LMFDB - Embedded newform 1536.2.d.g.769.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(769,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, 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("1536.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.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:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.3
Root			$0.500000 + 1.44392i$ of defining polynomial
Character	$\chi$	$=$	1536.769
Dual form			1536.2.d.g.769.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.08402i q^{5} -5.03127 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.08402i q^{5} -5.03127 q^{7} -1.00000 q^{9} -0.828427i q^{11} +2.94725i q^{13} +2.08402 q^{15} +4.82843 q^{17} -2.82843i q^{19} +5.03127i q^{21} +4.16804 q^{23} +0.656854 q^{25} +1.00000i q^{27} -7.97852i q^{29} -5.03127 q^{31} -0.828427 q^{33} -10.4853i q^{35} -7.11529i q^{37} +2.94725 q^{39} -8.82843 q^{41} -12.4853i q^{43} -2.08402i q^{45} -4.16804 q^{47} +18.3137 q^{49} -4.82843i q^{51} -12.1466i q^{53} +1.72646 q^{55} -2.82843 q^{57} -1.65685i q^{59} +7.11529i q^{61} +5.03127 q^{63} -6.14214 q^{65} +2.34315i q^{67} -4.16804i q^{69} +10.0625 q^{71} -4.00000 q^{73} -0.656854i q^{75} +4.16804i q^{77} +5.03127 q^{79} +1.00000 q^{81} -3.17157i q^{83} +10.0625i q^{85} -7.97852 q^{87} -10.0000 q^{89} -14.8284i q^{91} +5.03127i q^{93} +5.89450 q^{95} +0.343146 q^{97} +0.828427i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 16 q^{17} - 40 q^{25} + 16 q^{33} - 48 q^{41} + 56 q^{49} + 64 q^{65} - 32 q^{73} + 8 q^{81} - 80 q^{89} + 48 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^17 - 40 * q^25 + 16 * q^33 - 48 * q^41 + 56 * q^49 + 64 * q^65 - 32 * q^73 + 8 * q^81 - 80 * q^89 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$.

$n$	$511$	$517$	$1025$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	2.08402i	0.932003i	0.884784	+	0.466001i	$0.154306\pi$
$5$	2.08402i	0.932003i	−0.884784	+	0.466001i	$0.845694\pi$
$6$	0	0
$7$	−5.03127	−1.90164	−0.950821	−	0.309740i	$-0.899758\pi$
$7$	−5.03127	−1.90164	−0.950821	+	0.309740i	$0.899758\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 0.828427i	− 0.249780i	−0.992171	−	0.124890i	$-0.960142\pi$
$11$	− 0.828427i	− 0.249780i	0.992171	−	0.124890i	$-0.0398578\pi$
$12$	0	0
$13$	2.94725i	0.817420i	0.912664	+	0.408710i	$0.134021\pi$
$13$	2.94725i	0.817420i	−0.912664	+	0.408710i	$0.865979\pi$
$14$	0	0
$15$	2.08402	0.538092
$16$	0	0
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	− 2.82843i	− 0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$	− 2.82843i	− 0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$	0	0
$21$	5.03127i	1.09791i
$22$	0	0
$23$	4.16804	0.869097	0.434549	−	0.900648i	$-0.356908\pi$
$23$	4.16804	0.869097	0.434549	+	0.900648i	$0.356908\pi$
$24$	0	0
$25$	0.656854	0.131371
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 7.97852i	− 1.48157i	−0.671740	−	0.740787i	$-0.734453\pi$
$29$	− 7.97852i	− 1.48157i	0.671740	−	0.740787i	$-0.265547\pi$
$30$	0	0
$31$	−5.03127	−0.903643	−0.451822	−	0.892108i	$-0.649226\pi$
$31$	−5.03127	−0.903643	−0.451822	+	0.892108i	$0.649226\pi$
$32$	0	0
$33$	−0.828427	−0.144211
$34$	0	0
$35$	− 10.4853i	− 1.77234i
$36$	0	0
$37$	− 7.11529i	− 1.16975i	−0.811124	−	0.584874i	$-0.801144\pi$
$37$	− 7.11529i	− 1.16975i	0.811124	−	0.584874i	$-0.198856\pi$
$38$	0	0
$39$	2.94725	0.471938
$40$	0	0
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	− 12.4853i	− 1.90399i	−0.306117	−	0.951994i	$-0.599030\pi$
$43$	− 12.4853i	− 1.90399i	0.306117	−	0.951994i	$-0.400970\pi$
$44$	0	0
$45$	− 2.08402i	− 0.310668i
$46$	0	0
$47$	−4.16804	−0.607972	−0.303986	−	0.952677i	$-0.598318\pi$
$47$	−4.16804	−0.607972	−0.303986	+	0.952677i	$0.598318\pi$
$48$	0	0
$49$	18.3137	2.61624
$50$	0	0
$51$	− 4.82843i	− 0.676115i
$52$	0	0
$53$	− 12.1466i	− 1.66846i	−0.551417	−	0.834230i	$-0.685913\pi$
$53$	− 12.1466i	− 1.66846i	0.551417	−	0.834230i	$-0.314087\pi$
$54$	0	0
$55$	1.72646	0.232796
$56$	0	0
$57$	−2.82843	−0.374634
$58$	0	0
$59$	− 1.65685i	− 0.215704i	−0.994167	−	0.107852i	$-0.965603\pi$
$59$	− 1.65685i	− 0.215704i	0.994167	−	0.107852i	$-0.0343973\pi$
$60$	0	0
$61$	7.11529i	0.911020i	0.890231	+	0.455510i	$0.150543\pi$
$61$	7.11529i	0.911020i	−0.890231	+	0.455510i	$0.849457\pi$
$62$	0	0
$63$	5.03127	0.633881
$64$	0	0
$65$	−6.14214	−0.761838
$66$	0	0
$67$	2.34315i	0.286261i	0.989704	+	0.143130i	$0.0457168\pi$
$67$	2.34315i	0.286261i	−0.989704	+	0.143130i	$0.954283\pi$
$68$	0	0
$69$	− 4.16804i	− 0.501773i
$70$	0	0
$71$	10.0625	1.19420	0.597102	−	0.802165i	$-0.296319\pi$
$71$	10.0625	1.19420	0.597102	+	0.802165i	$0.296319\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	− 0.656854i	− 0.0758470i
$76$	0	0
$77$	4.16804i	0.474993i
$78$	0	0
$79$	5.03127	0.566062	0.283031	−	0.959111i	$-0.408660\pi$
$79$	5.03127	0.566062	0.283031	+	0.959111i	$0.408660\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 3.17157i	− 0.348125i	−0.984735	−	0.174063i	$-0.944310\pi$
$83$	− 3.17157i	− 0.348125i	0.984735	−	0.174063i	$-0.0556895\pi$
$84$	0	0
$85$	10.0625i	1.09144i
$86$	0	0
$87$	−7.97852	−0.855388
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	− 14.8284i	− 1.55444i
$92$	0	0
$93$	5.03127i	0.521719i
$94$	0	0
$95$	5.89450	0.604763
$96$	0	0
$97$	0.343146	0.0348412	0.0174206	−	0.999848i	$-0.494455\pi$
$97$	0.343146	0.0348412	0.0174206	+	0.999848i	$0.494455\pi$
$98$	0	0
$99$	0.828427i	0.0832601i
$100$	0	0
$101$	12.1466i	1.20863i	0.796746	+	0.604314i	$0.206553\pi$
$101$	12.1466i	1.20863i	−0.796746	+	0.604314i	$0.793447\pi$
$102$	0	0
$103$	3.30481	0.325633	0.162816	−	0.986656i	$-0.447942\pi$
$103$	3.30481	0.325633	0.162816	+	0.986656i	$0.447942\pi$
$104$	0	0
$105$	−10.4853	−1.02326
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	− 8.84175i	− 0.846886i	−0.905923	−	0.423443i	$-0.860821\pi$
$109$	− 8.84175i	− 0.846886i	0.905923	−	0.423443i	$-0.139179\pi$
$110$	0	0
$111$	−7.11529	−0.675354
$112$	0	0
$113$	11.6569	1.09658	0.548292	−	0.836287i	$-0.315278\pi$
$113$	11.6569	1.09658	0.548292	+	0.836287i	$0.315278\pi$
$114$	0	0
$115$	8.68629i	0.810001i
$116$	0	0
$117$	− 2.94725i	− 0.272473i
$118$	0	0
$119$	−24.2931	−2.22695
$120$	0	0
$121$	10.3137	0.937610
$122$	0	0
$123$	8.82843i	0.796032i
$124$	0	0
$125$	11.7890i	1.05444i
$126$	0	0
$127$	13.3674	1.18616	0.593081	−	0.805143i	$-0.297912\pi$
$127$	13.3674	1.18616	0.593081	+	0.805143i	$0.297912\pi$
$128$	0	0
$129$	−12.4853	−1.09927
$130$	0	0
$131$	− 15.3137i	− 1.33796i	−0.743278	−	0.668982i	$-0.766730\pi$
$131$	− 15.3137i	− 1.33796i	0.743278	−	0.668982i	$-0.233270\pi$
$132$	0	0
$133$	14.2306i	1.23395i
$134$	0	0
$135$	−2.08402	−0.179364
$136$	0	0
$137$	−12.1421	−1.03737	−0.518686	−	0.854965i	$-0.673579\pi$
$137$	−12.1421	−1.03737	−0.518686	+	0.854965i	$0.673579\pi$
$138$	0	0
$139$	− 8.00000i	− 0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$	− 8.00000i	− 0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$	0	0
$141$	4.16804i	0.351013i
$142$	0	0
$143$	2.44158	0.204175
$144$	0	0
$145$	16.6274	1.38083
$146$	0	0
$147$	− 18.3137i	− 1.51049i
$148$	0	0
$149$	2.08402i	0.170730i	0.996350	+	0.0853648i	$0.0272056\pi$
$149$	2.08402i	0.170730i	−0.996350	+	0.0853648i	$0.972794\pi$
$150$	0	0
$151$	13.3674	1.08782	0.543910	−	0.839143i	$-0.316943\pi$
$151$	13.3674	1.08782	0.543910	+	0.839143i	$0.316943\pi$
$152$	0	0
$153$	−4.82843	−0.390355
$154$	0	0
$155$	− 10.4853i	− 0.842198i
$156$	0	0
$157$	− 15.4514i	− 1.23315i	−0.787294	−	0.616577i	$-0.788519\pi$
$157$	− 15.4514i	− 1.23315i	0.787294	−	0.616577i	$-0.211481\pi$
$158$	0	0
$159$	−12.1466	−0.963285
$160$	0	0
$161$	−20.9706	−1.65271
$162$	0	0
$163$	18.1421i	1.42100i	0.703696	+	0.710501i	$0.251532\pi$
$163$	18.1421i	1.42100i	−0.703696	+	0.710501i	$0.748468\pi$
$164$	0	0
$165$	− 1.72646i	− 0.134405i
$166$	0	0
$167$	−14.2306	−1.10120	−0.550598	−	0.834771i	$-0.685600\pi$
$167$	−14.2306	−1.10120	−0.550598	+	0.834771i	$0.685600\pi$
$168$	0	0
$169$	4.31371	0.331824
$170$	0	0
$171$	2.82843i	0.216295i
$172$	0	0
$173$	− 6.25206i	− 0.475336i	−0.971346	−	0.237668i	$-0.923617\pi$
$173$	− 6.25206i	− 0.475336i	0.971346	−	0.237668i	$-0.0763830\pi$
$174$	0	0
$175$	−3.30481	−0.249820
$176$	0	0
$177$	−1.65685	−0.124537
$178$	0	0
$179$	20.9706i	1.56741i	0.621131	+	0.783707i	$0.286673\pi$
$179$	20.9706i	1.56741i	−0.621131	+	0.783707i	$0.713327\pi$
$180$	0	0
$181$	− 8.84175i	− 0.657202i	−0.944469	−	0.328601i	$-0.893423\pi$
$181$	− 8.84175i	− 0.657202i	0.944469	−	0.328601i	$-0.106577\pi$
$182$	0	0
$183$	7.11529	0.525978
$184$	0	0
$185$	14.8284	1.09021
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	− 5.03127i	− 0.365971i
$190$	0	0
$191$	−26.0196	−1.88271	−0.941356	−	0.337415i	$-0.890447\pi$
$191$	−26.0196	−1.88271	−0.941356	+	0.337415i	$0.890447\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	6.14214i	0.439847i
$196$	0	0
$197$	− 20.4827i	− 1.45933i	−0.683806	−	0.729664i	$-0.739676\pi$
$197$	− 20.4827i	− 1.45933i	0.683806	−	0.729664i	$-0.260324\pi$
$198$	0	0
$199$	−5.03127	−0.356657	−0.178329	−	0.983971i	$-0.557069\pi$
$199$	−5.03127	−0.356657	−0.178329	+	0.983971i	$0.557069\pi$
$200$	0	0
$201$	2.34315	0.165273
$202$	0	0
$203$	40.1421i	2.81743i
$204$	0	0
$205$	− 18.3986i	− 1.28502i
$206$	0	0
$207$	−4.16804	−0.289699
$208$	0	0
$209$	−2.34315	−0.162079
$210$	0	0
$211$	− 8.00000i	− 0.550743i	−0.961338	−	0.275371i	$-0.911199\pi$
$211$	− 8.00000i	− 0.550743i	0.961338	−	0.275371i	$-0.0888008\pi$
$212$	0	0
$213$	− 10.0625i	− 0.689474i
$214$	0	0
$215$	26.0196	1.77452
$216$	0	0
$217$	25.3137	1.71841
$218$	0	0
$219$	4.00000i	0.270295i
$220$	0	0
$221$	14.2306i	0.957253i
$222$	0	0
$223$	−13.3674	−0.895145	−0.447572	−	0.894248i	$-0.647711\pi$
$223$	−13.3674	−0.895145	−0.447572	+	0.894248i	$0.647711\pi$
$224$	0	0
$225$	−0.656854	−0.0437903
$226$	0	0
$227$	− 16.1421i	− 1.07139i	−0.844411	−	0.535696i	$-0.820049\pi$
$227$	− 16.1421i	− 1.07139i	0.844411	−	0.535696i	$-0.179951\pi$
$228$	0	0
$229$	− 5.38883i	− 0.356104i	−0.984021	−	0.178052i	$-0.943020\pi$
$229$	− 5.38883i	− 0.356104i	0.984021	−	0.178052i	$-0.0569796\pi$
$230$	0	0
$231$	4.16804	0.274237
$232$	0	0
$233$	−1.31371	−0.0860639	−0.0430320	−	0.999074i	$-0.513702\pi$
$233$	−1.31371	−0.0860639	−0.0430320	+	0.999074i	$0.513702\pi$
$234$	0	0
$235$	− 8.68629i	− 0.566631i
$236$	0	0
$237$	− 5.03127i	− 0.326816i
$238$	0	0
$239$	14.2306	0.920500	0.460250	−	0.887789i	$-0.347760\pi$
$239$	14.2306	0.920500	0.460250	+	0.887789i	$0.347760\pi$
$240$	0	0
$241$	16.9706	1.09317	0.546585	−	0.837404i	$-0.315928\pi$
$241$	16.9706	1.09317	0.546585	+	0.837404i	$0.315928\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	38.1662i	2.43835i
$246$	0	0
$247$	8.33609	0.530412
$248$	0	0
$249$	−3.17157	−0.200990
$250$	0	0
$251$	15.1716i	0.957621i	0.877918	+	0.478811i	$0.158932\pi$
$251$	15.1716i	0.957621i	−0.877918	+	0.478811i	$0.841068\pi$
$252$	0	0
$253$	− 3.45292i	− 0.217083i
$254$	0	0
$255$	10.0625	0.630141
$256$	0	0
$257$	−8.34315	−0.520431	−0.260216	−	0.965551i	$-0.583794\pi$
$257$	−8.34315	−0.520431	−0.260216	+	0.965551i	$0.583794\pi$
$258$	0	0
$259$	35.7990i	2.22444i
$260$	0	0
$261$	7.97852i	0.493858i
$262$	0	0
$263$	−8.33609	−0.514025	−0.257013	−	0.966408i	$-0.582738\pi$
$263$	−8.33609	−0.514025	−0.257013	+	0.966408i	$0.582738\pi$
$264$	0	0
$265$	25.3137	1.55501
$266$	0	0
$267$	10.0000i	0.611990i
$268$	0	0
$269$	7.97852i	0.486459i	0.969969	+	0.243230i	$0.0782069\pi$
$269$	7.97852i	0.486459i	−0.969969	+	0.243230i	$0.921793\pi$
$270$	0	0
$271$	3.30481	0.200753	0.100377	−	0.994950i	$-0.467995\pi$
$271$	3.30481	0.200753	0.100377	+	0.994950i	$0.467995\pi$
$272$	0	0
$273$	−14.8284	−0.897457
$274$	0	0
$275$	− 0.544156i	− 0.0328138i
$276$	0	0
$277$	− 25.5139i	− 1.53298i	−0.642254	−	0.766492i	$-0.722001\pi$
$277$	− 25.5139i	− 1.53298i	0.642254	−	0.766492i	$-0.277999\pi$
$278$	0	0
$279$	5.03127	0.301214
$280$	0	0
$281$	−11.6569	−0.695390	−0.347695	−	0.937608i	$-0.613035\pi$
$281$	−11.6569	−0.695390	−0.347695	+	0.937608i	$0.613035\pi$
$282$	0	0
$283$	4.97056i	0.295469i	0.989027	+	0.147735i	$0.0471982\pi$
$283$	4.97056i	0.295469i	−0.989027	+	0.147735i	$0.952802\pi$
$284$	0	0
$285$	− 5.89450i	− 0.349160i
$286$	0	0
$287$	44.4182	2.62193
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	− 0.343146i	− 0.0201156i
$292$	0	0
$293$	− 12.1466i	− 0.709610i	−0.934940	−	0.354805i	$-0.884547\pi$
$293$	− 12.1466i	− 0.709610i	0.934940	−	0.354805i	$-0.115453\pi$
$294$	0	0
$295$	3.45292	0.201037
$296$	0	0
$297$	0.828427	0.0480702
$298$	0	0
$299$	12.2843i	0.710418i
$300$	0	0
$301$	62.8169i	3.62070i
$302$	0	0
$303$	12.1466	0.697802
$304$	0	0
$305$	−14.8284	−0.849073
$306$	0	0
$307$	10.3431i	0.590315i	0.955449	+	0.295157i	$0.0953720\pi$
$307$	10.3431i	0.590315i	−0.955449	+	0.295157i	$0.904628\pi$
$308$	0	0
$309$	− 3.30481i	− 0.188004i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	15.3137	0.865582	0.432791	−	0.901494i	$-0.357529\pi$
$313$	15.3137	0.865582	0.432791	+	0.901494i	$0.357529\pi$
$314$	0	0
$315$	10.4853i	0.590779i
$316$	0	0
$317$	− 23.9356i	− 1.34436i	−0.740390	−	0.672178i	$-0.765359\pi$
$317$	− 23.9356i	− 1.34436i	0.740390	−	0.672178i	$-0.234641\pi$
$318$	0	0
$319$	−6.60963	−0.370068
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 13.6569i	− 0.759888i
$324$	0	0
$325$	1.93591i	0.107385i
$326$	0	0
$327$	−8.84175	−0.488950
$328$	0	0
$329$	20.9706	1.15614
$330$	0	0
$331$	21.6569i	1.19037i	0.803589	+	0.595184i	$0.202921\pi$
$331$	21.6569i	1.19037i	−0.803589	+	0.595184i	$0.797079\pi$
$332$	0	0
$333$	7.11529i	0.389916i
$334$	0	0
$335$	−4.88317	−0.266796
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	0	0
$339$	− 11.6569i	− 0.633113i
$340$	0	0
$341$	4.16804i	0.225712i
$342$	0	0
$343$	−56.9224	−3.07352
$344$	0	0
$345$	8.68629	0.467654
$346$	0	0
$347$	− 4.82843i	− 0.259204i	−0.991566	−	0.129602i	$-0.958630\pi$
$347$	− 4.82843i	− 0.259204i	0.991566	−	0.129602i	$-0.0413699\pi$
$348$	0	0
$349$	27.2404i	1.45814i	0.684437	+	0.729072i	$0.260048\pi$
$349$	27.2404i	1.45814i	−0.684437	+	0.729072i	$0.739952\pi$
$350$	0	0
$351$	−2.94725	−0.157313
$352$	0	0
$353$	−21.3137	−1.13441	−0.567207	−	0.823575i	$-0.691976\pi$
$353$	−21.3137	−1.13441	−0.567207	+	0.823575i	$0.691976\pi$
$354$	0	0
$355$	20.9706i	1.11300i
$356$	0	0
$357$	24.2931i	1.28573i
$358$	0	0
$359$	6.60963	0.348843	0.174421	−	0.984671i	$-0.444195\pi$
$359$	6.60963	0.348843	0.174421	+	0.984671i	$0.444195\pi$
$360$	0	0
$361$	11.0000	0.578947
$362$	0	0
$363$	− 10.3137i	− 0.541329i
$364$	0	0
$365$	− 8.33609i	− 0.436331i
$366$	0	0
$367$	−16.8203	−0.878011	−0.439006	−	0.898484i	$-0.644669\pi$
$367$	−16.8203	−0.878011	−0.439006	+	0.898484i	$0.644669\pi$
$368$	0	0
$369$	8.82843	0.459590
$370$	0	0
$371$	61.1127i	3.17281i
$372$	0	0
$373$	18.9043i	0.978828i	0.872052	+	0.489414i	$0.162789\pi$
$373$	18.9043i	0.978828i	−0.872052	+	0.489414i	$0.837211\pi$
$374$	0	0
$375$	11.7890	0.608782
$376$	0	0
$377$	23.5147	1.21107
$378$	0	0
$379$	16.4853i	0.846792i	0.905945	+	0.423396i	$0.139162\pi$
$379$	16.4853i	0.846792i	−0.905945	+	0.423396i	$0.860838\pi$
$380$	0	0
$381$	− 13.3674i	− 0.684831i
$382$	0	0
$383$	−5.89450	−0.301195	−0.150598	−	0.988595i	$-0.548120\pi$
$383$	−5.89450	−0.301195	−0.150598	+	0.988595i	$0.548120\pi$
$384$	0	0
$385$	−8.68629	−0.442694
$386$	0	0
$387$	12.4853i	0.634663i
$388$	0	0
$389$	− 2.08402i	− 0.105664i	−0.998603	−	0.0528320i	$-0.983175\pi$
$389$	− 2.08402i	− 0.105664i	0.998603	−	0.0528320i	$-0.0168248\pi$
$390$	0	0
$391$	20.1251	1.01777
$392$	0	0
$393$	−15.3137	−0.772474
$394$	0	0
$395$	10.4853i	0.527572i
$396$	0	0
$397$	− 15.4514i	− 0.775483i	−0.921768	−	0.387741i	$-0.873255\pi$
$397$	− 15.4514i	− 0.775483i	0.921768	−	0.387741i	$-0.126745\pi$
$398$	0	0
$399$	14.2306	0.712421
$400$	0	0
$401$	−4.14214	−0.206848	−0.103424	−	0.994637i	$-0.532980\pi$
$401$	−4.14214	−0.206848	−0.103424	+	0.994637i	$0.532980\pi$
$402$	0	0
$403$	− 14.8284i	− 0.738657i
$404$	0	0
$405$	2.08402i	0.103556i
$406$	0	0
$407$	−5.89450	−0.292180
$408$	0	0
$409$	−7.65685	−0.378607	−0.189304	−	0.981919i	$-0.560623\pi$
$409$	−7.65685	−0.378607	−0.189304	+	0.981919i	$0.560623\pi$
$410$	0	0
$411$	12.1421i	0.598927i
$412$	0	0
$413$	8.33609i	0.410192i
$414$	0	0
$415$	6.60963	0.324454
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	− 20.8284i	− 1.01754i	−0.860904	−	0.508768i	$-0.830101\pi$
$419$	− 20.8284i	− 1.01754i	0.860904	−	0.508768i	$-0.169899\pi$
$420$	0	0
$421$	− 5.38883i	− 0.262636i	−0.991340	−	0.131318i	$-0.958079\pi$
$421$	− 5.38883i	− 0.262636i	0.991340	−	0.131318i	$-0.0419209\pi$
$422$	0	0
$423$	4.16804	0.202657
$424$	0	0
$425$	3.17157	0.153844
$426$	0	0
$427$	− 35.7990i	− 1.73243i
$428$	0	0
$429$	− 2.44158i	− 0.117881i
$430$	0	0
$431$	−15.9570	−0.768624	−0.384312	−	0.923203i	$-0.625561\pi$
$431$	−15.9570	−0.768624	−0.384312	+	0.923203i	$0.625561\pi$
$432$	0	0
$433$	−20.6274	−0.991290	−0.495645	−	0.868525i	$-0.665068\pi$
$433$	−20.6274	−0.991290	−0.495645	+	0.868525i	$0.665068\pi$
$434$	0	0
$435$	− 16.6274i	− 0.797224i
$436$	0	0
$437$	− 11.7890i	− 0.563945i
$438$	0	0
$439$	−8.48419	−0.404928	−0.202464	−	0.979290i	$-0.564895\pi$
$439$	−8.48419	−0.404928	−0.202464	+	0.979290i	$0.564895\pi$
$440$	0	0
$441$	−18.3137	−0.872081
$442$	0	0
$443$	− 1.51472i	− 0.0719665i	−0.999352	−	0.0359832i	$-0.988544\pi$
$443$	− 1.51472i	− 0.0719665i	0.999352	−	0.0359832i	$-0.0114563\pi$
$444$	0	0
$445$	− 20.8402i	− 0.987921i
$446$	0	0
$447$	2.08402	0.0985708
$448$	0	0
$449$	34.4853	1.62746	0.813731	−	0.581242i	$-0.197433\pi$
$449$	34.4853	1.62746	0.813731	+	0.581242i	$0.197433\pi$
$450$	0	0
$451$	7.31371i	0.344389i
$452$	0	0
$453$	− 13.3674i	− 0.628053i
$454$	0	0
$455$	30.9028	1.44874
$456$	0	0
$457$	12.3431	0.577388	0.288694	−	0.957421i	$-0.406779\pi$
$457$	12.3431	0.577388	0.288694	+	0.957421i	$0.406779\pi$
$458$	0	0
$459$	4.82843i	0.225372i
$460$	0	0
$461$	− 29.8301i	− 1.38933i	−0.719336	−	0.694663i	$-0.755554\pi$
$461$	− 29.8301i	− 1.38933i	0.719336	−	0.694663i	$-0.244446\pi$
$462$	0	0
$463$	−23.4299	−1.08888	−0.544440	−	0.838800i	$-0.683258\pi$
$463$	−23.4299	−1.08888	−0.544440	+	0.838800i	$0.683258\pi$
$464$	0	0
$465$	−10.4853	−0.486243
$466$	0	0
$467$	1.79899i	0.0832473i	0.999133	+	0.0416237i	$0.0132531\pi$
$467$	1.79899i	0.0832473i	−0.999133	+	0.0416237i	$0.986747\pi$
$468$	0	0
$469$	− 11.7890i	− 0.544366i
$470$	0	0
$471$	−15.4514	−0.711962
$472$	0	0
$473$	−10.3431	−0.475578
$474$	0	0
$475$	− 1.85786i	− 0.0852447i
$476$	0	0
$477$	12.1466i	0.556153i
$478$	0	0
$479$	−41.9766	−1.91796	−0.958981	−	0.283471i	$-0.908514\pi$
$479$	−41.9766	−1.91796	−0.958981	+	0.283471i	$0.908514\pi$
$480$	0	0
$481$	20.9706	0.956175
$482$	0	0
$483$	20.9706i	0.954194i
$484$	0	0
$485$	0.715123i	0.0324721i
$486$	0	0
$487$	6.75773	0.306222	0.153111	−	0.988209i	$-0.451071\pi$
$487$	6.75773	0.306222	0.153111	+	0.988209i	$0.451071\pi$
$488$	0	0
$489$	18.1421	0.820416
$490$	0	0
$491$	− 9.65685i	− 0.435808i	−0.975970	−	0.217904i	$-0.930078\pi$
$491$	− 9.65685i	− 0.435808i	0.975970	−	0.217904i	$-0.0699219\pi$
$492$	0	0
$493$	− 38.5237i	− 1.73502i
$494$	0	0
$495$	−1.72646	−0.0775986
$496$	0	0
$497$	−50.6274	−2.27095
$498$	0	0
$499$	41.6569i	1.86482i	0.361406	+	0.932408i	$0.382297\pi$
$499$	41.6569i	1.86482i	−0.361406	+	0.932408i	$0.617703\pi$
$500$	0	0
$501$	14.2306i	0.635776i
$502$	0	0
$503$	18.3986	0.820354	0.410177	−	0.912006i	$-0.365467\pi$
$503$	18.3986	0.820354	0.410177	+	0.912006i	$0.365467\pi$
$504$	0	0
$505$	−25.3137	−1.12645
$506$	0	0
$507$	− 4.31371i	− 0.191579i
$508$	0	0
$509$	− 16.3146i	− 0.723132i	−0.932346	−	0.361566i	$-0.882242\pi$
$509$	− 16.3146i	− 0.723132i	0.932346	−	0.361566i	$-0.117758\pi$
$510$	0	0
$511$	20.1251	0.890282
$512$	0	0
$513$	2.82843	0.124878
$514$	0	0
$515$	6.88730i	0.303491i
$516$	0	0
$517$	3.45292i	0.151859i
$518$	0	0
$519$	−6.25206	−0.274435
$520$	0	0
$521$	−20.8284	−0.912510	−0.456255	−	0.889849i	$-0.650810\pi$
$521$	−20.8284	−0.912510	−0.456255	+	0.889849i	$0.650810\pi$
$522$	0	0
$523$	18.8284i	0.823310i	0.911340	+	0.411655i	$0.135049\pi$
$523$	18.8284i	0.823310i	−0.911340	+	0.411655i	$0.864951\pi$
$524$	0	0
$525$	3.30481i	0.144234i
$526$	0	0
$527$	−24.2931	−1.05823
$528$	0	0
$529$	−5.62742	−0.244670
$530$	0	0
$531$	1.65685i	0.0719014i
$532$	0	0
$533$	− 26.0196i	− 1.12703i
$534$	0	0
$535$	8.33609	0.360400
$536$	0	0
$537$	20.9706	0.904947
$538$	0	0
$539$	− 15.1716i	− 0.653486i
$540$	0	0
$541$	− 11.2833i	− 0.485109i	−0.970138	−	0.242554i	$-0.922015\pi$
$541$	− 11.2833i	− 0.485109i	0.970138	−	0.242554i	$-0.0779853\pi$
$542$	0	0
$543$	−8.84175	−0.379436
$544$	0	0
$545$	18.4264	0.789301
$546$	0	0
$547$	− 31.7990i	− 1.35963i	−0.733385	−	0.679813i	$-0.762061\pi$
$547$	− 31.7990i	− 1.35963i	0.733385	−	0.679813i	$-0.237939\pi$
$548$	0	0
$549$	− 7.11529i	− 0.303673i
$550$	0	0
$551$	−22.5667	−0.961373
$552$	0	0
$553$	−25.3137	−1.07645
$554$	0	0
$555$	− 14.8284i	− 0.629432i
$556$	0	0
$557$	14.5882i	0.618120i	0.951043	+	0.309060i	$0.100014\pi$
$557$	14.5882i	0.618120i	−0.951043	+	0.309060i	$0.899986\pi$
$558$	0	0
$559$	36.7973	1.55636
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	0.828427i	0.0349140i	0.999848	+	0.0174570i	$0.00555702\pi$
$563$	0.828427i	0.0349140i	−0.999848	+	0.0174570i	$0.994443\pi$
$564$	0	0
$565$	24.2931i	1.02202i
$566$	0	0
$567$	−5.03127	−0.211294
$568$	0	0
$569$	−23.4558	−0.983320	−0.491660	−	0.870787i	$-0.663610\pi$
$569$	−23.4558	−0.983320	−0.491660	+	0.870787i	$0.663610\pi$
$570$	0	0
$571$	23.3137i	0.975648i	0.872942	+	0.487824i	$0.162209\pi$
$571$	23.3137i	0.975648i	−0.872942	+	0.487824i	$0.837791\pi$
$572$	0	0
$573$	26.0196i	1.08698i
$574$	0	0
$575$	2.73780	0.114174
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	4.00000i	0.166234i
$580$	0	0
$581$	15.9570i	0.662010i
$582$	0	0
$583$	−10.0625	−0.416748
$584$	0	0
$585$	6.14214	0.253946
$586$	0	0
$587$	− 4.97056i	− 0.205157i	−0.994725	−	0.102579i	$-0.967291\pi$
$587$	− 4.97056i	− 0.205157i	0.994725	−	0.102579i	$-0.0327093\pi$
$588$	0	0
$589$	14.2306i	0.586361i
$590$	0	0
$591$	−20.4827	−0.842544
$592$	0	0
$593$	−2.97056	−0.121986	−0.0609932	−	0.998138i	$-0.519427\pi$
$593$	−2.97056	−0.121986	−0.0609932	+	0.998138i	$0.519427\pi$
$594$	0	0
$595$	− 50.6274i	− 2.07552i
$596$	0	0
$597$	5.03127i	0.205916i
$598$	0	0
$599$	−0.715123	−0.0292191	−0.0146096	−	0.999893i	$-0.504651\pi$
$599$	−0.715123	−0.0292191	−0.0146096	+	0.999893i	$0.504651\pi$
$600$	0	0
$601$	4.68629	0.191158	0.0955789	−	0.995422i	$-0.469530\pi$
$601$	4.68629	0.191158	0.0955789	+	0.995422i	$0.469530\pi$
$602$	0	0
$603$	− 2.34315i	− 0.0954203i
$604$	0	0
$605$	21.4940i	0.873855i
$606$	0	0
$607$	25.1564	1.02107	0.510533	−	0.859858i	$-0.329448\pi$
$607$	25.1564	1.02107	0.510533	+	0.859858i	$0.329448\pi$
$608$	0	0
$609$	40.1421	1.62664
$610$	0	0
$611$	− 12.2843i	− 0.496968i
$612$	0	0
$613$	10.5682i	0.426846i	0.976960	+	0.213423i	$0.0684613\pi$
$613$	10.5682i	0.426846i	−0.976960	+	0.213423i	$0.931539\pi$
$614$	0	0
$615$	−18.3986	−0.741904
$616$	0	0
$617$	43.9411	1.76900	0.884502	−	0.466537i	$-0.154499\pi$
$617$	43.9411	1.76900	0.884502	+	0.466537i	$0.154499\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	4.16804i	0.167258i
$622$	0	0
$623$	50.3127	2.01574
$624$	0	0
$625$	−21.2843	−0.851371
$626$	0	0
$627$	2.34315i	0.0935762i
$628$	0	0
$629$	− 34.3557i	− 1.36985i
$630$	0	0
$631$	35.2189	1.40204	0.701021	−	0.713140i	$-0.252728\pi$
$631$	35.2189	1.40204	0.701021	+	0.713140i	$0.252728\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	27.8579i	1.10551i
$636$	0	0
$637$	53.9751i	2.13857i
$638$	0	0
$639$	−10.0625	−0.398068
$640$	0	0
$641$	32.8284	1.29664	0.648322	−	0.761366i	$-0.275471\pi$
$641$	32.8284	1.29664	0.648322	+	0.761366i	$0.275471\pi$
$642$	0	0
$643$	− 3.51472i	− 0.138607i	−0.997596	−	0.0693035i	$-0.977922\pi$
$643$	− 3.51472i	− 0.138607i	0.997596	−	0.0693035i	$-0.0220777\pi$
$644$	0	0
$645$	− 26.0196i	− 1.02452i
$646$	0	0
$647$	−44.4182	−1.74626	−0.873130	−	0.487487i	$-0.837914\pi$
$647$	−44.4182	−1.74626	−0.873130	+	0.487487i	$0.837914\pi$
$648$	0	0
$649$	−1.37258	−0.0538786
$650$	0	0
$651$	− 25.3137i	− 0.992122i
$652$	0	0
$653$	30.5452i	1.19533i	0.801747	+	0.597663i	$0.203904\pi$
$653$	30.5452i	1.19533i	−0.801747	+	0.597663i	$0.796096\pi$
$654$	0	0
$655$	31.9141	1.24699
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	9.65685i	0.376178i	0.982152	+	0.188089i	$0.0602293\pi$
$659$	9.65685i	0.376178i	−0.982152	+	0.188089i	$0.939771\pi$
$660$	0	0
$661$	27.2404i	1.05953i	0.848145	+	0.529764i	$0.177720\pi$
$661$	27.2404i	1.05953i	−0.848145	+	0.529764i	$0.822280\pi$
$662$	0	0
$663$	14.2306	0.552670
$664$	0	0
$665$	−29.6569	−1.15004
$666$	0	0
$667$	− 33.2548i	− 1.28763i
$668$	0	0
$669$	13.3674i	0.516812i
$670$	0	0
$671$	5.89450	0.227555
$672$	0	0
$673$	37.3137	1.43834	0.719169	−	0.694835i	$-0.244523\pi$
$673$	37.3137	1.43834	0.719169	+	0.694835i	$0.244523\pi$
$674$	0	0
$675$	0.656854i	0.0252823i
$676$	0	0
$677$	22.2091i	0.853566i	0.904354	+	0.426783i	$0.140353\pi$
$677$	22.2091i	0.853566i	−0.904354	+	0.426783i	$0.859647\pi$
$678$	0	0
$679$	−1.72646	−0.0662555
$680$	0	0
$681$	−16.1421	−0.618568
$682$	0	0
$683$	− 22.4853i	− 0.860375i	−0.902739	−	0.430188i	$-0.858447\pi$
$683$	− 22.4853i	− 0.860375i	0.902739	−	0.430188i	$-0.141553\pi$
$684$	0	0
$685$	− 25.3045i	− 0.966834i
$686$	0	0
$687$	−5.38883	−0.205597
$688$	0	0
$689$	35.7990	1.36383
$690$	0	0
$691$	− 33.1716i	− 1.26191i	−0.775821	−	0.630953i	$-0.782664\pi$
$691$	− 33.1716i	− 1.26191i	0.775821	−	0.630953i	$-0.217336\pi$
$692$	0	0
$693$	− 4.16804i	− 0.158331i
$694$	0	0
$695$	16.6722	0.632412
$696$	0	0
$697$	−42.6274	−1.61463
$698$	0	0
$699$	1.31371i	0.0496890i
$700$	0	0
$701$	4.52560i	0.170930i	0.996341	+	0.0854649i	$0.0272375\pi$
$701$	4.52560i	0.170930i	−0.996341	+	0.0854649i	$0.972762\pi$
$702$	0	0
$703$	−20.1251	−0.759032
$704$	0	0
$705$	−8.68629	−0.327145
$706$	0	0
$707$	− 61.1127i	− 2.29838i
$708$	0	0
$709$	19.6194i	0.736823i	0.929663	+	0.368411i	$0.120098\pi$
$709$	19.6194i	0.736823i	−0.929663	+	0.368411i	$0.879902\pi$
$710$	0	0
$711$	−5.03127	−0.188687
$712$	0	0
$713$	−20.9706	−0.785354
$714$	0	0
$715$	5.08831i	0.190292i
$716$	0	0
$717$	− 14.2306i	− 0.531451i
$718$	0	0
$719$	6.60963	0.246497	0.123249	−	0.992376i	$-0.460669\pi$
$719$	6.60963	0.246497	0.123249	+	0.992376i	$0.460669\pi$
$720$	0	0
$721$	−16.6274	−0.619237
$722$	0	0
$723$	− 16.9706i	− 0.631142i
$724$	0	0
$725$	− 5.24073i	− 0.194636i
$726$	0	0
$727$	36.9454	1.37023	0.685114	−	0.728436i	$-0.259752\pi$
$727$	36.9454	1.37023	0.685114	+	0.728436i	$0.259752\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 60.2843i	− 2.22969i
$732$	0	0
$733$	− 43.1974i	− 1.59553i	−0.602966	−	0.797767i	$-0.706015\pi$
$733$	− 43.1974i	− 1.59553i	0.602966	−	0.797767i	$-0.293985\pi$
$734$	0	0
$735$	38.1662	1.40778
$736$	0	0
$737$	1.94113	0.0715023
$738$	0	0
$739$	4.00000i	0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$	4.00000i	0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$	0	0
$741$	− 8.33609i	− 0.306234i
$742$	0	0
$743$	17.6835	0.648745	0.324373	−	0.945929i	$-0.394847\pi$
$743$	17.6835	0.648745	0.324373	+	0.945929i	$0.394847\pi$
$744$	0	0
$745$	−4.34315	−0.159121
$746$	0	0
$747$	3.17157i	0.116042i
$748$	0	0
$749$	20.1251i	0.735355i
$750$	0	0
$751$	5.03127	0.183594	0.0917969	−	0.995778i	$-0.470739\pi$
$751$	5.03127	0.183594	0.0917969	+	0.995778i	$0.470739\pi$
$752$	0	0
$753$	15.1716	0.552883
$754$	0	0
$755$	27.8579i	1.01385i
$756$	0	0
$757$	− 17.1778i	− 0.624339i	−0.950026	−	0.312170i	$-0.898944\pi$
$757$	− 17.1778i	− 0.624339i	0.950026	−	0.312170i	$-0.101056\pi$
$758$	0	0
$759$	−3.45292	−0.125333
$760$	0	0
$761$	42.7696	1.55040	0.775198	−	0.631719i	$-0.217650\pi$
$761$	42.7696	1.55040	0.775198	+	0.631719i	$0.217650\pi$
$762$	0	0
$763$	44.4853i	1.61048i
$764$	0	0
$765$	− 10.0625i	− 0.363812i
$766$	0	0
$767$	4.88317	0.176321
$768$	0	0
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	8.34315i	0.300471i
$772$	0	0
$773$	32.2717i	1.16073i	0.814356	+	0.580365i	$0.197090\pi$
$773$	32.2717i	1.16073i	−0.814356	+	0.580365i	$0.802910\pi$
$774$	0	0
$775$	−3.30481	−0.118712
$776$	0	0
$777$	35.7990	1.28428
$778$	0	0
$779$	24.9706i	0.894663i
$780$	0	0
$781$	− 8.33609i	− 0.298289i
$782$	0	0
$783$	7.97852	0.285129
$784$	0	0
$785$	32.2010	1.14930
$786$	0	0
$787$	− 52.7696i	− 1.88103i	−0.339750	−	0.940516i	$-0.610343\pi$
$787$	− 52.7696i	− 1.88103i	0.339750	−	0.940516i	$-0.389657\pi$
$788$	0	0
$789$	8.33609i	0.296773i
$790$	0	0
$791$	−58.6488	−2.08531
$792$	0	0
$793$	−20.9706	−0.744687
$794$	0	0
$795$	− 25.3137i	− 0.897785i
$796$	0	0
$797$	− 30.5452i	− 1.08197i	−0.841033	−	0.540983i	$-0.818052\pi$
$797$	− 30.5452i	− 1.08197i	0.841033	−	0.540983i	$-0.181948\pi$
$798$	0	0
$799$	−20.1251	−0.711975
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	3.31371i	0.116938i
$804$	0	0
$805$	− 43.7031i	− 1.54033i
$806$	0	0
$807$	7.97852	0.280857
$808$	0	0
$809$	3.45584	0.121501	0.0607505	−	0.998153i	$-0.480651\pi$
$809$	3.45584	0.121501	0.0607505	+	0.998153i	$0.480651\pi$
$810$	0	0
$811$	49.4558i	1.73663i	0.496014	+	0.868315i	$0.334797\pi$
$811$	49.4558i	1.73663i	−0.496014	+	0.868315i	$0.665203\pi$
$812$	0	0
$813$	− 3.30481i	− 0.115905i
$814$	0	0
$815$	−37.8086	−1.32438
$816$	0	0
$817$	−35.3137	−1.23547
$818$	0	0
$819$	14.8284i	0.518147i
$820$	0	0
$821$	− 36.4397i	− 1.27175i	−0.771790	−	0.635877i	$-0.780638\pi$
$821$	− 36.4397i	− 1.27175i	0.771790	−	0.635877i	$-0.219362\pi$
$822$	0	0
$823$	−13.3674	−0.465957	−0.232978	−	0.972482i	$-0.574847\pi$
$823$	−13.3674	−0.465957	−0.232978	+	0.972482i	$0.574847\pi$
$824$	0	0
$825$	−0.544156	−0.0189451
$826$	0	0
$827$	− 53.9411i	− 1.87572i	−0.347018	−	0.937858i	$-0.612806\pi$
$827$	− 53.9411i	− 1.87572i	0.347018	−	0.937858i	$-0.387194\pi$
$828$	0	0
$829$	2.94725i	0.102362i	0.998689	+	0.0511811i	$0.0162986\pi$
$829$	2.94725i	0.102362i	−0.998689	+	0.0511811i	$0.983701\pi$
$830$	0	0
$831$	−25.5139	−0.885068
$832$	0	0
$833$	88.4264	3.06379
$834$	0	0
$835$	− 29.6569i	− 1.02632i
$836$	0	0
$837$	− 5.03127i	− 0.173906i
$838$	0	0
$839$	41.9766	1.44919	0.724597	−	0.689172i	$-0.242026\pi$
$839$	41.9766	1.44919	0.724597	+	0.689172i	$0.242026\pi$
$840$	0	0
$841$	−34.6569	−1.19506
$842$	0	0
$843$	11.6569i	0.401483i
$844$	0	0
$845$	8.98986i	0.309261i
$846$	0	0
$847$	−51.8911	−1.78300
$848$	0	0
$849$	4.97056	0.170589
$850$	0	0
$851$	− 29.6569i	− 1.01662i
$852$	0	0
$853$	15.4514i	0.529045i	0.964379	+	0.264523i	$0.0852144\pi$
$853$	15.4514i	0.529045i	−0.964379	+	0.264523i	$0.914786\pi$
$854$	0	0
$855$	−5.89450	−0.201588
$856$	0	0
$857$	−30.4853	−1.04136	−0.520679	−	0.853753i	$-0.674321\pi$
$857$	−30.4853	−1.04136	−0.520679	+	0.853753i	$0.674321\pi$
$858$	0	0
$859$	21.4558i	0.732064i	0.930602	+	0.366032i	$0.119284\pi$
$859$	21.4558i	0.732064i	−0.930602	+	0.366032i	$0.880716\pi$
$860$	0	0
$861$	− 44.4182i	− 1.51377i
$862$	0	0
$863$	−14.2306	−0.484415	−0.242207	−	0.970224i	$-0.577871\pi$
$863$	−14.2306	−0.484415	−0.242207	+	0.970224i	$0.577871\pi$
$864$	0	0
$865$	13.0294	0.443014
$866$	0	0
$867$	− 6.31371i	− 0.214425i
$868$	0	0
$869$	− 4.16804i	− 0.141391i
$870$	0	0
$871$	−6.90584	−0.233995
$872$	0	0
$873$	−0.343146	−0.0116137
$874$	0	0
$875$	− 59.3137i	− 2.00517i
$876$	0	0
$877$	22.3572i	0.754950i	0.926020	+	0.377475i	$0.123208\pi$
$877$	22.3572i	0.754950i	−0.926020	+	0.377475i	$0.876792\pi$
$878$	0	0
$879$	−12.1466	−0.409694
$880$	0	0
$881$	−38.9706	−1.31295	−0.656476	−	0.754347i	$-0.727954\pi$
$881$	−38.9706	−1.31295	−0.656476	+	0.754347i	$0.727954\pi$
$882$	0	0
$883$	− 24.7696i	− 0.833562i	−0.909007	−	0.416781i	$-0.863158\pi$
$883$	− 24.7696i	− 0.833562i	0.909007	−	0.416781i	$-0.136842\pi$
$884$	0	0
$885$	− 3.45292i	− 0.116069i
$886$	0	0
$887$	−8.33609	−0.279898	−0.139949	−	0.990159i	$-0.544694\pi$
$887$	−8.33609	−0.279898	−0.139949	+	0.990159i	$0.544694\pi$
$888$	0	0
$889$	−67.2548	−2.25565
$890$	0	0
$891$	− 0.828427i	− 0.0277534i
$892$	0	0
$893$	11.7890i	0.394504i
$894$	0	0
$895$	−43.7031	−1.46083
$896$	0	0
$897$	12.2843	0.410160
$898$	0	0
$899$	40.1421i	1.33882i
$900$	0	0
$901$	− 58.6488i	− 1.95388i
$902$	0	0
$903$	62.8169	2.09041
$904$	0	0
$905$	18.4264	0.612514
$906$	0	0
$907$	− 23.7990i	− 0.790232i	−0.918631	−	0.395116i	$-0.870704\pi$
$907$	− 23.7990i	− 0.790232i	0.918631	−	0.395116i	$-0.129296\pi$
$908$	0	0
$909$	− 12.1466i	− 0.402876i
$910$	0	0
$911$	34.3557	1.13825	0.569127	−	0.822249i	$-0.307281\pi$
$911$	34.3557	1.13825	0.569127	+	0.822249i	$0.307281\pi$
$912$	0	0
$913$	−2.62742	−0.0869548
$914$	0	0
$915$	14.8284i	0.490213i
$916$	0	0
$917$	77.0474i	2.54433i
$918$	0	0
$919$	15.0938	0.497899	0.248950	−	0.968516i	$-0.419915\pi$
$919$	15.0938	0.497899	0.248950	+	0.968516i	$0.419915\pi$
$920$	0	0
$921$	10.3431	0.340818
$922$	0	0
$923$	29.6569i	0.976167i
$924$	0	0
$925$	− 4.67371i	− 0.153671i
$926$	0	0
$927$	−3.30481	−0.108544
$928$	0	0
$929$	−17.7990	−0.583966	−0.291983	−	0.956424i	$-0.594315\pi$
$929$	−17.7990	−0.583966	−0.291983	+	0.956424i	$0.594315\pi$
$930$	0	0
$931$	− 51.7990i	− 1.69764i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.33609	0.272619
$936$	0	0
$937$	−4.62742	−0.151171	−0.0755856	−	0.997139i	$-0.524083\pi$
$937$	−4.62742	−0.151171	−0.0755856	+	0.997139i	$0.524083\pi$
$938$	0	0
$939$	− 15.3137i	− 0.499744i
$940$	0	0
$941$	− 51.3854i	− 1.67512i	−0.546348	−	0.837558i	$-0.683982\pi$
$941$	− 51.3854i	− 1.67512i	0.546348	−	0.837558i	$-0.316018\pi$
$942$	0	0
$943$	−36.7973	−1.19828
$944$	0	0
$945$	10.4853	0.341086
$946$	0	0
$947$	− 4.97056i	− 0.161522i	−0.996734	−	0.0807608i	$-0.974265\pi$
$947$	− 4.97056i	− 0.161522i	0.996734	−	0.0807608i	$-0.0257350\pi$
$948$	0	0
$949$	− 11.7890i	− 0.382687i
$950$	0	0
$951$	−23.9356	−0.776164
$952$	0	0
$953$	9.79899	0.317420	0.158710	−	0.987325i	$-0.449266\pi$
$953$	9.79899	0.317420	0.158710	+	0.987325i	$0.449266\pi$
$954$	0	0
$955$	− 54.2254i	− 1.75469i
$956$	0	0
$957$	6.60963i	0.213659i
$958$	0	0
$959$	61.0904	1.97271
$960$	0	0
$961$	−5.68629	−0.183429
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	− 8.33609i	− 0.268348i
$966$	0	0
$967$	−18.5467	−0.596423	−0.298211	−	0.954500i	$-0.596390\pi$
$967$	−18.5467	−0.596423	−0.298211	+	0.954500i	$0.596390\pi$
$968$	0	0
$969$	−13.6569	−0.438721
$970$	0	0
$971$	35.4558i	1.13783i	0.822396	+	0.568916i	$0.192637\pi$
$971$	35.4558i	1.13783i	−0.822396	+	0.568916i	$0.807363\pi$
$972$	0	0
$973$	40.2502i	1.29036i
$974$	0	0
$975$	1.93591	0.0619989
$976$	0	0
$977$	30.7696	0.984405	0.492203	−	0.870481i	$-0.336192\pi$
$977$	30.7696	0.984405	0.492203	+	0.870481i	$0.336192\pi$
$978$	0	0
$979$	8.28427i	0.264766i
$980$	0	0
$981$	8.84175i	0.282295i
$982$	0	0
$983$	−3.45292	−0.110131	−0.0550655	−	0.998483i	$-0.517537\pi$
$983$	−3.45292	−0.110131	−0.0550655	+	0.998483i	$0.517537\pi$
$984$	0	0
$985$	42.6863	1.36010
$986$	0	0
$987$	− 20.9706i	− 0.667500i
$988$	0	0
$989$	− 52.0392i	− 1.65475i
$990$	0	0
$991$	28.6093	0.908804	0.454402	−	0.890797i	$-0.349853\pi$
$991$	28.6093	0.908804	0.454402	+	0.890797i	$0.349853\pi$
$992$	0	0
$993$	21.6569	0.687259
$994$	0	0
$995$	− 10.4853i	− 0.332406i
$996$	0	0
$997$	9.55688i	0.302669i	0.988483	+	0.151335i	$0.0483571\pi$
$997$	9.55688i	0.302669i	−0.988483	+	0.151335i	$0.951643\pi$
$998$	0	0
$999$	7.11529	0.225118

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.d.g.769.3		8
3.2	odd	2		4608.2.d.p.2305.3		8
4.3	odd	2	inner	1536.2.d.g.769.7		8
8.3	odd	2	inner	1536.2.d.g.769.2		8
8.5	even	2	inner	1536.2.d.g.769.6		8
12.11	even	2		4608.2.d.p.2305.4		8
16.3	odd	4		1536.2.a.m.1.3	yes	4
16.5	even	4		1536.2.a.m.1.2	✓	4
16.11	odd	4		1536.2.a.n.1.2	yes	4
16.13	even	4		1536.2.a.n.1.3	yes	4
24.5	odd	2		4608.2.d.p.2305.5		8
24.11	even	2		4608.2.d.p.2305.6		8
48.5	odd	4		4608.2.a.ba.1.3		4
48.11	even	4		4608.2.a.t.1.3		4
48.29	odd	4		4608.2.a.t.1.2		4
48.35	even	4		4608.2.a.ba.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.m.1.2	✓	4	16.5	even	4
1536.2.a.m.1.3	yes	4	16.3	odd	4
1536.2.a.n.1.2	yes	4	16.11	odd	4
1536.2.a.n.1.3	yes	4	16.13	even	4
1536.2.d.g.769.2		8	8.3	odd	2	inner
1536.2.d.g.769.3		8	1.1	even	1	trivial
1536.2.d.g.769.6		8	8.5	even	2	inner
1536.2.d.g.769.7		8	4.3	odd	2	inner
4608.2.a.t.1.2		4	48.29	odd	4
4608.2.a.t.1.3		4	48.11	even	4
4608.2.a.ba.1.2		4	48.35	even	4
4608.2.a.ba.1.3		4	48.5	odd	4
4608.2.d.p.2305.3		8	3.2	odd	2
4608.2.d.p.2305.4		8	12.11	even	2
4608.2.d.p.2305.5		8	24.5	odd	2
4608.2.d.p.2305.6		8	24.11	even	2

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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.08402i q^{5} -5.03127 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.08402i q^{5} -5.03127 q^{7} -1.00000 q^{9} -0.828427i q^{11} +2.94725i q^{13} +2.08402 q^{15} +4.82843 q^{17} -2.82843i q^{19} +5.03127i q^{21} +4.16804 q^{23} +0.656854 q^{25} +1.00000i q^{27} -7.97852i q^{29} -5.03127 q^{31} -0.828427 q^{33} -10.4853i q^{35} -7.11529i q^{37} +2.94725 q^{39} -8.82843 q^{41} -12.4853i q^{43} -2.08402i q^{45} -4.16804 q^{47} +18.3137 q^{49} -4.82843i q^{51} -12.1466i q^{53} +1.72646 q^{55} -2.82843 q^{57} -1.65685i q^{59} +7.11529i q^{61} +5.03127 q^{63} -6.14214 q^{65} +2.34315i q^{67} -4.16804i q^{69} +10.0625 q^{71} -4.00000 q^{73} -0.656854i q^{75} +4.16804i q^{77} +5.03127 q^{79} +1.00000 q^{81} -3.17157i q^{83} +10.0625i q^{85} -7.97852 q^{87} -10.0000 q^{89} -14.8284i q^{91} +5.03127i q^{93} +5.89450 q^{95} +0.343146 q^{97} +0.828427i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 16 q^{17} - 40 q^{25} + 16 q^{33} - 48 q^{41} + 56 q^{49} + 64 q^{65} - 32 q^{73} + 8 q^{81} - 80 q^{89} + 48 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^17 - 40 * q^25 + 16 * q^33 - 48 * q^41 + 56 * q^49 + 64 * q^65 - 32 * q^73 + 8 * q^81 - 80 * q^89 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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