Properties

Label 3822.2.a.bs
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + \beta q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + \beta q^{5} + q^{6} + q^{8} + q^{9} + \beta q^{10} + ( - \beta - 1) q^{11} + q^{12} + q^{13} + \beta q^{15} + q^{16} + ( - 3 \beta + 1) q^{17} + q^{18} + 3 q^{19} + \beta q^{20} + ( - \beta - 1) q^{22} + (5 \beta - 2) q^{23} + q^{24} - 3 q^{25} + q^{26} + q^{27} + 5 q^{29} + \beta q^{30} + ( - 2 \beta + 4) q^{31} + q^{32} + ( - \beta - 1) q^{33} + ( - 3 \beta + 1) q^{34} + q^{36} + (5 \beta + 2) q^{37} + 3 q^{38} + q^{39} + \beta q^{40} + (\beta + 6) q^{41} + ( - 2 \beta + 2) q^{43} + ( - \beta - 1) q^{44} + \beta q^{45} + (5 \beta - 2) q^{46} + 9 q^{47} + q^{48} - 3 q^{50} + ( - 3 \beta + 1) q^{51} + q^{52} + ( - 8 \beta + 1) q^{53} + q^{54} + ( - \beta - 2) q^{55} + 3 q^{57} + 5 q^{58} + (\beta + 9) q^{59} + \beta q^{60} + (3 \beta - 1) q^{61} + ( - 2 \beta + 4) q^{62} + q^{64} + \beta q^{65} + ( - \beta - 1) q^{66} + (4 \beta - 9) q^{67} + ( - 3 \beta + 1) q^{68} + (5 \beta - 2) q^{69} - q^{71} + q^{72} + (5 \beta + 6) q^{73} + (5 \beta + 2) q^{74} - 3 q^{75} + 3 q^{76} + q^{78} + ( - 8 \beta + 2) q^{79} + \beta q^{80} + q^{81} + (\beta + 6) q^{82} + (4 \beta - 2) q^{83} + (\beta - 6) q^{85} + ( - 2 \beta + 2) q^{86} + 5 q^{87} + ( - \beta - 1) q^{88} + (3 \beta - 2) q^{89} + \beta q^{90} + (5 \beta - 2) q^{92} + ( - 2 \beta + 4) q^{93} + 9 q^{94} + 3 \beta q^{95} + q^{96} + ( - 7 \beta - 2) q^{97} + ( - \beta - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 6 q^{19} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 6 q^{25} + 2 q^{26} + 2 q^{27} + 10 q^{29} + 8 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{34} + 2 q^{36} + 4 q^{37} + 6 q^{38} + 2 q^{39} + 12 q^{41} + 4 q^{43} - 2 q^{44} - 4 q^{46} + 18 q^{47} + 2 q^{48} - 6 q^{50} + 2 q^{51} + 2 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} + 6 q^{57} + 10 q^{58} + 18 q^{59} - 2 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{66} - 18 q^{67} + 2 q^{68} - 4 q^{69} - 2 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} - 6 q^{75} + 6 q^{76} + 2 q^{78} + 4 q^{79} + 2 q^{81} + 12 q^{82} - 4 q^{83} - 12 q^{85} + 4 q^{86} + 10 q^{87} - 2 q^{88} - 4 q^{89} - 4 q^{92} + 8 q^{93} + 18 q^{94} + 2 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
1.00000 1.00000 1.00000 −1.41421 1.00000 0 1.00000 1.00000 −1.41421
1.2 1.00000 1.00000 1.00000 1.41421 1.00000 0 1.00000 1.00000 1.41421
\(n\): 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.bs 2
7.b odd 2 1 3822.2.a.bp 2
7.c even 3 2 546.2.i.h 4
21.h odd 6 2 1638.2.j.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.h 4 7.c even 3 2
1638.2.j.n 4 21.h odd 6 2
3822.2.a.bp 2 7.b odd 2 1
3822.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\):

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 17 \) Copy content Toggle raw display
\( T_{29} - 5 \) 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
$5$ \( T^{2} - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$19$ \( (T - 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 46 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 46 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 34 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$47$ \( (T - 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 127 \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 79 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$67$ \( T^{2} + 18T + 49 \) Copy content Toggle raw display
$71$ \( (T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 14 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 94 \) Copy content Toggle raw display
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