LMFDB - Embedded newform 1216.2.c.d.609.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(609,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.c (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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1216.609
Dual form			1216.2.c.d.609.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.00000 q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.00000 q^{7} +2.00000 q^{9} -3.00000i q^{13} +3.00000 q^{17} +1.00000i q^{19} -3.00000i q^{21} -9.00000 q^{23} +5.00000 q^{25} -5.00000i q^{27} +9.00000i q^{29} +6.00000 q^{31} +6.00000i q^{37} -3.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} +2.00000 q^{49} -3.00000i q^{51} -9.00000i q^{53} +1.00000 q^{57} -3.00000i q^{59} -6.00000i q^{61} +6.00000 q^{63} -5.00000i q^{67} +9.00000i q^{69} +11.0000 q^{73} -5.00000i q^{75} -12.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +9.00000 q^{87} -9.00000i q^{91} -6.00000i q^{93} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q + 6 * q^7 + 4 * q^9	$ 2 q + 6 q^{7} + 4 q^{9} + 6 q^{17} - 18 q^{23} + 10 q^{25} + 12 q^{31} - 6 q^{39} + 12 q^{41} + 4 q^{49} + 2 q^{57} + 12 q^{63} + 22 q^{73} - 24 q^{79} + 2 q^{81} + 18 q^{87} - 16 q^{97}+O(q^{100}) $ 2 * q + 6 * q^7 + 4 * q^9 + 6 * q^17 - 18 * q^23 + 10 * q^25 + 12 * q^31 - 6 * q^39 + 12 * q^41 + 4 * q^49 + 2 * q^57 + 12 * q^63 + 22 * q^73 - 24 * q^79 + 2 * q^81 + 18 * q^87 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	− 3.00000i	− 0.654654i
$22$	0	0
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	9.00000i	1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$	9.00000i	1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	− 3.00000i	− 0.420084i
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	− 3.00000i	− 0.390567i	−0.980747	−	0.195283i	$-0.937437\pi$
$59$	− 3.00000i	− 0.390567i	0.980747	−	0.195283i	$-0.0625627\pi$
$60$	0	0
$61$	− 6.00000i	− 0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$	− 6.00000i	− 0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	0	0
$69$	9.00000i	1.08347i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	− 5.00000i	− 0.577350i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.00000	0.964901
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	− 9.00000i	− 0.943456i
$92$	0	0
$93$	− 6.00000i	− 0.622171i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	− 9.00000i	− 0.862044i	−0.902342	−	0.431022i	$-0.858153\pi$
$109$	− 9.00000i	− 0.862044i	0.902342	−	0.431022i	$-0.141847\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	9.00000	0.825029
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	12.0000i	1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$	12.0000i	1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$	0	0
$133$	3.00000i	0.260133i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	14.0000i	1.18746i	0.804663	+	0.593732i	$0.202346\pi$
$139$	14.0000i	1.18746i	−0.804663	+	0.593732i	$0.797654\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.00000i	− 0.164957i
$148$	0	0
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	24.0000i	1.91541i	0.287754	+	0.957704i	$0.407091\pi$
$157$	24.0000i	1.91541i	−0.287754	+	0.957704i	$0.592909\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	−27.0000	−2.12790
$162$	0	0
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	2.00000i	0.152944i
$172$	0	0
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	15.0000	1.13389
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	0	0
$181$	− 18.0000i	− 1.33793i	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	− 18.0000i	− 1.33793i	0.743294	−	0.668965i	$-0.233262\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 15.0000i	− 1.09109i
$190$	0	0
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	27.0000i	1.89503i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.0000	−1.25109
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 5.00000i	− 0.344214i	−0.985078	−	0.172107i	$-0.944942\pi$
$211$	− 5.00000i	− 0.344214i	0.985078	−	0.172107i	$-0.0550575\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	− 11.0000i	− 0.743311i
$220$	0	0
$221$	− 9.00000i	− 0.605406i
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	0	0
$225$	10.0000	0.666667
$226$	0	0
$227$	27.0000i	1.79205i	0.444001	+	0.896026i	$0.353559\pi$
$227$	27.0000i	1.79205i	−0.444001	+	0.896026i	$0.646441\pi$
$228$	0	0
$229$	− 6.00000i	− 0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$	− 6.00000i	− 0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$	0	0
$231$	0	0
$232$	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
$237$	12.0000i	0.779484i
$238$	0	0
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.00000	0.190885
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	24.0000i	1.51487i	0.652913	+	0.757433i	$0.273547\pi$
$251$	24.0000i	1.51487i	−0.652913	+	0.757433i	$0.726453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	18.0000i	1.11847i
$260$	0	0
$261$	18.0000i	1.11417i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 18.0000i	− 1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$	− 18.0000i	− 1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$	0	0
$271$	9.00000	0.546711	0.273356	−	0.961913i	$-0.411866\pi$
$271$	9.00000	0.546711	0.273356	+	0.961913i	$0.411866\pi$
$272$	0	0
$273$	−9.00000	−0.544705
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	12.0000	0.718421
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.0000	1.06251
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	8.00000i	0.468968i
$292$	0	0
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	27.0000i	1.56145i
$300$	0	0
$301$	− 24.0000i	− 1.38334i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	− 6.00000i	− 0.341328i
$310$	0	0
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	0	0
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.00000i	− 0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$	− 9.00000i	− 0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	3.00000	0.167444
$322$	0	0
$323$	3.00000i	0.166924i
$324$	0	0
$325$	− 15.0000i	− 0.832050i
$326$	0	0
$327$	−9.00000	−0.497701
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 1.00000i	− 0.0549650i	−0.999622	−	0.0274825i	$-0.991251\pi$
$331$	− 1.00000i	− 0.0549650i	0.999622	−	0.0274825i	$-0.00874905\pi$
$332$	0	0
$333$	12.0000i	0.657596i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	0	0
$339$	12.0000i	0.651751i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	0	0
$349$	24.0000i	1.28469i	0.766415	+	0.642345i	$0.222038\pi$
$349$	24.0000i	1.28469i	−0.766415	+	0.642345i	$0.777962\pi$
$350$	0	0
$351$	−15.0000	−0.800641
$352$	0	0
$353$	−27.0000	−1.43706	−0.718532	−	0.695493i	$-0.755186\pi$
$353$	−27.0000	−1.43706	−0.718532	+	0.695493i	$0.755186\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 9.00000i	− 0.476331i
$358$	0	0
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	− 27.0000i	− 1.40177i
$372$	0	0
$373$	9.00000i	0.466002i	0.972476	+	0.233001i	$0.0748546\pi$
$373$	9.00000i	0.466002i	−0.972476	+	0.233001i	$0.925145\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	27.0000	1.39057
$378$	0	0
$379$	25.0000i	1.28416i	0.766636	+	0.642082i	$0.221929\pi$
$379$	25.0000i	1.28416i	−0.766636	+	0.642082i	$0.778071\pi$
$380$	0	0
$381$	18.0000i	0.922168i
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 16.0000i	− 0.813326i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	−27.0000	−1.36545
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	0	0
$399$	3.00000	0.150188
$400$	0	0
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	− 18.0000i	− 0.896644i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	− 9.00000i	− 0.443937i
$412$	0	0
$413$	− 9.00000i	− 0.442861i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	36.0000i	1.75872i	0.476162	+	0.879358i	$0.342028\pi$
$419$	36.0000i	1.75872i	−0.476162	+	0.879358i	$0.657972\pi$
$420$	0	0
$421$	27.0000i	1.31590i	0.753062	+	0.657950i	$0.228576\pi$
$421$	27.0000i	1.31590i	−0.753062	+	0.657950i	$0.771424\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.0000	0.727607
$426$	0	0
$427$	− 18.0000i	− 0.871081i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	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
$435$	0	0
$436$	0	0
$437$	− 9.00000i	− 0.430528i
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.0000	1.63723	0.818615	−	0.574342i	$-0.194742\pi$
$457$	35.0000	1.63723	0.818615	+	0.574342i	$0.194742\pi$
$458$	0	0
$459$	− 15.0000i	− 0.700140i
$460$	0	0
$461$	− 36.0000i	− 1.67669i	−0.545142	−	0.838344i	$-0.683524\pi$
$461$	− 36.0000i	− 1.67669i	0.545142	−	0.838344i	$-0.316476\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 24.0000i	− 1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$	− 24.0000i	− 1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	0	0
$469$	− 15.0000i	− 0.692636i
$470$	0	0
$471$	24.0000	1.10586
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.00000i	0.229416i
$476$	0	0
$477$	− 18.0000i	− 0.824163i
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	27.0000i	1.22854i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	24.0000i	1.08310i	0.840667	+	0.541552i	$0.182163\pi$
$491$	24.0000i	1.08310i	−0.840667	+	0.541552i	$0.817837\pi$
$492$	0	0
$493$	27.0000i	1.21602i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 32.0000i	− 1.43252i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$	− 32.0000i	− 1.43252i	0.697835	−	0.716258i	$-0.254147\pi$
$500$	0	0
$501$	18.0000i	0.804181i
$502$	0	0
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.00000i	− 0.177646i
$508$	0	0
$509$	18.0000i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$	18.0000i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$	0	0
$511$	33.0000	1.45983
$512$	0	0
$513$	5.00000	0.220755
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	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
$523$	− 29.0000i	− 1.26808i	−0.773300	−	0.634041i	$-0.781395\pi$
$523$	− 29.0000i	− 1.26808i	0.773300	−	0.634041i	$-0.218605\pi$
$524$	0	0
$525$	− 15.0000i	− 0.654654i
$526$	0	0
$527$	18.0000	0.784092
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	− 6.00000i	− 0.260378i
$532$	0	0
$533$	− 18.0000i	− 0.779667i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 30.0000i	− 1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$	− 30.0000i	− 1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	− 12.0000i	− 0.512148i
$550$	0	0
$551$	−9.00000	−0.383413
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.0000i	1.52537i	0.646771	+	0.762684i	$0.276119\pi$
$557$	36.0000i	1.52537i	−0.646771	+	0.762684i	$0.723881\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	14.0000i	0.585882i	0.956131	+	0.292941i	$0.0946339\pi$
$571$	14.0000i	0.585882i	−0.956131	+	0.292941i	$0.905366\pi$
$572$	0	0
$573$	− 9.00000i	− 0.375980i
$574$	0	0
$575$	−45.0000	−1.87663
$576$	0	0
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	0	0
$579$	4.00000i	0.166234i
$580$	0	0
$581$	− 18.0000i	− 0.746766i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	6.00000i	0.247226i
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 15.0000i	− 0.613909i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	− 10.0000i	− 0.407231i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	27.0000	1.09410
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 18.0000i	− 0.727013i	−0.931592	−	0.363507i	$-0.881579\pi$
$613$	− 18.0000i	− 0.727013i	0.931592	−	0.363507i	$-0.118421\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	− 10.0000i	− 0.401934i	−0.979598	−	0.200967i	$-0.935592\pi$
$619$	− 10.0000i	− 0.401934i	0.979598	−	0.200967i	$-0.0644084\pi$
$620$	0	0
$621$	45.0000i	1.80579i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.0000i	0.717707i
$630$	0	0
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	0	0
$633$	−5.00000	−0.198732
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 18.0000i	− 0.705476i
$652$	0	0
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	22.0000	0.858302
$658$	0	0
$659$	9.00000i	0.350590i	0.984516	+	0.175295i	$0.0560880\pi$
$659$	9.00000i	0.350590i	−0.984516	+	0.175295i	$0.943912\pi$
$660$	0	0
$661$	45.0000i	1.75030i	0.483854	+	0.875149i	$0.339236\pi$
$661$	45.0000i	1.75030i	−0.483854	+	0.875149i	$0.660764\pi$
$662$	0	0
$663$	−9.00000	−0.349531
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 81.0000i	− 3.13633i
$668$	0	0
$669$	6.00000i	0.231973i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	0	0
$675$	− 25.0000i	− 0.962250i
$676$	0	0
$677$	9.00000i	0.345898i	0.984931	+	0.172949i	$0.0553296\pi$
$677$	9.00000i	0.345898i	−0.984931	+	0.172949i	$0.944670\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	0	0
$681$	27.0000	1.03464
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	−27.0000	−1.02862
$690$	0	0
$691$	− 46.0000i	− 1.74992i	−0.484193	−	0.874961i	$-0.660887\pi$
$691$	− 46.0000i	− 1.74992i	0.484193	−	0.874961i	$-0.339113\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.0000	0.681799
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	36.0000i	1.35970i	0.733351	+	0.679851i	$0.237955\pi$
$701$	36.0000i	1.35970i	−0.733351	+	0.679851i	$0.762045\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 42.0000i	− 1.57734i	−0.614815	−	0.788672i	$-0.710769\pi$
$709$	− 42.0000i	− 1.57734i	0.614815	−	0.788672i	$-0.289231\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	0	0
$713$	−54.0000	−2.02232
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 9.00000i	− 0.336111i
$718$	0	0
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	10.0000i	0.371904i
$724$	0	0
$725$	45.0000i	1.67126i
$726$	0	0
$727$	3.00000	0.111264	0.0556319	−	0.998451i	$-0.482283\pi$
$727$	3.00000	0.111264	0.0556319	+	0.998451i	$0.482283\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 24.0000i	− 0.887672i
$732$	0	0
$733$	− 36.0000i	− 1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$	− 36.0000i	− 1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	0	0
$741$	− 3.00000i	− 0.110208i
$742$	0	0
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	9.00000i	0.328853i
$750$	0	0
$751$	−30.0000	−1.09472	−0.547358	−	0.836899i	$-0.684366\pi$
$751$	−30.0000	−1.09472	−0.547358	+	0.836899i	$0.684366\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.00000i	0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$	6.00000i	0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.0000	1.19625	0.598125	−	0.801403i	$-0.295913\pi$
$761$	33.0000	1.19625	0.598125	+	0.801403i	$0.295913\pi$
$762$	0	0
$763$	− 27.0000i	− 0.977466i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.00000	−0.324971
$768$	0	0
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	0	0
$771$	24.0000i	0.864339i
$772$	0	0
$773$	9.00000i	0.323708i	0.986815	+	0.161854i	$0.0517473\pi$
$773$	9.00000i	0.323708i	−0.986815	+	0.161854i	$0.948253\pi$
$774$	0	0
$775$	30.0000	1.07763
$776$	0	0
$777$	18.0000	0.645746
$778$	0	0
$779$	6.00000i	0.214972i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	45.0000	1.60817
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23.0000i	0.819861i	0.912117	+	0.409931i	$0.134447\pi$
$787$	23.0000i	0.819861i	−0.912117	+	0.409931i	$0.865553\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	−18.0000	−0.639199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 27.0000i	− 0.956389i	−0.878254	−	0.478195i	$-0.841291\pi$
$797$	− 27.0000i	− 0.956389i	0.878254	−	0.478195i	$-0.158709\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	43.0000i	1.50993i	0.655763	+	0.754967i	$0.272347\pi$
$811$	43.0000i	1.50993i	−0.655763	+	0.754967i	$0.727653\pi$
$812$	0	0
$813$	− 9.00000i	− 0.315644i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	− 18.0000i	− 0.628971i
$820$	0	0
$821$	18.0000i	0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$	18.0000i	0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$	0	0
$823$	9.00000	0.313720	0.156860	−	0.987621i	$-0.449863\pi$
$823$	9.00000	0.313720	0.156860	+	0.987621i	$0.449863\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 39.0000i	− 1.35616i	−0.734987	−	0.678081i	$-0.762812\pi$
$827$	− 39.0000i	− 1.35616i	0.734987	−	0.678081i	$-0.237188\pi$
$828$	0	0
$829$	− 21.0000i	− 0.729360i	−0.931133	−	0.364680i	$-0.881178\pi$
$829$	− 21.0000i	− 0.729360i	0.931133	−	0.364680i	$-0.118822\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 30.0000i	− 1.03695i
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−52.0000	−1.79310
$842$	0	0
$843$	24.0000i	0.826604i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	33.0000	1.13389
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	− 54.0000i	− 1.85110i
$852$	0	0
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	22.0000i	0.750630i	0.926897	+	0.375315i	$0.122466\pi$
$859$	22.0000i	0.750630i	−0.926897	+	0.375315i	$0.877534\pi$
$860$	0	0
$861$	− 18.0000i	− 0.613438i
$862$	0	0
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−15.0000	−0.508256
$872$	0	0
$873$	−16.0000	−0.541518
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 21.0000i	− 0.709120i	−0.935033	−	0.354560i	$-0.884631\pi$
$877$	− 21.0000i	− 0.709120i	0.935033	−	0.354560i	$-0.115369\pi$
$878$	0	0
$879$	9.00000	0.303562
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	− 34.0000i	− 1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$	− 34.0000i	− 1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	−54.0000	−1.81110
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	27.0000	0.901504
$898$	0	0
$899$	54.0000i	1.80100i
$900$	0	0
$901$	− 27.0000i	− 0.899500i
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 17.0000i	− 0.564476i	−0.959344	−	0.282238i	$-0.908923\pi$
$907$	− 17.0000i	− 0.564476i	0.959344	−	0.282238i	$-0.0910767\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.0000i	1.18882i
$918$	0	0
$919$	−3.00000	−0.0989609	−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000	−0.0989609	−0.0494804	+	0.998775i	$0.515757\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	0	0
$924$	0	0
$925$	30.0000i	0.986394i
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	2.00000i	0.0655474i
$932$	0	0
$933$	9.00000i	0.294647i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	17.0000i	0.554774i
$940$	0	0
$941$	− 27.0000i	− 0.880175i	−0.897955	−	0.440087i	$-0.854947\pi$
$941$	− 27.0000i	− 0.880175i	0.897955	−	0.440087i	$-0.145053\pi$
$942$	0	0
$943$	−54.0000	−1.75848
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	0	0
$949$	− 33.0000i	− 1.07123i
$950$	0	0
$951$	−9.00000	−0.291845
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	27.0000	0.871875
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	− 24.0000i	− 0.770197i	−0.922876	−	0.385098i	$-0.874168\pi$
$971$	− 24.0000i	− 0.770197i	0.922876	−	0.385098i	$-0.125832\pi$
$972$	0	0
$973$	42.0000i	1.34646i
$974$	0	0
$975$	−15.0000	−0.480384
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 18.0000i	− 0.574696i
$982$	0	0
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	72.0000i	2.28947i
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	−1.00000	−0.0317340
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	0	0
$999$	30.0000	0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.c.d.609.1	yes	2
4.3	odd	2		1216.2.c.a.609.2	yes	2
8.3	odd	2		1216.2.c.a.609.1	✓	2
8.5	even	2	inner	1216.2.c.d.609.2	yes	2
16.3	odd	4		4864.2.a.e.1.1		1
16.5	even	4		4864.2.a.d.1.1		1
16.11	odd	4		4864.2.a.m.1.1		1
16.13	even	4		4864.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.a.609.1	✓	2	8.3	odd	2
1216.2.c.a.609.2	yes	2	4.3	odd	2
1216.2.c.d.609.1	yes	2	1.1	even	1	trivial
1216.2.c.d.609.2	yes	2	8.5	even	2	inner
4864.2.a.d.1.1		1	16.5	even	4
4864.2.a.e.1.1		1	16.3	odd	4
4864.2.a.l.1.1		1	16.13	even	4
4864.2.a.m.1.1		1	16.11	odd	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.00000 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.00000 q^{7} +2.00000 q^{9} -3.00000i q^{13} +3.00000 q^{17} +1.00000i q^{19} -3.00000i q^{21} -9.00000 q^{23} +5.00000 q^{25} -5.00000i q^{27} +9.00000i q^{29} +6.00000 q^{31} +6.00000i q^{37} -3.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} +2.00000 q^{49} -3.00000i q^{51} -9.00000i q^{53} +1.00000 q^{57} -3.00000i q^{59} -6.00000i q^{61} +6.00000 q^{63} -5.00000i q^{67} +9.00000i q^{69} +11.0000 q^{73} -5.00000i q^{75} -12.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +9.00000 q^{87} -9.00000i q^{91} -6.00000i q^{93} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q + 6 * q^7 + 4 * q^9	\( 2 q + 6 q^{7} + 4 q^{9} + 6 q^{17} - 18 q^{23} + 10 q^{25} + 12 q^{31} - 6 q^{39} + 12 q^{41} + 4 q^{49} + 2 q^{57} + 12 q^{63} + 22 q^{73} - 24 q^{79} + 2 q^{81} + 18 q^{87} - 16 q^{97}+O(q^{100}) \) 2 * q + 6 * q^7 + 4 * q^9 + 6 * q^17 - 18 * q^23 + 10 * q^25 + 12 * q^31 - 6 * q^39 + 12 * q^41 + 4 * q^49 + 2 * q^57 + 12 * q^63 + 22 * q^73 - 24 * q^79 + 2 * q^81 + 18 * q^87 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more