LMFDB - Embedded newform 5625.2.a.u.1.3

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:	$0$
Dimension:	$8$
Coefficient field:	8.8.13366265625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 $ x^8 - x^7 - 12x^6 + 10x^5 + 41x^4 - 20x^3 - 48x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.3
Root			$1.31354$ 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)

$f(q)$	$=$	$q-1.31354 q^{2} -0.274605 q^{4} +4.19091 q^{7} +2.98779 q^{8} +O(q^{10})$	$q-1.31354 q^{2} -0.274605 q^{4} +4.19091 q^{7} +2.98779 q^{8} +0.167760 q^{11} -3.39519 q^{13} -5.50493 q^{14} -3.37538 q^{16} -4.57476 q^{17} +3.78258 q^{19} -0.220360 q^{22} -8.31374 q^{23} +4.45973 q^{26} -1.15084 q^{28} -2.74553 q^{29} -7.71771 q^{31} -1.54187 q^{32} +6.00914 q^{34} +2.07700 q^{37} -4.96858 q^{38} +1.28043 q^{41} +11.6226 q^{43} -0.0460679 q^{44} +10.9204 q^{46} +9.99490 q^{47} +10.5637 q^{49} +0.932338 q^{52} +1.07165 q^{53} +12.5216 q^{56} +3.60638 q^{58} +4.95536 q^{59} +2.36706 q^{61} +10.1375 q^{62} +8.77608 q^{64} +12.2369 q^{67} +1.25625 q^{68} -14.1746 q^{71} -8.67832 q^{73} -2.72823 q^{74} -1.03872 q^{76} +0.703068 q^{77} -2.64466 q^{79} -1.68189 q^{82} +14.3131 q^{83} -15.2668 q^{86} +0.501233 q^{88} +4.06867 q^{89} -14.2289 q^{91} +2.28300 q^{92} -13.1287 q^{94} -2.78196 q^{97} -13.8759 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8	$ 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8} - 12 q^{11} + 14 q^{13} - 16 q^{14} + 15 q^{16} + q^{17} + 16 q^{19} + 18 q^{22} + 4 q^{23} + 34 q^{26} - 21 q^{28} - 2 q^{29} + 13 q^{31} + 18 q^{32} - 37 q^{34} - 8 q^{37} + 24 q^{38} + 12 q^{41} + 20 q^{43} - 47 q^{44} + 33 q^{46} + 15 q^{47} + 30 q^{49} - q^{52} + 4 q^{53} - 60 q^{56} + 2 q^{58} - 14 q^{59} + 10 q^{61} - 4 q^{62} + 41 q^{64} + 19 q^{67} + 33 q^{68} - 21 q^{71} - 19 q^{73} + 9 q^{74} - q^{76} + 11 q^{77} + 10 q^{79} + 24 q^{82} + 27 q^{83} - 42 q^{86} + 53 q^{88} + 9 q^{89} - 12 q^{91} + 63 q^{92} + 14 q^{94} + 24 q^{97} + 24 q^{98}+O(q^{100}) $ 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8 - 12 * q^11 + 14 * q^13 - 16 * q^14 + 15 * q^16 + q^17 + 16 * q^19 + 18 * q^22 + 4 * q^23 + 34 * q^26 - 21 * q^28 - 2 * q^29 + 13 * q^31 + 18 * q^32 - 37 * q^34 - 8 * q^37 + 24 * q^38 + 12 * q^41 + 20 * q^43 - 47 * q^44 + 33 * q^46 + 15 * q^47 + 30 * q^49 - q^52 + 4 * q^53 - 60 * q^56 + 2 * q^58 - 14 * q^59 + 10 * q^61 - 4 * q^62 + 41 * q^64 + 19 * q^67 + 33 * q^68 - 21 * q^71 - 19 * q^73 + 9 * q^74 - q^76 + 11 * q^77 + 10 * q^79 + 24 * q^82 + 27 * q^83 - 42 * q^86 + 53 * q^88 + 9 * q^89 - 12 * q^91 + 63 * q^92 + 14 * q^94 + 24 * q^97 + 24 * 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 $

$2$	−1.31354	−0.928815	−0.464408	−	0.885622i	$-0.653733\pi$
$2$	−1.31354	−0.928815	−0.464408	+	0.885622i	$0.653733\pi$
$3$	0	0
$3$	0	0
$4$	−0.274605	−0.137303
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.19091	1.58401	0.792007	−	0.610512i	$-0.209036\pi$
$7$	4.19091	1.58401	0.792007	+	0.610512i	$0.209036\pi$
$8$	2.98779	1.05634
$9$	0	0
$10$	0	0
$11$	0.167760	0.0505816	0.0252908	−	0.999680i	$-0.491949\pi$
$11$	0.167760	0.0505816	0.0252908	+	0.999680i	$0.491949\pi$
$12$	0	0
$13$	−3.39519	−0.941658	−0.470829	−	0.882225i	$-0.656045\pi$
$13$	−3.39519	−0.941658	−0.470829	+	0.882225i	$0.656045\pi$
$14$	−5.50493	−1.47126
$15$	0	0
$16$	−3.37538	−0.843845
$17$	−4.57476	−1.10954	−0.554771	−	0.832003i	$-0.687194\pi$
$17$	−4.57476	−1.10954	−0.554771	+	0.832003i	$0.687194\pi$
$18$	0	0
$19$	3.78258	0.867784	0.433892	−	0.900965i	$-0.357140\pi$
$19$	3.78258	0.867784	0.433892	+	0.900965i	$0.357140\pi$
$20$	0	0
$21$	0	0
$22$	−0.220360	−0.0469810
$23$	−8.31374	−1.73353	−0.866767	−	0.498713i	$-0.833806\pi$
$23$	−8.31374	−1.73353	−0.866767	+	0.498713i	$0.833806\pi$
$24$	0	0
$25$	0	0
$26$	4.45973	0.874626
$27$	0	0
$28$	−1.15084	−0.217489
$29$	−2.74553	−0.509833	−0.254916	−	0.966963i	$-0.582048\pi$
$29$	−2.74553	−0.509833	−0.254916	+	0.966963i	$0.582048\pi$
$30$	0	0
$31$	−7.71771	−1.38614	−0.693070	−	0.720870i	$-0.743742\pi$
$31$	−7.71771	−1.38614	−0.693070	+	0.720870i	$0.743742\pi$
$32$	−1.54187	−0.272567
$33$	0	0
$34$	6.00914	1.03056
$35$	0	0
$36$	0	0
$37$	2.07700	0.341457	0.170728	−	0.985318i	$-0.445388\pi$
$37$	2.07700	0.341457	0.170728	+	0.985318i	$0.445388\pi$
$38$	−4.96858	−0.806011
$39$	0	0
$40$	0	0
$41$	1.28043	0.199969	0.0999845	−	0.994989i	$-0.468121\pi$
$41$	1.28043	0.199969	0.0999845	+	0.994989i	$0.468121\pi$
$42$	0	0
$43$	11.6226	1.77243	0.886217	−	0.463270i	$-0.153324\pi$
$43$	11.6226	1.77243	0.886217	+	0.463270i	$0.153324\pi$
$44$	−0.0460679	−0.00694499
$45$	0	0
$46$	10.9204	1.61013
$47$	9.99490	1.45791	0.728953	−	0.684564i	$-0.240007\pi$
$47$	9.99490	1.45791	0.728953	+	0.684564i	$0.240007\pi$
$48$	0	0
$49$	10.5637	1.50910
$50$	0	0
$51$	0	0
$52$	0.932338	0.129292
$53$	1.07165	0.147203	0.0736015	−	0.997288i	$-0.476551\pi$
$53$	1.07165	0.147203	0.0736015	+	0.997288i	$0.476551\pi$
$54$	0	0
$55$	0	0
$56$	12.5216	1.67326
$57$	0	0
$58$	3.60638	0.473540
$59$	4.95536	0.645133	0.322566	−	0.946547i	$-0.395454\pi$
$59$	4.95536	0.645133	0.322566	+	0.946547i	$0.395454\pi$
$60$	0	0
$61$	2.36706	0.303071	0.151536	−	0.988452i	$-0.451578\pi$
$61$	2.36706	0.303071	0.151536	+	0.988452i	$0.451578\pi$
$62$	10.1375	1.28747
$63$	0	0
$64$	8.77608	1.09701
$65$	0	0
$66$	0	0
$67$	12.2369	1.49497	0.747486	−	0.664277i	$-0.231261\pi$
$67$	12.2369	1.49497	0.747486	+	0.664277i	$0.231261\pi$
$68$	1.25625	0.152343
$69$	0	0
$70$	0	0
$71$	−14.1746	−1.68222	−0.841109	−	0.540866i	$-0.818097\pi$
$71$	−14.1746	−1.68222	−0.841109	+	0.540866i	$0.818097\pi$
$72$	0	0
$73$	−8.67832	−1.01572	−0.507860	−	0.861439i	$-0.669563\pi$
$73$	−8.67832	−1.01572	−0.507860	+	0.861439i	$0.669563\pi$
$74$	−2.72823	−0.317150
$75$	0	0
$76$	−1.03872	−0.119149
$77$	0.703068	0.0801220
$78$	0	0
$79$	−2.64466	−0.297547	−0.148774	−	0.988871i	$-0.547533\pi$
$79$	−2.64466	−0.297547	−0.148774	+	0.988871i	$0.547533\pi$
$80$	0	0
$81$	0	0
$82$	−1.68189	−0.185734
$83$	14.3131	1.57106	0.785532	−	0.618821i	$-0.212389\pi$
$83$	14.3131	1.57106	0.785532	+	0.618821i	$0.212389\pi$
$84$	0	0
$85$	0	0
$86$	−15.2668	−1.64626
$87$	0	0
$88$	0.501233	0.0534316
$89$	4.06867	0.431278	0.215639	−	0.976473i	$-0.430817\pi$
$89$	4.06867	0.431278	0.215639	+	0.976473i	$0.430817\pi$
$90$	0	0
$91$	−14.2289	−1.49160
$92$	2.28300	0.238019
$93$	0	0
$94$	−13.1287	−1.35412
$95$	0	0
$96$	0	0
$97$	−2.78196	−0.282466	−0.141233	−	0.989976i	$-0.545107\pi$
$97$	−2.78196	−0.282466	−0.141233	+	0.989976i	$0.545107\pi$
$98$	−13.8759	−1.40167
$99$	0	0
$100$	0	0
$101$	10.7004	1.06473	0.532366	−	0.846514i	$-0.321303\pi$
$101$	10.7004	1.06473	0.532366	+	0.846514i	$0.321303\pi$
$102$	0	0
$103$	−3.61627	−0.356322	−0.178161	−	0.984001i	$-0.557015\pi$
$103$	−3.61627	−0.356322	−0.178161	+	0.984001i	$0.557015\pi$
$104$	−10.1441	−0.994714
$105$	0	0
$106$	−1.40766	−0.136724
$107$	−0.522034	−0.0504669	−0.0252335	−	0.999682i	$-0.508033\pi$
$107$	−0.522034	−0.0504669	−0.0252335	+	0.999682i	$0.508033\pi$
$108$	0	0
$109$	11.5033	1.10181	0.550907	−	0.834566i	$-0.314282\pi$
$109$	11.5033	1.10181	0.550907	+	0.834566i	$0.314282\pi$
$110$	0	0
$111$	0	0
$112$	−14.1459	−1.33666
$113$	20.1421	1.89481	0.947404	−	0.320041i	$-0.103697\pi$
$113$	20.1421	1.89481	0.947404	+	0.320041i	$0.103697\pi$
$114$	0	0
$115$	0	0
$116$	0.753938	0.0700014
$117$	0	0
$118$	−6.50908	−0.599209
$119$	−19.1724	−1.75753
$120$	0	0
$121$	−10.9719	−0.997441
$122$	−3.10924	−0.281497
$123$	0	0
$124$	2.11932	0.190321
$125$	0	0
$126$	0	0
$127$	16.3345	1.44945	0.724726	−	0.689037i	$-0.241966\pi$
$127$	16.3345	1.44945	0.724726	+	0.689037i	$0.241966\pi$
$128$	−8.44401	−0.746352
$129$	0	0
$130$	0	0
$131$	−18.9147	−1.65259	−0.826294	−	0.563239i	$-0.809555\pi$
$131$	−18.9147	−1.65259	−0.826294	+	0.563239i	$0.809555\pi$
$132$	0	0
$133$	15.8524	1.37458
$134$	−16.0737	−1.38855
$135$	0	0
$136$	−13.6684	−1.17206
$137$	13.5975	1.16171	0.580856	−	0.814006i	$-0.302718\pi$
$137$	13.5975	1.16171	0.580856	+	0.814006i	$0.302718\pi$
$138$	0	0
$139$	−3.36103	−0.285079	−0.142540	−	0.989789i	$-0.545527\pi$
$139$	−3.36103	−0.285079	−0.142540	+	0.989789i	$0.545527\pi$
$140$	0	0
$141$	0	0
$142$	18.6190	1.56247
$143$	−0.569579	−0.0476306
$144$	0	0
$145$	0	0
$146$	11.3993	0.943417
$147$	0	0
$148$	−0.570355	−0.0468829
$149$	6.33430	0.518926	0.259463	−	0.965753i	$-0.416454\pi$
$149$	6.33430	0.518926	0.259463	+	0.965753i	$0.416454\pi$
$150$	0	0
$151$	0.375265	0.0305386	0.0152693	−	0.999883i	$-0.495139\pi$
$151$	0.375265	0.0305386	0.0152693	+	0.999883i	$0.495139\pi$
$152$	11.3016	0.916678
$153$	0	0
$154$	−0.923510	−0.0744185
$155$	0	0
$156$	0	0
$157$	10.5002	0.838004	0.419002	−	0.907985i	$-0.362380\pi$
$157$	10.5002	0.838004	0.419002	+	0.907985i	$0.362380\pi$
$158$	3.47387	0.276366
$159$	0	0
$160$	0	0
$161$	−34.8421	−2.74594
$162$	0	0
$163$	−13.8457	−1.08448	−0.542241	−	0.840223i	$-0.682424\pi$
$163$	−13.8457	−1.08448	−0.542241	+	0.840223i	$0.682424\pi$
$164$	−0.351612	−0.0274563
$165$	0	0
$166$	−18.8008	−1.45923
$167$	1.08114	0.0836611	0.0418306	−	0.999125i	$-0.486681\pi$
$167$	1.08114	0.0836611	0.0418306	+	0.999125i	$0.486681\pi$
$168$	0	0
$169$	−1.47265	−0.113281
$170$	0	0
$171$	0	0
$172$	−3.19163	−0.243360
$173$	18.2635	1.38854	0.694272	−	0.719712i	$-0.255726\pi$
$173$	18.2635	1.38854	0.694272	+	0.719712i	$0.255726\pi$
$174$	0	0
$175$	0	0
$176$	−0.566255	−0.0426831
$177$	0	0
$178$	−5.34437	−0.400578
$179$	6.93140	0.518077	0.259039	−	0.965867i	$-0.416594\pi$
$179$	6.93140	0.518077	0.259039	+	0.965867i	$0.416594\pi$
$180$	0	0
$181$	13.1858	0.980091	0.490045	−	0.871697i	$-0.336980\pi$
$181$	13.1858	0.980091	0.490045	+	0.871697i	$0.336980\pi$
$182$	18.6903	1.38542
$183$	0	0
$184$	−24.8397	−1.83121
$185$	0	0
$186$	0	0
$187$	−0.767463	−0.0561224
$188$	−2.74465	−0.200174
$189$	0	0
$190$	0	0
$191$	−23.3938	−1.69272	−0.846358	−	0.532614i	$-0.821210\pi$
$191$	−23.3938	−1.69272	−0.846358	+	0.532614i	$0.821210\pi$
$192$	0	0
$193$	−9.06435	−0.652466	−0.326233	−	0.945289i	$-0.605779\pi$
$193$	−9.06435	−0.652466	−0.326233	+	0.945289i	$0.605779\pi$
$194$	3.65423	0.262358
$195$	0	0
$196$	−2.90084	−0.207203
$197$	13.2823	0.946327	0.473163	−	0.880975i	$-0.343112\pi$
$197$	13.2823	0.946327	0.473163	+	0.880975i	$0.343112\pi$
$198$	0	0
$199$	23.9985	1.70121	0.850606	−	0.525804i	$-0.176235\pi$
$199$	23.9985	1.70121	0.850606	+	0.525804i	$0.176235\pi$
$200$	0	0
$201$	0	0
$202$	−14.0555	−0.988938
$203$	−11.5063	−0.807582
$204$	0	0
$205$	0	0
$206$	4.75013	0.330957
$207$	0	0
$208$	11.4601	0.794613
$209$	0.634567	0.0438939
$210$	0	0
$211$	−7.75260	−0.533711	−0.266855	−	0.963737i	$-0.585985\pi$
$211$	−7.75260	−0.533711	−0.266855	+	0.963737i	$0.585985\pi$
$212$	−0.294282	−0.0202114
$213$	0	0
$214$	0.685714	0.0468744
$215$	0	0
$216$	0	0
$217$	−32.3442	−2.19567
$218$	−15.1101	−1.02338
$219$	0	0
$220$	0	0
$221$	15.5322	1.04481
$222$	0	0
$223$	−22.1577	−1.48379	−0.741895	−	0.670517i	$-0.766073\pi$
$223$	−22.1577	−1.48379	−0.741895	+	0.670517i	$0.766073\pi$
$224$	−6.46185	−0.431751
$225$	0	0
$226$	−26.4575	−1.75993
$227$	11.9286	0.791729	0.395864	−	0.918309i	$-0.370445\pi$
$227$	11.9286	0.791729	0.395864	+	0.918309i	$0.370445\pi$
$228$	0	0
$229$	−7.73897	−0.511406	−0.255703	−	0.966755i	$-0.582307\pi$
$229$	−7.73897	−0.511406	−0.255703	+	0.966755i	$0.582307\pi$
$230$	0	0
$231$	0	0
$232$	−8.20308	−0.538559
$233$	2.85168	0.186820	0.0934099	−	0.995628i	$-0.470223\pi$
$233$	2.85168	0.186820	0.0934099	+	0.995628i	$0.470223\pi$
$234$	0	0
$235$	0	0
$236$	−1.36077	−0.0885784
$237$	0	0
$238$	25.1837	1.63242
$239$	19.9126	1.28804	0.644020	−	0.765009i	$-0.277265\pi$
$239$	19.9126	1.28804	0.644020	+	0.765009i	$0.277265\pi$
$240$	0	0
$241$	6.61379	0.426032	0.213016	−	0.977049i	$-0.431671\pi$
$241$	6.61379	0.426032	0.213016	+	0.977049i	$0.431671\pi$
$242$	14.4120	0.926439
$243$	0	0
$244$	−0.650008	−0.0416125
$245$	0	0
$246$	0	0
$247$	−12.8426	−0.817155
$248$	−23.0589	−1.46424
$249$	0	0
$250$	0	0
$251$	12.0644	0.761501	0.380750	−	0.924678i	$-0.375666\pi$
$251$	12.0644	0.761501	0.380750	+	0.924678i	$0.375666\pi$
$252$	0	0
$253$	−1.39472	−0.0876850
$254$	−21.4561	−1.34627
$255$	0	0
$256$	−6.46059	−0.403787
$257$	12.7951	0.798136	0.399068	−	0.916921i	$-0.369334\pi$
$257$	12.7951	0.798136	0.399068	+	0.916921i	$0.369334\pi$
$258$	0	0
$259$	8.70451	0.540872
$260$	0	0
$261$	0	0
$262$	24.8453	1.53495
$263$	10.5225	0.648843	0.324422	−	0.945913i	$-0.394830\pi$
$263$	10.5225	0.648843	0.324422	+	0.945913i	$0.394830\pi$
$264$	0	0
$265$	0	0
$266$	−20.8229	−1.27673
$267$	0	0
$268$	−3.36031	−0.205264
$269$	−6.80128	−0.414681	−0.207341	−	0.978269i	$-0.566481\pi$
$269$	−6.80128	−0.414681	−0.207341	+	0.978269i	$0.566481\pi$
$270$	0	0
$271$	5.66981	0.344416	0.172208	−	0.985061i	$-0.444910\pi$
$271$	5.66981	0.344416	0.172208	+	0.985061i	$0.444910\pi$
$272$	15.4415	0.936281
$273$	0	0
$274$	−17.8609	−1.07902
$275$	0	0
$276$	0	0
$277$	0.874982	0.0525726	0.0262863	−	0.999654i	$-0.491632\pi$
$277$	0.874982	0.0525726	0.0262863	+	0.999654i	$0.491632\pi$
$278$	4.41486	0.264786
$279$	0	0
$280$	0	0
$281$	4.85806	0.289808	0.144904	−	0.989446i	$-0.453713\pi$
$281$	4.85806	0.289808	0.144904	+	0.989446i	$0.453713\pi$
$282$	0	0
$283$	11.4230	0.679024	0.339512	−	0.940602i	$-0.389738\pi$
$283$	11.4230	0.679024	0.339512	+	0.940602i	$0.389738\pi$
$284$	3.89242	0.230973
$285$	0	0
$286$	0.748167	0.0442400
$287$	5.36615	0.316754
$288$	0	0
$289$	3.92839	0.231082
$290$	0	0
$291$	0	0
$292$	2.38311	0.139461
$293$	−0.0808664	−0.00472427	−0.00236213	−	0.999997i	$-0.500752\pi$
$293$	−0.0808664	−0.00472427	−0.00236213	+	0.999997i	$0.500752\pi$
$294$	0	0
$295$	0	0
$296$	6.20564	0.360696
$297$	0	0
$298$	−8.32038	−0.481987
$299$	28.2268	1.63240
$300$	0	0
$301$	48.7093	2.80756
$302$	−0.492927	−0.0283647
$303$	0	0
$304$	−12.7677	−0.732275
$305$	0	0
$306$	0	0
$307$	−7.77807	−0.443918	−0.221959	−	0.975056i	$-0.571245\pi$
$307$	−7.77807	−0.443918	−0.221959	+	0.975056i	$0.571245\pi$
$308$	−0.193066	−0.0110010
$309$	0	0
$310$	0	0
$311$	13.6181	0.772213	0.386106	−	0.922454i	$-0.373820\pi$
$311$	13.6181	0.772213	0.386106	+	0.922454i	$0.373820\pi$
$312$	0	0
$313$	4.85037	0.274159	0.137079	−	0.990560i	$-0.456228\pi$
$313$	4.85037	0.274159	0.137079	+	0.990560i	$0.456228\pi$
$314$	−13.7924	−0.778351
$315$	0	0
$316$	0.726237	0.0408540
$317$	22.3020	1.25260	0.626302	−	0.779580i	$-0.284568\pi$
$317$	22.3020	1.25260	0.626302	+	0.779580i	$0.284568\pi$
$318$	0	0
$319$	−0.460592	−0.0257882
$320$	0	0
$321$	0	0
$322$	45.7666	2.55047
$323$	−17.3044	−0.962842
$324$	0	0
$325$	0	0
$326$	18.1870	1.00728
$327$	0	0
$328$	3.82565	0.211236
$329$	41.8877	2.30934
$330$	0	0
$331$	0.144235	0.00792788	0.00396394	−	0.999992i	$-0.498738\pi$
$331$	0.144235	0.00792788	0.00396394	+	0.999992i	$0.498738\pi$
$332$	−3.93045	−0.215711
$333$	0	0
$334$	−1.42012	−0.0777057
$335$	0	0
$336$	0	0
$337$	10.1832	0.554717	0.277358	−	0.960767i	$-0.410541\pi$
$337$	10.1832	0.554717	0.277358	+	0.960767i	$0.410541\pi$
$338$	1.93439	0.105217
$339$	0	0
$340$	0	0
$341$	−1.29472	−0.0701133
$342$	0	0
$343$	14.9351	0.806418
$344$	34.7260	1.87230
$345$	0	0
$346$	−23.9898	−1.28970
$347$	−10.5782	−0.567870	−0.283935	−	0.958844i	$-0.591640\pi$
$347$	−10.5782	−0.567870	−0.283935	+	0.958844i	$0.591640\pi$
$348$	0	0
$349$	−29.1511	−1.56042	−0.780210	−	0.625517i	$-0.784888\pi$
$349$	−29.1511	−1.56042	−0.780210	+	0.625517i	$0.784888\pi$
$350$	0	0
$351$	0	0
$352$	−0.258665	−0.0137869
$353$	−3.81907	−0.203268	−0.101634	−	0.994822i	$-0.532407\pi$
$353$	−3.81907	−0.203268	−0.101634	+	0.994822i	$0.532407\pi$
$354$	0	0
$355$	0	0
$356$	−1.11728	−0.0592156
$357$	0	0
$358$	−9.10469	−0.481198
$359$	10.9849	0.579759	0.289879	−	0.957063i	$-0.406385\pi$
$359$	10.9849	0.579759	0.289879	+	0.957063i	$0.406385\pi$
$360$	0	0
$361$	−4.69207	−0.246951
$362$	−17.3201	−0.910323
$363$	0	0
$364$	3.90734	0.204800
$365$	0	0
$366$	0	0
$367$	4.07169	0.212541	0.106270	−	0.994337i	$-0.466109\pi$
$367$	4.07169	0.212541	0.106270	+	0.994337i	$0.466109\pi$
$368$	28.0620	1.46283
$369$	0	0
$370$	0	0
$371$	4.49120	0.233171
$372$	0	0
$373$	1.92973	0.0999175	0.0499588	−	0.998751i	$-0.484091\pi$
$373$	1.92973	0.0999175	0.0499588	+	0.998751i	$0.484091\pi$
$374$	1.00810	0.0521274
$375$	0	0
$376$	29.8627	1.54005
$377$	9.32162	0.480088
$378$	0	0
$379$	14.6050	0.750210	0.375105	−	0.926982i	$-0.377607\pi$
$379$	14.6050	0.750210	0.375105	+	0.926982i	$0.377607\pi$
$380$	0	0
$381$	0	0
$382$	30.7288	1.57222
$383$	31.5290	1.61106	0.805530	−	0.592556i	$-0.201881\pi$
$383$	31.5290	1.61106	0.805530	+	0.592556i	$0.201881\pi$
$384$	0	0
$385$	0	0
$386$	11.9064	0.606021
$387$	0	0
$388$	0.763942	0.0387833
$389$	−14.6046	−0.740485	−0.370242	−	0.928935i	$-0.620725\pi$
$389$	−14.6046	−0.740485	−0.370242	+	0.928935i	$0.620725\pi$
$390$	0	0
$391$	38.0333	1.92343
$392$	31.5621	1.59413
$393$	0	0
$394$	−17.4469	−0.878962
$395$	0	0
$396$	0	0
$397$	15.2793	0.766845	0.383422	−	0.923573i	$-0.374745\pi$
$397$	15.2793	0.766845	0.383422	+	0.923573i	$0.374745\pi$
$398$	−31.5231	−1.58011
$399$	0	0
$400$	0	0
$401$	15.7148	0.784760	0.392380	−	0.919803i	$-0.371652\pi$
$401$	15.7148	0.784760	0.392380	+	0.919803i	$0.371652\pi$
$402$	0	0
$403$	26.2031	1.30527
$404$	−2.93839	−0.146190
$405$	0	0
$406$	15.1140	0.750094
$407$	0.348438	0.0172714
$408$	0	0
$409$	12.4187	0.614064	0.307032	−	0.951699i	$-0.400664\pi$
$409$	12.4187	0.614064	0.307032	+	0.951699i	$0.400664\pi$
$410$	0	0
$411$	0	0
$412$	0.993047	0.0489239
$413$	20.7674	1.02190
$414$	0	0
$415$	0	0
$416$	5.23496	0.256665
$417$	0	0
$418$	−0.833531	−0.0407693
$419$	9.08687	0.443923	0.221961	−	0.975055i	$-0.428754\pi$
$419$	9.08687	0.443923	0.221961	+	0.975055i	$0.428754\pi$
$420$	0	0
$421$	5.15620	0.251298	0.125649	−	0.992075i	$-0.459899\pi$
$421$	5.15620	0.251298	0.125649	+	0.992075i	$0.459899\pi$
$422$	10.1834	0.495719
$423$	0	0
$424$	3.20188	0.155497
$425$	0	0
$426$	0	0
$427$	9.92014	0.480069
$428$	0.143353	0.00692924
$429$	0	0
$430$	0	0
$431$	−3.66953	−0.176755	−0.0883776	−	0.996087i	$-0.528168\pi$
$431$	−3.66953	−0.176755	−0.0883776	+	0.996087i	$0.528168\pi$
$432$	0	0
$433$	−17.7340	−0.852242	−0.426121	−	0.904666i	$-0.640120\pi$
$433$	−17.7340	−0.852242	−0.426121	+	0.904666i	$0.640120\pi$
$434$	42.4855	2.03937
$435$	0	0
$436$	−3.15886	−0.151282
$437$	−31.4474	−1.50433
$438$	0	0
$439$	−26.1823	−1.24961	−0.624806	−	0.780780i	$-0.714822\pi$
$439$	−26.1823	−1.24961	−0.624806	+	0.780780i	$0.714822\pi$
$440$	0	0
$441$	0	0
$442$	−20.4022	−0.970433
$443$	−19.2199	−0.913166	−0.456583	−	0.889681i	$-0.650927\pi$
$443$	−19.2199	−0.913166	−0.456583	+	0.889681i	$0.650927\pi$
$444$	0	0
$445$	0	0
$446$	29.1051	1.37817
$447$	0	0
$448$	36.7797	1.73768
$449$	−33.6407	−1.58760	−0.793802	−	0.608177i	$-0.791901\pi$
$449$	−33.6407	−1.58760	−0.793802	+	0.608177i	$0.791901\pi$
$450$	0	0
$451$	0.214805	0.0101148
$452$	−5.53112	−0.260162
$453$	0	0
$454$	−15.6687	−0.735369
$455$	0	0
$456$	0	0
$457$	−24.0255	−1.12387	−0.561933	−	0.827183i	$-0.689942\pi$
$457$	−24.0255	−1.12387	−0.561933	+	0.827183i	$0.689942\pi$
$458$	10.1655	0.475001
$459$	0	0
$460$	0	0
$461$	14.4282	0.671990	0.335995	−	0.941864i	$-0.390927\pi$
$461$	14.4282	0.671990	0.335995	+	0.941864i	$0.390927\pi$
$462$	0	0
$463$	−2.95599	−0.137377	−0.0686883	−	0.997638i	$-0.521881\pi$
$463$	−2.95599	−0.137377	−0.0686883	+	0.997638i	$0.521881\pi$
$464$	9.26722	0.430220
$465$	0	0
$466$	−3.74581	−0.173521
$467$	−37.1933	−1.72110	−0.860551	−	0.509364i	$-0.829881\pi$
$467$	−37.1933	−1.72110	−0.860551	+	0.509364i	$0.829881\pi$
$468$	0	0
$469$	51.2836	2.36806
$470$	0	0
$471$	0	0
$472$	14.8056	0.681482
$473$	1.94982	0.0896526
$474$	0	0
$475$	0	0
$476$	5.26483	0.241313
$477$	0	0
$478$	−26.1561	−1.19635
$479$	8.65394	0.395409	0.197704	−	0.980262i	$-0.436651\pi$
$479$	8.65394	0.395409	0.197704	+	0.980262i	$0.436651\pi$
$480$	0	0
$481$	−7.05182	−0.321535
$482$	−8.68750	−0.395705
$483$	0	0
$484$	3.01293	0.136951
$485$	0	0
$486$	0	0
$487$	38.4645	1.74299	0.871496	−	0.490402i	$-0.163150\pi$
$487$	38.4645	1.74299	0.871496	+	0.490402i	$0.163150\pi$
$488$	7.07229	0.320148
$489$	0	0
$490$	0	0
$491$	−30.0853	−1.35773	−0.678865	−	0.734263i	$-0.737528\pi$
$491$	−30.0853	−1.35773	−0.678865	+	0.734263i	$0.737528\pi$
$492$	0	0
$493$	12.5601	0.565680
$494$	16.8693	0.758986
$495$	0	0
$496$	26.0502	1.16969
$497$	−59.4045	−2.66466
$498$	0	0
$499$	−19.6343	−0.878951	−0.439476	−	0.898254i	$-0.644836\pi$
$499$	−19.6343	−0.878951	−0.439476	+	0.898254i	$0.644836\pi$
$500$	0	0
$501$	0	0
$502$	−15.8472	−0.707293
$503$	30.4802	1.35904	0.679522	−	0.733655i	$-0.262187\pi$
$503$	30.4802	1.35904	0.679522	+	0.733655i	$0.262187\pi$
$504$	0	0
$505$	0	0
$506$	1.83202	0.0814431
$507$	0	0
$508$	−4.48554	−0.199014
$509$	−11.4371	−0.506942	−0.253471	−	0.967343i	$-0.581572\pi$
$509$	−11.4371	−0.506942	−0.253471	+	0.967343i	$0.581572\pi$
$510$	0	0
$511$	−36.3700	−1.60892
$512$	25.3743	1.12140
$513$	0	0
$514$	−16.8069	−0.741321
$515$	0	0
$516$	0	0
$517$	1.67675	0.0737433
$518$	−11.4337	−0.502370
$519$	0	0
$520$	0	0
$521$	−16.6733	−0.730471	−0.365236	−	0.930915i	$-0.619012\pi$
$521$	−16.6733	−0.730471	−0.365236	+	0.930915i	$0.619012\pi$
$522$	0	0
$523$	2.54272	0.111185	0.0555926	−	0.998454i	$-0.482295\pi$
$523$	2.54272	0.111185	0.0555926	+	0.998454i	$0.482295\pi$
$524$	5.19409	0.226905
$525$	0	0
$526$	−13.8217	−0.602656
$527$	35.3066	1.53798
$528$	0	0
$529$	46.1182	2.00514
$530$	0	0
$531$	0	0
$532$	−4.35316	−0.188734
$533$	−4.34730	−0.188302
$534$	0	0
$535$	0	0
$536$	36.5612	1.57921
$537$	0	0
$538$	8.93377	0.385162
$539$	1.77217	0.0763327
$540$	0	0
$541$	26.3649	1.13352	0.566758	−	0.823884i	$-0.308197\pi$
$541$	26.3649	1.13352	0.566758	+	0.823884i	$0.308197\pi$
$542$	−7.44753	−0.319899
$543$	0	0
$544$	7.05370	0.302425
$545$	0	0
$546$	0	0
$547$	−14.8319	−0.634165	−0.317082	−	0.948398i	$-0.602703\pi$
$547$	−14.8319	−0.634165	−0.317082	+	0.948398i	$0.602703\pi$
$548$	−3.73394	−0.159506
$549$	0	0
$550$	0	0
$551$	−10.3852	−0.442425
$552$	0	0
$553$	−11.0835	−0.471319
$554$	−1.14933	−0.0488302
$555$	0	0
$556$	0.922957	0.0391421
$557$	18.7407	0.794069	0.397034	−	0.917804i	$-0.370039\pi$
$557$	18.7407	0.794069	0.397034	+	0.917804i	$0.370039\pi$
$558$	0	0
$559$	−39.4611	−1.66903
$560$	0	0
$561$	0	0
$562$	−6.38127	−0.269178
$563$	−32.1447	−1.35474	−0.677368	−	0.735644i	$-0.736879\pi$
$563$	−32.1447	−1.35474	−0.677368	+	0.735644i	$0.736879\pi$
$564$	0	0
$565$	0	0
$566$	−15.0045	−0.630688
$567$	0	0
$568$	−42.3508	−1.77700
$569$	2.14141	0.0897725	0.0448862	−	0.998992i	$-0.485707\pi$
$569$	2.14141	0.0897725	0.0448862	+	0.998992i	$0.485707\pi$
$570$	0	0
$571$	11.8476	0.495805	0.247902	−	0.968785i	$-0.420259\pi$
$571$	11.8476	0.495805	0.247902	+	0.968785i	$0.420259\pi$
$572$	0.156409	0.00653981
$573$	0	0
$574$	−7.04866	−0.294205
$575$	0	0
$576$	0	0
$577$	15.6985	0.653539	0.326769	−	0.945104i	$-0.394040\pi$
$577$	15.6985	0.653539	0.326769	+	0.945104i	$0.394040\pi$
$578$	−5.16011	−0.214632
$579$	0	0
$580$	0	0
$581$	59.9847	2.48859
$582$	0	0
$583$	0.179781	0.00744577
$584$	−25.9290	−1.07295
$585$	0	0
$586$	0.106221	0.00438797
$587$	−2.32845	−0.0961055	−0.0480527	−	0.998845i	$-0.515302\pi$
$587$	−2.32845	−0.0961055	−0.0480527	+	0.998845i	$0.515302\pi$
$588$	0	0
$589$	−29.1929	−1.20287
$590$	0	0
$591$	0	0
$592$	−7.01067	−0.288137
$593$	−34.0275	−1.39734	−0.698672	−	0.715442i	$-0.746225\pi$
$593$	−34.0275	−1.39734	−0.698672	+	0.715442i	$0.746225\pi$
$594$	0	0
$595$	0	0
$596$	−1.73943	−0.0712499
$597$	0	0
$598$	−37.0771	−1.51619
$599$	2.93932	0.120097	0.0600486	−	0.998195i	$-0.480874\pi$
$599$	2.93932	0.120097	0.0600486	+	0.998195i	$0.480874\pi$
$600$	0	0
$601$	5.17826	0.211225	0.105613	−	0.994407i	$-0.466320\pi$
$601$	5.17826	0.211225	0.105613	+	0.994407i	$0.466320\pi$
$602$	−63.9818	−2.60770
$603$	0	0
$604$	−0.103050	−0.00419303
$605$	0	0
$606$	0	0
$607$	1.45215	0.0589409	0.0294705	−	0.999566i	$-0.490618\pi$
$607$	1.45215	0.0589409	0.0294705	+	0.999566i	$0.490618\pi$
$608$	−5.83227	−0.236530
$609$	0	0
$610$	0	0
$611$	−33.9346	−1.37285
$612$	0	0
$613$	41.1889	1.66361	0.831803	−	0.555072i	$-0.187309\pi$
$613$	41.1889	1.66361	0.831803	+	0.555072i	$0.187309\pi$
$614$	10.2168	0.412318
$615$	0	0
$616$	2.10062	0.0846364
$617$	33.9582	1.36711	0.683553	−	0.729901i	$-0.260434\pi$
$617$	33.9582	1.36711	0.683553	+	0.729901i	$0.260434\pi$
$618$	0	0
$619$	−31.1879	−1.25355	−0.626774	−	0.779201i	$-0.715625\pi$
$619$	−31.1879	−1.25355	−0.626774	+	0.779201i	$0.715625\pi$
$620$	0	0
$621$	0	0
$622$	−17.8880	−0.717243
$623$	17.0514	0.683151
$624$	0	0
$625$	0	0
$626$	−6.37117	−0.254643
$627$	0	0
$628$	−2.88340	−0.115060
$629$	−9.50177	−0.378860
$630$	0	0
$631$	21.9274	0.872917	0.436459	−	0.899724i	$-0.356233\pi$
$631$	21.9274	0.872917	0.436459	+	0.899724i	$0.356233\pi$
$632$	−7.90168	−0.314312
$633$	0	0
$634$	−29.2946	−1.16344
$635$	0	0
$636$	0	0
$637$	−35.8658	−1.42105
$638$	0.605007	0.0239524
$639$	0	0
$640$	0	0
$641$	19.2185	0.759083	0.379542	−	0.925175i	$-0.376082\pi$
$641$	19.2185	0.759083	0.379542	+	0.925175i	$0.376082\pi$
$642$	0	0
$643$	−13.6611	−0.538739	−0.269370	−	0.963037i	$-0.586815\pi$
$643$	−13.6611	−0.538739	−0.269370	+	0.963037i	$0.586815\pi$
$644$	9.56782	0.377025
$645$	0	0
$646$	22.7301	0.894302
$647$	30.7060	1.20718	0.603588	−	0.797296i	$-0.293737\pi$
$647$	30.7060	1.20718	0.603588	+	0.797296i	$0.293737\pi$
$648$	0	0
$649$	0.831313	0.0326319
$650$	0	0
$651$	0	0
$652$	3.80211	0.148902
$653$	5.75846	0.225346	0.112673	−	0.993632i	$-0.464059\pi$
$653$	5.75846	0.225346	0.112673	+	0.993632i	$0.464059\pi$
$654$	0	0
$655$	0	0
$656$	−4.32193	−0.168743
$657$	0	0
$658$	−55.0212	−2.14495
$659$	−15.9750	−0.622298	−0.311149	−	0.950361i	$-0.600714\pi$
$659$	−15.9750	−0.622298	−0.311149	+	0.950361i	$0.600714\pi$
$660$	0	0
$661$	31.1613	1.21203	0.606017	−	0.795451i	$-0.292766\pi$
$661$	31.1613	1.21203	0.606017	+	0.795451i	$0.292766\pi$
$662$	−0.189459	−0.00736354
$663$	0	0
$664$	42.7645	1.65958
$665$	0	0
$666$	0	0
$667$	22.8256	0.883812
$668$	−0.296887	−0.0114869
$669$	0	0
$670$	0	0
$671$	0.397099	0.0153298
$672$	0	0
$673$	19.1758	0.739174	0.369587	−	0.929196i	$-0.379499\pi$
$673$	19.1758	0.739174	0.369587	+	0.929196i	$0.379499\pi$
$674$	−13.3761	−0.515229
$675$	0	0
$676$	0.404398	0.0155538
$677$	25.2153	0.969105	0.484552	−	0.874762i	$-0.338983\pi$
$677$	25.2153	0.969105	0.484552	+	0.874762i	$0.338983\pi$
$678$	0	0
$679$	−11.6590	−0.447429
$680$	0	0
$681$	0	0
$682$	1.70068	0.0651223
$683$	−28.1160	−1.07583	−0.537914	−	0.843000i	$-0.680788\pi$
$683$	−28.1160	−1.07583	−0.537914	+	0.843000i	$0.680788\pi$
$684$	0	0
$685$	0	0
$686$	−19.6179	−0.749013
$687$	0	0
$688$	−39.2308	−1.49566
$689$	−3.63847	−0.138615
$690$	0	0
$691$	11.8688	0.451511	0.225756	−	0.974184i	$-0.427515\pi$
$691$	11.8688	0.451511	0.225756	+	0.974184i	$0.427515\pi$
$692$	−5.01524	−0.190651
$693$	0	0
$694$	13.8950	0.527446
$695$	0	0
$696$	0	0
$697$	−5.85764	−0.221874
$698$	38.2912	1.44934
$699$	0	0
$700$	0	0
$701$	−26.1748	−0.988608	−0.494304	−	0.869289i	$-0.664577\pi$
$701$	−26.1748	−0.988608	−0.494304	+	0.869289i	$0.664577\pi$
$702$	0	0
$703$	7.85642	0.296311
$704$	1.47228	0.0554886
$705$	0	0
$706$	5.01651	0.188799
$707$	44.8444	1.68655
$708$	0	0
$709$	19.8934	0.747111	0.373556	−	0.927608i	$-0.378138\pi$
$709$	19.8934	0.747111	0.373556	+	0.927608i	$0.378138\pi$
$710$	0	0
$711$	0	0
$712$	12.1563	0.455578
$713$	64.1630	2.40292
$714$	0	0
$715$	0	0
$716$	−1.90340	−0.0711333
$717$	0	0
$718$	−14.4291	−0.538489
$719$	−53.3070	−1.98802	−0.994008	−	0.109309i	$-0.965136\pi$
$719$	−53.3070	−1.98802	−0.994008	+	0.109309i	$0.965136\pi$
$720$	0	0
$721$	−15.1555	−0.564419
$722$	6.16324	0.229372
$723$	0	0
$724$	−3.62088	−0.134569
$725$	0	0
$726$	0	0
$727$	21.0186	0.779537	0.389768	−	0.920913i	$-0.372555\pi$
$727$	21.0186	0.779537	0.389768	+	0.920913i	$0.372555\pi$
$728$	−42.5131	−1.57564
$729$	0	0
$730$	0	0
$731$	−53.1707	−1.96659
$732$	0	0
$733$	−24.4993	−0.904901	−0.452451	−	0.891789i	$-0.649450\pi$
$733$	−24.4993	−0.904901	−0.452451	+	0.891789i	$0.649450\pi$
$734$	−5.34834	−0.197411
$735$	0	0
$736$	12.8187	0.472505
$737$	2.05286	0.0756182
$738$	0	0
$739$	−11.3415	−0.417203	−0.208601	−	0.978001i	$-0.566891\pi$
$739$	−11.3415	−0.417203	−0.208601	+	0.978001i	$0.566891\pi$
$740$	0	0
$741$	0	0
$742$	−5.89938	−0.216573
$743$	20.0546	0.735732	0.367866	−	0.929879i	$-0.380088\pi$
$743$	20.0546	0.735732	0.367866	+	0.929879i	$0.380088\pi$
$744$	0	0
$745$	0	0
$746$	−2.53478	−0.0928049
$747$	0	0
$748$	0.210749	0.00770576
$749$	−2.18779	−0.0799403
$750$	0	0
$751$	−10.7620	−0.392711	−0.196355	−	0.980533i	$-0.562911\pi$
$751$	−10.7620	−0.392711	−0.196355	+	0.980533i	$0.562911\pi$
$752$	−33.7366	−1.23025
$753$	0	0
$754$	−12.2443	−0.445913
$755$	0	0
$756$	0	0
$757$	36.8528	1.33944	0.669718	−	0.742615i	$-0.266415\pi$
$757$	36.8528	1.33944	0.669718	+	0.742615i	$0.266415\pi$
$758$	−19.1843	−0.696806
$759$	0	0
$760$	0	0
$761$	15.5973	0.565403	0.282701	−	0.959208i	$-0.408769\pi$
$761$	15.5973	0.565403	0.282701	+	0.959208i	$0.408769\pi$
$762$	0	0
$763$	48.2092	1.74529
$764$	6.42406	0.232414
$765$	0	0
$766$	−41.4148	−1.49638
$767$	−16.8244	−0.607494
$768$	0	0
$769$	−10.4385	−0.376423	−0.188212	−	0.982128i	$-0.560269\pi$
$769$	−10.4385	−0.376423	−0.188212	+	0.982128i	$0.560269\pi$
$770$	0	0
$771$	0	0
$772$	2.48912	0.0895853
$773$	29.1248	1.04755	0.523773	−	0.851858i	$-0.324524\pi$
$773$	29.1248	1.04755	0.523773	+	0.851858i	$0.324524\pi$
$774$	0	0
$775$	0	0
$776$	−8.31193	−0.298381
$777$	0	0
$778$	19.1838	0.687773
$779$	4.84332	0.173530
$780$	0	0
$781$	−2.37794	−0.0850894
$782$	−49.9584	−1.78651
$783$	0	0
$784$	−35.6565	−1.27345
$785$	0	0
$786$	0	0
$787$	−50.9945	−1.81776	−0.908879	−	0.417060i	$-0.863060\pi$
$787$	−50.9945	−1.81776	−0.908879	+	0.417060i	$0.863060\pi$
$788$	−3.64740	−0.129933
$789$	0	0
$790$	0	0
$791$	84.4135	3.00140
$792$	0	0
$793$	−8.03664	−0.285389
$794$	−20.0700	−0.712257
$795$	0	0
$796$	−6.59013	−0.233581
$797$	−36.6176	−1.29706	−0.648532	−	0.761188i	$-0.724617\pi$
$797$	−36.6176	−1.29706	−0.648532	+	0.761188i	$0.724617\pi$
$798$	0	0
$799$	−45.7242	−1.61761
$800$	0	0
$801$	0	0
$802$	−20.6421	−0.728897
$803$	−1.45588	−0.0513768
$804$	0	0
$805$	0	0
$806$	−34.4189	−1.21235
$807$	0	0
$808$	31.9706	1.12472
$809$	−34.4639	−1.21168	−0.605842	−	0.795585i	$-0.707164\pi$
$809$	−34.4639	−1.21168	−0.605842	+	0.795585i	$0.707164\pi$
$810$	0	0
$811$	35.3834	1.24248	0.621240	−	0.783620i	$-0.286629\pi$
$811$	35.3834	1.24248	0.621240	+	0.783620i	$0.286629\pi$
$812$	3.15968	0.110883
$813$	0	0
$814$	−0.457689	−0.0160420
$815$	0	0
$816$	0	0
$817$	43.9635	1.53809
$818$	−16.3125	−0.570352
$819$	0	0
$820$	0	0
$821$	2.15613	0.0752493	0.0376246	−	0.999292i	$-0.488021\pi$
$821$	2.15613	0.0752493	0.0376246	+	0.999292i	$0.488021\pi$
$822$	0	0
$823$	34.5492	1.20431	0.602154	−	0.798380i	$-0.294309\pi$
$823$	34.5492	1.20431	0.602154	+	0.798380i	$0.294309\pi$
$824$	−10.8047	−0.376398
$825$	0	0
$826$	−27.2789	−0.949155
$827$	21.2543	0.739083	0.369542	−	0.929214i	$-0.379515\pi$
$827$	21.2543	0.739083	0.369542	+	0.929214i	$0.379515\pi$
$828$	0	0
$829$	28.8519	1.00207	0.501035	−	0.865427i	$-0.332953\pi$
$829$	28.8519	1.00207	0.501035	+	0.865427i	$0.332953\pi$
$830$	0	0
$831$	0	0
$832$	−29.7965	−1.03301
$833$	−48.3263	−1.67441
$834$	0	0
$835$	0	0
$836$	−0.174255	−0.00602675
$837$	0	0
$838$	−11.9360	−0.412322
$839$	−0.443198	−0.0153009	−0.00765045	−	0.999971i	$-0.502435\pi$
$839$	−0.443198	−0.0153009	−0.00765045	+	0.999971i	$0.502435\pi$
$840$	0	0
$841$	−21.4620	−0.740071
$842$	−6.77289	−0.233409
$843$	0	0
$844$	2.12890	0.0732799
$845$	0	0
$846$	0	0
$847$	−45.9820	−1.57996
$848$	−3.61724	−0.124217
$849$	0	0
$850$	0	0
$851$	−17.2676	−0.591927
$852$	0	0
$853$	29.0351	0.994142	0.497071	−	0.867710i	$-0.334409\pi$
$853$	29.0351	0.994142	0.497071	+	0.867710i	$0.334409\pi$
$854$	−13.0305	−0.445895
$855$	0	0
$856$	−1.55973	−0.0533104
$857$	−44.1483	−1.50808	−0.754039	−	0.656830i	$-0.771897\pi$
$857$	−44.1483	−1.50808	−0.754039	+	0.656830i	$0.771897\pi$
$858$	0	0
$859$	51.1393	1.74485	0.872426	−	0.488747i	$-0.162546\pi$
$859$	51.1393	1.74485	0.872426	+	0.488747i	$0.162546\pi$
$860$	0	0
$861$	0	0
$862$	4.82009	0.164173
$863$	2.01515	0.0685967	0.0342983	−	0.999412i	$-0.489080\pi$
$863$	2.01515	0.0685967	0.0342983	+	0.999412i	$0.489080\pi$
$864$	0	0
$865$	0	0
$866$	23.2944	0.791575
$867$	0	0
$868$	8.88188	0.301471
$869$	−0.443669	−0.0150504
$870$	0	0
$871$	−41.5466	−1.40775
$872$	34.3694	1.16390
$873$	0	0
$874$	41.3075	1.39725
$875$	0	0
$876$	0	0
$877$	−5.02192	−0.169578	−0.0847891	−	0.996399i	$-0.527022\pi$
$877$	−5.02192	−0.169578	−0.0847891	+	0.996399i	$0.527022\pi$
$878$	34.3916	1.16066
$879$	0	0
$880$	0	0
$881$	42.2541	1.42358	0.711789	−	0.702393i	$-0.247885\pi$
$881$	42.2541	1.42358	0.711789	+	0.702393i	$0.247885\pi$
$882$	0	0
$883$	56.8687	1.91379	0.956893	−	0.290442i	$-0.0938024\pi$
$883$	56.8687	1.91379	0.956893	+	0.290442i	$0.0938024\pi$
$884$	−4.26522	−0.143455
$885$	0	0
$886$	25.2462	0.848162
$887$	17.1287	0.575127	0.287563	−	0.957762i	$-0.407155\pi$
$887$	17.1287	0.575127	0.287563	+	0.957762i	$0.407155\pi$
$888$	0	0
$889$	68.4563	2.29595
$890$	0	0
$891$	0	0
$892$	6.08462	0.203728
$893$	37.8065	1.26515
$894$	0	0
$895$	0	0
$896$	−35.3880	−1.18223
$897$	0	0
$898$	44.1885	1.47459
$899$	21.1892	0.706700
$900$	0	0
$901$	−4.90255	−0.163328
$902$	−0.282155	−0.00939474
$903$	0	0
$904$	60.1803	2.00157
$905$	0	0
$906$	0	0
$907$	−57.9111	−1.92291	−0.961454	−	0.274967i	$-0.911333\pi$
$907$	−57.9111	−1.92291	−0.961454	+	0.274967i	$0.911333\pi$
$908$	−3.27565	−0.108706
$909$	0	0
$910$	0	0
$911$	−7.89167	−0.261463	−0.130731	−	0.991418i	$-0.541733\pi$
$911$	−7.89167	−0.261463	−0.130731	+	0.991418i	$0.541733\pi$
$912$	0	0
$913$	2.40117	0.0794670
$914$	31.5585	1.04386
$915$	0	0
$916$	2.12516	0.0702173
$917$	−79.2699	−2.61772
$918$	0	0
$919$	−3.66345	−0.120846	−0.0604230	−	0.998173i	$-0.519245\pi$
$919$	−3.66345	−0.120846	−0.0604230	+	0.998173i	$0.519245\pi$
$920$	0	0
$921$	0	0
$922$	−18.9521	−0.624155
$923$	48.1256	1.58407
$924$	0	0
$925$	0	0
$926$	3.88282	0.127597
$927$	0	0
$928$	4.23327	0.138964
$929$	49.7787	1.63319	0.816594	−	0.577213i	$-0.195860\pi$
$929$	49.7787	1.63319	0.816594	+	0.577213i	$0.195860\pi$
$930$	0	0
$931$	39.9580	1.30957
$932$	−0.783087	−0.0256509
$933$	0	0
$934$	48.8550	1.59859
$935$	0	0
$936$	0	0
$937$	−9.69363	−0.316677	−0.158338	−	0.987385i	$-0.550614\pi$
$937$	−9.69363	−0.316677	−0.158338	+	0.987385i	$0.550614\pi$
$938$	−67.3632	−2.19949
$939$	0	0
$940$	0	0
$941$	−27.4206	−0.893886	−0.446943	−	0.894562i	$-0.647487\pi$
$941$	−27.4206	−0.893886	−0.446943	+	0.894562i	$0.647487\pi$
$942$	0	0
$943$	−10.6451	−0.346653
$944$	−16.7262	−0.544392
$945$	0	0
$946$	−2.56117	−0.0832707
$947$	12.7622	0.414716	0.207358	−	0.978265i	$-0.433514\pi$
$947$	12.7622	0.414716	0.207358	+	0.978265i	$0.433514\pi$
$948$	0	0
$949$	29.4646	0.956461
$950$	0	0
$951$	0	0
$952$	−57.2830	−1.85655
$953$	−22.3816	−0.725012	−0.362506	−	0.931981i	$-0.618079\pi$
$953$	−22.3816	−0.725012	−0.362506	+	0.931981i	$0.618079\pi$
$954$	0	0
$955$	0	0
$956$	−5.46811	−0.176851
$957$	0	0
$958$	−11.3673	−0.367262
$959$	56.9858	1.84017
$960$	0	0
$961$	28.5630	0.921386
$962$	9.26287	0.298647
$963$	0	0
$964$	−1.81618	−0.0584953
$965$	0	0
$966$	0	0
$967$	−41.4419	−1.33268	−0.666342	−	0.745647i	$-0.732141\pi$
$967$	−41.4419	−1.33268	−0.666342	+	0.745647i	$0.732141\pi$
$968$	−32.7816	−1.05364
$969$	0	0
$970$	0	0
$971$	−40.7475	−1.30765	−0.653824	−	0.756646i	$-0.726836\pi$
$971$	−40.7475	−1.30765	−0.653824	+	0.756646i	$0.726836\pi$
$972$	0	0
$973$	−14.0858	−0.451569
$974$	−50.5248	−1.61892
$975$	0	0
$976$	−7.98974	−0.255745
$977$	21.8246	0.698231	0.349116	−	0.937080i	$-0.386482\pi$
$977$	21.8246	0.698231	0.349116	+	0.937080i	$0.386482\pi$
$978$	0	0
$979$	0.682562	0.0218148
$980$	0	0
$981$	0	0
$982$	39.5183	1.26108
$983$	15.8023	0.504016	0.252008	−	0.967725i	$-0.418909\pi$
$983$	15.8023	0.504016	0.252008	+	0.967725i	$0.418909\pi$
$984$	0	0
$985$	0	0
$986$	−16.4983	−0.525412
$987$	0	0
$988$	3.52665	0.112198
$989$	−96.6275	−3.07257
$990$	0	0
$991$	7.78545	0.247313	0.123657	−	0.992325i	$-0.460538\pi$
$991$	7.78545	0.247313	0.123657	+	0.992325i	$0.460538\pi$
$992$	11.8997	0.377817
$993$	0	0
$994$	78.0303	2.47497
$995$	0	0
$996$	0	0
$997$	20.9919	0.664820	0.332410	−	0.943135i	$-0.392138\pi$
$997$	20.9919	0.664820	0.332410	+	0.943135i	$0.392138\pi$
$998$	25.7905	0.816383
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.u.1.3		8
3.2	odd	2		1875.2.a.o.1.6	yes	8
5.4	even	2		5625.2.a.bc.1.6		8
15.2	even	4		1875.2.b.g.1249.12		16
15.8	even	4		1875.2.b.g.1249.5		16
15.14	odd	2		1875.2.a.n.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.3	✓	8	15.14	odd	2
1875.2.a.o.1.6	yes	8	3.2	odd	2
1875.2.b.g.1249.5		16	15.8	even	4
1875.2.b.g.1249.12		16	15.2	even	4
5625.2.a.u.1.3		8	1.1	even	1	trivial
5625.2.a.bc.1.6		8	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-1.31354 q^{2} -0.274605 q^{4} +4.19091 q^{7} +2.98779 q^{8} +O(q^{10})\)	\(q-1.31354 q^{2} -0.274605 q^{4} +4.19091 q^{7} +2.98779 q^{8} +0.167760 q^{11} -3.39519 q^{13} -5.50493 q^{14} -3.37538 q^{16} -4.57476 q^{17} +3.78258 q^{19} -0.220360 q^{22} -8.31374 q^{23} +4.45973 q^{26} -1.15084 q^{28} -2.74553 q^{29} -7.71771 q^{31} -1.54187 q^{32} +6.00914 q^{34} +2.07700 q^{37} -4.96858 q^{38} +1.28043 q^{41} +11.6226 q^{43} -0.0460679 q^{44} +10.9204 q^{46} +9.99490 q^{47} +10.5637 q^{49} +0.932338 q^{52} +1.07165 q^{53} +12.5216 q^{56} +3.60638 q^{58} +4.95536 q^{59} +2.36706 q^{61} +10.1375 q^{62} +8.77608 q^{64} +12.2369 q^{67} +1.25625 q^{68} -14.1746 q^{71} -8.67832 q^{73} -2.72823 q^{74} -1.03872 q^{76} +0.703068 q^{77} -2.64466 q^{79} -1.68189 q^{82} +14.3131 q^{83} -15.2668 q^{86} +0.501233 q^{88} +4.06867 q^{89} -14.2289 q^{91} +2.28300 q^{92} -13.1287 q^{94} -2.78196 q^{97} -13.8759 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8} - 12 q^{11} + 14 q^{13} - 16 q^{14} + 15 q^{16} + q^{17} + 16 q^{19} + 18 q^{22} + 4 q^{23} + 34 q^{26} - 21 q^{28} - 2 q^{29} + 13 q^{31} + 18 q^{32} - 37 q^{34} - 8 q^{37} + 24 q^{38} + 12 q^{41} + 20 q^{43} - 47 q^{44} + 33 q^{46} + 15 q^{47} + 30 q^{49} - q^{52} + 4 q^{53} - 60 q^{56} + 2 q^{58} - 14 q^{59} + 10 q^{61} - 4 q^{62} + 41 q^{64} + 19 q^{67} + 33 q^{68} - 21 q^{71} - 19 q^{73} + 9 q^{74} - q^{76} + 11 q^{77} + 10 q^{79} + 24 q^{82} + 27 q^{83} - 42 q^{86} + 53 q^{88} + 9 q^{89} - 12 q^{91} + 63 q^{92} + 14 q^{94} + 24 q^{97} + 24 q^{98}+O(q^{100}) \) 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8 - 12 * q^11 + 14 * q^13 - 16 * q^14 + 15 * q^16 + q^17 + 16 * q^19 + 18 * q^22 + 4 * q^23 + 34 * q^26 - 21 * q^28 - 2 * q^29 + 13 * q^31 + 18 * q^32 - 37 * q^34 - 8 * q^37 + 24 * q^38 + 12 * q^41 + 20 * q^43 - 47 * q^44 + 33 * q^46 + 15 * q^47 + 30 * q^49 - q^52 + 4 * q^53 - 60 * q^56 + 2 * q^58 - 14 * q^59 + 10 * q^61 - 4 * q^62 + 41 * q^64 + 19 * q^67 + 33 * q^68 - 21 * q^71 - 19 * q^73 + 9 * q^74 - q^76 + 11 * q^77 + 10 * q^79 + 24 * q^82 + 27 * q^83 - 42 * q^86 + 53 * q^88 + 9 * q^89 - 12 * q^91 + 63 * q^92 + 14 * q^94 + 24 * q^97 + 24 * q^98

Introduction

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