LMFDB - Embedded newform 1152.5.b.m.703.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,5,Mod(703,1152)] 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("1152.703"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1152.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-480] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$119.082197473$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + \cdots + 362673936 $ x^16 + 46x^14 + 1311x^12 + 24382x^10 + 338077x^8 + 3338772x^6 + 24662556x^4 + 120434256*x^2 + 362673936
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{88}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.3
Root			$-2.31712 + 3.01338i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.5.b.m.703.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-23.5183i q^{5} -40.3412i q^{7} -227.579 q^{11} -310.305i q^{13} +380.831 q^{17} -343.152 q^{19} -113.116i q^{23} +71.8887 q^{25} -621.827i q^{29} -1528.61i q^{31} -948.758 q^{35} -1385.54i q^{37} -2765.92 q^{41} +1018.24 q^{43} -1335.48i q^{47} +773.586 q^{49} +4253.04i q^{53} +5352.28i q^{55} -77.6265 q^{59} -1154.74i q^{61} -7297.86 q^{65} -3477.85 q^{67} -2055.93i q^{71} +7351.49 q^{73} +9180.83i q^{77} -3277.77i q^{79} -7747.77 q^{83} -8956.49i q^{85} +4675.62 q^{89} -12518.1 q^{91} +8070.35i q^{95} +408.505 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 480 q^{17} + 144 q^{25} + 3552 q^{41} - 20080 q^{49} - 23040 q^{65} + 34400 q^{73} + 15648 q^{89} + 5088 q^{97}+O(q^{100}) $ 16 * q - 480 * q^17 + 144 * q^25 + 3552 * q^41 - 20080 * q^49 - 23040 * q^65 + 34400 * q^73 + 15648 * q^89 + 5088 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 23.5183i	− 0.940733i	−0.882471	−	0.470366i	$-0.844122\pi$
$5$	− 23.5183i	− 0.940733i	0.882471	−	0.470366i	$-0.155878\pi$
$6$	0	0
$7$	− 40.3412i	− 0.823290i	−0.911344	−	0.411645i	$-0.864954\pi$
$7$	− 40.3412i	− 0.823290i	0.911344	−	0.411645i	$-0.135046\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−227.579	−1.88082	−0.940411	−	0.340041i	$-0.889559\pi$
$11$	−227.579	−1.88082	−0.940411	+	0.340041i	$0.889559\pi$
$12$	0	0
$13$	− 310.305i	− 1.83613i	−0.396435	−	0.918063i	$-0.629753\pi$
$13$	− 310.305i	− 1.83613i	0.396435	−	0.918063i	$-0.370247\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	380.831	1.31775	0.658876	−	0.752251i	$-0.271032\pi$
$17$	380.831	1.31775	0.658876	+	0.752251i	$0.271032\pi$
$18$	0	0
$19$	−343.152	−0.950559	−0.475279	−	0.879835i	$-0.657653\pi$
$19$	−343.152	−0.950559	−0.475279	+	0.879835i	$0.657653\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 113.116i	− 0.213831i	−0.994268	−	0.106915i	$-0.965903\pi$
$23$	− 113.116i	− 0.213831i	0.994268	−	0.106915i	$-0.0340974\pi$
$24$	0	0
$25$	71.8887	0.115022
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 621.827i	− 0.739390i	−0.929153	−	0.369695i	$-0.879462\pi$
$29$	− 621.827i	− 0.739390i	0.929153	−	0.369695i	$-0.120538\pi$
$30$	0	0
$31$	− 1528.61i	− 1.59064i	−0.606189	−	0.795321i	$-0.707302\pi$
$31$	− 1528.61i	− 1.59064i	0.606189	−	0.795321i	$-0.292698\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−948.758	−0.774496
$36$	0	0
$37$	− 1385.54i	− 1.01208i	−0.862509	−	0.506042i	$-0.831108\pi$
$37$	− 1385.54i	− 1.01208i	0.862509	−	0.506042i	$-0.168892\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2765.92	−1.64540	−0.822702	−	0.568474i	$-0.807534\pi$
$41$	−2765.92	−1.64540	−0.822702	+	0.568474i	$0.807534\pi$
$42$	0	0
$43$	1018.24	0.550698	0.275349	−	0.961344i	$-0.411207\pi$
$43$	1018.24	0.550698	0.275349	+	0.961344i	$0.411207\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1335.48i	− 0.604564i	−0.953219	−	0.302282i	$-0.902252\pi$
$47$	− 1335.48i	− 0.604564i	0.953219	−	0.302282i	$-0.0977484\pi$
$48$	0	0
$49$	773.586	0.322193
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4253.04i	1.51408i	0.653371	+	0.757038i	$0.273354\pi$
$53$	4253.04i	1.51408i	−0.653371	+	0.757038i	$0.726646\pi$
$54$	0	0
$55$	5352.28i	1.76935i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−77.6265	−0.0223001	−0.0111500	−	0.999938i	$-0.503549\pi$
$59$	−77.6265	−0.0223001	−0.0111500	+	0.999938i	$0.503549\pi$
$60$	0	0
$61$	− 1154.74i	− 0.310332i	−0.987888	−	0.155166i	$-0.950409\pi$
$61$	− 1154.74i	− 0.310332i	0.987888	−	0.155166i	$-0.0495912\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7297.86	−1.72730
$66$	0	0
$67$	−3477.85	−0.774748	−0.387374	−	0.921923i	$-0.626618\pi$
$67$	−3477.85	−0.774748	−0.387374	+	0.921923i	$0.626618\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 2055.93i	− 0.407841i	−0.978987	−	0.203921i	$-0.934632\pi$
$71$	− 2055.93i	− 0.407841i	0.978987	−	0.203921i	$-0.0653684\pi$
$72$	0	0
$73$	7351.49	1.37953	0.689763	−	0.724035i	$-0.257715\pi$
$73$	7351.49	1.37953	0.689763	+	0.724035i	$0.257715\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9180.83i	1.54846i
$78$	0	0
$79$	− 3277.77i	− 0.525199i	−0.964905	−	0.262600i	$-0.915420\pi$
$79$	− 3277.77i	− 0.525199i	0.964905	−	0.262600i	$-0.0845798\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7747.77	−1.12466	−0.562329	−	0.826913i	$-0.690095\pi$
$83$	−7747.77	−1.12466	−0.562329	+	0.826913i	$0.690095\pi$
$84$	0	0
$85$	− 8956.49i	− 1.23965i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4675.62	0.590281	0.295141	−	0.955454i	$-0.404633\pi$
$89$	4675.62	0.590281	0.295141	+	0.955454i	$0.404633\pi$
$90$	0	0
$91$	−12518.1	−1.51166
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8070.35i	0.894222i
$96$	0	0
$97$	408.505	0.0434164	0.0217082	−	0.999764i	$-0.493090\pi$
$97$	408.505	0.0434164	0.0217082	+	0.999764i	$0.493090\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6780.48i	0.664688i	0.943158	+	0.332344i	$0.107839\pi$
$101$	6780.48i	0.664688i	−0.943158	+	0.332344i	$0.892161\pi$
$102$	0	0
$103$	1694.72i	0.159744i	0.996805	+	0.0798719i	$0.0254511\pi$
$103$	1694.72i	0.159744i	−0.996805	+	0.0798719i	$0.974549\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11254.2	0.982983	0.491492	−	0.870882i	$-0.336452\pi$
$107$	11254.2	0.982983	0.491492	+	0.870882i	$0.336452\pi$
$108$	0	0
$109$	− 3445.49i	− 0.290000i	−0.989432	−	0.145000i	$-0.953682\pi$
$109$	− 3445.49i	− 0.290000i	0.989432	−	0.145000i	$-0.0463182\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8660.33	−0.678231	−0.339116	−	0.940745i	$-0.610128\pi$
$113$	−8660.33	−0.678231	−0.339116	+	0.940745i	$0.610128\pi$
$114$	0	0
$115$	−2660.31	−0.201158
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 15363.2i	− 1.08489i
$120$	0	0
$121$	37151.4	2.53749
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 16389.7i	− 1.04894i
$126$	0	0
$127$	16133.1i	1.00026i	0.865952	+	0.500128i	$0.166714\pi$
$127$	16133.1i	1.00026i	−0.865952	+	0.500128i	$0.833286\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2698.97	0.157274	0.0786368	−	0.996903i	$-0.474943\pi$
$131$	2698.97	0.157274	0.0786368	+	0.996903i	$0.474943\pi$
$132$	0	0
$133$	13843.2i	0.782586i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5527.88	0.294522	0.147261	−	0.989098i	$-0.452954\pi$
$137$	5527.88	0.294522	0.147261	+	0.989098i	$0.452954\pi$
$138$	0	0
$139$	23618.9	1.22245	0.611224	−	0.791458i	$-0.290678\pi$
$139$	23618.9	1.22245	0.611224	+	0.791458i	$0.290678\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	70619.1i	3.45342i
$144$	0	0
$145$	−14624.3	−0.695568
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3606.61i	− 0.162453i	−0.996696	−	0.0812264i	$-0.974116\pi$
$149$	− 3606.61i	− 0.162453i	0.996696	−	0.0812264i	$-0.0258837\pi$
$150$	0	0
$151$	43619.5i	1.91305i	0.291645	+	0.956527i	$0.405797\pi$
$151$	43619.5i	1.91305i	−0.291645	+	0.956527i	$0.594203\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−35950.3	−1.49637
$156$	0	0
$157$	− 39000.2i	− 1.58222i	−0.611673	−	0.791111i	$-0.709503\pi$
$157$	− 39000.2i	− 1.58222i	0.611673	−	0.791111i	$-0.290497\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4563.26	−0.176045
$162$	0	0
$163$	−13933.2	−0.524415	−0.262208	−	0.965012i	$-0.584451\pi$
$163$	−13933.2	−0.524415	−0.262208	+	0.965012i	$0.584451\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9886.95i	0.354511i	0.984165	+	0.177255i	$0.0567218\pi$
$167$	9886.95i	0.354511i	−0.984165	+	0.177255i	$0.943278\pi$
$168$	0	0
$169$	−67728.4	−2.37136
$170$	0	0
$171$	0	0
$172$	0	0
$173$	26467.6i	0.884346i	0.896930	+	0.442173i	$0.145792\pi$
$173$	26467.6i	0.884346i	−0.896930	+	0.442173i	$0.854208\pi$
$174$	0	0
$175$	− 2900.08i	− 0.0946964i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	33711.5	1.05213	0.526067	−	0.850443i	$-0.323666\pi$
$179$	33711.5	1.05213	0.526067	+	0.850443i	$0.323666\pi$
$180$	0	0
$181$	36239.3i	1.10617i	0.833124	+	0.553087i	$0.186550\pi$
$181$	36239.3i	1.10617i	−0.833124	+	0.553087i	$0.813450\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−32585.6	−0.952100
$186$	0	0
$187$	−86669.2	−2.47846
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 20931.9i	− 0.573774i	−0.957964	−	0.286887i	$-0.907379\pi$
$191$	− 20931.9i	− 0.573774i	0.957964	−	0.286887i	$-0.0926205\pi$
$192$	0	0
$193$	126.993	0.00340929	0.00170465	−	0.999999i	$-0.499457\pi$
$193$	126.993	0.00340929	0.00170465	+	0.999999i	$0.499457\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8881.31i	0.228847i	0.993432	+	0.114423i	$0.0365020\pi$
$197$	8881.31i	0.228847i	−0.993432	+	0.114423i	$0.963498\pi$
$198$	0	0
$199$	76018.6i	1.91961i	0.280664	+	0.959806i	$0.409445\pi$
$199$	76018.6i	1.91961i	−0.280664	+	0.959806i	$0.590555\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−25085.3	−0.608732
$204$	0	0
$205$	65049.8i	1.54788i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	78094.3	1.78783
$210$	0	0
$211$	−54603.0	−1.22645	−0.613227	−	0.789907i	$-0.710129\pi$
$211$	−54603.0	−1.22645	−0.613227	+	0.789907i	$0.710129\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 23947.3i	− 0.518060i
$216$	0	0
$217$	−61665.9	−1.30956
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 118174.i	− 2.41956i
$222$	0	0
$223$	− 80149.6i	− 1.61173i	−0.592102	−	0.805863i	$-0.701702\pi$
$223$	− 80149.6i	− 1.61173i	0.592102	−	0.805863i	$-0.298298\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−86867.2	−1.68579	−0.842896	−	0.538077i	$-0.819151\pi$
$227$	−86867.2	−1.68579	−0.842896	+	0.538077i	$0.819151\pi$
$228$	0	0
$229$	− 36536.4i	− 0.696714i	−0.937362	−	0.348357i	$-0.886740\pi$
$229$	− 36536.4i	− 0.696714i	0.937362	−	0.348357i	$-0.113260\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	45963.5	0.846646	0.423323	−	0.905979i	$-0.360864\pi$
$233$	45963.5	0.846646	0.423323	+	0.905979i	$0.360864\pi$
$234$	0	0
$235$	−31408.3	−0.568733
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 8705.16i	− 0.152399i	−0.997093	−	0.0761993i	$-0.975721\pi$
$239$	− 8705.16i	− 0.152399i	0.997093	−	0.0761993i	$-0.0242785\pi$
$240$	0	0
$241$	−4184.78	−0.0720508	−0.0360254	−	0.999351i	$-0.511470\pi$
$241$	−4184.78	−0.0720508	−0.0360254	+	0.999351i	$0.511470\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 18193.4i	− 0.303098i
$246$	0	0
$247$	106482.i	1.74535i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23956.2	−0.380251	−0.190125	−	0.981760i	$-0.560889\pi$
$251$	−23956.2	−0.380251	−0.190125	+	0.981760i	$0.560889\pi$
$252$	0	0
$253$	25743.0i	0.402177i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	32912.7	0.498307	0.249154	−	0.968464i	$-0.419848\pi$
$257$	32912.7	0.498307	0.249154	+	0.968464i	$0.419848\pi$
$258$	0	0
$259$	−55894.5	−0.833239
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 28615.0i	− 0.413696i	−0.978373	−	0.206848i	$-0.933679\pi$
$263$	− 28615.0i	− 0.413696i	0.978373	−	0.206848i	$-0.0663206\pi$
$264$	0	0
$265$	100024.	1.42434
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 105926.i	− 1.46385i	−0.681383	−	0.731927i	$-0.738621\pi$
$269$	− 105926.i	− 1.46385i	0.681383	−	0.731927i	$-0.261379\pi$
$270$	0	0
$271$	105722.i	1.43955i	0.694206	+	0.719776i	$0.255756\pi$
$271$	105722.i	1.43955i	−0.694206	+	0.719776i	$0.744244\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16360.4	−0.216336
$276$	0	0
$277$	132880.i	1.73181i	0.500212	+	0.865903i	$0.333256\pi$
$277$	132880.i	1.73181i	−0.500212	+	0.865903i	$0.666744\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	75828.4	0.960327	0.480163	−	0.877179i	$-0.340577\pi$
$281$	75828.4	0.960327	0.480163	+	0.877179i	$0.340577\pi$
$282$	0	0
$283$	−82382.2	−1.02863	−0.514317	−	0.857600i	$-0.671954\pi$
$283$	−82382.2	−1.02863	−0.514317	+	0.857600i	$0.671954\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	111581.i	1.35464i
$288$	0	0
$289$	61510.9	0.736473
$290$	0	0
$291$	0	0
$292$	0	0
$293$	50845.8i	0.592270i	0.955146	+	0.296135i	$0.0956978\pi$
$293$	50845.8i	0.592270i	−0.955146	+	0.296135i	$0.904302\pi$
$294$	0	0
$295$	1825.65i	0.0209784i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−35100.6	−0.392620
$300$	0	0
$301$	− 41077.1i	− 0.453384i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−27157.6	−0.291939
$306$	0	0
$307$	22007.7	0.233505	0.116753	−	0.993161i	$-0.462752\pi$
$307$	22007.7	0.233505	0.116753	+	0.993161i	$0.462752\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	161735.i	1.67218i	0.548589	+	0.836092i	$0.315165\pi$
$311$	161735.i	1.67218i	−0.548589	+	0.836092i	$0.684835\pi$
$312$	0	0
$313$	−131522.	−1.34249	−0.671245	−	0.741236i	$-0.734240\pi$
$313$	−131522.	−1.34249	−0.671245	+	0.741236i	$0.734240\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	65811.6i	0.654913i	0.944866	+	0.327456i	$0.106191\pi$
$317$	65811.6i	0.654913i	−0.944866	+	0.327456i	$0.893809\pi$
$318$	0	0
$319$	141515.i	1.39066i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−130683.	−1.25260
$324$	0	0
$325$	− 22307.4i	− 0.211195i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−53874.9	−0.497731
$330$	0	0
$331$	6965.35	0.0635751	0.0317875	−	0.999495i	$-0.489880\pi$
$331$	6965.35	0.0635751	0.0317875	+	0.999495i	$0.489880\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	81793.1i	0.728831i
$336$	0	0
$337$	−95799.4	−0.843535	−0.421767	−	0.906704i	$-0.638590\pi$
$337$	−95799.4	−0.843535	−0.421767	+	0.906704i	$0.638590\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	347879.i	2.99171i
$342$	0	0
$343$	− 128067.i	− 1.08855i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	65514.1	0.544096	0.272048	−	0.962284i	$-0.412299\pi$
$347$	65514.1	0.544096	0.272048	+	0.962284i	$0.412299\pi$
$348$	0	0
$349$	− 124174.i	− 1.01948i	−0.860329	−	0.509740i	$-0.829742\pi$
$349$	− 124174.i	− 1.01948i	0.860329	−	0.509740i	$-0.170258\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−144384.	−1.15870	−0.579350	−	0.815079i	$-0.696694\pi$
$353$	−144384.	−1.15870	−0.579350	+	0.815079i	$0.696694\pi$
$354$	0	0
$355$	−48351.9	−0.383670
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 125622.i	− 0.974715i	−0.873203	−	0.487357i	$-0.837961\pi$
$359$	− 125622.i	− 0.974715i	0.873203	−	0.487357i	$-0.162039\pi$
$360$	0	0
$361$	−12567.9	−0.0964377
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 172895.i	− 1.29776i
$366$	0	0
$367$	132797.i	0.985950i	0.870044	+	0.492975i	$0.164091\pi$
$367$	132797.i	0.985950i	−0.870044	+	0.492975i	$0.835909\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	171573.	1.24652
$372$	0	0
$373$	83521.7i	0.600319i	0.953889	+	0.300159i	$0.0970399\pi$
$373$	83521.7i	0.600319i	−0.953889	+	0.300159i	$0.902960\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−192956.	−1.35761
$378$	0	0
$379$	85671.5	0.596428	0.298214	−	0.954499i	$-0.403609\pi$
$379$	85671.5	0.596428	0.298214	+	0.954499i	$0.403609\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 270564.i	− 1.84448i	−0.386621	−	0.922239i	$-0.626358\pi$
$383$	− 270564.i	− 1.84448i	0.386621	−	0.922239i	$-0.373642\pi$
$384$	0	0
$385$	215918.	1.45669
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28603.7i	0.189027i	0.995524	+	0.0945133i	$0.0301295\pi$
$389$	28603.7i	0.189027i	−0.995524	+	0.0945133i	$0.969870\pi$
$390$	0	0
$391$	− 43078.2i	− 0.281776i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−77087.6	−0.494072
$396$	0	0
$397$	− 215256.i	− 1.36576i	−0.730530	−	0.682880i	$-0.760727\pi$
$397$	− 215256.i	− 1.36576i	0.730530	−	0.682880i	$-0.239273\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−79995.2	−0.497479	−0.248740	−	0.968570i	$-0.580016\pi$
$401$	−79995.2	−0.497479	−0.248740	+	0.968570i	$0.580016\pi$
$402$	0	0
$403$	−474335.	−2.92062
$404$	0	0
$405$	0	0
$406$	0	0
$407$	315321.i	1.90355i
$408$	0	0
$409$	205058.	1.22583	0.612915	−	0.790149i	$-0.289997\pi$
$409$	205058.	1.22583	0.612915	+	0.790149i	$0.289997\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3131.55i	0.0183594i
$414$	0	0
$415$	182215.i	1.05800i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−56983.1	−0.324577	−0.162289	−	0.986743i	$-0.551888\pi$
$419$	−56983.1	−0.324577	−0.162289	+	0.986743i	$0.551888\pi$
$420$	0	0
$421$	− 243587.i	− 1.37432i	−0.726504	−	0.687162i	$-0.758856\pi$
$421$	− 243587.i	− 1.37432i	0.726504	−	0.687162i	$-0.241144\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	27377.4	0.151570
$426$	0	0
$427$	−46583.8	−0.255493
$428$	0	0
$429$	0	0
$430$	0	0
$431$	67838.5i	0.365192i	0.983188	+	0.182596i	$0.0584501\pi$
$431$	67838.5i	0.365192i	−0.983188	+	0.182596i	$0.941550\pi$
$432$	0	0
$433$	103864.	0.553974	0.276987	−	0.960874i	$-0.410664\pi$
$433$	103864.	0.553974	0.276987	+	0.960874i	$0.410664\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	38816.1i	0.203259i
$438$	0	0
$439$	− 124230.i	− 0.644612i	−0.946636	−	0.322306i	$-0.895542\pi$
$439$	− 124230.i	− 0.644612i	0.946636	−	0.322306i	$-0.104458\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	313124.	1.59554	0.797772	−	0.602960i	$-0.206012\pi$
$443$	313124.	1.59554	0.797772	+	0.602960i	$0.206012\pi$
$444$	0	0
$445$	− 109963.i	− 0.555297i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	220308.	1.09279	0.546395	−	0.837528i	$-0.316000\pi$
$449$	220308.	1.09279	0.546395	+	0.837528i	$0.316000\pi$
$450$	0	0
$451$	629467.	3.09471
$452$	0	0
$453$	0	0
$454$	0	0
$455$	294405.i	1.42207i
$456$	0	0
$457$	143104.	0.685202	0.342601	−	0.939481i	$-0.388692\pi$
$457$	143104.	0.685202	0.342601	+	0.939481i	$0.388692\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 82138.4i	− 0.386496i	−0.981150	−	0.193248i	$-0.938098\pi$
$461$	− 82138.4i	− 0.386496i	0.981150	−	0.193248i	$-0.0619021\pi$
$462$	0	0
$463$	102147.i	0.476501i	0.971204	+	0.238251i	$0.0765740\pi$
$463$	102147.i	0.476501i	−0.971204	+	0.238251i	$0.923426\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8057.50	−0.0369459	−0.0184730	−	0.999829i	$-0.505880\pi$
$467$	−8057.50	−0.0369459	−0.0184730	+	0.999829i	$0.505880\pi$
$468$	0	0
$469$	140301.i	0.637843i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−231731.	−1.03576
$474$	0	0
$475$	−24668.7	−0.109335
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 112208.i	− 0.489051i	−0.969643	−	0.244525i	$-0.921368\pi$
$479$	− 112208.i	− 0.489051i	0.969643	−	0.244525i	$-0.0786321\pi$
$480$	0	0
$481$	−429941.	−1.85831
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9607.35i	− 0.0408432i
$486$	0	0
$487$	220829.i	0.931102i	0.885021	+	0.465551i	$0.154144\pi$
$487$	220829.i	0.931102i	−0.885021	+	0.465551i	$0.845856\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−181363.	−0.752290	−0.376145	−	0.926561i	$-0.622751\pi$
$491$	−181363.	−0.752290	−0.376145	+	0.926561i	$0.622751\pi$
$492$	0	0
$493$	− 236811.i	− 0.974333i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−82938.6	−0.335772
$498$	0	0
$499$	−1211.21	−0.00486429	−0.00243214	−	0.999997i	$-0.500774\pi$
$499$	−1211.21	−0.00486429	−0.00243214	+	0.999997i	$0.500774\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 316513.i	− 1.25099i	−0.780227	−	0.625497i	$-0.784896\pi$
$503$	− 316513.i	− 1.25099i	0.780227	−	0.625497i	$-0.215104\pi$
$504$	0	0
$505$	159466.	0.625294
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 60847.8i	− 0.234860i	−0.993081	−	0.117430i	$-0.962534\pi$
$509$	− 60847.8i	− 0.234860i	0.993081	−	0.117430i	$-0.0374656\pi$
$510$	0	0
$511$	− 296568.i	− 1.13575i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	39857.0	0.150276
$516$	0	0
$517$	303928.i	1.13708i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	224829.	0.828278	0.414139	−	0.910214i	$-0.364083\pi$
$521$	224829.	0.828278	0.414139	+	0.910214i	$0.364083\pi$
$522$	0	0
$523$	−93536.3	−0.341961	−0.170981	−	0.985274i	$-0.554694\pi$
$523$	−93536.3	−0.341961	−0.170981	+	0.985274i	$0.554694\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 582140.i	− 2.09607i
$528$	0	0
$529$	267046.	0.954276
$530$	0	0
$531$	0	0
$532$	0	0
$533$	858280.i	3.02117i
$534$	0	0
$535$	− 264679.i	− 0.924724i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−176052.	−0.605988
$540$	0	0
$541$	23161.3i	0.0791351i	0.999217	+	0.0395675i	$0.0125980\pi$
$541$	23161.3i	0.0791351i	−0.999217	+	0.0395675i	$0.987402\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−81032.1	−0.272813
$546$	0	0
$547$	355721.	1.18887	0.594436	−	0.804143i	$-0.297375\pi$
$547$	355721.	1.18887	0.594436	+	0.804143i	$0.297375\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	213381.i	0.702834i
$552$	0	0
$553$	−132229.	−0.432391
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 350706.i	− 1.13040i	−0.824954	−	0.565200i	$-0.808799\pi$
$557$	− 350706.i	− 1.13040i	0.824954	−	0.565200i	$-0.191201\pi$
$558$	0	0
$559$	− 315966.i	− 1.01115i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	220322.	0.695089	0.347545	−	0.937663i	$-0.387016\pi$
$563$	220322.	0.695089	0.347545	+	0.937663i	$0.387016\pi$
$564$	0	0
$565$	203676.i	0.638034i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	540989.	1.67095	0.835476	−	0.549526i	$-0.185192\pi$
$569$	540989.	1.67095	0.835476	+	0.549526i	$0.185192\pi$
$570$	0	0
$571$	479447.	1.47051	0.735255	−	0.677790i	$-0.237062\pi$
$571$	479447.	1.47051	0.735255	+	0.677790i	$0.237062\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 8131.80i	− 0.0245952i
$576$	0	0
$577$	−299723.	−0.900261	−0.450131	−	0.892963i	$-0.648623\pi$
$577$	−299723.	−0.900261	−0.450131	+	0.892963i	$0.648623\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	312555.i	0.925920i
$582$	0	0
$583$	− 967904.i	− 2.84771i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−205529.	−0.596480	−0.298240	−	0.954491i	$-0.596400\pi$
$587$	−205529.	−0.596480	−0.298240	+	0.954491i	$0.596400\pi$
$588$	0	0
$589$	524544.i	1.51200i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−566721.	−1.61161	−0.805805	−	0.592181i	$-0.798267\pi$
$593$	−566721.	−1.61161	−0.805805	+	0.592181i	$0.798267\pi$
$594$	0	0
$595$	−361316.	−1.02059
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 224984.i	− 0.627044i	−0.949581	−	0.313522i	$-0.898491\pi$
$599$	− 224984.i	− 0.627044i	0.949581	−	0.313522i	$-0.101509\pi$
$600$	0	0
$601$	50655.9	0.140243	0.0701215	−	0.997538i	$-0.477661\pi$
$601$	50655.9	0.140243	0.0701215	+	0.997538i	$0.477661\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 873738.i	− 2.38710i
$606$	0	0
$607$	− 225688.i	− 0.612534i	−0.951946	−	0.306267i	$-0.900920\pi$
$607$	− 225688.i	− 0.612534i	0.951946	−	0.306267i	$-0.0990801\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−414407.	−1.11005
$612$	0	0
$613$	200659.i	0.533994i	0.963697	+	0.266997i	$0.0860314\pi$
$613$	200659.i	0.533994i	−0.963697	+	0.266997i	$0.913969\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14910.4	−0.0391669	−0.0195835	−	0.999808i	$-0.506234\pi$
$617$	−14910.4	−0.0391669	−0.0195835	+	0.999808i	$0.506234\pi$
$618$	0	0
$619$	−160175.	−0.418037	−0.209018	−	0.977912i	$-0.567027\pi$
$619$	−160175.	−0.418037	−0.209018	+	0.977912i	$0.567027\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 188620.i	− 0.485973i
$624$	0	0
$625$	−340527.	−0.871748
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 527657.i	− 1.33368i
$630$	0	0
$631$	238216.i	0.598291i	0.954208	+	0.299145i	$0.0967016\pi$
$631$	238216.i	0.598291i	−0.954208	+	0.299145i	$0.903298\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	379424.	0.940973
$636$	0	0
$637$	− 240048.i	− 0.591587i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	100397.	0.244346	0.122173	−	0.992509i	$-0.461014\pi$
$641$	100397.	0.244346	0.122173	+	0.992509i	$0.461014\pi$
$642$	0	0
$643$	265567.	0.642321	0.321160	−	0.947025i	$-0.395927\pi$
$643$	265567.	0.642321	0.321160	+	0.947025i	$0.395927\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 210025.i	− 0.501721i	−0.968023	−	0.250860i	$-0.919287\pi$
$647$	− 210025.i	− 0.501721i	0.968023	−	0.250860i	$-0.0807135\pi$
$648$	0	0
$649$	17666.2	0.0419424
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 44076.3i	− 0.103366i	−0.998664	−	0.0516831i	$-0.983541\pi$
$653$	− 44076.3i	− 0.103366i	0.998664	−	0.0516831i	$-0.0164586\pi$
$654$	0	0
$655$	− 63475.3i	− 0.147952i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	706672.	1.62722	0.813611	−	0.581409i	$-0.197498\pi$
$659$	706672.	1.62722	0.813611	+	0.581409i	$0.197498\pi$
$660$	0	0
$661$	506522.i	1.15930i	0.814866	+	0.579649i	$0.196810\pi$
$661$	506522.i	1.15930i	−0.814866	+	0.579649i	$0.803190\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	325568.	0.736204
$666$	0	0
$667$	−70338.8	−0.158104
$668$	0	0
$669$	0	0
$670$	0	0
$671$	262796.i	0.583678i
$672$	0	0
$673$	−72766.9	−0.160659	−0.0803293	−	0.996768i	$-0.525597\pi$
$673$	−72766.9	−0.160659	−0.0803293	+	0.996768i	$0.525597\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 95915.9i	− 0.209273i	−0.994511	−	0.104637i	$-0.966632\pi$
$677$	− 95915.9i	− 0.209273i	0.994511	−	0.104637i	$-0.0333679\pi$
$678$	0	0
$679$	− 16479.6i	− 0.0357443i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−83040.2	−0.178011	−0.0890055	−	0.996031i	$-0.528369\pi$
$683$	−83040.2	−0.178011	−0.0890055	+	0.996031i	$0.528369\pi$
$684$	0	0
$685$	− 130006.i	− 0.277066i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.31974e6	2.78004
$690$	0	0
$691$	337325.	0.706469	0.353234	−	0.935535i	$-0.385082\pi$
$691$	337325.	0.706469	0.353234	+	0.935535i	$0.385082\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 555477.i	− 1.15000i
$696$	0	0
$697$	−1.05335e6	−2.16823
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 442010.i	− 0.899490i	−0.893157	−	0.449745i	$-0.851515\pi$
$701$	− 442010.i	− 0.899490i	0.893157	−	0.449745i	$-0.148485\pi$
$702$	0	0
$703$	475451.i	0.962045i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	273533.	0.547231
$708$	0	0
$709$	− 380072.i	− 0.756089i	−0.925787	−	0.378045i	$-0.876597\pi$
$709$	− 380072.i	− 0.756089i	0.925787	−	0.378045i	$-0.123403\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−172911.	−0.340128
$714$	0	0
$715$	1.66084e6	3.24875
$716$	0	0
$717$	0	0
$718$	0	0
$719$	564351.i	1.09167i	0.837893	+	0.545835i	$0.183787\pi$
$719$	564351.i	1.09167i	−0.837893	+	0.545835i	$0.816213\pi$
$720$	0	0
$721$	68367.1	0.131515
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 44702.3i	− 0.0850460i
$726$	0	0
$727$	25822.2i	0.0488568i	0.999702	+	0.0244284i	$0.00777658\pi$
$727$	25822.2i	0.0488568i	−0.999702	+	0.0244284i	$0.992223\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	387777.	0.725684
$732$	0	0
$733$	− 191223.i	− 0.355904i	−0.984039	−	0.177952i	$-0.943053\pi$
$733$	− 191223.i	− 0.355904i	0.984039	−	0.177952i	$-0.0569471\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	791486.	1.45716
$738$	0	0
$739$	954938.	1.74858	0.874292	−	0.485400i	$-0.161326\pi$
$739$	954938.	1.74858	0.874292	+	0.485400i	$0.161326\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 417501.i	− 0.756275i	−0.925749	−	0.378138i	$-0.876565\pi$
$743$	− 417501.i	− 0.756275i	0.925749	−	0.378138i	$-0.123435\pi$
$744$	0	0
$745$	−84821.5	−0.152825
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 454007.i	− 0.809280i
$750$	0	0
$751$	− 776902.i	− 1.37748i	−0.725007	−	0.688742i	$-0.758163\pi$
$751$	− 776902.i	− 1.37748i	0.725007	−	0.688742i	$-0.241837\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.02586e6	1.79967
$756$	0	0
$757$	− 522866.i	− 0.912429i	−0.889870	−	0.456214i	$-0.849205\pi$
$757$	− 522866.i	− 0.912429i	0.889870	−	0.456214i	$-0.150795\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−929896.	−1.60570	−0.802851	−	0.596180i	$-0.796684\pi$
$761$	−929896.	−1.60570	−0.802851	+	0.596180i	$0.796684\pi$
$762$	0	0
$763$	−138995.	−0.238754
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24087.9i	0.0409457i
$768$	0	0
$769$	890030.	1.50505	0.752527	−	0.658562i	$-0.228835\pi$
$769$	890030.	1.50505	0.752527	+	0.658562i	$0.228835\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	774804.i	1.29668i	0.761351	+	0.648340i	$0.224536\pi$
$773$	774804.i	1.29668i	−0.761351	+	0.648340i	$0.775464\pi$
$774$	0	0
$775$	− 109890.i	− 0.182959i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	949131.	1.56405
$780$	0	0
$781$	467887.i	0.767076i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−917219.	−1.48845
$786$	0	0
$787$	−797643.	−1.28783	−0.643916	−	0.765096i	$-0.722691\pi$
$787$	−797643.	−1.28783	−0.643916	+	0.765096i	$0.722691\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	349368.i	0.558381i
$792$	0	0
$793$	−358323.	−0.569808
$794$	0	0
$795$	0	0
$796$	0	0
$797$	51158.3i	0.0805378i	0.999189	+	0.0402689i	$0.0128215\pi$
$797$	51158.3i	0.0805378i	−0.999189	+	0.0402689i	$0.987179\pi$
$798$	0	0
$799$	− 508592.i	− 0.796666i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.67305e6	−2.59464
$804$	0	0
$805$	107320.i	0.165611i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	97436.9	0.148877	0.0744383	−	0.997226i	$-0.476284\pi$
$809$	97436.9	0.148877	0.0744383	+	0.997226i	$0.476284\pi$
$810$	0	0
$811$	−52742.9	−0.0801904	−0.0400952	−	0.999196i	$-0.512766\pi$
$811$	−52742.9	−0.0801904	−0.0400952	+	0.999196i	$0.512766\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	327685.i	0.493334i
$816$	0	0
$817$	−349411.	−0.523471
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 435598.i	− 0.646248i	−0.946357	−	0.323124i	$-0.895267\pi$
$821$	− 435598.i	− 0.646248i	0.946357	−	0.323124i	$-0.104733\pi$
$822$	0	0
$823$	− 164224.i	− 0.242459i	−0.992625	−	0.121229i	$-0.961316\pi$
$823$	− 164224.i	− 0.242459i	0.992625	−	0.121229i	$-0.0386837\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	652007.	0.953326	0.476663	−	0.879086i	$-0.341846\pi$
$827$	652007.	0.953326	0.476663	+	0.879086i	$0.341846\pi$
$828$	0	0
$829$	− 112061.i	− 0.163060i	−0.996671	−	0.0815299i	$-0.974019\pi$
$829$	− 112061.i	− 0.163060i	0.996671	−	0.0815299i	$-0.0259806\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	294605.	0.424571
$834$	0	0
$835$	232524.	0.333500
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 890284.i	− 1.26475i	−0.774663	−	0.632375i	$-0.782080\pi$
$839$	− 890284.i	− 1.26475i	0.774663	−	0.632375i	$-0.217920\pi$
$840$	0	0
$841$	320612.	0.453303
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.59286e6i	2.23081i
$846$	0	0
$847$	− 1.49873e6i	− 2.08909i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−156728.	−0.216415
$852$	0	0
$853$	850265.i	1.16857i	0.811547	+	0.584287i	$0.198626\pi$
$853$	850265.i	1.16857i	−0.811547	+	0.584287i	$0.801374\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.00134e6	−1.36339	−0.681694	−	0.731638i	$-0.738756\pi$
$857$	−1.00134e6	−1.36339	−0.681694	+	0.731638i	$0.738756\pi$
$858$	0	0
$859$	−1.14307e6	−1.54913	−0.774564	−	0.632495i	$-0.782031\pi$
$859$	−1.14307e6	−1.54913	−0.774564	+	0.632495i	$0.782031\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.01524e6i	1.36317i	0.731741	+	0.681583i	$0.238708\pi$
$863$	1.01524e6i	1.36317i	−0.731741	+	0.681583i	$0.761292\pi$
$864$	0	0
$865$	622473.	0.831933
$866$	0	0
$867$	0	0
$868$	0	0
$869$	745952.i	0.987806i
$870$	0	0
$871$	1.07919e6i	1.42254i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−661179.	−0.863580
$876$	0	0
$877$	− 1.03625e6i	− 1.34730i	−0.739049	−	0.673652i	$-0.764725\pi$
$877$	− 1.03625e6i	− 1.34730i	0.739049	−	0.673652i	$-0.235275\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.33421e6	−1.71899	−0.859494	−	0.511146i	$-0.829221\pi$
$881$	−1.33421e6	−1.71899	−0.859494	+	0.511146i	$0.829221\pi$
$882$	0	0
$883$	922293.	1.18290	0.591449	−	0.806342i	$-0.298556\pi$
$883$	922293.	1.18290	0.591449	+	0.806342i	$0.298556\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 870179.i	− 1.10602i	−0.833176	−	0.553008i	$-0.813480\pi$
$887$	− 870179.i	− 1.10602i	0.833176	−	0.553008i	$-0.186520\pi$
$888$	0	0
$889$	650830.	0.823501
$890$	0	0
$891$	0	0
$892$	0	0
$893$	458273.i	0.574673i
$894$	0	0
$895$	− 792837.i	− 0.989778i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−950529.	−1.17610
$900$	0	0
$901$	1.61969e6i	1.99518i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	852288.	1.04061
$906$	0	0
$907$	1.42374e6	1.73068	0.865341	−	0.501184i	$-0.167102\pi$
$907$	1.42374e6	1.73068	0.865341	+	0.501184i	$0.167102\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 784773.i	− 0.945600i	−0.881170	−	0.472800i	$-0.843243\pi$
$911$	− 784773.i	− 0.945600i	0.881170	−	0.472800i	$-0.156757\pi$
$912$	0	0
$913$	1.76323e6	2.11528
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 108880.i	− 0.129482i
$918$	0	0
$919$	252716.i	0.299227i	0.988745	+	0.149614i	$0.0478030\pi$
$919$	252716.i	0.299227i	−0.988745	+	0.149614i	$0.952197\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−637965.	−0.748848
$924$	0	0
$925$	− 99604.9i	− 0.116412i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.07539e6	−1.24605	−0.623025	−	0.782202i	$-0.714097\pi$
$929$	−1.07539e6	−1.24605	−0.623025	+	0.782202i	$0.714097\pi$
$930$	0	0
$931$	−265457.	−0.306264
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.03831e6i	2.33157i
$936$	0	0
$937$	503408.	0.573378	0.286689	−	0.958024i	$-0.407445\pi$
$937$	503408.	0.573378	0.286689	+	0.958024i	$0.407445\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	85259.4i	0.0962860i	0.998840	+	0.0481430i	$0.0153303\pi$
$941$	85259.4i	0.0962860i	−0.998840	+	0.0481430i	$0.984670\pi$
$942$	0	0
$943$	312871.i	0.351838i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	874018.	0.974587	0.487293	−	0.873238i	$-0.337984\pi$
$947$	874018.	0.974587	0.487293	+	0.873238i	$0.337984\pi$
$948$	0	0
$949$	− 2.28121e6i	− 2.53298i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−653547.	−0.719600	−0.359800	−	0.933029i	$-0.617155\pi$
$953$	−653547.	−0.719600	−0.359800	+	0.933029i	$0.617155\pi$
$954$	0	0
$955$	−492282.	−0.539768
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 223002.i	− 0.242477i
$960$	0	0
$961$	−1.41312e6	−1.53014
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2986.66i	− 0.00320723i
$966$	0	0
$967$	1.57343e6i	1.68265i	0.540527	+	0.841326i	$0.318225\pi$
$967$	1.57343e6i	1.68265i	−0.540527	+	0.841326i	$0.681775\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−280331.	−0.297326	−0.148663	−	0.988888i	$-0.547497\pi$
$971$	−280331.	−0.297326	−0.148663	+	0.988888i	$0.547497\pi$
$972$	0	0
$973$	− 952816.i	− 1.00643i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.19063e6	1.24735	0.623673	−	0.781685i	$-0.285640\pi$
$977$	1.19063e6	1.24735	0.623673	+	0.781685i	$0.285640\pi$
$978$	0	0
$979$	−1.06407e6	−1.11021
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.13386e6i	− 1.17341i	−0.809800	−	0.586706i	$-0.800424\pi$
$983$	− 1.13386e6i	− 1.17341i	0.809800	−	0.586706i	$-0.199576\pi$
$984$	0	0
$985$	208874.	0.215284
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 115180.i	− 0.117756i
$990$	0	0
$991$	− 535087.i	− 0.544850i	−0.962177	−	0.272425i	$-0.912174\pi$
$991$	− 535087.i	− 0.544850i	0.962177	−	0.272425i	$-0.0878256\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.78783e6	1.80584
$996$	0	0
$997$	− 1.02178e6i	− 1.02794i	−0.857808	−	0.513970i	$-0.828174\pi$
$997$	− 1.02178e6i	− 1.02794i	0.857808	−	0.513970i	$-0.171826\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.5.b.m.703.3		16
3.2	odd	2		384.5.b.d.319.15	yes	16
4.3	odd	2	inner	1152.5.b.m.703.4		16
8.3	odd	2	inner	1152.5.b.m.703.14		16
8.5	even	2	inner	1152.5.b.m.703.13		16
12.11	even	2		384.5.b.d.319.7	yes	16
24.5	odd	2		384.5.b.d.319.2	✓	16
24.11	even	2		384.5.b.d.319.10	yes	16
48.5	odd	4		768.5.g.h.511.2		8
48.11	even	4		768.5.g.h.511.6		8
48.29	odd	4		768.5.g.j.511.7		8
48.35	even	4		768.5.g.j.511.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.d.319.2	✓	16	24.5	odd	2
384.5.b.d.319.7	yes	16	12.11	even	2
384.5.b.d.319.10	yes	16	24.11	even	2
384.5.b.d.319.15	yes	16	3.2	odd	2
768.5.g.h.511.2		8	48.5	odd	4
768.5.g.h.511.6		8	48.11	even	4
768.5.g.j.511.3		8	48.35	even	4
768.5.g.j.511.7		8	48.29	odd	4
1152.5.b.m.703.3		16	1.1	even	1	trivial
1152.5.b.m.703.4		16	4.3	odd	2	inner
1152.5.b.m.703.13		16	8.5	even	2	inner
1152.5.b.m.703.14		16	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1152.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-23.5183i q^{5} -40.3412i q^{7} -227.579 q^{11} -310.305i q^{13} +380.831 q^{17} -343.152 q^{19} -113.116i q^{23} +71.8887 q^{25} -621.827i q^{29} -1528.61i q^{31} -948.758 q^{35} -1385.54i q^{37} -2765.92 q^{41} +1018.24 q^{43} -1335.48i q^{47} +773.586 q^{49} +4253.04i q^{53} +5352.28i q^{55} -77.6265 q^{59} -1154.74i q^{61} -7297.86 q^{65} -3477.85 q^{67} -2055.93i q^{71} +7351.49 q^{73} +9180.83i q^{77} -3277.77i q^{79} -7747.77 q^{83} -8956.49i q^{85} +4675.62 q^{89} -12518.1 q^{91} +8070.35i q^{95} +408.505 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 480 q^{17} + 144 q^{25} + 3552 q^{41} - 20080 q^{49} - 23040 q^{65} + 34400 q^{73} + 15648 q^{89} + 5088 q^{97}+O(q^{100}) \) 16 * q - 480 * q^17 + 144 * q^25 + 3552 * q^41 - 20080 * q^49 - 23040 * q^65 + 34400 * q^73 + 15648 * q^89 + 5088 * q^97

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