LMFDB - Embedded newform 5625.2.a.p.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5625,2,Mod(1,5625)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,10,0,0,-6,3,0,0,-3,0,-6,22,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.44400625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 $ x^6 - 11x^4 - x^3 + 29x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.68704$ 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)

$f(q)$	$=$	$q+2.68704 q^{2} +5.22020 q^{4} +1.68704 q^{7} +8.65280 q^{8} -1.07882 q^{11} +2.67723 q^{13} +4.53315 q^{14} +12.8101 q^{16} +3.93167 q^{17} -1.17755 q^{19} -2.89883 q^{22} +4.06295 q^{23} +7.19384 q^{26} +8.80669 q^{28} -5.95595 q^{29} -7.10310 q^{31} +17.1155 q^{32} +10.5646 q^{34} -4.58187 q^{37} -3.16412 q^{38} +11.2614 q^{41} +2.58587 q^{43} -5.63164 q^{44} +10.9173 q^{46} +1.91782 q^{47} -4.15389 q^{49} +13.9757 q^{52} +2.54861 q^{53} +14.5976 q^{56} -16.0039 q^{58} -1.33599 q^{59} -7.28106 q^{61} -19.0863 q^{62} +20.3701 q^{64} +12.4451 q^{67} +20.5241 q^{68} +5.98480 q^{71} -3.30065 q^{73} -12.3117 q^{74} -6.14702 q^{76} -1.82001 q^{77} -4.00404 q^{79} +30.2600 q^{82} -8.87482 q^{83} +6.94834 q^{86} -9.33480 q^{88} -15.4436 q^{89} +4.51660 q^{91} +21.2094 q^{92} +5.15327 q^{94} +10.7682 q^{97} -11.1617 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37}+ \cdots - 41 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8 - 3 * q^11 - 6 * q^13 + 22 * q^14 + 18 * q^16 + 13 * q^17 + 11 * q^19 - 16 * q^22 + 13 * q^23 + 28 * q^26 - 7 * q^28 + 3 * q^29 - 11 * q^31 + 16 * q^32 + 15 * q^34 - 21 * q^37 - 9 * q^38 + q^41 - 2 * q^43 - 9 * q^44 + 19 * q^46 + 14 * q^47 - 14 * q^49 - 13 * q^52 + 23 * q^53 + 35 * q^56 - 22 * q^58 - 9 * q^59 + 11 * q^61 - 23 * q^62 - 23 * q^64 - 8 * q^67 + 50 * q^68 + 8 * q^71 - 13 * q^73 + 22 * q^74 - 26 * q^76 - 13 * q^77 - 5 * q^79 + 13 * q^82 - 20 * q^83 + 37 * q^86 - 28 * q^88 + 4 * q^89 + 34 * q^91 + 61 * q^92 + 41 * q^94 + 7 * q^97 - 41 * q^98

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 $

$2$	2.68704	1.90003	0.950013	−	0.312211i	$-0.101069\pi$
$2$	2.68704	1.90003	0.950013	+	0.312211i	$0.101069\pi$
$3$	0	0
$3$	0	0
$4$	5.22020	2.61010
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.68704	0.637642	0.318821	−	0.947815i	$-0.396713\pi$
$7$	1.68704	0.637642	0.318821	+	0.947815i	$0.396713\pi$
$8$	8.65280	3.05923
$9$	0	0
$10$	0	0
$11$	−1.07882	−0.325276	−0.162638	−	0.986686i	$-0.552000\pi$
$11$	−1.07882	−0.325276	−0.162638	+	0.986686i	$0.552000\pi$
$12$	0	0
$13$	2.67723	0.742531	0.371265	−	0.928527i	$-0.378924\pi$
$13$	2.67723	0.742531	0.371265	+	0.928527i	$0.378924\pi$
$14$	4.53315	1.21154
$15$	0	0
$16$	12.8101	3.20251
$17$	3.93167	0.953570	0.476785	−	0.879020i	$-0.341802\pi$
$17$	3.93167	0.953570	0.476785	+	0.879020i	$0.341802\pi$
$18$	0	0
$19$	−1.17755	−0.270148	−0.135074	−	0.990836i	$-0.543127\pi$
$19$	−1.17755	−0.270148	−0.135074	+	0.990836i	$0.543127\pi$
$20$	0	0
$21$	0	0
$22$	−2.89883	−0.618032
$23$	4.06295	0.847183	0.423591	−	0.905853i	$-0.360769\pi$
$23$	4.06295	0.847183	0.423591	+	0.905853i	$0.360769\pi$
$24$	0	0
$25$	0	0
$26$	7.19384	1.41083
$27$	0	0
$28$	8.80669	1.66431
$29$	−5.95595	−1.10599	−0.552996	−	0.833184i	$-0.686516\pi$
$29$	−5.95595	−1.10599	−0.552996	+	0.833184i	$0.686516\pi$
$30$	0	0
$31$	−7.10310	−1.27575	−0.637877	−	0.770138i	$-0.720187\pi$
$31$	−7.10310	−1.27575	−0.637877	+	0.770138i	$0.720187\pi$
$32$	17.1155	3.02563
$33$	0	0
$34$	10.5646	1.81181
$35$	0	0
$36$	0	0
$37$	−4.58187	−0.753254	−0.376627	−	0.926365i	$-0.622916\pi$
$37$	−4.58187	−0.753254	−0.376627	+	0.926365i	$0.622916\pi$
$38$	−3.16412	−0.513287
$39$	0	0
$40$	0	0
$41$	11.2614	1.75874	0.879371	−	0.476137i	$-0.157963\pi$
$41$	11.2614	1.75874	0.879371	+	0.476137i	$0.157963\pi$
$42$	0	0
$43$	2.58587	0.394342	0.197171	−	0.980369i	$-0.436825\pi$
$43$	2.58587	0.394342	0.197171	+	0.980369i	$0.436825\pi$
$44$	−5.63164	−0.849002
$45$	0	0
$46$	10.9173	1.60967
$47$	1.91782	0.279743	0.139871	−	0.990170i	$-0.455331\pi$
$47$	1.91782	0.279743	0.139871	+	0.990170i	$0.455331\pi$
$48$	0	0
$49$	−4.15389	−0.593413
$50$	0	0
$51$	0	0
$52$	13.9757	1.93808
$53$	2.54861	0.350078	0.175039	−	0.984561i	$-0.443995\pi$
$53$	2.54861	0.350078	0.175039	+	0.984561i	$0.443995\pi$
$54$	0	0
$55$	0	0
$56$	14.5976	1.95069
$57$	0	0
$58$	−16.0039	−2.10141
$59$	−1.33599	−0.173931	−0.0869654	−	0.996211i	$-0.527717\pi$
$59$	−1.33599	−0.173931	−0.0869654	+	0.996211i	$0.527717\pi$
$60$	0	0
$61$	−7.28106	−0.932245	−0.466122	−	0.884720i	$-0.654349\pi$
$61$	−7.28106	−0.932245	−0.466122	+	0.884720i	$0.654349\pi$
$62$	−19.0863	−2.42397
$63$	0	0
$64$	20.3701	2.54626
$65$	0	0
$66$	0	0
$67$	12.4451	1.52041	0.760203	−	0.649686i	$-0.225100\pi$
$67$	12.4451	1.52041	0.760203	+	0.649686i	$0.225100\pi$
$68$	20.5241	2.48891
$69$	0	0
$70$	0	0
$71$	5.98480	0.710266	0.355133	−	0.934816i	$-0.384436\pi$
$71$	5.98480	0.710266	0.355133	+	0.934816i	$0.384436\pi$
$72$	0	0
$73$	−3.30065	−0.386312	−0.193156	−	0.981168i	$-0.561872\pi$
$73$	−3.30065	−0.386312	−0.193156	+	0.981168i	$0.561872\pi$
$74$	−12.3117	−1.43120
$75$	0	0
$76$	−6.14702	−0.705112
$77$	−1.82001	−0.207410
$78$	0	0
$79$	−4.00404	−0.450490	−0.225245	−	0.974302i	$-0.572318\pi$
$79$	−4.00404	−0.450490	−0.225245	+	0.974302i	$0.572318\pi$
$80$	0	0
$81$	0	0
$82$	30.2600	3.34166
$83$	−8.87482	−0.974138	−0.487069	−	0.873364i	$-0.661934\pi$
$83$	−8.87482	−0.974138	−0.487069	+	0.873364i	$0.661934\pi$
$84$	0	0
$85$	0	0
$86$	6.94834	0.749259
$87$	0	0
$88$	−9.33480	−0.995093
$89$	−15.4436	−1.63702	−0.818509	−	0.574493i	$-0.805199\pi$
$89$	−15.4436	−1.63702	−0.818509	+	0.574493i	$0.805199\pi$
$90$	0	0
$91$	4.51660	0.473469
$92$	21.2094	2.21123
$93$	0	0
$94$	5.15327	0.531519
$95$	0	0
$96$	0	0
$97$	10.7682	1.09335	0.546673	−	0.837346i	$-0.315894\pi$
$97$	10.7682	1.09335	0.546673	+	0.837346i	$0.315894\pi$
$98$	−11.1617	−1.12750
$99$	0	0
$100$	0	0
$101$	13.5654	1.34981	0.674905	−	0.737905i	$-0.264185\pi$
$101$	13.5654	1.34981	0.674905	+	0.737905i	$0.264185\pi$
$102$	0	0
$103$	1.36904	0.134895	0.0674476	−	0.997723i	$-0.478514\pi$
$103$	1.36904	0.134895	0.0674476	+	0.997723i	$0.478514\pi$
$104$	23.1656	2.27157
$105$	0	0
$106$	6.84822	0.665158
$107$	−11.0115	−1.06452	−0.532262	−	0.846579i	$-0.678658\pi$
$107$	−11.0115	−1.06452	−0.532262	+	0.846579i	$0.678658\pi$
$108$	0	0
$109$	3.44890	0.330344	0.165172	−	0.986265i	$-0.447182\pi$
$109$	3.44890	0.330344	0.165172	+	0.986265i	$0.447182\pi$
$110$	0	0
$111$	0	0
$112$	21.6111	2.04206
$113$	−6.05194	−0.569319	−0.284660	−	0.958629i	$-0.591881\pi$
$113$	−6.05194	−0.569319	−0.284660	+	0.958629i	$0.591881\pi$
$114$	0	0
$115$	0	0
$116$	−31.0912	−2.88675
$117$	0	0
$118$	−3.58986	−0.330473
$119$	6.63289	0.608036
$120$	0	0
$121$	−9.83615	−0.894196
$122$	−19.5645	−1.77129
$123$	0	0
$124$	−37.0796	−3.32984
$125$	0	0
$126$	0	0
$127$	−20.2362	−1.79567	−0.897835	−	0.440332i	$-0.854861\pi$
$127$	−20.2362	−1.79567	−0.897835	+	0.440332i	$0.854861\pi$
$128$	20.5042	1.81233
$129$	0	0
$130$	0	0
$131$	3.05011	0.266490	0.133245	−	0.991083i	$-0.457460\pi$
$131$	3.05011	0.266490	0.133245	+	0.991083i	$0.457460\pi$
$132$	0	0
$133$	−1.98657	−0.172257
$134$	33.4404	2.88881
$135$	0	0
$136$	34.0200	2.91719
$137$	20.5100	1.75229	0.876145	−	0.482048i	$-0.160107\pi$
$137$	20.5100	1.75229	0.876145	+	0.482048i	$0.160107\pi$
$138$	0	0
$139$	18.3184	1.55374	0.776871	−	0.629659i	$-0.216806\pi$
$139$	18.3184	1.55374	0.776871	+	0.629659i	$0.216806\pi$
$140$	0	0
$141$	0	0
$142$	16.0814	1.34952
$143$	−2.88825	−0.241527
$144$	0	0
$145$	0	0
$146$	−8.86899	−0.734003
$147$	0	0
$148$	−23.9182	−1.96607
$149$	−0.705938	−0.0578327	−0.0289163	−	0.999582i	$-0.509206\pi$
$149$	−0.705938	−0.0578327	−0.0289163	+	0.999582i	$0.509206\pi$
$150$	0	0
$151$	7.04538	0.573345	0.286673	−	0.958029i	$-0.407451\pi$
$151$	7.04538	0.573345	0.286673	+	0.958029i	$0.407451\pi$
$152$	−10.1891	−0.826443
$153$	0	0
$154$	−4.89045	−0.394083
$155$	0	0
$156$	0	0
$157$	−5.12880	−0.409323	−0.204662	−	0.978833i	$-0.565609\pi$
$157$	−5.12880	−0.409323	−0.204662	+	0.978833i	$0.565609\pi$
$158$	−10.7590	−0.855942
$159$	0	0
$160$	0	0
$161$	6.85436	0.540199
$162$	0	0
$163$	−24.2613	−1.90029	−0.950146	−	0.311806i	$-0.899066\pi$
$163$	−24.2613	−1.90029	−0.950146	+	0.311806i	$0.899066\pi$
$164$	58.7869	4.59049
$165$	0	0
$166$	−23.8470	−1.85089
$167$	−4.54890	−0.352004	−0.176002	−	0.984390i	$-0.556317\pi$
$167$	−4.54890	−0.352004	−0.176002	+	0.984390i	$0.556317\pi$
$168$	0	0
$169$	−5.83243	−0.448648
$170$	0	0
$171$	0	0
$172$	13.4988	1.02927
$173$	2.67729	0.203550	0.101775	−	0.994807i	$-0.467548\pi$
$173$	2.67729	0.203550	0.101775	+	0.994807i	$0.467548\pi$
$174$	0	0
$175$	0	0
$176$	−13.8197	−1.04170
$177$	0	0
$178$	−41.4976	−3.11038
$179$	15.6266	1.16799	0.583995	−	0.811757i	$-0.301489\pi$
$179$	15.6266	1.16799	0.583995	+	0.811757i	$0.301489\pi$
$180$	0	0
$181$	0.202843	0.0150772	0.00753859	−	0.999972i	$-0.497600\pi$
$181$	0.202843	0.0150772	0.00753859	+	0.999972i	$0.497600\pi$
$182$	12.1363	0.899603
$183$	0	0
$184$	35.1559	2.59172
$185$	0	0
$186$	0	0
$187$	−4.24155	−0.310173
$188$	10.0114	0.730156
$189$	0	0
$190$	0	0
$191$	7.91975	0.573053	0.286526	−	0.958072i	$-0.407499\pi$
$191$	7.91975	0.573053	0.286526	+	0.958072i	$0.407499\pi$
$192$	0	0
$193$	−16.4269	−1.18243	−0.591216	−	0.806513i	$-0.701352\pi$
$193$	−16.4269	−1.18243	−0.591216	+	0.806513i	$0.701352\pi$
$194$	28.9346	2.07738
$195$	0	0
$196$	−21.6841	−1.54887
$197$	17.1369	1.22095	0.610475	−	0.792035i	$-0.290978\pi$
$197$	17.1369	1.22095	0.610475	+	0.792035i	$0.290978\pi$
$198$	0	0
$199$	−2.01848	−0.143086	−0.0715430	−	0.997438i	$-0.522792\pi$
$199$	−2.01848	−0.143086	−0.0715430	+	0.997438i	$0.522792\pi$
$200$	0	0
$201$	0	0
$202$	36.4509	2.56467
$203$	−10.0479	−0.705227
$204$	0	0
$205$	0	0
$206$	3.67866	0.256304
$207$	0	0
$208$	34.2955	2.37796
$209$	1.27036	0.0878725
$210$	0	0
$211$	−23.8213	−1.63993	−0.819963	−	0.572416i	$-0.806006\pi$
$211$	−23.8213	−1.63993	−0.819963	+	0.572416i	$0.806006\pi$
$212$	13.3042	0.913738
$213$	0	0
$214$	−29.5884	−2.02262
$215$	0	0
$216$	0	0
$217$	−11.9832	−0.813475
$218$	9.26733	0.627663
$219$	0	0
$220$	0	0
$221$	10.5260	0.708055
$222$	0	0
$223$	13.1112	0.877989	0.438995	−	0.898490i	$-0.355335\pi$
$223$	13.1112	0.877989	0.438995	+	0.898490i	$0.355335\pi$
$224$	28.8746	1.92927
$225$	0	0
$226$	−16.2618	−1.08172
$227$	26.2807	1.74431	0.872155	−	0.489229i	$-0.162722\pi$
$227$	26.2807	1.74431	0.872155	+	0.489229i	$0.162722\pi$
$228$	0	0
$229$	1.99605	0.131903	0.0659513	−	0.997823i	$-0.478992\pi$
$229$	1.99605	0.131903	0.0659513	+	0.997823i	$0.478992\pi$
$230$	0	0
$231$	0	0
$232$	−51.5357	−3.38348
$233$	0.525548	0.0344298	0.0172149	−	0.999852i	$-0.494520\pi$
$233$	0.525548	0.0344298	0.0172149	+	0.999852i	$0.494520\pi$
$234$	0	0
$235$	0	0
$236$	−6.97412	−0.453976
$237$	0	0
$238$	17.8229	1.15528
$239$	7.00971	0.453421	0.226710	−	0.973962i	$-0.427203\pi$
$239$	7.00971	0.453421	0.226710	+	0.973962i	$0.427203\pi$
$240$	0	0
$241$	2.31554	0.149157	0.0745786	−	0.997215i	$-0.476239\pi$
$241$	2.31554	0.149157	0.0745786	+	0.997215i	$0.476239\pi$
$242$	−26.4302	−1.69899
$243$	0	0
$244$	−38.0086	−2.43325
$245$	0	0
$246$	0	0
$247$	−3.15257	−0.200593
$248$	−61.4617	−3.90282
$249$	0	0
$250$	0	0
$251$	6.17885	0.390005	0.195003	−	0.980803i	$-0.437528\pi$
$251$	6.17885	0.390005	0.195003	+	0.980803i	$0.437528\pi$
$252$	0	0
$253$	−4.38318	−0.275568
$254$	−54.3755	−3.41182
$255$	0	0
$256$	14.3555	0.897217
$257$	−13.2239	−0.824882	−0.412441	−	0.910984i	$-0.635324\pi$
$257$	−13.2239	−0.824882	−0.412441	+	0.910984i	$0.635324\pi$
$258$	0	0
$259$	−7.72980	−0.480306
$260$	0	0
$261$	0	0
$262$	8.19578	0.506337
$263$	6.97643	0.430185	0.215092	−	0.976594i	$-0.430995\pi$
$263$	6.97643	0.430185	0.215092	+	0.976594i	$0.430995\pi$
$264$	0	0
$265$	0	0
$266$	−5.33800	−0.327294
$267$	0	0
$268$	64.9656	3.96841
$269$	−23.7199	−1.44623	−0.723113	−	0.690730i	$-0.757289\pi$
$269$	−23.7199	−1.44623	−0.723113	+	0.690730i	$0.757289\pi$
$270$	0	0
$271$	28.9910	1.76108	0.880539	−	0.473975i	$-0.157181\pi$
$271$	28.9910	1.76108	0.880539	+	0.473975i	$0.157181\pi$
$272$	50.3649	3.05382
$273$	0	0
$274$	55.1113	3.32940
$275$	0	0
$276$	0	0
$277$	−8.58667	−0.515923	−0.257962	−	0.966155i	$-0.583051\pi$
$277$	−8.58667	−0.515923	−0.257962	+	0.966155i	$0.583051\pi$
$278$	49.2222	2.95215
$279$	0	0
$280$	0	0
$281$	−17.1661	−1.02405	−0.512023	−	0.858972i	$-0.671104\pi$
$281$	−17.1661	−1.02405	−0.512023	+	0.858972i	$0.671104\pi$
$282$	0	0
$283$	−9.62005	−0.571853	−0.285926	−	0.958252i	$-0.592301\pi$
$283$	−9.62005	−0.571853	−0.285926	+	0.958252i	$0.592301\pi$
$284$	31.2419	1.85386
$285$	0	0
$286$	−7.76084	−0.458908
$287$	18.9985	1.12145
$288$	0	0
$289$	−1.54198	−0.0907048
$290$	0	0
$291$	0	0
$292$	−17.2301	−1.00831
$293$	−16.8202	−0.982649	−0.491324	−	0.870977i	$-0.663487\pi$
$293$	−16.8202	−0.982649	−0.491324	+	0.870977i	$0.663487\pi$
$294$	0	0
$295$	0	0
$296$	−39.6460	−2.30438
$297$	0	0
$298$	−1.89688	−0.109884
$299$	10.8775	0.629059
$300$	0	0
$301$	4.36247	0.251449
$302$	18.9312	1.08937
$303$	0	0
$304$	−15.0844	−0.865151
$305$	0	0
$306$	0	0
$307$	11.3212	0.646134	0.323067	−	0.946376i	$-0.395286\pi$
$307$	11.3212	0.646134	0.323067	+	0.946376i	$0.395286\pi$
$308$	−9.50081	−0.541359
$309$	0	0
$310$	0	0
$311$	−18.5635	−1.05264	−0.526318	−	0.850288i	$-0.676428\pi$
$311$	−18.5635	−1.05264	−0.526318	+	0.850288i	$0.676428\pi$
$312$	0	0
$313$	−9.69951	−0.548249	−0.274124	−	0.961694i	$-0.588388\pi$
$313$	−9.69951	−0.548249	−0.274124	+	0.961694i	$0.588388\pi$
$314$	−13.7813	−0.777724
$315$	0	0
$316$	−20.9019	−1.17582
$317$	−7.47647	−0.419920	−0.209960	−	0.977710i	$-0.567333\pi$
$317$	−7.47647	−0.419920	−0.209960	+	0.977710i	$0.567333\pi$
$318$	0	0
$319$	6.42538	0.359752
$320$	0	0
$321$	0	0
$322$	18.4180	1.02639
$323$	−4.62972	−0.257605
$324$	0	0
$325$	0	0
$326$	−65.1911	−3.61060
$327$	0	0
$328$	97.4430	5.38039
$329$	3.23544	0.178376
$330$	0	0
$331$	−25.2570	−1.38825	−0.694125	−	0.719854i	$-0.744209\pi$
$331$	−25.2570	−1.38825	−0.694125	+	0.719854i	$0.744209\pi$
$332$	−46.3283	−2.54259
$333$	0	0
$334$	−12.2231	−0.668817
$335$	0	0
$336$	0	0
$337$	−13.0240	−0.709463	−0.354731	−	0.934968i	$-0.615428\pi$
$337$	−13.0240	−0.709463	−0.354731	+	0.934968i	$0.615428\pi$
$338$	−15.6720	−0.852443
$339$	0	0
$340$	0	0
$341$	7.66295	0.414972
$342$	0	0
$343$	−18.8171	−1.01603
$344$	22.3750	1.20638
$345$	0	0
$346$	7.19398	0.386751
$347$	6.00149	0.322177	0.161089	−	0.986940i	$-0.448500\pi$
$347$	6.00149	0.322177	0.161089	+	0.986940i	$0.448500\pi$
$348$	0	0
$349$	−22.7183	−1.21608	−0.608042	−	0.793905i	$-0.708045\pi$
$349$	−22.7183	−1.21608	−0.608042	+	0.793905i	$0.708045\pi$
$350$	0	0
$351$	0	0
$352$	−18.4646	−0.984164
$353$	4.85493	0.258402	0.129201	−	0.991618i	$-0.458759\pi$
$353$	4.85493	0.258402	0.129201	+	0.991618i	$0.458759\pi$
$354$	0	0
$355$	0	0
$356$	−80.6186	−4.27278
$357$	0	0
$358$	41.9895	2.21921
$359$	−28.8910	−1.52481	−0.762403	−	0.647102i	$-0.775981\pi$
$359$	−28.8910	−1.52481	−0.762403	+	0.647102i	$0.775981\pi$
$360$	0	0
$361$	−17.6134	−0.927020
$362$	0.545047	0.0286470
$363$	0	0
$364$	23.5776	1.23580
$365$	0	0
$366$	0	0
$367$	−14.7089	−0.767800	−0.383900	−	0.923375i	$-0.625419\pi$
$367$	−14.7089	−0.767800	−0.383900	+	0.923375i	$0.625419\pi$
$368$	52.0465	2.71311
$369$	0	0
$370$	0	0
$371$	4.29961	0.223225
$372$	0	0
$373$	−33.0391	−1.71070	−0.855350	−	0.518051i	$-0.826658\pi$
$373$	−33.0391	−1.71070	−0.855350	+	0.518051i	$0.826658\pi$
$374$	−11.3972	−0.589337
$375$	0	0
$376$	16.5945	0.855797
$377$	−15.9455	−0.821233
$378$	0	0
$379$	−15.1556	−0.778493	−0.389247	−	0.921134i	$-0.627265\pi$
$379$	−15.1556	−0.778493	−0.389247	+	0.921134i	$0.627265\pi$
$380$	0	0
$381$	0	0
$382$	21.2807	1.08882
$383$	8.00811	0.409195	0.204598	−	0.978846i	$-0.434411\pi$
$383$	8.00811	0.409195	0.204598	+	0.978846i	$0.434411\pi$
$384$	0	0
$385$	0	0
$386$	−44.1397	−2.24665
$387$	0	0
$388$	56.2121	2.85374
$389$	−1.07463	−0.0544858	−0.0272429	−	0.999629i	$-0.508673\pi$
$389$	−1.07463	−0.0544858	−0.0272429	+	0.999629i	$0.508673\pi$
$390$	0	0
$391$	15.9742	0.807848
$392$	−35.9428	−1.81538
$393$	0	0
$394$	46.0475	2.31984
$395$	0	0
$396$	0	0
$397$	−24.2139	−1.21526	−0.607630	−	0.794220i	$-0.707880\pi$
$397$	−24.2139	−1.21526	−0.607630	+	0.794220i	$0.707880\pi$
$398$	−5.42373	−0.271867
$399$	0	0
$400$	0	0
$401$	1.99317	0.0995340	0.0497670	−	0.998761i	$-0.484152\pi$
$401$	1.99317	0.0995340	0.0497670	+	0.998761i	$0.484152\pi$
$402$	0	0
$403$	−19.0166	−0.947287
$404$	70.8142	3.52314
$405$	0	0
$406$	−26.9992	−1.33995
$407$	4.94300	0.245015
$408$	0	0
$409$	6.66478	0.329552	0.164776	−	0.986331i	$-0.447310\pi$
$409$	6.66478	0.329552	0.164776	+	0.986331i	$0.447310\pi$
$410$	0	0
$411$	0	0
$412$	7.14664	0.352090
$413$	−2.25387	−0.110906
$414$	0	0
$415$	0	0
$416$	45.8223	2.24662
$417$	0	0
$418$	3.41351	0.166960
$419$	−21.3540	−1.04321	−0.521606	−	0.853186i	$-0.674667\pi$
$419$	−21.3540	−1.04321	−0.521606	+	0.853186i	$0.674667\pi$
$420$	0	0
$421$	22.3367	1.08863	0.544313	−	0.838882i	$-0.316790\pi$
$421$	22.3367	1.08863	0.544313	+	0.838882i	$0.316790\pi$
$422$	−64.0088	−3.11590
$423$	0	0
$424$	22.0526	1.07097
$425$	0	0
$426$	0	0
$427$	−12.2835	−0.594438
$428$	−57.4823	−2.77851
$429$	0	0
$430$	0	0
$431$	30.4129	1.46494	0.732469	−	0.680800i	$-0.238368\pi$
$431$	30.4129	1.46494	0.732469	+	0.680800i	$0.238368\pi$
$432$	0	0
$433$	0.576639	0.0277115	0.0138558	−	0.999904i	$-0.495589\pi$
$433$	0.576639	0.0277115	0.0138558	+	0.999904i	$0.495589\pi$
$434$	−32.1994	−1.54562
$435$	0	0
$436$	18.0039	0.862231
$437$	−4.78431	−0.228864
$438$	0	0
$439$	12.8045	0.611125	0.305563	−	0.952172i	$-0.401155\pi$
$439$	12.8045	0.611125	0.305563	+	0.952172i	$0.401155\pi$
$440$	0	0
$441$	0	0
$442$	28.2838	1.34532
$443$	−14.3147	−0.680110	−0.340055	−	0.940405i	$-0.610446\pi$
$443$	−14.3147	−0.680110	−0.340055	+	0.940405i	$0.610446\pi$
$444$	0	0
$445$	0	0
$446$	35.2303	1.66820
$447$	0	0
$448$	34.3652	1.62360
$449$	11.9663	0.564724	0.282362	−	0.959308i	$-0.408882\pi$
$449$	11.9663	0.564724	0.282362	+	0.959308i	$0.408882\pi$
$450$	0	0
$451$	−12.1490	−0.572076
$452$	−31.5923	−1.48598
$453$	0	0
$454$	70.6173	3.31424
$455$	0	0
$456$	0	0
$457$	−8.22154	−0.384587	−0.192294	−	0.981337i	$-0.561593\pi$
$457$	−8.22154	−0.384587	−0.192294	+	0.981337i	$0.561593\pi$
$458$	5.36346	0.250618
$459$	0	0
$460$	0	0
$461$	−1.27216	−0.0592505	−0.0296253	−	0.999561i	$-0.509431\pi$
$461$	−1.27216	−0.0592505	−0.0296253	+	0.999561i	$0.509431\pi$
$462$	0	0
$463$	32.6764	1.51860	0.759300	−	0.650740i	$-0.225541\pi$
$463$	32.6764	1.51860	0.759300	+	0.650740i	$0.225541\pi$
$464$	−76.2960	−3.54195
$465$	0	0
$466$	1.41217	0.0654175
$467$	−14.4347	−0.667958	−0.333979	−	0.942581i	$-0.608391\pi$
$467$	−14.4347	−0.667958	−0.333979	+	0.942581i	$0.608391\pi$
$468$	0	0
$469$	20.9953	0.969474
$470$	0	0
$471$	0	0
$472$	−11.5600	−0.532094
$473$	−2.78968	−0.128270
$474$	0	0
$475$	0	0
$476$	34.6250	1.58703
$477$	0	0
$478$	18.8354	0.861511
$479$	16.8618	0.770436	0.385218	−	0.922826i	$-0.374126\pi$
$479$	16.8618	0.770436	0.385218	+	0.922826i	$0.374126\pi$
$480$	0	0
$481$	−12.2667	−0.559314
$482$	6.22196	0.283403
$483$	0	0
$484$	−51.3466	−2.33394
$485$	0	0
$486$	0	0
$487$	39.8235	1.80457	0.902287	−	0.431135i	$-0.141887\pi$
$487$	39.8235	1.80457	0.902287	+	0.431135i	$0.141887\pi$
$488$	−63.0016	−2.85195
$489$	0	0
$490$	0	0
$491$	−14.0255	−0.632961	−0.316480	−	0.948599i	$-0.602501\pi$
$491$	−14.0255	−0.632961	−0.316480	+	0.948599i	$0.602501\pi$
$492$	0	0
$493$	−23.4168	−1.05464
$494$	−8.47108	−0.381132
$495$	0	0
$496$	−90.9911	−4.08562
$497$	10.0966	0.452895
$498$	0	0
$499$	15.2315	0.681857	0.340929	−	0.940089i	$-0.389259\pi$
$499$	15.2315	0.681857	0.340929	+	0.940089i	$0.389259\pi$
$500$	0	0
$501$	0	0
$502$	16.6028	0.741020
$503$	−23.7330	−1.05820	−0.529101	−	0.848559i	$-0.677471\pi$
$503$	−23.7330	−1.05820	−0.529101	+	0.848559i	$0.677471\pi$
$504$	0	0
$505$	0	0
$506$	−11.7778	−0.523586
$507$	0	0
$508$	−105.637	−4.68688
$509$	12.5097	0.554482	0.277241	−	0.960800i	$-0.410580\pi$
$509$	12.5097	0.554482	0.277241	+	0.960800i	$0.410580\pi$
$510$	0	0
$511$	−5.56834	−0.246329
$512$	−2.43464	−0.107597
$513$	0	0
$514$	−35.5331	−1.56730
$515$	0	0
$516$	0	0
$517$	−2.06898	−0.0909936
$518$	−20.7703	−0.912595
$519$	0	0
$520$	0	0
$521$	27.9638	1.22512	0.612559	−	0.790425i	$-0.290140\pi$
$521$	27.9638	1.22512	0.612559	+	0.790425i	$0.290140\pi$
$522$	0	0
$523$	15.7013	0.686569	0.343285	−	0.939231i	$-0.388460\pi$
$523$	15.7013	0.686569	0.343285	+	0.939231i	$0.388460\pi$
$524$	15.9222	0.695564
$525$	0	0
$526$	18.7460	0.817362
$527$	−27.9270	−1.21652
$528$	0	0
$529$	−6.49247	−0.282281
$530$	0	0
$531$	0	0
$532$	−10.3703	−0.449609
$533$	30.1495	1.30592
$534$	0	0
$535$	0	0
$536$	107.685	4.65127
$537$	0	0
$538$	−63.7363	−2.74787
$539$	4.48129	0.193023
$540$	0	0
$541$	−23.9783	−1.03091	−0.515455	−	0.856917i	$-0.672377\pi$
$541$	−23.9783	−1.03091	−0.515455	+	0.856917i	$0.672377\pi$
$542$	77.9000	3.34609
$543$	0	0
$544$	67.2927	2.88515
$545$	0	0
$546$	0	0
$547$	−5.62681	−0.240585	−0.120293	−	0.992738i	$-0.538383\pi$
$547$	−5.62681	−0.240585	−0.120293	+	0.992738i	$0.538383\pi$
$548$	107.066	4.57365
$549$	0	0
$550$	0	0
$551$	7.01341	0.298781
$552$	0	0
$553$	−6.75498	−0.287251
$554$	−23.0728	−0.980268
$555$	0	0
$556$	95.6254	4.05542
$557$	42.4247	1.79759	0.898796	−	0.438366i	$-0.144443\pi$
$557$	42.4247	1.79759	0.898796	+	0.438366i	$0.144443\pi$
$558$	0	0
$559$	6.92298	0.292811
$560$	0	0
$561$	0	0
$562$	−46.1261	−1.94571
$563$	−5.78597	−0.243850	−0.121925	−	0.992539i	$-0.538907\pi$
$563$	−5.78597	−0.243850	−0.121925	+	0.992539i	$0.538907\pi$
$564$	0	0
$565$	0	0
$566$	−25.8495	−1.08653
$567$	0	0
$568$	51.7853	2.17286
$569$	22.0056	0.922522	0.461261	−	0.887265i	$-0.347397\pi$
$569$	22.0056	0.922522	0.461261	+	0.887265i	$0.347397\pi$
$570$	0	0
$571$	33.6143	1.40672	0.703358	−	0.710835i	$-0.251683\pi$
$571$	33.6143	1.40672	0.703358	+	0.710835i	$0.251683\pi$
$572$	−15.0772	−0.630410
$573$	0	0
$574$	51.0499	2.13078
$575$	0	0
$576$	0	0
$577$	40.2350	1.67501	0.837503	−	0.546433i	$-0.184015\pi$
$577$	40.2350	1.67501	0.837503	+	0.546433i	$0.184015\pi$
$578$	−4.14337	−0.172341
$579$	0	0
$580$	0	0
$581$	−14.9722	−0.621151
$582$	0	0
$583$	−2.74948	−0.113872
$584$	−28.5599	−1.18182
$585$	0	0
$586$	−45.1967	−1.86706
$587$	2.82299	0.116517	0.0582587	−	0.998302i	$-0.481445\pi$
$587$	2.82299	0.116517	0.0582587	+	0.998302i	$0.481445\pi$
$588$	0	0
$589$	8.36423	0.344642
$590$	0	0
$591$	0	0
$592$	−58.6939	−2.41231
$593$	24.6805	1.01351	0.506754	−	0.862091i	$-0.330845\pi$
$593$	24.6805	1.01351	0.506754	+	0.862091i	$0.330845\pi$
$594$	0	0
$595$	0	0
$596$	−3.68513	−0.150949
$597$	0	0
$598$	29.2282	1.19523
$599$	7.71547	0.315245	0.157623	−	0.987499i	$-0.449617\pi$
$599$	7.71547	0.315245	0.157623	+	0.987499i	$0.449617\pi$
$600$	0	0
$601$	22.6506	0.923938	0.461969	−	0.886896i	$-0.347143\pi$
$601$	22.6506	0.923938	0.461969	+	0.886896i	$0.347143\pi$
$602$	11.7222	0.477759
$603$	0	0
$604$	36.7783	1.49649
$605$	0	0
$606$	0	0
$607$	−24.6423	−1.00020	−0.500099	−	0.865968i	$-0.666703\pi$
$607$	−24.6423	−1.00020	−0.500099	+	0.865968i	$0.666703\pi$
$608$	−20.1543	−0.817367
$609$	0	0
$610$	0	0
$611$	5.13445	0.207718
$612$	0	0
$613$	−41.8034	−1.68842	−0.844211	−	0.536011i	$-0.819931\pi$
$613$	−41.8034	−1.68842	−0.844211	+	0.536011i	$0.819931\pi$
$614$	30.4205	1.22767
$615$	0	0
$616$	−15.7482	−0.634513
$617$	−15.9390	−0.641680	−0.320840	−	0.947133i	$-0.603965\pi$
$617$	−15.9390	−0.641680	−0.320840	+	0.947133i	$0.603965\pi$
$618$	0	0
$619$	−47.1700	−1.89592	−0.947961	−	0.318385i	$-0.896859\pi$
$619$	−47.1700	−1.89592	−0.947961	+	0.318385i	$0.896859\pi$
$620$	0	0
$621$	0	0
$622$	−49.8808	−2.00004
$623$	−26.0540	−1.04383
$624$	0	0
$625$	0	0
$626$	−26.0630	−1.04169
$627$	0	0
$628$	−26.7734	−1.06837
$629$	−18.0144	−0.718280
$630$	0	0
$631$	32.3634	1.28836	0.644182	−	0.764872i	$-0.277198\pi$
$631$	32.3634	1.28836	0.644182	+	0.764872i	$0.277198\pi$
$632$	−34.6462	−1.37815
$633$	0	0
$634$	−20.0896	−0.797859
$635$	0	0
$636$	0	0
$637$	−11.1209	−0.440627
$638$	17.2653	0.683539
$639$	0	0
$640$	0	0
$641$	−18.0042	−0.711122	−0.355561	−	0.934653i	$-0.615710\pi$
$641$	−18.0042	−0.711122	−0.355561	+	0.934653i	$0.615710\pi$
$642$	0	0
$643$	21.8891	0.863224	0.431612	−	0.902059i	$-0.357945\pi$
$643$	21.8891	0.863224	0.431612	+	0.902059i	$0.357945\pi$
$644$	35.7811	1.40997
$645$	0	0
$646$	−12.4403	−0.489455
$647$	−19.0263	−0.748003	−0.374001	−	0.927428i	$-0.622015\pi$
$647$	−19.0263	−0.748003	−0.374001	+	0.927428i	$0.622015\pi$
$648$	0	0
$649$	1.44129	0.0565755
$650$	0	0
$651$	0	0
$652$	−126.649	−4.95995
$653$	2.31971	0.0907774	0.0453887	−	0.998969i	$-0.485547\pi$
$653$	2.31971	0.0907774	0.0453887	+	0.998969i	$0.485547\pi$
$654$	0	0
$655$	0	0
$656$	144.260	5.63239
$657$	0	0
$658$	8.69378	0.338919
$659$	−7.19180	−0.280153	−0.140076	−	0.990141i	$-0.544735\pi$
$659$	−7.19180	−0.280153	−0.140076	+	0.990141i	$0.544735\pi$
$660$	0	0
$661$	43.0214	1.67334	0.836669	−	0.547709i	$-0.184500\pi$
$661$	43.0214	1.67334	0.836669	+	0.547709i	$0.184500\pi$
$662$	−67.8666	−2.63771
$663$	0	0
$664$	−76.7920	−2.98011
$665$	0	0
$666$	0	0
$667$	−24.1987	−0.936977
$668$	−23.7461	−0.918765
$669$	0	0
$670$	0	0
$671$	7.85494	0.303237
$672$	0	0
$673$	−29.8031	−1.14882	−0.574412	−	0.818566i	$-0.694769\pi$
$673$	−29.8031	−1.14882	−0.574412	+	0.818566i	$0.694769\pi$
$674$	−34.9961	−1.34800
$675$	0	0
$676$	−30.4464	−1.17102
$677$	26.7346	1.02749	0.513746	−	0.857942i	$-0.328257\pi$
$677$	26.7346	1.02749	0.513746	+	0.857942i	$0.328257\pi$
$678$	0	0
$679$	18.1664	0.697163
$680$	0	0
$681$	0	0
$682$	20.5907	0.788457
$683$	−28.4950	−1.09033	−0.545165	−	0.838329i	$-0.683533\pi$
$683$	−28.4950	−1.09033	−0.545165	+	0.838329i	$0.683533\pi$
$684$	0	0
$685$	0	0
$686$	−50.5623	−1.93048
$687$	0	0
$688$	33.1251	1.26288
$689$	6.82321	0.259944
$690$	0	0
$691$	34.6469	1.31803	0.659015	−	0.752130i	$-0.270973\pi$
$691$	34.6469	1.31803	0.659015	+	0.752130i	$0.270973\pi$
$692$	13.9760	0.531286
$693$	0	0
$694$	16.1263	0.612145
$695$	0	0
$696$	0	0
$697$	44.2763	1.67708
$698$	−61.0451	−2.31059
$699$	0	0
$700$	0	0
$701$	−0.973305	−0.0367612	−0.0183806	−	0.999831i	$-0.505851\pi$
$701$	−0.973305	−0.0367612	−0.0183806	+	0.999831i	$0.505851\pi$
$702$	0	0
$703$	5.39536	0.203490
$704$	−21.9756	−0.828237
$705$	0	0
$706$	13.0454	0.490971
$707$	22.8854	0.860696
$708$	0	0
$709$	−11.3242	−0.425291	−0.212645	−	0.977129i	$-0.568208\pi$
$709$	−11.3242	−0.425291	−0.212645	+	0.977129i	$0.568208\pi$
$710$	0	0
$711$	0	0
$712$	−133.630	−5.00801
$713$	−28.8595	−1.08080
$714$	0	0
$715$	0	0
$716$	81.5742	3.04857
$717$	0	0
$718$	−77.6312	−2.89717
$719$	30.7320	1.14611	0.573056	−	0.819516i	$-0.305758\pi$
$719$	30.7320	1.14611	0.573056	+	0.819516i	$0.305758\pi$
$720$	0	0
$721$	2.30962	0.0860149
$722$	−47.3279	−1.76136
$723$	0	0
$724$	1.05888	0.0393529
$725$	0	0
$726$	0	0
$727$	−6.56836	−0.243607	−0.121803	−	0.992554i	$-0.538868\pi$
$727$	−6.56836	−0.243607	−0.121803	+	0.992554i	$0.538868\pi$
$728$	39.0813	1.44845
$729$	0	0
$730$	0	0
$731$	10.1668	0.376032
$732$	0	0
$733$	−31.9276	−1.17927	−0.589636	−	0.807669i	$-0.700729\pi$
$733$	−31.9276	−1.17927	−0.589636	+	0.807669i	$0.700729\pi$
$734$	−39.5235	−1.45884
$735$	0	0
$736$	69.5395	2.56326
$737$	−13.4259	−0.494551
$738$	0	0
$739$	−15.3072	−0.563083	−0.281542	−	0.959549i	$-0.590846\pi$
$739$	−15.3072	−0.563083	−0.281542	+	0.959549i	$0.590846\pi$
$740$	0	0
$741$	0	0
$742$	11.5532	0.424132
$743$	−16.5455	−0.606995	−0.303498	−	0.952832i	$-0.598154\pi$
$743$	−16.5455	−0.606995	−0.303498	+	0.952832i	$0.598154\pi$
$744$	0	0
$745$	0	0
$746$	−88.7774	−3.25037
$747$	0	0
$748$	−22.1417	−0.809582
$749$	−18.5769	−0.678786
$750$	0	0
$751$	46.0279	1.67958	0.839791	−	0.542911i	$-0.182678\pi$
$751$	46.0279	1.67958	0.839791	+	0.542911i	$0.182678\pi$
$752$	24.5674	0.895880
$753$	0	0
$754$	−42.8461	−1.56036
$755$	0	0
$756$	0	0
$757$	−26.5282	−0.964184	−0.482092	−	0.876121i	$-0.660123\pi$
$757$	−26.5282	−0.964184	−0.482092	+	0.876121i	$0.660123\pi$
$758$	−40.7239	−1.47916
$759$	0	0
$760$	0	0
$761$	28.4241	1.03037	0.515187	−	0.857078i	$-0.327723\pi$
$761$	28.4241	1.03037	0.515187	+	0.857078i	$0.327723\pi$
$762$	0	0
$763$	5.81843	0.210641
$764$	41.3426	1.49572
$765$	0	0
$766$	21.5181	0.777481
$767$	−3.57675	−0.129149
$768$	0	0
$769$	−8.95952	−0.323089	−0.161544	−	0.986865i	$-0.551647\pi$
$769$	−8.95952	−0.323089	−0.161544	+	0.986865i	$0.551647\pi$
$770$	0	0
$771$	0	0
$772$	−85.7515	−3.08626
$773$	−23.1042	−0.831001	−0.415500	−	0.909593i	$-0.636393\pi$
$773$	−23.1042	−0.831001	−0.415500	+	0.909593i	$0.636393\pi$
$774$	0	0
$775$	0	0
$776$	93.1752	3.34479
$777$	0	0
$778$	−2.88757	−0.103524
$779$	−13.2609	−0.475120
$780$	0	0
$781$	−6.45651	−0.231032
$782$	42.9232	1.53493
$783$	0	0
$784$	−53.2115	−1.90041
$785$	0	0
$786$	0	0
$787$	−2.03390	−0.0725009	−0.0362504	−	0.999343i	$-0.511541\pi$
$787$	−2.03390	−0.0725009	−0.0362504	+	0.999343i	$0.511541\pi$
$788$	89.4578	3.18680
$789$	0	0
$790$	0	0
$791$	−10.2099	−0.363022
$792$	0	0
$793$	−19.4931	−0.692220
$794$	−65.0637	−2.30902
$795$	0	0
$796$	−10.5368	−0.373469
$797$	12.1923	0.431874	0.215937	−	0.976407i	$-0.430719\pi$
$797$	12.1923	0.431874	0.215937	+	0.976407i	$0.430719\pi$
$798$	0	0
$799$	7.54024	0.266754
$800$	0	0
$801$	0	0
$802$	5.35572	0.189117
$803$	3.56080	0.125658
$804$	0	0
$805$	0	0
$806$	−51.0985	−1.79987
$807$	0	0
$808$	117.379	4.12938
$809$	−8.83355	−0.310571	−0.155286	−	0.987870i	$-0.549630\pi$
$809$	−8.83355	−0.310571	−0.155286	+	0.987870i	$0.549630\pi$
$810$	0	0
$811$	−13.9184	−0.488741	−0.244370	−	0.969682i	$-0.578581\pi$
$811$	−13.9184	−0.488741	−0.244370	+	0.969682i	$0.578581\pi$
$812$	−52.4522	−1.84071
$813$	0	0
$814$	13.2820	0.465535
$815$	0	0
$816$	0	0
$817$	−3.04498	−0.106530
$818$	17.9085	0.626158
$819$	0	0
$820$	0	0
$821$	29.4930	1.02931	0.514656	−	0.857397i	$-0.327920\pi$
$821$	29.4930	1.02931	0.514656	+	0.857397i	$0.327920\pi$
$822$	0	0
$823$	−41.9661	−1.46285	−0.731423	−	0.681924i	$-0.761143\pi$
$823$	−41.9661	−1.46285	−0.731423	+	0.681924i	$0.761143\pi$
$824$	11.8460	0.412675
$825$	0	0
$826$	−6.05624	−0.210724
$827$	−37.4674	−1.30287	−0.651435	−	0.758704i	$-0.725833\pi$
$827$	−37.4674	−1.30287	−0.651435	+	0.758704i	$0.725833\pi$
$828$	0	0
$829$	−9.55614	−0.331899	−0.165949	−	0.986134i	$-0.553069\pi$
$829$	−9.55614	−0.331899	−0.165949	+	0.986134i	$0.553069\pi$
$830$	0	0
$831$	0	0
$832$	54.5355	1.89068
$833$	−16.3317	−0.565860
$834$	0	0
$835$	0	0
$836$	6.63152	0.229356
$837$	0	0
$838$	−57.3792	−1.98213
$839$	−54.7310	−1.88952	−0.944761	−	0.327759i	$-0.893707\pi$
$839$	−54.7310	−1.88952	−0.944761	+	0.327759i	$0.893707\pi$
$840$	0	0
$841$	6.47334	0.223219
$842$	60.0197	2.06842
$843$	0	0
$844$	−124.352	−4.28037
$845$	0	0
$846$	0	0
$847$	−16.5940	−0.570177
$848$	32.6478	1.12113
$849$	0	0
$850$	0	0
$851$	−18.6159	−0.638144
$852$	0	0
$853$	−15.3067	−0.524092	−0.262046	−	0.965055i	$-0.584397\pi$
$853$	−15.3067	−0.524092	−0.262046	+	0.965055i	$0.584397\pi$
$854$	−33.0062	−1.12945
$855$	0	0
$856$	−95.2806	−3.25662
$857$	8.66910	0.296131	0.148065	−	0.988978i	$-0.452695\pi$
$857$	8.66910	0.296131	0.148065	+	0.988978i	$0.452695\pi$
$858$	0	0
$859$	−30.4078	−1.03750	−0.518750	−	0.854926i	$-0.673603\pi$
$859$	−30.4078	−1.03750	−0.518750	+	0.854926i	$0.673603\pi$
$860$	0	0
$861$	0	0
$862$	81.7208	2.78342
$863$	18.0425	0.614175	0.307088	−	0.951681i	$-0.400646\pi$
$863$	18.0425	0.614175	0.307088	+	0.951681i	$0.400646\pi$
$864$	0	0
$865$	0	0
$866$	1.54945	0.0526526
$867$	0	0
$868$	−62.5548	−2.12325
$869$	4.31963	0.146533
$870$	0	0
$871$	33.3183	1.12895
$872$	29.8426	1.01060
$873$	0	0
$874$	−12.8556	−0.434848
$875$	0	0
$876$	0	0
$877$	22.3057	0.753211	0.376605	−	0.926374i	$-0.377091\pi$
$877$	22.3057	0.753211	0.376605	+	0.926374i	$0.377091\pi$
$878$	34.4062	1.16115
$879$	0	0
$880$	0	0
$881$	−7.75956	−0.261426	−0.130713	−	0.991420i	$-0.541727\pi$
$881$	−7.75956	−0.261426	−0.130713	+	0.991420i	$0.541727\pi$
$882$	0	0
$883$	−19.9527	−0.671462	−0.335731	−	0.941958i	$-0.608983\pi$
$883$	−19.9527	−0.671462	−0.335731	+	0.941958i	$0.608983\pi$
$884$	54.9477	1.84809
$885$	0	0
$886$	−38.4641	−1.29223
$887$	−8.60289	−0.288857	−0.144428	−	0.989515i	$-0.546134\pi$
$887$	−8.60289	−0.288857	−0.144428	+	0.989515i	$0.546134\pi$
$888$	0	0
$889$	−34.1393	−1.14499
$890$	0	0
$891$	0	0
$892$	68.4429	2.29164
$893$	−2.25832	−0.0755719
$894$	0	0
$895$	0	0
$896$	34.5915	1.15562
$897$	0	0
$898$	32.1539	1.07299
$899$	42.3057	1.41097
$900$	0	0
$901$	10.0203	0.333824
$902$	−32.6450	−1.08696
$903$	0	0
$904$	−52.3663	−1.74168
$905$	0	0
$906$	0	0
$907$	6.26125	0.207902	0.103951	−	0.994582i	$-0.466852\pi$
$907$	6.26125	0.207902	0.103951	+	0.994582i	$0.466852\pi$
$908$	137.190	4.55282
$909$	0	0
$910$	0	0
$911$	−0.905575	−0.0300030	−0.0150015	−	0.999887i	$-0.504775\pi$
$911$	−0.905575	−0.0300030	−0.0150015	+	0.999887i	$0.504775\pi$
$912$	0	0
$913$	9.57431	0.316863
$914$	−22.0916	−0.730726
$915$	0	0
$916$	10.4198	0.344279
$917$	5.14567	0.169925
$918$	0	0
$919$	43.9933	1.45120	0.725602	−	0.688114i	$-0.241561\pi$
$919$	43.9933	1.45120	0.725602	+	0.688114i	$0.241561\pi$
$920$	0	0
$921$	0	0
$922$	−3.41836	−0.112578
$923$	16.0227	0.527394
$924$	0	0
$925$	0	0
$926$	87.8029	2.88538
$927$	0	0
$928$	−101.939	−3.34632
$929$	36.7447	1.20555	0.602777	−	0.797910i	$-0.294061\pi$
$929$	36.7447	1.20555	0.602777	+	0.797910i	$0.294061\pi$
$930$	0	0
$931$	4.89140	0.160309
$932$	2.74347	0.0898652
$933$	0	0
$934$	−38.7866	−1.26914
$935$	0	0
$936$	0	0
$937$	24.6490	0.805248	0.402624	−	0.915366i	$-0.368098\pi$
$937$	24.6490	0.805248	0.402624	+	0.915366i	$0.368098\pi$
$938$	56.4153	1.84203
$939$	0	0
$940$	0	0
$941$	59.1479	1.92817	0.964083	−	0.265602i	$-0.0855706\pi$
$941$	59.1479	1.92817	0.964083	+	0.265602i	$0.0855706\pi$
$942$	0	0
$943$	45.7546	1.48998
$944$	−17.1141	−0.557016
$945$	0	0
$946$	−7.49600	−0.243716
$947$	30.0960	0.977989	0.488995	−	0.872287i	$-0.337364\pi$
$947$	30.0960	0.977989	0.488995	+	0.872287i	$0.337364\pi$
$948$	0	0
$949$	−8.83661	−0.286849
$950$	0	0
$951$	0	0
$952$	57.3931	1.86012
$953$	20.6393	0.668572	0.334286	−	0.942472i	$-0.391505\pi$
$953$	20.6393	0.668572	0.334286	+	0.942472i	$0.391505\pi$
$954$	0	0
$955$	0	0
$956$	36.5921	1.18347
$957$	0	0
$958$	45.3084	1.46385
$959$	34.6013	1.11733
$960$	0	0
$961$	19.4540	0.627549
$962$	−32.9612	−1.06271
$963$	0	0
$964$	12.0876	0.389315
$965$	0	0
$966$	0	0
$967$	−22.6016	−0.726817	−0.363408	−	0.931630i	$-0.618387\pi$
$967$	−22.6016	−0.726817	−0.363408	+	0.931630i	$0.618387\pi$
$968$	−85.1103	−2.73555
$969$	0	0
$970$	0	0
$971$	−24.1381	−0.774628	−0.387314	−	0.921948i	$-0.626597\pi$
$971$	−24.1381	−0.774628	−0.387314	+	0.921948i	$0.626597\pi$
$972$	0	0
$973$	30.9038	0.990732
$974$	107.007	3.42874
$975$	0	0
$976$	−93.2708	−2.98553
$977$	−23.0506	−0.737454	−0.368727	−	0.929538i	$-0.620206\pi$
$977$	−23.0506	−0.737454	−0.368727	+	0.929538i	$0.620206\pi$
$978$	0	0
$979$	16.6608	0.532483
$980$	0	0
$981$	0	0
$982$	−37.6870	−1.20264
$983$	−29.8713	−0.952745	−0.476373	−	0.879243i	$-0.658049\pi$
$983$	−29.8713	−0.952745	−0.476373	+	0.879243i	$0.658049\pi$
$984$	0	0
$985$	0	0
$986$	−62.9220	−2.00384
$987$	0	0
$988$	−16.4570	−0.523567
$989$	10.5063	0.334079
$990$	0	0
$991$	16.6560	0.529096	0.264548	−	0.964372i	$-0.414777\pi$
$991$	16.6560	0.529096	0.264548	+	0.964372i	$0.414777\pi$
$992$	−121.573	−3.85996
$993$	0	0
$994$	27.1300	0.860513
$995$	0	0
$996$	0	0
$997$	12.4934	0.395669	0.197835	−	0.980235i	$-0.436609\pi$
$997$	12.4934	0.395669	0.197835	+	0.980235i	$0.436609\pi$
$998$	40.9278	1.29555
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.p.1.6		6
3.2	odd	2		1875.2.a.j.1.1		6
5.4	even	2		5625.2.a.q.1.1		6
15.2	even	4		1875.2.b.f.1249.1		12
15.8	even	4		1875.2.b.f.1249.12		12
15.14	odd	2		1875.2.a.k.1.6		6
25.11	even	5		225.2.h.d.46.3		12
25.16	even	5		225.2.h.d.181.3		12
75.2	even	20		375.2.i.d.274.1		24
75.11	odd	10		75.2.g.c.46.1	yes	12
75.14	odd	10		375.2.g.c.226.3		12
75.23	even	20		375.2.i.d.274.6		24
75.38	even	20		375.2.i.d.349.1		24
75.41	odd	10		75.2.g.c.31.1	✓	12
75.59	odd	10		375.2.g.c.151.3		12
75.62	even	20		375.2.i.d.349.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.c.31.1	✓	12	75.41	odd	10
75.2.g.c.46.1	yes	12	75.11	odd	10
225.2.h.d.46.3		12	25.11	even	5
225.2.h.d.181.3		12	25.16	even	5
375.2.g.c.151.3		12	75.59	odd	10
375.2.g.c.226.3		12	75.14	odd	10
375.2.i.d.274.1		24	75.2	even	20
375.2.i.d.274.6		24	75.23	even	20
375.2.i.d.349.1		24	75.38	even	20
375.2.i.d.349.6		24	75.62	even	20
1875.2.a.j.1.1		6	3.2	odd	2
1875.2.a.k.1.6		6	15.14	odd	2
1875.2.b.f.1249.1		12	15.2	even	4
1875.2.b.f.1249.12		12	15.8	even	4
5625.2.a.p.1.6		6	1.1	even	1	trivial
5625.2.a.q.1.1		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68704 q^{2} +5.22020 q^{4} +1.68704 q^{7} +8.65280 q^{8} -1.07882 q^{11} +2.67723 q^{13} +4.53315 q^{14} +12.8101 q^{16} +3.93167 q^{17} -1.17755 q^{19} -2.89883 q^{22} +4.06295 q^{23} +7.19384 q^{26} +8.80669 q^{28} -5.95595 q^{29} -7.10310 q^{31} +17.1155 q^{32} +10.5646 q^{34} -4.58187 q^{37} -3.16412 q^{38} +11.2614 q^{41} +2.58587 q^{43} -5.63164 q^{44} +10.9173 q^{46} +1.91782 q^{47} -4.15389 q^{49} +13.9757 q^{52} +2.54861 q^{53} +14.5976 q^{56} -16.0039 q^{58} -1.33599 q^{59} -7.28106 q^{61} -19.0863 q^{62} +20.3701 q^{64} +12.4451 q^{67} +20.5241 q^{68} +5.98480 q^{71} -3.30065 q^{73} -12.3117 q^{74} -6.14702 q^{76} -1.82001 q^{77} -4.00404 q^{79} +30.2600 q^{82} -8.87482 q^{83} +6.94834 q^{86} -9.33480 q^{88} -15.4436 q^{89} +4.51660 q^{91} +21.2094 q^{92} +5.15327 q^{94} +10.7682 q^{97} -11.1617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37}+ \cdots - 41 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8 - 3 * q^11 - 6 * q^13 + 22 * q^14 + 18 * q^16 + 13 * q^17 + 11 * q^19 - 16 * q^22 + 13 * q^23 + 28 * q^26 - 7 * q^28 + 3 * q^29 - 11 * q^31 + 16 * q^32 + 15 * q^34 - 21 * q^37 - 9 * q^38 + q^41 - 2 * q^43 - 9 * q^44 + 19 * q^46 + 14 * q^47 - 14 * q^49 - 13 * q^52 + 23 * q^53 + 35 * q^56 - 22 * q^58 - 9 * q^59 + 11 * q^61 - 23 * q^62 - 23 * q^64 - 8 * q^67 + 50 * q^68 + 8 * q^71 - 13 * q^73 + 22 * q^74 - 26 * q^76 - 13 * q^77 - 5 * q^79 + 13 * q^82 - 20 * q^83 + 37 * q^86 - 28 * q^88 + 4 * q^89 + 34 * q^91 + 61 * q^92 + 41 * q^94 + 7 * q^97 - 41 * q^98

Introduction

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