -8.35890 q^{4} +2.64110 q^{6} +15.3238i q^{8} +8.43560 q^{9} +O(q^{10})\)	\(q-3.51552i q^{2} +0.751268i q^{3} -8.35890 q^{4} +2.64110 q^{6} +15.3238i q^{8} +8.43560 q^{9} +17.4356 q^{11} -6.27978i q^{12} -25.8704i q^{13} +20.4356 q^{16} -29.6555i q^{18} -19.0000 q^{19} -61.2952i q^{22} -11.5123 q^{24} -90.9479 q^{26} +13.0988i q^{27} -10.5466i q^{32} +13.0988i q^{33} -70.5123 q^{36} -44.9793i q^{37} +66.7949i q^{38} +19.4356 q^{39} -145.742 q^{44} +15.3526i q^{48} -49.0000 q^{49} +216.248i q^{52} -105.735i q^{53} +46.0492 q^{54} -14.2741i q^{57} +17.4356 q^{61} +44.6657 q^{64} +46.0492 q^{66} -124.844i q^{67} +129.265i q^{72} -158.126 q^{74} +158.819 q^{76} -68.3263i q^{78} +66.0796 q^{81} +267.180i q^{88} +7.92330 q^{96} +172.077i q^{97} +172.261i q^{98} +147.080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} + 28 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 16 * q^4 + 28 * q^6 - 36 * q^9	\( 4 q - 16 q^{4} + 28 q^{6} - 36 q^{9} + 12 q^{16} - 76 q^{19} + 76 q^{24} - 172 q^{26} - 160 q^{36} + 8 q^{39} - 304 q^{44} - 196 q^{49} - 304 q^{54} - 48 q^{64} - 304 q^{66} - 92 q^{74} + 304 q^{76} + 892 q^{81} + 84 q^{96} + 1216 q^{99}+O(q^{100}) \) 4 * q - 16 * q^4 + 28 * q^6 - 36 * q^9 + 12 * q^16 - 76 * q^19 + 76 * q^24 - 172 * q^26 - 160 * q^36 + 8 * q^39 - 304 * q^44 - 196 * q^49 - 304 * q^54 - 48 * q^64 - 304 * q^66 - 92 * q^74 + 304 * q^76 + 892 * q^81 + 84 * q^96 + 1216 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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