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Properties

Label 49.9.d.d
Level $49$
Weight $9$
Character orbit 49.d
Analytic conductor $19.962$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,9,Mod(19,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.9615518930\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{2} - 2420 q^{4} + 10776 q^{8} + 52264 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{2} - 2420 q^{4} + 10776 q^{8} + 52264 q^{9} - 16776 q^{11} - 546064 q^{15} - 78996 q^{16} - 448660 q^{18} - 2856256 q^{22} + 2113368 q^{23} + 621352 q^{25} - 9648960 q^{29} + 7700664 q^{30} - 10839636 q^{32} - 41980520 q^{36} + 1343872 q^{37} + 1874824 q^{39} - 9516464 q^{43} + 21867960 q^{44} + 5284952 q^{46} + 121880088 q^{50} - 28087368 q^{51} - 1599336 q^{53} - 33561760 q^{57} - 77994984 q^{58} + 37037784 q^{60} + 234558872 q^{64} - 36849120 q^{65} + 96050160 q^{67} + 200642688 q^{71} - 108909532 q^{72} - 123556488 q^{74} + 411093872 q^{78} - 27694928 q^{79} + 208772304 q^{81} - 366534736 q^{85} + 267125232 q^{86} - 22093456 q^{88} - 1173975504 q^{92} - 139768784 q^{93} - 108859368 q^{95} - 1416024000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −15.3244 + 26.5426i −91.1463 + 52.6233i −341.672 − 591.793i 250.596 + 144.682i − 3225.67i 0 13097.5 2257.93 − 3910.85i −7680.46 + 4434.31i
19.2 −15.3244 + 26.5426i 91.1463 − 52.6233i −341.672 − 591.793i −250.596 − 144.682i 3225.67i 0 13097.5 2257.93 − 3910.85i 7680.46 − 4434.31i
19.3 −9.61862 + 16.6599i −54.9500 + 31.7254i −57.0355 − 98.7884i 668.131 + 385.746i − 1220.62i 0 −2730.32 −1267.50 + 2195.37i −12853.0 + 7420.68i
19.4 −9.61862 + 16.6599i 54.9500 − 31.7254i −57.0355 − 98.7884i −668.131 − 385.746i 1220.62i 0 −2730.32 −1267.50 + 2195.37i 12853.0 − 7420.68i
19.5 −6.75812 + 11.7054i −120.901 + 69.8023i 36.6556 + 63.4893i 603.205 + 348.261i − 1886.93i 0 −4451.05 6464.23 − 11196.4i −8153.07 + 4707.18i
19.6 −6.75812 + 11.7054i 120.901 − 69.8023i 36.6556 + 63.4893i −603.205 − 348.261i 1886.93i 0 −4451.05 6464.23 − 11196.4i 8153.07 − 4707.18i
19.7 −4.31300 + 7.47033i −36.6168 + 21.1407i 90.7961 + 157.263i 9.49664 + 5.48289i − 364.720i 0 −3774.67 −2386.64 + 4133.78i −81.9179 + 47.2953i
19.8 −4.31300 + 7.47033i 36.6168 − 21.1407i 90.7961 + 157.263i −9.49664 − 5.48289i 364.720i 0 −3774.67 −2386.64 + 4133.78i 81.9179 − 47.2953i
19.9 5.39397 − 9.34264i −45.7886 + 26.4361i 69.8101 + 120.915i −443.760 − 256.205i 570.382i 0 4267.93 −1882.77 + 3261.05i −4787.26 + 2763.92i
19.10 5.39397 − 9.34264i 45.7886 − 26.4361i 69.8101 + 120.915i 443.760 + 256.205i − 570.382i 0 4267.93 −1882.77 + 3261.05i 4787.26 − 2763.92i
19.11 7.87015 − 13.6315i −83.0871 + 47.9704i 4.12144 + 7.13854i 175.588 + 101.376i 1510.14i 0 4159.26 1321.81 − 2289.45i 2763.80 − 1595.68i
19.12 7.87015 − 13.6315i 83.0871 − 47.9704i 4.12144 + 7.13854i −175.588 − 101.376i − 1510.14i 0 4159.26 1321.81 − 2289.45i −2763.80 + 1595.68i
19.13 12.4828 − 21.6208i −92.6479 + 53.4903i −183.639 − 318.072i −1049.93 − 606.178i 2670.83i 0 −2778.10 2441.92 − 4229.53i −26212.1 + 15133.6i
19.14 12.4828 − 21.6208i 92.6479 − 53.4903i −183.639 − 318.072i 1049.93 + 606.178i − 2670.83i 0 −2778.10 2441.92 − 4229.53i 26212.1 − 15133.6i
19.15 13.2672 − 22.9795i −118.728 + 68.5474i −224.037 − 388.044i 611.149 + 352.847i 3637.73i 0 −5096.58 6117.01 − 10595.0i 16216.5 − 9362.58i
19.16 13.2672 − 22.9795i 118.728 − 68.5474i −224.037 − 388.044i −611.149 − 352.847i − 3637.73i 0 −5096.58 6117.01 − 10595.0i −16216.5 + 9362.58i
31.1 −15.3244 − 26.5426i −91.1463 − 52.6233i −341.672 + 591.793i 250.596 − 144.682i 3225.67i 0 13097.5 2257.93 + 3910.85i −7680.46 − 4434.31i
31.2 −15.3244 − 26.5426i 91.1463 + 52.6233i −341.672 + 591.793i −250.596 + 144.682i − 3225.67i 0 13097.5 2257.93 + 3910.85i 7680.46 + 4434.31i
31.3 −9.61862 − 16.6599i −54.9500 − 31.7254i −57.0355 + 98.7884i 668.131 − 385.746i 1220.62i 0 −2730.32 −1267.50 − 2195.37i −12853.0 − 7420.68i
31.4 −9.61862 − 16.6599i 54.9500 + 31.7254i −57.0355 + 98.7884i −668.131 + 385.746i − 1220.62i 0 −2730.32 −1267.50 − 2195.37i 12853.0 + 7420.68i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

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 49.9.d.d 32
7.b odd 2 1 inner 49.9.d.d 32
7.c even 3 1 49.9.b.b ✓ 16
7.c even 3 1 inner 49.9.d.d 32
7.d odd 6 1 49.9.b.b ✓ 16
7.d odd 6 1 inner 49.9.d.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.9.b.b ✓ 16 7.c even 3 1
49.9.b.b ✓ 16 7.d odd 6 1
49.9.d.d 32 1.a even 1 1 trivial
49.9.d.d 32 7.b odd 2 1 inner
49.9.d.d 32 7.c even 3 1 inner
49.9.d.d 32 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 6 T_{2}^{15} + 1647 T_{2}^{14} - 9558 T_{2}^{13} + 1859023 T_{2}^{12} + \cdots + 59\!\cdots\!76 \) acting on \(S_{9}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display