LMFDB - Embedded newform 5625.2.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.33826$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82709 q^{2} +1.33826 q^{4} +1.44512 q^{7} -1.20906 q^{8} +O(q^{10})$	$q+1.82709 q^{2} +1.33826 q^{4} +1.44512 q^{7} -1.20906 q^{8} +2.12920 q^{11} -5.70353 q^{13} +2.64037 q^{14} -4.88558 q^{16} -4.15622 q^{17} +1.70353 q^{19} +3.89025 q^{22} +0.323478 q^{23} -10.4209 q^{26} +1.93395 q^{28} +8.74724 q^{29} -8.45991 q^{31} -6.50828 q^{32} -7.59378 q^{34} +1.75170 q^{37} +3.11251 q^{38} +6.87802 q^{41} -11.1411 q^{43} +2.84943 q^{44} +0.591023 q^{46} -12.5982 q^{47} -4.91161 q^{49} -7.63282 q^{52} -8.34451 q^{53} -1.74724 q^{56} +15.9820 q^{58} +2.12474 q^{59} -5.38952 q^{61} -15.4570 q^{62} -2.12007 q^{64} -7.13078 q^{67} -5.56210 q^{68} -2.67461 q^{71} +6.28253 q^{73} +3.20052 q^{74} +2.27977 q^{76} +3.07697 q^{77} +8.37092 q^{79} +12.5668 q^{82} +14.5872 q^{83} -20.3558 q^{86} -2.57433 q^{88} -2.68119 q^{89} -8.24232 q^{91} +0.432897 q^{92} -23.0181 q^{94} +8.55105 q^{97} -8.97397 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8	$ 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8} + 6 q^{11} - 7 q^{13} + 10 q^{14} - 9 q^{16} - 7 q^{17} - 9 q^{19} - 6 q^{22} + 10 q^{23} + 2 q^{26} - 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} + 10 q^{37} - 6 q^{38} - q^{43} + 9 q^{44} + 5 q^{46} - 23 q^{47} - 3 q^{49} - 13 q^{52} + 2 q^{58} - 4 q^{59} - 43 q^{61} - 10 q^{62} - 7 q^{64} - 8 q^{67} - 3 q^{68} + 27 q^{71} - 15 q^{73} - 5 q^{74} + 9 q^{76} - 15 q^{77} + 10 q^{79} + 20 q^{82} + 3 q^{83} - 24 q^{86} + 3 q^{88} + 9 q^{89} + 5 q^{91} - 15 q^{92} - 22 q^{94} - 13 q^{97} - 42 q^{98}+O(q^{100}) $ 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8 + 6 * q^11 - 7 * q^13 + 10 * q^14 - 9 * q^16 - 7 * q^17 - 9 * q^19 - 6 * q^22 + 10 * q^23 + 2 * q^26 - 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 + 10 * q^37 - 6 * q^38 - q^43 + 9 * q^44 + 5 * q^46 - 23 * q^47 - 3 * q^49 - 13 * q^52 + 2 * q^58 - 4 * q^59 - 43 * q^61 - 10 * q^62 - 7 * q^64 - 8 * q^67 - 3 * q^68 + 27 * q^71 - 15 * q^73 - 5 * q^74 + 9 * q^76 - 15 * q^77 + 10 * q^79 + 20 * q^82 + 3 * q^83 - 24 * q^86 + 3 * q^88 + 9 * q^89 + 5 * q^91 - 15 * q^92 - 22 * q^94 - 13 * q^97 - 42 * q^98

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 $

$2$	1.82709	1.29195	0.645974	−	0.763359i	$-0.276451\pi$
$2$	1.82709	1.29195	0.645974	+	0.763359i	$0.276451\pi$
$3$	0	0
$3$	0	0
$4$	1.33826	0.669131
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.44512	0.546206	0.273103	−	0.961985i	$-0.411950\pi$
$7$	1.44512	0.546206	0.273103	+	0.961985i	$0.411950\pi$
$8$	−1.20906	−0.427466
$9$	0	0
$10$	0	0
$11$	2.12920	0.641979	0.320990	−	0.947083i	$-0.395985\pi$
$11$	2.12920	0.641979	0.320990	+	0.947083i	$0.395985\pi$
$12$	0	0
$13$	−5.70353	−1.58188	−0.790938	−	0.611897i	$-0.790407\pi$
$13$	−5.70353	−1.58188	−0.790938	+	0.611897i	$0.790407\pi$
$14$	2.64037	0.705670
$15$	0	0
$16$	−4.88558	−1.22139
$17$	−4.15622	−1.00803	−0.504015	−	0.863695i	$-0.668144\pi$
$17$	−4.15622	−1.00803	−0.504015	+	0.863695i	$0.668144\pi$
$18$	0	0
$19$	1.70353	0.390817	0.195409	−	0.980722i	$-0.437397\pi$
$19$	1.70353	0.390817	0.195409	+	0.980722i	$0.437397\pi$
$20$	0	0
$21$	0	0
$22$	3.89025	0.829404
$23$	0.323478	0.0674497	0.0337249	−	0.999431i	$-0.489263\pi$
$23$	0.323478	0.0674497	0.0337249	+	0.999431i	$0.489263\pi$
$24$	0	0
$25$	0	0
$26$	−10.4209	−2.04370
$27$	0	0
$28$	1.93395	0.365483
$29$	8.74724	1.62432	0.812161	−	0.583434i	$-0.198291\pi$
$29$	8.74724	1.62432	0.812161	+	0.583434i	$0.198291\pi$
$30$	0	0
$31$	−8.45991	−1.51944	−0.759722	−	0.650248i	$-0.774665\pi$
$31$	−8.45991	−1.51944	−0.759722	+	0.650248i	$0.774665\pi$
$32$	−6.50828	−1.15051
$33$	0	0
$34$	−7.59378	−1.30232
$35$	0	0
$36$	0	0
$37$	1.75170	0.287978	0.143989	−	0.989579i	$-0.454007\pi$
$37$	1.75170	0.287978	0.143989	+	0.989579i	$0.454007\pi$
$38$	3.11251	0.504916
$39$	0	0
$40$	0	0
$41$	6.87802	1.07417	0.537083	−	0.843529i	$-0.319526\pi$
$41$	6.87802	1.07417	0.537083	+	0.843529i	$0.319526\pi$
$42$	0	0
$43$	−11.1411	−1.69900	−0.849501	−	0.527587i	$-0.823097\pi$
$43$	−11.1411	−1.69900	−0.849501	+	0.527587i	$0.823097\pi$
$44$	2.84943	0.429568
$45$	0	0
$46$	0.591023	0.0871416
$47$	−12.5982	−1.83764	−0.918822	−	0.394673i	$-0.870858\pi$
$47$	−12.5982	−1.83764	−0.918822	+	0.394673i	$0.870858\pi$
$48$	0	0
$49$	−4.91161	−0.701659
$50$	0	0
$51$	0	0
$52$	−7.63282	−1.05848
$53$	−8.34451	−1.14621	−0.573103	−	0.819483i	$-0.694261\pi$
$53$	−8.34451	−1.14621	−0.573103	+	0.819483i	$0.694261\pi$
$54$	0	0
$55$	0	0
$56$	−1.74724	−0.233485
$57$	0	0
$58$	15.9820	2.09854
$59$	2.12474	0.276617	0.138309	−	0.990389i	$-0.455833\pi$
$59$	2.12474	0.276617	0.138309	+	0.990389i	$0.455833\pi$
$60$	0	0
$61$	−5.38952	−0.690058	−0.345029	−	0.938592i	$-0.612131\pi$
$61$	−5.38952	−0.690058	−0.345029	+	0.938592i	$0.612131\pi$
$62$	−15.4570	−1.96304
$63$	0	0
$64$	−2.12007	−0.265008
$65$	0	0
$66$	0	0
$67$	−7.13078	−0.871164	−0.435582	−	0.900149i	$-0.643457\pi$
$67$	−7.13078	−0.871164	−0.435582	+	0.900149i	$0.643457\pi$
$68$	−5.56210	−0.674504
$69$	0	0
$70$	0	0
$71$	−2.67461	−0.317418	−0.158709	−	0.987325i	$-0.550733\pi$
$71$	−2.67461	−0.317418	−0.158709	+	0.987325i	$0.550733\pi$
$72$	0	0
$73$	6.28253	0.735315	0.367657	−	0.929961i	$-0.380160\pi$
$73$	6.28253	0.735315	0.367657	+	0.929961i	$0.380160\pi$
$74$	3.20052	0.372053
$75$	0	0
$76$	2.27977	0.261508
$77$	3.07697	0.350653
$78$	0	0
$79$	8.37092	0.941802	0.470901	−	0.882186i	$-0.343929\pi$
$79$	8.37092	0.941802	0.470901	+	0.882186i	$0.343929\pi$
$80$	0	0
$81$	0	0
$82$	12.5668	1.38777
$83$	14.5872	1.60115	0.800577	−	0.599230i	$-0.204527\pi$
$83$	14.5872	1.60115	0.800577	+	0.599230i	$0.204527\pi$
$84$	0	0
$85$	0	0
$86$	−20.3558	−2.19502
$87$	0	0
$88$	−2.57433	−0.274424
$89$	−2.68119	−0.284206	−0.142103	−	0.989852i	$-0.545386\pi$
$89$	−2.68119	−0.284206	−0.142103	+	0.989852i	$0.545386\pi$
$90$	0	0
$91$	−8.24232	−0.864030
$92$	0.432897	0.0451327
$93$	0	0
$94$	−23.0181	−2.37414
$95$	0	0
$96$	0	0
$97$	8.55105	0.868228	0.434114	−	0.900858i	$-0.357061\pi$
$97$	8.55105	0.868228	0.434114	+	0.900858i	$0.357061\pi$
$98$	−8.97397	−0.906507
$99$	0	0
$100$	0	0
$101$	−9.85377	−0.980487	−0.490243	−	0.871586i	$-0.663092\pi$
$101$	−9.85377	−0.980487	−0.490243	+	0.871586i	$0.663092\pi$
$102$	0	0
$103$	−16.1279	−1.58913	−0.794564	−	0.607180i	$-0.792301\pi$
$103$	−16.1279	−1.58913	−0.794564	+	0.607180i	$0.792301\pi$
$104$	6.89590	0.676198
$105$	0	0
$106$	−15.2462	−1.48084
$107$	1.81198	0.175170	0.0875852	−	0.996157i	$-0.472085\pi$
$107$	1.81198	0.175170	0.0875852	+	0.996157i	$0.472085\pi$
$108$	0	0
$109$	−18.4646	−1.76859	−0.884293	−	0.466933i	$-0.845359\pi$
$109$	−18.4646	−1.76859	−0.884293	+	0.466933i	$0.845359\pi$
$110$	0	0
$111$	0	0
$112$	−7.06027	−0.667133
$113$	−13.3653	−1.25730	−0.628650	−	0.777689i	$-0.716392\pi$
$113$	−13.3653	−1.25730	−0.628650	+	0.777689i	$0.716392\pi$
$114$	0	0
$115$	0	0
$116$	11.7061	1.08688
$117$	0	0
$118$	3.88209	0.357375
$119$	−6.00625	−0.550592
$120$	0	0
$121$	−6.46649	−0.587863
$122$	−9.84715	−0.891519
$123$	0	0
$124$	−11.3216	−1.01671
$125$	0	0
$126$	0	0
$127$	1.01381	0.0899608	0.0449804	−	0.998988i	$-0.485677\pi$
$127$	1.01381	0.0899608	0.0449804	+	0.998988i	$0.485677\pi$
$128$	9.14301	0.808136
$129$	0	0
$130$	0	0
$131$	−2.08550	−0.182211	−0.0911055	−	0.995841i	$-0.529040\pi$
$131$	−2.08550	−0.182211	−0.0911055	+	0.995841i	$0.529040\pi$
$132$	0	0
$133$	2.46182	0.213467
$134$	−13.0286	−1.12550
$135$	0	0
$136$	5.02510	0.430899
$137$	16.9365	1.44698	0.723492	−	0.690333i	$-0.242536\pi$
$137$	16.9365	1.44698	0.723492	+	0.690333i	$0.242536\pi$
$138$	0	0
$139$	14.8883	1.26281	0.631406	−	0.775452i	$-0.282478\pi$
$139$	14.8883	1.26281	0.631406	+	0.775452i	$0.282478\pi$
$140$	0	0
$141$	0	0
$142$	−4.88676	−0.410088
$143$	−12.1440	−1.01553
$144$	0	0
$145$	0	0
$146$	11.4788	0.949989
$147$	0	0
$148$	2.34424	0.192695
$149$	4.70991	0.385851	0.192925	−	0.981213i	$-0.438202\pi$
$149$	4.70991	0.385851	0.192925	+	0.981213i	$0.438202\pi$
$150$	0	0
$151$	−9.63091	−0.783752	−0.391876	−	0.920018i	$-0.628174\pi$
$151$	−9.63091	−0.783752	−0.391876	+	0.920018i	$0.628174\pi$
$152$	−2.05967	−0.167061
$153$	0	0
$154$	5.62190	0.453025
$155$	0	0
$156$	0	0
$157$	−18.5557	−1.48091	−0.740454	−	0.672107i	$-0.765389\pi$
$157$	−18.5557	−1.48091	−0.740454	+	0.672107i	$0.765389\pi$
$158$	15.2944	1.21676
$159$	0	0
$160$	0	0
$161$	0.467465	0.0368414
$162$	0	0
$163$	−0.451705	−0.0353803	−0.0176902	−	0.999844i	$-0.505631\pi$
$163$	−0.451705	−0.0353803	−0.0176902	+	0.999844i	$0.505631\pi$
$164$	9.20459	0.718758
$165$	0	0
$166$	26.6521	2.06861
$167$	4.72648	0.365746	0.182873	−	0.983137i	$-0.441460\pi$
$167$	4.72648	0.365746	0.182873	+	0.983137i	$0.441460\pi$
$168$	0	0
$169$	19.5303	1.50233
$170$	0	0
$171$	0	0
$172$	−14.9097	−1.13685
$173$	2.29489	0.174477	0.0872385	−	0.996187i	$-0.472196\pi$
$173$	2.29489	0.174477	0.0872385	+	0.996187i	$0.472196\pi$
$174$	0	0
$175$	0	0
$176$	−10.4024	−0.784110
$177$	0	0
$178$	−4.89878	−0.367179
$179$	−2.68757	−0.200878	−0.100439	−	0.994943i	$-0.532025\pi$
$179$	−2.68757	−0.200878	−0.100439	+	0.994943i	$0.532025\pi$
$180$	0	0
$181$	−23.9088	−1.77712	−0.888562	−	0.458756i	$-0.848295\pi$
$181$	−23.9088	−1.77712	−0.888562	+	0.458756i	$0.848295\pi$
$182$	−15.0595	−1.11628
$183$	0	0
$184$	−0.391103	−0.0288325
$185$	0	0
$186$	0	0
$187$	−8.84943	−0.647135
$188$	−16.8597	−1.22962
$189$	0	0
$190$	0	0
$191$	3.91824	0.283514	0.141757	−	0.989902i	$-0.454725\pi$
$191$	3.91824	0.283514	0.141757	+	0.989902i	$0.454725\pi$
$192$	0	0
$193$	8.54752	0.615264	0.307632	−	0.951505i	$-0.400463\pi$
$193$	8.54752	0.615264	0.307632	+	0.951505i	$0.400463\pi$
$194$	15.6236	1.12171
$195$	0	0
$196$	−6.57302	−0.469502
$197$	2.45235	0.174723	0.0873614	−	0.996177i	$-0.472157\pi$
$197$	2.45235	0.174723	0.0873614	+	0.996177i	$0.472157\pi$
$198$	0	0
$199$	−10.4345	−0.739680	−0.369840	−	0.929095i	$-0.620588\pi$
$199$	−10.4345	−0.739680	−0.369840	+	0.929095i	$0.620588\pi$
$200$	0	0
$201$	0	0
$202$	−18.0037	−1.26674
$203$	12.6409	0.887214
$204$	0	0
$205$	0	0
$206$	−29.4671	−2.05307
$207$	0	0
$208$	27.8651	1.93209
$209$	3.62717	0.250897
$210$	0	0
$211$	19.2618	1.32604	0.663018	−	0.748604i	$-0.269275\pi$
$211$	19.2618	1.32604	0.663018	+	0.748604i	$0.269275\pi$
$212$	−11.1671	−0.766962
$213$	0	0
$214$	3.31065	0.226311
$215$	0	0
$216$	0	0
$217$	−12.2256	−0.829929
$218$	−33.7365	−2.28492
$219$	0	0
$220$	0	0
$221$	23.7051	1.59458
$222$	0	0
$223$	−7.61706	−0.510076	−0.255038	−	0.966931i	$-0.582088\pi$
$223$	−7.61706	−0.510076	−0.255038	+	0.966931i	$0.582088\pi$
$224$	−9.40528	−0.628417
$225$	0	0
$226$	−24.4196	−1.62437
$227$	−1.31803	−0.0874810	−0.0437405	−	0.999043i	$-0.513927\pi$
$227$	−1.31803	−0.0874810	−0.0437405	+	0.999043i	$0.513927\pi$
$228$	0	0
$229$	1.12632	0.0744292	0.0372146	−	0.999307i	$-0.488151\pi$
$229$	1.12632	0.0744292	0.0372146	+	0.999307i	$0.488151\pi$
$230$	0	0
$231$	0	0
$232$	−10.5759	−0.694342
$233$	26.6071	1.74309	0.871543	−	0.490319i	$-0.163120\pi$
$233$	26.6071	1.74309	0.871543	+	0.490319i	$0.163120\pi$
$234$	0	0
$235$	0	0
$236$	2.84345	0.185093
$237$	0	0
$238$	−10.9740	−0.711337
$239$	24.8692	1.60865	0.804326	−	0.594188i	$-0.202527\pi$
$239$	24.8692	1.60865	0.804326	+	0.594188i	$0.202527\pi$
$240$	0	0
$241$	−12.3532	−0.795743	−0.397871	−	0.917441i	$-0.630251\pi$
$241$	−12.3532	−0.795743	−0.397871	+	0.917441i	$0.630251\pi$
$242$	−11.8149	−0.759488
$243$	0	0
$244$	−7.21259	−0.461739
$245$	0	0
$246$	0	0
$247$	−9.71616	−0.618224
$248$	10.2285	0.649511
$249$	0	0
$250$	0	0
$251$	29.0986	1.83669	0.918343	−	0.395786i	$-0.129528\pi$
$251$	29.0986	1.83669	0.918343	+	0.395786i	$0.129528\pi$
$252$	0	0
$253$	0.688750	0.0433013
$254$	1.85232	0.116225
$255$	0	0
$256$	20.9452	1.30908
$257$	−12.0500	−0.751656	−0.375828	−	0.926690i	$-0.622642\pi$
$257$	−12.0500	−0.751656	−0.375828	+	0.926690i	$0.622642\pi$
$258$	0	0
$259$	2.53143	0.157296
$260$	0	0
$261$	0	0
$262$	−3.81040	−0.235407
$263$	−1.02892	−0.0634460	−0.0317230	−	0.999497i	$-0.510099\pi$
$263$	−1.02892	−0.0634460	−0.0317230	+	0.999497i	$0.510099\pi$
$264$	0	0
$265$	0	0
$266$	4.49797	0.275788
$267$	0	0
$268$	−9.54285	−0.582922
$269$	−23.9072	−1.45765	−0.728824	−	0.684701i	$-0.759933\pi$
$269$	−23.9072	−1.45765	−0.728824	+	0.684701i	$0.759933\pi$
$270$	0	0
$271$	3.72214	0.226104	0.113052	−	0.993589i	$-0.463937\pi$
$271$	3.72214	0.226104	0.113052	+	0.993589i	$0.463937\pi$
$272$	20.3055	1.23120
$273$	0	0
$274$	30.9445	1.86943
$275$	0	0
$276$	0	0
$277$	17.5802	1.05629	0.528147	−	0.849153i	$-0.322887\pi$
$277$	17.5802	1.05629	0.528147	+	0.849153i	$0.322887\pi$
$278$	27.2024	1.63149
$279$	0	0
$280$	0	0
$281$	−13.2536	−0.790644	−0.395322	−	0.918543i	$-0.629367\pi$
$281$	−13.2536	−0.790644	−0.395322	+	0.918543i	$0.629367\pi$
$282$	0	0
$283$	−12.0996	−0.719249	−0.359624	−	0.933097i	$-0.617095\pi$
$283$	−12.0996	−0.719249	−0.359624	+	0.933097i	$0.617095\pi$
$284$	−3.57933	−0.212394
$285$	0	0
$286$	−22.1882	−1.31201
$287$	9.93960	0.586716
$288$	0	0
$289$	0.274126	0.0161251
$290$	0	0
$291$	0	0
$292$	8.40767	0.492022
$293$	3.61673	0.211291	0.105646	−	0.994404i	$-0.466309\pi$
$293$	3.61673	0.211291	0.105646	+	0.994404i	$0.466309\pi$
$294$	0	0
$295$	0	0
$296$	−2.11791	−0.123101
$297$	0	0
$298$	8.60543	0.498499
$299$	−1.84497	−0.106697
$300$	0	0
$301$	−16.1003	−0.928005
$302$	−17.5965	−1.01257
$303$	0	0
$304$	−8.32275	−0.477342
$305$	0	0
$306$	0	0
$307$	−14.8273	−0.846238	−0.423119	−	0.906074i	$-0.639065\pi$
$307$	−14.8273	−0.846238	−0.423119	+	0.906074i	$0.639065\pi$
$308$	4.11778	0.234633
$309$	0	0
$310$	0	0
$311$	6.61579	0.375147	0.187574	−	0.982251i	$-0.439938\pi$
$311$	6.61579	0.375147	0.187574	+	0.982251i	$0.439938\pi$
$312$	0	0
$313$	7.17823	0.405737	0.202869	−	0.979206i	$-0.434974\pi$
$313$	7.17823	0.405737	0.202869	+	0.979206i	$0.434974\pi$
$314$	−33.9030	−1.91326
$315$	0	0
$316$	11.2025	0.630189
$317$	−9.75255	−0.547758	−0.273879	−	0.961764i	$-0.588307\pi$
$317$	−9.75255	−0.547758	−0.273879	+	0.961764i	$0.588307\pi$
$318$	0	0
$319$	18.6247	1.04278
$320$	0	0
$321$	0	0
$322$	0.854102	0.0475972
$323$	−7.08025	−0.393956
$324$	0	0
$325$	0	0
$326$	−0.825307	−0.0457095
$327$	0	0
$328$	−8.31592	−0.459170
$329$	−18.2060	−1.00373
$330$	0	0
$331$	6.68822	0.367618	0.183809	−	0.982962i	$-0.441157\pi$
$331$	6.68822	0.367618	0.183809	+	0.982962i	$0.441157\pi$
$332$	19.5215	1.07138
$333$	0	0
$334$	8.63570	0.472525
$335$	0	0
$336$	0	0
$337$	20.8191	1.13409	0.567045	−	0.823687i	$-0.308086\pi$
$337$	20.8191	1.13409	0.567045	+	0.823687i	$0.308086\pi$
$338$	35.6836	1.94093
$339$	0	0
$340$	0	0
$341$	−18.0129	−0.975452
$342$	0	0
$343$	−17.2138	−0.929456
$344$	13.4702	0.726266
$345$	0	0
$346$	4.19297	0.225415
$347$	−30.9907	−1.66367	−0.831835	−	0.555023i	$-0.812709\pi$
$347$	−30.9907	−1.66367	−0.831835	+	0.555023i	$0.812709\pi$
$348$	0	0
$349$	−7.45484	−0.399048	−0.199524	−	0.979893i	$-0.563940\pi$
$349$	−7.45484	−0.399048	−0.199524	+	0.979893i	$0.563940\pi$
$350$	0	0
$351$	0	0
$352$	−13.8575	−0.738605
$353$	−3.68948	−0.196371	−0.0981856	−	0.995168i	$-0.531304\pi$
$353$	−3.68948	−0.196371	−0.0981856	+	0.995168i	$0.531304\pi$
$354$	0	0
$355$	0	0
$356$	−3.58814	−0.190171
$357$	0	0
$358$	−4.91043	−0.259524
$359$	9.92989	0.524079	0.262040	−	0.965057i	$-0.415605\pi$
$359$	9.92989	0.524079	0.262040	+	0.965057i	$0.415605\pi$
$360$	0	0
$361$	−16.0980	−0.847262
$362$	−43.6835	−2.29595
$363$	0	0
$364$	−11.0304	−0.578149
$365$	0	0
$366$	0	0
$367$	12.5507	0.655142	0.327571	−	0.944826i	$-0.393770\pi$
$367$	12.5507	0.655142	0.327571	+	0.944826i	$0.393770\pi$
$368$	−1.58038	−0.0823828
$369$	0	0
$370$	0	0
$371$	−12.0589	−0.626065
$372$	0	0
$373$	−14.8576	−0.769299	−0.384650	−	0.923063i	$-0.625678\pi$
$373$	−14.8576	−0.769299	−0.384650	+	0.923063i	$0.625678\pi$
$374$	−16.1687	−0.836064
$375$	0	0
$376$	15.2320	0.785530
$377$	−49.8902	−2.56947
$378$	0	0
$379$	11.2957	0.580223	0.290112	−	0.956993i	$-0.406308\pi$
$379$	11.2957	0.580223	0.290112	+	0.956993i	$0.406308\pi$
$380$	0	0
$381$	0	0
$382$	7.15898	0.366285
$383$	−3.04968	−0.155831	−0.0779157	−	0.996960i	$-0.524827\pi$
$383$	−3.04968	−0.155831	−0.0779157	+	0.996960i	$0.524827\pi$
$384$	0	0
$385$	0	0
$386$	15.6171	0.794889
$387$	0	0
$388$	11.4435	0.580958
$389$	21.3939	1.08472	0.542358	−	0.840148i	$-0.317532\pi$
$389$	21.3939	1.08472	0.542358	+	0.840148i	$0.317532\pi$
$390$	0	0
$391$	−1.34444	−0.0679914
$392$	5.93842	0.299936
$393$	0	0
$394$	4.48067	0.225733
$395$	0	0
$396$	0	0
$397$	−34.4305	−1.72802	−0.864009	−	0.503476i	$-0.832054\pi$
$397$	−34.4305	−1.72802	−0.864009	+	0.503476i	$0.832054\pi$
$398$	−19.0647	−0.955629
$399$	0	0
$400$	0	0
$401$	−36.2976	−1.81262	−0.906309	−	0.422616i	$-0.861112\pi$
$401$	−36.2976	−1.81262	−0.906309	+	0.422616i	$0.861112\pi$
$402$	0	0
$403$	48.2514	2.40357
$404$	−13.1869	−0.656074
$405$	0	0
$406$	23.0960	1.14623
$407$	3.72974	0.184876
$408$	0	0
$409$	24.9896	1.23566	0.617828	−	0.786313i	$-0.288013\pi$
$409$	24.9896	1.23566	0.617828	+	0.786313i	$0.288013\pi$
$410$	0	0
$411$	0	0
$412$	−21.5833	−1.06333
$413$	3.07051	0.151090
$414$	0	0
$415$	0	0
$416$	37.1202	1.81997
$417$	0	0
$418$	6.62717	0.324146
$419$	−2.70313	−0.132057	−0.0660284	−	0.997818i	$-0.521033\pi$
$419$	−2.70313	−0.132057	−0.0660284	+	0.997818i	$0.521033\pi$
$420$	0	0
$421$	26.6496	1.29882	0.649411	−	0.760438i	$-0.275016\pi$
$421$	26.6496	1.29882	0.649411	+	0.760438i	$0.275016\pi$
$422$	35.1930	1.71317
$423$	0	0
$424$	10.0890	0.489965
$425$	0	0
$426$	0	0
$427$	−7.78853	−0.376914
$428$	2.42490	0.117212
$429$	0	0
$430$	0	0
$431$	−33.6242	−1.61962	−0.809811	−	0.586691i	$-0.800430\pi$
$431$	−33.6242	−1.61962	−0.809811	+	0.586691i	$0.800430\pi$
$432$	0	0
$433$	−5.34260	−0.256749	−0.128375	−	0.991726i	$-0.540976\pi$
$433$	−5.34260	−0.256749	−0.128375	+	0.991726i	$0.540976\pi$
$434$	−22.3373	−1.07223
$435$	0	0
$436$	−24.7104	−1.18341
$437$	0.551055	0.0263605
$438$	0	0
$439$	26.9690	1.28716	0.643580	−	0.765379i	$-0.277448\pi$
$439$	26.9690	1.28716	0.643580	+	0.765379i	$0.277448\pi$
$440$	0	0
$441$	0	0
$442$	43.3114	2.06011
$443$	16.8670	0.801374	0.400687	−	0.916215i	$-0.368771\pi$
$443$	16.8670	0.801374	0.400687	+	0.916215i	$0.368771\pi$
$444$	0	0
$445$	0	0
$446$	−13.9171	−0.658992
$447$	0	0
$448$	−3.06376	−0.144749
$449$	−21.5942	−1.01909	−0.509546	−	0.860443i	$-0.670187\pi$
$449$	−21.5942	−1.01909	−0.509546	+	0.860443i	$0.670187\pi$
$450$	0	0
$451$	14.6447	0.689593
$452$	−17.8862	−0.841297
$453$	0	0
$454$	−2.40817	−0.113021
$455$	0	0
$456$	0	0
$457$	5.60699	0.262284	0.131142	−	0.991364i	$-0.458136\pi$
$457$	5.60699	0.262284	0.131142	+	0.991364i	$0.458136\pi$
$458$	2.05788	0.0961587
$459$	0	0
$460$	0	0
$461$	9.54504	0.444557	0.222278	−	0.974983i	$-0.428651\pi$
$461$	9.54504	0.444557	0.222278	+	0.974983i	$0.428651\pi$
$462$	0	0
$463$	23.5262	1.09336	0.546678	−	0.837343i	$-0.315892\pi$
$463$	23.5262	1.09336	0.546678	+	0.837343i	$0.315892\pi$
$464$	−42.7353	−1.98394
$465$	0	0
$466$	48.6135	2.25198
$467$	−36.5834	−1.69288	−0.846439	−	0.532486i	$-0.821258\pi$
$467$	−36.5834	−1.69288	−0.846439	+	0.532486i	$0.821258\pi$
$468$	0	0
$469$	−10.3049	−0.475835
$470$	0	0
$471$	0	0
$472$	−2.56893	−0.118245
$473$	−23.7217	−1.09072
$474$	0	0
$475$	0	0
$476$	−8.03793	−0.368418
$477$	0	0
$478$	45.4382	2.07830
$479$	−29.2750	−1.33761	−0.668804	−	0.743439i	$-0.733193\pi$
$479$	−29.2750	−1.33761	−0.668804	+	0.743439i	$0.733193\pi$
$480$	0	0
$481$	−9.99091	−0.455546
$482$	−22.5705	−1.02806
$483$	0	0
$484$	−8.65385	−0.393357
$485$	0	0
$486$	0	0
$487$	21.2869	0.964600	0.482300	−	0.876006i	$-0.339801\pi$
$487$	21.2869	0.964600	0.482300	+	0.876006i	$0.339801\pi$
$488$	6.51624	0.294976
$489$	0	0
$490$	0	0
$491$	33.1940	1.49803	0.749013	−	0.662556i	$-0.230528\pi$
$491$	33.1940	1.49803	0.749013	+	0.662556i	$0.230528\pi$
$492$	0	0
$493$	−36.3554	−1.63737
$494$	−17.7523	−0.798714
$495$	0	0
$496$	41.3316	1.85584
$497$	−3.86515	−0.173376
$498$	0	0
$499$	4.72872	0.211686	0.105843	−	0.994383i	$-0.466246\pi$
$499$	4.72872	0.211686	0.105843	+	0.994383i	$0.466246\pi$
$500$	0	0
$501$	0	0
$502$	53.1657	2.37290
$503$	−0.400444	−0.0178549	−0.00892745	−	0.999960i	$-0.502842\pi$
$503$	−0.400444	−0.0178549	−0.00892745	+	0.999960i	$0.502842\pi$
$504$	0	0
$505$	0	0
$506$	1.25841	0.0559431
$507$	0	0
$508$	1.35674	0.0601956
$509$	27.6104	1.22381	0.611904	−	0.790932i	$-0.290404\pi$
$509$	27.6104	1.22381	0.611904	+	0.790932i	$0.290404\pi$
$510$	0	0
$511$	9.07905	0.401633
$512$	19.9828	0.883126
$513$	0	0
$514$	−22.0164	−0.971100
$515$	0	0
$516$	0	0
$517$	−26.8242	−1.17973
$518$	4.62516	0.203218
$519$	0	0
$520$	0	0
$521$	28.7374	1.25901	0.629505	−	0.776996i	$-0.283258\pi$
$521$	28.7374	1.25901	0.629505	+	0.776996i	$0.283258\pi$
$522$	0	0
$523$	−18.0660	−0.789969	−0.394985	−	0.918688i	$-0.629250\pi$
$523$	−18.0660	−0.789969	−0.394985	+	0.918688i	$0.629250\pi$
$524$	−2.79094	−0.121923
$525$	0	0
$526$	−1.87993	−0.0819690
$527$	35.1612	1.53165
$528$	0	0
$529$	−22.8954	−0.995451
$530$	0	0
$531$	0	0
$532$	3.29456	0.142837
$533$	−39.2290	−1.69920
$534$	0	0
$535$	0	0
$536$	8.62152	0.372393
$537$	0	0
$538$	−43.6807	−1.88321
$539$	−10.4578	−0.450451
$540$	0	0
$541$	−5.26032	−0.226159	−0.113079	−	0.993586i	$-0.536071\pi$
$541$	−5.26032	−0.226159	−0.113079	+	0.993586i	$0.536071\pi$
$542$	6.80068	0.292114
$543$	0	0
$544$	27.0498	1.15975
$545$	0	0
$546$	0	0
$547$	36.7888	1.57297	0.786487	−	0.617607i	$-0.211898\pi$
$547$	36.7888	1.57297	0.786487	+	0.617607i	$0.211898\pi$
$548$	22.6655	0.968221
$549$	0	0
$550$	0	0
$551$	14.9012	0.634813
$552$	0	0
$553$	12.0970	0.514418
$554$	32.1207	1.36468
$555$	0	0
$556$	19.9245	0.844986
$557$	23.3475	0.989266	0.494633	−	0.869102i	$-0.335302\pi$
$557$	23.3475	0.989266	0.494633	+	0.869102i	$0.335302\pi$
$558$	0	0
$559$	63.5436	2.68761
$560$	0	0
$561$	0	0
$562$	−24.2156	−1.02147
$563$	−29.2369	−1.23219	−0.616095	−	0.787672i	$-0.711286\pi$
$563$	−29.2369	−1.23219	−0.616095	+	0.787672i	$0.711286\pi$
$564$	0	0
$565$	0	0
$566$	−22.1071	−0.929232
$567$	0	0
$568$	3.23376	0.135685
$569$	20.2530	0.849051	0.424526	−	0.905416i	$-0.360441\pi$
$569$	20.2530	0.849051	0.424526	+	0.905416i	$0.360441\pi$
$570$	0	0
$571$	−4.57564	−0.191484	−0.0957422	−	0.995406i	$-0.530522\pi$
$571$	−4.57564	−0.191484	−0.0957422	+	0.995406i	$0.530522\pi$
$572$	−16.2518	−0.679523
$573$	0	0
$574$	18.1606	0.758007
$575$	0	0
$576$	0	0
$577$	31.3502	1.30513	0.652564	−	0.757734i	$-0.273694\pi$
$577$	31.3502	1.30513	0.652564	+	0.757734i	$0.273694\pi$
$578$	0.500853	0.0208327
$579$	0	0
$580$	0	0
$581$	21.0803	0.874559
$582$	0	0
$583$	−17.7672	−0.735841
$584$	−7.59594	−0.314322
$585$	0	0
$586$	6.60809	0.272978
$587$	−4.67218	−0.192842	−0.0964208	−	0.995341i	$-0.530739\pi$
$587$	−4.67218	−0.192842	−0.0964208	+	0.995341i	$0.530739\pi$
$588$	0	0
$589$	−14.4117	−0.593825
$590$	0	0
$591$	0	0
$592$	−8.55809	−0.351735
$593$	−7.37978	−0.303051	−0.151526	−	0.988453i	$-0.548419\pi$
$593$	−7.37978	−0.303051	−0.151526	+	0.988453i	$0.548419\pi$
$594$	0	0
$595$	0	0
$596$	6.30309	0.258185
$597$	0	0
$598$	−3.37092	−0.137847
$599$	46.7505	1.91018	0.955088	−	0.296323i	$-0.0957605\pi$
$599$	46.7505	1.91018	0.955088	+	0.296323i	$0.0957605\pi$
$600$	0	0
$601$	32.3225	1.31846	0.659230	−	0.751941i	$-0.270882\pi$
$601$	32.3225	1.31846	0.659230	+	0.751941i	$0.270882\pi$
$602$	−29.4167	−1.19893
$603$	0	0
$604$	−12.8887	−0.524433
$605$	0	0
$606$	0	0
$607$	14.0891	0.571858	0.285929	−	0.958251i	$-0.407698\pi$
$607$	14.0891	0.571858	0.285929	+	0.958251i	$0.407698\pi$
$608$	−11.0871	−0.449640
$609$	0	0
$610$	0	0
$611$	71.8545	2.90692
$612$	0	0
$613$	12.2959	0.496625	0.248313	−	0.968680i	$-0.420124\pi$
$613$	12.2959	0.496625	0.248313	+	0.968680i	$0.420124\pi$
$614$	−27.0908	−1.09330
$615$	0	0
$616$	−3.72023	−0.149892
$617$	−1.96969	−0.0792969	−0.0396485	−	0.999214i	$-0.512624\pi$
$617$	−1.96969	−0.0792969	−0.0396485	+	0.999214i	$0.512624\pi$
$618$	0	0
$619$	14.9714	0.601749	0.300875	−	0.953664i	$-0.402721\pi$
$619$	14.9714	0.601749	0.300875	+	0.953664i	$0.402721\pi$
$620$	0	0
$621$	0	0
$622$	12.0877	0.484671
$623$	−3.87466	−0.155235
$624$	0	0
$625$	0	0
$626$	13.1153	0.524192
$627$	0	0
$628$	−24.8324	−0.990921
$629$	−7.28046	−0.290291
$630$	0	0
$631$	18.0729	0.719472	0.359736	−	0.933054i	$-0.382867\pi$
$631$	18.0729	0.719472	0.359736	+	0.933054i	$0.382867\pi$
$632$	−10.1209	−0.402588
$633$	0	0
$634$	−17.8188	−0.707675
$635$	0	0
$636$	0	0
$637$	28.0136	1.10994
$638$	34.0289	1.34722
$639$	0	0
$640$	0	0
$641$	−12.2381	−0.483374	−0.241687	−	0.970354i	$-0.577701\pi$
$641$	−12.2381	−0.483374	−0.241687	+	0.970354i	$0.577701\pi$
$642$	0	0
$643$	16.4453	0.648540	0.324270	−	0.945964i	$-0.394881\pi$
$643$	16.4453	0.648540	0.324270	+	0.945964i	$0.394881\pi$
$644$	0.625591	0.0246517
$645$	0	0
$646$	−12.9363	−0.508971
$647$	17.3991	0.684028	0.342014	−	0.939695i	$-0.388891\pi$
$647$	17.3991	0.684028	0.342014	+	0.939695i	$0.388891\pi$
$648$	0	0
$649$	4.52400	0.177583
$650$	0	0
$651$	0	0
$652$	−0.604500	−0.0236740
$653$	−28.4645	−1.11390	−0.556950	−	0.830546i	$-0.688029\pi$
$653$	−28.4645	−1.11390	−0.556950	+	0.830546i	$0.688029\pi$
$654$	0	0
$655$	0	0
$656$	−33.6031	−1.31198
$657$	0	0
$658$	−33.2641	−1.29677
$659$	1.51333	0.0589509	0.0294754	−	0.999566i	$-0.490616\pi$
$659$	1.51333	0.0589509	0.0294754	+	0.999566i	$0.490616\pi$
$660$	0	0
$661$	26.7842	1.04179	0.520893	−	0.853622i	$-0.325599\pi$
$661$	26.7842	1.04179	0.520893	+	0.853622i	$0.325599\pi$
$662$	12.2200	0.474943
$663$	0	0
$664$	−17.6368	−0.684439
$665$	0	0
$666$	0	0
$667$	2.82954	0.109560
$668$	6.32526	0.244732
$669$	0	0
$670$	0	0
$671$	−11.4754	−0.443003
$672$	0	0
$673$	−9.31141	−0.358929	−0.179464	−	0.983764i	$-0.557436\pi$
$673$	−9.31141	−0.358929	−0.179464	+	0.983764i	$0.557436\pi$
$674$	38.0385	1.46519
$675$	0	0
$676$	26.1366	1.00526
$677$	24.7854	0.952579	0.476290	−	0.879288i	$-0.341981\pi$
$677$	24.7854	0.952579	0.476290	+	0.879288i	$0.341981\pi$
$678$	0	0
$679$	12.3573	0.474231
$680$	0	0
$681$	0	0
$682$	−32.9112	−1.26023
$683$	−37.3864	−1.43055	−0.715276	−	0.698842i	$-0.753699\pi$
$683$	−37.3864	−1.43055	−0.715276	+	0.698842i	$0.753699\pi$
$684$	0	0
$685$	0	0
$686$	−31.4511	−1.20081
$687$	0	0
$688$	54.4307	2.07515
$689$	47.5932	1.81316
$690$	0	0
$691$	−32.8909	−1.25123	−0.625615	−	0.780132i	$-0.715152\pi$
$691$	−32.8909	−1.25123	−0.625615	+	0.780132i	$0.715152\pi$
$692$	3.07116	0.116748
$693$	0	0
$694$	−56.6229	−2.14938
$695$	0	0
$696$	0	0
$697$	−28.5865	−1.08279
$698$	−13.6207	−0.515550
$699$	0	0
$700$	0	0
$701$	42.3336	1.59892	0.799459	−	0.600721i	$-0.205120\pi$
$701$	42.3336	1.59892	0.799459	+	0.600721i	$0.205120\pi$
$702$	0	0
$703$	2.98409	0.112547
$704$	−4.51406	−0.170130
$705$	0	0
$706$	−6.74101	−0.253701
$707$	−14.2399	−0.535548
$708$	0	0
$709$	18.3593	0.689498	0.344749	−	0.938695i	$-0.387964\pi$
$709$	18.3593	0.689498	0.344749	+	0.938695i	$0.387964\pi$
$710$	0	0
$711$	0	0
$712$	3.24171	0.121488
$713$	−2.73659	−0.102486
$714$	0	0
$715$	0	0
$716$	−3.59667	−0.134414
$717$	0	0
$718$	18.1428	0.677084
$719$	19.2706	0.718671	0.359336	−	0.933208i	$-0.383003\pi$
$719$	19.2706	0.718671	0.359336	+	0.933208i	$0.383003\pi$
$720$	0	0
$721$	−23.3068	−0.867992
$722$	−29.4125	−1.09462
$723$	0	0
$724$	−31.9962	−1.18913
$725$	0	0
$726$	0	0
$727$	−17.4238	−0.646214	−0.323107	−	0.946362i	$-0.604727\pi$
$727$	−17.4238	−0.646214	−0.323107	+	0.946362i	$0.604727\pi$
$728$	9.96543	0.369343
$729$	0	0
$730$	0	0
$731$	46.3048	1.71265
$732$	0	0
$733$	−24.2010	−0.893883	−0.446942	−	0.894563i	$-0.647487\pi$
$733$	−24.2010	−0.893883	−0.446942	+	0.894563i	$0.647487\pi$
$734$	22.9313	0.846410
$735$	0	0
$736$	−2.10528	−0.0776018
$737$	−15.1829	−0.559269
$738$	0	0
$739$	−38.5798	−1.41918	−0.709591	−	0.704613i	$-0.751120\pi$
$739$	−38.5798	−1.41918	−0.709591	+	0.704613i	$0.751120\pi$
$740$	0	0
$741$	0	0
$742$	−22.0326	−0.808844
$743$	−33.8580	−1.24213	−0.621065	−	0.783759i	$-0.713300\pi$
$743$	−33.8580	−1.24213	−0.621065	+	0.783759i	$0.713300\pi$
$744$	0	0
$745$	0	0
$746$	−27.1462	−0.993895
$747$	0	0
$748$	−11.8429	−0.433018
$749$	2.61853	0.0956791
$750$	0	0
$751$	−45.9138	−1.67542	−0.837710	−	0.546115i	$-0.816106\pi$
$751$	−45.9138	−1.67542	−0.837710	+	0.546115i	$0.816106\pi$
$752$	61.5497	2.24449
$753$	0	0
$754$	−91.1539	−3.31963
$755$	0	0
$756$	0	0
$757$	−10.3561	−0.376400	−0.188200	−	0.982131i	$-0.560265\pi$
$757$	−10.3561	−0.376400	−0.188200	+	0.982131i	$0.560265\pi$
$758$	20.6383	0.749618
$759$	0	0
$760$	0	0
$761$	−49.2652	−1.78586	−0.892931	−	0.450193i	$-0.851355\pi$
$761$	−49.2652	−1.78586	−0.892931	+	0.450193i	$0.851355\pi$
$762$	0	0
$763$	−26.6836	−0.966012
$764$	5.24362	0.189708
$765$	0	0
$766$	−5.57205	−0.201326
$767$	−12.1185	−0.437574
$768$	0	0
$769$	−1.26208	−0.0455116	−0.0227558	−	0.999741i	$-0.507244\pi$
$769$	−1.26208	−0.0455116	−0.0227558	+	0.999741i	$0.507244\pi$
$770$	0	0
$771$	0	0
$772$	11.4388	0.411692
$773$	−26.9903	−0.970772	−0.485386	−	0.874300i	$-0.661321\pi$
$773$	−26.9903	−0.970772	−0.485386	+	0.874300i	$0.661321\pi$
$774$	0	0
$775$	0	0
$776$	−10.3387	−0.371138
$777$	0	0
$778$	39.0887	1.40140
$779$	11.7169	0.419803
$780$	0	0
$781$	−5.69480	−0.203776
$782$	−2.45642	−0.0878413
$783$	0	0
$784$	23.9961	0.857003
$785$	0	0
$786$	0	0
$787$	−7.62303	−0.271732	−0.135866	−	0.990727i	$-0.543382\pi$
$787$	−7.62303	−0.271732	−0.135866	+	0.990727i	$0.543382\pi$
$788$	3.28189	0.116912
$789$	0	0
$790$	0	0
$791$	−19.3145	−0.686744
$792$	0	0
$793$	30.7393	1.09159
$794$	−62.9077	−2.23251
$795$	0	0
$796$	−13.9641	−0.494943
$797$	4.98266	0.176495	0.0882474	−	0.996099i	$-0.471873\pi$
$797$	4.98266	0.176495	0.0882474	+	0.996099i	$0.471873\pi$
$798$	0	0
$799$	52.3610	1.85240
$800$	0	0
$801$	0	0
$802$	−66.3191	−2.34181
$803$	13.3768	0.472057
$804$	0	0
$805$	0	0
$806$	88.1596	3.10529
$807$	0	0
$808$	11.9138	0.419125
$809$	−5.74271	−0.201903	−0.100952	−	0.994891i	$-0.532189\pi$
$809$	−5.74271	−0.201903	−0.100952	+	0.994891i	$0.532189\pi$
$810$	0	0
$811$	38.8021	1.36253	0.681264	−	0.732038i	$-0.261431\pi$
$811$	38.8021	1.36253	0.681264	+	0.732038i	$0.261431\pi$
$812$	16.9168	0.593662
$813$	0	0
$814$	6.81457	0.238851
$815$	0	0
$816$	0	0
$817$	−18.9792	−0.664000
$818$	45.6583	1.59640
$819$	0	0
$820$	0	0
$821$	41.8919	1.46204	0.731019	−	0.682357i	$-0.239045\pi$
$821$	41.8919	1.46204	0.731019	+	0.682357i	$0.239045\pi$
$822$	0	0
$823$	29.2212	1.01859	0.509294	−	0.860593i	$-0.329907\pi$
$823$	29.2212	1.01859	0.509294	+	0.860593i	$0.329907\pi$
$824$	19.4995	0.679299
$825$	0	0
$826$	5.61010	0.195200
$827$	7.41934	0.257996	0.128998	−	0.991645i	$-0.458824\pi$
$827$	7.41934	0.257996	0.128998	+	0.991645i	$0.458824\pi$
$828$	0	0
$829$	29.7586	1.03356	0.516779	−	0.856119i	$-0.327131\pi$
$829$	29.7586	1.03356	0.516779	+	0.856119i	$0.327131\pi$
$830$	0	0
$831$	0	0
$832$	12.0919	0.419210
$833$	20.4137	0.707294
$834$	0	0
$835$	0	0
$836$	4.85410	0.167883
$837$	0	0
$838$	−4.93887	−0.170610
$839$	−29.5793	−1.02119	−0.510595	−	0.859821i	$-0.670575\pi$
$839$	−29.5793	−1.02119	−0.510595	+	0.859821i	$0.670575\pi$
$840$	0	0
$841$	47.5142	1.63842
$842$	48.6912	1.67801
$843$	0	0
$844$	25.7773	0.887291
$845$	0	0
$846$	0	0
$847$	−9.34488	−0.321094
$848$	40.7678	1.39997
$849$	0	0
$850$	0	0
$851$	0.566637	0.0194241
$852$	0	0
$853$	−37.5298	−1.28499	−0.642497	−	0.766288i	$-0.722102\pi$
$853$	−37.5298	−1.28499	−0.642497	+	0.766288i	$0.722102\pi$
$854$	−14.2304	−0.486953
$855$	0	0
$856$	−2.19078	−0.0748794
$857$	49.4333	1.68861	0.844305	−	0.535864i	$-0.180014\pi$
$857$	49.4333	1.68861	0.844305	+	0.535864i	$0.180014\pi$
$858$	0	0
$859$	2.82651	0.0964394	0.0482197	−	0.998837i	$-0.484645\pi$
$859$	2.82651	0.0964394	0.0482197	+	0.998837i	$0.484645\pi$
$860$	0	0
$861$	0	0
$862$	−61.4345	−2.09247
$863$	−33.3977	−1.13687	−0.568436	−	0.822727i	$-0.692451\pi$
$863$	−33.3977	−1.13687	−0.568436	+	0.822727i	$0.692451\pi$
$864$	0	0
$865$	0	0
$866$	−9.76142	−0.331707
$867$	0	0
$868$	−16.3611	−0.555331
$869$	17.8234	0.604617
$870$	0	0
$871$	40.6707	1.37807
$872$	22.3247	0.756011
$873$	0	0
$874$	1.00683	0.0340564
$875$	0	0
$876$	0	0
$877$	−9.28115	−0.313402	−0.156701	−	0.987646i	$-0.550086\pi$
$877$	−9.28115	−0.313402	−0.156701	+	0.987646i	$0.550086\pi$
$878$	49.2748	1.66294
$879$	0	0
$880$	0	0
$881$	−6.58736	−0.221934	−0.110967	−	0.993824i	$-0.535395\pi$
$881$	−6.58736	−0.221934	−0.110967	+	0.993824i	$0.535395\pi$
$882$	0	0
$883$	4.18883	0.140965	0.0704827	−	0.997513i	$-0.477546\pi$
$883$	4.18883	0.140965	0.0704827	+	0.997513i	$0.477546\pi$
$884$	31.7236	1.06698
$885$	0	0
$886$	30.8175	1.03533
$887$	26.8716	0.902260	0.451130	−	0.892458i	$-0.351021\pi$
$887$	26.8716	0.902260	0.451130	+	0.892458i	$0.351021\pi$
$888$	0	0
$889$	1.46508	0.0491371
$890$	0	0
$891$	0	0
$892$	−10.1936	−0.341307
$893$	−21.4615	−0.718183
$894$	0	0
$895$	0	0
$896$	13.2128	0.441408
$897$	0	0
$898$	−39.4545	−1.31661
$899$	−74.0008	−2.46807
$900$	0	0
$901$	34.6816	1.15541
$902$	26.7572	0.890918
$903$	0	0
$904$	16.1594	0.537453
$905$	0	0
$906$	0	0
$907$	39.1821	1.30102	0.650511	−	0.759497i	$-0.274555\pi$
$907$	39.1821	1.30102	0.650511	+	0.759497i	$0.274555\pi$
$908$	−1.76387	−0.0585362
$909$	0	0
$910$	0	0
$911$	−54.2926	−1.79880	−0.899398	−	0.437132i	$-0.855994\pi$
$911$	−54.2926	−1.79880	−0.899398	+	0.437132i	$0.855994\pi$
$912$	0	0
$913$	31.0591	1.02791
$914$	10.2445	0.338857
$915$	0	0
$916$	1.50731	0.0498028
$917$	−3.01381	−0.0995247
$918$	0	0
$919$	−28.0959	−0.926798	−0.463399	−	0.886150i	$-0.653370\pi$
$919$	−28.0959	−0.926798	−0.463399	+	0.886150i	$0.653370\pi$
$920$	0	0
$921$	0	0
$922$	17.4396	0.574344
$923$	15.2547	0.502116
$924$	0	0
$925$	0	0
$926$	42.9846	1.41256
$927$	0	0
$928$	−56.9295	−1.86880
$929$	31.3772	1.02945	0.514727	−	0.857354i	$-0.327893\pi$
$929$	31.3772	1.02945	0.514727	+	0.857354i	$0.327893\pi$
$930$	0	0
$931$	−8.36710	−0.274221
$932$	35.6072	1.16635
$933$	0	0
$934$	−66.8412	−2.18711
$935$	0	0
$936$	0	0
$937$	49.7286	1.62456	0.812281	−	0.583266i	$-0.198226\pi$
$937$	49.7286	1.62456	0.812281	+	0.583266i	$0.198226\pi$
$938$	−18.8279	−0.614754
$939$	0	0
$940$	0	0
$941$	4.49598	0.146565	0.0732824	−	0.997311i	$-0.476653\pi$
$941$	4.49598	0.146565	0.0732824	+	0.997311i	$0.476653\pi$
$942$	0	0
$943$	2.22489	0.0724523
$944$	−10.3806	−0.337859
$945$	0	0
$946$	−43.3417	−1.40916
$947$	−28.8993	−0.939101	−0.469551	−	0.882906i	$-0.655584\pi$
$947$	−28.8993	−0.939101	−0.469551	+	0.882906i	$0.655584\pi$
$948$	0	0
$949$	−35.8326	−1.16318
$950$	0	0
$951$	0	0
$952$	7.26190	0.235359
$953$	−46.1922	−1.49631	−0.748156	−	0.663523i	$-0.769060\pi$
$953$	−46.1922	−1.49631	−0.748156	+	0.663523i	$0.769060\pi$
$954$	0	0
$955$	0	0
$956$	33.2814	1.07640
$957$	0	0
$958$	−53.4880	−1.72812
$959$	24.4754	0.790351
$960$	0	0
$961$	40.5701	1.30871
$962$	−18.2543	−0.588542
$963$	0	0
$964$	−16.5319	−0.532456
$965$	0	0
$966$	0	0
$967$	−8.41820	−0.270711	−0.135356	−	0.990797i	$-0.543218\pi$
$967$	−8.41820	−0.270711	−0.135356	+	0.990797i	$0.543218\pi$
$968$	7.81835	0.251291
$969$	0	0
$970$	0	0
$971$	−26.4421	−0.848567	−0.424284	−	0.905529i	$-0.639474\pi$
$971$	−26.4421	−0.848567	−0.424284	+	0.905529i	$0.639474\pi$
$972$	0	0
$973$	21.5155	0.689756
$974$	38.8931	1.24621
$975$	0	0
$976$	26.3309	0.842833
$977$	39.8235	1.27407	0.637034	−	0.770836i	$-0.280161\pi$
$977$	39.8235	1.27407	0.637034	+	0.770836i	$0.280161\pi$
$978$	0	0
$979$	−5.70881	−0.182454
$980$	0	0
$981$	0	0
$982$	60.6485	1.93537
$983$	−24.1179	−0.769242	−0.384621	−	0.923075i	$-0.625668\pi$
$983$	−24.1179	−0.769242	−0.384621	+	0.923075i	$0.625668\pi$
$984$	0	0
$985$	0	0
$986$	−66.4246	−2.11539
$987$	0	0
$988$	−13.0028	−0.413673
$989$	−3.60390	−0.114597
$990$	0	0
$991$	−10.4044	−0.330506	−0.165253	−	0.986251i	$-0.552844\pi$
$991$	−10.4044	−0.330506	−0.165253	+	0.986251i	$0.552844\pi$
$992$	55.0595	1.74814
$993$	0	0
$994$	−7.06198	−0.223992
$995$	0	0
$996$	0	0
$997$	13.3642	0.423249	0.211624	−	0.977351i	$-0.432125\pi$
$997$	13.3642	0.423249	0.211624	+	0.977351i	$0.432125\pi$
$998$	8.63980	0.273488
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.m.1.4		4
3.2	odd	2		1875.2.a.f.1.1	✓	4
5.4	even	2		5625.2.a.j.1.1		4
15.2	even	4		1875.2.b.d.1249.2		8
15.8	even	4		1875.2.b.d.1249.7		8
15.14	odd	2		1875.2.a.g.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.f.1.1	✓	4	3.2	odd	2
1875.2.a.g.1.4	yes	4	15.14	odd	2
1875.2.b.d.1249.2		8	15.2	even	4
1875.2.b.d.1249.7		8	15.8	even	4
5625.2.a.j.1.1		4	5.4	even	2
5625.2.a.m.1.4		4	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82709 q^{2} +1.33826 q^{4} +1.44512 q^{7} -1.20906 q^{8} +O(q^{10})\)	\(q+1.82709 q^{2} +1.33826 q^{4} +1.44512 q^{7} -1.20906 q^{8} +2.12920 q^{11} -5.70353 q^{13} +2.64037 q^{14} -4.88558 q^{16} -4.15622 q^{17} +1.70353 q^{19} +3.89025 q^{22} +0.323478 q^{23} -10.4209 q^{26} +1.93395 q^{28} +8.74724 q^{29} -8.45991 q^{31} -6.50828 q^{32} -7.59378 q^{34} +1.75170 q^{37} +3.11251 q^{38} +6.87802 q^{41} -11.1411 q^{43} +2.84943 q^{44} +0.591023 q^{46} -12.5982 q^{47} -4.91161 q^{49} -7.63282 q^{52} -8.34451 q^{53} -1.74724 q^{56} +15.9820 q^{58} +2.12474 q^{59} -5.38952 q^{61} -15.4570 q^{62} -2.12007 q^{64} -7.13078 q^{67} -5.56210 q^{68} -2.67461 q^{71} +6.28253 q^{73} +3.20052 q^{74} +2.27977 q^{76} +3.07697 q^{77} +8.37092 q^{79} +12.5668 q^{82} +14.5872 q^{83} -20.3558 q^{86} -2.57433 q^{88} -2.68119 q^{89} -8.24232 q^{91} +0.432897 q^{92} -23.0181 q^{94} +8.55105 q^{97} -8.97397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8	\( 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8} + 6 q^{11} - 7 q^{13} + 10 q^{14} - 9 q^{16} - 7 q^{17} - 9 q^{19} - 6 q^{22} + 10 q^{23} + 2 q^{26} - 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} + 10 q^{37} - 6 q^{38} - q^{43} + 9 q^{44} + 5 q^{46} - 23 q^{47} - 3 q^{49} - 13 q^{52} + 2 q^{58} - 4 q^{59} - 43 q^{61} - 10 q^{62} - 7 q^{64} - 8 q^{67} - 3 q^{68} + 27 q^{71} - 15 q^{73} - 5 q^{74} + 9 q^{76} - 15 q^{77} + 10 q^{79} + 20 q^{82} + 3 q^{83} - 24 q^{86} + 3 q^{88} + 9 q^{89} + 5 q^{91} - 15 q^{92} - 22 q^{94} - 13 q^{97} - 42 q^{98}+O(q^{100}) \) 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8 + 6 * q^11 - 7 * q^13 + 10 * q^14 - 9 * q^16 - 7 * q^17 - 9 * q^19 - 6 * q^22 + 10 * q^23 + 2 * q^26 - 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 + 10 * q^37 - 6 * q^38 - q^43 + 9 * q^44 + 5 * q^46 - 23 * q^47 - 3 * q^49 - 13 * q^52 + 2 * q^58 - 4 * q^59 - 43 * q^61 - 10 * q^62 - 7 * q^64 - 8 * q^67 - 3 * q^68 + 27 * q^71 - 15 * q^73 - 5 * q^74 + 9 * q^76 - 15 * q^77 + 10 * q^79 + 20 * q^82 + 3 * q^83 - 24 * q^86 + 3 * q^88 + 9 * q^89 + 5 * q^91 - 15 * q^92 - 22 * q^94 - 13 * q^97 - 42 * q^98

Introduction

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