LMFDB - Embedded newform 3072.2.d.j.1537.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$8$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.3
Root			$0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.j.1537.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -0.331821i q^{5} +3.08239 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -0.331821i q^{5} +3.08239 q^{7} -1.00000 q^{9} -3.69552i q^{11} -4.64047i q^{13} -0.331821 q^{15} +6.52395 q^{17} +0.867091i q^{19} -3.08239i q^{21} +4.00000 q^{23} +4.88989 q^{25} +1.00000i q^{27} +4.89443i q^{29} -6.14386 q^{31} -3.69552 q^{33} -1.02280i q^{35} -3.64725i q^{37} -4.64047 q^{39} -3.92856 q^{41} +3.92856i q^{43} +0.331821i q^{45} +1.65685 q^{47} +2.50114 q^{49} -6.52395i q^{51} +0.564862i q^{53} -1.22625 q^{55} +0.867091 q^{57} -6.59539i q^{59} +14.8052i q^{61} -3.08239 q^{63} -1.53981 q^{65} -13.9864i q^{67} -4.00000i q^{69} +7.49207 q^{71} -5.62408 q^{73} -4.88989i q^{75} -11.3910i q^{77} -14.9040 q^{79} +1.00000 q^{81} -9.35237i q^{83} -2.16478i q^{85} +4.89443 q^{87} +18.1094 q^{89} -14.3037i q^{91} +6.14386i q^{93} +0.287719 q^{95} -17.0479 q^{97} +3.69552i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q + 16 * q^7 - 8 * q^9	$ 8 q + 16 q^{7} - 8 q^{9} + 32 q^{23} - 8 q^{25} - 16 q^{31} + 16 q^{39} - 32 q^{47} + 8 q^{49} + 32 q^{55} - 16 q^{63} - 16 q^{65} + 32 q^{71} + 16 q^{73} - 48 q^{79} + 8 q^{81} + 16 q^{89} - 64 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 - 8 * q^9 + 32 * q^23 - 8 * q^25 - 16 * q^31 + 16 * q^39 - 32 * q^47 + 8 * q^49 + 32 * q^55 - 16 * q^63 - 16 * q^65 + 32 * q^71 + 16 * q^73 - 48 * q^79 + 8 * q^81 + 16 * q^89 - 64 * q^95 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 0.331821i	− 0.148395i	−0.997244	−	0.0741975i	$-0.976360\pi$
$5$	− 0.331821i	− 0.148395i	0.997244	−	0.0741975i	$-0.0236395\pi$
$6$	0	0
$7$	3.08239	1.16503	0.582517	−	0.812818i	$-0.302068\pi$
$7$	3.08239	1.16503	0.582517	+	0.812818i	$0.302068\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 3.69552i	− 1.11424i	−0.830432	−	0.557120i	$-0.811906\pi$
$11$	− 3.69552i	− 1.11424i	0.830432	−	0.557120i	$-0.188094\pi$
$12$	0	0
$13$	− 4.64047i	− 1.28703i	−0.765432	−	0.643517i	$-0.777475\pi$
$13$	− 4.64047i	− 1.28703i	0.765432	−	0.643517i	$-0.222525\pi$
$14$	0	0
$15$	−0.331821	−0.0856759
$16$	0	0
$17$	6.52395	1.58229	0.791145	−	0.611629i	$-0.209486\pi$
$17$	6.52395	1.58229	0.791145	+	0.611629i	$0.209486\pi$
$18$	0	0
$19$	0.867091i	0.198924i	0.995041	+	0.0994622i	$0.0317122\pi$
$19$	0.867091i	0.198924i	−0.995041	+	0.0994622i	$0.968288\pi$
$20$	0	0
$21$	− 3.08239i	− 0.672633i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	4.88989	0.977979
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	4.89443i	0.908873i	0.890779	+	0.454436i	$0.150159\pi$
$29$	4.89443i	0.908873i	−0.890779	+	0.454436i	$0.849841\pi$
$30$	0	0
$31$	−6.14386	−1.10347	−0.551735	−	0.834020i	$-0.686034\pi$
$31$	−6.14386	−1.10347	−0.551735	+	0.834020i	$0.686034\pi$
$32$	0	0
$33$	−3.69552	−0.643307
$34$	0	0
$35$	− 1.02280i	− 0.172885i
$36$	0	0
$37$	− 3.64725i	− 0.599605i	−0.954001	−	0.299802i	$-0.903079\pi$
$37$	− 3.64725i	− 0.599605i	0.954001	−	0.299802i	$-0.0969208\pi$
$38$	0	0
$39$	−4.64047	−0.743069
$40$	0	0
$41$	−3.92856	−0.613538	−0.306769	−	0.951784i	$-0.599248\pi$
$41$	−3.92856	−0.613538	−0.306769	+	0.951784i	$0.599248\pi$
$42$	0	0
$43$	3.92856i	0.599100i	0.954081	+	0.299550i	$0.0968365\pi$
$43$	3.92856i	0.599100i	−0.954081	+	0.299550i	$0.903164\pi$
$44$	0	0
$45$	0.331821i	0.0494650i
$46$	0	0
$47$	1.65685	0.241677	0.120839	−	0.992672i	$-0.461442\pi$
$47$	1.65685	0.241677	0.120839	+	0.992672i	$0.461442\pi$
$48$	0	0
$49$	2.50114	0.357306
$50$	0	0
$51$	− 6.52395i	− 0.913535i
$52$	0	0
$53$	0.564862i	0.0775897i	0.999247	+	0.0387949i	$0.0123519\pi$
$53$	0.564862i	0.0775897i	−0.999247	+	0.0387949i	$0.987648\pi$
$54$	0	0
$55$	−1.22625	−0.165348
$56$	0	0
$57$	0.867091	0.114849
$58$	0	0
$59$	− 6.59539i	− 0.858646i	−0.903151	−	0.429323i	$-0.858752\pi$
$59$	− 6.59539i	− 0.858646i	0.903151	−	0.429323i	$-0.141248\pi$
$60$	0	0
$61$	14.8052i	1.89562i	0.318839	+	0.947809i	$0.396707\pi$
$61$	14.8052i	1.89562i	−0.318839	+	0.947809i	$0.603293\pi$
$62$	0	0
$63$	−3.08239	−0.388345
$64$	0	0
$65$	−1.53981	−0.190989
$66$	0	0
$67$	− 13.9864i	− 1.70871i	−0.519687	−	0.854357i	$-0.673951\pi$
$67$	− 13.9864i	− 1.70871i	0.519687	−	0.854357i	$-0.326049\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	0	0
$71$	7.49207	0.889145	0.444573	−	0.895743i	$-0.353356\pi$
$71$	7.49207	0.889145	0.444573	+	0.895743i	$0.353356\pi$
$72$	0	0
$73$	−5.62408	−0.658248	−0.329124	−	0.944287i	$-0.606753\pi$
$73$	−5.62408	−0.658248	−0.329124	+	0.944287i	$0.606753\pi$
$74$	0	0
$75$	− 4.88989i	− 0.564636i
$76$	0	0
$77$	− 11.3910i	− 1.29813i
$78$	0	0
$79$	−14.9040	−1.67683	−0.838417	−	0.545029i	$-0.816519\pi$
$79$	−14.9040	−1.67683	−0.838417	+	0.545029i	$0.816519\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 9.35237i	− 1.02656i	−0.858222	−	0.513278i	$-0.828431\pi$
$83$	− 9.35237i	− 1.02656i	0.858222	−	0.513278i	$-0.171569\pi$
$84$	0	0
$85$	− 2.16478i	− 0.234804i
$86$	0	0
$87$	4.89443	0.524738
$88$	0	0
$89$	18.1094	1.91959	0.959794	−	0.280705i	$-0.0905683\pi$
$89$	18.1094	1.91959	0.959794	+	0.280705i	$0.0905683\pi$
$90$	0	0
$91$	− 14.3037i	− 1.49944i
$92$	0	0
$93$	6.14386i	0.637089i
$94$	0	0
$95$	0.287719	0.0295194
$96$	0	0
$97$	−17.0479	−1.73095	−0.865476	−	0.500951i	$-0.832984\pi$
$97$	−17.0479	−1.73095	−0.865476	+	0.500951i	$0.832984\pi$
$98$	0	0
$99$	3.69552i	0.371414i
$100$	0	0
$101$	12.1535i	1.20931i	0.796486	+	0.604657i	$0.206690\pi$
$101$	12.1535i	1.20931i	−0.796486	+	0.604657i	$0.793310\pi$
$102$	0	0
$103$	17.1043	1.68534	0.842668	−	0.538433i	$-0.180984\pi$
$103$	17.1043	1.68534	0.842668	+	0.538433i	$0.180984\pi$
$104$	0	0
$105$	−1.02280	−0.0998154
$106$	0	0
$107$	− 18.3752i	− 1.77640i	−0.459462	−	0.888198i	$-0.651958\pi$
$107$	− 18.3752i	− 1.77640i	0.459462	−	0.888198i	$-0.348042\pi$
$108$	0	0
$109$	7.84482i	0.751397i	0.926742	+	0.375699i	$0.122597\pi$
$109$	7.84482i	0.751397i	−0.926742	+	0.375699i	$0.877403\pi$
$110$	0	0
$111$	−3.64725	−0.346182
$112$	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
$115$	− 1.32729i	− 0.123770i
$116$	0	0
$117$	4.64047i	0.429011i
$118$	0	0
$119$	20.1094	1.84342
$120$	0	0
$121$	−2.65685	−0.241532
$122$	0	0
$123$	3.92856i	0.354226i
$124$	0	0
$125$	− 3.28168i	− 0.293522i
$126$	0	0
$127$	0.638213	0.0566322	0.0283161	−	0.999599i	$-0.490985\pi$
$127$	0.638213	0.0566322	0.0283161	+	0.999599i	$0.490985\pi$
$128$	0	0
$129$	3.92856	0.345890
$130$	0	0
$131$	9.51397i	0.831240i	0.909538	+	0.415620i	$0.136435\pi$
$131$	9.51397i	0.831240i	−0.909538	+	0.415620i	$0.863565\pi$
$132$	0	0
$133$	2.67271i	0.231754i
$134$	0	0
$135$	0.331821	0.0285586
$136$	0	0
$137$	13.9150	1.18884	0.594419	−	0.804156i	$-0.297382\pi$
$137$	13.9150	1.18884	0.594419	+	0.804156i	$0.297382\pi$
$138$	0	0
$139$	0.0773278i	0.00655886i	0.999995	+	0.00327943i	$0.00104388\pi$
$139$	0.0773278i	0.00655886i	−0.999995	+	0.00327943i	$0.998956\pi$
$140$	0	0
$141$	− 1.65685i	− 0.139532i
$142$	0	0
$143$	−17.1489	−1.43407
$144$	0	0
$145$	1.62408	0.134872
$146$	0	0
$147$	− 2.50114i	− 0.206291i
$148$	0	0
$149$	13.9887i	1.14600i	0.819556	+	0.572998i	$0.194220\pi$
$149$	13.9887i	1.14600i	−0.819556	+	0.572998i	$0.805780\pi$
$150$	0	0
$151$	−18.5828	−1.51225	−0.756123	−	0.654430i	$-0.772909\pi$
$151$	−18.5828	−1.51225	−0.756123	+	0.654430i	$0.772909\pi$
$152$	0	0
$153$	−6.52395	−0.527430
$154$	0	0
$155$	2.03866i	0.163749i
$156$	0	0
$157$	− 5.30411i	− 0.423314i	−0.977344	−	0.211657i	$-0.932114\pi$
$157$	− 5.30411i	− 0.423314i	0.977344	−	0.211657i	$-0.0678860\pi$
$158$	0	0
$159$	0.564862	0.0447964
$160$	0	0
$161$	12.3296	0.971706
$162$	0	0
$163$	− 13.3060i	− 1.04221i	−0.853493	−	0.521104i	$-0.825520\pi$
$163$	− 13.3060i	− 1.04221i	0.853493	−	0.521104i	$-0.174480\pi$
$164$	0	0
$165$	1.22625i	0.0954636i
$166$	0	0
$167$	−15.9310	−1.23278	−0.616389	−	0.787442i	$-0.711405\pi$
$167$	−15.9310	−1.23278	−0.616389	+	0.787442i	$0.711405\pi$
$168$	0	0
$169$	−8.53392	−0.656455
$170$	0	0
$171$	− 0.867091i	− 0.0663081i
$172$	0	0
$173$	− 19.7229i	− 1.49950i	−0.661721	−	0.749751i	$-0.730173\pi$
$173$	− 19.7229i	− 1.49950i	0.661721	−	0.749751i	$-0.269827\pi$
$174$	0	0
$175$	15.0726	1.13938
$176$	0	0
$177$	−6.59539	−0.495740
$178$	0	0
$179$	− 5.98642i	− 0.447446i	−0.974653	−	0.223723i	$-0.928179\pi$
$179$	− 5.98642i	− 0.447446i	0.974653	−	0.223723i	$-0.0718211\pi$
$180$	0	0
$181$	− 9.81204i	− 0.729323i	−0.931140	−	0.364662i	$-0.881185\pi$
$181$	− 9.81204i	− 0.729323i	0.931140	−	0.364662i	$-0.118815\pi$
$182$	0	0
$183$	14.8052	1.09444
$184$	0	0
$185$	−1.21024	−0.0889784
$186$	0	0
$187$	− 24.1094i	− 1.76305i
$188$	0	0
$189$	3.08239i	0.224211i
$190$	0	0
$191$	−19.2900	−1.39578	−0.697888	−	0.716207i	$-0.745877\pi$
$191$	−19.2900	−1.39578	−0.697888	+	0.716207i	$0.745877\pi$
$192$	0	0
$193$	−0.155713	−0.0112084	−0.00560422	−	0.999984i	$-0.501784\pi$
$193$	−0.155713	−0.0112084	−0.00560422	+	0.999984i	$0.501784\pi$
$194$	0	0
$195$	1.53981i	0.110268i
$196$	0	0
$197$	− 13.2014i	− 0.940557i	−0.882518	−	0.470279i	$-0.844153\pi$
$197$	− 13.2014i	− 0.940557i	0.882518	−	0.470279i	$-0.155847\pi$
$198$	0	0
$199$	23.1144	1.63854	0.819269	−	0.573409i	$-0.194380\pi$
$199$	23.1144	1.63854	0.819269	+	0.573409i	$0.194380\pi$
$200$	0	0
$201$	−13.9864	−0.986526
$202$	0	0
$203$	15.0866i	1.05887i
$204$	0	0
$205$	1.30358i	0.0910459i
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	3.20435	0.221650
$210$	0	0
$211$	3.39104i	0.233449i	0.993164	+	0.116724i	$0.0372394\pi$
$211$	3.39104i	0.233449i	−0.993164	+	0.116724i	$0.962761\pi$
$212$	0	0
$213$	− 7.49207i	− 0.513348i
$214$	0	0
$215$	1.30358	0.0889034
$216$	0	0
$217$	−18.9378	−1.28558
$218$	0	0
$219$	5.62408i	0.380040i
$220$	0	0
$221$	− 30.2741i	− 2.03646i
$222$	0	0
$223$	−10.5326	−0.705316	−0.352658	−	0.935752i	$-0.614722\pi$
$223$	−10.5326	−0.705316	−0.352658	+	0.935752i	$0.614722\pi$
$224$	0	0
$225$	−4.88989	−0.325993
$226$	0	0
$227$	− 23.4753i	− 1.55811i	−0.626955	−	0.779055i	$-0.715699\pi$
$227$	− 23.4753i	− 1.55811i	0.626955	−	0.779055i	$-0.284301\pi$
$228$	0	0
$229$	3.13932i	0.207452i	0.994606	+	0.103726i	$0.0330766\pi$
$229$	3.13932i	0.207452i	−0.994606	+	0.103726i	$0.966923\pi$
$230$	0	0
$231$	−11.3910	−0.749475
$232$	0	0
$233$	8.98414	0.588571	0.294285	−	0.955718i	$-0.404918\pi$
$233$	8.98414	0.588571	0.294285	+	0.955718i	$0.404918\pi$
$234$	0	0
$235$	− 0.549780i	− 0.0358637i
$236$	0	0
$237$	14.9040i	0.968121i
$238$	0	0
$239$	1.69870	0.109880	0.0549400	−	0.998490i	$-0.482503\pi$
$239$	1.69870	0.109880	0.0549400	+	0.998490i	$0.482503\pi$
$240$	0	0
$241$	−14.3288	−0.923001	−0.461500	−	0.887140i	$-0.652689\pi$
$241$	−14.3288	−0.923001	−0.461500	+	0.887140i	$0.652689\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 0.829932i	− 0.0530224i
$246$	0	0
$247$	4.02371	0.256022
$248$	0	0
$249$	−9.35237	−0.592683
$250$	0	0
$251$	11.6955i	0.738215i	0.929387	+	0.369107i	$0.120337\pi$
$251$	11.6955i	0.738215i	−0.929387	+	0.369107i	$0.879663\pi$
$252$	0	0
$253$	− 14.7821i	− 0.929341i
$254$	0	0
$255$	−2.16478	−0.135564
$256$	0	0
$257$	3.98642	0.248666	0.124333	−	0.992241i	$-0.460321\pi$
$257$	3.98642	0.248666	0.124333	+	0.992241i	$0.460321\pi$
$258$	0	0
$259$	− 11.2423i	− 0.698560i
$260$	0	0
$261$	− 4.89443i	− 0.302958i
$262$	0	0
$263$	11.8635	0.731534	0.365767	−	0.930706i	$-0.380807\pi$
$263$	11.8635	0.731534	0.365767	+	0.930706i	$0.380807\pi$
$264$	0	0
$265$	0.187433	0.0115139
$266$	0	0
$267$	− 18.1094i	− 1.10827i
$268$	0	0
$269$	21.3880i	1.30405i	0.758197	+	0.652026i	$0.226081\pi$
$269$	21.3880i	1.30405i	−0.758197	+	0.652026i	$0.773919\pi$
$270$	0	0
$271$	−12.0530	−0.732165	−0.366082	−	0.930582i	$-0.619301\pi$
$271$	−12.0530	−0.732165	−0.366082	+	0.930582i	$0.619301\pi$
$272$	0	0
$273$	−14.3037	−0.865701
$274$	0	0
$275$	− 18.0707i	− 1.08970i
$276$	0	0
$277$	4.81432i	0.289265i	0.989485	+	0.144632i	$0.0461999\pi$
$277$	4.81432i	0.289265i	−0.989485	+	0.144632i	$0.953800\pi$
$278$	0	0
$279$	6.14386	0.367823
$280$	0	0
$281$	−18.5754	−1.10812	−0.554059	−	0.832477i	$-0.686922\pi$
$281$	−18.5754	−1.10812	−0.554059	+	0.832477i	$0.686922\pi$
$282$	0	0
$283$	− 1.40461i	− 0.0834956i	−0.999128	−	0.0417478i	$-0.986707\pi$
$283$	− 1.40461i	− 0.0834956i	0.999128	−	0.0417478i	$-0.0132926\pi$
$284$	0	0
$285$	− 0.287719i	− 0.0170430i
$286$	0	0
$287$	−12.1094	−0.714793
$288$	0	0
$289$	25.5619	1.50364
$290$	0	0
$291$	17.0479i	0.999365i
$292$	0	0
$293$	27.8786i	1.62868i	0.580386	+	0.814342i	$0.302902\pi$
$293$	27.8786i	1.62868i	−0.580386	+	0.814342i	$0.697098\pi$
$294$	0	0
$295$	−2.18849	−0.127419
$296$	0	0
$297$	3.69552	0.214436
$298$	0	0
$299$	− 18.5619i	− 1.07346i
$300$	0	0
$301$	12.1094i	0.697972i
$302$	0	0
$303$	12.1535	0.698198
$304$	0	0
$305$	4.91270	0.281300
$306$	0	0
$307$	17.4548i	0.996197i	0.867120	+	0.498099i	$0.165968\pi$
$307$	17.4548i	0.996197i	−0.867120	+	0.498099i	$0.834032\pi$
$308$	0	0
$309$	− 17.1043i	− 0.973029i
$310$	0	0
$311$	−0.466081	−0.0264290	−0.0132145	−	0.999913i	$-0.504206\pi$
$311$	−0.466081	−0.0264290	−0.0132145	+	0.999913i	$0.504206\pi$
$312$	0	0
$313$	−2.49886	−0.141244	−0.0706219	−	0.997503i	$-0.522498\pi$
$313$	−2.49886	−0.141244	−0.0706219	+	0.997503i	$0.522498\pi$
$314$	0	0
$315$	1.02280i	0.0576285i
$316$	0	0
$317$	− 10.7425i	− 0.603358i	−0.953410	−	0.301679i	$-0.902453\pi$
$317$	− 10.7425i	− 0.603358i	0.953410	−	0.301679i	$-0.0975470\pi$
$318$	0	0
$319$	18.0875	1.01270
$320$	0	0
$321$	−18.3752	−1.02560
$322$	0	0
$323$	5.65685i	0.314756i
$324$	0	0
$325$	− 22.6914i	− 1.25869i
$326$	0	0
$327$	7.84482	0.433819
$328$	0	0
$329$	5.10707	0.281562
$330$	0	0
$331$	− 5.26810i	− 0.289561i	−0.989464	−	0.144781i	$-0.953752\pi$
$331$	− 5.26810i	− 0.289561i	0.989464	−	0.144781i	$-0.0462476\pi$
$332$	0	0
$333$	3.64725i	0.199868i
$334$	0	0
$335$	−4.64099	−0.253565
$336$	0	0
$337$	−27.4740	−1.49660	−0.748302	−	0.663359i	$-0.769130\pi$
$337$	−27.4740	−1.49660	−0.748302	+	0.663359i	$0.769130\pi$
$338$	0	0
$339$	− 3.65685i	− 0.198613i
$340$	0	0
$341$	22.7047i	1.22953i
$342$	0	0
$343$	−13.8672	−0.748761
$344$	0	0
$345$	−1.32729	−0.0714586
$346$	0	0
$347$	8.08427i	0.433986i	0.976173	+	0.216993i	$0.0696249\pi$
$347$	8.08427i	0.433986i	−0.976173	+	0.216993i	$0.930375\pi$
$348$	0	0
$349$	4.97454i	0.266281i	0.991097	+	0.133140i	$0.0425062\pi$
$349$	4.97454i	0.266281i	−0.991097	+	0.133140i	$0.957494\pi$
$350$	0	0
$351$	4.64047	0.247690
$352$	0	0
$353$	23.0799	1.22842	0.614210	−	0.789143i	$-0.289475\pi$
$353$	23.0799	1.22842	0.614210	+	0.789143i	$0.289475\pi$
$354$	0	0
$355$	− 2.48603i	− 0.131945i
$356$	0	0
$357$	− 20.1094i	− 1.06430i
$358$	0	0
$359$	−29.2583	−1.54419	−0.772097	−	0.635505i	$-0.780792\pi$
$359$	−29.2583	−1.54419	−0.772097	+	0.635505i	$0.780792\pi$
$360$	0	0
$361$	18.2482	0.960429
$362$	0	0
$363$	2.65685i	0.139449i
$364$	0	0
$365$	1.86619i	0.0976808i
$366$	0	0
$367$	13.1698	0.687461	0.343730	−	0.939068i	$-0.388309\pi$
$367$	13.1698	0.687461	0.343730	+	0.939068i	$0.388309\pi$
$368$	0	0
$369$	3.92856	0.204513
$370$	0	0
$371$	1.74113i	0.0903947i
$372$	0	0
$373$	15.2905i	0.791714i	0.918312	+	0.395857i	$0.129552\pi$
$373$	15.2905i	0.791714i	−0.918312	+	0.395857i	$0.870448\pi$
$374$	0	0
$375$	−3.28168	−0.169465
$376$	0	0
$377$	22.7124	1.16975
$378$	0	0
$379$	6.41459i	0.329495i	0.986336	+	0.164748i	$0.0526810\pi$
$379$	6.41459i	0.329495i	−0.986336	+	0.164748i	$0.947319\pi$
$380$	0	0
$381$	− 0.638213i	− 0.0326966i
$382$	0	0
$383$	34.0721	1.74100	0.870501	−	0.492167i	$-0.163795\pi$
$383$	34.0721	1.74100	0.870501	+	0.492167i	$0.163795\pi$
$384$	0	0
$385$	−3.77979	−0.192636
$386$	0	0
$387$	− 3.92856i	− 0.199700i
$388$	0	0
$389$	32.5185i	1.64875i	0.566041	+	0.824377i	$0.308474\pi$
$389$	32.5185i	1.64875i	−0.566041	+	0.824377i	$0.691526\pi$
$390$	0	0
$391$	26.0958	1.31972
$392$	0	0
$393$	9.51397	0.479916
$394$	0	0
$395$	4.94548i	0.248834i
$396$	0	0
$397$	4.48926i	0.225309i	0.993634	+	0.112655i	$0.0359354\pi$
$397$	4.48926i	0.225309i	−0.993634	+	0.112655i	$0.964065\pi$
$398$	0	0
$399$	2.67271	0.133803
$400$	0	0
$401$	−3.21024	−0.160312	−0.0801558	−	0.996782i	$-0.525542\pi$
$401$	−3.21024	−0.160312	−0.0801558	+	0.996782i	$0.525542\pi$
$402$	0	0
$403$	28.5104i	1.42020i
$404$	0	0
$405$	− 0.331821i	− 0.0164883i
$406$	0	0
$407$	−13.4785	−0.668104
$408$	0	0
$409$	−14.0456	−0.694511	−0.347255	−	0.937771i	$-0.612886\pi$
$409$	−14.0456	−0.694511	−0.347255	+	0.937771i	$0.612886\pi$
$410$	0	0
$411$	− 13.9150i	− 0.686375i
$412$	0	0
$413$	− 20.3296i	− 1.00035i
$414$	0	0
$415$	−3.10332	−0.152336
$416$	0	0
$417$	0.0773278	0.00378676
$418$	0	0
$419$	26.9437i	1.31628i	0.752894	+	0.658142i	$0.228657\pi$
$419$	26.9437i	1.31628i	−0.752894	+	0.658142i	$0.771343\pi$
$420$	0	0
$421$	− 5.72188i	− 0.278867i	−0.990231	−	0.139434i	$-0.955472\pi$
$421$	− 5.72188i	− 0.278867i	0.990231	−	0.139434i	$-0.0445282\pi$
$422$	0	0
$423$	−1.65685	−0.0805590
$424$	0	0
$425$	31.9014	1.54745
$426$	0	0
$427$	45.6356i	2.20846i
$428$	0	0
$429$	17.1489i	0.827958i
$430$	0	0
$431$	27.9547	1.34653	0.673265	−	0.739401i	$-0.264891\pi$
$431$	27.9547	1.34653	0.673265	+	0.739401i	$0.264891\pi$
$432$	0	0
$433$	−1.81151	−0.0870556	−0.0435278	−	0.999052i	$-0.513860\pi$
$433$	−1.81151	−0.0870556	−0.0435278	+	0.999052i	$0.513860\pi$
$434$	0	0
$435$	− 1.62408i	− 0.0778685i
$436$	0	0
$437$	3.46836i	0.165914i
$438$	0	0
$439$	4.45985	0.212857	0.106429	−	0.994320i	$-0.466058\pi$
$439$	4.45985	0.212857	0.106429	+	0.994320i	$0.466058\pi$
$440$	0	0
$441$	−2.50114	−0.119102
$442$	0	0
$443$	2.62047i	0.124502i	0.998061	+	0.0622512i	$0.0198280\pi$
$443$	2.62047i	0.124502i	−0.998061	+	0.0622512i	$0.980172\pi$
$444$	0	0
$445$	− 6.00907i	− 0.284857i
$446$	0	0
$447$	13.9887	0.661642
$448$	0	0
$449$	27.9787	1.32040	0.660199	−	0.751091i	$-0.270472\pi$
$449$	27.9787	1.32040	0.660199	+	0.751091i	$0.270472\pi$
$450$	0	0
$451$	14.5181i	0.683629i
$452$	0	0
$453$	18.5828i	0.873095i
$454$	0	0
$455$	−4.74628	−0.222509
$456$	0	0
$457$	14.3933	0.673291	0.336646	−	0.941631i	$-0.390708\pi$
$457$	14.3933	0.673291	0.336646	+	0.941631i	$0.390708\pi$
$458$	0	0
$459$	6.52395i	0.304512i
$460$	0	0
$461$	7.78841i	0.362743i	0.983415	+	0.181371i	$0.0580536\pi$
$461$	7.78841i	0.362743i	−0.983415	+	0.181371i	$0.941946\pi$
$462$	0	0
$463$	−15.6642	−0.727977	−0.363989	−	0.931403i	$-0.618585\pi$
$463$	−15.6642	−0.727977	−0.363989	+	0.931403i	$0.618585\pi$
$464$	0	0
$465$	2.03866	0.0945408
$466$	0	0
$467$	5.27504i	0.244100i	0.992524	+	0.122050i	$0.0389468\pi$
$467$	5.27504i	0.244100i	−0.992524	+	0.122050i	$0.961053\pi$
$468$	0	0
$469$	− 43.1116i	− 1.99071i
$470$	0	0
$471$	−5.30411	−0.244400
$472$	0	0
$473$	14.5181	0.667541
$474$	0	0
$475$	4.23998i	0.194544i
$476$	0	0
$477$	− 0.564862i	− 0.0258632i
$478$	0	0
$479$	−11.9310	−0.545141	−0.272571	−	0.962136i	$-0.587874\pi$
$479$	−11.9310	−0.545141	−0.272571	+	0.962136i	$0.587874\pi$
$480$	0	0
$481$	−16.9250	−0.771712
$482$	0	0
$483$	− 12.3296i	− 0.561015i
$484$	0	0
$485$	5.65685i	0.256865i
$486$	0	0
$487$	−37.1782	−1.68470	−0.842352	−	0.538928i	$-0.818830\pi$
$487$	−37.1782	−1.68470	−0.842352	+	0.538928i	$0.818830\pi$
$488$	0	0
$489$	−13.3060	−0.601719
$490$	0	0
$491$	25.0981i	1.13266i	0.824179	+	0.566330i	$0.191637\pi$
$491$	25.0981i	1.13266i	−0.824179	+	0.566330i	$0.808363\pi$
$492$	0	0
$493$	31.9310i	1.43810i
$494$	0	0
$495$	1.22625	0.0551159
$496$	0	0
$497$	23.0935	1.03588
$498$	0	0
$499$	− 32.8458i	− 1.47038i	−0.677861	−	0.735190i	$-0.737093\pi$
$499$	− 32.8458i	− 1.47038i	0.677861	−	0.735190i	$-0.262907\pi$
$500$	0	0
$501$	15.9310i	0.711744i
$502$	0	0
$503$	8.35327	0.372454	0.186227	−	0.982507i	$-0.440374\pi$
$503$	8.35327	0.372454	0.186227	+	0.982507i	$0.440374\pi$
$504$	0	0
$505$	4.03278	0.179456
$506$	0	0
$507$	8.53392i	0.379005i
$508$	0	0
$509$	9.33786i	0.413893i	0.978352	+	0.206947i	$0.0663527\pi$
$509$	9.33786i	0.413893i	−0.978352	+	0.206947i	$0.933647\pi$
$510$	0	0
$511$	−17.3356	−0.766882
$512$	0	0
$513$	−0.867091	−0.0382830
$514$	0	0
$515$	− 5.67557i	− 0.250095i
$516$	0	0
$517$	− 6.12293i	− 0.269286i
$518$	0	0
$519$	−19.7229	−0.865737
$520$	0	0
$521$	4.38287	0.192017	0.0960084	−	0.995381i	$-0.469392\pi$
$521$	4.38287	0.192017	0.0960084	+	0.995381i	$0.469392\pi$
$522$	0	0
$523$	9.22869i	0.403542i	0.979433	+	0.201771i	$0.0646697\pi$
$523$	9.22869i	0.403542i	−0.979433	+	0.201771i	$0.935330\pi$
$524$	0	0
$525$	− 15.0726i	− 0.657821i
$526$	0	0
$527$	−40.0822	−1.74601
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.59539i	0.286215i
$532$	0	0
$533$	18.2303i	0.789644i
$534$	0	0
$535$	−6.09728	−0.263608
$536$	0	0
$537$	−5.98642	−0.258333
$538$	0	0
$539$	− 9.24301i	− 0.398125i
$540$	0	0
$541$	36.9700i	1.58947i	0.606959	+	0.794733i	$0.292389\pi$
$541$	36.9700i	1.58947i	−0.606959	+	0.794733i	$0.707611\pi$
$542$	0	0
$543$	−9.81204	−0.421075
$544$	0	0
$545$	2.60308	0.111504
$546$	0	0
$547$	40.6789i	1.73930i	0.493665	+	0.869652i	$0.335657\pi$
$547$	40.6789i	1.73930i	−0.493665	+	0.869652i	$0.664343\pi$
$548$	0	0
$549$	− 14.8052i	− 0.631873i
$550$	0	0
$551$	−4.24392	−0.180797
$552$	0	0
$553$	−45.9401	−1.95357
$554$	0	0
$555$	1.21024i	0.0513717i
$556$	0	0
$557$	− 39.6184i	− 1.67868i	−0.543603	−	0.839342i	$-0.682940\pi$
$557$	− 39.6184i	− 1.67868i	0.543603	−	0.839342i	$-0.317060\pi$
$558$	0	0
$559$	18.2303	0.771061
$560$	0	0
$561$	−24.1094	−1.01790
$562$	0	0
$563$	20.8090i	0.876993i	0.898733	+	0.438497i	$0.144489\pi$
$563$	20.8090i	0.876993i	−0.898733	+	0.438497i	$0.855511\pi$
$564$	0	0
$565$	− 1.21342i	− 0.0510491i
$566$	0	0
$567$	3.08239	0.129448
$568$	0	0
$569$	15.7585	0.660632	0.330316	−	0.943870i	$-0.392845\pi$
$569$	15.7585	0.660632	0.330316	+	0.943870i	$0.392845\pi$
$570$	0	0
$571$	24.6912i	1.03329i	0.856199	+	0.516647i	$0.172820\pi$
$571$	24.6912i	1.03329i	−0.856199	+	0.516647i	$0.827180\pi$
$572$	0	0
$573$	19.2900i	0.805851i
$574$	0	0
$575$	19.5596	0.815691
$576$	0	0
$577$	−21.0924	−0.878090	−0.439045	−	0.898465i	$-0.644683\pi$
$577$	−21.0924	−0.878090	−0.439045	+	0.898465i	$0.644683\pi$
$578$	0	0
$579$	0.155713i	0.00647119i
$580$	0	0
$581$	− 28.8277i	− 1.19597i
$582$	0	0
$583$	2.08746	0.0864536
$584$	0	0
$585$	1.53981	0.0636631
$586$	0	0
$587$	− 14.3933i	− 0.594076i	−0.954866	−	0.297038i	$-0.904001\pi$
$587$	− 14.3933i	− 0.594076i	0.954866	−	0.297038i	$-0.0959988\pi$
$588$	0	0
$589$	− 5.32729i	− 0.219507i
$590$	0	0
$591$	−13.2014	−0.543031
$592$	0	0
$593$	22.1931	0.911360	0.455680	−	0.890144i	$-0.349396\pi$
$593$	22.1931	0.911360	0.455680	+	0.890144i	$0.349396\pi$
$594$	0	0
$595$	− 6.67271i	− 0.273555i
$596$	0	0
$597$	− 23.1144i	− 0.946010i
$598$	0	0
$599$	−37.4366	−1.52962	−0.764810	−	0.644256i	$-0.777167\pi$
$599$	−37.4366	−1.52962	−0.764810	+	0.644256i	$0.777167\pi$
$600$	0	0
$601$	−1.68963	−0.0689215	−0.0344608	−	0.999406i	$-0.510971\pi$
$601$	−1.68963	−0.0689215	−0.0344608	+	0.999406i	$0.510971\pi$
$602$	0	0
$603$	13.9864i	0.569571i
$604$	0	0
$605$	0.881601i	0.0358422i
$606$	0	0
$607$	36.8841	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$607$	36.8841	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$608$	0	0
$609$	15.0866	0.611338
$610$	0	0
$611$	− 7.68857i	− 0.311046i
$612$	0	0
$613$	− 13.4598i	− 0.543637i	−0.962349	−	0.271819i	$-0.912375\pi$
$613$	− 13.4598i	− 0.543637i	0.962349	−	0.271819i	$-0.0876250\pi$
$614$	0	0
$615$	1.30358	0.0525654
$616$	0	0
$617$	20.3160	0.817891	0.408946	−	0.912559i	$-0.365897\pi$
$617$	20.3160	0.817891	0.408946	+	0.912559i	$0.365897\pi$
$618$	0	0
$619$	− 39.0279i	− 1.56867i	−0.620340	−	0.784333i	$-0.713006\pi$
$619$	− 39.0279i	− 1.56867i	0.620340	−	0.784333i	$-0.286994\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	55.8201	2.23639
$624$	0	0
$625$	23.3605	0.934422
$626$	0	0
$627$	− 3.20435i	− 0.127969i
$628$	0	0
$629$	− 23.7945i	− 0.948748i
$630$	0	0
$631$	−0.629888	−0.0250755	−0.0125377	−	0.999921i	$-0.503991\pi$
$631$	−0.629888	−0.0250755	−0.0125377	+	0.999921i	$0.503991\pi$
$632$	0	0
$633$	3.39104	0.134782
$634$	0	0
$635$	− 0.211773i	− 0.00840394i
$636$	0	0
$637$	− 11.6065i	− 0.459865i
$638$	0	0
$639$	−7.49207	−0.296382
$640$	0	0
$641$	3.52166	0.139097	0.0695486	−	0.997579i	$-0.477844\pi$
$641$	3.52166	0.139097	0.0695486	+	0.997579i	$0.477844\pi$
$642$	0	0
$643$	9.83765i	0.387959i	0.981006	+	0.193980i	$0.0621396\pi$
$643$	9.83765i	0.387959i	−0.981006	+	0.193980i	$0.937860\pi$
$644$	0	0
$645$	− 1.30358i	− 0.0513284i
$646$	0	0
$647$	30.7549	1.20910	0.604550	−	0.796567i	$-0.293353\pi$
$647$	30.7549	1.20910	0.604550	+	0.796567i	$0.293353\pi$
$648$	0	0
$649$	−24.3734	−0.956739
$650$	0	0
$651$	18.9378i	0.742230i
$652$	0	0
$653$	− 9.22745i	− 0.361098i	−0.983566	−	0.180549i	$-0.942213\pi$
$653$	− 9.22745i	− 0.361098i	0.983566	−	0.180549i	$-0.0577874\pi$
$654$	0	0
$655$	3.15694	0.123352
$656$	0	0
$657$	5.62408	0.219416
$658$	0	0
$659$	46.5619i	1.81379i	0.421354	+	0.906896i	$0.361555\pi$
$659$	46.5619i	1.81379i	−0.421354	+	0.906896i	$0.638445\pi$
$660$	0	0
$661$	− 27.9315i	− 1.08641i	−0.839600	−	0.543205i	$-0.817211\pi$
$661$	− 27.9315i	− 1.08641i	0.839600	−	0.543205i	$-0.182789\pi$
$662$	0	0
$663$	−30.2741	−1.17575
$664$	0	0
$665$	0.886864	0.0343911
$666$	0	0
$667$	19.5777i	0.758052i
$668$	0	0
$669$	10.5326i	0.407214i
$670$	0	0
$671$	54.7131	2.11217
$672$	0	0
$673$	6.34315	0.244510	0.122255	−	0.992499i	$-0.460987\pi$
$673$	6.34315	0.244510	0.122255	+	0.992499i	$0.460987\pi$
$674$	0	0
$675$	4.88989i	0.188212i
$676$	0	0
$677$	− 31.4918i	− 1.21033i	−0.796101	−	0.605164i	$-0.793108\pi$
$677$	− 31.4918i	− 1.21033i	0.796101	−	0.605164i	$-0.206892\pi$
$678$	0	0
$679$	−52.5483	−2.01662
$680$	0	0
$681$	−23.4753	−0.899576
$682$	0	0
$683$	28.0069i	1.07166i	0.844327	+	0.535828i	$0.180000\pi$
$683$	28.0069i	1.07166i	−0.844327	+	0.535828i	$0.820000\pi$
$684$	0	0
$685$	− 4.61729i	− 0.176418i
$686$	0	0
$687$	3.13932	0.119773
$688$	0	0
$689$	2.62122	0.0998606
$690$	0	0
$691$	38.2264i	1.45420i	0.686531	+	0.727101i	$0.259133\pi$
$691$	38.2264i	1.45420i	−0.686531	+	0.727101i	$0.740867\pi$
$692$	0	0
$693$	11.3910i	0.432710i
$694$	0	0
$695$	0.0256590	0.000973301 0
$696$	0	0
$697$	−25.6297	−0.970794
$698$	0	0
$699$	− 8.98414i	− 0.339811i
$700$	0	0
$701$	− 9.03731i	− 0.341335i	−0.985329	−	0.170667i	$-0.945408\pi$
$701$	− 9.03731i	− 0.341335i	0.985329	−	0.170667i	$-0.0545923\pi$
$702$	0	0
$703$	3.16250	0.119276
$704$	0	0
$705$	−0.549780	−0.0207059
$706$	0	0
$707$	37.4617i	1.40889i
$708$	0	0
$709$	39.9270i	1.49949i	0.661726	+	0.749745i	$0.269824\pi$
$709$	39.9270i	1.49949i	−0.661726	+	0.749745i	$0.730176\pi$
$710$	0	0
$711$	14.9040	0.558945
$712$	0	0
$713$	−24.5754	−0.920357
$714$	0	0
$715$	5.69038i	0.212808i
$716$	0	0
$717$	− 1.69870i	− 0.0634393i
$718$	0	0
$719$	29.3036	1.09284	0.546420	−	0.837512i	$-0.315990\pi$
$719$	29.3036	1.09284	0.546420	+	0.837512i	$0.315990\pi$
$720$	0	0
$721$	52.7221	1.96348
$722$	0	0
$723$	14.3288i	0.532895i
$724$	0	0
$725$	23.9332i	0.888859i
$726$	0	0
$727$	−12.6201	−0.468052	−0.234026	−	0.972230i	$-0.575190\pi$
$727$	−12.6201	−0.468052	−0.234026	+	0.972230i	$0.575190\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	25.6297i	0.947949i
$732$	0	0
$733$	9.92140i	0.366455i	0.983071	+	0.183228i	$0.0586545\pi$
$733$	9.92140i	0.366455i	−0.983071	+	0.183228i	$0.941345\pi$
$734$	0	0
$735$	−0.829932	−0.0306125
$736$	0	0
$737$	−51.6871	−1.90392
$738$	0	0
$739$	− 20.0592i	− 0.737889i	−0.929451	−	0.368945i	$-0.879719\pi$
$739$	− 20.0592i	− 0.737889i	0.929451	−	0.368945i	$-0.120281\pi$
$740$	0	0
$741$	− 4.02371i	− 0.147815i
$742$	0	0
$743$	11.0969	0.407107	0.203554	−	0.979064i	$-0.434751\pi$
$743$	11.0969	0.407107	0.203554	+	0.979064i	$0.434751\pi$
$744$	0	0
$745$	4.64174	0.170060
$746$	0	0
$747$	9.35237i	0.342185i
$748$	0	0
$749$	− 56.6395i	− 2.06956i
$750$	0	0
$751$	37.3294	1.36217	0.681084	−	0.732205i	$-0.261509\pi$
$751$	37.3294	1.36217	0.681084	+	0.732205i	$0.261509\pi$
$752$	0	0
$753$	11.6955	0.426208
$754$	0	0
$755$	6.16617i	0.224410i
$756$	0	0
$757$	26.7362i	0.971745i	0.874030	+	0.485873i	$0.161498\pi$
$757$	26.7362i	0.971745i	−0.874030	+	0.485873i	$0.838502\pi$
$758$	0	0
$759$	−14.7821	−0.536555
$760$	0	0
$761$	27.1767	0.985155	0.492578	−	0.870269i	$-0.336055\pi$
$761$	27.1767	0.985155	0.492578	+	0.870269i	$0.336055\pi$
$762$	0	0
$763$	24.1808i	0.875404i
$764$	0	0
$765$	2.16478i	0.0782679i
$766$	0	0
$767$	−30.6057	−1.10511
$768$	0	0
$769$	−8.34877	−0.301064	−0.150532	−	0.988605i	$-0.548099\pi$
$769$	−8.34877	−0.301064	−0.150532	+	0.988605i	$0.548099\pi$
$770$	0	0
$771$	− 3.98642i	− 0.143568i
$772$	0	0
$773$	20.7289i	0.745567i	0.927918	+	0.372783i	$0.121597\pi$
$773$	20.7289i	0.745567i	−0.927918	+	0.372783i	$0.878403\pi$
$774$	0	0
$775$	−30.0428	−1.07917
$776$	0	0
$777$	−11.2423	−0.403314
$778$	0	0
$779$	− 3.40642i	− 0.122048i
$780$	0	0
$781$	− 27.6871i	− 0.990722i
$782$	0	0
$783$	−4.89443	−0.174913
$784$	0	0
$785$	−1.76002	−0.0628177
$786$	0	0
$787$	− 21.7129i	− 0.773982i	−0.922084	−	0.386991i	$-0.873514\pi$
$787$	− 21.7129i	− 0.773982i	0.922084	−	0.386991i	$-0.126486\pi$
$788$	0	0
$789$	− 11.8635i	− 0.422351i
$790$	0	0
$791$	11.2719	0.400781
$792$	0	0
$793$	68.7032	2.43972
$794$	0	0
$795$	− 0.187433i	− 0.00664757i
$796$	0	0
$797$	27.5528i	0.975971i	0.872852	+	0.487985i	$0.162268\pi$
$797$	27.5528i	0.975971i	−0.872852	+	0.487985i	$0.837732\pi$
$798$	0	0
$799$	10.8092	0.382403
$800$	0	0
$801$	−18.1094	−0.639863
$802$	0	0
$803$	20.7839i	0.733447i
$804$	0	0
$805$	− 4.09121i	− 0.144196i
$806$	0	0
$807$	21.3880	0.752895
$808$	0	0
$809$	−18.8535	−0.662854	−0.331427	−	0.943481i	$-0.607530\pi$
$809$	−18.8535	−0.662854	−0.331427	+	0.943481i	$0.607530\pi$
$810$	0	0
$811$	− 6.78796i	− 0.238357i	−0.992873	−	0.119179i	$-0.961974\pi$
$811$	− 6.78796i	− 0.238357i	0.992873	−	0.119179i	$-0.0380262\pi$
$812$	0	0
$813$	12.0530i	0.422716i
$814$	0	0
$815$	−4.41522	−0.154658
$816$	0	0
$817$	−3.40642	−0.119175
$818$	0	0
$819$	14.3037i	0.499813i
$820$	0	0
$821$	− 16.0169i	− 0.558995i	−0.960146	−	0.279498i	$-0.909832\pi$
$821$	− 16.0169i	− 0.558995i	0.960146	−	0.279498i	$-0.0901679\pi$
$822$	0	0
$823$	28.5609	0.995570	0.497785	−	0.867301i	$-0.334147\pi$
$823$	28.5609	0.995570	0.497785	+	0.867301i	$0.334147\pi$
$824$	0	0
$825$	−18.0707	−0.629141
$826$	0	0
$827$	− 19.1388i	− 0.665521i	−0.943011	−	0.332761i	$-0.892020\pi$
$827$	− 19.1388i	− 0.665521i	0.943011	−	0.332761i	$-0.107980\pi$
$828$	0	0
$829$	36.0907i	1.25348i	0.779228	+	0.626741i	$0.215611\pi$
$829$	36.0907i	1.25348i	−0.779228	+	0.626741i	$0.784389\pi$
$830$	0	0
$831$	4.81432	0.167007
$832$	0	0
$833$	16.3173	0.565361
$834$	0	0
$835$	5.28624i	0.182938i
$836$	0	0
$837$	− 6.14386i	− 0.212363i
$838$	0	0
$839$	−18.8767	−0.651697	−0.325849	−	0.945422i	$-0.605650\pi$
$839$	−18.8767	−0.651697	−0.325849	+	0.945422i	$0.605650\pi$
$840$	0	0
$841$	5.04455	0.173950
$842$	0	0
$843$	18.5754i	0.639772i
$844$	0	0
$845$	2.83174i	0.0974147i
$846$	0	0
$847$	−8.18947	−0.281393
$848$	0	0
$849$	−1.40461	−0.0482062
$850$	0	0
$851$	− 14.5890i	− 0.500105i
$852$	0	0
$853$	26.3018i	0.900557i	0.892888	+	0.450279i	$0.148675\pi$
$853$	26.3018i	0.900557i	−0.892888	+	0.450279i	$0.851325\pi$
$854$	0	0
$855$	−0.287719	−0.00983979
$856$	0	0
$857$	24.8671	0.849444	0.424722	−	0.905324i	$-0.360372\pi$
$857$	24.8671	0.849444	0.424722	+	0.905324i	$0.360372\pi$
$858$	0	0
$859$	15.5990i	0.532231i	0.963941	+	0.266115i	$0.0857402\pi$
$859$	15.5990i	0.532231i	−0.963941	+	0.266115i	$0.914260\pi$
$860$	0	0
$861$	12.1094i	0.412686i
$862$	0	0
$863$	−43.8654	−1.49320	−0.746598	−	0.665275i	$-0.768314\pi$
$863$	−43.8654	−1.49320	−0.746598	+	0.665275i	$0.768314\pi$
$864$	0	0
$865$	−6.54447	−0.222519
$866$	0	0
$867$	− 25.5619i	− 0.868126i
$868$	0	0
$869$	55.0781i	1.86840i
$870$	0	0
$871$	−64.9035	−2.19917
$872$	0	0
$873$	17.0479	0.576984
$874$	0	0
$875$	− 10.1154i	− 0.341964i
$876$	0	0
$877$	− 45.5410i	− 1.53781i	−0.639364	−	0.768905i	$-0.720802\pi$
$877$	− 45.5410i	− 1.53781i	0.639364	−	0.768905i	$-0.279198\pi$
$878$	0	0
$879$	27.8786	0.940321
$880$	0	0
$881$	−19.8589	−0.669064	−0.334532	−	0.942384i	$-0.608578\pi$
$881$	−19.8589	−0.669064	−0.334532	+	0.942384i	$0.608578\pi$
$882$	0	0
$883$	− 25.2740i	− 0.850537i	−0.905067	−	0.425269i	$-0.860180\pi$
$883$	− 25.2740i	− 0.850537i	0.905067	−	0.425269i	$-0.139820\pi$
$884$	0	0
$885$	2.18849i	0.0735653i
$886$	0	0
$887$	2.07012	0.0695079	0.0347539	−	0.999396i	$-0.488935\pi$
$887$	2.07012	0.0695079	0.0347539	+	0.999396i	$0.488935\pi$
$888$	0	0
$889$	1.96722	0.0659785
$890$	0	0
$891$	− 3.69552i	− 0.123805i
$892$	0	0
$893$	1.43664i	0.0480754i
$894$	0	0
$895$	−1.98642	−0.0663988
$896$	0	0
$897$	−18.5619	−0.619763
$898$	0	0
$899$	− 30.0707i	− 1.00291i
$900$	0	0
$901$	3.68513i	0.122769i
$902$	0	0
$903$	12.1094	0.402974
$904$	0	0
$905$	−3.25584	−0.108228
$906$	0	0
$907$	8.10347i	0.269071i	0.990909	+	0.134536i	$0.0429543\pi$
$907$	8.10347i	0.269071i	−0.990909	+	0.134536i	$0.957046\pi$
$908$	0	0
$909$	− 12.1535i	− 0.403105i
$910$	0	0
$911$	−6.23034	−0.206420	−0.103210	−	0.994660i	$-0.532911\pi$
$911$	−6.23034	−0.206420	−0.103210	+	0.994660i	$0.532911\pi$
$912$	0	0
$913$	−34.5619	−1.14383
$914$	0	0
$915$	− 4.91270i	− 0.162409i
$916$	0	0
$917$	29.3258i	0.968423i
$918$	0	0
$919$	15.1680	0.500348	0.250174	−	0.968201i	$-0.419512\pi$
$919$	15.1680	0.500348	0.250174	+	0.968201i	$0.419512\pi$
$920$	0	0
$921$	17.4548	0.575155
$922$	0	0
$923$	− 34.7667i	− 1.14436i
$924$	0	0
$925$	− 17.8347i	− 0.586401i
$926$	0	0
$927$	−17.1043	−0.561779
$928$	0	0
$929$	−9.16951	−0.300842	−0.150421	−	0.988622i	$-0.548063\pi$
$929$	−9.16951	−0.300842	−0.150421	+	0.988622i	$0.548063\pi$
$930$	0	0
$931$	2.16872i	0.0710768i
$932$	0	0
$933$	0.466081i	0.0152588i
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	19.3183	0.631101	0.315550	−	0.948909i	$-0.397811\pi$
$937$	19.3183	0.631101	0.315550	+	0.948909i	$0.397811\pi$
$938$	0	0
$939$	2.49886i	0.0815472i
$940$	0	0
$941$	− 10.4385i	− 0.340285i	−0.985419	−	0.170142i	$-0.945577\pi$
$941$	− 10.4385i	− 0.340285i	0.985419	−	0.170142i	$-0.0544228\pi$
$942$	0	0
$943$	−15.7142	−0.511726
$944$	0	0
$945$	1.02280	0.0332718
$946$	0	0
$947$	36.2640i	1.17842i	0.807979	+	0.589211i	$0.200561\pi$
$947$	36.2640i	1.17842i	−0.807979	+	0.589211i	$0.799439\pi$
$948$	0	0
$949$	26.0983i	0.847188i
$950$	0	0
$951$	−10.7425	−0.348349
$952$	0	0
$953$	−47.9271	−1.55251	−0.776255	−	0.630419i	$-0.782883\pi$
$953$	−47.9271	−1.55251	−0.776255	+	0.630419i	$0.782883\pi$
$954$	0	0
$955$	6.40083i	0.207126i
$956$	0	0
$957$	− 18.0875i	− 0.584684i
$958$	0	0
$959$	42.8914	1.38504
$960$	0	0
$961$	6.74701	0.217646
$962$	0	0
$963$	18.3752i	0.592132i
$964$	0	0
$965$	0.0516688i	0.00166328i
$966$	0	0
$967$	20.7211	0.666346	0.333173	−	0.942866i	$-0.391881\pi$
$967$	20.7211	0.666346	0.333173	+	0.942866i	$0.391881\pi$
$968$	0	0
$969$	5.65685	0.181724
$970$	0	0
$971$	38.0888i	1.22233i	0.791504	+	0.611164i	$0.209299\pi$
$971$	38.0888i	1.22233i	−0.791504	+	0.611164i	$0.790701\pi$
$972$	0	0
$973$	0.238354i	0.00764129i
$974$	0	0
$975$	−22.6914	−0.726706
$976$	0	0
$977$	42.7363	1.36726	0.683628	−	0.729831i	$-0.260401\pi$
$977$	42.7363	1.36726	0.683628	+	0.729831i	$0.260401\pi$
$978$	0	0
$979$	− 66.9235i	− 2.13888i
$980$	0	0
$981$	− 7.84482i	− 0.250466i
$982$	0	0
$983$	−42.0031	−1.33969	−0.669845	−	0.742501i	$-0.733639\pi$
$983$	−42.0031	−1.33969	−0.669845	+	0.742501i	$0.733639\pi$
$984$	0	0
$985$	−4.38049	−0.139574
$986$	0	0
$987$	− 5.10707i	− 0.162560i
$988$	0	0
$989$	15.7142i	0.499684i
$990$	0	0
$991$	−11.1918	−0.355518	−0.177759	−	0.984074i	$-0.556885\pi$
$991$	−11.1918	−0.355518	−0.177759	+	0.984074i	$0.556885\pi$
$992$	0	0
$993$	−5.26810	−0.167178
$994$	0	0
$995$	− 7.66986i	− 0.243151i
$996$	0	0
$997$	− 52.3230i	− 1.65709i	−0.559924	−	0.828544i	$-0.689170\pi$
$997$	− 52.3230i	− 1.65709i	0.559924	−	0.828544i	$-0.310830\pi$
$998$	0	0
$999$	3.64725	0.115394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.j.1537.3		8
4.3	odd	2		3072.2.d.e.1537.7		8
8.3	odd	2		3072.2.d.e.1537.2		8
8.5	even	2	inner	3072.2.d.j.1537.6		8
16.3	odd	4		3072.2.a.m.1.3		4
16.5	even	4		3072.2.a.j.1.2		4
16.11	odd	4		3072.2.a.s.1.2		4
16.13	even	4		3072.2.a.p.1.3		4
32.3	odd	8		1536.2.j.j.385.1	yes	8
32.5	even	8		1536.2.j.i.1153.3	yes	8
32.11	odd	8		1536.2.j.e.1153.4	yes	8
32.13	even	8		1536.2.j.f.385.2	yes	8
32.19	odd	8		1536.2.j.e.385.4	✓	8
32.21	even	8		1536.2.j.f.1153.2	yes	8
32.27	odd	8		1536.2.j.j.1153.1	yes	8
32.29	even	8		1536.2.j.i.385.3	yes	8
48.5	odd	4		9216.2.a.ba.1.3		4
48.11	even	4		9216.2.a.bm.1.3		4
48.29	odd	4		9216.2.a.z.1.2		4
48.35	even	4		9216.2.a.bl.1.2		4
96.5	odd	8		4608.2.k.bc.1153.3		8
96.11	even	8		4608.2.k.bh.1153.2		8
96.29	odd	8		4608.2.k.bc.3457.3		8
96.35	even	8		4608.2.k.be.3457.3		8
96.53	odd	8		4608.2.k.bj.1153.2		8
96.59	even	8		4608.2.k.be.1153.3		8
96.77	odd	8		4608.2.k.bj.3457.2		8
96.83	even	8		4608.2.k.bh.3457.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.4	✓	8	32.19	odd	8
1536.2.j.e.1153.4	yes	8	32.11	odd	8
1536.2.j.f.385.2	yes	8	32.13	even	8
1536.2.j.f.1153.2	yes	8	32.21	even	8
1536.2.j.i.385.3	yes	8	32.29	even	8
1536.2.j.i.1153.3	yes	8	32.5	even	8
1536.2.j.j.385.1	yes	8	32.3	odd	8
1536.2.j.j.1153.1	yes	8	32.27	odd	8
3072.2.a.j.1.2		4	16.5	even	4
3072.2.a.m.1.3		4	16.3	odd	4
3072.2.a.p.1.3		4	16.13	even	4
3072.2.a.s.1.2		4	16.11	odd	4
3072.2.d.e.1537.2		8	8.3	odd	2
3072.2.d.e.1537.7		8	4.3	odd	2
3072.2.d.j.1537.3		8	1.1	even	1	trivial
3072.2.d.j.1537.6		8	8.5	even	2	inner
4608.2.k.bc.1153.3		8	96.5	odd	8
4608.2.k.bc.3457.3		8	96.29	odd	8
4608.2.k.be.1153.3		8	96.59	even	8
4608.2.k.be.3457.3		8	96.35	even	8
4608.2.k.bh.1153.2		8	96.11	even	8
4608.2.k.bh.3457.2		8	96.83	even	8
4608.2.k.bj.1153.2		8	96.53	odd	8
4608.2.k.bj.3457.2		8	96.77	odd	8
9216.2.a.z.1.2		4	48.29	odd	4
9216.2.a.ba.1.3		4	48.5	odd	4
9216.2.a.bl.1.2		4	48.35	even	4
9216.2.a.bm.1.3		4	48.11	even	4

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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -0.331821i q^{5} +3.08239 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -0.331821i q^{5} +3.08239 q^{7} -1.00000 q^{9} -3.69552i q^{11} -4.64047i q^{13} -0.331821 q^{15} +6.52395 q^{17} +0.867091i q^{19} -3.08239i q^{21} +4.00000 q^{23} +4.88989 q^{25} +1.00000i q^{27} +4.89443i q^{29} -6.14386 q^{31} -3.69552 q^{33} -1.02280i q^{35} -3.64725i q^{37} -4.64047 q^{39} -3.92856 q^{41} +3.92856i q^{43} +0.331821i q^{45} +1.65685 q^{47} +2.50114 q^{49} -6.52395i q^{51} +0.564862i q^{53} -1.22625 q^{55} +0.867091 q^{57} -6.59539i q^{59} +14.8052i q^{61} -3.08239 q^{63} -1.53981 q^{65} -13.9864i q^{67} -4.00000i q^{69} +7.49207 q^{71} -5.62408 q^{73} -4.88989i q^{75} -11.3910i q^{77} -14.9040 q^{79} +1.00000 q^{81} -9.35237i q^{83} -2.16478i q^{85} +4.89443 q^{87} +18.1094 q^{89} -14.3037i q^{91} +6.14386i q^{93} +0.287719 q^{95} -17.0479 q^{97} +3.69552i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q + 16 * q^7 - 8 * q^9	\( 8 q + 16 q^{7} - 8 q^{9} + 32 q^{23} - 8 q^{25} - 16 q^{31} + 16 q^{39} - 32 q^{47} + 8 q^{49} + 32 q^{55} - 16 q^{63} - 16 q^{65} + 32 q^{71} + 16 q^{73} - 48 q^{79} + 8 q^{81} + 16 q^{89} - 64 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 - 8 * q^9 + 32 * q^23 - 8 * q^25 - 16 * q^31 + 16 * q^39 - 32 * q^47 + 8 * q^49 + 32 * q^55 - 16 * q^63 - 16 * q^65 + 32 * q^71 + 16 * q^73 - 48 * q^79 + 8 * q^81 + 16 * q^89 - 64 * q^95 - 32 * q^97

Introduction

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