Properties

Label 3822.2.a.bm
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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−3.27492
4.27492
−1.00000 1.00000 1.00000 −3.27492 −1.00000 0 −1.00000 1.00000 3.27492
1.2 −1.00000 1.00000 1.00000 4.27492 −1.00000 0 −1.00000 1.00000 −4.27492
\(n\): 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.bm 2
7.b odd 2 1 546.2.a.h 2
21.c even 2 1 1638.2.a.y 2
28.d even 2 1 4368.2.a.bh 2
91.b odd 2 1 7098.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.h 2 7.b odd 2 1
1638.2.a.y 2 21.c even 2 1
3822.2.a.bm 2 1.a even 1 1 trivial
4368.2.a.bh 2 28.d even 2 1
7098.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\):

\( T_{5}^{2} - T_{5} - 14 \)
\( T_{11}^{2} - 3 T_{11} - 12 \)
\( T_{17}^{2} - 7 T_{17} - 2 \)
\( T_{29}^{2} - 9 T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -14 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -12 - 3 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -2 - 7 T + T^{2} \)
$19$ \( -12 + 3 T + T^{2} \)
$23$ \( -12 + 3 T + T^{2} \)
$29$ \( 6 - 9 T + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( -14 + T + T^{2} \)
$41$ \( -56 - 2 T + T^{2} \)
$43$ \( -12 - 3 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( 58 - 17 T + T^{2} \)
$67$ \( -32 - 10 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -126 - 3 T + T^{2} \)
$79$ \( -32 - 10 T + T^{2} \)
$83$ \( -48 + 6 T + T^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( -228 + T^{2} \)
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