LMFDB - Embedded newform 4608.2.k.bi.1153.1

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			1153.1
Root			$0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	4608.1153
Dual form			4608.2.k.bi.3457.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.414214 + 0.414214i) q^{5} -3.69552i q^{7} +(2.93015 - 2.93015i) q^{11} +(2.41421 + 2.41421i) q^{13} +2.82843 q^{17} +(-4.46088 - 4.46088i) q^{19} +6.75699i q^{23} +4.65685i q^{25} +(5.24264 + 5.24264i) q^{29} +3.06147 q^{31} +(1.53073 + 1.53073i) q^{35} +(6.41421 - 6.41421i) q^{37} +4.00000i q^{41} +(0.765367 - 0.765367i) q^{43} +3.06147 q^{47} -6.65685 q^{49} +(3.24264 - 3.24264i) q^{53} +2.42742i q^{55} +(-0.765367 + 0.765367i) q^{59} +(0.757359 + 0.757359i) q^{61} -2.00000 q^{65} +(-1.39942 - 1.39942i) q^{67} -8.02509i q^{71} -6.48528i q^{73} +(-10.8284 - 10.8284i) q^{77} +14.7821 q^{79} +(-9.68714 - 9.68714i) q^{83} +(-1.17157 + 1.17157i) q^{85} +4.82843i q^{89} +(8.92177 - 8.92177i) q^{91} +3.69552 q^{95} +5.17157 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

Character values

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

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$5$	−0.414214	+	0.414214i	−0.185242	+	0.185242i	−0.793635	−	0.608394i	$-0.791814\pi$
$5$	−0.414214	+	0.414214i	−0.185242	+	0.185242i	0.608394	+	0.793635i	$0.291814\pi$
$6$		0			0
$7$		−	3.69552i		−	1.39677i	−0.715720	−	0.698387i	$-0.753901\pi$
$7$		−	3.69552i		−	1.39677i	0.715720	−	0.698387i	$-0.246099\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.93015	−	2.93015i	0.883474	−	0.883474i	−0.110412	−	0.993886i	$-0.535217\pi$
$11$	2.93015	−	2.93015i	0.883474	−	0.883474i	0.993886	+	0.110412i	$0.0352171\pi$
$12$		0			0
$13$	2.41421	+	2.41421i	0.669582	+	0.669582i	0.957619	−	0.288037i	$-0.0930026\pi$
$13$	2.41421	+	2.41421i	0.669582	+	0.669582i	−0.288037	+	0.957619i	$0.593003\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.82843			0.685994			0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843			0.685994			0.342997	+	0.939336i	$0.388558\pi$
$18$		0			0
$19$	−4.46088	−	4.46088i	−1.02340	−	1.02340i	−0.999720	−	0.0236776i	$-0.992462\pi$
$19$	−4.46088	−	4.46088i	−1.02340	−	1.02340i	−0.0236776	−	0.999720i	$-0.507538\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.75699i			1.40893i	0.709739	+	0.704464i	$0.248813\pi$
$23$			6.75699i			1.40893i	−0.709739	+	0.704464i	$0.751187\pi$
$24$		0			0
$25$			4.65685i			0.931371i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.24264	+	5.24264i	0.973534	+	0.973534i	0.999659	−	0.0261248i	$-0.00831671\pi$
$29$	5.24264	+	5.24264i	0.973534	+	0.973534i	−0.0261248	+	0.999659i	$0.508317\pi$
$30$		0			0
$31$	3.06147			0.549856			0.274928	−	0.961465i	$-0.411346\pi$
$31$	3.06147			0.549856			0.274928	+	0.961465i	$0.411346\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.53073	+	1.53073i	0.258741	+	0.258741i
$36$		0			0
$37$	6.41421	−	6.41421i	1.05449	−	1.05449i	0.0560630	−	0.998427i	$-0.482145\pi$
$37$	6.41421	−	6.41421i	1.05449	−	1.05449i	0.998427	−	0.0560630i	$-0.0178548\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.00000i			0.624695i	0.949968	+	0.312348i	$0.101115\pi$
$41$			4.00000i			0.624695i	−0.949968	+	0.312348i	$0.898885\pi$
$42$		0			0
$43$	0.765367	−	0.765367i	0.116717	−	0.116717i	−0.646336	−	0.763053i	$-0.723699\pi$
$43$	0.765367	−	0.765367i	0.116717	−	0.116717i	0.763053	+	0.646336i	$0.223699\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.06147			0.446561			0.223280	−	0.974754i	$-0.428323\pi$
$47$	3.06147			0.446561			0.223280	+	0.974754i	$0.428323\pi$
$48$		0			0
$49$	−6.65685			−0.950979
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.24264	−	3.24264i	0.445411	−	0.445411i	−0.448415	−	0.893826i	$-0.648011\pi$
$53$	3.24264	−	3.24264i	0.445411	−	0.445411i	0.893826	+	0.448415i	$0.148011\pi$
$54$		0			0
$55$			2.42742i			0.327313i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.765367	+	0.765367i	−0.0996423	+	0.0996423i	−0.755171	−	0.655528i	$-0.772446\pi$
$59$	−0.765367	+	0.765367i	−0.0996423	+	0.0996423i	0.655528	+	0.755171i	$0.272446\pi$
$60$		0			0
$61$	0.757359	+	0.757359i	0.0969699	+	0.0969699i	0.753928	−	0.656958i	$-0.228157\pi$
$61$	0.757359	+	0.757359i	0.0969699	+	0.0969699i	−0.656958	+	0.753928i	$0.728157\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000			−0.248069
$66$		0			0
$67$	−1.39942	−	1.39942i	−0.170966	−	0.170966i	0.616438	−	0.787404i	$-0.288575\pi$
$67$	−1.39942	−	1.39942i	−0.170966	−	0.170966i	−0.787404	+	0.616438i	$0.788575\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.02509i		−	0.952403i	−0.879336	−	0.476201i	$-0.842013\pi$
$71$		−	8.02509i		−	0.952403i	0.879336	−	0.476201i	$-0.157987\pi$
$72$		0			0
$73$		−	6.48528i		−	0.759045i	−0.925183	−	0.379522i	$-0.876088\pi$
$73$		−	6.48528i		−	0.759045i	0.925183	−	0.379522i	$-0.123912\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.8284	−	10.8284i	−1.23401	−	1.23401i
$78$		0			0
$79$	14.7821			1.66311			0.831557	−	0.555440i	$-0.187450\pi$
$79$	14.7821			1.66311			0.831557	+	0.555440i	$0.187450\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.68714	−	9.68714i	−1.06330	−	1.06330i	−0.997856	−	0.0654452i	$-0.979153\pi$
$83$	−9.68714	−	9.68714i	−1.06330	−	1.06330i	−0.0654452	−	0.997856i	$-0.520847\pi$
$84$		0			0
$85$	−1.17157	+	1.17157i	−0.127075	+	0.127075i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			4.82843i			0.511812i	0.966702	+	0.255906i	$0.0823738\pi$
$89$			4.82843i			0.511812i	−0.966702	+	0.255906i	$0.917626\pi$
$90$		0			0
$91$	8.92177	−	8.92177i	0.935256	−	0.935256i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.69552			0.379152
$96$		0			0
$97$	5.17157			0.525094			0.262547	−	0.964919i	$-0.415438\pi$
$97$	5.17157			0.525094			0.262547	+	0.964919i	$0.415438\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.0711	+	12.0711i	−1.20112	+	1.20112i	−0.227289	+	0.973827i	$0.572986\pi$
$101$	−12.0711	+	12.0711i	−1.20112	+	1.20112i	−0.973827	+	0.227289i	$0.927014\pi$
$102$		0			0
$103$			14.1480i			1.39405i	0.717049	+	0.697023i	$0.245492\pi$
$103$			14.1480i			1.39405i	−0.717049	+	0.697023i	$0.754508\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.39942	+	1.39942i	−0.135287	+	0.135287i	−0.771507	−	0.636220i	$-0.780497\pi$
$107$	−1.39942	+	1.39942i	−0.135287	+	0.135287i	0.636220	+	0.771507i	$0.280497\pi$
$108$		0			0
$109$	−9.24264	−	9.24264i	−0.885284	−	0.885284i	0.108781	−	0.994066i	$-0.465305\pi$
$109$	−9.24264	−	9.24264i	−0.885284	−	0.885284i	−0.994066	+	0.108781i	$0.965305\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.65685			−0.720296			−0.360148	−	0.932895i	$-0.617274\pi$
$113$	−7.65685			−0.720296			−0.360148	+	0.932895i	$0.617274\pi$
$114$		0			0
$115$	−2.79884	−	2.79884i	−0.260993	−	0.260993i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	10.4525i		−	0.958179i
$120$		0			0
$121$		−	6.17157i		−	0.561052i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.00000	−	4.00000i	−0.357771	−	0.357771i
$126$		0			0
$127$	11.7206			1.04004			0.520018	−	0.854155i	$-0.325925\pi$
$127$	11.7206			1.04004			0.520018	+	0.854155i	$0.325925\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.3212	−	10.3212i	−0.901766	−	0.901766i	0.0938226	−	0.995589i	$-0.470091\pi$
$131$	−10.3212	−	10.3212i	−0.901766	−	0.901766i	−0.995589	+	0.0938226i	$0.970091\pi$
$132$		0			0
$133$	−16.4853	+	16.4853i	−1.42946	+	1.42946i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	12.9706i		−	1.10815i	−0.832467	−	0.554075i	$-0.813072\pi$
$137$		−	12.9706i		−	1.10815i	0.832467	−	0.554075i	$-0.186928\pi$
$138$		0			0
$139$	14.2793	−	14.2793i	1.21116	−	1.21116i	0.240511	−	0.970646i	$-0.422685\pi$
$139$	14.2793	−	14.2793i	1.21116	−	1.21116i	0.970646	−	0.240511i	$-0.0773151\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.1480			1.18312
$144$		0			0
$145$	−4.34315			−0.360679
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.24264	−	1.24264i	0.101801	−	0.101801i	−0.654372	−	0.756173i	$-0.727067\pi$
$149$	1.24264	−	1.24264i	0.101801	−	0.101801i	0.756173	+	0.654372i	$0.227067\pi$
$150$		0			0
$151$		−	0.634051i		−	0.0515983i	−0.999667	−	0.0257992i	$-0.991787\pi$
$151$		−	0.634051i		−	0.0515983i	0.999667	−	0.0257992i	$-0.00821304\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.26810	+	1.26810i	−0.101856	+	0.101856i
$156$		0			0
$157$	−1.58579	−	1.58579i	−0.126560	−	0.126560i	0.640990	−	0.767549i	$-0.278524\pi$
$157$	−1.58579	−	1.58579i	−0.126560	−	0.126560i	−0.767549	+	0.640990i	$0.778524\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	24.9706			1.96796
$162$		0			0
$163$	−3.82683	−	3.82683i	−0.299741	−	0.299741i	0.541171	−	0.840912i	$-0.317981\pi$
$163$	−3.82683	−	3.82683i	−0.299741	−	0.299741i	−0.840912	+	0.541171i	$0.817981\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.69552i			0.285968i	0.989725	+	0.142984i	$0.0456697\pi$
$167$			3.69552i			0.285968i	−0.989725	+	0.142984i	$0.954330\pi$
$168$		0			0
$169$		−	1.34315i		−	0.103319i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.5858	+	13.5858i	1.03291	+	1.03291i	0.999440	+	0.0334684i	$0.0106553\pi$
$173$	13.5858	+	13.5858i	1.03291	+	1.03291i	0.0334684	+	0.999440i	$0.489345\pi$
$174$		0			0
$175$	17.2095			1.30092
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.15640	+	8.15640i	0.609638	+	0.609638i	0.942851	−	0.333213i	$-0.108133\pi$
$179$	8.15640	+	8.15640i	0.609638	+	0.609638i	−0.333213	+	0.942851i	$0.608133\pi$
$180$		0			0
$181$	10.0711	−	10.0711i	0.748577	−	0.748577i	−0.225635	−	0.974212i	$-0.572446\pi$
$181$	10.0711	−	10.0711i	0.748577	−	0.748577i	0.974212	+	0.225635i	$0.0724458\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5.31371i			0.390672i
$186$		0			0
$187$	8.28772	−	8.28772i	0.606058	−	0.606058i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.7206			0.848073			0.424037	−	0.905645i	$-0.360613\pi$
$191$	11.7206			0.848073			0.424037	+	0.905645i	$0.360613\pi$
$192$		0			0
$193$	0.485281			0.0349313			0.0174657	−	0.999847i	$-0.494440\pi$
$193$	0.485281			0.0349313			0.0174657	+	0.999847i	$0.494440\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.58579	−	7.58579i	0.540465	−	0.540465i	−0.383200	−	0.923665i	$-0.625178\pi$
$197$	7.58579	−	7.58579i	0.540465	−	0.540465i	0.923665	+	0.383200i	$0.125178\pi$
$198$		0			0
$199$		−	12.3547i		−	0.875798i	−0.899024	−	0.437899i	$-0.855723\pi$
$199$		−	12.3547i		−	0.875798i	0.899024	−	0.437899i	$-0.144277\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	19.3743	−	19.3743i	1.35981	−	1.35981i
$204$		0			0
$205$	−1.65685	−	1.65685i	−0.115720	−	0.115720i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−26.1421			−1.80829
$210$		0			0
$211$	−9.42450	−	9.42450i	−0.648810	−	0.648810i	0.303896	−	0.952705i	$-0.401713\pi$
$211$	−9.42450	−	9.42450i	−0.648810	−	0.648810i	−0.952705	+	0.303896i	$0.901713\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.634051i			0.0432419i
$216$		0			0
$217$		−	11.3137i		−	0.768025i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.82843	+	6.82843i	0.459330	+	0.459330i
$222$		0			0
$223$	3.06147			0.205011			0.102506	−	0.994732i	$-0.467314\pi$
$223$	3.06147			0.205011			0.102506	+	0.994732i	$0.467314\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.765367	+	0.765367i	0.0507992	+	0.0507992i	0.732050	−	0.681251i	$-0.238564\pi$
$227$	0.765367	+	0.765367i	0.0507992	+	0.0507992i	−0.681251	+	0.732050i	$0.738564\pi$
$228$		0			0
$229$	2.75736	−	2.75736i	0.182211	−	0.182211i	−0.610107	−	0.792319i	$-0.708874\pi$
$229$	2.75736	−	2.75736i	0.182211	−	0.182211i	0.792319	+	0.610107i	$0.208874\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.8284i		−	1.36452i	−0.731112	−	0.682258i	$-0.760998\pi$
$233$		−	20.8284i		−	1.36452i	0.731112	−	0.682258i	$-0.239002\pi$
$234$		0			0
$235$	−1.26810	+	1.26810i	−0.0827218	+	0.0827218i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.2459			−0.792119			−0.396060	−	0.918225i	$-0.629623\pi$
$239$	−12.2459			−0.792119			−0.396060	+	0.918225i	$0.629623\pi$
$240$		0			0
$241$	13.1716			0.848456			0.424228	−	0.905556i	$-0.360546\pi$
$241$	13.1716			0.848456			0.424228	+	0.905556i	$0.360546\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.75736	−	2.75736i	0.176161	−	0.176161i
$246$		0			0
$247$		−	21.5391i		−	1.37050i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.7486	−	12.7486i	0.804685	−	0.804685i	−0.179139	−	0.983824i	$-0.557331\pi$
$251$	12.7486	−	12.7486i	0.804685	−	0.804685i	0.983824	+	0.179139i	$0.0573312\pi$
$252$		0			0
$253$	19.7990	+	19.7990i	1.24475	+	1.24475i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.9706			0.933838			0.466919	−	0.884300i	$-0.345364\pi$
$257$	14.9706			0.933838			0.466919	+	0.884300i	$0.345364\pi$
$258$		0			0
$259$	−23.7038	−	23.7038i	−1.47289	−	1.47289i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			12.8799i			0.794210i	0.917773	+	0.397105i	$0.129985\pi$
$263$			12.8799i			0.794210i	−0.917773	+	0.397105i	$0.870015\pi$
$264$		0			0
$265$			2.68629i			0.165018i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.0711	−	18.0711i	−1.10181	−	1.10181i	−0.994192	−	0.107620i	$-0.965677\pi$
$269$	−18.0711	−	18.0711i	−1.10181	−	1.10181i	−0.107620	−	0.994192i	$-0.534323\pi$
$270$		0			0
$271$	5.59767			0.340034			0.170017	−	0.985441i	$-0.445618\pi$
$271$	5.59767			0.340034			0.170017	+	0.985441i	$0.445618\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.6453	+	13.6453i	0.822842	+	0.822842i
$276$		0			0
$277$	2.41421	−	2.41421i	0.145056	−	0.145056i	−0.630849	−	0.775905i	$-0.717293\pi$
$277$	2.41421	−	2.41421i	0.145056	−	0.145056i	0.775905	+	0.630849i	$0.217293\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			17.7990i			1.06180i	0.847435	+	0.530899i	$0.178146\pi$
$281$			17.7990i			1.06180i	−0.847435	+	0.530899i	$0.821854\pi$
$282$		0			0
$283$	−10.3212	+	10.3212i	−0.613531	+	0.613531i	−0.943864	−	0.330333i	$-0.892839\pi$
$283$	−10.3212	+	10.3212i	−0.613531	+	0.613531i	0.330333	+	0.943864i	$0.392839\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	14.7821			0.872558
$288$		0			0
$289$	−9.00000			−0.529412
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.414214	+	0.414214i	−0.0241986	+	0.0241986i	−0.719103	−	0.694904i	$-0.755447\pi$
$293$	−0.414214	+	0.414214i	−0.0241986	+	0.0241986i	0.694904	+	0.719103i	$0.255447\pi$
$294$		0			0
$295$		−	0.634051i		−	0.0369159i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.3128	+	16.3128i	−0.943394	+	0.943394i
$300$		0			0
$301$	−2.82843	−	2.82843i	−0.163028	−	0.163028i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.627417			−0.0359258
$306$		0			0
$307$	−3.19278	−	3.19278i	−0.182222	−	0.182222i	0.610101	−	0.792323i	$-0.291129\pi$
$307$	−3.19278	−	3.19278i	−0.182222	−	0.182222i	−0.792323	+	0.610101i	$0.791129\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			24.6005i			1.39497i	0.716600	+	0.697484i	$0.245697\pi$
$311$			24.6005i			1.39497i	−0.716600	+	0.697484i	$0.754303\pi$
$312$		0			0
$313$			31.3137i			1.76996i	0.465633	+	0.884978i	$0.345827\pi$
$313$			31.3137i			1.76996i	−0.465633	+	0.884978i	$0.654173\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.8995	+	10.8995i	0.612177	+	0.612177i	0.943513	−	0.331336i	$-0.107499\pi$
$317$	10.8995	+	10.8995i	0.612177	+	0.612177i	−0.331336	+	0.943513i	$0.607499\pi$
$318$		0			0
$319$	30.7235			1.72018
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.6173	−	12.6173i	−0.702045	−	0.702045i
$324$		0			0
$325$	−11.2426	+	11.2426i	−0.623629	+	0.623629i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	11.3137i		−	0.623745i
$330$		0			0
$331$	−5.99162	+	5.99162i	−0.329329	+	0.329329i	−0.852331	−	0.523002i	$-0.824812\pi$
$331$	−5.99162	+	5.99162i	−0.329329	+	0.329329i	0.523002	+	0.852331i	$0.324812\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.15932			0.0633402
$336$		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
$341$	8.97056	−	8.97056i	0.485783	−	0.485783i
$342$		0			0
$343$		−	1.26810i		−	0.0684710i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.9552	−	10.9552i	0.588108	−	0.588108i	−0.349011	−	0.937119i	$-0.613482\pi$
$347$	10.9552	−	10.9552i	0.588108	−	0.588108i	0.937119	+	0.349011i	$0.113482\pi$
$348$		0			0
$349$	−1.92893	−	1.92893i	−0.103253	−	0.103253i	0.653593	−	0.756846i	$-0.273261\pi$
$349$	−1.92893	−	1.92893i	−0.103253	−	0.103253i	−0.756846	+	0.653593i	$0.773261\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.9706			−1.22260			−0.611300	−	0.791399i	$-0.709353\pi$
$353$	−22.9706			−1.22260			−0.611300	+	0.791399i	$0.709353\pi$
$354$		0			0
$355$	3.32410	+	3.32410i	0.176425	+	0.176425i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			0.634051i			0.0334639i	0.999860	+	0.0167320i	$0.00532620\pi$
$359$			0.634051i			0.0334639i	−0.999860	+	0.0167320i	$0.994674\pi$
$360$		0			0
$361$			20.7990i			1.09468i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.68629	+	2.68629i	0.140607	+	0.140607i
$366$		0			0
$367$	9.18440			0.479422			0.239711	−	0.970844i	$-0.422947\pi$
$367$	9.18440			0.479422			0.239711	+	0.970844i	$0.422947\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.9832	−	11.9832i	−0.622139	−	0.622139i
$372$		0			0
$373$	4.75736	−	4.75736i	0.246327	−	0.246327i	−0.573135	−	0.819461i	$-0.694273\pi$
$373$	4.75736	−	4.75736i	0.246327	−	0.246327i	0.819461	+	0.573135i	$0.194273\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			25.3137i			1.30372i
$378$		0			0
$379$	18.6089	−	18.6089i	0.955875	−	0.955875i	−0.0431915	−	0.999067i	$-0.513753\pi$
$379$	18.6089	−	18.6089i	0.955875	−	0.955875i	0.999067	+	0.0431915i	$0.0137526\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	20.9050			1.06820			0.534098	−	0.845423i	$-0.320651\pi$
$383$	20.9050			1.06820			0.534098	+	0.845423i	$0.320651\pi$
$384$		0			0
$385$	8.97056			0.457182
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.4142	+	14.4142i	−0.730830	+	0.730830i	−0.970784	−	0.239954i	$-0.922867\pi$
$389$	−14.4142	+	14.4142i	−0.730830	+	0.730830i	0.239954	+	0.970784i	$0.422867\pi$
$390$		0			0
$391$			19.1116i			0.966517i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.12293	+	6.12293i	−0.308078	+	0.308078i
$396$		0			0
$397$	−3.58579	−	3.58579i	−0.179965	−	0.179965i	0.611375	−	0.791341i	$-0.290617\pi$
$397$	−3.58579	−	3.58579i	−0.179965	−	0.179965i	−0.791341	+	0.611375i	$0.790617\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.51472			0.375267			0.187634	−	0.982239i	$-0.439918\pi$
$401$	7.51472			0.375267			0.187634	+	0.982239i	$0.439918\pi$
$402$		0			0
$403$	7.39104	+	7.39104i	0.368174	+	0.368174i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	37.5892i		−	1.86323i
$408$		0			0
$409$			31.3137i			1.54836i	0.632964	+	0.774182i	$0.281838\pi$
$409$			31.3137i			1.54836i	−0.632964	+	0.774182i	$0.718162\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.82843	+	2.82843i	0.139178	+	0.139178i
$414$		0			0
$415$	8.02509			0.393936
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.79045	+	8.79045i	0.429442	+	0.429442i	0.888438	−	0.458996i	$-0.151791\pi$
$419$	8.79045	+	8.79045i	0.429442	+	0.429442i	−0.458996	+	0.888438i	$0.651791\pi$
$420$		0			0
$421$	6.07107	−	6.07107i	0.295886	−	0.295886i	−0.543514	−	0.839400i	$-0.682907\pi$
$421$	6.07107	−	6.07107i	0.295886	−	0.295886i	0.839400	+	0.543514i	$0.182907\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			13.1716i			0.638915i
$426$		0			0
$427$	2.79884	−	2.79884i	0.135445	−	0.135445i
$428$		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	$-0.709500\pi$
$433$	−25.4558			−1.22333			−0.611665	+	0.791117i	$0.709500\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	30.1421	−	30.1421i	1.44189	−	1.44189i
$438$		0			0
$439$			38.1145i			1.81911i	0.415588	+	0.909553i	$0.363576\pi$
$439$			38.1145i			1.81911i	−0.415588	+	0.909553i	$0.636424\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.39942	+	1.39942i	−0.0664883	+	0.0664883i	−0.739569	−	0.673081i	$-0.764971\pi$
$443$	−1.39942	+	1.39942i	−0.0664883	+	0.0664883i	0.673081	+	0.739569i	$0.264971\pi$
$444$		0			0
$445$	−2.00000	−	2.00000i	−0.0948091	−	0.0948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.14214			0.289865			0.144933	−	0.989442i	$-0.453703\pi$
$449$	6.14214			0.289865			0.144933	+	0.989442i	$0.453703\pi$
$450$		0			0
$451$	11.7206	+	11.7206i	0.551902	+	0.551902i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			7.39104i			0.346497i
$456$		0			0
$457$		−	26.6274i		−	1.24558i	−0.782390	−	0.622789i	$-0.785999\pi$
$457$		−	26.6274i		−	1.24558i	0.782390	−	0.622789i	$-0.214001\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−27.0416	−	27.0416i	−1.25945	−	1.25945i	−0.951354	−	0.308100i	$-0.900307\pi$
$461$	−27.0416	−	27.0416i	−1.25945	−	1.25945i	−0.308100	−	0.951354i	$-0.599693\pi$
$462$		0			0
$463$	−35.6871			−1.65852			−0.829260	−	0.558864i	$-0.811238\pi$
$463$	−35.6871			−1.65852			−0.829260	+	0.558864i	$0.811238\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−26.8966	−	26.8966i	−1.24463	−	1.24463i	−0.958060	−	0.286567i	$-0.907486\pi$
$467$	−26.8966	−	26.8966i	−1.24463	−	1.24463i	−0.286567	−	0.958060i	$-0.592514\pi$
$468$		0			0
$469$	−5.17157	+	5.17157i	−0.238801	+	0.238801i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4.48528i		−	0.206233i
$474$		0			0
$475$	20.7737	−	20.7737i	0.953162	−	0.953162i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−35.1618			−1.60658			−0.803292	−	0.595585i	$-0.796920\pi$
$479$	−35.1618			−1.60658			−0.803292	+	0.595585i	$0.796920\pi$
$480$		0			0
$481$	30.9706			1.41214
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.14214	+	2.14214i	−0.0972694	+	0.0972694i
$486$		0			0
$487$		−	6.23172i		−	0.282386i	−0.989982	−	0.141193i	$-0.954906\pi$
$487$		−	6.23172i		−	0.282386i	0.989982	−	0.141193i	$-0.0450938\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.56420	−	3.56420i	0.160850	−	0.160850i	−0.622093	−	0.782943i	$-0.713717\pi$
$491$	3.56420	−	3.56420i	0.160850	−	0.160850i	0.782943	+	0.622093i	$0.213717\pi$
$492$		0			0
$493$	14.8284	+	14.8284i	0.667839	+	0.667839i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−29.6569			−1.33029
$498$		0			0
$499$	−12.4860	−	12.4860i	−0.558949	−	0.558949i	0.370059	−	0.929008i	$-0.379337\pi$
$499$	−12.4860	−	12.4860i	−0.558949	−	0.558949i	−0.929008	+	0.370059i	$0.879337\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	22.8072i		−	1.01692i	−0.861085	−	0.508460i	$-0.830215\pi$
$503$		−	22.8072i		−	1.01692i	0.861085	−	0.508460i	$-0.169785\pi$
$504$		0			0
$505$		−	10.0000i		−	0.444994i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.5563	+	18.5563i	0.822496	+	0.822496i	0.986465	−	0.163970i	$-0.0524299\pi$
$509$	18.5563	+	18.5563i	0.822496	+	0.822496i	−0.163970	+	0.986465i	$0.552430\pi$
$510$		0			0
$511$	−23.9665			−1.06021
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.86030	−	5.86030i	−0.258236	−	0.258236i
$516$		0			0
$517$	8.97056	−	8.97056i	0.394525	−	0.394525i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	6.34315i		−	0.277898i	−0.990300	−	0.138949i	$-0.955628\pi$
$521$		−	6.34315i		−	0.277898i	0.990300	−	0.138949i	$-0.0443725\pi$
$522$		0			0
$523$	10.0586	−	10.0586i	0.439830	−	0.439830i	−0.452125	−	0.891955i	$-0.649334\pi$
$523$	10.0586	−	10.0586i	0.439830	−	0.439830i	0.891955	+	0.452125i	$0.149334\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.65914			0.377198
$528$		0			0
$529$	−22.6569			−0.985081
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.65685	+	9.65685i	−0.418285	+	0.418285i
$534$		0			0
$535$		−	1.15932i		−	0.0501216i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−19.5056	+	19.5056i	−0.840165	+	0.840165i
$540$		0			0
$541$	−7.24264	−	7.24264i	−0.311385	−	0.311385i	0.534061	−	0.845446i	$-0.320665\pi$
$541$	−7.24264	−	7.24264i	−0.311385	−	0.311385i	−0.845446	+	0.534061i	$0.820665\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.65685			0.327984
$546$		0			0
$547$	13.3827	+	13.3827i	0.572201	+	0.572201i	0.932743	−	0.360542i	$-0.117408\pi$
$547$	13.3827	+	13.3827i	0.572201	+	0.572201i	−0.360542	+	0.932743i	$0.617408\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	46.7736i		−	1.99262i
$552$		0			0
$553$		−	54.6274i		−	2.32299i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.07107	−	8.07107i	−0.341982	−	0.341982i	0.515130	−	0.857112i	$-0.327744\pi$
$557$	−8.07107	−	8.07107i	−0.341982	−	0.341982i	−0.857112	+	0.515130i	$0.827744\pi$
$558$		0			0
$559$	3.69552			0.156304
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.5838	+	10.5838i	0.446055	+	0.446055i	0.894041	−	0.447986i	$-0.147859\pi$
$563$	10.5838	+	10.5838i	0.446055	+	0.446055i	−0.447986	+	0.894041i	$0.647859\pi$
$564$		0			0
$565$	3.17157	−	3.17157i	0.133429	−	0.133429i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	25.6569i		−	1.07559i	−0.843075	−	0.537796i	$-0.819257\pi$
$569$		−	25.6569i		−	1.07559i	0.843075	−	0.537796i	$-0.180743\pi$
$570$		0			0
$571$	−19.5056	+	19.5056i	−0.816284	+	0.816284i	−0.985567	−	0.169284i	$-0.945855\pi$
$571$	−19.5056	+	19.5056i	−0.816284	+	0.816284i	0.169284	+	0.985567i	$0.445855\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−31.4663			−1.31224
$576$		0			0
$577$	17.3137			0.720779			0.360390	−	0.932802i	$-0.382644\pi$
$577$	17.3137			0.720779			0.360390	+	0.932802i	$0.382644\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−35.7990	+	35.7990i	−1.48519	+	1.48519i
$582$		0			0
$583$		−	19.0029i		−	0.787018i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.1200	+	13.1200i	−0.541521	+	0.541521i	−0.923975	−	0.382453i	$-0.875079\pi$
$587$	−13.1200	+	13.1200i	−0.541521	+	0.541521i	0.382453	+	0.923975i	$0.375079\pi$
$588$		0			0
$589$	−13.6569	−	13.6569i	−0.562721	−	0.562721i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.31371			0.382468			0.191234	−	0.981544i	$-0.438751\pi$
$593$	9.31371			0.382468			0.191234	+	0.981544i	$0.438751\pi$
$594$		0			0
$595$	4.32957	+	4.32957i	0.177495	+	0.177495i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			45.5055i			1.85931i	0.368437	+	0.929653i	$0.379893\pi$
$599$			45.5055i			1.85931i	−0.368437	+	0.929653i	$0.620107\pi$
$600$		0			0
$601$		−	20.8284i		−	0.849609i	−0.905285	−	0.424805i	$-0.860343\pi$
$601$		−	20.8284i		−	0.849609i	0.905285	−	0.424805i	$-0.139657\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.55635	+	2.55635i	0.103930	+	0.103930i
$606$		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
$611$	7.39104	+	7.39104i	0.299009	+	0.299009i
$612$		0			0
$613$	−27.8701	+	27.8701i	−1.12566	+	1.12566i	−0.134786	+	0.990875i	$0.543035\pi$
$613$	−27.8701	+	27.8701i	−1.12566	+	1.12566i	−0.990875	+	0.134786i	$0.956965\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	38.4853i		−	1.54936i	−0.632354	−	0.774680i	$-0.717911\pi$
$617$		−	38.4853i		−	1.54936i	0.632354	−	0.774680i	$-0.282089\pi$
$618$		0			0
$619$	−2.93015	+	2.93015i	−0.117773	+	0.117773i	−0.763537	−	0.645764i	$-0.776539\pi$
$619$	−2.93015	+	2.93015i	−0.117773	+	0.117773i	0.645764	+	0.763537i	$0.276539\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17.8435			0.714886
$624$		0			0
$625$	−19.9706			−0.798823
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18.1421	−	18.1421i	0.723374	−	0.723374i
$630$		0			0
$631$			14.1480i			0.563224i	0.959528	+	0.281612i	$0.0908691\pi$
$631$			14.1480i			0.563224i	−0.959528	+	0.281612i	$0.909131\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.85483	+	4.85483i	−0.192658	+	0.192658i
$636$		0			0
$637$	−16.0711	−	16.0711i	−0.636759	−	0.636759i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	38.1421			1.50652			0.753262	−	0.657721i	$-0.228479\pi$
$641$	38.1421			1.50652			0.753262	+	0.657721i	$0.228479\pi$
$642$		0			0
$643$	13.3827	+	13.3827i	0.527760	+	0.527760i	0.919904	−	0.392144i	$-0.128266\pi$
$643$	13.3827	+	13.3827i	0.527760	+	0.527760i	−0.392144	+	0.919904i	$0.628266\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			21.5391i			0.846788i	0.905946	+	0.423394i	$0.139161\pi$
$647$			21.5391i			0.846788i	−0.905946	+	0.423394i	$0.860839\pi$
$648$		0			0
$649$			4.48528i			0.176063i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.55635	+	4.55635i	0.178304	+	0.178304i	0.790616	−	0.612312i	$-0.209760\pi$
$653$	4.55635	+	4.55635i	0.178304	+	0.178304i	−0.612312	+	0.790616i	$0.709760\pi$
$654$		0			0
$655$	8.55035			0.334090
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.02800	−	1.02800i	−0.0400452	−	0.0400452i	0.686801	−	0.726846i	$-0.259015\pi$
$659$	−1.02800	−	1.02800i	−0.0400452	−	0.0400452i	−0.726846	+	0.686801i	$0.759015\pi$
$660$		0			0
$661$	−20.5563	+	20.5563i	−0.799549	+	0.799549i	−0.983024	−	0.183475i	$-0.941265\pi$
$661$	−20.5563	+	20.5563i	−0.799549	+	0.799549i	0.183475	+	0.983024i	$0.441265\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	13.6569i		−	0.529590i
$666$		0			0
$667$	−35.4244	+	35.4244i	−1.37164	+	1.37164i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.43835			0.171341
$672$		0			0
$673$	−26.8284			−1.03416			−0.517080	−	0.855937i	$-0.672981\pi$
$673$	−26.8284			−1.03416			−0.517080	+	0.855937i	$0.672981\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.58579	−	9.58579i	0.368412	−	0.368412i	−0.498486	−	0.866898i	$-0.666110\pi$
$677$	9.58579	−	9.58579i	0.368412	−	0.368412i	0.866898	+	0.498486i	$0.166110\pi$
$678$		0			0
$679$		−	19.1116i		−	0.733437i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	23.2011	−	23.2011i	0.887766	−	0.887766i	−0.106542	−	0.994308i	$-0.533978\pi$
$683$	23.2011	−	23.2011i	0.887766	−	0.887766i	0.994308	+	0.106542i	$0.0339780\pi$
$684$		0			0
$685$	5.37258	+	5.37258i	0.205276	+	0.205276i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.6569			0.596479
$690$		0			0
$691$	28.7988	+	28.7988i	1.09556	+	1.09556i	0.994924	+	0.100634i	$0.0320870\pi$
$691$	28.7988	+	28.7988i	1.09556	+	1.09556i	0.100634	+	0.994924i	$0.467913\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			11.8294i			0.448714i
$696$		0			0
$697$			11.3137i			0.428537i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.5858	+	17.5858i	0.664206	+	0.664206i	0.956369	−	0.292163i	$-0.0943749\pi$
$701$	17.5858	+	17.5858i	0.664206	+	0.664206i	−0.292163	+	0.956369i	$0.594375\pi$
$702$		0			0
$703$	−57.2261			−2.15832
$704$		0			0
$705$		0			0
$706$		0			0
$707$	44.6088	+	44.6088i	1.67769	+	1.67769i
$708$		0			0
$709$	16.7574	−	16.7574i	0.629336	−	0.629336i	−0.318565	−	0.947901i	$-0.603201\pi$
$709$	16.7574	−	16.7574i	0.629336	−	0.629336i	0.947901	+	0.318565i	$0.103201\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			20.6863i			0.774708i
$714$		0			0
$715$	−5.86030	+	5.86030i	−0.219163	+	0.219163i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29.5641			−1.10256			−0.551278	−	0.834321i	$-0.685860\pi$
$719$	−29.5641			−1.10256			−0.551278	+	0.834321i	$0.685860\pi$
$720$		0			0
$721$	52.2843			1.94717
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−24.4142	+	24.4142i	−0.906721	+	0.906721i
$726$		0			0
$727$		−	22.0643i		−	0.818320i	−0.912463	−	0.409160i	$-0.865822\pi$
$727$		−	22.0643i		−	0.818320i	0.912463	−	0.409160i	$-0.134178\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.16478	−	2.16478i	0.0800674	−	0.0800674i
$732$		0			0
$733$	13.1005	+	13.1005i	0.483878	+	0.483878i	0.906368	−	0.422490i	$-0.138844\pi$
$733$	13.1005	+	13.1005i	0.483878	+	0.483878i	−0.422490	+	0.906368i	$0.638844\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.20101			−0.302088
$738$		0			0
$739$	−35.1843	−	35.1843i	−1.29428	−	1.29428i	−0.932115	−	0.362162i	$-0.882039\pi$
$739$	−35.1843	−	35.1843i	−1.29428	−	1.29428i	−0.362162	−	0.932115i	$-0.617961\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			27.1367i			0.995550i	0.867306	+	0.497775i	$0.165849\pi$
$743$			27.1367i			0.995550i	−0.867306	+	0.497775i	$0.834151\pi$
$744$		0			0
$745$			1.02944i			0.0377157i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.17157	+	5.17157i	0.188965	+	0.188965i
$750$		0			0
$751$	−14.7821			−0.539405			−0.269703	−	0.962944i	$-0.586925\pi$
$751$	−14.7821			−0.539405			−0.269703	+	0.962944i	$0.586925\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.262632	+	0.262632i	0.00955817	+	0.00955817i
$756$		0			0
$757$	13.3848	−	13.3848i	0.486478	−	0.486478i	−0.420715	−	0.907193i	$-0.638221\pi$
$757$	13.3848	−	13.3848i	0.486478	−	0.486478i	0.907193	+	0.420715i	$0.138221\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			34.6274i			1.25524i	0.778519	+	0.627621i	$0.215971\pi$
$761$			34.6274i			1.25524i	−0.778519	+	0.627621i	$0.784029\pi$
$762$		0			0
$763$	−34.1563	+	34.1563i	−1.23654	+	1.23654i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.69552			−0.133437
$768$		0			0
$769$	5.17157			0.186492			0.0932458	−	0.995643i	$-0.470276\pi$
$769$	5.17157			0.186492			0.0932458	+	0.995643i	$0.470276\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−28.0711	+	28.0711i	−1.00965	+	1.00965i	−0.00969311	+	0.999953i	$0.503085\pi$
$773$	−28.0711	+	28.0711i	−1.00965	+	1.00965i	−0.999953	+	0.00969311i	$0.996915\pi$
$774$		0			0
$775$			14.2568i			0.512120i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17.8435	−	17.8435i	0.639311	−	0.639311i
$780$		0			0
$781$	−23.5147	−	23.5147i	−0.841423	−	0.841423i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.31371			0.0468883
$786$		0			0
$787$	−5.62020	−	5.62020i	−0.200339	−	0.200339i	0.599807	−	0.800145i	$-0.295244\pi$
$787$	−5.62020	−	5.62020i	−0.200339	−	0.200339i	−0.800145	+	0.599807i	$0.795244\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			28.2960i			1.00609i
$792$		0			0
$793$			3.65685i			0.129859i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.0711	−	28.0711i	−0.994328	−	0.994328i	0.00565577	−	0.999984i	$-0.498200\pi$
$797$	−28.0711	−	28.0711i	−0.994328	−	0.994328i	−0.999984	+	0.00565577i	$0.998200\pi$
$798$		0			0
$799$	8.65914			0.306338
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.0029	−	19.0029i	−0.670596	−	0.670596i
$804$		0			0
$805$	−10.3431	+	10.3431i	−0.364548	+	0.364548i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			24.6863i			0.867924i	0.900931	+	0.433962i	$0.142885\pi$
$809$			24.6863i			0.867924i	−0.900931	+	0.433962i	$0.857115\pi$
$810$		0			0
$811$	−1.66205	+	1.66205i	−0.0583625	+	0.0583625i	−0.735686	−	0.677323i	$-0.763140\pi$
$811$	−1.66205	+	1.66205i	−0.0583625	+	0.0583625i	0.677323	+	0.735686i	$0.263140\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.17025			0.111049
$816$		0			0
$817$	−6.82843			−0.238896
$818$		0			0
$819$		0			0
$820$		0			0
$821$	27.2426	−	27.2426i	0.950775	−	0.950775i	−0.0480693	−	0.998844i	$-0.515307\pi$
$821$	27.2426	−	27.2426i	0.950775	−	0.950775i	0.998844	+	0.0480693i	$0.0153068\pi$
$822$		0			0
$823$		−	22.0643i		−	0.769114i	−0.923101	−	0.384557i	$-0.874354\pi$
$823$		−	22.0643i		−	0.769114i	0.923101	−	0.384557i	$-0.125646\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13.1200	+	13.1200i	−0.456228	+	0.456228i	−0.897415	−	0.441187i	$-0.854557\pi$
$827$	−13.1200	+	13.1200i	−0.456228	+	0.456228i	0.441187	+	0.897415i	$0.354557\pi$
$828$		0			0
$829$	8.75736	+	8.75736i	0.304156	+	0.304156i	0.842637	−	0.538482i	$-0.181002\pi$
$829$	8.75736	+	8.75736i	0.304156	+	0.304156i	−0.538482	+	0.842637i	$0.681002\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.8284			−0.652366
$834$		0			0
$835$	−1.53073	−	1.53073i	−0.0529732	−	0.0529732i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	29.4554i		−	1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$839$		−	29.4554i		−	1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$840$		0			0
$841$			25.9706i			0.895537i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.556349	+	0.556349i	0.0191390	+	0.0191390i
$846$		0			0
$847$	−22.8072			−0.783663
$848$		0			0
$849$		0			0
$850$		0			0
$851$	43.3407	+	43.3407i	1.48570	+	1.48570i
$852$		0			0
$853$	22.6985	−	22.6985i	0.777181	−	0.777181i	−0.202169	−	0.979351i	$-0.564799\pi$
$853$	22.6985	−	22.6985i	0.777181	−	0.777181i	0.979351	+	0.202169i	$0.0647991\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			24.2843i			0.829535i	0.909927	+	0.414767i	$0.136137\pi$
$857$			24.2843i			0.829535i	−0.909927	+	0.414767i	$0.863863\pi$
$858$		0			0
$859$	−8.41904	+	8.41904i	−0.287254	+	0.287254i	−0.835993	−	0.548740i	$-0.815108\pi$
$859$	−8.41904	+	8.41904i	−0.287254	+	0.287254i	0.548740	+	0.835993i	$0.315108\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.525265			−0.0178802			−0.00894011	−	0.999960i	$-0.502846\pi$
$863$	−0.525265			−0.0178802			−0.00894011	+	0.999960i	$0.502846\pi$
$864$		0			0
$865$	−11.2548			−0.382676
$866$		0			0
$867$		0			0
$868$		0			0
$869$	43.3137	−	43.3137i	1.46932	−	1.46932i
$870$		0			0
$871$		−	6.75699i		−	0.228952i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.7821	+	14.7821i	−0.499725	+	0.499725i
$876$		0			0
$877$	35.0416	+	35.0416i	1.18327	+	1.18327i	0.978892	+	0.204380i	$0.0655179\pi$
$877$	35.0416	+	35.0416i	1.18327	+	1.18327i	0.204380	+	0.978892i	$0.434482\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39.9411			−1.34565			−0.672825	−	0.739801i	$-0.734919\pi$
$881$	−39.9411			−1.34565			−0.672825	+	0.739801i	$0.734919\pi$
$882$		0			0
$883$	20.7737	+	20.7737i	0.699090	+	0.699090i	0.964214	−	0.265124i	$-0.0854129\pi$
$883$	20.7737	+	20.7737i	0.699090	+	0.699090i	−0.265124	+	0.964214i	$0.585413\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	17.2095i		−	0.577838i	−0.957354	−	0.288919i	$-0.906704\pi$
$887$		−	17.2095i		−	0.577838i	0.957354	−	0.288919i	$-0.0932958\pi$
$888$		0			0
$889$		−	43.3137i		−	1.45270i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.6569	−	13.6569i	−0.457009	−	0.457009i
$894$		0			0
$895$	−6.75699			−0.225861
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.0502	+	16.0502i	0.535303	+	0.535303i
$900$		0			0
$901$	9.17157	−	9.17157i	0.305549	−	0.305549i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			8.34315i			0.277336i
$906$		0			0
$907$	−31.8602	+	31.8602i	−1.05790	+	1.05790i	−0.0596848	+	0.998217i	$0.519010\pi$
$907$	−31.8602	+	31.8602i	−1.05790	+	1.05790i	−0.998217	+	0.0596848i	$0.980990\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	23.9665			0.794045			0.397022	−	0.917809i	$-0.370044\pi$
$911$	23.9665			0.794045			0.397022	+	0.917809i	$0.370044\pi$
$912$		0			0
$913$	−56.7696			−1.87880
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−38.1421	+	38.1421i	−1.25956	+	1.25956i
$918$		0			0
$919$		−	19.0029i		−	0.626846i	−0.949614	−	0.313423i	$-0.898524\pi$
$919$		−	19.0029i		−	0.626846i	0.949614	−	0.313423i	$-0.101476\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	19.3743	−	19.3743i	0.637712	−	0.637712i
$924$		0			0
$925$	29.8701	+	29.8701i	0.982121	+	0.982121i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	42.8284			1.40516			0.702578	−	0.711607i	$-0.252032\pi$
$929$	42.8284			1.40516			0.702578	+	0.711607i	$0.252032\pi$
$930$		0			0
$931$	29.6955	+	29.6955i	0.973229	+	0.973229i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			6.86577i			0.224535i
$936$		0			0
$937$			43.4558i			1.41964i	0.704383	+	0.709820i	$0.251224\pi$
$937$			43.4558i			1.41964i	−0.704383	+	0.709820i	$0.748776\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.07107	−	2.07107i	−0.0675149	−	0.0675149i	0.672543	−	0.740058i	$-0.265202\pi$
$941$	−2.07107	−	2.07107i	−0.0675149	−	0.0675149i	−0.740058	+	0.672543i	$0.765202\pi$
$942$		0			0
$943$	−27.0279			−0.880151
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.1760	−	15.1760i	−0.493154	−	0.493154i	0.416144	−	0.909299i	$-0.363381\pi$
$947$	−15.1760	−	15.1760i	−0.493154	−	0.493154i	−0.909299	+	0.416144i	$0.863381\pi$
$948$		0			0
$949$	15.6569	−	15.6569i	0.508243	−	0.508243i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			11.0294i			0.357279i	0.983915	+	0.178639i	$0.0571695\pi$
$953$			11.0294i			0.357279i	−0.983915	+	0.178639i	$0.942830\pi$
$954$		0			0
$955$	−4.85483	+	4.85483i	−0.157099	+	0.157099i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−47.9329			−1.54784
$960$		0			0
$961$	−21.6274			−0.697659
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.201010	+	0.201010i	−0.00647074	+	0.00647074i
$966$		0			0
$967$			11.0866i			0.356520i	0.983983	+	0.178260i	$0.0570467\pi$
$967$			11.0866i			0.356520i	−0.983983	+	0.178260i	$0.942953\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−40.7820	+	40.7820i	−1.30876	+	1.30876i	−0.386445	+	0.922313i	$0.626297\pi$
$971$	−40.7820	+	40.7820i	−1.30876	+	1.30876i	−0.922313	+	0.386445i	$0.873703\pi$
$972$		0			0
$973$	−52.7696	−	52.7696i	−1.69171	−	1.69171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−47.1127			−1.50727			−0.753634	−	0.657294i	$-0.771701\pi$
$977$	−47.1127			−1.50727			−0.753634	+	0.657294i	$0.771701\pi$
$978$		0			0
$979$	14.1480	+	14.1480i	0.452173	+	0.452173i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	46.7736i		−	1.49185i	−0.666031	−	0.745924i	$-0.732008\pi$
$983$		−	46.7736i		−	1.49185i	0.666031	−	0.745924i	$-0.267992\pi$
$984$		0			0
$985$			6.28427i			0.200234i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.17157	+	5.17157i	0.164446	+	0.164446i
$990$		0			0
$991$	−27.0279			−0.858571			−0.429285	−	0.903169i	$-0.641235\pi$
$991$	−27.0279			−0.858571			−0.429285	+	0.903169i	$0.641235\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.11747	+	5.11747i	0.162235	+	0.162235i
$996$		0			0
$997$	20.0711	−	20.0711i	0.635657	−	0.635657i	−0.313824	−	0.949481i	$-0.601610\pi$
$997$	20.0711	−	20.0711i	0.635657	−	0.635657i	0.949481	+	0.313824i	$0.101610\pi$
$998$		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.1153.1		8
3.2	odd	2		512.2.e.i.129.2	✓	8
4.3	odd	2	inner	4608.2.k.bi.1153.2		8
8.3	odd	2		4608.2.k.bd.1153.4		8
8.5	even	2		4608.2.k.bd.1153.3		8
12.11	even	2		512.2.e.i.129.3	yes	8
16.3	odd	4		4608.2.k.bd.3457.3		8
16.5	even	4	inner	4608.2.k.bi.3457.2		8
16.11	odd	4	inner	4608.2.k.bi.3457.1		8
16.13	even	4		4608.2.k.bd.3457.4		8
24.5	odd	2		512.2.e.j.129.3	yes	8
24.11	even	2		512.2.e.j.129.2	yes	8
32.5	even	8		9216.2.a.w.1.4		4
32.11	odd	8		9216.2.a.bp.1.1		4
32.21	even	8		9216.2.a.bp.1.2		4
32.27	odd	8		9216.2.a.w.1.3		4
48.5	odd	4		512.2.e.i.385.2	yes	8
48.11	even	4		512.2.e.i.385.3	yes	8
48.29	odd	4		512.2.e.j.385.3	yes	8
48.35	even	4		512.2.e.j.385.2	yes	8
96.5	odd	8		1024.2.a.i.1.3		4
96.11	even	8		1024.2.a.h.1.3		4
96.29	odd	8		1024.2.b.g.513.4		8
96.35	even	8		1024.2.b.g.513.6		8
96.53	odd	8		1024.2.a.h.1.2		4
96.59	even	8		1024.2.a.i.1.2		4
96.77	odd	8		1024.2.b.g.513.5		8
96.83	even	8		1024.2.b.g.513.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.i.129.2	✓	8	3.2	odd	2
512.2.e.i.129.3	yes	8	12.11	even	2
512.2.e.i.385.2	yes	8	48.5	odd	4
512.2.e.i.385.3	yes	8	48.11	even	4
512.2.e.j.129.2	yes	8	24.11	even	2
512.2.e.j.129.3	yes	8	24.5	odd	2
512.2.e.j.385.2	yes	8	48.35	even	4
512.2.e.j.385.3	yes	8	48.29	odd	4
1024.2.a.h.1.2		4	96.53	odd	8
1024.2.a.h.1.3		4	96.11	even	8
1024.2.a.i.1.2		4	96.59	even	8
1024.2.a.i.1.3		4	96.5	odd	8
1024.2.b.g.513.3		8	96.83	even	8
1024.2.b.g.513.4		8	96.29	odd	8
1024.2.b.g.513.5		8	96.77	odd	8
1024.2.b.g.513.6		8	96.35	even	8
4608.2.k.bd.1153.3		8	8.5	even	2
4608.2.k.bd.1153.4		8	8.3	odd	2
4608.2.k.bd.3457.3		8	16.3	odd	4
4608.2.k.bd.3457.4		8	16.13	even	4
4608.2.k.bi.1153.1		8	1.1	even	1	trivial
4608.2.k.bi.1153.2		8	4.3	odd	2	inner
4608.2.k.bi.3457.1		8	16.11	odd	4	inner
4608.2.k.bi.3457.2		8	16.5	even	4	inner
9216.2.a.w.1.3		4	32.27	odd	8
9216.2.a.w.1.4		4	32.5	even	8
9216.2.a.bp.1.1		4	32.11	odd	8
9216.2.a.bp.1.2		4	32.21	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)

\(f(q)\)	\(=\)	\(q+(-0.414214 + 0.414214i) q^{5} -3.69552i q^{7} +(2.93015 - 2.93015i) q^{11} +(2.41421 + 2.41421i) q^{13} +2.82843 q^{17} +(-4.46088 - 4.46088i) q^{19} +6.75699i q^{23} +4.65685i q^{25} +(5.24264 + 5.24264i) q^{29} +3.06147 q^{31} +(1.53073 + 1.53073i) q^{35} +(6.41421 - 6.41421i) q^{37} +4.00000i q^{41} +(0.765367 - 0.765367i) q^{43} +3.06147 q^{47} -6.65685 q^{49} +(3.24264 - 3.24264i) q^{53} +2.42742i q^{55} +(-0.765367 + 0.765367i) q^{59} +(0.757359 + 0.757359i) q^{61} -2.00000 q^{65} +(-1.39942 - 1.39942i) q^{67} -8.02509i q^{71} -6.48528i q^{73} +(-10.8284 - 10.8284i) q^{77} +14.7821 q^{79} +(-9.68714 - 9.68714i) q^{83} +(-1.17157 + 1.17157i) q^{85} +4.82843i q^{89} +(8.92177 - 8.92177i) q^{91} +3.69552 q^{95} +5.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} + 8 q^{13} + 8 q^{29} + 40 q^{37} - 8 q^{49} - 8 q^{53} + 40 q^{61} - 16 q^{65} - 64 q^{77} - 32 q^{85} + 64 q^{97}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

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