LMFDB - Embedded newform 3042.2.b.n.1351.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,2,Mod(1351,3042)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [6,0,0,-6,0,0,0,0,0,-4,0,0,0,8,0,6,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 338)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.3
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.n.1351.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.49396i q^{5} -1.60388i q^{7} +1.00000i q^{8} +2.49396 q^{10} -2.04892i q^{11} -1.60388 q^{14} +1.00000 q^{16} -4.54288 q^{17} +4.85086i q^{19} -2.49396i q^{20} -2.04892 q^{22} +2.71379 q^{23} -1.21983 q^{25} +1.60388i q^{28} +9.20775 q^{29} -5.10992i q^{31} -1.00000i q^{32} +4.54288i q^{34} +4.00000 q^{35} +7.60388i q^{37} +4.85086 q^{38} -2.49396 q^{40} -3.46681i q^{41} -11.3448 q^{43} +2.04892i q^{44} -2.71379i q^{46} -0.219833i q^{47} +4.42758 q^{49} +1.21983i q^{50} +2.71379 q^{53} +5.10992 q^{55} +1.60388 q^{56} -9.20775i q^{58} +4.07606i q^{59} +10.4155 q^{61} -5.10992 q^{62} -1.00000 q^{64} +12.0761i q^{67} +4.54288 q^{68} -4.00000i q^{70} +1.28621i q^{71} +3.62565i q^{73} +7.60388 q^{74} -4.85086i q^{76} -3.28621 q^{77} -5.32975 q^{79} +2.49396i q^{80} -3.46681 q^{82} +4.85086i q^{83} -11.3297i q^{85} +11.3448i q^{86} +2.04892 q^{88} +16.5700i q^{89} -2.71379 q^{92} -0.219833 q^{94} -12.0978 q^{95} -4.64071i q^{97} -4.42758i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 4 q^{10} + 8 q^{14} + 6 q^{16} + 10 q^{17} + 6 q^{22} - 10 q^{25} + 20 q^{29} + 24 q^{35} + 2 q^{38} + 4 q^{40} - 22 q^{43} - 6 q^{49} + 32 q^{55} - 8 q^{56} - 8 q^{61} - 32 q^{62} - 6 q^{64}+ \cdots - 36 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 - 4 * q^10 + 8 * q^14 + 6 * q^16 + 10 * q^17 + 6 * q^22 - 10 * q^25 + 20 * q^29 + 24 * q^35 + 2 * q^38 + 4 * q^40 - 22 * q^43 - 6 * q^49 + 32 * q^55 - 8 * q^56 - 8 * q^61 - 32 * q^62 - 6 * q^64 - 10 * q^68 + 28 * q^74 - 36 * q^77 - 36 * q^79 - 14 * q^82 - 6 * q^88 - 4 * q^94 - 36 * 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
$5$	2.49396i	1.11533i	0.830065	+	0.557666i	$0.188303\pi$
$5$	2.49396i	1.11533i	−0.830065	+	0.557666i	$0.811697\pi$
$6$	0	0
$7$	− 1.60388i	− 0.606208i	−0.952957	−	0.303104i	$-0.901977\pi$
$7$	− 1.60388i	− 0.606208i	0.952957	−	0.303104i	$-0.0980229\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	2.49396	0.788659
$11$	− 2.04892i	− 0.617772i	−0.951099	−	0.308886i	$-0.900044\pi$
$11$	− 2.04892i	− 0.617772i	0.951099	−	0.308886i	$-0.0999561\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.60388	−0.428654
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.54288	−1.10181	−0.550905	−	0.834568i	$-0.685717\pi$
$17$	−4.54288	−1.10181	−0.550905	+	0.834568i	$0.685717\pi$
$18$	0	0
$19$	4.85086i	1.11286i	0.830894	+	0.556431i	$0.187830\pi$
$19$	4.85086i	1.11286i	−0.830894	+	0.556431i	$0.812170\pi$
$20$	− 2.49396i	− 0.557666i
$21$	0	0
$22$	−2.04892	−0.436831
$23$	2.71379	0.565865	0.282932	−	0.959140i	$-0.408693\pi$
$23$	2.71379	0.565865	0.282932	+	0.959140i	$0.408693\pi$
$24$	0	0
$25$	−1.21983	−0.243967
$26$	0	0
$27$	0	0
$28$	1.60388i	0.303104i
$29$	9.20775	1.70984	0.854918	−	0.518763i	$-0.173607\pi$
$29$	9.20775	1.70984	0.854918	+	0.518763i	$0.173607\pi$
$30$	0	0
$31$	− 5.10992i	− 0.917768i	−0.888496	−	0.458884i	$-0.848249\pi$
$31$	− 5.10992i	− 0.917768i	0.888496	−	0.458884i	$-0.151751\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	4.54288i	0.779097i
$35$	4.00000	0.676123
$36$	0	0
$37$	7.60388i	1.25007i	0.780597	+	0.625035i	$0.214915\pi$
$37$	7.60388i	1.25007i	−0.780597	+	0.625035i	$0.785085\pi$
$38$	4.85086	0.786913
$39$	0	0
$40$	−2.49396	−0.394330
$41$	− 3.46681i	− 0.541425i	−0.962660	−	0.270713i	$-0.912741\pi$
$41$	− 3.46681i	− 0.541425i	0.962660	−	0.270713i	$-0.0872593\pi$
$42$	0	0
$43$	−11.3448	−1.73007	−0.865034	−	0.501713i	$-0.832703\pi$
$43$	−11.3448	−1.73007	−0.865034	+	0.501713i	$0.832703\pi$
$44$	2.04892i	0.308886i
$45$	0	0
$46$	− 2.71379i	− 0.400127i
$47$	− 0.219833i	− 0.0320659i	−0.999871	−	0.0160329i	$-0.994896\pi$
$47$	− 0.219833i	− 0.0320659i	0.999871	−	0.0160329i	$-0.00510366\pi$
$48$	0	0
$49$	4.42758	0.632512
$50$	1.21983i	0.172510i
$51$	0	0
$52$	0	0
$53$	2.71379	0.372768	0.186384	−	0.982477i	$-0.440323\pi$
$53$	2.71379	0.372768	0.186384	+	0.982477i	$0.440323\pi$
$54$	0	0
$55$	5.10992	0.689021
$56$	1.60388	0.214327
$57$	0	0
$58$	− 9.20775i	− 1.20904i
$59$	4.07606i	0.530658i	0.964158	+	0.265329i	$0.0854806\pi$
$59$	4.07606i	0.530658i	−0.964158	+	0.265329i	$0.914519\pi$
$60$	0	0
$61$	10.4155	1.33357	0.666784	−	0.745251i	$-0.267670\pi$
$61$	10.4155	1.33357	0.666784	+	0.745251i	$0.267670\pi$
$62$	−5.10992	−0.648960
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	12.0761i	1.47533i	0.675169	+	0.737663i	$0.264071\pi$
$67$	12.0761i	1.47533i	−0.675169	+	0.737663i	$0.735929\pi$
$68$	4.54288	0.550905
$69$	0	0
$70$	− 4.00000i	− 0.478091i
$71$	1.28621i	0.152645i	0.997083	+	0.0763224i	$0.0243178\pi$
$71$	1.28621i	0.152645i	−0.997083	+	0.0763224i	$0.975682\pi$
$72$	0	0
$73$	3.62565i	0.424350i	0.977232	+	0.212175i	$0.0680546\pi$
$73$	3.62565i	0.424350i	−0.977232	+	0.212175i	$0.931945\pi$
$74$	7.60388	0.883933
$75$	0	0
$76$	− 4.85086i	− 0.556431i
$77$	−3.28621	−0.374498
$78$	0	0
$79$	−5.32975	−0.599644	−0.299822	−	0.953995i	$-0.596927\pi$
$79$	−5.32975	−0.599644	−0.299822	+	0.953995i	$0.596927\pi$
$80$	2.49396i	0.278833i
$81$	0	0
$82$	−3.46681	−0.382845
$83$	4.85086i	0.532451i	0.963911	+	0.266225i	$0.0857765\pi$
$83$	4.85086i	0.532451i	−0.963911	+	0.266225i	$0.914223\pi$
$84$	0	0
$85$	− 11.3297i	− 1.22888i
$86$	11.3448i	1.22334i
$87$	0	0
$88$	2.04892	0.218415
$89$	16.5700i	1.75642i	0.478276	+	0.878209i	$0.341262\pi$
$89$	16.5700i	1.75642i	−0.478276	+	0.878209i	$0.658738\pi$
$90$	0	0
$91$	0	0
$92$	−2.71379	−0.282932
$93$	0	0
$94$	−0.219833	−0.0226740
$95$	−12.0978	−1.24121
$96$	0	0
$97$	− 4.64071i	− 0.471193i	−0.971851	−	0.235596i	$-0.924296\pi$
$97$	− 4.64071i	− 0.471193i	0.971851	−	0.235596i	$-0.0757043\pi$
$98$	− 4.42758i	− 0.447253i
$99$	0	0
$100$	1.21983	0.121983
$101$	7.42758	0.739072	0.369536	−	0.929216i	$-0.379517\pi$
$101$	7.42758	0.739072	0.369536	+	0.929216i	$0.379517\pi$
$102$	0	0
$103$	−0.518122	−0.0510521	−0.0255261	−	0.999674i	$-0.508126\pi$
$103$	−0.518122	−0.0510521	−0.0255261	+	0.999674i	$0.508126\pi$
$104$	0	0
$105$	0	0
$106$	− 2.71379i	− 0.263587i
$107$	−3.51035	−0.339359	−0.169679	−	0.985499i	$-0.554273\pi$
$107$	−3.51035	−0.339359	−0.169679	+	0.985499i	$0.554273\pi$
$108$	0	0
$109$	3.38404i	0.324133i	0.986780	+	0.162066i	$0.0518158\pi$
$109$	3.38404i	0.324133i	−0.986780	+	0.162066i	$0.948184\pi$
$110$	− 5.10992i	− 0.487211i
$111$	0	0
$112$	− 1.60388i	− 0.151552i
$113$	5.44935	0.512632	0.256316	−	0.966593i	$-0.417491\pi$
$113$	5.44935	0.512632	0.256316	+	0.966593i	$0.417491\pi$
$114$	0	0
$115$	6.76809i	0.631127i
$116$	−9.20775	−0.854918
$117$	0	0
$118$	4.07606	0.375232
$119$	7.28621i	0.667926i
$120$	0	0
$121$	6.80194	0.618358
$122$	− 10.4155i	− 0.942975i
$123$	0	0
$124$	5.10992i	0.458884i
$125$	9.42758i	0.843229i
$126$	0	0
$127$	6.19567	0.549777	0.274888	−	0.961476i	$-0.411359\pi$
$127$	6.19567	0.549777	0.274888	+	0.961476i	$0.411359\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	4.13706	0.361457	0.180728	−	0.983533i	$-0.442154\pi$
$131$	4.13706	0.361457	0.180728	+	0.983533i	$0.442154\pi$
$132$	0	0
$133$	7.78017	0.674626
$134$	12.0761	1.04321
$135$	0	0
$136$	− 4.54288i	− 0.389548i
$137$	19.3817i	1.65589i	0.560812	+	0.827943i	$0.310489\pi$
$137$	19.3817i	1.65589i	−0.560812	+	0.827943i	$0.689511\pi$
$138$	0	0
$139$	18.5864	1.57648	0.788240	−	0.615368i	$-0.210993\pi$
$139$	18.5864	1.57648	0.788240	+	0.615368i	$0.210993\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	1.28621	0.107936
$143$	0	0
$144$	0	0
$145$	22.9638i	1.90704i
$146$	3.62565	0.300061
$147$	0	0
$148$	− 7.60388i	− 0.625035i
$149$	− 3.65817i	− 0.299689i	−0.988710	−	0.149844i	$-0.952123\pi$
$149$	− 3.65817i	− 0.299689i	0.988710	−	0.149844i	$-0.0478773\pi$
$150$	0	0
$151$	14.5918i	1.18746i	0.804663	+	0.593732i	$0.202346\pi$
$151$	14.5918i	1.18746i	−0.804663	+	0.593732i	$0.797654\pi$
$152$	−4.85086	−0.393456
$153$	0	0
$154$	3.28621i	0.264810i
$155$	12.7439	1.02362
$156$	0	0
$157$	21.6039	1.72418	0.862088	−	0.506758i	$-0.169156\pi$
$157$	21.6039	1.72418	0.862088	+	0.506758i	$0.169156\pi$
$158$	5.32975i	0.424012i
$159$	0	0
$160$	2.49396	0.197165
$161$	− 4.35258i	− 0.343032i
$162$	0	0
$163$	13.6093i	1.06596i	0.846128	+	0.532979i	$0.178928\pi$
$163$	13.6093i	1.06596i	−0.846128	+	0.532979i	$0.821072\pi$
$164$	3.46681i	0.270713i
$165$	0	0
$166$	4.85086	0.376499
$167$	− 14.0000i	− 1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$	− 14.0000i	− 1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	0	0
$169$	0	0
$170$	−11.3297	−0.868952
$171$	0	0
$172$	11.3448	0.865034
$173$	−4.21983	−0.320828	−0.160414	−	0.987050i	$-0.551283\pi$
$173$	−4.21983	−0.320828	−0.160414	+	0.987050i	$0.551283\pi$
$174$	0	0
$175$	1.95646i	0.147894i
$176$	− 2.04892i	− 0.154443i
$177$	0	0
$178$	16.5700	1.24198
$179$	15.8291	1.18312	0.591561	−	0.806260i	$-0.298512\pi$
$179$	15.8291	1.18312	0.591561	+	0.806260i	$0.298512\pi$
$180$	0	0
$181$	15.3056	1.13766	0.568828	−	0.822457i	$-0.307397\pi$
$181$	15.3056	1.13766	0.568828	+	0.822457i	$0.307397\pi$
$182$	0	0
$183$	0	0
$184$	2.71379i	0.200063i
$185$	−18.9638	−1.39424
$186$	0	0
$187$	9.30798i	0.680667i
$188$	0.219833i	0.0160329i
$189$	0	0
$190$	12.0978i	0.877669i
$191$	−3.60388	−0.260767	−0.130384	−	0.991464i	$-0.541621\pi$
$191$	−3.60388	−0.260767	−0.130384	+	0.991464i	$0.541621\pi$
$192$	0	0
$193$	− 11.2228i	− 0.807836i	−0.914795	−	0.403918i	$-0.867648\pi$
$193$	− 11.2228i	− 0.807836i	0.914795	−	0.403918i	$-0.132352\pi$
$194$	−4.64071	−0.333184
$195$	0	0
$196$	−4.42758	−0.316256
$197$	23.1051i	1.64617i	0.567916	+	0.823086i	$0.307750\pi$
$197$	23.1051i	1.64617i	−0.567916	+	0.823086i	$0.692250\pi$
$198$	0	0
$199$	−21.2620	−1.50723	−0.753613	−	0.657318i	$-0.771691\pi$
$199$	−21.2620	−1.50723	−0.753613	+	0.657318i	$0.771691\pi$
$200$	− 1.21983i	− 0.0862552i
$201$	0	0
$202$	− 7.42758i	− 0.522603i
$203$	− 14.7681i	− 1.03652i
$204$	0	0
$205$	8.64609	0.603869
$206$	0.518122i	0.0360993i
$207$	0	0
$208$	0	0
$209$	9.93900	0.687495
$210$	0	0
$211$	1.70709	0.117521	0.0587604	−	0.998272i	$-0.481285\pi$
$211$	1.70709	0.117521	0.0587604	+	0.998272i	$0.481285\pi$
$212$	−2.71379	−0.186384
$213$	0	0
$214$	3.51035i	0.239963i
$215$	− 28.2935i	− 1.92960i
$216$	0	0
$217$	−8.19567	−0.556358
$218$	3.38404	0.229196
$219$	0	0
$220$	−5.10992	−0.344510
$221$	0	0
$222$	0	0
$223$	− 6.21983i	− 0.416511i	−0.978074	−	0.208255i	$-0.933221\pi$
$223$	− 6.21983i	− 0.416511i	0.978074	−	0.208255i	$-0.0667785\pi$
$224$	−1.60388	−0.107163
$225$	0	0
$226$	− 5.44935i	− 0.362486i
$227$	− 0.955395i	− 0.0634118i	−0.999497	−	0.0317059i	$-0.989906\pi$
$227$	− 0.955395i	− 0.0634118i	0.999497	−	0.0317059i	$-0.0100940\pi$
$228$	0	0
$229$	− 22.4155i	− 1.48126i	−0.671914	−	0.740629i	$-0.734528\pi$
$229$	− 22.4155i	− 1.48126i	0.671914	−	0.740629i	$-0.265472\pi$
$230$	6.76809	0.446274
$231$	0	0
$232$	9.20775i	0.604518i
$233$	2.99031	0.195902	0.0979509	−	0.995191i	$-0.468771\pi$
$233$	2.99031	0.195902	0.0979509	+	0.995191i	$0.468771\pi$
$234$	0	0
$235$	0.548253	0.0357641
$236$	− 4.07606i	− 0.265329i
$237$	0	0
$238$	7.28621	0.472295
$239$	− 11.1293i	− 0.719894i	−0.932973	−	0.359947i	$-0.882795\pi$
$239$	− 11.1293i	− 0.719894i	0.932973	−	0.359947i	$-0.117205\pi$
$240$	0	0
$241$	5.20775i	0.335461i	0.985833	+	0.167730i	$0.0536438\pi$
$241$	5.20775i	0.335461i	−0.985833	+	0.167730i	$0.946356\pi$
$242$	− 6.80194i	− 0.437245i
$243$	0	0
$244$	−10.4155	−0.666784
$245$	11.0422i	0.705461i
$246$	0	0
$247$	0	0
$248$	5.10992	0.324480
$249$	0	0
$250$	9.42758	0.596253
$251$	−22.6950	−1.43250	−0.716248	−	0.697846i	$-0.754142\pi$
$251$	−22.6950	−1.43250	−0.716248	+	0.697846i	$0.754142\pi$
$252$	0	0
$253$	− 5.56033i	− 0.349575i
$254$	− 6.19567i	− 0.388751i
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.4306	0.650641	0.325320	−	0.945604i	$-0.394528\pi$
$257$	10.4306	0.650641	0.325320	+	0.945604i	$0.394528\pi$
$258$	0	0
$259$	12.1957	0.757802
$260$	0	0
$261$	0	0
$262$	− 4.13706i	− 0.255589i
$263$	7.10992	0.438416	0.219208	−	0.975678i	$-0.429653\pi$
$263$	7.10992	0.438416	0.219208	+	0.975678i	$0.429653\pi$
$264$	0	0
$265$	6.76809i	0.415760i
$266$	− 7.78017i	− 0.477033i
$267$	0	0
$268$	− 12.0761i	− 0.737663i
$269$	−2.02416	−0.123415	−0.0617077	−	0.998094i	$-0.519655\pi$
$269$	−2.02416	−0.123415	−0.0617077	+	0.998094i	$0.519655\pi$
$270$	0	0
$271$	− 15.9651i	− 0.969810i	−0.874567	−	0.484905i	$-0.838854\pi$
$271$	− 15.9651i	− 0.969810i	0.874567	−	0.484905i	$-0.161146\pi$
$272$	−4.54288	−0.275452
$273$	0	0
$274$	19.3817	1.17089
$275$	2.49934i	0.150716i
$276$	0	0
$277$	−13.7017	−0.823256	−0.411628	−	0.911352i	$-0.635040\pi$
$277$	−13.7017	−0.823256	−0.411628	+	0.911352i	$0.635040\pi$
$278$	− 18.5864i	− 1.11474i
$279$	0	0
$280$	4.00000i	0.239046i
$281$	− 15.2024i	− 0.906898i	−0.891282	−	0.453449i	$-0.850193\pi$
$281$	− 15.2024i	− 0.906898i	0.891282	−	0.453449i	$-0.149807\pi$
$282$	0	0
$283$	−20.6069	−1.22495	−0.612475	−	0.790490i	$-0.709826\pi$
$283$	−20.6069	−1.22495	−0.612475	+	0.790490i	$0.709826\pi$
$284$	− 1.28621i	− 0.0763224i
$285$	0	0
$286$	0	0
$287$	−5.56033	−0.328216
$288$	0	0
$289$	3.63773	0.213984
$290$	22.9638	1.34848
$291$	0	0
$292$	− 3.62565i	− 0.212175i
$293$	− 2.17629i	− 0.127140i	−0.997977	−	0.0635702i	$-0.979751\pi$
$293$	− 2.17629i	− 0.127140i	0.997977	−	0.0635702i	$-0.0202487\pi$
$294$	0	0
$295$	−10.1655	−0.591861
$296$	−7.60388	−0.441966
$297$	0	0
$298$	−3.65817	−0.211912
$299$	0	0
$300$	0	0
$301$	18.1957i	1.04878i
$302$	14.5918	0.839663
$303$	0	0
$304$	4.85086i	0.278216i
$305$	25.9758i	1.48737i
$306$	0	0
$307$	17.0127i	0.970965i	0.874247	+	0.485482i	$0.161356\pi$
$307$	17.0127i	0.970965i	−0.874247	+	0.485482i	$0.838644\pi$
$308$	3.28621	0.187249
$309$	0	0
$310$	− 12.7439i	− 0.723806i
$311$	−4.71379	−0.267295	−0.133647	−	0.991029i	$-0.542669\pi$
$311$	−4.71379	−0.267295	−0.133647	+	0.991029i	$0.542669\pi$
$312$	0	0
$313$	1.67696	0.0947872	0.0473936	−	0.998876i	$-0.484909\pi$
$313$	1.67696	0.0947872	0.0473936	+	0.998876i	$0.484909\pi$
$314$	− 21.6039i	− 1.21918i
$315$	0	0
$316$	5.32975	0.299822
$317$	29.1400i	1.63667i	0.574743	+	0.818334i	$0.305102\pi$
$317$	29.1400i	1.63667i	−0.574743	+	0.818334i	$0.694898\pi$
$318$	0	0
$319$	− 18.8659i	− 1.05629i
$320$	− 2.49396i	− 0.139417i
$321$	0	0
$322$	−4.35258	−0.242560
$323$	− 22.0368i	− 1.22616i
$324$	0	0
$325$	0	0
$326$	13.6093	0.753747
$327$	0	0
$328$	3.46681	0.191423
$329$	−0.352584	−0.0194386
$330$	0	0
$331$	− 8.74392i	− 0.480609i	−0.970698	−	0.240305i	$-0.922753\pi$
$331$	− 8.74392i	− 0.480609i	0.970698	−	0.240305i	$-0.0772474\pi$
$332$	− 4.85086i	− 0.266225i
$333$	0	0
$334$	−14.0000	−0.766046
$335$	−30.1172	−1.64548
$336$	0	0
$337$	3.10560	0.169173	0.0845865	−	0.996416i	$-0.473043\pi$
$337$	3.10560	0.169173	0.0845865	+	0.996416i	$0.473043\pi$
$338$	0	0
$339$	0	0
$340$	11.3297i	0.614442i
$341$	−10.4698	−0.566971
$342$	0	0
$343$	− 18.3284i	− 0.989642i
$344$	− 11.3448i	− 0.611671i
$345$	0	0
$346$	4.21983i	0.226860i
$347$	−17.9758	−0.964993	−0.482497	−	0.875898i	$-0.660270\pi$
$347$	−17.9758	−0.964993	−0.482497	+	0.875898i	$0.660270\pi$
$348$	0	0
$349$	15.1642i	0.811722i	0.913935	+	0.405861i	$0.133028\pi$
$349$	15.1642i	0.811722i	−0.913935	+	0.405861i	$0.866972\pi$
$350$	1.95646	0.104577
$351$	0	0
$352$	−2.04892	−0.109208
$353$	− 3.30021i	− 0.175652i	−0.996136	−	0.0878262i	$-0.972008\pi$
$353$	− 3.30021i	− 0.175652i	0.996136	−	0.0878262i	$-0.0279920\pi$
$354$	0	0
$355$	−3.20775	−0.170250
$356$	− 16.5700i	− 0.878209i
$357$	0	0
$358$	− 15.8291i	− 0.836593i
$359$	− 18.8901i	− 0.996980i	−0.866895	−	0.498490i	$-0.833888\pi$
$359$	− 18.8901i	− 0.996980i	0.866895	−	0.498490i	$-0.166112\pi$
$360$	0	0
$361$	−4.53079	−0.238463
$362$	− 15.3056i	− 0.804444i
$363$	0	0
$364$	0	0
$365$	−9.04221	−0.473291
$366$	0	0
$367$	−0.195669	−0.0102139	−0.00510693	−	0.999987i	$-0.501626\pi$
$367$	−0.195669	−0.0102139	−0.00510693	+	0.999987i	$0.501626\pi$
$368$	2.71379	0.141466
$369$	0	0
$370$	18.9638i	0.985879i
$371$	− 4.35258i	− 0.225975i
$372$	0	0
$373$	−10.7681	−0.557550	−0.278775	−	0.960356i	$-0.589928\pi$
$373$	−10.7681	−0.557550	−0.278775	+	0.960356i	$0.589928\pi$
$374$	9.30798	0.481304
$375$	0	0
$376$	0.219833	0.0113370
$377$	0	0
$378$	0	0
$379$	− 27.9627i	− 1.43635i	−0.695864	−	0.718173i	$-0.744978\pi$
$379$	− 27.9627i	− 1.43635i	0.695864	−	0.718173i	$-0.255022\pi$
$380$	12.0978	0.620606
$381$	0	0
$382$	3.60388i	0.184390i
$383$	32.8310i	1.67759i	0.544451	+	0.838793i	$0.316738\pi$
$383$	32.8310i	1.67759i	−0.544451	+	0.838793i	$0.683262\pi$
$384$	0	0
$385$	− 8.19567i	− 0.417690i
$386$	−11.2228	−0.571226
$387$	0	0
$388$	4.64071i	0.235596i
$389$	8.90946	0.451728	0.225864	−	0.974159i	$-0.427480\pi$
$389$	8.90946	0.451728	0.225864	+	0.974159i	$0.427480\pi$
$390$	0	0
$391$	−12.3284	−0.623475
$392$	4.42758i	0.223627i
$393$	0	0
$394$	23.1051	1.16402
$395$	− 13.2922i	− 0.668802i
$396$	0	0
$397$	11.7888i	0.591662i	0.955240	+	0.295831i	$0.0955966\pi$
$397$	11.7888i	0.591662i	−0.955240	+	0.295831i	$0.904403\pi$
$398$	21.2620i	1.06577i
$399$	0	0
$400$	−1.21983	−0.0609916
$401$	− 3.21313i	− 0.160456i	−0.996777	−	0.0802280i	$-0.974435\pi$
$401$	− 3.21313i	− 0.160456i	0.996777	−	0.0802280i	$-0.0255648\pi$
$402$	0	0
$403$	0	0
$404$	−7.42758	−0.369536
$405$	0	0
$406$	−14.7681	−0.732928
$407$	15.5797	0.772258
$408$	0	0
$409$	− 12.0218i	− 0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$409$	− 12.0218i	− 0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$410$	− 8.64609i	− 0.427000i
$411$	0	0
$412$	0.518122	0.0255261
$413$	6.53750	0.321689
$414$	0	0
$415$	−12.0978	−0.593859
$416$	0	0
$417$	0	0
$418$	− 9.93900i	− 0.486132i
$419$	3.56033	0.173934	0.0869669	−	0.996211i	$-0.472283\pi$
$419$	3.56033	0.173934	0.0869669	+	0.996211i	$0.472283\pi$
$420$	0	0
$421$	1.28621i	0.0626860i	0.999509	+	0.0313430i	$0.00997841\pi$
$421$	1.28621i	0.0626860i	−0.999509	+	0.0313430i	$0.990022\pi$
$422$	− 1.70709i	− 0.0830997i
$423$	0	0
$424$	2.71379i	0.131793i
$425$	5.54155	0.268805
$426$	0	0
$427$	− 16.7052i	− 0.808420i
$428$	3.51035	0.169679
$429$	0	0
$430$	−28.2935	−1.36443
$431$	26.9879i	1.29996i	0.759950	+	0.649981i	$0.225223\pi$
$431$	26.9879i	1.29996i	−0.759950	+	0.649981i	$0.774777\pi$
$432$	0	0
$433$	16.0170	0.769727	0.384864	−	0.922973i	$-0.374248\pi$
$433$	16.0170	0.769727	0.384864	+	0.922973i	$0.374248\pi$
$434$	8.19567i	0.393405i
$435$	0	0
$436$	− 3.38404i	− 0.162066i
$437$	13.1642i	0.629730i
$438$	0	0
$439$	37.1400	1.77260	0.886299	−	0.463114i	$-0.153268\pi$
$439$	37.1400	1.77260	0.886299	+	0.463114i	$0.153268\pi$
$440$	5.10992i	0.243606i
$441$	0	0
$442$	0	0
$443$	41.2083	1.95787	0.978934	−	0.204178i	$-0.0654522\pi$
$443$	41.2083	1.95787	0.978934	+	0.204178i	$0.0654522\pi$
$444$	0	0
$445$	−41.3250	−1.95899
$446$	−6.21983	−0.294518
$447$	0	0
$448$	1.60388i	0.0757760i
$449$	10.1263i	0.477890i	0.971033	+	0.238945i	$0.0768016\pi$
$449$	10.1263i	0.477890i	−0.971033	+	0.238945i	$0.923198\pi$
$450$	0	0
$451$	−7.10321	−0.334477
$452$	−5.44935	−0.256316
$453$	0	0
$454$	−0.955395	−0.0448389
$455$	0	0
$456$	0	0
$457$	− 21.7560i	− 1.01770i	−0.860854	−	0.508851i	$-0.830070\pi$
$457$	− 21.7560i	− 1.01770i	0.860854	−	0.508851i	$-0.169930\pi$
$458$	−22.4155	−1.04741
$459$	0	0
$460$	− 6.76809i	− 0.315564i
$461$	− 38.5676i	− 1.79627i	−0.439716	−	0.898137i	$-0.644921\pi$
$461$	− 38.5676i	− 1.79627i	0.439716	−	0.898137i	$-0.355079\pi$
$462$	0	0
$463$	− 33.0073i	− 1.53398i	−0.641660	−	0.766990i	$-0.721754\pi$
$463$	− 33.0073i	− 1.53398i	0.641660	−	0.766990i	$-0.278246\pi$
$464$	9.20775	0.427459
$465$	0	0
$466$	− 2.99031i	− 0.138523i
$467$	6.53989	0.302630	0.151315	−	0.988486i	$-0.451649\pi$
$467$	6.53989	0.302630	0.151315	+	0.988486i	$0.451649\pi$
$468$	0	0
$469$	19.3685	0.894354
$470$	− 0.548253i	− 0.0252890i
$471$	0	0
$472$	−4.07606	−0.187616
$473$	23.2446i	1.06879i
$474$	0	0
$475$	− 5.91723i	− 0.271501i
$476$	− 7.28621i	− 0.333963i
$477$	0	0
$478$	−11.1293	−0.509042
$479$	− 8.30691i	− 0.379553i	−0.981827	−	0.189776i	$-0.939224\pi$
$479$	− 8.30691i	− 0.379553i	0.981827	−	0.189776i	$-0.0607763\pi$
$480$	0	0
$481$	0	0
$482$	5.20775	0.237207
$483$	0	0
$484$	−6.80194	−0.309179
$485$	11.5737	0.525537
$486$	0	0
$487$	31.6534i	1.43435i	0.696892	+	0.717176i	$0.254566\pi$
$487$	31.6534i	1.43435i	−0.696892	+	0.717176i	$0.745434\pi$
$488$	10.4155i	0.471488i
$489$	0	0
$490$	11.0422	0.498836
$491$	−36.6631	−1.65458	−0.827291	−	0.561774i	$-0.810119\pi$
$491$	−36.6631	−1.65458	−0.827291	+	0.561774i	$0.810119\pi$
$492$	0	0
$493$	−41.8297	−1.88391
$494$	0	0
$495$	0	0
$496$	− 5.10992i	− 0.229442i
$497$	2.06292	0.0925345
$498$	0	0
$499$	29.8920i	1.33815i	0.743195	+	0.669075i	$0.233309\pi$
$499$	29.8920i	1.33815i	−0.743195	+	0.669075i	$0.766691\pi$
$500$	− 9.42758i	− 0.421614i
$501$	0	0
$502$	22.6950i	1.01293i
$503$	−15.5905	−0.695145	−0.347572	−	0.937653i	$-0.612994\pi$
$503$	−15.5905	−0.695145	−0.347572	+	0.937653i	$0.612994\pi$
$504$	0	0
$505$	18.5241i	0.824311i
$506$	−5.56033	−0.247187
$507$	0	0
$508$	−6.19567	−0.274888
$509$	− 17.1642i	− 0.760790i	−0.924824	−	0.380395i	$-0.875788\pi$
$509$	− 17.1642i	− 0.760790i	0.924824	−	0.380395i	$-0.124212\pi$
$510$	0	0
$511$	5.81508	0.257244
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 10.4306i	− 0.460073i
$515$	− 1.29218i	− 0.0569401i
$516$	0	0
$517$	−0.450419	−0.0198094
$518$	− 12.1957i	− 0.535847i
$519$	0	0
$520$	0	0
$521$	−6.15452	−0.269634	−0.134817	−	0.990870i	$-0.543045\pi$
$521$	−6.15452	−0.269634	−0.134817	+	0.990870i	$0.543045\pi$
$522$	0	0
$523$	6.93900	0.303421	0.151711	−	0.988425i	$-0.451522\pi$
$523$	6.93900	0.303421	0.151711	+	0.988425i	$0.451522\pi$
$524$	−4.13706	−0.180728
$525$	0	0
$526$	− 7.10992i	− 0.310007i
$527$	23.2137i	1.01121i
$528$	0	0
$529$	−15.6353	−0.679797
$530$	6.76809	0.293987
$531$	0	0
$532$	−7.78017	−0.337313
$533$	0	0
$534$	0	0
$535$	− 8.75468i	− 0.378498i
$536$	−12.0761	−0.521607
$537$	0	0
$538$	2.02416i	0.0872679i
$539$	− 9.07175i	− 0.390748i
$540$	0	0
$541$	− 0.426256i	− 0.0183262i	−0.999958	−	0.00916308i	$-0.997083\pi$
$541$	− 0.426256i	− 0.0183262i	0.999958	−	0.00916308i	$-0.00291674\pi$
$542$	−15.9651	−0.685759
$543$	0	0
$544$	4.54288i	0.194774i
$545$	−8.43967	−0.361516
$546$	0	0
$547$	3.72348	0.159205	0.0796023	−	0.996827i	$-0.474635\pi$
$547$	3.72348	0.159205	0.0796023	+	0.996827i	$0.474635\pi$
$548$	− 19.3817i	− 0.827943i
$549$	0	0
$550$	2.49934	0.106572
$551$	44.6655i	1.90281i
$552$	0	0
$553$	8.54825i	0.363509i
$554$	13.7017i	0.582130i
$555$	0	0
$556$	−18.5864	−0.788240
$557$	42.7961i	1.81333i	0.421853	+	0.906664i	$0.361380\pi$
$557$	42.7961i	1.81333i	−0.421853	+	0.906664i	$0.638620\pi$
$558$	0	0
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	−15.2024	−0.641273
$563$	2.92500	0.123274	0.0616370	−	0.998099i	$-0.480368\pi$
$563$	2.92500	0.123274	0.0616370	+	0.998099i	$0.480368\pi$
$564$	0	0
$565$	13.5905i	0.571755i
$566$	20.6069i	0.866171i
$567$	0	0
$568$	−1.28621	−0.0539681
$569$	−26.5894	−1.11469	−0.557343	−	0.830282i	$-0.688179\pi$
$569$	−26.5894	−1.11469	−0.557343	+	0.830282i	$0.688179\pi$
$570$	0	0
$571$	2.34422	0.0981027	0.0490513	−	0.998796i	$-0.484380\pi$
$571$	2.34422	0.0981027	0.0490513	+	0.998796i	$0.484380\pi$
$572$	0	0
$573$	0	0
$574$	5.56033i	0.232084i
$575$	−3.31037	−0.138052
$576$	0	0
$577$	− 40.4064i	− 1.68214i	−0.540926	−	0.841070i	$-0.681926\pi$
$577$	− 40.4064i	− 1.68214i	0.540926	−	0.841070i	$-0.318074\pi$
$578$	− 3.63773i	− 0.151310i
$579$	0	0
$580$	− 22.9638i	− 0.953518i
$581$	7.78017	0.322776
$582$	0	0
$583$	− 5.56033i	− 0.230286i
$584$	−3.62565	−0.150030
$585$	0	0
$586$	−2.17629	−0.0899018
$587$	− 43.2180i	− 1.78380i	−0.452234	−	0.891900i	$-0.649373\pi$
$587$	− 43.2180i	− 1.78380i	0.452234	−	0.891900i	$-0.350627\pi$
$588$	0	0
$589$	24.7875	1.02135
$590$	10.1655i	0.418509i
$591$	0	0
$592$	7.60388i	0.312517i
$593$	7.01746i	0.288172i	0.989565	+	0.144086i	$0.0460243\pi$
$593$	7.01746i	0.288172i	−0.989565	+	0.144086i	$0.953976\pi$
$594$	0	0
$595$	−18.1715	−0.744959
$596$	3.65817i	0.149844i
$597$	0	0
$598$	0	0
$599$	−14.9772	−0.611950	−0.305975	−	0.952039i	$-0.598982\pi$
$599$	−14.9772	−0.611950	−0.305975	+	0.952039i	$0.598982\pi$
$600$	0	0
$601$	−37.2771	−1.52057	−0.760283	−	0.649593i	$-0.774940\pi$
$601$	−37.2771	−1.52057	−0.760283	+	0.649593i	$0.774940\pi$
$602$	18.1957	0.741600
$603$	0	0
$604$	− 14.5918i	− 0.593732i
$605$	16.9638i	0.689675i
$606$	0	0
$607$	0.803003	0.0325929	0.0162964	−	0.999867i	$-0.494812\pi$
$607$	0.803003	0.0325929	0.0162964	+	0.999867i	$0.494812\pi$
$608$	4.85086	0.196728
$609$	0	0
$610$	25.9758	1.05173
$611$	0	0
$612$	0	0
$613$	− 4.87071i	− 0.196726i	−0.995151	−	0.0983630i	$-0.968639\pi$
$613$	− 4.87071i	− 0.196726i	0.995151	−	0.0983630i	$-0.0313606\pi$
$614$	17.0127	0.686576
$615$	0	0
$616$	− 3.28621i	− 0.132405i
$617$	3.34290i	0.134580i	0.997733	+	0.0672899i	$0.0214353\pi$
$617$	3.34290i	0.134580i	−0.997733	+	0.0672899i	$0.978565\pi$
$618$	0	0
$619$	− 34.1715i	− 1.37347i	−0.726908	−	0.686734i	$-0.759044\pi$
$619$	− 34.1715i	− 1.37347i	0.726908	−	0.686734i	$-0.240956\pi$
$620$	−12.7439	−0.511808
$621$	0	0
$622$	4.71379i	0.189006i
$623$	26.5763	1.06476
$624$	0	0
$625$	−29.6112	−1.18445
$626$	− 1.67696i	− 0.0670246i
$627$	0	0
$628$	−21.6039	−0.862088
$629$	− 34.5435i	− 1.37734i
$630$	0	0
$631$	39.8297i	1.58559i	0.609486	+	0.792797i	$0.291376\pi$
$631$	39.8297i	1.58559i	−0.609486	+	0.792797i	$0.708624\pi$
$632$	− 5.32975i	− 0.212006i
$633$	0	0
$634$	29.1400	1.15730
$635$	15.4517i	0.613184i
$636$	0	0
$637$	0	0
$638$	−18.8659	−0.746909
$639$	0	0
$640$	−2.49396	−0.0985824
$641$	−13.4910	−0.532861	−0.266431	−	0.963854i	$-0.585844\pi$
$641$	−13.4910	−0.532861	−0.266431	+	0.963854i	$0.585844\pi$
$642$	0	0
$643$	16.6638i	0.657156i	0.944477	+	0.328578i	$0.106569\pi$
$643$	16.6638i	0.657156i	−0.944477	+	0.328578i	$0.893431\pi$
$644$	4.35258i	0.171516i
$645$	0	0
$646$	−22.0368	−0.867028
$647$	48.7332	1.91590	0.957949	−	0.286938i	$-0.0926372\pi$
$647$	48.7332	1.91590	0.957949	+	0.286938i	$0.0926372\pi$
$648$	0	0
$649$	8.35152	0.327826
$650$	0	0
$651$	0	0
$652$	− 13.6093i	− 0.532979i
$653$	−47.0702	−1.84200	−0.921000	−	0.389563i	$-0.872626\pi$
$653$	−47.0702	−1.84200	−0.921000	+	0.389563i	$0.872626\pi$
$654$	0	0
$655$	10.3177i	0.403145i
$656$	− 3.46681i	− 0.135356i
$657$	0	0
$658$	0.352584i	0.0137452i
$659$	22.2524	0.866829	0.433414	−	0.901195i	$-0.357309\pi$
$659$	22.2524	0.866829	0.433414	+	0.901195i	$0.357309\pi$
$660$	0	0
$661$	− 2.57242i	− 0.100055i	−0.998748	−	0.0500277i	$-0.984069\pi$
$661$	− 2.57242i	− 0.100055i	0.998748	−	0.0500277i	$-0.0159310\pi$
$662$	−8.74392	−0.339842
$663$	0	0
$664$	−4.85086	−0.188250
$665$	19.4034i	0.752432i
$666$	0	0
$667$	24.9879	0.967536
$668$	14.0000i	0.541676i
$669$	0	0
$670$	30.1172i	1.16353i
$671$	− 21.3405i	− 0.823841i
$672$	0	0
$673$	−24.1691	−0.931651	−0.465825	−	0.884877i	$-0.654243\pi$
$673$	−24.1691	−0.931651	−0.465825	+	0.884877i	$0.654243\pi$
$674$	− 3.10560i	− 0.119623i
$675$	0	0
$676$	0	0
$677$	4.91425	0.188870	0.0944349	−	0.995531i	$-0.469896\pi$
$677$	4.91425	0.188870	0.0944349	+	0.995531i	$0.469896\pi$
$678$	0	0
$679$	−7.44312	−0.285641
$680$	11.3297	0.434476
$681$	0	0
$682$	10.4698i	0.400909i
$683$	1.96556i	0.0752100i	0.999293	+	0.0376050i	$0.0119729\pi$
$683$	1.96556i	0.0752100i	−0.999293	+	0.0376050i	$0.988027\pi$
$684$	0	0
$685$	−48.3370	−1.84686
$686$	−18.3284	−0.699782
$687$	0	0
$688$	−11.3448	−0.432517
$689$	0	0
$690$	0	0
$691$	− 25.5077i	− 0.970359i	−0.874415	−	0.485179i	$-0.838754\pi$
$691$	− 25.5077i	− 0.970359i	0.874415	−	0.485179i	$-0.161246\pi$
$692$	4.21983	0.160414
$693$	0	0
$694$	17.9758i	0.682353i
$695$	46.3538i	1.75830i
$696$	0	0
$697$	15.7493i	0.596547i
$698$	15.1642	0.573974
$699$	0	0
$700$	− 1.95646i	− 0.0739472i
$701$	−41.8491	−1.58062	−0.790308	−	0.612709i	$-0.790080\pi$
$701$	−41.8491	−1.58062	−0.790308	+	0.612709i	$0.790080\pi$
$702$	0	0
$703$	−36.8853	−1.39116
$704$	2.04892i	0.0772215i
$705$	0	0
$706$	−3.30021	−0.124205
$707$	− 11.9129i	− 0.448031i
$708$	0	0
$709$	33.3056i	1.25082i	0.780297	+	0.625409i	$0.215068\pi$
$709$	33.3056i	1.25082i	−0.780297	+	0.625409i	$0.784932\pi$
$710$	3.20775i	0.120385i
$711$	0	0
$712$	−16.5700	−0.620988
$713$	− 13.8672i	− 0.519333i
$714$	0	0
$715$	0	0
$716$	−15.8291	−0.591561
$717$	0	0
$718$	−18.8901	−0.704972
$719$	6.37196	0.237634	0.118817	−	0.992916i	$-0.462090\pi$
$719$	6.37196	0.237634	0.118817	+	0.992916i	$0.462090\pi$
$720$	0	0
$721$	0.831004i	0.0309482i
$722$	4.53079i	0.168619i
$723$	0	0
$724$	−15.3056	−0.568828
$725$	−11.2319	−0.417143
$726$	0	0
$727$	−23.7995	−0.882676	−0.441338	−	0.897341i	$-0.645496\pi$
$727$	−23.7995	−0.882676	−0.441338	+	0.897341i	$0.645496\pi$
$728$	0	0
$729$	0	0
$730$	9.04221i	0.334667i
$731$	51.5381	1.90621
$732$	0	0
$733$	27.4142i	1.01257i	0.862368	+	0.506283i	$0.168981\pi$
$733$	27.4142i	1.01257i	−0.862368	+	0.506283i	$0.831019\pi$
$734$	0.195669i	0.00722229i
$735$	0	0
$736$	− 2.71379i	− 0.100032i
$737$	24.7429	0.911415
$738$	0	0
$739$	35.3274i	1.29954i	0.760132	+	0.649769i	$0.225134\pi$
$739$	35.3274i	1.29954i	−0.760132	+	0.649769i	$0.774866\pi$
$740$	18.9638	0.697122
$741$	0	0
$742$	−4.35258	−0.159788
$743$	− 23.9758i	− 0.879588i	−0.898099	−	0.439794i	$-0.855051\pi$
$743$	− 23.9758i	− 0.879588i	0.898099	−	0.439794i	$-0.144949\pi$
$744$	0	0
$745$	9.12333	0.334253
$746$	10.7681i	0.394248i
$747$	0	0
$748$	− 9.30798i	− 0.340333i
$749$	5.63017i	0.205722i
$750$	0	0
$751$	24.8659	0.907370	0.453685	−	0.891162i	$-0.350109\pi$
$751$	24.8659	0.907370	0.453685	+	0.891162i	$0.350109\pi$
$752$	− 0.219833i	− 0.00801647i
$753$	0	0
$754$	0	0
$755$	−36.3913	−1.32442
$756$	0	0
$757$	4.93362	0.179316	0.0896578	−	0.995973i	$-0.471423\pi$
$757$	4.93362	0.179316	0.0896578	+	0.995973i	$0.471423\pi$
$758$	−27.9627	−1.01565
$759$	0	0
$760$	− 12.0978i	− 0.438835i
$761$	15.0067i	0.543993i	0.962298	+	0.271996i	$0.0876839\pi$
$761$	15.0067i	0.543993i	−0.962298	+	0.271996i	$0.912316\pi$
$762$	0	0
$763$	5.42758	0.196492
$764$	3.60388	0.130384
$765$	0	0
$766$	32.8310	1.18623
$767$	0	0
$768$	0	0
$769$	− 22.2640i	− 0.802859i	−0.915890	−	0.401430i	$-0.868513\pi$
$769$	− 22.2640i	− 0.802859i	0.915890	−	0.401430i	$-0.131487\pi$
$770$	−8.19567	−0.295351
$771$	0	0
$772$	11.2228i	0.403918i
$773$	− 11.7366i	− 0.422137i	−0.977471	−	0.211069i	$-0.932306\pi$
$773$	− 11.7366i	− 0.422137i	0.977471	−	0.211069i	$-0.0676943\pi$
$774$	0	0
$775$	6.23324i	0.223905i
$776$	4.64071	0.166592
$777$	0	0
$778$	− 8.90946i	− 0.319420i
$779$	16.8170	0.602532
$780$	0	0
$781$	2.63533	0.0942997
$782$	12.3284i	0.440863i
$783$	0	0
$784$	4.42758	0.158128
$785$	53.8792i	1.92303i
$786$	0	0
$787$	− 14.9498i	− 0.532901i	−0.963849	−	0.266451i	$-0.914149\pi$
$787$	− 14.9498i	− 0.532901i	0.963849	−	0.266451i	$-0.0858509\pi$
$788$	− 23.1051i	− 0.823086i
$789$	0	0
$790$	−13.2922	−0.472914
$791$	− 8.74008i	− 0.310762i
$792$	0	0
$793$	0	0
$794$	11.7888	0.418369
$795$	0	0
$796$	21.2620	0.753613
$797$	35.1728	1.24589	0.622943	−	0.782267i	$-0.285937\pi$
$797$	35.1728	1.24589	0.622943	+	0.782267i	$0.285937\pi$
$798$	0	0
$799$	0.998672i	0.0353305i
$800$	1.21983i	0.0431276i
$801$	0	0
$802$	−3.21313	−0.113459
$803$	7.42865	0.262151
$804$	0	0
$805$	10.8552	0.382594
$806$	0	0
$807$	0	0
$808$	7.42758i	0.261301i
$809$	−42.9487	−1.51000	−0.754998	−	0.655727i	$-0.772362\pi$
$809$	−42.9487	−1.51000	−0.754998	+	0.655727i	$0.772362\pi$
$810$	0	0
$811$	20.4644i	0.718603i	0.933222	+	0.359301i	$0.116985\pi$
$811$	20.4644i	0.718603i	−0.933222	+	0.359301i	$0.883015\pi$
$812$	14.7681i	0.518258i
$813$	0	0
$814$	− 15.5797i	− 0.546069i
$815$	−33.9409	−1.18890
$816$	0	0
$817$	− 55.0320i	− 1.92533i
$818$	−12.0218	−0.420331
$819$	0	0
$820$	−8.64609	−0.301934
$821$	− 36.9987i	− 1.29126i	−0.763649	−	0.645631i	$-0.776594\pi$
$821$	− 36.9987i	− 1.29126i	0.763649	−	0.645631i	$-0.223406\pi$
$822$	0	0
$823$	−9.27545	−0.323322	−0.161661	−	0.986846i	$-0.551685\pi$
$823$	−9.27545	−0.323322	−0.161661	+	0.986846i	$0.551685\pi$
$824$	− 0.518122i	− 0.0180496i
$825$	0	0
$826$	− 6.53750i	− 0.227469i
$827$	− 6.05669i	− 0.210612i	−0.994440	−	0.105306i	$-0.966418\pi$
$827$	− 6.05669i	− 0.210612i	0.994440	−	0.105306i	$-0.0335821\pi$
$828$	0	0
$829$	−9.94092	−0.345262	−0.172631	−	0.984987i	$-0.555227\pi$
$829$	−9.94092	−0.345262	−0.172631	+	0.984987i	$0.555227\pi$
$830$	12.0978i	0.419922i
$831$	0	0
$832$	0	0
$833$	−20.1140	−0.696908
$834$	0	0
$835$	34.9154	1.20830
$836$	−9.93900	−0.343748
$837$	0	0
$838$	− 3.56033i	− 0.122990i
$839$	− 34.4698i	− 1.19003i	−0.803715	−	0.595015i	$-0.797146\pi$
$839$	− 34.4698i	− 1.19003i	0.803715	−	0.595015i	$-0.202854\pi$
$840$	0	0
$841$	55.7827	1.92354
$842$	1.28621	0.0443257
$843$	0	0
$844$	−1.70709	−0.0587604
$845$	0	0
$846$	0	0
$847$	− 10.9095i	− 0.374854i
$848$	2.71379	0.0931920
$849$	0	0
$850$	− 5.54155i	− 0.190074i
$851$	20.6353i	0.707370i
$852$	0	0
$853$	22.1521i	0.758474i	0.925299	+	0.379237i	$0.123814\pi$
$853$	22.1521i	0.758474i	−0.925299	+	0.379237i	$0.876186\pi$
$854$	−16.7052	−0.571639
$855$	0	0
$856$	− 3.51035i	− 0.119981i
$857$	−30.5187	−1.04250	−0.521250	−	0.853404i	$-0.674534\pi$
$857$	−30.5187	−1.04250	−0.521250	+	0.853404i	$0.674534\pi$
$858$	0	0
$859$	−20.9071	−0.713340	−0.356670	−	0.934230i	$-0.616088\pi$
$859$	−20.9071	−0.713340	−0.356670	+	0.934230i	$0.616088\pi$
$860$	28.2935i	0.964800i
$861$	0	0
$862$	26.9879	0.919212
$863$	− 55.6969i	− 1.89595i	−0.318352	−	0.947973i	$-0.603129\pi$
$863$	− 55.6969i	− 1.89595i	0.318352	−	0.947973i	$-0.396871\pi$
$864$	0	0
$865$	− 10.5241i	− 0.357830i
$866$	− 16.0170i	− 0.544279i
$867$	0	0
$868$	8.19567	0.278179
$869$	10.9202i	0.370443i
$870$	0	0
$871$	0	0
$872$	−3.38404	−0.114598
$873$	0	0
$874$	13.1642	0.445286
$875$	15.1207	0.511172
$876$	0	0
$877$	32.7922i	1.10732i	0.832744	+	0.553658i	$0.186768\pi$
$877$	32.7922i	1.10732i	−0.832744	+	0.553658i	$0.813232\pi$
$878$	− 37.1400i	− 1.25342i
$879$	0	0
$880$	5.10992	0.172255
$881$	22.4101	0.755016	0.377508	−	0.926006i	$-0.376781\pi$
$881$	22.4101	0.755016	0.377508	+	0.926006i	$0.376781\pi$
$882$	0	0
$883$	38.9670	1.31134	0.655672	−	0.755046i	$-0.272385\pi$
$883$	38.9670	1.31134	0.655672	+	0.755046i	$0.272385\pi$
$884$	0	0
$885$	0	0
$886$	− 41.2083i	− 1.38442i
$887$	34.8745	1.17097	0.585486	−	0.810682i	$-0.300904\pi$
$887$	34.8745	1.17097	0.585486	+	0.810682i	$0.300904\pi$
$888$	0	0
$889$	− 9.93708i	− 0.333279i
$890$	41.3250i	1.38522i
$891$	0	0
$892$	6.21983i	0.208255i
$893$	1.06638	0.0356849
$894$	0	0
$895$	39.4771i	1.31957i
$896$	1.60388	0.0535817
$897$	0	0
$898$	10.1263	0.337919
$899$	− 47.0508i	− 1.56923i
$900$	0	0
$901$	−12.3284	−0.410719
$902$	7.10321i	0.236511i
$903$	0	0
$904$	5.44935i	0.181243i
$905$	38.1715i	1.26886i
$906$	0	0
$907$	−15.1317	−0.502439	−0.251220	−	0.967930i	$-0.580832\pi$
$907$	−15.1317	−0.502439	−0.251220	+	0.967930i	$0.580832\pi$
$908$	0.955395i	0.0317059i
$909$	0	0
$910$	0	0
$911$	−23.0810	−0.764707	−0.382353	−	0.924016i	$-0.624886\pi$
$911$	−23.0810	−0.764707	−0.382353	+	0.924016i	$0.624886\pi$
$912$	0	0
$913$	9.93900	0.328933
$914$	−21.7560	−0.719625
$915$	0	0
$916$	22.4155i	0.740629i
$917$	− 6.63533i	− 0.219118i
$918$	0	0
$919$	−39.9275	−1.31709	−0.658544	−	0.752543i	$-0.728827\pi$
$919$	−39.9275	−1.31709	−0.658544	+	0.752543i	$0.728827\pi$
$920$	−6.76809	−0.223137
$921$	0	0
$922$	−38.5676	−1.27016
$923$	0	0
$924$	0	0
$925$	− 9.27545i	− 0.304975i
$926$	−33.0073	−1.08469
$927$	0	0
$928$	− 9.20775i	− 0.302259i
$929$	− 45.1771i	− 1.48221i	−0.671387	−	0.741107i	$-0.734301\pi$
$929$	− 45.1771i	− 1.48221i	0.671387	−	0.741107i	$-0.265699\pi$
$930$	0	0
$931$	21.4776i	0.703899i
$932$	−2.99031	−0.0979509
$933$	0	0
$934$	− 6.53989i	− 0.213992i
$935$	−23.2137	−0.759170
$936$	0	0
$937$	29.0901	0.950331	0.475165	−	0.879896i	$-0.342388\pi$
$937$	29.0901	0.950331	0.475165	+	0.879896i	$0.342388\pi$
$938$	− 19.3685i	− 0.632404i
$939$	0	0
$940$	−0.548253	−0.0178821
$941$	− 26.1220i	− 0.851553i	−0.904828	−	0.425776i	$-0.860001\pi$
$941$	− 26.1220i	− 0.851553i	0.904828	−	0.425776i	$-0.139999\pi$
$942$	0	0
$943$	− 9.40821i	− 0.306373i
$944$	4.07606i	0.132665i
$945$	0	0
$946$	23.2446	0.755747
$947$	− 42.6698i	− 1.38658i	−0.720658	−	0.693291i	$-0.756160\pi$
$947$	− 42.6698i	− 1.38658i	0.720658	−	0.693291i	$-0.243840\pi$
$948$	0	0
$949$	0	0
$950$	−5.91723	−0.191980
$951$	0	0
$952$	−7.28621	−0.236147
$953$	−29.9038	−0.968680	−0.484340	−	0.874880i	$-0.660940\pi$
$953$	−29.9038	−0.968680	−0.484340	+	0.874880i	$0.660940\pi$
$954$	0	0
$955$	− 8.98792i	− 0.290842i
$956$	11.1293i	0.359947i
$957$	0	0
$958$	−8.30691	−0.268384
$959$	31.0858	1.00381
$960$	0	0
$961$	4.88876	0.157702
$962$	0	0
$963$	0	0
$964$	− 5.20775i	− 0.167730i
$965$	27.9892	0.901006
$966$	0	0
$967$	5.16900i	0.166224i	0.996540	+	0.0831119i	$0.0264859\pi$
$967$	5.16900i	0.166224i	−0.996540	+	0.0831119i	$0.973514\pi$
$968$	6.80194i	0.218623i
$969$	0	0
$970$	− 11.5737i	− 0.371611i
$971$	−35.9715	−1.15438	−0.577191	−	0.816609i	$-0.695851\pi$
$971$	−35.9715	−1.15438	−0.577191	+	0.816609i	$0.695851\pi$
$972$	0	0
$973$	− 29.8103i	− 0.955674i
$974$	31.6534	1.01424
$975$	0	0
$976$	10.4155	0.333392
$977$	34.2435i	1.09555i	0.836627	+	0.547774i	$0.184525\pi$
$977$	34.2435i	1.09555i	−0.836627	+	0.547774i	$0.815475\pi$
$978$	0	0
$979$	33.9506	1.08507
$980$	− 11.0422i	− 0.352731i
$981$	0	0
$982$	36.6631i	1.16997i
$983$	54.8939i	1.75084i	0.483359	+	0.875422i	$0.339416\pi$
$983$	54.8939i	1.75084i	−0.483359	+	0.875422i	$0.660584\pi$
$984$	0	0
$985$	−57.6233	−1.83603
$986$	41.8297i	1.33213i
$987$	0	0
$988$	0	0
$989$	−30.7875	−0.978984
$990$	0	0
$991$	−16.6655	−0.529396	−0.264698	−	0.964331i	$-0.585272\pi$
$991$	−16.6655	−0.529396	−0.264698	+	0.964331i	$0.585272\pi$
$992$	−5.10992	−0.162240
$993$	0	0
$994$	− 2.06292i	− 0.0654318i
$995$	− 53.0267i	− 1.68106i
$996$	0	0
$997$	16.3961	0.519270	0.259635	−	0.965707i	$-0.416398\pi$
$997$	16.3961	0.519270	0.259635	+	0.965707i	$0.416398\pi$
$998$	29.8920	0.946215
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.n.1351.3		6
3.2	odd	2		338.2.b.d.337.6		6
12.11	even	2		2704.2.f.m.337.1		6
13.5	odd	4		3042.2.a.z.1.3		3
13.8	odd	4		3042.2.a.bi.1.1		3
13.12	even	2	inner	3042.2.b.n.1351.4		6
39.2	even	12		338.2.c.h.191.1		6
39.5	even	4		338.2.a.h.1.3	yes	3
39.8	even	4		338.2.a.g.1.3	✓	3
39.11	even	12		338.2.c.i.191.1		6
39.17	odd	6		338.2.e.e.23.1		12
39.20	even	12		338.2.c.i.315.1		6
39.23	odd	6		338.2.e.e.147.4		12
39.29	odd	6		338.2.e.e.147.1		12
39.32	even	12		338.2.c.h.315.1		6
39.35	odd	6		338.2.e.e.23.4		12
39.38	odd	2		338.2.b.d.337.3		6
156.47	odd	4		2704.2.a.v.1.1		3
156.83	odd	4		2704.2.a.w.1.1		3
156.155	even	2		2704.2.f.m.337.2		6
195.44	even	4		8450.2.a.bn.1.1		3
195.164	even	4		8450.2.a.bx.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.3	✓	3	39.8	even	4
338.2.a.h.1.3	yes	3	39.5	even	4
338.2.b.d.337.3		6	39.38	odd	2
338.2.b.d.337.6		6	3.2	odd	2
338.2.c.h.191.1		6	39.2	even	12
338.2.c.h.315.1		6	39.32	even	12
338.2.c.i.191.1		6	39.11	even	12
338.2.c.i.315.1		6	39.20	even	12
338.2.e.e.23.1		12	39.17	odd	6
338.2.e.e.23.4		12	39.35	odd	6
338.2.e.e.147.1		12	39.29	odd	6
338.2.e.e.147.4		12	39.23	odd	6
2704.2.a.v.1.1		3	156.47	odd	4
2704.2.a.w.1.1		3	156.83	odd	4
2704.2.f.m.337.1		6	12.11	even	2
2704.2.f.m.337.2		6	156.155	even	2
3042.2.a.z.1.3		3	13.5	odd	4
3042.2.a.bi.1.1		3	13.8	odd	4
3042.2.b.n.1351.3		6	1.1	even	1	trivial
3042.2.b.n.1351.4		6	13.12	even	2	inner
8450.2.a.bn.1.1		3	195.44	even	4
8450.2.a.bx.1.1		3	195.164	even	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.49396i q^{5} -1.60388i q^{7} +1.00000i q^{8} +2.49396 q^{10} -2.04892i q^{11} -1.60388 q^{14} +1.00000 q^{16} -4.54288 q^{17} +4.85086i q^{19} -2.49396i q^{20} -2.04892 q^{22} +2.71379 q^{23} -1.21983 q^{25} +1.60388i q^{28} +9.20775 q^{29} -5.10992i q^{31} -1.00000i q^{32} +4.54288i q^{34} +4.00000 q^{35} +7.60388i q^{37} +4.85086 q^{38} -2.49396 q^{40} -3.46681i q^{41} -11.3448 q^{43} +2.04892i q^{44} -2.71379i q^{46} -0.219833i q^{47} +4.42758 q^{49} +1.21983i q^{50} +2.71379 q^{53} +5.10992 q^{55} +1.60388 q^{56} -9.20775i q^{58} +4.07606i q^{59} +10.4155 q^{61} -5.10992 q^{62} -1.00000 q^{64} +12.0761i q^{67} +4.54288 q^{68} -4.00000i q^{70} +1.28621i q^{71} +3.62565i q^{73} +7.60388 q^{74} -4.85086i q^{76} -3.28621 q^{77} -5.32975 q^{79} +2.49396i q^{80} -3.46681 q^{82} +4.85086i q^{83} -11.3297i q^{85} +11.3448i q^{86} +2.04892 q^{88} +16.5700i q^{89} -2.71379 q^{92} -0.219833 q^{94} -12.0978 q^{95} -4.64071i q^{97} -4.42758i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 4 q^{10} + 8 q^{14} + 6 q^{16} + 10 q^{17} + 6 q^{22} - 10 q^{25} + 20 q^{29} + 24 q^{35} + 2 q^{38} + 4 q^{40} - 22 q^{43} - 6 q^{49} + 32 q^{55} - 8 q^{56} - 8 q^{61} - 32 q^{62} - 6 q^{64}+ \cdots - 36 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 - 4 * q^10 + 8 * q^14 + 6 * q^16 + 10 * q^17 + 6 * q^22 - 10 * q^25 + 20 * q^29 + 24 * q^35 + 2 * q^38 + 4 * q^40 - 22 * q^43 - 6 * q^49 + 32 * q^55 - 8 * q^56 - 8 * q^61 - 32 * q^62 - 6 * q^64 - 10 * q^68 + 28 * q^74 - 36 * q^77 - 36 * q^79 - 14 * q^82 - 6 * q^88 - 4 * q^94 - 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more