LMFDB - Embedded newform 1536.2.d.g.769.4

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			769.4
Root			$0.500000 - 2.10607i$ of defining polynomial
Character	$\chi$	$=$	1536.769
Dual form			1536.2.d.g.769.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.95687i q^{5} +1.63899 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.95687i q^{5} +1.63899 q^{7} -1.00000 q^{9} +4.82843i q^{11} -5.59587i q^{13} +3.95687 q^{15} -0.828427 q^{17} +2.82843i q^{19} -1.63899i q^{21} +7.91375 q^{23} -10.6569 q^{25} +1.00000i q^{27} +7.23486i q^{29} +1.63899 q^{31} +4.82843 q^{33} +6.48528i q^{35} -2.31788i q^{37} -5.59587 q^{39} -3.17157 q^{41} +4.48528i q^{43} -3.95687i q^{45} -7.91375 q^{47} -4.31371 q^{49} +0.828427i q^{51} -0.678892i q^{53} -19.1055 q^{55} +2.82843 q^{57} +9.65685i q^{59} +2.31788i q^{61} -1.63899 q^{63} +22.1421 q^{65} +13.6569i q^{67} -7.91375i q^{69} -3.27798 q^{71} -4.00000 q^{73} +10.6569i q^{75} +7.91375i q^{77} -1.63899 q^{79} +1.00000 q^{81} -8.82843i q^{83} -3.27798i q^{85} +7.23486 q^{87} -10.0000 q^{89} -9.17157i q^{91} -1.63899i q^{93} -11.1917 q^{95} +11.6569 q^{97} -4.82843i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.95687i	1.76957i	0.466001	+	0.884784i	$0.345694\pi$
$5$	3.95687i	1.76957i	−0.466001	+	0.884784i	$0.654306\pi$
$6$	0	0
$7$	1.63899	0.619480	0.309740	−	0.950821i	$-0.399758\pi$
$7$	1.63899	0.619480	0.309740	+	0.950821i	$0.399758\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.82843i	1.45583i	0.685670	+	0.727913i	$0.259509\pi$
$11$	4.82843i	1.45583i	−0.685670	+	0.727913i	$0.740491\pi$
$12$	0	0
$13$	− 5.59587i	− 1.55201i	−0.630724	−	0.776007i	$-0.717242\pi$
$13$	− 5.59587i	− 1.55201i	0.630724	−	0.776007i	$-0.282758\pi$
$14$	0	0
$15$	3.95687	1.02166
$16$	0	0
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	2.82843i	0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$	2.82843i	0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$	0	0
$21$	− 1.63899i	− 0.357657i
$22$	0	0
$23$	7.91375	1.65013	0.825065	−	0.565037i	$-0.191138\pi$
$23$	7.91375	1.65013	0.825065	+	0.565037i	$0.191138\pi$
$24$	0	0
$25$	−10.6569	−2.13137
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	7.23486i	1.34348i	0.740787	+	0.671740i	$0.234453\pi$
$29$	7.23486i	1.34348i	−0.740787	+	0.671740i	$0.765547\pi$
$30$	0	0
$31$	1.63899	0.294371	0.147186	−	0.989109i	$-0.452978\pi$
$31$	1.63899	0.294371	0.147186	+	0.989109i	$0.452978\pi$
$32$	0	0
$33$	4.82843	0.840521
$34$	0	0
$35$	6.48528i	1.09621i
$36$	0	0
$37$	− 2.31788i	− 0.381058i	−0.981682	−	0.190529i	$-0.938980\pi$
$37$	− 2.31788i	− 0.381058i	0.981682	−	0.190529i	$-0.0610203\pi$
$38$	0	0
$39$	−5.59587	−0.896056
$40$	0	0
$41$	−3.17157	−0.495316	−0.247658	−	0.968847i	$-0.579661\pi$
$41$	−3.17157	−0.495316	−0.247658	+	0.968847i	$0.579661\pi$
$42$	0	0
$43$	4.48528i	0.683999i	0.939700	+	0.341999i	$0.111104\pi$
$43$	4.48528i	0.683999i	−0.939700	+	0.341999i	$0.888896\pi$
$44$	0	0
$45$	− 3.95687i	− 0.589856i
$46$	0	0
$47$	−7.91375	−1.15434	−0.577169	−	0.816624i	$-0.695843\pi$
$47$	−7.91375	−1.15434	−0.577169	+	0.816624i	$0.695843\pi$
$48$	0	0
$49$	−4.31371	−0.616244
$50$	0	0
$51$	0.828427i	0.116003i
$52$	0	0
$53$	− 0.678892i	− 0.0932530i	−0.998912	−	0.0466265i	$-0.985153\pi$
$53$	− 0.678892i	− 0.0932530i	0.998912	−	0.0466265i	$-0.0148471\pi$
$54$	0	0
$55$	−19.1055	−2.57618
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	9.65685i	1.25722i	0.777723	+	0.628608i	$0.216375\pi$
$59$	9.65685i	1.25722i	−0.777723	+	0.628608i	$0.783625\pi$
$60$	0	0
$61$	2.31788i	0.296775i	0.988929	+	0.148387i	$0.0474082\pi$
$61$	2.31788i	0.296775i	−0.988929	+	0.148387i	$0.952592\pi$
$62$	0	0
$63$	−1.63899	−0.206493
$64$	0	0
$65$	22.1421	2.74639
$66$	0	0
$67$	13.6569i	1.66845i	0.551424	+	0.834225i	$0.314085\pi$
$67$	13.6569i	1.66845i	−0.551424	+	0.834225i	$0.685915\pi$
$68$	0	0
$69$	− 7.91375i	− 0.952703i
$70$	0	0
$71$	−3.27798	−0.389025	−0.194512	−	0.980900i	$-0.562312\pi$
$71$	−3.27798	−0.389025	−0.194512	+	0.980900i	$0.562312\pi$
$72$	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
$75$	10.6569i	1.23055i
$76$	0	0
$77$	7.91375i	0.901855i
$78$	0	0
$79$	−1.63899	−0.184401	−0.0922004	−	0.995740i	$-0.529390\pi$
$79$	−1.63899	−0.184401	−0.0922004	+	0.995740i	$0.529390\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 8.82843i	− 0.969046i	−0.874779	−	0.484523i	$-0.838993\pi$
$83$	− 8.82843i	− 0.969046i	0.874779	−	0.484523i	$-0.161007\pi$
$84$	0	0
$85$	− 3.27798i	− 0.355547i
$86$	0	0
$87$	7.23486	0.775658
$88$	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
$91$	− 9.17157i	− 0.961442i
$92$	0	0
$93$	− 1.63899i	− 0.169955i
$94$	0	0
$95$	−11.1917	−1.14825
$96$	0	0
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	0	0
$99$	− 4.82843i	− 0.485275i
$100$	0	0
$101$	0.678892i	0.0675523i	0.999429	+	0.0337762i	$0.0107533\pi$
$101$	0.678892i	0.0675523i	−0.999429	+	0.0337762i	$0.989247\pi$
$102$	0	0
$103$	17.4665	1.72102	0.860512	−	0.509430i	$-0.170144\pi$
$103$	17.4665	1.72102	0.860512	+	0.509430i	$0.170144\pi$
$104$	0	0
$105$	6.48528	0.632899
$106$	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
$109$	16.7876i	1.60796i	0.594656	+	0.803980i	$0.297288\pi$
$109$	16.7876i	1.60796i	−0.594656	+	0.803980i	$0.702712\pi$
$110$	0	0
$111$	−2.31788	−0.220004
$112$	0	0
$113$	0.343146	0.0322804	0.0161402	−	0.999870i	$-0.494862\pi$
$113$	0.343146	0.0322804	0.0161402	+	0.999870i	$0.494862\pi$
$114$	0	0
$115$	31.3137i	2.92002i
$116$	0	0
$117$	5.59587i	0.517338i
$118$	0	0
$119$	−1.35778	−0.124468
$120$	0	0
$121$	−12.3137	−1.11943
$122$	0	0
$123$	3.17157i	0.285971i
$124$	0	0
$125$	− 22.3835i	− 2.00204i
$126$	0	0
$127$	14.1885	1.25903	0.629513	−	0.776990i	$-0.283254\pi$
$127$	14.1885	1.25903	0.629513	+	0.776990i	$0.283254\pi$
$128$	0	0
$129$	4.48528	0.394907
$130$	0	0
$131$	7.31371i	0.639002i	0.947586	+	0.319501i	$0.103515\pi$
$131$	7.31371i	0.639002i	−0.947586	+	0.319501i	$0.896485\pi$
$132$	0	0
$133$	4.63577i	0.401972i
$134$	0	0
$135$	−3.95687	−0.340554
$136$	0	0
$137$	16.1421	1.37912	0.689558	−	0.724231i	$-0.257805\pi$
$137$	16.1421	1.37912	0.689558	+	0.724231i	$0.257805\pi$
$138$	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
$141$	7.91375i	0.666458i
$142$	0	0
$143$	27.0192	2.25946
$144$	0	0
$145$	−28.6274	−2.37738
$146$	0	0
$147$	4.31371i	0.355789i
$148$	0	0
$149$	3.95687i	0.324160i	0.986778	+	0.162080i	$0.0518202\pi$
$149$	3.95687i	0.324160i	−0.986778	+	0.162080i	$0.948180\pi$
$150$	0	0
$151$	14.1885	1.15464	0.577322	−	0.816516i	$-0.304098\pi$
$151$	14.1885	1.15464	0.577322	+	0.816516i	$0.304098\pi$
$152$	0	0
$153$	0.828427	0.0669744
$154$	0	0
$155$	6.48528i	0.520910i
$156$	0	0
$157$	− 18.1454i	− 1.44816i	−0.689717	−	0.724080i	$-0.742265\pi$
$157$	− 18.1454i	− 1.44816i	0.689717	−	0.724080i	$-0.257735\pi$
$158$	0	0
$159$	−0.678892	−0.0538397
$160$	0	0
$161$	12.9706	1.02222
$162$	0	0
$163$	− 10.1421i	− 0.794393i	−0.917734	−	0.397197i	$-0.869983\pi$
$163$	− 10.1421i	− 0.794393i	0.917734	−	0.397197i	$-0.130017\pi$
$164$	0	0
$165$	19.1055i	1.48736i
$166$	0	0
$167$	−4.63577	−0.358726	−0.179363	−	0.983783i	$-0.557404\pi$
$167$	−4.63577	−0.358726	−0.179363	+	0.983783i	$0.557404\pi$
$168$	0	0
$169$	−18.3137	−1.40875
$170$	0	0
$171$	− 2.82843i	− 0.216295i
$172$	0	0
$173$	− 11.8706i	− 0.902507i	−0.892396	−	0.451253i	$-0.850977\pi$
$173$	− 11.8706i	− 0.902507i	0.892396	−	0.451253i	$-0.149023\pi$
$174$	0	0
$175$	−17.4665	−1.32034
$176$	0	0
$177$	9.65685	0.725854
$178$	0	0
$179$	− 12.9706i	− 0.969465i	−0.874662	−	0.484733i	$-0.838917\pi$
$179$	− 12.9706i	− 0.969465i	0.874662	−	0.484733i	$-0.161083\pi$
$180$	0	0
$181$	16.7876i	1.24781i	0.781499	+	0.623906i	$0.214455\pi$
$181$	16.7876i	1.24781i	−0.781499	+	0.623906i	$0.785545\pi$
$182$	0	0
$183$	2.31788	0.171343
$184$	0	0
$185$	9.17157	0.674307
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	1.63899i	0.119219i
$190$	0	0
$191$	17.7477	1.28418	0.642089	−	0.766630i	$-0.278068\pi$
$191$	17.7477	1.28418	0.642089	+	0.766630i	$0.278068\pi$
$192$	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
$195$	− 22.1421i	− 1.58563i
$196$	0	0
$197$	− 16.5064i	− 1.17603i	−0.808849	−	0.588016i	$-0.799909\pi$
$197$	− 16.5064i	− 1.17603i	0.808849	−	0.588016i	$-0.200091\pi$
$198$	0	0
$199$	1.63899	0.116185	0.0580925	−	0.998311i	$-0.481498\pi$
$199$	1.63899	0.116185	0.0580925	+	0.998311i	$0.481498\pi$
$200$	0	0
$201$	13.6569	0.963280
$202$	0	0
$203$	11.8579i	0.832259i
$204$	0	0
$205$	− 12.5495i	− 0.876496i
$206$	0	0
$207$	−7.91375	−0.550044
$208$	0	0
$209$	−13.6569	−0.944664
$210$	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
$213$	3.27798i	0.224604i
$214$	0	0
$215$	−17.7477	−1.21038
$216$	0	0
$217$	2.68629	0.182357
$218$	0	0
$219$	4.00000i	0.270295i
$220$	0	0
$221$	4.63577i	0.311835i
$222$	0	0
$223$	−14.1885	−0.950133	−0.475066	−	0.879950i	$-0.657576\pi$
$223$	−14.1885	−0.950133	−0.475066	+	0.879950i	$0.657576\pi$
$224$	0	0
$225$	10.6569	0.710457
$226$	0	0
$227$	12.1421i	0.805902i	0.915222	+	0.402951i	$0.132015\pi$
$227$	12.1421i	0.805902i	−0.915222	+	0.402951i	$0.867985\pi$
$228$	0	0
$229$	− 21.4234i	− 1.41570i	−0.706365	−	0.707848i	$-0.749666\pi$
$229$	− 21.4234i	− 1.41570i	0.706365	−	0.707848i	$-0.250334\pi$
$230$	0	0
$231$	7.91375	0.520686
$232$	0	0
$233$	21.3137	1.39631	0.698154	−	0.715948i	$-0.254005\pi$
$233$	21.3137	1.39631	0.698154	+	0.715948i	$0.254005\pi$
$234$	0	0
$235$	− 31.3137i	− 2.04268i
$236$	0	0
$237$	1.63899i	0.106464i
$238$	0	0
$239$	4.63577	0.299863	0.149931	−	0.988696i	$-0.452095\pi$
$239$	4.63577	0.299863	0.149931	+	0.988696i	$0.452095\pi$
$240$	0	0
$241$	−16.9706	−1.09317	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	−16.9706	−1.09317	−0.546585	+	0.837404i	$0.684072\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 17.0688i	− 1.09049i
$246$	0	0
$247$	15.8275	1.00708
$248$	0	0
$249$	−8.82843	−0.559479
$250$	0	0
$251$	20.8284i	1.31468i	0.753595	+	0.657339i	$0.228318\pi$
$251$	20.8284i	1.31468i	−0.753595	+	0.657339i	$0.771682\pi$
$252$	0	0
$253$	38.2110i	2.40230i
$254$	0	0
$255$	−3.27798	−0.205275
$256$	0	0
$257$	−19.6569	−1.22616	−0.613080	−	0.790020i	$-0.710070\pi$
$257$	−19.6569	−1.22616	−0.613080	+	0.790020i	$0.710070\pi$
$258$	0	0
$259$	− 3.79899i	− 0.236058i
$260$	0	0
$261$	− 7.23486i	− 0.447826i
$262$	0	0
$263$	−15.8275	−0.975965	−0.487983	−	0.872853i	$-0.662267\pi$
$263$	−15.8275	−0.975965	−0.487983	+	0.872853i	$0.662267\pi$
$264$	0	0
$265$	2.68629	0.165018
$266$	0	0
$267$	10.0000i	0.611990i
$268$	0	0
$269$	− 7.23486i	− 0.441117i	−0.975374	−	0.220558i	$-0.929212\pi$
$269$	− 7.23486i	− 0.441117i	0.975374	−	0.220558i	$-0.0707880\pi$
$270$	0	0
$271$	17.4665	1.06101	0.530507	−	0.847681i	$-0.322002\pi$
$271$	17.4665	1.06101	0.530507	+	0.847681i	$0.322002\pi$
$272$	0	0
$273$	−9.17157	−0.555089
$274$	0	0
$275$	− 51.4558i	− 3.10290i
$276$	0	0
$277$	− 14.8674i	− 0.893295i	−0.894710	−	0.446648i	$-0.852618\pi$
$277$	− 14.8674i	− 0.893295i	0.894710	−	0.446648i	$-0.147382\pi$
$278$	0	0
$279$	−1.63899	−0.0981238
$280$	0	0
$281$	−0.343146	−0.0204704	−0.0102352	−	0.999948i	$-0.503258\pi$
$281$	−0.343146	−0.0204704	−0.0102352	+	0.999948i	$0.503258\pi$
$282$	0	0
$283$	− 28.9706i	− 1.72212i	−0.508502	−	0.861061i	$-0.669801\pi$
$283$	− 28.9706i	− 1.72212i	0.508502	−	0.861061i	$-0.330199\pi$
$284$	0	0
$285$	11.1917i	0.662941i
$286$	0	0
$287$	−5.19818	−0.306839
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	− 11.6569i	− 0.683337i
$292$	0	0
$293$	− 0.678892i	− 0.0396613i	−0.999803	−	0.0198307i	$-0.993687\pi$
$293$	− 0.678892i	− 0.0396613i	0.999803	−	0.0198307i	$-0.00631271\pi$
$294$	0	0
$295$	−38.2110	−2.22473
$296$	0	0
$297$	−4.82843	−0.280174
$298$	0	0
$299$	− 44.2843i	− 2.56103i
$300$	0	0
$301$	7.35134i	0.423724i
$302$	0	0
$303$	0.678892	0.0390013
$304$	0	0
$305$	−9.17157	−0.525163
$306$	0	0
$307$	21.6569i	1.23602i	0.786169	+	0.618011i	$0.212061\pi$
$307$	21.6569i	1.23602i	−0.786169	+	0.618011i	$0.787939\pi$
$308$	0	0
$309$	− 17.4665i	− 0.993634i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−7.31371	−0.413395	−0.206698	−	0.978405i	$-0.566272\pi$
$313$	−7.31371	−0.413395	−0.206698	+	0.978405i	$0.566272\pi$
$314$	0	0
$315$	− 6.48528i	− 0.365404i
$316$	0	0
$317$	21.7046i	1.21905i	0.792767	+	0.609525i	$0.208640\pi$
$317$	21.7046i	1.21905i	−0.792767	+	0.609525i	$0.791360\pi$
$318$	0	0
$319$	−34.9330	−1.95587
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 2.34315i	− 0.130376i
$324$	0	0
$325$	59.6343i	3.30792i
$326$	0	0
$327$	16.7876	0.928356
$328$	0	0
$329$	−12.9706	−0.715090
$330$	0	0
$331$	10.3431i	0.568511i	0.958749	+	0.284255i	$0.0917463\pi$
$331$	10.3431i	0.568511i	−0.958749	+	0.284255i	$0.908254\pi$
$332$	0	0
$333$	2.31788i	0.127019i
$334$	0	0
$335$	−54.0385	−2.95244
$336$	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
$339$	− 0.343146i	− 0.0186371i
$340$	0	0
$341$	7.91375i	0.428554i
$342$	0	0
$343$	−18.5431	−1.00123
$344$	0	0
$345$	31.3137	1.68587
$346$	0	0
$347$	0.828427i	0.0444723i	0.999753	+	0.0222361i	$0.00707857\pi$
$347$	0.828427i	0.0444723i	−0.999753	+	0.0222361i	$0.992921\pi$
$348$	0	0
$349$	− 4.23808i	− 0.226859i	−0.993546	−	0.113430i	$-0.963816\pi$
$349$	− 4.23808i	− 0.226859i	0.993546	−	0.113430i	$-0.0361836\pi$
$350$	0	0
$351$	5.59587	0.298685
$352$	0	0
$353$	1.31371	0.0699216	0.0349608	−	0.999389i	$-0.488869\pi$
$353$	1.31371	0.0699216	0.0349608	+	0.999389i	$0.488869\pi$
$354$	0	0
$355$	− 12.9706i	− 0.688406i
$356$	0	0
$357$	1.35778i	0.0718616i
$358$	0	0
$359$	34.9330	1.84369	0.921846	−	0.387556i	$-0.126681\pi$
$359$	34.9330	1.84369	0.921846	+	0.387556i	$0.126681\pi$
$360$	0	0
$361$	11.0000	0.578947
$362$	0	0
$363$	12.3137i	0.646302i
$364$	0	0
$365$	− 15.8275i	− 0.828449i
$366$	0	0
$367$	24.0225	1.25396	0.626981	−	0.779035i	$-0.284290\pi$
$367$	24.0225	1.25396	0.626981	+	0.779035i	$0.284290\pi$
$368$	0	0
$369$	3.17157	0.165105
$370$	0	0
$371$	− 1.11270i	− 0.0577684i
$372$	0	0
$373$	− 20.0656i	− 1.03896i	−0.854484	−	0.519478i	$-0.826126\pi$
$373$	− 20.0656i	− 1.03896i	0.854484	−	0.519478i	$-0.173874\pi$
$374$	0	0
$375$	−22.3835	−1.15588
$376$	0	0
$377$	40.4853	2.08510
$378$	0	0
$379$	− 0.485281i	− 0.0249272i	−0.999922	−	0.0124636i	$-0.996033\pi$
$379$	− 0.485281i	− 0.0249272i	0.999922	−	0.0124636i	$-0.00396739\pi$
$380$	0	0
$381$	− 14.1885i	− 0.726899i
$382$	0	0
$383$	11.1917	0.571871	0.285935	−	0.958249i	$-0.407696\pi$
$383$	11.1917	0.571871	0.285935	+	0.958249i	$0.407696\pi$
$384$	0	0
$385$	−31.3137	−1.59589
$386$	0	0
$387$	− 4.48528i	− 0.228000i
$388$	0	0
$389$	− 3.95687i	− 0.200621i	−0.994956	−	0.100311i	$-0.968016\pi$
$389$	− 3.95687i	− 0.200621i	0.994956	−	0.100311i	$-0.0319837\pi$
$390$	0	0
$391$	−6.55596	−0.331549
$392$	0	0
$393$	7.31371	0.368928
$394$	0	0
$395$	− 6.48528i	− 0.326310i
$396$	0	0
$397$	− 18.1454i	− 0.910691i	−0.890315	−	0.455345i	$-0.849516\pi$
$397$	− 18.1454i	− 0.910691i	0.890315	−	0.455345i	$-0.150484\pi$
$398$	0	0
$399$	4.63577	0.232079
$400$	0	0
$401$	24.1421	1.20560	0.602800	−	0.797892i	$-0.294052\pi$
$401$	24.1421	1.20560	0.602800	+	0.797892i	$0.294052\pi$
$402$	0	0
$403$	− 9.17157i	− 0.456869i
$404$	0	0
$405$	3.95687i	0.196619i
$406$	0	0
$407$	11.1917	0.554753
$408$	0	0
$409$	3.65685	0.180820	0.0904099	−	0.995905i	$-0.471182\pi$
$409$	3.65685	0.180820	0.0904099	+	0.995905i	$0.471182\pi$
$410$	0	0
$411$	− 16.1421i	− 0.796233i
$412$	0	0
$413$	15.8275i	0.778820i
$414$	0	0
$415$	34.9330	1.71479
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	− 15.1716i	− 0.741180i	−0.928797	−	0.370590i	$-0.879156\pi$
$419$	− 15.1716i	− 0.741180i	0.928797	−	0.370590i	$-0.120844\pi$
$420$	0	0
$421$	− 21.4234i	− 1.04411i	−0.852912	−	0.522055i	$-0.825165\pi$
$421$	− 21.4234i	− 1.04411i	0.852912	−	0.522055i	$-0.174835\pi$
$422$	0	0
$423$	7.91375	0.384780
$424$	0	0
$425$	8.82843	0.428242
$426$	0	0
$427$	3.79899i	0.183846i
$428$	0	0
$429$	− 27.0192i	− 1.30450i
$430$	0	0
$431$	14.4697	0.696982	0.348491	−	0.937312i	$-0.386694\pi$
$431$	14.4697	0.696982	0.348491	+	0.937312i	$0.386694\pi$
$432$	0	0
$433$	24.6274	1.18352	0.591759	−	0.806115i	$-0.298434\pi$
$433$	24.6274	1.18352	0.591759	+	0.806115i	$0.298434\pi$
$434$	0	0
$435$	28.6274i	1.37258i
$436$	0	0
$437$	22.3835i	1.07075i
$438$	0	0
$439$	39.8499	1.90193	0.950967	−	0.309292i	$-0.100092\pi$
$439$	39.8499	1.90193	0.950967	+	0.309292i	$0.100092\pi$
$440$	0	0
$441$	4.31371	0.205415
$442$	0	0
$443$	− 18.4853i	− 0.878262i	−0.898423	−	0.439131i	$-0.855286\pi$
$443$	− 18.4853i	− 0.878262i	0.898423	−	0.439131i	$-0.144714\pi$
$444$	0	0
$445$	− 39.5687i	− 1.87574i
$446$	0	0
$447$	3.95687	0.187154
$448$	0	0
$449$	17.5147	0.826571	0.413285	−	0.910602i	$-0.364381\pi$
$449$	17.5147	0.826571	0.413285	+	0.910602i	$0.364381\pi$
$450$	0	0
$451$	− 15.3137i	− 0.721094i
$452$	0	0
$453$	− 14.1885i	− 0.666634i
$454$	0	0
$455$	36.2908	1.70134
$456$	0	0
$457$	23.6569	1.10662	0.553310	−	0.832975i	$-0.313364\pi$
$457$	23.6569	1.10662	0.553310	+	0.832975i	$0.313364\pi$
$458$	0	0
$459$	− 0.828427i	− 0.0386677i
$460$	0	0
$461$	32.8963i	1.53213i	0.642761	+	0.766067i	$0.277789\pi$
$461$	32.8963i	1.53213i	−0.642761	+	0.766067i	$0.722211\pi$
$462$	0	0
$463$	−10.9105	−0.507055	−0.253528	−	0.967328i	$-0.581591\pi$
$463$	−10.9105	−0.507055	−0.253528	+	0.967328i	$0.581591\pi$
$464$	0	0
$465$	6.48528	0.300748
$466$	0	0
$467$	− 37.7990i	− 1.74913i	−0.484910	−	0.874564i	$-0.661147\pi$
$467$	− 37.7990i	− 1.74913i	0.484910	−	0.874564i	$-0.338853\pi$
$468$	0	0
$469$	22.3835i	1.03357i
$470$	0	0
$471$	−18.1454	−0.836095
$472$	0	0
$473$	−21.6569	−0.995783
$474$	0	0
$475$	− 30.1421i	− 1.38302i
$476$	0	0
$477$	0.678892i	0.0310843i
$478$	0	0
$479$	32.2174	1.47205	0.736025	−	0.676954i	$-0.236700\pi$
$479$	32.2174	1.47205	0.736025	+	0.676954i	$0.236700\pi$
$480$	0	0
$481$	−12.9706	−0.591407
$482$	0	0
$483$	− 12.9706i	− 0.590181i
$484$	0	0
$485$	46.1247i	2.09442i
$486$	0	0
$487$	−20.7445	−0.940022	−0.470011	−	0.882661i	$-0.655750\pi$
$487$	−20.7445	−0.940022	−0.470011	+	0.882661i	$0.655750\pi$
$488$	0	0
$489$	−10.1421	−0.458643
$490$	0	0
$491$	1.65685i	0.0747728i	0.999301	+	0.0373864i	$0.0119032\pi$
$491$	1.65685i	0.0747728i	−0.999301	+	0.0373864i	$0.988097\pi$
$492$	0	0
$493$	− 5.99355i	− 0.269936i
$494$	0	0
$495$	19.1055	0.858727
$496$	0	0
$497$	−5.37258	−0.240993
$498$	0	0
$499$	30.3431i	1.35835i	0.733978	+	0.679173i	$0.237661\pi$
$499$	30.3431i	1.35835i	−0.733978	+	0.679173i	$0.762339\pi$
$500$	0	0
$501$	4.63577i	0.207111i
$502$	0	0
$503$	12.5495	0.559555	0.279778	−	0.960065i	$-0.409739\pi$
$503$	12.5495	0.559555	0.279778	+	0.960065i	$0.409739\pi$
$504$	0	0
$505$	−2.68629	−0.119538
$506$	0	0
$507$	18.3137i	0.813340i
$508$	0	0
$509$	− 8.59264i	− 0.380862i	−0.981701	−	0.190431i	$-0.939011\pi$
$509$	− 8.59264i	− 0.380862i	0.981701	−	0.190431i	$-0.0609886\pi$
$510$	0	0
$511$	−6.55596	−0.290019
$512$	0	0
$513$	−2.82843	−0.124878
$514$	0	0
$515$	69.1127i	3.04547i
$516$	0	0
$517$	− 38.2110i	− 1.68052i
$518$	0	0
$519$	−11.8706	−0.521063
$520$	0	0
$521$	−15.1716	−0.664679	−0.332339	−	0.943160i	$-0.607838\pi$
$521$	−15.1716	−0.664679	−0.332339	+	0.943160i	$0.607838\pi$
$522$	0	0
$523$	13.1716i	0.575953i	0.957638	+	0.287976i	$0.0929824\pi$
$523$	13.1716i	0.575953i	−0.957638	+	0.287976i	$0.907018\pi$
$524$	0	0
$525$	17.4665i	0.762300i
$526$	0	0
$527$	−1.35778	−0.0591460
$528$	0	0
$529$	39.6274	1.72293
$530$	0	0
$531$	− 9.65685i	− 0.419072i
$532$	0	0
$533$	17.7477i	0.768738i
$534$	0	0
$535$	15.8275	0.684282
$536$	0	0
$537$	−12.9706	−0.559721
$538$	0	0
$539$	− 20.8284i	− 0.897144i
$540$	0	0
$541$	− 10.2316i	− 0.439892i	−0.975512	−	0.219946i	$-0.929412\pi$
$541$	− 10.2316i	− 0.439892i	0.975512	−	0.219946i	$-0.0705882\pi$
$542$	0	0
$543$	16.7876	0.720425
$544$	0	0
$545$	−66.4264	−2.84539
$546$	0	0
$547$	7.79899i	0.333461i	0.986003	+	0.166730i	$0.0533209\pi$
$547$	7.79899i	0.333461i	−0.986003	+	0.166730i	$0.946679\pi$
$548$	0	0
$549$	− 2.31788i	− 0.0989248i
$550$	0	0
$551$	−20.4633	−0.871764
$552$	0	0
$553$	−2.68629	−0.114233
$554$	0	0
$555$	− 9.17157i	− 0.389312i
$556$	0	0
$557$	27.6981i	1.17361i	0.809729	+	0.586804i	$0.199614\pi$
$557$	27.6981i	1.17361i	−0.809729	+	0.586804i	$0.800386\pi$
$558$	0	0
$559$	25.0990	1.06158
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	− 4.82843i	− 0.203494i	−0.994810	−	0.101747i	$-0.967557\pi$
$563$	− 4.82843i	− 0.203494i	0.994810	−	0.101747i	$-0.0324432\pi$
$564$	0	0
$565$	1.35778i	0.0571224i
$566$	0	0
$567$	1.63899	0.0688312
$568$	0	0
$569$	27.4558	1.15101	0.575504	−	0.817799i	$-0.304806\pi$
$569$	27.4558	1.15101	0.575504	+	0.817799i	$0.304806\pi$
$570$	0	0
$571$	0.686292i	0.0287204i	0.999897	+	0.0143602i	$0.00457115\pi$
$571$	0.686292i	0.0287204i	−0.999897	+	0.0143602i	$0.995429\pi$
$572$	0	0
$573$	− 17.7477i	− 0.741421i
$574$	0	0
$575$	−84.3357	−3.51704
$576$	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
$581$	− 14.4697i	− 0.600305i
$582$	0	0
$583$	3.27798	0.135760
$584$	0	0
$585$	−22.1421	−0.915465
$586$	0	0
$587$	28.9706i	1.19574i	0.801592	+	0.597872i	$0.203987\pi$
$587$	28.9706i	1.19574i	−0.801592	+	0.597872i	$0.796013\pi$
$588$	0	0
$589$	4.63577i	0.191013i
$590$	0	0
$591$	−16.5064	−0.678982
$592$	0	0
$593$	30.9706	1.27181	0.635904	−	0.771768i	$-0.280627\pi$
$593$	30.9706	1.27181	0.635904	+	0.771768i	$0.280627\pi$
$594$	0	0
$595$	− 5.37258i	− 0.220254i
$596$	0	0
$597$	− 1.63899i	− 0.0670794i
$598$	0	0
$599$	−46.1247	−1.88460	−0.942302	−	0.334763i	$-0.891344\pi$
$599$	−46.1247	−1.88460	−0.942302	+	0.334763i	$0.891344\pi$
$600$	0	0
$601$	27.3137	1.11415	0.557075	−	0.830462i	$-0.311924\pi$
$601$	27.3137	1.11415	0.557075	+	0.830462i	$0.311924\pi$
$602$	0	0
$603$	− 13.6569i	− 0.556150i
$604$	0	0
$605$	− 48.7238i	− 1.98090i
$606$	0	0
$607$	−8.19496	−0.332623	−0.166311	−	0.986073i	$-0.553186\pi$
$607$	−8.19496	−0.332623	−0.166311	+	0.986073i	$0.553186\pi$
$608$	0	0
$609$	11.8579	0.480505
$610$	0	0
$611$	44.2843i	1.79155i
$612$	0	0
$613$	− 35.8931i	− 1.44971i	−0.688903	−	0.724854i	$-0.741907\pi$
$613$	− 35.8931i	− 1.44971i	0.688903	−	0.724854i	$-0.258093\pi$
$614$	0	0
$615$	−12.5495	−0.506045
$616$	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
$619$	− 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
$621$	7.91375i	0.317568i
$622$	0	0
$623$	−16.3899	−0.656648
$624$	0	0
$625$	35.2843	1.41137
$626$	0	0
$627$	13.6569i	0.545402i
$628$	0	0
$629$	1.92020i	0.0765633i
$630$	0	0
$631$	−11.4729	−0.456730	−0.228365	−	0.973576i	$-0.573338\pi$
$631$	−11.4729	−0.456730	−0.228365	+	0.973576i	$0.573338\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	56.1421i	2.22793i
$636$	0	0
$637$	24.1389i	0.956419i
$638$	0	0
$639$	3.27798	0.129675
$640$	0	0
$641$	27.1716	1.07321	0.536606	−	0.843833i	$-0.319706\pi$
$641$	27.1716	1.07321	0.536606	+	0.843833i	$0.319706\pi$
$642$	0	0
$643$	− 20.4853i	− 0.807861i	−0.914790	−	0.403930i	$-0.867644\pi$
$643$	− 20.4853i	− 0.807861i	0.914790	−	0.403930i	$-0.132356\pi$
$644$	0	0
$645$	17.7477i	0.698815i
$646$	0	0
$647$	5.19818	0.204362	0.102181	−	0.994766i	$-0.467418\pi$
$647$	5.19818	0.204362	0.102181	+	0.994766i	$0.467418\pi$
$648$	0	0
$649$	−46.6274	−1.83029
$650$	0	0
$651$	− 2.68629i	− 0.105284i
$652$	0	0
$653$	13.2284i	0.517668i	0.965922	+	0.258834i	$0.0833382\pi$
$653$	13.2284i	0.517668i	−0.965922	+	0.258834i	$0.916662\pi$
$654$	0	0
$655$	−28.9394	−1.13076
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	− 1.65685i	− 0.0645419i	−0.999479	−	0.0322709i	$-0.989726\pi$
$659$	− 1.65685i	− 0.0645419i	0.999479	−	0.0322709i	$-0.0102739\pi$
$660$	0	0
$661$	− 4.23808i	− 0.164842i	−0.996598	−	0.0824211i	$-0.973735\pi$
$661$	− 4.23808i	− 0.164842i	0.996598	−	0.0824211i	$-0.0262653\pi$
$662$	0	0
$663$	4.63577	0.180038
$664$	0	0
$665$	−18.3431	−0.711317
$666$	0	0
$667$	57.2548i	2.21692i
$668$	0	0
$669$	14.1885i	0.548559i
$670$	0	0
$671$	−11.1917	−0.432052
$672$	0	0
$673$	14.6863	0.566115	0.283057	−	0.959103i	$-0.408651\pi$
$673$	14.6863	0.566115	0.283057	+	0.959103i	$0.408651\pi$
$674$	0	0
$675$	− 10.6569i	− 0.410183i
$676$	0	0
$677$	− 2.59909i	− 0.0998911i	−0.998752	−	0.0499456i	$-0.984095\pi$
$677$	− 2.59909i	− 0.0998911i	0.998752	−	0.0499456i	$-0.0159048\pi$
$678$	0	0
$679$	19.1055	0.733201
$680$	0	0
$681$	12.1421	0.465288
$682$	0	0
$683$	− 5.51472i	− 0.211015i	−0.994419	−	0.105507i	$-0.966353\pi$
$683$	− 5.51472i	− 0.211015i	0.994419	−	0.105507i	$-0.0336467\pi$
$684$	0	0
$685$	63.8724i	2.44044i
$686$	0	0
$687$	−21.4234	−0.817352
$688$	0	0
$689$	−3.79899	−0.144730
$690$	0	0
$691$	− 38.8284i	− 1.47710i	−0.674197	−	0.738551i	$-0.735510\pi$
$691$	− 38.8284i	− 1.47710i	0.674197	−	0.738551i	$-0.264490\pi$
$692$	0	0
$693$	− 7.91375i	− 0.300618i
$694$	0	0
$695$	31.6550	1.20074
$696$	0	0
$697$	2.62742	0.0995205
$698$	0	0
$699$	− 21.3137i	− 0.806158i
$700$	0	0
$701$	30.9761i	1.16995i	0.811051	+	0.584976i	$0.198896\pi$
$701$	30.9761i	1.16995i	−0.811051	+	0.584976i	$0.801104\pi$
$702$	0	0
$703$	6.55596	0.247263
$704$	0	0
$705$	−31.3137	−1.17934
$706$	0	0
$707$	1.11270i	0.0418473i
$708$	0	0
$709$	26.0591i	0.978671i	0.872096	+	0.489336i	$0.162761\pi$
$709$	26.0591i	0.978671i	−0.872096	+	0.489336i	$0.837239\pi$
$710$	0	0
$711$	1.63899	0.0614670
$712$	0	0
$713$	12.9706	0.485751
$714$	0	0
$715$	106.912i	3.99827i
$716$	0	0
$717$	− 4.63577i	− 0.173126i
$718$	0	0
$719$	34.9330	1.30278	0.651390	−	0.758743i	$-0.274186\pi$
$719$	34.9330	1.30278	0.651390	+	0.758743i	$0.274186\pi$
$720$	0	0
$721$	28.6274	1.06614
$722$	0	0
$723$	16.9706i	0.631142i
$724$	0	0
$725$	− 77.1008i	− 2.86345i
$726$	0	0
$727$	−30.5784	−1.13409	−0.567045	−	0.823687i	$-0.691914\pi$
$727$	−30.5784	−1.13409	−0.567045	+	0.823687i	$0.691914\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 3.71573i	− 0.137431i
$732$	0	0
$733$	18.7078i	0.690988i	0.938421	+	0.345494i	$0.112289\pi$
$733$	18.7078i	0.690988i	−0.938421	+	0.345494i	$0.887711\pi$
$734$	0	0
$735$	−17.0688	−0.629592
$736$	0	0
$737$	−65.9411	−2.42897
$738$	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
$741$	− 15.8275i	− 0.581438i
$742$	0	0
$743$	−33.5752	−1.23175	−0.615877	−	0.787842i	$-0.711198\pi$
$743$	−33.5752	−1.23175	−0.615877	+	0.787842i	$0.711198\pi$
$744$	0	0
$745$	−15.6569	−0.573623
$746$	0	0
$747$	8.82843i	0.323015i
$748$	0	0
$749$	− 6.55596i	− 0.239550i
$750$	0	0
$751$	−1.63899	−0.0598076	−0.0299038	−	0.999553i	$-0.509520\pi$
$751$	−1.63899	−0.0598076	−0.0299038	+	0.999553i	$0.509520\pi$
$752$	0	0
$753$	20.8284	0.759030
$754$	0	0
$755$	56.1421i	2.04322i
$756$	0	0
$757$	0.960099i	0.0348954i	0.999848	+	0.0174477i	$0.00555405\pi$
$757$	0.960099i	0.0348954i	−0.999848	+	0.0174477i	$0.994446\pi$
$758$	0	0
$759$	38.2110	1.38697
$760$	0	0
$761$	−30.7696	−1.11540	−0.557698	−	0.830044i	$-0.688315\pi$
$761$	−30.7696	−1.11540	−0.557698	+	0.830044i	$0.688315\pi$
$762$	0	0
$763$	27.5147i	0.996100i
$764$	0	0
$765$	3.27798i	0.118516i
$766$	0	0
$767$	54.0385	1.95122
$768$	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
$771$	19.6569i	0.707924i
$772$	0	0
$773$	− 5.87707i	− 0.211384i	−0.994399	−	0.105692i	$-0.966294\pi$
$773$	− 5.87707i	− 0.211384i	0.994399	−	0.105692i	$-0.0337057\pi$
$774$	0	0
$775$	−17.4665	−0.627415
$776$	0	0
$777$	−3.79899	−0.136288
$778$	0	0
$779$	− 8.97056i	− 0.321404i
$780$	0	0
$781$	− 15.8275i	− 0.566352i
$782$	0	0
$783$	−7.23486	−0.258553
$784$	0	0
$785$	71.7990	2.56262
$786$	0	0
$787$	20.7696i	0.740355i	0.928961	+	0.370177i	$0.120703\pi$
$787$	20.7696i	0.740355i	−0.928961	+	0.370177i	$0.879297\pi$
$788$	0	0
$789$	15.8275i	0.563474i
$790$	0	0
$791$	0.562413	0.0199971
$792$	0	0
$793$	12.9706	0.460598
$794$	0	0
$795$	− 2.68629i	− 0.0952729i
$796$	0	0
$797$	− 13.2284i	− 0.468574i	−0.972167	−	0.234287i	$-0.924724\pi$
$797$	− 13.2284i	− 0.468574i	0.972167	−	0.234287i	$-0.0752756\pi$
$798$	0	0
$799$	6.55596	0.231933
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	− 19.3137i	− 0.681566i
$804$	0	0
$805$	51.3229i	1.80889i
$806$	0	0
$807$	−7.23486	−0.254679
$808$	0	0
$809$	−47.4558	−1.66846	−0.834229	−	0.551418i	$-0.814087\pi$
$809$	−47.4558	−1.66846	−0.834229	+	0.551418i	$0.814087\pi$
$810$	0	0
$811$	− 1.45584i	− 0.0511216i	−0.999673	−	0.0255608i	$-0.991863\pi$
$811$	− 1.45584i	− 0.0511216i	0.999673	−	0.0255608i	$-0.00813714\pi$
$812$	0	0
$813$	− 17.4665i	− 0.612576i
$814$	0	0
$815$	40.1312	1.40573
$816$	0	0
$817$	−12.6863	−0.443837
$818$	0	0
$819$	9.17157i	0.320481i
$820$	0	0
$821$	− 2.03668i	− 0.0710805i	−0.999368	−	0.0355403i	$-0.988685\pi$
$821$	− 2.03668i	− 0.0710805i	0.999368	−	0.0355403i	$-0.0113152\pi$
$822$	0	0
$823$	−14.1885	−0.494580	−0.247290	−	0.968941i	$-0.579540\pi$
$823$	−14.1885	−0.494580	−0.247290	+	0.968941i	$0.579540\pi$
$824$	0	0
$825$	−51.4558	−1.79146
$826$	0	0
$827$	13.9411i	0.484780i	0.970179	+	0.242390i	$0.0779314\pi$
$827$	13.9411i	0.484780i	−0.970179	+	0.242390i	$0.922069\pi$
$828$	0	0
$829$	− 5.59587i	− 0.194352i	−0.995267	−	0.0971762i	$-0.969019\pi$
$829$	− 5.59587i	− 0.194352i	0.995267	−	0.0971762i	$-0.0309810\pi$
$830$	0	0
$831$	−14.8674	−0.515744
$832$	0	0
$833$	3.57359	0.123818
$834$	0	0
$835$	− 18.3431i	− 0.634791i
$836$	0	0
$837$	1.63899i	0.0566518i
$838$	0	0
$839$	−32.2174	−1.11227	−0.556134	−	0.831092i	$-0.687716\pi$
$839$	−32.2174	−1.11227	−0.556134	+	0.831092i	$0.687716\pi$
$840$	0	0
$841$	−23.3431	−0.804936
$842$	0	0
$843$	0.343146i	0.0118186i
$844$	0	0
$845$	− 72.4650i	− 2.49287i
$846$	0	0
$847$	−20.1821	−0.693464
$848$	0	0
$849$	−28.9706	−0.994267
$850$	0	0
$851$	− 18.3431i	− 0.628795i
$852$	0	0
$853$	18.1454i	0.621286i	0.950527	+	0.310643i	$0.100544\pi$
$853$	18.1454i	0.621286i	−0.950527	+	0.310643i	$0.899456\pi$
$854$	0	0
$855$	11.1917	0.382749
$856$	0	0
$857$	−13.5147	−0.461654	−0.230827	−	0.972995i	$-0.574143\pi$
$857$	−13.5147	−0.461654	−0.230827	+	0.972995i	$0.574143\pi$
$858$	0	0
$859$	− 29.4558i	− 1.00502i	−0.864571	−	0.502510i	$-0.832410\pi$
$859$	− 29.4558i	− 1.00502i	0.864571	−	0.502510i	$-0.167590\pi$
$860$	0	0
$861$	5.19818i	0.177153i
$862$	0	0
$863$	−4.63577	−0.157803	−0.0789017	−	0.996882i	$-0.525141\pi$
$863$	−4.63577	−0.157803	−0.0789017	+	0.996882i	$0.525141\pi$
$864$	0	0
$865$	46.9706	1.59705
$866$	0	0
$867$	16.3137i	0.554043i
$868$	0	0
$869$	− 7.91375i	− 0.268456i
$870$	0	0
$871$	76.4219	2.58946
$872$	0	0
$873$	−11.6569	−0.394525
$874$	0	0
$875$	− 36.6863i	− 1.24022i
$876$	0	0
$877$	− 58.2765i	− 1.96786i	−0.178558	−	0.983929i	$-0.557143\pi$
$877$	− 58.2765i	− 1.96786i	0.178558	−	0.983929i	$-0.442857\pi$
$878$	0	0
$879$	−0.678892	−0.0228985
$880$	0	0
$881$	−5.02944	−0.169446	−0.0847230	−	0.996405i	$-0.527001\pi$
$881$	−5.02944	−0.169446	−0.0847230	+	0.996405i	$0.527001\pi$
$882$	0	0
$883$	48.7696i	1.64123i	0.571484	+	0.820613i	$0.306368\pi$
$883$	48.7696i	1.64123i	−0.571484	+	0.820613i	$0.693632\pi$
$884$	0	0
$885$	38.2110i	1.28445i
$886$	0	0
$887$	−15.8275	−0.531435	−0.265718	−	0.964051i	$-0.585609\pi$
$887$	−15.8275	−0.531435	−0.265718	+	0.964051i	$0.585609\pi$
$888$	0	0
$889$	23.2548	0.779942
$890$	0	0
$891$	4.82843i	0.161758i
$892$	0	0
$893$	− 22.3835i	− 0.749034i
$894$	0	0
$895$	51.3229	1.71553
$896$	0	0
$897$	−44.2843	−1.47861
$898$	0	0
$899$	11.8579i	0.395482i
$900$	0	0
$901$	0.562413i	0.0187367i
$902$	0	0
$903$	7.35134	0.244637
$904$	0	0
$905$	−66.4264	−2.20809
$906$	0	0
$907$	15.7990i	0.524597i	0.964987	+	0.262298i	$0.0844805\pi$
$907$	15.7990i	0.524597i	−0.964987	+	0.262298i	$0.915520\pi$
$908$	0	0
$909$	− 0.678892i	− 0.0225174i
$910$	0	0
$911$	−1.92020	−0.0636190	−0.0318095	−	0.999494i	$-0.510127\pi$
$911$	−1.92020	−0.0636190	−0.0318095	+	0.999494i	$0.510127\pi$
$912$	0	0
$913$	42.6274	1.41076
$914$	0	0
$915$	9.17157i	0.303203i
$916$	0	0
$917$	11.9871i	0.395849i
$918$	0	0
$919$	−4.91697	−0.162196	−0.0810980	−	0.996706i	$-0.525843\pi$
$919$	−4.91697	−0.162196	−0.0810980	+	0.996706i	$0.525843\pi$
$920$	0	0
$921$	21.6569	0.713618
$922$	0	0
$923$	18.3431i	0.603772i
$924$	0	0
$925$	24.7013i	0.812175i
$926$	0	0
$927$	−17.4665	−0.573675
$928$	0	0
$929$	21.7990	0.715202	0.357601	−	0.933875i	$-0.383595\pi$
$929$	21.7990	0.715202	0.357601	+	0.933875i	$0.383595\pi$
$930$	0	0
$931$	− 12.2010i	− 0.399872i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15.8275	0.517615
$936$	0	0
$937$	40.6274	1.32724	0.663620	−	0.748070i	$-0.269019\pi$
$937$	40.6274	1.32724	0.663620	+	0.748070i	$0.269019\pi$
$938$	0	0
$939$	7.31371i	0.238674i
$940$	0	0
$941$	− 52.7972i	− 1.72114i	−0.509334	−	0.860569i	$-0.670108\pi$
$941$	− 52.7972i	− 1.72114i	0.509334	−	0.860569i	$-0.329892\pi$
$942$	0	0
$943$	−25.0990	−0.817337
$944$	0	0
$945$	−6.48528	−0.210966
$946$	0	0
$947$	28.9706i	0.941417i	0.882289	+	0.470708i	$0.156002\pi$
$947$	28.9706i	0.941417i	−0.882289	+	0.470708i	$0.843998\pi$
$948$	0	0
$949$	22.3835i	0.726598i
$950$	0	0
$951$	21.7046	0.703819
$952$	0	0
$953$	−29.7990	−0.965284	−0.482642	−	0.875818i	$-0.660323\pi$
$953$	−29.7990	−0.965284	−0.482642	+	0.875818i	$0.660323\pi$
$954$	0	0
$955$	70.2254i	2.27244i
$956$	0	0
$957$	34.9330i	1.12922i
$958$	0	0
$959$	26.4568	0.854335
$960$	0	0
$961$	−28.3137	−0.913345
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	− 15.8275i	− 0.509505i
$966$	0	0
$967$	43.1279	1.38690	0.693450	−	0.720504i	$-0.256090\pi$
$967$	43.1279	1.38690	0.693450	+	0.720504i	$0.256090\pi$
$968$	0	0
$969$	−2.34315	−0.0752727
$970$	0	0
$971$	− 15.4558i	− 0.496002i	−0.968760	−	0.248001i	$-0.920226\pi$
$971$	− 15.4558i	− 0.496002i	0.968760	−	0.248001i	$-0.0797736\pi$
$972$	0	0
$973$	− 13.1119i	− 0.420349i
$974$	0	0
$975$	59.6343	1.90983
$976$	0	0
$977$	−42.7696	−1.36832	−0.684160	−	0.729332i	$-0.739831\pi$
$977$	−42.7696	−1.36832	−0.684160	+	0.729332i	$0.739831\pi$
$978$	0	0
$979$	− 48.2843i	− 1.54317i
$980$	0	0
$981$	− 16.7876i	− 0.535987i
$982$	0	0
$983$	38.2110	1.21874	0.609370	−	0.792886i	$-0.291422\pi$
$983$	38.2110	1.21874	0.609370	+	0.792886i	$0.291422\pi$
$984$	0	0
$985$	65.3137	2.08107
$986$	0	0
$987$	12.9706i	0.412858i
$988$	0	0
$989$	35.4954i	1.12869i
$990$	0	0
$991$	−46.4059	−1.47413	−0.737066	−	0.675821i	$-0.763789\pi$
$991$	−46.4059	−1.47413	−0.737066	+	0.675821i	$0.763789\pi$
$992$	0	0
$993$	10.3431	0.328230
$994$	0	0
$995$	6.48528i	0.205597i
$996$	0	0
$997$	29.3371i	0.929116i	0.885543	+	0.464558i	$0.153787\pi$
$997$	29.3371i	0.929116i	−0.885543	+	0.464558i	$0.846213\pi$
$998$	0	0
$999$	2.31788	0.0733346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.d.g.769.4		8
3.2	odd	2		4608.2.d.p.2305.2		8
4.3	odd	2	inner	1536.2.d.g.769.8		8
8.3	odd	2	inner	1536.2.d.g.769.1		8
8.5	even	2	inner	1536.2.d.g.769.5		8
12.11	even	2		4608.2.d.p.2305.1		8
16.3	odd	4		1536.2.a.m.1.4	yes	4
16.5	even	4		1536.2.a.m.1.1	✓	4
16.11	odd	4		1536.2.a.n.1.1	yes	4
16.13	even	4		1536.2.a.n.1.4	yes	4
24.5	odd	2		4608.2.d.p.2305.8		8
24.11	even	2		4608.2.d.p.2305.7		8
48.5	odd	4		4608.2.a.ba.1.4		4
48.11	even	4		4608.2.a.t.1.4		4
48.29	odd	4		4608.2.a.t.1.1		4
48.35	even	4		4608.2.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.m.1.1	✓	4	16.5	even	4
1536.2.a.m.1.4	yes	4	16.3	odd	4
1536.2.a.n.1.1	yes	4	16.11	odd	4
1536.2.a.n.1.4	yes	4	16.13	even	4
1536.2.d.g.769.1		8	8.3	odd	2	inner
1536.2.d.g.769.4		8	1.1	even	1	trivial
1536.2.d.g.769.5		8	8.5	even	2	inner
1536.2.d.g.769.8		8	4.3	odd	2	inner
4608.2.a.t.1.1		4	48.29	odd	4
4608.2.a.t.1.4		4	48.11	even	4
4608.2.a.ba.1.1		4	48.35	even	4
4608.2.a.ba.1.4		4	48.5	odd	4
4608.2.d.p.2305.1		8	12.11	even	2
4608.2.d.p.2305.2		8	3.2	odd	2
4608.2.d.p.2305.7		8	24.11	even	2
4608.2.d.p.2305.8		8	24.5	odd	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.95687i q^{5} +1.63899 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.95687i q^{5} +1.63899 q^{7} -1.00000 q^{9} +4.82843i q^{11} -5.59587i q^{13} +3.95687 q^{15} -0.828427 q^{17} +2.82843i q^{19} -1.63899i q^{21} +7.91375 q^{23} -10.6569 q^{25} +1.00000i q^{27} +7.23486i q^{29} +1.63899 q^{31} +4.82843 q^{33} +6.48528i q^{35} -2.31788i q^{37} -5.59587 q^{39} -3.17157 q^{41} +4.48528i q^{43} -3.95687i q^{45} -7.91375 q^{47} -4.31371 q^{49} +0.828427i q^{51} -0.678892i q^{53} -19.1055 q^{55} +2.82843 q^{57} +9.65685i q^{59} +2.31788i q^{61} -1.63899 q^{63} +22.1421 q^{65} +13.6569i q^{67} -7.91375i q^{69} -3.27798 q^{71} -4.00000 q^{73} +10.6569i q^{75} +7.91375i q^{77} -1.63899 q^{79} +1.00000 q^{81} -8.82843i q^{83} -3.27798i q^{85} +7.23486 q^{87} -10.0000 q^{89} -9.17157i q^{91} -1.63899i q^{93} -11.1917 q^{95} +11.6569 q^{97} -4.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 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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