LMFDB - Embedded newform 4925.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4925,2,Mod(1,4925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4925, 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("4925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.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:	$39.3263229955$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	10.10.21886214112361.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 16x^{7} + 26x^{6} - 38x^{5} - 27x^{4} + 32x^{3} + 6x^{2} - 7x + 1 $ x^10 - 2x^9 - 9x^8 + 16x^7 + 26x^6 - 38x^5 - 27x^4 + 32x^3 + 6x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.285611$ of defining polynomial
Character	$\chi$	$=$	4925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.285611 q^{2} +1.26844 q^{3} -1.91843 q^{4} +0.362280 q^{6} +2.73433 q^{7} -1.11915 q^{8} -1.39107 q^{9} +3.99518 q^{11} -2.43340 q^{12} -6.64477 q^{13} +0.780956 q^{14} +3.51721 q^{16} -1.76947 q^{17} -0.397305 q^{18} -4.25268 q^{19} +3.46833 q^{21} +1.14107 q^{22} +7.54738 q^{23} -1.41957 q^{24} -1.89782 q^{26} -5.56979 q^{27} -5.24562 q^{28} -0.649084 q^{29} -9.61490 q^{31} +3.24285 q^{32} +5.06763 q^{33} -0.505380 q^{34} +2.66866 q^{36} +9.93994 q^{37} -1.21461 q^{38} -8.42847 q^{39} +5.28489 q^{41} +0.990594 q^{42} +7.01525 q^{43} -7.66446 q^{44} +2.15562 q^{46} -5.51975 q^{47} +4.46136 q^{48} +0.476575 q^{49} -2.24446 q^{51} +12.7475 q^{52} -8.56114 q^{53} -1.59080 q^{54} -3.06012 q^{56} -5.39426 q^{57} -0.185386 q^{58} -14.6857 q^{59} -11.4206 q^{61} -2.74612 q^{62} -3.80364 q^{63} -6.10823 q^{64} +1.44737 q^{66} -7.62193 q^{67} +3.39459 q^{68} +9.57338 q^{69} -2.65014 q^{71} +1.55681 q^{72} -5.54362 q^{73} +2.83896 q^{74} +8.15845 q^{76} +10.9241 q^{77} -2.40727 q^{78} +7.00903 q^{79} -2.89173 q^{81} +1.50942 q^{82} +0.0317081 q^{83} -6.65373 q^{84} +2.00363 q^{86} -0.823322 q^{87} -4.47119 q^{88} +1.24794 q^{89} -18.1690 q^{91} -14.4791 q^{92} -12.1959 q^{93} -1.57650 q^{94} +4.11335 q^{96} +18.6425 q^{97} +0.136115 q^{98} -5.55756 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 9 q^{6} + 6 q^{7} + 6 q^{8} - 5 q^{9} - 11 q^{11} + 5 q^{13} - 9 q^{14} - 2 q^{16} + 4 q^{17} - 15 q^{18} - 28 q^{19} - 7 q^{21} + 12 q^{22} + 24 q^{23} - 3 q^{24} + 7 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) $ 10 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 9 * q^6 + 6 * q^7 + 6 * q^8 - 5 * q^9 - 11 * q^11 + 5 * q^13 - 9 * q^14 - 2 * q^16 + 4 * q^17 - 15 * q^18 - 28 * q^19 - 7 * q^21 + 12 * q^22 + 24 * q^23 - 3 * q^24 + 7 * q^26 + 9 * q^27 - 3 * q^28 - 16 * q^29 - 34 * q^31 + 4 * q^32 + 10 * q^33 + q^34 - q^36 - 5 * q^37 - 13 * q^38 - 15 * q^39 - 15 * q^41 - 7 * q^42 + 5 * q^43 - 11 * q^44 + 6 * q^46 + 14 * q^47 - 12 * q^48 + 4 * q^49 - q^51 + 7 * q^52 + 15 * q^53 - 4 * q^54 - 5 * q^56 - 9 * q^57 - 28 * q^58 - 49 * q^59 - 16 * q^61 - 20 * q^62 - 8 * q^63 - 16 * q^64 + 40 * q^66 + 2 * q^67 - 19 * q^68 - 16 * q^69 - 64 * q^71 - q^72 - 18 * q^73 - 37 * q^74 - 19 * q^76 - 8 * q^77 - 4 * q^78 - 11 * q^79 - 14 * q^81 - 10 * q^82 - 4 * q^83 - 11 * q^84 + 5 * q^86 + q^87 - 19 * q^88 - 5 * q^89 - 30 * q^91 + 32 * q^92 - 41 * q^93 - 37 * q^94 + 26 * q^96 - 20 * q^97 + 37 * q^98 - 16 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.285611	0.201958	0.100979	−	0.994889i	$-0.467803\pi$
$2$	0.285611	0.201958	0.100979	+	0.994889i	$0.467803\pi$
$3$	1.26844	0.732333	0.366166	−	0.930549i	$-0.380670\pi$
$3$	1.26844	0.732333	0.366166	+	0.930549i	$0.380670\pi$
$4$	−1.91843	−0.959213
$5$	0	0
$5$	0	0
$6$	0.362280	0.147900
$7$	2.73433	1.03348	0.516740	−	0.856142i	$-0.327145\pi$
$7$	2.73433	1.03348	0.516740	+	0.856142i	$0.327145\pi$
$8$	−1.11915	−0.395678
$9$	−1.39107	−0.463689
$10$	0	0
$11$	3.99518	1.20459	0.602296	−	0.798273i	$-0.294253\pi$
$11$	3.99518	1.20459	0.602296	+	0.798273i	$0.294253\pi$
$12$	−2.43340	−0.702463
$13$	−6.64477	−1.84293	−0.921463	−	0.388466i	$-0.873005\pi$
$13$	−6.64477	−1.84293	−0.921463	+	0.388466i	$0.873005\pi$
$14$	0.780956	0.208719
$15$	0	0
$16$	3.51721	0.879303
$17$	−1.76947	−0.429159	−0.214579	−	0.976707i	$-0.568838\pi$
$17$	−1.76947	−0.429159	−0.214579	+	0.976707i	$0.568838\pi$
$18$	−0.397305	−0.0936456
$19$	−4.25268	−0.975631	−0.487816	−	0.872947i	$-0.662206\pi$
$19$	−4.25268	−0.975631	−0.487816	+	0.872947i	$0.662206\pi$
$20$	0	0
$21$	3.46833	0.756851
$22$	1.14107	0.243277
$23$	7.54738	1.57374	0.786869	−	0.617121i	$-0.211701\pi$
$23$	7.54738	1.57374	0.786869	+	0.617121i	$0.211701\pi$
$24$	−1.41957	−0.289768
$25$	0	0
$26$	−1.89782	−0.372193
$27$	−5.56979	−1.07191
$28$	−5.24562	−0.991328
$29$	−0.649084	−0.120532	−0.0602659	−	0.998182i	$-0.519195\pi$
$29$	−0.649084	−0.120532	−0.0602659	+	0.998182i	$0.519195\pi$
$30$	0	0
$31$	−9.61490	−1.72689	−0.863443	−	0.504446i	$-0.831697\pi$
$31$	−9.61490	−1.72689	−0.863443	+	0.504446i	$0.831697\pi$
$32$	3.24285	0.573260
$33$	5.06763	0.882162
$34$	−0.505380	−0.0866719
$35$	0	0
$36$	2.66866	0.444777
$37$	9.93994	1.63412	0.817058	−	0.576555i	$-0.195603\pi$
$37$	9.93994	1.63412	0.817058	+	0.576555i	$0.195603\pi$
$38$	−1.21461	−0.197036
$39$	−8.42847	−1.34963
$40$	0	0
$41$	5.28489	0.825361	0.412681	−	0.910876i	$-0.364593\pi$
$41$	5.28489	0.825361	0.412681	+	0.910876i	$0.364593\pi$
$42$	0.990594	0.152852
$43$	7.01525	1.06982	0.534908	−	0.844910i	$-0.320346\pi$
$43$	7.01525	1.06982	0.534908	+	0.844910i	$0.320346\pi$
$44$	−7.66446	−1.15546
$45$	0	0
$46$	2.15562	0.317828
$47$	−5.51975	−0.805138	−0.402569	−	0.915390i	$-0.631883\pi$
$47$	−5.51975	−0.805138	−0.402569	+	0.915390i	$0.631883\pi$
$48$	4.46136	0.643942
$49$	0.476575	0.0680822
$50$	0	0
$51$	−2.24446	−0.314287
$52$	12.7475	1.76776
$53$	−8.56114	−1.17596	−0.587982	−	0.808874i	$-0.700077\pi$
$53$	−8.56114	−1.17596	−0.587982	+	0.808874i	$0.700077\pi$
$54$	−1.59080	−0.216480
$55$	0	0
$56$	−3.06012	−0.408926
$57$	−5.39426	−0.714487
$58$	−0.185386	−0.0243423
$59$	−14.6857	−1.91192	−0.955961	−	0.293495i	$-0.905182\pi$
$59$	−14.6857	−1.91192	−0.955961	+	0.293495i	$0.905182\pi$
$60$	0	0
$61$	−11.4206	−1.46226	−0.731130	−	0.682238i	$-0.761007\pi$
$61$	−11.4206	−1.46226	−0.731130	+	0.682238i	$0.761007\pi$
$62$	−2.74612	−0.348758
$63$	−3.80364	−0.479214
$64$	−6.10823	−0.763529
$65$	0	0
$66$	1.44737	0.178159
$67$	−7.62193	−0.931167	−0.465583	−	0.885004i	$-0.654155\pi$
$67$	−7.62193	−0.931167	−0.465583	+	0.885004i	$0.654155\pi$
$68$	3.39459	0.411655
$69$	9.57338	1.15250
$70$	0	0
$71$	−2.65014	−0.314514	−0.157257	−	0.987558i	$-0.550265\pi$
$71$	−2.65014	−0.314514	−0.157257	+	0.987558i	$0.550265\pi$
$72$	1.55681	0.183472
$73$	−5.54362	−0.648831	−0.324416	−	0.945915i	$-0.605168\pi$
$73$	−5.54362	−0.648831	−0.324416	+	0.945915i	$0.605168\pi$
$74$	2.83896	0.330022
$75$	0	0
$76$	8.15845	0.935838
$77$	10.9241	1.24492
$78$	−2.40727	−0.272569
$79$	7.00903	0.788578	0.394289	−	0.918987i	$-0.370991\pi$
$79$	7.00903	0.788578	0.394289	+	0.918987i	$0.370991\pi$
$80$	0	0
$81$	−2.89173	−0.321303
$82$	1.50942	0.166688
$83$	0.0317081	0.00348041	0.00174021	−	0.999998i	$-0.499446\pi$
$83$	0.0317081	0.00348041	0.00174021	+	0.999998i	$0.499446\pi$
$84$	−6.65373	−0.725982
$85$	0	0
$86$	2.00363	0.216057
$87$	−0.823322	−0.0882694
$88$	−4.47119	−0.476631
$89$	1.24794	0.132282	0.0661408	−	0.997810i	$-0.478931\pi$
$89$	1.24794	0.132282	0.0661408	+	0.997810i	$0.478931\pi$
$90$	0	0
$91$	−18.1690	−1.90463
$92$	−14.4791	−1.50955
$93$	−12.1959	−1.26466
$94$	−1.57650	−0.162604
$95$	0	0
$96$	4.11335	0.419817
$97$	18.6425	1.89286	0.946430	−	0.322910i	$-0.104661\pi$
$97$	18.6425	1.89286	0.946430	+	0.322910i	$0.104661\pi$
$98$	0.136115	0.0137497
$99$	−5.55756	−0.558556
$100$	0	0
$101$	−19.0353	−1.89408	−0.947041	−	0.321111i	$-0.895944\pi$
$101$	−19.0353	−1.89408	−0.947041	+	0.321111i	$0.895944\pi$
$102$	−0.641042	−0.0634726
$103$	10.0537	0.990624	0.495312	−	0.868715i	$-0.335054\pi$
$103$	10.0537	0.990624	0.495312	+	0.868715i	$0.335054\pi$
$104$	7.43647	0.729206
$105$	0	0
$106$	−2.44516	−0.237495
$107$	−6.01815	−0.581797	−0.290898	−	0.956754i	$-0.593954\pi$
$107$	−6.01815	−0.581797	−0.290898	+	0.956754i	$0.593954\pi$
$108$	10.6852	1.02819
$109$	−13.6223	−1.30478	−0.652392	−	0.757882i	$-0.726234\pi$
$109$	−13.6223	−1.30478	−0.652392	+	0.757882i	$0.726234\pi$
$110$	0	0
$111$	12.6082	1.19672
$112$	9.61723	0.908742
$113$	−11.7108	−1.10166	−0.550830	−	0.834617i	$-0.685689\pi$
$113$	−11.7108	−1.10166	−0.550830	+	0.834617i	$0.685689\pi$
$114$	−1.54066	−0.144296
$115$	0	0
$116$	1.24522	0.115616
$117$	9.24332	0.854545
$118$	−4.19441	−0.386127
$119$	−4.83831	−0.443527
$120$	0	0
$121$	4.96146	0.451041
$122$	−3.26186	−0.295315
$123$	6.70355	0.604439
$124$	18.4455	1.65645
$125$	0	0
$126$	−1.08636	−0.0967809
$127$	16.0949	1.42819	0.714097	−	0.700047i	$-0.246838\pi$
$127$	16.0949	1.42819	0.714097	+	0.700047i	$0.246838\pi$
$128$	−8.23028	−0.727461
$129$	8.89840	0.783461
$130$	0	0
$131$	−0.0711362	−0.00621520	−0.00310760	−	0.999995i	$-0.500989\pi$
$131$	−0.0711362	−0.00621520	−0.00310760	+	0.999995i	$0.500989\pi$
$132$	−9.72188	−0.846181
$133$	−11.6282	−1.00830
$134$	−2.17691	−0.188056
$135$	0	0
$136$	1.98029	0.169809
$137$	−3.92717	−0.335521	−0.167760	−	0.985828i	$-0.553654\pi$
$137$	−3.92717	−0.335521	−0.167760	+	0.985828i	$0.553654\pi$
$138$	2.73426	0.232756
$139$	1.18876	0.100829	0.0504146	−	0.998728i	$-0.483946\pi$
$139$	1.18876	0.100829	0.0504146	+	0.998728i	$0.483946\pi$
$140$	0	0
$141$	−7.00145	−0.589629
$142$	−0.756911	−0.0635185
$143$	−26.5470	−2.21997
$144$	−4.89268	−0.407723
$145$	0	0
$146$	−1.58332	−0.131036
$147$	0.604506	0.0498588
$148$	−19.0690	−1.56747
$149$	−16.2732	−1.33315	−0.666577	−	0.745436i	$-0.732241\pi$
$149$	−16.2732	−1.33315	−0.666577	+	0.745436i	$0.732241\pi$
$150$	0	0
$151$	15.9502	1.29800	0.649002	−	0.760786i	$-0.275186\pi$
$151$	15.9502	1.29800	0.649002	+	0.760786i	$0.275186\pi$
$152$	4.75937	0.386036
$153$	2.46145	0.198996
$154$	3.12006	0.251422
$155$	0	0
$156$	16.1694	1.29459
$157$	−15.9697	−1.27452	−0.637260	−	0.770649i	$-0.719932\pi$
$157$	−15.9697	−1.27452	−0.637260	+	0.770649i	$0.719932\pi$
$158$	2.00186	0.159259
$159$	−10.8593	−0.861196
$160$	0	0
$161$	20.6370	1.62643
$162$	−0.825911	−0.0648897
$163$	−3.02367	−0.236832	−0.118416	−	0.992964i	$-0.537782\pi$
$163$	−3.02367	−0.236832	−0.118416	+	0.992964i	$0.537782\pi$
$164$	−10.1387	−0.791697
$165$	0	0
$166$	0.00905618	0.000702896 0
$167$	13.1045	1.01405	0.507026	−	0.861931i	$-0.330745\pi$
$167$	13.1045	1.01405	0.507026	+	0.861931i	$0.330745\pi$
$168$	−3.88157	−0.299470
$169$	31.1529	2.39638
$170$	0	0
$171$	5.91576	0.452390
$172$	−13.4582	−1.02618
$173$	4.14632	0.315239	0.157619	−	0.987500i	$-0.449618\pi$
$173$	4.14632	0.315239	0.157619	+	0.987500i	$0.449618\pi$
$174$	−0.235150	−0.0178267
$175$	0	0
$176$	14.0519	1.05920
$177$	−18.6279	−1.40016
$178$	0.356427	0.0267153
$179$	−11.2236	−0.838892	−0.419446	−	0.907780i	$-0.637776\pi$
$179$	−11.2236	−0.838892	−0.419446	+	0.907780i	$0.637776\pi$
$180$	0	0
$181$	−1.42131	−0.105645	−0.0528225	−	0.998604i	$-0.516822\pi$
$181$	−1.42131	−0.105645	−0.0528225	+	0.998604i	$0.516822\pi$
$182$	−5.18927	−0.384654
$183$	−14.4863	−1.07086
$184$	−8.44662	−0.622693
$185$	0	0
$186$	−3.48329	−0.255407
$187$	−7.06933	−0.516961
$188$	10.5892	0.772299
$189$	−15.2297	−1.10780
$190$	0	0
$191$	−12.2250	−0.884569	−0.442285	−	0.896875i	$-0.645832\pi$
$191$	−12.2250	−0.884569	−0.442285	+	0.896875i	$0.645832\pi$
$192$	−7.74790	−0.559157
$193$	−2.35887	−0.169795	−0.0848975	−	0.996390i	$-0.527056\pi$
$193$	−2.35887	−0.169795	−0.0848975	+	0.996390i	$0.527056\pi$
$194$	5.32451	0.382277
$195$	0	0
$196$	−0.914274	−0.0653053
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	−1.58730	−0.112805
$199$	20.0757	1.42313	0.711566	−	0.702620i	$-0.247986\pi$
$199$	20.0757	1.42313	0.711566	+	0.702620i	$0.247986\pi$
$200$	0	0
$201$	−9.66794	−0.681924
$202$	−5.43670	−0.382525
$203$	−1.77481	−0.124567
$204$	4.30582	0.301468
$205$	0	0
$206$	2.87146	0.200064
$207$	−10.4989	−0.729725
$208$	−23.3710	−1.62049
$209$	−16.9902	−1.17524
$210$	0	0
$211$	1.16375	0.0801161	0.0400581	−	0.999197i	$-0.487246\pi$
$211$	1.16375	0.0801161	0.0400581	+	0.999197i	$0.487246\pi$
$212$	16.4239	1.12800
$213$	−3.36154	−0.230329
$214$	−1.71885	−0.117498
$215$	0	0
$216$	6.23342	0.424130
$217$	−26.2903	−1.78470
$218$	−3.89070	−0.263511
$219$	−7.03173	−0.475160
$220$	0	0
$221$	11.7577	0.790908
$222$	3.60104	0.241686
$223$	−6.55642	−0.439050	−0.219525	−	0.975607i	$-0.570451\pi$
$223$	−6.55642	−0.439050	−0.219525	+	0.975607i	$0.570451\pi$
$224$	8.86703	0.592453
$225$	0	0
$226$	−3.34474	−0.222489
$227$	−13.3074	−0.883243	−0.441622	−	0.897201i	$-0.645597\pi$
$227$	−13.3074	−0.883243	−0.441622	+	0.897201i	$0.645597\pi$
$228$	10.3485	0.685345
$229$	−5.36545	−0.354559	−0.177279	−	0.984161i	$-0.556730\pi$
$229$	−5.36545	−0.354559	−0.177279	+	0.984161i	$0.556730\pi$
$230$	0	0
$231$	13.8566	0.911697
$232$	0.726420	0.0476918
$233$	12.0023	0.786295	0.393148	−	0.919475i	$-0.371386\pi$
$233$	12.0023	0.786295	0.393148	+	0.919475i	$0.371386\pi$
$234$	2.64000	0.172582
$235$	0	0
$236$	28.1735	1.83394
$237$	8.89052	0.577501
$238$	−1.38188	−0.0895737
$239$	−5.98330	−0.387028	−0.193514	−	0.981098i	$-0.561988\pi$
$239$	−5.98330	−0.387028	−0.193514	+	0.981098i	$0.561988\pi$
$240$	0	0
$241$	−14.8072	−0.953818	−0.476909	−	0.878953i	$-0.658243\pi$
$241$	−14.8072	−0.953818	−0.476909	+	0.878953i	$0.658243\pi$
$242$	1.41705	0.0910913
$243$	13.0414	0.836606
$244$	21.9096	1.40262
$245$	0	0
$246$	1.91461	0.122071
$247$	28.2581	1.79802
$248$	10.7605	0.683291
$249$	0.0402197	0.00254882
$250$	0	0
$251$	−24.7192	−1.56026	−0.780130	−	0.625618i	$-0.784847\pi$
$251$	−24.7192	−1.56026	−0.780130	+	0.625618i	$0.784847\pi$
$252$	7.29700	0.459668
$253$	30.1531	1.89571
$254$	4.59689	0.288435
$255$	0	0
$256$	9.86580	0.616612
$257$	−5.56346	−0.347039	−0.173519	−	0.984830i	$-0.555514\pi$
$257$	−5.56346	−0.347039	−0.173519	+	0.984830i	$0.555514\pi$
$258$	2.54148	0.158226
$259$	27.1791	1.68883
$260$	0	0
$261$	0.902919	0.0558893
$262$	−0.0203173	−0.00125521
$263$	21.0618	1.29873	0.649363	−	0.760479i	$-0.275036\pi$
$263$	21.0618	1.29873	0.649363	+	0.760479i	$0.275036\pi$
$264$	−5.67143	−0.349052
$265$	0	0
$266$	−3.32116	−0.203633
$267$	1.58294	0.0968742
$268$	14.6221	0.893187
$269$	−2.82590	−0.172298	−0.0861490	−	0.996282i	$-0.527456\pi$
$269$	−2.82590	−0.172298	−0.0861490	+	0.996282i	$0.527456\pi$
$270$	0	0
$271$	−6.32775	−0.384384	−0.192192	−	0.981357i	$-0.561560\pi$
$271$	−6.32775	−0.384384	−0.192192	+	0.981357i	$0.561560\pi$
$272$	−6.22359	−0.377360
$273$	−23.0462	−1.39482
$274$	−1.12164	−0.0677610
$275$	0	0
$276$	−18.3658	−1.10549
$277$	−23.9716	−1.44031	−0.720156	−	0.693812i	$-0.755930\pi$
$277$	−23.9716	−1.44031	−0.720156	+	0.693812i	$0.755930\pi$
$278$	0.339523	0.0203632
$279$	13.3750	0.800739
$280$	0	0
$281$	−12.8708	−0.767806	−0.383903	−	0.923373i	$-0.625420\pi$
$281$	−12.8708	−0.767806	−0.383903	+	0.923373i	$0.625420\pi$
$282$	−1.99969	−0.119080
$283$	18.7422	1.11411	0.557054	−	0.830476i	$-0.311932\pi$
$283$	18.7422	1.11411	0.557054	+	0.830476i	$0.311932\pi$
$284$	5.08410	0.301686
$285$	0	0
$286$	−7.58213	−0.448341
$287$	14.4506	0.852995
$288$	−4.51102	−0.265814
$289$	−13.8690	−0.815823
$290$	0	0
$291$	23.6468	1.38620
$292$	10.6350	0.622368
$293$	−17.6868	−1.03328	−0.516638	−	0.856204i	$-0.672817\pi$
$293$	−17.6868	−1.03328	−0.516638	+	0.856204i	$0.672817\pi$
$294$	0.172654	0.0100694
$295$	0	0
$296$	−11.1243	−0.646584
$297$	−22.2523	−1.29121
$298$	−4.64781	−0.269241
$299$	−50.1506	−2.90028
$300$	0	0
$301$	19.1820	1.10563
$302$	4.55554	0.262142
$303$	−24.1451	−1.38710
$304$	−14.9576	−0.857875
$305$	0	0
$306$	0.703017	0.0401888
$307$	−28.8607	−1.64717	−0.823584	−	0.567194i	$-0.808029\pi$
$307$	−28.8607	−1.64717	−0.823584	+	0.567194i	$0.808029\pi$
$308$	−20.9572	−1.19415
$309$	12.7525	0.725466
$310$	0	0
$311$	17.7931	1.00896	0.504478	−	0.863424i	$-0.331685\pi$
$311$	17.7931	1.00896	0.504478	+	0.863424i	$0.331685\pi$
$312$	9.43269	0.534021
$313$	−19.7809	−1.11808	−0.559042	−	0.829140i	$-0.688831\pi$
$313$	−19.7809	−1.11808	−0.559042	+	0.829140i	$0.688831\pi$
$314$	−4.56113	−0.257399
$315$	0	0
$316$	−13.4463	−0.756414
$317$	−8.49694	−0.477236	−0.238618	−	0.971114i	$-0.576694\pi$
$317$	−8.49694	−0.477236	−0.238618	+	0.971114i	$0.576694\pi$
$318$	−3.10153	−0.173925
$319$	−2.59321	−0.145192
$320$	0	0
$321$	−7.63365	−0.426069
$322$	5.89417	0.328469
$323$	7.52497	0.418701
$324$	5.54757	0.308198
$325$	0	0
$326$	−0.863593	−0.0478300
$327$	−17.2791	−0.955536
$328$	−5.91457	−0.326577
$329$	−15.0928	−0.832094
$330$	0	0
$331$	0.830686	0.0456586	0.0228293	−	0.999739i	$-0.492733\pi$
$331$	0.830686	0.0456586	0.0228293	+	0.999739i	$0.492733\pi$
$332$	−0.0608296	−0.00333846
$333$	−13.8271	−0.757722
$334$	3.74278	0.204796
$335$	0	0
$336$	12.1988	0.665502
$337$	−16.1561	−0.880080	−0.440040	−	0.897978i	$-0.645036\pi$
$337$	−16.1561	−0.880080	−0.440040	+	0.897978i	$0.645036\pi$
$338$	8.89762	0.483967
$339$	−14.8544	−0.806782
$340$	0	0
$341$	−38.4132	−2.08019
$342$	1.68961	0.0913636
$343$	−17.8372	−0.963119
$344$	−7.85109	−0.423303
$345$	0	0
$346$	1.18424	0.0636649
$347$	25.5274	1.37038	0.685192	−	0.728362i	$-0.259718\pi$
$347$	25.5274	1.37038	0.685192	+	0.728362i	$0.259718\pi$
$348$	1.57948	0.0846692
$349$	−34.5894	−1.85153	−0.925764	−	0.378101i	$-0.876577\pi$
$349$	−34.5894	−1.85153	−0.925764	+	0.378101i	$0.876577\pi$
$350$	0	0
$351$	37.0100	1.97545
$352$	12.9558	0.690544
$353$	−28.4853	−1.51612	−0.758059	−	0.652186i	$-0.773852\pi$
$353$	−28.4853	−1.51612	−0.758059	+	0.652186i	$0.773852\pi$
$354$	−5.32035	−0.282773
$355$	0	0
$356$	−2.39409	−0.126886
$357$	−6.13709	−0.324809
$358$	−3.20559	−0.169421
$359$	−0.149001	−0.00786397	−0.00393198	−	0.999992i	$-0.501252\pi$
$359$	−0.149001	−0.00786397	−0.00393198	+	0.999992i	$0.501252\pi$
$360$	0	0
$361$	−0.914725	−0.0481434
$362$	−0.405942	−0.0213358
$363$	6.29329	0.330312
$364$	34.8559	1.82694
$365$	0	0
$366$	−4.13746	−0.216269
$367$	−4.91740	−0.256686	−0.128343	−	0.991730i	$-0.540966\pi$
$367$	−4.91740	−0.256686	−0.128343	+	0.991730i	$0.540966\pi$
$368$	26.5457	1.38379
$369$	−7.35164	−0.382711
$370$	0	0
$371$	−23.4090	−1.21534
$372$	23.3969	1.21307
$373$	11.4854	0.594694	0.297347	−	0.954769i	$-0.403898\pi$
$373$	11.4854	0.594694	0.297347	+	0.954769i	$0.403898\pi$
$374$	−2.01908	−0.104404
$375$	0	0
$376$	6.17741	0.318576
$377$	4.31301	0.222131
$378$	−4.34977	−0.223728
$379$	−11.6098	−0.596355	−0.298177	−	0.954511i	$-0.596379\pi$
$379$	−11.6098	−0.596355	−0.298177	+	0.954511i	$0.596379\pi$
$380$	0	0
$381$	20.4154	1.04591
$382$	−3.49160	−0.178646
$383$	23.7069	1.21137	0.605684	−	0.795705i	$-0.292900\pi$
$383$	23.7069	1.21137	0.605684	+	0.795705i	$0.292900\pi$
$384$	−10.4396	−0.532743
$385$	0	0
$386$	−0.673720	−0.0342914
$387$	−9.75868	−0.496062
$388$	−35.7643	−1.81566
$389$	−0.0130692	−0.000662637 0	−0.000331318	−	1.00000i	$-0.500105\pi$
$389$	−0.0130692	−0.000662637 0	−0.000331318		1.00000i	$0.500105\pi$
$390$	0	0
$391$	−13.3548	−0.675383
$392$	−0.533358	−0.0269386
$393$	−0.0902318	−0.00455159
$394$	−0.285611	−0.0143889
$395$	0	0
$396$	10.6618	0.535774
$397$	0.543246	0.0272648	0.0136324	−	0.999907i	$-0.495661\pi$
$397$	0.543246	0.0272648	0.0136324	+	0.999907i	$0.495661\pi$
$398$	5.73386	0.287412
$399$	−14.7497	−0.738408
$400$	0	0
$401$	12.1782	0.608148	0.304074	−	0.952648i	$-0.401653\pi$
$401$	12.1782	0.608148	0.304074	+	0.952648i	$0.401653\pi$
$402$	−2.76127	−0.137720
$403$	63.8887	3.18253
$404$	36.5178	1.81683
$405$	0	0
$406$	−0.506906	−0.0251573
$407$	39.7119	1.96844
$408$	2.51188	0.124356
$409$	−18.0409	−0.892063	−0.446032	−	0.895017i	$-0.647163\pi$
$409$	−18.0409	−0.892063	−0.446032	+	0.895017i	$0.647163\pi$
$410$	0	0
$411$	−4.98137	−0.245713
$412$	−19.2873	−0.950220
$413$	−40.1557	−1.97593
$414$	−2.99861	−0.147374
$415$	0	0
$416$	−21.5480	−1.05648
$417$	1.50787	0.0738405
$418$	−4.85260	−0.237348
$419$	16.8784	0.824565	0.412282	−	0.911056i	$-0.364732\pi$
$419$	16.8784	0.824565	0.412282	+	0.911056i	$0.364732\pi$
$420$	0	0
$421$	15.4061	0.750847	0.375423	−	0.926853i	$-0.377497\pi$
$421$	15.4061	0.750847	0.375423	+	0.926853i	$0.377497\pi$
$422$	0.332381	0.0161801
$423$	7.67834	0.373334
$424$	9.58118	0.465303
$425$	0	0
$426$	−0.960093	−0.0465167
$427$	−31.2278	−1.51122
$428$	11.5454	0.558067
$429$	−33.6732	−1.62576
$430$	0	0
$431$	−10.3040	−0.496325	−0.248162	−	0.968718i	$-0.579827\pi$
$431$	−10.3040	−0.496325	−0.248162	+	0.968718i	$0.579827\pi$
$432$	−19.5901	−0.942531
$433$	1.47200	0.0707397	0.0353698	−	0.999374i	$-0.488739\pi$
$433$	1.47200	0.0707397	0.0353698	+	0.999374i	$0.488739\pi$
$434$	−7.50882	−0.360435
$435$	0	0
$436$	26.1335	1.25157
$437$	−32.0966	−1.53539
$438$	−2.00834	−0.0959623
$439$	0.161040	0.00768601	0.00384301	−	0.999993i	$-0.498777\pi$
$439$	0.161040	0.00768601	0.00384301	+	0.999993i	$0.498777\pi$
$440$	0	0
$441$	−0.662948	−0.0315690
$442$	3.35813	0.159730
$443$	26.8949	1.27781	0.638906	−	0.769284i	$-0.279387\pi$
$443$	26.8949	1.27781	0.638906	+	0.769284i	$0.279387\pi$
$444$	−24.1879	−1.14791
$445$	0	0
$446$	−1.87259	−0.0886696
$447$	−20.6415	−0.976312
$448$	−16.7019	−0.789092
$449$	40.2914	1.90147	0.950734	−	0.310007i	$-0.100331\pi$
$449$	40.2914	1.90147	0.950734	+	0.310007i	$0.100331\pi$
$450$	0	0
$451$	21.1141	0.994223
$452$	22.4663	1.05673
$453$	20.2318	0.950571
$454$	−3.80074	−0.178378
$455$	0	0
$456$	6.03696	0.282707
$457$	−20.6919	−0.967925	−0.483963	−	0.875089i	$-0.660803\pi$
$457$	−20.6919	−0.967925	−0.483963	+	0.875089i	$0.660803\pi$
$458$	−1.53243	−0.0716059
$459$	9.85556	0.460018
$460$	0	0
$461$	27.8487	1.29704	0.648522	−	0.761196i	$-0.275388\pi$
$461$	27.8487	1.29704	0.648522	+	0.761196i	$0.275388\pi$
$462$	3.95760	0.184124
$463$	19.6045	0.911099	0.455550	−	0.890210i	$-0.349443\pi$
$463$	19.6045	0.911099	0.455550	+	0.890210i	$0.349443\pi$
$464$	−2.28297	−0.105984
$465$	0	0
$466$	3.42799	0.158798
$467$	32.8789	1.52146	0.760728	−	0.649071i	$-0.224842\pi$
$467$	32.8789	1.52146	0.760728	+	0.649071i	$0.224842\pi$
$468$	−17.7326	−0.819691
$469$	−20.8409	−0.962343
$470$	0	0
$471$	−20.2566	−0.933373
$472$	16.4355	0.756505
$473$	28.0272	1.28869
$474$	2.53923	0.116631
$475$	0	0
$476$	9.28194	0.425437
$477$	11.9091	0.545281
$478$	−1.70890	−0.0781632
$479$	6.51052	0.297473	0.148737	−	0.988877i	$-0.452479\pi$
$479$	6.51052	0.297473	0.148737	+	0.988877i	$0.452479\pi$
$480$	0	0
$481$	−66.0486	−3.01156
$482$	−4.22912	−0.192631
$483$	26.1768	1.19109
$484$	−9.51819	−0.432645
$485$	0	0
$486$	3.72477	0.168959
$487$	−14.3214	−0.648965	−0.324482	−	0.945892i	$-0.605190\pi$
$487$	−14.3214	−0.648965	−0.324482	+	0.945892i	$0.605190\pi$
$488$	12.7813	0.578584
$489$	−3.83533	−0.173440
$490$	0	0
$491$	12.1320	0.547509	0.273754	−	0.961800i	$-0.411734\pi$
$491$	12.1320	0.547509	0.273754	+	0.961800i	$0.411734\pi$
$492$	−12.8603	−0.579786
$493$	1.14853	0.0517273
$494$	8.07082	0.363123
$495$	0	0
$496$	−33.8176	−1.51846
$497$	−7.24637	−0.325044
$498$	0.0114872	0.000514754 0
$499$	25.0458	1.12120	0.560601	−	0.828086i	$-0.310570\pi$
$499$	25.0458	1.12120	0.560601	+	0.828086i	$0.310570\pi$
$500$	0	0
$501$	16.6222	0.742624
$502$	−7.06007	−0.315106
$503$	2.63461	0.117471	0.0587356	−	0.998274i	$-0.481293\pi$
$503$	2.63461	0.117471	0.0587356	+	0.998274i	$0.481293\pi$
$504$	4.25683	0.189614
$505$	0	0
$506$	8.61207	0.382853
$507$	39.5155	1.75495
$508$	−30.8769	−1.36994
$509$	36.9906	1.63958	0.819791	−	0.572663i	$-0.194090\pi$
$509$	36.9906	1.63958	0.819791	+	0.572663i	$0.194090\pi$
$510$	0	0
$511$	−15.1581	−0.670555
$512$	19.2783	0.851990
$513$	23.6865	1.04579
$514$	−1.58899	−0.0700872
$515$	0	0
$516$	−17.0709	−0.751506
$517$	−22.0524	−0.969863
$518$	7.76266	0.341072
$519$	5.25934	0.230860
$520$	0	0
$521$	−26.6209	−1.16628	−0.583141	−	0.812371i	$-0.698177\pi$
$521$	−26.6209	−1.16628	−0.583141	+	0.812371i	$0.698177\pi$
$522$	0.257884	0.0112873
$523$	−9.11852	−0.398725	−0.199362	−	0.979926i	$-0.563887\pi$
$523$	−9.11852	−0.398725	−0.199362	+	0.979926i	$0.563887\pi$
$524$	0.136470	0.00596170
$525$	0	0
$526$	6.01549	0.262288
$527$	17.0132	0.741108
$528$	17.8239	0.775687
$529$	33.9629	1.47665
$530$	0	0
$531$	20.4289	0.886537
$532$	22.3079	0.967171
$533$	−35.1169	−1.52108
$534$	0.452105	0.0195645
$535$	0	0
$536$	8.53006	0.368442
$537$	−14.2364	−0.614348
$538$	−0.807108	−0.0347969
$539$	1.90400	0.0820112
$540$	0	0
$541$	−10.5060	−0.451687	−0.225844	−	0.974164i	$-0.572514\pi$
$541$	−10.5060	−0.451687	−0.225844	+	0.974164i	$0.572514\pi$
$542$	−1.80728	−0.0776292
$543$	−1.80284	−0.0773673
$544$	−5.73811	−0.246020
$545$	0	0
$546$	−6.58227	−0.281695
$547$	−10.9285	−0.467268	−0.233634	−	0.972325i	$-0.575062\pi$
$547$	−10.9285	−0.467268	−0.233634	+	0.972325i	$0.575062\pi$
$548$	7.53399	0.321836
$549$	15.8868	0.678034
$550$	0	0
$551$	2.76035	0.117595
$552$	−10.7140	−0.456019
$553$	19.1650	0.814980
$554$	−6.84655	−0.290882
$555$	0	0
$556$	−2.28055	−0.0967167
$557$	23.5988	0.999914	0.499957	−	0.866050i	$-0.333349\pi$
$557$	23.5988	0.999914	0.499957	+	0.866050i	$0.333349\pi$
$558$	3.82004	0.161715
$559$	−46.6147	−1.97159
$560$	0	0
$561$	−8.96701	−0.378587
$562$	−3.67604	−0.155064
$563$	9.18846	0.387247	0.193624	−	0.981076i	$-0.437976\pi$
$563$	9.18846	0.387247	0.193624	+	0.981076i	$0.437976\pi$
$564$	13.4318	0.565580
$565$	0	0
$566$	5.35298	0.225003
$567$	−7.90695	−0.332061
$568$	2.96590	0.124446
$569$	−11.3862	−0.477333	−0.238667	−	0.971102i	$-0.576710\pi$
$569$	−11.3862	−0.477333	−0.238667	+	0.971102i	$0.576710\pi$
$570$	0	0
$571$	20.3067	0.849808	0.424904	−	0.905238i	$-0.360308\pi$
$571$	20.3067	0.849808	0.424904	+	0.905238i	$0.360308\pi$
$572$	50.9285	2.12943
$573$	−15.5066	−0.647799
$574$	4.12727	0.172269
$575$	0	0
$576$	8.49696	0.354040
$577$	27.7811	1.15654	0.578272	−	0.815844i	$-0.303727\pi$
$577$	27.7811	1.15654	0.578272	+	0.815844i	$0.303727\pi$
$578$	−3.96114	−0.164762
$579$	−2.99208	−0.124346
$580$	0	0
$581$	0.0867004	0.00359694
$582$	6.75381	0.279954
$583$	−34.2033	−1.41656
$584$	6.20412	0.256728
$585$	0	0
$586$	−5.05156	−0.208678
$587$	38.1125	1.57307	0.786536	−	0.617544i	$-0.211872\pi$
$587$	38.1125	1.57307	0.786536	+	0.617544i	$0.211872\pi$
$588$	−1.15970	−0.0478252
$589$	40.8891	1.68480
$590$	0	0
$591$	−1.26844	−0.0521765
$592$	34.9609	1.43688
$593$	40.7949	1.67524	0.837622	−	0.546250i	$-0.183945\pi$
$593$	40.7949	1.67524	0.837622	+	0.546250i	$0.183945\pi$
$594$	−6.35551	−0.260770
$595$	0	0
$596$	31.2190	1.27878
$597$	25.4648	1.04221
$598$	−14.3236	−0.585734
$599$	−4.03027	−0.164673	−0.0823363	−	0.996605i	$-0.526238\pi$
$599$	−4.03027	−0.164673	−0.0823363	+	0.996605i	$0.526238\pi$
$600$	0	0
$601$	41.3797	1.68791	0.843956	−	0.536412i	$-0.180221\pi$
$601$	41.3797	1.68791	0.843956	+	0.536412i	$0.180221\pi$
$602$	5.47860	0.223291
$603$	10.6026	0.431772
$604$	−30.5992	−1.24506
$605$	0	0
$606$	−6.89611	−0.280135
$607$	37.8690	1.53706	0.768528	−	0.639817i	$-0.220990\pi$
$607$	37.8690	1.53706	0.768528	+	0.639817i	$0.220990\pi$
$608$	−13.7908	−0.559291
$609$	−2.25124	−0.0912247
$610$	0	0
$611$	36.6774	1.48381
$612$	−4.72210	−0.190880
$613$	−0.263324	−0.0106356	−0.00531778	−	0.999986i	$-0.501693\pi$
$613$	−0.263324	−0.0106356	−0.00531778	+	0.999986i	$0.501693\pi$
$614$	−8.24295	−0.332658
$615$	0	0
$616$	−12.2257	−0.492589
$617$	1.40544	0.0565808	0.0282904	−	0.999600i	$-0.490994\pi$
$617$	1.40544	0.0565808	0.0282904	+	0.999600i	$0.490994\pi$
$618$	3.64227	0.146513
$619$	34.5556	1.38891	0.694453	−	0.719538i	$-0.255647\pi$
$619$	34.5556	1.38891	0.694453	+	0.719538i	$0.255647\pi$
$620$	0	0
$621$	−42.0373	−1.68690
$622$	5.08192	0.203767
$623$	3.41229	0.136711
$624$	−29.6447	−1.18674
$625$	0	0
$626$	−5.64965	−0.225806
$627$	−21.5510	−0.860665
$628$	30.6367	1.22254
$629$	−17.5884	−0.701295
$630$	0	0
$631$	−11.8111	−0.470191	−0.235096	−	0.971972i	$-0.575540\pi$
$631$	−11.8111	−0.470191	−0.235096	+	0.971972i	$0.575540\pi$
$632$	−7.84414	−0.312023
$633$	1.47615	0.0586716
$634$	−2.42682	−0.0963814
$635$	0	0
$636$	20.8327	0.826071
$637$	−3.16673	−0.125470
$638$	−0.740649	−0.0293226
$639$	3.68653	0.145837
$640$	0	0
$641$	21.2055	0.837569	0.418784	−	0.908086i	$-0.362456\pi$
$641$	21.2055	0.837569	0.418784	+	0.908086i	$0.362456\pi$
$642$	−2.18026	−0.0860479
$643$	−12.3809	−0.488254	−0.244127	−	0.969743i	$-0.578501\pi$
$643$	−12.3809	−0.488254	−0.244127	+	0.969743i	$0.578501\pi$
$644$	−39.5906	−1.56009
$645$	0	0
$646$	2.14922	0.0845598
$647$	−7.87708	−0.309680	−0.154840	−	0.987940i	$-0.549486\pi$
$647$	−7.87708	−0.309680	−0.154840	+	0.987940i	$0.549486\pi$
$648$	3.23627	0.127133
$649$	−58.6722	−2.30308
$650$	0	0
$651$	−33.3476	−1.30700
$652$	5.80068	0.227172
$653$	−17.3244	−0.677955	−0.338977	−	0.940794i	$-0.610081\pi$
$653$	−17.3244	−0.677955	−0.338977	+	0.940794i	$0.610081\pi$
$654$	−4.93510	−0.192978
$655$	0	0
$656$	18.5881	0.725742
$657$	7.71154	0.300856
$658$	−4.31068	−0.168048
$659$	18.1979	0.708890	0.354445	−	0.935077i	$-0.384670\pi$
$659$	18.1979	0.708890	0.354445	+	0.935077i	$0.384670\pi$
$660$	0	0
$661$	21.3092	0.828831	0.414415	−	0.910088i	$-0.363986\pi$
$661$	21.3092	0.828831	0.414415	+	0.910088i	$0.363986\pi$
$662$	0.237253	0.00922111
$663$	14.9139	0.579207
$664$	−0.0354860	−0.00137712
$665$	0	0
$666$	−3.94918	−0.153028
$667$	−4.89888	−0.189685
$668$	−25.1399	−0.972693
$669$	−8.31641	−0.321531
$670$	0	0
$671$	−45.6274	−1.76143
$672$	11.2473	0.433873
$673$	11.9703	0.461420	0.230710	−	0.973023i	$-0.425895\pi$
$673$	11.9703	0.461420	0.230710	+	0.973023i	$0.425895\pi$
$674$	−4.61437	−0.177739
$675$	0	0
$676$	−59.7646	−2.29864
$677$	−33.6112	−1.29178	−0.645891	−	0.763430i	$-0.723514\pi$
$677$	−33.6112	−1.29178	−0.645891	+	0.763430i	$0.723514\pi$
$678$	−4.24259	−0.162936
$679$	50.9748	1.95623
$680$	0	0
$681$	−16.8796	−0.646828
$682$	−10.9713	−0.420111
$683$	−23.7259	−0.907845	−0.453923	−	0.891041i	$-0.649976\pi$
$683$	−23.7259	−0.907845	−0.453923	+	0.891041i	$0.649976\pi$
$684$	−11.3490	−0.433938
$685$	0	0
$686$	−5.09451	−0.194509
$687$	−6.80574	−0.259655
$688$	24.6741	0.940692
$689$	56.8868	2.16721
$690$	0	0
$691$	−5.76655	−0.219370	−0.109685	−	0.993966i	$-0.534984\pi$
$691$	−5.76655	−0.219370	−0.109685	+	0.993966i	$0.534984\pi$
$692$	−7.95441	−0.302381
$693$	−15.1962	−0.577257
$694$	7.29093	0.276760
$695$	0	0
$696$	0.921419	0.0349263
$697$	−9.35143	−0.354211
$698$	−9.87913	−0.373930
$699$	15.2241	0.575830
$700$	0	0
$701$	41.2611	1.55841	0.779206	−	0.626768i	$-0.215623\pi$
$701$	41.2611	1.55841	0.779206	+	0.626768i	$0.215623\pi$
$702$	10.5705	0.398957
$703$	−42.2714	−1.59430
$704$	−24.4035	−0.919740
$705$	0	0
$706$	−8.13571	−0.306192
$707$	−52.0488	−1.95750
$708$	35.7363	1.34305
$709$	−47.7296	−1.79252	−0.896262	−	0.443525i	$-0.853728\pi$
$709$	−47.7296	−1.79252	−0.896262	+	0.443525i	$0.853728\pi$
$710$	0	0
$711$	−9.75003	−0.365655
$712$	−1.39663	−0.0523410
$713$	−72.5673	−2.71767
$714$	−1.75282	−0.0655977
$715$	0	0
$716$	21.5317	0.804676
$717$	−7.58944	−0.283433
$718$	−0.0425563	−0.00158819
$719$	15.0241	0.560305	0.280152	−	0.959956i	$-0.409615\pi$
$719$	15.0241	0.560305	0.280152	+	0.959956i	$0.409615\pi$
$720$	0	0
$721$	27.4903	1.02379
$722$	−0.261256	−0.00972293
$723$	−18.7821	−0.698512
$724$	2.72668	0.101336
$725$	0	0
$726$	1.79744	0.0667091
$727$	−14.4682	−0.536597	−0.268298	−	0.963336i	$-0.586461\pi$
$727$	−14.4682	−0.536597	−0.268298	+	0.963336i	$0.586461\pi$
$728$	20.3338	0.753620
$729$	25.2174	0.933977
$730$	0	0
$731$	−12.4132	−0.459120
$732$	27.7910	1.02718
$733$	30.6712	1.13287	0.566434	−	0.824107i	$-0.308323\pi$
$733$	30.6712	1.13287	0.566434	+	0.824107i	$0.308323\pi$
$734$	−1.40447	−0.0518398
$735$	0	0
$736$	24.4750	0.902161
$737$	−30.4510	−1.12168
$738$	−2.09971	−0.0772914
$739$	41.3433	1.52084	0.760418	−	0.649434i	$-0.224994\pi$
$739$	41.3433	1.52084	0.760418	+	0.649434i	$0.224994\pi$
$740$	0	0
$741$	35.8436	1.31675
$742$	−6.68588	−0.245446
$743$	−23.0461	−0.845480	−0.422740	−	0.906251i	$-0.638932\pi$
$743$	−23.0461	−0.845480	−0.422740	+	0.906251i	$0.638932\pi$
$744$	13.6490	0.500397
$745$	0	0
$746$	3.28037	0.120103
$747$	−0.0441080	−0.00161383
$748$	13.5620	0.495876
$749$	−16.4556	−0.601276
$750$	0	0
$751$	26.5531	0.968935	0.484468	−	0.874809i	$-0.339013\pi$
$751$	26.5531	0.968935	0.484468	+	0.874809i	$0.339013\pi$
$752$	−19.4141	−0.707960
$753$	−31.3547	−1.14263
$754$	1.23184	0.0448611
$755$	0	0
$756$	29.2170	1.06261
$757$	6.96437	0.253124	0.126562	−	0.991959i	$-0.459606\pi$
$757$	6.96437	0.253124	0.126562	+	0.991959i	$0.459606\pi$
$758$	−3.31589	−0.120438
$759$	38.2473	1.38829
$760$	0	0
$761$	−28.6154	−1.03731	−0.518654	−	0.854984i	$-0.673567\pi$
$761$	−28.6154	−1.03731	−0.518654	+	0.854984i	$0.673567\pi$
$762$	5.83087	0.211230
$763$	−37.2480	−1.34847
$764$	23.4527	0.848491
$765$	0	0
$766$	6.77097	0.244645
$767$	97.5833	3.52353
$768$	12.5141	0.451565
$769$	15.3002	0.551739	0.275870	−	0.961195i	$-0.411034\pi$
$769$	15.3002	0.551739	0.275870	+	0.961195i	$0.411034\pi$
$770$	0	0
$771$	−7.05690	−0.254148
$772$	4.52531	0.162870
$773$	23.6016	0.848891	0.424446	−	0.905453i	$-0.360469\pi$
$773$	23.6016	0.848891	0.424446	+	0.905453i	$0.360469\pi$
$774$	−2.78719	−0.100183
$775$	0	0
$776$	−20.8637	−0.748963
$777$	34.4750	1.23678
$778$	−0.00373272	−0.000133825 0
$779$	−22.4749	−0.805248
$780$	0	0
$781$	−10.5878	−0.378861
$782$	−3.81429	−0.136399
$783$	3.61526	0.129199
$784$	1.67622	0.0598648
$785$	0	0
$786$	−0.0257712	−0.000919229 0
$787$	−10.6041	−0.377994	−0.188997	−	0.981978i	$-0.560524\pi$
$787$	−10.6041	−0.377994	−0.188997	+	0.981978i	$0.560524\pi$
$788$	1.91843	0.0683411
$789$	26.7156	0.951099
$790$	0	0
$791$	−32.0213	−1.13855
$792$	6.21973	0.221008
$793$	75.8873	2.69484
$794$	0.155157	0.00550633
$795$	0	0
$796$	−38.5138	−1.36509
$797$	24.6779	0.874137	0.437069	−	0.899428i	$-0.356017\pi$
$797$	24.6779	0.874137	0.437069	+	0.899428i	$0.356017\pi$
$798$	−4.21268	−0.149127
$799$	9.76701	0.345532
$800$	0	0
$801$	−1.73597	−0.0613376
$802$	3.47822	0.122820
$803$	−22.1477	−0.781577
$804$	18.5472	0.654110
$805$	0	0
$806$	18.2473	0.642735
$807$	−3.58447	−0.126179
$808$	21.3033	0.749447
$809$	−43.2934	−1.52211	−0.761057	−	0.648685i	$-0.775319\pi$
$809$	−43.2934	−1.52211	−0.761057	+	0.648685i	$0.775319\pi$
$810$	0	0
$811$	−8.96333	−0.314745	−0.157373	−	0.987539i	$-0.550302\pi$
$811$	−8.96333	−0.314745	−0.157373	+	0.987539i	$0.550302\pi$
$812$	3.40485	0.119487
$813$	−8.02636	−0.281497
$814$	11.3422	0.397542
$815$	0	0
$816$	−7.89423	−0.276353
$817$	−29.8336	−1.04375
$818$	−5.15268	−0.180159
$819$	25.2743	0.883156
$820$	0	0
$821$	−23.3168	−0.813762	−0.406881	−	0.913481i	$-0.633384\pi$
$821$	−23.3168	−0.813762	−0.406881	+	0.913481i	$0.633384\pi$
$822$	−1.42274	−0.0496236
$823$	−44.7664	−1.56046	−0.780229	−	0.625495i	$-0.784897\pi$
$823$	−44.7664	−1.56046	−0.780229	+	0.625495i	$0.784897\pi$
$824$	−11.2516	−0.391968
$825$	0	0
$826$	−11.4689	−0.399055
$827$	21.1244	0.734568	0.367284	−	0.930109i	$-0.380288\pi$
$827$	21.1244	0.734568	0.367284	+	0.930109i	$0.380288\pi$
$828$	20.1414	0.699962
$829$	4.23189	0.146980	0.0734899	−	0.997296i	$-0.476586\pi$
$829$	4.23189	0.146980	0.0734899	+	0.997296i	$0.476586\pi$
$830$	0	0
$831$	−30.4064	−1.05479
$832$	40.5877	1.40713
$833$	−0.843284	−0.0292180
$834$	0.430664	0.0149127
$835$	0	0
$836$	32.5945	1.12730
$837$	53.5530	1.85106
$838$	4.82067	0.166527
$839$	47.2870	1.63253	0.816265	−	0.577678i	$-0.196041\pi$
$839$	47.2870	1.63253	0.816265	+	0.577678i	$0.196041\pi$
$840$	0	0
$841$	−28.5787	−0.985472
$842$	4.40015	0.151639
$843$	−16.3258	−0.562289
$844$	−2.23258	−0.0768484
$845$	0	0
$846$	2.19302	0.0753976
$847$	13.5663	0.466143
$848$	−30.1114	−1.03403
$849$	23.7733	0.815898
$850$	0	0
$851$	75.0205	2.57167
$852$	6.44886	0.220934
$853$	−27.7277	−0.949380	−0.474690	−	0.880153i	$-0.657440\pi$
$853$	−27.7277	−0.949380	−0.474690	+	0.880153i	$0.657440\pi$
$854$	−8.91900	−0.305202
$855$	0	0
$856$	6.73520	0.230204
$857$	9.63331	0.329068	0.164534	−	0.986371i	$-0.447388\pi$
$857$	9.63331	0.329068	0.164534	+	0.986371i	$0.447388\pi$
$858$	−9.61746	−0.328335
$859$	−5.78218	−0.197285	−0.0986427	−	0.995123i	$-0.531450\pi$
$859$	−5.78218	−0.197285	−0.0986427	+	0.995123i	$0.531450\pi$
$860$	0	0
$861$	18.3297	0.624676
$862$	−2.94293	−0.100237
$863$	10.7342	0.365397	0.182698	−	0.983169i	$-0.441517\pi$
$863$	10.7342	0.365397	0.182698	+	0.983169i	$0.441517\pi$
$864$	−18.0620	−0.614482
$865$	0	0
$866$	0.420419	0.0142864
$867$	−17.5919	−0.597454
$868$	50.4361	1.71191
$869$	28.0023	0.949914
$870$	0	0
$871$	50.6459	1.71607
$872$	15.2454	0.516275
$873$	−25.9330	−0.877698
$874$	−9.16714	−0.310083
$875$	0	0
$876$	13.4899	0.455780
$877$	−19.2178	−0.648939	−0.324470	−	0.945896i	$-0.605186\pi$
$877$	−19.2178	−0.648939	−0.324470	+	0.945896i	$0.605186\pi$
$878$	0.0459948	0.00155225
$879$	−22.4346	−0.756701
$880$	0	0
$881$	−44.0829	−1.48519	−0.742596	−	0.669740i	$-0.766405\pi$
$881$	−44.0829	−1.48519	−0.742596	+	0.669740i	$0.766405\pi$
$882$	−0.189345	−0.00637559
$883$	22.3382	0.751742	0.375871	−	0.926672i	$-0.377344\pi$
$883$	22.3382	0.751742	0.375871	+	0.926672i	$0.377344\pi$
$884$	−22.5563	−0.758649
$885$	0	0
$886$	7.68147	0.258064
$887$	−35.1042	−1.17868	−0.589342	−	0.807884i	$-0.700613\pi$
$887$	−35.1042	−1.17868	−0.589342	+	0.807884i	$0.700613\pi$
$888$	−14.1104	−0.473515
$889$	44.0089	1.47601
$890$	0	0
$891$	−11.5530	−0.387039
$892$	12.5780	0.421143
$893$	23.4737	0.785518
$894$	−5.89546	−0.197174
$895$	0	0
$896$	−22.5043	−0.751816
$897$	−63.6128	−2.12397
$898$	11.5077	0.384016
$899$	6.24088	0.208145
$900$	0	0
$901$	15.1487	0.504675
$902$	6.03042	0.200791
$903$	24.3312	0.809691
$904$	13.1061	0.435903
$905$	0	0
$906$	5.77842	0.191975
$907$	−5.31596	−0.176513	−0.0882567	−	0.996098i	$-0.528130\pi$
$907$	−5.31596	−0.176513	−0.0882567	+	0.996098i	$0.528130\pi$
$908$	25.5293	0.847218
$909$	26.4794	0.878266
$910$	0	0
$911$	−19.9370	−0.660541	−0.330271	−	0.943886i	$-0.607140\pi$
$911$	−19.9370	−0.660541	−0.330271	+	0.943886i	$0.607140\pi$
$912$	−18.9727	−0.628250
$913$	0.126679	0.00419248
$914$	−5.90984	−0.195480
$915$	0	0
$916$	10.2932	0.340098
$917$	−0.194510	−0.00642329
$918$	2.81486	0.0929042
$919$	23.3818	0.771294	0.385647	−	0.922646i	$-0.373978\pi$
$919$	23.3818	0.771294	0.385647	+	0.922646i	$0.373978\pi$
$920$	0	0
$921$	−36.6080	−1.20628
$922$	7.95390	0.261948
$923$	17.6096	0.579626
$924$	−26.5829	−0.874512
$925$	0	0
$926$	5.59927	0.184004
$927$	−13.9854	−0.459342
$928$	−2.10488	−0.0690961
$929$	−1.14512	−0.0375700	−0.0187850	−	0.999824i	$-0.505980\pi$
$929$	−1.14512	−0.0375700	−0.0187850	+	0.999824i	$0.505980\pi$
$930$	0	0
$931$	−2.02672	−0.0664231
$932$	−23.0255	−0.754225
$933$	22.5695	0.738892
$934$	9.39059	0.307270
$935$	0	0
$936$	−10.3446	−0.338125
$937$	−41.4063	−1.35268	−0.676342	−	0.736588i	$-0.736436\pi$
$937$	−41.4063	−1.35268	−0.676342	+	0.736588i	$0.736436\pi$
$938$	−5.95239	−0.194353
$939$	−25.0908	−0.818809
$940$	0	0
$941$	−26.6383	−0.868383	−0.434191	−	0.900821i	$-0.642966\pi$
$941$	−26.6383	−0.868383	−0.434191	+	0.900821i	$0.642966\pi$
$942$	−5.78550	−0.188502
$943$	39.8871	1.29890
$944$	−51.6529	−1.68116
$945$	0	0
$946$	8.00488	0.260261
$947$	−31.7820	−1.03278	−0.516389	−	0.856354i	$-0.672724\pi$
$947$	−31.7820	−1.03278	−0.516389	+	0.856354i	$0.672724\pi$
$948$	−17.0558	−0.553947
$949$	36.8360	1.19575
$950$	0	0
$951$	−10.7778	−0.349495
$952$	5.41478	0.175494
$953$	28.7458	0.931168	0.465584	−	0.885004i	$-0.345844\pi$
$953$	28.7458	0.931168	0.465584	+	0.885004i	$0.345844\pi$
$954$	3.40138	0.110124
$955$	0	0
$956$	11.4785	0.371242
$957$	−3.28932	−0.106329
$958$	1.85948	0.0600770
$959$	−10.7382	−0.346754
$960$	0	0
$961$	61.4463	1.98214
$962$	−18.8642	−0.608207
$963$	8.37166	0.269773
$964$	28.4066	0.914915
$965$	0	0
$966$	7.47639	0.240549
$967$	39.3136	1.26424	0.632120	−	0.774870i	$-0.282185\pi$
$967$	39.3136	1.26424	0.632120	+	0.774870i	$0.282185\pi$
$968$	−5.55260	−0.178467
$969$	9.54495	0.306628
$970$	0	0
$971$	49.7035	1.59506	0.797531	−	0.603278i	$-0.206139\pi$
$971$	49.7035	1.59506	0.797531	+	0.603278i	$0.206139\pi$
$972$	−25.0190	−0.802484
$973$	3.25046	0.104205
$974$	−4.09036	−0.131063
$975$	0	0
$976$	−40.1687	−1.28577
$977$	7.36078	0.235492	0.117746	−	0.993044i	$-0.462433\pi$
$977$	7.36078	0.235492	0.117746	+	0.993044i	$0.462433\pi$
$978$	−1.09541	−0.0350275
$979$	4.98576	0.159345
$980$	0	0
$981$	18.9496	0.605014
$982$	3.46503	0.110574
$983$	17.8926	0.570685	0.285343	−	0.958426i	$-0.407893\pi$
$983$	17.8926	0.570685	0.285343	+	0.958426i	$0.407893\pi$
$984$	−7.50226	−0.239163
$985$	0	0
$986$	0.328034	0.0104467
$987$	−19.1443	−0.609370
$988$	−54.2110	−1.72468
$989$	52.9467	1.68361
$990$	0	0
$991$	−43.9817	−1.39713	−0.698563	−	0.715548i	$-0.746177\pi$
$991$	−43.9817	−1.39713	−0.698563	+	0.715548i	$0.746177\pi$
$992$	−31.1797	−0.989955
$993$	1.05367	0.0334373
$994$	−2.06965	−0.0656452
$995$	0	0
$996$	−0.0771585	−0.00244486
$997$	−4.13475	−0.130949	−0.0654745	−	0.997854i	$-0.520856\pi$
$997$	−4.13475	−0.130949	−0.0654745	+	0.997854i	$0.520856\pi$
$998$	7.15336	0.226436
$999$	−55.3634	−1.75162

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.j.1.6		10
5.4	even	2		985.2.a.e.1.5	✓	10
15.14	odd	2		8865.2.a.t.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.a.e.1.5	✓	10	5.4	even	2
4925.2.a.j.1.6		10	1.1	even	1	trivial
8865.2.a.t.1.6		10	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:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q+0.285611 q^{2} +1.26844 q^{3} -1.91843 q^{4} +0.362280 q^{6} +2.73433 q^{7} -1.11915 q^{8} -1.39107 q^{9} +3.99518 q^{11} -2.43340 q^{12} -6.64477 q^{13} +0.780956 q^{14} +3.51721 q^{16} -1.76947 q^{17} -0.397305 q^{18} -4.25268 q^{19} +3.46833 q^{21} +1.14107 q^{22} +7.54738 q^{23} -1.41957 q^{24} -1.89782 q^{26} -5.56979 q^{27} -5.24562 q^{28} -0.649084 q^{29} -9.61490 q^{31} +3.24285 q^{32} +5.06763 q^{33} -0.505380 q^{34} +2.66866 q^{36} +9.93994 q^{37} -1.21461 q^{38} -8.42847 q^{39} +5.28489 q^{41} +0.990594 q^{42} +7.01525 q^{43} -7.66446 q^{44} +2.15562 q^{46} -5.51975 q^{47} +4.46136 q^{48} +0.476575 q^{49} -2.24446 q^{51} +12.7475 q^{52} -8.56114 q^{53} -1.59080 q^{54} -3.06012 q^{56} -5.39426 q^{57} -0.185386 q^{58} -14.6857 q^{59} -11.4206 q^{61} -2.74612 q^{62} -3.80364 q^{63} -6.10823 q^{64} +1.44737 q^{66} -7.62193 q^{67} +3.39459 q^{68} +9.57338 q^{69} -2.65014 q^{71} +1.55681 q^{72} -5.54362 q^{73} +2.83896 q^{74} +8.15845 q^{76} +10.9241 q^{77} -2.40727 q^{78} +7.00903 q^{79} -2.89173 q^{81} +1.50942 q^{82} +0.0317081 q^{83} -6.65373 q^{84} +2.00363 q^{86} -0.823322 q^{87} -4.47119 q^{88} +1.24794 q^{89} -18.1690 q^{91} -14.4791 q^{92} -12.1959 q^{93} -1.57650 q^{94} +4.11335 q^{96} +18.6425 q^{97} +0.136115 q^{98} -5.55756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 9 q^{6} + 6 q^{7} + 6 q^{8} - 5 q^{9} - 11 q^{11} + 5 q^{13} - 9 q^{14} - 2 q^{16} + 4 q^{17} - 15 q^{18} - 28 q^{19} - 7 q^{21} + 12 q^{22} + 24 q^{23} - 3 q^{24} + 7 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) \) 10 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 9 * q^6 + 6 * q^7 + 6 * q^8 - 5 * q^9 - 11 * q^11 + 5 * q^13 - 9 * q^14 - 2 * q^16 + 4 * q^17 - 15 * q^18 - 28 * q^19 - 7 * q^21 + 12 * q^22 + 24 * q^23 - 3 * q^24 + 7 * q^26 + 9 * q^27 - 3 * q^28 - 16 * q^29 - 34 * q^31 + 4 * q^32 + 10 * q^33 + q^34 - q^36 - 5 * q^37 - 13 * q^38 - 15 * q^39 - 15 * q^41 - 7 * q^42 + 5 * q^43 - 11 * q^44 + 6 * q^46 + 14 * q^47 - 12 * q^48 + 4 * q^49 - q^51 + 7 * q^52 + 15 * q^53 - 4 * q^54 - 5 * q^56 - 9 * q^57 - 28 * q^58 - 49 * q^59 - 16 * q^61 - 20 * q^62 - 8 * q^63 - 16 * q^64 + 40 * q^66 + 2 * q^67 - 19 * q^68 - 16 * q^69 - 64 * q^71 - q^72 - 18 * q^73 - 37 * q^74 - 19 * q^76 - 8 * q^77 - 4 * q^78 - 11 * q^79 - 14 * q^81 - 10 * q^82 - 4 * q^83 - 11 * q^84 + 5 * q^86 + q^87 - 19 * q^88 - 5 * q^89 - 30 * q^91 + 32 * q^92 - 41 * q^93 - 37 * q^94 + 26 * q^96 - 20 * q^97 + 37 * q^98 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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