LMFDB - Embedded newform 2304.3.h.j.2177.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(2177,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.2177");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.959512576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 81 $ x^8 + 7*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2177.8
Root			$-0.819051 - 1.52616i$ of defining polynomial
Character	$\chi$	$=$	2304.2177
Dual form			2304.3.h.j.2177.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.04746 q^{5} -2.00000 q^{7} +O(q^{10})$	$q+8.04746 q^{5} -2.00000 q^{7} -21.7518 q^{11} +17.3808i q^{13} -11.8523i q^{17} -10.7617i q^{19} +35.0183i q^{23} +39.7617 q^{25} +11.7515 q^{29} -35.5233 q^{31} -16.0949 q^{35} -26.0000i q^{37} -2.28985i q^{41} -65.5233i q^{43} -27.2071i q^{47} -45.0000 q^{49} -49.5982 q^{53} -175.047 q^{55} -73.5391 q^{59} -7.52333i q^{61} +139.872i q^{65} +65.2383i q^{67} -84.1787i q^{71} -84.5700 q^{73} +43.5036 q^{77} +75.5233 q^{79} -48.2848 q^{83} -95.3808i q^{85} -146.067i q^{89} -34.7617i q^{91} -86.6041i q^{95} -106.762 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^7	$ 8 q - 16 q^{7} + 168 q^{25} + 16 q^{31} - 360 q^{49} - 800 q^{55} + 224 q^{73} + 304 q^{79} - 704 q^{97}+O(q^{100}) $ 8 * q - 16 * q^7 + 168 * q^25 + 16 * q^31 - 360 * q^49 - 800 * q^55 + 224 * q^73 + 304 * q^79 - 704 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	8.04746	1.60949	0.804746	−	0.593619i	$-0.202301\pi$
$5$	8.04746	1.60949	0.804746	+	0.593619i	$0.202301\pi$
$6$	0	0
$7$	−2.00000	−0.285714	−0.142857	−	0.989743i	$-0.545629\pi$
$7$	−2.00000	−0.285714	−0.142857	+	0.989743i	$0.545629\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−21.7518	−1.97743	−0.988717	−	0.149794i	$-0.952139\pi$
$11$	−21.7518	−1.97743	−0.988717	+	0.149794i	$0.952139\pi$
$12$	0	0
$13$	17.3808i	1.33699i	0.743718	+	0.668494i	$0.233061\pi$
$13$	17.3808i	1.33699i	−0.743718	+	0.668494i	$0.766939\pi$
$14$	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
$19$	− 10.7617i	− 0.566403i	−0.959060	−	0.283202i	$-0.908603\pi$
$19$	− 10.7617i	− 0.566403i	0.959060	−	0.283202i	$-0.0913966\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	35.0183i	1.52253i	0.648439	+	0.761267i	$0.275422\pi$
$23$	35.0183i	1.52253i	−0.648439	+	0.761267i	$0.724578\pi$
$24$	0	0
$25$	39.7617	1.59047
$26$	0	0
$27$	0	0
$28$	0	0
$29$	11.7515	0.405225	0.202613	−	0.979259i	$-0.435057\pi$
$29$	11.7515	0.405225	0.202613	+	0.979259i	$0.435057\pi$
$30$	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
$35$	−16.0949	−0.459855
$36$	0	0
$37$	− 26.0000i	− 0.702703i	−0.936244	−	0.351351i	$-0.885722\pi$
$37$	− 26.0000i	− 0.702703i	0.936244	−	0.351351i	$-0.114278\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.28985i	− 0.0558500i	−0.999610	−	0.0279250i	$-0.991110\pi$
$41$	− 2.28985i	− 0.0558500i	0.999610	−	0.0279250i	$-0.00888996\pi$
$42$	0	0
$43$	− 65.5233i	− 1.52380i	−0.647696	−	0.761899i	$-0.724267\pi$
$43$	− 65.5233i	− 1.52380i	0.647696	−	0.761899i	$-0.275733\pi$
$44$	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
$53$	−49.5982	−0.935816	−0.467908	−	0.883777i	$-0.654992\pi$
$53$	−49.5982	−0.935816	−0.467908	+	0.883777i	$0.654992\pi$
$54$	0	0
$55$	−175.047	−3.18267
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−73.5391	−1.24643	−0.623213	−	0.782052i	$-0.714173\pi$
$59$	−73.5391	−1.24643	−0.623213	+	0.782052i	$0.714173\pi$
$60$	0	0
$61$	− 7.52333i	− 0.123333i	−0.998097	−	0.0616666i	$-0.980358\pi$
$61$	− 7.52333i	− 0.123333i	0.998097	−	0.0616666i	$-0.0196416\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	139.872i	2.15187i
$66$	0	0
$67$	65.2383i	0.973707i	0.873484	+	0.486853i	$0.161855\pi$
$67$	65.2383i	0.973707i	−0.873484	+	0.486853i	$0.838145\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 84.1787i	− 1.18562i	−0.805344	−	0.592808i	$-0.798019\pi$
$71$	− 84.1787i	− 1.18562i	0.805344	−	0.592808i	$-0.201981\pi$
$72$	0	0
$73$	−84.5700	−1.15849	−0.579246	−	0.815152i	$-0.696653\pi$
$73$	−84.5700	−1.15849	−0.579246	+	0.815152i	$0.696653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	43.5036	0.564981
$78$	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
$83$	−48.2848	−0.581744	−0.290872	−	0.956762i	$-0.593945\pi$
$83$	−48.2848	−0.581744	−0.290872	+	0.956762i	$0.593945\pi$
$84$	0	0
$85$	− 95.3808i	− 1.12213i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 146.067i	− 1.64120i	−0.571501	−	0.820601i	$-0.693639\pi$
$89$	− 146.067i	− 1.64120i	0.571501	−	0.820601i	$-0.306361\pi$
$90$	0	0
$91$	− 34.7617i	− 0.381996i
$92$	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
$101$	−144.889	−1.43455	−0.717273	−	0.696792i	$-0.754610\pi$
$101$	−144.889	−1.43455	−0.717273	+	0.696792i	$0.754610\pi$
$102$	0	0
$103$	26.5700	0.257961	0.128980	−	0.991647i	$-0.458830\pi$
$103$	26.5700	0.257961	0.128980	+	0.991647i	$0.458830\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−71.3848	−0.667148	−0.333574	−	0.942724i	$-0.608255\pi$
$107$	−71.3848	−0.667148	−0.333574	+	0.942724i	$0.608255\pi$
$108$	0	0
$109$	118.619i	1.08825i	0.839004	+	0.544125i	$0.183138\pi$
$109$	118.619i	1.08825i	−0.839004	+	0.544125i	$0.816862\pi$
$110$	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
$115$	281.808i	2.45051i
$116$	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
$125$	118.794	0.950352
$126$	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
$131$	−5.25382	−0.0401055	−0.0200527	−	0.999799i	$-0.506383\pi$
$131$	−5.25382	−0.0401055	−0.0200527	+	0.999799i	$0.506383\pi$
$132$	0	0
$133$	21.5233i	0.161830i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	120.139i	0.876924i	0.898750	+	0.438462i	$0.144477\pi$
$137$	120.139i	0.876924i	−0.898750	+	0.438462i	$0.855523\pi$
$138$	0	0
$139$	242.378i	1.74373i	0.489747	+	0.871864i	$0.337089\pi$
$139$	242.378i	1.74373i	−0.489747	+	0.871864i	$0.662911\pi$
$140$	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
$149$	192.500	1.29194	0.645972	−	0.763361i	$-0.276452\pi$
$149$	192.500	1.29194	0.645972	+	0.763361i	$0.276452\pi$
$150$	0	0
$151$	−17.4300	−0.115431	−0.0577153	−	0.998333i	$-0.518382\pi$
$151$	−17.4300	−0.115431	−0.0577153	+	0.998333i	$0.518382\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−285.873	−1.84434
$156$	0	0
$157$	213.617i	1.36062i	0.732927	+	0.680308i	$0.238154\pi$
$157$	213.617i	1.36062i	−0.732927	+	0.680308i	$0.761846\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 70.0366i	− 0.435010i
$162$	0	0
$163$	231.047i	1.41746i	0.705478	+	0.708732i	$0.250732\pi$
$163$	231.047i	1.41746i	−0.705478	+	0.708732i	$0.749268\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 126.202i	− 0.755701i	−0.925867	−	0.377850i	$-0.876663\pi$
$167$	− 126.202i	− 0.755701i	0.925867	−	0.377850i	$-0.123337\pi$
$168$	0	0
$169$	−133.093	−0.787534
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−199.032	−1.15048	−0.575238	−	0.817986i	$-0.695090\pi$
$173$	−199.032	−1.15048	−0.575238	+	0.817986i	$0.695090\pi$
$174$	0	0
$175$	−79.5233	−0.454419
$176$	0	0
$177$	0	0
$178$	0	0
$179$	170.915	0.954831	0.477415	−	0.878678i	$-0.341574\pi$
$179$	170.915	0.954831	0.477415	+	0.878678i	$0.341574\pi$
$180$	0	0
$181$	− 38.6192i	− 0.213366i	−0.994293	−	0.106683i	$-0.965977\pi$
$181$	− 38.6192i	− 0.213366i	0.994293	−	0.106683i	$-0.0340229\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 209.234i	− 1.13099i
$186$	0	0
$187$	257.808i	1.37865i
$188$	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
$197$	80.0368	0.406278	0.203139	−	0.979150i	$-0.434886\pi$
$197$	80.0368	0.406278	0.203139	+	0.979150i	$0.434886\pi$
$198$	0	0
$199$	−265.047	−1.33189	−0.665946	−	0.746000i	$-0.731972\pi$
$199$	−265.047	−1.33189	−0.665946	+	0.746000i	$0.731972\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−23.5031	−0.115779
$204$	0	0
$205$	− 18.4275i	− 0.0898902i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	234.085i	1.12003i
$210$	0	0
$211$	− 159.332i	− 0.755126i	−0.925984	−	0.377563i	$-0.876762\pi$
$211$	− 159.332i	− 0.755126i	0.925984	−	0.377563i	$-0.123238\pi$
$212$	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
$221$	206.003	0.932138
$222$	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
$227$	−322.371	−1.42014	−0.710069	−	0.704133i	$-0.751336\pi$
$227$	−322.371	−1.42014	−0.710069	+	0.704133i	$0.751336\pi$
$228$	0	0
$229$	− 180.997i	− 0.790382i	−0.918599	−	0.395191i	$-0.870678\pi$
$229$	− 180.997i	− 0.790382i	0.918599	−	0.395191i	$-0.129322\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	329.241i	1.41305i	0.707688	+	0.706525i	$0.249738\pi$
$233$	329.241i	1.41305i	−0.707688	+	0.706525i	$0.750262\pi$
$234$	0	0
$235$	− 218.948i	− 0.931695i
$236$	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
$245$	−362.136	−1.47811
$246$	0	0
$247$	187.047	0.757274
$248$	0	0
$249$	0	0
$250$	0	0
$251$	245.329	0.977408	0.488704	−	0.872450i	$-0.337470\pi$
$251$	245.329	0.977408	0.488704	+	0.872450i	$0.337470\pi$
$252$	0	0
$253$	− 761.710i	− 3.01071i
$254$	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
$259$	52.0000i	0.200772i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	217.921i	0.828596i	0.910141	+	0.414298i	$0.135973\pi$
$263$	217.921i	0.828596i	−0.910141	+	0.414298i	$0.864027\pi$
$264$	0	0
$265$	−399.140	−1.50619
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−205.433	−0.763691	−0.381845	−	0.924226i	$-0.624711\pi$
$269$	−205.433	−0.763691	−0.381845	+	0.924226i	$0.624711\pi$
$270$	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
$275$	−864.887	−3.14504
$276$	0	0
$277$	185.951i	0.671303i	0.941986	+	0.335651i	$0.108956\pi$
$277$	185.951i	0.671303i	−0.941986	+	0.335651i	$0.891044\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 87.7472i	− 0.312268i	−0.987736	−	0.156134i	$-0.950097\pi$
$281$	− 87.7472i	− 0.312268i	0.987736	−	0.156134i	$-0.0499031\pi$
$282$	0	0
$283$	− 46.0933i	− 0.162874i	−0.996678	−	0.0814369i	$-0.974049\pi$
$283$	− 46.0933i	− 0.162874i	0.996678	−	0.0814369i	$-0.0259509\pi$
$284$	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
$293$	−118.287	−0.403708	−0.201854	−	0.979416i	$-0.564697\pi$
$293$	−118.287	−0.403708	−0.201854	+	0.979416i	$0.564697\pi$
$294$	0	0
$295$	−591.803	−2.00611
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−608.647	−2.03561
$300$	0	0
$301$	131.047i	0.435371i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 60.5437i	− 0.198504i
$306$	0	0
$307$	116.285i	0.378778i	0.981902	+	0.189389i	$0.0606508\pi$
$307$	116.285i	0.378778i	−0.981902	+	0.189389i	$0.939349\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 261.564i	− 0.841040i	−0.907283	−	0.420520i	$-0.861848\pi$
$311$	− 261.564i	− 0.841040i	0.907283	−	0.420520i	$-0.138152\pi$
$312$	0	0
$313$	−310.000	−0.990415	−0.495208	−	0.868775i	$-0.664908\pi$
$313$	−310.000	−0.990415	−0.495208	+	0.868775i	$0.664908\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	223.279	0.704350	0.352175	−	0.935934i	$-0.385442\pi$
$317$	223.279	0.704350	0.352175	+	0.935934i	$0.385442\pi$
$318$	0	0
$319$	−255.617	−0.801306
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−127.550	−0.394893
$324$	0	0
$325$	691.091i	2.12643i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	54.4142i	0.165393i
$330$	0	0
$331$	353.238i	1.06719i	0.845742	+	0.533593i	$0.179158\pi$
$331$	353.238i	1.06719i	−0.845742	+	0.533593i	$0.820842\pi$
$332$	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
$341$	772.696	2.26597
$342$	0	0
$343$	188.000	0.548105
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−230.916	−0.665465	−0.332732	−	0.943021i	$-0.607971\pi$
$347$	−230.916	−0.665465	−0.332732	+	0.943021i	$0.607971\pi$
$348$	0	0
$349$	189.233i	0.542216i	0.962549	+	0.271108i	$0.0873900\pi$
$349$	189.233i	0.542216i	−0.962549	+	0.271108i	$0.912610\pi$
$350$	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
$355$	− 677.425i	− 1.90824i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 217.789i	− 0.606654i	−0.952886	−	0.303327i	$-0.901903\pi$
$359$	− 217.789i	− 0.606654i	0.952886	−	0.303327i	$-0.0980975\pi$
$360$	0	0
$361$	245.187	0.679187
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−680.574	−1.86459
$366$	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
$371$	99.1965	0.267376
$372$	0	0
$373$	− 124.663i	− 0.334218i	−0.985938	−	0.167109i	$-0.946557\pi$
$373$	− 124.663i	− 0.334218i	0.985938	−	0.167109i	$-0.0534432\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	204.251i	0.541781i
$378$	0	0
$379$	419.332i	1.10642i	0.833043	+	0.553208i	$0.186597\pi$
$379$	419.332i	1.10642i	−0.833043	+	0.553208i	$0.813403\pi$
$380$	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
$389$	214.988	0.552669	0.276334	−	0.961062i	$-0.410880\pi$
$389$	214.988	0.552669	0.276334	+	0.961062i	$0.410880\pi$
$390$	0	0
$391$	415.047	1.06150
$392$	0	0
$393$	0	0
$394$	0	0
$395$	607.771	1.53866
$396$	0	0
$397$	506.187i	1.27503i	0.770438	+	0.637515i	$0.220037\pi$
$397$	506.187i	1.27503i	−0.770438	+	0.637515i	$0.779963\pi$
$398$	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
$403$	− 617.425i	− 1.53207i
$404$	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	$-0.488620\pi$
$409$	29.2383	0.0714874	0.0357437	+	0.999361i	$0.488620\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	147.078	0.356122
$414$	0	0
$415$	−388.570	−0.936313
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.22387	−0.00530757	−0.00265379	−	0.999996i	$-0.500845\pi$
$419$	−2.22387	−0.00530757	−0.00265379	+	0.999996i	$0.500845\pi$
$420$	0	0
$421$	117.194i	0.278371i	0.990266	+	0.139186i	$0.0444484\pi$
$421$	117.194i	0.278371i	−0.990266	+	0.139186i	$0.955552\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 471.267i	− 1.10886i
$426$	0	0
$427$	15.0467i	0.0352381i
$428$	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
$437$	376.855	0.862368
$438$	0	0
$439$	231.140	0.526515	0.263257	−	0.964726i	$-0.415203\pi$
$439$	231.140	0.526515	0.263257	+	0.964726i	$0.415203\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.08983	−0.0205188	−0.0102594	−	0.999947i	$-0.503266\pi$
$443$	−9.08983	−0.0205188	−0.0102594	+	0.999947i	$0.503266\pi$
$444$	0	0
$445$	− 1175.47i	− 2.64150i
$446$	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
$451$	49.8083i	0.110440i
$452$	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	$-0.381282\pi$
$457$	333.042	0.728756	0.364378	+	0.931251i	$0.381282\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−132.971	−0.288440	−0.144220	−	0.989546i	$-0.546067\pi$
$461$	−132.971	−0.288440	−0.144220	+	0.989546i	$0.546067\pi$
$462$	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
$467$	469.310	1.00495	0.502473	−	0.864593i	$-0.332423\pi$
$467$	469.310	1.00495	0.502473	+	0.864593i	$0.332423\pi$
$468$	0	0
$469$	− 130.477i	− 0.278202i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1425.25i	3.01321i
$474$	0	0
$475$	− 427.902i	− 0.900846i
$476$	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
$485$	−859.161	−1.77147
$486$	0	0
$487$	839.897	1.72463	0.862317	−	0.506369i	$-0.169013\pi$
$487$	839.897	1.72463	0.862317	+	0.506369i	$0.169013\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	388.572	0.791388	0.395694	−	0.918382i	$-0.370504\pi$
$491$	388.572	0.791388	0.395694	+	0.918382i	$0.370504\pi$
$492$	0	0
$493$	− 139.282i	− 0.282520i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	168.357i	0.338747i
$498$	0	0
$499$	488.373i	0.978704i	0.872086	+	0.489352i	$0.162767\pi$
$499$	488.373i	0.978704i	−0.872086	+	0.489352i	$0.837233\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 533.488i	− 1.06061i	−0.847806	−	0.530307i	$-0.822077\pi$
$503$	− 533.488i	− 1.06061i	0.847806	−	0.530307i	$-0.177923\pi$
$504$	0	0
$505$	−1165.99	−2.30889
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−179.094	−0.351855	−0.175927	−	0.984403i	$-0.556292\pi$
$509$	−179.094	−0.351855	−0.175927	+	0.984403i	$0.556292\pi$
$510$	0	0
$511$	169.140	0.330998
$512$	0	0
$513$	0	0
$514$	0	0
$515$	213.821	0.415186
$516$	0	0
$517$	591.803i	1.14469i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 397.456i	− 0.762872i	−0.924395	−	0.381436i	$-0.875430\pi$
$521$	− 397.456i	− 0.762872i	0.924395	−	0.381436i	$-0.124570\pi$
$522$	0	0
$523$	987.135i	1.88745i	0.330735	+	0.943724i	$0.392703\pi$
$523$	987.135i	1.88745i	−0.330735	+	0.943724i	$0.607297\pi$
$524$	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
$533$	39.7995	0.0746707
$534$	0	0
$535$	−574.467	−1.07377
$536$	0	0
$537$	0	0
$538$	0	0
$539$	978.830	1.81601
$540$	0	0
$541$	214.992i	0.397398i	0.980061	+	0.198699i	$0.0636716\pi$
$541$	214.992i	0.397398i	−0.980061	+	0.198699i	$0.936328\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	954.583i	1.75153i
$546$	0	0
$547$	381.710i	0.697824i	0.937155	+	0.348912i	$0.113449\pi$
$547$	381.710i	0.697824i	−0.937155	+	0.348912i	$0.886551\pi$
$548$	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
$557$	−924.388	−1.65958	−0.829792	−	0.558073i	$-0.811541\pi$
$557$	−924.388	−1.65958	−0.829792	+	0.558073i	$0.811541\pi$
$558$	0	0
$559$	1138.85	2.03730
$560$	0	0
$561$	0	0
$562$	0	0
$563$	70.5092	0.125238	0.0626191	−	0.998037i	$-0.480055\pi$
$563$	70.5092	0.125238	0.0626191	+	0.998037i	$0.480055\pi$
$564$	0	0
$565$	− 697.474i	− 1.23447i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 607.844i	− 1.06827i	−0.845400	−	0.534134i	$-0.820638\pi$
$569$	− 607.844i	− 1.06827i	0.845400	−	0.534134i	$-0.179362\pi$
$570$	0	0
$571$	− 455.430i	− 0.797601i	−0.917038	−	0.398800i	$-0.869427\pi$
$571$	− 455.430i	− 0.797601i	0.917038	−	0.398800i	$-0.130573\pi$
$572$	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
$581$	96.5696	0.166213
$582$	0	0
$583$	1078.85	1.85051
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−178.059	−0.303337	−0.151669	−	0.988431i	$-0.548465\pi$
$587$	−178.059	−0.303337	−0.151669	+	0.988431i	$0.548465\pi$
$588$	0	0
$589$	382.290i	0.649049i
$590$	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
$595$	190.762i	0.320608i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	680.706i	1.13640i	0.822889	+	0.568202i	$0.192361\pi$
$599$	680.706i	1.13640i	−0.822889	+	0.568202i	$0.807639\pi$
$600$	0	0
$601$	−239.327	−0.398214	−0.199107	−	0.979978i	$-0.563804\pi$
$601$	−239.327	−0.398214	−0.199107	+	0.979978i	$0.563804\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2833.83	4.68402
$606$	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
$611$	472.882	0.773948
$612$	0	0
$613$	− 1106.56i	− 1.80515i	−0.430528	−	0.902577i	$-0.641673\pi$
$613$	− 1106.56i	− 1.80515i	0.430528	−	0.902577i	$-0.358327\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 271.185i	− 0.439522i	−0.975554	−	0.219761i	$-0.929472\pi$
$617$	− 271.185i	− 0.439522i	0.975554	−	0.219761i	$-0.0705277\pi$
$618$	0	0
$619$	− 447.430i	− 0.722827i	−0.932406	−	0.361414i	$-0.882294\pi$
$619$	− 447.430i	− 0.722827i	0.932406	−	0.361414i	$-0.117706\pi$
$620$	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
$629$	−308.159	−0.489920
$630$	0	0
$631$	−369.047	−0.584860	−0.292430	−	0.956287i	$-0.594464\pi$
$631$	−369.047	−0.584860	−0.292430	+	0.956287i	$0.594464\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	555.650	0.875040
$636$	0	0
$637$	− 782.137i	− 1.22785i
$638$	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
$643$	41.3266i	0.0642715i	0.999484	+	0.0321357i	$0.0102309\pi$
$643$	41.3266i	0.0642715i	−0.999484	+	0.0321357i	$0.989769\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	160.011i	0.247313i	0.992325	+	0.123656i	$0.0394620\pi$
$647$	160.011i	0.247313i	−0.992325	+	0.123656i	$0.960538\pi$
$648$	0	0
$649$	1599.61	2.46472
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1111.54	−1.70220	−0.851100	−	0.525003i	$-0.824064\pi$
$653$	−1111.54	−1.70220	−0.851100	+	0.525003i	$0.824064\pi$
$654$	0	0
$655$	−42.2799	−0.0645495
$656$	0	0
$657$	0	0
$658$	0	0
$659$	800.313	1.21444	0.607218	−	0.794536i	$-0.292286\pi$
$659$	800.313	1.21444	0.607218	+	0.794536i	$0.292286\pi$
$660$	0	0
$661$	1142.18i	1.72795i	0.503533	+	0.863976i	$0.332033\pi$
$661$	1142.18i	1.72795i	−0.503533	+	0.863976i	$0.667967\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	173.208i	0.260463i
$666$	0	0
$667$	411.518i	0.616969i
$668$	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
$677$	−533.801	−0.788481	−0.394240	−	0.919007i	$-0.628992\pi$
$677$	−533.801	−0.788481	−0.394240	+	0.919007i	$0.628992\pi$
$678$	0	0
$679$	213.523	0.314467
$680$	0	0
$681$	0	0
$682$	0	0
$683$	408.711	0.598406	0.299203	−	0.954189i	$-0.403279\pi$
$683$	408.711	0.598406	0.299203	+	0.954189i	$0.403279\pi$
$684$	0	0
$685$	966.811i	1.41140i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 862.059i	− 1.25117i
$690$	0	0
$691$	− 331.233i	− 0.479353i	−0.970853	−	0.239677i	$-0.922959\pi$
$691$	− 331.233i	− 0.479353i	0.970853	−	0.239677i	$-0.0770415\pi$
$692$	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
$701$	116.814	0.166638	0.0833192	−	0.996523i	$-0.473448\pi$
$701$	116.814	0.166638	0.0833192	+	0.996523i	$0.473448\pi$
$702$	0	0
$703$	−279.803	−0.398013
$704$	0	0
$705$	0	0
$706$	0	0
$707$	289.778	0.409870
$708$	0	0
$709$	737.469i	1.04015i	0.854119	+	0.520077i	$0.174097\pi$
$709$	737.469i	1.04015i	−0.854119	+	0.520077i	$0.825903\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1243.97i	− 1.74469i
$714$	0	0
$715$	− 3042.46i	− 4.25518i
$716$	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
$725$	467.260	0.644497
$726$	0	0
$727$	−277.430	−0.381609	−0.190805	−	0.981628i	$-0.561110\pi$
$727$	−277.430	−0.381609	−0.190805	+	0.981628i	$0.561110\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−776.601	−1.06238
$732$	0	0
$733$	405.371i	0.553030i	0.961010	+	0.276515i	$0.0891795\pi$
$733$	405.371i	0.553030i	−0.961010	+	0.276515i	$0.910821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1419.05i	− 1.92544i
$738$	0	0
$739$	− 295.036i	− 0.399237i	−0.979874	−	0.199619i	$-0.936030\pi$
$739$	− 295.036i	− 0.399237i	0.979874	−	0.199619i	$-0.0639704\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	486.350i	0.654576i	0.944925	+	0.327288i	$0.106135\pi$
$743$	486.350i	0.654576i	−0.944925	+	0.327288i	$0.893865\pi$
$744$	0	0
$745$	1549.13	2.07938
$746$	0	0
$747$	0	0
$748$	0	0
$749$	142.770	0.190614
$750$	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
$755$	−140.267	−0.185785
$756$	0	0
$757$	− 1362.89i	− 1.80039i	−0.435489	−	0.900194i	$-0.643424\pi$
$757$	− 1362.89i	− 1.80039i	0.435489	−	0.900194i	$-0.356576\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	625.746i	0.822268i	0.911575	+	0.411134i	$0.134867\pi$
$761$	625.746i	0.822268i	−0.911575	+	0.411134i	$0.865133\pi$
$762$	0	0
$763$	− 237.238i	− 0.310928i
$764$	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
$773$	536.282	0.693767	0.346884	−	0.937908i	$-0.387240\pi$
$773$	536.282	0.693767	0.346884	+	0.937908i	$0.387240\pi$
$774$	0	0
$775$	−1412.47	−1.82254
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.6426	−0.0316336
$780$	0	0
$781$	1831.04i	2.34448i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1719.07i	2.18990i
$786$	0	0
$787$	− 769.425i	− 0.977668i	−0.872377	−	0.488834i	$-0.837422\pi$
$787$	− 769.425i	− 0.977668i	0.872377	−	0.488834i	$-0.162578\pi$
$788$	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
$797$	765.593	0.960594	0.480297	−	0.877106i	$-0.340529\pi$
$797$	765.593	0.960594	0.480297	+	0.877106i	$0.340529\pi$
$798$	0	0
$799$	−322.467	−0.403588
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1839.55	2.29084
$804$	0	0
$805$	− 563.617i	− 0.700145i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	865.488i	1.06982i	0.844908	+	0.534912i	$0.179655\pi$
$809$	865.488i	1.06982i	−0.844908	+	0.534912i	$0.820345\pi$
$810$	0	0
$811$	292.472i	0.360631i	0.983609	+	0.180315i	$0.0577118\pi$
$811$	292.472i	0.360631i	−0.983609	+	0.180315i	$0.942288\pi$
$812$	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
$821$	343.616	0.418533	0.209266	−	0.977859i	$-0.432892\pi$
$821$	343.616	0.418533	0.209266	+	0.977859i	$0.432892\pi$
$822$	0	0
$823$	773.430	0.939769	0.469885	−	0.882728i	$-0.344296\pi$
$823$	773.430	0.939769	0.469885	+	0.882728i	$0.344296\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−854.310	−1.03302	−0.516511	−	0.856280i	$-0.672770\pi$
$827$	−854.310	−1.03302	−0.516511	+	0.856280i	$0.672770\pi$
$828$	0	0
$829$	− 85.5674i	− 0.103218i	−0.998667	−	0.0516088i	$-0.983565\pi$
$829$	− 85.5674i	− 0.103218i	0.998667	−	0.0516088i	$-0.0164349\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	533.353i	0.640280i
$834$	0	0
$835$	− 1015.61i	− 1.21630i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1653.28i	1.97054i	0.171011	+	0.985269i	$0.445297\pi$
$839$	1653.28i	1.97054i	−0.171011	+	0.985269i	$0.554703\pi$
$840$	0	0
$841$	−702.902	−0.835793
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1071.06	−1.26753
$846$	0	0
$847$	−704.280	−0.831499
$848$	0	0
$849$	0	0
$850$	0	0
$851$	910.475	1.06989
$852$	0	0
$853$	− 1045.62i	− 1.22581i	−0.790156	−	0.612905i	$-0.790001\pi$
$853$	− 1045.62i	− 1.22581i	0.790156	−	0.612905i	$-0.209999\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 939.437i	− 1.09619i	−0.836415	−	0.548096i	$-0.815353\pi$
$857$	− 939.437i	− 1.09619i	0.836415	−	0.548096i	$-0.184647\pi$
$858$	0	0
$859$	− 1249.52i	− 1.45463i	−0.686306	−	0.727313i	$-0.740769\pi$
$859$	− 1249.52i	− 1.45463i	0.686306	−	0.727313i	$-0.259231\pi$
$860$	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
$869$	−1642.77	−1.89041
$870$	0	0
$871$	−1133.90	−1.30183
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−237.588	−0.271529
$876$	0	0
$877$	195.720i	0.223170i	0.993755	+	0.111585i	$0.0355927\pi$
$877$	195.720i	0.223170i	−0.993755	+	0.111585i	$0.964407\pi$
$878$	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
$883$	− 963.803i	− 1.09151i	−0.837945	−	0.545755i	$-0.816243\pi$
$883$	− 963.803i	− 1.09151i	0.837945	−	0.545755i	$-0.183757\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 507.769i	− 0.572456i	−0.958162	−	0.286228i	$-0.907598\pi$
$887$	− 507.769i	− 0.572456i	0.958162	−	0.286228i	$-0.0924015\pi$
$888$	0	0
$889$	−138.093	−0.155336
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−292.794	−0.327877
$894$	0	0
$895$	1375.43	1.53679
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−417.453	−0.464353
$900$	0	0
$901$	587.852i	0.652444i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 310.786i	− 0.343410i
$906$	0	0
$907$	− 48.3834i	− 0.0533444i	−0.999644	−	0.0266722i	$-0.991509\pi$
$907$	− 48.3834i	− 0.0533444i	0.999644	−	0.0266722i	$-0.00849103\pi$
$908$	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
$917$	10.5076	0.0114587
$918$	0	0
$919$	583.513	0.634944	0.317472	−	0.948268i	$-0.397166\pi$
$919$	583.513	0.634944	0.317472	+	0.948268i	$0.397166\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1463.10	1.58515
$924$	0	0
$925$	− 1033.80i	− 1.11763i
$926$	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
$931$	484.275i	0.520166i
$932$	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	$-0.580425\pi$
$937$	−468.467	−0.499964	−0.249982	+	0.968250i	$0.580425\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−469.964	−0.499430	−0.249715	−	0.968319i	$-0.580337\pi$
$941$	−469.964	−0.499430	−0.249715	+	0.968319i	$0.580337\pi$
$942$	0	0
$943$	80.1866	0.0850335
$944$	0	0
$945$	0	0
$946$	0	0
$947$	200.686	0.211918	0.105959	−	0.994370i	$-0.466209\pi$
$947$	200.686	0.211918	0.105959	+	0.994370i	$0.466209\pi$
$948$	0	0
$949$	− 1469.90i	− 1.54889i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 493.081i	− 0.517398i	−0.965958	−	0.258699i	$-0.916706\pi$
$953$	− 493.081i	− 0.517398i	0.965958	−	0.258699i	$-0.0832939\pi$
$954$	0	0
$955$	− 2171.03i	− 2.27333i
$956$	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
$965$	−2117.61	−2.19441
$966$	0	0
$967$	−204.467	−0.211444	−0.105722	−	0.994396i	$-0.533715\pi$
$967$	−204.467	−0.211444	−0.105722	+	0.994396i	$0.533715\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−250.986	−0.258482	−0.129241	−	0.991613i	$-0.541254\pi$
$971$	−250.986	−0.258482	−0.129241	+	0.991613i	$0.541254\pi$
$972$	0	0
$973$	− 484.757i	− 0.498208i
$974$	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
$979$	3177.22i	3.24537i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 569.174i	− 0.579017i	−0.957175	−	0.289508i	$-0.906508\pi$
$983$	− 569.174i	− 0.579017i	0.957175	−	0.289508i	$-0.0934918\pi$
$984$	0	0
$985$	644.093	0.653902
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2294.51	2.32003
$990$	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
$995$	−2132.95	−2.14367
$996$	0	0
$997$	112.290i	0.112628i	0.998413	+	0.0563140i	$0.0179348\pi$
$997$	112.290i	0.112628i	−0.998413	+	0.0563140i	$0.982065\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.04746 q^{5} -2.00000 q^{7} +O(q^{10})\)	\(q+8.04746 q^{5} -2.00000 q^{7} -21.7518 q^{11} +17.3808i q^{13} -11.8523i q^{17} -10.7617i q^{19} +35.0183i q^{23} +39.7617 q^{25} +11.7515 q^{29} -35.5233 q^{31} -16.0949 q^{35} -26.0000i q^{37} -2.28985i q^{41} -65.5233i q^{43} -27.2071i q^{47} -45.0000 q^{49} -49.5982 q^{53} -175.047 q^{55} -73.5391 q^{59} -7.52333i q^{61} +139.872i q^{65} +65.2383i q^{67} -84.1787i q^{71} -84.5700 q^{73} +43.5036 q^{77} +75.5233 q^{79} -48.2848 q^{83} -95.3808i q^{85} -146.067i q^{89} -34.7617i q^{91} -86.6041i q^{95} -106.762 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^7	\( 8 q - 16 q^{7} + 168 q^{25} + 16 q^{31} - 360 q^{49} - 800 q^{55} + 224 q^{73} + 304 q^{79} - 704 q^{97}+O(q^{100}) \) 8 * q - 16 * q^7 + 168 * q^25 + 16 * q^31 - 360 * q^49 - 800 * q^55 + 224 * q^73 + 304 * q^79 - 704 * q^97

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