Properties

Label 49.26.a.e
Level $49$
Weight $26$
Character orbit 49.a
Self dual yes
Analytic conductor $194.038$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 6 x^{11} - 1893235651143 x^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: multiple of \( 2^{48}\cdot 3^{20}\cdot 5^{5}\cdot 7^{15} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 705) q^{2} - \beta_1 q^{3} + (\beta_{3} - 414 \beta_{2} + 13487358) q^{4} + (\beta_{4} - 24 \beta_1) q^{5} + ( - \beta_{5} - 2 \beta_{4} - 782 \beta_1) q^{6} + ( - \beta_{8} + \beta_{7} - 766 \beta_{3} + 13042881 \beta_{2} + \cdots - 5116252380) q^{8}+ \cdots + (2 \beta_{8} - \beta_{7} + 9 \beta_{6} - 9028 \beta_{3} + \cdots + 414821946565) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 705) q^{2} - \beta_1 q^{3} + (\beta_{3} - 414 \beta_{2} + 13487358) q^{4} + (\beta_{4} - 24 \beta_1) q^{5} + ( - \beta_{5} - 2 \beta_{4} - 782 \beta_1) q^{6} + ( - \beta_{8} + \beta_{7} - 766 \beta_{3} + 13042881 \beta_{2} + \cdots - 5116252380) q^{8}+ \cdots + (50908319360787 \beta_{8} + \cdots - 12\!\cdots\!24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8460 q^{2} + 161848296 q^{4} - 61395028560 q^{8} + 4977863358780 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8460 q^{2} + 161848296 q^{4} - 61395028560 q^{8} + 4977863358780 q^{9} + 39593455677648 q^{11} + 357546272706144 q^{15} + 18\!\cdots\!56 q^{16}+ \cdots - 14\!\cdots\!88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 1893235651143 x^{10} + \cdots + 18\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 31\!\cdots\!47 \nu^{11} + \cdots - 35\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31\!\cdots\!47 \nu^{11} + \cdots + 35\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70\!\cdots\!49 \nu^{11} + \cdots - 40\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58\!\cdots\!49 \nu^{11} + \cdots - 10\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 75\!\cdots\!41 \nu^{11} + \cdots - 85\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!67 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\!\cdots\!83 \nu^{11} + \cdots + 74\!\cdots\!00 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 82\!\cdots\!91 \nu^{11} + \cdots - 23\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53\!\cdots\!81 \nu^{11} + \cdots + 50\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46\!\cdots\!87 \nu^{11} + \cdots + 54\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!48 \nu^{11} + \cdots + 18\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{8} - \beta_{7} + 9 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 9027 \beta_{3} + 40334030 \beta_{2} + 2976 \beta _1 + 1262157100774 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4602 \beta_{11} + 16075 \beta_{10} - 36707 \beta_{9} + 63518 \beta_{8} - 43712 \beta_{7} - 44490 \beta_{6} + 42385872 \beta_{5} + 209125458 \beta_{4} - 42387172 \beta_{3} + \cdots + 56\!\cdots\!20 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2032056 \beta_{11} + 51256304 \beta_{10} - 1739060632 \beta_{9} + 3078364325175 \beta_{8} - 2309268638457 \beta_{7} + 10132611743536 \beta_{6} + \cdots + 11\!\cdots\!76 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 26\!\cdots\!35 \beta_{11} + \cdots + 67\!\cdots\!24 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 56\!\cdots\!09 \beta_{11} + \cdots + 29\!\cdots\!94 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 14\!\cdots\!43 \beta_{11} + \cdots + 59\!\cdots\!80 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 13\!\cdots\!84 \beta_{11} + \cdots + 32\!\cdots\!76 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 80\!\cdots\!06 \beta_{11} + \cdots + 46\!\cdots\!84 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 59\!\cdots\!94 \beta_{11} + \cdots + 93\!\cdots\!74 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 45\!\cdots\!32 \beta_{11} + \cdots + 34\!\cdots\!00 ) / 8 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
365993.
−375379.
568120.
−574437.
245972.
−247870.
765345.
−764338.
767021.
−760822.
452270.
−441867.
−10092.2 −741372. 6.82987e7 −7.48736e8 7.48209e9 0 −3.50647e11 −2.97656e11 7.55642e12
1.2 −10092.2 741372. 6.82987e7 7.48736e8 −7.48209e9 0 −3.50647e11 −2.97656e11 −7.55642e12
1.3 −7023.80 −1.14256e6 1.57794e7 4.38325e8 8.02510e9 0 1.24848e11 4.58148e11 −3.07871e12
1.4 −7023.80 1.14256e6 1.57794e7 −4.38325e8 −8.02510e9 0 1.24848e11 4.58148e11 3.07871e12
1.5 −2603.49 −493842. −2.67763e7 3.09471e8 1.28571e9 0 1.57070e11 −6.03409e11 −8.05703e11
1.6 −2603.49 493842. −2.67763e7 −3.09471e8 −1.28571e9 0 1.57070e11 −6.03409e11 8.05703e11
1.7 300.357 −1.52968e6 −3.34642e7 −9.44412e8 −4.59451e8 0 −2.01295e10 1.49264e12 −2.83661e11
1.8 300.357 1.52968e6 −3.34642e7 9.44412e8 4.59451e8 0 −2.01295e10 1.49264e12 2.83661e11
1.9 5492.18 −1.52784e6 −3.39037e6 1.00981e9 −8.39119e9 0 −2.02908e11 1.48702e12 5.54607e12
1.10 5492.18 1.52784e6 −3.39037e6 −1.00981e9 8.39119e9 0 −2.02908e11 1.48702e12 −5.54607e12
1.11 9696.98 −894137. 6.04770e7 −4.19963e8 −8.67043e9 0 2.61068e11 −4.78083e10 −4.07237e12
1.12 9696.98 894137. 6.04770e7 4.19963e8 8.67043e9 0 2.61068e11 −4.78083e10 4.07237e12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 49.26.a.e 12
7.b odd 2 1 inner 49.26.a.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.26.a.e 12 1.a even 1 1 trivial
49.26.a.e 12 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 \(S_{26}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{6} + 4230 T_{2}^{5} - 132178920 T_{2}^{4} - 479489817600 T_{2}^{3} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
\( T_{3}^{12} - 7572663336048 T_{3}^{10} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 4230 T^{5} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 7572663336048 T^{10} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 19796727838824 T^{5} + \cdots + 92\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 83\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
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