Properties

Label 7098.2.a.ca
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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−3.27492
4.27492
1.00000 1.00000 1.00000 −3.27492 1.00000 1.00000 1.00000 1.00000 −3.27492
1.2 1.00000 1.00000 1.00000 4.27492 1.00000 1.00000 1.00000 1.00000 4.27492
\(n\): 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.ca 2
13.b even 2 1 7098.2.a.bm 2
13.c even 3 2 546.2.l.j 4
39.i odd 6 2 1638.2.r.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.j 4 13.c even 3 2
1638.2.r.x 4 39.i odd 6 2
7098.2.a.bm 2 13.b even 2 1
7098.2.a.ca 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{2} - T_{5} - 14 \)
\( T_{11} + 4 \)
\( T_{17} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -14 - T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( -56 + 2 T + T^{2} \)
$23$ \( -12 + 3 T + T^{2} \)
$29$ \( 6 - 9 T + T^{2} \)
$31$ \( -8 - 5 T + T^{2} \)
$37$ \( -2 + 7 T + T^{2} \)
$41$ \( 6 - 9 T + T^{2} \)
$43$ \( -128 + T + T^{2} \)
$47$ \( -56 - 2 T + T^{2} \)
$53$ \( -41 - 8 T + T^{2} \)
$59$ \( -128 + T + T^{2} \)
$61$ \( ( 9 + T )^{2} \)
$67$ \( 28 + 13 T + T^{2} \)
$71$ \( -12 - 3 T + T^{2} \)
$73$ \( 76 - 19 T + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( -128 + T + T^{2} \)
$89$ \( 6 - 9 T + T^{2} \)
$97$ \( -8 - 14 T + T^{2} \)
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