LMFDB - Embedded newform 5225.2.a.l.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7281497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.497517$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51246 q^{2} -0.895533 q^{3} +4.31247 q^{4} -2.24999 q^{6} +0.393051 q^{7} +5.80998 q^{8} -2.19802 q^{9} +O(q^{10})$	\(q+2.51246 q^{2} -0.895533 q^{3} +4.31247 q^{4} -2.24999 q^{6} +0.393051 q^{7} +5.80998 q^{8} -2.19802 q^{9} -1.00000 q^{11} -3.86196 q^{12} -2.40800 q^{13} +0.987526 q^{14} +5.97243 q^{16} -6.11195 q^{17} -5.52244 q^{18} -1.00000 q^{19} -0.351990 q^{21} -2.51246 q^{22} +5.08108 q^{23} -5.20303 q^{24} -6.05000 q^{26} +4.65500 q^{27} +1.69502 q^{28} -2.85297 q^{29} -5.10948 q^{31} +3.38554 q^{32} +0.895533 q^{33} -15.3560 q^{34} -9.47889 q^{36} -10.9804 q^{37} -2.51246 q^{38} +2.15644 q^{39} -8.92443 q^{41} -0.884362 q^{42} +0.585538 q^{43} -4.31247 q^{44} +12.7660 q^{46} +10.7199 q^{47} -5.34851 q^{48} -6.84551 q^{49} +5.47346 q^{51} -10.3844 q^{52} +3.62189 q^{53} +11.6955 q^{54} +2.28362 q^{56} +0.895533 q^{57} -7.16798 q^{58} -4.71198 q^{59} +15.3869 q^{61} -12.8374 q^{62} -0.863934 q^{63} -3.43882 q^{64} +2.24999 q^{66} -8.26896 q^{67} -26.3576 q^{68} -4.55028 q^{69} -1.66748 q^{71} -12.7705 q^{72} -1.82246 q^{73} -27.5879 q^{74} -4.31247 q^{76} -0.393051 q^{77} +5.41798 q^{78} -8.29828 q^{79} +2.42535 q^{81} -22.4223 q^{82} +7.26577 q^{83} -1.51795 q^{84} +1.47114 q^{86} +2.55493 q^{87} -5.80998 q^{88} -15.1675 q^{89} -0.946465 q^{91} +21.9120 q^{92} +4.57571 q^{93} +26.9335 q^{94} -3.03187 q^{96} +15.0484 q^{97} -17.1991 q^{98} +2.19802 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9	$ 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9} - 6 q^{11} - q^{12} + 5 q^{13} - 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{18} - 6 q^{19} - 21 q^{21} - 2 q^{22} - 4 q^{23} - q^{24} - 14 q^{26} + 16 q^{27} - 10 q^{28} - 9 q^{29} - 21 q^{31} + q^{32} - q^{33} - 28 q^{36} + 3 q^{37} - 2 q^{38} + 20 q^{39} - 23 q^{41} - q^{42} - 7 q^{43} - 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} - 16 q^{51} - 13 q^{52} + 17 q^{53} + q^{54} - 2 q^{56} - q^{57} - 23 q^{58} - 29 q^{59} + 17 q^{61} - 2 q^{62} - 6 q^{63} - 18 q^{64} - 8 q^{67} + q^{68} - 38 q^{69} - 12 q^{71} - 13 q^{72} - 2 q^{73} - 37 q^{74} - 4 q^{76} + 5 q^{77} - q^{78} + 3 q^{79} - 2 q^{81} - 24 q^{82} + 11 q^{83} - 3 q^{84} - 12 q^{86} + 12 q^{87} - 12 q^{88} - 22 q^{89} - 18 q^{91} + 15 q^{92} - 18 q^{93} + 22 q^{94} - 17 q^{96} + 2 q^{97} + q^{98} - q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9 - 6 * q^11 - q^12 + 5 * q^13 - 8 * q^14 + 4 * q^16 - q^17 - 6 * q^18 - 6 * q^19 - 21 * q^21 - 2 * q^22 - 4 * q^23 - q^24 - 14 * q^26 + 16 * q^27 - 10 * q^28 - 9 * q^29 - 21 * q^31 + q^32 - q^33 - 28 * q^36 + 3 * q^37 - 2 * q^38 + 20 * q^39 - 23 * q^41 - q^42 - 7 * q^43 - 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 - 16 * q^51 - 13 * q^52 + 17 * q^53 + q^54 - 2 * q^56 - q^57 - 23 * q^58 - 29 * q^59 + 17 * q^61 - 2 * q^62 - 6 * q^63 - 18 * q^64 - 8 * q^67 + q^68 - 38 * q^69 - 12 * q^71 - 13 * q^72 - 2 * q^73 - 37 * q^74 - 4 * q^76 + 5 * q^77 - q^78 + 3 * q^79 - 2 * q^81 - 24 * q^82 + 11 * q^83 - 3 * q^84 - 12 * q^86 + 12 * q^87 - 12 * q^88 - 22 * q^89 - 18 * q^91 + 15 * q^92 - 18 * q^93 + 22 * q^94 - 17 * q^96 + 2 * q^97 + q^98 - 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 $

$2$	2.51246	1.77658	0.888290	−	0.459284i	$-0.151894\pi$
$2$	2.51246	1.77658	0.888290	+	0.459284i	$0.151894\pi$
$3$	−0.895533	−0.517036	−0.258518	−	0.966006i	$-0.583234\pi$
$3$	−0.895533	−0.517036	−0.258518	+	0.966006i	$0.583234\pi$
$4$	4.31247	2.15623
$5$	0	0
$5$	0	0
$6$	−2.24999	−0.918556
$7$	0.393051	0.148559	0.0742796	−	0.997237i	$-0.476334\pi$
$7$	0.393051	0.148559	0.0742796	+	0.997237i	$0.476334\pi$
$8$	5.80998	2.05414
$9$	−2.19802	−0.732673
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.86196	−1.11485
$13$	−2.40800	−0.667858	−0.333929	−	0.942598i	$-0.608375\pi$
$13$	−2.40800	−0.667858	−0.333929	+	0.942598i	$0.608375\pi$
$14$	0.987526	0.263927
$15$	0	0
$16$	5.97243	1.49311
$17$	−6.11195	−1.48237	−0.741183	−	0.671303i	$-0.765735\pi$
$17$	−6.11195	−1.48237	−0.741183	+	0.671303i	$0.765735\pi$
$18$	−5.52244	−1.30165
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.351990	−0.0768106
$22$	−2.51246	−0.535659
$23$	5.08108	1.05948	0.529739	−	0.848161i	$-0.322290\pi$
$23$	5.08108	1.05948	0.529739	+	0.848161i	$0.322290\pi$
$24$	−5.20303	−1.06206
$25$	0	0
$26$	−6.05000	−1.18650
$27$	4.65500	0.895855
$28$	1.69502	0.320328
$29$	−2.85297	−0.529783	−0.264892	−	0.964278i	$-0.585336\pi$
$29$	−2.85297	−0.529783	−0.264892	+	0.964278i	$0.585336\pi$
$30$	0	0
$31$	−5.10948	−0.917690	−0.458845	−	0.888516i	$-0.651737\pi$
$31$	−5.10948	−0.917690	−0.458845	+	0.888516i	$0.651737\pi$
$32$	3.38554	0.598485
$33$	0.895533	0.155892
$34$	−15.3560	−2.63354
$35$	0	0
$36$	−9.47889	−1.57981
$37$	−10.9804	−1.80517	−0.902586	−	0.430509i	$-0.858334\pi$
$37$	−10.9804	−1.80517	−0.902586	+	0.430509i	$0.858334\pi$
$38$	−2.51246	−0.407575
$39$	2.15644	0.345307
$40$	0	0
$41$	−8.92443	−1.39376	−0.696881	−	0.717187i	$-0.745429\pi$
$41$	−8.92443	−1.39376	−0.696881	+	0.717187i	$0.745429\pi$
$42$	−0.884362	−0.136460
$43$	0.585538	0.0892936	0.0446468	−	0.999003i	$-0.485784\pi$
$43$	0.585538	0.0892936	0.0446468	+	0.999003i	$0.485784\pi$
$44$	−4.31247	−0.650129
$45$	0	0
$46$	12.7660	1.88225
$47$	10.7199	1.56366	0.781832	−	0.623489i	$-0.214285\pi$
$47$	10.7199	1.56366	0.781832	+	0.623489i	$0.214285\pi$
$48$	−5.34851	−0.771991
$49$	−6.84551	−0.977930
$50$	0	0
$51$	5.47346	0.766437
$52$	−10.3844	−1.44006
$53$	3.62189	0.497505	0.248753	−	0.968567i	$-0.419979\pi$
$53$	3.62189	0.497505	0.248753	+	0.968567i	$0.419979\pi$
$54$	11.6955	1.59156
$55$	0	0
$56$	2.28362	0.305161
$57$	0.895533	0.118616
$58$	−7.16798	−0.941202
$59$	−4.71198	−0.613448	−0.306724	−	0.951798i	$-0.599233\pi$
$59$	−4.71198	−0.613448	−0.306724	+	0.951798i	$0.599233\pi$
$60$	0	0
$61$	15.3869	1.97009	0.985043	−	0.172307i	$-0.0551220\pi$
$61$	15.3869	1.97009	0.985043	+	0.172307i	$0.0551220\pi$
$62$	−12.8374	−1.63035
$63$	−0.863934	−0.108845
$64$	−3.43882	−0.429852
$65$	0	0
$66$	2.24999	0.276955
$67$	−8.26896	−1.01021	−0.505107	−	0.863057i	$-0.668547\pi$
$67$	−8.26896	−1.01021	−0.505107	+	0.863057i	$0.668547\pi$
$68$	−26.3576	−3.19633
$69$	−4.55028	−0.547789
$70$	0	0
$71$	−1.66748	−0.197893	−0.0989464	−	0.995093i	$-0.531547\pi$
$71$	−1.66748	−0.197893	−0.0989464	+	0.995093i	$0.531547\pi$
$72$	−12.7705	−1.50501
$73$	−1.82246	−0.213303	−0.106651	−	0.994296i	$-0.534013\pi$
$73$	−1.82246	−0.213303	−0.106651	+	0.994296i	$0.534013\pi$
$74$	−27.5879	−3.20703
$75$	0	0
$76$	−4.31247	−0.494674
$77$	−0.393051	−0.0447923
$78$	5.41798	0.613465
$79$	−8.29828	−0.933630	−0.466815	−	0.884355i	$-0.654599\pi$
$79$	−8.29828	−0.933630	−0.466815	+	0.884355i	$0.654599\pi$
$80$	0	0
$81$	2.42535	0.269483
$82$	−22.4223	−2.47613
$83$	7.26577	0.797522	0.398761	−	0.917055i	$-0.369440\pi$
$83$	7.26577	0.797522	0.398761	+	0.917055i	$0.369440\pi$
$84$	−1.51795	−0.165621
$85$	0	0
$86$	1.47114	0.158637
$87$	2.55493	0.273917
$88$	−5.80998	−0.619346
$89$	−15.1675	−1.60775	−0.803875	−	0.594798i	$-0.797232\pi$
$89$	−15.1675	−1.60775	−0.803875	+	0.594798i	$0.797232\pi$
$90$	0	0
$91$	−0.946465	−0.0992165
$92$	21.9120	2.28448
$93$	4.57571	0.474479
$94$	26.9335	2.77797
$95$	0	0
$96$	−3.03187	−0.309438
$97$	15.0484	1.52794	0.763969	−	0.645253i	$-0.223248\pi$
$97$	15.0484	1.52794	0.763969	+	0.645253i	$0.223248\pi$
$98$	−17.1991	−1.73737
$99$	2.19802	0.220909
$100$	0	0
$101$	1.51854	0.151101	0.0755504	−	0.997142i	$-0.475929\pi$
$101$	1.51854	0.151101	0.0755504	+	0.997142i	$0.475929\pi$
$102$	13.7519	1.36164
$103$	12.8186	1.26305	0.631525	−	0.775356i	$-0.282429\pi$
$103$	12.8186	1.26305	0.631525	+	0.775356i	$0.282429\pi$
$104$	−13.9904	−1.37187
$105$	0	0
$106$	9.09987	0.883858
$107$	−0.382403	−0.0369683	−0.0184842	−	0.999829i	$-0.505884\pi$
$107$	−0.382403	−0.0369683	−0.0184842	+	0.999829i	$0.505884\pi$
$108$	20.0745	1.93167
$109$	−13.3758	−1.28117	−0.640586	−	0.767886i	$-0.721309\pi$
$109$	−13.3758	−1.28117	−0.640586	+	0.767886i	$0.721309\pi$
$110$	0	0
$111$	9.83335	0.933340
$112$	2.34747	0.221815
$113$	2.15943	0.203142	0.101571	−	0.994828i	$-0.467613\pi$
$113$	2.15943	0.203142	0.101571	+	0.994828i	$0.467613\pi$
$114$	2.24999	0.210731
$115$	0	0
$116$	−12.3033	−1.14234
$117$	5.29282	0.489322
$118$	−11.8387	−1.08984
$119$	−2.40231	−0.220219
$120$	0	0
$121$	1.00000	0.0909091
$122$	38.6589	3.50001
$123$	7.99213	0.720626
$124$	−22.0345	−1.97875
$125$	0	0
$126$	−2.17060	−0.193372
$127$	−6.10615	−0.541833	−0.270917	−	0.962603i	$-0.587327\pi$
$127$	−6.10615	−0.541833	−0.270917	+	0.962603i	$0.587327\pi$
$128$	−15.4110	−1.36215
$129$	−0.524368	−0.0461681
$130$	0	0
$131$	−5.85495	−0.511549	−0.255775	−	0.966736i	$-0.582330\pi$
$131$	−5.85495	−0.511549	−0.255775	+	0.966736i	$0.582330\pi$
$132$	3.86196	0.336140
$133$	−0.393051	−0.0340818
$134$	−20.7754	−1.79473
$135$	0	0
$136$	−35.5103	−3.04499
$137$	−19.7338	−1.68598	−0.842988	−	0.537933i	$-0.819205\pi$
$137$	−19.7338	−1.68598	−0.842988	+	0.537933i	$0.819205\pi$
$138$	−11.4324	−0.973190
$139$	7.59903	0.644541	0.322271	−	0.946648i	$-0.395554\pi$
$139$	7.59903	0.644541	0.322271	+	0.946648i	$0.395554\pi$
$140$	0	0
$141$	−9.60007	−0.808472
$142$	−4.18947	−0.351572
$143$	2.40800	0.201367
$144$	−13.1275	−1.09396
$145$	0	0
$146$	−4.57886	−0.378949
$147$	6.13038	0.505626
$148$	−47.3528	−3.89237
$149$	−13.7580	−1.12710	−0.563551	−	0.826081i	$-0.690565\pi$
$149$	−13.7580	−1.12710	−0.563551	+	0.826081i	$0.690565\pi$
$150$	0	0
$151$	−5.95309	−0.484456	−0.242228	−	0.970219i	$-0.577878\pi$
$151$	−5.95309	−0.484456	−0.242228	+	0.970219i	$0.577878\pi$
$152$	−5.80998	−0.471252
$153$	13.4342	1.08609
$154$	−0.987526	−0.0795771
$155$	0	0
$156$	9.29958	0.744562
$157$	16.5678	1.32226	0.661129	−	0.750272i	$-0.270078\pi$
$157$	16.5678	1.32226	0.661129	+	0.750272i	$0.270078\pi$
$158$	−20.8491	−1.65867
$159$	−3.24353	−0.257228
$160$	0	0
$161$	1.99712	0.157395
$162$	6.09360	0.478759
$163$	18.0594	1.41452	0.707261	−	0.706952i	$-0.249931\pi$
$163$	18.0594	1.41452	0.707261	+	0.706952i	$0.249931\pi$
$164$	−38.4863	−3.00528
$165$	0	0
$166$	18.2550	1.41686
$167$	18.1669	1.40580	0.702900	−	0.711289i	$-0.251888\pi$
$167$	18.1669	1.40580	0.702900	+	0.711289i	$0.251888\pi$
$168$	−2.04506	−0.157780
$169$	−7.20156	−0.553966
$170$	0	0
$171$	2.19802	0.168087
$172$	2.52511	0.192538
$173$	−15.3509	−1.16711	−0.583553	−	0.812075i	$-0.698338\pi$
$173$	−15.3509	−1.16711	−0.583553	+	0.812075i	$0.698338\pi$
$174$	6.41917	0.486636
$175$	0	0
$176$	−5.97243	−0.450189
$177$	4.21974	0.317175
$178$	−38.1077	−2.85629
$179$	2.98617	0.223197	0.111598	−	0.993753i	$-0.464403\pi$
$179$	2.98617	0.223197	0.111598	+	0.993753i	$0.464403\pi$
$180$	0	0
$181$	−9.39129	−0.698049	−0.349025	−	0.937114i	$-0.613487\pi$
$181$	−9.39129	−0.698049	−0.349025	+	0.937114i	$0.613487\pi$
$182$	−2.37796	−0.176266
$183$	−13.7795	−1.01861
$184$	29.5210	2.17632
$185$	0	0
$186$	11.4963	0.842950
$187$	6.11195	0.446950
$188$	46.2294	3.37162
$189$	1.82965	0.133088
$190$	0	0
$191$	3.95452	0.286139	0.143069	−	0.989713i	$-0.454303\pi$
$191$	3.95452	0.286139	0.143069	+	0.989713i	$0.454303\pi$
$192$	3.07958	0.222249
$193$	21.3183	1.53453	0.767263	−	0.641332i	$-0.221618\pi$
$193$	21.3183	1.53453	0.767263	+	0.641332i	$0.221618\pi$
$194$	37.8086	2.71450
$195$	0	0
$196$	−29.5210	−2.10865
$197$	−9.09357	−0.647890	−0.323945	−	0.946076i	$-0.605009\pi$
$197$	−9.09357	−0.647890	−0.323945	+	0.946076i	$0.605009\pi$
$198$	5.52244	0.392463
$199$	15.4196	1.09306	0.546532	−	0.837438i	$-0.315948\pi$
$199$	15.4196	1.09306	0.546532	+	0.837438i	$0.315948\pi$
$200$	0	0
$201$	7.40513	0.522318
$202$	3.81528	0.268442
$203$	−1.12136	−0.0787042
$204$	23.6041	1.65262
$205$	0	0
$206$	32.2061	2.24391
$207$	−11.1683	−0.776251
$208$	−14.3816	−0.997184
$209$	1.00000	0.0691714
$210$	0	0
$211$	−5.04618	−0.347394	−0.173697	−	0.984799i	$-0.555571\pi$
$211$	−5.04618	−0.347394	−0.173697	+	0.984799i	$0.555571\pi$
$212$	15.6193	1.07274
$213$	1.49328	0.102318
$214$	−0.960774	−0.0656772
$215$	0	0
$216$	27.0455	1.84021
$217$	−2.00829	−0.136331
$218$	−33.6063	−2.27610
$219$	1.63207	0.110285
$220$	0	0
$221$	14.7176	0.990010
$222$	24.7059	1.65815
$223$	−17.8194	−1.19327	−0.596637	−	0.802511i	$-0.703497\pi$
$223$	−17.8194	−1.19327	−0.596637	+	0.802511i	$0.703497\pi$
$224$	1.33069	0.0889105
$225$	0	0
$226$	5.42548	0.360898
$227$	−4.39589	−0.291766	−0.145883	−	0.989302i	$-0.546602\pi$
$227$	−4.39589	−0.291766	−0.145883	+	0.989302i	$0.546602\pi$
$228$	3.86196	0.255764
$229$	−9.42520	−0.622835	−0.311417	−	0.950273i	$-0.600804\pi$
$229$	−9.42520	−0.622835	−0.311417	+	0.950273i	$0.600804\pi$
$230$	0	0
$231$	0.351990	0.0231593
$232$	−16.5757	−1.08825
$233$	−29.1795	−1.91161	−0.955805	−	0.294001i	$-0.905013\pi$
$233$	−29.1795	−1.91161	−0.955805	+	0.294001i	$0.905013\pi$
$234$	13.2980	0.869318
$235$	0	0
$236$	−20.3203	−1.32274
$237$	7.43139	0.482721
$238$	−6.03571	−0.391237
$239$	29.3674	1.89962	0.949810	−	0.312827i	$-0.101276\pi$
$239$	29.3674	1.89962	0.949810	+	0.312827i	$0.101276\pi$
$240$	0	0
$241$	5.76217	0.371174	0.185587	−	0.982628i	$-0.440581\pi$
$241$	5.76217	0.371174	0.185587	+	0.982628i	$0.440581\pi$
$242$	2.51246	0.161507
$243$	−16.1370	−1.03519
$244$	66.3554	4.24797
$245$	0	0
$246$	20.0799	1.28025
$247$	2.40800	0.153217
$248$	−29.6860	−1.88506
$249$	−6.50674	−0.412348
$250$	0	0
$251$	2.11320	0.133384	0.0666922	−	0.997774i	$-0.478755\pi$
$251$	2.11320	0.133384	0.0666922	+	0.997774i	$0.478755\pi$
$252$	−3.72568	−0.234696
$253$	−5.08108	−0.319445
$254$	−15.3415	−0.962610
$255$	0	0
$256$	−31.8419	−1.99012
$257$	2.36217	0.147348	0.0736740	−	0.997282i	$-0.476528\pi$
$257$	2.36217	0.147348	0.0736740	+	0.997282i	$0.476528\pi$
$258$	−1.31746	−0.0820212
$259$	−4.31587	−0.268175
$260$	0	0
$261$	6.27089	0.388158
$262$	−14.7103	−0.908807
$263$	26.7079	1.64688	0.823440	−	0.567403i	$-0.192052\pi$
$263$	26.7079	1.64688	0.823440	+	0.567403i	$0.192052\pi$
$264$	5.20303	0.320225
$265$	0	0
$266$	−0.987526	−0.0605491
$267$	13.5830	0.831265
$268$	−35.6596	−2.17826
$269$	6.15301	0.375156	0.187578	−	0.982250i	$-0.439936\pi$
$269$	6.15301	0.375156	0.187578	+	0.982250i	$0.439936\pi$
$270$	0	0
$271$	27.0433	1.64276	0.821380	−	0.570381i	$-0.193204\pi$
$271$	27.0433	1.64276	0.821380	+	0.570381i	$0.193204\pi$
$272$	−36.5032	−2.21333
$273$	0.847591	0.0512985
$274$	−49.5805	−2.99527
$275$	0	0
$276$	−19.6229	−1.18116
$277$	4.16147	0.250038	0.125019	−	0.992154i	$-0.460101\pi$
$277$	4.16147	0.250038	0.125019	+	0.992154i	$0.460101\pi$
$278$	19.0923	1.14508
$279$	11.2307	0.672367
$280$	0	0
$281$	−5.46692	−0.326129	−0.163065	−	0.986615i	$-0.552138\pi$
$281$	−5.46692	−0.326129	−0.163065	+	0.986615i	$0.552138\pi$
$282$	−24.1198	−1.43631
$283$	10.7672	0.640046	0.320023	−	0.947410i	$-0.396309\pi$
$283$	10.7672	0.640046	0.320023	+	0.947410i	$0.396309\pi$
$284$	−7.19093	−0.426703
$285$	0	0
$286$	6.05000	0.357744
$287$	−3.50776	−0.207056
$288$	−7.44149	−0.438494
$289$	20.3559	1.19741
$290$	0	0
$291$	−13.4764	−0.789999
$292$	−7.85929	−0.459930
$293$	20.9240	1.22240	0.611198	−	0.791478i	$-0.290688\pi$
$293$	20.9240	1.22240	0.611198	+	0.791478i	$0.290688\pi$
$294$	15.4024	0.898284
$295$	0	0
$296$	−63.7961	−3.70808
$297$	−4.65500	−0.270111
$298$	−34.5665	−2.00239
$299$	−12.2352	−0.707581
$300$	0	0
$301$	0.230146	0.0132654
$302$	−14.9569	−0.860674
$303$	−1.35991	−0.0781246
$304$	−5.97243	−0.342542
$305$	0	0
$306$	33.7529	1.92952
$307$	6.74306	0.384847	0.192423	−	0.981312i	$-0.438365\pi$
$307$	6.74306	0.384847	0.192423	+	0.981312i	$0.438365\pi$
$308$	−1.69502	−0.0965827
$309$	−11.4794	−0.653043
$310$	0	0
$311$	20.6800	1.17266	0.586328	−	0.810074i	$-0.300573\pi$
$311$	20.6800	1.17266	0.586328	+	0.810074i	$0.300573\pi$
$312$	12.5289	0.709308
$313$	−27.6570	−1.56326	−0.781632	−	0.623740i	$-0.785613\pi$
$313$	−27.6570	−1.56326	−0.781632	+	0.623740i	$0.785613\pi$
$314$	41.6261	2.34910
$315$	0	0
$316$	−35.7861	−2.01312
$317$	−25.0739	−1.40829	−0.704146	−	0.710055i	$-0.748670\pi$
$317$	−25.0739	−1.40829	−0.704146	+	0.710055i	$0.748670\pi$
$318$	−8.14924	−0.456987
$319$	2.85297	0.159736
$320$	0	0
$321$	0.342455	0.0191140
$322$	5.01769	0.279625
$323$	6.11195	0.340078
$324$	10.4592	0.581069
$325$	0	0
$326$	45.3736	2.51301
$327$	11.9785	0.662413
$328$	−51.8508	−2.86298
$329$	4.21348	0.232297
$330$	0	0
$331$	−25.8106	−1.41868	−0.709338	−	0.704868i	$-0.751006\pi$
$331$	−25.8106	−1.41868	−0.709338	+	0.704868i	$0.751006\pi$
$332$	31.3334	1.71964
$333$	24.1352	1.32260
$334$	45.6437	2.49751
$335$	0	0
$336$	−2.10224	−0.114686
$337$	6.22788	0.339254	0.169627	−	0.985508i	$-0.445744\pi$
$337$	6.22788	0.339254	0.169627	+	0.985508i	$0.445744\pi$
$338$	−18.0936	−0.984164
$339$	−1.93384	−0.105032
$340$	0	0
$341$	5.10948	0.276694
$342$	5.52244	0.298619
$343$	−5.44199	−0.293840
$344$	3.40196	0.183422
$345$	0	0
$346$	−38.5685	−2.07346
$347$	18.8989	1.01455	0.507273	−	0.861785i	$-0.330653\pi$
$347$	18.8989	1.01455	0.507273	+	0.861785i	$0.330653\pi$
$348$	11.0181	0.590630
$349$	15.3598	0.822192	0.411096	−	0.911592i	$-0.365146\pi$
$349$	15.3598	0.822192	0.411096	+	0.911592i	$0.365146\pi$
$350$	0	0
$351$	−11.2092	−0.598304
$352$	−3.38554	−0.180450
$353$	−14.6667	−0.780632	−0.390316	−	0.920681i	$-0.627634\pi$
$353$	−14.6667	−0.780632	−0.390316	+	0.920681i	$0.627634\pi$
$354$	10.6019	0.563486
$355$	0	0
$356$	−65.4093	−3.46668
$357$	2.15135	0.113861
$358$	7.50263	0.396526
$359$	−14.3322	−0.756424	−0.378212	−	0.925719i	$-0.623461\pi$
$359$	−14.3322	−0.756424	−0.378212	+	0.925719i	$0.623461\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−23.5953	−1.24014
$363$	−0.895533	−0.0470033
$364$	−4.08160	−0.213934
$365$	0	0
$366$	−34.6204	−1.80964
$367$	−22.0345	−1.15019	−0.575096	−	0.818086i	$-0.695035\pi$
$367$	−22.0345	−1.15019	−0.575096	+	0.818086i	$0.695035\pi$
$368$	30.3464	1.58192
$369$	19.6161	1.02117
$370$	0	0
$371$	1.42359	0.0739090
$372$	19.7326	1.02309
$373$	−3.73971	−0.193635	−0.0968174	−	0.995302i	$-0.530866\pi$
$373$	−3.73971	−0.193635	−0.0968174	+	0.995302i	$0.530866\pi$
$374$	15.3560	0.794042
$375$	0	0
$376$	62.2827	3.21198
$377$	6.86994	0.353820
$378$	4.59693	0.236441
$379$	−24.0832	−1.23707	−0.618536	−	0.785757i	$-0.712274\pi$
$379$	−24.0832	−1.23707	−0.618536	+	0.785757i	$0.712274\pi$
$380$	0	0
$381$	5.46826	0.280148
$382$	9.93558	0.508348
$383$	−12.7578	−0.651895	−0.325948	−	0.945388i	$-0.605683\pi$
$383$	−12.7578	−0.651895	−0.325948	+	0.945388i	$0.605683\pi$
$384$	13.8011	0.704282
$385$	0	0
$386$	53.5615	2.72621
$387$	−1.28702	−0.0654231
$388$	64.8959	3.29459
$389$	24.2926	1.23169	0.615843	−	0.787869i	$-0.288816\pi$
$389$	24.2926	1.23169	0.615843	+	0.787869i	$0.288816\pi$
$390$	0	0
$391$	−31.0553	−1.57053
$392$	−39.7723	−2.00880
$393$	5.24330	0.264490
$394$	−22.8473	−1.15103
$395$	0	0
$396$	9.47889	0.476332
$397$	−18.3098	−0.918945	−0.459472	−	0.888192i	$-0.651962\pi$
$397$	−18.3098	−0.918945	−0.459472	+	0.888192i	$0.651962\pi$
$398$	38.7411	1.94191
$399$	0.351990	0.0176216
$400$	0	0
$401$	−35.5628	−1.77592	−0.887960	−	0.459921i	$-0.847878\pi$
$401$	−35.5628	−1.77592	−0.887960	+	0.459921i	$0.847878\pi$
$402$	18.6051	0.927939
$403$	12.3036	0.612886
$404$	6.54867	0.325808
$405$	0	0
$406$	−2.81738	−0.139824
$407$	10.9804	0.544280
$408$	31.8007	1.57437
$409$	−8.90544	−0.440345	−0.220173	−	0.975461i	$-0.570662\pi$
$409$	−8.90544	−0.440345	−0.220173	+	0.975461i	$0.570662\pi$
$410$	0	0
$411$	17.6723	0.871711
$412$	55.2796	2.72343
$413$	−1.85205	−0.0911334
$414$	−28.0600	−1.37907
$415$	0	0
$416$	−8.15237	−0.399703
$417$	−6.80519	−0.333251
$418$	2.51246	0.122889
$419$	−17.3759	−0.848870	−0.424435	−	0.905458i	$-0.639527\pi$
$419$	−17.3759	−0.848870	−0.424435	+	0.905458i	$0.639527\pi$
$420$	0	0
$421$	−1.93051	−0.0940871	−0.0470436	−	0.998893i	$-0.514980\pi$
$421$	−1.93051	−0.0940871	−0.0470436	+	0.998893i	$0.514980\pi$
$422$	−12.6783	−0.617172
$423$	−23.5626	−1.14566
$424$	21.0431	1.02195
$425$	0	0
$426$	3.75181	0.181776
$427$	6.04782	0.292675
$428$	−1.64910	−0.0797123
$429$	−2.15644	−0.104114
$430$	0	0
$431$	6.59444	0.317643	0.158821	−	0.987307i	$-0.449231\pi$
$431$	6.59444	0.317643	0.158821	+	0.987307i	$0.449231\pi$
$432$	27.8017	1.33761
$433$	9.80105	0.471009	0.235504	−	0.971873i	$-0.424326\pi$
$433$	9.80105	0.471009	0.235504	+	0.971873i	$0.424326\pi$
$434$	−5.04574	−0.242203
$435$	0	0
$436$	−57.6828	−2.76251
$437$	−5.08108	−0.243061
$438$	4.10052	0.195930
$439$	−9.33721	−0.445641	−0.222820	−	0.974860i	$-0.571526\pi$
$439$	−9.33721	−0.445641	−0.222820	+	0.974860i	$0.571526\pi$
$440$	0	0
$441$	15.0466	0.716503
$442$	36.9773	1.75883
$443$	−14.4161	−0.684932	−0.342466	−	0.939530i	$-0.611262\pi$
$443$	−14.4161	−0.684932	−0.342466	+	0.939530i	$0.611262\pi$
$444$	42.4060	2.01250
$445$	0	0
$446$	−44.7706	−2.11995
$447$	12.3208	0.582753
$448$	−1.35163	−0.0638585
$449$	−7.81495	−0.368810	−0.184405	−	0.982850i	$-0.559036\pi$
$449$	−7.81495	−0.368810	−0.184405	+	0.982850i	$0.559036\pi$
$450$	0	0
$451$	8.92443	0.420235
$452$	9.31247	0.438022
$453$	5.33119	0.250481
$454$	−11.0445	−0.518344
$455$	0	0
$456$	5.20303	0.243654
$457$	−4.96675	−0.232335	−0.116167	−	0.993230i	$-0.537061\pi$
$457$	−4.96675	−0.232335	−0.116167	+	0.993230i	$0.537061\pi$
$458$	−23.6805	−1.10652
$459$	−28.4511	−1.32799
$460$	0	0
$461$	6.08665	0.283484	0.141742	−	0.989904i	$-0.454730\pi$
$461$	6.08665	0.283484	0.141742	+	0.989904i	$0.454730\pi$
$462$	0.884362	0.0411442
$463$	9.56761	0.444644	0.222322	−	0.974973i	$-0.428636\pi$
$463$	9.56761	0.444644	0.222322	+	0.974973i	$0.428636\pi$
$464$	−17.0392	−0.791024
$465$	0	0
$466$	−73.3123	−3.39613
$467$	−26.0909	−1.20734	−0.603671	−	0.797234i	$-0.706296\pi$
$467$	−26.0909	−1.20734	−0.603671	+	0.797234i	$0.706296\pi$
$468$	22.8251	1.05509
$469$	−3.25012	−0.150077
$470$	0	0
$471$	−14.8371	−0.683656
$472$	−27.3765	−1.26011
$473$	−0.585538	−0.0269230
$474$	18.6711	0.857591
$475$	0	0
$476$	−10.3599	−0.474844
$477$	−7.96099	−0.364509
$478$	73.7845	3.37483
$479$	−2.29386	−0.104809	−0.0524046	−	0.998626i	$-0.516689\pi$
$479$	−2.29386	−0.104809	−0.0524046	+	0.998626i	$0.516689\pi$
$480$	0	0
$481$	26.4408	1.20560
$482$	14.4772	0.659420
$483$	−1.78849	−0.0813791
$484$	4.31247	0.196021
$485$	0	0
$486$	−40.5436	−1.83909
$487$	9.76343	0.442423	0.221212	−	0.975226i	$-0.428999\pi$
$487$	9.76343	0.442423	0.221212	+	0.975226i	$0.428999\pi$
$488$	89.3975	4.04683
$489$	−16.1728	−0.731359
$490$	0	0
$491$	19.7257	0.890210	0.445105	−	0.895478i	$-0.353166\pi$
$491$	19.7257	0.890210	0.445105	+	0.895478i	$0.353166\pi$
$492$	34.4658	1.55384
$493$	17.4372	0.785333
$494$	6.05000	0.272202
$495$	0	0
$496$	−30.5160	−1.37021
$497$	−0.655403	−0.0293988
$498$	−16.3479	−0.732569
$499$	−11.8920	−0.532358	−0.266179	−	0.963924i	$-0.585761\pi$
$499$	−11.8920	−0.532358	−0.266179	+	0.963924i	$0.585761\pi$
$500$	0	0
$501$	−16.2691	−0.726850
$502$	5.30935	0.236968
$503$	27.1187	1.20916	0.604582	−	0.796543i	$-0.293340\pi$
$503$	27.1187	1.20916	0.604582	+	0.796543i	$0.293340\pi$
$504$	−5.01944	−0.223584
$505$	0	0
$506$	−12.7660	−0.567519
$507$	6.44924	0.286421
$508$	−26.3326	−1.16832
$509$	23.2095	1.02874	0.514371	−	0.857568i	$-0.328026\pi$
$509$	23.2095	1.02874	0.514371	+	0.857568i	$0.328026\pi$
$510$	0	0
$511$	−0.716319	−0.0316881
$512$	−49.1795	−2.17345
$513$	−4.65500	−0.205523
$514$	5.93486	0.261775
$515$	0	0
$516$	−2.26132	−0.0995491
$517$	−10.7199	−0.471463
$518$	−10.8435	−0.476434
$519$	13.7472	0.603436
$520$	0	0
$521$	21.4122	0.938084	0.469042	−	0.883176i	$-0.344599\pi$
$521$	21.4122	0.938084	0.469042	+	0.883176i	$0.344599\pi$
$522$	15.7554	0.689594
$523$	−37.7739	−1.65174	−0.825868	−	0.563863i	$-0.809314\pi$
$523$	−37.7739	−1.65174	−0.825868	+	0.563863i	$0.809314\pi$
$524$	−25.2493	−1.10302
$525$	0	0
$526$	67.1026	2.92581
$527$	31.2289	1.36035
$528$	5.34851	0.232764
$529$	2.81736	0.122494
$530$	0	0
$531$	10.3570	0.449457
$532$	−1.69502	−0.0734884
$533$	21.4900	0.930835
$534$	34.1267	1.47681
$535$	0	0
$536$	−48.0425	−2.07512
$537$	−2.67421	−0.115401
$538$	15.4592	0.666494
$539$	6.84551	0.294857
$540$	0	0
$541$	41.1230	1.76802	0.884008	−	0.467471i	$-0.154835\pi$
$541$	41.1230	1.76802	0.884008	+	0.467471i	$0.154835\pi$
$542$	67.9452	2.91849
$543$	8.41021	0.360917
$544$	−20.6923	−0.887174
$545$	0	0
$546$	2.12954	0.0911359
$547$	−2.62583	−0.112272	−0.0561362	−	0.998423i	$-0.517878\pi$
$547$	−2.62583	−0.112272	−0.0561362	+	0.998423i	$0.517878\pi$
$548$	−85.1015	−3.63536
$549$	−33.8206	−1.44343
$550$	0	0
$551$	2.85297	0.121541
$552$	−26.4370	−1.12523
$553$	−3.26165	−0.138699
$554$	10.4555	0.444213
$555$	0	0
$556$	32.7706	1.38978
$557$	−8.24917	−0.349529	−0.174764	−	0.984610i	$-0.555916\pi$
$557$	−8.24917	−0.349529	−0.174764	+	0.984610i	$0.555916\pi$
$558$	28.2168	1.19451
$559$	−1.40997	−0.0596355
$560$	0	0
$561$	−5.47346	−0.231090
$562$	−13.7354	−0.579394
$563$	−36.6270	−1.54364	−0.771822	−	0.635839i	$-0.780654\pi$
$563$	−36.6270	−1.54364	−0.771822	+	0.635839i	$0.780654\pi$
$564$	−41.4000	−1.74325
$565$	0	0
$566$	27.0523	1.13709
$567$	0.953286	0.0400343
$568$	−9.68800	−0.406500
$569$	12.9209	0.541672	0.270836	−	0.962625i	$-0.412700\pi$
$569$	12.9209	0.541672	0.270836	+	0.962625i	$0.412700\pi$
$570$	0	0
$571$	4.98632	0.208671	0.104335	−	0.994542i	$-0.466728\pi$
$571$	4.98632	0.208671	0.104335	+	0.994542i	$0.466728\pi$
$572$	10.3844	0.434194
$573$	−3.54140	−0.147944
$574$	−8.81310	−0.367852
$575$	0	0
$576$	7.55859	0.314941
$577$	−5.45951	−0.227283	−0.113641	−	0.993522i	$-0.536251\pi$
$577$	−5.45951	−0.227283	−0.113641	+	0.993522i	$0.536251\pi$
$578$	51.1436	2.12729
$579$	−19.0913	−0.793406
$580$	0	0
$581$	2.85582	0.118479
$582$	−33.8589	−1.40350
$583$	−3.62189	−0.150004
$584$	−10.5885	−0.438153
$585$	0	0
$586$	52.5708	2.17168
$587$	−11.7076	−0.483223	−0.241611	−	0.970373i	$-0.577676\pi$
$587$	−11.7076	−0.483223	−0.241611	+	0.970373i	$0.577676\pi$
$588$	26.4371	1.09025
$589$	5.10948	0.210533
$590$	0	0
$591$	8.14360	0.334983
$592$	−65.5799	−2.69532
$593$	46.4536	1.90762	0.953810	−	0.300409i	$-0.0971232\pi$
$593$	46.4536	1.90762	0.953810	+	0.300409i	$0.0971232\pi$
$594$	−11.6955	−0.479873
$595$	0	0
$596$	−59.3311	−2.43029
$597$	−13.8087	−0.565154
$598$	−30.7405	−1.25707
$599$	−34.8106	−1.42232	−0.711161	−	0.703029i	$-0.751830\pi$
$599$	−34.8106	−1.42232	−0.711161	+	0.703029i	$0.751830\pi$
$600$	0	0
$601$	20.7514	0.846467	0.423234	−	0.906021i	$-0.360895\pi$
$601$	20.7514	0.846467	0.423234	+	0.906021i	$0.360895\pi$
$602$	0.578233	0.0235670
$603$	18.1753	0.740157
$604$	−25.6725	−1.04460
$605$	0	0
$606$	−3.41671	−0.138795
$607$	−48.3127	−1.96095	−0.980476	−	0.196639i	$-0.936997\pi$
$607$	−48.3127	−1.96095	−0.980476	+	0.196639i	$0.936997\pi$
$608$	−3.38554	−0.137302
$609$	1.00422	0.0406930
$610$	0	0
$611$	−25.8136	−1.04431
$612$	57.9345	2.34186
$613$	7.19281	0.290515	0.145257	−	0.989394i	$-0.453599\pi$
$613$	7.19281	0.290515	0.145257	+	0.989394i	$0.453599\pi$
$614$	16.9417	0.683711
$615$	0	0
$616$	−2.28362	−0.0920096
$617$	−28.1591	−1.13364	−0.566822	−	0.823840i	$-0.691827\pi$
$617$	−28.1591	−1.13364	−0.566822	+	0.823840i	$0.691827\pi$
$618$	−28.8417	−1.16018
$619$	−41.9363	−1.68556	−0.842781	−	0.538257i	$-0.819083\pi$
$619$	−41.9363	−1.68556	−0.842781	+	0.538257i	$0.819083\pi$
$620$	0	0
$621$	23.6524	0.949139
$622$	51.9578	2.08332
$623$	−5.96159	−0.238846
$624$	12.8792	0.515580
$625$	0	0
$626$	−69.4871	−2.77726
$627$	−0.895533	−0.0357642
$628$	71.4483	2.85110
$629$	67.1119	2.67593
$630$	0	0
$631$	13.9367	0.554811	0.277405	−	0.960753i	$-0.410526\pi$
$631$	13.9367	0.554811	0.277405	+	0.960753i	$0.410526\pi$
$632$	−48.2129	−1.91781
$633$	4.51903	0.179615
$634$	−62.9973	−2.50194
$635$	0	0
$636$	−13.9876	−0.554644
$637$	16.4840	0.653118
$638$	7.16798	0.283783
$639$	3.66514	0.144991
$640$	0	0
$641$	30.0930	1.18860	0.594302	−	0.804242i	$-0.297428\pi$
$641$	30.0930	1.18860	0.594302	+	0.804242i	$0.297428\pi$
$642$	0.860405	0.0339575
$643$	27.1236	1.06965	0.534825	−	0.844963i	$-0.320377\pi$
$643$	27.1236	1.06965	0.534825	+	0.844963i	$0.320377\pi$
$644$	8.61252	0.339381
$645$	0	0
$646$	15.3560	0.604176
$647$	−22.4779	−0.883697	−0.441848	−	0.897090i	$-0.645677\pi$
$647$	−22.4779	−0.883697	−0.441848	+	0.897090i	$0.645677\pi$
$648$	14.0913	0.553557
$649$	4.71198	0.184962
$650$	0	0
$651$	1.79849	0.0704883
$652$	77.8806	3.05004
$653$	20.3218	0.795252	0.397626	−	0.917548i	$-0.369834\pi$
$653$	20.3218	0.795252	0.397626	+	0.917548i	$0.369834\pi$
$654$	30.0955	1.17683
$655$	0	0
$656$	−53.3005	−2.08104
$657$	4.00580	0.156281
$658$	10.5862	0.412694
$659$	31.2540	1.21748	0.608742	−	0.793368i	$-0.291675\pi$
$659$	31.2540	1.21748	0.608742	+	0.793368i	$0.291675\pi$
$660$	0	0
$661$	39.9155	1.55253	0.776266	−	0.630405i	$-0.217111\pi$
$661$	39.9155	1.55253	0.776266	+	0.630405i	$0.217111\pi$
$662$	−64.8480	−2.52039
$663$	−13.1801	−0.511871
$664$	42.2140	1.63822
$665$	0	0
$666$	60.6388	2.34971
$667$	−14.4962	−0.561294
$668$	78.3443	3.03123
$669$	15.9579	0.616967
$670$	0	0
$671$	−15.3869	−0.594004
$672$	−1.19168	−0.0459700
$673$	−4.79686	−0.184905	−0.0924526	−	0.995717i	$-0.529471\pi$
$673$	−4.79686	−0.184905	−0.0924526	+	0.995717i	$0.529471\pi$
$674$	15.6473	0.602712
$675$	0	0
$676$	−31.0565	−1.19448
$677$	−1.99692	−0.0767479	−0.0383739	−	0.999263i	$-0.512218\pi$
$677$	−1.99692	−0.0767479	−0.0383739	+	0.999263i	$0.512218\pi$
$678$	−4.85870	−0.186597
$679$	5.91480	0.226989
$680$	0	0
$681$	3.93667	0.150853
$682$	12.8374	0.491569
$683$	19.7294	0.754923	0.377462	−	0.926025i	$-0.376797\pi$
$683$	19.7294	0.754923	0.377462	+	0.926025i	$0.376797\pi$
$684$	9.47889	0.362434
$685$	0	0
$686$	−13.6728	−0.522030
$687$	8.44058	0.322028
$688$	3.49708	0.133325
$689$	−8.72150	−0.332263
$690$	0	0
$691$	−17.8975	−0.680853	−0.340427	−	0.940271i	$-0.610572\pi$
$691$	−17.8975	−0.680853	−0.340427	+	0.940271i	$0.610572\pi$
$692$	−66.2002	−2.51655
$693$	0.863934	0.0328181
$694$	47.4828	1.80242
$695$	0	0
$696$	14.8441	0.562664
$697$	54.5457	2.06606
$698$	38.5910	1.46069
$699$	26.1312	0.988372
$700$	0	0
$701$	32.4459	1.22546	0.612732	−	0.790291i	$-0.290071\pi$
$701$	32.4459	1.22546	0.612732	+	0.790291i	$0.290071\pi$
$702$	−28.1627	−1.06293
$703$	10.9804	0.414135
$704$	3.43882	0.129605
$705$	0	0
$706$	−36.8496	−1.38685
$707$	0.596865	0.0224474
$708$	18.1975	0.683903
$709$	21.3057	0.800151	0.400076	−	0.916482i	$-0.368984\pi$
$709$	21.3057	0.800151	0.400076	+	0.916482i	$0.368984\pi$
$710$	0	0
$711$	18.2398	0.684045
$712$	−88.1228	−3.30254
$713$	−25.9617	−0.972272
$714$	5.40518	0.202284
$715$	0	0
$716$	12.8777	0.481264
$717$	−26.2995	−0.982173
$718$	−36.0091	−1.34385
$719$	−4.98470	−0.185898	−0.0929489	−	0.995671i	$-0.529629\pi$
$719$	−4.98470	−0.185898	−0.0929489	+	0.995671i	$0.529629\pi$
$720$	0	0
$721$	5.03834	0.187638
$722$	2.51246	0.0935042
$723$	−5.16022	−0.191911
$724$	−40.4996	−1.50516
$725$	0	0
$726$	−2.24999	−0.0835051
$727$	36.3622	1.34860	0.674300	−	0.738458i	$-0.264446\pi$
$727$	36.3622	1.34860	0.674300	+	0.738458i	$0.264446\pi$
$728$	−5.49895	−0.203804
$729$	7.17516	0.265747
$730$	0	0
$731$	−3.57878	−0.132366
$732$	−59.4234	−2.19635
$733$	−13.6271	−0.503329	−0.251665	−	0.967814i	$-0.580978\pi$
$733$	−13.6271	−0.503329	−0.251665	+	0.967814i	$0.580978\pi$
$734$	−55.3609	−2.04341
$735$	0	0
$736$	17.2022	0.634082
$737$	8.26896	0.304591
$738$	49.2846	1.81419
$739$	7.55834	0.278038	0.139019	−	0.990290i	$-0.455605\pi$
$739$	7.55834	0.278038	0.139019	+	0.990290i	$0.455605\pi$
$740$	0	0
$741$	−2.15644	−0.0792188
$742$	3.57671	0.131305
$743$	−5.79027	−0.212425	−0.106212	−	0.994343i	$-0.533872\pi$
$743$	−5.79027	−0.212425	−0.106212	+	0.994343i	$0.533872\pi$
$744$	26.5848	0.974646
$745$	0	0
$746$	−9.39588	−0.344008
$747$	−15.9703	−0.584323
$748$	26.3576	0.963729
$749$	−0.150304	−0.00549199
$750$	0	0
$751$	−15.7138	−0.573406	−0.286703	−	0.958019i	$-0.592559\pi$
$751$	−15.7138	−0.573406	−0.286703	+	0.958019i	$0.592559\pi$
$752$	64.0241	2.33472
$753$	−1.89245	−0.0689646
$754$	17.2605	0.628589
$755$	0	0
$756$	7.89031	0.286968
$757$	22.9951	0.835773	0.417886	−	0.908499i	$-0.362771\pi$
$757$	22.9951	0.835773	0.417886	+	0.908499i	$0.362771\pi$
$758$	−60.5082	−2.19776
$759$	4.55028	0.165165
$760$	0	0
$761$	4.29367	0.155645	0.0778227	−	0.996967i	$-0.475203\pi$
$761$	4.29367	0.155645	0.0778227	+	0.996967i	$0.475203\pi$
$762$	13.7388	0.497704
$763$	−5.25738	−0.190330
$764$	17.0537	0.616982
$765$	0	0
$766$	−32.0536	−1.15814
$767$	11.3464	0.409696
$768$	28.5155	1.02896
$769$	−29.8296	−1.07568	−0.537842	−	0.843046i	$-0.680760\pi$
$769$	−29.8296	−1.07568	−0.537842	+	0.843046i	$0.680760\pi$
$770$	0	0
$771$	−2.11540	−0.0761843
$772$	91.9345	3.30880
$773$	41.3201	1.48618	0.743090	−	0.669192i	$-0.233360\pi$
$773$	41.3201	1.48618	0.743090	+	0.669192i	$0.233360\pi$
$774$	−3.23360	−0.116229
$775$	0	0
$776$	87.4312	3.13860
$777$	3.86501	0.138656
$778$	61.0344	2.18819
$779$	8.92443	0.319751
$780$	0	0
$781$	1.66748	0.0596669
$782$	−78.0253	−2.79018
$783$	−13.2806	−0.474609
$784$	−40.8843	−1.46016
$785$	0	0
$786$	13.1736	0.469887
$787$	45.9086	1.63646	0.818232	−	0.574889i	$-0.194955\pi$
$787$	45.9086	1.63646	0.818232	+	0.574889i	$0.194955\pi$
$788$	−39.2157	−1.39700
$789$	−23.9178	−0.851497
$790$	0	0
$791$	0.848766	0.0301786
$792$	12.7705	0.453779
$793$	−37.0515	−1.31574
$794$	−46.0028	−1.63258
$795$	0	0
$796$	66.4963	2.35690
$797$	29.6572	1.05051	0.525256	−	0.850944i	$-0.323969\pi$
$797$	29.6572	1.05051	0.525256	+	0.850944i	$0.323969\pi$
$798$	0.884362	0.0313061
$799$	−65.5198	−2.31792
$800$	0	0
$801$	33.3384	1.17796
$802$	−89.3501	−3.15506
$803$	1.82246	0.0643131
$804$	31.9344	1.12624
$805$	0	0
$806$	30.9123	1.08884
$807$	−5.51023	−0.193969
$808$	8.82272	0.310382
$809$	−36.7733	−1.29288	−0.646440	−	0.762965i	$-0.723743\pi$
$809$	−36.7733	−1.29288	−0.646440	+	0.762965i	$0.723743\pi$
$810$	0	0
$811$	−31.4798	−1.10541	−0.552703	−	0.833378i	$-0.686404\pi$
$811$	−31.4798	−1.10541	−0.552703	+	0.833378i	$0.686404\pi$
$812$	−4.83584	−0.169705
$813$	−24.2181	−0.849367
$814$	27.5879	0.966956
$815$	0	0
$816$	32.6898	1.14437
$817$	−0.585538	−0.0204854
$818$	−22.3746	−0.782308
$819$	2.08035	0.0726933
$820$	0	0
$821$	−40.8925	−1.42716	−0.713580	−	0.700574i	$-0.752927\pi$
$821$	−40.8925	−1.42716	−0.713580	+	0.700574i	$0.752927\pi$
$822$	44.4010	1.54866
$823$	−1.58148	−0.0551268	−0.0275634	−	0.999620i	$-0.508775\pi$
$823$	−1.58148	−0.0551268	−0.0275634	+	0.999620i	$0.508775\pi$
$824$	74.4756	2.59448
$825$	0	0
$826$	−4.65320	−0.161906
$827$	34.9721	1.21610	0.608050	−	0.793899i	$-0.291952\pi$
$827$	34.9721	1.21610	0.608050	+	0.793899i	$0.291952\pi$
$828$	−48.1630	−1.67378
$829$	−5.53605	−0.192275	−0.0961375	−	0.995368i	$-0.530649\pi$
$829$	−5.53605	−0.192275	−0.0961375	+	0.995368i	$0.530649\pi$
$830$	0	0
$831$	−3.72674	−0.129279
$832$	8.28066	0.287080
$833$	41.8394	1.44965
$834$	−17.0978	−0.592047
$835$	0	0
$836$	4.31247	0.149150
$837$	−23.7846	−0.822117
$838$	−43.6564	−1.50808
$839$	54.4122	1.87852	0.939259	−	0.343208i	$-0.111514\pi$
$839$	54.4122	1.87852	0.939259	+	0.343208i	$0.111514\pi$
$840$	0	0
$841$	−20.8606	−0.719330
$842$	−4.85032	−0.167153
$843$	4.89581	0.168621
$844$	−21.7615	−0.749062
$845$	0	0
$846$	−59.2003	−2.03535
$847$	0.393051	0.0135054
$848$	21.6315	0.742829
$849$	−9.64243	−0.330927
$850$	0	0
$851$	−55.7925	−1.91254
$852$	6.43972	0.220621
$853$	−13.1887	−0.451574	−0.225787	−	0.974177i	$-0.572495\pi$
$853$	−13.1887	−0.451574	−0.225787	+	0.974177i	$0.572495\pi$
$854$	15.1949	0.519960
$855$	0	0
$856$	−2.22176	−0.0759381
$857$	−15.1757	−0.518390	−0.259195	−	0.965825i	$-0.583457\pi$
$857$	−15.1757	−0.518390	−0.259195	+	0.965825i	$0.583457\pi$
$858$	−5.41798	−0.184967
$859$	14.1708	0.483500	0.241750	−	0.970339i	$-0.422279\pi$
$859$	14.1708	0.483500	0.241750	+	0.970339i	$0.422279\pi$
$860$	0	0
$861$	3.14131	0.107056
$862$	16.5683	0.564318
$863$	41.8781	1.42555	0.712773	−	0.701394i	$-0.247439\pi$
$863$	41.8781	1.42555	0.712773	+	0.701394i	$0.247439\pi$
$864$	15.7597	0.536156
$865$	0	0
$866$	24.6248	0.836784
$867$	−18.2294	−0.619104
$868$	−8.66067	−0.293962
$869$	8.29828	0.281500
$870$	0	0
$871$	19.9116	0.674679
$872$	−77.7134	−2.63171
$873$	−33.0768	−1.11948
$874$	−12.7660	−0.431817
$875$	0	0
$876$	7.03826	0.237801
$877$	25.2042	0.851084	0.425542	−	0.904939i	$-0.360083\pi$
$877$	25.2042	0.851084	0.425542	+	0.904939i	$0.360083\pi$
$878$	−23.4594	−0.791716
$879$	−18.7382	−0.632023
$880$	0	0
$881$	−13.4877	−0.454412	−0.227206	−	0.973847i	$-0.572959\pi$
$881$	−13.4877	−0.454412	−0.227206	+	0.973847i	$0.572959\pi$
$882$	37.8039	1.27292
$883$	−14.5766	−0.490543	−0.245271	−	0.969454i	$-0.578877\pi$
$883$	−14.5766	−0.490543	−0.245271	+	0.969454i	$0.578877\pi$
$884$	63.4689	2.13469
$885$	0	0
$886$	−36.2200	−1.21684
$887$	32.5010	1.09128	0.545638	−	0.838021i	$-0.316287\pi$
$887$	32.5010	1.09128	0.545638	+	0.838021i	$0.316287\pi$
$888$	57.1316	1.91721
$889$	−2.40003	−0.0804944
$890$	0	0
$891$	−2.42535	−0.0812523
$892$	−76.8455	−2.57298
$893$	−10.7199	−0.358729
$894$	30.9555	1.03531
$895$	0	0
$896$	−6.05730	−0.202360
$897$	10.9570	0.365845
$898$	−19.6348	−0.655221
$899$	14.5772	0.486177
$900$	0	0
$901$	−22.1368	−0.737485
$902$	22.4223	0.746581
$903$	−0.206103	−0.00685869
$904$	12.5462	0.417282
$905$	0	0
$906$	13.3944	0.445000
$907$	7.14326	0.237188	0.118594	−	0.992943i	$-0.462161\pi$
$907$	7.14326	0.237188	0.118594	+	0.992943i	$0.462161\pi$
$908$	−18.9571	−0.629114
$909$	−3.33779	−0.110707
$910$	0	0
$911$	−26.7545	−0.886418	−0.443209	−	0.896418i	$-0.646160\pi$
$911$	−26.7545	−0.886418	−0.443209	+	0.896418i	$0.646160\pi$
$912$	5.34851	0.177107
$913$	−7.26577	−0.240462
$914$	−12.4788	−0.412761
$915$	0	0
$916$	−40.6459	−1.34298
$917$	−2.30129	−0.0759954
$918$	−71.4824	−2.35927
$919$	−41.4968	−1.36885	−0.684426	−	0.729082i	$-0.739947\pi$
$919$	−41.4968	−1.36885	−0.684426	+	0.729082i	$0.739947\pi$
$920$	0	0
$921$	−6.03864	−0.198980
$922$	15.2925	0.503631
$923$	4.01527	0.132164
$924$	1.51795	0.0499368
$925$	0	0
$926$	24.0383	0.789946
$927$	−28.1754	−0.925403
$928$	−9.65885	−0.317067
$929$	1.12610	0.0369461	0.0184731	−	0.999829i	$-0.494120\pi$
$929$	1.12610	0.0369461	0.0184731	+	0.999829i	$0.494120\pi$
$930$	0	0
$931$	6.84551	0.224353
$932$	−125.835	−4.12188
$933$	−18.5196	−0.606306
$934$	−65.5523	−2.14494
$935$	0	0
$936$	30.7512	1.00513
$937$	40.0060	1.30694	0.653470	−	0.756952i	$-0.273313\pi$
$937$	40.0060	1.30694	0.653470	+	0.756952i	$0.273313\pi$
$938$	−8.16581	−0.266623
$939$	24.7677	0.808265
$940$	0	0
$941$	19.0713	0.621706	0.310853	−	0.950458i	$-0.399385\pi$
$941$	19.0713	0.621706	0.310853	+	0.950458i	$0.399385\pi$
$942$	−37.2775	−1.21457
$943$	−45.3457	−1.47666
$944$	−28.1420	−0.915944
$945$	0	0
$946$	−1.47114	−0.0478309
$947$	−17.6474	−0.573464	−0.286732	−	0.958011i	$-0.592569\pi$
$947$	−17.6474	−0.573464	−0.286732	+	0.958011i	$0.592569\pi$
$948$	32.0476	1.04086
$949$	4.38847	0.142456
$950$	0	0
$951$	22.4545	0.728138
$952$	−13.9574	−0.452361
$953$	−25.3932	−0.822566	−0.411283	−	0.911508i	$-0.634919\pi$
$953$	−25.3932	−0.822566	−0.411283	+	0.911508i	$0.634919\pi$
$954$	−20.0017	−0.647579
$955$	0	0
$956$	126.646	4.09602
$957$	−2.55493	−0.0825892
$958$	−5.76324	−0.186202
$959$	−7.75640	−0.250467
$960$	0	0
$961$	−4.89320	−0.157845
$962$	66.4316	2.14184
$963$	0.840530	0.0270857
$964$	24.8492	0.800338
$965$	0	0
$966$	−4.49351	−0.144576
$967$	−28.2497	−0.908450	−0.454225	−	0.890887i	$-0.650084\pi$
$967$	−28.2497	−0.908450	−0.454225	+	0.890887i	$0.650084\pi$
$968$	5.80998	0.186740
$969$	−5.47346	−0.175833
$970$	0	0
$971$	−4.11498	−0.132056	−0.0660279	−	0.997818i	$-0.521033\pi$
$971$	−4.11498	−0.132056	−0.0660279	+	0.997818i	$0.521033\pi$
$972$	−69.5902	−2.23211
$973$	2.98681	0.0957526
$974$	24.5303	0.786000
$975$	0	0
$976$	91.8970	2.94155
$977$	−28.5510	−0.913428	−0.456714	−	0.889613i	$-0.650974\pi$
$977$	−28.5510	−0.913428	−0.456714	+	0.889613i	$0.650974\pi$
$978$	−40.6336	−1.29932
$979$	15.1675	0.484755
$980$	0	0
$981$	29.4003	0.938681
$982$	49.5602	1.58153
$983$	11.0911	0.353753	0.176876	−	0.984233i	$-0.443401\pi$
$983$	11.0911	0.353753	0.176876	+	0.984233i	$0.443401\pi$
$984$	46.4341	1.48027
$985$	0	0
$986$	43.8104	1.39521
$987$	−3.77332	−0.120106
$988$	10.3844	0.330372
$989$	2.97516	0.0946047
$990$	0	0
$991$	10.2116	0.324381	0.162191	−	0.986759i	$-0.448144\pi$
$991$	10.2116	0.324381	0.162191	+	0.986759i	$0.448144\pi$
$992$	−17.2984	−0.549223
$993$	23.1142	0.733508
$994$	−1.64667	−0.0522293
$995$	0	0
$996$	−28.0601	−0.889118
$997$	−14.3457	−0.454334	−0.227167	−	0.973856i	$-0.572946\pi$
$997$	−14.3457	−0.454334	−0.227167	+	0.973856i	$0.572946\pi$
$998$	−29.8781	−0.945775
$999$	−51.1139	−1.61717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.l.1.6		6
5.4	even	2		1045.2.a.f.1.1	✓	6
15.14	odd	2		9405.2.a.z.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.1	✓	6	5.4	even	2
5225.2.a.l.1.6		6	1.1	even	1	trivial
9405.2.a.z.1.6		6	15.14	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:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51246 q^{2} -0.895533 q^{3} +4.31247 q^{4} -2.24999 q^{6} +0.393051 q^{7} +5.80998 q^{8} -2.19802 q^{9} +O(q^{10})\)	\(q+2.51246 q^{2} -0.895533 q^{3} +4.31247 q^{4} -2.24999 q^{6} +0.393051 q^{7} +5.80998 q^{8} -2.19802 q^{9} -1.00000 q^{11} -3.86196 q^{12} -2.40800 q^{13} +0.987526 q^{14} +5.97243 q^{16} -6.11195 q^{17} -5.52244 q^{18} -1.00000 q^{19} -0.351990 q^{21} -2.51246 q^{22} +5.08108 q^{23} -5.20303 q^{24} -6.05000 q^{26} +4.65500 q^{27} +1.69502 q^{28} -2.85297 q^{29} -5.10948 q^{31} +3.38554 q^{32} +0.895533 q^{33} -15.3560 q^{34} -9.47889 q^{36} -10.9804 q^{37} -2.51246 q^{38} +2.15644 q^{39} -8.92443 q^{41} -0.884362 q^{42} +0.585538 q^{43} -4.31247 q^{44} +12.7660 q^{46} +10.7199 q^{47} -5.34851 q^{48} -6.84551 q^{49} +5.47346 q^{51} -10.3844 q^{52} +3.62189 q^{53} +11.6955 q^{54} +2.28362 q^{56} +0.895533 q^{57} -7.16798 q^{58} -4.71198 q^{59} +15.3869 q^{61} -12.8374 q^{62} -0.863934 q^{63} -3.43882 q^{64} +2.24999 q^{66} -8.26896 q^{67} -26.3576 q^{68} -4.55028 q^{69} -1.66748 q^{71} -12.7705 q^{72} -1.82246 q^{73} -27.5879 q^{74} -4.31247 q^{76} -0.393051 q^{77} +5.41798 q^{78} -8.29828 q^{79} +2.42535 q^{81} -22.4223 q^{82} +7.26577 q^{83} -1.51795 q^{84} +1.47114 q^{86} +2.55493 q^{87} -5.80998 q^{88} -15.1675 q^{89} -0.946465 q^{91} +21.9120 q^{92} +4.57571 q^{93} +26.9335 q^{94} -3.03187 q^{96} +15.0484 q^{97} -17.1991 q^{98} +2.19802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9	\( 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9} - 6 q^{11} - q^{12} + 5 q^{13} - 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{18} - 6 q^{19} - 21 q^{21} - 2 q^{22} - 4 q^{23} - q^{24} - 14 q^{26} + 16 q^{27} - 10 q^{28} - 9 q^{29} - 21 q^{31} + q^{32} - q^{33} - 28 q^{36} + 3 q^{37} - 2 q^{38} + 20 q^{39} - 23 q^{41} - q^{42} - 7 q^{43} - 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} - 16 q^{51} - 13 q^{52} + 17 q^{53} + q^{54} - 2 q^{56} - q^{57} - 23 q^{58} - 29 q^{59} + 17 q^{61} - 2 q^{62} - 6 q^{63} - 18 q^{64} - 8 q^{67} + q^{68} - 38 q^{69} - 12 q^{71} - 13 q^{72} - 2 q^{73} - 37 q^{74} - 4 q^{76} + 5 q^{77} - q^{78} + 3 q^{79} - 2 q^{81} - 24 q^{82} + 11 q^{83} - 3 q^{84} - 12 q^{86} + 12 q^{87} - 12 q^{88} - 22 q^{89} - 18 q^{91} + 15 q^{92} - 18 q^{93} + 22 q^{94} - 17 q^{96} + 2 q^{97} + q^{98} - q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9 - 6 * q^11 - q^12 + 5 * q^13 - 8 * q^14 + 4 * q^16 - q^17 - 6 * q^18 - 6 * q^19 - 21 * q^21 - 2 * q^22 - 4 * q^23 - q^24 - 14 * q^26 + 16 * q^27 - 10 * q^28 - 9 * q^29 - 21 * q^31 + q^32 - q^33 - 28 * q^36 + 3 * q^37 - 2 * q^38 + 20 * q^39 - 23 * q^41 - q^42 - 7 * q^43 - 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 - 16 * q^51 - 13 * q^52 + 17 * q^53 + q^54 - 2 * q^56 - q^57 - 23 * q^58 - 29 * q^59 + 17 * q^61 - 2 * q^62 - 6 * q^63 - 18 * q^64 - 8 * q^67 + q^68 - 38 * q^69 - 12 * q^71 - 13 * q^72 - 2 * q^73 - 37 * q^74 - 4 * q^76 + 5 * q^77 - q^78 + 3 * q^79 - 2 * q^81 - 24 * q^82 + 11 * q^83 - 3 * q^84 - 12 * q^86 + 12 * q^87 - 12 * q^88 - 22 * q^89 - 18 * q^91 + 15 * q^92 - 18 * q^93 + 22 * q^94 - 17 * q^96 + 2 * q^97 + q^98 - q^99

Introduction

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Database

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