Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,22,Mod(1,25)] 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("25.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1312] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.8693360718\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 54559x - 2833496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-200.579\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-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)
 
\(f(q)\) \(=\) \(q+938.952 q^{2} -156099. q^{3} -1.21552e6 q^{4} -1.46569e8 q^{6} +2.53688e8 q^{7} -3.11044e9 q^{8} +1.39065e10 q^{9} -8.13050e10 q^{11} +1.89741e11 q^{12} +3.79465e11 q^{13} +2.38201e11 q^{14} -3.71421e11 q^{16} +9.55753e12 q^{17} +1.30575e13 q^{18} -1.33763e13 q^{19} -3.96004e13 q^{21} -7.63415e13 q^{22} +3.84421e14 q^{23} +4.85536e14 q^{24} +3.56300e14 q^{26} -5.37934e14 q^{27} -3.08363e14 q^{28} +4.23297e15 q^{29} -6.80305e15 q^{31} +6.17432e15 q^{32} +1.26916e16 q^{33} +8.97406e15 q^{34} -1.69036e16 q^{36} -3.29427e16 q^{37} -1.25597e16 q^{38} -5.92341e16 q^{39} -4.24223e16 q^{41} -3.71828e16 q^{42} -1.83781e17 q^{43} +9.88280e16 q^{44} +3.60953e17 q^{46} -3.37760e17 q^{47} +5.79783e16 q^{48} -4.94188e17 q^{49} -1.49192e18 q^{51} -4.61248e17 q^{52} -2.51810e17 q^{53} -5.05094e17 q^{54} -7.89081e17 q^{56} +2.08803e18 q^{57} +3.97456e18 q^{58} +4.47696e18 q^{59} +2.42664e18 q^{61} -6.38774e18 q^{62} +3.52790e18 q^{63} +6.57632e18 q^{64} +1.19168e19 q^{66} +5.38129e18 q^{67} -1.16174e19 q^{68} -6.00076e19 q^{69} -1.93416e18 q^{71} -4.32552e19 q^{72} +3.84631e19 q^{73} -3.09316e19 q^{74} +1.62592e19 q^{76} -2.06261e19 q^{77} -5.56179e19 q^{78} -9.76074e19 q^{79} -6.14957e19 q^{81} -3.98325e19 q^{82} -7.76469e17 q^{83} +4.81351e19 q^{84} -1.72562e20 q^{86} -6.60762e20 q^{87} +2.52894e20 q^{88} +6.27796e19 q^{89} +9.62657e19 q^{91} -4.67272e20 q^{92} +1.06195e21 q^{93} -3.17141e20 q^{94} -9.63804e20 q^{96} +5.50682e20 q^{97} -4.64019e20 q^{98} -1.13067e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 1312 q^{2} - 52194 q^{3} + 991136 q^{4} + 24240576 q^{6} - 684416558 q^{7} - 1119275520 q^{8} + 2005625619 q^{9} + 3188512736 q^{11} + 360872818752 q^{12} + 806175066506 q^{13} - 2283397948608 q^{14}+ \cdots - 13\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 938.952 0.648378 0.324189 − 0.945992i \(-0.394909\pi\)
0.324189 + 0.945992i \(0.394909\pi\)
\(3\) −156099. −1.52625 −0.763126 − 0.646250i \(-0.776336\pi\)
−0.763126 + 0.646250i \(0.776336\pi\)
\(4\) −1.21552e6 −0.579606
\(5\) 0 0
\(6\) −1.46569e8 −0.989588
\(7\) 2.53688e8 0.339446 0.169723 − 0.985492i \(-0.445713\pi\)
0.169723 + 0.985492i \(0.445713\pi\)
\(8\) −3.11044e9 −1.02418
\(9\) 1.39065e10 1.32945
\(10\) 0 0
\(11\) −8.13050e10 −0.945136 −0.472568 − 0.881294i \(-0.656673\pi\)
−0.472568 + 0.881294i \(0.656673\pi\)
\(12\) 1.89741e11 0.884625
\(13\) 3.79465e11 0.763426 0.381713 − 0.924281i \(-0.375334\pi\)
0.381713 + 0.924281i \(0.375334\pi\)
\(14\) 2.38201e11 0.220089
\(15\) 0 0
\(16\) −3.71421e11 −0.0844513
\(17\) 9.55753e12 1.14983 0.574913 − 0.818215i \(-0.305036\pi\)
0.574913 + 0.818215i \(0.305036\pi\)
\(18\) 1.30575e13 0.861983
\(19\) −1.33763e13 −0.500524 −0.250262 − 0.968178i \(-0.580517\pi\)
−0.250262 + 0.968178i \(0.580517\pi\)
\(20\) 0 0
\(21\) −3.96004e13 −0.518080
\(22\) −7.63415e13 −0.612805
\(23\) 3.84421e14 1.93493 0.967463 − 0.253011i \(-0.0814210\pi\)
0.967463 + 0.253011i \(0.0814210\pi\)
\(24\) 4.85536e14 1.56316
\(25\) 0 0
\(26\) 3.56300e14 0.494989
\(27\) −5.37934e14 −0.502817
\(28\) −3.08363e14 −0.196745
\(29\) 4.23297e15 1.86839 0.934193 − 0.356768i \(-0.116121\pi\)
0.934193 + 0.356768i \(0.116121\pi\)
\(30\) 0 0
\(31\) −6.80305e15 −1.49075 −0.745377 − 0.666643i \(-0.767731\pi\)
−0.745377 + 0.666643i \(0.767731\pi\)
\(32\) 6.17432e15 0.969425
\(33\) 1.26916e16 1.44252
\(34\) 8.97406e15 0.745522
\(35\) 0 0
\(36\) −1.69036e16 −0.770554
\(37\) −3.29427e16 −1.12627 −0.563133 − 0.826366i \(-0.690404\pi\)
−0.563133 + 0.826366i \(0.690404\pi\)
\(38\) −1.25597e16 −0.324529
\(39\) −5.92341e16 −1.16518
\(40\) 0 0
\(41\) −4.24223e16 −0.493587 −0.246794 − 0.969068i \(-0.579377\pi\)
−0.246794 + 0.969068i \(0.579377\pi\)
\(42\) −3.71828e16 −0.335912
\(43\) −1.83781e17 −1.29683 −0.648414 − 0.761288i \(-0.724567\pi\)
−0.648414 + 0.761288i \(0.724567\pi\)
\(44\) 9.88280e16 0.547806
\(45\) 0 0
\(46\) 3.60953e17 1.25456
\(47\) −3.37760e17 −0.936658 −0.468329 − 0.883554i \(-0.655144\pi\)
−0.468329 + 0.883554i \(0.655144\pi\)
\(48\) 5.79783e16 0.128894
\(49\) −4.94188e17 −0.884777
\(50\) 0 0
\(51\) −1.49192e18 −1.75492
\(52\) −4.61248e17 −0.442486
\(53\) −2.51810e17 −0.197777 −0.0988887 − 0.995099i \(-0.531529\pi\)
−0.0988887 + 0.995099i \(0.531529\pi\)
\(54\) −5.05094e17 −0.326015
\(55\) 0 0
\(56\) −7.89081e17 −0.347654
\(57\) 2.08803e18 0.763926
\(58\) 3.97456e18 1.21142
\(59\) 4.47696e18 1.14035 0.570174 − 0.821524i \(-0.306876\pi\)
0.570174 + 0.821524i \(0.306876\pi\)
\(60\) 0 0
\(61\) 2.42664e18 0.435554 0.217777 − 0.975999i \(-0.430119\pi\)
0.217777 + 0.975999i \(0.430119\pi\)
\(62\) −6.38774e18 −0.966572
\(63\) 3.52790e18 0.451274
\(64\) 6.57632e18 0.713006
\(65\) 0 0
\(66\) 1.19168e19 0.935295
\(67\) 5.38129e18 0.360662 0.180331 − 0.983606i \(-0.442283\pi\)
0.180331 + 0.983606i \(0.442283\pi\)
\(68\) −1.16174e19 −0.666446
\(69\) −6.00076e19 −2.95319
\(70\) 0 0
\(71\) −1.93416e18 −0.0705148 −0.0352574 − 0.999378i \(-0.511225\pi\)
−0.0352574 + 0.999378i \(0.511225\pi\)
\(72\) −4.32552e19 −1.36159
\(73\) 3.84631e19 1.04750 0.523751 − 0.851872i \(-0.324532\pi\)
0.523751 + 0.851872i \(0.324532\pi\)
\(74\) −3.09316e19 −0.730246
\(75\) 0 0
\(76\) 1.62592e19 0.290107
\(77\) −2.06261e19 −0.320822
\(78\) −5.56179e19 −0.755477
\(79\) −9.76074e19 −1.15984 −0.579920 − 0.814673i \(-0.696916\pi\)
−0.579920 + 0.814673i \(0.696916\pi\)
\(80\) 0 0
\(81\) −6.14957e19 −0.562020
\(82\) −3.98325e19 −0.320031
\(83\) −7.76469e17 −0.00549293 −0.00274646 − 0.999996i \(-0.500874\pi\)
−0.00274646 + 0.999996i \(0.500874\pi\)
\(84\) 4.81351e19 0.300282
\(85\) 0 0
\(86\) −1.72562e20 −0.840834
\(87\) −6.60762e20 −2.85163
\(88\) 2.52894e20 0.967991
\(89\) 6.27796e19 0.213414 0.106707 − 0.994291i \(-0.465969\pi\)
0.106707 + 0.994291i \(0.465969\pi\)
\(90\) 0 0
\(91\) 9.62657e19 0.259142
\(92\) −4.67272e20 −1.12149
\(93\) 1.06195e21 2.27527
\(94\) −3.17141e20 −0.607309
\(95\) 0 0
\(96\) −9.63804e20 −1.47959
\(97\) 5.50682e20 0.758224 0.379112 − 0.925351i \(-0.376229\pi\)
0.379112 + 0.925351i \(0.376229\pi\)
\(98\) −4.64019e20 −0.573670
\(99\) −1.13067e21 −1.25651
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 25.22.a.b.1.2 3
5.2 odd 4 25.22.b.b.24.4 6
5.3 odd 4 25.22.b.b.24.3 6
5.4 even 2 5.22.a.a.1.2 ✓ 3
15.14 odd 2 45.22.a.d.1.2 3
20.19 odd 2 80.22.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.22.a.a.1.2 ✓ 3 5.4 even 2
25.22.a.b.1.2 3 1.1 even 1 trivial
25.22.b.b.24.3 6 5.3 odd 4
25.22.b.b.24.4 6 5.2 odd 4
45.22.a.d.1.2 3 15.14 odd 2
80.22.a.e.1.1 3 20.19 odd 2