LMFDB - Embedded newform 1152.3.e.g.1025.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(1025,1152)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1152.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.e (of order $2$, degree $1$, 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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 11x^{2} - 10x + 3 $ x^4 - 2x^3 + 11x^2 - 10*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.4
Root			$0.500000 - 3.07253i$ of defining polynomial
Character	$\chi$	$=$	1152.1025
Dual form			1152.3.e.g.1025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.04746i q^{5} +2.00000 q^{7} +O(q^{10})$	$q+8.04746i q^{5} +2.00000 q^{7} -21.7518i q^{11} +17.3808 q^{13} -11.8523i q^{17} -10.7617 q^{19} -35.0183i q^{23} -39.7617 q^{25} -11.7515i q^{29} -35.5233 q^{31} +16.0949i q^{35} +26.0000 q^{37} +2.28985i q^{41} +65.5233 q^{43} -27.2071i q^{47} -45.0000 q^{49} -49.5982i q^{53} +175.047 q^{55} -73.5391i q^{59} -7.52333 q^{61} +139.872i q^{65} +65.2383 q^{67} +84.1787i q^{71} +84.5700 q^{73} -43.5036i q^{77} +75.5233 q^{79} +48.2848i q^{83} +95.3808 q^{85} +146.067i q^{89} +34.7617 q^{91} -86.6041i q^{95} -106.762 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} + 32 q^{13} + 32 q^{19} - 84 q^{25} + 8 q^{31} + 104 q^{37} + 112 q^{43} - 180 q^{49} + 400 q^{55} + 120 q^{61} + 336 q^{67} - 112 q^{73} + 152 q^{79} + 344 q^{85} + 64 q^{91} - 352 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 + 32 * q^13 + 32 * q^19 - 84 * q^25 + 8 * q^31 + 104 * q^37 + 112 * q^43 - 180 * q^49 + 400 * q^55 + 120 * q^61 + 336 * q^67 - 112 * q^73 + 152 * q^79 + 344 * q^85 + 64 * q^91 - 352 * q^97

Display 100 coefficients 10 coefficients

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
$5$	8.04746i	1.60949i	0.593619	+	0.804746i	$0.297699\pi$
$5$	8.04746i	1.60949i	−0.593619	+	0.804746i	$0.702301\pi$
$6$	0	0
$7$	2.00000	0.285714	0.142857	−	0.989743i	$-0.454371\pi$
$7$	2.00000	0.285714	0.142857	+	0.989743i	$0.454371\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 21.7518i	− 1.97743i	−0.149794	−	0.988717i	$-0.547861\pi$
$11$	− 21.7518i	− 1.97743i	0.149794	−	0.988717i	$-0.452139\pi$
$12$	0	0
$13$	17.3808	1.33699	0.668494	−	0.743718i	$-0.266939\pi$
$13$	17.3808	1.33699	0.668494	+	0.743718i	$0.266939\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 11.8523i	− 0.697193i	−0.937273	−	0.348597i	$-0.886658\pi$
$17$	− 11.8523i	− 0.697193i	0.937273	−	0.348597i	$-0.113342\pi$
$18$	0	0
$19$	−10.7617	−0.566403	−0.283202	−	0.959060i	$-0.591397\pi$
$19$	−10.7617	−0.566403	−0.283202	+	0.959060i	$0.591397\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 35.0183i	− 1.52253i	−0.648439	−	0.761267i	$-0.724578\pi$
$23$	− 35.0183i	− 1.52253i	0.648439	−	0.761267i	$-0.275422\pi$
$24$	0	0
$25$	−39.7617	−1.59047
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 11.7515i	− 0.405225i	−0.979259	−	0.202613i	$-0.935057\pi$
$29$	− 11.7515i	− 0.405225i	0.979259	−	0.202613i	$-0.0649432\pi$
$30$	0	0
$31$	−35.5233	−1.14591	−0.572957	−	0.819586i	$-0.694204\pi$
$31$	−35.5233	−1.14591	−0.572957	+	0.819586i	$0.694204\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.0949i	0.459855i
$36$	0	0
$37$	26.0000	0.702703	0.351351	−	0.936244i	$-0.385722\pi$
$37$	26.0000	0.702703	0.351351	+	0.936244i	$0.385722\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.28985i	0.0558500i	0.999610	+	0.0279250i	$0.00888996\pi$
$41$	2.28985i	0.0558500i	−0.999610	+	0.0279250i	$0.991110\pi$
$42$	0	0
$43$	65.5233	1.52380	0.761899	−	0.647696i	$-0.224267\pi$
$43$	65.5233	1.52380	0.761899	+	0.647696i	$0.224267\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 27.2071i	− 0.578875i	−0.957197	−	0.289437i	$-0.906532\pi$
$47$	− 27.2071i	− 0.578875i	0.957197	−	0.289437i	$-0.0934682\pi$
$48$	0	0
$49$	−45.0000	−0.918367
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 49.5982i	− 0.935816i	−0.883777	−	0.467908i	$-0.845008\pi$
$53$	− 49.5982i	− 0.935816i	0.883777	−	0.467908i	$-0.154992\pi$
$54$	0	0
$55$	175.047	3.18267
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 73.5391i	− 1.24643i	−0.782052	−	0.623213i	$-0.785827\pi$
$59$	− 73.5391i	− 1.24643i	0.782052	−	0.623213i	$-0.214173\pi$
$60$	0	0
$61$	−7.52333	−0.123333	−0.0616666	−	0.998097i	$-0.519642\pi$
$61$	−7.52333	−0.123333	−0.0616666	+	0.998097i	$0.519642\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	139.872i	2.15187i
$66$	0	0
$67$	65.2383	0.973707	0.486853	−	0.873484i	$-0.338145\pi$
$67$	65.2383	0.973707	0.486853	+	0.873484i	$0.338145\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	84.1787i	1.18562i	0.805344	+	0.592808i	$0.201981\pi$
$71$	84.1787i	1.18562i	−0.805344	+	0.592808i	$0.798019\pi$
$72$	0	0
$73$	84.5700	1.15849	0.579246	−	0.815152i	$-0.303347\pi$
$73$	84.5700	1.15849	0.579246	+	0.815152i	$0.303347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 43.5036i	− 0.564981i
$78$	0	0
$79$	75.5233	0.955991	0.477996	−	0.878362i	$-0.341363\pi$
$79$	75.5233	0.955991	0.477996	+	0.878362i	$0.341363\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	48.2848i	0.581744i	0.956762	+	0.290872i	$0.0939454\pi$
$83$	48.2848i	0.581744i	−0.956762	+	0.290872i	$0.906055\pi$
$84$	0	0
$85$	95.3808	1.12213
$86$	0	0
$87$	0	0
$88$	0	0
$89$	146.067i	1.64120i	0.571501	+	0.820601i	$0.306361\pi$
$89$	146.067i	1.64120i	−0.571501	+	0.820601i	$0.693639\pi$
$90$	0	0
$91$	34.7617	0.381996
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 86.6041i	− 0.911622i
$96$	0	0
$97$	−106.762	−1.10064	−0.550318	−	0.834955i	$-0.685493\pi$
$97$	−106.762	−1.10064	−0.550318	+	0.834955i	$0.685493\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 144.889i	− 1.43455i	−0.696792	−	0.717273i	$-0.745390\pi$
$101$	− 144.889i	− 1.43455i	0.696792	−	0.717273i	$-0.254610\pi$
$102$	0	0
$103$	−26.5700	−0.257961	−0.128980	−	0.991647i	$-0.541170\pi$
$103$	−26.5700	−0.257961	−0.128980	+	0.991647i	$0.541170\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 71.3848i	− 0.667148i	−0.942724	−	0.333574i	$-0.891745\pi$
$107$	− 71.3848i	− 0.667148i	0.942724	−	0.333574i	$-0.108255\pi$
$108$	0	0
$109$	118.619	1.08825	0.544125	−	0.839004i	$-0.316862\pi$
$109$	118.619	1.08825	0.544125	+	0.839004i	$0.316862\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 86.6701i	− 0.766992i	−0.923543	−	0.383496i	$-0.874720\pi$
$113$	− 86.6701i	− 0.766992i	0.923543	−	0.383496i	$-0.125280\pi$
$114$	0	0
$115$	281.808	2.45051
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 23.7046i	− 0.199198i
$120$	0	0
$121$	−352.140	−2.91025
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 118.794i	− 0.950352i
$126$	0	0
$127$	69.0467	0.543674	0.271837	−	0.962343i	$-0.412369\pi$
$127$	69.0467	0.543674	0.271837	+	0.962343i	$0.412369\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.25382i	0.0401055i	0.999799	+	0.0200527i	$0.00638341\pi$
$131$	5.25382i	0.0401055i	−0.999799	+	0.0200527i	$0.993617\pi$
$132$	0	0
$133$	−21.5233	−0.161830
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 120.139i	− 0.876924i	−0.898750	−	0.438462i	$-0.855523\pi$
$137$	− 120.139i	− 0.876924i	0.898750	−	0.438462i	$-0.144477\pi$
$138$	0	0
$139$	−242.378	−1.74373	−0.871864	−	0.489747i	$-0.837089\pi$
$139$	−242.378	−1.74373	−0.871864	+	0.489747i	$0.837089\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 378.064i	− 2.64380i
$144$	0	0
$145$	94.5700	0.652207
$146$	0	0
$147$	0	0
$148$	0	0
$149$	192.500i	1.29194i	0.763361	+	0.645972i	$0.223548\pi$
$149$	192.500i	1.29194i	−0.763361	+	0.645972i	$0.776452\pi$
$150$	0	0
$151$	17.4300	0.115431	0.0577153	−	0.998333i	$-0.481618\pi$
$151$	17.4300	0.115431	0.0577153	+	0.998333i	$0.481618\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 285.873i	− 1.84434i
$156$	0	0
$157$	213.617	1.36062	0.680308	−	0.732927i	$-0.261846\pi$
$157$	213.617	1.36062	0.680308	+	0.732927i	$0.261846\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 70.0366i	− 0.435010i
$162$	0	0
$163$	231.047	1.41746	0.708732	−	0.705478i	$-0.249268\pi$
$163$	231.047	1.41746	0.708732	+	0.705478i	$0.249268\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	126.202i	0.755701i	0.925867	+	0.377850i	$0.123337\pi$
$167$	126.202i	0.755701i	−0.925867	+	0.377850i	$0.876663\pi$
$168$	0	0
$169$	133.093	0.787534
$170$	0	0
$171$	0	0
$172$	0	0
$173$	199.032i	1.15048i	0.817986	+	0.575238i	$0.195090\pi$
$173$	199.032i	1.15048i	−0.817986	+	0.575238i	$0.804910\pi$
$174$	0	0
$175$	−79.5233	−0.454419
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 170.915i	− 0.954831i	−0.878678	−	0.477415i	$-0.841574\pi$
$179$	− 170.915i	− 0.954831i	0.878678	−	0.477415i	$-0.158426\pi$
$180$	0	0
$181$	38.6192	0.213366	0.106683	−	0.994293i	$-0.465977\pi$
$181$	38.6192	0.213366	0.106683	+	0.994293i	$0.465977\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	209.234i	1.13099i
$186$	0	0
$187$	−257.808	−1.37865
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 269.778i	− 1.41245i	−0.707988	−	0.706224i	$-0.750397\pi$
$191$	− 269.778i	− 1.41245i	0.707988	−	0.706224i	$-0.249603\pi$
$192$	0	0
$193$	−263.140	−1.36342	−0.681710	−	0.731623i	$-0.738763\pi$
$193$	−263.140	−1.36342	−0.681710	+	0.731623i	$0.738763\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	80.0368i	0.406278i	0.979150	+	0.203139i	$0.0651144\pi$
$197$	80.0368i	0.406278i	−0.979150	+	0.203139i	$0.934886\pi$
$198$	0	0
$199$	265.047	1.33189	0.665946	−	0.746000i	$-0.268028\pi$
$199$	265.047	1.33189	0.665946	+	0.746000i	$0.268028\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 23.5031i	− 0.115779i
$204$	0	0
$205$	−18.4275	−0.0898902
$206$	0	0
$207$	0	0
$208$	0	0
$209$	234.085i	1.12003i
$210$	0	0
$211$	−159.332	−0.755126	−0.377563	−	0.925984i	$-0.623238\pi$
$211$	−159.332	−0.755126	−0.377563	+	0.925984i	$0.623238\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	527.297i	2.45254i
$216$	0	0
$217$	−71.0467	−0.327404
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 206.003i	− 0.932138i
$222$	0	0
$223$	−90.9533	−0.407863	−0.203931	−	0.978985i	$-0.565372\pi$
$223$	−90.9533	−0.407863	−0.203931	+	0.978985i	$0.565372\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	322.371i	1.42014i	0.704133	+	0.710069i	$0.251336\pi$
$227$	322.371i	1.42014i	−0.704133	+	0.710069i	$0.748664\pi$
$228$	0	0
$229$	180.997	0.790382	0.395191	−	0.918599i	$-0.370678\pi$
$229$	180.997	0.790382	0.395191	+	0.918599i	$0.370678\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 329.241i	− 1.41305i	−0.707688	−	0.706525i	$-0.750262\pi$
$233$	− 329.241i	− 1.41305i	0.707688	−	0.706525i	$-0.249738\pi$
$234$	0	0
$235$	218.948	0.931695
$236$	0	0
$237$	0	0
$238$	0	0
$239$	197.045i	0.824455i	0.911081	+	0.412227i	$0.135249\pi$
$239$	197.045i	0.824455i	−0.911081	+	0.412227i	$0.864751\pi$
$240$	0	0
$241$	215.332	0.893492	0.446746	−	0.894661i	$-0.352583\pi$
$241$	215.332	0.893492	0.446746	+	0.894661i	$0.352583\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 362.136i	− 1.47811i
$246$	0	0
$247$	−187.047	−0.757274
$248$	0	0
$249$	0	0
$250$	0	0
$251$	245.329i	0.977408i	0.872450	+	0.488704i	$0.162530\pi$
$251$	245.329i	0.977408i	−0.872450	+	0.488704i	$0.837470\pi$
$252$	0	0
$253$	−761.710	−3.01071
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 39.5320i	− 0.153821i	−0.997038	−	0.0769105i	$-0.975494\pi$
$257$	− 39.5320i	− 0.153821i	0.997038	−	0.0769105i	$-0.0245056\pi$
$258$	0	0
$259$	52.0000	0.200772
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 217.921i	− 0.828596i	−0.910141	−	0.414298i	$-0.864027\pi$
$263$	− 217.921i	− 0.828596i	0.910141	−	0.414298i	$-0.135973\pi$
$264$	0	0
$265$	399.140	1.50619
$266$	0	0
$267$	0	0
$268$	0	0
$269$	205.433i	0.763691i	0.924226	+	0.381845i	$0.124711\pi$
$269$	205.433i	0.763691i	−0.924226	+	0.381845i	$0.875289\pi$
$270$	0	0
$271$	−441.047	−1.62748	−0.813739	−	0.581230i	$-0.802572\pi$
$271$	−441.047	−1.62748	−0.813739	+	0.581230i	$0.802572\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	864.887i	3.14504i
$276$	0	0
$277$	−185.951	−0.671303	−0.335651	−	0.941986i	$-0.608956\pi$
$277$	−185.951	−0.671303	−0.335651	+	0.941986i	$0.608956\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	87.7472i	0.312268i	0.987736	+	0.156134i	$0.0499031\pi$
$281$	87.7472i	0.312268i	−0.987736	+	0.156134i	$0.950097\pi$
$282$	0	0
$283$	46.0933	0.162874	0.0814369	−	0.996678i	$-0.474049\pi$
$283$	46.0933	0.162874	0.0814369	+	0.996678i	$0.474049\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.57970i	0.0159571i
$288$	0	0
$289$	148.523	0.513922
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 118.287i	− 0.403708i	−0.979416	−	0.201854i	$-0.935303\pi$
$293$	− 118.287i	− 0.403708i	0.979416	−	0.201854i	$-0.0646967\pi$
$294$	0	0
$295$	591.803	2.00611
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 608.647i	− 2.03561i
$300$	0	0
$301$	131.047	0.435371
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 60.5437i	− 0.198504i
$306$	0	0
$307$	116.285	0.378778	0.189389	−	0.981902i	$-0.439349\pi$
$307$	116.285	0.378778	0.189389	+	0.981902i	$0.439349\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	261.564i	0.841040i	0.907283	+	0.420520i	$0.138152\pi$
$311$	261.564i	0.841040i	−0.907283	+	0.420520i	$0.861848\pi$
$312$	0	0
$313$	310.000	0.990415	0.495208	−	0.868775i	$-0.335092\pi$
$313$	310.000	0.990415	0.495208	+	0.868775i	$0.335092\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 223.279i	− 0.704350i	−0.935934	−	0.352175i	$-0.885442\pi$
$317$	− 223.279i	− 0.704350i	0.935934	−	0.352175i	$-0.114558\pi$
$318$	0	0
$319$	−255.617	−0.801306
$320$	0	0
$321$	0	0
$322$	0	0
$323$	127.550i	0.394893i
$324$	0	0
$325$	−691.091	−2.12643
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 54.4142i	− 0.165393i
$330$	0	0
$331$	−353.238	−1.06719	−0.533593	−	0.845742i	$-0.679158\pi$
$331$	−353.238	−1.06719	−0.533593	+	0.845742i	$0.679158\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	525.003i	1.56717i
$336$	0	0
$337$	316.000	0.937685	0.468843	−	0.883282i	$-0.344671\pi$
$337$	316.000	0.937685	0.468843	+	0.883282i	$0.344671\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	772.696i	2.26597i
$342$	0	0
$343$	−188.000	−0.548105
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 230.916i	− 0.665465i	−0.943021	−	0.332732i	$-0.892029\pi$
$347$	− 230.916i	− 0.665465i	0.943021	−	0.332732i	$-0.107971\pi$
$348$	0	0
$349$	189.233	0.542216	0.271108	−	0.962549i	$-0.412610\pi$
$349$	189.233	0.542216	0.271108	+	0.962549i	$0.412610\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 151.050i	− 0.427903i	−0.976844	−	0.213952i	$-0.931367\pi$
$353$	− 151.050i	− 0.427903i	0.976844	−	0.213952i	$-0.0686335\pi$
$354$	0	0
$355$	−677.425	−1.90824
$356$	0	0
$357$	0	0
$358$	0	0
$359$	217.789i	0.606654i	0.952886	+	0.303327i	$0.0980975\pi$
$359$	217.789i	0.606654i	−0.952886	+	0.303327i	$0.901903\pi$
$360$	0	0
$361$	−245.187	−0.679187
$362$	0	0
$363$	0	0
$364$	0	0
$365$	680.574i	1.86459i
$366$	0	0
$367$	−444.093	−1.21006	−0.605032	−	0.796201i	$-0.706840\pi$
$367$	−444.093	−1.21006	−0.605032	+	0.796201i	$0.706840\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 99.1965i	− 0.267376i
$372$	0	0
$373$	124.663	0.334218	0.167109	−	0.985938i	$-0.446557\pi$
$373$	124.663	0.334218	0.167109	+	0.985938i	$0.446557\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 204.251i	− 0.541781i
$378$	0	0
$379$	−419.332	−1.10642	−0.553208	−	0.833043i	$-0.686597\pi$
$379$	−419.332	−1.10642	−0.553208	+	0.833043i	$0.686597\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	218.060i	0.569347i	0.958625	+	0.284674i	$0.0918852\pi$
$383$	218.060i	0.569347i	−0.958625	+	0.284674i	$0.908115\pi$
$384$	0	0
$385$	350.093	0.909333
$386$	0	0
$387$	0	0
$388$	0	0
$389$	214.988i	0.552669i	0.961062	+	0.276334i	$0.0891197\pi$
$389$	214.988i	0.552669i	−0.961062	+	0.276334i	$0.910880\pi$
$390$	0	0
$391$	−415.047	−1.06150
$392$	0	0
$393$	0	0
$394$	0	0
$395$	607.771i	1.53866i
$396$	0	0
$397$	506.187	1.27503	0.637515	−	0.770438i	$-0.279963\pi$
$397$	506.187	1.27503	0.637515	+	0.770438i	$0.279963\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 718.820i	− 1.79257i	−0.443480	−	0.896284i	$-0.646256\pi$
$401$	− 718.820i	− 1.79257i	0.443480	−	0.896284i	$-0.353744\pi$
$402$	0	0
$403$	−617.425	−1.53207
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 565.546i	− 1.38955i
$408$	0	0
$409$	−29.2383	−0.0714874	−0.0357437	−	0.999361i	$-0.511380\pi$
$409$	−29.2383	−0.0714874	−0.0357437	+	0.999361i	$0.511380\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 147.078i	− 0.356122i
$414$	0	0
$415$	−388.570	−0.936313
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.22387i	0.00530757i	0.999996	+	0.00265379i	$0.000844728\pi$
$419$	2.22387i	0.00530757i	−0.999996	+	0.00265379i	$0.999155\pi$
$420$	0	0
$421$	−117.194	−0.278371	−0.139186	−	0.990266i	$-0.544448\pi$
$421$	−117.194	−0.278371	−0.139186	+	0.990266i	$0.544448\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	471.267i	1.10886i
$426$	0	0
$427$	−15.0467	−0.0352381
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 755.315i	− 1.75247i	−0.481884	−	0.876235i	$-0.660047\pi$
$431$	− 755.315i	− 1.75247i	0.481884	−	0.876235i	$-0.339953\pi$
$432$	0	0
$433$	−433.233	−1.00054	−0.500269	−	0.865870i	$-0.666766\pi$
$433$	−433.233	−1.00054	−0.500269	+	0.865870i	$0.666766\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	376.855i	0.862368i
$438$	0	0
$439$	−231.140	−0.526515	−0.263257	−	0.964726i	$-0.584797\pi$
$439$	−231.140	−0.526515	−0.263257	+	0.964726i	$0.584797\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9.08983i	− 0.0205188i	−0.999947	−	0.0102594i	$-0.996734\pi$
$443$	− 9.08983i	− 0.0205188i	0.999947	−	0.0102594i	$-0.00326573\pi$
$444$	0	0
$445$	−1175.47	−2.64150
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 290.921i	− 0.647932i	−0.946069	−	0.323966i	$-0.894984\pi$
$449$	− 290.921i	− 0.647932i	0.946069	−	0.323966i	$-0.105016\pi$
$450$	0	0
$451$	49.8083	0.110440
$452$	0	0
$453$	0	0
$454$	0	0
$455$	279.743i	0.614820i
$456$	0	0
$457$	−333.042	−0.728756	−0.364378	−	0.931251i	$-0.618718\pi$
$457$	−333.042	−0.728756	−0.364378	+	0.931251i	$0.618718\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	132.971i	0.288440i	0.989546	+	0.144220i	$0.0460673\pi$
$461$	132.971i	0.288440i	−0.989546	+	0.144220i	$0.953933\pi$
$462$	0	0
$463$	391.140	0.844795	0.422397	−	0.906411i	$-0.361189\pi$
$463$	391.140	0.844795	0.422397	+	0.906411i	$0.361189\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 469.310i	− 1.00495i	−0.864593	−	0.502473i	$-0.832423\pi$
$467$	− 469.310i	− 1.00495i	0.864593	−	0.502473i	$-0.167577\pi$
$468$	0	0
$469$	130.477	0.278202
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1425.25i	− 3.01321i
$474$	0	0
$475$	427.902	0.900846
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 355.708i	− 0.742605i	−0.928512	−	0.371302i	$-0.878911\pi$
$479$	− 355.708i	− 0.742605i	0.928512	−	0.371302i	$-0.121089\pi$
$480$	0	0
$481$	451.902	0.939504
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 859.161i	− 1.77147i
$486$	0	0
$487$	−839.897	−1.72463	−0.862317	−	0.506369i	$-0.830987\pi$
$487$	−839.897	−1.72463	−0.862317	+	0.506369i	$0.830987\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	388.572i	0.791388i	0.918382	+	0.395694i	$0.129496\pi$
$491$	388.572i	0.791388i	−0.918382	+	0.395694i	$0.870504\pi$
$492$	0	0
$493$	−139.282	−0.282520
$494$	0	0
$495$	0	0
$496$	0	0
$497$	168.357i	0.338747i
$498$	0	0
$499$	488.373	0.978704	0.489352	−	0.872086i	$-0.337233\pi$
$499$	488.373	0.978704	0.489352	+	0.872086i	$0.337233\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	533.488i	1.06061i	0.847806	+	0.530307i	$0.177923\pi$
$503$	533.488i	1.06061i	−0.847806	+	0.530307i	$0.822077\pi$
$504$	0	0
$505$	1165.99	2.30889
$506$	0	0
$507$	0	0
$508$	0	0
$509$	179.094i	0.351855i	0.984403	+	0.175927i	$0.0562924\pi$
$509$	179.094i	0.351855i	−0.984403	+	0.175927i	$0.943708\pi$
$510$	0	0
$511$	169.140	0.330998
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 213.821i	− 0.415186i
$516$	0	0
$517$	−591.803	−1.14469
$518$	0	0
$519$	0	0
$520$	0	0
$521$	397.456i	0.762872i	0.924395	+	0.381436i	$0.124570\pi$
$521$	397.456i	0.762872i	−0.924395	+	0.381436i	$0.875430\pi$
$522$	0	0
$523$	−987.135	−1.88745	−0.943724	−	0.330735i	$-0.892703\pi$
$523$	−987.135	−1.88745	−0.943724	+	0.330735i	$0.892703\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	421.033i	0.798923i
$528$	0	0
$529$	−697.280	−1.31811
$530$	0	0
$531$	0	0
$532$	0	0
$533$	39.7995i	0.0746707i
$534$	0	0
$535$	574.467	1.07377
$536$	0	0
$537$	0	0
$538$	0	0
$539$	978.830i	1.81601i
$540$	0	0
$541$	214.992	0.397398	0.198699	−	0.980061i	$-0.436328\pi$
$541$	214.992	0.397398	0.198699	+	0.980061i	$0.436328\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	954.583i	1.75153i
$546$	0	0
$547$	381.710	0.697824	0.348912	−	0.937155i	$-0.386551\pi$
$547$	381.710	0.697824	0.348912	+	0.937155i	$0.386551\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	126.466i	0.229521i
$552$	0	0
$553$	151.047	0.273140
$554$	0	0
$555$	0	0
$556$	0	0
$557$	924.388i	1.65958i	0.558073	+	0.829792i	$0.311541\pi$
$557$	924.388i	1.65958i	−0.558073	+	0.829792i	$0.688459\pi$
$558$	0	0
$559$	1138.85	2.03730
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 70.5092i	− 0.125238i	−0.998037	−	0.0626191i	$-0.980055\pi$
$563$	− 70.5092i	− 0.125238i	0.998037	−	0.0626191i	$-0.0199453\pi$
$564$	0	0
$565$	697.474	1.23447
$566$	0	0
$567$	0	0
$568$	0	0
$569$	607.844i	1.06827i	0.845400	+	0.534134i	$0.179362\pi$
$569$	607.844i	1.06827i	−0.845400	+	0.534134i	$0.820638\pi$
$570$	0	0
$571$	455.430	0.797601	0.398800	−	0.917038i	$-0.369427\pi$
$571$	455.430	0.797601	0.398800	+	0.917038i	$0.369427\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1392.39i	2.42154i
$576$	0	0
$577$	501.233	0.868688	0.434344	−	0.900747i	$-0.356980\pi$
$577$	501.233	0.868688	0.434344	+	0.900747i	$0.356980\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	96.5696i	0.166213i
$582$	0	0
$583$	−1078.85	−1.85051
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 178.059i	− 0.303337i	−0.988431	−	0.151669i	$-0.951535\pi$
$587$	− 178.059i	− 0.303337i	0.988431	−	0.151669i	$-0.0484647\pi$
$588$	0	0
$589$	382.290	0.649049
$590$	0	0
$591$	0	0
$592$	0	0
$593$	960.244i	1.61930i	0.586914	+	0.809649i	$0.300343\pi$
$593$	960.244i	1.61930i	−0.586914	+	0.809649i	$0.699657\pi$
$594$	0	0
$595$	190.762	0.320608
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 680.706i	− 1.13640i	−0.822889	−	0.568202i	$-0.807639\pi$
$599$	− 680.706i	− 1.13640i	0.822889	−	0.568202i	$-0.192361\pi$
$600$	0	0
$601$	239.327	0.398214	0.199107	−	0.979978i	$-0.436196\pi$
$601$	239.327	0.398214	0.199107	+	0.979978i	$0.436196\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2833.83i	− 4.68402i
$606$	0	0
$607$	1177.80	1.94037	0.970184	−	0.242370i	$-0.0779248\pi$
$607$	1177.80	1.94037	0.970184	+	0.242370i	$0.0779248\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 472.882i	− 0.773948i
$612$	0	0
$613$	1106.56	1.80515	0.902577	−	0.430528i	$-0.141673\pi$
$613$	1106.56	1.80515	0.902577	+	0.430528i	$0.141673\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	271.185i	0.439522i	0.975554	+	0.219761i	$0.0705277\pi$
$617$	271.185i	0.439522i	−0.975554	+	0.219761i	$0.929472\pi$
$618$	0	0
$619$	447.430	0.722827	0.361414	−	0.932406i	$-0.382294\pi$
$619$	447.430	0.722827	0.361414	+	0.932406i	$0.382294\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	292.134i	0.468915i
$624$	0	0
$625$	−38.0517	−0.0608828
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 308.159i	− 0.489920i
$630$	0	0
$631$	369.047	0.584860	0.292430	−	0.956287i	$-0.405536\pi$
$631$	369.047	0.584860	0.292430	+	0.956287i	$0.405536\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	555.650i	0.875040i
$636$	0	0
$637$	−782.137	−1.22785
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 319.198i	− 0.497970i	−0.968507	−	0.248985i	$-0.919903\pi$
$641$	− 319.198i	− 0.497970i	0.968507	−	0.248985i	$-0.0800969\pi$
$642$	0	0
$643$	41.3266	0.0642715	0.0321357	−	0.999484i	$-0.489769\pi$
$643$	41.3266	0.0642715	0.0321357	+	0.999484i	$0.489769\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 160.011i	− 0.247313i	−0.992325	−	0.123656i	$-0.960538\pi$
$647$	− 160.011i	− 0.247313i	0.992325	−	0.123656i	$-0.0394620\pi$
$648$	0	0
$649$	−1599.61	−2.46472
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1111.54i	1.70220i	0.525003	+	0.851100i	$0.324064\pi$
$653$	1111.54i	1.70220i	−0.525003	+	0.851100i	$0.675936\pi$
$654$	0	0
$655$	−42.2799	−0.0645495
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 800.313i	− 1.21444i	−0.794536	−	0.607218i	$-0.792286\pi$
$659$	− 800.313i	− 1.21444i	0.794536	−	0.607218i	$-0.207714\pi$
$660$	0	0
$661$	−1142.18	−1.72795	−0.863976	−	0.503533i	$-0.832033\pi$
$661$	−1142.18	−1.72795	−0.863976	+	0.503533i	$0.832033\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 173.208i	− 0.260463i
$666$	0	0
$667$	−411.518	−0.616969
$668$	0	0
$669$	0	0
$670$	0	0
$671$	163.646i	0.243883i
$672$	0	0
$673$	−169.047	−0.251184	−0.125592	−	0.992082i	$-0.540083\pi$
$673$	−169.047	−0.251184	−0.125592	+	0.992082i	$0.540083\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 533.801i	− 0.788481i	−0.919007	−	0.394240i	$-0.871008\pi$
$677$	− 533.801i	− 0.788481i	0.919007	−	0.394240i	$-0.128992\pi$
$678$	0	0
$679$	−213.523	−0.314467
$680$	0	0
$681$	0	0
$682$	0	0
$683$	408.711i	0.598406i	0.954189	+	0.299203i	$0.0967208\pi$
$683$	408.711i	0.598406i	−0.954189	+	0.299203i	$0.903279\pi$
$684$	0	0
$685$	966.811	1.41140
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 862.059i	− 1.25117i
$690$	0	0
$691$	−331.233	−0.479353	−0.239677	−	0.970853i	$-0.577041\pi$
$691$	−331.233	−0.479353	−0.239677	+	0.970853i	$0.577041\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1950.53i	− 2.80652i
$696$	0	0
$697$	27.1400	0.0389382
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 116.814i	− 0.166638i	−0.996523	−	0.0833192i	$-0.973448\pi$
$701$	− 116.814i	− 0.166638i	0.996523	−	0.0833192i	$-0.0265521\pi$
$702$	0	0
$703$	−279.803	−0.398013
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 289.778i	− 0.409870i
$708$	0	0
$709$	−737.469	−1.04015	−0.520077	−	0.854119i	$-0.674097\pi$
$709$	−737.469	−1.04015	−0.520077	+	0.854119i	$0.674097\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1243.97i	1.74469i
$714$	0	0
$715$	3042.46	4.25518
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1017.28i	− 1.41486i	−0.706785	−	0.707428i	$-0.749855\pi$
$719$	− 1017.28i	− 1.41486i	0.706785	−	0.707428i	$-0.250145\pi$
$720$	0	0
$721$	−53.1400	−0.0737031
$722$	0	0
$723$	0	0
$724$	0	0
$725$	467.260i	0.644497i
$726$	0	0
$727$	277.430	0.381609	0.190805	−	0.981628i	$-0.438890\pi$
$727$	277.430	0.381609	0.190805	+	0.981628i	$0.438890\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 776.601i	− 1.06238i
$732$	0	0
$733$	405.371	0.553030	0.276515	−	0.961010i	$-0.410821\pi$
$733$	405.371	0.553030	0.276515	+	0.961010i	$0.410821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1419.05i	− 1.92544i
$738$	0	0
$739$	−295.036	−0.399237	−0.199619	−	0.979874i	$-0.563970\pi$
$739$	−295.036	−0.399237	−0.199619	+	0.979874i	$0.563970\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 486.350i	− 0.654576i	−0.944925	−	0.327288i	$-0.893865\pi$
$743$	− 486.350i	− 0.654576i	0.944925	−	0.327288i	$-0.106135\pi$
$744$	0	0
$745$	−1549.13	−2.07938
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 142.770i	− 0.190614i
$750$	0	0
$751$	705.430	0.939321	0.469660	−	0.882847i	$-0.344376\pi$
$751$	705.430	0.939321	0.469660	+	0.882847i	$0.344376\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	140.267i	0.185785i
$756$	0	0
$757$	1362.89	1.80039	0.900194	−	0.435489i	$-0.143424\pi$
$757$	1362.89	1.80039	0.900194	+	0.435489i	$0.143424\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 625.746i	− 0.822268i	−0.911575	−	0.411134i	$-0.865133\pi$
$761$	− 625.746i	− 0.822268i	0.911575	−	0.411134i	$-0.134867\pi$
$762$	0	0
$763$	237.238	0.310928
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1278.17i	− 1.66645i
$768$	0	0
$769$	−637.813	−0.829406	−0.414703	−	0.909957i	$-0.636115\pi$
$769$	−637.813	−0.829406	−0.414703	+	0.909957i	$0.636115\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	536.282i	0.693767i	0.937908	+	0.346884i	$0.112760\pi$
$773$	536.282i	0.693767i	−0.937908	+	0.346884i	$0.887240\pi$
$774$	0	0
$775$	1412.47	1.82254
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 24.6426i	− 0.0316336i
$780$	0	0
$781$	1831.04	2.34448
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1719.07i	2.18990i
$786$	0	0
$787$	−769.425	−0.977668	−0.488834	−	0.872377i	$-0.662578\pi$
$787$	−769.425	−0.977668	−0.488834	+	0.872377i	$0.662578\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 173.340i	− 0.219140i
$792$	0	0
$793$	−130.762	−0.164895
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 765.593i	− 0.960594i	−0.877106	−	0.480297i	$-0.840529\pi$
$797$	− 765.593i	− 0.960594i	0.877106	−	0.480297i	$-0.159471\pi$
$798$	0	0
$799$	−322.467	−0.403588
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1839.55i	− 2.29084i
$804$	0	0
$805$	563.617	0.700145
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 865.488i	− 1.06982i	−0.844908	−	0.534912i	$-0.820345\pi$
$809$	− 865.488i	− 1.06982i	0.844908	−	0.534912i	$-0.179655\pi$
$810$	0	0
$811$	−292.472	−0.360631	−0.180315	−	0.983609i	$-0.557712\pi$
$811$	−292.472	−0.360631	−0.180315	+	0.983609i	$0.557712\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1859.34i	2.28140i
$816$	0	0
$817$	−705.140	−0.863084
$818$	0	0
$819$	0	0
$820$	0	0
$821$	343.616i	0.418533i	0.977859	+	0.209266i	$0.0671076\pi$
$821$	343.616i	0.418533i	−0.977859	+	0.209266i	$0.932892\pi$
$822$	0	0
$823$	−773.430	−0.939769	−0.469885	−	0.882728i	$-0.655704\pi$
$823$	−773.430	−0.939769	−0.469885	+	0.882728i	$0.655704\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 854.310i	− 1.03302i	−0.856280	−	0.516511i	$-0.827230\pi$
$827$	− 854.310i	− 1.03302i	0.856280	−	0.516511i	$-0.172770\pi$
$828$	0	0
$829$	−85.5674	−0.103218	−0.0516088	−	0.998667i	$-0.516435\pi$
$829$	−85.5674	−0.103218	−0.0516088	+	0.998667i	$0.516435\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	533.353i	0.640280i
$834$	0	0
$835$	−1015.61	−1.21630
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1653.28i	− 1.97054i	−0.171011	−	0.985269i	$-0.554703\pi$
$839$	− 1653.28i	− 1.97054i	0.171011	−	0.985269i	$-0.445297\pi$
$840$	0	0
$841$	702.902	0.835793
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1071.06i	1.26753i
$846$	0	0
$847$	−704.280	−0.831499
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 910.475i	− 1.06989i
$852$	0	0
$853$	1045.62	1.22581	0.612905	−	0.790156i	$-0.290001\pi$
$853$	1045.62	1.22581	0.612905	+	0.790156i	$0.290001\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	939.437i	1.09619i	0.836415	+	0.548096i	$0.184647\pi$
$857$	939.437i	1.09619i	−0.836415	+	0.548096i	$0.815353\pi$
$858$	0	0
$859$	1249.52	1.45463	0.727313	−	0.686306i	$-0.240769\pi$
$859$	1249.52	1.45463	0.727313	+	0.686306i	$0.240769\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1470.64i	1.70411i	0.523456	+	0.852053i	$0.324642\pi$
$863$	1470.64i	1.70411i	−0.523456	+	0.852053i	$0.675358\pi$
$864$	0	0
$865$	−1601.70	−1.85168
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1642.77i	− 1.89041i
$870$	0	0
$871$	1133.90	1.30183
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 237.588i	− 0.271529i
$876$	0	0
$877$	195.720	0.223170	0.111585	−	0.993755i	$-0.464407\pi$
$877$	195.720	0.223170	0.111585	+	0.993755i	$0.464407\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	533.686i	0.605773i	0.953027	+	0.302887i	$0.0979504\pi$
$881$	533.686i	0.605773i	−0.953027	+	0.302887i	$0.902050\pi$
$882$	0	0
$883$	−963.803	−1.09151	−0.545755	−	0.837945i	$-0.683757\pi$
$883$	−963.803	−1.09151	−0.545755	+	0.837945i	$0.683757\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	507.769i	0.572456i	0.958162	+	0.286228i	$0.0924015\pi$
$887$	507.769i	0.572456i	−0.958162	+	0.286228i	$0.907598\pi$
$888$	0	0
$889$	138.093	0.155336
$890$	0	0
$891$	0	0
$892$	0	0
$893$	292.794i	0.327877i
$894$	0	0
$895$	1375.43	1.53679
$896$	0	0
$897$	0	0
$898$	0	0
$899$	417.453i	0.464353i
$900$	0	0
$901$	−587.852	−0.652444
$902$	0	0
$903$	0	0
$904$	0	0
$905$	310.786i	0.343410i
$906$	0	0
$907$	48.3834	0.0533444	0.0266722	−	0.999644i	$-0.491509\pi$
$907$	48.3834	0.0533444	0.0266722	+	0.999644i	$0.491509\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	318.257i	0.349349i	0.984626	+	0.174674i	$0.0558873\pi$
$911$	318.257i	0.349349i	−0.984626	+	0.174674i	$0.944113\pi$
$912$	0	0
$913$	1050.28	1.15036
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.5076i	0.0114587i
$918$	0	0
$919$	−583.513	−0.634944	−0.317472	−	0.948268i	$-0.602834\pi$
$919$	−583.513	−0.634944	−0.317472	+	0.948268i	$0.602834\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1463.10i	1.58515i
$924$	0	0
$925$	−1033.80	−1.11763
$926$	0	0
$927$	0	0
$928$	0	0
$929$	322.847i	0.347521i	0.984788	+	0.173761i	$0.0555919\pi$
$929$	322.847i	0.347521i	−0.984788	+	0.173761i	$0.944408\pi$
$930$	0	0
$931$	484.275	0.520166
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2074.70i	− 2.21893i
$936$	0	0
$937$	468.467	0.499964	0.249982	−	0.968250i	$-0.419575\pi$
$937$	468.467	0.499964	0.249982	+	0.968250i	$0.419575\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	469.964i	0.499430i	0.968319	+	0.249715i	$0.0803370\pi$
$941$	469.964i	0.499430i	−0.968319	+	0.249715i	$0.919663\pi$
$942$	0	0
$943$	80.1866	0.0850335
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 200.686i	− 0.211918i	−0.994370	−	0.105959i	$-0.966209\pi$
$947$	− 200.686i	− 0.211918i	0.994370	−	0.105959i	$-0.0337912\pi$
$948$	0	0
$949$	1469.90	1.54889
$950$	0	0
$951$	0	0
$952$	0	0
$953$	493.081i	0.517398i	0.965958	+	0.258699i	$0.0832939\pi$
$953$	493.081i	0.517398i	−0.965958	+	0.258699i	$0.916706\pi$
$954$	0	0
$955$	2171.03	2.27333
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 240.277i	− 0.250550i
$960$	0	0
$961$	300.907	0.313118
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2117.61i	− 2.19441i
$966$	0	0
$967$	204.467	0.211444	0.105722	−	0.994396i	$-0.466285\pi$
$967$	204.467	0.211444	0.105722	+	0.994396i	$0.466285\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 250.986i	− 0.258482i	−0.991613	−	0.129241i	$-0.958746\pi$
$971$	− 250.986i	− 0.258482i	0.991613	−	0.129241i	$-0.0412541\pi$
$972$	0	0
$973$	−484.757	−0.498208
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 520.823i	− 0.533084i	−0.963823	−	0.266542i	$-0.914119\pi$
$977$	− 520.823i	− 0.533084i	0.963823	−	0.266542i	$-0.0858811\pi$
$978$	0	0
$979$	3177.22	3.24537
$980$	0	0
$981$	0	0
$982$	0	0
$983$	569.174i	0.579017i	0.957175	+	0.289508i	$0.0934918\pi$
$983$	569.174i	0.579017i	−0.957175	+	0.289508i	$0.906508\pi$
$984$	0	0
$985$	−644.093	−0.653902
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2294.51i	− 2.32003i
$990$	0	0
$991$	−97.2434	−0.0981266	−0.0490633	−	0.998796i	$-0.515624\pi$
$991$	−97.2434	−0.0981266	−0.0490633	+	0.998796i	$0.515624\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2132.95i	2.14367i
$996$	0	0
$997$	−112.290	−0.112628	−0.0563140	−	0.998413i	$-0.517935\pi$
$997$	−112.290	−0.112628	−0.0563140	+	0.998413i	$0.517935\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.e.g.1025.4	yes	4
3.2	odd	2	inner	1152.3.e.g.1025.1	yes	4
4.3	odd	2		1152.3.e.c.1025.4	yes	4
8.3	odd	2		1152.3.e.a.1025.1	✓	4
8.5	even	2		1152.3.e.e.1025.1	yes	4
12.11	even	2		1152.3.e.c.1025.1	yes	4
16.3	odd	4		2304.3.h.l.2177.8		8
16.5	even	4		2304.3.h.j.2177.1		8
16.11	odd	4		2304.3.h.l.2177.1		8
16.13	even	4		2304.3.h.j.2177.8		8
24.5	odd	2		1152.3.e.e.1025.4	yes	4
24.11	even	2		1152.3.e.a.1025.4	yes	4
48.5	odd	4		2304.3.h.j.2177.7		8
48.11	even	4		2304.3.h.l.2177.7		8
48.29	odd	4		2304.3.h.j.2177.2		8
48.35	even	4		2304.3.h.l.2177.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.e.a.1025.1	✓	4	8.3	odd	2
1152.3.e.a.1025.4	yes	4	24.11	even	2
1152.3.e.c.1025.1	yes	4	12.11	even	2
1152.3.e.c.1025.4	yes	4	4.3	odd	2
1152.3.e.e.1025.1	yes	4	8.5	even	2
1152.3.e.e.1025.4	yes	4	24.5	odd	2
1152.3.e.g.1025.1	yes	4	3.2	odd	2	inner
1152.3.e.g.1025.4	yes	4	1.1	even	1	trivial
2304.3.h.j.2177.1		8	16.5	even	4
2304.3.h.j.2177.2		8	48.29	odd	4
2304.3.h.j.2177.7		8	48.5	odd	4
2304.3.h.j.2177.8		8	16.13	even	4
2304.3.h.l.2177.1		8	16.11	odd	4
2304.3.h.l.2177.2		8	48.35	even	4
2304.3.h.l.2177.7		8	48.11	even	4
2304.3.h.l.2177.8		8	16.3	odd	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.04746i q^{5} +2.00000 q^{7} +O(q^{10})\)	\(q+8.04746i q^{5} +2.00000 q^{7} -21.7518i q^{11} +17.3808 q^{13} -11.8523i q^{17} -10.7617 q^{19} -35.0183i q^{23} -39.7617 q^{25} -11.7515i q^{29} -35.5233 q^{31} +16.0949i q^{35} +26.0000 q^{37} +2.28985i q^{41} +65.5233 q^{43} -27.2071i q^{47} -45.0000 q^{49} -49.5982i q^{53} +175.047 q^{55} -73.5391i q^{59} -7.52333 q^{61} +139.872i q^{65} +65.2383 q^{67} +84.1787i q^{71} +84.5700 q^{73} -43.5036i q^{77} +75.5233 q^{79} +48.2848i q^{83} +95.3808 q^{85} +146.067i q^{89} +34.7617 q^{91} -86.6041i q^{95} -106.762 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} + 32 q^{13} + 32 q^{19} - 84 q^{25} + 8 q^{31} + 104 q^{37} + 112 q^{43} - 180 q^{49} + 400 q^{55} + 120 q^{61} + 336 q^{67} - 112 q^{73} + 152 q^{79} + 344 q^{85} + 64 q^{91} - 352 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 + 32 * q^13 + 32 * q^19 - 84 * q^25 + 8 * q^31 + 104 * q^37 + 112 * q^43 - 180 * q^49 + 400 * q^55 + 120 * q^61 + 336 * q^67 - 112 * q^73 + 152 * q^79 + 344 * q^85 + 64 * q^91 - 352 * q^97

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