LMFDB - Embedded newform 2034.3.d.a.2033.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2034,3,Mod(2033,2034)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2034.2033"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2034, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2034 = 2 \cdot 3^{2} \cdot 113 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2034.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$55.4224857709$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2033.20
Character	$\chi$	$=$	2034.2033
Dual form			2034.3.d.a.2033.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} +4.78233 q^{5} -4.28693 q^{7} -2.82843i q^{8} +6.76323i q^{10} +2.61042i q^{11} -21.6969 q^{13} -6.06264i q^{14} +4.00000 q^{16} +16.9553 q^{17} -14.8684i q^{19} -9.56466 q^{20} -3.69169 q^{22} +33.3382 q^{23} -2.12933 q^{25} -30.6841i q^{26} +8.57387 q^{28} +6.75614 q^{29} +34.8616 q^{31} +5.65685i q^{32} +23.9784i q^{34} -20.5015 q^{35} -7.05401i q^{37} +21.0271 q^{38} -13.5265i q^{40} +59.9148i q^{41} +23.0449i q^{43} -5.22083i q^{44} +47.1473i q^{46} -9.91720 q^{47} -30.6222 q^{49} -3.01132i q^{50} +43.3938 q^{52} -19.5523i q^{53} +12.4839i q^{55} +12.1253i q^{56} +9.55462i q^{58} -4.01614 q^{59} +42.7166 q^{61} +49.3017i q^{62} -8.00000 q^{64} -103.762 q^{65} +123.748i q^{67} -33.9105 q^{68} -28.9935i q^{70} +61.3106 q^{71} -35.4491i q^{73} +9.97587 q^{74} +29.7368i q^{76} -11.1907i q^{77} +92.5692i q^{79} +19.1293 q^{80} -84.7324 q^{82} -27.2279i q^{83} +81.0857 q^{85} -32.5904 q^{86} +7.38338 q^{88} +133.262 q^{89} +93.0132 q^{91} -66.6763 q^{92} -14.0250i q^{94} -71.1055i q^{95} +164.696 q^{97} -43.3063i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4} - 8 q^{7} + 144 q^{16} + 36 q^{25} + 16 q^{28} + 64 q^{31} + 108 q^{49} - 80 q^{61} - 288 q^{64} + 248 q^{82} - 164 q^{85} + 516 q^{91} - 420 q^{97}+O(q^{100}) $ 36 * q - 72 * q^4 - 8 * q^7 + 144 * q^16 + 36 * q^25 + 16 * q^28 + 64 * q^31 + 108 * q^49 - 80 * q^61 - 288 * q^64 + 248 * q^82 - 164 * q^85 + 516 * q^91 - 420 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2034\mathbb{Z}\right)^\times$.

$n$	$227$	$1585$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$		0			0
$3$		0			0
$4$	−2.00000			−0.500000
$5$	4.78233			0.956466			0.478233	−	0.878233i	$-0.341277\pi$
$5$	4.78233			0.956466			0.478233	+	0.878233i	$0.341277\pi$
$6$		0			0
$7$	−4.28693			−0.612419			−0.306210	−	0.951964i	$-0.599061\pi$
$7$	−4.28693			−0.612419			−0.306210	+	0.951964i	$0.599061\pi$
$8$		−	2.82843i		−	0.353553i
$9$		0			0
$10$			6.76323i			0.676323i
$11$			2.61042i			0.237311i	0.992936	+	0.118655i	$0.0378584\pi$
$11$			2.61042i			0.237311i	−0.992936	+	0.118655i	$0.962142\pi$
$12$		0			0
$13$	−21.6969			−1.66899			−0.834496	−	0.551014i	$-0.814241\pi$
$13$	−21.6969			−1.66899			−0.834496	+	0.551014i	$0.814241\pi$
$14$		−	6.06264i		−	0.433046i
$15$		0			0
$16$	4.00000			0.250000
$17$	16.9553			0.997369			0.498684	−	0.866784i	$-0.333817\pi$
$17$	16.9553			0.997369			0.498684	+	0.866784i	$0.333817\pi$
$18$		0			0
$19$		−	14.8684i		−	0.782547i	−0.920275	−	0.391273i	$-0.872035\pi$
$19$		−	14.8684i		−	0.782547i	0.920275	−	0.391273i	$-0.127965\pi$
$20$	−9.56466			−0.478233
$21$		0			0
$22$	−3.69169			−0.167804
$23$	33.3382			1.44949			0.724743	−	0.689020i	$-0.241959\pi$
$23$	33.3382			1.44949			0.724743	+	0.689020i	$0.241959\pi$
$24$		0			0
$25$	−2.12933			−0.0851730
$26$		−	30.6841i		−	1.18016i
$27$		0			0
$28$	8.57387			0.306210
$29$	6.75614			0.232970			0.116485	−	0.993192i	$-0.462837\pi$
$29$	6.75614			0.232970			0.116485	+	0.993192i	$0.462837\pi$
$30$		0			0
$31$	34.8616			1.12457			0.562284	−	0.826944i	$-0.309923\pi$
$31$	34.8616			1.12457			0.562284	+	0.826944i	$0.309923\pi$
$32$			5.65685i			0.176777i
$33$		0			0
$34$			23.9784i			0.705246i
$35$	−20.5015			−0.585758
$36$		0			0
$37$		−	7.05401i		−	0.190649i	−0.995446	−	0.0953244i	$-0.969611\pi$
$37$		−	7.05401i		−	0.190649i	0.995446	−	0.0953244i	$-0.0303888\pi$
$38$	21.0271			0.553344
$39$		0			0
$40$		−	13.5265i		−	0.338162i
$41$			59.9148i			1.46134i	0.682732	+	0.730669i	$0.260792\pi$
$41$			59.9148i			1.46134i	−0.682732	+	0.730669i	$0.739208\pi$
$42$		0			0
$43$			23.0449i			0.535928i	0.963429	+	0.267964i	$0.0863508\pi$
$43$			23.0449i			0.535928i	−0.963429	+	0.267964i	$0.913649\pi$
$44$		−	5.22083i		−	0.118655i
$45$		0			0
$46$			47.1473i			1.02494i
$47$	−9.91720			−0.211004			−0.105502	−	0.994419i	$-0.533645\pi$
$47$	−9.91720			−0.211004			−0.105502	+	0.994419i	$0.533645\pi$
$48$		0			0
$49$	−30.6222			−0.624943
$50$		−	3.01132i		−	0.0602264i
$51$		0			0
$52$	43.3938			0.834496
$53$		−	19.5523i		−	0.368911i	−0.982841	−	0.184456i	$-0.940948\pi$
$53$		−	19.5523i		−	0.368911i	0.982841	−	0.184456i	$-0.0590522\pi$
$54$		0			0
$55$			12.4839i			0.226980i
$56$			12.1253i			0.216523i
$57$		0			0
$58$			9.55462i			0.164735i
$59$	−4.01614			−0.0680701			−0.0340351	−	0.999421i	$-0.510836\pi$
$59$	−4.01614			−0.0680701			−0.0340351	+	0.999421i	$0.510836\pi$
$60$		0			0
$61$	42.7166			0.700272			0.350136	−	0.936699i	$-0.386135\pi$
$61$	42.7166			0.700272			0.350136	+	0.936699i	$0.386135\pi$
$62$			49.3017i			0.795189i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	−103.762			−1.59633
$66$		0			0
$67$			123.748i			1.84698i	0.383621	+	0.923491i	$0.374677\pi$
$67$			123.748i			1.84698i	−0.383621	+	0.923491i	$0.625323\pi$
$68$	−33.9105			−0.498684
$69$		0			0
$70$		−	28.9935i		−	0.414193i
$71$	61.3106			0.863529			0.431765	−	0.901986i	$-0.357891\pi$
$71$	61.3106			0.863529			0.431765	+	0.901986i	$0.357891\pi$
$72$		0			0
$73$		−	35.4491i		−	0.485604i	−0.970076	−	0.242802i	$-0.921933\pi$
$73$		−	35.4491i		−	0.485604i	0.970076	−	0.242802i	$-0.0780666\pi$
$74$	9.97587			0.134809
$75$		0			0
$76$			29.7368i			0.391273i
$77$		−	11.1907i		−	0.145334i
$78$		0			0
$79$			92.5692i			1.17176i	0.810397	+	0.585881i	$0.199252\pi$
$79$			92.5692i			1.17176i	−0.810397	+	0.585881i	$0.800748\pi$
$80$	19.1293			0.239116
$81$		0			0
$82$	−84.7324			−1.03332
$83$		−	27.2279i		−	0.328047i	−0.986456	−	0.164023i	$-0.947553\pi$
$83$		−	27.2279i		−	0.328047i	0.986456	−	0.164023i	$-0.0524473\pi$
$84$		0			0
$85$	81.0857			0.953949
$86$	−32.5904			−0.378958
$87$		0			0
$88$	7.38338			0.0839020
$89$	133.262			1.49733			0.748664	−	0.662950i	$-0.230696\pi$
$89$	133.262			1.49733			0.748664	+	0.662950i	$0.230696\pi$
$90$		0			0
$91$	93.0132			1.02212
$92$	−66.6763			−0.724743
$93$		0			0
$94$		−	14.0250i		−	0.149203i
$95$		−	71.1055i		−	0.748479i
$96$		0			0
$97$	164.696			1.69790			0.848950	−	0.528472i	$-0.177235\pi$
$97$	164.696			1.69790			0.848950	+	0.528472i	$0.177235\pi$
$98$		−	43.3063i		−	0.441901i
$99$		0			0
$100$	4.25865			0.0425865
$101$	−80.7404			−0.799410			−0.399705	−	0.916644i	$-0.630887\pi$
$101$	−80.7404			−0.799410			−0.399705	+	0.916644i	$0.630887\pi$
$102$		0			0
$103$		−	17.6842i		−	0.171691i	−0.996308	−	0.0858455i	$-0.972641\pi$
$103$		−	17.6842i		−	0.171691i	0.996308	−	0.0858455i	$-0.0273591\pi$
$104$			61.3681i			0.590078i
$105$		0			0
$106$	27.6511			0.260860
$107$	−37.9739			−0.354897			−0.177448	−	0.984130i	$-0.556784\pi$
$107$	−37.9739			−0.354897			−0.177448	+	0.984130i	$0.556784\pi$
$108$		0			0
$109$	2.55319			0.0234237			0.0117119	−	0.999931i	$-0.496272\pi$
$109$	2.55319			0.0234237			0.0117119	+	0.999931i	$0.496272\pi$
$110$	−17.6549			−0.160499
$111$		0			0
$112$	−17.1477			−0.153105
$113$	64.9644	+	92.4588i	0.574906	+	0.818219i
$113$	64.9644	+	92.4588i	0.574906	+	0.818219i
$114$		0			0
$115$	159.434			1.38638
$116$	−13.5123			−0.116485
$117$		0			0
$118$		−	5.67968i		−	0.0481329i
$119$	−72.6861			−0.610808
$120$		0			0
$121$	114.186			0.943684
$122$			60.4104i			0.495167i
$123$		0			0
$124$	−69.7232			−0.562284
$125$	−129.741			−1.03793
$126$		0			0
$127$	46.9636			0.369792			0.184896	−	0.982758i	$-0.440805\pi$
$127$	46.9636			0.369792			0.184896	+	0.982758i	$0.440805\pi$
$128$		−	11.3137i		−	0.0883883i
$129$		0			0
$130$		−	146.741i		−	1.12878i
$131$			185.810i			1.41839i	0.705010	+	0.709197i	$0.250942\pi$
$131$			185.810i			1.41839i	−0.705010	+	0.709197i	$0.749058\pi$
$132$		0			0
$133$			63.7398i			0.479246i
$134$	−175.006			−1.30601
$135$		0			0
$136$		−	47.9567i		−	0.352623i
$137$	168.520			1.23007			0.615037	−	0.788498i	$-0.289141\pi$
$137$	168.520			1.23007			0.615037	+	0.788498i	$0.289141\pi$
$138$		0			0
$139$	0.332327			0.00239084			0.00119542	−	0.999999i	$-0.499619\pi$
$139$	0.332327			0.00239084			0.00119542	+	0.999999i	$0.499619\pi$
$140$	41.0031			0.292879
$141$		0			0
$142$			86.7062i			0.610607i
$143$		−	56.6380i		−	0.396070i
$144$		0			0
$145$	32.3101			0.222828
$146$	50.1326			0.343374
$147$		0			0
$148$			14.1080i			0.0953244i
$149$			204.457i			1.37220i	0.727509	+	0.686098i	$0.240678\pi$
$149$			204.457i			1.37220i	−0.727509	+	0.686098i	$0.759322\pi$
$150$		0			0
$151$		−	43.3642i		−	0.287180i	−0.989637	−	0.143590i	$-0.954135\pi$
$151$		−	43.3642i		−	0.287180i	0.989637	−	0.143590i	$-0.0458647\pi$
$152$	−42.0541			−0.276672
$153$		0			0
$154$	15.8260			0.102766
$155$	166.720			1.07561
$156$		0			0
$157$	45.3556			0.288889			0.144444	−	0.989513i	$-0.453860\pi$
$157$	45.3556			0.288889			0.144444	+	0.989513i	$0.453860\pi$
$158$	−130.913			−0.828561
$159$		0			0
$160$			27.0529i			0.169081i
$161$	−142.918			−0.887692
$162$		0			0
$163$	−17.7282			−0.108762			−0.0543808	−	0.998520i	$-0.517319\pi$
$163$	−17.7282			−0.108762			−0.0543808	+	0.998520i	$0.517319\pi$
$164$		−	119.830i		−	0.730669i
$165$		0			0
$166$	38.5060			0.231964
$167$	44.6081			0.267115			0.133557	−	0.991041i	$-0.457360\pi$
$167$	44.6081			0.267115			0.133557	+	0.991041i	$0.457360\pi$
$168$		0			0
$169$	301.755			1.78554
$170$			114.672i			0.674544i
$171$		0			0
$172$		−	46.0898i		−	0.267964i
$173$		−	23.5228i		−	0.135970i	−0.997686	−	0.0679850i	$-0.978343\pi$
$173$		−	23.5228i		−	0.135970i	0.997686	−	0.0679850i	$-0.0216570\pi$
$174$		0			0
$175$	9.12828			0.0521616
$176$			10.4417i			0.0593277i
$177$		0			0
$178$			188.461i			1.05877i
$179$	−298.526			−1.66774			−0.833871	−	0.551959i	$-0.813881\pi$
$179$	−298.526			−1.66774			−0.833871	+	0.551959i	$0.813881\pi$
$180$		0			0
$181$			149.275i			0.824723i	0.911020	+	0.412362i	$0.135296\pi$
$181$			149.275i			0.824723i	−0.911020	+	0.412362i	$0.864704\pi$
$182$			131.540i			0.722750i
$183$		0			0
$184$		−	94.2946i		−	0.512470i
$185$		−	33.7346i		−	0.182349i
$186$		0			0
$187$			44.2603i			0.236686i
$188$	19.8344			0.105502
$189$		0			0
$190$	100.558			0.529255
$191$	−351.745			−1.84159			−0.920797	−	0.390042i	$-0.872461\pi$
$191$	−351.745			−1.84159			−0.920797	+	0.390042i	$0.872461\pi$
$192$		0			0
$193$			165.323i			0.856597i	0.903637	+	0.428299i	$0.140887\pi$
$193$			165.323i			0.856597i	−0.903637	+	0.428299i	$0.859113\pi$
$194$			232.916i			1.20060i
$195$		0			0
$196$	61.2444			0.312471
$197$	189.756			0.963229			0.481614	−	0.876383i	$-0.340051\pi$
$197$	189.756			0.963229			0.481614	+	0.876383i	$0.340051\pi$
$198$		0			0
$199$		−	20.0124i		−	0.100565i	−0.998735	−	0.0502823i	$-0.983988\pi$
$199$		−	20.0124i		−	0.100565i	0.998735	−	0.0502823i	$-0.0160121\pi$
$200$			6.02264i			0.0301132i
$201$		0			0
$202$		−	114.184i		−	0.565268i
$203$	−28.9631			−0.142675
$204$		0			0
$205$			286.532i			1.39772i
$206$	25.0092			0.121404
$207$		0			0
$208$	−86.7876			−0.417248
$209$	38.8127			0.185707
$210$		0			0
$211$	57.7762			0.273821			0.136910	−	0.990583i	$-0.456283\pi$
$211$	57.7762			0.273821			0.136910	+	0.990583i	$0.456283\pi$
$212$			39.1046i			0.184456i
$213$		0			0
$214$		−	53.7033i		−	0.250950i
$215$			110.208i			0.512596i
$216$		0			0
$217$	−149.449			−0.688707
$218$			3.61075i			0.0165631i
$219$		0			0
$220$		−	24.9678i		−	0.113490i
$221$	−367.877			−1.66460
$222$		0			0
$223$			94.7696i			0.424976i	0.977164	+	0.212488i	$0.0681566\pi$
$223$			94.7696i			0.424976i	−0.977164	+	0.212488i	$0.931843\pi$
$224$		−	24.2506i		−	0.108261i
$225$		0			0
$226$	−130.756	+	91.8735i	−0.578569	+	0.406520i
$227$			116.202i			0.511902i	0.966690	+	0.255951i	$0.0823885\pi$
$227$			116.202i			0.511902i	−0.966690	+	0.255951i	$0.917611\pi$
$228$		0			0
$229$		−	250.845i		−	1.09539i	−0.836676	−	0.547697i	$-0.815504\pi$
$229$		−	250.845i		−	1.09539i	0.836676	−	0.547697i	$-0.184496\pi$
$230$			225.474i			0.980321i
$231$		0			0
$232$		−	19.1092i		−	0.0823674i
$233$			201.921i			0.866613i	0.901247	+	0.433307i	$0.142653\pi$
$233$			201.921i			0.866613i	−0.901247	+	0.433307i	$0.857347\pi$
$234$		0			0
$235$	−47.4273			−0.201818
$236$	8.03228			0.0340351
$237$		0			0
$238$		−	102.794i		−	0.431906i
$239$		−	250.281i		−	1.04720i	−0.851964	−	0.523600i	$-0.824589\pi$
$239$		−	250.281i		−	1.04720i	0.851964	−	0.523600i	$-0.175411\pi$
$240$		0			0
$241$	145.425			0.603423			0.301712	−	0.953399i	$-0.402442\pi$
$241$	145.425			0.603423			0.301712	+	0.953399i	$0.402442\pi$
$242$			161.483i			0.667285i
$243$		0			0
$244$	−85.4332			−0.350136
$245$	−146.445			−0.597737
$246$		0			0
$247$			322.598i			1.30606i
$248$		−	98.6035i		−	0.397595i
$249$		0			0
$250$		−	183.482i		−	0.733928i
$251$			151.472i			0.603475i	0.953391	+	0.301737i	$0.0975666\pi$
$251$			151.472i			0.603475i	−0.953391	+	0.301737i	$0.902433\pi$
$252$		0			0
$253$			87.0265i			0.343978i
$254$			66.4166i			0.261483i
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	29.8406i		−	0.116111i	−0.998313	−	0.0580557i	$-0.981510\pi$
$257$		−	29.8406i		−	0.116111i	0.998313	−	0.0580557i	$-0.0184901\pi$
$258$		0			0
$259$			30.2401i			0.116757i
$260$	207.523			0.798167
$261$		0			0
$262$	−262.774			−1.00296
$263$	−44.0729			−0.167578			−0.0837888	−	0.996484i	$-0.526702\pi$
$263$	−44.0729			−0.167578			−0.0837888	+	0.996484i	$0.526702\pi$
$264$		0			0
$265$		−	93.5055i		−	0.352851i
$266$	−90.1416			−0.338878
$267$		0			0
$268$		−	247.496i		−	0.923491i
$269$	410.011			1.52420			0.762101	−	0.647458i	$-0.224168\pi$
$269$	410.011			1.52420			0.762101	+	0.647458i	$0.224168\pi$
$270$		0			0
$271$			124.888i			0.460840i	0.973091	+	0.230420i	$0.0740101\pi$
$271$			124.888i			0.460840i	−0.973091	+	0.230420i	$0.925990\pi$
$272$	67.8211			0.249342
$273$		0			0
$274$			238.324i			0.869794i
$275$		−	5.55843i		−	0.0202125i
$276$		0			0
$277$	−37.3166			−0.134717			−0.0673584	−	0.997729i	$-0.521457\pi$
$277$	−37.3166			−0.134717			−0.0673584	+	0.997729i	$0.521457\pi$
$278$			0.469981i			0.00169058i
$279$		0			0
$280$			57.9871i			0.207097i
$281$	173.254			0.616562			0.308281	−	0.951295i	$-0.400246\pi$
$281$	173.254			0.616562			0.308281	+	0.951295i	$0.400246\pi$
$282$		0			0
$283$	−458.213			−1.61913			−0.809563	−	0.587033i	$-0.800296\pi$
$283$	−458.213			−1.61913			−0.809563	+	0.587033i	$0.800296\pi$
$284$	−122.621			−0.431765
$285$		0			0
$286$	80.0982			0.280064
$287$		−	256.851i		−	0.894951i
$288$		0			0
$289$	−1.51878			−0.00525528
$290$			45.6933i			0.157563i
$291$		0			0
$292$			70.8982i			0.242802i
$293$	399.154			1.36230			0.681151	−	0.732143i	$-0.261480\pi$
$293$	399.154			1.36230			0.681151	+	0.732143i	$0.261480\pi$
$294$		0			0
$295$	−19.2065			−0.0651068
$296$	−19.9517			−0.0674045
$297$		0			0
$298$	−289.146			−0.970289
$299$	−723.335			−2.41918
$300$		0			0
$301$		−	98.7919i		−	0.328212i
$302$	61.3263			0.203067
$303$		0			0
$304$		−	59.4735i		−	0.195637i
$305$	204.285			0.669786
$306$		0			0
$307$	−301.988			−0.983675			−0.491838	−	0.870687i	$-0.663675\pi$
$307$	−301.988			−0.983675			−0.491838	+	0.870687i	$0.663675\pi$
$308$			22.3814i			0.0726668i
$309$		0			0
$310$			235.777i			0.760572i
$311$			161.873i			0.520494i	0.965542	+	0.260247i	$0.0838039\pi$
$311$			161.873i			0.520494i	−0.965542	+	0.260247i	$0.916196\pi$
$312$		0			0
$313$	400.548			1.27971			0.639853	−	0.768497i	$-0.278995\pi$
$313$	400.548			1.27971			0.639853	+	0.768497i	$0.278995\pi$
$314$			64.1425i			0.204275i
$315$		0			0
$316$		−	185.138i		−	0.585881i
$317$		−	192.010i		−	0.605711i	−0.953037	−	0.302855i	$-0.902060\pi$
$317$		−	192.010i		−	0.605711i	0.953037	−	0.302855i	$-0.0979399\pi$
$318$		0			0
$319$			17.6363i			0.0552863i
$320$	−38.2586			−0.119558
$321$		0			0
$322$		−	202.117i		−	0.627693i
$323$		−	252.097i		−	0.780488i
$324$		0			0
$325$	46.1998			0.142153
$326$		−	25.0714i		−	0.0769061i
$327$		0			0
$328$	169.465			0.516661
$329$	42.5144			0.129223
$330$		0			0
$331$	−333.921			−1.00882			−0.504412	−	0.863463i	$-0.668291\pi$
$331$	−333.921			−1.00882			−0.504412	+	0.863463i	$0.668291\pi$
$332$			54.4558i			0.164023i
$333$		0			0
$334$			63.0854i			0.188879i
$335$			591.803i			1.76657i
$336$		0			0
$337$	35.9196			0.106586			0.0532931	−	0.998579i	$-0.483028\pi$
$337$	35.9196			0.106586			0.0532931	+	0.998579i	$0.483028\pi$
$338$			426.747i			1.26256i
$339$		0			0
$340$	−162.171			−0.476975
$341$			91.0033i			0.266872i
$342$		0			0
$343$	341.335			0.995146
$344$	65.1808			0.189479
$345$		0			0
$346$	33.2663			0.0961453
$347$		−	9.25373i		−	0.0266678i	−0.999911	−	0.0133339i	$-0.995756\pi$
$347$		−	9.25373i		−	0.0266678i	0.999911	−	0.0133339i	$-0.00424444\pi$
$348$		0			0
$349$			56.7279i			0.162544i	0.996692	+	0.0812721i	$0.0258983\pi$
$349$			56.7279i			0.162544i	−0.996692	+	0.0812721i	$0.974102\pi$
$350$			12.9093i			0.0368838i
$351$		0			0
$352$	−14.7668			−0.0419510
$353$		−	552.577i		−	1.56537i	−0.622416	−	0.782687i	$-0.713849\pi$
$353$		−	552.577i		−	1.56537i	0.622416	−	0.782687i	$-0.286151\pi$
$354$		0			0
$355$	293.207			0.825936
$356$	−266.524			−0.748664
$357$		0			0
$358$		−	422.179i		−	1.17927i
$359$	302.972			0.843932			0.421966	−	0.906612i	$-0.361340\pi$
$359$	302.972			0.843932			0.421966	+	0.906612i	$0.361340\pi$
$360$		0			0
$361$	139.931			0.387621
$362$	−211.107			−0.583167
$363$		0			0
$364$	−186.026			−0.511061
$365$		−	169.529i		−	0.464464i
$366$		0			0
$367$	534.252			1.45573			0.727864	−	0.685721i	$-0.240513\pi$
$367$	534.252			1.45573			0.727864	+	0.685721i	$0.240513\pi$
$368$	133.353			0.362371
$369$		0			0
$370$	47.7079			0.128940
$371$			83.8193i			0.225928i
$372$		0			0
$373$			574.967i			1.54147i	0.637158	+	0.770734i	$0.280110\pi$
$373$			574.967i			1.54147i	−0.637158	+	0.770734i	$0.719890\pi$
$374$	−62.5936			−0.167362
$375$		0			0
$376$			28.0501i			0.0746013i
$377$	−146.587			−0.388826
$378$		0			0
$379$		−	655.400i		−	1.72929i	−0.502385	−	0.864644i	$-0.667544\pi$
$379$		−	655.400i		−	1.72929i	0.502385	−	0.864644i	$-0.332456\pi$
$380$			142.211i			0.374240i
$381$		0			0
$382$		−	497.442i		−	1.30220i
$383$		−	418.637i		−	1.09305i	−0.837444	−	0.546524i	$-0.815951\pi$
$383$		−	418.637i		−	1.09305i	0.837444	−	0.546524i	$-0.184049\pi$
$384$		0			0
$385$		−	53.5175i		−	0.139007i
$386$	−233.802			−0.605706
$387$		0			0
$388$	−329.393			−0.848950
$389$			98.2947i			0.252686i	0.991987	+	0.126343i	$0.0403239\pi$
$389$			98.2947i			0.252686i	−0.991987	+	0.126343i	$0.959676\pi$
$390$		0			0
$391$	565.258			1.44567
$392$			86.6127i			0.220951i
$393$		0			0
$394$			268.356i			0.681105i
$395$			442.697i			1.12075i
$396$		0			0
$397$			588.597i			1.48261i	0.671168	+	0.741306i	$0.265793\pi$
$397$			588.597i			1.48261i	−0.671168	+	0.741306i	$0.734207\pi$
$398$	28.3018			0.0711099
$399$		0			0
$400$	−8.51730			−0.0212933
$401$			655.042i			1.63352i	0.576977	+	0.816760i	$0.304232\pi$
$401$			655.042i			1.63352i	−0.576977	+	0.816760i	$0.695768\pi$
$402$		0			0
$403$	−756.389			−1.87689
$404$	161.481			0.399705
$405$		0			0
$406$		−	40.9600i		−	0.100887i
$407$	18.4139			0.0452430
$408$		0			0
$409$			613.209i			1.49929i	0.661841	+	0.749644i	$0.269775\pi$
$409$			613.209i			1.49929i	−0.661841	+	0.749644i	$0.730225\pi$
$410$	−405.218			−0.988337
$411$		0			0
$412$			35.3683i			0.0858455i
$413$	17.2169			0.0416875
$414$		0			0
$415$		−	130.213i		−	0.313766i
$416$		−	122.736i		−	0.295039i
$417$		0			0
$418$			54.8894i			0.131314i
$419$	−368.494			−0.879461			−0.439731	−	0.898130i	$-0.644926\pi$
$419$	−368.494			−0.879461			−0.439731	+	0.898130i	$0.644926\pi$
$420$		0			0
$421$	−116.678			−0.277144			−0.138572	−	0.990352i	$-0.544251\pi$
$421$	−116.678			−0.277144			−0.138572	+	0.990352i	$0.544251\pi$
$422$			81.7078i			0.193620i
$423$		0			0
$424$	−55.3022			−0.130430
$425$	−36.1033			−0.0849489
$426$		0			0
$427$	−183.123			−0.428860
$428$	75.9479			0.177448
$429$		0			0
$430$	−155.858			−0.362460
$431$	248.188			0.575841			0.287921	−	0.957654i	$-0.407036\pi$
$431$	248.188			0.575841			0.287921	+	0.957654i	$0.407036\pi$
$432$		0			0
$433$		−	381.341i		−	0.880695i	−0.897827	−	0.440348i	$-0.854855\pi$
$433$		−	381.341i		−	0.880695i	0.897827	−	0.440348i	$-0.145145\pi$
$434$		−	211.353i		−	0.486989i
$435$		0			0
$436$	−5.10637			−0.0117119
$437$		−	495.685i		−	1.13429i
$438$		0			0
$439$	−706.752			−1.60991			−0.804956	−	0.593334i	$-0.797811\pi$
$439$	−706.752			−1.60991			−0.804956	+	0.593334i	$0.797811\pi$
$440$	35.3097			0.0802494
$441$		0			0
$442$		−	520.256i		−	1.17705i
$443$			425.622i			0.960771i	0.877057	+	0.480386i	$0.159503\pi$
$443$			425.622i			0.960771i	−0.877057	+	0.480386i	$0.840497\pi$
$444$		0			0
$445$	637.304			1.43214
$446$	−134.024			−0.300503
$447$		0			0
$448$	34.2955			0.0765524
$449$	−247.871			−0.552051			−0.276025	−	0.961150i	$-0.589017\pi$
$449$	−247.871			−0.552051			−0.276025	+	0.961150i	$0.589017\pi$
$450$		0			0
$451$	−156.403			−0.346791
$452$	−129.929	−	184.918i	−0.287453	−	0.409110i
$453$		0			0
$454$	−164.334			−0.361969
$455$	444.820			0.977625
$456$		0			0
$457$		−	803.585i		−	1.75839i	−0.476460	−	0.879196i	$-0.658080\pi$
$457$		−	803.585i		−	1.75839i	0.476460	−	0.879196i	$-0.341920\pi$
$458$	354.749			0.774561
$459$		0			0
$460$	−318.868			−0.693192
$461$		−	226.687i		−	0.491729i	−0.969304	−	0.245865i	$-0.920928\pi$
$461$		−	226.687i		−	0.491729i	0.969304	−	0.245865i	$-0.0790718\pi$
$462$		0			0
$463$	101.395			0.218997			0.109498	−	0.993987i	$-0.465076\pi$
$463$	101.395			0.218997			0.109498	+	0.993987i	$0.465076\pi$
$464$	27.0246			0.0582426
$465$		0			0
$466$	−285.559			−0.612788
$467$		−	719.470i		−	1.54062i	−0.637669	−	0.770310i	$-0.720101\pi$
$467$		−	719.470i		−	1.54062i	0.637669	−	0.770310i	$-0.279899\pi$
$468$		0			0
$469$		−	530.498i		−	1.13113i
$470$		−	67.0723i		−	0.142707i
$471$		0			0
$472$			11.3594i			0.0240664i
$473$	−60.1568			−0.127181
$474$		0			0
$475$			31.6596i			0.0666519i
$476$	145.372			0.305404
$477$		0			0
$478$	353.951			0.740482
$479$	−303.865			−0.634374			−0.317187	−	0.948363i	$-0.602738\pi$
$479$	−303.865			−0.634374			−0.317187	+	0.948363i	$0.602738\pi$
$480$		0			0
$481$			153.050i			0.318191i
$482$			205.662i			0.426685i
$483$		0			0
$484$	−228.371			−0.471842
$485$	787.632			1.62398
$486$		0			0
$487$		−	612.269i		−	1.25723i	−0.777718	−	0.628613i	$-0.783623\pi$
$487$		−	612.269i		−	1.25723i	0.777718	−	0.628613i	$-0.216377\pi$
$488$		−	120.821i		−	0.247584i
$489$		0			0
$490$		−	207.105i		−	0.422664i
$491$	325.265			0.662454			0.331227	−	0.943551i	$-0.392537\pi$
$491$	325.265			0.662454			0.331227	+	0.943551i	$0.392537\pi$
$492$		0			0
$493$	114.552			0.232357
$494$	−456.222			−0.923527
$495$		0			0
$496$	139.446			0.281142
$497$	−262.834			−0.528842
$498$		0			0
$499$		−	374.732i		−	0.750966i	−0.926829	−	0.375483i	$-0.877477\pi$
$499$		−	374.732i		−	0.750966i	0.926829	−	0.375483i	$-0.122523\pi$
$500$	259.483			0.518965
$501$		0			0
$502$	−214.214			−0.426721
$503$		−	184.219i		−	0.366240i	−0.983091	−	0.183120i	$-0.941380\pi$
$503$		−	184.219i		−	0.366240i	0.983091	−	0.183120i	$-0.0586197\pi$
$504$		0			0
$505$	−386.127			−0.764608
$506$	−123.074			−0.243229
$507$		0			0
$508$	−93.9272			−0.184896
$509$			248.911i			0.489019i	0.969647	+	0.244510i	$0.0786270\pi$
$509$			248.911i			0.489019i	−0.969647	+	0.244510i	$0.921373\pi$
$510$		0			0
$511$			151.968i			0.297393i
$512$			22.6274i			0.0441942i
$513$		0			0
$514$	42.2010			0.0821031
$515$		−	84.5715i		−	0.164217i
$516$		0			0
$517$		−	25.8880i		−	0.0500736i
$518$	−42.7659			−0.0825596
$519$		0			0
$520$			293.482i			0.564389i
$521$		−	603.346i		−	1.15805i	−0.815308	−	0.579027i	$-0.803433\pi$
$521$		−	603.346i		−	1.15805i	0.815308	−	0.579027i	$-0.196567\pi$
$522$		0			0
$523$		−	140.214i		−	0.268095i	−0.990975	−	0.134048i	$-0.957202\pi$
$523$		−	140.214i		−	0.268095i	0.990975	−	0.134048i	$-0.0427975\pi$
$524$		−	371.619i		−	0.709197i
$525$		0			0
$526$		−	62.3285i		−	0.118495i
$527$	591.088			1.12161
$528$		0			0
$529$	582.433			1.10101
$530$	132.237			0.249503
$531$		0			0
$532$		−	127.480i		−	0.239623i
$533$		−	1299.97i		−	2.43896i
$534$		0			0
$535$	−181.604			−0.339446
$536$	350.012			0.653007
$537$		0			0
$538$			579.843i			1.07777i
$539$		−	79.9367i		−	0.148306i
$540$		0			0
$541$			712.625i			1.31724i	0.752477	+	0.658618i	$0.228859\pi$
$541$			712.625i			1.31724i	−0.752477	+	0.658618i	$0.771141\pi$
$542$	−176.618			−0.325863
$543$		0			0
$544$			95.9135i			0.176312i
$545$	12.2102			0.0224040
$546$		0			0
$547$	−334.655			−0.611800			−0.305900	−	0.952064i	$-0.598957\pi$
$547$	−334.655			−0.611800			−0.305900	+	0.952064i	$0.598957\pi$
$548$	−337.040			−0.615037
$549$		0			0
$550$	7.86081			0.0142924
$551$		−	100.453i		−	0.182310i
$552$		0			0
$553$		−	396.838i		−	0.717610i
$554$		−	52.7736i		−	0.0952592i
$555$		0			0
$556$	−0.664653			−0.00119542
$557$		−	337.054i		−	0.605125i	−0.953130	−	0.302562i	$-0.902158\pi$
$557$		−	337.054i		−	0.605125i	0.953130	−	0.302562i	$-0.0978421\pi$
$558$		0			0
$559$		−	500.003i		−	0.894459i
$560$	−82.0061			−0.146439
$561$		0			0
$562$			245.018i			0.435975i
$563$			234.004i			0.415638i	0.978167	+	0.207819i	$0.0666365\pi$
$563$			234.004i			0.415638i	−0.978167	+	0.207819i	$0.933364\pi$
$564$		0			0
$565$	310.681	+	442.168i	0.549878	+	0.782599i
$566$		−	648.011i		−	1.14490i
$567$		0			0
$568$		−	173.412i		−	0.305304i
$569$		−	651.311i		−	1.14466i	−0.820024	−	0.572329i	$-0.806040\pi$
$569$		−	651.311i		−	1.14466i	0.820024	−	0.572329i	$-0.193960\pi$
$570$		0			0
$571$		−	727.882i		−	1.27475i	−0.770554	−	0.637374i	$-0.780020\pi$
$571$		−	727.882i		−	1.27475i	0.770554	−	0.637374i	$-0.219980\pi$
$572$			113.276i			0.198035i
$573$		0			0
$574$	363.242			0.632826
$575$	−70.9878			−0.123457
$576$		0			0
$577$		−	41.1833i		−	0.0713748i	−0.999363	−	0.0356874i	$-0.988638\pi$
$577$		−	41.1833i		−	0.0713748i	0.999363	−	0.0356874i	$-0.0113621\pi$
$578$		−	2.14788i		−	0.00371605i
$579$		0			0
$580$	−64.6202			−0.111414
$581$			116.724i			0.200902i
$582$		0			0
$583$	51.0396			0.0875465
$584$	−100.265			−0.171687
$585$		0			0
$586$			564.490i			0.963293i
$587$			81.1730i			0.138285i	0.997607	+	0.0691423i	$0.0220262\pi$
$587$			81.1730i			0.138285i	−0.997607	+	0.0691423i	$0.977974\pi$
$588$		0			0
$589$		−	518.336i		−	0.880027i
$590$		−	27.1621i		−	0.0460374i
$591$		0			0
$592$		−	28.2160i		−	0.0476622i
$593$			77.3160i			0.130381i	0.997873	+	0.0651906i	$0.0207655\pi$
$593$			77.3160i			0.130381i	−0.997873	+	0.0651906i	$0.979234\pi$
$594$		0			0
$595$	−347.609			−0.584217
$596$		−	408.914i		−	0.686098i
$597$		0			0
$598$		−	1022.95i		−	1.71062i
$599$	−209.698			−0.350080			−0.175040	−	0.984561i	$-0.556005\pi$
$599$	−209.698			−0.350080			−0.175040	+	0.984561i	$0.556005\pi$
$600$		0			0
$601$	123.967			0.206268			0.103134	−	0.994667i	$-0.467113\pi$
$601$	123.967			0.206268			0.103134	+	0.994667i	$0.467113\pi$
$602$	139.713			0.232081
$603$		0			0
$604$			86.7284i			0.143590i
$605$	546.074			0.902601
$606$		0			0
$607$			1044.75i			1.72117i	0.509309	+	0.860584i	$0.329901\pi$
$607$			1044.75i			1.72117i	−0.509309	+	0.860584i	$0.670099\pi$
$608$	84.1083			0.138336
$609$		0			0
$610$			288.902i			0.473610i
$611$	215.172			0.352164
$612$		0			0
$613$		−	728.098i		−	1.18776i	−0.804553	−	0.593881i	$-0.797595\pi$
$613$		−	728.098i		−	1.18776i	0.804553	−	0.593881i	$-0.202405\pi$
$614$		−	427.076i		−	0.695563i
$615$		0			0
$616$	−31.6520			−0.0513832
$617$		−	324.372i		−	0.525724i	−0.964833	−	0.262862i	$-0.915334\pi$
$617$		−	324.372i		−	0.525724i	0.964833	−	0.262862i	$-0.0846664\pi$
$618$		0			0
$619$			215.105i			0.347503i	0.984790	+	0.173752i	$0.0555890\pi$
$619$			215.105i			0.347503i	−0.984790	+	0.173752i	$0.944411\pi$
$620$	−333.439			−0.537805
$621$		0			0
$622$	−228.924			−0.368045
$623$	−571.286			−0.916992
$624$		0			0
$625$	−567.233			−0.907573
$626$			566.460i			0.904889i
$627$		0			0
$628$	−90.7111			−0.144444
$629$		−	119.603i		−	0.190147i
$630$		0			0
$631$			869.843i			1.37851i	0.724517	+	0.689257i	$0.242063\pi$
$631$			869.843i			1.37851i	−0.724517	+	0.689257i	$0.757937\pi$
$632$	261.825			0.414281
$633$		0			0
$634$	271.544			0.428302
$635$	224.595			0.353694
$636$		0			0
$637$	664.407			1.04302
$638$	−24.9415			−0.0390933
$639$		0			0
$640$		−	54.1059i		−	0.0845404i
$641$	1217.38			1.89919			0.949596	−	0.313478i	$-0.101494\pi$
$641$	1217.38			1.89919			0.949596	+	0.313478i	$0.101494\pi$
$642$		0			0
$643$		−	27.9082i		−	0.0434031i	−0.999764	−	0.0217015i	$-0.993092\pi$
$643$		−	27.9082i		−	0.0434031i	0.999764	−	0.0217015i	$-0.00690836\pi$
$644$	285.837			0.443846
$645$		0			0
$646$	356.520			0.551888
$647$		−	368.570i		−	0.569661i	−0.958578	−	0.284830i	$-0.908063\pi$
$647$		−	368.570i		−	0.569661i	0.958578	−	0.284830i	$-0.0919372\pi$
$648$		0			0
$649$		−	10.4838i		−	0.0161538i
$650$			65.3364i			0.100517i
$651$		0			0
$652$	35.4563			0.0543808
$653$			413.673i			0.633496i	0.948510	+	0.316748i	$0.102591\pi$
$653$			413.673i			0.633496i	−0.948510	+	0.316748i	$0.897409\pi$
$654$		0			0
$655$			888.603i			1.35665i
$656$			239.659i			0.365334i
$657$		0			0
$658$			60.1244i			0.0913745i
$659$	235.713			0.357683			0.178842	−	0.983878i	$-0.442765\pi$
$659$	235.713			0.357683			0.178842	+	0.983878i	$0.442765\pi$
$660$		0			0
$661$			353.474i			0.534756i	0.963592	+	0.267378i	$0.0861572\pi$
$661$			353.474i			0.534756i	−0.963592	+	0.267378i	$0.913843\pi$
$662$		−	472.235i		−	0.713346i
$663$		0			0
$664$	−77.0121			−0.115982
$665$			304.825i			0.458383i
$666$		0			0
$667$	225.237			0.337687
$668$	−89.2163			−0.133557
$669$		0			0
$670$	−836.935			−1.24916
$671$			111.508i			0.166182i
$672$		0			0
$673$		−	583.249i		−	0.866640i	−0.901240	−	0.433320i	$-0.857342\pi$
$673$		−	583.249i		−	0.866640i	0.901240	−	0.433320i	$-0.142658\pi$
$674$			50.7979i			0.0753679i
$675$		0			0
$676$	−603.511			−0.892768
$677$			688.358i			1.01678i	0.861128	+	0.508389i	$0.169759\pi$
$677$			688.358i			1.01678i	−0.861128	+	0.508389i	$0.830241\pi$
$678$		0			0
$679$	−706.042			−1.03983
$680$		−	229.345i		−	0.337272i
$681$		0			0
$682$	−128.698			−0.188707
$683$	841.455			1.23200			0.616000	−	0.787746i	$-0.288752\pi$
$683$	841.455			1.23200			0.616000	+	0.787746i	$0.288752\pi$
$684$		0			0
$685$	805.919			1.17652
$686$			482.721i			0.703674i
$687$		0			0
$688$			92.1796i			0.133982i
$689$			424.224i			0.615710i
$690$		0			0
$691$	910.864			1.31818			0.659091	−	0.752063i	$-0.270941\pi$
$691$	910.864			1.31818			0.659091	+	0.752063i	$0.270941\pi$
$692$			47.0456i			0.0679850i
$693$		0			0
$694$	13.0867			0.0188570
$695$	1.58930			0.00228676
$696$		0			0
$697$			1015.87i			1.45749i
$698$	−80.2254			−0.114936
$699$		0			0
$700$	−18.2566			−0.0260808
$701$	435.611			0.621413			0.310707	−	0.950506i	$-0.399434\pi$
$701$	435.611			0.621413			0.310707	+	0.950506i	$0.399434\pi$
$702$		0			0
$703$	−104.882			−0.149192
$704$		−	20.8833i		−	0.0296638i
$705$		0			0
$706$	781.462			1.10689
$707$	346.129			0.489574
$708$		0			0
$709$	206.833			0.291725			0.145862	−	0.989305i	$-0.453404\pi$
$709$	206.833			0.291725			0.145862	+	0.989305i	$0.453404\pi$
$710$			414.658i			0.584025i
$711$		0			0
$712$		−	376.922i		−	0.529385i
$713$	1162.22			1.63004
$714$		0			0
$715$		−	270.861i		−	0.378827i
$716$	597.052			0.833871
$717$		0			0
$718$			428.467i			0.596750i
$719$		−	1143.77i		−	1.59079i	−0.606095	−	0.795393i	$-0.707265\pi$
$719$		−	1143.77i		−	1.59079i	0.606095	−	0.795393i	$-0.292735\pi$
$720$		0			0
$721$			75.8109i			0.105147i
$722$			197.893i			0.274089i
$723$		0			0
$724$		−	298.550i		−	0.412362i
$725$	−14.3860			−0.0198428
$726$		0			0
$727$	1210.10			1.66451			0.832255	−	0.554392i	$-0.187049\pi$
$727$	1210.10			1.66451			0.832255	+	0.554392i	$0.187049\pi$
$728$		−	263.081i		−	0.361375i
$729$		0			0
$730$	239.751			0.328426
$731$			390.732i			0.534518i
$732$		0			0
$733$			583.236i			0.795683i	0.917454	+	0.397842i	$0.130241\pi$
$733$			583.236i			0.795683i	−0.917454	+	0.397842i	$0.869759\pi$
$734$			755.547i			1.02936i
$735$		0			0
$736$			188.589i			0.256235i
$737$	−323.033			−0.438308
$738$		0			0
$739$	−1031.13			−1.39530			−0.697652	−	0.716437i	$-0.745772\pi$
$739$	−1031.13			−1.39530			−0.697652	+	0.716437i	$0.745772\pi$
$740$			67.4692i			0.0911746i
$741$		0			0
$742$	−118.538			−0.159755
$743$	−1067.07			−1.43616			−0.718081	−	0.695960i	$-0.754979\pi$
$743$	−1067.07			−1.43616			−0.718081	+	0.695960i	$0.754979\pi$
$744$		0			0
$745$			977.782i			1.31246i
$746$	−813.127			−1.08998
$747$		0			0
$748$		−	88.5207i		−	0.118343i
$749$	162.792			0.217345
$750$		0			0
$751$			797.207i			1.06153i	0.847520	+	0.530763i	$0.178095\pi$
$751$			797.207i			1.06153i	−0.847520	+	0.530763i	$0.821905\pi$
$752$	−39.6688			−0.0527511
$753$		0			0
$754$		−	207.306i		−	0.274941i
$755$		−	207.382i		−	0.274678i
$756$		0			0
$757$		−	331.601i		−	0.438047i	−0.975720	−	0.219023i	$-0.929713\pi$
$757$		−	331.601i		−	0.438047i	0.975720	−	0.219023i	$-0.0702871\pi$
$758$	926.876			1.22279
$759$		0			0
$760$	−201.117			−0.264627
$761$		−	857.953i		−	1.12740i	−0.825979	−	0.563701i	$-0.809377\pi$
$761$		−	857.953i		−	1.12740i	0.825979	−	0.563701i	$-0.190623\pi$
$762$		0			0
$763$	−10.9453			−0.0143451
$764$	703.489			0.920797
$765$		0			0
$766$	592.042			0.772901
$767$	87.1378			0.113609
$768$		0			0
$769$	−477.211			−0.620561			−0.310281	−	0.950645i	$-0.600423\pi$
$769$	−477.211			−0.620561			−0.310281	+	0.950645i	$0.600423\pi$
$770$	75.6852			0.0982925
$771$		0			0
$772$		−	330.647i		−	0.428299i
$773$		−	963.778i		−	1.24680i	−0.781902	−	0.623401i	$-0.785751\pi$
$773$		−	963.778i		−	1.24680i	0.781902	−	0.623401i	$-0.214249\pi$
$774$		0			0
$775$	−74.2317			−0.0957829
$776$		−	465.832i		−	0.600299i
$777$		0			0
$778$	−139.010			−0.178676
$779$	890.837			1.14356
$780$		0			0
$781$			160.046i			0.204925i
$782$			799.395i			1.02224i
$783$		0			0
$784$	−122.489			−0.156236
$785$	216.905			0.276312
$786$		0			0
$787$	1229.87			1.56273			0.781365	−	0.624074i	$-0.214524\pi$
$787$	1229.87			1.56273			0.781365	+	0.624074i	$0.214524\pi$
$788$	−379.512			−0.481614
$789$		0			0
$790$	−626.068			−0.792491
$791$	−278.498	−	396.365i	−0.352083	−	0.501093i
$792$		0			0
$793$	−926.818			−1.16875
$794$	−832.401			−1.04836
$795$		0			0
$796$			40.0247i			0.0502823i
$797$	−102.871			−0.129073			−0.0645367	−	0.997915i	$-0.520557\pi$
$797$	−102.871			−0.129073			−0.0645367	+	0.997915i	$0.520557\pi$
$798$		0			0
$799$	−168.149			−0.210449
$800$		−	12.0453i		−	0.0150566i
$801$		0			0
$802$	−926.369			−1.15507
$803$	92.5370			0.115239
$804$		0			0
$805$	−683.483			−0.849047
$806$		−	1069.70i		−	1.32717i
$807$		0			0
$808$			228.368i			0.282634i
$809$			23.3365i			0.0288461i	0.999896	+	0.0144230i	$0.00459116\pi$
$809$			23.3365i			0.0288461i	−0.999896	+	0.0144230i	$0.995409\pi$
$810$		0			0
$811$		−	761.616i		−	0.939108i	−0.882904	−	0.469554i	$-0.844415\pi$
$811$		−	761.616i		−	0.939108i	0.882904	−	0.469554i	$-0.155585\pi$
$812$	57.9262			0.0713377
$813$		0			0
$814$			26.0412i			0.0319916i
$815$	−84.7819			−0.104027
$816$		0			0
$817$	342.640			0.419388
$818$	−867.208			−1.06016
$819$		0			0
$820$		−	573.065i		−	0.698860i
$821$		−	1316.60i		−	1.60366i	−0.597554	−	0.801829i	$-0.703861\pi$
$821$		−	1316.60i		−	1.60366i	0.597554	−	0.801829i	$-0.296139\pi$
$822$		0			0
$823$	−285.112			−0.346430			−0.173215	−	0.984884i	$-0.555416\pi$
$823$	−285.112			−0.346430			−0.173215	+	0.984884i	$0.555416\pi$
$824$	−50.0184			−0.0607019
$825$		0			0
$826$			24.3484i			0.0294775i
$827$			475.326i			0.574759i	0.957817	+	0.287379i	$0.0927840\pi$
$827$			475.326i			0.574759i	−0.957817	+	0.287379i	$0.907216\pi$
$828$		0			0
$829$		−	269.455i		−	0.325036i	−0.986706	−	0.162518i	$-0.948038\pi$
$829$		−	269.455i		−	0.325036i	0.986706	−	0.162518i	$-0.0519615\pi$
$830$	184.149			0.221866
$831$		0			0
$832$	173.575			0.208624
$833$	−519.208			−0.623299
$834$		0			0
$835$	213.331			0.255486
$836$	−77.6254			−0.0928533
$837$		0			0
$838$		−	521.130i		−	0.621873i
$839$	653.011			0.778320			0.389160	−	0.921170i	$-0.372765\pi$
$839$	653.011			0.778320			0.389160	+	0.921170i	$0.372765\pi$
$840$		0			0
$841$	−795.355			−0.945725
$842$		−	165.007i		−	0.195970i
$843$		0			0
$844$	−115.552			−0.136910
$845$	1443.09			1.70780
$846$		0			0
$847$	−489.507			−0.577930
$848$		−	78.2091i		−	0.0922278i
$849$		0			0
$850$		−	51.0578i		−	0.0600680i
$851$		−	235.168i		−	0.276343i
$852$		0			0
$853$	−624.206			−0.731777			−0.365889	−	0.930659i	$-0.619235\pi$
$853$	−624.206			−0.731777			−0.365889	+	0.930659i	$0.619235\pi$
$854$		−	258.975i		−	0.303250i
$855$		0			0
$856$			107.407i			0.125475i
$857$	−961.461			−1.12189			−0.560946	−	0.827852i	$-0.689562\pi$
$857$	−961.461			−1.12189			−0.560946	+	0.827852i	$0.689562\pi$
$858$		0			0
$859$		−	1042.43i		−	1.21354i	−0.794879	−	0.606768i	$-0.792466\pi$
$859$		−	1042.43i		−	1.21354i	0.794879	−	0.606768i	$-0.207534\pi$
$860$		−	220.416i		−	0.256298i
$861$		0			0
$862$			350.990i			0.407181i
$863$		−	688.236i		−	0.797492i	−0.917061	−	0.398746i	$-0.869445\pi$
$863$		−	688.236i		−	0.797492i	0.917061	−	0.398746i	$-0.130555\pi$
$864$		0			0
$865$		−	112.494i		−	0.130051i
$866$	539.298			0.622746
$867$		0			0
$868$	298.899			0.344353
$869$	−241.644			−0.278072
$870$		0			0
$871$		−	2684.94i		−	3.08260i
$872$		−	7.22150i		−	0.00828153i
$873$		0			0
$874$	701.004			0.802064
$875$	556.193			0.635649
$876$		0			0
$877$		−	1685.88i		−	1.92233i	−0.275977	−	0.961164i	$-0.589001\pi$
$877$		−	1685.88i		−	1.92233i	0.275977	−	0.961164i	$-0.410999\pi$
$878$		−	999.498i		−	1.13838i
$879$		0			0
$880$			49.9355i			0.0567449i
$881$	−1170.78			−1.32892			−0.664461	−	0.747323i	$-0.731339\pi$
$881$	−1170.78			−1.32892			−0.664461	+	0.747323i	$0.731339\pi$
$882$		0			0
$883$		−	760.596i		−	0.861377i	−0.902501	−	0.430689i	$-0.858271\pi$
$883$		−	760.596i		−	0.861377i	0.902501	−	0.430689i	$-0.141729\pi$
$884$	735.754			0.832301
$885$		0			0
$886$	−601.920			−0.679368
$887$	−38.0266			−0.0428710			−0.0214355	−	0.999770i	$-0.506824\pi$
$887$	−38.0266			−0.0428710			−0.0214355	+	0.999770i	$0.506824\pi$
$888$		0			0
$889$	−201.330			−0.226468
$890$			901.284i			1.01268i
$891$		0			0
$892$		−	189.539i		−	0.212488i
$893$			147.453i			0.165121i
$894$		0			0
$895$	−1427.65			−1.59514
$896$			48.5011i			0.0541307i
$897$		0			0
$898$		−	350.542i		−	0.390359i
$899$	235.530			0.261991
$900$		0			0
$901$		−	331.514i		−	0.367940i
$902$		−	221.187i		−	0.245218i
$903$		0			0
$904$	261.513	−	183.747i	0.289284	−	0.203260i
$905$			713.882i			0.788819i
$906$		0			0
$907$			1632.97i			1.80041i	0.435468	+	0.900204i	$0.356583\pi$
$907$			1632.97i			1.80041i	−0.435468	+	0.900204i	$0.643417\pi$
$908$		−	232.403i		−	0.255951i
$909$		0			0
$910$			629.070i			0.691286i
$911$		−	865.703i		−	0.950278i	−0.879911	−	0.475139i	$-0.842398\pi$
$911$		−	865.703i		−	0.950278i	0.879911	−	0.475139i	$-0.157602\pi$
$912$		0			0
$913$	71.0761			0.0778490
$914$	1136.44			1.24337
$915$		0			0
$916$			501.691i			0.547697i
$917$		−	796.553i		−	0.868651i
$918$		0			0
$919$	687.911			0.748543			0.374272	−	0.927319i	$-0.377893\pi$
$919$	687.911			0.748543			0.374272	+	0.927319i	$0.377893\pi$
$920$		−	450.948i		−	0.490160i
$921$		0			0
$922$	320.584			0.347705
$923$	−1330.25			−1.44122
$924$		0			0
$925$			15.0203i			0.0162381i
$926$			143.395i			0.154854i
$927$		0			0
$928$			38.2185i			0.0411837i
$929$		−	80.0055i		−	0.0861200i	−0.999072	−	0.0430600i	$-0.986289\pi$
$929$		−	80.0055i		−	0.0861200i	0.999072	−	0.0430600i	$-0.0137107\pi$
$930$		0			0
$931$			455.303i			0.489047i
$932$		−	403.842i		−	0.433307i
$933$		0			0
$934$	1017.48			1.08938
$935$			211.667i			0.226382i
$936$		0			0
$937$			633.675i			0.676281i	0.941096	+	0.338141i	$0.109798\pi$
$937$			633.675i			0.676281i	−0.941096	+	0.338141i	$0.890202\pi$
$938$	750.238			0.799827
$939$		0			0
$940$	94.8546			0.100909
$941$	−172.479			−0.183293			−0.0916466	−	0.995792i	$-0.529213\pi$
$941$	−172.479			−0.183293			−0.0916466	+	0.995792i	$0.529213\pi$
$942$		0			0
$943$			1997.45i			2.11819i
$944$	−16.0646			−0.0170175
$945$		0			0
$946$		−	85.0745i		−	0.0899308i
$947$	−628.992			−0.664194			−0.332097	−	0.943245i	$-0.607756\pi$
$947$	−628.992			−0.664194			−0.332097	+	0.943245i	$0.607756\pi$
$948$		0			0
$949$			769.136i			0.810470i
$950$	−44.7735			−0.0471300
$951$		0			0
$952$			205.587i			0.215953i
$953$		−	920.153i		−	0.965533i	−0.875749	−	0.482767i	$-0.839632\pi$
$953$		−	920.153i		−	0.965533i	0.875749	−	0.482767i	$-0.160368\pi$
$954$		0			0
$955$	−1682.16			−1.76142
$956$			500.562i			0.523600i
$957$		0			0
$958$		−	429.730i		−	0.448570i
$959$	−722.435			−0.753321
$960$		0			0
$961$	254.331			0.264652
$962$	−216.446			−0.224995
$963$		0			0
$964$	−290.850			−0.301712
$965$			790.630i			0.819306i
$966$		0			0
$967$	−898.472			−0.929133			−0.464567	−	0.885538i	$-0.653790\pi$
$967$	−898.472			−0.929133			−0.464567	+	0.885538i	$0.653790\pi$
$968$		−	322.966i		−	0.333643i
$969$		0			0
$970$			1113.88i			1.14833i
$971$	−260.558			−0.268340			−0.134170	−	0.990958i	$-0.542837\pi$
$971$	−260.558			−0.268340			−0.134170	+	0.990958i	$0.542837\pi$
$972$		0			0
$973$	−1.42466			−0.00146420
$974$	865.880			0.888994
$975$		0			0
$976$	170.866			0.175068
$977$	−884.936			−0.905768			−0.452884	−	0.891569i	$-0.649605\pi$
$977$	−884.936			−0.905768			−0.452884	+	0.891569i	$0.649605\pi$
$978$		0			0
$979$			347.870i			0.355332i
$980$	292.891			0.298868
$981$		0			0
$982$			459.994i			0.468426i
$983$	1471.31			1.49676			0.748379	−	0.663271i	$-0.230832\pi$
$983$	1471.31			1.49676			0.748379	+	0.663271i	$0.230832\pi$
$984$		0			0
$985$	907.476			0.921295
$986$			162.001i			0.164301i
$987$		0			0
$988$		−	645.196i		−	0.653032i
$989$			768.274i			0.776819i
$990$		0			0
$991$	473.236			0.477534			0.238767	−	0.971077i	$-0.423257\pi$
$991$	473.236			0.477534			0.238767	+	0.971077i	$0.423257\pi$
$992$			197.207i			0.198797i
$993$		0			0
$994$		−	371.704i		−	0.373947i
$995$		−	95.7057i		−	0.0961867i
$996$		0			0
$997$		−	203.087i		−	0.203698i	−0.994800	−	0.101849i	$-0.967524\pi$
$997$		−	203.087i		−	0.203698i	0.994800	−	0.101849i	$-0.0324759\pi$
$998$	529.951			0.531013
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2034.3.d.a.2033.20	yes	36
3.2	odd	2	inner	2034.3.d.a.2033.15	✓	36
113.112	even	2	inner	2034.3.d.a.2033.16	yes	36
339.338	odd	2	inner	2034.3.d.a.2033.19	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2034.3.d.a.2033.15	✓	36	3.2	odd	2	inner
2034.3.d.a.2033.16	yes	36	113.112	even	2	inner
2034.3.d.a.2033.19	yes	36	339.338	odd	2	inner
2034.3.d.a.2033.20	yes	36	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2034.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +4.78233 q^{5} -4.28693 q^{7} -2.82843i q^{8} +6.76323i q^{10} +2.61042i q^{11} -21.6969 q^{13} -6.06264i q^{14} +4.00000 q^{16} +16.9553 q^{17} -14.8684i q^{19} -9.56466 q^{20} -3.69169 q^{22} +33.3382 q^{23} -2.12933 q^{25} -30.6841i q^{26} +8.57387 q^{28} +6.75614 q^{29} +34.8616 q^{31} +5.65685i q^{32} +23.9784i q^{34} -20.5015 q^{35} -7.05401i q^{37} +21.0271 q^{38} -13.5265i q^{40} +59.9148i q^{41} +23.0449i q^{43} -5.22083i q^{44} +47.1473i q^{46} -9.91720 q^{47} -30.6222 q^{49} -3.01132i q^{50} +43.3938 q^{52} -19.5523i q^{53} +12.4839i q^{55} +12.1253i q^{56} +9.55462i q^{58} -4.01614 q^{59} +42.7166 q^{61} +49.3017i q^{62} -8.00000 q^{64} -103.762 q^{65} +123.748i q^{67} -33.9105 q^{68} -28.9935i q^{70} +61.3106 q^{71} -35.4491i q^{73} +9.97587 q^{74} +29.7368i q^{76} -11.1907i q^{77} +92.5692i q^{79} +19.1293 q^{80} -84.7324 q^{82} -27.2279i q^{83} +81.0857 q^{85} -32.5904 q^{86} +7.38338 q^{88} +133.262 q^{89} +93.0132 q^{91} -66.6763 q^{92} -14.0250i q^{94} -71.1055i q^{95} +164.696 q^{97} -43.3063i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} - 8 q^{7} + 144 q^{16} + 36 q^{25} + 16 q^{28} + 64 q^{31} + 108 q^{49} - 80 q^{61} - 288 q^{64} + 248 q^{82} - 164 q^{85} + 516 q^{91} - 420 q^{97}+O(q^{100}) \) 36 * q - 72 * q^4 - 8 * q^7 + 144 * q^16 + 36 * q^25 + 16 * q^28 + 64 * q^31 + 108 * q^49 - 80 * q^61 - 288 * q^64 + 248 * q^82 - 164 * q^85 + 516 * q^91 - 420 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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