LMFDB - Embedded newform 252.12.t.a.17.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(17,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("252.17");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.t (of order $6$, degree $2$, 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:	$193.622481501$
Analytic rank:	$0$
Dimension:	$60$
Relative dimension:	$30$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.4
Character	$\chi$	$=$	252.17
Dual form			252.12.t.a.89.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5207.12 - 9018.99i) q^{5} +(-35022.7 + 27399.6i) q^{7} +O(q^{10})$	$q+(-5207.12 - 9018.99i) q^{5} +(-35022.7 + 27399.6i) q^{7} +(-168067. - 97033.6i) q^{11} -1.37914e6i q^{13} +(3.27549e6 - 5.67332e6i) q^{17} +(1.51156e7 - 8.72702e6i) q^{19} +(1.35590e7 - 7.82828e6i) q^{23} +(-2.98141e7 + 5.16396e7i) q^{25} +1.83694e8i q^{29} +(9.50443e7 + 5.48738e7i) q^{31} +(4.29484e8 + 1.73196e8i) q^{35} +(-2.71429e8 - 4.70128e8i) q^{37} -4.83612e8 q^{41} +4.53850e8 q^{43} +(1.27874e9 + 2.21484e9i) q^{47} +(4.75848e8 - 1.91922e9i) q^{49} +(4.42190e9 + 2.55299e9i) q^{53} +2.02106e9i q^{55} +(3.14281e9 - 5.44351e9i) q^{59} +(9.92150e9 - 5.72818e9i) q^{61} +(-1.24384e10 + 7.18133e9i) q^{65} +(7.48676e9 - 1.29675e10i) q^{67} -8.90698e9i q^{71} +(1.91848e10 + 1.10764e10i) q^{73} +(8.54484e9 - 1.20660e9i) q^{77} +(-1.81139e10 - 3.13742e10i) q^{79} -2.60508e9 q^{83} -6.82235e10 q^{85} +(3.01748e10 + 5.22643e10i) q^{89} +(3.77879e10 + 4.83011e10i) q^{91} +(-1.57418e11 - 9.08853e10i) q^{95} +1.12008e11i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 60 q - 31278 q^{7}+O(q^{10}) $ 60 * q - 31278 * q^7	$ 60 q - 31278 q^{7} - 34858710 q^{19} - 323828862 q^{25} + 1186783866 q^{31} - 706751058 q^{37} + 1104997284 q^{43} + 7568925402 q^{49} - 5130550116 q^{61} - 22208949354 q^{67} + 54154878858 q^{73} - 61996342614 q^{79} + 107647163952 q^{85} - 2229336678 q^{91}+O(q^{100}) $ 60 * q - 31278 * q^7 - 34858710 * q^19 - 323828862 * q^25 + 1186783866 * q^31 - 706751058 * q^37 + 1104997284 * q^43 + 7568925402 * q^49 - 5130550116 * q^61 - 22208949354 * q^67 + 54154878858 * q^73 - 61996342614 * q^79 + 107647163952 * q^85 - 2229336678 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−5207.12	−	9018.99i	−0.745182	−	1.29069i	−0.950110	−	0.311916i	$-0.899029\pi$
$5$	−5207.12	−	9018.99i	−0.745182	−	1.29069i	0.204927	−	0.978777i	$-0.434304\pi$
$6$		0			0
$7$	−35022.7	+	27399.6i	−0.787608	+	0.616177i
$7$	−35022.7	+	27399.6i	−0.787608	+	0.616177i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−168067.	−	97033.6i	−0.314646	−	0.181661i	0.334357	−	0.942446i	$-0.391481\pi$
$11$	−168067.	−	97033.6i	−0.314646	−	0.181661i	−0.649004	+	0.760785i	$0.724814\pi$
$12$		0			0
$13$		−	1.37914e6i		−	1.03019i	−0.857132	−	0.515097i	$-0.827756\pi$
$13$		−	1.37914e6i		−	1.03019i	0.857132	−	0.515097i	$-0.172244\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.27549e6	−	5.67332e6i	0.559509	−	0.969098i	−0.438028	−	0.898961i	$-0.644323\pi$
$17$	3.27549e6	−	5.67332e6i	0.559509	−	0.969098i	0.997537	−	0.0701372i	$-0.0223437\pi$
$18$		0			0
$19$	1.51156e7	−	8.72702e6i	1.40050	−	0.808577i	0.406053	−	0.913850i	$-0.366905\pi$
$19$	1.51156e7	−	8.72702e6i	1.40050	−	0.808577i	0.994443	+	0.105273i	$0.0335715\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.35590e7	−	7.82828e6i	0.439262	−	0.253608i	−0.264022	−	0.964517i	$-0.585049\pi$
$23$	1.35590e7	−	7.82828e6i	0.439262	−	0.253608i	0.703285	+	0.710908i	$0.251716\pi$
$24$		0			0
$25$	−2.98141e7	+	5.16396e7i	−0.610593	+	1.05758i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.83694e8i			1.66305i	0.555488	+	0.831524i	$0.312531\pi$
$29$			1.83694e8i			1.66305i	−0.555488	+	0.831524i	$0.687469\pi$
$30$		0			0
$31$	9.50443e7	+	5.48738e7i	0.596261	+	0.344252i	0.767569	−	0.640966i	$-0.221466\pi$
$31$	9.50443e7	+	5.48738e7i	0.596261	+	0.344252i	−0.171308	+	0.985218i	$0.554799\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.29484e8	+	1.73196e8i	1.38221	+	0.557396i
$36$		0			0
$37$	−2.71429e8	−	4.70128e8i	−0.643496	−	1.11457i	−0.984647	−	0.174559i	$-0.944150\pi$
$37$	−2.71429e8	−	4.70128e8i	−0.643496	−	1.11457i	0.341150	−	0.940009i	$-0.389183\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.83612e8			−0.651908			−0.325954	−	0.945386i	$-0.605685\pi$
$41$	−4.83612e8			−0.651908			−0.325954	+	0.945386i	$0.605685\pi$
$42$		0			0
$43$	4.53850e8			0.470799			0.235399	−	0.971899i	$-0.424360\pi$
$43$	4.53850e8			0.470799			0.235399	+	0.971899i	$0.424360\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.27874e9	+	2.21484e9i	0.813286	+	1.40865i	0.910552	+	0.413394i	$0.135657\pi$
$47$	1.27874e9	+	2.21484e9i	0.813286	+	1.40865i	−0.0972661	+	0.995258i	$0.531010\pi$
$48$		0			0
$49$	4.75848e8	−	1.91922e9i	0.240652	−	0.970611i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.42190e9	+	2.55299e9i	1.45242	+	0.838554i	0.998618	−	0.0525498i	$-0.0167348\pi$
$53$	4.42190e9	+	2.55299e9i	1.45242	+	0.838554i	0.453800	+	0.891104i	$0.350068\pi$
$54$		0			0
$55$			2.02106e9i			0.541483i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.14281e9	−	5.44351e9i	0.572311	−	0.991272i	−0.424017	−	0.905654i	$-0.639380\pi$
$59$	3.14281e9	−	5.44351e9i	0.572311	−	0.991272i	0.996328	−	0.0856179i	$-0.0272864\pi$
$60$		0			0
$61$	9.92150e9	−	5.72818e9i	1.50405	−	0.868366i	0.504065	−	0.863666i	$-0.331837\pi$
$61$	9.92150e9	−	5.72818e9i	1.50405	−	0.868366i	0.999989	−	0.00469974i	$-0.00149598\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.24384e10	+	7.18133e9i	−1.32967	+	0.767683i
$66$		0			0
$67$	7.48676e9	−	1.29675e10i	0.677458	−	1.17339i	−0.298285	−	0.954477i	$-0.596415\pi$
$67$	7.48676e9	−	1.29675e10i	0.677458	−	1.17339i	0.975744	−	0.218916i	$-0.0702520\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.90698e9i		−	0.585881i	−0.956131	−	0.292941i	$-0.905366\pi$
$71$		−	8.90698e9i		−	0.585881i	0.956131	−	0.292941i	$-0.0946338\pi$
$72$		0			0
$73$	1.91848e10	+	1.10764e10i	1.08313	+	0.625347i	0.931740	−	0.363126i	$-0.118291\pi$
$73$	1.91848e10	+	1.10764e10i	1.08313	+	0.625347i	0.151393	+	0.988474i	$0.451624\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.54484e9	−	1.20660e9i	0.359753	−	0.0508001i
$78$		0			0
$79$	−1.81139e10	−	3.13742e10i	−0.662312	−	1.14716i	−0.980007	−	0.198965i	$-0.936242\pi$
$79$	−1.81139e10	−	3.13742e10i	−0.662312	−	1.14716i	0.317695	−	0.948193i	$-0.397091\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.60508e9			−0.0725926			−0.0362963	−	0.999341i	$-0.511556\pi$
$83$	−2.60508e9			−0.0725926			−0.0362963	+	0.999341i	$0.511556\pi$
$84$		0			0
$85$	−6.82235e10			−1.66775
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.01748e10	+	5.22643e10i	0.572795	+	0.992111i	0.996277	+	0.0862063i	$0.0274744\pi$
$89$	3.01748e10	+	5.22643e10i	0.572795	+	0.992111i	−0.423482	+	0.905905i	$0.639192\pi$
$90$		0			0
$91$	3.77879e10	+	4.83011e10i	0.634782	+	0.811389i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.57418e11	−	9.08853e10i	−2.08725	−	1.20507i
$96$		0			0
$97$			1.12008e11i			1.32435i	0.749349	+	0.662175i	$0.230366\pi$
$97$			1.12008e11i			1.32435i	−0.749349	+	0.662175i	$0.769634\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.56805e10	−	1.31082e11i	0.716500	−	1.24101i	−0.245878	−	0.969301i	$-0.579076\pi$
$101$	7.56805e10	−	1.31082e11i	0.716500	−	1.24101i	0.962378	−	0.271714i	$-0.0875903\pi$
$102$		0			0
$103$	−9.98044e10	+	5.76221e10i	−0.848291	+	0.489761i	−0.860074	−	0.510169i	$-0.829583\pi$
$103$	−9.98044e10	+	5.76221e10i	−0.848291	+	0.489761i	0.0117826	+	0.999931i	$0.496249\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.14547e10	+	2.97074e10i	−0.354661	+	0.204764i	−0.666736	−	0.745294i	$-0.732309\pi$
$107$	−5.14547e10	+	2.97074e10i	−0.354661	+	0.204764i	0.312075	+	0.950057i	$0.398976\pi$
$108$		0			0
$109$	−2.57839e10	+	4.46591e10i	−0.160511	+	0.278012i	−0.935052	−	0.354511i	$-0.884647\pi$
$109$	−2.57839e10	+	4.46591e10i	−0.160511	+	0.278012i	0.774541	+	0.632523i	$0.217981\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			2.83587e10i			0.144795i	0.997376	+	0.0723976i	$0.0230651\pi$
$113$			2.83587e10i			0.144795i	−0.997376	+	0.0723976i	$0.976935\pi$
$114$		0			0
$115$	−1.41206e11	−	8.15256e10i	−0.654661	−	0.377969i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.07303e10	+	2.88442e11i	0.156462	+	1.10803i
$120$		0			0
$121$	−1.23825e11	−	2.14471e11i	−0.433998	−	0.751707i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.12475e11			0.329648
$126$		0			0
$127$	3.46701e11			0.931181			0.465590	−	0.885000i	$-0.345842\pi$
$127$	3.46701e11			0.931181			0.465590	+	0.885000i	$0.345842\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.79175e10	−	6.56750e10i	−0.0858712	−	0.148733i	0.819891	−	0.572520i	$-0.194034\pi$
$131$	−3.79175e10	−	6.56750e10i	−0.0858712	−	0.148733i	−0.905762	+	0.423786i	$0.860701\pi$
$132$		0			0
$133$	−2.90273e11	+	7.19807e11i	−0.604815	+	1.49980i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.72293e11	−	2.72678e11i	−0.836081	−	0.482711i	0.0198494	−	0.999803i	$-0.493681\pi$
$137$	−4.72293e11	−	2.72678e11i	−0.836081	−	0.482711i	−0.855930	+	0.517092i	$0.827015\pi$
$138$		0			0
$139$		−	7.85601e11i		−	1.28416i	−0.766636	−	0.642082i	$-0.778071\pi$
$139$		−	7.85601e11i		−	1.28416i	0.766636	−	0.642082i	$-0.221929\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.33823e11	+	2.31788e11i	−0.187146	+	0.324147i
$144$		0			0
$145$	1.65673e12	−	9.56514e11i	2.14649	−	1.23927i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.86506e10	+	2.80885e10i	−0.0542705	+	0.0313331i	−0.526890	−	0.849934i	$-0.676642\pi$
$149$	−4.86506e10	+	2.80885e10i	−0.0542705	+	0.0313331i	0.472619	+	0.881267i	$0.343309\pi$
$150$		0			0
$151$	5.26092e11	−	9.11218e11i	0.545367	−	0.944603i	−0.453217	−	0.891400i	$-0.649724\pi$
$151$	5.26092e11	−	9.11218e11i	0.545367	−	0.944603i	0.998584	−	0.0532027i	$-0.0169430\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	1.14294e12i		−	1.02612i
$156$		0			0
$157$	4.51539e11	+	2.60696e11i	0.377787	+	0.218115i	0.676855	−	0.736116i	$-0.263342\pi$
$157$	4.51539e11	+	2.60696e11i	0.377787	+	0.218115i	−0.299068	+	0.954232i	$0.596676\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.60380e11	+	6.45678e11i	−0.189699	+	0.470407i
$162$		0			0
$163$	−4.34151e11	−	7.51971e11i	−0.295535	−	0.511881i	0.679574	−	0.733607i	$-0.262164\pi$
$163$	−4.34151e11	−	7.51971e11i	−0.295535	−	0.511881i	−0.975109	+	0.221725i	$0.928831\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.96368e11			0.414857			0.207428	−	0.978250i	$-0.433491\pi$
$167$	6.96368e11			0.414857			0.207428	+	0.978250i	$0.433491\pi$
$168$		0			0
$169$	−1.09861e11			−0.0613008
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.99390e11	−	1.21138e12i	−0.343136	−	0.594329i	0.641878	−	0.766807i	$-0.278156\pi$
$173$	−6.99390e11	−	1.21138e12i	−0.343136	−	0.594329i	−0.985013	+	0.172479i	$0.944822\pi$
$174$		0			0
$175$	−3.70735e11	−	2.62545e12i	−0.170747	−	1.20919i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.68469e12	+	9.72658e11i	0.685219	+	0.395611i	0.801818	−	0.597568i	$-0.203866\pi$
$179$	1.68469e12	+	9.72658e11i	0.685219	+	0.395611i	−0.116600	+	0.993179i	$0.537199\pi$
$180$		0			0
$181$			2.70280e12i			1.03415i	0.855941	+	0.517074i	$0.172979\pi$
$181$			2.70280e12i			1.03415i	−0.855941	+	0.517074i	$0.827021\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.82672e12	+	4.89602e12i	−0.959044	+	1.66111i
$186$		0			0
$187$	−1.10100e12	+	6.35665e11i	−0.352095	+	0.203282i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.01455e12	+	2.89515e12i	−1.42741	+	0.824115i	−0.996916	−	0.0784795i	$-0.974993\pi$
$191$	−5.01455e12	+	2.89515e12i	−1.42741	+	0.824115i	−0.430493	+	0.902594i	$0.641660\pi$
$192$		0			0
$193$	−2.21186e11	+	3.83105e11i	−0.0594555	+	0.102980i	−0.894221	−	0.447626i	$-0.852270\pi$
$193$	−2.21186e11	+	3.83105e11i	−0.0594555	+	0.102980i	0.834766	+	0.550605i	$0.185603\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.70428e12i		−	1.60986i	−0.593370	−	0.804930i	$-0.702203\pi$
$197$		−	6.70428e12i		−	1.60986i	0.593370	−	0.804930i	$-0.297797\pi$
$198$		0			0
$199$	−5.38677e12	−	3.11006e12i	−1.22359	−	0.706441i	−0.257911	−	0.966169i	$-0.583034\pi$
$199$	−5.38677e12	−	3.11006e12i	−1.22359	−	0.706441i	−0.965682	+	0.259727i	$0.916367\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.03314e12	−	6.43344e12i	−1.02473	−	1.30983i
$204$		0			0
$205$	2.51823e12	+	4.36170e12i	0.485790	+	0.841413i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.38726e12			−0.587548
$210$		0			0
$211$	1.81885e12			0.299395			0.149697	−	0.988732i	$-0.452170\pi$
$211$	1.81885e12			0.299395			0.149697	+	0.988732i	$0.452170\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.36325e12	−	4.09327e12i	−0.350831	−	0.607657i
$216$		0			0
$217$	−4.83223e12	+	6.82349e11i	−0.681740	+	0.0962672i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.82428e12	−	4.51735e12i	−0.998360	−	0.576403i
$222$		0			0
$223$		−	1.00250e13i		−	1.21733i	−0.793428	−	0.608664i	$-0.791706\pi$
$223$		−	1.00250e13i		−	1.21733i	0.793428	−	0.608664i	$-0.208294\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.68912e12	−	6.38974e12i	0.406238	−	0.703624i	−0.588227	−	0.808696i	$-0.700174\pi$
$227$	3.68912e12	−	6.38974e12i	0.406238	−	0.703624i	0.994465	+	0.105072i	$0.0335072\pi$
$228$		0			0
$229$	−3.97029e12	+	2.29225e12i	−0.416608	+	0.240529i	−0.693625	−	0.720336i	$-0.743987\pi$
$229$	−3.97029e12	+	2.29225e12i	−0.416608	+	0.240529i	0.277017	+	0.960865i	$0.410654\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.35489e12	+	3.09165e12i	−0.510849	+	0.294939i	−0.733183	−	0.680032i	$-0.761966\pi$
$233$	−5.35489e12	+	3.09165e12i	−0.510849	+	0.294939i	0.222333	+	0.974971i	$0.428633\pi$
$234$		0			0
$235$	1.33171e13	−	2.30659e13i	1.21209	−	2.09941i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.69238e12i			0.223330i	0.993746	+	0.111665i	$0.0356184\pi$
$239$			2.69238e12i			0.223330i	−0.993746	+	0.111665i	$0.964382\pi$
$240$		0			0
$241$	1.86755e13	+	1.07823e13i	1.47971	+	0.854313i	0.999736	−	0.0229671i	$-0.00731131\pi$
$241$	1.86755e13	+	1.07823e13i	1.47971	+	0.854313i	0.479978	+	0.877281i	$0.340645\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.97872e13	+	5.70192e12i	−1.43209	+	0.412674i
$246$		0			0
$247$	−1.20358e13	−	2.08466e13i	−0.832992	−	1.44278i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.85222e13			−1.80708			−0.903539	−	0.428505i	$-0.859040\pi$
$251$	−2.85222e13			−1.80708			−0.903539	+	0.428505i	$0.859040\pi$
$252$		0			0
$253$	−3.03842e12			−0.184283
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.04241e13	−	1.80550e13i	−0.579969	−	1.00454i	−0.995482	−	0.0949482i	$-0.969731\pi$
$257$	−1.04241e13	−	1.80550e13i	−0.579969	−	1.00454i	0.415514	−	0.909587i	$-0.363602\pi$
$258$		0			0
$259$	2.23875e13	+	9.02810e12i	1.19359	+	0.481335i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.03565e13	+	5.97932e12i	0.507523	+	0.293018i	0.731815	−	0.681504i	$-0.238674\pi$
$263$	1.03565e13	+	5.97932e12i	0.507523	+	0.293018i	−0.224292	+	0.974522i	$0.572007\pi$
$264$		0			0
$265$		−	5.31748e13i		−	2.49950i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.46727e12	+	1.46657e13i	−0.366527	+	0.634843i	−0.989020	−	0.147782i	$-0.952787\pi$
$269$	−8.46727e12	+	1.46657e13i	−0.366527	+	0.634843i	0.622493	+	0.782625i	$0.286120\pi$
$270$		0			0
$271$	1.60700e13	−	9.27803e12i	0.667860	−	0.385589i	−0.127405	−	0.991851i	$-0.540665\pi$
$271$	1.60700e13	−	9.27803e12i	0.667860	−	0.385589i	0.795265	+	0.606262i	$0.207332\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00215e13	−	5.78594e12i	0.384242	−	0.221842i
$276$		0			0
$277$	1.86414e13	−	3.22878e13i	0.686815	−	1.18960i	−0.286048	−	0.958215i	$-0.592342\pi$
$277$	1.86414e13	−	3.22878e13i	0.686815	−	1.18960i	0.972863	−	0.231383i	$-0.0743251\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.15344e13i			1.07374i	0.843665	+	0.536870i	$0.180394\pi$
$281$			3.15344e13i			1.07374i	−0.843665	+	0.536870i	$0.819606\pi$
$282$		0			0
$283$	1.11304e13	+	6.42611e12i	0.364488	+	0.210437i	0.671048	−	0.741414i	$-0.265845\pi$
$283$	1.11304e13	+	6.42611e12i	0.364488	+	0.210437i	−0.306560	+	0.951851i	$0.599178\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.69374e13	−	1.32508e13i	0.513448	−	0.401691i
$288$		0			0
$289$	−4.32173e12	−	7.48545e12i	−0.126101	−	0.218414i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.86062e13			−0.773905			−0.386953	−	0.922100i	$-0.626472\pi$
$293$	−2.86062e13			−0.773905			−0.386953	+	0.922100i	$0.626472\pi$
$294$		0			0
$295$	−6.54600e13			−1.70590
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.07963e13	−	1.86997e13i	−0.261266	−	0.452526i
$300$		0			0
$301$	−1.58950e13	+	1.24353e13i	−0.370805	+	0.290095i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.03325e14	−	5.96546e13i	−2.24159	−	1.29418i
$306$		0			0
$307$			1.30334e13i			0.272770i	0.990656	+	0.136385i	$0.0435484\pi$
$307$			1.30334e13i			0.272770i	−0.990656	+	0.136385i	$0.956452\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.13793e13	+	5.43505e13i	−0.611590	+	1.05931i	0.379382	+	0.925240i	$0.376137\pi$
$311$	−3.13793e13	+	5.43505e13i	−0.611590	+	1.05931i	−0.990972	+	0.134066i	$0.957197\pi$
$312$		0			0
$313$	4.31485e13	−	2.49118e13i	0.811842	−	0.468717i	−0.0357529	−	0.999361i	$-0.511383\pi$
$313$	4.31485e13	−	2.49118e13i	0.811842	−	0.468717i	0.847595	+	0.530643i	$0.178050\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.92295e12	+	3.41962e12i	−0.103923	+	0.0600000i	−0.551061	−	0.834465i	$-0.685777\pi$
$317$	−5.92295e12	+	3.41962e12i	−0.103923	+	0.0600000i	0.447138	+	0.894465i	$0.352443\pi$
$318$		0			0
$319$	1.78244e13	−	3.08728e13i	0.302111	−	0.523272i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.14341e14i		−	1.80963i
$324$		0			0
$325$	7.12181e13	+	4.11178e13i	1.08951	+	0.629030i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.05471e14	−	4.25326e13i	−1.50853	−	0.608338i
$330$		0			0
$331$	−6.85881e13	−	1.18798e14i	−0.948844	−	1.64345i	−0.747866	−	0.663850i	$-0.768921\pi$
$331$	−6.85881e13	−	1.18798e14i	−0.948844	−	1.64345i	−0.200978	−	0.979596i	$-0.564412\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.55938e14			−2.01932
$336$		0			0
$337$	−1.22261e14			−1.53222			−0.766112	−	0.642707i	$-0.777811\pi$
$337$	−1.22261e14			−1.53222			−0.766112	+	0.642707i	$0.777811\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.06492e13	−	1.84450e13i	−0.125074	−	0.216635i
$342$		0			0
$343$	3.59203e13	+	8.02541e13i	0.408529	+	0.912745i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.04567e14	+	6.03720e13i	1.11579	+	0.644204i	0.940324	−	0.340281i	$-0.110522\pi$
$347$	1.04567e14	+	6.03720e13i	1.11579	+	0.644204i	0.175470	+	0.984485i	$0.443855\pi$
$348$		0			0
$349$		−	1.25933e13i		−	0.130196i	−0.997879	−	0.0650982i	$-0.979264\pi$
$349$		−	1.25933e13i		−	0.130196i	0.997879	−	0.0650982i	$-0.0207361\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.72065e13	+	6.44436e13i	−0.361292	+	0.625776i	−0.988174	−	0.153338i	$-0.950998\pi$
$353$	−3.72065e13	+	6.44436e13i	−0.361292	+	0.625776i	0.626882	+	0.779114i	$0.284331\pi$
$354$		0			0
$355$	−8.03320e13	+	4.63797e13i	−0.756193	+	0.436588i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.09816e12	−	1.21138e12i	0.0185703	−	0.0107216i	−0.490686	−	0.871336i	$-0.663254\pi$
$359$	2.09816e12	−	1.21138e12i	0.0185703	−	0.0107216i	0.509256	+	0.860615i	$0.329920\pi$
$360$		0			0
$361$	9.40768e13	−	1.62946e14i	0.807594	−	1.39879i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2.30704e14i		−	1.86399i
$366$		0			0
$367$	1.10314e14	+	6.36899e13i	0.864904	+	0.499352i	0.865651	−	0.500647i	$-0.166905\pi$
$367$	1.10314e14	+	6.36899e13i	0.864904	+	0.499352i	−0.000747579		1.00000i	$0.500238\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.24818e14	+	3.17460e13i	−1.66063	+	0.234495i
$372$		0			0
$373$	−4.11880e13	−	7.13396e13i	−0.295374	−	0.511602i	0.679698	−	0.733492i	$-0.262111\pi$
$373$	−4.11880e13	−	7.13396e13i	−0.295374	−	0.511602i	−0.975072	+	0.221890i	$0.928777\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.53339e14			1.71326
$378$		0			0
$379$	2.43006e14			1.59625			0.798126	−	0.602491i	$-0.205825\pi$
$379$	2.43006e14			1.59625			0.798126	+	0.602491i	$0.205825\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.30410e14	−	2.25877e14i	−0.808570	−	1.40048i	−0.913854	−	0.406043i	$-0.866908\pi$
$383$	−1.30410e14	−	2.25877e14i	−0.808570	−	1.40048i	0.105284	−	0.994442i	$-0.466425\pi$
$384$		0			0
$385$	−5.53763e13	−	7.07830e13i	−0.333649	−	0.426476i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.63809e14	+	1.52310e14i	1.50164	+	0.866974i	0.999998	+	0.00190082i	$0.000605050\pi$
$389$	2.63809e14	+	1.52310e14i	1.50164	+	0.866974i	0.501645	+	0.865073i	$0.332728\pi$
$390$		0			0
$391$		−	1.02566e14i		−	0.567584i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.88642e14	+	3.26738e14i	−0.987086	+	1.70968i
$396$		0			0
$397$	−1.73660e14	+	1.00263e14i	−0.883796	+	0.510260i	−0.871908	−	0.489670i	$-0.837117\pi$
$397$	−1.73660e14	+	1.00263e14i	−0.883796	+	0.510260i	−0.0118877	+	0.999929i	$0.503784\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.72733e13	−	3.30668e13i	0.275841	−	0.159257i	−0.355698	−	0.934601i	$-0.615757\pi$
$401$	5.72733e13	−	3.30668e13i	0.275841	−	0.159257i	0.631539	+	0.775344i	$0.282424\pi$
$402$		0			0
$403$	7.56786e13	−	1.31079e14i	0.354646	−	0.614265i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.05351e14i			0.467593i
$408$		0			0
$409$	1.31382e14	+	7.58535e13i	0.567620	+	0.327716i	0.756198	−	0.654342i	$-0.227055\pi$
$409$	1.31382e14	+	7.58535e13i	0.567620	+	0.327716i	−0.188578	+	0.982058i	$0.560388\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.90805e13	+	2.76758e14i	0.160042	+	1.13338i
$414$		0			0
$415$	1.35650e13	+	2.34952e13i	0.0540947	+	0.0936948i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.03707e14			1.52718			0.763589	−	0.645703i	$-0.223436\pi$
$419$	4.03707e14			1.52718			0.763589	+	0.645703i	$0.223436\pi$
$420$		0			0
$421$	−2.12933e14			−0.784678			−0.392339	−	0.919821i	$-0.628334\pi$
$421$	−2.12933e14			−0.784678			−0.392339	+	0.919821i	$0.628334\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.95312e14	+	3.38290e14i	0.683265	+	1.18345i
$426$		0			0
$427$	−1.90527e14	+	4.72462e14i	−0.649537	+	1.61069i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.45938e14	+	1.41992e14i	0.796527	+	0.459875i	0.842255	−	0.539079i	$-0.181228\pi$
$431$	2.45938e14	+	1.41992e14i	0.796527	+	0.459875i	−0.0457284	+	0.998954i	$0.514561\pi$
$432$		0			0
$433$		−	1.19296e14i		−	0.376655i	−0.982106	−	0.188327i	$-0.939693\pi$
$433$		−	1.19296e14i		−	0.376655i	0.982106	−	0.188327i	$-0.0603066\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.36635e14	−	2.36659e14i	0.410123	−	0.710355i
$438$		0			0
$439$	1.23314e14	−	7.11956e13i	0.360960	−	0.208400i	−0.308542	−	0.951211i	$-0.599841\pi$
$439$	1.23314e14	−	7.11956e13i	0.360960	−	0.208400i	0.669502	+	0.742811i	$0.266508\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.74277e14	+	1.58354e14i	−0.763782	+	0.440970i	−0.830652	−	0.556792i	$-0.812032\pi$
$443$	−2.74277e14	+	1.58354e14i	−0.763782	+	0.440970i	0.0668700	+	0.997762i	$0.478699\pi$
$444$		0			0
$445$	3.14248e14	−	5.44293e14i	0.853674	−	1.47861i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.15459e14i		−	1.07442i	−0.843449	−	0.537209i	$-0.819479\pi$
$449$		−	4.15459e14i		−	1.07442i	0.843449	−	0.537209i	$-0.180521\pi$
$450$		0			0
$451$	8.12793e13	+	4.69266e13i	0.205121	+	0.118426i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.38861e14	−	5.92318e14i	0.574227	−	1.42394i
$456$		0			0
$457$	1.50909e14	+	2.61382e14i	0.354141	+	0.613390i	0.986971	−	0.160901i	$-0.0514400\pi$
$457$	1.50909e14	+	2.61382e14i	0.354141	+	0.613390i	−0.632830	+	0.774291i	$0.718107\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.05989e13			−0.202660			−0.101330	−	0.994853i	$-0.532310\pi$
$461$	−9.05989e13			−0.202660			−0.101330	+	0.994853i	$0.532310\pi$
$462$		0			0
$463$	−5.80552e14			−1.26808			−0.634038	−	0.773302i	$-0.718604\pi$
$463$	−5.80552e14			−1.26808			−0.634038	+	0.773302i	$0.718604\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.15748e14	−	5.46892e14i	−0.657807	−	1.13935i	−0.981182	−	0.193084i	$-0.938151\pi$
$467$	−3.15748e14	−	5.46892e14i	−0.657807	−	1.13935i	0.323376	−	0.946271i	$-0.395182\pi$
$468$		0			0
$469$	9.30970e13	+	6.59289e14i	0.189446	+	1.34161i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.62772e13	−	4.40386e13i	−0.148135	−	0.0855259i
$474$		0			0
$475$			1.04075e15i			1.97485i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.84599e14	+	6.66145e14i	−0.696887	+	1.20704i	0.272653	+	0.962112i	$0.412099\pi$
$479$	−3.84599e14	+	6.66145e14i	−0.696887	+	1.20704i	−0.969540	+	0.244932i	$0.921234\pi$
$480$		0			0
$481$	−6.48371e14	+	3.74337e14i	−1.14822	+	0.662926i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.01020e15	−	5.83237e14i	1.70933	−	0.986882i
$486$		0			0
$487$	−1.58760e14	+	2.74980e14i	−0.262622	+	0.454875i	−0.966938	−	0.255012i	$-0.917921\pi$
$487$	−1.58760e14	+	2.74980e14i	−0.262622	+	0.454875i	0.704316	+	0.709887i	$0.251254\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			1.91403e14i			0.302692i	0.988481	+	0.151346i	$0.0483608\pi$
$491$			1.91403e14i			0.302692i	−0.988481	+	0.151346i	$0.951639\pi$
$492$		0			0
$493$	1.04215e15	+	6.01687e14i	1.61166	+	0.930491i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.44048e14	+	3.11946e14i	0.361007	+	0.461445i
$498$		0			0
$499$	−1.92632e14	−	3.33648e14i	−0.278725	−	0.482765i	0.692343	−	0.721568i	$-0.256578\pi$
$499$	−1.92632e14	−	3.33648e14i	−0.278725	−	0.482765i	−0.971068	+	0.238803i	$0.923245\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.66152e14			0.230081			0.115041	−	0.993361i	$-0.463300\pi$
$503$	1.66152e14			0.230081			0.115041	+	0.993361i	$0.463300\pi$
$504$		0			0
$505$	−1.57631e15			−2.13569
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.88507e14	+	8.46119e14i	0.633757	+	1.09770i	0.986777	+	0.162084i	$0.0518215\pi$
$509$	4.88507e14	+	8.46119e14i	0.633757	+	1.09770i	−0.353020	+	0.935616i	$0.614845\pi$
$510$		0			0
$511$	−9.75392e14	+	1.37733e14i	−1.23841	+	0.174873i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.03939e15	+	6.00090e14i	1.26426	+	0.729923i
$516$		0			0
$517$		−	4.96322e14i		−	0.590970i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.39601e14	−	7.61411e14i	0.501708	−	0.868983i	−0.498290	−	0.867010i	$-0.666039\pi$
$521$	4.39601e14	−	7.61411e14i	0.501708	−	0.868983i	0.999998	−	0.00197301i	$-0.000628028\pi$
$522$		0			0
$523$	2.13255e14	−	1.23123e14i	0.238308	−	0.137587i	−0.376091	−	0.926583i	$-0.622732\pi$
$523$	2.13255e14	−	1.23123e14i	0.238308	−	0.137587i	0.614399	+	0.788995i	$0.289399\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.22633e14	−	3.59477e14i	0.667227	−	0.385224i
$528$		0			0
$529$	−3.53841e14	+	6.12871e14i	−0.371366	+	0.643224i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			6.66968e14i			0.671592i
$534$		0			0
$535$	5.35861e14	+	3.09380e14i	0.528575	+	0.305173i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.66203e14	+	2.76384e14i	−0.252043	+	0.261682i
$540$		0			0
$541$	4.88280e14	+	8.45725e14i	0.452985	+	0.784593i	0.998570	−	0.0534626i	$-0.0170258\pi$
$541$	4.88280e14	+	8.45725e14i	0.452985	+	0.784593i	−0.545585	+	0.838056i	$0.683692\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.37040e14			0.478438
$546$		0			0
$547$	9.07245e14			0.792126			0.396063	−	0.918223i	$-0.370376\pi$
$547$	9.07245e14			0.792126			0.396063	+	0.918223i	$0.370376\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.60310e15	+	2.77665e15i	1.34470	+	2.32909i
$552$		0			0
$553$	1.49404e15	+	6.02494e14i	1.22849	+	0.495409i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.01071e14	+	2.89294e14i	0.396001	+	0.228631i	0.684757	−	0.728772i	$-0.259908\pi$
$557$	5.01071e14	+	2.89294e14i	0.396001	+	0.228631i	−0.288756	+	0.957403i	$0.593242\pi$
$558$		0			0
$559$		−	6.25921e14i		−	0.485014i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.15844e15	−	2.00648e15i	0.863132	−	1.49499i	−0.00575714	−	0.999983i	$-0.501833\pi$
$563$	1.15844e15	−	2.00648e15i	0.863132	−	1.49499i	0.868890	−	0.495006i	$-0.164834\pi$
$564$		0			0
$565$	2.55767e14	−	1.47667e14i	0.186886	−	0.107899i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.18151e14	+	3.56890e14i	−0.434488	+	0.250851i	−0.701257	−	0.712909i	$-0.747377\pi$
$569$	−6.18151e14	+	3.56890e14i	−0.434488	+	0.250851i	0.266769	+	0.963760i	$0.414044\pi$
$570$		0			0
$571$	−7.89354e14	+	1.36720e15i	−0.544218	+	0.942614i	0.454437	+	0.890779i	$0.349840\pi$
$571$	−7.89354e14	+	1.36720e15i	−0.544218	+	0.942614i	−0.998656	+	0.0518352i	$0.983493\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			9.33573e14i			0.619406i
$576$		0			0
$577$	−1.53166e15	−	8.84305e14i	−0.997002	−	0.575619i	−0.0896420	−	0.995974i	$-0.528572\pi$
$577$	−1.53166e15	−	8.84305e14i	−0.997002	−	0.575619i	−0.907360	+	0.420355i	$0.861906\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	9.12370e13	−	7.13783e13i	0.0571745	−	0.0447299i
$582$		0			0
$583$	−4.95450e14	−	8.58145e14i	−0.304665	−	0.527696i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.08009e15			1.23189			0.615945	−	0.787789i	$-0.288774\pi$
$587$	2.08009e15			1.23189			0.615945	+	0.787789i	$0.288774\pi$
$588$		0			0
$589$	1.91554e15			1.11342
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.79001e14	−	1.34927e15i	−0.436252	−	0.755610i	0.561145	−	0.827717i	$-0.310361\pi$
$593$	−7.79001e14	−	1.34927e15i	−0.436252	−	0.755610i	−0.997397	+	0.0721072i	$0.977028\pi$
$594$		0			0
$595$	2.38937e15	−	1.86930e15i	1.31353	−	1.02763i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.03668e15	+	5.98529e14i	0.549286	+	0.317130i	0.748834	−	0.662758i	$-0.230614\pi$
$599$	1.03668e15	+	5.98529e14i	0.549286	+	0.317130i	−0.199548	+	0.979888i	$0.563947\pi$
$600$		0			0
$601$			1.28904e15i			0.670588i	0.942114	+	0.335294i	$0.108836\pi$
$601$			1.28904e15i			0.670588i	−0.942114	+	0.335294i	$0.891164\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.28954e15	+	2.23355e15i	−0.646816	+	1.12032i
$606$		0			0
$607$	−7.04075e14	+	4.06498e14i	−0.346801	+	0.200226i	−0.663276	−	0.748375i	$-0.730834\pi$
$607$	−7.04075e14	+	4.06498e14i	−0.346801	+	0.200226i	0.316474	+	0.948601i	$0.397501\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.05457e15	−	1.76356e15i	1.45119	−	0.837843i
$612$		0			0
$613$	−4.23385e14	+	7.33325e14i	−0.197562	+	0.342187i	−0.947737	−	0.319052i	$-0.896636\pi$
$613$	−4.23385e14	+	7.33325e14i	−0.197562	+	0.342187i	0.750175	+	0.661239i	$0.229969\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	7.44868e14i		−	0.335360i	−0.985841	−	0.167680i	$-0.946372\pi$
$617$		−	7.44868e14i		−	0.335360i	0.985841	−	0.167680i	$-0.0536275\pi$
$618$		0			0
$619$	−3.11345e15	−	1.79755e15i	−1.37703	−	0.795027i	−0.385227	−	0.922822i	$-0.625877\pi$
$619$	−3.11345e15	−	1.79755e15i	−1.37703	−	0.795027i	−0.991801	+	0.127794i	$0.959210\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.48883e15	−	1.00366e15i	−1.06245	−	0.428451i
$624$		0			0
$625$	8.70097e14	+	1.50705e15i	0.364945	+	0.632104i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.55625e15			−1.44017
$630$		0			0
$631$	5.35268e14			0.213015			0.106507	−	0.994312i	$-0.466033\pi$
$631$	5.35268e14			0.213015			0.106507	+	0.994312i	$0.466033\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.80531e15	−	3.12689e15i	−0.693899	−	1.20187i
$636$		0			0
$637$	−2.64686e15	−	6.56260e14i	−0.999919	−	0.247918i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.55985e15	−	9.00580e14i	−0.569330	−	0.328703i	0.187552	−	0.982255i	$-0.439945\pi$
$641$	−1.55985e15	−	9.00580e14i	−0.569330	−	0.328703i	−0.756882	+	0.653552i	$0.773278\pi$
$642$		0			0
$643$		−	2.64079e14i		−	0.0947487i	−0.998877	−	0.0473744i	$-0.984915\pi$
$643$		−	2.64079e14i		−	0.0947487i	0.998877	−	0.0473744i	$-0.0150854\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.52268e15	+	2.63736e15i	−0.528001	+	0.914524i	0.471466	+	0.881884i	$0.343725\pi$
$647$	−1.52268e15	+	2.63736e15i	−0.528001	+	0.914524i	−0.999467	+	0.0326401i	$0.989608\pi$
$648$		0			0
$649$	−1.05641e15	+	6.09916e14i	−0.360151	+	0.207934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.93586e15	+	1.69502e15i	−0.967639	+	0.558667i	−0.898516	−	0.438942i	$-0.855354\pi$
$653$	−2.93586e15	+	1.69502e15i	−0.967639	+	0.558667i	−0.0691233	+	0.997608i	$0.522020\pi$
$654$		0			0
$655$	−3.94882e14	+	6.83955e14i	−0.127979	+	0.221667i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1.38274e15i		−	0.433383i	−0.976240	−	0.216691i	$-0.930473\pi$
$659$		−	1.38274e15i		−	0.433383i	0.976240	−	0.216691i	$-0.0695266\pi$
$660$		0			0
$661$	2.23861e15	+	1.29246e15i	0.690034	+	0.398392i	0.803625	−	0.595136i	$-0.202902\pi$
$661$	2.23861e15	+	1.29246e15i	0.690034	+	0.398392i	−0.113591	+	0.993528i	$0.536235\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.00342e15	−	1.13015e15i	2.38647	−	0.336989i
$666$		0			0
$667$	1.43800e15	+	2.49070e15i	0.421763	+	0.730515i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.22330e15			−0.630993
$672$		0			0
$673$	1.92098e15			0.536340			0.268170	−	0.963372i	$-0.413581\pi$
$673$	1.92098e15			0.536340			0.268170	+	0.963372i	$0.413581\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.11517e14	−	7.12768e14i	−0.111212	−	0.192624i	0.805047	−	0.593210i	$-0.202140\pi$
$677$	−4.11517e14	−	7.12768e14i	−0.111212	−	0.192624i	−0.916259	+	0.400586i	$0.868806\pi$
$678$		0			0
$679$	−3.06896e15	−	3.92280e15i	−0.816034	−	1.04307i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.95754e15	−	2.28488e15i	−1.01885	−	0.588234i	−0.105081	−	0.994464i	$-0.533510\pi$
$683$	−3.95754e15	−	2.28488e15i	−1.01885	−	0.588234i	−0.913771	+	0.406229i	$0.866843\pi$
$684$		0			0
$685$			5.67947e15i			1.43883i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.52092e15	−	6.09841e15i	0.863874	−	1.49627i
$690$		0			0
$691$	1.82242e15	−	1.05218e15i	0.440068	−	0.254073i	−0.263558	−	0.964643i	$-0.584896\pi$
$691$	1.82242e15	−	1.05218e15i	0.440068	−	0.254073i	0.703627	+	0.710570i	$0.251563\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.08533e15	+	4.09072e15i	−1.65746	+	0.956936i
$696$		0			0
$697$	−1.58407e15	+	2.74369e15i	−0.364749	+	0.631763i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			2.98802e15i			0.666706i	0.942802	+	0.333353i	$0.108180\pi$
$701$			2.98802e15i			0.666706i	−0.942802	+	0.333353i	$0.891820\pi$
$702$		0			0
$703$	−8.20564e15	−	4.73753e15i	−1.80243	−	1.04063i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.41077e14	+	6.66447e15i	0.200363	+	1.41892i
$708$		0			0
$709$	−2.62214e15	−	4.54167e15i	−0.549669	−	0.952054i	−0.998297	−	0.0583357i	$-0.981421\pi$
$709$	−2.62214e15	−	4.54167e15i	−0.549669	−	0.952054i	0.448628	−	0.893718i	$-0.351913\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.71827e15			0.349220
$714$		0			0
$715$	2.78732e15			0.557833
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.07375e14	+	3.59184e14i	0.0402483	+	0.0697121i	0.885448	−	0.464739i	$-0.153852\pi$
$719$	2.07375e14	+	3.59184e14i	0.0402483	+	0.0697121i	−0.845200	+	0.534451i	$0.820518\pi$
$720$		0			0
$721$	1.91659e15	−	4.75268e15i	0.366341	−	0.908437i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.48586e15	−	5.47666e15i	−1.75880	−	1.01545i
$726$		0			0
$727$		−	7.72747e15i		−	1.41123i	−0.708595	−	0.705616i	$-0.750671\pi$
$727$		−	7.72747e15i		−	1.41123i	0.708595	−	0.705616i	$-0.249329\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.48658e15	−	2.57483e15i	0.263416	−	0.456250i
$732$		0			0
$733$	9.32485e15	−	5.38370e15i	1.62768	−	0.939743i	0.642901	−	0.765950i	$-0.277731\pi$
$733$	9.32485e15	−	5.38370e15i	1.62768	−	0.939743i	0.984782	−	0.173794i	$-0.0556025\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.51656e15	+	1.45293e15i	−0.426320	+	0.246136i
$738$		0			0
$739$	3.43877e15	−	5.95612e15i	0.573929	−	0.994075i	−0.422228	−	0.906490i	$-0.638752\pi$
$739$	3.43877e15	−	5.95612e15i	0.573929	−	0.994075i	0.996157	−	0.0875849i	$-0.0279149\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			3.82914e15i			0.620387i	0.950673	+	0.310194i	$0.100394\pi$
$743$			3.82914e15i			0.620387i	−0.950673	+	0.310194i	$0.899606\pi$
$744$		0			0
$745$	5.06659e14	+	2.92520e14i	0.0808828	+	0.0466977i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.88109e14	−	2.45027e15i	0.153163	−	0.379808i
$750$		0			0
$751$	3.48621e15	+	6.03830e15i	0.532518	+	0.922348i	0.999279	+	0.0379646i	$0.0120874\pi$
$751$	3.48621e15	+	6.03830e15i	0.532518	+	0.922348i	−0.466761	+	0.884383i	$0.654579\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.09577e16			−1.62559
$756$		0			0
$757$	9.63464e15			1.40867			0.704333	−	0.709869i	$-0.251246\pi$
$757$	9.63464e15			1.40867			0.704333	+	0.709869i	$0.251246\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.03750e15	+	3.52905e15i	0.289389	+	0.501236i	0.973664	−	0.227988i	$-0.0732147\pi$
$761$	2.03750e15	+	3.52905e15i	0.289389	+	0.501236i	−0.684275	+	0.729224i	$0.739881\pi$
$762$		0			0
$763$	−3.20620e14	−	2.27055e15i	−0.0448855	−	0.317868i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.50735e15	−	4.33437e15i	−1.02120	−	0.589592i
$768$		0			0
$769$		−	1.17374e16i		−	1.57390i	−0.617017	−	0.786950i	$-0.711659\pi$
$769$		−	1.17374e16i		−	1.57390i	0.617017	−	0.786950i	$-0.288341\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.78721e15	−	4.82759e15i	0.363231	−	0.629134i	−0.625260	−	0.780417i	$-0.715007\pi$
$773$	2.78721e15	−	4.82759e15i	0.363231	−	0.629134i	0.988491	+	0.151283i	$0.0483404\pi$
$774$		0			0
$775$	−5.66732e15	+	3.27203e15i	−0.728146	+	0.420395i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.31012e15	+	4.22050e15i	−0.912995	+	0.527118i
$780$		0			0
$781$	−8.64276e14	+	1.49697e15i	−0.106432	+	0.184345i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	5.42990e15i		−	0.650143i
$786$		0			0
$787$	6.79102e13	+	3.92080e13i	0.00801814	+	0.00462928i	0.504004	−	0.863701i	$-0.331860\pi$
$787$	6.79102e13	+	3.92080e13i	0.00801814	+	0.00462928i	−0.495986	+	0.868331i	$0.665193\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−7.77017e14	−	9.93196e14i	−0.0892195	−	0.114042i
$792$		0			0
$793$	−7.89995e15	−	1.36831e16i	−0.894586	−	1.54947i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.59703e15			0.726654			0.363327	−	0.931662i	$-0.381641\pi$
$797$	6.59703e15			0.726654			0.363327	+	0.931662i	$0.381641\pi$
$798$		0			0
$799$	1.67540e16			1.82016
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.14956e15	−	3.72314e15i	−0.227203	−	0.393527i
$804$		0			0
$805$	7.17920e15	−	1.01376e15i	0.748511	−	0.105696i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.55880e15	−	3.20938e15i	−0.563981	−	0.325614i	0.190761	−	0.981637i	$-0.438904\pi$
$809$	−5.55880e15	−	3.20938e15i	−0.563981	−	0.325614i	−0.754742	+	0.656022i	$0.772238\pi$
$810$		0			0
$811$			5.14403e14i			0.0514859i	0.999669	+	0.0257430i	$0.00819515\pi$
$811$			5.14403e14i			0.0514859i	−0.999669	+	0.0257430i	$0.991805\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.52135e15	+	7.83120e15i	−0.440455	+	0.762890i
$816$		0			0
$817$	6.86023e15	−	3.96076e15i	0.659352	−	0.380677i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.21813e15	−	3.59004e15i	0.581798	−	0.335901i	−0.180050	−	0.983658i	$-0.557626\pi$
$821$	6.21813e15	−	3.59004e15i	0.581798	−	0.335901i	0.761848	+	0.647756i	$0.224292\pi$
$822$		0			0
$823$	−2.54802e15	+	4.41330e15i	−0.235236	+	0.407440i	−0.959341	−	0.282249i	$-0.908920\pi$
$823$	−2.54802e15	+	4.41330e15i	−0.235236	+	0.407440i	0.724105	+	0.689689i	$0.242253\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.50841e15i			0.225486i	0.993624	+	0.112743i	$0.0359636\pi$
$827$			2.50841e15i			0.225486i	−0.993624	+	0.112743i	$0.964036\pi$
$828$		0			0
$829$	−1.02077e16	−	5.89340e15i	−0.905475	−	0.522776i	−0.0265027	−	0.999649i	$-0.508437\pi$
$829$	−1.02077e16	−	5.89340e15i	−0.905475	−	0.522776i	−0.878973	+	0.476872i	$0.841770\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−9.32968e15	−	8.98601e15i	−0.805971	−	0.776282i
$834$		0			0
$835$	−3.62607e15	−	6.28054e15i	−0.309144	−	0.535453i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.41285e16			−1.17329			−0.586647	−	0.809843i	$-0.699552\pi$
$839$	−1.41285e16			−1.17329			−0.586647	+	0.809843i	$0.699552\pi$
$840$		0			0
$841$	−2.15428e16			−1.76573
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.72059e14	+	9.90835e14i	0.0456803	+	0.0791205i
$846$		0			0
$847$	1.02131e16	+	4.11859e15i	0.805005	+	0.324631i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.36059e15	−	4.24964e15i	−0.565327	−	0.326392i
$852$		0			0
$853$			2.36491e16i			1.79306i	0.442984	+	0.896530i	$0.353920\pi$
$853$			2.36491e16i			1.79306i	−0.442984	+	0.896530i	$0.646080\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.71032e15	−	1.50867e16i	0.643635	−	1.11481i	−0.340980	−	0.940070i	$-0.610759\pi$
$857$	8.71032e15	−	1.50867e16i	0.643635	−	1.11481i	0.984615	−	0.174738i	$-0.0559077\pi$
$858$		0			0
$859$	−1.02196e16	+	5.90031e15i	−0.745544	+	0.430440i	−0.824081	−	0.566471i	$-0.808308\pi$
$859$	−1.02196e16	+	5.90031e15i	−0.745544	+	0.430440i	0.0785378	+	0.996911i	$0.474975\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.26239e16	+	7.28842e15i	−0.897707	+	0.518291i	−0.876456	−	0.481483i	$-0.840098\pi$
$863$	−1.26239e16	+	7.28842e15i	−0.897707	+	0.518291i	−0.0212514	+	0.999774i	$0.506765\pi$
$864$		0			0
$865$	−7.28362e15	+	1.26156e16i	−0.511397	+	0.885766i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			7.03062e15i			0.481266i
$870$		0			0
$871$	−1.78839e16	−	1.03253e16i	−1.20882	−	0.697914i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.93917e15	+	3.08177e15i	−0.259633	+	0.203121i
$876$		0			0
$877$	1.31863e16	+	2.28394e16i	0.858275	+	1.48658i	0.873573	+	0.486694i	$0.161797\pi$
$877$	1.31863e16	+	2.28394e16i	0.858275	+	1.48658i	−0.0152974	+	0.999883i	$0.504869\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8.62363e15			−0.547423			−0.273711	−	0.961812i	$-0.588251\pi$
$881$	−8.62363e15			−0.547423			−0.273711	+	0.961812i	$0.588251\pi$
$882$		0			0
$883$	−2.22578e16			−1.39540			−0.697701	−	0.716389i	$-0.745793\pi$
$883$	−2.22578e16			−1.39540			−0.697701	+	0.716389i	$0.745793\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.10857e15	+	1.23124e16i	0.434713	+	0.752945i	0.997272	−	0.0738121i	$-0.0235165\pi$
$887$	7.10857e15	+	1.23124e16i	0.434713	+	0.752945i	−0.562559	+	0.826757i	$0.690183\pi$
$888$		0			0
$889$	−1.21424e16	+	9.49946e15i	−0.733405	+	0.573772i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.86579e16	+	2.23192e16i	2.27801	+	1.31521i
$894$		0			0
$895$		−	2.02590e16i		−	1.17921i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.00800e16	+	1.74590e16i	−0.572507	+	0.991612i
$900$		0			0
$901$	2.89678e16	−	1.67246e16i	1.62528	−	0.938357i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.43766e16	−	1.40738e16i	1.33477	−	0.770628i
$906$		0			0
$907$	−8.55954e15	+	1.48256e16i	−0.463031	+	0.801994i	−0.999110	−	0.0421742i	$-0.986572\pi$
$907$	−8.55954e15	+	1.48256e16i	−0.463031	+	0.801994i	0.536079	+	0.844168i	$0.319905\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	3.25019e16i		−	1.71616i	−0.513519	−	0.858079i	$-0.671658\pi$
$911$		−	3.25019e16i		−	1.71616i	0.513519	−	0.858079i	$-0.328342\pi$
$912$		0			0
$913$	4.37829e14	+	2.52781e14i	0.0228410	+	0.0131873i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.12744e15	+	1.26119e15i	0.159279	+	0.0642316i
$918$		0			0
$919$	1.23791e16	+	2.14412e16i	0.622949	+	1.07898i	0.988934	+	0.148358i	$0.0473989\pi$
$919$	1.23791e16	+	2.14412e16i	0.622949	+	1.07898i	−0.365985	+	0.930621i	$0.619268\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.22840e16			−0.603572
$924$		0			0
$925$	3.23696e16			1.57166
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.31956e16	−	2.28554e16i	−0.625664	−	1.08368i	−0.988412	−	0.151795i	$-0.951495\pi$
$929$	−1.31956e16	−	2.28554e16i	−0.625664	−	1.08368i	0.362748	−	0.931887i	$-0.381839\pi$
$930$		0			0
$931$	−9.55630e15	−	3.31629e16i	−0.447782	−	1.55392i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.14661e16	+	6.61997e15i	0.524750	+	0.302965i
$936$		0			0
$937$		−	1.18800e16i		−	0.537339i	−0.963232	−	0.268670i	$-0.913416\pi$
$937$		−	1.18800e16i		−	0.537339i	0.963232	−	0.268670i	$-0.0865840\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.00838e15	+	1.38709e16i	−0.353836	+	0.612861i	−0.986918	−	0.161223i	$-0.948456\pi$
$941$	−8.00838e15	+	1.38709e16i	−0.353836	+	0.612861i	0.633082	+	0.774084i	$0.281789\pi$
$942$		0			0
$943$	−6.55729e15	+	3.78585e15i	−0.286359	+	0.165329i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.59497e16	−	1.49821e16i	1.10715	−	0.639215i	0.169062	−	0.985605i	$-0.445926\pi$
$947$	2.59497e16	−	1.49821e16i	1.10715	−	0.639215i	0.938090	+	0.346391i	$0.112593\pi$
$948$		0			0
$949$	1.52758e16	−	2.64585e16i	0.644229	−	1.11584i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	9.21103e15i		−	0.379575i	−0.981825	−	0.189787i	$-0.939220\pi$
$953$		−	9.21103e15i		−	0.379575i	0.981825	−	0.189787i	$-0.0607798\pi$
$954$		0			0
$955$	5.22227e16	+	3.01508e16i	2.12736	+	1.22823i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.40122e16	−	3.39072e15i	0.955939	−	0.134986i
$960$		0			0
$961$	−6.68196e15	−	1.15735e16i	−0.262982	−	0.455498i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.60696e15			0.177221
$966$		0			0
$967$	−1.77937e16			−0.676739			−0.338369	−	0.941013i	$-0.609875\pi$
$967$	−1.77937e16			−0.676739			−0.338369	+	0.941013i	$0.609875\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.41488e16	+	2.45065e16i	0.526035	+	0.911120i	0.999540	+	0.0303284i	$0.00965532\pi$
$971$	1.41488e16	+	2.45065e16i	0.526035	+	0.911120i	−0.473505	+	0.880791i	$0.657011\pi$
$972$		0			0
$973$	2.15252e16	+	2.75138e16i	0.791272	+	1.01142i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.66447e16	+	9.60981e15i	0.598213	+	0.345378i	0.768338	−	0.640044i	$-0.221084\pi$
$977$	1.66447e16	+	9.60981e15i	0.598213	+	0.345378i	−0.170126	+	0.985422i	$0.554417\pi$
$978$		0			0
$979$		−	1.17119e16i		−	0.416219i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.49533e16	+	2.58999e16i	−0.519629	+	0.900025i	0.480110	+	0.877208i	$0.340597\pi$
$983$	−1.49533e16	+	2.58999e16i	−0.519629	+	0.900025i	−0.999740	+	0.0228164i	$0.992737\pi$
$984$		0			0
$985$	−6.04659e16	+	3.49100e16i	−2.07784	+	1.19964i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.15374e15	−	3.55286e15i	0.206804	−	0.119398i
$990$		0			0
$991$	2.37179e16	−	4.10806e16i	0.788263	−	1.36531i	−0.138767	−	0.990325i	$-0.544314\pi$
$991$	2.37179e16	−	4.10806e16i	0.788263	−	1.36531i	0.927030	−	0.374987i	$-0.122353\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			6.47777e16i			2.10571i
$996$		0			0
$997$	3.51108e16	+	2.02712e16i	1.12880	+	0.651713i	0.943633	−	0.330994i	$-0.107384\pi$
$997$	3.51108e16	+	2.02712e16i	1.12880	+	0.651713i	0.185167	+	0.982707i	$0.440717\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.t.a.17.4	✓	60
3.2	odd	2	inner	252.12.t.a.17.27	yes	60
7.5	odd	6	inner	252.12.t.a.89.27	yes	60
21.5	even	6	inner	252.12.t.a.89.4	yes	60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.12.t.a.17.4	✓	60	1.1	even	1	trivial
252.12.t.a.17.27	yes	60	3.2	odd	2	inner
252.12.t.a.89.4	yes	60	21.5	even	6	inner
252.12.t.a.89.27	yes	60	7.5	odd	6	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-5207.12 - 9018.99i) q^{5} +(-35022.7 + 27399.6i) q^{7} +O(q^{10})\)	\(q+(-5207.12 - 9018.99i) q^{5} +(-35022.7 + 27399.6i) q^{7} +(-168067. - 97033.6i) q^{11} -1.37914e6i q^{13} +(3.27549e6 - 5.67332e6i) q^{17} +(1.51156e7 - 8.72702e6i) q^{19} +(1.35590e7 - 7.82828e6i) q^{23} +(-2.98141e7 + 5.16396e7i) q^{25} +1.83694e8i q^{29} +(9.50443e7 + 5.48738e7i) q^{31} +(4.29484e8 + 1.73196e8i) q^{35} +(-2.71429e8 - 4.70128e8i) q^{37} -4.83612e8 q^{41} +4.53850e8 q^{43} +(1.27874e9 + 2.21484e9i) q^{47} +(4.75848e8 - 1.91922e9i) q^{49} +(4.42190e9 + 2.55299e9i) q^{53} +2.02106e9i q^{55} +(3.14281e9 - 5.44351e9i) q^{59} +(9.92150e9 - 5.72818e9i) q^{61} +(-1.24384e10 + 7.18133e9i) q^{65} +(7.48676e9 - 1.29675e10i) q^{67} -8.90698e9i q^{71} +(1.91848e10 + 1.10764e10i) q^{73} +(8.54484e9 - 1.20660e9i) q^{77} +(-1.81139e10 - 3.13742e10i) q^{79} -2.60508e9 q^{83} -6.82235e10 q^{85} +(3.01748e10 + 5.22643e10i) q^{89} +(3.77879e10 + 4.83011e10i) q^{91} +(-1.57418e11 - 9.08853e10i) q^{95} +1.12008e11i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 31278 q^{7}+O(q^{10}) \) 60 * q - 31278 * q^7	\( 60 q - 31278 q^{7} - 34858710 q^{19} - 323828862 q^{25} + 1186783866 q^{31} - 706751058 q^{37} + 1104997284 q^{43} + 7568925402 q^{49} - 5130550116 q^{61} - 22208949354 q^{67} + 54154878858 q^{73} - 61996342614 q^{79} + 107647163952 q^{85} - 2229336678 q^{91}+O(q^{100}) \) 60 * q - 31278 * q^7 - 34858710 * q^19 - 323828862 * q^25 + 1186783866 * q^31 - 706751058 * q^37 + 1104997284 * q^43 + 7568925402 * q^49 - 5130550116 * q^61 - 22208949354 * q^67 + 54154878858 * q^73 - 61996342614 * q^79 + 107647163952 * q^85 - 2229336678 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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