LMFDB - Embedded newform 768.4.d.m.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 96)
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.m.385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +2.00000i q^{5} +12.0000 q^{7} -9.00000 q^{9} -60.0000i q^{11} +42.0000i q^{13} -6.00000 q^{15} +10.0000 q^{17} +132.000i q^{19} +36.0000i q^{21} -48.0000 q^{23} +121.000 q^{25} -27.0000i q^{27} -226.000i q^{29} +252.000 q^{31} +180.000 q^{33} +24.0000i q^{35} -362.000i q^{37} -126.000 q^{39} +94.0000 q^{41} +228.000i q^{43} -18.0000i q^{45} +408.000 q^{47} -199.000 q^{49} +30.0000i q^{51} +346.000i q^{53} +120.000 q^{55} -396.000 q^{57} +300.000i q^{59} +466.000i q^{61} -108.000 q^{63} -84.0000 q^{65} +204.000i q^{67} -144.000i q^{69} +1056.00 q^{71} -330.000 q^{73} +363.000i q^{75} -720.000i q^{77} -612.000 q^{79} +81.0000 q^{81} +564.000i q^{83} +20.0000i q^{85} +678.000 q^{87} +1510.00 q^{89} +504.000i q^{91} +756.000i q^{93} -264.000 q^{95} +594.000 q^{97} +540.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{7} - 18 q^{9} - 12 q^{15} + 20 q^{17} - 96 q^{23} + 242 q^{25} + 504 q^{31} + 360 q^{33} - 252 q^{39} + 188 q^{41} + 816 q^{47} - 398 q^{49} + 240 q^{55} - 792 q^{57} - 216 q^{63} - 168 q^{65}+ \cdots + 1188 q^{97}+O(q^{100}) $ 2 * q + 24 * q^7 - 18 * q^9 - 12 * q^15 + 20 * q^17 - 96 * q^23 + 242 * q^25 + 504 * q^31 + 360 * q^33 - 252 * q^39 + 188 * q^41 + 816 * q^47 - 398 * q^49 + 240 * q^55 - 792 * q^57 - 216 * q^63 - 168 * q^65 + 2112 * q^71 - 660 * q^73 - 1224 * q^79 + 162 * q^81 + 1356 * q^87 + 3020 * q^89 - 528 * q^95 + 1188 * q^97

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
$5$	2.00000i	0.178885i	0.995992	+	0.0894427i	$0.0285086\pi$
$5$	2.00000i	0.178885i	−0.995992	+	0.0894427i	$0.971491\pi$
$6$	0	0
$7$	12.0000	0.647939	0.323970	−	0.946068i	$-0.394982\pi$
$7$	12.0000	0.647939	0.323970	+	0.946068i	$0.394982\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 60.0000i	− 1.64461i	−0.569049	−	0.822304i	$-0.692689\pi$
$11$	− 60.0000i	− 1.64461i	0.569049	−	0.822304i	$-0.307311\pi$
$12$	0	0
$13$	42.0000i	0.896054i	0.894020	+	0.448027i	$0.147873\pi$
$13$	42.0000i	0.896054i	−0.894020	+	0.448027i	$0.852127\pi$
$14$	0	0
$15$	−6.00000	−0.103280
$16$	0	0
$17$	10.0000	0.142668	0.0713340	−	0.997452i	$-0.477274\pi$
$17$	10.0000	0.142668	0.0713340	+	0.997452i	$0.477274\pi$
$18$	0	0
$19$	132.000i	1.59384i	0.604088	+	0.796918i	$0.293538\pi$
$19$	132.000i	1.59384i	−0.604088	+	0.796918i	$0.706462\pi$
$20$	0	0
$21$	36.0000i	0.374088i
$22$	0	0
$23$	−48.0000	−0.435161	−0.217580	−	0.976042i	$-0.569816\pi$
$23$	−48.0000	−0.435161	−0.217580	+	0.976042i	$0.569816\pi$
$24$	0	0
$25$	121.000	0.968000
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 226.000i	− 1.44714i	−0.690249	−	0.723571i	$-0.742499\pi$
$29$	− 226.000i	− 1.44714i	0.690249	−	0.723571i	$-0.257501\pi$
$30$	0	0
$31$	252.000	1.46002	0.730009	−	0.683438i	$-0.239516\pi$
$31$	252.000	1.46002	0.730009	+	0.683438i	$0.239516\pi$
$32$	0	0
$33$	180.000	0.949514
$34$	0	0
$35$	24.0000i	0.115907i
$36$	0	0
$37$	− 362.000i	− 1.60844i	−0.594329	−	0.804222i	$-0.702582\pi$
$37$	− 362.000i	− 1.60844i	0.594329	−	0.804222i	$-0.297418\pi$
$38$	0	0
$39$	−126.000	−0.517337
$40$	0	0
$41$	94.0000	0.358057	0.179028	−	0.983844i	$-0.442705\pi$
$41$	94.0000	0.358057	0.179028	+	0.983844i	$0.442705\pi$
$42$	0	0
$43$	228.000i	0.808597i	0.914627	+	0.404299i	$0.132484\pi$
$43$	228.000i	0.808597i	−0.914627	+	0.404299i	$0.867516\pi$
$44$	0	0
$45$	− 18.0000i	− 0.0596285i
$46$	0	0
$47$	408.000	1.26623	0.633116	−	0.774057i	$-0.281776\pi$
$47$	408.000	1.26623	0.633116	+	0.774057i	$0.281776\pi$
$48$	0	0
$49$	−199.000	−0.580175
$50$	0	0
$51$	30.0000i	0.0823694i
$52$	0	0
$53$	346.000i	0.896731i	0.893850	+	0.448366i	$0.147994\pi$
$53$	346.000i	0.896731i	−0.893850	+	0.448366i	$0.852006\pi$
$54$	0	0
$55$	120.000	0.294196
$56$	0	0
$57$	−396.000	−0.920201
$58$	0	0
$59$	300.000i	0.661978i	0.943635	+	0.330989i	$0.107382\pi$
$59$	300.000i	0.661978i	−0.943635	+	0.330989i	$0.892618\pi$
$60$	0	0
$61$	466.000i	0.978118i	0.872251	+	0.489059i	$0.162660\pi$
$61$	466.000i	0.978118i	−0.872251	+	0.489059i	$0.837340\pi$
$62$	0	0
$63$	−108.000	−0.215980
$64$	0	0
$65$	−84.0000	−0.160291
$66$	0	0
$67$	204.000i	0.371979i	0.982552	+	0.185989i	$0.0595490\pi$
$67$	204.000i	0.371979i	−0.982552	+	0.185989i	$0.940451\pi$
$68$	0	0
$69$	− 144.000i	− 0.251240i
$70$	0	0
$71$	1056.00	1.76513	0.882564	−	0.470192i	$-0.155815\pi$
$71$	1056.00	1.76513	0.882564	+	0.470192i	$0.155815\pi$
$72$	0	0
$73$	−330.000	−0.529090	−0.264545	−	0.964373i	$-0.585222\pi$
$73$	−330.000	−0.529090	−0.264545	+	0.964373i	$0.585222\pi$
$74$	0	0
$75$	363.000i	0.558875i
$76$	0	0
$77$	− 720.000i	− 1.06561i
$78$	0	0
$79$	−612.000	−0.871587	−0.435794	−	0.900047i	$-0.643532\pi$
$79$	−612.000	−0.871587	−0.435794	+	0.900047i	$0.643532\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	564.000i	0.745868i	0.927858	+	0.372934i	$0.121648\pi$
$83$	564.000i	0.745868i	−0.927858	+	0.372934i	$0.878352\pi$
$84$	0	0
$85$	20.0000i	0.0255212i
$86$	0	0
$87$	678.000	0.835508
$88$	0	0
$89$	1510.00	1.79842	0.899212	−	0.437514i	$-0.144141\pi$
$89$	1510.00	1.79842	0.899212	+	0.437514i	$0.144141\pi$
$90$	0	0
$91$	504.000i	0.580589i
$92$	0	0
$93$	756.000i	0.842941i
$94$	0	0
$95$	−264.000	−0.285114
$96$	0	0
$97$	594.000	0.621769	0.310884	−	0.950448i	$-0.399375\pi$
$97$	594.000	0.621769	0.310884	+	0.950448i	$0.399375\pi$
$98$	0	0
$99$	540.000i	0.548202i
$100$	0	0
$101$	554.000i	0.545793i	0.962043	+	0.272896i	$0.0879816\pi$
$101$	554.000i	0.545793i	−0.962043	+	0.272896i	$0.912018\pi$
$102$	0	0
$103$	1284.00	1.22831	0.614157	−	0.789184i	$-0.289496\pi$
$103$	1284.00	1.22831	0.614157	+	0.789184i	$0.289496\pi$
$104$	0	0
$105$	−72.0000	−0.0669189
$106$	0	0
$107$	1356.00i	1.22514i	0.790418	+	0.612568i	$0.209863\pi$
$107$	1356.00i	1.22514i	−0.790418	+	0.612568i	$0.790137\pi$
$108$	0	0
$109$	− 390.000i	− 0.342708i	−0.985209	−	0.171354i	$-0.945186\pi$
$109$	− 390.000i	− 0.342708i	0.985209	−	0.171354i	$-0.0548143\pi$
$110$	0	0
$111$	1086.00	0.928636
$112$	0	0
$113$	−766.000	−0.637692	−0.318846	−	0.947807i	$-0.603295\pi$
$113$	−766.000	−0.637692	−0.318846	+	0.947807i	$0.603295\pi$
$114$	0	0
$115$	− 96.0000i	− 0.0778439i
$116$	0	0
$117$	− 378.000i	− 0.298685i
$118$	0	0
$119$	120.000	0.0924402
$120$	0	0
$121$	−2269.00	−1.70473
$122$	0	0
$123$	282.000i	0.206724i
$124$	0	0
$125$	492.000i	0.352047i
$126$	0	0
$127$	2388.00	1.66851	0.834255	−	0.551379i	$-0.185898\pi$
$127$	2388.00	1.66851	0.834255	+	0.551379i	$0.185898\pi$
$128$	0	0
$129$	−684.000	−0.466844
$130$	0	0
$131$	396.000i	0.264112i	0.991242	+	0.132056i	$0.0421579\pi$
$131$	396.000i	0.264112i	−0.991242	+	0.132056i	$0.957842\pi$
$132$	0	0
$133$	1584.00i	1.03271i
$134$	0	0
$135$	54.0000	0.0344265
$136$	0	0
$137$	110.000	0.0685981	0.0342990	−	0.999412i	$-0.489080\pi$
$137$	110.000	0.0685981	0.0342990	+	0.999412i	$0.489080\pi$
$138$	0	0
$139$	732.000i	0.446672i	0.974742	+	0.223336i	$0.0716947\pi$
$139$	732.000i	0.446672i	−0.974742	+	0.223336i	$0.928305\pi$
$140$	0	0
$141$	1224.00i	0.731060i
$142$	0	0
$143$	2520.00	1.47366
$144$	0	0
$145$	452.000	0.258873
$146$	0	0
$147$	− 597.000i	− 0.334964i
$148$	0	0
$149$	− 1934.00i	− 1.06335i	−0.846948	−	0.531676i	$-0.821562\pi$
$149$	− 1934.00i	− 1.06335i	0.846948	−	0.531676i	$-0.178438\pi$
$150$	0	0
$151$	−1092.00	−0.588515	−0.294257	−	0.955726i	$-0.595072\pi$
$151$	−1092.00	−0.588515	−0.294257	+	0.955726i	$0.595072\pi$
$152$	0	0
$153$	−90.0000	−0.0475560
$154$	0	0
$155$	504.000i	0.261176i
$156$	0	0
$157$	578.000i	0.293818i	0.989150	+	0.146909i	$0.0469324\pi$
$157$	578.000i	0.293818i	−0.989150	+	0.146909i	$0.953068\pi$
$158$	0	0
$159$	−1038.00	−0.517728
$160$	0	0
$161$	−576.000	−0.281958
$162$	0	0
$163$	2532.00i	1.21670i	0.793670	+	0.608348i	$0.208168\pi$
$163$	2532.00i	1.21670i	−0.793670	+	0.608348i	$0.791832\pi$
$164$	0	0
$165$	360.000i	0.169854i
$166$	0	0
$167$	−648.000	−0.300262	−0.150131	−	0.988666i	$-0.547970\pi$
$167$	−648.000	−0.300262	−0.150131	+	0.988666i	$0.547970\pi$
$168$	0	0
$169$	433.000	0.197087
$170$	0	0
$171$	− 1188.00i	− 0.531279i
$172$	0	0
$173$	− 3338.00i	− 1.46696i	−0.679713	−	0.733478i	$-0.737896\pi$
$173$	− 3338.00i	− 1.46696i	0.679713	−	0.733478i	$-0.262104\pi$
$174$	0	0
$175$	1452.00	0.627205
$176$	0	0
$177$	−900.000	−0.382193
$178$	0	0
$179$	3804.00i	1.58840i	0.607654	+	0.794202i	$0.292111\pi$
$179$	3804.00i	1.58840i	−0.607654	+	0.794202i	$0.707889\pi$
$180$	0	0
$181$	1854.00i	0.761363i	0.924706	+	0.380682i	$0.124311\pi$
$181$	1854.00i	0.761363i	−0.924706	+	0.380682i	$0.875689\pi$
$182$	0	0
$183$	−1398.00	−0.564717
$184$	0	0
$185$	724.000	0.287727
$186$	0	0
$187$	− 600.000i	− 0.234633i
$188$	0	0
$189$	− 324.000i	− 0.124696i
$190$	0	0
$191$	−1344.00	−0.509154	−0.254577	−	0.967052i	$-0.581936\pi$
$191$	−1344.00	−0.509154	−0.254577	+	0.967052i	$0.581936\pi$
$192$	0	0
$193$	−1262.00	−0.470677	−0.235339	−	0.971913i	$-0.575620\pi$
$193$	−1262.00	−0.470677	−0.235339	+	0.971913i	$0.575620\pi$
$194$	0	0
$195$	− 252.000i	− 0.0925441i
$196$	0	0
$197$	− 4294.00i	− 1.55297i	−0.630137	−	0.776484i	$-0.717001\pi$
$197$	− 4294.00i	− 1.55297i	0.630137	−	0.776484i	$-0.282999\pi$
$198$	0	0
$199$	4308.00	1.53460	0.767302	−	0.641286i	$-0.221599\pi$
$199$	4308.00	1.53460	0.767302	+	0.641286i	$0.221599\pi$
$200$	0	0
$201$	−612.000	−0.214762
$202$	0	0
$203$	− 2712.00i	− 0.937661i
$204$	0	0
$205$	188.000i	0.0640512i
$206$	0	0
$207$	432.000	0.145054
$208$	0	0
$209$	7920.00	2.62123
$210$	0	0
$211$	1212.00i	0.395438i	0.980259	+	0.197719i	$0.0633534\pi$
$211$	1212.00i	0.395438i	−0.980259	+	0.197719i	$0.936647\pi$
$212$	0	0
$213$	3168.00i	1.01910i
$214$	0	0
$215$	−456.000	−0.144646
$216$	0	0
$217$	3024.00	0.946002
$218$	0	0
$219$	− 990.000i	− 0.305470i
$220$	0	0
$221$	420.000i	0.127838i
$222$	0	0
$223$	−2172.00	−0.652233	−0.326116	−	0.945330i	$-0.605740\pi$
$223$	−2172.00	−0.652233	−0.326116	+	0.945330i	$0.605740\pi$
$224$	0	0
$225$	−1089.00	−0.322667
$226$	0	0
$227$	− 3948.00i	− 1.15435i	−0.816620	−	0.577176i	$-0.804155\pi$
$227$	− 3948.00i	− 1.15435i	0.816620	−	0.577176i	$-0.195845\pi$
$228$	0	0
$229$	− 3522.00i	− 1.01633i	−0.861259	−	0.508167i	$-0.830323\pi$
$229$	− 3522.00i	− 1.01633i	0.861259	−	0.508167i	$-0.169677\pi$
$230$	0	0
$231$	2160.00	0.615228
$232$	0	0
$233$	2774.00	0.779960	0.389980	−	0.920823i	$-0.372482\pi$
$233$	2774.00	0.779960	0.389980	+	0.920823i	$0.372482\pi$
$234$	0	0
$235$	816.000i	0.226511i
$236$	0	0
$237$	− 1836.00i	− 0.503211i
$238$	0	0
$239$	−2784.00	−0.753481	−0.376741	−	0.926319i	$-0.622955\pi$
$239$	−2784.00	−0.753481	−0.376741	+	0.926319i	$0.622955\pi$
$240$	0	0
$241$	−4686.00	−1.25250	−0.626249	−	0.779623i	$-0.715410\pi$
$241$	−4686.00	−1.25250	−0.626249	+	0.779623i	$0.715410\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 398.000i	− 0.103785i
$246$	0	0
$247$	−5544.00	−1.42816
$248$	0	0
$249$	−1692.00	−0.430627
$250$	0	0
$251$	2484.00i	0.624656i	0.949974	+	0.312328i	$0.101109\pi$
$251$	2484.00i	0.624656i	−0.949974	+	0.312328i	$0.898891\pi$
$252$	0	0
$253$	2880.00i	0.715668i
$254$	0	0
$255$	−60.0000	−0.0147347
$256$	0	0
$257$	6658.00	1.61601	0.808005	−	0.589175i	$-0.200547\pi$
$257$	6658.00	1.61601	0.808005	+	0.589175i	$0.200547\pi$
$258$	0	0
$259$	− 4344.00i	− 1.04217i
$260$	0	0
$261$	2034.00i	0.482381i
$262$	0	0
$263$	2904.00	0.680868	0.340434	−	0.940268i	$-0.389426\pi$
$263$	2904.00	0.680868	0.340434	+	0.940268i	$0.389426\pi$
$264$	0	0
$265$	−692.000	−0.160412
$266$	0	0
$267$	4530.00i	1.03832i
$268$	0	0
$269$	1006.00i	0.228018i	0.993480	+	0.114009i	$0.0363693\pi$
$269$	1006.00i	0.228018i	−0.993480	+	0.114009i	$0.963631\pi$
$270$	0	0
$271$	−876.000	−0.196359	−0.0981794	−	0.995169i	$-0.531302\pi$
$271$	−876.000	−0.196359	−0.0981794	+	0.995169i	$0.531302\pi$
$272$	0	0
$273$	−1512.00	−0.335203
$274$	0	0
$275$	− 7260.00i	− 1.59198i
$276$	0	0
$277$	2718.00i	0.589562i	0.955565	+	0.294781i	$0.0952468\pi$
$277$	2718.00i	0.589562i	−0.955565	+	0.294781i	$0.904753\pi$
$278$	0	0
$279$	−2268.00	−0.486672
$280$	0	0
$281$	−5354.00	−1.13663	−0.568315	−	0.822811i	$-0.692404\pi$
$281$	−5354.00	−1.13663	−0.568315	+	0.822811i	$0.692404\pi$
$282$	0	0
$283$	780.000i	0.163838i	0.996639	+	0.0819191i	$0.0261049\pi$
$283$	780.000i	0.163838i	−0.996639	+	0.0819191i	$0.973895\pi$
$284$	0	0
$285$	− 792.000i	− 0.164611i
$286$	0	0
$287$	1128.00	0.231999
$288$	0	0
$289$	−4813.00	−0.979646
$290$	0	0
$291$	1782.00i	0.358978i
$292$	0	0
$293$	− 3350.00i	− 0.667949i	−0.942582	−	0.333975i	$-0.891610\pi$
$293$	− 3350.00i	− 0.667949i	0.942582	−	0.333975i	$-0.108390\pi$
$294$	0	0
$295$	−600.000	−0.118418
$296$	0	0
$297$	−1620.00	−0.316505
$298$	0	0
$299$	− 2016.00i	− 0.389927i
$300$	0	0
$301$	2736.00i	0.523922i
$302$	0	0
$303$	−1662.00	−0.315114
$304$	0	0
$305$	−932.000	−0.174971
$306$	0	0
$307$	− 9636.00i	− 1.79139i	−0.444673	−	0.895693i	$-0.646680\pi$
$307$	− 9636.00i	− 1.79139i	0.444673	−	0.895693i	$-0.353320\pi$
$308$	0	0
$309$	3852.00i	0.709167i
$310$	0	0
$311$	−7560.00	−1.37842	−0.689209	−	0.724562i	$-0.742042\pi$
$311$	−7560.00	−1.37842	−0.689209	+	0.724562i	$0.742042\pi$
$312$	0	0
$313$	3526.00	0.636745	0.318373	−	0.947966i	$-0.396864\pi$
$313$	3526.00	0.636745	0.318373	+	0.947966i	$0.396864\pi$
$314$	0	0
$315$	− 216.000i	− 0.0386356i
$316$	0	0
$317$	− 7634.00i	− 1.35258i	−0.736635	−	0.676290i	$-0.763586\pi$
$317$	− 7634.00i	− 1.35258i	0.736635	−	0.676290i	$-0.236414\pi$
$318$	0	0
$319$	−13560.0	−2.37998
$320$	0	0
$321$	−4068.00	−0.707332
$322$	0	0
$323$	1320.00i	0.227389i
$324$	0	0
$325$	5082.00i	0.867380i
$326$	0	0
$327$	1170.00	0.197863
$328$	0	0
$329$	4896.00	0.820441
$330$	0	0
$331$	− 7572.00i	− 1.25739i	−0.777654	−	0.628693i	$-0.783590\pi$
$331$	− 7572.00i	− 1.25739i	0.777654	−	0.628693i	$-0.216410\pi$
$332$	0	0
$333$	3258.00i	0.536148i
$334$	0	0
$335$	−408.000	−0.0665416
$336$	0	0
$337$	162.000	0.0261861	0.0130930	−	0.999914i	$-0.495832\pi$
$337$	162.000	0.0261861	0.0130930	+	0.999914i	$0.495832\pi$
$338$	0	0
$339$	− 2298.00i	− 0.368172i
$340$	0	0
$341$	− 15120.0i	− 2.40116i
$342$	0	0
$343$	−6504.00	−1.02386
$344$	0	0
$345$	288.000	0.0449432
$346$	0	0
$347$	− 6636.00i	− 1.02663i	−0.858202	−	0.513313i	$-0.828418\pi$
$347$	− 6636.00i	− 1.02663i	0.858202	−	0.513313i	$-0.171582\pi$
$348$	0	0
$349$	− 4430.00i	− 0.679463i	−0.940523	−	0.339731i	$-0.889664\pi$
$349$	− 4430.00i	− 0.679463i	0.940523	−	0.339731i	$-0.110336\pi$
$350$	0	0
$351$	1134.00	0.172446
$352$	0	0
$353$	8402.00	1.26684	0.633418	−	0.773810i	$-0.281651\pi$
$353$	8402.00	1.26684	0.633418	+	0.773810i	$0.281651\pi$
$354$	0	0
$355$	2112.00i	0.315756i
$356$	0	0
$357$	360.000i	0.0533704i
$358$	0	0
$359$	11520.0	1.69360	0.846800	−	0.531912i	$-0.178526\pi$
$359$	11520.0	1.69360	0.846800	+	0.531912i	$0.178526\pi$
$360$	0	0
$361$	−10565.0	−1.54031
$362$	0	0
$363$	− 6807.00i	− 0.984228i
$364$	0	0
$365$	− 660.000i	− 0.0946465i
$366$	0	0
$367$	7404.00	1.05309	0.526547	−	0.850146i	$-0.323486\pi$
$367$	7404.00	1.05309	0.526547	+	0.850146i	$0.323486\pi$
$368$	0	0
$369$	−846.000	−0.119352
$370$	0	0
$371$	4152.00i	0.581027i
$372$	0	0
$373$	1910.00i	0.265137i	0.991174	+	0.132568i	$0.0423224\pi$
$373$	1910.00i	0.265137i	−0.991174	+	0.132568i	$0.957678\pi$
$374$	0	0
$375$	−1476.00	−0.203254
$376$	0	0
$377$	9492.00	1.29672
$378$	0	0
$379$	− 10332.0i	− 1.40031i	−0.713989	−	0.700157i	$-0.753113\pi$
$379$	− 10332.0i	− 1.40031i	0.713989	−	0.700157i	$-0.246887\pi$
$380$	0	0
$381$	7164.00i	0.963315i
$382$	0	0
$383$	−6624.00	−0.883735	−0.441868	−	0.897080i	$-0.645684\pi$
$383$	−6624.00	−0.883735	−0.441868	+	0.897080i	$0.645684\pi$
$384$	0	0
$385$	1440.00	0.190621
$386$	0	0
$387$	− 2052.00i	− 0.269532i
$388$	0	0
$389$	10210.0i	1.33076i	0.746503	+	0.665382i	$0.231732\pi$
$389$	10210.0i	1.33076i	−0.746503	+	0.665382i	$0.768268\pi$
$390$	0	0
$391$	−480.000	−0.0620835
$392$	0	0
$393$	−1188.00	−0.152485
$394$	0	0
$395$	− 1224.00i	− 0.155914i
$396$	0	0
$397$	4066.00i	0.514022i	0.966408	+	0.257011i	$0.0827376\pi$
$397$	4066.00i	0.514022i	−0.966408	+	0.257011i	$0.917262\pi$
$398$	0	0
$399$	−4752.00	−0.596234
$400$	0	0
$401$	−5510.00	−0.686175	−0.343088	−	0.939303i	$-0.611473\pi$
$401$	−5510.00	−0.686175	−0.343088	+	0.939303i	$0.611473\pi$
$402$	0	0
$403$	10584.0i	1.30825i
$404$	0	0
$405$	162.000i	0.0198762i
$406$	0	0
$407$	−21720.0	−2.64526
$408$	0	0
$409$	−15450.0	−1.86786	−0.933928	−	0.357460i	$-0.883643\pi$
$409$	−15450.0	−1.86786	−0.933928	+	0.357460i	$0.883643\pi$
$410$	0	0
$411$	330.000i	0.0396051i
$412$	0	0
$413$	3600.00i	0.428921i
$414$	0	0
$415$	−1128.00	−0.133425
$416$	0	0
$417$	−2196.00	−0.257886
$418$	0	0
$419$	− 3084.00i	− 0.359578i	−0.983705	−	0.179789i	$-0.942458\pi$
$419$	− 3084.00i	− 0.359578i	0.983705	−	0.179789i	$-0.0575415\pi$
$420$	0	0
$421$	10446.0i	1.20928i	0.796499	+	0.604640i	$0.206683\pi$
$421$	10446.0i	1.20928i	−0.796499	+	0.604640i	$0.793317\pi$
$422$	0	0
$423$	−3672.00	−0.422077
$424$	0	0
$425$	1210.00	0.138103
$426$	0	0
$427$	5592.00i	0.633761i
$428$	0	0
$429$	7560.00i	0.850816i
$430$	0	0
$431$	2184.00	0.244083	0.122041	−	0.992525i	$-0.461056\pi$
$431$	2184.00	0.244083	0.122041	+	0.992525i	$0.461056\pi$
$432$	0	0
$433$	−110.000	−0.0122085	−0.00610423	−	0.999981i	$-0.501943\pi$
$433$	−110.000	−0.0122085	−0.00610423	+	0.999981i	$0.501943\pi$
$434$	0	0
$435$	1356.00i	0.149460i
$436$	0	0
$437$	− 6336.00i	− 0.693574i
$438$	0	0
$439$	−2412.00	−0.262229	−0.131114	−	0.991367i	$-0.541856\pi$
$439$	−2412.00	−0.262229	−0.131114	+	0.991367i	$0.541856\pi$
$440$	0	0
$441$	1791.00	0.193392
$442$	0	0
$443$	− 6540.00i	− 0.701410i	−0.936486	−	0.350705i	$-0.885942\pi$
$443$	− 6540.00i	− 0.701410i	0.936486	−	0.350705i	$-0.114058\pi$
$444$	0	0
$445$	3020.00i	0.321712i
$446$	0	0
$447$	5802.00	0.613927
$448$	0	0
$449$	−9670.00	−1.01638	−0.508191	−	0.861244i	$-0.669686\pi$
$449$	−9670.00	−1.01638	−0.508191	+	0.861244i	$0.669686\pi$
$450$	0	0
$451$	− 5640.00i	− 0.588863i
$452$	0	0
$453$	− 3276.00i	− 0.339779i
$454$	0	0
$455$	−1008.00	−0.103859
$456$	0	0
$457$	6774.00	0.693379	0.346690	−	0.937980i	$-0.387306\pi$
$457$	6774.00	0.693379	0.346690	+	0.937980i	$0.387306\pi$
$458$	0	0
$459$	− 270.000i	− 0.0274565i
$460$	0	0
$461$	− 14602.0i	− 1.47523i	−0.675219	−	0.737617i	$-0.735951\pi$
$461$	− 14602.0i	− 1.47523i	0.675219	−	0.737617i	$-0.264049\pi$
$462$	0	0
$463$	13620.0	1.36712	0.683558	−	0.729896i	$-0.260431\pi$
$463$	13620.0	1.36712	0.683558	+	0.729896i	$0.260431\pi$
$464$	0	0
$465$	−1512.00	−0.150790
$466$	0	0
$467$	− 8508.00i	− 0.843048i	−0.906817	−	0.421524i	$-0.861495\pi$
$467$	− 8508.00i	− 0.843048i	0.906817	−	0.421524i	$-0.138505\pi$
$468$	0	0
$469$	2448.00i	0.241019i
$470$	0	0
$471$	−1734.00	−0.169636
$472$	0	0
$473$	13680.0	1.32982
$474$	0	0
$475$	15972.0i	1.54283i
$476$	0	0
$477$	− 3114.00i	− 0.298910i
$478$	0	0
$479$	−6312.00	−0.602093	−0.301047	−	0.953609i	$-0.597336\pi$
$479$	−6312.00	−0.602093	−0.301047	+	0.953609i	$0.597336\pi$
$480$	0	0
$481$	15204.0	1.44125
$482$	0	0
$483$	− 1728.00i	− 0.162788i
$484$	0	0
$485$	1188.00i	0.111225i
$486$	0	0
$487$	10572.0	0.983702	0.491851	−	0.870679i	$-0.336320\pi$
$487$	10572.0	0.983702	0.491851	+	0.870679i	$0.336320\pi$
$488$	0	0
$489$	−7596.00	−0.702460
$490$	0	0
$491$	4332.00i	0.398168i	0.979982	+	0.199084i	$0.0637966\pi$
$491$	4332.00i	0.398168i	−0.979982	+	0.199084i	$0.936203\pi$
$492$	0	0
$493$	− 2260.00i	− 0.206461i
$494$	0	0
$495$	−1080.00	−0.0980654
$496$	0	0
$497$	12672.0	1.14370
$498$	0	0
$499$	− 3684.00i	− 0.330498i	−0.986252	−	0.165249i	$-0.947157\pi$
$499$	− 3684.00i	− 0.330498i	0.986252	−	0.165249i	$-0.0528428\pi$
$500$	0	0
$501$	− 1944.00i	− 0.173356i
$502$	0	0
$503$	−11184.0	−0.991391	−0.495696	−	0.868496i	$-0.665087\pi$
$503$	−11184.0	−0.991391	−0.495696	+	0.868496i	$0.665087\pi$
$504$	0	0
$505$	−1108.00	−0.0976344
$506$	0	0
$507$	1299.00i	0.113788i
$508$	0	0
$509$	− 12946.0i	− 1.12735i	−0.825997	−	0.563675i	$-0.809387\pi$
$509$	− 12946.0i	− 1.12735i	0.825997	−	0.563675i	$-0.190613\pi$
$510$	0	0
$511$	−3960.00	−0.342818
$512$	0	0
$513$	3564.00	0.306734
$514$	0	0
$515$	2568.00i	0.219727i
$516$	0	0
$517$	− 24480.0i	− 2.08245i
$518$	0	0
$519$	10014.0	0.846948
$520$	0	0
$521$	17150.0	1.44214	0.721070	−	0.692862i	$-0.243651\pi$
$521$	17150.0	1.44214	0.721070	+	0.692862i	$0.243651\pi$
$522$	0	0
$523$	− 7884.00i	− 0.659165i	−0.944127	−	0.329582i	$-0.893092\pi$
$523$	− 7884.00i	− 0.659165i	0.944127	−	0.329582i	$-0.106908\pi$
$524$	0	0
$525$	4356.00i	0.362117i
$526$	0	0
$527$	2520.00	0.208298
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	− 2700.00i	− 0.220659i
$532$	0	0
$533$	3948.00i	0.320838i
$534$	0	0
$535$	−2712.00	−0.219159
$536$	0	0
$537$	−11412.0	−0.917065
$538$	0	0
$539$	11940.0i	0.954160i
$540$	0	0
$541$	− 5910.00i	− 0.469669i	−0.972035	−	0.234834i	$-0.924545\pi$
$541$	− 5910.00i	− 0.469669i	0.972035	−	0.234834i	$-0.0754548\pi$
$542$	0	0
$543$	−5562.00	−0.439573
$544$	0	0
$545$	780.000	0.0613056
$546$	0	0
$547$	− 972.000i	− 0.0759775i	−0.999278	−	0.0379888i	$-0.987905\pi$
$547$	− 972.000i	− 0.0759775i	0.999278	−	0.0379888i	$-0.0120951\pi$
$548$	0	0
$549$	− 4194.00i	− 0.326039i
$550$	0	0
$551$	29832.0	2.30651
$552$	0	0
$553$	−7344.00	−0.564735
$554$	0	0
$555$	2172.00i	0.166119i
$556$	0	0
$557$	− 2458.00i	− 0.186982i	−0.995620	−	0.0934908i	$-0.970197\pi$
$557$	− 2458.00i	− 0.186982i	0.995620	−	0.0934908i	$-0.0298026\pi$
$558$	0	0
$559$	−9576.00	−0.724547
$560$	0	0
$561$	1800.00	0.135465
$562$	0	0
$563$	11316.0i	0.847092i	0.905875	+	0.423546i	$0.139215\pi$
$563$	11316.0i	0.847092i	−0.905875	+	0.423546i	$0.860785\pi$
$564$	0	0
$565$	− 1532.00i	− 0.114074i
$566$	0	0
$567$	972.000	0.0719932
$568$	0	0
$569$	−1810.00	−0.133355	−0.0666776	−	0.997775i	$-0.521240\pi$
$569$	−1810.00	−0.133355	−0.0666776	+	0.997775i	$0.521240\pi$
$570$	0	0
$571$	− 10500.0i	− 0.769547i	−0.923011	−	0.384773i	$-0.874280\pi$
$571$	− 10500.0i	− 0.769547i	0.923011	−	0.384773i	$-0.125720\pi$
$572$	0	0
$573$	− 4032.00i	− 0.293960i
$574$	0	0
$575$	−5808.00	−0.421235
$576$	0	0
$577$	−19438.0	−1.40245	−0.701226	−	0.712939i	$-0.747363\pi$
$577$	−19438.0	−1.40245	−0.701226	+	0.712939i	$0.747363\pi$
$578$	0	0
$579$	− 3786.00i	− 0.271746i
$580$	0	0
$581$	6768.00i	0.483277i
$582$	0	0
$583$	20760.0	1.47477
$584$	0	0
$585$	756.000	0.0534303
$586$	0	0
$587$	15084.0i	1.06062i	0.847804	+	0.530309i	$0.177924\pi$
$587$	15084.0i	1.06062i	−0.847804	+	0.530309i	$0.822076\pi$
$588$	0	0
$589$	33264.0i	2.32703i
$590$	0	0
$591$	12882.0	0.896607
$592$	0	0
$593$	5794.00	0.401233	0.200616	−	0.979670i	$-0.435706\pi$
$593$	5794.00	0.401233	0.200616	+	0.979670i	$0.435706\pi$
$594$	0	0
$595$	240.000i	0.0165362i
$596$	0	0
$597$	12924.0i	0.886004i
$598$	0	0
$599$	25152.0	1.71566	0.857832	−	0.513930i	$-0.171811\pi$
$599$	25152.0	1.71566	0.857832	+	0.513930i	$0.171811\pi$
$600$	0	0
$601$	11846.0	0.804007	0.402004	−	0.915638i	$-0.368314\pi$
$601$	11846.0	0.804007	0.402004	+	0.915638i	$0.368314\pi$
$602$	0	0
$603$	− 1836.00i	− 0.123993i
$604$	0	0
$605$	− 4538.00i	− 0.304952i
$606$	0	0
$607$	−8940.00	−0.597798	−0.298899	−	0.954285i	$-0.596619\pi$
$607$	−8940.00	−0.597798	−0.298899	+	0.954285i	$0.596619\pi$
$608$	0	0
$609$	8136.00	0.541359
$610$	0	0
$611$	17136.0i	1.13461i
$612$	0	0
$613$	− 4570.00i	− 0.301110i	−0.988602	−	0.150555i	$-0.951894\pi$
$613$	− 4570.00i	− 0.301110i	0.988602	−	0.150555i	$-0.0481061\pi$
$614$	0	0
$615$	−564.000	−0.0369800
$616$	0	0
$617$	−17786.0	−1.16051	−0.580257	−	0.814433i	$-0.697048\pi$
$617$	−17786.0	−1.16051	−0.580257	+	0.814433i	$0.697048\pi$
$618$	0	0
$619$	15804.0i	1.02620i	0.858330	+	0.513099i	$0.171503\pi$
$619$	15804.0i	1.02620i	−0.858330	+	0.513099i	$0.828497\pi$
$620$	0	0
$621$	1296.00i	0.0837467i
$622$	0	0
$623$	18120.0	1.16527
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	23760.0i	1.51337i
$628$	0	0
$629$	− 3620.00i	− 0.229474i
$630$	0	0
$631$	−18468.0	−1.16513	−0.582567	−	0.812783i	$-0.697952\pi$
$631$	−18468.0	−1.16513	−0.582567	+	0.812783i	$0.697952\pi$
$632$	0	0
$633$	−3636.00	−0.228307
$634$	0	0
$635$	4776.00i	0.298472i
$636$	0	0
$637$	− 8358.00i	− 0.519868i
$638$	0	0
$639$	−9504.00	−0.588376
$640$	0	0
$641$	−7814.00	−0.481489	−0.240744	−	0.970589i	$-0.577392\pi$
$641$	−7814.00	−0.481489	−0.240744	+	0.970589i	$0.577392\pi$
$642$	0	0
$643$	5364.00i	0.328982i	0.986379	+	0.164491i	$0.0525982\pi$
$643$	5364.00i	0.328982i	−0.986379	+	0.164491i	$0.947402\pi$
$644$	0	0
$645$	− 1368.00i	− 0.0835115i
$646$	0	0
$647$	−3936.00	−0.239166	−0.119583	−	0.992824i	$-0.538156\pi$
$647$	−3936.00	−0.239166	−0.119583	+	0.992824i	$0.538156\pi$
$648$	0	0
$649$	18000.0	1.08869
$650$	0	0
$651$	9072.00i	0.546175i
$652$	0	0
$653$	− 7610.00i	− 0.456053i	−0.973655	−	0.228026i	$-0.926773\pi$
$653$	− 7610.00i	− 0.456053i	0.973655	−	0.228026i	$-0.0732272\pi$
$654$	0	0
$655$	−792.000	−0.0472458
$656$	0	0
$657$	2970.00	0.176363
$658$	0	0
$659$	− 13620.0i	− 0.805098i	−0.915398	−	0.402549i	$-0.868124\pi$
$659$	− 13620.0i	− 0.805098i	0.915398	−	0.402549i	$-0.131876\pi$
$660$	0	0
$661$	8710.00i	0.512526i	0.966607	+	0.256263i	$0.0824913\pi$
$661$	8710.00i	0.512526i	−0.966607	+	0.256263i	$0.917509\pi$
$662$	0	0
$663$	−1260.00	−0.0738075
$664$	0	0
$665$	−3168.00	−0.184736
$666$	0	0
$667$	10848.0i	0.629739i
$668$	0	0
$669$	− 6516.00i	− 0.376567i
$670$	0	0
$671$	27960.0	1.60862
$672$	0	0
$673$	−12094.0	−0.692703	−0.346352	−	0.938105i	$-0.612580\pi$
$673$	−12094.0	−0.692703	−0.346352	+	0.938105i	$0.612580\pi$
$674$	0	0
$675$	− 3267.00i	− 0.186292i
$676$	0	0
$677$	16466.0i	0.934771i	0.884054	+	0.467385i	$0.154804\pi$
$677$	16466.0i	0.934771i	−0.884054	+	0.467385i	$0.845196\pi$
$678$	0	0
$679$	7128.00	0.402868
$680$	0	0
$681$	11844.0	0.666466
$682$	0	0
$683$	− 16428.0i	− 0.920351i	−0.887828	−	0.460176i	$-0.847786\pi$
$683$	− 16428.0i	− 0.920351i	0.887828	−	0.460176i	$-0.152214\pi$
$684$	0	0
$685$	220.000i	0.0122712i
$686$	0	0
$687$	10566.0	0.586780
$688$	0	0
$689$	−14532.0	−0.803520
$690$	0	0
$691$	13332.0i	0.733970i	0.930227	+	0.366985i	$0.119610\pi$
$691$	13332.0i	0.733970i	−0.930227	+	0.366985i	$0.880390\pi$
$692$	0	0
$693$	6480.00i	0.355202i
$694$	0	0
$695$	−1464.00	−0.0799031
$696$	0	0
$697$	940.000	0.0510833
$698$	0	0
$699$	8322.00i	0.450310i
$700$	0	0
$701$	19118.0i	1.03007i	0.857170	+	0.515033i	$0.172221\pi$
$701$	19118.0i	1.03007i	−0.857170	+	0.515033i	$0.827779\pi$
$702$	0	0
$703$	47784.0	2.56360
$704$	0	0
$705$	−2448.00	−0.130776
$706$	0	0
$707$	6648.00i	0.353640i
$708$	0	0
$709$	798.000i	0.0422701i	0.999777	+	0.0211351i	$0.00672800\pi$
$709$	798.000i	0.0422701i	−0.999777	+	0.0211351i	$0.993272\pi$
$710$	0	0
$711$	5508.00	0.290529
$712$	0	0
$713$	−12096.0	−0.635342
$714$	0	0
$715$	5040.00i	0.263616i
$716$	0	0
$717$	− 8352.00i	− 0.435023i
$718$	0	0
$719$	8856.00	0.459351	0.229675	−	0.973267i	$-0.426234\pi$
$719$	8856.00	0.459351	0.229675	+	0.973267i	$0.426234\pi$
$720$	0	0
$721$	15408.0	0.795872
$722$	0	0
$723$	− 14058.0i	− 0.723130i
$724$	0	0
$725$	− 27346.0i	− 1.40083i
$726$	0	0
$727$	−13764.0	−0.702171	−0.351086	−	0.936343i	$-0.614187\pi$
$727$	−13764.0	−0.702171	−0.351086	+	0.936343i	$0.614187\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	2280.00i	0.115361i
$732$	0	0
$733$	20538.0i	1.03491i	0.855711	+	0.517455i	$0.173120\pi$
$733$	20538.0i	1.03491i	−0.855711	+	0.517455i	$0.826880\pi$
$734$	0	0
$735$	1194.00	0.0599202
$736$	0	0
$737$	12240.0	0.611759
$738$	0	0
$739$	15900.0i	0.791463i	0.918366	+	0.395731i	$0.129509\pi$
$739$	15900.0i	0.791463i	−0.918366	+	0.395731i	$0.870491\pi$
$740$	0	0
$741$	− 16632.0i	− 0.824550i
$742$	0	0
$743$	−20856.0	−1.02979	−0.514894	−	0.857254i	$-0.672169\pi$
$743$	−20856.0	−1.02979	−0.514894	+	0.857254i	$0.672169\pi$
$744$	0	0
$745$	3868.00	0.190218
$746$	0	0
$747$	− 5076.00i	− 0.248623i
$748$	0	0
$749$	16272.0i	0.793813i
$750$	0	0
$751$	10332.0	0.502024	0.251012	−	0.967984i	$-0.419237\pi$
$751$	10332.0	0.502024	0.251012	+	0.967984i	$0.419237\pi$
$752$	0	0
$753$	−7452.00	−0.360645
$754$	0	0
$755$	− 2184.00i	− 0.105277i
$756$	0	0
$757$	13806.0i	0.662863i	0.943479	+	0.331432i	$0.107532\pi$
$757$	13806.0i	0.662863i	−0.943479	+	0.331432i	$0.892468\pi$
$758$	0	0
$759$	−8640.00	−0.413191
$760$	0	0
$761$	−15554.0	−0.740909	−0.370455	−	0.928851i	$-0.620798\pi$
$761$	−15554.0	−0.740909	−0.370455	+	0.928851i	$0.620798\pi$
$762$	0	0
$763$	− 4680.00i	− 0.222054i
$764$	0	0
$765$	− 180.000i	− 0.00850708i
$766$	0	0
$767$	−12600.0	−0.593168
$768$	0	0
$769$	13106.0	0.614583	0.307292	−	0.951615i	$-0.400577\pi$
$769$	13106.0	0.614583	0.307292	+	0.951615i	$0.400577\pi$
$770$	0	0
$771$	19974.0i	0.933004i
$772$	0	0
$773$	18874.0i	0.878203i	0.898438	+	0.439101i	$0.144703\pi$
$773$	18874.0i	0.878203i	−0.898438	+	0.439101i	$0.855297\pi$
$774$	0	0
$775$	30492.0	1.41330
$776$	0	0
$777$	13032.0	0.601699
$778$	0	0
$779$	12408.0i	0.570684i
$780$	0	0
$781$	− 63360.0i	− 2.90294i
$782$	0	0
$783$	−6102.00	−0.278503
$784$	0	0
$785$	−1156.00	−0.0525598
$786$	0	0
$787$	15444.0i	0.699516i	0.936840	+	0.349758i	$0.113736\pi$
$787$	15444.0i	0.699516i	−0.936840	+	0.349758i	$0.886264\pi$
$788$	0	0
$789$	8712.00i	0.393099i
$790$	0	0
$791$	−9192.00	−0.413186
$792$	0	0
$793$	−19572.0	−0.876447
$794$	0	0
$795$	− 2076.00i	− 0.0926140i
$796$	0	0
$797$	39286.0i	1.74602i	0.487698	+	0.873012i	$0.337837\pi$
$797$	39286.0i	1.74602i	−0.487698	+	0.873012i	$0.662163\pi$
$798$	0	0
$799$	4080.00	0.180651
$800$	0	0
$801$	−13590.0	−0.599474
$802$	0	0
$803$	19800.0i	0.870145i
$804$	0	0
$805$	− 1152.00i	− 0.0504381i
$806$	0	0
$807$	−3018.00	−0.131646
$808$	0	0
$809$	−20018.0	−0.869957	−0.434979	−	0.900441i	$-0.643244\pi$
$809$	−20018.0	−0.869957	−0.434979	+	0.900441i	$0.643244\pi$
$810$	0	0
$811$	8388.00i	0.363184i	0.983374	+	0.181592i	$0.0581251\pi$
$811$	8388.00i	0.363184i	−0.983374	+	0.181592i	$0.941875\pi$
$812$	0	0
$813$	− 2628.00i	− 0.113368i
$814$	0	0
$815$	−5064.00	−0.217649
$816$	0	0
$817$	−30096.0	−1.28877
$818$	0	0
$819$	− 4536.00i	− 0.193530i
$820$	0	0
$821$	− 37942.0i	− 1.61289i	−0.591307	−	0.806446i	$-0.701388\pi$
$821$	− 37942.0i	− 1.61289i	0.591307	−	0.806446i	$-0.298612\pi$
$822$	0	0
$823$	11628.0	0.492499	0.246249	−	0.969206i	$-0.420802\pi$
$823$	11628.0	0.492499	0.246249	+	0.969206i	$0.420802\pi$
$824$	0	0
$825$	21780.0	0.919130
$826$	0	0
$827$	− 32388.0i	− 1.36184i	−0.732358	−	0.680920i	$-0.761580\pi$
$827$	− 32388.0i	− 1.36184i	0.732358	−	0.680920i	$-0.238420\pi$
$828$	0	0
$829$	− 9846.00i	− 0.412504i	−0.978499	−	0.206252i	$-0.933873\pi$
$829$	− 9846.00i	− 0.412504i	0.978499	−	0.206252i	$-0.0661266\pi$
$830$	0	0
$831$	−8154.00	−0.340384
$832$	0	0
$833$	−1990.00	−0.0827724
$834$	0	0
$835$	− 1296.00i	− 0.0537125i
$836$	0	0
$837$	− 6804.00i	− 0.280980i
$838$	0	0
$839$	16848.0	0.693275	0.346637	−	0.937999i	$-0.387323\pi$
$839$	16848.0	0.693275	0.346637	+	0.937999i	$0.387323\pi$
$840$	0	0
$841$	−26687.0	−1.09422
$842$	0	0
$843$	− 16062.0i	− 0.656233i
$844$	0	0
$845$	866.000i	0.0352560i
$846$	0	0
$847$	−27228.0	−1.10456
$848$	0	0
$849$	−2340.00	−0.0945920
$850$	0	0
$851$	17376.0i	0.699931i
$852$	0	0
$853$	18214.0i	0.731108i	0.930790	+	0.365554i	$0.119121\pi$
$853$	18214.0i	0.731108i	−0.930790	+	0.365554i	$0.880879\pi$
$854$	0	0
$855$	2376.00	0.0950380
$856$	0	0
$857$	2446.00	0.0974956	0.0487478	−	0.998811i	$-0.484477\pi$
$857$	2446.00	0.0974956	0.0487478	+	0.998811i	$0.484477\pi$
$858$	0	0
$859$	26244.0i	1.04241i	0.853430	+	0.521207i	$0.174518\pi$
$859$	26244.0i	1.04241i	−0.853430	+	0.521207i	$0.825482\pi$
$860$	0	0
$861$	3384.00i	0.133945i
$862$	0	0
$863$	−25248.0	−0.995889	−0.497944	−	0.867209i	$-0.665912\pi$
$863$	−25248.0	−0.995889	−0.497944	+	0.867209i	$0.665912\pi$
$864$	0	0
$865$	6676.00	0.262417
$866$	0	0
$867$	− 14439.0i	− 0.565599i
$868$	0	0
$869$	36720.0i	1.43342i
$870$	0	0
$871$	−8568.00	−0.333313
$872$	0	0
$873$	−5346.00	−0.207256
$874$	0	0
$875$	5904.00i	0.228105i
$876$	0	0
$877$	34.0000i	0.00130912i	1.00000		0.000654560i	$0.000208353\pi$
$877$	34.0000i	0.00130912i	−1.00000		0.000654560i	$0.999792\pi$
$878$	0	0
$879$	10050.0	0.385641
$880$	0	0
$881$	−19022.0	−0.727432	−0.363716	−	0.931510i	$-0.618492\pi$
$881$	−19022.0	−0.727432	−0.363716	+	0.931510i	$0.618492\pi$
$882$	0	0
$883$	12852.0i	0.489812i	0.969547	+	0.244906i	$0.0787571\pi$
$883$	12852.0i	0.489812i	−0.969547	+	0.244906i	$0.921243\pi$
$884$	0	0
$885$	− 1800.00i	− 0.0683687i
$886$	0	0
$887$	40104.0	1.51811	0.759053	−	0.651028i	$-0.225662\pi$
$887$	40104.0	1.51811	0.759053	+	0.651028i	$0.225662\pi$
$888$	0	0
$889$	28656.0	1.08109
$890$	0	0
$891$	− 4860.00i	− 0.182734i
$892$	0	0
$893$	53856.0i	2.01817i
$894$	0	0
$895$	−7608.00	−0.284142
$896$	0	0
$897$	6048.00	0.225125
$898$	0	0
$899$	− 56952.0i	− 2.11285i
$900$	0	0
$901$	3460.00i	0.127935i
$902$	0	0
$903$	−8208.00	−0.302486
$904$	0	0
$905$	−3708.00	−0.136197
$906$	0	0
$907$	− 42540.0i	− 1.55735i	−0.627427	−	0.778676i	$-0.715892\pi$
$907$	− 42540.0i	− 1.55735i	0.627427	−	0.778676i	$-0.284108\pi$
$908$	0	0
$909$	− 4986.00i	− 0.181931i
$910$	0	0
$911$	18528.0	0.673831	0.336915	−	0.941535i	$-0.390616\pi$
$911$	18528.0	0.673831	0.336915	+	0.941535i	$0.390616\pi$
$912$	0	0
$913$	33840.0	1.22666
$914$	0	0
$915$	− 2796.00i	− 0.101020i
$916$	0	0
$917$	4752.00i	0.171129i
$918$	0	0
$919$	15756.0	0.565552	0.282776	−	0.959186i	$-0.408745\pi$
$919$	15756.0	0.565552	0.282776	+	0.959186i	$0.408745\pi$
$920$	0	0
$921$	28908.0	1.03426
$922$	0	0
$923$	44352.0i	1.58165i
$924$	0	0
$925$	− 43802.0i	− 1.55697i
$926$	0	0
$927$	−11556.0	−0.409438
$928$	0	0
$929$	−15542.0	−0.548887	−0.274444	−	0.961603i	$-0.588494\pi$
$929$	−15542.0	−0.548887	−0.274444	+	0.961603i	$0.588494\pi$
$930$	0	0
$931$	− 26268.0i	− 0.924703i
$932$	0	0
$933$	− 22680.0i	− 0.795831i
$934$	0	0
$935$	1200.00	0.0419724
$936$	0	0
$937$	29702.0	1.03556	0.517781	−	0.855513i	$-0.326758\pi$
$937$	29702.0	1.03556	0.517781	+	0.855513i	$0.326758\pi$
$938$	0	0
$939$	10578.0i	0.367625i
$940$	0	0
$941$	− 2890.00i	− 0.100118i	−0.998746	−	0.0500591i	$-0.984059\pi$
$941$	− 2890.00i	− 0.100118i	0.998746	−	0.0500591i	$-0.0159410\pi$
$942$	0	0
$943$	−4512.00	−0.155812
$944$	0	0
$945$	648.000	0.0223063
$946$	0	0
$947$	9180.00i	0.315005i	0.987519	+	0.157503i	$0.0503443\pi$
$947$	9180.00i	0.315005i	−0.987519	+	0.157503i	$0.949656\pi$
$948$	0	0
$949$	− 13860.0i	− 0.474093i
$950$	0	0
$951$	22902.0	0.780913
$952$	0	0
$953$	−37906.0	−1.28845	−0.644227	−	0.764835i	$-0.722821\pi$
$953$	−37906.0	−1.28845	−0.644227	+	0.764835i	$0.722821\pi$
$954$	0	0
$955$	− 2688.00i	− 0.0910802i
$956$	0	0
$957$	− 40680.0i	− 1.37408i
$958$	0	0
$959$	1320.00	0.0444474
$960$	0	0
$961$	33713.0	1.13165
$962$	0	0
$963$	− 12204.0i	− 0.408378i
$964$	0	0
$965$	− 2524.00i	− 0.0841973i
$966$	0	0
$967$	−41916.0	−1.39393	−0.696964	−	0.717106i	$-0.745466\pi$
$967$	−41916.0	−1.39393	−0.696964	+	0.717106i	$0.745466\pi$
$968$	0	0
$969$	−3960.00	−0.131283
$970$	0	0
$971$	7764.00i	0.256600i	0.991735	+	0.128300i	$0.0409520\pi$
$971$	7764.00i	0.256600i	−0.991735	+	0.128300i	$0.959048\pi$
$972$	0	0
$973$	8784.00i	0.289416i
$974$	0	0
$975$	−15246.0	−0.500782
$976$	0	0
$977$	32666.0	1.06968	0.534840	−	0.844953i	$-0.320372\pi$
$977$	32666.0	1.06968	0.534840	+	0.844953i	$0.320372\pi$
$978$	0	0
$979$	− 90600.0i	− 2.95770i
$980$	0	0
$981$	3510.00i	0.114236i
$982$	0	0
$983$	−53016.0	−1.72019	−0.860096	−	0.510133i	$-0.829596\pi$
$983$	−53016.0	−1.72019	−0.860096	+	0.510133i	$0.829596\pi$
$984$	0	0
$985$	8588.00	0.277803
$986$	0	0
$987$	14688.0i	0.473682i
$988$	0	0
$989$	− 10944.0i	− 0.351870i
$990$	0	0
$991$	−17844.0	−0.571981	−0.285991	−	0.958232i	$-0.592323\pi$
$991$	−17844.0	−0.571981	−0.285991	+	0.958232i	$0.592323\pi$
$992$	0	0
$993$	22716.0	0.725952
$994$	0	0
$995$	8616.00i	0.274518i
$996$	0	0
$997$	− 55834.0i	− 1.77360i	−0.462152	−	0.886801i	$-0.652923\pi$
$997$	− 55834.0i	− 1.77360i	0.462152	−	0.886801i	$-0.347077\pi$
$998$	0	0
$999$	−9774.00	−0.309545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.d.m.385.2		2
4.3	odd	2		768.4.d.d.385.1		2
8.3	odd	2		768.4.d.d.385.2		2
8.5	even	2	inner	768.4.d.m.385.1		2
16.3	odd	4		96.4.a.e.1.1	yes	1
16.5	even	4		192.4.a.j.1.1		1
16.11	odd	4		192.4.a.d.1.1		1
16.13	even	4		96.4.a.b.1.1	✓	1
48.5	odd	4		576.4.a.n.1.1		1
48.11	even	4		576.4.a.o.1.1		1
48.29	odd	4		288.4.a.e.1.1		1
48.35	even	4		288.4.a.f.1.1		1
80.19	odd	4		2400.4.a.c.1.1		1
80.29	even	4		2400.4.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.4.a.b.1.1	✓	1	16.13	even	4
96.4.a.e.1.1	yes	1	16.3	odd	4
192.4.a.d.1.1		1	16.11	odd	4
192.4.a.j.1.1		1	16.5	even	4
288.4.a.e.1.1		1	48.29	odd	4
288.4.a.f.1.1		1	48.35	even	4
576.4.a.n.1.1		1	48.5	odd	4
576.4.a.o.1.1		1	48.11	even	4
768.4.d.d.385.1		2	4.3	odd	2
768.4.d.d.385.2		2	8.3	odd	2
768.4.d.m.385.1		2	8.5	even	2	inner
768.4.d.m.385.2		2	1.1	even	1	trivial
2400.4.a.c.1.1		1	80.19	odd	4
2400.4.a.t.1.1		1	80.29	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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +2.00000i q^{5} +12.0000 q^{7} -9.00000 q^{9} -60.0000i q^{11} +42.0000i q^{13} -6.00000 q^{15} +10.0000 q^{17} +132.000i q^{19} +36.0000i q^{21} -48.0000 q^{23} +121.000 q^{25} -27.0000i q^{27} -226.000i q^{29} +252.000 q^{31} +180.000 q^{33} +24.0000i q^{35} -362.000i q^{37} -126.000 q^{39} +94.0000 q^{41} +228.000i q^{43} -18.0000i q^{45} +408.000 q^{47} -199.000 q^{49} +30.0000i q^{51} +346.000i q^{53} +120.000 q^{55} -396.000 q^{57} +300.000i q^{59} +466.000i q^{61} -108.000 q^{63} -84.0000 q^{65} +204.000i q^{67} -144.000i q^{69} +1056.00 q^{71} -330.000 q^{73} +363.000i q^{75} -720.000i q^{77} -612.000 q^{79} +81.0000 q^{81} +564.000i q^{83} +20.0000i q^{85} +678.000 q^{87} +1510.00 q^{89} +504.000i q^{91} +756.000i q^{93} -264.000 q^{95} +594.000 q^{97} +540.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{7} - 18 q^{9} - 12 q^{15} + 20 q^{17} - 96 q^{23} + 242 q^{25} + 504 q^{31} + 360 q^{33} - 252 q^{39} + 188 q^{41} + 816 q^{47} - 398 q^{49} + 240 q^{55} - 792 q^{57} - 216 q^{63} - 168 q^{65}+ \cdots + 1188 q^{97}+O(q^{100}) \) 2 * q + 24 * q^7 - 18 * q^9 - 12 * q^15 + 20 * q^17 - 96 * q^23 + 242 * q^25 + 504 * q^31 + 360 * q^33 - 252 * q^39 + 188 * q^41 + 816 * q^47 - 398 * q^49 + 240 * q^55 - 792 * q^57 - 216 * q^63 - 168 * q^65 + 2112 * q^71 - 660 * q^73 - 1224 * q^79 + 162 * q^81 + 1356 * q^87 + 3020 * q^89 - 528 * q^95 + 1188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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