LMFDB - Embedded newform 156.2.b.a.25.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [156,2,Mod(25,156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(156, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("156.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 156 = 2^{2} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	156.b (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:	$1.24566627153$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			25.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	156.25
Dual form			156.2.b.a.25.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.46410i q^{5} +1.00000 q^{9} -3.46410i q^{11} +(-1.00000 - 3.46410i) q^{13} +3.46410i q^{15} +6.00000 q^{17} +6.92820i q^{19} -7.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +6.92820i q^{31} +3.46410i q^{33} +(1.00000 + 3.46410i) q^{39} +3.46410i q^{41} +8.00000 q^{43} -3.46410i q^{45} -3.46410i q^{47} +7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} -12.0000 q^{55} -6.92820i q^{57} +3.46410i q^{59} +10.0000 q^{61} +(-12.0000 + 3.46410i) q^{65} -13.8564i q^{67} -10.3923i q^{71} +6.92820i q^{73} +7.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} +3.46410i q^{83} -20.7846i q^{85} +6.00000 q^{87} +17.3205i q^{89} -6.92820i q^{93} +24.0000 q^{95} -6.92820i q^{97} -3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} - 2 q^{13} + 12 q^{17} - 14 q^{25} - 2 q^{27} - 12 q^{29} + 2 q^{39} + 16 q^{43} + 14 q^{49} - 12 q^{51} + 12 q^{53} - 24 q^{55} + 20 q^{61} - 24 q^{65} + 14 q^{75} - 16 q^{79} + 2 q^{81}+ \cdots + 48 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 2 * q^13 + 12 * q^17 - 14 * q^25 - 2 * q^27 - 12 * q^29 + 2 * q^39 + 16 * q^43 + 14 * q^49 - 12 * q^51 + 12 * q^53 - 24 * q^55 + 20 * q^61 - 24 * q^65 + 14 * q^75 - 16 * q^79 + 2 * q^81 + 12 * q^87 + 48 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/156\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		−	3.46410i		−	1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$		−	3.46410i		−	1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$14$		0			0
$15$			3.46410i			0.894427i
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$			6.92820i			1.58944i	0.606977	+	0.794719i	$0.292382\pi$
$19$			6.92820i			1.58944i	−0.606977	+	0.794719i	$0.707618\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$			6.92820i			1.24434i	0.782881	+	0.622171i	$0.213749\pi$
$31$			6.92820i			1.24434i	−0.782881	+	0.622171i	$0.786251\pi$
$32$		0			0
$33$			3.46410i			0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	1.00000	+	3.46410i	0.160128	+	0.554700i
$40$		0			0
$41$			3.46410i			0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$			3.46410i			0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		−	3.46410i		−	0.516398i
$46$		0			0
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−6.00000			−0.840168
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$		0			0
$57$		−	6.92820i		−	0.917663i
$58$		0			0
$59$			3.46410i			0.450988i	0.974245	+	0.225494i	$0.0723995\pi$
$59$			3.46410i			0.450988i	−0.974245	+	0.225494i	$0.927600\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.0000	+	3.46410i	−1.48842	+	0.429669i
$66$		0			0
$67$		−	13.8564i		−	1.69283i	−0.532524	−	0.846415i	$-0.678756\pi$
$67$		−	13.8564i		−	1.69283i	0.532524	−	0.846415i	$-0.321244\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		0			0
$75$	7.00000			0.808290
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$		−	20.7846i		−	2.25441i
$86$		0			0
$87$	6.00000			0.643268
$88$		0			0
$89$			17.3205i			1.83597i	0.396615	+	0.917985i	$0.370185\pi$
$89$			17.3205i			1.83597i	−0.396615	+	0.917985i	$0.629815\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	6.92820i		−	0.718421i
$94$		0			0
$95$	24.0000			2.46235
$96$		0			0
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$		0			0
$99$		−	3.46410i		−	0.348155i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$			6.92820i			0.663602i	0.943349	+	0.331801i	$0.107656\pi$
$109$			6.92820i			0.663602i	−0.943349	+	0.331801i	$0.892344\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.00000	−	3.46410i	−0.0924500	−	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	3.46410i		−	0.312348i
$124$		0			0
$125$			6.92820i			0.619677i
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			3.46410i			0.298142i
$136$		0			0
$137$			10.3923i			0.887875i	0.896058	+	0.443937i	$0.146419\pi$
$137$			10.3923i			0.887875i	−0.896058	+	0.443937i	$0.853581\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$			3.46410i			0.291730i
$142$		0			0
$143$	−12.0000	+	3.46410i	−1.00349	+	0.289683i
$144$		0			0
$145$			20.7846i			1.72607i
$146$		0			0
$147$	−7.00000			−0.577350
$148$		0			0
$149$			17.3205i			1.41895i	0.704730	+	0.709476i	$0.251068\pi$
$149$			17.3205i			1.41895i	−0.704730	+	0.709476i	$0.748932\pi$
$150$		0			0
$151$		−	6.92820i		−	0.563809i	−0.959442	−	0.281905i	$-0.909034\pi$
$151$		−	6.92820i		−	0.563809i	0.959442	−	0.281905i	$-0.0909662\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$	24.0000			1.92773
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$		0			0
$162$		0			0
$163$			20.7846i			1.62798i	0.580881	+	0.813988i	$0.302708\pi$
$163$			20.7846i			1.62798i	−0.580881	+	0.813988i	$0.697292\pi$
$164$		0			0
$165$	12.0000			0.934199
$166$		0			0
$167$		−	17.3205i		−	1.34030i	−0.742225	−	0.670151i	$-0.766230\pi$
$167$		−	17.3205i		−	1.34030i	0.742225	−	0.670151i	$-0.233770\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$			6.92820i			0.529813i
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	3.46410i		−	0.260378i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	20.7846i		−	1.51992i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	13.8564i		−	0.997406i	−0.866773	−	0.498703i	$-0.833810\pi$
$193$		−	13.8564i		−	0.997406i	0.866773	−	0.498703i	$-0.166190\pi$
$194$		0			0
$195$	12.0000	−	3.46410i	0.859338	−	0.248069i
$196$		0			0
$197$		−	17.3205i		−	1.23404i	−0.786949	−	0.617018i	$-0.788341\pi$
$197$		−	17.3205i		−	1.23404i	0.786949	−	0.617018i	$-0.211659\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$			13.8564i			0.977356i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	12.0000			0.838116
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$			10.3923i			0.712069i
$214$		0			0
$215$		−	27.7128i		−	1.89000i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	6.92820i		−	0.468165i
$220$		0			0
$221$	−6.00000	−	20.7846i	−0.403604	−	1.39812i
$222$		0			0
$223$			6.92820i			0.463947i	0.972722	+	0.231973i	$0.0745182\pi$
$223$			6.92820i			0.463947i	−0.972722	+	0.231973i	$0.925482\pi$
$224$		0			0
$225$	−7.00000			−0.466667
$226$		0			0
$227$			10.3923i			0.689761i	0.938647	+	0.344881i	$0.112081\pi$
$227$			10.3923i			0.689761i	−0.938647	+	0.344881i	$0.887919\pi$
$228$		0			0
$229$		−	6.92820i		−	0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$		−	6.92820i		−	0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$		0			0
$237$	8.00000			0.519656
$238$		0			0
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$			20.7846i			1.33885i	0.742878	+	0.669427i	$0.233460\pi$
$241$			20.7846i			1.33885i	−0.742878	+	0.669427i	$0.766540\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	24.2487i		−	1.54919i
$246$		0			0
$247$	24.0000	−	6.92820i	1.52708	−	0.440831i
$248$		0			0
$249$		−	3.46410i		−	0.219529i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$			20.7846i			1.30158i
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		−	20.7846i		−	1.27679i
$266$		0			0
$267$		−	17.3205i		−	1.06000i
$268$		0			0
$269$	−30.0000			−1.82913			−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000			−1.82913			−0.914566	+	0.404436i	$0.867468\pi$
$270$		0			0
$271$			27.7128i			1.68343i	0.539919	+	0.841717i	$0.318455\pi$
$271$			27.7128i			1.68343i	−0.539919	+	0.841717i	$0.681545\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			24.2487i			1.46225i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$			6.92820i			0.414781i
$280$		0			0
$281$		−	17.3205i		−	1.03325i	−0.856210	−	0.516627i	$-0.827187\pi$
$281$		−	17.3205i		−	1.03325i	0.856210	−	0.516627i	$-0.172813\pi$
$282$		0			0
$283$	−20.0000			−1.18888			−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000			−1.18888			−0.594438	+	0.804141i	$0.702626\pi$
$284$		0			0
$285$	−24.0000			−1.42164
$286$		0			0
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$			6.92820i			0.406138i
$292$		0			0
$293$		−	3.46410i		−	0.202375i	−0.994867	−	0.101187i	$-0.967736\pi$
$293$		−	3.46410i		−	0.202375i	0.994867	−	0.101187i	$-0.0322642\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$		0			0
$297$			3.46410i			0.201008i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−6.00000			−0.344691
$304$		0			0
$305$		−	34.6410i		−	1.98354i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	4.00000			0.227552
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			24.2487i			1.36194i	0.732310	+	0.680972i	$0.238442\pi$
$317$			24.2487i			1.36194i	−0.732310	+	0.680972i	$0.761558\pi$
$318$		0			0
$319$			20.7846i			1.16371i
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$			41.5692i			2.31297i
$324$		0			0
$325$	7.00000	+	24.2487i	0.388290	+	1.34508i
$326$		0			0
$327$		−	6.92820i		−	0.383131i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	13.8564i		−	0.761617i	−0.924654	−	0.380808i	$-0.875646\pi$
$331$		−	13.8564i		−	0.761617i	0.924654	−	0.380808i	$-0.124354\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−48.0000			−2.62252
$336$		0			0
$337$	10.0000			0.544735			0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000			0.544735			0.272367	+	0.962193i	$0.412193\pi$
$338$		0			0
$339$	18.0000			0.977626
$340$		0			0
$341$	24.0000			1.29967
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	1.00000	+	3.46410i	0.0533761	+	0.184900i
$352$		0			0
$353$		−	10.3923i		−	0.553127i	−0.960996	−	0.276563i	$-0.910804\pi$
$353$		−	10.3923i		−	0.553127i	0.960996	−	0.276563i	$-0.0891955\pi$
$354$		0			0
$355$	−36.0000			−1.91068
$356$		0			0
$357$		0			0
$358$		0			0
$359$			24.2487i			1.27980i	0.768459	+	0.639899i	$0.221024\pi$
$359$			24.2487i			1.27980i	−0.768459	+	0.639899i	$0.778976\pi$
$360$		0			0
$361$	−29.0000			−1.52632
$362$		0			0
$363$	1.00000			0.0524864
$364$		0			0
$365$	24.0000			1.25622
$366$		0			0
$367$	−28.0000			−1.46159			−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000			−1.46159			−0.730794	+	0.682598i	$0.760850\pi$
$368$		0			0
$369$			3.46410i			0.180334i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.00000			−0.103556			−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000			−0.103556			−0.0517780	+	0.998659i	$0.516489\pi$
$374$		0			0
$375$		−	6.92820i		−	0.357771i
$376$		0			0
$377$	6.00000	+	20.7846i	0.309016	+	1.07046i
$378$		0			0
$379$		−	20.7846i		−	1.06763i	−0.845600	−	0.533817i	$-0.820757\pi$
$379$		−	20.7846i		−	1.06763i	0.845600	−	0.533817i	$-0.179243\pi$
$380$		0			0
$381$	−16.0000			−0.819705
$382$		0			0
$383$			24.2487i			1.23905i	0.784976	+	0.619526i	$0.212675\pi$
$383$			24.2487i			1.23905i	−0.784976	+	0.619526i	$0.787325\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	8.00000			0.406663
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$			27.7128i			1.39438i
$396$		0			0
$397$		−	13.8564i		−	0.695433i	−0.937600	−	0.347717i	$-0.886957\pi$
$397$		−	13.8564i		−	0.695433i	0.937600	−	0.347717i	$-0.113043\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	10.3923i		−	0.518967i	−0.965748	−	0.259483i	$-0.916448\pi$
$401$		−	10.3923i		−	0.518967i	0.965748	−	0.259483i	$-0.0835523\pi$
$402$		0			0
$403$	24.0000	−	6.92820i	1.19553	−	0.345118i
$404$		0			0
$405$		−	3.46410i		−	0.172133i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	20.7846i		−	1.02773i	−0.857870	−	0.513866i	$-0.828213\pi$
$409$		−	20.7846i		−	1.02773i	0.857870	−	0.513866i	$-0.171787\pi$
$410$		0			0
$411$		−	10.3923i		−	0.512615i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	12.0000			0.589057
$416$		0			0
$417$	−8.00000			−0.391762
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$			20.7846i			1.01298i	0.862246	+	0.506490i	$0.169057\pi$
$421$			20.7846i			1.01298i	−0.862246	+	0.506490i	$0.830943\pi$
$422$		0			0
$423$		−	3.46410i		−	0.168430i
$424$		0			0
$425$	−42.0000			−2.03730
$426$		0			0
$427$		0			0
$428$		0			0
$429$	12.0000	−	3.46410i	0.579365	−	0.167248i
$430$		0			0
$431$		−	31.1769i		−	1.50174i	−0.660451	−	0.750870i	$-0.729635\pi$
$431$		−	31.1769i		−	1.50174i	0.660451	−	0.750870i	$-0.270365\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		−	20.7846i		−	0.996546i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$	60.0000			2.84427
$446$		0			0
$447$		−	17.3205i		−	0.819232i
$448$		0			0
$449$			3.46410i			0.163481i	0.996654	+	0.0817405i	$0.0260479\pi$
$449$			3.46410i			0.163481i	−0.996654	+	0.0817405i	$0.973952\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$			6.92820i			0.325515i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	6.92820i		−	0.324088i	−0.986784	−	0.162044i	$-0.948191\pi$
$457$		−	6.92820i		−	0.324088i	0.986784	−	0.162044i	$-0.0518086\pi$
$458$		0			0
$459$	−6.00000			−0.280056
$460$		0			0
$461$			10.3923i			0.484018i	0.970274	+	0.242009i	$0.0778063\pi$
$461$			10.3923i			0.484018i	−0.970274	+	0.242009i	$0.922194\pi$
$462$		0			0
$463$		−	13.8564i		−	0.643962i	−0.946746	−	0.321981i	$-0.895651\pi$
$463$		−	13.8564i		−	0.643962i	0.946746	−	0.321981i	$-0.104349\pi$
$464$		0			0
$465$	−24.0000			−1.11297
$466$		0			0
$467$	36.0000			1.66588			0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000			1.66588			0.832941	+	0.553362i	$0.186655\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−2.00000			−0.0921551
$472$		0			0
$473$		−	27.7128i		−	1.27424i
$474$		0			0
$475$		−	48.4974i		−	2.22521i
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$			3.46410i			0.158279i	0.996864	+	0.0791394i	$0.0252172\pi$
$479$			3.46410i			0.158279i	−0.996864	+	0.0791394i	$0.974783\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.0000			−1.08978
$486$		0			0
$487$		−	6.92820i		−	0.313947i	−0.987603	−	0.156973i	$-0.949826\pi$
$487$		−	6.92820i		−	0.313947i	0.987603	−	0.156973i	$-0.0501737\pi$
$488$		0			0
$489$		−	20.7846i		−	0.939913i
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$	−36.0000			−1.62136
$494$		0			0
$495$	−12.0000			−0.539360
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$			17.3205i			0.773823i
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$		−	20.7846i		−	0.924903i
$506$		0			0
$507$	11.0000	−	6.92820i	0.488527	−	0.307692i
$508$		0			0
$509$			24.2487i			1.07481i	0.843326	+	0.537403i	$0.180594\pi$
$509$			24.2487i			1.07481i	−0.843326	+	0.537403i	$0.819406\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	6.92820i		−	0.305888i
$514$		0			0
$515$			13.8564i			0.610586i
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$	8.00000			0.349816			0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000			0.349816			0.174908	+	0.984585i	$0.444037\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			41.5692i			1.81078i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$			3.46410i			0.150329i
$532$		0			0
$533$	12.0000	−	3.46410i	0.519778	−	0.150047i
$534$		0			0
$535$			41.5692i			1.79719i
$536$		0			0
$537$	12.0000			0.517838
$538$		0			0
$539$		−	24.2487i		−	1.04447i
$540$		0			0
$541$			20.7846i			0.893600i	0.894634	+	0.446800i	$0.147436\pi$
$541$			20.7846i			0.893600i	−0.894634	+	0.446800i	$0.852564\pi$
$542$		0			0
$543$	22.0000			0.944110
$544$		0			0
$545$	24.0000			1.02805
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$		−	41.5692i		−	1.77091i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			24.2487i			1.02745i	0.857955	+	0.513725i	$0.171735\pi$
$557$			24.2487i			1.02745i	−0.857955	+	0.513725i	$0.828265\pi$
$558$		0			0
$559$	−8.00000	−	27.7128i	−0.338364	−	1.17213i
$560$		0			0
$561$			20.7846i			0.877527i
$562$		0			0
$563$	−36.0000			−1.51722			−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000			−1.51722			−0.758610	+	0.651546i	$0.774121\pi$
$564$		0			0
$565$			62.3538i			2.62325i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			13.8564i			0.576850i	0.957503	+	0.288425i	$0.0931316\pi$
$577$			13.8564i			0.576850i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$			13.8564i			0.575853i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	20.7846i		−	0.860811i
$584$		0			0
$585$	−12.0000	+	3.46410i	−0.496139	+	0.143223i
$586$		0			0
$587$			31.1769i			1.28681i	0.765526	+	0.643404i	$0.222479\pi$
$587$			31.1769i			1.28681i	−0.765526	+	0.643404i	$0.777521\pi$
$588$		0			0
$589$	−48.0000			−1.97781
$590$		0			0
$591$			17.3205i			0.712470i
$592$		0			0
$593$		−	24.2487i		−	0.995775i	−0.867242	−	0.497888i	$-0.834109\pi$
$593$		−	24.2487i		−	0.995775i	0.867242	−	0.497888i	$-0.165891\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	8.00000			0.327418
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		−	13.8564i		−	0.564276i
$604$		0			0
$605$			3.46410i			0.140836i
$606$		0			0
$607$	−28.0000			−1.13648			−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000			−1.13648			−0.568242	+	0.822861i	$0.692376\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	+	3.46410i	−0.485468	+	0.140143i
$612$		0			0
$613$		−	13.8564i		−	0.559655i	−0.960050	−	0.279827i	$-0.909723\pi$
$613$		−	13.8564i		−	0.559655i	0.960050	−	0.279827i	$-0.0902773\pi$
$614$		0			0
$615$	−12.0000			−0.483887
$616$		0			0
$617$			31.1769i			1.25514i	0.778562	+	0.627568i	$0.215949\pi$
$617$			31.1769i			1.25514i	−0.778562	+	0.627568i	$0.784051\pi$
$618$		0			0
$619$			27.7128i			1.11387i	0.830555	+	0.556936i	$0.188023\pi$
$619$			27.7128i			1.11387i	−0.830555	+	0.556936i	$0.811977\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	−24.0000			−0.958468
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	20.7846i		−	0.827422i	−0.910408	−	0.413711i	$-0.864232\pi$
$631$		−	20.7846i		−	0.827422i	0.910408	−	0.413711i	$-0.135768\pi$
$632$		0			0
$633$	−4.00000			−0.158986
$634$		0			0
$635$		−	55.4256i		−	2.19950i
$636$		0			0
$637$	−7.00000	−	24.2487i	−0.277350	−	0.960769i
$638$		0			0
$639$		−	10.3923i		−	0.411113i
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$			41.5692i			1.63933i	0.572843	+	0.819665i	$0.305840\pi$
$643$			41.5692i			1.63933i	−0.572843	+	0.819665i	$0.694160\pi$
$644$		0			0
$645$			27.7128i			1.09119i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$			41.5692i			1.62424i
$656$		0			0
$657$			6.92820i			0.270295i
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$			13.8564i			0.538952i	0.963007	+	0.269476i	$0.0868504\pi$
$661$			13.8564i			0.538952i	−0.963007	+	0.269476i	$0.913150\pi$
$662$		0			0
$663$	6.00000	+	20.7846i	0.233021	+	0.807207i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	6.92820i		−	0.267860i
$670$		0			0
$671$		−	34.6410i		−	1.33730i
$672$		0			0
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$		0			0
$675$	7.00000			0.269430
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	10.3923i		−	0.398234i
$682$		0			0
$683$		−	3.46410i		−	0.132550i	−0.997801	−	0.0662751i	$-0.978889\pi$
$683$		−	3.46410i		−	0.132550i	0.997801	−	0.0662751i	$-0.0211115\pi$
$684$		0			0
$685$	36.0000			1.37549
$686$		0			0
$687$			6.92820i			0.264327i
$688$		0			0
$689$	−6.00000	−	20.7846i	−0.228582	−	0.791831i
$690$		0			0
$691$		−	27.7128i		−	1.05425i	−0.849789	−	0.527123i	$-0.823271\pi$
$691$		−	27.7128i		−	1.05425i	0.849789	−	0.527123i	$-0.176729\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	27.7128i		−	1.05121i
$696$		0			0
$697$			20.7846i			0.787273i
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	12.0000			0.451946
$706$		0			0
$707$		0			0
$708$		0			0
$709$			6.92820i			0.260194i	0.991501	+	0.130097i	$0.0415289\pi$
$709$			6.92820i			0.260194i	−0.991501	+	0.130097i	$0.958471\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		0			0
$714$		0			0
$715$	12.0000	+	41.5692i	0.448775	+	1.55460i
$716$		0			0
$717$			10.3923i			0.388108i
$718$		0			0
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	20.7846i		−	0.772988i
$724$		0			0
$725$	42.0000			1.55984
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	48.0000			1.77534
$732$		0			0
$733$		−	6.92820i		−	0.255899i	−0.991781	−	0.127950i	$-0.959160\pi$
$733$		−	6.92820i		−	0.255899i	0.991781	−	0.127950i	$-0.0408395\pi$
$734$		0			0
$735$			24.2487i			0.894427i
$736$		0			0
$737$	−48.0000			−1.76810
$738$		0			0
$739$		−	27.7128i		−	1.01943i	−0.860343	−	0.509716i	$-0.829750\pi$
$739$		−	27.7128i		−	1.01943i	0.860343	−	0.509716i	$-0.170250\pi$
$740$		0			0
$741$	−24.0000	+	6.92820i	−0.881662	+	0.254514i
$742$		0			0
$743$		−	10.3923i		−	0.381257i	−0.981662	−	0.190628i	$-0.938947\pi$
$743$		−	10.3923i		−	0.381257i	0.981662	−	0.190628i	$-0.0610525\pi$
$744$		0			0
$745$	60.0000			2.19823
$746$		0			0
$747$			3.46410i			0.126745i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−28.0000			−1.02173			−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000			−1.02173			−0.510867	+	0.859660i	$0.670676\pi$
$752$		0			0
$753$	−12.0000			−0.437304
$754$		0			0
$755$	−24.0000			−0.873449
$756$		0			0
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	31.1769i		−	1.13016i	−0.825035	−	0.565081i	$-0.808845\pi$
$761$		−	31.1769i		−	1.13016i	0.825035	−	0.565081i	$-0.191155\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		−	20.7846i		−	0.751469i
$766$		0			0
$767$	12.0000	−	3.46410i	0.433295	−	0.125081i
$768$		0			0
$769$		−	27.7128i		−	0.999350i	−0.866213	−	0.499675i	$-0.833453\pi$
$769$		−	27.7128i		−	0.999350i	0.866213	−	0.499675i	$-0.166547\pi$
$770$		0			0
$771$	6.00000			0.216085
$772$		0			0
$773$		−	51.9615i		−	1.86893i	−0.356060	−	0.934463i	$-0.615880\pi$
$773$		−	51.9615i		−	1.86893i	0.356060	−	0.934463i	$-0.384120\pi$
$774$		0			0
$775$		−	48.4974i		−	1.74208i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000			−0.859889
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$	6.00000			0.214423
$784$		0			0
$785$		−	6.92820i		−	0.247278i
$786$		0			0
$787$			27.7128i			0.987855i	0.869503	+	0.493928i	$0.164439\pi$
$787$			27.7128i			0.987855i	−0.869503	+	0.493928i	$0.835561\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−10.0000	−	34.6410i	−0.355110	−	1.23014i
$794$		0			0
$795$			20.7846i			0.737154i
$796$		0			0
$797$	−30.0000			−1.06265			−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000			−1.06265			−0.531327	+	0.847167i	$0.678307\pi$
$798$		0			0
$799$		−	20.7846i		−	0.735307i
$800$		0			0
$801$			17.3205i			0.611990i
$802$		0			0
$803$	24.0000			0.846942
$804$		0			0
$805$		0			0
$806$		0			0
$807$	30.0000			1.05605
$808$		0			0
$809$	−18.0000			−0.632846			−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000			−0.632846			−0.316423	+	0.948618i	$0.602482\pi$
$810$		0			0
$811$		−	6.92820i		−	0.243282i	−0.992574	−	0.121641i	$-0.961184\pi$
$811$		−	6.92820i		−	0.243282i	0.992574	−	0.121641i	$-0.0388157\pi$
$812$		0			0
$813$		−	27.7128i		−	0.971931i
$814$		0			0
$815$	72.0000			2.52205
$816$		0			0
$817$			55.4256i			1.93910i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			3.46410i			0.120898i	0.998171	+	0.0604490i	$0.0192532\pi$
$821$			3.46410i			0.120898i	−0.998171	+	0.0604490i	$0.980747\pi$
$822$		0			0
$823$	−4.00000			−0.139431			−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000			−0.139431			−0.0697156	+	0.997567i	$0.522209\pi$
$824$		0			0
$825$		−	24.2487i		−	0.844232i
$826$		0			0
$827$		−	17.3205i		−	0.602293i	−0.953578	−	0.301147i	$-0.902631\pi$
$827$		−	17.3205i		−	0.602293i	0.953578	−	0.301147i	$-0.0973693\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$		0			0
$833$	42.0000			1.45521
$834$		0			0
$835$	−60.0000			−2.07639
$836$		0			0
$837$		−	6.92820i		−	0.239474i
$838$		0			0
$839$		−	10.3923i		−	0.358782i	−0.983778	−	0.179391i	$-0.942587\pi$
$839$		−	10.3923i		−	0.358782i	0.983778	−	0.179391i	$-0.0574128\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			17.3205i			0.596550i
$844$		0			0
$845$	24.0000	+	38.1051i	0.825625	+	1.31086i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	20.0000			0.686398
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	13.8564i		−	0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$		−	13.8564i		−	0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$		0			0
$855$	24.0000			0.820783
$856$		0			0
$857$	−6.00000			−0.204956			−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000			−0.204956			−0.102478	+	0.994735i	$0.532677\pi$
$858$		0			0
$859$	28.0000			0.955348			0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000			0.955348			0.477674	+	0.878537i	$0.341480\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			38.1051i			1.29711i	0.761166	+	0.648557i	$0.224627\pi$
$863$			38.1051i			1.29711i	−0.761166	+	0.648557i	$0.775373\pi$
$864$		0			0
$865$		−	20.7846i		−	0.706698i
$866$		0			0
$867$	−19.0000			−0.645274
$868$		0			0
$869$			27.7128i			0.940093i
$870$		0			0
$871$	−48.0000	+	13.8564i	−1.62642	+	0.469506i
$872$		0			0
$873$		−	6.92820i		−	0.234484i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	41.5692i		−	1.40369i	−0.712328	−	0.701846i	$-0.752359\pi$
$877$		−	41.5692i		−	1.40369i	0.712328	−	0.701846i	$-0.247641\pi$
$878$		0			0
$879$			3.46410i			0.116841i
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$	−12.0000			−0.403376
$886$		0			0
$887$	−48.0000			−1.61168			−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000			−1.61168			−0.805841	+	0.592132i	$0.798286\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	3.46410i		−	0.116052i
$892$		0			0
$893$	24.0000			0.803129
$894$		0			0
$895$			41.5692i			1.38951i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	41.5692i		−	1.38641i
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$			76.2102i			2.53331i
$906$		0			0
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$	12.0000			0.397142
$914$		0			0
$915$			34.6410i			1.14520i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−32.0000			−1.05558			−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000			−1.05558			−0.527791	+	0.849374i	$0.676980\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−36.0000	+	10.3923i	−1.18495	+	0.342067i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−4.00000			−0.131377
$928$		0			0
$929$			31.1769i			1.02288i	0.859319	+	0.511441i	$0.170888\pi$
$929$			31.1769i			1.02288i	−0.859319	+	0.511441i	$0.829112\pi$
$930$		0			0
$931$			48.4974i			1.58944i
$932$		0			0
$933$	−24.0000			−0.785725
$934$		0			0
$935$	−72.0000			−2.35465
$936$		0			0
$937$	22.0000			0.718709			0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000			0.718709			0.359354	+	0.933201i	$0.382997\pi$
$938$		0			0
$939$	−10.0000			−0.326338
$940$		0			0
$941$		−	10.3923i		−	0.338779i	−0.985549	−	0.169390i	$-0.945820\pi$
$941$		−	10.3923i		−	0.338779i	0.985549	−	0.169390i	$-0.0541797\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			45.0333i			1.46339i	0.681634	+	0.731693i	$0.261270\pi$
$947$			45.0333i			1.46339i	−0.681634	+	0.731693i	$0.738730\pi$
$948$		0			0
$949$	24.0000	−	6.92820i	0.779073	−	0.224899i
$950$		0			0
$951$		−	24.2487i		−	0.786318i
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	20.7846i		−	0.671871i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−17.0000			−0.548387
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$	−48.0000			−1.54517
$966$		0			0
$967$			34.6410i			1.11398i	0.830519	+	0.556990i	$0.188044\pi$
$967$			34.6410i			1.11398i	−0.830519	+	0.556990i	$0.811956\pi$
$968$		0			0
$969$		−	41.5692i		−	1.33540i
$970$		0			0
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−7.00000	−	24.2487i	−0.224179	−	0.776580i
$976$		0			0
$977$			24.2487i			0.775785i	0.921705	+	0.387893i	$0.126797\pi$
$977$			24.2487i			0.775785i	−0.921705	+	0.387893i	$0.873203\pi$
$978$		0			0
$979$	60.0000			1.91761
$980$		0			0
$981$			6.92820i			0.221201i
$982$		0			0
$983$			31.1769i			0.994389i	0.867639	+	0.497195i	$0.165636\pi$
$983$			31.1769i			0.994389i	−0.867639	+	0.497195i	$0.834364\pi$
$984$		0			0
$985$	−60.0000			−1.91176
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$			13.8564i			0.439720i
$994$		0			0
$995$			27.7128i			0.878555i
$996$		0			0
$997$	38.0000			1.20347			0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000			1.20347			0.601736	+	0.798695i	$0.294476\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.2.b.a.25.1	✓	2
3.2	odd	2		468.2.b.a.181.2		2
4.3	odd	2		624.2.c.f.337.1		2
5.2	odd	4		3900.2.j.h.649.3		4
5.3	odd	4		3900.2.j.h.649.1		4
5.4	even	2		3900.2.c.c.3301.1		2
7.6	odd	2		7644.2.e.g.4705.2		2
8.3	odd	2		2496.2.c.e.961.2		2
8.5	even	2		2496.2.c.l.961.2		2
12.11	even	2		1872.2.c.c.1585.2		2
13.2	odd	12		2028.2.i.i.529.1		4
13.3	even	3		2028.2.q.c.1837.1		2
13.4	even	6		2028.2.q.c.361.1		2
13.5	odd	4		2028.2.a.g.1.1		2
13.6	odd	12		2028.2.i.i.2005.1		4
13.7	odd	12		2028.2.i.i.2005.2		4
13.8	odd	4		2028.2.a.g.1.2		2
13.9	even	3		2028.2.q.b.361.1		2
13.10	even	6		2028.2.q.b.1837.1		2
13.11	odd	12		2028.2.i.i.529.2		4
13.12	even	2	inner	156.2.b.a.25.2	yes	2
39.5	even	4		6084.2.a.v.1.2		2
39.8	even	4		6084.2.a.v.1.1		2
39.38	odd	2		468.2.b.a.181.1		2
52.31	even	4		8112.2.a.bs.1.1		2
52.47	even	4		8112.2.a.bs.1.2		2
52.51	odd	2		624.2.c.f.337.2		2
65.12	odd	4		3900.2.j.h.649.4		4
65.38	odd	4		3900.2.j.h.649.2		4
65.64	even	2		3900.2.c.c.3301.2		2
91.90	odd	2		7644.2.e.g.4705.1		2
104.51	odd	2		2496.2.c.e.961.1		2
104.77	even	2		2496.2.c.l.961.1		2
156.155	even	2		1872.2.c.c.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.b.a.25.1	✓	2	1.1	even	1	trivial
156.2.b.a.25.2	yes	2	13.12	even	2	inner
468.2.b.a.181.1		2	39.38	odd	2
468.2.b.a.181.2		2	3.2	odd	2
624.2.c.f.337.1		2	4.3	odd	2
624.2.c.f.337.2		2	52.51	odd	2
1872.2.c.c.1585.1		2	156.155	even	2
1872.2.c.c.1585.2		2	12.11	even	2
2028.2.a.g.1.1		2	13.5	odd	4
2028.2.a.g.1.2		2	13.8	odd	4
2028.2.i.i.529.1		4	13.2	odd	12
2028.2.i.i.529.2		4	13.11	odd	12
2028.2.i.i.2005.1		4	13.6	odd	12
2028.2.i.i.2005.2		4	13.7	odd	12
2028.2.q.b.361.1		2	13.9	even	3
2028.2.q.b.1837.1		2	13.10	even	6
2028.2.q.c.361.1		2	13.4	even	6
2028.2.q.c.1837.1		2	13.3	even	3
2496.2.c.e.961.1		2	104.51	odd	2
2496.2.c.e.961.2		2	8.3	odd	2
2496.2.c.l.961.1		2	104.77	even	2
2496.2.c.l.961.2		2	8.5	even	2
3900.2.c.c.3301.1		2	5.4	even	2
3900.2.c.c.3301.2		2	65.64	even	2
3900.2.j.h.649.1		4	5.3	odd	4
3900.2.j.h.649.2		4	65.38	odd	4
3900.2.j.h.649.3		4	5.2	odd	4
3900.2.j.h.649.4		4	65.12	odd	4
6084.2.a.v.1.1		2	39.8	even	4
6084.2.a.v.1.2		2	39.5	even	4
7644.2.e.g.4705.1		2	91.90	odd	2
7644.2.e.g.4705.2		2	7.6	odd	2
8112.2.a.bs.1.1		2	52.31	even	4
8112.2.a.bs.1.2		2	52.47	even	4

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:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	156.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.46410i q^{5} +1.00000 q^{9} -3.46410i q^{11} +(-1.00000 - 3.46410i) q^{13} +3.46410i q^{15} +6.00000 q^{17} +6.92820i q^{19} -7.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +6.92820i q^{31} +3.46410i q^{33} +(1.00000 + 3.46410i) q^{39} +3.46410i q^{41} +8.00000 q^{43} -3.46410i q^{45} -3.46410i q^{47} +7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} -12.0000 q^{55} -6.92820i q^{57} +3.46410i q^{59} +10.0000 q^{61} +(-12.0000 + 3.46410i) q^{65} -13.8564i q^{67} -10.3923i q^{71} +6.92820i q^{73} +7.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} +3.46410i q^{83} -20.7846i q^{85} +6.00000 q^{87} +17.3205i q^{89} -6.92820i q^{93} +24.0000 q^{95} -6.92820i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} - 2 q^{13} + 12 q^{17} - 14 q^{25} - 2 q^{27} - 12 q^{29} + 2 q^{39} + 16 q^{43} + 14 q^{49} - 12 q^{51} + 12 q^{53} - 24 q^{55} + 20 q^{61} - 24 q^{65} + 14 q^{75} - 16 q^{79} + 2 q^{81}+ \cdots + 48 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 2 * q^13 + 12 * q^17 - 14 * q^25 - 2 * q^27 - 12 * q^29 + 2 * q^39 + 16 * q^43 + 14 * q^49 - 12 * q^51 + 12 * q^53 - 24 * q^55 + 20 * q^61 - 24 * q^65 + 14 * q^75 - 16 * q^79 + 2 * q^81 + 12 * q^87 + 48 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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