Properties

Label 7098.2.a.bv
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: \(\Q(\sqrt{17}) \)
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.

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

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
−1.56155
2.56155
1.00000 −1.00000 1.00000 −0.561553 −1.00000 1.00000 1.00000 1.00000 −0.561553
1.2 1.00000 −1.00000 1.00000 3.56155 −1.00000 1.00000 1.00000 1.00000 3.56155
\(n\): 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.bv 2
13.b even 2 1 7098.2.a.bg 2
13.d odd 4 2 546.2.c.e 4
39.f even 4 2 1638.2.c.h 4
52.f even 4 2 4368.2.h.n 4
91.i even 4 2 3822.2.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.e 4 13.d odd 4 2
1638.2.c.h 4 39.f even 4 2
3822.2.c.h 4 91.i even 4 2
4368.2.h.n 4 52.f even 4 2
7098.2.a.bg 2 13.b even 2 1
7098.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{2} - 3 T_{5} - 2 \)
\( T_{11}^{2} + 7 T_{11} + 8 \)
\( T_{17}^{2} - T_{17} - 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -2 - 3 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 8 + 7 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -38 - T + T^{2} \)
$19$ \( -4 - T + T^{2} \)
$23$ \( -38 - T + T^{2} \)
$29$ \( -4 + T + T^{2} \)
$31$ \( -64 - 4 T + T^{2} \)
$37$ \( -18 + 9 T + T^{2} \)
$41$ \( ( -4 + T )^{2} \)
$43$ \( 68 - 17 T + T^{2} \)
$47$ \( -64 - 4 T + T^{2} \)
$53$ \( 64 - 18 T + T^{2} \)
$59$ \( -52 - 8 T + T^{2} \)
$61$ \( -4 - T + T^{2} \)
$67$ \( -8 - 6 T + T^{2} \)
$71$ \( -144 + 6 T + T^{2} \)
$73$ \( 86 + 19 T + T^{2} \)
$79$ \( ( -16 + T )^{2} \)
$83$ \( ( 2 + T )^{2} \)
$89$ \( ( -8 + T )^{2} \)
$97$ \( ( -10 + T )^{2} \)
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