Properties

Label 7098.2.a.co
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.13968.1
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 4 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 6 \nu + 2 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 2\)

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
1.38651
−2.27743
−0.386509
3.27743
1.00000 −1.00000 1.00000 −1.78801 −1.00000 1.00000 1.00000 1.00000 −1.78801
1.2 1.00000 −1.00000 1.00000 0.332808 −1.00000 1.00000 1.00000 1.00000 0.332808
1.3 1.00000 −1.00000 1.00000 3.05596 −1.00000 1.00000 1.00000 1.00000 3.05596
1.4 1.00000 −1.00000 1.00000 4.39924 −1.00000 1.00000 1.00000 1.00000 4.39924
\(n\): 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.co 4
13.b even 2 1 7098.2.a.cn 4
13.f odd 12 2 546.2.s.e 8
39.k even 12 2 1638.2.bj.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.e 8 13.f odd 12 2
1638.2.bj.f 8 39.k even 12 2
7098.2.a.cn 4 13.b even 2 1
7098.2.a.co 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{4} - 6 T_{5}^{3} + 2 T_{5}^{2} + 24 T_{5} - 8 \)
\( T_{11}^{4} - 4 T_{11}^{3} - 7 T_{11}^{2} + 40 T_{11} - 32 \)
\( T_{17}^{4} - 2 T_{17}^{3} - 30 T_{17}^{2} + 22 T_{17} + 193 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( -8 + 24 T + 2 T^{2} - 6 T^{3} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( -32 + 40 T - 7 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 193 + 22 T - 30 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 484 + 176 T - 67 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( -428 + 232 T - 15 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( 64 + 16 T - 31 T^{2} + 2 T^{3} + T^{4} \)
$31$ \( -218 + 254 T - 25 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( -72 + 144 T - 30 T^{2} - 6 T^{3} + T^{4} \)
$41$ \( 16 + 32 T - 3 T^{2} - 10 T^{3} + T^{4} \)
$43$ \( 142 - 74 T - 25 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( 7732 + 152 T - 187 T^{2} - 2 T^{3} + T^{4} \)
$53$ \( 2461 - 426 T - 106 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( 2116 + 460 T - 127 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( -1511 + 1112 T - 118 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( -3452 - 516 T + 125 T^{2} + 24 T^{3} + T^{4} \)
$71$ \( -18 + 18 T + 27 T^{2} - 12 T^{3} + T^{4} \)
$73$ \( 256 - 704 T + 248 T^{2} - 28 T^{3} + T^{4} \)
$79$ \( -104 - 148 T - 39 T^{2} + 2 T^{3} + T^{4} \)
$83$ \( -4706 + 1202 T + 51 T^{2} - 22 T^{3} + T^{4} \)
$89$ \( 4537 - 2630 T + 498 T^{2} - 38 T^{3} + T^{4} \)
$97$ \( -1136 + 920 T + 30 T^{2} - 22 T^{3} + T^{4} \)
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