Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content 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);
 
Copy content 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

Copy content comment:select newform
 
Copy content sage:traces = [7,1,-2,15,0,-2,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.61330\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.61330 q^{2} -1.19599 q^{3} +4.82936 q^{4} -3.12549 q^{6} -3.61829 q^{7} +7.39397 q^{8} -1.56960 q^{9} -1.00000 q^{11} -5.77587 q^{12} +1.47857 q^{13} -9.45570 q^{14} +9.66398 q^{16} +3.27003 q^{17} -4.10185 q^{18} +1.00000 q^{19} +4.32745 q^{21} -2.61330 q^{22} +7.45793 q^{23} -8.84313 q^{24} +3.86395 q^{26} +5.46521 q^{27} -17.4740 q^{28} +1.02535 q^{29} +1.64921 q^{31} +10.4670 q^{32} +1.19599 q^{33} +8.54558 q^{34} -7.58018 q^{36} +6.71293 q^{37} +2.61330 q^{38} -1.76836 q^{39} -3.92451 q^{41} +11.3089 q^{42} -5.38113 q^{43} -4.82936 q^{44} +19.4898 q^{46} +3.71597 q^{47} -11.5580 q^{48} +6.09205 q^{49} -3.91093 q^{51} +7.14054 q^{52} +0.102902 q^{53} +14.2823 q^{54} -26.7536 q^{56} -1.19599 q^{57} +2.67955 q^{58} +13.2986 q^{59} -6.49664 q^{61} +4.30989 q^{62} +5.67929 q^{63} +8.02543 q^{64} +3.12549 q^{66} +3.70989 q^{67} +15.7921 q^{68} -8.91962 q^{69} +6.32968 q^{71} -11.6056 q^{72} +1.37759 q^{73} +17.5429 q^{74} +4.82936 q^{76} +3.61829 q^{77} -4.62125 q^{78} +13.6725 q^{79} -1.82753 q^{81} -10.2559 q^{82} -5.44061 q^{83} +20.8988 q^{84} -14.0625 q^{86} -1.22631 q^{87} -7.39397 q^{88} +12.1357 q^{89} -5.34990 q^{91} +36.0170 q^{92} -1.97244 q^{93} +9.71096 q^{94} -12.5184 q^{96} +13.7910 q^{97} +15.9204 q^{98} +1.56960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9} - 7 q^{11} + 16 q^{12} + 4 q^{13} + 6 q^{14} + 27 q^{16} - 2 q^{17} - 9 q^{18} + 7 q^{19} - 14 q^{21} - q^{22} - 10 q^{23} - 2 q^{24}+ \cdots - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61330 1.84789 0.923943 − 0.382531i \(-0.124948\pi\)
0.923943 + 0.382531i \(0.124948\pi\)
\(3\) −1.19599 −0.690506 −0.345253 − 0.938510i \(-0.612207\pi\)
−0.345253 + 0.938510i \(0.612207\pi\)
\(4\) 4.82936 2.41468
\(5\) 0 0
\(6\) −3.12549 −1.27598
\(7\) −3.61829 −1.36759 −0.683793 − 0.729676i \(-0.739671\pi\)
−0.683793 + 0.729676i \(0.739671\pi\)
\(8\) 7.39397 2.61416
\(9\) −1.56960 −0.523201
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −5.77587 −1.66735
\(13\) 1.47857 0.410081 0.205041 − 0.978753i \(-0.434267\pi\)
0.205041 + 0.978753i \(0.434267\pi\)
\(14\) −9.45570 −2.52714
\(15\) 0 0
\(16\) 9.66398 2.41600
\(17\) 3.27003 0.793099 0.396549 − 0.918013i \(-0.370208\pi\)
0.396549 + 0.918013i \(0.370208\pi\)
\(18\) −4.10185 −0.966816
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.32745 0.944327
\(22\) −2.61330 −0.557158
\(23\) 7.45793 1.55509 0.777543 − 0.628830i \(-0.216466\pi\)
0.777543 + 0.628830i \(0.216466\pi\)
\(24\) −8.84313 −1.80510
\(25\) 0 0
\(26\) 3.86395 0.757783
\(27\) 5.46521 1.05178
\(28\) −17.4740 −3.30228
\(29\) 1.02535 0.190403 0.0952013 − 0.995458i \(-0.469651\pi\)
0.0952013 + 0.995458i \(0.469651\pi\)
\(30\) 0 0
\(31\) 1.64921 0.296207 0.148104 − 0.988972i \(-0.452683\pi\)
0.148104 + 0.988972i \(0.452683\pi\)
\(32\) 10.4670 1.85032
\(33\) 1.19599 0.208195
\(34\) 8.54558 1.46556
\(35\) 0 0
\(36\) −7.58018 −1.26336
\(37\) 6.71293 1.10360 0.551799 − 0.833977i \(-0.313941\pi\)
0.551799 + 0.833977i \(0.313941\pi\)
\(38\) 2.61330 0.423934
\(39\) −1.76836 −0.283164
\(40\) 0 0
\(41\) −3.92451 −0.612905 −0.306453 − 0.951886i \(-0.599142\pi\)
−0.306453 + 0.951886i \(0.599142\pi\)
\(42\) 11.3089 1.74501
\(43\) −5.38113 −0.820614 −0.410307 − 0.911947i \(-0.634578\pi\)
−0.410307 + 0.911947i \(0.634578\pi\)
\(44\) −4.82936 −0.728053
\(45\) 0 0
\(46\) 19.4898 2.87362
\(47\) 3.71597 0.542030 0.271015 − 0.962575i \(-0.412641\pi\)
0.271015 + 0.962575i \(0.412641\pi\)
\(48\) −11.5580 −1.66826
\(49\) 6.09205 0.870292
\(50\) 0 0
\(51\) −3.91093 −0.547640
\(52\) 7.14054 0.990215
\(53\) 0.102902 0.0141347 0.00706733 − 0.999975i \(-0.497750\pi\)
0.00706733 + 0.999975i \(0.497750\pi\)
\(54\) 14.2823 1.94357
\(55\) 0 0
\(56\) −26.7536 −3.57509
\(57\) −1.19599 −0.158413
\(58\) 2.67955 0.351842
\(59\) 13.2986 1.73134 0.865668 − 0.500619i \(-0.166894\pi\)
0.865668 + 0.500619i \(0.166894\pi\)
\(60\) 0 0
\(61\) −6.49664 −0.831809 −0.415905 − 0.909408i \(-0.636535\pi\)
−0.415905 + 0.909408i \(0.636535\pi\)
\(62\) 4.30989 0.547357
\(63\) 5.67929 0.715523
\(64\) 8.02543 1.00318
\(65\) 0 0
\(66\) 3.12549 0.384721
\(67\) 3.70989 0.453235 0.226618 − 0.973984i \(-0.427233\pi\)
0.226618 + 0.973984i \(0.427233\pi\)
\(68\) 15.7921 1.91508
\(69\) −8.91962 −1.07380
\(70\) 0 0
\(71\) 6.32968 0.751194 0.375597 − 0.926783i \(-0.377438\pi\)
0.375597 + 0.926783i \(0.377438\pi\)
\(72\) −11.6056 −1.36773
\(73\) 1.37759 0.161235 0.0806173 − 0.996745i \(-0.474311\pi\)
0.0806173 + 0.996745i \(0.474311\pi\)
\(74\) 17.5429 2.03932
\(75\) 0 0
\(76\) 4.82936 0.553965
\(77\) 3.61829 0.412343
\(78\) −4.62125 −0.523254
\(79\) 13.6725 1.53828 0.769141 − 0.639079i \(-0.220684\pi\)
0.769141 + 0.639079i \(0.220684\pi\)
\(80\) 0 0
\(81\) −1.82753 −0.203059
\(82\) −10.2559 −1.13258
\(83\) −5.44061 −0.597184 −0.298592 − 0.954381i \(-0.596517\pi\)
−0.298592 + 0.954381i \(0.596517\pi\)
\(84\) 20.8988 2.28025
\(85\) 0 0
\(86\) −14.0625 −1.51640
\(87\) −1.22631 −0.131474
\(88\) −7.39397 −0.788200
\(89\) 12.1357 1.28638 0.643191 − 0.765706i \(-0.277610\pi\)
0.643191 + 0.765706i \(0.277610\pi\)
\(90\) 0 0
\(91\) −5.34990 −0.560822
\(92\) 36.0170 3.75503
\(93\) −1.97244 −0.204533
\(94\) 9.71096 1.00161
\(95\) 0 0
\(96\) −12.5184 −1.27766
\(97\) 13.7910 1.40026 0.700131 − 0.714014i \(-0.253125\pi\)
0.700131 + 0.714014i \(0.253125\pi\)
\(98\) 15.9204 1.60820
\(99\) 1.56960 0.157751
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5225.2.a.n.1.6 7
5.4 even 2 209.2.a.d.1.2 ✓ 7
15.14 odd 2 1881.2.a.p.1.6 7
20.19 odd 2 3344.2.a.ba.1.4 7
55.54 odd 2 2299.2.a.q.1.6 7
95.94 odd 2 3971.2.a.i.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.2 ✓ 7 5.4 even 2
1881.2.a.p.1.6 7 15.14 odd 2
2299.2.a.q.1.6 7 55.54 odd 2
3344.2.a.ba.1.4 7 20.19 odd 2
3971.2.a.i.1.6 7 95.94 odd 2
5225.2.a.n.1.6 7 1.1 even 1 trivial