LMFDB - Embedded newform 1925.2.a.ba.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1925,2,Mod(1,1925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,-1,0,13,0,-1,-7,-6,13,0,-7,21,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 10x^{4} + 47x^{3} - 25x^{2} - 35x + 20 $ x^7 - x^6 - 13x^5 + 10x^4 + 47x^3 - 25x^2 - 35*x + 20
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.815390$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.815390 q^{2} -3.26294 q^{3} -1.33514 q^{4} +2.66057 q^{6} -1.00000 q^{7} +2.71944 q^{8} +7.64680 q^{9} -1.00000 q^{11} +4.35648 q^{12} +4.02935 q^{13} +0.815390 q^{14} +0.452875 q^{16} -7.77478 q^{17} -6.23513 q^{18} +6.63172 q^{19} +3.26294 q^{21} +0.815390 q^{22} +1.29268 q^{23} -8.87338 q^{24} -3.28549 q^{26} -15.1623 q^{27} +1.33514 q^{28} -6.31390 q^{29} +6.71702 q^{31} -5.80815 q^{32} +3.26294 q^{33} +6.33948 q^{34} -10.2095 q^{36} -9.84650 q^{37} -5.40744 q^{38} -13.1476 q^{39} -4.73667 q^{41} -2.66057 q^{42} +0.544889 q^{43} +1.33514 q^{44} -1.05404 q^{46} +2.66057 q^{47} -1.47770 q^{48} +1.00000 q^{49} +25.3687 q^{51} -5.37975 q^{52} -3.65910 q^{53} +12.3632 q^{54} -2.71944 q^{56} -21.6389 q^{57} +5.14829 q^{58} +4.02040 q^{59} -5.19496 q^{61} -5.47699 q^{62} -7.64680 q^{63} +3.83016 q^{64} -2.66057 q^{66} +11.8802 q^{67} +10.3804 q^{68} -4.21795 q^{69} -13.4894 q^{71} +20.7950 q^{72} -2.60702 q^{73} +8.02873 q^{74} -8.85428 q^{76} +1.00000 q^{77} +10.7204 q^{78} -8.11865 q^{79} +26.5332 q^{81} +3.86223 q^{82} +5.75966 q^{83} -4.35648 q^{84} -0.444297 q^{86} +20.6019 q^{87} -2.71944 q^{88} +7.73546 q^{89} -4.02935 q^{91} -1.72591 q^{92} -21.9173 q^{93} -2.16940 q^{94} +18.9517 q^{96} +16.9417 q^{97} -0.815390 q^{98} -7.64680 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9} - 7 q^{11} + 21 q^{12} + 3 q^{13} + q^{14} + 29 q^{16} - 14 q^{18} + 18 q^{19} + q^{22} + 7 q^{23} - 22 q^{24} + 13 q^{26} - 6 q^{27} - 13 q^{28}+ \cdots - 13 q^{99}+O(q^{100}) $ 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9 - 7 * q^11 + 21 * q^12 + 3 * q^13 + q^14 + 29 * q^16 - 14 * q^18 + 18 * q^19 + q^22 + 7 * q^23 - 22 * q^24 + 13 * q^26 - 6 * q^27 - 13 * q^28 - 2 * q^29 + 24 * q^31 - 33 * q^32 + 33 * q^34 + 44 * q^36 - 21 * q^37 - 9 * q^38 + 10 * q^39 - 10 * q^41 + q^42 + 2 * q^43 - 13 * q^44 + 3 * q^46 - q^47 + 77 * q^48 + 7 * q^49 + 29 * q^51 - 9 * q^52 + 11 * q^53 - 47 * q^54 + 6 * q^56 - 7 * q^57 + 33 * q^58 + q^59 + 28 * q^61 - 16 * q^62 - 13 * q^63 + 48 * q^64 + q^66 + 46 * q^68 + 33 * q^69 - 10 * q^71 - 36 * q^72 + 11 * q^73 - 6 * q^74 + 20 * q^76 + 7 * q^77 + 31 * q^78 + 19 * q^79 + 27 * q^81 - 8 * q^82 + 19 * q^83 - 21 * q^84 + 55 * q^86 + 12 * q^87 + 6 * q^88 - 2 * q^89 - 3 * q^91 - 40 * q^92 - 58 * q^93 + 21 * q^94 - 23 * q^96 - 8 * q^97 - q^98 - 13 * 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 $

$2$	−0.815390	−0.576568	−0.288284	−	0.957545i	$-0.593085\pi$
$2$	−0.815390	−0.576568	−0.288284	+	0.957545i	$0.593085\pi$
$3$	−3.26294	−1.88386	−0.941931	−	0.335807i	$-0.890991\pi$
$3$	−3.26294	−1.88386	−0.941931	+	0.335807i	$0.890991\pi$
$4$	−1.33514	−0.667570
$5$	0	0
$5$	0	0
$6$	2.66057	1.08617
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.71944	0.961467
$9$	7.64680	2.54893
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	4.35648	1.25761
$13$	4.02935	1.11754	0.558771	−	0.829322i	$-0.311273\pi$
$13$	4.02935	1.11754	0.558771	+	0.829322i	$0.311273\pi$
$14$	0.815390	0.217922
$15$	0	0
$16$	0.452875	0.113219
$17$	−7.77478	−1.88566	−0.942831	−	0.333271i	$-0.891848\pi$
$17$	−7.77478	−1.88566	−0.942831	+	0.333271i	$0.891848\pi$
$18$	−6.23513	−1.46963
$19$	6.63172	1.52142	0.760711	−	0.649091i	$-0.224850\pi$
$19$	6.63172	1.52142	0.760711	+	0.649091i	$0.224850\pi$
$20$	0	0
$21$	3.26294	0.712033
$22$	0.815390	0.173842
$23$	1.29268	0.269543	0.134771	−	0.990877i	$-0.456970\pi$
$23$	1.29268	0.269543	0.134771	+	0.990877i	$0.456970\pi$
$24$	−8.87338	−1.81127
$25$	0	0
$26$	−3.28549	−0.644338
$27$	−15.1623	−2.91798
$28$	1.33514	0.252318
$29$	−6.31390	−1.17246	−0.586231	−	0.810144i	$-0.699389\pi$
$29$	−6.31390	−1.17246	−0.586231	+	0.810144i	$0.699389\pi$
$30$	0	0
$31$	6.71702	1.20641	0.603207	−	0.797585i	$-0.293889\pi$
$31$	6.71702	1.20641	0.603207	+	0.797585i	$0.293889\pi$
$32$	−5.80815	−1.02675
$33$	3.26294	0.568006
$34$	6.33948	1.08721
$35$	0	0
$36$	−10.2095	−1.70159
$37$	−9.84650	−1.61875	−0.809377	−	0.587290i	$-0.800195\pi$
$37$	−9.84650	−1.61875	−0.809377	+	0.587290i	$0.800195\pi$
$38$	−5.40744	−0.877203
$39$	−13.1476	−2.10529
$40$	0	0
$41$	−4.73667	−0.739744	−0.369872	−	0.929083i	$-0.620598\pi$
$41$	−4.73667	−0.739744	−0.369872	+	0.929083i	$0.620598\pi$
$42$	−2.66057	−0.410535
$43$	0.544889	0.0830948	0.0415474	−	0.999137i	$-0.486771\pi$
$43$	0.544889	0.0830948	0.0415474	+	0.999137i	$0.486771\pi$
$44$	1.33514	0.201280
$45$	0	0
$46$	−1.05404	−0.155410
$47$	2.66057	0.388084	0.194042	−	0.980993i	$-0.437840\pi$
$47$	2.66057	0.388084	0.194042	+	0.980993i	$0.437840\pi$
$48$	−1.47770	−0.213288
$49$	1.00000	0.142857
$50$	0	0
$51$	25.3687	3.55233
$52$	−5.37975	−0.746037
$53$	−3.65910	−0.502616	−0.251308	−	0.967907i	$-0.580861\pi$
$53$	−3.65910	−0.502616	−0.251308	+	0.967907i	$0.580861\pi$
$54$	12.3632	1.68241
$55$	0	0
$56$	−2.71944	−0.363400
$57$	−21.6389	−2.86615
$58$	5.14829	0.676004
$59$	4.02040	0.523411	0.261706	−	0.965148i	$-0.415715\pi$
$59$	4.02040	0.523411	0.261706	+	0.965148i	$0.415715\pi$
$60$	0	0
$61$	−5.19496	−0.665147	−0.332573	−	0.943077i	$-0.607917\pi$
$61$	−5.19496	−0.665147	−0.332573	+	0.943077i	$0.607917\pi$
$62$	−5.47699	−0.695579
$63$	−7.64680	−0.963407
$64$	3.83016	0.478770
$65$	0	0
$66$	−2.66057	−0.327494
$67$	11.8802	1.45140	0.725701	−	0.688010i	$-0.241516\pi$
$67$	11.8802	1.45140	0.725701	+	0.688010i	$0.241516\pi$
$68$	10.3804	1.25881
$69$	−4.21795	−0.507782
$70$	0	0
$71$	−13.4894	−1.60090	−0.800448	−	0.599402i	$-0.795405\pi$
$71$	−13.4894	−1.60090	−0.800448	+	0.599402i	$0.795405\pi$
$72$	20.7950	2.45072
$73$	−2.60702	−0.305129	−0.152565	−	0.988294i	$-0.548753\pi$
$73$	−2.60702	−0.305129	−0.152565	+	0.988294i	$0.548753\pi$
$74$	8.02873	0.933321
$75$	0	0
$76$	−8.85428	−1.01566
$77$	1.00000	0.113961
$78$	10.7204	1.21384
$79$	−8.11865	−0.913420	−0.456710	−	0.889616i	$-0.650972\pi$
$79$	−8.11865	−0.913420	−0.456710	+	0.889616i	$0.650972\pi$
$80$	0	0
$81$	26.5332	2.94813
$82$	3.86223	0.426512
$83$	5.75966	0.632205	0.316102	−	0.948725i	$-0.397626\pi$
$83$	5.75966	0.632205	0.316102	+	0.948725i	$0.397626\pi$
$84$	−4.35648	−0.475331
$85$	0	0
$86$	−0.444297	−0.0479098
$87$	20.6019	2.20876
$88$	−2.71944	−0.289893
$89$	7.73546	0.819958	0.409979	−	0.912095i	$-0.365536\pi$
$89$	7.73546	0.819958	0.409979	+	0.912095i	$0.365536\pi$
$90$	0	0
$91$	−4.02935	−0.422391
$92$	−1.72591	−0.179939
$93$	−21.9173	−2.27272
$94$	−2.16940	−0.223757
$95$	0	0
$96$	18.9517	1.93425
$97$	16.9417	1.72017	0.860085	−	0.510151i	$-0.170411\pi$
$97$	16.9417	1.72017	0.860085	+	0.510151i	$0.170411\pi$
$98$	−0.815390	−0.0823668
$99$	−7.64680	−0.768533
$100$	0	0
$101$	1.54184	0.153419	0.0767093	−	0.997053i	$-0.475559\pi$
$101$	1.54184	0.153419	0.0767093	+	0.997053i	$0.475559\pi$
$102$	−20.6854	−2.04816
$103$	−6.39290	−0.629911	−0.314956	−	0.949106i	$-0.601990\pi$
$103$	−6.39290	−0.629911	−0.314956	+	0.949106i	$0.601990\pi$
$104$	10.9576	1.07448
$105$	0	0
$106$	2.98359	0.289792
$107$	−0.521233	−0.0503895	−0.0251948	−	0.999683i	$-0.508021\pi$
$107$	−0.521233	−0.0503895	−0.0251948	+	0.999683i	$0.508021\pi$
$108$	20.2437	1.94795
$109$	−10.7962	−1.03409	−0.517043	−	0.855959i	$-0.672967\pi$
$109$	−10.7962	−1.03409	−0.517043	+	0.855959i	$0.672967\pi$
$110$	0	0
$111$	32.1286	3.04951
$112$	−0.452875	−0.0427926
$113$	19.1949	1.80570	0.902852	−	0.429952i	$-0.141470\pi$
$113$	19.1949	1.80570	0.902852	+	0.429952i	$0.141470\pi$
$114$	17.6442	1.65253
$115$	0	0
$116$	8.42994	0.782700
$117$	30.8117	2.84854
$118$	−3.27819	−0.301782
$119$	7.77478	0.712713
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.23592	0.383502
$123$	15.4555	1.39357
$124$	−8.96816	−0.805365
$125$	0	0
$126$	6.23513	0.555469
$127$	5.93032	0.526231	0.263116	−	0.964764i	$-0.415250\pi$
$127$	5.93032	0.526231	0.263116	+	0.964764i	$0.415250\pi$
$128$	8.49322	0.750702
$129$	−1.77794	−0.156539
$130$	0	0
$131$	5.49332	0.479953	0.239977	−	0.970779i	$-0.422860\pi$
$131$	5.49332	0.479953	0.239977	+	0.970779i	$0.422860\pi$
$132$	−4.35648	−0.379183
$133$	−6.63172	−0.575043
$134$	−9.68703	−0.836832
$135$	0	0
$136$	−21.1431	−1.81300
$137$	−12.3717	−1.05698	−0.528491	−	0.848939i	$-0.677242\pi$
$137$	−12.3717	−1.05698	−0.528491	+	0.848939i	$0.677242\pi$
$138$	3.43928	0.292771
$139$	20.7319	1.75846	0.879230	−	0.476397i	$-0.158058\pi$
$139$	20.7319	1.75846	0.879230	+	0.476397i	$0.158058\pi$
$140$	0	0
$141$	−8.68130	−0.731097
$142$	10.9991	0.923026
$143$	−4.02935	−0.336951
$144$	3.46304	0.288587
$145$	0	0
$146$	2.12574	0.175928
$147$	−3.26294	−0.269123
$148$	13.1464	1.08063
$149$	−20.5472	−1.68329	−0.841644	−	0.540032i	$-0.818412\pi$
$149$	−20.5472	−1.68329	−0.841644	+	0.540032i	$0.818412\pi$
$150$	0	0
$151$	2.12945	0.173292	0.0866461	−	0.996239i	$-0.472385\pi$
$151$	2.12945	0.173292	0.0866461	+	0.996239i	$0.472385\pi$
$152$	18.0346	1.46280
$153$	−59.4523	−4.80643
$154$	−0.815390	−0.0657060
$155$	0	0
$156$	17.5538	1.40543
$157$	−11.7630	−0.938793	−0.469397	−	0.882987i	$-0.655529\pi$
$157$	−11.7630	−0.938793	−0.469397	+	0.882987i	$0.655529\pi$
$158$	6.61987	0.526648
$159$	11.9394	0.946859
$160$	0	0
$161$	−1.29268	−0.101878
$162$	−21.6349	−1.69980
$163$	11.8065	0.924754	0.462377	−	0.886684i	$-0.346997\pi$
$163$	11.8065	0.924754	0.462377	+	0.886684i	$0.346997\pi$
$164$	6.32411	0.493830
$165$	0	0
$166$	−4.69637	−0.364509
$167$	7.12202	0.551118	0.275559	−	0.961284i	$-0.411137\pi$
$167$	7.12202	0.551118	0.275559	+	0.961284i	$0.411137\pi$
$168$	8.87338	0.684596
$169$	3.23568	0.248899
$170$	0	0
$171$	50.7115	3.87801
$172$	−0.727503	−0.0554716
$173$	6.62215	0.503473	0.251736	−	0.967796i	$-0.418998\pi$
$173$	6.62215	0.503473	0.251736	+	0.967796i	$0.418998\pi$
$174$	−16.7986	−1.27350
$175$	0	0
$176$	−0.452875	−0.0341367
$177$	−13.1183	−0.986035
$178$	−6.30742	−0.472761
$179$	−0.219295	−0.0163908	−0.00819542	−	0.999966i	$-0.502609\pi$
$179$	−0.219295	−0.0163908	−0.00819542	+	0.999966i	$0.502609\pi$
$180$	0	0
$181$	0.735466	0.0546667	0.0273334	−	0.999626i	$-0.491298\pi$
$181$	0.735466	0.0546667	0.0273334	+	0.999626i	$0.491298\pi$
$182$	3.28549	0.243537
$183$	16.9509	1.25304
$184$	3.51537	0.259157
$185$	0	0
$186$	17.8711	1.31037
$187$	7.77478	0.568549
$188$	−3.55223	−0.259073
$189$	15.1623	1.10289
$190$	0	0
$191$	10.7645	0.778894	0.389447	−	0.921049i	$-0.372666\pi$
$191$	10.7645	0.778894	0.389447	+	0.921049i	$0.372666\pi$
$192$	−12.4976	−0.901936
$193$	6.79618	0.489200	0.244600	−	0.969624i	$-0.421343\pi$
$193$	6.79618	0.489200	0.244600	+	0.969624i	$0.421343\pi$
$194$	−13.8141	−0.991794
$195$	0	0
$196$	−1.33514	−0.0953671
$197$	−4.56293	−0.325095	−0.162548	−	0.986701i	$-0.551971\pi$
$197$	−4.56293	−0.325095	−0.162548	+	0.986701i	$0.551971\pi$
$198$	6.23513	0.443111
$199$	−15.8062	−1.12047	−0.560236	−	0.828333i	$-0.689290\pi$
$199$	−15.8062	−1.12047	−0.560236	+	0.828333i	$0.689290\pi$
$200$	0	0
$201$	−38.7645	−2.73424
$202$	−1.25720	−0.0884562
$203$	6.31390	0.443149
$204$	−33.8707	−2.37143
$205$	0	0
$206$	5.21271	0.363187
$207$	9.88489	0.687047
$208$	1.82479	0.126527
$209$	−6.63172	−0.458726
$210$	0	0
$211$	−2.19451	−0.151076	−0.0755381	−	0.997143i	$-0.524067\pi$
$211$	−2.19451	−0.151076	−0.0755381	+	0.997143i	$0.524067\pi$
$212$	4.88541	0.335531
$213$	44.0151	3.01587
$214$	0.425008	0.0290530
$215$	0	0
$216$	−41.2329	−2.80554
$217$	−6.71702	−0.455981
$218$	8.80310	0.596221
$219$	8.50657	0.574821
$220$	0	0
$221$	−31.3273	−2.10731
$222$	−26.1973	−1.75825
$223$	−16.0124	−1.07227	−0.536134	−	0.844133i	$-0.680116\pi$
$223$	−16.0124	−1.07227	−0.536134	+	0.844133i	$0.680116\pi$
$224$	5.80815	0.388073
$225$	0	0
$226$	−15.6513	−1.04111
$227$	9.72207	0.645277	0.322638	−	0.946522i	$-0.395430\pi$
$227$	9.72207	0.645277	0.322638	+	0.946522i	$0.395430\pi$
$228$	28.8910	1.91335
$229$	0.871601	0.0575970	0.0287985	−	0.999585i	$-0.490832\pi$
$229$	0.871601	0.0575970	0.0287985	+	0.999585i	$0.490832\pi$
$230$	0	0
$231$	−3.26294	−0.214686
$232$	−17.1703	−1.12728
$233$	12.7523	0.835431	0.417715	−	0.908578i	$-0.362831\pi$
$233$	12.7523	0.835431	0.417715	+	0.908578i	$0.362831\pi$
$234$	−25.1235	−1.64238
$235$	0	0
$236$	−5.36779	−0.349413
$237$	26.4907	1.72076
$238$	−6.33948	−0.410928
$239$	−15.9745	−1.03330	−0.516652	−	0.856196i	$-0.672822\pi$
$239$	−15.9745	−1.03330	−0.516652	+	0.856196i	$0.672822\pi$
$240$	0	0
$241$	10.3643	0.667624	0.333812	−	0.942640i	$-0.391665\pi$
$241$	10.3643	0.667624	0.333812	+	0.942640i	$0.391665\pi$
$242$	−0.815390	−0.0524153
$243$	−41.0896	−2.63590
$244$	6.93600	0.444032
$245$	0	0
$246$	−12.6023	−0.803490
$247$	26.7216	1.70025
$248$	18.2665	1.15993
$249$	−18.7934	−1.19099
$250$	0	0
$251$	21.0924	1.33134	0.665670	−	0.746246i	$-0.268146\pi$
$251$	21.0924	1.33134	0.665670	+	0.746246i	$0.268146\pi$
$252$	10.2095	0.643141
$253$	−1.29268	−0.0812703
$254$	−4.83553	−0.303408
$255$	0	0
$256$	−14.5856	−0.911600
$257$	−21.8799	−1.36483	−0.682415	−	0.730965i	$-0.739070\pi$
$257$	−21.8799	−1.36483	−0.682415	+	0.730965i	$0.739070\pi$
$258$	1.44972	0.0902554
$259$	9.84650	0.611831
$260$	0	0
$261$	−48.2812	−2.98853
$262$	−4.47920	−0.276726
$263$	−0.267493	−0.0164943	−0.00824716	−	0.999966i	$-0.502625\pi$
$263$	−0.267493	−0.0164943	−0.00824716	+	0.999966i	$0.502625\pi$
$264$	8.87338	0.546119
$265$	0	0
$266$	5.40744	0.331552
$267$	−25.2404	−1.54469
$268$	−15.8618	−0.968912
$269$	13.5421	0.825678	0.412839	−	0.910804i	$-0.364537\pi$
$269$	13.5421	0.825678	0.412839	+	0.910804i	$0.364537\pi$
$270$	0	0
$271$	−2.52073	−0.153123	−0.0765616	−	0.997065i	$-0.524394\pi$
$271$	−2.52073	−0.153123	−0.0765616	+	0.997065i	$0.524394\pi$
$272$	−3.52100	−0.213492
$273$	13.1476	0.795726
$274$	10.0877	0.609422
$275$	0	0
$276$	5.63155	0.338980
$277$	20.2545	1.21697	0.608487	−	0.793564i	$-0.291777\pi$
$277$	20.2545	1.21697	0.608487	+	0.793564i	$0.291777\pi$
$278$	−16.9046	−1.01387
$279$	51.3638	3.07507
$280$	0	0
$281$	2.64588	0.157840	0.0789199	−	0.996881i	$-0.474853\pi$
$281$	2.64588	0.157840	0.0789199	+	0.996881i	$0.474853\pi$
$282$	7.07864	0.421527
$283$	9.86395	0.586351	0.293176	−	0.956059i	$-0.405288\pi$
$283$	9.86395	0.586351	0.293176	+	0.956059i	$0.405288\pi$
$284$	18.0102	1.06871
$285$	0	0
$286$	3.28549	0.194275
$287$	4.73667	0.279597
$288$	−44.4138	−2.61711
$289$	43.4473	2.55572
$290$	0	0
$291$	−55.2798	−3.24056
$292$	3.48074	0.203695
$293$	29.0419	1.69664	0.848322	−	0.529481i	$-0.177613\pi$
$293$	29.0419	1.69664	0.848322	+	0.529481i	$0.177613\pi$
$294$	2.66057	0.155168
$295$	0	0
$296$	−26.7769	−1.55638
$297$	15.1623	0.879804
$298$	16.7539	0.970530
$299$	5.20867	0.301225
$300$	0	0
$301$	−0.544889	−0.0314069
$302$	−1.73633	−0.0999147
$303$	−5.03093	−0.289019
$304$	3.00334	0.172253
$305$	0	0
$306$	48.4768	2.77123
$307$	5.33047	0.304226	0.152113	−	0.988363i	$-0.451392\pi$
$307$	5.33047	0.304226	0.152113	+	0.988363i	$0.451392\pi$
$308$	−1.33514	−0.0760766
$309$	20.8597	1.18667
$310$	0	0
$311$	−10.1987	−0.578314	−0.289157	−	0.957282i	$-0.593375\pi$
$311$	−10.1987	−0.578314	−0.289157	+	0.957282i	$0.593375\pi$
$312$	−35.7540	−2.02417
$313$	−19.2336	−1.08715	−0.543574	−	0.839361i	$-0.682929\pi$
$313$	−19.2336	−1.08715	−0.543574	+	0.839361i	$0.682929\pi$
$314$	9.59147	0.541278
$315$	0	0
$316$	10.8395	0.609771
$317$	−1.10991	−0.0623389	−0.0311694	−	0.999514i	$-0.509923\pi$
$317$	−1.10991	−0.0623389	−0.0311694	+	0.999514i	$0.509923\pi$
$318$	−9.73530	−0.545929
$319$	6.31390	0.353511
$320$	0	0
$321$	1.70075	0.0949269
$322$	1.05404	0.0587394
$323$	−51.5602	−2.86889
$324$	−35.4255	−1.96808
$325$	0	0
$326$	−9.62687	−0.533183
$327$	35.2273	1.94808
$328$	−12.8811	−0.711239
$329$	−2.66057	−0.146682
$330$	0	0
$331$	28.5253	1.56789	0.783946	−	0.620829i	$-0.213204\pi$
$331$	28.5253	1.56789	0.783946	+	0.620829i	$0.213204\pi$
$332$	−7.68995	−0.422041
$333$	−75.2942	−4.12610
$334$	−5.80722	−0.317757
$335$	0	0
$336$	1.47770	0.0806154
$337$	−19.3128	−1.05204	−0.526018	−	0.850473i	$-0.676316\pi$
$337$	−19.3128	−1.05204	−0.526018	+	0.850473i	$0.676316\pi$
$338$	−2.63834	−0.143507
$339$	−62.6319	−3.40170
$340$	0	0
$341$	−6.71702	−0.363747
$342$	−41.3497	−2.23593
$343$	−1.00000	−0.0539949
$344$	1.48179	0.0798929
$345$	0	0
$346$	−5.39963	−0.290286
$347$	−4.76310	−0.255696	−0.127848	−	0.991794i	$-0.540807\pi$
$347$	−4.76310	−0.255696	−0.127848	+	0.991794i	$0.540807\pi$
$348$	−27.5064	−1.47450
$349$	10.3582	0.554462	0.277231	−	0.960803i	$-0.410583\pi$
$349$	10.3582	0.554462	0.277231	+	0.960803i	$0.410583\pi$
$350$	0	0
$351$	−61.0941	−3.26096
$352$	5.80815	0.309575
$353$	1.32582	0.0705661	0.0352831	−	0.999377i	$-0.488767\pi$
$353$	1.32582	0.0705661	0.0352831	+	0.999377i	$0.488767\pi$
$354$	10.6966	0.568516
$355$	0	0
$356$	−10.3279	−0.547379
$357$	−25.3687	−1.34265
$358$	0.178811	0.00945043
$359$	16.2176	0.855934	0.427967	−	0.903794i	$-0.359230\pi$
$359$	16.2176	0.855934	0.427967	+	0.903794i	$0.359230\pi$
$360$	0	0
$361$	24.9798	1.31472
$362$	−0.599691	−0.0315191
$363$	−3.26294	−0.171260
$364$	5.37975	0.281975
$365$	0	0
$366$	−13.8216	−0.722465
$367$	32.3068	1.68640	0.843200	−	0.537600i	$-0.180669\pi$
$367$	32.3068	1.68640	0.843200	+	0.537600i	$0.180669\pi$
$368$	0.585423	0.0305173
$369$	−36.2204	−1.88556
$370$	0	0
$371$	3.65910	0.189971
$372$	29.2626	1.51720
$373$	0.697093	0.0360941	0.0180471	−	0.999837i	$-0.494255\pi$
$373$	0.697093	0.0360941	0.0180471	+	0.999837i	$0.494255\pi$
$374$	−6.33948	−0.327807
$375$	0	0
$376$	7.23526	0.373130
$377$	−25.4409	−1.31028
$378$	−12.3632	−0.635892
$379$	−13.8037	−0.709047	−0.354524	−	0.935047i	$-0.615357\pi$
$379$	−13.8037	−0.709047	−0.354524	+	0.935047i	$0.615357\pi$
$380$	0	0
$381$	−19.3503	−0.991347
$382$	−8.77729	−0.449085
$383$	−10.4300	−0.532948	−0.266474	−	0.963842i	$-0.585859\pi$
$383$	−10.4300	−0.532948	−0.266474	+	0.963842i	$0.585859\pi$
$384$	−27.7129	−1.41422
$385$	0	0
$386$	−5.54154	−0.282057
$387$	4.16666	0.211803
$388$	−22.6195	−1.14833
$389$	8.74919	0.443602	0.221801	−	0.975092i	$-0.428807\pi$
$389$	8.74919	0.443602	0.221801	+	0.975092i	$0.428807\pi$
$390$	0	0
$391$	−10.0503	−0.508267
$392$	2.71944	0.137352
$393$	−17.9244	−0.904166
$394$	3.72057	0.187439
$395$	0	0
$396$	10.2095	0.513049
$397$	8.35598	0.419375	0.209687	−	0.977769i	$-0.432755\pi$
$397$	8.35598	0.419375	0.209687	+	0.977769i	$0.432755\pi$
$398$	12.8882	0.646028
$399$	21.6389	1.08330
$400$	0	0
$401$	−3.89701	−0.194608	−0.0973038	−	0.995255i	$-0.531022\pi$
$401$	−3.89701	−0.194608	−0.0973038	+	0.995255i	$0.531022\pi$
$402$	31.6082	1.57647
$403$	27.0653	1.34822
$404$	−2.05857	−0.102418
$405$	0	0
$406$	−5.14829	−0.255505
$407$	9.84650	0.488073
$408$	68.9886	3.41544
$409$	8.74787	0.432554	0.216277	−	0.976332i	$-0.430608\pi$
$409$	8.74787	0.432554	0.216277	+	0.976332i	$0.430608\pi$
$410$	0	0
$411$	40.3680	1.99121
$412$	8.53541	0.420510
$413$	−4.02040	−0.197831
$414$	−8.06004	−0.396129
$415$	0	0
$416$	−23.4031	−1.14743
$417$	−67.6472	−3.31270
$418$	5.40744	0.264487
$419$	−24.1879	−1.18166	−0.590829	−	0.806797i	$-0.701199\pi$
$419$	−24.1879	−1.18166	−0.590829	+	0.806797i	$0.701199\pi$
$420$	0	0
$421$	32.6675	1.59212	0.796059	−	0.605219i	$-0.206915\pi$
$421$	32.6675	1.59212	0.796059	+	0.605219i	$0.206915\pi$
$422$	1.78938	0.0871057
$423$	20.3449	0.989202
$424$	−9.95070	−0.483249
$425$	0	0
$426$	−35.8895	−1.73885
$427$	5.19496	0.251402
$428$	0.695919	0.0336385
$429$	13.1476	0.634770
$430$	0	0
$431$	−11.9307	−0.574683	−0.287342	−	0.957828i	$-0.592772\pi$
$431$	−11.9307	−0.574683	−0.287342	+	0.957828i	$0.592772\pi$
$432$	−6.86660	−0.330370
$433$	15.6872	0.753879	0.376940	−	0.926238i	$-0.376976\pi$
$433$	15.6872	0.753879	0.376940	+	0.926238i	$0.376976\pi$
$434$	5.47699	0.262904
$435$	0	0
$436$	14.4144	0.690325
$437$	8.57272	0.410089
$438$	−6.93618	−0.331423
$439$	−9.84516	−0.469884	−0.234942	−	0.972009i	$-0.575490\pi$
$439$	−9.84516	−0.469884	−0.234942	+	0.972009i	$0.575490\pi$
$440$	0	0
$441$	7.64680	0.364134
$442$	25.5440	1.21500
$443$	29.2874	1.39149	0.695744	−	0.718290i	$-0.255075\pi$
$443$	29.2874	1.39149	0.695744	+	0.718290i	$0.255075\pi$
$444$	−42.8961	−2.03576
$445$	0	0
$446$	13.0563	0.618235
$447$	67.0442	3.17108
$448$	−3.83016	−0.180958
$449$	9.18694	0.433559	0.216779	−	0.976221i	$-0.430445\pi$
$449$	9.18694	0.433559	0.216779	+	0.976221i	$0.430445\pi$
$450$	0	0
$451$	4.73667	0.223041
$452$	−25.6278	−1.20543
$453$	−6.94828	−0.326459
$454$	−7.92728	−0.372046
$455$	0	0
$456$	−58.8458	−2.75571
$457$	35.9831	1.68322	0.841609	−	0.540087i	$-0.181609\pi$
$457$	35.9831	1.68322	0.841609	+	0.540087i	$0.181609\pi$
$458$	−0.710695	−0.0332086
$459$	117.883	5.50232
$460$	0	0
$461$	32.3468	1.50654	0.753271	−	0.657711i	$-0.228475\pi$
$461$	32.3468	1.50654	0.753271	+	0.657711i	$0.228475\pi$
$462$	2.66057	0.123781
$463$	−3.43774	−0.159765	−0.0798827	−	0.996804i	$-0.525455\pi$
$463$	−3.43774	−0.159765	−0.0798827	+	0.996804i	$0.525455\pi$
$464$	−2.85941	−0.132745
$465$	0	0
$466$	−10.3981	−0.481683
$467$	−8.28467	−0.383369	−0.191684	−	0.981457i	$-0.561395\pi$
$467$	−8.28467	−0.383369	−0.191684	+	0.981457i	$0.561395\pi$
$468$	−41.1379	−1.90160
$469$	−11.8802	−0.548578
$470$	0	0
$471$	38.3822	1.76856
$472$	10.9332	0.503243
$473$	−0.544889	−0.0250540
$474$	−21.6003	−0.992133
$475$	0	0
$476$	−10.3804	−0.475786
$477$	−27.9804	−1.28114
$478$	13.0254	0.595770
$479$	−1.03162	−0.0471360	−0.0235680	−	0.999722i	$-0.507503\pi$
$479$	−1.03162	−0.0471360	−0.0235680	+	0.999722i	$0.507503\pi$
$480$	0	0
$481$	−39.6750	−1.80902
$482$	−8.45096	−0.384931
$483$	4.21795	0.191923
$484$	−1.33514	−0.0606881
$485$	0	0
$486$	33.5040	1.51977
$487$	−2.77182	−0.125603	−0.0628015	−	0.998026i	$-0.520003\pi$
$487$	−2.77182	−0.125603	−0.0628015	+	0.998026i	$0.520003\pi$
$488$	−14.1274	−0.639517
$489$	−38.5238	−1.74211
$490$	0	0
$491$	−9.12276	−0.411704	−0.205852	−	0.978583i	$-0.565997\pi$
$491$	−9.12276	−0.411704	−0.205852	+	0.978583i	$0.565997\pi$
$492$	−20.6352	−0.930308
$493$	49.0892	2.21087
$494$	−21.7885	−0.980311
$495$	0	0
$496$	3.04197	0.136588
$497$	13.4894	0.605082
$498$	15.3240	0.686684
$499$	−5.05695	−0.226380	−0.113190	−	0.993573i	$-0.536107\pi$
$499$	−5.05695	−0.226380	−0.113190	+	0.993573i	$0.536107\pi$
$500$	0	0
$501$	−23.2387	−1.03823
$502$	−17.1985	−0.767608
$503$	31.9577	1.42492	0.712462	−	0.701711i	$-0.247580\pi$
$503$	31.9577	1.42492	0.712462	+	0.701711i	$0.247580\pi$
$504$	−20.7950	−0.926284
$505$	0	0
$506$	1.05404	0.0468578
$507$	−10.5579	−0.468891
$508$	−7.91781	−0.351296
$509$	32.6503	1.44720	0.723601	−	0.690219i	$-0.242486\pi$
$509$	32.6503	1.44720	0.723601	+	0.690219i	$0.242486\pi$
$510$	0	0
$511$	2.60702	0.115328
$512$	−5.09349	−0.225103
$513$	−100.552	−4.43948
$514$	17.8407	0.786918
$515$	0	0
$516$	2.37380	0.104501
$517$	−2.66057	−0.117012
$518$	−8.02873	−0.352762
$519$	−21.6077	−0.948473
$520$	0	0
$521$	0.00976971	0.000428019 0	0.000214009	−	1.00000i	$-0.499932\pi$
$521$	0.00976971	0.000428019 0	0.000214009		1.00000i	$0.499932\pi$
$522$	39.3680	1.72309
$523$	16.8422	0.736457	0.368229	−	0.929735i	$-0.379964\pi$
$523$	16.8422	0.736457	0.368229	+	0.929735i	$0.379964\pi$
$524$	−7.33434	−0.320402
$525$	0	0
$526$	0.218111	0.00951009
$527$	−52.2234	−2.27489
$528$	1.47770	0.0643088
$529$	−21.3290	−0.927347
$530$	0	0
$531$	30.7432	1.33414
$532$	8.85428	0.383882
$533$	−19.0857	−0.826694
$534$	20.5808	0.890617
$535$	0	0
$536$	32.3076	1.39547
$537$	0.715546	0.0308781
$538$	−11.0421	−0.476059
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	34.9770	1.50378	0.751888	−	0.659290i	$-0.229143\pi$
$541$	34.9770	1.50378	0.751888	+	0.659290i	$0.229143\pi$
$542$	2.05538	0.0882859
$543$	−2.39978	−0.102985
$544$	45.1571	1.93609
$545$	0	0
$546$	−10.7204	−0.458790
$547$	−19.6157	−0.838709	−0.419354	−	0.907823i	$-0.637744\pi$
$547$	−19.6157	−0.838709	−0.419354	+	0.907823i	$0.637744\pi$
$548$	16.5179	0.705609
$549$	−39.7249	−1.69542
$550$	0	0
$551$	−41.8721	−1.78381
$552$	−11.4705	−0.488215
$553$	8.11865	0.345240
$554$	−16.5153	−0.701668
$555$	0	0
$556$	−27.6800	−1.17389
$557$	23.0665	0.977360	0.488680	−	0.872463i	$-0.337479\pi$
$557$	23.0665	0.977360	0.488680	+	0.872463i	$0.337479\pi$
$558$	−41.8815	−1.77299
$559$	2.19555	0.0928619
$560$	0	0
$561$	−25.3687	−1.07107
$562$	−2.15742	−0.0910053
$563$	−12.0630	−0.508395	−0.254198	−	0.967152i	$-0.581811\pi$
$563$	−12.0630	−0.508395	−0.254198	+	0.967152i	$0.581811\pi$
$564$	11.5907	0.488058
$565$	0	0
$566$	−8.04297	−0.338071
$567$	−26.5332	−1.11429
$568$	−36.6836	−1.53921
$569$	−32.6016	−1.36673	−0.683365	−	0.730077i	$-0.739484\pi$
$569$	−32.6016	−1.36673	−0.683365	+	0.730077i	$0.739484\pi$
$570$	0	0
$571$	6.12144	0.256174	0.128087	−	0.991763i	$-0.459116\pi$
$571$	6.12144	0.256174	0.128087	+	0.991763i	$0.459116\pi$
$572$	5.37975	0.224938
$573$	−35.1241	−1.46733
$574$	−3.86223	−0.161207
$575$	0	0
$576$	29.2885	1.22035
$577$	34.0884	1.41912	0.709558	−	0.704647i	$-0.248894\pi$
$577$	34.0884	1.41912	0.709558	+	0.704647i	$0.248894\pi$
$578$	−35.4265	−1.47355
$579$	−22.1756	−0.921585
$580$	0	0
$581$	−5.75966	−0.238951
$582$	45.0746	1.86840
$583$	3.65910	0.151544
$584$	−7.08964	−0.293372
$585$	0	0
$586$	−23.6804	−0.978230
$587$	−17.5484	−0.724299	−0.362150	−	0.932120i	$-0.617957\pi$
$587$	−17.5484	−0.724299	−0.362150	+	0.932120i	$0.617957\pi$
$588$	4.35648	0.179658
$589$	44.5455	1.83546
$590$	0	0
$591$	14.8886	0.612434
$592$	−4.45923	−0.183273
$593$	15.5286	0.637683	0.318842	−	0.947808i	$-0.396706\pi$
$593$	15.5286	0.637683	0.318842	+	0.947808i	$0.396706\pi$
$594$	−12.3632	−0.507267
$595$	0	0
$596$	27.4333	1.12371
$597$	51.5748	2.11081
$598$	−4.24710	−0.173677
$599$	−27.2580	−1.11373	−0.556866	−	0.830603i	$-0.687996\pi$
$599$	−27.2580	−1.11373	−0.556866	+	0.830603i	$0.687996\pi$
$600$	0	0
$601$	−16.2109	−0.661257	−0.330628	−	0.943761i	$-0.607261\pi$
$601$	−16.2109	−0.661257	−0.330628	+	0.943761i	$0.607261\pi$
$602$	0.444297	0.0181082
$603$	90.8458	3.69953
$604$	−2.84311	−0.115685
$605$	0	0
$606$	4.10217	0.166639
$607$	14.0320	0.569541	0.284770	−	0.958596i	$-0.408083\pi$
$607$	14.0320	0.569541	0.284770	+	0.958596i	$0.408083\pi$
$608$	−38.5180	−1.56211
$609$	−20.6019	−0.834832
$610$	0	0
$611$	10.7204	0.433700
$612$	79.3770	3.20863
$613$	−24.7487	−0.999591	−0.499795	−	0.866143i	$-0.666591\pi$
$613$	−24.7487	−0.999591	−0.499795	+	0.866143i	$0.666591\pi$
$614$	−4.34642	−0.175407
$615$	0	0
$616$	2.71944	0.109569
$617$	44.1300	1.77661	0.888304	−	0.459255i	$-0.151884\pi$
$617$	44.1300	1.77661	0.888304	+	0.459255i	$0.151884\pi$
$618$	−17.0088	−0.684193
$619$	43.8551	1.76269	0.881343	−	0.472477i	$-0.156640\pi$
$619$	43.8551	1.76269	0.881343	+	0.472477i	$0.156640\pi$
$620$	0	0
$621$	−19.6000	−0.786521
$622$	8.31591	0.333437
$623$	−7.73546	−0.309915
$624$	−5.95419	−0.238358
$625$	0	0
$626$	15.6829	0.626814
$627$	21.6389	0.864176
$628$	15.7053	0.626710
$629$	76.5544	3.05242
$630$	0	0
$631$	34.3085	1.36580	0.682899	−	0.730513i	$-0.260719\pi$
$631$	34.3085	1.36580	0.682899	+	0.730513i	$0.260719\pi$
$632$	−22.0782	−0.878223
$633$	7.16056	0.284607
$634$	0.905011	0.0359426
$635$	0	0
$636$	−15.9408	−0.632095
$637$	4.02935	0.159649
$638$	−5.14829	−0.203823
$639$	−103.151	−4.08058
$640$	0	0
$641$	−2.34685	−0.0926951	−0.0463476	−	0.998925i	$-0.514758\pi$
$641$	−2.34685	−0.0926951	−0.0463476	+	0.998925i	$0.514758\pi$
$642$	−1.38678	−0.0547318
$643$	42.4998	1.67603	0.838013	−	0.545650i	$-0.183717\pi$
$643$	42.4998	1.67603	0.838013	+	0.545650i	$0.183717\pi$
$644$	1.72591	0.0680104
$645$	0	0
$646$	42.0417	1.65411
$647$	−24.3230	−0.956235	−0.478117	−	0.878296i	$-0.658681\pi$
$647$	−24.3230	−0.956235	−0.478117	+	0.878296i	$0.658681\pi$
$648$	72.1555	2.83453
$649$	−4.02040	−0.157814
$650$	0	0
$651$	21.9173	0.859006
$652$	−15.7633	−0.617337
$653$	−5.06091	−0.198049	−0.0990243	−	0.995085i	$-0.531572\pi$
$653$	−5.06091	−0.198049	−0.0990243	+	0.995085i	$0.531572\pi$
$654$	−28.7240	−1.12320
$655$	0	0
$656$	−2.14512	−0.0837528
$657$	−19.9354	−0.777754
$658$	2.16940	0.0845722
$659$	49.1062	1.91291	0.956453	−	0.291888i	$-0.0942833\pi$
$659$	49.1062	1.91291	0.956453	+	0.291888i	$0.0942833\pi$
$660$	0	0
$661$	22.7932	0.886555	0.443277	−	0.896385i	$-0.353816\pi$
$661$	22.7932	0.886555	0.443277	+	0.896385i	$0.353816\pi$
$662$	−23.2593	−0.903996
$663$	102.219	3.96987
$664$	15.6630	0.607844
$665$	0	0
$666$	61.3942	2.37898
$667$	−8.16187	−0.316029
$668$	−9.50888	−0.367910
$669$	52.2475	2.02001
$670$	0	0
$671$	5.19496	0.200549
$672$	−18.9517	−0.731076
$673$	−27.0906	−1.04426	−0.522132	−	0.852865i	$-0.674863\pi$
$673$	−27.0906	−1.04426	−0.522132	+	0.852865i	$0.674863\pi$
$674$	15.7475	0.606570
$675$	0	0
$676$	−4.32009	−0.166157
$677$	29.0859	1.11786	0.558931	−	0.829214i	$-0.311212\pi$
$677$	29.0859	1.11786	0.558931	+	0.829214i	$0.311212\pi$
$678$	51.0694	1.96131
$679$	−16.9417	−0.650163
$680$	0	0
$681$	−31.7226	−1.21561
$682$	5.47699	0.209725
$683$	−11.8696	−0.454178	−0.227089	−	0.973874i	$-0.572921\pi$
$683$	−11.8696	−0.454178	−0.227089	+	0.973874i	$0.572921\pi$
$684$	−67.7069	−2.58884
$685$	0	0
$686$	0.815390	0.0311317
$687$	−2.84398	−0.108505
$688$	0.246766	0.00940788
$689$	−14.7438	−0.561694
$690$	0	0
$691$	−16.9378	−0.644344	−0.322172	−	0.946681i	$-0.604413\pi$
$691$	−16.9378	−0.644344	−0.322172	+	0.946681i	$0.604413\pi$
$692$	−8.84149	−0.336103
$693$	7.64680	0.290478
$694$	3.88378	0.147426
$695$	0	0
$696$	56.0256	2.12365
$697$	36.8266	1.39491
$698$	−8.44598	−0.319685
$699$	−41.6100	−1.57384
$700$	0	0
$701$	−10.7466	−0.405892	−0.202946	−	0.979190i	$-0.565052\pi$
$701$	−10.7466	−0.405892	−0.202946	+	0.979190i	$0.565052\pi$
$702$	49.8155	1.88017
$703$	−65.2992	−2.46281
$704$	−3.83016	−0.144354
$705$	0	0
$706$	−1.08106	−0.0406862
$707$	−1.54184	−0.0579868
$708$	17.5148	0.658247
$709$	24.9141	0.935669	0.467835	−	0.883816i	$-0.345034\pi$
$709$	24.9141	0.935669	0.467835	+	0.883816i	$0.345034\pi$
$710$	0	0
$711$	−62.0818	−2.32825
$712$	21.0361	0.788362
$713$	8.68298	0.325180
$714$	20.6854	0.774131
$715$	0	0
$716$	0.292789	0.0109420
$717$	52.1239	1.94660
$718$	−13.2237	−0.493504
$719$	−16.9111	−0.630677	−0.315338	−	0.948979i	$-0.602118\pi$
$719$	−16.9111	−0.630677	−0.315338	+	0.948979i	$0.602118\pi$
$720$	0	0
$721$	6.39290	0.238084
$722$	−20.3683	−0.758028
$723$	−33.8182	−1.25771
$724$	−0.981949	−0.0364938
$725$	0	0
$726$	2.66057	0.0987431
$727$	19.5713	0.725861	0.362930	−	0.931816i	$-0.381776\pi$
$727$	19.5713	0.725861	0.362930	+	0.931816i	$0.381776\pi$
$728$	−10.9576	−0.406115
$729$	54.4734	2.01753
$730$	0	0
$731$	−4.23640	−0.156689
$732$	−22.6318	−0.836494
$733$	−33.6072	−1.24131	−0.620654	−	0.784084i	$-0.713133\pi$
$733$	−33.6072	−1.24131	−0.620654	+	0.784084i	$0.713133\pi$
$734$	−26.3426	−0.972324
$735$	0	0
$736$	−7.50809	−0.276752
$737$	−11.8802	−0.437614
$738$	29.5337	1.08715
$739$	10.9017	0.401024	0.200512	−	0.979691i	$-0.435739\pi$
$739$	10.9017	0.401024	0.200512	+	0.979691i	$0.435739\pi$
$740$	0	0
$741$	−87.1910	−3.20304
$742$	−2.98359	−0.109531
$743$	4.54503	0.166741	0.0833705	−	0.996519i	$-0.473432\pi$
$743$	4.54503	0.166741	0.0833705	+	0.996519i	$0.473432\pi$
$744$	−59.6027	−2.18514
$745$	0	0
$746$	−0.568403	−0.0208107
$747$	44.0430	1.61145
$748$	−10.3804	−0.379546
$749$	0.521233	0.0190454
$750$	0	0
$751$	−2.67127	−0.0974761	−0.0487381	−	0.998812i	$-0.515520\pi$
$751$	−2.67127	−0.0974761	−0.0487381	+	0.998812i	$0.515520\pi$
$752$	1.20491	0.0439384
$753$	−68.8233	−2.50806
$754$	20.7443	0.755462
$755$	0	0
$756$	−20.2437	−0.736257
$757$	−41.8574	−1.52133	−0.760667	−	0.649142i	$-0.775128\pi$
$757$	−41.8574	−1.52133	−0.760667	+	0.649142i	$0.775128\pi$
$758$	11.2554	0.408814
$759$	4.21795	0.153102
$760$	0	0
$761$	−18.7435	−0.679452	−0.339726	−	0.940524i	$-0.610334\pi$
$761$	−18.7435	−0.679452	−0.339726	+	0.940524i	$0.610334\pi$
$762$	15.7781	0.571578
$763$	10.7962	0.390848
$764$	−14.3721	−0.519966
$765$	0	0
$766$	8.50451	0.307280
$767$	16.1996	0.584934
$768$	47.5920	1.71733
$769$	32.5398	1.17342	0.586708	−	0.809799i	$-0.300424\pi$
$769$	32.5398	1.17342	0.586708	+	0.809799i	$0.300424\pi$
$770$	0	0
$771$	71.3929	2.57115
$772$	−9.07385	−0.326575
$773$	17.5454	0.631065	0.315532	−	0.948915i	$-0.397817\pi$
$773$	17.5454	0.631065	0.315532	+	0.948915i	$0.397817\pi$
$774$	−3.39745	−0.122119
$775$	0	0
$776$	46.0719	1.65389
$777$	−32.1286	−1.15261
$778$	−7.13400	−0.255766
$779$	−31.4123	−1.12546
$780$	0	0
$781$	13.4894	0.482689
$782$	8.19494	0.293050
$783$	95.7331	3.42122
$784$	0.452875	0.0161741
$785$	0	0
$786$	14.6154	0.521313
$787$	15.1921	0.541539	0.270770	−	0.962644i	$-0.412722\pi$
$787$	15.1921	0.541539	0.270770	+	0.962644i	$0.412722\pi$
$788$	6.09214	0.217024
$789$	0.872814	0.0310730
$790$	0	0
$791$	−19.1949	−0.682492
$792$	−20.7950	−0.738919
$793$	−20.9323	−0.743329
$794$	−6.81338	−0.241798
$795$	0	0
$796$	21.1035	0.747993
$797$	7.73366	0.273940	0.136970	−	0.990575i	$-0.456264\pi$
$797$	7.73366	0.273940	0.136970	+	0.990575i	$0.456264\pi$
$798$	−17.6442	−0.624597
$799$	−20.6854	−0.731796
$800$	0	0
$801$	59.1516	2.09002
$802$	3.17759	0.112204
$803$	2.60702	0.0919999
$804$	51.7561	1.82530
$805$	0	0
$806$	−22.0687	−0.777338
$807$	−44.1872	−1.55546
$808$	4.19294	0.147507
$809$	−1.13534	−0.0399163	−0.0199582	−	0.999801i	$-0.506353\pi$
$809$	−1.13534	−0.0399163	−0.0199582	+	0.999801i	$0.506353\pi$
$810$	0	0
$811$	−55.9646	−1.96518	−0.982592	−	0.185775i	$-0.940520\pi$
$811$	−55.9646	−1.96518	−0.982592	+	0.185775i	$0.940520\pi$
$812$	−8.42994	−0.295833
$813$	8.22499	0.288463
$814$	−8.02873	−0.281407
$815$	0	0
$816$	11.4888	0.402190
$817$	3.61355	0.126422
$818$	−7.13293	−0.249397
$819$	−30.8117	−1.07665
$820$	0	0
$821$	−38.9449	−1.35919	−0.679593	−	0.733590i	$-0.737843\pi$
$821$	−38.9449	−1.35919	−0.679593	+	0.733590i	$0.737843\pi$
$822$	−32.9157	−1.14807
$823$	−30.6608	−1.06877	−0.534385	−	0.845241i	$-0.679457\pi$
$823$	−30.6608	−1.06877	−0.534385	+	0.845241i	$0.679457\pi$
$824$	−17.3851	−0.605639
$825$	0	0
$826$	3.27819	0.114063
$827$	−16.4794	−0.573047	−0.286523	−	0.958073i	$-0.592500\pi$
$827$	−16.4794	−0.573047	−0.286523	+	0.958073i	$0.592500\pi$
$828$	−13.1977	−0.458652
$829$	24.4987	0.850876	0.425438	−	0.904988i	$-0.360120\pi$
$829$	24.4987	0.850876	0.425438	+	0.904988i	$0.360120\pi$
$830$	0	0
$831$	−66.0892	−2.29261
$832$	15.4331	0.535045
$833$	−7.77478	−0.269380
$834$	55.1588	1.90999
$835$	0	0
$836$	8.85428	0.306232
$837$	−101.845	−3.52029
$838$	19.7226	0.681306
$839$	−13.1212	−0.452993	−0.226497	−	0.974012i	$-0.572727\pi$
$839$	−13.1212	−0.452993	−0.226497	+	0.974012i	$0.572727\pi$
$840$	0	0
$841$	10.8654	0.374668
$842$	−26.6368	−0.917964
$843$	−8.63335	−0.297348
$844$	2.92998	0.100854
$845$	0	0
$846$	−16.5890	−0.570342
$847$	−1.00000	−0.0343604
$848$	−1.65711	−0.0569055
$849$	−32.1855	−1.10460
$850$	0	0
$851$	−12.7284	−0.436324
$852$	−58.7663	−2.01330
$853$	46.6526	1.59736	0.798678	−	0.601759i	$-0.205533\pi$
$853$	46.6526	1.59736	0.798678	+	0.601759i	$0.205533\pi$
$854$	−4.23592	−0.144950
$855$	0	0
$856$	−1.41746	−0.0484478
$857$	43.3400	1.48046	0.740232	−	0.672351i	$-0.234716\pi$
$857$	43.3400	1.48046	0.740232	+	0.672351i	$0.234716\pi$
$858$	−10.7204	−0.365988
$859$	38.8066	1.32406	0.662031	−	0.749476i	$-0.269695\pi$
$859$	38.8066	1.32406	0.662031	+	0.749476i	$0.269695\pi$
$860$	0	0
$861$	−15.4555	−0.526722
$862$	9.72821	0.331344
$863$	29.3228	0.998159	0.499079	−	0.866556i	$-0.333672\pi$
$863$	29.3228	0.998159	0.499079	+	0.866556i	$0.333672\pi$
$864$	88.0647	2.99602
$865$	0	0
$866$	−12.7912	−0.434663
$867$	−141.766	−4.81463
$868$	8.96816	0.304399
$869$	8.11865	0.275406
$870$	0	0
$871$	47.8697	1.62200
$872$	−29.3596	−0.994240
$873$	129.550	4.38460
$874$	−6.99011	−0.236444
$875$	0	0
$876$	−11.3575	−0.383733
$877$	−29.1181	−0.983249	−0.491624	−	0.870807i	$-0.663597\pi$
$877$	−29.1181	−0.983249	−0.491624	+	0.870807i	$0.663597\pi$
$878$	8.02764	0.270920
$879$	−94.7620	−3.19624
$880$	0	0
$881$	40.9563	1.37985	0.689927	−	0.723879i	$-0.257643\pi$
$881$	40.9563	1.37985	0.689927	+	0.723879i	$0.257643\pi$
$882$	−6.23513	−0.209948
$883$	15.7921	0.531446	0.265723	−	0.964049i	$-0.414389\pi$
$883$	15.7921	0.531446	0.265723	+	0.964049i	$0.414389\pi$
$884$	41.8264	1.40677
$885$	0	0
$886$	−23.8807	−0.802287
$887$	−0.283468	−0.00951794	−0.00475897	−	0.999989i	$-0.501515\pi$
$887$	−0.283468	−0.00951794	−0.00475897	+	0.999989i	$0.501515\pi$
$888$	87.3717	2.93200
$889$	−5.93032	−0.198897
$890$	0	0
$891$	−26.5332	−0.888896
$892$	21.3788	0.715814
$893$	17.6442	0.590440
$894$	−54.6672	−1.82834
$895$	0	0
$896$	−8.49322	−0.283739
$897$	−16.9956	−0.567467
$898$	−7.49094	−0.249976
$899$	−42.4106	−1.41447
$900$	0	0
$901$	28.4487	0.947764
$902$	−3.86223	−0.128598
$903$	1.77794	0.0591662
$904$	52.1993	1.73612
$905$	0	0
$906$	5.66556	0.188225
$907$	16.4371	0.545784	0.272892	−	0.962045i	$-0.412020\pi$
$907$	16.4371	0.545784	0.272892	+	0.962045i	$0.412020\pi$
$908$	−12.9803	−0.430767
$909$	11.7901	0.391054
$910$	0	0
$911$	−4.21284	−0.139578	−0.0697888	−	0.997562i	$-0.522233\pi$
$911$	−4.21284	−0.139578	−0.0697888	+	0.997562i	$0.522233\pi$
$912$	−9.79973	−0.324501
$913$	−5.75966	−0.190617
$914$	−29.3403	−0.970489
$915$	0	0
$916$	−1.16371	−0.0384500
$917$	−5.49332	−0.181405
$918$	−96.1209	−3.17246
$919$	−28.9593	−0.955281	−0.477640	−	0.878556i	$-0.658508\pi$
$919$	−28.9593	−0.955281	−0.477640	+	0.878556i	$0.658508\pi$
$920$	0	0
$921$	−17.3930	−0.573120
$922$	−26.3753	−0.868623
$923$	−54.3535	−1.78907
$924$	4.35648	0.143318
$925$	0	0
$926$	2.80310	0.0921156
$927$	−48.8853	−1.60560
$928$	36.6721	1.20382
$929$	6.15879	0.202063	0.101032	−	0.994883i	$-0.467786\pi$
$929$	6.15879	0.202063	0.101032	+	0.994883i	$0.467786\pi$
$930$	0	0
$931$	6.63172	0.217346
$932$	−17.0261	−0.557708
$933$	33.2777	1.08946
$934$	6.75524	0.221038
$935$	0	0
$936$	83.7905	2.73878
$937$	−22.0888	−0.721608	−0.360804	−	0.932642i	$-0.617498\pi$
$937$	−22.0888	−0.721608	−0.360804	+	0.932642i	$0.617498\pi$
$938$	9.68703	0.316293
$939$	62.7582	2.04804
$940$	0	0
$941$	−37.6613	−1.22772	−0.613861	−	0.789414i	$-0.710384\pi$
$941$	−37.6613	−1.22772	−0.613861	+	0.789414i	$0.710384\pi$
$942$	−31.2964	−1.01969
$943$	−6.12301	−0.199393
$944$	1.82074	0.0592599
$945$	0	0
$946$	0.444297	0.0144453
$947$	8.68278	0.282153	0.141076	−	0.989999i	$-0.454944\pi$
$947$	8.68278	0.282153	0.141076	+	0.989999i	$0.454944\pi$
$948$	−35.3688	−1.14872
$949$	−10.5046	−0.340994
$950$	0	0
$951$	3.62158	0.117438
$952$	21.1431	0.685250
$953$	−41.5075	−1.34456	−0.672280	−	0.740297i	$-0.734685\pi$
$953$	−41.5075	−1.34456	−0.672280	+	0.740297i	$0.734685\pi$
$954$	22.8150	0.738662
$955$	0	0
$956$	21.3282	0.689802
$957$	−20.6019	−0.665965
$958$	0.841174	0.0271771
$959$	12.3717	0.399502
$960$	0	0
$961$	14.1184	0.455432
$962$	32.3506	1.04303
$963$	−3.98577	−0.128440
$964$	−13.8378	−0.445685
$965$	0	0
$966$	−3.43928	−0.110657
$967$	−15.1795	−0.488141	−0.244071	−	0.969757i	$-0.578483\pi$
$967$	−15.1795	−0.488141	−0.244071	+	0.969757i	$0.578483\pi$
$968$	2.71944	0.0874061
$969$	168.238	5.40459
$970$	0	0
$971$	24.5127	0.786651	0.393326	−	0.919399i	$-0.371325\pi$
$971$	24.5127	0.786651	0.393326	+	0.919399i	$0.371325\pi$
$972$	54.8603	1.75965
$973$	−20.7319	−0.664636
$974$	2.26011	0.0724186
$975$	0	0
$976$	−2.35267	−0.0753070
$977$	44.7550	1.43184	0.715920	−	0.698182i	$-0.246008\pi$
$977$	44.7550	1.43184	0.715920	+	0.698182i	$0.246008\pi$
$978$	31.4119	1.00444
$979$	−7.73546	−0.247227
$980$	0	0
$981$	−82.5563	−2.63582
$982$	7.43861	0.237376
$983$	−45.5363	−1.45238	−0.726191	−	0.687493i	$-0.758711\pi$
$983$	−45.5363	−1.45238	−0.726191	+	0.687493i	$0.758711\pi$
$984$	42.0303	1.33988
$985$	0	0
$986$	−40.0269	−1.27472
$987$	8.68130	0.276329
$988$	−35.6770	−1.13504
$989$	0.704369	0.0223976
$990$	0	0
$991$	40.2154	1.27749	0.638743	−	0.769421i	$-0.279455\pi$
$991$	40.2154	1.27749	0.638743	+	0.769421i	$0.279455\pi$
$992$	−39.0135	−1.23868
$993$	−93.0765	−2.95369
$994$	−10.9991	−0.348871
$995$	0	0
$996$	25.0919	0.795066
$997$	−60.0595	−1.90210	−0.951051	−	0.309033i	$-0.899995\pi$
$997$	−60.0595	−1.90210	−0.951051	+	0.309033i	$0.899995\pi$
$998$	4.12339	0.130524
$999$	149.295	4.72349

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.ba.1.3	✓	7
5.2	odd	4		1925.2.b.q.1849.6		14
5.3	odd	4		1925.2.b.q.1849.9		14
5.4	even	2		1925.2.a.bc.1.5	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.ba.1.3	✓	7	1.1	even	1	trivial
1925.2.a.bc.1.5	yes	7	5.4	even	2
1925.2.b.q.1849.6		14	5.2	odd	4
1925.2.b.q.1849.9		14	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.815390 q^{2} -3.26294 q^{3} -1.33514 q^{4} +2.66057 q^{6} -1.00000 q^{7} +2.71944 q^{8} +7.64680 q^{9} -1.00000 q^{11} +4.35648 q^{12} +4.02935 q^{13} +0.815390 q^{14} +0.452875 q^{16} -7.77478 q^{17} -6.23513 q^{18} +6.63172 q^{19} +3.26294 q^{21} +0.815390 q^{22} +1.29268 q^{23} -8.87338 q^{24} -3.28549 q^{26} -15.1623 q^{27} +1.33514 q^{28} -6.31390 q^{29} +6.71702 q^{31} -5.80815 q^{32} +3.26294 q^{33} +6.33948 q^{34} -10.2095 q^{36} -9.84650 q^{37} -5.40744 q^{38} -13.1476 q^{39} -4.73667 q^{41} -2.66057 q^{42} +0.544889 q^{43} +1.33514 q^{44} -1.05404 q^{46} +2.66057 q^{47} -1.47770 q^{48} +1.00000 q^{49} +25.3687 q^{51} -5.37975 q^{52} -3.65910 q^{53} +12.3632 q^{54} -2.71944 q^{56} -21.6389 q^{57} +5.14829 q^{58} +4.02040 q^{59} -5.19496 q^{61} -5.47699 q^{62} -7.64680 q^{63} +3.83016 q^{64} -2.66057 q^{66} +11.8802 q^{67} +10.3804 q^{68} -4.21795 q^{69} -13.4894 q^{71} +20.7950 q^{72} -2.60702 q^{73} +8.02873 q^{74} -8.85428 q^{76} +1.00000 q^{77} +10.7204 q^{78} -8.11865 q^{79} +26.5332 q^{81} +3.86223 q^{82} +5.75966 q^{83} -4.35648 q^{84} -0.444297 q^{86} +20.6019 q^{87} -2.71944 q^{88} +7.73546 q^{89} -4.02935 q^{91} -1.72591 q^{92} -21.9173 q^{93} -2.16940 q^{94} +18.9517 q^{96} +16.9417 q^{97} -0.815390 q^{98} -7.64680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9} - 7 q^{11} + 21 q^{12} + 3 q^{13} + q^{14} + 29 q^{16} - 14 q^{18} + 18 q^{19} + q^{22} + 7 q^{23} - 22 q^{24} + 13 q^{26} - 6 q^{27} - 13 q^{28}+ \cdots - 13 q^{99}+O(q^{100}) \) 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9 - 7 * q^11 + 21 * q^12 + 3 * q^13 + q^14 + 29 * q^16 - 14 * q^18 + 18 * q^19 + q^22 + 7 * q^23 - 22 * q^24 + 13 * q^26 - 6 * q^27 - 13 * q^28 - 2 * q^29 + 24 * q^31 - 33 * q^32 + 33 * q^34 + 44 * q^36 - 21 * q^37 - 9 * q^38 + 10 * q^39 - 10 * q^41 + q^42 + 2 * q^43 - 13 * q^44 + 3 * q^46 - q^47 + 77 * q^48 + 7 * q^49 + 29 * q^51 - 9 * q^52 + 11 * q^53 - 47 * q^54 + 6 * q^56 - 7 * q^57 + 33 * q^58 + q^59 + 28 * q^61 - 16 * q^62 - 13 * q^63 + 48 * q^64 + q^66 + 46 * q^68 + 33 * q^69 - 10 * q^71 - 36 * q^72 + 11 * q^73 - 6 * q^74 + 20 * q^76 + 7 * q^77 + 31 * q^78 + 19 * q^79 + 27 * q^81 - 8 * q^82 + 19 * q^83 - 21 * q^84 + 55 * q^86 + 12 * q^87 + 6 * q^88 - 2 * q^89 - 3 * q^91 - 40 * q^92 - 58 * q^93 + 21 * q^94 - 23 * q^96 - 8 * q^97 - q^98 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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