LMFDB - Embedded newform 2268.4.f.a.1133.12

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			1133.12
Character	$\chi$	$=$	2268.1133
Dual form			2268.4.f.a.1133.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.9938 q^{5} +(11.4708 + 14.5403i) q^{7} +O(q^{10})$	$q-10.9938 q^{5} +(11.4708 + 14.5403i) q^{7} +15.9659i q^{11} +89.4072i q^{13} -106.953 q^{17} -8.31634i q^{19} +143.059i q^{23} -4.13623 q^{25} +149.789i q^{29} +42.9493i q^{31} +(-126.108 - 159.853i) q^{35} +390.706 q^{37} -344.973 q^{41} -57.1185 q^{43} -16.2656 q^{47} +(-79.8415 + 333.578i) q^{49} +445.230i q^{53} -175.526i q^{55} -386.700 q^{59} -485.964i q^{61} -982.925i q^{65} +503.642 q^{67} -751.418i q^{71} +507.533i q^{73} +(-232.149 + 183.141i) q^{77} -763.141 q^{79} +1214.48 q^{83} +1175.82 q^{85} +425.727 q^{89} +(-1300.01 + 1025.57i) q^{91} +91.4282i q^{95} +571.094i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−10.9938			−0.983316			−0.491658	−	0.870788i	$-0.663609\pi$
$5$	−10.9938			−0.983316			−0.491658	+	0.870788i	$0.663609\pi$
$6$		0			0
$7$	11.4708	+	14.5403i	0.619365	+	0.785103i
$7$	11.4708	+	14.5403i	0.619365	+	0.785103i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			15.9659i			0.437626i	0.975767	+	0.218813i	$0.0702185\pi$
$11$			15.9659i			0.437626i	−0.975767	+	0.218813i	$0.929781\pi$
$12$		0			0
$13$			89.4072i			1.90747i	0.300650	+	0.953734i	$0.402796\pi$
$13$			89.4072i			1.90747i	−0.300650	+	0.953734i	$0.597204\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−106.953			−1.52588			−0.762939	−	0.646471i	$-0.776244\pi$
$17$	−106.953			−1.52588			−0.762939	+	0.646471i	$0.776244\pi$
$18$		0			0
$19$		−	8.31634i		−	0.100416i	−0.998739	−	0.0502079i	$-0.984012\pi$
$19$		−	8.31634i		−	0.100416i	0.998739	−	0.0502079i	$-0.0159884\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			143.059i			1.29695i	0.761237	+	0.648474i	$0.224593\pi$
$23$			143.059i			1.29695i	−0.761237	+	0.648474i	$0.775407\pi$
$24$		0			0
$25$	−4.13623			−0.0330899
$26$		0			0
$27$		0			0
$28$		0			0
$29$			149.789i			0.959140i	0.877504	+	0.479570i	$0.159207\pi$
$29$			149.789i			0.959140i	−0.877504	+	0.479570i	$0.840793\pi$
$30$		0			0
$31$			42.9493i			0.248836i	0.992230	+	0.124418i	$0.0397064\pi$
$31$			42.9493i			0.248836i	−0.992230	+	0.124418i	$0.960294\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−126.108	−	159.853i	−0.609031	−	0.772004i
$36$		0			0
$37$	390.706			1.73599			0.867997	−	0.496570i	$-0.165407\pi$
$37$	390.706			1.73599			0.867997	+	0.496570i	$0.165407\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−344.973			−1.31404			−0.657021	−	0.753873i	$-0.728184\pi$
$41$	−344.973			−1.31404			−0.657021	+	0.753873i	$0.728184\pi$
$42$		0			0
$43$	−57.1185			−0.202570			−0.101285	−	0.994857i	$-0.532295\pi$
$43$	−57.1185			−0.202570			−0.101285	+	0.994857i	$0.532295\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−16.2656			−0.0504806			−0.0252403	−	0.999681i	$-0.508035\pi$
$47$	−16.2656			−0.0504806			−0.0252403	+	0.999681i	$0.508035\pi$
$48$		0			0
$49$	−79.8415	+	333.578i	−0.232774	+	0.972531i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			445.230i			1.15391i	0.816777	+	0.576954i	$0.195759\pi$
$53$			445.230i			1.15391i	−0.816777	+	0.576954i	$0.804241\pi$
$54$		0			0
$55$		−	175.526i		−	0.430325i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−386.700			−0.853288			−0.426644	−	0.904420i	$-0.640304\pi$
$59$	−386.700			−0.853288			−0.426644	+	0.904420i	$0.640304\pi$
$60$		0			0
$61$		−	485.964i		−	1.02002i	−0.860168	−	0.510011i	$-0.829641\pi$
$61$		−	485.964i		−	1.02002i	0.860168	−	0.510011i	$-0.170359\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	982.925i		−	1.87564i
$66$		0			0
$67$	503.642			0.918354			0.459177	−	0.888345i	$-0.348144\pi$
$67$	503.642			0.918354			0.459177	+	0.888345i	$0.348144\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	751.418i		−	1.25601i	−0.778208	−	0.628007i	$-0.783871\pi$
$71$		−	751.418i		−	1.25601i	0.778208	−	0.628007i	$-0.216129\pi$
$72$		0			0
$73$			507.533i			0.813729i	0.913489	+	0.406865i	$0.133378\pi$
$73$			507.533i			0.813729i	−0.913489	+	0.406865i	$0.866622\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−232.149	+	183.141i	−0.343582	+	0.271050i
$78$		0			0
$79$	−763.141			−1.08684			−0.543418	−	0.839462i	$-0.682870\pi$
$79$	−763.141			−1.08684			−0.543418	+	0.839462i	$0.682870\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1214.48			1.60610			0.803052	−	0.595908i	$-0.203208\pi$
$83$	1214.48			1.60610			0.803052	+	0.595908i	$0.203208\pi$
$84$		0			0
$85$	1175.82			1.50042
$86$		0			0
$87$		0			0
$88$		0			0
$89$	425.727			0.507045			0.253523	−	0.967329i	$-0.418411\pi$
$89$	425.727			0.507045			0.253523	+	0.967329i	$0.418411\pi$
$90$		0			0
$91$	−1300.01	+	1025.57i	−1.49756	+	1.18142i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			91.4282i			0.0987404i
$96$		0			0
$97$			571.094i			0.597792i	0.954286	+	0.298896i	$0.0966184\pi$
$97$			571.094i			0.597792i	−0.954286	+	0.298896i	$0.903382\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−706.770			−0.696299			−0.348150	−	0.937439i	$-0.613190\pi$
$101$	−706.770			−0.696299			−0.348150	+	0.937439i	$0.613190\pi$
$102$		0			0
$103$			251.271i			0.240373i	0.992751	+	0.120187i	$0.0383493\pi$
$103$			251.271i			0.240373i	−0.992751	+	0.120187i	$0.961651\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	739.962i		−	0.668549i	−0.942476	−	0.334275i	$-0.891509\pi$
$107$		−	739.962i		−	0.668549i	0.942476	−	0.334275i	$-0.108491\pi$
$108$		0			0
$109$	1497.38			1.31580			0.657902	−	0.753104i	$-0.271444\pi$
$109$	1497.38			1.31580			0.657902	+	0.753104i	$0.271444\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			995.254i			0.828546i	0.910153	+	0.414273i	$0.135964\pi$
$113$			995.254i			0.828546i	−0.910153	+	0.414273i	$0.864036\pi$
$114$		0			0
$115$		−	1572.76i		−	1.27531i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1226.84	−	1555.13i	−0.945075	−	1.19797i
$120$		0			0
$121$	1076.09			0.808483
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1419.70			1.01585
$126$		0			0
$127$	2469.41			1.72539			0.862696	−	0.505724i	$-0.168774\pi$
$127$	2469.41			1.72539			0.862696	+	0.505724i	$0.168774\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1539.57			−1.02682			−0.513409	−	0.858144i	$-0.671618\pi$
$131$	−1539.57			−1.02682			−0.513409	+	0.858144i	$0.671618\pi$
$132$		0			0
$133$	120.922	−	95.3951i	0.0788367	−	0.0621940i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1639.44i		−	1.02239i	−0.859465	−	0.511194i	$-0.829203\pi$
$137$		−	1639.44i		−	1.02239i	0.859465	−	0.511194i	$-0.170797\pi$
$138$		0			0
$139$			1038.69i			0.633817i	0.948456	+	0.316909i	$0.102645\pi$
$139$			1038.69i			0.633817i	−0.948456	+	0.316909i	$0.897355\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1427.46			−0.834759
$144$		0			0
$145$		−	1646.75i		−	0.943137i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			1758.80i			0.967025i	0.875337	+	0.483512i	$0.160639\pi$
$149$			1758.80i			0.967025i	−0.875337	+	0.483512i	$0.839361\pi$
$150$		0			0
$151$	−704.801			−0.379841			−0.189920	−	0.981800i	$-0.560823\pi$
$151$	−704.801			−0.379841			−0.189920	+	0.981800i	$0.560823\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	472.177i		−	0.244685i
$156$		0			0
$157$		−	668.631i		−	0.339889i	−0.985454	−	0.169944i	$-0.945641\pi$
$157$		−	668.631i		−	0.339889i	0.985454	−	0.169944i	$-0.0543588\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2080.12	+	1641.00i	−1.01824	+	0.803285i
$162$		0			0
$163$	−3926.19			−1.88665			−0.943323	−	0.331877i	$-0.892318\pi$
$163$	−3926.19			−1.88665			−0.943323	+	0.331877i	$0.892318\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1933.60			0.895967			0.447984	−	0.894042i	$-0.352142\pi$
$167$	1933.60			0.895967			0.447984	+	0.894042i	$0.352142\pi$
$168$		0			0
$169$	−5796.65			−2.63844
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1700.04			0.747118			0.373559	−	0.927606i	$-0.378137\pi$
$173$	1700.04			0.747118			0.373559	+	0.927606i	$0.378137\pi$
$174$		0			0
$175$	−47.4459	−	60.1421i	−0.0204947	−	0.0259790i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1175.15i		−	0.490697i	−0.969435	−	0.245349i	$-0.921098\pi$
$179$		−	1175.15i		−	0.490697i	0.969435	−	0.245349i	$-0.0789024\pi$
$180$		0			0
$181$			3506.89i			1.44014i	0.693902	+	0.720070i	$0.255890\pi$
$181$			3506.89i			1.44014i	−0.693902	+	0.720070i	$0.744110\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4295.35			−1.70703
$186$		0			0
$187$		−	1707.60i		−	0.667764i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1418.14i			0.537239i	0.963246	+	0.268620i	$0.0865674\pi$
$191$			1418.14i			0.537239i	−0.963246	+	0.268620i	$0.913433\pi$
$192$		0			0
$193$	−1493.64			−0.557070			−0.278535	−	0.960426i	$-0.589849\pi$
$193$	−1493.64			−0.557070			−0.278535	+	0.960426i	$0.589849\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2384.42i		−	0.862351i	−0.902268	−	0.431175i	$-0.858099\pi$
$197$		−	2384.42i		−	0.862351i	0.902268	−	0.431175i	$-0.141901\pi$
$198$		0			0
$199$		−	4960.24i		−	1.76695i	−0.468482	−	0.883473i	$-0.655199\pi$
$199$		−	4960.24i		−	1.76695i	0.468482	−	0.883473i	$-0.344801\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2177.97	+	1718.20i	−0.753024	+	0.594057i
$204$		0			0
$205$	3792.56			1.29212
$206$		0			0
$207$		0			0
$208$		0			0
$209$	132.778			0.0439446
$210$		0			0
$211$	−3947.29			−1.28788			−0.643939	−	0.765077i	$-0.722701\pi$
$211$	−3947.29			−1.28788			−0.643939	+	0.765077i	$0.722701\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	627.950			0.199190
$216$		0			0
$217$	−624.497	+	492.663i	−0.195362	+	0.154121i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	9562.37i		−	2.91056i
$222$		0			0
$223$		−	1613.65i		−	0.484565i	−0.970206	−	0.242283i	$-0.922104\pi$
$223$		−	1613.65i		−	0.484565i	0.970206	−	0.242283i	$-0.0778961\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3499.91			−1.02334			−0.511668	−	0.859183i	$-0.670972\pi$
$227$	−3499.91			−1.02334			−0.511668	+	0.859183i	$0.670972\pi$
$228$		0			0
$229$			1910.16i			0.551208i	0.961271	+	0.275604i	$0.0888778\pi$
$229$			1910.16i			0.551208i	−0.961271	+	0.275604i	$0.911122\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4408.26i		−	1.23946i	−0.784814	−	0.619731i	$-0.787242\pi$
$233$		−	4408.26i		−	1.23946i	0.784814	−	0.619731i	$-0.212758\pi$
$234$		0			0
$235$	178.821			0.0496383
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4290.45i		−	1.16120i	−0.814190	−	0.580599i	$-0.802818\pi$
$239$		−	4290.45i		−	1.16120i	0.814190	−	0.580599i	$-0.197182\pi$
$240$		0			0
$241$		−	1632.09i		−	0.436234i	−0.975923	−	0.218117i	$-0.930009\pi$
$241$		−	1632.09i		−	0.436234i	0.975923	−	0.218117i	$-0.0699914\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	877.762	−	3667.29i	0.228891	−	0.956305i
$246$		0			0
$247$	743.541			0.191540
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3912.60			−0.983908			−0.491954	−	0.870621i	$-0.663717\pi$
$251$	−3912.60			−0.983908			−0.491954	+	0.870621i	$0.663717\pi$
$252$		0			0
$253$	−2284.06			−0.567579
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2109.22			−0.511943			−0.255971	−	0.966684i	$-0.582395\pi$
$257$	−2109.22			−0.511943			−0.255971	+	0.966684i	$0.582395\pi$
$258$		0			0
$259$	4481.72	+	5681.00i	1.07521	+	1.36293i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4419.90i			1.03628i	0.855295	+	0.518142i	$0.173376\pi$
$263$			4419.90i			1.03628i	−0.855295	+	0.518142i	$0.826624\pi$
$264$		0			0
$265$		−	4894.78i		−	1.13466i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4557.94			−1.03309			−0.516547	−	0.856259i	$-0.672783\pi$
$269$	−4557.94			−1.03309			−0.516547	+	0.856259i	$0.672783\pi$
$270$		0			0
$271$			128.189i			0.0287341i	0.999897	+	0.0143671i	$0.00457334\pi$
$271$			128.189i			0.0287341i	−0.999897	+	0.0143671i	$0.995427\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	66.0385i		−	0.0144810i
$276$		0			0
$277$	−2040.02			−0.442502			−0.221251	−	0.975217i	$-0.571014\pi$
$277$	−2040.02			−0.442502			−0.221251	+	0.975217i	$0.571014\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1739.64i			0.369318i	0.982803	+	0.184659i	$0.0591180\pi$
$281$			1739.64i			0.369318i	−0.982803	+	0.184659i	$0.940882\pi$
$282$		0			0
$283$		−	1247.67i		−	0.262072i	−0.991378	−	0.131036i	$-0.958170\pi$
$283$		−	1247.67i		−	0.262072i	0.991378	−	0.131036i	$-0.0418303\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3957.11	−	5016.01i	−0.813871	−	1.03166i
$288$		0			0
$289$	6525.95			1.32830
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4791.18			0.955302			0.477651	−	0.878550i	$-0.341488\pi$
$293$	4791.18			0.955302			0.477651	+	0.878550i	$0.341488\pi$
$294$		0			0
$295$	4251.30			0.839052
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12790.5			−2.47389
$300$		0			0
$301$	−655.195	−	830.521i	−0.125465	−	0.159038i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			5342.59i			1.00300i
$306$		0			0
$307$		−	4679.41i		−	0.869929i	−0.900448	−	0.434965i	$-0.856761\pi$
$307$		−	4679.41i		−	0.869929i	0.900448	−	0.434965i	$-0.143239\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1838.33			−0.335183			−0.167592	−	0.985856i	$-0.553599\pi$
$311$	−1838.33			−0.335183			−0.167592	+	0.985856i	$0.553599\pi$
$312$		0			0
$313$			5958.40i			1.07600i	0.842944	+	0.538001i	$0.180821\pi$
$313$			5958.40i			1.07600i	−0.842944	+	0.538001i	$0.819179\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	10376.3i		−	1.83846i	−0.393716	−	0.919232i	$-0.628811\pi$
$317$		−	10376.3i		−	1.83846i	0.393716	−	0.919232i	$-0.371189\pi$
$318$		0			0
$319$	−2391.51			−0.419745
$320$		0			0
$321$		0			0
$322$		0			0
$323$			889.458i			0.153222i
$324$		0			0
$325$		−	369.809i		−	0.0631179i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−186.580	−	236.507i	−0.0312659	−	0.0396324i
$330$		0			0
$331$	4843.30			0.804265			0.402132	−	0.915582i	$-0.368269\pi$
$331$	4843.30			0.804265			0.402132	+	0.915582i	$0.368269\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5536.94			−0.903032
$336$		0			0
$337$	1118.03			0.180720			0.0903602	−	0.995909i	$-0.471198\pi$
$337$	1118.03			0.180720			0.0903602	+	0.995909i	$0.471198\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−685.723			−0.108897
$342$		0			0
$343$	−5766.18	+	2665.49i	−0.907709	+	0.419600i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4428.11i		−	0.685052i	−0.939508	−	0.342526i	$-0.888717\pi$
$347$		−	4428.11i		−	0.685052i	0.939508	−	0.342526i	$-0.111283\pi$
$348$		0			0
$349$			2286.45i			0.350691i	0.984507	+	0.175345i	$0.0561042\pi$
$349$			2286.45i			0.350691i	−0.984507	+	0.175345i	$0.943896\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8055.77			−1.21463			−0.607316	−	0.794460i	$-0.707754\pi$
$353$	−8055.77			−1.21463			−0.607316	+	0.794460i	$0.707754\pi$
$354$		0			0
$355$			8260.95i			1.23506i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5537.25i			0.814053i	0.913416	+	0.407027i	$0.133434\pi$
$359$			5537.25i			0.814053i	−0.913416	+	0.407027i	$0.866566\pi$
$360$		0			0
$361$	6789.84			0.989917
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	5579.72i		−	0.800153i
$366$		0			0
$367$			5391.15i			0.766800i	0.923582	+	0.383400i	$0.125247\pi$
$367$			5391.15i			0.766800i	−0.923582	+	0.383400i	$0.874753\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6473.79	+	5107.15i	−0.905936	+	0.714690i
$372$		0			0
$373$	−2124.17			−0.294866			−0.147433	−	0.989072i	$-0.547101\pi$
$373$	−2124.17			−0.294866			−0.147433	+	0.989072i	$0.547101\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13392.2			−1.82953
$378$		0			0
$379$	−5358.80			−0.726288			−0.363144	−	0.931733i	$-0.618297\pi$
$379$	−5358.80			−0.726288			−0.363144	+	0.931733i	$0.618297\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5458.46			−0.728235			−0.364118	−	0.931353i	$-0.618629\pi$
$383$	−5458.46			−0.728235			−0.364118	+	0.931353i	$0.618629\pi$
$384$		0			0
$385$	2552.20	−	2013.42i	0.337850	−	0.266528i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			1635.34i			0.213149i	0.994305	+	0.106575i	$0.0339883\pi$
$389$			1635.34i			0.213149i	−0.994305	+	0.106575i	$0.966012\pi$
$390$		0			0
$391$		−	15300.6i		−	1.97899i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8389.82			1.06870
$396$		0			0
$397$			7694.41i			0.972724i	0.873757	+	0.486362i	$0.161676\pi$
$397$			7694.41i			0.972724i	−0.873757	+	0.486362i	$0.838324\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	5543.10i		−	0.690297i	−0.938548	−	0.345148i	$-0.887829\pi$
$401$		−	5543.10i		−	0.690297i	0.938548	−	0.345148i	$-0.112171\pi$
$402$		0			0
$403$	−3839.98			−0.474648
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6237.97i			0.759716i
$408$		0			0
$409$		−	7345.86i		−	0.888091i	−0.896004	−	0.444045i	$-0.853543\pi$
$409$		−	7345.86i		−	0.888091i	0.896004	−	0.444045i	$-0.146457\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4435.75	−	5622.74i	−0.528497	−	0.669920i
$414$		0			0
$415$	−13351.8			−1.57931
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3434.91			−0.400492			−0.200246	−	0.979746i	$-0.564174\pi$
$419$	−3434.91			−0.400492			−0.200246	+	0.979746i	$0.564174\pi$
$420$		0			0
$421$	14482.3			1.67654			0.838269	−	0.545257i	$-0.183568\pi$
$421$	14482.3			1.67654			0.838269	+	0.545257i	$0.183568\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	442.383			0.0504911
$426$		0			0
$427$	7066.07	−	5574.40i	0.800822	−	0.631766i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	13632.5i		−	1.52356i	−0.647839	−	0.761778i	$-0.724327\pi$
$431$		−	13632.5i		−	1.52356i	0.647839	−	0.761778i	$-0.275673\pi$
$432$		0			0
$433$			6735.48i			0.747544i	0.927521	+	0.373772i	$0.121936\pi$
$433$			6735.48i			0.747544i	−0.927521	+	0.373772i	$0.878064\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1189.73			0.130234
$438$		0			0
$439$			10601.5i			1.15258i	0.817246	+	0.576289i	$0.195500\pi$
$439$			10601.5i			1.15258i	−0.817246	+	0.576289i	$0.804500\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	3593.27i		−	0.385376i	−0.981260	−	0.192688i	$-0.938280\pi$
$443$		−	3593.27i		−	0.385376i	0.981260	−	0.192688i	$-0.0617205\pi$
$444$		0			0
$445$	−4680.36			−0.498586
$446$		0			0
$447$		0			0
$448$		0			0
$449$			8675.98i			0.911904i	0.890004	+	0.455952i	$0.150701\pi$
$449$			8675.98i			0.911904i	−0.890004	+	0.455952i	$0.849299\pi$
$450$		0			0
$451$		−	5507.79i		−	0.575059i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14292.0	−	11274.9i	1.47257	−	1.16171i
$456$		0			0
$457$	2932.69			0.300187			0.150094	−	0.988672i	$-0.452042\pi$
$457$	2932.69			0.300187			0.150094	+	0.988672i	$0.452042\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6031.61			−0.609371			−0.304686	−	0.952453i	$-0.598551\pi$
$461$	−6031.61			−0.609371			−0.304686	+	0.952453i	$0.598551\pi$
$462$		0			0
$463$	−591.276			−0.0593497			−0.0296748	−	0.999560i	$-0.509447\pi$
$463$	−591.276			−0.0593497			−0.0296748	+	0.999560i	$0.509447\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16674.6			1.65227			0.826134	−	0.563473i	$-0.190535\pi$
$467$	16674.6			1.65227			0.826134	+	0.563473i	$0.190535\pi$
$468$		0			0
$469$	5777.18	+	7323.12i	0.568796	+	0.721002i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	911.947i		−	0.0886498i
$474$		0			0
$475$			34.3983i			0.00332274i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17217.5			1.64235			0.821175	−	0.570677i	$-0.193319\pi$
$479$	17217.5			1.64235			0.821175	+	0.570677i	$0.193319\pi$
$480$		0			0
$481$			34932.0i			3.31135i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	6278.50i		−	0.587818i
$486$		0			0
$487$	8745.94			0.813792			0.406896	−	0.913475i	$-0.366611\pi$
$487$	8745.94			0.813792			0.406896	+	0.913475i	$0.366611\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	754.388i		−	0.0693382i	−0.999399	−	0.0346691i	$-0.988962\pi$
$491$		−	754.388i		−	0.0693382i	0.999399	−	0.0346691i	$-0.0110377\pi$
$492$		0			0
$493$		−	16020.3i		−	1.46353i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10925.9	−	8619.37i	0.986100	−	0.777931i
$498$		0			0
$499$	4960.47			0.445013			0.222506	−	0.974931i	$-0.428576\pi$
$499$	4960.47			0.445013			0.222506	+	0.974931i	$0.428576\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5299.83			−0.469797			−0.234898	−	0.972020i	$-0.575476\pi$
$503$	−5299.83			−0.469797			−0.234898	+	0.972020i	$0.575476\pi$
$504$		0			0
$505$	7770.09			0.684682
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18689.6			−1.62751			−0.813755	−	0.581207i	$-0.802580\pi$
$509$	−18689.6			−1.62751			−0.813755	+	0.581207i	$0.802580\pi$
$510$		0			0
$511$	−7379.69	+	5821.81i	−0.638861	+	0.503995i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	2762.42i		−	0.236363i
$516$		0			0
$517$		−	259.695i		−	0.0220916i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8964.93			−0.753859			−0.376930	−	0.926242i	$-0.623020\pi$
$521$	−8964.93			−0.753859			−0.376930	+	0.926242i	$0.623020\pi$
$522$		0			0
$523$		−	20614.1i		−	1.72350i	−0.507330	−	0.861752i	$-0.669367\pi$
$523$		−	20614.1i		−	1.72350i	0.507330	−	0.861752i	$-0.330633\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	4593.56i		−	0.379694i
$528$		0			0
$529$	−8298.82			−0.682076
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	30843.0i		−	2.50649i
$534$		0			0
$535$			8134.99i			0.657395i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5325.86	−	1274.74i	−0.425605	−	0.101868i
$540$		0			0
$541$	9376.65			0.745164			0.372582	−	0.927999i	$-0.378473\pi$
$541$	9376.65			0.745164			0.372582	+	0.927999i	$0.378473\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−16461.9			−1.29385
$546$		0			0
$547$	−2344.50			−0.183261			−0.0916305	−	0.995793i	$-0.529208\pi$
$547$	−2344.50			−0.183261			−0.0916305	+	0.995793i	$0.529208\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1245.69			0.0963127
$552$		0			0
$553$	−8753.83	−	11096.3i	−0.673148	−	0.853278i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	15368.4i		−	1.16908i	−0.811364	−	0.584541i	$-0.801275\pi$
$557$		−	15368.4i		−	1.16908i	0.811364	−	0.584541i	$-0.198725\pi$
$558$		0			0
$559$		−	5106.81i		−	0.386395i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12895.5			0.965327			0.482664	−	0.875806i	$-0.339669\pi$
$563$	12895.5			0.965327			0.482664	+	0.875806i	$0.339669\pi$
$564$		0			0
$565$		−	10941.6i		−	0.814722i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			24236.2i			1.78565i	0.450406	+	0.892824i	$0.351279\pi$
$569$			24236.2i			1.78565i	−0.450406	+	0.892824i	$0.648721\pi$
$570$		0			0
$571$	7356.76			0.539178			0.269589	−	0.962975i	$-0.413112\pi$
$571$	7356.76			0.539178			0.269589	+	0.962975i	$0.413112\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	591.724i		−	0.0429158i
$576$		0			0
$577$		−	11928.2i		−	0.860617i	−0.902682	−	0.430308i	$-0.858405\pi$
$577$		−	11928.2i		−	0.860617i	0.902682	−	0.430308i	$-0.141595\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	13931.1	+	17658.9i	0.994765	+	1.26096i
$582$		0			0
$583$	−7108.49			−0.504980
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5276.72			0.371028			0.185514	−	0.982642i	$-0.440605\pi$
$587$	5276.72			0.371028			0.185514	+	0.982642i	$0.440605\pi$
$588$		0			0
$589$	357.181			0.0249871
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5290.53			0.366367			0.183184	−	0.983079i	$-0.441360\pi$
$593$	5290.53			0.366367			0.183184	+	0.983079i	$0.441360\pi$
$594$		0			0
$595$	13487.6	+	17096.8i	0.929308	+	1.17798i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			9894.37i			0.674913i	0.941341	+	0.337457i	$0.109567\pi$
$599$			9894.37i			0.674913i	−0.941341	+	0.337457i	$0.890433\pi$
$600$		0			0
$601$			2662.24i			0.180691i	0.995910	+	0.0903454i	$0.0287971\pi$
$601$			2662.24i			0.180691i	−0.995910	+	0.0903454i	$0.971203\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11830.3			−0.794994
$606$		0			0
$607$			11313.4i			0.756499i	0.925704	+	0.378249i	$0.123474\pi$
$607$			11313.4i			0.756499i	−0.925704	+	0.378249i	$0.876526\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1454.26i		−	0.0962901i
$612$		0			0
$613$	−23984.5			−1.58030			−0.790151	−	0.612912i	$-0.789998\pi$
$613$	−23984.5			−1.58030			−0.790151	+	0.612912i	$0.789998\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			16238.6i			1.05955i	0.848138	+	0.529775i	$0.177724\pi$
$617$			16238.6i			1.05955i	−0.848138	+	0.529775i	$0.822276\pi$
$618$		0			0
$619$		−	17287.8i		−	1.12254i	−0.827632	−	0.561271i	$-0.810313\pi$
$619$		−	17287.8i		−	1.12254i	0.827632	−	0.561271i	$-0.189687\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4883.43	+	6190.21i	0.314046	+	0.398083i
$624$		0			0
$625$	−15090.9			−0.965815
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−41787.2			−2.64891
$630$		0			0
$631$	348.827			0.0220073			0.0110036	−	0.999939i	$-0.496497\pi$
$631$	348.827			0.0220073			0.0110036	+	0.999939i	$0.496497\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27148.2			−1.69660
$636$		0			0
$637$	−29824.3	−	7138.41i	−1.85507	−	0.444009i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	16753.3i		−	1.03232i	−0.856493	−	0.516158i	$-0.827362\pi$
$641$		−	16753.3i		−	1.03232i	0.856493	−	0.516158i	$-0.172638\pi$
$642$		0			0
$643$		−	12309.7i		−	0.754975i	−0.926015	−	0.377487i	$-0.876788\pi$
$643$		−	12309.7i		−	0.754975i	0.926015	−	0.377487i	$-0.123212\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32197.6			1.95644			0.978220	−	0.207570i	$-0.0665555\pi$
$647$	32197.6			1.95644			0.978220	+	0.207570i	$0.0665555\pi$
$648$		0			0
$649$		−	6174.00i		−	0.373422i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			17824.6i			1.06820i	0.845422	+	0.534098i	$0.179349\pi$
$653$			17824.6i			1.06820i	−0.845422	+	0.534098i	$0.820651\pi$
$654$		0			0
$655$	16925.8			1.00969
$656$		0			0
$657$		0			0
$658$		0			0
$659$			20533.9i			1.21379i	0.794782	+	0.606895i	$0.207585\pi$
$659$			20533.9i			1.21379i	−0.794782	+	0.606895i	$0.792415\pi$
$660$		0			0
$661$		−	10184.0i		−	0.599261i	−0.954055	−	0.299630i	$-0.903137\pi$
$661$		−	10184.0i		−	0.599261i	0.954055	−	0.299630i	$-0.0968634\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1329.40	+	1048.75i	−0.0775214	+	0.0611563i
$666$		0			0
$667$	−21428.6			−1.24395
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7758.84			0.446388
$672$		0			0
$673$	−17547.2			−1.00504			−0.502521	−	0.864565i	$-0.667594\pi$
$673$	−17547.2			−1.00504			−0.502521	+	0.864565i	$0.667594\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9665.66			−0.548717			−0.274358	−	0.961627i	$-0.588465\pi$
$677$	−9665.66			−0.548717			−0.274358	+	0.961627i	$0.588465\pi$
$678$		0			0
$679$	−8303.89	+	6550.91i	−0.469329	+	0.370251i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4376.97i			0.245213i	0.992455	+	0.122606i	$0.0391253\pi$
$683$			4376.97i			0.245213i	−0.992455	+	0.122606i	$0.960875\pi$
$684$		0			0
$685$			18023.7i			1.00533i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−39806.8			−2.20104
$690$		0			0
$691$		−	20238.2i		−	1.11418i	−0.830453	−	0.557088i	$-0.811918\pi$
$691$		−	20238.2i		−	1.11418i	0.830453	−	0.557088i	$-0.188082\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	11419.2i		−	0.623243i
$696$		0			0
$697$	36895.9			2.00507
$698$		0			0
$699$		0			0
$700$		0			0
$701$			33130.3i			1.78504i	0.451004	+	0.892522i	$0.351066\pi$
$701$			33130.3i			1.78504i	−0.451004	+	0.892522i	$0.648934\pi$
$702$		0			0
$703$		−	3249.25i		−	0.174321i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8107.22	−	10276.7i	−0.431263	−	0.546667i
$708$		0			0
$709$	−35213.2			−1.86525			−0.932623	−	0.360852i	$-0.882486\pi$
$709$	−35213.2			−1.86525			−0.932623	+	0.360852i	$0.882486\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6144.28			−0.322728
$714$		0			0
$715$	15693.3			0.820831
$716$		0			0
$717$		0			0
$718$		0			0
$719$	21789.0			1.13017			0.565086	−	0.825032i	$-0.308843\pi$
$719$	21789.0			1.13017			0.565086	+	0.825032i	$0.308843\pi$
$720$		0			0
$721$	−3653.55	+	2882.28i	−0.188718	+	0.148879i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	619.560i		−	0.0317378i
$726$		0			0
$727$			791.178i			0.0403620i	0.999796	+	0.0201810i	$0.00642425\pi$
$727$			791.178i			0.0403620i	−0.999796	+	0.0201810i	$0.993576\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6109.00			0.309097
$732$		0			0
$733$		−	225.319i		−	0.0113538i	−0.999984	−	0.00567691i	$-0.998193\pi$
$733$		−	225.319i		−	0.0113538i	0.999984	−	0.00567691i	$-0.00180703\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			8041.08i			0.401896i
$738$		0			0
$739$	−2564.36			−0.127648			−0.0638238	−	0.997961i	$-0.520330\pi$
$739$	−2564.36			−0.127648			−0.0638238	+	0.997961i	$0.520330\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			35125.9i			1.73438i	0.497977	+	0.867190i	$0.334076\pi$
$743$			35125.9i			1.73438i	−0.497977	+	0.867190i	$0.665924\pi$
$744$		0			0
$745$		−	19335.9i		−	0.950891i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10759.3	−	8487.95i	0.524880	−	0.414076i
$750$		0			0
$751$	10427.5			0.506666			0.253333	−	0.967379i	$-0.418473\pi$
$751$	10427.5			0.506666			0.253333	+	0.967379i	$0.418473\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7748.45			0.373503
$756$		0			0
$757$	−14031.2			−0.673677			−0.336838	−	0.941562i	$-0.609358\pi$
$757$	−14031.2			−0.673677			−0.336838	+	0.941562i	$0.609358\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10871.3			−0.517849			−0.258924	−	0.965898i	$-0.583368\pi$
$761$	−10871.3			−0.517849			−0.258924	+	0.965898i	$0.583368\pi$
$762$		0			0
$763$	17176.1	+	21772.3i	0.814963	+	1.03304i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	34573.7i		−	1.62762i
$768$		0			0
$769$		−	20740.7i		−	0.972599i	−0.873792	−	0.486299i	$-0.838346\pi$
$769$		−	20740.7i		−	0.972599i	0.873792	−	0.486299i	$-0.161654\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7445.62			0.346443			0.173221	−	0.984883i	$-0.444582\pi$
$773$	7445.62			0.346443			0.173221	+	0.984883i	$0.444582\pi$
$774$		0			0
$775$		−	177.648i		−	0.00823396i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			2868.91i			0.131950i
$780$		0			0
$781$	11997.0			0.549665
$782$		0			0
$783$		0			0
$784$		0			0
$785$			7350.79i			0.334218i
$786$		0			0
$787$			16648.0i			0.754049i	0.926203	+	0.377024i	$0.123053\pi$
$787$			16648.0i			0.754049i	−0.926203	+	0.377024i	$0.876947\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14471.3	+	11416.4i	−0.650494	+	0.513172i
$792$		0			0
$793$	43448.7			1.94566
$794$		0			0
$795$		0			0
$796$		0			0
$797$	28637.6			1.27277			0.636383	−	0.771373i	$-0.280430\pi$
$797$	28637.6			1.27277			0.636383	+	0.771373i	$0.280430\pi$
$798$		0			0
$799$	1739.66			0.0770272
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8103.20			−0.356109
$804$		0			0
$805$	22868.4	−	18040.8i	1.00125	−	0.789882i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	49.4260i		−	0.00214799i	−0.999999	−	0.00107400i	$-0.999658\pi$
$809$		−	49.4260i		−	0.00214799i	0.999999	−	0.00107400i	$-0.000341863\pi$
$810$		0			0
$811$			37292.7i			1.61470i	0.590072	+	0.807350i	$0.299099\pi$
$811$			37292.7i			1.61470i	−0.590072	+	0.807350i	$0.700901\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	43163.8			1.85517
$816$		0			0
$817$			475.017i			0.0203412i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	807.586i		−	0.0343300i	−0.999853	−	0.0171650i	$-0.994536\pi$
$821$		−	807.586i		−	0.0343300i	0.999853	−	0.0171650i	$-0.00546406\pi$
$822$		0			0
$823$	−20680.5			−0.875913			−0.437957	−	0.898996i	$-0.644298\pi$
$823$	−20680.5			−0.875913			−0.437957	+	0.898996i	$0.644298\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	44485.5i		−	1.87051i	−0.353976	−	0.935255i	$-0.615170\pi$
$827$		−	44485.5i		−	1.87051i	0.353976	−	0.935255i	$-0.384830\pi$
$828$		0			0
$829$		−	7797.19i		−	0.326668i	−0.986571	−	0.163334i	$-0.947775\pi$
$829$		−	7797.19i		−	0.326668i	0.986571	−	0.163334i	$-0.0522248\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8539.30	−	35677.2i	0.355185	−	1.48396i
$834$		0			0
$835$	−21257.6			−0.881019
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29475.4			1.21288			0.606439	−	0.795130i	$-0.292598\pi$
$839$	29475.4			1.21288			0.606439	+	0.795130i	$0.292598\pi$
$840$		0			0
$841$	1952.37			0.0800512
$842$		0			0
$843$		0			0
$844$		0			0
$845$	63727.2			2.59442
$846$		0			0
$847$	12343.6	+	15646.7i	0.500746	+	0.634743i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			55894.0i			2.25149i
$852$		0			0
$853$		−	9768.10i		−	0.392091i	−0.980595	−	0.196045i	$-0.937190\pi$
$853$		−	9768.10i		−	0.392091i	0.980595	−	0.196045i	$-0.0628100\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22812.3			0.909281			0.454641	−	0.890675i	$-0.349768\pi$
$857$	22812.3			0.909281			0.454641	+	0.890675i	$0.349768\pi$
$858$		0			0
$859$			35568.7i			1.41279i	0.707817	+	0.706396i	$0.249680\pi$
$859$			35568.7i			1.41279i	−0.707817	+	0.706396i	$0.750320\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			34594.9i			1.36457i	0.731087	+	0.682285i	$0.239014\pi$
$863$			34594.9i			1.36457i	−0.731087	+	0.682285i	$0.760986\pi$
$864$		0			0
$865$	−18689.9			−0.734653
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	12184.2i		−	0.475628i
$870$		0			0
$871$			45029.2i			1.75173i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	16285.1	+	20642.9i	0.629184	+	0.797550i
$876$		0			0
$877$	−23303.7			−0.897275			−0.448637	−	0.893714i	$-0.648091\pi$
$877$	−23303.7			−0.897275			−0.448637	+	0.893714i	$0.648091\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	8387.34			0.320745			0.160373	−	0.987057i	$-0.448730\pi$
$881$	8387.34			0.320745			0.160373	+	0.987057i	$0.448730\pi$
$882$		0			0
$883$	4287.17			0.163391			0.0816957	−	0.996657i	$-0.473966\pi$
$883$	4287.17			0.163391			0.0816957	+	0.996657i	$0.473966\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13461.3			0.509569			0.254784	−	0.966998i	$-0.417996\pi$
$887$	13461.3			0.509569			0.254784	+	0.966998i	$0.417996\pi$
$888$		0			0
$889$	28326.1	+	35906.0i	1.06865	+	1.35461i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			135.271i			0.00506904i
$894$		0			0
$895$			12919.4i			0.482510i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6433.32			−0.238669
$900$		0			0
$901$		−	47618.7i		−	1.76072i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	38554.1i		−	1.41611i
$906$		0			0
$907$	13884.6			0.508304			0.254152	−	0.967164i	$-0.418204\pi$
$907$	13884.6			0.508304			0.254152	+	0.967164i	$0.418204\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			52493.2i			1.90909i	0.298073	+	0.954543i	$0.403656\pi$
$911$			52493.2i			1.90909i	−0.298073	+	0.954543i	$0.596344\pi$
$912$		0			0
$913$			19390.3i			0.702874i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−17660.1	−	22385.9i	−0.635975	−	0.806158i
$918$		0			0
$919$	29789.5			1.06928			0.534639	−	0.845081i	$-0.320448\pi$
$919$	29789.5			1.06928			0.534639	+	0.845081i	$0.320448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	67182.2			2.39581
$924$		0			0
$925$	−1616.05			−0.0574438
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18314.2			0.646792			0.323396	−	0.946264i	$-0.395175\pi$
$929$	18314.2			0.646792			0.323396	+	0.946264i	$0.395175\pi$
$930$		0			0
$931$	2774.15	+	663.989i	0.0976574	+	0.0233742i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			18773.0i			0.656623i
$936$		0			0
$937$		−	4717.18i		−	0.164465i	−0.996613	−	0.0822325i	$-0.973795\pi$
$937$		−	4717.18i		−	0.164465i	0.996613	−	0.0822325i	$-0.0262050\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−40928.9			−1.41790			−0.708950	−	0.705259i	$-0.750831\pi$
$941$	−40928.9			−1.41790			−0.708950	+	0.705259i	$0.750831\pi$
$942$		0			0
$943$		−	49351.4i		−	1.70424i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			24971.8i			0.856890i	0.903568	+	0.428445i	$0.140938\pi$
$947$			24971.8i			0.856890i	−0.903568	+	0.428445i	$0.859062\pi$
$948$		0			0
$949$	−45377.1			−1.55216
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	14607.3i		−	0.496512i	−0.968694	−	0.248256i	$-0.920143\pi$
$953$		−	14607.3i		−	0.496512i	0.968694	−	0.248256i	$-0.0798574\pi$
$954$		0			0
$955$		−	15590.7i		−	0.528276i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	23838.0	−	18805.7i	0.802680	−	0.633231i
$960$		0			0
$961$	27946.4			0.938080
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16420.8			0.547775
$966$		0			0
$967$	−27656.3			−0.919717			−0.459859	−	0.887992i	$-0.652100\pi$
$967$	−27656.3			−0.919717			−0.459859	+	0.887992i	$0.652100\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5235.20			0.173023			0.0865117	−	0.996251i	$-0.472428\pi$
$971$	5235.20			0.173023			0.0865117	+	0.996251i	$0.472428\pi$
$972$		0			0
$973$	−15102.9	+	11914.6i	−0.497612	+	0.392564i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	4586.77i		−	0.150198i	−0.997176	−	0.0750991i	$-0.976073\pi$
$977$		−	4586.77i		−	0.150198i	0.997176	−	0.0750991i	$-0.0239273\pi$
$978$		0			0
$979$			6797.11i			0.221896i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3934.28			−0.127654			−0.0638270	−	0.997961i	$-0.520331\pi$
$983$	−3934.28			−0.127654			−0.0638270	+	0.997961i	$0.520331\pi$
$984$		0			0
$985$			26213.9i			0.847963i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	8171.31i		−	0.262722i
$990$		0			0
$991$	−17434.1			−0.558841			−0.279421	−	0.960169i	$-0.590142\pi$
$991$	−17434.1			−0.558841			−0.279421	+	0.960169i	$0.590142\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			54531.9i			1.73747i
$996$		0			0
$997$			14149.8i			0.449476i	0.974419	+	0.224738i	$0.0721526\pi$
$997$			14149.8i			0.449476i	−0.974419	+	0.224738i	$0.927847\pi$
$998$		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.1133.12		48
3.2	odd	2	inner	2268.4.f.a.1133.37		48
7.6	odd	2	inner	2268.4.f.a.1133.38		48
9.2	odd	6		756.4.x.a.125.6		48
9.4	even	3		756.4.x.a.629.19		48
9.5	odd	6		252.4.x.a.209.11	yes	48
9.7	even	3		252.4.x.a.41.14	yes	48
21.20	even	2	inner	2268.4.f.a.1133.11		48
63.13	odd	6		756.4.x.a.629.6		48
63.20	even	6		756.4.x.a.125.19		48
63.34	odd	6		252.4.x.a.41.11	✓	48
63.41	even	6		252.4.x.a.209.14	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.11	✓	48	63.34	odd	6
252.4.x.a.41.14	yes	48	9.7	even	3
252.4.x.a.209.11	yes	48	9.5	odd	6
252.4.x.a.209.14	yes	48	63.41	even	6
756.4.x.a.125.6		48	9.2	odd	6
756.4.x.a.125.19		48	63.20	even	6
756.4.x.a.629.6		48	63.13	odd	6
756.4.x.a.629.19		48	9.4	even	3
2268.4.f.a.1133.11		48	21.20	even	2	inner
2268.4.f.a.1133.12		48	1.1	even	1	trivial
2268.4.f.a.1133.37		48	3.2	odd	2	inner
2268.4.f.a.1133.38		48	7.6	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)

\(f(q)\)	\(=\)	\(q-10.9938 q^{5} +(11.4708 + 14.5403i) q^{7} +O(q^{10})\)	\(q-10.9938 q^{5} +(11.4708 + 14.5403i) q^{7} +15.9659i q^{11} +89.4072i q^{13} -106.953 q^{17} -8.31634i q^{19} +143.059i q^{23} -4.13623 q^{25} +149.789i q^{29} +42.9493i q^{31} +(-126.108 - 159.853i) q^{35} +390.706 q^{37} -344.973 q^{41} -57.1185 q^{43} -16.2656 q^{47} +(-79.8415 + 333.578i) q^{49} +445.230i q^{53} -175.526i q^{55} -386.700 q^{59} -485.964i q^{61} -982.925i q^{65} +503.642 q^{67} -751.418i q^{71} +507.533i q^{73} +(-232.149 + 183.141i) q^{77} -763.141 q^{79} +1214.48 q^{83} +1175.82 q^{85} +425.727 q^{89} +(-1300.01 + 1025.57i) q^{91} +91.4282i q^{95} +571.094i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{7}+O(q^{10}) \) 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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