LMFDB - Embedded newform 1458.3.b.c.1457.2

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			1457.2
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -8.36153i q^{5} +9.16954 q^{7} +2.82843i q^{8} -11.8250 q^{10} +6.57978i q^{11} +6.00695 q^{13} -12.9677i q^{14} +4.00000 q^{16} -1.01616i q^{17} -21.1186 q^{19} +16.7231i q^{20} +9.30521 q^{22} -21.4417i q^{23} -44.9152 q^{25} -8.49512i q^{26} -18.3391 q^{28} -48.4543i q^{29} +6.60708 q^{31} -5.65685i q^{32} -1.43707 q^{34} -76.6714i q^{35} +66.5988 q^{37} +29.8663i q^{38} +23.6500 q^{40} -53.0349i q^{41} -47.4986 q^{43} -13.1596i q^{44} -30.3231 q^{46} +6.52510i q^{47} +35.0805 q^{49} +63.5197i q^{50} -12.0139 q^{52} +39.7705i q^{53} +55.0170 q^{55} +25.9354i q^{56} -68.5247 q^{58} -28.0401i q^{59} -41.0906 q^{61} -9.34383i q^{62} -8.00000 q^{64} -50.2274i q^{65} -14.8401 q^{67} +2.03232i q^{68} -108.430 q^{70} -75.8063i q^{71} +53.0280 q^{73} -94.1849i q^{74} +42.2373 q^{76} +60.3336i q^{77} -81.8603 q^{79} -33.4461i q^{80} -75.0026 q^{82} +46.9255i q^{83} -8.49667 q^{85} +67.1732i q^{86} -18.6104 q^{88} -23.4400i q^{89} +55.0810 q^{91} +42.8833i q^{92} +9.22788 q^{94} +176.584i q^{95} +35.0019 q^{97} -49.6113i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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

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
$5$	− 8.36153i	− 1.67231i	−0.548496	−	0.836153i	$-0.684799\pi$
$5$	− 8.36153i	− 1.67231i	0.548496	−	0.836153i	$-0.315201\pi$
$6$	0	0
$7$	9.16954	1.30993	0.654967	−	0.755657i	$-0.272682\pi$
$7$	9.16954	1.30993	0.654967	+	0.755657i	$0.272682\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−11.8250	−1.18250
$11$	6.57978i	0.598162i	0.954228	+	0.299081i	$0.0966800\pi$
$11$	6.57978i	0.598162i	−0.954228	+	0.299081i	$0.903320\pi$
$12$	0	0
$13$	6.00695	0.462073	0.231037	−	0.972945i	$-0.425788\pi$
$13$	6.00695	0.462073	0.231037	+	0.972945i	$0.425788\pi$
$14$	− 12.9677i	− 0.926264i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 1.01616i	− 0.0597742i	−0.999553	−	0.0298871i	$-0.990485\pi$
$17$	− 1.01616i	− 0.0597742i	0.999553	−	0.0298871i	$-0.00951478\pi$
$18$	0	0
$19$	−21.1186	−1.11151	−0.555754	−	0.831347i	$-0.687570\pi$
$19$	−21.1186	−1.11151	−0.555754	+	0.831347i	$0.687570\pi$
$20$	16.7231i	0.836153i
$21$	0	0
$22$	9.30521	0.422964
$23$	− 21.4417i	− 0.932247i	−0.884720	−	0.466123i	$-0.845650\pi$
$23$	− 21.4417i	− 0.932247i	0.884720	−	0.466123i	$-0.154350\pi$
$24$	0	0
$25$	−44.9152	−1.79661
$26$	− 8.49512i	− 0.326735i
$27$	0	0
$28$	−18.3391	−0.654967
$29$	− 48.4543i	− 1.67084i	−0.549615	−	0.835418i	$-0.685226\pi$
$29$	− 48.4543i	− 1.67084i	0.549615	−	0.835418i	$-0.314774\pi$
$30$	0	0
$31$	6.60708	0.213132	0.106566	−	0.994306i	$-0.466014\pi$
$31$	6.60708	0.213132	0.106566	+	0.994306i	$0.466014\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−1.43707	−0.0422667
$35$	− 76.6714i	− 2.19061i
$36$	0	0
$37$	66.5988	1.79997	0.899984	−	0.435923i	$-0.143578\pi$
$37$	66.5988	1.79997	0.899984	+	0.435923i	$0.143578\pi$
$38$	29.8663i	0.785955i
$39$	0	0
$40$	23.6500	0.591250
$41$	− 53.0349i	− 1.29353i	−0.762688	−	0.646767i	$-0.776121\pi$
$41$	− 53.0349i	− 1.29353i	0.762688	−	0.646767i	$-0.223879\pi$
$42$	0	0
$43$	−47.4986	−1.10462	−0.552310	−	0.833639i	$-0.686253\pi$
$43$	−47.4986	−1.10462	−0.552310	+	0.833639i	$0.686253\pi$
$44$	− 13.1596i	− 0.299081i
$45$	0	0
$46$	−30.3231	−0.659198
$47$	6.52510i	0.138832i	0.997588	+	0.0694160i	$0.0221136\pi$
$47$	6.52510i	0.138832i	−0.997588	+	0.0694160i	$0.977886\pi$
$48$	0	0
$49$	35.0805	0.715929
$50$	63.5197i	1.27039i
$51$	0	0
$52$	−12.0139	−0.231037
$53$	39.7705i	0.750386i	0.926947	+	0.375193i	$0.122424\pi$
$53$	39.7705i	0.750386i	−0.926947	+	0.375193i	$0.877576\pi$
$54$	0	0
$55$	55.0170	1.00031
$56$	25.9354i	0.463132i
$57$	0	0
$58$	−68.5247	−1.18146
$59$	− 28.0401i	− 0.475255i	−0.971356	−	0.237628i	$-0.923630\pi$
$59$	− 28.0401i	− 0.475255i	0.971356	−	0.237628i	$-0.0763698\pi$
$60$	0	0
$61$	−41.0906	−0.673616	−0.336808	−	0.941573i	$-0.609347\pi$
$61$	−41.0906	−0.673616	−0.336808	+	0.941573i	$0.609347\pi$
$62$	− 9.34383i	− 0.150707i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 50.2274i	− 0.772728i
$66$	0	0
$67$	−14.8401	−0.221493	−0.110747	−	0.993849i	$-0.535324\pi$
$67$	−14.8401	−0.221493	−0.110747	+	0.993849i	$0.535324\pi$
$68$	2.03232i	0.0298871i
$69$	0	0
$70$	−108.430	−1.54900
$71$	− 75.8063i	− 1.06769i	−0.845581	−	0.533847i	$-0.820746\pi$
$71$	− 75.8063i	− 1.06769i	0.845581	−	0.533847i	$-0.179254\pi$
$72$	0	0
$73$	53.0280	0.726411	0.363205	−	0.931709i	$-0.381682\pi$
$73$	53.0280	0.726411	0.363205	+	0.931709i	$0.381682\pi$
$74$	− 94.1849i	− 1.27277i
$75$	0	0
$76$	42.2373	0.555754
$77$	60.3336i	0.783553i
$78$	0	0
$79$	−81.8603	−1.03621	−0.518103	−	0.855318i	$-0.673362\pi$
$79$	−81.8603	−1.03621	−0.518103	+	0.855318i	$0.673362\pi$
$80$	− 33.4461i	− 0.418077i
$81$	0	0
$82$	−75.0026	−0.914666
$83$	46.9255i	0.565367i	0.959213	+	0.282683i	$0.0912246\pi$
$83$	46.9255i	0.565367i	−0.959213	+	0.282683i	$0.908775\pi$
$84$	0	0
$85$	−8.49667	−0.0999608
$86$	67.1732i	0.781084i
$87$	0	0
$88$	−18.6104	−0.211482
$89$	− 23.4400i	− 0.263370i	−0.991292	−	0.131685i	$-0.957961\pi$
$89$	− 23.4400i	− 0.263370i	0.991292	−	0.131685i	$-0.0420388\pi$
$90$	0	0
$91$	55.0810	0.605286
$92$	42.8833i	0.466123i
$93$	0	0
$94$	9.22788	0.0981690
$95$	176.584i	1.85878i
$96$	0	0
$97$	35.0019	0.360844	0.180422	−	0.983589i	$-0.442254\pi$
$97$	35.0019	0.360844	0.180422	+	0.983589i	$0.442254\pi$
$98$	− 49.6113i	− 0.506238i
$99$	0	0
$100$	89.8305	0.898305
$101$	121.268i	1.20067i	0.799747	+	0.600337i	$0.204967\pi$
$101$	121.268i	1.20067i	−0.799747	+	0.600337i	$0.795033\pi$
$102$	0	0
$103$	−188.927	−1.83424	−0.917121	−	0.398609i	$-0.869493\pi$
$103$	−188.927	−1.83424	−0.917121	+	0.398609i	$0.869493\pi$
$104$	16.9902i	0.163368i
$105$	0	0
$106$	56.2439	0.530603
$107$	− 76.4765i	− 0.714733i	−0.933964	−	0.357367i	$-0.883675\pi$
$107$	− 76.4765i	− 0.714733i	0.933964	−	0.357367i	$-0.116325\pi$
$108$	0	0
$109$	43.7808	0.401659	0.200829	−	0.979626i	$-0.435636\pi$
$109$	43.7808	0.401659	0.200829	+	0.979626i	$0.435636\pi$
$110$	− 77.8059i	− 0.707326i
$111$	0	0
$112$	36.6782	0.327484
$113$	− 0.207620i	− 0.00183734i	−1.00000		0.000918672i	$-0.999708\pi$
$113$	− 0.207620i	− 0.00183734i	1.00000		0.000918672i	$-0.000292422\pi$
$114$	0	0
$115$	−179.285	−1.55900
$116$	96.9085i	0.835418i
$117$	0	0
$118$	−39.6546	−0.336056
$119$	− 9.31774i	− 0.0783003i
$120$	0	0
$121$	77.7065	0.642203
$122$	58.1108i	0.476318i
$123$	0	0
$124$	−13.2142	−0.106566
$125$	166.522i	1.33218i
$126$	0	0
$127$	−155.090	−1.22118	−0.610589	−	0.791948i	$-0.709067\pi$
$127$	−155.090	−1.22118	−0.610589	+	0.791948i	$0.709067\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−71.0322	−0.546402
$131$	− 135.408i	− 1.03365i	−0.856092	−	0.516824i	$-0.827114\pi$
$131$	− 135.408i	− 1.03365i	0.856092	−	0.516824i	$-0.172886\pi$
$132$	0	0
$133$	−193.648	−1.45600
$134$	20.9870i	0.156619i
$135$	0	0
$136$	2.87414	0.0211334
$137$	47.0098i	0.343138i	0.985172	+	0.171569i	$0.0548836\pi$
$137$	47.0098i	0.343138i	−0.985172	+	0.171569i	$0.945116\pi$
$138$	0	0
$139$	48.4431	0.348512	0.174256	−	0.984700i	$-0.444248\pi$
$139$	48.4431	0.348512	0.174256	+	0.984700i	$0.444248\pi$
$140$	153.343i	1.09531i
$141$	0	0
$142$	−107.206	−0.754974
$143$	39.5244i	0.276395i
$144$	0	0
$145$	−405.152	−2.79415
$146$	− 74.9929i	− 0.513650i
$147$	0	0
$148$	−133.198	−0.899984
$149$	− 83.9193i	− 0.563217i	−0.959529	−	0.281608i	$-0.909132\pi$
$149$	− 83.9193i	− 0.563217i	0.959529	−	0.281608i	$-0.0908679\pi$
$150$	0	0
$151$	−129.080	−0.854835	−0.427417	−	0.904054i	$-0.640577\pi$
$151$	−129.080	−0.854835	−0.427417	+	0.904054i	$0.640577\pi$
$152$	− 59.7326i	− 0.392977i
$153$	0	0
$154$	85.3245	0.554055
$155$	− 55.2454i	− 0.356422i
$156$	0	0
$157$	207.631	1.32249	0.661244	−	0.750171i	$-0.270029\pi$
$157$	207.631	1.32249	0.661244	+	0.750171i	$0.270029\pi$
$158$	115.768i	0.732708i
$159$	0	0
$160$	−47.3000	−0.295625
$161$	− 196.610i	− 1.22118i
$162$	0	0
$163$	−171.033	−1.04928	−0.524641	−	0.851324i	$-0.675800\pi$
$163$	−171.033	−1.04928	−0.524641	+	0.851324i	$0.675800\pi$
$164$	106.070i	0.646767i
$165$	0	0
$166$	66.3626	0.399775
$167$	− 157.706i	− 0.944348i	−0.881505	−	0.472174i	$-0.843469\pi$
$167$	− 157.706i	− 0.944348i	0.881505	−	0.472174i	$-0.156531\pi$
$168$	0	0
$169$	−132.916	−0.786488
$170$	12.0161i	0.0706830i
$171$	0	0
$172$	94.9973	0.552310
$173$	16.7551i	0.0968505i	0.998827	+	0.0484252i	$0.0154203\pi$
$173$	16.7551i	0.0968505i	−0.998827	+	0.0484252i	$0.984580\pi$
$174$	0	0
$175$	−411.852	−2.35344
$176$	26.3191i	0.149540i
$177$	0	0
$178$	−33.1491	−0.186231
$179$	− 108.838i	− 0.608031i	−0.952667	−	0.304016i	$-0.901673\pi$
$179$	− 108.838i	− 0.608031i	0.952667	−	0.304016i	$-0.0983275\pi$
$180$	0	0
$181$	37.1568	0.205286	0.102643	−	0.994718i	$-0.467270\pi$
$181$	37.1568	0.205286	0.102643	+	0.994718i	$0.467270\pi$
$182$	− 77.8963i	− 0.428002i
$183$	0	0
$184$	60.6462	0.329599
$185$	− 556.868i	− 3.01010i
$186$	0	0
$187$	6.68612	0.0357546
$188$	− 13.0502i	− 0.0694160i
$189$	0	0
$190$	249.728	1.31436
$191$	214.731i	1.12425i	0.827053	+	0.562123i	$0.190015\pi$
$191$	214.731i	1.12425i	−0.827053	+	0.562123i	$0.809985\pi$
$192$	0	0
$193$	283.820	1.47057	0.735284	−	0.677759i	$-0.237049\pi$
$193$	283.820	1.47057	0.735284	+	0.677759i	$0.237049\pi$
$194$	− 49.5001i	− 0.255155i
$195$	0	0
$196$	−70.1610	−0.357964
$197$	178.058i	0.903849i	0.892056	+	0.451925i	$0.149262\pi$
$197$	178.058i	0.903849i	−0.892056	+	0.451925i	$0.850738\pi$
$198$	0	0
$199$	145.687	0.732096	0.366048	−	0.930596i	$-0.380711\pi$
$199$	145.687	0.732096	0.366048	+	0.930596i	$0.380711\pi$
$200$	− 127.039i	− 0.635197i
$201$	0	0
$202$	171.499	0.849005
$203$	− 444.303i	− 2.18869i
$204$	0	0
$205$	−443.453	−2.16318
$206$	267.183i	1.29700i
$207$	0	0
$208$	24.0278	0.115518
$209$	− 138.956i	− 0.664861i
$210$	0	0
$211$	−297.265	−1.40884	−0.704419	−	0.709784i	$-0.748792\pi$
$211$	−297.265	−1.40884	−0.704419	+	0.709784i	$0.748792\pi$
$212$	− 79.5409i	− 0.375193i
$213$	0	0
$214$	−108.154	−0.505393
$215$	397.161i	1.84726i
$216$	0	0
$217$	60.5839	0.279189
$218$	− 61.9154i	− 0.284016i
$219$	0	0
$220$	−110.034	−0.500155
$221$	− 6.10404i	− 0.0276201i
$222$	0	0
$223$	−211.291	−0.947493	−0.473747	−	0.880661i	$-0.657099\pi$
$223$	−211.291	−0.947493	−0.473747	+	0.880661i	$0.657099\pi$
$224$	− 51.8708i	− 0.231566i
$225$	0	0
$226$	−0.293619	−0.00129920
$227$	− 16.0908i	− 0.0708847i	−0.999372	−	0.0354424i	$-0.988716\pi$
$227$	− 16.0908i	− 0.0708847i	0.999372	−	0.0354424i	$-0.0112840\pi$
$228$	0	0
$229$	261.364	1.14133	0.570664	−	0.821183i	$-0.306686\pi$
$229$	261.364	1.14133	0.570664	+	0.821183i	$0.306686\pi$
$230$	253.548i	1.10238i
$231$	0	0
$232$	137.049	0.590730
$233$	321.716i	1.38075i	0.723450	+	0.690377i	$0.242555\pi$
$233$	321.716i	1.38075i	−0.723450	+	0.690377i	$0.757445\pi$
$234$	0	0
$235$	54.5598	0.232170
$236$	56.0801i	0.237628i
$237$	0	0
$238$	−13.1773	−0.0553667
$239$	78.8788i	0.330037i	0.986290	+	0.165019i	$0.0527684\pi$
$239$	78.8788i	0.330037i	−0.986290	+	0.165019i	$0.947232\pi$
$240$	0	0
$241$	302.958	1.25709	0.628543	−	0.777775i	$-0.283652\pi$
$241$	302.958	1.25709	0.628543	+	0.777775i	$0.283652\pi$
$242$	− 109.894i	− 0.454106i
$243$	0	0
$244$	82.1811	0.336808
$245$	− 293.327i	− 1.19725i
$246$	0	0
$247$	−126.859	−0.513598
$248$	18.6877i	0.0753535i
$249$	0	0
$250$	235.498	0.941991
$251$	82.4240i	0.328382i	0.986429	+	0.164191i	$0.0525014\pi$
$251$	82.4240i	0.328382i	−0.986429	+	0.164191i	$0.947499\pi$
$252$	0	0
$253$	141.081	0.557634
$254$	219.330i	0.863503i
$255$	0	0
$256$	16.0000	0.0625000
$257$	21.5924i	0.0840170i	0.999117	+	0.0420085i	$0.0133757\pi$
$257$	21.5924i	0.0840170i	−0.999117	+	0.0420085i	$0.986624\pi$
$258$	0	0
$259$	610.680	2.35784
$260$	100.455i	0.386364i
$261$	0	0
$262$	−191.496	−0.730900
$263$	50.1675i	0.190751i	0.995441	+	0.0953755i	$0.0304052\pi$
$263$	50.1675i	0.190751i	−0.995441	+	0.0953755i	$0.969595\pi$
$264$	0	0
$265$	332.542	1.25488
$266$	273.860i	1.02955i
$267$	0	0
$268$	29.6801	0.110747
$269$	119.776i	0.445263i	0.974903	+	0.222632i	$0.0714648\pi$
$269$	119.776i	0.445263i	−0.974903	+	0.222632i	$0.928535\pi$
$270$	0	0
$271$	−146.752	−0.541522	−0.270761	−	0.962647i	$-0.587275\pi$
$271$	−146.752	−0.541522	−0.270761	+	0.962647i	$0.587275\pi$
$272$	− 4.06465i	− 0.0149436i
$273$	0	0
$274$	66.4820	0.242635
$275$	− 295.532i	− 1.07466i
$276$	0	0
$277$	326.132	1.17737	0.588685	−	0.808362i	$-0.299646\pi$
$277$	326.132	1.17737	0.588685	+	0.808362i	$0.299646\pi$
$278$	− 68.5089i	− 0.246435i
$279$	0	0
$280$	216.860	0.774498
$281$	13.6658i	0.0486327i	0.999704	+	0.0243164i	$0.00774090\pi$
$281$	13.6658i	0.0486327i	−0.999704	+	0.0243164i	$0.992259\pi$
$282$	0	0
$283$	455.600	1.60990	0.804948	−	0.593346i	$-0.202193\pi$
$283$	455.600	1.60990	0.804948	+	0.593346i	$0.202193\pi$
$284$	151.613i	0.533847i
$285$	0	0
$286$	55.8960	0.195441
$287$	− 486.305i	− 1.69444i
$288$	0	0
$289$	287.967	0.996427
$290$	572.971i	1.97576i
$291$	0	0
$292$	−106.056	−0.363205
$293$	487.823i	1.66493i	0.554080	+	0.832463i	$0.313070\pi$
$293$	487.823i	1.66493i	−0.554080	+	0.832463i	$0.686930\pi$
$294$	0	0
$295$	−234.458	−0.794773
$296$	188.370i	0.636385i
$297$	0	0
$298$	−118.680	−0.398254
$299$	− 128.799i	− 0.430766i
$300$	0	0
$301$	−435.541	−1.44698
$302$	182.547i	0.604460i
$303$	0	0
$304$	−84.4746	−0.277877
$305$	343.580i	1.12649i
$306$	0	0
$307$	−8.74703	−0.0284920	−0.0142460	−	0.999899i	$-0.504535\pi$
$307$	−8.74703	−0.0284920	−0.0142460	+	0.999899i	$0.504535\pi$
$308$	− 120.667i	− 0.391776i
$309$	0	0
$310$	−78.1287	−0.252028
$311$	− 51.8808i	− 0.166819i	−0.996515	−	0.0834097i	$-0.973419\pi$
$311$	− 51.8808i	− 0.166819i	0.996515	−	0.0834097i	$-0.0265810\pi$
$312$	0	0
$313$	−365.478	−1.16766	−0.583831	−	0.811875i	$-0.698447\pi$
$313$	−365.478	−1.16766	−0.583831	+	0.811875i	$0.698447\pi$
$314$	− 293.634i	− 0.935140i
$315$	0	0
$316$	163.721	0.518103
$317$	− 458.443i	− 1.44619i	−0.690747	−	0.723097i	$-0.742718\pi$
$317$	− 458.443i	− 1.44619i	0.690747	−	0.723097i	$-0.257282\pi$
$318$	0	0
$319$	318.818	0.999430
$320$	66.8923i	0.209038i
$321$	0	0
$322$	−278.049	−0.863506
$323$	21.4600i	0.0664395i
$324$	0	0
$325$	−269.804	−0.830166
$326$	241.877i	0.741954i
$327$	0	0
$328$	150.005	0.457333
$329$	59.8322i	0.181861i
$330$	0	0
$331$	444.349	1.34244	0.671222	−	0.741257i	$-0.265770\pi$
$331$	444.349	1.34244	0.671222	+	0.741257i	$0.265770\pi$
$332$	− 93.8509i	− 0.282683i
$333$	0	0
$334$	−223.030	−0.667755
$335$	124.086i	0.370405i
$336$	0	0
$337$	−89.9513	−0.266918	−0.133459	−	0.991054i	$-0.542608\pi$
$337$	−89.9513	−0.266918	−0.133459	+	0.991054i	$0.542608\pi$
$338$	187.972i	0.556131i
$339$	0	0
$340$	16.9933	0.0499804
$341$	43.4732i	0.127487i
$342$	0	0
$343$	−127.635	−0.372115
$344$	− 134.346i	− 0.390542i
$345$	0	0
$346$	23.6953	0.0684836
$347$	496.609i	1.43115i	0.698535	+	0.715576i	$0.253836\pi$
$347$	496.609i	1.43115i	−0.698535	+	0.715576i	$0.746164\pi$
$348$	0	0
$349$	339.160	0.971806	0.485903	−	0.874013i	$-0.338491\pi$
$349$	339.160	0.971806	0.485903	+	0.874013i	$0.338491\pi$
$350$	582.447i	1.66413i
$351$	0	0
$352$	37.2209	0.105741
$353$	− 220.805i	− 0.625511i	−0.949834	−	0.312756i	$-0.898748\pi$
$353$	− 220.805i	− 0.625511i	0.949834	−	0.312756i	$-0.101252\pi$
$354$	0	0
$355$	−633.857	−1.78551
$356$	46.8799i	0.131685i
$357$	0	0
$358$	−153.920	−0.429943
$359$	− 570.658i	− 1.58958i	−0.606887	−	0.794788i	$-0.707582\pi$
$359$	− 570.658i	− 1.58958i	0.606887	−	0.794788i	$-0.292418\pi$
$360$	0	0
$361$	84.9973	0.235450
$362$	− 52.5477i	− 0.145159i
$363$	0	0
$364$	−110.162	−0.302643
$365$	− 443.395i	− 1.21478i
$366$	0	0
$367$	−211.449	−0.576155	−0.288078	−	0.957607i	$-0.593016\pi$
$367$	−211.449	−0.576155	−0.288078	+	0.957607i	$0.593016\pi$
$368$	− 85.7667i	− 0.233062i
$369$	0	0
$370$	−787.530	−2.12846
$371$	364.677i	0.982957i
$372$	0	0
$373$	−69.4639	−0.186230	−0.0931151	−	0.995655i	$-0.529682\pi$
$373$	−69.4639	−0.186230	−0.0931151	+	0.995655i	$0.529682\pi$
$374$	− 9.45560i	− 0.0252824i
$375$	0	0
$376$	−18.4558	−0.0490845
$377$	− 291.063i	− 0.772049i
$378$	0	0
$379$	390.211	1.02958	0.514791	−	0.857316i	$-0.327870\pi$
$379$	390.211	1.02958	0.514791	+	0.857316i	$0.327870\pi$
$380$	− 353.169i	− 0.929391i
$381$	0	0
$382$	303.676	0.794962
$383$	− 181.340i	− 0.473473i	−0.971574	−	0.236736i	$-0.923922\pi$
$383$	− 181.340i	− 0.473473i	0.971574	−	0.236736i	$-0.0760777\pi$
$384$	0	0
$385$	504.481	1.31034
$386$	− 401.382i	− 1.03985i
$387$	0	0
$388$	−70.0038	−0.180422
$389$	672.929i	1.72989i	0.501864	+	0.864947i	$0.332648\pi$
$389$	672.929i	1.72989i	−0.501864	+	0.864947i	$0.667352\pi$
$390$	0	0
$391$	−21.7882	−0.0557243
$392$	99.2226i	0.253119i
$393$	0	0
$394$	251.812	0.639118
$395$	684.478i	1.73285i
$396$	0	0
$397$	−453.644	−1.14268	−0.571340	−	0.820713i	$-0.693576\pi$
$397$	−453.644	−1.14268	−0.571340	+	0.820713i	$0.693576\pi$
$398$	− 206.033i	− 0.517670i
$399$	0	0
$400$	−179.661	−0.449152
$401$	− 313.979i	− 0.782989i	−0.920180	−	0.391495i	$-0.871958\pi$
$401$	− 313.979i	− 0.782989i	0.920180	−	0.391495i	$-0.128042\pi$
$402$	0	0
$403$	39.6885	0.0984825
$404$	− 242.536i	− 0.600337i
$405$	0	0
$406$	−628.340	−1.54763
$407$	438.205i	1.07667i
$408$	0	0
$409$	1.70186	0.00416104	0.00208052	−	0.999998i	$-0.499338\pi$
$409$	1.70186	0.00416104	0.00208052	+	0.999998i	$0.499338\pi$
$410$	627.137i	1.52960i
$411$	0	0
$412$	377.854	0.917121
$413$	− 257.115i	− 0.622553i
$414$	0	0
$415$	392.369	0.945467
$416$	− 33.9805i	− 0.0816838i
$417$	0	0
$418$	−196.514	−0.470128
$419$	100.066i	0.238822i	0.992845	+	0.119411i	$0.0381006\pi$
$419$	100.066i	0.238822i	−0.992845	+	0.119411i	$0.961899\pi$
$420$	0	0
$421$	635.229	1.50886	0.754429	−	0.656381i	$-0.227914\pi$
$421$	635.229	1.50886	0.754429	+	0.656381i	$0.227914\pi$
$422$	420.396i	0.996199i
$423$	0	0
$424$	−112.488	−0.265302
$425$	45.6411i	0.107391i
$426$	0	0
$427$	−376.782	−0.882393
$428$	152.953i	0.357367i
$429$	0	0
$430$	561.671	1.30621
$431$	− 180.242i	− 0.418195i	−0.977895	−	0.209098i	$-0.932947\pi$
$431$	− 180.242i	− 0.418195i	0.977895	−	0.209098i	$-0.0670526\pi$
$432$	0	0
$433$	161.992	0.374115	0.187058	−	0.982349i	$-0.440105\pi$
$433$	161.992	0.374115	0.187058	+	0.982349i	$0.440105\pi$
$434$	− 85.6786i	− 0.197416i
$435$	0	0
$436$	−87.5616	−0.200829
$437$	452.819i	1.03620i
$438$	0	0
$439$	−321.743	−0.732899	−0.366450	−	0.930438i	$-0.619427\pi$
$439$	−321.743	−0.732899	−0.366450	+	0.930438i	$0.619427\pi$
$440$	155.612i	0.353663i
$441$	0	0
$442$	−8.63241	−0.0195303
$443$	− 505.673i	− 1.14147i	−0.821133	−	0.570737i	$-0.806657\pi$
$443$	− 505.673i	− 1.14147i	0.821133	−	0.570737i	$-0.193343\pi$
$444$	0	0
$445$	−195.994	−0.440436
$446$	298.811i	0.669979i
$447$	0	0
$448$	−73.3563	−0.163742
$449$	− 717.640i	− 1.59831i	−0.601127	−	0.799153i	$-0.705281\pi$
$449$	− 717.640i	− 1.59831i	0.601127	−	0.799153i	$-0.294719\pi$
$450$	0	0
$451$	348.958	0.773742
$452$	0.415240i	0 0.000918672i
$453$	0	0
$454$	−22.7559	−0.0501231
$455$	− 460.562i	− 1.01222i
$456$	0	0
$457$	417.465	0.913490	0.456745	−	0.889598i	$-0.349015\pi$
$457$	417.465	0.913490	0.456745	+	0.889598i	$0.349015\pi$
$458$	− 369.625i	− 0.807041i
$459$	0	0
$460$	358.570	0.779501
$461$	− 819.949i	− 1.77863i	−0.457293	−	0.889316i	$-0.651181\pi$
$461$	− 819.949i	− 1.77863i	0.457293	−	0.889316i	$-0.348819\pi$
$462$	0	0
$463$	851.724	1.83958	0.919789	−	0.392414i	$-0.128360\pi$
$463$	851.724	1.83958	0.919789	+	0.392414i	$0.128360\pi$
$464$	− 193.817i	− 0.417709i
$465$	0	0
$466$	454.974	0.976340
$467$	643.170i	1.37724i	0.725124	+	0.688619i	$0.241783\pi$
$467$	643.170i	1.37724i	−0.725124	+	0.688619i	$0.758217\pi$
$468$	0	0
$469$	−136.077	−0.290142
$470$	− 77.1593i	− 0.164169i
$471$	0	0
$472$	79.3093	0.168028
$473$	− 312.531i	− 0.660741i
$474$	0	0
$475$	948.549	1.99695
$476$	18.6355i	0.0391501i
$477$	0	0
$478$	111.552	0.233371
$479$	302.308i	0.631124i	0.948905	+	0.315562i	$0.102193\pi$
$479$	302.308i	0.631124i	−0.948905	+	0.315562i	$0.897807\pi$
$480$	0	0
$481$	400.056	0.831717
$482$	− 428.447i	− 0.888894i
$483$	0	0
$484$	−155.413	−0.321101
$485$	− 292.669i	− 0.603442i
$486$	0	0
$487$	584.450	1.20010	0.600052	−	0.799961i	$-0.295147\pi$
$487$	584.450	1.20010	0.600052	+	0.799961i	$0.295147\pi$
$488$	− 116.222i	− 0.238159i
$489$	0	0
$490$	−414.827	−0.846585
$491$	− 275.382i	− 0.560858i	−0.959875	−	0.280429i	$-0.909523\pi$
$491$	− 275.382i	− 0.560858i	0.959875	−	0.280429i	$-0.0904768\pi$
$492$	0	0
$493$	−49.2373	−0.0998729
$494$	179.405i	0.363169i
$495$	0	0
$496$	26.4283	0.0532829
$497$	− 695.109i	− 1.39861i
$498$	0	0
$499$	−170.516	−0.341716	−0.170858	−	0.985296i	$-0.554654\pi$
$499$	−170.516	−0.341716	−0.170858	+	0.985296i	$0.554654\pi$
$500$	− 333.044i	− 0.666088i
$501$	0	0
$502$	116.565	0.232201
$503$	− 15.2624i	− 0.0303428i	−0.999885	−	0.0151714i	$-0.995171\pi$
$503$	− 15.2624i	− 0.0303428i	0.999885	−	0.0151714i	$-0.00482939\pi$
$504$	0	0
$505$	1013.99	2.00790
$506$	− 199.519i	− 0.394307i
$507$	0	0
$508$	310.179	0.610589
$509$	− 517.546i	− 1.01679i	−0.861124	−	0.508395i	$-0.830239\pi$
$509$	− 517.546i	− 1.01679i	0.861124	−	0.508395i	$-0.169761\pi$
$510$	0	0
$511$	486.242	0.951550
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	30.5362	0.0594090
$515$	1579.72i	3.06741i
$516$	0	0
$517$	−42.9337	−0.0830439
$518$	− 863.633i	− 1.66724i
$519$	0	0
$520$	142.064	0.273201
$521$	− 974.047i	− 1.86957i	−0.355211	−	0.934786i	$-0.615591\pi$
$521$	− 974.047i	− 1.86957i	0.355211	−	0.934786i	$-0.384409\pi$
$522$	0	0
$523$	−319.090	−0.610115	−0.305058	−	0.952334i	$-0.598676\pi$
$523$	−319.090	−0.610115	−0.305058	+	0.952334i	$0.598676\pi$
$524$	270.816i	0.516824i
$525$	0	0
$526$	70.9476	0.134881
$527$	− 6.71387i	− 0.0127398i
$528$	0	0
$529$	69.2548	0.130916
$530$	− 470.286i	− 0.887331i
$531$	0	0
$532$	387.297	0.728001
$533$	− 318.578i	− 0.597707i
$534$	0	0
$535$	−639.460	−1.19525
$536$	− 41.9740i	− 0.0783097i
$537$	0	0
$538$	169.389	0.314849
$539$	230.822i	0.428241i
$540$	0	0
$541$	−604.682	−1.11771	−0.558856	−	0.829265i	$-0.688759\pi$
$541$	−604.682	−1.11771	−0.558856	+	0.829265i	$0.688759\pi$
$542$	207.539i	0.382914i
$543$	0	0
$544$	−5.74828	−0.0105667
$545$	− 366.075i	− 0.671697i
$546$	0	0
$547$	839.957	1.53557	0.767785	−	0.640707i	$-0.221359\pi$
$547$	839.957	1.53557	0.767785	+	0.640707i	$0.221359\pi$
$548$	− 94.0197i	− 0.171569i
$549$	0	0
$550$	−417.946	−0.759902
$551$	1023.29i	1.85715i
$552$	0	0
$553$	−750.621	−1.35736
$554$	− 461.220i	− 0.832527i
$555$	0	0
$556$	−96.8862	−0.174256
$557$	− 907.425i	− 1.62913i	−0.580072	−	0.814565i	$-0.696976\pi$
$557$	− 907.425i	− 1.62913i	0.580072	−	0.814565i	$-0.303024\pi$
$558$	0	0
$559$	−285.322	−0.510415
$560$	− 306.686i	− 0.547653i
$561$	0	0
$562$	19.3263	0.0343885
$563$	68.6119i	0.121868i	0.998142	+	0.0609342i	$0.0194080\pi$
$563$	68.6119i	0.121868i	−0.998142	+	0.0609342i	$0.980592\pi$
$564$	0	0
$565$	−1.73602	−0.00307260
$566$	− 644.316i	− 1.13837i
$567$	0	0
$568$	214.413	0.377487
$569$	− 466.167i	− 0.819274i	−0.912249	−	0.409637i	$-0.865655\pi$
$569$	− 466.167i	− 0.819274i	0.912249	−	0.409637i	$-0.134345\pi$
$570$	0	0
$571$	527.916	0.924546	0.462273	−	0.886738i	$-0.347034\pi$
$571$	527.916	0.924546	0.462273	+	0.886738i	$0.347034\pi$
$572$	− 79.0489i	− 0.138197i
$573$	0	0
$574$	−687.740	−1.19815
$575$	963.058i	1.67488i
$576$	0	0
$577$	−343.674	−0.595622	−0.297811	−	0.954625i	$-0.596257\pi$
$577$	−343.674	−0.595622	−0.297811	+	0.954625i	$0.596257\pi$
$578$	− 407.247i	− 0.704580i
$579$	0	0
$580$	810.304	1.39708
$581$	430.285i	0.740594i
$582$	0	0
$583$	−261.681	−0.448852
$584$	149.986i	0.256825i
$585$	0	0
$586$	689.887	1.17728
$587$	164.396i	0.280061i	0.990147	+	0.140031i	$0.0447201\pi$
$587$	164.396i	0.280061i	−0.990147	+	0.140031i	$0.955280\pi$
$588$	0	0
$589$	−139.533	−0.236898
$590$	331.574i	0.561989i
$591$	0	0
$592$	266.395	0.449992
$593$	− 349.285i	− 0.589014i	−0.955649	−	0.294507i	$-0.904845\pi$
$593$	− 349.285i	− 0.589014i	0.955649	−	0.294507i	$-0.0951555\pi$
$594$	0	0
$595$	−77.9106	−0.130942
$596$	167.839i	0.281608i
$597$	0	0
$598$	−182.149	−0.304598
$599$	− 602.848i	− 1.00642i	−0.864163	−	0.503212i	$-0.832152\pi$
$599$	− 602.848i	− 1.00642i	0.864163	−	0.503212i	$-0.167848\pi$
$600$	0	0
$601$	−40.0570	−0.0666505	−0.0333253	−	0.999445i	$-0.510610\pi$
$601$	−40.0570	−0.0666505	−0.0333253	+	0.999445i	$0.510610\pi$
$602$	615.948i	1.02317i
$603$	0	0
$604$	258.160	0.427417
$605$	− 649.746i	− 1.07396i
$606$	0	0
$607$	1001.73	1.65030	0.825150	−	0.564914i	$-0.191091\pi$
$607$	1001.73	1.65030	0.825150	+	0.564914i	$0.191091\pi$
$608$	119.465i	0.196489i
$609$	0	0
$610$	485.896	0.796550
$611$	39.1960i	0.0641505i
$612$	0	0
$613$	90.5651	0.147741	0.0738704	−	0.997268i	$-0.476465\pi$
$613$	90.5651	0.147741	0.0738704	+	0.997268i	$0.476465\pi$
$614$	12.3702i	0.0201469i
$615$	0	0
$616$	−170.649	−0.277028
$617$	− 986.293i	− 1.59853i	−0.600979	−	0.799265i	$-0.705222\pi$
$617$	− 986.293i	− 1.59853i	0.600979	−	0.799265i	$-0.294778\pi$
$618$	0	0
$619$	974.221	1.57386	0.786931	−	0.617041i	$-0.211669\pi$
$619$	974.221	1.57386	0.786931	+	0.617041i	$0.211669\pi$
$620$	110.491i	0.178211i
$621$	0	0
$622$	−73.3706	−0.117959
$623$	− 214.934i	− 0.344998i
$624$	0	0
$625$	269.498	0.431197
$626$	516.864i	0.825661i
$627$	0	0
$628$	−415.261	−0.661244
$629$	− 67.6751i	− 0.107592i
$630$	0	0
$631$	−10.0349	−0.0159032	−0.00795160	−	0.999968i	$-0.502531\pi$
$631$	−10.0349	−0.0159032	−0.00795160	+	0.999968i	$0.502531\pi$
$632$	− 231.536i	− 0.366354i
$633$	0	0
$634$	−648.337	−1.02261
$635$	1296.79i	2.04218i
$636$	0	0
$637$	210.727	0.330812
$638$	− 450.877i	− 0.706704i
$639$	0	0
$640$	94.6000	0.147812
$641$	236.054i	0.368259i	0.982902	+	0.184130i	$0.0589466\pi$
$641$	236.054i	0.368259i	−0.982902	+	0.184130i	$0.941053\pi$
$642$	0	0
$643$	−28.7071	−0.0446456	−0.0223228	−	0.999751i	$-0.507106\pi$
$643$	−28.7071	−0.0446456	−0.0223228	+	0.999751i	$0.507106\pi$
$644$	393.221i	0.610591i
$645$	0	0
$646$	30.3490	0.0469798
$647$	− 230.995i	− 0.357024i	−0.983938	−	0.178512i	$-0.942872\pi$
$647$	− 230.995i	− 0.357024i	0.983938	−	0.178512i	$-0.0571284\pi$
$648$	0	0
$649$	184.497	0.284280
$650$	381.560i	0.587016i
$651$	0	0
$652$	342.066	0.524641
$653$	1095.46i	1.67759i	0.544451	+	0.838793i	$0.316738\pi$
$653$	1095.46i	1.67759i	−0.544451	+	0.838793i	$0.683262\pi$
$654$	0	0
$655$	−1132.22	−1.72858
$656$	− 212.139i	− 0.323383i
$657$	0	0
$658$	84.6155	0.128595
$659$	− 360.182i	− 0.546558i	−0.961935	−	0.273279i	$-0.911892\pi$
$659$	− 360.182i	− 0.546558i	0.961935	−	0.273279i	$-0.0881083\pi$
$660$	0	0
$661$	−156.404	−0.236617	−0.118308	−	0.992977i	$-0.537747\pi$
$661$	−156.404	−0.236617	−0.118308	+	0.992977i	$0.537747\pi$
$662$	− 628.404i	− 0.949251i
$663$	0	0
$664$	−132.725	−0.199887
$665$	1619.20i	2.43488i
$666$	0	0
$667$	−1038.94	−1.55763
$668$	315.412i	0.472174i
$669$	0	0
$670$	175.484	0.261916
$671$	− 270.367i	− 0.402931i
$672$	0	0
$673$	−758.880	−1.12761	−0.563804	−	0.825909i	$-0.690663\pi$
$673$	−758.880	−1.12761	−0.563804	+	0.825909i	$0.690663\pi$
$674$	127.210i	0.188739i
$675$	0	0
$676$	265.833	0.393244
$677$	1060.04i	1.56579i	0.622153	+	0.782896i	$0.286258\pi$
$677$	1060.04i	1.56579i	−0.622153	+	0.782896i	$0.713742\pi$
$678$	0	0
$679$	320.951	0.472682
$680$	− 24.0322i	− 0.0353415i
$681$	0	0
$682$	61.4803	0.0901471
$683$	− 1071.02i	− 1.56811i	−0.620694	−	0.784053i	$-0.713149\pi$
$683$	− 1071.02i	− 1.56811i	0.620694	−	0.784053i	$-0.286851\pi$
$684$	0	0
$685$	393.074	0.573831
$686$	180.504i	0.263125i
$687$	0	0
$688$	−189.995	−0.276155
$689$	238.899i	0.346733i
$690$	0	0
$691$	442.462	0.640321	0.320161	−	0.947363i	$-0.396263\pi$
$691$	442.462	0.640321	0.320161	+	0.947363i	$0.396263\pi$
$692$	− 33.5103i	− 0.0484252i
$693$	0	0
$694$	702.312	1.01198
$695$	− 405.059i	− 0.582818i
$696$	0	0
$697$	−53.8920	−0.0773199
$698$	− 479.645i	− 0.687170i
$699$	0	0
$700$	823.704	1.17672
$701$	77.6184i	0.110725i	0.998466	+	0.0553626i	$0.0176315\pi$
$701$	77.6184i	0.110725i	−0.998466	+	0.0553626i	$0.982369\pi$
$702$	0	0
$703$	−1406.48	−2.00068
$704$	− 52.6382i	− 0.0747702i
$705$	0	0
$706$	−312.266	−0.442303
$707$	1111.97i	1.57281i
$708$	0	0
$709$	148.778	0.209842	0.104921	−	0.994481i	$-0.466541\pi$
$709$	148.778	0.209842	0.104921	+	0.994481i	$0.466541\pi$
$710$	896.409i	1.26255i
$711$	0	0
$712$	66.2982	0.0931155
$713$	− 141.667i	− 0.198691i
$714$	0	0
$715$	330.485	0.462217
$716$	217.675i	0.304016i
$717$	0	0
$718$	−807.032	−1.12400
$719$	− 26.1111i	− 0.0363159i	−0.999835	−	0.0181580i	$-0.994220\pi$
$719$	− 26.1111i	− 0.0363159i	0.999835	−	0.0181580i	$-0.00578018\pi$
$720$	0	0
$721$	−1732.37	−2.40274
$722$	− 120.204i	− 0.166488i
$723$	0	0
$724$	−74.3136	−0.102643
$725$	2176.33i	3.00184i
$726$	0	0
$727$	−919.807	−1.26521	−0.632605	−	0.774475i	$-0.718014\pi$
$727$	−919.807	−1.26521	−0.632605	+	0.774475i	$0.718014\pi$
$728$	155.793i	0.214001i
$729$	0	0
$730$	−627.056	−0.858980
$731$	48.2663i	0.0660277i
$732$	0	0
$733$	−354.173	−0.483183	−0.241592	−	0.970378i	$-0.577669\pi$
$733$	−354.173	−0.483183	−0.241592	+	0.970378i	$0.577669\pi$
$734$	299.034i	0.407403i
$735$	0	0
$736$	−121.292	−0.164799
$737$	− 97.6443i	− 0.132489i
$738$	0	0
$739$	−820.692	−1.11054	−0.555272	−	0.831669i	$-0.687386\pi$
$739$	−820.692	−1.11054	−0.555272	+	0.831669i	$0.687386\pi$
$740$	1113.74i	1.50505i
$741$	0	0
$742$	515.731	0.695055
$743$	134.956i	0.181636i	0.995867	+	0.0908182i	$0.0289482\pi$
$743$	134.956i	0.181636i	−0.995867	+	0.0908182i	$0.971052\pi$
$744$	0	0
$745$	−701.694	−0.941871
$746$	98.2368i	0.131685i
$747$	0	0
$748$	−13.3722	−0.0178773
$749$	− 701.254i	− 0.936254i
$750$	0	0
$751$	1224.90	1.63102	0.815511	−	0.578741i	$-0.196456\pi$
$751$	1224.90	1.63102	0.815511	+	0.578741i	$0.196456\pi$
$752$	26.1004i	0.0347080i
$753$	0	0
$754$	−411.625	−0.545921
$755$	1079.31i	1.42955i
$756$	0	0
$757$	915.003	1.20872	0.604361	−	0.796710i	$-0.293428\pi$
$757$	915.003	1.20872	0.604361	+	0.796710i	$0.293428\pi$
$758$	− 551.842i	− 0.728024i
$759$	0	0
$760$	−499.456	−0.657179
$761$	− 749.875i	− 0.985381i	−0.870205	−	0.492690i	$-0.836013\pi$
$761$	− 749.875i	− 0.985381i	0.870205	−	0.492690i	$-0.163987\pi$
$762$	0	0
$763$	401.450	0.526147
$764$	− 429.462i	− 0.562123i
$765$	0	0
$766$	−256.454	−0.334796
$767$	− 168.435i	− 0.219603i
$768$	0	0
$769$	−919.543	−1.19576	−0.597882	−	0.801584i	$-0.703991\pi$
$769$	−919.543	−1.19576	−0.597882	+	0.801584i	$0.703991\pi$
$770$	− 713.444i	− 0.926551i
$771$	0	0
$772$	−567.639	−0.735284
$773$	− 278.063i	− 0.359720i	−0.983692	−	0.179860i	$-0.942436\pi$
$773$	− 278.063i	− 0.359720i	0.983692	−	0.179860i	$-0.0575644\pi$
$774$	0	0
$775$	−296.759	−0.382915
$776$	99.0003i	0.127578i
$777$	0	0
$778$	951.665	1.22322
$779$	1120.02i	1.43777i
$780$	0	0
$781$	498.789	0.638654
$782$	30.8132i	0.0394030i
$783$	0	0
$784$	140.322	0.178982
$785$	− 1736.11i	− 2.21161i
$786$	0	0
$787$	819.583	1.04140	0.520701	−	0.853739i	$-0.325671\pi$
$787$	819.583	1.04140	0.520701	+	0.853739i	$0.325671\pi$
$788$	− 356.117i	− 0.451925i
$789$	0	0
$790$	967.997	1.22531
$791$	− 1.90378i	− 0.00240680i
$792$	0	0
$793$	−246.829	−0.311260
$794$	641.550i	0.807997i
$795$	0	0
$796$	−291.374	−0.366048
$797$	1128.82i	1.41634i	0.706043	+	0.708169i	$0.250478\pi$
$797$	1128.82i	1.41634i	−0.706043	+	0.708169i	$0.749522\pi$
$798$	0	0
$799$	6.63055	0.00829857
$800$	254.079i	0.317599i
$801$	0	0
$802$	−444.033	−0.553657
$803$	348.912i	0.434511i
$804$	0	0
$805$	−1643.96	−2.04219
$806$	− 56.1280i	− 0.0696377i
$807$	0	0
$808$	−342.998	−0.424503
$809$	1010.80i	1.24945i	0.780846	+	0.624724i	$0.214788\pi$
$809$	1010.80i	1.24945i	−0.780846	+	0.624724i	$0.785212\pi$
$810$	0	0
$811$	481.599	0.593834	0.296917	−	0.954903i	$-0.404042\pi$
$811$	481.599	0.593834	0.296917	+	0.954903i	$0.404042\pi$
$812$	888.607i	1.09434i
$813$	0	0
$814$	619.716	0.761322
$815$	1430.10i	1.75472i
$816$	0	0
$817$	1003.11	1.22779
$818$	− 2.40680i	− 0.00294230i
$819$	0	0
$820$	886.906	1.08159
$821$	− 1066.52i	− 1.29905i	−0.760340	−	0.649525i	$-0.774968\pi$
$821$	− 1066.52i	− 1.29905i	0.760340	−	0.649525i	$-0.225032\pi$
$822$	0	0
$823$	−81.3728	−0.0988734	−0.0494367	−	0.998777i	$-0.515743\pi$
$823$	−81.3728	−0.0988734	−0.0494367	+	0.998777i	$0.515743\pi$
$824$	− 534.366i	− 0.648502i
$825$	0	0
$826$	−363.615	−0.440212
$827$	107.815i	0.130369i	0.997873	+	0.0651845i	$0.0207636\pi$
$827$	107.815i	0.130369i	−0.997873	+	0.0651845i	$0.979236\pi$
$828$	0	0
$829$	1315.48	1.58683	0.793415	−	0.608681i	$-0.208301\pi$
$829$	1315.48	1.58683	0.793415	+	0.608681i	$0.208301\pi$
$830$	− 554.893i	− 0.668546i
$831$	0	0
$832$	−48.0556	−0.0577592
$833$	− 35.6475i	− 0.0427941i
$834$	0	0
$835$	−1318.66	−1.57924
$836$	277.912i	0.332431i
$837$	0	0
$838$	141.515	0.168873
$839$	− 1312.55i	− 1.56442i	−0.623015	−	0.782210i	$-0.714093\pi$
$839$	− 1312.55i	− 1.56442i	0.623015	−	0.782210i	$-0.285907\pi$
$840$	0	0
$841$	−1506.81	−1.79169
$842$	− 898.350i	− 1.06692i
$843$	0	0
$844$	594.530	0.704419
$845$	1111.39i	1.31525i
$846$	0	0
$847$	712.533	0.841243
$848$	159.082i	0.187597i
$849$	0	0
$850$	64.5463	0.0759369
$851$	− 1427.99i	− 1.67801i
$852$	0	0
$853$	1104.43	1.29476	0.647381	−	0.762167i	$-0.275864\pi$
$853$	1104.43	1.29476	0.647381	+	0.762167i	$0.275864\pi$
$854$	532.850i	0.623946i
$855$	0	0
$856$	216.308	0.252696
$857$	1599.58i	1.86649i	0.359245	+	0.933243i	$0.383034\pi$
$857$	1599.58i	1.86649i	−0.359245	+	0.933243i	$0.616966\pi$
$858$	0	0
$859$	1113.15	1.29586	0.647931	−	0.761699i	$-0.275635\pi$
$859$	1113.15	1.29586	0.647931	+	0.761699i	$0.275635\pi$
$860$	− 794.323i	− 0.923631i
$861$	0	0
$862$	−254.901	−0.295709
$863$	172.993i	0.200455i	0.994965	+	0.100227i	$0.0319570\pi$
$863$	172.993i	0.200455i	−0.994965	+	0.100227i	$0.968043\pi$
$864$	0	0
$865$	140.099	0.161964
$866$	− 229.091i	− 0.264539i
$867$	0	0
$868$	−121.168	−0.139594
$869$	− 538.623i	− 0.619819i
$870$	0	0
$871$	−89.1436	−0.102346
$872$	123.831i	0.142008i
$873$	0	0
$874$	640.383	0.732704
$875$	1526.93i	1.74506i
$876$	0	0
$877$	1070.17	1.22027	0.610134	−	0.792299i	$-0.291116\pi$
$877$	1070.17	1.22027	0.610134	+	0.792299i	$0.291116\pi$
$878$	455.013i	0.518238i
$879$	0	0
$880$	220.068	0.250077
$881$	401.070i	0.455244i	0.973750	+	0.227622i	$0.0730950\pi$
$881$	401.070i	0.455244i	−0.973750	+	0.227622i	$0.926905\pi$
$882$	0	0
$883$	331.125	0.375000	0.187500	−	0.982265i	$-0.439961\pi$
$883$	331.125	0.375000	0.187500	+	0.982265i	$0.439961\pi$
$884$	12.2081i	0.0138100i
$885$	0	0
$886$	−715.130	−0.807145
$887$	237.408i	0.267653i	0.991005	+	0.133826i	$0.0427265\pi$
$887$	237.408i	0.267653i	−0.991005	+	0.133826i	$0.957274\pi$
$888$	0	0
$889$	−1422.10	−1.59966
$890$	277.177i	0.311435i
$891$	0	0
$892$	422.582	0.473747
$893$	− 137.801i	− 0.154313i
$894$	0	0
$895$	−910.049	−1.01681
$896$	103.742i	0.115783i
$897$	0	0
$898$	−1014.90	−1.13017
$899$	− 320.141i	− 0.356108i
$900$	0	0
$901$	40.4132	0.0448537
$902$	− 493.501i	− 0.547118i
$903$	0	0
$904$	0.587238	0.000649599 0
$905$	− 310.688i	− 0.343302i
$906$	0	0
$907$	−1058.30	−1.16681	−0.583406	−	0.812181i	$-0.698280\pi$
$907$	−1058.30	−1.16681	−0.583406	+	0.812181i	$0.698280\pi$
$908$	32.1817i	0.0354424i
$909$	0	0
$910$	−651.333	−0.715750
$911$	1803.13i	1.97928i	0.143562	+	0.989641i	$0.454144\pi$
$911$	1803.13i	1.97928i	−0.143562	+	0.989641i	$0.545856\pi$
$912$	0	0
$913$	−308.759	−0.338181
$914$	− 590.384i	− 0.645935i
$915$	0	0
$916$	−522.729	−0.570664
$917$	− 1241.63i	− 1.35401i
$918$	0	0
$919$	185.717	0.202086	0.101043	−	0.994882i	$-0.467782\pi$
$919$	185.717	0.202086	0.101043	+	0.994882i	$0.467782\pi$
$920$	− 507.095i	− 0.551190i
$921$	0	0
$922$	−1159.58	−1.25768
$923$	− 455.365i	− 0.493353i
$924$	0	0
$925$	−2991.30	−3.23384
$926$	− 1204.52i	− 1.30078i
$927$	0	0
$928$	−274.099	−0.295365
$929$	− 270.218i	− 0.290870i	−0.989368	−	0.145435i	$-0.953542\pi$
$929$	− 270.218i	− 0.290870i	0.989368	−	0.145435i	$-0.0464582\pi$
$930$	0	0
$931$	−740.853	−0.795760
$932$	− 643.431i	− 0.690377i
$933$	0	0
$934$	909.580	0.973854
$935$	− 55.9062i	− 0.0597927i
$936$	0	0
$937$	−193.983	−0.207025	−0.103513	−	0.994628i	$-0.533008\pi$
$937$	−193.983	−0.207025	−0.103513	+	0.994628i	$0.533008\pi$
$938$	192.441i	0.205161i
$939$	0	0
$940$	−109.120	−0.116085
$941$	54.6376i	0.0580633i	0.999578	+	0.0290317i	$0.00924236\pi$
$941$	54.6376i	0.0580633i	−0.999578	+	0.0290317i	$0.990758\pi$
$942$	0	0
$943$	−1137.16	−1.20589
$944$	− 112.160i	− 0.118814i
$945$	0	0
$946$	−441.985	−0.467214
$947$	52.6925i	0.0556415i	0.999613	+	0.0278207i	$0.00885676\pi$
$947$	52.6925i	0.0556415i	−0.999613	+	0.0278207i	$0.991143\pi$
$948$	0	0
$949$	318.537	0.335655
$950$	− 1341.45i	− 1.41205i
$951$	0	0
$952$	26.3545	0.0276833
$953$	290.679i	0.305015i	0.988302	+	0.152508i	$0.0487348\pi$
$953$	290.679i	0.305015i	−0.988302	+	0.152508i	$0.951265\pi$
$954$	0	0
$955$	1795.48	1.88008
$956$	− 157.758i	− 0.165019i
$957$	0	0
$958$	427.528	0.446272
$959$	431.059i	0.449488i
$960$	0	0
$961$	−917.346	−0.954575
$962$	− 565.765i	− 0.588113i
$963$	0	0
$964$	−605.916	−0.628543
$965$	− 2373.17i	− 2.45924i
$966$	0	0
$967$	443.361	0.458492	0.229246	−	0.973369i	$-0.426374\pi$
$967$	443.361	0.458492	0.229246	+	0.973369i	$0.426374\pi$
$968$	219.787i	0.227053i
$969$	0	0
$970$	−413.897	−0.426698
$971$	− 437.137i	− 0.450193i	−0.974337	−	0.225096i	$-0.927730\pi$
$971$	− 437.137i	− 0.450193i	0.974337	−	0.225096i	$-0.0722697\pi$
$972$	0	0
$973$	444.201	0.456527
$974$	− 826.538i	− 0.848601i
$975$	0	0
$976$	−164.362	−0.168404
$977$	38.9780i	0.0398956i	0.999801	+	0.0199478i	$0.00635000\pi$
$977$	38.9780i	0.0398956i	−0.999801	+	0.0199478i	$0.993650\pi$
$978$	0	0
$979$	154.230	0.157538
$980$	586.654i	0.598626i
$981$	0	0
$982$	−389.448	−0.396587
$983$	1776.23i	1.80695i	0.428643	+	0.903474i	$0.358992\pi$
$983$	1776.23i	1.80695i	−0.428643	+	0.903474i	$0.641008\pi$
$984$	0	0
$985$	1488.84	1.51151
$986$	69.6321i	0.0706208i
$987$	0	0
$988$	253.718	0.256799
$989$	1018.45i	1.02978i
$990$	0	0
$991$	1388.38	1.40098	0.700492	−	0.713660i	$-0.252964\pi$
$991$	1388.38	1.40098	0.700492	+	0.713660i	$0.252964\pi$
$992$	− 37.3753i	− 0.0376767i
$993$	0	0
$994$	−983.032	−0.988966
$995$	− 1218.17i	− 1.22429i
$996$	0	0
$997$	−242.014	−0.242742	−0.121371	−	0.992607i	$-0.538729\pi$
$997$	−242.014	−0.242742	−0.121371	+	0.992607i	$0.538729\pi$
$998$	241.147i	0.241630i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.2		36
3.2	odd	2	inner	1458.3.b.c.1457.35		36
27.5	odd	18		162.3.f.a.89.6		36
27.11	odd	18		54.3.f.a.41.2	yes	36
27.16	even	9		162.3.f.a.71.6		36
27.22	even	9		54.3.f.a.29.2	✓	36
108.11	even	18		432.3.bc.c.257.5		36
108.103	odd	18		432.3.bc.c.353.5		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.29.2	✓	36	27.22	even	9
54.3.f.a.41.2	yes	36	27.11	odd	18
162.3.f.a.71.6		36	27.16	even	9
162.3.f.a.89.6		36	27.5	odd	18
432.3.bc.c.257.5		36	108.11	even	18
432.3.bc.c.353.5		36	108.103	odd	18
1458.3.b.c.1457.2		36	1.1	even	1	trivial
1458.3.b.c.1457.35		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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -8.36153i q^{5} +9.16954 q^{7} +2.82843i q^{8} -11.8250 q^{10} +6.57978i q^{11} +6.00695 q^{13} -12.9677i q^{14} +4.00000 q^{16} -1.01616i q^{17} -21.1186 q^{19} +16.7231i q^{20} +9.30521 q^{22} -21.4417i q^{23} -44.9152 q^{25} -8.49512i q^{26} -18.3391 q^{28} -48.4543i q^{29} +6.60708 q^{31} -5.65685i q^{32} -1.43707 q^{34} -76.6714i q^{35} +66.5988 q^{37} +29.8663i q^{38} +23.6500 q^{40} -53.0349i q^{41} -47.4986 q^{43} -13.1596i q^{44} -30.3231 q^{46} +6.52510i q^{47} +35.0805 q^{49} +63.5197i q^{50} -12.0139 q^{52} +39.7705i q^{53} +55.0170 q^{55} +25.9354i q^{56} -68.5247 q^{58} -28.0401i q^{59} -41.0906 q^{61} -9.34383i q^{62} -8.00000 q^{64} -50.2274i q^{65} -14.8401 q^{67} +2.03232i q^{68} -108.430 q^{70} -75.8063i q^{71} +53.0280 q^{73} -94.1849i q^{74} +42.2373 q^{76} +60.3336i q^{77} -81.8603 q^{79} -33.4461i q^{80} -75.0026 q^{82} +46.9255i q^{83} -8.49667 q^{85} +67.1732i q^{86} -18.6104 q^{88} -23.4400i q^{89} +55.0810 q^{91} +42.8833i q^{92} +9.22788 q^{94} +176.584i q^{95} +35.0019 q^{97} -49.6113i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 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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