Properties

Label 7098.2.a.cl
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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
1.80194
−1.24698
0.445042
1.00000 1.00000 1.00000 −4.24698 1.00000 1.00000 1.00000 1.00000 −4.24698
1.2 1.00000 1.00000 1.00000 −2.55496 1.00000 1.00000 1.00000 1.00000 −2.55496
1.3 1.00000 1.00000 1.00000 −1.19806 1.00000 1.00000 1.00000 1.00000 −1.19806
\(n\): e.g. 2-40 or 990-1000
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.cl yes 3
13.b even 2 1 7098.2.a.cg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cg 3 13.b even 2 1
7098.2.a.cl yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{3} + 8 T_{5}^{2} + 19 T_{5} + 13 \)
\( T_{11}^{3} + 5 T_{11}^{2} + 6 T_{11} + 1 \)
\( T_{17}^{3} + 3 T_{17}^{2} - 18 T_{17} - 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( 13 + 19 T + 8 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 1 + 6 T + 5 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( -13 - 18 T + 3 T^{2} + T^{3} \)
$19$ \( -41 - 8 T + 5 T^{2} + T^{3} \)
$23$ \( -7 - 14 T + 7 T^{2} + T^{3} \)
$29$ \( -71 - 25 T + 4 T^{2} + T^{3} \)
$31$ \( 29 - 67 T - 4 T^{2} + T^{3} \)
$37$ \( 71 - 29 T - 2 T^{2} + T^{3} \)
$41$ \( -351 + 45 T + 18 T^{2} + T^{3} \)
$43$ \( 41 - 8 T - 5 T^{2} + T^{3} \)
$47$ \( 377 + 167 T + 23 T^{2} + T^{3} \)
$53$ \( -97 - 43 T + 12 T^{2} + T^{3} \)
$59$ \( -463 - 15 T + 16 T^{2} + T^{3} \)
$61$ \( 41 + 31 T - 17 T^{2} + T^{3} \)
$67$ \( -349 - 39 T + 10 T^{2} + T^{3} \)
$71$ \( 56 + 28 T - 14 T^{2} + T^{3} \)
$73$ \( 349 + 160 T + 23 T^{2} + T^{3} \)
$79$ \( 29 - 67 T - 4 T^{2} + T^{3} \)
$83$ \( 1693 + 440 T + 37 T^{2} + T^{3} \)
$89$ \( -113 - 81 T - 10 T^{2} + T^{3} \)
$97$ \( 29 + 47 T + 20 T^{2} + T^{3} \)
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