LMFDB - Embedded newform 3528.2.bl.b.1097.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(521,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3528.521");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.bl (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1097.8
Root			$0.130526 - 0.991445i$ of defining polynomial
Character	$\chi$	$=$	3528.1097
Dual form			3528.2.bl.b.521.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.84776 - 3.20041i) q^{5} +O(q^{10})$	$q+(1.84776 - 3.20041i) q^{5} +(2.95680 - 1.70711i) q^{11} +5.22625i q^{13} +(-3.37849 - 5.85172i) q^{17} +(1.87476 + 1.08239i) q^{19} +(-5.40629 - 3.12132i) q^{23} +(-4.32843 - 7.49706i) q^{25} -2.58579i q^{29} +(9.05213 - 5.22625i) q^{31} +(-2.00000 + 3.46410i) q^{37} -0.634051 q^{41} -6.48528 q^{43} +(1.53073 - 2.65131i) q^{47} +(2.23936 - 1.29289i) q^{53} -12.6173i q^{55} +(16.7262 + 9.65685i) q^{65} +(-0.828427 - 1.43488i) q^{67} -7.41421i q^{71} +(-0.776550 + 0.448342i) q^{73} +(6.82843 - 11.8272i) q^{79} +11.7206 q^{83} -24.9706 q^{85} +(-3.37849 + 5.85172i) q^{89} +(6.92820 - 4.00000i) q^{95} +9.55582i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{67} + 64 q^{79} - 128 q^{85}+O(q^{100}) $ 16 * q - 24 * q^25 - 32 * q^37 + 32 * q^43 + 32 * q^67 + 64 * q^79 - 128 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	1.84776	−	3.20041i	0.826343	−	1.43127i	−0.0745456	−	0.997218i	$-0.523751\pi$
$5$	1.84776	−	3.20041i	0.826343	−	1.43127i	0.900889	−	0.434050i	$-0.142916\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.95680	−	1.70711i	0.891507	−	0.514712i	0.0170722	−	0.999854i	$-0.494565\pi$
$11$	2.95680	−	1.70711i	0.891507	−	0.514712i	0.874435	+	0.485142i	$0.161232\pi$
$12$		0			0
$13$			5.22625i			1.44950i	0.689011	+	0.724751i	$0.258045\pi$
$13$			5.22625i			1.44950i	−0.689011	+	0.724751i	$0.741955\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.37849	−	5.85172i	−0.819405	−	1.41925i	−0.906121	−	0.423018i	$-0.860971\pi$
$17$	−3.37849	−	5.85172i	−0.819405	−	1.41925i	0.0867165	−	0.996233i	$-0.472363\pi$
$18$		0			0
$19$	1.87476	+	1.08239i	0.430099	+	0.248318i	0.699389	−	0.714741i	$-0.253456\pi$
$19$	1.87476	+	1.08239i	0.430099	+	0.248318i	−0.269290	+	0.963059i	$0.586789\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.40629	−	3.12132i	−1.12729	−	0.650840i	−0.184037	−	0.982919i	$-0.558917\pi$
$23$	−5.40629	−	3.12132i	−1.12729	−	0.650840i	−0.943252	+	0.332079i	$0.892250\pi$
$24$		0			0
$25$	−4.32843	−	7.49706i	−0.865685	−	1.49941i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.58579i		−	0.480168i	−0.970752	−	0.240084i	$-0.922825\pi$
$29$		−	2.58579i		−	0.480168i	0.970752	−	0.240084i	$-0.0771751\pi$
$30$		0			0
$31$	9.05213	−	5.22625i	1.62581	−	0.938663i	0.640487	−	0.767969i	$-0.278733\pi$
$31$	9.05213	−	5.22625i	1.62581	−	0.938663i	0.985324	−	0.170694i	$-0.0546008\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.00000	+	3.46410i	−0.328798	+	0.569495i	−0.982274	−	0.187453i	$-0.939977\pi$
$37$	−2.00000	+	3.46410i	−0.328798	+	0.569495i	0.653476	+	0.756948i	$0.273310\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.634051			−0.0990221			−0.0495110	−	0.998774i	$-0.515766\pi$
$41$	−0.634051			−0.0990221			−0.0495110	+	0.998774i	$0.515766\pi$
$42$		0			0
$43$	−6.48528			−0.988996			−0.494498	−	0.869179i	$-0.664648\pi$
$43$	−6.48528			−0.988996			−0.494498	+	0.869179i	$0.664648\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.53073	−	2.65131i	0.223280	−	0.386733i	−0.732522	−	0.680744i	$-0.761657\pi$
$47$	1.53073	−	2.65131i	0.223280	−	0.386733i	0.955802	+	0.294011i	$0.0949901\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.23936	−	1.29289i	0.307599	−	0.177593i	−0.338252	−	0.941055i	$-0.609836\pi$
$53$	2.23936	−	1.29289i	0.307599	−	0.177593i	0.645852	+	0.763463i	$0.276502\pi$
$54$		0			0
$55$		−	12.6173i		−	1.70131i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	16.7262	+	9.65685i	2.07463	+	1.19779i
$66$		0			0
$67$	−0.828427	−	1.43488i	−0.101208	−	0.175298i	0.810974	−	0.585082i	$-0.198938\pi$
$67$	−0.828427	−	1.43488i	−0.101208	−	0.175298i	−0.912183	+	0.409784i	$0.865604\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.41421i		−	0.879905i	−0.898021	−	0.439953i	$-0.854995\pi$
$71$		−	7.41421i		−	0.879905i	0.898021	−	0.439953i	$-0.145005\pi$
$72$		0			0
$73$	−0.776550	+	0.448342i	−0.0908883	+	0.0524744i	−0.544755	−	0.838595i	$-0.683377\pi$
$73$	−0.776550	+	0.448342i	−0.0908883	+	0.0524744i	0.453867	+	0.891069i	$0.350044\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.82843	−	11.8272i	0.768258	−	1.33066i	−0.170249	−	0.985401i	$-0.554457\pi$
$79$	6.82843	−	11.8272i	0.768258	−	1.33066i	0.938507	−	0.345261i	$-0.112210\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.7206			1.28650			0.643252	−	0.765655i	$-0.277585\pi$
$83$	11.7206			1.28650			0.643252	+	0.765655i	$0.277585\pi$
$84$		0			0
$85$	−24.9706			−2.70844
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.37849	+	5.85172i	−0.358120	+	0.620281i	−0.987647	−	0.156697i	$-0.949915\pi$
$89$	−3.37849	+	5.85172i	−0.358120	+	0.620281i	0.629527	+	0.776978i	$0.283249\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.92820	−	4.00000i	0.710819	−	0.410391i
$96$		0			0
$97$			9.55582i			0.970247i	0.874446	+	0.485123i	$0.161225\pi$
$97$			9.55582i			0.970247i	−0.874446	+	0.485123i	$0.838775\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.21371	+	2.10220i	0.120768	+	0.209177i	0.920071	−	0.391752i	$-0.128131\pi$
$101$	1.21371	+	2.10220i	0.120768	+	0.209177i	−0.799302	+	0.600929i	$0.794797\pi$
$102$		0			0
$103$	5.30262	+	3.06147i	0.522482	+	0.301655i	0.737950	−	0.674856i	$-0.235794\pi$
$103$	5.30262	+	3.06147i	0.522482	+	0.301655i	−0.215467	+	0.976511i	$0.569127\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.927572	−	0.535534i	−0.0896718	−	0.0517720i	0.454494	−	0.890750i	$-0.349820\pi$
$107$	−0.927572	−	0.535534i	−0.0896718	−	0.0517720i	−0.544165	+	0.838978i	$0.683154\pi$
$108$		0			0
$109$	−0.828427	−	1.43488i	−0.0793489	−	0.137436i	0.823620	−	0.567142i	$-0.191951\pi$
$109$	−0.828427	−	1.43488i	−0.0793489	−	0.137436i	−0.902969	+	0.429705i	$0.858617\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.7574i		−	1.10604i	−0.833168	−	0.553020i	$-0.813475\pi$
$113$		−	11.7574i		−	1.10604i	0.833168	−	0.553020i	$-0.186525\pi$
$114$		0			0
$115$	−19.9790	+	11.5349i	−1.86305	+	1.07563i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.328427	−	0.568852i	0.0298570	−	0.0517139i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−13.5140			−1.20873
$126$		0			0
$127$	−16.9706			−1.50589			−0.752947	−	0.658081i	$-0.771368\pi$
$127$	−16.9706			−1.50589			−0.752947	+	0.658081i	$0.771368\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.53073	+	2.65131i	−0.133741	+	0.231646i	−0.925116	−	0.379685i	$-0.876032\pi$
$131$	−1.53073	+	2.65131i	−0.133741	+	0.231646i	0.791375	+	0.611331i	$0.209366\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.57321	+	4.94975i	−0.732459	+	0.422885i	−0.819321	−	0.573335i	$-0.805649\pi$
$137$	−8.57321	+	4.94975i	−0.732459	+	0.422885i	0.0868620	+	0.996220i	$0.472316\pi$
$138$		0			0
$139$			4.32957i			0.367229i	0.982998	+	0.183615i	$0.0587798\pi$
$139$			4.32957i			0.367229i	−0.982998	+	0.183615i	$0.941220\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.92177	+	15.4530i	0.746076	+	1.29224i
$144$		0			0
$145$	−8.27558	−	4.77791i	−0.687250	−	0.396784i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.71807	−	3.87868i	−0.550366	−	0.317754i	0.198904	−	0.980019i	$-0.436262\pi$
$149$	−6.71807	−	3.87868i	−0.550366	−	0.317754i	−0.749270	+	0.662265i	$0.769595\pi$
$150$		0			0
$151$	−8.07107	−	13.9795i	−0.656814	−	1.13764i	−0.981436	−	0.191792i	$-0.938570\pi$
$151$	−8.07107	−	13.9795i	−0.656814	−	1.13764i	0.324622	−	0.945844i	$-0.394763\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	38.6274i		−	3.10263i
$156$		0			0
$157$	−16.5512	+	9.55582i	−1.32093	+	0.762638i	−0.983877	−	0.178849i	$-0.942763\pi$
$157$	−16.5512	+	9.55582i	−1.32093	+	0.762638i	−0.337050	+	0.941487i	$0.609429\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.82843	−	15.2913i	0.691496	−	1.19771i	−0.279852	−	0.960043i	$-0.590285\pi$
$163$	8.82843	−	15.2913i	0.691496	−	1.19771i	0.971348	−	0.237663i	$-0.0763813\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.7206			−0.906968			−0.453484	−	0.891265i	$-0.649819\pi$
$167$	−11.7206			−0.906968			−0.453484	+	0.891265i	$0.649819\pi$
$168$		0			0
$169$	−14.3137			−1.10105
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.17733	+	10.6994i	−0.469654	+	0.813464i	−0.999398	−	0.0346935i	$-0.988954\pi$
$173$	−6.17733	+	10.6994i	−0.469654	+	0.813464i	0.529744	+	0.848157i	$0.322288\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.7694	−	7.94975i	1.02917	−	0.594192i	0.112424	−	0.993660i	$-0.464139\pi$
$179$	13.7694	−	7.94975i	1.02917	−	0.594192i	0.916747	+	0.399468i	$0.130805\pi$
$180$		0			0
$181$		−	24.3379i		−	1.80902i	−0.426451	−	0.904511i	$-0.640236\pi$
$181$		−	24.3379i		−	1.80902i	0.426451	−	0.904511i	$-0.359764\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.39104	+	12.8017i	0.543400	+	0.941196i
$186$		0			0
$187$	−19.9790	−	11.5349i	−1.46101	−	0.843515i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.36245	−	1.36396i	−0.170941	−	0.0986928i	0.412089	−	0.911144i	$-0.364799\pi$
$191$	−2.36245	−	1.36396i	−0.170941	−	0.0986928i	−0.583029	+	0.812451i	$0.698133\pi$
$192$		0			0
$193$	−2.82843	−	4.89898i	−0.203595	−	0.352636i	0.746089	−	0.665846i	$-0.231929\pi$
$193$	−2.82843	−	4.89898i	−0.203595	−	0.352636i	−0.949684	+	0.313210i	$0.898596\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			22.3848i			1.59485i	0.603419	+	0.797425i	$0.293805\pi$
$197$			22.3848i			1.59485i	−0.603419	+	0.797425i	$0.706195\pi$
$198$		0			0
$199$	19.9790	−	11.5349i	1.41628	−	0.817687i	0.420306	−	0.907382i	$-0.361923\pi$
$199$	19.9790	−	11.5349i	1.41628	−	0.817687i	0.995969	+	0.0896952i	$0.0285893\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.17157	+	2.02922i	−0.0818262	+	0.141727i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.39104			0.511249
$210$		0			0
$211$	19.1716			1.31983			0.659913	−	0.751342i	$-0.270593\pi$
$211$	19.1716			1.31983			0.659913	+	0.751342i	$0.270593\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.9832	+	20.7556i	−0.817250	+	1.41552i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	30.5826	−	17.6569i	2.05721	−	1.18773i
$222$		0			0
$223$			23.0698i			1.54487i	0.635095	+	0.772434i	$0.280961\pi$
$223$			23.0698i			1.54487i	−0.635095	+	0.772434i	$0.719039\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.39104	−	12.8017i	−0.490560	−	0.849675i	0.509381	−	0.860541i	$-0.329875\pi$
$227$	−7.39104	−	12.8017i	−0.490560	−	0.849675i	−0.999941	+	0.0108659i	$0.996541\pi$
$228$		0			0
$229$	2.97297	+	1.71644i	0.196459	+	0.113426i	0.595003	−	0.803724i	$-0.297151\pi$
$229$	2.97297	+	1.71644i	0.196459	+	0.113426i	−0.398544	+	0.917149i	$0.630484\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	17.9509	+	10.3640i	1.17600	+	0.678966i	0.955087	−	0.296327i	$-0.0957618\pi$
$233$	17.9509	+	10.3640i	1.17600	+	0.678966i	0.220916	+	0.975293i	$0.429095\pi$
$234$		0			0
$235$	−5.65685	−	9.79796i	−0.369012	−	0.639148i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2.92893i		−	0.189457i	−0.995503	−	0.0947284i	$-0.969802\pi$
$239$		−	2.92893i		−	0.189457i	0.995503	−	0.0947284i	$-0.0301983\pi$
$240$		0			0
$241$	9.82868	−	5.67459i	0.633121	−	0.365533i	−0.148839	−	0.988861i	$-0.547554\pi$
$241$	9.82868	−	5.67459i	0.633121	−	0.365533i	0.781960	+	0.623329i	$0.214220\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5.65685	+	9.79796i	−0.359937	+	0.623429i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.59767			−0.353322			−0.176661	−	0.984272i	$-0.556530\pi$
$251$	−5.59767			−0.353322			−0.176661	+	0.984272i	$0.556530\pi$
$252$		0			0
$253$	−21.3137			−1.33998
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.4036	−	19.7516i	0.711336	−	1.23207i	−0.253020	−	0.967461i	$-0.581424\pi$
$257$	11.4036	−	19.7516i	0.711336	−	1.23207i	0.964356	−	0.264609i	$-0.0852428\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.2189	+	9.36396i	−1.00010	+	0.577407i	−0.908277	−	0.418369i	$-0.862602\pi$
$263$	−16.2189	+	9.36396i	−1.00010	+	0.577407i	−0.0918204	+	0.995776i	$0.529269\pi$
$264$		0			0
$265$		−	9.55582i		−	0.587009i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.54328	+	9.60124i	0.337980	+	0.585398i	0.984053	−	0.177878i	$-0.0569232\pi$
$269$	5.54328	+	9.60124i	0.337980	+	0.585398i	−0.646073	+	0.763276i	$0.723590\pi$
$270$		0			0
$271$	16.5512	+	9.55582i	1.00541	+	0.580475i	0.909845	−	0.414948i	$-0.136200\pi$
$271$	16.5512	+	9.55582i	1.00541	+	0.580475i	0.0955668	+	0.995423i	$0.469534\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−25.5965	−	14.7782i	−1.54353	−	0.891157i
$276$		0			0
$277$	−9.82843	−	17.0233i	−0.590533	−	1.02283i	−0.994161	−	0.107910i	$-0.965584\pi$
$277$	−9.82843	−	17.0233i	−0.590533	−	1.02283i	0.403628	−	0.914923i	$-0.367749\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.89949i			0.590554i	0.955412	+	0.295277i	$0.0954120\pi$
$281$			9.89949i			0.590554i	−0.955412	+	0.295277i	$0.904588\pi$
$282$		0			0
$283$	−12.4800	+	7.20533i	−0.741859	+	0.428312i	−0.822745	−	0.568411i	$-0.807558\pi$
$283$	−12.4800	+	7.20533i	−0.741859	+	0.428312i	0.0808861	+	0.996723i	$0.474225\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−14.3284	+	24.8176i	−0.842849	+	1.45986i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	25.8686			1.51126			0.755631	−	0.654998i	$-0.227331\pi$
$293$	25.8686			1.51126			0.755631	+	0.654998i	$0.227331\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.3128	−	28.2546i	0.943394	−	1.63401i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	10.8239i		−	0.617754i	−0.951102	−	0.308877i	$-0.900047\pi$
$307$		−	10.8239i		−	0.617754i	0.951102	−	0.308877i	$-0.0999531\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.12293	−	10.6052i	−0.347200	−	0.601368i	0.638551	−	0.769579i	$-0.279534\pi$
$311$	−6.12293	−	10.6052i	−0.347200	−	0.601368i	−0.985751	+	0.168212i	$0.946201\pi$
$312$		0			0
$313$	18.1043	+	10.4525i	1.02331	+	0.590810i	0.915062	−	0.403313i	$-0.132141\pi$
$313$	18.1043	+	10.4525i	1.02331	+	0.590810i	0.108252	+	0.994124i	$0.465475\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.3858	+	11.1924i	1.08881	+	0.628627i	0.933260	−	0.359200i	$-0.116951\pi$
$317$	19.3858	+	11.1924i	1.08881	+	0.628627i	0.155554	+	0.987827i	$0.450284\pi$
$318$		0			0
$319$	−4.41421	−	7.64564i	−0.247149	−	0.428074i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	14.6274i		−	0.813891i
$324$		0			0
$325$	39.1815	−	22.6215i	2.17340	−	1.25481i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.41421	+	4.18154i	−0.132697	+	0.229838i	−0.924715	−	0.380659i	$-0.875697\pi$
$331$	−2.41421	+	4.18154i	−0.132697	+	0.229838i	0.792018	+	0.610497i	$0.209030\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.12293			−0.334532
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	17.8435	−	30.9059i	0.966282	−	1.67365i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.7401	−	6.77817i	0.630244	−	0.363871i	−0.150603	−	0.988594i	$-0.548121\pi$
$347$	11.7401	−	6.77817i	0.630244	−	0.363871i	0.780847	+	0.624723i	$0.214788\pi$
$348$		0			0
$349$			19.1116i			1.02302i	0.859277	+	0.511511i	$0.170914\pi$
$349$			19.1116i			1.02302i	−0.859277	+	0.511511i	$0.829086\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.7695	−	18.6534i	−0.573204	−	0.992819i	−0.996234	−	0.0867031i	$-0.972367\pi$
$353$	−10.7695	−	18.6534i	−0.573204	−	0.992819i	0.423030	−	0.906116i	$-0.360966\pi$
$354$		0			0
$355$	−23.7285	−	13.6997i	−1.25938	−	0.727104i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.26143	−	4.19239i	−0.383244	−	0.221266i	0.295985	−	0.955193i	$-0.404352\pi$
$359$	−7.26143	−	4.19239i	−0.383244	−	0.221266i	−0.679229	+	0.733927i	$0.737685\pi$
$360$		0			0
$361$	−7.15685	−	12.3960i	−0.376677	−	0.652423i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3.31371i			0.173447i
$366$		0			0
$367$	−5.62427	+	3.24718i	−0.293585	+	0.169501i	−0.639557	−	0.768743i	$-0.720882\pi$
$367$	−5.62427	+	3.24718i	−0.293585	+	0.169501i	0.345973	+	0.938245i	$0.387549\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	8.65685	−	14.9941i	0.448235	−	0.776366i	−0.550036	−	0.835141i	$-0.685386\pi$
$373$	8.65685	−	14.9941i	0.448235	−	0.776366i	0.998271	+	0.0587751i	$0.0187195\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13.5140			0.696005
$378$		0			0
$379$	11.1716			0.573845			0.286923	−	0.957954i	$-0.407368\pi$
$379$	11.1716			0.573845			0.286923	+	0.957954i	$0.407368\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.79884	+	4.84772i	−0.143014	+	0.247707i	−0.928630	−	0.371007i	$-0.879013\pi$
$383$	−2.79884	+	4.84772i	−0.143014	+	0.247707i	0.785616	+	0.618714i	$0.212346\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−21.4150	+	12.3640i	−1.08578	+	0.626878i	−0.932451	−	0.361297i	$-0.882334\pi$
$389$	−21.4150	+	12.3640i	−1.08578	+	0.626878i	−0.153333	+	0.988175i	$0.549001\pi$
$390$		0			0
$391$			42.1814i			2.13321i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−25.2346	−	43.7076i	−1.26969	−	2.19917i
$396$		0			0
$397$	18.1043	+	10.4525i	0.908627	+	0.524596i	0.879989	−	0.474994i	$-0.157550\pi$
$397$	18.1043	+	10.4525i	0.908627	+	0.524596i	0.0286380	+	0.999590i	$0.490883\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.384213	−	0.221825i	−0.0191867	−	0.0110774i	0.490376	−	0.871511i	$-0.336859\pi$
$401$	−0.384213	−	0.221825i	−0.0191867	−	0.0110774i	−0.509563	+	0.860434i	$0.670193\pi$
$402$		0			0
$403$	27.3137	+	47.3087i	1.36059	+	2.35662i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			13.6569i			0.676945i
$408$		0			0
$409$	−0.776550	+	0.448342i	−0.0383979	+	0.0221691i	−0.519076	−	0.854728i	$-0.673724\pi$
$409$	−0.776550	+	0.448342i	−0.0383979	+	0.0221691i	0.480678	+	0.876897i	$0.340391\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	21.6569	−	37.5108i	1.06309	−	1.84133i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.9050			−1.02128			−0.510638	−	0.859796i	$-0.670591\pi$
$419$	−20.9050			−1.02128			−0.510638	+	0.859796i	$0.670591\pi$
$420$		0			0
$421$	18.9706			0.924569			0.462284	−	0.886732i	$-0.347030\pi$
$421$	18.9706			0.924569			0.462284	+	0.886732i	$0.347030\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−29.2471	+	50.6575i	−1.41869	+	2.45725i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.52192	−	0.878680i	0.0733082	−	0.0423245i	−0.462898	−	0.886412i	$-0.653190\pi$
$431$	1.52192	−	0.878680i	0.0733082	−	0.0423245i	0.536206	+	0.844087i	$0.319857\pi$
$432$		0			0
$433$		−	22.6984i		−	1.09081i	−0.838171	−	0.545407i	$-0.816375\pi$
$433$		−	22.6984i		−	1.09081i	0.838171	−	0.545407i	$-0.183625\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.75699	−	11.7034i	−0.323230	−	0.559852i
$438$		0			0
$439$	19.9790	+	11.5349i	0.953547	+	0.550531i	0.894181	−	0.447706i	$-0.147759\pi$
$439$	19.9790	+	11.5349i	0.953547	+	0.550531i	0.0593659	+	0.998236i	$0.481092\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.88500	+	5.70711i	0.469650	+	0.271153i	0.716093	−	0.698004i	$-0.245928\pi$
$443$	9.88500	+	5.70711i	0.469650	+	0.271153i	−0.246443	+	0.969157i	$0.579262\pi$
$444$		0			0
$445$	12.4853	+	21.6251i	0.591859	+	1.02513i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			6.58579i			0.310802i	0.987851	+	0.155401i	$0.0496670\pi$
$449$			6.58579i			0.310802i	−0.987851	+	0.155401i	$0.950333\pi$
$450$		0			0
$451$	−1.87476	+	1.08239i	−0.0882789	+	0.0509679i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.65685	+	9.79796i	−0.264616	+	0.458329i	−0.967463	−	0.253012i	$-0.918579\pi$
$457$	−5.65685	+	9.79796i	−0.264616	+	0.458329i	0.702847	+	0.711341i	$0.251912\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.8686			−1.20482			−0.602411	−	0.798186i	$-0.705793\pi$
$461$	−25.8686			−1.20482			−0.602411	+	0.798186i	$0.705793\pi$
$462$		0			0
$463$	−18.3431			−0.852478			−0.426239	−	0.904611i	$-0.640162\pi$
$463$	−18.3431			−0.852478			−0.426239	+	0.904611i	$0.640162\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.0447	−	26.0582i	0.696186	−	1.20583i	−0.273593	−	0.961845i	$-0.588212\pi$
$467$	15.0447	−	26.0582i	0.696186	−	1.20583i	0.969779	−	0.243984i	$-0.0784544\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−19.1757	+	11.0711i	−0.881697	+	0.509048i
$474$		0			0
$475$		−	18.7402i		−	0.859860i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.5140	+	23.4069i	0.617469	+	1.06949i	0.989946	+	0.141446i	$0.0451753\pi$
$479$	13.5140	+	23.4069i	0.617469	+	1.06949i	−0.372477	+	0.928041i	$0.621491\pi$
$480$		0			0
$481$	−18.1043	−	10.4525i	−0.825484	−	0.476593i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	30.5826	+	17.6569i	1.38868	+	0.801756i
$486$		0			0
$487$	5.72792	+	9.92105i	0.259557	+	0.449566i	0.966123	−	0.258081i	$-0.0830901\pi$
$487$	5.72792	+	9.92105i	0.259557	+	0.449566i	−0.706566	+	0.707647i	$0.749757\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	38.5269i		−	1.73870i	−0.494201	−	0.869348i	$-0.664539\pi$
$491$		−	38.5269i		−	1.73870i	0.494201	−	0.869348i	$-0.335461\pi$
$492$		0			0
$493$	−15.1313	+	8.73606i	−0.681480	+	0.393452i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.41421	−	11.1097i	0.287140	−	0.497340i	−0.685986	−	0.727615i	$-0.740629\pi$
$499$	6.41421	−	11.1097i	0.287140	−	0.497340i	0.973126	+	0.230274i	$0.0739623\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	26.5027			1.18170			0.590848	−	0.806783i	$-0.298793\pi$
$503$	26.5027			1.18170			0.590848	+	0.806783i	$0.298793\pi$
$504$		0			0
$505$	8.97056			0.399185
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.60474	+	14.9039i	−0.381399	+	0.660602i	−0.991262	−	0.131904i	$-0.957891\pi$
$509$	−8.60474	+	14.9039i	−0.381399	+	0.660602i	0.609864	+	0.792506i	$0.291224\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.5959	−	11.3137i	0.863499	−	0.498542i
$516$		0			0
$517$		−	10.4525i		−	0.459701i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.37849	+	5.85172i	0.148014	+	0.256369i	0.930494	−	0.366308i	$-0.119378\pi$
$521$	3.37849	+	5.85172i	0.148014	+	0.256369i	−0.782479	+	0.622677i	$0.786045\pi$
$522$		0			0
$523$	18.1043	+	10.4525i	0.791644	+	0.457056i	0.840541	−	0.541748i	$-0.182237\pi$
$523$	18.1043	+	10.4525i	0.791644	+	0.457056i	−0.0488968	+	0.998804i	$0.515571\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−61.1651	−	35.3137i	−2.66440	−	1.53829i
$528$		0			0
$529$	7.98528	+	13.8309i	0.347186	+	0.601344i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	3.31371i		−	0.143533i
$534$		0			0
$535$	−3.42786	+	1.97908i	−0.148199	+	0.0855629i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−13.8284	+	23.9515i	−0.594531	+	1.02976i	0.399082	+	0.916915i	$0.369329\pi$
$541$	−13.8284	+	23.9515i	−0.594531	+	1.02976i	−0.993613	+	0.112842i	$0.964005\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.12293			−0.262278
$546$		0			0
$547$	17.6569			0.754953			0.377476	−	0.926019i	$-0.376792\pi$
$547$	17.6569			0.754953			0.377476	+	0.926019i	$0.376792\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.79884	−	4.84772i	0.119234	−	0.206520i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−39.7501	+	22.9497i	−1.68427	+	0.972412i	−0.725499	+	0.688224i	$0.758391\pi$
$557$	−39.7501	+	22.9497i	−1.68427	+	0.972412i	−0.958768	+	0.284188i	$0.908276\pi$
$558$		0			0
$559$		−	33.8937i		−	1.43355i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.1062	+	31.3608i	0.763084	+	1.32170i	0.941254	+	0.337701i	$0.109649\pi$
$563$	18.1062	+	31.3608i	0.763084	+	1.32170i	−0.178169	+	0.984000i	$0.557017\pi$
$564$		0			0
$565$	−37.6284	−	21.7248i	−1.58304	−	0.913968i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.67423	−	2.12132i	−0.154032	−	0.0889304i	0.421003	−	0.907059i	$-0.361678\pi$
$569$	−3.67423	−	2.12132i	−0.154032	−	0.0889304i	−0.575035	+	0.818129i	$0.695012\pi$
$570$		0			0
$571$	−6.48528	−	11.2328i	−0.271401	−	0.470080i	0.697820	−	0.716273i	$-0.254153\pi$
$571$	−6.48528	−	11.2328i	−0.271401	−	0.470080i	−0.969221	+	0.246193i	$0.920820\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			54.0416i			2.25369i
$576$		0			0
$577$	−9.05213	+	5.22625i	−0.376845	+	0.217572i	−0.676445	−	0.736493i	$-0.736480\pi$
$577$	−9.05213	+	5.22625i	−0.376845	+	0.217572i	0.299600	+	0.954065i	$0.403147\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.41421	−	7.64564i	0.182818	−	0.316650i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	32.6256			1.34660			0.673302	−	0.739368i	$-0.264876\pi$
$587$	32.6256			1.34660			0.673302	+	0.739368i	$0.264876\pi$
$588$		0			0
$589$	22.6274			0.932346
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.4650	−	25.0542i	0.594008	−	1.02885i	−0.399678	−	0.916656i	$-0.630878\pi$
$593$	14.4650	−	25.0542i	0.594008	−	1.02885i	0.993686	−	0.112197i	$-0.0357887\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−27.0314	+	15.6066i	−1.10447	+	0.637668i	−0.937393	−	0.348275i	$-0.886768\pi$
$599$	−27.0314	+	15.6066i	−1.10447	+	0.637668i	−0.167082	+	0.985943i	$0.553434\pi$
$600$		0			0
$601$			22.6984i			0.925886i	0.886388	+	0.462943i	$0.153207\pi$
$601$			22.6984i			0.925886i	−0.886388	+	0.462943i	$0.846793\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.21371	−	2.10220i	−0.0493443	−	0.0854668i
$606$		0			0
$607$	1.87476	+	1.08239i	0.0760941	+	0.0439329i	0.537564	−	0.843223i	$-0.319345\pi$
$607$	1.87476	+	1.08239i	0.0760941	+	0.0439329i	−0.461470	+	0.887156i	$0.652678\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.8564	+	8.00000i	0.560570	+	0.323645i
$612$		0			0
$613$	10.6569	+	18.4582i	0.430426	+	0.745520i	0.996910	−	0.0785528i	$-0.0250299\pi$
$613$	10.6569	+	18.4582i	0.430426	+	0.745520i	−0.566484	+	0.824073i	$0.691697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	3.75736i		−	0.151266i	−0.997136	−	0.0756328i	$-0.975902\pi$
$617$		−	3.75736i		−	0.151266i	0.997136	−	0.0756328i	$-0.0240977\pi$
$618$		0			0
$619$	14.3548	−	8.28772i	0.576966	−	0.333112i	−0.182961	−	0.983120i	$-0.558568\pi$
$619$	14.3548	−	8.28772i	0.576966	−	0.333112i	0.759927	+	0.650009i	$0.225235\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.32843	+	5.76500i	−0.133137	+	0.230600i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	27.0279			1.07767
$630$		0			0
$631$	39.5980			1.57637			0.788185	−	0.615438i	$-0.211021\pi$
$631$	39.5980			1.57637			0.788185	+	0.615438i	$0.211021\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−31.3575	+	54.3128i	−1.24438	+	2.15534i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	30.1984	−	17.4350i	1.19276	−	0.688642i	0.233831	−	0.972277i	$-0.424874\pi$
$641$	30.1984	−	17.4350i	1.19276	−	0.688642i	0.958932	+	0.283635i	$0.0915404\pi$
$642$		0			0
$643$		−	18.7402i		−	0.739042i	−0.929222	−	0.369521i	$-0.879522\pi$
$643$		−	18.7402i		−	0.739042i	0.929222	−	0.369521i	$-0.120478\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	25.2346	+	43.7076i	0.992074	+	1.71832i	0.604860	+	0.796332i	$0.293229\pi$
$647$	25.2346	+	43.7076i	0.992074	+	1.71832i	0.387213	+	0.921990i	$0.373438\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.05066	+	0.606602i	0.0411157	+	0.0237382i	0.520417	−	0.853912i	$-0.325777\pi$
$653$	1.05066	+	0.606602i	0.0411157	+	0.0237382i	−0.479301	+	0.877650i	$0.659110\pi$
$654$		0			0
$655$	5.65685	+	9.79796i	0.221032	+	0.382838i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			44.5858i			1.73682i	0.495851	+	0.868408i	$0.334856\pi$
$659$			44.5858i			1.73682i	−0.495851	+	0.868408i	$0.665144\pi$
$660$		0			0
$661$	−24.0502	+	13.8854i	−0.935444	+	0.540079i	−0.888529	−	0.458820i	$-0.848272\pi$
$661$	−24.0502	+	13.8854i	−0.935444	+	0.540079i	−0.0469150	+	0.998899i	$0.514939\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.07107	+	13.9795i	−0.312513	+	0.541288i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−2.34315			−0.0903216			−0.0451608	−	0.998980i	$-0.514380\pi$
$673$	−2.34315			−0.0903216			−0.0451608	+	0.998980i	$0.514380\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.33664	+	12.7074i	−0.281970	+	0.488387i	−0.971870	−	0.235518i	$-0.924321\pi$
$677$	−7.33664	+	12.7074i	−0.281970	+	0.488387i	0.689900	+	0.723905i	$0.257655\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.02986	+	4.63604i	−0.307254	+	0.177393i	−0.645697	−	0.763594i	$-0.723433\pi$
$683$	−8.02986	+	4.63604i	−0.307254	+	0.177393i	0.338443	+	0.940987i	$0.390100\pi$
$684$		0			0
$685$			36.5838i			1.39779i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.75699	+	11.7034i	0.257421	+	0.445866i
$690$		0			0
$691$	11.2485	+	6.49435i	0.427915	+	0.247057i	0.698458	−	0.715651i	$-0.253870\pi$
$691$	11.2485	+	6.49435i	0.427915	+	0.247057i	−0.270543	+	0.962708i	$0.587203\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.8564	+	8.00000i	0.525603	+	0.303457i
$696$		0			0
$697$	2.14214	+	3.71029i	0.0811392	+	0.140537i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			43.0711i			1.62677i	0.581725	+	0.813386i	$0.302378\pi$
$701$			43.0711i			1.62677i	−0.581725	+	0.813386i	$0.697622\pi$
$702$		0			0
$703$	−7.49903	+	4.32957i	−0.282831	+	0.163293i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−16.1421	+	27.9590i	−0.606231	+	1.05002i	0.385625	+	0.922656i	$0.373986\pi$
$709$	−16.1421	+	27.9590i	−0.606231	+	1.05002i	−0.991856	+	0.127367i	$0.959348\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−65.2512			−2.44368
$714$		0			0
$715$	65.9411			2.46606
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.32957	+	7.49903i	−0.161466	+	0.279667i	−0.935395	−	0.353606i	$-0.884955\pi$
$719$	−4.32957	+	7.49903i	−0.161466	+	0.279667i	0.773929	+	0.633273i	$0.218289\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−19.3858	+	11.1924i	−0.719970	+	0.415675i
$726$		0			0
$727$		−	14.7821i		−	0.548237i	−0.961696	−	0.274118i	$-0.911614\pi$
$727$		−	14.7821i		−	0.548237i	0.961696	−	0.274118i	$-0.0883860\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.9105	+	37.9501i	0.810388	+	1.40363i
$732$		0			0
$733$	13.5782	+	7.83938i	0.501522	+	0.289554i	0.729342	−	0.684149i	$-0.239826\pi$
$733$	13.5782	+	7.83938i	0.501522	+	0.289554i	−0.227820	+	0.973703i	$0.573160\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.89898	−	2.82843i	−0.180456	−	0.104186i
$738$		0			0
$739$	8.14214	+	14.1026i	0.299513	+	0.518772i	0.976025	−	0.217660i	$-0.0698424\pi$
$739$	8.14214	+	14.1026i	0.299513	+	0.518772i	−0.676511	+	0.736432i	$0.736509\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			11.6985i			0.429176i	0.976705	+	0.214588i	$0.0688408\pi$
$743$			11.6985i			0.429176i	−0.976705	+	0.214588i	$0.931159\pi$
$744$		0			0
$745$	−24.8268	+	14.3337i	−0.909582	+	0.525147i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−15.2426	+	26.4010i	−0.556212	+	0.963387i	0.441596	+	0.897214i	$0.354412\pi$
$751$	−15.2426	+	26.4010i	−0.556212	+	0.963387i	−0.997808	+	0.0661733i	$0.978921\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−59.6536			−2.17102
$756$		0			0
$757$	51.5980			1.87536			0.937680	−	0.347499i	$-0.112969\pi$
$757$	51.5980			1.87536			0.937680	+	0.347499i	$0.112969\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.8925	−	29.2586i	0.612351	−	1.06062i	−0.378492	−	0.925605i	$-0.623557\pi$
$761$	16.8925	−	29.2586i	0.612351	−	1.06062i	0.990843	−	0.135019i	$-0.0431095\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			22.6984i			0.818524i	0.912417	+	0.409262i	$0.134214\pi$
$769$			22.6984i			0.818524i	−0.912417	+	0.409262i	$0.865786\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.21371	+	2.10220i	0.0436541	+	0.0756110i	0.887027	−	0.461718i	$-0.152767\pi$
$773$	1.21371	+	2.10220i	0.0436541	+	0.0756110i	−0.843373	+	0.537329i	$0.819433\pi$
$774$		0			0
$775$	−78.3630	−	45.2429i	−2.81488	−	1.62517i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.18869	−	0.686292i	−0.0425893	−	0.0245889i
$780$		0			0
$781$	−12.6569	−	21.9223i	−0.452898	−	0.784442i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			70.6274i			2.52080i
$786$		0			0
$787$	28.7095	−	16.5754i	1.02338	−	0.590851i	0.108301	−	0.994118i	$-0.465459\pi$
$787$	28.7095	−	16.5754i	1.02338	−	0.590851i	0.915082	+	0.403268i	$0.132126\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.3547			−0.437624			−0.218812	−	0.975767i	$-0.570218\pi$
$797$	−12.3547			−0.437624			−0.218812	+	0.975767i	$0.570218\pi$
$798$		0			0
$799$	−20.6863			−0.731828
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.53073	+	2.65131i	−0.0540184	+	0.0935626i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.12372	−	3.53553i	0.215299	−	0.124303i	−0.388473	−	0.921460i	$-0.626997\pi$
$809$	6.12372	−	3.53553i	0.215299	−	0.124303i	0.603772	+	0.797157i	$0.293664\pi$
$810$		0			0
$811$			29.5641i			1.03814i	0.854732	+	0.519069i	$0.173721\pi$
$811$			29.5641i			1.03814i	−0.854732	+	0.519069i	$0.826279\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−32.6256	−	56.5092i	−1.14283	−	1.97943i
$816$		0			0
$817$	−12.1583	−	7.01962i	−0.425366	−	0.245585i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−27.1545	−	15.6777i	−0.947699	−	0.547154i	−0.0553338	−	0.998468i	$-0.517622\pi$
$821$	−27.1545	−	15.6777i	−0.947699	−	0.547154i	−0.892365	+	0.451313i	$0.850956\pi$
$822$		0			0
$823$	−5.17157	−	8.95743i	−0.180270	−	0.312236i	0.761703	−	0.647927i	$-0.224364\pi$
$823$	−5.17157	−	8.95743i	−0.180270	−	0.312236i	−0.941972	+	0.335690i	$0.891030\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.0122i		−	0.939306i	−0.882851	−	0.469653i	$-0.844379\pi$
$827$		−	27.0122i		−	0.939306i	0.882851	−	0.469653i	$-0.155621\pi$
$828$		0			0
$829$	−22.6303	+	13.0656i	−0.785984	+	0.453788i	−0.838547	−	0.544830i	$-0.816594\pi$
$829$	−22.6303	+	13.0656i	−0.785984	+	0.453788i	0.0525628	+	0.998618i	$0.483261\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−21.6569	+	37.5108i	−0.749466	+	1.29811i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.4303			0.739855			0.369928	−	0.929061i	$-0.379382\pi$
$839$	21.4303			0.739855			0.369928	+	0.929061i	$0.379382\pi$
$840$		0			0
$841$	22.3137			0.769438
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.4483	+	45.8098i	−0.909849	+	1.57590i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	21.6251	−	12.4853i	0.741300	−	0.427990i
$852$		0			0
$853$		−	12.2459i		−	0.419291i	−0.977778	−	0.209645i	$-0.932769\pi$
$853$		−	12.2459i		−	0.419291i	0.977778	−	0.209645i	$-0.0672309\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.37849	+	5.85172i	0.115407	+	0.199891i	0.917942	−	0.396714i	$-0.129849\pi$
$857$	3.37849	+	5.85172i	0.115407	+	0.199891i	−0.802535	+	0.596605i	$0.796516\pi$
$858$		0			0
$859$	16.2295	+	9.37011i	0.553744	+	0.319704i	0.750631	−	0.660722i	$-0.229750\pi$
$859$	16.2295	+	9.37011i	0.553744	+	0.319704i	−0.196887	+	0.980426i	$0.563083\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.2773	+	11.7071i	0.690247	+	0.398515i	0.803705	−	0.595028i	$-0.202859\pi$
$863$	20.2773	+	11.7071i	0.690247	+	0.398515i	−0.113457	+	0.993543i	$0.536193\pi$
$864$		0			0
$865$	22.8284	+	39.5400i	0.776190	+	1.34440i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	46.6274i		−	1.58173i
$870$		0			0
$871$	7.49903	−	4.32957i	0.254095	−	0.146702i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	−0.723298	−	0.690536i	$-0.757375\pi$
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	0.959671	+	0.281126i	$0.0907079\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	31.4663			1.06013			0.530063	−	0.847958i	$-0.322168\pi$
$881$	31.4663			1.06013			0.530063	+	0.847958i	$0.322168\pi$
$882$		0			0
$883$	24.1421			0.812448			0.406224	−	0.913774i	$-0.366845\pi$
$883$	24.1421			0.812448			0.406224	+	0.913774i	$0.366845\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.9832	−	20.7556i	0.402358	−	0.696904i	−0.591652	−	0.806193i	$-0.701524\pi$
$887$	11.9832	−	20.7556i	0.402358	−	0.696904i	0.994010	+	0.109289i	$0.0348575\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.73951	−	3.31371i	0.192065	−	0.110889i
$894$		0			0
$895$		−	58.7569i		−	1.96403i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−13.5140	−	23.4069i	−0.450716	−	0.780663i
$900$		0			0
$901$	−15.1313	−	8.73606i	−0.504097	−	0.291040i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−77.8913	−	44.9706i	−2.58919	−	1.49487i
$906$		0			0
$907$	24.8995	+	43.1272i	0.826774	+	1.43201i	0.900556	+	0.434740i	$0.143160\pi$
$907$	24.8995	+	43.1272i	0.826774	+	1.43201i	−0.0737816	+	0.997274i	$0.523507\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			56.1838i			1.86145i	0.365718	+	0.930726i	$0.380823\pi$
$911$			56.1838i			1.86145i	−0.365718	+	0.930726i	$0.619177\pi$
$912$		0			0
$913$	34.6554	−	20.0083i	1.14693	−	0.662179i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.899495	+	1.55797i	−0.0296716	+	0.0513927i	−0.880480	−	0.474083i	$-0.842779\pi$
$919$	−0.899495	+	1.55797i	−0.0296716	+	0.0513927i	0.850808	+	0.525476i	$0.176113\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	38.7485			1.27542
$924$		0			0
$925$	34.6274			1.13854
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.9790	−	48.4611i	0.917962	−	1.58996i	0.115456	−	0.993313i	$-0.463167\pi$
$929$	27.9790	−	48.4611i	0.917962	−	1.58996i	0.802506	−	0.596644i	$-0.203499\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−73.8329	+	42.6274i	−2.41459	+	1.39407i
$936$		0			0
$937$			50.4692i			1.64876i	0.566040	+	0.824378i	$0.308475\pi$
$937$			50.4692i			1.64876i	−0.566040	+	0.824378i	$0.691525\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.17733	−	10.6994i	−0.201375	−	0.348792i	0.747597	−	0.664153i	$-0.231208\pi$
$941$	−6.17733	−	10.6994i	−0.201375	−	0.348792i	−0.948972	+	0.315361i	$0.897874\pi$
$942$		0			0
$943$	3.42786	+	1.97908i	0.111626	+	0.0644476i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23.7414	−	13.7071i	−0.771492	−	0.445421i	0.0619146	−	0.998081i	$-0.480279\pi$
$947$	−23.7414	−	13.7071i	−0.771492	−	0.445421i	−0.833407	+	0.552660i	$0.813613\pi$
$948$		0			0
$949$	−2.34315	−	4.05845i	−0.0760617	−	0.131743i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			10.7868i			0.349419i	0.984620	+	0.174709i	$0.0558986\pi$
$953$			10.7868i			0.349419i	−0.984620	+	0.174709i	$0.944101\pi$
$954$		0			0
$955$	−8.73048	+	5.04054i	−0.282512	+	0.163108i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	39.1274	−	67.7707i	1.26217	−	2.18615i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−20.9050			−0.672956
$966$		0			0
$967$	−18.3431			−0.589876			−0.294938	−	0.955516i	$-0.595299\pi$
$967$	−18.3431			−0.589876			−0.294938	+	0.955516i	$0.595299\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.5140	+	23.4069i	−0.433684	+	0.751163i	−0.997187	−	0.0749512i	$-0.976120\pi$
$971$	−13.5140	+	23.4069i	−0.433684	+	0.751163i	0.563503	+	0.826114i	$0.309453\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.64501	−	0.949747i	0.0526286	−	0.0303851i	−0.473455	−	0.880818i	$-0.656993\pi$
$977$	1.64501	−	0.949747i	0.0526286	−	0.0303851i	0.526083	+	0.850433i	$0.323660\pi$
$978$		0			0
$979$			23.0698i			0.737314i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−29.8268	−	51.6615i	−0.951326	−	1.64775i	−0.742559	−	0.669780i	$-0.766388\pi$
$983$	−29.8268	−	51.6615i	−0.951326	−	1.64775i	−0.208767	−	0.977965i	$-0.566945\pi$
$984$		0			0
$985$	71.6405	+	41.3617i	2.28266	+	1.31789i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	35.0613	+	20.2426i	1.11488	+	0.643679i
$990$		0			0
$991$	−28.0711	−	48.6205i	−0.891707	−	1.54448i	−0.837828	−	0.545934i	$-0.816175\pi$
$991$	−28.0711	−	48.6205i	−0.891707	−	1.54448i	−0.0538787	−	0.998547i	$-0.517158\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	85.2548i		−	2.70276i
$996$		0			0
$997$	40.6014	−	23.4412i	1.28586	−	0.742391i	0.307945	−	0.951404i	$-0.400359\pi$
$997$	40.6014	−	23.4412i	1.28586	−	0.742391i	0.977913	+	0.209013i	$0.0670252\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.bl.b.1097.8		16
3.2	odd	2	inner	3528.2.bl.b.1097.1		16
7.2	even	3		504.2.k.a.377.1	✓	8
7.3	odd	6	inner	3528.2.bl.b.521.1		16
7.4	even	3	inner	3528.2.bl.b.521.7		16
7.5	odd	6		504.2.k.a.377.8	yes	8
7.6	odd	2	inner	3528.2.bl.b.1097.2		16
21.2	odd	6		504.2.k.a.377.7	yes	8
21.5	even	6		504.2.k.a.377.2	yes	8
21.11	odd	6	inner	3528.2.bl.b.521.2		16
21.17	even	6	inner	3528.2.bl.b.521.8		16
21.20	even	2	inner	3528.2.bl.b.1097.7		16
28.19	even	6		1008.2.k.c.881.7		8
28.23	odd	6		1008.2.k.c.881.2		8
56.5	odd	6		4032.2.k.f.3905.2		8
56.19	even	6		4032.2.k.e.3905.1		8
56.37	even	6		4032.2.k.f.3905.7		8
56.51	odd	6		4032.2.k.e.3905.8		8
84.23	even	6		1008.2.k.c.881.8		8
84.47	odd	6		1008.2.k.c.881.1		8
168.5	even	6		4032.2.k.f.3905.8		8
168.107	even	6		4032.2.k.e.3905.2		8
168.131	odd	6		4032.2.k.e.3905.7		8
168.149	odd	6		4032.2.k.f.3905.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.k.a.377.1	✓	8	7.2	even	3
504.2.k.a.377.2	yes	8	21.5	even	6
504.2.k.a.377.7	yes	8	21.2	odd	6
504.2.k.a.377.8	yes	8	7.5	odd	6
1008.2.k.c.881.1		8	84.47	odd	6
1008.2.k.c.881.2		8	28.23	odd	6
1008.2.k.c.881.7		8	28.19	even	6
1008.2.k.c.881.8		8	84.23	even	6
3528.2.bl.b.521.1		16	7.3	odd	6	inner
3528.2.bl.b.521.2		16	21.11	odd	6	inner
3528.2.bl.b.521.7		16	7.4	even	3	inner
3528.2.bl.b.521.8		16	21.17	even	6	inner
3528.2.bl.b.1097.1		16	3.2	odd	2	inner
3528.2.bl.b.1097.2		16	7.6	odd	2	inner
3528.2.bl.b.1097.7		16	21.20	even	2	inner
3528.2.bl.b.1097.8		16	1.1	even	1	trivial
4032.2.k.e.3905.1		8	56.19	even	6
4032.2.k.e.3905.2		8	168.107	even	6
4032.2.k.e.3905.7		8	168.131	odd	6
4032.2.k.e.3905.8		8	56.51	odd	6
4032.2.k.f.3905.1		8	168.149	odd	6
4032.2.k.f.3905.2		8	56.5	odd	6
4032.2.k.f.3905.7		8	56.37	even	6
4032.2.k.f.3905.8		8	168.5	even	6

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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.84776 - 3.20041i) q^{5} +O(q^{10})\)	\(q+(1.84776 - 3.20041i) q^{5} +(2.95680 - 1.70711i) q^{11} +5.22625i q^{13} +(-3.37849 - 5.85172i) q^{17} +(1.87476 + 1.08239i) q^{19} +(-5.40629 - 3.12132i) q^{23} +(-4.32843 - 7.49706i) q^{25} -2.58579i q^{29} +(9.05213 - 5.22625i) q^{31} +(-2.00000 + 3.46410i) q^{37} -0.634051 q^{41} -6.48528 q^{43} +(1.53073 - 2.65131i) q^{47} +(2.23936 - 1.29289i) q^{53} -12.6173i q^{55} +(16.7262 + 9.65685i) q^{65} +(-0.828427 - 1.43488i) q^{67} -7.41421i q^{71} +(-0.776550 + 0.448342i) q^{73} +(6.82843 - 11.8272i) q^{79} +11.7206 q^{83} -24.9706 q^{85} +(-3.37849 + 5.85172i) q^{89} +(6.92820 - 4.00000i) q^{95} +9.55582i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 24 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{67} + 64 q^{79} - 128 q^{85}+O(q^{100}) \) 16 * q - 24 * q^25 - 32 * q^37 + 32 * q^43 + 32 * q^67 + 64 * q^79 - 128 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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