Properties

Label 7098.2.a.br
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{129}) \)
Defining polynomial: \(x^{2} - x - 32\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{129})\). We also show the integral \(q\)-expansion of the trace form.

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

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
6.17891
−5.17891
1.00000 −1.00000 1.00000 −2.00000 −1.00000 −1.00000 1.00000 1.00000 −2.00000
1.2 1.00000 −1.00000 1.00000 −2.00000 −1.00000 −1.00000 1.00000 1.00000 −2.00000
\(n\): 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.br 2
13.b even 2 1 7098.2.a.bk 2
13.c even 3 2 546.2.l.k 4
39.i odd 6 2 1638.2.r.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.k 4 13.c even 3 2
1638.2.r.ba 4 39.i odd 6 2
7098.2.a.bk 2 13.b even 2 1
7098.2.a.br 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5} + 2 \)
\( T_{11}^{2} - T_{11} - 32 \)
\( T_{17} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -32 - T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -5 + T )^{2} \)
$19$ \( -26 + 5 T + T^{2} \)
$23$ \( -26 + 5 T + T^{2} \)
$29$ \( -26 - 5 T + T^{2} \)
$31$ \( -30 + 3 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -30 + 3 T + T^{2} \)
$43$ \( 10 - 13 T + T^{2} \)
$47$ \( -30 + 3 T + T^{2} \)
$53$ \( ( 3 + T )^{2} \)
$59$ \( 10 + 13 T + T^{2} \)
$61$ \( ( 3 + T )^{2} \)
$67$ \( 58 + 19 T + T^{2} \)
$71$ \( -20 - 7 T + T^{2} \)
$73$ \( ( -4 + T )^{2} \)
$79$ \( 24 - 15 T + T^{2} \)
$83$ \( -32 - T + T^{2} \)
$89$ \( ( -9 + T )^{2} \)
$97$ \( -128 - 2 T + T^{2} \)
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