LMFDB - Embedded newform 1536.2.d.a.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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1536.769
Dual form			1536.2.d.a.769.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -3.41421i q^{5} -0.585786 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -3.41421i q^{5} -0.585786 q^{7} -1.00000 q^{9} -2.00000i q^{11} -2.82843i q^{13} +3.41421 q^{15} -7.65685 q^{17} +5.65685i q^{19} -0.585786i q^{21} -6.82843 q^{23} -6.65685 q^{25} -1.00000i q^{27} +3.41421i q^{29} +7.41421 q^{31} +2.00000 q^{33} +2.00000i q^{35} -1.65685i q^{37} +2.82843 q^{39} +0.343146 q^{41} +9.65685i q^{43} +3.41421i q^{45} -4.48528 q^{47} -6.65685 q^{49} -7.65685i q^{51} +7.89949i q^{53} -6.82843 q^{55} -5.65685 q^{57} +4.00000i q^{59} +1.65685i q^{61} +0.585786 q^{63} -9.65685 q^{65} -8.00000i q^{67} -6.82843i q^{69} -14.8284 q^{71} -9.65685 q^{73} -6.65685i q^{75} +1.17157i q^{77} +14.2426 q^{79} +1.00000 q^{81} -13.3137i q^{83} +26.1421i q^{85} -3.41421 q^{87} -2.00000 q^{89} +1.65685i q^{91} +7.41421i q^{93} +19.3137 q^{95} -9.31371 q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q - 8 * q^7 - 4 * q^9	$ 4 q - 8 q^{7} - 4 q^{9} + 8 q^{15} - 8 q^{17} - 16 q^{23} - 4 q^{25} + 24 q^{31} + 8 q^{33} + 24 q^{41} + 16 q^{47} - 4 q^{49} - 16 q^{55} + 8 q^{63} - 16 q^{65} - 48 q^{71} - 16 q^{73} + 40 q^{79} + 4 q^{81} - 8 q^{87} - 8 q^{89} + 32 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 - 4 * q^9 + 8 * q^15 - 8 * q^17 - 16 * q^23 - 4 * q^25 + 24 * q^31 + 8 * q^33 + 24 * q^41 + 16 * q^47 - 4 * q^49 - 16 * q^55 + 8 * q^63 - 16 * q^65 - 48 * q^71 - 16 * q^73 + 40 * q^79 + 4 * q^81 - 8 * q^87 - 8 * q^89 + 32 * q^95 + 8 * 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
$5$	− 3.41421i	− 1.52688i	−0.645877	−	0.763441i	$-0.723508\pi$
$5$	− 3.41421i	− 1.52688i	0.645877	−	0.763441i	$-0.276492\pi$
$6$	0	0
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	− 2.82843i	− 0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$13$	− 2.82843i	− 0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$14$	0	0
$15$	3.41421	0.881546
$16$	0	0
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0	0
$19$	5.65685i	1.29777i	0.760886	+	0.648886i	$0.224765\pi$
$19$	5.65685i	1.29777i	−0.760886	+	0.648886i	$0.775235\pi$
$20$	0	0
$21$	− 0.585786i	− 0.127829i
$22$	0	0
$23$	−6.82843	−1.42383	−0.711913	−	0.702268i	$-0.752171\pi$
$23$	−6.82843	−1.42383	−0.711913	+	0.702268i	$0.752171\pi$
$24$	0	0
$25$	−6.65685	−1.33137
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	3.41421i	0.634004i	0.948425	+	0.317002i	$0.102676\pi$
$29$	3.41421i	0.634004i	−0.948425	+	0.317002i	$0.897324\pi$
$30$	0	0
$31$	7.41421	1.33163	0.665816	−	0.746116i	$-0.268084\pi$
$31$	7.41421	1.33163	0.665816	+	0.746116i	$0.268084\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	2.00000i	0.338062i
$36$	0	0
$37$	− 1.65685i	− 0.272385i	−0.990682	−	0.136193i	$-0.956513\pi$
$37$	− 1.65685i	− 0.272385i	0.990682	−	0.136193i	$-0.0434866\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	0.343146	0.0535904	0.0267952	−	0.999641i	$-0.491470\pi$
$41$	0.343146	0.0535904	0.0267952	+	0.999641i	$0.491470\pi$
$42$	0	0
$43$	9.65685i	1.47266i	0.676625	+	0.736328i	$0.263442\pi$
$43$	9.65685i	1.47266i	−0.676625	+	0.736328i	$0.736558\pi$
$44$	0	0
$45$	3.41421i	0.508961i
$46$	0	0
$47$	−4.48528	−0.654246	−0.327123	−	0.944982i	$-0.606079\pi$
$47$	−4.48528	−0.654246	−0.327123	+	0.944982i	$0.606079\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	− 7.65685i	− 1.07217i
$52$	0	0
$53$	7.89949i	1.08508i	0.840030	+	0.542540i	$0.182537\pi$
$53$	7.89949i	1.08508i	−0.840030	+	0.542540i	$0.817463\pi$
$54$	0	0
$55$	−6.82843	−0.920745
$56$	0	0
$57$	−5.65685	−0.749269
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	1.65685i	0.212138i	0.994359	+	0.106069i	$0.0338265\pi$
$61$	1.65685i	0.212138i	−0.994359	+	0.106069i	$0.966173\pi$
$62$	0	0
$63$	0.585786	0.0738022
$64$	0	0
$65$	−9.65685	−1.19779
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	− 6.82843i	− 0.822046i
$70$	0	0
$71$	−14.8284	−1.75981	−0.879905	−	0.475149i	$-0.842394\pi$
$71$	−14.8284	−1.75981	−0.879905	+	0.475149i	$0.842394\pi$
$72$	0	0
$73$	−9.65685	−1.13025	−0.565125	−	0.825006i	$-0.691172\pi$
$73$	−9.65685	−1.13025	−0.565125	+	0.825006i	$0.691172\pi$
$74$	0	0
$75$	− 6.65685i	− 0.768667i
$76$	0	0
$77$	1.17157i	0.133513i
$78$	0	0
$79$	14.2426	1.60242	0.801211	−	0.598382i	$-0.204189\pi$
$79$	14.2426	1.60242	0.801211	+	0.598382i	$0.204189\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 13.3137i	− 1.46137i	−0.682715	−	0.730685i	$-0.739201\pi$
$83$	− 13.3137i	− 1.46137i	0.682715	−	0.730685i	$-0.260799\pi$
$84$	0	0
$85$	26.1421i	2.83551i
$86$	0	0
$87$	−3.41421	−0.366042
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	1.65685i	0.173686i
$92$	0	0
$93$	7.41421i	0.768818i
$94$	0	0
$95$	19.3137	1.98154
$96$	0	0
$97$	−9.31371	−0.945664	−0.472832	−	0.881153i	$-0.656768\pi$
$97$	−9.31371	−0.945664	−0.472832	+	0.881153i	$0.656768\pi$
$98$	0	0
$99$	2.00000i	0.201008i
$100$	0	0
$101$	− 13.5563i	− 1.34891i	−0.738317	−	0.674454i	$-0.764379\pi$
$101$	− 13.5563i	− 1.34891i	0.738317	−	0.674454i	$-0.235621\pi$
$102$	0	0
$103$	5.07107	0.499667	0.249834	−	0.968289i	$-0.419624\pi$
$103$	5.07107	0.499667	0.249834	+	0.968289i	$0.419624\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	− 7.31371i	− 0.707043i	−0.935426	−	0.353521i	$-0.884984\pi$
$107$	− 7.31371i	− 0.707043i	0.935426	−	0.353521i	$-0.115016\pi$
$108$	0	0
$109$	− 18.8284i	− 1.80344i	−0.432324	−	0.901718i	$-0.642306\pi$
$109$	− 18.8284i	− 1.80344i	0.432324	−	0.901718i	$-0.357694\pi$
$110$	0	0
$111$	1.65685	0.157262
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	23.3137i	2.17401i
$116$	0	0
$117$	2.82843i	0.261488i
$118$	0	0
$119$	4.48528	0.411165
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0.343146i	0.0309404i
$124$	0	0
$125$	5.65685i	0.505964i
$126$	0	0
$127$	−18.7279	−1.66183	−0.830917	−	0.556396i	$-0.812184\pi$
$127$	−18.7279	−1.66183	−0.830917	+	0.556396i	$0.812184\pi$
$128$	0	0
$129$	−9.65685	−0.850239
$130$	0	0
$131$	− 4.00000i	− 0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$	− 4.00000i	− 0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$	0	0
$133$	− 3.31371i	− 0.287335i
$134$	0	0
$135$	−3.41421	−0.293849
$136$	0	0
$137$	−15.6569	−1.33766	−0.668828	−	0.743417i	$-0.733204\pi$
$137$	−15.6569	−1.33766	−0.668828	+	0.743417i	$0.733204\pi$
$138$	0	0
$139$	3.31371i	0.281065i	0.990076	+	0.140533i	$0.0448815\pi$
$139$	3.31371i	0.281065i	−0.990076	+	0.140533i	$0.955119\pi$
$140$	0	0
$141$	− 4.48528i	− 0.377729i
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	11.6569	0.968049
$146$	0	0
$147$	− 6.65685i	− 0.549048i
$148$	0	0
$149$	− 5.75736i	− 0.471661i	−0.971794	−	0.235831i	$-0.924219\pi$
$149$	− 5.75736i	− 0.471661i	0.971794	−	0.235831i	$-0.0757811\pi$
$150$	0	0
$151$	−7.41421	−0.603360	−0.301680	−	0.953409i	$-0.597547\pi$
$151$	−7.41421	−0.603360	−0.301680	+	0.953409i	$0.597547\pi$
$152$	0	0
$153$	7.65685	0.619020
$154$	0	0
$155$	− 25.3137i	− 2.03325i
$156$	0	0
$157$	− 20.0000i	− 1.59617i	−0.602542	−	0.798087i	$-0.705846\pi$
$157$	− 20.0000i	− 1.59617i	0.602542	−	0.798087i	$-0.294154\pi$
$158$	0	0
$159$	−7.89949	−0.626471
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	17.6569i	1.38299i	0.722380	+	0.691496i	$0.243048\pi$
$163$	17.6569i	1.38299i	−0.722380	+	0.691496i	$0.756952\pi$
$164$	0	0
$165$	− 6.82843i	− 0.531592i
$166$	0	0
$167$	10.3431	0.800377	0.400188	−	0.916433i	$-0.368945\pi$
$167$	10.3431	0.800377	0.400188	+	0.916433i	$0.368945\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	− 5.65685i	− 0.432590i
$172$	0	0
$173$	− 6.72792i	− 0.511514i	−0.966741	−	0.255757i	$-0.917675\pi$
$173$	− 6.72792i	− 0.511514i	0.966741	−	0.255757i	$-0.0823248\pi$
$174$	0	0
$175$	3.89949	0.294774
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	− 4.00000i	− 0.298974i	−0.988764	−	0.149487i	$-0.952238\pi$
$179$	− 4.00000i	− 0.298974i	0.988764	−	0.149487i	$-0.0477622\pi$
$180$	0	0
$181$	8.48528i	0.630706i	0.948974	+	0.315353i	$0.102123\pi$
$181$	8.48528i	0.630706i	−0.948974	+	0.315353i	$0.897877\pi$
$182$	0	0
$183$	−1.65685	−0.122478
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	15.3137i	1.11985i
$188$	0	0
$189$	0.585786i	0.0426097i
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	−10.3431	−0.744516	−0.372258	−	0.928129i	$-0.621416\pi$
$193$	−10.3431	−0.744516	−0.372258	+	0.928129i	$0.621416\pi$
$194$	0	0
$195$	− 9.65685i	− 0.691542i
$196$	0	0
$197$	− 17.0711i	− 1.21626i	−0.793836	−	0.608132i	$-0.791919\pi$
$197$	− 17.0711i	− 1.21626i	0.793836	−	0.608132i	$-0.208081\pi$
$198$	0	0
$199$	−17.5563	−1.24454	−0.622268	−	0.782804i	$-0.713789\pi$
$199$	−17.5563	−1.24454	−0.622268	+	0.782804i	$0.713789\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	− 1.17157i	− 0.0818262i
$206$	0	0
$207$	6.82843	0.474608
$208$	0	0
$209$	11.3137	0.782586
$210$	0	0
$211$	3.31371i	0.228125i	0.993474	+	0.114063i	$0.0363864\pi$
$211$	3.31371i	0.228125i	−0.993474	+	0.114063i	$0.963614\pi$
$212$	0	0
$213$	− 14.8284i	− 1.01603i
$214$	0	0
$215$	32.9706	2.24857
$216$	0	0
$217$	−4.34315	−0.294832
$218$	0	0
$219$	− 9.65685i	− 0.652550i
$220$	0	0
$221$	21.6569i	1.45680i
$222$	0	0
$223$	−11.8995	−0.796849	−0.398425	−	0.917201i	$-0.630443\pi$
$223$	−11.8995	−0.796849	−0.398425	+	0.917201i	$0.630443\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	6.68629i	0.443785i	0.975071	+	0.221892i	$0.0712234\pi$
$227$	6.68629i	0.443785i	−0.975071	+	0.221892i	$0.928777\pi$
$228$	0	0
$229$	− 0.485281i	− 0.0320683i	−0.999871	−	0.0160341i	$-0.994896\pi$
$229$	− 0.485281i	− 0.0320683i	0.999871	−	0.0160341i	$-0.00510405\pi$
$230$	0	0
$231$	−1.17157	−0.0770838
$232$	0	0
$233$	13.3137	0.872210	0.436105	−	0.899896i	$-0.356358\pi$
$233$	13.3137	0.872210	0.436105	+	0.899896i	$0.356358\pi$
$234$	0	0
$235$	15.3137i	0.998956i
$236$	0	0
$237$	14.2426i	0.925159i
$238$	0	0
$239$	−16.9706	−1.09773	−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706	−1.09773	−0.548867	+	0.835910i	$0.684941\pi$
$240$	0	0
$241$	−6.34315	−0.408598	−0.204299	−	0.978909i	$-0.565491\pi$
$241$	−6.34315	−0.408598	−0.204299	+	0.978909i	$0.565491\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	22.7279i	1.45203i
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	13.3137	0.843722
$250$	0	0
$251$	22.0000i	1.38863i	0.719672	+	0.694314i	$0.244292\pi$
$251$	22.0000i	1.38863i	−0.719672	+	0.694314i	$0.755708\pi$
$252$	0	0
$253$	13.6569i	0.858599i
$254$	0	0
$255$	−26.1421	−1.63708
$256$	0	0
$257$	21.3137	1.32951	0.664756	−	0.747060i	$-0.268535\pi$
$257$	21.3137	1.32951	0.664756	+	0.747060i	$0.268535\pi$
$258$	0	0
$259$	0.970563i	0.0603078i
$260$	0	0
$261$	− 3.41421i	− 0.211335i
$262$	0	0
$263$	2.34315	0.144485	0.0722423	−	0.997387i	$-0.476985\pi$
$263$	2.34315	0.144485	0.0722423	+	0.997387i	$0.476985\pi$
$264$	0	0
$265$	26.9706	1.65679
$266$	0	0
$267$	− 2.00000i	− 0.122398i
$268$	0	0
$269$	2.24264i	0.136736i	0.997660	+	0.0683681i	$0.0217792\pi$
$269$	2.24264i	0.136736i	−0.997660	+	0.0683681i	$0.978221\pi$
$270$	0	0
$271$	−16.5858	−1.00751	−0.503757	−	0.863845i	$-0.668049\pi$
$271$	−16.5858	−1.00751	−0.503757	+	0.863845i	$0.668049\pi$
$272$	0	0
$273$	−1.65685	−0.100277
$274$	0	0
$275$	13.3137i	0.802847i
$276$	0	0
$277$	24.4853i	1.47118i	0.677428	+	0.735589i	$0.263094\pi$
$277$	24.4853i	1.47118i	−0.677428	+	0.735589i	$0.736906\pi$
$278$	0	0
$279$	−7.41421	−0.443877
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	7.31371i	0.434755i	0.976088	+	0.217377i	$0.0697502\pi$
$283$	7.31371i	0.434755i	−0.976088	+	0.217377i	$0.930250\pi$
$284$	0	0
$285$	19.3137i	1.14405i
$286$	0	0
$287$	−0.201010	−0.0118653
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	− 9.31371i	− 0.545979i
$292$	0	0
$293$	− 8.10051i	− 0.473237i	−0.971603	−	0.236618i	$-0.923961\pi$
$293$	− 8.10051i	− 0.473237i	0.971603	−	0.236618i	$-0.0760391\pi$
$294$	0	0
$295$	13.6569	0.795133
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	19.3137i	1.11694i
$300$	0	0
$301$	− 5.65685i	− 0.326056i
$302$	0	0
$303$	13.5563	0.778792
$304$	0	0
$305$	5.65685	0.323911
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	5.07107i	0.288483i
$310$	0	0
$311$	11.3137	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$311$	11.3137	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$312$	0	0
$313$	16.9706	0.959233	0.479616	−	0.877478i	$-0.340776\pi$
$313$	16.9706	0.959233	0.479616	+	0.877478i	$0.340776\pi$
$314$	0	0
$315$	− 2.00000i	− 0.112687i
$316$	0	0
$317$	− 9.07107i	− 0.509482i	−0.967009	−	0.254741i	$-0.918010\pi$
$317$	− 9.07107i	− 0.509482i	0.967009	−	0.254741i	$-0.0819902\pi$
$318$	0	0
$319$	6.82843	0.382319
$320$	0	0
$321$	7.31371	0.408211
$322$	0	0
$323$	− 43.3137i	− 2.41004i
$324$	0	0
$325$	18.8284i	1.04441i
$326$	0	0
$327$	18.8284	1.04121
$328$	0	0
$329$	2.62742	0.144854
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	1.65685i	0.0907951i
$334$	0	0
$335$	−27.3137	−1.49231
$336$	0	0
$337$	−12.9706	−0.706552	−0.353276	−	0.935519i	$-0.614932\pi$
$337$	−12.9706	−0.706552	−0.353276	+	0.935519i	$0.614932\pi$
$338$	0	0
$339$	6.00000i	0.325875i
$340$	0	0
$341$	− 14.8284i	− 0.803004i
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−23.3137	−1.25517
$346$	0	0
$347$	− 2.68629i	− 0.144208i	−0.997397	−	0.0721038i	$-0.977029\pi$
$347$	− 2.68629i	− 0.144208i	0.997397	−	0.0721038i	$-0.0229713\pi$
$348$	0	0
$349$	− 23.3137i	− 1.24795i	−0.781443	−	0.623977i	$-0.785516\pi$
$349$	− 23.3137i	− 1.24795i	0.781443	−	0.623977i	$-0.214484\pi$
$350$	0	0
$351$	−2.82843	−0.150970
$352$	0	0
$353$	9.31371	0.495719	0.247859	−	0.968796i	$-0.420273\pi$
$353$	9.31371	0.495719	0.247859	+	0.968796i	$0.420273\pi$
$354$	0	0
$355$	50.6274i	2.68702i
$356$	0	0
$357$	4.48528i	0.237386i
$358$	0	0
$359$	−23.7990	−1.25606	−0.628031	−	0.778188i	$-0.716139\pi$
$359$	−23.7990	−1.25606	−0.628031	+	0.778188i	$0.716139\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	32.9706i	1.72576i
$366$	0	0
$367$	2.72792	0.142396	0.0711982	−	0.997462i	$-0.477318\pi$
$367$	2.72792	0.142396	0.0711982	+	0.997462i	$0.477318\pi$
$368$	0	0
$369$	−0.343146	−0.0178635
$370$	0	0
$371$	− 4.62742i	− 0.240244i
$372$	0	0
$373$	− 32.2843i	− 1.67162i	−0.549022	−	0.835808i	$-0.685000\pi$
$373$	− 32.2843i	− 1.67162i	0.549022	−	0.835808i	$-0.315000\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	9.65685	0.497353
$378$	0	0
$379$	29.6569i	1.52337i	0.647947	+	0.761685i	$0.275628\pi$
$379$	29.6569i	1.52337i	−0.647947	+	0.761685i	$0.724372\pi$
$380$	0	0
$381$	− 18.7279i	− 0.959461i
$382$	0	0
$383$	3.31371	0.169323	0.0846613	−	0.996410i	$-0.473019\pi$
$383$	3.31371	0.169323	0.0846613	+	0.996410i	$0.473019\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	− 9.65685i	− 0.490885i
$388$	0	0
$389$	13.7574i	0.697526i	0.937211	+	0.348763i	$0.113398\pi$
$389$	13.7574i	0.697526i	−0.937211	+	0.348763i	$0.886602\pi$
$390$	0	0
$391$	52.2843	2.64413
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	− 48.6274i	− 2.44671i
$396$	0	0
$397$	− 4.00000i	− 0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$	− 4.00000i	− 0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$	0	0
$399$	3.31371	0.165893
$400$	0	0
$401$	−20.3431	−1.01589	−0.507944	−	0.861390i	$-0.669594\pi$
$401$	−20.3431	−1.01589	−0.507944	+	0.861390i	$0.669594\pi$
$402$	0	0
$403$	− 20.9706i	− 1.04462i
$404$	0	0
$405$	− 3.41421i	− 0.169654i
$406$	0	0
$407$	−3.31371	−0.164254
$408$	0	0
$409$	21.3137	1.05390	0.526948	−	0.849898i	$-0.323336\pi$
$409$	21.3137	1.05390	0.526948	+	0.849898i	$0.323336\pi$
$410$	0	0
$411$	− 15.6569i	− 0.772296i
$412$	0	0
$413$	− 2.34315i	− 0.115299i
$414$	0	0
$415$	−45.4558	−2.23134
$416$	0	0
$417$	−3.31371	−0.162273
$418$	0	0
$419$	21.3137i	1.04124i	0.853788	+	0.520621i	$0.174300\pi$
$419$	21.3137i	1.04124i	−0.853788	+	0.520621i	$0.825700\pi$
$420$	0	0
$421$	6.14214i	0.299349i	0.988735	+	0.149675i	$0.0478226\pi$
$421$	6.14214i	0.299349i	−0.988735	+	0.149675i	$0.952177\pi$
$422$	0	0
$423$	4.48528	0.218082
$424$	0	0
$425$	50.9706	2.47244
$426$	0	0
$427$	− 0.970563i	− 0.0469688i
$428$	0	0
$429$	− 5.65685i	− 0.273115i
$430$	0	0
$431$	−27.1127	−1.30597	−0.652986	−	0.757370i	$-0.726484\pi$
$431$	−27.1127	−1.30597	−0.652986	+	0.757370i	$0.726484\pi$
$432$	0	0
$433$	28.6274	1.37575	0.687873	−	0.725831i	$-0.258545\pi$
$433$	28.6274	1.37575	0.687873	+	0.725831i	$0.258545\pi$
$434$	0	0
$435$	11.6569i	0.558903i
$436$	0	0
$437$	− 38.6274i	− 1.84780i
$438$	0	0
$439$	−3.89949	−0.186113	−0.0930564	−	0.995661i	$-0.529664\pi$
$439$	−3.89949	−0.186113	−0.0930564	+	0.995661i	$0.529664\pi$
$440$	0	0
$441$	6.65685	0.316993
$442$	0	0
$443$	− 39.9411i	− 1.89766i	−0.315787	−	0.948830i	$-0.602269\pi$
$443$	− 39.9411i	− 1.89766i	0.315787	−	0.948830i	$-0.397731\pi$
$444$	0	0
$445$	6.82843i	0.323698i
$446$	0	0
$447$	5.75736	0.272314
$448$	0	0
$449$	3.65685	0.172578	0.0862888	−	0.996270i	$-0.472499\pi$
$449$	3.65685	0.172578	0.0862888	+	0.996270i	$0.472499\pi$
$450$	0	0
$451$	− 0.686292i	− 0.0323162i
$452$	0	0
$453$	− 7.41421i	− 0.348350i
$454$	0	0
$455$	5.65685	0.265197
$456$	0	0
$457$	−5.31371	−0.248565	−0.124282	−	0.992247i	$-0.539663\pi$
$457$	−5.31371	−0.248565	−0.124282	+	0.992247i	$0.539663\pi$
$458$	0	0
$459$	7.65685i	0.357391i
$460$	0	0
$461$	− 25.0711i	− 1.16768i	−0.811870	−	0.583838i	$-0.801550\pi$
$461$	− 25.0711i	− 1.16768i	0.811870	−	0.583838i	$-0.198450\pi$
$462$	0	0
$463$	33.5563	1.55950	0.779748	−	0.626094i	$-0.215347\pi$
$463$	33.5563	1.55950	0.779748	+	0.626094i	$0.215347\pi$
$464$	0	0
$465$	25.3137	1.17390
$466$	0	0
$467$	13.3137i	0.616085i	0.951373	+	0.308042i	$0.0996739\pi$
$467$	13.3137i	0.616085i	−0.951373	+	0.308042i	$0.900326\pi$
$468$	0	0
$469$	4.68629i	0.216393i
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	19.3137	0.888045
$474$	0	0
$475$	− 37.6569i	− 1.72781i
$476$	0	0
$477$	− 7.89949i	− 0.361693i
$478$	0	0
$479$	12.4853	0.570467	0.285234	−	0.958458i	$-0.407929\pi$
$479$	12.4853	0.570467	0.285234	+	0.958458i	$0.407929\pi$
$480$	0	0
$481$	−4.68629	−0.213676
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	31.7990i	1.44392i
$486$	0	0
$487$	−25.5563	−1.15807	−0.579034	−	0.815303i	$-0.696570\pi$
$487$	−25.5563	−1.15807	−0.579034	+	0.815303i	$0.696570\pi$
$488$	0	0
$489$	−17.6569	−0.798471
$490$	0	0
$491$	23.3137i	1.05213i	0.850443	+	0.526066i	$0.176334\pi$
$491$	23.3137i	1.05213i	−0.850443	+	0.526066i	$0.823666\pi$
$492$	0	0
$493$	− 26.1421i	− 1.17738i
$494$	0	0
$495$	6.82843	0.306915
$496$	0	0
$497$	8.68629	0.389633
$498$	0	0
$499$	0.686292i	0.0307226i	0.999882	+	0.0153613i	$0.00488985\pi$
$499$	0.686292i	0.0307226i	−0.999882	+	0.0153613i	$0.995110\pi$
$500$	0	0
$501$	10.3431i	0.462098i
$502$	0	0
$503$	−12.4853	−0.556691	−0.278346	−	0.960481i	$-0.589786\pi$
$503$	−12.4853	−0.556691	−0.278346	+	0.960481i	$0.589786\pi$
$504$	0	0
$505$	−46.2843	−2.05962
$506$	0	0
$507$	5.00000i	0.222058i
$508$	0	0
$509$	− 19.2132i	− 0.851610i	−0.904815	−	0.425805i	$-0.859991\pi$
$509$	− 19.2132i	− 0.851610i	0.904815	−	0.425805i	$-0.140009\pi$
$510$	0	0
$511$	5.65685	0.250244
$512$	0	0
$513$	5.65685	0.249756
$514$	0	0
$515$	− 17.3137i	− 0.762933i
$516$	0	0
$517$	8.97056i	0.394525i
$518$	0	0
$519$	6.72792	0.295323
$520$	0	0
$521$	−22.9706	−1.00636	−0.503179	−	0.864182i	$-0.667836\pi$
$521$	−22.9706	−1.00636	−0.503179	+	0.864182i	$0.667836\pi$
$522$	0	0
$523$	− 37.6569i	− 1.64662i	−0.567593	−	0.823310i	$-0.692125\pi$
$523$	− 37.6569i	− 1.64662i	0.567593	−	0.823310i	$-0.307875\pi$
$524$	0	0
$525$	3.89949i	0.170188i
$526$	0	0
$527$	−56.7696	−2.47292
$528$	0	0
$529$	23.6274	1.02728
$530$	0	0
$531$	− 4.00000i	− 0.173585i
$532$	0	0
$533$	− 0.970563i	− 0.0420397i
$534$	0	0
$535$	−24.9706	−1.07957
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	13.3137i	0.573462i
$540$	0	0
$541$	− 10.8284i	− 0.465550i	−0.972531	−	0.232775i	$-0.925219\pi$
$541$	− 10.8284i	− 0.465550i	0.972531	−	0.232775i	$-0.0747806\pi$
$542$	0	0
$543$	−8.48528	−0.364138
$544$	0	0
$545$	−64.2843	−2.75364
$546$	0	0
$547$	− 4.97056i	− 0.212526i	−0.994338	−	0.106263i	$-0.966111\pi$
$547$	− 4.97056i	− 0.212526i	0.994338	−	0.106263i	$-0.0338885\pi$
$548$	0	0
$549$	− 1.65685i	− 0.0707128i
$550$	0	0
$551$	−19.3137	−0.822792
$552$	0	0
$553$	−8.34315	−0.354787
$554$	0	0
$555$	− 5.65685i	− 0.240120i
$556$	0	0
$557$	1.07107i	0.0453826i	0.999743	+	0.0226913i	$0.00722349\pi$
$557$	1.07107i	0.0453826i	−0.999743	+	0.0226913i	$0.992777\pi$
$558$	0	0
$559$	27.3137	1.15525
$560$	0	0
$561$	−15.3137	−0.646545
$562$	0	0
$563$	16.6274i	0.700762i	0.936607	+	0.350381i	$0.113948\pi$
$563$	16.6274i	0.700762i	−0.936607	+	0.350381i	$0.886052\pi$
$564$	0	0
$565$	− 20.4853i	− 0.861822i
$566$	0	0
$567$	−0.585786	−0.0246007
$568$	0	0
$569$	14.9706	0.627599	0.313799	−	0.949489i	$-0.398398\pi$
$569$	14.9706	0.627599	0.313799	+	0.949489i	$0.398398\pi$
$570$	0	0
$571$	20.0000i	0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$	20.0000i	0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$	0	0
$573$	16.9706i	0.708955i
$574$	0	0
$575$	45.4558	1.89564
$576$	0	0
$577$	−28.2843	−1.17749	−0.588745	−	0.808319i	$-0.700378\pi$
$577$	−28.2843	−1.17749	−0.588745	+	0.808319i	$0.700378\pi$
$578$	0	0
$579$	− 10.3431i	− 0.429846i
$580$	0	0
$581$	7.79899i	0.323557i
$582$	0	0
$583$	15.7990	0.654327
$584$	0	0
$585$	9.65685	0.399262
$586$	0	0
$587$	31.3137i	1.29246i	0.763145	+	0.646228i	$0.223654\pi$
$587$	31.3137i	1.29246i	−0.763145	+	0.646228i	$0.776346\pi$
$588$	0	0
$589$	41.9411i	1.72815i
$590$	0	0
$591$	17.0711	0.702210
$592$	0	0
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	− 15.3137i	− 0.627801i
$596$	0	0
$597$	− 17.5563i	− 0.718534i
$598$	0	0
$599$	−11.5147	−0.470479	−0.235239	−	0.971937i	$-0.575587\pi$
$599$	−11.5147	−0.470479	−0.235239	+	0.971937i	$0.575587\pi$
$600$	0	0
$601$	5.65685	0.230748	0.115374	−	0.993322i	$-0.463193\pi$
$601$	5.65685	0.230748	0.115374	+	0.993322i	$0.463193\pi$
$602$	0	0
$603$	8.00000i	0.325785i
$604$	0	0
$605$	− 23.8995i	− 0.971653i
$606$	0	0
$607$	3.89949	0.158276	0.0791378	−	0.996864i	$-0.474783\pi$
$607$	3.89949	0.158276	0.0791378	+	0.996864i	$0.474783\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	12.6863i	0.513232i
$612$	0	0
$613$	10.6274i	0.429237i	0.976698	+	0.214619i	$0.0688509\pi$
$613$	10.6274i	0.429237i	−0.976698	+	0.214619i	$0.931149\pi$
$614$	0	0
$615$	1.17157	0.0472424
$616$	0	0
$617$	−23.9411	−0.963833	−0.481917	−	0.876217i	$-0.660059\pi$
$617$	−23.9411	−0.963833	−0.481917	+	0.876217i	$0.660059\pi$
$618$	0	0
$619$	− 4.00000i	− 0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$	− 4.00000i	− 0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$	0	0
$621$	6.82843i	0.274015i
$622$	0	0
$623$	1.17157	0.0469381
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	11.3137i	0.451826i
$628$	0	0
$629$	12.6863i	0.505836i
$630$	0	0
$631$	−18.9289	−0.753549	−0.376774	−	0.926305i	$-0.622967\pi$
$631$	−18.9289	−0.753549	−0.376774	+	0.926305i	$0.622967\pi$
$632$	0	0
$633$	−3.31371	−0.131708
$634$	0	0
$635$	63.9411i	2.53743i
$636$	0	0
$637$	18.8284i	0.746009i
$638$	0	0
$639$	14.8284	0.586604
$640$	0	0
$641$	9.02944	0.356641	0.178321	−	0.983972i	$-0.442934\pi$
$641$	9.02944	0.356641	0.178321	+	0.983972i	$0.442934\pi$
$642$	0	0
$643$	− 32.2843i	− 1.27317i	−0.771208	−	0.636584i	$-0.780347\pi$
$643$	− 32.2843i	− 1.27317i	0.771208	−	0.636584i	$-0.219653\pi$
$644$	0	0
$645$	32.9706i	1.29821i
$646$	0	0
$647$	22.8284	0.897478	0.448739	−	0.893663i	$-0.351873\pi$
$647$	22.8284	0.897478	0.448739	+	0.893663i	$0.351873\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	− 4.34315i	− 0.170221i
$652$	0	0
$653$	20.5858i	0.805584i	0.915292	+	0.402792i	$0.131960\pi$
$653$	20.5858i	0.805584i	−0.915292	+	0.402792i	$0.868040\pi$
$654$	0	0
$655$	−13.6569	−0.533617
$656$	0	0
$657$	9.65685	0.376750
$658$	0	0
$659$	− 39.3137i	− 1.53144i	−0.643171	−	0.765722i	$-0.722382\pi$
$659$	− 39.3137i	− 1.53144i	0.643171	−	0.765722i	$-0.277618\pi$
$660$	0	0
$661$	− 39.3137i	− 1.52913i	−0.644549	−	0.764563i	$-0.722955\pi$
$661$	− 39.3137i	− 1.52913i	0.644549	−	0.764563i	$-0.277045\pi$
$662$	0	0
$663$	−21.6569	−0.841083
$664$	0	0
$665$	−11.3137	−0.438727
$666$	0	0
$667$	− 23.3137i	− 0.902710i
$668$	0	0
$669$	− 11.8995i	− 0.460061i
$670$	0	0
$671$	3.31371	0.127924
$672$	0	0
$673$	−8.62742	−0.332562	−0.166281	−	0.986078i	$-0.553176\pi$
$673$	−8.62742	−0.332562	−0.166281	+	0.986078i	$0.553176\pi$
$674$	0	0
$675$	6.65685i	0.256222i
$676$	0	0
$677$	35.2132i	1.35335i	0.736280	+	0.676677i	$0.236580\pi$
$677$	35.2132i	1.35335i	−0.736280	+	0.676677i	$0.763420\pi$
$678$	0	0
$679$	5.45584	0.209376
$680$	0	0
$681$	−6.68629	−0.256219
$682$	0	0
$683$	2.68629i	0.102788i	0.998678	+	0.0513940i	$0.0163664\pi$
$683$	2.68629i	0.102788i	−0.998678	+	0.0513940i	$0.983634\pi$
$684$	0	0
$685$	53.4558i	2.04244i
$686$	0	0
$687$	0.485281	0.0185146
$688$	0	0
$689$	22.3431	0.851206
$690$	0	0
$691$	− 6.34315i	− 0.241305i	−0.992695	−	0.120652i	$-0.961501\pi$
$691$	− 6.34315i	− 0.241305i	0.992695	−	0.120652i	$-0.0384986\pi$
$692$	0	0
$693$	− 1.17157i	− 0.0445044i
$694$	0	0
$695$	11.3137	0.429153
$696$	0	0
$697$	−2.62742	−0.0995205
$698$	0	0
$699$	13.3137i	0.503571i
$700$	0	0
$701$	4.58579i	0.173203i	0.996243	+	0.0866014i	$0.0276006\pi$
$701$	4.58579i	0.173203i	−0.996243	+	0.0866014i	$0.972399\pi$
$702$	0	0
$703$	9.37258	0.353494
$704$	0	0
$705$	−15.3137	−0.576748
$706$	0	0
$707$	7.94113i	0.298657i
$708$	0	0
$709$	31.1127i	1.16846i	0.811587	+	0.584231i	$0.198604\pi$
$709$	31.1127i	1.16846i	−0.811587	+	0.584231i	$0.801396\pi$
$710$	0	0
$711$	−14.2426	−0.534141
$712$	0	0
$713$	−50.6274	−1.89601
$714$	0	0
$715$	19.3137i	0.722292i
$716$	0	0
$717$	− 16.9706i	− 0.633777i
$718$	0	0
$719$	30.8284	1.14971	0.574853	−	0.818257i	$-0.305059\pi$
$719$	30.8284	1.14971	0.574853	+	0.818257i	$0.305059\pi$
$720$	0	0
$721$	−2.97056	−0.110630
$722$	0	0
$723$	− 6.34315i	− 0.235904i
$724$	0	0
$725$	− 22.7279i	− 0.844094i
$726$	0	0
$727$	22.2426	0.824934	0.412467	−	0.910973i	$-0.364667\pi$
$727$	22.2426	0.824934	0.412467	+	0.910973i	$0.364667\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 73.9411i	− 2.73481i
$732$	0	0
$733$	23.5147i	0.868536i	0.900784	+	0.434268i	$0.142993\pi$
$733$	23.5147i	0.868536i	−0.900784	+	0.434268i	$0.857007\pi$
$734$	0	0
$735$	−22.7279	−0.838332
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	− 10.6274i	− 0.390936i	−0.980710	−	0.195468i	$-0.937377\pi$
$739$	− 10.6274i	− 0.390936i	0.980710	−	0.195468i	$-0.0626226\pi$
$740$	0	0
$741$	16.0000i	0.587775i
$742$	0	0
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	0	0
$745$	−19.6569	−0.720171
$746$	0	0
$747$	13.3137i	0.487123i
$748$	0	0
$749$	4.28427i	0.156544i
$750$	0	0
$751$	5.27208	0.192381	0.0961904	−	0.995363i	$-0.469334\pi$
$751$	5.27208	0.192381	0.0961904	+	0.995363i	$0.469334\pi$
$752$	0	0
$753$	−22.0000	−0.801725
$754$	0	0
$755$	25.3137i	0.921260i
$756$	0	0
$757$	− 32.4853i	− 1.18070i	−0.807148	−	0.590349i	$-0.798990\pi$
$757$	− 32.4853i	− 1.18070i	0.807148	−	0.590349i	$-0.201010\pi$
$758$	0	0
$759$	−13.6569	−0.495712
$760$	0	0
$761$	−35.6569	−1.29256	−0.646280	−	0.763100i	$-0.723676\pi$
$761$	−35.6569	−1.29256	−0.646280	+	0.763100i	$0.723676\pi$
$762$	0	0
$763$	11.0294i	0.399292i
$764$	0	0
$765$	− 26.1421i	− 0.945171i
$766$	0	0
$767$	11.3137	0.408514
$768$	0	0
$769$	−2.34315	−0.0844960	−0.0422480	−	0.999107i	$-0.513452\pi$
$769$	−2.34315	−0.0844960	−0.0422480	+	0.999107i	$0.513452\pi$
$770$	0	0
$771$	21.3137i	0.767594i
$772$	0	0
$773$	6.72792i	0.241987i	0.992653	+	0.120993i	$0.0386079\pi$
$773$	6.72792i	0.241987i	−0.992653	+	0.120993i	$0.961392\pi$
$774$	0	0
$775$	−49.3553	−1.77290
$776$	0	0
$777$	−0.970563	−0.0348187
$778$	0	0
$779$	1.94113i	0.0695480i
$780$	0	0
$781$	29.6569i	1.06121i
$782$	0	0
$783$	3.41421	0.122014
$784$	0	0
$785$	−68.2843	−2.43717
$786$	0	0
$787$	21.6569i	0.771983i	0.922502	+	0.385992i	$0.126141\pi$
$787$	21.6569i	0.771983i	−0.922502	+	0.385992i	$0.873859\pi$
$788$	0	0
$789$	2.34315i	0.0834182i
$790$	0	0
$791$	−3.51472	−0.124969
$792$	0	0
$793$	4.68629	0.166415
$794$	0	0
$795$	26.9706i	0.956547i
$796$	0	0
$797$	35.4142i	1.25444i	0.778844	+	0.627218i	$0.215806\pi$
$797$	35.4142i	1.25444i	−0.778844	+	0.627218i	$0.784194\pi$
$798$	0	0
$799$	34.3431	1.21497
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	19.3137i	0.681566i
$804$	0	0
$805$	− 13.6569i	− 0.481341i
$806$	0	0
$807$	−2.24264	−0.0789447
$808$	0	0
$809$	5.02944	0.176826	0.0884128	−	0.996084i	$-0.471821\pi$
$809$	5.02944	0.176826	0.0884128	+	0.996084i	$0.471821\pi$
$810$	0	0
$811$	18.3431i	0.644115i	0.946720	+	0.322057i	$0.104374\pi$
$811$	18.3431i	0.644115i	−0.946720	+	0.322057i	$0.895626\pi$
$812$	0	0
$813$	− 16.5858i	− 0.581689i
$814$	0	0
$815$	60.2843	2.11167
$816$	0	0
$817$	−54.6274	−1.91117
$818$	0	0
$819$	− 1.65685i	− 0.0578952i
$820$	0	0
$821$	− 8.87006i	− 0.309567i	−0.987948	−	0.154784i	$-0.950532\pi$
$821$	− 8.87006i	− 0.309567i	0.987948	−	0.154784i	$-0.0494680\pi$
$822$	0	0
$823$	−39.2132	−1.36689	−0.683443	−	0.730004i	$-0.739518\pi$
$823$	−39.2132	−1.36689	−0.683443	+	0.730004i	$0.739518\pi$
$824$	0	0
$825$	−13.3137	−0.463524
$826$	0	0
$827$	29.9411i	1.04115i	0.853814	+	0.520577i	$0.174283\pi$
$827$	29.9411i	1.04115i	−0.853814	+	0.520577i	$0.825717\pi$
$828$	0	0
$829$	− 52.7696i	− 1.83276i	−0.400307	−	0.916381i	$-0.631096\pi$
$829$	− 52.7696i	− 1.83276i	0.400307	−	0.916381i	$-0.368904\pi$
$830$	0	0
$831$	−24.4853	−0.849385
$832$	0	0
$833$	50.9706	1.76603
$834$	0	0
$835$	− 35.3137i	− 1.22208i
$836$	0	0
$837$	− 7.41421i	− 0.256273i
$838$	0	0
$839$	−28.4853	−0.983421	−0.491711	−	0.870759i	$-0.663628\pi$
$839$	−28.4853	−0.983421	−0.491711	+	0.870759i	$0.663628\pi$
$840$	0	0
$841$	17.3431	0.598040
$842$	0	0
$843$	2.00000i	0.0688837i
$844$	0	0
$845$	− 17.0711i	− 0.587263i
$846$	0	0
$847$	−4.10051	−0.140895
$848$	0	0
$849$	−7.31371	−0.251006
$850$	0	0
$851$	11.3137i	0.387829i
$852$	0	0
$853$	− 34.6274i	− 1.18562i	−0.805342	−	0.592810i	$-0.798018\pi$
$853$	− 34.6274i	− 1.18562i	0.805342	−	0.592810i	$-0.201982\pi$
$854$	0	0
$855$	−19.3137	−0.660515
$856$	0	0
$857$	−41.5980	−1.42096	−0.710480	−	0.703717i	$-0.751522\pi$
$857$	−41.5980	−1.42096	−0.710480	+	0.703717i	$0.751522\pi$
$858$	0	0
$859$	− 44.9706i	− 1.53438i	−0.641422	−	0.767188i	$-0.721655\pi$
$859$	− 44.9706i	− 1.53438i	0.641422	−	0.767188i	$-0.278345\pi$
$860$	0	0
$861$	− 0.201010i	− 0.00685041i
$862$	0	0
$863$	15.0294	0.511608	0.255804	−	0.966729i	$-0.417660\pi$
$863$	15.0294	0.511608	0.255804	+	0.966729i	$0.417660\pi$
$864$	0	0
$865$	−22.9706	−0.781023
$866$	0	0
$867$	41.6274i	1.41374i
$868$	0	0
$869$	− 28.4853i	− 0.966297i
$870$	0	0
$871$	−22.6274	−0.766701
$872$	0	0
$873$	9.31371	0.315221
$874$	0	0
$875$	− 3.31371i	− 0.112024i
$876$	0	0
$877$	− 23.3137i	− 0.787248i	−0.919272	−	0.393624i	$-0.871221\pi$
$877$	− 23.3137i	− 0.787248i	0.919272	−	0.393624i	$-0.128779\pi$
$878$	0	0
$879$	8.10051	0.273223
$880$	0	0
$881$	45.3137	1.52666	0.763329	−	0.646010i	$-0.223564\pi$
$881$	45.3137	1.52666	0.763329	+	0.646010i	$0.223564\pi$
$882$	0	0
$883$	30.3431i	1.02113i	0.859840	+	0.510564i	$0.170563\pi$
$883$	30.3431i	1.02113i	−0.859840	+	0.510564i	$0.829437\pi$
$884$	0	0
$885$	13.6569i	0.459070i
$886$	0	0
$887$	29.6569	0.995780	0.497890	−	0.867240i	$-0.334108\pi$
$887$	29.6569	0.995780	0.497890	+	0.867240i	$0.334108\pi$
$888$	0	0
$889$	10.9706	0.367941
$890$	0	0
$891$	− 2.00000i	− 0.0670025i
$892$	0	0
$893$	− 25.3726i	− 0.849061i
$894$	0	0
$895$	−13.6569	−0.456498
$896$	0	0
$897$	−19.3137	−0.644866
$898$	0	0
$899$	25.3137i	0.844259i
$900$	0	0
$901$	− 60.4853i	− 2.01506i
$902$	0	0
$903$	5.65685	0.188248
$904$	0	0
$905$	28.9706	0.963014
$906$	0	0
$907$	− 51.5980i	− 1.71328i	−0.515912	−	0.856641i	$-0.672547\pi$
$907$	− 51.5980i	− 1.71328i	0.515912	−	0.856641i	$-0.327453\pi$
$908$	0	0
$909$	13.5563i	0.449636i
$910$	0	0
$911$	28.6863	0.950419	0.475210	−	0.879873i	$-0.342372\pi$
$911$	28.6863	0.950419	0.475210	+	0.879873i	$0.342372\pi$
$912$	0	0
$913$	−26.6274	−0.881239
$914$	0	0
$915$	5.65685i	0.187010i
$916$	0	0
$917$	2.34315i	0.0773775i
$918$	0	0
$919$	17.3553	0.572500	0.286250	−	0.958155i	$-0.407591\pi$
$919$	17.3553	0.572500	0.286250	+	0.958155i	$0.407591\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	41.9411i	1.38051i
$924$	0	0
$925$	11.0294i	0.362646i
$926$	0	0
$927$	−5.07107	−0.166556
$928$	0	0
$929$	29.5980	0.971078	0.485539	−	0.874215i	$-0.338623\pi$
$929$	29.5980	0.971078	0.485539	+	0.874215i	$0.338623\pi$
$930$	0	0
$931$	− 37.6569i	− 1.23415i
$932$	0	0
$933$	11.3137i	0.370394i
$934$	0	0
$935$	52.2843	1.70988
$936$	0	0
$937$	−36.6274	−1.19657	−0.598283	−	0.801285i	$-0.704150\pi$
$937$	−36.6274	−1.19657	−0.598283	+	0.801285i	$0.704150\pi$
$938$	0	0
$939$	16.9706i	0.553813i
$940$	0	0
$941$	− 20.3848i	− 0.664525i	−0.943187	−	0.332262i	$-0.892188\pi$
$941$	− 20.3848i	− 0.664525i	0.943187	−	0.332262i	$-0.107812\pi$
$942$	0	0
$943$	−2.34315	−0.0763033
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	− 34.6274i	− 1.12524i	−0.826716	−	0.562620i	$-0.809793\pi$
$947$	− 34.6274i	− 1.12524i	0.826716	−	0.562620i	$-0.190207\pi$
$948$	0	0
$949$	27.3137i	0.886640i
$950$	0	0
$951$	9.07107	0.294150
$952$	0	0
$953$	−30.9706	−1.00323	−0.501617	−	0.865090i	$-0.667261\pi$
$953$	−30.9706	−1.00323	−0.501617	+	0.865090i	$0.667261\pi$
$954$	0	0
$955$	− 57.9411i	− 1.87493i
$956$	0	0
$957$	6.82843i	0.220732i
$958$	0	0
$959$	9.17157	0.296166
$960$	0	0
$961$	23.9706	0.773244
$962$	0	0
$963$	7.31371i	0.235681i
$964$	0	0
$965$	35.3137i	1.13679i
$966$	0	0
$967$	−1.75736	−0.0565129	−0.0282564	−	0.999601i	$-0.508995\pi$
$967$	−1.75736	−0.0565129	−0.0282564	+	0.999601i	$0.508995\pi$
$968$	0	0
$969$	43.3137	1.39144
$970$	0	0
$971$	− 29.3137i	− 0.940722i	−0.882474	−	0.470361i	$-0.844124\pi$
$971$	− 29.3137i	− 0.940722i	0.882474	−	0.470361i	$-0.155876\pi$
$972$	0	0
$973$	− 1.94113i	− 0.0622296i
$974$	0	0
$975$	−18.8284	−0.602992
$976$	0	0
$977$	−30.2843	−0.968880	−0.484440	−	0.874825i	$-0.660977\pi$
$977$	−30.2843	−0.968880	−0.484440	+	0.874825i	$0.660977\pi$
$978$	0	0
$979$	4.00000i	0.127841i
$980$	0	0
$981$	18.8284i	0.601145i
$982$	0	0
$983$	4.28427	0.136647	0.0683235	−	0.997663i	$-0.478235\pi$
$983$	4.28427	0.136647	0.0683235	+	0.997663i	$0.478235\pi$
$984$	0	0
$985$	−58.2843	−1.85709
$986$	0	0
$987$	2.62742i	0.0836316i
$988$	0	0
$989$	− 65.9411i	− 2.09681i
$990$	0	0
$991$	11.8995	0.378000	0.189000	−	0.981977i	$-0.439475\pi$
$991$	11.8995	0.378000	0.189000	+	0.981977i	$0.439475\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	59.9411i	1.90026i
$996$	0	0
$997$	− 12.9706i	− 0.410782i	−0.978680	−	0.205391i	$-0.934153\pi$
$997$	− 12.9706i	− 0.410782i	0.978680	−	0.205391i	$-0.0658466\pi$
$998$	0	0
$999$	−1.65685	−0.0524205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.d.a.769.3		4
3.2	odd	2		4608.2.d.c.2305.4		4
4.3	odd	2		1536.2.d.f.769.1		4
8.3	odd	2		1536.2.d.f.769.4		4
8.5	even	2	inner	1536.2.d.a.769.2		4
12.11	even	2		4608.2.d.o.2305.4		4
16.3	odd	4		1536.2.a.g.1.1	yes	2
16.5	even	4		1536.2.a.l.1.2	yes	2
16.11	odd	4		1536.2.a.e.1.2	yes	2
16.13	even	4		1536.2.a.b.1.1	✓	2
24.5	odd	2		4608.2.d.c.2305.1		4
24.11	even	2		4608.2.d.o.2305.1		4
48.5	odd	4		4608.2.a.e.1.1		2
48.11	even	4		4608.2.a.a.1.1		2
48.29	odd	4		4608.2.a.r.1.2		2
48.35	even	4		4608.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.b.1.1	✓	2	16.13	even	4
1536.2.a.e.1.2	yes	2	16.11	odd	4
1536.2.a.g.1.1	yes	2	16.3	odd	4
1536.2.a.l.1.2	yes	2	16.5	even	4
1536.2.d.a.769.2		4	8.5	even	2	inner
1536.2.d.a.769.3		4	1.1	even	1	trivial
1536.2.d.f.769.1		4	4.3	odd	2
1536.2.d.f.769.4		4	8.3	odd	2
4608.2.a.a.1.1		2	48.11	even	4
4608.2.a.e.1.1		2	48.5	odd	4
4608.2.a.n.1.2		2	48.35	even	4
4608.2.a.r.1.2		2	48.29	odd	4
4608.2.d.c.2305.1		4	24.5	odd	2
4608.2.d.c.2305.4		4	3.2	odd	2
4608.2.d.o.2305.1		4	24.11	even	2
4608.2.d.o.2305.4		4	12.11	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -3.41421i q^{5} -0.585786 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -3.41421i q^{5} -0.585786 q^{7} -1.00000 q^{9} -2.00000i q^{11} -2.82843i q^{13} +3.41421 q^{15} -7.65685 q^{17} +5.65685i q^{19} -0.585786i q^{21} -6.82843 q^{23} -6.65685 q^{25} -1.00000i q^{27} +3.41421i q^{29} +7.41421 q^{31} +2.00000 q^{33} +2.00000i q^{35} -1.65685i q^{37} +2.82843 q^{39} +0.343146 q^{41} +9.65685i q^{43} +3.41421i q^{45} -4.48528 q^{47} -6.65685 q^{49} -7.65685i q^{51} +7.89949i q^{53} -6.82843 q^{55} -5.65685 q^{57} +4.00000i q^{59} +1.65685i q^{61} +0.585786 q^{63} -9.65685 q^{65} -8.00000i q^{67} -6.82843i q^{69} -14.8284 q^{71} -9.65685 q^{73} -6.65685i q^{75} +1.17157i q^{77} +14.2426 q^{79} +1.00000 q^{81} -13.3137i q^{83} +26.1421i q^{85} -3.41421 q^{87} -2.00000 q^{89} +1.65685i q^{91} +7.41421i q^{93} +19.3137 q^{95} -9.31371 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q - 8 * q^7 - 4 * q^9	\( 4 q - 8 q^{7} - 4 q^{9} + 8 q^{15} - 8 q^{17} - 16 q^{23} - 4 q^{25} + 24 q^{31} + 8 q^{33} + 24 q^{41} + 16 q^{47} - 4 q^{49} - 16 q^{55} + 8 q^{63} - 16 q^{65} - 48 q^{71} - 16 q^{73} + 40 q^{79} + 4 q^{81} - 8 q^{87} - 8 q^{89} + 32 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 - 4 * q^9 + 8 * q^15 - 8 * q^17 - 16 * q^23 - 4 * q^25 + 24 * q^31 + 8 * q^33 + 24 * q^41 + 16 * q^47 - 4 * q^49 - 16 * q^55 + 8 * q^63 - 16 * q^65 - 48 * q^71 - 16 * q^73 + 40 * q^79 + 4 * q^81 - 8 * q^87 - 8 * q^89 + 32 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more