Properties

Label 49.22.a.f
Level $49$
Weight $22$
Character orbit 49.a
Self dual yes
Analytic conductor $136.944$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,22,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

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: yes
Analytic conductor: \(136.943898701\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4879679 x^{11} - 12560597 x^{10} + 8757989673832 x^{9} + 34815675420589 x^{8} + \cdots + 22\!\cdots\!69 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{43}\cdot 3^{9}\cdot 5^{3}\cdot 7^{19} \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 22) q^{2} + (\beta_{2} + 6 \beta_1 - 9084) q^{3} + (\beta_{3} - 4 \beta_{2} + 51 \beta_1 + 906210) q^{4} + ( - \beta_{5} - \beta_{3} - 6 \beta_{2} + 1187 \beta_1 - 1484379) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 95 \beta_{2} + 24508 \beta_1 - 17588503) q^{6} + ( - \beta_{6} + 45 \beta_{5} - 5 \beta_{4} - 82 \beta_{3} + \cdots - 129113969) q^{8}+ \cdots + ( - \beta_{7} - 17 \beta_{5} - 3 \beta_{4} + 100 \beta_{3} + \cdots + 2789252598) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 22) q^{2} + (\beta_{2} + 6 \beta_1 - 9084) q^{3} + (\beta_{3} - 4 \beta_{2} + 51 \beta_1 + 906210) q^{4} + ( - \beta_{5} - \beta_{3} - 6 \beta_{2} + 1187 \beta_1 - 1484379) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 95 \beta_{2} + 24508 \beta_1 - 17588503) q^{6} + ( - \beta_{6} + 45 \beta_{5} - 5 \beta_{4} - 82 \beta_{3} + \cdots - 129113969) q^{8}+ \cdots + (1367743832 \beta_{12} + 24403911 \beta_{11} + \cdots - 91\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 286 q^{2} - 118097 q^{3} + 11780748 q^{4} - 19296893 q^{5} - 228651010 q^{6} - 1678492320 q^{8} + 36260337262 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 286 q^{2} - 118097 q^{3} + 11780748 q^{4} - 19296893 q^{5} - 228651010 q^{6} - 1678492320 q^{8} + 36260337262 q^{9} - 45908292458 q^{10} + 96908527507 q^{11} - 703726516612 q^{12} + 286575277674 q^{13} - 625196966663 q^{15} + 13121838202992 q^{16} - 3631296873225 q^{17} + 26768119563764 q^{18} - 56849486179647 q^{19} - 105617876046508 q^{20} - 282103228670978 q^{22} + 56010101087361 q^{23} + 151975129265904 q^{24} + 672811740581052 q^{25} + 13\!\cdots\!76 q^{26}+ \cdots - 11\!\cdots\!26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4879679 x^{11} - 12560597 x^{10} + 8757989673832 x^{9} + 34815675420589 x^{8} + \cdots + 22\!\cdots\!69 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!77 \nu^{12} + \cdots + 11\!\cdots\!75 ) / 65\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!77 \nu^{12} + \cdots + 61\!\cdots\!11 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!69 \nu^{12} + \cdots + 86\!\cdots\!55 ) / 43\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!91 \nu^{12} + \cdots - 81\!\cdots\!29 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26\!\cdots\!01 \nu^{12} + \cdots - 17\!\cdots\!91 ) / 65\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!83 \nu^{12} + \cdots + 69\!\cdots\!13 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 38\!\cdots\!71 \nu^{12} + \cdots + 20\!\cdots\!05 ) / 65\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!85 \nu^{12} + \cdots + 88\!\cdots\!03 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 67\!\cdots\!11 \nu^{12} + \cdots - 40\!\cdots\!41 ) / 43\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!09 \nu^{12} + \cdots - 93\!\cdots\!55 ) / 65\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 50\!\cdots\!79 \nu^{12} + \cdots + 33\!\cdots\!33 ) / 32\!\cdots\!76 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 4\beta_{2} + 7\beta _1 + 3002878 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} - 45\beta_{5} + 5\beta_{4} + 16\beta_{3} - 1896\beta_{2} + 5160267\beta _1 + 23188061 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7 \beta_{12} + 30 \beta_{11} - 44 \beta_{10} + 9 \beta_{9} - 84 \beta_{8} - 101 \beta_{7} + 207 \beta_{6} + 26818 \beta_{5} + 770 \beta_{4} + 7089098 \beta_{3} - 37670196 \beta_{2} + 102681483 \beta _1 + 15496079641053 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1484 \beta_{12} + 8032 \beta_{11} - 1472 \beta_{10} - 11436 \beta_{9} + 10416 \beta_{8} - 5780 \beta_{7} + 2021819 \beta_{6} - 135413331 \beta_{5} + 17378939 \beta_{4} + \cdots + 81463228144635 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 21669025 \beta_{12} + 78664290 \beta_{11} - 150576340 \beta_{10} - 15415521 \beta_{9} - 223627852 \beta_{8} - 281038019 \beta_{7} + 462475555 \beta_{6} + \cdots + 23\!\cdots\!81 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7164201205 \beta_{12} + 47314592614 \beta_{11} + 2202892292 \beta_{10} - 75169558715 \beta_{9} + 79158455292 \beta_{8} - 1685424817 \beta_{7} + \cdots + 73\!\cdots\!83 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 51978429565237 \beta_{12} + 165045758057418 \beta_{11} - 354958556804836 \beta_{10} - 96053779583029 \beta_{9} - 460314335906236 \beta_{8} + \cdots + 37\!\cdots\!09 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12\!\cdots\!61 \beta_{12} + \cdots + 20\!\cdots\!77 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 55\!\cdots\!06 \beta_{12} + \cdots + 30\!\cdots\!39 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 19\!\cdots\!29 \beta_{12} + \cdots + 45\!\cdots\!13 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 11\!\cdots\!81 \beta_{12} + \cdots + 52\!\cdots\!65 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1347.46
1200.86
846.595
782.007
453.062
345.792
−101.684
−357.683
−362.499
−629.250
−1082.21
−1100.84
−1341.60
−2716.92 −130903. 5.28448e6 1.84186e6 3.55654e8 0 −8.65972e9 6.67535e9 −5.00417e9
1.2 −2423.71 110187. 3.77722e6 1.52335e7 −2.67062e8 0 −4.07200e9 1.68089e9 −3.69216e10
1.3 −1715.19 92396.6 844728. −4.02673e7 −1.58478e8 0 2.14815e9 −1.92321e9 6.90662e10
1.4 −1586.01 −54205.1 418287. −519009. 8.59700e7 0 2.66270e9 −7.52216e9 8.23155e8
1.5 −928.125 −139849. −1.23574e6 1.09421e6 1.29797e8 0 3.09334e9 9.09732e9 −1.01556e9
1.6 −713.583 73254.7 −1.58795e6 3.44439e7 −5.22734e7 0 2.62963e9 −5.09410e9 −2.45786e10
1.7 181.368 178970. −2.06426e6 −7.58931e6 3.24596e7 0 −7.54748e8 2.15700e10 −1.37646e9
1.8 693.366 −70860.0 −1.61640e6 −3.51441e7 −4.91319e7 0 −2.57485e9 −5.43922e9 −2.43677e10
1.9 702.999 11606.3 −1.60294e6 918916. 8.15919e6 0 −2.60116e9 −1.03256e10 6.45997e8
1.10 1236.50 −177130. −568220. 3.17380e7 −2.19021e8 0 −3.29573e9 2.09148e10 3.92440e10
1.11 2142.43 1538.81 2.49284e6 1.96425e7 3.29679e6 0 8.47744e8 −1.04580e10 4.20827e10
1.12 2179.67 131142. 2.65382e6 −7.84677e6 2.85847e8 0 1.21336e9 6.73789e9 −1.71034e10
1.13 2661.21 −144246. 4.98486e6 −3.28434e7 −3.83867e8 0 7.68480e9 1.03465e10 −8.74029e10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 49.22.a.f 13
7.b odd 2 1 49.22.a.g 13
7.c even 3 2 7.22.c.a 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.22.c.a 26 7.c even 3 2
49.22.a.f 13 1.a even 1 1 trivial
49.22.a.g 13 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{13} + 286 T_{2}^{12} - 19480964 T_{2}^{11} - 4619999168 T_{2}^{10} + 139630520705152 T_{2}^{9} + \cdots - 16\!\cdots\!88 \) Copy content Toggle raw display
\( T_{3}^{13} + 118097 T_{3}^{12} - 79149013746 T_{3}^{11} + \cdots - 56\!\cdots\!83 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 286 T^{12} + \cdots - 16\!\cdots\!88 \) Copy content Toggle raw display
$3$ \( T^{13} + 118097 T^{12} + \cdots - 56\!\cdots\!83 \) Copy content Toggle raw display
$5$ \( T^{13} + 19296893 T^{12} + \cdots - 87\!\cdots\!75 \) Copy content Toggle raw display
$7$ \( T^{13} \) Copy content Toggle raw display
$11$ \( T^{13} - 96908527507 T^{12} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{13} - 286575277674 T^{12} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{13} + 3631296873225 T^{12} + \cdots - 93\!\cdots\!39 \) Copy content Toggle raw display
$19$ \( T^{13} + 56849486179647 T^{12} + \cdots - 41\!\cdots\!73 \) Copy content Toggle raw display
$23$ \( T^{13} - 56010101087361 T^{12} + \cdots - 21\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots - 48\!\cdots\!43 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 51\!\cdots\!75 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 78\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 99\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 22\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 16\!\cdots\!77 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 14\!\cdots\!87 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 11\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 28\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 38\!\cdots\!57 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 28\!\cdots\!71 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 43\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 22\!\cdots\!87 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
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