LMFDB - Embedded newform 3042.2.b.r.1351.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, 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("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (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:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1014)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.5
Root			$1.80194i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.r.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.356896i q^{5} +4.04892i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +0.356896i q^{5} +4.04892i q^{7} -1.00000i q^{8} -0.356896 q^{10} +0.911854i q^{11} -4.04892 q^{14} +1.00000 q^{16} -2.09783 q^{17} +4.98792i q^{19} -0.356896i q^{20} -0.911854 q^{22} +8.49396 q^{23} +4.87263 q^{25} -4.04892i q^{28} -8.51573 q^{29} +10.7899i q^{31} +1.00000i q^{32} -2.09783i q^{34} -1.44504 q^{35} -0.615957i q^{37} -4.98792 q^{38} +0.356896 q^{40} -7.60388i q^{41} +6.27413 q^{43} -0.911854i q^{44} +8.49396i q^{46} -1.78017i q^{47} -9.39373 q^{49} +4.87263i q^{50} -10.4112 q^{53} -0.325437 q^{55} +4.04892 q^{56} -8.51573i q^{58} +6.04892i q^{59} -3.10992 q^{61} -10.7899 q^{62} -1.00000 q^{64} -13.5797i q^{67} +2.09783 q^{68} -1.44504i q^{70} +11.4819i q^{71} +0.533188i q^{73} +0.615957 q^{74} -4.98792i q^{76} -3.69202 q^{77} -11.7071 q^{79} +0.356896i q^{80} +7.60388 q^{82} -6.49934i q^{83} -0.748709i q^{85} +6.27413i q^{86} +0.911854 q^{88} +6.49396i q^{89} -8.49396 q^{92} +1.78017 q^{94} -1.78017 q^{95} +1.96077i q^{97} -9.39373i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4}+O(q^{10}) $ 6 * q - 6 * q^4	$ 6 q - 6 q^{4} + 6 q^{10} - 6 q^{14} + 6 q^{16} + 24 q^{17} + 2 q^{22} + 32 q^{23} - 4 q^{25} - 26 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} + 8 q^{49} - 30 q^{53} - 44 q^{55} + 6 q^{56} - 20 q^{61} - 18 q^{62} - 6 q^{64} - 24 q^{68} + 24 q^{74} - 12 q^{77} - 10 q^{79} + 28 q^{82} - 2 q^{88} - 32 q^{92} + 8 q^{94} - 8 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 + 6 * q^10 - 6 * q^14 + 6 * q^16 + 24 * q^17 + 2 * q^22 + 32 * q^23 - 4 * q^25 - 26 * q^29 - 8 * q^35 + 8 * q^38 - 6 * q^40 + 16 * q^43 + 8 * q^49 - 30 * q^53 - 44 * q^55 + 6 * q^56 - 20 * q^61 - 18 * q^62 - 6 * q^64 - 24 * q^68 + 24 * q^74 - 12 * q^77 - 10 * q^79 + 28 * q^82 - 2 * q^88 - 32 * q^92 + 8 * q^94 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0.356896i	0.159609i	0.996811	+	0.0798043i	$0.0254296\pi$
$5$	0.356896i	0.159609i	−0.996811	+	0.0798043i	$0.974570\pi$
$6$	0	0
$7$	4.04892i	1.53035i	0.643824	+	0.765173i	$0.277347\pi$
$7$	4.04892i	1.53035i	−0.643824	+	0.765173i	$0.722653\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	−0.356896	−0.112860
$11$	0.911854i	0.274934i	0.990506	+	0.137467i	$0.0438962\pi$
$11$	0.911854i	0.274934i	−0.990506	+	0.137467i	$0.956104\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−4.04892	−1.08212
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.09783	−0.508800	−0.254400	−	0.967099i	$-0.581878\pi$
$17$	−2.09783	−0.508800	−0.254400	+	0.967099i	$0.581878\pi$
$18$	0	0
$19$	4.98792i	1.14431i	0.820147	+	0.572153i	$0.193892\pi$
$19$	4.98792i	1.14431i	−0.820147	+	0.572153i	$0.806108\pi$
$20$	− 0.356896i	− 0.0798043i
$21$	0	0
$22$	−0.911854	−0.194408
$23$	8.49396	1.77111	0.885556	−	0.464532i	$-0.153777\pi$
$23$	8.49396	1.77111	0.885556	+	0.464532i	$0.153777\pi$
$24$	0	0
$25$	4.87263	0.974525
$26$	0	0
$27$	0	0
$28$	− 4.04892i	− 0.765173i
$29$	−8.51573	−1.58133	−0.790666	−	0.612248i	$-0.790265\pi$
$29$	−8.51573	−1.58133	−0.790666	+	0.612248i	$0.790265\pi$
$30$	0	0
$31$	10.7899i	1.93792i	0.247227	+	0.968958i	$0.420481\pi$
$31$	10.7899i	1.93792i	−0.247227	+	0.968958i	$0.579519\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 2.09783i	− 0.359776i
$35$	−1.44504	−0.244257
$36$	0	0
$37$	− 0.615957i	− 0.101263i	−0.998717	−	0.0506314i	$-0.983877\pi$
$37$	− 0.615957i	− 0.101263i	0.998717	−	0.0506314i	$-0.0161234\pi$
$38$	−4.98792	−0.809147
$39$	0	0
$40$	0.356896	0.0564302
$41$	− 7.60388i	− 1.18753i	−0.804640	−	0.593763i	$-0.797642\pi$
$41$	− 7.60388i	− 1.18753i	0.804640	−	0.593763i	$-0.202358\pi$
$42$	0	0
$43$	6.27413	0.956795	0.478398	−	0.878143i	$-0.341218\pi$
$43$	6.27413	0.956795	0.478398	+	0.878143i	$0.341218\pi$
$44$	− 0.911854i	− 0.137467i
$45$	0	0
$46$	8.49396i	1.25237i
$47$	− 1.78017i	− 0.259664i	−0.991536	−	0.129832i	$-0.958556\pi$
$47$	− 1.78017i	− 0.259664i	0.991536	−	0.129832i	$-0.0414438\pi$
$48$	0	0
$49$	−9.39373	−1.34196
$50$	4.87263i	0.689093i
$51$	0	0
$52$	0	0
$53$	−10.4112	−1.43009	−0.715043	−	0.699080i	$-0.753593\pi$
$53$	−10.4112	−1.43009	−0.715043	+	0.699080i	$0.753593\pi$
$54$	0	0
$55$	−0.325437	−0.0438819
$56$	4.04892	0.541059
$57$	0	0
$58$	− 8.51573i	− 1.11817i
$59$	6.04892i	0.787502i	0.919217	+	0.393751i	$0.128823\pi$
$59$	6.04892i	0.787502i	−0.919217	+	0.393751i	$0.871177\pi$
$60$	0	0
$61$	−3.10992	−0.398184	−0.199092	−	0.979981i	$-0.563799\pi$
$61$	−3.10992	−0.398184	−0.199092	+	0.979981i	$0.563799\pi$
$62$	−10.7899	−1.37031
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 13.5797i	− 1.65903i	−0.558487	−	0.829513i	$-0.688618\pi$
$67$	− 13.5797i	− 1.65903i	0.558487	−	0.829513i	$-0.311382\pi$
$68$	2.09783	0.254400
$69$	0	0
$70$	− 1.44504i	− 0.172716i
$71$	11.4819i	1.36265i	0.731982	+	0.681324i	$0.238596\pi$
$71$	11.4819i	1.36265i	−0.731982	+	0.681324i	$0.761404\pi$
$72$	0	0
$73$	0.533188i	0.0624049i	0.999513	+	0.0312025i	$0.00993366\pi$
$73$	0.533188i	0.0624049i	−0.999513	+	0.0312025i	$0.990066\pi$
$74$	0.615957	0.0716036
$75$	0	0
$76$	− 4.98792i	− 0.572153i
$77$	−3.69202	−0.420745
$78$	0	0
$79$	−11.7071	−1.31715	−0.658575	−	0.752515i	$-0.728841\pi$
$79$	−11.7071	−1.31715	−0.658575	+	0.752515i	$0.728841\pi$
$80$	0.356896i	0.0399022i
$81$	0	0
$82$	7.60388	0.839708
$83$	− 6.49934i	− 0.713395i	−0.934220	−	0.356697i	$-0.883903\pi$
$83$	− 6.49934i	− 0.713395i	0.934220	−	0.356697i	$-0.116097\pi$
$84$	0	0
$85$	− 0.748709i	− 0.0812088i
$86$	6.27413i	0.676556i
$87$	0	0
$88$	0.911854	0.0972040
$89$	6.49396i	0.688358i	0.938904	+	0.344179i	$0.111843\pi$
$89$	6.49396i	0.688358i	−0.938904	+	0.344179i	$0.888157\pi$
$90$	0	0
$91$	0	0
$92$	−8.49396	−0.885556
$93$	0	0
$94$	1.78017	0.183610
$95$	−1.78017	−0.182641
$96$	0	0
$97$	1.96077i	0.199086i	0.995033	+	0.0995431i	$0.0317381\pi$
$97$	1.96077i	0.199086i	−0.995033	+	0.0995431i	$0.968262\pi$
$98$	− 9.39373i	− 0.948910i
$99$	0	0
$100$	−4.87263	−0.487263
$101$	6.98254	0.694789	0.347394	−	0.937719i	$-0.387067\pi$
$101$	6.98254	0.694789	0.347394	+	0.937719i	$0.387067\pi$
$102$	0	0
$103$	−4.94869	−0.487609	−0.243804	−	0.969824i	$-0.578396\pi$
$103$	−4.94869	−0.487609	−0.243804	+	0.969824i	$0.578396\pi$
$104$	0	0
$105$	0	0
$106$	− 10.4112i	− 1.01122i
$107$	4.26875	0.412676	0.206338	−	0.978481i	$-0.433845\pi$
$107$	4.26875	0.412676	0.206338	+	0.978481i	$0.433845\pi$
$108$	0	0
$109$	6.21983i	0.595752i	0.954605	+	0.297876i	$0.0962782\pi$
$109$	6.21983i	0.595752i	−0.954605	+	0.297876i	$0.903722\pi$
$110$	− 0.325437i	− 0.0310292i
$111$	0	0
$112$	4.04892i	0.382587i
$113$	−12.9879	−1.22180	−0.610900	−	0.791708i	$-0.709192\pi$
$113$	−12.9879	−1.22180	−0.610900	+	0.791708i	$0.709192\pi$
$114$	0	0
$115$	3.03146i	0.282685i
$116$	8.51573	0.790666
$117$	0	0
$118$	−6.04892	−0.556848
$119$	− 8.49396i	− 0.778640i
$120$	0	0
$121$	10.1685	0.924411
$122$	− 3.10992i	− 0.281559i
$123$	0	0
$124$	− 10.7899i	− 0.968958i
$125$	3.52350i	0.315151i
$126$	0	0
$127$	−9.22282	−0.818393	−0.409196	−	0.912446i	$-0.634191\pi$
$127$	−9.22282	−0.818393	−0.409196	+	0.912446i	$0.634191\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	14.5526	1.27146	0.635732	−	0.771910i	$-0.280698\pi$
$131$	14.5526	1.27146	0.635732	+	0.771910i	$0.280698\pi$
$132$	0	0
$133$	−20.1957	−1.75119
$134$	13.5797	1.17311
$135$	0	0
$136$	2.09783i	0.179888i
$137$	15.4034i	1.31600i	0.753017	+	0.658002i	$0.228598\pi$
$137$	15.4034i	1.31600i	−0.753017	+	0.658002i	$0.771402\pi$
$138$	0	0
$139$	2.71379	0.230181	0.115090	−	0.993355i	$-0.463284\pi$
$139$	2.71379	0.230181	0.115090	+	0.993355i	$0.463284\pi$
$140$	1.44504	0.122128
$141$	0	0
$142$	−11.4819	−0.963538
$143$	0	0
$144$	0	0
$145$	− 3.03923i	− 0.252394i
$146$	−0.533188	−0.0441269
$147$	0	0
$148$	0.615957i	0.0506314i
$149$	− 14.7356i	− 1.20718i	−0.797293	−	0.603592i	$-0.793736\pi$
$149$	− 14.7356i	− 1.20718i	0.797293	−	0.603592i	$-0.206264\pi$
$150$	0	0
$151$	− 15.8213i	− 1.28752i	−0.765227	−	0.643760i	$-0.777373\pi$
$151$	− 15.8213i	− 1.28752i	0.765227	−	0.643760i	$-0.222627\pi$
$152$	4.98792	0.404574
$153$	0	0
$154$	− 3.69202i	− 0.297512i
$155$	−3.85086	−0.309308
$156$	0	0
$157$	−4.27413	−0.341112	−0.170556	−	0.985348i	$-0.554556\pi$
$157$	−4.27413	−0.341112	−0.170556	+	0.985348i	$0.554556\pi$
$158$	− 11.7071i	− 0.931366i
$159$	0	0
$160$	−0.356896	−0.0282151
$161$	34.3913i	2.71042i
$162$	0	0
$163$	− 0.317667i	− 0.0248816i	−0.999923	−	0.0124408i	$-0.996040\pi$
$163$	− 0.317667i	− 0.0248816i	0.999923	−	0.0124408i	$-0.00396013\pi$
$164$	7.60388i	0.593763i
$165$	0	0
$166$	6.49934	0.504446
$167$	− 12.3612i	− 0.956539i	−0.878213	−	0.478269i	$-0.841264\pi$
$167$	− 12.3612i	− 0.956539i	0.878213	−	0.478269i	$-0.158736\pi$
$168$	0	0
$169$	0	0
$170$	0.748709	0.0574233
$171$	0	0
$172$	−6.27413	−0.478398
$173$	−17.0640	−1.29735	−0.648675	−	0.761065i	$-0.724677\pi$
$173$	−17.0640	−1.29735	−0.648675	+	0.761065i	$0.724677\pi$
$174$	0	0
$175$	19.7289i	1.49136i
$176$	0.911854i	0.0687336i
$177$	0	0
$178$	−6.49396	−0.486743
$179$	−24.9681	−1.86620	−0.933100	−	0.359616i	$-0.882908\pi$
$179$	−24.9681	−1.86620	−0.933100	+	0.359616i	$0.882908\pi$
$180$	0	0
$181$	−5.26205	−0.391125	−0.195562	−	0.980691i	$-0.562653\pi$
$181$	−5.26205	−0.391125	−0.195562	+	0.980691i	$0.562653\pi$
$182$	0	0
$183$	0	0
$184$	− 8.49396i	− 0.626183i
$185$	0.219833	0.0161624
$186$	0	0
$187$	− 1.91292i	− 0.139886i
$188$	1.78017i	0.129832i
$189$	0	0
$190$	− 1.78017i	− 0.129147i
$191$	−10.5375	−0.762467	−0.381233	−	0.924479i	$-0.624501\pi$
$191$	−10.5375	−0.762467	−0.381233	+	0.924479i	$0.624501\pi$
$192$	0	0
$193$	3.42758i	0.246723i	0.992362	+	0.123361i	$0.0393674\pi$
$193$	3.42758i	0.246723i	−0.992362	+	0.123361i	$0.960633\pi$
$194$	−1.96077	−0.140775
$195$	0	0
$196$	9.39373	0.670981
$197$	3.77479i	0.268943i	0.990917	+	0.134471i	$0.0429336\pi$
$197$	3.77479i	0.268943i	−0.990917	+	0.134471i	$0.957066\pi$
$198$	0	0
$199$	−17.9541	−1.27273	−0.636365	−	0.771388i	$-0.719563\pi$
$199$	−17.9541	−1.27273	−0.636365	+	0.771388i	$0.719563\pi$
$200$	− 4.87263i	− 0.344547i
$201$	0	0
$202$	6.98254i	0.491290i
$203$	− 34.4795i	− 2.41999i
$204$	0	0
$205$	2.71379	0.189539
$206$	− 4.94869i	− 0.344792i
$207$	0	0
$208$	0	0
$209$	−4.54825	−0.314609
$210$	0	0
$211$	−12.5375	−0.863117	−0.431559	−	0.902085i	$-0.642036\pi$
$211$	−12.5375	−0.863117	−0.431559	+	0.902085i	$0.642036\pi$
$212$	10.4112	0.715043
$213$	0	0
$214$	4.26875i	0.291806i
$215$	2.23921i	0.152713i
$216$	0	0
$217$	−43.6872	−2.96568
$218$	−6.21983	−0.421260
$219$	0	0
$220$	0.325437	0.0219410
$221$	0	0
$222$	0	0
$223$	− 5.42758i	− 0.363458i	−0.983349	−	0.181729i	$-0.941831\pi$
$223$	− 5.42758i	− 0.363458i	0.983349	−	0.181729i	$-0.0581693\pi$
$224$	−4.04892	−0.270530
$225$	0	0
$226$	− 12.9879i	− 0.863943i
$227$	16.5767i	1.10024i	0.835087	+	0.550118i	$0.185417\pi$
$227$	16.5767i	1.10024i	−0.835087	+	0.550118i	$0.814583\pi$
$228$	0	0
$229$	− 23.8780i	− 1.57790i	−0.614456	−	0.788951i	$-0.710624\pi$
$229$	− 23.8780i	− 1.57790i	0.614456	−	0.788951i	$-0.289376\pi$
$230$	−3.03146	−0.199888
$231$	0	0
$232$	8.51573i	0.559085i
$233$	13.9952	0.916857	0.458428	−	0.888731i	$-0.348413\pi$
$233$	13.9952	0.916857	0.458428	+	0.888731i	$0.348413\pi$
$234$	0	0
$235$	0.635334	0.0414446
$236$	− 6.04892i	− 0.393751i
$237$	0	0
$238$	8.49396	0.550582
$239$	13.2862i	0.859413i	0.902969	+	0.429707i	$0.141383\pi$
$239$	13.2862i	0.859413i	−0.902969	+	0.429707i	$0.858617\pi$
$240$	0	0
$241$	10.4789i	0.675005i	0.941325	+	0.337502i	$0.109582\pi$
$241$	10.4789i	0.675005i	−0.941325	+	0.337502i	$0.890418\pi$
$242$	10.1685i	0.653657i
$243$	0	0
$244$	3.10992	0.199092
$245$	− 3.35258i	− 0.214189i
$246$	0	0
$247$	0	0
$248$	10.7899	0.685157
$249$	0	0
$250$	−3.52350	−0.222846
$251$	3.48725	0.220114	0.110057	−	0.993925i	$-0.464897\pi$
$251$	3.48725	0.220114	0.110057	+	0.993925i	$0.464897\pi$
$252$	0	0
$253$	7.74525i	0.486940i
$254$	− 9.22282i	− 0.578691i
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.53750	0.407798	0.203899	−	0.978992i	$-0.434639\pi$
$257$	6.53750	0.407798	0.203899	+	0.978992i	$0.434639\pi$
$258$	0	0
$259$	2.49396	0.154967
$260$	0	0
$261$	0	0
$262$	14.5526i	0.899060i
$263$	−8.01938	−0.494496	−0.247248	−	0.968952i	$-0.579526\pi$
$263$	−8.01938	−0.494496	−0.247248	+	0.968952i	$0.579526\pi$
$264$	0	0
$265$	− 3.71571i	− 0.228254i
$266$	− 20.1957i	− 1.23828i
$267$	0	0
$268$	13.5797i	0.829513i
$269$	27.6732	1.68727	0.843633	−	0.536920i	$-0.180412\pi$
$269$	27.6732	1.68727	0.843633	+	0.536920i	$0.180412\pi$
$270$	0	0
$271$	14.7289i	0.894714i	0.894355	+	0.447357i	$0.147635\pi$
$271$	14.7289i	0.894714i	−0.894355	+	0.447357i	$0.852365\pi$
$272$	−2.09783	−0.127200
$273$	0	0
$274$	−15.4034	−0.930555
$275$	4.44312i	0.267930i
$276$	0	0
$277$	−3.26205	−0.195997	−0.0979986	−	0.995187i	$-0.531244\pi$
$277$	−3.26205	−0.195997	−0.0979986	+	0.995187i	$0.531244\pi$
$278$	2.71379i	0.162762i
$279$	0	0
$280$	1.44504i	0.0863578i
$281$	− 7.72587i	− 0.460887i	−0.973086	−	0.230443i	$-0.925982\pi$
$281$	− 7.72587i	− 0.460887i	0.973086	−	0.230443i	$-0.0740177\pi$
$282$	0	0
$283$	19.7802	1.17581	0.587904	−	0.808930i	$-0.299953\pi$
$283$	19.7802	1.17581	0.587904	+	0.808930i	$0.299953\pi$
$284$	− 11.4819i	− 0.681324i
$285$	0	0
$286$	0	0
$287$	30.7875	1.81733
$288$	0	0
$289$	−12.5991	−0.741123
$290$	3.03923	0.178470
$291$	0	0
$292$	− 0.533188i	− 0.0312025i
$293$	12.9119i	0.754319i	0.926148	+	0.377159i	$0.123099\pi$
$293$	12.9119i	0.754319i	−0.926148	+	0.377159i	$0.876901\pi$
$294$	0	0
$295$	−2.15883	−0.125692
$296$	−0.615957	−0.0358018
$297$	0	0
$298$	14.7356	0.853608
$299$	0	0
$300$	0	0
$301$	25.4034i	1.46423i
$302$	15.8213	0.910414
$303$	0	0
$304$	4.98792i	0.286077i
$305$	− 1.10992i	− 0.0635536i
$306$	0	0
$307$	19.9651i	1.13947i	0.821829	+	0.569734i	$0.192954\pi$
$307$	19.9651i	1.13947i	−0.821829	+	0.569734i	$0.807046\pi$
$308$	3.69202	0.210372
$309$	0	0
$310$	− 3.85086i	− 0.218714i
$311$	13.4819	0.764487	0.382244	−	0.924062i	$-0.375152\pi$
$311$	13.4819	0.764487	0.382244	+	0.924062i	$0.375152\pi$
$312$	0	0
$313$	12.9245	0.730537	0.365269	−	0.930902i	$-0.380977\pi$
$313$	12.9245	0.730537	0.365269	+	0.930902i	$0.380977\pi$
$314$	− 4.27413i	− 0.241203i
$315$	0	0
$316$	11.7071	0.658575
$317$	− 11.8726i	− 0.666833i	−0.942780	−	0.333417i	$-0.891798\pi$
$317$	− 11.8726i	− 0.666833i	0.942780	−	0.333417i	$-0.108202\pi$
$318$	0	0
$319$	− 7.76510i	− 0.434762i
$320$	− 0.356896i	− 0.0199511i
$321$	0	0
$322$	−34.3913	−1.91655
$323$	− 10.4638i	− 0.582223i
$324$	0	0
$325$	0	0
$326$	0.317667	0.0175940
$327$	0	0
$328$	−7.60388	−0.419854
$329$	7.20775	0.397376
$330$	0	0
$331$	− 10.2392i	− 0.562798i	−0.959591	−	0.281399i	$-0.909202\pi$
$331$	− 10.2392i	− 0.562798i	0.959591	−	0.281399i	$-0.0907984\pi$
$332$	6.49934i	0.356697i
$333$	0	0
$334$	12.3612	0.676375
$335$	4.84654	0.264795
$336$	0	0
$337$	−1.44935	−0.0789513	−0.0394757	−	0.999221i	$-0.512569\pi$
$337$	−1.44935	−0.0789513	−0.0394757	+	0.999221i	$0.512569\pi$
$338$	0	0
$339$	0	0
$340$	0.748709i	0.0406044i
$341$	−9.83877	−0.532799
$342$	0	0
$343$	− 9.69202i	− 0.523320i
$344$	− 6.27413i	− 0.338278i
$345$	0	0
$346$	− 17.0640i	− 0.917365i
$347$	−6.84117	−0.367253	−0.183627	−	0.982996i	$-0.558784\pi$
$347$	−6.84117	−0.367253	−0.183627	+	0.982996i	$0.558784\pi$
$348$	0	0
$349$	− 34.3370i	− 1.83802i	−0.394234	−	0.919010i	$-0.628990\pi$
$349$	− 34.3370i	− 1.83802i	0.394234	−	0.919010i	$-0.371010\pi$
$350$	−19.7289	−1.05455
$351$	0	0
$352$	−0.911854	−0.0486020
$353$	26.0495i	1.38648i	0.720709	+	0.693238i	$0.243816\pi$
$353$	26.0495i	1.38648i	−0.720709	+	0.693238i	$0.756184\pi$
$354$	0	0
$355$	−4.09783	−0.217490
$356$	− 6.49396i	− 0.344179i
$357$	0	0
$358$	− 24.9681i	− 1.31960i
$359$	− 8.49396i	− 0.448294i	−0.974555	−	0.224147i	$-0.928040\pi$
$359$	− 8.49396i	− 0.448294i	0.974555	−	0.224147i	$-0.0719596\pi$
$360$	0	0
$361$	−5.87933	−0.309438
$362$	− 5.26205i	− 0.276567i
$363$	0	0
$364$	0	0
$365$	−0.190293	−0.00996037
$366$	0	0
$367$	27.4523	1.43300	0.716500	−	0.697587i	$-0.245743\pi$
$367$	27.4523	1.43300	0.716500	+	0.697587i	$0.245743\pi$
$368$	8.49396	0.442778
$369$	0	0
$370$	0.219833i	0.0114285i
$371$	− 42.1540i	− 2.18853i
$372$	0	0
$373$	26.6219	1.37843	0.689216	−	0.724556i	$-0.257955\pi$
$373$	26.6219	1.37843	0.689216	+	0.724556i	$0.257955\pi$
$374$	1.91292	0.0989147
$375$	0	0
$376$	−1.78017	−0.0918051
$377$	0	0
$378$	0	0
$379$	− 11.6474i	− 0.598288i	−0.954208	−	0.299144i	$-0.903299\pi$
$379$	− 11.6474i	− 0.598288i	0.954208	−	0.299144i	$-0.0967010\pi$
$380$	1.78017	0.0913207
$381$	0	0
$382$	− 10.5375i	− 0.539145i
$383$	10.5181i	0.537451i	0.963217	+	0.268725i	$0.0866024\pi$
$383$	10.5181i	0.537451i	−0.963217	+	0.268725i	$0.913398\pi$
$384$	0	0
$385$	− 1.31767i	− 0.0671545i
$386$	−3.42758	−0.174459
$387$	0	0
$388$	− 1.96077i	− 0.0995431i
$389$	−9.25965	−0.469483	−0.234742	−	0.972058i	$-0.575424\pi$
$389$	−9.25965	−0.469483	−0.234742	+	0.972058i	$0.575424\pi$
$390$	0	0
$391$	−17.8189	−0.901142
$392$	9.39373i	0.474455i
$393$	0	0
$394$	−3.77479	−0.190171
$395$	− 4.17821i	− 0.210229i
$396$	0	0
$397$	14.5133i	0.728403i	0.931320	+	0.364202i	$0.118658\pi$
$397$	14.5133i	0.728403i	−0.931320	+	0.364202i	$0.881342\pi$
$398$	− 17.9541i	− 0.899956i
$399$	0	0
$400$	4.87263	0.243631
$401$	38.8418i	1.93966i	0.243773	+	0.969832i	$0.421615\pi$
$401$	38.8418i	1.93966i	−0.243773	+	0.969832i	$0.578385\pi$
$402$	0	0
$403$	0	0
$404$	−6.98254	−0.347394
$405$	0	0
$406$	34.4795	1.71119
$407$	0.561663	0.0278406
$408$	0	0
$409$	− 33.9221i	− 1.67734i	−0.544639	−	0.838671i	$-0.683333\pi$
$409$	− 33.9221i	− 1.67734i	0.544639	−	0.838671i	$-0.316667\pi$
$410$	2.71379i	0.134025i
$411$	0	0
$412$	4.94869	0.243804
$413$	−24.4916	−1.20515
$414$	0	0
$415$	2.31959	0.113864
$416$	0	0
$417$	0	0
$418$	− 4.54825i	− 0.222462i
$419$	−0.955395	−0.0466741	−0.0233370	−	0.999728i	$-0.507429\pi$
$419$	−0.955395	−0.0466741	−0.0233370	+	0.999728i	$0.507429\pi$
$420$	0	0
$421$	− 5.68233i	− 0.276940i	−0.990367	−	0.138470i	$-0.955782\pi$
$421$	− 5.68233i	− 0.276940i	0.990367	−	0.138470i	$-0.0442184\pi$
$422$	− 12.5375i	− 0.610316i
$423$	0	0
$424$	10.4112i	0.505612i
$425$	−10.2220	−0.495838
$426$	0	0
$427$	− 12.5918i	− 0.609360i
$428$	−4.26875	−0.206338
$429$	0	0
$430$	−2.23921	−0.107984
$431$	14.8465i	0.715133i	0.933888	+	0.357566i	$0.116393\pi$
$431$	14.8465i	0.715133i	−0.933888	+	0.357566i	$0.883607\pi$
$432$	0	0
$433$	26.1497	1.25668	0.628338	−	0.777940i	$-0.283735\pi$
$433$	26.1497	1.25668	0.628338	+	0.777940i	$0.283735\pi$
$434$	− 43.6872i	− 2.09705i
$435$	0	0
$436$	− 6.21983i	− 0.297876i
$437$	42.3672i	2.02670i
$438$	0	0
$439$	23.5502	1.12399	0.561994	−	0.827141i	$-0.310034\pi$
$439$	23.5502	1.12399	0.561994	+	0.827141i	$0.310034\pi$
$440$	0.325437i	0.0155146i
$441$	0	0
$442$	0	0
$443$	−21.9433	−1.04256	−0.521279	−	0.853386i	$-0.674545\pi$
$443$	−21.9433	−1.04256	−0.521279	+	0.853386i	$0.674545\pi$
$444$	0	0
$445$	−2.31767	−0.109868
$446$	5.42758	0.257004
$447$	0	0
$448$	− 4.04892i	− 0.191293i
$449$	11.4034i	0.538161i	0.963118	+	0.269080i	$0.0867197\pi$
$449$	11.4034i	0.538161i	−0.963118	+	0.269080i	$0.913280\pi$
$450$	0	0
$451$	6.93362	0.326492
$452$	12.9879	0.610900
$453$	0	0
$454$	−16.5767	−0.777984
$455$	0	0
$456$	0	0
$457$	7.66919i	0.358749i	0.983781	+	0.179375i	$0.0574074\pi$
$457$	7.66919i	0.358749i	−0.983781	+	0.179375i	$0.942593\pi$
$458$	23.8780	1.11575
$459$	0	0
$460$	− 3.03146i	− 0.141343i
$461$	28.5080i	1.32775i	0.747844	+	0.663874i	$0.231089\pi$
$461$	28.5080i	1.32775i	−0.747844	+	0.663874i	$0.768911\pi$
$462$	0	0
$463$	14.3284i	0.665898i	0.942945	+	0.332949i	$0.108044\pi$
$463$	14.3284i	0.665898i	−0.942945	+	0.332949i	$0.891956\pi$
$464$	−8.51573	−0.395333
$465$	0	0
$466$	13.9952i	0.648316i
$467$	33.3207	1.54190	0.770948	−	0.636898i	$-0.219783\pi$
$467$	33.3207	1.54190	0.770948	+	0.636898i	$0.219783\pi$
$468$	0	0
$469$	54.9831	2.53889
$470$	0.635334i	0.0293058i
$471$	0	0
$472$	6.04892	0.278424
$473$	5.72109i	0.263056i
$474$	0	0
$475$	24.3043i	1.11516i
$476$	8.49396i	0.389320i
$477$	0	0
$478$	−13.2862	−0.607697
$479$	− 22.1280i	− 1.01105i	−0.862811	−	0.505526i	$-0.831298\pi$
$479$	− 22.1280i	− 1.01105i	0.862811	−	0.505526i	$-0.168702\pi$
$480$	0	0
$481$	0	0
$482$	−10.4789	−0.477301
$483$	0	0
$484$	−10.1685	−0.462206
$485$	−0.699791	−0.0317759
$486$	0	0
$487$	− 0.126310i	− 0.00572364i	−0.999996	−	0.00286182i	$-0.999089\pi$
$487$	− 0.126310i	− 0.00572364i	0.999996	−	0.00286182i	$-0.000910947\pi$
$488$	3.10992i	0.140779i
$489$	0	0
$490$	3.35258	0.151454
$491$	13.9433	0.629253	0.314626	−	0.949216i	$-0.398121\pi$
$491$	13.9433	0.629253	0.314626	+	0.949216i	$0.398121\pi$
$492$	0	0
$493$	17.8646	0.804581
$494$	0	0
$495$	0	0
$496$	10.7899i	0.484479i
$497$	−46.4892	−2.08532
$498$	0	0
$499$	28.3913i	1.27097i	0.772113	+	0.635485i	$0.219200\pi$
$499$	28.3913i	1.27097i	−0.772113	+	0.635485i	$0.780800\pi$
$500$	− 3.52350i	− 0.157576i
$501$	0	0
$502$	3.48725i	0.155644i
$503$	−12.5676	−0.560363	−0.280181	−	0.959947i	$-0.590395\pi$
$503$	−12.5676	−0.560363	−0.280181	+	0.959947i	$0.590395\pi$
$504$	0	0
$505$	2.49204i	0.110894i
$506$	−7.74525	−0.344318
$507$	0	0
$508$	9.22282	0.409196
$509$	− 4.37675i	− 0.193996i	−0.995285	−	0.0969980i	$-0.969076\pi$
$509$	− 4.37675i	− 0.193996i	0.995285	−	0.0969980i	$-0.0309240\pi$
$510$	0	0
$511$	−2.15883	−0.0955012
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	6.53750i	0.288357i
$515$	− 1.76617i	− 0.0778266i
$516$	0	0
$517$	1.62325	0.0713906
$518$	2.49396i	0.109578i
$519$	0	0
$520$	0	0
$521$	23.2707	1.01951	0.509753	−	0.860321i	$-0.329737\pi$
$521$	23.2707	1.01951	0.509753	+	0.860321i	$0.329737\pi$
$522$	0	0
$523$	37.9952	1.66141	0.830707	−	0.556709i	$-0.187936\pi$
$523$	37.9952	1.66141	0.830707	+	0.556709i	$0.187936\pi$
$524$	−14.5526	−0.635732
$525$	0	0
$526$	− 8.01938i	− 0.349661i
$527$	− 22.6353i	− 0.986011i
$528$	0	0
$529$	49.1473	2.13684
$530$	3.71571	0.161400
$531$	0	0
$532$	20.1957	0.875593
$533$	0	0
$534$	0	0
$535$	1.52350i	0.0658666i
$536$	−13.5797	−0.586554
$537$	0	0
$538$	27.6732i	1.19308i
$539$	− 8.56571i	− 0.368951i
$540$	0	0
$541$	− 3.16421i	− 0.136040i	−0.997684	−	0.0680200i	$-0.978332\pi$
$541$	− 3.16421i	− 0.136040i	0.997684	−	0.0680200i	$-0.0216682\pi$
$542$	−14.7289	−0.632659
$543$	0	0
$544$	− 2.09783i	− 0.0899439i
$545$	−2.21983	−0.0950872
$546$	0	0
$547$	7.56033	0.323257	0.161628	−	0.986852i	$-0.448325\pi$
$547$	7.56033	0.323257	0.161628	+	0.986852i	$0.448325\pi$
$548$	− 15.4034i	− 0.658002i
$549$	0	0
$550$	−4.44312	−0.189455
$551$	− 42.4758i	− 1.80953i
$552$	0	0
$553$	− 47.4010i	− 2.01570i
$554$	− 3.26205i	− 0.138591i
$555$	0	0
$556$	−2.71379	−0.115090
$557$	− 0.415502i	− 0.0176054i	−0.999961	−	0.00880269i	$-0.997198\pi$
$557$	− 0.415502i	− 0.0176054i	0.999961	−	0.00880269i	$-0.00280202\pi$
$558$	0	0
$559$	0	0
$560$	−1.44504	−0.0610642
$561$	0	0
$562$	7.72587	0.325896
$563$	−29.0465	−1.22417	−0.612083	−	0.790794i	$-0.709668\pi$
$563$	−29.0465	−1.22417	−0.612083	+	0.790794i	$0.709668\pi$
$564$	0	0
$565$	− 4.63533i	− 0.195010i
$566$	19.7802i	0.831422i
$567$	0	0
$568$	11.4819	0.481769
$569$	39.6862	1.66373	0.831865	−	0.554977i	$-0.187273\pi$
$569$	39.6862	1.66373	0.831865	+	0.554977i	$0.187273\pi$
$570$	0	0
$571$	7.09651	0.296980	0.148490	−	0.988914i	$-0.452559\pi$
$571$	7.09651	0.296980	0.148490	+	0.988914i	$0.452559\pi$
$572$	0	0
$573$	0	0
$574$	30.7875i	1.28504i
$575$	41.3879	1.72599
$576$	0	0
$577$	− 8.78687i	− 0.365802i	−0.983131	−	0.182901i	$-0.941451\pi$
$577$	− 8.78687i	− 0.365802i	0.983131	−	0.182901i	$-0.0585488\pi$
$578$	− 12.5991i	− 0.524053i
$579$	0	0
$580$	3.03923i	0.126197i
$581$	26.3153	1.09174
$582$	0	0
$583$	− 9.49349i	− 0.393180i
$584$	0.533188	0.0220635
$585$	0	0
$586$	−12.9119	−0.533384
$587$	36.7066i	1.51504i	0.652810	+	0.757522i	$0.273590\pi$
$587$	36.7066i	1.51504i	−0.652810	+	0.757522i	$0.726410\pi$
$588$	0	0
$589$	−53.8189	−2.21757
$590$	− 2.15883i	− 0.0888778i
$591$	0	0
$592$	− 0.615957i	− 0.0253157i
$593$	− 10.8310i	− 0.444776i	−0.974958	−	0.222388i	$-0.928615\pi$
$593$	− 10.8310i	− 0.444776i	0.974958	−	0.222388i	$-0.0713852\pi$
$594$	0	0
$595$	3.03146	0.124278
$596$	14.7356i	0.603592i
$597$	0	0
$598$	0	0
$599$	−23.5254	−0.961223	−0.480611	−	0.876934i	$-0.659585\pi$
$599$	−23.5254	−0.961223	−0.480611	+	0.876934i	$0.659585\pi$
$600$	0	0
$601$	27.8213	1.13486	0.567428	−	0.823423i	$-0.307939\pi$
$601$	27.8213	1.13486	0.567428	+	0.823423i	$0.307939\pi$
$602$	−25.4034	−1.03537
$603$	0	0
$604$	15.8213i	0.643760i
$605$	3.62910i	0.147544i
$606$	0	0
$607$	0.0972437	0.00394700	0.00197350	−	0.999998i	$-0.499372\pi$
$607$	0.0972437	0.00394700	0.00197350	+	0.999998i	$0.499372\pi$
$608$	−4.98792	−0.202287
$609$	0	0
$610$	1.10992	0.0449392
$611$	0	0
$612$	0	0
$613$	8.06505i	0.325744i	0.986647	+	0.162872i	$0.0520758\pi$
$613$	8.06505i	0.325744i	−0.986647	+	0.162872i	$0.947924\pi$
$614$	−19.9651	−0.805725
$615$	0	0
$616$	3.69202i	0.148756i
$617$	19.2185i	0.773708i	0.922141	+	0.386854i	$0.126438\pi$
$617$	19.2185i	0.773708i	−0.922141	+	0.386854i	$0.873562\pi$
$618$	0	0
$619$	− 12.3827i	− 0.497703i	−0.968542	−	0.248852i	$-0.919947\pi$
$619$	− 12.3827i	− 0.497703i	0.968542	−	0.248852i	$-0.0800532\pi$
$620$	3.85086	0.154654
$621$	0	0
$622$	13.4819i	0.540574i
$623$	−26.2935	−1.05343
$624$	0	0
$625$	23.1056	0.924224
$626$	12.9245i	0.516568i
$627$	0	0
$628$	4.27413	0.170556
$629$	1.29218i	0.0515224i
$630$	0	0
$631$	4.74333i	0.188829i	0.995533	+	0.0944145i	$0.0300979\pi$
$631$	4.74333i	0.188829i	−0.995533	+	0.0944145i	$0.969902\pi$
$632$	11.7071i	0.465683i
$633$	0	0
$634$	11.8726	0.471522
$635$	− 3.29159i	− 0.130623i
$636$	0	0
$637$	0	0
$638$	7.76510	0.307423
$639$	0	0
$640$	0.356896	0.0141075
$641$	−16.4456	−0.649563	−0.324782	−	0.945789i	$-0.605291\pi$
$641$	−16.4456	−0.649563	−0.324782	+	0.945789i	$0.605291\pi$
$642$	0	0
$643$	1.74525i	0.0688260i	0.999408	+	0.0344130i	$0.0109562\pi$
$643$	1.74525i	0.0688260i	−0.999408	+	0.0344130i	$0.989044\pi$
$644$	− 34.3913i	− 1.35521i
$645$	0	0
$646$	10.4638	0.411694
$647$	28.7633	1.13080	0.565401	−	0.824816i	$-0.308721\pi$
$647$	28.7633	1.13080	0.565401	+	0.824816i	$0.308721\pi$
$648$	0	0
$649$	−5.51573	−0.216511
$650$	0	0
$651$	0	0
$652$	0.317667i	0.0124408i
$653$	16.1661	0.632630	0.316315	−	0.948654i	$-0.397554\pi$
$653$	16.1661	0.632630	0.316315	+	0.948654i	$0.397554\pi$
$654$	0	0
$655$	5.19375i	0.202937i
$656$	− 7.60388i	− 0.296881i
$657$	0	0
$658$	7.20775i	0.280987i
$659$	16.3558	0.637133	0.318566	−	0.947901i	$-0.396799\pi$
$659$	16.3558	0.637133	0.318566	+	0.947901i	$0.396799\pi$
$660$	0	0
$661$	33.1159i	1.28806i	0.765001	+	0.644029i	$0.222739\pi$
$661$	33.1159i	1.28806i	−0.765001	+	0.644029i	$0.777261\pi$
$662$	10.2392	0.397958
$663$	0	0
$664$	−6.49934	−0.252223
$665$	− 7.20775i	− 0.279505i
$666$	0	0
$667$	−72.3323	−2.80072
$668$	12.3612i	0.478269i
$669$	0	0
$670$	4.84654i	0.187238i
$671$	− 2.83579i	− 0.109474i
$672$	0	0
$673$	35.1540	1.35509	0.677544	−	0.735482i	$-0.263044\pi$
$673$	35.1540	1.35509	0.677544	+	0.735482i	$0.263044\pi$
$674$	− 1.44935i	− 0.0558270i
$675$	0	0
$676$	0	0
$677$	−23.7855	−0.914153	−0.457076	−	0.889427i	$-0.651103\pi$
$677$	−23.7855	−0.914153	−0.457076	+	0.889427i	$0.651103\pi$
$678$	0	0
$679$	−7.93900	−0.304671
$680$	−0.748709	−0.0287117
$681$	0	0
$682$	− 9.83877i	− 0.376746i
$683$	2.99223i	0.114495i	0.998360	+	0.0572473i	$0.0182323\pi$
$683$	2.99223i	0.114495i	−0.998360	+	0.0572473i	$0.981768\pi$
$684$	0	0
$685$	−5.49742	−0.210046
$686$	9.69202	0.370043
$687$	0	0
$688$	6.27413	0.239199
$689$	0	0
$690$	0	0
$691$	11.6233i	0.442169i	0.975255	+	0.221085i	$0.0709597\pi$
$691$	11.6233i	0.442169i	−0.975255	+	0.221085i	$0.929040\pi$
$692$	17.0640	0.648675
$693$	0	0
$694$	− 6.84117i	− 0.259687i
$695$	0.968541i	0.0367389i
$696$	0	0
$697$	15.9517i	0.604213i
$698$	34.3370	1.29968
$699$	0	0
$700$	− 19.7289i	− 0.745681i
$701$	−33.8431	−1.27824	−0.639118	−	0.769109i	$-0.720700\pi$
$701$	−33.8431	−1.27824	−0.639118	+	0.769109i	$0.720700\pi$
$702$	0	0
$703$	3.07234	0.115876
$704$	− 0.911854i	− 0.0343668i
$705$	0	0
$706$	−26.0495	−0.980386
$707$	28.2717i	1.06327i
$708$	0	0
$709$	26.1909i	0.983619i	0.870703	+	0.491810i	$0.163664\pi$
$709$	26.1909i	0.983619i	−0.870703	+	0.491810i	$0.836336\pi$
$710$	− 4.09783i	− 0.153789i
$711$	0	0
$712$	6.49396	0.243371
$713$	91.6486i	3.43227i
$714$	0	0
$715$	0	0
$716$	24.9681	0.933100
$717$	0	0
$718$	8.49396	0.316992
$719$	−21.7345	−0.810560	−0.405280	−	0.914193i	$-0.632826\pi$
$719$	−21.7345	−0.810560	−0.405280	+	0.914193i	$0.632826\pi$
$720$	0	0
$721$	− 20.0368i	− 0.746211i
$722$	− 5.87933i	− 0.218806i
$723$	0	0
$724$	5.26205	0.195562
$725$	−41.4940	−1.54105
$726$	0	0
$727$	2.01400	0.0746951	0.0373476	−	0.999302i	$-0.488109\pi$
$727$	2.01400	0.0746951	0.0373476	+	0.999302i	$0.488109\pi$
$728$	0	0
$729$	0	0
$730$	− 0.190293i	− 0.00704304i
$731$	−13.1621	−0.486817
$732$	0	0
$733$	13.5013i	0.498680i	0.968416	+	0.249340i	$0.0802137\pi$
$733$	13.5013i	0.498680i	−0.968416	+	0.249340i	$0.919786\pi$
$734$	27.4523i	1.01328i
$735$	0	0
$736$	8.49396i	0.313091i
$737$	12.3827	0.456123
$738$	0	0
$739$	16.5918i	0.610339i	0.952298	+	0.305170i	$0.0987131\pi$
$739$	16.5918i	0.610339i	−0.952298	+	0.305170i	$0.901287\pi$
$740$	−0.219833	−0.00808120
$741$	0	0
$742$	42.1540	1.54752
$743$	19.8479i	0.728148i	0.931370	+	0.364074i	$0.118614\pi$
$743$	19.8479i	0.728148i	−0.931370	+	0.364074i	$0.881386\pi$
$744$	0	0
$745$	5.25906	0.192677
$746$	26.6219i	0.974698i
$747$	0	0
$748$	1.91292i	0.0699432i
$749$	17.2838i	0.631537i
$750$	0	0
$751$	−27.9347	−1.01935	−0.509676	−	0.860367i	$-0.670235\pi$
$751$	−27.9347	−1.01935	−0.509676	+	0.860367i	$0.670235\pi$
$752$	− 1.78017i	− 0.0649160i
$753$	0	0
$754$	0	0
$755$	5.64656	0.205499
$756$	0	0
$757$	−0.548253	−0.0199266	−0.00996330	−	0.999950i	$-0.503171\pi$
$757$	−0.548253	−0.0199266	−0.00996330	+	0.999950i	$0.503171\pi$
$758$	11.6474	0.423053
$759$	0	0
$760$	1.78017i	0.0645735i
$761$	1.97584i	0.0716240i	0.999359	+	0.0358120i	$0.0114018\pi$
$761$	1.97584i	0.0716240i	−0.999359	+	0.0358120i	$0.988598\pi$
$762$	0	0
$763$	−25.1836	−0.911707
$764$	10.5375	0.381233
$765$	0	0
$766$	−10.5181	−0.380035
$767$	0	0
$768$	0	0
$769$	28.6112i	1.03175i	0.856665	+	0.515873i	$0.172532\pi$
$769$	28.6112i	1.03175i	−0.856665	+	0.515873i	$0.827468\pi$
$770$	1.31767	0.0474854
$771$	0	0
$772$	− 3.42758i	− 0.123361i
$773$	− 52.5080i	− 1.88858i	−0.329114	−	0.944290i	$-0.606750\pi$
$773$	− 52.5080i	− 1.88858i	0.329114	−	0.944290i	$-0.393250\pi$
$774$	0	0
$775$	52.5749i	1.88855i
$776$	1.96077	0.0703876
$777$	0	0
$778$	− 9.25965i	− 0.331975i
$779$	37.9275	1.35889
$780$	0	0
$781$	−10.4698	−0.374639
$782$	− 17.8189i	− 0.637203i
$783$	0	0
$784$	−9.39373	−0.335490
$785$	− 1.52542i	− 0.0544445i
$786$	0	0
$787$	− 23.6426i	− 0.842769i	−0.906882	−	0.421384i	$-0.861544\pi$
$787$	− 23.6426i	− 0.842769i	0.906882	−	0.421384i	$-0.138456\pi$
$788$	− 3.77479i	− 0.134471i
$789$	0	0
$790$	4.17821	0.148654
$791$	− 52.5870i	− 1.86978i
$792$	0	0
$793$	0	0
$794$	−14.5133	−0.515059
$795$	0	0
$796$	17.9541	0.636365
$797$	41.6558	1.47552	0.737762	−	0.675061i	$-0.235883\pi$
$797$	41.6558	1.47552	0.737762	+	0.675061i	$0.235883\pi$
$798$	0	0
$799$	3.73450i	0.132117i
$800$	4.87263i	0.172273i
$801$	0	0
$802$	−38.8418	−1.37155
$803$	−0.486189	−0.0171573
$804$	0	0
$805$	−12.2741	−0.432606
$806$	0	0
$807$	0	0
$808$	− 6.98254i	− 0.245645i
$809$	44.2392	1.55537	0.777684	−	0.628656i	$-0.216394\pi$
$809$	44.2392	1.55537	0.777684	+	0.628656i	$0.216394\pi$
$810$	0	0
$811$	− 52.3913i	− 1.83971i	−0.392260	−	0.919854i	$-0.628307\pi$
$811$	− 52.3913i	− 1.83971i	0.392260	−	0.919854i	$-0.371693\pi$
$812$	34.4795i	1.20999i
$813$	0	0
$814$	0.561663i	0.0196863i
$815$	0.113374	0.00397132
$816$	0	0
$817$	31.2948i	1.09487i
$818$	33.9221	1.18606
$819$	0	0
$820$	−2.71379	−0.0947697
$821$	− 25.6276i	− 0.894408i	−0.894432	−	0.447204i	$-0.852420\pi$
$821$	− 25.6276i	− 0.894408i	0.894432	−	0.447204i	$-0.147580\pi$
$822$	0	0
$823$	40.2553	1.40321	0.701606	−	0.712565i	$-0.252466\pi$
$823$	40.2553	1.40321	0.701606	+	0.712565i	$0.252466\pi$
$824$	4.94869i	0.172396i
$825$	0	0
$826$	− 24.4916i	− 0.852171i
$827$	18.0519i	0.627726i	0.949468	+	0.313863i	$0.101623\pi$
$827$	18.0519i	0.627726i	−0.949468	+	0.313863i	$0.898377\pi$
$828$	0	0
$829$	−22.6655	−0.787204	−0.393602	−	0.919281i	$-0.628771\pi$
$829$	−22.6655	−0.787204	−0.393602	+	0.919281i	$0.628771\pi$
$830$	2.31959i	0.0805140i
$831$	0	0
$832$	0	0
$833$	19.7065	0.682790
$834$	0	0
$835$	4.41166	0.152672
$836$	4.54825	0.157305
$837$	0	0
$838$	− 0.955395i	− 0.0330036i
$839$	− 22.0823i	− 0.762365i	−0.924500	−	0.381183i	$-0.875517\pi$
$839$	− 22.0823i	− 0.762365i	0.924500	−	0.381183i	$-0.124483\pi$
$840$	0	0
$841$	43.5176	1.50061
$842$	5.68233	0.195826
$843$	0	0
$844$	12.5375	0.431559
$845$	0	0
$846$	0	0
$847$	41.1715i	1.41467i
$848$	−10.4112	−0.357522
$849$	0	0
$850$	− 10.2220i	− 0.350610i
$851$	− 5.23191i	− 0.179348i
$852$	0	0
$853$	28.9831i	0.992364i	0.868219	+	0.496182i	$0.165265\pi$
$853$	28.9831i	0.992364i	−0.868219	+	0.496182i	$0.834735\pi$
$854$	12.5918	0.430882
$855$	0	0
$856$	− 4.26875i	− 0.145903i
$857$	−42.9047	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$857$	−42.9047	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$858$	0	0
$859$	−47.0616	−1.60572	−0.802860	−	0.596167i	$-0.796690\pi$
$859$	−47.0616	−1.60572	−0.802860	+	0.596167i	$0.796690\pi$
$860$	− 2.23921i	− 0.0763564i
$861$	0	0
$862$	−14.8465	−0.505675
$863$	42.6064i	1.45034i	0.688571	+	0.725169i	$0.258238\pi$
$863$	42.6064i	1.45034i	−0.688571	+	0.725169i	$0.741762\pi$
$864$	0	0
$865$	− 6.09006i	− 0.207068i
$866$	26.1497i	0.888604i
$867$	0	0
$868$	43.6872	1.48284
$869$	− 10.6752i	− 0.362130i
$870$	0	0
$871$	0	0
$872$	6.21983	0.210630
$873$	0	0
$874$	−42.3672	−1.43309
$875$	−14.2664	−0.482291
$876$	0	0
$877$	27.4082i	0.925509i	0.886486	+	0.462755i	$0.153139\pi$
$877$	27.4082i	0.925509i	−0.886486	+	0.462755i	$0.846861\pi$
$878$	23.5502i	0.794780i
$879$	0	0
$880$	−0.325437	−0.0109705
$881$	24.3177	0.819283	0.409642	−	0.912247i	$-0.365654\pi$
$881$	24.3177	0.819283	0.409642	+	0.912247i	$0.365654\pi$
$882$	0	0
$883$	32.2306	1.08465	0.542323	−	0.840170i	$-0.317545\pi$
$883$	32.2306	1.08465	0.542323	+	0.840170i	$0.317545\pi$
$884$	0	0
$885$	0	0
$886$	− 21.9433i	− 0.737200i
$887$	35.1642	1.18070	0.590349	−	0.807148i	$-0.298990\pi$
$887$	35.1642	1.18070	0.590349	+	0.807148i	$0.298990\pi$
$888$	0	0
$889$	− 37.3424i	− 1.25242i
$890$	− 2.31767i	− 0.0776884i
$891$	0	0
$892$	5.42758i	0.181729i
$893$	8.87933	0.297135
$894$	0	0
$895$	− 8.91100i	− 0.297862i
$896$	4.04892	0.135265
$897$	0	0
$898$	−11.4034	−0.380537
$899$	− 91.8835i	− 3.06449i
$900$	0	0
$901$	21.8410	0.727628
$902$	6.93362i	0.230864i
$903$	0	0
$904$	12.9879i	0.431972i
$905$	− 1.87800i	− 0.0624269i
$906$	0	0
$907$	13.1207	0.435665	0.217832	−	0.975986i	$-0.430101\pi$
$907$	13.1207	0.435665	0.217832	+	0.975986i	$0.430101\pi$
$908$	− 16.5767i	− 0.550118i
$909$	0	0
$910$	0	0
$911$	−45.7453	−1.51561	−0.757804	−	0.652482i	$-0.773728\pi$
$911$	−45.7453	−1.51561	−0.757804	+	0.652482i	$0.773728\pi$
$912$	0	0
$913$	5.92645	0.196137
$914$	−7.66919	−0.253674
$915$	0	0
$916$	23.8780i	0.788951i
$917$	58.9221i	1.94578i
$918$	0	0
$919$	17.9849	0.593268	0.296634	−	0.954991i	$-0.404136\pi$
$919$	17.9849	0.593268	0.296634	+	0.954991i	$0.404136\pi$
$920$	3.03146	0.0999442
$921$	0	0
$922$	−28.5080	−0.938860
$923$	0	0
$924$	0	0
$925$	− 3.00133i	− 0.0986831i
$926$	−14.3284	−0.470861
$927$	0	0
$928$	− 8.51573i	− 0.279543i
$929$	− 31.6883i	− 1.03966i	−0.854270	−	0.519830i	$-0.825995\pi$
$929$	− 31.6883i	− 1.03966i	0.854270	−	0.519830i	$-0.174005\pi$
$930$	0	0
$931$	− 46.8552i	− 1.53562i
$932$	−13.9952	−0.458428
$933$	0	0
$934$	33.3207i	1.09029i
$935$	0.682713	0.0223271
$936$	0	0
$937$	−53.0484	−1.73302	−0.866509	−	0.499162i	$-0.833641\pi$
$937$	−53.0484	−1.73302	−0.866509	+	0.499162i	$0.833641\pi$
$938$	54.9831i	1.79526i
$939$	0	0
$940$	−0.635334	−0.0207223
$941$	− 21.9433i	− 0.715332i	−0.933850	−	0.357666i	$-0.883573\pi$
$941$	− 21.9433i	− 0.715332i	0.933850	−	0.357666i	$-0.116427\pi$
$942$	0	0
$943$	− 64.5870i	− 2.10324i
$944$	6.04892i	0.196875i
$945$	0	0
$946$	−5.72109	−0.186009
$947$	12.2241i	0.397231i	0.980077	+	0.198616i	$0.0636445\pi$
$947$	12.2241i	0.397231i	−0.980077	+	0.198616i	$0.936355\pi$
$948$	0	0
$949$	0	0
$950$	−24.3043	−0.788534
$951$	0	0
$952$	−8.49396	−0.275291
$953$	−3.85862	−0.124993	−0.0624966	−	0.998045i	$-0.519906\pi$
$953$	−3.85862	−0.124993	−0.0624966	+	0.998045i	$0.519906\pi$
$954$	0	0
$955$	− 3.76079i	− 0.121696i
$956$	− 13.2862i	− 0.429707i
$957$	0	0
$958$	22.1280	0.714922
$959$	−62.3672	−2.01394
$960$	0	0
$961$	−85.4210	−2.75552
$962$	0	0
$963$	0	0
$964$	− 10.4789i	− 0.337502i
$965$	−1.22329	−0.0393791
$966$	0	0
$967$	− 0.613564i	− 0.0197309i	−0.999951	−	0.00986545i	$-0.996860\pi$
$967$	− 0.613564i	− 0.0197309i	0.999951	−	0.00986545i	$-0.00314032\pi$
$968$	− 10.1685i	− 0.326829i
$969$	0	0
$970$	− 0.699791i	− 0.0224689i
$971$	12.9769	0.416449	0.208224	−	0.978081i	$-0.433232\pi$
$971$	12.9769	0.416449	0.208224	+	0.978081i	$0.433232\pi$
$972$	0	0
$973$	10.9879i	0.352256i
$974$	0.126310	0.00404722
$975$	0	0
$976$	−3.10992	−0.0995460
$977$	37.0616i	1.18571i	0.805311	+	0.592853i	$0.201998\pi$
$977$	37.0616i	1.18571i	−0.805311	+	0.592853i	$0.798002\pi$
$978$	0	0
$979$	−5.92154	−0.189253
$980$	3.35258i	0.107094i
$981$	0	0
$982$	13.9433i	0.444949i
$983$	− 14.1193i	− 0.450337i	−0.974320	−	0.225169i	$-0.927707\pi$
$983$	− 14.1193i	− 0.450337i	0.974320	−	0.225169i	$-0.0722933\pi$
$984$	0	0
$985$	−1.34721	−0.0429256
$986$	17.8646i	0.568925i
$987$	0	0
$988$	0	0
$989$	53.2922	1.69459
$990$	0	0
$991$	28.5392	0.906576	0.453288	−	0.891364i	$-0.350251\pi$
$991$	28.5392	0.906576	0.453288	+	0.891364i	$0.350251\pi$
$992$	−10.7899	−0.342578
$993$	0	0
$994$	− 46.4892i	− 1.47455i
$995$	− 6.40773i	− 0.203139i
$996$	0	0
$997$	−18.8853	−0.598103	−0.299052	−	0.954237i	$-0.596670\pi$
$997$	−18.8853	−0.598103	−0.299052	+	0.954237i	$0.596670\pi$
$998$	−28.3913	−0.898712
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.r.1351.5		6
3.2	odd	2		1014.2.b.g.337.2		6
13.5	odd	4		3042.2.a.be.1.2		3
13.8	odd	4		3042.2.a.bd.1.2		3
13.12	even	2	inner	3042.2.b.r.1351.2		6
39.2	even	12		1014.2.e.m.529.2		6
39.5	even	4		1014.2.a.m.1.2	✓	3
39.8	even	4		1014.2.a.o.1.2	yes	3
39.11	even	12		1014.2.e.k.529.2		6
39.17	odd	6		1014.2.i.g.361.5		12
39.20	even	12		1014.2.e.k.991.2		6
39.23	odd	6		1014.2.i.g.823.2		12
39.29	odd	6		1014.2.i.g.823.5		12
39.32	even	12		1014.2.e.m.991.2		6
39.35	odd	6		1014.2.i.g.361.2		12
39.38	odd	2		1014.2.b.g.337.5		6
156.47	odd	4		8112.2.a.bz.1.2		3
156.83	odd	4		8112.2.a.ce.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.m.1.2	✓	3	39.5	even	4
1014.2.a.o.1.2	yes	3	39.8	even	4
1014.2.b.g.337.2		6	3.2	odd	2
1014.2.b.g.337.5		6	39.38	odd	2
1014.2.e.k.529.2		6	39.11	even	12
1014.2.e.k.991.2		6	39.20	even	12
1014.2.e.m.529.2		6	39.2	even	12
1014.2.e.m.991.2		6	39.32	even	12
1014.2.i.g.361.2		12	39.35	odd	6
1014.2.i.g.361.5		12	39.17	odd	6
1014.2.i.g.823.2		12	39.23	odd	6
1014.2.i.g.823.5		12	39.29	odd	6
3042.2.a.bd.1.2		3	13.8	odd	4
3042.2.a.be.1.2		3	13.5	odd	4
3042.2.b.r.1351.2		6	13.12	even	2	inner
3042.2.b.r.1351.5		6	1.1	even	1	trivial
8112.2.a.bz.1.2		3	156.47	odd	4
8112.2.a.ce.1.2		3	156.83	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.356896i q^{5} +4.04892i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.356896i q^{5} +4.04892i q^{7} -1.00000i q^{8} -0.356896 q^{10} +0.911854i q^{11} -4.04892 q^{14} +1.00000 q^{16} -2.09783 q^{17} +4.98792i q^{19} -0.356896i q^{20} -0.911854 q^{22} +8.49396 q^{23} +4.87263 q^{25} -4.04892i q^{28} -8.51573 q^{29} +10.7899i q^{31} +1.00000i q^{32} -2.09783i q^{34} -1.44504 q^{35} -0.615957i q^{37} -4.98792 q^{38} +0.356896 q^{40} -7.60388i q^{41} +6.27413 q^{43} -0.911854i q^{44} +8.49396i q^{46} -1.78017i q^{47} -9.39373 q^{49} +4.87263i q^{50} -10.4112 q^{53} -0.325437 q^{55} +4.04892 q^{56} -8.51573i q^{58} +6.04892i q^{59} -3.10992 q^{61} -10.7899 q^{62} -1.00000 q^{64} -13.5797i q^{67} +2.09783 q^{68} -1.44504i q^{70} +11.4819i q^{71} +0.533188i q^{73} +0.615957 q^{74} -4.98792i q^{76} -3.69202 q^{77} -11.7071 q^{79} +0.356896i q^{80} +7.60388 q^{82} -6.49934i q^{83} -0.748709i q^{85} +6.27413i q^{86} +0.911854 q^{88} +6.49396i q^{89} -8.49396 q^{92} +1.78017 q^{94} -1.78017 q^{95} +1.96077i q^{97} -9.39373i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4}+O(q^{10}) \) 6 * q - 6 * q^4	\( 6 q - 6 q^{4} + 6 q^{10} - 6 q^{14} + 6 q^{16} + 24 q^{17} + 2 q^{22} + 32 q^{23} - 4 q^{25} - 26 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} + 8 q^{49} - 30 q^{53} - 44 q^{55} + 6 q^{56} - 20 q^{61} - 18 q^{62} - 6 q^{64} - 24 q^{68} + 24 q^{74} - 12 q^{77} - 10 q^{79} + 28 q^{82} - 2 q^{88} - 32 q^{92} + 8 q^{94} - 8 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 + 6 * q^10 - 6 * q^14 + 6 * q^16 + 24 * q^17 + 2 * q^22 + 32 * q^23 - 4 * q^25 - 26 * q^29 - 8 * q^35 + 8 * q^38 - 6 * q^40 + 16 * q^43 + 8 * q^49 - 30 * q^53 - 44 * q^55 + 6 * q^56 - 20 * q^61 - 18 * q^62 - 6 * q^64 - 24 * q^68 + 24 * q^74 - 12 * q^77 - 10 * q^79 + 28 * q^82 - 2 * q^88 - 32 * q^92 + 8 * q^94 - 8 * q^95

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