LMFDB - Embedded newform 768.4.d.n.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:	$1$
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 6)
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.n.385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +6.00000i q^{5} +16.0000 q^{7} -9.00000 q^{9} +12.0000i q^{11} -38.0000i q^{13} -18.0000 q^{15} -126.000 q^{17} -20.0000i q^{19} +48.0000i q^{21} -168.000 q^{23} +89.0000 q^{25} -27.0000i q^{27} -30.0000i q^{29} -88.0000 q^{31} -36.0000 q^{33} +96.0000i q^{35} +254.000i q^{37} +114.000 q^{39} -42.0000 q^{41} -52.0000i q^{43} -54.0000i q^{45} -96.0000 q^{47} -87.0000 q^{49} -378.000i q^{51} +198.000i q^{53} -72.0000 q^{55} +60.0000 q^{57} -660.000i q^{59} +538.000i q^{61} -144.000 q^{63} +228.000 q^{65} -884.000i q^{67} -504.000i q^{69} -792.000 q^{71} -218.000 q^{73} +267.000i q^{75} +192.000i q^{77} -520.000 q^{79} +81.0000 q^{81} +492.000i q^{83} -756.000i q^{85} +90.0000 q^{87} -810.000 q^{89} -608.000i q^{91} -264.000i q^{93} +120.000 q^{95} +1154.00 q^{97} -108.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{7} - 18 q^{9} - 36 q^{15} - 252 q^{17} - 336 q^{23} + 178 q^{25} - 176 q^{31} - 72 q^{33} + 228 q^{39} - 84 q^{41} - 192 q^{47} - 174 q^{49} - 144 q^{55} + 120 q^{57} - 288 q^{63} + 456 q^{65}+ \cdots + 2308 q^{97}+O(q^{100}) $ 2 * q + 32 * q^7 - 18 * q^9 - 36 * q^15 - 252 * q^17 - 336 * q^23 + 178 * q^25 - 176 * q^31 - 72 * q^33 + 228 * q^39 - 84 * q^41 - 192 * q^47 - 174 * q^49 - 144 * q^55 + 120 * q^57 - 288 * q^63 + 456 * q^65 - 1584 * q^71 - 436 * q^73 - 1040 * q^79 + 162 * q^81 + 180 * q^87 - 1620 * q^89 + 240 * q^95 + 2308 * 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
$5$	6.00000i	0.536656i	0.963328	+	0.268328i	$0.0864711\pi$
$5$	6.00000i	0.536656i	−0.963328	+	0.268328i	$0.913529\pi$
$6$	0	0
$7$	16.0000	0.863919	0.431959	−	0.901893i	$-0.357822\pi$
$7$	16.0000	0.863919	0.431959	+	0.901893i	$0.357822\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	12.0000i	0.328921i	0.986384	+	0.164461i	$0.0525884\pi$
$11$	12.0000i	0.328921i	−0.986384	+	0.164461i	$0.947412\pi$
$12$	0	0
$13$	− 38.0000i	− 0.810716i	−0.914158	−	0.405358i	$-0.867147\pi$
$13$	− 38.0000i	− 0.810716i	0.914158	−	0.405358i	$-0.132853\pi$
$14$	0	0
$15$	−18.0000	−0.309839
$16$	0	0
$17$	−126.000	−1.79762	−0.898808	−	0.438342i	$-0.855566\pi$
$17$	−126.000	−1.79762	−0.898808	+	0.438342i	$0.855566\pi$
$18$	0	0
$19$	− 20.0000i	− 0.241490i	−0.992684	−	0.120745i	$-0.961472\pi$
$19$	− 20.0000i	− 0.241490i	0.992684	−	0.120745i	$-0.0385284\pi$
$20$	0	0
$21$	48.0000i	0.498784i
$22$	0	0
$23$	−168.000	−1.52306	−0.761531	−	0.648129i	$-0.775552\pi$
$23$	−168.000	−1.52306	−0.761531	+	0.648129i	$0.775552\pi$
$24$	0	0
$25$	89.0000	0.712000
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 30.0000i	− 0.192099i	−0.995377	−	0.0960493i	$-0.969379\pi$
$29$	− 30.0000i	− 0.192099i	0.995377	−	0.0960493i	$-0.0306207\pi$
$30$	0	0
$31$	−88.0000	−0.509847	−0.254924	−	0.966961i	$-0.582050\pi$
$31$	−88.0000	−0.509847	−0.254924	+	0.966961i	$0.582050\pi$
$32$	0	0
$33$	−36.0000	−0.189903
$34$	0	0
$35$	96.0000i	0.463627i
$36$	0	0
$37$	254.000i	1.12858i	0.825578	+	0.564288i	$0.190849\pi$
$37$	254.000i	1.12858i	−0.825578	+	0.564288i	$0.809151\pi$
$38$	0	0
$39$	114.000	0.468067
$40$	0	0
$41$	−42.0000	−0.159983	−0.0799914	−	0.996796i	$-0.525489\pi$
$41$	−42.0000	−0.159983	−0.0799914	+	0.996796i	$0.525489\pi$
$42$	0	0
$43$	− 52.0000i	− 0.184417i	−0.995740	−	0.0922084i	$-0.970607\pi$
$43$	− 52.0000i	− 0.184417i	0.995740	−	0.0922084i	$-0.0293926\pi$
$44$	0	0
$45$	− 54.0000i	− 0.178885i
$46$	0	0
$47$	−96.0000	−0.297937	−0.148969	−	0.988842i	$-0.547595\pi$
$47$	−96.0000	−0.297937	−0.148969	+	0.988842i	$0.547595\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	− 378.000i	− 1.03785i
$52$	0	0
$53$	198.000i	0.513158i	0.966523	+	0.256579i	$0.0825954\pi$
$53$	198.000i	0.513158i	−0.966523	+	0.256579i	$0.917405\pi$
$54$	0	0
$55$	−72.0000	−0.176518
$56$	0	0
$57$	60.0000	0.139424
$58$	0	0
$59$	− 660.000i	− 1.45635i	−0.685391	−	0.728175i	$-0.740369\pi$
$59$	− 660.000i	− 1.45635i	0.685391	−	0.728175i	$-0.259631\pi$
$60$	0	0
$61$	538.000i	1.12924i	0.825350	+	0.564622i	$0.190978\pi$
$61$	538.000i	1.12924i	−0.825350	+	0.564622i	$0.809022\pi$
$62$	0	0
$63$	−144.000	−0.287973
$64$	0	0
$65$	228.000	0.435076
$66$	0	0
$67$	− 884.000i	− 1.61191i	−0.591979	−	0.805954i	$-0.701653\pi$
$67$	− 884.000i	− 1.61191i	0.591979	−	0.805954i	$-0.298347\pi$
$68$	0	0
$69$	− 504.000i	− 0.879340i
$70$	0	0
$71$	−792.000	−1.32385	−0.661923	−	0.749572i	$-0.730260\pi$
$71$	−792.000	−1.32385	−0.661923	+	0.749572i	$0.730260\pi$
$72$	0	0
$73$	−218.000	−0.349520	−0.174760	−	0.984611i	$-0.555915\pi$
$73$	−218.000	−0.349520	−0.174760	+	0.984611i	$0.555915\pi$
$74$	0	0
$75$	267.000i	0.411073i
$76$	0	0
$77$	192.000i	0.284161i
$78$	0	0
$79$	−520.000	−0.740564	−0.370282	−	0.928919i	$-0.620739\pi$
$79$	−520.000	−0.740564	−0.370282	+	0.928919i	$0.620739\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	492.000i	0.650651i	0.945602	+	0.325325i	$0.105474\pi$
$83$	492.000i	0.650651i	−0.945602	+	0.325325i	$0.894526\pi$
$84$	0	0
$85$	− 756.000i	− 0.964703i
$86$	0	0
$87$	90.0000	0.110908
$88$	0	0
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	0	0
$91$	− 608.000i	− 0.700393i
$92$	0	0
$93$	− 264.000i	− 0.294360i
$94$	0	0
$95$	120.000	0.129597
$96$	0	0
$97$	1154.00	1.20795	0.603974	−	0.797004i	$-0.293583\pi$
$97$	1154.00	1.20795	0.603974	+	0.797004i	$0.293583\pi$
$98$	0	0
$99$	− 108.000i	− 0.109640i
$100$	0	0
$101$	− 618.000i	− 0.608845i	−0.952537	−	0.304422i	$-0.901537\pi$
$101$	− 618.000i	− 0.608845i	0.952537	−	0.304422i	$-0.0984634\pi$
$102$	0	0
$103$	−128.000	−0.122449	−0.0612243	−	0.998124i	$-0.519501\pi$
$103$	−128.000	−0.122449	−0.0612243	+	0.998124i	$0.519501\pi$
$104$	0	0
$105$	−288.000	−0.267675
$106$	0	0
$107$	− 1476.00i	− 1.33355i	−0.745257	−	0.666777i	$-0.767673\pi$
$107$	− 1476.00i	− 1.33355i	0.745257	−	0.666777i	$-0.232327\pi$
$108$	0	0
$109$	− 1190.00i	− 1.04570i	−0.852425	−	0.522850i	$-0.824869\pi$
$109$	− 1190.00i	− 1.04570i	0.852425	−	0.522850i	$-0.175131\pi$
$110$	0	0
$111$	−762.000	−0.651584
$112$	0	0
$113$	−462.000	−0.384613	−0.192307	−	0.981335i	$-0.561597\pi$
$113$	−462.000	−0.384613	−0.192307	+	0.981335i	$0.561597\pi$
$114$	0	0
$115$	− 1008.00i	− 0.817361i
$116$	0	0
$117$	342.000i	0.270239i
$118$	0	0
$119$	−2016.00	−1.55300
$120$	0	0
$121$	1187.00	0.891811
$122$	0	0
$123$	− 126.000i	− 0.0923662i
$124$	0	0
$125$	1284.00i	0.918756i
$126$	0	0
$127$	−2536.00	−1.77192	−0.885959	−	0.463763i	$-0.846499\pi$
$127$	−2536.00	−1.77192	−0.885959	+	0.463763i	$0.846499\pi$
$128$	0	0
$129$	156.000	0.106473
$130$	0	0
$131$	− 2292.00i	− 1.52865i	−0.644832	−	0.764324i	$-0.723073\pi$
$131$	− 2292.00i	− 1.52865i	0.644832	−	0.764324i	$-0.276927\pi$
$132$	0	0
$133$	− 320.000i	− 0.208628i
$134$	0	0
$135$	162.000	0.103280
$136$	0	0
$137$	726.000	0.452747	0.226374	−	0.974041i	$-0.427313\pi$
$137$	726.000	0.452747	0.226374	+	0.974041i	$0.427313\pi$
$138$	0	0
$139$	380.000i	0.231879i	0.993256	+	0.115939i	$0.0369879\pi$
$139$	380.000i	0.231879i	−0.993256	+	0.115939i	$0.963012\pi$
$140$	0	0
$141$	− 288.000i	− 0.172014i
$142$	0	0
$143$	456.000	0.266662
$144$	0	0
$145$	180.000	0.103091
$146$	0	0
$147$	− 261.000i	− 0.146442i
$148$	0	0
$149$	1590.00i	0.874214i	0.899410	+	0.437107i	$0.143997\pi$
$149$	1590.00i	0.874214i	−0.899410	+	0.437107i	$0.856003\pi$
$150$	0	0
$151$	−2432.00	−1.31068	−0.655342	−	0.755332i	$-0.727476\pi$
$151$	−2432.00	−1.31068	−0.655342	+	0.755332i	$0.727476\pi$
$152$	0	0
$153$	1134.00	0.599206
$154$	0	0
$155$	− 528.000i	− 0.273613i
$156$	0	0
$157$	− 614.000i	− 0.312118i	−0.987748	−	0.156059i	$-0.950121\pi$
$157$	− 614.000i	− 0.312118i	0.987748	−	0.156059i	$-0.0498790\pi$
$158$	0	0
$159$	−594.000	−0.296272
$160$	0	0
$161$	−2688.00	−1.31580
$162$	0	0
$163$	1852.00i	0.889938i	0.895546	+	0.444969i	$0.146785\pi$
$163$	1852.00i	0.889938i	−0.895546	+	0.444969i	$0.853215\pi$
$164$	0	0
$165$	− 216.000i	− 0.101913i
$166$	0	0
$167$	2136.00	0.989752	0.494876	−	0.868964i	$-0.335213\pi$
$167$	2136.00	0.989752	0.494876	+	0.868964i	$0.335213\pi$
$168$	0	0
$169$	753.000	0.342740
$170$	0	0
$171$	180.000i	0.0804967i
$172$	0	0
$173$	− 1758.00i	− 0.772591i	−0.922375	−	0.386296i	$-0.873754\pi$
$173$	− 1758.00i	− 0.772591i	0.922375	−	0.386296i	$-0.126246\pi$
$174$	0	0
$175$	1424.00	0.615110
$176$	0	0
$177$	1980.00	0.840824
$178$	0	0
$179$	540.000i	0.225483i	0.993624	+	0.112742i	$0.0359632\pi$
$179$	540.000i	0.225483i	−0.993624	+	0.112742i	$0.964037\pi$
$180$	0	0
$181$	1982.00i	0.813928i	0.913444	+	0.406964i	$0.133412\pi$
$181$	1982.00i	0.813928i	−0.913444	+	0.406964i	$0.866588\pi$
$182$	0	0
$183$	−1614.00	−0.651969
$184$	0	0
$185$	−1524.00	−0.605658
$186$	0	0
$187$	− 1512.00i	− 0.591275i
$188$	0	0
$189$	− 432.000i	− 0.166261i
$190$	0	0
$191$	−2688.00	−1.01831	−0.509154	−	0.860675i	$-0.670042\pi$
$191$	−2688.00	−1.01831	−0.509154	+	0.860675i	$0.670042\pi$
$192$	0	0
$193$	−2302.00	−0.858557	−0.429279	−	0.903172i	$-0.641232\pi$
$193$	−2302.00	−0.858557	−0.429279	+	0.903172i	$0.641232\pi$
$194$	0	0
$195$	684.000i	0.251191i
$196$	0	0
$197$	4374.00i	1.58190i	0.611880	+	0.790951i	$0.290414\pi$
$197$	4374.00i	1.58190i	−0.611880	+	0.790951i	$0.709586\pi$
$198$	0	0
$199$	1600.00	0.569955	0.284977	−	0.958534i	$-0.408014\pi$
$199$	1600.00	0.569955	0.284977	+	0.958534i	$0.408014\pi$
$200$	0	0
$201$	2652.00	0.930635
$202$	0	0
$203$	− 480.000i	− 0.165958i
$204$	0	0
$205$	− 252.000i	− 0.0858558i
$206$	0	0
$207$	1512.00	0.507687
$208$	0	0
$209$	240.000	0.0794313
$210$	0	0
$211$	− 3332.00i	− 1.08713i	−0.839367	−	0.543565i	$-0.817074\pi$
$211$	− 3332.00i	− 1.08713i	0.839367	−	0.543565i	$-0.182926\pi$
$212$	0	0
$213$	− 2376.00i	− 0.764323i
$214$	0	0
$215$	312.000	0.0989685
$216$	0	0
$217$	−1408.00	−0.440467
$218$	0	0
$219$	− 654.000i	− 0.201796i
$220$	0	0
$221$	4788.00i	1.45736i
$222$	0	0
$223$	2648.00	0.795171	0.397586	−	0.917565i	$-0.369848\pi$
$223$	2648.00	0.795171	0.397586	+	0.917565i	$0.369848\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	0	0
$227$	− 2244.00i	− 0.656121i	−0.944657	−	0.328061i	$-0.893605\pi$
$227$	− 2244.00i	− 0.656121i	0.944657	−	0.328061i	$-0.106395\pi$
$228$	0	0
$229$	− 5650.00i	− 1.63040i	−0.579177	−	0.815202i	$-0.696626\pi$
$229$	− 5650.00i	− 1.63040i	0.579177	−	0.815202i	$-0.303374\pi$
$230$	0	0
$231$	−576.000	−0.164061
$232$	0	0
$233$	−4698.00	−1.32093	−0.660464	−	0.750858i	$-0.729640\pi$
$233$	−4698.00	−1.32093	−0.660464	+	0.750858i	$0.729640\pi$
$234$	0	0
$235$	− 576.000i	− 0.159890i
$236$	0	0
$237$	− 1560.00i	− 0.427565i
$238$	0	0
$239$	−1200.00	−0.324776	−0.162388	−	0.986727i	$-0.551920\pi$
$239$	−1200.00	−0.324776	−0.162388	+	0.986727i	$0.551920\pi$
$240$	0	0
$241$	−718.000	−0.191911	−0.0959553	−	0.995386i	$-0.530591\pi$
$241$	−718.000	−0.191911	−0.0959553	+	0.995386i	$0.530591\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 522.000i	− 0.136120i
$246$	0	0
$247$	−760.000	−0.195780
$248$	0	0
$249$	−1476.00	−0.375653
$250$	0	0
$251$	6012.00i	1.51185i	0.654659	+	0.755924i	$0.272812\pi$
$251$	6012.00i	1.51185i	−0.654659	+	0.755924i	$0.727188\pi$
$252$	0	0
$253$	− 2016.00i	− 0.500968i
$254$	0	0
$255$	2268.00	0.556971
$256$	0	0
$257$	−2046.00	−0.496599	−0.248300	−	0.968683i	$-0.579872\pi$
$257$	−2046.00	−0.496599	−0.248300	+	0.968683i	$0.579872\pi$
$258$	0	0
$259$	4064.00i	0.974999i
$260$	0	0
$261$	270.000i	0.0640329i
$262$	0	0
$263$	6072.00	1.42363	0.711817	−	0.702365i	$-0.247873\pi$
$263$	6072.00	1.42363	0.711817	+	0.702365i	$0.247873\pi$
$264$	0	0
$265$	−1188.00	−0.275390
$266$	0	0
$267$	− 2430.00i	− 0.556980i
$268$	0	0
$269$	6930.00i	1.57074i	0.619025	+	0.785371i	$0.287528\pi$
$269$	6930.00i	1.57074i	−0.619025	+	0.785371i	$0.712472\pi$
$270$	0	0
$271$	1352.00	0.303056	0.151528	−	0.988453i	$-0.451581\pi$
$271$	1352.00	0.303056	0.151528	+	0.988453i	$0.451581\pi$
$272$	0	0
$273$	1824.00	0.404372
$274$	0	0
$275$	1068.00i	0.234192i
$276$	0	0
$277$	− 1186.00i	− 0.257256i	−0.991693	−	0.128628i	$-0.958943\pi$
$277$	− 1186.00i	− 0.257256i	0.991693	−	0.128628i	$-0.0410573\pi$
$278$	0	0
$279$	792.000	0.169949
$280$	0	0
$281$	−2442.00	−0.518425	−0.259213	−	0.965820i	$-0.583463\pi$
$281$	−2442.00	−0.518425	−0.259213	+	0.965820i	$0.583463\pi$
$282$	0	0
$283$	2828.00i	0.594018i	0.954875	+	0.297009i	$0.0959892\pi$
$283$	2828.00i	0.594018i	−0.954875	+	0.297009i	$0.904011\pi$
$284$	0	0
$285$	360.000i	0.0748230i
$286$	0	0
$287$	−672.000	−0.138212
$288$	0	0
$289$	10963.0	2.23143
$290$	0	0
$291$	3462.00i	0.697409i
$292$	0	0
$293$	4758.00i	0.948687i	0.880340	+	0.474344i	$0.157315\pi$
$293$	4758.00i	0.948687i	−0.880340	+	0.474344i	$0.842685\pi$
$294$	0	0
$295$	3960.00	0.781560
$296$	0	0
$297$	324.000	0.0633010
$298$	0	0
$299$	6384.00i	1.23477i
$300$	0	0
$301$	− 832.000i	− 0.159321i
$302$	0	0
$303$	1854.00	0.351517
$304$	0	0
$305$	−3228.00	−0.606016
$306$	0	0
$307$	8476.00i	1.57574i	0.615844	+	0.787868i	$0.288815\pi$
$307$	8476.00i	1.57574i	−0.615844	+	0.787868i	$0.711185\pi$
$308$	0	0
$309$	− 384.000i	− 0.0706958i
$310$	0	0
$311$	−4632.00	−0.844555	−0.422278	−	0.906467i	$-0.638769\pi$
$311$	−4632.00	−0.844555	−0.422278	+	0.906467i	$0.638769\pi$
$312$	0	0
$313$	4822.00	0.870785	0.435392	−	0.900241i	$-0.356610\pi$
$313$	4822.00	0.870785	0.435392	+	0.900241i	$0.356610\pi$
$314$	0	0
$315$	− 864.000i	− 0.154542i
$316$	0	0
$317$	3426.00i	0.607014i	0.952829	+	0.303507i	$0.0981575\pi$
$317$	3426.00i	0.607014i	−0.952829	+	0.303507i	$0.901842\pi$
$318$	0	0
$319$	360.000	0.0631854
$320$	0	0
$321$	4428.00	0.769928
$322$	0	0
$323$	2520.00i	0.434107i
$324$	0	0
$325$	− 3382.00i	− 0.577230i
$326$	0	0
$327$	3570.00	0.603735
$328$	0	0
$329$	−1536.00	−0.257393
$330$	0	0
$331$	− 2788.00i	− 0.462968i	−0.972839	−	0.231484i	$-0.925642\pi$
$331$	− 2788.00i	− 0.462968i	0.972839	−	0.231484i	$-0.0743581\pi$
$332$	0	0
$333$	− 2286.00i	− 0.376192i
$334$	0	0
$335$	5304.00	0.865040
$336$	0	0
$337$	434.000	0.0701528	0.0350764	−	0.999385i	$-0.488833\pi$
$337$	434.000	0.0701528	0.0350764	+	0.999385i	$0.488833\pi$
$338$	0	0
$339$	− 1386.00i	− 0.222057i
$340$	0	0
$341$	− 1056.00i	− 0.167700i
$342$	0	0
$343$	−6880.00	−1.08305
$344$	0	0
$345$	3024.00	0.471903
$346$	0	0
$347$	6684.00i	1.03405i	0.855970	+	0.517026i	$0.172961\pi$
$347$	6684.00i	1.03405i	−0.855970	+	0.517026i	$0.827039\pi$
$348$	0	0
$349$	− 2630.00i	− 0.403383i	−0.979449	−	0.201692i	$-0.935356\pi$
$349$	− 2630.00i	− 0.403383i	0.979449	−	0.201692i	$-0.0646438\pi$
$350$	0	0
$351$	−1026.00	−0.156022
$352$	0	0
$353$	−7422.00	−1.11907	−0.559537	−	0.828805i	$-0.689021\pi$
$353$	−7422.00	−1.11907	−0.559537	+	0.828805i	$0.689021\pi$
$354$	0	0
$355$	− 4752.00i	− 0.710451i
$356$	0	0
$357$	− 6048.00i	− 0.896622i
$358$	0	0
$359$	10440.0	1.53482	0.767412	−	0.641154i	$-0.221544\pi$
$359$	10440.0	1.53482	0.767412	+	0.641154i	$0.221544\pi$
$360$	0	0
$361$	6459.00	0.941682
$362$	0	0
$363$	3561.00i	0.514887i
$364$	0	0
$365$	− 1308.00i	− 0.187572i
$366$	0	0
$367$	10424.0	1.48264	0.741319	−	0.671153i	$-0.234200\pi$
$367$	10424.0	1.48264	0.741319	+	0.671153i	$0.234200\pi$
$368$	0	0
$369$	378.000	0.0533276
$370$	0	0
$371$	3168.00i	0.443327i
$372$	0	0
$373$	3278.00i	0.455036i	0.973774	+	0.227518i	$0.0730610\pi$
$373$	3278.00i	0.455036i	−0.973774	+	0.227518i	$0.926939\pi$
$374$	0	0
$375$	−3852.00	−0.530444
$376$	0	0
$377$	−1140.00	−0.155737
$378$	0	0
$379$	6140.00i	0.832165i	0.909327	+	0.416083i	$0.136597\pi$
$379$	6140.00i	0.832165i	−0.909327	+	0.416083i	$0.863403\pi$
$380$	0	0
$381$	− 7608.00i	− 1.02302i
$382$	0	0
$383$	−3072.00	−0.409848	−0.204924	−	0.978778i	$-0.565695\pi$
$383$	−3072.00	−0.409848	−0.204924	+	0.978778i	$0.565695\pi$
$384$	0	0
$385$	−1152.00	−0.152497
$386$	0	0
$387$	468.000i	0.0614723i
$388$	0	0
$389$	6150.00i	0.801587i	0.916168	+	0.400794i	$0.131266\pi$
$389$	6150.00i	0.801587i	−0.916168	+	0.400794i	$0.868734\pi$
$390$	0	0
$391$	21168.0	2.73788
$392$	0	0
$393$	6876.00	0.882566
$394$	0	0
$395$	− 3120.00i	− 0.397428i
$396$	0	0
$397$	106.000i	0.0134005i	0.999978	+	0.00670024i	$0.00213277\pi$
$397$	106.000i	0.0134005i	−0.999978	+	0.00670024i	$0.997867\pi$
$398$	0	0
$399$	960.000	0.120451
$400$	0	0
$401$	−1758.00	−0.218929	−0.109464	−	0.993991i	$-0.534914\pi$
$401$	−1758.00	−0.218929	−0.109464	+	0.993991i	$0.534914\pi$
$402$	0	0
$403$	3344.00i	0.413341i
$404$	0	0
$405$	486.000i	0.0596285i
$406$	0	0
$407$	−3048.00	−0.371213
$408$	0	0
$409$	3670.00	0.443691	0.221846	−	0.975082i	$-0.428792\pi$
$409$	3670.00	0.443691	0.221846	+	0.975082i	$0.428792\pi$
$410$	0	0
$411$	2178.00i	0.261394i
$412$	0	0
$413$	− 10560.0i	− 1.25817i
$414$	0	0
$415$	−2952.00	−0.349176
$416$	0	0
$417$	−1140.00	−0.133875
$418$	0	0
$419$	9660.00i	1.12631i	0.826353	+	0.563153i	$0.190412\pi$
$419$	9660.00i	1.12631i	−0.826353	+	0.563153i	$0.809588\pi$
$420$	0	0
$421$	8462.00i	0.979602i	0.871834	+	0.489801i	$0.162931\pi$
$421$	8462.00i	0.979602i	−0.871834	+	0.489801i	$0.837069\pi$
$422$	0	0
$423$	864.000	0.0993123
$424$	0	0
$425$	−11214.0	−1.27990
$426$	0	0
$427$	8608.00i	0.975575i
$428$	0	0
$429$	1368.00i	0.153957i
$430$	0	0
$431$	9792.00	1.09435	0.547174	−	0.837019i	$-0.315704\pi$
$431$	9792.00	1.09435	0.547174	+	0.837019i	$0.315704\pi$
$432$	0	0
$433$	−7342.00	−0.814859	−0.407430	−	0.913237i	$-0.633575\pi$
$433$	−7342.00	−0.814859	−0.407430	+	0.913237i	$0.633575\pi$
$434$	0	0
$435$	540.000i	0.0595196i
$436$	0	0
$437$	3360.00i	0.367805i
$438$	0	0
$439$	−10640.0	−1.15676	−0.578382	−	0.815766i	$-0.696316\pi$
$439$	−10640.0	−1.15676	−0.578382	+	0.815766i	$0.696316\pi$
$440$	0	0
$441$	783.000	0.0845481
$442$	0	0
$443$	− 17412.0i	− 1.86742i	−0.358024	−	0.933712i	$-0.616549\pi$
$443$	− 17412.0i	− 1.86742i	0.358024	−	0.933712i	$-0.383451\pi$
$444$	0	0
$445$	− 4860.00i	− 0.517722i
$446$	0	0
$447$	−4770.00	−0.504728
$448$	0	0
$449$	−1710.00	−0.179732	−0.0898662	−	0.995954i	$-0.528644\pi$
$449$	−1710.00	−0.179732	−0.0898662	+	0.995954i	$0.528644\pi$
$450$	0	0
$451$	− 504.000i	− 0.0526218i
$452$	0	0
$453$	− 7296.00i	− 0.756724i
$454$	0	0
$455$	3648.00	0.375870
$456$	0	0
$457$	646.000	0.0661239	0.0330619	−	0.999453i	$-0.489474\pi$
$457$	646.000	0.0661239	0.0330619	+	0.999453i	$0.489474\pi$
$458$	0	0
$459$	3402.00i	0.345952i
$460$	0	0
$461$	6018.00i	0.607996i	0.952673	+	0.303998i	$0.0983216\pi$
$461$	6018.00i	0.607996i	−0.952673	+	0.303998i	$0.901678\pi$
$462$	0	0
$463$	−6712.00	−0.673722	−0.336861	−	0.941554i	$-0.609365\pi$
$463$	−6712.00	−0.673722	−0.336861	+	0.941554i	$0.609365\pi$
$464$	0	0
$465$	1584.00	0.157970
$466$	0	0
$467$	− 5364.00i	− 0.531512i	−0.964040	−	0.265756i	$-0.914378\pi$
$467$	− 5364.00i	− 0.531512i	0.964040	−	0.265756i	$-0.0856216\pi$
$468$	0	0
$469$	− 14144.0i	− 1.39256i
$470$	0	0
$471$	1842.00	0.180201
$472$	0	0
$473$	624.000	0.0606587
$474$	0	0
$475$	− 1780.00i	− 0.171941i
$476$	0	0
$477$	− 1782.00i	− 0.171053i
$478$	0	0
$479$	9840.00	0.938624	0.469312	−	0.883032i	$-0.344502\pi$
$479$	9840.00	0.938624	0.469312	+	0.883032i	$0.344502\pi$
$480$	0	0
$481$	9652.00	0.914955
$482$	0	0
$483$	− 8064.00i	− 0.759678i
$484$	0	0
$485$	6924.00i	0.648253i
$486$	0	0
$487$	−1424.00	−0.132500	−0.0662501	−	0.997803i	$-0.521104\pi$
$487$	−1424.00	−0.132500	−0.0662501	+	0.997803i	$0.521104\pi$
$488$	0	0
$489$	−5556.00	−0.513806
$490$	0	0
$491$	− 4548.00i	− 0.418021i	−0.977913	−	0.209011i	$-0.932976\pi$
$491$	− 4548.00i	− 0.418021i	0.977913	−	0.209011i	$-0.0670243\pi$
$492$	0	0
$493$	3780.00i	0.345320i
$494$	0	0
$495$	648.000	0.0588393
$496$	0	0
$497$	−12672.0	−1.14370
$498$	0	0
$499$	− 6500.00i	− 0.583126i	−0.956552	−	0.291563i	$-0.905825\pi$
$499$	− 6500.00i	− 0.583126i	0.956552	−	0.291563i	$-0.0941753\pi$
$500$	0	0
$501$	6408.00i	0.571434i
$502$	0	0
$503$	−12168.0	−1.07862	−0.539308	−	0.842108i	$-0.681314\pi$
$503$	−12168.0	−1.07862	−0.539308	+	0.842108i	$0.681314\pi$
$504$	0	0
$505$	3708.00	0.326740
$506$	0	0
$507$	2259.00i	0.197881i
$508$	0	0
$509$	21090.0i	1.83654i	0.395957	+	0.918269i	$0.370413\pi$
$509$	21090.0i	1.83654i	−0.395957	+	0.918269i	$0.629587\pi$
$510$	0	0
$511$	−3488.00	−0.301957
$512$	0	0
$513$	−540.000	−0.0464748
$514$	0	0
$515$	− 768.000i	− 0.0657129i
$516$	0	0
$517$	− 1152.00i	− 0.0979979i
$518$	0	0
$519$	5274.00	0.446056
$520$	0	0
$521$	5238.00	0.440462	0.220231	−	0.975448i	$-0.429319\pi$
$521$	5238.00	0.440462	0.220231	+	0.975448i	$0.429319\pi$
$522$	0	0
$523$	8588.00i	0.718025i	0.933333	+	0.359012i	$0.116886\pi$
$523$	8588.00i	0.718025i	−0.933333	+	0.359012i	$0.883114\pi$
$524$	0	0
$525$	4272.00i	0.355134i
$526$	0	0
$527$	11088.0	0.916510
$528$	0	0
$529$	16057.0	1.31972
$530$	0	0
$531$	5940.00i	0.485450i
$532$	0	0
$533$	1596.00i	0.129701i
$534$	0	0
$535$	8856.00	0.715660
$536$	0	0
$537$	−1620.00	−0.130183
$538$	0	0
$539$	− 1044.00i	− 0.0834291i
$540$	0	0
$541$	− 3062.00i	− 0.243338i	−0.992571	−	0.121669i	$-0.961175\pi$
$541$	− 3062.00i	− 0.243338i	0.992571	−	0.121669i	$-0.0388246\pi$
$542$	0	0
$543$	−5946.00	−0.469921
$544$	0	0
$545$	7140.00	0.561182
$546$	0	0
$547$	8476.00i	0.662537i	0.943537	+	0.331268i	$0.107477\pi$
$547$	8476.00i	0.662537i	−0.943537	+	0.331268i	$0.892523\pi$
$548$	0	0
$549$	− 4842.00i	− 0.376414i
$550$	0	0
$551$	−600.000	−0.0463899
$552$	0	0
$553$	−8320.00	−0.639787
$554$	0	0
$555$	− 4572.00i	− 0.349677i
$556$	0	0
$557$	12546.0i	0.954383i	0.878799	+	0.477191i	$0.158345\pi$
$557$	12546.0i	0.954383i	−0.878799	+	0.477191i	$0.841655\pi$
$558$	0	0
$559$	−1976.00	−0.149510
$560$	0	0
$561$	4536.00	0.341373
$562$	0	0
$563$	12.0000i	0 0.000898294i	1.00000		0.000449147i	$0.000142968\pi$
$563$	12.0000i	0 0.000898294i	−1.00000		0.000449147i	$0.999857\pi$
$564$	0	0
$565$	− 2772.00i	− 0.206405i
$566$	0	0
$567$	1296.00	0.0959910
$568$	0	0
$569$	−19290.0	−1.42123	−0.710614	−	0.703582i	$-0.751583\pi$
$569$	−19290.0	−1.42123	−0.710614	+	0.703582i	$0.751583\pi$
$570$	0	0
$571$	− 12148.0i	− 0.890329i	−0.895449	−	0.445165i	$-0.853145\pi$
$571$	− 12148.0i	− 0.890329i	0.895449	−	0.445165i	$-0.146855\pi$
$572$	0	0
$573$	− 8064.00i	− 0.587920i
$574$	0	0
$575$	−14952.0	−1.08442
$576$	0	0
$577$	−10366.0	−0.747907	−0.373953	−	0.927447i	$-0.621998\pi$
$577$	−10366.0	−0.747907	−0.373953	+	0.927447i	$0.621998\pi$
$578$	0	0
$579$	− 6906.00i	− 0.495688i
$580$	0	0
$581$	7872.00i	0.562109i
$582$	0	0
$583$	−2376.00	−0.168789
$584$	0	0
$585$	−2052.00	−0.145025
$586$	0	0
$587$	7644.00i	0.537482i	0.963213	+	0.268741i	$0.0866075\pi$
$587$	7644.00i	0.537482i	−0.963213	+	0.268741i	$0.913393\pi$
$588$	0	0
$589$	1760.00i	0.123123i
$590$	0	0
$591$	−13122.0	−0.913311
$592$	0	0
$593$	8658.00	0.599564	0.299782	−	0.954008i	$-0.403086\pi$
$593$	8658.00	0.599564	0.299782	+	0.954008i	$0.403086\pi$
$594$	0	0
$595$	− 12096.0i	− 0.833425i
$596$	0	0
$597$	4800.00i	0.329064i
$598$	0	0
$599$	−25800.0	−1.75987	−0.879933	−	0.475098i	$-0.842413\pi$
$599$	−25800.0	−1.75987	−0.879933	+	0.475098i	$0.842413\pi$
$600$	0	0
$601$	−16202.0	−1.09966	−0.549828	−	0.835278i	$-0.685307\pi$
$601$	−16202.0	−1.09966	−0.549828	+	0.835278i	$0.685307\pi$
$602$	0	0
$603$	7956.00i	0.537302i
$604$	0	0
$605$	7122.00i	0.478596i
$606$	0	0
$607$	−24136.0	−1.61392	−0.806960	−	0.590605i	$-0.798889\pi$
$607$	−24136.0	−1.61392	−0.806960	+	0.590605i	$0.798889\pi$
$608$	0	0
$609$	1440.00	0.0958157
$610$	0	0
$611$	3648.00i	0.241542i
$612$	0	0
$613$	− 4642.00i	− 0.305854i	−0.988237	−	0.152927i	$-0.951130\pi$
$613$	− 4642.00i	− 0.305854i	0.988237	−	0.152927i	$-0.0488700\pi$
$614$	0	0
$615$	756.000	0.0495689
$616$	0	0
$617$	6726.00	0.438863	0.219432	−	0.975628i	$-0.429580\pi$
$617$	6726.00	0.438863	0.219432	+	0.975628i	$0.429580\pi$
$618$	0	0
$619$	− 21220.0i	− 1.37787i	−0.724821	−	0.688937i	$-0.758078\pi$
$619$	− 21220.0i	− 1.37787i	0.724821	−	0.688937i	$-0.241922\pi$
$620$	0	0
$621$	4536.00i	0.293113i
$622$	0	0
$623$	−12960.0	−0.833437
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	720.000i	0.0458597i
$628$	0	0
$629$	− 32004.0i	− 2.02875i
$630$	0	0
$631$	−29792.0	−1.87956	−0.939779	−	0.341783i	$-0.888969\pi$
$631$	−29792.0	−1.87956	−0.939779	+	0.341783i	$0.888969\pi$
$632$	0	0
$633$	9996.00	0.627655
$634$	0	0
$635$	− 15216.0i	− 0.950911i
$636$	0	0
$637$	3306.00i	0.205633i
$638$	0	0
$639$	7128.00	0.441282
$640$	0	0
$641$	−10158.0	−0.625923	−0.312962	−	0.949766i	$-0.601321\pi$
$641$	−10158.0	−0.625923	−0.312962	+	0.949766i	$0.601321\pi$
$642$	0	0
$643$	− 29828.0i	− 1.82940i	−0.404138	−	0.914698i	$-0.632429\pi$
$643$	− 29828.0i	− 1.82940i	0.404138	−	0.914698i	$-0.367571\pi$
$644$	0	0
$645$	936.000i	0.0571395i
$646$	0	0
$647$	−1944.00	−0.118124	−0.0590622	−	0.998254i	$-0.518811\pi$
$647$	−1944.00	−0.118124	−0.0590622	+	0.998254i	$0.518811\pi$
$648$	0	0
$649$	7920.00	0.479025
$650$	0	0
$651$	− 4224.00i	− 0.254304i
$652$	0	0
$653$	− 26718.0i	− 1.60116i	−0.599227	−	0.800579i	$-0.704525\pi$
$653$	− 26718.0i	− 1.60116i	0.599227	−	0.800579i	$-0.295475\pi$
$654$	0	0
$655$	13752.0	0.820359
$656$	0	0
$657$	1962.00	0.116507
$658$	0	0
$659$	− 4260.00i	− 0.251815i	−0.992042	−	0.125907i	$-0.959816\pi$
$659$	− 4260.00i	− 0.251815i	0.992042	−	0.125907i	$-0.0401842\pi$
$660$	0	0
$661$	22862.0i	1.34528i	0.739971	+	0.672639i	$0.234839\pi$
$661$	22862.0i	1.34528i	−0.739971	+	0.672639i	$0.765161\pi$
$662$	0	0
$663$	−14364.0	−0.841405
$664$	0	0
$665$	1920.00	0.111962
$666$	0	0
$667$	5040.00i	0.292578i
$668$	0	0
$669$	7944.00i	0.459092i
$670$	0	0
$671$	−6456.00	−0.371432
$672$	0	0
$673$	−32542.0	−1.86390	−0.931948	−	0.362592i	$-0.881892\pi$
$673$	−32542.0	−1.86390	−0.931948	+	0.362592i	$0.881892\pi$
$674$	0	0
$675$	− 2403.00i	− 0.137024i
$676$	0	0
$677$	14214.0i	0.806925i	0.914996	+	0.403463i	$0.132193\pi$
$677$	14214.0i	0.806925i	−0.914996	+	0.403463i	$0.867807\pi$
$678$	0	0
$679$	18464.0	1.04357
$680$	0	0
$681$	6732.00	0.378812
$682$	0	0
$683$	− 7092.00i	− 0.397317i	−0.980069	−	0.198659i	$-0.936341\pi$
$683$	− 7092.00i	− 0.397317i	0.980069	−	0.198659i	$-0.0636585\pi$
$684$	0	0
$685$	4356.00i	0.242970i
$686$	0	0
$687$	16950.0	0.941314
$688$	0	0
$689$	7524.00	0.416026
$690$	0	0
$691$	13228.0i	0.728244i	0.931351	+	0.364122i	$0.118631\pi$
$691$	13228.0i	0.728244i	−0.931351	+	0.364122i	$0.881369\pi$
$692$	0	0
$693$	− 1728.00i	− 0.0947205i
$694$	0	0
$695$	−2280.00	−0.124439
$696$	0	0
$697$	5292.00	0.287588
$698$	0	0
$699$	− 14094.0i	− 0.762638i
$700$	0	0
$701$	− 28062.0i	− 1.51196i	−0.654592	−	0.755982i	$-0.727160\pi$
$701$	− 28062.0i	− 1.51196i	0.654592	−	0.755982i	$-0.272840\pi$
$702$	0	0
$703$	5080.00	0.272540
$704$	0	0
$705$	1728.00	0.0923124
$706$	0	0
$707$	− 9888.00i	− 0.525992i
$708$	0	0
$709$	− 27250.0i	− 1.44343i	−0.692188	−	0.721717i	$-0.743353\pi$
$709$	− 27250.0i	− 1.44343i	0.692188	−	0.721717i	$-0.256647\pi$
$710$	0	0
$711$	4680.00	0.246855
$712$	0	0
$713$	14784.0	0.776529
$714$	0	0
$715$	2736.00i	0.143106i
$716$	0	0
$717$	− 3600.00i	− 0.187510i
$718$	0	0
$719$	−14400.0	−0.746912	−0.373456	−	0.927648i	$-0.621827\pi$
$719$	−14400.0	−0.746912	−0.373456	+	0.927648i	$0.621827\pi$
$720$	0	0
$721$	−2048.00	−0.105786
$722$	0	0
$723$	− 2154.00i	− 0.110800i
$724$	0	0
$725$	− 2670.00i	− 0.136774i
$726$	0	0
$727$	−17984.0	−0.917455	−0.458727	−	0.888577i	$-0.651695\pi$
$727$	−17984.0	−0.917455	−0.458727	+	0.888577i	$0.651695\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	6552.00i	0.331511i
$732$	0	0
$733$	− 16598.0i	− 0.836373i	−0.908361	−	0.418186i	$-0.862666\pi$
$733$	− 16598.0i	− 0.836373i	0.908361	−	0.418186i	$-0.137334\pi$
$734$	0	0
$735$	1566.00	0.0785888
$736$	0	0
$737$	10608.0	0.530191
$738$	0	0
$739$	− 1460.00i	− 0.0726752i	−0.999340	−	0.0363376i	$-0.988431\pi$
$739$	− 1460.00i	− 0.0726752i	0.999340	−	0.0363376i	$-0.0115692\pi$
$740$	0	0
$741$	− 2280.00i	− 0.113034i
$742$	0	0
$743$	30072.0	1.48484	0.742419	−	0.669936i	$-0.233678\pi$
$743$	30072.0	1.48484	0.742419	+	0.669936i	$0.233678\pi$
$744$	0	0
$745$	−9540.00	−0.469152
$746$	0	0
$747$	− 4428.00i	− 0.216884i
$748$	0	0
$749$	− 23616.0i	− 1.15208i
$750$	0	0
$751$	−18088.0	−0.878882	−0.439441	−	0.898271i	$-0.644823\pi$
$751$	−18088.0	−0.878882	−0.439441	+	0.898271i	$0.644823\pi$
$752$	0	0
$753$	−18036.0	−0.872866
$754$	0	0
$755$	− 14592.0i	− 0.703387i
$756$	0	0
$757$	24734.0i	1.18755i	0.804633	+	0.593773i	$0.202362\pi$
$757$	24734.0i	1.18755i	−0.804633	+	0.593773i	$0.797638\pi$
$758$	0	0
$759$	6048.00	0.289234
$760$	0	0
$761$	22278.0	1.06120	0.530602	−	0.847621i	$-0.321966\pi$
$761$	22278.0	1.06120	0.530602	+	0.847621i	$0.321966\pi$
$762$	0	0
$763$	− 19040.0i	− 0.903400i
$764$	0	0
$765$	6804.00i	0.321568i
$766$	0	0
$767$	−25080.0	−1.18069
$768$	0	0
$769$	16130.0	0.756388	0.378194	−	0.925726i	$-0.376545\pi$
$769$	16130.0	0.756388	0.378194	+	0.925726i	$0.376545\pi$
$770$	0	0
$771$	− 6138.00i	− 0.286712i
$772$	0	0
$773$	29718.0i	1.38277i	0.722486	+	0.691386i	$0.242999\pi$
$773$	29718.0i	1.38277i	−0.722486	+	0.691386i	$0.757001\pi$
$774$	0	0
$775$	−7832.00	−0.363011
$776$	0	0
$777$	−12192.0	−0.562916
$778$	0	0
$779$	840.000i	0.0386343i
$780$	0	0
$781$	− 9504.00i	− 0.435442i
$782$	0	0
$783$	−810.000	−0.0369694
$784$	0	0
$785$	3684.00	0.167500
$786$	0	0
$787$	− 9524.00i	− 0.431377i	−0.976462	−	0.215689i	$-0.930800\pi$
$787$	− 9524.00i	− 0.431377i	0.976462	−	0.215689i	$-0.0691996\pi$
$788$	0	0
$789$	18216.0i	0.821935i
$790$	0	0
$791$	−7392.00	−0.332275
$792$	0	0
$793$	20444.0	0.915495
$794$	0	0
$795$	− 3564.00i	− 0.158996i
$796$	0	0
$797$	33906.0i	1.50692i	0.657496	+	0.753458i	$0.271616\pi$
$797$	33906.0i	1.50692i	−0.657496	+	0.753458i	$0.728384\pi$
$798$	0	0
$799$	12096.0	0.535577
$800$	0	0
$801$	7290.00	0.321572
$802$	0	0
$803$	− 2616.00i	− 0.114965i
$804$	0	0
$805$	− 16128.0i	− 0.706133i
$806$	0	0
$807$	−20790.0	−0.906868
$808$	0	0
$809$	630.000	0.0273790	0.0136895	−	0.999906i	$-0.495642\pi$
$809$	630.000	0.0273790	0.0136895	+	0.999906i	$0.495642\pi$
$810$	0	0
$811$	− 20788.0i	− 0.900081i	−0.893008	−	0.450040i	$-0.851410\pi$
$811$	− 20788.0i	− 0.900081i	0.893008	−	0.450040i	$-0.148590\pi$
$812$	0	0
$813$	4056.00i	0.174969i
$814$	0	0
$815$	−11112.0	−0.477591
$816$	0	0
$817$	−1040.00	−0.0445349
$818$	0	0
$819$	5472.00i	0.233464i
$820$	0	0
$821$	− 43098.0i	− 1.83207i	−0.401097	−	0.916036i	$-0.631371\pi$
$821$	− 43098.0i	− 1.83207i	0.401097	−	0.916036i	$-0.368629\pi$
$822$	0	0
$823$	14272.0	0.604484	0.302242	−	0.953231i	$-0.402265\pi$
$823$	14272.0	0.604484	0.302242	+	0.953231i	$0.402265\pi$
$824$	0	0
$825$	−3204.00	−0.135211
$826$	0	0
$827$	13644.0i	0.573698i	0.957976	+	0.286849i	$0.0926078\pi$
$827$	13644.0i	0.573698i	−0.957976	+	0.286849i	$0.907392\pi$
$828$	0	0
$829$	2410.00i	0.100968i	0.998725	+	0.0504842i	$0.0160764\pi$
$829$	2410.00i	0.100968i	−0.998725	+	0.0504842i	$0.983924\pi$
$830$	0	0
$831$	3558.00	0.148527
$832$	0	0
$833$	10962.0	0.455955
$834$	0	0
$835$	12816.0i	0.531157i
$836$	0	0
$837$	2376.00i	0.0981202i
$838$	0	0
$839$	−23160.0	−0.953006	−0.476503	−	0.879173i	$-0.658096\pi$
$839$	−23160.0	−0.953006	−0.476503	+	0.879173i	$0.658096\pi$
$840$	0	0
$841$	23489.0	0.963098
$842$	0	0
$843$	− 7326.00i	− 0.299313i
$844$	0	0
$845$	4518.00i	0.183934i
$846$	0	0
$847$	18992.0	0.770452
$848$	0	0
$849$	−8484.00	−0.342957
$850$	0	0
$851$	− 42672.0i	− 1.71889i
$852$	0	0
$853$	32078.0i	1.28761i	0.765190	+	0.643804i	$0.222645\pi$
$853$	32078.0i	1.28761i	−0.765190	+	0.643804i	$0.777355\pi$
$854$	0	0
$855$	−1080.00	−0.0431991
$856$	0	0
$857$	14406.0	0.574212	0.287106	−	0.957899i	$-0.407307\pi$
$857$	14406.0	0.574212	0.287106	+	0.957899i	$0.407307\pi$
$858$	0	0
$859$	30620.0i	1.21623i	0.793849	+	0.608115i	$0.208074\pi$
$859$	30620.0i	1.21623i	−0.793849	+	0.608115i	$0.791926\pi$
$860$	0	0
$861$	− 2016.00i	− 0.0797969i
$862$	0	0
$863$	17568.0	0.692957	0.346478	−	0.938058i	$-0.387377\pi$
$863$	17568.0	0.692957	0.346478	+	0.938058i	$0.387377\pi$
$864$	0	0
$865$	10548.0	0.414616
$866$	0	0
$867$	32889.0i	1.28831i
$868$	0	0
$869$	− 6240.00i	− 0.243587i
$870$	0	0
$871$	−33592.0	−1.30680
$872$	0	0
$873$	−10386.0	−0.402649
$874$	0	0
$875$	20544.0i	0.793730i
$876$	0	0
$877$	21706.0i	0.835758i	0.908503	+	0.417879i	$0.137226\pi$
$877$	21706.0i	0.835758i	−0.908503	+	0.417879i	$0.862774\pi$
$878$	0	0
$879$	−14274.0	−0.547725
$880$	0	0
$881$	−14958.0	−0.572018	−0.286009	−	0.958227i	$-0.592329\pi$
$881$	−14958.0	−0.572018	−0.286009	+	0.958227i	$0.592329\pi$
$882$	0	0
$883$	32812.0i	1.25052i	0.780415	+	0.625261i	$0.215008\pi$
$883$	32812.0i	1.25052i	−0.780415	+	0.625261i	$0.784992\pi$
$884$	0	0
$885$	11880.0i	0.451234i
$886$	0	0
$887$	38856.0	1.47086	0.735432	−	0.677598i	$-0.236979\pi$
$887$	38856.0	1.47086	0.735432	+	0.677598i	$0.236979\pi$
$888$	0	0
$889$	−40576.0	−1.53079
$890$	0	0
$891$	972.000i	0.0365468i
$892$	0	0
$893$	1920.00i	0.0719489i
$894$	0	0
$895$	−3240.00	−0.121007
$896$	0	0
$897$	−19152.0	−0.712895
$898$	0	0
$899$	2640.00i	0.0979410i
$900$	0	0
$901$	− 24948.0i	− 0.922462i
$902$	0	0
$903$	2496.00	0.0919841
$904$	0	0
$905$	−11892.0	−0.436799
$906$	0	0
$907$	− 28276.0i	− 1.03516i	−0.855635	−	0.517579i	$-0.826833\pi$
$907$	− 28276.0i	− 1.03516i	0.855635	−	0.517579i	$-0.173167\pi$
$908$	0	0
$909$	5562.00i	0.202948i
$910$	0	0
$911$	8112.00	0.295019	0.147510	−	0.989061i	$-0.452874\pi$
$911$	8112.00	0.295019	0.147510	+	0.989061i	$0.452874\pi$
$912$	0	0
$913$	−5904.00	−0.214013
$914$	0	0
$915$	− 9684.00i	− 0.349883i
$916$	0	0
$917$	− 36672.0i	− 1.32063i
$918$	0	0
$919$	26080.0	0.936126	0.468063	−	0.883695i	$-0.344952\pi$
$919$	26080.0	0.936126	0.468063	+	0.883695i	$0.344952\pi$
$920$	0	0
$921$	−25428.0	−0.909751
$922$	0	0
$923$	30096.0i	1.07326i
$924$	0	0
$925$	22606.0i	0.803547i
$926$	0	0
$927$	1152.00	0.0408162
$928$	0	0
$929$	49170.0	1.73651	0.868254	−	0.496120i	$-0.165243\pi$
$929$	49170.0	1.73651	0.868254	+	0.496120i	$0.165243\pi$
$930$	0	0
$931$	1740.00i	0.0612526i
$932$	0	0
$933$	− 13896.0i	− 0.487604i
$934$	0	0
$935$	9072.00	0.317311
$936$	0	0
$937$	−48314.0	−1.68447	−0.842236	−	0.539110i	$-0.818761\pi$
$937$	−48314.0	−1.68447	−0.842236	+	0.539110i	$0.818761\pi$
$938$	0	0
$939$	14466.0i	0.502748i
$940$	0	0
$941$	− 34782.0i	− 1.20495i	−0.798137	−	0.602477i	$-0.794181\pi$
$941$	− 34782.0i	− 1.20495i	0.798137	−	0.602477i	$-0.205819\pi$
$942$	0	0
$943$	7056.00	0.243664
$944$	0	0
$945$	2592.00	0.0892251
$946$	0	0
$947$	25116.0i	0.861838i	0.902391	+	0.430919i	$0.141810\pi$
$947$	25116.0i	0.861838i	−0.902391	+	0.430919i	$0.858190\pi$
$948$	0	0
$949$	8284.00i	0.283361i
$950$	0	0
$951$	−10278.0	−0.350460
$952$	0	0
$953$	15462.0	0.525565	0.262782	−	0.964855i	$-0.415360\pi$
$953$	15462.0	0.525565	0.262782	+	0.964855i	$0.415360\pi$
$954$	0	0
$955$	− 16128.0i	− 0.546481i
$956$	0	0
$957$	1080.00i	0.0364801i
$958$	0	0
$959$	11616.0	0.391137
$960$	0	0
$961$	−22047.0	−0.740056
$962$	0	0
$963$	13284.0i	0.444518i
$964$	0	0
$965$	− 13812.0i	− 0.460750i
$966$	0	0
$967$	736.000	0.0244759	0.0122379	−	0.999925i	$-0.496104\pi$
$967$	736.000	0.0244759	0.0122379	+	0.999925i	$0.496104\pi$
$968$	0	0
$969$	−7560.00	−0.250632
$970$	0	0
$971$	− 29268.0i	− 0.967307i	−0.875260	−	0.483653i	$-0.839310\pi$
$971$	− 29268.0i	− 0.967307i	0.875260	−	0.483653i	$-0.160690\pi$
$972$	0	0
$973$	6080.00i	0.200325i
$974$	0	0
$975$	10146.0	0.333264
$976$	0	0
$977$	16674.0	0.546007	0.273003	−	0.962013i	$-0.411983\pi$
$977$	16674.0	0.546007	0.273003	+	0.962013i	$0.411983\pi$
$978$	0	0
$979$	− 9720.00i	− 0.317316i
$980$	0	0
$981$	10710.0i	0.348567i
$982$	0	0
$983$	31272.0	1.01467	0.507336	−	0.861749i	$-0.330630\pi$
$983$	31272.0	1.01467	0.507336	+	0.861749i	$0.330630\pi$
$984$	0	0
$985$	−26244.0	−0.848937
$986$	0	0
$987$	− 4608.00i	− 0.148606i
$988$	0	0
$989$	8736.00i	0.280878i
$990$	0	0
$991$	−15928.0	−0.510565	−0.255282	−	0.966867i	$-0.582168\pi$
$991$	−15928.0	−0.510565	−0.255282	+	0.966867i	$0.582168\pi$
$992$	0	0
$993$	8364.00	0.267295
$994$	0	0
$995$	9600.00i	0.305870i
$996$	0	0
$997$	42014.0i	1.33460i	0.744789	+	0.667300i	$0.232550\pi$
$997$	42014.0i	1.33460i	−0.744789	+	0.667300i	$0.767450\pi$
$998$	0	0
$999$	6858.00	0.217195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.d.n.385.2		2
4.3	odd	2		768.4.d.c.385.1		2
8.3	odd	2		768.4.d.c.385.2		2
8.5	even	2	inner	768.4.d.n.385.1		2
16.3	odd	4		48.4.a.c.1.1		1
16.5	even	4		192.4.a.i.1.1		1
16.11	odd	4		192.4.a.c.1.1		1
16.13	even	4		6.4.a.a.1.1	✓	1
48.5	odd	4		576.4.a.q.1.1		1
48.11	even	4		576.4.a.r.1.1		1
48.29	odd	4		18.4.a.a.1.1		1
48.35	even	4		144.4.a.c.1.1		1
80.3	even	4		1200.4.f.j.49.2		2
80.13	odd	4		150.4.c.d.49.2		2
80.19	odd	4		1200.4.a.b.1.1		1
80.29	even	4		150.4.a.i.1.1		1
80.67	even	4		1200.4.f.j.49.1		2
80.77	odd	4		150.4.c.d.49.1		2
112.13	odd	4		294.4.a.e.1.1		1
112.45	odd	12		294.4.e.g.79.1		2
112.61	odd	12		294.4.e.g.67.1		2
112.83	even	4		2352.4.a.e.1.1		1
112.93	even	12		294.4.e.h.67.1		2
112.109	even	12		294.4.e.h.79.1		2
144.13	even	12		162.4.c.f.55.1		2
144.29	odd	12		162.4.c.c.109.1		2
144.61	even	12		162.4.c.f.109.1		2
144.77	odd	12		162.4.c.c.55.1		2
176.109	odd	4		726.4.a.f.1.1		1
208.77	even	4		1014.4.a.g.1.1		1
208.109	odd	4		1014.4.b.d.337.2		2
208.125	odd	4		1014.4.b.d.337.1		2
240.29	odd	4		450.4.a.h.1.1		1
240.77	even	4		450.4.c.e.199.2		2
240.173	even	4		450.4.c.e.199.1		2
272.237	even	4		1734.4.a.d.1.1		1
304.189	odd	4		2166.4.a.i.1.1		1
336.125	even	4		882.4.a.n.1.1		1
336.173	even	12		882.4.g.f.361.1		2
336.221	odd	12		882.4.g.i.667.1		2
336.269	even	12		882.4.g.f.667.1		2
336.317	odd	12		882.4.g.i.361.1		2
528.461	even	4		2178.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.4.a.a.1.1	✓	1	16.13	even	4
18.4.a.a.1.1		1	48.29	odd	4
48.4.a.c.1.1		1	16.3	odd	4
144.4.a.c.1.1		1	48.35	even	4
150.4.a.i.1.1		1	80.29	even	4
150.4.c.d.49.1		2	80.77	odd	4
150.4.c.d.49.2		2	80.13	odd	4
162.4.c.c.55.1		2	144.77	odd	12
162.4.c.c.109.1		2	144.29	odd	12
162.4.c.f.55.1		2	144.13	even	12
162.4.c.f.109.1		2	144.61	even	12
192.4.a.c.1.1		1	16.11	odd	4
192.4.a.i.1.1		1	16.5	even	4
294.4.a.e.1.1		1	112.13	odd	4
294.4.e.g.67.1		2	112.61	odd	12
294.4.e.g.79.1		2	112.45	odd	12
294.4.e.h.67.1		2	112.93	even	12
294.4.e.h.79.1		2	112.109	even	12
450.4.a.h.1.1		1	240.29	odd	4
450.4.c.e.199.1		2	240.173	even	4
450.4.c.e.199.2		2	240.77	even	4
576.4.a.q.1.1		1	48.5	odd	4
576.4.a.r.1.1		1	48.11	even	4
726.4.a.f.1.1		1	176.109	odd	4
768.4.d.c.385.1		2	4.3	odd	2
768.4.d.c.385.2		2	8.3	odd	2
768.4.d.n.385.1		2	8.5	even	2	inner
768.4.d.n.385.2		2	1.1	even	1	trivial
882.4.a.n.1.1		1	336.125	even	4
882.4.g.f.361.1		2	336.173	even	12
882.4.g.f.667.1		2	336.269	even	12
882.4.g.i.361.1		2	336.317	odd	12
882.4.g.i.667.1		2	336.221	odd	12
1014.4.a.g.1.1		1	208.77	even	4
1014.4.b.d.337.1		2	208.125	odd	4
1014.4.b.d.337.2		2	208.109	odd	4
1200.4.a.b.1.1		1	80.19	odd	4
1200.4.f.j.49.1		2	80.67	even	4
1200.4.f.j.49.2		2	80.3	even	4
1734.4.a.d.1.1		1	272.237	even	4
2166.4.a.i.1.1		1	304.189	odd	4
2178.4.a.e.1.1		1	528.461	even	4
2352.4.a.e.1.1		1	112.83	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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +6.00000i q^{5} +16.0000 q^{7} -9.00000 q^{9} +12.0000i q^{11} -38.0000i q^{13} -18.0000 q^{15} -126.000 q^{17} -20.0000i q^{19} +48.0000i q^{21} -168.000 q^{23} +89.0000 q^{25} -27.0000i q^{27} -30.0000i q^{29} -88.0000 q^{31} -36.0000 q^{33} +96.0000i q^{35} +254.000i q^{37} +114.000 q^{39} -42.0000 q^{41} -52.0000i q^{43} -54.0000i q^{45} -96.0000 q^{47} -87.0000 q^{49} -378.000i q^{51} +198.000i q^{53} -72.0000 q^{55} +60.0000 q^{57} -660.000i q^{59} +538.000i q^{61} -144.000 q^{63} +228.000 q^{65} -884.000i q^{67} -504.000i q^{69} -792.000 q^{71} -218.000 q^{73} +267.000i q^{75} +192.000i q^{77} -520.000 q^{79} +81.0000 q^{81} +492.000i q^{83} -756.000i q^{85} +90.0000 q^{87} -810.000 q^{89} -608.000i q^{91} -264.000i q^{93} +120.000 q^{95} +1154.00 q^{97} -108.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{7} - 18 q^{9} - 36 q^{15} - 252 q^{17} - 336 q^{23} + 178 q^{25} - 176 q^{31} - 72 q^{33} + 228 q^{39} - 84 q^{41} - 192 q^{47} - 174 q^{49} - 144 q^{55} + 120 q^{57} - 288 q^{63} + 456 q^{65}+ \cdots + 2308 q^{97}+O(q^{100}) \) 2 * q + 32 * q^7 - 18 * q^9 - 36 * q^15 - 252 * q^17 - 336 * q^23 + 178 * q^25 - 176 * q^31 - 72 * q^33 + 228 * q^39 - 84 * q^41 - 192 * q^47 - 174 * q^49 - 144 * q^55 + 120 * q^57 - 288 * q^63 + 456 * q^65 - 1584 * q^71 - 436 * q^73 - 1040 * q^79 + 162 * q^81 + 180 * q^87 - 1620 * q^89 + 240 * q^95 + 2308 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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