LMFDB - Embedded newform 3042.2.b.p.1351.4

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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.p.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.80194i q^{5} +4.85086i q^{7} -1.00000i q^{8} +1.80194 q^{10} +6.04892i q^{11} -4.85086 q^{14} +1.00000 q^{16} -4.89008 q^{17} +4.49396i q^{19} +1.80194i q^{20} -6.04892 q^{22} -6.71379 q^{23} +1.75302 q^{25} -4.85086i q^{28} +3.55496 q^{29} -3.82908i q^{31} +1.00000i q^{32} -4.89008i q^{34} +8.74094 q^{35} -3.78017i q^{37} -4.49396 q^{38} -1.80194 q^{40} -5.70171i q^{41} +3.78017 q^{43} -6.04892i q^{44} -6.71379i q^{46} -3.82371i q^{47} -16.5308 q^{49} +1.75302i q^{50} -2.78986 q^{53} +10.8998 q^{55} +4.85086 q^{56} +3.55496i q^{58} -8.41119i q^{59} +0.219833 q^{61} +3.82908 q^{62} -1.00000 q^{64} +11.4819i q^{67} +4.89008 q^{68} +8.74094i q^{70} +13.5254i q^{71} -0.417895i q^{73} +3.78017 q^{74} -4.49396i q^{76} -29.3424 q^{77} -10.5526 q^{79} -1.80194i q^{80} +5.70171 q^{82} -3.42327i q^{83} +8.81163i q^{85} +3.78017i q^{86} +6.04892 q^{88} -17.3599i q^{89} +6.71379 q^{92} +3.82371 q^{94} +8.09783 q^{95} +1.00969i q^{97} -16.5308i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} + 2 q^{10} - 2 q^{14} + 6 q^{16} - 28 q^{17} - 18 q^{22} - 24 q^{23} + 20 q^{25} + 22 q^{29} + 24 q^{35} - 8 q^{38} - 2 q^{40} + 20 q^{43} - 24 q^{49} + 30 q^{53} + 20 q^{55} + 2 q^{56} + 4 q^{61}+ \cdots + 12 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 + 2 * q^10 - 2 * q^14 + 6 * q^16 - 28 * q^17 - 18 * q^22 - 24 * q^23 + 20 * q^25 + 22 * q^29 + 24 * q^35 - 8 * q^38 - 2 * q^40 + 20 * q^43 - 24 * q^49 + 30 * q^53 + 20 * q^55 + 2 * q^56 + 4 * q^61 + 2 * q^62 - 6 * q^64 + 28 * q^68 + 20 * q^74 - 48 * q^77 + 18 * q^79 - 20 * q^82 + 18 * q^88 + 24 * q^92 + 8 * q^94 + 12 * q^95

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
$5$	− 1.80194i	− 0.805851i	−0.915233	−	0.402926i	$-0.867993\pi$
$5$	− 1.80194i	− 0.805851i	0.915233	−	0.402926i	$-0.132007\pi$
$6$	0	0
$7$	4.85086i	1.83345i	0.399518	+	0.916725i	$0.369178\pi$
$7$	4.85086i	1.83345i	−0.399518	+	0.916725i	$0.630822\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	1.80194	0.569823
$11$	6.04892i	1.82382i	0.410393	+	0.911909i	$0.365391\pi$
$11$	6.04892i	1.82382i	−0.410393	+	0.911909i	$0.634609\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−4.85086	−1.29645
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.89008	−1.18602	−0.593010	−	0.805195i	$-0.702060\pi$
$17$	−4.89008	−1.18602	−0.593010	+	0.805195i	$0.702060\pi$
$18$	0	0
$19$	4.49396i	1.03098i	0.856894	+	0.515492i	$0.172391\pi$
$19$	4.49396i	1.03098i	−0.856894	+	0.515492i	$0.827609\pi$
$20$	1.80194i	0.402926i
$21$	0	0
$22$	−6.04892	−1.28963
$23$	−6.71379	−1.39992	−0.699961	−	0.714181i	$-0.746799\pi$
$23$	−6.71379	−1.39992	−0.699961	+	0.714181i	$0.746799\pi$
$24$	0	0
$25$	1.75302	0.350604
$26$	0	0
$27$	0	0
$28$	− 4.85086i	− 0.916725i
$29$	3.55496	0.660139	0.330070	−	0.943957i	$-0.392928\pi$
$29$	3.55496	0.660139	0.330070	+	0.943957i	$0.392928\pi$
$30$	0	0
$31$	− 3.82908i	− 0.687724i	−0.939020	−	0.343862i	$-0.888265\pi$
$31$	− 3.82908i	− 0.687724i	0.939020	−	0.343862i	$-0.111735\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 4.89008i	− 0.838642i
$35$	8.74094	1.47749
$36$	0	0
$37$	− 3.78017i	− 0.621456i	−0.950499	−	0.310728i	$-0.899427\pi$
$37$	− 3.78017i	− 0.621456i	0.950499	−	0.310728i	$-0.100573\pi$
$38$	−4.49396	−0.729016
$39$	0	0
$40$	−1.80194	−0.284911
$41$	− 5.70171i	− 0.890458i	−0.895417	−	0.445229i	$-0.853122\pi$
$41$	− 5.70171i	− 0.890458i	0.895417	−	0.445229i	$-0.146878\pi$
$42$	0	0
$43$	3.78017	0.576470	0.288235	−	0.957560i	$-0.406932\pi$
$43$	3.78017	0.576470	0.288235	+	0.957560i	$0.406932\pi$
$44$	− 6.04892i	− 0.911909i
$45$	0	0
$46$	− 6.71379i	− 0.989895i
$47$	− 3.82371i	− 0.557745i	−0.960328	−	0.278873i	$-0.910039\pi$
$47$	− 3.82371i	− 0.557745i	0.960328	−	0.278873i	$-0.0899607\pi$
$48$	0	0
$49$	−16.5308	−2.36154
$50$	1.75302i	0.247915i
$51$	0	0
$52$	0	0
$53$	−2.78986	−0.383216	−0.191608	−	0.981472i	$-0.561370\pi$
$53$	−2.78986	−0.383216	−0.191608	+	0.981472i	$0.561370\pi$
$54$	0	0
$55$	10.8998	1.46973
$56$	4.85086	0.648223
$57$	0	0
$58$	3.55496i	0.466789i
$59$	− 8.41119i	− 1.09504i	−0.836791	−	0.547522i	$-0.815571\pi$
$59$	− 8.41119i	− 1.09504i	0.836791	−	0.547522i	$-0.184429\pi$
$60$	0	0
$61$	0.219833	0.0281467	0.0140733	−	0.999901i	$-0.495520\pi$
$61$	0.219833	0.0281467	0.0140733	+	0.999901i	$0.495520\pi$
$62$	3.82908	0.486294
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	11.4819i	1.40273i	0.712800	+	0.701367i	$0.247427\pi$
$67$	11.4819i	1.40273i	−0.712800	+	0.701367i	$0.752573\pi$
$68$	4.89008	0.593010
$69$	0	0
$70$	8.74094i	1.04474i
$71$	13.5254i	1.60517i	0.596537	+	0.802586i	$0.296543\pi$
$71$	13.5254i	1.60517i	−0.596537	+	0.802586i	$0.703457\pi$
$72$	0	0
$73$	− 0.417895i	− 0.0489109i	−0.999701	−	0.0244554i	$-0.992215\pi$
$73$	− 0.417895i	− 0.0489109i	0.999701	−	0.0244554i	$-0.00778519\pi$
$74$	3.78017	0.439436
$75$	0	0
$76$	− 4.49396i	− 0.515492i
$77$	−29.3424	−3.34388
$78$	0	0
$79$	−10.5526	−1.18726	−0.593628	−	0.804739i	$-0.702305\pi$
$79$	−10.5526	−1.18726	−0.593628	+	0.804739i	$0.702305\pi$
$80$	− 1.80194i	− 0.201463i
$81$	0	0
$82$	5.70171	0.629649
$83$	− 3.42327i	− 0.375753i	−0.982193	−	0.187876i	$-0.939840\pi$
$83$	− 3.42327i	− 0.375753i	0.982193	−	0.187876i	$-0.0601605\pi$
$84$	0	0
$85$	8.81163i	0.955755i
$86$	3.78017i	0.407626i
$87$	0	0
$88$	6.04892	0.644817
$89$	− 17.3599i	− 1.84014i	−0.391750	−	0.920072i	$-0.628130\pi$
$89$	− 17.3599i	− 1.84014i	0.391750	−	0.920072i	$-0.371870\pi$
$90$	0	0
$91$	0	0
$92$	6.71379	0.699961
$93$	0	0
$94$	3.82371	0.394385
$95$	8.09783	0.830820
$96$	0	0
$97$	1.00969i	0.102518i	0.998685	+	0.0512592i	$0.0163235\pi$
$97$	1.00969i	0.102518i	−0.998685	+	0.0512592i	$0.983677\pi$
$98$	− 16.5308i	− 1.66986i
$99$	0	0
$100$	−1.75302	−0.175302
$101$	−5.68664	−0.565842	−0.282921	−	0.959143i	$-0.591303\pi$
$101$	−5.68664	−0.565842	−0.282921	+	0.959143i	$0.591303\pi$
$102$	0	0
$103$	6.73556	0.663675	0.331837	−	0.943337i	$-0.392331\pi$
$103$	6.73556	0.663675	0.331837	+	0.943337i	$0.392331\pi$
$104$	0	0
$105$	0	0
$106$	− 2.78986i	− 0.270975i
$107$	−0.0586060	−0.00566566	−0.00283283	−	0.999996i	$-0.500902\pi$
$107$	−0.0586060	−0.00566566	−0.00283283	+	0.999996i	$0.500902\pi$
$108$	0	0
$109$	15.5254i	1.48707i	0.668700	+	0.743533i	$0.266851\pi$
$109$	15.5254i	1.48707i	−0.668700	+	0.743533i	$0.733149\pi$
$110$	10.8998i	1.03925i
$111$	0	0
$112$	4.85086i	0.458363i
$113$	1.95646	0.184048	0.0920241	−	0.995757i	$-0.470666\pi$
$113$	1.95646	0.184048	0.0920241	+	0.995757i	$0.470666\pi$
$114$	0	0
$115$	12.0978i	1.12813i
$116$	−3.55496	−0.330070
$117$	0	0
$118$	8.41119	0.774313
$119$	− 23.7211i	− 2.17451i
$120$	0	0
$121$	−25.5894	−2.32631
$122$	0.219833i	0.0199027i
$123$	0	0
$124$	3.82908i	0.343862i
$125$	− 12.1685i	− 1.08839i
$126$	0	0
$127$	8.11960	0.720498	0.360249	−	0.932856i	$-0.382692\pi$
$127$	8.11960	0.720498	0.360249	+	0.932856i	$0.382692\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−12.6679	−1.10680	−0.553398	−	0.832917i	$-0.686669\pi$
$131$	−12.6679	−1.10680	−0.553398	+	0.832917i	$0.686669\pi$
$132$	0	0
$133$	−21.7995	−1.89026
$134$	−11.4819	−0.991883
$135$	0	0
$136$	4.89008i	0.419321i
$137$	− 15.3599i	− 1.31228i	−0.754638	−	0.656142i	$-0.772187\pi$
$137$	− 15.3599i	− 1.31228i	0.754638	−	0.656142i	$-0.227813\pi$
$138$	0	0
$139$	−11.7995	−1.00082	−0.500412	−	0.865787i	$-0.666818\pi$
$139$	−11.7995	−1.00082	−0.500412	+	0.865787i	$0.666818\pi$
$140$	−8.74094	−0.738744
$141$	0	0
$142$	−13.5254	−1.13503
$143$	0	0
$144$	0	0
$145$	− 6.40581i	− 0.531974i
$146$	0.417895	0.0345852
$147$	0	0
$148$	3.78017i	0.310728i
$149$	− 3.21313i	− 0.263230i	−0.991301	−	0.131615i	$-0.957984\pi$
$149$	− 3.21313i	− 0.263230i	0.991301	−	0.131615i	$-0.0420162\pi$
$150$	0	0
$151$	9.15883i	0.745335i	0.927965	+	0.372668i	$0.121557\pi$
$151$	9.15883i	0.745335i	−0.927965	+	0.372668i	$0.878443\pi$
$152$	4.49396	0.364508
$153$	0	0
$154$	− 29.3424i	− 2.36448i
$155$	−6.89977	−0.554203
$156$	0	0
$157$	−17.9323	−1.43115	−0.715577	−	0.698534i	$-0.753836\pi$
$157$	−17.9323	−1.43115	−0.715577	+	0.698534i	$0.753836\pi$
$158$	− 10.5526i	− 0.839517i
$159$	0	0
$160$	1.80194	0.142456
$161$	− 32.5676i	− 2.56669i
$162$	0	0
$163$	− 0.591794i	− 0.0463529i	−0.999731	−	0.0231764i	$-0.992622\pi$
$163$	− 0.591794i	− 0.0463529i	0.999731	−	0.0231764i	$-0.00737795\pi$
$164$	5.70171i	0.445229i
$165$	0	0
$166$	3.42327	0.265697
$167$	14.0000i	1.08335i	0.840587	+	0.541676i	$0.182210\pi$
$167$	14.0000i	1.08335i	−0.840587	+	0.541676i	$0.817790\pi$
$168$	0	0
$169$	0	0
$170$	−8.81163	−0.675821
$171$	0	0
$172$	−3.78017	−0.288235
$173$	0.982542	0.0747013	0.0373506	−	0.999302i	$-0.488108\pi$
$173$	0.982542	0.0747013	0.0373506	+	0.999302i	$0.488108\pi$
$174$	0	0
$175$	8.50365i	0.642815i
$176$	6.04892i	0.455954i
$177$	0	0
$178$	17.3599	1.30118
$179$	−9.94331	−0.743198	−0.371599	−	0.928393i	$-0.621190\pi$
$179$	−9.94331	−0.743198	−0.371599	+	0.928393i	$0.621190\pi$
$180$	0	0
$181$	5.87800	0.436908	0.218454	−	0.975847i	$-0.429899\pi$
$181$	5.87800	0.436908	0.218454	+	0.975847i	$0.429899\pi$
$182$	0	0
$183$	0	0
$184$	6.71379i	0.494947i
$185$	−6.81163	−0.500801
$186$	0	0
$187$	− 29.5797i	− 2.16308i
$188$	3.82371i	0.278873i
$189$	0	0
$190$	8.09783i	0.587479i
$191$	11.0422	0.798986	0.399493	−	0.916736i	$-0.369186\pi$
$191$	11.0422	0.798986	0.399493	+	0.916736i	$0.369186\pi$
$192$	0	0
$193$	20.9638i	1.50900i	0.656298	+	0.754502i	$0.272122\pi$
$193$	20.9638i	1.50900i	−0.656298	+	0.754502i	$0.727878\pi$
$194$	−1.00969	−0.0724914
$195$	0	0
$196$	16.5308	1.18077
$197$	15.7409i	1.12150i	0.827987	+	0.560748i	$0.189486\pi$
$197$	15.7409i	1.12150i	−0.827987	+	0.560748i	$0.810514\pi$
$198$	0	0
$199$	1.51142	0.107142	0.0535708	−	0.998564i	$-0.482940\pi$
$199$	1.51142	0.107142	0.0535708	+	0.998564i	$0.482940\pi$
$200$	− 1.75302i	− 0.123957i
$201$	0	0
$202$	− 5.68664i	− 0.400111i
$203$	17.2446i	1.21033i
$204$	0	0
$205$	−10.2741	−0.717576
$206$	6.73556i	0.469289i
$207$	0	0
$208$	0	0
$209$	−27.1836	−1.88033
$210$	0	0
$211$	5.65817	0.389524	0.194762	−	0.980850i	$-0.437606\pi$
$211$	5.65817	0.389524	0.194762	+	0.980850i	$0.437606\pi$
$212$	2.78986	0.191608
$213$	0	0
$214$	− 0.0586060i	− 0.00400622i
$215$	− 6.81163i	− 0.464549i
$216$	0	0
$217$	18.5743	1.26091
$218$	−15.5254	−1.05151
$219$	0	0
$220$	−10.8998	−0.734863
$221$	0	0
$222$	0	0
$223$	1.97584i	0.132312i	0.997809	+	0.0661559i	$0.0210735\pi$
$223$	1.97584i	0.132312i	−0.997809	+	0.0661559i	$0.978927\pi$
$224$	−4.85086	−0.324111
$225$	0	0
$226$	1.95646i	0.130142i
$227$	25.0640i	1.66355i	0.555109	+	0.831777i	$0.312676\pi$
$227$	25.0640i	1.66355i	−0.555109	+	0.831777i	$0.687324\pi$
$228$	0	0
$229$	− 24.5133i	− 1.61989i	−0.586508	−	0.809943i	$-0.699498\pi$
$229$	− 24.5133i	− 1.61989i	0.586508	−	0.809943i	$-0.300502\pi$
$230$	−12.0978	−0.797708
$231$	0	0
$232$	− 3.55496i	− 0.233394i
$233$	8.37196	0.548465	0.274233	−	0.961663i	$-0.411576\pi$
$233$	8.37196	0.548465	0.274233	+	0.961663i	$0.411576\pi$
$234$	0	0
$235$	−6.89008	−0.449460
$236$	8.41119i	0.547522i
$237$	0	0
$238$	23.7211	1.53761
$239$	− 14.0194i	− 0.906838i	−0.891297	−	0.453419i	$-0.850204\pi$
$239$	− 14.0194i	− 0.906838i	0.891297	−	0.453419i	$-0.149796\pi$
$240$	0	0
$241$	− 21.1183i	− 1.36035i	−0.733051	−	0.680174i	$-0.761904\pi$
$241$	− 21.1183i	− 1.36035i	0.733051	−	0.680174i	$-0.238096\pi$
$242$	− 25.5894i	− 1.64495i
$243$	0	0
$244$	−0.219833	−0.0140733
$245$	29.7875i	1.90305i
$246$	0	0
$247$	0	0
$248$	−3.82908	−0.243147
$249$	0	0
$250$	12.1685	0.769605
$251$	−26.1129	−1.64823	−0.824116	−	0.566421i	$-0.808328\pi$
$251$	−26.1129	−1.64823	−0.824116	+	0.566421i	$0.808328\pi$
$252$	0	0
$253$	− 40.6112i	− 2.55320i
$254$	8.11960i	0.509469i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.5918	−1.03497	−0.517484	−	0.855693i	$-0.673131\pi$
$257$	−16.5918	−1.03497	−0.517484	+	0.855693i	$0.673131\pi$
$258$	0	0
$259$	18.3370	1.13941
$260$	0	0
$261$	0	0
$262$	− 12.6679i	− 0.782623i
$263$	9.64742	0.594885	0.297443	−	0.954740i	$-0.403866\pi$
$263$	9.64742	0.594885	0.297443	+	0.954740i	$0.403866\pi$
$264$	0	0
$265$	5.02715i	0.308815i
$266$	− 21.7995i	− 1.33662i
$267$	0	0
$268$	− 11.4819i	− 0.701367i
$269$	10.1981	0.621787	0.310893	−	0.950445i	$-0.399372\pi$
$269$	10.1981	0.621787	0.310893	+	0.950445i	$0.399372\pi$
$270$	0	0
$271$	27.6799i	1.68144i	0.541473	+	0.840718i	$0.317867\pi$
$271$	27.6799i	1.68144i	−0.541473	+	0.840718i	$0.682133\pi$
$272$	−4.89008	−0.296505
$273$	0	0
$274$	15.3599	0.927924
$275$	10.6039i	0.639438i
$276$	0	0
$277$	−6.32842	−0.380238	−0.190119	−	0.981761i	$-0.560887\pi$
$277$	−6.32842	−0.380238	−0.190119	+	0.981761i	$0.560887\pi$
$278$	− 11.7995i	− 0.707690i
$279$	0	0
$280$	− 8.74094i	− 0.522371i
$281$	9.26205i	0.552527i	0.961082	+	0.276264i	$0.0890963\pi$
$281$	9.26205i	0.552527i	−0.961082	+	0.276264i	$0.910904\pi$
$282$	0	0
$283$	−0.561663	−0.0333874	−0.0166937	−	0.999861i	$-0.505314\pi$
$283$	−0.561663	−0.0333874	−0.0166937	+	0.999861i	$0.505314\pi$
$284$	− 13.5254i	− 0.802586i
$285$	0	0
$286$	0	0
$287$	27.6582	1.63261
$288$	0	0
$289$	6.91292	0.406642
$290$	6.40581	0.376162
$291$	0	0
$292$	0.417895i	0.0244554i
$293$	11.7506i	0.686479i	0.939248	+	0.343239i	$0.111524\pi$
$293$	11.7506i	0.686479i	−0.939248	+	0.343239i	$0.888476\pi$
$294$	0	0
$295$	−15.1564	−0.882442
$296$	−3.78017	−0.219718
$297$	0	0
$298$	3.21313	0.186131
$299$	0	0
$300$	0	0
$301$	18.3370i	1.05693i
$302$	−9.15883	−0.527032
$303$	0	0
$304$	4.49396i	0.257746i
$305$	− 0.396125i	− 0.0226820i
$306$	0	0
$307$	1.09054i	0.0622403i	0.999516	+	0.0311202i	$0.00990746\pi$
$307$	1.09054i	0.0622403i	−0.999516	+	0.0311202i	$0.990093\pi$
$308$	29.3424	1.67194
$309$	0	0
$310$	− 6.89977i	− 0.391881i
$311$	−8.09783	−0.459186	−0.229593	−	0.973287i	$-0.573740\pi$
$311$	−8.09783	−0.459186	−0.229593	+	0.973287i	$0.573740\pi$
$312$	0	0
$313$	19.4252	1.09798	0.548988	−	0.835830i	$-0.315013\pi$
$313$	19.4252	1.09798	0.548988	+	0.835830i	$0.315013\pi$
$314$	− 17.9323i	− 1.01198i
$315$	0	0
$316$	10.5526	0.593628
$317$	24.5719i	1.38010i	0.723763	+	0.690049i	$0.242411\pi$
$317$	24.5719i	1.38010i	−0.723763	+	0.690049i	$0.757589\pi$
$318$	0	0
$319$	21.5036i	1.20397i
$320$	1.80194i	0.100731i
$321$	0	0
$322$	32.5676	1.81492
$323$	− 21.9758i	− 1.22277i
$324$	0	0
$325$	0	0
$326$	0.591794	0.0327764
$327$	0	0
$328$	−5.70171	−0.314824
$329$	18.5483	1.02260
$330$	0	0
$331$	− 25.9758i	− 1.42776i	−0.700267	−	0.713881i	$-0.746936\pi$
$331$	− 25.9758i	− 1.42776i	0.700267	−	0.713881i	$-0.253064\pi$
$332$	3.42327i	0.187876i
$333$	0	0
$334$	−14.0000	−0.766046
$335$	20.6896	1.13040
$336$	0	0
$337$	−1.87263	−0.102008	−0.0510042	−	0.998698i	$-0.516242\pi$
$337$	−1.87263	−0.102008	−0.0510042	+	0.998698i	$0.516242\pi$
$338$	0	0
$339$	0	0
$340$	− 8.81163i	− 0.477878i
$341$	23.1618	1.25428
$342$	0	0
$343$	− 46.2325i	− 2.49632i
$344$	− 3.78017i	− 0.203813i
$345$	0	0
$346$	0.982542i	0.0528218i
$347$	−16.6853	−0.895715	−0.447857	−	0.894105i	$-0.647813\pi$
$347$	−16.6853	−0.895715	−0.447857	+	0.894105i	$0.647813\pi$
$348$	0	0
$349$	4.01938i	0.215152i	0.994197	+	0.107576i	$0.0343090\pi$
$349$	4.01938i	0.215152i	−0.994197	+	0.107576i	$0.965691\pi$
$350$	−8.50365	−0.454539
$351$	0	0
$352$	−6.04892	−0.322408
$353$	− 21.6039i	− 1.14986i	−0.818203	−	0.574929i	$-0.805030\pi$
$353$	− 21.6039i	− 1.14986i	0.818203	−	0.574929i	$-0.194970\pi$
$354$	0	0
$355$	24.3720	1.29353
$356$	17.3599i	0.920072i
$357$	0	0
$358$	− 9.94331i	− 0.525520i
$359$	0.835790i	0.0441113i	0.999757	+	0.0220556i	$0.00702110\pi$
$359$	0.835790i	0.0441113i	−0.999757	+	0.0220556i	$0.992979\pi$
$360$	0	0
$361$	−1.19567	−0.0629300
$362$	5.87800i	0.308941i
$363$	0	0
$364$	0	0
$365$	−0.753020	−0.0394149
$366$	0	0
$367$	−21.1250	−1.10272	−0.551358	−	0.834269i	$-0.685890\pi$
$367$	−21.1250	−1.10272	−0.551358	+	0.834269i	$0.685890\pi$
$368$	−6.71379	−0.349981
$369$	0	0
$370$	− 6.81163i	− 0.354120i
$371$	− 13.5332i	− 0.702608i
$372$	0	0
$373$	21.3840	1.10722	0.553612	−	0.832775i	$-0.313249\pi$
$373$	21.3840	1.10722	0.553612	+	0.832775i	$0.313249\pi$
$374$	29.5797	1.52953
$375$	0	0
$376$	−3.82371	−0.197193
$377$	0	0
$378$	0	0
$379$	− 13.5496i	− 0.695995i	−0.937495	−	0.347998i	$-0.886862\pi$
$379$	− 13.5496i	− 0.695995i	0.937495	−	0.347998i	$-0.113138\pi$
$380$	−8.09783	−0.415410
$381$	0	0
$382$	11.0422i	0.564969i
$383$	− 9.10992i	− 0.465495i	−0.972537	−	0.232747i	$-0.925228\pi$
$383$	− 9.10992i	− 0.465495i	0.972537	−	0.232747i	$-0.0747716\pi$
$384$	0	0
$385$	52.8732i	2.69467i
$386$	−20.9638	−1.06703
$387$	0	0
$388$	− 1.00969i	− 0.0512592i
$389$	14.8498	0.752914	0.376457	−	0.926434i	$-0.377142\pi$
$389$	14.8498	0.752914	0.376457	+	0.926434i	$0.377142\pi$
$390$	0	0
$391$	32.8310	1.66034
$392$	16.5308i	0.834931i
$393$	0	0
$394$	−15.7409	−0.793017
$395$	19.0151i	0.956752i
$396$	0	0
$397$	33.7453i	1.69363i	0.531891	+	0.846813i	$0.321482\pi$
$397$	33.7453i	1.69363i	−0.531891	+	0.846813i	$0.678518\pi$
$398$	1.51142i	0.0757605i
$399$	0	0
$400$	1.75302	0.0876510
$401$	− 19.7017i	− 0.983856i	−0.870636	−	0.491928i	$-0.836292\pi$
$401$	− 19.7017i	− 0.983856i	0.870636	−	0.491928i	$-0.163708\pi$
$402$	0	0
$403$	0	0
$404$	5.68664	0.282921
$405$	0	0
$406$	−17.2446	−0.855834
$407$	22.8659	1.13342
$408$	0	0
$409$	35.9366i	1.77695i	0.458924	+	0.888475i	$0.348235\pi$
$409$	35.9366i	1.77695i	−0.458924	+	0.888475i	$0.651765\pi$
$410$	− 10.2741i	− 0.507403i
$411$	0	0
$412$	−6.73556	−0.331837
$413$	40.8015	2.00771
$414$	0	0
$415$	−6.16852	−0.302801
$416$	0	0
$417$	0	0
$418$	− 27.1836i	− 1.32959i
$419$	18.7928	0.918090	0.459045	−	0.888413i	$-0.348192\pi$
$419$	18.7928	0.918090	0.459045	+	0.888413i	$0.348192\pi$
$420$	0	0
$421$	− 3.90217i	− 0.190180i	−0.995469	−	0.0950900i	$-0.969686\pi$
$421$	− 3.90217i	− 0.190180i	0.995469	−	0.0950900i	$-0.0303139\pi$
$422$	5.65817i	0.275435i
$423$	0	0
$424$	2.78986i	0.135487i
$425$	−8.57242	−0.415823
$426$	0	0
$427$	1.06638i	0.0516055i
$428$	0.0586060	0.00283283
$429$	0	0
$430$	6.81163	0.328486
$431$	18.8310i	0.907058i	0.891242	+	0.453529i	$0.149835\pi$
$431$	18.8310i	0.907058i	−0.891242	+	0.453529i	$0.850165\pi$
$432$	0	0
$433$	−24.8364	−1.19356	−0.596780	−	0.802405i	$-0.703554\pi$
$433$	−24.8364	−1.19356	−0.596780	+	0.802405i	$0.703554\pi$
$434$	18.5743i	0.891597i
$435$	0	0
$436$	− 15.5254i	− 0.743533i
$437$	− 30.1715i	− 1.44330i
$438$	0	0
$439$	−22.2784	−1.06329	−0.531646	−	0.846967i	$-0.678426\pi$
$439$	−22.2784	−1.06329	−0.531646	+	0.846967i	$0.678426\pi$
$440$	− 10.8998i	− 0.519626i
$441$	0	0
$442$	0	0
$443$	26.6359	1.26551	0.632756	−	0.774352i	$-0.281924\pi$
$443$	26.6359	1.26551	0.632756	+	0.774352i	$0.281924\pi$
$444$	0	0
$445$	−31.2814	−1.48288
$446$	−1.97584	−0.0935586
$447$	0	0
$448$	− 4.85086i	− 0.229181i
$449$	21.8538i	1.03135i	0.856785	+	0.515673i	$0.172458\pi$
$449$	21.8538i	1.03135i	−0.856785	+	0.515673i	$0.827542\pi$
$450$	0	0
$451$	34.4892	1.62403
$452$	−1.95646	−0.0920241
$453$	0	0
$454$	−25.0640	−1.17631
$455$	0	0
$456$	0	0
$457$	20.7278i	0.969605i	0.874624	+	0.484803i	$0.161109\pi$
$457$	20.7278i	0.969605i	−0.874624	+	0.484803i	$0.838891\pi$
$458$	24.5133	1.14543
$459$	0	0
$460$	− 12.0978i	− 0.564064i
$461$	10.7832i	0.502221i	0.967958	+	0.251111i	$0.0807958\pi$
$461$	10.7832i	0.502221i	−0.967958	+	0.251111i	$0.919204\pi$
$462$	0	0
$463$	− 23.1594i	− 1.07631i	−0.842846	−	0.538155i	$-0.819122\pi$
$463$	− 23.1594i	− 1.07631i	0.842846	−	0.538155i	$-0.180878\pi$
$464$	3.55496	0.165035
$465$	0	0
$466$	8.37196i	0.387824i
$467$	−13.3515	−0.617835	−0.308917	−	0.951089i	$-0.599967\pi$
$467$	−13.3515	−0.617835	−0.308917	+	0.951089i	$0.599967\pi$
$468$	0	0
$469$	−55.6969	−2.57185
$470$	− 6.89008i	− 0.317816i
$471$	0	0
$472$	−8.41119	−0.387156
$473$	22.8659i	1.05138i
$474$	0	0
$475$	7.87800i	0.361468i
$476$	23.7211i	1.08725i
$477$	0	0
$478$	14.0194	0.641231
$479$	− 4.38537i	− 0.200373i	−0.994969	−	0.100186i	$-0.968056\pi$
$479$	− 4.38537i	− 0.200373i	0.994969	−	0.100186i	$-0.0319439\pi$
$480$	0	0
$481$	0	0
$482$	21.1183	0.961911
$483$	0	0
$484$	25.5894	1.16315
$485$	1.81940	0.0826145
$486$	0	0
$487$	− 7.33214i	− 0.332251i	−0.986105	−	0.166126i	$-0.946874\pi$
$487$	− 7.33214i	− 0.332251i	0.986105	−	0.166126i	$-0.0531257\pi$
$488$	− 0.219833i	− 0.00995135i
$489$	0	0
$490$	−29.7875	−1.34566
$491$	16.8062	0.758455	0.379228	−	0.925303i	$-0.376190\pi$
$491$	16.8062	0.758455	0.379228	+	0.925303i	$0.376190\pi$
$492$	0	0
$493$	−17.3840	−0.782938
$494$	0	0
$495$	0	0
$496$	− 3.82908i	− 0.171931i
$497$	−65.6098	−2.94300
$498$	0	0
$499$	6.59658i	0.295303i	0.989039	+	0.147652i	$0.0471715\pi$
$499$	6.59658i	0.295303i	−0.989039	+	0.147652i	$0.952829\pi$
$500$	12.1685i	0.544193i
$501$	0	0
$502$	− 26.1129i	− 1.16548i
$503$	−27.8297	−1.24086	−0.620432	−	0.784260i	$-0.713043\pi$
$503$	−27.8297	−1.24086	−0.620432	+	0.784260i	$0.713043\pi$
$504$	0	0
$505$	10.2470i	0.455985i
$506$	40.6112	1.80539
$507$	0	0
$508$	−8.11960	−0.360249
$509$	− 4.13275i	− 0.183181i	−0.995797	−	0.0915905i	$-0.970805\pi$
$509$	− 4.13275i	− 0.183181i	0.995797	−	0.0915905i	$-0.0291951\pi$
$510$	0	0
$511$	2.02715	0.0896757
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 16.5918i	− 0.731833i
$515$	− 12.1371i	− 0.534823i
$516$	0	0
$517$	23.1293	1.01723
$518$	18.3370i	0.805683i
$519$	0	0
$520$	0	0
$521$	11.7995	0.516947	0.258474	−	0.966018i	$-0.416780\pi$
$521$	11.7995	0.516947	0.258474	+	0.966018i	$0.416780\pi$
$522$	0	0
$523$	38.2887	1.67425	0.837124	−	0.547013i	$-0.184235\pi$
$523$	38.2887	1.67425	0.837124	+	0.547013i	$0.184235\pi$
$524$	12.6679	0.553398
$525$	0	0
$526$	9.64742i	0.420647i
$527$	18.7245i	0.815654i
$528$	0	0
$529$	22.0750	0.959783
$530$	−5.02715	−0.218365
$531$	0	0
$532$	21.7995	0.945130
$533$	0	0
$534$	0	0
$535$	0.105604i	0.00456568i
$536$	11.4819	0.495942
$537$	0	0
$538$	10.1981i	0.439670i
$539$	− 99.9934i	− 4.30702i
$540$	0	0
$541$	17.4142i	0.748694i	0.927289	+	0.374347i	$0.122133\pi$
$541$	17.4142i	0.748694i	−0.927289	+	0.374347i	$0.877867\pi$
$542$	−27.6799	−1.18896
$543$	0	0
$544$	− 4.89008i	− 0.209661i
$545$	27.9758	1.19835
$546$	0	0
$547$	16.9444	0.724489	0.362245	−	0.932083i	$-0.382010\pi$
$547$	16.9444	0.724489	0.362245	+	0.932083i	$0.382010\pi$
$548$	15.3599i	0.656142i
$549$	0	0
$550$	−10.6039	−0.452151
$551$	15.9758i	0.680594i
$552$	0	0
$553$	− 51.1890i	− 2.17678i
$554$	− 6.32842i	− 0.268869i
$555$	0	0
$556$	11.7995	0.500412
$557$	11.9758i	0.507432i	0.967279	+	0.253716i	$0.0816529\pi$
$557$	11.9758i	0.507432i	−0.967279	+	0.253716i	$0.918347\pi$
$558$	0	0
$559$	0	0
$560$	8.74094	0.369372
$561$	0	0
$562$	−9.26205	−0.390696
$563$	−42.2911	−1.78236	−0.891179	−	0.453652i	$-0.850121\pi$
$563$	−42.2911	−1.78236	−0.891179	+	0.453652i	$0.850121\pi$
$564$	0	0
$565$	− 3.52542i	− 0.148315i
$566$	− 0.561663i	− 0.0236085i
$567$	0	0
$568$	13.5254	0.567514
$569$	−8.26934	−0.346669	−0.173334	−	0.984863i	$-0.555454\pi$
$569$	−8.26934	−0.346669	−0.173334	+	0.984863i	$0.555454\pi$
$570$	0	0
$571$	16.8552	0.705367	0.352683	−	0.935743i	$-0.385269\pi$
$571$	16.8552	0.705367	0.352683	+	0.935743i	$0.385269\pi$
$572$	0	0
$573$	0	0
$574$	27.6582i	1.15443i
$575$	−11.7694	−0.490818
$576$	0	0
$577$	37.9995i	1.58194i	0.611854	+	0.790970i	$0.290424\pi$
$577$	37.9995i	1.58194i	−0.611854	+	0.790970i	$0.709576\pi$
$578$	6.91292i	0.287540i
$579$	0	0
$580$	6.40581i	0.265987i
$581$	16.6058	0.688924
$582$	0	0
$583$	− 16.8756i	− 0.698916i
$584$	−0.417895	−0.0172926
$585$	0	0
$586$	−11.7506	−0.485414
$587$	31.4282i	1.29718i	0.761138	+	0.648590i	$0.224641\pi$
$587$	31.4282i	1.29718i	−0.761138	+	0.648590i	$0.775359\pi$
$588$	0	0
$589$	17.2078	0.709033
$590$	− 15.1564i	− 0.623981i
$591$	0	0
$592$	− 3.78017i	− 0.155364i
$593$	41.8866i	1.72008i	0.510229	+	0.860039i	$0.329561\pi$
$593$	41.8866i	1.72008i	−0.510229	+	0.860039i	$0.670439\pi$
$594$	0	0
$595$	−42.7439	−1.75233
$596$	3.21313i	0.131615i
$597$	0	0
$598$	0	0
$599$	−36.0194	−1.47171	−0.735856	−	0.677138i	$-0.763220\pi$
$599$	−36.0194	−1.47171	−0.735856	+	0.677138i	$0.763220\pi$
$600$	0	0
$601$	5.13946	0.209643	0.104821	−	0.994491i	$-0.466573\pi$
$601$	5.13946	0.209643	0.104821	+	0.994491i	$0.466573\pi$
$602$	−18.3370	−0.747362
$603$	0	0
$604$	− 9.15883i	− 0.372668i
$605$	46.1105i	1.87466i
$606$	0	0
$607$	−17.6517	−0.716462	−0.358231	−	0.933633i	$-0.616620\pi$
$607$	−17.6517	−0.716462	−0.358231	+	0.933633i	$0.616620\pi$
$608$	−4.49396	−0.182254
$609$	0	0
$610$	0.396125	0.0160386
$611$	0	0
$612$	0	0
$613$	45.8974i	1.85378i	0.375336	+	0.926889i	$0.377527\pi$
$613$	45.8974i	1.85378i	−0.375336	+	0.926889i	$0.622473\pi$
$614$	−1.09054	−0.0440106
$615$	0	0
$616$	29.3424i	1.18224i
$617$	− 5.12929i	− 0.206498i	−0.994656	−	0.103249i	$-0.967076\pi$
$617$	− 5.12929i	− 0.206498i	0.994656	−	0.103249i	$-0.0329238\pi$
$618$	0	0
$619$	− 43.5448i	− 1.75021i	−0.483930	−	0.875107i	$-0.660791\pi$
$619$	− 43.5448i	− 1.75021i	0.483930	−	0.875107i	$-0.339209\pi$
$620$	6.89977	0.277102
$621$	0	0
$622$	− 8.09783i	− 0.324694i
$623$	84.2103	3.37381
$624$	0	0
$625$	−13.1618	−0.526473
$626$	19.4252i	0.776387i
$627$	0	0
$628$	17.9323	0.715577
$629$	18.4853i	0.737059i
$630$	0	0
$631$	30.4655i	1.21281i	0.795155	+	0.606406i	$0.207389\pi$
$631$	30.4655i	1.21281i	−0.795155	+	0.606406i	$0.792611\pi$
$632$	10.5526i	0.419759i
$633$	0	0
$634$	−24.5719	−0.975877
$635$	− 14.6310i	− 0.580614i
$636$	0	0
$637$	0	0
$638$	−21.5036	−0.851338
$639$	0	0
$640$	−1.80194	−0.0712278
$641$	32.3370	1.27724	0.638618	−	0.769524i	$-0.279506\pi$
$641$	32.3370	1.27724	0.638618	+	0.769524i	$0.279506\pi$
$642$	0	0
$643$	− 1.79092i	− 0.0706270i	−0.999376	−	0.0353135i	$-0.988757\pi$
$643$	− 1.79092i	− 0.0706270i	0.999376	−	0.0353135i	$-0.0112430\pi$
$644$	32.5676i	1.28334i
$645$	0	0
$646$	21.9758	0.864628
$647$	−41.4685	−1.63029	−0.815147	−	0.579254i	$-0.803344\pi$
$647$	−41.4685	−1.63029	−0.815147	+	0.579254i	$0.803344\pi$
$648$	0	0
$649$	50.8786	1.99716
$650$	0	0
$651$	0	0
$652$	0.591794i	0.0231764i
$653$	−2.39181	−0.0935989	−0.0467994	−	0.998904i	$-0.514902\pi$
$653$	−2.39181	−0.0935989	−0.0467994	+	0.998904i	$0.514902\pi$
$654$	0	0
$655$	22.8267i	0.891913i
$656$	− 5.70171i	− 0.222614i
$657$	0	0
$658$	18.5483i	0.723086i
$659$	1.76032	0.0685722	0.0342861	−	0.999412i	$-0.489084\pi$
$659$	1.76032	0.0685722	0.0342861	+	0.999412i	$0.489084\pi$
$660$	0	0
$661$	− 44.7138i	− 1.73916i	−0.493788	−	0.869582i	$-0.664388\pi$
$661$	− 44.7138i	− 1.73916i	0.493788	−	0.869582i	$-0.335612\pi$
$662$	25.9758	1.00958
$663$	0	0
$664$	−3.42327	−0.132849
$665$	39.2814i	1.52327i
$666$	0	0
$667$	−23.8672	−0.924144
$668$	− 14.0000i	− 0.541676i
$669$	0	0
$670$	20.6896i	0.799310i
$671$	1.32975i	0.0513344i
$672$	0	0
$673$	5.84356	0.225253	0.112626	−	0.993637i	$-0.464074\pi$
$673$	5.84356	0.225253	0.112626	+	0.993637i	$0.464074\pi$
$674$	− 1.87263i	− 0.0721308i
$675$	0	0
$676$	0	0
$677$	27.8883	1.07183	0.535917	−	0.844271i	$-0.319966\pi$
$677$	27.8883	1.07183	0.535917	+	0.844271i	$0.319966\pi$
$678$	0	0
$679$	−4.89785	−0.187962
$680$	8.81163	0.337910
$681$	0	0
$682$	23.1618i	0.886912i
$683$	28.3803i	1.08594i	0.839751	+	0.542971i	$0.182701\pi$
$683$	28.3803i	1.08594i	−0.839751	+	0.542971i	$0.817299\pi$
$684$	0	0
$685$	−27.6775	−1.05750
$686$	46.2325	1.76517
$687$	0	0
$688$	3.78017	0.144118
$689$	0	0
$690$	0	0
$691$	− 31.3142i	− 1.19125i	−0.803263	−	0.595624i	$-0.796905\pi$
$691$	− 31.3142i	− 1.19125i	0.803263	−	0.595624i	$-0.203095\pi$
$692$	−0.982542	−0.0373506
$693$	0	0
$694$	− 16.6853i	− 0.633366i
$695$	21.2620i	0.806515i
$696$	0	0
$697$	27.8818i	1.05610i
$698$	−4.01938	−0.152136
$699$	0	0
$700$	− 8.50365i	− 0.321408i
$701$	22.6568	0.855737	0.427869	−	0.903841i	$-0.359265\pi$
$701$	22.6568	0.855737	0.427869	+	0.903841i	$0.359265\pi$
$702$	0	0
$703$	16.9879	0.640711
$704$	− 6.04892i	− 0.227977i
$705$	0	0
$706$	21.6039	0.813073
$707$	− 27.5851i	− 1.03744i
$708$	0	0
$709$	− 24.4155i	− 0.916943i	−0.888709	−	0.458472i	$-0.848397\pi$
$709$	− 24.4155i	− 0.916943i	0.888709	−	0.458472i	$-0.151603\pi$
$710$	24.3720i	0.914663i
$711$	0	0
$712$	−17.3599	−0.650589
$713$	25.7077i	0.962760i
$714$	0	0
$715$	0	0
$716$	9.94331	0.371599
$717$	0	0
$718$	−0.835790	−0.0311914
$719$	−13.2707	−0.494912	−0.247456	−	0.968899i	$-0.579595\pi$
$719$	−13.2707	−0.494912	−0.247456	+	0.968899i	$0.579595\pi$
$720$	0	0
$721$	32.6732i	1.21681i
$722$	− 1.19567i	− 0.0444982i
$723$	0	0
$724$	−5.87800	−0.218454
$725$	6.23191	0.231447
$726$	0	0
$727$	−25.7904	−0.956515	−0.478257	−	0.878220i	$-0.658731\pi$
$727$	−25.7904	−0.956515	−0.478257	+	0.878220i	$0.658731\pi$
$728$	0	0
$729$	0	0
$730$	− 0.753020i	− 0.0278705i
$731$	−18.4853	−0.683705
$732$	0	0
$733$	− 13.1400i	− 0.485339i	−0.970109	−	0.242669i	$-0.921977\pi$
$733$	− 13.1400i	− 0.485339i	0.970109	−	0.242669i	$-0.0780230\pi$
$734$	− 21.1250i	− 0.779737i
$735$	0	0
$736$	− 6.71379i	− 0.247474i
$737$	−69.4529	−2.55833
$738$	0	0
$739$	11.8130i	0.434547i	0.976111	+	0.217273i	$0.0697163\pi$
$739$	11.8130i	0.434547i	−0.976111	+	0.217273i	$0.930284\pi$
$740$	6.81163	0.250400
$741$	0	0
$742$	13.5332	0.496819
$743$	3.50125i	0.128449i	0.997935	+	0.0642243i	$0.0204573\pi$
$743$	3.50125i	0.128449i	−0.997935	+	0.0642243i	$0.979543\pi$
$744$	0	0
$745$	−5.78986	−0.212124
$746$	21.3840i	0.782925i
$747$	0	0
$748$	29.5797i	1.08154i
$749$	− 0.284289i	− 0.0103877i
$750$	0	0
$751$	11.6722	0.425924	0.212962	−	0.977061i	$-0.431689\pi$
$751$	11.6722	0.425924	0.212962	+	0.977061i	$0.431689\pi$
$752$	− 3.82371i	− 0.139436i
$753$	0	0
$754$	0	0
$755$	16.5036	0.600629
$756$	0	0
$757$	0.444451	0.0161538	0.00807692	−	0.999967i	$-0.497429\pi$
$757$	0.444451	0.0161538	0.00807692	+	0.999967i	$0.497429\pi$
$758$	13.5496	0.492143
$759$	0	0
$760$	− 8.09783i	− 0.293739i
$761$	− 29.4905i	− 1.06903i	−0.845159	−	0.534515i	$-0.820494\pi$
$761$	− 29.4905i	− 1.06903i	0.845159	−	0.534515i	$-0.179506\pi$
$762$	0	0
$763$	−75.3116	−2.72646
$764$	−11.0422	−0.399493
$765$	0	0
$766$	9.10992	0.329155
$767$	0	0
$768$	0	0
$769$	13.4517i	0.485082i	0.970141	+	0.242541i	$0.0779810\pi$
$769$	13.4517i	0.485082i	−0.970141	+	0.242541i	$0.922019\pi$
$770$	−52.8732	−1.90542
$771$	0	0
$772$	− 20.9638i	− 0.754502i
$773$	− 24.1021i	− 0.866894i	−0.901179	−	0.433447i	$-0.857297\pi$
$773$	− 24.1021i	− 0.866894i	0.901179	−	0.433447i	$-0.142703\pi$
$774$	0	0
$775$	− 6.71246i	− 0.241119i
$776$	1.00969	0.0362457
$777$	0	0
$778$	14.8498i	0.532391i
$779$	25.6233	0.918048
$780$	0	0
$781$	−81.8141	−2.92754
$782$	32.8310i	1.17403i
$783$	0	0
$784$	−16.5308	−0.590386
$785$	32.3129i	1.15330i
$786$	0	0
$787$	16.2258i	0.578387i	0.957271	+	0.289194i	$0.0933872\pi$
$787$	16.2258i	0.578387i	−0.957271	+	0.289194i	$0.906613\pi$
$788$	− 15.7409i	− 0.560748i
$789$	0	0
$790$	−19.0151	−0.676526
$791$	9.49050i	0.337443i
$792$	0	0
$793$	0	0
$794$	−33.7453	−1.19757
$795$	0	0
$796$	−1.51142	−0.0535708
$797$	−37.2760	−1.32039	−0.660193	−	0.751096i	$-0.729525\pi$
$797$	−37.2760	−1.32039	−0.660193	+	0.751096i	$0.729525\pi$
$798$	0	0
$799$	18.6983i	0.661497i
$800$	1.75302i	0.0619786i
$801$	0	0
$802$	19.7017	0.695692
$803$	2.52781	0.0892045
$804$	0	0
$805$	−58.6848	−2.06837
$806$	0	0
$807$	0	0
$808$	5.68664i	0.200055i
$809$	24.0844	0.846763	0.423382	−	0.905951i	$-0.360843\pi$
$809$	24.0844	0.846763	0.423382	+	0.905951i	$0.360843\pi$
$810$	0	0
$811$	− 9.87800i	− 0.346864i	−0.984846	−	0.173432i	$-0.944514\pi$
$811$	− 9.87800i	− 0.346864i	0.984846	−	0.173432i	$-0.0554856\pi$
$812$	− 17.2446i	− 0.605166i
$813$	0	0
$814$	22.8659i	0.801450i
$815$	−1.06638	−0.0373535
$816$	0	0
$817$	16.9879i	0.594332i
$818$	−35.9366	−1.25649
$819$	0	0
$820$	10.2741	0.358788
$821$	− 31.2731i	− 1.09144i	−0.837968	−	0.545719i	$-0.816257\pi$
$821$	− 31.2731i	− 1.09144i	0.837968	−	0.545719i	$-0.183743\pi$
$822$	0	0
$823$	−33.8388	−1.17955	−0.589773	−	0.807569i	$-0.700783\pi$
$823$	−33.8388	−1.17955	−0.589773	+	0.807569i	$0.700783\pi$
$824$	− 6.73556i	− 0.234644i
$825$	0	0
$826$	40.8015i	1.41966i
$827$	17.4028i	0.605156i	0.953125	+	0.302578i	$0.0978472\pi$
$827$	17.4028i	0.605156i	−0.953125	+	0.302578i	$0.902153\pi$
$828$	0	0
$829$	50.9724	1.77034	0.885172	−	0.465264i	$-0.154041\pi$
$829$	50.9724	1.77034	0.885172	+	0.465264i	$0.154041\pi$
$830$	− 6.16852i	− 0.214113i
$831$	0	0
$832$	0	0
$833$	80.8370	2.80084
$834$	0	0
$835$	25.2271	0.873021
$836$	27.1836	0.940164
$837$	0	0
$838$	18.7928i	0.649188i
$839$	34.6983i	1.19792i	0.800780	+	0.598958i	$0.204419\pi$
$839$	34.6983i	1.19792i	−0.800780	+	0.598958i	$0.795581\pi$
$840$	0	0
$841$	−16.3623	−0.564216
$842$	3.90217	0.134477
$843$	0	0
$844$	−5.65817	−0.194762
$845$	0	0
$846$	0	0
$847$	− 124.130i	− 4.26517i
$848$	−2.78986	−0.0958041
$849$	0	0
$850$	− 8.57242i	− 0.294031i
$851$	25.3793i	0.869990i
$852$	0	0
$853$	− 8.37675i	− 0.286814i	−0.989664	−	0.143407i	$-0.954194\pi$
$853$	− 8.37675i	− 0.286814i	0.989664	−	0.143407i	$-0.0458059\pi$
$854$	−1.06638	−0.0364906
$855$	0	0
$856$	0.0586060i	0.00200311i
$857$	17.7888	0.607654	0.303827	−	0.952727i	$-0.401736\pi$
$857$	17.7888	0.607654	0.303827	+	0.952727i	$0.401736\pi$
$858$	0	0
$859$	50.1473	1.71101	0.855503	−	0.517798i	$-0.173248\pi$
$859$	50.1473	1.71101	0.855503	+	0.517798i	$0.173248\pi$
$860$	6.81163i	0.232275i
$861$	0	0
$862$	−18.8310	−0.641387
$863$	50.1172i	1.70601i	0.521903	+	0.853005i	$0.325222\pi$
$863$	50.1172i	1.70601i	−0.521903	+	0.853005i	$0.674778\pi$
$864$	0	0
$865$	− 1.77048i	− 0.0601981i
$866$	− 24.8364i	− 0.843975i
$867$	0	0
$868$	−18.5743	−0.630454
$869$	− 63.8316i	− 2.16534i
$870$	0	0
$871$	0	0
$872$	15.5254	0.525757
$873$	0	0
$874$	30.1715	1.02057
$875$	59.0277	1.99550
$876$	0	0
$877$	38.3827i	1.29609i	0.761601	+	0.648046i	$0.224414\pi$
$877$	38.3827i	1.29609i	−0.761601	+	0.648046i	$0.775586\pi$
$878$	− 22.2784i	− 0.751861i
$879$	0	0
$880$	10.8998	0.367431
$881$	31.8383	1.07266	0.536330	−	0.844009i	$-0.319810\pi$
$881$	31.8383	1.07266	0.536330	+	0.844009i	$0.319810\pi$
$882$	0	0
$883$	−4.74871	−0.159807	−0.0799034	−	0.996803i	$-0.525461\pi$
$883$	−4.74871	−0.159807	−0.0799034	+	0.996803i	$0.525461\pi$
$884$	0	0
$885$	0	0
$886$	26.6359i	0.894851i
$887$	−22.9288	−0.769875	−0.384938	−	0.922943i	$-0.625777\pi$
$887$	−22.9288	−0.769875	−0.384938	+	0.922943i	$0.625777\pi$
$888$	0	0
$889$	39.3870i	1.32100i
$890$	− 31.2814i	− 1.04856i
$891$	0	0
$892$	− 1.97584i	− 0.0661559i
$893$	17.1836	0.575027
$894$	0	0
$895$	17.9172i	0.598907i
$896$	4.85086	0.162056
$897$	0	0
$898$	−21.8538	−0.729272
$899$	− 13.6122i	− 0.453993i
$900$	0	0
$901$	13.6426	0.454502
$902$	34.4892i	1.14836i
$903$	0	0
$904$	− 1.95646i	− 0.0650709i
$905$	− 10.5918i	− 0.352083i
$906$	0	0
$907$	−43.0267	−1.42868	−0.714339	−	0.699800i	$-0.753272\pi$
$907$	−43.0267	−1.42868	−0.714339	+	0.699800i	$0.753272\pi$
$908$	− 25.0640i	− 0.831777i
$909$	0	0
$910$	0	0
$911$	−6.65087	−0.220353	−0.110177	−	0.993912i	$-0.535142\pi$
$911$	−6.65087	−0.220353	−0.110177	+	0.993912i	$0.535142\pi$
$912$	0	0
$913$	20.7071	0.685305
$914$	−20.7278	−0.685614
$915$	0	0
$916$	24.5133i	0.809943i
$917$	− 61.4499i	− 2.02926i
$918$	0	0
$919$	−9.11231	−0.300587	−0.150294	−	0.988641i	$-0.548022\pi$
$919$	−9.11231	−0.300587	−0.150294	+	0.988641i	$0.548022\pi$
$920$	12.0978	0.398854
$921$	0	0
$922$	−10.7832	−0.355124
$923$	0	0
$924$	0	0
$925$	− 6.62671i	− 0.217885i
$926$	23.1594	0.761066
$927$	0	0
$928$	3.55496i	0.116697i
$929$	11.8672i	0.389352i	0.980868	+	0.194676i	$0.0623655\pi$
$929$	11.8672i	0.389352i	−0.980868	+	0.194676i	$0.937634\pi$
$930$	0	0
$931$	− 74.2887i	− 2.43471i
$932$	−8.37196	−0.274233
$933$	0	0
$934$	− 13.3515i	− 0.436875i
$935$	−53.3008	−1.74312
$936$	0	0
$937$	19.1153	0.624469	0.312235	−	0.950005i	$-0.398922\pi$
$937$	19.1153	0.624469	0.312235	+	0.950005i	$0.398922\pi$
$938$	− 55.6969i	− 1.81857i
$939$	0	0
$940$	6.89008	0.224730
$941$	− 0.445042i	− 0.0145080i	−0.999974	−	0.00725398i	$-0.997691\pi$
$941$	− 0.445042i	− 0.0145080i	0.999974	−	0.00725398i	$-0.00230903\pi$
$942$	0	0
$943$	38.2801i	1.24657i
$944$	− 8.41119i	− 0.273761i
$945$	0	0
$946$	−22.8659	−0.743435
$947$	− 8.99894i	− 0.292426i	−0.989253	−	0.146213i	$-0.953291\pi$
$947$	− 8.99894i	− 0.292426i	0.989253	−	0.146213i	$-0.0467085\pi$
$948$	0	0
$949$	0	0
$950$	−7.87800	−0.255596
$951$	0	0
$952$	−23.7211	−0.768805
$953$	−37.2573	−1.20688	−0.603441	−	0.797408i	$-0.706204\pi$
$953$	−37.2573	−1.20688	−0.603441	+	0.797408i	$0.706204\pi$
$954$	0	0
$955$	− 19.8974i	− 0.643864i
$956$	14.0194i	0.453419i
$957$	0	0
$958$	4.38537	0.141685
$959$	74.5086	2.40601
$960$	0	0
$961$	16.3381	0.527036
$962$	0	0
$963$	0	0
$964$	21.1183i	0.680174i
$965$	37.7754	1.21603
$966$	0	0
$967$	− 37.2567i	− 1.19809i	−0.800714	−	0.599047i	$-0.795546\pi$
$967$	− 37.2567i	− 1.19809i	0.800714	−	0.599047i	$-0.204454\pi$
$968$	25.5894i	0.822474i
$969$	0	0
$970$	1.81940i	0.0584173i
$971$	−54.0393	−1.73421	−0.867103	−	0.498130i	$-0.834020\pi$
$971$	−54.0393	−1.73421	−0.867103	+	0.498130i	$0.834020\pi$
$972$	0	0
$973$	− 57.2379i	− 1.83496i
$974$	7.33214	0.234937
$975$	0	0
$976$	0.219833	0.00703667
$977$	11.4470i	0.366221i	0.983092	+	0.183110i	$0.0586166\pi$
$977$	11.4470i	0.366221i	−0.983092	+	0.183110i	$0.941383\pi$
$978$	0	0
$979$	105.008	3.35609
$980$	− 29.7875i	− 0.951526i
$981$	0	0
$982$	16.8062i	0.536309i
$983$	− 0.111244i	− 0.00354814i	−0.999998	−	0.00177407i	$-0.999435\pi$
$983$	− 0.111244i	− 0.00354814i	0.999998	−	0.00177407i	$-0.000564704\pi$
$984$	0	0
$985$	28.3642	0.903758
$986$	− 17.3840i	− 0.553621i
$987$	0	0
$988$	0	0
$989$	−25.3793	−0.807013
$990$	0	0
$991$	11.3709	0.361208	0.180604	−	0.983556i	$-0.442195\pi$
$991$	11.3709	0.361208	0.180604	+	0.983556i	$0.442195\pi$
$992$	3.82908	0.121574
$993$	0	0
$994$	− 65.6098i	− 2.08102i
$995$	− 2.72348i	− 0.0863401i
$996$	0	0
$997$	6.45042	0.204287	0.102143	−	0.994770i	$-0.467430\pi$
$997$	6.45042	0.204287	0.102143	+	0.994770i	$0.467430\pi$
$998$	−6.59658	−0.208811
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.p.1351.4		6
3.2	odd	2		3042.2.b.q.1351.3		6
13.5	odd	4		3042.2.a.bf.1.1	yes	3
13.8	odd	4		3042.2.a.bc.1.3	yes	3
13.12	even	2	inner	3042.2.b.p.1351.3		6
39.5	even	4		3042.2.a.bb.1.3	✓	3
39.8	even	4		3042.2.a.bg.1.1	yes	3
39.38	odd	2		3042.2.b.q.1351.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3042.2.a.bb.1.3	✓	3	39.5	even	4
3042.2.a.bc.1.3	yes	3	13.8	odd	4
3042.2.a.bf.1.1	yes	3	13.5	odd	4
3042.2.a.bg.1.1	yes	3	39.8	even	4
3042.2.b.p.1351.3		6	13.12	even	2	inner
3042.2.b.p.1351.4		6	1.1	even	1	trivial
3042.2.b.q.1351.3		6	3.2	odd	2
3042.2.b.q.1351.4		6	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.80194i q^{5} +4.85086i q^{7} -1.00000i q^{8} +1.80194 q^{10} +6.04892i q^{11} -4.85086 q^{14} +1.00000 q^{16} -4.89008 q^{17} +4.49396i q^{19} +1.80194i q^{20} -6.04892 q^{22} -6.71379 q^{23} +1.75302 q^{25} -4.85086i q^{28} +3.55496 q^{29} -3.82908i q^{31} +1.00000i q^{32} -4.89008i q^{34} +8.74094 q^{35} -3.78017i q^{37} -4.49396 q^{38} -1.80194 q^{40} -5.70171i q^{41} +3.78017 q^{43} -6.04892i q^{44} -6.71379i q^{46} -3.82371i q^{47} -16.5308 q^{49} +1.75302i q^{50} -2.78986 q^{53} +10.8998 q^{55} +4.85086 q^{56} +3.55496i q^{58} -8.41119i q^{59} +0.219833 q^{61} +3.82908 q^{62} -1.00000 q^{64} +11.4819i q^{67} +4.89008 q^{68} +8.74094i q^{70} +13.5254i q^{71} -0.417895i q^{73} +3.78017 q^{74} -4.49396i q^{76} -29.3424 q^{77} -10.5526 q^{79} -1.80194i q^{80} +5.70171 q^{82} -3.42327i q^{83} +8.81163i q^{85} +3.78017i q^{86} +6.04892 q^{88} -17.3599i q^{89} +6.71379 q^{92} +3.82371 q^{94} +8.09783 q^{95} +1.00969i q^{97} -16.5308i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} + 2 q^{10} - 2 q^{14} + 6 q^{16} - 28 q^{17} - 18 q^{22} - 24 q^{23} + 20 q^{25} + 22 q^{29} + 24 q^{35} - 8 q^{38} - 2 q^{40} + 20 q^{43} - 24 q^{49} + 30 q^{53} + 20 q^{55} + 2 q^{56} + 4 q^{61}+ \cdots + 12 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 + 2 * q^10 - 2 * q^14 + 6 * q^16 - 28 * q^17 - 18 * q^22 - 24 * q^23 + 20 * q^25 + 22 * q^29 + 24 * q^35 - 8 * q^38 - 2 * q^40 + 20 * q^43 - 24 * q^49 + 30 * q^53 + 20 * q^55 + 2 * q^56 + 4 * q^61 + 2 * q^62 - 6 * q^64 + 28 * q^68 + 20 * q^74 - 48 * q^77 + 18 * q^79 - 20 * q^82 + 18 * q^88 + 24 * q^92 + 8 * q^94 + 12 * q^95

Introduction

L-functions

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Database

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