Properties

Label 7098.2.a.bn
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.00000 1.00000 1.00000 −3.46410 −1.00000 1.00000 −1.00000 1.00000 3.46410
1.2 −1.00000 1.00000 1.00000 3.46410 −1.00000 1.00000 −1.00000 1.00000 −3.46410
\(n\): 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.bn 2
13.b even 2 1 7098.2.a.bz 2
13.f odd 12 2 546.2.s.c 4
39.k even 12 2 1638.2.bj.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.c 4 13.f odd 12 2
1638.2.bj.e 4 39.k even 12 2
7098.2.a.bn 2 1.a even 1 1 trivial
7098.2.a.bz 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{2} - 12 \)
\( T_{11} - 4 \)
\( T_{17}^{2} - 6 T_{17} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -12 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( -3 - 6 T + T^{2} \)
$19$ \( 4 - 8 T + T^{2} \)
$23$ \( 13 - 8 T + T^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( -3 + 6 T + T^{2} \)
$37$ \( -44 - 4 T + T^{2} \)
$41$ \( -48 + T^{2} \)
$43$ \( -11 - 2 T + T^{2} \)
$47$ \( 4 + 8 T + T^{2} \)
$53$ \( 97 + 20 T + T^{2} \)
$59$ \( 61 - 16 T + T^{2} \)
$61$ \( -27 + T^{2} \)
$67$ \( -23 + 4 T + T^{2} \)
$71$ \( -23 + 10 T + T^{2} \)
$73$ \( -32 + 8 T + T^{2} \)
$79$ \( 24 + 12 T + T^{2} \)
$83$ \( -3 + T^{2} \)
$89$ \( 97 + 20 T + T^{2} \)
$97$ \( -104 - 4 T + T^{2} \)
show more
show less