LMFDB - Embedded newform 5625.2.a.q.1.4

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)

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")

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);

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			1.4
Root			$-0.246759$ 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)

$f(q)$	$=$	$q+0.246759 q^{2} -1.93911 q^{4} +1.24676 q^{7} -0.972011 q^{8} -2.56971 q^{11} +4.68126 q^{13} +0.307649 q^{14} +3.63837 q^{16} +5.83914 q^{17} +4.16857 q^{19} -0.634099 q^{22} +1.60547 q^{23} +1.15514 q^{26} -2.41760 q^{28} +3.21573 q^{29} -9.19312 q^{31} +2.84182 q^{32} +1.44086 q^{34} +1.27778 q^{37} +1.02863 q^{38} -8.69592 q^{41} +3.88086 q^{43} +4.98295 q^{44} +0.396163 q^{46} -3.20952 q^{47} -5.44559 q^{49} -9.07748 q^{52} -13.3343 q^{53} -1.21186 q^{56} +0.793511 q^{58} +6.39279 q^{59} +3.82692 q^{61} -2.26848 q^{62} -6.57549 q^{64} +10.1238 q^{67} -11.3227 q^{68} -13.8791 q^{71} +11.3814 q^{73} +0.315305 q^{74} -8.08332 q^{76} -3.20381 q^{77} +11.4832 q^{79} -2.14580 q^{82} -2.83229 q^{83} +0.957636 q^{86} +2.49779 q^{88} +1.38077 q^{89} +5.83641 q^{91} -3.11318 q^{92} -0.791979 q^{94} -0.0305081 q^{97} -1.34375 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.246759	0.174485	0.0872425	−	0.996187i	$-0.472195\pi$
$2$	0.246759	0.174485	0.0872425	+	0.996187i	$0.472195\pi$
$3$	0	0
$3$	0	0
$4$	−1.93911	−0.969555
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.24676	0.471231	0.235615	−	0.971846i	$-0.424289\pi$
$7$	1.24676	0.471231	0.235615	+	0.971846i	$0.424289\pi$
$8$	−0.972011	−0.343658
$9$	0	0
$10$	0	0
$11$	−2.56971	−0.774797	−0.387398	−	0.921912i	$-0.626626\pi$
$11$	−2.56971	−0.774797	−0.387398	+	0.921912i	$0.626626\pi$
$12$	0	0
$13$	4.68126	1.29835	0.649174	−	0.760640i	$-0.275115\pi$
$13$	4.68126	1.29835	0.649174	+	0.760640i	$0.275115\pi$
$14$	0.307649	0.0822226
$15$	0	0
$16$	3.63837	0.909592
$17$	5.83914	1.41620	0.708100	−	0.706112i	$-0.249553\pi$
$17$	5.83914	1.41620	0.708100	+	0.706112i	$0.249553\pi$
$18$	0	0
$19$	4.16857	0.956336	0.478168	−	0.878268i	$-0.341301\pi$
$19$	4.16857	0.956336	0.478168	+	0.878268i	$0.341301\pi$
$20$	0	0
$21$	0	0
$22$	−0.634099	−0.135190
$23$	1.60547	0.334763	0.167382	−	0.985892i	$-0.446469\pi$
$23$	1.60547	0.334763	0.167382	+	0.985892i	$0.446469\pi$
$24$	0	0
$25$	0	0
$26$	1.15514	0.226542
$27$	0	0
$28$	−2.41760	−0.456884
$29$	3.21573	0.597147	0.298573	−	0.954387i	$-0.403489\pi$
$29$	3.21573	0.597147	0.298573	+	0.954387i	$0.403489\pi$
$30$	0	0
$31$	−9.19312	−1.65113	−0.825566	−	0.564305i	$-0.809144\pi$
$31$	−9.19312	−1.65113	−0.825566	+	0.564305i	$0.809144\pi$
$32$	2.84182	0.502368
$33$	0	0
$34$	1.44086	0.247106
$35$	0	0
$36$	0	0
$37$	1.27778	0.210066	0.105033	−	0.994469i	$-0.466505\pi$
$37$	1.27778	0.210066	0.105033	+	0.994469i	$0.466505\pi$
$38$	1.02863	0.166866
$39$	0	0
$40$	0	0
$41$	−8.69592	−1.35808	−0.679038	−	0.734104i	$-0.737603\pi$
$41$	−8.69592	−1.35808	−0.679038	+	0.734104i	$0.737603\pi$
$42$	0	0
$43$	3.88086	0.591825	0.295913	−	0.955215i	$-0.404376\pi$
$43$	3.88086	0.591825	0.295913	+	0.955215i	$0.404376\pi$
$44$	4.98295	0.751208
$45$	0	0
$46$	0.396163	0.0584111
$47$	−3.20952	−0.468157	−0.234079	−	0.972218i	$-0.575207\pi$
$47$	−3.20952	−0.468157	−0.234079	+	0.972218i	$0.575207\pi$
$48$	0	0
$49$	−5.44559	−0.777942
$50$	0	0
$51$	0	0
$52$	−9.07748	−1.25882
$53$	−13.3343	−1.83161	−0.915803	−	0.401628i	$-0.868444\pi$
$53$	−13.3343	−1.83161	−0.915803	+	0.401628i	$0.868444\pi$
$54$	0	0
$55$	0	0
$56$	−1.21186	−0.161942
$57$	0	0
$58$	0.793511	0.104193
$59$	6.39279	0.832270	0.416135	−	0.909303i	$-0.363384\pi$
$59$	6.39279	0.832270	0.416135	+	0.909303i	$0.363384\pi$
$60$	0	0
$61$	3.82692	0.489986	0.244993	−	0.969525i	$-0.421214\pi$
$61$	3.82692	0.489986	0.244993	+	0.969525i	$0.421214\pi$
$62$	−2.26848	−0.288098
$63$	0	0
$64$	−6.57549	−0.821936
$65$	0	0
$66$	0	0
$67$	10.1238	1.23682	0.618411	−	0.785855i	$-0.287777\pi$
$67$	10.1238	1.23682	0.618411	+	0.785855i	$0.287777\pi$
$68$	−11.3227	−1.37308
$69$	0	0
$70$	0	0
$71$	−13.8791	−1.64715	−0.823574	−	0.567209i	$-0.808023\pi$
$71$	−13.8791	−1.64715	−0.823574	+	0.567209i	$0.808023\pi$
$72$	0	0
$73$	11.3814	1.33209	0.666045	−	0.745912i	$-0.267986\pi$
$73$	11.3814	1.33209	0.666045	+	0.745912i	$0.267986\pi$
$74$	0.315305	0.0366534
$75$	0	0
$76$	−8.08332	−0.927220
$77$	−3.20381	−0.365108
$78$	0	0
$79$	11.4832	1.29196	0.645979	−	0.763355i	$-0.276449\pi$
$79$	11.4832	1.29196	0.645979	+	0.763355i	$0.276449\pi$
$80$	0	0
$81$	0	0
$82$	−2.14580	−0.236964
$83$	−2.83229	−0.310884	−0.155442	−	0.987845i	$-0.549680\pi$
$83$	−2.83229	−0.310884	−0.155442	+	0.987845i	$0.549680\pi$
$84$	0	0
$85$	0	0
$86$	0.957636	0.103265
$87$	0	0
$88$	2.49779	0.266265
$89$	1.38077	0.146362	0.0731808	−	0.997319i	$-0.476685\pi$
$89$	1.38077	0.146362	0.0731808	+	0.997319i	$0.476685\pi$
$90$	0	0
$91$	5.83641	0.611822
$92$	−3.11318	−0.324571
$93$	0	0
$94$	−0.791979	−0.0816864
$95$	0	0
$96$	0	0
$97$	−0.0305081	−0.00309763	−0.00154881	−	0.999999i	$-0.500493\pi$
$97$	−0.0305081	−0.00309763	−0.00154881	+	0.999999i	$0.500493\pi$
$98$	−1.34375	−0.135739
$99$	0	0
$100$	0	0
$101$	11.2308	1.11750	0.558751	−	0.829336i	$-0.311281\pi$
$101$	11.2308	1.11750	0.558751	+	0.829336i	$0.311281\pi$
$102$	0	0
$103$	0.720983	0.0710406	0.0355203	−	0.999369i	$-0.488691\pi$
$103$	0.720983	0.0710406	0.0355203	+	0.999369i	$0.488691\pi$
$104$	−4.55024	−0.446187
$105$	0	0
$106$	−3.29036	−0.319588
$107$	2.11668	0.204627	0.102313	−	0.994752i	$-0.467376\pi$
$107$	2.11668	0.204627	0.102313	+	0.994752i	$0.467376\pi$
$108$	0	0
$109$	2.23483	0.214058	0.107029	−	0.994256i	$-0.465866\pi$
$109$	2.23483	0.214058	0.107029	+	0.994256i	$0.465866\pi$
$110$	0	0
$111$	0	0
$112$	4.53617	0.428628
$113$	−3.14940	−0.296270	−0.148135	−	0.988967i	$-0.547327\pi$
$113$	−3.14940	−0.296270	−0.148135	+	0.988967i	$0.547327\pi$
$114$	0	0
$115$	0	0
$116$	−6.23566	−0.578967
$117$	0	0
$118$	1.57748	0.145219
$119$	7.28000	0.667357
$120$	0	0
$121$	−4.39659	−0.399690
$122$	0.944326	0.0854952
$123$	0	0
$124$	17.8265	1.60086
$125$	0	0
$126$	0	0
$127$	15.6603	1.38962	0.694812	−	0.719191i	$-0.255487\pi$
$127$	15.6603	1.38962	0.694812	+	0.719191i	$0.255487\pi$
$128$	−7.30620	−0.645783
$129$	0	0
$130$	0	0
$131$	14.9564	1.30675	0.653374	−	0.757036i	$-0.273353\pi$
$131$	14.9564	1.30675	0.653374	+	0.757036i	$0.273353\pi$
$132$	0	0
$133$	5.19720	0.450655
$134$	2.49814	0.215807
$135$	0	0
$136$	−5.67571	−0.486688
$137$	−10.4946	−0.896611	−0.448305	−	0.893881i	$-0.647972\pi$
$137$	−10.4946	−0.896611	−0.448305	+	0.893881i	$0.647972\pi$
$138$	0	0
$139$	12.4508	1.05606	0.528030	−	0.849226i	$-0.322931\pi$
$139$	12.4508	1.05606	0.528030	+	0.849226i	$0.322931\pi$
$140$	0	0
$141$	0	0
$142$	−3.42480	−0.287402
$143$	−12.0295	−1.00596
$144$	0	0
$145$	0	0
$146$	2.80846	0.232430
$147$	0	0
$148$	−2.47776	−0.203671
$149$	17.0680	1.39826	0.699131	−	0.714994i	$-0.253570\pi$
$149$	17.0680	1.39826	0.699131	+	0.714994i	$0.253570\pi$
$150$	0	0
$151$	−1.57516	−0.128185	−0.0640923	−	0.997944i	$-0.520415\pi$
$151$	−1.57516	−0.128185	−0.0640923	+	0.997944i	$0.520415\pi$
$152$	−4.05190	−0.328652
$153$	0	0
$154$	−0.790569	−0.0637059
$155$	0	0
$156$	0	0
$157$	9.20058	0.734286	0.367143	−	0.930164i	$-0.380336\pi$
$157$	9.20058	0.734286	0.367143	+	0.930164i	$0.380336\pi$
$158$	2.83358	0.225427
$159$	0	0
$160$	0	0
$161$	2.00163	0.157751
$162$	0	0
$163$	4.58076	0.358793	0.179396	−	0.983777i	$-0.442586\pi$
$163$	4.58076	0.358793	0.179396	+	0.983777i	$0.442586\pi$
$164$	16.8624	1.31673
$165$	0	0
$166$	−0.698893	−0.0542446
$167$	0.142531	0.0110294	0.00551469	−	0.999985i	$-0.498245\pi$
$167$	0.142531	0.0110294	0.00551469	+	0.999985i	$0.498245\pi$
$168$	0	0
$169$	8.91422	0.685709
$170$	0	0
$171$	0	0
$172$	−7.52541	−0.573807
$173$	24.6228	1.87203	0.936017	−	0.351954i	$-0.114483\pi$
$173$	24.6228	1.87203	0.936017	+	0.351954i	$0.114483\pi$
$174$	0	0
$175$	0	0
$176$	−9.34955	−0.704749
$177$	0	0
$178$	0.340718	0.0255379
$179$	24.1553	1.80545	0.902726	−	0.430216i	$-0.141563\pi$
$179$	24.1553	1.80545	0.902726	+	0.430216i	$0.141563\pi$
$180$	0	0
$181$	1.96805	0.146284	0.0731419	−	0.997322i	$-0.476697\pi$
$181$	1.96805	0.146284	0.0731419	+	0.997322i	$0.476697\pi$
$182$	1.44019	0.106754
$183$	0	0
$184$	−1.56053	−0.115044
$185$	0	0
$186$	0	0
$187$	−15.0049	−1.09727
$188$	6.22362	0.453904
$189$	0	0
$190$	0	0
$191$	7.76854	0.562112	0.281056	−	0.959691i	$-0.409315\pi$
$191$	7.76854	0.562112	0.281056	+	0.959691i	$0.409315\pi$
$192$	0	0
$193$	−14.3458	−1.03264	−0.516318	−	0.856397i	$-0.672698\pi$
$193$	−14.3458	−1.03264	−0.516318	+	0.856397i	$0.672698\pi$
$194$	−0.00752814	−0.000540489 0
$195$	0	0
$196$	10.5596	0.754257
$197$	0.163540	0.0116517	0.00582587	−	0.999983i	$-0.498146\pi$
$197$	0.163540	0.0116517	0.00582587	+	0.999983i	$0.498146\pi$
$198$	0	0
$199$	4.96974	0.352295	0.176148	−	0.984364i	$-0.443636\pi$
$199$	4.96974	0.352295	0.176148	+	0.984364i	$0.443636\pi$
$200$	0	0
$201$	0	0
$202$	2.77129	0.194987
$203$	4.00925	0.281394
$204$	0	0
$205$	0	0
$206$	0.177909	0.0123955
$207$	0	0
$208$	17.0322	1.18097
$209$	−10.7120	−0.740966
$210$	0	0
$211$	3.29381	0.226755	0.113378	−	0.993552i	$-0.463833\pi$
$211$	3.29381	0.226755	0.113378	+	0.993552i	$0.463833\pi$
$212$	25.8567	1.77584
$213$	0	0
$214$	0.522309	0.0357043
$215$	0	0
$216$	0	0
$217$	−11.4616	−0.778064
$218$	0.551465	0.0373500
$219$	0	0
$220$	0	0
$221$	27.3346	1.83872
$222$	0	0
$223$	15.7732	1.05625	0.528127	−	0.849166i	$-0.322895\pi$
$223$	15.7732	1.05625	0.528127	+	0.849166i	$0.322895\pi$
$224$	3.54307	0.236731
$225$	0	0
$226$	−0.777142	−0.0516947
$227$	13.5406	0.898721	0.449360	−	0.893351i	$-0.351652\pi$
$227$	13.5406	0.898721	0.449360	+	0.893351i	$0.351652\pi$
$228$	0	0
$229$	−14.7829	−0.976879	−0.488440	−	0.872598i	$-0.662434\pi$
$229$	−14.7829	−0.976879	−0.488440	+	0.872598i	$0.662434\pi$
$230$	0	0
$231$	0	0
$232$	−3.12573	−0.205214
$233$	0.518980	0.0339995	0.0169998	−	0.999855i	$-0.494589\pi$
$233$	0.518980	0.0339995	0.0169998	+	0.999855i	$0.494589\pi$
$234$	0	0
$235$	0	0
$236$	−12.3963	−0.806932
$237$	0	0
$238$	1.79641	0.116444
$239$	−8.23032	−0.532375	−0.266188	−	0.963921i	$-0.585764\pi$
$239$	−8.23032	−0.532375	−0.266188	+	0.963921i	$0.585764\pi$
$240$	0	0
$241$	−1.28118	−0.0825280	−0.0412640	−	0.999148i	$-0.513138\pi$
$241$	−1.28118	−0.0825280	−0.0412640	+	0.999148i	$0.513138\pi$
$242$	−1.08490	−0.0697398
$243$	0	0
$244$	−7.42081	−0.475069
$245$	0	0
$246$	0	0
$247$	19.5142	1.24166
$248$	8.93581	0.567424
$249$	0	0
$250$	0	0
$251$	−4.63494	−0.292555	−0.146277	−	0.989244i	$-0.546729\pi$
$251$	−4.63494	−0.292555	−0.146277	+	0.989244i	$0.546729\pi$
$252$	0	0
$253$	−4.12559	−0.259373
$254$	3.86431	0.242469
$255$	0	0
$256$	11.3481	0.709257
$257$	−18.3001	−1.14153	−0.570766	−	0.821113i	$-0.693353\pi$
$257$	−18.3001	−1.14153	−0.570766	+	0.821113i	$0.693353\pi$
$258$	0	0
$259$	1.59309	0.0989897
$260$	0	0
$261$	0	0
$262$	3.69063	0.228008
$263$	16.6666	1.02771	0.513854	−	0.857877i	$-0.328217\pi$
$263$	16.6666	1.02771	0.513854	+	0.857877i	$0.328217\pi$
$264$	0	0
$265$	0	0
$266$	1.28246	0.0786324
$267$	0	0
$268$	−19.6312	−1.19917
$269$	−2.63211	−0.160483	−0.0802414	−	0.996775i	$-0.525569\pi$
$269$	−2.63211	−0.160483	−0.0802414	+	0.996775i	$0.525569\pi$
$270$	0	0
$271$	4.56346	0.277210	0.138605	−	0.990348i	$-0.455738\pi$
$271$	4.56346	0.277210	0.138605	+	0.990348i	$0.455738\pi$
$272$	21.2449	1.28816
$273$	0	0
$274$	−2.58963	−0.156445
$275$	0	0
$276$	0	0
$277$	−1.25926	−0.0756614	−0.0378307	−	0.999284i	$-0.512045\pi$
$277$	−1.25926	−0.0756614	−0.0378307	+	0.999284i	$0.512045\pi$
$278$	3.07234	0.184266
$279$	0	0
$280$	0	0
$281$	−13.4167	−0.800370	−0.400185	−	0.916434i	$-0.631054\pi$
$281$	−13.4167	−0.800370	−0.400185	+	0.916434i	$0.631054\pi$
$282$	0	0
$283$	−24.0890	−1.43194	−0.715971	−	0.698130i	$-0.754016\pi$
$283$	−24.0890	−1.43194	−0.715971	+	0.698130i	$0.754016\pi$
$284$	26.9131	1.59700
$285$	0	0
$286$	−2.96838	−0.175524
$287$	−10.8417	−0.639966
$288$	0	0
$289$	17.0956	1.00562
$290$	0	0
$291$	0	0
$292$	−22.0697	−1.29153
$293$	−32.1727	−1.87955	−0.939776	−	0.341792i	$-0.888966\pi$
$293$	−32.1727	−1.87955	−0.939776	+	0.341792i	$0.888966\pi$
$294$	0	0
$295$	0	0
$296$	−1.24202	−0.0721909
$297$	0	0
$298$	4.21167	0.243976
$299$	7.51561	0.434639
$300$	0	0
$301$	4.83849	0.278886
$302$	−0.388685	−0.0223663
$303$	0	0
$304$	15.1668	0.869875
$305$	0	0
$306$	0	0
$307$	−20.2962	−1.15837	−0.579183	−	0.815198i	$-0.696628\pi$
$307$	−20.2962	−1.15837	−0.579183	+	0.815198i	$0.696628\pi$
$308$	6.21254	0.353992
$309$	0	0
$310$	0	0
$311$	18.7224	1.06165	0.530825	−	0.847481i	$-0.321882\pi$
$311$	18.7224	1.06165	0.530825	+	0.847481i	$0.321882\pi$
$312$	0	0
$313$	−24.8839	−1.40652	−0.703261	−	0.710932i	$-0.748274\pi$
$313$	−24.8839	−1.40652	−0.703261	+	0.710932i	$0.748274\pi$
$314$	2.27033	0.128122
$315$	0	0
$316$	−22.2671	−1.25262
$317$	−25.1880	−1.41470	−0.707350	−	0.706863i	$-0.750110\pi$
$317$	−25.1880	−1.41470	−0.707350	+	0.706863i	$0.750110\pi$
$318$	0	0
$319$	−8.26351	−0.462668
$320$	0	0
$321$	0	0
$322$	0.493920	0.0275251
$323$	24.3409	1.35436
$324$	0	0
$325$	0	0
$326$	1.13034	0.0626039
$327$	0	0
$328$	8.45253	0.466713
$329$	−4.00150	−0.220610
$330$	0	0
$331$	30.4193	1.67199	0.835997	−	0.548735i	$-0.184890\pi$
$331$	30.4193	1.67199	0.835997	+	0.548735i	$0.184890\pi$
$332$	5.49212	0.301419
$333$	0	0
$334$	0.0351708	0.00192446
$335$	0	0
$336$	0	0
$337$	4.57455	0.249192	0.124596	−	0.992208i	$-0.460237\pi$
$337$	4.57455	0.249192	0.124596	+	0.992208i	$0.460237\pi$
$338$	2.19966	0.119646
$339$	0	0
$340$	0	0
$341$	23.6237	1.27929
$342$	0	0
$343$	−15.5167	−0.837821
$344$	−3.77224	−0.203385
$345$	0	0
$346$	6.07589	0.326642
$347$	−16.8153	−0.902691	−0.451346	−	0.892349i	$-0.649056\pi$
$347$	−16.8153	−0.902691	−0.451346	+	0.892349i	$0.649056\pi$
$348$	0	0
$349$	0.373581	0.0199973	0.00999866	−	0.999950i	$-0.496817\pi$
$349$	0.373581	0.0199973	0.00999866	+	0.999950i	$0.496817\pi$
$350$	0	0
$351$	0	0
$352$	−7.30266	−0.389233
$353$	−23.9685	−1.27572	−0.637858	−	0.770154i	$-0.720179\pi$
$353$	−23.9685	−1.27572	−0.637858	+	0.770154i	$0.720179\pi$
$354$	0	0
$355$	0	0
$356$	−2.67747	−0.141906
$357$	0	0
$358$	5.96054	0.315024
$359$	4.75020	0.250706	0.125353	−	0.992112i	$-0.459994\pi$
$359$	4.75020	0.250706	0.125353	+	0.992112i	$0.459994\pi$
$360$	0	0
$361$	−1.62302	−0.0854221
$362$	0.485633	0.0255243
$363$	0	0
$364$	−11.3174	−0.593195
$365$	0	0
$366$	0	0
$367$	17.3700	0.906706	0.453353	−	0.891331i	$-0.350228\pi$
$367$	17.3700	0.906706	0.453353	+	0.891331i	$0.350228\pi$
$368$	5.84128	0.304498
$369$	0	0
$370$	0	0
$371$	−16.6246	−0.863109
$372$	0	0
$373$	−2.10877	−0.109188	−0.0545939	−	0.998509i	$-0.517386\pi$
$373$	−2.10877	−0.109188	−0.0545939	+	0.998509i	$0.517386\pi$
$374$	−3.70260	−0.191457
$375$	0	0
$376$	3.11969	0.160886
$377$	15.0537	0.775305
$378$	0	0
$379$	12.4090	0.637405	0.318703	−	0.947855i	$-0.396753\pi$
$379$	12.4090	0.637405	0.318703	+	0.947855i	$0.396753\pi$
$380$	0	0
$381$	0	0
$382$	1.91696	0.0980801
$383$	33.3056	1.70184	0.850919	−	0.525297i	$-0.176046\pi$
$383$	33.3056	1.70184	0.850919	+	0.525297i	$0.176046\pi$
$384$	0	0
$385$	0	0
$386$	−3.53996	−0.180179
$387$	0	0
$388$	0.0591585	0.00300332
$389$	16.9777	0.860802	0.430401	−	0.902638i	$-0.358372\pi$
$389$	16.9777	0.860802	0.430401	+	0.902638i	$0.358372\pi$
$390$	0	0
$391$	9.37455	0.474091
$392$	5.29317	0.267346
$393$	0	0
$394$	0.0403549	0.00203305
$395$	0	0
$396$	0	0
$397$	0.866570	0.0434919	0.0217460	−	0.999764i	$-0.493078\pi$
$397$	0.866570	0.0434919	0.0217460	+	0.999764i	$0.493078\pi$
$398$	1.22633	0.0614702
$399$	0	0
$400$	0	0
$401$	−9.68680	−0.483736	−0.241868	−	0.970309i	$-0.577760\pi$
$401$	−9.68680	−0.483736	−0.241868	+	0.970309i	$0.577760\pi$
$402$	0	0
$403$	−43.0354	−2.14375
$404$	−21.7777	−1.08348
$405$	0	0
$406$	0.989317	0.0490990
$407$	−3.28353	−0.162759
$408$	0	0
$409$	1.51785	0.0750531	0.0375265	−	0.999296i	$-0.488052\pi$
$409$	1.51785	0.0750531	0.0375265	+	0.999296i	$0.488052\pi$
$410$	0	0
$411$	0	0
$412$	−1.39807	−0.0688778
$413$	7.97027	0.392191
$414$	0	0
$415$	0	0
$416$	13.3033	0.652249
$417$	0	0
$418$	−2.64329	−0.129287
$419$	30.5620	1.49305	0.746526	−	0.665356i	$-0.231720\pi$
$419$	30.5620	1.49305	0.746526	+	0.665356i	$0.231720\pi$
$420$	0	0
$421$	−6.97162	−0.339776	−0.169888	−	0.985463i	$-0.554341\pi$
$421$	−6.97162	−0.339776	−0.169888	+	0.985463i	$0.554341\pi$
$422$	0.812777	0.0395654
$423$	0	0
$424$	12.9611	0.629445
$425$	0	0
$426$	0	0
$427$	4.77124	0.230897
$428$	−4.10447	−0.198397
$429$	0	0
$430$	0	0
$431$	−10.9472	−0.527310	−0.263655	−	0.964617i	$-0.584928\pi$
$431$	−10.9472	−0.527310	−0.263655	+	0.964617i	$0.584928\pi$
$432$	0	0
$433$	−10.4393	−0.501683	−0.250841	−	0.968028i	$-0.580707\pi$
$433$	−10.4393	−0.501683	−0.250841	+	0.968028i	$0.580707\pi$
$434$	−2.82825	−0.135761
$435$	0	0
$436$	−4.33359	−0.207541
$437$	6.69250	0.320146
$438$	0	0
$439$	15.4334	0.736594	0.368297	−	0.929708i	$-0.379941\pi$
$439$	15.4334	0.736594	0.368297	+	0.929708i	$0.379941\pi$
$440$	0	0
$441$	0	0
$442$	6.74505	0.320829
$443$	18.9105	0.898463	0.449231	−	0.893415i	$-0.351698\pi$
$443$	18.9105	0.898463	0.449231	+	0.893415i	$0.351698\pi$
$444$	0	0
$445$	0	0
$446$	3.89218	0.184300
$447$	0	0
$448$	−8.19805	−0.387322
$449$	11.4152	0.538714	0.269357	−	0.963040i	$-0.413189\pi$
$449$	11.4152	0.538714	0.269357	+	0.963040i	$0.413189\pi$
$450$	0	0
$451$	22.3460	1.05223
$452$	6.10703	0.287250
$453$	0	0
$454$	3.34126	0.156813
$455$	0	0
$456$	0	0
$457$	33.9739	1.58923	0.794616	−	0.607112i	$-0.207672\pi$
$457$	33.9739	1.58923	0.794616	+	0.607112i	$0.207672\pi$
$458$	−3.64780	−0.170451
$459$	0	0
$460$	0	0
$461$	24.6907	1.14996	0.574981	−	0.818167i	$-0.305009\pi$
$461$	24.6907	1.14996	0.574981	+	0.818167i	$0.305009\pi$
$462$	0	0
$463$	13.6315	0.633510	0.316755	−	0.948507i	$-0.397407\pi$
$463$	13.6315	0.633510	0.316755	+	0.948507i	$0.397407\pi$
$464$	11.7000	0.543160
$465$	0	0
$466$	0.128063	0.00593241
$467$	9.47115	0.438272	0.219136	−	0.975694i	$-0.429676\pi$
$467$	9.47115	0.438272	0.219136	+	0.975694i	$0.429676\pi$
$468$	0	0
$469$	12.6220	0.582828
$470$	0	0
$471$	0	0
$472$	−6.21386	−0.286016
$473$	−9.97268	−0.458544
$474$	0	0
$475$	0	0
$476$	−14.1167	−0.647039
$477$	0	0
$478$	−2.03091	−0.0928914
$479$	25.7286	1.17557	0.587784	−	0.809018i	$-0.300000\pi$
$479$	25.7286	1.17557	0.587784	+	0.809018i	$0.300000\pi$
$480$	0	0
$481$	5.98164	0.272739
$482$	−0.316142	−0.0143999
$483$	0	0
$484$	8.52546	0.387521
$485$	0	0
$486$	0	0
$487$	28.4046	1.28714	0.643568	−	0.765389i	$-0.277453\pi$
$487$	28.4046	1.28714	0.643568	+	0.765389i	$0.277453\pi$
$488$	−3.71980	−0.168388
$489$	0	0
$490$	0	0
$491$	6.10299	0.275424	0.137712	−	0.990472i	$-0.456025\pi$
$491$	6.10299	0.275424	0.137712	+	0.990472i	$0.456025\pi$
$492$	0	0
$493$	18.7771	0.845679
$494$	4.81530	0.216650
$495$	0	0
$496$	−33.4479	−1.50186
$497$	−17.3039	−0.776186
$498$	0	0
$499$	−20.3163	−0.909481	−0.454740	−	0.890624i	$-0.650268\pi$
$499$	−20.3163	−0.909481	−0.454740	+	0.890624i	$0.650268\pi$
$500$	0	0
$501$	0	0
$502$	−1.14371	−0.0510464
$503$	14.6267	0.652171	0.326085	−	0.945340i	$-0.394270\pi$
$503$	14.6267	0.652171	0.326085	+	0.945340i	$0.394270\pi$
$504$	0	0
$505$	0	0
$506$	−1.01803	−0.0452567
$507$	0	0
$508$	−30.3670	−1.34732
$509$	4.30584	0.190853	0.0954265	−	0.995436i	$-0.469578\pi$
$509$	4.30584	0.190853	0.0954265	+	0.995436i	$0.469578\pi$
$510$	0	0
$511$	14.1898	0.627721
$512$	17.4127	0.769538
$513$	0	0
$514$	−4.51572	−0.199180
$515$	0	0
$516$	0	0
$517$	8.24755	0.362727
$518$	0.393109	0.0172722
$519$	0	0
$520$	0	0
$521$	39.6413	1.73672	0.868358	−	0.495938i	$-0.165176\pi$
$521$	39.6413	1.73672	0.868358	+	0.495938i	$0.165176\pi$
$522$	0	0
$523$	−36.2874	−1.58674	−0.793369	−	0.608741i	$-0.791675\pi$
$523$	−36.2874	−1.58674	−0.793369	+	0.608741i	$0.791675\pi$
$524$	−29.0021	−1.26696
$525$	0	0
$526$	4.11264	0.179320
$527$	−53.6799	−2.33833
$528$	0	0
$529$	−20.4225	−0.887934
$530$	0	0
$531$	0	0
$532$	−10.0779	−0.436934
$533$	−40.7079	−1.76325
$534$	0	0
$535$	0	0
$536$	−9.84046	−0.425043
$537$	0	0
$538$	−0.649497	−0.0280018
$539$	13.9936	0.602747
$540$	0	0
$541$	−38.6044	−1.65973	−0.829866	−	0.557962i	$-0.811583\pi$
$541$	−38.6044	−1.65973	−0.829866	+	0.557962i	$0.811583\pi$
$542$	1.12607	0.0483690
$543$	0	0
$544$	16.5938	0.711453
$545$	0	0
$546$	0	0
$547$	−10.3620	−0.443047	−0.221524	−	0.975155i	$-0.571103\pi$
$547$	−10.3620	−0.443047	−0.221524	+	0.975155i	$0.571103\pi$
$548$	20.3501	0.869313
$549$	0	0
$550$	0	0
$551$	13.4050	0.571073
$552$	0	0
$553$	14.3167	0.608810
$554$	−0.310733	−0.0132018
$555$	0	0
$556$	−24.1434	−1.02391
$557$	12.6830	0.537398	0.268699	−	0.963224i	$-0.413406\pi$
$557$	12.6830	0.537398	0.268699	+	0.963224i	$0.413406\pi$
$558$	0	0
$559$	18.1673	0.768396
$560$	0	0
$561$	0	0
$562$	−3.31068	−0.139653
$563$	13.0892	0.551645	0.275823	−	0.961209i	$-0.411050\pi$
$563$	13.0892	0.551645	0.275823	+	0.961209i	$0.411050\pi$
$564$	0	0
$565$	0	0
$566$	−5.94417	−0.249852
$567$	0	0
$568$	13.4906	0.566055
$569$	2.58803	0.108496	0.0542478	−	0.998528i	$-0.482724\pi$
$569$	2.58803	0.108496	0.0542478	+	0.998528i	$0.482724\pi$
$570$	0	0
$571$	−10.5754	−0.442568	−0.221284	−	0.975209i	$-0.571025\pi$
$571$	−10.5754	−0.442568	−0.221284	+	0.975209i	$0.571025\pi$
$572$	23.3265	0.975330
$573$	0	0
$574$	−2.67529	−0.111665
$575$	0	0
$576$	0	0
$577$	0.0142904	0.000594917 0	0.000297458	−	1.00000i	$-0.499905\pi$
$577$	0.0142904	0.000594917 0	0.000297458		1.00000i	$0.499905\pi$
$578$	4.21849	0.175466
$579$	0	0
$580$	0	0
$581$	−3.53118	−0.146498
$582$	0	0
$583$	34.2653	1.41912
$584$	−11.0628	−0.457783
$585$	0	0
$586$	−7.93891	−0.327953
$587$	−22.6389	−0.934408	−0.467204	−	0.884149i	$-0.654739\pi$
$587$	−22.6389	−0.934408	−0.467204	+	0.884149i	$0.654739\pi$
$588$	0	0
$589$	−38.3222	−1.57904
$590$	0	0
$591$	0	0
$592$	4.64905	0.191075
$593$	12.7423	0.523264	0.261632	−	0.965168i	$-0.415739\pi$
$593$	12.7423	0.523264	0.261632	+	0.965168i	$0.415739\pi$
$594$	0	0
$595$	0	0
$596$	−33.0967	−1.35569
$597$	0	0
$598$	1.85454	0.0758380
$599$	6.83599	0.279311	0.139656	−	0.990200i	$-0.455400\pi$
$599$	6.83599	0.279311	0.139656	+	0.990200i	$0.455400\pi$
$600$	0	0
$601$	31.8422	1.29887	0.649435	−	0.760417i	$-0.275005\pi$
$601$	31.8422	1.29887	0.649435	+	0.760417i	$0.275005\pi$
$602$	1.19394	0.0486614
$603$	0	0
$604$	3.05441	0.124282
$605$	0	0
$606$	0	0
$607$	−27.7763	−1.12741	−0.563703	−	0.825977i	$-0.690624\pi$
$607$	−27.7763	−1.12741	−0.563703	+	0.825977i	$0.690624\pi$
$608$	11.8463	0.480432
$609$	0	0
$610$	0	0
$611$	−15.0246	−0.607831
$612$	0	0
$613$	3.61694	0.146087	0.0730433	−	0.997329i	$-0.476729\pi$
$613$	3.61694	0.146087	0.0730433	+	0.997329i	$0.476729\pi$
$614$	−5.00827	−0.202117
$615$	0	0
$616$	3.11414	0.125472
$617$	44.6425	1.79724	0.898621	−	0.438727i	$-0.144570\pi$
$617$	44.6425	1.79724	0.898621	+	0.438727i	$0.144570\pi$
$618$	0	0
$619$	20.5085	0.824307	0.412153	−	0.911115i	$-0.364777\pi$
$619$	20.5085	0.824307	0.412153	+	0.911115i	$0.364777\pi$
$620$	0	0
$621$	0	0
$622$	4.61992	0.185242
$623$	1.72149	0.0689701
$624$	0	0
$625$	0	0
$626$	−6.14033	−0.245417
$627$	0	0
$628$	−17.8409	−0.711931
$629$	7.46116	0.297496
$630$	0	0
$631$	−37.7357	−1.50223	−0.751117	−	0.660169i	$-0.770485\pi$
$631$	−37.7357	−1.50223	−0.751117	+	0.660169i	$0.770485\pi$
$632$	−11.1618	−0.443991
$633$	0	0
$634$	−6.21537	−0.246844
$635$	0	0
$636$	0	0
$637$	−25.4922	−1.01004
$638$	−2.03909	−0.0807285
$639$	0	0
$640$	0	0
$641$	9.18606	0.362828	0.181414	−	0.983407i	$-0.441933\pi$
$641$	9.18606	0.362828	0.181414	+	0.983407i	$0.441933\pi$
$642$	0	0
$643$	−3.42111	−0.134915	−0.0674577	−	0.997722i	$-0.521489\pi$
$643$	−3.42111	−0.134915	−0.0674577	+	0.997722i	$0.521489\pi$
$644$	−3.88138	−0.152948
$645$	0	0
$646$	6.00633	0.236316
$647$	3.26735	0.128453	0.0642264	−	0.997935i	$-0.479542\pi$
$647$	3.26735	0.128453	0.0642264	+	0.997935i	$0.479542\pi$
$648$	0	0
$649$	−16.4276	−0.644840
$650$	0	0
$651$	0	0
$652$	−8.88259	−0.347869
$653$	12.6362	0.494494	0.247247	−	0.968952i	$-0.420474\pi$
$653$	12.6362	0.494494	0.247247	+	0.968952i	$0.420474\pi$
$654$	0	0
$655$	0	0
$656$	−31.6390	−1.23529
$657$	0	0
$658$	−0.987407	−0.0384931
$659$	9.14896	0.356393	0.178197	−	0.983995i	$-0.442974\pi$
$659$	9.14896	0.356393	0.178197	+	0.983995i	$0.442974\pi$
$660$	0	0
$661$	6.39083	0.248574	0.124287	−	0.992246i	$-0.460336\pi$
$661$	6.39083	0.248574	0.124287	+	0.992246i	$0.460336\pi$
$662$	7.50622	0.291738
$663$	0	0
$664$	2.75302	0.106838
$665$	0	0
$666$	0	0
$667$	5.16276	0.199903
$668$	−0.276383	−0.0106936
$669$	0	0
$670$	0	0
$671$	−9.83407	−0.379640
$672$	0	0
$673$	33.7063	1.29928	0.649641	−	0.760241i	$-0.274919\pi$
$673$	33.7063	1.29928	0.649641	+	0.760241i	$0.274919\pi$
$674$	1.12881	0.0434802
$675$	0	0
$676$	−17.2857	−0.664833
$677$	9.20340	0.353715	0.176858	−	0.984236i	$-0.443407\pi$
$677$	9.20340	0.353715	0.176858	+	0.984236i	$0.443407\pi$
$678$	0	0
$679$	−0.0380362	−0.00145970
$680$	0	0
$681$	0	0
$682$	5.82935	0.223217
$683$	27.2216	1.04161	0.520803	−	0.853677i	$-0.325633\pi$
$683$	27.2216	1.04161	0.520803	+	0.853677i	$0.325633\pi$
$684$	0	0
$685$	0	0
$686$	−3.82887	−0.146187
$687$	0	0
$688$	14.1200	0.538320
$689$	−62.4213	−2.37806
$690$	0	0
$691$	29.8238	1.13455	0.567276	−	0.823528i	$-0.307997\pi$
$691$	29.8238	1.13455	0.567276	+	0.823528i	$0.307997\pi$
$692$	−47.7463	−1.81504
$693$	0	0
$694$	−4.14932	−0.157506
$695$	0	0
$696$	0	0
$697$	−50.7767	−1.92331
$698$	0.0921844	0.00348923
$699$	0	0
$700$	0	0
$701$	19.1081	0.721704	0.360852	−	0.932623i	$-0.382486\pi$
$701$	19.1081	0.721704	0.360852	+	0.932623i	$0.382486\pi$
$702$	0	0
$703$	5.32653	0.200894
$704$	16.8971	0.636834
$705$	0	0
$706$	−5.91445	−0.222593
$707$	14.0020	0.526601
$708$	0	0
$709$	14.5146	0.545107	0.272553	−	0.962141i	$-0.412132\pi$
$709$	14.5146	0.545107	0.272553	+	0.962141i	$0.412132\pi$
$710$	0	0
$711$	0	0
$712$	−1.34213	−0.0502983
$713$	−14.7592	−0.552738
$714$	0	0
$715$	0	0
$716$	−46.8398	−1.75048
$717$	0	0
$718$	1.17215	0.0437444
$719$	−34.7507	−1.29598	−0.647992	−	0.761647i	$-0.724391\pi$
$719$	−34.7507	−1.29598	−0.647992	+	0.761647i	$0.724391\pi$
$720$	0	0
$721$	0.898892	0.0334765
$722$	−0.400494	−0.0149049
$723$	0	0
$724$	−3.81626	−0.141830
$725$	0	0
$726$	0	0
$727$	−42.6279	−1.58098	−0.790490	−	0.612475i	$-0.790174\pi$
$727$	−42.6279	−1.58098	−0.790490	+	0.612475i	$0.790174\pi$
$728$	−5.67305	−0.210257
$729$	0	0
$730$	0	0
$731$	22.6609	0.838143
$732$	0	0
$733$	−1.96230	−0.0724793	−0.0362397	−	0.999343i	$-0.511538\pi$
$733$	−1.96230	−0.0724793	−0.0362397	+	0.999343i	$0.511538\pi$
$734$	4.28620	0.158207
$735$	0	0
$736$	4.56245	0.168174
$737$	−26.0153	−0.958286
$738$	0	0
$739$	12.3601	0.454672	0.227336	−	0.973816i	$-0.426998\pi$
$739$	12.3601	0.454672	0.227336	+	0.973816i	$0.426998\pi$
$740$	0	0
$741$	0	0
$742$	−4.10228	−0.150599
$743$	8.63742	0.316876	0.158438	−	0.987369i	$-0.449354\pi$
$743$	8.63742	0.316876	0.158438	+	0.987369i	$0.449354\pi$
$744$	0	0
$745$	0	0
$746$	−0.520357	−0.0190516
$747$	0	0
$748$	29.0962	1.06386
$749$	2.63898	0.0964264
$750$	0	0
$751$	−24.3654	−0.889105	−0.444552	−	0.895753i	$-0.646637\pi$
$751$	−24.3654	−0.889105	−0.444552	+	0.895753i	$0.646637\pi$
$752$	−11.6774	−0.425832
$753$	0	0
$754$	3.71463	0.135279
$755$	0	0
$756$	0	0
$757$	5.18352	0.188398	0.0941991	−	0.995553i	$-0.469971\pi$
$757$	5.18352	0.188398	0.0941991	+	0.995553i	$0.469971\pi$
$758$	3.06202	0.111218
$759$	0	0
$760$	0	0
$761$	−31.2636	−1.13331	−0.566653	−	0.823957i	$-0.691762\pi$
$761$	−31.2636	−1.13331	−0.566653	+	0.823957i	$0.691762\pi$
$762$	0	0
$763$	2.78630	0.100871
$764$	−15.0641	−0.544999
$765$	0	0
$766$	8.21846	0.296945
$767$	29.9263	1.08058
$768$	0	0
$769$	−44.1891	−1.59350	−0.796750	−	0.604309i	$-0.793449\pi$
$769$	−44.1891	−1.59350	−0.796750	+	0.604309i	$0.793449\pi$
$770$	0	0
$771$	0	0
$772$	27.8182	1.00120
$773$	−21.5881	−0.776471	−0.388236	−	0.921560i	$-0.626915\pi$
$773$	−21.5881	−0.776471	−0.388236	+	0.921560i	$0.626915\pi$
$774$	0	0
$775$	0	0
$776$	0.0296542	0.00106452
$777$	0	0
$778$	4.18939	0.150197
$779$	−36.2496	−1.29878
$780$	0	0
$781$	35.6653	1.27621
$782$	2.31325	0.0827218
$783$	0	0
$784$	−19.8131	−0.707610
$785$	0	0
$786$	0	0
$787$	−14.4212	−0.514059	−0.257029	−	0.966404i	$-0.582744\pi$
$787$	−14.4212	−0.514059	−0.257029	+	0.966404i	$0.582744\pi$
$788$	−0.317122	−0.0112970
$789$	0	0
$790$	0	0
$791$	−3.92654	−0.139612
$792$	0	0
$793$	17.9148	0.636173
$794$	0.213834	0.00758868
$795$	0	0
$796$	−9.63687	−0.341570
$797$	−39.4567	−1.39763	−0.698814	−	0.715304i	$-0.746288\pi$
$797$	−39.4567	−1.39763	−0.698814	+	0.715304i	$0.746288\pi$
$798$	0	0
$799$	−18.7409	−0.663004
$800$	0	0
$801$	0	0
$802$	−2.39031	−0.0844046
$803$	−29.2469	−1.03210
$804$	0	0
$805$	0	0
$806$	−10.6194	−0.374051
$807$	0	0
$808$	−10.9164	−0.384038
$809$	5.96204	0.209614	0.104807	−	0.994493i	$-0.466577\pi$
$809$	5.96204	0.209614	0.104807	+	0.994493i	$0.466577\pi$
$810$	0	0
$811$	−39.1764	−1.37567	−0.687835	−	0.725867i	$-0.741439\pi$
$811$	−39.1764	−1.37567	−0.687835	+	0.725867i	$0.741439\pi$
$812$	−7.77437	−0.272827
$813$	0	0
$814$	−0.810242	−0.0283990
$815$	0	0
$816$	0	0
$817$	16.1776	0.565984
$818$	0.374544	0.0130956
$819$	0	0
$820$	0	0
$821$	−20.6329	−0.720092	−0.360046	−	0.932934i	$-0.617239\pi$
$821$	−20.6329	−0.720092	−0.360046	+	0.932934i	$0.617239\pi$
$822$	0	0
$823$	−39.0162	−1.36002	−0.680009	−	0.733203i	$-0.738024\pi$
$823$	−39.0162	−1.36002	−0.680009	+	0.733203i	$0.738024\pi$
$824$	−0.700803	−0.0244136
$825$	0	0
$826$	1.96673	0.0684314
$827$	19.5976	0.681474	0.340737	−	0.940159i	$-0.389323\pi$
$827$	19.5976	0.681474	0.340737	+	0.940159i	$0.389323\pi$
$828$	0	0
$829$	−8.67511	−0.301299	−0.150650	−	0.988587i	$-0.548136\pi$
$829$	−8.67511	−0.301299	−0.150650	+	0.988587i	$0.548136\pi$
$830$	0	0
$831$	0	0
$832$	−30.7816	−1.06716
$833$	−31.7976	−1.10172
$834$	0	0
$835$	0	0
$836$	20.7718	0.718407
$837$	0	0
$838$	7.54145	0.260515
$839$	3.85921	0.133235	0.0666173	−	0.997779i	$-0.478779\pi$
$839$	3.85921	0.133235	0.0666173	+	0.997779i	$0.478779\pi$
$840$	0	0
$841$	−18.6591	−0.643416
$842$	−1.72031	−0.0592858
$843$	0	0
$844$	−6.38706	−0.219852
$845$	0	0
$846$	0	0
$847$	−5.48148	−0.188346
$848$	−48.5151	−1.66601
$849$	0	0
$850$	0	0
$851$	2.05144	0.0703224
$852$	0	0
$853$	29.7898	1.01998	0.509991	−	0.860180i	$-0.329649\pi$
$853$	29.7898	1.01998	0.509991	+	0.860180i	$0.329649\pi$
$854$	1.17735	0.0402880
$855$	0	0
$856$	−2.05743	−0.0703216
$857$	36.2041	1.23671	0.618354	−	0.785899i	$-0.287800\pi$
$857$	36.2041	1.23671	0.618354	+	0.785899i	$0.287800\pi$
$858$	0	0
$859$	−37.6533	−1.28471	−0.642357	−	0.766406i	$-0.722043\pi$
$859$	−37.6533	−1.28471	−0.642357	+	0.766406i	$0.722043\pi$
$860$	0	0
$861$	0	0
$862$	−2.70133	−0.0920077
$863$	−2.49913	−0.0850715	−0.0425357	−	0.999095i	$-0.513544\pi$
$863$	−2.49913	−0.0850715	−0.0425357	+	0.999095i	$0.513544\pi$
$864$	0	0
$865$	0	0
$866$	−2.57600	−0.0875361
$867$	0	0
$868$	22.2253	0.754376
$869$	−29.5084	−1.00100
$870$	0	0
$871$	47.3923	1.60583
$872$	−2.17228	−0.0735628
$873$	0	0
$874$	1.65143	0.0558606
$875$	0	0
$876$	0	0
$877$	32.4491	1.09573	0.547864	−	0.836567i	$-0.315441\pi$
$877$	32.4491	1.09573	0.547864	+	0.836567i	$0.315441\pi$
$878$	3.80832	0.128525
$879$	0	0
$880$	0	0
$881$	33.2625	1.12064	0.560321	−	0.828275i	$-0.310678\pi$
$881$	33.2625	1.12064	0.560321	+	0.828275i	$0.310678\pi$
$882$	0	0
$883$	10.9730	0.369269	0.184635	−	0.982807i	$-0.440890\pi$
$883$	10.9730	0.369269	0.184635	+	0.982807i	$0.440890\pi$
$884$	−53.0047	−1.78274
$885$	0	0
$886$	4.66632	0.156768
$887$	−26.0052	−0.873168	−0.436584	−	0.899664i	$-0.643812\pi$
$887$	−26.0052	−0.873168	−0.436584	+	0.899664i	$0.643812\pi$
$888$	0	0
$889$	19.5246	0.654834
$890$	0	0
$891$	0	0
$892$	−30.5860	−1.02410
$893$	−13.3791	−0.447715
$894$	0	0
$895$	0	0
$896$	−9.10907	−0.304313
$897$	0	0
$898$	2.81679	0.0939975
$899$	−29.5626	−0.985969
$900$	0	0
$901$	−77.8608	−2.59392
$902$	5.51408	0.183599
$903$	0	0
$904$	3.06125	0.101816
$905$	0	0
$906$	0	0
$907$	55.0108	1.82660	0.913302	−	0.407283i	$-0.133524\pi$
$907$	55.0108	1.82660	0.913302	+	0.407283i	$0.133524\pi$
$908$	−26.2567	−0.871359
$909$	0	0
$910$	0	0
$911$	12.1075	0.401139	0.200569	−	0.979679i	$-0.435721\pi$
$911$	12.1075	0.401139	0.200569	+	0.979679i	$0.435721\pi$
$912$	0	0
$913$	7.27817	0.240872
$914$	8.38337	0.277297
$915$	0	0
$916$	28.6656	0.947138
$917$	18.6470	0.615779
$918$	0	0
$919$	5.97191	0.196995	0.0984975	−	0.995137i	$-0.468596\pi$
$919$	5.97191	0.196995	0.0984975	+	0.995137i	$0.468596\pi$
$920$	0	0
$921$	0	0
$922$	6.09266	0.200651
$923$	−64.9718	−2.13857
$924$	0	0
$925$	0	0
$926$	3.36370	0.110538
$927$	0	0
$928$	9.13854	0.299987
$929$	−46.8756	−1.53794	−0.768970	−	0.639285i	$-0.779230\pi$
$929$	−46.8756	−1.53794	−0.768970	+	0.639285i	$0.779230\pi$
$930$	0	0
$931$	−22.7003	−0.743973
$932$	−1.00636	−0.0329644
$933$	0	0
$934$	2.33709	0.0764719
$935$	0	0
$936$	0	0
$937$	19.4335	0.634864	0.317432	−	0.948281i	$-0.397179\pi$
$937$	19.4335	0.634864	0.317432	+	0.948281i	$0.397179\pi$
$938$	3.11458	0.101695
$939$	0	0
$940$	0	0
$941$	−21.7097	−0.707717	−0.353858	−	0.935299i	$-0.615130\pi$
$941$	−21.7097	−0.707717	−0.353858	+	0.935299i	$0.615130\pi$
$942$	0	0
$943$	−13.9610	−0.454633
$944$	23.2593	0.757026
$945$	0	0
$946$	−2.46085	−0.0800091
$947$	−6.31560	−0.205229	−0.102615	−	0.994721i	$-0.532721\pi$
$947$	−6.31560	−0.205229	−0.102615	+	0.994721i	$0.532721\pi$
$948$	0	0
$949$	53.2792	1.72952
$950$	0	0
$951$	0	0
$952$	−7.07624	−0.229342
$953$	21.8017	0.706227	0.353114	−	0.935580i	$-0.385123\pi$
$953$	21.8017	0.706227	0.353114	+	0.935580i	$0.385123\pi$
$954$	0	0
$955$	0	0
$956$	15.9595	0.516167
$957$	0	0
$958$	6.34876	0.205119
$959$	−13.0842	−0.422510
$960$	0	0
$961$	53.5134	1.72624
$962$	1.47602	0.0475889
$963$	0	0
$964$	2.48435	0.0800154
$965$	0	0
$966$	0	0
$967$	7.01268	0.225513	0.112756	−	0.993623i	$-0.464032\pi$
$967$	7.01268	0.225513	0.112756	+	0.993623i	$0.464032\pi$
$968$	4.27353	0.137356
$969$	0	0
$970$	0	0
$971$	−7.41028	−0.237807	−0.118904	−	0.992906i	$-0.537938\pi$
$971$	−7.41028	−0.237807	−0.118904	+	0.992906i	$0.537938\pi$
$972$	0	0
$973$	15.5231	0.497647
$974$	7.00909	0.224586
$975$	0	0
$976$	13.9237	0.445688
$977$	39.2585	1.25599	0.627995	−	0.778217i	$-0.283876\pi$
$977$	39.2585	1.25599	0.627995	+	0.778217i	$0.283876\pi$
$978$	0	0
$979$	−3.54819	−0.113401
$980$	0	0
$981$	0	0
$982$	1.50597	0.0480573
$983$	−18.3827	−0.586317	−0.293159	−	0.956064i	$-0.594706\pi$
$983$	−18.3827	−0.586317	−0.293159	+	0.956064i	$0.594706\pi$
$984$	0	0
$985$	0	0
$986$	4.63343	0.147558
$987$	0	0
$988$	−37.8401	−1.20385
$989$	6.23059	0.198121
$990$	0	0
$991$	52.4658	1.66663	0.833316	−	0.552797i	$-0.186440\pi$
$991$	52.4658	1.66663	0.833316	+	0.552797i	$0.186440\pi$
$992$	−26.1252	−0.829476
$993$	0	0
$994$	−4.26989	−0.135433
$995$	0	0
$996$	0	0
$997$	36.3720	1.15191	0.575956	−	0.817480i	$-0.304630\pi$
$997$	36.3720	1.15191	0.575956	+	0.817480i	$0.304630\pi$
$998$	−5.01322	−0.158691
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.q.1.4		6
3.2	odd	2		1875.2.a.k.1.3		6
5.4	even	2		5625.2.a.p.1.3		6
15.2	even	4		1875.2.b.f.1249.5		12
15.8	even	4		1875.2.b.f.1249.8		12
15.14	odd	2		1875.2.a.j.1.4		6
25.9	even	10		225.2.h.d.181.2		12
25.14	even	10		225.2.h.d.46.2		12
75.2	even	20		375.2.i.d.274.3		24
75.11	odd	10		375.2.g.c.226.2		12
75.14	odd	10		75.2.g.c.46.2	yes	12
75.23	even	20		375.2.i.d.274.4		24
75.38	even	20		375.2.i.d.349.3		24
75.41	odd	10		375.2.g.c.151.2		12
75.59	odd	10		75.2.g.c.31.2	✓	12
75.62	even	20		375.2.i.d.349.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.c.31.2	✓	12	75.59	odd	10
75.2.g.c.46.2	yes	12	75.14	odd	10
225.2.h.d.46.2		12	25.14	even	10
225.2.h.d.181.2		12	25.9	even	10
375.2.g.c.151.2		12	75.41	odd	10
375.2.g.c.226.2		12	75.11	odd	10
375.2.i.d.274.3		24	75.2	even	20
375.2.i.d.274.4		24	75.23	even	20
375.2.i.d.349.3		24	75.38	even	20
375.2.i.d.349.4		24	75.62	even	20
1875.2.a.j.1.4		6	15.14	odd	2
1875.2.a.k.1.3		6	3.2	odd	2
1875.2.b.f.1249.5		12	15.2	even	4
1875.2.b.f.1249.8		12	15.8	even	4
5625.2.a.p.1.3		6	5.4	even	2
5625.2.a.q.1.4		6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.246759 q^{2} -1.93911 q^{4} +1.24676 q^{7} -0.972011 q^{8} -2.56971 q^{11} +4.68126 q^{13} +0.307649 q^{14} +3.63837 q^{16} +5.83914 q^{17} +4.16857 q^{19} -0.634099 q^{22} +1.60547 q^{23} +1.15514 q^{26} -2.41760 q^{28} +3.21573 q^{29} -9.19312 q^{31} +2.84182 q^{32} +1.44086 q^{34} +1.27778 q^{37} +1.02863 q^{38} -8.69592 q^{41} +3.88086 q^{43} +4.98295 q^{44} +0.396163 q^{46} -3.20952 q^{47} -5.44559 q^{49} -9.07748 q^{52} -13.3343 q^{53} -1.21186 q^{56} +0.793511 q^{58} +6.39279 q^{59} +3.82692 q^{61} -2.26848 q^{62} -6.57549 q^{64} +10.1238 q^{67} -11.3227 q^{68} -13.8791 q^{71} +11.3814 q^{73} +0.315305 q^{74} -8.08332 q^{76} -3.20381 q^{77} +11.4832 q^{79} -2.14580 q^{82} -2.83229 q^{83} +0.957636 q^{86} +2.49779 q^{88} +1.38077 q^{89} +5.83641 q^{91} -3.11318 q^{92} -0.791979 q^{94} -0.0305081 q^{97} -1.34375 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} + 6 q^{7} - 3 q^{8} - 3 q^{11} + 6 q^{13} + 22 q^{14} + 18 q^{16} - 13 q^{17} + 11 q^{19} + 16 q^{22} - 13 q^{23} + 28 q^{26} + 7 q^{28} + 3 q^{29} - 11 q^{31} - 16 q^{32} + 15 q^{34} + 21 q^{37}+ \cdots + 41 q^{98}+O(q^{100}) \) 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

L-functions

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