Properties

Label 7098.2.a.cd
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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
0.445042
1.80194
−1.24698
−1.00000 −1.00000 1.00000 −1.69202 1.00000 1.00000 −1.00000 1.00000 1.69202
1.2 −1.00000 −1.00000 1.00000 −1.35690 1.00000 1.00000 −1.00000 1.00000 1.35690
1.3 −1.00000 −1.00000 1.00000 3.04892 1.00000 1.00000 −1.00000 1.00000 −3.04892
\(n\): 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.cd 3
13.b even 2 1 7098.2.a.ci yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cd 3 1.a even 1 1 trivial
7098.2.a.ci yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{3} - 7 T_{5} - 7 \)
\( T_{11}^{3} + T_{11}^{2} - 16 T_{11} - 29 \)
\( T_{17}^{3} - 9 T_{17}^{2} + 6 T_{17} + 29 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -7 - 7 T + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( -29 - 16 T + T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 29 + 6 T - 9 T^{2} + T^{3} \)
$19$ \( -29 - 16 T + T^{2} + T^{3} \)
$23$ \( -83 - 44 T - T^{2} + T^{3} \)
$29$ \( -71 - 29 T + 2 T^{2} + T^{3} \)
$31$ \( -169 - 39 T + 4 T^{2} + T^{3} \)
$37$ \( -7 + 35 T + 14 T^{2} + T^{3} \)
$41$ \( -41 - 37 T + 6 T^{2} + T^{3} \)
$43$ \( 881 + 278 T + 29 T^{2} + T^{3} \)
$47$ \( -91 - 21 T + 7 T^{2} + T^{3} \)
$53$ \( 43 + 55 T - 16 T^{2} + T^{3} \)
$59$ \( -29 + 31 T - 10 T^{2} + T^{3} \)
$61$ \( 71 + 3 T - 11 T^{2} + T^{3} \)
$67$ \( 251 - 85 T + 2 T^{2} + T^{3} \)
$71$ \( -56 + 28 T + 14 T^{2} + T^{3} \)
$73$ \( -1267 - 210 T + 7 T^{2} + T^{3} \)
$79$ \( -1261 - 211 T + 2 T^{2} + T^{3} \)
$83$ \( 419 - 78 T - 5 T^{2} + T^{3} \)
$89$ \( 497 - 147 T + T^{3} \)
$97$ \( 757 - 205 T + 6 T^{2} + T^{3} \)
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