LMFDB - Embedded newform 768.4.d.l.385.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(385,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.385");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			385.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	768.385
Dual form			768.4.d.l.385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -8.00000i q^{5} +10.0000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -8.00000i q^{5} +10.0000 q^{7} -9.00000 q^{9} +68.0000i q^{11} -46.0000i q^{13} +24.0000 q^{15} -74.0000 q^{17} -16.0000i q^{19} +30.0000i q^{21} +20.0000 q^{23} +61.0000 q^{25} -27.0000i q^{27} +228.000i q^{29} -162.000 q^{31} -204.000 q^{33} -80.0000i q^{35} -262.000i q^{37} +138.000 q^{39} -30.0000 q^{41} +264.000i q^{43} +72.0000i q^{45} +124.000 q^{47} -243.000 q^{49} -222.000i q^{51} +204.000i q^{53} +544.000 q^{55} +48.0000 q^{57} +340.000i q^{59} +950.000i q^{61} -90.0000 q^{63} -368.000 q^{65} +436.000i q^{67} +60.0000i q^{69} +780.000 q^{71} -518.000 q^{73} +183.000i q^{75} +680.000i q^{77} -1010.00 q^{79} +81.0000 q^{81} -852.000i q^{83} +592.000i q^{85} -684.000 q^{87} +686.000 q^{89} -460.000i q^{91} -486.000i q^{93} -128.000 q^{95} -806.000 q^{97} -612.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{7} - 18 q^{9}+O(q^{10}) $ 2 * q + 20 * q^7 - 18 * q^9	$ 2 q + 20 q^{7} - 18 q^{9} + 48 q^{15} - 148 q^{17} + 40 q^{23} + 122 q^{25} - 324 q^{31} - 408 q^{33} + 276 q^{39} - 60 q^{41} + 248 q^{47} - 486 q^{49} + 1088 q^{55} + 96 q^{57} - 180 q^{63} - 736 q^{65} + 1560 q^{71} - 1036 q^{73} - 2020 q^{79} + 162 q^{81} - 1368 q^{87} + 1372 q^{89} - 256 q^{95} - 1612 q^{97}+O(q^{100}) $ 2 * q + 20 * q^7 - 18 * q^9 + 48 * q^15 - 148 * q^17 + 40 * q^23 + 122 * q^25 - 324 * q^31 - 408 * q^33 + 276 * q^39 - 60 * q^41 + 248 * q^47 - 486 * q^49 + 1088 * q^55 + 96 * q^57 - 180 * q^63 - 736 * q^65 + 1560 * q^71 - 1036 * q^73 - 2020 * q^79 + 162 * q^81 - 1368 * q^87 + 1372 * q^89 - 256 * q^95 - 1612 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	− 8.00000i	− 0.715542i	−0.933809	−	0.357771i	$-0.883537\pi$
$5$	− 8.00000i	− 0.715542i	0.933809	−	0.357771i	$-0.116463\pi$
$6$	0	0
$7$	10.0000	0.539949	0.269975	−	0.962867i	$-0.412985\pi$
$7$	10.0000	0.539949	0.269975	+	0.962867i	$0.412985\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	68.0000i	1.86389i	0.362602	+	0.931944i	$0.381889\pi$
$11$	68.0000i	1.86389i	−0.362602	+	0.931944i	$0.618111\pi$
$12$	0	0
$13$	− 46.0000i	− 0.981393i	−0.871331	−	0.490696i	$-0.836742\pi$
$13$	− 46.0000i	− 0.981393i	0.871331	−	0.490696i	$-0.163258\pi$
$14$	0	0
$15$	24.0000	0.413118
$16$	0	0
$17$	−74.0000	−1.05574	−0.527872	−	0.849324i	$-0.677010\pi$
$17$	−74.0000	−1.05574	−0.527872	+	0.849324i	$0.677010\pi$
$18$	0	0
$19$	− 16.0000i	− 0.193192i	−0.995324	−	0.0965961i	$-0.969204\pi$
$19$	− 16.0000i	− 0.193192i	0.995324	−	0.0965961i	$-0.0307955\pi$
$20$	0	0
$21$	30.0000i	0.311740i
$22$	0	0
$23$	20.0000	0.181317	0.0906584	−	0.995882i	$-0.471103\pi$
$23$	20.0000	0.181317	0.0906584	+	0.995882i	$0.471103\pi$
$24$	0	0
$25$	61.0000	0.488000
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	228.000i	1.45995i	0.683474	+	0.729975i	$0.260468\pi$
$29$	228.000i	1.45995i	−0.683474	+	0.729975i	$0.739532\pi$
$30$	0	0
$31$	−162.000	−0.938583	−0.469291	−	0.883043i	$-0.655491\pi$
$31$	−162.000	−0.938583	−0.469291	+	0.883043i	$0.655491\pi$
$32$	0	0
$33$	−204.000	−1.07612
$34$	0	0
$35$	− 80.0000i	− 0.386356i
$36$	0	0
$37$	− 262.000i	− 1.16412i	−0.813145	−	0.582061i	$-0.802246\pi$
$37$	− 262.000i	− 1.16412i	0.813145	−	0.582061i	$-0.197754\pi$
$38$	0	0
$39$	138.000	0.566607
$40$	0	0
$41$	−30.0000	−0.114273	−0.0571367	−	0.998366i	$-0.518197\pi$
$41$	−30.0000	−0.114273	−0.0571367	+	0.998366i	$0.518197\pi$
$42$	0	0
$43$	264.000i	0.936270i	0.883657	+	0.468135i	$0.155074\pi$
$43$	264.000i	0.936270i	−0.883657	+	0.468135i	$0.844926\pi$
$44$	0	0
$45$	72.0000i	0.238514i
$46$	0	0
$47$	124.000	0.384835	0.192418	−	0.981313i	$-0.438367\pi$
$47$	124.000	0.384835	0.192418	+	0.981313i	$0.438367\pi$
$48$	0	0
$49$	−243.000	−0.708455
$50$	0	0
$51$	− 222.000i	− 0.609534i
$52$	0	0
$53$	204.000i	0.528709i	0.964426	+	0.264354i	$0.0851588\pi$
$53$	204.000i	0.528709i	−0.964426	+	0.264354i	$0.914841\pi$
$54$	0	0
$55$	544.000	1.33369
$56$	0	0
$57$	48.0000	0.111540
$58$	0	0
$59$	340.000i	0.750241i	0.926976	+	0.375121i	$0.122399\pi$
$59$	340.000i	0.750241i	−0.926976	+	0.375121i	$0.877601\pi$
$60$	0	0
$61$	950.000i	1.99402i	0.0772921	+	0.997008i	$0.475373\pi$
$61$	950.000i	1.99402i	−0.0772921	+	0.997008i	$0.524627\pi$
$62$	0	0
$63$	−90.0000	−0.179983
$64$	0	0
$65$	−368.000	−0.702227
$66$	0	0
$67$	436.000i	0.795013i	0.917599	+	0.397507i	$0.130124\pi$
$67$	436.000i	0.795013i	−0.917599	+	0.397507i	$0.869876\pi$
$68$	0	0
$69$	60.0000i	0.104683i
$70$	0	0
$71$	780.000	1.30379	0.651894	−	0.758310i	$-0.273975\pi$
$71$	780.000	1.30379	0.651894	+	0.758310i	$0.273975\pi$
$72$	0	0
$73$	−518.000	−0.830511	−0.415256	−	0.909705i	$-0.636308\pi$
$73$	−518.000	−0.830511	−0.415256	+	0.909705i	$0.636308\pi$
$74$	0	0
$75$	183.000i	0.281747i
$76$	0	0
$77$	680.000i	1.00641i
$78$	0	0
$79$	−1010.00	−1.43840	−0.719202	−	0.694801i	$-0.755492\pi$
$79$	−1010.00	−1.43840	−0.719202	+	0.694801i	$0.755492\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 852.000i	− 1.12674i	−0.826206	−	0.563368i	$-0.809505\pi$
$83$	− 852.000i	− 1.12674i	0.826206	−	0.563368i	$-0.190495\pi$
$84$	0	0
$85$	592.000i	0.755428i
$86$	0	0
$87$	−684.000	−0.842902
$88$	0	0
$89$	686.000	0.817032	0.408516	−	0.912751i	$-0.366046\pi$
$89$	686.000	0.817032	0.408516	+	0.912751i	$0.366046\pi$
$90$	0	0
$91$	− 460.000i	− 0.529902i
$92$	0	0
$93$	− 486.000i	− 0.541891i
$94$	0	0
$95$	−128.000	−0.138237
$96$	0	0
$97$	−806.000	−0.843679	−0.421840	−	0.906670i	$-0.638615\pi$
$97$	−806.000	−0.843679	−0.421840	+	0.906670i	$0.638615\pi$
$98$	0	0
$99$	− 612.000i	− 0.621296i
$100$	0	0
$101$	636.000i	0.626578i	0.949658	+	0.313289i	$0.101431\pi$
$101$	636.000i	0.626578i	−0.949658	+	0.313289i	$0.898569\pi$
$102$	0	0
$103$	−1586.00	−1.51722	−0.758608	−	0.651547i	$-0.774120\pi$
$103$	−1586.00	−1.51722	−0.758608	+	0.651547i	$0.774120\pi$
$104$	0	0
$105$	240.000	0.223063
$106$	0	0
$107$	92.0000i	0.0831213i	0.999136	+	0.0415606i	$0.0132330\pi$
$107$	92.0000i	0.0831213i	−0.999136	+	0.0415606i	$0.986767\pi$
$108$	0	0
$109$	2010.00i	1.76627i	0.469122	+	0.883133i	$0.344570\pi$
$109$	2010.00i	1.76627i	−0.469122	+	0.883133i	$0.655430\pi$
$110$	0	0
$111$	786.000	0.672106
$112$	0	0
$113$	−2062.00	−1.71661	−0.858304	−	0.513142i	$-0.828481\pi$
$113$	−2062.00	−1.71661	−0.858304	+	0.513142i	$0.828481\pi$
$114$	0	0
$115$	− 160.000i	− 0.129740i
$116$	0	0
$117$	414.000i	0.327131i
$118$	0	0
$119$	−740.000	−0.570048
$120$	0	0
$121$	−3293.00	−2.47408
$122$	0	0
$123$	− 90.0000i	− 0.0659758i
$124$	0	0
$125$	− 1488.00i	− 1.06473i
$126$	0	0
$127$	−1046.00	−0.730846	−0.365423	−	0.930841i	$-0.619076\pi$
$127$	−1046.00	−0.730846	−0.365423	+	0.930841i	$0.619076\pi$
$128$	0	0
$129$	−792.000	−0.540556
$130$	0	0
$131$	1884.00i	1.25653i	0.777998	+	0.628267i	$0.216235\pi$
$131$	1884.00i	1.25653i	−0.777998	+	0.628267i	$0.783765\pi$
$132$	0	0
$133$	− 160.000i	− 0.104314i
$134$	0	0
$135$	−216.000	−0.137706
$136$	0	0
$137$	−2374.00	−1.48047	−0.740235	−	0.672348i	$-0.765286\pi$
$137$	−2374.00	−1.48047	−0.740235	+	0.672348i	$0.765286\pi$
$138$	0	0
$139$	− 2156.00i	− 1.31561i	−0.753189	−	0.657804i	$-0.771485\pi$
$139$	− 2156.00i	− 1.31561i	0.753189	−	0.657804i	$-0.228515\pi$
$140$	0	0
$141$	372.000i	0.222185i
$142$	0	0
$143$	3128.00	1.82921
$144$	0	0
$145$	1824.00	1.04465
$146$	0	0
$147$	− 729.000i	− 0.409027i
$148$	0	0
$149$	2952.00i	1.62307i	0.584305	+	0.811534i	$0.301367\pi$
$149$	2952.00i	1.62307i	−0.584305	+	0.811534i	$0.698633\pi$
$150$	0	0
$151$	3410.00	1.83776	0.918880	−	0.394536i	$-0.129095\pi$
$151$	3410.00	1.83776	0.918880	+	0.394536i	$0.129095\pi$
$152$	0	0
$153$	666.000	0.351914
$154$	0	0
$155$	1296.00i	0.671595i
$156$	0	0
$157$	− 666.000i	− 0.338552i	−0.985569	−	0.169276i	$-0.945857\pi$
$157$	− 666.000i	− 0.338552i	0.985569	−	0.169276i	$-0.0541428\pi$
$158$	0	0
$159$	−612.000	−0.305250
$160$	0	0
$161$	200.000	0.0979019
$162$	0	0
$163$	704.000i	0.338292i	0.985591	+	0.169146i	$0.0541009\pi$
$163$	704.000i	0.338292i	−0.985591	+	0.169146i	$0.945899\pi$
$164$	0	0
$165$	1632.00i	0.770006i
$166$	0	0
$167$	−1648.00	−0.763629	−0.381815	−	0.924239i	$-0.624701\pi$
$167$	−1648.00	−0.763629	−0.381815	+	0.924239i	$0.624701\pi$
$168$	0	0
$169$	81.0000	0.0368685
$170$	0	0
$171$	144.000i	0.0643974i
$172$	0	0
$173$	− 360.000i	− 0.158210i	−0.996866	−	0.0791049i	$-0.974794\pi$
$173$	− 360.000i	− 0.158210i	0.996866	−	0.0791049i	$-0.0252062\pi$
$174$	0	0
$175$	610.000	0.263495
$176$	0	0
$177$	−1020.00	−0.433152
$178$	0	0
$179$	1956.00i	0.816750i	0.912814	+	0.408375i	$0.133904\pi$
$179$	1956.00i	0.816750i	−0.912814	+	0.408375i	$0.866096\pi$
$180$	0	0
$181$	− 218.000i	− 0.0895238i	−0.998998	−	0.0447619i	$-0.985747\pi$
$181$	− 218.000i	− 0.0895238i	0.998998	−	0.0447619i	$-0.0142529\pi$
$182$	0	0
$183$	−2850.00	−1.15125
$184$	0	0
$185$	−2096.00	−0.832978
$186$	0	0
$187$	− 5032.00i	− 1.96779i
$188$	0	0
$189$	− 270.000i	− 0.103913i
$190$	0	0
$191$	680.000	0.257608	0.128804	−	0.991670i	$-0.458886\pi$
$191$	680.000	0.257608	0.128804	+	0.991670i	$0.458886\pi$
$192$	0	0
$193$	4510.00	1.68206	0.841028	−	0.540991i	$-0.181951\pi$
$193$	4510.00	1.68206	0.841028	+	0.540991i	$0.181951\pi$
$194$	0	0
$195$	− 1104.00i	− 0.405431i
$196$	0	0
$197$	− 796.000i	− 0.287881i	−0.989586	−	0.143941i	$-0.954023\pi$
$197$	− 796.000i	− 0.287881i	0.989586	−	0.143941i	$-0.0459775\pi$
$198$	0	0
$199$	−986.000	−0.351235	−0.175617	−	0.984459i	$-0.556192\pi$
$199$	−986.000	−0.351235	−0.175617	+	0.984459i	$0.556192\pi$
$200$	0	0
$201$	−1308.00	−0.459001
$202$	0	0
$203$	2280.00i	0.788299i
$204$	0	0
$205$	240.000i	0.0817674i
$206$	0	0
$207$	−180.000	−0.0604390
$208$	0	0
$209$	1088.00	0.360089
$210$	0	0
$211$	4196.00i	1.36903i	0.729001	+	0.684513i	$0.239985\pi$
$211$	4196.00i	1.36903i	−0.729001	+	0.684513i	$0.760015\pi$
$212$	0	0
$213$	2340.00i	0.752743i
$214$	0	0
$215$	2112.00	0.669940
$216$	0	0
$217$	−1620.00	−0.506787
$218$	0	0
$219$	− 1554.00i	− 0.479496i
$220$	0	0
$221$	3404.00i	1.03610i
$222$	0	0
$223$	−2502.00	−0.751329	−0.375664	−	0.926756i	$-0.622585\pi$
$223$	−2502.00	−0.751329	−0.375664	+	0.926756i	$0.622585\pi$
$224$	0	0
$225$	−549.000	−0.162667
$226$	0	0
$227$	− 324.000i	− 0.0947341i	−0.998878	−	0.0473670i	$-0.984917\pi$
$227$	− 324.000i	− 0.0947341i	0.998878	−	0.0473670i	$-0.0150830\pi$
$228$	0	0
$229$	− 2194.00i	− 0.633116i	−0.948573	−	0.316558i	$-0.897473\pi$
$229$	− 2194.00i	− 0.633116i	0.948573	−	0.316558i	$-0.102527\pi$
$230$	0	0
$231$	−2040.00	−0.581048
$232$	0	0
$233$	4230.00	1.18934	0.594671	−	0.803969i	$-0.297282\pi$
$233$	4230.00	1.18934	0.594671	+	0.803969i	$0.297282\pi$
$234$	0	0
$235$	− 992.000i	− 0.275366i
$236$	0	0
$237$	− 3030.00i	− 0.830463i
$238$	0	0
$239$	6672.00	1.80576	0.902878	−	0.429896i	$-0.141450\pi$
$239$	6672.00	1.80576	0.902878	+	0.429896i	$0.141450\pi$
$240$	0	0
$241$	−1490.00	−0.398255	−0.199127	−	0.979974i	$-0.563811\pi$
$241$	−1490.00	−0.398255	−0.199127	+	0.979974i	$0.563811\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	1944.00i	0.506929i
$246$	0	0
$247$	−736.000	−0.189597
$248$	0	0
$249$	2556.00	0.650522
$250$	0	0
$251$	7380.00i	1.85586i	0.372751	+	0.927931i	$0.378414\pi$
$251$	7380.00i	1.85586i	−0.372751	+	0.927931i	$0.621586\pi$
$252$	0	0
$253$	1360.00i	0.337954i
$254$	0	0
$255$	−1776.00	−0.436147
$256$	0	0
$257$	−1998.00	−0.484949	−0.242474	−	0.970158i	$-0.577959\pi$
$257$	−1998.00	−0.484949	−0.242474	+	0.970158i	$0.577959\pi$
$258$	0	0
$259$	− 2620.00i	− 0.628567i
$260$	0	0
$261$	− 2052.00i	− 0.486650i
$262$	0	0
$263$	3768.00	0.883440	0.441720	−	0.897153i	$-0.354368\pi$
$263$	3768.00	0.883440	0.441720	+	0.897153i	$0.354368\pi$
$264$	0	0
$265$	1632.00	0.378313
$266$	0	0
$267$	2058.00i	0.471714i
$268$	0	0
$269$	− 7964.00i	− 1.80511i	−0.430578	−	0.902553i	$-0.641690\pi$
$269$	− 7964.00i	− 1.80511i	0.430578	−	0.902553i	$-0.358310\pi$
$270$	0	0
$271$	2330.00	0.522278	0.261139	−	0.965301i	$-0.415902\pi$
$271$	2330.00	0.522278	0.261139	+	0.965301i	$0.415902\pi$
$272$	0	0
$273$	1380.00	0.305939
$274$	0	0
$275$	4148.00i	0.909577i
$276$	0	0
$277$	− 2154.00i	− 0.467225i	−0.972330	−	0.233612i	$-0.924945\pi$
$277$	− 2154.00i	− 0.467225i	0.972330	−	0.233612i	$-0.0750547\pi$
$278$	0	0
$279$	1458.00	0.312861
$280$	0	0
$281$	5598.00	1.18843	0.594215	−	0.804306i	$-0.297463\pi$
$281$	5598.00	1.18843	0.594215	+	0.804306i	$0.297463\pi$
$282$	0	0
$283$	1884.00i	0.395732i	0.980229	+	0.197866i	$0.0634011\pi$
$283$	1884.00i	0.395732i	−0.980229	+	0.197866i	$0.936599\pi$
$284$	0	0
$285$	− 384.000i	− 0.0798112i
$286$	0	0
$287$	−300.000	−0.0617019
$288$	0	0
$289$	563.000	0.114594
$290$	0	0
$291$	− 2418.00i	− 0.487099i
$292$	0	0
$293$	− 2788.00i	− 0.555893i	−0.960597	−	0.277947i	$-0.910346\pi$
$293$	− 2788.00i	− 0.555893i	0.960597	−	0.277947i	$-0.0896538\pi$
$294$	0	0
$295$	2720.00	0.536829
$296$	0	0
$297$	1836.00	0.358705
$298$	0	0
$299$	− 920.000i	− 0.177943i
$300$	0	0
$301$	2640.00i	0.505538i
$302$	0	0
$303$	−1908.00	−0.361755
$304$	0	0
$305$	7600.00	1.42680
$306$	0	0
$307$	− 9180.00i	− 1.70661i	−0.521409	−	0.853307i	$-0.674594\pi$
$307$	− 9180.00i	− 1.70661i	0.521409	−	0.853307i	$-0.325406\pi$
$308$	0	0
$309$	− 4758.00i	− 0.875965i
$310$	0	0
$311$	8032.00	1.46448	0.732239	−	0.681047i	$-0.238475\pi$
$311$	8032.00	1.46448	0.732239	+	0.681047i	$0.238475\pi$
$312$	0	0
$313$	−214.000	−0.0386454	−0.0193227	−	0.999813i	$-0.506151\pi$
$313$	−214.000	−0.0386454	−0.0193227	+	0.999813i	$0.506151\pi$
$314$	0	0
$315$	720.000i	0.128785i
$316$	0	0
$317$	− 1428.00i	− 0.253011i	−0.991966	−	0.126505i	$-0.959624\pi$
$317$	− 1428.00i	− 0.253011i	0.991966	−	0.126505i	$-0.0403761\pi$
$318$	0	0
$319$	−15504.0	−2.72118
$320$	0	0
$321$	−276.000	−0.0479901
$322$	0	0
$323$	1184.00i	0.203961i
$324$	0	0
$325$	− 2806.00i	− 0.478920i
$326$	0	0
$327$	−6030.00	−1.01975
$328$	0	0
$329$	1240.00	0.207792
$330$	0	0
$331$	− 2708.00i	− 0.449683i	−0.974395	−	0.224842i	$-0.927814\pi$
$331$	− 2708.00i	− 0.449683i	0.974395	−	0.224842i	$-0.0721865\pi$
$332$	0	0
$333$	2358.00i	0.388041i
$334$	0	0
$335$	3488.00	0.568865
$336$	0	0
$337$	−6338.00	−1.02449	−0.512245	−	0.858840i	$-0.671186\pi$
$337$	−6338.00	−1.02449	−0.512245	+	0.858840i	$0.671186\pi$
$338$	0	0
$339$	− 6186.00i	− 0.991084i
$340$	0	0
$341$	− 11016.0i	− 1.74941i
$342$	0	0
$343$	−5860.00	−0.922479
$344$	0	0
$345$	480.000	0.0749053
$346$	0	0
$347$	1980.00i	0.306317i	0.988202	+	0.153158i	$0.0489445\pi$
$347$	1980.00i	0.306317i	−0.988202	+	0.153158i	$0.951056\pi$
$348$	0	0
$349$	− 7418.00i	− 1.13775i	−0.822422	−	0.568877i	$-0.807378\pi$
$349$	− 7418.00i	− 1.13775i	0.822422	−	0.568877i	$-0.192622\pi$
$350$	0	0
$351$	−1242.00	−0.188869
$352$	0	0
$353$	−3438.00	−0.518375	−0.259187	−	0.965827i	$-0.583455\pi$
$353$	−3438.00	−0.518375	−0.259187	+	0.965827i	$0.583455\pi$
$354$	0	0
$355$	− 6240.00i	− 0.932915i
$356$	0	0
$357$	− 2220.00i	− 0.329117i
$358$	0	0
$359$	−2372.00	−0.348717	−0.174358	−	0.984682i	$-0.555785\pi$
$359$	−2372.00	−0.348717	−0.174358	+	0.984682i	$0.555785\pi$
$360$	0	0
$361$	6603.00	0.962677
$362$	0	0
$363$	− 9879.00i	− 1.42841i
$364$	0	0
$365$	4144.00i	0.594265i
$366$	0	0
$367$	6046.00	0.859942	0.429971	−	0.902843i	$-0.358524\pi$
$367$	6046.00	0.859942	0.429971	+	0.902843i	$0.358524\pi$
$368$	0	0
$369$	270.000	0.0380912
$370$	0	0
$371$	2040.00i	0.285476i
$372$	0	0
$373$	− 9238.00i	− 1.28237i	−0.767385	−	0.641187i	$-0.778442\pi$
$373$	− 9238.00i	− 1.28237i	0.767385	−	0.641187i	$-0.221558\pi$
$374$	0	0
$375$	4464.00	0.614720
$376$	0	0
$377$	10488.0	1.43278
$378$	0	0
$379$	2936.00i	0.397921i	0.980007	+	0.198961i	$0.0637566\pi$
$379$	2936.00i	0.397921i	−0.980007	+	0.198961i	$0.936243\pi$
$380$	0	0
$381$	− 3138.00i	− 0.421954i
$382$	0	0
$383$	−12600.0	−1.68102	−0.840509	−	0.541798i	$-0.817744\pi$
$383$	−12600.0	−1.68102	−0.840509	+	0.541798i	$0.817744\pi$
$384$	0	0
$385$	5440.00	0.720125
$386$	0	0
$387$	− 2376.00i	− 0.312090i
$388$	0	0
$389$	− 3112.00i	− 0.405616i	−0.979219	−	0.202808i	$-0.934993\pi$
$389$	− 3112.00i	− 0.405616i	0.979219	−	0.202808i	$-0.0650067\pi$
$390$	0	0
$391$	−1480.00	−0.191424
$392$	0	0
$393$	−5652.00	−0.725460
$394$	0	0
$395$	8080.00i	1.02924i
$396$	0	0
$397$	− 2130.00i	− 0.269274i	−0.990895	−	0.134637i	$-0.957013\pi$
$397$	− 2130.00i	− 0.269274i	0.990895	−	0.134637i	$-0.0429868\pi$
$398$	0	0
$399$	480.000	0.0602257
$400$	0	0
$401$	−1554.00	−0.193524	−0.0967619	−	0.995308i	$-0.530849\pi$
$401$	−1554.00	−0.193524	−0.0967619	+	0.995308i	$0.530849\pi$
$402$	0	0
$403$	7452.00i	0.921118i
$404$	0	0
$405$	− 648.000i	− 0.0795046i
$406$	0	0
$407$	17816.0	2.16979
$408$	0	0
$409$	8942.00	1.08106	0.540530	−	0.841325i	$-0.318224\pi$
$409$	8942.00	1.08106	0.540530	+	0.841325i	$0.318224\pi$
$410$	0	0
$411$	− 7122.00i	− 0.854750i
$412$	0	0
$413$	3400.00i	0.405092i
$414$	0	0
$415$	−6816.00	−0.806227
$416$	0	0
$417$	6468.00	0.759567
$418$	0	0
$419$	6908.00i	0.805436i	0.915324	+	0.402718i	$0.131935\pi$
$419$	6908.00i	0.805436i	−0.915324	+	0.402718i	$0.868065\pi$
$420$	0	0
$421$	7862.00i	0.910144i	0.890455	+	0.455072i	$0.150386\pi$
$421$	7862.00i	0.910144i	−0.890455	+	0.455072i	$0.849614\pi$
$422$	0	0
$423$	−1116.00	−0.128278
$424$	0	0
$425$	−4514.00	−0.515203
$426$	0	0
$427$	9500.00i	1.07667i
$428$	0	0
$429$	9384.00i	1.05609i
$430$	0	0
$431$	12100.0	1.35229	0.676144	−	0.736769i	$-0.263650\pi$
$431$	12100.0	1.35229	0.676144	+	0.736769i	$0.263650\pi$
$432$	0	0
$433$	12882.0	1.42972	0.714861	−	0.699267i	$-0.246490\pi$
$433$	12882.0	1.42972	0.714861	+	0.699267i	$0.246490\pi$
$434$	0	0
$435$	5472.00i	0.603132i
$436$	0	0
$437$	− 320.000i	− 0.0350290i
$438$	0	0
$439$	−10818.0	−1.17612	−0.588058	−	0.808819i	$-0.700107\pi$
$439$	−10818.0	−1.17612	−0.588058	+	0.808819i	$0.700107\pi$
$440$	0	0
$441$	2187.00	0.236152
$442$	0	0
$443$	− 852.000i	− 0.0913764i	−0.998956	−	0.0456882i	$-0.985452\pi$
$443$	− 852.000i	− 0.0913764i	0.998956	−	0.0456882i	$-0.0145481\pi$
$444$	0	0
$445$	− 5488.00i	− 0.584621i
$446$	0	0
$447$	−8856.00	−0.937079
$448$	0	0
$449$	5222.00	0.548867	0.274434	−	0.961606i	$-0.411510\pi$
$449$	5222.00	0.548867	0.274434	+	0.961606i	$0.411510\pi$
$450$	0	0
$451$	− 2040.00i	− 0.212993i
$452$	0	0
$453$	10230.0i	1.06103i
$454$	0	0
$455$	−3680.00	−0.379167
$456$	0	0
$457$	11598.0	1.18716	0.593579	−	0.804775i	$-0.297714\pi$
$457$	11598.0	1.18716	0.593579	+	0.804775i	$0.297714\pi$
$458$	0	0
$459$	1998.00i	0.203178i
$460$	0	0
$461$	− 4088.00i	− 0.413009i	−0.978446	−	0.206504i	$-0.933791\pi$
$461$	− 4088.00i	− 0.413009i	0.978446	−	0.206504i	$-0.0662088\pi$
$462$	0	0
$463$	15394.0	1.54518	0.772592	−	0.634903i	$-0.218960\pi$
$463$	15394.0	1.54518	0.772592	+	0.634903i	$0.218960\pi$
$464$	0	0
$465$	−3888.00	−0.387746
$466$	0	0
$467$	− 3092.00i	− 0.306383i	−0.988197	−	0.153191i	$-0.951045\pi$
$467$	− 3092.00i	− 0.306383i	0.988197	−	0.153191i	$-0.0489551\pi$
$468$	0	0
$469$	4360.00i	0.429267i
$470$	0	0
$471$	1998.00	0.195463
$472$	0	0
$473$	−17952.0	−1.74510
$474$	0	0
$475$	− 976.000i	− 0.0942778i
$476$	0	0
$477$	− 1836.00i	− 0.176236i
$478$	0	0
$479$	9612.00	0.916876	0.458438	−	0.888726i	$-0.348409\pi$
$479$	9612.00	0.916876	0.458438	+	0.888726i	$0.348409\pi$
$480$	0	0
$481$	−12052.0	−1.14246
$482$	0	0
$483$	600.000i	0.0565237i
$484$	0	0
$485$	6448.00i	0.603688i
$486$	0	0
$487$	−13606.0	−1.26601	−0.633005	−	0.774148i	$-0.718179\pi$
$487$	−13606.0	−1.26601	−0.633005	+	0.774148i	$0.718179\pi$
$488$	0	0
$489$	−2112.00	−0.195313
$490$	0	0
$491$	2020.00i	0.185665i	0.995682	+	0.0928323i	$0.0295920\pi$
$491$	2020.00i	0.185665i	−0.995682	+	0.0928323i	$0.970408\pi$
$492$	0	0
$493$	− 16872.0i	− 1.54133i
$494$	0	0
$495$	−4896.00	−0.444563
$496$	0	0
$497$	7800.00	0.703980
$498$	0	0
$499$	− 19348.0i	− 1.73574i	−0.496789	−	0.867871i	$-0.665488\pi$
$499$	− 19348.0i	− 1.73574i	0.496789	−	0.867871i	$-0.334512\pi$
$500$	0	0
$501$	− 4944.00i	− 0.440881i
$502$	0	0
$503$	−596.000	−0.0528317	−0.0264158	−	0.999651i	$-0.508409\pi$
$503$	−596.000	−0.0528317	−0.0264158	+	0.999651i	$0.508409\pi$
$504$	0	0
$505$	5088.00	0.448343
$506$	0	0
$507$	243.000i	0.0212860i
$508$	0	0
$509$	4300.00i	0.374448i	0.982317	+	0.187224i	$0.0599491\pi$
$509$	4300.00i	0.374448i	−0.982317	+	0.187224i	$0.940051\pi$
$510$	0	0
$511$	−5180.00	−0.448434
$512$	0	0
$513$	−432.000	−0.0371799
$514$	0	0
$515$	12688.0i	1.08563i
$516$	0	0
$517$	8432.00i	0.717290i
$518$	0	0
$519$	1080.00	0.0913425
$520$	0	0
$521$	−9158.00	−0.770095	−0.385047	−	0.922897i	$-0.625815\pi$
$521$	−9158.00	−0.770095	−0.385047	+	0.922897i	$0.625815\pi$
$522$	0	0
$523$	− 10040.0i	− 0.839424i	−0.907657	−	0.419712i	$-0.862131\pi$
$523$	− 10040.0i	− 0.839424i	0.907657	−	0.419712i	$-0.137869\pi$
$524$	0	0
$525$	1830.00i	0.152129i
$526$	0	0
$527$	11988.0	0.990902
$528$	0	0
$529$	−11767.0	−0.967124
$530$	0	0
$531$	− 3060.00i	− 0.250080i
$532$	0	0
$533$	1380.00i	0.112147i
$534$	0	0
$535$	736.000	0.0594767
$536$	0	0
$537$	−5868.00	−0.471551
$538$	0	0
$539$	− 16524.0i	− 1.32048i
$540$	0	0
$541$	1906.00i	0.151470i	0.997128	+	0.0757351i	$0.0241303\pi$
$541$	1906.00i	0.151470i	−0.997128	+	0.0757351i	$0.975870\pi$
$542$	0	0
$543$	654.000	0.0516866
$544$	0	0
$545$	16080.0	1.26384
$546$	0	0
$547$	10264.0i	0.802298i	0.916013	+	0.401149i	$0.131389\pi$
$547$	10264.0i	0.802298i	−0.916013	+	0.401149i	$0.868611\pi$
$548$	0	0
$549$	− 8550.00i	− 0.664672i
$550$	0	0
$551$	3648.00	0.282051
$552$	0	0
$553$	−10100.0	−0.776665
$554$	0	0
$555$	− 6288.00i	− 0.480920i
$556$	0	0
$557$	− 15856.0i	− 1.20618i	−0.797674	−	0.603088i	$-0.793937\pi$
$557$	− 15856.0i	− 1.20618i	0.797674	−	0.603088i	$-0.206063\pi$
$558$	0	0
$559$	12144.0	0.918849
$560$	0	0
$561$	15096.0	1.13610
$562$	0	0
$563$	19764.0i	1.47949i	0.672887	+	0.739745i	$0.265054\pi$
$563$	19764.0i	1.47949i	−0.672887	+	0.739745i	$0.734946\pi$
$564$	0	0
$565$	16496.0i	1.22830i
$566$	0	0
$567$	810.000	0.0599944
$568$	0	0
$569$	−6302.00	−0.464312	−0.232156	−	0.972679i	$-0.574578\pi$
$569$	−6302.00	−0.464312	−0.232156	+	0.972679i	$0.574578\pi$
$570$	0	0
$571$	− 764.000i	− 0.0559937i	−0.999608	−	0.0279969i	$-0.991087\pi$
$571$	− 764.000i	− 0.0559937i	0.999608	−	0.0279969i	$-0.00891284\pi$
$572$	0	0
$573$	2040.00i	0.148730i
$574$	0	0
$575$	1220.00	0.0884826
$576$	0	0
$577$	−10618.0	−0.766089	−0.383044	−	0.923730i	$-0.625124\pi$
$577$	−10618.0	−0.766089	−0.383044	+	0.923730i	$0.625124\pi$
$578$	0	0
$579$	13530.0i	0.971136i
$580$	0	0
$581$	− 8520.00i	− 0.608381i
$582$	0	0
$583$	−13872.0	−0.985454
$584$	0	0
$585$	3312.00	0.234076
$586$	0	0
$587$	4036.00i	0.283788i	0.989882	+	0.141894i	$0.0453192\pi$
$587$	4036.00i	0.283788i	−0.989882	+	0.141894i	$0.954681\pi$
$588$	0	0
$589$	2592.00i	0.181327i
$590$	0	0
$591$	2388.00	0.166208
$592$	0	0
$593$	−24750.0	−1.71393	−0.856965	−	0.515374i	$-0.827653\pi$
$593$	−24750.0	−1.71393	−0.856965	+	0.515374i	$0.827653\pi$
$594$	0	0
$595$	5920.00i	0.407893i
$596$	0	0
$597$	− 2958.00i	− 0.202785i
$598$	0	0
$599$	4908.00	0.334784	0.167392	−	0.985890i	$-0.446465\pi$
$599$	4908.00	0.334784	0.167392	+	0.985890i	$0.446465\pi$
$600$	0	0
$601$	−22062.0	−1.49738	−0.748692	−	0.662918i	$-0.769318\pi$
$601$	−22062.0	−1.49738	−0.748692	+	0.662918i	$0.769318\pi$
$602$	0	0
$603$	− 3924.00i	− 0.265004i
$604$	0	0
$605$	26344.0i	1.77031i
$606$	0	0
$607$	12986.0	0.868345	0.434173	−	0.900830i	$-0.357041\pi$
$607$	12986.0	0.868345	0.434173	+	0.900830i	$0.357041\pi$
$608$	0	0
$609$	−6840.00	−0.455124
$610$	0	0
$611$	− 5704.00i	− 0.377675i
$612$	0	0
$613$	24986.0i	1.64629i	0.567832	+	0.823144i	$0.307782\pi$
$613$	24986.0i	1.64629i	−0.567832	+	0.823144i	$0.692218\pi$
$614$	0	0
$615$	−720.000	−0.0472085
$616$	0	0
$617$	3230.00	0.210753	0.105377	−	0.994432i	$-0.466395\pi$
$617$	3230.00	0.210753	0.105377	+	0.994432i	$0.466395\pi$
$618$	0	0
$619$	15500.0i	1.00646i	0.864153	+	0.503229i	$0.167855\pi$
$619$	15500.0i	1.00646i	−0.864153	+	0.503229i	$0.832145\pi$
$620$	0	0
$621$	− 540.000i	− 0.0348945i
$622$	0	0
$623$	6860.00	0.441156
$624$	0	0
$625$	−4279.00	−0.273856
$626$	0	0
$627$	3264.00i	0.207897i
$628$	0	0
$629$	19388.0i	1.22901i
$630$	0	0
$631$	16874.0	1.06457	0.532285	−	0.846565i	$-0.321334\pi$
$631$	16874.0	1.06457	0.532285	+	0.846565i	$0.321334\pi$
$632$	0	0
$633$	−12588.0	−0.790408
$634$	0	0
$635$	8368.00i	0.522951i
$636$	0	0
$637$	11178.0i	0.695272i
$638$	0	0
$639$	−7020.00	−0.434596
$640$	0	0
$641$	−978.000	−0.0602631	−0.0301316	−	0.999546i	$-0.509593\pi$
$641$	−978.000	−0.0602631	−0.0301316	+	0.999546i	$0.509593\pi$
$642$	0	0
$643$	− 1992.00i	− 0.122172i	−0.998132	−	0.0610862i	$-0.980544\pi$
$643$	− 1992.00i	− 0.122172i	0.998132	−	0.0610862i	$-0.0194564\pi$
$644$	0	0
$645$	6336.00i	0.386790i
$646$	0	0
$647$	3588.00	0.218020	0.109010	−	0.994041i	$-0.465232\pi$
$647$	3588.00	0.218020	0.109010	+	0.994041i	$0.465232\pi$
$648$	0	0
$649$	−23120.0	−1.39837
$650$	0	0
$651$	− 4860.00i	− 0.292594i
$652$	0	0
$653$	− 7536.00i	− 0.451618i	−0.974172	−	0.225809i	$-0.927497\pi$
$653$	− 7536.00i	− 0.451618i	0.974172	−	0.225809i	$-0.0725025\pi$
$654$	0	0
$655$	15072.0	0.899102
$656$	0	0
$657$	4662.00	0.276837
$658$	0	0
$659$	3684.00i	0.217767i	0.994055	+	0.108883i	$0.0347275\pi$
$659$	3684.00i	0.217767i	−0.994055	+	0.108883i	$0.965272\pi$
$660$	0	0
$661$	4418.00i	0.259970i	0.991516	+	0.129985i	$0.0414929\pi$
$661$	4418.00i	0.259970i	−0.991516	+	0.129985i	$0.958507\pi$
$662$	0	0
$663$	−10212.0	−0.598192
$664$	0	0
$665$	−1280.00	−0.0746410
$666$	0	0
$667$	4560.00i	0.264714i
$668$	0	0
$669$	− 7506.00i	− 0.433780i
$670$	0	0
$671$	−64600.0	−3.71662
$672$	0	0
$673$	−158.000	−0.00904971	−0.00452485	−	0.999990i	$-0.501440\pi$
$673$	−158.000	−0.00904971	−0.00452485	+	0.999990i	$0.501440\pi$
$674$	0	0
$675$	− 1647.00i	− 0.0939156i
$676$	0	0
$677$	− 10736.0i	− 0.609480i	−0.952436	−	0.304740i	$-0.901430\pi$
$677$	− 10736.0i	− 0.609480i	0.952436	−	0.304740i	$-0.0985696\pi$
$678$	0	0
$679$	−8060.00	−0.455544
$680$	0	0
$681$	972.000	0.0546947
$682$	0	0
$683$	− 6132.00i	− 0.343535i	−0.985138	−	0.171768i	$-0.945052\pi$
$683$	− 6132.00i	− 0.343535i	0.985138	−	0.171768i	$-0.0549478\pi$
$684$	0	0
$685$	18992.0i	1.05934i
$686$	0	0
$687$	6582.00	0.365530
$688$	0	0
$689$	9384.00	0.518871
$690$	0	0
$691$	− 28320.0i	− 1.55911i	−0.626335	−	0.779554i	$-0.715446\pi$
$691$	− 28320.0i	− 1.55911i	0.626335	−	0.779554i	$-0.284554\pi$
$692$	0	0
$693$	− 6120.00i	− 0.335468i
$694$	0	0
$695$	−17248.0	−0.941373
$696$	0	0
$697$	2220.00	0.120643
$698$	0	0
$699$	12690.0i	0.686666i
$700$	0	0
$701$	23420.0i	1.26186i	0.775841	+	0.630928i	$0.217326\pi$
$701$	23420.0i	1.26186i	−0.775841	+	0.630928i	$0.782674\pi$
$702$	0	0
$703$	−4192.00	−0.224899
$704$	0	0
$705$	2976.00	0.158982
$706$	0	0
$707$	6360.00i	0.338320i
$708$	0	0
$709$	− 13634.0i	− 0.722194i	−0.932528	−	0.361097i	$-0.882402\pi$
$709$	− 13634.0i	− 0.722194i	0.932528	−	0.361097i	$-0.117598\pi$
$710$	0	0
$711$	9090.00	0.479468
$712$	0	0
$713$	−3240.00	−0.170181
$714$	0	0
$715$	− 25024.0i	− 1.30887i
$716$	0	0
$717$	20016.0i	1.04255i
$718$	0	0
$719$	−4044.00	−0.209758	−0.104879	−	0.994485i	$-0.533445\pi$
$719$	−4044.00	−0.209758	−0.104879	+	0.994485i	$0.533445\pi$
$720$	0	0
$721$	−15860.0	−0.819220
$722$	0	0
$723$	− 4470.00i	− 0.229932i
$724$	0	0
$725$	13908.0i	0.712455i
$726$	0	0
$727$	25842.0	1.31833	0.659166	−	0.751998i	$-0.270910\pi$
$727$	25842.0	1.31833	0.659166	+	0.751998i	$0.270910\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 19536.0i	− 0.988461i
$732$	0	0
$733$	− 33414.0i	− 1.68373i	−0.539688	−	0.841865i	$-0.681458\pi$
$733$	− 33414.0i	− 1.68373i	0.539688	−	0.841865i	$-0.318542\pi$
$734$	0	0
$735$	−5832.00	−0.292676
$736$	0	0
$737$	−29648.0	−1.48182
$738$	0	0
$739$	− 21708.0i	− 1.08057i	−0.841482	−	0.540285i	$-0.818316\pi$
$739$	− 21708.0i	− 1.08057i	0.841482	−	0.540285i	$-0.181684\pi$
$740$	0	0
$741$	− 2208.00i	− 0.109464i
$742$	0	0
$743$	−12488.0	−0.616609	−0.308304	−	0.951288i	$-0.599762\pi$
$743$	−12488.0	−0.616609	−0.308304	+	0.951288i	$0.599762\pi$
$744$	0	0
$745$	23616.0	1.16137
$746$	0	0
$747$	7668.00i	0.375579i
$748$	0	0
$749$	920.000i	0.0448813i
$750$	0	0
$751$	−13522.0	−0.657024	−0.328512	−	0.944500i	$-0.606547\pi$
$751$	−13522.0	−0.657024	−0.328512	+	0.944500i	$0.606547\pi$
$752$	0	0
$753$	−22140.0	−1.07148
$754$	0	0
$755$	− 27280.0i	− 1.31499i
$756$	0	0
$757$	− 10178.0i	− 0.488673i	−0.969690	−	0.244337i	$-0.921430\pi$
$757$	− 10178.0i	− 0.488673i	0.969690	−	0.244337i	$-0.0785702\pi$
$758$	0	0
$759$	−4080.00	−0.195118
$760$	0	0
$761$	482.000	0.0229599	0.0114800	−	0.999934i	$-0.496346\pi$
$761$	482.000	0.0229599	0.0114800	+	0.999934i	$0.496346\pi$
$762$	0	0
$763$	20100.0i	0.953694i
$764$	0	0
$765$	− 5328.00i	− 0.251809i
$766$	0	0
$767$	15640.0	0.736281
$768$	0	0
$769$	−6706.00	−0.314466	−0.157233	−	0.987562i	$-0.550257\pi$
$769$	−6706.00	−0.314466	−0.157233	+	0.987562i	$0.550257\pi$
$770$	0	0
$771$	− 5994.00i	− 0.279985i
$772$	0	0
$773$	40788.0i	1.89786i	0.315493	+	0.948928i	$0.397830\pi$
$773$	40788.0i	1.89786i	−0.315493	+	0.948928i	$0.602170\pi$
$774$	0	0
$775$	−9882.00	−0.458028
$776$	0	0
$777$	7860.00	0.362903
$778$	0	0
$779$	480.000i	0.0220767i
$780$	0	0
$781$	53040.0i	2.43012i
$782$	0	0
$783$	6156.00	0.280967
$784$	0	0
$785$	−5328.00	−0.242248
$786$	0	0
$787$	29720.0i	1.34613i	0.739584	+	0.673065i	$0.235022\pi$
$787$	29720.0i	1.34613i	−0.739584	+	0.673065i	$0.764978\pi$
$788$	0	0
$789$	11304.0i	0.510055i
$790$	0	0
$791$	−20620.0	−0.926881
$792$	0	0
$793$	43700.0	1.95691
$794$	0	0
$795$	4896.00i	0.218419i
$796$	0	0
$797$	20736.0i	0.921589i	0.887507	+	0.460795i	$0.152436\pi$
$797$	20736.0i	0.921589i	−0.887507	+	0.460795i	$0.847564\pi$
$798$	0	0
$799$	−9176.00	−0.406287
$800$	0	0
$801$	−6174.00	−0.272344
$802$	0	0
$803$	− 35224.0i	− 1.54798i
$804$	0	0
$805$	− 1600.00i	− 0.0700529i
$806$	0	0
$807$	23892.0	1.04218
$808$	0	0
$809$	9834.00	0.427373	0.213687	−	0.976902i	$-0.431453\pi$
$809$	9834.00	0.427373	0.213687	+	0.976902i	$0.431453\pi$
$810$	0	0
$811$	4176.00i	0.180813i	0.995905	+	0.0904064i	$0.0288166\pi$
$811$	4176.00i	0.180813i	−0.995905	+	0.0904064i	$0.971183\pi$
$812$	0	0
$813$	6990.00i	0.301538i
$814$	0	0
$815$	5632.00	0.242062
$816$	0	0
$817$	4224.00	0.180880
$818$	0	0
$819$	4140.00i	0.176634i
$820$	0	0
$821$	− 1996.00i	− 0.0848488i	−0.999100	−	0.0424244i	$-0.986492\pi$
$821$	− 1996.00i	− 0.0848488i	0.999100	−	0.0424244i	$-0.0135082\pi$
$822$	0	0
$823$	14386.0	0.609313	0.304656	−	0.952462i	$-0.401458\pi$
$823$	14386.0	0.609313	0.304656	+	0.952462i	$0.401458\pi$
$824$	0	0
$825$	−12444.0	−0.525145
$826$	0	0
$827$	33836.0i	1.42272i	0.702826	+	0.711362i	$0.251921\pi$
$827$	33836.0i	1.42272i	−0.702826	+	0.711362i	$0.748079\pi$
$828$	0	0
$829$	− 16358.0i	− 0.685328i	−0.939458	−	0.342664i	$-0.888671\pi$
$829$	− 16358.0i	− 0.685328i	0.939458	−	0.342664i	$-0.111329\pi$
$830$	0	0
$831$	6462.00	0.269752
$832$	0	0
$833$	17982.0	0.747946
$834$	0	0
$835$	13184.0i	0.546409i
$836$	0	0
$837$	4374.00i	0.180630i
$838$	0	0
$839$	26244.0	1.07991	0.539954	−	0.841694i	$-0.318441\pi$
$839$	26244.0	1.07991	0.539954	+	0.841694i	$0.318441\pi$
$840$	0	0
$841$	−27595.0	−1.13145
$842$	0	0
$843$	16794.0i	0.686140i
$844$	0	0
$845$	− 648.000i	− 0.0263809i
$846$	0	0
$847$	−32930.0	−1.33588
$848$	0	0
$849$	−5652.00	−0.228476
$850$	0	0
$851$	− 5240.00i	− 0.211075i
$852$	0	0
$853$	− 39854.0i	− 1.59974i	−0.600176	−	0.799868i	$-0.704903\pi$
$853$	− 39854.0i	− 1.59974i	0.600176	−	0.799868i	$-0.295097\pi$
$854$	0	0
$855$	1152.00	0.0460790
$856$	0	0
$857$	2706.00	0.107859	0.0539295	−	0.998545i	$-0.482825\pi$
$857$	2706.00	0.107859	0.0539295	+	0.998545i	$0.482825\pi$
$858$	0	0
$859$	− 17728.0i	− 0.704158i	−0.935970	−	0.352079i	$-0.885475\pi$
$859$	− 17728.0i	− 0.704158i	0.935970	−	0.352079i	$-0.114525\pi$
$860$	0	0
$861$	− 900.000i	− 0.0356236i
$862$	0	0
$863$	20056.0	0.791094	0.395547	−	0.918446i	$-0.370555\pi$
$863$	20056.0	0.791094	0.395547	+	0.918446i	$0.370555\pi$
$864$	0	0
$865$	−2880.00	−0.113206
$866$	0	0
$867$	1689.00i	0.0661608i
$868$	0	0
$869$	− 68680.0i	− 2.68102i
$870$	0	0
$871$	20056.0	0.780220
$872$	0	0
$873$	7254.00	0.281226
$874$	0	0
$875$	− 14880.0i	− 0.574898i
$876$	0	0
$877$	26534.0i	1.02165i	0.859684	+	0.510826i	$0.170661\pi$
$877$	26534.0i	1.02165i	−0.859684	+	0.510826i	$0.829339\pi$
$878$	0	0
$879$	8364.00	0.320945
$880$	0	0
$881$	−45838.0	−1.75292	−0.876459	−	0.481476i	$-0.840101\pi$
$881$	−45838.0	−1.75292	−0.876459	+	0.481476i	$0.840101\pi$
$882$	0	0
$883$	23200.0i	0.884193i	0.896968	+	0.442096i	$0.145765\pi$
$883$	23200.0i	0.884193i	−0.896968	+	0.442096i	$0.854235\pi$
$884$	0	0
$885$	8160.00i	0.309938i
$886$	0	0
$887$	25272.0	0.956652	0.478326	−	0.878182i	$-0.341244\pi$
$887$	25272.0	0.956652	0.478326	+	0.878182i	$0.341244\pi$
$888$	0	0
$889$	−10460.0	−0.394620
$890$	0	0
$891$	5508.00i	0.207099i
$892$	0	0
$893$	− 1984.00i	− 0.0743472i
$894$	0	0
$895$	15648.0	0.584419
$896$	0	0
$897$	2760.00	0.102735
$898$	0	0
$899$	− 36936.0i	− 1.37028i
$900$	0	0
$901$	− 15096.0i	− 0.558181i
$902$	0	0
$903$	−7920.00	−0.291873
$904$	0	0
$905$	−1744.00	−0.0640580
$906$	0	0
$907$	42448.0i	1.55398i	0.629511	+	0.776992i	$0.283255\pi$
$907$	42448.0i	1.55398i	−0.629511	+	0.776992i	$0.716745\pi$
$908$	0	0
$909$	− 5724.00i	− 0.208859i
$910$	0	0
$911$	−49776.0	−1.81027	−0.905133	−	0.425128i	$-0.860229\pi$
$911$	−49776.0	−1.81027	−0.905133	+	0.425128i	$0.860229\pi$
$912$	0	0
$913$	57936.0	2.10011
$914$	0	0
$915$	22800.0i	0.823765i
$916$	0	0
$917$	18840.0i	0.678464i
$918$	0	0
$919$	3042.00	0.109191	0.0545954	−	0.998509i	$-0.482613\pi$
$919$	3042.00	0.109191	0.0545954	+	0.998509i	$0.482613\pi$
$920$	0	0
$921$	27540.0	0.985314
$922$	0	0
$923$	− 35880.0i	− 1.27953i
$924$	0	0
$925$	− 15982.0i	− 0.568092i
$926$	0	0
$927$	14274.0	0.505739
$928$	0	0
$929$	−11490.0	−0.405785	−0.202893	−	0.979201i	$-0.565034\pi$
$929$	−11490.0	−0.405785	−0.202893	+	0.979201i	$0.565034\pi$
$930$	0	0
$931$	3888.00i	0.136868i
$932$	0	0
$933$	24096.0i	0.845517i
$934$	0	0
$935$	−40256.0	−1.40803
$936$	0	0
$937$	−19882.0	−0.693187	−0.346594	−	0.938015i	$-0.612662\pi$
$937$	−19882.0	−0.693187	−0.346594	+	0.938015i	$0.612662\pi$
$938$	0	0
$939$	− 642.000i	− 0.0223119i
$940$	0	0
$941$	− 41336.0i	− 1.43200i	−0.698099	−	0.716002i	$-0.745970\pi$
$941$	− 41336.0i	− 1.43200i	0.698099	−	0.716002i	$-0.254030\pi$
$942$	0	0
$943$	−600.000	−0.0207197
$944$	0	0
$945$	−2160.00	−0.0743543
$946$	0	0
$947$	− 35036.0i	− 1.20224i	−0.799160	−	0.601118i	$-0.794722\pi$
$947$	− 35036.0i	− 1.20224i	0.799160	−	0.601118i	$-0.205278\pi$
$948$	0	0
$949$	23828.0i	0.815058i
$950$	0	0
$951$	4284.00	0.146076
$952$	0	0
$953$	−9814.00	−0.333585	−0.166793	−	0.985992i	$-0.553341\pi$
$953$	−9814.00	−0.333585	−0.166793	+	0.985992i	$0.553341\pi$
$954$	0	0
$955$	− 5440.00i	− 0.184329i
$956$	0	0
$957$	− 46512.0i	− 1.57108i
$958$	0	0
$959$	−23740.0	−0.799379
$960$	0	0
$961$	−3547.00	−0.119063
$962$	0	0
$963$	− 828.000i	− 0.0277071i
$964$	0	0
$965$	− 36080.0i	− 1.20358i
$966$	0	0
$967$	26006.0	0.864836	0.432418	−	0.901673i	$-0.357660\pi$
$967$	26006.0	0.864836	0.432418	+	0.901673i	$0.357660\pi$
$968$	0	0
$969$	−3552.00	−0.117757
$970$	0	0
$971$	17404.0i	0.575202i	0.957750	+	0.287601i	$0.0928577\pi$
$971$	17404.0i	0.575202i	−0.957750	+	0.287601i	$0.907142\pi$
$972$	0	0
$973$	− 21560.0i	− 0.710362i
$974$	0	0
$975$	8418.00	0.276504
$976$	0	0
$977$	44526.0	1.45805	0.729024	−	0.684488i	$-0.239974\pi$
$977$	44526.0	1.45805	0.729024	+	0.684488i	$0.239974\pi$
$978$	0	0
$979$	46648.0i	1.52286i
$980$	0	0
$981$	− 18090.0i	− 0.588756i
$982$	0	0
$983$	−47480.0	−1.54057	−0.770283	−	0.637702i	$-0.779885\pi$
$983$	−47480.0	−1.54057	−0.770283	+	0.637702i	$0.779885\pi$
$984$	0	0
$985$	−6368.00	−0.205991
$986$	0	0
$987$	3720.00i	0.119968i
$988$	0	0
$989$	5280.00i	0.169762i
$990$	0	0
$991$	−18866.0	−0.604741	−0.302370	−	0.953190i	$-0.597778\pi$
$991$	−18866.0	−0.604741	−0.302370	+	0.953190i	$0.597778\pi$
$992$	0	0
$993$	8124.00	0.259625
$994$	0	0
$995$	7888.00i	0.251323i
$996$	0	0
$997$	− 17550.0i	− 0.557487i	−0.960366	−	0.278743i	$-0.910082\pi$
$997$	− 17550.0i	− 0.557487i	0.960366	−	0.278743i	$-0.0899179\pi$
$998$	0	0
$999$	−7074.00	−0.224035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.d.l.385.2		2
4.3	odd	2		768.4.d.e.385.1		2
8.3	odd	2		768.4.d.e.385.2		2
8.5	even	2	inner	768.4.d.l.385.1		2
16.3	odd	4		384.4.a.e.1.1	yes	1
16.5	even	4		384.4.a.h.1.1	yes	1
16.11	odd	4		384.4.a.d.1.1	yes	1
16.13	even	4		384.4.a.a.1.1	✓	1
48.5	odd	4		1152.4.a.a.1.1		1
48.11	even	4		1152.4.a.b.1.1		1
48.29	odd	4		1152.4.a.k.1.1		1
48.35	even	4		1152.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.a.a.1.1	✓	1	16.13	even	4
384.4.a.d.1.1	yes	1	16.11	odd	4
384.4.a.e.1.1	yes	1	16.3	odd	4
384.4.a.h.1.1	yes	1	16.5	even	4
768.4.d.e.385.1		2	4.3	odd	2
768.4.d.e.385.2		2	8.3	odd	2
768.4.d.l.385.1		2	8.5	even	2	inner
768.4.d.l.385.2		2	1.1	even	1	trivial
1152.4.a.a.1.1		1	48.5	odd	4
1152.4.a.b.1.1		1	48.11	even	4
1152.4.a.k.1.1		1	48.29	odd	4
1152.4.a.l.1.1		1	48.35	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -8.00000i q^{5} +10.0000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -8.00000i q^{5} +10.0000 q^{7} -9.00000 q^{9} +68.0000i q^{11} -46.0000i q^{13} +24.0000 q^{15} -74.0000 q^{17} -16.0000i q^{19} +30.0000i q^{21} +20.0000 q^{23} +61.0000 q^{25} -27.0000i q^{27} +228.000i q^{29} -162.000 q^{31} -204.000 q^{33} -80.0000i q^{35} -262.000i q^{37} +138.000 q^{39} -30.0000 q^{41} +264.000i q^{43} +72.0000i q^{45} +124.000 q^{47} -243.000 q^{49} -222.000i q^{51} +204.000i q^{53} +544.000 q^{55} +48.0000 q^{57} +340.000i q^{59} +950.000i q^{61} -90.0000 q^{63} -368.000 q^{65} +436.000i q^{67} +60.0000i q^{69} +780.000 q^{71} -518.000 q^{73} +183.000i q^{75} +680.000i q^{77} -1010.00 q^{79} +81.0000 q^{81} -852.000i q^{83} +592.000i q^{85} -684.000 q^{87} +686.000 q^{89} -460.000i q^{91} -486.000i q^{93} -128.000 q^{95} -806.000 q^{97} -612.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{7} - 18 q^{9}+O(q^{10}) \) 2 * q + 20 * q^7 - 18 * q^9	\( 2 q + 20 q^{7} - 18 q^{9} + 48 q^{15} - 148 q^{17} + 40 q^{23} + 122 q^{25} - 324 q^{31} - 408 q^{33} + 276 q^{39} - 60 q^{41} + 248 q^{47} - 486 q^{49} + 1088 q^{55} + 96 q^{57} - 180 q^{63} - 736 q^{65} + 1560 q^{71} - 1036 q^{73} - 2020 q^{79} + 162 q^{81} - 1368 q^{87} + 1372 q^{89} - 256 q^{95} - 1612 q^{97}+O(q^{100}) \) 2 * q + 20 * q^7 - 18 * q^9 + 48 * q^15 - 148 * q^17 + 40 * q^23 + 122 * q^25 - 324 * q^31 - 408 * q^33 + 276 * q^39 - 60 * q^41 + 248 * q^47 - 486 * q^49 + 1088 * q^55 + 96 * q^57 - 180 * q^63 - 736 * q^65 + 1560 * q^71 - 1036 * q^73 - 2020 * q^79 + 162 * q^81 - 1368 * q^87 + 1372 * q^89 - 256 * q^95 - 1612 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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