Properties

Label 7098.2.a.cm
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.

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

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 −2.24698 1.00000 −1.00000 1.00000 1.00000 −2.24698
1.2 1.00000 1.00000 1.00000 −0.554958 1.00000 −1.00000 1.00000 1.00000 −0.554958
1.3 1.00000 1.00000 1.00000 0.801938 1.00000 −1.00000 1.00000 1.00000 0.801938
\(n\): 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.cm yes 3
13.b even 2 1 7098.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cf 3 13.b even 2 1
7098.2.a.cm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{3} + 2 T_{5}^{2} - T_{5} - 1 \)
\( T_{11}^{3} + 9 T_{11}^{2} + 20 T_{11} + 13 \)
\( T_{17}^{3} - 5 T_{17}^{2} - 8 T_{17} + 41 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -1 - T + 2 T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 13 + 20 T + 9 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 41 - 8 T - 5 T^{2} + T^{3} \)
$19$ \( -7 + 7 T^{2} + T^{3} \)
$23$ \( 13 - 16 T + T^{2} + T^{3} \)
$29$ \( -13 + 5 T + 6 T^{2} + T^{3} \)
$31$ \( 13 - 15 T + 2 T^{2} + T^{3} \)
$37$ \( 7 + 49 T + 14 T^{2} + T^{3} \)
$41$ \( 13 + 19 T + 8 T^{2} + T^{3} \)
$43$ \( 127 - 60 T + 3 T^{2} + T^{3} \)
$47$ \( 49 - 49 T + 7 T^{2} + T^{3} \)
$53$ \( 41 + 17 T - 10 T^{2} + T^{3} \)
$59$ \( -1 + 5 T + 8 T^{2} + T^{3} \)
$61$ \( ( 1 + T )^{3} \)
$67$ \( 7 - 7 T + T^{3} \)
$71$ \( 104 + 124 T + 22 T^{2} + T^{3} \)
$73$ \( 49 + 98 T + 21 T^{2} + T^{3} \)
$79$ \( 13 - 99 T + 2 T^{2} + T^{3} \)
$83$ \( -349 - 256 T + 3 T^{2} + T^{3} \)
$89$ \( 351 + 171 T + 24 T^{2} + T^{3} \)
$97$ \( -727 - 183 T - 2 T^{2} + T^{3} \)
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