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Properties

Label 49.7.d.e
Level $49$
Weight $7$
Character orbit 49.d
Analytic conductor $11.273$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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] = [49,7,Mod(19,49)] 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("49.19"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-20] 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: no
Analytic conductor: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{2} - 564 q^{4} + 1880 q^{8} + 1252 q^{9} + 3872 q^{11} + 44864 q^{15} - 37908 q^{16} + 19436 q^{18} + 159744 q^{22} - 40032 q^{23} + 28860 q^{25} + 68032 q^{29} - 21192 q^{30} + 82060 q^{32}+ \cdots - 8949984 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −7.22006 + 12.5055i −5.48749 + 3.16820i −72.2584 − 125.155i −169.953 − 98.1223i − 91.4984i 0 1162.67 −344.425 + 596.562i 2454.14 − 1416.90i
19.2 −7.22006 + 12.5055i 5.48749 − 3.16820i −72.2584 − 125.155i 169.953 + 98.1223i 91.4984i 0 1162.67 −344.425 + 596.562i −2454.14 + 1416.90i
19.3 −6.49614 + 11.2516i −40.4001 + 23.3250i −52.3997 − 90.7589i 26.6652 + 15.3952i − 606.091i 0 530.077 723.614 − 1253.34i −346.442 + 200.018i
19.4 −6.49614 + 11.2516i 40.4001 − 23.3250i −52.3997 − 90.7589i −26.6652 − 15.3952i 606.091i 0 530.077 723.614 − 1253.34i 346.442 − 200.018i
19.5 −2.33452 + 4.04351i −32.1003 + 18.5331i 21.1000 + 36.5463i −189.226 − 109.250i − 173.064i 0 −495.852 322.454 − 558.506i 883.503 − 510.091i
19.6 −2.33452 + 4.04351i 32.1003 − 18.5331i 21.1000 + 36.5463i 189.226 + 109.250i 173.064i 0 −495.852 322.454 − 558.506i −883.503 + 510.091i
19.7 0.171081 − 0.296322i −8.41775 + 4.85999i 31.9415 + 55.3242i −69.2835 − 40.0009i 3.32582i 0 43.7568 −317.261 + 549.512i −23.7063 + 13.6868i
19.8 0.171081 − 0.296322i 8.41775 − 4.85999i 31.9415 + 55.3242i 69.2835 + 40.0009i − 3.32582i 0 43.7568 −317.261 + 549.512i 23.7063 − 13.6868i
19.9 3.50223 − 6.06605i −10.6752 + 6.16334i 7.46871 + 12.9362i −100.778 − 58.1845i 86.3419i 0 552.915 −288.526 + 499.742i −705.899 + 407.551i
19.10 3.50223 − 6.06605i 10.6752 − 6.16334i 7.46871 + 12.9362i 100.778 + 58.1845i − 86.3419i 0 552.915 −288.526 + 499.742i 705.899 − 407.551i
19.11 7.37740 − 12.7780i −29.5376 + 17.0535i −76.8521 − 133.112i −27.8768 − 16.0947i 503.243i 0 −1323.57 217.145 − 376.106i −411.317 + 237.474i
19.12 7.37740 − 12.7780i 29.5376 − 17.0535i −76.8521 − 133.112i 27.8768 + 16.0947i − 503.243i 0 −1323.57 217.145 − 376.106i 411.317 − 237.474i
31.1 −7.22006 − 12.5055i −5.48749 − 3.16820i −72.2584 + 125.155i −169.953 + 98.1223i 91.4984i 0 1162.67 −344.425 − 596.562i 2454.14 + 1416.90i
31.2 −7.22006 − 12.5055i 5.48749 + 3.16820i −72.2584 + 125.155i 169.953 − 98.1223i − 91.4984i 0 1162.67 −344.425 − 596.562i −2454.14 − 1416.90i
31.3 −6.49614 − 11.2516i −40.4001 − 23.3250i −52.3997 + 90.7589i 26.6652 − 15.3952i 606.091i 0 530.077 723.614 + 1253.34i −346.442 − 200.018i
31.4 −6.49614 − 11.2516i 40.4001 + 23.3250i −52.3997 + 90.7589i −26.6652 + 15.3952i − 606.091i 0 530.077 723.614 + 1253.34i 346.442 + 200.018i
31.5 −2.33452 − 4.04351i −32.1003 − 18.5331i 21.1000 − 36.5463i −189.226 + 109.250i 173.064i 0 −495.852 322.454 + 558.506i 883.503 + 510.091i
31.6 −2.33452 − 4.04351i 32.1003 + 18.5331i 21.1000 − 36.5463i 189.226 − 109.250i − 173.064i 0 −495.852 322.454 + 558.506i −883.503 − 510.091i
31.7 0.171081 + 0.296322i −8.41775 − 4.85999i 31.9415 − 55.3242i −69.2835 + 40.0009i − 3.32582i 0 43.7568 −317.261 − 549.512i −23.7063 − 13.6868i
31.8 0.171081 + 0.296322i 8.41775 + 4.85999i 31.9415 − 55.3242i 69.2835 − 40.0009i 3.32582i 0 43.7568 −317.261 − 549.512i 23.7063 + 13.6868i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.7.d.e 24
7.b odd 2 1 inner 49.7.d.e 24
7.c even 3 1 49.7.b.c ✓ 12
7.c even 3 1 inner 49.7.d.e 24
7.d odd 6 1 49.7.b.c ✓ 12
7.d odd 6 1 inner 49.7.d.e 24
21.g even 6 1 441.7.d.e 12
21.h odd 6 1 441.7.d.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.7.b.c ✓ 12 7.c even 3 1
49.7.b.c ✓ 12 7.d odd 6 1
49.7.d.e 24 1.a even 1 1 trivial
49.7.d.e 24 7.b odd 2 1 inner
49.7.d.e 24 7.c even 3 1 inner
49.7.d.e 24 7.d odd 6 1 inner
441.7.d.e 12 21.g even 6 1
441.7.d.e 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 10 T_{2}^{11} + 383 T_{2}^{10} + 2330 T_{2}^{9} + 91407 T_{2}^{8} + 526388 T_{2}^{7} + \cdots + 959512576 \) acting on \(S_{7}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display