LMFDB - Embedded newform 432.8.i.e.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,8,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("432.145");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	432.i (of order $3$, degree $2$, not 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:	$134.950331009$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 332929 x^{18} - 2996076 x^{17} + 44578211685 x^{16} - 356557783716 x^{15} + \cdots + 79\!\cdots\!67 $ x^20 - 10x^19 + 332929x^18 - 2996076x^17 + 44578211685x^16 - 356557783716x^15 + 3097483289700118x^14 - 21676143504368032x^13 + 119894285805340530265x^12 - 719083941188982737718x^11 + 2563232515499525310010527x^10 - 12809571491275660354252428x^9 + 27741346343998271210179496359x^8 - 110888535856083235142883207868x^7 + 121645138768923372291810301132558x^6 - 364547360223565095382569624026928x^5 + 164798224123857530371927578258953667x^4 - 328988998658352429191508377804789922x^3 + 18304547169687091044739805564504070921x^2 - 18140113409773705053928159582048276332*x + 79919855971481307718190429644423027167
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{48}\cdot 3^{45} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$0.500000 + 187.726i$ of defining polynomial
Character	$\chi$	$=$	432.289
Dual form			432.8.i.e.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-156.325 - 270.763i) q^{5} +(-626.107 + 1084.45i) q^{7} +(992.406 - 1718.90i) q^{11} +(6039.81 + 10461.3i) q^{13} +5453.94 q^{17} -51911.6 q^{19} +(-45567.2 - 78924.8i) q^{23} +(-9812.59 + 16995.9i) q^{25} +(-56370.4 + 97636.4i) q^{29} +(165765. + 287113. i) q^{31} +391505. q^{35} +313951. q^{37} +(95942.4 + 166177. i) q^{41} +(-84339.4 + 146080. i) q^{43} +(-240497. + 416554. i) q^{47} +(-372249. - 644754. i) q^{49} +562959. q^{53} -620552. q^{55} +(523339. + 906449. i) q^{59} +(587842. - 1.01817e6i) q^{61} +(1.88835e6 - 3.27072e6i) q^{65} +(-1.63387e6 - 2.82995e6i) q^{67} +2.19874e6 q^{71} +366453. q^{73} +(1.24270e6 + 2.15243e6i) q^{77} +(1.01679e6 - 1.76114e6i) q^{79} +(-1.79755e6 + 3.11344e6i) q^{83} +(-852588. - 1.47673e6i) q^{85} -5.04034e6 q^{89} -1.51263e7 q^{91} +(8.11508e6 + 1.40557e7i) q^{95} +(4.80998e6 - 8.33112e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 125 q^{5} - 1245 q^{7} + 6106 q^{11} + 4937 q^{13} - 48722 q^{17} + 26882 q^{19} + 19387 q^{23} - 218957 q^{25} + 46791 q^{29} + 185039 q^{31} + 83094 q^{35} + 108420 q^{37} + 638112 q^{41} - 892628 q^{43}+ \cdots + 17951260 q^{97}+O(q^{100}) $ 20 * q + 125 * q^5 - 1245 * q^7 + 6106 * q^11 + 4937 * q^13 - 48722 * q^17 + 26882 * q^19 + 19387 * q^23 - 218957 * q^25 + 46791 * q^29 + 185039 * q^31 + 83094 * q^35 + 108420 * q^37 + 638112 * q^41 - 892628 * q^43 - 230883 * q^47 - 1034741 * q^49 + 2872940 * q^53 - 1089998 * q^55 + 2172454 * q^59 - 1878325 * q^61 - 1239133 * q^65 - 531496 * q^67 + 3723056 * q^71 - 1804522 * q^73 - 6276543 * q^77 - 3607847 * q^79 + 10794491 * q^83 + 3597658 * q^85 - 32214888 * q^89 + 16117530 * q^91 - 756868 * q^95 + 17951260 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−156.325	−	270.763i	−0.559286	−	0.968711i	−0.997556	−	0.0698682i	$-0.977742\pi$
$5$	−156.325	−	270.763i	−0.559286	−	0.968711i	0.438270	−	0.898843i	$-0.355591\pi$
$6$		0			0
$7$	−626.107	+	1084.45i	−0.689931	+	1.19500i	0.281929	+	0.959435i	$0.409026\pi$
$7$	−626.107	+	1084.45i	−0.689931	+	1.19500i	−0.971860	+	0.235560i	$0.924308\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	992.406	−	1718.90i	0.224810	−	0.389382i	−0.731453	−	0.681892i	$-0.761157\pi$
$11$	992.406	−	1718.90i	0.224810	−	0.389382i	0.956262	+	0.292511i	$0.0944907\pi$
$12$		0			0
$13$	6039.81	+	10461.3i	0.762468	+	1.32063i	0.941575	+	0.336804i	$0.109346\pi$
$13$	6039.81	+	10461.3i	0.762468	+	1.32063i	−0.179106	+	0.983830i	$0.557321\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5453.94			0.269240			0.134620	−	0.990897i	$-0.457019\pi$
$17$	5453.94			0.269240			0.134620	+	0.990897i	$0.457019\pi$
$18$		0			0
$19$	−51911.6			−1.73631			−0.868154	−	0.496295i	$-0.834693\pi$
$19$	−51911.6			−1.73631			−0.868154	+	0.496295i	$0.834693\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−45567.2	−	78924.8i	−0.780918	−	1.35259i	−0.931408	−	0.363977i	$-0.881419\pi$
$23$	−45567.2	−	78924.8i	−0.780918	−	1.35259i	0.150490	−	0.988611i	$-0.451915\pi$
$24$		0			0
$25$	−9812.59	+	16995.9i	−0.125601	+	0.217547i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−56370.4	+	97636.4i	−0.429199	+	0.743394i	−0.996802	−	0.0799082i	$-0.974537\pi$
$29$	−56370.4	+	97636.4i	−0.429199	+	0.743394i	0.567604	+	0.823302i	$0.307871\pi$
$30$		0			0
$31$	165765.	+	287113.i	0.999370	+	1.73096i	0.530411	+	0.847741i	$0.322038\pi$
$31$	165765.	+	287113.i	0.999370	+	1.73096i	0.468960	+	0.883220i	$0.344629\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	391505.			1.54347
$36$		0			0
$37$	313951.			1.01896			0.509478	−	0.860484i	$-0.329838\pi$
$37$	313951.			1.01896			0.509478	+	0.860484i	$0.329838\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	95942.4	+	166177.i	0.217404	+	0.376554i	0.954013	−	0.299764i	$-0.0969079\pi$
$41$	95942.4	+	166177.i	0.217404	+	0.376554i	−0.736610	+	0.676318i	$0.763575\pi$
$42$		0			0
$43$	−84339.4	+	146080.i	−0.161767	+	0.280189i	−0.935503	−	0.353320i	$-0.885053\pi$
$43$	−84339.4	+	146080.i	−0.161767	+	0.280189i	0.773735	+	0.633509i	$0.218386\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−240497.	+	416554.i	−0.337884	+	0.585233i	−0.984035	−	0.177978i	$-0.943045\pi$
$47$	−240497.	+	416554.i	−0.337884	+	0.585233i	0.646150	+	0.763210i	$0.276378\pi$
$48$		0			0
$49$	−372249.	−	644754.i	−0.452009	−	0.782902i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	562959.			0.519411			0.259706	−	0.965688i	$-0.416375\pi$
$53$	562959.			0.519411			0.259706	+	0.965688i	$0.416375\pi$
$54$		0			0
$55$	−620552.			−0.502931
$56$		0			0
$57$		0			0
$58$		0			0
$59$	523339.	+	906449.i	0.331742	+	0.574594i	0.982854	−	0.184388i	$-0.0590302\pi$
$59$	523339.	+	906449.i	0.331742	+	0.574594i	−0.651111	+	0.758982i	$0.725697\pi$
$60$		0			0
$61$	587842.	−	1.01817e6i	0.331594	−	0.574337i	−0.651231	−	0.758880i	$-0.725747\pi$
$61$	587842.	−	1.01817e6i	0.331594	−	0.574337i	0.982825	+	0.184542i	$0.0590803\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.88835e6	−	3.27072e6i	0.852875	−	1.47722i
$66$		0			0
$67$	−1.63387e6	−	2.82995e6i	−0.663677	−	1.14952i	−0.979642	−	0.200752i	$-0.935661\pi$
$67$	−1.63387e6	−	2.82995e6i	−0.663677	−	1.14952i	0.315965	−	0.948771i	$-0.397672\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.19874e6			0.729069			0.364535	−	0.931190i	$-0.381228\pi$
$71$	2.19874e6			0.729069			0.364535	+	0.931190i	$0.381228\pi$
$72$		0			0
$73$	366453.			0.110252			0.0551262	−	0.998479i	$-0.482444\pi$
$73$	366453.			0.110252			0.0551262	+	0.998479i	$0.482444\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.24270e6	+	2.15243e6i	0.310206	+	0.537293i
$78$		0			0
$79$	1.01679e6	−	1.76114e6i	0.232027	−	0.401883i	−0.726377	−	0.687296i	$-0.758798\pi$
$79$	1.01679e6	−	1.76114e6i	0.232027	−	0.401883i	0.958405	+	0.285413i	$0.0921309\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.79755e6	+	3.11344e6i	−0.345070	+	0.597679i	−0.985366	−	0.170449i	$-0.945478\pi$
$83$	−1.79755e6	+	3.11344e6i	−0.345070	+	0.597679i	0.640297	+	0.768128i	$0.278811\pi$
$84$		0			0
$85$	−852588.	−	1.47673e6i	−0.150582	−	0.260816i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.04034e6			−0.757870			−0.378935	−	0.925423i	$-0.623710\pi$
$89$	−5.04034e6			−0.757870			−0.378935	+	0.925423i	$0.623710\pi$
$90$		0			0
$91$	−1.51263e7			−2.10420
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.11508e6	+	1.40557e7i	0.971092	+	1.68198i
$96$		0			0
$97$	4.80998e6	−	8.33112e6i	0.535109	−	0.926835i	−0.464050	−	0.885809i	$-0.653604\pi$
$97$	4.80998e6	−	8.33112e6i	0.535109	−	0.926835i	0.999158	−	0.0410259i	$-0.0130626\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.05741e6	−	1.56879e7i	0.874740	−	1.51509i	0.0177006	−	0.999843i	$-0.494365\pi$
$101$	9.05741e6	−	1.56879e7i	0.874740	−	1.51509i	0.857039	−	0.515251i	$-0.172301\pi$
$102$		0			0
$103$	−2.84928e6	−	4.93510e6i	−0.256924	−	0.445006i	0.708492	−	0.705719i	$-0.249376\pi$
$103$	−2.84928e6	−	4.93510e6i	−0.256924	−	0.445006i	−0.965416	+	0.260713i	$0.916042\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.24222e7			−1.76943			−0.884717	−	0.466129i	$-0.845648\pi$
$107$	−2.24222e7			−1.76943			−0.884717	+	0.466129i	$0.845648\pi$
$108$		0			0
$109$	−1.56382e7			−1.15663			−0.578313	−	0.815815i	$-0.696289\pi$
$109$	−1.56382e7			−1.15663			−0.578313	+	0.815815i	$0.696289\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.83473e6	−	8.37400e6i	−0.315208	−	0.545957i	0.664273	−	0.747490i	$-0.268741\pi$
$113$	−4.83473e6	−	8.37400e6i	−0.315208	−	0.545957i	−0.979482	+	0.201533i	$0.935408\pi$
$114$		0			0
$115$	−1.42466e7	+	2.46758e7i	−0.873512	+	1.51297i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.41475e6	+	5.91452e6i	−0.185757	+	0.321740i
$120$		0			0
$121$	7.77385e6	+	1.34647e7i	0.398921	+	0.690952i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.82900e7			−0.837584
$126$		0			0
$127$	−1.24333e7			−0.538607			−0.269303	−	0.963055i	$-0.586793\pi$
$127$	−1.24333e7			−0.538607			−0.269303	+	0.963055i	$0.586793\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.94757e6	−	3.37329e6i	−0.0756908	−	0.131100i	0.825696	−	0.564116i	$-0.190783\pi$
$131$	−1.94757e6	−	3.37329e6i	−0.0756908	−	0.131100i	−0.901386	+	0.433015i	$0.857450\pi$
$132$		0			0
$133$	3.25022e7	−	5.62955e7i	1.19793	−	2.07488i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.51354e6	+	2.62153e6i	−0.0502889	+	0.0871029i	−0.890074	−	0.455816i	$-0.849348\pi$
$137$	−1.51354e6	+	2.62153e6i	−0.0502889	+	0.0871029i	0.839785	+	0.542919i	$0.182681\pi$
$138$		0			0
$139$	1.09738e7	+	1.90072e7i	0.346581	+	0.600296i	0.985640	−	0.168862i	$-0.0540092\pi$
$139$	1.09738e7	+	1.90072e7i	0.346581	+	0.600296i	−0.639059	+	0.769158i	$0.720676\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.39758e7			0.685641
$144$		0			0
$145$	3.52484e7			0.960179
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.40718e7	−	5.90140e7i	−0.843806	−	1.46151i	−0.886654	−	0.462433i	$-0.846977\pi$
$149$	−3.40718e7	−	5.90140e7i	−0.843806	−	1.46151i	0.0428483	−	0.999082i	$-0.486357\pi$
$150$		0			0
$151$	1.57918e7	−	2.73521e7i	0.373260	−	0.646505i	−0.616805	−	0.787116i	$-0.711573\pi$
$151$	1.57918e7	−	2.73521e7i	0.373260	−	0.646505i	0.990065	+	0.140611i	$0.0449067\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.18264e7	−	8.97660e7i	1.11787	−	1.93620i
$156$		0			0
$157$	−82154.5	−	142296.i	−0.00169427	−	0.00293456i	0.865177	−	0.501467i	$-0.167206\pi$
$157$	−82154.5	−	142296.i	−0.00169427	−	0.00293456i	−0.866871	+	0.498532i	$0.833873\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.14120e8			2.15512
$162$		0			0
$163$	−5.92828e7			−1.07219			−0.536096	−	0.844157i	$-0.680101\pi$
$163$	−5.92828e7			−1.07219			−0.536096	+	0.844157i	$0.680101\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.28547e7	−	3.95856e7i	−0.379725	−	0.657702i	0.611297	−	0.791401i	$-0.290648\pi$
$167$	−2.28547e7	−	3.95856e7i	−0.379725	−	0.657702i	−0.991022	+	0.133699i	$0.957315\pi$
$168$		0			0
$169$	−4.15844e7	+	7.20264e7i	−0.662716	+	1.14786i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.29464e7	−	1.09026e8i	0.924293	−	1.60092i	0.131599	−	0.991303i	$-0.457989\pi$
$173$	6.29464e7	−	1.09026e8i	0.924293	−	1.60092i	0.792694	−	0.609620i	$-0.208678\pi$
$174$		0			0
$175$	−1.22875e7	−	2.12825e7i	−0.173312	−	0.300185i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.62701e7			−0.342354			−0.171177	−	0.985240i	$-0.554757\pi$
$179$	−2.62701e7			−0.342354			−0.171177	+	0.985240i	$0.554757\pi$
$180$		0			0
$181$	−7.44262e7			−0.932934			−0.466467	−	0.884539i	$-0.654473\pi$
$181$	−7.44262e7			−0.932934			−0.466467	+	0.884539i	$0.654473\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.90784e7	−	8.50063e7i	−0.569888	−	0.987074i
$186$		0			0
$187$	5.41252e6	−	9.37476e6i	0.0605277	−	0.104837i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.01302e7	+	8.68281e7i	−0.520574	+	0.901661i	0.479139	+	0.877739i	$0.340949\pi$
$191$	−5.01302e7	+	8.68281e7i	−0.520574	+	0.901661i	−0.999714	+	0.0239225i	$0.992384\pi$
$192$		0			0
$193$	3.28715e6	+	5.69351e6i	0.0329131	+	0.0570072i	0.882013	−	0.471226i	$-0.156188\pi$
$193$	3.28715e6	+	5.69351e6i	0.0329131	+	0.0570072i	−0.849100	+	0.528233i	$0.822855\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.71132e8			−1.59478			−0.797389	−	0.603466i	$-0.793786\pi$
$197$	−1.71132e8			−1.59478			−0.797389	+	0.603466i	$0.793786\pi$
$198$		0			0
$199$	6.93519e7			0.623839			0.311920	−	0.950109i	$-0.399028\pi$
$199$	6.93519e7			0.623839			0.311920	+	0.950109i	$0.399028\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.05878e7	−	1.22262e8i	−0.592235	−	1.02578i
$204$		0			0
$205$	2.99964e7	−	5.19553e7i	0.243182	−	0.421203i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.15173e7	+	8.92307e7i	−0.390339	+	0.676086i
$210$		0			0
$211$	−7.71886e7	−	1.33695e8i	−0.565672	−	0.979773i	−0.996987	−	0.0775711i	$-0.975284\pi$
$211$	−7.71886e7	−	1.33695e8i	−0.565672	−	0.979773i	0.431315	−	0.902201i	$-0.358050\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.27375e7			0.361897
$216$		0			0
$217$	−4.15146e8			−2.75799
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.29408e7	+	5.70551e7i	0.205287	+	0.355567i
$222$		0			0
$223$	1.75831e6	−	3.04549e6i	0.0106177	−	0.0183903i	−0.860668	−	0.509167i	$-0.829954\pi$
$223$	1.75831e6	−	3.04549e6i	0.0106177	−	0.0183903i	0.871285	+	0.490777i	$0.163287\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.30268e7	−	1.26486e8i	0.414373	−	0.717716i	−0.580989	−	0.813911i	$-0.697334\pi$
$227$	7.30268e7	−	1.26486e8i	0.414373	−	0.717716i	0.995362	+	0.0961956i	$0.0306674\pi$
$228$		0			0
$229$	−7.37847e7	−	1.27799e8i	−0.406015	−	0.703239i	0.588424	−	0.808553i	$-0.299749\pi$
$229$	−7.37847e7	−	1.27799e8i	−0.406015	−	0.703239i	−0.994439	+	0.105314i	$0.966415\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.17721e7			0.475297			0.237648	−	0.971351i	$-0.423623\pi$
$233$	9.17721e7			0.475297			0.237648	+	0.971351i	$0.423623\pi$
$234$		0			0
$235$	1.50383e8			0.755895
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.98194e8	−	3.43282e8i	−0.939070	−	1.62652i	−0.767210	−	0.641396i	$-0.778356\pi$
$239$	−1.98194e8	−	3.43282e8i	−0.939070	−	1.62652i	−0.171860	−	0.985121i	$-0.554978\pi$
$240$		0			0
$241$	3.21419e7	−	5.56713e7i	0.147915	−	0.256196i	−0.782542	−	0.622598i	$-0.786077\pi$
$241$	3.21419e7	−	5.56713e7i	0.147915	−	0.256196i	0.930457	+	0.366402i	$0.119411\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.16384e8	+	2.01582e8i	−0.505604	+	0.875732i
$246$		0			0
$247$	−3.13536e8	−	5.43061e8i	−1.32388	−	2.29303i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.44658e8			0.976565			0.488282	−	0.872686i	$-0.337624\pi$
$251$	2.44658e8			0.976565			0.488282	+	0.872686i	$0.337624\pi$
$252$		0			0
$253$	−1.80885e8			−0.702231
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.52502e8	+	2.64142e8i	0.560416	+	0.970669i	0.997460	+	0.0712287i	$0.0226920\pi$
$257$	1.52502e8	+	2.64142e8i	0.560416	+	0.970669i	−0.437044	+	0.899440i	$0.643975\pi$
$258$		0			0
$259$	−1.96567e8	+	3.40464e8i	−0.703009	+	1.21765i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.71110e8	−	2.96371e8i	0.580003	−	1.00460i	−0.415475	−	0.909605i	$-0.636385\pi$
$263$	1.71110e8	−	2.96371e8i	0.580003	−	1.00460i	0.995478	−	0.0949905i	$-0.0302821\pi$
$264$		0			0
$265$	−8.80046e7	−	1.52428e8i	−0.290499	−	0.503159i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.36577e7			0.168073			0.0840367	−	0.996463i	$-0.473219\pi$
$269$	5.36577e7			0.168073			0.0840367	+	0.996463i	$0.473219\pi$
$270$		0			0
$271$	−7.20909e7			−0.220033			−0.110017	−	0.993930i	$-0.535090\pi$
$271$	−7.20909e7			−0.220033			−0.110017	+	0.993930i	$0.535090\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.94761e7	+	3.37336e7i	0.0564727	+	0.0978135i
$276$		0			0
$277$	−8.46674e6	+	1.46648e7i	−0.0239352	+	0.0414570i	−0.877745	−	0.479128i	$-0.840953\pi$
$277$	−8.46674e6	+	1.46648e7i	−0.0239352	+	0.0414570i	0.853810	+	0.520585i	$0.174286\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.83198e8	−	4.90513e8i	0.761409	−	1.31880i	−0.180715	−	0.983535i	$-0.557841\pi$
$281$	2.83198e8	−	4.90513e8i	0.761409	−	1.31880i	0.942124	−	0.335264i	$-0.108825\pi$
$282$		0			0
$283$	−1.27769e8	−	2.21303e8i	−0.335099	−	0.580409i	0.648405	−	0.761296i	$-0.275437\pi$
$283$	−1.27769e8	−	2.21303e8i	−0.335099	−	0.580409i	−0.983504	+	0.180887i	$0.942103\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.40281e8			−0.599974
$288$		0			0
$289$	−3.80593e8			−0.927510
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.90411e7	+	1.02262e8i	0.137125	+	0.237508i	0.926407	−	0.376523i	$-0.122880\pi$
$293$	5.90411e7	+	1.02262e8i	0.137125	+	0.237508i	−0.789282	+	0.614031i	$0.789547\pi$
$294$		0			0
$295$	1.63622e8	−	2.83402e8i	0.371077	−	0.642725i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.50435e8	−	9.53382e8i	1.19085	−	2.06261i
$300$		0			0
$301$	−1.05611e8	−	1.82924e8i	−0.223217	−	0.386623i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.67578e8			−0.741823
$306$		0			0
$307$	6.74541e8			1.33053			0.665264	−	0.746608i	$-0.268319\pi$
$307$	6.74541e8			1.33053			0.665264	+	0.746608i	$0.268319\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.54229e8	+	7.86747e8i	0.856275	+	1.48311i	0.875457	+	0.483295i	$0.160560\pi$
$311$	4.54229e8	+	7.86747e8i	0.856275	+	1.48311i	−0.0191826	+	0.999816i	$0.506106\pi$
$312$		0			0
$313$	9.89750e7	−	1.71430e8i	0.182440	−	0.315995i	−0.760271	−	0.649606i	$-0.774934\pi$
$313$	9.89750e7	−	1.71430e8i	0.182440	−	0.315995i	0.942711	+	0.333611i	$0.108267\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.02010e8	−	8.69506e8i	0.885125	−	1.53308i	0.0395543	−	0.999217i	$-0.487406\pi$
$317$	5.02010e8	−	8.69506e8i	0.885125	−	1.53308i	0.845570	−	0.533864i	$-0.179260\pi$
$318$		0			0
$319$	1.11885e8	+	1.93790e8i	0.192976	+	0.334244i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.83123e8			−0.467483
$324$		0			0
$325$	−2.37065e8			−0.383067
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.01154e8	−	5.21614e8i	−0.466233	−	0.807540i
$330$		0			0
$331$	−2.00562e8	+	3.47384e8i	−0.303984	+	0.526516i	−0.977035	−	0.213081i	$-0.931650\pi$
$331$	−2.00562e8	+	3.47384e8i	−0.303984	+	0.526516i	0.673050	+	0.739597i	$0.264984\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.10831e8	+	8.84786e8i	−0.742371	+	1.28582i
$336$		0			0
$337$	4.64377e8	+	8.04325e8i	0.660947	+	1.14479i	0.980367	+	0.197180i	$0.0631784\pi$
$337$	4.64377e8	+	8.04325e8i	0.660947	+	1.14479i	−0.319421	+	0.947613i	$0.603488\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.58024e8			0.898672
$342$		0			0
$343$	−9.89821e7			−0.132442
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.01563e6	+	5.22323e6i	0.00387459	+	0.00671098i	0.867956	−	0.496641i	$-0.165433\pi$
$347$	3.01563e6	+	5.22323e6i	0.00387459	+	0.00671098i	−0.864082	+	0.503352i	$0.832100\pi$
$348$		0			0
$349$	7.35229e7	−	1.27345e8i	0.0925835	−	0.160359i	−0.816014	−	0.578032i	$-0.803821\pi$
$349$	7.35229e7	−	1.27345e8i	0.0925835	−	0.160359i	0.908598	+	0.417673i	$0.137154\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.76816e8	+	9.99075e8i	−0.697953	+	1.20889i	0.271223	+	0.962517i	$0.412572\pi$
$353$	−5.76816e8	+	9.99075e8i	−0.697953	+	1.20889i	−0.969175	+	0.246373i	$0.920761\pi$
$354$		0			0
$355$	−3.43718e8	−	5.95336e8i	−0.407758	−	0.706258i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.18824e8			−1.04810			−0.524049	−	0.851688i	$-0.675579\pi$
$359$	−9.18824e8			−1.04810			−0.524049	+	0.851688i	$0.675579\pi$
$360$		0			0
$361$	1.80094e9			2.01476
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.72857e7	−	9.92218e7i	−0.0616626	−	0.106803i
$366$		0			0
$367$	−4.51396e8	+	7.81840e8i	−0.476679	+	0.825632i	−0.999643	−	0.0267225i	$-0.991493\pi$
$367$	−4.51396e8	+	7.81840e8i	−0.476679	+	0.825632i	0.522964	+	0.852355i	$0.324826\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.52473e8	+	6.10500e8i	−0.358358	+	0.620694i
$372$		0			0
$373$	8.32346e8	+	1.44166e9i	0.830468	+	1.43841i	0.897668	+	0.440673i	$0.145260\pi$
$373$	8.32346e8	+	1.44166e9i	0.830468	+	1.43841i	−0.0672001	+	0.997740i	$0.521407\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.36187e9			−1.30900
$378$		0			0
$379$	−1.59369e8			−0.150372			−0.0751862	−	0.997170i	$-0.523955\pi$
$379$	−1.59369e8			−0.150372			−0.0751862	+	0.997170i	$0.523955\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.63721e7	+	2.83573e7i	0.0148905	+	0.0257910i	0.873375	−	0.487049i	$-0.161927\pi$
$383$	1.63721e7	+	2.83573e7i	0.0148905	+	0.0257910i	−0.858484	+	0.512840i	$0.828593\pi$
$384$		0			0
$385$	3.88532e8	−	6.72957e8i	0.346988	−	0.601000i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.80917e8	−	8.32973e8i	0.414235	−	0.717476i	−0.581113	−	0.813823i	$-0.697382\pi$
$389$	4.80917e8	−	8.32973e8i	0.414235	−	0.717476i	0.995348	+	0.0963469i	$0.0307158\pi$
$390$		0			0
$391$	−2.48521e8	−	4.30451e8i	−0.210254	−	0.364171i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.35802e8			−0.519078
$396$		0			0
$397$	−1.54506e9			−1.23931			−0.619653	−	0.784876i	$-0.712726\pi$
$397$	−1.54506e9			−1.23931			−0.619653	+	0.784876i	$0.712726\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.63954e8	−	1.49641e9i	−0.669091	−	1.15890i	−0.978159	−	0.207860i	$-0.933350\pi$
$401$	−8.63954e8	−	1.49641e9i	−0.669091	−	1.15890i	0.309067	−	0.951040i	$-0.399983\pi$
$402$		0			0
$403$	−2.00238e9	+	3.46822e9i	−1.52398	+	2.63960i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.11566e8	−	5.39649e8i	0.229071	−	0.396763i
$408$		0			0
$409$	−4.09724e8	−	7.09662e8i	−0.296114	−	0.512885i	0.679129	−	0.734019i	$-0.262358\pi$
$409$	−4.09724e8	−	7.09662e8i	−0.296114	−	0.512885i	−0.975244	+	0.221134i	$0.929024\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.31066e9			−0.915517
$414$		0			0
$415$	1.12401e9			0.771971
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.59438e8	+	6.22565e8i	0.238712	+	0.413462i	0.960345	−	0.278814i	$-0.0899414\pi$
$419$	3.59438e8	+	6.22565e8i	0.238712	+	0.413462i	−0.721633	+	0.692276i	$0.756608\pi$
$420$		0			0
$421$	1.35748e9	−	2.35122e9i	0.886638	−	1.53570i	0.0428124	−	0.999083i	$-0.486368\pi$
$421$	1.35748e9	−	2.35122e9i	0.886638	−	1.53570i	0.843825	−	0.536618i	$-0.180298\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.35173e7	+	9.26946e7i	−0.0338168	+	0.0585724i
$426$		0			0
$427$	7.36105e8	+	1.27497e9i	0.457554	+	0.792506i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.70358e8			0.222819			0.111409	−	0.993775i	$-0.464464\pi$
$431$	3.70358e8			0.222819			0.111409	+	0.993775i	$0.464464\pi$
$432$		0			0
$433$	−4.91018e8			−0.290663			−0.145332	−	0.989383i	$-0.546425\pi$
$433$	−4.91018e8			−0.290663			−0.145332	+	0.989383i	$0.546425\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.36547e9	+	4.09711e9i	1.35591	+	2.34851i
$438$		0			0
$439$	1.50342e9	−	2.60401e9i	0.848117	−	1.46898i	−0.0347701	−	0.999395i	$-0.511070\pi$
$439$	1.50342e9	−	2.60401e9i	0.848117	−	1.46898i	0.882887	−	0.469586i	$-0.155597\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.03867e8	+	8.72724e8i	−0.275361	+	0.476940i	−0.970226	−	0.242200i	$-0.922131\pi$
$443$	−5.03867e8	+	8.72724e8i	−0.275361	+	0.476940i	0.694865	+	0.719140i	$0.255464\pi$
$444$		0			0
$445$	7.87931e8	+	1.36474e9i	0.423866	+	0.734157i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.97900e9			1.03177			0.515886	−	0.856657i	$-0.327463\pi$
$449$	1.97900e9			1.03177			0.515886	+	0.856657i	$0.327463\pi$
$450$		0			0
$451$	3.80855e8			0.195498
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.36462e9	+	4.09564e9i	1.17685	+	2.03836i
$456$		0			0
$457$	6.27954e8	−	1.08765e9i	0.307767	−	0.533067i	−0.670107	−	0.742265i	$-0.733752\pi$
$457$	6.27954e8	−	1.08765e9i	0.307767	−	0.533067i	0.977873	+	0.209197i	$0.0670851\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.23226e9	−	2.13433e9i	0.585798	−	1.01463i	−0.408978	−	0.912544i	$-0.634115\pi$
$461$	1.23226e9	−	2.13433e9i	0.585798	−	1.01463i	0.994775	−	0.102087i	$-0.0325520\pi$
$462$		0			0
$463$	6.66460e8	+	1.15434e9i	0.312062	+	0.540507i	0.978809	−	0.204777i	$-0.0656470\pi$
$463$	6.66460e8	+	1.15434e9i	0.312062	+	0.540507i	−0.666747	+	0.745284i	$0.732314\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.35293e9			0.614706			0.307353	−	0.951596i	$-0.400557\pi$
$467$	1.35293e9			0.614706			0.307353	+	0.951596i	$0.400557\pi$
$468$		0			0
$469$	4.09192e9			1.83157
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.67398e8	+	2.89942e8i	0.0727337	+	0.125979i
$474$		0			0
$475$	5.09387e8	−	8.82284e8i	0.218082	−	0.377729i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.80465e8	+	1.17860e9i	−0.282899	+	0.489995i	−0.972097	−	0.234577i	$-0.924629\pi$
$479$	−6.80465e8	+	1.17860e9i	−0.282899	+	0.489995i	0.689199	+	0.724572i	$0.257963\pi$
$480$		0			0
$481$	1.89620e9	+	3.28432e9i	0.776922	+	1.34567i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.00768e9			−1.19711
$486$		0			0
$487$	1.72113e8			0.0675247			0.0337623	−	0.999430i	$-0.489251\pi$
$487$	1.72113e8			0.0675247			0.0337623	+	0.999430i	$0.489251\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.48430e9	−	2.57089e9i	−0.565897	−	0.980162i	−0.996966	−	0.0778432i	$-0.975197\pi$
$491$	−1.48430e9	−	2.57089e9i	−0.565897	−	0.980162i	0.431069	−	0.902319i	$-0.358137\pi$
$492$		0			0
$493$	−3.07441e8	+	5.32503e8i	−0.115557	+	0.200151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.37664e9	+	2.38442e9i	−0.503007	+	0.871234i
$498$		0			0
$499$	−1.62065e9	−	2.80704e9i	−0.583897	−	1.01134i	−0.995012	−	0.0997561i	$-0.968194\pi$
$499$	−1.62065e9	−	2.80704e9i	−0.583897	−	1.01134i	0.411115	−	0.911584i	$-0.365140\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.38952e9			−0.486831			−0.243415	−	0.969922i	$-0.578268\pi$
$503$	−1.38952e9			−0.486831			−0.243415	+	0.969922i	$0.578268\pi$
$504$		0			0
$505$	−5.66360e9			−1.95692
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.05258e9	+	3.55518e9i	0.689904	+	1.19495i	0.971868	+	0.235524i	$0.0756807\pi$
$509$	2.05258e9	+	3.55518e9i	0.689904	+	1.19495i	−0.281964	+	0.959425i	$0.590986\pi$
$510$		0			0
$511$	−2.29439e8	+	3.97399e8i	−0.0760665	+	0.131751i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.90828e8	+	1.54296e9i	−0.287388	+	0.497771i
$516$		0			0
$517$	4.77342e8	+	8.26780e8i	0.151919	+	0.263132i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.64582e8			0.0819651			0.0409825	−	0.999160i	$-0.486951\pi$
$521$	2.64582e8			0.0819651			0.0409825	+	0.999160i	$0.486951\pi$
$522$		0			0
$523$	3.53766e9			1.08133			0.540667	−	0.841237i	$-0.318172\pi$
$523$	3.53766e9			1.08133			0.540667	+	0.841237i	$0.318172\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.04072e8	+	1.56590e9i	0.269070	+	0.466043i
$528$		0			0
$529$	−2.45033e9	+	4.24410e9i	−0.719664	+	1.24650i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.15895e9	+	2.00736e9i	−0.331527	+	0.574221i
$534$		0			0
$535$	3.50515e9	+	6.07109e9i	0.989619	+	1.71407i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.47769e9			−0.406464
$540$		0			0
$541$	−1.60977e8			−0.0437092			−0.0218546	−	0.999761i	$-0.506957\pi$
$541$	−1.60977e8			−0.0437092			−0.0218546	+	0.999761i	$0.506957\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.44464e9	+	4.23424e9i	0.646885	+	1.12044i
$546$		0			0
$547$	2.74652e9	−	4.75712e9i	0.717510	−	1.24276i	−0.244474	−	0.969656i	$-0.578615\pi$
$547$	2.74652e9	−	4.75712e9i	0.717510	−	1.24276i	0.961983	−	0.273108i	$-0.0880515\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.92628e9	−	5.06846e9i	0.745221	−	1.29076i
$552$		0			0
$553$	1.27325e9	+	2.20533e9i	0.320165	+	0.554542i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.53683e9			0.622012			0.311006	−	0.950408i	$-0.399334\pi$
$557$	2.53683e9			0.622012			0.311006	+	0.950408i	$0.399334\pi$
$558$		0			0
$559$	−2.03758e9			−0.493370
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3.67329e9	+	6.36232e9i	0.867513	+	1.50258i	0.864531	+	0.502580i	$0.167616\pi$
$563$	3.67329e9	+	6.36232e9i	0.867513	+	1.50258i	0.00298204	+	0.999996i	$0.499051\pi$
$564$		0			0
$565$	−1.51158e9	+	2.61813e9i	−0.352583	+	0.610692i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.84835e9	+	4.93349e9i	−0.648188	+	1.12269i	0.335368	+	0.942087i	$0.391139\pi$
$569$	−2.84835e9	+	4.93349e9i	−0.648188	+	1.12269i	−0.983555	+	0.180607i	$0.942194\pi$
$570$		0			0
$571$	1.78543e9	+	3.09245e9i	0.401343	+	0.695147i	0.993888	−	0.110391i	$-0.0352102\pi$
$571$	1.78543e9	+	3.09245e9i	0.401343	+	0.695147i	−0.592545	+	0.805537i	$0.701877\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.78853e9			0.392336
$576$		0			0
$577$	4.44708e9			0.963739			0.481869	−	0.876243i	$-0.339958\pi$
$577$	4.44708e9			0.963739			0.481869	+	0.876243i	$0.339958\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.25091e9	−	3.89870e9i	−0.476149	−	0.824714i
$582$		0			0
$583$	5.58684e8	−	9.67668e8i	0.116769	−	0.202249i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.39248e8	−	1.45362e9i	0.171260	−	0.296632i	−0.767600	−	0.640929i	$-0.778549\pi$
$587$	8.39248e8	−	1.45362e9i	0.171260	−	0.296632i	0.938861	+	0.344297i	$0.111883\pi$
$588$		0			0
$589$	−8.60511e9	−	1.49045e10i	−1.73521	−	3.00548i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.89473e9			−0.766984			−0.383492	−	0.923544i	$-0.625279\pi$
$593$	−3.89473e9			−0.766984			−0.383492	+	0.923544i	$0.625279\pi$
$594$		0			0
$595$	2.13525e9			0.415565
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.40239e9	−	7.62517e9i	−0.836941	−	1.44962i	−0.892440	−	0.451166i	$-0.851008\pi$
$599$	−4.40239e9	−	7.62517e9i	−0.836941	−	1.44962i	0.0554990	−	0.998459i	$-0.482325\pi$
$600$		0			0
$601$	−1.79514e9	+	3.10928e9i	−0.337317	+	0.584251i	−0.983927	−	0.178570i	$-0.942853\pi$
$601$	−1.79514e9	+	3.10928e9i	−0.337317	+	0.584251i	0.646610	+	0.762821i	$0.276186\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.43050e9	−	4.20974e9i	0.446222	−	0.772879i
$606$		0			0
$607$	−1.40497e9	−	2.43347e9i	−0.254980	−	0.441638i	0.709910	−	0.704292i	$-0.248735\pi$
$607$	−1.40497e9	−	2.43347e9i	−0.254980	−	0.441638i	−0.964890	+	0.262654i	$0.915402\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.81024e9			−1.03050
$612$		0			0
$613$	−8.63009e8			−0.151323			−0.0756613	−	0.997134i	$-0.524107\pi$
$613$	−8.63009e8			−0.151323			−0.0756613	+	0.997134i	$0.524107\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.67139e9	+	6.35904e9i	0.629263	+	1.08992i	0.987700	+	0.156362i	$0.0499767\pi$
$617$	3.67139e9	+	6.35904e9i	0.629263	+	1.08992i	−0.358436	+	0.933554i	$0.616690\pi$
$618$		0			0
$619$	5.18346e7	−	8.97801e7i	0.00878420	−	0.0152147i	−0.861600	−	0.507588i	$-0.830537\pi$
$619$	5.18346e7	−	8.97801e7i	0.00878420	−	0.0152147i	0.870384	+	0.492373i	$0.163871\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.15579e9	−	5.46599e9i	0.522878	−	0.905651i
$624$		0			0
$625$	3.62579e9	+	6.28006e9i	0.594050	+	1.02892i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.71227e9			0.274344
$630$		0			0
$631$	−2.99264e9			−0.474189			−0.237094	−	0.971487i	$-0.576195\pi$
$631$	−2.99264e9			−0.474189			−0.237094	+	0.971487i	$0.576195\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.94363e9	+	3.36647e9i	0.301235	+	0.521755i
$636$		0			0
$637$	4.49662e9	−	7.78838e9i	0.689285	−	1.19388i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.47265e8	+	1.64071e9i	−0.142059	+	0.246053i	−0.928272	−	0.371902i	$-0.878706\pi$
$641$	−9.47265e8	+	1.64071e9i	−0.142059	+	0.246053i	0.786213	+	0.617956i	$0.212039\pi$
$642$		0			0
$643$	−4.39965e9	−	7.62041e9i	−0.652648	−	1.13042i	−0.982478	−	0.186380i	$-0.940325\pi$
$643$	−4.39965e9	−	7.62041e9i	−0.652648	−	1.13042i	0.329829	−	0.944041i	$-0.393009\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.07161e10			−1.55550			−0.777750	−	0.628574i	$-0.783639\pi$
$647$	−1.07161e10			−1.55550			−0.777750	+	0.628574i	$0.783639\pi$
$648$		0			0
$649$	2.07746e9			0.298315
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.84818e8	−	1.53255e9i	−0.124353	−	0.215386i	0.797127	−	0.603812i	$-0.206352\pi$
$653$	−8.84818e8	−	1.53255e9i	−0.124353	−	0.215386i	−0.921480	+	0.388426i	$0.873019\pi$
$654$		0			0
$655$	−6.08908e8	+	1.05466e9i	−0.0846656	+	0.146645i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.51603e9	−	7.82199e9i	0.614692	−	1.06468i	−0.375746	−	0.926723i	$-0.622613\pi$
$659$	4.51603e9	−	7.82199e9i	0.614692	−	1.06468i	0.990438	−	0.137956i	$-0.0440532\pi$
$660$		0			0
$661$	−1.21392e9	−	2.10257e9i	−0.163488	−	0.283169i	0.772629	−	0.634857i	$-0.218941\pi$
$661$	−1.21392e9	−	2.10257e9i	−0.163488	−	0.283169i	−0.936117	+	0.351688i	$0.885608\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.03236e10			−2.67994
$666$		0			0
$667$	1.02746e10			1.34067
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.16676e9	−	2.02088e9i	−0.149091	−	0.258233i
$672$		0			0
$673$	−5.08131e9	+	8.80109e9i	−0.642574	+	1.11297i	0.342282	+	0.939597i	$0.388800\pi$
$673$	−5.08131e9	+	8.80109e9i	−0.642574	+	1.11297i	−0.984856	+	0.173373i	$0.944533\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.89120e9	−	6.73975e9i	0.481973	−	0.834802i	−0.517813	−	0.855494i	$-0.673254\pi$
$677$	3.89120e9	−	6.73975e9i	0.481973	−	0.834802i	0.999786	+	0.0206918i	$0.00658689\pi$
$678$		0			0
$679$	6.02312e9	+	1.04324e10i	0.738376	+	1.27890i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.85965e9			−0.583624			−0.291812	−	0.956476i	$-0.594258\pi$
$683$	−4.85965e9			−0.583624			−0.291812	+	0.956476i	$0.594258\pi$
$684$		0			0
$685$	9.46417e8			0.112503
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.40017e9	+	5.88926e9i	0.396034	+	0.685952i
$690$		0			0
$691$	−2.02680e9	+	3.51052e9i	−0.233689	+	0.404760i	−0.958891	−	0.283776i	$-0.908413\pi$
$691$	−2.02680e9	+	3.51052e9i	−0.233689	+	0.404760i	0.725202	+	0.688536i	$0.241746\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.43096e9	−	5.94259e9i	0.387676	−	0.671474i
$696$		0			0
$697$	5.23264e8	+	9.06320e8i	0.0585337	+	0.101383i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.65648e9			−0.181624			−0.0908118	−	0.995868i	$-0.528946\pi$
$701$	−1.65648e9			−0.181624			−0.0908118	+	0.995868i	$0.528946\pi$
$702$		0			0
$703$	−1.62977e10			−1.76922
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.13418e10	+	1.96446e10i	1.20702	+	2.09062i
$708$		0			0
$709$	4.42534e8	−	7.66491e8i	0.0466321	−	0.0807691i	−0.841767	−	0.539841i	$-0.818485\pi$
$709$	4.42534e8	−	7.66491e8i	0.0466321	−	0.0807691i	0.888399	+	0.459072i	$0.151818\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.51069e10	−	2.61659e10i	1.56085	−	2.70347i
$714$		0			0
$715$	−3.74802e9	−	6.49175e9i	−0.383469	−	0.664188i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.53057e9			0.253902			0.126951	−	0.991909i	$-0.459481\pi$
$719$	2.53057e9			0.253902			0.126951	+	0.991909i	$0.459481\pi$
$720$		0			0
$721$	7.13582e9			0.709039
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.10628e9	−	1.91613e9i	−0.107816	−	0.186742i
$726$		0			0
$727$	−3.01834e9	+	5.22791e9i	−0.291338	+	0.504613i	−0.974126	−	0.226004i	$-0.927434\pi$
$727$	−3.01834e9	+	5.22791e9i	−0.291338	+	0.504613i	0.682788	+	0.730616i	$0.260767\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.59982e8	+	7.96713e8i	−0.0435542	+	0.0754381i
$732$		0			0
$733$	−5.72318e8	−	9.91284e8i	−0.0536752	−	0.0929681i	0.837939	−	0.545763i	$-0.183760\pi$
$733$	−5.72318e8	−	9.91284e8i	−0.0536752	−	0.0929681i	−0.891615	+	0.452795i	$0.850427\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.48587e9			−0.596804
$738$		0			0
$739$	−1.09705e10			−0.999929			−0.499964	−	0.866046i	$-0.666654\pi$
$739$	−1.09705e10			−0.999929			−0.499964	+	0.866046i	$0.666654\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.20870e9	−	5.55763e9i	−0.286991	−	0.497083i	0.686099	−	0.727508i	$-0.259322\pi$
$743$	−3.20870e9	−	5.55763e9i	−0.286991	−	0.497083i	−0.973090	+	0.230425i	$0.925988\pi$
$744$		0			0
$745$	−1.06525e10	+	1.84507e10i	−0.943857	+	1.63481i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.40387e10	−	2.43157e10i	1.22079	−	2.11446i
$750$		0			0
$751$	1.88052e9	+	3.25716e9i	0.162009	+	0.280608i	0.935589	−	0.353091i	$-0.114869\pi$
$751$	1.88052e9	+	3.25716e9i	0.162009	+	0.280608i	−0.773580	+	0.633698i	$0.781536\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.87459e9			−0.835035
$756$		0			0
$757$	1.67431e10			1.40281			0.701406	−	0.712762i	$-0.252556\pi$
$757$	1.67431e10			1.40281			0.701406	+	0.712762i	$0.252556\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.05428e9	−	1.22184e10i	−0.580238	−	1.00500i	−0.995451	−	0.0952776i	$-0.969626\pi$
$761$	−7.05428e9	−	1.22184e10i	−0.580238	−	1.00500i	0.415213	−	0.909724i	$-0.363707\pi$
$762$		0			0
$763$	9.79117e9	−	1.69588e10i	0.797992	−	1.38216i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.32174e9	+	1.09496e10i	−0.505886	+	0.876220i
$768$		0			0
$769$	−1.15935e10	−	2.00806e10i	−0.919336	−	1.59234i	−0.800426	−	0.599431i	$-0.795393\pi$
$769$	−1.15935e10	−	2.00806e10i	−0.919336	−	1.59234i	−0.118910	−	0.992905i	$-0.537940\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.85122e10			1.44155			0.720776	−	0.693168i	$-0.243786\pi$
$773$	1.85122e10			1.44155			0.720776	+	0.693168i	$0.243786\pi$
$774$		0			0
$775$	−6.50633e9			−0.502088
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.98052e9	−	8.62651e9i	−0.377480	−	0.653814i
$780$		0			0
$781$	2.18204e9	−	3.77940e9i	0.163902	−	0.283886i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.56856e7	+	4.44888e7i	−0.00189516	+	0.00328252i
$786$		0			0
$787$	6.54533e9	+	1.13369e10i	0.478652	+	0.829050i	0.999700	−	0.0244770i	$-0.00779205\pi$
$787$	6.54533e9	+	1.13369e10i	0.478652	+	0.829050i	−0.521048	+	0.853527i	$0.674459\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.21082e10			0.869888
$792$		0			0
$793$	1.42018e10			1.01132
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.90232e8	−	1.19552e9i	−0.0482938	−	0.0836473i	0.840868	−	0.541240i	$-0.182045\pi$
$797$	−6.90232e8	−	1.19552e9i	−0.0482938	−	0.0836473i	−0.889162	+	0.457593i	$0.848712\pi$
$798$		0			0
$799$	−1.31166e9	+	2.27186e9i	−0.0909719	+	0.157568i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.63670e8	−	6.29894e8i	0.0247858	−	0.0429302i
$804$		0			0
$805$	−1.78398e10	−	3.08994e10i	−1.20533	−	2.08769i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.61305e10			−1.07109			−0.535547	−	0.844506i	$-0.679894\pi$
$809$	−1.61305e10			−1.07109			−0.535547	+	0.844506i	$0.679894\pi$
$810$		0			0
$811$	−2.56158e10			−1.68630			−0.843149	−	0.537680i	$-0.819301\pi$
$811$	−2.56158e10			−1.68630			−0.843149	+	0.537680i	$0.819301\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.26740e9	+	1.60516e10i	0.599661	+	1.03864i
$816$		0			0
$817$	4.37819e9	−	7.58325e9i	0.280878	−	0.486495i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.05197e9	+	1.56785e10i	−0.570876	+	0.988787i	0.425600	+	0.904911i	$0.360063\pi$
$821$	−9.05197e9	+	1.56785e10i	−0.570876	+	0.988787i	−0.996476	+	0.0838753i	$0.973270\pi$
$822$		0			0
$823$	−1.11889e10	−	1.93797e10i	−0.699659	−	1.21184i	−0.968585	−	0.248684i	$-0.920002\pi$
$823$	−1.11889e10	−	1.93797e10i	−0.699659	−	1.21184i	0.268926	−	0.963161i	$-0.413331\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.84371e10			1.13350			0.566751	−	0.823889i	$-0.308200\pi$
$827$	1.84371e10			1.13350			0.566751	+	0.823889i	$0.308200\pi$
$828$		0			0
$829$	−1.97903e10			−1.20646			−0.603229	−	0.797568i	$-0.706120\pi$
$829$	−1.97903e10			−1.20646			−0.603229	+	0.797568i	$0.706120\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.03022e9	−	3.51645e9i	−0.121699	−	0.210788i
$834$		0			0
$835$	−7.14554e9	+	1.23764e10i	−0.424749	+	0.735687i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.88842e9	−	1.01990e10i	0.344217	−	0.596201i	−0.640994	−	0.767546i	$-0.721478\pi$
$839$	5.88842e9	−	1.01990e10i	0.344217	−	0.596201i	0.985211	+	0.171345i	$0.0548111\pi$
$840$		0			0
$841$	2.26969e9	+	3.93122e9i	0.131577	+	0.227898i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.60028e10			1.48259
$846$		0			0
$847$	−1.94690e10			−1.10091
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.43059e10	−	2.47785e10i	−0.795721	−	1.37823i
$852$		0			0
$853$	1.51855e10	−	2.63021e10i	0.837736	−	1.45100i	−0.0540465	−	0.998538i	$-0.517212\pi$
$853$	1.51855e10	−	2.63021e10i	0.837736	−	1.45100i	0.891783	−	0.452464i	$-0.149455\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.53340e10	+	2.65593e10i	−0.832191	+	1.44140i	0.0641056	+	0.997943i	$0.479581\pi$
$857$	−1.53340e10	+	2.65593e10i	−0.832191	+	1.44140i	−0.896297	+	0.443454i	$0.853753\pi$
$858$		0			0
$859$	−1.02801e10	−	1.78057e10i	−0.553379	−	0.958481i	−0.998028	−	0.0627758i	$-0.980005\pi$
$859$	−1.02801e10	−	1.78057e10i	−0.553379	−	0.958481i	0.444648	−	0.895705i	$-0.353329\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.22477e9			0.223751			0.111876	−	0.993722i	$-0.464314\pi$
$863$	4.22477e9			0.223751			0.111876	+	0.993722i	$0.464314\pi$
$864$		0			0
$865$	−3.93604e10			−2.06778
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.01815e9	−	3.49553e9i	−0.104324	−	0.180694i
$870$		0			0
$871$	1.97366e10	−	3.41848e10i	1.01207	−	1.75295i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.14515e10	−	1.98346e10i	0.577875	−	1.00091i
$876$		0			0
$877$	6.71489e9	+	1.16305e10i	0.336155	+	0.582238i	0.983706	−	0.179784i	$-0.0575399\pi$
$877$	6.71489e9	+	1.16305e10i	0.336155	+	0.582238i	−0.647551	+	0.762022i	$0.724207\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.77155e9			−0.382906			−0.191453	−	0.981502i	$-0.561320\pi$
$881$	−7.77155e9			−0.382906			−0.191453	+	0.981502i	$0.561320\pi$
$882$		0			0
$883$	−2.88850e10			−1.41192			−0.705960	−	0.708252i	$-0.749484\pi$
$883$	−2.88850e10			−1.41192			−0.705960	+	0.708252i	$0.749484\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.10173e9	+	5.37236e9i	0.149235	+	0.258483i	0.930945	−	0.365159i	$-0.118985\pi$
$887$	3.10173e9	+	5.37236e9i	0.149235	+	0.258483i	−0.781710	+	0.623642i	$0.785652\pi$
$888$		0			0
$889$	7.78455e9	−	1.34832e10i	0.371601	−	0.643633i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.24846e10	−	2.16240e10i	0.586671	−	1.01614i
$894$		0			0
$895$	4.10668e9	+	7.11297e9i	0.191474	+	0.331643i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.73769e10			−1.71571
$900$		0			0
$901$	3.07035e9			0.139846
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.16347e10	+	2.01519e10i	0.521777	+	0.903744i
$906$		0			0
$907$	−5.90280e9	+	1.02240e10i	−0.262684	+	0.454981i	−0.966954	−	0.254950i	$-0.917941\pi$
$907$	−5.90280e9	+	1.02240e10i	−0.262684	+	0.454981i	0.704271	+	0.709932i	$0.251274\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.59334e9	+	9.68794e9i	−0.245107	+	0.424539i	−0.962162	−	0.272479i	$-0.912157\pi$
$911$	−5.59334e9	+	9.68794e9i	−0.245107	+	0.424539i	0.717054	+	0.697017i	$0.245490\pi$
$912$		0			0
$913$	3.56779e9	+	6.17960e9i	0.155150	+	0.268728i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.87755e9			0.208886
$918$		0			0
$919$	4.12146e10			1.75165			0.875824	−	0.482630i	$-0.160318\pi$
$919$	4.12146e10			1.75165			0.875824	+	0.482630i	$0.160318\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.32800e10	+	2.30015e10i	0.555892	+	0.962833i
$924$		0			0
$925$	−3.08067e9	+	5.33587e9i	−0.127982	+	0.221671i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.07376e9	−	8.78801e9i	0.207623	−	0.359613i	−0.743342	−	0.668911i	$-0.766761\pi$
$929$	5.07376e9	−	8.78801e9i	0.207623	−	0.359613i	0.950965	+	0.309298i	$0.100094\pi$
$930$		0			0
$931$	1.93240e10	+	3.34702e10i	0.784826	+	1.35936i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.38445e9			−0.135409
$936$		0			0
$937$	−5.92808e8			−0.0235410			−0.0117705	−	0.999931i	$-0.503747\pi$
$937$	−5.92808e8			−0.0235410			−0.0117705	+	0.999931i	$0.503747\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.34877e9	+	9.26433e9i	0.209262	+	0.362452i	0.951482	−	0.307704i	$-0.0995606\pi$
$941$	5.34877e9	+	9.26433e9i	0.209262	+	0.362452i	−0.742220	+	0.670156i	$0.766227\pi$
$942$		0			0
$943$	8.74365e9	−	1.51445e10i	0.339549	−	0.588116i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.37884e9	+	1.10485e10i	−0.244071	+	0.422744i	−0.961870	−	0.273507i	$-0.911816\pi$
$947$	−6.37884e9	+	1.10485e10i	−0.244071	+	0.422744i	0.717799	+	0.696250i	$0.245150\pi$
$948$		0			0
$949$	2.21331e9	+	3.83356e9i	0.0840639	+	0.145603i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.65154e10			−1.74089			−0.870446	−	0.492264i	$-0.836170\pi$
$953$	−4.65154e10			−1.74089			−0.870446	+	0.492264i	$0.836170\pi$
$954$		0			0
$955$	3.13465e10			1.16460
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.89528e9	−	3.28271e9i	−0.0693917	−	0.120190i
$960$		0			0
$961$	−4.11997e10	+	7.13599e10i	−1.49748	+	2.59372i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.02773e9	−	1.78008e9i	0.0368157	−	0.0637666i
$966$		0			0
$967$	−6.31824e9	−	1.09435e10i	−0.224700	−	0.389192i	0.731529	−	0.681810i	$-0.238807\pi$
$967$	−6.31824e9	−	1.09435e10i	−0.224700	−	0.389192i	−0.956229	+	0.292618i	$0.905474\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.15235e10			−0.754477			−0.377239	−	0.926116i	$-0.623126\pi$
$971$	−2.15235e10			−0.754477			−0.377239	+	0.926116i	$0.623126\pi$
$972$		0			0
$973$	−2.74831e10			−0.956468
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.00079e10	−	1.73343e10i	−0.343332	−	0.594668i	0.641717	−	0.766941i	$-0.278222\pi$
$977$	−1.00079e10	−	1.73343e10i	−0.343332	−	0.594668i	−0.985049	+	0.172273i	$0.944889\pi$
$978$		0			0
$979$	−5.00206e9	+	8.66382e9i	−0.170376	+	0.295101i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.40447e10	−	4.16466e10i	0.807386	−	1.39843i	−0.107282	−	0.994229i	$-0.534215\pi$
$983$	2.40447e10	−	4.16466e10i	0.807386	−	1.39843i	0.914668	−	0.404206i	$-0.132452\pi$
$984$		0			0
$985$	2.67523e10	+	4.63363e10i	0.891936	+	1.54488i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.53725e10			0.505308
$990$		0			0
$991$	3.43048e10			1.11969			0.559844	−	0.828598i	$-0.310861\pi$
$991$	3.43048e10			1.11969			0.559844	+	0.828598i	$0.310861\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.08414e10	−	1.87779e10i	−0.348904	−	0.604320i
$996$		0			0
$997$	−1.00711e9	+	1.74437e9i	−0.0321843	+	0.0557449i	−0.881669	−	0.471869i	$-0.843580\pi$
$997$	−1.00711e9	+	1.74437e9i	−0.0321843	+	0.0557449i	0.849485	+	0.527613i	$0.176913\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.8.i.e.289.2		20
3.2	odd	2		144.8.i.e.97.5		20
4.3	odd	2		216.8.i.a.73.2		20
9.4	even	3	inner	432.8.i.e.145.2		20
9.5	odd	6		144.8.i.e.49.5		20
12.11	even	2		72.8.i.a.25.6	✓	20
36.23	even	6		72.8.i.a.49.6	yes	20
36.31	odd	6		216.8.i.a.145.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.i.a.25.6	✓	20	12.11	even	2
72.8.i.a.49.6	yes	20	36.23	even	6
144.8.i.e.49.5		20	9.5	odd	6
144.8.i.e.97.5		20	3.2	odd	2
216.8.i.a.73.2		20	4.3	odd	2
216.8.i.a.145.2		20	36.31	odd	6
432.8.i.e.145.2		20	9.4	even	3	inner
432.8.i.e.289.2		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	432.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-156.325 - 270.763i) q^{5} +(-626.107 + 1084.45i) q^{7} +(992.406 - 1718.90i) q^{11} +(6039.81 + 10461.3i) q^{13} +5453.94 q^{17} -51911.6 q^{19} +(-45567.2 - 78924.8i) q^{23} +(-9812.59 + 16995.9i) q^{25} +(-56370.4 + 97636.4i) q^{29} +(165765. + 287113. i) q^{31} +391505. q^{35} +313951. q^{37} +(95942.4 + 166177. i) q^{41} +(-84339.4 + 146080. i) q^{43} +(-240497. + 416554. i) q^{47} +(-372249. - 644754. i) q^{49} +562959. q^{53} -620552. q^{55} +(523339. + 906449. i) q^{59} +(587842. - 1.01817e6i) q^{61} +(1.88835e6 - 3.27072e6i) q^{65} +(-1.63387e6 - 2.82995e6i) q^{67} +2.19874e6 q^{71} +366453. q^{73} +(1.24270e6 + 2.15243e6i) q^{77} +(1.01679e6 - 1.76114e6i) q^{79} +(-1.79755e6 + 3.11344e6i) q^{83} +(-852588. - 1.47673e6i) q^{85} -5.04034e6 q^{89} -1.51263e7 q^{91} +(8.11508e6 + 1.40557e7i) q^{95} +(4.80998e6 - 8.33112e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 125 q^{5} - 1245 q^{7} + 6106 q^{11} + 4937 q^{13} - 48722 q^{17} + 26882 q^{19} + 19387 q^{23} - 218957 q^{25} + 46791 q^{29} + 185039 q^{31} + 83094 q^{35} + 108420 q^{37} + 638112 q^{41} - 892628 q^{43}+ \cdots + 17951260 q^{97}+O(q^{100}) \) 20 * q + 125 * q^5 - 1245 * q^7 + 6106 * q^11 + 4937 * q^13 - 48722 * q^17 + 26882 * q^19 + 19387 * q^23 - 218957 * q^25 + 46791 * q^29 + 185039 * q^31 + 83094 * q^35 + 108420 * q^37 + 638112 * q^41 - 892628 * q^43 - 230883 * q^47 - 1034741 * q^49 + 2872940 * q^53 - 1089998 * q^55 + 2172454 * q^59 - 1878325 * q^61 - 1239133 * q^65 - 531496 * q^67 + 3723056 * q^71 - 1804522 * q^73 - 6276543 * q^77 - 3607847 * q^79 + 10794491 * q^83 + 3597658 * q^85 - 32214888 * q^89 + 16117530 * q^91 - 756868 * q^95 + 17951260 * q^97

Introduction

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