Properties

Label 7098.2.a.ch
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.

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

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
0.445042
−1.24698
1.00000 −1.00000 1.00000 −2.80194 −1.00000 1.00000 1.00000 1.00000 −2.80194
1.2 1.00000 −1.00000 1.00000 −1.44504 −1.00000 1.00000 1.00000 1.00000 −1.44504
1.3 1.00000 −1.00000 1.00000 0.246980 −1.00000 1.00000 1.00000 1.00000 0.246980
\(n\): 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.ch yes 3
13.b even 2 1 7098.2.a.ce 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.ce 3 13.b even 2 1
7098.2.a.ch 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{3} + 4 T_{5}^{2} + 3 T_{5} - 1 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 18 T_{11} + 27 \)
\( T_{17}^{3} + 3 T_{17}^{2} - 4 T_{17} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -1 + 3 T + 4 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 27 - 18 T - 3 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 1 - 4 T + 3 T^{2} + T^{3} \)
$19$ \( 7 + 14 T + 7 T^{2} + T^{3} \)
$23$ \( 43 - 36 T + 5 T^{2} + T^{3} \)
$29$ \( -127 - 43 T + 2 T^{2} + T^{3} \)
$31$ \( -13 + 3 T + 10 T^{2} + T^{3} \)
$37$ \( 7 - 7 T + T^{3} \)
$41$ \( 1 + 3 T - 4 T^{2} + T^{3} \)
$43$ \( 203 - 28 T - 7 T^{2} + T^{3} \)
$47$ \( -83 - 25 T + 3 T^{2} + T^{3} \)
$53$ \( -113 - 71 T + 2 T^{2} + T^{3} \)
$59$ \( -13 - 29 T - 2 T^{2} + T^{3} \)
$61$ \( ( 7 + T )^{3} \)
$67$ \( -461 - 79 T + 6 T^{2} + T^{3} \)
$71$ \( 8 - 36 T - 2 T^{2} + T^{3} \)
$73$ \( -29 + 24 T + 11 T^{2} + T^{3} \)
$79$ \( 377 + 159 T + 22 T^{2} + T^{3} \)
$83$ \( 1079 + 318 T + 31 T^{2} + T^{3} \)
$89$ \( -889 - 273 T + T^{3} \)
$97$ \( 679 + 245 T + 28 T^{2} + T^{3} \)
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