Properties

Label 7.22.c.a
Level $7$
Weight $22$
Character orbit 7.c
Analytic conductor $19.563$
Analytic rank $0$
Dimension $26$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7,22,Mod(2,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.2");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 7.c (of order \(3\), degree \(2\), 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: \(19.5634141001\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 286 q^{2} + 118097 q^{3} - 11780748 q^{4} + 19296893 q^{5} - 457302020 q^{6} + 1244099388 q^{7} - 3356984640 q^{8} - 36260337262 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 286 q^{2} + 118097 q^{3} - 11780748 q^{4} + 19296893 q^{5} - 457302020 q^{6} + 1244099388 q^{7} - 3356984640 q^{8} - 36260337262 q^{9} + 45908292458 q^{10} - 96908527507 q^{11} + 703726516612 q^{12} + 573150555348 q^{13} - 519063760642 q^{14} - 1250393933326 q^{15} - 13121838202992 q^{16} + 3631296873225 q^{17} - 26768119563764 q^{18} + 56849486179647 q^{19} - 211235752093016 q^{20} + 287520772660135 q^{21} - 564206457341956 q^{22} - 56010101087361 q^{23} - 151975129265904 q^{24} - 672811740581052 q^{25} - 13\!\cdots\!76 q^{26}+ \cdots - 23\!\cdots\!52 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1330.60 + 2304.67i 72122.8 + 124920.i −2.49243e6 4.31702e6i 1.64217e7 2.84432e7i −3.83867e8 6.58019e8 + 3.54341e8i 7.68480e9 −5.17323e9 + 8.96030e9i 4.37015e10 + 7.56932e10i
2.2 −1089.84 + 1887.65i −65571.0 113572.i −1.32691e6 2.29828e6i 3.92338e6 6.79550e6i 2.85847e8 4.31432e8 6.10256e8i 1.21336e9 −3.36895e9 + 5.83519e9i 8.55170e9 + 1.48120e10i
2.3 −1071.21 + 1855.40i −769.406 1332.65i −1.24642e6 2.15887e6i −9.82127e6 + 1.70109e7i 3.29679e6 −6.30322e8 + 4.01546e8i 8.47744e8 5.22899e9 9.05688e9i −2.10414e10 3.64447e10i
2.4 −618.250 + 1070.84i 88565.1 + 153399.i 284110. + 492093.i −1.58690e7 + 2.74859e7i −2.19021e8 4.82606e7 7.45799e8i −3.29573e9 −1.04574e10 + 1.81127e10i −1.96220e10 3.39863e10i
2.5 −351.499 + 608.815i −5803.13 10051.3i 801472. + 1.38819e6i −459458. + 795804.i 8.15919e6 6.00451e8 + 4.44977e8i −2.60116e9 5.16282e9 8.94227e9i −3.22998e8 5.59450e8i
2.6 −346.683 + 600.472i 35430.0 + 61366.5i 808198. + 1.39984e6i 1.75720e7 3.04357e7i −4.91319e7 −5.82353e8 4.68414e8i −2.57485e9 2.71961e9 4.71050e9i 1.21838e10 + 2.11030e10i
2.7 −90.6842 + 157.070i −89485.1 154993.i 1.03213e6 + 1.78770e6i 3.79465e6 6.57253e6i 3.24596e7 −6.07843e8 + 4.34825e8i −7.54748e8 −1.07850e10 + 1.86802e10i 6.88230e8 + 1.19205e9i
2.8 356.792 617.981i −36627.4 63440.5i 793975. + 1.37521e6i −1.72220e7 + 2.98293e7i −5.22734e7 2.12395e8 7.16543e8i 2.62963e9 2.54705e9 4.41162e9i 1.22893e10 + 2.12857e10i
2.9 464.062 803.780i 69924.4 + 121113.i 617868. + 1.07018e6i −547104. + 947613.i 1.29797e8 3.31595e8 + 6.69769e8i 3.09334e9 −4.54866e9 + 7.87851e9i 5.07781e8 + 8.79503e8i
2.10 793.007 1373.53i 27102.5 + 46943.0i −209144. 362248.i 259504. 449475.i 8.59700e7 −7.41186e8 9.58592e7i 2.66270e9 3.76108e9 6.51438e9i −4.11578e8 7.12873e8i
2.11 857.595 1485.40i −46198.3 80017.8i −422364. 731556.i 2.01337e7 3.48725e7i −1.58478e8 6.24130e8 4.11105e8i 2.14815e9 9.61607e8 1.66555e9i −3.45331e10 5.98130e10i
2.12 1211.86 2098.99i −55093.7 95425.0i −1.88861e6 3.27117e6i −7.61674e6 + 1.31926e7i −2.67062e8 −1.78785e8 + 7.25660e8i −4.07200e9 −8.40444e8 + 1.45569e9i 1.84608e10 + 3.19750e10i
2.13 1358.46 2352.92i 65451.7 + 113366.i −2.64224e6 4.57650e6i −920929. + 1.59510e6i 3.55654e8 4.56257e8 5.91925e8i −8.65972e9 −3.33767e9 + 5.78102e9i 2.50209e9 + 4.33374e9i
4.1 −1330.60 2304.67i 72122.8 124920.i −2.49243e6 + 4.31702e6i 1.64217e7 + 2.84432e7i −3.83867e8 6.58019e8 3.54341e8i 7.68480e9 −5.17323e9 8.96030e9i 4.37015e10 7.56932e10i
4.2 −1089.84 1887.65i −65571.0 + 113572.i −1.32691e6 + 2.29828e6i 3.92338e6 + 6.79550e6i 2.85847e8 4.31432e8 + 6.10256e8i 1.21336e9 −3.36895e9 5.83519e9i 8.55170e9 1.48120e10i
4.3 −1071.21 1855.40i −769.406 + 1332.65i −1.24642e6 + 2.15887e6i −9.82127e6 1.70109e7i 3.29679e6 −6.30322e8 4.01546e8i 8.47744e8 5.22899e9 + 9.05688e9i −2.10414e10 + 3.64447e10i
4.4 −618.250 1070.84i 88565.1 153399.i 284110. 492093.i −1.58690e7 2.74859e7i −2.19021e8 4.82606e7 + 7.45799e8i −3.29573e9 −1.04574e10 1.81127e10i −1.96220e10 + 3.39863e10i
4.5 −351.499 608.815i −5803.13 + 10051.3i 801472. 1.38819e6i −459458. 795804.i 8.15919e6 6.00451e8 4.44977e8i −2.60116e9 5.16282e9 + 8.94227e9i −3.22998e8 + 5.59450e8i
4.6 −346.683 600.472i 35430.0 61366.5i 808198. 1.39984e6i 1.75720e7 + 3.04357e7i −4.91319e7 −5.82353e8 + 4.68414e8i −2.57485e9 2.71961e9 + 4.71050e9i 1.21838e10 2.11030e10i
4.7 −90.6842 157.070i −89485.1 + 154993.i 1.03213e6 1.78770e6i 3.79465e6 + 6.57253e6i 3.24596e7 −6.07843e8 4.34825e8i −7.54748e8 −1.07850e10 1.86802e10i 6.88230e8 1.19205e9i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.22.c.a 26
7.c even 3 1 inner 7.22.c.a 26
7.c even 3 1 49.22.a.f 13
7.d odd 6 1 49.22.a.g 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.22.c.a 26 1.a even 1 1 trivial
7.22.c.a 26 7.c even 3 1 inner
49.22.a.f 13 7.c even 3 1
49.22.a.g 13 7.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(7, [\chi])\).