LMFDB - Embedded newform 5225.2.a.p.1.7

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 = [9,-3,-3,9,0,0,-13] 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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 $ x^9 - 3x^8 - 9x^7 + 29x^6 + 23x^5 - 84x^4 - 23x^3 + 89x^2 + 8x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.24377$ 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)

$f(q)$	$=$	\(q+1.24377 q^{2} -3.09454 q^{3} -0.453038 q^{4} -3.84889 q^{6} -3.83759 q^{7} -3.05101 q^{8} +6.57616 q^{9} +1.00000 q^{11} +1.40194 q^{12} +2.44695 q^{13} -4.77307 q^{14} -2.88868 q^{16} -4.44695 q^{17} +8.17922 q^{18} +1.00000 q^{19} +11.8755 q^{21} +1.24377 q^{22} +5.34873 q^{23} +9.44147 q^{24} +3.04344 q^{26} -11.0665 q^{27} +1.73857 q^{28} -9.42505 q^{29} -2.25347 q^{31} +2.50917 q^{32} -3.09454 q^{33} -5.53098 q^{34} -2.97925 q^{36} +9.50113 q^{37} +1.24377 q^{38} -7.57217 q^{39} +6.98607 q^{41} +14.7704 q^{42} -3.25340 q^{43} -0.453038 q^{44} +6.65259 q^{46} +4.99452 q^{47} +8.93913 q^{48} +7.72706 q^{49} +13.7612 q^{51} -1.10856 q^{52} +6.72067 q^{53} -13.7642 q^{54} +11.7085 q^{56} -3.09454 q^{57} -11.7226 q^{58} +3.81487 q^{59} +7.33227 q^{61} -2.80280 q^{62} -25.2366 q^{63} +8.89820 q^{64} -3.84889 q^{66} +8.95031 q^{67} +2.01463 q^{68} -16.5518 q^{69} +4.94182 q^{71} -20.0639 q^{72} +3.88580 q^{73} +11.8172 q^{74} -0.453038 q^{76} -3.83759 q^{77} -9.41804 q^{78} -13.5466 q^{79} +14.5174 q^{81} +8.68906 q^{82} -8.59991 q^{83} -5.38007 q^{84} -4.04648 q^{86} +29.1661 q^{87} -3.05101 q^{88} -3.34023 q^{89} -9.39038 q^{91} -2.42317 q^{92} +6.97346 q^{93} +6.21203 q^{94} -7.76472 q^{96} +8.42974 q^{97} +9.61068 q^{98} +6.57616 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9} + 9 q^{11} + 5 q^{12} - 5 q^{13} - 2 q^{14} + q^{16} - 13 q^{17} - q^{18} + 9 q^{19} + q^{21} - 3 q^{22} - 8 q^{23} - 7 q^{24} + 8 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) $ 9 * q - 3 * q^2 - 3 * q^3 + 9 * q^4 - 13 * q^7 - 9 * q^8 + 12 * q^9 + 9 * q^11 + 5 * q^12 - 5 * q^13 - 2 * q^14 + q^16 - 13 * q^17 - q^18 + 9 * q^19 + q^21 - 3 * q^22 - 8 * q^23 - 7 * q^24 + 8 * q^26 - 6 * q^27 - 10 * q^28 - 3 * q^29 - 9 * q^31 - 6 * q^32 - 3 * q^33 - 2 * q^34 - 15 * q^36 + 7 * q^37 - 3 * q^38 - 2 * q^39 + 9 * q^41 + 9 * q^42 - 23 * q^43 + 9 * q^44 - 32 * q^46 - 20 * q^47 + 18 * q^48 - 4 * q^49 + 8 * q^51 - 9 * q^52 + 5 * q^53 + 9 * q^54 + 4 * q^56 - 3 * q^57 - 3 * q^58 + 19 * q^59 + q^61 - 18 * q^62 - 24 * q^63 + 23 * q^64 + 10 * q^67 - 9 * q^68 - 28 * q^69 - 24 * q^72 - 12 * q^73 + 5 * q^74 + 9 * q^76 - 13 * q^77 - 19 * q^78 - 21 * q^79 - 3 * q^81 + 14 * q^82 - 47 * q^83 - 11 * q^84 + 12 * q^86 + 2 * q^87 - 9 * q^88 - 2 * q^89 + 2 * q^91 + 19 * q^92 + 2 * q^93 + 26 * q^94 - 55 * q^96 + 32 * q^97 + 12 * 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 $

$2$	1.24377	0.879478	0.439739	−	0.898126i	$-0.355071\pi$
$2$	1.24377	0.879478	0.439739	+	0.898126i	$0.355071\pi$
$3$	−3.09454	−1.78663	−0.893316	−	0.449430i	$-0.851627\pi$
$3$	−3.09454	−1.78663	−0.893316	+	0.449430i	$0.851627\pi$
$4$	−0.453038	−0.226519
$5$	0	0
$5$	0	0
$6$	−3.84889	−1.57130
$7$	−3.83759	−1.45047	−0.725235	−	0.688501i	$-0.758269\pi$
$7$	−3.83759	−1.45047	−0.725235	+	0.688501i	$0.758269\pi$
$8$	−3.05101	−1.07870
$9$	6.57616	2.19205
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.40194	0.404706
$13$	2.44695	0.678661	0.339331	−	0.940667i	$-0.389799\pi$
$13$	2.44695	0.678661	0.339331	+	0.940667i	$0.389799\pi$
$14$	−4.77307	−1.27566
$15$	0	0
$16$	−2.88868	−0.722170
$17$	−4.44695	−1.07854	−0.539272	−	0.842132i	$-0.681300\pi$
$17$	−4.44695	−1.07854	−0.539272	+	0.842132i	$0.681300\pi$
$18$	8.17922	1.92786
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	11.8755	2.59146
$22$	1.24377	0.265173
$23$	5.34873	1.11529	0.557644	−	0.830081i	$-0.311706\pi$
$23$	5.34873	1.11529	0.557644	+	0.830081i	$0.311706\pi$
$24$	9.44147	1.92723
$25$	0	0
$26$	3.04344	0.596868
$27$	−11.0665	−2.12976
$28$	1.73857	0.328559
$29$	−9.42505	−1.75019	−0.875094	−	0.483954i	$-0.839200\pi$
$29$	−9.42505	−1.75019	−0.875094	+	0.483954i	$0.839200\pi$
$30$	0	0
$31$	−2.25347	−0.404736	−0.202368	−	0.979310i	$-0.564864\pi$
$31$	−2.25347	−0.404736	−0.202368	+	0.979310i	$0.564864\pi$
$32$	2.50917	0.443563
$33$	−3.09454	−0.538690
$34$	−5.53098	−0.948555
$35$	0	0
$36$	−2.97925	−0.496541
$37$	9.50113	1.56198	0.780988	−	0.624546i	$-0.214716\pi$
$37$	9.50113	1.56198	0.780988	+	0.624546i	$0.214716\pi$
$38$	1.24377	0.201766
$39$	−7.57217	−1.21252
$40$	0	0
$41$	6.98607	1.09104	0.545521	−	0.838097i	$-0.316332\pi$
$41$	6.98607	1.09104	0.545521	+	0.838097i	$0.316332\pi$
$42$	14.7704	2.27913
$43$	−3.25340	−0.496138	−0.248069	−	0.968742i	$-0.579796\pi$
$43$	−3.25340	−0.496138	−0.248069	+	0.968742i	$0.579796\pi$
$44$	−0.453038	−0.0682980
$45$	0	0
$46$	6.65259	0.980870
$47$	4.99452	0.728526	0.364263	−	0.931296i	$-0.381321\pi$
$47$	4.99452	0.728526	0.364263	+	0.931296i	$0.381321\pi$
$48$	8.93913	1.29025
$49$	7.72706	1.10387
$50$	0	0
$51$	13.7612	1.92696
$52$	−1.10856	−0.153730
$53$	6.72067	0.923154	0.461577	−	0.887100i	$-0.347284\pi$
$53$	6.72067	0.923154	0.461577	+	0.887100i	$0.347284\pi$
$54$	−13.7642	−1.87307
$55$	0	0
$56$	11.7085	1.56462
$57$	−3.09454	−0.409881
$58$	−11.7226	−1.53925
$59$	3.81487	0.496654	0.248327	−	0.968676i	$-0.420119\pi$
$59$	3.81487	0.496654	0.248327	+	0.968676i	$0.420119\pi$
$60$	0	0
$61$	7.33227	0.938801	0.469400	−	0.882985i	$-0.344470\pi$
$61$	7.33227	0.938801	0.469400	+	0.882985i	$0.344470\pi$
$62$	−2.80280	−0.355956
$63$	−25.2366	−3.17951
$64$	8.89820	1.11227
$65$	0	0
$66$	−3.84889	−0.473766
$67$	8.95031	1.09345	0.546727	−	0.837311i	$-0.315873\pi$
$67$	8.95031	1.09345	0.546727	+	0.837311i	$0.315873\pi$
$68$	2.01463	0.244310
$69$	−16.5518	−1.99261
$70$	0	0
$71$	4.94182	0.586486	0.293243	−	0.956038i	$-0.405266\pi$
$71$	4.94182	0.586486	0.293243	+	0.956038i	$0.405266\pi$
$72$	−20.0639	−2.36456
$73$	3.88580	0.454799	0.227400	−	0.973802i	$-0.426978\pi$
$73$	3.88580	0.454799	0.227400	+	0.973802i	$0.426978\pi$
$74$	11.8172	1.37372
$75$	0	0
$76$	−0.453038	−0.0519670
$77$	−3.83759	−0.437333
$78$	−9.41804	−1.06638
$79$	−13.5466	−1.52411	−0.762053	−	0.647514i	$-0.775809\pi$
$79$	−13.5466	−1.52411	−0.762053	+	0.647514i	$0.775809\pi$
$80$	0	0
$81$	14.5174	1.61304
$82$	8.68906	0.959546
$83$	−8.59991	−0.943963	−0.471982	−	0.881608i	$-0.656461\pi$
$83$	−8.59991	−0.943963	−0.471982	+	0.881608i	$0.656461\pi$
$84$	−5.38007	−0.587014
$85$	0	0
$86$	−4.04648	−0.436343
$87$	29.1661	3.12694
$88$	−3.05101	−0.325239
$89$	−3.34023	−0.354063	−0.177032	−	0.984205i	$-0.556649\pi$
$89$	−3.34023	−0.354063	−0.177032	+	0.984205i	$0.556649\pi$
$90$	0	0
$91$	−9.39038	−0.984379
$92$	−2.42317	−0.252633
$93$	6.97346	0.723114
$94$	6.21203	0.640722
$95$	0	0
$96$	−7.76472	−0.792484
$97$	8.42974	0.855911	0.427955	−	0.903800i	$-0.359234\pi$
$97$	8.42974	0.855911	0.427955	+	0.903800i	$0.359234\pi$
$98$	9.61068	0.970826
$99$	6.57616	0.660929
$100$	0	0
$101$	14.9671	1.48928	0.744639	−	0.667467i	$-0.232621\pi$
$101$	14.9671	1.48928	0.744639	+	0.667467i	$0.232621\pi$
$102$	17.1158	1.69472
$103$	−0.113124	−0.0111464	−0.00557320	−	0.999984i	$-0.501774\pi$
$103$	−0.113124	−0.0111464	−0.00557320	+	0.999984i	$0.501774\pi$
$104$	−7.46567	−0.732069
$105$	0	0
$106$	8.35896	0.811894
$107$	−16.5759	−1.60246	−0.801229	−	0.598358i	$-0.795820\pi$
$107$	−16.5759	−1.60246	−0.801229	+	0.598358i	$0.795820\pi$
$108$	5.01356	0.482430
$109$	−7.56098	−0.724211	−0.362105	−	0.932137i	$-0.617942\pi$
$109$	−7.56098	−0.724211	−0.362105	+	0.932137i	$0.617942\pi$
$110$	0	0
$111$	−29.4016	−2.79068
$112$	11.0856	1.04749
$113$	−3.29190	−0.309676	−0.154838	−	0.987940i	$-0.549486\pi$
$113$	−3.29190	−0.309676	−0.154838	+	0.987940i	$0.549486\pi$
$114$	−3.84889	−0.360482
$115$	0	0
$116$	4.26990	0.396450
$117$	16.0915	1.48766
$118$	4.74482	0.436796
$119$	17.0655	1.56440
$120$	0	0
$121$	1.00000	0.0909091
$122$	9.11965	0.825655
$123$	−21.6187	−1.94929
$124$	1.02091	0.0916803
$125$	0	0
$126$	−31.3885	−2.79631
$127$	−21.6432	−1.92053	−0.960264	−	0.279094i	$-0.909966\pi$
$127$	−21.6432	−1.92053	−0.960264	+	0.279094i	$0.909966\pi$
$128$	6.04896	0.534658
$129$	10.0678	0.886416
$130$	0	0
$131$	−17.0780	−1.49211	−0.746057	−	0.665882i	$-0.768055\pi$
$131$	−17.0780	−1.49211	−0.746057	+	0.665882i	$0.768055\pi$
$132$	1.40194	0.122023
$133$	−3.83759	−0.332761
$134$	11.1321	0.961669
$135$	0	0
$136$	13.5677	1.16342
$137$	−14.0843	−1.20330	−0.601650	−	0.798760i	$-0.705490\pi$
$137$	−14.0843	−1.20330	−0.601650	+	0.798760i	$0.705490\pi$
$138$	−20.5867	−1.75245
$139$	−14.6799	−1.24513	−0.622567	−	0.782567i	$-0.713910\pi$
$139$	−14.6799	−1.24513	−0.622567	+	0.782567i	$0.713910\pi$
$140$	0	0
$141$	−15.4557	−1.30161
$142$	6.14648	0.515801
$143$	2.44695	0.204624
$144$	−18.9964	−1.58304
$145$	0	0
$146$	4.83305	0.399986
$147$	−23.9117	−1.97220
$148$	−4.30437	−0.353817
$149$	11.0069	0.901718	0.450859	−	0.892595i	$-0.351118\pi$
$149$	11.0069	0.901718	0.450859	+	0.892595i	$0.351118\pi$
$150$	0	0
$151$	−4.88477	−0.397517	−0.198758	−	0.980049i	$-0.563691\pi$
$151$	−4.88477	−0.397517	−0.198758	+	0.980049i	$0.563691\pi$
$152$	−3.05101	−0.247470
$153$	−29.2438	−2.36422
$154$	−4.77307	−0.384625
$155$	0	0
$156$	3.43048	0.274658
$157$	−3.30059	−0.263416	−0.131708	−	0.991289i	$-0.542046\pi$
$157$	−3.30059	−0.263416	−0.131708	+	0.991289i	$0.542046\pi$
$158$	−16.8488	−1.34042
$159$	−20.7973	−1.64934
$160$	0	0
$161$	−20.5262	−1.61769
$162$	18.0562	1.41863
$163$	−5.58288	−0.437285	−0.218642	−	0.975805i	$-0.570163\pi$
$163$	−5.58288	−0.437285	−0.218642	+	0.975805i	$0.570163\pi$
$164$	−3.16495	−0.247141
$165$	0	0
$166$	−10.6963	−0.830195
$167$	21.7614	1.68395	0.841973	−	0.539520i	$-0.181394\pi$
$167$	21.7614	1.68395	0.841973	+	0.539520i	$0.181394\pi$
$168$	−36.2325	−2.79539
$169$	−7.01244	−0.539419
$170$	0	0
$171$	6.57616	0.502891
$172$	1.47391	0.112385
$173$	−19.7081	−1.49838	−0.749188	−	0.662357i	$-0.769556\pi$
$173$	−19.7081	−1.49838	−0.749188	+	0.662357i	$0.769556\pi$
$174$	36.2760	2.75007
$175$	0	0
$176$	−2.88868	−0.217743
$177$	−11.8053	−0.887337
$178$	−4.15447	−0.311391
$179$	−23.0630	−1.72381	−0.861905	−	0.507070i	$-0.830728\pi$
$179$	−23.0630	−1.72381	−0.861905	+	0.507070i	$0.830728\pi$
$180$	0	0
$181$	7.03753	0.523096	0.261548	−	0.965190i	$-0.415767\pi$
$181$	7.03753	0.523096	0.261548	+	0.965190i	$0.415767\pi$
$182$	−11.6795	−0.865739
$183$	−22.6900	−1.67729
$184$	−16.3190	−1.20306
$185$	0	0
$186$	8.67337	0.635963
$187$	−4.44695	−0.325193
$188$	−2.26271	−0.165025
$189$	42.4688	3.08915
$190$	0	0
$191$	13.6102	0.984798	0.492399	−	0.870370i	$-0.336120\pi$
$191$	13.6102	0.984798	0.492399	+	0.870370i	$0.336120\pi$
$192$	−27.5358	−1.98722
$193$	−0.756294	−0.0544392	−0.0272196	−	0.999629i	$-0.508665\pi$
$193$	−0.756294	−0.0544392	−0.0272196	+	0.999629i	$0.508665\pi$
$194$	10.4847	0.752755
$195$	0	0
$196$	−3.50065	−0.250046
$197$	−3.15982	−0.225128	−0.112564	−	0.993644i	$-0.535906\pi$
$197$	−3.15982	−0.225128	−0.112564	+	0.993644i	$0.535906\pi$
$198$	8.17922	0.581272
$199$	−12.5193	−0.887471	−0.443736	−	0.896158i	$-0.646347\pi$
$199$	−12.5193	−0.887471	−0.443736	+	0.896158i	$0.646347\pi$
$200$	0	0
$201$	−27.6971	−1.95360
$202$	18.6156	1.30979
$203$	36.1694	2.53860
$204$	−6.23436	−0.436493
$205$	0	0
$206$	−0.140700	−0.00980301
$207$	35.1741	2.44477
$208$	−7.06846	−0.490109
$209$	1.00000	0.0691714
$210$	0	0
$211$	14.8799	1.02437	0.512186	−	0.858875i	$-0.328836\pi$
$211$	14.8799	1.02437	0.512186	+	0.858875i	$0.328836\pi$
$212$	−3.04471	−0.209112
$213$	−15.2926	−1.04783
$214$	−20.6167	−1.40933
$215$	0	0
$216$	33.7642	2.29736
$217$	8.64790	0.587058
$218$	−9.40412	−0.636927
$219$	−12.0248	−0.812558
$220$	0	0
$221$	−10.8815	−0.731966
$222$	−36.5688	−2.45434
$223$	6.74009	0.451350	0.225675	−	0.974203i	$-0.427541\pi$
$223$	6.74009	0.451350	0.225675	+	0.974203i	$0.427541\pi$
$224$	−9.62916	−0.643375
$225$	0	0
$226$	−4.09437	−0.272353
$227$	−9.13145	−0.606075	−0.303038	−	0.952979i	$-0.598001\pi$
$227$	−9.13145	−0.606075	−0.303038	+	0.952979i	$0.598001\pi$
$228$	1.40194	0.0928458
$229$	−20.5688	−1.35922	−0.679612	−	0.733572i	$-0.737852\pi$
$229$	−20.5688	−1.35922	−0.679612	+	0.733572i	$0.737852\pi$
$230$	0	0
$231$	11.8755	0.781354
$232$	28.7559	1.88792
$233$	−6.63624	−0.434754	−0.217377	−	0.976088i	$-0.569750\pi$
$233$	−6.63624	−0.434754	−0.217377	+	0.976088i	$0.569750\pi$
$234$	20.0141	1.30837
$235$	0	0
$236$	−1.72828	−0.112501
$237$	41.9203	2.72302
$238$	21.2256	1.37585
$239$	13.8790	0.897758	0.448879	−	0.893593i	$-0.351823\pi$
$239$	13.8790	0.897758	0.448879	+	0.893593i	$0.351823\pi$
$240$	0	0
$241$	12.3764	0.797232	0.398616	−	0.917118i	$-0.369491\pi$
$241$	12.3764	0.797232	0.398616	+	0.917118i	$0.369491\pi$
$242$	1.24377	0.0799525
$243$	−11.7249	−0.752150
$244$	−3.32179	−0.212656
$245$	0	0
$246$	−26.8886	−1.71436
$247$	2.44695	0.155696
$248$	6.87538	0.436587
$249$	26.6127	1.68651
$250$	0	0
$251$	26.9081	1.69842	0.849212	−	0.528051i	$-0.177077\pi$
$251$	26.9081	1.69842	0.849212	+	0.528051i	$0.177077\pi$
$252$	11.4331	0.720218
$253$	5.34873	0.336272
$254$	−26.9192	−1.68906
$255$	0	0
$256$	−10.2729	−0.642055
$257$	23.3930	1.45921	0.729607	−	0.683867i	$-0.239703\pi$
$257$	23.3930	1.45921	0.729607	+	0.683867i	$0.239703\pi$
$258$	12.5220	0.779584
$259$	−36.4614	−2.26560
$260$	0	0
$261$	−61.9806	−3.83650
$262$	−21.2411	−1.31228
$263$	27.2224	1.67860	0.839302	−	0.543665i	$-0.182964\pi$
$263$	27.2224	1.67860	0.839302	+	0.543665i	$0.182964\pi$
$264$	9.44147	0.581082
$265$	0	0
$266$	−4.77307	−0.292656
$267$	10.3364	0.632580
$268$	−4.05483	−0.247688
$269$	14.5520	0.887249	0.443624	−	0.896213i	$-0.353692\pi$
$269$	14.5520	0.887249	0.443624	+	0.896213i	$0.353692\pi$
$270$	0	0
$271$	−31.7669	−1.92970	−0.964851	−	0.262798i	$-0.915355\pi$
$271$	−31.7669	−1.92970	−0.964851	+	0.262798i	$0.915355\pi$
$272$	12.8458	0.778892
$273$	29.0589	1.75872
$274$	−17.5176	−1.05828
$275$	0	0
$276$	7.49860	0.451363
$277$	13.9031	0.835358	0.417679	−	0.908595i	$-0.362844\pi$
$277$	13.9031	0.835358	0.417679	+	0.908595i	$0.362844\pi$
$278$	−18.2584	−1.09507
$279$	−14.8192	−0.887202
$280$	0	0
$281$	−9.22180	−0.550126	−0.275063	−	0.961426i	$-0.588699\pi$
$281$	−9.22180	−0.550126	−0.275063	+	0.961426i	$0.588699\pi$
$282$	−19.2234	−1.14473
$283$	15.2824	0.908444	0.454222	−	0.890888i	$-0.349917\pi$
$283$	15.2824	0.908444	0.454222	+	0.890888i	$0.349917\pi$
$284$	−2.23883	−0.132850
$285$	0	0
$286$	3.04344	0.179962
$287$	−26.8096	−1.58252
$288$	16.5007	0.972313
$289$	2.77535	0.163256
$290$	0	0
$291$	−26.0861	−1.52920
$292$	−1.76042	−0.103021
$293$	−1.81359	−0.105951	−0.0529756	−	0.998596i	$-0.516871\pi$
$293$	−1.81359	−0.105951	−0.0529756	+	0.998596i	$0.516871\pi$
$294$	−29.7406	−1.73451
$295$	0	0
$296$	−28.9881	−1.68490
$297$	−11.0665	−0.642146
$298$	13.6900	0.793041
$299$	13.0881	0.756902
$300$	0	0
$301$	12.4852	0.719634
$302$	−6.07552	−0.349607
$303$	−46.3161	−2.66079
$304$	−2.88868	−0.165677
$305$	0	0
$306$	−36.3726	−2.07928
$307$	−17.7976	−1.01576	−0.507880	−	0.861428i	$-0.669571\pi$
$307$	−17.7976	−1.01576	−0.507880	+	0.861428i	$0.669571\pi$
$308$	1.73857	0.0990642
$309$	0.350065	0.0199145
$310$	0	0
$311$	−17.2822	−0.979982	−0.489991	−	0.871727i	$-0.663000\pi$
$311$	−17.2822	−0.979982	−0.489991	+	0.871727i	$0.663000\pi$
$312$	23.1028	1.30794
$313$	29.6212	1.67429	0.837145	−	0.546981i	$-0.184223\pi$
$313$	29.6212	1.67429	0.837145	+	0.546981i	$0.184223\pi$
$314$	−4.10518	−0.231669
$315$	0	0
$316$	6.13710	0.345239
$317$	−31.6170	−1.77579	−0.887893	−	0.460049i	$-0.847832\pi$
$317$	−31.6170	−1.77579	−0.887893	+	0.460049i	$0.847832\pi$
$318$	−25.8671	−1.45056
$319$	−9.42505	−0.527701
$320$	0	0
$321$	51.2949	2.86300
$322$	−25.5299	−1.42272
$323$	−4.44695	−0.247435
$324$	−6.57691	−0.365384
$325$	0	0
$326$	−6.94381	−0.384582
$327$	23.3977	1.29390
$328$	−21.3146	−1.17690
$329$	−19.1669	−1.05671
$330$	0	0
$331$	−2.20361	−0.121121	−0.0605607	−	0.998165i	$-0.519289\pi$
$331$	−2.20361	−0.121121	−0.0605607	+	0.998165i	$0.519289\pi$
$332$	3.89608	0.213825
$333$	62.4809	3.42393
$334$	27.0661	1.48099
$335$	0	0
$336$	−34.3047	−1.87147
$337$	−9.15267	−0.498577	−0.249289	−	0.968429i	$-0.580197\pi$
$337$	−9.15267	−0.498577	−0.249289	+	0.968429i	$0.580197\pi$
$338$	−8.72186	−0.474407
$339$	10.1869	0.553277
$340$	0	0
$341$	−2.25347	−0.122032
$342$	8.17922	0.442282
$343$	−2.79017	−0.150655
$344$	9.92616	0.535183
$345$	0	0
$346$	−24.5123	−1.31779
$347$	−10.9854	−0.589726	−0.294863	−	0.955540i	$-0.595274\pi$
$347$	−10.9854	−0.589726	−0.294863	+	0.955540i	$0.595274\pi$
$348$	−13.2134	−0.708310
$349$	−18.1845	−0.973396	−0.486698	−	0.873570i	$-0.661799\pi$
$349$	−18.1845	−0.973396	−0.486698	+	0.873570i	$0.661799\pi$
$350$	0	0
$351$	−27.0793	−1.44538
$352$	2.50917	0.133739
$353$	0.0447307	0.00238077	0.00119039	−	0.999999i	$-0.499621\pi$
$353$	0.0447307	0.00238077	0.00119039	+	0.999999i	$0.499621\pi$
$354$	−14.6830	−0.780393
$355$	0	0
$356$	1.51325	0.0802019
$357$	−52.8100	−2.79500
$358$	−28.6851	−1.51605
$359$	−16.6983	−0.881303	−0.440652	−	0.897678i	$-0.645253\pi$
$359$	−16.6983	−0.881303	−0.440652	+	0.897678i	$0.645253\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	8.75307	0.460051
$363$	−3.09454	−0.162421
$364$	4.25419	0.222980
$365$	0	0
$366$	−28.2211	−1.47514
$367$	−6.08480	−0.317624	−0.158812	−	0.987309i	$-0.550766\pi$
$367$	−6.08480	−0.317624	−0.158812	+	0.987309i	$0.550766\pi$
$368$	−15.4508	−0.805427
$369$	45.9415	2.39162
$370$	0	0
$371$	−25.7911	−1.33901
$372$	−3.15924	−0.163799
$373$	−6.64786	−0.344213	−0.172107	−	0.985078i	$-0.555057\pi$
$373$	−6.64786	−0.344213	−0.172107	+	0.985078i	$0.555057\pi$
$374$	−5.53098	−0.286000
$375$	0	0
$376$	−15.2384	−0.785858
$377$	−23.0626	−1.18778
$378$	52.8214	2.71684
$379$	16.3481	0.839745	0.419873	−	0.907583i	$-0.362075\pi$
$379$	16.3481	0.839745	0.419873	+	0.907583i	$0.362075\pi$
$380$	0	0
$381$	66.9758	3.43128
$382$	16.9279	0.866108
$383$	3.93415	0.201026	0.100513	−	0.994936i	$-0.467952\pi$
$383$	3.93415	0.201026	0.100513	+	0.994936i	$0.467952\pi$
$384$	−18.7187	−0.955236
$385$	0	0
$386$	−0.940655	−0.0478781
$387$	−21.3948	−1.08756
$388$	−3.81899	−0.193880
$389$	38.6765	1.96098	0.980489	−	0.196572i	$-0.0629811\pi$
$389$	38.6765	1.96098	0.980489	+	0.196572i	$0.0629811\pi$
$390$	0	0
$391$	−23.7855	−1.20289
$392$	−23.5754	−1.19074
$393$	52.8486	2.66586
$394$	−3.93009	−0.197995
$395$	0	0
$396$	−2.97925	−0.149713
$397$	−8.45961	−0.424576	−0.212288	−	0.977207i	$-0.568091\pi$
$397$	−8.45961	−0.424576	−0.212288	+	0.977207i	$0.568091\pi$
$398$	−15.5712	−0.780511
$399$	11.8755	0.594521
$400$	0	0
$401$	−28.4450	−1.42047	−0.710237	−	0.703963i	$-0.751412\pi$
$401$	−28.4450	−1.42047	−0.710237	+	0.703963i	$0.751412\pi$
$402$	−34.4488	−1.71815
$403$	−5.51414	−0.274679
$404$	−6.78064	−0.337350
$405$	0	0
$406$	44.9864	2.23264
$407$	9.50113	0.470954
$408$	−41.9857	−2.07860
$409$	−24.9510	−1.23375	−0.616873	−	0.787063i	$-0.711601\pi$
$409$	−24.9510	−1.23375	−0.616873	+	0.787063i	$0.711601\pi$
$410$	0	0
$411$	43.5843	2.14985
$412$	0.0512492	0.00252487
$413$	−14.6399	−0.720382
$414$	43.7484	2.15012
$415$	0	0
$416$	6.13982	0.301029
$417$	45.4275	2.22459
$418$	1.24377	0.0608348
$419$	−11.7203	−0.572575	−0.286287	−	0.958144i	$-0.592421\pi$
$419$	−11.7203	−0.572575	−0.286287	+	0.958144i	$0.592421\pi$
$420$	0	0
$421$	18.5710	0.905095	0.452547	−	0.891740i	$-0.350515\pi$
$421$	18.5710	0.905095	0.452547	+	0.891740i	$0.350515\pi$
$422$	18.5071	0.900913
$423$	32.8448	1.59697
$424$	−20.5048	−0.995803
$425$	0	0
$426$	−19.0205	−0.921547
$427$	−28.1382	−1.36170
$428$	7.50953	0.362987
$429$	−7.57217	−0.365588
$430$	0	0
$431$	−13.5731	−0.653792	−0.326896	−	0.945060i	$-0.606003\pi$
$431$	−13.5731	−0.653792	−0.326896	+	0.945060i	$0.606003\pi$
$432$	31.9677	1.53805
$433$	33.2974	1.60017	0.800086	−	0.599885i	$-0.204787\pi$
$433$	33.2974	1.60017	0.800086	+	0.599885i	$0.204787\pi$
$434$	10.7560	0.516304
$435$	0	0
$436$	3.42541	0.164047
$437$	5.34873	0.255864
$438$	−14.9560	−0.714627
$439$	18.5043	0.883163	0.441581	−	0.897221i	$-0.354418\pi$
$439$	18.5043	0.883163	0.441581	+	0.897221i	$0.354418\pi$
$440$	0	0
$441$	50.8144	2.41973
$442$	−13.5340	−0.643748
$443$	26.8846	1.27733	0.638664	−	0.769486i	$-0.279488\pi$
$443$	26.8846	1.27733	0.638664	+	0.769486i	$0.279488\pi$
$444$	13.3200	0.632141
$445$	0	0
$446$	8.38312	0.396952
$447$	−34.0612	−1.61104
$448$	−34.1476	−1.61332
$449$	9.74064	0.459689	0.229845	−	0.973227i	$-0.426178\pi$
$449$	9.74064	0.459689	0.229845	+	0.973227i	$0.426178\pi$
$450$	0	0
$451$	6.98607	0.328961
$452$	1.49136	0.0701475
$453$	15.1161	0.710216
$454$	−11.3574	−0.533030
$455$	0	0
$456$	9.44147	0.442137
$457$	6.33245	0.296219	0.148110	−	0.988971i	$-0.452681\pi$
$457$	6.33245	0.296219	0.148110	+	0.988971i	$0.452681\pi$
$458$	−25.5828	−1.19541
$459$	49.2124	2.29704
$460$	0	0
$461$	17.9981	0.838253	0.419127	−	0.907928i	$-0.362336\pi$
$461$	17.9981	0.838253	0.419127	+	0.907928i	$0.362336\pi$
$462$	14.7704	0.687183
$463$	−23.2596	−1.08097	−0.540483	−	0.841355i	$-0.681758\pi$
$463$	−23.2596	−1.08097	−0.540483	+	0.841355i	$0.681758\pi$
$464$	27.2260	1.26393
$465$	0	0
$466$	−8.25395	−0.382357
$467$	−27.9976	−1.29557	−0.647787	−	0.761822i	$-0.724305\pi$
$467$	−27.9976	−1.29557	−0.647787	+	0.761822i	$0.724305\pi$
$468$	−7.29006	−0.336983
$469$	−34.3476	−1.58602
$470$	0	0
$471$	10.2138	0.470627
$472$	−11.6392	−0.535738
$473$	−3.25340	−0.149591
$474$	52.1392	2.39483
$475$	0	0
$476$	−7.73133	−0.354365
$477$	44.1961	2.02360
$478$	17.2623	0.789558
$479$	18.8574	0.861615	0.430808	−	0.902444i	$-0.358229\pi$
$479$	18.8574	0.861615	0.430808	+	0.902444i	$0.358229\pi$
$480$	0	0
$481$	23.2488	1.06005
$482$	15.3934	0.701148
$483$	63.5191	2.89022
$484$	−0.453038	−0.0205926
$485$	0	0
$486$	−14.5830	−0.661499
$487$	17.3588	0.786601	0.393301	−	0.919410i	$-0.371333\pi$
$487$	17.3588	0.786601	0.393301	+	0.919410i	$0.371333\pi$
$488$	−22.3709	−1.01268
$489$	17.2764	0.781267
$490$	0	0
$491$	−31.7460	−1.43268	−0.716338	−	0.697754i	$-0.754183\pi$
$491$	−31.7460	−1.43268	−0.716338	+	0.697754i	$0.754183\pi$
$492$	9.79406	0.441550
$493$	41.9127	1.88765
$494$	3.04344	0.136931
$495$	0	0
$496$	6.50957	0.292288
$497$	−18.9646	−0.850680
$498$	33.1001	1.48325
$499$	−4.37020	−0.195637	−0.0978185	−	0.995204i	$-0.531186\pi$
$499$	−4.37020	−0.195637	−0.0978185	+	0.995204i	$0.531186\pi$
$500$	0	0
$501$	−67.3414	−3.00859
$502$	33.4675	1.49373
$503$	−37.9393	−1.69163	−0.845816	−	0.533475i	$-0.820886\pi$
$503$	−37.9393	−1.69163	−0.845816	+	0.533475i	$0.820886\pi$
$504$	76.9971	3.42972
$505$	0	0
$506$	6.65259	0.295744
$507$	21.7003	0.963742
$508$	9.80520	0.435036
$509$	42.2367	1.87211	0.936055	−	0.351853i	$-0.114448\pi$
$509$	42.2367	1.87211	0.936055	+	0.351853i	$0.114448\pi$
$510$	0	0
$511$	−14.9121	−0.659673
$512$	−24.8750	−1.09933
$513$	−11.0665	−0.488600
$514$	29.0955	1.28335
$515$	0	0
$516$	−4.56107	−0.200790
$517$	4.99452	0.219659
$518$	−45.3496	−1.99255
$519$	60.9873	2.67705
$520$	0	0
$521$	−23.2491	−1.01856	−0.509280	−	0.860601i	$-0.670088\pi$
$521$	−23.2491	−1.01856	−0.509280	+	0.860601i	$0.670088\pi$
$522$	−77.0895	−3.37412
$523$	1.46476	0.0640496	0.0320248	−	0.999487i	$-0.489804\pi$
$523$	1.46476	0.0640496	0.0320248	+	0.999487i	$0.489804\pi$
$524$	7.73699	0.337992
$525$	0	0
$526$	33.8584	1.47630
$527$	10.0211	0.436525
$528$	8.93913	0.389026
$529$	5.60890	0.243865
$530$	0	0
$531$	25.0872	1.08869
$532$	1.73857	0.0753766
$533$	17.0946	0.740447
$534$	12.8562	0.556340
$535$	0	0
$536$	−27.3075	−1.17951
$537$	71.3693	3.07981
$538$	18.0993	0.780316
$539$	7.72706	0.332828
$540$	0	0
$541$	−21.5910	−0.928268	−0.464134	−	0.885765i	$-0.653634\pi$
$541$	−21.5910	−0.928268	−0.464134	+	0.885765i	$0.653634\pi$
$542$	−39.5107	−1.69713
$543$	−21.7779	−0.934579
$544$	−11.1582	−0.478402
$545$	0	0
$546$	36.1425	1.54676
$547$	−33.5992	−1.43660	−0.718300	−	0.695734i	$-0.755079\pi$
$547$	−33.5992	−1.43660	−0.718300	+	0.695734i	$0.755079\pi$
$548$	6.38070	0.272570
$549$	48.2181	2.05790
$550$	0	0
$551$	−9.42505	−0.401520
$552$	50.4999	2.14942
$553$	51.9861	2.21067
$554$	17.2923	0.734678
$555$	0	0
$556$	6.65055	0.282046
$557$	−37.7230	−1.59838	−0.799188	−	0.601082i	$-0.794737\pi$
$557$	−37.7230	−1.59838	−0.799188	+	0.601082i	$0.794737\pi$
$558$	−18.4317	−0.780275
$559$	−7.96090	−0.336710
$560$	0	0
$561$	13.7612	0.581000
$562$	−11.4698	−0.483824
$563$	−0.353129	−0.0148826	−0.00744131	−	0.999972i	$-0.502369\pi$
$563$	−0.353129	−0.0148826	−0.00744131	+	0.999972i	$0.502369\pi$
$564$	7.00203	0.294839
$565$	0	0
$566$	19.0078	0.798957
$567$	−55.7116	−2.33967
$568$	−15.0775	−0.632640
$569$	−6.10053	−0.255747	−0.127874	−	0.991790i	$-0.540815\pi$
$569$	−6.10053	−0.255747	−0.127874	+	0.991790i	$0.540815\pi$
$570$	0	0
$571$	10.7153	0.448420	0.224210	−	0.974541i	$-0.428020\pi$
$571$	10.7153	0.448420	0.224210	+	0.974541i	$0.428020\pi$
$572$	−1.10856	−0.0463512
$573$	−42.1172	−1.75947
$574$	−33.3450	−1.39179
$575$	0	0
$576$	58.5159	2.43816
$577$	22.8110	0.949636	0.474818	−	0.880084i	$-0.342514\pi$
$577$	22.8110	0.949636	0.474818	+	0.880084i	$0.342514\pi$
$578$	3.45190	0.143580
$579$	2.34038	0.0972628
$580$	0	0
$581$	33.0029	1.36919
$582$	−32.4452	−1.34489
$583$	6.72067	0.278342
$584$	−11.8556	−0.490590
$585$	0	0
$586$	−2.25569	−0.0931817
$587$	−11.2009	−0.462311	−0.231156	−	0.972917i	$-0.574251\pi$
$587$	−11.2009	−0.462311	−0.231156	+	0.972917i	$0.574251\pi$
$588$	10.8329	0.446741
$589$	−2.25347	−0.0928528
$590$	0	0
$591$	9.77819	0.402221
$592$	−27.4458	−1.12801
$593$	−21.1463	−0.868376	−0.434188	−	0.900822i	$-0.642965\pi$
$593$	−21.1463	−0.868376	−0.434188	+	0.900822i	$0.642965\pi$
$594$	−13.7642	−0.564753
$595$	0	0
$596$	−4.98653	−0.204256
$597$	38.7415	1.58558
$598$	16.2785	0.665679
$599$	5.65347	0.230995	0.115497	−	0.993308i	$-0.463154\pi$
$599$	5.65347	0.230995	0.115497	+	0.993308i	$0.463154\pi$
$600$	0	0
$601$	−10.2942	−0.419911	−0.209955	−	0.977711i	$-0.567332\pi$
$601$	−10.2942	−0.419911	−0.209955	+	0.977711i	$0.567332\pi$
$602$	15.5287	0.632902
$603$	58.8586	2.39691
$604$	2.21298	0.0900450
$605$	0	0
$606$	−57.6066	−2.34011
$607$	7.90688	0.320930	0.160465	−	0.987041i	$-0.448701\pi$
$607$	7.90688	0.320930	0.160465	+	0.987041i	$0.448701\pi$
$608$	2.50917	0.101760
$609$	−111.928	−4.53553
$610$	0	0
$611$	12.2213	0.494423
$612$	13.2486	0.535541
$613$	−19.1079	−0.771761	−0.385880	−	0.922549i	$-0.626102\pi$
$613$	−19.1079	−0.771761	−0.385880	+	0.922549i	$0.626102\pi$
$614$	−22.1361	−0.893339
$615$	0	0
$616$	11.7085	0.471750
$617$	−9.68005	−0.389704	−0.194852	−	0.980833i	$-0.562423\pi$
$617$	−9.68005	−0.389704	−0.194852	+	0.980833i	$0.562423\pi$
$618$	0.435400	0.0175144
$619$	46.7198	1.87783	0.938913	−	0.344155i	$-0.111835\pi$
$619$	46.7198	1.87783	0.938913	+	0.344155i	$0.111835\pi$
$620$	0	0
$621$	−59.1919	−2.37529
$622$	−21.4950	−0.861873
$623$	12.8184	0.513558
$624$	21.8736	0.875645
$625$	0	0
$626$	36.8420	1.47250
$627$	−3.09454	−0.123584
$628$	1.49529	0.0596687
$629$	−42.2511	−1.68466
$630$	0	0
$631$	21.8178	0.868551	0.434276	−	0.900780i	$-0.357004\pi$
$631$	21.8178	0.868551	0.434276	+	0.900780i	$0.357004\pi$
$632$	41.3307	1.64405
$633$	−46.0463	−1.83018
$634$	−39.3242	−1.56177
$635$	0	0
$636$	9.42198	0.373606
$637$	18.9077	0.749151
$638$	−11.7226	−0.464102
$639$	32.4982	1.28561
$640$	0	0
$641$	23.1143	0.912960	0.456480	−	0.889734i	$-0.349110\pi$
$641$	23.1143	0.912960	0.456480	+	0.889734i	$0.349110\pi$
$642$	63.7990	2.51795
$643$	−25.6612	−1.01198	−0.505989	−	0.862540i	$-0.668872\pi$
$643$	−25.6612	−1.01198	−0.505989	+	0.862540i	$0.668872\pi$
$644$	9.29914	0.366438
$645$	0	0
$646$	−5.53098	−0.217613
$647$	−21.4078	−0.841628	−0.420814	−	0.907147i	$-0.638255\pi$
$647$	−21.4078	−0.841628	−0.420814	+	0.907147i	$0.638255\pi$
$648$	−44.2927	−1.73998
$649$	3.81487	0.149747
$650$	0	0
$651$	−26.7612	−1.04886
$652$	2.52925	0.0990532
$653$	−32.4660	−1.27049	−0.635246	−	0.772310i	$-0.719101\pi$
$653$	−32.4660	−1.27049	−0.635246	+	0.772310i	$0.719101\pi$
$654$	29.1014	1.13795
$655$	0	0
$656$	−20.1805	−0.787918
$657$	25.5537	0.996943
$658$	−23.8392	−0.929349
$659$	−17.6170	−0.686259	−0.343130	−	0.939288i	$-0.611487\pi$
$659$	−17.6170	−0.686259	−0.343130	+	0.939288i	$0.611487\pi$
$660$	0	0
$661$	47.7091	1.85567	0.927834	−	0.372993i	$-0.121669\pi$
$661$	47.7091	1.85567	0.927834	+	0.372993i	$0.121669\pi$
$662$	−2.74078	−0.106524
$663$	33.6731	1.30775
$664$	26.2384	1.01825
$665$	0	0
$666$	77.7119	3.01127
$667$	−50.4120	−1.95196
$668$	−9.85872	−0.381445
$669$	−20.8575	−0.806396
$670$	0	0
$671$	7.33227	0.283059
$672$	29.7978	1.14947
$673$	−43.3750	−1.67199	−0.835993	−	0.548741i	$-0.815107\pi$
$673$	−43.3750	−1.67199	−0.835993	+	0.548741i	$0.815107\pi$
$674$	−11.3838	−0.438488
$675$	0	0
$676$	3.17690	0.122188
$677$	32.9976	1.26820	0.634100	−	0.773251i	$-0.281371\pi$
$677$	32.9976	1.26820	0.634100	+	0.773251i	$0.281371\pi$
$678$	12.6702	0.486595
$679$	−32.3499	−1.24147
$680$	0	0
$681$	28.2576	1.08283
$682$	−2.80280	−0.107325
$683$	−11.0439	−0.422585	−0.211292	−	0.977423i	$-0.567767\pi$
$683$	−11.0439	−0.422585	−0.211292	+	0.977423i	$0.567767\pi$
$684$	−2.97925	−0.113914
$685$	0	0
$686$	−3.47032	−0.132498
$687$	63.6509	2.42843
$688$	9.39803	0.358297
$689$	16.4451	0.626509
$690$	0	0
$691$	−10.1642	−0.386665	−0.193332	−	0.981133i	$-0.561930\pi$
$691$	−10.1642	−0.386665	−0.193332	+	0.981133i	$0.561930\pi$
$692$	8.92849	0.339410
$693$	−25.2366	−0.958658
$694$	−13.6633	−0.518651
$695$	0	0
$696$	−88.9863	−3.37302
$697$	−31.0667	−1.17674
$698$	−22.6174	−0.856080
$699$	20.5361	0.776746
$700$	0	0
$701$	18.7655	0.708762	0.354381	−	0.935101i	$-0.384692\pi$
$701$	18.7655	0.708762	0.354381	+	0.935101i	$0.384692\pi$
$702$	−33.6804	−1.27118
$703$	9.50113	0.358342
$704$	8.89820	0.335363
$705$	0	0
$706$	0.0556347	0.00209384
$707$	−57.4374	−2.16016
$708$	5.34822	0.200998
$709$	−0.593064	−0.0222730	−0.0111365	−	0.999938i	$-0.503545\pi$
$709$	−0.593064	−0.0222730	−0.0111365	+	0.999938i	$0.503545\pi$
$710$	0	0
$711$	−89.0843	−3.34092
$712$	10.1911	0.381927
$713$	−12.0532	−0.451397
$714$	−65.6834	−2.45814
$715$	0	0
$716$	10.4484	0.390475
$717$	−42.9491	−1.60396
$718$	−20.7688	−0.775087
$719$	−39.7948	−1.48410	−0.742049	−	0.670346i	$-0.766146\pi$
$719$	−39.7948	−1.48410	−0.742049	+	0.670346i	$0.766146\pi$
$720$	0	0
$721$	0.434121	0.0161675
$722$	1.24377	0.0462883
$723$	−38.2991	−1.42436
$724$	−3.18827	−0.118491
$725$	0	0
$726$	−3.84889	−0.142846
$727$	−10.2940	−0.381782	−0.190891	−	0.981611i	$-0.561138\pi$
$727$	−10.2940	−0.381782	−0.190891	+	0.981611i	$0.561138\pi$
$728$	28.6502	1.06185
$729$	−7.26907	−0.269225
$730$	0	0
$731$	14.4677	0.535107
$732$	10.2794	0.379938
$733$	−11.7380	−0.433551	−0.216776	−	0.976221i	$-0.569554\pi$
$733$	−11.7380	−0.433551	−0.216776	+	0.976221i	$0.569554\pi$
$734$	−7.56809	−0.279343
$735$	0	0
$736$	13.4209	0.494700
$737$	8.95031	0.329689
$738$	57.1406	2.10338
$739$	24.1090	0.886865	0.443433	−	0.896308i	$-0.353761\pi$
$739$	24.1090	0.886865	0.443433	+	0.896308i	$0.353761\pi$
$740$	0	0
$741$	−7.57217	−0.278171
$742$	−32.0782	−1.17763
$743$	9.25055	0.339370	0.169685	−	0.985498i	$-0.445725\pi$
$743$	9.25055	0.339370	0.169685	+	0.985498i	$0.445725\pi$
$744$	−21.2761	−0.780020
$745$	0	0
$746$	−8.26841	−0.302728
$747$	−56.5544	−2.06922
$748$	2.01463	0.0736623
$749$	63.6116	2.32432
$750$	0	0
$751$	−34.6758	−1.26534	−0.632669	−	0.774422i	$-0.718041\pi$
$751$	−34.6758	−1.26534	−0.632669	+	0.774422i	$0.718041\pi$
$752$	−14.4276	−0.526120
$753$	−83.2681	−3.03446
$754$	−28.6846	−1.04463
$755$	0	0
$756$	−19.2400	−0.699751
$757$	23.2943	0.846647	0.423323	−	0.905979i	$-0.360863\pi$
$757$	23.2943	0.846647	0.423323	+	0.905979i	$0.360863\pi$
$758$	20.3333	0.738537
$759$	−16.5518	−0.600794
$760$	0	0
$761$	−1.10679	−0.0401210	−0.0200605	−	0.999799i	$-0.506386\pi$
$761$	−1.10679	−0.0401210	−0.0200605	+	0.999799i	$0.506386\pi$
$762$	83.3025	3.01773
$763$	29.0159	1.05045
$764$	−6.16592	−0.223075
$765$	0	0
$766$	4.89317	0.176798
$767$	9.33479	0.337060
$768$	31.7898	1.14712
$769$	3.11121	0.112193	0.0560965	−	0.998425i	$-0.482135\pi$
$769$	3.11121	0.112193	0.0560965	+	0.998425i	$0.482135\pi$
$770$	0	0
$771$	−72.3904	−2.60708
$772$	0.342630	0.0123315
$773$	11.7199	0.421536	0.210768	−	0.977536i	$-0.432404\pi$
$773$	11.7199	0.421536	0.210768	+	0.977536i	$0.432404\pi$
$774$	−26.6103	−0.956486
$775$	0	0
$776$	−25.7193	−0.923268
$777$	112.831	4.04780
$778$	48.1047	1.72464
$779$	6.98607	0.250302
$780$	0	0
$781$	4.94182	0.176832
$782$	−29.5837	−1.05791
$783$	104.303	3.72747
$784$	−22.3210	−0.797180
$785$	0	0
$786$	65.7315	2.34456
$787$	41.2119	1.46904	0.734522	−	0.678584i	$-0.237406\pi$
$787$	41.2119	1.46904	0.734522	+	0.678584i	$0.237406\pi$
$788$	1.43152	0.0509957
$789$	−84.2407	−2.99905
$790$	0	0
$791$	12.6330	0.449176
$792$	−20.0639	−0.712941
$793$	17.9417	0.637128
$794$	−10.5218	−0.373405
$795$	0	0
$796$	5.67172	0.201029
$797$	25.6880	0.909915	0.454958	−	0.890513i	$-0.349654\pi$
$797$	25.6880	0.909915	0.454958	+	0.890513i	$0.349654\pi$
$798$	14.7704	0.522868
$799$	−22.2104	−0.785747
$800$	0	0
$801$	−21.9658	−0.776125
$802$	−35.3790	−1.24927
$803$	3.88580	0.137127
$804$	12.5478	0.442527
$805$	0	0
$806$	−6.85831	−0.241574
$807$	−45.0316	−1.58519
$808$	−45.6647	−1.60648
$809$	−16.4577	−0.578623	−0.289311	−	0.957235i	$-0.593426\pi$
$809$	−16.4577	−0.578623	−0.289311	+	0.957235i	$0.593426\pi$
$810$	0	0
$811$	−4.99059	−0.175243	−0.0876217	−	0.996154i	$-0.527927\pi$
$811$	−4.99059	−0.175243	−0.0876217	+	0.996154i	$0.527927\pi$
$812$	−16.3861	−0.575040
$813$	98.3038	3.44767
$814$	11.8172	0.414193
$815$	0	0
$816$	−39.7519	−1.39159
$817$	−3.25340	−0.113822
$818$	−31.0333	−1.08505
$819$	−61.7526	−2.15781
$820$	0	0
$821$	−33.8114	−1.18003	−0.590013	−	0.807393i	$-0.700878\pi$
$821$	−33.8114	−1.18003	−0.590013	+	0.807393i	$0.700878\pi$
$822$	54.2088	1.89075
$823$	−50.7096	−1.76762	−0.883812	−	0.467842i	$-0.845032\pi$
$823$	−50.7096	−1.76762	−0.883812	+	0.467842i	$0.845032\pi$
$824$	0.345141	0.0120236
$825$	0	0
$826$	−18.2086	−0.633560
$827$	−10.5942	−0.368396	−0.184198	−	0.982889i	$-0.558969\pi$
$827$	−10.5942	−0.368396	−0.184198	+	0.982889i	$0.558969\pi$
$828$	−15.9352	−0.553786
$829$	−43.2271	−1.50134	−0.750670	−	0.660677i	$-0.770269\pi$
$829$	−43.2271	−1.50134	−0.750670	+	0.660677i	$0.770269\pi$
$830$	0	0
$831$	−43.0237	−1.49248
$832$	21.7734	0.754858
$833$	−34.3619	−1.19057
$834$	56.5014	1.95648
$835$	0	0
$836$	−0.453038	−0.0156686
$837$	24.9382	0.861989
$838$	−14.5774	−0.503567
$839$	−34.4342	−1.18880	−0.594401	−	0.804169i	$-0.702611\pi$
$839$	−34.4342	−1.18880	−0.594401	+	0.804169i	$0.702611\pi$
$840$	0	0
$841$	59.8315	2.06315
$842$	23.0980	0.796011
$843$	28.5372	0.982873
$844$	−6.74114	−0.232040
$845$	0	0
$846$	40.8513	1.40450
$847$	−3.83759	−0.131861
$848$	−19.4139	−0.666675
$849$	−47.2919	−1.62306
$850$	0	0
$851$	50.8190	1.74205
$852$	6.92814	0.237354
$853$	0.0569549	0.00195010	0.000975050	−	1.00000i	$-0.499690\pi$
$853$	0.0569549	0.00195010	0.000975050		1.00000i	$0.499690\pi$
$854$	−34.9974	−1.19759
$855$	0	0
$856$	50.5734	1.72856
$857$	−7.28367	−0.248805	−0.124403	−	0.992232i	$-0.539701\pi$
$857$	−7.28367	−0.248805	−0.124403	+	0.992232i	$0.539701\pi$
$858$	−9.41804	−0.321526
$859$	1.89445	0.0646379	0.0323189	−	0.999478i	$-0.489711\pi$
$859$	1.89445	0.0646379	0.0323189	+	0.999478i	$0.489711\pi$
$860$	0	0
$861$	82.9634	2.82739
$862$	−16.8818	−0.574996
$863$	1.45592	0.0495602	0.0247801	−	0.999693i	$-0.492111\pi$
$863$	1.45592	0.0495602	0.0247801	+	0.999693i	$0.492111\pi$
$864$	−27.7679	−0.944682
$865$	0	0
$866$	41.4143	1.40732
$867$	−8.58843	−0.291678
$868$	−3.91782	−0.132980
$869$	−13.5466	−0.459535
$870$	0	0
$871$	21.9010	0.742085
$872$	23.0687	0.781203
$873$	55.4353	1.87620
$874$	6.65259	0.225027
$875$	0	0
$876$	5.44767	0.184060
$877$	−9.00209	−0.303979	−0.151990	−	0.988382i	$-0.548568\pi$
$877$	−9.00209	−0.303979	−0.151990	+	0.988382i	$0.548568\pi$
$878$	23.0151	0.776722
$879$	5.61222	0.189296
$880$	0	0
$881$	−49.4226	−1.66509	−0.832545	−	0.553957i	$-0.813117\pi$
$881$	−49.4226	−1.66509	−0.832545	+	0.553957i	$0.813117\pi$
$882$	63.2014	2.12810
$883$	11.5155	0.387529	0.193764	−	0.981048i	$-0.437930\pi$
$883$	11.5155	0.387529	0.193764	+	0.981048i	$0.437930\pi$
$884$	4.92971	0.165804
$885$	0	0
$886$	33.4383	1.12338
$887$	5.16282	0.173350	0.0866752	−	0.996237i	$-0.472376\pi$
$887$	5.16282	0.173350	0.0866752	+	0.996237i	$0.472376\pi$
$888$	89.7047	3.01029
$889$	83.0578	2.78567
$890$	0	0
$891$	14.5174	0.486350
$892$	−3.05351	−0.102239
$893$	4.99452	0.167135
$894$	−42.3642	−1.41687
$895$	0	0
$896$	−23.2134	−0.775505
$897$	−40.5015	−1.35231
$898$	12.1151	0.404286
$899$	21.2391	0.708363
$900$	0	0
$901$	−29.8865	−0.995662
$902$	8.68906	0.289314
$903$	−38.6359	−1.28572
$904$	10.0436	0.334046
$905$	0	0
$906$	18.8009	0.624619
$907$	−36.5420	−1.21336	−0.606679	−	0.794947i	$-0.707499\pi$
$907$	−36.5420	−1.21336	−0.606679	+	0.794947i	$0.707499\pi$
$908$	4.13689	0.137287
$909$	98.4258	3.26458
$910$	0	0
$911$	9.84473	0.326171	0.163085	−	0.986612i	$-0.447855\pi$
$911$	9.84473	0.326171	0.163085	+	0.986612i	$0.447855\pi$
$912$	8.93913	0.296004
$913$	−8.59991	−0.284616
$914$	7.87610	0.260518
$915$	0	0
$916$	9.31843	0.307890
$917$	65.5384	2.16427
$918$	61.2088	2.02019
$919$	8.16405	0.269307	0.134654	−	0.990893i	$-0.457008\pi$
$919$	8.16405	0.269307	0.134654	+	0.990893i	$0.457008\pi$
$920$	0	0
$921$	55.0752	1.81479
$922$	22.3854	0.737225
$923$	12.0924	0.398025
$924$	−5.38007	−0.176991
$925$	0	0
$926$	−28.9296	−0.950685
$927$	−0.743918	−0.0244335
$928$	−23.6491	−0.776318
$929$	33.3986	1.09577	0.547887	−	0.836552i	$-0.315432\pi$
$929$	33.3986	1.09577	0.547887	+	0.836552i	$0.315432\pi$
$930$	0	0
$931$	7.72706	0.253244
$932$	3.00646	0.0984800
$933$	53.4803	1.75087
$934$	−34.8225	−1.13943
$935$	0	0
$936$	−49.0954	−1.60473
$937$	8.19016	0.267561	0.133780	−	0.991011i	$-0.457288\pi$
$937$	8.19016	0.267561	0.133780	+	0.991011i	$0.457288\pi$
$938$	−42.7205	−1.39487
$939$	−91.6639	−2.99134
$940$	0	0
$941$	−15.7134	−0.512243	−0.256122	−	0.966645i	$-0.582445\pi$
$941$	−15.7134	−0.512243	−0.256122	+	0.966645i	$0.582445\pi$
$942$	12.7036	0.413906
$943$	37.3666	1.21682
$944$	−11.0199	−0.358669
$945$	0	0
$946$	−4.04648	−0.131562
$947$	−51.1369	−1.66172	−0.830862	−	0.556478i	$-0.812152\pi$
$947$	−51.1369	−1.66172	−0.830862	+	0.556478i	$0.812152\pi$
$948$	−18.9915	−0.616814
$949$	9.50837	0.308655
$950$	0	0
$951$	97.8399	3.17268
$952$	−52.0672	−1.68751
$953$	40.0172	1.29628	0.648142	−	0.761520i	$-0.275546\pi$
$953$	40.0172	1.29628	0.648142	+	0.761520i	$0.275546\pi$
$954$	54.9698	1.77971
$955$	0	0
$956$	−6.28771	−0.203359
$957$	29.1661	0.942808
$958$	23.4542	0.757772
$959$	54.0495	1.74535
$960$	0	0
$961$	−25.9219	−0.836189
$962$	28.9161	0.932293
$963$	−109.006	−3.51267
$964$	−5.60696	−0.180588
$965$	0	0
$966$	79.0031	2.54188
$967$	32.3043	1.03884	0.519418	−	0.854520i	$-0.326149\pi$
$967$	32.3043	1.03884	0.519418	+	0.854520i	$0.326149\pi$
$968$	−3.05101	−0.0980633
$969$	13.7612	0.442075
$970$	0	0
$971$	−15.9162	−0.510775	−0.255387	−	0.966839i	$-0.582203\pi$
$971$	−15.9162	−0.510775	−0.255387	+	0.966839i	$0.582203\pi$
$972$	5.31180	0.170376
$973$	56.3354	1.80603
$974$	21.5903	0.691798
$975$	0	0
$976$	−21.1806	−0.677974
$977$	−16.2932	−0.521264	−0.260632	−	0.965438i	$-0.583931\pi$
$977$	−16.2932	−0.521264	−0.260632	+	0.965438i	$0.583931\pi$
$978$	21.4879	0.687107
$979$	−3.34023	−0.106754
$980$	0	0
$981$	−49.7222	−1.58751
$982$	−39.4847	−1.26001
$983$	−12.4417	−0.396830	−0.198415	−	0.980118i	$-0.563579\pi$
$983$	−12.4417	−0.396830	−0.198415	+	0.980118i	$0.563579\pi$
$984$	65.9588	2.10269
$985$	0	0
$986$	52.1297	1.66015
$987$	59.3127	1.88794
$988$	−1.10856	−0.0352680
$989$	−17.4015	−0.553337
$990$	0	0
$991$	15.7073	0.498960	0.249480	−	0.968380i	$-0.419740\pi$
$991$	15.7073	0.498960	0.249480	+	0.968380i	$0.419740\pi$
$992$	−5.65435	−0.179526
$993$	6.81915	0.216399
$994$	−23.5876	−0.748155
$995$	0	0
$996$	−12.0566	−0.382027
$997$	46.9517	1.48698	0.743488	−	0.668749i	$-0.233170\pi$
$997$	46.9517	1.48698	0.743488	+	0.668749i	$0.233170\pi$
$998$	−5.43552	−0.172058
$999$	−105.145	−3.32663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.p.1.7		9
5.4	even	2		1045.2.a.k.1.3	✓	9
15.14	odd	2		9405.2.a.bh.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.k.1.3	✓	9	5.4	even	2
5225.2.a.p.1.7		9	1.1	even	1	trivial
9405.2.a.bh.1.7		9	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)

\(f(q)\)	\(=\)	\(q+1.24377 q^{2} -3.09454 q^{3} -0.453038 q^{4} -3.84889 q^{6} -3.83759 q^{7} -3.05101 q^{8} +6.57616 q^{9} +1.00000 q^{11} +1.40194 q^{12} +2.44695 q^{13} -4.77307 q^{14} -2.88868 q^{16} -4.44695 q^{17} +8.17922 q^{18} +1.00000 q^{19} +11.8755 q^{21} +1.24377 q^{22} +5.34873 q^{23} +9.44147 q^{24} +3.04344 q^{26} -11.0665 q^{27} +1.73857 q^{28} -9.42505 q^{29} -2.25347 q^{31} +2.50917 q^{32} -3.09454 q^{33} -5.53098 q^{34} -2.97925 q^{36} +9.50113 q^{37} +1.24377 q^{38} -7.57217 q^{39} +6.98607 q^{41} +14.7704 q^{42} -3.25340 q^{43} -0.453038 q^{44} +6.65259 q^{46} +4.99452 q^{47} +8.93913 q^{48} +7.72706 q^{49} +13.7612 q^{51} -1.10856 q^{52} +6.72067 q^{53} -13.7642 q^{54} +11.7085 q^{56} -3.09454 q^{57} -11.7226 q^{58} +3.81487 q^{59} +7.33227 q^{61} -2.80280 q^{62} -25.2366 q^{63} +8.89820 q^{64} -3.84889 q^{66} +8.95031 q^{67} +2.01463 q^{68} -16.5518 q^{69} +4.94182 q^{71} -20.0639 q^{72} +3.88580 q^{73} +11.8172 q^{74} -0.453038 q^{76} -3.83759 q^{77} -9.41804 q^{78} -13.5466 q^{79} +14.5174 q^{81} +8.68906 q^{82} -8.59991 q^{83} -5.38007 q^{84} -4.04648 q^{86} +29.1661 q^{87} -3.05101 q^{88} -3.34023 q^{89} -9.39038 q^{91} -2.42317 q^{92} +6.97346 q^{93} +6.21203 q^{94} -7.76472 q^{96} +8.42974 q^{97} +9.61068 q^{98} +6.57616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9} + 9 q^{11} + 5 q^{12} - 5 q^{13} - 2 q^{14} + q^{16} - 13 q^{17} - q^{18} + 9 q^{19} + q^{21} - 3 q^{22} - 8 q^{23} - 7 q^{24} + 8 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) 9 * q - 3 * q^2 - 3 * q^3 + 9 * q^4 - 13 * q^7 - 9 * q^8 + 12 * q^9 + 9 * q^11 + 5 * q^12 - 5 * q^13 - 2 * q^14 + q^16 - 13 * q^17 - q^18 + 9 * q^19 + q^21 - 3 * q^22 - 8 * q^23 - 7 * q^24 + 8 * q^26 - 6 * q^27 - 10 * q^28 - 3 * q^29 - 9 * q^31 - 6 * q^32 - 3 * q^33 - 2 * q^34 - 15 * q^36 + 7 * q^37 - 3 * q^38 - 2 * q^39 + 9 * q^41 + 9 * q^42 - 23 * q^43 + 9 * q^44 - 32 * q^46 - 20 * q^47 + 18 * q^48 - 4 * q^49 + 8 * q^51 - 9 * q^52 + 5 * q^53 + 9 * q^54 + 4 * q^56 - 3 * q^57 - 3 * q^58 + 19 * q^59 + q^61 - 18 * q^62 - 24 * q^63 + 23 * q^64 + 10 * q^67 - 9 * q^68 - 28 * q^69 - 24 * q^72 - 12 * q^73 + 5 * q^74 + 9 * q^76 - 13 * q^77 - 19 * q^78 - 21 * q^79 - 3 * q^81 + 14 * q^82 - 47 * q^83 - 11 * q^84 + 12 * q^86 + 2 * q^87 - 9 * q^88 - 2 * q^89 + 2 * q^91 + 19 * q^92 + 2 * q^93 + 26 * q^94 - 55 * q^96 + 32 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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