LMFDB - Embedded newform 5225.2.a.k.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 = [6,0,-3,8,0,2,-5] 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:	$6$
Coefficient field:	6.6.131947641.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 25x^{2} - 3x - 9 $ x^6 - 10x^4 + 25x^2 - 3*x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			$2.56745$ 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)

$f(q)$	$=$	\(q-2.56745 q^{2} -0.903918 q^{3} +4.59179 q^{4} +2.32076 q^{6} -2.16848 q^{7} -6.65430 q^{8} -2.18293 q^{9} -1.00000 q^{11} -4.15060 q^{12} +4.47137 q^{13} +5.56745 q^{14} +7.90099 q^{16} -2.71222 q^{17} +5.60457 q^{18} -1.00000 q^{19} +1.96012 q^{21} +2.56745 q^{22} +2.85635 q^{23} +6.01494 q^{24} -11.4800 q^{26} +4.68494 q^{27} -9.95719 q^{28} -2.17663 q^{29} -6.53387 q^{31} -6.97678 q^{32} +0.903918 q^{33} +6.96349 q^{34} -10.0236 q^{36} -0.861702 q^{37} +2.56745 q^{38} -4.04175 q^{39} +10.7487 q^{41} -5.03252 q^{42} -8.75038 q^{43} -4.59179 q^{44} -7.33354 q^{46} +0.665870 q^{47} -7.14184 q^{48} -2.29772 q^{49} +2.45163 q^{51} +20.5316 q^{52} +13.9364 q^{53} -12.0284 q^{54} +14.4297 q^{56} +0.903918 q^{57} +5.58839 q^{58} +8.66353 q^{59} +5.59008 q^{61} +16.7754 q^{62} +4.73364 q^{63} +2.11055 q^{64} -2.32076 q^{66} -15.2043 q^{67} -12.4540 q^{68} -2.58191 q^{69} -5.99426 q^{71} +14.5259 q^{72} +10.8568 q^{73} +2.21237 q^{74} -4.59179 q^{76} +2.16848 q^{77} +10.3770 q^{78} +5.74408 q^{79} +2.31399 q^{81} -27.5967 q^{82} +12.3220 q^{83} +9.00048 q^{84} +22.4662 q^{86} +1.96750 q^{87} +6.65430 q^{88} +4.83970 q^{89} -9.69605 q^{91} +13.1158 q^{92} +5.90608 q^{93} -1.70959 q^{94} +6.30643 q^{96} -14.8355 q^{97} +5.89927 q^{98} +2.18293 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * 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 $

$2$	−2.56745	−1.81546	−0.907730	−	0.419554i	$-0.862186\pi$
$2$	−2.56745	−1.81546	−0.907730	+	0.419554i	$0.862186\pi$
$3$	−0.903918	−0.521877	−0.260939	−	0.965355i	$-0.584032\pi$
$3$	−0.903918	−0.521877	−0.260939	+	0.965355i	$0.584032\pi$
$4$	4.59179	2.29590
$5$	0	0
$5$	0	0
$6$	2.32076	0.947447
$7$	−2.16848	−0.819607	−0.409803	−	0.912174i	$-0.634403\pi$
$7$	−2.16848	−0.819607	−0.409803	+	0.912174i	$0.634403\pi$
$8$	−6.65430	−2.35265
$9$	−2.18293	−0.727644
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.15060	−1.19818
$13$	4.47137	1.24013	0.620067	−	0.784549i	$-0.287105\pi$
$13$	4.47137	1.24013	0.620067	+	0.784549i	$0.287105\pi$
$14$	5.56745	1.48796
$15$	0	0
$16$	7.90099	1.97525
$17$	−2.71222	−0.657810	−0.328905	−	0.944363i	$-0.606680\pi$
$17$	−2.71222	−0.657810	−0.328905	+	0.944363i	$0.606680\pi$
$18$	5.60457	1.32101
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.96012	0.427734
$22$	2.56745	0.547382
$23$	2.85635	0.595590	0.297795	−	0.954630i	$-0.403749\pi$
$23$	2.85635	0.595590	0.297795	+	0.954630i	$0.403749\pi$
$24$	6.01494	1.22779
$25$	0	0
$26$	−11.4800	−2.25141
$27$	4.68494	0.901618
$28$	−9.95719	−1.88173
$29$	−2.17663	−0.404191	−0.202095	−	0.979366i	$-0.564775\pi$
$29$	−2.17663	−0.404191	−0.202095	+	0.979366i	$0.564775\pi$
$30$	0	0
$31$	−6.53387	−1.17352	−0.586759	−	0.809762i	$-0.699596\pi$
$31$	−6.53387	−1.17352	−0.586759	+	0.809762i	$0.699596\pi$
$32$	−6.97678	−1.23333
$33$	0.903918	0.157352
$34$	6.96349	1.19423
$35$	0	0
$36$	−10.0236	−1.67060
$37$	−0.861702	−0.141663	−0.0708314	−	0.997488i	$-0.522565\pi$
$37$	−0.861702	−0.141663	−0.0708314	+	0.997488i	$0.522565\pi$
$38$	2.56745	0.416495
$39$	−4.04175	−0.647198
$40$	0	0
$41$	10.7487	1.67866	0.839332	−	0.543619i	$-0.182946\pi$
$41$	10.7487	1.67866	0.839332	+	0.543619i	$0.182946\pi$
$42$	−5.03252	−0.776534
$43$	−8.75038	−1.33442	−0.667210	−	0.744869i	$-0.732512\pi$
$43$	−8.75038	−1.33442	−0.667210	+	0.744869i	$0.732512\pi$
$44$	−4.59179	−0.692239
$45$	0	0
$46$	−7.33354	−1.08127
$47$	0.665870	0.0971272	0.0485636	−	0.998820i	$-0.484536\pi$
$47$	0.665870	0.0971272	0.0485636	+	0.998820i	$0.484536\pi$
$48$	−7.14184	−1.03084
$49$	−2.29772	−0.328245
$50$	0	0
$51$	2.45163	0.343296
$52$	20.5316	2.84722
$53$	13.9364	1.91431	0.957155	−	0.289576i	$-0.0935142\pi$
$53$	13.9364	1.91431	0.957155	+	0.289576i	$0.0935142\pi$
$54$	−12.0284	−1.63685
$55$	0	0
$56$	14.4297	1.92825
$57$	0.903918	0.119727
$58$	5.58839	0.733792
$59$	8.66353	1.12790	0.563948	−	0.825810i	$-0.309282\pi$
$59$	8.66353	1.12790	0.563948	+	0.825810i	$0.309282\pi$
$60$	0	0
$61$	5.59008	0.715736	0.357868	−	0.933772i	$-0.383504\pi$
$61$	5.59008	0.715736	0.357868	+	0.933772i	$0.383504\pi$
$62$	16.7754	2.13048
$63$	4.73364	0.596382
$64$	2.11055	0.263819
$65$	0	0
$66$	−2.32076	−0.285666
$67$	−15.2043	−1.85751	−0.928753	−	0.370698i	$-0.879118\pi$
$67$	−15.2043	−1.85751	−0.928753	+	0.370698i	$0.879118\pi$
$68$	−12.4540	−1.51027
$69$	−2.58191	−0.310825
$70$	0	0
$71$	−5.99426	−0.711388	−0.355694	−	0.934603i	$-0.615755\pi$
$71$	−5.99426	−0.711388	−0.355694	+	0.934603i	$0.615755\pi$
$72$	14.5259	1.71189
$73$	10.8568	1.27069	0.635347	−	0.772226i	$-0.280857\pi$
$73$	10.8568	1.27069	0.635347	+	0.772226i	$0.280857\pi$
$74$	2.21237	0.257183
$75$	0	0
$76$	−4.59179	−0.526715
$77$	2.16848	0.247121
$78$	10.3770	1.17496
$79$	5.74408	0.646260	0.323130	−	0.946355i	$-0.395265\pi$
$79$	5.74408	0.646260	0.323130	+	0.946355i	$0.395265\pi$
$80$	0	0
$81$	2.31399	0.257110
$82$	−27.5967	−3.04755
$83$	12.3220	1.35252	0.676259	−	0.736664i	$-0.263600\pi$
$83$	12.3220	1.35252	0.676259	+	0.736664i	$0.263600\pi$
$84$	9.00048	0.982033
$85$	0	0
$86$	22.4662	2.42259
$87$	1.96750	0.210938
$88$	6.65430	0.709351
$89$	4.83970	0.513007	0.256503	−	0.966543i	$-0.417430\pi$
$89$	4.83970	0.513007	0.256503	+	0.966543i	$0.417430\pi$
$90$	0	0
$91$	−9.69605	−1.01642
$92$	13.1158	1.36741
$93$	5.90608	0.612432
$94$	−1.70959	−0.176331
$95$	0	0
$96$	6.30643	0.643648
$97$	−14.8355	−1.50631	−0.753157	−	0.657841i	$-0.771470\pi$
$97$	−14.8355	−1.50631	−0.753157	+	0.657841i	$0.771470\pi$
$98$	5.89927	0.595916
$99$	2.18293	0.219393
$100$	0	0
$101$	9.35188	0.930546	0.465273	−	0.885167i	$-0.345956\pi$
$101$	9.35188	0.930546	0.465273	+	0.885167i	$0.345956\pi$
$102$	−6.29442	−0.623241
$103$	−8.08853	−0.796987	−0.398493	−	0.917171i	$-0.630467\pi$
$103$	−8.08853	−0.796987	−0.398493	+	0.917171i	$0.630467\pi$
$104$	−29.7538	−2.91760
$105$	0	0
$106$	−35.7810	−3.47535
$107$	2.71050	0.262034	0.131017	−	0.991380i	$-0.458176\pi$
$107$	2.71050	0.262034	0.131017	+	0.991380i	$0.458176\pi$
$108$	21.5123	2.07002
$109$	0.495056	0.0474178	0.0237089	−	0.999719i	$-0.492453\pi$
$109$	0.495056	0.0474178	0.0237089	+	0.999719i	$0.492453\pi$
$110$	0	0
$111$	0.778907	0.0739306
$112$	−17.1331	−1.61892
$113$	−20.6374	−1.94140	−0.970701	−	0.240292i	$-0.922757\pi$
$113$	−20.6374	−1.94140	−0.970701	+	0.240292i	$0.922757\pi$
$114$	−2.32076	−0.217359
$115$	0	0
$116$	−9.99465	−0.927980
$117$	−9.76069	−0.902376
$118$	−22.2432	−2.04765
$119$	5.88139	0.539146
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.3522	−1.29939
$123$	−9.71594	−0.876057
$124$	−30.0022	−2.69428
$125$	0	0
$126$	−12.1534	−1.08271
$127$	15.2674	1.35477	0.677383	−	0.735631i	$-0.263114\pi$
$127$	15.2674	1.35477	0.677383	+	0.735631i	$0.263114\pi$
$128$	8.53482	0.754379
$129$	7.90962	0.696404
$130$	0	0
$131$	−14.5918	−1.27489	−0.637444	−	0.770497i	$-0.720008\pi$
$131$	−14.5918	−1.27489	−0.637444	+	0.770497i	$0.720008\pi$
$132$	4.15060	0.361264
$133$	2.16848	0.188031
$134$	39.0364	3.37223
$135$	0	0
$136$	18.0479	1.54760
$137$	11.9896	1.02434	0.512169	−	0.858885i	$-0.328842\pi$
$137$	11.9896	1.02434	0.512169	+	0.858885i	$0.328842\pi$
$138$	6.62891	0.564291
$139$	−7.16027	−0.607326	−0.303663	−	0.952779i	$-0.598210\pi$
$139$	−7.16027	−0.607326	−0.303663	+	0.952779i	$0.598210\pi$
$140$	0	0
$141$	−0.601892	−0.0506885
$142$	15.3900	1.29150
$143$	−4.47137	−0.373914
$144$	−17.2473	−1.43728
$145$	0	0
$146$	−27.8743	−2.30690
$147$	2.07695	0.171304
$148$	−3.95676	−0.325243
$149$	3.73998	0.306391	0.153196	−	0.988196i	$-0.451044\pi$
$149$	3.73998	0.306391	0.153196	+	0.988196i	$0.451044\pi$
$150$	0	0
$151$	−7.70787	−0.627258	−0.313629	−	0.949546i	$-0.601545\pi$
$151$	−7.70787	−0.627258	−0.313629	+	0.949546i	$0.601545\pi$
$152$	6.65430	0.539735
$153$	5.92060	0.478652
$154$	−5.56745	−0.448638
$155$	0	0
$156$	−18.5589	−1.48590
$157$	7.78415	0.621242	0.310621	−	0.950534i	$-0.399463\pi$
$157$	7.78415	0.621242	0.310621	+	0.950534i	$0.399463\pi$
$158$	−14.7476	−1.17326
$159$	−12.5973	−0.999035
$160$	0	0
$161$	−6.19393	−0.488150
$162$	−5.94106	−0.466774
$163$	3.60748	0.282560	0.141280	−	0.989970i	$-0.454878\pi$
$163$	3.60748	0.282560	0.141280	+	0.989970i	$0.454878\pi$
$164$	49.3558	3.85404
$165$	0	0
$166$	−31.6362	−2.45544
$167$	0.588302	0.0455242	0.0227621	−	0.999741i	$-0.492754\pi$
$167$	0.588302	0.0455242	0.0227621	+	0.999741i	$0.492754\pi$
$168$	−13.0432	−1.00631
$169$	6.99312	0.537932
$170$	0	0
$171$	2.18293	0.166933
$172$	−40.1800	−3.06369
$173$	−23.1505	−1.76010	−0.880050	−	0.474882i	$-0.842491\pi$
$173$	−23.1505	−1.76010	−0.880050	+	0.474882i	$0.842491\pi$
$174$	−5.05145	−0.382949
$175$	0	0
$176$	−7.90099	−0.595559
$177$	−7.83112	−0.588623
$178$	−12.4257	−0.931343
$179$	11.2340	0.839666	0.419833	−	0.907601i	$-0.362089\pi$
$179$	11.2340	0.839666	0.419833	+	0.907601i	$0.362089\pi$
$180$	0	0
$181$	−7.18465	−0.534031	−0.267015	−	0.963692i	$-0.586037\pi$
$181$	−7.18465	−0.534031	−0.267015	+	0.963692i	$0.586037\pi$
$182$	24.8941	1.84527
$183$	−5.05297	−0.373526
$184$	−19.0070	−1.40122
$185$	0	0
$186$	−15.1636	−1.11185
$187$	2.71222	0.198337
$188$	3.05754	0.222994
$189$	−10.1592	−0.738972
$190$	0	0
$191$	−13.9736	−1.01109	−0.505546	−	0.862800i	$-0.668709\pi$
$191$	−13.9736	−1.01109	−0.505546	+	0.862800i	$0.668709\pi$
$192$	−1.90777	−0.137681
$193$	12.8048	0.921710	0.460855	−	0.887475i	$-0.347543\pi$
$193$	12.8048	0.921710	0.460855	+	0.887475i	$0.347543\pi$
$194$	38.0893	2.73465
$195$	0	0
$196$	−10.5506	−0.753617
$197$	−3.78816	−0.269895	−0.134948	−	0.990853i	$-0.543087\pi$
$197$	−3.78816	−0.269895	−0.134948	+	0.990853i	$0.543087\pi$
$198$	−5.60457	−0.398299
$199$	15.9061	1.12755	0.563777	−	0.825927i	$-0.309348\pi$
$199$	15.9061	1.12755	0.563777	+	0.825927i	$0.309348\pi$
$200$	0	0
$201$	13.7435	0.969390
$202$	−24.0105	−1.68937
$203$	4.71997	0.331277
$204$	11.2574	0.788173
$205$	0	0
$206$	20.7669	1.44690
$207$	−6.23522	−0.433378
$208$	35.3282	2.44957
$209$	1.00000	0.0691714
$210$	0	0
$211$	2.20422	0.151745	0.0758723	−	0.997118i	$-0.475826\pi$
$211$	2.20422	0.151745	0.0758723	+	0.997118i	$0.475826\pi$
$212$	63.9930	4.39506
$213$	5.41832	0.371257
$214$	−6.95908	−0.475713
$215$	0	0
$216$	−31.1750	−2.12119
$217$	14.1685	0.961823
$218$	−1.27103	−0.0860851
$219$	−9.81367	−0.663147
$220$	0	0
$221$	−12.1273	−0.815773
$222$	−1.99980	−0.134218
$223$	14.3670	0.962086	0.481043	−	0.876697i	$-0.340258\pi$
$223$	14.3670	0.962086	0.481043	+	0.876697i	$0.340258\pi$
$224$	15.1290	1.01085
$225$	0	0
$226$	52.9854	3.52454
$227$	−21.6909	−1.43968	−0.719838	−	0.694142i	$-0.755784\pi$
$227$	−21.6909	−1.43968	−0.719838	+	0.694142i	$0.755784\pi$
$228$	4.15060	0.274880
$229$	9.39735	0.620995	0.310497	−	0.950574i	$-0.399504\pi$
$229$	9.39735	0.620995	0.310497	+	0.950574i	$0.399504\pi$
$230$	0	0
$231$	−1.96012	−0.128967
$232$	14.4840	0.950919
$233$	22.6514	1.48394	0.741971	−	0.670432i	$-0.233891\pi$
$233$	22.6514	1.48394	0.741971	+	0.670432i	$0.233891\pi$
$234$	25.0601	1.63823
$235$	0	0
$236$	39.7812	2.58953
$237$	−5.19218	−0.337268
$238$	−15.1002	−0.978798
$239$	12.3190	0.796853	0.398426	−	0.917200i	$-0.369556\pi$
$239$	12.3190	0.796853	0.398426	+	0.917200i	$0.369556\pi$
$240$	0	0
$241$	−27.5506	−1.77469	−0.887345	−	0.461106i	$-0.847453\pi$
$241$	−27.5506	−1.77469	−0.887345	+	0.461106i	$0.847453\pi$
$242$	−2.56745	−0.165042
$243$	−16.1465	−1.03580
$244$	25.6685	1.64326
$245$	0	0
$246$	24.9452	1.59045
$247$	−4.47137	−0.284506
$248$	43.4783	2.76088
$249$	−11.1381	−0.705848
$250$	0	0
$251$	11.9051	0.751443	0.375722	−	0.926733i	$-0.377395\pi$
$251$	11.9051	0.751443	0.375722	+	0.926733i	$0.377395\pi$
$252$	21.7359	1.36923
$253$	−2.85635	−0.179577
$254$	−39.1983	−2.45952
$255$	0	0
$256$	−26.1338	−1.63336
$257$	8.19766	0.511356	0.255678	−	0.966762i	$-0.417701\pi$
$257$	8.19766	0.511356	0.255678	+	0.966762i	$0.417701\pi$
$258$	−20.3076	−1.26429
$259$	1.86858	0.116108
$260$	0	0
$261$	4.75144	0.294107
$262$	37.4636	2.31451
$263$	13.1875	0.813175	0.406588	−	0.913612i	$-0.366719\pi$
$263$	13.1875	0.813175	0.406588	+	0.913612i	$0.366719\pi$
$264$	−6.01494	−0.370194
$265$	0	0
$266$	−5.56745	−0.341362
$267$	−4.37469	−0.267726
$268$	−69.8152	−4.26464
$269$	19.3988	1.18277	0.591384	−	0.806390i	$-0.298582\pi$
$269$	19.3988	1.18277	0.591384	+	0.806390i	$0.298582\pi$
$270$	0	0
$271$	3.57069	0.216904	0.108452	−	0.994102i	$-0.465411\pi$
$271$	3.57069	0.216904	0.108452	+	0.994102i	$0.465411\pi$
$272$	−21.4292	−1.29934
$273$	8.76443	0.530447
$274$	−30.7826	−1.85964
$275$	0	0
$276$	−11.8556	−0.713622
$277$	20.6671	1.24176	0.620882	−	0.783904i	$-0.286775\pi$
$277$	20.6671	1.24176	0.620882	+	0.783904i	$0.286775\pi$
$278$	18.3836	1.10258
$279$	14.2630	0.853904
$280$	0	0
$281$	−6.89894	−0.411556	−0.205778	−	0.978599i	$-0.565973\pi$
$281$	−6.89894	−0.411556	−0.205778	+	0.978599i	$0.565973\pi$
$282$	1.54533	0.0920229
$283$	−23.5344	−1.39898	−0.699488	−	0.714644i	$-0.746589\pi$
$283$	−23.5344	−1.39898	−0.699488	+	0.714644i	$0.746589\pi$
$284$	−27.5244	−1.63327
$285$	0	0
$286$	11.4800	0.678827
$287$	−23.3083	−1.37584
$288$	15.2298	0.897427
$289$	−9.64385	−0.567285
$290$	0	0
$291$	13.4100	0.786111
$292$	49.8523	2.91738
$293$	−8.70291	−0.508429	−0.254215	−	0.967148i	$-0.581817\pi$
$293$	−8.70291	−0.508429	−0.254215	+	0.967148i	$0.581817\pi$
$294$	−5.33245	−0.310995
$295$	0	0
$296$	5.73402	0.333283
$297$	−4.68494	−0.271848
$298$	−9.60221	−0.556241
$299$	12.7718	0.738612
$300$	0	0
$301$	18.9750	1.09370
$302$	19.7896	1.13876
$303$	−8.45333	−0.485631
$304$	−7.90099	−0.453153
$305$	0	0
$306$	−15.2008	−0.868974
$307$	0.0345724	0.00197315	0.000986576	−	1.00000i	$-0.499686\pi$
$307$	0.0345724	0.00197315	0.000986576		1.00000i	$0.499686\pi$
$308$	9.95719	0.567364
$309$	7.31137	0.415929
$310$	0	0
$311$	−16.8869	−0.957570	−0.478785	−	0.877932i	$-0.658923\pi$
$311$	−16.8869	−0.957570	−0.478785	+	0.877932i	$0.658923\pi$
$312$	26.8950	1.52263
$313$	1.47670	0.0834682	0.0417341	−	0.999129i	$-0.486712\pi$
$313$	1.47670	0.0834682	0.0417341	+	0.999129i	$0.486712\pi$
$314$	−19.9854	−1.12784
$315$	0	0
$316$	26.3756	1.48375
$317$	−24.5279	−1.37763	−0.688813	−	0.724939i	$-0.741868\pi$
$317$	−24.5279	−1.37763	−0.688813	+	0.724939i	$0.741868\pi$
$318$	32.3430	1.81371
$319$	2.17663	0.121868
$320$	0	0
$321$	−2.45007	−0.136750
$322$	15.9026	0.886217
$323$	2.71222	0.150912
$324$	10.6254	0.590299
$325$	0	0
$326$	−9.26202	−0.512976
$327$	−0.447490	−0.0247463
$328$	−71.5251	−3.94931
$329$	−1.44392	−0.0796061
$330$	0	0
$331$	19.8183	1.08931	0.544657	−	0.838659i	$-0.316660\pi$
$331$	19.8183	1.08931	0.544657	+	0.838659i	$0.316660\pi$
$332$	56.5802	3.10524
$333$	1.88104	0.103080
$334$	−1.51044	−0.0826474
$335$	0	0
$336$	15.4869	0.844880
$337$	−12.6702	−0.690188	−0.345094	−	0.938568i	$-0.612153\pi$
$337$	−12.6702	−0.690188	−0.345094	+	0.938568i	$0.612153\pi$
$338$	−17.9545	−0.976595
$339$	18.6545	1.01317
$340$	0	0
$341$	6.53387	0.353829
$342$	−5.60457	−0.303060
$343$	20.1619	1.08864
$344$	58.2277	3.13943
$345$	0	0
$346$	59.4377	3.19539
$347$	−6.29698	−0.338040	−0.169020	−	0.985613i	$-0.554060\pi$
$347$	−6.29698	−0.338040	−0.169020	+	0.985613i	$0.554060\pi$
$348$	9.03434	0.484292
$349$	−4.90512	−0.262565	−0.131283	−	0.991345i	$-0.541910\pi$
$349$	−4.90512	−0.262565	−0.131283	+	0.991345i	$0.541910\pi$
$350$	0	0
$351$	20.9481	1.11813
$352$	6.97678	0.371864
$353$	−22.7868	−1.21282	−0.606409	−	0.795153i	$-0.707391\pi$
$353$	−22.7868	−1.21282	−0.606409	+	0.795153i	$0.707391\pi$
$354$	20.1060	1.06862
$355$	0	0
$356$	22.2229	1.17781
$357$	−5.31629	−0.281368
$358$	−28.8426	−1.52438
$359$	26.4835	1.39774	0.698872	−	0.715246i	$-0.253686\pi$
$359$	26.4835	1.39774	0.698872	+	0.715246i	$0.253686\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.4462	0.969512
$363$	−0.903918	−0.0474434
$364$	−44.5223	−2.33360
$365$	0	0
$366$	12.9732	0.678122
$367$	−14.4411	−0.753818	−0.376909	−	0.926250i	$-0.623013\pi$
$367$	−14.4411	−0.753818	−0.376909	+	0.926250i	$0.623013\pi$
$368$	22.5680	1.17644
$369$	−23.4637	−1.22147
$370$	0	0
$371$	−30.2207	−1.56898
$372$	27.1195	1.40608
$373$	3.81445	0.197505	0.0987525	−	0.995112i	$-0.468515\pi$
$373$	3.81445	0.197505	0.0987525	+	0.995112i	$0.468515\pi$
$374$	−6.96349	−0.360074
$375$	0	0
$376$	−4.43090	−0.228506
$377$	−9.73252	−0.501250
$378$	26.0832	1.34157
$379$	−8.00109	−0.410988	−0.205494	−	0.978658i	$-0.565880\pi$
$379$	−8.00109	−0.410988	−0.205494	+	0.978658i	$0.565880\pi$
$380$	0	0
$381$	−13.8005	−0.707021
$382$	35.8764	1.83560
$383$	−13.5609	−0.692932	−0.346466	−	0.938063i	$-0.612618\pi$
$383$	−13.5609	−0.692932	−0.346466	+	0.938063i	$0.612618\pi$
$384$	−7.71478	−0.393693
$385$	0	0
$386$	−32.8757	−1.67333
$387$	19.1015	0.970984
$388$	−68.1214	−3.45834
$389$	−1.63547	−0.0829216	−0.0414608	−	0.999140i	$-0.513201\pi$
$389$	−1.63547	−0.0829216	−0.0414608	+	0.999140i	$0.513201\pi$
$390$	0	0
$391$	−7.74706	−0.391786
$392$	15.2897	0.772246
$393$	13.1898	0.665335
$394$	9.72590	0.489984
$395$	0	0
$396$	10.0236	0.503704
$397$	−9.24683	−0.464085	−0.232043	−	0.972706i	$-0.574541\pi$
$397$	−9.24683	−0.464085	−0.232043	+	0.972706i	$0.574541\pi$
$398$	−40.8381	−2.04703
$399$	−1.96012	−0.0981289
$400$	0	0
$401$	−0.299914	−0.0149770	−0.00748851	−	0.999972i	$-0.502384\pi$
$401$	−0.299914	−0.0149770	−0.00748851	+	0.999972i	$0.502384\pi$
$402$	−35.2857	−1.75989
$403$	−29.2153	−1.45532
$404$	42.9419	2.13644
$405$	0	0
$406$	−12.1183	−0.601421
$407$	0.861702	0.0427130
$408$	−16.3138	−0.807656
$409$	9.28447	0.459088	0.229544	−	0.973298i	$-0.426277\pi$
$409$	9.28447	0.459088	0.229544	+	0.973298i	$0.426277\pi$
$410$	0	0
$411$	−10.8376	−0.534578
$412$	−37.1409	−1.82980
$413$	−18.7867	−0.924431
$414$	16.0086	0.786781
$415$	0	0
$416$	−31.1957	−1.52950
$417$	6.47230	0.316950
$418$	−2.56745	−0.125578
$419$	−9.33196	−0.455896	−0.227948	−	0.973673i	$-0.573202\pi$
$419$	−9.33196	−0.455896	−0.227948	+	0.973673i	$0.573202\pi$
$420$	0	0
$421$	−25.1456	−1.22552	−0.612760	−	0.790269i	$-0.709941\pi$
$421$	−25.1456	−1.22552	−0.612760	+	0.790269i	$0.709941\pi$
$422$	−5.65921	−0.275486
$423$	−1.45355	−0.0706740
$424$	−92.7369	−4.50370
$425$	0	0
$426$	−13.9113	−0.674003
$427$	−12.1219	−0.586622
$428$	12.4461	0.601604
$429$	4.04175	0.195137
$430$	0	0
$431$	37.5660	1.80949	0.904744	−	0.425955i	$-0.140062\pi$
$431$	37.5660	1.80949	0.904744	+	0.425955i	$0.140062\pi$
$432$	37.0157	1.78092
$433$	−9.24282	−0.444181	−0.222091	−	0.975026i	$-0.571288\pi$
$433$	−9.24282	−0.444181	−0.222091	+	0.975026i	$0.571288\pi$
$434$	−36.3770	−1.74615
$435$	0	0
$436$	2.27320	0.108866
$437$	−2.85635	−0.136638
$438$	25.1961	1.20392
$439$	0.941826	0.0449509	0.0224754	−	0.999747i	$-0.492845\pi$
$439$	0.941826	0.0449509	0.0224754	+	0.999747i	$0.492845\pi$
$440$	0	0
$441$	5.01576	0.238846
$442$	31.1363	1.48100
$443$	−17.2727	−0.820649	−0.410324	−	0.911940i	$-0.634585\pi$
$443$	−17.2727	−0.820649	−0.410324	+	0.911940i	$0.634585\pi$
$444$	3.57658	0.169737
$445$	0	0
$446$	−36.8866	−1.74663
$447$	−3.38063	−0.159899
$448$	−4.57668	−0.216228
$449$	−20.9917	−0.990661	−0.495330	−	0.868705i	$-0.664953\pi$
$449$	−20.9917	−0.990661	−0.495330	+	0.868705i	$0.664953\pi$
$450$	0	0
$451$	−10.7487	−0.506136
$452$	−94.7626	−4.45726
$453$	6.96728	0.327351
$454$	55.6903	2.61368
$455$	0	0
$456$	−6.01494	−0.281675
$457$	28.9893	1.35606	0.678032	−	0.735032i	$-0.262833\pi$
$457$	28.9893	1.35606	0.678032	+	0.735032i	$0.262833\pi$
$458$	−24.1272	−1.12739
$459$	−12.7066	−0.593094
$460$	0	0
$461$	−24.7915	−1.15465	−0.577327	−	0.816513i	$-0.695904\pi$
$461$	−24.7915	−1.15465	−0.577327	+	0.816513i	$0.695904\pi$
$462$	5.03252	0.234134
$463$	0.757122	0.0351864	0.0175932	−	0.999845i	$-0.494400\pi$
$463$	0.757122	0.0351864	0.0175932	+	0.999845i	$0.494400\pi$
$464$	−17.1975	−0.798376
$465$	0	0
$466$	−58.1563	−2.69404
$467$	5.54461	0.256574	0.128287	−	0.991737i	$-0.459052\pi$
$467$	5.54461	0.256574	0.128287	+	0.991737i	$0.459052\pi$
$468$	−44.8191	−2.07176
$469$	32.9702	1.52242
$470$	0	0
$471$	−7.03623	−0.324212
$472$	−57.6497	−2.65354
$473$	8.75038	0.402343
$474$	13.3307	0.612297
$475$	0	0
$476$	27.0061	1.23782
$477$	−30.4222	−1.39294
$478$	−31.6285	−1.44665
$479$	−8.78759	−0.401515	−0.200758	−	0.979641i	$-0.564340\pi$
$479$	−8.78759	−0.401515	−0.200758	+	0.979641i	$0.564340\pi$
$480$	0	0
$481$	−3.85298	−0.175681
$482$	70.7348	3.22188
$483$	5.59880	0.254754
$484$	4.59179	0.208718
$485$	0	0
$486$	41.4553	1.88045
$487$	−36.8929	−1.67178	−0.835889	−	0.548898i	$-0.815047\pi$
$487$	−36.8929	−1.67178	−0.835889	+	0.548898i	$0.815047\pi$
$488$	−37.1980	−1.68388
$489$	−3.26086	−0.147461
$490$	0	0
$491$	−6.12358	−0.276353	−0.138177	−	0.990408i	$-0.544124\pi$
$491$	−6.12358	−0.276353	−0.138177	+	0.990408i	$0.544124\pi$
$492$	−44.6136	−2.01134
$493$	5.90351	0.265881
$494$	11.4800	0.516510
$495$	0	0
$496$	−51.6240	−2.31799
$497$	12.9984	0.583058
$498$	28.5965	1.28144
$499$	−16.9154	−0.757239	−0.378619	−	0.925552i	$-0.623601\pi$
$499$	−16.9154	−0.757239	−0.378619	+	0.925552i	$0.623601\pi$
$500$	0	0
$501$	−0.531777	−0.0237580
$502$	−30.5657	−1.36422
$503$	−4.84680	−0.216108	−0.108054	−	0.994145i	$-0.534462\pi$
$503$	−4.84680	−0.216108	−0.108054	+	0.994145i	$0.534462\pi$
$504$	−31.4990	−1.40308
$505$	0	0
$506$	7.33354	0.326015
$507$	−6.32120	−0.280735
$508$	70.1049	3.11040
$509$	−18.1418	−0.804122	−0.402061	−	0.915613i	$-0.631706\pi$
$509$	−18.1418	−0.804122	−0.402061	+	0.915613i	$0.631706\pi$
$510$	0	0
$511$	−23.5427	−1.04147
$512$	50.0276	2.21093
$513$	−4.68494	−0.206845
$514$	−21.0471	−0.928346
$515$	0	0
$516$	36.3194	1.59887
$517$	−0.665870	−0.0292849
$518$	−4.79748	−0.210789
$519$	20.9261	0.918556
$520$	0	0
$521$	36.0419	1.57902	0.789511	−	0.613736i	$-0.210334\pi$
$521$	36.0419	1.57902	0.789511	+	0.613736i	$0.210334\pi$
$522$	−12.1991	−0.533940
$523$	−38.6617	−1.69056	−0.845279	−	0.534325i	$-0.820566\pi$
$523$	−38.6617	−1.69056	−0.845279	+	0.534325i	$0.820566\pi$
$524$	−67.0024	−2.92701
$525$	0	0
$526$	−33.8582	−1.47629
$527$	17.7213	0.771952
$528$	7.14184	0.310809
$529$	−14.8413	−0.645272
$530$	0	0
$531$	−18.9119	−0.820707
$532$	9.95719	0.431699
$533$	48.0614	2.08177
$534$	11.2318	0.486047
$535$	0	0
$536$	101.174	4.37006
$537$	−10.1546	−0.438202
$538$	−49.8056	−2.14727
$539$	2.29772	0.0989696
$540$	0	0
$541$	−39.8208	−1.71203	−0.856015	−	0.516951i	$-0.827067\pi$
$541$	−39.8208	−1.71203	−0.856015	+	0.516951i	$0.827067\pi$
$542$	−9.16756	−0.393780
$543$	6.49433	0.278698
$544$	18.9226	0.811299
$545$	0	0
$546$	−22.5022	−0.963006
$547$	−34.9397	−1.49392	−0.746958	−	0.664871i	$-0.768486\pi$
$547$	−34.9397	−1.49392	−0.746958	+	0.664871i	$0.768486\pi$
$548$	55.0536	2.35177
$549$	−12.2028	−0.520801
$550$	0	0
$551$	2.17663	0.0927277
$552$	17.1808	0.731263
$553$	−12.4559	−0.529679
$554$	−53.0616	−2.25437
$555$	0	0
$556$	−32.8785	−1.39436
$557$	1.09011	0.0461896	0.0230948	−	0.999733i	$-0.492648\pi$
$557$	1.09011	0.0461896	0.0230948	+	0.999733i	$0.492648\pi$
$558$	−36.6195	−1.55023
$559$	−39.1262	−1.65486
$560$	0	0
$561$	−2.45163	−0.103508
$562$	17.7127	0.747164
$563$	2.39110	0.100773	0.0503864	−	0.998730i	$-0.483955\pi$
$563$	2.39110	0.100773	0.0503864	+	0.998730i	$0.483955\pi$
$564$	−2.76376	−0.116375
$565$	0	0
$566$	60.4234	2.53979
$567$	−5.01784	−0.210729
$568$	39.8876	1.67365
$569$	−27.9054	−1.16985	−0.584927	−	0.811086i	$-0.698877\pi$
$569$	−27.9054	−1.16985	−0.584927	+	0.811086i	$0.698877\pi$
$570$	0	0
$571$	9.33966	0.390853	0.195426	−	0.980718i	$-0.437391\pi$
$571$	9.33966	0.390853	0.195426	+	0.980718i	$0.437391\pi$
$572$	−20.5316	−0.858469
$573$	12.6310	0.527666
$574$	59.8428	2.49779
$575$	0	0
$576$	−4.60719	−0.191966
$577$	−24.2224	−1.00839	−0.504195	−	0.863590i	$-0.668211\pi$
$577$	−24.2224	−1.00839	−0.504195	+	0.863590i	$0.668211\pi$
$578$	24.7601	1.02988
$579$	−11.5745	−0.481020
$580$	0	0
$581$	−26.7200	−1.10853
$582$	−34.4296	−1.42715
$583$	−13.9364	−0.577186
$584$	−72.2445	−2.98950
$585$	0	0
$586$	22.3443	0.923033
$587$	−3.72615	−0.153795	−0.0768973	−	0.997039i	$-0.524501\pi$
$587$	−3.72615	−0.153795	−0.0768973	+	0.997039i	$0.524501\pi$
$588$	9.53691	0.393296
$589$	6.53387	0.269224
$590$	0	0
$591$	3.42418	0.140852
$592$	−6.80829	−0.279819
$593$	−14.1469	−0.580944	−0.290472	−	0.956883i	$-0.593812\pi$
$593$	−14.1469	−0.580944	−0.290472	+	0.956883i	$0.593812\pi$
$594$	12.0284	0.493529
$595$	0	0
$596$	17.1732	0.703442
$597$	−14.3778	−0.588444
$598$	−32.7909	−1.34092
$599$	−37.7856	−1.54388	−0.771940	−	0.635696i	$-0.780713\pi$
$599$	−37.7856	−1.54388	−0.771940	+	0.635696i	$0.780713\pi$
$600$	0	0
$601$	−29.5251	−1.20436	−0.602178	−	0.798362i	$-0.705700\pi$
$601$	−29.5251	−1.20436	−0.602178	+	0.798362i	$0.705700\pi$
$602$	−48.7173	−1.98557
$603$	33.1901	1.35160
$604$	−35.3930	−1.44012
$605$	0	0
$606$	21.7035	0.881644
$607$	5.19284	0.210771	0.105385	−	0.994431i	$-0.466392\pi$
$607$	5.19284	0.210771	0.105385	+	0.994431i	$0.466392\pi$
$608$	6.97678	0.282946
$609$	−4.26647	−0.172886
$610$	0	0
$611$	2.97735	0.120451
$612$	27.1862	1.09894
$613$	44.9254	1.81452	0.907260	−	0.420569i	$-0.138170\pi$
$613$	44.9254	1.81452	0.907260	+	0.420569i	$0.138170\pi$
$614$	−0.0887629	−0.00358218
$615$	0	0
$616$	−14.4297	−0.581388
$617$	8.39687	0.338045	0.169023	−	0.985612i	$-0.445939\pi$
$617$	8.39687	0.338045	0.169023	+	0.985612i	$0.445939\pi$
$618$	−18.7716	−0.755103
$619$	4.27492	0.171823	0.0859117	−	0.996303i	$-0.472620\pi$
$619$	4.27492	0.171823	0.0859117	+	0.996303i	$0.472620\pi$
$620$	0	0
$621$	13.3818	0.536995
$622$	43.3563	1.73843
$623$	−10.4948	−0.420464
$624$	−31.9338	−1.27837
$625$	0	0
$626$	−3.79136	−0.151533
$627$	−0.903918	−0.0360990
$628$	35.7432	1.42631
$629$	2.33713	0.0931873
$630$	0	0
$631$	7.33731	0.292094	0.146047	−	0.989278i	$-0.453345\pi$
$631$	7.33731	0.292094	0.146047	+	0.989278i	$0.453345\pi$
$632$	−38.2228	−1.52042
$633$	−1.99243	−0.0791920
$634$	62.9742	2.50103
$635$	0	0
$636$	−57.8444	−2.29368
$637$	−10.2739	−0.407068
$638$	−5.58839	−0.221247
$639$	13.0851	0.517637
$640$	0	0
$641$	13.0203	0.514272	0.257136	−	0.966375i	$-0.417221\pi$
$641$	13.0203	0.514272	0.257136	+	0.966375i	$0.417221\pi$
$642$	6.29044	0.248264
$643$	−33.7766	−1.33202	−0.666010	−	0.745943i	$-0.731999\pi$
$643$	−33.7766	−1.33202	−0.666010	+	0.745943i	$0.731999\pi$
$644$	−28.4412	−1.12074
$645$	0	0
$646$	−6.96349	−0.273975
$647$	−36.6351	−1.44028	−0.720138	−	0.693831i	$-0.755921\pi$
$647$	−36.6351	−1.44028	−0.720138	+	0.693831i	$0.755921\pi$
$648$	−15.3980	−0.604891
$649$	−8.66353	−0.340073
$650$	0	0
$651$	−12.8072	−0.501953
$652$	16.5648	0.648728
$653$	−0.637706	−0.0249553	−0.0124777	−	0.999922i	$-0.503972\pi$
$653$	−0.637706	−0.0249553	−0.0124777	+	0.999922i	$0.503972\pi$
$654$	1.14891	0.0449259
$655$	0	0
$656$	84.9253	3.31578
$657$	−23.6997	−0.924614
$658$	3.70720	0.144522
$659$	−5.35534	−0.208614	−0.104307	−	0.994545i	$-0.533263\pi$
$659$	−5.35534	−0.208614	−0.104307	+	0.994545i	$0.533263\pi$
$660$	0	0
$661$	−38.4599	−1.49592	−0.747958	−	0.663746i	$-0.768966\pi$
$661$	−38.4599	−1.49592	−0.747958	+	0.663746i	$0.768966\pi$
$662$	−50.8825	−1.97761
$663$	10.9621	0.425733
$664$	−81.9945	−3.18200
$665$	0	0
$666$	−4.82947	−0.187138
$667$	−6.21723	−0.240732
$668$	2.70136	0.104519
$669$	−12.9866	−0.502091
$670$	0	0
$671$	−5.59008	−0.215803
$672$	−13.6753	−0.527538
$673$	34.8302	1.34261	0.671303	−	0.741183i	$-0.265735\pi$
$673$	34.8302	1.34261	0.671303	+	0.741183i	$0.265735\pi$
$674$	32.5300	1.25301
$675$	0	0
$676$	32.1110	1.23504
$677$	−24.7488	−0.951174	−0.475587	−	0.879669i	$-0.657764\pi$
$677$	−24.7488	−0.951174	−0.475587	+	0.879669i	$0.657764\pi$
$678$	−47.8945	−1.83938
$679$	32.1703	1.23458
$680$	0	0
$681$	19.6068	0.751334
$682$	−16.7754	−0.642363
$683$	−7.74417	−0.296322	−0.148161	−	0.988963i	$-0.547335\pi$
$683$	−7.74417	−0.296322	−0.148161	+	0.988963i	$0.547335\pi$
$684$	10.0236	0.383261
$685$	0	0
$686$	−51.7646	−1.97638
$687$	−8.49443	−0.324083
$688$	−69.1366	−2.63581
$689$	62.3147	2.37400
$690$	0	0
$691$	−29.3966	−1.11830	−0.559150	−	0.829066i	$-0.688873\pi$
$691$	−29.3966	−1.11830	−0.559150	+	0.829066i	$0.688873\pi$
$692$	−106.302	−4.04101
$693$	−4.73364	−0.179816
$694$	16.1672	0.613698
$695$	0	0
$696$	−13.0923	−0.496263
$697$	−29.1529	−1.10424
$698$	12.5936	0.476677
$699$	−20.4750	−0.774435
$700$	0	0
$701$	−16.3181	−0.616324	−0.308162	−	0.951334i	$-0.599714\pi$
$701$	−16.3181	−0.616324	−0.308162	+	0.951334i	$0.599714\pi$
$702$	−53.7832	−2.02992
$703$	0.861702	0.0324997
$704$	−2.11055	−0.0795444
$705$	0	0
$706$	58.5039	2.20182
$707$	−20.2793	−0.762682
$708$	−35.9589	−1.35142
$709$	−17.0214	−0.639251	−0.319625	−	0.947544i	$-0.603557\pi$
$709$	−17.0214	−0.639251	−0.319625	+	0.947544i	$0.603557\pi$
$710$	0	0
$711$	−12.5389	−0.470247
$712$	−32.2048	−1.20693
$713$	−18.6630	−0.698936
$714$	13.6493	0.510812
$715$	0	0
$716$	51.5840	1.92779
$717$	−11.1354	−0.415859
$718$	−67.9950	−2.53755
$719$	−13.3760	−0.498842	−0.249421	−	0.968395i	$-0.580240\pi$
$719$	−13.3760	−0.498842	−0.249421	+	0.968395i	$0.580240\pi$
$720$	0	0
$721$	17.5398	0.653216
$722$	−2.56745	−0.0955506
$723$	24.9035	0.926170
$724$	−32.9904	−1.22608
$725$	0	0
$726$	2.32076	0.0861316
$727$	−34.6984	−1.28689	−0.643446	−	0.765492i	$-0.722496\pi$
$727$	−34.6984	−1.28689	−0.643446	+	0.765492i	$0.722496\pi$
$728$	64.5204	2.39129
$729$	7.65312	0.283449
$730$	0	0
$731$	23.7330	0.877796
$732$	−23.2022	−0.857578
$733$	8.67688	0.320488	0.160244	−	0.987077i	$-0.448772\pi$
$733$	8.67688	0.320488	0.160244	+	0.987077i	$0.448772\pi$
$734$	37.0767	1.36853
$735$	0	0
$736$	−19.9281	−0.734561
$737$	15.2043	0.560059
$738$	60.2418	2.21753
$739$	50.7472	1.86677	0.933383	−	0.358882i	$-0.116842\pi$
$739$	50.7472	1.86677	0.933383	+	0.358882i	$0.116842\pi$
$740$	0	0
$741$	4.04175	0.148477
$742$	77.5901	2.84842
$743$	−24.0074	−0.880745	−0.440372	−	0.897815i	$-0.645154\pi$
$743$	−24.0074	−0.880745	−0.440372	+	0.897815i	$0.645154\pi$
$744$	−39.3008	−1.44084
$745$	0	0
$746$	−9.79342	−0.358563
$747$	−26.8982	−0.984152
$748$	12.4540	0.455362
$749$	−5.87766	−0.214765
$750$	0	0
$751$	−6.31971	−0.230610	−0.115305	−	0.993330i	$-0.536784\pi$
$751$	−6.31971	−0.230610	−0.115305	+	0.993330i	$0.536784\pi$
$752$	5.26103	0.191850
$753$	−10.7612	−0.392161
$754$	24.9878	0.910000
$755$	0	0
$756$	−46.6489	−1.69660
$757$	−1.33647	−0.0485748	−0.0242874	−	0.999705i	$-0.507732\pi$
$757$	−1.33647	−0.0485748	−0.0242874	+	0.999705i	$0.507732\pi$
$758$	20.5424	0.746133
$759$	2.58191	0.0937173
$760$	0	0
$761$	26.0134	0.942986	0.471493	−	0.881870i	$-0.343715\pi$
$761$	26.0134	0.942986	0.471493	+	0.881870i	$0.343715\pi$
$762$	35.4321	1.28357
$763$	−1.07352	−0.0388639
$764$	−64.1637	−2.32136
$765$	0	0
$766$	34.8170	1.25799
$767$	38.7378	1.39874
$768$	23.6228	0.852415
$769$	14.6805	0.529394	0.264697	−	0.964332i	$-0.414728\pi$
$769$	14.6805	0.529394	0.264697	+	0.964332i	$0.414728\pi$
$770$	0	0
$771$	−7.41001	−0.266865
$772$	58.7970	2.11615
$773$	47.7853	1.71872	0.859358	−	0.511374i	$-0.170863\pi$
$773$	47.7853	1.71872	0.859358	+	0.511374i	$0.170863\pi$
$774$	−49.0421	−1.76278
$775$	0	0
$776$	98.7197	3.54383
$777$	−1.68904	−0.0605940
$778$	4.19898	0.150541
$779$	−10.7487	−0.385112
$780$	0	0
$781$	5.99426	0.214491
$782$	19.8902	0.711271
$783$	−10.1974	−0.364426
$784$	−18.1542	−0.648365
$785$	0	0
$786$	−33.8640	−1.20789
$787$	11.9838	0.427177	0.213589	−	0.976924i	$-0.431485\pi$
$787$	11.9838	0.427177	0.213589	+	0.976924i	$0.431485\pi$
$788$	−17.3944	−0.619651
$789$	−11.9204	−0.424377
$790$	0	0
$791$	44.7516	1.59119
$792$	−14.5259	−0.516155
$793$	24.9953	0.887609
$794$	23.7408	0.842528
$795$	0	0
$796$	73.0375	2.58875
$797$	−33.0078	−1.16920	−0.584598	−	0.811323i	$-0.698748\pi$
$797$	−33.0078	−1.16920	−0.584598	+	0.811323i	$0.698748\pi$
$798$	5.03252	0.178149
$799$	−1.80599	−0.0638913
$800$	0	0
$801$	−10.5647	−0.373286
$802$	0.770015	0.0271902
$803$	−10.8568	−0.383129
$804$	63.1072	2.22562
$805$	0	0
$806$	75.0089	2.64208
$807$	−17.5350	−0.617260
$808$	−62.2302	−2.18925
$809$	−29.9288	−1.05224	−0.526120	−	0.850411i	$-0.676354\pi$
$809$	−29.9288	−1.05224	−0.526120	+	0.850411i	$0.676354\pi$
$810$	0	0
$811$	−27.3858	−0.961647	−0.480824	−	0.876817i	$-0.659662\pi$
$811$	−27.3858	−0.961647	−0.480824	+	0.876817i	$0.659662\pi$
$812$	21.6731	0.760578
$813$	−3.22761	−0.113197
$814$	−2.21237	−0.0775437
$815$	0	0
$816$	19.3703	0.678095
$817$	8.75038	0.306137
$818$	−23.8374	−0.833456
$819$	21.1658	0.739594
$820$	0	0
$821$	17.1910	0.599971	0.299986	−	0.953944i	$-0.403018\pi$
$821$	17.1910	0.599971	0.299986	+	0.953944i	$0.403018\pi$
$822$	27.8249	0.970506
$823$	−5.25893	−0.183315	−0.0916575	−	0.995791i	$-0.529216\pi$
$823$	−5.25893	−0.183315	−0.0916575	+	0.995791i	$0.529216\pi$
$824$	53.8235	1.87503
$825$	0	0
$826$	48.2338	1.67827
$827$	18.3379	0.637672	0.318836	−	0.947810i	$-0.396708\pi$
$827$	18.3379	0.637672	0.318836	+	0.947810i	$0.396708\pi$
$828$	−28.6309	−0.994991
$829$	37.5846	1.30537	0.652684	−	0.757630i	$-0.273643\pi$
$829$	37.5846	1.30537	0.652684	+	0.757630i	$0.273643\pi$
$830$	0	0
$831$	−18.6813	−0.648048
$832$	9.43705	0.327171
$833$	6.23192	0.215923
$834$	−16.6173	−0.575410
$835$	0	0
$836$	4.59179	0.158811
$837$	−30.6108	−1.05807
$838$	23.9593	0.827662
$839$	−52.4722	−1.81154	−0.905770	−	0.423769i	$-0.860707\pi$
$839$	−52.4722	−1.81154	−0.905770	+	0.423769i	$0.860707\pi$
$840$	0	0
$841$	−24.2623	−0.836630
$842$	64.5599	2.22488
$843$	6.23607	0.214782
$844$	10.1213	0.348390
$845$	0	0
$846$	3.73192	0.128306
$847$	−2.16848	−0.0745097
$848$	110.111	3.78123
$849$	21.2732	0.730093
$850$	0	0
$851$	−2.46132	−0.0843730
$852$	24.8798	0.852368
$853$	29.5852	1.01298	0.506489	−	0.862246i	$-0.330943\pi$
$853$	29.5852	1.01298	0.506489	+	0.862246i	$0.330943\pi$
$854$	31.1225	1.06499
$855$	0	0
$856$	−18.0365	−0.616475
$857$	23.6715	0.808601	0.404301	−	0.914626i	$-0.367515\pi$
$857$	23.6715	0.808601	0.404301	+	0.914626i	$0.367515\pi$
$858$	−10.3770	−0.354264
$859$	−29.0072	−0.989714	−0.494857	−	0.868974i	$-0.664780\pi$
$859$	−29.0072	−0.989714	−0.494857	+	0.868974i	$0.664780\pi$
$860$	0	0
$861$	21.0688	0.718022
$862$	−96.4487	−3.28505
$863$	36.3796	1.23837	0.619187	−	0.785243i	$-0.287462\pi$
$863$	36.3796	1.23837	0.619187	+	0.785243i	$0.287462\pi$
$864$	−32.6858	−1.11199
$865$	0	0
$866$	23.7305	0.806394
$867$	8.71725	0.296053
$868$	65.0590	2.20825
$869$	−5.74408	−0.194855
$870$	0	0
$871$	−67.9842	−2.30356
$872$	−3.29425	−0.111557
$873$	32.3848	1.09606
$874$	7.33354	0.248061
$875$	0	0
$876$	−45.0624	−1.52252
$877$	56.9075	1.92163	0.960815	−	0.277192i	$-0.0894037\pi$
$877$	56.9075	1.92163	0.960815	+	0.277192i	$0.0894037\pi$
$878$	−2.41809	−0.0816066
$879$	7.86671	0.265338
$880$	0	0
$881$	53.1897	1.79201	0.896004	−	0.444047i	$-0.146458\pi$
$881$	53.1897	1.79201	0.896004	+	0.444047i	$0.146458\pi$
$882$	−12.8777	−0.433615
$883$	−22.9447	−0.772151	−0.386075	−	0.922467i	$-0.626169\pi$
$883$	−22.9447	−0.772151	−0.386075	+	0.922467i	$0.626169\pi$
$884$	−55.6862	−1.87293
$885$	0	0
$886$	44.3467	1.48986
$887$	−35.0993	−1.17852	−0.589260	−	0.807943i	$-0.700581\pi$
$887$	−35.0993	−1.17852	−0.589260	+	0.807943i	$0.700581\pi$
$888$	−5.18308	−0.173933
$889$	−33.1070	−1.11037
$890$	0	0
$891$	−2.31399	−0.0775217
$892$	65.9704	2.20885
$893$	−0.665870	−0.0222825
$894$	8.67961	0.290289
$895$	0	0
$896$	−18.5075	−0.618294
$897$	−11.5447	−0.385465
$898$	53.8952	1.79851
$899$	14.2218	0.474325
$900$	0	0
$901$	−37.7986	−1.25925
$902$	27.5967	0.918871
$903$	−17.1518	−0.570777
$904$	137.327	4.56744
$905$	0	0
$906$	−17.8881	−0.594294
$907$	−49.6640	−1.64906	−0.824532	−	0.565815i	$-0.808562\pi$
$907$	−49.6640	−1.64906	−0.824532	+	0.565815i	$0.808562\pi$
$908$	−99.6002	−3.30535
$909$	−20.4145	−0.677107
$910$	0	0
$911$	44.2360	1.46560	0.732802	−	0.680442i	$-0.238212\pi$
$911$	44.2360	1.46560	0.732802	+	0.680442i	$0.238212\pi$
$912$	7.14184	0.236490
$913$	−12.3220	−0.407800
$914$	−74.4286	−2.46188
$915$	0	0
$916$	43.1507	1.42574
$917$	31.6419	1.04491
$918$	32.6236	1.07674
$919$	−36.2925	−1.19718	−0.598590	−	0.801055i	$-0.704272\pi$
$919$	−36.2925	−1.19718	−0.598590	+	0.801055i	$0.704272\pi$
$920$	0	0
$921$	−0.0312506	−0.00102974
$922$	63.6508	2.09623
$923$	−26.8025	−0.882216
$924$	−9.00048	−0.296094
$925$	0	0
$926$	−1.94387	−0.0638796
$927$	17.6567	0.579923
$928$	15.1859	0.498501
$929$	20.6900	0.678818	0.339409	−	0.940639i	$-0.389773\pi$
$929$	20.6900	0.678818	0.339409	+	0.940639i	$0.389773\pi$
$930$	0	0
$931$	2.29772	0.0753046
$932$	104.010	3.40698
$933$	15.2644	0.499734
$934$	−14.2355	−0.465800
$935$	0	0
$936$	64.9506	2.12298
$937$	58.8084	1.92119	0.960594	−	0.277957i	$-0.0896573\pi$
$937$	58.8084	1.92119	0.960594	+	0.277957i	$0.0896573\pi$
$938$	−84.6494	−2.76390
$939$	−1.33482	−0.0435602
$940$	0	0
$941$	−18.1745	−0.592471	−0.296235	−	0.955115i	$-0.595731\pi$
$941$	−18.1745	−0.592471	−0.296235	+	0.955115i	$0.595731\pi$
$942$	18.0652	0.588595
$943$	30.7021	0.999797
$944$	68.4504	2.22787
$945$	0	0
$946$	−22.4662	−0.730438
$947$	−9.22596	−0.299803	−0.149902	−	0.988701i	$-0.547896\pi$
$947$	−9.22596	−0.299803	−0.149902	+	0.988701i	$0.547896\pi$
$948$	−23.8414	−0.774333
$949$	48.5448	1.57583
$950$	0	0
$951$	22.1712	0.718951
$952$	−39.1365	−1.26842
$953$	−34.8382	−1.12852	−0.564260	−	0.825597i	$-0.690838\pi$
$953$	−34.8382	−1.12852	−0.564260	+	0.825597i	$0.690838\pi$
$954$	78.1074	2.52882
$955$	0	0
$956$	56.5665	1.82949
$957$	−1.96750	−0.0636001
$958$	22.5617	0.728935
$959$	−25.9991	−0.839554
$960$	0	0
$961$	11.6915	0.377145
$962$	9.89234	0.318942
$963$	−5.91685	−0.190668
$964$	−126.507	−4.07451
$965$	0	0
$966$	−14.3746	−0.462496
$967$	21.5696	0.693631	0.346816	−	0.937933i	$-0.387263\pi$
$967$	21.5696	0.693631	0.346816	+	0.937933i	$0.387263\pi$
$968$	−6.65430	−0.213877
$969$	−2.45163	−0.0787576
$970$	0	0
$971$	52.0792	1.67130	0.835650	−	0.549262i	$-0.185091\pi$
$971$	52.0792	1.67130	0.835650	+	0.549262i	$0.185091\pi$
$972$	−74.1414	−2.37809
$973$	15.5269	0.497769
$974$	94.7207	3.03505
$975$	0	0
$976$	44.1671	1.41376
$977$	14.7908	0.473200	0.236600	−	0.971607i	$-0.423967\pi$
$977$	14.7908	0.473200	0.236600	+	0.971607i	$0.423967\pi$
$978$	8.37210	0.267710
$979$	−4.83970	−0.154677
$980$	0	0
$981$	−1.08067	−0.0345033
$982$	15.7220	0.501708
$983$	35.1347	1.12062	0.560311	−	0.828282i	$-0.310682\pi$
$983$	35.1347	1.12062	0.560311	+	0.828282i	$0.310682\pi$
$984$	64.6528	2.06105
$985$	0	0
$986$	−15.1570	−0.482696
$987$	1.30519	0.0415446
$988$	−20.5316	−0.653197
$989$	−24.9942	−0.794768
$990$	0	0
$991$	−50.5723	−1.60648	−0.803241	−	0.595654i	$-0.796893\pi$
$991$	−50.5723	−1.60648	−0.803241	+	0.595654i	$0.796893\pi$
$992$	45.5854	1.44734
$993$	−17.9141	−0.568488
$994$	−33.3727	−1.05852
$995$	0	0
$996$	−51.1439	−1.62056
$997$	16.9360	0.536370	0.268185	−	0.963367i	$-0.413576\pi$
$997$	16.9360	0.536370	0.268185	+	0.963367i	$0.413576\pi$
$998$	43.4295	1.37474
$999$	−4.03702	−0.127726

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.k.1.1		6
5.4	even	2		1045.2.a.g.1.6	✓	6
15.14	odd	2		9405.2.a.w.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.g.1.6	✓	6	5.4	even	2
5225.2.a.k.1.1		6	1.1	even	1	trivial
9405.2.a.w.1.1		6	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)

\(f(q)\)	\(=\)	\(q-2.56745 q^{2} -0.903918 q^{3} +4.59179 q^{4} +2.32076 q^{6} -2.16848 q^{7} -6.65430 q^{8} -2.18293 q^{9} -1.00000 q^{11} -4.15060 q^{12} +4.47137 q^{13} +5.56745 q^{14} +7.90099 q^{16} -2.71222 q^{17} +5.60457 q^{18} -1.00000 q^{19} +1.96012 q^{21} +2.56745 q^{22} +2.85635 q^{23} +6.01494 q^{24} -11.4800 q^{26} +4.68494 q^{27} -9.95719 q^{28} -2.17663 q^{29} -6.53387 q^{31} -6.97678 q^{32} +0.903918 q^{33} +6.96349 q^{34} -10.0236 q^{36} -0.861702 q^{37} +2.56745 q^{38} -4.04175 q^{39} +10.7487 q^{41} -5.03252 q^{42} -8.75038 q^{43} -4.59179 q^{44} -7.33354 q^{46} +0.665870 q^{47} -7.14184 q^{48} -2.29772 q^{49} +2.45163 q^{51} +20.5316 q^{52} +13.9364 q^{53} -12.0284 q^{54} +14.4297 q^{56} +0.903918 q^{57} +5.58839 q^{58} +8.66353 q^{59} +5.59008 q^{61} +16.7754 q^{62} +4.73364 q^{63} +2.11055 q^{64} -2.32076 q^{66} -15.2043 q^{67} -12.4540 q^{68} -2.58191 q^{69} -5.99426 q^{71} +14.5259 q^{72} +10.8568 q^{73} +2.21237 q^{74} -4.59179 q^{76} +2.16848 q^{77} +10.3770 q^{78} +5.74408 q^{79} +2.31399 q^{81} -27.5967 q^{82} +12.3220 q^{83} +9.00048 q^{84} +22.4662 q^{86} +1.96750 q^{87} +6.65430 q^{88} +4.83970 q^{89} -9.69605 q^{91} +13.1158 q^{92} +5.90608 q^{93} -1.70959 q^{94} +6.30643 q^{96} -14.8355 q^{97} +5.89927 q^{98} +2.18293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more