LMFDB - Embedded newform 2736.2.f.g.1025.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1025,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.f (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 37x^{4} + 22x^{2} + 4 $ x^8 + 18x^6 + 37x^4 + 22*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$0.676777i$ of defining polynomial
Character	$\chi$	$=$	2736.1025
Dual form			2736.2.f.g.1025.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60938i q^{5} +0.723074 q^{7} +O(q^{10})$	$q-2.60938i q^{5} +0.723074 q^{7} -5.10683i q^{11} +5.91037i q^{13} -2.27841i q^{17} +(4.22215 - 1.08324i) q^{19} -0.0606600i q^{23} -1.80886 q^{25} -2.46807 q^{29} -8.86012i q^{31} -1.88678i q^{35} +1.03678i q^{37} +10.9982 q^{41} -0.413294 q^{43} -1.94744i q^{47} -6.47716 q^{49} +6.82659 q^{53} -13.3257 q^{55} -5.46623 q^{59} +7.54966 q^{61} +15.4224 q^{65} -12.6040i q^{67} -12.3585 q^{71} -14.3038 q^{73} -3.69262i q^{77} +9.40074i q^{79} -8.01619i q^{83} -5.94523 q^{85} -16.2928 q^{89} +4.27364i q^{91} +(-2.82659 - 11.0172i) q^{95} +0.783264i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 4 q^{19} - 8 q^{25} - 32 q^{29} + 24 q^{41} + 28 q^{43} + 4 q^{49} - 8 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} + 24 q^{65} - 24 q^{71} + 4 q^{73} - 4 q^{85} - 16 q^{89} + 40 q^{95}+O(q^{100}) $ 8 * q - 4 * q^19 - 8 * q^25 - 32 * q^29 + 24 * q^41 + 28 * q^43 + 4 * q^49 - 8 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 + 24 * q^65 - 24 * q^71 + 4 * q^73 - 4 * q^85 - 16 * q^89 + 40 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$		−	2.60938i		−	1.16695i	−0.812131	−	0.583475i	$-0.801693\pi$
$5$		−	2.60938i		−	1.16695i	0.812131	−	0.583475i	$-0.198307\pi$
$6$		0			0
$7$	0.723074			0.273296			0.136648	−	0.990620i	$-0.456367\pi$
$7$	0.723074			0.273296			0.136648	+	0.990620i	$0.456367\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.10683i		−	1.53977i	−0.638184	−	0.769884i	$-0.720314\pi$
$11$		−	5.10683i		−	1.53977i	0.638184	−	0.769884i	$-0.279686\pi$
$12$		0			0
$13$			5.91037i			1.63924i	0.572907	+	0.819621i	$0.305816\pi$
$13$			5.91037i			1.63924i	−0.572907	+	0.819621i	$0.694184\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.27841i		−	0.552595i	−0.961072	−	0.276297i	$-0.910893\pi$
$17$		−	2.27841i		−	0.552595i	0.961072	−	0.276297i	$-0.0891075\pi$
$18$		0			0
$19$	4.22215	−	1.08324i	0.968629	−	0.248513i
$19$	4.22215	−	1.08324i	0.968629	−	0.248513i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.0606600i		−	0.0126485i	−0.999980	−	0.00632424i	$-0.997987\pi$
$23$		−	0.0606600i		−	0.0126485i	0.999980	−	0.00632424i	$-0.00201308\pi$
$24$		0			0
$25$	−1.80886			−0.361772
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.46807			−0.458308			−0.229154	−	0.973390i	$-0.573596\pi$
$29$	−2.46807			−0.458308			−0.229154	+	0.973390i	$0.573596\pi$
$30$		0			0
$31$		−	8.86012i		−	1.59132i	−0.605741	−	0.795662i	$-0.707123\pi$
$31$		−	8.86012i		−	1.59132i	0.605741	−	0.795662i	$-0.292877\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.88678i		−	0.318923i
$36$		0			0
$37$			1.03678i			0.170445i	0.996362	+	0.0852226i	$0.0271601\pi$
$37$			1.03678i			0.170445i	−0.996362	+	0.0852226i	$0.972840\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.9982			1.71762			0.858812	−	0.512291i	$-0.171203\pi$
$41$	10.9982			1.71762			0.858812	+	0.512291i	$0.171203\pi$
$42$		0			0
$43$	−0.413294			−0.0630267			−0.0315134	−	0.999503i	$-0.510033\pi$
$43$	−0.413294			−0.0630267			−0.0315134	+	0.999503i	$0.510033\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	1.94744i		−	0.284063i	−0.989862	−	0.142031i	$-0.954637\pi$
$47$		−	1.94744i		−	0.284063i	0.989862	−	0.142031i	$-0.0453634\pi$
$48$		0			0
$49$	−6.47716			−0.925309
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.82659			0.937704			0.468852	−	0.883277i	$-0.344668\pi$
$53$	6.82659			0.937704			0.468852	+	0.883277i	$0.344668\pi$
$54$		0			0
$55$	−13.3257			−1.79683
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.46623			−0.711642			−0.355821	−	0.934554i	$-0.615799\pi$
$59$	−5.46623			−0.711642			−0.355821	+	0.934554i	$0.615799\pi$
$60$		0			0
$61$	7.54966			0.966635			0.483318	−	0.875445i	$-0.339432\pi$
$61$	7.54966			0.966635			0.483318	+	0.875445i	$0.339432\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	15.4224			1.91291
$66$		0			0
$67$		−	12.6040i		−	1.53982i	−0.638150	−	0.769912i	$-0.720300\pi$
$67$		−	12.6040i		−	1.53982i	0.638150	−	0.769912i	$-0.279700\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.3585			−1.46669			−0.733343	−	0.679858i	$-0.762041\pi$
$71$	−12.3585			−1.46669			−0.733343	+	0.679858i	$0.762041\pi$
$72$		0			0
$73$	−14.3038			−1.67413			−0.837064	−	0.547105i	$-0.815730\pi$
$73$	−14.3038			−1.67413			−0.837064	+	0.547105i	$0.815730\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.69262i		−	0.420813i
$78$		0			0
$79$			9.40074i			1.05767i	0.848726	+	0.528833i	$0.177370\pi$
$79$			9.40074i			1.05767i	−0.848726	+	0.528833i	$0.822630\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	8.01619i		−	0.879891i	−0.898024	−	0.439946i	$-0.854998\pi$
$83$		−	8.01619i		−	0.879891i	0.898024	−	0.439946i	$-0.145002\pi$
$84$		0			0
$85$	−5.94523			−0.644851
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−16.2928			−1.72703			−0.863517	−	0.504319i	$-0.831744\pi$
$89$	−16.2928			−1.72703			−0.863517	+	0.504319i	$0.831744\pi$
$90$		0			0
$91$			4.27364i			0.447999i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.82659	−	11.0172i	−0.290002	−	1.13034i
$96$		0			0
$97$			0.783264i			0.0795284i	0.999209	+	0.0397642i	$0.0126607\pi$
$97$			0.783264i			0.0795284i	−0.999209	+	0.0397642i	$0.987339\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	2.32967i		−	0.231811i	−0.993260	−	0.115905i	$-0.963023\pi$
$101$		−	2.32967i		−	0.231811i	0.993260	−	0.115905i	$-0.0369770\pi$
$102$		0			0
$103$		−	10.7840i		−	1.06257i	−0.847192	−	0.531287i	$-0.821708\pi$
$103$		−	10.7840i		−	1.06257i	0.847192	−	0.531287i	$-0.178292\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.4224			1.10424			0.552122	−	0.833763i	$-0.313818\pi$
$107$	11.4224			1.10424			0.552122	+	0.833763i	$0.313818\pi$
$108$		0			0
$109$			11.5672i			1.10794i	0.832537	+	0.553969i	$0.186888\pi$
$109$			11.5672i			1.10794i	−0.832537	+	0.553969i	$0.813112\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.76272			−0.730255			−0.365128	−	0.930958i	$-0.618975\pi$
$113$	−7.76272			−0.730255			−0.365128	+	0.930958i	$0.618975\pi$
$114$		0			0
$115$	−0.158285			−0.0147601
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	1.64746i		−	0.151022i
$120$		0			0
$121$	−15.0798			−1.37089
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	8.32689i		−	0.744780i
$126$		0			0
$127$		−	11.5672i		−	1.02643i	−0.858262	−	0.513213i	$-0.828455\pi$
$127$		−	11.5672i		−	1.02643i	0.858262	−	0.513213i	$-0.171545\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.39843i			0.733774i	0.930266	+	0.366887i	$0.119576\pi$
$131$			8.39843i			0.733774i	−0.930266	+	0.366887i	$0.880424\pi$
$132$		0			0
$133$	3.05293	−	0.783264i	0.264723	−	0.0679176i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	18.4166i		−	1.57344i	−0.617310	−	0.786720i	$-0.711778\pi$
$137$		−	18.4166i		−	1.57344i	0.617310	−	0.786720i	$-0.288222\pi$
$138$		0			0
$139$	−13.2532			−1.12412			−0.562060	−	0.827097i	$-0.689991\pi$
$139$	−13.2532			−1.12412			−0.562060	+	0.827097i	$0.689991\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	30.1833			2.52405
$144$		0			0
$145$			6.44012i			0.534823i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			14.6289i			1.19845i	0.800582	+	0.599223i	$0.204524\pi$
$149$			14.6289i			1.19845i	−0.800582	+	0.599223i	$0.795476\pi$
$150$		0			0
$151$			6.00330i			0.488541i	0.969707	+	0.244271i	$0.0785485\pi$
$151$			6.00330i			0.488541i	−0.969707	+	0.244271i	$0.921451\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−23.1194			−1.85699
$156$		0			0
$157$	20.3585			1.62479			0.812393	−	0.583110i	$-0.198164\pi$
$157$	20.3585			1.62479			0.812393	+	0.583110i	$0.198164\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.0438617i		−	0.00345678i
$162$		0			0
$163$	2.31657			0.181448			0.0907239	−	0.995876i	$-0.471082\pi$
$163$	2.31657			0.181448			0.0907239	+	0.995876i	$0.471082\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.40236			−0.185900			−0.0929499	−	0.995671i	$-0.529630\pi$
$167$	−2.40236			−0.185900			−0.0929499	+	0.995671i	$0.529630\pi$
$168$		0			0
$169$	−21.9325			−1.68711
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.93429			0.603233			0.301616	−	0.953429i	$-0.402474\pi$
$173$	7.93429			0.603233			0.301616	+	0.953429i	$0.402474\pi$
$174$		0			0
$175$	−1.30794			−0.0988710
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.82475			0.435362			0.217681	−	0.976020i	$-0.430151\pi$
$179$	5.82475			0.435362			0.217681	+	0.976020i	$0.430151\pi$
$180$		0			0
$181$		−	1.12971i		−	0.0839703i	−0.999118	−	0.0419852i	$-0.986632\pi$
$181$		−	1.12971i		−	0.0839703i	0.999118	−	0.0419852i	$-0.0133682\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.70535			0.198901
$186$		0			0
$187$	−11.6354			−0.850868
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	17.1560i		−	1.24137i	−0.784062	−	0.620683i	$-0.786855\pi$
$191$		−	17.1560i		−	1.24137i	0.784062	−	0.620683i	$-0.213145\pi$
$192$		0			0
$193$		−	20.4273i		−	1.47039i	−0.677855	−	0.735196i	$-0.737090\pi$
$193$		−	20.4273i		−	1.47039i	0.677855	−	0.735196i	$-0.262910\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.76065i		−	0.125441i	−0.998031	−	0.0627207i	$-0.980022\pi$
$197$		−	1.76065i		−	0.125441i	0.998031	−	0.0627207i	$-0.0199777\pi$
$198$		0			0
$199$	−4.41329			−0.312850			−0.156425	−	0.987690i	$-0.549997\pi$
$199$	−4.41329			−0.312850			−0.156425	+	0.987690i	$0.549997\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.78459			−0.125254
$204$		0			0
$205$		−	28.6984i		−	2.00438i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.53193	−	21.5618i	−0.382652	−	1.49146i
$210$		0			0
$211$			6.70451i			0.461557i	0.973006	+	0.230779i	$0.0741273\pi$
$211$			6.70451i			0.461557i	−0.973006	+	0.230779i	$0.925873\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			1.07844i			0.0735490i
$216$		0			0
$217$		−	6.40652i		−	0.434903i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.4662			0.905836
$222$		0			0
$223$			1.54381i			0.103381i	0.998663	+	0.0516904i	$0.0164609\pi$
$223$			1.54381i			0.103381i	−0.998663	+	0.0516904i	$0.983539\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	25.6532			1.70266			0.851331	−	0.524630i	$-0.175796\pi$
$227$	25.6532			1.70266			0.851331	+	0.524630i	$0.175796\pi$
$228$		0			0
$229$	−9.59350			−0.633956			−0.316978	−	0.948433i	$-0.602668\pi$
$229$	−9.59350			−0.633956			−0.316978	+	0.948433i	$0.602668\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.12811i		−	0.270442i	−0.990815	−	0.135221i	$-0.956826\pi$
$233$		−	4.12811i		−	0.270442i	0.990815	−	0.135221i	$-0.0431744\pi$
$234$		0			0
$235$	−5.08160			−0.331487
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	9.87330i		−	0.638651i	−0.947645	−	0.319325i	$-0.896544\pi$
$239$		−	9.87330i		−	0.638651i	0.947645	−	0.319325i	$-0.103456\pi$
$240$		0			0
$241$			15.3111i			0.986275i	0.869951	+	0.493138i	$0.164150\pi$
$241$			15.3111i			0.986275i	−0.869951	+	0.493138i	$0.835850\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			16.9014i			1.07979i
$246$		0			0
$247$	6.40236	+	24.9545i	0.407372	+	1.58782i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	13.0611i		−	0.824409i	−0.911091	−	0.412204i	$-0.864759\pi$
$251$		−	13.0611i		−	0.824409i	0.911091	−	0.412204i	$-0.135241\pi$
$252$		0			0
$253$	−0.309780			−0.0194757
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.4005			−1.08541			−0.542707	−	0.839922i	$-0.682601\pi$
$257$	−17.4005			−1.08541			−0.542707	+	0.839922i	$0.682601\pi$
$258$		0			0
$259$			0.749667i			0.0465821i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			14.7502i			0.909537i	0.890610	+	0.454769i	$0.150278\pi$
$263$			14.7502i			0.909537i	−0.890610	+	0.454769i	$0.849722\pi$
$264$		0			0
$265$		−	17.8132i		−	1.09425i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.1259			−1.04418			−0.522092	−	0.852889i	$-0.674848\pi$
$269$	−17.1259			−1.04418			−0.522092	+	0.852889i	$0.674848\pi$
$270$		0			0
$271$	25.6532			1.55832			0.779160	−	0.626825i	$-0.215646\pi$
$271$	25.6532			1.55832			0.779160	+	0.626825i	$0.215646\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			9.23755i			0.557045i
$276$		0			0
$277$	21.5460			1.29457			0.647286	−	0.762247i	$-0.275904\pi$
$277$	21.5460			1.29457			0.647286	+	0.762247i	$0.275904\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	19.7152			1.17611			0.588055	−	0.808821i	$-0.299894\pi$
$281$	19.7152			1.17611			0.588055	+	0.808821i	$0.299894\pi$
$282$		0			0
$283$	−18.5192			−1.10085			−0.550425	−	0.834885i	$-0.685534\pi$
$283$	−18.5192			−1.10085			−0.550425	+	0.834885i	$0.685534\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.95249			0.469420
$288$		0			0
$289$	11.8089			0.694639
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.16973			−0.418860			−0.209430	−	0.977824i	$-0.567161\pi$
$293$	−7.16973			−0.418860			−0.209430	+	0.977824i	$0.567161\pi$
$294$		0			0
$295$			14.2635i			0.830450i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.358523			0.0207339
$300$		0			0
$301$	−0.298842			−0.0172250
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	19.6999i		−	1.12801i
$306$		0			0
$307$			20.9344i			1.19479i	0.801948	+	0.597394i	$0.203797\pi$
$307$			20.9344i			1.19479i	−0.801948	+	0.597394i	$0.796203\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			11.9469i			0.677444i	0.940886	+	0.338722i	$0.109995\pi$
$311$			11.9469i			0.677444i	−0.940886	+	0.338722i	$0.890005\pi$
$312$		0			0
$313$	11.0639			0.625367			0.312683	−	0.949857i	$-0.398772\pi$
$313$	11.0639			0.625367			0.312683	+	0.949857i	$0.398772\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.82659			−0.158757			−0.0793785	−	0.996845i	$-0.525294\pi$
$317$	−2.82659			−0.158757			−0.0793785	+	0.996845i	$0.525294\pi$
$318$		0			0
$319$			12.6040i			0.705689i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.46807	−	9.61979i	−0.137327	−	0.535259i
$324$		0			0
$325$		−	10.6910i		−	0.593032i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	1.40814i		−	0.0776333i
$330$		0			0
$331$		−	21.7512i		−	1.19556i	−0.801662	−	0.597778i	$-0.796051\pi$
$331$		−	21.7512i		−	1.19556i	0.801662	−	0.597778i	$-0.203949\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−32.8886			−1.79690
$336$		0			0
$337$			11.8543i			0.645747i	0.946442	+	0.322873i	$0.104649\pi$
$337$			11.8543i			0.645747i	−0.946442	+	0.322873i	$0.895351\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−45.2471			−2.45027
$342$		0			0
$343$	−9.74499			−0.526180
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	0.0344660i		−	0.00185023i	−1.00000		0.000925117i	$-0.999706\pi$
$347$		−	0.0344660i		−	0.00185023i	1.00000		0.000925117i	$-0.000294474\pi$
$348$		0			0
$349$	26.5374			1.42051			0.710256	−	0.703943i	$-0.248579\pi$
$349$	26.5374			1.42051			0.710256	+	0.703943i	$0.248579\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	24.8130i		−	1.32067i	−0.750973	−	0.660333i	$-0.770415\pi$
$353$		−	24.8130i		−	1.32067i	0.750973	−	0.660333i	$-0.229585\pi$
$354$		0			0
$355$			32.2481i			1.71155i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.40421i			0.285223i	0.989779	+	0.142612i	$0.0455500\pi$
$359$			5.40421i			0.285223i	−0.989779	+	0.142612i	$0.954450\pi$
$360$		0			0
$361$	16.6532	−	9.14723i	0.876483	−	0.481433i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			37.3239i			1.95362i
$366$		0			0
$367$	28.5856			1.49216			0.746079	−	0.665858i	$-0.231934\pi$
$367$	28.5856			1.49216			0.746079	+	0.665858i	$0.231934\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.93613			0.256271
$372$		0			0
$373$		−	1.54381i		−	0.0799352i	−0.999201	−	0.0399676i	$-0.987275\pi$
$373$		−	1.54381i		−	0.0799352i	0.999201	−	0.0399676i	$-0.0127255\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.5872i		−	0.751278i
$378$		0			0
$379$			5.14983i			0.264529i	0.991214	+	0.132264i	$0.0422248\pi$
$379$			5.14983i			0.264529i	−0.991214	+	0.132264i	$0.957775\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.2508			0.779280			0.389640	−	0.920967i	$-0.372599\pi$
$383$	15.2508			0.779280			0.389640	+	0.920967i	$0.372599\pi$
$384$		0			0
$385$	−9.63545			−0.491068
$386$		0			0
$387$		0			0
$388$		0			0
$389$			29.6420i			1.50291i	0.659784	+	0.751455i	$0.270648\pi$
$389$			29.6420i			1.50291i	−0.659784	+	0.751455i	$0.729352\pi$
$390$		0			0
$391$	−0.138208			−0.00698948
$392$		0			0
$393$		0			0
$394$		0			0
$395$	24.5301			1.23424
$396$		0			0
$397$	−21.0026			−1.05409			−0.527045	−	0.849837i	$-0.676700\pi$
$397$	−21.0026			−1.05409			−0.527045	+	0.849837i	$0.676700\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.9982			0.748972			0.374486	−	0.927232i	$-0.377819\pi$
$401$	14.9982			0.748972			0.374486	+	0.927232i	$0.377819\pi$
$402$		0			0
$403$	52.3665			2.60856
$404$		0			0
$405$		0			0
$406$		0			0
$407$	5.29465			0.262446
$408$		0			0
$409$		−	0.0929271i		−	0.00459495i	−0.999997	−	0.00229747i	$-0.999269\pi$
$409$		−	0.0929271i		−	0.00459495i	0.999997	−	0.00229747i	$-0.000731310\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.95249			−0.194489
$414$		0			0
$415$	−20.9173			−1.02679
$416$		0			0
$417$		0			0
$418$		0			0
$419$			27.6887i			1.35268i	0.736590	+	0.676340i	$0.236435\pi$
$419$			27.6887i			1.35268i	−0.736590	+	0.676340i	$0.763565\pi$
$420$		0			0
$421$			11.8692i			0.578469i	0.957258	+	0.289235i	$0.0934007\pi$
$421$			11.8692i			0.578469i	−0.957258	+	0.289235i	$0.906599\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			4.12132i			0.199913i
$426$		0			0
$427$	5.45897			0.264178
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.5455			0.796967			0.398484	−	0.917175i	$-0.369537\pi$
$431$	16.5455			0.796967			0.398484	+	0.917175i	$0.369537\pi$
$432$		0			0
$433$		−	37.3979i		−	1.79723i	−0.438740	−	0.898614i	$-0.644575\pi$
$433$		−	37.3979i		−	1.79723i	0.438740	−	0.898614i	$-0.355425\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.0657094	−	0.256116i	−0.00314331	−	0.0122517i
$438$		0			0
$439$		−	22.5454i		−	1.07603i	−0.842935	−	0.538016i	$-0.819174\pi$
$439$		−	22.5454i		−	1.07603i	0.842935	−	0.538016i	$-0.180826\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5.86503i		−	0.278656i	−0.990246	−	0.139328i	$-0.955506\pi$
$443$		−	5.86503i		−	0.278656i	0.990246	−	0.139328i	$-0.0444942\pi$
$444$		0			0
$445$			42.5141i			2.01536i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	13.3604			0.630514			0.315257	−	0.949006i	$-0.397909\pi$
$449$	13.3604			0.630514			0.315257	+	0.949006i	$0.397909\pi$
$450$		0			0
$451$		−	56.1658i		−	2.64474i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	11.1515			0.522792
$456$		0			0
$457$	4.65058			0.217545			0.108772	−	0.994067i	$-0.465308\pi$
$457$	4.65058			0.217545			0.108772	+	0.994067i	$0.465308\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			17.3953i			0.810181i	0.914277	+	0.405091i	$0.132760\pi$
$461$			17.3953i			0.810181i	−0.914277	+	0.405091i	$0.867240\pi$
$462$		0			0
$463$	−30.8931			−1.43572			−0.717861	−	0.696186i	$-0.754879\pi$
$463$	−30.8931			−1.43572			−0.717861	+	0.696186i	$0.754879\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	30.8891i		−	1.42938i	−0.699444	−	0.714688i	$-0.746569\pi$
$467$		−	30.8891i		−	1.42938i	0.699444	−	0.714688i	$-0.253431\pi$
$468$		0			0
$469$		−	9.11363i		−	0.420828i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2.11062i			0.0970466i
$474$		0			0
$475$	−7.63729	+	1.95943i	−0.350423	+	0.0899049i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			23.1951i			1.05981i	0.848057	+	0.529906i	$0.177773\pi$
$479$			23.1951i			1.05981i	−0.848057	+	0.529906i	$0.822227\pi$
$480$		0			0
$481$	−6.12774			−0.279401
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.04383			0.0928057
$486$		0			0
$487$			6.41740i			0.290800i	0.989373	+	0.145400i	$0.0464469\pi$
$487$			6.41740i			0.290800i	−0.989373	+	0.145400i	$0.953553\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			35.8720i			1.61888i	0.587203	+	0.809440i	$0.300229\pi$
$491$			35.8720i			1.61888i	−0.587203	+	0.809440i	$0.699771\pi$
$492$		0			0
$493$			5.62326i			0.253259i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.93613			−0.400840
$498$		0			0
$499$	17.5916			0.787509			0.393754	−	0.919216i	$-0.371176\pi$
$499$	17.5916			0.787509			0.393754	+	0.919216i	$0.371176\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			42.3605i			1.88876i	0.328852	+	0.944382i	$0.393338\pi$
$503$			42.3605i			1.88876i	−0.328852	+	0.944382i	$0.606662\pi$
$504$		0			0
$505$	−6.07900			−0.270512
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−23.4881			−1.04109			−0.520546	−	0.853834i	$-0.674272\pi$
$509$	−23.4881			−1.04109			−0.520546	+	0.853834i	$0.674272\pi$
$510$		0			0
$511$	−10.3427			−0.457533
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28.1394			−1.23997
$516$		0			0
$517$	−9.94523			−0.437391
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.0774			−0.441499			−0.220750	−	0.975331i	$-0.570850\pi$
$521$	−10.0774			−0.441499			−0.220750	+	0.975331i	$0.570850\pi$
$522$		0			0
$523$			30.9578i			1.35369i	0.736126	+	0.676845i	$0.236653\pi$
$523$			30.9578i			1.35369i	−0.736126	+	0.676845i	$0.763347\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.1870			−0.879357
$528$		0			0
$529$	22.9963			0.999840
$530$		0			0
$531$		0			0
$532$		0			0
$533$			65.0032i			2.81560i
$534$		0			0
$535$		−	29.8054i		−	1.28860i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			33.0778i			1.42476i
$540$		0			0
$541$	7.34942			0.315976			0.157988	−	0.987441i	$-0.449499\pi$
$541$	7.34942			0.315976			0.157988	+	0.987441i	$0.449499\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	30.1833			1.29291
$546$		0			0
$547$			29.6675i			1.26849i	0.773132	+	0.634245i	$0.218689\pi$
$547$			29.6675i			1.26849i	−0.773132	+	0.634245i	$0.781311\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.4206	+	2.67351i	−0.443930	+	0.113895i
$552$		0			0
$553$			6.79743i			0.289056i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			11.2849i			0.478157i	0.971000	+	0.239079i	$0.0768453\pi$
$557$			11.2849i			0.478157i	−0.971000	+	0.239079i	$0.923155\pi$
$558$		0			0
$559$		−	2.44272i		−	0.103316i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−39.5940			−1.66869			−0.834343	−	0.551245i	$-0.814153\pi$
$563$	−39.5940			−1.66869			−0.834343	+	0.551245i	$0.814153\pi$
$564$		0			0
$565$			20.2559i			0.852171i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.18511			0.133527			0.0667634	−	0.997769i	$-0.478733\pi$
$569$	3.18511			0.133527			0.0667634	+	0.997769i	$0.478733\pi$
$570$		0			0
$571$	−24.5856			−1.02888			−0.514438	−	0.857527i	$-0.672000\pi$
$571$	−24.5856			−1.02888			−0.514438	+	0.857527i	$0.672000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			0.109725i			0.00457587i
$576$		0			0
$577$	−23.5697			−0.981221			−0.490611	−	0.871379i	$-0.663226\pi$
$577$	−23.5697			−0.981221			−0.490611	+	0.871379i	$0.663226\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	5.79630i		−	0.240471i
$582$		0			0
$583$		−	34.8623i		−	1.44385i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	23.9676i		−	0.989251i	−0.869106	−	0.494625i	$-0.835305\pi$
$587$		−	23.9676i		−	0.989251i	0.869106	−	0.494625i	$-0.164695\pi$
$588$		0			0
$589$	−9.59764	−	37.4088i	−0.395464	−	1.54140i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			2.67611i			0.109895i	0.998489	+	0.0549474i	$0.0174991\pi$
$593$			2.67611i			0.109895i	−0.998489	+	0.0549474i	$0.982501\pi$
$594$		0			0
$595$	−4.29884			−0.176235
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−28.3110			−1.15676			−0.578378	−	0.815769i	$-0.696314\pi$
$599$	−28.3110			−1.15676			−0.578378	+	0.815769i	$0.696314\pi$
$600$		0			0
$601$			30.1745i			1.23084i	0.788198	+	0.615422i	$0.211014\pi$
$601$			30.1745i			1.23084i	−0.788198	+	0.615422i	$0.788986\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			39.3488i			1.59976i
$606$		0			0
$607$			31.9352i			1.29621i	0.761551	+	0.648105i	$0.224438\pi$
$607$			31.9352i			1.29621i	−0.761551	+	0.648105i	$0.775562\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.5101			0.465647
$612$		0			0
$613$	41.8844			1.69170			0.845848	−	0.533424i	$-0.179095\pi$
$613$	41.8844			1.69170			0.845848	+	0.533424i	$0.179095\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	5.71971i		−	0.230267i	−0.993350	−	0.115133i	$-0.963270\pi$
$617$		−	5.71971i		−	0.230267i	0.993350	−	0.115133i	$-0.0367296\pi$
$618$		0			0
$619$	22.1376			0.889787			0.444893	−	0.895584i	$-0.353242\pi$
$619$	22.1376			0.889787			0.444893	+	0.895584i	$0.353242\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.7809			−0.471992
$624$		0			0
$625$	−30.7723			−1.23089
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.36220			0.0941872
$630$		0			0
$631$	1.02607			0.0408470			0.0204235	−	0.999791i	$-0.493499\pi$
$631$	1.02607			0.0408470			0.0204235	+	0.999791i	$0.493499\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−30.1833			−1.19779
$636$		0			0
$637$		−	38.2824i		−	1.51680i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	40.0737			1.58282			0.791409	−	0.611288i	$-0.209348\pi$
$641$	40.0737			1.58282			0.791409	+	0.611288i	$0.209348\pi$
$642$		0			0
$643$	13.7144			0.540845			0.270422	−	0.962742i	$-0.412837\pi$
$643$	13.7144			0.540845			0.270422	+	0.962742i	$0.412837\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			8.25771i			0.324644i	0.986738	+	0.162322i	$0.0518983\pi$
$647$			8.25771i			0.324644i	−0.986738	+	0.162322i	$0.948102\pi$
$648$		0			0
$649$			27.9151i			1.09576i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.23831i		−	0.0484587i	−0.999706	−	0.0242293i	$-0.992287\pi$
$653$		−	1.23831i		−	0.0484587i	0.999706	−	0.0242293i	$-0.00771320\pi$
$654$		0			0
$655$	21.9147			0.856278
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−42.8849			−1.67056			−0.835280	−	0.549825i	$-0.814694\pi$
$659$	−42.8849			−1.67056			−0.835280	+	0.549825i	$0.814694\pi$
$660$		0			0
$661$			50.6359i			1.96951i	0.173950	+	0.984754i	$0.444347\pi$
$661$			50.6359i			1.96951i	−0.173950	+	0.984754i	$0.555653\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.04383	−	7.96626i	−0.0792564	−	0.308918i
$666$		0			0
$667$			0.149713i			0.00579690i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	38.5549i		−	1.48839i
$672$		0			0
$673$			41.5450i			1.60144i	0.599037	+	0.800721i	$0.295550\pi$
$673$			41.5450i			1.60144i	−0.599037	+	0.800721i	$0.704450\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	34.8266			1.33849			0.669247	−	0.743040i	$-0.266617\pi$
$677$	34.8266			1.33849			0.669247	+	0.743040i	$0.266617\pi$
$678$		0			0
$679$			0.566358i			0.0217348i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	42.8448			1.63941			0.819705	−	0.572786i	$-0.194137\pi$
$683$	42.8448			1.63941			0.819705	+	0.572786i	$0.194137\pi$
$684$		0			0
$685$	−48.0560			−1.83613
$686$		0			0
$687$		0			0
$688$		0			0
$689$			40.3476i			1.53712i
$690$		0			0
$691$	32.4278			1.23361			0.616806	−	0.787115i	$-0.288426\pi$
$691$	32.4278			1.23361			0.616806	+	0.787115i	$0.288426\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			34.5825i			1.31179i
$696$		0			0
$697$		−	25.0583i		−	0.949150i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	14.3983i		−	0.543814i	−0.962323	−	0.271907i	$-0.912346\pi$
$701$		−	14.3983i		−	0.543814i	0.962323	−	0.271907i	$-0.0876544\pi$
$702$		0			0
$703$	1.12308	+	4.37744i	0.0423578	+	0.165098i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1.68453i		−	0.0633531i
$708$		0			0
$709$	47.6458			1.78938			0.894688	−	0.446691i	$-0.147398\pi$
$709$	47.6458			1.78938			0.894688	+	0.446691i	$0.147398\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.537454			−0.0201278
$714$		0			0
$715$		−	78.7596i		−	2.94544i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	14.7314i		−	0.549389i	−0.961532	−	0.274695i	$-0.911423\pi$
$719$		−	14.7314i		−	0.549389i	0.961532	−	0.274695i	$-0.0885768\pi$
$720$		0			0
$721$		−	7.79760i		−	0.290398i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.46439			0.165803
$726$		0			0
$727$	24.1723			0.896502			0.448251	−	0.893908i	$-0.352047\pi$
$727$	24.1723			0.896502			0.448251	+	0.893908i	$0.352047\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			0.941652i			0.0348283i
$732$		0			0
$733$	36.7134			1.35604			0.678020	−	0.735044i	$-0.262838\pi$
$733$	36.7134			1.35604			0.678020	+	0.735044i	$0.262838\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−64.3665			−2.37097
$738$		0			0
$739$	15.0425			0.553346			0.276673	−	0.960964i	$-0.410768\pi$
$739$	15.0425			0.553346			0.276673	+	0.960964i	$0.410768\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.4709			−0.384139			−0.192070	−	0.981381i	$-0.561520\pi$
$743$	−10.4709			−0.384139			−0.192070	+	0.981381i	$0.561520\pi$
$744$		0			0
$745$	38.1723			1.39853
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.25924			0.301786
$750$		0			0
$751$		−	24.9209i		−	0.909376i	−0.890651	−	0.454688i	$-0.849751\pi$
$751$		−	24.9209i		−	0.909376i	0.890651	−	0.454688i	$-0.150249\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.6649			0.570103
$756$		0			0
$757$	6.66104			0.242100			0.121050	−	0.992646i	$-0.461374\pi$
$757$	6.66104			0.242100			0.121050	+	0.992646i	$0.461374\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	38.3148i		−	1.38891i	−0.719535	−	0.694456i	$-0.755645\pi$
$761$		−	38.3148i		−	1.38891i	0.719535	−	0.694456i	$-0.244355\pi$
$762$		0			0
$763$			8.36396i			0.302796i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	32.3074i		−	1.16655i
$768$		0			0
$769$	20.6373			0.744200			0.372100	−	0.928193i	$-0.378638\pi$
$769$	20.6373			0.744200			0.372100	+	0.928193i	$0.378638\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.83493			−0.0659977			−0.0329988	−	0.999455i	$-0.510506\pi$
$773$	−1.83493			−0.0659977			−0.0329988	+	0.999455i	$0.510506\pi$
$774$		0			0
$775$			16.0267i			0.575696i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	46.4359	−	11.9137i	1.66374	−	0.426851i
$780$		0			0
$781$			63.1129i			2.25836i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	53.1231i		−	1.89605i
$786$		0			0
$787$		−	12.9732i		−	0.462443i	−0.972901	−	0.231222i	$-0.925728\pi$
$787$		−	12.9732i		−	0.462443i	0.972901	−	0.231222i	$-0.0742723\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5.61302			−0.199576
$792$		0			0
$793$			44.6213i			1.58455i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	23.7306			0.840580			0.420290	−	0.907390i	$-0.361928\pi$
$797$	23.7306			0.840580			0.420290	+	0.907390i	$0.361928\pi$
$798$		0			0
$799$	−4.43705			−0.156972
$800$		0			0
$801$		0			0
$802$		0			0
$803$			73.0469i			2.57777i
$804$		0			0
$805$	−0.114452			−0.00403389
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	20.1252i		−	0.707563i	−0.935328	−	0.353781i	$-0.884896\pi$
$809$		−	20.1252i		−	0.707563i	0.935328	−	0.353781i	$-0.115104\pi$
$810$		0			0
$811$		−	26.3268i		−	0.924460i	−0.886760	−	0.462230i	$-0.847049\pi$
$811$		−	26.3268i		−	0.924460i	0.886760	−	0.462230i	$-0.152951\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	6.04481i		−	0.211740i
$816$		0			0
$817$	−1.74499	+	0.447697i	−0.0610495	+	0.0156629i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	2.43567i		−	0.0850055i	−0.999096	−	0.0425028i	$-0.986467\pi$
$821$		−	2.43567i		−	0.0850055i	0.999096	−	0.0425028i	$-0.0135331\pi$
$822$		0			0
$823$	41.4803			1.44591			0.722956	−	0.690894i	$-0.242783\pi$
$823$	41.4803			1.44591			0.722956	+	0.690894i	$0.242783\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−27.4779			−0.955501			−0.477751	−	0.878496i	$-0.658548\pi$
$827$	−27.4779			−0.955501			−0.477751	+	0.878496i	$0.658548\pi$
$828$		0			0
$829$			53.5625i			1.86030i	0.367178	+	0.930151i	$0.380324\pi$
$829$			53.5625i			1.86030i	−0.367178	+	0.930151i	$0.619676\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			14.7576i			0.511321i
$834$		0			0
$835$			6.26866i			0.216936i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	50.4104			1.74036			0.870180	−	0.492734i	$-0.164003\pi$
$839$	50.4104			1.74036			0.870180	+	0.492734i	$0.164003\pi$
$840$		0			0
$841$	−22.9087			−0.789954
$842$		0			0
$843$		0			0
$844$		0			0
$845$			57.2301i			1.96877i
$846$		0			0
$847$	−10.9038			−0.374658
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.0628909			0.00215587
$852$		0			0
$853$	36.4943			1.24954			0.624770	−	0.780809i	$-0.285193\pi$
$853$	36.4943			1.24954			0.624770	+	0.780809i	$0.285193\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.50822			−0.222317			−0.111158	−	0.993803i	$-0.535456\pi$
$857$	−6.50822			−0.222317			−0.111158	+	0.993803i	$0.535456\pi$
$858$		0			0
$859$	−23.7194			−0.809294			−0.404647	−	0.914473i	$-0.632606\pi$
$859$	−23.7194			−0.809294			−0.404647	+	0.914473i	$0.632606\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27.6970			−0.942817			−0.471409	−	0.881915i	$-0.656254\pi$
$863$	−27.6970			−0.942817			−0.471409	+	0.881915i	$0.656254\pi$
$864$		0			0
$865$		−	20.7036i		−	0.703943i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	48.0080			1.62856
$870$		0			0
$871$	74.4943			2.52414
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	6.02096i		−	0.203546i
$876$		0			0
$877$		−	37.1701i		−	1.25515i	−0.778558	−	0.627573i	$-0.784048\pi$
$877$		−	37.1701i		−	1.25515i	0.778558	−	0.627573i	$-0.215952\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			54.4435i			1.83425i	0.398603	+	0.917124i	$0.369495\pi$
$881$			54.4435i			1.83425i	−0.398603	+	0.917124i	$0.630505\pi$
$882$		0			0
$883$	33.2886			1.12025			0.560126	−	0.828408i	$-0.310753\pi$
$883$	33.2886			1.12025			0.560126	+	0.828408i	$0.310753\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	37.0281			1.24328			0.621640	−	0.783303i	$-0.286467\pi$
$887$	37.0281			1.24328			0.621640	+	0.783303i	$0.286467\pi$
$888$		0			0
$889$		−	8.36396i		−	0.280518i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.10954	−	8.22237i	−0.0705931	−	0.275151i
$894$		0			0
$895$		−	15.1990i		−	0.508046i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			21.8673i			0.729317i
$900$		0			0
$901$		−	15.5537i		−	0.518170i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.94783			−0.0979891
$906$		0			0
$907$		−	21.1538i		−	0.702401i	−0.936300	−	0.351200i	$-0.885774\pi$
$907$		−	21.1538i		−	0.702401i	0.936300	−	0.351200i	$-0.114226\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−42.7689			−1.41700			−0.708498	−	0.705712i	$-0.750627\pi$
$911$	−42.7689			−1.41700			−0.708498	+	0.705712i	$0.750627\pi$
$912$		0			0
$913$	−40.9374			−1.35483
$914$		0			0
$915$		0			0
$916$		0			0
$917$			6.07269i			0.200538i
$918$		0			0
$919$	−9.36506			−0.308925			−0.154462	−	0.987999i	$-0.549365\pi$
$919$	−9.36506			−0.308925			−0.154462	+	0.987999i	$0.549365\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	73.0434i		−	2.40425i
$924$		0			0
$925$		−	1.87539i		−	0.0616623i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	10.7164i		−	0.351592i	−0.984427	−	0.175796i	$-0.943750\pi$
$929$		−	10.7164i		−	0.351592i	0.984427	−	0.175796i	$-0.0562499\pi$
$930$		0			0
$931$	−27.3476	+	7.01633i	−0.896281	+	0.229951i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			30.3613i			0.992921i
$936$		0			0
$937$	18.3256			0.598672			0.299336	−	0.954148i	$-0.403235\pi$
$937$	18.3256			0.598672			0.299336	+	0.954148i	$0.403235\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.9865			−0.423347			−0.211673	−	0.977340i	$-0.567891\pi$
$941$	−12.9865			−0.423347			−0.211673	+	0.977340i	$0.567891\pi$
$942$		0			0
$943$		−	0.667148i		−	0.0217253i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.34528i			0.0437158i	0.999761	+	0.0218579i	$0.00695814\pi$
$947$			1.34528i			0.0437158i	−0.999761	+	0.0218579i	$0.993042\pi$
$948$		0			0
$949$		−	84.5404i		−	2.74430i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.85030			0.319083			0.159541	−	0.987191i	$-0.448998\pi$
$953$	9.85030			0.319083			0.159541	+	0.987191i	$0.448998\pi$
$954$		0			0
$955$	−44.7665			−1.44861
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	13.3166i		−	0.430015i
$960$		0			0
$961$	−47.5016			−1.53231
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−53.3027			−1.71587
$966$		0			0
$967$	−31.6951			−1.01925			−0.509623	−	0.860398i	$-0.670215\pi$
$967$	−31.6951			−1.01925			−0.509623	+	0.860398i	$0.670215\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.6769			−0.920285			−0.460143	−	0.887845i	$-0.652202\pi$
$971$	−28.6769			−0.920285			−0.460143	+	0.887845i	$0.652202\pi$
$972$		0			0
$973$	−9.58303			−0.307218
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.8905			0.828309			0.414155	−	0.910207i	$-0.364077\pi$
$977$	25.8905			0.828309			0.414155	+	0.910207i	$0.364077\pi$
$978$		0			0
$979$			83.2047i			2.65923i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−58.8849			−1.87814			−0.939069	−	0.343729i	$-0.888310\pi$
$983$	−58.8849			−1.87814			−0.939069	+	0.343729i	$0.888310\pi$
$984$		0			0
$985$	−4.59422			−0.146384
$986$		0			0
$987$		0			0
$988$		0			0
$989$			0.0250704i			0.000797192i
$990$		0			0
$991$		−	22.5118i		−	0.715110i	−0.933892	−	0.357555i	$-0.883610\pi$
$991$		−	22.5118i		−	0.715110i	0.933892	−	0.357555i	$-0.116390\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			11.5160i			0.365080i
$996$		0			0
$997$	23.8928			0.756693			0.378346	−	0.925664i	$-0.376493\pi$
$997$	23.8928			0.756693			0.378346	+	0.925664i	$0.376493\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.f.g.1025.2		8
3.2	odd	2		2736.2.f.h.1025.7		8
4.3	odd	2		1368.2.f.c.1025.2	✓	8
12.11	even	2		1368.2.f.d.1025.7	yes	8
19.18	odd	2		2736.2.f.h.1025.2		8
57.56	even	2	inner	2736.2.f.g.1025.7		8
76.75	even	2		1368.2.f.d.1025.2	yes	8
228.227	odd	2		1368.2.f.c.1025.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.f.c.1025.2	✓	8	4.3	odd	2
1368.2.f.c.1025.7	yes	8	228.227	odd	2
1368.2.f.d.1025.2	yes	8	76.75	even	2
1368.2.f.d.1025.7	yes	8	12.11	even	2
2736.2.f.g.1025.2		8	1.1	even	1	trivial
2736.2.f.g.1025.7		8	57.56	even	2	inner
2736.2.f.h.1025.2		8	19.18	odd	2
2736.2.f.h.1025.7		8	3.2	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.60938i q^{5} +0.723074 q^{7} +O(q^{10})\)	\(q-2.60938i q^{5} +0.723074 q^{7} -5.10683i q^{11} +5.91037i q^{13} -2.27841i q^{17} +(4.22215 - 1.08324i) q^{19} -0.0606600i q^{23} -1.80886 q^{25} -2.46807 q^{29} -8.86012i q^{31} -1.88678i q^{35} +1.03678i q^{37} +10.9982 q^{41} -0.413294 q^{43} -1.94744i q^{47} -6.47716 q^{49} +6.82659 q^{53} -13.3257 q^{55} -5.46623 q^{59} +7.54966 q^{61} +15.4224 q^{65} -12.6040i q^{67} -12.3585 q^{71} -14.3038 q^{73} -3.69262i q^{77} +9.40074i q^{79} -8.01619i q^{83} -5.94523 q^{85} -16.2928 q^{89} +4.27364i q^{91} +(-2.82659 - 11.0172i) q^{95} +0.783264i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 4 q^{19} - 8 q^{25} - 32 q^{29} + 24 q^{41} + 28 q^{43} + 4 q^{49} - 8 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} + 24 q^{65} - 24 q^{71} + 4 q^{73} - 4 q^{85} - 16 q^{89} + 40 q^{95}+O(q^{100}) \) 8 * q - 4 * q^19 - 8 * q^25 - 32 * q^29 + 24 * q^41 + 28 * q^43 + 4 * q^49 - 8 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 + 24 * q^65 - 24 * q^71 + 4 * q^73 - 4 * q^85 - 16 * q^89 + 40 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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