LMFDB - Embedded newform 3025.2.a.bi.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 $ x^8 - 3x^7 - 7x^6 + 18x^5 + 16x^4 - 30x^3 - 12x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.95882$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.958815 q^{2} -0.899342 q^{3} -1.08067 q^{4} -0.862303 q^{6} +0.761645 q^{7} -2.95380 q^{8} -2.19118 q^{9} +O(q^{10})$	\(q+0.958815 q^{2} -0.899342 q^{3} -1.08067 q^{4} -0.862303 q^{6} +0.761645 q^{7} -2.95380 q^{8} -2.19118 q^{9} +0.971895 q^{12} +3.25819 q^{13} +0.730277 q^{14} -0.670799 q^{16} +0.782845 q^{17} -2.10094 q^{18} +4.59787 q^{19} -0.684980 q^{21} -2.51720 q^{23} +2.65647 q^{24} +3.12400 q^{26} +4.66865 q^{27} -0.823090 q^{28} +3.37985 q^{29} -4.90481 q^{31} +5.26442 q^{32} +0.750604 q^{34} +2.36795 q^{36} +5.69215 q^{37} +4.40851 q^{38} -2.93022 q^{39} -5.44874 q^{41} -0.656769 q^{42} -10.5601 q^{43} -2.41353 q^{46} +3.00667 q^{47} +0.603278 q^{48} -6.41990 q^{49} -0.704046 q^{51} -3.52103 q^{52} -13.8424 q^{53} +4.47637 q^{54} -2.24975 q^{56} -4.13506 q^{57} +3.24066 q^{58} -14.2468 q^{59} +9.44700 q^{61} -4.70281 q^{62} -1.66890 q^{63} +6.38921 q^{64} -11.8555 q^{67} -0.846000 q^{68} +2.26383 q^{69} -2.18239 q^{71} +6.47231 q^{72} -14.4991 q^{73} +5.45772 q^{74} -4.96880 q^{76} -2.80954 q^{78} +7.34554 q^{79} +2.37484 q^{81} -5.22433 q^{82} -0.788913 q^{83} +0.740239 q^{84} -10.1252 q^{86} -3.03965 q^{87} +2.41895 q^{89} +2.48158 q^{91} +2.72027 q^{92} +4.41110 q^{93} +2.88284 q^{94} -4.73452 q^{96} +1.80480 q^{97} -6.15550 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} - q^{3} + 9 q^{4} + q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 - q^3 + 9 * q^4 + q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	$ 8 q - 5 q^{2} - q^{3} + 9 q^{4} + q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} - 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} + q^{19} + 5 q^{21} - 2 q^{23} + q^{24} - 2 q^{26} + 2 q^{27} - 9 q^{28} + 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} + 8 q^{37} + 37 q^{38} - q^{39} - 41 q^{42} - 14 q^{43} - 20 q^{46} - 11 q^{47} + 27 q^{48} - 12 q^{49} - 25 q^{51} + 7 q^{52} - 11 q^{53} + 30 q^{54} - 10 q^{56} + 2 q^{57} - 27 q^{58} + 17 q^{59} - 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} - 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} - 6 q^{74} - 31 q^{76} + 17 q^{78} + 23 q^{79} - 4 q^{81} + 17 q^{82} - 41 q^{83} + 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} - 33 q^{92} + 59 q^{93} + 50 q^{94} - 61 q^{96} - 2 q^{97} - 26 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 - q^3 + 9 * q^4 + q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 - 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 + q^19 + 5 * q^21 - 2 * q^23 + q^24 - 2 * q^26 + 2 * q^27 - 9 * q^28 + 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 + 8 * q^37 + 37 * q^38 - q^39 - 41 * q^42 - 14 * q^43 - 20 * q^46 - 11 * q^47 + 27 * q^48 - 12 * q^49 - 25 * q^51 + 7 * q^52 - 11 * q^53 + 30 * q^54 - 10 * q^56 + 2 * q^57 - 27 * q^58 + 17 * q^59 - 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 - 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 - 6 * q^74 - 31 * q^76 + 17 * q^78 + 23 * q^79 - 4 * q^81 + 17 * q^82 - 41 * q^83 + 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 - 33 * q^92 + 59 * q^93 + 50 * q^94 - 61 * q^96 - 2 * q^97 - 26 * q^98

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 $

$2$	0.958815	0.677985	0.338992	−	0.940789i	$-0.389914\pi$
$2$	0.958815	0.677985	0.338992	+	0.940789i	$0.389914\pi$
$3$	−0.899342	−0.519235	−0.259618	−	0.965711i	$-0.583597\pi$
$3$	−0.899342	−0.519235	−0.259618	+	0.965711i	$0.583597\pi$
$4$	−1.08067	−0.540337
$5$	0	0
$5$	0	0
$6$	−0.862303	−0.352034
$7$	0.761645	0.287875	0.143937	−	0.989587i	$-0.454024\pi$
$7$	0.761645	0.287875	0.143937	+	0.989587i	$0.454024\pi$
$8$	−2.95380	−1.04432
$9$	−2.19118	−0.730395
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.971895	0.280562
$13$	3.25819	0.903658	0.451829	−	0.892104i	$-0.350772\pi$
$13$	3.25819	0.903658	0.451829	+	0.892104i	$0.350772\pi$
$14$	0.730277	0.195175
$15$	0	0
$16$	−0.670799	−0.167700
$17$	0.782845	0.189868	0.0949339	−	0.995484i	$-0.469736\pi$
$17$	0.782845	0.189868	0.0949339	+	0.995484i	$0.469736\pi$
$18$	−2.10094	−0.495196
$19$	4.59787	1.05482	0.527412	−	0.849609i	$-0.323162\pi$
$19$	4.59787	1.05482	0.527412	+	0.849609i	$0.323162\pi$
$20$	0	0
$21$	−0.684980	−0.149475
$22$	0	0
$23$	−2.51720	−0.524873	−0.262436	−	0.964949i	$-0.584526\pi$
$23$	−2.51720	−0.524873	−0.262436	+	0.964949i	$0.584526\pi$
$24$	2.65647	0.542251
$25$	0	0
$26$	3.12400	0.612667
$27$	4.66865	0.898482
$28$	−0.823090	−0.155549
$29$	3.37985	0.627623	0.313812	−	0.949485i	$-0.398394\pi$
$29$	3.37985	0.627623	0.313812	+	0.949485i	$0.398394\pi$
$30$	0	0
$31$	−4.90481	−0.880930	−0.440465	−	0.897770i	$-0.645186\pi$
$31$	−4.90481	−0.880930	−0.440465	+	0.897770i	$0.645186\pi$
$32$	5.26442	0.930627
$33$	0	0
$34$	0.750604	0.128728
$35$	0	0
$36$	2.36795	0.394659
$37$	5.69215	0.935784	0.467892	−	0.883786i	$-0.345014\pi$
$37$	5.69215	0.935784	0.467892	+	0.883786i	$0.345014\pi$
$38$	4.40851	0.715155
$39$	−2.93022	−0.469211
$40$	0	0
$41$	−5.44874	−0.850950	−0.425475	−	0.904970i	$-0.639893\pi$
$41$	−5.44874	−0.850950	−0.425475	+	0.904970i	$0.639893\pi$
$42$	−0.656769	−0.101342
$43$	−10.5601	−1.61040	−0.805200	−	0.593003i	$-0.797942\pi$
$43$	−10.5601	−1.61040	−0.805200	+	0.593003i	$0.797942\pi$
$44$	0	0
$45$	0	0
$46$	−2.41353	−0.355856
$47$	3.00667	0.438568	0.219284	−	0.975661i	$-0.429628\pi$
$47$	3.00667	0.438568	0.219284	+	0.975661i	$0.429628\pi$
$48$	0.603278	0.0870757
$49$	−6.41990	−0.917128
$50$	0	0
$51$	−0.704046	−0.0985861
$52$	−3.52103	−0.488280
$53$	−13.8424	−1.90140	−0.950699	−	0.310115i	$-0.899632\pi$
$53$	−13.8424	−1.90140	−0.950699	+	0.310115i	$0.899632\pi$
$54$	4.47637	0.609157
$55$	0	0
$56$	−2.24975	−0.300635
$57$	−4.13506	−0.547702
$58$	3.24066	0.425519
$59$	−14.2468	−1.85478	−0.927389	−	0.374100i	$-0.877952\pi$
$59$	−14.2468	−1.85478	−0.927389	+	0.374100i	$0.877952\pi$
$60$	0	0
$61$	9.44700	1.20956	0.604782	−	0.796391i	$-0.293260\pi$
$61$	9.44700	1.20956	0.604782	+	0.796391i	$0.293260\pi$
$62$	−4.70281	−0.597257
$63$	−1.66890	−0.210262
$64$	6.38921	0.798651
$65$	0	0
$66$	0	0
$67$	−11.8555	−1.44838	−0.724189	−	0.689601i	$-0.757786\pi$
$67$	−11.8555	−1.44838	−0.724189	+	0.689601i	$0.757786\pi$
$68$	−0.846000	−0.102593
$69$	2.26383	0.272533
$70$	0	0
$71$	−2.18239	−0.259002	−0.129501	−	0.991579i	$-0.541338\pi$
$71$	−2.18239	−0.259002	−0.129501	+	0.991579i	$0.541338\pi$
$72$	6.47231	0.762769
$73$	−14.4991	−1.69699	−0.848494	−	0.529205i	$-0.822490\pi$
$73$	−14.4991	−1.69699	−0.848494	+	0.529205i	$0.822490\pi$
$74$	5.45772	0.634447
$75$	0	0
$76$	−4.96880	−0.569960
$77$	0	0
$78$	−2.80954	−0.318118
$79$	7.34554	0.826438	0.413219	−	0.910632i	$-0.364404\pi$
$79$	7.34554	0.826438	0.413219	+	0.910632i	$0.364404\pi$
$80$	0	0
$81$	2.37484	0.263871
$82$	−5.22433	−0.576931
$83$	−0.788913	−0.0865944	−0.0432972	−	0.999062i	$-0.513786\pi$
$83$	−0.788913	−0.0865944	−0.0432972	+	0.999062i	$0.513786\pi$
$84$	0.740239	0.0807667
$85$	0	0
$86$	−10.1252	−1.09183
$87$	−3.03965	−0.325884
$88$	0	0
$89$	2.41895	0.256408	0.128204	−	0.991748i	$-0.459079\pi$
$89$	2.41895	0.256408	0.128204	+	0.991748i	$0.459079\pi$
$90$	0	0
$91$	2.48158	0.260141
$92$	2.72027	0.283608
$93$	4.41110	0.457410
$94$	2.88284	0.297342
$95$	0	0
$96$	−4.73452	−0.483214
$97$	1.80480	0.183249	0.0916247	−	0.995794i	$-0.470794\pi$
$97$	1.80480	0.183249	0.0916247	+	0.995794i	$0.470794\pi$
$98$	−6.15550	−0.621799
$99$	0	0
$100$	0	0
$101$	11.0478	1.09930	0.549648	−	0.835396i	$-0.314762\pi$
$101$	11.0478	1.09930	0.549648	+	0.835396i	$0.314762\pi$
$102$	−0.675050	−0.0668399
$103$	14.1500	1.39424	0.697119	−	0.716956i	$-0.254465\pi$
$103$	14.1500	1.39424	0.697119	+	0.716956i	$0.254465\pi$
$104$	−9.62402	−0.943713
$105$	0	0
$106$	−13.2723	−1.28912
$107$	−11.3577	−1.09799	−0.548997	−	0.835824i	$-0.684990\pi$
$107$	−11.3577	−1.09799	−0.548997	+	0.835824i	$0.684990\pi$
$108$	−5.04529	−0.485483
$109$	−10.6108	−1.01634	−0.508168	−	0.861258i	$-0.669677\pi$
$109$	−10.6108	−1.01634	−0.508168	+	0.861258i	$0.669677\pi$
$110$	0	0
$111$	−5.11919	−0.485892
$112$	−0.510911	−0.0482766
$113$	−10.6727	−1.00400	−0.502001	−	0.864867i	$-0.667402\pi$
$113$	−10.6727	−1.00400	−0.502001	+	0.864867i	$0.667402\pi$
$114$	−3.96476	−0.371334
$115$	0	0
$116$	−3.65252	−0.339128
$117$	−7.13928	−0.660027
$118$	−13.6601	−1.25751
$119$	0.596251	0.0546582
$120$	0	0
$121$	0	0
$122$	9.05793	0.820066
$123$	4.90028	0.441843
$124$	5.30050	0.475999
$125$	0	0
$126$	−1.60017	−0.142555
$127$	−6.91796	−0.613869	−0.306935	−	0.951731i	$-0.599303\pi$
$127$	−6.91796	−0.613869	−0.306935	+	0.951731i	$0.599303\pi$
$128$	−4.40277	−0.389154
$129$	9.49714	0.836177
$130$	0	0
$131$	−7.01744	−0.613117	−0.306558	−	0.951852i	$-0.599177\pi$
$131$	−7.01744	−0.613117	−0.306558	+	0.951852i	$0.599177\pi$
$132$	0	0
$133$	3.50195	0.303658
$134$	−11.3672	−0.981979
$135$	0	0
$136$	−2.31237	−0.198284
$137$	7.95932	0.680011	0.340005	−	0.940423i	$-0.389571\pi$
$137$	7.95932	0.680011	0.340005	+	0.940423i	$0.389571\pi$
$138$	2.17059	0.184773
$139$	13.8450	1.17432	0.587158	−	0.809473i	$-0.300247\pi$
$139$	13.8450	1.17432	0.587158	+	0.809473i	$0.300247\pi$
$140$	0	0
$141$	−2.70403	−0.227720
$142$	−2.09251	−0.175600
$143$	0	0
$144$	1.46984	0.122487
$145$	0	0
$146$	−13.9019	−1.15053
$147$	5.77368	0.476205
$148$	−6.15136	−0.505638
$149$	−6.94056	−0.568593	−0.284297	−	0.958736i	$-0.591760\pi$
$149$	−6.94056	−0.568593	−0.284297	+	0.958736i	$0.591760\pi$
$150$	0	0
$151$	−12.0162	−0.977862	−0.488931	−	0.872322i	$-0.662613\pi$
$151$	−12.0162	−0.977862	−0.488931	+	0.872322i	$0.662613\pi$
$152$	−13.5812	−1.10158
$153$	−1.71536	−0.138678
$154$	0	0
$155$	0	0
$156$	3.16662	0.253532
$157$	−17.7526	−1.41681	−0.708405	−	0.705806i	$-0.750585\pi$
$157$	−17.7526	−1.41681	−0.708405	+	0.705806i	$0.750585\pi$
$158$	7.04302	0.560313
$159$	12.4490	0.987273
$160$	0	0
$161$	−1.91721	−0.151098
$162$	2.27703	0.178900
$163$	−8.59091	−0.672892	−0.336446	−	0.941703i	$-0.609225\pi$
$163$	−8.59091	−0.672892	−0.336446	+	0.941703i	$0.609225\pi$
$164$	5.88831	0.459799
$165$	0	0
$166$	−0.756422	−0.0587097
$167$	0.747673	0.0578567	0.0289283	−	0.999581i	$-0.490791\pi$
$167$	0.747673	0.0578567	0.0289283	+	0.999581i	$0.490791\pi$
$168$	2.02329	0.156100
$169$	−2.38422	−0.183402
$170$	0	0
$171$	−10.0748	−0.770438
$172$	11.4120	0.870158
$173$	−25.9204	−1.97069	−0.985346	−	0.170565i	$-0.945441\pi$
$173$	−25.9204	−1.97069	−0.985346	+	0.170565i	$0.945441\pi$
$174$	−2.91446	−0.220945
$175$	0	0
$176$	0	0
$177$	12.8128	0.963066
$178$	2.31933	0.173841
$179$	−6.44765	−0.481920	−0.240960	−	0.970535i	$-0.577462\pi$
$179$	−6.44765	−0.481920	−0.240960	+	0.970535i	$0.577462\pi$
$180$	0	0
$181$	21.8187	1.62177	0.810887	−	0.585202i	$-0.198985\pi$
$181$	21.8187	1.62177	0.810887	+	0.585202i	$0.198985\pi$
$182$	2.37938	0.176371
$183$	−8.49609	−0.628049
$184$	7.43530	0.548138
$185$	0	0
$186$	4.22943	0.310117
$187$	0	0
$188$	−3.24923	−0.236974
$189$	3.55586	0.258650
$190$	0	0
$191$	16.9374	1.22555	0.612775	−	0.790258i	$-0.290053\pi$
$191$	16.9374	1.22555	0.612775	+	0.790258i	$0.290053\pi$
$192$	−5.74608	−0.414688
$193$	7.46517	0.537354	0.268677	−	0.963230i	$-0.413414\pi$
$193$	7.46517	0.537354	0.268677	+	0.963230i	$0.413414\pi$
$194$	1.73047	0.124240
$195$	0	0
$196$	6.93781	0.495558
$197$	−18.8091	−1.34009	−0.670045	−	0.742320i	$-0.733725\pi$
$197$	−18.8091	−1.34009	−0.670045	+	0.742320i	$0.733725\pi$
$198$	0	0
$199$	13.7708	0.976187	0.488094	−	0.872791i	$-0.337692\pi$
$199$	13.7708	0.976187	0.488094	+	0.872791i	$0.337692\pi$
$200$	0	0
$201$	10.6621	0.752049
$202$	10.5928	0.745306
$203$	2.57425	0.180677
$204$	0.760843	0.0532697
$205$	0	0
$206$	13.5672	0.945272
$207$	5.51565	0.383364
$208$	−2.18559	−0.151543
$209$	0	0
$210$	0	0
$211$	−17.0376	−1.17291	−0.586457	−	0.809980i	$-0.699478\pi$
$211$	−17.0376	−1.17291	−0.586457	+	0.809980i	$0.699478\pi$
$212$	14.9591	1.02739
$213$	1.96272	0.134483
$214$	−10.8900	−0.744424
$215$	0	0
$216$	−13.7902	−0.938307
$217$	−3.73573	−0.253598
$218$	−10.1738	−0.689060
$219$	13.0396	0.881136
$220$	0	0
$221$	2.55066	0.171576
$222$	−4.90836	−0.329428
$223$	7.19470	0.481793	0.240896	−	0.970551i	$-0.422559\pi$
$223$	7.19470	0.481793	0.240896	+	0.970551i	$0.422559\pi$
$224$	4.00962	0.267904
$225$	0	0
$226$	−10.2331	−0.680698
$227$	−13.5046	−0.896330	−0.448165	−	0.893951i	$-0.647922\pi$
$227$	−13.5046	−0.896330	−0.448165	+	0.893951i	$0.647922\pi$
$228$	4.46865	0.295944
$229$	24.5209	1.62038	0.810192	−	0.586164i	$-0.199363\pi$
$229$	24.5209	1.62038	0.810192	+	0.586164i	$0.199363\pi$
$230$	0	0
$231$	0	0
$232$	−9.98340	−0.655442
$233$	−6.17558	−0.404576	−0.202288	−	0.979326i	$-0.564838\pi$
$233$	−6.17558	−0.404576	−0.202288	+	0.979326i	$0.564838\pi$
$234$	−6.84526	−0.447488
$235$	0	0
$236$	15.3961	1.00220
$237$	−6.60616	−0.429116
$238$	0.571694	0.0370574
$239$	−12.0519	−0.779574	−0.389787	−	0.920905i	$-0.627451\pi$
$239$	−12.0519	−0.779574	−0.389787	+	0.920905i	$0.627451\pi$
$240$	0	0
$241$	−5.38650	−0.346975	−0.173488	−	0.984836i	$-0.555504\pi$
$241$	−5.38650	−0.346975	−0.173488	+	0.984836i	$0.555504\pi$
$242$	0	0
$243$	−16.1417	−1.03549
$244$	−10.2091	−0.653572
$245$	0	0
$246$	4.69846	0.299563
$247$	14.9807	0.953201
$248$	14.4878	0.919977
$249$	0.709502	0.0449629
$250$	0	0
$251$	−4.54538	−0.286902	−0.143451	−	0.989657i	$-0.545820\pi$
$251$	−4.54538	−0.286902	−0.143451	+	0.989657i	$0.545820\pi$
$252$	1.80354	0.113612
$253$	0	0
$254$	−6.63304	−0.416194
$255$	0	0
$256$	−16.9999	−1.06249
$257$	−21.9515	−1.36930	−0.684648	−	0.728874i	$-0.740044\pi$
$257$	−21.9515	−1.36930	−0.684648	+	0.728874i	$0.740044\pi$
$258$	9.10601	0.566915
$259$	4.33540	0.269389
$260$	0	0
$261$	−7.40588	−0.458413
$262$	−6.72843	−0.415684
$263$	−1.42036	−0.0875834	−0.0437917	−	0.999041i	$-0.513944\pi$
$263$	−1.42036	−0.0875834	−0.0437917	+	0.999041i	$0.513944\pi$
$264$	0	0
$265$	0	0
$266$	3.35772	0.205875
$267$	−2.17546	−0.133136
$268$	12.8119	0.782612
$269$	6.86809	0.418755	0.209377	−	0.977835i	$-0.432856\pi$
$269$	6.86809	0.418755	0.209377	+	0.977835i	$0.432856\pi$
$270$	0	0
$271$	16.3452	0.992900	0.496450	−	0.868065i	$-0.334637\pi$
$271$	16.3452	0.992900	0.496450	+	0.868065i	$0.334637\pi$
$272$	−0.525132	−0.0318408
$273$	−2.23179	−0.135074
$274$	7.63152	0.461037
$275$	0	0
$276$	−2.44646	−0.147259
$277$	9.95773	0.598302	0.299151	−	0.954206i	$-0.403297\pi$
$277$	9.95773	0.598302	0.299151	+	0.954206i	$0.403297\pi$
$278$	13.2748	0.796168
$279$	10.7473	0.643426
$280$	0	0
$281$	21.6466	1.29133	0.645663	−	0.763623i	$-0.276581\pi$
$281$	21.6466	1.29133	0.645663	+	0.763623i	$0.276581\pi$
$282$	−2.59266	−0.154391
$283$	6.61565	0.393260	0.196630	−	0.980478i	$-0.437000\pi$
$283$	6.61565	0.393260	0.196630	+	0.980478i	$0.437000\pi$
$284$	2.35845	0.139948
$285$	0	0
$286$	0	0
$287$	−4.15001	−0.244967
$288$	−11.5353	−0.679725
$289$	−16.3872	−0.963950
$290$	0	0
$291$	−1.62313	−0.0951495
$292$	15.6688	0.916944
$293$	10.4738	0.611885	0.305943	−	0.952050i	$-0.401028\pi$
$293$	10.4738	0.611885	0.305943	+	0.952050i	$0.401028\pi$
$294$	5.53590	0.322860
$295$	0	0
$296$	−16.8135	−0.977263
$297$	0	0
$298$	−6.65472	−0.385498
$299$	−8.20151	−0.474306
$300$	0	0
$301$	−8.04305	−0.463594
$302$	−11.5213	−0.662976
$303$	−9.93574	−0.570793
$304$	−3.08425	−0.176894
$305$	0	0
$306$	−1.64471	−0.0940219
$307$	−13.0191	−0.743041	−0.371520	−	0.928425i	$-0.621163\pi$
$307$	−13.0191	−0.743041	−0.371520	+	0.928425i	$0.621163\pi$
$308$	0	0
$309$	−12.7257	−0.723938
$310$	0	0
$311$	17.2791	0.979807	0.489904	−	0.871777i	$-0.337032\pi$
$311$	17.2791	0.979807	0.489904	+	0.871777i	$0.337032\pi$
$312$	8.65529	0.490009
$313$	10.0540	0.568286	0.284143	−	0.958782i	$-0.408291\pi$
$313$	10.0540	0.568286	0.284143	+	0.958782i	$0.408291\pi$
$314$	−17.0214	−0.960575
$315$	0	0
$316$	−7.93813	−0.446555
$317$	1.99347	0.111964	0.0559822	−	0.998432i	$-0.482171\pi$
$317$	1.99347	0.111964	0.0559822	+	0.998432i	$0.482171\pi$
$318$	11.9363	0.669356
$319$	0	0
$320$	0	0
$321$	10.2145	0.570118
$322$	−1.83826	−0.102442
$323$	3.59942	0.200277
$324$	−2.56642	−0.142579
$325$	0	0
$326$	−8.23710	−0.456211
$327$	9.54279	0.527717
$328$	16.0945	0.888668
$329$	2.29002	0.126253
$330$	0	0
$331$	−10.4300	−0.573283	−0.286642	−	0.958038i	$-0.592539\pi$
$331$	−10.4300	−0.573283	−0.286642	+	0.958038i	$0.592539\pi$
$332$	0.852557	0.0467901
$333$	−12.4726	−0.683492
$334$	0.716881	0.0392260
$335$	0	0
$336$	0.459484	0.0250669
$337$	−11.6395	−0.634044	−0.317022	−	0.948418i	$-0.602683\pi$
$337$	−11.6395	−0.634044	−0.317022	+	0.948418i	$0.602683\pi$
$338$	−2.28603	−0.124343
$339$	9.59839	0.521313
$340$	0	0
$341$	0	0
$342$	−9.65986	−0.522345
$343$	−10.2212	−0.551893
$344$	31.1924	1.68178
$345$	0	0
$346$	−24.8529	−1.33610
$347$	−34.1016	−1.83067	−0.915334	−	0.402695i	$-0.868074\pi$
$347$	−34.1016	−1.83067	−0.915334	+	0.402695i	$0.868074\pi$
$348$	3.28486	0.176087
$349$	−6.32525	−0.338583	−0.169291	−	0.985566i	$-0.554148\pi$
$349$	−6.32525	−0.338583	−0.169291	+	0.985566i	$0.554148\pi$
$350$	0	0
$351$	15.2113	0.811921
$352$	0	0
$353$	30.5776	1.62748	0.813742	−	0.581226i	$-0.197427\pi$
$353$	30.5776	1.62748	0.813742	+	0.581226i	$0.197427\pi$
$354$	12.2851	0.652944
$355$	0	0
$356$	−2.61409	−0.138547
$357$	−0.536233	−0.0283805
$358$	−6.18211	−0.326734
$359$	−14.4505	−0.762668	−0.381334	−	0.924437i	$-0.624535\pi$
$359$	−14.4505	−0.762668	−0.381334	+	0.924437i	$0.624535\pi$
$360$	0	0
$361$	2.14045	0.112655
$362$	20.9201	1.09954
$363$	0	0
$364$	−2.68178	−0.140563
$365$	0	0
$366$	−8.14618	−0.425808
$367$	−29.0792	−1.51792	−0.758962	−	0.651135i	$-0.774293\pi$
$367$	−29.0792	−1.51792	−0.758962	+	0.651135i	$0.774293\pi$
$368$	1.68854	0.0880211
$369$	11.9392	0.621529
$370$	0	0
$371$	−10.5430	−0.547365
$372$	−4.76696	−0.247155
$373$	12.1992	0.631653	0.315826	−	0.948817i	$-0.397718\pi$
$373$	12.1992	0.631653	0.315826	+	0.948817i	$0.397718\pi$
$374$	0	0
$375$	0	0
$376$	−8.88109	−0.458007
$377$	11.0122	0.567157
$378$	3.40941	0.175361
$379$	27.0350	1.38869	0.694346	−	0.719641i	$-0.255694\pi$
$379$	27.0350	1.38869	0.694346	+	0.719641i	$0.255694\pi$
$380$	0	0
$381$	6.22161	0.318743
$382$	16.2399	0.830904
$383$	26.9866	1.37895	0.689477	−	0.724308i	$-0.257841\pi$
$383$	26.9866	1.37895	0.689477	+	0.724308i	$0.257841\pi$
$384$	3.95960	0.202062
$385$	0	0
$386$	7.15772	0.364318
$387$	23.1391	1.17623
$388$	−1.95040	−0.0990163
$389$	10.3178	0.523133	0.261567	−	0.965185i	$-0.415761\pi$
$389$	10.3178	0.523133	0.261567	+	0.965185i	$0.415761\pi$
$390$	0	0
$391$	−1.97058	−0.0996565
$392$	18.9631	0.957780
$393$	6.31108	0.318352
$394$	−18.0344	−0.908561
$395$	0	0
$396$	0	0
$397$	−7.63508	−0.383194	−0.191597	−	0.981474i	$-0.561367\pi$
$397$	−7.63508	−0.383194	−0.191597	+	0.981474i	$0.561367\pi$
$398$	13.2037	0.661840
$399$	−3.14945	−0.157670
$400$	0	0
$401$	−20.3039	−1.01393	−0.506964	−	0.861967i	$-0.669232\pi$
$401$	−20.3039	−1.01393	−0.506964	+	0.861967i	$0.669232\pi$
$402$	10.2230	0.509878
$403$	−15.9808	−0.796060
$404$	−11.9390	−0.593990
$405$	0	0
$406$	2.46823	0.122496
$407$	0	0
$408$	2.07961	0.102956
$409$	−16.4461	−0.813208	−0.406604	−	0.913605i	$-0.633287\pi$
$409$	−16.4461	−0.813208	−0.406604	+	0.913605i	$0.633287\pi$
$410$	0	0
$411$	−7.15816	−0.353086
$412$	−15.2915	−0.753358
$413$	−10.8510	−0.533944
$414$	5.28849	0.259915
$415$	0	0
$416$	17.1525	0.840969
$417$	−12.4514	−0.609746
$418$	0	0
$419$	0.536621	0.0262157	0.0131078	−	0.999914i	$-0.495828\pi$
$419$	0.536621	0.0262157	0.0131078	+	0.999914i	$0.495828\pi$
$420$	0	0
$421$	−7.94084	−0.387013	−0.193506	−	0.981099i	$-0.561986\pi$
$421$	−7.94084	−0.387013	−0.193506	+	0.981099i	$0.561986\pi$
$422$	−16.3359	−0.795219
$423$	−6.58817	−0.320328
$424$	40.8876	1.98568
$425$	0	0
$426$	1.88188	0.0911776
$427$	7.19527	0.348203
$428$	12.2740	0.593287
$429$	0	0
$430$	0	0
$431$	4.97717	0.239742	0.119871	−	0.992789i	$-0.461752\pi$
$431$	4.97717	0.239742	0.119871	+	0.992789i	$0.461752\pi$
$432$	−3.13173	−0.150675
$433$	−15.5489	−0.747230	−0.373615	−	0.927584i	$-0.621882\pi$
$433$	−15.5489	−0.747230	−0.373615	+	0.927584i	$0.621882\pi$
$434$	−3.58187	−0.171935
$435$	0	0
$436$	11.4669	0.549163
$437$	−11.5738	−0.553649
$438$	12.5026	0.597397
$439$	−13.4404	−0.641476	−0.320738	−	0.947168i	$-0.603931\pi$
$439$	−13.4404	−0.641476	−0.320738	+	0.947168i	$0.603931\pi$
$440$	0	0
$441$	14.0672	0.669865
$442$	2.44561	0.116326
$443$	−17.6173	−0.837023	−0.418511	−	0.908212i	$-0.637448\pi$
$443$	−17.6173	−0.837023	−0.418511	+	0.908212i	$0.637448\pi$
$444$	5.53217	0.262545
$445$	0	0
$446$	6.89839	0.326648
$447$	6.24194	0.295234
$448$	4.86631	0.229912
$449$	1.16048	0.0547665	0.0273833	−	0.999625i	$-0.491283\pi$
$449$	1.16048	0.0547665	0.0273833	+	0.999625i	$0.491283\pi$
$450$	0	0
$451$	0	0
$452$	11.5337	0.542499
$453$	10.8067	0.507741
$454$	−12.9484	−0.607698
$455$	0	0
$456$	12.2141	0.571979
$457$	21.7498	1.01741	0.508705	−	0.860941i	$-0.330124\pi$
$457$	21.7498	1.01741	0.508705	+	0.860941i	$0.330124\pi$
$458$	23.5110	1.09860
$459$	3.65483	0.170593
$460$	0	0
$461$	34.7416	1.61808	0.809040	−	0.587754i	$-0.199988\pi$
$461$	34.7416	1.61808	0.809040	+	0.587754i	$0.199988\pi$
$462$	0	0
$463$	−4.01443	−0.186566	−0.0932832	−	0.995640i	$-0.529736\pi$
$463$	−4.01443	−0.186566	−0.0932832	+	0.995640i	$0.529736\pi$
$464$	−2.26720	−0.105252
$465$	0	0
$466$	−5.92124	−0.274296
$467$	2.64795	0.122533	0.0612663	−	0.998121i	$-0.480486\pi$
$467$	2.64795	0.122533	0.0612663	+	0.998121i	$0.480486\pi$
$468$	7.71523	0.356637
$469$	−9.02968	−0.416952
$470$	0	0
$471$	15.9656	0.735658
$472$	42.0822	1.93699
$473$	0	0
$474$	−6.33409	−0.290934
$475$	0	0
$476$	−0.644352	−0.0295338
$477$	30.3312	1.38877
$478$	−11.5556	−0.528539
$479$	−15.9127	−0.727068	−0.363534	−	0.931581i	$-0.618430\pi$
$479$	−15.9127	−0.727068	−0.363534	+	0.931581i	$0.618430\pi$
$480$	0	0
$481$	18.5461	0.845629
$482$	−5.16466	−0.235244
$483$	1.72423	0.0784553
$484$	0	0
$485$	0	0
$486$	−15.4770	−0.702049
$487$	−5.71235	−0.258851	−0.129426	−	0.991589i	$-0.541313\pi$
$487$	−5.71235	−0.258851	−0.129426	+	0.991589i	$0.541313\pi$
$488$	−27.9045	−1.26318
$489$	7.72617	0.349389
$490$	0	0
$491$	5.24698	0.236793	0.118396	−	0.992966i	$-0.462225\pi$
$491$	5.24698	0.236793	0.118396	+	0.992966i	$0.462225\pi$
$492$	−5.29560	−0.238744
$493$	2.64590	0.119165
$494$	14.3638	0.646256
$495$	0	0
$496$	3.29014	0.147732
$497$	−1.66221	−0.0745603
$498$	0.680282	0.0304842
$499$	15.8703	0.710451	0.355225	−	0.934781i	$-0.384404\pi$
$499$	15.8703	0.710451	0.355225	+	0.934781i	$0.384404\pi$
$500$	0	0
$501$	−0.672414	−0.0300412
$502$	−4.35818	−0.194515
$503$	36.2385	1.61579	0.807897	−	0.589323i	$-0.200606\pi$
$503$	36.2385	1.61579	0.807897	+	0.589323i	$0.200606\pi$
$504$	4.92961	0.219582
$505$	0	0
$506$	0	0
$507$	2.14423	0.0952286
$508$	7.47605	0.331696
$509$	17.8023	0.789075	0.394537	−	0.918880i	$-0.370905\pi$
$509$	17.8023	0.789075	0.394537	+	0.918880i	$0.370905\pi$
$510$	0	0
$511$	−11.0431	−0.488520
$512$	−7.49418	−0.331199
$513$	21.4659	0.947741
$514$	−21.0474	−0.928362
$515$	0	0
$516$	−10.2633	−0.451817
$517$	0	0
$518$	4.15685	0.182642
$519$	23.3113	1.02325
$520$	0	0
$521$	31.2701	1.36997	0.684984	−	0.728558i	$-0.259809\pi$
$521$	31.2701	1.36997	0.684984	+	0.728558i	$0.259809\pi$
$522$	−7.10087	−0.310797
$523$	−3.14554	−0.137545	−0.0687723	−	0.997632i	$-0.521908\pi$
$523$	−3.14554	−0.137545	−0.0687723	+	0.997632i	$0.521908\pi$
$524$	7.58356	0.331289
$525$	0	0
$526$	−1.36187	−0.0593802
$527$	−3.83971	−0.167260
$528$	0	0
$529$	−16.6637	−0.724509
$530$	0	0
$531$	31.2174	1.35472
$532$	−3.78446	−0.164077
$533$	−17.7530	−0.768968
$534$	−2.08587	−0.0902643
$535$	0	0
$536$	35.0187	1.51258
$537$	5.79864	0.250230
$538$	6.58523	0.283909
$539$	0	0
$540$	0	0
$541$	−26.2469	−1.12844	−0.564221	−	0.825624i	$-0.690823\pi$
$541$	−26.2469	−1.12844	−0.564221	+	0.825624i	$0.690823\pi$
$542$	15.6720	0.673171
$543$	−19.6225	−0.842083
$544$	4.12123	0.176696
$545$	0	0
$546$	−2.13988	−0.0915783
$547$	34.1455	1.45996	0.729979	−	0.683470i	$-0.239530\pi$
$547$	34.1455	1.45996	0.729979	+	0.683470i	$0.239530\pi$
$548$	−8.60143	−0.367435
$549$	−20.7001	−0.883459
$550$	0	0
$551$	15.5401	0.662032
$552$	−6.68688	−0.284613
$553$	5.59470	0.237911
$554$	9.54762	0.405640
$555$	0	0
$556$	−14.9619	−0.634525
$557$	34.6084	1.46640	0.733202	−	0.680010i	$-0.238025\pi$
$557$	34.6084	1.46640	0.733202	+	0.680010i	$0.238025\pi$
$558$	10.3047	0.436233
$559$	−34.4068	−1.45525
$560$	0	0
$561$	0	0
$562$	20.7550	0.875499
$563$	−25.9394	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$563$	−25.9394	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$564$	2.92217	0.123045
$565$	0	0
$566$	6.34319	0.266624
$567$	1.80878	0.0759617
$568$	6.44634	0.270482
$569$	−16.3801	−0.686691	−0.343346	−	0.939209i	$-0.611560\pi$
$569$	−16.3801	−0.686691	−0.343346	+	0.939209i	$0.611560\pi$
$570$	0	0
$571$	14.8503	0.621465	0.310733	−	0.950497i	$-0.399426\pi$
$571$	14.8503	0.621465	0.310733	+	0.950497i	$0.399426\pi$
$572$	0	0
$573$	−15.2325	−0.636349
$574$	−3.97909	−0.166084
$575$	0	0
$576$	−13.9999	−0.583330
$577$	34.2715	1.42674	0.713370	−	0.700788i	$-0.247168\pi$
$577$	34.2715	1.42674	0.713370	+	0.700788i	$0.247168\pi$
$578$	−15.7123	−0.653544
$579$	−6.71374	−0.279013
$580$	0	0
$581$	−0.600872	−0.0249284
$582$	−1.55628	−0.0645100
$583$	0	0
$584$	42.8273	1.77221
$585$	0	0
$586$	10.0424	0.414849
$587$	−6.32594	−0.261100	−0.130550	−	0.991442i	$-0.541674\pi$
$587$	−6.32594	−0.261100	−0.130550	+	0.991442i	$0.541674\pi$
$588$	−6.23946	−0.257311
$589$	−22.5517	−0.929227
$590$	0	0
$591$	16.9158	0.695822
$592$	−3.81829	−0.156931
$593$	−18.3276	−0.752623	−0.376311	−	0.926493i	$-0.622808\pi$
$593$	−18.3276	−0.752623	−0.376311	+	0.926493i	$0.622808\pi$
$594$	0	0
$595$	0	0
$596$	7.50048	0.307232
$597$	−12.3847	−0.506871
$598$	−7.86374	−0.321572
$599$	−18.9096	−0.772627	−0.386313	−	0.922368i	$-0.626252\pi$
$599$	−18.9096	−0.772627	−0.386313	+	0.922368i	$0.626252\pi$
$600$	0	0
$601$	45.9804	1.87558	0.937790	−	0.347202i	$-0.112868\pi$
$601$	45.9804	1.87558	0.937790	+	0.347202i	$0.112868\pi$
$602$	−7.71180	−0.314310
$603$	25.9776	1.05789
$604$	12.9856	0.528375
$605$	0	0
$606$	−9.52654	−0.386989
$607$	−35.0051	−1.42081	−0.710406	−	0.703792i	$-0.751489\pi$
$607$	−35.0051	−1.42081	−0.710406	+	0.703792i	$0.751489\pi$
$608$	24.2051	0.981648
$609$	−2.31513	−0.0938139
$610$	0	0
$611$	9.79629	0.396316
$612$	1.85374	0.0749330
$613$	41.0877	1.65951	0.829757	−	0.558124i	$-0.188479\pi$
$613$	41.0877	1.65951	0.829757	+	0.558124i	$0.188479\pi$
$614$	−12.4829	−0.503770
$615$	0	0
$616$	0	0
$617$	11.1082	0.447201	0.223600	−	0.974681i	$-0.428219\pi$
$617$	11.1082	0.447201	0.223600	+	0.974681i	$0.428219\pi$
$618$	−12.2016	−0.490819
$619$	−24.0771	−0.967741	−0.483870	−	0.875140i	$-0.660769\pi$
$619$	−24.0771	−0.967741	−0.483870	+	0.875140i	$0.660769\pi$
$620$	0	0
$621$	−11.7519	−0.471589
$622$	16.5675	0.664295
$623$	1.84238	0.0738135
$624$	1.96559	0.0786867
$625$	0	0
$626$	9.63993	0.385289
$627$	0	0
$628$	19.1847	0.765554
$629$	4.45608	0.177675
$630$	0	0
$631$	−43.6944	−1.73945	−0.869724	−	0.493538i	$-0.835703\pi$
$631$	−43.6944	−1.73945	−0.869724	+	0.493538i	$0.835703\pi$
$632$	−21.6972	−0.863070
$633$	15.3226	0.609019
$634$	1.91137	0.0759101
$635$	0	0
$636$	−13.4533	−0.533460
$637$	−20.9172	−0.828770
$638$	0	0
$639$	4.78202	0.189174
$640$	0	0
$641$	−16.7544	−0.661760	−0.330880	−	0.943673i	$-0.607346\pi$
$641$	−16.7544	−0.661760	−0.330880	+	0.943673i	$0.607346\pi$
$642$	9.79382	0.386531
$643$	4.18868	0.165186	0.0825928	−	0.996583i	$-0.473680\pi$
$643$	4.18868	0.165186	0.0825928	+	0.996583i	$0.473680\pi$
$644$	2.07188	0.0816436
$645$	0	0
$646$	3.45118	0.135785
$647$	14.6389	0.575516	0.287758	−	0.957703i	$-0.407090\pi$
$647$	14.6389	0.575516	0.287758	+	0.957703i	$0.407090\pi$
$648$	−7.01478	−0.275567
$649$	0	0
$650$	0	0
$651$	3.35970	0.131677
$652$	9.28397	0.363588
$653$	−22.5629	−0.882954	−0.441477	−	0.897273i	$-0.645545\pi$
$653$	−22.5629	−0.882954	−0.441477	+	0.897273i	$0.645545\pi$
$654$	9.14977	0.357784
$655$	0	0
$656$	3.65501	0.142704
$657$	31.7701	1.23947
$658$	2.19570	0.0855974
$659$	−42.3114	−1.64822	−0.824110	−	0.566430i	$-0.808324\pi$
$659$	−42.3114	−1.64822	−0.824110	+	0.566430i	$0.808324\pi$
$660$	0	0
$661$	−10.6525	−0.414334	−0.207167	−	0.978306i	$-0.566424\pi$
$661$	−10.6525	−0.414334	−0.207167	+	0.978306i	$0.566424\pi$
$662$	−10.0004	−0.388677
$663$	−2.29391	−0.0890882
$664$	2.33029	0.0904327
$665$	0	0
$666$	−11.9589	−0.463397
$667$	−8.50777	−0.329422
$668$	−0.807990	−0.0312621
$669$	−6.47050	−0.250164
$670$	0	0
$671$	0	0
$672$	−3.60602	−0.139105
$673$	18.7484	0.722696	0.361348	−	0.932431i	$-0.382317\pi$
$673$	18.7484	0.722696	0.361348	+	0.932431i	$0.382317\pi$
$674$	−11.1601	−0.429872
$675$	0	0
$676$	2.57656	0.0990986
$677$	−46.0553	−1.77005	−0.885025	−	0.465543i	$-0.845859\pi$
$677$	−46.0553	−1.77005	−0.885025	+	0.465543i	$0.845859\pi$
$678$	9.20308	0.353442
$679$	1.37461	0.0527529
$680$	0	0
$681$	12.1452	0.465406
$682$	0	0
$683$	−7.99721	−0.306005	−0.153002	−	0.988226i	$-0.548894\pi$
$683$	−7.99721	−0.306005	−0.153002	+	0.988226i	$0.548894\pi$
$684$	10.8876	0.416296
$685$	0	0
$686$	−9.80025	−0.374175
$687$	−22.0527	−0.841361
$688$	7.08371	0.270064
$689$	−45.1011	−1.71821
$690$	0	0
$691$	6.14408	0.233732	0.116866	−	0.993148i	$-0.462715\pi$
$691$	6.14408	0.233732	0.116866	+	0.993148i	$0.462715\pi$
$692$	28.0115	1.06484
$693$	0	0
$694$	−32.6971	−1.24117
$695$	0	0
$696$	8.97850	0.340329
$697$	−4.26552	−0.161568
$698$	−6.06475	−0.229554
$699$	5.55396	0.210070
$700$	0	0
$701$	22.5075	0.850096	0.425048	−	0.905171i	$-0.360257\pi$
$701$	22.5075	0.850096	0.425048	+	0.905171i	$0.360257\pi$
$702$	14.5849	0.550470
$703$	26.1718	0.987088
$704$	0	0
$705$	0	0
$706$	29.3183	1.10341
$707$	8.41449	0.316460
$708$	−13.8464	−0.520380
$709$	−6.00326	−0.225457	−0.112729	−	0.993626i	$-0.535959\pi$
$709$	−6.00326	−0.225457	−0.112729	+	0.993626i	$0.535959\pi$
$710$	0	0
$711$	−16.0954	−0.603626
$712$	−7.14509	−0.267773
$713$	12.3464	0.462376
$714$	−0.514149	−0.0192415
$715$	0	0
$716$	6.96780	0.260399
$717$	10.8388	0.404782
$718$	−13.8554	−0.517077
$719$	−26.2179	−0.977762	−0.488881	−	0.872351i	$-0.662595\pi$
$719$	−26.2179	−0.977762	−0.488881	+	0.872351i	$0.662595\pi$
$720$	0	0
$721$	10.7773	0.401366
$722$	2.05230	0.0763786
$723$	4.84431	0.180162
$724$	−23.5789	−0.876304
$725$	0	0
$726$	0	0
$727$	−4.86669	−0.180495	−0.0902477	−	0.995919i	$-0.528766\pi$
$727$	−4.86669	−0.180495	−0.0902477	+	0.995919i	$0.528766\pi$
$728$	−7.33009	−0.271671
$729$	7.39244	0.273794
$730$	0	0
$731$	−8.26693	−0.305763
$732$	9.18149	0.339358
$733$	37.0900	1.36995	0.684975	−	0.728566i	$-0.259813\pi$
$733$	37.0900	1.36995	0.684975	+	0.728566i	$0.259813\pi$
$734$	−27.8816	−1.02913
$735$	0	0
$736$	−13.2516	−0.488461
$737$	0	0
$738$	11.4475	0.421387
$739$	−0.723750	−0.0266236	−0.0133118	−	0.999911i	$-0.504237\pi$
$739$	−0.723750	−0.0266236	−0.0133118	+	0.999911i	$0.504237\pi$
$740$	0	0
$741$	−13.4728	−0.494936
$742$	−10.1088	−0.371105
$743$	−34.6512	−1.27123	−0.635614	−	0.772007i	$-0.719253\pi$
$743$	−34.6512	−1.27123	−0.635614	+	0.772007i	$0.719253\pi$
$744$	−13.0295	−0.477685
$745$	0	0
$746$	11.6968	0.428251
$747$	1.72865	0.0632481
$748$	0	0
$749$	−8.65057	−0.316085
$750$	0	0
$751$	6.49256	0.236917	0.118458	−	0.992959i	$-0.462205\pi$
$751$	6.49256	0.236917	0.118458	+	0.992959i	$0.462205\pi$
$752$	−2.01687	−0.0735478
$753$	4.08785	0.148970
$754$	10.5587	0.384524
$755$	0	0
$756$	−3.84272	−0.139758
$757$	34.8773	1.26764	0.633818	−	0.773482i	$-0.281487\pi$
$757$	34.8773	1.26764	0.633818	+	0.773482i	$0.281487\pi$
$758$	25.9215	0.941512
$759$	0	0
$760$	0	0
$761$	44.3602	1.60806	0.804029	−	0.594591i	$-0.202686\pi$
$761$	44.3602	1.60806	0.804029	+	0.594591i	$0.202686\pi$
$762$	5.96538	0.216103
$763$	−8.08170	−0.292577
$764$	−18.3038	−0.662209
$765$	0	0
$766$	25.8752	0.934909
$767$	−46.4188	−1.67608
$768$	15.2887	0.551683
$769$	−17.0854	−0.616117	−0.308058	−	0.951367i	$-0.599679\pi$
$769$	−17.0854	−0.616117	−0.308058	+	0.951367i	$0.599679\pi$
$770$	0	0
$771$	19.7419	0.710987
$772$	−8.06740	−0.290352
$773$	12.6914	0.456477	0.228239	−	0.973605i	$-0.426703\pi$
$773$	12.6914	0.456477	0.228239	+	0.973605i	$0.426703\pi$
$774$	22.1861	0.797464
$775$	0	0
$776$	−5.33100	−0.191372
$777$	−3.89901	−0.139876
$778$	9.89286	0.354676
$779$	−25.0526	−0.897603
$780$	0	0
$781$	0	0
$782$	−1.88942	−0.0675656
$783$	15.7794	0.563908
$784$	4.30646	0.153802
$785$	0	0
$786$	6.05116	0.215838
$787$	35.8290	1.27717	0.638584	−	0.769552i	$-0.279521\pi$
$787$	35.8290	1.27717	0.638584	+	0.769552i	$0.279521\pi$
$788$	20.3264	0.724100
$789$	1.27739	0.0454764
$790$	0	0
$791$	−8.12880	−0.289027
$792$	0	0
$793$	30.7801	1.09303
$794$	−7.32063	−0.259800
$795$	0	0
$796$	−14.8817	−0.527470
$797$	4.70921	0.166809	0.0834043	−	0.996516i	$-0.473421\pi$
$797$	4.70921	0.166809	0.0834043	+	0.996516i	$0.473421\pi$
$798$	−3.01974	−0.106898
$799$	2.35376	0.0832700
$800$	0	0
$801$	−5.30036	−0.187279
$802$	−19.4677	−0.687428
$803$	0	0
$804$	−11.5223	−0.406360
$805$	0	0
$806$	−15.3226	−0.539717
$807$	−6.17676	−0.217432
$808$	−32.6329	−1.14802
$809$	−18.6284	−0.654940	−0.327470	−	0.944862i	$-0.606196\pi$
$809$	−18.6284	−0.654940	−0.327470	+	0.944862i	$0.606196\pi$
$810$	0	0
$811$	−15.8626	−0.557012	−0.278506	−	0.960435i	$-0.589839\pi$
$811$	−15.8626	−0.557012	−0.278506	+	0.960435i	$0.589839\pi$
$812$	−2.78192	−0.0976264
$813$	−14.6999	−0.515549
$814$	0	0
$815$	0	0
$816$	0.472274	0.0165329
$817$	−48.5540	−1.69869
$818$	−15.7688	−0.551342
$819$	−5.43760	−0.190005
$820$	0	0
$821$	0.685709	0.0239314	0.0119657	−	0.999928i	$-0.496191\pi$
$821$	0.685709	0.0239314	0.0119657	+	0.999928i	$0.496191\pi$
$822$	−6.86335	−0.239387
$823$	24.5414	0.855460	0.427730	−	0.903907i	$-0.359313\pi$
$823$	24.5414	0.855460	0.427730	+	0.903907i	$0.359313\pi$
$824$	−41.7961	−1.45604
$825$	0	0
$826$	−10.4041	−0.362006
$827$	−30.4322	−1.05823	−0.529116	−	0.848550i	$-0.677476\pi$
$827$	−30.4322	−1.05823	−0.529116	+	0.848550i	$0.677476\pi$
$828$	−5.96062	−0.207146
$829$	−10.0715	−0.349797	−0.174899	−	0.984586i	$-0.555960\pi$
$829$	−10.0715	−0.349797	−0.174899	+	0.984586i	$0.555960\pi$
$830$	0	0
$831$	−8.95541	−0.310660
$832$	20.8172	0.721707
$833$	−5.02579	−0.174133
$834$	−11.9386	−0.413399
$835$	0	0
$836$	0	0
$837$	−22.8988	−0.791500
$838$	0.514521	0.0177738
$839$	34.8027	1.20152	0.600762	−	0.799428i	$-0.294864\pi$
$839$	34.8027	1.20152	0.600762	+	0.799428i	$0.294864\pi$
$840$	0	0
$841$	−17.5766	−0.606089
$842$	−7.61380	−0.262389
$843$	−19.4677	−0.670502
$844$	18.4120	0.633769
$845$	0	0
$846$	−6.31683	−0.217177
$847$	0	0
$848$	9.28547	0.318864
$849$	−5.94973	−0.204194
$850$	0	0
$851$	−14.3283	−0.491168
$852$	−2.12106	−0.0726662
$853$	−27.1822	−0.930699	−0.465350	−	0.885127i	$-0.654071\pi$
$853$	−27.1822	−0.930699	−0.465350	+	0.885127i	$0.654071\pi$
$854$	6.89893	0.236077
$855$	0	0
$856$	33.5485	1.14666
$857$	31.7263	1.08375	0.541875	−	0.840459i	$-0.317715\pi$
$857$	31.7263	1.08375	0.541875	+	0.840459i	$0.317715\pi$
$858$	0	0
$859$	−0.649871	−0.0221733	−0.0110867	−	0.999939i	$-0.503529\pi$
$859$	−0.649871	−0.0221733	−0.0110867	+	0.999939i	$0.503529\pi$
$860$	0	0
$861$	3.73228	0.127196
$862$	4.77219	0.162541
$863$	−29.6096	−1.00792	−0.503961	−	0.863727i	$-0.668124\pi$
$863$	−29.6096	−1.00792	−0.503961	+	0.863727i	$0.668124\pi$
$864$	24.5777	0.836152
$865$	0	0
$866$	−14.9085	−0.506611
$867$	14.7377	0.500517
$868$	4.03710	0.137028
$869$	0	0
$870$	0	0
$871$	−38.6274	−1.30884
$872$	31.3423	1.06138
$873$	−3.95464	−0.133844
$874$	−11.0971	−0.375366
$875$	0	0
$876$	−14.0916	−0.476110
$877$	11.1676	0.377104	0.188552	−	0.982063i	$-0.439621\pi$
$877$	11.1676	0.377104	0.188552	+	0.982063i	$0.439621\pi$
$878$	−12.8869	−0.434911
$879$	−9.41952	−0.317713
$880$	0	0
$881$	−35.2369	−1.18716	−0.593581	−	0.804774i	$-0.702286\pi$
$881$	−35.2369	−1.18716	−0.593581	+	0.804774i	$0.702286\pi$
$882$	13.4878	0.454159
$883$	8.91151	0.299896	0.149948	−	0.988694i	$-0.452089\pi$
$883$	8.91151	0.299896	0.149948	+	0.988694i	$0.452089\pi$
$884$	−2.75643	−0.0927086
$885$	0	0
$886$	−16.8917	−0.567489
$887$	−35.3317	−1.18632	−0.593161	−	0.805084i	$-0.702120\pi$
$887$	−35.3317	−1.18632	−0.593161	+	0.805084i	$0.702120\pi$
$888$	15.1211	0.507429
$889$	−5.26903	−0.176718
$890$	0	0
$891$	0	0
$892$	−7.77512	−0.260330
$893$	13.8243	0.462612
$894$	5.98487	0.200164
$895$	0	0
$896$	−3.35335	−0.112028
$897$	7.37597	0.246276
$898$	1.11269	0.0371309
$899$	−16.5775	−0.552892
$900$	0	0
$901$	−10.8364	−0.361014
$902$	0	0
$903$	7.23346	0.240714
$904$	31.5249	1.04850
$905$	0	0
$906$	10.3616	0.344241
$907$	12.0851	0.401280	0.200640	−	0.979665i	$-0.435698\pi$
$907$	12.0851	0.401280	0.200640	+	0.979665i	$0.435698\pi$
$908$	14.5940	0.484320
$909$	−24.2077	−0.802919
$910$	0	0
$911$	39.3913	1.30509	0.652546	−	0.757749i	$-0.273701\pi$
$911$	39.3913	1.30509	0.652546	+	0.757749i	$0.273701\pi$
$912$	2.77380	0.0918496
$913$	0	0
$914$	20.8540	0.689789
$915$	0	0
$916$	−26.4990	−0.875553
$917$	−5.34480	−0.176501
$918$	3.50431	0.115659
$919$	−12.2124	−0.402849	−0.201424	−	0.979504i	$-0.564557\pi$
$919$	−12.2124	−0.402849	−0.201424	+	0.979504i	$0.564557\pi$
$920$	0	0
$921$	11.7087	0.385813
$922$	33.3108	1.09703
$923$	−7.11064	−0.234050
$924$	0	0
$925$	0	0
$926$	−3.84910	−0.126489
$927$	−31.0052	−1.01834
$928$	17.7930	0.584083
$929$	36.5892	1.20045	0.600227	−	0.799830i	$-0.295077\pi$
$929$	36.5892	1.20045	0.600227	+	0.799830i	$0.295077\pi$
$930$	0	0
$931$	−29.5179	−0.967409
$932$	6.67378	0.218607
$933$	−15.5398	−0.508751
$934$	2.53890	0.0830753
$935$	0	0
$936$	21.0880	0.689283
$937$	38.6598	1.26296	0.631481	−	0.775391i	$-0.282447\pi$
$937$	38.6598	1.26296	0.631481	+	0.775391i	$0.282447\pi$
$938$	−8.65779	−0.282687
$939$	−9.04199	−0.295074
$940$	0	0
$941$	25.4244	0.828811	0.414405	−	0.910092i	$-0.363990\pi$
$941$	25.4244	0.828811	0.414405	+	0.910092i	$0.363990\pi$
$942$	15.3081	0.498765
$943$	13.7156	0.446641
$944$	9.55675	0.311046
$945$	0	0
$946$	0	0
$947$	49.6579	1.61367	0.806833	−	0.590780i	$-0.201180\pi$
$947$	49.6579	1.61367	0.806833	+	0.590780i	$0.201180\pi$
$948$	7.13910	0.231867
$949$	−47.2407	−1.53350
$950$	0	0
$951$	−1.79281	−0.0581358
$952$	−1.76120	−0.0570809
$953$	34.4689	1.11656	0.558279	−	0.829654i	$-0.311462\pi$
$953$	34.4689	1.11656	0.558279	+	0.829654i	$0.311462\pi$
$954$	29.0820	0.941565
$955$	0	0
$956$	13.0242	0.421232
$957$	0	0
$958$	−15.2573	−0.492941
$959$	6.06218	0.195758
$960$	0	0
$961$	−6.94283	−0.223962
$962$	17.7823	0.573324
$963$	24.8869	0.801969
$964$	5.82105	0.187483
$965$	0	0
$966$	1.65322	0.0531915
$967$	−9.89492	−0.318199	−0.159100	−	0.987263i	$-0.550859\pi$
$967$	−9.89492	−0.318199	−0.159100	+	0.987263i	$0.550859\pi$
$968$	0	0
$969$	−3.23711	−0.103991
$970$	0	0
$971$	−7.28258	−0.233709	−0.116855	−	0.993149i	$-0.537281\pi$
$971$	−7.28258	−0.233709	−0.116855	+	0.993149i	$0.537281\pi$
$972$	17.4439	0.559515
$973$	10.5450	0.338056
$974$	−5.47709	−0.175497
$975$	0	0
$976$	−6.33704	−0.202844
$977$	−51.9849	−1.66315	−0.831573	−	0.555416i	$-0.812559\pi$
$977$	−51.9849	−1.66315	−0.831573	+	0.555416i	$0.812559\pi$
$978$	7.40797	0.236881
$979$	0	0
$980$	0	0
$981$	23.2503	0.742326
$982$	5.03088	0.160542
$983$	−51.2712	−1.63530	−0.817648	−	0.575719i	$-0.804722\pi$
$983$	−51.2712	−1.63530	−0.817648	+	0.575719i	$0.804722\pi$
$984$	−14.4744	−0.461428
$985$	0	0
$986$	2.53693	0.0807924
$987$	−2.05951	−0.0655549
$988$	−16.1893	−0.515049
$989$	26.5819	0.845255
$990$	0	0
$991$	−5.14987	−0.163591	−0.0817955	−	0.996649i	$-0.526065\pi$
$991$	−5.14987	−0.163591	−0.0817955	+	0.996649i	$0.526065\pi$
$992$	−25.8210	−0.819817
$993$	9.38012	0.297669
$994$	−1.59375	−0.0505507
$995$	0	0
$996$	−0.766740	−0.0242951
$997$	29.4832	0.933743	0.466871	−	0.884325i	$-0.345381\pi$
$997$	29.4832	0.933743	0.466871	+	0.884325i	$0.345381\pi$
$998$	15.2167	0.481675
$999$	26.5747	0.840785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bi.1.7		8
5.4	even	2		3025.2.a.bn.1.2		8
11.3	even	5		275.2.h.c.251.4	yes	16
11.4	even	5		275.2.h.c.126.4	✓	16
11.10	odd	2		3025.2.a.bm.1.2		8
55.3	odd	20		275.2.z.c.174.3		32
55.4	even	10		275.2.h.e.126.1	yes	16
55.14	even	10		275.2.h.e.251.1	yes	16
55.37	odd	20		275.2.z.c.49.3		32
55.47	odd	20		275.2.z.c.174.6		32
55.48	odd	20		275.2.z.c.49.6		32
55.54	odd	2		3025.2.a.bj.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.h.c.126.4	✓	16	11.4	even	5
275.2.h.c.251.4	yes	16	11.3	even	5
275.2.h.e.126.1	yes	16	55.4	even	10
275.2.h.e.251.1	yes	16	55.14	even	10
275.2.z.c.49.3		32	55.37	odd	20
275.2.z.c.49.6		32	55.48	odd	20
275.2.z.c.174.3		32	55.3	odd	20
275.2.z.c.174.6		32	55.47	odd	20
3025.2.a.bi.1.7		8	1.1	even	1	trivial
3025.2.a.bj.1.7		8	55.54	odd	2
3025.2.a.bm.1.2		8	11.10	odd	2
3025.2.a.bn.1.2		8	5.4	even	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.958815 q^{2} -0.899342 q^{3} -1.08067 q^{4} -0.862303 q^{6} +0.761645 q^{7} -2.95380 q^{8} -2.19118 q^{9} +O(q^{10})\)	\(q+0.958815 q^{2} -0.899342 q^{3} -1.08067 q^{4} -0.862303 q^{6} +0.761645 q^{7} -2.95380 q^{8} -2.19118 q^{9} +0.971895 q^{12} +3.25819 q^{13} +0.730277 q^{14} -0.670799 q^{16} +0.782845 q^{17} -2.10094 q^{18} +4.59787 q^{19} -0.684980 q^{21} -2.51720 q^{23} +2.65647 q^{24} +3.12400 q^{26} +4.66865 q^{27} -0.823090 q^{28} +3.37985 q^{29} -4.90481 q^{31} +5.26442 q^{32} +0.750604 q^{34} +2.36795 q^{36} +5.69215 q^{37} +4.40851 q^{38} -2.93022 q^{39} -5.44874 q^{41} -0.656769 q^{42} -10.5601 q^{43} -2.41353 q^{46} +3.00667 q^{47} +0.603278 q^{48} -6.41990 q^{49} -0.704046 q^{51} -3.52103 q^{52} -13.8424 q^{53} +4.47637 q^{54} -2.24975 q^{56} -4.13506 q^{57} +3.24066 q^{58} -14.2468 q^{59} +9.44700 q^{61} -4.70281 q^{62} -1.66890 q^{63} +6.38921 q^{64} -11.8555 q^{67} -0.846000 q^{68} +2.26383 q^{69} -2.18239 q^{71} +6.47231 q^{72} -14.4991 q^{73} +5.45772 q^{74} -4.96880 q^{76} -2.80954 q^{78} +7.34554 q^{79} +2.37484 q^{81} -5.22433 q^{82} -0.788913 q^{83} +0.740239 q^{84} -10.1252 q^{86} -3.03965 q^{87} +2.41895 q^{89} +2.48158 q^{91} +2.72027 q^{92} +4.41110 q^{93} +2.88284 q^{94} -4.73452 q^{96} +1.80480 q^{97} -6.15550 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - q^{3} + 9 q^{4} + q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 - q^3 + 9 * q^4 + q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} - q^{3} + 9 q^{4} + q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} - 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} + q^{19} + 5 q^{21} - 2 q^{23} + q^{24} - 2 q^{26} + 2 q^{27} - 9 q^{28} + 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} + 8 q^{37} + 37 q^{38} - q^{39} - 41 q^{42} - 14 q^{43} - 20 q^{46} - 11 q^{47} + 27 q^{48} - 12 q^{49} - 25 q^{51} + 7 q^{52} - 11 q^{53} + 30 q^{54} - 10 q^{56} + 2 q^{57} - 27 q^{58} + 17 q^{59} - 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} - 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} - 6 q^{74} - 31 q^{76} + 17 q^{78} + 23 q^{79} - 4 q^{81} + 17 q^{82} - 41 q^{83} + 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} - 33 q^{92} + 59 q^{93} + 50 q^{94} - 61 q^{96} - 2 q^{97} - 26 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 - q^3 + 9 * q^4 + q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 - 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 + q^19 + 5 * q^21 - 2 * q^23 + q^24 - 2 * q^26 + 2 * q^27 - 9 * q^28 + 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 + 8 * q^37 + 37 * q^38 - q^39 - 41 * q^42 - 14 * q^43 - 20 * q^46 - 11 * q^47 + 27 * q^48 - 12 * q^49 - 25 * q^51 + 7 * q^52 - 11 * q^53 + 30 * q^54 - 10 * q^56 + 2 * q^57 - 27 * q^58 + 17 * q^59 - 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 - 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 - 6 * q^74 - 31 * q^76 + 17 * q^78 + 23 * q^79 - 4 * q^81 + 17 * q^82 - 41 * q^83 + 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 - 33 * q^92 + 59 * q^93 + 50 * q^94 - 61 * q^96 - 2 * q^97 - 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more