Properties

Label 7098.2.a.bs
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
1.00000 −1.00000 1.00000 −2.73205 −1.00000 −1.00000 1.00000 1.00000 −2.73205
1.2 1.00000 −1.00000 1.00000 0.732051 −1.00000 −1.00000 1.00000 1.00000 0.732051
\(n\): 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.bs 2
13.b even 2 1 7098.2.a.bj 2
13.f odd 12 2 546.2.s.d 4
39.k even 12 2 1638.2.bj.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.d 4 13.f odd 12 2
1638.2.bj.d 4 39.k even 12 2
7098.2.a.bj 2 13.b even 2 1
7098.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -2 + 2 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + 4 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 1 - 4 T + T^{2} \)
$19$ \( -11 - 2 T + T^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( -74 - 2 T + T^{2} \)
$37$ \( 6 - 6 T + T^{2} \)
$41$ \( ( -7 + T )^{2} \)
$43$ \( -2 + 2 T + T^{2} \)
$47$ \( -11 + 2 T + T^{2} \)
$53$ \( 37 + 14 T + T^{2} \)
$59$ \( -12 - 12 T + T^{2} \)
$61$ \( 97 + 20 T + T^{2} \)
$67$ \( -12 - 12 T + T^{2} \)
$71$ \( -18 + 6 T + T^{2} \)
$73$ \( 88 + 20 T + T^{2} \)
$79$ \( -183 + 6 T + T^{2} \)
$83$ \( -66 - 6 T + T^{2} \)
$89$ \( -3 - 6 T + T^{2} \)
$97$ \( -2 + 2 T + T^{2} \)
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