LMFDB - Embedded newform 5225.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,-4,2,0,4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.41421 q^{2} -2.00000 q^{3} +3.82843 q^{4} +4.82843 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} -1.00000 q^{11} -7.65685 q^{12} -6.82843 q^{13} -6.82843 q^{14} +3.00000 q^{16} +4.82843 q^{17} -2.41421 q^{18} -1.00000 q^{19} -5.65685 q^{21} +2.41421 q^{22} -4.00000 q^{23} +8.82843 q^{24} +16.4853 q^{26} +4.00000 q^{27} +10.8284 q^{28} +3.17157 q^{29} +1.17157 q^{31} +1.58579 q^{32} +2.00000 q^{33} -11.6569 q^{34} +3.82843 q^{36} +1.17157 q^{37} +2.41421 q^{38} +13.6569 q^{39} -8.82843 q^{41} +13.6569 q^{42} +6.82843 q^{43} -3.82843 q^{44} +9.65685 q^{46} -6.00000 q^{48} +1.00000 q^{49} -9.65685 q^{51} -26.1421 q^{52} +2.82843 q^{53} -9.65685 q^{54} -12.4853 q^{56} +2.00000 q^{57} -7.65685 q^{58} +12.4853 q^{59} -3.65685 q^{61} -2.82843 q^{62} +2.82843 q^{63} -9.82843 q^{64} -4.82843 q^{66} -0.343146 q^{67} +18.4853 q^{68} +8.00000 q^{69} +9.17157 q^{71} -4.41421 q^{72} +10.4853 q^{73} -2.82843 q^{74} -3.82843 q^{76} -2.82843 q^{77} -32.9706 q^{78} -12.0000 q^{79} -11.0000 q^{81} +21.3137 q^{82} -2.82843 q^{83} -21.6569 q^{84} -16.4853 q^{86} -6.34315 q^{87} +4.41421 q^{88} -13.3137 q^{89} -19.3137 q^{91} -15.3137 q^{92} -2.34315 q^{93} -3.17157 q^{96} -10.1421 q^{97} -2.41421 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} - 6 q^{8} + 2 q^{9} - 2 q^{11} - 4 q^{12} - 8 q^{13} - 8 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} - 2 q^{19} + 2 q^{22} - 8 q^{23} + 12 q^{24} + 16 q^{26} + 8 q^{27}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 - 6 * q^8 + 2 * q^9 - 2 * q^11 - 4 * q^12 - 8 * q^13 - 8 * q^14 + 6 * q^16 + 4 * q^17 - 2 * q^18 - 2 * q^19 + 2 * q^22 - 8 * q^23 + 12 * q^24 + 16 * q^26 + 8 * q^27 + 16 * q^28 + 12 * q^29 + 8 * q^31 + 6 * q^32 + 4 * q^33 - 12 * q^34 + 2 * q^36 + 8 * q^37 + 2 * q^38 + 16 * q^39 - 12 * q^41 + 16 * q^42 + 8 * q^43 - 2 * q^44 + 8 * q^46 - 12 * q^48 + 2 * q^49 - 8 * q^51 - 24 * q^52 - 8 * q^54 - 8 * q^56 + 4 * q^57 - 4 * q^58 + 8 * q^59 + 4 * q^61 - 14 * q^64 - 4 * q^66 - 12 * q^67 + 20 * q^68 + 16 * q^69 + 24 * q^71 - 6 * q^72 + 4 * q^73 - 2 * q^76 - 32 * q^78 - 24 * q^79 - 22 * q^81 + 20 * q^82 - 32 * q^84 - 16 * q^86 - 24 * q^87 + 6 * q^88 - 4 * q^89 - 16 * q^91 - 8 * q^92 - 16 * q^93 - 12 * q^96 + 8 * q^97 - 2 * q^98 - 2 * 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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	4.82843	1.97120
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	−4.41421	−1.56066
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−7.65685	−2.21034
$13$	−6.82843	−1.89386	−0.946932	−	0.321433i	$-0.895836\pi$
$13$	−6.82843	−1.89386	−0.946932	+	0.321433i	$0.895836\pi$
$14$	−6.82843	−1.82497
$15$	0	0
$16$	3.00000	0.750000
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	−2.41421	−0.569036
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−5.65685	−1.23443
$22$	2.41421	0.514712
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	8.82843	1.80210
$25$	0	0
$26$	16.4853	3.23303
$27$	4.00000	0.769800
$28$	10.8284	2.04638
$29$	3.17157	0.588946	0.294473	−	0.955660i	$-0.404856\pi$
$29$	3.17157	0.588946	0.294473	+	0.955660i	$0.404856\pi$
$30$	0	0
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	1.58579	0.280330
$33$	2.00000	0.348155
$34$	−11.6569	−1.99913
$35$	0	0
$36$	3.82843	0.638071
$37$	1.17157	0.192605	0.0963027	−	0.995352i	$-0.469298\pi$
$37$	1.17157	0.192605	0.0963027	+	0.995352i	$0.469298\pi$
$38$	2.41421	0.391637
$39$	13.6569	2.18685
$40$	0	0
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	13.6569	2.10730
$43$	6.82843	1.04133	0.520663	−	0.853762i	$-0.325685\pi$
$43$	6.82843	1.04133	0.520663	+	0.853762i	$0.325685\pi$
$44$	−3.82843	−0.577157
$45$	0	0
$46$	9.65685	1.42383
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−6.00000	−0.866025
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.65685	−1.35223
$52$	−26.1421	−3.62526
$53$	2.82843	0.388514	0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843	0.388514	0.194257	+	0.980951i	$0.437770\pi$
$54$	−9.65685	−1.31413
$55$	0	0
$56$	−12.4853	−1.66842
$57$	2.00000	0.264906
$58$	−7.65685	−1.00539
$59$	12.4853	1.62545	0.812723	−	0.582651i	$-0.197984\pi$
$59$	12.4853	1.62545	0.812723	+	0.582651i	$0.197984\pi$
$60$	0	0
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	−2.82843	−0.359211
$63$	2.82843	0.356348
$64$	−9.82843	−1.22855
$65$	0	0
$66$	−4.82843	−0.594338
$67$	−0.343146	−0.0419219	−0.0209610	−	0.999780i	$-0.506673\pi$
$67$	−0.343146	−0.0419219	−0.0209610	+	0.999780i	$0.506673\pi$
$68$	18.4853	2.24167
$69$	8.00000	0.963087
$70$	0	0
$71$	9.17157	1.08847	0.544233	−	0.838934i	$-0.316821\pi$
$71$	9.17157	1.08847	0.544233	+	0.838934i	$0.316821\pi$
$72$	−4.41421	−0.520220
$73$	10.4853	1.22721	0.613605	−	0.789613i	$-0.289719\pi$
$73$	10.4853	1.22721	0.613605	+	0.789613i	$0.289719\pi$
$74$	−2.82843	−0.328798
$75$	0	0
$76$	−3.82843	−0.439151
$77$	−2.82843	−0.322329
$78$	−32.9706	−3.73318
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	21.3137	2.35371
$83$	−2.82843	−0.310460	−0.155230	−	0.987878i	$-0.549612\pi$
$83$	−2.82843	−0.310460	−0.155230	+	0.987878i	$0.549612\pi$
$84$	−21.6569	−2.36296
$85$	0	0
$86$	−16.4853	−1.77765
$87$	−6.34315	−0.680057
$88$	4.41421	0.470557
$89$	−13.3137	−1.41125	−0.705625	−	0.708585i	$-0.749334\pi$
$89$	−13.3137	−1.41125	−0.705625	+	0.708585i	$0.749334\pi$
$90$	0	0
$91$	−19.3137	−2.02463
$92$	−15.3137	−1.59656
$93$	−2.34315	−0.242973
$94$	0	0
$95$	0	0
$96$	−3.17157	−0.323697
$97$	−10.1421	−1.02978	−0.514889	−	0.857257i	$-0.672167\pi$
$97$	−10.1421	−1.02978	−0.514889	+	0.857257i	$0.672167\pi$
$98$	−2.41421	−0.243872
$99$	−1.00000	−0.100504
$100$	0	0
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	23.3137	2.30840
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	30.1421	2.95568
$105$	0	0
$106$	−6.82843	−0.663235
$107$	−13.3137	−1.28708	−0.643542	−	0.765410i	$-0.722536\pi$
$107$	−13.3137	−1.28708	−0.643542	+	0.765410i	$0.722536\pi$
$108$	15.3137	1.47356
$109$	−0.828427	−0.0793489	−0.0396745	−	0.999213i	$-0.512632\pi$
$109$	−0.828427	−0.0793489	−0.0396745	+	0.999213i	$0.512632\pi$
$110$	0	0
$111$	−2.34315	−0.222402
$112$	8.48528	0.801784
$113$	12.4853	1.17452	0.587258	−	0.809400i	$-0.300207\pi$
$113$	12.4853	1.17452	0.587258	+	0.809400i	$0.300207\pi$
$114$	−4.82843	−0.452224
$115$	0	0
$116$	12.1421	1.12737
$117$	−6.82843	−0.631288
$118$	−30.1421	−2.77481
$119$	13.6569	1.25192
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.82843	0.799288
$123$	17.6569	1.59206
$124$	4.48528	0.402790
$125$	0	0
$126$	−6.82843	−0.608325
$127$	−14.9706	−1.32842	−0.664211	−	0.747545i	$-0.731232\pi$
$127$	−14.9706	−1.32842	−0.664211	+	0.747545i	$0.731232\pi$
$128$	20.5563	1.81694
$129$	−13.6569	−1.20242
$130$	0	0
$131$	9.65685	0.843723	0.421862	−	0.906660i	$-0.361377\pi$
$131$	9.65685	0.843723	0.421862	+	0.906660i	$0.361377\pi$
$132$	7.65685	0.666444
$133$	−2.82843	−0.245256
$134$	0.828427	0.0715652
$135$	0	0
$136$	−21.3137	−1.82764
$137$	−9.31371	−0.795724	−0.397862	−	0.917445i	$-0.630248\pi$
$137$	−9.31371	−0.795724	−0.397862	+	0.917445i	$0.630248\pi$
$138$	−19.3137	−1.64409
$139$	20.9706	1.77870	0.889350	−	0.457227i	$-0.151157\pi$
$139$	20.9706	1.77870	0.889350	+	0.457227i	$0.151157\pi$
$140$	0	0
$141$	0	0
$142$	−22.1421	−1.85813
$143$	6.82843	0.571022
$144$	3.00000	0.250000
$145$	0	0
$146$	−25.3137	−2.09498
$147$	−2.00000	−0.164957
$148$	4.48528	0.368688
$149$	7.65685	0.627274	0.313637	−	0.949543i	$-0.398453\pi$
$149$	7.65685	0.627274	0.313637	+	0.949543i	$0.398453\pi$
$150$	0	0
$151$	9.65685	0.785864	0.392932	−	0.919568i	$-0.371461\pi$
$151$	9.65685	0.785864	0.392932	+	0.919568i	$0.371461\pi$
$152$	4.41421	0.358040
$153$	4.82843	0.390355
$154$	6.82843	0.550250
$155$	0	0
$156$	52.2843	4.18609
$157$	17.3137	1.38178	0.690892	−	0.722958i	$-0.257218\pi$
$157$	17.3137	1.38178	0.690892	+	0.722958i	$0.257218\pi$
$158$	28.9706	2.30477
$159$	−5.65685	−0.448618
$160$	0	0
$161$	−11.3137	−0.891645
$162$	26.5563	2.08646
$163$	−17.6569	−1.38299	−0.691496	−	0.722380i	$-0.743048\pi$
$163$	−17.6569	−1.38299	−0.691496	+	0.722380i	$0.743048\pi$
$164$	−33.7990	−2.63926
$165$	0	0
$166$	6.82843	0.529989
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	24.9706	1.92652
$169$	33.6274	2.58672
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	26.1421	1.99332
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	15.3137	1.16093
$175$	0	0
$176$	−3.00000	−0.226134
$177$	−24.9706	−1.87690
$178$	32.1421	2.40915
$179$	12.4853	0.933194	0.466597	−	0.884470i	$-0.345480\pi$
$179$	12.4853	0.933194	0.466597	+	0.884470i	$0.345480\pi$
$180$	0	0
$181$	−5.31371	−0.394965	−0.197482	−	0.980306i	$-0.563277\pi$
$181$	−5.31371	−0.394965	−0.197482	+	0.980306i	$0.563277\pi$
$182$	46.6274	3.45625
$183$	7.31371	0.540645
$184$	17.6569	1.30168
$185$	0	0
$186$	5.65685	0.414781
$187$	−4.82843	−0.353090
$188$	0	0
$189$	11.3137	0.822951
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	19.6569	1.41861
$193$	−15.7990	−1.13724	−0.568618	−	0.822602i	$-0.692522\pi$
$193$	−15.7990	−1.13724	−0.568618	+	0.822602i	$0.692522\pi$
$194$	24.4853	1.75794
$195$	0	0
$196$	3.82843	0.273459
$197$	13.7990	0.983137	0.491569	−	0.870839i	$-0.336424\pi$
$197$	13.7990	0.983137	0.491569	+	0.870839i	$0.336424\pi$
$198$	2.41421	0.171571
$199$	2.34315	0.166101	0.0830506	−	0.996545i	$-0.473534\pi$
$199$	2.34315	0.166101	0.0830506	+	0.996545i	$0.473534\pi$
$200$	0	0
$201$	0.686292	0.0484073
$202$	−18.4853	−1.30062
$203$	8.97056	0.629610
$204$	−36.9706	−2.58846
$205$	0	0
$206$	24.1421	1.68206
$207$	−4.00000	−0.278019
$208$	−20.4853	−1.42040
$209$	1.00000	0.0691714
$210$	0	0
$211$	−24.0000	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$211$	−24.0000	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$212$	10.8284	0.743699
$213$	−18.3431	−1.25685
$214$	32.1421	2.19719
$215$	0	0
$216$	−17.6569	−1.20140
$217$	3.31371	0.224949
$218$	2.00000	0.135457
$219$	−20.9706	−1.41706
$220$	0	0
$221$	−32.9706	−2.21784
$222$	5.65685	0.379663
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	4.48528	0.299685
$225$	0	0
$226$	−30.1421	−2.00503
$227$	−21.3137	−1.41464	−0.707320	−	0.706893i	$-0.750096\pi$
$227$	−21.3137	−1.41464	−0.707320	+	0.706893i	$0.750096\pi$
$228$	7.65685	0.507088
$229$	16.6274	1.09877	0.549385	−	0.835569i	$-0.314862\pi$
$229$	16.6274	1.09877	0.549385	+	0.835569i	$0.314862\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	−14.0000	−0.919145
$233$	−10.4853	−0.686914	−0.343457	−	0.939168i	$-0.611598\pi$
$233$	−10.4853	−0.686914	−0.343457	+	0.939168i	$0.611598\pi$
$234$	16.4853	1.07768
$235$	0	0
$236$	47.7990	3.11145
$237$	24.0000	1.55897
$238$	−32.9706	−2.13716
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−16.8284	−1.08401	−0.542007	−	0.840374i	$-0.682335\pi$
$241$	−16.8284	−1.08401	−0.542007	+	0.840374i	$0.682335\pi$
$242$	−2.41421	−0.155192
$243$	10.0000	0.641500
$244$	−14.0000	−0.896258
$245$	0	0
$246$	−42.6274	−2.71782
$247$	6.82843	0.434482
$248$	−5.17157	−0.328395
$249$	5.65685	0.358489
$250$	0	0
$251$	7.31371	0.461637	0.230819	−	0.972997i	$-0.425860\pi$
$251$	7.31371	0.461637	0.230819	+	0.972997i	$0.425860\pi$
$252$	10.8284	0.682127
$253$	4.00000	0.251478
$254$	36.1421	2.26776
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−1.85786	−0.115890	−0.0579452	−	0.998320i	$-0.518455\pi$
$257$	−1.85786	−0.115890	−0.0579452	+	0.998320i	$0.518455\pi$
$258$	32.9706	2.05266
$259$	3.31371	0.205904
$260$	0	0
$261$	3.17157	0.196315
$262$	−23.3137	−1.44033
$263$	12.4853	0.769875	0.384938	−	0.922943i	$-0.374223\pi$
$263$	12.4853	0.769875	0.384938	+	0.922943i	$0.374223\pi$
$264$	−8.82843	−0.543352
$265$	0	0
$266$	6.82843	0.418678
$267$	26.6274	1.62957
$268$	−1.31371	−0.0802475
$269$	−23.6569	−1.44238	−0.721192	−	0.692735i	$-0.756405\pi$
$269$	−23.6569	−1.44238	−0.721192	+	0.692735i	$0.756405\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	14.4853	0.878299
$273$	38.6274	2.33784
$274$	22.4853	1.35839
$275$	0	0
$276$	30.6274	1.84355
$277$	−8.14214	−0.489214	−0.244607	−	0.969622i	$-0.578659\pi$
$277$	−8.14214	−0.489214	−0.244607	+	0.969622i	$0.578659\pi$
$278$	−50.6274	−3.03643
$279$	1.17157	0.0701402
$280$	0	0
$281$	10.4853	0.625499	0.312750	−	0.949836i	$-0.398750\pi$
$281$	10.4853	0.625499	0.312750	+	0.949836i	$0.398750\pi$
$282$	0	0
$283$	16.4853	0.979948	0.489974	−	0.871737i	$-0.337006\pi$
$283$	16.4853	0.979948	0.489974	+	0.871737i	$0.337006\pi$
$284$	35.1127	2.08356
$285$	0	0
$286$	−16.4853	−0.974795
$287$	−24.9706	−1.47397
$288$	1.58579	0.0934434
$289$	6.31371	0.371395
$290$	0	0
$291$	20.2843	1.18909
$292$	40.1421	2.34914
$293$	5.17157	0.302127	0.151063	−	0.988524i	$-0.451730\pi$
$293$	5.17157	0.302127	0.151063	+	0.988524i	$0.451730\pi$
$294$	4.82843	0.281600
$295$	0	0
$296$	−5.17157	−0.300592
$297$	−4.00000	−0.232104
$298$	−18.4853	−1.07082
$299$	27.3137	1.57959
$300$	0	0
$301$	19.3137	1.11322
$302$	−23.3137	−1.34155
$303$	−15.3137	−0.879750
$304$	−3.00000	−0.172062
$305$	0	0
$306$	−11.6569	−0.666378
$307$	33.3137	1.90131	0.950657	−	0.310244i	$-0.100411\pi$
$307$	33.3137	1.90131	0.950657	+	0.310244i	$0.100411\pi$
$308$	−10.8284	−0.617007
$309$	20.0000	1.13776
$310$	0	0
$311$	13.6569	0.774409	0.387205	−	0.921994i	$-0.373441\pi$
$311$	13.6569	0.774409	0.387205	+	0.921994i	$0.373441\pi$
$312$	−60.2843	−3.41292
$313$	−26.9706	−1.52447	−0.762233	−	0.647303i	$-0.775897\pi$
$313$	−26.9706	−1.52447	−0.762233	+	0.647303i	$0.775897\pi$
$314$	−41.7990	−2.35885
$315$	0	0
$316$	−45.9411	−2.58439
$317$	15.7990	0.887360	0.443680	−	0.896185i	$-0.353673\pi$
$317$	15.7990	0.887360	0.443680	+	0.896185i	$0.353673\pi$
$318$	13.6569	0.765838
$319$	−3.17157	−0.177574
$320$	0	0
$321$	26.6274	1.48620
$322$	27.3137	1.52213
$323$	−4.82843	−0.268661
$324$	−42.1127	−2.33959
$325$	0	0
$326$	42.6274	2.36091
$327$	1.65685	0.0916242
$328$	38.9706	2.15179
$329$	0	0
$330$	0	0
$331$	−23.7990	−1.30811	−0.654055	−	0.756447i	$-0.726934\pi$
$331$	−23.7990	−1.30811	−0.654055	+	0.756447i	$0.726934\pi$
$332$	−10.8284	−0.594287
$333$	1.17157	0.0642018
$334$	−33.7990	−1.84940
$335$	0	0
$336$	−16.9706	−0.925820
$337$	11.5147	0.627247	0.313623	−	0.949547i	$-0.398457\pi$
$337$	11.5147	0.627247	0.313623	+	0.949547i	$0.398457\pi$
$338$	−81.1838	−4.41581
$339$	−24.9706	−1.35621
$340$	0	0
$341$	−1.17157	−0.0634442
$342$	2.41421	0.130546
$343$	−16.9706	−0.916324
$344$	−30.1421	−1.62516
$345$	0	0
$346$	6.82843	0.367099
$347$	31.1127	1.67022	0.835109	−	0.550085i	$-0.185405\pi$
$347$	31.1127	1.67022	0.835109	+	0.550085i	$0.185405\pi$
$348$	−24.2843	−1.30177
$349$	−24.6274	−1.31828	−0.659138	−	0.752022i	$-0.729079\pi$
$349$	−24.6274	−1.31828	−0.659138	+	0.752022i	$0.729079\pi$
$350$	0	0
$351$	−27.3137	−1.45790
$352$	−1.58579	−0.0845227
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	60.2843	3.20407
$355$	0	0
$356$	−50.9706	−2.70143
$357$	−27.3137	−1.44559
$358$	−30.1421	−1.59306
$359$	19.3137	1.01934	0.509669	−	0.860370i	$-0.329768\pi$
$359$	19.3137	1.01934	0.509669	+	0.860370i	$0.329768\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	12.8284	0.674247
$363$	−2.00000	−0.104973
$364$	−73.9411	−3.87557
$365$	0	0
$366$	−17.6569	−0.922939
$367$	−35.3137	−1.84336	−0.921680	−	0.387950i	$-0.873183\pi$
$367$	−35.3137	−1.84336	−0.921680	+	0.387950i	$0.873183\pi$
$368$	−12.0000	−0.625543
$369$	−8.82843	−0.459590
$370$	0	0
$371$	8.00000	0.415339
$372$	−8.97056	−0.465102
$373$	13.1716	0.681998	0.340999	−	0.940064i	$-0.389235\pi$
$373$	13.1716	0.681998	0.340999	+	0.940064i	$0.389235\pi$
$374$	11.6569	0.602762
$375$	0	0
$376$	0	0
$377$	−21.6569	−1.11538
$378$	−27.3137	−1.40487
$379$	−18.8284	−0.967151	−0.483576	−	0.875303i	$-0.660662\pi$
$379$	−18.8284	−0.967151	−0.483576	+	0.875303i	$0.660662\pi$
$380$	0	0
$381$	29.9411	1.53393
$382$	−40.9706	−2.09624
$383$	13.3137	0.680299	0.340149	−	0.940371i	$-0.389522\pi$
$383$	13.3137	0.680299	0.340149	+	0.940371i	$0.389522\pi$
$384$	−41.1127	−2.09802
$385$	0	0
$386$	38.1421	1.94138
$387$	6.82843	0.347108
$388$	−38.8284	−1.97121
$389$	−16.6274	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$390$	0	0
$391$	−19.3137	−0.976736
$392$	−4.41421	−0.222951
$393$	−19.3137	−0.974248
$394$	−33.3137	−1.67832
$395$	0	0
$396$	−3.82843	−0.192386
$397$	−13.3137	−0.668196	−0.334098	−	0.942538i	$-0.608432\pi$
$397$	−13.3137	−0.668196	−0.334098	+	0.942538i	$0.608432\pi$
$398$	−5.65685	−0.283552
$399$	5.65685	0.283197
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−1.65685	−0.0826364
$403$	−8.00000	−0.398508
$404$	29.3137	1.45841
$405$	0	0
$406$	−21.6569	−1.07481
$407$	−1.17157	−0.0580727
$408$	42.6274	2.11037
$409$	−7.17157	−0.354611	−0.177306	−	0.984156i	$-0.556738\pi$
$409$	−7.17157	−0.354611	−0.177306	+	0.984156i	$0.556738\pi$
$410$	0	0
$411$	18.6274	0.918823
$412$	−38.2843	−1.88613
$413$	35.3137	1.73767
$414$	9.65685	0.474608
$415$	0	0
$416$	−10.8284	−0.530907
$417$	−41.9411	−2.05387
$418$	−2.41421	−0.118083
$419$	−12.9706	−0.633653	−0.316827	−	0.948483i	$-0.602617\pi$
$419$	−12.9706	−0.633653	−0.316827	+	0.948483i	$0.602617\pi$
$420$	0	0
$421$	−10.6863	−0.520818	−0.260409	−	0.965498i	$-0.583857\pi$
$421$	−10.6863	−0.520818	−0.260409	+	0.965498i	$0.583857\pi$
$422$	57.9411	2.82053
$423$	0	0
$424$	−12.4853	−0.606339
$425$	0	0
$426$	44.2843	2.14558
$427$	−10.3431	−0.500540
$428$	−50.9706	−2.46376
$429$	−13.6569	−0.659359
$430$	0	0
$431$	−26.6274	−1.28260	−0.641299	−	0.767291i	$-0.721604\pi$
$431$	−26.6274	−1.28260	−0.641299	+	0.767291i	$0.721604\pi$
$432$	12.0000	0.577350
$433$	−37.4558	−1.80001	−0.900006	−	0.435876i	$-0.856438\pi$
$433$	−37.4558	−1.80001	−0.900006	+	0.435876i	$0.856438\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−3.17157	−0.151891
$437$	4.00000	0.191346
$438$	50.6274	2.41907
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	79.5980	3.78609
$443$	−26.3431	−1.25160	−0.625800	−	0.779983i	$-0.715228\pi$
$443$	−26.3431	−1.25160	−0.625800	+	0.779983i	$0.715228\pi$
$444$	−8.97056	−0.425724
$445$	0	0
$446$	4.82843	0.228633
$447$	−15.3137	−0.724314
$448$	−27.7990	−1.31338
$449$	23.6569	1.11644	0.558218	−	0.829694i	$-0.311485\pi$
$449$	23.6569	1.11644	0.558218	+	0.829694i	$0.311485\pi$
$450$	0	0
$451$	8.82843	0.415714
$452$	47.7990	2.24828
$453$	−19.3137	−0.907437
$454$	51.4558	2.41494
$455$	0	0
$456$	−8.82843	−0.413429
$457$	23.4558	1.09722	0.548609	−	0.836079i	$-0.315158\pi$
$457$	23.4558	1.09722	0.548609	+	0.836079i	$0.315158\pi$
$458$	−40.1421	−1.87572
$459$	19.3137	0.901487
$460$	0	0
$461$	38.9706	1.81504	0.907520	−	0.420009i	$-0.137973\pi$
$461$	38.9706	1.81504	0.907520	+	0.420009i	$0.137973\pi$
$462$	−13.6569	−0.635374
$463$	−26.6274	−1.23748	−0.618741	−	0.785595i	$-0.712357\pi$
$463$	−26.6274	−1.23748	−0.618741	+	0.785595i	$0.712357\pi$
$464$	9.51472	0.441710
$465$	0	0
$466$	25.3137	1.17263
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	−26.1421	−1.20842
$469$	−0.970563	−0.0448164
$470$	0	0
$471$	−34.6274	−1.59555
$472$	−55.1127	−2.53677
$473$	−6.82843	−0.313971
$474$	−57.9411	−2.66132
$475$	0	0
$476$	52.2843	2.39645
$477$	2.82843	0.129505
$478$	19.3137	0.883388
$479$	−24.9706	−1.14093	−0.570467	−	0.821320i	$-0.693238\pi$
$479$	−24.9706	−1.14093	−0.570467	+	0.821320i	$0.693238\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	40.6274	1.85053
$483$	22.6274	1.02958
$484$	3.82843	0.174019
$485$	0	0
$486$	−24.1421	−1.09511
$487$	9.31371	0.422044	0.211022	−	0.977481i	$-0.432321\pi$
$487$	9.31371	0.422044	0.211022	+	0.977481i	$0.432321\pi$
$488$	16.1421	0.730720
$489$	35.3137	1.59694
$490$	0	0
$491$	−23.3137	−1.05213	−0.526066	−	0.850443i	$-0.676334\pi$
$491$	−23.3137	−1.05213	−0.526066	+	0.850443i	$0.676334\pi$
$492$	67.5980	3.04755
$493$	15.3137	0.689695
$494$	−16.4853	−0.741708
$495$	0	0
$496$	3.51472	0.157816
$497$	25.9411	1.16362
$498$	−13.6569	−0.611978
$499$	−34.6274	−1.55014	−0.775068	−	0.631878i	$-0.782284\pi$
$499$	−34.6274	−1.55014	−0.775068	+	0.631878i	$0.782284\pi$
$500$	0	0
$501$	−28.0000	−1.25095
$502$	−17.6569	−0.788064
$503$	−7.79899	−0.347740	−0.173870	−	0.984769i	$-0.555627\pi$
$503$	−7.79899	−0.347740	−0.173870	+	0.984769i	$0.555627\pi$
$504$	−12.4853	−0.556139
$505$	0	0
$506$	−9.65685	−0.429300
$507$	−67.2548	−2.98689
$508$	−57.3137	−2.54288
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	29.6569	1.31194
$512$	31.2426	1.38074
$513$	−4.00000	−0.176604
$514$	4.48528	0.197837
$515$	0	0
$516$	−52.2843	−2.30169
$517$	0	0
$518$	−8.00000	−0.351500
$519$	5.65685	0.248308
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−7.65685	−0.335131
$523$	−27.6569	−1.20935	−0.604675	−	0.796472i	$-0.706697\pi$
$523$	−27.6569	−1.20935	−0.604675	+	0.796472i	$0.706697\pi$
$524$	36.9706	1.61507
$525$	0	0
$526$	−30.1421	−1.31426
$527$	5.65685	0.246416
$528$	6.00000	0.261116
$529$	−7.00000	−0.304348
$530$	0	0
$531$	12.4853	0.541815
$532$	−10.8284	−0.469472
$533$	60.2843	2.61120
$534$	−64.2843	−2.78185
$535$	0	0
$536$	1.51472	0.0654259
$537$	−24.9706	−1.07756
$538$	57.1127	2.46230
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−39.9411	−1.71720	−0.858602	−	0.512644i	$-0.828666\pi$
$541$	−39.9411	−1.71720	−0.858602	+	0.512644i	$0.828666\pi$
$542$	57.9411	2.48878
$543$	10.6274	0.456066
$544$	7.65685	0.328285
$545$	0	0
$546$	−93.2548	−3.99094
$547$	34.2843	1.46589	0.732945	−	0.680288i	$-0.238145\pi$
$547$	34.2843	1.46589	0.732945	+	0.680288i	$0.238145\pi$
$548$	−35.6569	−1.52319
$549$	−3.65685	−0.156071
$550$	0	0
$551$	−3.17157	−0.135114
$552$	−35.3137	−1.50305
$553$	−33.9411	−1.44332
$554$	19.6569	0.835140
$555$	0	0
$556$	80.2843	3.40481
$557$	−35.4558	−1.50231	−0.751156	−	0.660125i	$-0.770503\pi$
$557$	−35.4558	−1.50231	−0.751156	+	0.660125i	$0.770503\pi$
$558$	−2.82843	−0.119737
$559$	−46.6274	−1.97213
$560$	0	0
$561$	9.65685	0.407713
$562$	−25.3137	−1.06779
$563$	12.3431	0.520202	0.260101	−	0.965581i	$-0.416244\pi$
$563$	12.3431	0.520202	0.260101	+	0.965581i	$0.416244\pi$
$564$	0	0
$565$	0	0
$566$	−39.7990	−1.67288
$567$	−31.1127	−1.30661
$568$	−40.4853	−1.69872
$569$	20.1421	0.844402	0.422201	−	0.906502i	$-0.361258\pi$
$569$	20.1421	0.844402	0.422201	+	0.906502i	$0.361258\pi$
$570$	0	0
$571$	−30.3431	−1.26982	−0.634911	−	0.772586i	$-0.718963\pi$
$571$	−30.3431	−1.26982	−0.634911	+	0.772586i	$0.718963\pi$
$572$	26.1421	1.09306
$573$	−33.9411	−1.41791
$574$	60.2843	2.51622
$575$	0	0
$576$	−9.82843	−0.409518
$577$	17.3137	0.720779	0.360390	−	0.932802i	$-0.382644\pi$
$577$	17.3137	0.720779	0.360390	+	0.932802i	$0.382644\pi$
$578$	−15.2426	−0.634010
$579$	31.5980	1.31317
$580$	0	0
$581$	−8.00000	−0.331896
$582$	−48.9706	−2.02990
$583$	−2.82843	−0.117141
$584$	−46.2843	−1.91526
$585$	0	0
$586$	−12.4853	−0.515762
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	−7.65685	−0.315763
$589$	−1.17157	−0.0482738
$590$	0	0
$591$	−27.5980	−1.13523
$592$	3.51472	0.144454
$593$	−36.8284	−1.51236	−0.756181	−	0.654362i	$-0.772937\pi$
$593$	−36.8284	−1.51236	−0.756181	+	0.654362i	$0.772937\pi$
$594$	9.65685	0.396226
$595$	0	0
$596$	29.3137	1.20074
$597$	−4.68629	−0.191797
$598$	−65.9411	−2.69653
$599$	−27.7990	−1.13584	−0.567918	−	0.823085i	$-0.692251\pi$
$599$	−27.7990	−1.13584	−0.567918	+	0.823085i	$0.692251\pi$
$600$	0	0
$601$	36.8284	1.50226	0.751131	−	0.660153i	$-0.229508\pi$
$601$	36.8284	1.50226	0.751131	+	0.660153i	$0.229508\pi$
$602$	−46.6274	−1.90039
$603$	−0.343146	−0.0139740
$604$	36.9706	1.50431
$605$	0	0
$606$	36.9706	1.50183
$607$	6.97056	0.282926	0.141463	−	0.989944i	$-0.454819\pi$
$607$	6.97056	0.282926	0.141463	+	0.989944i	$0.454819\pi$
$608$	−1.58579	−0.0643121
$609$	−17.9411	−0.727011
$610$	0	0
$611$	0	0
$612$	18.4853	0.747223
$613$	28.1421	1.13665	0.568325	−	0.822804i	$-0.307592\pi$
$613$	28.1421	1.13665	0.568325	+	0.822804i	$0.307592\pi$
$614$	−80.4264	−3.24575
$615$	0	0
$616$	12.4853	0.503046
$617$	−22.9706	−0.924760	−0.462380	−	0.886682i	$-0.653004\pi$
$617$	−22.9706	−0.924760	−0.462380	+	0.886682i	$0.653004\pi$
$618$	−48.2843	−1.94228
$619$	47.3137	1.90170	0.950849	−	0.309654i	$-0.100213\pi$
$619$	47.3137	1.90170	0.950849	+	0.309654i	$0.100213\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	−32.9706	−1.32200
$623$	−37.6569	−1.50869
$624$	40.9706	1.64014
$625$	0	0
$626$	65.1127	2.60243
$627$	−2.00000	−0.0798723
$628$	66.2843	2.64503
$629$	5.65685	0.225554
$630$	0	0
$631$	−5.65685	−0.225196	−0.112598	−	0.993641i	$-0.535917\pi$
$631$	−5.65685	−0.225196	−0.112598	+	0.993641i	$0.535917\pi$
$632$	52.9706	2.10706
$633$	48.0000	1.90783
$634$	−38.1421	−1.51482
$635$	0	0
$636$	−21.6569	−0.858750
$637$	−6.82843	−0.270552
$638$	7.65685	0.303138
$639$	9.17157	0.362822
$640$	0	0
$641$	−19.9411	−0.787627	−0.393814	−	0.919190i	$-0.628844\pi$
$641$	−19.9411	−0.787627	−0.393814	+	0.919190i	$0.628844\pi$
$642$	−64.2843	−2.53710
$643$	21.6569	0.854063	0.427031	−	0.904237i	$-0.359559\pi$
$643$	21.6569	0.854063	0.427031	+	0.904237i	$0.359559\pi$
$644$	−43.3137	−1.70680
$645$	0	0
$646$	11.6569	0.458633
$647$	−20.9706	−0.824438	−0.412219	−	0.911085i	$-0.635246\pi$
$647$	−20.9706	−0.824438	−0.412219	+	0.911085i	$0.635246\pi$
$648$	48.5563	1.90747
$649$	−12.4853	−0.490090
$650$	0	0
$651$	−6.62742	−0.259749
$652$	−67.5980	−2.64734
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	−26.4853	−1.03408
$657$	10.4853	0.409070
$658$	0	0
$659$	7.02944	0.273828	0.136914	−	0.990583i	$-0.456282\pi$
$659$	7.02944	0.273828	0.136914	+	0.990583i	$0.456282\pi$
$660$	0	0
$661$	42.9706	1.67136	0.835681	−	0.549216i	$-0.185073\pi$
$661$	42.9706	1.67136	0.835681	+	0.549216i	$0.185073\pi$
$662$	57.4558	2.23308
$663$	65.9411	2.56094
$664$	12.4853	0.484523
$665$	0	0
$666$	−2.82843	−0.109599
$667$	−12.6863	−0.491215
$668$	53.5980	2.07377
$669$	4.00000	0.154649
$670$	0	0
$671$	3.65685	0.141171
$672$	−8.97056	−0.346047
$673$	33.4558	1.28963	0.644814	−	0.764340i	$-0.276935\pi$
$673$	33.4558	1.28963	0.644814	+	0.764340i	$0.276935\pi$
$674$	−27.7990	−1.07078
$675$	0	0
$676$	128.740	4.95154
$677$	1.85786	0.0714035	0.0357018	−	0.999362i	$-0.488633\pi$
$677$	1.85786	0.0714035	0.0357018	+	0.999362i	$0.488633\pi$
$678$	60.2843	2.31520
$679$	−28.6863	−1.10088
$680$	0	0
$681$	42.6274	1.63349
$682$	2.82843	0.108306
$683$	5.31371	0.203323	0.101662	−	0.994819i	$-0.467584\pi$
$683$	5.31371	0.203323	0.101662	+	0.994819i	$0.467584\pi$
$684$	−3.82843	−0.146384
$685$	0	0
$686$	40.9706	1.56426
$687$	−33.2548	−1.26875
$688$	20.4853	0.780994
$689$	−19.3137	−0.735794
$690$	0	0
$691$	10.6274	0.404286	0.202143	−	0.979356i	$-0.435209\pi$
$691$	10.6274	0.404286	0.202143	+	0.979356i	$0.435209\pi$
$692$	−10.8284	−0.411635
$693$	−2.82843	−0.107443
$694$	−75.1127	−2.85124
$695$	0	0
$696$	28.0000	1.06134
$697$	−42.6274	−1.61463
$698$	59.4558	2.25044
$699$	20.9706	0.793180
$700$	0	0
$701$	−39.9411	−1.50856	−0.754278	−	0.656555i	$-0.772013\pi$
$701$	−39.9411	−1.50856	−0.754278	+	0.656555i	$0.772013\pi$
$702$	65.9411	2.48879
$703$	−1.17157	−0.0441867
$704$	9.82843	0.370423
$705$	0	0
$706$	−24.1421	−0.908601
$707$	21.6569	0.814490
$708$	−95.5980	−3.59279
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	58.7696	2.20248
$713$	−4.68629	−0.175503
$714$	65.9411	2.46778
$715$	0	0
$716$	47.7990	1.78633
$717$	16.0000	0.597531
$718$	−46.6274	−1.74012
$719$	−12.6863	−0.473119	−0.236559	−	0.971617i	$-0.576020\pi$
$719$	−12.6863	−0.473119	−0.236559	+	0.971617i	$0.576020\pi$
$720$	0	0
$721$	−28.2843	−1.05336
$722$	−2.41421	−0.0898477
$723$	33.6569	1.25171
$724$	−20.3431	−0.756047
$725$	0	0
$726$	4.82843	0.179200
$727$	−24.9706	−0.926107	−0.463053	−	0.886330i	$-0.653246\pi$
$727$	−24.9706	−0.926107	−0.463053	+	0.886330i	$0.653246\pi$
$728$	85.2548	3.15975
$729$	13.0000	0.481481
$730$	0	0
$731$	32.9706	1.21946
$732$	28.0000	1.03491
$733$	−6.48528	−0.239539	−0.119770	−	0.992802i	$-0.538216\pi$
$733$	−6.48528	−0.239539	−0.119770	+	0.992802i	$0.538216\pi$
$734$	85.2548	3.14681
$735$	0	0
$736$	−6.34315	−0.233811
$737$	0.343146	0.0126399
$738$	21.3137	0.784568
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	−13.6569	−0.501697
$742$	−19.3137	−0.709029
$743$	−27.6569	−1.01463	−0.507316	−	0.861760i	$-0.669362\pi$
$743$	−27.6569	−1.01463	−0.507316	+	0.861760i	$0.669362\pi$
$744$	10.3431	0.379198
$745$	0	0
$746$	−31.7990	−1.16424
$747$	−2.82843	−0.103487
$748$	−18.4853	−0.675889
$749$	−37.6569	−1.37595
$750$	0	0
$751$	−51.7990	−1.89017	−0.945086	−	0.326822i	$-0.894022\pi$
$751$	−51.7990	−1.89017	−0.945086	+	0.326822i	$0.894022\pi$
$752$	0	0
$753$	−14.6274	−0.533053
$754$	52.2843	1.90408
$755$	0	0
$756$	43.3137	1.57530
$757$	6.68629	0.243017	0.121509	−	0.992590i	$-0.461227\pi$
$757$	6.68629	0.243017	0.121509	+	0.992590i	$0.461227\pi$
$758$	45.4558	1.65103
$759$	−8.00000	−0.290382
$760$	0	0
$761$	48.9117	1.77305	0.886524	−	0.462683i	$-0.153113\pi$
$761$	48.9117	1.77305	0.886524	+	0.462683i	$0.153113\pi$
$762$	−72.2843	−2.61858
$763$	−2.34315	−0.0848276
$764$	64.9706	2.35055
$765$	0	0
$766$	−32.1421	−1.16134
$767$	−85.2548	−3.07837
$768$	59.9411	2.16294
$769$	−1.02944	−0.0371225	−0.0185612	−	0.999828i	$-0.505909\pi$
$769$	−1.02944	−0.0371225	−0.0185612	+	0.999828i	$0.505909\pi$
$770$	0	0
$771$	3.71573	0.133819
$772$	−60.4853	−2.17691
$773$	31.7990	1.14373	0.571865	−	0.820348i	$-0.306220\pi$
$773$	31.7990	1.14373	0.571865	+	0.820348i	$0.306220\pi$
$774$	−16.4853	−0.592551
$775$	0	0
$776$	44.7696	1.60713
$777$	−6.62742	−0.237757
$778$	40.1421	1.43917
$779$	8.82843	0.316311
$780$	0	0
$781$	−9.17157	−0.328185
$782$	46.6274	1.66739
$783$	12.6863	0.453371
$784$	3.00000	0.107143
$785$	0	0
$786$	46.6274	1.66314
$787$	14.9706	0.533643	0.266821	−	0.963746i	$-0.414027\pi$
$787$	14.9706	0.533643	0.266821	+	0.963746i	$0.414027\pi$
$788$	52.8284	1.88193
$789$	−24.9706	−0.888976
$790$	0	0
$791$	35.3137	1.25561
$792$	4.41421	0.156852
$793$	24.9706	0.886731
$794$	32.1421	1.14068
$795$	0	0
$796$	8.97056	0.317953
$797$	10.8284	0.383563	0.191781	−	0.981438i	$-0.438574\pi$
$797$	10.8284	0.383563	0.191781	+	0.981438i	$0.438574\pi$
$798$	−13.6569	−0.483447
$799$	0	0
$800$	0	0
$801$	−13.3137	−0.470417
$802$	−43.4558	−1.53448
$803$	−10.4853	−0.370018
$804$	2.62742	0.0926619
$805$	0	0
$806$	19.3137	0.680296
$807$	47.3137	1.66552
$808$	−33.7990	−1.18904
$809$	−43.6569	−1.53489	−0.767447	−	0.641113i	$-0.778473\pi$
$809$	−43.6569	−1.53489	−0.767447	+	0.641113i	$0.778473\pi$
$810$	0	0
$811$	−7.31371	−0.256819	−0.128410	−	0.991721i	$-0.540987\pi$
$811$	−7.31371	−0.256819	−0.128410	+	0.991721i	$0.540987\pi$
$812$	34.3431	1.20521
$813$	48.0000	1.68343
$814$	2.82843	0.0991363
$815$	0	0
$816$	−28.9706	−1.01417
$817$	−6.82843	−0.238896
$818$	17.3137	0.605360
$819$	−19.3137	−0.674876
$820$	0	0
$821$	44.6274	1.55751	0.778754	−	0.627330i	$-0.215852\pi$
$821$	44.6274	1.55751	0.778754	+	0.627330i	$0.215852\pi$
$822$	−44.9706	−1.56853
$823$	−5.65685	−0.197186	−0.0985928	−	0.995128i	$-0.531434\pi$
$823$	−5.65685	−0.197186	−0.0985928	+	0.995128i	$0.531434\pi$
$824$	44.1421	1.53776
$825$	0	0
$826$	−85.2548	−2.96640
$827$	−35.6569	−1.23991	−0.619955	−	0.784637i	$-0.712849\pi$
$827$	−35.6569	−1.23991	−0.619955	+	0.784637i	$0.712849\pi$
$828$	−15.3137	−0.532188
$829$	−26.9706	−0.936726	−0.468363	−	0.883536i	$-0.655156\pi$
$829$	−26.9706	−0.936726	−0.468363	+	0.883536i	$0.655156\pi$
$830$	0	0
$831$	16.2843	0.564895
$832$	67.1127	2.32671
$833$	4.82843	0.167295
$834$	101.255	3.50617
$835$	0	0
$836$	3.82843	0.132409
$837$	4.68629	0.161982
$838$	31.3137	1.08171
$839$	−7.51472	−0.259437	−0.129718	−	0.991551i	$-0.541407\pi$
$839$	−7.51472	−0.259437	−0.129718	+	0.991551i	$0.541407\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	25.7990	0.889092
$843$	−20.9706	−0.722265
$844$	−91.8823	−3.16272
$845$	0	0
$846$	0	0
$847$	2.82843	0.0971859
$848$	8.48528	0.291386
$849$	−32.9706	−1.13155
$850$	0	0
$851$	−4.68629	−0.160644
$852$	−70.2254	−2.40588
$853$	−42.7696	−1.46440	−0.732201	−	0.681089i	$-0.761507\pi$
$853$	−42.7696	−1.46440	−0.732201	+	0.681089i	$0.761507\pi$
$854$	24.9706	0.854475
$855$	0	0
$856$	58.7696	2.00870
$857$	9.45584	0.323005	0.161503	−	0.986872i	$-0.448366\pi$
$857$	9.45584	0.323005	0.161503	+	0.986872i	$0.448366\pi$
$858$	32.9706	1.12560
$859$	31.3137	1.06841	0.534205	−	0.845355i	$-0.320611\pi$
$859$	31.3137	1.06841	0.534205	+	0.845355i	$0.320611\pi$
$860$	0	0
$861$	49.9411	1.70199
$862$	64.2843	2.18953
$863$	−13.0294	−0.443527	−0.221764	−	0.975100i	$-0.571181\pi$
$863$	−13.0294	−0.443527	−0.221764	+	0.975100i	$0.571181\pi$
$864$	6.34315	0.215798
$865$	0	0
$866$	90.4264	3.07281
$867$	−12.6274	−0.428850
$868$	12.6863	0.430601
$869$	12.0000	0.407072
$870$	0	0
$871$	2.34315	0.0793945
$872$	3.65685	0.123837
$873$	−10.1421	−0.343259
$874$	−9.65685	−0.326648
$875$	0	0
$876$	−80.2843	−2.71255
$877$	−15.5147	−0.523895	−0.261947	−	0.965082i	$-0.584365\pi$
$877$	−15.5147	−0.523895	−0.261947	+	0.965082i	$0.584365\pi$
$878$	−48.2843	−1.62952
$879$	−10.3431	−0.348866
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−2.41421	−0.0812908
$883$	−48.2843	−1.62490	−0.812448	−	0.583034i	$-0.801865\pi$
$883$	−48.2843	−1.62490	−0.812448	+	0.583034i	$0.801865\pi$
$884$	−126.225	−4.24542
$885$	0	0
$886$	63.5980	2.13662
$887$	−25.3137	−0.849951	−0.424976	−	0.905205i	$-0.639717\pi$
$887$	−25.3137	−0.849951	−0.424976	+	0.905205i	$0.639717\pi$
$888$	10.3431	0.347093
$889$	−42.3431	−1.42014
$890$	0	0
$891$	11.0000	0.368514
$892$	−7.65685	−0.256370
$893$	0	0
$894$	36.9706	1.23648
$895$	0	0
$896$	58.1421	1.94239
$897$	−54.6274	−1.82396
$898$	−57.1127	−1.90588
$899$	3.71573	0.123926
$900$	0	0
$901$	13.6569	0.454976
$902$	−21.3137	−0.709669
$903$	−38.6274	−1.28544
$904$	−55.1127	−1.83302
$905$	0	0
$906$	46.6274	1.54909
$907$	3.65685	0.121424	0.0607119	−	0.998155i	$-0.480663\pi$
$907$	3.65685	0.121424	0.0607119	+	0.998155i	$0.480663\pi$
$908$	−81.5980	−2.70792
$909$	7.65685	0.253962
$910$	0	0
$911$	17.4558	0.578338	0.289169	−	0.957278i	$-0.406621\pi$
$911$	17.4558	0.578338	0.289169	+	0.957278i	$0.406621\pi$
$912$	6.00000	0.198680
$913$	2.82843	0.0936073
$914$	−56.6274	−1.87307
$915$	0	0
$916$	63.6569	2.10328
$917$	27.3137	0.901978
$918$	−46.6274	−1.53893
$919$	−0.970563	−0.0320159	−0.0160080	−	0.999872i	$-0.505096\pi$
$919$	−0.970563	−0.0320159	−0.0160080	+	0.999872i	$0.505096\pi$
$920$	0	0
$921$	−66.6274	−2.19545
$922$	−94.0833	−3.09847
$923$	−62.6274	−2.06141
$924$	21.6569	0.712458
$925$	0	0
$926$	64.2843	2.11251
$927$	−10.0000	−0.328443
$928$	5.02944	0.165099
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−40.1421	−1.31490
$933$	−27.3137	−0.894211
$934$	57.9411	1.89589
$935$	0	0
$936$	30.1421	0.985227
$937$	17.5147	0.572181	0.286090	−	0.958203i	$-0.407644\pi$
$937$	17.5147	0.572181	0.286090	+	0.958203i	$0.407644\pi$
$938$	2.34315	0.0765064
$939$	53.9411	1.76030
$940$	0	0
$941$	−4.14214	−0.135030	−0.0675149	−	0.997718i	$-0.521507\pi$
$941$	−4.14214	−0.135030	−0.0675149	+	0.997718i	$0.521507\pi$
$942$	83.5980	2.72377
$943$	35.3137	1.14997
$944$	37.4558	1.21908
$945$	0	0
$946$	16.4853	0.535983
$947$	−19.3137	−0.627611	−0.313806	−	0.949487i	$-0.601604\pi$
$947$	−19.3137	−0.627611	−0.313806	+	0.949487i	$0.601604\pi$
$948$	91.8823	2.98420
$949$	−71.5980	−2.32417
$950$	0	0
$951$	−31.5980	−1.02463
$952$	−60.2843	−1.95382
$953$	38.1421	1.23554	0.617772	−	0.786357i	$-0.288035\pi$
$953$	38.1421	1.23554	0.617772	+	0.786357i	$0.288035\pi$
$954$	−6.82843	−0.221078
$955$	0	0
$956$	−30.6274	−0.990561
$957$	6.34315	0.205045
$958$	60.2843	1.94770
$959$	−26.3431	−0.850665
$960$	0	0
$961$	−29.6274	−0.955723
$962$	19.3137	0.622699
$963$	−13.3137	−0.429028
$964$	−64.4264	−2.07503
$965$	0	0
$966$	−54.6274	−1.75761
$967$	24.7696	0.796535	0.398268	−	0.917269i	$-0.369612\pi$
$967$	24.7696	0.796535	0.398268	+	0.917269i	$0.369612\pi$
$968$	−4.41421	−0.141878
$969$	9.65685	0.310223
$970$	0	0
$971$	7.51472	0.241159	0.120579	−	0.992704i	$-0.461525\pi$
$971$	7.51472	0.241159	0.120579	+	0.992704i	$0.461525\pi$
$972$	38.2843	1.22797
$973$	59.3137	1.90151
$974$	−22.4853	−0.720475
$975$	0	0
$976$	−10.9706	−0.351159
$977$	−21.1716	−0.677339	−0.338669	−	0.940905i	$-0.609977\pi$
$977$	−21.1716	−0.677339	−0.338669	+	0.940905i	$0.609977\pi$
$978$	−85.2548	−2.72615
$979$	13.3137	0.425508
$980$	0	0
$981$	−0.828427	−0.0264496
$982$	56.2843	1.79610
$983$	−51.6569	−1.64760	−0.823799	−	0.566882i	$-0.808149\pi$
$983$	−51.6569	−1.64760	−0.823799	+	0.566882i	$0.808149\pi$
$984$	−77.9411	−2.48467
$985$	0	0
$986$	−36.9706	−1.17738
$987$	0	0
$988$	26.1421	0.831692
$989$	−27.3137	−0.868525
$990$	0	0
$991$	−40.0833	−1.27329	−0.636643	−	0.771158i	$-0.719678\pi$
$991$	−40.0833	−1.27329	−0.636643	+	0.771158i	$0.719678\pi$
$992$	1.85786	0.0589873
$993$	47.5980	1.51048
$994$	−62.6274	−1.98642
$995$	0	0
$996$	21.6569	0.686224
$997$	4.14214	0.131183	0.0655914	−	0.997847i	$-0.479107\pi$
$997$	4.14214	0.131183	0.0655914	+	0.997847i	$0.479107\pi$
$998$	83.5980	2.64625
$999$	4.68629	0.148268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.e.1.1		2
5.4	even	2		1045.2.a.c.1.2	✓	2
15.14	odd	2		9405.2.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.c.1.2	✓	2	5.4	even	2
5225.2.a.e.1.1		2	1.1	even	1	trivial
9405.2.a.o.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} -2.00000 q^{3} +3.82843 q^{4} +4.82843 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} -1.00000 q^{11} -7.65685 q^{12} -6.82843 q^{13} -6.82843 q^{14} +3.00000 q^{16} +4.82843 q^{17} -2.41421 q^{18} -1.00000 q^{19} -5.65685 q^{21} +2.41421 q^{22} -4.00000 q^{23} +8.82843 q^{24} +16.4853 q^{26} +4.00000 q^{27} +10.8284 q^{28} +3.17157 q^{29} +1.17157 q^{31} +1.58579 q^{32} +2.00000 q^{33} -11.6569 q^{34} +3.82843 q^{36} +1.17157 q^{37} +2.41421 q^{38} +13.6569 q^{39} -8.82843 q^{41} +13.6569 q^{42} +6.82843 q^{43} -3.82843 q^{44} +9.65685 q^{46} -6.00000 q^{48} +1.00000 q^{49} -9.65685 q^{51} -26.1421 q^{52} +2.82843 q^{53} -9.65685 q^{54} -12.4853 q^{56} +2.00000 q^{57} -7.65685 q^{58} +12.4853 q^{59} -3.65685 q^{61} -2.82843 q^{62} +2.82843 q^{63} -9.82843 q^{64} -4.82843 q^{66} -0.343146 q^{67} +18.4853 q^{68} +8.00000 q^{69} +9.17157 q^{71} -4.41421 q^{72} +10.4853 q^{73} -2.82843 q^{74} -3.82843 q^{76} -2.82843 q^{77} -32.9706 q^{78} -12.0000 q^{79} -11.0000 q^{81} +21.3137 q^{82} -2.82843 q^{83} -21.6569 q^{84} -16.4853 q^{86} -6.34315 q^{87} +4.41421 q^{88} -13.3137 q^{89} -19.3137 q^{91} -15.3137 q^{92} -2.34315 q^{93} -3.17157 q^{96} -10.1421 q^{97} -2.41421 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} - 6 q^{8} + 2 q^{9} - 2 q^{11} - 4 q^{12} - 8 q^{13} - 8 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} - 2 q^{19} + 2 q^{22} - 8 q^{23} + 12 q^{24} + 16 q^{26} + 8 q^{27}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 - 6 * q^8 + 2 * q^9 - 2 * q^11 - 4 * q^12 - 8 * q^13 - 8 * q^14 + 6 * q^16 + 4 * q^17 - 2 * q^18 - 2 * q^19 + 2 * q^22 - 8 * q^23 + 12 * q^24 + 16 * q^26 + 8 * q^27 + 16 * q^28 + 12 * q^29 + 8 * q^31 + 6 * q^32 + 4 * q^33 - 12 * q^34 + 2 * q^36 + 8 * q^37 + 2 * q^38 + 16 * q^39 - 12 * q^41 + 16 * q^42 + 8 * q^43 - 2 * q^44 + 8 * q^46 - 12 * q^48 + 2 * q^49 - 8 * q^51 - 24 * q^52 - 8 * q^54 - 8 * q^56 + 4 * q^57 - 4 * q^58 + 8 * q^59 + 4 * q^61 - 14 * q^64 - 4 * q^66 - 12 * q^67 + 20 * q^68 + 16 * q^69 + 24 * q^71 - 6 * q^72 + 4 * q^73 - 2 * q^76 - 32 * q^78 - 24 * q^79 - 22 * q^81 + 20 * q^82 - 32 * q^84 - 16 * q^86 - 24 * q^87 + 6 * q^88 - 4 * q^89 - 16 * q^91 - 8 * q^92 - 16 * q^93 - 12 * q^96 + 8 * q^97 - 2 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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