LMFDB - Embedded newform 2496.2.d.m.1535.8

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:	$8$
Coefficient field:	8.0.121550625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 $ x^8 - x^7 - 4x^6 - 9x^5 + 23x^4 + 18x^3 - 16x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1535.8
Root			$0.553538 - 0.676408i$ of defining polynomial
Character	$\chi$	$=$	2496.1535
Dual form			2496.2.d.m.1535.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.70466 + 0.306808i) q^{3} -2.09201i q^{5} +0.613616i q^{7} +(2.81174 + 1.04601i) q^{9} +O(q^{10})$	$q+(1.70466 + 0.306808i) q^{3} -2.09201i q^{5} +0.613616i q^{7} +(2.81174 + 1.04601i) q^{9} -1.00000 q^{13} +(0.641847 - 3.56618i) q^{15} -2.09201i q^{17} +5.29150i q^{19} +(-0.188262 + 1.04601i) q^{21} +6.81864 q^{23} +0.623475 q^{25} +(4.47214 + 2.64575i) q^{27} +6.92820i q^{29} +5.29150i q^{31} +1.28369 q^{35} +5.62348 q^{37} +(-1.70466 - 0.306808i) q^{39} -11.1122i q^{41} -11.1966i q^{43} +(2.18826 - 5.88220i) q^{45} +3.40932 q^{47} +6.62348 q^{49} +(0.641847 - 3.56618i) q^{51} -11.1122i q^{53} +(-1.62348 + 9.02022i) q^{57} +2.00000 q^{61} +(-0.641847 + 1.72533i) q^{63} +2.09201i q^{65} +5.29150i q^{67} +(11.6235 + 2.09201i) q^{69} +3.40932 q^{71} -13.2470 q^{73} +(1.06281 + 0.191287i) q^{75} -6.51873i q^{79} +(6.81174 + 5.88220i) q^{81} +13.6373 q^{83} -4.37652 q^{85} +(-2.12563 + 11.8102i) q^{87} +2.74417i q^{89} -0.613616i q^{91} +(-1.62348 + 9.02022i) q^{93} +11.0699 q^{95} -5.24695 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{9}+O(q^{10}) $ 8 * q + 2 * q^9	$ 8 q + 2 q^{9} - 8 q^{13} - 22 q^{21} - 36 q^{25} + 4 q^{37} + 38 q^{45} + 12 q^{49} + 28 q^{57} + 16 q^{61} + 52 q^{69} - 24 q^{73} + 34 q^{81} - 76 q^{85} + 28 q^{93} + 40 q^{97}+O(q^{100}) $ 8 * q + 2 * q^9 - 8 * q^13 - 22 * q^21 - 36 * q^25 + 4 * q^37 + 38 * q^45 + 12 * q^49 + 28 * q^57 + 16 * q^61 + 52 * q^69 - 24 * q^73 + 34 * q^81 - 76 * q^85 + 28 * q^93 + 40 * q^97

Display 100 coefficients 10 coefficients

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
$3$	1.70466	+	0.306808i	0.984186	+	0.177136i
$3$	1.70466	+	0.306808i	0.984186	+	0.177136i
$4$		0			0
$5$		−	2.09201i		−	0.935577i	−0.883840	−	0.467789i	$-0.845051\pi$
$5$		−	2.09201i		−	0.935577i	0.883840	−	0.467789i	$-0.154949\pi$
$6$		0			0
$7$			0.613616i			0.231925i	0.993254	+	0.115963i	$0.0369953\pi$
$7$			0.613616i			0.231925i	−0.993254	+	0.115963i	$0.963005\pi$
$8$		0			0
$9$	2.81174	+	1.04601i	0.937246	+	0.348669i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	0.641847	−	3.56618i	0.165724	−	0.920783i
$16$		0			0
$17$		−	2.09201i		−	0.507388i	−0.967285	−	0.253694i	$-0.918354\pi$
$17$		−	2.09201i		−	0.507388i	0.967285	−	0.253694i	$-0.0816456\pi$
$18$		0			0
$19$			5.29150i			1.21395i	0.794719	+	0.606977i	$0.207618\pi$
$19$			5.29150i			1.21395i	−0.794719	+	0.606977i	$0.792382\pi$
$20$		0			0
$21$	−0.188262	+	1.04601i	−0.0410822	+	0.228257i
$22$		0			0
$23$	6.81864			1.42179			0.710893	−	0.703300i	$-0.248291\pi$
$23$	6.81864			1.42179			0.710893	+	0.703300i	$0.248291\pi$
$24$		0			0
$25$	0.623475			0.124695
$26$		0			0
$27$	4.47214	+	2.64575i	0.860663	+	0.509175i
$28$		0			0
$29$			6.92820i			1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$			6.92820i			1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$		0			0
$31$			5.29150i			0.950382i	0.879883	+	0.475191i	$0.157621\pi$
$31$			5.29150i			0.950382i	−0.879883	+	0.475191i	$0.842379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.28369			0.216984
$36$		0			0
$37$	5.62348			0.924494			0.462247	−	0.886751i	$-0.347043\pi$
$37$	5.62348			0.924494			0.462247	+	0.886751i	$0.347043\pi$
$38$		0			0
$39$	−1.70466	−	0.306808i	−0.272964	−	0.0491286i
$40$		0			0
$41$		−	11.1122i		−	1.73544i	−0.497054	−	0.867720i	$-0.665585\pi$
$41$		−	11.1122i		−	1.73544i	0.497054	−	0.867720i	$-0.334415\pi$
$42$		0			0
$43$		−	11.1966i		−	1.70747i	−0.520709	−	0.853734i	$-0.674332\pi$
$43$		−	11.1966i		−	1.70747i	0.520709	−	0.853734i	$-0.325668\pi$
$44$		0			0
$45$	2.18826	−	5.88220i	0.326207	−	0.876866i
$46$		0			0
$47$	3.40932			0.497301			0.248650	−	0.968593i	$-0.420013\pi$
$47$	3.40932			0.497301			0.248650	+	0.968593i	$0.420013\pi$
$48$		0			0
$49$	6.62348			0.946211
$50$		0			0
$51$	0.641847	−	3.56618i	0.0898765	−	0.499364i
$52$		0			0
$53$		−	11.1122i		−	1.52638i	−0.646173	−	0.763191i	$-0.723631\pi$
$53$		−	11.1122i		−	1.52638i	0.646173	−	0.763191i	$-0.276369\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.62348	+	9.02022i	−0.215035	+	1.19476i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$	−0.641847	+	1.72533i	−0.0808651	+	0.217371i
$64$		0			0
$65$			2.09201i			0.259482i
$66$		0			0
$67$			5.29150i			0.646460i	0.946320	+	0.323230i	$0.104769\pi$
$67$			5.29150i			0.646460i	−0.946320	+	0.323230i	$0.895231\pi$
$68$		0			0
$69$	11.6235	+	2.09201i	1.39930	+	0.251849i
$70$		0			0
$71$	3.40932			0.404612			0.202306	−	0.979322i	$-0.435156\pi$
$71$	3.40932			0.404612			0.202306	+	0.979322i	$0.435156\pi$
$72$		0			0
$73$	−13.2470			−1.55044			−0.775219	−	0.631692i	$-0.782361\pi$
$73$	−13.2470			−1.55044			−0.775219	+	0.631692i	$0.782361\pi$
$74$		0			0
$75$	1.06281	+	0.191287i	0.122723	+	0.0220879i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	6.51873i		−	0.733415i	−0.930336	−	0.366707i	$-0.880485\pi$
$79$		−	6.51873i		−	0.733415i	0.930336	−	0.366707i	$-0.119515\pi$
$80$		0			0
$81$	6.81174	+	5.88220i	0.756860	+	0.653577i
$82$		0			0
$83$	13.6373			1.49689			0.748443	−	0.663199i	$-0.230802\pi$
$83$	13.6373			1.49689			0.748443	+	0.663199i	$0.230802\pi$
$84$		0			0
$85$	−4.37652			−0.474701
$86$		0			0
$87$	−2.12563	+	11.8102i	−0.227891	+	1.26619i
$88$		0			0
$89$			2.74417i			0.290882i	0.989367	+	0.145441i	$0.0464601\pi$
$89$			2.74417i			0.290882i	−0.989367	+	0.145441i	$0.953540\pi$
$90$		0			0
$91$		−	0.613616i		−	0.0643244i
$92$		0			0
$93$	−1.62348	+	9.02022i	−0.168347	+	0.935353i
$94$		0			0
$95$	11.0699			1.13575
$96$		0			0
$97$	−5.24695			−0.532747			−0.266374	−	0.963870i	$-0.585825\pi$
$97$	−5.24695			−0.532747			−0.266374	+	0.963870i	$0.585825\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			6.92820i			0.689382i	0.938716	+	0.344691i	$0.112016\pi$
$101$			6.92820i			0.689382i	−0.938716	+	0.344691i	$0.887984\pi$
$102$		0			0
$103$		−	8.97320i		−	0.884156i	−0.896977	−	0.442078i	$-0.854242\pi$
$103$		−	8.97320i		−	0.884156i	0.896977	−	0.442078i	$-0.145758\pi$
$104$		0			0
$105$	2.18826	+	0.393847i	0.213553	+	0.0384356i
$106$		0			0
$107$	−4.69302			−0.453691			−0.226845	−	0.973931i	$-0.572841\pi$
$107$	−4.69302			−0.453691			−0.226845	+	0.973931i	$0.572841\pi$
$108$		0			0
$109$	−9.62348			−0.921762			−0.460881	−	0.887462i	$-0.652466\pi$
$109$	−9.62348			−0.921762			−0.460881	+	0.887462i	$0.652466\pi$
$110$		0			0
$111$	9.58612	+	1.72533i	0.909874	+	0.163761i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	14.2647i		−	1.33019i
$116$		0			0
$117$	−2.81174	−	1.04601i	−0.259945	−	0.0967034i
$118$		0			0
$119$	1.28369			0.117676
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	3.40932	−	18.9426i	0.307408	−	1.70800i
$124$		0			0
$125$		−	11.7644i		−	1.05224i
$126$		0			0
$127$		−	20.7834i		−	1.84423i	−0.386914	−	0.922116i	$-0.626459\pi$
$127$		−	20.7834i		−	1.84423i	0.386914	−	0.922116i	$-0.373541\pi$
$128$		0			0
$129$	3.43521	−	19.0864i	0.302454	−	1.68047i
$130$		0			0
$131$	3.40932			0.297874			0.148937	−	0.988847i	$-0.452415\pi$
$131$	3.40932			0.297874			0.148937	+	0.988847i	$0.452415\pi$
$132$		0			0
$133$	−3.24695			−0.281546
$134$		0			0
$135$	5.53495	−	9.35577i	0.476373	−	0.805217i
$136$		0			0
$137$			11.1122i			0.949382i	0.880152	+	0.474691i	$0.157440\pi$
$137$			11.1122i			0.949382i	−0.880152	+	0.474691i	$0.842560\pi$
$138$		0			0
$139$			12.4239i			1.05378i	0.849934	+	0.526889i	$0.176642\pi$
$139$			12.4239i			1.05378i	−0.849934	+	0.526889i	$0.823358\pi$
$140$		0			0
$141$	5.81174	+	1.04601i	0.489437	+	0.0880897i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	14.4939			1.20365
$146$		0			0
$147$	11.2908	+	2.03214i	0.931248	+	0.167608i
$148$		0			0
$149$			13.8564i			1.13516i	0.823318	+	0.567581i	$0.192120\pi$
$149$			13.8564i			1.13516i	−0.823318	+	0.567581i	$0.807880\pi$
$150$		0			0
$151$			14.8783i			1.21078i	0.795929	+	0.605390i	$0.206983\pi$
$151$			14.8783i			1.21078i	−0.795929	+	0.605390i	$0.793017\pi$
$152$		0			0
$153$	2.18826	−	5.88220i	0.176911	−	0.475547i
$154$		0			0
$155$	11.0699			0.889156
$156$		0			0
$157$	−21.2470			−1.69569			−0.847846	−	0.530243i	$-0.822101\pi$
$157$	−21.2470			−1.69569			−0.847846	+	0.530243i	$0.822101\pi$
$158$		0			0
$159$	3.40932	−	18.9426i	0.270377	−	1.50224i
$160$		0			0
$161$			4.18403i			0.329748i
$162$		0			0
$163$			5.29150i			0.414462i	0.978292	+	0.207231i	$0.0664452\pi$
$163$			5.29150i			0.414462i	−0.978292	+	0.207231i	$0.933555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.69302			−0.363156			−0.181578	−	0.983377i	$-0.558121\pi$
$167$	−4.69302			−0.363156			−0.181578	+	0.983377i	$0.558121\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−5.53495	+	14.8783i	−0.423268	+	1.13777i
$172$		0			0
$173$			1.43985i			0.109470i	0.998501	+	0.0547351i	$0.0174314\pi$
$173$			1.43985i			0.109470i	−0.998501	+	0.0547351i	$0.982569\pi$
$174$		0			0
$175$			0.382574i			0.0289199i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.2979			1.59188			0.795939	−	0.605377i	$-0.206978\pi$
$179$	21.2979			1.59188			0.795939	+	0.605377i	$0.206978\pi$
$180$		0			0
$181$	1.24695			0.0926851			0.0463426	−	0.998926i	$-0.485243\pi$
$181$	1.24695			0.0926851			0.0463426	+	0.998926i	$0.485243\pi$
$182$		0			0
$183$	3.40932	+	0.613616i	0.252024	+	0.0453598i
$184$		0			0
$185$		−	11.7644i		−	0.864935i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.62348	+	2.74417i	−0.118090	+	0.199609i
$190$		0			0
$191$	9.38603			0.679149			0.339575	−	0.940579i	$-0.389717\pi$
$191$	9.38603			0.679149			0.339575	+	0.940579i	$0.389717\pi$
$192$		0			0
$193$	−21.2470			−1.52939			−0.764694	−	0.644393i	$-0.777110\pi$
$193$	−21.2470			−1.52939			−0.764694	+	0.644393i	$0.777110\pi$
$194$		0			0
$195$	−0.641847	+	3.56618i	−0.0459636	+	0.255379i
$196$		0			0
$197$		−	10.4601i		−	0.745249i	−0.927982	−	0.372625i	$-0.878458\pi$
$197$		−	10.4601i		−	0.745249i	0.927982	−	0.372625i	$-0.121542\pi$
$198$		0			0
$199$			5.29150i			0.375105i	0.982255	+	0.187552i	$0.0600554\pi$
$199$			5.29150i			0.375105i	−0.982255	+	0.187552i	$0.939945\pi$
$200$		0			0
$201$	−1.62348	+	9.02022i	−0.114511	+	0.636237i
$202$		0			0
$203$	−4.25126			−0.298380
$204$		0			0
$205$	−23.2470			−1.62364
$206$		0			0
$207$	19.1722	+	7.13235i	1.33256	+	0.495733i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	4.29531i		−	0.295701i	−0.989010	−	0.147851i	$-0.952764\pi$
$211$		−	4.29531i		−	0.295701i	0.989010	−	0.147851i	$-0.0472355\pi$
$212$		0			0
$213$	5.81174	+	1.04601i	0.398214	+	0.0716712i
$214$		0			0
$215$	−23.4235			−1.59747
$216$		0			0
$217$	−3.24695			−0.220417
$218$		0			0
$219$	−22.5816	−	4.06427i	−1.52592	−	0.274638i
$220$		0			0
$221$			2.09201i			0.140724i
$222$		0			0
$223$			17.3328i			1.16069i	0.814371	+	0.580344i	$0.197082\pi$
$223$			17.3328i			1.16069i	−0.814371	+	0.580344i	$0.802918\pi$
$224$		0			0
$225$	1.75305	+	0.652160i	0.116870	+	0.0434773i
$226$		0			0
$227$	2.56739			0.170403			0.0852017	−	0.996364i	$-0.472847\pi$
$227$	2.56739			0.170403			0.0852017	+	0.996364i	$0.472847\pi$
$228$		0			0
$229$	−1.62348			−0.107282			−0.0536411	−	0.998560i	$-0.517083\pi$
$229$	−1.62348			−0.107282			−0.0536411	+	0.998560i	$0.517083\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7.58036i			0.496606i	0.968682	+	0.248303i	$0.0798729\pi$
$233$			7.58036i			0.496606i	−0.968682	+	0.248303i	$0.920127\pi$
$234$		0			0
$235$		−	7.13235i		−	0.465263i
$236$		0			0
$237$	2.00000	−	11.1122i	0.129914	−	0.721817i
$238$		0			0
$239$	12.7954			0.827663			0.413831	−	0.910354i	$-0.364190\pi$
$239$	12.7954			0.827663			0.413831	+	0.910354i	$0.364190\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$	9.80700	+	12.1170i	0.629119	+	0.777309i
$244$		0			0
$245$		−	13.8564i		−	0.885253i
$246$		0			0
$247$		−	5.29150i		−	0.336690i
$248$		0			0
$249$	23.2470	+	4.18403i	1.47322	+	0.265152i
$250$		0			0
$251$	−8.94427			−0.564557			−0.282279	−	0.959332i	$-0.591090\pi$
$251$	−8.94427			−0.564557			−0.282279	+	0.959332i	$0.591090\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−7.46049	−	1.34275i	−0.467194	−	0.0840864i
$256$		0			0
$257$			10.4601i			0.652481i	0.945287	+	0.326241i	$0.105782\pi$
$257$			10.4601i			0.652481i	−0.945287	+	0.326241i	$0.894218\pi$
$258$		0			0
$259$			3.45065i			0.214413i
$260$		0			0
$261$	−7.24695	+	19.4803i	−0.448575	+	1.20580i
$262$		0			0
$263$	−16.2047			−0.999223			−0.499612	−	0.866250i	$-0.666524\pi$
$263$	−16.2047			−0.999223			−0.499612	+	0.866250i	$0.666524\pi$
$264$		0			0
$265$	−23.2470			−1.42805
$266$		0			0
$267$	−0.841935	+	4.67789i	−0.0515256	+	0.286282i
$268$		0			0
$269$			11.1122i			0.677525i	0.940872	+	0.338762i	$0.110008\pi$
$269$			11.1122i			0.677525i	−0.940872	+	0.338762i	$0.889992\pi$
$270$		0			0
$271$			12.4239i			0.754695i	0.926072	+	0.377348i	$0.123164\pi$
$271$			12.4239i			0.754695i	−0.926072	+	0.377348i	$0.876836\pi$
$272$		0			0
$273$	0.188262	−	1.04601i	0.0113942	−	0.0633072i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	24.4939			1.47170			0.735848	−	0.677147i	$-0.236784\pi$
$277$	24.4939			1.47170			0.735848	+	0.677147i	$0.236784\pi$
$278$		0			0
$279$	−5.53495	+	14.8783i	−0.331369	+	0.890742i
$280$		0			0
$281$			11.1122i			0.662900i	0.943473	+	0.331450i	$0.107538\pi$
$281$			11.1122i			0.662900i	−0.943473	+	0.331450i	$0.892462\pi$
$282$		0			0
$283$			10.2004i			0.606353i	0.952934	+	0.303176i	$0.0980471\pi$
$283$			10.2004i			0.606353i	−0.952934	+	0.303176i	$0.901953\pi$
$284$		0			0
$285$	18.8704	+	3.39633i	1.11779	+	0.201181i
$286$		0			0
$287$	6.81864			0.402492
$288$		0			0
$289$	12.6235			0.742557
$290$		0			0
$291$	−8.94427	−	1.60981i	−0.524323	−	0.0943685i
$292$		0			0
$293$		−	7.58036i		−	0.442850i	−0.975177	−	0.221425i	$-0.928929\pi$
$293$		−	7.58036i		−	0.442850i	0.975177	−	0.221425i	$-0.0710707\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.81864			−0.394332
$300$		0			0
$301$	6.87043			0.396005
$302$		0			0
$303$	−2.12563	+	11.8102i	−0.122114	+	0.678480i
$304$		0			0
$305$		−	4.18403i		−	0.239577i
$306$		0			0
$307$		−	27.6847i		−	1.58005i	−0.613073	−	0.790026i	$-0.710067\pi$
$307$		−	27.6847i		−	1.58005i	0.613073	−	0.790026i	$-0.289933\pi$
$308$		0			0
$309$	2.75305	−	15.2963i	0.156615	−	0.870174i
$310$		0			0
$311$	−6.81864			−0.386650			−0.193325	−	0.981135i	$-0.561927\pi$
$311$	−6.81864			−0.386650			−0.193325	+	0.981135i	$0.561927\pi$
$312$		0			0
$313$	−17.6235			−0.996138			−0.498069	−	0.867137i	$-0.665957\pi$
$313$	−17.6235			−0.996138			−0.498069	+	0.867137i	$0.665957\pi$
$314$		0			0
$315$	3.60941	+	1.34275i	0.203367	+	0.0756555i
$316$		0			0
$317$			16.7361i			0.939994i	0.882668	+	0.469997i	$0.155745\pi$
$317$			16.7361i			0.939994i	−0.882668	+	0.469997i	$0.844255\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−8.00000	−	1.43985i	−0.446516	−	0.0803649i
$322$		0			0
$323$	11.0699			0.615946
$324$		0			0
$325$	−0.623475			−0.0345842
$326$		0			0
$327$	−16.4048	−	2.95256i	−0.907185	−	0.163277i
$328$		0			0
$329$			2.09201i			0.115336i
$330$		0			0
$331$			17.1017i			0.939997i	0.882667	+	0.469998i	$0.155745\pi$
$331$			17.1017i			0.939997i	−0.882667	+	0.469998i	$0.844255\pi$
$332$		0			0
$333$	15.8117	+	5.88220i	0.866478	+	0.322342i
$334$		0			0
$335$	11.0699			0.604813
$336$		0			0
$337$	20.8704			1.13688			0.568442	−	0.822723i	$-0.307546\pi$
$337$	20.8704			1.13688			0.568442	+	0.822723i	$0.307546\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			8.35958i			0.451375i
$344$		0			0
$345$	4.37652	−	24.3165i	0.235624	−	1.30916i
$346$		0			0
$347$	−14.4792			−0.777285			−0.388643	−	0.921389i	$-0.627056\pi$
$347$	−14.4792			−0.777285			−0.388643	+	0.921389i	$0.627056\pi$
$348$		0			0
$349$	−2.37652			−0.127212			−0.0636062	−	0.997975i	$-0.520260\pi$
$349$	−2.37652			−0.127212			−0.0636062	+	0.997975i	$0.520260\pi$
$350$		0			0
$351$	−4.47214	−	2.64575i	−0.238705	−	0.141220i
$352$		0			0
$353$			6.92820i			0.368751i	0.982856	+	0.184376i	$0.0590263\pi$
$353$			6.92820i			0.368751i	−0.982856	+	0.184376i	$0.940974\pi$
$354$		0			0
$355$		−	7.13235i		−	0.378546i
$356$		0			0
$357$	2.18826	+	0.393847i	0.115815	+	0.0208446i
$358$		0			0
$359$	−18.3303			−0.967436			−0.483718	−	0.875224i	$-0.660714\pi$
$359$	−18.3303			−0.967436			−0.483718	+	0.875224i	$0.660714\pi$
$360$		0			0
$361$	−9.00000			−0.473684
$362$		0			0
$363$	−18.7513	−	3.37489i	−0.984186	−	0.177136i
$364$		0			0
$365$			27.7128i			1.45055i
$366$		0			0
$367$			7.74597i			0.404336i	0.979351	+	0.202168i	$0.0647987\pi$
$367$			7.74597i			0.404336i	−0.979351	+	0.202168i	$0.935201\pi$
$368$		0			0
$369$	11.6235	−	31.2447i	0.605094	−	1.62653i
$370$		0			0
$371$	6.81864			0.354006
$372$		0			0
$373$	−13.2470			−0.685901			−0.342951	−	0.939353i	$-0.611426\pi$
$373$	−13.2470			−0.685901			−0.342951	+	0.939353i	$0.611426\pi$
$374$		0			0
$375$	3.60941	−	20.0543i	0.186389	−	1.03560i
$376$		0			0
$377$		−	6.92820i		−	0.356821i
$378$		0			0
$379$			12.6549i			0.650038i	0.945707	+	0.325019i	$0.105371\pi$
$379$			12.6549i			0.650038i	−0.945707	+	0.325019i	$0.894629\pi$
$380$		0			0
$381$	6.37652	−	35.4287i	0.326679	−	1.81507i
$382$		0			0
$383$	34.9352			1.78510			0.892551	−	0.450946i	$-0.148913\pi$
$383$	34.9352			1.78510			0.892551	+	0.450946i	$0.148913\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	11.7117	−	31.4820i	0.595342	−	1.60032i
$388$		0			0
$389$		−	29.1527i		−	1.47810i	−0.673651	−	0.739049i	$-0.735275\pi$
$389$		−	29.1527i		−	1.47810i	0.673651	−	0.739049i	$-0.264725\pi$
$390$		0			0
$391$		−	14.2647i		−	0.721397i
$392$		0			0
$393$	5.81174	+	1.04601i	0.293163	+	0.0527641i
$394$		0			0
$395$	−13.6373			−0.686166
$396$		0			0
$397$	−12.4939			−0.627051			−0.313525	−	0.949580i	$-0.601510\pi$
$397$	−12.4939			−0.627051			−0.313525	+	0.949580i	$0.601510\pi$
$398$		0			0
$399$	−5.53495	−	0.996190i	−0.277094	−	0.0498719i
$400$		0			0
$401$		−	11.1122i		−	0.554918i	−0.960737	−	0.277459i	$-0.910508\pi$
$401$		−	11.1122i		−	0.554918i	0.960737	−	0.277459i	$-0.0894923\pi$
$402$		0			0
$403$		−	5.29150i		−	0.263589i
$404$		0			0
$405$	12.3056	−	14.2503i	0.611472	−	0.708101i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	8.49390			0.419996			0.209998	−	0.977702i	$-0.432654\pi$
$409$	8.49390			0.419996			0.209998	+	0.977702i	$0.432654\pi$
$410$		0			0
$411$	−3.40932	+	18.9426i	−0.168169	+	0.934369i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	28.5294i		−	1.40045i
$416$		0			0
$417$	−3.81174	+	21.1785i	−0.186662	+	1.03711i
$418$		0			0
$419$	−37.5025			−1.83212			−0.916059	−	0.401042i	$-0.868648\pi$
$419$	−37.5025			−1.83212			−0.916059	+	0.401042i	$0.868648\pi$
$420$		0			0
$421$	−17.6235			−0.858916			−0.429458	−	0.903087i	$-0.641295\pi$
$421$	−17.6235			−0.858916			−0.429458	+	0.903087i	$0.641295\pi$
$422$		0			0
$423$	9.58612	+	3.56618i	0.466093	+	0.173393i
$424$		0			0
$425$		−	1.30432i		−	0.0632688i
$426$		0			0
$427$			1.22723i			0.0593899i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−28.1165			−1.35432			−0.677162	−	0.735834i	$-0.736791\pi$
$431$	−28.1165			−1.35432			−0.677162	+	0.735834i	$0.736791\pi$
$432$		0			0
$433$	6.37652			0.306436			0.153218	−	0.988192i	$-0.451036\pi$
$433$	6.37652			0.306436			0.153218	+	0.988192i	$0.451036\pi$
$434$		0			0
$435$	24.7072	+	4.44685i	1.18462	+	0.213210i
$436$		0			0
$437$			36.0809i			1.72598i
$438$		0			0
$439$		−	4.06427i		−	0.193977i	−0.995285	−	0.0969885i	$-0.969079\pi$
$439$		−	4.06427i		−	0.193977i	0.995285	−	0.0969885i	$-0.0309210\pi$
$440$		0			0
$441$	18.6235	+	6.92820i	0.886832	+	0.329914i
$442$		0			0
$443$	−19.6140			−0.931889			−0.465945	−	0.884814i	$-0.654285\pi$
$443$	−19.6140			−0.931889			−0.465945	+	0.884814i	$0.654285\pi$
$444$		0			0
$445$	5.74085			0.272142
$446$		0			0
$447$	−4.25126	+	23.6205i	−0.201078	+	1.11721i
$448$		0			0
$449$			1.43985i			0.0679509i	0.999423	+	0.0339755i	$0.0108168\pi$
$449$			1.43985i			0.0679509i	−0.999423	+	0.0339755i	$0.989183\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−4.56479	+	25.3625i	−0.214472	+	1.19163i
$454$		0			0
$455$	−1.28369			−0.0601805
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$	5.53495	−	9.35577i	0.258349	−	0.436690i
$460$		0			0
$461$		−	15.9484i		−	0.742792i	−0.928475	−	0.371396i	$-0.878879\pi$
$461$		−	15.9484i		−	0.742792i	0.928475	−	0.371396i	$-0.121121\pi$
$462$		0			0
$463$		−	13.4200i		−	0.623682i	−0.950134	−	0.311841i	$-0.899054\pi$
$463$		−	13.4200i		−	0.623682i	0.950134	−	0.311841i	$-0.100946\pi$
$464$		0			0
$465$	18.8704	+	3.39633i	0.875095	+	0.157501i
$466$		0			0
$467$	22.5816			1.04495			0.522475	−	0.852655i	$-0.325009\pi$
$467$	22.5816			1.04495			0.522475	+	0.852655i	$0.325009\pi$
$468$		0			0
$469$	−3.24695			−0.149930
$470$		0			0
$471$	−36.2188	−	6.51873i	−1.66888	−	0.300367i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			3.29912i			0.151374i
$476$		0			0
$477$	11.6235	−	31.2447i	0.532202	−	1.43060i
$478$		0			0
$479$	−34.9352			−1.59623			−0.798114	−	0.602507i	$-0.794169\pi$
$479$	−34.9352			−1.59623			−0.798114	+	0.602507i	$0.794169\pi$
$480$		0			0
$481$	−5.62348			−0.256408
$482$		0			0
$483$	−1.28369	+	7.13235i	−0.0584101	+	0.324533i
$484$		0			0
$485$			10.9767i			0.498426i
$486$		0			0
$487$		−	20.7834i		−	0.941787i	−0.882190	−	0.470894i	$-0.843932\pi$
$487$		−	20.7834i		−	0.941787i	0.882190	−	0.470894i	$-0.156068\pi$
$488$		0			0
$489$	−1.62348	+	9.02022i	−0.0734161	+	0.407908i
$490$		0			0
$491$	−7.66058			−0.345717			−0.172859	−	0.984947i	$-0.555300\pi$
$491$	−7.66058			−0.345717			−0.172859	+	0.984947i	$0.555300\pi$
$492$		0			0
$493$	14.4939			0.652772
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.09201i			0.0938397i
$498$		0			0
$499$			28.9120i			1.29428i	0.762372	+	0.647139i	$0.224035\pi$
$499$			28.9120i			1.29428i	−0.762372	+	0.647139i	$0.775965\pi$
$500$		0			0
$501$	−8.00000	−	1.43985i	−0.357414	−	0.0643280i
$502$		0			0
$503$	11.0699			0.493582			0.246791	−	0.969069i	$-0.420624\pi$
$503$	11.0699			0.493582			0.246791	+	0.969069i	$0.420624\pi$
$504$		0			0
$505$	14.4939			0.644970
$506$		0			0
$507$	1.70466	+	0.306808i	0.0757066	+	0.0136258i
$508$		0			0
$509$		−	16.7361i		−	0.741815i	−0.928670	−	0.370908i	$-0.879047\pi$
$509$		−	16.7361i		−	0.741815i	0.928670	−	0.370908i	$-0.120953\pi$
$510$		0			0
$511$		−	8.12854i		−	0.359585i
$512$		0			0
$513$	−14.0000	+	23.6643i	−0.618115	+	1.04481i
$514$		0			0
$515$	−18.7721			−0.827196
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.441759	+	2.45446i	−0.0193911	+	0.107739i
$520$		0			0
$521$		−	18.8281i		−	0.824875i	−0.910986	−	0.412438i	$-0.864677\pi$
$521$		−	18.8281i		−	0.824875i	0.910986	−	0.412438i	$-0.135323\pi$
$522$		0			0
$523$		−	13.4200i		−	0.586818i	−0.955987	−	0.293409i	$-0.905210\pi$
$523$		−	13.4200i		−	0.586818i	0.955987	−	0.293409i	$-0.0947897\pi$
$524$		0			0
$525$	−0.117377	+	0.652160i	−0.00512275	+	0.0284626i
$526$		0			0
$527$	11.0699			0.482212
$528$		0			0
$529$	23.4939			1.02147
$530$		0			0
$531$		0			0
$532$		0			0
$533$			11.1122i			0.481324i
$534$		0			0
$535$			9.81786i			0.424463i
$536$		0			0
$537$	36.3056	+	6.53436i	1.56670	+	0.281978i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9.62348			−0.413746			−0.206873	−	0.978368i	$-0.566329\pi$
$541$	−9.62348			−0.413746			−0.206873	+	0.978368i	$0.566329\pi$
$542$		0			0
$543$	2.12563	+	0.382574i	0.0912194	+	0.0164178i
$544$		0			0
$545$			20.1324i			0.862379i
$546$		0			0
$547$			24.2341i			1.03617i	0.855328	+	0.518087i	$0.173356\pi$
$547$			24.2341i			1.03617i	−0.855328	+	0.518087i	$0.826644\pi$
$548$		0			0
$549$	5.62348	+	2.09201i	0.240004	+	0.0892850i
$550$		0			0
$551$	−36.6606			−1.56179
$552$		0			0
$553$	4.00000			0.170097
$554$		0			0
$555$	3.60941	−	20.0543i	0.153211	−	0.851258i
$556$		0			0
$557$		−	10.4601i		−	0.443207i	−0.975137	−	0.221604i	$-0.928871\pi$
$557$		−	10.4601i		−	0.443207i	0.975137	−	0.221604i	$-0.0711291\pi$
$558$		0			0
$559$			11.1966i			0.473567i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.5491			−1.07677			−0.538384	−	0.842700i	$-0.680965\pi$
$563$	−25.5491			−1.07677			−0.538384	+	0.842700i	$0.680965\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−3.60941	+	4.17979i	−0.151581	+	0.175535i
$568$		0			0
$569$		−	20.1324i		−	0.843996i	−0.906597	−	0.421998i	$-0.861329\pi$
$569$		−	20.1324i		−	0.843996i	0.906597	−	0.421998i	$-0.138671\pi$
$570$		0			0
$571$		−	39.7260i		−	1.66248i	−0.555912	−	0.831241i	$-0.687631\pi$
$571$		−	39.7260i		−	1.66248i	0.555912	−	0.831241i	$-0.312369\pi$
$572$		0			0
$573$	16.0000	+	2.87971i	0.668410	+	0.120302i
$574$		0			0
$575$	4.25126			0.177290
$576$		0			0
$577$	−37.2470			−1.55061			−0.775305	−	0.631586i	$-0.782404\pi$
$577$	−37.2470			−1.55061			−0.775305	+	0.631586i	$0.782404\pi$
$578$		0			0
$579$	−36.2188	−	6.51873i	−1.50520	−	0.270909i
$580$		0			0
$581$			8.36806i			0.347166i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−2.18826	+	5.88220i	−0.0904735	+	0.243199i
$586$		0			0
$587$	−28.9584			−1.19524			−0.597621	−	0.801778i	$-0.703887\pi$
$587$	−28.9584			−1.19524			−0.597621	+	0.801778i	$0.703887\pi$
$588$		0			0
$589$	−28.0000			−1.15372
$590$		0			0
$591$	3.20923	−	17.8309i	0.132010	−	0.733464i
$592$		0			0
$593$		−	23.6643i		−	0.971777i	−0.874021	−	0.485889i	$-0.838496\pi$
$593$		−	23.6643i		−	0.971777i	0.874021	−	0.485889i	$-0.161504\pi$
$594$		0			0
$595$		−	2.68551i		−	0.110095i
$596$		0			0
$597$	−1.62348	+	9.02022i	−0.0664444	+	0.369173i
$598$		0			0
$599$	−27.2746			−1.11441			−0.557204	−	0.830375i	$-0.688126\pi$
$599$	−27.2746			−1.11441			−0.557204	+	0.830375i	$0.688126\pi$
$600$		0			0
$601$	−26.3765			−1.07592			−0.537960	−	0.842970i	$-0.680805\pi$
$601$	−26.3765			−1.07592			−0.537960	+	0.842970i	$0.680805\pi$
$602$		0			0
$603$	−5.53495	+	14.8783i	−0.225401	+	0.605892i
$604$		0			0
$605$			23.0122i			0.935577i
$606$		0			0
$607$			35.8133i			1.45362i	0.686840	+	0.726808i	$0.258997\pi$
$607$			35.8133i			1.45362i	−0.686840	+	0.726808i	$0.741003\pi$
$608$		0			0
$609$	−7.24695	−	1.30432i	−0.293661	−	0.0528537i
$610$		0			0
$611$	−3.40932			−0.137926
$612$		0			0
$613$	24.4939			0.989299			0.494650	−	0.869092i	$-0.335296\pi$
$613$	24.4939			0.989299			0.494650	+	0.869092i	$0.335296\pi$
$614$		0			0
$615$	−39.6282	−	7.13235i	−1.59796	−	0.287604i
$616$		0			0
$617$		−	43.0091i		−	1.73148i	−0.500494	−	0.865740i	$-0.666848\pi$
$617$		−	43.0091i		−	1.73148i	0.500494	−	0.865740i	$-0.333152\pi$
$618$		0			0
$619$			36.2754i			1.45803i	0.684498	+	0.729015i	$0.260022\pi$
$619$			36.2754i			1.45803i	−0.684498	+	0.729015i	$0.739978\pi$
$620$		0			0
$621$	30.4939	+	18.0404i	1.22368	+	0.723938i
$622$		0			0
$623$	−1.68387			−0.0674628
$624$		0			0
$625$	−21.4939			−0.859756
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	11.7644i		−	0.469077i
$630$		0			0
$631$			43.4077i			1.72803i	0.503463	+	0.864017i	$0.332059\pi$
$631$			43.4077i			1.72803i	−0.503463	+	0.864017i	$0.667941\pi$
$632$		0			0
$633$	1.31784	−	7.32205i	0.0523793	−	0.291025i
$634$		0			0
$635$	−43.4792			−1.72542
$636$		0			0
$637$	−6.62348			−0.262432
$638$		0			0
$639$	9.58612	+	3.56618i	0.379221	+	0.141076i
$640$		0			0
$641$		−	5.48835i		−	0.216777i	−0.994109	−	0.108388i	$-0.965431\pi$
$641$		−	5.48835i		−	0.216777i	0.994109	−	0.108388i	$-0.0345690\pi$
$642$		0			0
$643$		−	37.0405i		−	1.46074i	−0.683054	−	0.730368i	$-0.739349\pi$
$643$		−	37.0405i		−	1.46074i	0.683054	−	0.730368i	$-0.260651\pi$
$644$		0			0
$645$	−39.9291	−	7.18652i	−1.57221	−	0.282969i
$646$		0			0
$647$	23.0233			0.905140			0.452570	−	0.891729i	$-0.350507\pi$
$647$	23.0233			0.905140			0.452570	+	0.891729i	$0.350507\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−5.53495	−	0.996190i	−0.216932	−	0.0390438i
$652$		0			0
$653$		−	29.1527i		−	1.14083i	−0.821356	−	0.570416i	$-0.806782\pi$
$653$		−	29.1527i		−	1.14083i	0.821356	−	0.570416i	$-0.193218\pi$
$654$		0			0
$655$		−	7.13235i		−	0.278684i
$656$		0			0
$657$	−37.2470	−	13.8564i	−1.45314	−	0.540590i
$658$		0			0
$659$	31.9676			1.24528			0.622640	−	0.782508i	$-0.286060\pi$
$659$	31.9676			1.24528			0.622640	+	0.782508i	$0.286060\pi$
$660$		0			0
$661$	−4.49390			−0.174793			−0.0873963	−	0.996174i	$-0.527855\pi$
$661$	−4.49390			−0.174793			−0.0873963	+	0.996174i	$0.527855\pi$
$662$		0			0
$663$	−0.641847	+	3.56618i	−0.0249273	+	0.138499i
$664$		0			0
$665$			6.79267i			0.263408i
$666$		0			0
$667$			47.2409i			1.82918i
$668$		0			0
$669$	−5.31784	+	29.5465i	−0.205599	+	1.14233i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−31.3643			−1.20901			−0.604503	−	0.796603i	$-0.706628\pi$
$673$	−31.3643			−1.20901			−0.604503	+	0.796603i	$0.706628\pi$
$674$		0			0
$675$	2.78827	+	1.64956i	0.107320	+	0.0634916i
$676$		0			0
$677$		−	15.2963i		−	0.587883i	−0.955823	−	0.293942i	$-0.905033\pi$
$677$		−	15.2963i		−	0.587883i	0.955823	−	0.293942i	$-0.0949671\pi$
$678$		0			0
$679$		−	3.21961i		−	0.123557i
$680$		0			0
$681$	4.37652	+	0.787695i	0.167709	+	0.0301845i
$682$		0			0
$683$	22.1398			0.847156			0.423578	−	0.905860i	$-0.360774\pi$
$683$	22.1398			0.847156			0.423578	+	0.905860i	$0.360774\pi$
$684$		0			0
$685$	23.2470			0.888220
$686$		0			0
$687$	−2.76748	−	0.498095i	−0.105586	−	0.0190035i
$688$		0			0
$689$			11.1122i			0.423342i
$690$		0			0
$691$		−	27.6847i		−	1.05318i	−0.850120	−	0.526589i	$-0.823471\pi$
$691$		−	27.6847i		−	1.05318i	0.850120	−	0.526589i	$-0.176529\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	25.9909			0.985890
$696$		0			0
$697$	−23.2470			−0.880541
$698$		0			0
$699$	−2.32572	+	12.9219i	−0.0879667	+	0.488753i
$700$		0			0
$701$		−	19.4803i		−	0.735760i	−0.929873	−	0.367880i	$-0.880084\pi$
$701$		−	19.4803i		−	0.735760i	0.929873	−	0.367880i	$-0.119916\pi$
$702$		0			0
$703$			29.7566i			1.12229i
$704$		0			0
$705$	2.18826	−	12.1582i	0.0824147	−	0.457906i
$706$		0			0
$707$	−4.25126			−0.159885
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$	6.81864	−	18.3290i	0.255719	−	0.687390i
$712$		0			0
$713$			36.0809i			1.35124i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	21.8117	+	3.92572i	0.814574	+	0.146609i
$718$		0			0
$719$	−11.0699			−0.412838			−0.206419	−	0.978464i	$-0.566181\pi$
$719$	−11.0699			−0.412838			−0.206419	+	0.978464i	$0.566181\pi$
$720$		0			0
$721$	5.50610			0.205058
$722$		0			0
$723$	−23.8653	−	4.29531i	−0.887558	−	0.159744i
$724$		0			0
$725$			4.31956i			0.160425i
$726$		0			0
$727$			14.6473i			0.543237i	0.962405	+	0.271619i	$0.0875590\pi$
$727$			14.6473i			0.543237i	−0.962405	+	0.271619i	$0.912441\pi$
$728$		0			0
$729$	13.0000	+	23.6643i	0.481481	+	0.876456i
$730$		0			0
$731$	−23.4235			−0.866349
$732$		0			0
$733$	−32.8704			−1.21410			−0.607048	−	0.794665i	$-0.707647\pi$
$733$	−32.8704			−1.21410			−0.607048	+	0.794665i	$0.707647\pi$
$734$		0			0
$735$	4.25126	−	23.6205i	0.156810	−	0.871254i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			10.2004i			0.375229i	0.982243	+	0.187614i	$0.0600756\pi$
$739$			10.2004i			0.375229i	−0.982243	+	0.187614i	$0.939924\pi$
$740$		0			0
$741$	1.62348	−	9.02022i	0.0596399	−	0.331366i
$742$		0			0
$743$	−23.8653			−0.875531			−0.437766	−	0.899089i	$-0.644230\pi$
$743$	−23.8653			−0.875531			−0.437766	+	0.899089i	$0.644230\pi$
$744$		0			0
$745$	28.9878			1.06203
$746$		0			0
$747$	38.3445	+	14.2647i	1.40295	+	0.521918i
$748$		0			0
$749$		−	2.87971i		−	0.105222i
$750$		0			0
$751$			21.5486i			0.786319i	0.919470	+	0.393160i	$0.128618\pi$
$751$			21.5486i			0.786319i	−0.919470	+	0.393160i	$0.871382\pi$
$752$		0			0
$753$	−15.2470	−	2.74417i	−0.555630	−	0.100003i
$754$		0			0
$755$	31.1257			1.13278
$756$		0			0
$757$	−4.49390			−0.163334			−0.0816668	−	0.996660i	$-0.526024\pi$
$757$	−4.49390			−0.163334			−0.0816668	+	0.996660i	$0.526024\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			43.0091i			1.55908i	0.626354	+	0.779539i	$0.284547\pi$
$761$			43.0091i			1.55908i	−0.626354	+	0.779539i	$0.715453\pi$
$762$		0			0
$763$		−	5.90512i		−	0.213780i
$764$		0			0
$765$	−12.3056	−	4.57788i	−0.444911	−	0.165513i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−21.2470			−0.766185			−0.383092	−	0.923710i	$-0.625141\pi$
$769$	−21.2470			−0.766185			−0.383092	+	0.923710i	$0.625141\pi$
$770$		0			0
$771$	−3.20923	+	17.8309i	−0.115578	+	0.642163i
$772$		0			0
$773$		−	23.0122i		−	0.827690i	−0.910347	−	0.413845i	$-0.864186\pi$
$773$		−	23.0122i		−	0.827690i	0.910347	−	0.413845i	$-0.135814\pi$
$774$		0			0
$775$			3.29912i			0.118508i
$776$		0			0
$777$	−1.05869	+	5.88220i	−0.0379802	+	0.211023i
$778$		0			0
$779$	58.8004			2.10674
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−18.3303	+	30.9839i	−0.655072	+	1.10727i
$784$		0			0
$785$			44.4489i			1.58645i
$786$		0			0
$787$			0.382574i			0.0136373i	0.999977	+	0.00681865i	$0.00217046\pi$
$787$			0.382574i			0.0136373i	−0.999977	+	0.00681865i	$0.997830\pi$
$788$		0			0
$789$	−27.6235	−	4.97172i	−0.983422	−	0.176998i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.00000			−0.0710221
$794$		0			0
$795$	−39.6282	−	7.13235i	−1.40547	−	0.252958i
$796$		0			0
$797$			33.3367i			1.18085i	0.807094	+	0.590423i	$0.201039\pi$
$797$			33.3367i			1.18085i	−0.807094	+	0.590423i	$0.798961\pi$
$798$		0			0
$799$		−	7.13235i		−	0.252324i
$800$		0			0
$801$	−2.87043	+	7.71590i	−0.101422	+	0.272628i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	8.75305			0.308504
$806$		0			0
$807$	−3.40932	+	18.9426i	−0.120014	+	0.666811i
$808$		0			0
$809$			13.0687i			0.459471i	0.973253	+	0.229736i	$0.0737862\pi$
$809$			13.0687i			0.459471i	−0.973253	+	0.229736i	$0.926214\pi$
$810$		0			0
$811$		−	35.0481i		−	1.23071i	−0.788251	−	0.615353i	$-0.789013\pi$
$811$		−	35.0481i		−	1.23071i	0.788251	−	0.615353i	$-0.210987\pi$
$812$		0			0
$813$	−3.81174	+	21.1785i	−0.133683	+	0.742761i
$814$		0			0
$815$	11.0699			0.387762
$816$		0			0
$817$	59.2470			2.07279
$818$		0			0
$819$	0.641847	−	1.72533i	0.0224279	−	0.0602878i
$820$		0			0
$821$			6.27604i			0.219035i	0.993985	+	0.109518i	$0.0349306\pi$
$821$			6.27604i			0.219035i	−0.993985	+	0.109518i	$0.965069\pi$
$822$		0			0
$823$		−	4.06427i		−	0.141672i	−0.997488	−	0.0708358i	$-0.977433\pi$
$823$		−	4.06427i		−	0.141672i	0.997488	−	0.0708358i	$-0.0225666\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.7072			−0.859153			−0.429577	−	0.903030i	$-0.641337\pi$
$827$	−24.7072			−0.859153			−0.429577	+	0.903030i	$0.641337\pi$
$828$		0			0
$829$	30.9878			1.07625			0.538125	−	0.842865i	$-0.319133\pi$
$829$	30.9878			1.07625			0.538125	+	0.842865i	$0.319133\pi$
$830$		0			0
$831$	41.7538	+	7.51493i	1.44842	+	0.260690i
$832$		0			0
$833$		−	13.8564i		−	0.480096i
$834$		0			0
$835$			9.81786i			0.339761i
$836$		0			0
$837$	−14.0000	+	23.6643i	−0.483911	+	0.817959i
$838$		0			0
$839$	−13.1955			−0.455560			−0.227780	−	0.973713i	$-0.573147\pi$
$839$	−13.1955			−0.455560			−0.227780	+	0.973713i	$0.573147\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$		0			0
$843$	−3.40932	+	18.9426i	−0.117423	+	0.652417i
$844$		0			0
$845$		−	2.09201i		−	0.0719675i
$846$		0			0
$847$		−	6.74978i		−	0.231925i
$848$		0			0
$849$	−3.12957	+	17.3883i	−0.107407	+	0.596764i
$850$		0			0
$851$	38.3445			1.31443
$852$		0			0
$853$	−32.1174			−1.09968			−0.549839	−	0.835271i	$-0.685311\pi$
$853$	−32.1174			−1.09968			−0.549839	+	0.835271i	$0.685311\pi$
$854$		0			0
$855$	31.1257	+	11.5792i	1.06447	+	0.396000i
$856$		0			0
$857$		−	36.0809i		−	1.23250i	−0.787551	−	0.616250i	$-0.788651\pi$
$857$		−	36.0809i		−	1.23250i	0.787551	−	0.616250i	$-0.211349\pi$
$858$		0			0
$859$		−	41.9494i		−	1.43130i	−0.698461	−	0.715648i	$-0.746131\pi$
$859$		−	41.9494i		−	1.43130i	0.698461	−	0.715648i	$-0.253869\pi$
$860$		0			0
$861$	11.6235	+	2.09201i	0.396127	+	0.0712957i
$862$		0			0
$863$	−25.5491			−0.869702			−0.434851	−	0.900502i	$-0.643199\pi$
$863$	−25.5491			−0.869702			−0.434851	+	0.900502i	$0.643199\pi$
$864$		0			0
$865$	3.01220			0.102418
$866$		0			0
$867$	21.5187	+	3.87298i	0.730815	+	0.131533i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	5.29150i		−	0.179296i
$872$		0			0
$873$	−14.7530	−	5.48835i	−0.499315	−	0.185752i
$874$		0			0
$875$	7.21882			0.244041
$876$		0			0
$877$	−31.3643			−1.05910			−0.529549	−	0.848279i	$-0.677639\pi$
$877$	−31.3643			−1.05910			−0.529549	+	0.848279i	$0.677639\pi$
$878$		0			0
$879$	2.32572	−	12.9219i	0.0784445	−	0.435847i
$880$		0			0
$881$			2.09201i			0.0704818i	0.999379	+	0.0352409i	$0.0112198\pi$
$881$			2.09201i			0.0704818i	−0.999379	+	0.0352409i	$0.988780\pi$
$882$		0			0
$883$			47.8546i			1.61043i	0.592980	+	0.805217i	$0.297951\pi$
$883$			47.8546i			1.61043i	−0.592980	+	0.805217i	$0.702049\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.0699			−0.371691			−0.185845	−	0.982579i	$-0.559502\pi$
$887$	−11.0699			−0.371691			−0.185845	+	0.982579i	$0.559502\pi$
$888$		0			0
$889$	12.7530			0.427724
$890$		0			0
$891$		0			0
$892$		0			0
$893$			18.0404i			0.603700i
$894$		0			0
$895$		−	44.5554i		−	1.48932i
$896$		0			0
$897$	−11.6235	−	2.09201i	−0.388097	−	0.0698503i
$898$		0			0
$899$	−36.6606			−1.22270
$900$		0			0
$901$	−23.2470			−0.774468
$902$		0			0
$903$	11.7117	+	2.10790i	0.389743	+	0.0701466i
$904$		0			0
$905$		−	2.60864i		−	0.0867141i
$906$		0			0
$907$		−	1.84085i		−	0.0611244i	−0.999533	−	0.0305622i	$-0.990270\pi$
$907$		−	1.84085i		−	0.0611244i	0.999533	−	0.0305622i	$-0.00972976\pi$
$908$		0			0
$909$	−7.24695	+	19.4803i	−0.240366	+	0.646120i
$910$		0			0
$911$	31.5258			1.04450			0.522249	−	0.852793i	$-0.325093\pi$
$911$	31.5258			1.04450			0.522249	+	0.852793i	$0.325093\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	1.28369	−	7.13235i	0.0424376	−	0.235788i
$916$		0			0
$917$			2.09201i			0.0690844i
$918$		0			0
$919$		−	56.2141i		−	1.85433i	−0.374649	−	0.927167i	$-0.622237\pi$
$919$		−	56.2141i		−	1.85433i	0.374649	−	0.927167i	$-0.377763\pi$
$920$		0			0
$921$	8.49390	−	47.1931i	0.279884	−	1.55507i
$922$		0			0
$923$	−3.40932			−0.112219
$924$		0			0
$925$	3.50610			0.115280
$926$		0			0
$927$	9.38603	−	25.2303i	0.308278	−	0.828671i
$928$		0			0
$929$		−	9.80791i		−	0.321787i	−0.986972	−	0.160894i	$-0.948562\pi$
$929$		−	9.80791i		−	0.321787i	0.986972	−	0.160894i	$-0.0514376\pi$
$930$		0			0
$931$			35.0481i			1.14866i
$932$		0			0
$933$	−11.6235	−	2.09201i	−0.380536	−	0.0684895i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−20.4939			−0.669507			−0.334753	−	0.942306i	$-0.608653\pi$
$937$	−20.4939			−0.669507			−0.334753	+	0.942306i	$0.608653\pi$
$938$		0			0
$939$	−30.0420	−	5.40702i	−0.980385	−	0.176452i
$940$		0			0
$941$		−	28.5005i		−	0.929090i	−0.885550	−	0.464545i	$-0.846218\pi$
$941$		−	28.5005i		−	0.929090i	0.885550	−	0.464545i	$-0.153782\pi$
$942$		0			0
$943$		−	75.7704i		−	2.46742i
$944$		0			0
$945$	5.74085	+	3.39633i	0.186750	+	0.110483i
$946$		0			0
$947$	45.1631			1.46760			0.733802	−	0.679363i	$-0.237744\pi$
$947$	45.1631			1.46760			0.733802	+	0.679363i	$0.237744\pi$
$948$		0			0
$949$	13.2470			0.430014
$950$		0			0
$951$	−5.13477	+	28.5294i	−0.166506	+	0.925129i
$952$		0			0
$953$		−	43.6612i		−	1.41433i	−0.707051	−	0.707163i	$-0.749975\pi$
$953$		−	43.6612i		−	1.41433i	0.707051	−	0.707163i	$-0.250025\pi$
$954$		0			0
$955$		−	19.6357i		−	0.635397i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.81864			−0.220185
$960$		0			0
$961$	3.00000			0.0967742
$962$		0			0
$963$	−13.1955	−	4.90893i	−0.425220	−	0.158188i
$964$		0			0
$965$			44.4489i			1.43086i
$966$		0			0
$967$		−	18.5600i		−	0.596850i	−0.954433	−	0.298425i	$-0.903539\pi$
$967$		−	18.5600i		−	0.596850i	0.954433	−	0.298425i	$-0.0964612\pi$
$968$		0			0
$969$	18.8704	+	3.39633i	0.606205	+	0.109106i
$970$		0			0
$971$	5.09319			0.163448			0.0817241	−	0.996655i	$-0.473957\pi$
$971$	5.09319			0.163448			0.0817241	+	0.996655i	$0.473957\pi$
$972$		0			0
$973$	−7.62348			−0.244397
$974$		0			0
$975$	−1.06281	−	0.191287i	−0.0340373	−	0.00612609i
$976$		0			0
$977$		−	50.0728i		−	1.60197i	−0.598684	−	0.800986i	$-0.704309\pi$
$977$		−	50.0728i		−	1.60197i	0.598684	−	0.800986i	$-0.295691\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−27.0587	−	10.0662i	−0.863917	−	0.321390i
$982$		0			0
$983$	−50.2563			−1.60293			−0.801464	−	0.598043i	$-0.795945\pi$
$983$	−50.2563			−1.60293			−0.801464	+	0.598043i	$0.795945\pi$
$984$		0			0
$985$	−21.8826			−0.697238
$986$		0			0
$987$	−0.641847	+	3.56618i	−0.0204302	+	0.113513i
$988$		0			0
$989$		−	76.3458i		−	2.42765i
$990$		0			0
$991$			2.83704i			0.0901215i	0.998984	+	0.0450607i	$0.0143481\pi$
$991$			2.83704i			0.0901215i	−0.998984	+	0.0450607i	$0.985652\pi$
$992$		0			0
$993$	−5.24695	+	29.1527i	−0.166507	+	0.925132i
$994$		0			0
$995$	11.0699			0.350939
$996$		0			0
$997$	31.7409			1.00524			0.502621	−	0.864507i	$-0.332369\pi$
$997$	31.7409			1.00524			0.502621	+	0.864507i	$0.332369\pi$
$998$		0			0
$999$	25.1489	+	14.8783i	0.795677	+	0.470729i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.d.m.1535.8		8
3.2	odd	2	inner	2496.2.d.m.1535.2		8
4.3	odd	2	inner	2496.2.d.m.1535.1		8
8.3	odd	2		156.2.c.c.131.4	yes	8
8.5	even	2		156.2.c.c.131.7	yes	8
12.11	even	2	inner	2496.2.d.m.1535.7		8
24.5	odd	2		156.2.c.c.131.2	✓	8
24.11	even	2		156.2.c.c.131.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.c.c.131.2	✓	8	24.5	odd	2
156.2.c.c.131.4	yes	8	8.3	odd	2
156.2.c.c.131.5	yes	8	24.11	even	2
156.2.c.c.131.7	yes	8	8.5	even	2
2496.2.d.m.1535.1		8	4.3	odd	2	inner
2496.2.d.m.1535.2		8	3.2	odd	2	inner
2496.2.d.m.1535.7		8	12.11	even	2	inner
2496.2.d.m.1535.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.70466 + 0.306808i) q^{3} -2.09201i q^{5} +0.613616i q^{7} +(2.81174 + 1.04601i) q^{9} +O(q^{10})\)	\(q+(1.70466 + 0.306808i) q^{3} -2.09201i q^{5} +0.613616i q^{7} +(2.81174 + 1.04601i) q^{9} -1.00000 q^{13} +(0.641847 - 3.56618i) q^{15} -2.09201i q^{17} +5.29150i q^{19} +(-0.188262 + 1.04601i) q^{21} +6.81864 q^{23} +0.623475 q^{25} +(4.47214 + 2.64575i) q^{27} +6.92820i q^{29} +5.29150i q^{31} +1.28369 q^{35} +5.62348 q^{37} +(-1.70466 - 0.306808i) q^{39} -11.1122i q^{41} -11.1966i q^{43} +(2.18826 - 5.88220i) q^{45} +3.40932 q^{47} +6.62348 q^{49} +(0.641847 - 3.56618i) q^{51} -11.1122i q^{53} +(-1.62348 + 9.02022i) q^{57} +2.00000 q^{61} +(-0.641847 + 1.72533i) q^{63} +2.09201i q^{65} +5.29150i q^{67} +(11.6235 + 2.09201i) q^{69} +3.40932 q^{71} -13.2470 q^{73} +(1.06281 + 0.191287i) q^{75} -6.51873i q^{79} +(6.81174 + 5.88220i) q^{81} +13.6373 q^{83} -4.37652 q^{85} +(-2.12563 + 11.8102i) q^{87} +2.74417i q^{89} -0.613616i q^{91} +(-1.62348 + 9.02022i) q^{93} +11.0699 q^{95} -5.24695 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{9}+O(q^{10}) \) 8 * q + 2 * q^9	\( 8 q + 2 q^{9} - 8 q^{13} - 22 q^{21} - 36 q^{25} + 4 q^{37} + 38 q^{45} + 12 q^{49} + 28 q^{57} + 16 q^{61} + 52 q^{69} - 24 q^{73} + 34 q^{81} - 76 q^{85} + 28 q^{93} + 40 q^{97}+O(q^{100}) \) 8 * q + 2 * q^9 - 8 * q^13 - 22 * q^21 - 36 * q^25 + 4 * q^37 + 38 * q^45 + 12 * q^49 + 28 * q^57 + 16 * q^61 + 52 * q^69 - 24 * q^73 + 34 * q^81 - 76 * q^85 + 28 * q^93 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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