Properties

Label 798.2.a.k
Level $798$
Weight $2$
Character orbit 798.a
Self dual yes
Analytic conductor $6.372$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - \beta q^{10} + (\beta - 2) q^{11} - q^{12} + \beta q^{13} + q^{14} - \beta q^{15} + q^{16} + ( - \beta + 4) q^{17} - q^{18} + q^{19} + \beta q^{20} + q^{21} + ( - \beta + 2) q^{22} + (\beta - 2) q^{23} + q^{24} + 7 q^{25} - \beta q^{26} - q^{27} - q^{28} - 6 q^{29} + \beta q^{30} - 2 \beta q^{31} - q^{32} + ( - \beta + 2) q^{33} + (\beta - 4) q^{34} - \beta q^{35} + q^{36} + 10 q^{37} - q^{38} - \beta q^{39} - \beta q^{40} - 2 q^{41} - q^{42} + 2 \beta q^{43} + (\beta - 2) q^{44} + \beta q^{45} + ( - \beta + 2) q^{46} + ( - 2 \beta + 4) q^{47} - q^{48} + q^{49} - 7 q^{50} + (\beta - 4) q^{51} + \beta q^{52} + 2 q^{53} + q^{54} + ( - 2 \beta + 12) q^{55} + q^{56} - q^{57} + 6 q^{58} + 4 q^{59} - \beta q^{60} + (2 \beta + 2) q^{61} + 2 \beta q^{62} - q^{63} + q^{64} + 12 q^{65} + (\beta - 2) q^{66} + ( - \beta + 6) q^{67} + ( - \beta + 4) q^{68} + ( - \beta + 2) q^{69} + \beta q^{70} + ( - 2 \beta - 4) q^{71} - q^{72} + 10 q^{73} - 10 q^{74} - 7 q^{75} + q^{76} + ( - \beta + 2) q^{77} + \beta q^{78} + (3 \beta - 2) q^{79} + \beta q^{80} + q^{81} + 2 q^{82} + 8 q^{83} + q^{84} + (4 \beta - 12) q^{85} - 2 \beta q^{86} + 6 q^{87} + ( - \beta + 2) q^{88} - 2 q^{89} - \beta q^{90} - \beta q^{91} + (\beta - 2) q^{92} + 2 \beta q^{93} + (2 \beta - 4) q^{94} + \beta q^{95} + q^{96} + (\beta + 8) q^{97} - q^{98} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} + 2 q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} + 2 q^{21} + 4 q^{22} - 4 q^{23} + 2 q^{24} + 14 q^{25} - 2 q^{27} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{33} - 8 q^{34} + 2 q^{36} + 20 q^{37} - 2 q^{38} - 4 q^{41} - 2 q^{42} - 4 q^{44} + 4 q^{46} + 8 q^{47} - 2 q^{48} + 2 q^{49} - 14 q^{50} - 8 q^{51} + 4 q^{53} + 2 q^{54} + 24 q^{55} + 2 q^{56} - 2 q^{57} + 12 q^{58} + 8 q^{59} + 4 q^{61} - 2 q^{63} + 2 q^{64} + 24 q^{65} - 4 q^{66} + 12 q^{67} + 8 q^{68} + 4 q^{69} - 8 q^{71} - 2 q^{72} + 20 q^{73} - 20 q^{74} - 14 q^{75} + 2 q^{76} + 4 q^{77} - 4 q^{79} + 2 q^{81} + 4 q^{82} + 16 q^{83} + 2 q^{84} - 24 q^{85} + 12 q^{87} + 4 q^{88} - 4 q^{89} - 4 q^{92} - 8 q^{94} + 2 q^{96} + 16 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(1\)
\(19\) \(-1\)

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 798.2.a.k 2
3.b odd 2 1 2394.2.a.x 2
4.b odd 2 1 6384.2.a.br 2
7.b odd 2 1 5586.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.k 2 1.a even 1 1 trivial
2394.2.a.x 2 3.b odd 2 1
5586.2.a.bd 2 7.b odd 2 1
6384.2.a.br 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(798))\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 8 \) Copy content Toggle raw display
\( T_{17}^{2} - 8T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$13$ \( T^{2} - 12 \) Copy content Toggle raw display
$17$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 48 \) Copy content Toggle raw display
$37$ \( (T - 10)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 48 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 104 \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 52 \) Copy content Toggle raw display
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