LMFDB - Embedded newform 768.4.d.o.385.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,4,Mod(385,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,0,0,0,48,0,-18,0,0,0,0,0,84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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 24)
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.o.385.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -14.0000i q^{5} +24.0000 q^{7} -9.00000 q^{9} +28.0000i q^{11} -74.0000i q^{13} +42.0000 q^{15} +82.0000 q^{17} +92.0000i q^{19} +72.0000i q^{21} -8.00000 q^{23} -71.0000 q^{25} -27.0000i q^{27} -138.000i q^{29} +80.0000 q^{31} -84.0000 q^{33} -336.000i q^{35} -30.0000i q^{37} +222.000 q^{39} -282.000 q^{41} -4.00000i q^{43} +126.000i q^{45} +240.000 q^{47} +233.000 q^{49} +246.000i q^{51} +130.000i q^{53} +392.000 q^{55} -276.000 q^{57} -596.000i q^{59} -218.000i q^{61} -216.000 q^{63} -1036.00 q^{65} -436.000i q^{67} -24.0000i q^{69} -856.000 q^{71} +998.000 q^{73} -213.000i q^{75} +672.000i q^{77} -32.0000 q^{79} +81.0000 q^{81} -1508.00i q^{83} -1148.00i q^{85} +414.000 q^{87} +246.000 q^{89} -1776.00i q^{91} +240.000i q^{93} +1288.00 q^{95} +866.000 q^{97} -252.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 48 q^{7} - 18 q^{9} + 84 q^{15} + 164 q^{17} - 16 q^{23} - 142 q^{25} + 160 q^{31} - 168 q^{33} + 444 q^{39} - 564 q^{41} + 480 q^{47} + 466 q^{49} + 784 q^{55} - 552 q^{57} - 432 q^{63} - 2072 q^{65}+ \cdots + 1732 q^{97}+O(q^{100}) $ 2 * q + 48 * q^7 - 18 * q^9 + 84 * q^15 + 164 * q^17 - 16 * q^23 - 142 * q^25 + 160 * q^31 - 168 * q^33 + 444 * q^39 - 564 * q^41 + 480 * q^47 + 466 * q^49 + 784 * q^55 - 552 * q^57 - 432 * q^63 - 2072 * q^65 - 1712 * q^71 + 1996 * q^73 - 64 * q^79 + 162 * q^81 + 828 * q^87 + 492 * q^89 + 2576 * q^95 + 1732 * 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
$5$	− 14.0000i	− 1.25220i	−0.779744	−	0.626099i	$-0.784651\pi$
$5$	− 14.0000i	− 1.25220i	0.779744	−	0.626099i	$-0.215349\pi$
$6$	0	0
$7$	24.0000	1.29588	0.647939	−	0.761692i	$-0.275631\pi$
$7$	24.0000	1.29588	0.647939	+	0.761692i	$0.275631\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	28.0000i	0.767483i	0.923440	+	0.383742i	$0.125365\pi$
$11$	28.0000i	0.767483i	−0.923440	+	0.383742i	$0.874635\pi$
$12$	0	0
$13$	− 74.0000i	− 1.57876i	−0.613904	−	0.789381i	$-0.710402\pi$
$13$	− 74.0000i	− 1.57876i	0.613904	−	0.789381i	$-0.289598\pi$
$14$	0	0
$15$	42.0000	0.722957
$16$	0	0
$17$	82.0000	1.16988	0.584939	−	0.811077i	$-0.301118\pi$
$17$	82.0000	1.16988	0.584939	+	0.811077i	$0.301118\pi$
$18$	0	0
$19$	92.0000i	1.11086i	0.831565	+	0.555428i	$0.187445\pi$
$19$	92.0000i	1.11086i	−0.831565	+	0.555428i	$0.812555\pi$
$20$	0	0
$21$	72.0000i	0.748176i
$22$	0	0
$23$	−8.00000	−0.0725268	−0.0362634	−	0.999342i	$-0.511546\pi$
$23$	−8.00000	−0.0725268	−0.0362634	+	0.999342i	$0.511546\pi$
$24$	0	0
$25$	−71.0000	−0.568000
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 138.000i	− 0.883654i	−0.897100	−	0.441827i	$-0.854331\pi$
$29$	− 138.000i	− 0.883654i	0.897100	−	0.441827i	$-0.145669\pi$
$30$	0	0
$31$	80.0000	0.463498	0.231749	−	0.972776i	$-0.425555\pi$
$31$	80.0000	0.463498	0.231749	+	0.972776i	$0.425555\pi$
$32$	0	0
$33$	−84.0000	−0.443107
$34$	0	0
$35$	− 336.000i	− 1.62270i
$36$	0	0
$37$	− 30.0000i	− 0.133296i	−0.997777	−	0.0666482i	$-0.978769\pi$
$37$	− 30.0000i	− 0.133296i	0.997777	−	0.0666482i	$-0.0212305\pi$
$38$	0	0
$39$	222.000	0.911499
$40$	0	0
$41$	−282.000	−1.07417	−0.537085	−	0.843528i	$-0.680475\pi$
$41$	−282.000	−1.07417	−0.537085	+	0.843528i	$0.680475\pi$
$42$	0	0
$43$	− 4.00000i	− 0.0141859i	−0.999975	−	0.00709296i	$-0.997742\pi$
$43$	− 4.00000i	− 0.0141859i	0.999975	−	0.00709296i	$-0.00225778\pi$
$44$	0	0
$45$	126.000i	0.417399i
$46$	0	0
$47$	240.000	0.744843	0.372421	−	0.928064i	$-0.378528\pi$
$47$	240.000	0.744843	0.372421	+	0.928064i	$0.378528\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	0	0
$51$	246.000i	0.675429i
$52$	0	0
$53$	130.000i	0.336922i	0.985708	+	0.168461i	$0.0538797\pi$
$53$	130.000i	0.336922i	−0.985708	+	0.168461i	$0.946120\pi$
$54$	0	0
$55$	392.000	0.961041
$56$	0	0
$57$	−276.000	−0.641353
$58$	0	0
$59$	− 596.000i	− 1.31513i	−0.753398	−	0.657564i	$-0.771587\pi$
$59$	− 596.000i	− 1.31513i	0.753398	−	0.657564i	$-0.228413\pi$
$60$	0	0
$61$	− 218.000i	− 0.457574i	−0.973476	−	0.228787i	$-0.926524\pi$
$61$	− 218.000i	− 0.457574i	0.973476	−	0.228787i	$-0.0734760\pi$
$62$	0	0
$63$	−216.000	−0.431959
$64$	0	0
$65$	−1036.00	−1.97692
$66$	0	0
$67$	− 436.000i	− 0.795013i	−0.917599	−	0.397507i	$-0.869876\pi$
$67$	− 436.000i	− 0.795013i	0.917599	−	0.397507i	$-0.130124\pi$
$68$	0	0
$69$	− 24.0000i	− 0.0418733i
$70$	0	0
$71$	−856.000	−1.43082	−0.715412	−	0.698703i	$-0.753761\pi$
$71$	−856.000	−1.43082	−0.715412	+	0.698703i	$0.753761\pi$
$72$	0	0
$73$	998.000	1.60010	0.800048	−	0.599935i	$-0.204807\pi$
$73$	998.000	1.60010	0.800048	+	0.599935i	$0.204807\pi$
$74$	0	0
$75$	− 213.000i	− 0.327935i
$76$	0	0
$77$	672.000i	0.994565i
$78$	0	0
$79$	−32.0000	−0.0455732	−0.0227866	−	0.999740i	$-0.507254\pi$
$79$	−32.0000	−0.0455732	−0.0227866	+	0.999740i	$0.507254\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1508.00i	− 1.99427i	−0.0756351	−	0.997136i	$-0.524098\pi$
$83$	− 1508.00i	− 1.99427i	0.0756351	−	0.997136i	$-0.475902\pi$
$84$	0	0
$85$	− 1148.00i	− 1.46492i
$86$	0	0
$87$	414.000	0.510178
$88$	0	0
$89$	246.000	0.292988	0.146494	−	0.989212i	$-0.453201\pi$
$89$	246.000	0.292988	0.146494	+	0.989212i	$0.453201\pi$
$90$	0	0
$91$	− 1776.00i	− 2.04588i
$92$	0	0
$93$	240.000i	0.267600i
$94$	0	0
$95$	1288.00	1.39101
$96$	0	0
$97$	866.000	0.906484	0.453242	−	0.891387i	$-0.350267\pi$
$97$	866.000	0.906484	0.453242	+	0.891387i	$0.350267\pi$
$98$	0	0
$99$	− 252.000i	− 0.255828i
$100$	0	0
$101$	− 270.000i	− 0.266000i	−0.991116	−	0.133000i	$-0.957539\pi$
$101$	− 270.000i	− 0.266000i	0.991116	−	0.133000i	$-0.0424610\pi$
$102$	0	0
$103$	1496.00	1.43112	0.715560	−	0.698552i	$-0.246172\pi$
$103$	1496.00	1.43112	0.715560	+	0.698552i	$0.246172\pi$
$104$	0	0
$105$	1008.00	0.936864
$106$	0	0
$107$	1692.00i	1.52871i	0.644797	+	0.764354i	$0.276942\pi$
$107$	1692.00i	1.52871i	−0.644797	+	0.764354i	$0.723058\pi$
$108$	0	0
$109$	406.000i	0.356768i	0.983961	+	0.178384i	$0.0570870\pi$
$109$	406.000i	0.356768i	−0.983961	+	0.178384i	$0.942913\pi$
$110$	0	0
$111$	90.0000	0.0769588
$112$	0	0
$113$	786.000	0.654342	0.327171	−	0.944965i	$-0.393905\pi$
$113$	786.000	0.654342	0.327171	+	0.944965i	$0.393905\pi$
$114$	0	0
$115$	112.000i	0.0908179i
$116$	0	0
$117$	666.000i	0.526254i
$118$	0	0
$119$	1968.00	1.51602
$120$	0	0
$121$	547.000	0.410969
$122$	0	0
$123$	− 846.000i	− 0.620173i
$124$	0	0
$125$	− 756.000i	− 0.540950i
$126$	0	0
$127$	1744.00	1.21854	0.609272	−	0.792962i	$-0.291462\pi$
$127$	1744.00	1.21854	0.609272	+	0.792962i	$0.291462\pi$
$128$	0	0
$129$	12.0000	0.00819024
$130$	0	0
$131$	652.000i	0.434851i	0.976077	+	0.217426i	$0.0697659\pi$
$131$	652.000i	0.434851i	−0.976077	+	0.217426i	$0.930234\pi$
$132$	0	0
$133$	2208.00i	1.43953i
$134$	0	0
$135$	−378.000	−0.240986
$136$	0	0
$137$	−1530.00	−0.954137	−0.477068	−	0.878866i	$-0.658301\pi$
$137$	−1530.00	−0.954137	−0.477068	+	0.878866i	$0.658301\pi$
$138$	0	0
$139$	− 516.000i	− 0.314867i	−0.987530	−	0.157434i	$-0.949678\pi$
$139$	− 516.000i	− 0.314867i	0.987530	−	0.157434i	$-0.0503220\pi$
$140$	0	0
$141$	720.000i	0.430035i
$142$	0	0
$143$	2072.00	1.21167
$144$	0	0
$145$	−1932.00	−1.10651
$146$	0	0
$147$	699.000i	0.392194i
$148$	0	0
$149$	− 1342.00i	− 0.737859i	−0.929457	−	0.368929i	$-0.879724\pi$
$149$	− 1342.00i	− 0.737859i	0.929457	−	0.368929i	$-0.120276\pi$
$150$	0	0
$151$	424.000	0.228507	0.114254	−	0.993452i	$-0.463552\pi$
$151$	424.000	0.228507	0.114254	+	0.993452i	$0.463552\pi$
$152$	0	0
$153$	−738.000	−0.389959
$154$	0	0
$155$	− 1120.00i	− 0.580391i
$156$	0	0
$157$	262.000i	0.133184i	0.997780	+	0.0665920i	$0.0212126\pi$
$157$	262.000i	0.133184i	−0.997780	+	0.0665920i	$0.978787\pi$
$158$	0	0
$159$	−390.000	−0.194522
$160$	0	0
$161$	−192.000	−0.0939858
$162$	0	0
$163$	− 2292.00i	− 1.10137i	−0.834713	−	0.550685i	$-0.814367\pi$
$163$	− 2292.00i	− 1.10137i	0.834713	−	0.550685i	$-0.185633\pi$
$164$	0	0
$165$	1176.00i	0.554857i
$166$	0	0
$167$	1896.00	0.878544	0.439272	−	0.898354i	$-0.355236\pi$
$167$	1896.00	0.878544	0.439272	+	0.898354i	$0.355236\pi$
$168$	0	0
$169$	−3279.00	−1.49249
$170$	0	0
$171$	− 828.000i	− 0.370285i
$172$	0	0
$173$	− 2874.00i	− 1.26304i	−0.775359	−	0.631521i	$-0.782431\pi$
$173$	− 2874.00i	− 1.26304i	0.775359	−	0.631521i	$-0.217569\pi$
$174$	0	0
$175$	−1704.00	−0.736059
$176$	0	0
$177$	1788.00	0.759290
$178$	0	0
$179$	− 1188.00i	− 0.496063i	−0.968752	−	0.248032i	$-0.920216\pi$
$179$	− 1188.00i	− 0.496063i	0.968752	−	0.248032i	$-0.0797837\pi$
$180$	0	0
$181$	3474.00i	1.42663i	0.700843	+	0.713316i	$0.252808\pi$
$181$	3474.00i	1.42663i	−0.700843	+	0.713316i	$0.747192\pi$
$182$	0	0
$183$	654.000	0.264181
$184$	0	0
$185$	−420.000	−0.166914
$186$	0	0
$187$	2296.00i	0.897862i
$188$	0	0
$189$	− 648.000i	− 0.249392i
$190$	0	0
$191$	192.000	0.0727363	0.0363681	−	0.999338i	$-0.488421\pi$
$191$	192.000	0.0727363	0.0363681	+	0.999338i	$0.488421\pi$
$192$	0	0
$193$	4802.00	1.79096	0.895481	−	0.445100i	$-0.146832\pi$
$193$	4802.00	1.79096	0.895481	+	0.445100i	$0.146832\pi$
$194$	0	0
$195$	− 3108.00i	− 1.14138i
$196$	0	0
$197$	− 1518.00i	− 0.549000i	−0.961587	−	0.274500i	$-0.911488\pi$
$197$	− 1518.00i	− 0.549000i	0.961587	−	0.274500i	$-0.0885123\pi$
$198$	0	0
$199$	−5128.00	−1.82670	−0.913352	−	0.407170i	$-0.866516\pi$
$199$	−5128.00	−1.82670	−0.913352	+	0.407170i	$0.866516\pi$
$200$	0	0
$201$	1308.00	0.459001
$202$	0	0
$203$	− 3312.00i	− 1.14511i
$204$	0	0
$205$	3948.00i	1.34507i
$206$	0	0
$207$	72.0000	0.0241756
$208$	0	0
$209$	−2576.00	−0.852563
$210$	0	0
$211$	1084.00i	0.353676i	0.984240	+	0.176838i	$0.0565869\pi$
$211$	1084.00i	0.353676i	−0.984240	+	0.176838i	$0.943413\pi$
$212$	0	0
$213$	− 2568.00i	− 0.826087i
$214$	0	0
$215$	−56.0000	−0.0177636
$216$	0	0
$217$	1920.00	0.600636
$218$	0	0
$219$	2994.00i	0.923816i
$220$	0	0
$221$	− 6068.00i	− 1.84696i
$222$	0	0
$223$	688.000	0.206600	0.103300	−	0.994650i	$-0.467060\pi$
$223$	688.000	0.206600	0.103300	+	0.994650i	$0.467060\pi$
$224$	0	0
$225$	639.000	0.189333
$226$	0	0
$227$	4812.00i	1.40698i	0.710707	+	0.703488i	$0.248375\pi$
$227$	4812.00i	1.40698i	−0.710707	+	0.703488i	$0.751625\pi$
$228$	0	0
$229$	− 2494.00i	− 0.719686i	−0.933013	−	0.359843i	$-0.882830\pi$
$229$	− 2494.00i	− 0.719686i	0.933013	−	0.359843i	$-0.117170\pi$
$230$	0	0
$231$	−2016.00	−0.574212
$232$	0	0
$233$	−698.000	−0.196255	−0.0981277	−	0.995174i	$-0.531285\pi$
$233$	−698.000	−0.196255	−0.0981277	+	0.995174i	$0.531285\pi$
$234$	0	0
$235$	− 3360.00i	− 0.932690i
$236$	0	0
$237$	− 96.0000i	− 0.0263117i
$238$	0	0
$239$	−6320.00	−1.71049	−0.855244	−	0.518225i	$-0.826593\pi$
$239$	−6320.00	−1.71049	−0.855244	+	0.518225i	$0.826593\pi$
$240$	0	0
$241$	−6510.00	−1.74002	−0.870012	−	0.493030i	$-0.835889\pi$
$241$	−6510.00	−1.74002	−0.870012	+	0.493030i	$0.835889\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 3262.00i	− 0.850619i
$246$	0	0
$247$	6808.00	1.75378
$248$	0	0
$249$	4524.00	1.15139
$250$	0	0
$251$	− 628.000i	− 0.157924i	−0.996878	−	0.0789622i	$-0.974839\pi$
$251$	− 628.000i	− 0.157924i	0.996878	−	0.0789622i	$-0.0251606\pi$
$252$	0	0
$253$	− 224.000i	− 0.0556631i
$254$	0	0
$255$	3444.00	0.845771
$256$	0	0
$257$	−4862.00	−1.18009	−0.590045	−	0.807370i	$-0.700890\pi$
$257$	−4862.00	−1.18009	−0.590045	+	0.807370i	$0.700890\pi$
$258$	0	0
$259$	− 720.000i	− 0.172736i
$260$	0	0
$261$	1242.00i	0.294551i
$262$	0	0
$263$	−5816.00	−1.36361	−0.681806	−	0.731533i	$-0.738805\pi$
$263$	−5816.00	−1.36361	−0.681806	+	0.731533i	$0.738805\pi$
$264$	0	0
$265$	1820.00	0.421893
$266$	0	0
$267$	738.000i	0.169157i
$268$	0	0
$269$	3526.00i	0.799197i	0.916690	+	0.399599i	$0.130850\pi$
$269$	3526.00i	0.799197i	−0.916690	+	0.399599i	$0.869150\pi$
$270$	0	0
$271$	−256.000	−0.0573834	−0.0286917	−	0.999588i	$-0.509134\pi$
$271$	−256.000	−0.0573834	−0.0286917	+	0.999588i	$0.509134\pi$
$272$	0	0
$273$	5328.00	1.18119
$274$	0	0
$275$	− 1988.00i	− 0.435931i
$276$	0	0
$277$	− 142.000i	− 0.0308013i	−0.999881	−	0.0154006i	$-0.995098\pi$
$277$	− 142.000i	− 0.0308013i	0.999881	−	0.0154006i	$-0.00490237\pi$
$278$	0	0
$279$	−720.000	−0.154499
$280$	0	0
$281$	−8842.00	−1.87712	−0.938558	−	0.345122i	$-0.887838\pi$
$281$	−8842.00	−1.87712	−0.938558	+	0.345122i	$0.887838\pi$
$282$	0	0
$283$	7180.00i	1.50815i	0.656788	+	0.754075i	$0.271915\pi$
$283$	7180.00i	1.50815i	−0.656788	+	0.754075i	$0.728085\pi$
$284$	0	0
$285$	3864.00i	0.803100i
$286$	0	0
$287$	−6768.00	−1.39199
$288$	0	0
$289$	1811.00	0.368614
$290$	0	0
$291$	2598.00i	0.523359i
$292$	0	0
$293$	− 7374.00i	− 1.47029i	−0.677912	−	0.735143i	$-0.737115\pi$
$293$	− 7374.00i	− 1.47029i	0.677912	−	0.735143i	$-0.262885\pi$
$294$	0	0
$295$	−8344.00	−1.64680
$296$	0	0
$297$	756.000	0.147702
$298$	0	0
$299$	592.000i	0.114502i
$300$	0	0
$301$	− 96.0000i	− 0.0183832i
$302$	0	0
$303$	810.000	0.153575
$304$	0	0
$305$	−3052.00	−0.572974
$306$	0	0
$307$	1500.00i	0.278858i	0.990232	+	0.139429i	$0.0445268\pi$
$307$	1500.00i	0.278858i	−0.990232	+	0.139429i	$0.955473\pi$
$308$	0	0
$309$	4488.00i	0.826257i
$310$	0	0
$311$	7608.00	1.38717	0.693585	−	0.720374i	$-0.256030\pi$
$311$	7608.00	1.38717	0.693585	+	0.720374i	$0.256030\pi$
$312$	0	0
$313$	4758.00	0.859227	0.429614	−	0.903013i	$-0.358650\pi$
$313$	4758.00	0.859227	0.429614	+	0.903013i	$0.358650\pi$
$314$	0	0
$315$	3024.00i	0.540899i
$316$	0	0
$317$	4374.00i	0.774979i	0.921874	+	0.387489i	$0.126658\pi$
$317$	4374.00i	0.774979i	−0.921874	+	0.387489i	$0.873342\pi$
$318$	0	0
$319$	3864.00	0.678190
$320$	0	0
$321$	−5076.00	−0.882600
$322$	0	0
$323$	7544.00i	1.29956i
$324$	0	0
$325$	5254.00i	0.896737i
$326$	0	0
$327$	−1218.00	−0.205980
$328$	0	0
$329$	5760.00	0.965225
$330$	0	0
$331$	7804.00i	1.29591i	0.761678	+	0.647956i	$0.224376\pi$
$331$	7804.00i	1.29591i	−0.761678	+	0.647956i	$0.775624\pi$
$332$	0	0
$333$	270.000i	0.0444322i
$334$	0	0
$335$	−6104.00	−0.995514
$336$	0	0
$337$	5106.00	0.825346	0.412673	−	0.910879i	$-0.364595\pi$
$337$	5106.00	0.825346	0.412673	+	0.910879i	$0.364595\pi$
$338$	0	0
$339$	2358.00i	0.377785i
$340$	0	0
$341$	2240.00i	0.355727i
$342$	0	0
$343$	−2640.00	−0.415588
$344$	0	0
$345$	−336.000	−0.0524337
$346$	0	0
$347$	4716.00i	0.729591i	0.931088	+	0.364796i	$0.118861\pi$
$347$	4716.00i	0.729591i	−0.931088	+	0.364796i	$0.881139\pi$
$348$	0	0
$349$	7302.00i	1.11996i	0.828505	+	0.559982i	$0.189192\pi$
$349$	7302.00i	1.11996i	−0.828505	+	0.559982i	$0.810808\pi$
$350$	0	0
$351$	−1998.00	−0.303833
$352$	0	0
$353$	−4382.00	−0.660709	−0.330355	−	0.943857i	$-0.607168\pi$
$353$	−4382.00	−0.660709	−0.330355	+	0.943857i	$0.607168\pi$
$354$	0	0
$355$	11984.0i	1.79168i
$356$	0	0
$357$	5904.00i	0.875274i
$358$	0	0
$359$	−7224.00	−1.06203	−0.531014	−	0.847363i	$-0.678189\pi$
$359$	−7224.00	−1.06203	−0.531014	+	0.847363i	$0.678189\pi$
$360$	0	0
$361$	−1605.00	−0.233999
$362$	0	0
$363$	1641.00i	0.237273i
$364$	0	0
$365$	− 13972.0i	− 2.00364i
$366$	0	0
$367$	1408.00	0.200264	0.100132	−	0.994974i	$-0.468073\pi$
$367$	1408.00	0.200264	0.100132	+	0.994974i	$0.468073\pi$
$368$	0	0
$369$	2538.00	0.358057
$370$	0	0
$371$	3120.00i	0.436610i
$372$	0	0
$373$	1714.00i	0.237929i	0.992899	+	0.118965i	$0.0379575\pi$
$373$	1714.00i	0.237929i	−0.992899	+	0.118965i	$0.962043\pi$
$374$	0	0
$375$	2268.00	0.312317
$376$	0	0
$377$	−10212.0	−1.39508
$378$	0	0
$379$	− 884.000i	− 0.119810i	−0.998204	−	0.0599051i	$-0.980920\pi$
$379$	− 884.000i	− 0.119810i	0.998204	−	0.0599051i	$-0.0190798\pi$
$380$	0	0
$381$	5232.00i	0.703526i
$382$	0	0
$383$	10368.0	1.38324	0.691619	−	0.722263i	$-0.256898\pi$
$383$	10368.0	1.38324	0.691619	+	0.722263i	$0.256898\pi$
$384$	0	0
$385$	9408.00	1.24539
$386$	0	0
$387$	36.0000i	0.00472864i
$388$	0	0
$389$	− 398.000i	− 0.0518751i	−0.999664	−	0.0259375i	$-0.991743\pi$
$389$	− 398.000i	− 0.0518751i	0.999664	−	0.0259375i	$-0.00825710\pi$
$390$	0	0
$391$	−656.000	−0.0848474
$392$	0	0
$393$	−1956.00	−0.251061
$394$	0	0
$395$	448.000i	0.0570666i
$396$	0	0
$397$	− 5098.00i	− 0.644487i	−0.946657	−	0.322243i	$-0.895563\pi$
$397$	− 5098.00i	− 0.644487i	0.946657	−	0.322243i	$-0.104437\pi$
$398$	0	0
$399$	−6624.00	−0.831115
$400$	0	0
$401$	10002.0	1.24558	0.622788	−	0.782391i	$-0.286000\pi$
$401$	10002.0	1.24558	0.622788	+	0.782391i	$0.286000\pi$
$402$	0	0
$403$	− 5920.00i	− 0.731752i
$404$	0	0
$405$	− 1134.00i	− 0.139133i
$406$	0	0
$407$	840.000	0.102303
$408$	0	0
$409$	9270.00	1.12071	0.560357	−	0.828251i	$-0.310664\pi$
$409$	9270.00	1.12071	0.560357	+	0.828251i	$0.310664\pi$
$410$	0	0
$411$	− 4590.00i	− 0.550871i
$412$	0	0
$413$	− 14304.0i	− 1.70425i
$414$	0	0
$415$	−21112.0	−2.49722
$416$	0	0
$417$	1548.00	0.181789
$418$	0	0
$419$	− 6516.00i	− 0.759731i	−0.925042	−	0.379866i	$-0.875970\pi$
$419$	− 6516.00i	− 0.759731i	0.925042	−	0.379866i	$-0.124030\pi$
$420$	0	0
$421$	2626.00i	0.303999i	0.988381	+	0.151999i	$0.0485711\pi$
$421$	2626.00i	0.303999i	−0.988381	+	0.151999i	$0.951429\pi$
$422$	0	0
$423$	−2160.00	−0.248281
$424$	0	0
$425$	−5822.00	−0.664491
$426$	0	0
$427$	− 5232.00i	− 0.592961i
$428$	0	0
$429$	6216.00i	0.699560i
$430$	0	0
$431$	−4304.00	−0.481012	−0.240506	−	0.970648i	$-0.577313\pi$
$431$	−4304.00	−0.481012	−0.240506	+	0.970648i	$0.577313\pi$
$432$	0	0
$433$	11794.0	1.30897	0.654484	−	0.756076i	$-0.272886\pi$
$433$	11794.0	1.30897	0.654484	+	0.756076i	$0.272886\pi$
$434$	0	0
$435$	− 5796.00i	− 0.638844i
$436$	0	0
$437$	− 736.000i	− 0.0805667i
$438$	0	0
$439$	5544.00	0.602735	0.301368	−	0.953508i	$-0.402557\pi$
$439$	5544.00	0.602735	0.301368	+	0.953508i	$0.402557\pi$
$440$	0	0
$441$	−2097.00	−0.226433
$442$	0	0
$443$	3788.00i	0.406260i	0.979152	+	0.203130i	$0.0651115\pi$
$443$	3788.00i	0.406260i	−0.979152	+	0.203130i	$0.934889\pi$
$444$	0	0
$445$	− 3444.00i	− 0.366879i
$446$	0	0
$447$	4026.00	0.426003
$448$	0	0
$449$	−13342.0	−1.40233	−0.701167	−	0.712997i	$-0.747337\pi$
$449$	−13342.0	−1.40233	−0.701167	+	0.712997i	$0.747337\pi$
$450$	0	0
$451$	− 7896.00i	− 0.824408i
$452$	0	0
$453$	1272.00i	0.131929i
$454$	0	0
$455$	−24864.0	−2.56185
$456$	0	0
$457$	4390.00	0.449356	0.224678	−	0.974433i	$-0.427867\pi$
$457$	4390.00	0.449356	0.224678	+	0.974433i	$0.427867\pi$
$458$	0	0
$459$	− 2214.00i	− 0.225143i
$460$	0	0
$461$	5798.00i	0.585770i	0.956148	+	0.292885i	$0.0946152\pi$
$461$	5798.00i	0.585770i	−0.956148	+	0.292885i	$0.905385\pi$
$462$	0	0
$463$	−14656.0	−1.47111	−0.735553	−	0.677467i	$-0.763078\pi$
$463$	−14656.0	−1.47111	−0.735553	+	0.677467i	$0.763078\pi$
$464$	0	0
$465$	3360.00	0.335089
$466$	0	0
$467$	8412.00i	0.833535i	0.909013	+	0.416768i	$0.136837\pi$
$467$	8412.00i	0.833535i	−0.909013	+	0.416768i	$0.863163\pi$
$468$	0	0
$469$	− 10464.0i	− 1.03024i
$470$	0	0
$471$	−786.000	−0.0768938
$472$	0	0
$473$	112.000	0.0108875
$474$	0	0
$475$	− 6532.00i	− 0.630966i
$476$	0	0
$477$	− 1170.00i	− 0.112307i
$478$	0	0
$479$	14848.0	1.41633	0.708165	−	0.706047i	$-0.249523\pi$
$479$	14848.0	1.41633	0.708165	+	0.706047i	$0.249523\pi$
$480$	0	0
$481$	−2220.00	−0.210443
$482$	0	0
$483$	− 576.000i	− 0.0542627i
$484$	0	0
$485$	− 12124.0i	− 1.13510i
$486$	0	0
$487$	−18568.0	−1.72771	−0.863857	−	0.503738i	$-0.831958\pi$
$487$	−18568.0	−1.72771	−0.863857	+	0.503738i	$0.831958\pi$
$488$	0	0
$489$	6876.00	0.635876
$490$	0	0
$491$	14364.0i	1.32024i	0.751160	+	0.660120i	$0.229495\pi$
$491$	14364.0i	1.32024i	−0.751160	+	0.660120i	$0.770505\pi$
$492$	0	0
$493$	− 11316.0i	− 1.03377i
$494$	0	0
$495$	−3528.00	−0.320347
$496$	0	0
$497$	−20544.0	−1.85417
$498$	0	0
$499$	21660.0i	1.94316i	0.236720	+	0.971578i	$0.423928\pi$
$499$	21660.0i	1.94316i	−0.236720	+	0.971578i	$0.576072\pi$
$500$	0	0
$501$	5688.00i	0.507228i
$502$	0	0
$503$	17112.0	1.51687	0.758436	−	0.651748i	$-0.225964\pi$
$503$	17112.0	1.51687	0.758436	+	0.651748i	$0.225964\pi$
$504$	0	0
$505$	−3780.00	−0.333085
$506$	0	0
$507$	− 9837.00i	− 0.861689i
$508$	0	0
$509$	11478.0i	0.999516i	0.866165	+	0.499758i	$0.166578\pi$
$509$	11478.0i	0.999516i	−0.866165	+	0.499758i	$0.833422\pi$
$510$	0	0
$511$	23952.0	2.07353
$512$	0	0
$513$	2484.00	0.213784
$514$	0	0
$515$	− 20944.0i	− 1.79204i
$516$	0	0
$517$	6720.00i	0.571654i
$518$	0	0
$519$	8622.00	0.729217
$520$	0	0
$521$	−13114.0	−1.10275	−0.551377	−	0.834256i	$-0.685897\pi$
$521$	−13114.0	−1.10275	−0.551377	+	0.834256i	$0.685897\pi$
$522$	0	0
$523$	4508.00i	0.376905i	0.982082	+	0.188452i	$0.0603471\pi$
$523$	4508.00i	0.376905i	−0.982082	+	0.188452i	$0.939653\pi$
$524$	0	0
$525$	− 5112.00i	− 0.424964i
$526$	0	0
$527$	6560.00	0.542235
$528$	0	0
$529$	−12103.0	−0.994740
$530$	0	0
$531$	5364.00i	0.438376i
$532$	0	0
$533$	20868.0i	1.69586i
$534$	0	0
$535$	23688.0	1.91425
$536$	0	0
$537$	3564.00	0.286402
$538$	0	0
$539$	6524.00i	0.521352i
$540$	0	0
$541$	22950.0i	1.82384i	0.410368	+	0.911920i	$0.365400\pi$
$541$	22950.0i	1.82384i	−0.410368	+	0.911920i	$0.634600\pi$
$542$	0	0
$543$	−10422.0	−0.823666
$544$	0	0
$545$	5684.00	0.446745
$546$	0	0
$547$	− 6580.00i	− 0.514334i	−0.966367	−	0.257167i	$-0.917211\pi$
$547$	− 6580.00i	− 0.514334i	0.966367	−	0.257167i	$-0.0827890\pi$
$548$	0	0
$549$	1962.00i	0.152525i
$550$	0	0
$551$	12696.0	0.981611
$552$	0	0
$553$	−768.000	−0.0590573
$554$	0	0
$555$	− 1260.00i	− 0.0963676i
$556$	0	0
$557$	7046.00i	0.535994i	0.963420	+	0.267997i	$0.0863617\pi$
$557$	7046.00i	0.535994i	−0.963420	+	0.267997i	$0.913638\pi$
$558$	0	0
$559$	−296.000	−0.0223962
$560$	0	0
$561$	−6888.00	−0.518381
$562$	0	0
$563$	8252.00i	0.617727i	0.951106	+	0.308864i	$0.0999486\pi$
$563$	8252.00i	0.617727i	−0.951106	+	0.308864i	$0.900051\pi$
$564$	0	0
$565$	− 11004.0i	− 0.819366i
$566$	0	0
$567$	1944.00	0.143986
$568$	0	0
$569$	6838.00	0.503803	0.251901	−	0.967753i	$-0.418944\pi$
$569$	6838.00	0.503803	0.251901	+	0.967753i	$0.418944\pi$
$570$	0	0
$571$	− 23316.0i	− 1.70883i	−0.519588	−	0.854417i	$-0.673915\pi$
$571$	− 23316.0i	− 1.70883i	0.519588	−	0.854417i	$-0.326085\pi$
$572$	0	0
$573$	576.000i	0.0419943i
$574$	0	0
$575$	568.000	0.0411952
$576$	0	0
$577$	−10558.0	−0.761760	−0.380880	−	0.924625i	$-0.624379\pi$
$577$	−10558.0	−0.761760	−0.380880	+	0.924625i	$0.624379\pi$
$578$	0	0
$579$	14406.0i	1.03401i
$580$	0	0
$581$	− 36192.0i	− 2.58433i
$582$	0	0
$583$	−3640.00	−0.258582
$584$	0	0
$585$	9324.00	0.658974
$586$	0	0
$587$	− 1028.00i	− 0.0722830i	−0.999347	−	0.0361415i	$-0.988493\pi$
$587$	− 1028.00i	− 0.0722830i	0.999347	−	0.0361415i	$-0.0115067\pi$
$588$	0	0
$589$	7360.00i	0.514879i
$590$	0	0
$591$	4554.00	0.316965
$592$	0	0
$593$	1202.00	0.0832382	0.0416191	−	0.999134i	$-0.486748\pi$
$593$	1202.00	0.0832382	0.0416191	+	0.999134i	$0.486748\pi$
$594$	0	0
$595$	− 27552.0i	− 1.89836i
$596$	0	0
$597$	− 15384.0i	− 1.05465i
$598$	0	0
$599$	3576.00	0.243926	0.121963	−	0.992535i	$-0.461081\pi$
$599$	3576.00	0.243926	0.121963	+	0.992535i	$0.461081\pi$
$600$	0	0
$601$	−8650.00	−0.587090	−0.293545	−	0.955945i	$-0.594835\pi$
$601$	−8650.00	−0.587090	−0.293545	+	0.955945i	$0.594835\pi$
$602$	0	0
$603$	3924.00i	0.265004i
$604$	0	0
$605$	− 7658.00i	− 0.514615i
$606$	0	0
$607$	12656.0	0.846279	0.423139	−	0.906065i	$-0.360928\pi$
$607$	12656.0	0.846279	0.423139	+	0.906065i	$0.360928\pi$
$608$	0	0
$609$	9936.00	0.661128
$610$	0	0
$611$	− 17760.0i	− 1.17593i
$612$	0	0
$613$	3298.00i	0.217300i	0.994080	+	0.108650i	$0.0346528\pi$
$613$	3298.00i	0.217300i	−0.994080	+	0.108650i	$0.965347\pi$
$614$	0	0
$615$	−11844.0	−0.776579
$616$	0	0
$617$	−5370.00	−0.350386	−0.175193	−	0.984534i	$-0.556055\pi$
$617$	−5370.00	−0.350386	−0.175193	+	0.984534i	$0.556055\pi$
$618$	0	0
$619$	16220.0i	1.05321i	0.850110	+	0.526605i	$0.176535\pi$
$619$	16220.0i	1.05321i	−0.850110	+	0.526605i	$0.823465\pi$
$620$	0	0
$621$	216.000i	0.0139578i
$622$	0	0
$623$	5904.00	0.379677
$624$	0	0
$625$	−19459.0	−1.24538
$626$	0	0
$627$	− 7728.00i	− 0.492227i
$628$	0	0
$629$	− 2460.00i	− 0.155941i
$630$	0	0
$631$	20360.0	1.28450	0.642249	−	0.766496i	$-0.278001\pi$
$631$	20360.0	1.28450	0.642249	+	0.766496i	$0.278001\pi$
$632$	0	0
$633$	−3252.00	−0.204195
$634$	0	0
$635$	− 24416.0i	− 1.52586i
$636$	0	0
$637$	− 17242.0i	− 1.07245i
$638$	0	0
$639$	7704.00	0.476941
$640$	0	0
$641$	14498.0	0.893349	0.446674	−	0.894697i	$-0.352608\pi$
$641$	14498.0	0.893349	0.446674	+	0.894697i	$0.352608\pi$
$642$	0	0
$643$	21612.0i	1.32550i	0.748842	+	0.662748i	$0.230610\pi$
$643$	21612.0i	1.32550i	−0.748842	+	0.662748i	$0.769390\pi$
$644$	0	0
$645$	− 168.000i	− 0.0102558i
$646$	0	0
$647$	−12184.0	−0.740344	−0.370172	−	0.928963i	$-0.620701\pi$
$647$	−12184.0	−0.740344	−0.370172	+	0.928963i	$0.620701\pi$
$648$	0	0
$649$	16688.0	1.00934
$650$	0	0
$651$	5760.00i	0.346778i
$652$	0	0
$653$	− 28122.0i	− 1.68530i	−0.538464	−	0.842648i	$-0.680995\pi$
$653$	− 28122.0i	− 1.68530i	0.538464	−	0.842648i	$-0.319005\pi$
$654$	0	0
$655$	9128.00	0.544520
$656$	0	0
$657$	−8982.00	−0.533366
$658$	0	0
$659$	− 5700.00i	− 0.336935i	−0.985707	−	0.168468i	$-0.946118\pi$
$659$	− 5700.00i	− 0.336935i	0.985707	−	0.168468i	$-0.0538819\pi$
$660$	0	0
$661$	29458.0i	1.73341i	0.498822	+	0.866705i	$0.333766\pi$
$661$	29458.0i	1.73341i	−0.498822	+	0.866705i	$0.666234\pi$
$662$	0	0
$663$	18204.0	1.06634
$664$	0	0
$665$	30912.0	1.80258
$666$	0	0
$667$	1104.00i	0.0640885i
$668$	0	0
$669$	2064.00i	0.119281i
$670$	0	0
$671$	6104.00	0.351181
$672$	0	0
$673$	19810.0	1.13465	0.567325	−	0.823494i	$-0.307978\pi$
$673$	19810.0	1.13465	0.567325	+	0.823494i	$0.307978\pi$
$674$	0	0
$675$	1917.00i	0.109312i
$676$	0	0
$677$	10450.0i	0.593244i	0.954995	+	0.296622i	$0.0958601\pi$
$677$	10450.0i	0.593244i	−0.954995	+	0.296622i	$0.904140\pi$
$678$	0	0
$679$	20784.0	1.17469
$680$	0	0
$681$	−14436.0	−0.812318
$682$	0	0
$683$	− 23300.0i	− 1.30534i	−0.757641	−	0.652672i	$-0.773648\pi$
$683$	− 23300.0i	− 1.30534i	0.757641	−	0.652672i	$-0.226352\pi$
$684$	0	0
$685$	21420.0i	1.19477i
$686$	0	0
$687$	7482.00	0.415511
$688$	0	0
$689$	9620.00	0.531920
$690$	0	0
$691$	− 14212.0i	− 0.782417i	−0.920302	−	0.391208i	$-0.872057\pi$
$691$	− 14212.0i	− 0.782417i	0.920302	−	0.391208i	$-0.127943\pi$
$692$	0	0
$693$	− 6048.00i	− 0.331522i
$694$	0	0
$695$	−7224.00	−0.394276
$696$	0	0
$697$	−23124.0	−1.25665
$698$	0	0
$699$	− 2094.00i	− 0.113308i
$700$	0	0
$701$	− 15978.0i	− 0.860885i	−0.902618	−	0.430443i	$-0.858357\pi$
$701$	− 15978.0i	− 0.860885i	0.902618	−	0.430443i	$-0.141643\pi$
$702$	0	0
$703$	2760.00	0.148073
$704$	0	0
$705$	10080.0	0.538489
$706$	0	0
$707$	− 6480.00i	− 0.344704i
$708$	0	0
$709$	8866.00i	0.469633i	0.972040	+	0.234816i	$0.0754489\pi$
$709$	8866.00i	0.469633i	−0.972040	+	0.234816i	$0.924551\pi$
$710$	0	0
$711$	288.000	0.0151911
$712$	0	0
$713$	−640.000	−0.0336160
$714$	0	0
$715$	− 29008.0i	− 1.51726i
$716$	0	0
$717$	− 18960.0i	− 0.987551i
$718$	0	0
$719$	7760.00	0.402502	0.201251	−	0.979540i	$-0.435499\pi$
$719$	7760.00	0.402502	0.201251	+	0.979540i	$0.435499\pi$
$720$	0	0
$721$	35904.0	1.85456
$722$	0	0
$723$	− 19530.0i	− 1.00460i
$724$	0	0
$725$	9798.00i	0.501915i
$726$	0	0
$727$	−13080.0	−0.667277	−0.333638	−	0.942701i	$-0.608276\pi$
$727$	−13080.0	−0.667277	−0.333638	+	0.942701i	$0.608276\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 328.000i	− 0.0165958i
$732$	0	0
$733$	16934.0i	0.853304i	0.904416	+	0.426652i	$0.140307\pi$
$733$	16934.0i	0.853304i	−0.904416	+	0.426652i	$0.859693\pi$
$734$	0	0
$735$	9786.00	0.491105
$736$	0	0
$737$	12208.0	0.610159
$738$	0	0
$739$	− 7060.00i	− 0.351429i	−0.984441	−	0.175715i	$-0.943776\pi$
$739$	− 7060.00i	− 0.351429i	0.984441	−	0.175715i	$-0.0562236\pi$
$740$	0	0
$741$	20424.0i	1.01254i
$742$	0	0
$743$	12520.0	0.618189	0.309094	−	0.951031i	$-0.399974\pi$
$743$	12520.0	0.618189	0.309094	+	0.951031i	$0.399974\pi$
$744$	0	0
$745$	−18788.0	−0.923945
$746$	0	0
$747$	13572.0i	0.664757i
$748$	0	0
$749$	40608.0i	1.98102i
$750$	0	0
$751$	−9792.00	−0.475786	−0.237893	−	0.971291i	$-0.576457\pi$
$751$	−9792.00	−0.475786	−0.237893	+	0.971291i	$0.576457\pi$
$752$	0	0
$753$	1884.00	0.0911777
$754$	0	0
$755$	− 5936.00i	− 0.286137i
$756$	0	0
$757$	− 13166.0i	− 0.632135i	−0.948737	−	0.316068i	$-0.897637\pi$
$757$	− 13166.0i	− 0.632135i	0.948737	−	0.316068i	$-0.102363\pi$
$758$	0	0
$759$	672.000	0.0321371
$760$	0	0
$761$	23222.0	1.10617	0.553086	−	0.833124i	$-0.313450\pi$
$761$	23222.0	1.10617	0.553086	+	0.833124i	$0.313450\pi$
$762$	0	0
$763$	9744.00i	0.462328i
$764$	0	0
$765$	10332.0i	0.488306i
$766$	0	0
$767$	−44104.0	−2.07628
$768$	0	0
$769$	−39934.0	−1.87264	−0.936318	−	0.351154i	$-0.885789\pi$
$769$	−39934.0	−1.87264	−0.936318	+	0.351154i	$0.885789\pi$
$770$	0	0
$771$	− 14586.0i	− 0.681325i
$772$	0	0
$773$	17106.0i	0.795938i	0.917399	+	0.397969i	$0.130285\pi$
$773$	17106.0i	0.795938i	−0.917399	+	0.397969i	$0.869715\pi$
$774$	0	0
$775$	−5680.00	−0.263267
$776$	0	0
$777$	2160.00	0.0997292
$778$	0	0
$779$	− 25944.0i	− 1.19325i
$780$	0	0
$781$	− 23968.0i	− 1.09813i
$782$	0	0
$783$	−3726.00	−0.170059
$784$	0	0
$785$	3668.00	0.166773
$786$	0	0
$787$	− 9956.00i	− 0.450944i	−0.974250	−	0.225472i	$-0.927608\pi$
$787$	− 9956.00i	− 0.450944i	0.974250	−	0.225472i	$-0.0723924\pi$
$788$	0	0
$789$	− 17448.0i	− 0.787282i
$790$	0	0
$791$	18864.0	0.847948
$792$	0	0
$793$	−16132.0	−0.722401
$794$	0	0
$795$	5460.00i	0.243580i
$796$	0	0
$797$	− 9130.00i	− 0.405773i	−0.979202	−	0.202887i	$-0.934968\pi$
$797$	− 9130.00i	− 0.405773i	0.979202	−	0.202887i	$-0.0650323\pi$
$798$	0	0
$799$	19680.0	0.871375
$800$	0	0
$801$	−2214.00	−0.0976627
$802$	0	0
$803$	27944.0i	1.22805i
$804$	0	0
$805$	2688.00i	0.117689i
$806$	0	0
$807$	−10578.0	−0.461417
$808$	0	0
$809$	−11482.0	−0.498993	−0.249497	−	0.968376i	$-0.580265\pi$
$809$	−11482.0	−0.498993	−0.249497	+	0.968376i	$0.580265\pi$
$810$	0	0
$811$	− 4612.00i	− 0.199691i	−0.995003	−	0.0998454i	$-0.968165\pi$
$811$	− 4612.00i	− 0.199691i	0.995003	−	0.0998454i	$-0.0318348\pi$
$812$	0	0
$813$	− 768.000i	− 0.0331303i
$814$	0	0
$815$	−32088.0	−1.37913
$816$	0	0
$817$	368.000	0.0157585
$818$	0	0
$819$	15984.0i	0.681961i
$820$	0	0
$821$	35010.0i	1.48826i	0.668038	+	0.744128i	$0.267135\pi$
$821$	35010.0i	1.48826i	−0.668038	+	0.744128i	$0.732865\pi$
$822$	0	0
$823$	−13688.0	−0.579749	−0.289875	−	0.957065i	$-0.593614\pi$
$823$	−13688.0	−0.579749	−0.289875	+	0.957065i	$0.593614\pi$
$824$	0	0
$825$	5964.00	0.251685
$826$	0	0
$827$	− 11668.0i	− 0.490612i	−0.969446	−	0.245306i	$-0.921112\pi$
$827$	− 11668.0i	− 0.490612i	0.969446	−	0.245306i	$-0.0788884\pi$
$828$	0	0
$829$	− 29306.0i	− 1.22779i	−0.789387	−	0.613896i	$-0.789601\pi$
$829$	− 29306.0i	− 1.22779i	0.789387	−	0.613896i	$-0.210399\pi$
$830$	0	0
$831$	426.000	0.0177831
$832$	0	0
$833$	19106.0	0.794698
$834$	0	0
$835$	− 26544.0i	− 1.10011i
$836$	0	0
$837$	− 2160.00i	− 0.0892001i
$838$	0	0
$839$	2664.00	0.109620	0.0548102	−	0.998497i	$-0.482545\pi$
$839$	2664.00	0.109620	0.0548102	+	0.998497i	$0.482545\pi$
$840$	0	0
$841$	5345.00	0.219156
$842$	0	0
$843$	− 26526.0i	− 1.08375i
$844$	0	0
$845$	45906.0i	1.86889i
$846$	0	0
$847$	13128.0	0.532566
$848$	0	0
$849$	−21540.0	−0.870731
$850$	0	0
$851$	240.000i	0.00966756i
$852$	0	0
$853$	− 26030.0i	− 1.04484i	−0.852688	−	0.522421i	$-0.825029\pi$
$853$	− 26030.0i	− 1.04484i	0.852688	−	0.522421i	$-0.174971\pi$
$854$	0	0
$855$	−11592.0	−0.463670
$856$	0	0
$857$	−44202.0	−1.76186	−0.880929	−	0.473249i	$-0.843081\pi$
$857$	−44202.0	−1.76186	−0.880929	+	0.473249i	$0.843081\pi$
$858$	0	0
$859$	32748.0i	1.30075i	0.759612	+	0.650377i	$0.225389\pi$
$859$	32748.0i	1.30075i	−0.759612	+	0.650377i	$0.774611\pi$
$860$	0	0
$861$	− 20304.0i	− 0.803668i
$862$	0	0
$863$	45344.0	1.78856	0.894280	−	0.447507i	$-0.147688\pi$
$863$	45344.0	1.78856	0.894280	+	0.447507i	$0.147688\pi$
$864$	0	0
$865$	−40236.0	−1.58158
$866$	0	0
$867$	5433.00i	0.212819i
$868$	0	0
$869$	− 896.000i	− 0.0349767i
$870$	0	0
$871$	−32264.0	−1.25514
$872$	0	0
$873$	−7794.00	−0.302161
$874$	0	0
$875$	− 18144.0i	− 0.701005i
$876$	0	0
$877$	− 8778.00i	− 0.337984i	−0.985617	−	0.168992i	$-0.945949\pi$
$877$	− 8778.00i	− 0.337984i	0.985617	−	0.168992i	$-0.0540512\pi$
$878$	0	0
$879$	22122.0	0.848870
$880$	0	0
$881$	−4142.00	−0.158397	−0.0791984	−	0.996859i	$-0.525236\pi$
$881$	−4142.00	−0.158397	−0.0791984	+	0.996859i	$0.525236\pi$
$882$	0	0
$883$	22076.0i	0.841355i	0.907210	+	0.420678i	$0.138208\pi$
$883$	22076.0i	0.841355i	−0.907210	+	0.420678i	$0.861792\pi$
$884$	0	0
$885$	− 25032.0i	− 0.950781i
$886$	0	0
$887$	40376.0	1.52840	0.764201	−	0.644978i	$-0.223133\pi$
$887$	40376.0	1.52840	0.764201	+	0.644978i	$0.223133\pi$
$888$	0	0
$889$	41856.0	1.57908
$890$	0	0
$891$	2268.00i	0.0852759i
$892$	0	0
$893$	22080.0i	0.827412i
$894$	0	0
$895$	−16632.0	−0.621169
$896$	0	0
$897$	−1776.00	−0.0661080
$898$	0	0
$899$	− 11040.0i	− 0.409571i
$900$	0	0
$901$	10660.0i	0.394158i
$902$	0	0
$903$	288.000	0.0106136
$904$	0	0
$905$	48636.0	1.78643
$906$	0	0
$907$	26396.0i	0.966334i	0.875528	+	0.483167i	$0.160514\pi$
$907$	26396.0i	0.966334i	−0.875528	+	0.483167i	$0.839486\pi$
$908$	0	0
$909$	2430.00i	0.0886667i
$910$	0	0
$911$	24368.0	0.886222	0.443111	−	0.896467i	$-0.353875\pi$
$911$	24368.0	0.886222	0.443111	+	0.896467i	$0.353875\pi$
$912$	0	0
$913$	42224.0	1.53057
$914$	0	0
$915$	− 9156.00i	− 0.330807i
$916$	0	0
$917$	15648.0i	0.563514i
$918$	0	0
$919$	5096.00	0.182918	0.0914589	−	0.995809i	$-0.470847\pi$
$919$	5096.00	0.182918	0.0914589	+	0.995809i	$0.470847\pi$
$920$	0	0
$921$	−4500.00	−0.160999
$922$	0	0
$923$	63344.0i	2.25893i
$924$	0	0
$925$	2130.00i	0.0757124i
$926$	0	0
$927$	−13464.0	−0.477040
$928$	0	0
$929$	−18494.0	−0.653142	−0.326571	−	0.945173i	$-0.605893\pi$
$929$	−18494.0	−0.653142	−0.326571	+	0.945173i	$0.605893\pi$
$930$	0	0
$931$	21436.0i	0.754604i
$932$	0	0
$933$	22824.0i	0.800883i
$934$	0	0
$935$	32144.0	1.12430
$936$	0	0
$937$	33222.0	1.15829	0.579144	−	0.815225i	$-0.303387\pi$
$937$	33222.0	1.15829	0.579144	+	0.815225i	$0.303387\pi$
$938$	0	0
$939$	14274.0i	0.496075i
$940$	0	0
$941$	27846.0i	0.964669i	0.875987	+	0.482335i	$0.160211\pi$
$941$	27846.0i	0.964669i	−0.875987	+	0.482335i	$0.839789\pi$
$942$	0	0
$943$	2256.00	0.0779061
$944$	0	0
$945$	−9072.00	−0.312288
$946$	0	0
$947$	41052.0i	1.40867i	0.709868	+	0.704335i	$0.248755\pi$
$947$	41052.0i	1.40867i	−0.709868	+	0.704335i	$0.751245\pi$
$948$	0	0
$949$	− 73852.0i	− 2.52617i
$950$	0	0
$951$	−13122.0	−0.447434
$952$	0	0
$953$	−5706.00	−0.193951	−0.0969756	−	0.995287i	$-0.530917\pi$
$953$	−5706.00	−0.193951	−0.0969756	+	0.995287i	$0.530917\pi$
$954$	0	0
$955$	− 2688.00i	− 0.0910802i
$956$	0	0
$957$	11592.0i	0.391553i
$958$	0	0
$959$	−36720.0	−1.23644
$960$	0	0
$961$	−23391.0	−0.785170
$962$	0	0
$963$	− 15228.0i	− 0.509570i
$964$	0	0
$965$	− 67228.0i	− 2.24264i
$966$	0	0
$967$	39352.0	1.30866	0.654330	−	0.756209i	$-0.272951\pi$
$967$	39352.0	1.30866	0.654330	+	0.756209i	$0.272951\pi$
$968$	0	0
$969$	−22632.0	−0.750304
$970$	0	0
$971$	33180.0i	1.09660i	0.836282	+	0.548299i	$0.184724\pi$
$971$	33180.0i	1.09660i	−0.836282	+	0.548299i	$0.815276\pi$
$972$	0	0
$973$	− 12384.0i	− 0.408030i
$974$	0	0
$975$	−15762.0	−0.517731
$976$	0	0
$977$	−4014.00	−0.131442	−0.0657212	−	0.997838i	$-0.520935\pi$
$977$	−4014.00	−0.131442	−0.0657212	+	0.997838i	$0.520935\pi$
$978$	0	0
$979$	6888.00i	0.224864i
$980$	0	0
$981$	− 3654.00i	− 0.118923i
$982$	0	0
$983$	−20328.0	−0.659575	−0.329788	−	0.944055i	$-0.606977\pi$
$983$	−20328.0	−0.659575	−0.329788	+	0.944055i	$0.606977\pi$
$984$	0	0
$985$	−21252.0	−0.687457
$986$	0	0
$987$	17280.0i	0.557273i
$988$	0	0
$989$	32.0000i	0.00102886i
$990$	0	0
$991$	11728.0	0.375936	0.187968	−	0.982175i	$-0.439810\pi$
$991$	11728.0	0.375936	0.187968	+	0.982175i	$0.439810\pi$
$992$	0	0
$993$	−23412.0	−0.748195
$994$	0	0
$995$	71792.0i	2.28740i
$996$	0	0
$997$	− 50974.0i	− 1.61922i	−0.586968	−	0.809610i	$-0.699679\pi$
$997$	− 50974.0i	− 1.61922i	0.586968	−	0.809610i	$-0.300321\pi$
$998$	0	0
$999$	−810.000	−0.0256529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.d.o.385.2		2
4.3	odd	2		768.4.d.b.385.1		2
8.3	odd	2		768.4.d.b.385.2		2
8.5	even	2	inner	768.4.d.o.385.1		2
16.3	odd	4		192.4.a.g.1.1		1
16.5	even	4		24.4.a.a.1.1	✓	1
16.11	odd	4		48.4.a.b.1.1		1
16.13	even	4		192.4.a.a.1.1		1
48.5	odd	4		72.4.a.b.1.1		1
48.11	even	4		144.4.a.b.1.1		1
48.29	odd	4		576.4.a.u.1.1		1
48.35	even	4		576.4.a.v.1.1		1
80.27	even	4		1200.4.f.p.49.2		2
80.37	odd	4		600.4.f.b.49.1		2
80.43	even	4		1200.4.f.p.49.1		2
80.53	odd	4		600.4.f.b.49.2		2
80.59	odd	4		1200.4.a.u.1.1		1
80.69	even	4		600.4.a.h.1.1		1
112.27	even	4		2352.4.a.w.1.1		1
112.69	odd	4		1176.4.a.a.1.1		1
144.5	odd	12		648.4.i.k.217.1		2
144.85	even	12		648.4.i.b.217.1		2
144.101	odd	12		648.4.i.k.433.1		2
144.133	even	12		648.4.i.b.433.1		2
240.53	even	4		1800.4.f.q.649.2		2
240.149	odd	4		1800.4.a.bg.1.1		1
240.197	even	4		1800.4.f.q.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.a.a.1.1	✓	1	16.5	even	4
48.4.a.b.1.1		1	16.11	odd	4
72.4.a.b.1.1		1	48.5	odd	4
144.4.a.b.1.1		1	48.11	even	4
192.4.a.a.1.1		1	16.13	even	4
192.4.a.g.1.1		1	16.3	odd	4
576.4.a.u.1.1		1	48.29	odd	4
576.4.a.v.1.1		1	48.35	even	4
600.4.a.h.1.1		1	80.69	even	4
600.4.f.b.49.1		2	80.37	odd	4
600.4.f.b.49.2		2	80.53	odd	4
648.4.i.b.217.1		2	144.85	even	12
648.4.i.b.433.1		2	144.133	even	12
648.4.i.k.217.1		2	144.5	odd	12
648.4.i.k.433.1		2	144.101	odd	12
768.4.d.b.385.1		2	4.3	odd	2
768.4.d.b.385.2		2	8.3	odd	2
768.4.d.o.385.1		2	8.5	even	2	inner
768.4.d.o.385.2		2	1.1	even	1	trivial
1176.4.a.a.1.1		1	112.69	odd	4
1200.4.a.u.1.1		1	80.59	odd	4
1200.4.f.p.49.1		2	80.43	even	4
1200.4.f.p.49.2		2	80.27	even	4
1800.4.a.bg.1.1		1	240.149	odd	4
1800.4.f.q.649.1		2	240.197	even	4
1800.4.f.q.649.2		2	240.53	even	4
2352.4.a.w.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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -14.0000i q^{5} +24.0000 q^{7} -9.00000 q^{9} +28.0000i q^{11} -74.0000i q^{13} +42.0000 q^{15} +82.0000 q^{17} +92.0000i q^{19} +72.0000i q^{21} -8.00000 q^{23} -71.0000 q^{25} -27.0000i q^{27} -138.000i q^{29} +80.0000 q^{31} -84.0000 q^{33} -336.000i q^{35} -30.0000i q^{37} +222.000 q^{39} -282.000 q^{41} -4.00000i q^{43} +126.000i q^{45} +240.000 q^{47} +233.000 q^{49} +246.000i q^{51} +130.000i q^{53} +392.000 q^{55} -276.000 q^{57} -596.000i q^{59} -218.000i q^{61} -216.000 q^{63} -1036.00 q^{65} -436.000i q^{67} -24.0000i q^{69} -856.000 q^{71} +998.000 q^{73} -213.000i q^{75} +672.000i q^{77} -32.0000 q^{79} +81.0000 q^{81} -1508.00i q^{83} -1148.00i q^{85} +414.000 q^{87} +246.000 q^{89} -1776.00i q^{91} +240.000i q^{93} +1288.00 q^{95} +866.000 q^{97} -252.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 48 q^{7} - 18 q^{9} + 84 q^{15} + 164 q^{17} - 16 q^{23} - 142 q^{25} + 160 q^{31} - 168 q^{33} + 444 q^{39} - 564 q^{41} + 480 q^{47} + 466 q^{49} + 784 q^{55} - 552 q^{57} - 432 q^{63} - 2072 q^{65}+ \cdots + 1732 q^{97}+O(q^{100}) \) 2 * q + 48 * q^7 - 18 * q^9 + 84 * q^15 + 164 * q^17 - 16 * q^23 - 142 * q^25 + 160 * q^31 - 168 * q^33 + 444 * q^39 - 564 * q^41 + 480 * q^47 + 466 * q^49 + 784 * q^55 - 552 * q^57 - 432 * q^63 - 2072 * q^65 - 1712 * q^71 + 1996 * q^73 - 64 * q^79 + 162 * q^81 + 828 * q^87 + 492 * q^89 + 2576 * q^95 + 1732 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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