LMFDB - Embedded newform 2268.4.f.a.1133.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,4,Mod(1133,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2268.1133");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2268.f (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:	$133.816331893$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1133.5
Character	$\chi$	$=$	2268.1133
Dual form			2268.4.f.a.1133.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.5975 q^{5} +(-11.9446 - 14.1537i) q^{7} +O(q^{10})$	$q-16.5975 q^{5} +(-11.9446 - 14.1537i) q^{7} -53.4353i q^{11} +12.7565i q^{13} +96.7463 q^{17} -54.6996i q^{19} -64.2853i q^{23} +150.476 q^{25} +130.148i q^{29} -220.107i q^{31} +(198.250 + 234.915i) q^{35} +279.735 q^{37} -370.617 q^{41} +307.751 q^{43} +327.365 q^{47} +(-57.6532 + 338.120i) q^{49} -451.749i q^{53} +886.892i q^{55} +517.477 q^{59} +270.790i q^{61} -211.725i q^{65} +741.763 q^{67} -914.198i q^{71} +337.210i q^{73} +(-756.306 + 638.264i) q^{77} +997.165 q^{79} +34.0541 q^{83} -1605.74 q^{85} +208.953 q^{89} +(180.551 - 152.371i) q^{91} +907.875i q^{95} +1271.26i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 12 q^{7}+O(q^{10}) $ 48 * q - 12 * q^7	$ 48 q - 12 q^{7} + 1200 q^{25} + 336 q^{37} - 168 q^{43} - 636 q^{49} + 1176 q^{67} - 408 q^{79} + 720 q^{85} - 1080 q^{91}+O(q^{100}) $ 48 * q - 12 * q^7 + 1200 * q^25 + 336 * q^37 - 168 * q^43 - 636 * q^49 + 1176 * q^67 - 408 * q^79 + 720 * q^85 - 1080 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	−16.5975			−1.48452			−0.742262	−	0.670110i	$-0.766247\pi$
$5$	−16.5975			−1.48452			−0.742262	+	0.670110i	$0.766247\pi$
$6$		0			0
$7$	−11.9446	−	14.1537i	−0.644948	−	0.764227i
$7$	−11.9446	−	14.1537i	−0.644948	−	0.764227i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	53.4353i		−	1.46467i	−0.680945	−	0.732334i	$-0.738431\pi$
$11$		−	53.4353i		−	1.46467i	0.680945	−	0.732334i	$-0.261569\pi$
$12$		0			0
$13$			12.7565i			0.272155i	0.990698	+	0.136077i	$0.0434496\pi$
$13$			12.7565i			0.272155i	−0.990698	+	0.136077i	$0.956550\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	96.7463			1.38026			0.690130	−	0.723686i	$-0.257553\pi$
$17$	96.7463			1.38026			0.690130	+	0.723686i	$0.257553\pi$
$18$		0			0
$19$		−	54.6996i		−	0.660471i	−0.943899	−	0.330235i	$-0.892872\pi$
$19$		−	54.6996i		−	0.660471i	0.943899	−	0.330235i	$-0.107128\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	64.2853i		−	0.582800i	−0.956601	−	0.291400i	$-0.905879\pi$
$23$		−	64.2853i		−	0.582800i	0.956601	−	0.291400i	$-0.0941211\pi$
$24$		0			0
$25$	150.476			1.20381
$26$		0			0
$27$		0			0
$28$		0			0
$29$			130.148i			0.833375i	0.909050	+	0.416688i	$0.136809\pi$
$29$			130.148i			0.833375i	−0.909050	+	0.416688i	$0.863191\pi$
$30$		0			0
$31$		−	220.107i		−	1.27524i	−0.770353	−	0.637618i	$-0.779920\pi$
$31$		−	220.107i		−	1.27524i	0.770353	−	0.637618i	$-0.220080\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	198.250	+	234.915i	0.957440	+	1.13451i
$36$		0			0
$37$	279.735			1.24292			0.621462	−	0.783445i	$-0.286539\pi$
$37$	279.735			1.24292			0.621462	+	0.783445i	$0.286539\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−370.617			−1.41172			−0.705861	−	0.708351i	$-0.749440\pi$
$41$	−370.617			−1.41172			−0.705861	+	0.708351i	$0.749440\pi$
$42$		0			0
$43$	307.751			1.09143			0.545717	−	0.837970i	$-0.316258\pi$
$43$	307.751			1.09143			0.545717	+	0.837970i	$0.316258\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	327.365			1.01598			0.507990	−	0.861363i	$-0.330389\pi$
$47$	327.365			1.01598			0.507990	+	0.861363i	$0.330389\pi$
$48$		0			0
$49$	−57.6532	+	338.120i	−0.168085	+	0.985772i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	451.749i		−	1.17080i	−0.810744	−	0.585401i	$-0.800937\pi$
$53$		−	451.749i		−	1.17080i	0.810744	−	0.585401i	$-0.199063\pi$
$54$		0			0
$55$			886.892i			2.17434i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	517.477			1.14186			0.570930	−	0.820998i	$-0.306583\pi$
$59$	517.477			1.14186			0.570930	+	0.820998i	$0.306583\pi$
$60$		0			0
$61$			270.790i			0.568378i	0.958768	+	0.284189i	$0.0917243\pi$
$61$			270.790i			0.568378i	−0.958768	+	0.284189i	$0.908276\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	211.725i		−	0.404020i
$66$		0			0
$67$	741.763			1.35255			0.676274	−	0.736650i	$-0.263594\pi$
$67$	741.763			1.35255			0.676274	+	0.736650i	$0.263594\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	914.198i		−	1.52810i	−0.645155	−	0.764051i	$-0.723207\pi$
$71$		−	914.198i		−	1.52810i	0.645155	−	0.764051i	$-0.276793\pi$
$72$		0			0
$73$			337.210i			0.540650i	0.962769	+	0.270325i	$0.0871311\pi$
$73$			337.210i			0.540650i	−0.962769	+	0.270325i	$0.912869\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−756.306	+	638.264i	−1.11934	+	0.944635i
$78$		0			0
$79$	997.165			1.42012			0.710062	−	0.704139i	$-0.248667\pi$
$79$	997.165			1.42012			0.710062	+	0.704139i	$0.248667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	34.0541			0.0450352			0.0225176	−	0.999746i	$-0.492832\pi$
$83$	34.0541			0.0450352			0.0225176	+	0.999746i	$0.492832\pi$
$84$		0			0
$85$	−1605.74			−2.04903
$86$		0			0
$87$		0			0
$88$		0			0
$89$	208.953			0.248865			0.124433	−	0.992228i	$-0.460289\pi$
$89$	208.953			0.248865			0.124433	+	0.992228i	$0.460289\pi$
$90$		0			0
$91$	180.551	−	152.371i	0.207988	−	0.175526i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			907.875i			0.980484i
$96$		0			0
$97$			1271.26i			1.33069i	0.746536	+	0.665345i	$0.231715\pi$
$97$			1271.26i			1.33069i	−0.746536	+	0.665345i	$0.768285\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−614.247			−0.605148			−0.302574	−	0.953126i	$-0.597846\pi$
$101$	−614.247			−0.605148			−0.302574	+	0.953126i	$0.597846\pi$
$102$		0			0
$103$		−	1043.17i		−	0.997926i	−0.866623	−	0.498963i	$-0.833714\pi$
$103$		−	1043.17i		−	0.997926i	0.866623	−	0.498963i	$-0.166286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1128.83i		−	1.01989i	−0.860207	−	0.509945i	$-0.829666\pi$
$107$		−	1128.83i		−	1.01989i	0.860207	−	0.509945i	$-0.170334\pi$
$108$		0			0
$109$	1555.72			1.36707			0.683535	−	0.729918i	$-0.260442\pi$
$109$	1555.72			1.36707			0.683535	+	0.729918i	$0.260442\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			258.461i			0.215168i	0.994196	+	0.107584i	$0.0343115\pi$
$113$			258.461i			0.215168i	−0.994196	+	0.107584i	$0.965689\pi$
$114$		0			0
$115$			1066.97i			0.865180i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1155.60	−	1369.32i	−0.890195	−	1.05483i
$120$		0			0
$121$	−1524.33			−1.14526
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−422.841			−0.302560
$126$		0			0
$127$	−586.759			−0.409972			−0.204986	−	0.978765i	$-0.565715\pi$
$127$	−586.759			−0.409972			−0.204986	+	0.978765i	$0.565715\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2264.86			−1.51055			−0.755275	−	0.655408i	$-0.772497\pi$
$131$	−2264.86			−1.51055			−0.755275	+	0.655408i	$0.772497\pi$
$132$		0			0
$133$	−774.200	+	653.364i	−0.504749	+	0.425969i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1910.69i		−	1.19154i	−0.803154	−	0.595772i	$-0.796846\pi$
$137$		−	1910.69i		−	1.19154i	0.803154	−	0.595772i	$-0.203154\pi$
$138$		0			0
$139$		−	1697.17i		−	1.03563i	−0.855494	−	0.517813i	$-0.826746\pi$
$139$		−	1697.17i		−	1.03563i	0.855494	−	0.517813i	$-0.173254\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	681.647			0.398617
$144$		0			0
$145$		−	2160.13i		−	1.23717i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	161.997i		−	0.0890695i	−0.999008	−	0.0445347i	$-0.985819\pi$
$149$		−	161.997i		−	0.0890695i	0.999008	−	0.0445347i	$-0.0141805\pi$
$150$		0			0
$151$	−3072.80			−1.65603			−0.828017	−	0.560703i	$-0.810531\pi$
$151$	−3072.80			−1.65603			−0.828017	+	0.560703i	$0.810531\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3653.21i			1.89312i
$156$		0			0
$157$			1013.64i			0.515271i	0.966242	+	0.257636i	$0.0829434\pi$
$157$			1013.64i			0.515271i	−0.966242	+	0.257636i	$0.917057\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−909.873	+	767.862i	−0.445391	+	0.375876i
$162$		0			0
$163$	1297.23			0.623356			0.311678	−	0.950188i	$-0.399109\pi$
$163$	1297.23			0.623356			0.311678	+	0.950188i	$0.399109\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1432.47			−0.663758			−0.331879	−	0.943322i	$-0.607683\pi$
$167$	−1432.47			−0.663758			−0.331879	+	0.943322i	$0.607683\pi$
$168$		0			0
$169$	2034.27			0.925932
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1975.32			−0.868096			−0.434048	−	0.900890i	$-0.642915\pi$
$173$	−1975.32			−0.868096			−0.434048	+	0.900890i	$0.642915\pi$
$174$		0			0
$175$	−1797.38	−	2129.79i	−0.776394	−	0.919984i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2310.07i		−	0.964595i	−0.876007	−	0.482298i	$-0.839802\pi$
$179$		−	2310.07i		−	0.964595i	0.876007	−	0.482298i	$-0.160198\pi$
$180$		0			0
$181$			423.545i			0.173933i	0.996211	+	0.0869665i	$0.0277173\pi$
$181$			423.545i			0.173933i	−0.996211	+	0.0869665i	$0.972283\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4642.90			−1.84515
$186$		0			0
$187$		−	5169.67i		−	2.02162i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	2344.54i		−	0.888192i	−0.895979	−	0.444096i	$-0.853525\pi$
$191$		−	2344.54i		−	0.888192i	0.895979	−	0.444096i	$-0.146475\pi$
$192$		0			0
$193$	4050.71			1.51076			0.755380	−	0.655287i	$-0.227452\pi$
$193$	4050.71			1.51076			0.755380	+	0.655287i	$0.227452\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3659.21i		−	1.32339i	−0.749773	−	0.661695i	$-0.769837\pi$
$197$		−	3659.21i		−	1.32339i	0.749773	−	0.661695i	$-0.230163\pi$
$198$		0			0
$199$			557.050i			0.198433i	0.995066	+	0.0992166i	$0.0316337\pi$
$199$			557.050i			0.198433i	−0.995066	+	0.0992166i	$0.968366\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1842.07	−	1554.57i	0.636888	−	0.537483i
$204$		0			0
$205$	6151.30			2.09573
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2922.89			−0.967371
$210$		0			0
$211$	−4390.07			−1.43235			−0.716173	−	0.697923i	$-0.754108\pi$
$211$	−4390.07			−1.43235			−0.716173	+	0.697923i	$0.754108\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5107.89			−1.62026
$216$		0			0
$217$	−3115.32	+	2629.08i	−0.974569	+	0.822460i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1234.14i			0.375644i
$222$		0			0
$223$		−	2507.72i		−	0.753047i	−0.926407	−	0.376523i	$-0.877119\pi$
$223$		−	2507.72i		−	0.753047i	0.926407	−	0.376523i	$-0.122881\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2689.69			0.786435			0.393218	−	0.919445i	$-0.371362\pi$
$227$	2689.69			0.786435			0.393218	+	0.919445i	$0.371362\pi$
$228$		0			0
$229$		−	4113.08i		−	1.18690i	−0.804871	−	0.593449i	$-0.797766\pi$
$229$		−	4113.08i		−	1.18690i	0.804871	−	0.593449i	$-0.202234\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			2621.07i			0.736960i	0.929636	+	0.368480i	$0.120122\pi$
$233$			2621.07i			0.736960i	−0.929636	+	0.368480i	$0.879878\pi$
$234$		0			0
$235$	−5433.43			−1.50825
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5133.90i		−	1.38948i	−0.719263	−	0.694738i	$-0.755520\pi$
$239$		−	5133.90i		−	1.38948i	0.719263	−	0.694738i	$-0.244480\pi$
$240$		0			0
$241$			656.905i			0.175581i	0.996139	+	0.0877903i	$0.0279806\pi$
$241$			656.905i			0.175581i	−0.996139	+	0.0877903i	$0.972019\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	956.897	−	5611.94i	0.249526	−	1.46340i
$246$		0			0
$247$	697.774			0.179750
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1807.27			0.454479			0.227239	−	0.973839i	$-0.427030\pi$
$251$	1807.27			0.454479			0.227239	+	0.973839i	$0.427030\pi$
$252$		0			0
$253$	−3435.10			−0.853609
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2089.43			0.507140			0.253570	−	0.967317i	$-0.418395\pi$
$257$	2089.43			0.507140			0.253570	+	0.967317i	$0.418395\pi$
$258$		0			0
$259$	−3341.32	−	3959.28i	−0.801620	−	0.949875i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4872.09i			1.14230i	0.820844	+	0.571152i	$0.193503\pi$
$263$			4872.09i			1.14230i	−0.820844	+	0.571152i	$0.806497\pi$
$264$		0			0
$265$			7497.90i			1.73808i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4640.31			−1.05176			−0.525882	−	0.850557i	$-0.676265\pi$
$269$	−4640.31			−1.05176			−0.525882	+	0.850557i	$0.676265\pi$
$270$		0			0
$271$		−	5685.82i		−	1.27450i	−0.770658	−	0.637249i	$-0.780072\pi$
$271$		−	5685.82i		−	1.27450i	0.770658	−	0.637249i	$-0.219928\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	8040.75i		−	1.76318i
$276$		0			0
$277$	−5015.19			−1.08785			−0.543923	−	0.839135i	$-0.683062\pi$
$277$	−5015.19			−1.08785			−0.543923	+	0.839135i	$0.683062\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	5105.90i		−	1.08396i	−0.840392	−	0.541979i	$-0.817675\pi$
$281$		−	5105.90i		−	1.08396i	0.840392	−	0.541979i	$-0.182325\pi$
$282$		0			0
$283$			6350.15i			1.33384i	0.745129	+	0.666921i	$0.232388\pi$
$283$			6350.15i			1.33384i	−0.745129	+	0.666921i	$0.767612\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4426.87	+	5245.59i	0.910486	+	1.07888i
$288$		0			0
$289$	4446.84			0.905117
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1524.59			−0.303986			−0.151993	−	0.988382i	$-0.548569\pi$
$293$	−1524.59			−0.303986			−0.151993	+	0.988382i	$0.548569\pi$
$294$		0			0
$295$	−8588.81			−1.69512
$296$		0			0
$297$		0			0
$298$		0			0
$299$	820.054			0.158612
$300$		0			0
$301$	−3675.96	−	4355.81i	−0.703917	−	0.834102i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4494.43i		−	0.843771i
$306$		0			0
$307$			633.080i			0.117693i	0.998267	+	0.0588466i	$0.0187423\pi$
$307$			633.080i			0.117693i	−0.998267	+	0.0588466i	$0.981258\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4444.71			0.810407			0.405204	−	0.914226i	$-0.367201\pi$
$311$	4444.71			0.810407			0.405204	+	0.914226i	$0.367201\pi$
$312$		0			0
$313$			10268.7i			1.85438i	0.374593	+	0.927189i	$0.377782\pi$
$313$			10268.7i			1.85438i	−0.374593	+	0.927189i	$0.622218\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1877.40i		−	0.332635i	−0.986072	−	0.166318i	$-0.946812\pi$
$317$		−	1877.40i		−	0.332635i	0.986072	−	0.166318i	$-0.0531877\pi$
$318$		0			0
$319$	6954.50			1.22062
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5291.98i		−	0.911621i
$324$		0			0
$325$			1919.55i			0.327623i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3910.24	−	4633.42i	−0.655254	−	0.776440i
$330$		0			0
$331$	−3098.06			−0.514455			−0.257228	−	0.966351i	$-0.582809\pi$
$331$	−3098.06			−0.514455			−0.257228	+	0.966351i	$0.582809\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12311.4			−2.00789
$336$		0			0
$337$	10890.2			1.76032			0.880158	−	0.474681i	$-0.157436\pi$
$337$	10890.2			1.76032			0.880158	+	0.474681i	$0.157436\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−11761.5			−1.86780
$342$		0			0
$343$	5474.29	−	3222.70i	0.861760	−	0.507317i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3725.99i		−	0.576430i	−0.957566	−	0.288215i	$-0.906938\pi$
$347$		−	3725.99i		−	0.576430i	0.957566	−	0.288215i	$-0.0930619\pi$
$348$		0			0
$349$			611.863i			0.0938460i	0.998899	+	0.0469230i	$0.0149415\pi$
$349$			611.863i			0.0938460i	−0.998899	+	0.0469230i	$0.985058\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5184.79			−0.781752			−0.390876	−	0.920443i	$-0.627828\pi$
$353$	−5184.79			−0.781752			−0.390876	+	0.920443i	$0.627828\pi$
$354$		0			0
$355$			15173.4i			2.26850i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			10228.9i			1.50379i	0.659280	+	0.751897i	$0.270861\pi$
$359$			10228.9i			1.50379i	−0.659280	+	0.751897i	$0.729139\pi$
$360$		0			0
$361$	3866.96			0.563778
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	5596.83i		−	0.802607i
$366$		0			0
$367$			9972.52i			1.41842i	0.704996	+	0.709212i	$0.250949\pi$
$367$			9972.52i			1.41842i	−0.704996	+	0.709212i	$0.749051\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6393.91	+	5395.96i	−0.894759	+	0.755106i
$372$		0			0
$373$	8460.91			1.17450			0.587251	−	0.809405i	$-0.300210\pi$
$373$	8460.91			1.17450			0.587251	+	0.809405i	$0.300210\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1660.23			−0.226807
$378$		0			0
$379$	−1.71044			−0.000231819			−0.000115910	−	1.00000i	$-0.500037\pi$
$379$	−1.71044			−0.000231819			−0.000115910		1.00000i	$0.500037\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2556.55			0.341080			0.170540	−	0.985351i	$-0.445449\pi$
$383$	2556.55			0.341080			0.170540	+	0.985351i	$0.445449\pi$
$384$		0			0
$385$	12552.8	−	10593.6i	1.66169	−	1.40233i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	2380.29i		−	0.310245i	−0.987895	−	0.155123i	$-0.950423\pi$
$389$		−	2380.29i		−	0.310245i	0.987895	−	0.155123i	$-0.0495773\pi$
$390$		0			0
$391$		−	6219.36i		−	0.804415i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16550.4			−2.10821
$396$		0			0
$397$		−	12885.5i		−	1.62898i	−0.580177	−	0.814491i	$-0.697017\pi$
$397$		−	12885.5i		−	1.62898i	0.580177	−	0.814491i	$-0.302983\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			7376.69i			0.918640i	0.888271	+	0.459320i	$0.151907\pi$
$401$			7376.69i			0.918640i	−0.888271	+	0.459320i	$0.848093\pi$
$402$		0			0
$403$	2807.79			0.347061
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	14947.7i		−	1.82047i
$408$		0			0
$409$			6777.29i			0.819353i	0.912231	+	0.409676i	$0.134358\pi$
$409$			6777.29i			0.819353i	−0.912231	+	0.409676i	$0.865642\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6181.06	−	7324.20i	−0.736440	−	0.872641i
$414$		0			0
$415$	−565.212			−0.0668558
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5415.30			0.631395			0.315698	−	0.948860i	$-0.397761\pi$
$419$	5415.30			0.631395			0.315698	+	0.948860i	$0.397761\pi$
$420$		0			0
$421$	818.272			0.0947271			0.0473636	−	0.998878i	$-0.484918\pi$
$421$	818.272			0.0947271			0.0473636	+	0.998878i	$0.484918\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14558.0			1.66157
$426$		0			0
$427$	3832.67	−	3234.47i	0.434370	−	0.366574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	10300.0i		−	1.15113i	−0.817757	−	0.575564i	$-0.804783\pi$
$431$		−	10300.0i		−	1.15113i	0.817757	−	0.575564i	$-0.195217\pi$
$432$		0			0
$433$			16119.3i			1.78902i	0.447047	+	0.894511i	$0.352476\pi$
$433$			16119.3i			1.78902i	−0.447047	+	0.894511i	$0.647524\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3516.38			−0.384922
$438$		0			0
$439$		−	9460.41i		−	1.02852i	−0.857634	−	0.514260i	$-0.828066\pi$
$439$		−	9460.41i		−	1.02852i	0.857634	−	0.514260i	$-0.171934\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	3842.11i		−	0.412063i	−0.978545	−	0.206032i	$-0.933945\pi$
$443$		−	3842.11i		−	0.412063i	0.978545	−	0.206032i	$-0.0660550\pi$
$444$		0			0
$445$	−3468.10			−0.369446
$446$		0			0
$447$		0			0
$448$		0			0
$449$			15663.0i			1.64628i	0.567837	+	0.823141i	$0.307780\pi$
$449$			15663.0i			1.64628i	−0.567837	+	0.823141i	$0.692220\pi$
$450$		0			0
$451$			19804.0i			2.06770i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2996.69	+	2528.98i	−0.308763	+	0.260572i
$456$		0			0
$457$	4953.18			0.507002			0.253501	−	0.967335i	$-0.418418\pi$
$457$	4953.18			0.507002			0.253501	+	0.967335i	$0.418418\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3510.43			0.354657			0.177329	−	0.984152i	$-0.443254\pi$
$461$	3510.43			0.354657			0.177329	+	0.984152i	$0.443254\pi$
$462$		0			0
$463$	14173.5			1.42268			0.711338	−	0.702850i	$-0.248090\pi$
$463$	14173.5			1.42268			0.711338	+	0.702850i	$0.248090\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1922.47			−0.190495			−0.0952474	−	0.995454i	$-0.530364\pi$
$467$	−1922.47			−0.190495			−0.0952474	+	0.995454i	$0.530364\pi$
$468$		0			0
$469$	−8860.05	−	10498.7i	−0.872323	−	1.03365i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	16444.8i		−	1.59859i
$474$		0			0
$475$		−	8230.99i		−	0.795081i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17113.0			−1.63238			−0.816192	−	0.577781i	$-0.803919\pi$
$479$	−17113.0			−1.63238			−0.816192	+	0.577781i	$0.803919\pi$
$480$		0			0
$481$			3568.44i			0.338268i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	21099.7i		−	1.97544i
$486$		0			0
$487$	3067.21			0.285397			0.142699	−	0.989766i	$-0.454422\pi$
$487$	3067.21			0.285397			0.142699	+	0.989766i	$0.454422\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			8626.73i			0.792910i	0.918054	+	0.396455i	$0.129760\pi$
$491$			8626.73i			0.792910i	−0.918054	+	0.396455i	$0.870240\pi$
$492$		0			0
$493$			12591.3i			1.15027i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12939.3	+	10919.7i	−1.16782	+	0.985546i
$498$		0			0
$499$	−12284.4			−1.10205			−0.551026	−	0.834488i	$-0.685764\pi$
$499$	−12284.4			−1.10205			−0.551026	+	0.834488i	$0.685764\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3977.87			−0.352613			−0.176307	−	0.984335i	$-0.556415\pi$
$503$	−3977.87			−0.352613			−0.176307	+	0.984335i	$0.556415\pi$
$504$		0			0
$505$	10195.0			0.898356
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16724.6			−1.45640			−0.728199	−	0.685365i	$-0.759642\pi$
$509$	−16724.6			−1.45640			−0.728199	+	0.685365i	$0.759642\pi$
$510$		0			0
$511$	4772.76	−	4027.84i	0.413179	−	0.348691i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			17313.9i			1.48144i
$516$		0			0
$517$		−	17492.9i		−	1.48808i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9088.77			0.764273			0.382137	−	0.924106i	$-0.375188\pi$
$521$	9088.77			0.764273			0.382137	+	0.924106i	$0.375188\pi$
$522$		0			0
$523$		−	13218.7i		−	1.10519i	−0.833450	−	0.552596i	$-0.813637\pi$
$523$		−	13218.7i		−	1.10519i	0.833450	−	0.552596i	$-0.186363\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	21294.5i		−	1.76016i
$528$		0			0
$529$	8034.41			0.660344
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	4727.76i		−	0.384207i
$534$		0			0
$535$			18735.7i			1.51405i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18067.6	+	3080.72i	1.44383	+	0.246189i
$540$		0			0
$541$	−20011.9			−1.59035			−0.795174	−	0.606382i	$-0.792620\pi$
$541$	−20011.9			−1.59035			−0.795174	+	0.606382i	$0.792620\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−25821.0			−2.02945
$546$		0			0
$547$	−17451.2			−1.36409			−0.682046	−	0.731309i	$-0.738910\pi$
$547$	−17451.2			−1.36409			−0.682046	+	0.731309i	$0.738910\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7119.04			0.550420
$552$		0			0
$553$	−11910.7	−	14113.6i	−0.915906	−	1.08530i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			15169.7i			1.15397i	0.816754	+	0.576986i	$0.195771\pi$
$557$			15169.7i			1.15397i	−0.816754	+	0.576986i	$0.804229\pi$
$558$		0			0
$559$			3925.82i			0.297039i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6977.24			−0.522301			−0.261151	−	0.965298i	$-0.584102\pi$
$563$	−6977.24			−0.522301			−0.261151	+	0.965298i	$0.584102\pi$
$564$		0			0
$565$		−	4289.81i		−	0.319422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	20039.6i		−	1.47645i	−0.674552	−	0.738227i	$-0.735663\pi$
$569$		−	20039.6i		−	1.47645i	0.674552	−	0.738227i	$-0.264337\pi$
$570$		0			0
$571$	−16655.4			−1.22068			−0.610339	−	0.792140i	$-0.708967\pi$
$571$	−16655.4			−1.22068			−0.610339	+	0.792140i	$0.708967\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	9673.40i		−	0.701580i
$576$		0			0
$577$		−	8678.08i		−	0.626123i	−0.949733	−	0.313062i	$-0.898645\pi$
$577$		−	8678.08i		−	0.626123i	0.949733	−	0.313062i	$-0.101355\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−406.763	−	481.991i	−0.0290454	−	0.0344171i
$582$		0			0
$583$	−24139.4			−1.71484
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11883.7			−0.835593			−0.417796	−	0.908541i	$-0.637197\pi$
$587$	−11883.7			−0.835593			−0.417796	+	0.908541i	$0.637197\pi$
$588$		0			0
$589$	−12039.7			−0.842256
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12702.9			−0.879675			−0.439838	−	0.898077i	$-0.644964\pi$
$593$	−12702.9			−0.879675			−0.439838	+	0.898077i	$0.644964\pi$
$594$		0			0
$595$	19180.0	+	22727.2i	1.32152	+	1.56592i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			23873.0i			1.62842i	0.580572	+	0.814209i	$0.302829\pi$
$599$			23873.0i			1.62842i	−0.580572	+	0.814209i	$0.697171\pi$
$600$		0			0
$601$			10411.4i			0.706640i	0.935503	+	0.353320i	$0.114947\pi$
$601$			10411.4i			0.706640i	−0.935503	+	0.353320i	$0.885053\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	25300.1			1.70016
$606$		0			0
$607$		−	15527.0i		−	1.03826i	−0.854696	−	0.519129i	$-0.826256\pi$
$607$		−	15527.0i		−	1.03826i	0.854696	−	0.519129i	$-0.173744\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			4176.03i			0.276504i
$612$		0			0
$613$	−9275.88			−0.611173			−0.305587	−	0.952164i	$-0.598853\pi$
$613$	−9275.88			−0.611173			−0.305587	+	0.952164i	$0.598853\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			3102.94i			0.202463i	0.994863	+	0.101232i	$0.0322783\pi$
$617$			3102.94i			0.202463i	−0.994863	+	0.101232i	$0.967722\pi$
$618$		0			0
$619$			6765.73i			0.439318i	0.975577	+	0.219659i	$0.0704945\pi$
$619$			6765.73i			0.439318i	−0.975577	+	0.219659i	$0.929506\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2495.86	−	2957.46i	−0.160505	−	0.190190i
$624$		0			0
$625$	−11791.4			−0.754652
$626$		0			0
$627$		0			0
$628$		0			0
$629$	27063.3			1.71556
$630$		0			0
$631$	−4096.60			−0.258452			−0.129226	−	0.991615i	$-0.541249\pi$
$631$	−4096.60			−0.258452			−0.129226	+	0.991615i	$0.541249\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9738.72			0.608613
$636$		0			0
$637$	−4313.22	−	735.452i	−0.268283	−	0.0457452i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	17478.5i		−	1.07700i	−0.842624	−	0.538502i	$-0.818990\pi$
$641$		−	17478.5i		−	1.07700i	0.842624	−	0.538502i	$-0.181010\pi$
$642$		0			0
$643$		−	10352.1i		−	0.634908i	−0.948274	−	0.317454i	$-0.897172\pi$
$643$		−	10352.1i		−	0.634908i	0.948274	−	0.317454i	$-0.102828\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13536.7			0.822539			0.411269	−	0.911514i	$-0.365086\pi$
$647$	13536.7			0.822539			0.411269	+	0.911514i	$0.365086\pi$
$648$		0			0
$649$		−	27651.6i		−	1.67245i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			23295.9i			1.39608i	0.716060	+	0.698039i	$0.245944\pi$
$653$			23295.9i			1.39608i	−0.716060	+	0.698039i	$0.754056\pi$
$654$		0			0
$655$	37591.0			2.24245
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1192.51i			0.0704909i	0.999379	+	0.0352454i	$0.0112213\pi$
$659$			1192.51i			0.0704909i	−0.999379	+	0.0352454i	$0.988779\pi$
$660$		0			0
$661$			12612.5i			0.742162i	0.928601	+	0.371081i	$0.121013\pi$
$661$			12612.5i			0.742162i	−0.928601	+	0.371081i	$0.878987\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12849.8	−	10844.2i	0.749312	−	0.632361i
$666$		0			0
$667$	8366.60			0.485691
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14469.7			0.832486
$672$		0			0
$673$	25648.0			1.46903			0.734515	−	0.678592i	$-0.237410\pi$
$673$	25648.0			1.46903			0.734515	+	0.678592i	$0.237410\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14099.7			0.800439			0.400220	−	0.916419i	$-0.368934\pi$
$677$	14099.7			0.800439			0.400220	+	0.916419i	$0.368934\pi$
$678$		0			0
$679$	17993.0	−	15184.7i	1.01695	−	0.858225i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	6066.68i		−	0.339875i	−0.985455	−	0.169938i	$-0.945643\pi$
$683$		−	6066.68i		−	0.339875i	0.985455	−	0.169938i	$-0.0543567\pi$
$684$		0			0
$685$			31712.7i			1.76888i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5762.73			0.318640
$690$		0			0
$691$			31595.8i			1.73945i	0.493537	+	0.869725i	$0.335704\pi$
$691$			31595.8i			1.73945i	−0.493537	+	0.869725i	$0.664296\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			28168.7i			1.53741i
$696$		0			0
$697$	−35855.8			−1.94854
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	14828.2i		−	0.798936i	−0.916747	−	0.399468i	$-0.869195\pi$
$701$		−	14828.2i		−	0.798936i	0.916747	−	0.399468i	$-0.130805\pi$
$702$		0			0
$703$		−	15301.4i		−	0.820914i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7336.94	+	8693.86i	0.390288	+	0.462470i
$708$		0			0
$709$	−21027.1			−1.11381			−0.556903	−	0.830578i	$-0.688011\pi$
$709$	−21027.1			−1.11381			−0.556903	+	0.830578i	$0.688011\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−14149.6			−0.743207
$714$		0			0
$715$	−11313.6			−0.591756
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−36268.4			−1.88120			−0.940601	−	0.339514i	$-0.889738\pi$
$719$	−36268.4			−1.88120			−0.940601	+	0.339514i	$0.889738\pi$
$720$		0			0
$721$	−14764.7	+	12460.2i	−0.762642	+	0.643610i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			19584.2i			1.00323i
$726$		0			0
$727$		−	23568.7i		−	1.20236i	−0.799115	−	0.601178i	$-0.794698\pi$
$727$		−	23568.7i		−	1.20236i	0.799115	−	0.601178i	$-0.205302\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	29773.8			1.50646
$732$		0			0
$733$		−	18430.9i		−	0.928734i	−0.885643	−	0.464367i	$-0.846282\pi$
$733$		−	18430.9i		−	0.928734i	0.885643	−	0.464367i	$-0.153718\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	39636.3i		−	1.98104i
$738$		0			0
$739$	−2029.42			−0.101020			−0.0505098	−	0.998724i	$-0.516085\pi$
$739$	−2029.42			−0.101020			−0.0505098	+	0.998724i	$0.516085\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1839.15i			0.0908101i	0.998969	+	0.0454050i	$0.0144578\pi$
$743$			1839.15i			0.0908101i	−0.998969	+	0.0454050i	$0.985542\pi$
$744$		0			0
$745$			2688.75i			0.132226i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15977.1	+	13483.4i	−0.779427	+	0.657776i
$750$		0			0
$751$	−2182.60			−0.106051			−0.0530254	−	0.998593i	$-0.516886\pi$
$751$	−2182.60			−0.106051			−0.0530254	+	0.998593i	$0.516886\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	51000.8			2.45842
$756$		0			0
$757$	14605.5			0.701250			0.350625	−	0.936516i	$-0.385969\pi$
$757$	14605.5			0.701250			0.350625	+	0.936516i	$0.385969\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17942.2			0.854671			0.427336	−	0.904093i	$-0.359452\pi$
$761$	17942.2			0.854671			0.427336	+	0.904093i	$0.359452\pi$
$762$		0			0
$763$	−18582.4	−	22019.1i	−0.881688	−	1.04475i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			6601.19i			0.310763i
$768$		0			0
$769$			5311.86i			0.249091i	0.992214	+	0.124545i	$0.0397472\pi$
$769$			5311.86i			0.249091i	−0.992214	+	0.124545i	$0.960253\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−21593.6			−1.00475			−0.502373	−	0.864651i	$-0.667540\pi$
$773$	−21593.6			−1.00475			−0.502373	+	0.864651i	$0.667540\pi$
$774$		0			0
$775$		−	33120.8i		−	1.53514i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			20272.6i			0.932401i
$780$		0			0
$781$	−48850.5			−2.23817
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	16823.9i		−	0.764932i
$786$		0			0
$787$		−	34701.8i		−	1.57177i	−0.618371	−	0.785887i	$-0.712207\pi$
$787$		−	34701.8i		−	1.57177i	0.618371	−	0.785887i	$-0.287793\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3658.18	−	3087.22i	0.164437	−	0.138772i
$792$		0			0
$793$	−3454.33			−0.154687
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11120.7			0.494248			0.247124	−	0.968984i	$-0.420515\pi$
$797$	11120.7			0.494248			0.247124	+	0.968984i	$0.420515\pi$
$798$		0			0
$799$	31671.3			1.40232
$800$		0			0
$801$		0			0
$802$		0			0
$803$	18018.9			0.791873
$804$		0			0
$805$	15101.6	−	12744.6i	0.661194	−	0.557996i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			39619.1i			1.72180i	0.508777	+	0.860898i	$0.330098\pi$
$809$			39619.1i			1.72180i	−0.508777	+	0.860898i	$0.669902\pi$
$810$		0			0
$811$			2483.62i			0.107536i	0.998553	+	0.0537679i	$0.0171231\pi$
$811$			2483.62i			0.107536i	−0.998553	+	0.0537679i	$0.982877\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21530.8			−0.925387
$816$		0			0
$817$		−	16833.9i		−	0.720860i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			4898.20i			0.208220i	0.994566	+	0.104110i	$0.0331994\pi$
$821$			4898.20i			0.208220i	−0.994566	+	0.104110i	$0.966801\pi$
$822$		0			0
$823$	15936.4			0.674980			0.337490	−	0.941329i	$-0.390422\pi$
$823$	15936.4			0.674980			0.337490	+	0.941329i	$0.390422\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	11183.9i		−	0.470256i	−0.971964	−	0.235128i	$-0.924449\pi$
$827$		−	11183.9i		−	0.470256i	0.971964	−	0.235128i	$-0.0755509\pi$
$828$		0			0
$829$		−	6180.21i		−	0.258924i	−0.991584	−	0.129462i	$-0.958675\pi$
$829$		−	6180.21i		−	0.258924i	0.991584	−	0.129462i	$-0.0413250\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5577.73	+	32711.8i	−0.232001	+	1.36062i
$834$		0			0
$835$	23775.3			0.985364
$836$		0			0
$837$		0			0
$838$		0			0
$839$	20319.4			0.836119			0.418059	−	0.908420i	$-0.362710\pi$
$839$	20319.4			0.836119			0.418059	+	0.908420i	$0.362710\pi$
$840$		0			0
$841$	7450.49			0.305486
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−33763.8			−1.37457
$846$		0			0
$847$	18207.6	+	21574.9i	0.738630	+	0.875235i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	17982.8i		−	0.724376i
$852$		0			0
$853$			25093.0i			1.00723i	0.863927	+	0.503616i	$0.167997\pi$
$853$			25093.0i			1.00723i	−0.863927	+	0.503616i	$0.832003\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−16363.5			−0.652236			−0.326118	−	0.945329i	$-0.605741\pi$
$857$	−16363.5			−0.652236			−0.326118	+	0.945329i	$0.605741\pi$
$858$		0			0
$859$			35461.8i			1.40855i	0.709929	+	0.704273i	$0.248727\pi$
$859$			35461.8i			1.40855i	−0.709929	+	0.704273i	$0.751273\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			33166.4i			1.30822i	0.756398	+	0.654111i	$0.226957\pi$
$863$			33166.4i			1.30822i	−0.756398	+	0.654111i	$0.773043\pi$
$864$		0			0
$865$	32785.3			1.28871
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	53283.9i		−	2.08001i
$870$		0			0
$871$			9462.28i			0.368102i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5050.67	+	5984.76i	0.195136	+	0.231225i
$876$		0			0
$877$	11235.9			0.432621			0.216311	−	0.976325i	$-0.430598\pi$
$877$	11235.9			0.432621			0.216311	+	0.976325i	$0.430598\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	25537.5			0.976597			0.488298	−	0.872677i	$-0.337618\pi$
$881$	25537.5			0.976597			0.488298	+	0.872677i	$0.337618\pi$
$882$		0			0
$883$	17936.9			0.683606			0.341803	−	0.939772i	$-0.388962\pi$
$883$	17936.9			0.683606			0.341803	+	0.939772i	$0.388962\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5293.97			0.200399			0.100200	−	0.994967i	$-0.468052\pi$
$887$	5293.97			0.200399			0.100200	+	0.994967i	$0.468052\pi$
$888$		0			0
$889$	7008.60	+	8304.80i	0.264411	+	0.313312i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	17906.7i		−	0.671026i
$894$		0			0
$895$			38341.3i			1.43196i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	28646.4			1.06275
$900$		0			0
$901$		−	43705.0i		−	1.61601i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	7029.78i		−	0.258208i
$906$		0			0
$907$	13540.8			0.495715			0.247857	−	0.968797i	$-0.420274\pi$
$907$	13540.8			0.495715			0.247857	+	0.968797i	$0.420274\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			27141.4i			0.987084i	0.869722	+	0.493542i	$0.164298\pi$
$911$			27141.4i			0.987084i	−0.869722	+	0.493542i	$0.835702\pi$
$912$		0			0
$913$		−	1819.69i		−	0.0659617i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	27052.9	+	32056.2i	0.974226	+	1.15440i
$918$		0			0
$919$	−32872.5			−1.17994			−0.589969	−	0.807426i	$-0.700860\pi$
$919$	−32872.5			−1.17994			−0.589969	+	0.807426i	$0.700860\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11662.0			0.415881
$924$		0			0
$925$	42093.5			1.49624
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23656.4			0.835459			0.417729	−	0.908572i	$-0.362826\pi$
$929$	23656.4			0.835459			0.417729	+	0.908572i	$0.362826\pi$
$930$		0			0
$931$	18495.0	+	3153.61i	0.651074	+	0.111015i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			85803.5i			3.00115i
$936$		0			0
$937$			26716.0i			0.931456i	0.884928	+	0.465728i	$0.154207\pi$
$937$			26716.0i			0.931456i	−0.884928	+	0.465728i	$0.845793\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39531.9			1.36950			0.684752	−	0.728776i	$-0.259910\pi$
$941$	39531.9			1.36950			0.684752	+	0.728776i	$0.259910\pi$
$942$		0			0
$943$			23825.2i			0.822751i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			45015.8i			1.54469i	0.635206	+	0.772343i	$0.280915\pi$
$947$			45015.8i			1.54469i	−0.635206	+	0.772343i	$0.719085\pi$
$948$		0			0
$949$	−4301.61			−0.147140
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	9240.19i		−	0.314081i	−0.987592	−	0.157040i	$-0.949805\pi$
$953$		−	9240.19i		−	0.314081i	0.987592	−	0.157040i	$-0.0501953\pi$
$954$		0			0
$955$			38913.4i			1.31854i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−27043.3	+	22822.5i	−0.910610	+	0.768484i
$960$		0			0
$961$	−18655.9			−0.626225
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−67231.6			−2.24276
$966$		0			0
$967$	−24783.6			−0.824186			−0.412093	−	0.911142i	$-0.635202\pi$
$967$	−24783.6			−0.824186			−0.412093	+	0.911142i	$0.635202\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	37516.1			1.23991			0.619953	−	0.784639i	$-0.287152\pi$
$971$	37516.1			1.23991			0.619953	+	0.784639i	$0.287152\pi$
$972$		0			0
$973$	−24021.2	+	20272.0i	−0.791454	+	0.667925i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			16367.8i			0.535981i	0.963422	+	0.267991i	$0.0863596\pi$
$977$			16367.8i			0.535981i	−0.963422	+	0.267991i	$0.913640\pi$
$978$		0			0
$979$		−	11165.5i		−	0.364505i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	29516.2			0.957700			0.478850	−	0.877897i	$-0.341054\pi$
$983$	29516.2			0.957700			0.478850	+	0.877897i	$0.341054\pi$
$984$		0			0
$985$			60733.6i			1.96460i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	19783.9i		−	0.636087i
$990$		0			0
$991$	−13893.1			−0.445336			−0.222668	−	0.974894i	$-0.571477\pi$
$991$	−13893.1			−0.445336			−0.222668	+	0.974894i	$0.571477\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	9245.62i		−	0.294579i
$996$		0			0
$997$		−	19790.8i		−	0.628667i	−0.949313	−	0.314333i	$-0.898219\pi$
$997$		−	19790.8i		−	0.628667i	0.949313	−	0.314333i	$-0.101781\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.4.f.a.1133.5		48
3.2	odd	2	inner	2268.4.f.a.1133.44		48
7.6	odd	2	inner	2268.4.f.a.1133.43		48
9.2	odd	6		252.4.x.a.41.17	yes	48
9.4	even	3		252.4.x.a.209.8	yes	48
9.5	odd	6		756.4.x.a.629.3		48
9.7	even	3		756.4.x.a.125.22		48
21.20	even	2	inner	2268.4.f.a.1133.6		48
63.13	odd	6		252.4.x.a.209.17	yes	48
63.20	even	6		252.4.x.a.41.8	✓	48
63.34	odd	6		756.4.x.a.125.3		48
63.41	even	6		756.4.x.a.629.22		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.8	✓	48	63.20	even	6
252.4.x.a.41.17	yes	48	9.2	odd	6
252.4.x.a.209.8	yes	48	9.4	even	3
252.4.x.a.209.17	yes	48	63.13	odd	6
756.4.x.a.125.3		48	63.34	odd	6
756.4.x.a.125.22		48	9.7	even	3
756.4.x.a.629.3		48	9.5	odd	6
756.4.x.a.629.22		48	63.41	even	6
2268.4.f.a.1133.5		48	1.1	even	1	trivial
2268.4.f.a.1133.6		48	21.20	even	2	inner
2268.4.f.a.1133.43		48	7.6	odd	2	inner
2268.4.f.a.1133.44		48	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2268.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.5975 q^{5} +(-11.9446 - 14.1537i) q^{7} +O(q^{10})\)	\(q-16.5975 q^{5} +(-11.9446 - 14.1537i) q^{7} -53.4353i q^{11} +12.7565i q^{13} +96.7463 q^{17} -54.6996i q^{19} -64.2853i q^{23} +150.476 q^{25} +130.148i q^{29} -220.107i q^{31} +(198.250 + 234.915i) q^{35} +279.735 q^{37} -370.617 q^{41} +307.751 q^{43} +327.365 q^{47} +(-57.6532 + 338.120i) q^{49} -451.749i q^{53} +886.892i q^{55} +517.477 q^{59} +270.790i q^{61} -211.725i q^{65} +741.763 q^{67} -914.198i q^{71} +337.210i q^{73} +(-756.306 + 638.264i) q^{77} +997.165 q^{79} +34.0541 q^{83} -1605.74 q^{85} +208.953 q^{89} +(180.551 - 152.371i) q^{91} +907.875i q^{95} +1271.26i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{7}+O(q^{10}) \) 48 * q - 12 * q^7	\( 48 q - 12 q^{7} + 1200 q^{25} + 336 q^{37} - 168 q^{43} - 636 q^{49} + 1176 q^{67} - 408 q^{79} + 720 q^{85} - 1080 q^{91}+O(q^{100}) \) 48 * q - 12 * q^7 + 1200 * q^25 + 336 * q^37 - 168 * q^43 - 636 * q^49 + 1176 * q^67 - 408 * q^79 + 720 * q^85 - 1080 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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