Properties

Label 3822.2.a.bt
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \( x^{2} - x - 10 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). 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 \( q + q^{2} + q^{3} + q^{4} + \beta q^{5} + q^{6} + q^{8} + q^{9} + \beta q^{10} + (\beta + 2) q^{11} + q^{12} - q^{13} + \beta q^{15} + q^{16} - \beta q^{17} + q^{18} + ( - \beta - 2) q^{19} + \beta q^{20} + (\beta + 2) q^{22} + ( - \beta + 2) q^{23} + q^{24} + (\beta + 5) q^{25} - q^{26} + q^{27} - \beta q^{29} + \beta q^{30} + q^{32} + (\beta + 2) q^{33} - \beta q^{34} + q^{36} + ( - \beta + 8) q^{37} + ( - \beta - 2) q^{38} - q^{39} + \beta q^{40} + (2 \beta + 2) q^{41} + (3 \beta - 2) q^{43} + (\beta + 2) q^{44} + \beta q^{45} + ( - \beta + 2) q^{46} + 8 q^{47} + q^{48} + (\beta + 5) q^{50} - \beta q^{51} - q^{52} - 2 q^{53} + q^{54} + (3 \beta + 10) q^{55} + ( - \beta - 2) q^{57} - \beta q^{58} + ( - 4 \beta + 4) q^{59} + \beta q^{60} + ( - \beta - 4) q^{61} + q^{64} - \beta q^{65} + (\beta + 2) q^{66} - 2 \beta q^{67} - \beta q^{68} + ( - \beta + 2) q^{69} - 8 q^{71} + q^{72} + (\beta + 4) q^{73} + ( - \beta + 8) q^{74} + (\beta + 5) q^{75} + ( - \beta - 2) q^{76} - q^{78} + ( - 2 \beta + 4) q^{79} + \beta q^{80} + q^{81} + (2 \beta + 2) q^{82} + ( - 2 \beta + 8) q^{83} + ( - \beta - 10) q^{85} + (3 \beta - 2) q^{86} - \beta q^{87} + (\beta + 2) q^{88} + ( - 4 \beta - 2) q^{89} + \beta q^{90} + ( - \beta + 2) q^{92} + 8 q^{94} + ( - 3 \beta - 10) q^{95} + q^{96} + ( - 4 \beta - 2) q^{97} + (\beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + q^{10} + 5 q^{11} + 2 q^{12} - 2 q^{13} + q^{15} + 2 q^{16} - q^{17} + 2 q^{18} - 5 q^{19} + q^{20} + 5 q^{22} + 3 q^{23} + 2 q^{24} + 11 q^{25} - 2 q^{26} + 2 q^{27} - q^{29} + q^{30} + 2 q^{32} + 5 q^{33} - q^{34} + 2 q^{36} + 15 q^{37} - 5 q^{38} - 2 q^{39} + q^{40} + 6 q^{41} - q^{43} + 5 q^{44} + q^{45} + 3 q^{46} + 16 q^{47} + 2 q^{48} + 11 q^{50} - q^{51} - 2 q^{52} - 4 q^{53} + 2 q^{54} + 23 q^{55} - 5 q^{57} - q^{58} + 4 q^{59} + q^{60} - 9 q^{61} + 2 q^{64} - q^{65} + 5 q^{66} - 2 q^{67} - q^{68} + 3 q^{69} - 16 q^{71} + 2 q^{72} + 9 q^{73} + 15 q^{74} + 11 q^{75} - 5 q^{76} - 2 q^{78} + 6 q^{79} + q^{80} + 2 q^{81} + 6 q^{82} + 14 q^{83} - 21 q^{85} - q^{86} - q^{87} + 5 q^{88} - 8 q^{89} + q^{90} + 3 q^{92} + 16 q^{94} - 23 q^{95} + 2 q^{96} - 8 q^{97} + 5 q^{99}+O(q^{100}) \) 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
−2.70156
3.70156
1.00000 1.00000 1.00000 −2.70156 1.00000 0 1.00000 1.00000 −2.70156
1.2 1.00000 1.00000 1.00000 3.70156 1.00000 0 1.00000 1.00000 3.70156
\(n\): 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.bt 2
7.b odd 2 1 546.2.a.i 2
21.c even 2 1 1638.2.a.w 2
28.d even 2 1 4368.2.a.bg 2
91.b odd 2 1 7098.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.i 2 7.b odd 2 1
1638.2.a.w 2 21.c even 2 1
3822.2.a.bt 2 1.a even 1 1 trivial
4368.2.a.bg 2 28.d even 2 1
7098.2.a.bh 2 91.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\):

\( T_{5}^{2} - T_{5} - 10 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + T_{17} - 10 \) Copy content Toggle raw display
\( T_{29}^{2} + T_{29} - 10 \) 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
$5$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5T - 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 15T + 46 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 32 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 92 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 160 \) Copy content Toggle raw display
$61$ \( T^{2} + 9T + 10 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 40 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 9T + 10 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 8T - 148 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 148 \) Copy content Toggle raw display
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