Properties

Label 7098.2.a.cs
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.8569169.1
Defining polynomial: \(x^{6} - 2 x^{5} - 12 x^{4} + 11 x^{3} + 44 x^{2} - 9 x - 41\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( -\beta_{1} + \beta_{5} ) q^{5} - q^{6} + q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + ( -\beta_{1} + \beta_{5} ) q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + ( -\beta_{1} + \beta_{5} ) q^{10} + ( -2 \beta_{1} - \beta_{2} ) q^{11} - q^{12} + q^{14} + ( \beta_{1} - \beta_{5} ) q^{15} + q^{16} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{17} + q^{18} + ( -1 + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{19} + ( -\beta_{1} + \beta_{5} ) q^{20} - q^{21} + ( -2 \beta_{1} - \beta_{2} ) q^{22} + ( -1 - 2 \beta_{2} + \beta_{3} ) q^{23} - q^{24} + ( -2 + \beta_{1} + 3 \beta_{2} - 4 \beta_{4} - \beta_{5} ) q^{25} - q^{27} + q^{28} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{29} + ( \beta_{1} - \beta_{5} ) q^{30} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{31} + q^{32} + ( 2 \beta_{1} + \beta_{2} ) q^{33} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{34} + ( -\beta_{1} + \beta_{5} ) q^{35} + q^{36} + ( 2 - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{37} + ( -1 + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{38} + ( -\beta_{1} + \beta_{5} ) q^{40} + ( 4 - 5 \beta_{2} + 6 \beta_{4} - \beta_{5} ) q^{41} - q^{42} + ( \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{43} + ( -2 \beta_{1} - \beta_{2} ) q^{44} + ( -\beta_{1} + \beta_{5} ) q^{45} + ( -1 - 2 \beta_{2} + \beta_{3} ) q^{46} + ( 4 + 3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{47} - q^{48} + q^{49} + ( -2 + \beta_{1} + 3 \beta_{2} - 4 \beta_{4} - \beta_{5} ) q^{50} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{51} + ( 1 - 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} ) q^{53} - q^{54} + ( 2 + 2 \beta_{1} - 6 \beta_{4} - \beta_{5} ) q^{55} + q^{56} + ( 1 - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{57} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{58} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( \beta_{1} - \beta_{5} ) q^{60} + ( 4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{61} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{62} + q^{63} + q^{64} + ( 2 \beta_{1} + \beta_{2} ) q^{66} + ( 3 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{67} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{68} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{69} + ( -\beta_{1} + \beta_{5} ) q^{70} + ( -4 + \beta_{1} + 8 \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{71} + q^{72} + ( 3 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{73} + ( 2 - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{74} + ( 2 - \beta_{1} - 3 \beta_{2} + 4 \beta_{4} + \beta_{5} ) q^{75} + ( -1 + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{76} + ( -2 \beta_{1} - \beta_{2} ) q^{77} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{79} + ( -\beta_{1} + \beta_{5} ) q^{80} + q^{81} + ( 4 - 5 \beta_{2} + 6 \beta_{4} - \beta_{5} ) q^{82} + ( 3 - 6 \beta_{2} - \beta_{4} + \beta_{5} ) q^{83} - q^{84} + ( 1 + \beta_{1} - \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{85} + ( \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{86} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{87} + ( -2 \beta_{1} - \beta_{2} ) q^{88} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{89} + ( -\beta_{1} + \beta_{5} ) q^{90} + ( -1 - 2 \beta_{2} + \beta_{3} ) q^{92} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( 4 + 3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{94} + ( 7 - 6 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{95} - q^{96} + ( -3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{97} + q^{98} + ( -2 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - 6q^{3} + 6q^{4} - 3q^{5} - 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - 6q^{3} + 6q^{4} - 3q^{5} - 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} - 3q^{10} - 6q^{11} - 6q^{12} + 6q^{14} + 3q^{15} + 6q^{16} + 10q^{17} + 6q^{18} + 8q^{19} - 3q^{20} - 6q^{21} - 6q^{22} - 12q^{23} - 6q^{24} + 5q^{25} - 6q^{27} + 6q^{28} + 4q^{29} + 3q^{30} + 7q^{31} + 6q^{32} + 6q^{33} + 10q^{34} - 3q^{35} + 6q^{36} + 3q^{37} + 8q^{38} - 3q^{40} + 3q^{41} - 6q^{42} - 3q^{43} - 6q^{44} - 3q^{45} - 12q^{46} + 29q^{47} - 6q^{48} + 6q^{49} + 5q^{50} - 10q^{51} - 12q^{53} - 6q^{54} + 29q^{55} + 6q^{56} - 8q^{57} + 4q^{58} + 2q^{59} + 3q^{60} + 13q^{61} + 7q^{62} + 6q^{63} + 6q^{64} + 6q^{66} + 22q^{67} + 10q^{68} + 12q^{69} - 3q^{70} + q^{71} + 6q^{72} + 29q^{73} + 3q^{74} - 5q^{75} + 8q^{76} - 6q^{77} - 24q^{79} - 3q^{80} + 6q^{81} + 3q^{82} + 7q^{83} - 6q^{84} + 21q^{85} - 3q^{86} - 4q^{87} - 6q^{88} - 11q^{89} - 3q^{90} - 12q^{92} - 7q^{93} + 29q^{94} + 8q^{95} - 6q^{96} - 4q^{97} + 6q^{98} - 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 12 x^{4} + 11 x^{3} + 44 x^{2} - 9 x - 41\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 9 \nu^{3} + 20 \nu^{2} + 20 \nu - 25 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 9 \nu^{3} + 24 \nu^{2} + 16 \nu - 45 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} + 5 \nu^{3} - 23 \nu^{2} - 7 \nu + 28 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 7 \nu^{4} - 12 \nu^{3} + 37 \nu^{2} + 21 \nu - 41 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 6 \beta_{1} + 9\)
\(\nu^{4}\)\(=\)\(4 \beta_{5} + 6 \beta_{4} + 15 \beta_{3} - 19 \beta_{2} + 14 \beta_{1} + 48\)
\(\nu^{5}\)\(=\)\(21 \beta_{5} + 27 \beta_{4} + 52 \beta_{3} - 78 \beta_{2} + 56 \beta_{1} + 150\)

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
2.18325
−1.93682
3.67480
−1.42783
1.13488
−1.62829
1.00000 −1.00000 1.00000 −3.93408 −1.00000 1.00000 1.00000 1.00000 −3.93408
1.2 1.00000 −1.00000 1.00000 −2.41517 −1.00000 1.00000 1.00000 1.00000 −2.41517
1.3 1.00000 −1.00000 1.00000 −1.63544 −1.00000 1.00000 1.00000 1.00000 −1.63544
1.4 1.00000 −1.00000 1.00000 0.635442 −1.00000 1.00000 1.00000 1.00000 0.635442
1.5 1.00000 −1.00000 1.00000 1.41517 −1.00000 1.00000 1.00000 1.00000 1.41517
1.6 1.00000 −1.00000 1.00000 2.93408 −1.00000 1.00000 1.00000 1.00000 2.93408
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 7098.2.a.cs yes 6
13.b even 2 1 7098.2.a.cp 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cp 6 13.b even 2 1
7098.2.a.cs yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{6} + 3 T_{5}^{5} - 13 T_{5}^{4} - 31 T_{5}^{3} + 39 T_{5}^{2} + 55 T_{5} - 41 \)
\( T_{11}^{6} + 6 T_{11}^{5} - 35 T_{11}^{4} - 180 T_{11}^{3} + 476 T_{11}^{2} + 1328 T_{11} - 2857 \)
\( T_{17}^{6} - 10 T_{17}^{5} - 13 T_{17}^{4} + 290 T_{17}^{3} - 52 T_{17}^{2} - 2162 T_{17} + 379 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( -41 + 55 T + 39 T^{2} - 31 T^{3} - 13 T^{4} + 3 T^{5} + T^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( -2857 + 1328 T + 476 T^{2} - 180 T^{3} - 35 T^{4} + 6 T^{5} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( 379 - 2162 T - 52 T^{2} + 290 T^{3} - 13 T^{4} - 10 T^{5} + T^{6} \)
$19$ \( -1373 - 2368 T + 14 T^{2} + 436 T^{3} - 49 T^{4} - 8 T^{5} + T^{6} \)
$23$ \( -883 - 1682 T - 1088 T^{2} - 240 T^{3} + 17 T^{4} + 12 T^{5} + T^{6} \)
$29$ \( -35776 - 2752 T + 3812 T^{2} + 263 T^{3} - 111 T^{4} - 4 T^{5} + T^{6} \)
$31$ \( 637 - 1519 T - 1127 T^{2} + 861 T^{3} - 105 T^{4} - 7 T^{5} + T^{6} \)
$37$ \( -251 - 12263 T + 6409 T^{2} + 437 T^{3} - 167 T^{4} - 3 T^{5} + T^{6} \)
$41$ \( -1849 - 23351 T + 5111 T^{2} + 597 T^{3} - 149 T^{4} - 3 T^{5} + T^{6} \)
$43$ \( -568 - 1768 T - 1722 T^{2} - 649 T^{3} - 84 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( -776 + 12424 T - 4262 T^{2} - 323 T^{3} + 253 T^{4} - 29 T^{5} + T^{6} \)
$53$ \( 3032 + 7192 T - 2434 T^{2} - 2409 T^{3} - 185 T^{4} + 12 T^{5} + T^{6} \)
$59$ \( 344 - 380 T - 12 T^{2} + 123 T^{3} - 33 T^{4} - 2 T^{5} + T^{6} \)
$61$ \( 202376 - 73008 T - 630 T^{2} + 2357 T^{3} - 147 T^{4} - 13 T^{5} + T^{6} \)
$67$ \( 57016 - 7788 T - 7482 T^{2} + 1299 T^{3} + 67 T^{4} - 22 T^{5} + T^{6} \)
$71$ \( 33832 - 44612 T + 9206 T^{2} + 1021 T^{3} - 244 T^{4} - T^{5} + T^{6} \)
$73$ \( 650152 - 77380 T - 24472 T^{2} + 3663 T^{3} + 84 T^{4} - 29 T^{5} + T^{6} \)
$79$ \( 664 - 596 T - 2232 T^{2} - 149 T^{3} + 145 T^{4} + 24 T^{5} + T^{6} \)
$83$ \( 15016 - 43008 T + 8858 T^{2} + 1519 T^{3} - 228 T^{4} - 7 T^{5} + T^{6} \)
$89$ \( 16393 + 36621 T + 2975 T^{2} - 1577 T^{3} - 147 T^{4} + 11 T^{5} + T^{6} \)
$97$ \( -2584072 + 101532 T + 57904 T^{2} - 1353 T^{3} - 427 T^{4} + 4 T^{5} + T^{6} \)
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