LMFDB - Embedded newform 2034.3.d.a.2033.15

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			2033.15
Character	$\chi$	$=$	2034.2033
Dual form			2034.3.d.a.2033.16

$q$-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	$-0.658723\pi$
$5$	−4.78233			−0.956466			−0.478233	+	0.878233i	$0.658723\pi$
$6$		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	$-0.962142\pi$
$11$		−	2.61042i		−	0.237311i	0.992936	−	0.118655i	$-0.0378584\pi$
$12$		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	$-0.666183\pi$
$17$	−16.9553			−0.997369			−0.498684	+	0.866784i	$0.666183\pi$
$18$		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	$-0.758041\pi$
$23$	−33.3382			−1.44949			−0.724743	+	0.689020i	$0.758041\pi$
$24$		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	$-0.537163\pi$
$29$	−6.75614			−0.232970			−0.116485	+	0.993192i	$0.537163\pi$
$30$		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	$-0.739208\pi$
$41$		−	59.9148i		−	1.46134i	0.682732	−	0.730669i	$-0.260792\pi$
$42$		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	$-0.466355\pi$
$47$	9.91720			0.211004			0.105502	+	0.994419i	$0.466355\pi$
$48$		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	$0.0590522\pi$
$53$			19.5523i			0.368911i	−0.982841	+	0.184456i	$0.940948\pi$
$54$		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	$-0.489164\pi$
$59$	4.01614			0.0680701			0.0340351	+	0.999421i	$0.489164\pi$
$60$		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	$-0.642109\pi$
$71$	−61.3106			−0.863529			−0.431765	+	0.901986i	$0.642109\pi$
$72$		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	$0.0524473\pi$
$83$			27.2279i			0.328047i	−0.986456	+	0.164023i	$0.947553\pi$
$84$		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	$-0.769304\pi$
$89$	−133.262			−1.49733			−0.748664	+	0.662950i	$0.769304\pi$
$90$		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	$-0.369113\pi$
$101$	80.7404			0.799410			0.399705	+	0.916644i	$0.369113\pi$
$102$		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	$-0.443216\pi$
$107$	37.9739			0.354897			0.177448	+	0.984130i	$0.443216\pi$
$108$		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	$-0.749058\pi$
$131$		−	185.810i		−	1.41839i	0.705010	−	0.709197i	$-0.250942\pi$
$132$		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	$-0.710859\pi$
$137$	−168.520			−1.23007			−0.615037	+	0.788498i	$0.710859\pi$
$138$		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	$-0.759322\pi$
$149$		−	204.457i		−	1.37220i	0.727509	−	0.686098i	$-0.240678\pi$
$150$		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	$-0.542640\pi$
$167$	−44.6081			−0.267115			−0.133557	+	0.991041i	$0.542640\pi$
$168$		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	$0.0216570\pi$
$173$			23.5228i			0.135970i	−0.997686	+	0.0679850i	$0.978343\pi$
$174$		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	$-0.186119\pi$
$179$	298.526			1.66774			0.833871	+	0.551959i	$0.186119\pi$
$180$		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	$-0.127539\pi$
$191$	351.745			1.84159			0.920797	+	0.390042i	$0.127539\pi$
$192$		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	$-0.659949\pi$
$197$	−189.756			−0.963229			−0.481614	+	0.876383i	$0.659949\pi$
$198$		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	$-0.917611\pi$
$227$		−	116.202i		−	0.511902i	0.966690	−	0.255951i	$-0.0823885\pi$
$228$		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	$-0.857347\pi$
$233$		−	201.921i		−	0.866613i	0.901247	−	0.433307i	$-0.142653\pi$
$234$		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	$0.175411\pi$
$239$			250.281i			1.04720i	−0.851964	+	0.523600i	$0.824589\pi$
$240$		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	$-0.902433\pi$
$251$		−	151.472i		−	0.603475i	0.953391	−	0.301737i	$-0.0975666\pi$
$252$		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	$0.0184901\pi$
$257$			29.8406i			0.116111i	−0.998313	+	0.0580557i	$0.981510\pi$
$258$		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	$-0.473298\pi$
$263$	44.0729			0.167578			0.0837888	+	0.996484i	$0.473298\pi$
$264$		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	$-0.775832\pi$
$269$	−410.011			−1.52420			−0.762101	+	0.647458i	$0.775832\pi$
$270$		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	$-0.599754\pi$
$281$	−173.254			−0.616562			−0.308281	+	0.951295i	$0.599754\pi$
$282$		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	$-0.738520\pi$
$293$	−399.154			−1.36230			−0.681151	+	0.732143i	$0.738520\pi$
$294$		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	$-0.916196\pi$
$311$		−	161.873i		−	0.520494i	0.965542	−	0.260247i	$-0.0838039\pi$
$312$		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	$0.0979399\pi$
$317$			192.010i			0.605711i	−0.953037	+	0.302855i	$0.902060\pi$
$318$		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	$0.00424444\pi$
$347$			9.25373i			0.0266678i	−0.999911	+	0.0133339i	$0.995756\pi$
$348$		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	$0.286151\pi$
$353$			552.577i			1.56537i	−0.622416	+	0.782687i	$0.713849\pi$
$354$		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	$-0.638660\pi$
$359$	−302.972			−0.843932			−0.421966	+	0.906612i	$0.638660\pi$
$360$		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	$0.184049\pi$
$383$			418.637i			1.09305i	−0.837444	+	0.546524i	$0.815951\pi$
$384$		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	$-0.959676\pi$
$389$		−	98.2947i		−	0.252686i	0.991987	−	0.126343i	$-0.0403239\pi$
$390$		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	$-0.695768\pi$
$401$		−	655.042i		−	1.63352i	0.576977	−	0.816760i	$-0.304232\pi$
$402$		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	$-0.355074\pi$
$419$	368.494			0.879461			0.439731	+	0.898130i	$0.355074\pi$
$420$		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	$-0.592964\pi$
$431$	−248.188			−0.575841			−0.287921	+	0.957654i	$0.592964\pi$
$432$		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	$-0.840497\pi$
$443$		−	425.622i		−	0.960771i	0.877057	−	0.480386i	$-0.159503\pi$
$444$		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	$-0.410983\pi$
$449$	247.871			0.552051			0.276025	+	0.961150i	$0.410983\pi$
$450$		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	$0.0790718\pi$
$461$			226.687i			0.491729i	−0.969304	+	0.245865i	$0.920928\pi$
$462$		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	$0.279899\pi$
$467$			719.470i			1.54062i	−0.637669	+	0.770310i	$0.720101\pi$
$468$		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	$-0.397262\pi$
$479$	303.865			0.634374			0.317187	+	0.948363i	$0.397262\pi$
$480$		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	$-0.607463\pi$
$491$	−325.265			−0.662454			−0.331227	+	0.943551i	$0.607463\pi$
$492$		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	$0.0586197\pi$
$503$			184.219i			0.366240i	−0.983091	+	0.183120i	$0.941380\pi$
$504$		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	$-0.921373\pi$
$509$		−	248.911i		−	0.489019i	0.969647	−	0.244510i	$-0.0786270\pi$
$510$		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	$0.196567\pi$
$521$			603.346i			1.15805i	−0.815308	+	0.579027i	$0.803433\pi$
$522$		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	$0.0978421\pi$
$557$			337.054i			0.605125i	−0.953130	+	0.302562i	$0.902158\pi$
$558$		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	$-0.933364\pi$
$563$		−	234.004i		−	0.415638i	0.978167	−	0.207819i	$-0.0666365\pi$
$564$		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	$0.193960\pi$
$569$			651.311i			1.14466i	−0.820024	+	0.572329i	$0.806040\pi$
$570$		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	$-0.977974\pi$
$587$		−	81.1730i		−	0.138285i	0.997607	−	0.0691423i	$-0.0220262\pi$
$588$		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	$-0.979234\pi$
$593$		−	77.3160i		−	0.130381i	0.997873	−	0.0651906i	$-0.0207655\pi$
$594$		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	$-0.443995\pi$
$599$	209.698			0.350080			0.175040	+	0.984561i	$0.443995\pi$
$600$		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	$0.0846664\pi$
$617$			324.372i			0.525724i	−0.964833	+	0.262862i	$0.915334\pi$
$618$		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	$-0.898506\pi$
$641$	−1217.38			−1.89919			−0.949596	+	0.313478i	$0.898506\pi$
$642$		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	$0.0919372\pi$
$647$			368.570i			0.569661i	−0.958578	+	0.284830i	$0.908063\pi$
$648$		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	$-0.897409\pi$
$653$		−	413.673i		−	0.633496i	0.948510	−	0.316748i	$-0.102591\pi$
$654$		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	$-0.557235\pi$
$659$	−235.713			−0.357683			−0.178842	+	0.983878i	$0.557235\pi$
$660$		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	$-0.830241\pi$
$677$		−	688.358i		−	1.01678i	0.861128	−	0.508389i	$-0.169759\pi$
$678$		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	$-0.711248\pi$
$683$	−841.455			−1.23200			−0.616000	+	0.787746i	$0.711248\pi$
$684$		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	$-0.600566\pi$
$701$	−435.611			−0.621413			−0.310707	+	0.950506i	$0.600566\pi$
$702$		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	$0.292735\pi$
$719$			1143.77i			1.59079i	−0.606095	+	0.795393i	$0.707265\pi$
$720$		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	$-0.245021\pi$
$743$	1067.07			1.43616			0.718081	+	0.695960i	$0.245021\pi$
$744$		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	$0.190623\pi$
$761$			857.953i			1.12740i	−0.825979	+	0.563701i	$0.809377\pi$
$762$		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	$0.214249\pi$
$773$			963.778i			1.24680i	−0.781902	+	0.623401i	$0.785751\pi$
$774$		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	$-0.479443\pi$
$797$	102.871			0.129073			0.0645367	+	0.997915i	$0.479443\pi$
$798$		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	$-0.995409\pi$
$809$		−	23.3365i		−	0.0288461i	0.999896	−	0.0144230i	$-0.00459116\pi$
$810$		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	$0.296139\pi$
$821$			1316.60i			1.60366i	−0.597554	+	0.801829i	$0.703861\pi$
$822$		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	$-0.907216\pi$
$827$		−	475.326i		−	0.574759i	0.957817	−	0.287379i	$-0.0927840\pi$
$828$		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	$-0.627235\pi$
$839$	−653.011			−0.778320			−0.389160	+	0.921170i	$0.627235\pi$
$840$		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	$-0.310438\pi$
$857$	961.461			1.12189			0.560946	+	0.827852i	$0.310438\pi$
$858$		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	$0.130555\pi$
$863$			688.236i			0.797492i	−0.917061	+	0.398746i	$0.869445\pi$
$864$		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	$-0.268661\pi$
$881$	1170.78			1.32892			0.664461	+	0.747323i	$0.268661\pi$
$882$		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	$-0.493176\pi$
$887$	38.0266			0.0428710			0.0214355	+	0.999770i	$0.493176\pi$
$888$		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	$0.157602\pi$
$911$			865.703i			0.950278i	−0.879911	+	0.475139i	$0.842398\pi$
$912$		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	$0.0137107\pi$
$929$			80.0055i			0.0861200i	−0.999072	+	0.0430600i	$0.986289\pi$
$930$		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	$-0.470787\pi$
$941$	172.479			0.183293			0.0916466	+	0.995792i	$0.470787\pi$
$942$		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	$-0.392244\pi$
$947$	628.992			0.664194			0.332097	+	0.943245i	$0.392244\pi$
$948$		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	$0.160368\pi$
$953$			920.153i			0.965533i	−0.875749	+	0.482767i	$0.839632\pi$
$954$		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	$-0.457163\pi$
$971$	260.558			0.268340			0.134170	+	0.990958i	$0.457163\pi$
$972$		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	$-0.350395\pi$
$977$	884.936			0.905768			0.452884	+	0.891569i	$0.350395\pi$
$978$		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	$-0.769168\pi$
$983$	−1471.31			−1.49676			−0.748379	+	0.663271i	$0.769168\pi$
$984$		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.2033.15	✓	36
3.2	odd	2	inner	2034.3.d.a.2033.20	yes	36
113.112	even	2	inner	2034.3.d.a.2033.19	yes	36
339.338	odd	2	inner	2034.3.d.a.2033.16	yes	36

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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