LMFDB - Embedded newform 3750.2.c.g.1249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1249,3750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3750, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3750.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.c (of order $2$, degree $1$, not 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:	$29.9439007580$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$1.95630i$ of defining polynomial
Character	$\chi$	$=$	3750.1249
Dual form			3750.2.c.g.1249.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +0.511170i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.29456 q^{11} +1.00000i q^{12} -2.65418i q^{13} +0.511170 q^{14} +1.00000 q^{16} +5.57433i q^{17} +1.00000i q^{18} -0.374409 q^{19} +0.511170 q^{21} -3.29456i q^{22} -5.75638i q^{23} +1.00000 q^{24} -2.65418 q^{26} +1.00000i q^{27} -0.511170i q^{28} +3.89025 q^{29} -1.80008 q^{31} -1.00000i q^{32} -3.29456i q^{33} +5.57433 q^{34} +1.00000 q^{36} +10.3577i q^{37} +0.374409i q^{38} -2.65418 q^{39} +2.33728 q^{41} -0.511170i q^{42} +5.87802i q^{43} -3.29456 q^{44} -5.75638 q^{46} -8.19959i q^{47} -1.00000i q^{48} +6.73870 q^{49} +5.57433 q^{51} +2.65418i q^{52} -0.518727i q^{53} +1.00000 q^{54} -0.511170 q^{56} +0.374409i q^{57} -3.89025i q^{58} +7.62993 q^{59} +2.90345 q^{61} +1.80008i q^{62} -0.511170i q^{63} -1.00000 q^{64} -3.29456 q^{66} -1.78806i q^{67} -5.57433i q^{68} -5.75638 q^{69} +7.27786 q^{71} -1.00000i q^{72} +1.40995i q^{73} +10.3577 q^{74} +0.374409 q^{76} +1.68408i q^{77} +2.65418i q^{78} +12.1395 q^{79} +1.00000 q^{81} -2.33728i q^{82} -3.09306i q^{83} -0.511170 q^{84} +5.87802 q^{86} -3.89025i q^{87} +3.29456i q^{88} -3.42567 q^{89} +1.35674 q^{91} +5.75638i q^{92} +1.80008i q^{93} -8.19959 q^{94} -1.00000 q^{96} -15.1483i q^{97} -6.73870i q^{98} -3.29456 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{6} - 8 q^{9} + 8 q^{14} + 8 q^{16} + 22 q^{19} + 8 q^{21} + 8 q^{24} + 4 q^{26} - 12 q^{29} - 16 q^{31} + 18 q^{34} + 8 q^{36} + 4 q^{39} + 12 q^{41} - 30 q^{46} + 8 q^{49} + 18 q^{51}+ \cdots - 8 q^{96}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^6 - 8 * q^9 + 8 * q^14 + 8 * q^16 + 22 * q^19 + 8 * q^21 + 8 * q^24 + 4 * q^26 - 12 * q^29 - 16 * q^31 + 18 * q^34 + 8 * q^36 + 4 * q^39 + 12 * q^41 - 30 * q^46 + 8 * q^49 + 18 * q^51 + 8 * q^54 - 8 * q^56 - 30 * q^59 - 10 * q^61 - 8 * q^64 - 30 * q^69 + 26 * q^74 - 22 * q^76 + 46 * q^79 + 8 * q^81 - 8 * q^84 - 8 * q^86 - 54 * q^89 - 44 * q^91 + 30 * q^94 - 8 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times$.

$n$	$2501$	$3127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0.511170i	0.193204i	0.995323	+	0.0966021i	$0.0307974\pi$
$7$	0.511170i	0.193204i	−0.995323	+	0.0966021i	$0.969203\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.29456	0.993346	0.496673	−	0.867938i	$-0.334555\pi$
$11$	3.29456	0.993346	0.496673	+	0.867938i	$0.334555\pi$
$12$	1.00000i	0.288675i
$13$	− 2.65418i	− 0.736138i	−0.929799	−	0.368069i	$-0.880019\pi$
$13$	− 2.65418i	− 0.736138i	0.929799	−	0.368069i	$-0.119981\pi$
$14$	0.511170	0.136616
$15$	0	0
$16$	1.00000	0.250000
$17$	5.57433i	1.35197i	0.736914	+	0.675987i	$0.236282\pi$
$17$	5.57433i	1.35197i	−0.736914	+	0.675987i	$0.763718\pi$
$18$	1.00000i	0.235702i
$19$	−0.374409	−0.0858953	−0.0429477	−	0.999077i	$-0.513675\pi$
$19$	−0.374409	−0.0858953	−0.0429477	+	0.999077i	$0.513675\pi$
$20$	0	0
$21$	0.511170	0.111547
$22$	− 3.29456i	− 0.702402i
$23$	− 5.75638i	− 1.20029i	−0.799892	−	0.600144i	$-0.795110\pi$
$23$	− 5.75638i	− 1.20029i	0.799892	−	0.600144i	$-0.204890\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.65418	−0.520528
$27$	1.00000i	0.192450i
$28$	− 0.511170i	− 0.0966021i
$29$	3.89025	0.722401	0.361201	−	0.932488i	$-0.382367\pi$
$29$	3.89025	0.722401	0.361201	+	0.932488i	$0.382367\pi$
$30$	0	0
$31$	−1.80008	−0.323304	−0.161652	−	0.986848i	$-0.551682\pi$
$31$	−1.80008	−0.323304	−0.161652	+	0.986848i	$0.551682\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 3.29456i	− 0.573509i
$34$	5.57433	0.955990
$35$	0	0
$36$	1.00000	0.166667
$37$	10.3577i	1.70280i	0.524519	+	0.851399i	$0.324245\pi$
$37$	10.3577i	1.70280i	−0.524519	+	0.851399i	$0.675755\pi$
$38$	0.374409i	0.0607372i
$39$	−2.65418	−0.425009
$40$	0	0
$41$	2.33728	0.365023	0.182511	−	0.983204i	$-0.441577\pi$
$41$	2.33728	0.365023	0.182511	+	0.983204i	$0.441577\pi$
$42$	− 0.511170i	− 0.0788753i
$43$	5.87802i	0.896390i	0.893936	+	0.448195i	$0.147933\pi$
$43$	5.87802i	0.896390i	−0.893936	+	0.448195i	$0.852067\pi$
$44$	−3.29456	−0.496673
$45$	0	0
$46$	−5.75638	−0.848731
$47$	− 8.19959i	− 1.19603i	−0.801484	−	0.598017i	$-0.795956\pi$
$47$	− 8.19959i	− 1.19603i	0.801484	−	0.598017i	$-0.204044\pi$
$48$	− 1.00000i	− 0.144338i
$49$	6.73870	0.962672
$50$	0	0
$51$	5.57433	0.780562
$52$	2.65418i	0.368069i
$53$	− 0.518727i	− 0.0712527i	−0.999365	−	0.0356263i	$-0.988657\pi$
$53$	− 0.518727i	− 0.0712527i	0.999365	−	0.0356263i	$-0.0113426\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−0.511170	−0.0683080
$57$	0.374409i	0.0495917i
$58$	− 3.89025i	− 0.510815i
$59$	7.62993	0.993332	0.496666	−	0.867942i	$-0.334557\pi$
$59$	7.62993	0.993332	0.496666	+	0.867942i	$0.334557\pi$
$60$	0	0
$61$	2.90345	0.371749	0.185875	−	0.982573i	$-0.440488\pi$
$61$	2.90345	0.371749	0.185875	+	0.982573i	$0.440488\pi$
$62$	1.80008i	0.228610i
$63$	− 0.511170i	− 0.0644014i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.29456	−0.405532
$67$	− 1.78806i	− 0.218446i	−0.994017	−	0.109223i	$-0.965164\pi$
$67$	− 1.78806i	− 0.218446i	0.994017	−	0.109223i	$-0.0348362\pi$
$68$	− 5.57433i	− 0.675987i
$69$	−5.75638	−0.692986
$70$	0	0
$71$	7.27786	0.863723	0.431862	−	0.901940i	$-0.357857\pi$
$71$	7.27786	0.863723	0.431862	+	0.901940i	$0.357857\pi$
$72$	− 1.00000i	− 0.117851i
$73$	1.40995i	0.165023i	0.996590	+	0.0825113i	$0.0262941\pi$
$73$	1.40995i	0.165023i	−0.996590	+	0.0825113i	$0.973706\pi$
$74$	10.3577	1.20406
$75$	0	0
$76$	0.374409	0.0429477
$77$	1.68408i	0.191919i
$78$	2.65418i	0.300527i
$79$	12.1395	1.36580	0.682901	−	0.730511i	$-0.260718\pi$
$79$	12.1395	1.36580	0.682901	+	0.730511i	$0.260718\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.33728i	− 0.258110i
$83$	− 3.09306i	− 0.339507i	−0.985487	−	0.169754i	$-0.945703\pi$
$83$	− 3.09306i	− 0.339507i	0.985487	−	0.169754i	$-0.0542972\pi$
$84$	−0.511170	−0.0557733
$85$	0	0
$86$	5.87802	0.633843
$87$	− 3.89025i	− 0.417079i
$88$	3.29456i	0.351201i
$89$	−3.42567	−0.363120	−0.181560	−	0.983380i	$-0.558115\pi$
$89$	−3.42567	−0.363120	−0.181560	+	0.983380i	$0.558115\pi$
$90$	0	0
$91$	1.35674	0.142225
$92$	5.75638i	0.600144i
$93$	1.80008i	0.186660i
$94$	−8.19959	−0.845723
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 15.1483i	− 1.53808i	−0.639201	−	0.769040i	$-0.720735\pi$
$97$	− 15.1483i	− 1.53808i	0.639201	−	0.769040i	$-0.279265\pi$
$98$	− 6.73870i	− 0.680712i
$99$	−3.29456	−0.331115
$100$	0	0
$101$	5.58091	0.555321	0.277661	−	0.960679i	$-0.410441\pi$
$101$	5.58091	0.555321	0.277661	+	0.960679i	$0.410441\pi$
$102$	− 5.57433i	− 0.551941i
$103$	− 15.1228i	− 1.49010i	−0.667011	−	0.745048i	$-0.732426\pi$
$103$	− 15.1228i	− 1.49010i	0.667011	−	0.745048i	$-0.267574\pi$
$104$	2.65418	0.260264
$105$	0	0
$106$	−0.518727	−0.0503832
$107$	18.7290i	1.81060i	0.424768	+	0.905302i	$0.360356\pi$
$107$	18.7290i	1.81060i	−0.424768	+	0.905302i	$0.639644\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	18.5221	1.77410	0.887049	−	0.461675i	$-0.152751\pi$
$109$	18.5221	1.77410	0.887049	+	0.461675i	$0.152751\pi$
$110$	0	0
$111$	10.3577	0.983111
$112$	0.511170i	0.0483011i
$113$	− 16.1273i	− 1.51713i	−0.651598	−	0.758564i	$-0.725901\pi$
$113$	− 16.1273i	− 1.51713i	0.651598	−	0.758564i	$-0.274099\pi$
$114$	0.374409	0.0350666
$115$	0	0
$116$	−3.89025	−0.361201
$117$	2.65418i	0.245379i
$118$	− 7.62993i	− 0.702392i
$119$	−2.84943	−0.261207
$120$	0	0
$121$	−0.145898	−0.0132635
$122$	− 2.90345i	− 0.262866i
$123$	− 2.33728i	− 0.210746i
$124$	1.80008	0.161652
$125$	0	0
$126$	−0.511170	−0.0455387
$127$	− 3.62677i	− 0.321824i	−0.986969	−	0.160912i	$-0.948556\pi$
$127$	− 3.62677i	− 0.321824i	0.986969	−	0.160912i	$-0.0514435\pi$
$128$	1.00000i	0.0883883i
$129$	5.87802	0.517531
$130$	0	0
$131$	−17.4131	−1.52139	−0.760695	−	0.649109i	$-0.775142\pi$
$131$	−17.4131	−1.52139	−0.760695	+	0.649109i	$0.775142\pi$
$132$	3.29456i	0.286754i
$133$	− 0.191387i	− 0.0165953i
$134$	−1.78806	−0.154465
$135$	0	0
$136$	−5.57433	−0.477995
$137$	19.7230i	1.68505i	0.538658	+	0.842524i	$0.318931\pi$
$137$	19.7230i	1.68505i	−0.538658	+	0.842524i	$0.681069\pi$
$138$	5.75638i	0.490015i
$139$	−11.4357	−0.969960	−0.484980	−	0.874525i	$-0.661173\pi$
$139$	−11.4357	−0.969960	−0.484980	+	0.874525i	$0.661173\pi$
$140$	0	0
$141$	−8.19959	−0.690530
$142$	− 7.27786i	− 0.610745i
$143$	− 8.74435i	− 0.731239i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	1.40995	0.116689
$147$	− 6.73870i	− 0.555799i
$148$	− 10.3577i	− 0.851399i
$149$	−20.5095	−1.68021	−0.840103	−	0.542427i	$-0.817506\pi$
$149$	−20.5095	−1.68021	−0.840103	+	0.542427i	$0.817506\pi$
$150$	0	0
$151$	6.22542	0.506618	0.253309	−	0.967385i	$-0.418481\pi$
$151$	6.22542	0.506618	0.253309	+	0.967385i	$0.418481\pi$
$152$	− 0.374409i	− 0.0303686i
$153$	− 5.57433i	− 0.450658i
$154$	1.68408	0.135707
$155$	0	0
$156$	2.65418	0.212505
$157$	− 23.9795i	− 1.91377i	−0.290465	−	0.956886i	$-0.593810\pi$
$157$	− 23.9795i	− 1.91377i	0.290465	−	0.956886i	$-0.406190\pi$
$158$	− 12.1395i	− 0.965768i
$159$	−0.518727	−0.0411377
$160$	0	0
$161$	2.94249	0.231901
$162$	− 1.00000i	− 0.0785674i
$163$	− 24.1232i	− 1.88948i	−0.327825	−	0.944738i	$-0.606316\pi$
$163$	− 24.1232i	− 1.88948i	0.327825	−	0.944738i	$-0.393684\pi$
$164$	−2.33728	−0.182511
$165$	0	0
$166$	−3.09306	−0.240068
$167$	5.43441i	0.420527i	0.977645	+	0.210264i	$0.0674322\pi$
$167$	5.43441i	0.420527i	−0.977645	+	0.210264i	$0.932568\pi$
$168$	0.511170i	0.0394376i
$169$	5.95532	0.458101
$170$	0	0
$171$	0.374409	0.0286318
$172$	− 5.87802i	− 0.448195i
$173$	− 1.14183i	− 0.0868118i	−0.999058	−	0.0434059i	$-0.986179\pi$
$173$	− 1.14183i	− 0.0868118i	0.999058	−	0.0434059i	$-0.0138209\pi$
$174$	−3.89025	−0.294919
$175$	0	0
$176$	3.29456	0.248337
$177$	− 7.62993i	− 0.573501i
$178$	3.42567i	0.256765i
$179$	15.4030	1.15127	0.575637	−	0.817705i	$-0.304754\pi$
$179$	15.4030	1.15127	0.575637	+	0.817705i	$0.304754\pi$
$180$	0	0
$181$	20.5110	1.52457	0.762287	−	0.647239i	$-0.224077\pi$
$181$	20.5110	1.52457	0.762287	+	0.647239i	$0.224077\pi$
$182$	− 1.35674i	− 0.100568i
$183$	− 2.90345i	− 0.214629i
$184$	5.75638	0.424366
$185$	0	0
$186$	1.80008	0.131988
$187$	18.3649i	1.34298i
$188$	8.19959i	0.598017i
$189$	−0.511170	−0.0371822
$190$	0	0
$191$	7.27380	0.526313	0.263157	−	0.964753i	$-0.415236\pi$
$191$	7.27380	0.526313	0.263157	+	0.964753i	$0.415236\pi$
$192$	1.00000i	0.0721688i
$193$	− 11.4686i	− 0.825531i	−0.910837	−	0.412766i	$-0.864563\pi$
$193$	− 11.4686i	− 0.825531i	0.910837	−	0.412766i	$-0.135437\pi$
$194$	−15.1483	−1.08759
$195$	0	0
$196$	−6.73870	−0.481336
$197$	− 2.16068i	− 0.153942i	−0.997033	−	0.0769711i	$-0.975475\pi$
$197$	− 2.16068i	− 0.153942i	0.997033	−	0.0769711i	$-0.0245249\pi$
$198$	3.29456i	0.234134i
$199$	7.27998	0.516064	0.258032	−	0.966136i	$-0.416926\pi$
$199$	7.27998	0.516064	0.258032	+	0.966136i	$0.416926\pi$
$200$	0	0
$201$	−1.78806	−0.126120
$202$	− 5.58091i	− 0.392671i
$203$	1.98858i	0.139571i
$204$	−5.57433	−0.390281
$205$	0	0
$206$	−15.1228	−1.05366
$207$	5.75638i	0.400096i
$208$	− 2.65418i	− 0.184034i
$209$	−1.23351	−0.0853238
$210$	0	0
$211$	−28.7670	−1.98040	−0.990200	−	0.139658i	$-0.955400\pi$
$211$	−28.7670	−1.98040	−0.990200	+	0.139658i	$0.955400\pi$
$212$	0.518727i	0.0356263i
$213$	− 7.27786i	− 0.498671i
$214$	18.7290	1.28029
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 0.920147i	− 0.0624637i
$218$	− 18.5221i	− 1.25448i
$219$	1.40995	0.0952758
$220$	0	0
$221$	14.7953	0.995238
$222$	− 10.3577i	− 0.695164i
$223$	0.662342i	0.0443537i	0.999754	+	0.0221769i	$0.00705969\pi$
$223$	0.662342i	0.0443537i	−0.999754	+	0.0221769i	$0.992940\pi$
$224$	0.511170	0.0341540
$225$	0	0
$226$	−16.1273	−1.07277
$227$	− 15.5730i	− 1.03362i	−0.856101	−	0.516809i	$-0.827120\pi$
$227$	− 15.5730i	− 1.03362i	0.856101	−	0.516809i	$-0.172880\pi$
$228$	− 0.374409i	− 0.0247958i
$229$	27.7376	1.83296	0.916478	−	0.400086i	$-0.131020\pi$
$229$	27.7376	1.83296	0.916478	+	0.400086i	$0.131020\pi$
$230$	0	0
$231$	1.68408	0.110804
$232$	3.89025i	0.255407i
$233$	23.2480i	1.52303i	0.648149	+	0.761514i	$0.275544\pi$
$233$	23.2480i	1.52303i	−0.648149	+	0.761514i	$0.724456\pi$
$234$	2.65418	0.173509
$235$	0	0
$236$	−7.62993	−0.496666
$237$	− 12.1395i	− 0.788547i
$238$	2.84943i	0.184701i
$239$	−5.81933	−0.376421	−0.188211	−	0.982129i	$-0.560269\pi$
$239$	−5.81933	−0.376421	−0.188211	+	0.982129i	$0.560269\pi$
$240$	0	0
$241$	14.1483	0.911374	0.455687	−	0.890140i	$-0.349394\pi$
$241$	14.1483	0.911374	0.455687	+	0.890140i	$0.349394\pi$
$242$	0.145898i	0.00937868i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.90345	−0.185875
$245$	0	0
$246$	−2.33728	−0.149020
$247$	0.993750i	0.0632308i
$248$	− 1.80008i	− 0.114305i
$249$	−3.09306	−0.196014
$250$	0	0
$251$	3.98246	0.251370	0.125685	−	0.992070i	$-0.459887\pi$
$251$	3.98246	0.251370	0.125685	+	0.992070i	$0.459887\pi$
$252$	0.511170i	0.0322007i
$253$	− 18.9647i	− 1.19230i
$254$	−3.62677	−0.227564
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.5114i	0.780442i	0.920721	+	0.390221i	$0.127601\pi$
$257$	12.5114i	0.780442i	−0.920721	+	0.390221i	$0.872399\pi$
$258$	− 5.87802i	− 0.365950i
$259$	−5.29456	−0.328988
$260$	0	0
$261$	−3.89025	−0.240800
$262$	17.4131i	1.07579i
$263$	2.25374i	0.138971i	0.997583	+	0.0694857i	$0.0221358\pi$
$263$	2.25374i	0.138971i	−0.997583	+	0.0694857i	$0.977864\pi$
$264$	3.29456	0.202766
$265$	0	0
$266$	−0.191387	−0.0117347
$267$	3.42567i	0.209648i
$268$	1.78806i	0.109223i
$269$	−25.4934	−1.55436	−0.777179	−	0.629279i	$-0.783350\pi$
$269$	−25.4934	−1.55436	−0.777179	+	0.629279i	$0.783350\pi$
$270$	0	0
$271$	−19.2503	−1.16937	−0.584687	−	0.811259i	$-0.698783\pi$
$271$	−19.2503	−1.16937	−0.584687	+	0.811259i	$0.698783\pi$
$272$	5.57433i	0.337993i
$273$	− 1.35674i	− 0.0821136i
$274$	19.7230	1.19151
$275$	0	0
$276$	5.75638	0.346493
$277$	7.73948i	0.465020i	0.972594	+	0.232510i	$0.0746939\pi$
$277$	7.73948i	0.465020i	−0.972594	+	0.232510i	$0.925306\pi$
$278$	11.4357i	0.685865i
$279$	1.80008	0.107768
$280$	0	0
$281$	−10.2724	−0.612801	−0.306401	−	0.951903i	$-0.599125\pi$
$281$	−10.2724	−0.612801	−0.306401	+	0.951903i	$0.599125\pi$
$282$	8.19959i	0.488278i
$283$	− 9.86811i	− 0.586598i	−0.956021	−	0.293299i	$-0.905247\pi$
$283$	− 9.86811i	− 0.586598i	0.956021	−	0.293299i	$-0.0947532\pi$
$284$	−7.27786	−0.431862
$285$	0	0
$286$	−8.74435	−0.517064
$287$	1.19475i	0.0705239i
$288$	1.00000i	0.0589256i
$289$	−14.0731	−0.827832
$290$	0	0
$291$	−15.1483	−0.888011
$292$	− 1.40995i	− 0.0825113i
$293$	− 3.08103i	− 0.179996i	−0.995942	−	0.0899979i	$-0.971314\pi$
$293$	− 3.08103i	− 0.179996i	0.995942	−	0.0899979i	$-0.0286860\pi$
$294$	−6.73870	−0.393009
$295$	0	0
$296$	−10.3577	−0.602030
$297$	3.29456i	0.191170i
$298$	20.5095i	1.18809i
$299$	−15.2785	−0.883577
$300$	0	0
$301$	−3.00467	−0.173186
$302$	− 6.22542i	− 0.358233i
$303$	− 5.58091i	− 0.320615i
$304$	−0.374409	−0.0214738
$305$	0	0
$306$	−5.57433	−0.318663
$307$	6.43192i	0.367089i	0.983011	+	0.183545i	$0.0587572\pi$
$307$	6.43192i	0.367089i	−0.983011	+	0.183545i	$0.941243\pi$
$308$	− 1.68408i	− 0.0959593i
$309$	−15.1228	−0.860308
$310$	0	0
$311$	20.8818	1.18410	0.592048	−	0.805903i	$-0.298320\pi$
$311$	20.8818	1.18410	0.592048	+	0.805903i	$0.298320\pi$
$312$	− 2.65418i	− 0.150263i
$313$	11.7245i	0.662708i	0.943507	+	0.331354i	$0.107505\pi$
$313$	11.7245i	0.662708i	−0.943507	+	0.331354i	$0.892495\pi$
$314$	−23.9795	−1.35324
$315$	0	0
$316$	−12.1395	−0.682901
$317$	− 5.92233i	− 0.332631i	−0.986073	−	0.166316i	$-0.946813\pi$
$317$	− 5.92233i	− 0.332631i	0.986073	−	0.166316i	$-0.0531871\pi$
$318$	0.518727i	0.0290888i
$319$	12.8166	0.717594
$320$	0	0
$321$	18.7290	1.04535
$322$	− 2.94249i	− 0.163978i
$323$	− 2.08708i	− 0.116128i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−24.1232	−1.33606
$327$	− 18.5221i	− 1.02428i
$328$	2.33728i	0.129055i
$329$	4.19139	0.231079
$330$	0	0
$331$	27.3753	1.50468	0.752340	−	0.658775i	$-0.228925\pi$
$331$	27.3753	1.50468	0.752340	+	0.658775i	$0.228925\pi$
$332$	3.09306i	0.169754i
$333$	− 10.3577i	− 0.567599i
$334$	5.43441	0.297358
$335$	0	0
$336$	0.511170	0.0278866
$337$	5.86677i	0.319583i	0.987151	+	0.159792i	$0.0510823\pi$
$337$	5.86677i	0.319583i	−0.987151	+	0.159792i	$0.948918\pi$
$338$	− 5.95532i	− 0.323927i
$339$	−16.1273	−0.875914
$340$	0	0
$341$	−5.93046	−0.321153
$342$	− 0.374409i	− 0.0202457i
$343$	7.02282i	0.379197i
$344$	−5.87802	−0.316922
$345$	0	0
$346$	−1.14183	−0.0613852
$347$	− 11.8634i	− 0.636863i	−0.947946	−	0.318431i	$-0.896844\pi$
$347$	− 11.8634i	− 0.636863i	0.947946	−	0.318431i	$-0.103156\pi$
$348$	3.89025i	0.208539i
$349$	−28.3391	−1.51696	−0.758479	−	0.651698i	$-0.774057\pi$
$349$	−28.3391	−1.51696	−0.758479	+	0.651698i	$0.774057\pi$
$350$	0	0
$351$	2.65418	0.141670
$352$	− 3.29456i	− 0.175600i
$353$	5.83404i	0.310515i	0.987874	+	0.155257i	$0.0496207\pi$
$353$	5.83404i	0.310515i	−0.987874	+	0.155257i	$0.950379\pi$
$354$	−7.62993	−0.405526
$355$	0	0
$356$	3.42567	0.181560
$357$	2.84943i	0.150808i
$358$	− 15.4030i	− 0.814074i
$359$	−9.94932	−0.525105	−0.262552	−	0.964918i	$-0.584564\pi$
$359$	−9.94932	−0.525105	−0.262552	+	0.964918i	$0.584564\pi$
$360$	0	0
$361$	−18.8598	−0.992622
$362$	− 20.5110i	− 1.07804i
$363$	0.145898i	0.00765766i
$364$	−1.35674	−0.0711124
$365$	0	0
$366$	−2.90345	−0.151766
$367$	− 16.2990i	− 0.850802i	−0.905005	−	0.425401i	$-0.860133\pi$
$367$	− 16.2990i	− 0.850802i	0.905005	−	0.425401i	$-0.139867\pi$
$368$	− 5.75638i	− 0.300072i
$369$	−2.33728	−0.121674
$370$	0	0
$371$	0.265158	0.0137663
$372$	− 1.80008i	− 0.0933298i
$373$	− 8.85626i	− 0.458560i	−0.973361	−	0.229280i	$-0.926363\pi$
$373$	− 8.85626i	− 0.458560i	0.973361	−	0.229280i	$-0.0736371\pi$
$374$	18.3649	0.949629
$375$	0	0
$376$	8.19959	0.422862
$377$	− 10.3254i	− 0.531787i
$378$	0.511170i	0.0262918i
$379$	14.1431	0.726480	0.363240	−	0.931696i	$-0.381671\pi$
$379$	14.1431	0.726480	0.363240	+	0.931696i	$0.381671\pi$
$380$	0	0
$381$	−3.62677	−0.185805
$382$	− 7.27380i	− 0.372160i
$383$	32.9848i	1.68544i	0.538350	+	0.842721i	$0.319048\pi$
$383$	32.9848i	1.68544i	−0.538350	+	0.842721i	$0.680952\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−11.4686	−0.583739
$387$	− 5.87802i	− 0.298797i
$388$	15.1483i	0.769040i
$389$	−11.4919	−0.582663	−0.291332	−	0.956622i	$-0.594098\pi$
$389$	−11.4919	−0.582663	−0.291332	+	0.956622i	$0.594098\pi$
$390$	0	0
$391$	32.0879	1.62276
$392$	6.73870i	0.340356i
$393$	17.4131i	0.878375i
$394$	−2.16068	−0.108854
$395$	0	0
$396$	3.29456	0.165558
$397$	4.82958i	0.242390i	0.992629	+	0.121195i	$0.0386726\pi$
$397$	4.82958i	0.242390i	−0.992629	+	0.121195i	$0.961327\pi$
$398$	− 7.27998i	− 0.364912i
$399$	−0.191387	−0.00958132
$400$	0	0
$401$	−21.5779	−1.07755	−0.538775	−	0.842449i	$-0.681113\pi$
$401$	−21.5779	−1.07755	−0.538775	+	0.842449i	$0.681113\pi$
$402$	1.78806i	0.0891802i
$403$	4.77774i	0.237996i
$404$	−5.58091	−0.277661
$405$	0	0
$406$	1.98858	0.0986916
$407$	34.1241i	1.69147i
$408$	5.57433i	0.275970i
$409$	1.57998	0.0781248	0.0390624	−	0.999237i	$-0.487563\pi$
$409$	1.57998	0.0781248	0.0390624	+	0.999237i	$0.487563\pi$
$410$	0	0
$411$	19.7230	0.972863
$412$	15.1228i	0.745048i
$413$	3.90019i	0.191916i
$414$	5.75638	0.282910
$415$	0	0
$416$	−2.65418	−0.130132
$417$	11.4357i	0.560007i
$418$	1.23351i	0.0603330i
$419$	36.7568	1.79569	0.897843	−	0.440316i	$-0.145134\pi$
$419$	36.7568	1.79569	0.897843	+	0.440316i	$0.145134\pi$
$420$	0	0
$421$	6.96946	0.339671	0.169835	−	0.985472i	$-0.445676\pi$
$421$	6.96946	0.339671	0.169835	+	0.985472i	$0.445676\pi$
$422$	28.7670i	1.40035i
$423$	8.19959i	0.398678i
$424$	0.518727	0.0251916
$425$	0	0
$426$	−7.27786	−0.352614
$427$	1.48416i	0.0718235i
$428$	− 18.7290i	− 0.905302i
$429$	−8.74435	−0.422181
$430$	0	0
$431$	25.4098	1.22395	0.611974	−	0.790878i	$-0.290376\pi$
$431$	25.4098	1.22395	0.611974	+	0.790878i	$0.290376\pi$
$432$	1.00000i	0.0481125i
$433$	13.4595i	0.646823i	0.946258	+	0.323411i	$0.104830\pi$
$433$	13.4595i	0.646823i	−0.946258	+	0.323411i	$0.895170\pi$
$434$	−0.920147	−0.0441685
$435$	0	0
$436$	−18.5221	−0.887049
$437$	2.15524i	0.103099i
$438$	− 1.40995i	− 0.0673702i
$439$	37.8392	1.80597	0.902983	−	0.429676i	$-0.141372\pi$
$439$	37.8392	1.80597	0.902983	+	0.429676i	$0.141372\pi$
$440$	0	0
$441$	−6.73870	−0.320891
$442$	− 14.7953i	− 0.703740i
$443$	− 2.70138i	− 0.128346i	−0.997939	−	0.0641731i	$-0.979559\pi$
$443$	− 2.70138i	− 0.128346i	0.997939	−	0.0641731i	$-0.0204410\pi$
$444$	−10.3577	−0.491555
$445$	0	0
$446$	0.662342	0.0313628
$447$	20.5095i	0.970068i
$448$	− 0.511170i	− 0.0241505i
$449$	−18.9478	−0.894200	−0.447100	−	0.894484i	$-0.647543\pi$
$449$	−18.9478	−0.894200	−0.447100	+	0.894484i	$0.647543\pi$
$450$	0	0
$451$	7.70032	0.362594
$452$	16.1273i	0.758564i
$453$	− 6.22542i	− 0.292496i
$454$	−15.5730	−0.730878
$455$	0	0
$456$	−0.374409	−0.0175333
$457$	− 38.2879i	− 1.79103i	−0.445028	−	0.895517i	$-0.646806\pi$
$457$	− 38.2879i	− 1.79103i	0.445028	−	0.895517i	$-0.353194\pi$
$458$	− 27.7376i	− 1.29610i
$459$	−5.57433	−0.260187
$460$	0	0
$461$	17.3427	0.807731	0.403866	−	0.914818i	$-0.367666\pi$
$461$	17.3427	0.807731	0.403866	+	0.914818i	$0.367666\pi$
$462$	− 1.68408i	− 0.0783505i
$463$	− 17.6641i	− 0.820922i	−0.911878	−	0.410461i	$-0.865368\pi$
$463$	− 17.6641i	− 0.820922i	0.911878	−	0.410461i	$-0.134632\pi$
$464$	3.89025	0.180600
$465$	0	0
$466$	23.2480	1.07694
$467$	− 2.88972i	− 0.133720i	−0.997762	−	0.0668600i	$-0.978702\pi$
$467$	− 2.88972i	− 0.133720i	0.997762	−	0.0668600i	$-0.0212981\pi$
$468$	− 2.65418i	− 0.122690i
$469$	0.914001	0.0422047
$470$	0	0
$471$	−23.9795	−1.10492
$472$	7.62993i	0.351196i
$473$	19.3655i	0.890426i
$474$	−12.1395	−0.557587
$475$	0	0
$476$	2.84943	0.130603
$477$	0.518727i	0.0237509i
$478$	5.81933i	0.266170i
$479$	−27.5859	−1.26043	−0.630216	−	0.776420i	$-0.717034\pi$
$479$	−27.5859	−1.26043	−0.630216	+	0.776420i	$0.717034\pi$
$480$	0	0
$481$	27.4913	1.25349
$482$	− 14.1483i	− 0.644439i
$483$	− 2.94249i	− 0.133888i
$484$	0.145898	0.00663173
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	14.9338i	0.676713i	0.941018	+	0.338356i	$0.109871\pi$
$487$	14.9338i	0.676713i	−0.941018	+	0.338356i	$0.890129\pi$
$488$	2.90345i	0.131433i
$489$	−24.1232	−1.09089
$490$	0	0
$491$	26.4452	1.19346	0.596728	−	0.802444i	$-0.296467\pi$
$491$	26.4452	1.19346	0.596728	+	0.802444i	$0.296467\pi$
$492$	2.33728i	0.105373i
$493$	21.6855i	0.976667i
$494$	0.993750	0.0447109
$495$	0	0
$496$	−1.80008	−0.0808260
$497$	3.72023i	0.166875i
$498$	3.09306i	0.138603i
$499$	10.3375	0.462771	0.231386	−	0.972862i	$-0.425674\pi$
$499$	10.3375	0.462771	0.231386	+	0.972862i	$0.425674\pi$
$500$	0	0
$501$	5.43441	0.242791
$502$	− 3.98246i	− 0.177746i
$503$	− 16.9084i	− 0.753908i	−0.926232	−	0.376954i	$-0.876971\pi$
$503$	− 16.9084i	− 0.753908i	0.926232	−	0.376954i	$-0.123029\pi$
$504$	0.511170	0.0227693
$505$	0	0
$506$	−18.9647	−0.843084
$507$	− 5.95532i	− 0.264485i
$508$	3.62677i	0.160912i
$509$	−6.48318	−0.287362	−0.143681	−	0.989624i	$-0.545894\pi$
$509$	−6.48318	−0.287362	−0.143681	+	0.989624i	$0.545894\pi$
$510$	0	0
$511$	−0.720726	−0.0318831
$512$	− 1.00000i	− 0.0441942i
$513$	− 0.374409i	− 0.0165306i
$514$	12.5114	0.551856
$515$	0	0
$516$	−5.87802	−0.258766
$517$	− 27.0140i	− 1.18807i
$518$	5.29456i	0.232629i
$519$	−1.14183	−0.0501208
$520$	0	0
$521$	21.6371	0.947939	0.473970	−	0.880541i	$-0.342821\pi$
$521$	21.6371	0.947939	0.473970	+	0.880541i	$0.342821\pi$
$522$	3.89025i	0.170272i
$523$	− 15.6560i	− 0.684589i	−0.939593	−	0.342295i	$-0.888796\pi$
$523$	− 15.6560i	− 0.684589i	0.939593	−	0.342295i	$-0.111204\pi$
$524$	17.4131	0.760695
$525$	0	0
$526$	2.25374	0.0982677
$527$	− 10.0342i	− 0.437098i
$528$	− 3.29456i	− 0.143377i
$529$	−10.1359	−0.440689
$530$	0	0
$531$	−7.62993	−0.331111
$532$	0.191387i	0.00829767i
$533$	− 6.20358i	− 0.268707i
$534$	3.42567	0.148243
$535$	0	0
$536$	1.78806	0.0772323
$537$	− 15.4030i	− 0.664689i
$538$	25.4934i	1.09910i
$539$	22.2010	0.956267
$540$	0	0
$541$	14.2839	0.614113	0.307057	−	0.951691i	$-0.400656\pi$
$541$	14.2839	0.614113	0.307057	+	0.951691i	$0.400656\pi$
$542$	19.2503i	0.826872i
$543$	− 20.5110i	− 0.880213i
$544$	5.57433	0.238997
$545$	0	0
$546$	−1.35674	−0.0580631
$547$	42.8226i	1.83096i	0.402361	+	0.915481i	$0.368190\pi$
$547$	42.8226i	1.83096i	−0.402361	+	0.915481i	$0.631810\pi$
$548$	− 19.7230i	− 0.842524i
$549$	−2.90345	−0.123916
$550$	0	0
$551$	−1.45654	−0.0620509
$552$	− 5.75638i	− 0.245008i
$553$	6.20536i	0.263879i
$554$	7.73948	0.328819
$555$	0	0
$556$	11.4357	0.484980
$557$	14.5543i	0.616688i	0.951275	+	0.308344i	$0.0997747\pi$
$557$	14.5543i	0.616688i	−0.951275	+	0.308344i	$0.900225\pi$
$558$	− 1.80008i	− 0.0762035i
$559$	15.6013	0.659866
$560$	0	0
$561$	18.3649	0.775368
$562$	10.2724i	0.433316i
$563$	− 4.52778i	− 0.190823i	−0.995438	−	0.0954116i	$-0.969583\pi$
$563$	− 4.52778i	− 0.190823i	0.995438	−	0.0954116i	$-0.0304167\pi$
$564$	8.19959	0.345265
$565$	0	0
$566$	−9.86811	−0.414788
$567$	0.511170i	0.0214671i
$568$	7.27786i	0.305372i
$569$	−10.0904	−0.423010	−0.211505	−	0.977377i	$-0.567837\pi$
$569$	−10.0904	−0.423010	−0.211505	+	0.977377i	$0.567837\pi$
$570$	0	0
$571$	−20.3024	−0.849630	−0.424815	−	0.905280i	$-0.639661\pi$
$571$	−20.3024	−0.849630	−0.424815	+	0.905280i	$0.639661\pi$
$572$	8.74435i	0.365620i
$573$	− 7.27380i	− 0.303867i
$574$	1.19475	0.0498679
$575$	0	0
$576$	1.00000	0.0416667
$577$	19.3900i	0.807218i	0.914932	+	0.403609i	$0.132244\pi$
$577$	19.3900i	0.807218i	−0.914932	+	0.403609i	$0.867756\pi$
$578$	14.0731i	0.585366i
$579$	−11.4686	−0.476621
$580$	0	0
$581$	1.58108	0.0655942
$582$	15.1483i	0.627918i
$583$	− 1.70898i	− 0.0707786i
$584$	−1.40995	−0.0583443
$585$	0	0
$586$	−3.08103	−0.127276
$587$	− 43.8857i	− 1.81136i	−0.423964	−	0.905679i	$-0.639362\pi$
$587$	− 43.8857i	− 1.81136i	0.423964	−	0.905679i	$-0.360638\pi$
$588$	6.73870i	0.277900i
$589$	0.673966	0.0277703
$590$	0	0
$591$	−2.16068	−0.0888786
$592$	10.3577i	0.425699i
$593$	27.5651i	1.13196i	0.824418	+	0.565981i	$0.191502\pi$
$593$	27.5651i	1.13196i	−0.824418	+	0.565981i	$0.808498\pi$
$594$	3.29456	0.135177
$595$	0	0
$596$	20.5095	0.840103
$597$	− 7.27998i	− 0.297950i
$598$	15.2785i	0.624783i
$599$	−13.3970	−0.547385	−0.273692	−	0.961817i	$-0.588245\pi$
$599$	−13.3970	−0.547385	−0.273692	+	0.961817i	$0.588245\pi$
$600$	0	0
$601$	−32.3401	−1.31918	−0.659590	−	0.751625i	$-0.729270\pi$
$601$	−32.3401	−1.31918	−0.659590	+	0.751625i	$0.729270\pi$
$602$	3.00467i	0.122461i
$603$	1.78806i	0.0728153i
$604$	−6.22542	−0.253309
$605$	0	0
$606$	−5.58091	−0.226709
$607$	− 18.7755i	− 0.762074i	−0.924560	−	0.381037i	$-0.875567\pi$
$607$	− 18.7755i	− 0.762074i	0.924560	−	0.381037i	$-0.124433\pi$
$608$	0.374409i	0.0151843i
$609$	1.98858	0.0805813
$610$	0	0
$611$	−21.7632	−0.880445
$612$	5.57433i	0.225329i
$613$	5.12696i	0.207076i	0.994625	+	0.103538i	$0.0330164\pi$
$613$	5.12696i	0.207076i	−0.994625	+	0.103538i	$0.966984\pi$
$614$	6.43192	0.259571
$615$	0	0
$616$	−1.68408	−0.0678535
$617$	27.9234i	1.12415i	0.827085	+	0.562077i	$0.189998\pi$
$617$	27.9234i	1.12415i	−0.827085	+	0.562077i	$0.810002\pi$
$618$	15.1228i	0.608329i
$619$	36.9317	1.48441	0.742205	−	0.670172i	$-0.233780\pi$
$619$	36.9317	1.48441	0.742205	+	0.670172i	$0.233780\pi$
$620$	0	0
$621$	5.75638	0.230995
$622$	− 20.8818i	− 0.837282i
$623$	− 1.75110i	− 0.0701564i
$624$	−2.65418	−0.106252
$625$	0	0
$626$	11.7245	0.468605
$627$	1.23351i	0.0492617i
$628$	23.9795i	0.956886i
$629$	−57.7373	−2.30214
$630$	0	0
$631$	−7.04589	−0.280492	−0.140246	−	0.990117i	$-0.544789\pi$
$631$	−7.04589	−0.280492	−0.140246	+	0.990117i	$0.544789\pi$
$632$	12.1395i	0.482884i
$633$	28.7670i	1.14338i
$634$	−5.92233	−0.235206
$635$	0	0
$636$	0.518727	0.0205689
$637$	− 17.8857i	− 0.708659i
$638$	− 12.8166i	− 0.507416i
$639$	−7.27786	−0.287908
$640$	0	0
$641$	−41.2092	−1.62767	−0.813834	−	0.581098i	$-0.802623\pi$
$641$	−41.2092	−1.62767	−0.813834	+	0.581098i	$0.802623\pi$
$642$	− 18.7290i	− 0.739176i
$643$	− 4.40938i	− 0.173889i	−0.996213	−	0.0869444i	$-0.972290\pi$
$643$	− 4.40938i	− 0.173889i	0.996213	−	0.0869444i	$-0.0277103\pi$
$644$	−2.94249	−0.115950
$645$	0	0
$646$	−2.08708	−0.0821150
$647$	12.8764i	0.506223i	0.967437	+	0.253111i	$0.0814539\pi$
$647$	12.8764i	0.506223i	−0.967437	+	0.253111i	$0.918546\pi$
$648$	1.00000i	0.0392837i
$649$	25.1372	0.986723
$650$	0	0
$651$	−0.920147	−0.0360634
$652$	24.1232i	0.944738i
$653$	33.9475i	1.32847i	0.747525	+	0.664234i	$0.231242\pi$
$653$	33.9475i	1.32847i	−0.747525	+	0.664234i	$0.768758\pi$
$654$	−18.5221	−0.724273
$655$	0	0
$656$	2.33728	0.0912556
$657$	− 1.40995i	− 0.0550075i
$658$	− 4.19139i	− 0.163397i
$659$	12.6089	0.491171	0.245586	−	0.969375i	$-0.421020\pi$
$659$	12.6089	0.491171	0.245586	+	0.969375i	$0.421020\pi$
$660$	0	0
$661$	14.6188	0.568607	0.284304	−	0.958734i	$-0.408238\pi$
$661$	14.6188	0.568607	0.284304	+	0.958734i	$0.408238\pi$
$662$	− 27.3753i	− 1.06397i
$663$	− 14.7953i	− 0.574601i
$664$	3.09306	0.120034
$665$	0	0
$666$	−10.3577	−0.401353
$667$	− 22.3937i	− 0.867089i
$668$	− 5.43441i	− 0.210264i
$669$	0.662342	0.0256076
$670$	0	0
$671$	9.56559	0.369276
$672$	− 0.511170i	− 0.0197188i
$673$	− 13.4341i	− 0.517847i	−0.965898	−	0.258924i	$-0.916632\pi$
$673$	− 13.4341i	− 0.517847i	0.965898	−	0.258924i	$-0.0833678\pi$
$674$	5.86677	0.225980
$675$	0	0
$676$	−5.95532	−0.229051
$677$	31.9199i	1.22678i	0.789780	+	0.613391i	$0.210195\pi$
$677$	31.9199i	1.22678i	−0.789780	+	0.613391i	$0.789805\pi$
$678$	16.1273i	0.619365i
$679$	7.74338	0.297163
$680$	0	0
$681$	−15.5730	−0.596760
$682$	5.93046i	0.227089i
$683$	19.0698i	0.729685i	0.931069	+	0.364842i	$0.118877\pi$
$683$	19.0698i	0.729685i	−0.931069	+	0.364842i	$0.881123\pi$
$684$	−0.374409	−0.0143159
$685$	0	0
$686$	7.02282	0.268132
$687$	− 27.7376i	− 1.05826i
$688$	5.87802i	0.224098i
$689$	−1.37680	−0.0524518
$690$	0	0
$691$	21.5074	0.818181	0.409091	−	0.912494i	$-0.365846\pi$
$691$	21.5074	0.818181	0.409091	+	0.912494i	$0.365846\pi$
$692$	1.14183i	0.0434059i
$693$	− 1.68408i	− 0.0639729i
$694$	−11.8634	−0.450330
$695$	0	0
$696$	3.89025	0.147460
$697$	13.0288i	0.493501i
$698$	28.3391i	1.07265i
$699$	23.2480	0.879320
$700$	0	0
$701$	−5.65381	−0.213541	−0.106771	−	0.994284i	$-0.534051\pi$
$701$	−5.65381	−0.213541	−0.106771	+	0.994284i	$0.534051\pi$
$702$	− 2.65418i	− 0.100176i
$703$	− 3.87802i	− 0.146262i
$704$	−3.29456	−0.124168
$705$	0	0
$706$	5.83404	0.219567
$707$	2.85280i	0.107290i
$708$	7.62993i	0.286750i
$709$	−43.7198	−1.64193	−0.820967	−	0.570976i	$-0.806565\pi$
$709$	−43.7198	−1.64193	−0.820967	+	0.570976i	$0.806565\pi$
$710$	0	0
$711$	−12.1395	−0.455268
$712$	− 3.42567i	− 0.128382i
$713$	10.3619i	0.388058i
$714$	2.84943	0.106637
$715$	0	0
$716$	−15.4030	−0.575637
$717$	5.81933i	0.217327i
$718$	9.94932i	0.371305i
$719$	8.09200	0.301781	0.150890	−	0.988551i	$-0.451786\pi$
$719$	8.09200	0.301781	0.150890	+	0.988551i	$0.451786\pi$
$720$	0	0
$721$	7.73034	0.287893
$722$	18.8598i	0.701890i
$723$	− 14.1483i	− 0.526182i
$724$	−20.5110	−0.762287
$725$	0	0
$726$	0.145898	0.00541478
$727$	47.3889i	1.75756i	0.477228	+	0.878779i	$0.341641\pi$
$727$	47.3889i	1.75756i	−0.477228	+	0.878779i	$0.658359\pi$
$728$	1.35674i	0.0502841i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−32.7660	−1.21190
$732$	2.90345i	0.107315i
$733$	14.6145i	0.539801i	0.962888	+	0.269900i	$0.0869907\pi$
$733$	14.6145i	0.539801i	−0.962888	+	0.269900i	$0.913009\pi$
$734$	−16.2990	−0.601608
$735$	0	0
$736$	−5.75638	−0.212183
$737$	− 5.89085i	− 0.216992i
$738$	2.33728i	0.0860366i
$739$	12.1592	0.447283	0.223641	−	0.974672i	$-0.428206\pi$
$739$	12.1592	0.447283	0.223641	+	0.974672i	$0.428206\pi$
$740$	0	0
$741$	0.993750	0.0365063
$742$	− 0.265158i	− 0.00973426i
$743$	50.5368i	1.85402i	0.375041	+	0.927008i	$0.377629\pi$
$743$	50.5368i	1.85402i	−0.375041	+	0.927008i	$0.622371\pi$
$744$	−1.80008	−0.0659941
$745$	0	0
$746$	−8.85626	−0.324251
$747$	3.09306i	0.113169i
$748$	− 18.3649i	− 0.671489i
$749$	−9.57373	−0.349816
$750$	0	0
$751$	33.9598	1.23921	0.619605	−	0.784914i	$-0.287293\pi$
$751$	33.9598	1.23921	0.619605	+	0.784914i	$0.287293\pi$
$752$	− 8.19959i	− 0.299008i
$753$	− 3.98246i	− 0.145129i
$754$	−10.3254	−0.376030
$755$	0	0
$756$	0.511170	0.0185911
$757$	− 47.9954i	− 1.74442i	−0.489130	−	0.872211i	$-0.662685\pi$
$757$	− 47.9954i	− 1.74442i	0.489130	−	0.872211i	$-0.337315\pi$
$758$	− 14.1431i	− 0.513699i
$759$	−18.9647	−0.688375
$760$	0	0
$761$	27.6622	1.00275	0.501377	−	0.865229i	$-0.332827\pi$
$761$	27.6622	1.00275	0.501377	+	0.865229i	$0.332827\pi$
$762$	3.62677i	0.131384i
$763$	9.46796i	0.342763i
$764$	−7.27380	−0.263157
$765$	0	0
$766$	32.9848	1.19179
$767$	− 20.2512i	− 0.731229i
$768$	− 1.00000i	− 0.0360844i
$769$	15.0664	0.543310	0.271655	−	0.962395i	$-0.412429\pi$
$769$	15.0664	0.543310	0.271655	+	0.962395i	$0.412429\pi$
$770$	0	0
$771$	12.5114	0.450589
$772$	11.4686i	0.412766i
$773$	29.4391i	1.05885i	0.848356	+	0.529426i	$0.177593\pi$
$773$	29.4391i	1.05885i	−0.848356	+	0.529426i	$0.822407\pi$
$774$	−5.87802	−0.211281
$775$	0	0
$776$	15.1483	0.543793
$777$	5.29456i	0.189941i
$778$	11.4919i	0.412005i
$779$	−0.875101	−0.0313537
$780$	0	0
$781$	23.9773	0.857976
$782$	− 32.0879i	− 1.14746i
$783$	3.89025i	0.139026i
$784$	6.73870	0.240668
$785$	0	0
$786$	17.4131	0.621105
$787$	− 17.7230i	− 0.631756i	−0.948800	−	0.315878i	$-0.897701\pi$
$787$	− 17.7230i	− 0.631756i	0.948800	−	0.315878i	$-0.102299\pi$
$788$	2.16068i	0.0769711i
$789$	2.25374	0.0802352
$790$	0	0
$791$	8.24379	0.293116
$792$	− 3.29456i	− 0.117067i
$793$	− 7.70629i	− 0.273659i
$794$	4.82958	0.171395
$795$	0	0
$796$	−7.27998	−0.258032
$797$	− 24.6949i	− 0.874737i	−0.899282	−	0.437368i	$-0.855911\pi$
$797$	− 24.6949i	− 0.874737i	0.899282	−	0.437368i	$-0.144089\pi$
$798$	0.191387i	0.00677502i
$799$	45.7072	1.61700
$800$	0	0
$801$	3.42567	0.121040
$802$	21.5779i	0.761944i
$803$	4.64517i	0.163925i
$804$	1.78806	0.0630599
$805$	0	0
$806$	4.77774	0.168289
$807$	25.4934i	0.897409i
$808$	5.58091i	0.196336i
$809$	16.4366	0.577879	0.288939	−	0.957347i	$-0.406697\pi$
$809$	16.4366	0.577879	0.288939	+	0.957347i	$0.406697\pi$
$810$	0	0
$811$	−12.9773	−0.455696	−0.227848	−	0.973697i	$-0.573169\pi$
$811$	−12.9773	−0.455696	−0.227848	+	0.973697i	$0.573169\pi$
$812$	− 1.98858i	− 0.0697855i
$813$	19.2503i	0.675138i
$814$	34.1241	1.19605
$815$	0	0
$816$	5.57433	0.195141
$817$	− 2.20078i	− 0.0769957i
$818$	− 1.57998i	− 0.0552426i
$819$	−1.35674	−0.0474083
$820$	0	0
$821$	−31.7784	−1.10907	−0.554537	−	0.832159i	$-0.687105\pi$
$821$	−31.7784	−1.10907	−0.554537	+	0.832159i	$0.687105\pi$
$822$	− 19.7230i	− 0.687918i
$823$	38.3742i	1.33764i	0.743425	+	0.668820i	$0.233200\pi$
$823$	38.3742i	1.33764i	−0.743425	+	0.668820i	$0.766800\pi$
$824$	15.1228	0.526829
$825$	0	0
$826$	3.90019	0.135705
$827$	− 23.0395i	− 0.801162i	−0.916261	−	0.400581i	$-0.868808\pi$
$827$	− 23.0395i	− 0.801162i	0.916261	−	0.400581i	$-0.131192\pi$
$828$	− 5.75638i	− 0.200048i
$829$	17.5852	0.610758	0.305379	−	0.952231i	$-0.401217\pi$
$829$	17.5852	0.610758	0.305379	+	0.952231i	$0.401217\pi$
$830$	0	0
$831$	7.73948	0.268479
$832$	2.65418i	0.0920172i
$833$	37.5638i	1.30151i
$834$	11.4357	0.395984
$835$	0	0
$836$	1.23351	0.0426619
$837$	− 1.80008i	− 0.0622199i
$838$	− 36.7568i	− 1.26974i
$839$	44.0619	1.52119	0.760593	−	0.649229i	$-0.224908\pi$
$839$	44.0619	1.52119	0.760593	+	0.649229i	$0.224908\pi$
$840$	0	0
$841$	−13.8660	−0.478136
$842$	− 6.96946i	− 0.240183i
$843$	10.2724i	0.353801i
$844$	28.7670	0.990200
$845$	0	0
$846$	8.19959	0.281908
$847$	− 0.0745787i	− 0.00256256i
$848$	− 0.518727i	− 0.0178132i
$849$	−9.86811	−0.338673
$850$	0	0
$851$	59.6229	2.04385
$852$	7.27786i	0.249335i
$853$	38.1231i	1.30531i	0.757656	+	0.652655i	$0.226345\pi$
$853$	38.1231i	1.30531i	−0.757656	+	0.652655i	$0.773655\pi$
$854$	1.48416	0.0507869
$855$	0	0
$856$	−18.7290	−0.640145
$857$	− 40.2837i	− 1.37607i	−0.725679	−	0.688033i	$-0.758474\pi$
$857$	− 40.2837i	− 1.37607i	0.725679	−	0.688033i	$-0.241526\pi$
$858$	8.74435i	0.298527i
$859$	16.9314	0.577691	0.288845	−	0.957376i	$-0.406729\pi$
$859$	16.9314	0.577691	0.288845	+	0.957376i	$0.406729\pi$
$860$	0	0
$861$	1.19475	0.0407170
$862$	− 25.4098i	− 0.865462i
$863$	38.3696i	1.30612i	0.757308	+	0.653058i	$0.226514\pi$
$863$	38.3696i	1.30612i	−0.757308	+	0.653058i	$0.773486\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	13.4595	0.457373
$867$	14.0731i	0.477949i
$868$	0.920147i	0.0312318i
$869$	39.9943	1.35672
$870$	0	0
$871$	−4.74583	−0.160806
$872$	18.5221i	0.627239i
$873$	15.1483i	0.512693i
$874$	2.15524	0.0729020
$875$	0	0
$876$	−1.40995	−0.0476379
$877$	9.59661i	0.324055i	0.986786	+	0.162027i	$0.0518033\pi$
$877$	9.59661i	0.324055i	−0.986786	+	0.162027i	$0.948197\pi$
$878$	− 37.8392i	− 1.27701i
$879$	−3.08103	−0.103921
$880$	0	0
$881$	21.0921	0.710610	0.355305	−	0.934750i	$-0.384377\pi$
$881$	21.0921	0.710610	0.355305	+	0.934750i	$0.384377\pi$
$882$	6.73870i	0.226904i
$883$	1.26807i	0.0426740i	0.999772	+	0.0213370i	$0.00679229\pi$
$883$	1.26807i	0.0426740i	−0.999772	+	0.0213370i	$0.993208\pi$
$884$	−14.7953	−0.497619
$885$	0	0
$886$	−2.70138	−0.0907545
$887$	− 3.50065i	− 0.117540i	−0.998272	−	0.0587701i	$-0.981282\pi$
$887$	− 3.50065i	− 0.117540i	0.998272	−	0.0587701i	$-0.0187179\pi$
$888$	10.3577i	0.347582i
$889$	1.85390	0.0621777
$890$	0	0
$891$	3.29456	0.110372
$892$	− 0.662342i	− 0.0221769i
$893$	3.07000i	0.102734i
$894$	20.5095	0.685941
$895$	0	0
$896$	−0.511170	−0.0170770
$897$	15.2785i	0.510133i
$898$	18.9478i	0.632295i
$899$	−7.00276	−0.233555
$900$	0	0
$901$	2.89156	0.0963317
$902$	− 7.70032i	− 0.256393i
$903$	3.00467i	0.0999892i
$904$	16.1273	0.536386
$905$	0	0
$906$	−6.22542	−0.206826
$907$	− 20.2610i	− 0.672757i	−0.941727	−	0.336378i	$-0.890798\pi$
$907$	− 20.2610i	− 0.672757i	0.941727	−	0.336378i	$-0.109202\pi$
$908$	15.5730i	0.516809i
$909$	−5.58091	−0.185107
$910$	0	0
$911$	−5.90220	−0.195549	−0.0977744	−	0.995209i	$-0.531172\pi$
$911$	−5.90220	−0.195549	−0.0977744	+	0.995209i	$0.531172\pi$
$912$	0.374409i	0.0123979i
$913$	− 10.1902i	− 0.337248i
$914$	−38.2879	−1.26645
$915$	0	0
$916$	−27.7376	−0.916478
$917$	− 8.90107i	− 0.293939i
$918$	5.57433i	0.183980i
$919$	45.3645	1.49644	0.748219	−	0.663452i	$-0.230909\pi$
$919$	45.3645	1.49644	0.748219	+	0.663452i	$0.230909\pi$
$920$	0	0
$921$	6.43192	0.211939
$922$	− 17.3427i	− 0.571152i
$923$	− 19.3168i	− 0.635819i
$924$	−1.68408	−0.0554021
$925$	0	0
$926$	−17.6641	−0.580479
$927$	15.1228i	0.496699i
$928$	− 3.89025i	− 0.127704i
$929$	−46.2829	−1.51849	−0.759247	−	0.650803i	$-0.774432\pi$
$929$	−46.2829	−1.51849	−0.759247	+	0.650803i	$0.774432\pi$
$930$	0	0
$931$	−2.52303	−0.0826890
$932$	− 23.2480i	− 0.761514i
$933$	− 20.8818i	− 0.683638i
$934$	−2.88972	−0.0945544
$935$	0	0
$936$	−2.65418	−0.0867546
$937$	21.2179i	0.693158i	0.938021	+	0.346579i	$0.112657\pi$
$937$	21.2179i	0.693158i	−0.938021	+	0.346579i	$0.887343\pi$
$938$	− 0.914001i	− 0.0298432i
$939$	11.7245	0.382615
$940$	0	0
$941$	−43.4850	−1.41757	−0.708784	−	0.705425i	$-0.750756\pi$
$941$	−43.4850	−1.41757	−0.708784	+	0.705425i	$0.750756\pi$
$942$	23.9795i	0.781294i
$943$	− 13.4543i	− 0.438132i
$944$	7.62993	0.248333
$945$	0	0
$946$	19.3655	0.629626
$947$	− 34.6211i	− 1.12504i	−0.826785	−	0.562518i	$-0.809833\pi$
$947$	− 34.6211i	− 1.12504i	0.826785	−	0.562518i	$-0.190167\pi$
$948$	12.1395i	0.394273i
$949$	3.74227	0.121479
$950$	0	0
$951$	−5.92233	−0.192045
$952$	− 2.84943i	− 0.0923506i
$953$	21.1695i	0.685747i	0.939382	+	0.342874i	$0.111400\pi$
$953$	21.1695i	0.685747i	−0.939382	+	0.342874i	$0.888600\pi$
$954$	0.518727	0.0167944
$955$	0	0
$956$	5.81933	0.188211
$957$	− 12.8166i	− 0.414303i
$958$	27.5859i	0.891260i
$959$	−10.0818	−0.325558
$960$	0	0
$961$	−27.7597	−0.895475
$962$	− 27.4913i	− 0.886354i
$963$	− 18.7290i	− 0.603535i
$964$	−14.1483	−0.455687
$965$	0	0
$966$	−2.94249	−0.0946730
$967$	43.1011i	1.38604i	0.720919	+	0.693020i	$0.243720\pi$
$967$	43.1011i	1.38604i	−0.720919	+	0.693020i	$0.756280\pi$
$968$	− 0.145898i	− 0.00468934i
$969$	−2.08708	−0.0670466
$970$	0	0
$971$	−32.5128	−1.04338	−0.521692	−	0.853134i	$-0.674699\pi$
$971$	−32.5128	−1.04338	−0.521692	+	0.853134i	$0.674699\pi$
$972$	1.00000i	0.0320750i
$973$	− 5.84557i	− 0.187400i
$974$	14.9338	0.478508
$975$	0	0
$976$	2.90345	0.0929373
$977$	56.2425i	1.79936i	0.436552	+	0.899679i	$0.356200\pi$
$977$	56.2425i	1.79936i	−0.436552	+	0.899679i	$0.643800\pi$
$978$	24.1232i	0.771376i
$979$	−11.2861	−0.360704
$980$	0	0
$981$	−18.5221	−0.591366
$982$	− 26.4452i	− 0.843900i
$983$	− 31.7227i	− 1.01180i	−0.862593	−	0.505898i	$-0.831161\pi$
$983$	− 31.7227i	− 1.01180i	0.862593	−	0.505898i	$-0.168839\pi$
$984$	2.33728	0.0745099
$985$	0	0
$986$	21.6855	0.690608
$987$	− 4.19139i	− 0.133413i
$988$	− 0.993750i	− 0.0316154i
$989$	33.8361	1.07593
$990$	0	0
$991$	15.6911	0.498445	0.249222	−	0.968446i	$-0.419825\pi$
$991$	15.6911	0.498445	0.249222	+	0.968446i	$0.419825\pi$
$992$	1.80008i	0.0571526i
$993$	− 27.3753i	− 0.868728i
$994$	3.72023	0.117998
$995$	0	0
$996$	3.09306	0.0980072
$997$	34.2809i	1.08569i	0.839834	+	0.542843i	$0.182652\pi$
$997$	34.2809i	1.08569i	−0.839834	+	0.542843i	$0.817348\pi$
$998$	− 10.3375i	− 0.327229i
$999$	−10.3577	−0.327704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.c.g.1249.3		8
5.2	odd	4		3750.2.a.p.1.2	yes	4
5.3	odd	4		3750.2.a.n.1.3	✓	4
5.4	even	2	inner	3750.2.c.g.1249.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3750.2.a.n.1.3	✓	4	5.3	odd	4
3750.2.a.p.1.2	yes	4	5.2	odd	4
3750.2.c.g.1249.3		8	1.1	even	1	trivial
3750.2.c.g.1249.6		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +0.511170i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.29456 q^{11} +1.00000i q^{12} -2.65418i q^{13} +0.511170 q^{14} +1.00000 q^{16} +5.57433i q^{17} +1.00000i q^{18} -0.374409 q^{19} +0.511170 q^{21} -3.29456i q^{22} -5.75638i q^{23} +1.00000 q^{24} -2.65418 q^{26} +1.00000i q^{27} -0.511170i q^{28} +3.89025 q^{29} -1.80008 q^{31} -1.00000i q^{32} -3.29456i q^{33} +5.57433 q^{34} +1.00000 q^{36} +10.3577i q^{37} +0.374409i q^{38} -2.65418 q^{39} +2.33728 q^{41} -0.511170i q^{42} +5.87802i q^{43} -3.29456 q^{44} -5.75638 q^{46} -8.19959i q^{47} -1.00000i q^{48} +6.73870 q^{49} +5.57433 q^{51} +2.65418i q^{52} -0.518727i q^{53} +1.00000 q^{54} -0.511170 q^{56} +0.374409i q^{57} -3.89025i q^{58} +7.62993 q^{59} +2.90345 q^{61} +1.80008i q^{62} -0.511170i q^{63} -1.00000 q^{64} -3.29456 q^{66} -1.78806i q^{67} -5.57433i q^{68} -5.75638 q^{69} +7.27786 q^{71} -1.00000i q^{72} +1.40995i q^{73} +10.3577 q^{74} +0.374409 q^{76} +1.68408i q^{77} +2.65418i q^{78} +12.1395 q^{79} +1.00000 q^{81} -2.33728i q^{82} -3.09306i q^{83} -0.511170 q^{84} +5.87802 q^{86} -3.89025i q^{87} +3.29456i q^{88} -3.42567 q^{89} +1.35674 q^{91} +5.75638i q^{92} +1.80008i q^{93} -8.19959 q^{94} -1.00000 q^{96} -15.1483i q^{97} -6.73870i q^{98} -3.29456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{6} - 8 q^{9} + 8 q^{14} + 8 q^{16} + 22 q^{19} + 8 q^{21} + 8 q^{24} + 4 q^{26} - 12 q^{29} - 16 q^{31} + 18 q^{34} + 8 q^{36} + 4 q^{39} + 12 q^{41} - 30 q^{46} + 8 q^{49} + 18 q^{51}+ \cdots - 8 q^{96}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^6 - 8 * q^9 + 8 * q^14 + 8 * q^16 + 22 * q^19 + 8 * q^21 + 8 * q^24 + 4 * q^26 - 12 * q^29 - 16 * q^31 + 18 * q^34 + 8 * q^36 + 4 * q^39 + 12 * q^41 - 30 * q^46 + 8 * q^49 + 18 * q^51 + 8 * q^54 - 8 * q^56 - 30 * q^59 - 10 * q^61 - 8 * q^64 - 30 * q^69 + 26 * q^74 - 22 * q^76 + 46 * q^79 + 8 * q^81 - 8 * q^84 - 8 * q^86 - 54 * q^89 - 44 * q^91 + 30 * q^94 - 8 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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