LMFDB - Embedded newform 4608.2.k.bi.3457.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1153,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("4608.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.k (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3457.4
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	4608.3457
Dual form			4608.2.k.bi.1153.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.41421 + 2.41421i) q^{5} +1.53073i q^{7} +O(q^{10})$	$q+(2.41421 + 2.41421i) q^{5} +1.53073i q^{7} +(3.37849 + 3.37849i) q^{11} +(-0.414214 + 0.414214i) q^{13} -2.82843 q^{17} +(0.317025 - 0.317025i) q^{19} +5.86030i q^{23} +6.65685i q^{25} +(-3.24264 + 3.24264i) q^{29} -7.39104 q^{31} +(-3.69552 + 3.69552i) q^{35} +(3.58579 + 3.58579i) q^{37} -4.00000i q^{41} +(-1.84776 - 1.84776i) q^{43} -7.39104 q^{47} +4.65685 q^{49} +(-5.24264 - 5.24264i) q^{53} +16.3128i q^{55} +(1.84776 + 1.84776i) q^{59} +(9.24264 - 9.24264i) q^{61} -2.00000 q^{65} +(-7.07401 + 7.07401i) q^{67} +11.9832i q^{71} -10.4853i q^{73} +(-5.17157 + 5.17157i) q^{77} +6.12293 q^{79} +(2.48181 - 2.48181i) q^{83} +(-6.82843 - 6.82843i) q^{85} +0.828427i q^{89} +(-0.634051 - 0.634051i) q^{91} +1.53073 q^{95} +10.8284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} + 8 q^{13} + 8 q^{29} + 40 q^{37} - 8 q^{49} - 8 q^{53} + 40 q^{61} - 16 q^{65} - 64 q^{77} - 32 q^{85} + 64 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 + 8 * q^13 + 8 * q^29 + 40 * q^37 - 8 * q^49 - 8 * q^53 + 40 * q^61 - 16 * q^65 - 64 * q^77 - 32 * q^85 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$5$	2.41421	+	2.41421i	1.07967	+	1.07967i	0.996539	+	0.0831305i	$0.0264918\pi$
$5$	2.41421	+	2.41421i	1.07967	+	1.07967i	0.0831305	+	0.996539i	$0.473508\pi$
$6$		0			0
$7$			1.53073i			0.578563i	0.957244	+	0.289281i	$0.0934164\pi$
$7$			1.53073i			0.578563i	−0.957244	+	0.289281i	$0.906584\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.37849	+	3.37849i	1.01865	+	1.01865i	0.999823	+	0.0188312i	$0.00599452\pi$
$11$	3.37849	+	3.37849i	1.01865	+	1.01865i	0.0188312	+	0.999823i	$0.494005\pi$
$12$		0			0
$13$	−0.414214	+	0.414214i	−0.114882	+	0.114882i	−0.762211	−	0.647329i	$-0.775886\pi$
$13$	−0.414214	+	0.414214i	−0.114882	+	0.114882i	0.647329	+	0.762211i	$0.275886\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.82843			−0.685994			−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843			−0.685994			−0.342997	+	0.939336i	$0.611442\pi$
$18$		0			0
$19$	0.317025	−	0.317025i	0.0727306	−	0.0727306i	−0.669806	−	0.742536i	$-0.733623\pi$
$19$	0.317025	−	0.317025i	0.0727306	−	0.0727306i	0.742536	+	0.669806i	$0.233623\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.86030i			1.22196i	0.791647	+	0.610979i	$0.209224\pi$
$23$			5.86030i			1.22196i	−0.791647	+	0.610979i	$0.790776\pi$
$24$		0			0
$25$			6.65685i			1.33137i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.24264	+	3.24264i	−0.602143	+	0.602143i	−0.940881	−	0.338738i	$-0.890000\pi$
$29$	−3.24264	+	3.24264i	−0.602143	+	0.602143i	0.338738	+	0.940881i	$0.390000\pi$
$30$		0			0
$31$	−7.39104			−1.32747			−0.663735	−	0.747968i	$-0.731030\pi$
$31$	−7.39104			−1.32747			−0.663735	+	0.747968i	$0.731030\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.69552	+	3.69552i	−0.624657	+	0.624657i
$36$		0			0
$37$	3.58579	+	3.58579i	0.589500	+	0.589500i	0.937496	−	0.347996i	$-0.113138\pi$
$37$	3.58579	+	3.58579i	0.589500	+	0.589500i	−0.347996	+	0.937496i	$0.613138\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$	−1.84776	−	1.84776i	−0.281781	−	0.281781i	0.552038	−	0.833819i	$-0.313850\pi$
$43$	−1.84776	−	1.84776i	−0.281781	−	0.281781i	−0.833819	+	0.552038i	$0.813850\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.39104			−1.07809			−0.539047	−	0.842276i	$-0.681215\pi$
$47$	−7.39104			−1.07809			−0.539047	+	0.842276i	$0.681215\pi$
$48$		0			0
$49$	4.65685			0.665265
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.24264	−	5.24264i	−0.720132	−	0.720132i	0.248500	−	0.968632i	$-0.420062\pi$
$53$	−5.24264	−	5.24264i	−0.720132	−	0.720132i	−0.968632	+	0.248500i	$0.920062\pi$
$54$		0			0
$55$			16.3128i			2.19962i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.84776	+	1.84776i	0.240558	+	0.240558i	0.817081	−	0.576523i	$-0.195591\pi$
$59$	1.84776	+	1.84776i	0.240558	+	0.240558i	−0.576523	+	0.817081i	$0.695591\pi$
$60$		0			0
$61$	9.24264	−	9.24264i	1.18340	−	1.18340i	0.204541	−	0.978858i	$-0.434430\pi$
$61$	9.24264	−	9.24264i	1.18340	−	1.18340i	0.978858	−	0.204541i	$-0.0655701\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000			−0.248069
$66$		0			0
$67$	−7.07401	+	7.07401i	−0.864228	+	0.864228i	−0.991826	−	0.127598i	$-0.959273\pi$
$67$	−7.07401	+	7.07401i	−0.864228	+	0.864228i	0.127598	+	0.991826i	$0.459273\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.9832i			1.42215i	0.703117	+	0.711074i	$0.251791\pi$
$71$			11.9832i			1.42215i	−0.703117	+	0.711074i	$0.748209\pi$
$72$		0			0
$73$		−	10.4853i		−	1.22721i	−0.789613	−	0.613605i	$-0.789719\pi$
$73$		−	10.4853i		−	1.22721i	0.789613	−	0.613605i	$-0.210281\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.17157	+	5.17157i	−0.589355	+	0.589355i
$78$		0			0
$79$	6.12293			0.688884			0.344442	−	0.938808i	$-0.388068\pi$
$79$	6.12293			0.688884			0.344442	+	0.938808i	$0.388068\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.48181	−	2.48181i	0.272414	−	0.272414i	−0.557657	−	0.830071i	$-0.688300\pi$
$83$	2.48181	−	2.48181i	0.272414	−	0.272414i	0.830071	+	0.557657i	$0.188300\pi$
$84$		0			0
$85$	−6.82843	−	6.82843i	−0.740647	−	0.740647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			0.828427i			0.0878131i	0.999036	+	0.0439065i	$0.0139804\pi$
$89$			0.828427i			0.0878131i	−0.999036	+	0.0439065i	$0.986020\pi$
$90$		0			0
$91$	−0.634051	−	0.634051i	−0.0664666	−	0.0664666i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.53073			0.157050
$96$		0			0
$97$	10.8284			1.09946			0.549730	−	0.835342i	$-0.314731\pi$
$97$	10.8284			1.09946			0.549730	+	0.835342i	$0.314731\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.07107	+	2.07107i	0.206079	+	0.206079i	0.802599	−	0.596520i	$-0.203450\pi$
$101$	2.07107	+	2.07107i	0.206079	+	0.206079i	−0.596520	+	0.802599i	$0.703450\pi$
$102$		0			0
$103$			2.79884i			0.275777i	0.990448	+	0.137889i	$0.0440316\pi$
$103$			2.79884i			0.275777i	−0.990448	+	0.137889i	$0.955968\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.07401	−	7.07401i	−0.683870	−	0.683870i	0.277000	−	0.960870i	$-0.410660\pi$
$107$	−7.07401	−	7.07401i	−0.683870	−	0.683870i	−0.960870	+	0.277000i	$0.910660\pi$
$108$		0			0
$109$	−0.757359	+	0.757359i	−0.0725419	+	0.0725419i	−0.742447	−	0.669905i	$-0.766335\pi$
$109$	−0.757359	+	0.757359i	−0.0725419	+	0.0725419i	0.669905	+	0.742447i	$0.266335\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.65685			0.344008			0.172004	−	0.985096i	$-0.444976\pi$
$113$	3.65685			0.344008			0.172004	+	0.985096i	$0.444976\pi$
$114$		0			0
$115$	−14.1480	+	14.1480i	−1.31931	+	1.31931i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.32957i		−	0.396891i
$120$		0			0
$121$			11.8284i			1.07531i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.00000	+	4.00000i	−0.357771	+	0.357771i
$126$		0			0
$127$	13.5140			1.19917			0.599586	−	0.800311i	$-0.295332\pi$
$127$	13.5140			1.19917			0.599586	+	0.800311i	$0.295332\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.43996	+	6.43996i	−0.562662	+	0.562662i	−0.930063	−	0.367401i	$-0.880248\pi$
$131$	−6.43996	+	6.43996i	−0.562662	+	0.562662i	0.367401	+	0.930063i	$0.380248\pi$
$132$		0			0
$133$	0.485281	+	0.485281i	0.0420792	+	0.0420792i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	20.9706i		−	1.79164i	−0.444421	−	0.895818i	$-0.646591\pi$
$137$		−	20.9706i		−	1.79164i	0.444421	−	0.895818i	$-0.353409\pi$
$138$		0			0
$139$	−13.5684	−	13.5684i	−1.15085	−	1.15085i	−0.986382	−	0.164472i	$-0.947408\pi$
$139$	−13.5684	−	13.5684i	−1.15085	−	1.15085i	−0.164472	−	0.986382i	$-0.552592\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.79884			−0.234050
$144$		0			0
$145$	−15.6569			−1.30023
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.24264	−	7.24264i	−0.593340	−	0.593340i	0.345192	−	0.938532i	$-0.387814\pi$
$149$	−7.24264	−	7.24264i	−0.593340	−	0.593340i	−0.938532	+	0.345192i	$0.887814\pi$
$150$		0			0
$151$			8.92177i			0.726043i	0.931781	+	0.363022i	$0.118255\pi$
$151$			8.92177i			0.726043i	−0.931781	+	0.363022i	$0.881745\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−17.8435	−	17.8435i	−1.43323	−	1.43323i
$156$		0			0
$157$	−4.41421	+	4.41421i	−0.352293	+	0.352293i	−0.860962	−	0.508669i	$-0.830138\pi$
$157$	−4.41421	+	4.41421i	−0.352293	+	0.352293i	0.508669	+	0.860962i	$0.330138\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.97056			−0.706979
$162$		0			0
$163$	9.23880	−	9.23880i	0.723638	−	0.723638i	−0.245706	−	0.969344i	$-0.579020\pi$
$163$	9.23880	−	9.23880i	0.723638	−	0.723638i	0.969344	+	0.245706i	$0.0790198\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.53073i		−	0.118452i	−0.998245	−	0.0592259i	$-0.981137\pi$
$167$		−	1.53073i		−	0.118452i	0.998245	−	0.0592259i	$-0.0188632\pi$
$168$		0			0
$169$			12.6569i			0.973604i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.4142	−	16.4142i	1.24795	−	1.24795i	0.291326	−	0.956624i	$-0.405904\pi$
$173$	16.4142	−	16.4142i	1.24795	−	1.24795i	0.956624	−	0.291326i	$-0.0940963\pi$
$174$		0			0
$175$	−10.1899			−0.770282
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.21371	−	1.21371i	0.0907168	−	0.0907168i	−0.660292	−	0.751009i	$-0.729568\pi$
$179$	1.21371	−	1.21371i	0.0907168	−	0.0907168i	0.751009	+	0.660292i	$0.229568\pi$
$180$		0			0
$181$	−4.07107	−	4.07107i	−0.302600	−	0.302600i	0.539430	−	0.842030i	$-0.318640\pi$
$181$	−4.07107	−	4.07107i	−0.302600	−	0.302600i	−0.842030	+	0.539430i	$0.818640\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			17.3137i			1.27293i
$186$		0			0
$187$	−9.55582	−	9.55582i	−0.698791	−	0.698791i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.5140			0.977837			0.488918	−	0.872330i	$-0.337392\pi$
$191$	13.5140			0.977837			0.488918	+	0.872330i	$0.337392\pi$
$192$		0			0
$193$	−16.4853			−1.18664			−0.593318	−	0.804968i	$-0.702182\pi$
$193$	−16.4853			−1.18664			−0.593318	+	0.804968i	$0.702182\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.4142	+	10.4142i	0.741982	+	0.741982i	0.972959	−	0.230977i	$-0.0741923\pi$
$197$	10.4142	+	10.4142i	0.741982	+	0.741982i	−0.230977	+	0.972959i	$0.574192\pi$
$198$		0			0
$199$			22.4357i			1.59043i	0.606329	+	0.795214i	$0.292641\pi$
$199$			22.4357i			1.59043i	−0.606329	+	0.795214i	$0.707359\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.96362	−	4.96362i	−0.348378	−	0.348378i
$204$		0			0
$205$	9.65685	−	9.65685i	0.674464	−	0.674464i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.14214			0.148175
$210$		0			0
$211$	−19.0572	+	19.0572i	−1.31196	+	1.31196i	−0.391982	+	0.919973i	$0.628211\pi$
$211$	−19.0572	+	19.0572i	−1.31196	+	1.31196i	−0.919973	+	0.391982i	$0.871789\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	8.92177i		−	0.608460i
$216$		0			0
$217$		−	11.3137i		−	0.768025i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.17157	−	1.17157i	0.0788085	−	0.0788085i
$222$		0			0
$223$	−7.39104			−0.494940			−0.247470	−	0.968896i	$-0.579599\pi$
$223$	−7.39104			−0.494940			−0.247470	+	0.968896i	$0.579599\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.84776	+	1.84776i	−0.122640	+	0.122640i	−0.765763	−	0.643123i	$-0.777638\pi$
$227$	−1.84776	+	1.84776i	−0.122640	+	0.122640i	0.643123	+	0.765763i	$0.277638\pi$
$228$		0			0
$229$	11.2426	+	11.2426i	0.742935	+	0.742935i	0.973142	−	0.230207i	$-0.0739403\pi$
$229$	11.2426	+	11.2426i	0.742935	+	0.742935i	−0.230207	+	0.973142i	$0.573940\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.1716i			0.993923i	0.867773	+	0.496961i	$0.165551\pi$
$233$			15.1716i			0.993923i	−0.867773	+	0.496961i	$0.834449\pi$
$234$		0			0
$235$	−17.8435	−	17.8435i	−1.16398	−	1.16398i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	29.5641			1.91235			0.956173	−	0.292803i	$-0.0945880\pi$
$239$	29.5641			1.91235			0.956173	+	0.292803i	$0.0945880\pi$
$240$		0			0
$241$	18.8284			1.21285			0.606423	−	0.795142i	$-0.292604\pi$
$241$	18.8284			1.21285			0.606423	+	0.795142i	$0.292604\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.2426	+	11.2426i	0.718266	+	0.718266i
$246$		0			0
$247$			0.262632i			0.0167109i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.87285	−	9.87285i	−0.623169	−	0.623169i	0.323172	−	0.946340i	$-0.395251\pi$
$251$	−9.87285	−	9.87285i	−0.623169	−	0.623169i	−0.946340	+	0.323172i	$0.895251\pi$
$252$		0			0
$253$	−19.7990	+	19.7990i	−1.24475	+	1.24475i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.9706			−1.18335			−0.591676	−	0.806176i	$-0.701533\pi$
$257$	−18.9706			−1.18335			−0.591676	+	0.806176i	$0.701533\pi$
$258$		0			0
$259$	−5.48888	+	5.48888i	−0.341063	+	0.341063i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			20.6424i			1.27286i	0.771333	+	0.636432i	$0.219590\pi$
$263$			20.6424i			1.27286i	−0.771333	+	0.636432i	$0.780410\pi$
$264$		0			0
$265$		−	25.3137i		−	1.55501i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.92893	+	3.92893i	−0.239551	+	0.239551i	−0.816664	−	0.577113i	$-0.804179\pi$
$269$	−3.92893	+	3.92893i	−0.239551	+	0.239551i	0.577113	+	0.816664i	$0.304179\pi$
$270$		0			0
$271$	28.2960			1.71886			0.859431	−	0.511252i	$-0.170818\pi$
$271$	28.2960			1.71886			0.859431	+	0.511252i	$0.170818\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−22.4901	+	22.4901i	−1.35621	+	1.35621i
$276$		0			0
$277$	−0.414214	−	0.414214i	−0.0248877	−	0.0248877i	0.694553	−	0.719441i	$-0.255602\pi$
$277$	−0.414214	−	0.414214i	−0.0248877	−	0.0248877i	−0.719441	+	0.694553i	$0.755602\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			21.7990i			1.30042i	0.759755	+	0.650209i	$0.225319\pi$
$281$			21.7990i			1.30042i	−0.759755	+	0.650209i	$0.774681\pi$
$282$		0			0
$283$	−6.43996	−	6.43996i	−0.382816	−	0.382816i	0.489300	−	0.872116i	$-0.337252\pi$
$283$	−6.43996	−	6.43996i	−0.382816	−	0.382816i	−0.872116	+	0.489300i	$0.837252\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.12293			0.361425
$288$		0			0
$289$	−9.00000			−0.529412
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.41421	+	2.41421i	0.141040	+	0.141040i	0.774101	−	0.633062i	$-0.218202\pi$
$293$	2.41421	+	2.41421i	0.141040	+	0.141040i	−0.633062	+	0.774101i	$0.718202\pi$
$294$		0			0
$295$			8.92177i			0.519446i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.42742	−	2.42742i	−0.140381	−	0.140381i
$300$		0			0
$301$	2.82843	−	2.82843i	0.163028	−	0.163028i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	44.6274			2.55536
$306$		0			0
$307$	18.1606	−	18.1606i	1.03648	−	1.03648i	0.0371692	−	0.999309i	$-0.488166\pi$
$307$	18.1606	−	18.1606i	1.03648	−	1.03648i	0.999309	−	0.0371692i	$-0.0118341\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			7.12840i			0.404215i	0.979363	+	0.202107i	$0.0647790\pi$
$311$			7.12840i			0.404215i	−0.979363	+	0.202107i	$0.935221\pi$
$312$		0			0
$313$		−	8.68629i		−	0.490978i	−0.969399	−	0.245489i	$-0.921051\pi$
$313$		−	8.68629i		−	0.490978i	0.969399	−	0.245489i	$-0.0789486\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.89949	+	8.89949i	−0.499845	+	0.499845i	−0.911390	−	0.411544i	$-0.864990\pi$
$317$	−8.89949	+	8.89949i	−0.499845	+	0.499845i	0.411544	+	0.911390i	$0.364990\pi$
$318$		0			0
$319$	−21.9105			−1.22675
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.896683	+	0.896683i	−0.0498928	+	0.0498928i
$324$		0			0
$325$	−2.75736	−	2.75736i	−0.152951	−	0.152951i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	11.3137i		−	0.623745i
$330$		0			0
$331$	4.01254	+	4.01254i	0.220549	+	0.220549i	0.808730	−	0.588180i	$-0.200156\pi$
$331$	4.01254	+	4.01254i	0.220549	+	0.220549i	−0.588180	+	0.808730i	$0.700156\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−34.1563			−1.86616
$336$		0			0
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−24.9706	−	24.9706i	−1.35223	−	1.35223i
$342$		0			0
$343$			17.8435i			0.963461i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.3617	+	15.3617i	0.824661	+	0.824661i	0.986772	−	0.162112i	$-0.0518304\pi$
$347$	15.3617	+	15.3617i	0.824661	+	0.824661i	−0.162112	+	0.986772i	$0.551830\pi$
$348$		0			0
$349$	−16.0711	+	16.0711i	−0.860265	+	0.860265i	−0.991369	−	0.131104i	$-0.958148\pi$
$349$	−16.0711	+	16.0711i	−0.860265	+	0.860265i	0.131104	+	0.991369i	$0.458148\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.9706			0.583904			0.291952	−	0.956433i	$-0.405695\pi$
$353$	10.9706			0.583904			0.291952	+	0.956433i	$0.405695\pi$
$354$		0			0
$355$	−28.9301	+	28.9301i	−1.53545	+	1.53545i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.92177i		−	0.470873i	−0.971890	−	0.235437i	$-0.924348\pi$
$359$		−	8.92177i		−	0.470873i	0.971890	−	0.235437i	$-0.0756520\pi$
$360$		0			0
$361$			18.7990i			0.989421i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	25.3137	−	25.3137i	1.32498	−	1.32498i
$366$		0			0
$367$	−22.1731			−1.15743			−0.578713	−	0.815531i	$-0.696445\pi$
$367$	−22.1731			−1.15743			−0.578713	+	0.815531i	$0.696445\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.02509	−	8.02509i	0.416642	−	0.416642i
$372$		0			0
$373$	13.2426	+	13.2426i	0.685678	+	0.685678i	0.961274	−	0.275596i	$-0.0888751\pi$
$373$	13.2426	+	13.2426i	0.685678	+	0.685678i	−0.275596	+	0.961274i	$0.588875\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2.68629i		−	0.138351i
$378$		0			0
$379$	−3.11586	−	3.11586i	−0.160051	−	0.160051i	0.622538	−	0.782589i	$-0.286101\pi$
$379$	−3.11586	−	3.11586i	−0.160051	−	0.160051i	−0.782589	+	0.622538i	$0.786101\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.65914			−0.442461			−0.221231	−	0.975222i	$-0.571007\pi$
$383$	−8.65914			−0.442461			−0.221231	+	0.975222i	$0.571007\pi$
$384$		0			0
$385$	−24.9706			−1.27262
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.5858	−	11.5858i	−0.587423	−	0.587423i	0.349510	−	0.936933i	$-0.386348\pi$
$389$	−11.5858	−	11.5858i	−0.587423	−	0.587423i	−0.936933	+	0.349510i	$0.886348\pi$
$390$		0			0
$391$		−	16.5754i		−	0.838256i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.7821	+	14.7821i	0.743767	+	0.743767i
$396$		0			0
$397$	−6.41421	+	6.41421i	−0.321920	+	0.321920i	−0.849503	−	0.527583i	$-0.823098\pi$
$397$	−6.41421	+	6.41421i	−0.321920	+	0.321920i	0.527583	+	0.849503i	$0.323098\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	24.4853			1.22274			0.611368	−	0.791346i	$-0.290619\pi$
$401$	24.4853			1.22274			0.611368	+	0.791346i	$0.290619\pi$
$402$		0			0
$403$	3.06147	−	3.06147i	0.152503	−	0.152503i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			24.2291i			1.20099i
$408$		0			0
$409$		−	8.68629i		−	0.429509i	−0.976668	−	0.214755i	$-0.931105\pi$
$409$		−	8.68629i		−	0.429509i	0.976668	−	0.214755i	$-0.0688952\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.82843	+	2.82843i	−0.139178	+	0.139178i
$414$		0			0
$415$	11.9832			0.588234
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.1355	−	10.1355i	0.495151	−	0.495151i	−0.414774	−	0.909924i	$-0.636139\pi$
$419$	10.1355	−	10.1355i	0.495151	−	0.495151i	0.909924	+	0.414774i	$0.136139\pi$
$420$		0			0
$421$	−8.07107	−	8.07107i	−0.393360	−	0.393360i	0.482523	−	0.875883i	$-0.339720\pi$
$421$	−8.07107	−	8.07107i	−0.393360	−	0.393360i	−0.875883	+	0.482523i	$0.839720\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	18.8284i		−	0.913313i
$426$		0			0
$427$	14.1480	+	14.1480i	0.684671	+	0.684671i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	25.4558			1.22333			0.611665	−	0.791117i	$-0.290500\pi$
$433$	25.4558			1.22333			0.611665	+	0.791117i	$0.290500\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.85786	+	1.85786i	0.0888737	+	0.0888737i
$438$		0			0
$439$			18.8490i			0.899614i	0.893126	+	0.449807i	$0.148507\pi$
$439$			18.8490i			0.899614i	−0.893126	+	0.449807i	$0.851493\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.07401	−	7.07401i	−0.336096	−	0.336096i	0.518800	−	0.854896i	$-0.326379\pi$
$443$	−7.07401	−	7.07401i	−0.336096	−	0.336096i	−0.854896	+	0.518800i	$0.826379\pi$
$444$		0			0
$445$	−2.00000	+	2.00000i	−0.0948091	+	0.0948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−22.1421			−1.04495			−0.522476	−	0.852654i	$-0.674992\pi$
$449$	−22.1421			−1.04495			−0.522476	+	0.852654i	$0.674992\pi$
$450$		0			0
$451$	13.5140	−	13.5140i	0.636348	−	0.636348i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	3.06147i		−	0.143524i
$456$		0			0
$457$		−	18.6274i		−	0.871354i	−0.900103	−	0.435677i	$-0.856509\pi$
$457$		−	18.6274i		−	0.871354i	0.900103	−	0.435677i	$-0.143491\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	21.0416	−	21.0416i	0.980006	−	0.980006i	−0.0197976	−	0.999804i	$-0.506302\pi$
$461$	21.0416	−	21.0416i	0.980006	−	0.980006i	0.999804	+	0.0197976i	$0.00630217\pi$
$462$		0			0
$463$	2.53620			0.117867			0.0589337	−	0.998262i	$-0.481230\pi$
$463$	2.53620			0.117867			0.0589337	+	0.998262i	$0.481230\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.6717	−	12.6717i	0.586375	−	0.586375i	−0.350272	−	0.936648i	$-0.613911\pi$
$467$	12.6717	−	12.6717i	0.586375	−	0.586375i	0.936648	+	0.350272i	$0.113911\pi$
$468$		0			0
$469$	−10.8284	−	10.8284i	−0.500010	−	0.500010i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	12.4853i		−	0.574074i
$474$		0			0
$475$	2.11039	+	2.11039i	0.0968314	+	0.0968314i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−40.5419			−1.85241			−0.926204	−	0.377024i	$-0.876948\pi$
$479$	−40.5419			−1.85241			−0.926204	+	0.377024i	$0.876948\pi$
$480$		0			0
$481$	−2.97056			−0.135446
$482$		0			0
$483$		0			0
$484$		0			0
$485$	26.1421	+	26.1421i	1.18705	+	1.18705i
$486$		0			0
$487$			37.2178i			1.68650i	0.537521	+	0.843250i	$0.319361\pi$
$487$			37.2178i			1.68650i	−0.537521	+	0.843250i	$0.680639\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.3003	+	12.3003i	0.555103	+	0.555103i	0.927909	−	0.372806i	$-0.121604\pi$
$491$	12.3003	+	12.3003i	0.555103	+	0.555103i	−0.372806	+	0.927909i	$0.621604\pi$
$492$		0			0
$493$	9.17157	−	9.17157i	0.413067	−	0.413067i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18.3431			−0.822803
$498$		0			0
$499$	−11.6662	+	11.6662i	−0.522251	+	0.522251i	−0.918251	−	0.395999i	$-0.870398\pi$
$499$	−11.6662	+	11.6662i	−0.522251	+	0.522251i	0.395999	+	0.918251i	$0.370398\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.1062i			0.807314i	0.914910	+	0.403657i	$0.132261\pi$
$503$			18.1062i			0.807314i	−0.914910	+	0.403657i	$0.867739\pi$
$504$		0			0
$505$			10.0000i			0.444994i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.5563	+	12.5563i	−0.556550	+	0.556550i	−0.928324	−	0.371773i	$-0.878750\pi$
$509$	−12.5563	+	12.5563i	−0.556550	+	0.556550i	0.371773	+	0.928324i	$0.378750\pi$
$510$		0			0
$511$	16.0502			0.710018
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.75699	+	6.75699i	−0.297748	+	0.297748i
$516$		0			0
$517$	−24.9706	−	24.9706i	−1.09820	−	1.09820i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			17.6569i			0.773561i	0.922172	+	0.386780i	$0.126413\pi$
$521$			17.6569i			0.773561i	−0.922172	+	0.386780i	$0.873587\pi$
$522$		0			0
$523$	27.9790	+	27.9790i	1.22344	+	1.22344i	0.966402	+	0.257035i	$0.0827455\pi$
$523$	27.9790	+	27.9790i	1.22344	+	1.22344i	0.257035	+	0.966402i	$0.417254\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.9050			0.910636
$528$		0			0
$529$	−11.3431			−0.493180
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.65685	+	1.65685i	0.0717663	+	0.0717663i
$534$		0			0
$535$		−	34.1563i		−	1.47671i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	15.7331	+	15.7331i	0.677675	+	0.677675i
$540$		0			0
$541$	1.24264	−	1.24264i	0.0534253	−	0.0534253i	−0.679889	−	0.733315i	$-0.737972\pi$
$541$	1.24264	−	1.24264i	0.0534253	−	0.0534253i	0.733315	+	0.679889i	$0.237972\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.65685			−0.156642
$546$		0			0
$547$	−0.951076	+	0.951076i	−0.0406651	+	0.0406651i	−0.727147	−	0.686482i	$-0.759154\pi$
$547$	−0.951076	+	0.951076i	−0.0406651	+	0.0406651i	0.686482	+	0.727147i	$0.259154\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			2.05600i			0.0875885i
$552$		0			0
$553$			9.37258i			0.398563i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.07107	−	6.07107i	0.257239	−	0.257239i	−0.566691	−	0.823930i	$-0.691777\pi$
$557$	6.07107	−	6.07107i	0.257239	−	0.257239i	0.823930	+	0.566691i	$0.191777\pi$
$558$		0			0
$559$	1.53073			0.0647431
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.0991	+	15.0991i	−0.636351	+	0.636351i	−0.949653	−	0.313302i	$-0.898565\pi$
$563$	−15.0991	+	15.0991i	−0.636351	+	0.636351i	0.313302	+	0.949653i	$0.398565\pi$
$564$		0			0
$565$	8.82843	+	8.82843i	0.371415	+	0.371415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			14.3431i			0.601296i	0.953735	+	0.300648i	$0.0972029\pi$
$569$			14.3431i			0.601296i	−0.953735	+	0.300648i	$0.902797\pi$
$570$		0			0
$571$	15.7331	+	15.7331i	0.658412	+	0.658412i	0.955004	−	0.296592i	$-0.0958503\pi$
$571$	15.7331	+	15.7331i	0.658412	+	0.658412i	−0.296592	+	0.955004i	$0.595850\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−39.0112			−1.62688
$576$		0			0
$577$	−5.31371			−0.221213			−0.110606	−	0.993864i	$-0.535279\pi$
$577$	−5.31371			−0.221213			−0.110606	+	0.993864i	$0.535279\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.79899	+	3.79899i	0.157609	+	0.157609i
$582$		0			0
$583$		−	35.4244i		−	1.46713i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.5880	−	20.5880i	−0.849757	−	0.849757i	0.140346	−	0.990103i	$-0.455179\pi$
$587$	−20.5880	−	20.5880i	−0.849757	−	0.849757i	−0.990103	+	0.140346i	$0.955179\pi$
$588$		0			0
$589$	−2.34315	+	2.34315i	−0.0965476	+	0.0965476i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.3137			−0.546728			−0.273364	−	0.961911i	$-0.588136\pi$
$593$	−13.3137			−0.546728			−0.273364	+	0.961911i	$0.588136\pi$
$594$		0			0
$595$	10.4525	−	10.4525i	0.428511	−	0.428511i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			15.7875i			0.645061i	0.946559	+	0.322531i	$0.104534\pi$
$599$			15.7875i			0.645061i	−0.946559	+	0.322531i	$0.895466\pi$
$600$		0			0
$601$			15.1716i			0.618861i	0.950922	+	0.309431i	$0.100138\pi$
$601$			15.1716i			0.618861i	−0.950922	+	0.309431i	$0.899862\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−28.5563	+	28.5563i	−1.16098	+	1.16098i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.06147	−	3.06147i	0.123854	−	0.123854i
$612$		0			0
$613$	25.8701	+	25.8701i	1.04488	+	1.04488i	0.998944	+	0.0459375i	$0.0146275\pi$
$613$	25.8701	+	25.8701i	1.04488	+	1.04488i	0.0459375	+	0.998944i	$0.485373\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			21.5147i			0.866150i	0.901358	+	0.433075i	$0.142571\pi$
$617$			21.5147i			0.866150i	−0.901358	+	0.433075i	$0.857429\pi$
$618$		0			0
$619$	−3.37849	−	3.37849i	−0.135793	−	0.135793i	0.635943	−	0.771736i	$-0.280611\pi$
$619$	−3.37849	−	3.37849i	−0.135793	−	0.135793i	−0.771736	+	0.635943i	$0.780611\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.26810			−0.0508054
$624$		0			0
$625$	13.9706			0.558823
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−10.1421	−	10.1421i	−0.404393	−	0.404393i
$630$		0			0
$631$			2.79884i			0.111420i	0.998447	+	0.0557099i	$0.0177422\pi$
$631$			2.79884i			0.111420i	−0.998447	+	0.0557099i	$0.982258\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	32.6256	+	32.6256i	1.29471	+	1.29471i
$636$		0			0
$637$	−1.92893	+	1.92893i	−0.0764271	+	0.0764271i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.85786			0.389362			0.194681	−	0.980867i	$-0.437633\pi$
$641$	9.85786			0.389362			0.194681	+	0.980867i	$0.437633\pi$
$642$		0			0
$643$	−0.951076	+	0.951076i	−0.0375068	+	0.0375068i	−0.725611	−	0.688105i	$-0.758443\pi$
$643$	−0.951076	+	0.951076i	−0.0375068	+	0.0375068i	0.688105	+	0.725611i	$0.258443\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	0.262632i		−	0.0103251i	−0.999987	−	0.00516257i	$-0.998357\pi$
$647$		−	0.262632i		−	0.0103251i	0.999987	−	0.00516257i	$-0.00164331\pi$
$648$		0			0
$649$			12.4853i			0.490090i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−26.5563	+	26.5563i	−1.03923	+	1.03923i	−0.0400318	+	0.999198i	$0.512746\pi$
$653$	−26.5563	+	26.5563i	−1.03923	+	1.03923i	−0.999198	+	0.0400318i	$0.987254\pi$
$654$		0			0
$655$	−31.0949			−1.21498
$656$		0			0
$657$		0			0
$658$		0			0
$659$	23.3868	−	23.3868i	0.911021	−	0.911021i	−0.0853316	−	0.996353i	$-0.527195\pi$
$659$	23.3868	−	23.3868i	0.911021	−	0.911021i	0.996353	+	0.0853316i	$0.0271950\pi$
$660$		0			0
$661$	10.5563	+	10.5563i	0.410594	+	0.410594i	0.881946	−	0.471351i	$-0.156234\pi$
$661$	10.5563	+	10.5563i	0.410594	+	0.410594i	−0.471351	+	0.881946i	$0.656234\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			2.34315i			0.0908633i
$666$		0			0
$667$	−19.0029	−	19.0029i	−0.735794	−	0.735794i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	62.4524			2.41095
$672$		0			0
$673$	−21.1716			−0.816104			−0.408052	−	0.912959i	$-0.633792\pi$
$673$	−21.1716			−0.816104			−0.408052	+	0.912959i	$0.633792\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.4142	+	12.4142i	0.477117	+	0.477117i	0.904208	−	0.427091i	$-0.140462\pi$
$677$	12.4142	+	12.4142i	0.477117	+	0.477117i	−0.427091	+	0.904208i	$0.640462\pi$
$678$		0			0
$679$			16.5754i			0.636107i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14.2024	−	14.2024i	−0.543440	−	0.543440i	0.381095	−	0.924536i	$-0.375547\pi$
$683$	−14.2024	−	14.2024i	−0.543440	−	0.543440i	−0.924536	+	0.381095i	$0.875547\pi$
$684$		0			0
$685$	50.6274	−	50.6274i	1.93437	−	1.93437i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.34315			0.165461
$690$		0			0
$691$	14.0936	−	14.0936i	0.536147	−	0.536147i	−0.386248	−	0.922395i	$-0.626229\pi$
$691$	14.0936	−	14.0936i	0.536147	−	0.536147i	0.922395	+	0.386248i	$0.126229\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	65.5139i		−	2.48508i
$696$		0			0
$697$			11.3137i			0.428537i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20.4142	−	20.4142i	0.771034	−	0.771034i	−0.207253	−	0.978287i	$-0.566452\pi$
$701$	20.4142	−	20.4142i	0.771034	−	0.771034i	0.978287	+	0.207253i	$0.0664524\pi$
$702$		0			0
$703$	2.27357			0.0857493
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.17025	+	3.17025i	−0.119230	+	0.119230i
$708$		0			0
$709$	25.2426	+	25.2426i	0.948007	+	0.948007i	0.998714	−	0.0507063i	$-0.0161473\pi$
$709$	25.2426	+	25.2426i	0.948007	+	0.948007i	−0.0507063	+	0.998714i	$0.516147\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	43.3137i		−	1.62211i
$714$		0			0
$715$	−6.75699	−	6.75699i	−0.252697	−	0.252697i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.2459			−0.456694			−0.228347	−	0.973580i	$-0.573332\pi$
$719$	−12.2459			−0.456694			−0.228347	+	0.973580i	$0.573332\pi$
$720$		0			0
$721$	−4.28427			−0.159555
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.5858	−	21.5858i	−0.801676	−	0.801676i
$726$		0			0
$727$		−	42.8155i		−	1.58794i	−0.607958	−	0.793969i	$-0.708011\pi$
$727$		−	42.8155i		−	1.58794i	0.607958	−	0.793969i	$-0.291989\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.22625	+	5.22625i	0.193300	+	0.193300i
$732$		0			0
$733$	32.8995	−	32.8995i	1.21517	−	1.21517i	0.245867	−	0.969304i	$-0.420927\pi$
$733$	32.8995	−	32.8995i	1.21517	−	1.21517i	0.969304	−	0.245867i	$-0.0790726\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−47.7990			−1.76070
$738$		0			0
$739$	22.2275	−	22.2275i	0.817652	−	0.817652i	−0.168115	−	0.985767i	$-0.553768\pi$
$739$	22.2275	−	22.2275i	0.817652	−	0.817652i	0.985767	+	0.168115i	$0.0537681\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	28.5587i		−	1.04772i	−0.851806	−	0.523858i	$-0.824492\pi$
$743$		−	28.5587i		−	1.04772i	0.851806	−	0.523858i	$-0.175508\pi$
$744$		0			0
$745$		−	34.9706i		−	1.28122i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.8284	−	10.8284i	0.395662	−	0.395662i
$750$		0			0
$751$	−6.12293			−0.223429			−0.111715	−	0.993740i	$-0.535634\pi$
$751$	−6.12293			−0.223429			−0.111715	+	0.993740i	$0.535634\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.5391	+	21.5391i	−0.783887	+	0.783887i
$756$		0			0
$757$	−23.3848	−	23.3848i	−0.849934	−	0.849934i	0.140190	−	0.990125i	$-0.455229\pi$
$757$	−23.3848	−	23.3848i	−0.849934	−	0.849934i	−0.990125	+	0.140190i	$0.955229\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			10.6274i			0.385244i	0.981273	+	0.192622i	$0.0616990\pi$
$761$			10.6274i			0.385244i	−0.981273	+	0.192622i	$0.938301\pi$
$762$		0			0
$763$	−1.15932	−	1.15932i	−0.0419700	−	0.0419700i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.53073			−0.0552716
$768$		0			0
$769$	10.8284			0.390483			0.195242	−	0.980755i	$-0.437451\pi$
$769$	10.8284			0.390483			0.195242	+	0.980755i	$0.437451\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13.9289	−	13.9289i	−0.500989	−	0.500989i	0.410756	−	0.911745i	$-0.365265\pi$
$773$	−13.9289	−	13.9289i	−0.500989	−	0.500989i	−0.911745	+	0.410756i	$0.865265\pi$
$774$		0			0
$775$		−	49.2011i		−	1.76735i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.26810	−	1.26810i	−0.0454344	−	0.0454344i
$780$		0			0
$781$	−40.4853	+	40.4853i	−1.44868	+	1.44868i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−21.3137			−0.760719
$786$		0			0
$787$	34.4734	−	34.4734i	1.22884	−	1.22884i	0.264441	−	0.964402i	$-0.414813\pi$
$787$	34.4734	−	34.4734i	1.22884	−	1.22884i	0.964402	−	0.264441i	$-0.0851875\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			5.59767i			0.199030i
$792$		0			0
$793$			7.65685i			0.271903i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.9289	+	13.9289i	−0.493388	+	0.493388i	−0.909372	−	0.415984i	$-0.863437\pi$
$797$	−13.9289	+	13.9289i	−0.493388	+	0.493388i	0.415984	+	0.909372i	$0.363437\pi$
$798$		0			0
$799$	20.9050			0.739566
$800$		0			0
$801$		0			0
$802$		0			0
$803$	35.4244	−	35.4244i	1.25010	−	1.25010i
$804$		0			0
$805$	−21.6569	−	21.6569i	−0.763304	−	0.763304i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	47.3137i		−	1.66346i	−0.555179	−	0.831731i	$-0.687350\pi$
$809$		−	47.3137i		−	1.66346i	0.555179	−	0.831731i	$-0.312650\pi$
$810$		0			0
$811$	14.4650	+	14.4650i	0.507937	+	0.507937i	0.913893	−	0.405956i	$-0.133061\pi$
$811$	14.4650	+	14.4650i	0.507937	+	0.507937i	−0.405956	+	0.913893i	$0.633061\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	44.6088			1.56258
$816$		0			0
$817$	−1.17157			−0.0409881
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.7574	+	18.7574i	0.654636	+	0.654636i	0.954106	−	0.299470i	$-0.0968097\pi$
$821$	18.7574	+	18.7574i	0.654636	+	0.654636i	−0.299470	+	0.954106i	$0.596810\pi$
$822$		0			0
$823$		−	42.8155i		−	1.49245i	−0.665692	−	0.746227i	$-0.731863\pi$
$823$		−	42.8155i		−	1.49245i	0.665692	−	0.746227i	$-0.268137\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.5880	−	20.5880i	−0.715914	−	0.715914i	0.251852	−	0.967766i	$-0.418961\pi$
$827$	−20.5880	−	20.5880i	−0.715914	−	0.715914i	−0.967766	+	0.251852i	$0.918961\pi$
$828$		0			0
$829$	17.2426	−	17.2426i	0.598862	−	0.598862i	−0.341148	−	0.940010i	$-0.610816\pi$
$829$	17.2426	−	17.2426i	0.598862	−	0.598862i	0.940010	+	0.341148i	$0.110816\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.1716			−0.456368
$834$		0			0
$835$	3.69552	−	3.69552i	0.127889	−	0.127889i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	39.7540i		−	1.37246i	−0.727384	−	0.686231i	$-0.759264\pi$
$839$		−	39.7540i		−	1.37246i	0.727384	−	0.686231i	$-0.240736\pi$
$840$		0			0
$841$			7.97056i			0.274847i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−30.5563	+	30.5563i	−1.05117	+	1.05117i
$846$		0			0
$847$	−18.1062			−0.622135
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−21.0138	+	21.0138i	−0.720344	+	0.720344i
$852$		0			0
$853$	−36.6985	−	36.6985i	−1.25653	−	1.25653i	−0.952738	−	0.303795i	$-0.901746\pi$
$853$	−36.6985	−	36.6985i	−1.25653	−	1.25653i	−0.303795	−	0.952738i	$-0.598254\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			32.2843i			1.10281i	0.834238	+	0.551405i	$0.185908\pi$
$857$			32.2843i			1.10281i	−0.834238	+	0.551405i	$0.814092\pi$
$858$		0			0
$859$	20.3253	+	20.3253i	0.693492	+	0.693492i	0.962999	−	0.269507i	$-0.0868606\pi$
$859$	20.3253	+	20.3253i	0.693492	+	0.693492i	−0.269507	+	0.962999i	$0.586861\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	43.0781			1.46640			0.733198	−	0.680015i	$-0.238027\pi$
$863$	43.0781			1.46640			0.733198	+	0.680015i	$0.238027\pi$
$864$		0			0
$865$	79.2548			2.69475
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20.6863	+	20.6863i	0.701734	+	0.701734i
$870$		0			0
$871$		−	5.86030i		−	0.198569i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.12293	−	6.12293i	−0.206993	−	0.206993i
$876$		0			0
$877$	−13.0416	+	13.0416i	−0.440385	+	0.440385i	−0.892141	−	0.451757i	$-0.850798\pi$
$877$	−13.0416	+	13.0416i	−0.440385	+	0.440385i	0.451757	+	0.892141i	$0.350798\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27.9411			0.941360			0.470680	−	0.882304i	$-0.344009\pi$
$881$	27.9411			0.941360			0.470680	+	0.882304i	$0.344009\pi$
$882$		0			0
$883$	2.11039	−	2.11039i	0.0710203	−	0.0710203i	−0.670704	−	0.741725i	$-0.734008\pi$
$883$	2.11039	−	2.11039i	0.0710203	−	0.0710203i	0.741725	+	0.670704i	$0.234008\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	10.1899i		−	0.342142i	−0.985259	−	0.171071i	$-0.945277\pi$
$887$		−	10.1899i		−	0.342142i	0.985259	−	0.171071i	$-0.0547228\pi$
$888$		0			0
$889$			20.6863i			0.693796i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.34315	+	2.34315i	−0.0784104	+	0.0784104i
$894$		0			0
$895$	5.86030			0.195888
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23.9665	−	23.9665i	0.799327	−	0.799327i
$900$		0			0
$901$	14.8284	+	14.8284i	0.494007	+	0.494007i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	19.6569i		−	0.653416i
$906$		0			0
$907$	−6.70259	−	6.70259i	−0.222556	−	0.222556i	0.587018	−	0.809574i	$-0.300302\pi$
$907$	−6.70259	−	6.70259i	−0.222556	−	0.222556i	−0.809574	+	0.587018i	$0.800302\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.0502			−0.531766			−0.265883	−	0.964005i	$-0.585663\pi$
$911$	−16.0502			−0.531766			−0.265883	+	0.964005i	$0.585663\pi$
$912$		0			0
$913$	16.7696			0.554991
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.85786	−	9.85786i	−0.325535	−	0.325535i
$918$		0			0
$919$		−	35.4244i		−	1.16854i	−0.811558	−	0.584272i	$-0.801380\pi$
$919$		−	35.4244i		−	1.16854i	0.811558	−	0.584272i	$-0.198620\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.96362	−	4.96362i	−0.163380	−	0.163380i
$924$		0			0
$925$	−23.8701	+	23.8701i	−0.784843	+	0.784843i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	37.1716			1.21956			0.609780	−	0.792571i	$-0.291258\pi$
$929$	37.1716			1.21956			0.609780	+	0.792571i	$0.291258\pi$
$930$		0			0
$931$	1.47634	−	1.47634i	0.0483851	−	0.0483851i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	46.1396i		−	1.50893i
$936$		0			0
$937$			7.45584i			0.243572i	0.992556	+	0.121786i	$0.0388621\pi$
$937$			7.45584i			0.243572i	−0.992556	+	0.121786i	$0.961138\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.0711	−	12.0711i	0.393506	−	0.393506i	−0.482429	−	0.875935i	$-0.660245\pi$
$941$	12.0711	−	12.0711i	0.393506	−	0.393506i	0.875935	+	0.482429i	$0.160245\pi$
$942$		0			0
$943$	23.4412			0.763351
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26.1857	−	26.1857i	0.850919	−	0.850919i	−0.139327	−	0.990246i	$-0.544494\pi$
$947$	26.1857	−	26.1857i	0.850919	−	0.850919i	0.990246	+	0.139327i	$0.0444939\pi$
$948$		0			0
$949$	4.34315	+	4.34315i	0.140984	+	0.140984i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	44.9706i		−	1.45674i	−0.685184	−	0.728370i	$-0.740278\pi$
$953$		−	44.9706i		−	1.45674i	0.685184	−	0.728370i	$-0.259722\pi$
$954$		0			0
$955$	32.6256	+	32.6256i	1.05574	+	1.05574i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	32.1003			1.03657
$960$		0			0
$961$	23.6274			0.762175
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−39.7990	−	39.7990i	−1.28117	−	1.28117i
$966$		0			0
$967$		−	4.59220i		−	0.147675i	−0.997270	−	0.0738376i	$-0.976475\pi$
$967$		−	4.59220i		−	0.147675i	0.997270	−	0.0738376i	$-0.0235247\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.06854	−	6.06854i	−0.194749	−	0.194749i	0.602996	−	0.797744i	$-0.293974\pi$
$971$	−6.06854	−	6.06854i	−0.194749	−	0.194749i	−0.797744	+	0.602996i	$0.793974\pi$
$972$		0			0
$973$	20.7696	−	20.7696i	0.665841	−	0.665841i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.1127			0.483498			0.241749	−	0.970339i	$-0.422279\pi$
$977$	15.1127			0.483498			0.241749	+	0.970339i	$0.422279\pi$
$978$		0			0
$979$	−2.79884	+	2.79884i	−0.0894512	+	0.0894512i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2.05600i			0.0655762i	0.999462	+	0.0327881i	$0.0104386\pi$
$983$			2.05600i			0.0655762i	−0.999462	+	0.0327881i	$0.989561\pi$
$984$		0			0
$985$			50.2843i			1.60219i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.8284	−	10.8284i	0.344324	−	0.344324i
$990$		0			0
$991$	23.4412			0.744635			0.372317	−	0.928106i	$-0.378563\pi$
$991$	23.4412			0.744635			0.372317	+	0.928106i	$0.378563\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−54.1647	+	54.1647i	−1.71714	+	1.71714i
$996$		0			0
$997$	5.92893	+	5.92893i	0.187771	+	0.187771i	0.794732	−	0.606961i	$-0.207611\pi$
$997$	5.92893	+	5.92893i	0.187771	+	0.187771i	−0.606961	+	0.794732i	$0.707611\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.bi.3457.4		8
3.2	odd	2		512.2.e.i.385.4	yes	8
4.3	odd	2	inner	4608.2.k.bi.3457.3		8
8.3	odd	2		4608.2.k.bd.3457.1		8
8.5	even	2		4608.2.k.bd.3457.2		8
12.11	even	2		512.2.e.i.385.1	yes	8
16.3	odd	4	inner	4608.2.k.bi.1153.4		8
16.5	even	4		4608.2.k.bd.1153.1		8
16.11	odd	4		4608.2.k.bd.1153.2		8
16.13	even	4	inner	4608.2.k.bi.1153.3		8
24.5	odd	2		512.2.e.j.385.1	yes	8
24.11	even	2		512.2.e.j.385.4	yes	8
32.3	odd	8		9216.2.a.w.1.1		4
32.13	even	8		9216.2.a.bp.1.4		4
32.19	odd	8		9216.2.a.bp.1.3		4
32.29	even	8		9216.2.a.w.1.2		4
48.5	odd	4		512.2.e.j.129.1	yes	8
48.11	even	4		512.2.e.j.129.4	yes	8
48.29	odd	4		512.2.e.i.129.4	yes	8
48.35	even	4		512.2.e.i.129.1	✓	8
96.5	odd	8		1024.2.b.g.513.7		8
96.11	even	8		1024.2.b.g.513.8		8
96.29	odd	8		1024.2.a.i.1.4		4
96.35	even	8		1024.2.a.i.1.1		4
96.53	odd	8		1024.2.b.g.513.2		8
96.59	even	8		1024.2.b.g.513.1		8
96.77	odd	8		1024.2.a.h.1.1		4
96.83	even	8		1024.2.a.h.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.i.129.1	✓	8	48.35	even	4
512.2.e.i.129.4	yes	8	48.29	odd	4
512.2.e.i.385.1	yes	8	12.11	even	2
512.2.e.i.385.4	yes	8	3.2	odd	2
512.2.e.j.129.1	yes	8	48.5	odd	4
512.2.e.j.129.4	yes	8	48.11	even	4
512.2.e.j.385.1	yes	8	24.5	odd	2
512.2.e.j.385.4	yes	8	24.11	even	2
1024.2.a.h.1.1		4	96.77	odd	8
1024.2.a.h.1.4		4	96.83	even	8
1024.2.a.i.1.1		4	96.35	even	8
1024.2.a.i.1.4		4	96.29	odd	8
1024.2.b.g.513.1		8	96.59	even	8
1024.2.b.g.513.2		8	96.53	odd	8
1024.2.b.g.513.7		8	96.5	odd	8
1024.2.b.g.513.8		8	96.11	even	8
4608.2.k.bd.1153.1		8	16.5	even	4
4608.2.k.bd.1153.2		8	16.11	odd	4
4608.2.k.bd.3457.1		8	8.3	odd	2
4608.2.k.bd.3457.2		8	8.5	even	2
4608.2.k.bi.1153.3		8	16.13	even	4	inner
4608.2.k.bi.1153.4		8	16.3	odd	4	inner
4608.2.k.bi.3457.3		8	4.3	odd	2	inner
4608.2.k.bi.3457.4		8	1.1	even	1	trivial
9216.2.a.w.1.1		4	32.3	odd	8
9216.2.a.w.1.2		4	32.29	even	8
9216.2.a.bp.1.3		4	32.19	odd	8
9216.2.a.bp.1.4		4	32.13	even	8

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.41421 + 2.41421i) q^{5} +1.53073i q^{7} +O(q^{10})\)	\(q+(2.41421 + 2.41421i) q^{5} +1.53073i q^{7} +(3.37849 + 3.37849i) q^{11} +(-0.414214 + 0.414214i) q^{13} -2.82843 q^{17} +(0.317025 - 0.317025i) q^{19} +5.86030i q^{23} +6.65685i q^{25} +(-3.24264 + 3.24264i) q^{29} -7.39104 q^{31} +(-3.69552 + 3.69552i) q^{35} +(3.58579 + 3.58579i) q^{37} -4.00000i q^{41} +(-1.84776 - 1.84776i) q^{43} -7.39104 q^{47} +4.65685 q^{49} +(-5.24264 - 5.24264i) q^{53} +16.3128i q^{55} +(1.84776 + 1.84776i) q^{59} +(9.24264 - 9.24264i) q^{61} -2.00000 q^{65} +(-7.07401 + 7.07401i) q^{67} +11.9832i q^{71} -10.4853i q^{73} +(-5.17157 + 5.17157i) q^{77} +6.12293 q^{79} +(2.48181 - 2.48181i) q^{83} +(-6.82843 - 6.82843i) q^{85} +0.828427i q^{89} +(-0.634051 - 0.634051i) q^{91} +1.53073 q^{95} +10.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} + 8 q^{13} + 8 q^{29} + 40 q^{37} - 8 q^{49} - 8 q^{53} + 40 q^{61} - 16 q^{65} - 64 q^{77} - 32 q^{85} + 64 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 + 8 * q^13 + 8 * q^29 + 40 * q^37 - 8 * q^49 - 8 * q^53 + 40 * q^61 - 16 * q^65 - 64 * q^77 - 32 * q^85 + 64 * q^97

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