LMFDB - Embedded newform 8028.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, 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("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.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:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1710888.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 3x^{2} + 12x + 4 $ x^5 - x^4 - 10x^3 + 3x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2676)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.676546$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.95619 q^{5} -3.03905 q^{7} +O(q^{10})$	$q+2.95619 q^{5} -3.03905 q^{7} -4.44820 q^{17} +3.54229 q^{19} -8.07810 q^{23} +3.73907 q^{25} -6.58893 q^{29} -3.23584 q^{31} -8.98402 q^{35} -4.77166 q^{37} +7.03905 q^{41} +10.7010 q^{43} +12.0513 q^{47} +2.23584 q^{49} +12.7072 q^{53} +2.79380 q^{59} -5.81909 q^{61} +14.6045 q^{67} +15.3436 q^{71} -4.70943 q^{73} +7.33903 q^{79} +7.65157 q^{83} -13.1497 q^{85} -0.858147 q^{89} +10.4717 q^{95} +2.67141 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{5} + 3 q^{7}+O(q^{10}) $ 5 * q + 6 * q^5 + 3 * q^7	$ 5 q + 6 q^{5} + 3 q^{7} + 13 q^{17} - q^{19} - 4 q^{23} + 5 q^{25} - 11 q^{29} - 3 q^{31} + 8 q^{35} + 7 q^{37} + 17 q^{41} + 15 q^{43} + 25 q^{47} - 2 q^{49} + 3 q^{53} + 14 q^{59} - 18 q^{61} + 24 q^{67} + 14 q^{71} - 23 q^{73} + 14 q^{79} + 15 q^{83} + 6 q^{85} + 25 q^{89} + 26 q^{95} - 12 q^{97}+O(q^{100}) $ 5 * q + 6 * q^5 + 3 * q^7 + 13 * q^17 - q^19 - 4 * q^23 + 5 * q^25 - 11 * q^29 - 3 * q^31 + 8 * q^35 + 7 * q^37 + 17 * q^41 + 15 * q^43 + 25 * q^47 - 2 * q^49 + 3 * q^53 + 14 * q^59 - 18 * q^61 + 24 * q^67 + 14 * q^71 - 23 * q^73 + 14 * q^79 + 15 * q^83 + 6 * q^85 + 25 * q^89 + 26 * q^95 - 12 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.95619	1.32205	0.661025	−	0.750364i	$-0.270122\pi$
$5$	2.95619	1.32205	0.661025	+	0.750364i	$0.270122\pi$
$6$	0	0
$7$	−3.03905	−1.14865	−0.574327	−	0.818626i	$-0.694736\pi$
$7$	−3.03905	−1.14865	−0.574327	+	0.818626i	$0.694736\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.44820	−1.07885	−0.539424	−	0.842034i	$-0.681358\pi$
$17$	−4.44820	−1.07885	−0.539424	+	0.842034i	$0.681358\pi$
$18$	0	0
$19$	3.54229	0.812656	0.406328	−	0.913727i	$-0.366809\pi$
$19$	3.54229	0.812656	0.406328	+	0.913727i	$0.366809\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.07810	−1.68440	−0.842201	−	0.539164i	$-0.818740\pi$
$23$	−8.07810	−1.68440	−0.842201	+	0.539164i	$0.818740\pi$
$24$	0	0
$25$	3.73907	0.747814
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.58893	−1.22353	−0.611767	−	0.791038i	$-0.709541\pi$
$29$	−6.58893	−1.22353	−0.611767	+	0.791038i	$0.709541\pi$
$30$	0	0
$31$	−3.23584	−0.581174	−0.290587	−	0.956849i	$-0.593851\pi$
$31$	−3.23584	−0.581174	−0.290587	+	0.956849i	$0.593851\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.98402	−1.51858
$36$	0	0
$37$	−4.77166	−0.784456	−0.392228	−	0.919868i	$-0.628295\pi$
$37$	−4.77166	−0.784456	−0.392228	+	0.919868i	$0.628295\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.03905	1.09932	0.549658	−	0.835390i	$-0.314758\pi$
$41$	7.03905	1.09932	0.549658	+	0.835390i	$0.314758\pi$
$42$	0	0
$43$	10.7010	1.63189	0.815947	−	0.578126i	$-0.196216\pi$
$43$	10.7010	1.63189	0.815947	+	0.578126i	$0.196216\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0513	1.75786	0.878932	−	0.476948i	$-0.158257\pi$
$47$	12.0513	1.75786	0.878932	+	0.476948i	$0.158257\pi$
$48$	0	0
$49$	2.23584	0.319405
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.7072	1.74547	0.872735	−	0.488194i	$-0.162344\pi$
$53$	12.7072	1.74547	0.872735	+	0.488194i	$0.162344\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.79380	0.363722	0.181861	−	0.983324i	$-0.441788\pi$
$59$	2.79380	0.363722	0.181861	+	0.983324i	$0.441788\pi$
$60$	0	0
$61$	−5.81909	−0.745059	−0.372529	−	0.928020i	$-0.621509\pi$
$61$	−5.81909	−0.745059	−0.372529	+	0.928020i	$0.621509\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.6045	1.78422	0.892112	−	0.451814i	$-0.149223\pi$
$67$	14.6045	1.78422	0.892112	+	0.451814i	$0.149223\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.3436	1.82095	0.910474	−	0.413566i	$-0.135717\pi$
$71$	15.3436	1.82095	0.910474	+	0.413566i	$0.135717\pi$
$72$	0	0
$73$	−4.70943	−0.551197	−0.275599	−	0.961273i	$-0.588876\pi$
$73$	−4.70943	−0.551197	−0.275599	+	0.961273i	$0.588876\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.33903	0.825706	0.412853	−	0.910798i	$-0.364532\pi$
$79$	7.33903	0.825706	0.412853	+	0.910798i	$0.364532\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.65157	0.839868	0.419934	−	0.907555i	$-0.362053\pi$
$83$	7.65157	0.839868	0.419934	+	0.907555i	$0.362053\pi$
$84$	0	0
$85$	−13.1497	−1.42629
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.858147	−0.0909634	−0.0454817	−	0.998965i	$-0.514482\pi$
$89$	−0.858147	−0.0909634	−0.0454817	+	0.998965i	$0.514482\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.4717	1.07437
$96$	0	0
$97$	2.67141	0.271241	0.135620	−	0.990761i	$-0.456697\pi$
$97$	2.67141	0.271241	0.135620	+	0.990761i	$0.456697\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.2453	1.01944	0.509720	−	0.860340i	$-0.329749\pi$
$101$	10.2453	1.01944	0.509720	+	0.860340i	$0.329749\pi$
$102$	0	0
$103$	6.55817	0.646195	0.323098	−	0.946366i	$-0.395276\pi$
$103$	6.55817	0.646195	0.323098	+	0.946366i	$0.395276\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.61210	0.929237	0.464618	−	0.885511i	$-0.346191\pi$
$107$	9.61210	0.929237	0.464618	+	0.885511i	$0.346191\pi$
$108$	0	0
$109$	11.7336	1.12388	0.561939	−	0.827179i	$-0.310056\pi$
$109$	11.7336	1.12388	0.561939	+	0.827179i	$0.310056\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.8153	−1.29963	−0.649815	−	0.760092i	$-0.725154\pi$
$113$	−13.8153	−1.29963	−0.649815	+	0.760092i	$0.725154\pi$
$114$	0	0
$115$	−23.8804	−2.22686
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.5183	1.23922
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.72755	−0.333402
$126$	0	0
$127$	0.889704	0.0789485	0.0394742	−	0.999221i	$-0.487432\pi$
$127$	0.889704	0.0789485	0.0394742	+	0.999221i	$0.487432\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.60735	−0.140434	−0.0702172	−	0.997532i	$-0.522369\pi$
$131$	−1.60735	−0.140434	−0.0702172	+	0.997532i	$0.522369\pi$
$132$	0	0
$133$	−10.7652	−0.933460
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.00900	0.684255	0.342128	−	0.939654i	$-0.388852\pi$
$137$	8.00900	0.684255	0.342128	+	0.939654i	$0.388852\pi$
$138$	0	0
$139$	0.426157	0.0361462	0.0180731	−	0.999837i	$-0.494247\pi$
$139$	0.426157	0.0361462	0.0180731	+	0.999837i	$0.494247\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−19.4781	−1.61757
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.371127	−0.0304039	−0.0152020	−	0.999884i	$-0.504839\pi$
$149$	−0.371127	−0.0304039	−0.0152020	+	0.999884i	$0.504839\pi$
$150$	0	0
$151$	14.2506	1.15970	0.579850	−	0.814723i	$-0.303111\pi$
$151$	14.2506	1.15970	0.579850	+	0.814723i	$0.303111\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.56576	−0.768340
$156$	0	0
$157$	16.7691	1.33832	0.669160	−	0.743118i	$-0.266654\pi$
$157$	16.7691	1.33832	0.669160	+	0.743118i	$0.266654\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.5498	1.93479
$162$	0	0
$163$	0.726138	0.0568755	0.0284378	−	0.999596i	$-0.490947\pi$
$163$	0.726138	0.0568755	0.0284378	+	0.999596i	$0.490947\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.26627	0.484898	0.242449	−	0.970164i	$-0.422049\pi$
$167$	6.26627	0.484898	0.242449	+	0.970164i	$0.422049\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.5286	0.800475	0.400238	−	0.916411i	$-0.368928\pi$
$173$	10.5286	0.800475	0.400238	+	0.916411i	$0.368928\pi$
$174$	0	0
$175$	−11.3632	−0.858980
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.37588	0.177582	0.0887909	−	0.996050i	$-0.471700\pi$
$179$	2.37588	0.177582	0.0887909	+	0.996050i	$0.471700\pi$
$180$	0	0
$181$	−16.4929	−1.22591	−0.612955	−	0.790118i	$-0.710019\pi$
$181$	−16.4929	−1.22591	−0.612955	+	0.790118i	$0.710019\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−14.1059	−1.03709
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.84360	−0.278113	−0.139056	−	0.990284i	$-0.544407\pi$
$191$	−3.84360	−0.278113	−0.139056	+	0.990284i	$0.544407\pi$
$192$	0	0
$193$	−2.15339	−0.155004	−0.0775022	−	0.996992i	$-0.524694\pi$
$193$	−2.15339	−0.155004	−0.0775022	+	0.996992i	$0.524694\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.69243	0.334322	0.167161	−	0.985930i	$-0.446540\pi$
$197$	4.69243	0.334322	0.167161	+	0.985930i	$0.446540\pi$
$198$	0	0
$199$	−19.5918	−1.38882	−0.694412	−	0.719577i	$-0.744336\pi$
$199$	−19.5918	−1.38882	−0.694412	+	0.719577i	$0.744336\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.0241	1.40542
$204$	0	0
$205$	20.8088	1.45335
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.85129	0.471662	0.235831	−	0.971794i	$-0.424219\pi$
$211$	6.85129	0.471662	0.235831	+	0.971794i	$0.424219\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	31.6344	2.15744
$216$	0	0
$217$	9.83388	0.667567
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−28.0501	−1.86175	−0.930875	−	0.365339i	$-0.880953\pi$
$227$	−28.0501	−1.86175	−0.930875	+	0.365339i	$0.880953\pi$
$228$	0	0
$229$	13.3031	0.879096	0.439548	−	0.898219i	$-0.355139\pi$
$229$	13.3031	0.879096	0.439548	+	0.898219i	$0.355139\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.5678	1.54398	0.771989	−	0.635635i	$-0.219262\pi$
$233$	23.5678	1.54398	0.771989	+	0.635635i	$0.219262\pi$
$234$	0	0
$235$	35.6260	2.32398
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.7781	1.40871	0.704355	−	0.709848i	$-0.251236\pi$
$239$	21.7781	1.40871	0.704355	+	0.709848i	$0.251236\pi$
$240$	0	0
$241$	−28.1300	−1.81201	−0.906007	−	0.423263i	$-0.860885\pi$
$241$	−28.1300	−1.81201	−0.906007	+	0.423263i	$0.860885\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.60957	0.422270
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−26.3410	−1.66263	−0.831314	−	0.555803i	$-0.812411\pi$
$251$	−26.3410	−1.66263	−0.831314	+	0.555803i	$0.812411\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.42016	−0.338101	−0.169050	−	0.985607i	$-0.554070\pi$
$257$	−5.42016	−0.338101	−0.169050	+	0.985607i	$0.554070\pi$
$258$	0	0
$259$	14.5013	0.901068
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.1532	1.42769	0.713843	−	0.700306i	$-0.246953\pi$
$263$	23.1532	1.42769	0.713843	+	0.700306i	$0.246953\pi$
$264$	0	0
$265$	37.5650	2.30760
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.6804	1.74868	0.874338	−	0.485318i	$-0.161296\pi$
$269$	28.6804	1.74868	0.874338	+	0.485318i	$0.161296\pi$
$270$	0	0
$271$	25.5583	1.55256	0.776278	−	0.630390i	$-0.217105\pi$
$271$	25.5583	1.55256	0.776278	+	0.630390i	$0.217105\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.6214	−0.818432	−0.409216	−	0.912438i	$-0.634198\pi$
$277$	−13.6214	−0.818432	−0.409216	+	0.912438i	$0.634198\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.9536	0.832401	0.416201	−	0.909273i	$-0.363361\pi$
$281$	13.9536	0.832401	0.416201	+	0.909273i	$0.363361\pi$
$282$	0	0
$283$	−4.51719	−0.268519	−0.134260	−	0.990946i	$-0.542866\pi$
$283$	−4.51719	−0.268519	−0.134260	+	0.990946i	$0.542866\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−21.3920	−1.26273
$288$	0	0
$289$	2.78651	0.163912
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.0874	−1.52404	−0.762022	−	0.647551i	$-0.775793\pi$
$293$	−26.0874	−1.52404	−0.762022	+	0.647551i	$0.775793\pi$
$294$	0	0
$295$	8.25901	0.480858
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−32.5210	−1.87448
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−17.2024	−0.985004
$306$	0	0
$307$	7.87382	0.449383	0.224691	−	0.974430i	$-0.427863\pi$
$307$	7.87382	0.449383	0.224691	+	0.974430i	$0.427863\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.4443	1.04588	0.522941	−	0.852369i	$-0.324835\pi$
$311$	18.4443	1.04588	0.522941	+	0.852369i	$0.324835\pi$
$312$	0	0
$313$	−16.7307	−0.945674	−0.472837	−	0.881150i	$-0.656770\pi$
$313$	−16.7307	−0.945674	−0.472837	+	0.881150i	$0.656770\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.0673	0.565436	0.282718	−	0.959203i	$-0.408764\pi$
$317$	10.0673	0.565436	0.282718	+	0.959203i	$0.408764\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−15.7568	−0.876732
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−36.6245	−2.01918
$330$	0	0
$331$	−27.0235	−1.48535	−0.742673	−	0.669654i	$-0.766442\pi$
$331$	−27.0235	−1.48535	−0.742673	+	0.669654i	$0.766442\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	43.1737	2.35883
$336$	0	0
$337$	9.22579	0.502561	0.251280	−	0.967914i	$-0.419148\pi$
$337$	9.22579	0.502561	0.251280	+	0.967914i	$0.419148\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	14.4785	0.781768
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.9843	−1.07281	−0.536406	−	0.843960i	$-0.680218\pi$
$347$	−19.9843	−1.07281	−0.536406	+	0.843960i	$0.680218\pi$
$348$	0	0
$349$	−23.1128	−1.23720	−0.618600	−	0.785706i	$-0.712300\pi$
$349$	−23.1128	−1.23720	−0.618600	+	0.785706i	$0.712300\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	23.7364	1.26336	0.631681	−	0.775228i	$-0.282365\pi$
$353$	23.7364	1.26336	0.631681	+	0.775228i	$0.282365\pi$
$354$	0	0
$355$	45.3586	2.40738
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.4452	−0.604052	−0.302026	−	0.953300i	$-0.597663\pi$
$359$	−11.4452	−0.604052	−0.302026	+	0.953300i	$0.597663\pi$
$360$	0	0
$361$	−6.45221	−0.339590
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13.9220	−0.728710
$366$	0	0
$367$	−36.4243	−1.90133	−0.950667	−	0.310214i	$-0.899599\pi$
$367$	−36.4243	−1.90133	−0.950667	+	0.310214i	$0.899599\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−38.6179	−2.00494
$372$	0	0
$373$	15.8023	0.818213	0.409107	−	0.912487i	$-0.365840\pi$
$373$	15.8023	0.818213	0.409107	+	0.912487i	$0.365840\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−29.7609	−1.52872	−0.764358	−	0.644792i	$-0.776944\pi$
$379$	−29.7609	−1.52872	−0.764358	+	0.644792i	$0.776944\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.0706	1.53654	0.768269	−	0.640127i	$-0.221118\pi$
$383$	30.0706	1.53654	0.768269	+	0.640127i	$0.221118\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.77278	0.343394	0.171697	−	0.985150i	$-0.445075\pi$
$389$	6.77278	0.343394	0.171697	+	0.985150i	$0.445075\pi$
$390$	0	0
$391$	35.9330	1.81721
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.6956	1.09162
$396$	0	0
$397$	−9.36241	−0.469886	−0.234943	−	0.972009i	$-0.575490\pi$
$397$	−9.36241	−0.469886	−0.234943	+	0.972009i	$0.575490\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.84177	−0.341662	−0.170831	−	0.985300i	$-0.554645\pi$
$401$	−6.84177	−0.341662	−0.170831	+	0.985300i	$0.554645\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−13.2380	−0.654575	−0.327288	−	0.944925i	$-0.606135\pi$
$409$	−13.2380	−0.654575	−0.327288	+	0.944925i	$0.606135\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.49050	−0.417790
$414$	0	0
$415$	22.6195	1.11035
$416$	0	0
$417$	0	0
$418$	0	0
$419$	27.8080	1.35851	0.679255	−	0.733903i	$-0.262303\pi$
$419$	27.8080	1.35851	0.679255	+	0.733903i	$0.262303\pi$
$420$	0	0
$421$	−24.4745	−1.19281	−0.596407	−	0.802682i	$-0.703406\pi$
$421$	−24.4745	−1.19281	−0.596407	+	0.802682i	$0.703406\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16.6321	−0.806778
$426$	0	0
$427$	17.6845	0.855815
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.04390	0.0502830	0.0251415	−	0.999684i	$-0.491996\pi$
$431$	1.04390	0.0502830	0.0251415	+	0.999684i	$0.491996\pi$
$432$	0	0
$433$	23.0813	1.10921	0.554607	−	0.832112i	$-0.312869\pi$
$433$	23.0813	1.10921	0.554607	+	0.832112i	$0.312869\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−28.6150	−1.36884
$438$	0	0
$439$	−0.416947	−0.0198998	−0.00994990	−	0.999950i	$-0.503167\pi$
$439$	−0.416947	−0.0198998	−0.00994990	+	0.999950i	$0.503167\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−22.0416	−1.04723	−0.523614	−	0.851956i	$-0.675416\pi$
$443$	−22.0416	−1.04723	−0.523614	+	0.851956i	$0.675416\pi$
$444$	0	0
$445$	−2.53685	−0.120258
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.6989	0.504913	0.252457	−	0.967608i	$-0.418761\pi$
$449$	10.6989	0.504913	0.252457	+	0.967608i	$0.418761\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.1335	−1.50314	−0.751572	−	0.659651i	$-0.770704\pi$
$457$	−32.1335	−1.50314	−0.751572	+	0.659651i	$0.770704\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.7518	−1.24596	−0.622978	−	0.782240i	$-0.714077\pi$
$461$	−26.7518	−1.24596	−0.622978	+	0.782240i	$0.714077\pi$
$462$	0	0
$463$	−21.8261	−1.01434	−0.507172	−	0.861845i	$-0.669309\pi$
$463$	−21.8261	−1.01434	−0.507172	+	0.861845i	$0.669309\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.1583	1.67320	0.836602	−	0.547810i	$-0.184538\pi$
$467$	36.1583	1.67320	0.836602	+	0.547810i	$0.184538\pi$
$468$	0	0
$469$	−44.3839	−2.04946
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	13.2449	0.607716
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.1524	−0.829403	−0.414701	−	0.909958i	$-0.636114\pi$
$479$	−18.1524	−0.829403	−0.414701	+	0.909958i	$0.636114\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.89720	0.358593
$486$	0	0
$487$	28.9176	1.31038	0.655191	−	0.755463i	$-0.272588\pi$
$487$	28.9176	1.31038	0.655191	+	0.755463i	$0.272588\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−27.1157	−1.22372	−0.611858	−	0.790968i	$-0.709578\pi$
$491$	−27.1157	−1.22372	−0.611858	+	0.790968i	$0.709578\pi$
$492$	0	0
$493$	29.3089	1.32001
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−46.6299	−2.09164
$498$	0	0
$499$	−7.73005	−0.346045	−0.173022	−	0.984918i	$-0.555353\pi$
$499$	−7.73005	−0.346045	−0.173022	+	0.984918i	$0.555353\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10.8897	0.485550	0.242775	−	0.970083i	$-0.421942\pi$
$503$	10.8897	0.485550	0.242775	+	0.970083i	$0.421942\pi$
$504$	0	0
$505$	30.2869	1.34775
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.2034	1.73766	0.868830	−	0.495111i	$-0.164873\pi$
$509$	39.2034	1.73766	0.868830	+	0.495111i	$0.164873\pi$
$510$	0	0
$511$	14.3122	0.633135
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.3872	0.854302
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	33.1555	1.45257	0.726285	−	0.687394i	$-0.241245\pi$
$521$	33.1555	1.45257	0.726285	+	0.687394i	$0.241245\pi$
$522$	0	0
$523$	33.6155	1.46990	0.734952	−	0.678119i	$-0.237205\pi$
$523$	33.6155	1.46990	0.734952	+	0.678119i	$0.237205\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.3937	0.626998
$528$	0	0
$529$	42.2558	1.83721
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	28.4152	1.22850
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5.57467	0.239674	0.119837	−	0.992794i	$-0.461763\pi$
$541$	5.57467	0.239674	0.119837	+	0.992794i	$0.461763\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	34.6869	1.48582
$546$	0	0
$547$	2.51520	0.107542	0.0537711	−	0.998553i	$-0.482876\pi$
$547$	2.51520	0.107542	0.0537711	+	0.998553i	$0.482876\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23.3399	−0.994312
$552$	0	0
$553$	−22.3037	−0.948450
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.8047	1.09338	0.546689	−	0.837335i	$-0.315888\pi$
$557$	25.8047	1.09338	0.546689	+	0.837335i	$0.315888\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.2936	−0.475969	−0.237985	−	0.971269i	$-0.576487\pi$
$563$	−11.2936	−0.475969	−0.237985	+	0.971269i	$0.576487\pi$
$564$	0	0
$565$	−40.8406	−1.71817
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.1191	0.927279	0.463640	−	0.886024i	$-0.346543\pi$
$569$	22.1191	0.927279	0.463640	+	0.886024i	$0.346543\pi$
$570$	0	0
$571$	9.21459	0.385618	0.192809	−	0.981236i	$-0.438240\pi$
$571$	9.21459	0.385618	0.192809	+	0.981236i	$0.438240\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−30.2046	−1.25962
$576$	0	0
$577$	20.1590	0.839229	0.419614	−	0.907702i	$-0.362165\pi$
$577$	20.1590	0.839229	0.419614	+	0.907702i	$0.362165\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−23.2535	−0.964718
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.3551	0.881421	0.440711	−	0.897649i	$-0.354726\pi$
$587$	21.3551	0.881421	0.440711	+	0.897649i	$0.354726\pi$
$588$	0	0
$589$	−11.4623	−0.472294
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.700001	0.0287456	0.0143728	−	0.999897i	$-0.495425\pi$
$593$	0.700001	0.0287456	0.0143728	+	0.999897i	$0.495425\pi$
$594$	0	0
$595$	39.9628	1.63831
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.3697	−0.750566	−0.375283	−	0.926910i	$-0.622454\pi$
$599$	−18.3697	−0.750566	−0.375283	+	0.926910i	$0.622454\pi$
$600$	0	0
$601$	9.58376	0.390930	0.195465	−	0.980711i	$-0.437378\pi$
$601$	9.58376	0.390930	0.195465	+	0.980711i	$0.437378\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−32.5181	−1.32205
$606$	0	0
$607$	8.80219	0.357270	0.178635	−	0.983915i	$-0.442832\pi$
$607$	8.80219	0.357270	0.178635	+	0.983915i	$0.442832\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−26.2293	−1.05939	−0.529696	−	0.848188i	$-0.677694\pi$
$613$	−26.2293	−1.05939	−0.529696	+	0.848188i	$0.677694\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.5166	−1.14803	−0.574017	−	0.818843i	$-0.694616\pi$
$617$	−28.5166	−1.14803	−0.574017	+	0.818843i	$0.694616\pi$
$618$	0	0
$619$	−17.4952	−0.703193	−0.351597	−	0.936152i	$-0.614361\pi$
$619$	−17.4952	−0.703193	−0.351597	+	0.936152i	$0.614361\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.60795	0.104485
$624$	0	0
$625$	−29.7147	−1.18859
$626$	0	0
$627$	0	0
$628$	0	0
$629$	21.2253	0.846308
$630$	0	0
$631$	9.13724	0.363748	0.181874	−	0.983322i	$-0.441784\pi$
$631$	9.13724	0.363748	0.181874	+	0.983322i	$0.441784\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.63014	0.104374
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.75265	0.0692256	0.0346128	−	0.999401i	$-0.488980\pi$
$641$	1.75265	0.0692256	0.0346128	+	0.999401i	$0.488980\pi$
$642$	0	0
$643$	33.7975	1.33284	0.666421	−	0.745576i	$-0.267825\pi$
$643$	33.7975	1.33284	0.666421	+	0.745576i	$0.267825\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.0484	−0.512984	−0.256492	−	0.966546i	$-0.582567\pi$
$647$	−13.0484	−0.512984	−0.256492	+	0.966546i	$0.582567\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.0817210	−0.00319799	−0.00159900	−	0.999999i	$-0.500509\pi$
$653$	−0.0817210	−0.00319799	−0.00159900	+	0.999999i	$0.500509\pi$
$654$	0	0
$655$	−4.75162	−0.185661
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−35.0507	−1.36538	−0.682691	−	0.730707i	$-0.739190\pi$
$659$	−35.0507	−1.36538	−0.682691	+	0.730707i	$0.739190\pi$
$660$	0	0
$661$	−19.8173	−0.770804	−0.385402	−	0.922749i	$-0.625937\pi$
$661$	−19.8173	−0.770804	−0.385402	+	0.922749i	$0.625937\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−31.8240	−1.23408
$666$	0	0
$667$	53.2261	2.06092
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	46.2710	1.78362	0.891808	−	0.452415i	$-0.149437\pi$
$673$	46.2710	1.78362	0.891808	+	0.452415i	$0.149437\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.65311	0.370999	0.185500	−	0.982644i	$-0.440610\pi$
$677$	9.65311	0.370999	0.185500	+	0.982644i	$0.440610\pi$
$678$	0	0
$679$	−8.11855	−0.311561
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−36.9309	−1.41312	−0.706562	−	0.707652i	$-0.749755\pi$
$683$	−36.9309	−1.41312	−0.706562	+	0.707652i	$0.749755\pi$
$684$	0	0
$685$	23.6761	0.904619
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−25.7092	−0.978023	−0.489012	−	0.872277i	$-0.662642\pi$
$691$	−25.7092	−0.978023	−0.489012	+	0.872277i	$0.662642\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.25980	0.0477870
$696$	0	0
$697$	−31.3111	−1.18599
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.43682	0.129807	0.0649035	−	0.997892i	$-0.479326\pi$
$701$	3.43682	0.129807	0.0649035	+	0.997892i	$0.479326\pi$
$702$	0	0
$703$	−16.9026	−0.637493
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−31.1359	−1.17098
$708$	0	0
$709$	15.4126	0.578832	0.289416	−	0.957203i	$-0.406539\pi$
$709$	15.4126	0.578832	0.289416	+	0.957203i	$0.406539\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	26.1394	0.978930
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−35.2217	−1.31355	−0.656774	−	0.754087i	$-0.728079\pi$
$719$	−35.2217	−1.31355	−0.656774	+	0.754087i	$0.728079\pi$
$720$	0	0
$721$	−19.9306	−0.742255
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24.6365	−0.914976
$726$	0	0
$727$	40.9533	1.51887	0.759436	−	0.650582i	$-0.225475\pi$
$727$	40.9533	1.51887	0.759436	+	0.650582i	$0.225475\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−47.6004	−1.76057
$732$	0	0
$733$	6.75123	0.249362	0.124681	−	0.992197i	$-0.460209\pi$
$733$	6.75123	0.249362	0.124681	+	0.992197i	$0.460209\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−42.0659	−1.54742	−0.773709	−	0.633541i	$-0.781601\pi$
$739$	−42.0659	−1.54742	−0.773709	+	0.633541i	$0.781601\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.2130	0.851602	0.425801	−	0.904817i	$-0.359992\pi$
$743$	23.2130	0.851602	0.425801	+	0.904817i	$0.359992\pi$
$744$	0	0
$745$	−1.09712	−0.0401955
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−29.2117	−1.06737
$750$	0	0
$751$	−11.5611	−0.421871	−0.210936	−	0.977500i	$-0.567651\pi$
$751$	−11.5611	−0.421871	−0.210936	+	0.977500i	$0.567651\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	42.1276	1.53318
$756$	0	0
$757$	−1.25089	−0.0454645	−0.0227323	−	0.999742i	$-0.507237\pi$
$757$	−1.25089	−0.0454645	−0.0227323	+	0.999742i	$0.507237\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.3544	−1.39035	−0.695174	−	0.718842i	$-0.744673\pi$
$761$	−38.3544	−1.39035	−0.695174	+	0.718842i	$0.744673\pi$
$762$	0	0
$763$	−35.6591	−1.29095
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	29.5612	1.06600	0.533002	−	0.846114i	$-0.321064\pi$
$769$	29.5612	1.06600	0.533002	+	0.846114i	$0.321064\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.43846	0.267543	0.133771	−	0.991012i	$-0.457291\pi$
$773$	7.43846	0.267543	0.133771	+	0.991012i	$0.457291\pi$
$774$	0	0
$775$	−12.0990	−0.434610
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.9343	0.893365
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	49.5727	1.76933
$786$	0	0
$787$	34.5096	1.23014	0.615068	−	0.788474i	$-0.289129\pi$
$787$	34.5096	1.23014	0.615068	+	0.788474i	$0.289129\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	41.9853	1.49282
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.0599	−0.816824	−0.408412	−	0.912798i	$-0.633917\pi$
$797$	−23.0599	−0.816824	−0.408412	+	0.912798i	$0.633917\pi$
$798$	0	0
$799$	−53.6066	−1.89647
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	72.5739	2.55789
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43.9870	1.54650	0.773250	−	0.634102i	$-0.218630\pi$
$809$	43.9870	1.54650	0.773250	+	0.634102i	$0.218630\pi$
$810$	0	0
$811$	−15.5077	−0.544549	−0.272275	−	0.962220i	$-0.587776\pi$
$811$	−15.5077	−0.544549	−0.272275	+	0.962220i	$0.587776\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.14660	0.0751922
$816$	0	0
$817$	37.9062	1.32617
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.80316	0.307232	0.153616	−	0.988131i	$-0.450908\pi$
$821$	8.80316	0.307232	0.153616	+	0.988131i	$0.450908\pi$
$822$	0	0
$823$	16.6313	0.579729	0.289865	−	0.957068i	$-0.406390\pi$
$823$	16.6313	0.579729	0.289865	+	0.957068i	$0.406390\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.4578	0.954800	0.477400	−	0.878686i	$-0.341579\pi$
$827$	27.4578	0.954800	0.477400	+	0.878686i	$0.341579\pi$
$828$	0	0
$829$	38.2030	1.32684	0.663422	−	0.748246i	$-0.269104\pi$
$829$	38.2030	1.32684	0.663422	+	0.748246i	$0.269104\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.94546	−0.344590
$834$	0	0
$835$	18.5243	0.641060
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37.8596	1.30706	0.653529	−	0.756902i	$-0.273288\pi$
$839$	37.8596	1.30706	0.653529	+	0.756902i	$0.273288\pi$
$840$	0	0
$841$	14.4140	0.497034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−38.4305	−1.32205
$846$	0	0
$847$	33.4296	1.14865
$848$	0	0
$849$	0	0
$850$	0	0
$851$	38.5459	1.32134
$852$	0	0
$853$	−39.7706	−1.36172	−0.680859	−	0.732414i	$-0.738393\pi$
$853$	−39.7706	−1.36172	−0.680859	+	0.732414i	$0.738393\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.2942	−0.590759	−0.295380	−	0.955380i	$-0.595446\pi$
$857$	−17.2942	−0.590759	−0.295380	+	0.955380i	$0.595446\pi$
$858$	0	0
$859$	−34.8025	−1.18744	−0.593722	−	0.804670i	$-0.702342\pi$
$859$	−34.8025	−1.18744	−0.593722	+	0.804670i	$0.702342\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.6593	0.533048	0.266524	−	0.963828i	$-0.414125\pi$
$863$	15.6593	0.533048	0.266524	+	0.963828i	$0.414125\pi$
$864$	0	0
$865$	31.1246	1.05827
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.3282	0.382963
$876$	0	0
$877$	−9.00928	−0.304222	−0.152111	−	0.988363i	$-0.548607\pi$
$877$	−9.00928	−0.304222	−0.152111	+	0.988363i	$0.548607\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.6245	0.627475	0.313738	−	0.949510i	$-0.398419\pi$
$881$	18.6245	0.627475	0.313738	+	0.949510i	$0.398419\pi$
$882$	0	0
$883$	40.4979	1.36286	0.681431	−	0.731882i	$-0.261358\pi$
$883$	40.4979	1.36286	0.681431	+	0.731882i	$0.261358\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.0791	0.774921	0.387460	−	0.921886i	$-0.373352\pi$
$887$	23.0791	0.774921	0.387460	+	0.921886i	$0.373352\pi$
$888$	0	0
$889$	−2.70386	−0.0906845
$890$	0	0
$891$	0	0
$892$	0	0
$893$	42.6892	1.42854
$894$	0	0
$895$	7.02357	0.234772
$896$	0	0
$897$	0	0
$898$	0	0
$899$	21.3207	0.711086
$900$	0	0
$901$	−56.5243	−1.88310
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−48.7563	−1.62071
$906$	0	0
$907$	−6.21131	−0.206243	−0.103121	−	0.994669i	$-0.532883\pi$
$907$	−6.21131	−0.206243	−0.103121	+	0.994669i	$0.532883\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	18.7421	0.620953	0.310477	−	0.950581i	$-0.399511\pi$
$911$	18.7421	0.620953	0.310477	+	0.950581i	$0.399511\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.88481	0.161311
$918$	0	0
$919$	57.3450	1.89164	0.945819	−	0.324695i	$-0.105262\pi$
$919$	57.3450	1.89164	0.945819	+	0.324695i	$0.105262\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−17.8416	−0.586627
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.8849	0.685211	0.342605	−	0.939479i	$-0.388691\pi$
$929$	20.8849	0.685211	0.342605	+	0.939479i	$0.388691\pi$
$930$	0	0
$931$	7.91998	0.259567
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.7768	0.580744	0.290372	−	0.956914i	$-0.406221\pi$
$937$	17.7768	0.580744	0.290372	+	0.956914i	$0.406221\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.3181	0.434158	0.217079	−	0.976154i	$-0.430347\pi$
$941$	13.3181	0.434158	0.217079	+	0.976154i	$0.430347\pi$
$942$	0	0
$943$	−56.8622	−1.85169
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.8175	−1.35888	−0.679442	−	0.733729i	$-0.737778\pi$
$947$	−41.8175	−1.35888	−0.679442	+	0.733729i	$0.737778\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.5116	1.11794	0.558971	−	0.829187i	$-0.311196\pi$
$953$	34.5116	1.11794	0.558971	+	0.829187i	$0.311196\pi$
$954$	0	0
$955$	−11.3624	−0.367679
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.3398	−0.785972
$960$	0	0
$961$	−20.5294	−0.662237
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.36584	−0.204923
$966$	0	0
$967$	10.4140	0.334893	0.167447	−	0.985881i	$-0.446448\pi$
$967$	10.4140	0.334893	0.167447	+	0.985881i	$0.446448\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.6829	0.824205	0.412102	−	0.911138i	$-0.364795\pi$
$971$	25.6829	0.824205	0.412102	+	0.911138i	$0.364795\pi$
$972$	0	0
$973$	−1.29511	−0.0415195
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.7306	0.983158	0.491579	−	0.870833i	$-0.336420\pi$
$977$	30.7306	0.983158	0.491579	+	0.870833i	$0.336420\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.8498	−0.888272	−0.444136	−	0.895959i	$-0.646489\pi$
$983$	−27.8498	−0.888272	−0.444136	+	0.895959i	$0.646489\pi$
$984$	0	0
$985$	13.8717	0.441990
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−86.4442	−2.74876
$990$	0	0
$991$	−51.7906	−1.64518	−0.822591	−	0.568634i	$-0.807472\pi$
$991$	−51.7906	−1.64518	−0.822591	+	0.568634i	$0.807472\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−57.9171	−1.83609
$996$	0	0
$997$	11.4033	0.361145	0.180572	−	0.983562i	$-0.442205\pi$
$997$	11.4033	0.361145	0.180572	+	0.983562i	$0.442205\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.i.1.4		5
3.2	odd	2		2676.2.a.c.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2676.2.a.c.1.2	✓	5	3.2	odd	2
8028.2.a.i.1.4		5	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:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q+2.95619 q^{5} -3.03905 q^{7} +O(q^{10})\)	\(q+2.95619 q^{5} -3.03905 q^{7} -4.44820 q^{17} +3.54229 q^{19} -8.07810 q^{23} +3.73907 q^{25} -6.58893 q^{29} -3.23584 q^{31} -8.98402 q^{35} -4.77166 q^{37} +7.03905 q^{41} +10.7010 q^{43} +12.0513 q^{47} +2.23584 q^{49} +12.7072 q^{53} +2.79380 q^{59} -5.81909 q^{61} +14.6045 q^{67} +15.3436 q^{71} -4.70943 q^{73} +7.33903 q^{79} +7.65157 q^{83} -13.1497 q^{85} -0.858147 q^{89} +10.4717 q^{95} +2.67141 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 5 * q + 6 * q^5 + 3 * q^7	\( 5 q + 6 q^{5} + 3 q^{7} + 13 q^{17} - q^{19} - 4 q^{23} + 5 q^{25} - 11 q^{29} - 3 q^{31} + 8 q^{35} + 7 q^{37} + 17 q^{41} + 15 q^{43} + 25 q^{47} - 2 q^{49} + 3 q^{53} + 14 q^{59} - 18 q^{61} + 24 q^{67} + 14 q^{71} - 23 q^{73} + 14 q^{79} + 15 q^{83} + 6 q^{85} + 25 q^{89} + 26 q^{95} - 12 q^{97}+O(q^{100}) \) 5 * q + 6 * q^5 + 3 * q^7 + 13 * q^17 - q^19 - 4 * q^23 + 5 * q^25 - 11 * q^29 - 3 * q^31 + 8 * q^35 + 7 * q^37 + 17 * q^41 + 15 * q^43 + 25 * q^47 - 2 * q^49 + 3 * q^53 + 14 * q^59 - 18 * q^61 + 24 * q^67 + 14 * q^71 - 23 * q^73 + 14 * q^79 + 15 * q^83 + 6 * q^85 + 25 * q^89 + 26 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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