LMFDB - Embedded newform 768.4.d.g.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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
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.g.385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +18.0000i q^{5} -8.00000 q^{7} -9.00000 q^{9} -36.0000i q^{11} -10.0000i q^{13} -54.0000 q^{15} +18.0000 q^{17} -100.000i q^{19} -24.0000i q^{21} -72.0000 q^{23} -199.000 q^{25} -27.0000i q^{27} -234.000i q^{29} -16.0000 q^{31} +108.000 q^{33} -144.000i q^{35} +226.000i q^{37} +30.0000 q^{39} -90.0000 q^{41} -452.000i q^{43} -162.000i q^{45} +432.000 q^{47} -279.000 q^{49} +54.0000i q^{51} -414.000i q^{53} +648.000 q^{55} +300.000 q^{57} +684.000i q^{59} +422.000i q^{61} +72.0000 q^{63} +180.000 q^{65} +332.000i q^{67} -216.000i q^{69} +360.000 q^{71} -26.0000 q^{73} -597.000i q^{75} +288.000i q^{77} +512.000 q^{79} +81.0000 q^{81} -1188.00i q^{83} +324.000i q^{85} +702.000 q^{87} +630.000 q^{89} +80.0000i q^{91} -48.0000i q^{93} +1800.00 q^{95} -1054.00 q^{97} +324.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{7} - 18 q^{9} - 108 q^{15} + 36 q^{17} - 144 q^{23} - 398 q^{25} - 32 q^{31} + 216 q^{33} + 60 q^{39} - 180 q^{41} + 864 q^{47} - 558 q^{49} + 1296 q^{55} + 600 q^{57} + 144 q^{63} + 360 q^{65}+ \cdots - 2108 q^{97}+O(q^{100}) $ 2 * q - 16 * q^7 - 18 * q^9 - 108 * q^15 + 36 * q^17 - 144 * q^23 - 398 * q^25 - 32 * q^31 + 216 * q^33 + 60 * q^39 - 180 * q^41 + 864 * q^47 - 558 * q^49 + 1296 * q^55 + 600 * q^57 + 144 * q^63 + 360 * q^65 + 720 * q^71 - 52 * q^73 + 1024 * q^79 + 162 * q^81 + 1404 * q^87 + 1260 * q^89 + 3600 * q^95 - 2108 * 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
$5$	18.0000i	1.60997i	0.593296	+	0.804984i	$0.297826\pi$
$5$	18.0000i	1.60997i	−0.593296	+	0.804984i	$0.702174\pi$
$6$	0	0
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 36.0000i	− 0.986764i	−0.869813	−	0.493382i	$-0.835760\pi$
$11$	− 36.0000i	− 0.986764i	0.869813	−	0.493382i	$-0.164240\pi$
$12$	0	0
$13$	− 10.0000i	− 0.213346i	−0.994294	−	0.106673i	$-0.965980\pi$
$13$	− 10.0000i	− 0.213346i	0.994294	−	0.106673i	$-0.0340198\pi$
$14$	0	0
$15$	−54.0000	−0.929516
$16$	0	0
$17$	18.0000	0.256802	0.128401	−	0.991722i	$-0.459015\pi$
$17$	18.0000	0.256802	0.128401	+	0.991722i	$0.459015\pi$
$18$	0	0
$19$	− 100.000i	− 1.20745i	−0.797192	−	0.603726i	$-0.793682\pi$
$19$	− 100.000i	− 1.20745i	0.797192	−	0.603726i	$-0.206318\pi$
$20$	0	0
$21$	− 24.0000i	− 0.249392i
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	−199.000	−1.59200
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 234.000i	− 1.49837i	−0.662361	−	0.749185i	$-0.730446\pi$
$29$	− 234.000i	− 1.49837i	0.662361	−	0.749185i	$-0.269554\pi$
$30$	0	0
$31$	−16.0000	−0.0926995	−0.0463498	−	0.998925i	$-0.514759\pi$
$31$	−16.0000	−0.0926995	−0.0463498	+	0.998925i	$0.514759\pi$
$32$	0	0
$33$	108.000	0.569709
$34$	0	0
$35$	− 144.000i	− 0.695441i
$36$	0	0
$37$	226.000i	1.00417i	0.864819	+	0.502083i	$0.167433\pi$
$37$	226.000i	1.00417i	−0.864819	+	0.502083i	$0.832567\pi$
$38$	0	0
$39$	30.0000	0.123176
$40$	0	0
$41$	−90.0000	−0.342820	−0.171410	−	0.985200i	$-0.554832\pi$
$41$	−90.0000	−0.342820	−0.171410	+	0.985200i	$0.554832\pi$
$42$	0	0
$43$	− 452.000i	− 1.60301i	−0.597989	−	0.801504i	$-0.704033\pi$
$43$	− 452.000i	− 1.60301i	0.597989	−	0.801504i	$-0.295967\pi$
$44$	0	0
$45$	− 162.000i	− 0.536656i
$46$	0	0
$47$	432.000	1.34072	0.670358	−	0.742038i	$-0.266140\pi$
$47$	432.000	1.34072	0.670358	+	0.742038i	$0.266140\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	54.0000i	0.148265i
$52$	0	0
$53$	− 414.000i	− 1.07297i	−0.843911	−	0.536484i	$-0.819752\pi$
$53$	− 414.000i	− 1.07297i	0.843911	−	0.536484i	$-0.180248\pi$
$54$	0	0
$55$	648.000	1.58866
$56$	0	0
$57$	300.000	0.697122
$58$	0	0
$59$	684.000i	1.50931i	0.656123	+	0.754654i	$0.272195\pi$
$59$	684.000i	1.50931i	−0.656123	+	0.754654i	$0.727805\pi$
$60$	0	0
$61$	422.000i	0.885763i	0.896580	+	0.442882i	$0.146044\pi$
$61$	422.000i	0.885763i	−0.896580	+	0.442882i	$0.853956\pi$
$62$	0	0
$63$	72.0000	0.143986
$64$	0	0
$65$	180.000	0.343481
$66$	0	0
$67$	332.000i	0.605377i	0.953090	+	0.302688i	$0.0978842\pi$
$67$	332.000i	0.605377i	−0.953090	+	0.302688i	$0.902116\pi$
$68$	0	0
$69$	− 216.000i	− 0.376860i
$70$	0	0
$71$	360.000	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$72$	0	0
$73$	−26.0000	−0.0416859	−0.0208429	−	0.999783i	$-0.506635\pi$
$73$	−26.0000	−0.0416859	−0.0208429	+	0.999783i	$0.506635\pi$
$74$	0	0
$75$	− 597.000i	− 0.919142i
$76$	0	0
$77$	288.000i	0.426242i
$78$	0	0
$79$	512.000	0.729171	0.364585	−	0.931170i	$-0.381211\pi$
$79$	512.000	0.729171	0.364585	+	0.931170i	$0.381211\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1188.00i	− 1.57108i	−0.618809	−	0.785542i	$-0.712384\pi$
$83$	− 1188.00i	− 1.57108i	0.618809	−	0.785542i	$-0.287616\pi$
$84$	0	0
$85$	324.000i	0.413444i
$86$	0	0
$87$	702.000	0.865084
$88$	0	0
$89$	630.000	0.750336	0.375168	−	0.926957i	$-0.377585\pi$
$89$	630.000	0.750336	0.375168	+	0.926957i	$0.377585\pi$
$90$	0	0
$91$	80.0000i	0.0921569i
$92$	0	0
$93$	− 48.0000i	− 0.0535201i
$94$	0	0
$95$	1800.00	1.94396
$96$	0	0
$97$	−1054.00	−1.10327	−0.551637	−	0.834085i	$-0.685996\pi$
$97$	−1054.00	−1.10327	−0.551637	+	0.834085i	$0.685996\pi$
$98$	0	0
$99$	324.000i	0.328921i
$100$	0	0
$101$	− 558.000i	− 0.549733i	−0.961482	−	0.274867i	$-0.911366\pi$
$101$	− 558.000i	− 0.549733i	0.961482	−	0.274867i	$-0.0886337\pi$
$102$	0	0
$103$	−8.00000	−0.00765304	−0.00382652	−	0.999993i	$-0.501218\pi$
$103$	−8.00000	−0.00765304	−0.00382652	+	0.999993i	$0.501218\pi$
$104$	0	0
$105$	432.000	0.401513
$106$	0	0
$107$	− 1764.00i	− 1.59376i	−0.604138	−	0.796880i	$-0.706482\pi$
$107$	− 1764.00i	− 1.59376i	0.604138	−	0.796880i	$-0.293518\pi$
$108$	0	0
$109$	1622.00i	1.42532i	0.701512	+	0.712658i	$0.252509\pi$
$109$	1622.00i	1.42532i	−0.701512	+	0.712658i	$0.747491\pi$
$110$	0	0
$111$	−678.000	−0.579756
$112$	0	0
$113$	−1134.00	−0.944051	−0.472025	−	0.881585i	$-0.656477\pi$
$113$	−1134.00	−0.944051	−0.472025	+	0.881585i	$0.656477\pi$
$114$	0	0
$115$	− 1296.00i	− 1.05089i
$116$	0	0
$117$	90.0000i	0.0711154i
$118$	0	0
$119$	−144.000	−0.110928
$120$	0	0
$121$	35.0000	0.0262960
$122$	0	0
$123$	− 270.000i	− 0.197927i
$124$	0	0
$125$	− 1332.00i	− 0.953102i
$126$	0	0
$127$	−592.000	−0.413634	−0.206817	−	0.978380i	$-0.566310\pi$
$127$	−592.000	−0.413634	−0.206817	+	0.978380i	$0.566310\pi$
$128$	0	0
$129$	1356.00	0.925497
$130$	0	0
$131$	− 1908.00i	− 1.27254i	−0.771466	−	0.636270i	$-0.780476\pi$
$131$	− 1908.00i	− 1.27254i	0.771466	−	0.636270i	$-0.219524\pi$
$132$	0	0
$133$	800.000i	0.521570i
$134$	0	0
$135$	486.000	0.309839
$136$	0	0
$137$	−954.000	−0.594932	−0.297466	−	0.954732i	$-0.596142\pi$
$137$	−954.000	−0.594932	−0.297466	+	0.954732i	$0.596142\pi$
$138$	0	0
$139$	− 2564.00i	− 1.56457i	−0.622919	−	0.782286i	$-0.714053\pi$
$139$	− 2564.00i	− 1.56457i	0.622919	−	0.782286i	$-0.285947\pi$
$140$	0	0
$141$	1296.00i	0.774063i
$142$	0	0
$143$	−360.000	−0.210522
$144$	0	0
$145$	4212.00	2.41233
$146$	0	0
$147$	− 837.000i	− 0.469623i
$148$	0	0
$149$	738.000i	0.405767i	0.979203	+	0.202884i	$0.0650313\pi$
$149$	738.000i	0.405767i	−0.979203	+	0.202884i	$0.934969\pi$
$150$	0	0
$151$	2440.00	1.31500	0.657498	−	0.753456i	$-0.271615\pi$
$151$	2440.00	1.31500	0.657498	+	0.753456i	$0.271615\pi$
$152$	0	0
$153$	−162.000	−0.0856008
$154$	0	0
$155$	− 288.000i	− 0.149243i
$156$	0	0
$157$	− 2554.00i	− 1.29829i	−0.760665	−	0.649145i	$-0.775127\pi$
$157$	− 2554.00i	− 1.29829i	0.760665	−	0.649145i	$-0.224873\pi$
$158$	0	0
$159$	1242.00	0.619478
$160$	0	0
$161$	576.000	0.281958
$162$	0	0
$163$	− 820.000i	− 0.394033i	−0.980400	−	0.197016i	$-0.936875\pi$
$163$	− 820.000i	− 0.394033i	0.980400	−	0.197016i	$-0.0631252\pi$
$164$	0	0
$165$	1944.00i	0.917213i
$166$	0	0
$167$	−1944.00	−0.900786	−0.450393	−	0.892830i	$-0.648716\pi$
$167$	−1944.00	−0.900786	−0.450393	+	0.892830i	$0.648716\pi$
$168$	0	0
$169$	2097.00	0.954483
$170$	0	0
$171$	900.000i	0.402484i
$172$	0	0
$173$	− 1242.00i	− 0.545824i	−0.962039	−	0.272912i	$-0.912013\pi$
$173$	− 1242.00i	− 0.545824i	0.962039	−	0.272912i	$-0.0879867\pi$
$174$	0	0
$175$	1592.00	0.687679
$176$	0	0
$177$	−2052.00	−0.871400
$178$	0	0
$179$	1116.00i	0.465999i	0.972477	+	0.232999i	$0.0748540\pi$
$179$	1116.00i	0.465999i	−0.972477	+	0.232999i	$0.925146\pi$
$180$	0	0
$181$	− 1070.00i	− 0.439406i	−0.975567	−	0.219703i	$-0.929491\pi$
$181$	− 1070.00i	− 0.439406i	0.975567	−	0.219703i	$-0.0705088\pi$
$182$	0	0
$183$	−1266.00	−0.511396
$184$	0	0
$185$	−4068.00	−1.61668
$186$	0	0
$187$	− 648.000i	− 0.253403i
$188$	0	0
$189$	216.000i	0.0831306i
$190$	0	0
$191$	−576.000	−0.218209	−0.109104	−	0.994030i	$-0.534798\pi$
$191$	−576.000	−0.218209	−0.109104	+	0.994030i	$0.534798\pi$
$192$	0	0
$193$	−1342.00	−0.500514	−0.250257	−	0.968179i	$-0.580515\pi$
$193$	−1342.00	−0.500514	−0.250257	+	0.968179i	$0.580515\pi$
$194$	0	0
$195$	540.000i	0.198309i
$196$	0	0
$197$	− 1422.00i	− 0.514281i	−0.966374	−	0.257140i	$-0.917220\pi$
$197$	− 1422.00i	− 0.514281i	0.966374	−	0.257140i	$-0.0827803\pi$
$198$	0	0
$199$	−872.000	−0.310625	−0.155313	−	0.987865i	$-0.549639\pi$
$199$	−872.000	−0.310625	−0.155313	+	0.987865i	$0.549639\pi$
$200$	0	0
$201$	−996.000	−0.349515
$202$	0	0
$203$	1872.00i	0.647235i
$204$	0	0
$205$	− 1620.00i	− 0.551930i
$206$	0	0
$207$	648.000	0.217580
$208$	0	0
$209$	−3600.00	−1.19147
$210$	0	0
$211$	1340.00i	0.437201i	0.975814	+	0.218600i	$0.0701491\pi$
$211$	1340.00i	0.437201i	−0.975814	+	0.218600i	$0.929851\pi$
$212$	0	0
$213$	1080.00i	0.347420i
$214$	0	0
$215$	8136.00	2.58079
$216$	0	0
$217$	128.000	0.0400424
$218$	0	0
$219$	− 78.0000i	− 0.0240674i
$220$	0	0
$221$	− 180.000i	− 0.0547878i
$222$	0	0
$223$	4880.00	1.46542	0.732711	−	0.680540i	$-0.238255\pi$
$223$	4880.00	1.46542	0.732711	+	0.680540i	$0.238255\pi$
$224$	0	0
$225$	1791.00	0.530667
$226$	0	0
$227$	2700.00i	0.789451i	0.918799	+	0.394725i	$0.129160\pi$
$227$	2700.00i	0.789451i	−0.918799	+	0.394725i	$0.870840\pi$
$228$	0	0
$229$	− 254.000i	− 0.0732960i	−0.999328	−	0.0366480i	$-0.988332\pi$
$229$	− 254.000i	− 0.0732960i	0.999328	−	0.0366480i	$-0.0116680\pi$
$230$	0	0
$231$	−864.000	−0.246091
$232$	0	0
$233$	−4410.00	−1.23995	−0.619976	−	0.784621i	$-0.712858\pi$
$233$	−4410.00	−1.23995	−0.619976	+	0.784621i	$0.712858\pi$
$234$	0	0
$235$	7776.00i	2.15851i
$236$	0	0
$237$	1536.00i	0.420987i
$238$	0	0
$239$	−3888.00	−1.05228	−0.526138	−	0.850399i	$-0.676360\pi$
$239$	−3888.00	−1.05228	−0.526138	+	0.850399i	$0.676360\pi$
$240$	0	0
$241$	5138.00	1.37331	0.686655	−	0.726984i	$-0.259078\pi$
$241$	5138.00	1.37331	0.686655	+	0.726984i	$0.259078\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 5022.00i	− 1.30957i
$246$	0	0
$247$	−1000.00	−0.257605
$248$	0	0
$249$	3564.00	0.907066
$250$	0	0
$251$	− 4788.00i	− 1.20405i	−0.798478	−	0.602024i	$-0.794361\pi$
$251$	− 4788.00i	− 1.20405i	0.798478	−	0.602024i	$-0.205639\pi$
$252$	0	0
$253$	2592.00i	0.644101i
$254$	0	0
$255$	−972.000	−0.238702
$256$	0	0
$257$	−5886.00	−1.42863	−0.714316	−	0.699823i	$-0.753262\pi$
$257$	−5886.00	−1.42863	−0.714316	+	0.699823i	$0.753262\pi$
$258$	0	0
$259$	− 1808.00i	− 0.433759i
$260$	0	0
$261$	2106.00i	0.499456i
$262$	0	0
$263$	−2232.00	−0.523312	−0.261656	−	0.965161i	$-0.584269\pi$
$263$	−2232.00	−0.523312	−0.261656	+	0.965161i	$0.584269\pi$
$264$	0	0
$265$	7452.00	1.72744
$266$	0	0
$267$	1890.00i	0.433206i
$268$	0	0
$269$	− 666.000i	− 0.150954i	−0.997148	−	0.0754772i	$-0.975952\pi$
$269$	− 666.000i	− 0.150954i	0.997148	−	0.0754772i	$-0.0240480\pi$
$270$	0	0
$271$	−5536.00	−1.24092	−0.620458	−	0.784240i	$-0.713053\pi$
$271$	−5536.00	−1.24092	−0.620458	+	0.784240i	$0.713053\pi$
$272$	0	0
$273$	−240.000	−0.0532068
$274$	0	0
$275$	7164.00i	1.57093i
$276$	0	0
$277$	− 2126.00i	− 0.461151i	−0.973054	−	0.230576i	$-0.925939\pi$
$277$	− 2126.00i	− 0.461151i	0.973054	−	0.230576i	$-0.0740609\pi$
$278$	0	0
$279$	144.000	0.0308998
$280$	0	0
$281$	2934.00	0.622875	0.311437	−	0.950267i	$-0.399190\pi$
$281$	2934.00	0.622875	0.311437	+	0.950267i	$0.399190\pi$
$282$	0	0
$283$	− 2036.00i	− 0.427659i	−0.976871	−	0.213830i	$-0.931406\pi$
$283$	− 2036.00i	− 0.427659i	0.976871	−	0.213830i	$-0.0685938\pi$
$284$	0	0
$285$	5400.00i	1.12235i
$286$	0	0
$287$	720.000	0.148085
$288$	0	0
$289$	−4589.00	−0.934053
$290$	0	0
$291$	− 3162.00i	− 0.636975i
$292$	0	0
$293$	− 2286.00i	− 0.455800i	−0.973684	−	0.227900i	$-0.926814\pi$
$293$	− 2286.00i	− 0.455800i	0.973684	−	0.227900i	$-0.0731860\pi$
$294$	0	0
$295$	−12312.0	−2.42994
$296$	0	0
$297$	−972.000	−0.189903
$298$	0	0
$299$	720.000i	0.139260i
$300$	0	0
$301$	3616.00i	0.692434i
$302$	0	0
$303$	1674.00	0.317389
$304$	0	0
$305$	−7596.00	−1.42605
$306$	0	0
$307$	1244.00i	0.231267i	0.993292	+	0.115633i	$0.0368897\pi$
$307$	1244.00i	0.231267i	−0.993292	+	0.115633i	$0.963110\pi$
$308$	0	0
$309$	− 24.0000i	− 0.00441849i
$310$	0	0
$311$	−1224.00	−0.223173	−0.111586	−	0.993755i	$-0.535593\pi$
$311$	−1224.00	−0.223173	−0.111586	+	0.993755i	$0.535593\pi$
$312$	0	0
$313$	−1898.00	−0.342752	−0.171376	−	0.985206i	$-0.554821\pi$
$313$	−1898.00	−0.342752	−0.171376	+	0.985206i	$0.554821\pi$
$314$	0	0
$315$	1296.00i	0.231814i
$316$	0	0
$317$	− 9162.00i	− 1.62331i	−0.584137	−	0.811655i	$-0.698567\pi$
$317$	− 9162.00i	− 1.62331i	0.584137	−	0.811655i	$-0.301433\pi$
$318$	0	0
$319$	−8424.00	−1.47854
$320$	0	0
$321$	5292.00	0.920158
$322$	0	0
$323$	− 1800.00i	− 0.310076i
$324$	0	0
$325$	1990.00i	0.339647i
$326$	0	0
$327$	−4866.00	−0.822906
$328$	0	0
$329$	−3456.00	−0.579135
$330$	0	0
$331$	4348.00i	0.722017i	0.932562	+	0.361009i	$0.117568\pi$
$331$	4348.00i	0.722017i	−0.932562	+	0.361009i	$0.882432\pi$
$332$	0	0
$333$	− 2034.00i	− 0.334722i
$334$	0	0
$335$	−5976.00	−0.974638
$336$	0	0
$337$	7154.00	1.15639	0.578195	−	0.815899i	$-0.303757\pi$
$337$	7154.00	1.15639	0.578195	+	0.815899i	$0.303757\pi$
$338$	0	0
$339$	− 3402.00i	− 0.545048i
$340$	0	0
$341$	576.000i	0.0914726i
$342$	0	0
$343$	4976.00	0.783320
$344$	0	0
$345$	3888.00	0.606733
$346$	0	0
$347$	1836.00i	0.284039i	0.989864	+	0.142020i	$0.0453596\pi$
$347$	1836.00i	0.284039i	−0.989864	+	0.142020i	$0.954640\pi$
$348$	0	0
$349$	5894.00i	0.904007i	0.892016	+	0.452004i	$0.149291\pi$
$349$	5894.00i	0.904007i	−0.892016	+	0.452004i	$0.850709\pi$
$350$	0	0
$351$	−270.000	−0.0410585
$352$	0	0
$353$	11106.0	1.67454	0.837270	−	0.546789i	$-0.184150\pi$
$353$	11106.0	1.67454	0.837270	+	0.546789i	$0.184150\pi$
$354$	0	0
$355$	6480.00i	0.968796i
$356$	0	0
$357$	− 432.000i	− 0.0640444i
$358$	0	0
$359$	−13176.0	−1.93705	−0.968527	−	0.248907i	$-0.919929\pi$
$359$	−13176.0	−1.93705	−0.968527	+	0.248907i	$0.919929\pi$
$360$	0	0
$361$	−3141.00	−0.457938
$362$	0	0
$363$	105.000i	0.0151820i
$364$	0	0
$365$	− 468.000i	− 0.0671130i
$366$	0	0
$367$	−6112.00	−0.869329	−0.434665	−	0.900592i	$-0.643133\pi$
$367$	−6112.00	−0.869329	−0.434665	+	0.900592i	$0.643133\pi$
$368$	0	0
$369$	810.000	0.114273
$370$	0	0
$371$	3312.00i	0.463478i
$372$	0	0
$373$	13618.0i	1.89038i	0.326515	+	0.945192i	$0.394126\pi$
$373$	13618.0i	1.89038i	−0.326515	+	0.945192i	$0.605874\pi$
$374$	0	0
$375$	3996.00	0.550273
$376$	0	0
$377$	−2340.00	−0.319671
$378$	0	0
$379$	− 692.000i	− 0.0937880i	−0.998900	−	0.0468940i	$-0.985068\pi$
$379$	− 692.000i	− 0.0937880i	0.998900	−	0.0468940i	$-0.0149323\pi$
$380$	0	0
$381$	− 1776.00i	− 0.238812i
$382$	0	0
$383$	−8064.00	−1.07585	−0.537926	−	0.842992i	$-0.680792\pi$
$383$	−8064.00	−1.07585	−0.537926	+	0.842992i	$0.680792\pi$
$384$	0	0
$385$	−5184.00	−0.686237
$386$	0	0
$387$	4068.00i	0.534336i
$388$	0	0
$389$	− 12654.0i	− 1.64931i	−0.565633	−	0.824657i	$-0.691368\pi$
$389$	− 12654.0i	− 1.64931i	0.565633	−	0.824657i	$-0.308632\pi$
$390$	0	0
$391$	−1296.00	−0.167625
$392$	0	0
$393$	5724.00	0.734701
$394$	0	0
$395$	9216.00i	1.17394i
$396$	0	0
$397$	− 106.000i	− 0.0134005i	−0.999978	−	0.00670024i	$-0.997867\pi$
$397$	− 106.000i	− 0.0134005i	0.999978	−	0.00670024i	$-0.00213277\pi$
$398$	0	0
$399$	−2400.00	−0.301129
$400$	0	0
$401$	−4014.00	−0.499874	−0.249937	−	0.968262i	$-0.580410\pi$
$401$	−4014.00	−0.499874	−0.249937	+	0.968262i	$0.580410\pi$
$402$	0	0
$403$	160.000i	0.0197771i
$404$	0	0
$405$	1458.00i	0.178885i
$406$	0	0
$407$	8136.00	0.990876
$408$	0	0
$409$	−3914.00	−0.473190	−0.236595	−	0.971608i	$-0.576032\pi$
$409$	−3914.00	−0.473190	−0.236595	+	0.971608i	$0.576032\pi$
$410$	0	0
$411$	− 2862.00i	− 0.343484i
$412$	0	0
$413$	− 5472.00i	− 0.651960i
$414$	0	0
$415$	21384.0	2.52940
$416$	0	0
$417$	7692.00	0.903307
$418$	0	0
$419$	4428.00i	0.516282i	0.966107	+	0.258141i	$0.0831098\pi$
$419$	4428.00i	0.516282i	−0.966107	+	0.258141i	$0.916890\pi$
$420$	0	0
$421$	15490.0i	1.79320i	0.442843	+	0.896599i	$0.353970\pi$
$421$	15490.0i	1.79320i	−0.442843	+	0.896599i	$0.646030\pi$
$422$	0	0
$423$	−3888.00	−0.446906
$424$	0	0
$425$	−3582.00	−0.408829
$426$	0	0
$427$	− 3376.00i	− 0.382614i
$428$	0	0
$429$	− 1080.00i	− 0.121545i
$430$	0	0
$431$	6768.00	0.756388	0.378194	−	0.925726i	$-0.376545\pi$
$431$	6768.00	0.756388	0.378194	+	0.925726i	$0.376545\pi$
$432$	0	0
$433$	1298.00	0.144060	0.0720299	−	0.997402i	$-0.477052\pi$
$433$	1298.00	0.144060	0.0720299	+	0.997402i	$0.477052\pi$
$434$	0	0
$435$	12636.0i	1.39276i
$436$	0	0
$437$	7200.00i	0.788153i
$438$	0	0
$439$	2248.00	0.244399	0.122200	−	0.992506i	$-0.461005\pi$
$439$	2248.00	0.244399	0.122200	+	0.992506i	$0.461005\pi$
$440$	0	0
$441$	2511.00	0.271137
$442$	0	0
$443$	9612.00i	1.03088i	0.856926	+	0.515440i	$0.172372\pi$
$443$	9612.00i	1.03088i	−0.856926	+	0.515440i	$0.827628\pi$
$444$	0	0
$445$	11340.0i	1.20802i
$446$	0	0
$447$	−2214.00	−0.234270
$448$	0	0
$449$	162.000	0.0170273	0.00851364	−	0.999964i	$-0.497290\pi$
$449$	162.000	0.0170273	0.00851364	+	0.999964i	$0.497290\pi$
$450$	0	0
$451$	3240.00i	0.338283i
$452$	0	0
$453$	7320.00i	0.759213i
$454$	0	0
$455$	−1440.00	−0.148370
$456$	0	0
$457$	−1370.00	−0.140232	−0.0701159	−	0.997539i	$-0.522337\pi$
$457$	−1370.00	−0.140232	−0.0701159	+	0.997539i	$0.522337\pi$
$458$	0	0
$459$	− 486.000i	− 0.0494217i
$460$	0	0
$461$	− 15354.0i	− 1.55121i	−0.631220	−	0.775604i	$-0.717445\pi$
$461$	− 15354.0i	− 1.55121i	0.631220	−	0.775604i	$-0.282555\pi$
$462$	0	0
$463$	−13024.0	−1.30729	−0.653646	−	0.756800i	$-0.726762\pi$
$463$	−13024.0	−1.30729	−0.653646	+	0.756800i	$0.726762\pi$
$464$	0	0
$465$	864.000	0.0861657
$466$	0	0
$467$	− 14436.0i	− 1.43045i	−0.698896	−	0.715223i	$-0.746325\pi$
$467$	− 14436.0i	− 1.43045i	0.698896	−	0.715223i	$-0.253675\pi$
$468$	0	0
$469$	− 2656.00i	− 0.261498i
$470$	0	0
$471$	7662.00	0.749568
$472$	0	0
$473$	−16272.0	−1.58179
$474$	0	0
$475$	19900.0i	1.92226i
$476$	0	0
$477$	3726.00i	0.357656i
$478$	0	0
$479$	12096.0	1.15382	0.576911	−	0.816807i	$-0.304258\pi$
$479$	12096.0	1.15382	0.576911	+	0.816807i	$0.304258\pi$
$480$	0	0
$481$	2260.00	0.214235
$482$	0	0
$483$	1728.00i	0.162788i
$484$	0	0
$485$	− 18972.0i	− 1.77624i
$486$	0	0
$487$	−6056.00	−0.563498	−0.281749	−	0.959488i	$-0.590915\pi$
$487$	−6056.00	−0.563498	−0.281749	+	0.959488i	$0.590915\pi$
$488$	0	0
$489$	2460.00	0.227495
$490$	0	0
$491$	− 7524.00i	− 0.691555i	−0.938317	−	0.345777i	$-0.887615\pi$
$491$	− 7524.00i	− 0.691555i	0.938317	−	0.345777i	$-0.112385\pi$
$492$	0	0
$493$	− 4212.00i	− 0.384785i
$494$	0	0
$495$	−5832.00	−0.529553
$496$	0	0
$497$	−2880.00	−0.259931
$498$	0	0
$499$	5276.00i	0.473319i	0.971593	+	0.236660i	$0.0760526\pi$
$499$	5276.00i	0.473319i	−0.971593	+	0.236660i	$0.923947\pi$
$500$	0	0
$501$	− 5832.00i	− 0.520069i
$502$	0	0
$503$	−4968.00	−0.440382	−0.220191	−	0.975457i	$-0.570668\pi$
$503$	−4968.00	−0.440382	−0.220191	+	0.975457i	$0.570668\pi$
$504$	0	0
$505$	10044.0	0.885054
$506$	0	0
$507$	6291.00i	0.551071i
$508$	0	0
$509$	10998.0i	0.957717i	0.877892	+	0.478858i	$0.158949\pi$
$509$	10998.0i	0.957717i	−0.877892	+	0.478858i	$0.841051\pi$
$510$	0	0
$511$	208.000	0.0180066
$512$	0	0
$513$	−2700.00	−0.232374
$514$	0	0
$515$	− 144.000i	− 0.0123212i
$516$	0	0
$517$	− 15552.0i	− 1.32297i
$518$	0	0
$519$	3726.00	0.315131
$520$	0	0
$521$	8838.00	0.743186	0.371593	−	0.928396i	$-0.378812\pi$
$521$	8838.00	0.743186	0.371593	+	0.928396i	$0.378812\pi$
$522$	0	0
$523$	− 22436.0i	− 1.87583i	−0.346869	−	0.937914i	$-0.612755\pi$
$523$	− 22436.0i	− 1.87583i	0.346869	−	0.937914i	$-0.387245\pi$
$524$	0	0
$525$	4776.00i	0.397032i
$526$	0	0
$527$	−288.000	−0.0238055
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	− 6156.00i	− 0.503103i
$532$	0	0
$533$	900.000i	0.0731395i
$534$	0	0
$535$	31752.0	2.56590
$536$	0	0
$537$	−3348.00	−0.269044
$538$	0	0
$539$	10044.0i	0.802645i
$540$	0	0
$541$	− 4762.00i	− 0.378437i	−0.981935	−	0.189218i	$-0.939405\pi$
$541$	− 4762.00i	− 0.378437i	0.981935	−	0.189218i	$-0.0605954\pi$
$542$	0	0
$543$	3210.00	0.253691
$544$	0	0
$545$	−29196.0	−2.29471
$546$	0	0
$547$	− 6004.00i	− 0.469310i	−0.972079	−	0.234655i	$-0.924604\pi$
$547$	− 6004.00i	− 0.469310i	0.972079	−	0.234655i	$-0.0753960\pi$
$548$	0	0
$549$	− 3798.00i	− 0.295254i
$550$	0	0
$551$	−23400.0	−1.80921
$552$	0	0
$553$	−4096.00	−0.314972
$554$	0	0
$555$	− 12204.0i	− 0.933389i
$556$	0	0
$557$	− 5274.00i	− 0.401197i	−0.979674	−	0.200598i	$-0.935711\pi$
$557$	− 5274.00i	− 0.401197i	0.979674	−	0.200598i	$-0.0642886\pi$
$558$	0	0
$559$	−4520.00	−0.341996
$560$	0	0
$561$	1944.00	0.146303
$562$	0	0
$563$	− 12420.0i	− 0.929735i	−0.885380	−	0.464867i	$-0.846102\pi$
$563$	− 12420.0i	− 0.929735i	0.885380	−	0.464867i	$-0.153898\pi$
$564$	0	0
$565$	− 20412.0i	− 1.51989i
$566$	0	0
$567$	−648.000	−0.0479955
$568$	0	0
$569$	21366.0	1.57418	0.787091	−	0.616837i	$-0.211586\pi$
$569$	21366.0	1.57418	0.787091	+	0.616837i	$0.211586\pi$
$570$	0	0
$571$	− 21140.0i	− 1.54935i	−0.632357	−	0.774677i	$-0.717912\pi$
$571$	− 21140.0i	− 1.54935i	0.632357	−	0.774677i	$-0.282088\pi$
$572$	0	0
$573$	− 1728.00i	− 0.125983i
$574$	0	0
$575$	14328.0	1.03916
$576$	0	0
$577$	3266.00	0.235642	0.117821	−	0.993035i	$-0.462409\pi$
$577$	3266.00	0.235642	0.117821	+	0.993035i	$0.462409\pi$
$578$	0	0
$579$	− 4026.00i	− 0.288972i
$580$	0	0
$581$	9504.00i	0.678644i
$582$	0	0
$583$	−14904.0	−1.05877
$584$	0	0
$585$	−1620.00	−0.114494
$586$	0	0
$587$	− 17028.0i	− 1.19731i	−0.801007	−	0.598655i	$-0.795702\pi$
$587$	− 17028.0i	− 1.19731i	0.801007	−	0.598655i	$-0.204298\pi$
$588$	0	0
$589$	1600.00i	0.111930i
$590$	0	0
$591$	4266.00	0.296920
$592$	0	0
$593$	9522.00	0.659396	0.329698	−	0.944086i	$-0.393053\pi$
$593$	9522.00	0.659396	0.329698	+	0.944086i	$0.393053\pi$
$594$	0	0
$595$	− 2592.00i	− 0.178591i
$596$	0	0
$597$	− 2616.00i	− 0.179340i
$598$	0	0
$599$	10296.0	0.702309	0.351155	−	0.936318i	$-0.385789\pi$
$599$	10296.0	0.702309	0.351155	+	0.936318i	$0.385789\pi$
$600$	0	0
$601$	3382.00	0.229542	0.114771	−	0.993392i	$-0.463387\pi$
$601$	3382.00	0.229542	0.114771	+	0.993392i	$0.463387\pi$
$602$	0	0
$603$	− 2988.00i	− 0.201792i
$604$	0	0
$605$	630.000i	0.0423358i
$606$	0	0
$607$	−20656.0	−1.38122	−0.690611	−	0.723227i	$-0.742658\pi$
$607$	−20656.0	−1.38122	−0.690611	+	0.723227i	$0.742658\pi$
$608$	0	0
$609$	−5616.00	−0.373681
$610$	0	0
$611$	− 4320.00i	− 0.286037i
$612$	0	0
$613$	22114.0i	1.45706i	0.685015	+	0.728529i	$0.259795\pi$
$613$	22114.0i	1.45706i	−0.685015	+	0.728529i	$0.740205\pi$
$614$	0	0
$615$	4860.00	0.318657
$616$	0	0
$617$	−19962.0	−1.30250	−0.651248	−	0.758865i	$-0.725754\pi$
$617$	−19962.0	−1.30250	−0.651248	+	0.758865i	$0.725754\pi$
$618$	0	0
$619$	604.000i	0.0392194i	0.999808	+	0.0196097i	$0.00624236\pi$
$619$	604.000i	0.0392194i	−0.999808	+	0.0196097i	$0.993758\pi$
$620$	0	0
$621$	1944.00i	0.125620i
$622$	0	0
$623$	−5040.00	−0.324115
$624$	0	0
$625$	−899.000	−0.0575360
$626$	0	0
$627$	− 10800.0i	− 0.687895i
$628$	0	0
$629$	4068.00i	0.257872i
$630$	0	0
$631$	−152.000	−0.00958958	−0.00479479	−	0.999989i	$-0.501526\pi$
$631$	−152.000	−0.00958958	−0.00479479	+	0.999989i	$0.501526\pi$
$632$	0	0
$633$	−4020.00	−0.252418
$634$	0	0
$635$	− 10656.0i	− 0.665938i
$636$	0	0
$637$	2790.00i	0.173538i
$638$	0	0
$639$	−3240.00	−0.200583
$640$	0	0
$641$	4194.00	0.258429	0.129215	−	0.991617i	$-0.458754\pi$
$641$	4194.00	0.258429	0.129215	+	0.991617i	$0.458754\pi$
$642$	0	0
$643$	− 7252.00i	− 0.444776i	−0.974958	−	0.222388i	$-0.928615\pi$
$643$	− 7252.00i	− 0.444776i	0.974958	−	0.222388i	$-0.0713852\pi$
$644$	0	0
$645$	24408.0i	1.49002i
$646$	0	0
$647$	6696.00	0.406873	0.203437	−	0.979088i	$-0.434789\pi$
$647$	6696.00	0.406873	0.203437	+	0.979088i	$0.434789\pi$
$648$	0	0
$649$	24624.0	1.48933
$650$	0	0
$651$	384.000i	0.0231185i
$652$	0	0
$653$	28422.0i	1.70328i	0.524131	+	0.851638i	$0.324390\pi$
$653$	28422.0i	1.70328i	−0.524131	+	0.851638i	$0.675610\pi$
$654$	0	0
$655$	34344.0	2.04875
$656$	0	0
$657$	234.000	0.0138953
$658$	0	0
$659$	− 19908.0i	− 1.17679i	−0.808573	−	0.588396i	$-0.799760\pi$
$659$	− 19908.0i	− 1.17679i	0.808573	−	0.588396i	$-0.200240\pi$
$660$	0	0
$661$	− 14318.0i	− 0.842520i	−0.906940	−	0.421260i	$-0.861588\pi$
$661$	− 14318.0i	− 0.842520i	0.906940	−	0.421260i	$-0.138412\pi$
$662$	0	0
$663$	540.000	0.0316318
$664$	0	0
$665$	−14400.0	−0.839711
$666$	0	0
$667$	16848.0i	0.978047i
$668$	0	0
$669$	14640.0i	0.846061i
$670$	0	0
$671$	15192.0	0.874040
$672$	0	0
$673$	30050.0	1.72116	0.860581	−	0.509313i	$-0.170101\pi$
$673$	30050.0	1.72116	0.860581	+	0.509313i	$0.170101\pi$
$674$	0	0
$675$	5373.00i	0.306381i
$676$	0	0
$677$	− 22158.0i	− 1.25790i	−0.777444	−	0.628952i	$-0.783484\pi$
$677$	− 22158.0i	− 1.25790i	0.777444	−	0.628952i	$-0.216516\pi$
$678$	0	0
$679$	8432.00	0.476569
$680$	0	0
$681$	−8100.00	−0.455790
$682$	0	0
$683$	3132.00i	0.175465i	0.996144	+	0.0877325i	$0.0279621\pi$
$683$	3132.00i	0.175465i	−0.996144	+	0.0877325i	$0.972038\pi$
$684$	0	0
$685$	− 17172.0i	− 0.957822i
$686$	0	0
$687$	762.000	0.0423175
$688$	0	0
$689$	−4140.00	−0.228914
$690$	0	0
$691$	− 20932.0i	− 1.15237i	−0.817318	−	0.576187i	$-0.804540\pi$
$691$	− 20932.0i	− 1.15237i	0.817318	−	0.576187i	$-0.195460\pi$
$692$	0	0
$693$	− 2592.00i	− 0.142081i
$694$	0	0
$695$	46152.0	2.51891
$696$	0	0
$697$	−1620.00	−0.0880371
$698$	0	0
$699$	− 13230.0i	− 0.715886i
$700$	0	0
$701$	− 21834.0i	− 1.17640i	−0.808714	−	0.588202i	$-0.799836\pi$
$701$	− 21834.0i	− 1.17640i	0.808714	−	0.588202i	$-0.200164\pi$
$702$	0	0
$703$	22600.0	1.21248
$704$	0	0
$705$	−23328.0	−1.24622
$706$	0	0
$707$	4464.00i	0.237463i
$708$	0	0
$709$	− 12446.0i	− 0.659266i	−0.944109	−	0.329633i	$-0.893075\pi$
$709$	− 12446.0i	− 0.659266i	0.944109	−	0.329633i	$-0.106925\pi$
$710$	0	0
$711$	−4608.00	−0.243057
$712$	0	0
$713$	1152.00	0.0605088
$714$	0	0
$715$	− 6480.00i	− 0.338935i
$716$	0	0
$717$	− 11664.0i	− 0.607531i
$718$	0	0
$719$	−12528.0	−0.649813	−0.324907	−	0.945746i	$-0.605333\pi$
$719$	−12528.0	−0.649813	−0.324907	+	0.945746i	$0.605333\pi$
$720$	0	0
$721$	64.0000	0.00330580
$722$	0	0
$723$	15414.0i	0.792881i
$724$	0	0
$725$	46566.0i	2.38540i
$726$	0	0
$727$	−11576.0	−0.590550	−0.295275	−	0.955412i	$-0.595411\pi$
$727$	−11576.0	−0.590550	−0.295275	+	0.955412i	$0.595411\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 8136.00i	− 0.411656i
$732$	0	0
$733$	− 29338.0i	− 1.47834i	−0.673519	−	0.739170i	$-0.735218\pi$
$733$	− 29338.0i	− 1.47834i	0.673519	−	0.739170i	$-0.264782\pi$
$734$	0	0
$735$	15066.0	0.756079
$736$	0	0
$737$	11952.0	0.597364
$738$	0	0
$739$	2540.00i	0.126435i	0.998000	+	0.0632175i	$0.0201362\pi$
$739$	2540.00i	0.126435i	−0.998000	+	0.0632175i	$0.979864\pi$
$740$	0	0
$741$	− 3000.00i	− 0.148728i
$742$	0	0
$743$	18792.0	0.927876	0.463938	−	0.885868i	$-0.346436\pi$
$743$	18792.0	0.927876	0.463938	+	0.885868i	$0.346436\pi$
$744$	0	0
$745$	−13284.0	−0.653273
$746$	0	0
$747$	10692.0i	0.523695i
$748$	0	0
$749$	14112.0i	0.688440i
$750$	0	0
$751$	4832.00	0.234783	0.117392	−	0.993086i	$-0.462547\pi$
$751$	4832.00	0.234783	0.117392	+	0.993086i	$0.462547\pi$
$752$	0	0
$753$	14364.0	0.695157
$754$	0	0
$755$	43920.0i	2.11710i
$756$	0	0
$757$	20818.0i	0.999529i	0.866161	+	0.499764i	$0.166580\pi$
$757$	20818.0i	0.999529i	−0.866161	+	0.499764i	$0.833420\pi$
$758$	0	0
$759$	−7776.00	−0.371872
$760$	0	0
$761$	−12042.0	−0.573617	−0.286808	−	0.957988i	$-0.592594\pi$
$761$	−12042.0	−0.573617	−0.286808	+	0.957988i	$0.592594\pi$
$762$	0	0
$763$	− 12976.0i	− 0.615679i
$764$	0	0
$765$	− 2916.00i	− 0.137815i
$766$	0	0
$767$	6840.00	0.322005
$768$	0	0
$769$	13058.0	0.612332	0.306166	−	0.951978i	$-0.400954\pi$
$769$	13058.0	0.612332	0.306166	+	0.951978i	$0.400954\pi$
$770$	0	0
$771$	− 17658.0i	− 0.824821i
$772$	0	0
$773$	11826.0i	0.550261i	0.961407	+	0.275130i	$0.0887210\pi$
$773$	11826.0i	0.550261i	−0.961407	+	0.275130i	$0.911279\pi$
$774$	0	0
$775$	3184.00	0.147578
$776$	0	0
$777$	5424.00	0.250431
$778$	0	0
$779$	9000.00i	0.413939i
$780$	0	0
$781$	− 12960.0i	− 0.593784i
$782$	0	0
$783$	−6318.00	−0.288361
$784$	0	0
$785$	45972.0	2.09021
$786$	0	0
$787$	11996.0i	0.543343i	0.962390	+	0.271672i	$0.0875765\pi$
$787$	11996.0i	0.543343i	−0.962390	+	0.271672i	$0.912424\pi$
$788$	0	0
$789$	− 6696.00i	− 0.302134i
$790$	0	0
$791$	9072.00	0.407792
$792$	0	0
$793$	4220.00	0.188974
$794$	0	0
$795$	22356.0i	0.997340i
$796$	0	0
$797$	6966.00i	0.309596i	0.987946	+	0.154798i	$0.0494727\pi$
$797$	6966.00i	0.309596i	−0.987946	+	0.154798i	$0.950527\pi$
$798$	0	0
$799$	7776.00	0.344299
$800$	0	0
$801$	−5670.00	−0.250112
$802$	0	0
$803$	936.000i	0.0411342i
$804$	0	0
$805$	10368.0i	0.453943i
$806$	0	0
$807$	1998.00	0.0871536
$808$	0	0
$809$	40806.0	1.77338	0.886689	−	0.462367i	$-0.153000\pi$
$809$	40806.0	1.77338	0.886689	+	0.462367i	$0.153000\pi$
$810$	0	0
$811$	17980.0i	0.778500i	0.921132	+	0.389250i	$0.127266\pi$
$811$	17980.0i	0.778500i	−0.921132	+	0.389250i	$0.872734\pi$
$812$	0	0
$813$	− 16608.0i	− 0.716443i
$814$	0	0
$815$	14760.0	0.634381
$816$	0	0
$817$	−45200.0	−1.93555
$818$	0	0
$819$	− 720.000i	− 0.0307190i
$820$	0	0
$821$	12834.0i	0.545566i	0.962076	+	0.272783i	$0.0879441\pi$
$821$	12834.0i	0.545566i	−0.962076	+	0.272783i	$0.912056\pi$
$822$	0	0
$823$	37864.0	1.60371	0.801857	−	0.597516i	$-0.203846\pi$
$823$	37864.0	1.60371	0.801857	+	0.597516i	$0.203846\pi$
$824$	0	0
$825$	−21492.0	−0.906976
$826$	0	0
$827$	− 42516.0i	− 1.78770i	−0.448368	−	0.893849i	$-0.647995\pi$
$827$	− 42516.0i	− 1.78770i	0.448368	−	0.893849i	$-0.352005\pi$
$828$	0	0
$829$	45638.0i	1.91203i	0.293317	+	0.956015i	$0.405241\pi$
$829$	45638.0i	1.91203i	−0.293317	+	0.956015i	$0.594759\pi$
$830$	0	0
$831$	6378.00	0.266246
$832$	0	0
$833$	−5022.00	−0.208886
$834$	0	0
$835$	− 34992.0i	− 1.45024i
$836$	0	0
$837$	432.000i	0.0178400i
$838$	0	0
$839$	−17496.0	−0.719939	−0.359970	−	0.932964i	$-0.617213\pi$
$839$	−17496.0	−0.719939	−0.359970	+	0.932964i	$0.617213\pi$
$840$	0	0
$841$	−30367.0	−1.24511
$842$	0	0
$843$	8802.00i	0.359617i
$844$	0	0
$845$	37746.0i	1.53669i
$846$	0	0
$847$	−280.000	−0.0113588
$848$	0	0
$849$	6108.00	0.246909
$850$	0	0
$851$	− 16272.0i	− 0.655461i
$852$	0	0
$853$	− 32174.0i	− 1.29146i	−0.763565	−	0.645731i	$-0.776553\pi$
$853$	− 32174.0i	− 1.29146i	0.763565	−	0.645731i	$-0.223447\pi$
$854$	0	0
$855$	−16200.0	−0.647986
$856$	0	0
$857$	38934.0	1.55188	0.775939	−	0.630807i	$-0.217276\pi$
$857$	38934.0	1.55188	0.775939	+	0.630807i	$0.217276\pi$
$858$	0	0
$859$	− 29780.0i	− 1.18286i	−0.806355	−	0.591432i	$-0.798563\pi$
$859$	− 29780.0i	− 1.18286i	0.806355	−	0.591432i	$-0.201437\pi$
$860$	0	0
$861$	2160.00i	0.0854966i
$862$	0	0
$863$	−48096.0	−1.89711	−0.948556	−	0.316611i	$-0.897455\pi$
$863$	−48096.0	−1.89711	−0.948556	+	0.316611i	$0.897455\pi$
$864$	0	0
$865$	22356.0	0.878759
$866$	0	0
$867$	− 13767.0i	− 0.539275i
$868$	0	0
$869$	− 18432.0i	− 0.719520i
$870$	0	0
$871$	3320.00	0.129155
$872$	0	0
$873$	9486.00	0.367758
$874$	0	0
$875$	10656.0i	0.411701i
$876$	0	0
$877$	21302.0i	0.820202i	0.912040	+	0.410101i	$0.134507\pi$
$877$	21302.0i	0.820202i	−0.912040	+	0.410101i	$0.865493\pi$
$878$	0	0
$879$	6858.00	0.263157
$880$	0	0
$881$	−7470.00	−0.285665	−0.142832	−	0.989747i	$-0.545621\pi$
$881$	−7470.00	−0.285665	−0.142832	+	0.989747i	$0.545621\pi$
$882$	0	0
$883$	764.000i	0.0291174i	0.999894	+	0.0145587i	$0.00463434\pi$
$883$	764.000i	0.0291174i	−0.999894	+	0.0145587i	$0.995366\pi$
$884$	0	0
$885$	− 36936.0i	− 1.40293i
$886$	0	0
$887$	−32328.0	−1.22375	−0.611876	−	0.790954i	$-0.709585\pi$
$887$	−32328.0	−1.22375	−0.611876	+	0.790954i	$0.709585\pi$
$888$	0	0
$889$	4736.00	0.178673
$890$	0	0
$891$	− 2916.00i	− 0.109640i
$892$	0	0
$893$	− 43200.0i	− 1.61885i
$894$	0	0
$895$	−20088.0	−0.750243
$896$	0	0
$897$	−2160.00	−0.0804017
$898$	0	0
$899$	3744.00i	0.138898i
$900$	0	0
$901$	− 7452.00i	− 0.275541i
$902$	0	0
$903$	−10848.0	−0.399777
$904$	0	0
$905$	19260.0	0.707430
$906$	0	0
$907$	36316.0i	1.32950i	0.747068	+	0.664748i	$0.231461\pi$
$907$	36316.0i	1.32950i	−0.747068	+	0.664748i	$0.768539\pi$
$908$	0	0
$909$	5022.00i	0.183244i
$910$	0	0
$911$	−13392.0	−0.487044	−0.243522	−	0.969895i	$-0.578303\pi$
$911$	−13392.0	−0.487044	−0.243522	+	0.969895i	$0.578303\pi$
$912$	0	0
$913$	−42768.0	−1.55029
$914$	0	0
$915$	− 22788.0i	− 0.823331i
$916$	0	0
$917$	15264.0i	0.549686i
$918$	0	0
$919$	−38072.0	−1.36657	−0.683286	−	0.730151i	$-0.739450\pi$
$919$	−38072.0	−1.36657	−0.683286	+	0.730151i	$0.739450\pi$
$920$	0	0
$921$	−3732.00	−0.133522
$922$	0	0
$923$	− 3600.00i	− 0.128381i
$924$	0	0
$925$	− 44974.0i	− 1.59863i
$926$	0	0
$927$	72.0000	0.00255101
$928$	0	0
$929$	−12798.0	−0.451979	−0.225990	−	0.974130i	$-0.572562\pi$
$929$	−12798.0	−0.451979	−0.225990	+	0.974130i	$0.572562\pi$
$930$	0	0
$931$	27900.0i	0.982154i
$932$	0	0
$933$	− 3672.00i	− 0.128849i
$934$	0	0
$935$	11664.0	0.407972
$936$	0	0
$937$	−34874.0	−1.21588	−0.607942	−	0.793981i	$-0.708005\pi$
$937$	−34874.0	−1.21588	−0.607942	+	0.793981i	$0.708005\pi$
$938$	0	0
$939$	− 5694.00i	− 0.197888i
$940$	0	0
$941$	17190.0i	0.595513i	0.954642	+	0.297757i	$0.0962384\pi$
$941$	17190.0i	0.595513i	−0.954642	+	0.297757i	$0.903762\pi$
$942$	0	0
$943$	6480.00	0.223773
$944$	0	0
$945$	−3888.00	−0.133838
$946$	0	0
$947$	40284.0i	1.38232i	0.722703	+	0.691158i	$0.242899\pi$
$947$	40284.0i	1.38232i	−0.722703	+	0.691158i	$0.757101\pi$
$948$	0	0
$949$	260.000i	0.00889353i
$950$	0	0
$951$	27486.0	0.937218
$952$	0	0
$953$	−15498.0	−0.526789	−0.263394	−	0.964688i	$-0.584842\pi$
$953$	−15498.0	−0.526789	−0.263394	+	0.964688i	$0.584842\pi$
$954$	0	0
$955$	− 10368.0i	− 0.351310i
$956$	0	0
$957$	− 25272.0i	− 0.853634i
$958$	0	0
$959$	7632.00	0.256987
$960$	0	0
$961$	−29535.0	−0.991407
$962$	0	0
$963$	15876.0i	0.531253i
$964$	0	0
$965$	− 24156.0i	− 0.805813i
$966$	0	0
$967$	−37160.0	−1.23577	−0.617883	−	0.786270i	$-0.712009\pi$
$967$	−37160.0	−1.23577	−0.617883	+	0.786270i	$0.712009\pi$
$968$	0	0
$969$	5400.00	0.179023
$970$	0	0
$971$	− 18468.0i	− 0.610367i	−0.952294	−	0.305183i	$-0.901282\pi$
$971$	− 18468.0i	− 0.610367i	0.952294	−	0.305183i	$-0.0987178\pi$
$972$	0	0
$973$	20512.0i	0.675832i
$974$	0	0
$975$	−5970.00	−0.196095
$976$	0	0
$977$	10386.0	0.340100	0.170050	−	0.985435i	$-0.445607\pi$
$977$	10386.0	0.340100	0.170050	+	0.985435i	$0.445607\pi$
$978$	0	0
$979$	− 22680.0i	− 0.740404i
$980$	0	0
$981$	− 14598.0i	− 0.475105i
$982$	0	0
$983$	−44136.0	−1.43206	−0.716032	−	0.698067i	$-0.754044\pi$
$983$	−44136.0	−1.43206	−0.716032	+	0.698067i	$0.754044\pi$
$984$	0	0
$985$	25596.0	0.827976
$986$	0	0
$987$	− 10368.0i	− 0.334364i
$988$	0	0
$989$	32544.0i	1.04635i
$990$	0	0
$991$	−28432.0	−0.911375	−0.455687	−	0.890140i	$-0.650606\pi$
$991$	−28432.0	−0.911375	−0.455687	+	0.890140i	$0.650606\pi$
$992$	0	0
$993$	−13044.0	−0.416857
$994$	0	0
$995$	− 15696.0i	− 0.500097i
$996$	0	0
$997$	39778.0i	1.26357i	0.775143	+	0.631786i	$0.217678\pi$
$997$	39778.0i	1.26357i	−0.775143	+	0.631786i	$0.782322\pi$
$998$	0	0
$999$	6102.00	0.193252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.d.g.385.2		2
4.3	odd	2		768.4.d.j.385.1		2
8.3	odd	2		768.4.d.j.385.2		2
8.5	even	2	inner	768.4.d.g.385.1		2
16.3	odd	4		192.4.a.l.1.1		1
16.5	even	4		12.4.a.a.1.1	✓	1
16.11	odd	4		48.4.a.a.1.1		1
16.13	even	4		192.4.a.f.1.1		1
48.5	odd	4		36.4.a.a.1.1		1
48.11	even	4		144.4.a.g.1.1		1
48.29	odd	4		576.4.a.b.1.1		1
48.35	even	4		576.4.a.a.1.1		1
80.27	even	4		1200.4.f.d.49.2		2
80.37	odd	4		300.4.d.e.49.1		2
80.43	even	4		1200.4.f.d.49.1		2
80.53	odd	4		300.4.d.e.49.2		2
80.59	odd	4		1200.4.a.be.1.1		1
80.69	even	4		300.4.a.b.1.1		1
112.5	odd	12		588.4.i.e.361.1		2
112.27	even	4		2352.4.a.bk.1.1		1
112.37	even	12		588.4.i.d.361.1		2
112.53	even	12		588.4.i.d.373.1		2
112.69	odd	4		588.4.a.c.1.1		1
112.101	odd	12		588.4.i.e.373.1		2
144.5	odd	12		324.4.e.a.217.1		2
144.85	even	12		324.4.e.h.217.1		2
144.101	odd	12		324.4.e.a.109.1		2
144.133	even	12		324.4.e.h.109.1		2
176.21	odd	4		1452.4.a.d.1.1		1
208.5	odd	4		2028.4.b.c.337.2		2
208.21	odd	4		2028.4.b.c.337.1		2
208.181	even	4		2028.4.a.c.1.1		1
240.53	even	4		900.4.d.c.649.1		2
240.149	odd	4		900.4.a.g.1.1		1
240.197	even	4		900.4.d.c.649.2		2
336.5	even	12		1764.4.k.o.361.1		2
336.53	odd	12		1764.4.k.b.1549.1		2
336.101	even	12		1764.4.k.o.1549.1		2
336.149	odd	12		1764.4.k.b.361.1		2
336.293	even	4		1764.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.4.a.a.1.1	✓	1	16.5	even	4
36.4.a.a.1.1		1	48.5	odd	4
48.4.a.a.1.1		1	16.11	odd	4
144.4.a.g.1.1		1	48.11	even	4
192.4.a.f.1.1		1	16.13	even	4
192.4.a.l.1.1		1	16.3	odd	4
300.4.a.b.1.1		1	80.69	even	4
300.4.d.e.49.1		2	80.37	odd	4
300.4.d.e.49.2		2	80.53	odd	4
324.4.e.a.109.1		2	144.101	odd	12
324.4.e.a.217.1		2	144.5	odd	12
324.4.e.h.109.1		2	144.133	even	12
324.4.e.h.217.1		2	144.85	even	12
576.4.a.a.1.1		1	48.35	even	4
576.4.a.b.1.1		1	48.29	odd	4
588.4.a.c.1.1		1	112.69	odd	4
588.4.i.d.361.1		2	112.37	even	12
588.4.i.d.373.1		2	112.53	even	12
588.4.i.e.361.1		2	112.5	odd	12
588.4.i.e.373.1		2	112.101	odd	12
768.4.d.g.385.1		2	8.5	even	2	inner
768.4.d.g.385.2		2	1.1	even	1	trivial
768.4.d.j.385.1		2	4.3	odd	2
768.4.d.j.385.2		2	8.3	odd	2
900.4.a.g.1.1		1	240.149	odd	4
900.4.d.c.649.1		2	240.53	even	4
900.4.d.c.649.2		2	240.197	even	4
1200.4.a.be.1.1		1	80.59	odd	4
1200.4.f.d.49.1		2	80.43	even	4
1200.4.f.d.49.2		2	80.27	even	4
1452.4.a.d.1.1		1	176.21	odd	4
1764.4.a.b.1.1		1	336.293	even	4
1764.4.k.b.361.1		2	336.149	odd	12
1764.4.k.b.1549.1		2	336.53	odd	12
1764.4.k.o.361.1		2	336.5	even	12
1764.4.k.o.1549.1		2	336.101	even	12
2028.4.a.c.1.1		1	208.181	even	4
2028.4.b.c.337.1		2	208.21	odd	4
2028.4.b.c.337.2		2	208.5	odd	4
2352.4.a.bk.1.1		1	112.27	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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +18.0000i q^{5} -8.00000 q^{7} -9.00000 q^{9} -36.0000i q^{11} -10.0000i q^{13} -54.0000 q^{15} +18.0000 q^{17} -100.000i q^{19} -24.0000i q^{21} -72.0000 q^{23} -199.000 q^{25} -27.0000i q^{27} -234.000i q^{29} -16.0000 q^{31} +108.000 q^{33} -144.000i q^{35} +226.000i q^{37} +30.0000 q^{39} -90.0000 q^{41} -452.000i q^{43} -162.000i q^{45} +432.000 q^{47} -279.000 q^{49} +54.0000i q^{51} -414.000i q^{53} +648.000 q^{55} +300.000 q^{57} +684.000i q^{59} +422.000i q^{61} +72.0000 q^{63} +180.000 q^{65} +332.000i q^{67} -216.000i q^{69} +360.000 q^{71} -26.0000 q^{73} -597.000i q^{75} +288.000i q^{77} +512.000 q^{79} +81.0000 q^{81} -1188.00i q^{83} +324.000i q^{85} +702.000 q^{87} +630.000 q^{89} +80.0000i q^{91} -48.0000i q^{93} +1800.00 q^{95} -1054.00 q^{97} +324.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{7} - 18 q^{9} - 108 q^{15} + 36 q^{17} - 144 q^{23} - 398 q^{25} - 32 q^{31} + 216 q^{33} + 60 q^{39} - 180 q^{41} + 864 q^{47} - 558 q^{49} + 1296 q^{55} + 600 q^{57} + 144 q^{63} + 360 q^{65}+ \cdots - 2108 q^{97}+O(q^{100}) \) 2 * q - 16 * q^7 - 18 * q^9 - 108 * q^15 + 36 * q^17 - 144 * q^23 - 398 * q^25 - 32 * q^31 + 216 * q^33 + 60 * q^39 - 180 * q^41 + 864 * q^47 - 558 * q^49 + 1296 * q^55 + 600 * q^57 + 144 * q^63 + 360 * q^65 + 720 * q^71 - 52 * q^73 + 1024 * q^79 + 162 * q^81 + 1404 * q^87 + 1260 * q^89 + 3600 * q^95 - 2108 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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