Properties

Label 49.22.a.h
Level $49$
Weight $22$
Character orbit 49.a
Self dual yes
Analytic conductor $136.944$
Analytic rank $1$
Dimension $20$
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] = [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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6 x^{19} - 31156555 x^{18} + 473576540 x^{17} + 403264243134370 x^{16} + \cdots + 94\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{44}\cdot 3^{8}\cdot 5^{4}\cdot 7^{50} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 74) q^{2} - \beta_{3} q^{3} + (\beta_{2} - 162 \beta_1 + 1023984) q^{4} + (\beta_{6} + 15 \beta_{3}) q^{5} + ( - \beta_{8} + \beta_{7} + \beta_{6} + 128 \beta_{4} + 201 \beta_{3}) q^{6} + (\beta_{5} - 142 \beta_{2} + 1095359 \beta_1 - 426279819) q^{8} + ( - \beta_{10} - \beta_{9} - 1891 \beta_{2} - 520605 \beta_1 + 2131434816) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 74) q^{2} - \beta_{3} q^{3} + (\beta_{2} - 162 \beta_1 + 1023984) q^{4} + (\beta_{6} + 15 \beta_{3}) q^{5} + ( - \beta_{8} + \beta_{7} + \beta_{6} + 128 \beta_{4} + 201 \beta_{3}) q^{6} + (\beta_{5} - 142 \beta_{2} + 1095359 \beta_1 - 426279819) q^{8} + ( - \beta_{10} - \beta_{9} - 1891 \beta_{2} - 520605 \beta_1 + 2131434816) q^{9} + ( - \beta_{16} - 14 \beta_{12} - 158 \beta_{8} - 4 \beta_{7} - 229 \beta_{6} + \cdots - 17556 \beta_{3}) q^{10}+ \cdots + (1184018232 \beta_{17} - 3955985091 \beta_{14} + \cdots + 12\!\cdots\!15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 1474 q^{2} + 20478698 q^{4} - 8519022798 q^{8} + 42625591600 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 1474 q^{2} + 20478698 q^{4} - 8519022798 q^{8} + 42625591600 q^{9} - 245872023884 q^{11} - 3696433286656 q^{15} + 25937034707106 q^{16} - 35570311584670 q^{18} - 228840149911820 q^{22} - 680681299900256 q^{23} + 22\!\cdots\!04 q^{25}+ \cdots + 25\!\cdots\!36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} - 31156555 x^{18} + 473576540 x^{17} + 403264243134370 x^{16} + \cdots + 94\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 43\!\cdots\!76 \nu^{19} + \cdots - 23\!\cdots\!56 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 56\!\cdots\!19 \nu^{19} + \cdots + 17\!\cdots\!64 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!64 \nu^{19} + \cdots + 37\!\cdots\!84 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 51\!\cdots\!24 \nu^{19} + \cdots + 27\!\cdots\!44 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!56 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 63\!\cdots\!80 \nu^{19} + \cdots + 68\!\cdots\!44 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 97\!\cdots\!76 \nu^{19} + \cdots + 95\!\cdots\!56 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 76\!\cdots\!92 \nu^{19} + \cdots + 55\!\cdots\!52 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!73 \nu^{19} + \cdots + 76\!\cdots\!88 ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31\!\cdots\!01 \nu^{19} + \cdots + 20\!\cdots\!56 ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27\!\cdots\!41 \nu^{19} + \cdots + 17\!\cdots\!16 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45\!\cdots\!72 \nu^{19} + \cdots - 24\!\cdots\!72 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 25\!\cdots\!81 \nu^{19} + \cdots + 16\!\cdots\!36 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 32\!\cdots\!79 \nu^{19} + \cdots + 19\!\cdots\!24 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26\!\cdots\!48 \nu^{19} + \cdots + 11\!\cdots\!88 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 19\!\cdots\!68 \nu^{19} + \cdots + 61\!\cdots\!08 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 51\!\cdots\!09 \nu^{19} + \cdots - 32\!\cdots\!04 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 16\!\cdots\!04 \nu^{19} + \cdots + 85\!\cdots\!24 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 82\!\cdots\!44 \nu^{19} + \cdots - 57\!\cdots\!44 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + 117649\beta_1 ) / 117649 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -14\beta_{8} - 2\beta_{4} - 574\beta_{3} + 117649\beta_{2} - 1647086\beta _1 + 366554518638 ) / 117649 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1029 \beta_{12} - 1218 \beta_{8} + 12348 \beta_{6} + 117649 \beta_{5} + 9350339 \beta_{4} - 1430856 \beta_{3} + 9411920 \beta_{2} + 620025877317 \beta _1 - 5244389001579 ) / 117649 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 9604 \beta_{19} - 9604 \beta_{18} + 1529437 \beta_{17} - 9604 \beta_{16} + 4588311 \beta_{14} + 279888 \beta_{12} + 1529437 \beta_{11} + 14706125 \beta_{10} + \cdots + 19\!\cdots\!82 ) / 117649 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6926885 \beta_{19} - 22653435 \beta_{18} + 48118441 \beta_{17} + 113987475 \beta_{16} - 15294370 \beta_{15} + 1637085835 \beta_{14} - 26833383920 \beta_{13} + \cdots + 19\!\cdots\!52 ) / 117649 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 19755646834 \beta_{19} - 20074192306 \beta_{18} + 2574662867791 \beta_{17} - 19178225542 \beta_{16} - 1361972052 \beta_{15} + 7373537913469 \beta_{14} + \cdots + 17\!\cdots\!16 ) / 16807 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 89200007960721 \beta_{19} - 310813334738767 \beta_{18} - 402188462470725 \beta_{17} + \cdots + 14\!\cdots\!84 ) / 117649 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 14\!\cdots\!04 \beta_{19} + \cdots + 86\!\cdots\!84 ) / 117649 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 88\!\cdots\!25 \beta_{19} + \cdots + 38\!\cdots\!24 ) / 117649 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14\!\cdots\!14 \beta_{19} + \cdots + 63\!\cdots\!44 ) / 117649 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 80\!\cdots\!21 \beta_{19} + \cdots - 14\!\cdots\!72 ) / 117649 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 18\!\cdots\!52 \beta_{19} + \cdots + 68\!\cdots\!68 ) / 16807 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 71\!\cdots\!05 \beta_{19} + \cdots - 33\!\cdots\!88 ) / 117649 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 11\!\cdots\!10 \beta_{19} + \cdots + 36\!\cdots\!84 ) / 117649 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 62\!\cdots\!89 \beta_{19} + \cdots - 36\!\cdots\!36 ) / 117649 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 10\!\cdots\!96 \beta_{19} + \cdots + 27\!\cdots\!16 ) / 117649 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 54\!\cdots\!85 \beta_{19} + \cdots - 34\!\cdots\!56 ) / 117649 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 12\!\cdots\!98 \beta_{19} + \cdots + 30\!\cdots\!16 ) / 16807 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 46\!\cdots\!17 \beta_{19} + \cdots - 30\!\cdots\!48 ) / 117649 \) 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
−2767.78
−2764.95
−1949.06
−1951.89
−1483.33
−1480.50
−1430.63
−1433.46
−199.029
−196.200
527.803
524.975
890.980
893.809
1676.68
1679.50
1971.08
1968.25
2763.46
2766.29
−2840.37 −30071.7 5.97053e6 −3.26479e7 8.54147e7 0 −1.10018e10 −9.55605e9 9.27319e10
1.2 −2840.37 30071.7 5.97053e6 3.26479e7 −8.54147e7 0 −1.10018e10 −9.55605e9 −9.27319e10
1.3 −2024.47 −51418.7 2.00134e6 −2.44240e7 1.04096e8 0 1.93959e8 −7.81647e9 4.94458e10
1.4 −2024.47 51418.7 2.00134e6 2.44240e7 −1.04096e8 0 1.93959e8 −7.81647e9 −4.94458e10
1.5 −1555.92 −156109. 323722. 1.05199e6 2.42892e8 0 2.75931e9 1.39095e10 −1.63681e9
1.6 −1555.92 156109. 323722. −1.05199e6 −2.42892e8 0 2.75931e9 1.39095e10 1.63681e9
1.7 −1506.04 −188163. 171009. 2.77547e7 2.83381e8 0 2.90085e9 2.49448e10 −4.17997e10
1.8 −1506.04 188163. 171009. −2.77547e7 −2.83381e8 0 2.90085e9 2.49448e10 4.17997e10
1.9 −271.614 −2241.67 −2.02338e6 −1.52643e7 608871. 0 1.11920e9 −1.04553e10 4.14601e9
1.10 −271.614 2241.67 −2.02338e6 1.52643e7 −608871. 0 1.11920e9 −1.04553e10 −4.14601e9
1.11 452.389 −140042. −1.89250e6 −2.77867e7 −6.33535e7 0 −1.80487e9 9.15142e9 −1.25704e10
1.12 452.389 140042. −1.89250e6 2.77867e7 6.33535e7 0 −1.80487e9 9.15142e9 1.25704e10
1.13 818.394 −107446. −1.42738e6 3.73079e7 −8.79330e7 0 −2.88446e9 1.08424e9 3.05326e10
1.14 818.394 107446. −1.42738e6 −3.73079e7 8.79330e7 0 −2.88446e9 1.08424e9 −3.05326e10
1.15 1604.09 −137842. 475952. −2.38394e7 −2.21111e8 0 −2.60055e9 8.54009e9 −3.82405e10
1.16 1604.09 137842. 475952. 2.38394e7 2.21111e8 0 −2.60055e9 8.54009e9 3.82405e10
1.17 1895.67 −103887. 1.49641e6 1.08556e7 −1.96936e8 0 −1.13881e9 3.32250e8 2.05786e10
1.18 1895.67 103887. 1.49641e6 −1.08556e7 1.96936e8 0 −1.13881e9 3.32250e8 −2.05786e10
1.19 2690.87 −40480.0 5.14363e6 1.91327e7 −1.08926e8 0 8.19769e9 −8.82172e9 5.14835e10
1.20 2690.87 40480.0 5.14363e6 −1.91327e7 1.08926e8 0 8.19769e9 −8.82172e9 −5.14835e10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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 49.22.a.h 20
7.b odd 2 1 inner 49.22.a.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.22.a.h 20 1.a even 1 1 trivial
49.22.a.h 20 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 \(S_{22}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{10} + 737 T_{2}^{9} - 15333850 T_{2}^{8} - 8984248152 T_{2}^{7} + 78007251815072 T_{2}^{6} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
\( T_{3}^{20} - 125916327830 T_{3}^{18} + \cdots + 78\!\cdots\!04 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 737 T^{9} + \cdots - 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} - 125916327830 T^{18} + \cdots + 78\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} + 122936011942 T^{9} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{10} + 340340649950128 T^{9} + \cdots + 30\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 77\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 46\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 35\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 65\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
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