LMFDB - Embedded newform 4025.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, 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("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.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:	$32.1397868136$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.22545.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 5x + 4 $ x^4 - x^3 - 6x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.33866$ of defining polynomial
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46934 q^{2} -3.43628 q^{3} +4.09762 q^{4} -8.48532 q^{6} +1.00000 q^{7} +5.17972 q^{8} +8.80800 q^{9} +O(q^{10})$	\(q+2.46934 q^{2} -3.43628 q^{3} +4.09762 q^{4} -8.48532 q^{6} +1.00000 q^{7} +5.17972 q^{8} +8.80800 q^{9} +3.46934 q^{11} -14.0805 q^{12} +6.33866 q^{13} +2.46934 q^{14} +4.59522 q^{16} -4.14666 q^{17} +21.7499 q^{18} +0.869326 q^{19} -3.43628 q^{21} +8.56695 q^{22} -1.00000 q^{23} -17.7989 q^{24} +15.6523 q^{26} -19.9579 q^{27} +4.09762 q^{28} +6.67732 q^{29} +3.54665 q^{31} +0.987711 q^{32} -11.9216 q^{33} -10.2395 q^{34} +36.0918 q^{36} -3.20799 q^{37} +2.14666 q^{38} -21.7814 q^{39} -5.38770 q^{41} -8.48532 q^{42} +0.497606 q^{43} +14.2160 q^{44} -2.46934 q^{46} +3.13067 q^{47} -15.7905 q^{48} +1.00000 q^{49} +14.2491 q^{51} +25.9734 q^{52} -6.83627 q^{53} -49.2827 q^{54} +5.17972 q^{56} -2.98724 q^{57} +16.4885 q^{58} +3.66134 q^{59} -10.0363 q^{61} +8.75786 q^{62} +8.80800 q^{63} -6.75145 q^{64} -29.4384 q^{66} +12.1499 q^{67} -16.9914 q^{68} +3.43628 q^{69} +12.2646 q^{71} +45.6229 q^{72} -2.29763 q^{73} -7.92159 q^{74} +3.56216 q^{76} +3.46934 q^{77} -53.7856 q^{78} +6.30560 q^{79} +42.1568 q^{81} -13.3040 q^{82} +13.8885 q^{83} -14.0805 q^{84} +1.22876 q^{86} -22.9451 q^{87} +17.9702 q^{88} -10.6287 q^{89} +6.33866 q^{91} -4.09762 q^{92} -12.1873 q^{93} +7.73068 q^{94} -3.39405 q^{96} -4.42352 q^{97} +2.46934 q^{98} +30.5579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 11 q^{4} + 4 q^{7} + 18 q^{9}+O(q^{10}) $ 4 * q + q^2 + 11 * q^4 + 4 * q^7 + 18 * q^9	$ 4 q + q^{2} + 11 q^{4} + 4 q^{7} + 18 q^{9} + 5 q^{11} - 21 q^{12} + 17 q^{13} + q^{14} + 17 q^{16} + 9 q^{17} + 24 q^{18} + 4 q^{19} + 20 q^{22} - 4 q^{23} + 12 q^{24} + 5 q^{26} - 9 q^{27} + 11 q^{28} + 10 q^{29} - 2 q^{31} - 34 q^{32} - 18 q^{34} + 45 q^{36} - 5 q^{37} - 17 q^{38} - 9 q^{39} + 7 q^{41} + 6 q^{43} + 13 q^{44} - q^{46} + 12 q^{47} - 51 q^{48} + 4 q^{49} + 18 q^{51} + 43 q^{52} - 23 q^{53} - 60 q^{54} - 9 q^{57} + 4 q^{58} + 23 q^{59} - 17 q^{61} - 17 q^{62} + 18 q^{63} + 16 q^{64} - 21 q^{66} - 5 q^{67} + 33 q^{68} + 20 q^{71} + 57 q^{72} + 15 q^{73} + 16 q^{74} + 8 q^{76} + 5 q^{77} - 39 q^{78} + 12 q^{79} + 24 q^{81} - 20 q^{82} + 3 q^{83} - 21 q^{84} - 36 q^{86} - 18 q^{87} + 39 q^{88} - 11 q^{89} + 17 q^{91} - 11 q^{92} - 27 q^{93} + 21 q^{94} + 3 q^{96} - q^{97} + q^{98} + 42 q^{99}+O(q^{100}) $ 4 * q + q^2 + 11 * q^4 + 4 * q^7 + 18 * q^9 + 5 * q^11 - 21 * q^12 + 17 * q^13 + q^14 + 17 * q^16 + 9 * q^17 + 24 * q^18 + 4 * q^19 + 20 * q^22 - 4 * q^23 + 12 * q^24 + 5 * q^26 - 9 * q^27 + 11 * q^28 + 10 * q^29 - 2 * q^31 - 34 * q^32 - 18 * q^34 + 45 * q^36 - 5 * q^37 - 17 * q^38 - 9 * q^39 + 7 * q^41 + 6 * q^43 + 13 * q^44 - q^46 + 12 * q^47 - 51 * q^48 + 4 * q^49 + 18 * q^51 + 43 * q^52 - 23 * q^53 - 60 * q^54 - 9 * q^57 + 4 * q^58 + 23 * q^59 - 17 * q^61 - 17 * q^62 + 18 * q^63 + 16 * q^64 - 21 * q^66 - 5 * q^67 + 33 * q^68 + 20 * q^71 + 57 * q^72 + 15 * q^73 + 16 * q^74 + 8 * q^76 + 5 * q^77 - 39 * q^78 + 12 * q^79 + 24 * q^81 - 20 * q^82 + 3 * q^83 - 21 * q^84 - 36 * q^86 - 18 * q^87 + 39 * q^88 - 11 * q^89 + 17 * q^91 - 11 * q^92 - 27 * q^93 + 21 * q^94 + 3 * q^96 - q^97 + q^98 + 42 * 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 $

$2$	2.46934	1.74608	0.873042	−	0.487645i	$-0.162144\pi$
$2$	2.46934	1.74608	0.873042	+	0.487645i	$0.162144\pi$
$3$	−3.43628	−1.98394	−0.991968	−	0.126492i	$-0.959628\pi$
$3$	−3.43628	−1.98394	−0.991968	+	0.126492i	$0.959628\pi$
$4$	4.09762	2.04881
$5$	0	0
$5$	0	0
$6$	−8.48532	−3.46412
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	5.17972	1.83131
$9$	8.80800	2.93600
$10$	0	0
$11$	3.46934	1.04604	0.523022	−	0.852319i	$-0.324805\pi$
$11$	3.46934	1.04604	0.523022	+	0.852319i	$0.324805\pi$
$12$	−14.0805	−4.06470
$13$	6.33866	1.75803	0.879014	−	0.476796i	$-0.158202\pi$
$13$	6.33866	1.75803	0.879014	+	0.476796i	$0.158202\pi$
$14$	2.46934	0.659958
$15$	0	0
$16$	4.59522	1.14881
$17$	−4.14666	−1.00571	−0.502856	−	0.864370i	$-0.667717\pi$
$17$	−4.14666	−1.00571	−0.502856	+	0.864370i	$0.667717\pi$
$18$	21.7499	5.12650
$19$	0.869326	0.199437	0.0997185	−	0.995016i	$-0.468206\pi$
$19$	0.869326	0.199437	0.0997185	+	0.995016i	$0.468206\pi$
$20$	0	0
$21$	−3.43628	−0.749857
$22$	8.56695	1.82648
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−17.7989	−3.63319
$25$	0	0
$26$	15.6523	3.06966
$27$	−19.9579	−3.84090
$28$	4.09762	0.774377
$29$	6.67732	1.23995	0.619974	−	0.784623i	$-0.287143\pi$
$29$	6.67732	1.23995	0.619974	+	0.784623i	$0.287143\pi$
$30$	0	0
$31$	3.54665	0.636997	0.318498	−	0.947923i	$-0.396821\pi$
$31$	3.54665	0.636997	0.318498	+	0.947923i	$0.396821\pi$
$32$	0.987711	0.174604
$33$	−11.9216	−2.07528
$34$	−10.2395	−1.75606
$35$	0	0
$36$	36.0918	6.01530
$37$	−3.20799	−0.527390	−0.263695	−	0.964606i	$-0.584941\pi$
$37$	−3.20799	−0.527390	−0.263695	+	0.964606i	$0.584941\pi$
$38$	2.14666	0.348234
$39$	−21.7814	−3.48781
$40$	0	0
$41$	−5.38770	−0.841418	−0.420709	−	0.907196i	$-0.638219\pi$
$41$	−5.38770	−0.841418	−0.420709	+	0.907196i	$0.638219\pi$
$42$	−8.48532	−1.30931
$43$	0.497606	0.0758843	0.0379421	−	0.999280i	$-0.487920\pi$
$43$	0.497606	0.0758843	0.0379421	+	0.999280i	$0.487920\pi$
$44$	14.2160	2.14314
$45$	0	0
$46$	−2.46934	−0.364084
$47$	3.13067	0.456656	0.228328	−	0.973584i	$-0.426674\pi$
$47$	3.13067	0.456656	0.228328	+	0.973584i	$0.426674\pi$
$48$	−15.7905	−2.27916
$49$	1.00000	0.142857
$50$	0	0
$51$	14.2491	1.99527
$52$	25.9734	3.60186
$53$	−6.83627	−0.939034	−0.469517	−	0.882924i	$-0.655572\pi$
$53$	−6.83627	−0.939034	−0.469517	+	0.882924i	$0.655572\pi$
$54$	−49.2827	−6.70652
$55$	0	0
$56$	5.17972	0.692169
$57$	−2.98724	−0.395670
$58$	16.4885	2.16505
$59$	3.66134	0.476666	0.238333	−	0.971184i	$-0.423399\pi$
$59$	3.66134	0.476666	0.238333	+	0.971184i	$0.423399\pi$
$60$	0	0
$61$	−10.0363	−1.28501	−0.642507	−	0.766280i	$-0.722106\pi$
$61$	−10.0363	−1.28501	−0.642507	+	0.766280i	$0.722106\pi$
$62$	8.75786	1.11225
$63$	8.80800	1.10970
$64$	−6.75145	−0.843932
$65$	0	0
$66$	−29.4384	−3.62362
$67$	12.1499	1.48434	0.742172	−	0.670209i	$-0.233796\pi$
$67$	12.1499	1.48434	0.742172	+	0.670209i	$0.233796\pi$
$68$	−16.9914	−2.06051
$69$	3.43628	0.413679
$70$	0	0
$71$	12.2646	1.45554	0.727769	−	0.685823i	$-0.240557\pi$
$71$	12.2646	1.45554	0.727769	+	0.685823i	$0.240557\pi$
$72$	45.6229	5.37671
$73$	−2.29763	−0.268918	−0.134459	−	0.990919i	$-0.542930\pi$
$73$	−2.29763	−0.268918	−0.134459	+	0.990919i	$0.542930\pi$
$74$	−7.92159	−0.920867
$75$	0	0
$76$	3.56216	0.408608
$77$	3.46934	0.395367
$78$	−53.7856	−6.09001
$79$	6.30560	0.709436	0.354718	−	0.934973i	$-0.384577\pi$
$79$	6.30560	0.709436	0.354718	+	0.934973i	$0.384577\pi$
$80$	0	0
$81$	42.1568	4.68409
$82$	−13.3040	−1.46919
$83$	13.8885	1.52446	0.762232	−	0.647303i	$-0.224103\pi$
$83$	13.8885	1.52446	0.762232	+	0.647303i	$0.224103\pi$
$84$	−14.0805	−1.53631
$85$	0	0
$86$	1.22876	0.132500
$87$	−22.9451	−2.45998
$88$	17.9702	1.91563
$89$	−10.6287	−1.12665	−0.563323	−	0.826237i	$-0.690477\pi$
$89$	−10.6287	−1.12665	−0.563323	+	0.826237i	$0.690477\pi$
$90$	0	0
$91$	6.33866	0.664472
$92$	−4.09762	−0.427206
$93$	−12.1873	−1.26376
$94$	7.73068	0.797359
$95$	0	0
$96$	−3.39405	−0.346404
$97$	−4.42352	−0.449140	−0.224570	−	0.974458i	$-0.572098\pi$
$97$	−4.42352	−0.449140	−0.224570	+	0.974458i	$0.572098\pi$
$98$	2.46934	0.249441
$99$	30.5579	3.07118
$100$	0	0
$101$	4.35620	0.433458	0.216729	−	0.976232i	$-0.430461\pi$
$101$	4.35620	0.433458	0.216729	+	0.976232i	$0.430461\pi$
$102$	35.1857	3.48390
$103$	15.0347	1.48142	0.740708	−	0.671827i	$-0.234490\pi$
$103$	15.0347	1.48142	0.740708	+	0.671827i	$0.234490\pi$
$104$	32.8325	3.21949
$105$	0	0
$106$	−16.8810	−1.63963
$107$	−4.29441	−0.415156	−0.207578	−	0.978218i	$-0.566558\pi$
$107$	−4.29441	−0.415156	−0.207578	+	0.978218i	$0.566558\pi$
$108$	−81.7797	−7.86926
$109$	−8.97496	−0.859645	−0.429822	−	0.902913i	$-0.641424\pi$
$109$	−8.97496	−0.859645	−0.429822	+	0.902913i	$0.641424\pi$
$110$	0	0
$111$	11.0235	1.04631
$112$	4.59522	0.434208
$113$	4.71361	0.443419	0.221709	−	0.975113i	$-0.428836\pi$
$113$	4.71361	0.443419	0.221709	+	0.975113i	$0.428836\pi$
$114$	−7.37651	−0.690873
$115$	0	0
$116$	27.3611	2.54041
$117$	55.8309	5.16157
$118$	9.04107	0.832298
$119$	−4.14666	−0.380123
$120$	0	0
$121$	1.03629	0.0942078
$122$	−24.7830	−2.24374
$123$	18.5136	1.66932
$124$	14.5328	1.30508
$125$	0	0
$126$	21.7499	1.93763
$127$	−2.96741	−0.263315	−0.131657	−	0.991295i	$-0.542030\pi$
$127$	−2.96741	−0.263315	−0.131657	+	0.991295i	$0.542030\pi$
$128$	−18.6470	−1.64818
$129$	−1.70991	−0.150549
$130$	0	0
$131$	4.48209	0.391602	0.195801	−	0.980644i	$-0.437269\pi$
$131$	4.48209	0.391602	0.195801	+	0.980644i	$0.437269\pi$
$132$	−48.8501	−4.25186
$133$	0.869326	0.0753801
$134$	30.0021	2.59179
$135$	0	0
$136$	−21.4785	−1.84177
$137$	−12.7734	−1.09130	−0.545652	−	0.838012i	$-0.683718\pi$
$137$	−12.7734	−1.09130	−0.545652	+	0.838012i	$0.683718\pi$
$138$	8.48532	0.722318
$139$	12.8363	1.08876	0.544379	−	0.838839i	$-0.316765\pi$
$139$	12.8363	1.08876	0.544379	+	0.838839i	$0.316765\pi$
$140$	0	0
$141$	−10.7579	−0.905975
$142$	30.2853	2.54149
$143$	21.9909	1.83897
$144$	40.4747	3.37289
$145$	0	0
$146$	−5.67363	−0.469553
$147$	−3.43628	−0.283419
$148$	−13.1451	−1.08052
$149$	3.30607	0.270844	0.135422	−	0.990788i	$-0.456761\pi$
$149$	3.30607	0.270844	0.135422	+	0.990788i	$0.456761\pi$
$150$	0	0
$151$	4.00755	0.326130	0.163065	−	0.986615i	$-0.447862\pi$
$151$	4.00755	0.326130	0.163065	+	0.986615i	$0.447862\pi$
$152$	4.50286	0.365230
$153$	−36.5237	−2.95277
$154$	8.56695	0.690345
$155$	0	0
$156$	−89.2518	−7.14586
$157$	1.24427	0.0993038	0.0496519	−	0.998767i	$-0.484189\pi$
$157$	1.24427	0.0993038	0.0496519	+	0.998767i	$0.484189\pi$
$158$	15.5706	1.23873
$159$	23.4913	1.86298
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	104.099	8.17881
$163$	8.29331	0.649582	0.324791	−	0.945786i	$-0.394706\pi$
$163$	8.29331	0.649582	0.324791	+	0.945786i	$0.394706\pi$
$164$	−22.0767	−1.72390
$165$	0	0
$166$	34.2954	2.66184
$167$	17.0443	1.31893	0.659464	−	0.751736i	$-0.270783\pi$
$167$	17.0443	1.31893	0.659464	+	0.751736i	$0.270783\pi$
$168$	−17.7989	−1.37322
$169$	27.1786	2.09066
$170$	0	0
$171$	7.65702	0.585547
$172$	2.03900	0.155472
$173$	3.88531	0.295395	0.147697	−	0.989033i	$-0.452814\pi$
$173$	3.88531	0.295395	0.147697	+	0.989033i	$0.452814\pi$
$174$	−56.6592	−4.29532
$175$	0	0
$176$	15.9424	1.20170
$177$	−12.5814	−0.945674
$178$	−26.2459	−1.96722
$179$	−1.08533	−0.0811211	−0.0405606	−	0.999177i	$-0.512914\pi$
$179$	−1.08533	−0.0811211	−0.0405606	+	0.999177i	$0.512914\pi$
$180$	0	0
$181$	8.78295	0.652832	0.326416	−	0.945226i	$-0.394159\pi$
$181$	8.78295	0.652832	0.326416	+	0.945226i	$0.394159\pi$
$182$	15.6523	1.16022
$183$	34.4875	2.54939
$184$	−5.17972	−0.381854
$185$	0	0
$186$	−30.0944	−2.20663
$187$	−14.3861	−1.05202
$188$	12.8283	0.935600
$189$	−19.9579	−1.45172
$190$	0	0
$191$	−20.2129	−1.46255	−0.731276	−	0.682081i	$-0.761075\pi$
$191$	−20.2129	−1.46255	−0.731276	+	0.682081i	$0.761075\pi$
$192$	23.1999	1.67431
$193$	0.338661	0.0243773	0.0121887	−	0.999926i	$-0.496120\pi$
$193$	0.338661	0.0243773	0.0121887	+	0.999926i	$0.496120\pi$
$194$	−10.9232	−0.784237
$195$	0	0
$196$	4.09762	0.292687
$197$	6.12589	0.436451	0.218226	−	0.975898i	$-0.429973\pi$
$197$	6.12589	0.436451	0.218226	+	0.975898i	$0.429973\pi$
$198$	75.4577	5.36254
$199$	−27.1073	−1.92159	−0.960793	−	0.277268i	$-0.910571\pi$
$199$	−27.1073	−1.92159	−0.960793	+	0.277268i	$0.910571\pi$
$200$	0	0
$201$	−41.7504	−2.94484
$202$	10.7569	0.756855
$203$	6.67732	0.468656
$204$	58.3872	4.08792
$205$	0	0
$206$	37.1258	2.58668
$207$	−8.80800	−0.612198
$208$	29.1276	2.01963
$209$	3.01598	0.208620
$210$	0	0
$211$	−1.98568	−0.136700	−0.0683501	−	0.997661i	$-0.521773\pi$
$211$	−1.98568	−0.136700	−0.0683501	+	0.997661i	$0.521773\pi$
$212$	−28.0124	−1.92390
$213$	−42.1445	−2.88769
$214$	−10.6043	−0.724897
$215$	0	0
$216$	−103.376	−7.03386
$217$	3.54665	0.240762
$218$	−22.1622	−1.50101
$219$	7.89530	0.533515
$220$	0	0
$221$	−26.2843	−1.76807
$222$	27.2208	1.82694
$223$	12.1061	0.810681	0.405341	−	0.914166i	$-0.367153\pi$
$223$	12.1061	0.810681	0.405341	+	0.914166i	$0.367153\pi$
$224$	0.987711	0.0659942
$225$	0	0
$226$	11.6395	0.774246
$227$	−25.4570	−1.68964	−0.844822	−	0.535048i	$-0.820294\pi$
$227$	−25.4570	−1.68964	−0.844822	+	0.535048i	$0.820294\pi$
$228$	−12.2406	−0.810652
$229$	22.9296	1.51523	0.757616	−	0.652701i	$-0.226364\pi$
$229$	22.9296	1.51523	0.757616	+	0.652701i	$0.226364\pi$
$230$	0	0
$231$	−11.9216	−0.784383
$232$	34.5866	2.27072
$233$	17.4715	1.14459	0.572297	−	0.820046i	$-0.306052\pi$
$233$	17.4715	1.14459	0.572297	+	0.820046i	$0.306052\pi$
$234$	137.865	9.01253
$235$	0	0
$236$	15.0028	0.976596
$237$	−21.6678	−1.40747
$238$	−10.2395	−0.663727
$239$	−30.1818	−1.95230	−0.976151	−	0.217094i	$-0.930342\pi$
$239$	−30.1818	−1.95230	−0.976151	+	0.217094i	$0.930342\pi$
$240$	0	0
$241$	−17.6113	−1.13444	−0.567220	−	0.823566i	$-0.691981\pi$
$241$	−17.6113	−1.13444	−0.567220	+	0.823566i	$0.691981\pi$
$242$	2.55894	0.164495
$243$	−84.9888	−5.45203
$244$	−41.1248	−2.63275
$245$	0	0
$246$	45.7164	2.91477
$247$	5.51036	0.350616
$248$	18.3706	1.16654
$249$	−47.7248	−3.02444
$250$	0	0
$251$	−11.2144	−0.707849	−0.353925	−	0.935274i	$-0.615153\pi$
$251$	−11.2144	−0.707849	−0.353925	+	0.935274i	$0.615153\pi$
$252$	36.0918	2.27357
$253$	−3.46934	−0.218115
$254$	−7.32753	−0.459770
$255$	0	0
$256$	−32.5428	−2.03393
$257$	−5.89328	−0.367613	−0.183806	−	0.982962i	$-0.558842\pi$
$257$	−5.89328	−0.367613	−0.183806	+	0.982962i	$0.558842\pi$
$258$	−4.22235	−0.262872
$259$	−3.20799	−0.199335
$260$	0	0
$261$	58.8138	3.64048
$262$	11.0678	0.683770
$263$	28.3606	1.74879	0.874396	−	0.485214i	$-0.161258\pi$
$263$	28.3606	1.74879	0.874396	+	0.485214i	$0.161258\pi$
$264$	−61.7505	−3.80048
$265$	0	0
$266$	2.14666	0.131620
$267$	36.5233	2.23519
$268$	49.7856	3.04114
$269$	−24.5348	−1.49591	−0.747957	−	0.663747i	$-0.768965\pi$
$269$	−24.5348	−1.49591	−0.747957	+	0.663747i	$0.768965\pi$
$270$	0	0
$271$	28.2292	1.71480	0.857402	−	0.514648i	$-0.172077\pi$
$271$	28.2292	1.71480	0.857402	+	0.514648i	$0.172077\pi$
$272$	−19.0548	−1.15537
$273$	−21.7814	−1.31827
$274$	−31.5417	−1.90551
$275$	0	0
$276$	14.0805	0.847549
$277$	−7.58293	−0.455614	−0.227807	−	0.973706i	$-0.573156\pi$
$277$	−7.58293	−0.455614	−0.227807	+	0.973706i	$0.573156\pi$
$278$	31.6970	1.90106
$279$	31.2389	1.87022
$280$	0	0
$281$	12.0443	0.718500	0.359250	−	0.933241i	$-0.383033\pi$
$281$	12.0443	0.718500	0.359250	+	0.933241i	$0.383033\pi$
$282$	−26.5648	−1.58191
$283$	18.3627	1.09155	0.545773	−	0.837933i	$-0.316236\pi$
$283$	18.3627	1.09155	0.545773	+	0.837933i	$0.316236\pi$
$284$	50.2555	2.98212
$285$	0	0
$286$	54.3030	3.21100
$287$	−5.38770	−0.318026
$288$	8.69976	0.512638
$289$	0.194764	0.0114567
$290$	0	0
$291$	15.2004	0.891066
$292$	−9.41482	−0.550961
$293$	5.00953	0.292660	0.146330	−	0.989236i	$-0.453254\pi$
$293$	5.00953	0.292660	0.146330	+	0.989236i	$0.453254\pi$
$294$	−8.48532	−0.494874
$295$	0	0
$296$	−16.6165	−0.965812
$297$	−69.2406	−4.01775
$298$	8.16379	0.472916
$299$	−6.33866	−0.366574
$300$	0	0
$301$	0.497606	0.0286816
$302$	9.89598	0.569449
$303$	−14.9691	−0.859954
$304$	3.99475	0.229114
$305$	0	0
$306$	−90.1893	−5.15578
$307$	19.2069	1.09620	0.548098	−	0.836414i	$-0.315352\pi$
$307$	19.2069	1.09620	0.548098	+	0.836414i	$0.315352\pi$
$308$	14.2160	0.810032
$309$	−51.6635	−2.93903
$310$	0	0
$311$	−6.35943	−0.360610	−0.180305	−	0.983611i	$-0.557709\pi$
$311$	−6.35943	−0.360610	−0.180305	+	0.983611i	$0.557709\pi$
$312$	−112.821	−6.38725
$313$	10.9894	0.621156	0.310578	−	0.950548i	$-0.399477\pi$
$313$	10.9894	0.621156	0.310578	+	0.950548i	$0.399477\pi$
$314$	3.07253	0.173393
$315$	0	0
$316$	25.8379	1.45350
$317$	−34.7551	−1.95204	−0.976022	−	0.217673i	$-0.930153\pi$
$317$	−34.7551	−1.95204	−0.976022	+	0.217673i	$0.930153\pi$
$318$	58.0079	3.25292
$319$	23.1659	1.29704
$320$	0	0
$321$	14.7568	0.823643
$322$	−2.46934	−0.137611
$323$	−3.60480	−0.200576
$324$	172.742	9.59680
$325$	0	0
$326$	20.4790	1.13423
$327$	30.8404	1.70548
$328$	−27.9068	−1.54089
$329$	3.13067	0.172600
$330$	0	0
$331$	−25.7376	−1.41466	−0.707332	−	0.706881i	$-0.750101\pi$
$331$	−25.7376	−1.41466	−0.707332	+	0.706881i	$0.750101\pi$
$332$	56.9099	3.12334
$333$	−28.2559	−1.54842
$334$	42.0881	2.30296
$335$	0	0
$336$	−15.7905	−0.861440
$337$	4.72115	0.257178	0.128589	−	0.991698i	$-0.458955\pi$
$337$	4.72115	0.257178	0.128589	+	0.991698i	$0.458955\pi$
$338$	67.1131	3.65047
$339$	−16.1973	−0.879714
$340$	0	0
$341$	12.3045	0.666327
$342$	18.9077	1.02241
$343$	1.00000	0.0539949
$344$	2.57746	0.138967
$345$	0	0
$346$	9.59413	0.515784
$347$	−6.61552	−0.355140	−0.177570	−	0.984108i	$-0.556824\pi$
$347$	−6.61552	−0.355140	−0.177570	+	0.984108i	$0.556824\pi$
$348$	−94.0203	−5.04002
$349$	8.32866	0.445823	0.222912	−	0.974839i	$-0.428444\pi$
$349$	8.32866	0.445823	0.222912	+	0.974839i	$0.428444\pi$
$350$	0	0
$351$	−126.506	−6.75240
$352$	3.42670	0.182644
$353$	2.66410	0.141796	0.0708978	−	0.997484i	$-0.477414\pi$
$353$	2.66410	0.141796	0.0708978	+	0.997484i	$0.477414\pi$
$354$	−31.0676	−1.65123
$355$	0	0
$356$	−43.5525	−2.30828
$357$	14.2491	0.754140
$358$	−2.68004	−0.141644
$359$	8.62672	0.455301	0.227650	−	0.973743i	$-0.426896\pi$
$359$	8.62672	0.455301	0.227650	+	0.973743i	$0.426896\pi$
$360$	0	0
$361$	−18.2443	−0.960225
$362$	21.6881	1.13990
$363$	−3.56096	−0.186902
$364$	25.9734	1.36138
$365$	0	0
$366$	85.1611	4.45144
$367$	−12.2097	−0.637339	−0.318669	−	0.947866i	$-0.603236\pi$
$367$	−12.2097	−0.637339	−0.318669	+	0.947866i	$0.603236\pi$
$368$	−4.59522	−0.239543
$369$	−47.4549	−2.47040
$370$	0	0
$371$	−6.83627	−0.354921
$372$	−49.9387	−2.58920
$373$	17.5344	0.907895	0.453947	−	0.891029i	$-0.350015\pi$
$373$	17.5344	0.907895	0.453947	+	0.891029i	$0.350015\pi$
$374$	−35.5242	−1.83691
$375$	0	0
$376$	16.2160	0.836276
$377$	42.3253	2.17986
$378$	−49.2827	−2.53483
$379$	−10.8869	−0.559221	−0.279610	−	0.960114i	$-0.590205\pi$
$379$	−10.8869	−0.559221	−0.279610	+	0.960114i	$0.590205\pi$
$380$	0	0
$381$	10.1968	0.522400
$382$	−49.9124	−2.55374
$383$	12.2822	0.627592	0.313796	−	0.949490i	$-0.398399\pi$
$383$	12.2822	0.627592	0.313796	+	0.949490i	$0.398399\pi$
$384$	64.0763	3.26988
$385$	0	0
$386$	0.836267	0.0425649
$387$	4.38292	0.222796
$388$	−18.1259	−0.920202
$389$	1.84537	0.0935642	0.0467821	−	0.998905i	$-0.485103\pi$
$389$	1.84537	0.0935642	0.0467821	+	0.998905i	$0.485103\pi$
$390$	0	0
$391$	4.14666	0.209705
$392$	5.17972	0.261615
$393$	−15.4017	−0.776913
$394$	15.1269	0.762081
$395$	0	0
$396$	125.214	6.29226
$397$	−7.44262	−0.373535	−0.186767	−	0.982404i	$-0.559801\pi$
$397$	−7.44262	−0.373535	−0.186767	+	0.982404i	$0.559801\pi$
$398$	−66.9370	−3.35525
$399$	−2.98724	−0.149549
$400$	0	0
$401$	−5.63583	−0.281440	−0.140720	−	0.990049i	$-0.544942\pi$
$401$	−5.63583	−0.281440	−0.140720	+	0.990049i	$0.544942\pi$
$402$	−103.096	−5.14194
$403$	22.4810	1.11986
$404$	17.8500	0.888073
$405$	0	0
$406$	16.4885	0.818313
$407$	−11.1296	−0.551673
$408$	73.8061	3.65395
$409$	1.75297	0.0866787	0.0433393	−	0.999060i	$-0.486200\pi$
$409$	1.75297	0.0866787	0.0433393	+	0.999060i	$0.486200\pi$
$410$	0	0
$411$	43.8929	2.16507
$412$	61.6065	3.03514
$413$	3.66134	0.180163
$414$	−21.7499	−1.06895
$415$	0	0
$416$	6.26077	0.306959
$417$	−44.1090	−2.16003
$418$	7.44747	0.364268
$419$	−25.6268	−1.25195	−0.625975	−	0.779843i	$-0.715299\pi$
$419$	−25.6268	−1.25195	−0.625975	+	0.779843i	$0.715299\pi$
$420$	0	0
$421$	−33.3380	−1.62480	−0.812398	−	0.583103i	$-0.801838\pi$
$421$	−33.3380	−1.62480	−0.812398	+	0.583103i	$0.801838\pi$
$422$	−4.90332	−0.238690
$423$	27.5750	1.34074
$424$	−35.4099	−1.71966
$425$	0	0
$426$	−104.069	−5.04215
$427$	−10.0363	−0.485690
$428$	−17.5968	−0.850575
$429$	−75.5669	−3.64841
$430$	0	0
$431$	−19.8335	−0.955346	−0.477673	−	0.878538i	$-0.658520\pi$
$431$	−19.8335	−0.955346	−0.477673	+	0.878538i	$0.658520\pi$
$432$	−91.7109	−4.41244
$433$	−11.5495	−0.555031	−0.277516	−	0.960721i	$-0.589511\pi$
$433$	−11.5495	−0.555031	−0.277516	+	0.960721i	$0.589511\pi$
$434$	8.75786	0.420391
$435$	0	0
$436$	−36.7759	−1.76125
$437$	−0.869326	−0.0415855
$438$	19.4962	0.931562
$439$	13.5697	0.647647	0.323823	−	0.946118i	$-0.395032\pi$
$439$	13.5697	0.647647	0.323823	+	0.946118i	$0.395032\pi$
$440$	0	0
$441$	8.80800	0.419428
$442$	−64.9046	−3.08720
$443$	−30.9973	−1.47273	−0.736364	−	0.676585i	$-0.763459\pi$
$443$	−30.9973	−1.47273	−0.736364	+	0.676585i	$0.763459\pi$
$444$	45.1702	2.14368
$445$	0	0
$446$	29.8939	1.41552
$447$	−11.3606	−0.537336
$448$	−6.75145	−0.318976
$449$	8.67935	0.409604	0.204802	−	0.978803i	$-0.434345\pi$
$449$	8.67935	0.409604	0.204802	+	0.978803i	$0.434345\pi$
$450$	0	0
$451$	−18.6917	−0.880160
$452$	19.3146	0.908480
$453$	−13.7710	−0.647020
$454$	−62.8620	−2.95026
$455$	0	0
$456$	−15.4731	−0.724593
$457$	38.6161	1.80639	0.903194	−	0.429234i	$-0.141216\pi$
$457$	38.6161	1.80639	0.903194	+	0.429234i	$0.141216\pi$
$458$	56.6209	2.64572
$459$	82.7585	3.86284
$460$	0	0
$461$	−2.56653	−0.119535	−0.0597676	−	0.998212i	$-0.519036\pi$
$461$	−2.56653	−0.119535	−0.0597676	+	0.998212i	$0.519036\pi$
$462$	−29.4384	−1.36960
$463$	−30.9019	−1.43613	−0.718067	−	0.695974i	$-0.754973\pi$
$463$	−30.9019	−1.43613	−0.718067	+	0.695974i	$0.754973\pi$
$464$	30.6838	1.42446
$465$	0	0
$466$	43.1429	1.99856
$467$	−11.1067	−0.513956	−0.256978	−	0.966417i	$-0.582727\pi$
$467$	−11.1067	−0.513956	−0.256978	+	0.966417i	$0.582727\pi$
$468$	228.774	10.5751
$469$	12.1499	0.561030
$470$	0	0
$471$	−4.27566	−0.197012
$472$	18.9647	0.872921
$473$	1.72636	0.0793783
$474$	−53.5050	−2.45757
$475$	0	0
$476$	−16.9914	−0.778800
$477$	−60.2138	−2.75700
$478$	−74.5291	−3.40888
$479$	7.83793	0.358124	0.179062	−	0.983838i	$-0.442694\pi$
$479$	7.83793	0.358124	0.179062	+	0.983838i	$0.442694\pi$
$480$	0	0
$481$	−20.3343	−0.927166
$482$	−43.4881	−1.98083
$483$	3.43628	0.156356
$484$	4.24630	0.193014
$485$	0	0
$486$	−209.866	−9.51971
$487$	−13.1626	−0.596456	−0.298228	−	0.954495i	$-0.596396\pi$
$487$	−13.1626	−0.596456	−0.298228	+	0.954495i	$0.596396\pi$
$488$	−51.9851	−2.35326
$489$	−28.4981	−1.28873
$490$	0	0
$491$	4.13063	0.186413	0.0932063	−	0.995647i	$-0.470288\pi$
$491$	4.13063	0.186413	0.0932063	+	0.995647i	$0.470288\pi$
$492$	75.8618	3.42011
$493$	−27.6886	−1.24703
$494$	13.6069	0.612205
$495$	0	0
$496$	16.2976	0.731785
$497$	12.2646	0.550141
$498$	−117.849	−5.28092
$499$	−7.50714	−0.336066	−0.168033	−	0.985781i	$-0.553741\pi$
$499$	−7.50714	−0.336066	−0.168033	+	0.985781i	$0.553741\pi$
$500$	0	0
$501$	−58.5689	−2.61667
$502$	−27.6922	−1.23596
$503$	−27.8907	−1.24358	−0.621792	−	0.783183i	$-0.713595\pi$
$503$	−27.8907	−1.24358	−0.621792	+	0.783183i	$0.713595\pi$
$504$	45.6229	2.03221
$505$	0	0
$506$	−8.56695	−0.380847
$507$	−93.3933	−4.14774
$508$	−12.1593	−0.539482
$509$	−1.51749	−0.0672614	−0.0336307	−	0.999434i	$-0.510707\pi$
$509$	−1.51749	−0.0672614	−0.0336307	+	0.999434i	$0.510707\pi$
$510$	0	0
$511$	−2.29763	−0.101641
$512$	−43.0651	−1.90323
$513$	−17.3499	−0.766017
$514$	−14.5525	−0.641882
$515$	0	0
$516$	−7.00657	−0.308447
$517$	10.8614	0.477682
$518$	−7.92159	−0.348055
$519$	−13.3510	−0.586044
$520$	0	0
$521$	0.867192	0.0379924	0.0189962	−	0.999820i	$-0.493953\pi$
$521$	0.867192	0.0379924	0.0189962	+	0.999820i	$0.493953\pi$
$522$	145.231	6.35659
$523$	−11.8507	−0.518196	−0.259098	−	0.965851i	$-0.583425\pi$
$523$	−11.8507	−0.518196	−0.259098	+	0.965851i	$0.583425\pi$
$524$	18.3659	0.802317
$525$	0	0
$526$	70.0319	3.05354
$527$	−14.7067	−0.640635
$528$	−54.7824	−2.38410
$529$	1.00000	0.0434783
$530$	0	0
$531$	32.2491	1.39949
$532$	3.56216	0.154439
$533$	−34.1508	−1.47924
$534$	90.1883	3.90283
$535$	0	0
$536$	62.9329	2.71829
$537$	3.72948	0.160939
$538$	−60.5847	−2.61199
$539$	3.46934	0.149435
$540$	0	0
$541$	12.7718	0.549101	0.274550	−	0.961573i	$-0.411471\pi$
$541$	12.7718	0.549101	0.274550	+	0.961573i	$0.411471\pi$
$542$	69.7074	2.99419
$543$	−30.1807	−1.29518
$544$	−4.09570	−0.175602
$545$	0	0
$546$	−53.7856	−2.30181
$547$	−33.2564	−1.42194	−0.710971	−	0.703222i	$-0.751744\pi$
$547$	−33.2564	−1.42194	−0.710971	+	0.703222i	$0.751744\pi$
$548$	−52.3404	−2.23587
$549$	−88.3996	−3.77280
$550$	0	0
$551$	5.80477	0.247291
$552$	17.7989	0.757573
$553$	6.30560	0.268141
$554$	−18.7248	−0.795541
$555$	0	0
$556$	52.5981	2.23066
$557$	−27.9367	−1.18372	−0.591858	−	0.806042i	$-0.701605\pi$
$557$	−27.9367	−1.18372	−0.591858	+	0.806042i	$0.701605\pi$
$558$	77.1392	3.26556
$559$	3.15416	0.133407
$560$	0	0
$561$	49.4348	2.08714
$562$	29.7413	1.25456
$563$	9.91363	0.417810	0.208905	−	0.977936i	$-0.433010\pi$
$563$	9.91363	0.417810	0.208905	+	0.977936i	$0.433010\pi$
$564$	−44.0816	−1.85617
$565$	0	0
$566$	45.3436	1.90593
$567$	42.1568	1.77042
$568$	63.5270	2.66553
$569$	31.9089	1.33769	0.668845	−	0.743402i	$-0.266789\pi$
$569$	31.9089	1.33769	0.668845	+	0.743402i	$0.266789\pi$
$570$	0	0
$571$	−5.77696	−0.241758	−0.120879	−	0.992667i	$-0.538571\pi$
$571$	−5.77696	−0.241758	−0.120879	+	0.992667i	$0.538571\pi$
$572$	90.1104	3.76771
$573$	69.4570	2.90161
$574$	−13.3040	−0.555300
$575$	0	0
$576$	−59.4668	−2.47778
$577$	19.4715	0.810608	0.405304	−	0.914182i	$-0.367166\pi$
$577$	19.4715	0.810608	0.405304	+	0.914182i	$0.367166\pi$
$578$	0.480937	0.0200043
$579$	−1.16373	−0.0483631
$580$	0	0
$581$	13.8885	0.576194
$582$	37.5350	1.55587
$583$	−23.7173	−0.982270
$584$	−11.9011	−0.492471
$585$	0	0
$586$	12.3702	0.511008
$587$	−5.89598	−0.243353	−0.121676	−	0.992570i	$-0.538827\pi$
$587$	−5.89598	−0.243353	−0.121676	+	0.992570i	$0.538827\pi$
$588$	−14.0805	−0.580672
$589$	3.08319	0.127041
$590$	0	0
$591$	−21.0502	−0.865891
$592$	−14.7414	−0.605868
$593$	40.7639	1.67397	0.836986	−	0.547224i	$-0.184315\pi$
$593$	40.7639	1.67397	0.836986	+	0.547224i	$0.184315\pi$
$594$	−170.978	−7.01532
$595$	0	0
$596$	13.5470	0.554907
$597$	93.1482	3.81230
$598$	−15.6523	−0.640069
$599$	40.9894	1.67478	0.837390	−	0.546605i	$-0.184080\pi$
$599$	40.9894	1.67478	0.837390	+	0.546605i	$0.184080\pi$
$600$	0	0
$601$	20.5296	0.837419	0.418709	−	0.908120i	$-0.362483\pi$
$601$	20.5296	0.837419	0.418709	+	0.908120i	$0.362483\pi$
$602$	1.22876	0.0500804
$603$	107.016	4.35803
$604$	16.4214	0.668177
$605$	0	0
$606$	−36.9638	−1.50155
$607$	−1.28962	−0.0523440	−0.0261720	−	0.999657i	$-0.508332\pi$
$607$	−1.28962	−0.0523440	−0.0261720	+	0.999657i	$0.508332\pi$
$608$	0.858643	0.0348226
$609$	−22.9451	−0.929783
$610$	0	0
$611$	19.8443	0.802814
$612$	−149.660	−6.04966
$613$	−10.5136	−0.424642	−0.212321	−	0.977200i	$-0.568102\pi$
$613$	−10.5136	−0.424642	−0.212321	+	0.977200i	$0.568102\pi$
$614$	47.4283	1.91405
$615$	0	0
$616$	17.9702	0.724039
$617$	22.3516	0.899840	0.449920	−	0.893069i	$-0.351452\pi$
$617$	22.3516	0.899840	0.449920	+	0.893069i	$0.351452\pi$
$618$	−127.574	−5.13180
$619$	−25.3376	−1.01840	−0.509202	−	0.860647i	$-0.670060\pi$
$619$	−25.3376	−1.01840	−0.509202	+	0.860647i	$0.670060\pi$
$620$	0	0
$621$	19.9579	0.800882
$622$	−15.7036	−0.629656
$623$	−10.6287	−0.425832
$624$	−100.090	−4.00682
$625$	0	0
$626$	27.1365	1.08459
$627$	−10.3638	−0.413888
$628$	5.09855	0.203454
$629$	13.3024	0.530402
$630$	0	0
$631$	10.2758	0.409072	0.204536	−	0.978859i	$-0.434431\pi$
$631$	10.2758	0.409072	0.204536	+	0.978859i	$0.434431\pi$
$632$	32.6612	1.29919
$633$	6.82336	0.271204
$634$	−85.8221	−3.40843
$635$	0	0
$636$	96.2583	3.81689
$637$	6.33866	0.251147
$638$	57.2043	2.26474
$639$	108.026	4.27346
$640$	0	0
$641$	−0.362038	−0.0142997	−0.00714983	−	0.999974i	$-0.502276\pi$
$641$	−0.362038	−0.0142997	−0.00714983	+	0.999974i	$0.502276\pi$
$642$	36.4394	1.43815
$643$	21.8901	0.863263	0.431631	−	0.902050i	$-0.357938\pi$
$643$	21.8901	0.863263	0.431631	+	0.902050i	$0.357938\pi$
$644$	−4.09762	−0.161469
$645$	0	0
$646$	−8.90145	−0.350223
$647$	−29.3062	−1.15214	−0.576072	−	0.817399i	$-0.695415\pi$
$647$	−29.3062	−1.15214	−0.576072	+	0.817399i	$0.695415\pi$
$648$	218.360	8.57800
$649$	12.7024	0.498613
$650$	0	0
$651$	−12.1873	−0.477656
$652$	33.9828	1.33087
$653$	20.0060	0.782897	0.391448	−	0.920200i	$-0.371974\pi$
$653$	20.0060	0.782897	0.391448	+	0.920200i	$0.371974\pi$
$654$	76.1554	2.97791
$655$	0	0
$656$	−24.7577	−0.966625
$657$	−20.2375	−0.789542
$658$	7.73068	0.301373
$659$	9.84537	0.383521	0.191761	−	0.981442i	$-0.438580\pi$
$659$	9.84537	0.383521	0.191761	+	0.981442i	$0.438580\pi$
$660$	0	0
$661$	−20.9328	−0.814193	−0.407096	−	0.913385i	$-0.633459\pi$
$661$	−20.9328	−0.814193	−0.407096	+	0.913385i	$0.633459\pi$
$662$	−63.5547	−2.47012
$663$	90.3200	3.50774
$664$	71.9387	2.79176
$665$	0	0
$666$	−69.7734	−2.70366
$667$	−6.67732	−0.258547
$668$	69.8410	2.70223
$669$	−41.5997	−1.60834
$670$	0	0
$671$	−34.8192	−1.34418
$672$	−3.39405	−0.130928
$673$	7.54129	0.290695	0.145348	−	0.989381i	$-0.453570\pi$
$673$	7.54129	0.290695	0.145348	+	0.989381i	$0.453570\pi$
$674$	11.6581	0.449054
$675$	0	0
$676$	111.368	4.28337
$677$	−21.4614	−0.824827	−0.412414	−	0.910997i	$-0.635314\pi$
$677$	−21.4614	−0.824827	−0.412414	+	0.910997i	$0.635314\pi$
$678$	−39.9965	−1.53605
$679$	−4.42352	−0.169759
$680$	0	0
$681$	87.4774	3.35214
$682$	30.3840	1.16346
$683$	−34.5798	−1.32316	−0.661579	−	0.749876i	$-0.730113\pi$
$683$	−34.5798	−1.32316	−0.661579	+	0.749876i	$0.730113\pi$
$684$	31.3755	1.19967
$685$	0	0
$686$	2.46934	0.0942796
$687$	−78.7925	−3.00612
$688$	2.28661	0.0871763
$689$	−43.3328	−1.65085
$690$	0	0
$691$	34.2249	1.30198	0.650988	−	0.759088i	$-0.274355\pi$
$691$	34.2249	1.30198	0.650988	+	0.759088i	$0.274355\pi$
$692$	15.9205	0.605207
$693$	30.5579	1.16080
$694$	−16.3359	−0.620104
$695$	0	0
$696$	−118.849	−4.50497
$697$	22.3410	0.846224
$698$	20.5663	0.778445
$699$	−60.0368	−2.27080
$700$	0	0
$701$	−8.70283	−0.328701	−0.164351	−	0.986402i	$-0.552553\pi$
$701$	−8.70283	−0.328701	−0.164351	+	0.986402i	$0.552553\pi$
$702$	−312.386	−11.7903
$703$	−2.78879	−0.105181
$704$	−23.4231	−0.882790
$705$	0	0
$706$	6.57855	0.247587
$707$	4.35620	0.163832
$708$	−51.5536	−1.93750
$709$	−15.7861	−0.592861	−0.296430	−	0.955054i	$-0.595796\pi$
$709$	−15.7861	−0.592861	−0.296430	+	0.955054i	$0.595796\pi$
$710$	0	0
$711$	55.5397	2.08290
$712$	−55.0539	−2.06323
$713$	−3.54665	−0.132823
$714$	35.1857	1.31679
$715$	0	0
$716$	−4.44725	−0.166202
$717$	103.713	3.87324
$718$	21.3023	0.794993
$719$	21.8678	0.815530	0.407765	−	0.913087i	$-0.366308\pi$
$719$	21.8678	0.815530	0.407765	+	0.913087i	$0.366308\pi$
$720$	0	0
$721$	15.0347	0.559922
$722$	−45.0512	−1.67663
$723$	60.5171	2.25066
$724$	35.9892	1.33753
$725$	0	0
$726$	−8.79321	−0.326347
$727$	12.3829	0.459257	0.229628	−	0.973278i	$-0.426249\pi$
$727$	12.3829	0.459257	0.229628	+	0.973278i	$0.426249\pi$
$728$	32.8325	1.21685
$729$	165.575	6.13239
$730$	0	0
$731$	−2.06340	−0.0763177
$732$	141.316	5.22320
$733$	−12.9153	−0.477037	−0.238519	−	0.971138i	$-0.576662\pi$
$733$	−12.9153	−0.477037	−0.238519	+	0.971138i	$0.576662\pi$
$734$	−30.1497	−1.11285
$735$	0	0
$736$	−0.987711	−0.0364075
$737$	42.1520	1.55269
$738$	−117.182	−4.31353
$739$	−15.7343	−0.578797	−0.289398	−	0.957209i	$-0.593455\pi$
$739$	−15.7343	−0.578797	−0.289398	+	0.957209i	$0.593455\pi$
$740$	0	0
$741$	−18.9351	−0.695599
$742$	−16.8810	−0.619722
$743$	1.50511	0.0552170	0.0276085	−	0.999619i	$-0.491211\pi$
$743$	1.50511	0.0552170	0.0276085	+	0.999619i	$0.491211\pi$
$744$	−63.1266	−2.31433
$745$	0	0
$746$	43.2982	1.58526
$747$	122.330	4.47583
$748$	−58.9489	−2.15538
$749$	−4.29441	−0.156914
$750$	0	0
$751$	−22.1025	−0.806532	−0.403266	−	0.915083i	$-0.632125\pi$
$751$	−22.1025	−0.806532	−0.403266	+	0.915083i	$0.632125\pi$
$752$	14.3861	0.524609
$753$	38.5359	1.40433
$754$	104.515	3.80622
$755$	0	0
$756$	−81.7797	−2.97430
$757$	26.2801	0.955166	0.477583	−	0.878587i	$-0.341513\pi$
$757$	26.2801	0.955166	0.477583	+	0.878587i	$0.341513\pi$
$758$	−26.8833	−0.976446
$759$	11.9216	0.432726
$760$	0	0
$761$	−27.4085	−0.993556	−0.496778	−	0.867878i	$-0.665484\pi$
$761$	−27.4085	−0.993556	−0.496778	+	0.867878i	$0.665484\pi$
$762$	25.1794	0.912154
$763$	−8.97496	−0.324915
$764$	−82.8246	−2.99649
$765$	0	0
$766$	30.3289	1.09583
$767$	23.2080	0.837992
$768$	111.826	4.03518
$769$	46.5024	1.67692	0.838459	−	0.544964i	$-0.183457\pi$
$769$	46.5024	1.67692	0.838459	+	0.544964i	$0.183457\pi$
$770$	0	0
$771$	20.2509	0.729319
$772$	1.38770	0.0499445
$773$	−23.6544	−0.850790	−0.425395	−	0.905008i	$-0.639865\pi$
$773$	−23.6544	−0.850790	−0.425395	+	0.905008i	$0.639865\pi$
$774$	10.8229	0.389021
$775$	0	0
$776$	−22.9126	−0.822514
$777$	11.0235	0.395467
$778$	4.55685	0.163371
$779$	−4.68367	−0.167810
$780$	0	0
$781$	42.5499	1.52256
$782$	10.2395	0.366163
$783$	−133.265	−4.76251
$784$	4.59522	0.164115
$785$	0	0
$786$	−38.0320	−1.35656
$787$	45.3546	1.61672	0.808359	−	0.588690i	$-0.200356\pi$
$787$	45.3546	1.61672	0.808359	+	0.588690i	$0.200356\pi$
$788$	25.1015	0.894205
$789$	−97.4550	−3.46949
$790$	0	0
$791$	4.71361	0.167597
$792$	158.281	5.62428
$793$	−63.6166	−2.25909
$794$	−18.3783	−0.652223
$795$	0	0
$796$	−111.075	−3.93696
$797$	13.2231	0.468386	0.234193	−	0.972190i	$-0.424755\pi$
$797$	13.2231	0.468386	0.234193	+	0.972190i	$0.424755\pi$
$798$	−7.37651	−0.261126
$799$	−12.9818	−0.459264
$800$	0	0
$801$	−93.6180	−3.30783
$802$	−13.9167	−0.491417
$803$	−7.97126	−0.281300
$804$	−171.077	−6.03342
$805$	0	0
$806$	55.5131	1.95537
$807$	84.3084	2.96780
$808$	22.5639	0.793795
$809$	−17.7319	−0.623420	−0.311710	−	0.950177i	$-0.600902\pi$
$809$	−17.7319	−0.623420	−0.311710	+	0.950177i	$0.600902\pi$
$810$	0	0
$811$	20.8555	0.732336	0.366168	−	0.930549i	$-0.380669\pi$
$811$	20.8555	0.732336	0.366168	+	0.930549i	$0.380669\pi$
$812$	27.3611	0.960186
$813$	−97.0034	−3.40206
$814$	−27.4827	−0.963267
$815$	0	0
$816$	65.4776	2.29217
$817$	0.432582	0.0151341
$818$	4.32866	0.151348
$819$	55.8309	1.95089
$820$	0	0
$821$	−32.5700	−1.13670	−0.568351	−	0.822786i	$-0.692418\pi$
$821$	−32.5700	−1.13670	−0.568351	+	0.822786i	$0.692418\pi$
$822$	108.386	3.78040
$823$	28.2037	0.983120	0.491560	−	0.870844i	$-0.336427\pi$
$823$	28.2037	0.983120	0.491560	+	0.870844i	$0.336427\pi$
$824$	77.8756	2.71293
$825$	0	0
$826$	9.04107	0.314579
$827$	−41.4864	−1.44262	−0.721312	−	0.692610i	$-0.756461\pi$
$827$	−41.4864	−1.44262	−0.721312	+	0.692610i	$0.756461\pi$
$828$	−36.0918	−1.25428
$829$	−2.36927	−0.0822882	−0.0411441	−	0.999153i	$-0.513100\pi$
$829$	−2.36927	−0.0822882	−0.0411441	+	0.999153i	$0.513100\pi$
$830$	0	0
$831$	26.0571	0.903909
$832$	−42.7952	−1.48366
$833$	−4.14666	−0.143673
$834$	−108.920	−3.77158
$835$	0	0
$836$	12.3583	0.427422
$837$	−70.7836	−2.44664
$838$	−63.2811	−2.18601
$839$	44.6380	1.54108	0.770538	−	0.637395i	$-0.219988\pi$
$839$	44.6380	1.54108	0.770538	+	0.637395i	$0.219988\pi$
$840$	0	0
$841$	15.5866	0.537470
$842$	−82.3228	−2.83703
$843$	−41.3874	−1.42546
$844$	−8.13657	−0.280072
$845$	0	0
$846$	68.0918	2.34105
$847$	1.03629	0.0356072
$848$	−31.4142	−1.07877
$849$	−63.0992	−2.16556
$850$	0	0
$851$	3.20799	0.109968
$852$	−172.692	−5.91632
$853$	−52.4497	−1.79584	−0.897921	−	0.440157i	$-0.854923\pi$
$853$	−52.4497	−1.79584	−0.897921	+	0.440157i	$0.854923\pi$
$854$	−24.7830	−0.848055
$855$	0	0
$856$	−22.2438	−0.760278
$857$	−14.5119	−0.495718	−0.247859	−	0.968796i	$-0.579727\pi$
$857$	−14.5119	−0.495718	−0.247859	+	0.968796i	$0.579727\pi$
$858$	−186.600	−6.37042
$859$	12.5248	0.427342	0.213671	−	0.976906i	$-0.431458\pi$
$859$	12.5248	0.427342	0.213671	+	0.976906i	$0.431458\pi$
$860$	0	0
$861$	18.5136	0.630943
$862$	−48.9756	−1.66811
$863$	−23.8010	−0.810194	−0.405097	−	0.914274i	$-0.632762\pi$
$863$	−23.8010	−0.810194	−0.405097	+	0.914274i	$0.632762\pi$
$864$	−19.7126	−0.670637
$865$	0	0
$866$	−28.5195	−0.969131
$867$	−0.669262	−0.0227293
$868$	14.5328	0.493275
$869$	21.8762	0.742101
$870$	0	0
$871$	77.0140	2.60952
$872$	−46.4877	−1.57427
$873$	−38.9624	−1.31868
$874$	−2.14666	−0.0726118
$875$	0	0
$876$	32.3519	1.09307
$877$	7.65390	0.258454	0.129227	−	0.991615i	$-0.458750\pi$
$877$	7.65390	0.258454	0.129227	+	0.991615i	$0.458750\pi$
$878$	33.5082	1.13085
$879$	−17.2141	−0.580618
$880$	0	0
$881$	3.86625	0.130257	0.0651287	−	0.997877i	$-0.479254\pi$
$881$	3.86625	0.130257	0.0651287	+	0.997877i	$0.479254\pi$
$882$	21.7499	0.732357
$883$	−9.80138	−0.329843	−0.164921	−	0.986307i	$-0.552737\pi$
$883$	−9.80138	−0.329843	−0.164921	+	0.986307i	$0.552737\pi$
$884$	−107.703	−3.62244
$885$	0	0
$886$	−76.5428	−2.57151
$887$	−53.1071	−1.78316	−0.891582	−	0.452860i	$-0.850404\pi$
$887$	−53.1071	−1.78316	−0.891582	+	0.452860i	$0.850404\pi$
$888$	57.0987	1.91611
$889$	−2.96741	−0.0995237
$890$	0	0
$891$	146.256	4.89976
$892$	49.6059	1.66093
$893$	2.72158	0.0910741
$894$	−28.0531	−0.938234
$895$	0	0
$896$	−18.6470	−0.622953
$897$	21.7814	0.727259
$898$	21.4322	0.715202
$899$	23.6821	0.789843
$900$	0	0
$901$	28.3477	0.944397
$902$	−46.1562	−1.53683
$903$	−1.70991	−0.0569023
$904$	24.4151	0.812036
$905$	0	0
$906$	−34.0053	−1.12975
$907$	−48.3759	−1.60629	−0.803147	−	0.595780i	$-0.796843\pi$
$907$	−48.3759	−1.60629	−0.803147	+	0.595780i	$0.796843\pi$
$908$	−104.313	−3.46175
$909$	38.3694	1.27263
$910$	0	0
$911$	−50.5732	−1.67557	−0.837783	−	0.546003i	$-0.816149\pi$
$911$	−50.5732	−1.67557	−0.837783	+	0.546003i	$0.816149\pi$
$912$	−13.7270	−0.454548
$913$	48.1840	1.59466
$914$	95.3562	3.15410
$915$	0	0
$916$	93.9567	3.10442
$917$	4.48209	0.148012
$918$	204.358	6.74483
$919$	7.83106	0.258323	0.129161	−	0.991624i	$-0.458771\pi$
$919$	7.83106	0.258323	0.129161	+	0.991624i	$0.458771\pi$
$920$	0	0
$921$	−66.0002	−2.17478
$922$	−6.33762	−0.208718
$923$	77.7410	2.55888
$924$	−48.8501	−1.60705
$925$	0	0
$926$	−76.3072	−2.50761
$927$	132.426	4.34943
$928$	6.59527	0.216500
$929$	−48.9892	−1.60728	−0.803642	−	0.595113i	$-0.797107\pi$
$929$	−48.9892	−1.60728	−0.803642	+	0.595113i	$0.797107\pi$
$930$	0	0
$931$	0.869326	0.0284910
$932$	71.5914	2.34505
$933$	21.8528	0.715427
$934$	−27.4261	−0.897410
$935$	0	0
$936$	289.188	9.45241
$937$	−29.9542	−0.978562	−0.489281	−	0.872126i	$-0.662741\pi$
$937$	−29.9542	−0.978562	−0.489281	+	0.872126i	$0.662741\pi$
$938$	30.0021	0.979604
$939$	−37.7625	−1.23233
$940$	0	0
$941$	−10.6721	−0.347899	−0.173950	−	0.984755i	$-0.555653\pi$
$941$	−10.6721	−0.347899	−0.173950	+	0.984755i	$0.555653\pi$
$942$	−10.5580	−0.344000
$943$	5.38770	0.175448
$944$	16.8247	0.547596
$945$	0	0
$946$	4.26297	0.138601
$947$	2.71319	0.0881667	0.0440833	−	0.999028i	$-0.485963\pi$
$947$	2.71319	0.0881667	0.0440833	+	0.999028i	$0.485963\pi$
$948$	−88.7863	−2.88364
$949$	−14.5639	−0.472765
$950$	0	0
$951$	119.428	3.87273
$952$	−21.4785	−0.696122
$953$	−5.35584	−0.173493	−0.0867464	−	0.996230i	$-0.527647\pi$
$953$	−5.35584	−0.173493	−0.0867464	+	0.996230i	$0.527647\pi$
$954$	−148.688	−4.81395
$955$	0	0
$956$	−123.674	−3.99989
$957$	−79.6043	−2.57324
$958$	19.3545	0.625315
$959$	−12.7734	−0.412474
$960$	0	0
$961$	−18.4213	−0.594235
$962$	−50.2123	−1.61891
$963$	−37.8251	−1.21890
$964$	−72.1641	−2.32425
$965$	0	0
$966$	8.48532	0.273011
$967$	48.6703	1.56513	0.782565	−	0.622568i	$-0.213911\pi$
$967$	48.6703	1.56513	0.782565	+	0.622568i	$0.213911\pi$
$968$	5.36766	0.172523
$969$	12.3871	0.397930
$970$	0	0
$971$	41.8475	1.34295	0.671475	−	0.741027i	$-0.265661\pi$
$971$	41.8475	1.34295	0.671475	+	0.741027i	$0.265661\pi$
$972$	−348.251	−11.1702
$973$	12.8363	0.411512
$974$	−32.5030	−1.04146
$975$	0	0
$976$	−46.1190	−1.47623
$977$	30.2519	0.967843	0.483921	−	0.875111i	$-0.339212\pi$
$977$	30.2519	0.967843	0.483921	+	0.875111i	$0.339212\pi$
$978$	−70.3714	−2.25023
$979$	−36.8747	−1.17852
$980$	0	0
$981$	−79.0514	−2.52392
$982$	10.1999	0.325492
$983$	−52.5467	−1.67598	−0.837990	−	0.545686i	$-0.816269\pi$
$983$	−52.5467	−1.67598	−0.837990	+	0.545686i	$0.816269\pi$
$984$	95.8954	3.05703
$985$	0	0
$986$	−68.3723	−2.17742
$987$	−10.7579	−0.342427
$988$	22.5793	0.718345
$989$	−0.497606	−0.0158230
$990$	0	0
$991$	42.3825	1.34632	0.673162	−	0.739495i	$-0.264936\pi$
$991$	42.3825	1.34632	0.673162	+	0.739495i	$0.264936\pi$
$992$	3.50306	0.111222
$993$	88.4414	2.80660
$994$	30.2853	0.960593
$995$	0	0
$996$	−195.558	−6.19649
$997$	11.2981	0.357814	0.178907	−	0.983866i	$-0.442744\pi$
$997$	11.2981	0.357814	0.178907	+	0.983866i	$0.442744\pi$
$998$	−18.5376	−0.586798
$999$	64.0246	2.02565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.n.1.4		4
5.4	even	2		805.2.a.h.1.1	✓	4
15.14	odd	2		7245.2.a.be.1.4		4
35.34	odd	2		5635.2.a.t.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.a.h.1.1	✓	4	5.4	even	2
4025.2.a.n.1.4		4	1.1	even	1	trivial
5635.2.a.t.1.1		4	35.34	odd	2
7245.2.a.be.1.4		4	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:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.46934 q^{2} -3.43628 q^{3} +4.09762 q^{4} -8.48532 q^{6} +1.00000 q^{7} +5.17972 q^{8} +8.80800 q^{9} +O(q^{10})\)	\(q+2.46934 q^{2} -3.43628 q^{3} +4.09762 q^{4} -8.48532 q^{6} +1.00000 q^{7} +5.17972 q^{8} +8.80800 q^{9} +3.46934 q^{11} -14.0805 q^{12} +6.33866 q^{13} +2.46934 q^{14} +4.59522 q^{16} -4.14666 q^{17} +21.7499 q^{18} +0.869326 q^{19} -3.43628 q^{21} +8.56695 q^{22} -1.00000 q^{23} -17.7989 q^{24} +15.6523 q^{26} -19.9579 q^{27} +4.09762 q^{28} +6.67732 q^{29} +3.54665 q^{31} +0.987711 q^{32} -11.9216 q^{33} -10.2395 q^{34} +36.0918 q^{36} -3.20799 q^{37} +2.14666 q^{38} -21.7814 q^{39} -5.38770 q^{41} -8.48532 q^{42} +0.497606 q^{43} +14.2160 q^{44} -2.46934 q^{46} +3.13067 q^{47} -15.7905 q^{48} +1.00000 q^{49} +14.2491 q^{51} +25.9734 q^{52} -6.83627 q^{53} -49.2827 q^{54} +5.17972 q^{56} -2.98724 q^{57} +16.4885 q^{58} +3.66134 q^{59} -10.0363 q^{61} +8.75786 q^{62} +8.80800 q^{63} -6.75145 q^{64} -29.4384 q^{66} +12.1499 q^{67} -16.9914 q^{68} +3.43628 q^{69} +12.2646 q^{71} +45.6229 q^{72} -2.29763 q^{73} -7.92159 q^{74} +3.56216 q^{76} +3.46934 q^{77} -53.7856 q^{78} +6.30560 q^{79} +42.1568 q^{81} -13.3040 q^{82} +13.8885 q^{83} -14.0805 q^{84} +1.22876 q^{86} -22.9451 q^{87} +17.9702 q^{88} -10.6287 q^{89} +6.33866 q^{91} -4.09762 q^{92} -12.1873 q^{93} +7.73068 q^{94} -3.39405 q^{96} -4.42352 q^{97} +2.46934 q^{98} +30.5579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 11 q^{4} + 4 q^{7} + 18 q^{9}+O(q^{10}) \) 4 * q + q^2 + 11 * q^4 + 4 * q^7 + 18 * q^9	\( 4 q + q^{2} + 11 q^{4} + 4 q^{7} + 18 q^{9} + 5 q^{11} - 21 q^{12} + 17 q^{13} + q^{14} + 17 q^{16} + 9 q^{17} + 24 q^{18} + 4 q^{19} + 20 q^{22} - 4 q^{23} + 12 q^{24} + 5 q^{26} - 9 q^{27} + 11 q^{28} + 10 q^{29} - 2 q^{31} - 34 q^{32} - 18 q^{34} + 45 q^{36} - 5 q^{37} - 17 q^{38} - 9 q^{39} + 7 q^{41} + 6 q^{43} + 13 q^{44} - q^{46} + 12 q^{47} - 51 q^{48} + 4 q^{49} + 18 q^{51} + 43 q^{52} - 23 q^{53} - 60 q^{54} - 9 q^{57} + 4 q^{58} + 23 q^{59} - 17 q^{61} - 17 q^{62} + 18 q^{63} + 16 q^{64} - 21 q^{66} - 5 q^{67} + 33 q^{68} + 20 q^{71} + 57 q^{72} + 15 q^{73} + 16 q^{74} + 8 q^{76} + 5 q^{77} - 39 q^{78} + 12 q^{79} + 24 q^{81} - 20 q^{82} + 3 q^{83} - 21 q^{84} - 36 q^{86} - 18 q^{87} + 39 q^{88} - 11 q^{89} + 17 q^{91} - 11 q^{92} - 27 q^{93} + 21 q^{94} + 3 q^{96} - q^{97} + q^{98} + 42 q^{99}+O(q^{100}) \) 4 * q + q^2 + 11 * q^4 + 4 * q^7 + 18 * q^9 + 5 * q^11 - 21 * q^12 + 17 * q^13 + q^14 + 17 * q^16 + 9 * q^17 + 24 * q^18 + 4 * q^19 + 20 * q^22 - 4 * q^23 + 12 * q^24 + 5 * q^26 - 9 * q^27 + 11 * q^28 + 10 * q^29 - 2 * q^31 - 34 * q^32 - 18 * q^34 + 45 * q^36 - 5 * q^37 - 17 * q^38 - 9 * q^39 + 7 * q^41 + 6 * q^43 + 13 * q^44 - q^46 + 12 * q^47 - 51 * q^48 + 4 * q^49 + 18 * q^51 + 43 * q^52 - 23 * q^53 - 60 * q^54 - 9 * q^57 + 4 * q^58 + 23 * q^59 - 17 * q^61 - 17 * q^62 + 18 * q^63 + 16 * q^64 - 21 * q^66 - 5 * q^67 + 33 * q^68 + 20 * q^71 + 57 * q^72 + 15 * q^73 + 16 * q^74 + 8 * q^76 + 5 * q^77 - 39 * q^78 + 12 * q^79 + 24 * q^81 - 20 * q^82 + 3 * q^83 - 21 * q^84 - 36 * q^86 - 18 * q^87 + 39 * q^88 - 11 * q^89 + 17 * q^91 - 11 * q^92 - 27 * q^93 + 21 * q^94 + 3 * q^96 - q^97 + q^98 + 42 * q^99

Introduction

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