LMFDB - Embedded newform 2028.2.b.a.337.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(337,2028)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2028.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2028, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,0,0,2,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$16.1936615299$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2028.337
Dual form			2028.2.b.a.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.00000i q^{5} +2.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} +4.00000i q^{15} -2.00000 q^{17} -2.00000i q^{19} -2.00000i q^{21} -11.0000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -10.0000i q^{31} -4.00000i q^{33} +8.00000 q^{35} -10.0000i q^{37} +8.00000i q^{41} -4.00000 q^{43} -4.00000i q^{45} +4.00000i q^{47} +3.00000 q^{49} +2.00000 q^{51} -10.0000 q^{53} +16.0000 q^{55} +2.00000i q^{57} +8.00000i q^{59} -14.0000 q^{61} +2.00000i q^{63} +2.00000i q^{67} +16.0000i q^{71} +10.0000i q^{73} +11.0000 q^{75} -8.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} +8.00000i q^{85} +6.00000 q^{87} +4.00000i q^{89} +10.0000i q^{93} -8.00000 q^{95} -2.00000i q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} - 22 q^{25} - 2 q^{27} - 12 q^{29} + 16 q^{35} - 8 q^{43} + 6 q^{49} + 4 q^{51} - 20 q^{53} + 32 q^{55} - 28 q^{61} + 22 q^{75} - 16 q^{77} - 32 q^{79} + 2 q^{81} + 12 q^{87}+ \cdots - 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^17 - 22 * q^25 - 2 * q^27 - 12 * q^29 + 16 * q^35 - 8 * q^43 + 6 * q^49 + 4 * q^51 - 20 * q^53 + 32 * q^55 - 28 * q^61 + 22 * q^75 - 16 * q^77 - 32 * q^79 + 2 * q^81 + 12 * q^87 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 4.00000i	− 1.78885i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	− 4.00000i	− 1.78885i	0.447214	−	0.894427i	$-0.352416\pi$
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	4.00000i	1.03280i
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−11.0000	−2.20000
$26$	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
$31$	− 10.0000i	− 1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$	− 10.0000i	− 1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$	0	0
$33$	− 4.00000i	− 0.696311i
$34$	0	0
$35$	8.00000	1.35225
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.00000i	1.24939i	0.780869	+	0.624695i	$0.214777\pi$
$41$	8.00000i	1.24939i	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	− 4.00000i	− 0.596285i
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	16.0000	2.15744
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.0000i	1.89885i	0.313993	+	0.949425i	$0.398333\pi$
$71$	16.0000i	1.89885i	−0.313993	+	0.949425i	$0.601667\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	11.0000	1.27017
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	8.00000i	0.867722i
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	4.00000i	0.423999i	0.977270	+	0.212000i	$0.0679975\pi$
$89$	4.00000i	0.423999i	−0.977270	+	0.212000i	$0.932002\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	10.0000i	1.03695i
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	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
$105$	−8.00000	−0.780720
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	10.0000i	0.949158i
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 4.00000i	− 0.366679i
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 8.00000i	− 0.721336i
$124$	0	0
$125$	24.0000i	2.14663i
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	4.00000i	0.344265i
$136$	0	0
$137$	− 8.00000i	− 0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	− 8.00000i	− 0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	− 4.00000i	− 0.336861i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	24.0000i	1.99309i
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	− 18.0000i	− 1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$	− 18.0000i	− 1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−40.0000	−3.21288
$156$	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
$159$	10.0000	0.793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	0	0
$165$	−16.0000	−1.24560
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 2.00000i	− 0.152944i
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	− 22.0000i	− 1.66304i
$176$	0	0
$177$	− 8.00000i	− 0.601317i
$178$	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
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	−40.0000	−2.94086
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	− 2.00000i	− 0.141069i
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	32.0000	2.23498
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	− 16.0000i	− 1.09630i
$214$	0	0
$215$	16.0000i	1.09119i
$216$	0	0
$217$	20.0000	1.35769
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	0	0
$225$	−11.0000	−0.733333
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	2.00000i	0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$	2.00000i	0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	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
$235$	16.0000	1.04372
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	− 24.0000i	− 1.55243i	−0.630468	−	0.776215i	$-0.717137\pi$
$239$	− 24.0000i	− 1.55243i	0.630468	−	0.776215i	$-0.282863\pi$
$240$	0	0
$241$	2.00000i	0.128831i	0.997923	+	0.0644157i	$0.0205183\pi$
$241$	2.00000i	0.128831i	−0.997923	+	0.0644157i	$0.979482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	− 12.0000i	− 0.766652i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 8.00000i	− 0.500979i
$256$	0	0
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	20.0000	1.24274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	40.0000i	2.45718i
$266$	0	0
$267$	− 4.00000i	− 0.244796i
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	10.0000i	0.607457i	0.952759	+	0.303728i	$0.0982315\pi$
$271$	10.0000i	0.607457i	−0.952759	+	0.303728i	$0.901768\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 44.0000i	− 2.65330i
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	− 10.0000i	− 0.598684i
$280$	0	0
$281$	8.00000i	0.477240i	0.971113	+	0.238620i	$0.0766950\pi$
$281$	8.00000i	0.477240i	−0.971113	+	0.238620i	$0.923305\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	−16.0000	−0.944450
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.00000i	0.117242i
$292$	0	0
$293$	− 8.00000i	− 0.467365i	−0.972313	−	0.233682i	$-0.924922\pi$
$293$	− 8.00000i	− 0.467365i	0.972313	−	0.233682i	$-0.0750776\pi$
$294$	0	0
$295$	32.0000	1.86311
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	56.0000i	3.20655i
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	8.00000	0.450749
$316$	0	0
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	− 24.0000i	− 1.34374i
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.00000i	0.110600i
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	2.00000i	0.109930i	0.998488	+	0.0549650i	$0.0175047\pi$
$331$	2.00000i	0.109930i	−0.998488	+	0.0549650i	$0.982495\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	− 18.0000i	− 0.963518i	−0.876304	−	0.481759i	$-0.839998\pi$
$349$	− 18.0000i	− 0.963518i	0.876304	−	0.481759i	$-0.160002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 16.0000i	− 0.851594i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	− 16.0000i	− 0.851594i	0.904819	−	0.425797i	$-0.140006\pi$
$354$	0	0
$355$	64.0000	3.39677
$356$	0	0
$357$	4.00000i	0.211702i
$358$	0	0
$359$	12.0000i	0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$359$	12.0000i	0.633336i	−0.948536	+	0.316668i	$0.897436\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	40.0000	2.09370
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	8.00000i	0.416463i
$370$	0	0
$371$	− 20.0000i	− 1.03835i
$372$	0	0
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	0	0
$375$	− 24.0000i	− 1.23935i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	6.00000i	0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$	6.00000i	0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	0	0
$385$	32.0000i	1.63087i
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	64.0000i	3.22019i
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	12.0000i	0.599251i	0.954057	+	0.299626i	$0.0968618\pi$
$401$	12.0000i	0.599251i	−0.954057	+	0.299626i	$0.903138\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 4.00000i	− 0.198762i
$406$	0	0
$407$	40.0000	1.98273
$408$	0	0
$409$	14.0000i	0.692255i	0.938187	+	0.346128i	$0.112504\pi$
$409$	14.0000i	0.692255i	−0.938187	+	0.346128i	$0.887496\pi$
$410$	0	0
$411$	8.00000i	0.394611i
$412$	0	0
$413$	−16.0000	−0.787309
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	− 2.00000i	− 0.0974740i	−0.998812	−	0.0487370i	$-0.984480\pi$
$421$	− 2.00000i	− 0.0974740i	0.998812	−	0.0487370i	$-0.0155196\pi$
$422$	0	0
$423$	4.00000i	0.194487i
$424$	0	0
$425$	22.0000	1.06716
$426$	0	0
$427$	− 28.0000i	− 1.35501i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 20.0000i	− 0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$	− 20.0000i	− 0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	− 24.0000i	− 1.15071i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 12.0000i	− 0.566315i	−0.959073	−	0.283158i	$-0.908618\pi$
$449$	− 12.0000i	− 0.566315i	0.959073	−	0.283158i	$-0.0913819\pi$
$450$	0	0
$451$	−32.0000	−1.50682
$452$	0	0
$453$	18.0000i	0.845714i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	12.0000i	0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$	12.0000i	0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	0	0
$465$	40.0000	1.85496
$466$	0	0
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	22.0000i	1.00943i
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	16.0000i	0.731059i	0.930800	+	0.365529i	$0.119112\pi$
$479$	16.0000i	0.731059i	−0.930800	+	0.365529i	$0.880888\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.00000	−0.363261
$486$	0	0
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	0	0
$489$	2.00000i	0.0904431i
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	16.0000	0.719147
$496$	0	0
$497$	−32.0000	−1.43540
$498$	0	0
$499$	− 6.00000i	− 0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$	− 6.00000i	− 0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$	0	0
$501$	− 12.0000i	− 0.536120i
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	40.0000i	1.77998i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12.0000i	0.531891i	0.963988	+	0.265945i	$0.0856841\pi$
$509$	12.0000i	0.531891i	−0.963988	+	0.265945i	$0.914316\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	2.00000i	0.0883022i
$514$	0	0
$515$	− 32.0000i	− 1.41009i
$516$	0	0
$517$	−16.0000	−0.703679
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	22.0000i	0.960159i
$526$	0	0
$527$	20.0000i	0.871214i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000i	0.347170i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 48.0000i	− 2.07522i
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	12.0000i	0.516877i
$540$	0	0
$541$	− 38.0000i	− 1.63375i	−0.576816	−	0.816874i	$-0.695705\pi$
$541$	− 38.0000i	− 1.63375i	0.576816	−	0.816874i	$-0.304295\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	12.0000i	0.511217i
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	0	0
$555$	40.0000	1.69791
$556$	0	0
$557$	− 24.0000i	− 1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$	− 24.0000i	− 1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.00000i	0.337760i
$562$	0	0
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	− 24.0000i	− 1.00969i
$566$	0	0
$567$	2.00000i	0.0839921i
$568$	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
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	0	0
$579$	− 14.0000i	− 0.581820i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 40.0000i	− 1.65663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	12.0000i	0.493614i
$592$	0	0
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	4.00000	0.163709
$598$	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
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	2.00000i	0.0814463i
$604$	0	0
$605$	20.0000i	0.813116i
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	12.0000i	0.486265i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	0	0
$615$	−32.0000	−1.29036
$616$	0	0
$617$	− 48.0000i	− 1.93241i	−0.257780	−	0.966204i	$-0.582991\pi$
$617$	− 48.0000i	− 1.93241i	0.257780	−	0.966204i	$-0.417009\pi$
$618$	0	0
$619$	− 14.0000i	− 0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$	− 14.0000i	− 0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	20.0000i	0.797452i
$630$	0	0
$631$	30.0000i	1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$631$	30.0000i	1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	48.0000i	1.90482i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	16.0000i	0.632950i
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	26.0000i	1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$	26.0000i	1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$	0	0
$645$	− 16.0000i	− 0.629999i
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	−20.0000	−0.783862
$652$	0	0
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	− 16.0000i	− 0.625172i
$656$	0	0
$657$	10.0000i	0.390137i
$658$	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
$661$	− 10.0000i	− 0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$	− 10.0000i	− 0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 16.0000i	− 0.620453i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 14.0000i	− 0.541271i
$670$	0	0
$671$	− 56.0000i	− 2.16186i
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	11.0000	0.423390
$676$	0	0
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	4.00000i	0.153280i
$682$	0	0
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	0	0
$685$	−32.0000	−1.22266
$686$	0	0
$687$	− 2.00000i	− 0.0763048i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	10.0000i	0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$	10.0000i	0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 16.0000i	− 0.606043i
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	− 6.00000i	− 0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$	− 6.00000i	− 0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.0000i	0.896296i
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	16.0000i	0.595871i
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	66.0000	2.45118
$726$	0	0
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	0	0
$735$	12.0000i	0.442627i
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	− 22.0000i	− 0.809283i	−0.914475	−	0.404642i	$-0.867396\pi$
$739$	− 22.0000i	− 0.809283i	0.914475	−	0.404642i	$-0.132604\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	24.0000i	0.876941i
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	0	0
$755$	−72.0000	−2.62035
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 40.0000i	− 1.45000i	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	− 40.0000i	− 1.45000i	0.688749	−	0.724999i	$-0.258160\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	8.00000i	0.289241i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 22.0000i	− 0.793340i	−0.917961	−	0.396670i	$-0.870166\pi$
$769$	− 22.0000i	− 0.793340i	0.917961	−	0.396670i	$-0.129834\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	0	0
$773$	− 48.0000i	− 1.72644i	−0.504828	−	0.863220i	$-0.668444\pi$
$773$	− 48.0000i	− 1.72644i	0.504828	−	0.863220i	$-0.331556\pi$
$774$	0	0
$775$	110.000i	3.95132i
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	−64.0000	−2.29010
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	− 8.00000i	− 0.285532i
$786$	0	0
$787$	50.0000i	1.78231i	0.453701	+	0.891154i	$0.350103\pi$
$787$	50.0000i	1.78231i	−0.453701	+	0.891154i	$0.649897\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	12.0000i	0.426671i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 40.0000i	− 1.41865i
$796$	0	0
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	0	0
$799$	− 8.00000i	− 0.283020i
$800$	0	0
$801$	4.00000i	0.141333i
$802$	0	0
$803$	−40.0000	−1.41157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.0000	0.492823
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	6.00000i	0.210688i	0.994436	+	0.105344i	$0.0335944\pi$
$811$	6.00000i	0.210688i	−0.994436	+	0.105344i	$0.966406\pi$
$812$	0	0
$813$	− 10.0000i	− 0.350715i
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	8.00000i	0.279885i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.0000i	0.837606i	0.908077	+	0.418803i	$0.137550\pi$
$821$	24.0000i	0.837606i	−0.908077	+	0.418803i	$0.862450\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	44.0000i	1.53188i
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	48.0000	1.66111
$836$	0	0
$837$	10.0000i	0.345651i
$838$	0	0
$839$	16.0000i	0.552381i	0.961103	+	0.276191i	$0.0890721\pi$
$839$	16.0000i	0.552381i	−0.961103	+	0.276191i	$0.910928\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 8.00000i	− 0.275535i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	0	0
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	0	0
$865$	8.00000i	0.272008i
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	− 64.0000i	− 2.17105i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	−48.0000	−1.62270
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	8.00000i	0.269833i
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	32.0000	1.07689	0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000	1.07689	0.538443	+	0.842662i	$0.319013\pi$
$884$	0	0
$885$	−32.0000	−1.07567
$886$	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
$889$	− 24.0000i	− 0.804934i
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	48.0000i	1.60446i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	60.0000i	2.00111i
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	8.00000i	0.266223i
$904$	0	0
$905$	8.00000i	0.265929i
$906$	0	0
$907$	56.0000	1.85945	0.929725	−	0.368255i	$-0.120045\pi$
$907$	56.0000	1.85945	0.929725	+	0.368255i	$0.120045\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 56.0000i	− 1.85130i
$916$	0	0
$917$	8.00000i	0.264183i
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	22.0000i	0.724925i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	110.000i	3.61678i
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	16.0000i	0.524943i	0.964940	+	0.262471i	$0.0845376\pi$
$929$	16.0000i	0.524943i	−0.964940	+	0.262471i	$0.915462\pi$
$930$	0	0
$931$	− 6.00000i	− 0.196642i
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−32.0000	−1.04651
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	8.00000i	0.260793i	0.991462	+	0.130396i	$0.0416250\pi$
$941$	8.00000i	0.260793i	−0.991462	+	0.130396i	$0.958375\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	12.0000i	0.389127i
$952$	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
$955$	32.0000i	1.03550i
$956$	0	0
$957$	24.0000i	0.775810i
$958$	0	0
$959$	16.0000	0.516667
$960$	0	0
$961$	−69.0000	−2.22581
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	56.0000	1.80270
$966$	0	0
$967$	22.0000i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$	22.0000i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$	0	0
$969$	− 4.00000i	− 0.128499i
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	− 2.00000i	− 0.0638551i
$982$	0	0
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	−48.0000	−1.52941
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	0	0
$993$	− 2.00000i	− 0.0634681i
$994$	0	0
$995$	16.0000i	0.507234i
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	0	0
$999$	10.0000i	0.316386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.b.a.337.1		2
3.2	odd	2		6084.2.b.j.4393.2		2
13.2	odd	12		2028.2.i.e.529.1		2
13.3	even	3		2028.2.q.h.1837.1		4
13.4	even	6		2028.2.q.h.361.2		4
13.5	odd	4		156.2.a.a.1.1	✓	1
13.6	odd	12		2028.2.i.e.2005.1		2
13.7	odd	12		2028.2.i.g.2005.1		2
13.8	odd	4		2028.2.a.c.1.1		1
13.9	even	3		2028.2.q.h.361.1		4
13.10	even	6		2028.2.q.h.1837.2		4
13.11	odd	12		2028.2.i.g.529.1		2
13.12	even	2	inner	2028.2.b.a.337.2		2
39.5	even	4		468.2.a.d.1.1		1
39.8	even	4		6084.2.a.b.1.1		1
39.38	odd	2		6084.2.b.j.4393.1		2
52.31	even	4		624.2.a.e.1.1		1
52.47	even	4		8112.2.a.bi.1.1		1
65.18	even	4		3900.2.h.b.1249.1		2
65.44	odd	4		3900.2.a.m.1.1		1
65.57	even	4		3900.2.h.b.1249.2		2
91.83	even	4		7644.2.a.k.1.1		1
104.5	odd	4		2496.2.a.bc.1.1		1
104.83	even	4		2496.2.a.o.1.1		1
117.5	even	12		4212.2.i.b.2809.1		2
117.31	odd	12		4212.2.i.l.2809.1		2
117.70	odd	12		4212.2.i.l.1405.1		2
117.83	even	12		4212.2.i.b.1405.1		2
156.83	odd	4		1872.2.a.s.1.1		1
312.5	even	4		7488.2.a.c.1.1		1
312.83	odd	4		7488.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.a.a.1.1	✓	1	13.5	odd	4
468.2.a.d.1.1		1	39.5	even	4
624.2.a.e.1.1		1	52.31	even	4
1872.2.a.s.1.1		1	156.83	odd	4
2028.2.a.c.1.1		1	13.8	odd	4
2028.2.b.a.337.1		2	1.1	even	1	trivial
2028.2.b.a.337.2		2	13.12	even	2	inner
2028.2.i.e.529.1		2	13.2	odd	12
2028.2.i.e.2005.1		2	13.6	odd	12
2028.2.i.g.529.1		2	13.11	odd	12
2028.2.i.g.2005.1		2	13.7	odd	12
2028.2.q.h.361.1		4	13.9	even	3
2028.2.q.h.361.2		4	13.4	even	6
2028.2.q.h.1837.1		4	13.3	even	3
2028.2.q.h.1837.2		4	13.10	even	6
2496.2.a.o.1.1		1	104.83	even	4
2496.2.a.bc.1.1		1	104.5	odd	4
3900.2.a.m.1.1		1	65.44	odd	4
3900.2.h.b.1249.1		2	65.18	even	4
3900.2.h.b.1249.2		2	65.57	even	4
4212.2.i.b.1405.1		2	117.83	even	12
4212.2.i.b.2809.1		2	117.5	even	12
4212.2.i.l.1405.1		2	117.70	odd	12
4212.2.i.l.2809.1		2	117.31	odd	12
6084.2.a.b.1.1		1	39.8	even	4
6084.2.b.j.4393.1		2	39.38	odd	2
6084.2.b.j.4393.2		2	3.2	odd	2
7488.2.a.c.1.1		1	312.5	even	4
7488.2.a.d.1.1		1	312.83	odd	4
7644.2.a.k.1.1		1	91.83	even	4
8112.2.a.bi.1.1		1	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.00000i q^{5} +2.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} +4.00000i q^{15} -2.00000 q^{17} -2.00000i q^{19} -2.00000i q^{21} -11.0000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -10.0000i q^{31} -4.00000i q^{33} +8.00000 q^{35} -10.0000i q^{37} +8.00000i q^{41} -4.00000 q^{43} -4.00000i q^{45} +4.00000i q^{47} +3.00000 q^{49} +2.00000 q^{51} -10.0000 q^{53} +16.0000 q^{55} +2.00000i q^{57} +8.00000i q^{59} -14.0000 q^{61} +2.00000i q^{63} +2.00000i q^{67} +16.0000i q^{71} +10.0000i q^{73} +11.0000 q^{75} -8.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} +8.00000i q^{85} +6.00000 q^{87} +4.00000i q^{89} +10.0000i q^{93} -8.00000 q^{95} -2.00000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} - 22 q^{25} - 2 q^{27} - 12 q^{29} + 16 q^{35} - 8 q^{43} + 6 q^{49} + 4 q^{51} - 20 q^{53} + 32 q^{55} - 28 q^{61} + 22 q^{75} - 16 q^{77} - 32 q^{79} + 2 q^{81} + 12 q^{87}+ \cdots - 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^17 - 22 * q^25 - 2 * q^27 - 12 * q^29 + 16 * q^35 - 8 * q^43 + 6 * q^49 + 4 * q^51 - 20 * q^53 + 32 * q^55 - 28 * q^61 + 22 * q^75 - 16 * q^77 - 32 * q^79 + 2 * q^81 + 12 * q^87 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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