Properties

Label 7098.2.a.bq
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.

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

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
−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.bq 2
13.b even 2 1 7098.2.a.bw 2
13.e even 6 2 546.2.l.i 4
39.h odd 6 2 1638.2.r.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.i 4 13.e even 6 2
1638.2.r.z 4 39.h odd 6 2
7098.2.a.bq 2 1.a even 1 1 trivial
7098.2.a.bw 2 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 - 3 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -4 + T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 5 + T )^{2} \)
$19$ \( -36 + 3 T + T^{2} \)
$23$ \( 8 + 10 T + T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( -16 + 2 T + T^{2} \)
$37$ \( -106 + T + T^{2} \)
$41$ \( -17 + T^{2} \)
$43$ \( 32 + 14 T + T^{2} \)
$47$ \( -36 - 3 T + T^{2} \)
$53$ \( 47 + 16 T + T^{2} \)
$59$ \( 64 + 18 T + T^{2} \)
$61$ \( ( 1 + T )^{2} \)
$67$ \( -8 - 6 T + T^{2} \)
$71$ \( ( -4 + T )^{2} \)
$73$ \( 16 - 9 T + T^{2} \)
$79$ \( 8 - 7 T + T^{2} \)
$83$ \( -64 - 4 T + T^{2} \)
$89$ \( -26 - 7 T + T^{2} \)
$97$ \( -52 + 8 T + T^{2} \)
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