LMFDB - Embedded newform 2496.2.c.k.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,2,Mod(961,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2496.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.c (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:	$19.9306603445$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2496.961
Dual form			2496.2.c.k.961.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.46410i q^{7} +1.00000 q^{9} +3.46410i q^{11} +(1.00000 - 3.46410i) q^{13} -6.00000 q^{17} -3.46410i q^{19} -3.46410i q^{21} +5.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -3.46410i q^{31} +3.46410i q^{33} -6.92820i q^{37} +(1.00000 - 3.46410i) q^{39} -6.92820i q^{41} +4.00000 q^{43} +3.46410i q^{47} -5.00000 q^{49} -6.00000 q^{51} -6.00000 q^{53} -3.46410i q^{57} -10.3923i q^{59} +2.00000 q^{61} -3.46410i q^{63} +10.3923i q^{67} +3.46410i q^{71} +5.00000 q^{75} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +3.46410i q^{83} -6.00000 q^{87} -6.92820i q^{89} +(-12.0000 - 3.46410i) q^{91} -3.46410i q^{93} -13.8564i q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9} + 2 q^{13} - 12 q^{17} + 10 q^{25} + 2 q^{27} - 12 q^{29} + 2 q^{39} + 8 q^{43} - 10 q^{49} - 12 q^{51} - 12 q^{53} + 4 q^{61} + 10 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} - 12 q^{87}+ \cdots - 24 q^{91}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 2 * q^13 - 12 * q^17 + 10 * q^25 + 2 * q^27 - 12 * q^29 + 2 * q^39 + 8 * q^43 - 10 * q^49 - 12 * q^51 - 12 * q^53 + 4 * q^61 + 10 * q^75 + 24 * q^77 - 16 * q^79 + 2 * q^81 - 12 * q^87 - 24 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	3.46410i		−	1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$		−	3.46410i		−	1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	1.00000	−	3.46410i	0.277350	−	0.960769i
$13$	1.00000	−	3.46410i	0.277350	−	0.960769i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$		−	3.46410i		−	0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$		−	3.46410i		−	0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$		−	3.46410i		−	0.755929i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		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
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$			3.46410i			0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	6.92820i		−	1.13899i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$		−	6.92820i		−	1.13899i	0.821995	−	0.569495i	$-0.192861\pi$
$38$		0			0
$39$	1.00000	−	3.46410i	0.160128	−	0.554700i
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$	−6.00000			−0.840168
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	3.46410i		−	0.458831i
$58$		0			0
$59$		−	10.3923i		−	1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$		−	10.3923i		−	1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$		−	3.46410i		−	0.436436i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			10.3923i			1.26962i	0.772667	+	0.634811i	$0.218922\pi$
$67$			10.3923i			1.26962i	−0.772667	+	0.634811i	$0.781078\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	5.00000			0.577350
$76$		0			0
$77$	12.0000			1.36753
$78$		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
$85$		0			0
$86$		0			0
$87$	−6.00000			−0.643268
$88$		0			0
$89$		−	6.92820i		−	0.734388i	−0.930144	−	0.367194i	$-0.880318\pi$
$89$		−	6.92820i		−	0.734388i	0.930144	−	0.367194i	$-0.119682\pi$
$90$		0			0
$91$	−12.0000	−	3.46410i	−1.25794	−	0.363137i
$92$		0			0
$93$		−	3.46410i		−	0.359211i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	13.8564i		−	1.40690i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$		−	13.8564i		−	1.40690i	0.710742	−	0.703452i	$-0.248359\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		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	$-0.892344\pi$
$109$		−	6.92820i		−	0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$		0			0
$111$		−	6.92820i		−	0.657596i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.00000	−	3.46410i	0.0924500	−	0.320256i
$118$		0			0
$119$			20.7846i			1.90532i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	6.92820i		−	0.624695i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	−12.0000			−1.04053
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	20.7846i		−	1.77575i	−0.460086	−	0.887875i	$-0.652181\pi$
$137$		−	20.7846i		−	1.77575i	0.460086	−	0.887875i	$-0.347819\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$			3.46410i			0.291730i
$142$		0			0
$143$	12.0000	+	3.46410i	1.00349	+	0.289683i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.00000			−0.412393
$148$		0			0
$149$		−	13.8564i		−	1.13516i	−0.823318	−	0.567581i	$-0.807880\pi$
$149$		−	13.8564i		−	1.13516i	0.823318	−	0.567581i	$-0.192120\pi$
$150$		0			0
$151$			10.3923i			0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$			10.3923i			0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	3.46410i		−	0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$		−	3.46410i		−	0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.3205i			1.34030i	0.742225	+	0.670151i	$0.233770\pi$
$167$			17.3205i			1.34030i	−0.742225	+	0.670151i	$0.766230\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$		−	3.46410i		−	0.264906i
$172$		0			0
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		0			0
$175$		−	17.3205i		−	1.30931i
$176$		0			0
$177$		−	10.3923i		−	0.781133i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	20.7846i		−	1.51992i
$188$		0			0
$189$		−	3.46410i		−	0.251976i
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$		0			0
$201$			10.3923i			0.733017i
$202$		0			0
$203$			20.7846i			1.45879i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$			3.46410i			0.237356i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−12.0000			−0.814613
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	+	20.7846i	−0.403604	+	1.39812i
$222$		0			0
$223$		−	3.46410i		−	0.231973i	−0.993251	−	0.115987i	$-0.962997\pi$
$223$		−	3.46410i		−	0.231973i	0.993251	−	0.115987i	$-0.0370030\pi$
$224$		0			0
$225$	5.00000			0.333333
$226$		0			0
$227$			17.3205i			1.14960i	0.818293	+	0.574801i	$0.194921\pi$
$227$			17.3205i			1.14960i	−0.818293	+	0.574801i	$0.805079\pi$
$228$		0			0
$229$			6.92820i			0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$			6.92820i			0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$		0			0
$231$	12.0000			0.789542
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		0			0
$236$		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
$241$			13.8564i			0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$			13.8564i			0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−12.0000	−	3.46410i	−0.763542	−	0.220416i
$248$		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
$255$		0			0
$256$		0			0
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$	−24.0000			−1.49129
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	6.92820i		−	0.423999i
$268$		0			0
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$			10.3923i			0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$			10.3923i			0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$		0			0
$273$	−12.0000	−	3.46410i	−0.726273	−	0.209657i
$274$		0			0
$275$			17.3205i			1.04447i
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$		0			0
$279$		−	3.46410i		−	0.207390i
$280$		0			0
$281$			6.92820i			0.413302i	0.978415	+	0.206651i	$0.0662565\pi$
$281$			6.92820i			0.413302i	−0.978415	+	0.206651i	$0.933744\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.0000			−1.41668
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		−	13.8564i		−	0.812277i
$292$		0			0
$293$		−	27.7128i		−	1.61900i	−0.587120	−	0.809500i	$-0.699738\pi$
$293$		−	27.7128i		−	1.61900i	0.587120	−	0.809500i	$-0.300262\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			3.46410i			0.201008i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	13.8564i		−	0.798670i
$302$		0			0
$303$	−6.00000			−0.344691
$304$		0			0
$305$		0			0
$306$		0			0
$307$			10.3923i			0.593120i	0.955014	+	0.296560i	$0.0958395\pi$
$307$			10.3923i			0.593120i	−0.955014	+	0.296560i	$0.904160\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		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
$317$		−	13.8564i		−	0.778253i	−0.921184	−	0.389127i	$-0.872777\pi$
$317$		−	13.8564i		−	0.778253i	0.921184	−	0.389127i	$-0.127223\pi$
$318$		0			0
$319$		−	20.7846i		−	1.16371i
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$			20.7846i			1.15649i
$324$		0			0
$325$	5.00000	−	17.3205i	0.277350	−	0.960769i
$326$		0			0
$327$		−	6.92820i		−	0.383131i
$328$		0			0
$329$	12.0000			0.661581
$330$		0			0
$331$		−	3.46410i		−	0.190404i	−0.995458	−	0.0952021i	$-0.969650\pi$
$331$		−	3.46410i		−	0.190404i	0.995458	−	0.0952021i	$-0.0303497\pi$
$332$		0			0
$333$		−	6.92820i		−	0.379663i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$	12.0000			0.649836
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−36.0000			−1.93258			−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000			−1.93258			−0.966291	+	0.257454i	$0.917117\pi$
$348$		0			0
$349$			6.92820i			0.370858i	0.982658	+	0.185429i	$0.0593675\pi$
$349$			6.92820i			0.370858i	−0.982658	+	0.185429i	$0.940632\pi$
$350$		0			0
$351$	1.00000	−	3.46410i	0.0533761	−	0.184900i
$352$		0			0
$353$			34.6410i			1.84376i	0.387481	+	0.921878i	$0.373345\pi$
$353$			34.6410i			1.84376i	−0.387481	+	0.921878i	$0.626655\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			20.7846i			1.10004i
$358$		0			0
$359$			17.3205i			0.914141i	0.889430	+	0.457071i	$0.151101\pi$
$359$			17.3205i			0.914141i	−0.889430	+	0.457071i	$0.848899\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−16.0000			−0.835193			−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000			−0.835193			−0.417597	+	0.908633i	$0.637127\pi$
$368$		0			0
$369$		−	6.92820i		−	0.360668i
$370$		0			0
$371$			20.7846i			1.07908i
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.00000	+	20.7846i	−0.309016	+	1.07046i
$378$		0			0
$379$		−	17.3205i		−	0.889695i	−0.895606	−	0.444847i	$-0.853258\pi$
$379$		−	17.3205i		−	0.889695i	0.895606	−	0.444847i	$-0.146742\pi$
$380$		0			0
$381$	−8.00000			−0.409852
$382$		0			0
$383$			3.46410i			0.177007i	0.996076	+	0.0885037i	$0.0282085\pi$
$383$			3.46410i			0.177007i	−0.996076	+	0.0885037i	$0.971792\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.00000			0.203331
$388$		0			0
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$		0			0
$396$		0			0
$397$			34.6410i			1.73858i	0.494300	+	0.869291i	$0.335424\pi$
$397$			34.6410i			1.73858i	−0.494300	+	0.869291i	$0.664576\pi$
$398$		0			0
$399$	−12.0000			−0.600751
$400$		0			0
$401$			6.92820i			0.345978i	0.984924	+	0.172989i	$0.0553425\pi$
$401$			6.92820i			0.345978i	−0.984924	+	0.172989i	$0.944657\pi$
$402$		0			0
$403$	−12.0000	−	3.46410i	−0.597763	−	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$			27.7128i			1.37031i	0.728397	+	0.685155i	$0.240266\pi$
$409$			27.7128i			1.37031i	−0.728397	+	0.685155i	$0.759734\pi$
$410$		0			0
$411$		−	20.7846i		−	1.02523i
$412$		0			0
$413$	−36.0000			−1.77144
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.00000			0.195881
$418$		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
$421$			34.6410i			1.68830i	0.536107	+	0.844150i	$0.319894\pi$
$421$			34.6410i			1.68830i	−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$			3.46410i			0.168430i
$424$		0			0
$425$	−30.0000			−1.45521
$426$		0			0
$427$		−	6.92820i		−	0.335279i
$428$		0			0
$429$	12.0000	+	3.46410i	0.579365	+	0.167248i
$430$		0			0
$431$		−	24.2487i		−	1.16802i	−0.811747	−	0.584010i	$-0.801483\pi$
$431$		−	24.2487i		−	1.16802i	0.811747	−	0.584010i	$-0.198517\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$	−5.00000			−0.238095
$442$		0			0
$443$	36.0000			1.71041			0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000			1.71041			0.855206	+	0.518289i	$0.173431\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	13.8564i		−	0.655386i
$448$		0			0
$449$			6.92820i			0.326962i	0.986546	+	0.163481i	$0.0522723\pi$
$449$			6.92820i			0.326962i	−0.986546	+	0.163481i	$0.947728\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$		0			0
$453$			10.3923i			0.488273i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	27.7128i		−	1.29635i	−0.761491	−	0.648175i	$-0.775532\pi$
$457$		−	27.7128i		−	1.29635i	0.761491	−	0.648175i	$-0.224468\pi$
$458$		0			0
$459$	−6.00000			−0.280056
$460$		0			0
$461$			13.8564i			0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$			13.8564i			0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$		0			0
$463$		−	17.3205i		−	0.804952i	−0.915430	−	0.402476i	$-0.868150\pi$
$463$		−	17.3205i		−	0.804952i	0.915430	−	0.402476i	$-0.131850\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	36.0000			1.66233
$470$		0			0
$471$	−14.0000			−0.645086
$472$		0			0
$473$			13.8564i			0.637118i
$474$		0			0
$475$		−	17.3205i		−	0.794719i
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$		−	10.3923i		−	0.474837i	−0.971408	−	0.237418i	$-0.923699\pi$
$479$		−	10.3923i		−	0.474837i	0.971408	−	0.237418i	$-0.0763012\pi$
$480$		0			0
$481$	−24.0000	−	6.92820i	−1.09431	−	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			38.1051i			1.72671i	0.504599	+	0.863354i	$0.331640\pi$
$487$			38.1051i			1.72671i	−0.504599	+	0.863354i	$0.668360\pi$
$488$		0			0
$489$		−	3.46410i		−	0.156652i
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	36.0000			1.62136
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.0000			0.538274
$498$		0			0
$499$			10.3923i			0.465223i	0.972570	+	0.232612i	$0.0747271\pi$
$499$			10.3923i			0.465223i	−0.972570	+	0.232612i	$0.925273\pi$
$500$		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
$505$		0			0
$506$		0			0
$507$	−11.0000	−	6.92820i	−0.488527	−	0.307692i
$508$		0			0
$509$			41.5692i			1.84252i	0.388943	+	0.921262i	$0.372840\pi$
$509$			41.5692i			1.84252i	−0.388943	+	0.921262i	$0.627160\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	3.46410i		−	0.152944i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$	18.0000			0.790112
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$		0			0
$525$		−	17.3205i		−	0.755929i
$526$		0			0
$527$			20.7846i			0.905392i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		−	10.3923i		−	0.450988i
$532$		0			0
$533$	−24.0000	−	6.92820i	−1.03956	−	0.300094i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	12.0000			0.517838
$538$		0			0
$539$		−	17.3205i		−	0.746047i
$540$		0			0
$541$		−	6.92820i		−	0.297867i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$		−	6.92820i		−	0.297867i	0.988847	−	0.148933i	$-0.0475840\pi$
$542$		0			0
$543$	10.0000			0.429141
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−20.0000			−0.855138			−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000			−0.855138			−0.427569	+	0.903983i	$0.640630\pi$
$548$		0			0
$549$	2.00000			0.0853579
$550$		0			0
$551$			20.7846i			0.885454i
$552$		0			0
$553$			27.7128i			1.17847i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	13.8564i		−	0.587115i	−0.955941	−	0.293557i	$-0.905161\pi$
$557$		−	13.8564i		−	0.587115i	0.955941	−	0.293557i	$-0.0948392\pi$
$558$		0			0
$559$	4.00000	−	13.8564i	0.169182	−	0.586064i
$560$		0			0
$561$		−	20.7846i		−	0.877527i
$562$		0			0
$563$	12.0000			0.505740			0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000			0.505740			0.252870	+	0.967500i	$0.418626\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	3.46410i		−	0.145479i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$	24.0000			1.00261
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.0000			0.497844
$582$		0			0
$583$		−	20.7846i		−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	10.3923i		−	0.428936i	−0.976731	−	0.214468i	$-0.931198\pi$
$587$		−	10.3923i		−	0.428936i	0.976731	−	0.214468i	$-0.0688018\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$		0			0
$592$		0			0
$593$			6.92820i			0.284507i	0.989830	+	0.142254i	$0.0454349\pi$
$593$			6.92820i			0.284507i	−0.989830	+	0.142254i	$0.954565\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	16.0000			0.654836
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		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
$603$			10.3923i			0.423207i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		0			0
$609$			20.7846i			0.842235i
$610$		0			0
$611$	12.0000	+	3.46410i	0.485468	+	0.140143i
$612$		0			0
$613$			20.7846i			0.839482i	0.907644	+	0.419741i	$0.137879\pi$
$613$			20.7846i			0.839482i	−0.907644	+	0.419741i	$0.862121\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	6.92820i		−	0.278919i	−0.990228	−	0.139459i	$-0.955464\pi$
$617$		−	6.92820i		−	0.278919i	0.990228	−	0.139459i	$-0.0445365\pi$
$618$		0			0
$619$		−	31.1769i		−	1.25311i	−0.779379	−	0.626553i	$-0.784465\pi$
$619$		−	31.1769i		−	1.25311i	0.779379	−	0.626553i	$-0.215535\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−24.0000			−0.961540
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	12.0000			0.479234
$628$		0			0
$629$			41.5692i			1.65747i
$630$		0			0
$631$			38.1051i			1.51694i	0.651707	+	0.758470i	$0.274053\pi$
$631$			38.1051i			1.51694i	−0.651707	+	0.758470i	$0.725947\pi$
$632$		0			0
$633$	20.0000			0.794929
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.00000	+	17.3205i	−0.198107	+	0.686264i
$638$		0			0
$639$			3.46410i			0.137038i
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$			10.3923i			0.409832i	0.978780	+	0.204916i	$0.0656922\pi$
$643$			10.3923i			0.409832i	−0.978780	+	0.204916i	$0.934308\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$		0			0
$651$	−12.0000			−0.470317
$652$		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
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$			20.7846i			0.808428i	0.914665	+	0.404214i	$0.132455\pi$
$661$			20.7846i			0.808428i	−0.914665	+	0.404214i	$0.867545\pi$
$662$		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
$669$		−	3.46410i		−	0.133930i
$670$		0			0
$671$			6.92820i			0.267460i
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$	5.00000			0.192450
$676$		0			0
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		0			0
$679$	−48.0000			−1.84207
$680$		0			0
$681$			17.3205i			0.663723i
$682$		0			0
$683$			31.1769i			1.19295i	0.802631	+	0.596476i	$0.203433\pi$
$683$			31.1769i			1.19295i	−0.802631	+	0.596476i	$0.796567\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			6.92820i			0.264327i
$688$		0			0
$689$	−6.00000	+	20.7846i	−0.228582	+	0.791831i
$690$		0			0
$691$		−	45.0333i		−	1.71315i	−0.516024	−	0.856574i	$-0.672588\pi$
$691$		−	45.0333i		−	1.71315i	0.516024	−	0.856574i	$-0.327412\pi$
$692$		0			0
$693$	12.0000			0.455842
$694$		0			0
$695$		0			0
$696$		0			0
$697$			41.5692i			1.57455i
$698$		0			0
$699$	−6.00000			−0.226941
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$	−24.0000			−0.905177
$704$		0			0
$705$		0			0
$706$		0			0
$707$			20.7846i			0.781686i
$708$		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
$715$		0			0
$716$		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
$721$		−	27.7128i		−	1.03208i
$722$		0			0
$723$			13.8564i			0.515325i
$724$		0			0
$725$	−30.0000			−1.11417
$726$		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
$731$	−24.0000			−0.887672
$732$		0			0
$733$		−	34.6410i		−	1.27950i	−0.768585	−	0.639748i	$-0.779039\pi$
$733$		−	34.6410i		−	1.27950i	0.768585	−	0.639748i	$-0.220961\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−36.0000			−1.32608
$738$		0			0
$739$			38.1051i			1.40172i	0.713299	+	0.700860i	$0.247200\pi$
$739$			38.1051i			1.40172i	−0.713299	+	0.700860i	$0.752800\pi$
$740$		0			0
$741$	−12.0000	−	3.46410i	−0.440831	−	0.127257i
$742$		0			0
$743$			3.46410i			0.127086i	0.997979	+	0.0635428i	$0.0202399\pi$
$743$			3.46410i			0.127086i	−0.997979	+	0.0635428i	$0.979760\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			3.46410i			0.126745i
$748$		0			0
$749$			41.5692i			1.51891i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		0			0
$753$	12.0000			0.437304
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	48.4974i		−	1.75803i	−0.476794	−	0.879015i	$-0.658201\pi$
$761$		−	48.4974i		−	1.75803i	0.476794	−	0.879015i	$-0.341799\pi$
$762$		0			0
$763$	−24.0000			−0.868858
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−36.0000	−	10.3923i	−1.29988	−	0.375244i
$768$		0			0
$769$			27.7128i			0.999350i	0.866213	+	0.499675i	$0.166547\pi$
$769$			27.7128i			0.999350i	−0.866213	+	0.499675i	$0.833453\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		0			0
$773$			13.8564i			0.498380i	0.968455	+	0.249190i	$0.0801644\pi$
$773$			13.8564i			0.498380i	−0.968455	+	0.249190i	$0.919836\pi$
$774$		0			0
$775$		−	17.3205i		−	0.622171i
$776$		0			0
$777$	−24.0000			−0.860995
$778$		0			0
$779$	−24.0000			−0.859889
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$		0			0
$786$		0			0
$787$			10.3923i			0.370446i	0.982697	+	0.185223i	$0.0593007\pi$
$787$			10.3923i			0.370446i	−0.982697	+	0.185223i	$0.940699\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$			20.7846i			0.739016i
$792$		0			0
$793$	2.00000	−	6.92820i	0.0710221	−	0.246028i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	42.0000			1.48772			0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000			1.48772			0.743858	+	0.668338i	$0.232994\pi$
$798$		0			0
$799$		−	20.7846i		−	0.735307i
$800$		0			0
$801$		−	6.92820i		−	0.244796i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−6.00000			−0.211210
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$			38.1051i			1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$811$			38.1051i			1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$812$		0			0
$813$			10.3923i			0.364474i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$		0			0
$819$	−12.0000	−	3.46410i	−0.419314	−	0.121046i
$820$		0			0
$821$		−	13.8564i		−	0.483592i	−0.970327	−	0.241796i	$-0.922264\pi$
$821$		−	13.8564i		−	0.483592i	0.970327	−	0.241796i	$-0.0777365\pi$
$822$		0			0
$823$	−40.0000			−1.39431			−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000			−1.39431			−0.697156	+	0.716919i	$0.745552\pi$
$824$		0			0
$825$			17.3205i			0.603023i
$826$		0			0
$827$		−	24.2487i		−	0.843210i	−0.906780	−	0.421605i	$-0.861467\pi$
$827$		−	24.2487i		−	0.843210i	0.906780	−	0.421605i	$-0.138533\pi$
$828$		0			0
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$		0			0
$833$	30.0000			1.03944
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	3.46410i		−	0.119737i
$838$		0			0
$839$			3.46410i			0.119594i	0.998211	+	0.0597970i	$0.0190453\pi$
$839$			3.46410i			0.119594i	−0.998211	+	0.0597970i	$0.980955\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			6.92820i			0.238620i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			3.46410i			0.119028i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$		0			0
$852$		0			0
$853$			20.7846i			0.711651i	0.934552	+	0.355826i	$0.115800\pi$
$853$			20.7846i			0.711651i	−0.934552	+	0.355826i	$0.884200\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	42.0000			1.43469			0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000			1.43469			0.717346	+	0.696717i	$0.245357\pi$
$858$		0			0
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$	−24.0000			−0.817918
$862$		0			0
$863$			31.1769i			1.06127i	0.847599	+	0.530637i	$0.178047\pi$
$863$			31.1769i			1.06127i	−0.847599	+	0.530637i	$0.821953\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	19.0000			0.645274
$868$		0			0
$869$		−	27.7128i		−	0.940093i
$870$		0			0
$871$	36.0000	+	10.3923i	1.21981	+	0.352130i
$872$		0			0
$873$		−	13.8564i		−	0.468968i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	48.4974i		−	1.63764i	−0.574049	−	0.818821i	$-0.694628\pi$
$877$		−	48.4974i		−	1.63764i	0.574049	−	0.818821i	$-0.305372\pi$
$878$		0			0
$879$		−	27.7128i		−	0.934730i
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	48.0000			1.61168			0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000			1.61168			0.805841	+	0.592132i	$0.201714\pi$
$888$		0			0
$889$			27.7128i			0.929458i
$890$		0			0
$891$			3.46410i			0.116052i
$892$		0			0
$893$	12.0000			0.401565
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			20.7846i			0.693206i
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		−	13.8564i		−	0.461112i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	44.0000			1.46100			0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000			1.46100			0.730498	+	0.682915i	$0.239288\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	24.0000			0.795155			0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000			0.795155			0.397578	+	0.917568i	$0.369851\pi$
$912$		0			0
$913$	−12.0000			−0.397142
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	41.5692i		−	1.37274i
$918$		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
$921$			10.3923i			0.342438i
$922$		0			0
$923$	12.0000	+	3.46410i	0.394985	+	0.114022i
$924$		0			0
$925$		−	34.6410i		−	1.13899i
$926$		0			0
$927$	8.00000			0.262754
$928$		0			0
$929$		−	20.7846i		−	0.681921i	−0.940078	−	0.340960i	$-0.889248\pi$
$929$		−	20.7846i		−	0.681921i	0.940078	−	0.340960i	$-0.110752\pi$
$930$		0			0
$931$			17.3205i			0.567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$	10.0000			0.326338
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	51.9615i		−	1.68852i	−0.535932	−	0.844261i	$-0.680040\pi$
$947$		−	51.9615i		−	1.68852i	0.535932	−	0.844261i	$-0.319960\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		−	13.8564i		−	0.449325i
$952$		0			0
$953$	42.0000			1.36051			0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000			1.36051			0.680257	+	0.732974i	$0.261868\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	20.7846i		−	0.671871i
$958$		0			0
$959$	−72.0000			−2.32500
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$		0			0
$966$		0			0
$967$			10.3923i			0.334194i	0.985940	+	0.167097i	$0.0534393\pi$
$967$			10.3923i			0.334194i	−0.985940	+	0.167097i	$0.946561\pi$
$968$		0			0
$969$			20.7846i			0.667698i
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$		−	13.8564i		−	0.444216i
$974$		0			0
$975$	5.00000	−	17.3205i	0.160128	−	0.554700i
$976$		0			0
$977$			48.4974i			1.55157i	0.630997	+	0.775785i	$0.282646\pi$
$977$			48.4974i			1.55157i	−0.630997	+	0.775785i	$0.717354\pi$
$978$		0			0
$979$	24.0000			0.767043
$980$		0			0
$981$		−	6.92820i		−	0.221201i
$982$		0			0
$983$		−	51.9615i		−	1.65732i	−0.559756	−	0.828658i	$-0.689105\pi$
$983$		−	51.9615i		−	1.65732i	0.559756	−	0.828658i	$-0.310895\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	12.0000			0.381964
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$		−	3.46410i		−	0.109930i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−38.0000			−1.20347			−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000			−1.20347			−0.601736	+	0.798695i	$0.705524\pi$
$998$		0			0
$999$		−	6.92820i		−	0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.c.k.961.1		2
4.3	odd	2		2496.2.c.d.961.2		2
8.3	odd	2		624.2.c.e.337.2		2
8.5	even	2		39.2.b.a.25.2	yes	2
13.12	even	2	inner	2496.2.c.k.961.2		2
24.5	odd	2		117.2.b.a.64.1		2
24.11	even	2		1872.2.c.e.1585.2		2
40.13	odd	4		975.2.h.f.649.3		4
40.29	even	2		975.2.b.d.376.1		2
40.37	odd	4		975.2.h.f.649.2		4
52.51	odd	2		2496.2.c.d.961.1		2
56.13	odd	2		1911.2.c.d.883.2		2
104.5	odd	4		507.2.a.f.1.2		2
104.21	odd	4		507.2.a.f.1.1		2
104.29	even	6		507.2.j.a.316.1		2
104.37	odd	12		507.2.e.e.22.2		4
104.45	odd	12		507.2.e.e.484.1		4
104.51	odd	2		624.2.c.e.337.1		2
104.61	even	6		507.2.j.c.361.1		2
104.69	even	6		507.2.j.a.361.1		2
104.77	even	2		39.2.b.a.25.1	✓	2
104.83	even	4		8112.2.a.bv.1.1		2
104.85	odd	12		507.2.e.e.484.2		4
104.93	odd	12		507.2.e.e.22.1		4
104.99	even	4		8112.2.a.bv.1.2		2
104.101	even	6		507.2.j.c.316.1		2
312.5	even	4		1521.2.a.l.1.1		2
312.77	odd	2		117.2.b.a.64.2		2
312.125	even	4		1521.2.a.l.1.2		2
312.155	even	2		1872.2.c.e.1585.1		2
520.77	odd	4		975.2.h.f.649.4		4
520.389	even	2		975.2.b.d.376.2		2
520.493	odd	4		975.2.h.f.649.1		4
728.181	odd	2		1911.2.c.d.883.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.b.a.25.1	✓	2	104.77	even	2
39.2.b.a.25.2	yes	2	8.5	even	2
117.2.b.a.64.1		2	24.5	odd	2
117.2.b.a.64.2		2	312.77	odd	2
507.2.a.f.1.1		2	104.21	odd	4
507.2.a.f.1.2		2	104.5	odd	4
507.2.e.e.22.1		4	104.93	odd	12
507.2.e.e.22.2		4	104.37	odd	12
507.2.e.e.484.1		4	104.45	odd	12
507.2.e.e.484.2		4	104.85	odd	12
507.2.j.a.316.1		2	104.29	even	6
507.2.j.a.361.1		2	104.69	even	6
507.2.j.c.316.1		2	104.101	even	6
507.2.j.c.361.1		2	104.61	even	6
624.2.c.e.337.1		2	104.51	odd	2
624.2.c.e.337.2		2	8.3	odd	2
975.2.b.d.376.1		2	40.29	even	2
975.2.b.d.376.2		2	520.389	even	2
975.2.h.f.649.1		4	520.493	odd	4
975.2.h.f.649.2		4	40.37	odd	4
975.2.h.f.649.3		4	40.13	odd	4
975.2.h.f.649.4		4	520.77	odd	4
1521.2.a.l.1.1		2	312.5	even	4
1521.2.a.l.1.2		2	312.125	even	4
1872.2.c.e.1585.1		2	312.155	even	2
1872.2.c.e.1585.2		2	24.11	even	2
1911.2.c.d.883.1		2	728.181	odd	2
1911.2.c.d.883.2		2	56.13	odd	2
2496.2.c.d.961.1		2	52.51	odd	2
2496.2.c.d.961.2		2	4.3	odd	2
2496.2.c.k.961.1		2	1.1	even	1	trivial
2496.2.c.k.961.2		2	13.12	even	2	inner
8112.2.a.bv.1.1		2	104.83	even	4
8112.2.a.bv.1.2		2	104.99	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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.46410i q^{7} +1.00000 q^{9} +3.46410i q^{11} +(1.00000 - 3.46410i) q^{13} -6.00000 q^{17} -3.46410i q^{19} -3.46410i q^{21} +5.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -3.46410i q^{31} +3.46410i q^{33} -6.92820i q^{37} +(1.00000 - 3.46410i) q^{39} -6.92820i q^{41} +4.00000 q^{43} +3.46410i q^{47} -5.00000 q^{49} -6.00000 q^{51} -6.00000 q^{53} -3.46410i q^{57} -10.3923i q^{59} +2.00000 q^{61} -3.46410i q^{63} +10.3923i q^{67} +3.46410i q^{71} +5.00000 q^{75} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +3.46410i q^{83} -6.00000 q^{87} -6.92820i q^{89} +(-12.0000 - 3.46410i) q^{91} -3.46410i q^{93} -13.8564i q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9} + 2 q^{13} - 12 q^{17} + 10 q^{25} + 2 q^{27} - 12 q^{29} + 2 q^{39} + 8 q^{43} - 10 q^{49} - 12 q^{51} - 12 q^{53} + 4 q^{61} + 10 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} - 12 q^{87}+ \cdots - 24 q^{91}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 2 * q^13 - 12 * q^17 + 10 * q^25 + 2 * q^27 - 12 * q^29 + 2 * q^39 + 8 * q^43 - 10 * q^49 - 12 * q^51 - 12 * q^53 + 4 * q^61 + 10 * q^75 + 24 * q^77 - 16 * q^79 + 2 * q^81 - 12 * q^87 - 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more