Properties

Label 3822.2.a.bi
Level $3822$
Weight $2$
Character orbit 3822.a
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} - q^{8} + q^{9} + ( - \beta - 1) q^{10} + (\beta - 2) q^{11} - q^{12} - q^{13} + ( - \beta - 1) q^{15} + q^{16} + (\beta + 4) q^{17} - q^{18} - 5 q^{19} + (\beta + 1) q^{20} + ( - \beta + 2) q^{22} + (\beta - 5) q^{23} + q^{24} + (2 \beta + 3) q^{25} + q^{26} - q^{27} + (2 \beta - 1) q^{29} + (\beta + 1) q^{30} + ( - 2 \beta + 2) q^{31} - q^{32} + ( - \beta + 2) q^{33} + ( - \beta - 4) q^{34} + q^{36} + (\beta + 3) q^{37} + 5 q^{38} + q^{39} + ( - \beta - 1) q^{40} + ( - \beta + 5) q^{41} - 2 \beta q^{43} + (\beta - 2) q^{44} + (\beta + 1) q^{45} + ( - \beta + 5) q^{46} - 3 q^{47} - q^{48} + ( - 2 \beta - 3) q^{50} + ( - \beta - 4) q^{51} - q^{52} - 3 q^{53} + q^{54} + ( - \beta + 5) q^{55} + 5 q^{57} + ( - 2 \beta + 1) q^{58} + 3 \beta q^{59} + ( - \beta - 1) q^{60} + (3 \beta + 4) q^{61} + (2 \beta - 2) q^{62} + q^{64} + ( - \beta - 1) q^{65} + (\beta - 2) q^{66} + (4 \beta - 3) q^{67} + (\beta + 4) q^{68} + ( - \beta + 5) q^{69} + (2 \beta + 11) q^{71} - q^{72} + (\beta + 11) q^{73} + ( - \beta - 3) q^{74} + ( - 2 \beta - 3) q^{75} - 5 q^{76} - q^{78} - 10 q^{79} + (\beta + 1) q^{80} + q^{81} + (\beta - 5) q^{82} + (2 \beta + 8) q^{83} + (5 \beta + 11) q^{85} + 2 \beta q^{86} + ( - 2 \beta + 1) q^{87} + ( - \beta + 2) q^{88} + ( - 3 \beta - 9) q^{89} + ( - \beta - 1) q^{90} + (\beta - 5) q^{92} + (2 \beta - 2) q^{93} + 3 q^{94} + ( - 5 \beta - 5) q^{95} + q^{96} + ( - 3 \beta + 7) q^{97} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{13} - 2 q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{18} - 10 q^{19} + 2 q^{20} + 4 q^{22} - 10 q^{23} + 2 q^{24} + 6 q^{25} + 2 q^{26} - 2 q^{27} - 2 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 4 q^{33} - 8 q^{34} + 2 q^{36} + 6 q^{37} + 10 q^{38} + 2 q^{39} - 2 q^{40} + 10 q^{41} - 4 q^{44} + 2 q^{45} + 10 q^{46} - 6 q^{47} - 2 q^{48} - 6 q^{50} - 8 q^{51} - 2 q^{52} - 6 q^{53} + 2 q^{54} + 10 q^{55} + 10 q^{57} + 2 q^{58} - 2 q^{60} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 2 q^{65} - 4 q^{66} - 6 q^{67} + 8 q^{68} + 10 q^{69} + 22 q^{71} - 2 q^{72} + 22 q^{73} - 6 q^{74} - 6 q^{75} - 10 q^{76} - 2 q^{78} - 20 q^{79} + 2 q^{80} + 2 q^{81} - 10 q^{82} + 16 q^{83} + 22 q^{85} + 2 q^{87} + 4 q^{88} - 18 q^{89} - 2 q^{90} - 10 q^{92} - 4 q^{93} + 6 q^{94} - 10 q^{95} + 2 q^{96} + 14 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−2.64575
2.64575
−1.00000 −1.00000 1.00000 −1.64575 1.00000 0 −1.00000 1.00000 1.64575
1.2 −1.00000 −1.00000 1.00000 3.64575 1.00000 0 −1.00000 1.00000 −3.64575
\(n\): 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 3822.2.a.bi 2
7.b odd 2 1 3822.2.a.bk 2
7.d odd 6 2 546.2.i.j 4
21.g even 6 2 1638.2.j.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.j 4 7.d odd 6 2
1638.2.j.k 4 21.g even 6 2
3822.2.a.bi 2 1.a even 1 1 trivial
3822.2.a.bk 2 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\):

\( T_{5}^{2} - 2T_{5} - 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 8T_{17} + 9 \) Copy content Toggle raw display
\( T_{29}^{2} + 2T_{29} - 27 \) 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
$5$ \( T^{2} - 2T - 6 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 3 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 8T + 9 \) Copy content Toggle raw display
$19$ \( (T + 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10T + 18 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 27 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 24 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 10T + 18 \) Copy content Toggle raw display
$43$ \( T^{2} - 28 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 63 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 47 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T - 103 \) Copy content Toggle raw display
$71$ \( T^{2} - 22T + 93 \) Copy content Toggle raw display
$73$ \( T^{2} - 22T + 114 \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 16T + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 18 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T - 14 \) Copy content Toggle raw display
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