LMFDB - Embedded newform 4225.2.a.br.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.199374400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 10x^{2} - 1 $ x^6 - 8x^4 + 10x^2 - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.18733$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18733 q^{2} +0.345110 q^{3} -0.590239 q^{4} +0.409761 q^{6} +2.02956 q^{7} -3.07548 q^{8} -2.88090 q^{9} +O(q^{10})$	\(q+1.18733 q^{2} +0.345110 q^{3} -0.590239 q^{4} +0.409761 q^{6} +2.02956 q^{7} -3.07548 q^{8} -2.88090 q^{9} +3.88090 q^{11} -0.203698 q^{12} +2.40976 q^{14} -2.47114 q^{16} -5.45014 q^{17} -3.42059 q^{18} +5.88090 q^{19} +0.700420 q^{21} +4.60792 q^{22} -0.345110 q^{23} -1.06138 q^{24} -2.02956 q^{27} -1.19792 q^{28} +3.00000 q^{29} +1.18048 q^{31} +3.21689 q^{32} +1.33934 q^{33} -6.47114 q^{34} +1.70042 q^{36} +5.45014 q^{37} +6.98259 q^{38} +0.180479 q^{41} +0.831632 q^{42} -1.33934 q^{43} -2.29066 q^{44} -0.409761 q^{46} +12.2807 q^{47} -0.852815 q^{48} -2.88090 q^{49} -1.88090 q^{51} +2.42636 q^{53} -2.40976 q^{54} -6.24186 q^{56} +2.02956 q^{57} +3.56200 q^{58} -7.06138 q^{59} +6.76180 q^{61} +1.40162 q^{62} -5.84695 q^{63} +8.76180 q^{64} +1.59024 q^{66} -4.40422 q^{67} +3.21689 q^{68} -0.119101 q^{69} -1.88090 q^{71} +8.86014 q^{72} +8.86014 q^{73} +6.47114 q^{74} -3.47114 q^{76} +7.87651 q^{77} +11.1805 q^{79} +7.94228 q^{81} +0.214289 q^{82} +7.83540 q^{83} -0.413416 q^{84} -1.59024 q^{86} +1.03533 q^{87} -11.9356 q^{88} +12.2419 q^{89} +0.203698 q^{92} +0.407395 q^{93} +14.5813 q^{94} +1.11018 q^{96} -5.80585 q^{97} -3.42059 q^{98} -11.1805 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^4 + 10 * q^6 + 6 * q^9	$ 6 q + 4 q^{4} + 10 q^{6} + 6 q^{9} + 22 q^{14} + 16 q^{16} + 12 q^{19} - 4 q^{21} + 32 q^{24} + 18 q^{29} - 8 q^{31} - 8 q^{34} + 2 q^{36} - 14 q^{41} + 2 q^{44} - 10 q^{46} + 6 q^{49} + 12 q^{51} - 22 q^{54} + 16 q^{56} - 4 q^{59} - 6 q^{61} + 6 q^{64} + 2 q^{66} - 24 q^{69} + 12 q^{71} + 8 q^{74} + 10 q^{76} + 52 q^{79} - 14 q^{81} + 90 q^{84} - 2 q^{86} + 20 q^{89} + 56 q^{94} + 6 q^{96} - 52 q^{99}+O(q^{100}) $ 6 * q + 4 * q^4 + 10 * q^6 + 6 * q^9 + 22 * q^14 + 16 * q^16 + 12 * q^19 - 4 * q^21 + 32 * q^24 + 18 * q^29 - 8 * q^31 - 8 * q^34 + 2 * q^36 - 14 * q^41 + 2 * q^44 - 10 * q^46 + 6 * q^49 + 12 * q^51 - 22 * q^54 + 16 * q^56 - 4 * q^59 - 6 * q^61 + 6 * q^64 + 2 * q^66 - 24 * q^69 + 12 * q^71 + 8 * q^74 + 10 * q^76 + 52 * q^79 - 14 * q^81 + 90 * q^84 - 2 * q^86 + 20 * q^89 + 56 * q^94 + 6 * q^96 - 52 * q^99

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 $

$2$	1.18733	0.839571	0.419786	−	0.907623i	$-0.362105\pi$
$2$	1.18733	0.839571	0.419786	+	0.907623i	$0.362105\pi$
$3$	0.345110	0.199249	0.0996247	−	0.995025i	$-0.468236\pi$
$3$	0.345110	0.199249	0.0996247	+	0.995025i	$0.468236\pi$
$4$	−0.590239	−0.295120
$5$	0	0
$5$	0	0
$6$	0.409761	0.167284
$7$	2.02956	0.767100	0.383550	−	0.923520i	$-0.374701\pi$
$7$	2.02956	0.767100	0.383550	+	0.923520i	$0.374701\pi$
$8$	−3.07548	−1.08735
$9$	−2.88090	−0.960300
$10$	0	0
$11$	3.88090	1.17014	0.585068	−	0.810985i	$-0.301068\pi$
$11$	3.88090	1.17014	0.585068	+	0.810985i	$0.301068\pi$
$12$	−0.203698	−0.0588024
$13$	0	0
$13$	0	0
$14$	2.40976	0.644036
$15$	0	0
$16$	−2.47114	−0.617785
$17$	−5.45014	−1.32185	−0.660927	−	0.750450i	$-0.729837\pi$
$17$	−5.45014	−1.32185	−0.660927	+	0.750450i	$0.729837\pi$
$18$	−3.42059	−0.806240
$19$	5.88090	1.34917	0.674585	−	0.738197i	$-0.264322\pi$
$19$	5.88090	1.34917	0.674585	+	0.738197i	$0.264322\pi$
$20$	0	0
$21$	0.700420	0.152844
$22$	4.60792	0.982412
$23$	−0.345110	−0.0719604	−0.0359802	−	0.999353i	$-0.511455\pi$
$23$	−0.345110	−0.0719604	−0.0359802	+	0.999353i	$0.511455\pi$
$24$	−1.06138	−0.216653
$25$	0	0
$26$	0	0
$27$	−2.02956	−0.390588
$28$	−1.19792	−0.226386
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	1.18048	0.212020	0.106010	−	0.994365i	$-0.466192\pi$
$31$	1.18048	0.212020	0.106010	+	0.994365i	$0.466192\pi$
$32$	3.21689	0.568671
$33$	1.33934	0.233149
$34$	−6.47114	−1.10979
$35$	0	0
$36$	1.70042	0.283403
$37$	5.45014	0.895998	0.447999	−	0.894034i	$-0.352137\pi$
$37$	5.45014	0.895998	0.447999	+	0.894034i	$0.352137\pi$
$38$	6.98259	1.13273
$39$	0	0
$40$	0	0
$41$	0.180479	0.0281861	0.0140930	−	0.999901i	$-0.495514\pi$
$41$	0.180479	0.0281861	0.0140930	+	0.999901i	$0.495514\pi$
$42$	0.831632	0.128324
$43$	−1.33934	−0.204247	−0.102123	−	0.994772i	$-0.532564\pi$
$43$	−1.33934	−0.204247	−0.102123	+	0.994772i	$0.532564\pi$
$44$	−2.29066	−0.345330
$45$	0	0
$46$	−0.409761	−0.0604159
$47$	12.2807	1.79133	0.895664	−	0.444731i	$-0.146701\pi$
$47$	12.2807	1.79133	0.895664	+	0.444731i	$0.146701\pi$
$48$	−0.852815	−0.123093
$49$	−2.88090	−0.411557
$50$	0	0
$51$	−1.88090	−0.263379
$52$	0	0
$53$	2.42636	0.333286	0.166643	−	0.986017i	$-0.446707\pi$
$53$	2.42636	0.333286	0.166643	+	0.986017i	$0.446707\pi$
$54$	−2.40976	−0.327927
$55$	0	0
$56$	−6.24186	−0.834103
$57$	2.02956	0.268821
$58$	3.56200	0.467714
$59$	−7.06138	−0.919313	−0.459657	−	0.888097i	$-0.652027\pi$
$59$	−7.06138	−0.919313	−0.459657	+	0.888097i	$0.652027\pi$
$60$	0	0
$61$	6.76180	0.865760	0.432880	−	0.901452i	$-0.357497\pi$
$61$	6.76180	0.865760	0.432880	+	0.901452i	$0.357497\pi$
$62$	1.40162	0.178006
$63$	−5.84695	−0.736646
$64$	8.76180	1.09522
$65$	0	0
$66$	1.59024	0.195745
$67$	−4.40422	−0.538062	−0.269031	−	0.963132i	$-0.586703\pi$
$67$	−4.40422	−0.538062	−0.269031	+	0.963132i	$0.586703\pi$
$68$	3.21689	0.390105
$69$	−0.119101	−0.0143381
$70$	0	0
$71$	−1.88090	−0.223222	−0.111611	−	0.993752i	$-0.535601\pi$
$71$	−1.88090	−0.223222	−0.111611	+	0.993752i	$0.535601\pi$
$72$	8.86014	1.04418
$73$	8.86014	1.03700	0.518501	−	0.855077i	$-0.326490\pi$
$73$	8.86014	1.03700	0.518501	+	0.855077i	$0.326490\pi$
$74$	6.47114	0.752255
$75$	0	0
$76$	−3.47114	−0.398167
$77$	7.87651	0.897611
$78$	0	0
$79$	11.1805	1.25790	0.628951	−	0.777445i	$-0.283485\pi$
$79$	11.1805	1.25790	0.628951	+	0.777445i	$0.283485\pi$
$80$	0	0
$81$	7.94228	0.882475
$82$	0.214289	0.0236642
$83$	7.83540	0.860047	0.430024	−	0.902818i	$-0.358505\pi$
$83$	7.83540	0.860047	0.430024	+	0.902818i	$0.358505\pi$
$84$	−0.413416	−0.0451073
$85$	0	0
$86$	−1.59024	−0.171480
$87$	1.03533	0.110999
$88$	−11.9356	−1.27234
$89$	12.2419	1.29763	0.648817	−	0.760944i	$-0.275264\pi$
$89$	12.2419	1.29763	0.648817	+	0.760944i	$0.275264\pi$
$90$	0	0
$91$	0	0
$92$	0.203698	0.0212369
$93$	0.407395	0.0422449
$94$	14.5813	1.50395
$95$	0	0
$96$	1.11018	0.113307
$97$	−5.80585	−0.589494	−0.294747	−	0.955575i	$-0.595235\pi$
$97$	−5.80585	−0.589494	−0.294747	+	0.955575i	$0.595235\pi$
$98$	−3.42059	−0.345532
$99$	−11.1805	−1.12368
$100$	0	0
$101$	−5.94228	−0.591279	−0.295639	−	0.955300i	$-0.595533\pi$
$101$	−5.94228	−0.591279	−0.295639	+	0.955300i	$0.595533\pi$
$102$	−2.23325	−0.221125
$103$	−6.43378	−0.633939	−0.316970	−	0.948436i	$-0.602665\pi$
$103$	−6.43378	−0.633939	−0.316970	+	0.948436i	$0.602665\pi$
$104$	0	0
$105$	0	0
$106$	2.88090	0.279818
$107$	17.6792	1.70911	0.854555	−	0.519360i	$-0.173830\pi$
$107$	17.6792	1.70911	0.854555	+	0.519360i	$0.173830\pi$
$108$	1.19792	0.115270
$109$	5.76180	0.551880	0.275940	−	0.961175i	$-0.411011\pi$
$109$	5.76180	0.551880	0.275940	+	0.961175i	$0.411011\pi$
$110$	0	0
$111$	1.88090	0.178527
$112$	−5.01532	−0.473903
$113$	−4.75992	−0.447776	−0.223888	−	0.974615i	$-0.571875\pi$
$113$	−4.75992	−0.447776	−0.223888	+	0.974615i	$0.571875\pi$
$114$	2.40976	0.225695
$115$	0	0
$116$	−1.77072	−0.164407
$117$	0	0
$118$	−8.38421	−0.771829
$119$	−11.0614	−1.01399
$120$	0	0
$121$	4.06138	0.369216
$122$	8.02851	0.726867
$123$	0.0622851	0.00561605
$124$	−0.696765	−0.0625714
$125$	0	0
$126$	−6.94228	−0.618467
$127$	−16.7061	−1.48243	−0.741215	−	0.671268i	$-0.765750\pi$
$127$	−16.7061	−1.48243	−0.741215	+	0.671268i	$0.765750\pi$
$128$	3.96940	0.350848
$129$	−0.462218	−0.0406961
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	−0.790529	−0.0688068
$133$	11.9356	1.03495
$134$	−5.22928	−0.451741
$135$	0	0
$136$	16.7618	1.43731
$137$	1.97786	0.168980	0.0844901	−	0.996424i	$-0.473074\pi$
$137$	1.97786	0.168980	0.0844901	+	0.996424i	$0.473074\pi$
$138$	−0.141412	−0.0120378
$139$	8.70042	0.737960	0.368980	−	0.929437i	$-0.379707\pi$
$139$	8.70042	0.737960	0.368980	+	0.929437i	$0.379707\pi$
$140$	0	0
$141$	4.23820	0.356921
$142$	−2.23325	−0.187411
$143$	0	0
$144$	7.11910	0.593258
$145$	0	0
$146$	10.5199	0.870637
$147$	−0.994227	−0.0820025
$148$	−3.21689	−0.264427
$149$	22.3032	1.82715	0.913576	−	0.406668i	$-0.133309\pi$
$149$	22.3032	1.82715	0.913576	+	0.406668i	$0.133309\pi$
$150$	0	0
$151$	−19.1626	−1.55943	−0.779717	−	0.626132i	$-0.784637\pi$
$151$	−19.1626	−1.55943	−0.779717	+	0.626132i	$0.784637\pi$
$152$	−18.0866	−1.46701
$153$	15.7013	1.26938
$154$	9.35204	0.753609
$155$	0	0
$156$	0	0
$157$	−6.20265	−0.495025	−0.247513	−	0.968885i	$-0.579613\pi$
$157$	−6.20265	−0.495025	−0.247513	+	0.968885i	$0.579613\pi$
$158$	13.2750	1.05610
$159$	0.837361	0.0664071
$160$	0	0
$161$	−0.700420	−0.0552008
$162$	9.43013	0.740901
$163$	11.9356	0.934870	0.467435	−	0.884027i	$-0.345178\pi$
$163$	11.9356	0.934870	0.467435	+	0.884027i	$0.345178\pi$
$164$	−0.106526	−0.00831826
$165$	0	0
$166$	9.30323	0.722071
$167$	−2.02956	−0.157052	−0.0785259	−	0.996912i	$-0.525021\pi$
$167$	−2.02956	−0.157052	−0.0785259	+	0.996912i	$0.525021\pi$
$168$	−2.15413	−0.166195
$169$	0	0
$170$	0	0
$171$	−16.9423	−1.29561
$172$	0.790529	0.0602773
$173$	1.33934	0.101828	0.0509139	−	0.998703i	$-0.483787\pi$
$173$	1.33934	0.101828	0.0509139	+	0.998703i	$0.483787\pi$
$174$	1.22928	0.0931916
$175$	0	0
$176$	−9.59024	−0.722891
$177$	−2.43695	−0.183173
$178$	14.5352	1.08946
$179$	20.2240	1.51161	0.755807	−	0.654794i	$-0.227245\pi$
$179$	20.2240	1.51161	0.755807	+	0.654794i	$0.227245\pi$
$180$	0	0
$181$	19.8232	1.47345	0.736723	−	0.676195i	$-0.236372\pi$
$181$	19.8232	1.47345	0.736723	+	0.676195i	$0.236372\pi$
$182$	0	0
$183$	2.33356	0.172502
$184$	1.06138	0.0782458
$185$	0	0
$186$	0.483714	0.0354676
$187$	−21.1515	−1.54675
$188$	−7.24857	−0.528656
$189$	−4.11910	−0.299621
$190$	0	0
$191$	1.53778	0.111270	0.0556350	−	0.998451i	$-0.482282\pi$
$191$	1.53778	0.111270	0.0556350	+	0.998451i	$0.482282\pi$
$192$	3.02378	0.218223
$193$	21.2032	1.52624	0.763118	−	0.646259i	$-0.223667\pi$
$193$	21.2032	1.52624	0.763118	+	0.646259i	$0.223667\pi$
$194$	−6.89347	−0.494923
$195$	0	0
$196$	1.70042	0.121459
$197$	−9.25695	−0.659530	−0.329765	−	0.944063i	$-0.606970\pi$
$197$	−9.25695	−0.659530	−0.329765	+	0.944063i	$0.606970\pi$
$198$	−13.2750	−0.943410
$199$	−17.4045	−1.23377	−0.616886	−	0.787053i	$-0.711606\pi$
$199$	−17.4045	−1.23377	−0.616886	+	0.787053i	$0.711606\pi$
$200$	0	0
$201$	−1.51994	−0.107208
$202$	−7.05546	−0.496421
$203$	6.08867	0.427341
$204$	1.11018	0.0777282
$205$	0	0
$206$	−7.63904	−0.532237
$207$	0.994227	0.0691036
$208$	0	0
$209$	22.8232	1.57871
$210$	0	0
$211$	−7.28174	−0.501296	−0.250648	−	0.968078i	$-0.580644\pi$
$211$	−7.28174	−0.501296	−0.250648	+	0.968078i	$0.580644\pi$
$212$	−1.43213	−0.0983594
$213$	−0.649117	−0.0444768
$214$	20.9911	1.43492
$215$	0	0
$216$	6.24186	0.424705
$217$	2.39585	0.162641
$218$	6.84118	0.463343
$219$	3.05772	0.206622
$220$	0	0
$221$	0	0
$222$	2.23325	0.149886
$223$	−19.4670	−1.30361	−0.651804	−	0.758388i	$-0.725987\pi$
$223$	−19.4670	−1.30361	−0.651804	+	0.758388i	$0.725987\pi$
$224$	6.52886	0.436228
$225$	0	0
$226$	−5.65162	−0.375940
$227$	4.81162	0.319358	0.159679	−	0.987169i	$-0.448954\pi$
$227$	4.81162	0.319358	0.159679	+	0.987169i	$0.448954\pi$
$228$	−1.19792	−0.0793345
$229$	1.52360	0.100682	0.0503410	−	0.998732i	$-0.483969\pi$
$229$	1.52360	0.100682	0.0503410	+	0.998732i	$0.483969\pi$
$230$	0	0
$231$	2.71826	0.178848
$232$	−9.22643	−0.605745
$233$	−13.9652	−0.914889	−0.457445	−	0.889238i	$-0.651235\pi$
$233$	−13.9652	−0.914889	−0.457445	+	0.889238i	$0.651235\pi$
$234$	0	0
$235$	0	0
$236$	4.16790	0.271307
$237$	3.85849	0.250636
$238$	−13.1335	−0.851321
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−17.4659	−1.12508	−0.562538	−	0.826771i	$-0.690175\pi$
$241$	−17.4659	−1.12508	−0.562538	+	0.826771i	$0.690175\pi$
$242$	4.82221	0.309983
$243$	8.82963	0.566421
$244$	−3.99108	−0.255503
$245$	0	0
$246$	0.0739531	0.00471508
$247$	0	0
$248$	−3.63054	−0.230539
$249$	2.70408	0.171364
$250$	0	0
$251$	9.29958	0.586984	0.293492	−	0.955961i	$-0.405183\pi$
$251$	9.29958	0.586984	0.293492	+	0.955961i	$0.405183\pi$
$252$	3.45110	0.217399
$253$	−1.33934	−0.0842034
$254$	−19.8358	−1.24461
$255$	0	0
$256$	−12.8106	−0.800663
$257$	10.8897	0.679281	0.339640	−	0.940555i	$-0.389695\pi$
$257$	10.8897	0.679281	0.339640	+	0.940555i	$0.389695\pi$
$258$	−0.548807	−0.0341673
$259$	11.0614	0.687321
$260$	0	0
$261$	−8.64270	−0.534970
$262$	11.8733	0.733537
$263$	−13.4406	−0.828785	−0.414392	−	0.910098i	$-0.636006\pi$
$263$	−13.4406	−0.828785	−0.414392	+	0.910098i	$0.636006\pi$
$264$	−4.11910	−0.253513
$265$	0	0
$266$	14.1716	0.868914
$267$	4.22479	0.258553
$268$	2.59955	0.158793
$269$	3.66054	0.223187	0.111593	−	0.993754i	$-0.464405\pi$
$269$	3.66054	0.223187	0.111593	+	0.993754i	$0.464405\pi$
$270$	0	0
$271$	22.0037	1.33663	0.668313	−	0.743880i	$-0.267017\pi$
$271$	22.0037	1.33663	0.668313	+	0.743880i	$0.267017\pi$
$272$	13.4681	0.816621
$273$	0	0
$274$	2.34838	0.141871
$275$	0	0
$276$	0.0702980	0.00423144
$277$	9.89547	0.594561	0.297281	−	0.954790i	$-0.403920\pi$
$277$	9.89547	0.594561	0.297281	+	0.954790i	$0.403920\pi$
$278$	10.3303	0.619570
$279$	−3.40084	−0.203603
$280$	0	0
$281$	4.06138	0.242281	0.121141	−	0.992635i	$-0.461345\pi$
$281$	4.06138	0.242281	0.121141	+	0.992635i	$0.461345\pi$
$282$	5.03216	0.299661
$283$	−6.08867	−0.361934	−0.180967	−	0.983489i	$-0.557923\pi$
$283$	−6.08867	−0.361934	−0.180967	+	0.983489i	$0.557923\pi$
$284$	1.11018	0.0658771
$285$	0	0
$286$	0	0
$287$	0.366292	0.0216215
$288$	−9.26754	−0.546095
$289$	12.7041	0.747299
$290$	0	0
$291$	−2.00366	−0.117456
$292$	−5.22960	−0.306039
$293$	−9.79208	−0.572060	−0.286030	−	0.958221i	$-0.592336\pi$
$293$	−9.79208	−0.572060	−0.286030	+	0.958221i	$0.592336\pi$
$294$	−1.18048	−0.0688469
$295$	0	0
$296$	−16.7618	−0.974260
$297$	−7.87651	−0.457041
$298$	26.4814	1.53402
$299$	0	0
$300$	0	0
$301$	−2.71826	−0.156678
$302$	−22.7524	−1.30926
$303$	−2.05074	−0.117812
$304$	−14.5325	−0.833497
$305$	0	0
$306$	18.6427	1.06573
$307$	−22.1046	−1.26158	−0.630788	−	0.775955i	$-0.717268\pi$
$307$	−22.1046	−1.26158	−0.630788	+	0.775955i	$0.717268\pi$
$308$	−4.64902	−0.264903
$309$	−2.22036	−0.126312
$310$	0	0
$311$	7.63904	0.433170	0.216585	−	0.976264i	$-0.430508\pi$
$311$	7.63904	0.433170	0.216585	+	0.976264i	$0.430508\pi$
$312$	0	0
$313$	26.1425	1.47766	0.738831	−	0.673891i	$-0.235378\pi$
$313$	26.1425	1.47766	0.738831	+	0.673891i	$0.235378\pi$
$314$	−7.36461	−0.415609
$315$	0	0
$316$	−6.59916	−0.371232
$317$	11.8428	0.665159	0.332580	−	0.943075i	$-0.392081\pi$
$317$	11.8428	0.665159	0.332580	+	0.943075i	$0.392081\pi$
$318$	0.994227	0.0557535
$319$	11.6427	0.651866
$320$	0	0
$321$	6.10126	0.340539
$322$	−0.831632	−0.0463451
$323$	−32.0518	−1.78341
$324$	−4.68785	−0.260436
$325$	0	0
$326$	14.1716	0.784890
$327$	1.98845	0.109962
$328$	−0.555059	−0.0306480
$329$	24.9244	1.37413
$330$	0	0
$331$	−12.7004	−0.698078	−0.349039	−	0.937108i	$-0.613492\pi$
$331$	−12.7004	−0.698078	−0.349039	+	0.937108i	$0.613492\pi$
$332$	−4.62476	−0.253817
$333$	−15.7013	−0.860427
$334$	−2.40976	−0.131856
$335$	0	0
$336$	−1.73084	−0.0944248
$337$	−15.2939	−0.833113	−0.416556	−	0.909110i	$-0.636763\pi$
$337$	−15.2939	−0.833113	−0.416556	+	0.909110i	$0.636763\pi$
$338$	0	0
$339$	−1.64270	−0.0892191
$340$	0	0
$341$	4.58132	0.248092
$342$	−20.1161	−1.08776
$343$	−20.0538	−1.08281
$344$	4.11910	0.222087
$345$	0	0
$346$	1.59024	0.0854918
$347$	12.2396	0.657058	0.328529	−	0.944494i	$-0.393447\pi$
$347$	12.2396	0.657058	0.328529	+	0.944494i	$0.393447\pi$
$348$	−0.611093	−0.0327580
$349$	18.7004	1.00101	0.500505	−	0.865733i	$-0.333148\pi$
$349$	18.7004	1.00101	0.500505	+	0.865733i	$0.333148\pi$
$350$	0	0
$351$	0	0
$352$	12.4844	0.665422
$353$	1.30883	0.0696617	0.0348309	−	0.999393i	$-0.488911\pi$
$353$	1.30883	0.0696617	0.0348309	+	0.999393i	$0.488911\pi$
$354$	−2.89347	−0.153786
$355$	0	0
$356$	−7.22563	−0.382957
$357$	−3.81739	−0.202038
$358$	24.0127	1.26911
$359$	−29.4082	−1.55210	−0.776051	−	0.630670i	$-0.782780\pi$
$359$	−29.4082	−1.55210	−0.776051	+	0.630670i	$0.782780\pi$
$360$	0	0
$361$	15.5850	0.820262
$362$	23.5367	1.23706
$363$	1.40162	0.0735661
$364$	0	0
$365$	0	0
$366$	2.77072	0.144828
$367$	−33.4322	−1.74515	−0.872573	−	0.488484i	$-0.837550\pi$
$367$	−33.4322	−1.74515	−0.872573	+	0.488484i	$0.837550\pi$
$368$	0.852815	0.0444560
$369$	−0.519941	−0.0270671
$370$	0	0
$371$	4.92444	0.255664
$372$	−0.240461	−0.0124673
$373$	−34.4781	−1.78521	−0.892604	−	0.450841i	$-0.851124\pi$
$373$	−34.4781	−1.78521	−0.892604	+	0.450841i	$0.851124\pi$
$374$	−25.1138	−1.29861
$375$	0	0
$376$	−37.7691	−1.94779
$377$	0	0
$378$	−4.89075	−0.251553
$379$	−17.4045	−0.894009	−0.447004	−	0.894532i	$-0.647509\pi$
$379$	−17.4045	−0.894009	−0.447004	+	0.894532i	$0.647509\pi$
$380$	0	0
$381$	−5.76545	−0.295373
$382$	1.82586	0.0934191
$383$	1.74673	0.0892538	0.0446269	−	0.999004i	$-0.485790\pi$
$383$	1.74673	0.0892538	0.0446269	+	0.999004i	$0.485790\pi$
$384$	1.36988	0.0699063
$385$	0	0
$386$	25.1752	1.28138
$387$	3.85849	0.196138
$388$	3.42684	0.173971
$389$	22.0435	1.11765	0.558826	−	0.829285i	$-0.311252\pi$
$389$	22.0435	1.11765	0.558826	+	0.829285i	$0.311252\pi$
$390$	0	0
$391$	1.88090	0.0951212
$392$	8.86014	0.447505
$393$	3.45110	0.174085
$394$	−10.9911	−0.553723
$395$	0	0
$396$	6.59916	0.331620
$397$	−22.5319	−1.13084	−0.565422	−	0.824802i	$-0.691287\pi$
$397$	−22.5319	−1.13084	−0.565422	+	0.824802i	$0.691287\pi$
$398$	−20.6649	−1.03584
$399$	4.11910	0.206213
$400$	0	0
$401$	−3.70408	−0.184973	−0.0924863	−	0.995714i	$-0.529481\pi$
$401$	−3.70408	−0.184973	−0.0924863	+	0.995714i	$0.529481\pi$
$402$	−1.80468	−0.0900091
$403$	0	0
$404$	3.50737	0.174498
$405$	0	0
$406$	7.22928	0.358783
$407$	21.1515	1.04844
$408$	5.78466	0.286384
$409$	−13.4837	−0.666727	−0.333363	−	0.942798i	$-0.608184\pi$
$409$	−13.4837	−0.666727	−0.333363	+	0.942798i	$0.608184\pi$
$410$	0	0
$411$	0.682580	0.0336692
$412$	3.79747	0.187088
$413$	−14.3315	−0.705205
$414$	1.18048	0.0580174
$415$	0	0
$416$	0	0
$417$	3.00260	0.147038
$418$	27.0987	1.32544
$419$	16.8232	0.821866	0.410933	−	0.911666i	$-0.365203\pi$
$419$	16.8232	0.821866	0.410933	+	0.911666i	$0.365203\pi$
$420$	0	0
$421$	17.1013	0.833464	0.416732	−	0.909029i	$-0.363175\pi$
$421$	17.1013	0.833464	0.416732	+	0.909029i	$0.363175\pi$
$422$	−8.64585	−0.420874
$423$	−35.3795	−1.72021
$424$	−7.46222	−0.362397
$425$	0	0
$426$	−0.770718	−0.0373414
$427$	13.7235	0.664124
$428$	−10.4349	−0.504392
$429$	0	0
$430$	0	0
$431$	9.66054	0.465332	0.232666	−	0.972557i	$-0.425255\pi$
$431$	9.66054	0.465332	0.232666	+	0.972557i	$0.425255\pi$
$432$	5.01532	0.241300
$433$	24.7727	1.19050	0.595249	−	0.803541i	$-0.297053\pi$
$433$	24.7727	1.19050	0.595249	+	0.803541i	$0.297053\pi$
$434$	2.84467	0.136549
$435$	0	0
$436$	−3.40084	−0.162871
$437$	−2.02956	−0.0970869
$438$	3.63054	0.173474
$439$	7.06138	0.337021	0.168511	−	0.985700i	$-0.446104\pi$
$439$	7.06138	0.337021	0.168511	+	0.985700i	$0.446104\pi$
$440$	0	0
$441$	8.29958	0.395218
$442$	0	0
$443$	−38.2438	−1.81702	−0.908509	−	0.417865i	$-0.862778\pi$
$443$	−38.2438	−1.81702	−0.908509	+	0.417865i	$0.862778\pi$
$444$	−1.11018	−0.0526869
$445$	0	0
$446$	−23.1138	−1.09447
$447$	7.69707	0.364059
$448$	17.7826	0.840147
$449$	−12.4801	−0.588970	−0.294485	−	0.955656i	$-0.595148\pi$
$449$	−12.4801	−0.588970	−0.294485	+	0.955656i	$0.595148\pi$
$450$	0	0
$451$	0.700420	0.0329815
$452$	2.80950	0.132148
$453$	−6.61322	−0.310716
$454$	5.71300	0.268124
$455$	0	0
$456$	−6.24186	−0.292302
$457$	8.23221	0.385086	0.192543	−	0.981289i	$-0.438326\pi$
$457$	8.23221	0.385086	0.192543	+	0.981289i	$0.438326\pi$
$458$	1.80902	0.0845298
$459$	11.0614	0.516301
$460$	0	0
$461$	−4.54144	−0.211516	−0.105758	−	0.994392i	$-0.533727\pi$
$461$	−4.54144	−0.211516	−0.105758	+	0.994392i	$0.533727\pi$
$462$	3.22748	0.150156
$463$	−1.98845	−0.0924113	−0.0462056	−	0.998932i	$-0.514713\pi$
$463$	−1.98845	−0.0924113	−0.0462056	+	0.998932i	$0.514713\pi$
$464$	−7.41342	−0.344159
$465$	0	0
$466$	−16.5813	−0.768115
$467$	−32.8043	−1.51800	−0.759000	−	0.651091i	$-0.774312\pi$
$467$	−32.8043	−1.51800	−0.759000	+	0.651091i	$0.774312\pi$
$468$	0	0
$469$	−8.93862	−0.412747
$470$	0	0
$471$	−2.14060	−0.0986335
$472$	21.7171	0.999611
$473$	−5.19783	−0.238997
$474$	4.58132	0.210427
$475$	0	0
$476$	6.52886	0.299250
$477$	−6.99010	−0.320055
$478$	4.74933	0.217229
$479$	−30.8053	−1.40753	−0.703766	−	0.710432i	$-0.748500\pi$
$479$	−30.8053	−1.40753	−0.703766	+	0.710432i	$0.748500\pi$
$480$	0	0
$481$	0	0
$482$	−20.7378	−0.944582
$483$	−0.241722	−0.0109987
$484$	−2.39719	−0.108963
$485$	0	0
$486$	10.4837	0.475551
$487$	−22.3251	−1.01165	−0.505824	−	0.862637i	$-0.668811\pi$
$487$	−22.3251	−1.01165	−0.505824	+	0.862637i	$0.668811\pi$
$488$	−20.7958	−0.941380
$489$	4.11910	0.186272
$490$	0	0
$491$	10.6826	0.482098	0.241049	−	0.970513i	$-0.422509\pi$
$491$	10.6826	0.482098	0.241049	+	0.970513i	$0.422509\pi$
$492$	−0.0367631	−0.00165741
$493$	−16.3504	−0.736386
$494$	0	0
$495$	0	0
$496$	−2.91713	−0.130983
$497$	−3.81739	−0.171233
$498$	3.21064	0.143872
$499$	18.8195	0.842477	0.421239	−	0.906950i	$-0.361595\pi$
$499$	18.8195	0.842477	0.421239	+	0.906950i	$0.361595\pi$
$500$	0	0
$501$	−0.700420	−0.0312925
$502$	11.0417	0.492815
$503$	−5.68128	−0.253316	−0.126658	−	0.991946i	$-0.540425\pi$
$503$	−5.68128	−0.253316	−0.126658	+	0.991946i	$0.540425\pi$
$504$	17.9822	0.800989
$505$	0	0
$506$	−1.59024	−0.0706948
$507$	0	0
$508$	9.86062	0.437494
$509$	27.9244	1.23773	0.618864	−	0.785498i	$-0.287593\pi$
$509$	27.9244	1.23773	0.618864	+	0.785498i	$0.287593\pi$
$510$	0	0
$511$	17.9822	0.795484
$512$	−23.1492	−1.02306
$513$	−11.9356	−0.526970
$514$	12.9297	0.570305
$515$	0	0
$516$	0.272820	0.0120102
$517$	47.6603	2.09610
$518$	13.1335	0.577055
$519$	0.462218	0.0202891
$520$	0	0
$521$	6.29958	0.275990	0.137995	−	0.990433i	$-0.455934\pi$
$521$	6.29958	0.275990	0.137995	+	0.990433i	$0.455934\pi$
$522$	−10.2618	−0.449145
$523$	22.8571	0.999471	0.499735	−	0.866178i	$-0.333431\pi$
$523$	22.8571	0.999471	0.499735	+	0.866178i	$0.333431\pi$
$524$	−5.90239	−0.257847
$525$	0	0
$526$	−15.9585	−0.695824
$527$	−6.43378	−0.280260
$528$	−3.30969	−0.144036
$529$	−22.8809	−0.994822
$530$	0	0
$531$	20.3431	0.882816
$532$	−7.04487	−0.305434
$533$	0	0
$534$	5.01623	0.217074
$535$	0	0
$536$	13.5451	0.585059
$537$	6.97951	0.301188
$538$	4.34628	0.187381
$539$	−11.1805	−0.481577
$540$	0	0
$541$	−9.48006	−0.407580	−0.203790	−	0.979015i	$-0.565326\pi$
$541$	−9.48006	−0.407580	−0.203790	+	0.979015i	$0.565326\pi$
$542$	26.1257	1.12219
$543$	6.84118	0.293583
$544$	−17.5325	−0.751700
$545$	0	0
$546$	0	0
$547$	33.3911	1.42770	0.713850	−	0.700299i	$-0.246950\pi$
$547$	33.3911	1.42770	0.713850	+	0.700299i	$0.246950\pi$
$548$	−1.16741	−0.0498694
$549$	−19.4801	−0.831389
$550$	0	0
$551$	17.6427	0.751604
$552$	0.366292	0.0155904
$553$	22.6914	0.964937
$554$	11.7492	0.499177
$555$	0	0
$556$	−5.13533	−0.217787
$557$	−37.7648	−1.60015	−0.800073	−	0.599903i	$-0.795206\pi$
$557$	−37.7648	−1.60015	−0.800073	+	0.599903i	$0.795206\pi$
$558$	−4.03793	−0.170939
$559$	0	0
$560$	0	0
$561$	−7.29958	−0.308188
$562$	4.82221	0.203413
$563$	25.9008	1.09159	0.545794	−	0.837919i	$-0.316228\pi$
$563$	25.9008	1.09159	0.545794	+	0.837919i	$0.316228\pi$
$564$	−2.50155	−0.105334
$565$	0	0
$566$	−7.22928	−0.303869
$567$	16.1193	0.676947
$568$	5.78466	0.242719
$569$	−21.5451	−0.903217	−0.451609	−	0.892216i	$-0.649150\pi$
$569$	−21.5451	−0.903217	−0.451609	+	0.892216i	$0.649150\pi$
$570$	0	0
$571$	−2.22036	−0.0929192	−0.0464596	−	0.998920i	$-0.514794\pi$
$571$	−2.22036	−0.0929192	−0.0464596	+	0.998920i	$0.514794\pi$
$572$	0	0
$573$	0.530704	0.0221705
$574$	0.434911	0.0181528
$575$	0	0
$576$	−25.2419	−1.05174
$577$	6.20265	0.258220	0.129110	−	0.991630i	$-0.458788\pi$
$577$	6.20265	0.258220	0.129110	+	0.991630i	$0.458788\pi$
$578$	15.0840	0.627411
$579$	7.31742	0.304102
$580$	0	0
$581$	15.9024	0.659742
$582$	−2.37901	−0.0986130
$583$	9.41646	0.389990
$584$	−27.2492	−1.12758
$585$	0	0
$586$	−11.6265	−0.480285
$587$	1.82894	0.0754883	0.0377442	−	0.999287i	$-0.487983\pi$
$587$	1.82894	0.0754883	0.0377442	+	0.999287i	$0.487983\pi$
$588$	0.586832	0.0242005
$589$	6.94228	0.286052
$590$	0	0
$591$	−3.19466	−0.131411
$592$	−13.4681	−0.553534
$593$	0.0728761	0.00299266	0.00149633	−	0.999999i	$-0.499524\pi$
$593$	0.0728761	0.00299266	0.00149633	+	0.999999i	$0.499524\pi$
$594$	−9.35204	−0.383719
$595$	0	0
$596$	−13.1642	−0.539229
$597$	−6.00646	−0.245828
$598$	0	0
$599$	−14.5813	−0.595777	−0.297888	−	0.954601i	$-0.596282\pi$
$599$	−14.5813	−0.595777	−0.297888	+	0.954601i	$0.596282\pi$
$600$	0	0
$601$	−44.4082	−1.81145	−0.905723	−	0.423870i	$-0.860671\pi$
$601$	−44.4082	−1.81145	−0.905723	+	0.423870i	$0.860671\pi$
$602$	−3.22748	−0.131542
$603$	12.6881	0.516700
$604$	11.3105	0.460220
$605$	0	0
$606$	−2.43491	−0.0989115
$607$	−36.2354	−1.47075	−0.735375	−	0.677660i	$-0.762994\pi$
$607$	−36.2354	−1.47075	−0.735375	+	0.677660i	$0.762994\pi$
$608$	18.9182	0.767235
$609$	2.10126	0.0851474
$610$	0	0
$611$	0	0
$612$	−9.26754	−0.374618
$613$	−3.35830	−0.135641	−0.0678203	−	0.997698i	$-0.521604\pi$
$613$	−3.35830	−0.135641	−0.0678203	+	0.997698i	$0.521604\pi$
$614$	−26.2455	−1.05918
$615$	0	0
$616$	−24.2240	−0.976013
$617$	21.1820	0.852754	0.426377	−	0.904546i	$-0.359790\pi$
$617$	21.1820	0.852754	0.426377	+	0.904546i	$0.359790\pi$
$618$	−2.63631	−0.106048
$619$	25.4082	1.02124	0.510620	−	0.859807i	$-0.329416\pi$
$619$	25.4082	1.02124	0.510620	+	0.859807i	$0.329416\pi$
$620$	0	0
$621$	0.700420	0.0281069
$622$	9.07009	0.363677
$623$	24.8455	0.995416
$624$	0	0
$625$	0	0
$626$	31.0399	1.24060
$627$	7.87651	0.314557
$628$	3.66105	0.146092
$629$	−29.7041	−1.18438
$630$	0	0
$631$	43.5451	1.73350	0.866751	−	0.498740i	$-0.166204\pi$
$631$	43.5451	1.73350	0.866751	+	0.498740i	$0.166204\pi$
$632$	−34.3853	−1.36777
$633$	−2.51300	−0.0998828
$634$	14.0614	0.558449
$635$	0	0
$636$	−0.494244	−0.0195980
$637$	0	0
$638$	13.8238	0.547288
$639$	5.41868	0.214360
$640$	0	0
$641$	48.2854	1.90716	0.953579	−	0.301142i	$-0.0973679\pi$
$641$	48.2854	1.90716	0.953579	+	0.301142i	$0.0973679\pi$
$642$	7.24423	0.285907
$643$	−42.3440	−1.66989	−0.834943	−	0.550337i	$-0.814499\pi$
$643$	−42.3440	−1.66989	−0.834943	+	0.550337i	$0.814499\pi$
$644$	0.413416	0.0162909
$645$	0	0
$646$	−38.0561	−1.49730
$647$	−34.4052	−1.35261	−0.676305	−	0.736622i	$-0.736420\pi$
$647$	−34.4052	−1.35261	−0.676305	+	0.736622i	$0.736420\pi$
$648$	−24.4263	−0.959556
$649$	−27.4045	−1.07572
$650$	0	0
$651$	0.826831	0.0324061
$652$	−7.04487	−0.275899
$653$	−14.3315	−0.560834	−0.280417	−	0.959878i	$-0.590473\pi$
$653$	−14.3315	−0.560834	−0.280417	+	0.959878i	$0.590473\pi$
$654$	2.36096	0.0923208
$655$	0	0
$656$	−0.445988	−0.0174129
$657$	−25.5252	−0.995832
$658$	29.5936	1.15368
$659$	22.8232	0.889065	0.444532	−	0.895763i	$-0.353370\pi$
$659$	22.8232	0.889065	0.444532	+	0.895763i	$0.353370\pi$
$660$	0	0
$661$	14.4187	0.560822	0.280411	−	0.959880i	$-0.409529\pi$
$661$	14.4187	0.560822	0.280411	+	0.959880i	$0.409529\pi$
$662$	−15.0796	−0.586087
$663$	0	0
$664$	−24.0976	−0.935168
$665$	0	0
$666$	−18.6427	−0.722390
$667$	−1.03533	−0.0400881
$668$	1.19792	0.0463491
$669$	−6.71826	−0.259743
$670$	0	0
$671$	26.2419	1.01306
$672$	2.25318	0.0869181
$673$	−34.1741	−1.31731	−0.658657	−	0.752443i	$-0.728875\pi$
$673$	−34.1741	−1.31731	−0.658657	+	0.752443i	$0.728875\pi$
$674$	−18.1590	−0.699458
$675$	0	0
$676$	0	0
$677$	5.84695	0.224716	0.112358	−	0.993668i	$-0.464160\pi$
$677$	5.84695	0.224716	0.112358	+	0.993668i	$0.464160\pi$
$678$	−1.95043	−0.0749058
$679$	−11.7833	−0.452201
$680$	0	0
$681$	1.66054	0.0636319
$682$	5.43955	0.208291
$683$	−11.3488	−0.434249	−0.217125	−	0.976144i	$-0.569668\pi$
$683$	−11.3488	−0.434249	−0.217125	+	0.976144i	$0.569668\pi$
$684$	10.0000	0.382360
$685$	0	0
$686$	−23.8106	−0.909093
$687$	0.525808	0.0200608
$688$	3.30969	0.126181
$689$	0	0
$690$	0	0
$691$	−18.8232	−0.716067	−0.358034	−	0.933709i	$-0.616553\pi$
$691$	−18.8232	−0.716067	−0.358034	+	0.933709i	$0.616553\pi$
$692$	−0.790529	−0.0300514
$693$	−22.6914	−0.861976
$694$	14.5325	0.551647
$695$	0	0
$696$	−3.18413	−0.120694
$697$	−0.983636	−0.0372579
$698$	22.2036	0.840420
$699$	−4.81952	−0.182291
$700$	0	0
$701$	19.1626	0.723763	0.361881	−	0.932224i	$-0.382135\pi$
$701$	19.1626	0.723763	0.361881	+	0.932224i	$0.382135\pi$
$702$	0	0
$703$	32.0518	1.20885
$704$	34.0037	1.28156
$705$	0	0
$706$	1.55401	0.0584860
$707$	−12.0602	−0.453570
$708$	1.43839	0.0540578
$709$	23.4837	0.881949	0.440975	−	0.897520i	$-0.354633\pi$
$709$	23.4837	0.881949	0.440975	+	0.897520i	$0.354633\pi$
$710$	0	0
$711$	−32.2098	−1.20796
$712$	−37.6496	−1.41098
$713$	−0.407395	−0.0152571
$714$	−4.53252	−0.169625
$715$	0	0
$716$	−11.9370	−0.446107
$717$	1.38044	0.0515535
$718$	−34.9173	−1.30310
$719$	−14.1086	−0.526161	−0.263080	−	0.964774i	$-0.584738\pi$
$719$	−14.1086	−0.526161	−0.263080	+	0.964774i	$0.584738\pi$
$720$	0	0
$721$	−13.0577	−0.486295
$722$	18.5046	0.688668
$723$	−6.02765	−0.224171
$724$	−11.7004	−0.434843
$725$	0	0
$726$	1.66419	0.0617640
$727$	25.3762	0.941153	0.470576	−	0.882359i	$-0.344046\pi$
$727$	25.3762	0.941153	0.470576	+	0.882359i	$0.344046\pi$
$728$	0	0
$729$	−20.7796	−0.769616
$730$	0	0
$731$	7.29958	0.269985
$732$	−1.37736	−0.0509087
$733$	−10.6692	−0.394074	−0.197037	−	0.980396i	$-0.563132\pi$
$733$	−10.6692	−0.394074	−0.197037	+	0.980396i	$0.563132\pi$
$734$	−39.6952	−1.46517
$735$	0	0
$736$	−1.11018	−0.0409218
$737$	−17.0923	−0.629605
$738$	−0.617344	−0.0227247
$739$	−1.41503	−0.0520526	−0.0260263	−	0.999661i	$-0.508285\pi$
$739$	−1.41503	−0.0520526	−0.0260263	+	0.999661i	$0.508285\pi$
$740$	0	0
$741$	0	0
$742$	5.84695	0.214648
$743$	−29.8777	−1.09611	−0.548053	−	0.836443i	$-0.684631\pi$
$743$	−29.8777	−1.09611	−0.548053	+	0.836443i	$0.684631\pi$
$744$	−1.25293	−0.0459348
$745$	0	0
$746$	−40.9370	−1.49881
$747$	−22.5730	−0.825903
$748$	12.4844	0.456476
$749$	35.8809	1.31106
$750$	0	0
$751$	−19.9858	−0.729293	−0.364646	−	0.931146i	$-0.618810\pi$
$751$	−19.9858	−0.729293	−0.364646	+	0.931146i	$0.618810\pi$
$752$	−30.3474	−1.10666
$753$	3.20938	0.116956
$754$	0	0
$755$	0	0
$756$	2.43126	0.0884239
$757$	−17.0923	−0.621232	−0.310616	−	0.950535i	$-0.600535\pi$
$757$	−17.0923	−0.621232	−0.310616	+	0.950535i	$0.600535\pi$
$758$	−20.6649	−0.750584
$759$	−0.462218	−0.0167775
$760$	0	0
$761$	−42.2240	−1.53062	−0.765310	−	0.643662i	$-0.777414\pi$
$761$	−42.2240	−1.53062	−0.765310	+	0.643662i	$0.777414\pi$
$762$	−6.84552	−0.247987
$763$	11.6939	0.423347
$764$	−0.907659	−0.0328380
$765$	0	0
$766$	2.07395	0.0749350
$767$	0	0
$768$	−4.42107	−0.159531
$769$	−23.7655	−0.857004	−0.428502	−	0.903541i	$-0.640959\pi$
$769$	−23.7655	−0.857004	−0.428502	+	0.903541i	$0.640959\pi$
$770$	0	0
$771$	3.75814	0.135346
$772$	−12.5149	−0.450422
$773$	−0.284086	−0.0102179	−0.00510894	−	0.999987i	$-0.501626\pi$
$773$	−0.284086	−0.0102179	−0.00510894	+	0.999987i	$0.501626\pi$
$774$	4.58132	0.164672
$775$	0	0
$776$	17.8557	0.640984
$777$	3.81739	0.136948
$778$	26.1730	0.938349
$779$	1.06138	0.0380278
$780$	0	0
$781$	−7.29958	−0.261199
$782$	2.23325	0.0798610
$783$	−6.08867	−0.217591
$784$	7.11910	0.254254
$785$	0	0
$786$	4.09761	0.146157
$787$	24.1341	0.860289	0.430145	−	0.902760i	$-0.358463\pi$
$787$	24.1341	0.860289	0.430145	+	0.902760i	$0.358463\pi$
$788$	5.46381	0.194640
$789$	−4.63849	−0.165135
$790$	0	0
$791$	−9.66054	−0.343489
$792$	34.3853	1.22183
$793$	0	0
$794$	−26.7529	−0.949424
$795$	0	0
$796$	10.2728	0.364110
$797$	−34.3442	−1.21653	−0.608267	−	0.793732i	$-0.708135\pi$
$797$	−34.3442	−1.21653	−0.608267	+	0.793732i	$0.708135\pi$
$798$	4.89075	0.173131
$799$	−66.9317	−2.36787
$800$	0	0
$801$	−35.2676	−1.24612
$802$	−4.39797	−0.155298
$803$	34.3853	1.21343
$804$	0.897129	0.0316393
$805$	0	0
$806$	0	0
$807$	1.26329	0.0444698
$808$	18.2753	0.642924
$809$	−47.6862	−1.67656	−0.838279	−	0.545241i	$-0.816438\pi$
$809$	−47.6862	−1.67656	−0.838279	+	0.545241i	$0.816438\pi$
$810$	0	0
$811$	24.5992	0.863793	0.431897	−	0.901923i	$-0.357845\pi$
$811$	24.5992	0.863793	0.431897	+	0.901923i	$0.357845\pi$
$812$	−3.59377	−0.126117
$813$	7.59368	0.266322
$814$	25.1138	0.880239
$815$	0	0
$816$	4.64796	0.162711
$817$	−7.87651	−0.275564
$818$	−16.0097	−0.559765
$819$	0	0
$820$	0	0
$821$	17.2996	0.603759	0.301880	−	0.953346i	$-0.402386\pi$
$821$	17.2996	0.603759	0.301880	+	0.953346i	$0.402386\pi$
$822$	0.810450	0.0282677
$823$	−32.5625	−1.13506	−0.567529	−	0.823353i	$-0.692101\pi$
$823$	−32.5625	−1.13506	−0.567529	+	0.823353i	$0.692101\pi$
$824$	19.7869	0.689311
$825$	0	0
$826$	−17.0162	−0.592070
$827$	−15.4702	−0.537951	−0.268976	−	0.963147i	$-0.586685\pi$
$827$	−15.4702	−0.537951	−0.268976	+	0.963147i	$0.586685\pi$
$828$	−0.586832	−0.0203938
$829$	−14.5236	−0.504425	−0.252213	−	0.967672i	$-0.581158\pi$
$829$	−14.5236	−0.504425	−0.252213	+	0.967672i	$0.581158\pi$
$830$	0	0
$831$	3.41503	0.118466
$832$	0	0
$833$	15.7013	0.544018
$834$	3.56509	0.123449
$835$	0	0
$836$	−13.4711	−0.465909
$837$	−2.39585	−0.0828127
$838$	19.9747	0.690015
$839$	0.815866	0.0281668	0.0140834	−	0.999901i	$-0.495517\pi$
$839$	0.815866	0.0281668	0.0140834	+	0.999901i	$0.495517\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	20.3049	0.699753
$843$	1.40162	0.0482744
$844$	4.29797	0.147942
$845$	0	0
$846$	−42.0073	−1.44424
$847$	8.24280	0.283226
$848$	−5.99587	−0.205899
$849$	−2.10126	−0.0721151
$850$	0	0
$851$	−1.88090	−0.0644764
$852$	0.383134	0.0131260
$853$	20.0856	0.687719	0.343859	−	0.939021i	$-0.388266\pi$
$853$	20.0856	0.687719	0.343859	+	0.939021i	$0.388266\pi$
$854$	16.2943	0.557580
$855$	0	0
$856$	−54.3719	−1.85839
$857$	40.7886	1.39331	0.696656	−	0.717406i	$-0.254671\pi$
$857$	40.7886	1.39331	0.696656	+	0.717406i	$0.254671\pi$
$858$	0	0
$859$	−40.1301	−1.36922	−0.684610	−	0.728909i	$-0.740028\pi$
$859$	−40.1301	−1.36922	−0.684610	+	0.728909i	$0.740028\pi$
$860$	0	0
$861$	0.126411	0.00430808
$862$	11.4703	0.390679
$863$	−20.8275	−0.708977	−0.354489	−	0.935060i	$-0.615345\pi$
$863$	−20.8275	−0.708977	−0.354489	+	0.935060i	$0.615345\pi$
$864$	−6.52886	−0.222116
$865$	0	0
$866$	29.4134	0.999509
$867$	4.38430	0.148899
$868$	−1.41412	−0.0479985
$869$	43.3903	1.47192
$870$	0	0
$871$	0	0
$872$	−17.7203	−0.600084
$873$	16.7261	0.566091
$874$	−2.40976	−0.0815114
$875$	0	0
$876$	−1.80479	−0.0609782
$877$	46.5944	1.57338	0.786691	−	0.617347i	$-0.211793\pi$
$877$	46.5944	1.57338	0.786691	+	0.617347i	$0.211793\pi$
$878$	8.38421	0.282953
$879$	−3.37935	−0.113982
$880$	0	0
$881$	23.8447	0.803347	0.401674	−	0.915783i	$-0.368429\pi$
$881$	23.8447	0.803347	0.401674	+	0.915783i	$0.368429\pi$
$882$	9.85437	0.331814
$883$	−37.2496	−1.25355	−0.626774	−	0.779201i	$-0.715625\pi$
$883$	−37.2496	−1.25355	−0.626774	+	0.779201i	$0.715625\pi$
$884$	0	0
$885$	0	0
$886$	−45.4082	−1.52552
$887$	28.5795	0.959605	0.479802	−	0.877377i	$-0.340708\pi$
$887$	28.5795	0.959605	0.479802	+	0.877377i	$0.340708\pi$
$888$	−5.78466	−0.194121
$889$	−33.9060	−1.13717
$890$	0	0
$891$	30.8232	1.03262
$892$	11.4902	0.384720
$893$	72.2217	2.41681
$894$	9.13899	0.305653
$895$	0	0
$896$	8.05611	0.269136
$897$	0	0
$898$	−14.8180	−0.494483
$899$	3.54144	0.118114
$900$	0	0
$901$	−13.2240	−0.440556
$902$	0.831632	0.0276903
$903$	−0.938099	−0.0312180
$904$	14.6390	0.486887
$905$	0	0
$906$	−7.85209	−0.260868
$907$	6.41386	0.212969	0.106484	−	0.994314i	$-0.466041\pi$
$907$	6.41386	0.212969	0.106484	+	0.994314i	$0.466041\pi$
$908$	−2.84001	−0.0942489
$909$	17.1191	0.567805
$910$	0	0
$911$	22.2204	0.736193	0.368097	−	0.929788i	$-0.380010\pi$
$911$	22.2204	0.736193	0.368097	+	0.929788i	$0.380010\pi$
$912$	−5.01532	−0.166074
$913$	30.4084	1.00637
$914$	9.77437	0.323308
$915$	0	0
$916$	−0.899287	−0.0297133
$917$	20.2956	0.670219
$918$	13.1335	0.433472
$919$	26.1264	0.861831	0.430915	−	0.902392i	$-0.358191\pi$
$919$	26.1264	0.861831	0.430915	+	0.902392i	$0.358191\pi$
$920$	0	0
$921$	−7.62851	−0.251368
$922$	−5.39220	−0.177583
$923$	0	0
$924$	−1.60442	−0.0527817
$925$	0	0
$926$	−2.36096	−0.0775859
$927$	18.5351	0.608772
$928$	9.65067	0.316799
$929$	−23.9423	−0.785521	−0.392760	−	0.919641i	$-0.628480\pi$
$929$	−23.9423	−0.785521	−0.392760	+	0.919641i	$0.628480\pi$
$930$	0	0
$931$	−16.9423	−0.555261
$932$	8.24280	0.270002
$933$	2.63631	0.0863089
$934$	−38.9496	−1.27447
$935$	0	0
$936$	0	0
$937$	−18.5046	−0.604518	−0.302259	−	0.953226i	$-0.597741\pi$
$937$	−18.5046	−0.604518	−0.302259	+	0.953226i	$0.597741\pi$
$938$	−10.6131	−0.346531
$939$	9.02204	0.294423
$940$	0	0
$941$	−14.3788	−0.468735	−0.234368	−	0.972148i	$-0.575302\pi$
$941$	−14.3788	−0.468735	−0.234368	+	0.972148i	$0.575302\pi$
$942$	−2.54160	−0.0828098
$943$	−0.0622851	−0.00202828
$944$	17.4496	0.567938
$945$	0	0
$946$	−6.17156	−0.200655
$947$	58.3188	1.89511	0.947554	−	0.319596i	$-0.103547\pi$
$947$	58.3188	1.89511	0.947554	+	0.319596i	$0.103547\pi$
$948$	−2.27744	−0.0739677
$949$	0	0
$950$	0	0
$951$	4.08708	0.132533
$952$	34.0190	1.10256
$953$	−13.7995	−0.447010	−0.223505	−	0.974703i	$-0.571750\pi$
$953$	−13.7995	−0.447010	−0.223505	+	0.974703i	$0.571750\pi$
$954$	−8.29958	−0.268709
$955$	0	0
$956$	−2.36096	−0.0763588
$957$	4.01801	0.129884
$958$	−36.5762	−1.18172
$959$	4.01419	0.129625
$960$	0	0
$961$	−29.6065	−0.955047
$962$	0	0
$963$	−50.9319	−1.64126
$964$	10.3090	0.332032
$965$	0	0
$966$	−0.287005	−0.00923422
$967$	30.3474	0.975906	0.487953	−	0.872870i	$-0.337744\pi$
$967$	30.3474	0.975906	0.487953	+	0.872870i	$0.337744\pi$
$968$	−12.4907	−0.401466
$969$	−11.0614	−0.355343
$970$	0	0
$971$	44.1013	1.41528	0.707638	−	0.706575i	$-0.249761\pi$
$971$	44.1013	1.41528	0.707638	+	0.706575i	$0.249761\pi$
$972$	−5.21160	−0.167162
$973$	17.6580	0.566089
$974$	−26.5074	−0.849351
$975$	0	0
$976$	−16.7093	−0.534853
$977$	−22.3220	−0.714143	−0.357071	−	0.934077i	$-0.616225\pi$
$977$	−22.3220	−0.714143	−0.357071	+	0.934077i	$0.616225\pi$
$978$	4.89075	0.156389
$979$	47.5094	1.51841
$980$	0	0
$981$	−16.5992	−0.529970
$982$	12.6838	0.404756
$983$	4.03793	0.128790	0.0643950	−	0.997924i	$-0.479488\pi$
$983$	4.03793	0.128790	0.0643950	+	0.997924i	$0.479488\pi$
$984$	−0.191556	−0.00610659
$985$	0	0
$986$	−19.4134	−0.618249
$987$	8.60167	0.273794
$988$	0	0
$989$	0.462218	0.0146977
$990$	0	0
$991$	29.6500	0.941864	0.470932	−	0.882170i	$-0.343918\pi$
$991$	29.6500	0.941864	0.470932	+	0.882170i	$0.343918\pi$
$992$	3.79747	0.120570
$993$	−4.38304	−0.139092
$994$	−4.53252	−0.143763
$995$	0	0
$996$	−1.59605	−0.0505728
$997$	22.1762	0.702327	0.351164	−	0.936314i	$-0.385786\pi$
$997$	22.1762	0.702327	0.351164	+	0.936314i	$0.385786\pi$
$998$	22.3450	0.707320
$999$	−11.0614	−0.349967

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.br.1.5		6
5.2	odd	4		845.2.b.d.339.5		6
5.3	odd	4		845.2.b.d.339.2		6
5.4	even	2	inner	4225.2.a.br.1.2		6
13.3	even	3		325.2.e.e.126.2		12
13.9	even	3		325.2.e.e.276.2		12
13.12	even	2		4225.2.a.bq.1.2		6
65.2	even	12		845.2.l.f.654.4		24
65.3	odd	12		65.2.n.a.9.5	yes	12
65.7	even	12		845.2.l.f.699.9		24
65.8	even	4		845.2.d.d.844.10		12
65.9	even	6		325.2.e.e.276.5		12
65.12	odd	4		845.2.b.e.339.2		6
65.17	odd	12		845.2.n.e.484.2		12
65.18	even	4		845.2.d.d.844.4		12
65.22	odd	12		65.2.n.a.29.5	yes	12
65.23	odd	12		845.2.n.e.529.2		12
65.28	even	12		845.2.l.f.654.9		24
65.29	even	6		325.2.e.e.126.5		12
65.32	even	12		845.2.l.f.699.3		24
65.33	even	12		845.2.l.f.699.4		24
65.37	even	12		845.2.l.f.654.10		24
65.38	odd	4		845.2.b.e.339.5		6
65.42	odd	12		65.2.n.a.9.2	✓	12
65.43	odd	12		845.2.n.e.484.5		12
65.47	even	4		845.2.d.d.844.3		12
65.48	odd	12		65.2.n.a.29.2	yes	12
65.57	even	4		845.2.d.d.844.9		12
65.58	even	12		845.2.l.f.699.10		24
65.62	odd	12		845.2.n.e.529.5		12
65.63	even	12		845.2.l.f.654.3		24
65.64	even	2		4225.2.a.bq.1.5		6
195.68	even	12		585.2.bs.a.334.2		12
195.107	even	12		585.2.bs.a.334.5		12
195.113	even	12		585.2.bs.a.289.5		12
195.152	even	12		585.2.bs.a.289.2		12
260.3	even	12		1040.2.dh.a.529.4		12
260.87	even	12		1040.2.dh.a.289.4		12
260.107	even	12		1040.2.dh.a.529.3		12
260.243	even	12		1040.2.dh.a.289.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.n.a.9.2	✓	12	65.42	odd	12
65.2.n.a.9.5	yes	12	65.3	odd	12
65.2.n.a.29.2	yes	12	65.48	odd	12
65.2.n.a.29.5	yes	12	65.22	odd	12
325.2.e.e.126.2		12	13.3	even	3
325.2.e.e.126.5		12	65.29	even	6
325.2.e.e.276.2		12	13.9	even	3
325.2.e.e.276.5		12	65.9	even	6
585.2.bs.a.289.2		12	195.152	even	12
585.2.bs.a.289.5		12	195.113	even	12
585.2.bs.a.334.2		12	195.68	even	12
585.2.bs.a.334.5		12	195.107	even	12
845.2.b.d.339.2		6	5.3	odd	4
845.2.b.d.339.5		6	5.2	odd	4
845.2.b.e.339.2		6	65.12	odd	4
845.2.b.e.339.5		6	65.38	odd	4
845.2.d.d.844.3		12	65.47	even	4
845.2.d.d.844.4		12	65.18	even	4
845.2.d.d.844.9		12	65.57	even	4
845.2.d.d.844.10		12	65.8	even	4
845.2.l.f.654.3		24	65.63	even	12
845.2.l.f.654.4		24	65.2	even	12
845.2.l.f.654.9		24	65.28	even	12
845.2.l.f.654.10		24	65.37	even	12
845.2.l.f.699.3		24	65.32	even	12
845.2.l.f.699.4		24	65.33	even	12
845.2.l.f.699.9		24	65.7	even	12
845.2.l.f.699.10		24	65.58	even	12
845.2.n.e.484.2		12	65.17	odd	12
845.2.n.e.484.5		12	65.43	odd	12
845.2.n.e.529.2		12	65.23	odd	12
845.2.n.e.529.5		12	65.62	odd	12
1040.2.dh.a.289.3		12	260.243	even	12
1040.2.dh.a.289.4		12	260.87	even	12
1040.2.dh.a.529.3		12	260.107	even	12
1040.2.dh.a.529.4		12	260.3	even	12
4225.2.a.bq.1.2		6	13.12	even	2
4225.2.a.bq.1.5		6	65.64	even	2
4225.2.a.br.1.2		6	5.4	even	2	inner
4225.2.a.br.1.5		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18733 q^{2} +0.345110 q^{3} -0.590239 q^{4} +0.409761 q^{6} +2.02956 q^{7} -3.07548 q^{8} -2.88090 q^{9} +O(q^{10})\)	\(q+1.18733 q^{2} +0.345110 q^{3} -0.590239 q^{4} +0.409761 q^{6} +2.02956 q^{7} -3.07548 q^{8} -2.88090 q^{9} +3.88090 q^{11} -0.203698 q^{12} +2.40976 q^{14} -2.47114 q^{16} -5.45014 q^{17} -3.42059 q^{18} +5.88090 q^{19} +0.700420 q^{21} +4.60792 q^{22} -0.345110 q^{23} -1.06138 q^{24} -2.02956 q^{27} -1.19792 q^{28} +3.00000 q^{29} +1.18048 q^{31} +3.21689 q^{32} +1.33934 q^{33} -6.47114 q^{34} +1.70042 q^{36} +5.45014 q^{37} +6.98259 q^{38} +0.180479 q^{41} +0.831632 q^{42} -1.33934 q^{43} -2.29066 q^{44} -0.409761 q^{46} +12.2807 q^{47} -0.852815 q^{48} -2.88090 q^{49} -1.88090 q^{51} +2.42636 q^{53} -2.40976 q^{54} -6.24186 q^{56} +2.02956 q^{57} +3.56200 q^{58} -7.06138 q^{59} +6.76180 q^{61} +1.40162 q^{62} -5.84695 q^{63} +8.76180 q^{64} +1.59024 q^{66} -4.40422 q^{67} +3.21689 q^{68} -0.119101 q^{69} -1.88090 q^{71} +8.86014 q^{72} +8.86014 q^{73} +6.47114 q^{74} -3.47114 q^{76} +7.87651 q^{77} +11.1805 q^{79} +7.94228 q^{81} +0.214289 q^{82} +7.83540 q^{83} -0.413416 q^{84} -1.59024 q^{86} +1.03533 q^{87} -11.9356 q^{88} +12.2419 q^{89} +0.203698 q^{92} +0.407395 q^{93} +14.5813 q^{94} +1.11018 q^{96} -5.80585 q^{97} -3.42059 q^{98} -11.1805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^4 + 10 * q^6 + 6 * q^9	\( 6 q + 4 q^{4} + 10 q^{6} + 6 q^{9} + 22 q^{14} + 16 q^{16} + 12 q^{19} - 4 q^{21} + 32 q^{24} + 18 q^{29} - 8 q^{31} - 8 q^{34} + 2 q^{36} - 14 q^{41} + 2 q^{44} - 10 q^{46} + 6 q^{49} + 12 q^{51} - 22 q^{54} + 16 q^{56} - 4 q^{59} - 6 q^{61} + 6 q^{64} + 2 q^{66} - 24 q^{69} + 12 q^{71} + 8 q^{74} + 10 q^{76} + 52 q^{79} - 14 q^{81} + 90 q^{84} - 2 q^{86} + 20 q^{89} + 56 q^{94} + 6 q^{96} - 52 q^{99}+O(q^{100}) \) 6 * q + 4 * q^4 + 10 * q^6 + 6 * q^9 + 22 * q^14 + 16 * q^16 + 12 * q^19 - 4 * q^21 + 32 * q^24 + 18 * q^29 - 8 * q^31 - 8 * q^34 + 2 * q^36 - 14 * q^41 + 2 * q^44 - 10 * q^46 + 6 * q^49 + 12 * q^51 - 22 * q^54 + 16 * q^56 - 4 * q^59 - 6 * q^61 + 6 * q^64 + 2 * q^66 - 24 * q^69 + 12 * q^71 + 8 * q^74 + 10 * q^76 + 52 * q^79 - 14 * q^81 + 90 * q^84 - 2 * q^86 + 20 * q^89 + 56 * q^94 + 6 * q^96 - 52 * q^99

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