LMFDB - Embedded newform 1458.3.b.c.1457.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,3,Mod(1457,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1458.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1458.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$39.7276225437$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.14
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +4.50363i q^{5} +1.28890 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} +4.50363i q^{5} +1.28890 q^{7} +2.82843i q^{8} +6.36909 q^{10} +6.82513i q^{11} +11.0767 q^{13} -1.82278i q^{14} +4.00000 q^{16} -27.9243i q^{17} -29.0475 q^{19} -9.00726i q^{20} +9.65220 q^{22} -29.6636i q^{23} +4.71732 q^{25} -15.6648i q^{26} -2.57779 q^{28} +15.3611i q^{29} +46.5912 q^{31} -5.65685i q^{32} -39.4909 q^{34} +5.80472i q^{35} +13.2409 q^{37} +41.0794i q^{38} -12.7382 q^{40} +42.3235i q^{41} +51.1274 q^{43} -13.6503i q^{44} -41.9506 q^{46} +58.2328i q^{47} -47.3387 q^{49} -6.67130i q^{50} -22.1533 q^{52} +72.9927i q^{53} -30.7379 q^{55} +3.64555i q^{56} +21.7238 q^{58} +30.4810i q^{59} +15.1451 q^{61} -65.8899i q^{62} -8.00000 q^{64} +49.8852i q^{65} +112.646 q^{67} +55.8486i q^{68} +8.20911 q^{70} -8.69771i q^{71} +104.539 q^{73} -18.7254i q^{74} +58.0950 q^{76} +8.79689i q^{77} -60.2933 q^{79} +18.0145i q^{80} +59.8545 q^{82} -27.4791i q^{83} +125.761 q^{85} -72.3051i q^{86} -19.3044 q^{88} +123.113i q^{89} +14.2767 q^{91} +59.3272i q^{92} +82.3536 q^{94} -130.819i q^{95} -43.1095 q^{97} +66.9471i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4}+O(q^{10}) $ 36 * q - 72 * q^4	$ 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) $ 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times$.

$n$	$731$
$\chi(n)$	$-1$

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
$5$	4.50363i	0.900726i	0.892846	+	0.450363i	$0.148705\pi$
$5$	4.50363i	0.900726i	−0.892846	+	0.450363i	$0.851295\pi$
$6$	0	0
$7$	1.28890	0.184128	0.0920641	−	0.995753i	$-0.470654\pi$
$7$	1.28890	0.184128	0.0920641	+	0.995753i	$0.470654\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	6.36909	0.636909
$11$	6.82513i	0.620467i	0.950660	+	0.310233i	$0.100407\pi$
$11$	6.82513i	0.620467i	−0.950660	+	0.310233i	$0.899593\pi$
$12$	0	0
$13$	11.0767	0.852051	0.426025	−	0.904711i	$-0.359913\pi$
$13$	11.0767	0.852051	0.426025	+	0.904711i	$0.359913\pi$
$14$	− 1.82278i	− 0.130198i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 27.9243i	− 1.64261i	−0.570492	−	0.821303i	$-0.693248\pi$
$17$	− 27.9243i	− 1.64261i	0.570492	−	0.821303i	$-0.306752\pi$
$18$	0	0
$19$	−29.0475	−1.52882	−0.764408	−	0.644733i	$-0.776969\pi$
$19$	−29.0475	−1.52882	−0.764408	+	0.644733i	$0.776969\pi$
$20$	− 9.00726i	− 0.450363i
$21$	0	0
$22$	9.65220	0.438736
$23$	− 29.6636i	− 1.28972i	−0.764300	−	0.644861i	$-0.776915\pi$
$23$	− 29.6636i	− 1.28972i	0.764300	−	0.644861i	$-0.223085\pi$
$24$	0	0
$25$	4.71732	0.188693
$26$	− 15.6648i	− 0.602491i
$27$	0	0
$28$	−2.57779	−0.0920641
$29$	15.3611i	0.529692i	0.964291	+	0.264846i	$0.0853211\pi$
$29$	15.3611i	0.529692i	−0.964291	+	0.264846i	$0.914679\pi$
$30$	0	0
$31$	46.5912	1.50294	0.751470	−	0.659767i	$-0.229345\pi$
$31$	46.5912	1.50294	0.751470	+	0.659767i	$0.229345\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−39.4909	−1.16150
$35$	5.80472i	0.165849i
$36$	0	0
$37$	13.2409	0.357861	0.178931	−	0.983862i	$-0.442736\pi$
$37$	13.2409	0.357861	0.178931	+	0.983862i	$0.442736\pi$
$38$	41.0794i	1.08104i
$39$	0	0
$40$	−12.7382	−0.318455
$41$	42.3235i	1.03228i	0.856504	+	0.516141i	$0.172632\pi$
$41$	42.3235i	1.03228i	−0.856504	+	0.516141i	$0.827368\pi$
$42$	0	0
$43$	51.1274	1.18901	0.594505	−	0.804092i	$-0.297348\pi$
$43$	51.1274	1.18901	0.594505	+	0.804092i	$0.297348\pi$
$44$	− 13.6503i	− 0.310233i
$45$	0	0
$46$	−41.9506	−0.911971
$47$	58.2328i	1.23899i	0.784999	+	0.619497i	$0.212664\pi$
$47$	58.2328i	1.23899i	−0.784999	+	0.619497i	$0.787336\pi$
$48$	0	0
$49$	−47.3387	−0.966097
$50$	− 6.67130i	− 0.133426i
$51$	0	0
$52$	−22.1533	−0.426025
$53$	72.9927i	1.37722i	0.725131	+	0.688611i	$0.241779\pi$
$53$	72.9927i	1.37722i	−0.725131	+	0.688611i	$0.758221\pi$
$54$	0	0
$55$	−30.7379	−0.558870
$56$	3.64555i	0.0650991i
$57$	0	0
$58$	21.7238	0.374549
$59$	30.4810i	0.516627i	0.966061	+	0.258313i	$0.0831667\pi$
$59$	30.4810i	0.516627i	−0.966061	+	0.258313i	$0.916833\pi$
$60$	0	0
$61$	15.1451	0.248280	0.124140	−	0.992265i	$-0.460383\pi$
$61$	15.1451	0.248280	0.124140	+	0.992265i	$0.460383\pi$
$62$	− 65.8899i	− 1.06274i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	49.8852i	0.767464i
$66$	0	0
$67$	112.646	1.68128	0.840639	−	0.541595i	$-0.182179\pi$
$67$	112.646	1.68128	0.840639	+	0.541595i	$0.182179\pi$
$68$	55.8486i	0.821303i
$69$	0	0
$70$	8.20911	0.117273
$71$	− 8.69771i	− 0.122503i	−0.998122	−	0.0612515i	$-0.980491\pi$
$71$	− 8.69771i	− 0.122503i	0.998122	−	0.0612515i	$-0.0195092\pi$
$72$	0	0
$73$	104.539	1.43204	0.716020	−	0.698079i	$-0.245962\pi$
$73$	104.539	1.43204	0.716020	+	0.698079i	$0.245962\pi$
$74$	− 18.7254i	− 0.253046i
$75$	0	0
$76$	58.0950	0.764408
$77$	8.79689i	0.114245i
$78$	0	0
$79$	−60.2933	−0.763207	−0.381603	−	0.924326i	$-0.624628\pi$
$79$	−60.2933	−0.763207	−0.381603	+	0.924326i	$0.624628\pi$
$80$	18.0145i	0.225181i
$81$	0	0
$82$	59.8545	0.729933
$83$	− 27.4791i	− 0.331074i	−0.986204	−	0.165537i	$-0.947064\pi$
$83$	− 27.4791i	− 0.331074i	0.986204	−	0.165537i	$-0.0529357\pi$
$84$	0	0
$85$	125.761	1.47954
$86$	− 72.3051i	− 0.840757i
$87$	0	0
$88$	−19.3044	−0.219368
$89$	123.113i	1.38329i	0.722238	+	0.691645i	$0.243114\pi$
$89$	123.113i	1.38329i	−0.722238	+	0.691645i	$0.756886\pi$
$90$	0	0
$91$	14.2767	0.156887
$92$	59.3272i	0.644861i
$93$	0	0
$94$	82.3536	0.876102
$95$	− 130.819i	− 1.37704i
$96$	0	0
$97$	−43.1095	−0.444428	−0.222214	−	0.974998i	$-0.571328\pi$
$97$	−43.1095	−0.444428	−0.222214	+	0.974998i	$0.571328\pi$
$98$	66.9471i	0.683134i
$99$	0	0
$100$	−9.43464	−0.0943464
$101$	− 35.7657i	− 0.354116i	−0.984200	−	0.177058i	$-0.943342\pi$
$101$	− 35.7657i	− 0.354116i	0.984200	−	0.177058i	$-0.0566580\pi$
$102$	0	0
$103$	159.298	1.54658	0.773292	−	0.634050i	$-0.218609\pi$
$103$	159.298	1.54658	0.773292	+	0.634050i	$0.218609\pi$
$104$	31.3295i	0.301245i
$105$	0	0
$106$	103.227	0.973843
$107$	0.912530i	0.00852832i	0.999991	+	0.00426416i	$0.00135733\pi$
$107$	0.912530i	0.00852832i	−0.999991	+	0.00426416i	$0.998643\pi$
$108$	0	0
$109$	−58.9907	−0.541199	−0.270600	−	0.962692i	$-0.587222\pi$
$109$	−58.9907	−0.541199	−0.270600	+	0.962692i	$0.587222\pi$
$110$	43.4699i	0.395181i
$111$	0	0
$112$	5.15559	0.0460320
$113$	25.5941i	0.226497i	0.993567	+	0.113248i	$0.0361256\pi$
$113$	25.5941i	0.226497i	−0.993567	+	0.113248i	$0.963874\pi$
$114$	0	0
$115$	133.594	1.16169
$116$	− 30.7221i	− 0.264846i
$117$	0	0
$118$	43.1066	0.365310
$119$	− 35.9915i	− 0.302450i
$120$	0	0
$121$	74.4176	0.615021
$122$	− 21.4184i	− 0.175560i
$123$	0	0
$124$	−93.1823	−0.751470
$125$	133.836i	1.07069i
$126$	0	0
$127$	−46.1762	−0.363592	−0.181796	−	0.983336i	$-0.558191\pi$
$127$	−46.1762	−0.363592	−0.181796	+	0.983336i	$0.558191\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	70.5483	0.542679
$131$	28.5177i	0.217693i	0.994059	+	0.108846i	$0.0347157\pi$
$131$	28.5177i	0.217693i	−0.994059	+	0.108846i	$0.965284\pi$
$132$	0	0
$133$	−37.4392	−0.281498
$134$	− 159.305i	− 1.18884i
$135$	0	0
$136$	78.9818	0.580749
$137$	153.868i	1.12312i	0.827435	+	0.561561i	$0.189799\pi$
$137$	153.868i	1.12312i	−0.827435	+	0.561561i	$0.810201\pi$
$138$	0	0
$139$	−25.9653	−0.186801	−0.0934003	−	0.995629i	$-0.529774\pi$
$139$	−25.9653	−0.186801	−0.0934003	+	0.995629i	$0.529774\pi$
$140$	− 11.6094i	− 0.0829245i
$141$	0	0
$142$	−12.3004	−0.0866227
$143$	75.5997i	0.528669i
$144$	0	0
$145$	−69.1805	−0.477107
$146$	− 147.840i	− 1.01261i
$147$	0	0
$148$	−26.4817	−0.178931
$149$	− 53.4272i	− 0.358572i	−0.983797	−	0.179286i	$-0.942621\pi$
$149$	− 53.4272i	− 0.358572i	0.983797	−	0.179286i	$-0.0573787\pi$
$150$	0	0
$151$	−86.7106	−0.574243	−0.287121	−	0.957894i	$-0.592698\pi$
$151$	−86.7106	−0.574243	−0.287121	+	0.957894i	$0.592698\pi$
$152$	− 82.1587i	− 0.540518i
$153$	0	0
$154$	12.4407	0.0807837
$155$	209.829i	1.35374i
$156$	0	0
$157$	210.442	1.34039	0.670197	−	0.742184i	$-0.266210\pi$
$157$	210.442	1.34039	0.670197	+	0.742184i	$0.266210\pi$
$158$	85.2676i	0.539669i
$159$	0	0
$160$	25.4764	0.159227
$161$	− 38.2333i	− 0.237474i
$162$	0	0
$163$	95.6406	0.586752	0.293376	−	0.955997i	$-0.405221\pi$
$163$	95.6406	0.586752	0.293376	+	0.955997i	$0.405221\pi$
$164$	− 84.6471i	− 0.516141i
$165$	0	0
$166$	−38.8614	−0.234105
$167$	97.1890i	0.581970i	0.956728	+	0.290985i	$0.0939829\pi$
$167$	97.1890i	0.581970i	−0.956728	+	0.290985i	$0.906017\pi$
$168$	0	0
$169$	−46.3076	−0.274010
$170$	− 177.852i	− 1.04619i
$171$	0	0
$172$	−102.255	−0.594505
$173$	− 277.658i	− 1.60496i	−0.596680	−	0.802479i	$-0.703514\pi$
$173$	− 277.658i	− 1.60496i	0.596680	−	0.802479i	$-0.296486\pi$
$174$	0	0
$175$	6.08014	0.0347437
$176$	27.3005i	0.155117i
$177$	0	0
$178$	174.108	0.978133
$179$	76.7361i	0.428693i	0.976758	+	0.214347i	$0.0687622\pi$
$179$	76.7361i	0.428693i	−0.976758	+	0.214347i	$0.931238\pi$
$180$	0	0
$181$	285.099	1.57513	0.787567	−	0.616230i	$-0.211341\pi$
$181$	285.099	1.57513	0.787567	+	0.616230i	$0.211341\pi$
$182$	− 20.1903i	− 0.110936i
$183$	0	0
$184$	83.9013	0.455985
$185$	59.6320i	0.322335i
$186$	0	0
$187$	190.587	1.01918
$188$	− 116.466i	− 0.619497i
$189$	0	0
$190$	−185.006	−0.973717
$191$	− 63.5353i	− 0.332646i	−0.986071	−	0.166323i	$-0.946811\pi$
$191$	− 63.5353i	− 0.332646i	0.986071	−	0.166323i	$-0.0531894\pi$
$192$	0	0
$193$	270.290	1.40047	0.700233	−	0.713915i	$-0.253080\pi$
$193$	270.290	1.40047	0.700233	+	0.713915i	$0.253080\pi$
$194$	60.9661i	0.314258i
$195$	0	0
$196$	94.6775	0.483048
$197$	− 53.9116i	− 0.273663i	−0.990594	−	0.136831i	$-0.956308\pi$
$197$	− 53.9116i	− 0.273663i	0.990594	−	0.136831i	$-0.0436918\pi$
$198$	0	0
$199$	−134.982	−0.678300	−0.339150	−	0.940732i	$-0.610139\pi$
$199$	−134.982	−0.678300	−0.339150	+	0.940732i	$0.610139\pi$
$200$	13.3426i	0.0667130i
$201$	0	0
$202$	−50.5804	−0.250398
$203$	19.7988i	0.0975311i
$204$	0	0
$205$	−190.610	−0.929803
$206$	− 225.282i	− 1.09360i
$207$	0	0
$208$	44.3066	0.213013
$209$	− 198.253i	− 0.948579i
$210$	0	0
$211$	200.143	0.948546	0.474273	−	0.880378i	$-0.342711\pi$
$211$	200.143	0.948546	0.474273	+	0.880378i	$0.342711\pi$
$212$	− 145.985i	− 0.688611i
$213$	0	0
$214$	1.29051	0.00603043
$215$	230.259i	1.07097i
$216$	0	0
$217$	60.0512	0.276734
$218$	83.4254i	0.382686i
$219$	0	0
$220$	61.4757	0.279435
$221$	− 309.308i	− 1.39958i
$222$	0	0
$223$	−365.499	−1.63901	−0.819505	−	0.573071i	$-0.805752\pi$
$223$	−365.499	−1.63901	−0.819505	+	0.573071i	$0.805752\pi$
$224$	− 7.29110i	− 0.0325496i
$225$	0	0
$226$	36.1956	0.160157
$227$	− 167.358i	− 0.737262i	−0.929576	−	0.368631i	$-0.879827\pi$
$227$	− 167.358i	− 0.737262i	0.929576	−	0.368631i	$-0.120173\pi$
$228$	0	0
$229$	162.334	0.708883	0.354441	−	0.935078i	$-0.384671\pi$
$229$	162.334	0.708883	0.354441	+	0.935078i	$0.384671\pi$
$230$	− 188.930i	− 0.821436i
$231$	0	0
$232$	−43.4476	−0.187274
$233$	183.821i	0.788930i	0.918911	+	0.394465i	$0.129070\pi$
$233$	183.821i	0.788930i	−0.918911	+	0.394465i	$0.870930\pi$
$234$	0	0
$235$	−262.259	−1.11599
$236$	− 60.9619i	− 0.258313i
$237$	0	0
$238$	−50.8997	−0.213864
$239$	− 43.2100i	− 0.180795i	−0.995906	−	0.0903975i	$-0.971186\pi$
$239$	− 43.2100i	− 0.180795i	0.995906	−	0.0903975i	$-0.0288138\pi$
$240$	0	0
$241$	−65.1331	−0.270262	−0.135131	−	0.990828i	$-0.543145\pi$
$241$	−65.1331	−0.270262	−0.135131	+	0.990828i	$0.543145\pi$
$242$	− 105.242i	− 0.434886i
$243$	0	0
$244$	−30.2901	−0.124140
$245$	− 213.196i	− 0.870188i
$246$	0	0
$247$	−321.749	−1.30263
$248$	131.780i	0.531370i
$249$	0	0
$250$	189.272	0.757090
$251$	− 16.5637i	− 0.0659907i	−0.999456	−	0.0329953i	$-0.989495\pi$
$251$	− 16.5637i	− 0.0659907i	0.999456	−	0.0329953i	$-0.0105047\pi$
$252$	0	0
$253$	202.458	0.800229
$254$	65.3030i	0.257099i
$255$	0	0
$256$	16.0000	0.0625000
$257$	405.873i	1.57927i	0.613574	+	0.789637i	$0.289731\pi$
$257$	405.873i	1.57927i	−0.613574	+	0.789637i	$0.710269\pi$
$258$	0	0
$259$	17.0661	0.0658924
$260$	− 99.7703i	− 0.383732i
$261$	0	0
$262$	40.3302	0.153932
$263$	− 308.967i	− 1.17478i	−0.809304	−	0.587389i	$-0.800156\pi$
$263$	− 308.967i	− 1.17478i	0.809304	−	0.587389i	$-0.199844\pi$
$264$	0	0
$265$	−328.732	−1.24050
$266$	52.9471i	0.199049i
$267$	0	0
$268$	−225.291	−0.840639
$269$	− 243.504i	− 0.905221i	−0.891708	−	0.452611i	$-0.850493\pi$
$269$	− 243.504i	− 0.905221i	0.891708	−	0.452611i	$-0.149507\pi$
$270$	0	0
$271$	−138.215	−0.510018	−0.255009	−	0.966939i	$-0.582078\pi$
$271$	−138.215	−0.510018	−0.255009	+	0.966939i	$0.582078\pi$
$272$	− 111.697i	− 0.410651i
$273$	0	0
$274$	217.602	0.794167
$275$	32.1963i	0.117078i
$276$	0	0
$277$	417.396	1.50685	0.753423	−	0.657536i	$-0.228401\pi$
$277$	417.396	1.50685	0.753423	+	0.657536i	$0.228401\pi$
$278$	36.7205i	0.132088i
$279$	0	0
$280$	−16.4182	−0.0586365
$281$	390.348i	1.38914i	0.719426	+	0.694570i	$0.244405\pi$
$281$	390.348i	1.38914i	−0.719426	+	0.694570i	$0.755595\pi$
$282$	0	0
$283$	−13.1176	−0.0463518	−0.0231759	−	0.999731i	$-0.507378\pi$
$283$	−13.1176	−0.0463518	−0.0231759	+	0.999731i	$0.507378\pi$
$284$	17.3954i	0.0612515i
$285$	0	0
$286$	106.914	0.373825
$287$	54.5507i	0.190072i
$288$	0	0
$289$	−490.766	−1.69815
$290$	97.8360i	0.337366i
$291$	0	0
$292$	−209.078	−0.716020
$293$	164.654i	0.561959i	0.959714	+	0.280979i	$0.0906593\pi$
$293$	164.654i	0.561959i	−0.959714	+	0.280979i	$0.909341\pi$
$294$	0	0
$295$	−137.275	−0.465339
$296$	37.4508i	0.126523i
$297$	0	0
$298$	−75.5574	−0.253548
$299$	− 328.573i	− 1.09891i
$300$	0	0
$301$	65.8980	0.218930
$302$	122.627i	0.406051i
$303$	0	0
$304$	−116.190	−0.382204
$305$	68.2078i	0.223632i
$306$	0	0
$307$	−194.020	−0.631988	−0.315994	−	0.948761i	$-0.602338\pi$
$307$	−194.020	−0.631988	−0.315994	+	0.948761i	$0.602338\pi$
$308$	− 17.5938i	− 0.0571227i
$309$	0	0
$310$	296.744	0.957237
$311$	− 368.625i	− 1.18529i	−0.805465	−	0.592644i	$-0.798084\pi$
$311$	− 368.625i	− 1.18529i	0.805465	−	0.592644i	$-0.201916\pi$
$312$	0	0
$313$	435.207	1.39044	0.695219	−	0.718798i	$-0.255307\pi$
$313$	435.207	1.39044	0.695219	+	0.718798i	$0.255307\pi$
$314$	− 297.610i	− 0.947801i
$315$	0	0
$316$	120.587	0.381603
$317$	− 244.013i	− 0.769758i	−0.922967	−	0.384879i	$-0.874243\pi$
$317$	− 244.013i	− 0.769758i	0.922967	−	0.384879i	$-0.125757\pi$
$318$	0	0
$319$	−104.841	−0.328656
$320$	− 36.0290i	− 0.112591i
$321$	0	0
$322$	−54.0701	−0.167919
$323$	811.131i	2.51124i
$324$	0	0
$325$	52.2522	0.160776
$326$	− 135.256i	− 0.414896i
$327$	0	0
$328$	−119.709	−0.364967
$329$	75.0560i	0.228134i
$330$	0	0
$331$	−424.782	−1.28333	−0.641665	−	0.766985i	$-0.721756\pi$
$331$	−424.782	−1.28333	−0.641665	+	0.766985i	$0.721756\pi$
$332$	54.9583i	0.165537i
$333$	0	0
$334$	137.446	0.411515
$335$	507.314i	1.51437i
$336$	0	0
$337$	−150.158	−0.445574	−0.222787	−	0.974867i	$-0.571515\pi$
$337$	−150.158	−0.445574	−0.222787	+	0.974867i	$0.571515\pi$
$338$	65.4889i	0.193754i
$339$	0	0
$340$	−251.521	−0.739769
$341$	317.991i	0.932525i
$342$	0	0
$343$	−124.171	−0.362014
$344$	144.610i	0.420378i
$345$	0	0
$346$	−392.667	−1.13488
$347$	− 233.963i	− 0.674246i	−0.941461	−	0.337123i	$-0.890546\pi$
$347$	− 233.963i	− 0.674246i	0.941461	−	0.337123i	$-0.109454\pi$
$348$	0	0
$349$	−15.2426	−0.0436752	−0.0218376	−	0.999762i	$-0.506952\pi$
$349$	−15.2426	−0.0436752	−0.0218376	+	0.999762i	$0.506952\pi$
$350$	− 8.59862i	− 0.0245675i
$351$	0	0
$352$	38.6088	0.109684
$353$	236.725i	0.670608i	0.942110	+	0.335304i	$0.108839\pi$
$353$	236.725i	0.670608i	−0.942110	+	0.335304i	$0.891161\pi$
$354$	0	0
$355$	39.1713	0.110342
$356$	− 246.225i	− 0.691645i
$357$	0	0
$358$	108.521	0.303132
$359$	561.787i	1.56487i	0.622734	+	0.782433i	$0.286022\pi$
$359$	561.787i	1.56487i	−0.622734	+	0.782433i	$0.713978\pi$
$360$	0	0
$361$	482.757	1.33728
$362$	− 403.191i	− 1.11379i
$363$	0	0
$364$	−28.5534	−0.0784433
$365$	470.805i	1.28988i
$366$	0	0
$367$	573.144	1.56170	0.780850	−	0.624718i	$-0.214786\pi$
$367$	573.144	1.56170	0.780850	+	0.624718i	$0.214786\pi$
$368$	− 118.654i	− 0.322430i
$369$	0	0
$370$	84.3323	0.227925
$371$	94.0801i	0.253585i
$372$	0	0
$373$	−378.558	−1.01490	−0.507451	−	0.861681i	$-0.669412\pi$
$373$	−378.558	−1.01490	−0.507451	+	0.861681i	$0.669412\pi$
$374$	− 269.531i	− 0.720670i
$375$	0	0
$376$	−164.707	−0.438051
$377$	170.149i	0.451324i
$378$	0	0
$379$	424.202	1.11927	0.559633	−	0.828741i	$-0.310942\pi$
$379$	424.202	1.11927	0.559633	+	0.828741i	$0.310942\pi$
$380$	261.638i	0.688522i
$381$	0	0
$382$	−89.8525	−0.235216
$383$	− 515.396i	− 1.34568i	−0.739787	−	0.672841i	$-0.765074\pi$
$383$	− 515.396i	− 1.34568i	0.739787	−	0.672841i	$-0.234926\pi$
$384$	0	0
$385$	−39.6180	−0.102904
$386$	− 382.248i	− 0.990279i
$387$	0	0
$388$	86.2191	0.222214
$389$	292.238i	0.751256i	0.926771	+	0.375628i	$0.122573\pi$
$389$	292.238i	0.751256i	−0.926771	+	0.375628i	$0.877427\pi$
$390$	0	0
$391$	−828.335	−2.11850
$392$	− 133.894i	− 0.341567i
$393$	0	0
$394$	−76.2424	−0.193509
$395$	− 271.539i	− 0.687440i
$396$	0	0
$397$	501.614	1.26351	0.631755	−	0.775168i	$-0.282335\pi$
$397$	501.614	1.26351	0.631755	+	0.775168i	$0.282335\pi$
$398$	190.893i	0.479631i
$399$	0	0
$400$	18.8693	0.0471732
$401$	− 429.271i	− 1.07050i	−0.844693	−	0.535251i	$-0.820217\pi$
$401$	− 429.271i	− 1.07050i	0.844693	−	0.535251i	$-0.179783\pi$
$402$	0	0
$403$	516.075	1.28058
$404$	71.5314i	0.177058i
$405$	0	0
$406$	27.9998	0.0689649
$407$	90.3707i	0.222041i
$408$	0	0
$409$	−571.427	−1.39713	−0.698566	−	0.715545i	$-0.746178\pi$
$409$	−571.427	−1.39713	−0.698566	+	0.715545i	$0.746178\pi$
$410$	269.563i	0.657470i
$411$	0	0
$412$	−318.596	−0.773292
$413$	39.2868i	0.0951255i
$414$	0	0
$415$	123.756	0.298207
$416$	− 62.6590i	− 0.150623i
$417$	0	0
$418$	−280.372	−0.670747
$419$	− 13.7527i	− 0.0328227i	−0.999865	−	0.0164113i	$-0.994776\pi$
$419$	− 13.7527i	− 0.0328227i	0.999865	−	0.0164113i	$-0.00522412\pi$
$420$	0	0
$421$	−289.516	−0.687687	−0.343844	−	0.939027i	$-0.611729\pi$
$421$	−289.516	−0.687687	−0.343844	+	0.939027i	$0.611729\pi$
$422$	− 283.045i	− 0.670723i
$423$	0	0
$424$	−206.455	−0.486921
$425$	− 131.728i	− 0.309948i
$426$	0	0
$427$	19.5204	0.0457153
$428$	− 1.82506i	− 0.00426416i
$429$	0	0
$430$	325.635	0.757292
$431$	− 373.898i	− 0.867514i	−0.901030	−	0.433757i	$-0.857188\pi$
$431$	− 373.898i	− 0.867514i	0.901030	−	0.433757i	$-0.142812\pi$
$432$	0	0
$433$	−280.461	−0.647716	−0.323858	−	0.946106i	$-0.604980\pi$
$433$	−280.461	−0.647716	−0.323858	+	0.946106i	$0.604980\pi$
$434$	− 84.9253i	− 0.195680i
$435$	0	0
$436$	117.981	0.270600
$437$	861.653i	1.97175i
$438$	0	0
$439$	−357.226	−0.813728	−0.406864	−	0.913489i	$-0.633378\pi$
$439$	−357.226	−0.813728	−0.406864	+	0.913489i	$0.633378\pi$
$440$	− 86.9398i	− 0.197591i
$441$	0	0
$442$	−437.427	−0.989655
$443$	− 478.511i	− 1.08016i	−0.841614	−	0.540080i	$-0.818394\pi$
$443$	− 478.511i	− 1.08016i	0.841614	−	0.540080i	$-0.181606\pi$
$444$	0	0
$445$	−554.454	−1.24596
$446$	516.894i	1.15896i
$447$	0	0
$448$	−10.3112	−0.0230160
$449$	564.971i	1.25829i	0.777289	+	0.629143i	$0.216594\pi$
$449$	564.971i	1.25829i	−0.777289	+	0.629143i	$0.783406\pi$
$450$	0	0
$451$	−288.864	−0.640496
$452$	− 51.1882i	− 0.113248i
$453$	0	0
$454$	−236.681	−0.521323
$455$	64.2969i	0.141312i
$456$	0	0
$457$	−106.112	−0.232193	−0.116097	−	0.993238i	$-0.537038\pi$
$457$	−106.112	−0.232193	−0.116097	+	0.993238i	$0.537038\pi$
$458$	− 229.575i	− 0.501256i
$459$	0	0
$460$	−267.188	−0.580843
$461$	127.093i	0.275690i	0.990454	+	0.137845i	$0.0440176\pi$
$461$	127.093i	0.275690i	−0.990454	+	0.137845i	$0.955982\pi$
$462$	0	0
$463$	−249.329	−0.538508	−0.269254	−	0.963069i	$-0.586777\pi$
$463$	−249.329	−0.538508	−0.269254	+	0.963069i	$0.586777\pi$
$464$	61.4442i	0.132423i
$465$	0	0
$466$	259.962	0.557858
$467$	− 390.485i	− 0.836156i	−0.908411	−	0.418078i	$-0.862704\pi$
$467$	− 390.485i	− 0.836156i	0.908411	−	0.418078i	$-0.137296\pi$
$468$	0	0
$469$	145.189	0.309571
$470$	370.890i	0.789128i
$471$	0	0
$472$	−86.2132	−0.182655
$473$	348.951i	0.737741i
$474$	0	0
$475$	−137.026	−0.288477
$476$	71.9831i	0.151225i
$477$	0	0
$478$	−61.1082	−0.127841
$479$	784.256i	1.63728i	0.574308	+	0.818639i	$0.305271\pi$
$479$	784.256i	1.63728i	−0.574308	+	0.818639i	$0.694729\pi$
$480$	0	0
$481$	146.665	0.304916
$482$	92.1121i	0.191104i
$483$	0	0
$484$	−148.835	−0.307511
$485$	− 194.149i	− 0.400308i
$486$	0	0
$487$	386.211	0.793041	0.396521	−	0.918026i	$-0.370218\pi$
$487$	386.211	0.793041	0.396521	+	0.918026i	$0.370218\pi$
$488$	42.8367i	0.0877801i
$489$	0	0
$490$	−301.505	−0.615316
$491$	− 339.103i	− 0.690637i	−0.938485	−	0.345319i	$-0.887771\pi$
$491$	− 339.103i	− 0.690637i	0.938485	−	0.345319i	$-0.112229\pi$
$492$	0	0
$493$	428.947	0.870074
$494$	455.022i	0.921098i
$495$	0	0
$496$	186.365	0.375735
$497$	− 11.2105i	− 0.0225563i
$498$	0	0
$499$	729.314	1.46155	0.730776	−	0.682618i	$-0.239159\pi$
$499$	729.314	1.46155	0.730776	+	0.682618i	$0.239159\pi$
$500$	− 267.672i	− 0.535343i
$501$	0	0
$502$	−23.4246	−0.0466625
$503$	− 629.282i	− 1.25106i	−0.780201	−	0.625529i	$-0.784883\pi$
$503$	− 629.282i	− 1.25106i	0.780201	−	0.625529i	$-0.215117\pi$
$504$	0	0
$505$	161.076	0.318961
$506$	− 286.319i	− 0.565847i
$507$	0	0
$508$	92.3524	0.181796
$509$	946.299i	1.85913i	0.368655	+	0.929566i	$0.379818\pi$
$509$	946.299i	1.85913i	−0.368655	+	0.929566i	$0.620182\pi$
$510$	0	0
$511$	134.740	0.263679
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	573.992	1.11672
$515$	717.420i	1.39305i
$516$	0	0
$517$	−397.446	−0.768755
$518$	− 24.1351i	− 0.0465929i
$519$	0	0
$520$	−141.097	−0.271340
$521$	0.754538i	0.00144825i	1.00000		0.000724125i	$0.000230496\pi$
$521$	0.754538i	0.00144825i	−1.00000		0.000724125i	$0.999770\pi$
$522$	0	0
$523$	−419.594	−0.802282	−0.401141	−	0.916016i	$-0.631386\pi$
$523$	−419.594	−0.802282	−0.401141	+	0.916016i	$0.631386\pi$
$524$	− 57.0355i	− 0.108846i
$525$	0	0
$526$	−436.945	−0.830694
$527$	− 1301.03i	− 2.46874i
$528$	0	0
$529$	−350.928	−0.663381
$530$	464.898i	0.877165i
$531$	0	0
$532$	74.8785	0.140749
$533$	468.804i	0.879556i
$534$	0	0
$535$	−4.10970	−0.00768168
$536$	318.610i	0.594422i
$537$	0	0
$538$	−344.367	−0.640088
$539$	− 323.093i	− 0.599431i
$540$	0	0
$541$	−641.079	−1.18499	−0.592494	−	0.805575i	$-0.701857\pi$
$541$	−641.079	−1.18499	−0.592494	+	0.805575i	$0.701857\pi$
$542$	195.465i	0.360637i
$543$	0	0
$544$	−157.964	−0.290374
$545$	− 265.672i	− 0.487472i
$546$	0	0
$547$	673.102	1.23053	0.615266	−	0.788319i	$-0.289048\pi$
$547$	673.102	1.23053	0.615266	+	0.788319i	$0.289048\pi$
$548$	− 307.735i	− 0.561561i
$549$	0	0
$550$	45.5325	0.0827864
$551$	− 446.200i	− 0.809801i
$552$	0	0
$553$	−77.7119	−0.140528
$554$	− 590.287i	− 1.06550i
$555$	0	0
$556$	51.9306	0.0934003
$557$	− 166.976i	− 0.299777i	−0.988703	−	0.149888i	$-0.952109\pi$
$557$	− 166.976i	− 0.299777i	0.988703	−	0.149888i	$-0.0478914\pi$
$558$	0	0
$559$	566.321	1.01310
$560$	23.2189i	0.0414623i
$561$	0	0
$562$	552.036	0.982270
$563$	47.3972i	0.0841868i	0.999114	+	0.0420934i	$0.0134027\pi$
$563$	47.3972i	0.0841868i	−0.999114	+	0.0420934i	$0.986597\pi$
$564$	0	0
$565$	−115.266	−0.204011
$566$	18.5510i	0.0327757i
$567$	0	0
$568$	24.6008	0.0433113
$569$	− 297.257i	− 0.522419i	−0.965282	−	0.261210i	$-0.915879\pi$
$569$	− 297.257i	− 0.522419i	0.965282	−	0.261210i	$-0.0841214\pi$
$570$	0	0
$571$	−110.015	−0.192670	−0.0963350	−	0.995349i	$-0.530712\pi$
$571$	−110.015	−0.192670	−0.0963350	+	0.995349i	$0.530712\pi$
$572$	− 151.199i	− 0.264335i
$573$	0	0
$574$	77.1463	0.134401
$575$	− 139.933i	− 0.243361i
$576$	0	0
$577$	−1016.19	−1.76116	−0.880580	−	0.473898i	$-0.842846\pi$
$577$	−1016.19	−1.76116	−0.880580	+	0.473898i	$0.842846\pi$
$578$	694.048i	1.20078i
$579$	0	0
$580$	138.361	0.238553
$581$	− 35.4178i	− 0.0609601i
$582$	0	0
$583$	−498.185	−0.854520
$584$	295.681i	0.506303i
$585$	0	0
$586$	232.856	0.397365
$587$	− 137.285i	− 0.233876i	−0.993139	−	0.116938i	$-0.962692\pi$
$587$	− 137.285i	− 0.233876i	0.993139	−	0.116938i	$-0.0373079\pi$
$588$	0	0
$589$	−1353.36	−2.29772
$590$	194.136i	0.329044i
$591$	0	0
$592$	52.9635	0.0894653
$593$	− 697.717i	− 1.17659i	−0.808647	−	0.588295i	$-0.799800\pi$
$593$	− 697.717i	− 1.17659i	0.808647	−	0.588295i	$-0.200200\pi$
$594$	0	0
$595$	162.093	0.272425
$596$	106.854i	0.179286i
$597$	0	0
$598$	−464.673	−0.777045
$599$	48.4957i	0.0809611i	0.999180	+	0.0404806i	$0.0128889\pi$
$599$	48.4957i	0.0809611i	−0.999180	+	0.0404806i	$0.987111\pi$
$600$	0	0
$601$	432.807	0.720144	0.360072	−	0.932924i	$-0.382752\pi$
$601$	432.807	0.720144	0.360072	+	0.932924i	$0.382752\pi$
$602$	− 93.1938i	− 0.154807i
$603$	0	0
$604$	173.421	0.287121
$605$	335.149i	0.553966i
$606$	0	0
$607$	−608.899	−1.00313	−0.501564	−	0.865120i	$-0.667242\pi$
$607$	−608.899	−1.00313	−0.501564	+	0.865120i	$0.667242\pi$
$608$	164.317i	0.270259i
$609$	0	0
$610$	96.4603	0.158132
$611$	645.024i	1.05569i
$612$	0	0
$613$	−110.226	−0.179814	−0.0899071	−	0.995950i	$-0.528657\pi$
$613$	−110.226	−0.179814	−0.0899071	+	0.995950i	$0.528657\pi$
$614$	274.386i	0.446883i
$615$	0	0
$616$	−24.8814	−0.0403918
$617$	− 416.477i	− 0.675004i	−0.941325	−	0.337502i	$-0.890418\pi$
$617$	− 416.477i	− 0.675004i	0.941325	−	0.337502i	$-0.109582\pi$
$618$	0	0
$619$	−506.474	−0.818214	−0.409107	−	0.912487i	$-0.634160\pi$
$619$	−506.474	−0.818214	−0.409107	+	0.912487i	$0.634160\pi$
$620$	− 419.659i	− 0.676869i
$621$	0	0
$622$	−521.314	−0.838125
$623$	158.680i	0.254703i
$624$	0	0
$625$	−484.814	−0.775702
$626$	− 615.476i	− 0.983188i
$627$	0	0
$628$	−420.884	−0.670197
$629$	− 369.742i	− 0.587825i
$630$	0	0
$631$	292.628	0.463752	0.231876	−	0.972745i	$-0.425514\pi$
$631$	292.628	0.463752	0.231876	+	0.972745i	$0.425514\pi$
$632$	− 170.535i	− 0.269834i
$633$	0	0
$634$	−345.087	−0.544301
$635$	− 207.961i	− 0.327497i
$636$	0	0
$637$	−524.355	−0.823164
$638$	148.268i	0.232395i
$639$	0	0
$640$	−50.9528	−0.0796137
$641$	182.880i	0.285305i	0.989773	+	0.142652i	$0.0455631\pi$
$641$	182.880i	0.285305i	−0.989773	+	0.142652i	$0.954437\pi$
$642$	0	0
$643$	−339.348	−0.527758	−0.263879	−	0.964556i	$-0.585002\pi$
$643$	−339.348	−0.527758	−0.263879	+	0.964556i	$0.585002\pi$
$644$	76.4666i	0.118737i
$645$	0	0
$646$	1147.11	1.77572
$647$	489.141i	0.756014i	0.925803	+	0.378007i	$0.123390\pi$
$647$	489.141i	0.756014i	−0.925803	+	0.378007i	$0.876610\pi$
$648$	0	0
$649$	−208.037	−0.320550
$650$	− 73.8957i	− 0.113686i
$651$	0	0
$652$	−191.281	−0.293376
$653$	358.101i	0.548393i	0.961674	+	0.274197i	$0.0884119\pi$
$653$	358.101i	0.548393i	−0.961674	+	0.274197i	$0.911588\pi$
$654$	0	0
$655$	−128.433	−0.196081
$656$	169.294i	0.258070i
$657$	0	0
$658$	106.145	0.161315
$659$	− 184.844i	− 0.280491i	−0.990117	−	0.140246i	$-0.955211\pi$
$659$	− 184.844i	− 0.280491i	0.990117	−	0.140246i	$-0.0447892\pi$
$660$	0	0
$661$	65.9659	0.0997971	0.0498986	−	0.998754i	$-0.484110\pi$
$661$	65.9659	0.0997971	0.0498986	+	0.998754i	$0.484110\pi$
$662$	600.733i	0.907452i
$663$	0	0
$664$	77.7228	0.117052
$665$	− 168.612i	− 0.253553i
$666$	0	0
$667$	455.664	0.683154
$668$	− 194.378i	− 0.290985i
$669$	0	0
$670$	717.451	1.07082
$671$	103.367i	0.154049i
$672$	0	0
$673$	−103.543	−0.153853	−0.0769265	−	0.997037i	$-0.524511\pi$
$673$	−103.543	−0.153853	−0.0769265	+	0.997037i	$0.524511\pi$
$674$	212.356i	0.315068i
$675$	0	0
$676$	92.6152	0.137005
$677$	394.460i	0.582659i	0.956623	+	0.291329i	$0.0940976\pi$
$677$	394.460i	0.582659i	−0.956623	+	0.291329i	$0.905902\pi$
$678$	0	0
$679$	−55.5638	−0.0818318
$680$	355.705i	0.523095i
$681$	0	0
$682$	449.707	0.659394
$683$	− 175.768i	− 0.257348i	−0.991687	−	0.128674i	$-0.958928\pi$
$683$	− 175.768i	− 0.257348i	0.991687	−	0.128674i	$-0.0410720\pi$
$684$	0	0
$685$	−692.963	−1.01163
$686$	175.604i	0.255982i
$687$	0	0
$688$	204.510	0.297252
$689$	808.516i	1.17346i
$690$	0	0
$691$	−107.315	−0.155304	−0.0776518	−	0.996981i	$-0.524742\pi$
$691$	−107.315	−0.155304	−0.0776518	+	0.996981i	$0.524742\pi$
$692$	555.316i	0.802479i
$693$	0	0
$694$	−330.874	−0.476764
$695$	− 116.938i	− 0.168256i
$696$	0	0
$697$	1181.86	1.69563
$698$	21.5564i	0.0308830i
$699$	0	0
$700$	−12.1603	−0.0173718
$701$	1348.09i	1.92309i	0.274639	+	0.961547i	$0.411442\pi$
$701$	1348.09i	1.92309i	−0.274639	+	0.961547i	$0.588558\pi$
$702$	0	0
$703$	−384.614	−0.547104
$704$	− 54.6011i	− 0.0775583i
$705$	0	0
$706$	334.779	0.474191
$707$	− 46.0983i	− 0.0652027i
$708$	0	0
$709$	−215.483	−0.303925	−0.151963	−	0.988386i	$-0.548559\pi$
$709$	−215.483	−0.303925	−0.151963	+	0.988386i	$0.548559\pi$
$710$	− 55.3965i	− 0.0780233i
$711$	0	0
$712$	−348.215	−0.489067
$713$	− 1382.06i	− 1.93837i
$714$	0	0
$715$	−340.473	−0.476186
$716$	− 153.472i	− 0.214347i
$717$	0	0
$718$	794.487	1.10653
$719$	− 295.395i	− 0.410841i	−0.978674	−	0.205421i	$-0.934144\pi$
$719$	− 295.395i	− 0.410841i	0.978674	−	0.205421i	$-0.0658562\pi$
$720$	0	0
$721$	205.319	0.284770
$722$	− 682.722i	− 0.945598i
$723$	0	0
$724$	−570.198	−0.787567
$725$	72.4630i	0.0999490i
$726$	0	0
$727$	−313.938	−0.431826	−0.215913	−	0.976413i	$-0.569273\pi$
$727$	−313.938	−0.431826	−0.215913	+	0.976413i	$0.569273\pi$
$728$	40.3805i	0.0554678i
$729$	0	0
$730$	665.819	0.912080
$731$	− 1427.70i	− 1.95307i
$732$	0	0
$733$	1433.69	1.95592	0.977959	−	0.208798i	$-0.0669550\pi$
$733$	1433.69	1.95592	0.977959	+	0.208798i	$0.0669550\pi$
$734$	− 810.548i	− 1.10429i
$735$	0	0
$736$	−167.803	−0.227993
$737$	768.822i	1.04318i
$738$	0	0
$739$	919.895	1.24478	0.622392	−	0.782706i	$-0.286161\pi$
$739$	919.895	1.24478	0.622392	+	0.782706i	$0.286161\pi$
$740$	− 119.264i	− 0.161167i
$741$	0	0
$742$	133.049	0.179312
$743$	472.593i	0.636061i	0.948080	+	0.318031i	$0.103021\pi$
$743$	472.593i	0.636061i	−0.948080	+	0.318031i	$0.896979\pi$
$744$	0	0
$745$	240.616	0.322975
$746$	535.362i	0.717644i
$747$	0	0
$748$	−381.174	−0.509591
$749$	1.17616i	0.00157030i
$750$	0	0
$751$	−645.587	−0.859636	−0.429818	−	0.902915i	$-0.641422\pi$
$751$	−645.587	−0.859636	−0.429818	+	0.902915i	$0.641422\pi$
$752$	232.931i	0.309749i
$753$	0	0
$754$	240.627	0.319134
$755$	− 390.513i	− 0.517235i
$756$	0	0
$757$	−482.464	−0.637337	−0.318668	−	0.947866i	$-0.603236\pi$
$757$	−482.464	−0.637337	−0.318668	+	0.947866i	$0.603236\pi$
$758$	− 599.912i	− 0.791440i
$759$	0	0
$760$	370.013	0.486859
$761$	688.514i	0.904749i	0.891828	+	0.452375i	$0.149423\pi$
$761$	688.514i	0.904749i	−0.891828	+	0.452375i	$0.850577\pi$
$762$	0	0
$763$	−76.0330	−0.0996500
$764$	127.071i	0.166323i
$765$	0	0
$766$	−728.880	−0.951541
$767$	337.627i	0.440192i
$768$	0	0
$769$	979.350	1.27354	0.636768	−	0.771055i	$-0.280271\pi$
$769$	979.350	1.27354	0.636768	+	0.771055i	$0.280271\pi$
$770$	56.0283i	0.0727640i
$771$	0	0
$772$	−540.580	−0.700233
$773$	− 832.437i	− 1.07689i	−0.842660	−	0.538445i	$-0.819012\pi$
$773$	− 832.437i	− 1.07689i	0.842660	−	0.538445i	$-0.180988\pi$
$774$	0	0
$775$	219.785	0.283594
$776$	− 121.932i	− 0.157129i
$777$	0	0
$778$	413.288	0.531218
$779$	− 1229.39i	− 1.57817i
$780$	0	0
$781$	59.3630	0.0760090
$782$	1171.44i	1.49801i
$783$	0	0
$784$	−189.355	−0.241524
$785$	947.752i	1.20733i
$786$	0	0
$787$	297.079	0.377482	0.188741	−	0.982027i	$-0.439559\pi$
$787$	297.079	0.377482	0.188741	+	0.982027i	$0.439559\pi$
$788$	107.823i	0.136831i
$789$	0	0
$790$	−384.014	−0.486094
$791$	32.9882i	0.0417044i
$792$	0	0
$793$	167.757	0.211547
$794$	− 709.389i	− 0.893437i
$795$	0	0
$796$	269.963	0.339150
$797$	− 1168.69i	− 1.46636i	−0.680033	−	0.733182i	$-0.738035\pi$
$797$	− 1168.69i	− 1.46636i	0.680033	−	0.733182i	$-0.261965\pi$
$798$	0	0
$799$	1626.11	2.03518
$800$	− 26.6852i	− 0.0333565i
$801$	0	0
$802$	−607.081	−0.756959
$803$	713.492i	0.888533i
$804$	0	0
$805$	172.189	0.213899
$806$	− 729.840i	− 0.905508i
$807$	0	0
$808$	101.161	0.125199
$809$	− 947.214i	− 1.17085i	−0.810728	−	0.585423i	$-0.800929\pi$
$809$	− 947.214i	− 1.17085i	0.810728	−	0.585423i	$-0.199071\pi$
$810$	0	0
$811$	209.571	0.258411	0.129205	−	0.991618i	$-0.458757\pi$
$811$	209.571	0.258411	0.129205	+	0.991618i	$0.458757\pi$
$812$	− 39.5976i	− 0.0487656i
$813$	0	0
$814$	127.803	0.157007
$815$	430.730i	0.528503i
$816$	0	0
$817$	−1485.12	−1.81778
$818$	808.120i	0.987922i
$819$	0	0
$820$	381.219	0.464901
$821$	932.395i	1.13568i	0.823138	+	0.567841i	$0.192221\pi$
$821$	932.395i	1.13568i	−0.823138	+	0.567841i	$0.807779\pi$
$822$	0	0
$823$	597.351	0.725822	0.362911	−	0.931824i	$-0.381783\pi$
$823$	597.351	0.725822	0.362911	+	0.931824i	$0.381783\pi$
$824$	450.563i	0.546800i
$825$	0	0
$826$	55.5600	0.0672639
$827$	− 1036.61i	− 1.25345i	−0.779239	−	0.626727i	$-0.784394\pi$
$827$	− 1036.61i	− 1.25345i	0.779239	−	0.626727i	$-0.215606\pi$
$828$	0	0
$829$	−1122.48	−1.35401	−0.677006	−	0.735977i	$-0.736723\pi$
$829$	−1122.48	−1.35401	−0.677006	+	0.735977i	$0.736723\pi$
$830$	− 175.017i	− 0.210864i
$831$	0	0
$832$	−88.6133	−0.106506
$833$	1321.90i	1.58692i
$834$	0	0
$835$	−437.703	−0.524195
$836$	396.506i	0.474290i
$837$	0	0
$838$	−19.4492	−0.0232091
$839$	− 375.538i	− 0.447602i	−0.974635	−	0.223801i	$-0.928153\pi$
$839$	− 375.538i	− 0.447602i	0.974635	−	0.223801i	$-0.0718466\pi$
$840$	0	0
$841$	605.038	0.719427
$842$	409.438i	0.486268i
$843$	0	0
$844$	−400.286	−0.474273
$845$	− 208.552i	− 0.246807i
$846$	0	0
$847$	95.9166	0.113243
$848$	291.971i	0.344305i
$849$	0	0
$850$	−186.291	−0.219166
$851$	− 392.772i	− 0.461541i
$852$	0	0
$853$	342.332	0.401328	0.200664	−	0.979660i	$-0.435690\pi$
$853$	342.332	0.401328	0.200664	+	0.979660i	$0.435690\pi$
$854$	− 27.6061i	− 0.0323256i
$855$	0	0
$856$	−2.58103	−0.00301522
$857$	− 1632.39i	− 1.90478i	−0.304889	−	0.952388i	$-0.598619\pi$
$857$	− 1632.39i	− 1.90478i	0.304889	−	0.952388i	$-0.401381\pi$
$858$	0	0
$859$	794.565	0.924989	0.462494	−	0.886622i	$-0.346955\pi$
$859$	794.565	0.924989	0.462494	+	0.886622i	$0.346955\pi$
$860$	− 460.518i	− 0.535486i
$861$	0	0
$862$	−528.772	−0.613425
$863$	− 1059.53i	− 1.22772i	−0.789414	−	0.613862i	$-0.789615\pi$
$863$	− 1059.53i	− 1.22772i	0.789414	−	0.613862i	$-0.210385\pi$
$864$	0	0
$865$	1250.47	1.44563
$866$	396.632i	0.458004i
$867$	0	0
$868$	−120.102	−0.138367
$869$	− 411.510i	− 0.473544i
$870$	0	0
$871$	1247.74	1.43253
$872$	− 166.851i	− 0.191343i
$873$	0	0
$874$	1218.56	1.39424
$875$	172.501i	0.197144i
$876$	0	0
$877$	−10.6468	−0.0121401	−0.00607004	−	0.999982i	$-0.501932\pi$
$877$	−10.6468	−0.0121401	−0.00607004	+	0.999982i	$0.501932\pi$
$878$	505.194i	0.575392i
$879$	0	0
$880$	−122.951	−0.139718
$881$	− 998.400i	− 1.13326i	−0.823973	−	0.566629i	$-0.808247\pi$
$881$	− 998.400i	− 1.13326i	0.823973	−	0.566629i	$-0.191753\pi$
$882$	0	0
$883$	−469.371	−0.531564	−0.265782	−	0.964033i	$-0.585630\pi$
$883$	−469.371	−0.531564	−0.265782	+	0.964033i	$0.585630\pi$
$884$	618.616i	0.699792i
$885$	0	0
$886$	−676.716	−0.763788
$887$	− 1763.26i	− 1.98789i	−0.109891	−	0.993944i	$-0.535050\pi$
$887$	− 1763.26i	− 1.98789i	0.109891	−	0.993944i	$-0.464950\pi$
$888$	0	0
$889$	−59.5164	−0.0669476
$890$	784.117i	0.881030i
$891$	0	0
$892$	730.999	0.819505
$893$	− 1691.52i	− 1.89420i
$894$	0	0
$895$	−345.591	−0.386135
$896$	14.5822i	0.0162748i
$897$	0	0
$898$	798.989	0.889743
$899$	715.690i	0.796095i
$900$	0	0
$901$	2038.27	2.26223
$902$	408.515i	0.452899i
$903$	0	0
$904$	−72.3911	−0.0800787
$905$	1283.98i	1.41876i
$906$	0	0
$907$	−1483.12	−1.63519	−0.817594	−	0.575795i	$-0.804693\pi$
$907$	−1483.12	−1.63519	−0.817594	+	0.575795i	$0.804693\pi$
$908$	334.717i	0.368631i
$909$	0	0
$910$	90.9295	0.0999225
$911$	− 1497.94i	− 1.64428i	−0.569283	−	0.822142i	$-0.692779\pi$
$911$	− 1497.94i	− 1.64428i	0.569283	−	0.822142i	$-0.307221\pi$
$912$	0	0
$913$	187.549	0.205420
$914$	150.065i	0.164185i
$915$	0	0
$916$	−324.668	−0.354441
$917$	36.7564i	0.0400834i
$918$	0	0
$919$	−610.689	−0.664515	−0.332257	−	0.943189i	$-0.607810\pi$
$919$	−610.689	−0.664515	−0.332257	+	0.943189i	$0.607810\pi$
$920$	377.860i	0.410718i
$921$	0	0
$922$	179.737	0.194942
$923$	− 96.3416i	− 0.104379i
$924$	0	0
$925$	62.4614	0.0675259
$926$	352.605i	0.380782i
$927$	0	0
$928$	86.8953	0.0936371
$929$	− 904.149i	− 0.973250i	−0.873611	−	0.486625i	$-0.838228\pi$
$929$	− 904.149i	− 0.973250i	0.873611	−	0.486625i	$-0.161772\pi$
$930$	0	0
$931$	1375.07	1.47698
$932$	− 367.642i	− 0.394465i
$933$	0	0
$934$	−552.229	−0.591251
$935$	858.333i	0.918004i
$936$	0	0
$937$	−519.246	−0.554158	−0.277079	−	0.960847i	$-0.589366\pi$
$937$	−519.246	−0.554158	−0.277079	+	0.960847i	$0.589366\pi$
$938$	− 205.328i	− 0.218900i
$939$	0	0
$940$	524.518	0.557997
$941$	821.273i	0.872766i	0.899761	+	0.436383i	$0.143741\pi$
$941$	821.273i	0.872766i	−0.899761	+	0.436383i	$0.856259\pi$
$942$	0	0
$943$	1255.47	1.33136
$944$	121.924i	0.129157i
$945$	0	0
$946$	493.492	0.521662
$947$	116.317i	0.122827i	0.998112	+	0.0614134i	$0.0195608\pi$
$947$	116.317i	0.122827i	−0.998112	+	0.0614134i	$0.980439\pi$
$948$	0	0
$949$	1157.94	1.22017
$950$	193.785i	0.203984i
$951$	0	0
$952$	101.799	0.106932
$953$	− 792.524i	− 0.831609i	−0.909454	−	0.415805i	$-0.863500\pi$
$953$	− 792.524i	− 0.831609i	0.909454	−	0.415805i	$-0.136500\pi$
$954$	0	0
$955$	286.140	0.299623
$956$	86.4200i	0.0903975i
$957$	0	0
$958$	1109.11	1.15773
$959$	198.320i	0.206798i
$960$	0	0
$961$	1209.74	1.25883
$962$	− 207.415i	− 0.215608i
$963$	0	0
$964$	130.266	0.135131
$965$	1217.29i	1.26144i
$966$	0	0
$967$	648.585	0.670719	0.335360	−	0.942090i	$-0.391142\pi$
$967$	648.585	0.670719	0.335360	+	0.942090i	$0.391142\pi$
$968$	210.485i	0.217443i
$969$	0	0
$970$	−274.569	−0.283061
$971$	− 144.756i	− 0.149079i	−0.997218	−	0.0745395i	$-0.976251\pi$
$971$	− 144.756i	− 0.149079i	0.997218	−	0.0745395i	$-0.0237487\pi$
$972$	0	0
$973$	−33.4666	−0.0343953
$974$	− 546.185i	− 0.560765i
$975$	0	0
$976$	60.5802	0.0620699
$977$	899.786i	0.920969i	0.887668	+	0.460484i	$0.152324\pi$
$977$	899.786i	0.920969i	−0.887668	+	0.460484i	$0.847676\pi$
$978$	0	0
$979$	−840.261	−0.858285
$980$	426.392i	0.435094i
$981$	0	0
$982$	−479.564	−0.488354
$983$	1722.42i	1.75221i	0.482123	+	0.876104i	$0.339866\pi$
$983$	1722.42i	1.75221i	−0.482123	+	0.876104i	$0.660134\pi$
$984$	0	0
$985$	242.798	0.246495
$986$	− 606.622i	− 0.615235i
$987$	0	0
$988$	643.499	0.651314
$989$	− 1516.62i	− 1.53349i
$990$	0	0
$991$	151.393	0.152768	0.0763839	−	0.997078i	$-0.475663\pi$
$991$	151.393	0.152768	0.0763839	+	0.997078i	$0.475663\pi$
$992$	− 263.559i	− 0.265685i
$993$	0	0
$994$	−15.8540	−0.0159497
$995$	− 607.908i	− 0.610963i
$996$	0	0
$997$	−330.370	−0.331364	−0.165682	−	0.986179i	$-0.552982\pi$
$997$	−330.370	−0.331364	−0.165682	+	0.986179i	$0.552982\pi$
$998$	− 1031.41i	− 1.03347i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.14		36
3.2	odd	2	inner	1458.3.b.c.1457.23		36
27.4	even	9		162.3.f.a.35.6		36
27.7	even	9		54.3.f.a.5.2	✓	36
27.20	odd	18		162.3.f.a.125.6		36
27.23	odd	18		54.3.f.a.11.2	yes	36
108.7	odd	18		432.3.bc.c.113.4		36
108.23	even	18		432.3.bc.c.65.4		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.5.2	✓	36	27.7	even	9
54.3.f.a.11.2	yes	36	27.23	odd	18
162.3.f.a.35.6		36	27.4	even	9
162.3.f.a.125.6		36	27.20	odd	18
432.3.bc.c.65.4		36	108.23	even	18
432.3.bc.c.113.4		36	108.7	odd	18
1458.3.b.c.1457.14		36	1.1	even	1	trivial
1458.3.b.c.1457.23		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1458.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +4.50363i q^{5} +1.28890 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +4.50363i q^{5} +1.28890 q^{7} +2.82843i q^{8} +6.36909 q^{10} +6.82513i q^{11} +11.0767 q^{13} -1.82278i q^{14} +4.00000 q^{16} -27.9243i q^{17} -29.0475 q^{19} -9.00726i q^{20} +9.65220 q^{22} -29.6636i q^{23} +4.71732 q^{25} -15.6648i q^{26} -2.57779 q^{28} +15.3611i q^{29} +46.5912 q^{31} -5.65685i q^{32} -39.4909 q^{34} +5.80472i q^{35} +13.2409 q^{37} +41.0794i q^{38} -12.7382 q^{40} +42.3235i q^{41} +51.1274 q^{43} -13.6503i q^{44} -41.9506 q^{46} +58.2328i q^{47} -47.3387 q^{49} -6.67130i q^{50} -22.1533 q^{52} +72.9927i q^{53} -30.7379 q^{55} +3.64555i q^{56} +21.7238 q^{58} +30.4810i q^{59} +15.1451 q^{61} -65.8899i q^{62} -8.00000 q^{64} +49.8852i q^{65} +112.646 q^{67} +55.8486i q^{68} +8.20911 q^{70} -8.69771i q^{71} +104.539 q^{73} -18.7254i q^{74} +58.0950 q^{76} +8.79689i q^{77} -60.2933 q^{79} +18.0145i q^{80} +59.8545 q^{82} -27.4791i q^{83} +125.761 q^{85} -72.3051i q^{86} -19.3044 q^{88} +123.113i q^{89} +14.2767 q^{91} +59.3272i q^{92} +82.3536 q^{94} -130.819i q^{95} -43.1095 q^{97} +66.9471i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4}+O(q^{10}) \) 36 * q - 72 * q^4	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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