↑

Properties

Label 49.24.a.h
Level $49$
Weight $24$
Character orbit 49.a
Self dual yes
Analytic conductor $164.250$
Analytic rank $0$
Dimension $24$
Inner twists $2$

Related objects

Downloads

Learn more

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 = [24,6030,0] 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: \(24\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6030 q^{2} + 99000906 q^{4} + 31751706690 q^{8} + 975236583640 q^{9} + 3514223137536 q^{11} + 96662500006976 q^{15} + 850136746459362 q^{16} - 774764811988990 q^{18} + 81\!\cdots\!60 q^{22} + 34\!\cdots\!60 q^{23}+ \cdots + 70\!\cdots\!64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5639.35 −536798. 2.34137e7 −7.14765e7 3.02719e9 0 −8.47318e10 1.94009e11 4.03081e11
1.2 −5639.35 536798. 2.34137e7 7.14765e7 −3.02719e9 0 −8.47318e10 1.94009e11 −4.03081e11
1.3 −4540.89 −266199. 1.22311e7 2.18121e7 1.20878e9 0 −1.74484e10 −2.32812e10 −9.90464e10
1.4 −4540.89 266199. 1.22311e7 −2.18121e7 −1.20878e9 0 −1.74484e10 −2.32812e10 9.90464e10
1.5 −3184.55 −198271. 1.75275e6 1.30841e8 6.31403e8 0 2.11322e10 −5.48319e10 −4.16670e11
1.6 −3184.55 198271. 1.75275e6 −1.30841e8 −6.31403e8 0 2.11322e10 −5.48319e10 4.16670e11
1.7 −1835.09 −301132. −5.02104e6 −1.31871e8 5.52605e8 0 2.46080e10 −3.46277e9 2.41996e11
1.8 −1835.09 301132. −5.02104e6 1.31871e8 −5.52605e8 0 2.46080e10 −3.46277e9 −2.41996e11
1.9 −867.700 −393094. −7.63570e6 1.46240e8 3.41087e8 0 1.39043e10 6.03794e10 −1.26893e11
1.10 −867.700 393094. −7.63570e6 −1.46240e8 −3.41087e8 0 1.39043e10 6.03794e10 1.26893e11
1.11 −777.686 −501655. −7.78381e6 −2.08566e8 3.90130e8 0 1.25771e10 1.57515e11 1.62199e11
1.12 −777.686 501655. −7.78381e6 2.08566e8 −3.90130e8 0 1.25771e10 1.57515e11 −1.62199e11
1.13 717.223 −31440.9 −7.87420e6 2.53096e7 −2.25502e7 0 −1.16641e10 −9.31546e10 1.81526e10
1.14 717.223 31440.9 −7.87420e6 −2.53096e7 2.25502e7 0 −1.16641e10 −9.31546e10 −1.81526e10
1.15 1665.65 −536981. −5.61423e6 8.56876e7 −8.94421e8 0 −2.33238e10 1.94205e11 1.42725e11
1.16 1665.65 536981. −5.61423e6 −8.56876e7 8.94421e8 0 −2.33238e10 1.94205e11 −1.42725e11
1.17 2933.77 −142294. 218425. −6.28631e6 −4.17460e8 0 −2.39695e10 −7.38955e10 −1.84426e10
1.18 2933.77 142294. 218425. 6.28631e6 4.17460e8 0 −2.39695e10 −7.38955e10 1.84426e10
1.19 4465.27 −326247. 1.15500e7 −1.73494e8 −1.45678e9 0 1.41165e10 1.22937e10 −7.74697e11
1.20 4465.27 326247. 1.15500e7 1.73494e8 1.45678e9 0 1.41165e10 1.22937e10 7.74697e11
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.24.a.h ✓ 24
7.b odd 2 1 inner 49.24.a.h ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.24.a.h ✓ 24 1.a even 1 1 trivial
49.24.a.h ✓ 24 7.b odd 2 1 inner

Hecke kernels

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

\( T_{2}^{12} - 3015 T_{2}^{11} - 70536762 T_{2}^{10} + 199650960360 T_{2}^{9} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
\( T_{3}^{24} - 1617336437744 T_{3}^{22} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display