Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.48406561.1
Defining polynomial: \(x^{6} - 3 x^{5} - 17 x^{4} + 39 x^{3} + 111 x^{2} - 131 x - 281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 17 x^{4} + 39 x^{3} + 111 x^{2} - 131 x - 281\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 7 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 7 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 12 \nu^{2} + 13 \nu + 41 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 11 \nu^{3} + 27 \nu^{2} + 33 \nu - 47 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 8 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 14 \beta_{2} + 15 \beta_{1} + 57\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 3 \beta_{4} + 17 \beta_{3} + 26 \beta_{2} + 73 \beta_{1} + 106\)

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
3.41496
3.10863
2.83411
−1.83411
−2.10863
−2.41496
−1.00000 1.00000 1.00000 −3.41496 −1.00000 −1.00000 −1.00000 1.00000 3.41496
1.2 −1.00000 1.00000 1.00000 −3.10863 −1.00000 −1.00000 −1.00000 1.00000 3.10863
1.3 −1.00000 1.00000 1.00000 −2.83411 −1.00000 −1.00000 −1.00000 1.00000 2.83411
1.4 −1.00000 1.00000 1.00000 1.83411 −1.00000 −1.00000 −1.00000 1.00000 −1.83411
1.5 −1.00000 1.00000 1.00000 2.10863 −1.00000 −1.00000 −1.00000 1.00000 −2.10863
1.6 −1.00000 1.00000 1.00000 2.41496 −1.00000 −1.00000 −1.00000 1.00000 −2.41496
\(n\): 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.cr 6
13.b even 2 1 7098.2.a.ct yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cr 6 1.a even 1 1 trivial
7098.2.a.ct yes 6 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{6} + 3 T_{5}^{5} - 17 T_{5}^{4} - 39 T_{5}^{3} + 111 T_{5}^{2} + 131 T_{5} - 281 \)
\( T_{11}^{3} + 5 T_{11}^{2} + 6 T_{11} + 1 \)
\( T_{17}^{3} - T_{17}^{2} - 2 T_{17} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( ( -1 + T )^{6} \)
$5$ \( -281 + 131 T + 111 T^{2} - 39 T^{3} - 17 T^{4} + 3 T^{5} + T^{6} \)
$7$ \( ( 1 + T )^{6} \)
$11$ \( ( 1 + 6 T + 5 T^{2} + T^{3} )^{2} \)
$13$ \( T^{6} \)
$17$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$19$ \( -281 + 74 T + 248 T^{2} - 10 T^{3} - 41 T^{4} + 2 T^{5} + T^{6} \)
$23$ \( -10079 - 2508 T + 2020 T^{2} + 202 T^{3} - 89 T^{4} - 4 T^{5} + T^{6} \)
$29$ \( -6208 - 4736 T + 1556 T^{2} + 363 T^{3} - 101 T^{4} - 2 T^{5} + T^{6} \)
$31$ \( 875 + 525 T - 175 T^{2} - 141 T^{3} - T^{4} + 9 T^{5} + T^{6} \)
$37$ \( -35897 + 9751 T + 3317 T^{2} - 567 T^{3} - 103 T^{4} + 7 T^{5} + T^{6} \)
$41$ \( -4887 + 9963 T + 2799 T^{2} - 899 T^{3} - 101 T^{4} + 11 T^{5} + T^{6} \)
$43$ \( 11192 + 7976 T + 850 T^{2} - 431 T^{3} - 76 T^{4} + 5 T^{5} + T^{6} \)
$47$ \( 91624 + 49496 T + 5162 T^{2} - 1029 T^{3} - 163 T^{4} + 5 T^{5} + T^{6} \)
$53$ \( 14456 + 15840 T + 3270 T^{2} - 645 T^{3} - 125 T^{4} + 6 T^{5} + T^{6} \)
$59$ \( 70952 + 980 T - 7644 T^{2} - 637 T^{3} + 189 T^{4} + 28 T^{5} + T^{6} \)
$61$ \( 64168 + 4112 T - 14298 T^{2} + 2355 T^{3} + 29 T^{4} - 23 T^{5} + T^{6} \)
$67$ \( 1688 + 3404 T - 6058 T^{2} + 2241 T^{3} - 173 T^{4} - 10 T^{5} + T^{6} \)
$71$ \( -18824 + 16772 T - 1278 T^{2} - 1141 T^{3} + 32 T^{4} + 21 T^{5} + T^{6} \)
$73$ \( 104 - 28 T - 592 T^{2} + 651 T^{3} - 80 T^{4} - 7 T^{5} + T^{6} \)
$79$ \( -78184 + 22484 T + 18784 T^{2} - 2261 T^{3} - 241 T^{4} + 14 T^{5} + T^{6} \)
$83$ \( -7624 + 5192 T + 1070 T^{2} - 669 T^{3} - 38 T^{4} + 17 T^{5} + T^{6} \)
$89$ \( 200983 + 55109 T - 5209 T^{2} - 2307 T^{3} - 73 T^{4} + 17 T^{5} + T^{6} \)
$97$ \( -487096 - 15932 T + 23624 T^{2} + 119 T^{3} - 285 T^{4} + T^{6} \)
show more
show less