Properties

Label 49.26.a.d
Level $49$
Weight $26$
Character orbit 49.a
Self dual yes
Analytic conductor $194.038$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(194.038422177\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - x^{6} - 212249190 x^{5} + 97966970896 x^{4} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{10}\cdot 5^{4}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1196) q^{2} + (\beta_{2} - 3 \beta_1 + 85596) q^{3} + (\beta_{3} + 8 \beta_{2} + 1701 \beta_1 + 28518705) q^{4} + (\beta_{4} + 4 \beta_{3} + 154 \beta_{2} + 3430 \beta_1 - 69332099) q^{5} + (\beta_{5} + 9 \beta_{4} - 58 \beta_{3} + 1632 \beta_{2} + 246113 \beta_1 - 78430444) q^{6} + ( - \beta_{6} + 30 \beta_{5} - 179 \beta_{4} + 4381 \beta_{3} + \cdots + 97127645681) q^{8}+ \cdots + (20 \beta_{6} - 12 \beta_{5} + 1291 \beta_{4} + \cdots + 357024407710) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1196) q^{2} + (\beta_{2} - 3 \beta_1 + 85596) q^{3} + (\beta_{3} + 8 \beta_{2} + 1701 \beta_1 + 28518705) q^{4} + (\beta_{4} + 4 \beta_{3} + 154 \beta_{2} + 3430 \beta_1 - 69332099) q^{5} + (\beta_{5} + 9 \beta_{4} - 58 \beta_{3} + 1632 \beta_{2} + 246113 \beta_1 - 78430444) q^{6} + ( - \beta_{6} + 30 \beta_{5} - 179 \beta_{4} + 4381 \beta_{3} + \cdots + 97127645681) q^{8}+ \cdots + (193981206048288 \beta_{6} + \cdots + 42\!\cdots\!57) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 8373 q^{2} + 599172 q^{3} + 199632661 q^{4} - 485320794 q^{5} - 548762130 q^{6} + 679913241639 q^{8} + 2499178495563 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 8373 q^{2} + 599172 q^{3} + 199632661 q^{4} - 485320794 q^{5} - 548762130 q^{6} + 679913241639 q^{8} + 2499178495563 q^{9} + 876704815140 q^{10} - 7845139606524 q^{11} + 83731581305106 q^{12} + 75871445642734 q^{13} + 12\!\cdots\!84 q^{15}+ \cdots + 29\!\cdots\!00 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 212249190 x^{5} + 97966970896 x^{4} + \cdots + 48\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1036616647 \nu^{6} - 13865980076517 \nu^{5} + \cdots + 22\!\cdots\!76 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1036616647 \nu^{6} + 13865980076517 \nu^{5} + \cdots - 66\!\cdots\!44 ) / 71\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 752589740293 \nu^{6} + \cdots - 33\!\cdots\!44 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6160754048227 \nu^{6} + \cdots + 41\!\cdots\!16 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26076045920255 \nu^{6} + \cdots - 31\!\cdots\!56 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 8\beta_{2} - 691\beta _1 + 60642721 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 30\beta_{5} - 179\beta_{4} + 793\beta_{3} - 67150\beta_{2} + 85130293\beta _1 - 41907013459 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 34249 \beta_{6} - 156846 \beta_{5} - 305591 \beta_{4} + 119770063 \beta_{3} + 1250646434 \beta_{2} - 92519722561 \beta _1 + 51\!\cdots\!95 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 268445311 \beta_{6} + 4409193798 \beta_{5} - 25384934873 \beta_{4} - 13423327051 \beta_{3} - 14634945379154 \beta_{2} + \cdots - 56\!\cdots\!71 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5083574222981 \beta_{6} - 30471936583854 \beta_{5} - 73434644379475 \beta_{4} + \cdots + 49\!\cdots\!55 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10844.9
−8028.26
−6468.51
2316.35
4266.01
8655.05
10105.3
−9648.94 1.23076e6 5.95476e7 2.52272e8 −1.18755e10 0 −2.50807e11 6.67484e11 −2.43415e12
1.2 −6832.26 −1.31327e6 1.31253e7 4.35962e8 8.97259e9 0 1.39577e11 8.77387e11 −2.97861e12
1.3 −5272.51 291875. −5.75512e6 −1.09073e9 −1.53891e9 0 2.07260e11 −7.62098e11 5.75085e12
1.4 3512.35 741876. −2.12178e7 −1.79936e8 2.60573e9 0 −1.92379e11 −2.96909e11 −6.31998e11
1.5 5462.01 −1.29306e6 −3.72086e6 −2.23710e8 −7.06272e9 0 −2.03598e11 8.24720e11 −1.22191e12
1.6 9851.05 1.57513e6 6.34887e7 8.50288e8 1.55167e10 0 2.94883e11 1.63375e12 8.37622e12
1.7 11301.3 −634140. 9.41649e7 −5.29471e8 −7.16660e9 0 6.84977e11 −4.45155e11 −5.98371e12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 49.26.a.d 7
7.b odd 2 1 7.26.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.26.a.b 7 7.b odd 2 1
49.26.a.d 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{7} - 8373 T_{2}^{6} - 182203278 T_{2}^{5} + 1307318457096 T_{2}^{4} + \cdots + 74\!\cdots\!60 \) Copy content Toggle raw display
\( T_{3}^{7} - 599172 T_{3}^{6} - 4035595838040 T_{3}^{5} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 8373 T^{6} + \cdots + 74\!\cdots\!60 \) Copy content Toggle raw display
$3$ \( T^{7} - 599172 T^{6} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( T^{7} + 485320794 T^{6} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 7845139606524 T^{6} + \cdots - 12\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{7} - 75871445642734 T^{6} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 74\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 19\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 47\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 63\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 19\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
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