Properties

Label 49.26.a.c
Level $49$
Weight $26$
Character orbit 49.a
Self dual yes
Analytic conductor $194.038$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 3 x^{5} - 35625342 x^{4} - 2465469952 x^{3} + 282703727994240 x^{2} + \cdots - 21\!\cdots\!72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{15}\cdot 3^{4}\cdot 5\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 705) q^{2} + (\beta_{2} - 2 \beta_1 + 12017) q^{3} + (\beta_{3} - 15 \beta_{2} + 1621 \beta_1 + 14443059) q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 91 \beta_{2} - 24417 \beta_1 - 18360192) q^{5} + (9 \beta_{5} - 10 \beta_{4} + 56 \beta_{3} - 556 \beta_{2} + \cdots + 94160597) q^{6}+ \cdots + (717 \beta_{5} + 2431 \beta_{4} - 13589 \beta_{3} + \cdots + 211282296252) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 705) q^{2} + (\beta_{2} - 2 \beta_1 + 12017) q^{3} + (\beta_{3} - 15 \beta_{2} + 1621 \beta_1 + 14443059) q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 91 \beta_{2} - 24417 \beta_1 - 18360192) q^{5} + (9 \beta_{5} - 10 \beta_{4} + 56 \beta_{3} - 556 \beta_{2} + \cdots + 94160597) q^{6}+ \cdots + ( - 139599367960629 \beta_{5} + \cdots + 11\!\cdots\!49) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4230 q^{2} + 72104 q^{3} + 86658324 q^{4} - 110161332 q^{5} + 564962452 q^{6} - 381894066504 q^{8} + 1267694965630 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4230 q^{2} + 72104 q^{3} + 86658324 q^{4} - 110161332 q^{5} + 564962452 q^{6} - 381894066504 q^{8} + 1267694965630 q^{9} + 7032334098696 q^{10} - 1675999103976 q^{11} - 95344327788584 q^{12} - 5288670743748 q^{13} - 560616671505056 q^{15} + 13\!\cdots\!60 q^{16}+ \cdots + 68\!\cdots\!84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3 x^{5} - 35625342 x^{4} - 2465469952 x^{3} + 282703727994240 x^{2} + \cdots - 21\!\cdots\!72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12841 \nu^{5} - 5592603 \nu^{4} - 429451759182 \nu^{3} + 21098814385856 \nu^{2} + \cdots - 38\!\cdots\!48 ) / 414107157639168 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 64205 \nu^{5} - 27963015 \nu^{4} - 2147258795910 \nu^{3} + 657636948781504 \nu^{2} + \cdots - 84\!\cdots\!64 ) / 138035719213056 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 149371 \nu^{5} + 583943343 \nu^{4} - 7036761073386 \nu^{3} + \cdots - 58\!\cdots\!96 ) / 59158165377024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 164393 \nu^{5} - 178240539 \nu^{4} - 3218347502670 \nu^{3} + 189366713926336 \nu^{2} + \cdots + 18\!\cdots\!84 ) / 69017859606528 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 15\beta_{2} + 213\beta _1 + 47500467 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 142\beta_{5} + 20\beta_{4} + 1097\beta_{3} - 28991\beta_{2} + 39504739\beta _1 + 5144701685 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 446618 \beta_{5} + 427516 \beta_{4} + 22692105 \beta_{3} - 409498831 \beta_{2} + 26131832159 \beta _1 + 938364097587673 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4943532978 \beta_{5} + 855070668 \beta_{4} + 44927785993 \beta_{3} - 994275328623 \beta_{2} + 842934815460567 \beta _1 + 62\!\cdots\!69 ) / 4 \) 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
5095.68
3034.56
367.466
−20.7371
−4101.63
−4372.34
−10895.4 −750334. 8.51545e7 −3.86992e8 8.17517e9 0 −5.62202e11 −2.84287e11 4.21642e12
1.2 −6773.12 409558. 1.23207e7 2.95928e8 −2.77398e9 0 1.43819e11 −6.79551e11 −2.00436e12
1.3 −1438.93 1.81888e6 −3.14839e7 −1.62596e8 −2.61725e9 0 9.35858e10 2.46105e12 2.33964e11
1.4 −662.526 −1.07986e6 −3.31155e7 −3.68627e8 7.15434e8 0 4.41705e10 3.18807e11 2.44225e11
1.5 7499.26 576205. 2.26845e7 −4.14185e8 4.32111e9 0 −8.15167e10 −5.15276e11 −3.10608e12
1.6 8040.68 −902351. 3.10981e7 9.26311e8 −7.25551e9 0 −1.97508e10 −3.30517e10 7.44816e12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.c 6
7.b odd 2 1 7.26.a.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.26.a.a 6 7.b odd 2 1
49.26.a.c 6 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 \(S_{26}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{6} + 4230 T_{2}^{5} - 135046008 T_{2}^{4} - 374552082432 T_{2}^{3} + \cdots + 42\!\cdots\!40 \) Copy content Toggle raw display
\( T_{3}^{6} - 72104 T_{3}^{5} - 3173113817736 T_{3}^{4} + \cdots - 31\!\cdots\!76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 4230 T^{5} + \cdots + 42\!\cdots\!40 \) Copy content Toggle raw display
$3$ \( T^{6} - 72104 T^{5} + \cdots - 31\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{6} + 110161332 T^{5} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 1675999103976 T^{5} + \cdots + 56\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{6} + 5288670743748 T^{5} + \cdots + 73\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 84\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 50\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 63\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 24\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 95\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 25\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 57\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 25\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 30\!\cdots\!28 \) Copy content Toggle raw display
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