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Properties

Label 49.24.a.g
Level $49$
Weight $24$
Character orbit 49.a
Self dual yes
Analytic conductor $164.250$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,24,Mod(1,49)] 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("49.1"); S:= CuspForms(chi, 24); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 24, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-966,177148] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(164.249978299\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 7 x^{13} - 21591353 x^{12} - 1736098763 x^{11} + 177925612890704 x^{10} + \cdots - 50\!\cdots\!18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{49}\cdot 3^{13}\cdot 5^{3}\cdot 7^{23} \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 69) q^{2} + (\beta_{2} + 9 \beta_1 + 12653) q^{3} + (\beta_{3} + 3 \beta_{2} + \cdots + 3954081) q^{4} + ( - \beta_{4} + 21 \beta_{2} + \cdots + 5340778) q^{5} + (\beta_{5} + 27 \beta_{3} + \cdots + 107472093) q^{6}+ \cdots + ( - 558950183 \beta_{13} + \cdots - 16\!\cdots\!13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 966 q^{2} + 177148 q^{3} + 55357148 q^{4} + 74771022 q^{5} + 1504608254 q^{6} + 25222400616 q^{8} + 336909608980 q^{9} + 334296297894 q^{10} - 1355476566108 q^{11} + 4984668058916 q^{12} - 427040218556 q^{13}+ \cdots - 23\!\cdots\!84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 7 x^{13} - 21591353 x^{12} - 1736098763 x^{11} + 177925612890704 x^{10} + \cdots - 50\!\cdots\!18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\!\cdots\!75 \nu^{13} + \cdots + 80\!\cdots\!38 ) / 95\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 48\!\cdots\!75 \nu^{13} + \cdots - 47\!\cdots\!10 ) / 31\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 80\!\cdots\!41 \nu^{13} + \cdots - 53\!\cdots\!58 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42\!\cdots\!27 \nu^{13} + \cdots + 84\!\cdots\!62 ) / 95\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!71 \nu^{13} + \cdots - 11\!\cdots\!98 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 74\!\cdots\!67 \nu^{13} + \cdots + 24\!\cdots\!46 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 92\!\cdots\!41 \nu^{13} + \cdots - 25\!\cdots\!42 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!27 \nu^{13} + \cdots - 10\!\cdots\!06 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38\!\cdots\!69 \nu^{13} + \cdots - 26\!\cdots\!78 ) / 88\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!43 \nu^{13} + \cdots + 39\!\cdots\!34 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 69\!\cdots\!27 \nu^{13} + \cdots - 10\!\cdots\!14 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 90\!\cdots\!69 \nu^{13} + \cdots + 57\!\cdots\!22 ) / 23\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + 262\beta _1 + 12337929 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} + 9\beta_{5} - 52\beta_{4} + 871\beta_{3} + 16513\beta_{2} + 20448979\beta _1 + 3235258900 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{13} - 26 \beta_{12} + 78 \beta_{11} - 2 \beta_{10} + 94 \beta_{9} + 140 \beta_{8} + \cdots + 252255294973847 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2206 \beta_{13} - 9424 \beta_{12} - 2262 \beta_{11} + 9220 \beta_{10} + 277366 \beta_{9} + \cdots + 48\!\cdots\!86 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 18781945 \beta_{13} - 301691642 \beta_{12} + 822804626 \beta_{11} + 44725294 \beta_{10} + \cdots + 15\!\cdots\!15 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24568155439 \beta_{13} - 146898381802 \beta_{12} + 113648456514 \beta_{11} + 153510042366 \beta_{10} + \cdots + 52\!\cdots\!75 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 211661133081337 \beta_{13} + \cdots + 10\!\cdots\!23 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 19\!\cdots\!35 \beta_{13} + \cdots + 50\!\cdots\!23 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 19\!\cdots\!45 \beta_{13} + \cdots + 73\!\cdots\!71 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12\!\cdots\!87 \beta_{13} + \cdots + 45\!\cdots\!47 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 16\!\cdots\!17 \beta_{13} + \cdots + 54\!\cdots\!19 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 70\!\cdots\!83 \beta_{13} + \cdots + 41\!\cdots\!39 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2611.31
−2221.55
−1954.65
−1803.54
−1115.44
−979.571
−319.185
444.258
678.722
1291.79
1321.90
2115.04
2275.10
2885.44
−5292.63 260642. 1.96233e7 −1.96125e7 −1.37948e9 0 −5.94609e10 −2.62088e10 1.03801e11
1.2 −4513.09 −353177. 1.19794e7 −1.94265e8 1.59392e9 0 −1.62057e10 3.05906e10 8.76734e11
1.3 −3979.31 24569.1 7.44629e6 1.19361e8 −9.77681e7 0 3.74979e9 −9.35395e10 −4.74973e11
1.4 −3677.09 −465766. 5.13238e6 1.38328e8 1.71266e9 0 1.19734e10 1.22795e11 −5.08646e11
1.5 −2300.88 586391. −3.09457e6 4.69163e7 −1.34921e9 0 2.64214e10 2.49711e11 −1.07949e11
1.6 −2029.14 232905. −4.27119e6 −9.33995e7 −4.72597e8 0 2.56885e10 −3.98987e10 1.89521e11
1.7 −708.370 −198278. −7.88682e6 −3.63664e7 1.40454e8 0 1.15290e10 −5.48291e10 2.57609e10
1.8 818.517 27418.9 −7.71864e6 1.82942e8 2.24428e7 0 −1.31840e10 −9.33914e10 1.49741e11
1.9 1287.44 −536821. −6.73110e6 −5.01589e7 −6.91126e8 0 −1.94658e10 1.94033e11 −6.45768e10
1.10 2513.58 385363. −2.07052e6 7.20022e7 9.68640e8 0 −2.62899e10 5.43613e10 1.80983e11
1.11 2573.80 268374. −1.76418e6 −1.71375e8 6.90739e8 0 −2.61312e10 −2.21188e10 −4.41084e11
1.12 4160.08 −304576. 8.91769e6 1.02223e8 −1.26706e9 0 2.20102e9 −1.37649e9 4.25256e11
1.13 4480.20 −169729. 1.16836e7 −8.53191e7 −7.60422e8 0 1.47623e10 −6.53351e10 −3.82247e11
1.14 5700.89 419833. 2.41115e7 6.34941e7 2.39342e9 0 8.96344e10 8.21165e10 3.61973e11
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.24.a.g 14
7.b odd 2 1 49.24.a.f 14
7.c even 3 2 7.24.c.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.24.c.a ✓ 28 7.c even 3 2
49.24.a.f 14 7.b odd 2 1
49.24.a.g 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{14} + 966 T_{2}^{13} - 85932252 T_{2}^{12} - 86316234984 T_{2}^{11} + \cdots - 83\!\cdots\!32 \) Copy content Toggle raw display
\( T_{3}^{14} - 177148 T_{3}^{13} - 811766349327 T_{3}^{12} + \cdots + 94\!\cdots\!27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 83\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 94\!\cdots\!27 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 62\!\cdots\!75 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 11\!\cdots\!75 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 44\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 87\!\cdots\!27 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 46\!\cdots\!43 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 23\!\cdots\!13 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 59\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 66\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 35\!\cdots\!75 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 66\!\cdots\!47 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 20\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 10\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 23\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 57\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 17\!\cdots\!63 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 65\!\cdots\!07 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 11\!\cdots\!93 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
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