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 1.3
Root \(-55.0474\) 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)
 
\(f(q)\) \(=\) \(q+1965.76 q^{2} +94506.8 q^{3} +1.76707e6 q^{4} +1.85778e8 q^{6} -1.12850e9 q^{7} -6.48862e8 q^{8} -1.52881e9 q^{9} +7.88589e10 q^{11} +1.67000e11 q^{12} +5.16852e11 q^{13} -2.21836e12 q^{14} -4.98132e12 q^{16} -2.02632e12 q^{17} -3.00528e12 q^{18} -1.13845e13 q^{19} -1.06651e14 q^{21} +1.55018e14 q^{22} -2.85144e14 q^{23} -6.13219e13 q^{24} +1.01601e15 q^{26} -1.13306e15 q^{27} -1.99413e15 q^{28} -1.90734e15 q^{29} -5.14410e15 q^{31} -8.43134e15 q^{32} +7.45270e15 q^{33} -3.98327e15 q^{34} -2.70152e15 q^{36} +1.83092e16 q^{37} -2.23793e16 q^{38} +4.88461e16 q^{39} -1.54522e17 q^{41} -2.09650e17 q^{42} +6.97924e16 q^{43} +1.39349e17 q^{44} -5.60525e17 q^{46} -6.04535e17 q^{47} -4.70769e17 q^{48} +7.14960e17 q^{49} -1.91501e17 q^{51} +9.13314e17 q^{52} +2.24764e18 q^{53} -2.22732e18 q^{54} +7.32239e17 q^{56} -1.07592e18 q^{57} -3.74938e18 q^{58} -1.59427e18 q^{59} +3.25967e18 q^{61} -1.01121e19 q^{62} +1.72526e18 q^{63} -6.12741e18 q^{64} +1.46502e19 q^{66} +2.81533e19 q^{67} -3.58065e18 q^{68} -2.69480e19 q^{69} +6.22226e18 q^{71} +9.91988e17 q^{72} +4.27957e19 q^{73} +3.59916e19 q^{74} -2.01173e19 q^{76} -8.89920e19 q^{77} +9.60198e19 q^{78} -1.12715e20 q^{79} -9.10898e19 q^{81} -3.03754e20 q^{82} -5.32012e19 q^{83} -1.88459e20 q^{84} +1.37195e20 q^{86} -1.80257e20 q^{87} -5.11685e19 q^{88} +2.45025e20 q^{89} -5.83266e20 q^{91} -5.03869e20 q^{92} -4.86153e20 q^{93} -1.18837e21 q^{94} -7.96819e20 q^{96} -7.16085e20 q^{97} +1.40544e21 q^{98} -1.20560e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 1312 q^{2} - 52194 q^{3} + 991136 q^{4} + 24240576 q^{6} - 684416558 q^{7} - 1119275520 q^{8} + 2005625619 q^{9} + 3188512736 q^{11} + 360872818752 q^{12} + 806175066506 q^{13} - 2283397948608 q^{14}+ \cdots - 13\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1965.76 1.35743 0.678713 − 0.734404i \(-0.262538\pi\)
0.678713 + 0.734404i \(0.262538\pi\)
\(3\) 94506.8 0.924038 0.462019 − 0.886870i \(-0.347125\pi\)
0.462019 + 0.886870i \(0.347125\pi\)
\(4\) 1.76707e6 0.842605
\(5\) 0 0
\(6\) 1.85778e8 1.25431
\(7\) −1.12850e9 −1.50998 −0.754990 − 0.655737i \(-0.772358\pi\)
−0.754990 + 0.655737i \(0.772358\pi\)
\(8\) −6.48862e8 −0.213652
\(9\) −1.52881e9 −0.146153
\(10\) 0 0
\(11\) 7.88589e10 0.916700 0.458350 − 0.888772i \(-0.348441\pi\)
0.458350 + 0.888772i \(0.348441\pi\)
\(12\) 1.67000e11 0.778599
\(13\) 5.16852e11 1.03983 0.519914 − 0.854219i \(-0.325964\pi\)
0.519914 + 0.854219i \(0.325964\pi\)
\(14\) −2.21836e12 −2.04969
\(15\) 0 0
\(16\) −4.98132e12 −1.13262
\(17\) −2.02632e12 −0.243778 −0.121889 − 0.992544i \(-0.538895\pi\)
−0.121889 + 0.992544i \(0.538895\pi\)
\(18\) −3.00528e12 −0.198392
\(19\) −1.13845e13 −0.425993 −0.212997 − 0.977053i \(-0.568322\pi\)
−0.212997 + 0.977053i \(0.568322\pi\)
\(20\) 0 0
\(21\) −1.06651e14 −1.39528
\(22\) 1.55018e14 1.24435
\(23\) −2.85144e14 −1.43523 −0.717615 − 0.696440i \(-0.754766\pi\)
−0.717615 + 0.696440i \(0.754766\pi\)
\(24\) −6.13219e13 −0.197423
\(25\) 0 0
\(26\) 1.01601e15 1.41149
\(27\) −1.13306e15 −1.05909
\(28\) −1.99413e15 −1.27232
\(29\) −1.90734e15 −0.841878 −0.420939 − 0.907089i \(-0.638299\pi\)
−0.420939 + 0.907089i \(0.638299\pi\)
\(30\) 0 0
\(31\) −5.14410e15 −1.12723 −0.563614 − 0.826038i \(-0.690590\pi\)
−0.563614 + 0.826038i \(0.690590\pi\)
\(32\) −8.43134e15 −1.32380
\(33\) 7.45270e15 0.847067
\(34\) −3.98327e15 −0.330911
\(35\) 0 0
\(36\) −2.70152e15 −0.123149
\(37\) 1.83092e16 0.625968 0.312984 − 0.949758i \(-0.398671\pi\)
0.312984 + 0.949758i \(0.398671\pi\)
\(38\) −2.23793e16 −0.578254
\(39\) 4.88461e16 0.960841
\(40\) 0 0
\(41\) −1.54522e17 −1.79788 −0.898939 − 0.438074i \(-0.855661\pi\)
−0.898939 + 0.438074i \(0.855661\pi\)
\(42\) −2.09650e17 −1.89399
\(43\) 6.97924e16 0.492481 0.246240 − 0.969209i \(-0.420805\pi\)
0.246240 + 0.969209i \(0.420805\pi\)
\(44\) 1.39349e17 0.772416
\(45\) 0 0
\(46\) −5.60525e17 −1.94822
\(47\) −6.04535e17 −1.67646 −0.838231 − 0.545315i \(-0.816410\pi\)
−0.838231 + 0.545315i \(0.816410\pi\)
\(48\) −4.70769e17 −1.04659
\(49\) 7.14960e17 1.28004
\(50\) 0 0
\(51\) −1.91501e17 −0.225260
\(52\) 9.13314e17 0.876164
\(53\) 2.24764e18 1.76535 0.882675 − 0.469983i \(-0.155740\pi\)
0.882675 + 0.469983i \(0.155740\pi\)
\(54\) −2.22732e18 −1.43764
\(55\) 0 0
\(56\) 7.32239e17 0.322611
\(57\) −1.07592e18 −0.393634
\(58\) −3.74938e18 −1.14279
\(59\) −1.59427e18 −0.406084 −0.203042 − 0.979170i \(-0.565083\pi\)
−0.203042 + 0.979170i \(0.565083\pi\)
\(60\) 0 0
\(61\) 3.25967e18 0.585074 0.292537 − 0.956254i \(-0.405501\pi\)
0.292537 + 0.956254i \(0.405501\pi\)
\(62\) −1.01121e19 −1.53013
\(63\) 1.72526e18 0.220688
\(64\) −6.12741e18 −0.664336
\(65\) 0 0
\(66\) 1.46502e19 1.14983
\(67\) 2.81533e19 1.88688 0.943441 − 0.331541i \(-0.107569\pi\)
0.943441 + 0.331541i \(0.107569\pi\)
\(68\) −3.58065e18 −0.205409
\(69\) −2.69480e19 −1.32621
\(70\) 0 0
\(71\) 6.22226e18 0.226848 0.113424 − 0.993547i \(-0.463818\pi\)
0.113424 + 0.993547i \(0.463818\pi\)
\(72\) 9.91988e17 0.0312259
\(73\) 4.27957e19 1.16549 0.582746 − 0.812654i \(-0.301978\pi\)
0.582746 + 0.812654i \(0.301978\pi\)
\(74\) 3.59916e19 0.849705
\(75\) 0 0
\(76\) −2.01173e19 −0.358944
\(77\) −8.89920e19 −1.38420
\(78\) 9.60198e19 1.30427
\(79\) −1.12715e20 −1.33936 −0.669681 − 0.742649i \(-0.733569\pi\)
−0.669681 + 0.742649i \(0.733569\pi\)
\(80\) 0 0
\(81\) −9.10898e19 −0.832486
\(82\) −3.03754e20 −2.44049
\(83\) −5.32012e19 −0.376358 −0.188179 − 0.982135i \(-0.560259\pi\)
−0.188179 + 0.982135i \(0.560259\pi\)
\(84\) −1.88459e20 −1.17567
\(85\) 0 0
\(86\) 1.37195e20 0.668506
\(87\) −1.80257e20 −0.777928
\(88\) −5.11685e19 −0.195855
\(89\) 2.45025e20 0.832942 0.416471 − 0.909149i \(-0.363267\pi\)
0.416471 + 0.909149i \(0.363267\pi\)
\(90\) 0 0
\(91\) −5.83266e20 −1.57012
\(92\) −5.03869e20 −1.20933
\(93\) −4.86153e20 −1.04160
\(94\) −1.18837e21 −2.27567
\(95\) 0 0
\(96\) −7.96819e20 −1.22324
\(97\) −7.16085e20 −0.985964 −0.492982 − 0.870039i \(-0.664093\pi\)
−0.492982 + 0.870039i \(0.664093\pi\)
\(98\) 1.40544e21 1.73756
\(99\) −1.20560e20 −0.133978
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.1.3 3
5.2 odd 4 25.22.b.b.24.6 6
5.3 odd 4 25.22.b.b.24.1 6
5.4 even 2 5.22.a.a.1.1 ✓ 3
15.14 odd 2 45.22.a.d.1.3 3
20.19 odd 2 80.22.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.22.a.a.1.1 ✓ 3 5.4 even 2
25.22.a.b.1.3 3 1.1 even 1 trivial
25.22.b.b.24.1 6 5.3 odd 4
25.22.b.b.24.6 6 5.2 odd 4
45.22.a.d.1.3 3 15.14 odd 2
80.22.a.e.1.3 3 20.19 odd 2