LMFDB - Embedded newform 4225.2.a.bb.1.2

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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ 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)

$f(q)$	$=$	\(q-0.554958 q^{2} -0.801938 q^{3} -1.69202 q^{4} +0.445042 q^{6} -2.69202 q^{7} +2.04892 q^{8} -2.35690 q^{9} +1.19806 q^{11} +1.35690 q^{12} +1.49396 q^{14} +2.24698 q^{16} -1.13706 q^{17} +1.30798 q^{18} -1.93900 q^{19} +2.15883 q^{21} -0.664874 q^{22} +4.60388 q^{23} -1.64310 q^{24} +4.29590 q^{27} +4.55496 q^{28} -7.89977 q^{29} -5.89977 q^{31} -5.34481 q^{32} -0.960771 q^{33} +0.631023 q^{34} +3.98792 q^{36} +0.951083 q^{37} +1.07606 q^{38} -3.31767 q^{41} -1.19806 q^{42} -7.15883 q^{43} -2.02715 q^{44} -2.55496 q^{46} -7.69202 q^{47} -1.80194 q^{48} +0.246980 q^{49} +0.911854 q^{51} -5.87263 q^{53} -2.38404 q^{54} -5.51573 q^{56} +1.55496 q^{57} +4.38404 q^{58} +0.0120816 q^{59} -8.03684 q^{61} +3.27413 q^{62} +6.34481 q^{63} -1.52781 q^{64} +0.533188 q^{66} -9.25667 q^{67} +1.92394 q^{68} -3.69202 q^{69} +13.7409 q^{71} -4.82908 q^{72} +12.8170 q^{73} -0.527811 q^{74} +3.28083 q^{76} -3.22521 q^{77} +0.807315 q^{79} +3.62565 q^{81} +1.84117 q^{82} -16.3327 q^{83} -3.65279 q^{84} +3.97285 q^{86} +6.33513 q^{87} +2.45473 q^{88} +14.7289 q^{89} -7.78986 q^{92} +4.73125 q^{93} +4.26875 q^{94} +4.28621 q^{96} +3.13169 q^{97} -0.137063 q^{98} -2.82371 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 2 q^{3} + q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 8 q^{11} - 5 q^{14} + 2 q^{16} + 2 q^{17} + 9 q^{18} + 4 q^{19} - 2 q^{21} - 3 q^{22} + 5 q^{23} - 9 q^{24} - q^{27} + 14 q^{28} - q^{29}+ \cdots - q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 + 2 * q^3 + q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 8 * q^11 - 5 * q^14 + 2 * q^16 + 2 * q^17 + 9 * q^18 + 4 * q^19 - 2 * q^21 - 3 * q^22 + 5 * q^23 - 9 * q^24 - q^27 + 14 * q^28 - q^29 + 5 * q^31 + 7 * q^32 + 10 * q^33 - 13 * q^34 - 7 * q^36 + 12 * q^37 - 12 * q^38 + 7 * q^41 - 8 * q^42 - 13 * q^43 - 8 * q^46 - 18 * q^47 - q^48 - 4 * q^49 - q^51 - q^53 + 3 * q^54 - 4 * q^56 + 5 * q^57 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 + 21 * q^68 - 6 * q^69 + 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 21 * q^76 - 8 * q^77 - 5 * q^79 - q^81 + 14 * q^82 - 7 * q^83 + 7 * q^84 + 18 * q^86 + 18 * q^87 - 15 * q^88 + 11 * q^89 + 22 * q^93 + 5 * q^94 + 21 * q^96 + 7 * q^97 + 5 * q^98 - q^99

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 $

$2$	−0.554958	−0.392415	−0.196207	−	0.980562i	$-0.562863\pi$
$2$	−0.554958	−0.392415	−0.196207	+	0.980562i	$0.562863\pi$
$3$	−0.801938	−0.462999	−0.231499	−	0.972835i	$-0.574363\pi$
$3$	−0.801938	−0.462999	−0.231499	+	0.972835i	$0.574363\pi$
$4$	−1.69202	−0.846011
$5$	0	0
$5$	0	0
$6$	0.445042	0.181688
$7$	−2.69202	−1.01749	−0.508744	−	0.860918i	$-0.669890\pi$
$7$	−2.69202	−1.01749	−0.508744	+	0.860918i	$0.669890\pi$
$8$	2.04892	0.724402
$9$	−2.35690	−0.785632
$10$	0	0
$11$	1.19806	0.361229	0.180615	−	0.983554i	$-0.442191\pi$
$11$	1.19806	0.361229	0.180615	+	0.983554i	$0.442191\pi$
$12$	1.35690	0.391702
$13$	0	0
$13$	0	0
$14$	1.49396	0.399277
$15$	0	0
$16$	2.24698	0.561745
$17$	−1.13706	−0.275778	−0.137889	−	0.990448i	$-0.544032\pi$
$17$	−1.13706	−0.275778	−0.137889	+	0.990448i	$0.544032\pi$
$18$	1.30798	0.308293
$19$	−1.93900	−0.444837	−0.222419	−	0.974951i	$-0.571395\pi$
$19$	−1.93900	−0.444837	−0.222419	+	0.974951i	$0.571395\pi$
$20$	0	0
$21$	2.15883	0.471096
$22$	−0.664874	−0.141752
$23$	4.60388	0.959974	0.479987	−	0.877275i	$-0.340641\pi$
$23$	4.60388	0.959974	0.479987	+	0.877275i	$0.340641\pi$
$24$	−1.64310	−0.335397
$25$	0	0
$26$	0	0
$27$	4.29590	0.826746
$28$	4.55496	0.860806
$29$	−7.89977	−1.46695	−0.733475	−	0.679716i	$-0.762103\pi$
$29$	−7.89977	−1.46695	−0.733475	+	0.679716i	$0.762103\pi$
$30$	0	0
$31$	−5.89977	−1.05963	−0.529815	−	0.848113i	$-0.677739\pi$
$31$	−5.89977	−1.05963	−0.529815	+	0.848113i	$0.677739\pi$
$32$	−5.34481	−0.944839
$33$	−0.960771	−0.167249
$34$	0.631023	0.108219
$35$	0	0
$36$	3.98792	0.664653
$37$	0.951083	0.156357	0.0781785	−	0.996939i	$-0.475090\pi$
$37$	0.951083	0.156357	0.0781785	+	0.996939i	$0.475090\pi$
$38$	1.07606	0.174561
$39$	0	0
$40$	0	0
$41$	−3.31767	−0.518133	−0.259066	−	0.965860i	$-0.583415\pi$
$41$	−3.31767	−0.518133	−0.259066	+	0.965860i	$0.583415\pi$
$42$	−1.19806	−0.184865
$43$	−7.15883	−1.09171	−0.545856	−	0.837879i	$-0.683795\pi$
$43$	−7.15883	−1.09171	−0.545856	+	0.837879i	$0.683795\pi$
$44$	−2.02715	−0.305604
$45$	0	0
$46$	−2.55496	−0.376708
$47$	−7.69202	−1.12200	−0.560998	−	0.827817i	$-0.689583\pi$
$47$	−7.69202	−1.12200	−0.560998	+	0.827817i	$0.689583\pi$
$48$	−1.80194	−0.260087
$49$	0.246980	0.0352828
$50$	0	0
$51$	0.911854	0.127685
$52$	0	0
$53$	−5.87263	−0.806667	−0.403334	−	0.915053i	$-0.632149\pi$
$53$	−5.87263	−0.806667	−0.403334	+	0.915053i	$0.632149\pi$
$54$	−2.38404	−0.324427
$55$	0	0
$56$	−5.51573	−0.737070
$57$	1.55496	0.205959
$58$	4.38404	0.575653
$59$	0.0120816	0.00157289	0.000786444	−	1.00000i	$-0.499750\pi$
$59$	0.0120816	0.00157289	0.000786444		1.00000i	$0.499750\pi$
$60$	0	0
$61$	−8.03684	−1.02901	−0.514506	−	0.857487i	$-0.672025\pi$
$61$	−8.03684	−1.02901	−0.514506	+	0.857487i	$0.672025\pi$
$62$	3.27413	0.415815
$63$	6.34481	0.799371
$64$	−1.52781	−0.190976
$65$	0	0
$66$	0.533188	0.0656309
$67$	−9.25667	−1.13088	−0.565441	−	0.824789i	$-0.691294\pi$
$67$	−9.25667	−1.13088	−0.565441	+	0.824789i	$0.691294\pi$
$68$	1.92394	0.233311
$69$	−3.69202	−0.444467
$70$	0	0
$71$	13.7409	1.63075	0.815375	−	0.578934i	$-0.196531\pi$
$71$	13.7409	1.63075	0.815375	+	0.578934i	$0.196531\pi$
$72$	−4.82908	−0.569113
$73$	12.8170	1.50012	0.750058	−	0.661372i	$-0.230025\pi$
$73$	12.8170	1.50012	0.750058	+	0.661372i	$0.230025\pi$
$74$	−0.527811	−0.0613568
$75$	0	0
$76$	3.28083	0.376337
$77$	−3.22521	−0.367547
$78$	0	0
$79$	0.807315	0.0908300	0.0454150	−	0.998968i	$-0.485539\pi$
$79$	0.807315	0.0908300	0.0454150	+	0.998968i	$0.485539\pi$
$80$	0	0
$81$	3.62565	0.402850
$82$	1.84117	0.203323
$83$	−16.3327	−1.79275	−0.896375	−	0.443296i	$-0.853809\pi$
$83$	−16.3327	−1.79275	−0.896375	+	0.443296i	$0.853809\pi$
$84$	−3.65279	−0.398552
$85$	0	0
$86$	3.97285	0.428404
$87$	6.33513	0.679197
$88$	2.45473	0.261675
$89$	14.7289	1.56126	0.780628	−	0.624996i	$-0.214899\pi$
$89$	14.7289	1.56126	0.780628	+	0.624996i	$0.214899\pi$
$90$	0	0
$91$	0	0
$92$	−7.78986	−0.812149
$93$	4.73125	0.490608
$94$	4.26875	0.440288
$95$	0	0
$96$	4.28621	0.437459
$97$	3.13169	0.317975	0.158987	−	0.987281i	$-0.449177\pi$
$97$	3.13169	0.317975	0.158987	+	0.987281i	$0.449177\pi$
$98$	−0.137063	−0.0138455
$99$	−2.82371	−0.283793
$100$	0	0
$101$	5.29052	0.526426	0.263213	−	0.964738i	$-0.415218\pi$
$101$	5.29052	0.526426	0.263213	+	0.964738i	$0.415218\pi$
$102$	−0.506041	−0.0501055
$103$	13.5308	1.33323	0.666614	−	0.745403i	$-0.267743\pi$
$103$	13.5308	1.33323	0.666614	+	0.745403i	$0.267743\pi$
$104$	0	0
$105$	0	0
$106$	3.25906	0.316548
$107$	−5.63102	−0.544371	−0.272186	−	0.962245i	$-0.587747\pi$
$107$	−5.63102	−0.544371	−0.272186	+	0.962245i	$0.587747\pi$
$108$	−7.26875	−0.699436
$109$	−4.17629	−0.400016	−0.200008	−	0.979794i	$-0.564097\pi$
$109$	−4.17629	−0.400016	−0.200008	+	0.979794i	$0.564097\pi$
$110$	0	0
$111$	−0.762709	−0.0723931
$112$	−6.04892	−0.571569
$113$	−7.64310	−0.719003	−0.359501	−	0.933145i	$-0.617053\pi$
$113$	−7.64310	−0.719003	−0.359501	+	0.933145i	$0.617053\pi$
$114$	−0.862937	−0.0808214
$115$	0	0
$116$	13.3666	1.24106
$117$	0	0
$118$	−0.00670477	−0.000617224 0
$119$	3.06100	0.280601
$120$	0	0
$121$	−9.56465	−0.869513
$122$	4.46011	0.403799
$123$	2.66056	0.239895
$124$	9.98254	0.896459
$125$	0	0
$126$	−3.52111	−0.313685
$127$	−6.77777	−0.601430	−0.300715	−	0.953714i	$-0.597225\pi$
$127$	−6.77777	−0.601430	−0.300715	+	0.953714i	$0.597225\pi$
$128$	11.5375	1.01978
$129$	5.74094	0.505461
$130$	0	0
$131$	−13.6799	−1.19522	−0.597611	−	0.801786i	$-0.703883\pi$
$131$	−13.6799	−1.19522	−0.597611	+	0.801786i	$0.703883\pi$
$132$	1.62565	0.141494
$133$	5.21983	0.452617
$134$	5.13706	0.443775
$135$	0	0
$136$	−2.32975	−0.199774
$137$	12.9879	1.10963	0.554816	−	0.831973i	$-0.312789\pi$
$137$	12.9879	1.10963	0.554816	+	0.831973i	$0.312789\pi$
$138$	2.04892	0.174415
$139$	12.0465	1.02177	0.510886	−	0.859648i	$-0.329317\pi$
$139$	12.0465	1.02177	0.510886	+	0.859648i	$0.329317\pi$
$140$	0	0
$141$	6.16852	0.519483
$142$	−7.62565	−0.639930
$143$	0	0
$144$	−5.29590	−0.441325
$145$	0	0
$146$	−7.11290	−0.588668
$147$	−0.198062	−0.0163359
$148$	−1.60925	−0.132280
$149$	0.740939	0.0607001	0.0303500	−	0.999539i	$-0.490338\pi$
$149$	0.740939	0.0607001	0.0303500	+	0.999539i	$0.490338\pi$
$150$	0	0
$151$	19.0737	1.55219	0.776097	−	0.630614i	$-0.217197\pi$
$151$	19.0737	1.55219	0.776097	+	0.630614i	$0.217197\pi$
$152$	−3.97285	−0.322241
$153$	2.67994	0.216660
$154$	1.78986	0.144231
$155$	0	0
$156$	0	0
$157$	4.02177	0.320972	0.160486	−	0.987038i	$-0.448694\pi$
$157$	4.02177	0.320972	0.160486	+	0.987038i	$0.448694\pi$
$158$	−0.448026	−0.0356430
$159$	4.70948	0.373486
$160$	0	0
$161$	−12.3937	−0.976763
$162$	−2.01208	−0.158084
$163$	15.1371	1.18563	0.592813	−	0.805340i	$-0.298017\pi$
$163$	15.1371	1.18563	0.592813	+	0.805340i	$0.298017\pi$
$164$	5.61356	0.438346
$165$	0	0
$166$	9.06398	0.703502
$167$	−6.26337	−0.484674	−0.242337	−	0.970192i	$-0.577914\pi$
$167$	−6.26337	−0.484674	−0.242337	+	0.970192i	$0.577914\pi$
$168$	4.42327	0.341263
$169$	0	0
$170$	0	0
$171$	4.57002	0.349478
$172$	12.1129	0.923600
$173$	−16.3913	−1.24621	−0.623105	−	0.782138i	$-0.714129\pi$
$173$	−16.3913	−1.24621	−0.623105	+	0.782138i	$0.714129\pi$
$174$	−3.51573	−0.266527
$175$	0	0
$176$	2.69202	0.202919
$177$	−0.00968868	−0.000728246 0
$178$	−8.17390	−0.612660
$179$	−2.45473	−0.183475	−0.0917376	−	0.995783i	$-0.529242\pi$
$179$	−2.45473	−0.183475	−0.0917376	+	0.995783i	$0.529242\pi$
$180$	0	0
$181$	11.8073	0.877631	0.438815	−	0.898577i	$-0.355398\pi$
$181$	11.8073	0.877631	0.438815	+	0.898577i	$0.355398\pi$
$182$	0	0
$183$	6.44504	0.476431
$184$	9.43296	0.695407
$185$	0	0
$186$	−2.62565	−0.192522
$187$	−1.36227	−0.0996192
$188$	13.0151	0.949221
$189$	−11.5646	−0.841204
$190$	0	0
$191$	−8.99330	−0.650732	−0.325366	−	0.945588i	$-0.605488\pi$
$191$	−8.99330	−0.650732	−0.325366	+	0.945588i	$0.605488\pi$
$192$	1.22521	0.0884219
$193$	13.5254	0.973581	0.486790	−	0.873519i	$-0.338168\pi$
$193$	13.5254	0.973581	0.486790	+	0.873519i	$0.338168\pi$
$194$	−1.73795	−0.124778
$195$	0	0
$196$	−0.417895	−0.0298496
$197$	12.9758	0.924490	0.462245	−	0.886752i	$-0.347044\pi$
$197$	12.9758	0.924490	0.462245	+	0.886752i	$0.347044\pi$
$198$	1.56704	0.111365
$199$	−13.5864	−0.963116	−0.481558	−	0.876414i	$-0.659929\pi$
$199$	−13.5864	−0.963116	−0.481558	+	0.876414i	$0.659929\pi$
$200$	0	0
$201$	7.42327	0.523597
$202$	−2.93602	−0.206577
$203$	21.2664	1.49261
$204$	−1.54288	−0.108023
$205$	0	0
$206$	−7.50902	−0.523179
$207$	−10.8509	−0.754187
$208$	0	0
$209$	−2.32304	−0.160688
$210$	0	0
$211$	10.4601	0.720103	0.360052	−	0.932932i	$-0.382759\pi$
$211$	10.4601	0.720103	0.360052	+	0.932932i	$0.382759\pi$
$212$	9.93661	0.682449
$213$	−11.0194	−0.755035
$214$	3.12498	0.213619
$215$	0	0
$216$	8.80194	0.598896
$217$	15.8823	1.07816
$218$	2.31767	0.156972
$219$	−10.2784	−0.694553
$220$	0	0
$221$	0	0
$222$	0.423272	0.0284081
$223$	−11.4058	−0.763790	−0.381895	−	0.924206i	$-0.624728\pi$
$223$	−11.4058	−0.763790	−0.381895	+	0.924206i	$0.624728\pi$
$224$	14.3884	0.961362
$225$	0	0
$226$	4.24160	0.282147
$227$	10.6407	0.706249	0.353124	−	0.935576i	$-0.385119\pi$
$227$	10.6407	0.706249	0.353124	+	0.935576i	$0.385119\pi$
$228$	−2.63102	−0.174244
$229$	1.13946	0.0752974	0.0376487	−	0.999291i	$-0.488013\pi$
$229$	1.13946	0.0752974	0.0376487	+	0.999291i	$0.488013\pi$
$230$	0	0
$231$	2.58642	0.170174
$232$	−16.1860	−1.06266
$233$	10.8509	0.710863	0.355432	−	0.934702i	$-0.384334\pi$
$233$	10.8509	0.710863	0.355432	+	0.934702i	$0.384334\pi$
$234$	0	0
$235$	0	0
$236$	−0.0204423	−0.00133068
$237$	−0.647416	−0.0420542
$238$	−1.69873	−0.110112
$239$	11.9293	0.771643	0.385822	−	0.922573i	$-0.373918\pi$
$239$	11.9293	0.771643	0.385822	+	0.922573i	$0.373918\pi$
$240$	0	0
$241$	3.64848	0.235019	0.117510	−	0.993072i	$-0.462509\pi$
$241$	3.64848	0.235019	0.117510	+	0.993072i	$0.462509\pi$
$242$	5.30798	0.341210
$243$	−15.7952	−1.01326
$244$	13.5985	0.870555
$245$	0	0
$246$	−1.47650	−0.0941383
$247$	0	0
$248$	−12.0881	−0.767598
$249$	13.0978	0.830042
$250$	0	0
$251$	1.37329	0.0866813	0.0433406	−	0.999060i	$-0.486200\pi$
$251$	1.37329	0.0866813	0.0433406	+	0.999060i	$0.486200\pi$
$252$	−10.7356	−0.676277
$253$	5.51573	0.346771
$254$	3.76138	0.236010
$255$	0	0
$256$	−3.34721	−0.209200
$257$	−29.4359	−1.83616	−0.918082	−	0.396391i	$-0.870263\pi$
$257$	−29.4359	−1.83616	−0.918082	+	0.396391i	$0.870263\pi$
$258$	−3.18598	−0.198350
$259$	−2.56033	−0.159091
$260$	0	0
$261$	18.6189	1.15248
$262$	7.59179	0.469023
$263$	−10.6963	−0.659564	−0.329782	−	0.944057i	$-0.606975\pi$
$263$	−10.6963	−0.659564	−0.329782	+	0.944057i	$0.606975\pi$
$264$	−1.96854	−0.121155
$265$	0	0
$266$	−2.89679	−0.177613
$267$	−11.8116	−0.722860
$268$	15.6625	0.956738
$269$	−10.1860	−0.621050	−0.310525	−	0.950565i	$-0.600505\pi$
$269$	−10.1860	−0.621050	−0.310525	+	0.950565i	$0.600505\pi$
$270$	0	0
$271$	29.4523	1.78910	0.894551	−	0.446966i	$-0.147495\pi$
$271$	29.4523	1.78910	0.894551	+	0.446966i	$0.147495\pi$
$272$	−2.55496	−0.154917
$273$	0	0
$274$	−7.20775	−0.435436
$275$	0	0
$276$	6.24698	0.376024
$277$	10.2446	0.615538	0.307769	−	0.951461i	$-0.400418\pi$
$277$	10.2446	0.615538	0.307769	+	0.951461i	$0.400418\pi$
$278$	−6.68532	−0.400959
$279$	13.9051	0.832480
$280$	0	0
$281$	11.5646	0.689889	0.344944	−	0.938623i	$-0.387898\pi$
$281$	11.5646	0.689889	0.344944	+	0.938623i	$0.387898\pi$
$282$	−3.42327	−0.203853
$283$	30.7090	1.82546	0.912730	−	0.408562i	$-0.133970\pi$
$283$	30.7090	1.82546	0.912730	+	0.408562i	$0.133970\pi$
$284$	−23.2500	−1.37963
$285$	0	0
$286$	0	0
$287$	8.93123	0.527194
$288$	12.5972	0.742295
$289$	−15.7071	−0.923946
$290$	0	0
$291$	−2.51142	−0.147222
$292$	−21.6866	−1.26911
$293$	−18.6082	−1.08710	−0.543551	−	0.839376i	$-0.682921\pi$
$293$	−18.6082	−1.08710	−0.543551	+	0.839376i	$0.682921\pi$
$294$	0.109916	0.00641045
$295$	0	0
$296$	1.94869	0.113265
$297$	5.14675	0.298645
$298$	−0.411190	−0.0238196
$299$	0	0
$300$	0	0
$301$	19.2717	1.11080
$302$	−10.5851	−0.609103
$303$	−4.24267	−0.243735
$304$	−4.35690	−0.249885
$305$	0	0
$306$	−1.48725	−0.0850207
$307$	8.94438	0.510483	0.255241	−	0.966877i	$-0.417845\pi$
$307$	8.94438	0.510483	0.255241	+	0.966877i	$0.417845\pi$
$308$	5.45712	0.310948
$309$	−10.8509	−0.617284
$310$	0	0
$311$	21.0398	1.19306	0.596529	−	0.802591i	$-0.296546\pi$
$311$	21.0398	1.19306	0.596529	+	0.802591i	$0.296546\pi$
$312$	0	0
$313$	7.12737	0.402863	0.201432	−	0.979503i	$-0.435441\pi$
$313$	7.12737	0.402863	0.201432	+	0.979503i	$0.435441\pi$
$314$	−2.23191	−0.125954
$315$	0	0
$316$	−1.36599	−0.0768431
$317$	23.9651	1.34601	0.673007	−	0.739636i	$-0.265003\pi$
$317$	23.9651	1.34601	0.673007	+	0.739636i	$0.265003\pi$
$318$	−2.61356	−0.146561
$319$	−9.46442	−0.529906
$320$	0	0
$321$	4.51573	0.252043
$322$	6.87800	0.383296
$323$	2.20477	0.122677
$324$	−6.13467	−0.340815
$325$	0	0
$326$	−8.40044	−0.465257
$327$	3.34913	0.185207
$328$	−6.79763	−0.375336
$329$	20.7071	1.14162
$330$	0	0
$331$	−2.89546	−0.159149	−0.0795745	−	0.996829i	$-0.525356\pi$
$331$	−2.89546	−0.159149	−0.0795745	+	0.996829i	$0.525356\pi$
$332$	27.6353	1.51669
$333$	−2.24160	−0.122839
$334$	3.47591	0.190193
$335$	0	0
$336$	4.85086	0.264636
$337$	3.10560	0.169173	0.0845865	−	0.996416i	$-0.473043\pi$
$337$	3.10560	0.169173	0.0845865	+	0.996416i	$0.473043\pi$
$338$	0	0
$339$	6.12929	0.332898
$340$	0	0
$341$	−7.06829	−0.382770
$342$	−2.53617	−0.137140
$343$	18.1793	0.981589
$344$	−14.6679	−0.790838
$345$	0	0
$346$	9.09651	0.489031
$347$	11.3787	0.610839	0.305419	−	0.952218i	$-0.401203\pi$
$347$	11.3787	0.610839	0.305419	+	0.952218i	$0.401203\pi$
$348$	−10.7192	−0.574608
$349$	3.34721	0.179172	0.0895859	−	0.995979i	$-0.471446\pi$
$349$	3.34721	0.179172	0.0895859	+	0.995979i	$0.471446\pi$
$350$	0	0
$351$	0	0
$352$	−6.40342	−0.341303
$353$	0.637727	0.0339428	0.0169714	−	0.999856i	$-0.494598\pi$
$353$	0.637727	0.0339428	0.0169714	+	0.999856i	$0.494598\pi$
$354$	0.00537681	0.000285774 0
$355$	0	0
$356$	−24.9215	−1.32084
$357$	−2.45473	−0.129918
$358$	1.36227	0.0719983
$359$	21.4590	1.13256	0.566282	−	0.824211i	$-0.308381\pi$
$359$	21.4590	1.13256	0.566282	+	0.824211i	$0.308381\pi$
$360$	0	0
$361$	−15.2403	−0.802120
$362$	−6.55257	−0.344395
$363$	7.67025	0.402584
$364$	0	0
$365$	0	0
$366$	−3.57673	−0.186959
$367$	9.38703	0.489999	0.244999	−	0.969523i	$-0.421212\pi$
$367$	9.38703	0.489999	0.244999	+	0.969523i	$0.421212\pi$
$368$	10.3448	0.539261
$369$	7.81940	0.407062
$370$	0	0
$371$	15.8092	0.820775
$372$	−8.00538	−0.415059
$373$	−27.7265	−1.43562	−0.717811	−	0.696238i	$-0.754856\pi$
$373$	−27.7265	−1.43562	−0.717811	+	0.696238i	$0.754856\pi$
$374$	0.756004	0.0390921
$375$	0	0
$376$	−15.7603	−0.812776
$377$	0	0
$378$	6.41789	0.330101
$379$	−35.8702	−1.84253	−0.921265	−	0.388935i	$-0.872843\pi$
$379$	−35.8702	−1.84253	−0.921265	+	0.388935i	$0.872843\pi$
$380$	0	0
$381$	5.43535	0.278462
$382$	4.99090	0.255357
$383$	4.85517	0.248087	0.124044	−	0.992277i	$-0.460414\pi$
$383$	4.85517	0.248087	0.124044	+	0.992277i	$0.460414\pi$
$384$	−9.25236	−0.472157
$385$	0	0
$386$	−7.50604	−0.382047
$387$	16.8726	0.857684
$388$	−5.29888	−0.269010
$389$	2.38537	0.120943	0.0604716	−	0.998170i	$-0.480740\pi$
$389$	2.38537	0.120943	0.0604716	+	0.998170i	$0.480740\pi$
$390$	0	0
$391$	−5.23490	−0.264740
$392$	0.506041	0.0255589
$393$	10.9705	0.553387
$394$	−7.20105	−0.362783
$395$	0	0
$396$	4.77777	0.240092
$397$	−15.2664	−0.766196	−0.383098	−	0.923708i	$-0.625143\pi$
$397$	−15.2664	−0.766196	−0.383098	+	0.923708i	$0.625143\pi$
$398$	7.53989	0.377941
$399$	−4.18598	−0.209561
$400$	0	0
$401$	−12.7584	−0.637124	−0.318562	−	0.947902i	$-0.603200\pi$
$401$	−12.7584	−0.637124	−0.318562	+	0.947902i	$0.603200\pi$
$402$	−4.11960	−0.205467
$403$	0	0
$404$	−8.95167	−0.445362
$405$	0	0
$406$	−11.8019	−0.585720
$407$	1.13946	0.0564807
$408$	1.86831	0.0924953
$409$	25.3588	1.25391	0.626956	−	0.779054i	$-0.284300\pi$
$409$	25.3588	1.25391	0.626956	+	0.779054i	$0.284300\pi$
$410$	0	0
$411$	−10.4155	−0.513759
$412$	−22.8944	−1.12793
$413$	−0.0325239	−0.00160040
$414$	6.02177	0.295954
$415$	0	0
$416$	0	0
$417$	−9.66056	−0.473080
$418$	1.28919	0.0630565
$419$	−11.6673	−0.569983	−0.284992	−	0.958530i	$-0.591991\pi$
$419$	−11.6673	−0.569983	−0.284992	+	0.958530i	$0.591991\pi$
$420$	0	0
$421$	8.29291	0.404172	0.202086	−	0.979368i	$-0.435228\pi$
$421$	8.29291	0.404172	0.202086	+	0.979368i	$0.435228\pi$
$422$	−5.80492	−0.282579
$423$	18.1293	0.881476
$424$	−12.0325	−0.584351
$425$	0	0
$426$	6.11529	0.296287
$427$	21.6353	1.04701
$428$	9.52781	0.460544
$429$	0	0
$430$	0	0
$431$	−0.932296	−0.0449071	−0.0224536	−	0.999748i	$-0.507148\pi$
$431$	−0.932296	−0.0449071	−0.0224536	+	0.999748i	$0.507148\pi$
$432$	9.65279	0.464420
$433$	13.3502	0.641569	0.320785	−	0.947152i	$-0.396053\pi$
$433$	13.3502	0.641569	0.320785	+	0.947152i	$0.396053\pi$
$434$	−8.81402	−0.423086
$435$	0	0
$436$	7.06638	0.338418
$437$	−8.92692	−0.427032
$438$	5.70410	0.272553
$439$	13.9922	0.667813	0.333906	−	0.942606i	$-0.391633\pi$
$439$	13.9922	0.667813	0.333906	+	0.942606i	$0.391633\pi$
$440$	0	0
$441$	−0.582105	−0.0277193
$442$	0	0
$443$	−23.7017	−1.12610	−0.563051	−	0.826422i	$-0.690373\pi$
$443$	−23.7017	−1.12610	−0.563051	+	0.826422i	$0.690373\pi$
$444$	1.29052	0.0612454
$445$	0	0
$446$	6.32975	0.299722
$447$	−0.594187	−0.0281041
$448$	4.11290	0.194316
$449$	12.5864	0.593990	0.296995	−	0.954879i	$-0.404016\pi$
$449$	12.5864	0.593990	0.296995	+	0.954879i	$0.404016\pi$
$450$	0	0
$451$	−3.97477	−0.187165
$452$	12.9323	0.608284
$453$	−15.2959	−0.718664
$454$	−5.90515	−0.277142
$455$	0	0
$456$	3.18598	0.149197
$457$	−33.6383	−1.57353	−0.786767	−	0.617250i	$-0.788247\pi$
$457$	−33.6383	−1.57353	−0.786767	+	0.617250i	$0.788247\pi$
$458$	−0.632351	−0.0295478
$459$	−4.88471	−0.227999
$460$	0	0
$461$	1.40283	0.0653363	0.0326681	−	0.999466i	$-0.489600\pi$
$461$	1.40283	0.0653363	0.0326681	+	0.999466i	$0.489600\pi$
$462$	−1.43535	−0.0667787
$463$	−15.2010	−0.706453	−0.353226	−	0.935538i	$-0.614915\pi$
$463$	−15.2010	−0.706453	−0.353226	+	0.935538i	$0.614915\pi$
$464$	−17.7506	−0.824052
$465$	0	0
$466$	−6.02177	−0.278953
$467$	39.3414	1.82050	0.910250	−	0.414058i	$-0.135889\pi$
$467$	39.3414	1.82050	0.910250	+	0.414058i	$0.135889\pi$
$468$	0	0
$469$	24.9191	1.15066
$470$	0	0
$471$	−3.22521	−0.148610
$472$	0.0247542	0.00113940
$473$	−8.57673	−0.394358
$474$	0.359289	0.0165027
$475$	0	0
$476$	−5.17928	−0.237392
$477$	13.8412	0.633743
$478$	−6.62027	−0.302804
$479$	22.3690	1.02206	0.511032	−	0.859561i	$-0.329263\pi$
$479$	22.3690	1.02206	0.511032	+	0.859561i	$0.329263\pi$
$480$	0	0
$481$	0	0
$482$	−2.02475	−0.0922250
$483$	9.93900	0.452240
$484$	16.1836	0.735618
$485$	0	0
$486$	8.76569	0.397620
$487$	22.9205	1.03863	0.519313	−	0.854584i	$-0.326188\pi$
$487$	22.9205	1.03863	0.519313	+	0.854584i	$0.326188\pi$
$488$	−16.4668	−0.745418
$489$	−12.1390	−0.548944
$490$	0	0
$491$	1.84356	0.0831987	0.0415993	−	0.999134i	$-0.486755\pi$
$491$	1.84356	0.0831987	0.0415993	+	0.999134i	$0.486755\pi$
$492$	−4.50173	−0.202954
$493$	8.98254	0.404553
$494$	0	0
$495$	0	0
$496$	−13.2567	−0.595242
$497$	−36.9909	−1.65927
$498$	−7.26875	−0.325720
$499$	12.0344	0.538736	0.269368	−	0.963037i	$-0.413185\pi$
$499$	12.0344	0.538736	0.269368	+	0.963037i	$0.413185\pi$
$500$	0	0
$501$	5.02284	0.224404
$502$	−0.762118	−0.0340150
$503$	30.5056	1.36018	0.680088	−	0.733130i	$-0.261942\pi$
$503$	30.5056	1.36018	0.680088	+	0.733130i	$0.261942\pi$
$504$	13.0000	0.579066
$505$	0	0
$506$	−3.06100	−0.136078
$507$	0	0
$508$	11.4681	0.508816
$509$	1.51142	0.0669924	0.0334962	−	0.999439i	$-0.489336\pi$
$509$	1.51142	0.0669924	0.0334962	+	0.999439i	$0.489336\pi$
$510$	0	0
$511$	−34.5036	−1.52635
$512$	−21.2174	−0.937687
$513$	−8.32975	−0.367767
$514$	16.3357	0.720538
$515$	0	0
$516$	−9.71379	−0.427626
$517$	−9.21552	−0.405298
$518$	1.42088	0.0624298
$519$	13.1448	0.576994
$520$	0	0
$521$	−5.64012	−0.247098	−0.123549	−	0.992338i	$-0.539428\pi$
$521$	−5.64012	−0.247098	−0.123549	+	0.992338i	$0.539428\pi$
$522$	−10.3327	−0.452251
$523$	31.7506	1.38836	0.694179	−	0.719802i	$-0.255768\pi$
$523$	31.7506	1.38836	0.694179	+	0.719802i	$0.255768\pi$
$524$	23.1468	1.01117
$525$	0	0
$526$	5.93602	0.258823
$527$	6.70841	0.292223
$528$	−2.15883	−0.0939512
$529$	−1.80433	−0.0784492
$530$	0	0
$531$	−0.0284750	−0.00123571
$532$	−8.83207	−0.382919
$533$	0	0
$534$	6.55496	0.283661
$535$	0	0
$536$	−18.9661	−0.819213
$537$	1.96854	0.0849488
$538$	5.65279	0.243709
$539$	0.295897	0.0127452
$540$	0	0
$541$	−24.3297	−1.04602	−0.523009	−	0.852327i	$-0.675191\pi$
$541$	−24.3297	−1.04602	−0.523009	+	0.852327i	$0.675191\pi$
$542$	−16.3448	−0.702070
$543$	−9.46873	−0.406342
$544$	6.07739	0.260566
$545$	0	0
$546$	0	0
$547$	8.18896	0.350135	0.175067	−	0.984556i	$-0.443986\pi$
$547$	8.18896	0.350135	0.175067	+	0.984556i	$0.443986\pi$
$548$	−21.9758	−0.938761
$549$	18.9420	0.808424
$550$	0	0
$551$	15.3177	0.652555
$552$	−7.56465	−0.321973
$553$	−2.17331	−0.0924185
$554$	−5.68532	−0.241546
$555$	0	0
$556$	−20.3830	−0.864431
$557$	25.3327	1.07338	0.536691	−	0.843779i	$-0.319674\pi$
$557$	25.3327	1.07338	0.536691	+	0.843779i	$0.319674\pi$
$558$	−7.71678	−0.326677
$559$	0	0
$560$	0	0
$561$	1.09246	0.0461236
$562$	−6.41789	−0.270723
$563$	25.3937	1.07022	0.535109	−	0.844783i	$-0.320270\pi$
$563$	25.3937	1.07022	0.535109	+	0.844783i	$0.320270\pi$
$564$	−10.4373	−0.439488
$565$	0	0
$566$	−17.0422	−0.716338
$567$	−9.76032	−0.409895
$568$	28.1540	1.18132
$569$	−31.1347	−1.30523	−0.652617	−	0.757688i	$-0.726329\pi$
$569$	−31.1347	−1.30523	−0.652617	+	0.757688i	$0.726329\pi$
$570$	0	0
$571$	−20.5090	−0.858276	−0.429138	−	0.903239i	$-0.641183\pi$
$571$	−20.5090	−0.858276	−0.429138	+	0.903239i	$0.641183\pi$
$572$	0	0
$573$	7.21206	0.301288
$574$	−4.95646	−0.206879
$575$	0	0
$576$	3.60089	0.150037
$577$	15.6890	0.653143	0.326572	−	0.945172i	$-0.394107\pi$
$577$	15.6890	0.653143	0.326572	+	0.945172i	$0.394107\pi$
$578$	8.71678	0.362570
$579$	−10.8465	−0.450767
$580$	0	0
$581$	43.9681	1.82410
$582$	1.39373	0.0577720
$583$	−7.03577	−0.291392
$584$	26.2610	1.08669
$585$	0	0
$586$	10.3268	0.426595
$587$	30.5687	1.26171	0.630853	−	0.775903i	$-0.282705\pi$
$587$	30.5687	1.26171	0.630853	+	0.775903i	$0.282705\pi$
$588$	0.335126	0.0138203
$589$	11.4397	0.471363
$590$	0	0
$591$	−10.4058	−0.428038
$592$	2.13706	0.0878328
$593$	−29.6883	−1.21915	−0.609576	−	0.792727i	$-0.708660\pi$
$593$	−29.6883	−1.21915	−0.609576	+	0.792727i	$0.708660\pi$
$594$	−2.85623	−0.117193
$595$	0	0
$596$	−1.25368	−0.0513529
$597$	10.8955	0.445922
$598$	0	0
$599$	24.2325	0.990113	0.495057	−	0.868861i	$-0.335147\pi$
$599$	24.2325	0.990113	0.495057	+	0.868861i	$0.335147\pi$
$600$	0	0
$601$	16.4819	0.672310	0.336155	−	0.941807i	$-0.390873\pi$
$601$	16.4819	0.672310	0.336155	+	0.941807i	$0.390873\pi$
$602$	−10.6950	−0.435896
$603$	21.8170	0.888457
$604$	−32.2731	−1.31317
$605$	0	0
$606$	2.35450	0.0956451
$607$	−1.43190	−0.0581188	−0.0290594	−	0.999578i	$-0.509251\pi$
$607$	−1.43190	−0.0581188	−0.0290594	+	0.999578i	$0.509251\pi$
$608$	10.3636	0.420300
$609$	−17.0543	−0.691075
$610$	0	0
$611$	0	0
$612$	−4.53452	−0.183297
$613$	3.84846	0.155438	0.0777190	−	0.996975i	$-0.475236\pi$
$613$	3.84846	0.155438	0.0777190	+	0.996975i	$0.475236\pi$
$614$	−4.96376	−0.200321
$615$	0	0
$616$	−6.60819	−0.266251
$617$	−15.0388	−0.605437	−0.302719	−	0.953080i	$-0.597894\pi$
$617$	−15.0388	−0.605437	−0.302719	+	0.953080i	$0.597894\pi$
$618$	6.02177	0.242231
$619$	−12.8170	−0.515159	−0.257579	−	0.966257i	$-0.582925\pi$
$619$	−12.8170	−0.515159	−0.257579	+	0.966257i	$0.582925\pi$
$620$	0	0
$621$	19.7778	0.793655
$622$	−11.6762	−0.468174
$623$	−39.6504	−1.58856
$624$	0	0
$625$	0	0
$626$	−3.95539	−0.158089
$627$	1.86294	0.0743985
$628$	−6.80492	−0.271546
$629$	−1.08144	−0.0431199
$630$	0	0
$631$	−25.7517	−1.02516	−0.512579	−	0.858640i	$-0.671310\pi$
$631$	−25.7517	−1.02516	−0.512579	+	0.858640i	$0.671310\pi$
$632$	1.65412	0.0657974
$633$	−8.38835	−0.333407
$634$	−13.2996	−0.528195
$635$	0	0
$636$	−7.96854	−0.315973
$637$	0	0
$638$	5.25236	0.207943
$639$	−32.3860	−1.28117
$640$	0	0
$641$	24.4571	0.965998	0.482999	−	0.875621i	$-0.339547\pi$
$641$	24.4571	0.965998	0.482999	+	0.875621i	$0.339547\pi$
$642$	−2.50604	−0.0989055
$643$	−9.97344	−0.393314	−0.196657	−	0.980472i	$-0.563009\pi$
$643$	−9.97344	−0.393314	−0.196657	+	0.980472i	$0.563009\pi$
$644$	20.9705	0.826352
$645$	0	0
$646$	−1.22355	−0.0481401
$647$	−11.8431	−0.465600	−0.232800	−	0.972525i	$-0.574789\pi$
$647$	−11.8431	−0.465600	−0.232800	+	0.972525i	$0.574789\pi$
$648$	7.42865	0.291825
$649$	0.0144745	0.000568173 0
$650$	0	0
$651$	−12.7366	−0.499188
$652$	−25.6122	−1.00305
$653$	7.47411	0.292484	0.146242	−	0.989249i	$-0.453282\pi$
$653$	7.47411	0.292484	0.146242	+	0.989249i	$0.453282\pi$
$654$	−1.85862	−0.0726780
$655$	0	0
$656$	−7.45473	−0.291058
$657$	−30.2083	−1.17854
$658$	−11.4916	−0.447988
$659$	34.1739	1.33123	0.665613	−	0.746297i	$-0.268170\pi$
$659$	34.1739	1.33123	0.665613	+	0.746297i	$0.268170\pi$
$660$	0	0
$661$	−33.6088	−1.30723	−0.653615	−	0.756827i	$-0.726748\pi$
$661$	−33.6088	−1.30723	−0.653615	+	0.756827i	$0.726748\pi$
$662$	1.60686	0.0624524
$663$	0	0
$664$	−33.4644	−1.29867
$665$	0	0
$666$	1.24400	0.0482039
$667$	−36.3696	−1.40824
$668$	10.5978	0.410040
$669$	9.14675	0.353634
$670$	0	0
$671$	−9.62863	−0.371709
$672$	−11.5386	−0.445110
$673$	−48.0320	−1.85150	−0.925750	−	0.378137i	$-0.876565\pi$
$673$	−48.0320	−1.85150	−0.925750	+	0.378137i	$0.876565\pi$
$674$	−1.72348	−0.0663860
$675$	0	0
$676$	0	0
$677$	33.6582	1.29359	0.646794	−	0.762665i	$-0.276109\pi$
$677$	33.6582	1.29359	0.646794	+	0.762665i	$0.276109\pi$
$678$	−3.40150	−0.130634
$679$	−8.43057	−0.323535
$680$	0	0
$681$	−8.53319	−0.326992
$682$	3.92261	0.150204
$683$	15.9041	0.608553	0.304276	−	0.952584i	$-0.401585\pi$
$683$	15.9041	0.608553	0.304276	+	0.952584i	$0.401585\pi$
$684$	−7.73258	−0.295663
$685$	0	0
$686$	−10.0887	−0.385190
$687$	−0.913773	−0.0348626
$688$	−16.0858	−0.613264
$689$	0	0
$690$	0	0
$691$	−33.1903	−1.26262	−0.631309	−	0.775531i	$-0.717482\pi$
$691$	−33.1903	−1.26262	−0.631309	+	0.775531i	$0.717482\pi$
$692$	27.7345	1.05431
$693$	7.60148	0.288756
$694$	−6.31468	−0.239702
$695$	0	0
$696$	12.9801	0.492011
$697$	3.77240	0.142890
$698$	−1.85756	−0.0703097
$699$	−8.70171	−0.329129
$700$	0	0
$701$	−14.9129	−0.563253	−0.281627	−	0.959524i	$-0.590874\pi$
$701$	−14.9129	−0.563253	−0.281627	+	0.959524i	$0.590874\pi$
$702$	0	0
$703$	−1.84415	−0.0695534
$704$	−1.83041	−0.0689863
$705$	0	0
$706$	−0.353912	−0.0133197
$707$	−14.2422	−0.535633
$708$	0.0163935	0.000616104 0
$709$	−38.4312	−1.44331	−0.721656	−	0.692252i	$-0.756619\pi$
$709$	−38.4312	−1.44331	−0.721656	+	0.692252i	$0.756619\pi$
$710$	0	0
$711$	−1.90276	−0.0713589
$712$	30.1782	1.13098
$713$	−27.1618	−1.01722
$714$	1.36227	0.0509818
$715$	0	0
$716$	4.15346	0.155222
$717$	−9.56657	−0.357270
$718$	−11.9089	−0.444435
$719$	−11.4373	−0.426538	−0.213269	−	0.976993i	$-0.568411\pi$
$719$	−11.4373	−0.426538	−0.213269	+	0.976993i	$0.568411\pi$
$720$	0	0
$721$	−36.4252	−1.35654
$722$	8.45771	0.314764
$723$	−2.92585	−0.108814
$724$	−19.9782	−0.742485
$725$	0	0
$726$	−4.25667	−0.157980
$727$	−3.63640	−0.134867	−0.0674333	−	0.997724i	$-0.521481\pi$
$727$	−3.63640	−0.134867	−0.0674333	+	0.997724i	$0.521481\pi$
$728$	0	0
$729$	1.78986	0.0662910
$730$	0	0
$731$	8.14005	0.301071
$732$	−10.9051	−0.403066
$733$	−3.52217	−0.130094	−0.0650472	−	0.997882i	$-0.520720\pi$
$733$	−3.52217	−0.130094	−0.0650472	+	0.997882i	$0.520720\pi$
$734$	−5.20941	−0.192283
$735$	0	0
$736$	−24.6069	−0.907021
$737$	−11.0901	−0.408508
$738$	−4.33944	−0.159737
$739$	−0.420288	−0.0154605	−0.00773027	−	0.999970i	$-0.502461\pi$
$739$	−0.420288	−0.0154605	−0.00773027	+	0.999970i	$0.502461\pi$
$740$	0	0
$741$	0	0
$742$	−8.77346	−0.322084
$743$	−25.3623	−0.930452	−0.465226	−	0.885192i	$-0.654027\pi$
$743$	−25.3623	−0.930452	−0.465226	+	0.885192i	$0.654027\pi$
$744$	9.69394	0.355397
$745$	0	0
$746$	15.3870	0.563359
$747$	38.4946	1.40844
$748$	2.30499	0.0842790
$749$	15.1588	0.553892
$750$	0	0
$751$	0.650874	0.0237507	0.0118754	−	0.999929i	$-0.496220\pi$
$751$	0.650874	0.0237507	0.0118754	+	0.999929i	$0.496220\pi$
$752$	−17.2838	−0.630276
$753$	−1.10129	−0.0401333
$754$	0	0
$755$	0	0
$756$	19.5676	0.711668
$757$	16.7909	0.610276	0.305138	−	0.952308i	$-0.401297\pi$
$757$	16.7909	0.610276	0.305138	+	0.952308i	$0.401297\pi$
$758$	19.9065	0.723036
$759$	−4.42327	−0.160555
$760$	0	0
$761$	−30.9221	−1.12093	−0.560463	−	0.828179i	$-0.689377\pi$
$761$	−30.9221	−1.12093	−0.560463	+	0.828179i	$0.689377\pi$
$762$	−3.01639	−0.109272
$763$	11.2427	0.407012
$764$	15.2168	0.550526
$765$	0	0
$766$	−2.69441	−0.0973531
$767$	0	0
$768$	2.68425	0.0968596
$769$	43.7689	1.57835	0.789174	−	0.614169i	$-0.210509\pi$
$769$	43.7689	1.57835	0.789174	+	0.614169i	$0.210509\pi$
$770$	0	0
$771$	23.6058	0.850142
$772$	−22.8853	−0.823660
$773$	−42.4209	−1.52577	−0.762886	−	0.646532i	$-0.776218\pi$
$773$	−42.4209	−1.52577	−0.762886	+	0.646532i	$0.776218\pi$
$774$	−9.36360	−0.336568
$775$	0	0
$776$	6.41657	0.230341
$777$	2.05323	0.0736592
$778$	−1.32378	−0.0474598
$779$	6.43296	0.230485
$780$	0	0
$781$	16.4625	0.589075
$782$	2.90515	0.103888
$783$	−33.9366	−1.21280
$784$	0.554958	0.0198199
$785$	0	0
$786$	−6.08815	−0.217157
$787$	−36.0116	−1.28368	−0.641838	−	0.766841i	$-0.721828\pi$
$787$	−36.0116	−1.28368	−0.641838	+	0.766841i	$0.721828\pi$
$788$	−21.9554	−0.782129
$789$	8.57779	0.305378
$790$	0	0
$791$	20.5754	0.731577
$792$	−5.78554	−0.205580
$793$	0	0
$794$	8.47219	0.300667
$795$	0	0
$796$	22.9885	0.814806
$797$	31.7101	1.12323	0.561614	−	0.827399i	$-0.310181\pi$
$797$	31.7101	1.12323	0.561614	+	0.827399i	$0.310181\pi$
$798$	2.32304	0.0822349
$799$	8.74632	0.309422
$800$	0	0
$801$	−34.7144	−1.22657
$802$	7.08038	0.250017
$803$	15.3556	0.541886
$804$	−12.5603	−0.442969
$805$	0	0
$806$	0	0
$807$	8.16852	0.287546
$808$	10.8398	0.381344
$809$	−45.2814	−1.59201	−0.796005	−	0.605290i	$-0.793057\pi$
$809$	−45.2814	−1.59201	−0.796005	+	0.605290i	$0.793057\pi$
$810$	0	0
$811$	42.8635	1.50514	0.752571	−	0.658511i	$-0.228813\pi$
$811$	42.8635	1.50514	0.752571	+	0.658511i	$0.228813\pi$
$812$	−35.9831	−1.26276
$813$	−23.6189	−0.828352
$814$	−0.632351	−0.0221639
$815$	0	0
$816$	2.04892	0.0717265
$817$	13.8810	0.485634
$818$	−14.0731	−0.492054
$819$	0	0
$820$	0	0
$821$	7.82776	0.273191	0.136595	−	0.990627i	$-0.456384\pi$
$821$	7.82776	0.273191	0.136595	+	0.990627i	$0.456384\pi$
$822$	5.78017	0.201606
$823$	36.7754	1.28191	0.640955	−	0.767579i	$-0.278539\pi$
$823$	36.7754	1.28191	0.640955	+	0.767579i	$0.278539\pi$
$824$	27.7235	0.965793
$825$	0	0
$826$	0.0180494	0.000628019 0
$827$	47.3293	1.64580	0.822900	−	0.568186i	$-0.192355\pi$
$827$	47.3293	1.64580	0.822900	+	0.568186i	$0.192355\pi$
$828$	18.3599	0.638050
$829$	25.2687	0.877620	0.438810	−	0.898580i	$-0.355400\pi$
$829$	25.2687	0.877620	0.438810	+	0.898580i	$0.355400\pi$
$830$	0	0
$831$	−8.21552	−0.284993
$832$	0	0
$833$	−0.280831	−0.00973023
$834$	5.36121	0.185643
$835$	0	0
$836$	3.93064	0.135944
$837$	−25.3448	−0.876045
$838$	6.47484	0.223670
$839$	37.6883	1.30114	0.650572	−	0.759444i	$-0.274529\pi$
$839$	37.6883	1.30114	0.650572	+	0.759444i	$0.274529\pi$
$840$	0	0
$841$	33.4064	1.15194
$842$	−4.60222	−0.158603
$843$	−9.27413	−0.319418
$844$	−17.6987	−0.609215
$845$	0	0
$846$	−10.0610	−0.345904
$847$	25.7482	0.884720
$848$	−13.1957	−0.453141
$849$	−24.6267	−0.845187
$850$	0	0
$851$	4.37867	0.150099
$852$	18.6450	0.638768
$853$	−31.0121	−1.06183	−0.530917	−	0.847424i	$-0.678152\pi$
$853$	−31.0121	−1.06183	−0.530917	+	0.847424i	$0.678152\pi$
$854$	−12.0067	−0.410861
$855$	0	0
$856$	−11.5375	−0.394344
$857$	−12.4692	−0.425940	−0.212970	−	0.977059i	$-0.568314\pi$
$857$	−12.4692	−0.425940	−0.212970	+	0.977059i	$0.568314\pi$
$858$	0	0
$859$	−17.3163	−0.590826	−0.295413	−	0.955370i	$-0.595457\pi$
$859$	−17.3163	−0.590826	−0.295413	+	0.955370i	$0.595457\pi$
$860$	0	0
$861$	−7.16229	−0.244090
$862$	0.517385	0.0176222
$863$	−3.46383	−0.117910	−0.0589550	−	0.998261i	$-0.518777\pi$
$863$	−3.46383	−0.117910	−0.0589550	+	0.998261i	$0.518777\pi$
$864$	−22.9608	−0.781141
$865$	0	0
$866$	−7.40880	−0.251761
$867$	12.5961	0.427786
$868$	−26.8732	−0.912136
$869$	0.967213	0.0328105
$870$	0	0
$871$	0	0
$872$	−8.55688	−0.289772
$873$	−7.38106	−0.249811
$874$	4.95407	0.167574
$875$	0	0
$876$	17.3913	0.587599
$877$	57.2549	1.93336	0.966680	−	0.255989i	$-0.0824010\pi$
$877$	57.2549	1.93336	0.966680	+	0.255989i	$0.0824010\pi$
$878$	−7.76510	−0.262059
$879$	14.9226	0.503327
$880$	0	0
$881$	−43.1782	−1.45471	−0.727355	−	0.686261i	$-0.759251\pi$
$881$	−43.1782	−1.45471	−0.727355	+	0.686261i	$0.759251\pi$
$882$	0.323044	0.0108775
$883$	−49.9560	−1.68115	−0.840576	−	0.541693i	$-0.817784\pi$
$883$	−49.9560	−1.68115	−0.840576	+	0.541693i	$0.817784\pi$
$884$	0	0
$885$	0	0
$886$	13.1535	0.441899
$887$	−17.6746	−0.593454	−0.296727	−	0.954962i	$-0.595895\pi$
$887$	−17.6746	−0.593454	−0.296727	+	0.954962i	$0.595895\pi$
$888$	−1.56273	−0.0524417
$889$	18.2459	0.611948
$890$	0	0
$891$	4.34375	0.145521
$892$	19.2989	0.646174
$893$	14.9148	0.499106
$894$	0.329749	0.0110284
$895$	0	0
$896$	−31.0592	−1.03761
$897$	0	0
$898$	−6.98493	−0.233090
$899$	46.6069	1.55443
$900$	0	0
$901$	6.67755	0.222461
$902$	2.20583	0.0734462
$903$	−15.4547	−0.514301
$904$	−15.6601	−0.520847
$905$	0	0
$906$	8.48858	0.282014
$907$	−7.73423	−0.256811	−0.128406	−	0.991722i	$-0.540986\pi$
$907$	−7.73423	−0.256811	−0.128406	+	0.991722i	$0.540986\pi$
$908$	−18.0043	−0.597494
$909$	−12.4692	−0.413577
$910$	0	0
$911$	39.6179	1.31260	0.656299	−	0.754501i	$-0.272121\pi$
$911$	39.6179	1.31260	0.656299	+	0.754501i	$0.272121\pi$
$912$	3.49396	0.115697
$913$	−19.5676	−0.647594
$914$	18.6679	0.617478
$915$	0	0
$916$	−1.92798	−0.0637024
$917$	36.8267	1.21612
$918$	2.71081	0.0894700
$919$	14.6213	0.482313	0.241157	−	0.970486i	$-0.422473\pi$
$919$	14.6213	0.482313	0.241157	+	0.970486i	$0.422473\pi$
$920$	0	0
$921$	−7.17283	−0.236353
$922$	−0.778512	−0.0256389
$923$	0	0
$924$	−4.37627	−0.143969
$925$	0	0
$926$	8.43594	0.277222
$927$	−31.8907	−1.04743
$928$	42.2228	1.38603
$929$	−3.55735	−0.116713	−0.0583565	−	0.998296i	$-0.518586\pi$
$929$	−3.55735	−0.116713	−0.0583565	+	0.998296i	$0.518586\pi$
$930$	0	0
$931$	−0.478894	−0.0156951
$932$	−18.3599	−0.601398
$933$	−16.8726	−0.552385
$934$	−21.8328	−0.714391
$935$	0	0
$936$	0	0
$937$	−34.5526	−1.12878	−0.564392	−	0.825507i	$-0.690889\pi$
$937$	−34.5526	−1.12878	−0.564392	+	0.825507i	$0.690889\pi$
$938$	−13.8291	−0.451536
$939$	−5.71571	−0.186525
$940$	0	0
$941$	−20.6233	−0.672299	−0.336149	−	0.941809i	$-0.609125\pi$
$941$	−20.6233	−0.672299	−0.336149	+	0.941809i	$0.609125\pi$
$942$	1.78986	0.0583167
$943$	−15.2741	−0.497394
$944$	0.0271471	0.000883562 0
$945$	0	0
$946$	4.75973	0.154752
$947$	29.4999	0.958619	0.479309	−	0.877646i	$-0.340887\pi$
$947$	29.4999	0.958619	0.479309	+	0.877646i	$0.340887\pi$
$948$	1.09544	0.0355783
$949$	0	0
$950$	0	0
$951$	−19.2185	−0.623203
$952$	6.27173	0.203268
$953$	−26.2389	−0.849963	−0.424981	−	0.905202i	$-0.639719\pi$
$953$	−26.2389	−0.849963	−0.424981	+	0.905202i	$0.639719\pi$
$954$	−7.68127	−0.248690
$955$	0	0
$956$	−20.1847	−0.652818
$957$	7.58987	0.245346
$958$	−12.4138	−0.401073
$959$	−34.9638	−1.12904
$960$	0	0
$961$	3.80731	0.122817
$962$	0	0
$963$	13.2717	0.427676
$964$	−6.17331	−0.198829
$965$	0	0
$966$	−5.51573	−0.177466
$967$	17.5176	0.563330	0.281665	−	0.959513i	$-0.409113\pi$
$967$	17.5176	0.563330	0.281665	+	0.959513i	$0.409113\pi$
$968$	−19.5972	−0.629877
$969$	−1.76809	−0.0567991
$970$	0	0
$971$	20.5120	0.658262	0.329131	−	0.944284i	$-0.393244\pi$
$971$	20.5120	0.658262	0.329131	+	0.944284i	$0.393244\pi$
$972$	26.7259	0.857233
$973$	−32.4295	−1.03964
$974$	−12.7199	−0.407572
$975$	0	0
$976$	−18.0586	−0.578042
$977$	25.4450	0.814059	0.407030	−	0.913415i	$-0.366565\pi$
$977$	25.4450	0.814059	0.407030	+	0.913415i	$0.366565\pi$
$978$	6.73663	0.215414
$979$	17.6461	0.563971
$980$	0	0
$981$	9.84309	0.314266
$982$	−1.02310	−0.0326484
$983$	−39.5244	−1.26063	−0.630316	−	0.776339i	$-0.717074\pi$
$983$	−39.5244	−1.26063	−0.630316	+	0.776339i	$0.717074\pi$
$984$	5.45127	0.173780
$985$	0	0
$986$	−4.98493	−0.158753
$987$	−16.6058	−0.528568
$988$	0	0
$989$	−32.9584	−1.04802
$990$	0	0
$991$	−29.8377	−0.947826	−0.473913	−	0.880572i	$-0.657159\pi$
$991$	−29.8377	−0.947826	−0.473913	+	0.880572i	$0.657159\pi$
$992$	31.5332	1.00118
$993$	2.32198	0.0736858
$994$	20.5284	0.651121
$995$	0	0
$996$	−22.1618	−0.702224
$997$	4.93123	0.156174	0.0780868	−	0.996947i	$-0.475119\pi$
$997$	4.93123	0.156174	0.0780868	+	0.996947i	$0.475119\pi$
$998$	−6.67861	−0.211408
$999$	4.08575	0.129268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bb.1.2		3
5.4	even	2		169.2.a.c.1.2	yes	3
13.12	even	2		4225.2.a.bg.1.2		3
15.14	odd	2		1521.2.a.o.1.2		3
20.19	odd	2		2704.2.a.ba.1.1		3
35.34	odd	2		8281.2.a.bj.1.2		3
65.4	even	6		169.2.c.c.146.2		6
65.9	even	6		169.2.c.b.146.2		6
65.19	odd	12		169.2.e.b.23.3		12
65.24	odd	12		169.2.e.b.147.3		12
65.29	even	6		169.2.c.b.22.2		6
65.34	odd	4		169.2.b.b.168.4		6
65.44	odd	4		169.2.b.b.168.3		6
65.49	even	6		169.2.c.c.22.2		6
65.54	odd	12		169.2.e.b.147.4		12
65.59	odd	12		169.2.e.b.23.4		12
65.64	even	2		169.2.a.b.1.2	✓	3
195.44	even	4		1521.2.b.l.1351.4		6
195.164	even	4		1521.2.b.l.1351.3		6
195.194	odd	2		1521.2.a.r.1.2		3
260.99	even	4		2704.2.f.o.337.2		6
260.239	even	4		2704.2.f.o.337.1		6
260.259	odd	2		2704.2.a.z.1.1		3
455.454	odd	2		8281.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.2	✓	3	65.64	even	2
169.2.a.c.1.2	yes	3	5.4	even	2
169.2.b.b.168.3		6	65.44	odd	4
169.2.b.b.168.4		6	65.34	odd	4
169.2.c.b.22.2		6	65.29	even	6
169.2.c.b.146.2		6	65.9	even	6
169.2.c.c.22.2		6	65.49	even	6
169.2.c.c.146.2		6	65.4	even	6
169.2.e.b.23.3		12	65.19	odd	12
169.2.e.b.23.4		12	65.59	odd	12
169.2.e.b.147.3		12	65.24	odd	12
169.2.e.b.147.4		12	65.54	odd	12
1521.2.a.o.1.2		3	15.14	odd	2
1521.2.a.r.1.2		3	195.194	odd	2
1521.2.b.l.1351.3		6	195.164	even	4
1521.2.b.l.1351.4		6	195.44	even	4
2704.2.a.z.1.1		3	260.259	odd	2
2704.2.a.ba.1.1		3	20.19	odd	2
2704.2.f.o.337.1		6	260.239	even	4
2704.2.f.o.337.2		6	260.99	even	4
4225.2.a.bb.1.2		3	1.1	even	1	trivial
4225.2.a.bg.1.2		3	13.12	even	2
8281.2.a.bf.1.2		3	455.454	odd	2
8281.2.a.bj.1.2		3	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.554958 q^{2} -0.801938 q^{3} -1.69202 q^{4} +0.445042 q^{6} -2.69202 q^{7} +2.04892 q^{8} -2.35690 q^{9} +1.19806 q^{11} +1.35690 q^{12} +1.49396 q^{14} +2.24698 q^{16} -1.13706 q^{17} +1.30798 q^{18} -1.93900 q^{19} +2.15883 q^{21} -0.664874 q^{22} +4.60388 q^{23} -1.64310 q^{24} +4.29590 q^{27} +4.55496 q^{28} -7.89977 q^{29} -5.89977 q^{31} -5.34481 q^{32} -0.960771 q^{33} +0.631023 q^{34} +3.98792 q^{36} +0.951083 q^{37} +1.07606 q^{38} -3.31767 q^{41} -1.19806 q^{42} -7.15883 q^{43} -2.02715 q^{44} -2.55496 q^{46} -7.69202 q^{47} -1.80194 q^{48} +0.246980 q^{49} +0.911854 q^{51} -5.87263 q^{53} -2.38404 q^{54} -5.51573 q^{56} +1.55496 q^{57} +4.38404 q^{58} +0.0120816 q^{59} -8.03684 q^{61} +3.27413 q^{62} +6.34481 q^{63} -1.52781 q^{64} +0.533188 q^{66} -9.25667 q^{67} +1.92394 q^{68} -3.69202 q^{69} +13.7409 q^{71} -4.82908 q^{72} +12.8170 q^{73} -0.527811 q^{74} +3.28083 q^{76} -3.22521 q^{77} +0.807315 q^{79} +3.62565 q^{81} +1.84117 q^{82} -16.3327 q^{83} -3.65279 q^{84} +3.97285 q^{86} +6.33513 q^{87} +2.45473 q^{88} +14.7289 q^{89} -7.78986 q^{92} +4.73125 q^{93} +4.26875 q^{94} +4.28621 q^{96} +3.13169 q^{97} -0.137063 q^{98} -2.82371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 2 q^{3} + q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 8 q^{11} - 5 q^{14} + 2 q^{16} + 2 q^{17} + 9 q^{18} + 4 q^{19} - 2 q^{21} - 3 q^{22} + 5 q^{23} - 9 q^{24} - q^{27} + 14 q^{28} - q^{29}+ \cdots - q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 2 * q^3 + q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 8 * q^11 - 5 * q^14 + 2 * q^16 + 2 * q^17 + 9 * q^18 + 4 * q^19 - 2 * q^21 - 3 * q^22 + 5 * q^23 - 9 * q^24 - q^27 + 14 * q^28 - q^29 + 5 * q^31 + 7 * q^32 + 10 * q^33 - 13 * q^34 - 7 * q^36 + 12 * q^37 - 12 * q^38 + 7 * q^41 - 8 * q^42 - 13 * q^43 - 8 * q^46 - 18 * q^47 - q^48 - 4 * q^49 - q^51 - q^53 + 3 * q^54 - 4 * q^56 + 5 * q^57 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 + 21 * q^68 - 6 * q^69 + 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 21 * q^76 - 8 * q^77 - 5 * q^79 - q^81 + 14 * q^82 - 7 * q^83 + 7 * q^84 + 18 * q^86 + 18 * q^87 - 15 * q^88 + 11 * q^89 + 22 * q^93 + 5 * q^94 + 21 * q^96 + 7 * q^97 + 5 * q^98 - q^99

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