Properties

Label 7098.2.a.bl
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + ( -1 - \beta ) q^{5} - q^{6} + q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} + ( -1 - \beta ) q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + ( 1 + \beta ) q^{10} + ( -1 + \beta ) q^{11} + q^{12} - q^{14} + ( -1 - \beta ) q^{15} + q^{16} + ( 1 - 3 \beta ) q^{17} - q^{18} + ( -3 + 3 \beta ) q^{19} + ( -1 - \beta ) q^{20} + q^{21} + ( 1 - \beta ) q^{22} + ( -3 - \beta ) q^{23} - q^{24} + 3 \beta q^{25} + q^{27} + q^{28} + ( -1 + 3 \beta ) q^{29} + ( 1 + \beta ) q^{30} + ( 4 - 4 \beta ) q^{31} - q^{32} + ( -1 + \beta ) q^{33} + ( -1 + 3 \beta ) q^{34} + ( -1 - \beta ) q^{35} + q^{36} + ( 5 + \beta ) q^{37} + ( 3 - 3 \beta ) q^{38} + ( 1 + \beta ) q^{40} + ( -4 + 2 \beta ) q^{41} - q^{42} + ( -9 + \beta ) q^{43} + ( -1 + \beta ) q^{44} + ( -1 - \beta ) q^{45} + ( 3 + \beta ) q^{46} + q^{48} + q^{49} -3 \beta q^{50} + ( 1 - 3 \beta ) q^{51} + ( 2 + 4 \beta ) q^{53} - q^{54} + ( -3 - \beta ) q^{55} - q^{56} + ( -3 + 3 \beta ) q^{57} + ( 1 - 3 \beta ) q^{58} + ( 8 - 4 \beta ) q^{59} + ( -1 - \beta ) q^{60} + ( -1 + 3 \beta ) q^{61} + ( -4 + 4 \beta ) q^{62} + q^{63} + q^{64} + ( 1 - \beta ) q^{66} + ( 2 + 2 \beta ) q^{67} + ( 1 - 3 \beta ) q^{68} + ( -3 - \beta ) q^{69} + ( 1 + \beta ) q^{70} -8 q^{71} - q^{72} + ( 1 + \beta ) q^{73} + ( -5 - \beta ) q^{74} + 3 \beta q^{75} + ( -3 + 3 \beta ) q^{76} + ( -1 + \beta ) q^{77} + ( -6 - 2 \beta ) q^{79} + ( -1 - \beta ) q^{80} + q^{81} + ( 4 - 2 \beta ) q^{82} + ( -14 + 2 \beta ) q^{83} + q^{84} + ( 11 + 5 \beta ) q^{85} + ( 9 - \beta ) q^{86} + ( -1 + 3 \beta ) q^{87} + ( 1 - \beta ) q^{88} -10 q^{89} + ( 1 + \beta ) q^{90} + ( -3 - \beta ) q^{92} + ( 4 - 4 \beta ) q^{93} + ( -9 - 3 \beta ) q^{95} - q^{96} + ( 6 - 8 \beta ) q^{97} - q^{98} + ( -1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - q^{11} + 2 q^{12} - 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 3 q^{19} - 3 q^{20} + 2 q^{21} + q^{22} - 7 q^{23} - 2 q^{24} + 3 q^{25} + 2 q^{27} + 2 q^{28} + q^{29} + 3 q^{30} + 4 q^{31} - 2 q^{32} - q^{33} + q^{34} - 3 q^{35} + 2 q^{36} + 11 q^{37} + 3 q^{38} + 3 q^{40} - 6 q^{41} - 2 q^{42} - 17 q^{43} - q^{44} - 3 q^{45} + 7 q^{46} + 2 q^{48} + 2 q^{49} - 3 q^{50} - q^{51} + 8 q^{53} - 2 q^{54} - 7 q^{55} - 2 q^{56} - 3 q^{57} - q^{58} + 12 q^{59} - 3 q^{60} + q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} + q^{66} + 6 q^{67} - q^{68} - 7 q^{69} + 3 q^{70} - 16 q^{71} - 2 q^{72} + 3 q^{73} - 11 q^{74} + 3 q^{75} - 3 q^{76} - q^{77} - 14 q^{79} - 3 q^{80} + 2 q^{81} + 6 q^{82} - 26 q^{83} + 2 q^{84} + 27 q^{85} + 17 q^{86} + q^{87} + q^{88} - 20 q^{89} + 3 q^{90} - 7 q^{92} + 4 q^{93} - 21 q^{95} - 2 q^{96} + 4 q^{97} - 2 q^{98} - q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
−1.00000 1.00000 1.00000 −3.56155 −1.00000 1.00000 −1.00000 1.00000 3.56155
1.2 −1.00000 1.00000 1.00000 0.561553 −1.00000 1.00000 −1.00000 1.00000 −0.561553
\(n\): 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.bl 2
13.b even 2 1 546.2.a.j 2
39.d odd 2 1 1638.2.a.u 2
52.b odd 2 1 4368.2.a.be 2
91.b odd 2 1 3822.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.j 2 13.b even 2 1
1638.2.a.u 2 39.d odd 2 1
3822.2.a.bo 2 91.b odd 2 1
4368.2.a.be 2 52.b odd 2 1
7098.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{2} + 3 T_{5} - 2 \)
\( T_{11}^{2} + T_{11} - 4 \)
\( T_{17}^{2} + T_{17} - 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 + 3 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -4 + T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -38 + T + T^{2} \)
$19$ \( -36 + 3 T + T^{2} \)
$23$ \( 8 + 7 T + T^{2} \)
$29$ \( -38 - T + T^{2} \)
$31$ \( -64 - 4 T + T^{2} \)
$37$ \( 26 - 11 T + T^{2} \)
$41$ \( -8 + 6 T + T^{2} \)
$43$ \( 68 + 17 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( -32 - 12 T + T^{2} \)
$61$ \( -38 - T + T^{2} \)
$67$ \( -8 - 6 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( -2 - 3 T + T^{2} \)
$79$ \( 32 + 14 T + T^{2} \)
$83$ \( 152 + 26 T + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( -268 - 4 T + T^{2} \)
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