Properties

Label 49.26.a.g
Level $49$
Weight $26$
Character orbit 49.a
Self dual yes
Analytic conductor $194.038$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 7 x^{15} - 102767646 x^{14} - 8353831787 x^{13} + \cdots - 19\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{65}\cdot 3^{19}\cdot 5^{9}\cdot 7^{31} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 253) q^{2} + ( - \beta_{2} - 8 \beta_1 + 33214) q^{3} + (\beta_{3} - 3 \beta_{2} - 245 \beta_1 + 17893444) q^{4} + ( - \beta_{5} + 75 \beta_{2} + 1223 \beta_1 + 18010971) q^{5} + ( - \beta_{5} + \beta_{4} + 24 \beta_{3} - 46 \beta_{2} - 92067 \beta_1 + 408216329) q^{6} + ( - \beta_{6} + 3 \beta_{5} + 7 \beta_{4} - 179 \beta_{3} + \cdots + 8571694878) q^{8}+ \cdots + (\beta_{8} - 110 \beta_{5} + 38 \beta_{4} + 1719 \beta_{3} + \cdots + 321681529729) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 253) q^{2} + ( - \beta_{2} - 8 \beta_1 + 33214) q^{3} + (\beta_{3} - 3 \beta_{2} - 245 \beta_1 + 17893444) q^{4} + ( - \beta_{5} + 75 \beta_{2} + 1223 \beta_1 + 18010971) q^{5} + ( - \beta_{5} + \beta_{4} + 24 \beta_{3} - 46 \beta_{2} - 92067 \beta_1 + 408216329) q^{6} + ( - \beta_{6} + 3 \beta_{5} + 7 \beta_{4} - 179 \beta_{3} + \cdots + 8571694878) q^{8}+ \cdots + ( - 4510607345 \beta_{15} + \cdots + 13\!\cdots\!95) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4050 q^{2} + 531440 q^{3} + 286295596 q^{4} + 288173088 q^{5} + 6531645442 q^{6} + 137183373360 q^{8} + 5146896583216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4050 q^{2} + 531440 q^{3} + 286295596 q^{4} + 288173088 q^{5} + 6531645442 q^{6} + 137183373360 q^{8} + 5146896583216 q^{9} - 918803280822 q^{10} - 253661467680 q^{11} + 59498382182260 q^{12} - 68129645475920 q^{13} - 14\!\cdots\!08 q^{15}+ \cdots + 21\!\cdots\!28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7 x^{15} - 102767646 x^{14} - 8353831787 x^{13} + \cdots - 19\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21\!\cdots\!43 \nu^{15} + \cdots + 21\!\cdots\!09 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\!\cdots\!43 \nu^{15} + \cdots - 32\!\cdots\!71 ) / 66\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 63\!\cdots\!71 \nu^{15} + \cdots - 19\!\cdots\!25 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!05 \nu^{15} + \cdots - 10\!\cdots\!77 ) / 66\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43\!\cdots\!09 \nu^{15} + \cdots - 57\!\cdots\!15 ) / 19\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!21 \nu^{15} + \cdots - 46\!\cdots\!89 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!97 \nu^{15} + \cdots - 20\!\cdots\!37 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\!\cdots\!69 \nu^{15} + \cdots - 19\!\cdots\!13 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 75\!\cdots\!65 \nu^{15} + \cdots + 44\!\cdots\!41 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!43 \nu^{15} + \cdots + 11\!\cdots\!03 ) / 28\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!77 \nu^{15} + \cdots - 18\!\cdots\!03 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!53 \nu^{15} + \cdots - 27\!\cdots\!19 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 99\!\cdots\!17 \nu^{15} + \cdots - 49\!\cdots\!21 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 47\!\cdots\!93 \nu^{15} + \cdots + 31\!\cdots\!17 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3\beta_{2} + 263\beta _1 + 51383868 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} - 3\beta_{5} - 7\beta_{4} + 941\beta_{3} + 21965\beta_{2} + 85243557\beta _1 + 13620463462 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} - 19 \beta_{12} - 4 \beta_{11} + 7 \beta_{10} + 2 \beta_{9} - 63 \beta_{8} - 108 \beta_{7} + 1490 \beta_{6} + 108643 \beta_{5} - 6810 \beta_{4} + 121345656 \beta_{3} - 591407042 \beta_{2} + \cdots + 43\!\cdots\!11 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 93 \beta_{15} - 5741 \beta_{14} - 1011 \beta_{13} - 8163 \beta_{12} + 69214 \beta_{11} + 14607 \beta_{10} + 68822 \beta_{9} - 104631 \beta_{8} - 79252 \beta_{7} + 38647973 \beta_{6} + \cdots + 85\!\cdots\!39 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 39195651 \beta_{15} + 40815523 \beta_{14} - 46487883 \beta_{13} - 851762469 \beta_{12} + 65279168 \beta_{11} + 622844521 \beta_{10} + 376192326 \beta_{9} + \cdots + 10\!\cdots\!09 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 18189430247 \beta_{15} - 592389286503 \beta_{14} - 126130822553 \beta_{13} - 1258441117827 \beta_{12} + 7043336274116 \beta_{11} + \cdots + 78\!\cdots\!19 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12\!\cdots\!83 \beta_{15} + \cdots + 29\!\cdots\!01 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 85\!\cdots\!47 \beta_{15} + \cdots + 32\!\cdots\!39 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 17\!\cdots\!42 \beta_{15} + \cdots + 42\!\cdots\!28 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 29\!\cdots\!17 \beta_{15} + \cdots + 12\!\cdots\!77 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 50\!\cdots\!95 \beta_{15} + \cdots + 12\!\cdots\!85 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 42\!\cdots\!47 \beta_{15} + \cdots + 23\!\cdots\!67 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 14\!\cdots\!31 \beta_{15} + \cdots + 37\!\cdots\!65 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 10\!\cdots\!75 \beta_{15} + \cdots + 87\!\cdots\!25 ) / 8 \) 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
5778.37
4935.59
4265.61
3464.78
3201.28
1779.55
1607.81
−22.8492
−56.5992
−1268.52
−2334.92
−3350.76
−3416.50
−4128.77
−5183.35
−5263.74
−11302.7 854307. 9.41976e7 −4.59088e8 −9.65602e9 0 −6.85434e11 −1.17448e11 5.18896e12
1.2 −9617.18 −1.62258e6 5.89356e7 7.42240e8 1.56046e10 0 −2.44095e11 1.78548e12 −7.13825e12
1.3 −8277.22 −79373.8 3.49580e7 7.14800e7 6.56995e8 0 −1.16173e10 −8.40988e11 −5.91656e11
1.4 −6675.56 1.38614e6 1.10086e7 6.74635e8 −9.25329e9 0 1.50506e11 1.07411e12 −4.50356e12
1.5 −6148.57 −1.04706e6 4.25045e6 −8.38110e8 6.43792e9 0 1.80178e11 2.49046e11 5.15317e12
1.6 −3305.11 1.44999e6 −2.26307e7 −7.72886e8 −4.79237e9 0 1.85698e11 1.25518e12 2.55447e12
1.7 −2961.62 212163. −2.47832e7 3.66500e8 −6.28345e8 0 1.72774e11 −8.02276e11 −1.08543e12
1.8 299.698 −1.04757e6 −3.34646e7 8.64767e8 −3.13954e8 0 −2.00855e10 2.50105e11 2.59169e11
1.9 367.198 −1.08049e6 −3.34196e7 −3.08176e8 −3.96756e8 0 −2.45928e10 3.20179e11 −1.13162e11
1.10 2791.04 617931. −2.57645e7 −4.34224e8 1.72467e9 0 −1.65562e11 −4.65450e11 −1.21194e12
1.11 4923.84 1.40850e6 −9.31023e6 7.35037e8 6.93521e9 0 −2.11059e11 1.13657e12 3.61920e12
1.12 6955.51 218993. 1.48247e7 2.94492e8 1.52321e9 0 −1.30275e11 −7.99331e11 2.04834e12
1.13 7087.00 −1.56615e6 1.66712e7 −7.06414e7 −1.10993e10 0 −1.19652e11 1.60553e12 −5.00636e11
1.14 8511.53 −353422. 3.88917e7 −7.70711e8 −3.00816e9 0 4.54286e10 −7.22382e11 −6.55993e12
1.15 10620.7 −462798. 7.92447e7 7.26756e8 −4.91524e9 0 4.85262e11 −6.33106e11 7.71865e12
1.16 10781.5 1.64286e6 8.26859e7 −5.33897e8 1.77124e10 0 5.29710e11 1.85169e12 −5.75620e12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.g 16
7.b odd 2 1 49.26.a.f 16
7.d odd 6 2 7.26.c.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.26.c.a 32 7.d odd 6 2
49.26.a.f 16 7.b odd 2 1
49.26.a.g 16 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 \(S_{26}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{16} - 4050 T_{2}^{15} - 403382004 T_{2}^{14} + 1519515733680 T_{2}^{13} + \cdots - 26\!\cdots\!24 \) Copy content Toggle raw display
\( T_{3}^{16} - 531440 T_{3}^{15} - 9210542930352 T_{3}^{14} + \cdots + 44\!\cdots\!61 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4050 T^{15} + \cdots - 26\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{16} - 531440 T^{15} + \cdots + 44\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{16} - 288173088 T^{15} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 253661467680 T^{15} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{16} + 68129645475920 T^{15} + \cdots + 92\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 24\!\cdots\!59 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 73\!\cdots\!31 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 54\!\cdots\!79 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 42\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13\!\cdots\!29 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 30\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 59\!\cdots\!71 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 70\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 16\!\cdots\!79 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 75\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 83\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
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