Properties

Label 7098.2.a.bt
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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.

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

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
2.56155
−1.56155
1.00000 −1.00000 1.00000 −2.56155 −1.00000 1.00000 1.00000 1.00000 −2.56155
1.2 1.00000 −1.00000 1.00000 1.56155 −1.00000 1.00000 1.00000 1.00000 1.56155
\(n\): 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.bt 2
13.b even 2 1 7098.2.a.bi 2
13.c even 3 2 546.2.l.l 4
39.i odd 6 2 1638.2.r.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.l 4 13.c even 3 2
1638.2.r.y 4 39.i odd 6 2
7098.2.a.bi 2 13.b even 2 1
7098.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{2} + T_{5} - 4 \)
\( T_{11}^{2} + T_{11} - 4 \)
\( T_{17}^{2} + 8 T_{17} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -4 + T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -4 + T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -1 + 8 T + T^{2} \)
$19$ \( -4 + T + T^{2} \)
$23$ \( -8 - 6 T + T^{2} \)
$29$ \( -13 + 4 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 16 + 9 T + T^{2} \)
$41$ \( -59 + 6 T + T^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( 4 + 13 T + T^{2} \)
$53$ \( ( -7 + T )^{2} \)
$59$ \( -16 - 2 T + T^{2} \)
$61$ \( -59 + 6 T + T^{2} \)
$67$ \( 8 - 10 T + T^{2} \)
$71$ \( -152 - 2 T + T^{2} \)
$73$ \( 16 + 9 T + T^{2} \)
$79$ \( 16 + 9 T + T^{2} \)
$83$ \( -16 + 2 T + T^{2} \)
$89$ \( 26 - 11 T + T^{2} \)
$97$ \( 8 + 10 T + T^{2} \)
show more
show less