LMFDB - Embedded newform 6336.2.b.x.2177.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(2177,6336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6336.2177"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,4,0,0,0,0,0,0,0,-26,0,0,0,8, 0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.12781568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 17x^{2} + 8 $ x^6 + 8x^4 + 17x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2177.5
Root			$0.810603i$ of defining polynomial
Character	$\chi$	$=$	6336.2177
Dual form			6336.2.b.x.2177.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.52039i q^{5} +4.72761i q^{7} +(0.146365 - 3.31339i) q^{11} -2.10617i q^{13} +4.68585 q^{17} +0.206992i q^{19} +4.52062i q^{23} -7.39312 q^{25} -7.66442 q^{29} +2.97858 q^{31} -16.6430 q^{35} -6.97858 q^{37} -5.27131 q^{41} +5.86385i q^{43} -7.76303i q^{47} -15.3503 q^{49} +6.34881i q^{53} +(11.6644 + 0.515263i) q^{55} -2.41444i q^{59} -14.3898i q^{61} +7.41454 q^{65} -13.3717 q^{67} +2.10617i q^{71} +13.1116i q^{73} +(15.6644 + 0.691959i) q^{77} +12.7383i q^{79} +10.3931 q^{83} +16.4960i q^{85} +4.24264i q^{89} +9.95715 q^{91} -0.728692 q^{95} -10.3503 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{11} + 4 q^{17} - 26 q^{25} + 8 q^{29} - 12 q^{31} - 16 q^{35} - 12 q^{37} + 4 q^{41} - 14 q^{49} + 16 q^{55} + 56 q^{65} - 32 q^{67} + 40 q^{77} + 44 q^{83} - 40 q^{95} + 16 q^{97}+O(q^{100}) $ 6 * q - 2 * q^11 + 4 * q^17 - 26 * q^25 + 8 * q^29 - 12 * q^31 - 16 * q^35 - 12 * q^37 + 4 * q^41 - 14 * q^49 + 16 * q^55 + 56 * q^65 - 32 * q^67 + 40 * q^77 + 44 * q^83 - 40 * q^95 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times$.

$n$	$1729$	$3521$	$4159$	$4357$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$			3.52039i			1.57436i	0.616721	+	0.787182i	$0.288461\pi$
$5$			3.52039i			1.57436i	−0.616721	+	0.787182i	$0.711539\pi$
$6$		0			0
$7$			4.72761i			1.78687i	0.449195	+	0.893434i	$0.351711\pi$
$7$			4.72761i			1.78687i	−0.449195	+	0.893434i	$0.648289\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.146365	−	3.31339i	0.0441309	−	0.999026i
$11$	0.146365	−	3.31339i	0.0441309	−	0.999026i
$12$		0			0
$13$		−	2.10617i		−	0.584147i	−0.956396	−	0.292074i	$-0.905655\pi$
$13$		−	2.10617i		−	0.584147i	0.956396	−	0.292074i	$-0.0943452\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.68585			1.13648			0.568242	−	0.822861i	$-0.307624\pi$
$17$	4.68585			1.13648			0.568242	+	0.822861i	$0.307624\pi$
$18$		0			0
$19$			0.206992i			0.0474872i	0.999718	+	0.0237436i	$0.00755854\pi$
$19$			0.206992i			0.0474872i	−0.999718	+	0.0237436i	$0.992441\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.52062i			0.942613i	0.881969	+	0.471307i	$0.156218\pi$
$23$			4.52062i			0.942613i	−0.881969	+	0.471307i	$0.843782\pi$
$24$		0			0
$25$	−7.39312			−1.47862
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.66442			−1.42325			−0.711624	−	0.702561i	$-0.752040\pi$
$29$	−7.66442			−1.42325			−0.711624	+	0.702561i	$0.752040\pi$
$30$		0			0
$31$	2.97858			0.534968			0.267484	−	0.963562i	$-0.413808\pi$
$31$	2.97858			0.534968			0.267484	+	0.963562i	$0.413808\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−16.6430			−2.81318
$36$		0			0
$37$	−6.97858			−1.14727			−0.573636	−	0.819111i	$-0.694467\pi$
$37$	−6.97858			−1.14727			−0.573636	+	0.819111i	$0.694467\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.27131			−0.823240			−0.411620	−	0.911356i	$-0.635037\pi$
$41$	−5.27131			−0.823240			−0.411620	+	0.911356i	$0.635037\pi$
$42$		0			0
$43$			5.86385i			0.894228i	0.894477	+	0.447114i	$0.147548\pi$
$43$			5.86385i			0.894228i	−0.894477	+	0.447114i	$0.852452\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.76303i		−	1.13235i	−0.824284	−	0.566177i	$-0.808422\pi$
$47$		−	7.76303i		−	1.13235i	0.824284	−	0.566177i	$-0.191578\pi$
$48$		0			0
$49$	−15.3503			−2.19290
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.34881i			0.872077i	0.899928	+	0.436038i	$0.143619\pi$
$53$			6.34881i			0.872077i	−0.899928	+	0.436038i	$0.856381\pi$
$54$		0			0
$55$	11.6644	+	0.515263i	1.57283	+	0.0694780i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	2.41444i		−	0.314334i	−0.987572	−	0.157167i	$-0.949764\pi$
$59$		−	2.41444i		−	0.314334i	0.987572	−	0.157167i	$-0.0502360\pi$
$60$		0			0
$61$		−	14.3898i		−	1.84243i	−0.389058	−	0.921213i	$-0.627199\pi$
$61$		−	14.3898i		−	1.84243i	0.389058	−	0.921213i	$-0.372801\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.41454			0.919660
$66$		0			0
$67$	−13.3717			−1.63361			−0.816806	−	0.576912i	$-0.804257\pi$
$67$	−13.3717			−1.63361			−0.816806	+	0.576912i	$0.804257\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.10617i			0.249957i	0.992159	+	0.124978i	$0.0398861\pi$
$71$			2.10617i			0.249957i	−0.992159	+	0.124978i	$0.960114\pi$
$72$		0			0
$73$			13.1116i			1.53460i	0.641290	+	0.767299i	$0.278400\pi$
$73$			13.1116i			1.53460i	−0.641290	+	0.767299i	$0.721600\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	15.6644	+	0.691959i	1.78513	+	0.0788560i
$78$		0			0
$79$			12.7383i			1.43317i	0.697499	+	0.716586i	$0.254296\pi$
$79$			12.7383i			1.43317i	−0.697499	+	0.716586i	$0.745704\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.3931			1.14079			0.570396	−	0.821370i	$-0.306790\pi$
$83$	10.3931			1.14079			0.570396	+	0.821370i	$0.306790\pi$
$84$		0			0
$85$			16.4960i			1.78924i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			4.24264i			0.449719i	0.974391	+	0.224860i	$0.0721923\pi$
$89$			4.24264i			0.449719i	−0.974391	+	0.224860i	$0.927808\pi$
$90$		0			0
$91$	9.95715			1.04379
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.728692			−0.0747622
$96$		0			0
$97$	−10.3503			−1.05091			−0.525455	−	0.850821i	$-0.676105\pi$
$97$	−10.3503			−1.05091			−0.525455	+	0.850821i	$0.676105\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.27131			−0.922530			−0.461265	−	0.887263i	$-0.652604\pi$
$101$	−9.27131			−0.922530			−0.461265	+	0.887263i	$0.652604\pi$
$102$		0			0
$103$	−7.56404			−0.745307			−0.372653	−	0.927971i	$-0.621552\pi$
$103$	−7.56404			−0.745307			−0.372653	+	0.927971i	$0.621552\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.95715			0.575900			0.287950	−	0.957645i	$-0.407026\pi$
$107$	5.95715			0.575900			0.287950	+	0.957645i	$0.407026\pi$
$108$		0			0
$109$			13.4199i			1.28539i	0.766122	+	0.642696i	$0.222184\pi$
$109$			13.4199i			1.28539i	−0.766122	+	0.642696i	$0.777816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.97016i		−	0.185337i	−0.995697	−	0.0926686i	$-0.970460\pi$
$113$		−	1.97016i		−	0.185337i	0.995697	−	0.0926686i	$-0.0295397\pi$
$114$		0			0
$115$	−15.9143			−1.48402
$116$		0			0
$117$		0			0
$118$		0			0
$119$			22.1528i			2.03075i
$120$		0			0
$121$	−10.9572	−	0.969933i	−0.996105	−	0.0881757i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	8.42469i		−	0.753527i
$126$		0			0
$127$		−	12.7383i		−	1.13034i	−0.824974	−	0.565171i	$-0.808810\pi$
$127$		−	12.7383i		−	1.13034i	0.824974	−	0.565171i	$-0.191190\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.05754			0.703990			0.351995	−	0.936002i	$-0.385503\pi$
$131$	8.05754			0.703990			0.351995	+	0.936002i	$0.385503\pi$
$132$		0			0
$133$	−0.978577			−0.0848534
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	17.4962i		−	1.49480i	−0.664373	−	0.747401i	$-0.731301\pi$
$137$		−	17.4962i		−	1.49480i	0.664373	−	0.747401i	$-0.268699\pi$
$138$		0			0
$139$		−	7.30836i		−	0.619887i	−0.950755	−	0.309943i	$-0.899690\pi$
$139$		−	7.30836i		−	0.619887i	0.950755	−	0.309943i	$-0.100310\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.97858	−	0.308271i	−0.583578	−	0.0257789i
$144$		0			0
$145$		−	26.9817i		−	2.24071i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.68585			−0.383879			−0.191940	−	0.981407i	$-0.561478\pi$
$149$	−4.68585			−0.383879			−0.191940	+	0.981407i	$0.561478\pi$
$150$		0			0
$151$		−	5.14159i		−	0.418417i	−0.977871	−	0.209208i	$-0.932911\pi$
$151$		−	5.14159i		−	0.418417i	0.977871	−	0.209208i	$-0.0670887\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			10.4857i			0.842235i
$156$		0			0
$157$	−16.7434			−1.33627			−0.668134	−	0.744041i	$-0.732907\pi$
$157$	−16.7434			−1.33627			−0.668134	+	0.744041i	$0.732907\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−21.3717			−1.68433
$162$		0			0
$163$	4.58546			0.359161			0.179581	−	0.983743i	$-0.442526\pi$
$163$	4.58546			0.359161			0.179581	+	0.983743i	$0.442526\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.31415			0.720751			0.360375	−	0.932807i	$-0.382649\pi$
$167$	9.31415			0.720751			0.360375	+	0.932807i	$0.382649\pi$
$168$		0			0
$169$	8.56404			0.658772
$170$		0			0
$171$		0			0
$172$		0			0
$173$	20.4507			1.55483			0.777417	−	0.628985i	$-0.216529\pi$
$173$	20.4507			1.55483			0.777417	+	0.628985i	$0.216529\pi$
$174$		0			0
$175$		−	34.9517i		−	2.64210i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			9.86920i			0.737659i	0.929497	+	0.368829i	$0.120241\pi$
$179$			9.86920i			0.737659i	−0.929497	+	0.368829i	$0.879759\pi$
$180$		0			0
$181$	−5.80765			−0.431679			−0.215840	−	0.976429i	$-0.569249\pi$
$181$	−5.80765			−0.431679			−0.215840	+	0.976429i	$0.569249\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	24.5673i		−	1.80622i
$186$		0			0
$187$	0.685846	−	15.5261i	0.0501540	−	1.13538i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	9.70289i		−	0.702077i	−0.936361	−	0.351038i	$-0.885829\pi$
$191$		−	9.70289i		−	0.702077i	0.936361	−	0.351038i	$-0.114171\pi$
$192$		0			0
$193$			2.27248i			0.163577i	0.996650	+	0.0817883i	$0.0260631\pi$
$193$			2.27248i			0.163577i	−0.996650	+	0.0817883i	$0.973937\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.89962			−0.562824			−0.281412	−	0.959587i	$-0.590803\pi$
$197$	−7.89962			−0.562824			−0.281412	+	0.959587i	$0.590803\pi$
$198$		0			0
$199$	−23.3288			−1.65374			−0.826869	−	0.562395i	$-0.809880\pi$
$199$	−23.3288			−1.65374			−0.826869	+	0.562395i	$0.809880\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	36.2344i		−	2.54316i
$204$		0			0
$205$		−	18.5570i		−	1.29608i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.685846	+	0.0302965i	0.0474410	+	0.00209565i
$210$		0			0
$211$			24.7743i			1.70553i	0.522294	+	0.852766i	$0.325076\pi$
$211$			24.7743i			1.70553i	−0.522294	+	0.852766i	$0.674924\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−20.6430			−1.40784
$216$		0			0
$217$			14.0815i			0.955917i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	9.86920i		−	0.663874i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	21.9572			1.45735			0.728674	−	0.684861i	$-0.240137\pi$
$227$	21.9572			1.45735			0.728674	+	0.684861i	$0.240137\pi$
$228$		0			0
$229$	−18.7862			−1.24143			−0.620715	−	0.784037i	$-0.713157\pi$
$229$	−18.7862			−1.24143			−0.620715	+	0.784037i	$0.713157\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−24.0147			−1.57325			−0.786627	−	0.617428i	$-0.788175\pi$
$233$	−24.0147			−1.57325			−0.786627	+	0.617428i	$0.788175\pi$
$234$		0			0
$235$	27.3288			1.78274
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.6430			0.817808			0.408904	−	0.912577i	$-0.365911\pi$
$239$	12.6430			0.817808			0.408904	+	0.912577i	$0.365911\pi$
$240$		0			0
$241$		−	6.13143i		−	0.394960i	−0.980307	−	0.197480i	$-0.936724\pi$
$241$		−	6.13143i		−	0.394960i	0.980307	−	0.197480i	$-0.0632758\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	54.0389i		−	3.45242i
$246$		0			0
$247$	0.435961			0.0277395
$248$		0			0
$249$		0			0
$250$		0			0
$251$			6.15221i			0.388324i	0.980969	+	0.194162i	$0.0621988\pi$
$251$			6.15221i			0.388324i	−0.980969	+	0.194162i	$0.937801\pi$
$252$		0			0
$253$	14.9786	+	0.661662i	0.941695	+	0.0415983i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	16.1123i		−	1.00506i	−0.864560	−	0.502529i	$-0.832403\pi$
$257$		−	16.1123i		−	1.00506i	0.864560	−	0.502529i	$-0.167597\pi$
$258$		0			0
$259$		−	32.9920i		−	2.05002i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	22.6858			1.39887			0.699435	−	0.714697i	$-0.253435\pi$
$263$	22.6858			1.39887			0.699435	+	0.714697i	$0.253435\pi$
$264$		0			0
$265$	−22.3503			−1.37297
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	18.7744i		−	1.14470i	−0.820011	−	0.572348i	$-0.806033\pi$
$269$		−	18.7744i		−	1.14470i	0.820011	−	0.572348i	$-0.193967\pi$
$270$		0			0
$271$			17.2833i			1.04988i	0.851138	+	0.524942i	$0.175913\pi$
$271$			17.2833i			1.04988i	−0.851138	+	0.524942i	$0.824087\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.08210	+	24.4963i	−0.0652529	+	1.47718i
$276$		0			0
$277$			0.994274i			0.0597402i	0.999554	+	0.0298701i	$0.00950936\pi$
$277$			0.994274i			0.0597402i	−0.999554	+	0.0298701i	$0.990491\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.6858			−0.756774			−0.378387	−	0.925647i	$-0.623521\pi$
$281$	−12.6858			−0.756774			−0.378387	+	0.925647i	$0.623521\pi$
$282$		0			0
$283$			8.83424i			0.525141i	0.964913	+	0.262570i	$0.0845702\pi$
$283$			8.83424i			0.525141i	−0.964913	+	0.262570i	$0.915430\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	24.9207i		−	1.47102i
$288$		0			0
$289$	4.95715			0.291597
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.2499			0.715646			0.357823	−	0.933789i	$-0.383519\pi$
$293$	12.2499			0.715646			0.357823	+	0.933789i	$0.383519\pi$
$294$		0			0
$295$	8.49977			0.494876
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.52119			0.550625
$300$		0			0
$301$	−27.7220			−1.59787
$302$		0			0
$303$		0			0
$304$		0			0
$305$	50.6577			2.90065
$306$		0			0
$307$			3.09601i			0.176699i	0.996090	+	0.0883494i	$0.0281592\pi$
$307$			3.09601i			0.176699i	−0.996090	+	0.0883494i	$0.971841\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4.02526i		−	0.228252i	−0.993466	−	0.114126i	$-0.963593\pi$
$311$		−	4.02526i		−	0.228252i	0.993466	−	0.114126i	$-0.0364067\pi$
$312$		0			0
$313$	8.00000			0.452187			0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000			0.452187			0.226093	+	0.974106i	$0.427405\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.8854i		−	0.892213i	−0.894980	−	0.446106i	$-0.852810\pi$
$317$		−	15.8854i		−	0.892213i	0.894980	−	0.446106i	$-0.147190\pi$
$318$		0			0
$319$	−1.12181	+	25.3953i	−0.0628091	+	1.42186i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.969933i			0.0539685i
$324$		0			0
$325$			15.5712i			0.863733i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	36.7005			2.02337
$330$		0			0
$331$	0.200768			0.0110352			0.00551762	−	0.999985i	$-0.498244\pi$
$331$	0.200768			0.0110352			0.00551762	+	0.999985i	$0.498244\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	47.0735i		−	2.57190i
$336$		0			0
$337$		−	22.6482i		−	1.23373i	−0.787071	−	0.616863i	$-0.788403\pi$
$337$		−	22.6482i		−	1.23373i	0.787071	−	0.616863i	$-0.211597\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.435961	−	9.86920i	0.0236086	−	0.534447i
$342$		0			0
$343$		−	39.4768i		−	2.13155i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.35027			−0.448266			−0.224133	−	0.974559i	$-0.571955\pi$
$347$	−8.35027			−0.448266			−0.224133	+	0.974559i	$0.571955\pi$
$348$		0			0
$349$		−	6.46048i		−	0.345822i	−0.984937	−	0.172911i	$-0.944683\pi$
$349$		−	6.46048i		−	0.345822i	0.984937	−	0.172911i	$-0.0553172\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	21.7086i		−	1.15543i	−0.816239	−	0.577715i	$-0.803945\pi$
$353$		−	21.7086i		−	1.15543i	0.816239	−	0.577715i	$-0.196055\pi$
$354$		0			0
$355$	−7.41454			−0.393523
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11.2138			−0.591840			−0.295920	−	0.955213i	$-0.595626\pi$
$359$	−11.2138			−0.591840			−0.295920	+	0.955213i	$0.595626\pi$
$360$		0			0
$361$	18.9572			0.997745
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−46.1579			−2.41602
$366$		0			0
$367$	−30.1579			−1.57423			−0.787115	−	0.616806i	$-0.788426\pi$
$367$	−30.1579			−1.57423			−0.787115	+	0.616806i	$0.788426\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−30.0147			−1.55829
$372$		0			0
$373$			18.1068i			0.937535i	0.883322	+	0.468767i	$0.155302\pi$
$373$			18.1068i			0.937535i	−0.883322	+	0.468767i	$0.844698\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.1426i			0.831386i
$378$		0			0
$379$	−10.5426			−0.541538			−0.270769	−	0.962644i	$-0.587278\pi$
$379$	−10.5426			−0.541538			−0.270769	+	0.962644i	$0.587278\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			20.8746i			1.06664i	0.845912	+	0.533322i	$0.179057\pi$
$383$			20.8746i			1.06664i	−0.845912	+	0.533322i	$0.820943\pi$
$384$		0			0
$385$	−2.43596	+	55.1448i	−0.124148	+	2.81044i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			9.44926i			0.479096i	0.970884	+	0.239548i	$0.0769993\pi$
$389$			9.44926i			0.479096i	−0.970884	+	0.239548i	$0.923001\pi$
$390$		0			0
$391$			21.1829i			1.07127i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−44.8438			−2.25633
$396$		0			0
$397$	−11.7648			−0.590459			−0.295229	−	0.955426i	$-0.595396\pi$
$397$	−11.7648			−0.590459			−0.295229	+	0.955426i	$0.595396\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			6.24310i			0.311766i	0.987776	+	0.155883i	$0.0498222\pi$
$401$			6.24310i			0.311766i	−0.987776	+	0.155883i	$0.950178\pi$
$402$		0			0
$403$		−	6.27340i		−	0.312500i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.02142	+	23.1228i	−0.0506301	+	1.14615i
$408$		0			0
$409$		−	14.5561i		−	0.719754i	−0.933000	−	0.359877i	$-0.882819\pi$
$409$		−	14.5561i		−	0.719754i	0.933000	−	0.359877i	$-0.117181\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.4145			0.561673
$414$		0			0
$415$			36.5878i			1.79602i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	5.32424i		−	0.260106i	−0.991507	−	0.130053i	$-0.958485\pi$
$419$		−	5.32424i		−	0.260106i	0.991507	−	0.130053i	$-0.0415148\pi$
$420$		0			0
$421$	3.06427			0.149343			0.0746717	−	0.997208i	$-0.476209\pi$
$421$	3.06427			0.149343			0.0746717	+	0.997208i	$0.476209\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−34.6430			−1.68043
$426$		0			0
$427$	68.0294			3.29217
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−27.1856			−1.30949			−0.654743	−	0.755852i	$-0.727223\pi$
$431$	−27.1856			−1.30949			−0.654743	+	0.755852i	$0.727223\pi$
$432$		0			0
$433$	−3.91431			−0.188110			−0.0940548	−	0.995567i	$-0.529983\pi$
$433$	−3.91431			−0.188110			−0.0940548	+	0.995567i	$0.529983\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.935731			−0.0447621
$438$		0			0
$439$			19.9210i			0.950780i	0.879775	+	0.475390i	$0.157693\pi$
$439$			19.9210i			0.950780i	−0.879775	+	0.475390i	$0.842307\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			30.5775i			1.45278i	0.687282	+	0.726391i	$0.258804\pi$
$443$			30.5775i			1.45278i	−0.687282	+	0.726391i	$0.741196\pi$
$444$		0			0
$445$	−14.9357			−0.708022
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	19.2127i		−	0.906705i	−0.891331	−	0.453353i	$-0.850228\pi$
$449$		−	19.2127i		−	0.906705i	0.891331	−	0.453353i	$-0.149772\pi$
$450$		0			0
$451$	−0.771538	+	17.4659i	−0.0363303	+	0.822438i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			35.0530i			1.64331i
$456$		0			0
$457$			30.5775i			1.43036i	0.698942	+	0.715178i	$0.253654\pi$
$457$			30.5775i			1.43036i	−0.698942	+	0.715178i	$0.746346\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.46365			−0.347617			−0.173809	−	0.984779i	$-0.555607\pi$
$461$	−7.46365			−0.347617			−0.173809	+	0.984779i	$0.555607\pi$
$462$		0			0
$463$	−13.5212			−0.628383			−0.314192	−	0.949360i	$-0.601733\pi$
$463$	−13.5212			−0.628383			−0.314192	+	0.949360i	$0.601733\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	18.7685i		−	0.868501i	−0.900792	−	0.434250i	$-0.857013\pi$
$467$		−	18.7685i		−	0.868501i	0.900792	−	0.434250i	$-0.142987\pi$
$468$		0			0
$469$		−	63.2161i		−	2.91905i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	19.4292	+	0.858265i	0.893357	+	0.0394631i
$474$		0			0
$475$		−	1.53032i		−	0.0702157i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	38.2155			1.74611			0.873054	−	0.487623i	$-0.162136\pi$
$479$	38.2155			1.74611			0.873054	+	0.487623i	$0.162136\pi$
$480$		0			0
$481$			14.6981i			0.670175i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	36.4369i		−	1.65452i
$486$		0			0
$487$	35.9143			1.62743			0.813716	−	0.581262i	$-0.197441\pi$
$487$	35.9143			1.62743			0.813716	+	0.581262i	$0.197441\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.95715			0.268843			0.134421	−	0.990924i	$-0.457082\pi$
$491$	5.95715			0.268843			0.134421	+	0.990924i	$0.457082\pi$
$492$		0			0
$493$	−35.9143			−1.61750
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.95715			−0.446639
$498$		0			0
$499$	−29.2860			−1.31102			−0.655511	−	0.755186i	$-0.727547\pi$
$499$	−29.2860			−1.31102			−0.655511	+	0.755186i	$0.727547\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−11.4721			−0.511515			−0.255757	−	0.966741i	$-0.582325\pi$
$503$	−11.4721			−0.511515			−0.255757	+	0.966741i	$0.582325\pi$
$504$		0			0
$505$		−	32.6386i		−	1.45240i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			42.6646i			1.89107i	0.325515	+	0.945537i	$0.394462\pi$
$509$			42.6646i			1.89107i	−0.325515	+	0.945537i	$0.605538\pi$
$510$		0			0
$511$	−61.9865			−2.74212
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	26.6283i		−	1.17338i
$516$		0			0
$517$	−25.7220	−	1.13624i	−1.13125	−	0.0499717i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	2.30277i		−	0.100886i	−0.998727	−	0.0504432i	$-0.983937\pi$
$521$		−	2.30277i		−	0.100886i	0.998727	−	0.0504432i	$-0.0160634\pi$
$522$		0			0
$523$			25.6836i			1.12307i	0.827454	+	0.561533i	$0.189788\pi$
$523$			25.6836i			1.12307i	−0.827454	+	0.561533i	$0.810212\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	13.9572			0.607983
$528$		0			0
$529$	2.56404			0.111480
$530$		0			0
$531$		0			0
$532$		0			0
$533$			11.1023i			0.480893i
$534$		0			0
$535$			20.9715i			0.906676i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.24675	+	50.8615i	−0.0967744	+	2.19076i
$540$		0			0
$541$			12.8033i			0.550459i	0.961379	+	0.275229i	$0.0887538\pi$
$541$			12.8033i			0.550459i	−0.961379	+	0.275229i	$0.911246\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−47.2432			−2.02367
$546$		0			0
$547$			5.24730i			0.224359i	0.993688	+	0.112179i	$0.0357831\pi$
$547$			5.24730i			0.224359i	−0.993688	+	0.112179i	$0.964217\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.58647i		−	0.0675861i
$552$		0			0
$553$	−60.2217			−2.56089
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.62158			−0.238194			−0.119097	−	0.992883i	$-0.538000\pi$
$557$	−5.62158			−0.238194			−0.119097	+	0.992883i	$0.538000\pi$
$558$		0			0
$559$	12.3503			0.522361
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.86519			0.0786085			0.0393042	−	0.999227i	$-0.487486\pi$
$563$	1.86519			0.0786085			0.0393042	+	0.999227i	$0.487486\pi$
$564$		0			0
$565$	6.93573			0.291788
$566$		0			0
$567$		0			0
$568$		0			0
$569$	42.8438			1.79610			0.898052	−	0.439889i	$-0.144982\pi$
$569$	42.8438			1.79610			0.898052	+	0.439889i	$0.144982\pi$
$570$		0			0
$571$			30.0171i			1.25618i	0.778141	+	0.628089i	$0.216163\pi$
$571$			30.0171i			1.25618i	−0.778141	+	0.628089i	$0.783837\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	33.4214i		−	1.39377i
$576$		0			0
$577$	−6.43596			−0.267933			−0.133966	−	0.990986i	$-0.542771\pi$
$577$	−6.43596			−0.267933			−0.133966	+	0.990986i	$0.542771\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			49.1346i			2.03845i
$582$		0			0
$583$	21.0361	+	0.929247i	0.871227	+	0.0384855i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.1519i		−	0.749210i	−0.927184	−	0.374605i	$-0.877778\pi$
$587$		−	18.1519i		−	0.749210i	0.927184	−	0.374605i	$-0.122222\pi$
$588$		0			0
$589$			0.616542i			0.0254042i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.3864			−0.549713			−0.274856	−	0.961485i	$-0.588630\pi$
$593$	−13.3864			−0.549713			−0.274856	+	0.961485i	$0.588630\pi$
$594$		0			0
$595$	−77.9865			−3.19714
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1.42017i			0.0580265i	0.999579	+	0.0290133i	$0.00923650\pi$
$599$			1.42017i			0.0580265i	−0.999579	+	0.0290133i	$0.990763\pi$
$600$		0			0
$601$		−	21.6783i		−	0.884275i	−0.896947	−	0.442137i	$-0.854220\pi$
$601$		−	21.6783i		−	0.884275i	0.896947	−	0.442137i	$-0.145780\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.41454	−	38.5734i	0.138821	−	1.56823i
$606$		0			0
$607$		−	42.1345i		−	1.71019i	−0.518475	−	0.855093i	$-0.673500\pi$
$607$		−	42.1345i		−	1.71019i	0.518475	−	0.855093i	$-0.326500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.3503			−0.661461
$612$		0			0
$613$			16.9462i			0.684451i	0.939618	+	0.342226i	$0.111181\pi$
$613$			16.9462i			0.684451i	−0.939618	+	0.342226i	$0.888819\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			35.2947i			1.42091i	0.703741	+	0.710456i	$0.251511\pi$
$617$			35.2947i			1.42091i	−0.703741	+	0.710456i	$0.748489\pi$
$618$		0			0
$619$	35.5296			1.42806			0.714028	−	0.700117i	$-0.246869\pi$
$619$	35.5296			1.42806			0.714028	+	0.700117i	$0.246869\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−20.0575			−0.803588
$624$		0			0
$625$	−7.30742			−0.292297
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−32.7005			−1.30386
$630$		0			0
$631$	21.0508			0.838019			0.419010	−	0.907982i	$-0.362377\pi$
$631$	21.0508			0.838019			0.419010	+	0.907982i	$0.362377\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	44.8438			1.77957
$636$		0			0
$637$			32.3303i			1.28097i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			27.6374i			1.09161i	0.837911	+	0.545806i	$0.183777\pi$
$641$			27.6374i			1.09161i	−0.837911	+	0.545806i	$0.816223\pi$
$642$		0			0
$643$	−4.70054			−0.185371			−0.0926856	−	0.995695i	$-0.529545\pi$
$643$	−4.70054			−0.185371			−0.0926856	+	0.995695i	$0.529545\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3.40872i			0.134011i	0.997753	+	0.0670053i	$0.0213444\pi$
$647$			3.40872i			0.134011i	−0.997753	+	0.0670053i	$0.978656\pi$
$648$		0			0
$649$	−8.00000	−	0.353391i	−0.314027	−	0.0138718i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			28.6436i			1.12091i	0.828184	+	0.560456i	$0.189374\pi$
$653$			28.6436i			1.12091i	−0.828184	+	0.560456i	$0.810626\pi$
$654$		0			0
$655$			28.3656i			1.10834i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.878193			−0.0342095			−0.0171048	−	0.999854i	$-0.505445\pi$
$659$	−0.878193			−0.0342095			−0.0171048	+	0.999854i	$0.505445\pi$
$660$		0			0
$661$	−12.7434			−0.495660			−0.247830	−	0.968804i	$-0.579717\pi$
$661$	−12.7434			−0.495660			−0.247830	+	0.968804i	$0.579717\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	3.44497i		−	0.133590i
$666$		0			0
$667$		−	34.6479i		−	1.34157i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−47.6791	−	2.10617i	−1.84063	−	0.0813079i
$672$		0			0
$673$		−	8.21326i		−	0.316598i	−0.987391	−	0.158299i	$-0.949399\pi$
$673$		−	8.21326i		−	0.316598i	0.987391	−	0.158299i	$-0.0506010\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.4721			0.671506			0.335753	−	0.941950i	$-0.391009\pi$
$677$	17.4721			0.671506			0.335753	+	0.941950i	$0.391009\pi$
$678$		0			0
$679$		−	48.9320i		−	1.87784i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.32332i		−	0.0506356i	−0.999679	−	0.0253178i	$-0.991940\pi$
$683$		−	1.32332i		−	0.0506356i	0.999679	−	0.0253178i	$-0.00805977\pi$
$684$		0			0
$685$	61.5934			2.35336
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13.3717			0.509421
$690$		0			0
$691$	37.6707			1.43306			0.716530	−	0.697556i	$-0.245729\pi$
$691$	37.6707			1.43306			0.716530	+	0.697556i	$0.245729\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	25.7282			0.975928
$696$		0			0
$697$	−24.7005			−0.935600
$698$		0			0
$699$		0			0
$700$		0			0
$701$	38.3221			1.44741			0.723703	−	0.690111i	$-0.242438\pi$
$701$	38.3221			1.44741			0.723703	+	0.690111i	$0.242438\pi$
$702$		0			0
$703$		−	1.44451i		−	0.0544807i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	43.8311i		−	1.64844i
$708$		0			0
$709$	−7.17092			−0.269310			−0.134655	−	0.990893i	$-0.542993\pi$
$709$	−7.17092			−0.269310			−0.134655	+	0.990893i	$0.542993\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			13.4650i			0.504268i
$714$		0			0
$715$	1.08523	−	24.5673i	0.0405854	−	0.918764i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			44.4720i			1.65853i	0.558859	+	0.829263i	$0.311239\pi$
$719$			44.4720i			1.65853i	−0.558859	+	0.829263i	$0.688761\pi$
$720$		0			0
$721$		−	35.7598i		−	1.33176i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	56.6640			2.10445
$726$		0			0
$727$	17.5725			0.651726			0.325863	−	0.945417i	$-0.394345\pi$
$727$	17.5725			0.651726			0.325863	+	0.945417i	$0.394345\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			27.4771i			1.01628i
$732$		0			0
$733$			38.2711i			1.41358i	0.707426	+	0.706788i	$0.249856\pi$
$733$			38.2711i			1.41358i	−0.707426	+	0.706788i	$0.750144\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.95715	+	44.3057i	−0.0720927	+	1.63202i
$738$		0			0
$739$		−	11.8046i		−	0.434240i	−0.976145	−	0.217120i	$-0.930334\pi$
$739$		−	11.8046i		−	0.434240i	0.976145	−	0.217120i	$-0.0696663\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−30.3587			−1.11375			−0.556876	−	0.830596i	$-0.688000\pi$
$743$	−30.3587			−1.11375			−0.556876	+	0.830596i	$0.688000\pi$
$744$		0			0
$745$		−	16.4960i		−	0.604366i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			28.1631i			1.02906i
$750$		0			0
$751$	28.1495			1.02719			0.513595	−	0.858033i	$-0.328313\pi$
$751$	28.1495			1.02719			0.513595	+	0.858033i	$0.328313\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18.1004			0.658740
$756$		0			0
$757$	31.6791			1.15140			0.575699	−	0.817662i	$-0.304730\pi$
$757$	31.6791			1.15140			0.575699	+	0.817662i	$0.304730\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.570769			0.0206904			0.0103452	−	0.999946i	$-0.496707\pi$
$761$	0.570769			0.0206904			0.0103452	+	0.999946i	$0.496707\pi$
$762$		0			0
$763$	−63.4439			−2.29682
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.08523			−0.183617
$768$		0			0
$769$			26.0118i			0.938009i	0.883196	+	0.469005i	$0.155387\pi$
$769$			26.0118i			0.938009i	−0.883196	+	0.469005i	$0.844613\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			8.84463i			0.318119i	0.987269	+	0.159060i	$0.0508462\pi$
$773$			8.84463i			0.318119i	−0.987269	+	0.159060i	$0.949154\pi$
$774$		0			0
$775$	−22.0210			−0.791016
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1.09112i		−	0.0390934i
$780$		0			0
$781$	6.97858	+	0.308271i	0.249713	+	0.0110308i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	58.9432i		−	2.10377i
$786$		0			0
$787$		−	24.3603i		−	0.868351i	−0.900828	−	0.434175i	$-0.857040\pi$
$787$		−	24.3603i		−	0.868351i	0.900828	−	0.434175i	$-0.142960\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.31415			0.331173
$792$		0			0
$793$	−30.3074			−1.07625
$794$		0			0
$795$		0			0
$796$		0			0
$797$			17.3299i			0.613857i	0.951733	+	0.306928i	$0.0993012\pi$
$797$			17.3299i			0.613857i	−0.951733	+	0.306928i	$0.900699\pi$
$798$		0			0
$799$		−	36.3763i		−	1.28690i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	43.4439	+	1.91909i	1.53310	+	0.0677231i
$804$		0			0
$805$		−	75.2366i		−	2.65174i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17.1856			0.604214			0.302107	−	0.953274i	$-0.402310\pi$
$809$	17.1856			0.604214			0.302107	+	0.953274i	$0.402310\pi$
$810$		0			0
$811$			8.83424i			0.310212i	0.987898	+	0.155106i	$0.0495719\pi$
$811$			8.83424i			0.310212i	−0.987898	+	0.155106i	$0.950428\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			16.1426i			0.565450i
$816$		0			0
$817$	−1.21377			−0.0424644
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.8139			−0.412309			−0.206154	−	0.978519i	$-0.566095\pi$
$821$	−11.8139			−0.412309			−0.206154	+	0.978519i	$0.566095\pi$
$822$		0			0
$823$	44.8156			1.56217			0.781087	−	0.624422i	$-0.214665\pi$
$823$	44.8156			1.56217			0.781087	+	0.624422i	$0.214665\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.4360			0.432441			0.216220	−	0.976345i	$-0.430627\pi$
$827$	12.4360			0.432441			0.216220	+	0.976345i	$0.430627\pi$
$828$		0			0
$829$	−12.0428			−0.418265			−0.209133	−	0.977887i	$-0.567064\pi$
$829$	−12.0428			−0.418265			−0.209133	+	0.977887i	$0.567064\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−71.9290			−2.49219
$834$		0			0
$835$			32.7894i			1.13472i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	44.9466i		−	1.55173i	−0.630900	−	0.775864i	$-0.717314\pi$
$839$		−	44.9466i		−	1.55173i	0.630900	−	0.775864i	$-0.282686\pi$
$840$		0			0
$841$	29.7434			1.02563
$842$		0			0
$843$		0			0
$844$		0			0
$845$			30.1487i			1.03715i
$846$		0			0
$847$	4.58546	−	51.8011i	0.157558	−	1.77991i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	31.5475i		−	1.08143i
$852$		0			0
$853$		−	18.3182i		−	0.627204i	−0.949554	−	0.313602i	$-0.898464\pi$
$853$		−	18.3182i		−	0.627204i	0.949554	−	0.313602i	$-0.101536\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.68585			0.296703			0.148351	−	0.988935i	$-0.452603\pi$
$857$	8.68585			0.296703			0.148351	+	0.988935i	$0.452603\pi$
$858$		0			0
$859$	−17.1709			−0.585864			−0.292932	−	0.956133i	$-0.594631\pi$
$859$	−17.1709			−0.585864			−0.292932	+	0.956133i	$0.594631\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	24.2590i		−	0.825786i	−0.910779	−	0.412893i	$-0.864518\pi$
$863$		−	24.2590i		−	0.825786i	0.910779	−	0.412893i	$-0.135482\pi$
$864$		0			0
$865$			71.9942i			2.44788i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	42.2070	+	1.86445i	1.43178	+	0.0632471i
$870$		0			0
$871$			28.1631i			0.954270i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	39.8286			1.34645
$876$		0			0
$877$			22.8145i			0.770391i	0.922835	+	0.385195i	$0.125866\pi$
$877$			22.8145i			0.770391i	−0.922835	+	0.385195i	$0.874134\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			26.7489i			0.901192i	0.892728	+	0.450596i	$0.148789\pi$
$881$			26.7489i			0.901192i	−0.892728	+	0.450596i	$0.851211\pi$
$882$		0			0
$883$	−11.5296			−0.388002			−0.194001	−	0.981001i	$-0.562147\pi$
$883$	−11.5296			−0.388002			−0.194001	+	0.981001i	$0.562147\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	39.1281			1.31379			0.656896	−	0.753981i	$-0.271869\pi$
$887$	39.1281			1.31379			0.656896	+	0.753981i	$0.271869\pi$
$888$		0			0
$889$	60.2217			2.01977
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.60688			0.0537723
$894$		0			0
$895$	−34.7434			−1.16134
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22.8291			−0.761392
$900$		0			0
$901$			29.7496i			0.991102i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	20.4452i		−	0.679621i
$906$		0			0
$907$	2.62831			0.0872715			0.0436358	−	0.999048i	$-0.486106\pi$
$907$	2.62831			0.0872715			0.0436358	+	0.999048i	$0.486106\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			38.8151i			1.28600i	0.765865	+	0.643001i	$0.222311\pi$
$911$			38.8151i			1.28600i	−0.765865	+	0.643001i	$0.777689\pi$
$912$		0			0
$913$	1.52119	−	34.4365i	0.0503442	−	1.13968i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			38.0929i			1.25794i
$918$		0			0
$919$		−	19.1418i		−	0.631428i	−0.948854	−	0.315714i	$-0.897756\pi$
$919$		−	19.1418i		−	0.631428i	0.948854	−	0.315714i	$-0.102244\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.43596			0.146011
$924$		0			0
$925$	51.5934			1.69638
$926$		0			0
$927$		0			0
$928$		0			0
$929$			31.5778i			1.03603i	0.855371	+	0.518016i	$0.173329\pi$
$929$			31.5778i			1.03603i	−0.855371	+	0.518016i	$0.826671\pi$
$930$		0			0
$931$		−	3.17738i		−	0.104135i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	54.6577	+	2.41444i	1.78750	+	0.0789607i
$936$		0			0
$937$		−	18.9104i		−	0.617777i	−0.951098	−	0.308888i	$-0.900043\pi$
$937$		−	18.9104i		−	0.617777i	0.951098	−	0.308888i	$-0.0999570\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−40.8009			−1.33007			−0.665036	−	0.746811i	$-0.731584\pi$
$941$	−40.8009			−1.33007			−0.665036	+	0.746811i	$0.731584\pi$
$942$		0			0
$943$		−	23.8296i		−	0.775997i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			0.949154i			0.0308434i	0.999881	+	0.0154217i	$0.00490907\pi$
$947$			0.949154i			0.0308434i	−0.999881	+	0.0154217i	$0.995091\pi$
$948$		0			0
$949$	27.6153			0.896431
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21.4721			−0.695549			−0.347774	−	0.937578i	$-0.613062\pi$
$953$	−21.4721			−0.695549			−0.347774	+	0.937578i	$0.613062\pi$
$954$		0			0
$955$	34.1579			1.10532
$956$		0			0
$957$		0			0
$958$		0			0
$959$	82.7152			2.67101
$960$		0			0
$961$	−22.1281			−0.713809
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.00000			−0.257529
$966$		0			0
$967$		−	5.14159i		−	0.165342i	−0.996577	−	0.0826712i	$-0.973655\pi$
$967$		−	5.14159i		−	0.165342i	0.996577	−	0.0826712i	$-0.0263451\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	15.3841i		−	0.493699i	−0.969054	−	0.246849i	$-0.920605\pi$
$971$		−	15.3841i		−	0.493699i	0.969054	−	0.246849i	$-0.0793953\pi$
$972$		0			0
$973$	34.5510			1.10766
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	50.8208i		−	1.62590i	−0.582333	−	0.812951i	$-0.697860\pi$
$977$		−	50.8208i		−	1.62590i	0.582333	−	0.812951i	$-0.302140\pi$
$978$		0			0
$979$	14.0575	+	0.620976i	0.449281	+	0.0198465i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			54.4831i			1.73774i	0.495038	+	0.868871i	$0.335154\pi$
$983$			54.4831i			1.73774i	−0.495038	+	0.868871i	$0.664846\pi$
$984$		0			0
$985$		−	27.8097i		−	0.886091i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26.5082			−0.842911
$990$		0			0
$991$	−61.0508			−1.93934			−0.969671	−	0.244412i	$-0.921405\pi$
$991$	−61.0508			−1.93934			−0.969671	+	0.244412i	$0.921405\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	82.1265i		−	2.60359i
$996$		0			0
$997$		−	3.69265i		−	0.116947i	−0.998289	−	0.0584736i	$-0.981377\pi$
$997$		−	3.69265i		−	0.116947i	0.998289	−	0.0584736i	$-0.0186234\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.b.x.2177.5		6
3.2	odd	2		6336.2.b.y.2177.2		6
4.3	odd	2		6336.2.b.z.2177.5		6
8.3	odd	2		1584.2.b.f.593.2		6
8.5	even	2		792.2.b.b.593.2	yes	6
11.10	odd	2		6336.2.b.y.2177.5		6
12.11	even	2		6336.2.b.w.2177.2		6
24.5	odd	2		792.2.b.a.593.5	yes	6
24.11	even	2		1584.2.b.g.593.5		6
33.32	even	2	inner	6336.2.b.x.2177.2		6
44.43	even	2		6336.2.b.w.2177.5		6
88.21	odd	2		792.2.b.a.593.2	✓	6
88.43	even	2		1584.2.b.g.593.2		6
132.131	odd	2		6336.2.b.z.2177.2		6
264.131	odd	2		1584.2.b.f.593.5		6
264.197	even	2		792.2.b.b.593.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.b.a.593.2	✓	6	88.21	odd	2
792.2.b.a.593.5	yes	6	24.5	odd	2
792.2.b.b.593.2	yes	6	8.5	even	2
792.2.b.b.593.5	yes	6	264.197	even	2
1584.2.b.f.593.2		6	8.3	odd	2
1584.2.b.f.593.5		6	264.131	odd	2
1584.2.b.g.593.2		6	88.43	even	2
1584.2.b.g.593.5		6	24.11	even	2
6336.2.b.w.2177.2		6	12.11	even	2
6336.2.b.w.2177.5		6	44.43	even	2
6336.2.b.x.2177.2		6	33.32	even	2	inner
6336.2.b.x.2177.5		6	1.1	even	1	trivial
6336.2.b.y.2177.2		6	3.2	odd	2
6336.2.b.y.2177.5		6	11.10	odd	2
6336.2.b.z.2177.2		6	132.131	odd	2
6336.2.b.z.2177.5		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.52039i q^{5} +4.72761i q^{7} +(0.146365 - 3.31339i) q^{11} -2.10617i q^{13} +4.68585 q^{17} +0.206992i q^{19} +4.52062i q^{23} -7.39312 q^{25} -7.66442 q^{29} +2.97858 q^{31} -16.6430 q^{35} -6.97858 q^{37} -5.27131 q^{41} +5.86385i q^{43} -7.76303i q^{47} -15.3503 q^{49} +6.34881i q^{53} +(11.6644 + 0.515263i) q^{55} -2.41444i q^{59} -14.3898i q^{61} +7.41454 q^{65} -13.3717 q^{67} +2.10617i q^{71} +13.1116i q^{73} +(15.6644 + 0.691959i) q^{77} +12.7383i q^{79} +10.3931 q^{83} +16.4960i q^{85} +4.24264i q^{89} +9.95715 q^{91} -0.728692 q^{95} -10.3503 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{11} + 4 q^{17} - 26 q^{25} + 8 q^{29} - 12 q^{31} - 16 q^{35} - 12 q^{37} + 4 q^{41} - 14 q^{49} + 16 q^{55} + 56 q^{65} - 32 q^{67} + 40 q^{77} + 44 q^{83} - 40 q^{95} + 16 q^{97}+O(q^{100}) \) 6 * q - 2 * q^11 + 4 * q^17 - 26 * q^25 + 8 * q^29 - 12 * q^31 - 16 * q^35 - 12 * q^37 + 4 * q^41 - 14 * q^49 + 16 * q^55 + 56 * q^65 - 32 * q^67 + 40 * q^77 + 44 * q^83 - 40 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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