LMFDB - Embedded newform 4028.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.a (trivial)

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:	yes
Analytic conductor:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +2.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} +2.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +4.00000 q^{13} +2.00000 q^{15} +7.00000 q^{17} +1.00000 q^{19} -4.00000 q^{21} -8.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} -6.00000 q^{29} +9.00000 q^{31} -3.00000 q^{33} -8.00000 q^{35} -4.00000 q^{37} +4.00000 q^{39} +6.00000 q^{41} -11.0000 q^{43} -4.00000 q^{45} +4.00000 q^{47} +9.00000 q^{49} +7.00000 q^{51} -1.00000 q^{53} -6.00000 q^{55} +1.00000 q^{57} +4.00000 q^{59} -10.0000 q^{61} +8.00000 q^{63} +8.00000 q^{65} -13.0000 q^{67} -8.00000 q^{69} -15.0000 q^{71} -1.00000 q^{75} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +14.0000 q^{85} -6.00000 q^{87} +8.00000 q^{89} -16.0000 q^{91} +9.00000 q^{93} +2.00000 q^{95} +4.00000 q^{97} +6.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	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
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	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
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	−4.00000	−0.596285
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	7.00000	0.980196
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	8.00000	1.00791
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	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
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	14.0000	1.51851
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	−16.0000	−1.67726
$92$	0	0
$93$	9.00000	0.933257
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	0	0
$117$	−8.00000	−0.739600
$118$	0	0
$119$	−28.0000	−2.56676
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	−10.0000	−0.860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−14.0000	−1.13183
$154$	0	0
$155$	18.0000	1.44579
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	32.0000	2.52195
$162$	0	0
$163$	−21.0000	−1.64485	−0.822423	−	0.568876i	$-0.807379\pi$
$163$	−21.0000	−1.64485	−0.822423	+	0.568876i	$0.807379\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	25.0000	1.93456	0.967279	−	0.253715i	$-0.0816525\pi$
$167$	25.0000	1.93456	0.967279	+	0.253715i	$0.0816525\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−21.0000	−1.53567
$188$	0	0
$189$	20.0000	1.45479
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	0	0
$195$	8.00000	0.572892
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	0	0
$201$	−13.0000	−0.916949
$202$	0	0
$203$	24.0000	1.68447
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	16.0000	1.11208
$208$	0	0
$209$	−3.00000	−0.207514
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	0	0
$213$	−15.0000	−1.02778
$214$	0	0
$215$	−22.0000	−1.50039
$216$	0	0
$217$	−36.0000	−2.44384
$218$	0	0
$219$	0	0
$220$	0	0
$221$	28.0000	1.88348
$222$	0	0
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−28.0000	−1.81117	−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000	−1.81117	−0.905585	+	0.424165i	$0.860568\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	18.0000	1.14998
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	14.0000	0.876714
$256$	0	0
$257$	25.0000	1.55946	0.779729	−	0.626118i	$-0.215357\pi$
$257$	25.0000	1.55946	0.779729	+	0.626118i	$0.215357\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	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
$273$	−16.0000	−0.968364
$274$	0	0
$275$	3.00000	0.180907
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−24.0000	−1.41668
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	15.0000	0.870388
$298$	0	0
$299$	−32.0000	−1.85061
$300$	0	0
$301$	44.0000	2.53612
$302$	0	0
$303$	−16.0000	−0.919176
$304$	0	0
$305$	−20.0000	−1.14520
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	17.0000	0.963982	0.481991	−	0.876176i	$-0.339914\pi$
$311$	17.0000	0.963982	0.481991	+	0.876176i	$0.339914\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	16.0000	0.901498
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	7.00000	0.389490
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	−26.0000	−1.42053
$336$	0	0
$337$	−21.0000	−1.14394	−0.571971	−	0.820274i	$-0.693821\pi$
$337$	−21.0000	−1.14394	−0.571971	+	0.820274i	$0.693821\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−27.0000	−1.46213
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−30.0000	−1.59223
$356$	0	0
$357$	−28.0000	−1.48192
$358$	0	0
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	5.00000	0.258890	0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000	0.258890	0.129445	+	0.991587i	$0.458680\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	24.0000	1.22315
$386$	0	0
$387$	22.0000	1.11832
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−56.0000	−2.83204
$392$	0	0
$393$	−9.00000	−0.453990
$394$	0	0
$395$	−16.0000	−0.805047
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	35.0000	1.74782	0.873908	−	0.486091i	$-0.161578\pi$
$401$	35.0000	1.74782	0.873908	+	0.486091i	$0.161578\pi$
$402$	0	0
$403$	36.0000	1.79329
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	−16.0000	−0.787309
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−14.0000	−0.685583
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−21.0000	−1.02348	−0.511739	−	0.859141i	$-0.670998\pi$
$421$	−21.0000	−1.02348	−0.511739	+	0.859141i	$0.670998\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−7.00000	−0.339550
$426$	0	0
$427$	40.0000	1.93574
$428$	0	0
$429$	−12.0000	−0.579365
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	−32.0000	−1.50018
$456$	0	0
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	0	0
$459$	−35.0000	−1.63366
$460$	0	0
$461$	7.00000	0.326023	0.163011	−	0.986624i	$-0.447879\pi$
$461$	7.00000	0.326023	0.163011	+	0.986624i	$0.447879\pi$
$462$	0	0
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	0	0
$465$	18.0000	0.834730
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	52.0000	2.40114
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	33.0000	1.51734
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	32.0000	1.45605
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	−21.0000	−0.949653
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	−42.0000	−1.89158
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	60.0000	2.69137
$498$	0	0
$499$	−30.0000	−1.34298	−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000	−1.34298	−0.671492	+	0.741012i	$0.734346\pi$
$500$	0	0
$501$	25.0000	1.11692
$502$	0	0
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	0	0
$505$	−32.0000	−1.42398
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	14.0000	0.616914
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	−15.0000	−0.658427
$520$	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
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	63.0000	2.74432
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	5.00000	0.215766
$538$	0	0
$539$	−27.0000	−1.16297
$540$	0	0
$541$	−1.00000	−0.0429934	−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000	−0.0429934	−0.0214967	+	0.999769i	$0.506843\pi$
$542$	0	0
$543$	5.00000	0.214571
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	0	0
$549$	20.0000	0.853579
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	32.0000	1.36078
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	−44.0000	−1.86100
$560$	0	0
$561$	−21.0000	−0.886621
$562$	0	0
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	−20.0000	−0.835512
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	37.0000	1.54033	0.770165	−	0.637845i	$-0.220174\pi$
$577$	37.0000	1.54033	0.770165	+	0.637845i	$0.220174\pi$
$578$	0	0
$579$	17.0000	0.706496
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	−16.0000	−0.661519
$586$	0	0
$587$	48.0000	1.98117	0.990586	−	0.136892i	$-0.0437113\pi$
$587$	48.0000	1.98117	0.990586	+	0.136892i	$0.0437113\pi$
$588$	0	0
$589$	9.00000	0.370839
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	5.00000	0.205325	0.102663	−	0.994716i	$-0.467264\pi$
$593$	5.00000	0.205325	0.102663	+	0.994716i	$0.467264\pi$
$594$	0	0
$595$	−56.0000	−2.29578
$596$	0	0
$597$	25.0000	1.02318
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	26.0000	1.05880
$604$	0	0
$605$	−4.00000	−0.162623
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−23.0000	−0.924448	−0.462224	−	0.886763i	$-0.652948\pi$
$619$	−23.0000	−0.924448	−0.462224	+	0.886763i	$0.652948\pi$
$620$	0	0
$621$	40.0000	1.60514
$622$	0	0
$623$	−32.0000	−1.28205
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−3.00000	−0.119808
$628$	0	0
$629$	−28.0000	−1.11643
$630$	0	0
$631$	−26.0000	−1.03504	−0.517522	−	0.855670i	$-0.673145\pi$
$631$	−26.0000	−1.03504	−0.517522	+	0.855670i	$0.673145\pi$
$632$	0	0
$633$	24.0000	0.953914
$634$	0	0
$635$	−14.0000	−0.555573
$636$	0	0
$637$	36.0000	1.42637
$638$	0	0
$639$	30.0000	1.18678
$640$	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
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	0	0
$645$	−22.0000	−0.866249
$646$	0	0
$647$	−15.0000	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$647$	−15.0000	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−36.0000	−1.41095
$652$	0	0
$653$	11.0000	0.430463	0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000	0.430463	0.215232	+	0.976563i	$0.430949\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	28.0000	1.08743
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	18.0000	0.695920
$670$	0	0
$671$	30.0000	1.15814
$672$	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
$677$	33.0000	1.26829	0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000	1.26829	0.634147	+	0.773213i	$0.281352\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	2.00000	0.0766402
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	0	0
$687$	−11.0000	−0.419676
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	38.0000	1.44559	0.722794	−	0.691063i	$-0.242858\pi$
$691$	38.0000	1.44559	0.722794	+	0.691063i	$0.242858\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	−28.0000	−1.06210
$696$	0	0
$697$	42.0000	1.59086
$698$	0	0
$699$	16.0000	0.605176
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	64.0000	2.40697
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−72.0000	−2.69642
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	−28.0000	−1.04568
$718$	0	0
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	8.00000	0.297523
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−13.0000	−0.482143	−0.241072	−	0.970507i	$-0.577499\pi$
$727$	−13.0000	−0.482143	−0.241072	+	0.970507i	$0.577499\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−77.0000	−2.84795
$732$	0	0
$733$	49.0000	1.80986	0.904928	−	0.425564i	$-0.139924\pi$
$733$	49.0000	1.80986	0.904928	+	0.425564i	$0.139924\pi$
$734$	0	0
$735$	18.0000	0.663940
$736$	0	0
$737$	39.0000	1.43658
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	0	0
$745$	36.0000	1.31894
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	24.0000	0.876941
$750$	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
$753$	−2.00000	−0.0728841
$754$	0	0
$755$	32.0000	1.16460
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	0	0
$759$	24.0000	0.871145
$760$	0	0
$761$	4.00000	0.145000	0.0724999	−	0.997368i	$-0.476902\pi$
$761$	4.00000	0.145000	0.0724999	+	0.997368i	$0.476902\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	0	0
$765$	−28.0000	−1.01234
$766$	0	0
$767$	16.0000	0.577727
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	25.0000	0.900353
$772$	0	0
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	0	0
$775$	−9.00000	−0.323290
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	45.0000	1.61023
$782$	0	0
$783$	30.0000	1.07211
$784$	0	0
$785$	−8.00000	−0.285532
$786$	0	0
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	0	0
$789$	−10.0000	−0.356009
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	−45.0000	−1.59398	−0.796991	−	0.603991i	$-0.793576\pi$
$797$	−45.0000	−1.59398	−0.796991	+	0.603991i	$0.793576\pi$
$798$	0	0
$799$	28.0000	0.990569
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	0	0
$804$	0	0
$805$	64.0000	2.25570
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	0	0
$813$	9.00000	0.315644
$814$	0	0
$815$	−42.0000	−1.47120
$816$	0	0
$817$	−11.0000	−0.384841
$818$	0	0
$819$	32.0000	1.11817
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	3.00000	0.104447
$826$	0	0
$827$	23.0000	0.799788	0.399894	−	0.916561i	$-0.369047\pi$
$827$	23.0000	0.799788	0.399894	+	0.916561i	$0.369047\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	63.0000	2.18282
$834$	0	0
$835$	50.0000	1.73032
$836$	0	0
$837$	−45.0000	−1.55543
$838$	0	0
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−22.0000	−0.757720
$844$	0	0
$845$	6.00000	0.206406
$846$	0	0
$847$	8.00000	0.274883
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	−20.0000	−0.684787	−0.342393	−	0.939557i	$-0.611238\pi$
$853$	−20.0000	−0.684787	−0.342393	+	0.939557i	$0.611238\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−49.0000	−1.67186	−0.835929	−	0.548837i	$-0.815071\pi$
$859$	−49.0000	−1.67186	−0.835929	+	0.548837i	$0.815071\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	−30.0000	−1.02003
$866$	0	0
$867$	32.0000	1.08678
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	−52.0000	−1.76195
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	−44.0000	−1.48577	−0.742887	−	0.669417i	$-0.766544\pi$
$877$	−44.0000	−1.48577	−0.742887	+	0.669417i	$0.766544\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	0	0
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	28.0000	0.939090
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	4.00000	0.133855
$894$	0	0
$895$	10.0000	0.334263
$896$	0	0
$897$	−32.0000	−1.06845
$898$	0	0
$899$	−54.0000	−1.80100
$900$	0	0
$901$	−7.00000	−0.233204
$902$	0	0
$903$	44.0000	1.46423
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	0	0
$909$	32.0000	1.06137
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	−20.0000	−0.661180
$916$	0	0
$917$	36.0000	1.18882
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	−60.0000	−1.97492
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	7.00000	0.229663	0.114831	−	0.993385i	$-0.463367\pi$
$929$	7.00000	0.229663	0.114831	+	0.993385i	$0.463367\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	0	0
$933$	17.0000	0.556555
$934$	0	0
$935$	−42.0000	−1.37355
$936$	0	0
$937$	43.0000	1.40475	0.702374	−	0.711808i	$-0.252123\pi$
$937$	43.0000	1.40475	0.702374	+	0.711808i	$0.252123\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	0	0
$945$	40.0000	1.30120
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	−40.0000	−1.29437
$956$	0	0
$957$	18.0000	0.581857
$958$	0	0
$959$	48.0000	1.55000
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	34.0000	1.09450
$966$	0	0
$967$	−33.0000	−1.06121	−0.530604	−	0.847620i	$-0.678035\pi$
$967$	−33.0000	−1.06121	−0.530604	+	0.847620i	$0.678035\pi$
$968$	0	0
$969$	7.00000	0.224872
$970$	0	0
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	0	0
$973$	56.0000	1.79528
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	4.00000	0.127710
$982$	0	0
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	0	0
$985$	−44.0000	−1.40196
$986$	0	0
$987$	−16.0000	−0.509286
$988$	0	0
$989$	88.0000	2.79824
$990$	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
$993$	4.00000	0.126936
$994$	0	0
$995$	50.0000	1.58511
$996$	0	0
$997$	−33.0000	−1.04512	−0.522560	−	0.852602i	$-0.675023\pi$
$997$	−33.0000	−1.04512	−0.522560	+	0.852602i	$0.675023\pi$
$998$	0	0
$999$	20.0000	0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.b.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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