LMFDB - Embedded newform 2496.2.d.q.1535.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,2,Mod(1535,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2496.1535");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.d (of order $2$, degree $1$, 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:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$20$
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} - 4 x^{17} + 18 x^{16} + 8 x^{14} - 8 x^{13} + 241 x^{12} - 44 x^{11} - 112 x^{10} - 132 x^{9} + \cdots + 59049 $ x^20 - 4x^17 + 18x^16 + 8x^14 - 8x^13 + 241x^12 - 44x^11 - 112x^10 - 132x^9 + 2169x^8 - 216x^7 + 648x^6 + 13122x^4 - 8748*x^3 + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{23}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1248)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1535.13
Root			$-0.853779 + 1.50700i$ of defining polynomial
Character	$\chi$	$=$	2496.1535
Dual form			2496.2.d.q.1535.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.461900 - 1.66933i) q^{3} +1.99193i q^{5} +1.52766i q^{7} +(-2.57330 - 1.54212i) q^{9} +2.82843 q^{11} -1.00000 q^{13} +(3.32519 + 0.920073i) q^{15} -4.93184i q^{17} +0.312481i q^{19} +(2.55017 + 0.705628i) q^{21} +0.713524 q^{23} +1.03220 q^{25} +(-3.76291 + 3.58337i) q^{27} +2.82843i q^{29} -1.32156i q^{31} +(1.30645 - 4.72157i) q^{33} -3.04301 q^{35} +6.18986 q^{37} +(-0.461900 + 1.66933i) q^{39} -7.19538i q^{41} +9.95155i q^{43} +(3.07180 - 5.12584i) q^{45} +4.97738 q^{47} +4.66624 q^{49} +(-8.23285 - 2.27802i) q^{51} +0.271609i q^{53} +5.63404i q^{55} +(0.521633 + 0.144335i) q^{57} +2.82843 q^{59} +12.1758 q^{61} +(2.35585 - 3.93114i) q^{63} -1.99193i q^{65} +8.31248i q^{67} +(0.329576 - 1.19110i) q^{69} +7.19718 q^{71} +4.43037 q^{73} +(0.476774 - 1.72308i) q^{75} +4.32089i q^{77} -5.21299i q^{79} +(4.24372 + 7.93668i) q^{81} -1.49246 q^{83} +9.82390 q^{85} +(4.72157 + 1.30645i) q^{87} +4.43237i q^{89} -1.52766i q^{91} +(-2.20611 - 0.610426i) q^{93} -0.622442 q^{95} +16.0138 q^{97} +(-7.27838 - 4.36178i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{9} - 20 q^{13} + 12 q^{21} - 36 q^{25} - 16 q^{37} - 4 q^{45} - 76 q^{49} - 16 q^{57} + 56 q^{61} - 24 q^{69} + 88 q^{73} + 72 q^{81} - 56 q^{85} - 96 q^{93} - 72 q^{97}+O(q^{100}) $ 20 * q - 8 * q^9 - 20 * q^13 + 12 * q^21 - 36 * q^25 - 16 * q^37 - 4 * q^45 - 76 * q^49 - 16 * q^57 + 56 * q^61 - 24 * q^69 + 88 * q^73 + 72 * q^81 - 56 * q^85 - 96 * q^93 - 72 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.461900	−	1.66933i	0.266678	−	0.963786i
$3$	0.461900	−	1.66933i	0.266678	−	0.963786i
$4$		0			0
$5$			1.99193i			0.890820i	0.895327	+	0.445410i	$0.146942\pi$
$5$			1.99193i			0.890820i	−0.895327	+	0.445410i	$0.853058\pi$
$6$		0			0
$7$			1.52766i			0.577403i	0.957419	+	0.288702i	$0.0932235\pi$
$7$			1.52766i			0.577403i	−0.957419	+	0.288702i	$0.906776\pi$
$8$		0			0
$9$	−2.57330	−	1.54212i	−0.857766	−	0.514041i
$10$		0			0
$11$	2.82843			0.852803			0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843			0.852803			0.426401	+	0.904534i	$0.359781\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	3.32519	+	0.920073i	0.858559	+	0.237562i
$16$		0			0
$17$		−	4.93184i		−	1.19615i	−0.801441	−	0.598074i	$-0.795933\pi$
$17$		−	4.93184i		−	1.19615i	0.801441	−	0.598074i	$-0.204067\pi$
$18$		0			0
$19$			0.312481i			0.0716881i	0.999357	+	0.0358440i	$0.0114120\pi$
$19$			0.312481i			0.0716881i	−0.999357	+	0.0358440i	$0.988588\pi$
$20$		0			0
$21$	2.55017	+	0.705628i	0.556493	+	0.153981i
$22$		0			0
$23$	0.713524			0.148780			0.0743900	−	0.997229i	$-0.476299\pi$
$23$	0.713524			0.148780			0.0743900	+	0.997229i	$0.476299\pi$
$24$		0			0
$25$	1.03220			0.206440
$26$		0			0
$27$	−3.76291	+	3.58337i	−0.724172	+	0.689619i
$28$		0			0
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$		−	1.32156i		−	0.237359i	−0.992933	−	0.118679i	$-0.962134\pi$
$31$		−	1.32156i		−	0.237359i	0.992933	−	0.118679i	$-0.0378660\pi$
$32$		0			0
$33$	1.30645	−	4.72157i	0.227424	−	0.821919i
$34$		0			0
$35$	−3.04301			−0.514362
$36$		0			0
$37$	6.18986			1.01761			0.508803	−	0.860883i	$-0.330088\pi$
$37$	6.18986			1.01761			0.508803	+	0.860883i	$0.330088\pi$
$38$		0			0
$39$	−0.461900	+	1.66933i	−0.0739631	+	0.267306i
$40$		0			0
$41$		−	7.19538i		−	1.12373i	−0.827229	−	0.561865i	$-0.810084\pi$
$41$		−	7.19538i		−	1.12373i	0.827229	−	0.561865i	$-0.189916\pi$
$42$		0			0
$43$			9.95155i			1.51760i	0.651325	+	0.758799i	$0.274213\pi$
$43$			9.95155i			1.51760i	−0.651325	+	0.758799i	$0.725787\pi$
$44$		0			0
$45$	3.07180	−	5.12584i	0.457917	−	0.764115i
$46$		0			0
$47$	4.97738			0.726026			0.363013	−	0.931784i	$-0.381748\pi$
$47$	4.97738			0.726026			0.363013	+	0.931784i	$0.381748\pi$
$48$		0			0
$49$	4.66624			0.666606
$50$		0			0
$51$	−8.23285	−	2.27802i	−1.15283	−	0.318986i
$52$		0			0
$53$			0.271609i			0.0373083i	0.999826	+	0.0186542i	$0.00593815\pi$
$53$			0.271609i			0.0373083i	−0.999826	+	0.0186542i	$0.994062\pi$
$54$		0			0
$55$			5.63404i			0.759693i
$56$		0			0
$57$	0.521633	+	0.144335i	0.0690920	+	0.0191176i
$58$		0			0
$59$	2.82843			0.368230			0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843			0.368230			0.184115	+	0.982905i	$0.441058\pi$
$60$		0			0
$61$	12.1758			1.55895			0.779476	−	0.626432i	$-0.215485\pi$
$61$	12.1758			1.55895			0.779476	+	0.626432i	$0.215485\pi$
$62$		0			0
$63$	2.35585	−	3.93114i	0.296809	−	0.495277i
$64$		0			0
$65$		−	1.99193i		−	0.247069i
$66$		0			0
$67$			8.31248i			1.01553i	0.861495	+	0.507766i	$0.169528\pi$
$67$			8.31248i			1.01553i	−0.861495	+	0.507766i	$0.830472\pi$
$68$		0			0
$69$	0.329576	−	1.19110i	0.0396763	−	0.143392i
$70$		0			0
$71$	7.19718			0.854148			0.427074	−	0.904217i	$-0.359544\pi$
$71$	7.19718			0.854148			0.427074	+	0.904217i	$0.359544\pi$
$72$		0			0
$73$	4.43037			0.518535			0.259268	−	0.965806i	$-0.416519\pi$
$73$	4.43037			0.518535			0.259268	+	0.965806i	$0.416519\pi$
$74$		0			0
$75$	0.476774	−	1.72308i	0.0550531	−	0.198964i
$76$		0			0
$77$			4.32089i			0.492411i
$78$		0			0
$79$		−	5.21299i		−	0.586507i	−0.956035	−	0.293253i	$-0.905262\pi$
$79$		−	5.21299i		−	0.586507i	0.956035	−	0.293253i	$-0.0947380\pi$
$80$		0			0
$81$	4.24372	+	7.93668i	0.471525	+	0.881853i
$82$		0			0
$83$	−1.49246			−0.163819			−0.0819095	−	0.996640i	$-0.526102\pi$
$83$	−1.49246			−0.163819			−0.0819095	+	0.996640i	$0.526102\pi$
$84$		0			0
$85$	9.82390			1.06555
$86$		0			0
$87$	4.72157	+	1.30645i	0.506205	+	0.140066i
$88$		0			0
$89$			4.43237i			0.469830i	0.972016	+	0.234915i	$0.0754812\pi$
$89$			4.43237i			0.469830i	−0.972016	+	0.234915i	$0.924519\pi$
$90$		0			0
$91$		−	1.52766i		−	0.160143i
$92$		0			0
$93$	−2.20611	−	0.610426i	−0.228763	−	0.0632983i
$94$		0			0
$95$	−0.622442			−0.0638612
$96$		0			0
$97$	16.0138			1.62595			0.812975	−	0.582298i	$-0.197846\pi$
$97$	16.0138			1.62595			0.812975	+	0.582298i	$0.197846\pi$
$98$		0			0
$99$	−7.27838	−	4.36178i	−0.731505	−	0.438375i
$100$		0			0
$101$			7.14932i			0.711384i	0.934603	+	0.355692i	$0.115755\pi$
$101$			7.14932i			0.711384i	−0.934603	+	0.355692i	$0.884245\pi$
$102$		0			0
$103$			9.21299i			0.907783i	0.891057	+	0.453891i	$0.149965\pi$
$103$			9.21299i			0.907783i	−0.891057	+	0.453891i	$0.850035\pi$
$104$		0			0
$105$	−1.40556	+	5.07977i	−0.137169	+	0.495735i
$106$		0			0
$107$	18.1030			1.75009			0.875043	−	0.484046i	$-0.160833\pi$
$107$	18.1030			1.75009			0.875043	+	0.484046i	$0.160833\pi$
$108$		0			0
$109$	−10.1899			−0.976012			−0.488006	−	0.872840i	$-0.662276\pi$
$109$	−10.1899			−0.976012			−0.488006	+	0.872840i	$0.662276\pi$
$110$		0			0
$111$	2.85909	−	10.3329i	0.271373	−	0.980755i
$112$		0			0
$113$		−	12.9637i		−	1.21952i	−0.792585	−	0.609762i	$-0.791265\pi$
$113$		−	12.9637i		−	1.21952i	0.792585	−	0.609762i	$-0.208735\pi$
$114$		0			0
$115$			1.42129i			0.132536i
$116$		0			0
$117$	2.57330	+	1.54212i	0.237901	+	0.142569i
$118$		0			0
$119$	7.53420			0.690659
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$	−12.0114	−	3.32354i	−1.08304	−	0.299674i
$124$		0			0
$125$			12.0157i			1.07472i
$126$		0			0
$127$		−	3.79170i		−	0.336459i	−0.985748	−	0.168229i	$-0.946195\pi$
$127$		−	3.79170i		−	0.336459i	0.985748	−	0.168229i	$-0.0538049\pi$
$128$		0			0
$129$	16.6124	+	4.59662i	1.46264	+	0.404710i
$130$		0			0
$131$	−17.6695			−1.54379			−0.771895	−	0.635750i	$-0.780691\pi$
$131$	−17.6695			−1.54379			−0.771895	+	0.635750i	$0.780691\pi$
$132$		0			0
$133$	−0.477366			−0.0413929
$134$		0			0
$135$	−7.13783	−	7.49546i	−0.614326	−	0.645107i
$136$		0			0
$137$		−	12.3090i		−	1.05163i	−0.850599	−	0.525815i	$-0.823760\pi$
$137$		−	12.3090i		−	1.05163i	0.850599	−	0.525815i	$-0.176240\pi$
$138$		0			0
$139$			10.9606i			0.929668i	0.885398	+	0.464834i	$0.153886\pi$
$139$			10.9606i			0.929668i	−0.885398	+	0.464834i	$0.846114\pi$
$140$		0			0
$141$	2.29905	−	8.30887i	0.193615	−	0.699733i
$142$		0			0
$143$	−2.82843			−0.236525
$144$		0			0
$145$	−5.63404			−0.467881
$146$		0			0
$147$	2.15533	−	7.78948i	0.177769	−	0.642465i
$148$		0			0
$149$		−	22.1984i		−	1.81856i	−0.416183	−	0.909281i	$-0.636633\pi$
$149$		−	22.1984i		−	1.81856i	0.416183	−	0.909281i	$-0.363367\pi$
$150$		0			0
$151$			4.96711i			0.404217i	0.979363	+	0.202109i	$0.0647794\pi$
$151$			4.96711i			0.404217i	−0.979363	+	0.202109i	$0.935221\pi$
$152$		0			0
$153$	−7.60550	+	12.6911i	−0.614868	+	1.02601i
$154$		0			0
$155$	2.63245			0.211444
$156$		0			0
$157$	4.11140			0.328126			0.164063	−	0.986450i	$-0.447540\pi$
$157$	4.11140			0.328126			0.164063	+	0.986450i	$0.447540\pi$
$158$		0			0
$159$	0.453404	+	0.125456i	0.0359573	+	0.00994931i
$160$		0			0
$161$			1.09003i			0.0859060i
$162$		0			0
$163$		−	24.9049i		−	1.95071i	−0.220650	−	0.975353i	$-0.570818\pi$
$163$		−	24.9049i		−	1.95071i	0.220650	−	0.975353i	$-0.429182\pi$
$164$		0			0
$165$	9.40504	+	2.60236i	0.732182	+	0.202593i
$166$		0			0
$167$	−18.1684			−1.40592			−0.702958	−	0.711231i	$-0.748138\pi$
$167$	−18.1684			−1.40592			−0.702958	+	0.711231i	$0.748138\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	0.481884	−	0.804107i	0.0368506	−	0.0614916i
$172$		0			0
$173$			20.7739i			1.57941i	0.613486	+	0.789705i	$0.289767\pi$
$173$			20.7739i			1.57941i	−0.613486	+	0.789705i	$0.710233\pi$
$174$		0			0
$175$			1.57686i			0.119199i
$176$		0			0
$177$	1.30645	−	4.72157i	0.0981987	−	0.354895i
$178$		0			0
$179$	13.4627			1.00625			0.503123	−	0.864215i	$-0.332184\pi$
$179$	13.4627			1.00625			0.503123	+	0.864215i	$0.332184\pi$
$180$		0			0
$181$	9.74544			0.724373			0.362186	−	0.932106i	$-0.382030\pi$
$181$	9.74544			0.724373			0.362186	+	0.932106i	$0.382030\pi$
$182$		0			0
$183$	5.62400	−	20.3254i	0.415738	−	1.50250i
$184$		0			0
$185$			12.3298i			0.906504i
$186$		0			0
$187$		−	13.9494i		−	1.02008i
$188$		0			0
$189$	−5.47418	−	5.74846i	−0.398188	−	0.418139i
$190$		0			0
$191$	−21.4388			−1.55126			−0.775628	−	0.631190i	$-0.782567\pi$
$191$	−21.4388			−1.55126			−0.775628	+	0.631190i	$0.782567\pi$
$192$		0			0
$193$	5.20367			0.374568			0.187284	−	0.982306i	$-0.440031\pi$
$193$	5.20367			0.374568			0.187284	+	0.982306i	$0.440031\pi$
$194$		0			0
$195$	−3.32519	−	0.920073i	−0.238121	−	0.0658878i
$196$		0			0
$197$		−	20.9495i		−	1.49259i	−0.665614	−	0.746296i	$-0.731830\pi$
$197$		−	20.9495i		−	1.49259i	0.665614	−	0.746296i	$-0.268170\pi$
$198$		0			0
$199$			21.4257i			1.51883i	0.650607	+	0.759415i	$0.274515\pi$
$199$			21.4257i			1.51883i	−0.650607	+	0.759415i	$0.725485\pi$
$200$		0			0
$201$	13.8762	+	3.83953i	0.978754	+	0.270820i
$202$		0			0
$203$	−4.32089			−0.303267
$204$		0			0
$205$	14.3327			1.00104
$206$		0			0
$207$	−1.83611	−	1.10034i	−0.127618	−	0.0764790i
$208$		0			0
$209$			0.883830i			0.0611358i
$210$		0			0
$211$			2.04845i			0.141021i	0.997511	+	0.0705106i	$0.0224628\pi$
$211$			2.04845i			0.141021i	−0.997511	+	0.0705106i	$0.977537\pi$
$212$		0			0
$213$	3.32437	−	12.0144i	0.227782	−	0.823216i
$214$		0			0
$215$	−19.8228			−1.35191
$216$		0			0
$217$	2.01890			0.137052
$218$		0			0
$219$	2.04638	−	7.39573i	0.138282	−	0.499757i
$220$		0			0
$221$			4.93184i			0.331752i
$222$		0			0
$223$		−	1.91178i		−	0.128022i	−0.997949	−	0.0640111i	$-0.979611\pi$
$223$		−	1.91178i		−	0.128022i	0.997949	−	0.0640111i	$-0.0203893\pi$
$224$		0			0
$225$	−2.65616	−	1.59178i	−0.177078	−	0.106119i
$226$		0			0
$227$	19.8671			1.31863			0.659313	−	0.751869i	$-0.270847\pi$
$227$	19.8671			1.31863			0.659313	+	0.751869i	$0.270847\pi$
$228$		0			0
$229$	−15.8239			−1.04567			−0.522836	−	0.852433i	$-0.675126\pi$
$229$	−15.8239			−1.04567			−0.522836	+	0.852433i	$0.675126\pi$
$230$		0			0
$231$	7.21297	+	1.99582i	0.474579	+	0.131315i
$232$		0			0
$233$		−	10.2517i		−	0.671609i	−0.941932	−	0.335805i	$-0.890992\pi$
$233$		−	10.2517i		−	0.671609i	0.941932	−	0.335805i	$-0.109008\pi$
$234$		0			0
$235$			9.91461i			0.646758i
$236$		0			0
$237$	−8.70218	−	2.40788i	−0.565267	−	0.156408i
$238$		0			0
$239$	−17.6952			−1.14460			−0.572302	−	0.820043i	$-0.693950\pi$
$239$	−17.6952			−1.14460			−0.572302	+	0.820043i	$0.693950\pi$
$240$		0			0
$241$	−7.24992			−0.467009			−0.233504	−	0.972356i	$-0.575019\pi$
$241$	−7.24992			−0.467009			−0.233504	+	0.972356i	$0.575019\pi$
$242$		0			0
$243$	15.2091	−	3.41821i	0.975662	−	0.219278i
$244$		0			0
$245$			9.29484i			0.593825i
$246$		0			0
$247$		−	0.312481i		−	0.0198827i
$248$		0			0
$249$	−0.689367	+	2.49140i	−0.0436869	+	0.157886i
$250$		0			0
$251$	−0.985133			−0.0621810			−0.0310905	−	0.999517i	$-0.509898\pi$
$251$	−0.985133			−0.0621810			−0.0310905	+	0.999517i	$0.509898\pi$
$252$		0			0
$253$	2.01815			0.126880
$254$		0			0
$255$	4.53765	−	16.3993i	0.284159	−	1.02696i
$256$		0			0
$257$		−	13.1455i		−	0.819995i	−0.912087	−	0.409997i	$-0.865530\pi$
$257$		−	13.1455i		−	0.819995i	0.912087	−	0.409997i	$-0.134470\pi$
$258$		0			0
$259$			9.45603i			0.587569i
$260$		0			0
$261$	4.36178	−	7.27838i	0.269987	−	0.450521i
$262$		0			0
$263$	−29.6883			−1.83066			−0.915331	−	0.402703i	$-0.868071\pi$
$263$	−29.6883			−1.83066			−0.915331	+	0.402703i	$0.868071\pi$
$264$		0			0
$265$	−0.541027			−0.0332350
$266$		0			0
$267$	7.39907	+	2.04731i	0.452816	+	0.125293i
$268$		0			0
$269$		−	21.8771i		−	1.33387i	−0.745116	−	0.666935i	$-0.767606\pi$
$269$		−	21.8771i		−	1.33387i	0.745116	−	0.666935i	$-0.232394\pi$
$270$		0			0
$271$		−	13.1799i		−	0.800619i	−0.916380	−	0.400310i	$-0.868903\pi$
$271$		−	13.1799i		−	0.800619i	0.916380	−	0.400310i	$-0.131097\pi$
$272$		0			0
$273$	−2.55017	−	0.705628i	−0.154343	−	0.0427065i
$274$		0			0
$275$	2.91951			0.176053
$276$		0			0
$277$	7.63404			0.458685			0.229342	−	0.973346i	$-0.426342\pi$
$277$	7.63404			0.458685			0.229342	+	0.973346i	$0.426342\pi$
$278$		0			0
$279$	−2.03800	+	3.40076i	−0.122012	+	0.203598i
$280$		0			0
$281$		−	7.28647i		−	0.434674i	−0.976097	−	0.217337i	$-0.930263\pi$
$281$		−	7.28647i		−	0.434674i	0.976097	−	0.217337i	$-0.0697371\pi$
$282$		0			0
$283$			11.4764i			0.682200i	0.940027	+	0.341100i	$0.110799\pi$
$283$			11.4764i			0.682200i	−0.940027	+	0.341100i	$0.889201\pi$
$284$		0			0
$285$	−0.287505	+	1.03906i	−0.0170304	+	0.0615485i
$286$		0			0
$287$	10.9921			0.648845
$288$		0			0
$289$	−7.32306			−0.430768
$290$		0			0
$291$	7.39675	−	26.7322i	0.433605	−	1.56707i
$292$		0			0
$293$		−	1.44513i		−	0.0844251i	−0.999109	−	0.0422126i	$-0.986559\pi$
$293$		−	1.44513i		−	0.0844251i	0.999109	−	0.0422126i	$-0.0134407\pi$
$294$		0			0
$295$			5.63404i			0.328026i
$296$		0			0
$297$	−10.6431	+	10.1353i	−0.617576	+	0.588109i
$298$		0			0
$299$	−0.713524			−0.0412642
$300$		0			0
$301$	−15.2026			−0.876265
$302$		0			0
$303$	11.9345	+	3.30227i	0.685621	+	0.189710i
$304$		0			0
$305$			24.2534i			1.38875i
$306$		0			0
$307$		−	2.35874i		−	0.134620i	−0.997732	−	0.0673101i	$-0.978558\pi$
$307$		−	2.35874i		−	0.134620i	0.997732	−	0.0673101i	$-0.0214417\pi$
$308$		0			0
$309$	15.3795	+	4.25548i	0.874908	+	0.242086i
$310$		0			0
$311$	22.4509			1.27307			0.636537	−	0.771246i	$-0.280366\pi$
$311$	22.4509			1.27307			0.636537	+	0.771246i	$0.280366\pi$
$312$		0			0
$313$	−7.34728			−0.415293			−0.207646	−	0.978204i	$-0.566580\pi$
$313$	−7.34728			−0.415293			−0.207646	+	0.978204i	$0.566580\pi$
$314$		0			0
$315$	7.83056	+	4.69269i	0.441202	+	0.264403i
$316$		0			0
$317$		−	22.9646i		−	1.28982i	−0.764259	−	0.644909i	$-0.776895\pi$
$317$		−	22.9646i		−	1.28982i	0.764259	−	0.644909i	$-0.223105\pi$
$318$		0			0
$319$			8.00000i			0.447914i
$320$		0			0
$321$	8.36178	−	30.2198i	0.466709	−	1.68671i
$322$		0			0
$323$	1.54111			0.0857495
$324$		0			0
$325$	−1.03220			−0.0572563
$326$		0			0
$327$	−4.70669	+	17.0102i	−0.260281	+	0.940666i
$328$		0			0
$329$			7.60377i			0.419210i
$330$		0			0
$331$			12.7610i			0.701408i	0.936486	+	0.350704i	$0.114058\pi$
$331$			12.7610i			0.701408i	−0.936486	+	0.350704i	$0.885942\pi$
$332$		0			0
$333$	−15.9284	−	9.54552i	−0.872868	−	0.523091i
$334$		0			0
$335$	−16.5579			−0.904655
$336$		0			0
$337$	0.143607			0.00782275			0.00391138	−	0.999992i	$-0.498755\pi$
$337$	0.143607			0.00782275			0.00391138	+	0.999992i	$0.498755\pi$
$338$		0			0
$339$	−21.6407	−	5.98794i	−1.17536	−	0.325220i
$340$		0			0
$341$		−	3.73793i		−	0.202420i
$342$		0			0
$343$			17.8221i			0.962303i
$344$		0			0
$345$	2.37260	+	0.656494i	0.127736	+	0.0353445i
$346$		0			0
$347$	4.75843			0.255446			0.127723	−	0.991810i	$-0.459233\pi$
$347$	4.75843			0.255446			0.127723	+	0.991810i	$0.459233\pi$
$348$		0			0
$349$	−20.1111			−1.07652			−0.538262	−	0.842778i	$-0.680919\pi$
$349$	−20.1111			−1.07652			−0.538262	+	0.842778i	$0.680919\pi$
$350$		0			0
$351$	3.76291	−	3.58337i	0.200849	−	0.191266i
$352$		0			0
$353$			15.9329i			0.848022i	0.905657	+	0.424011i	$0.139378\pi$
$353$			15.9329i			0.848022i	−0.905657	+	0.424011i	$0.860622\pi$
$354$		0			0
$355$			14.3363i			0.760892i
$356$		0			0
$357$	3.48004	−	12.5770i	0.184183	−	0.665647i
$358$		0			0
$359$	17.4944			0.923319			0.461659	−	0.887057i	$-0.347254\pi$
$359$	17.4944			0.923319			0.461659	+	0.887057i	$0.347254\pi$
$360$		0			0
$361$	18.9024			0.994861
$362$		0			0
$363$	−1.38570	+	5.00798i	−0.0727303	+	0.262851i
$364$		0			0
$365$			8.82500i			0.461921i
$366$		0			0
$367$			23.6304i			1.23350i	0.787161	+	0.616748i	$0.211550\pi$
$367$			23.6304i			1.23350i	−0.787161	+	0.616748i	$0.788450\pi$
$368$		0			0
$369$	−11.0962	+	18.5159i	−0.577643	+	0.963897i
$370$		0			0
$371$	−0.414927			−0.0215420
$372$		0			0
$373$	−25.3325			−1.31167			−0.655833	−	0.754906i	$-0.727682\pi$
$373$	−25.3325			−1.31167			−0.655833	+	0.754906i	$0.727682\pi$
$374$		0			0
$375$	20.0582	+	5.55007i	1.03580	+	0.286604i
$376$		0			0
$377$		−	2.82843i		−	0.145671i
$378$		0			0
$379$			29.6224i			1.52160i	0.648986	+	0.760801i	$0.275194\pi$
$379$			29.6224i			1.52160i	−0.648986	+	0.760801i	$0.724806\pi$
$380$		0			0
$381$	−6.32958	−	1.75138i	−0.324274	−	0.0897261i
$382$		0			0
$383$	−11.0634			−0.565313			−0.282657	−	0.959221i	$-0.591216\pi$
$383$	−11.0634			−0.565313			−0.282657	+	0.959221i	$0.591216\pi$
$384$		0			0
$385$	−8.60692			−0.438649
$386$		0			0
$387$	15.3465	−	25.6083i	0.780107	−	1.30174i
$388$		0			0
$389$			22.8760i			1.15986i	0.814666	+	0.579931i	$0.196920\pi$
$389$			22.8760i			1.15986i	−0.814666	+	0.579931i	$0.803080\pi$
$390$		0			0
$391$		−	3.51899i		−	0.177963i
$392$		0			0
$393$	−8.16153	+	29.4961i	−0.411695	+	1.48788i
$394$		0			0
$395$	10.3839			0.522472
$396$		0			0
$397$	−0.204658			−0.0102715			−0.00513574	−	0.999987i	$-0.501635\pi$
$397$	−0.204658			−0.0102715			−0.00513574	+	0.999987i	$0.501635\pi$
$398$		0			0
$399$	−0.220495	+	0.796880i	−0.0110386	+	0.0398939i
$400$		0			0
$401$			24.3889i			1.21792i	0.793199	+	0.608962i	$0.208414\pi$
$401$			24.3889i			1.21792i	−0.793199	+	0.608962i	$0.791586\pi$
$402$		0			0
$403$			1.32156i			0.0658314i
$404$		0			0
$405$	−15.8093	+	8.45321i	−0.785572	+	0.420043i
$406$		0			0
$407$	17.5076			0.867818
$408$		0			0
$409$	25.9675			1.28401			0.642005	−	0.766700i	$-0.278103\pi$
$409$	25.9675			1.28401			0.642005	+	0.766700i	$0.278103\pi$
$410$		0			0
$411$	−20.5478	−	5.68553i	−1.01355	−	0.280447i
$412$		0			0
$413$			4.32089i			0.212617i
$414$		0			0
$415$		−	2.97288i		−	0.145933i
$416$		0			0
$417$	18.2969	+	5.06271i	0.896001	+	0.247922i
$418$		0			0
$419$	2.82300			0.137913			0.0689563	−	0.997620i	$-0.478033\pi$
$419$	2.82300			0.137913			0.0689563	+	0.997620i	$0.478033\pi$
$420$		0			0
$421$	−27.0920			−1.32038			−0.660191	−	0.751098i	$-0.729525\pi$
$421$	−27.0920			−1.32038			−0.660191	+	0.751098i	$0.729525\pi$
$422$		0			0
$423$	−12.8083	−	7.67573i	−0.622760	−	0.373207i
$424$		0			0
$425$		−	5.09066i		−	0.246933i
$426$		0			0
$427$			18.6006i			0.900144i
$428$		0			0
$429$	−1.30645	+	4.72157i	−0.0630760	+	0.227959i
$430$		0			0
$431$	34.2366			1.64912			0.824559	−	0.565776i	$-0.191423\pi$
$431$	34.2366			1.64912			0.824559	+	0.565776i	$0.191423\pi$
$432$		0			0
$433$	−35.5649			−1.70914			−0.854571	−	0.519334i	$-0.826180\pi$
$433$	−35.5649			−1.70914			−0.854571	+	0.519334i	$0.826180\pi$
$434$		0			0
$435$	−2.60236	+	9.40504i	−0.124774	+	0.450937i
$436$		0			0
$437$			0.222963i			0.0106658i
$438$		0			0
$439$			23.1623i			1.10548i	0.833354	+	0.552739i	$0.186417\pi$
$439$			23.1623i			1.10548i	−0.833354	+	0.552739i	$0.813583\pi$
$440$		0			0
$441$	−12.0076	−	7.19591i	−0.571792	−	0.342662i
$442$		0			0
$443$	−12.6867			−0.602763			−0.301381	−	0.953504i	$-0.597448\pi$
$443$	−12.6867			−0.602763			−0.301381	+	0.953504i	$0.597448\pi$
$444$		0			0
$445$	−8.82898			−0.418534
$446$		0			0
$447$	−37.0563	−	10.2534i	−1.75270	−	0.484970i
$448$		0			0
$449$			16.2929i			0.768909i	0.923144	+	0.384454i	$0.125610\pi$
$449$			16.2929i			0.768909i	−0.923144	+	0.384454i	$0.874390\pi$
$450$		0			0
$451$		−	20.3516i		−	0.958320i
$452$		0			0
$453$	8.29172	+	2.29430i	0.389579	+	0.107796i
$454$		0			0
$455$	3.04301			0.142658
$456$		0			0
$457$	2.93461			0.137275			0.0686376	−	0.997642i	$-0.478135\pi$
$457$	2.93461			0.137275			0.0686376	+	0.997642i	$0.478135\pi$
$458$		0			0
$459$	17.6726	+	18.5581i	0.824886	+	0.866217i
$460$		0			0
$461$			10.4118i			0.484926i	0.970161	+	0.242463i	$0.0779553\pi$
$461$			10.4118i			0.484926i	−0.970161	+	0.242463i	$0.922045\pi$
$462$		0			0
$463$		−	29.3491i		−	1.36397i	−0.731367	−	0.681984i	$-0.761118\pi$
$463$		−	29.3491i		−	1.36397i	0.731367	−	0.681984i	$-0.238882\pi$
$464$		0			0
$465$	1.21593	−	4.39442i	0.0563873	−	0.203786i
$466$		0			0
$467$	12.6395			0.584884			0.292442	−	0.956283i	$-0.405532\pi$
$467$	12.6395			0.584884			0.292442	+	0.956283i	$0.405532\pi$
$468$		0			0
$469$	−12.6987			−0.586371
$470$		0			0
$471$	1.89906	−	6.86327i	0.0875039	−	0.316243i
$472$		0			0
$473$			28.1472i			1.29421i
$474$		0			0
$475$			0.322544i			0.0147993i
$476$		0			0
$477$	0.418854	−	0.698930i	0.0191780	−	0.0320018i
$478$		0			0
$479$	−17.7181			−0.809562			−0.404781	−	0.914414i	$-0.632652\pi$
$479$	−17.7181			−0.809562			−0.404781	+	0.914414i	$0.632652\pi$
$480$		0			0
$481$	−6.18986			−0.282233
$482$		0			0
$483$	1.81961	+	0.503482i	0.0827950	+	0.0229092i
$484$		0			0
$485$			31.8983i			1.44843i
$486$		0			0
$487$			0.904949i			0.0410071i	0.999790	+	0.0205036i	$0.00652695\pi$
$487$			0.904949i			0.0410071i	−0.999790	+	0.0205036i	$0.993473\pi$
$488$		0			0
$489$	−41.5745	−	11.5036i	−1.88006	−	0.520210i
$490$		0			0
$491$	−39.8566			−1.79870			−0.899352	−	0.437225i	$-0.855961\pi$
$491$	−39.8566			−1.79870			−0.899352	+	0.437225i	$0.855961\pi$
$492$		0			0
$493$	13.9494			0.628247
$494$		0			0
$495$	8.68837	−	14.4981i	0.390513	−	0.651639i
$496$		0			0
$497$			10.9949i			0.493188i
$498$		0			0
$499$			22.1823i			0.993016i	0.868032	+	0.496508i	$0.165385\pi$
$499$			22.1823i			0.993016i	−0.868032	+	0.496508i	$0.834615\pi$
$500$		0			0
$501$	−8.39199	+	30.3290i	−0.374927	+	1.35500i
$502$		0			0
$503$	18.9576			0.845277			0.422639	−	0.906298i	$-0.361104\pi$
$503$	18.9576			0.845277			0.422639	+	0.906298i	$0.361104\pi$
$504$		0			0
$505$	−14.2410			−0.633714
$506$		0			0
$507$	0.461900	−	1.66933i	0.0205137	−	0.0741374i
$508$		0			0
$509$		−	13.6926i		−	0.606913i	−0.952845	−	0.303456i	$-0.901859\pi$
$509$		−	13.6926i		−	0.606913i	0.952845	−	0.303456i	$-0.0981407\pi$
$510$		0			0
$511$			6.76812i			0.299404i
$512$		0			0
$513$	−1.11973	−	1.17584i	−0.0494375	−	0.0519145i
$514$		0			0
$515$	−18.3517			−0.808671
$516$		0			0
$517$	14.0782			0.619157
$518$		0			0
$519$	34.6784	+	9.59546i	1.52221	+	0.421194i
$520$		0			0
$521$		−	33.0383i		−	1.44743i	−0.690097	−	0.723717i	$-0.742432\pi$
$521$		−	33.0383i		−	1.44743i	0.690097	−	0.723717i	$-0.257568\pi$
$522$		0			0
$523$			8.65243i			0.378344i	0.981944	+	0.189172i	$0.0605804\pi$
$523$			8.65243i			0.378344i	−0.981944	+	0.189172i	$0.939420\pi$
$524$		0			0
$525$	2.63229	+	0.728351i	0.114883	+	0.0317878i
$526$		0			0
$527$	−6.51771			−0.283916
$528$		0			0
$529$	−22.4909			−0.977865
$530$		0			0
$531$	−7.27838	−	4.36178i	−0.315855	−	0.189285i
$532$		0			0
$533$			7.19538i			0.311667i
$534$		0			0
$535$			36.0600i			1.55901i
$536$		0			0
$537$	6.21840	−	22.4736i	0.268344	−	0.969806i
$538$		0			0
$539$	13.1981			0.568483
$540$		0			0
$541$	31.6907			1.36249			0.681245	−	0.732056i	$-0.261439\pi$
$541$	31.6907			1.36249			0.681245	+	0.732056i	$0.261439\pi$
$542$		0			0
$543$	4.50142	−	16.2683i	0.193174	−	0.698140i
$544$		0			0
$545$		−	20.2975i		−	0.869450i
$546$		0			0
$547$		−	40.7835i		−	1.74378i	−0.489705	−	0.871888i	$-0.662896\pi$
$547$		−	40.7835i		−	1.74378i	0.489705	−	0.871888i	$-0.337104\pi$
$548$		0			0
$549$	−31.3320	−	18.7766i	−1.33722	−	0.801365i
$550$		0			0
$551$	−0.883830			−0.0376524
$552$		0			0
$553$	7.96370			0.338651
$554$		0			0
$555$	20.5824	+	5.69512i	0.873676	+	0.241745i
$556$		0			0
$557$			21.0442i			0.891671i	0.895115	+	0.445836i	$0.147093\pi$
$557$			21.0442i			0.891671i	−0.895115	+	0.445836i	$0.852907\pi$
$558$		0			0
$559$		−	9.95155i		−	0.420906i
$560$		0			0
$561$	−23.2860	−	6.44320i	−0.983136	−	0.272032i
$562$		0			0
$563$	−15.7735			−0.664776			−0.332388	−	0.943143i	$-0.607854\pi$
$563$	−15.7735			−0.664776			−0.332388	+	0.943143i	$0.607854\pi$
$564$		0			0
$565$	25.8229			1.08638
$566$		0			0
$567$	−12.1246	+	6.48298i	−0.509185	+	0.272260i
$568$		0			0
$569$		−	24.3284i		−	1.01990i	−0.860204	−	0.509950i	$-0.829664\pi$
$569$		−	24.3284i		−	1.01990i	0.860204	−	0.509950i	$-0.170336\pi$
$570$		0			0
$571$			25.2578i			1.05701i	0.848932	+	0.528503i	$0.177246\pi$
$571$			25.2578i			1.05701i	−0.848932	+	0.528503i	$0.822754\pi$
$572$		0			0
$573$	−9.90256	+	35.7883i	−0.413686	+	1.49508i
$574$		0			0
$575$	0.736501			0.0307142
$576$		0			0
$577$	3.79244			0.157881			0.0789407	−	0.996879i	$-0.474846\pi$
$577$	3.79244			0.157881			0.0789407	+	0.996879i	$0.474846\pi$
$578$		0			0
$579$	2.40357	−	8.68662i	0.0998891	−	0.361004i
$580$		0			0
$581$		−	2.27998i		−	0.0945896i
$582$		0			0
$583$			0.768226i			0.0318167i
$584$		0			0
$585$	−3.07180	+	5.12584i	−0.127003	+	0.211927i
$586$		0			0
$587$	5.15607			0.212814			0.106407	−	0.994323i	$-0.466065\pi$
$587$	5.15607			0.212814			0.106407	+	0.994323i	$0.466065\pi$
$588$		0			0
$589$	0.412962			0.0170158
$590$		0			0
$591$	−34.9716	−	9.67658i	−1.43854	−	0.398041i
$592$		0			0
$593$		−	8.50732i		−	0.349354i	−0.984626	−	0.174677i	$-0.944112\pi$
$593$		−	8.50732i		−	0.349354i	0.984626	−	0.174677i	$-0.0558881\pi$
$594$		0			0
$595$			15.0076i			0.615253i
$596$		0			0
$597$	35.7665	+	9.89654i	1.46383	+	0.405038i
$598$		0			0
$599$	47.6108			1.94533			0.972663	−	0.232221i	$-0.0745992\pi$
$599$	47.6108			1.94533			0.972663	+	0.232221i	$0.0745992\pi$
$600$		0			0
$601$	−37.4898			−1.52924			−0.764621	−	0.644480i	$-0.777074\pi$
$601$	−37.4898			−1.52924			−0.764621	+	0.644480i	$0.777074\pi$
$602$		0			0
$603$	12.8189	−	21.3905i	0.522024	−	0.871088i
$604$		0			0
$605$		−	5.97580i		−	0.242951i
$606$		0			0
$607$			16.3327i			0.662925i	0.943468	+	0.331462i	$0.107542\pi$
$607$			16.3327i			0.662925i	−0.943468	+	0.331462i	$0.892458\pi$
$608$		0			0
$609$	−1.99582	+	7.21297i	−0.0808746	+	0.292284i
$610$		0			0
$611$	−4.97738			−0.201363
$612$		0			0
$613$	−39.8244			−1.60849			−0.804245	−	0.594298i	$-0.797430\pi$
$613$	−39.8244			−1.60849			−0.804245	+	0.594298i	$0.797430\pi$
$614$		0			0
$615$	6.62028	−	23.9260i	0.266955	−	0.964789i
$616$		0			0
$617$			2.31346i			0.0931362i	0.998915	+	0.0465681i	$0.0148285\pi$
$617$			2.31346i			0.0931362i	−0.998915	+	0.0465681i	$0.985172\pi$
$618$		0			0
$619$			14.2618i			0.573232i	0.958046	+	0.286616i	$0.0925303\pi$
$619$			14.2618i			0.573232i	−0.958046	+	0.286616i	$0.907470\pi$
$620$		0			0
$621$	−2.68493	+	2.55682i	−0.107742	+	0.102602i
$622$		0			0
$623$	−6.77118			−0.271281
$624$		0			0
$625$	−18.7735			−0.750942
$626$		0			0
$627$	1.47540	+	0.408241i	0.0589218	+	0.0163036i
$628$		0			0
$629$		−	30.5274i		−	1.21721i
$630$		0			0
$631$		−	20.8339i		−	0.829385i	−0.909962	−	0.414692i	$-0.863889\pi$
$631$		−	20.8339i		−	0.829385i	0.909962	−	0.414692i	$-0.136111\pi$
$632$		0			0
$633$	3.41953	+	0.946178i	0.135914	+	0.0376072i
$634$		0			0
$635$	7.55281			0.299724
$636$		0			0
$637$	−4.66624			−0.184883
$638$		0			0
$639$	−18.5205	−	11.0989i	−0.732659	−	0.439067i
$640$		0			0
$641$		−	31.9762i		−	1.26299i	−0.775382	−	0.631493i	$-0.782443\pi$
$641$		−	31.9762i		−	1.26299i	0.775382	−	0.631493i	$-0.217557\pi$
$642$		0			0
$643$		−	33.9140i		−	1.33744i	−0.743515	−	0.668719i	$-0.766843\pi$
$643$		−	33.9140i		−	1.33744i	0.743515	−	0.668719i	$-0.233157\pi$
$644$		0			0
$645$	−9.15615	+	33.0907i	−0.360523	+	1.30295i
$646$		0			0
$647$	−23.7076			−0.932044			−0.466022	−	0.884773i	$-0.654313\pi$
$647$	−23.7076			−0.932044			−0.466022	+	0.884773i	$0.654313\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$	0.932527	−	3.37019i	0.0365486	−	0.132088i
$652$		0			0
$653$		−	20.0722i		−	0.785487i	−0.919648	−	0.392744i	$-0.871526\pi$
$653$		−	20.0722i		−	0.785487i	0.919648	−	0.392744i	$-0.128474\pi$
$654$		0			0
$655$		−	35.1964i		−	1.37524i
$656$		0			0
$657$	−11.4007	−	6.83217i	−0.444782	−	0.266548i
$658$		0			0
$659$	−39.4507			−1.53678			−0.768391	−	0.639981i	$-0.778942\pi$
$659$	−39.4507			−1.53678			−0.768391	+	0.639981i	$0.778942\pi$
$660$		0			0
$661$	−50.0275			−1.94584			−0.972922	−	0.231132i	$-0.925757\pi$
$661$	−50.0275			−1.94584			−0.972922	+	0.231132i	$0.925757\pi$
$662$		0			0
$663$	8.23285	+	2.27802i	0.319737	+	0.0884708i
$664$		0			0
$665$		−	0.950882i		−	0.0368736i
$666$		0			0
$667$			2.01815i			0.0781431i
$668$		0			0
$669$	−3.19138	−	0.883049i	−0.123386	−	0.0341407i
$670$		0			0
$671$	34.4384			1.32948
$672$		0			0
$673$	30.6051			1.17974			0.589871	−	0.807498i	$-0.299179\pi$
$673$	30.6051			1.17974			0.589871	+	0.807498i	$0.299179\pi$
$674$		0			0
$675$	−3.88408	+	3.69876i	−0.149498	+	0.142365i
$676$		0			0
$677$			20.4536i			0.786098i	0.919518	+	0.393049i	$0.128580\pi$
$677$			20.4536i			0.786098i	−0.919518	+	0.393049i	$0.871420\pi$
$678$		0			0
$679$			24.4637i			0.938829i
$680$		0			0
$681$	9.17660	−	33.1647i	0.351648	−	1.27087i
$682$		0			0
$683$	−34.9957			−1.33907			−0.669536	−	0.742779i	$-0.733507\pi$
$683$	−34.9957			−1.33907			−0.669536	+	0.742779i	$0.733507\pi$
$684$		0			0
$685$	24.5187			0.936813
$686$		0			0
$687$	−7.30905	+	26.4152i	−0.278858	+	1.00780i
$688$		0			0
$689$		−	0.271609i		−	0.0103475i
$690$		0			0
$691$			14.8587i			0.565251i	0.959230	+	0.282626i	$0.0912054\pi$
$691$			14.8587i			0.565251i	−0.959230	+	0.282626i	$0.908795\pi$
$692$		0			0
$693$	6.66334	−	11.1189i	0.253119	−	0.422373i
$694$		0			0
$695$	−21.8328			−0.828167
$696$		0			0
$697$	−35.4865			−1.34415
$698$		0			0
$699$	−17.1134	−	4.73524i	−0.647288	−	0.179103i
$700$		0			0
$701$		−	3.95820i		−	0.149499i	−0.997202	−	0.0747495i	$-0.976184\pi$
$701$		−	3.95820i		−	0.149499i	0.997202	−	0.0747495i	$-0.0238157\pi$
$702$		0			0
$703$			1.93421i			0.0729503i
$704$		0			0
$705$	16.5507	+	4.57956i	0.623336	+	0.172476i
$706$		0			0
$707$	−10.9218			−0.410755
$708$		0			0
$709$	7.98185			0.299765			0.149882	−	0.988704i	$-0.452111\pi$
$709$	7.98185			0.299765			0.149882	+	0.988704i	$0.452111\pi$
$710$		0			0
$711$	−8.03906	+	13.4146i	−0.301488	+	0.503086i
$712$		0			0
$713$		−	0.942962i		−	0.0353142i
$714$		0			0
$715$		−	5.63404i		−	0.210701i
$716$		0			0
$717$	−8.17339	+	29.5390i	−0.305241	+	1.10315i
$718$		0			0
$719$	17.7752			0.662902			0.331451	−	0.943472i	$-0.392462\pi$
$719$	17.7752			0.662902			0.331451	+	0.943472i	$0.392462\pi$
$720$		0			0
$721$	−14.0744			−0.524156
$722$		0			0
$723$	−3.34874	+	12.1025i	−0.124541	+	0.450096i
$724$		0			0
$725$			2.91951i			0.108428i
$726$		0			0
$727$			5.81424i			0.215638i	0.994171	+	0.107819i	$0.0343867\pi$
$727$			5.81424i			0.215638i	−0.994171	+	0.107819i	$0.965613\pi$
$728$		0			0
$729$	1.31896	−	26.9678i	0.0488505	−	0.998806i
$730$		0			0
$731$	49.0795			1.81527
$732$		0			0
$733$	18.1298			0.669641			0.334821	−	0.942282i	$-0.391324\pi$
$733$	18.1298			0.669641			0.334821	+	0.942282i	$0.391324\pi$
$734$		0			0
$735$	15.5161	+	4.29328i	0.572320	+	0.158360i
$736$		0			0
$737$			23.5112i			0.866048i
$738$		0			0
$739$		−	42.8348i		−	1.57570i	−0.615866	−	0.787851i	$-0.711194\pi$
$739$		−	42.8348i		−	1.57570i	0.615866	−	0.787851i	$-0.288806\pi$
$740$		0			0
$741$	−0.521633	−	0.144335i	−0.0191627	−	0.00530228i
$742$		0			0
$743$	34.0147			1.24788			0.623938	−	0.781474i	$-0.285532\pi$
$743$	34.0147			1.24788			0.623938	+	0.781474i	$0.285532\pi$
$744$		0			0
$745$	44.2177			1.62001
$746$		0			0
$747$	3.84055	+	2.30156i	0.140518	+	0.0842096i
$748$		0			0
$749$			27.6554i			1.01050i
$750$		0			0
$751$		−	8.41666i		−	0.307128i	−0.988139	−	0.153564i	$-0.950925\pi$
$751$		−	8.41666i		−	0.307128i	0.988139	−	0.153564i	$-0.0490752\pi$
$752$		0			0
$753$	−0.455032	+	1.64451i	−0.0165823	+	0.0599292i
$754$		0			0
$755$	−9.89415			−0.360085
$756$		0			0
$757$	−35.1198			−1.27645			−0.638225	−	0.769850i	$-0.720331\pi$
$757$	−35.1198			−1.27645			−0.638225	+	0.769850i	$0.720331\pi$
$758$		0			0
$759$	0.932183	−	3.36895i	0.0338361	−	0.122285i
$760$		0			0
$761$			13.7590i			0.498765i	0.968405	+	0.249383i	$0.0802276\pi$
$761$			13.7590i			0.498765i	−0.968405	+	0.249383i	$0.919772\pi$
$762$		0			0
$763$		−	15.5667i		−	0.563552i
$764$		0			0
$765$	−25.2798	−	15.1496i	−0.913994	−	0.547737i
$766$		0			0
$767$	−2.82843			−0.102129
$768$		0			0
$769$	−49.2675			−1.77663			−0.888316	−	0.459234i	$-0.848124\pi$
$769$	−49.2675			−1.77663			−0.888316	+	0.459234i	$0.848124\pi$
$770$		0			0
$771$	−21.9441	−	6.07191i	−0.790299	−	0.218674i
$772$		0			0
$773$		−	41.9899i		−	1.51027i	−0.655569	−	0.755136i	$-0.727571\pi$
$773$		−	41.9899i		−	1.51027i	0.655569	−	0.755136i	$-0.272429\pi$
$774$		0			0
$775$		−	1.36411i		−	0.0490004i
$776$		0			0
$777$	15.7852	+	4.36774i	0.566291	+	0.156692i
$778$		0			0
$779$	2.24842			0.0805581
$780$		0			0
$781$	20.3567			0.728420
$782$		0			0
$783$	−10.1353	−	10.6431i	−0.362206	−	0.380354i
$784$		0			0
$785$			8.18964i			0.292301i
$786$		0			0
$787$			0.0716318i			0.00255340i	0.999999	+	0.00127670i	$0.000406386\pi$
$787$			0.0716318i			0.00255340i	−0.999999	+	0.00127670i	$0.999594\pi$
$788$		0			0
$789$	−13.7130	+	49.5595i	−0.488197	+	1.76437i
$790$		0			0
$791$	19.8042			0.704157
$792$		0			0
$793$	−12.1758			−0.432376
$794$		0			0
$795$	−0.249900	+	0.903150i	−0.00886304	+	0.0320314i
$796$		0			0
$797$		−	15.3630i		−	0.544185i	−0.962271	−	0.272092i	$-0.912284\pi$
$797$		−	15.3630i		−	0.544185i	0.962271	−	0.272092i	$-0.0877157\pi$
$798$		0			0
$799$		−	24.5477i		−	0.868434i
$800$		0			0
$801$	6.83525	−	11.4058i	0.241512	−	0.403004i
$802$		0			0
$803$	12.5310			0.442208
$804$		0			0
$805$	−2.17126			−0.0765268
$806$		0			0
$807$	−36.5200	−	10.1050i	−1.28557	−	0.355714i
$808$		0			0
$809$			37.9238i			1.33333i	0.745358	+	0.666665i	$0.232279\pi$
$809$			37.9238i			1.33333i	−0.745358	+	0.666665i	$0.767721\pi$
$810$		0			0
$811$		−	2.19743i		−	0.0771622i	−0.999255	−	0.0385811i	$-0.987716\pi$
$811$		−	2.19743i		−	0.0771622i	0.999255	−	0.0385811i	$-0.0122838\pi$
$812$		0			0
$813$	−22.0015	−	6.08777i	−0.771625	−	0.213507i
$814$		0			0
$815$	49.6090			1.73773
$816$		0			0
$817$	−3.10967			−0.108794
$818$		0			0
$819$	−2.35585	+	3.93114i	−0.0823199	+	0.137365i
$820$		0			0
$821$			21.4504i			0.748624i	0.927303	+	0.374312i	$0.122121\pi$
$821$			21.4504i			0.748624i	−0.927303	+	0.374312i	$0.877879\pi$
$822$		0			0
$823$			6.53171i			0.227681i	0.993499	+	0.113841i	$0.0363153\pi$
$823$			6.53171i			0.227681i	−0.993499	+	0.113841i	$0.963685\pi$
$824$		0			0
$825$	1.34852	−	4.87361i	0.0469494	−	0.169677i
$826$		0			0
$827$	−46.3775			−1.61270			−0.806352	−	0.591436i	$-0.798561\pi$
$827$	−46.3775			−1.61270			−0.806352	+	0.591436i	$0.798561\pi$
$828$		0			0
$829$	14.6706			0.509530			0.254765	−	0.967003i	$-0.418002\pi$
$829$	14.6706			0.509530			0.254765	+	0.967003i	$0.418002\pi$
$830$		0			0
$831$	3.52616	−	12.7437i	0.122321	−	0.442074i
$832$		0			0
$833$		−	23.0132i		−	0.797359i
$834$		0			0
$835$		−	36.1903i		−	1.25242i
$836$		0			0
$837$	4.73562	+	4.97290i	0.163687	+	0.171888i
$838$		0			0
$839$	−4.64114			−0.160230			−0.0801150	−	0.996786i	$-0.525529\pi$
$839$	−4.64114			−0.160230			−0.0801150	+	0.996786i	$0.525529\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	−12.1635	−	3.36562i	−0.418933	−	0.115918i
$844$		0			0
$845$			1.99193i			0.0685246i
$846$		0			0
$847$		−	4.58299i		−	0.157474i
$848$		0			0
$849$	19.1578	+	5.30093i	0.657495	+	0.181928i
$850$		0			0
$851$	4.41661			0.151400
$852$		0			0
$853$	−40.5647			−1.38891			−0.694455	−	0.719537i	$-0.744354\pi$
$853$	−40.5647			−1.38891			−0.694455	+	0.719537i	$0.744354\pi$
$854$		0			0
$855$	1.60173	+	0.959881i	0.0547779	+	0.0328272i
$856$		0			0
$857$		−	8.55175i		−	0.292122i	−0.989276	−	0.146061i	$-0.953340\pi$
$857$		−	8.55175i		−	0.292122i	0.989276	−	0.146061i	$-0.0466596\pi$
$858$		0			0
$859$			28.4530i			0.970802i	0.874292	+	0.485401i	$0.161326\pi$
$859$			28.4530i			0.970802i	−0.874292	+	0.485401i	$0.838674\pi$
$860$		0			0
$861$	5.07726	−	18.3495i	0.173033	−	0.625348i
$862$		0			0
$863$	−5.51439			−0.187712			−0.0938560	−	0.995586i	$-0.529919\pi$
$863$	−5.51439			−0.187712			−0.0938560	+	0.995586i	$0.529919\pi$
$864$		0			0
$865$	−41.3802			−1.40697
$866$		0			0
$867$	−3.38252	+	12.2246i	−0.114876	+	0.415168i
$868$		0			0
$869$		−	14.7446i		−	0.500175i
$870$		0			0
$871$		−	8.31248i		−	0.281658i
$872$		0			0
$873$	−41.2082	−	24.6952i	−1.39469	−	0.835805i
$874$		0			0
$875$	−18.3560			−0.620547
$876$		0			0
$877$	46.4150			1.56732			0.783661	−	0.621188i	$-0.213350\pi$
$877$	46.4150			1.56732			0.783661	+	0.621188i	$0.213350\pi$
$878$		0			0
$879$	−2.41239	−	0.667503i	−0.0813677	−	0.0225143i
$880$		0			0
$881$		−	1.78585i		−	0.0601669i	−0.999547	−	0.0300834i	$-0.990423\pi$
$881$		−	1.78585i		−	0.0601669i	0.999547	−	0.0300834i	$-0.00957730\pi$
$882$		0			0
$883$			32.9419i			1.10858i	0.832323	+	0.554291i	$0.187011\pi$
$883$			32.9419i			1.10858i	−0.832323	+	0.554291i	$0.812989\pi$
$884$		0			0
$885$	9.40504	+	2.60236i	0.316147	+	0.0874774i
$886$		0			0
$887$	−55.4993			−1.86349			−0.931743	−	0.363119i	$-0.881712\pi$
$887$	−55.4993			−1.86349			−0.931743	+	0.363119i	$0.881712\pi$
$888$		0			0
$889$	5.79244			0.194272
$890$		0			0
$891$	12.0031	+	22.4483i	0.402118	+	0.752047i
$892$		0			0
$893$			1.55534i			0.0520474i
$894$		0			0
$895$			26.8167i			0.896384i
$896$		0			0
$897$	−0.329576	+	1.19110i	−0.0110042	+	0.0397698i
$898$		0			0
$899$	3.73793			0.124667
$900$		0			0
$901$	1.33953			0.0446263
$902$		0			0
$903$	−7.02209	+	25.3781i	−0.233681	+	0.844532i
$904$		0			0
$905$			19.4123i			0.645286i
$906$		0			0
$907$			5.09939i			0.169322i	0.996410	+	0.0846612i	$0.0269808\pi$
$907$			5.09939i			0.169322i	−0.996410	+	0.0846612i	$0.973019\pi$
$908$		0			0
$909$	11.0251	−	18.3973i	0.365680	−	0.610200i
$910$		0			0
$911$	2.91034			0.0964239			0.0482120	−	0.998837i	$-0.484648\pi$
$911$	2.91034			0.0964239			0.0482120	+	0.998837i	$0.484648\pi$
$912$		0			0
$913$	−4.22132			−0.139705
$914$		0			0
$915$	40.4868	+	11.2026i	1.33845	+	0.370348i
$916$		0			0
$917$		−	26.9931i		−	0.891389i
$918$		0			0
$919$		−	59.8779i		−	1.97519i	−0.157019	−	0.987596i	$-0.550188\pi$
$919$		−	59.8779i		−	1.97519i	0.157019	−	0.987596i	$-0.449812\pi$
$920$		0			0
$921$	−3.93750	−	1.08950i	−0.129745	−	0.0359002i
$922$		0			0
$923$	−7.19718			−0.236898
$924$		0			0
$925$	6.38919			0.210075
$926$		0			0
$927$	14.2075	−	23.7078i	0.466637	−	0.778665i
$928$		0			0
$929$			6.15670i			0.201995i	0.994887	+	0.100997i	$0.0322034\pi$
$929$			6.15670i			0.201995i	−0.994887	+	0.100997i	$0.967797\pi$
$930$		0			0
$931$			1.45811i			0.0477877i
$932$		0			0
$933$	10.3701	−	37.4779i	0.339501	−	1.22697i
$934$		0			0
$935$	27.7862			0.908705
$936$		0			0
$937$	25.8098			0.843171			0.421585	−	0.906789i	$-0.361474\pi$
$937$	25.8098			0.843171			0.421585	+	0.906789i	$0.361474\pi$
$938$		0			0
$939$	−3.39370	+	12.2650i	−0.110749	+	0.400253i
$940$		0			0
$941$		−	46.0736i		−	1.50196i	−0.660327	−	0.750978i	$-0.729582\pi$
$941$		−	46.0736i		−	1.50196i	0.660327	−	0.750978i	$-0.270418\pi$
$942$		0			0
$943$		−	5.13408i		−	0.167189i
$944$		0			0
$945$	11.4506	−	10.9042i	0.372487	−	0.354714i
$946$		0			0
$947$	45.5752			1.48099			0.740497	−	0.672060i	$-0.234590\pi$
$947$	45.5752			1.48099			0.740497	+	0.672060i	$0.234590\pi$
$948$		0			0
$949$	−4.43037			−0.143816
$950$		0			0
$951$	−38.3353	−	10.6073i	−1.24311	−	0.343966i
$952$		0			0
$953$			4.14268i			0.134195i	0.997746	+	0.0670973i	$0.0213738\pi$
$953$			4.14268i			0.134195i	−0.997746	+	0.0670973i	$0.978626\pi$
$954$		0			0
$955$		−	42.7046i		−	1.38189i
$956$		0			0
$957$	13.3546	+	3.69520i	0.431693	+	0.119449i
$958$		0			0
$959$	18.8041			0.607215
$960$		0			0
$961$	29.2535			0.943661
$962$		0			0
$963$	−46.5845	−	27.9171i	−1.50116	−	0.899615i
$964$		0			0
$965$			10.3654i			0.333673i
$966$		0			0
$967$			4.67699i			0.150402i	0.997168	+	0.0752010i	$0.0239598\pi$
$967$			4.67699i			0.150402i	−0.997168	+	0.0752010i	$0.976040\pi$
$968$		0			0
$969$	0.711837	−	2.57261i	0.0228675	−	0.0826442i
$970$		0			0
$971$	−42.4233			−1.36143			−0.680713	−	0.732550i	$-0.738330\pi$
$971$	−42.4233			−1.36143			−0.680713	+	0.732550i	$0.738330\pi$
$972$		0			0
$973$	−16.7442			−0.536793
$974$		0			0
$975$	−0.476774	+	1.72308i	−0.0152690	+	0.0551828i
$976$		0			0
$977$			44.4355i			1.42162i	0.703386	+	0.710808i	$0.251671\pi$
$977$			44.4355i			1.42162i	−0.703386	+	0.710808i	$0.748329\pi$
$978$		0			0
$979$			12.5366i			0.400673i
$980$		0			0
$981$	26.2215	+	15.7140i	0.837190	+	0.501710i
$982$		0			0
$983$	−32.2576			−1.02886			−0.514428	−	0.857534i	$-0.671996\pi$
$983$	−32.2576			−1.02886			−0.514428	+	0.857534i	$0.671996\pi$
$984$		0			0
$985$	41.7301			1.32963
$986$		0			0
$987$	12.6932	+	3.51218i	0.404028	+	0.111794i
$988$		0			0
$989$			7.10067i			0.225788i
$990$		0			0
$991$		−	8.04700i		−	0.255621i	−0.991799	−	0.127811i	$-0.959205\pi$
$991$		−	8.04700i		−	0.255621i	0.991799	−	0.127811i	$-0.0407950\pi$
$992$		0			0
$993$	21.3023	+	5.89430i	0.676007	+	0.187050i
$994$		0			0
$995$	−42.6786			−1.35300
$996$		0			0
$997$	−7.84865			−0.248569			−0.124285	−	0.992247i	$-0.539664\pi$
$997$	−7.84865			−0.248569			−0.124285	+	0.992247i	$0.539664\pi$
$998$		0			0
$999$	−23.2919	+	22.1805i	−0.736923	+	0.701761i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.d.q.1535.13		20
3.2	odd	2	inner	2496.2.d.q.1535.7		20
4.3	odd	2	inner	2496.2.d.q.1535.8		20
8.3	odd	2		1248.2.d.d.287.13	yes	20
8.5	even	2		1248.2.d.d.287.8	yes	20
12.11	even	2	inner	2496.2.d.q.1535.14		20
24.5	odd	2		1248.2.d.d.287.14	yes	20
24.11	even	2		1248.2.d.d.287.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.d.d.287.7	✓	20	24.11	even	2
1248.2.d.d.287.8	yes	20	8.5	even	2
1248.2.d.d.287.13	yes	20	8.3	odd	2
1248.2.d.d.287.14	yes	20	24.5	odd	2
2496.2.d.q.1535.7		20	3.2	odd	2	inner
2496.2.d.q.1535.8		20	4.3	odd	2	inner
2496.2.d.q.1535.13		20	1.1	even	1	trivial
2496.2.d.q.1535.14		20	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.461900 - 1.66933i) q^{3} +1.99193i q^{5} +1.52766i q^{7} +(-2.57330 - 1.54212i) q^{9} +2.82843 q^{11} -1.00000 q^{13} +(3.32519 + 0.920073i) q^{15} -4.93184i q^{17} +0.312481i q^{19} +(2.55017 + 0.705628i) q^{21} +0.713524 q^{23} +1.03220 q^{25} +(-3.76291 + 3.58337i) q^{27} +2.82843i q^{29} -1.32156i q^{31} +(1.30645 - 4.72157i) q^{33} -3.04301 q^{35} +6.18986 q^{37} +(-0.461900 + 1.66933i) q^{39} -7.19538i q^{41} +9.95155i q^{43} +(3.07180 - 5.12584i) q^{45} +4.97738 q^{47} +4.66624 q^{49} +(-8.23285 - 2.27802i) q^{51} +0.271609i q^{53} +5.63404i q^{55} +(0.521633 + 0.144335i) q^{57} +2.82843 q^{59} +12.1758 q^{61} +(2.35585 - 3.93114i) q^{63} -1.99193i q^{65} +8.31248i q^{67} +(0.329576 - 1.19110i) q^{69} +7.19718 q^{71} +4.43037 q^{73} +(0.476774 - 1.72308i) q^{75} +4.32089i q^{77} -5.21299i q^{79} +(4.24372 + 7.93668i) q^{81} -1.49246 q^{83} +9.82390 q^{85} +(4.72157 + 1.30645i) q^{87} +4.43237i q^{89} -1.52766i q^{91} +(-2.20611 - 0.610426i) q^{93} -0.622442 q^{95} +16.0138 q^{97} +(-7.27838 - 4.36178i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{9} - 20 q^{13} + 12 q^{21} - 36 q^{25} - 16 q^{37} - 4 q^{45} - 76 q^{49} - 16 q^{57} + 56 q^{61} - 24 q^{69} + 88 q^{73} + 72 q^{81} - 56 q^{85} - 96 q^{93} - 72 q^{97}+O(q^{100}) \) 20 * q - 8 * q^9 - 20 * q^13 + 12 * q^21 - 36 * q^25 - 16 * q^37 - 4 * q^45 - 76 * q^49 - 16 * q^57 + 56 * q^61 - 24 * q^69 + 88 * q^73 + 72 * q^81 - 56 * q^85 - 96 * q^93 - 72 * q^97

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