Properties

Label 3822.2.a.bo
Level $3822$
Weight $2$
Character orbit 3822.a
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). 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 \( q + q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + q^{8} + q^{9} + ( - \beta - 1) q^{10} + ( - \beta + 1) q^{11} - q^{12} - q^{13} + (\beta + 1) q^{15} + q^{16} + (3 \beta - 1) q^{17} + q^{18} + (3 \beta - 3) q^{19} + ( - \beta - 1) q^{20} + ( - \beta + 1) q^{22} + ( - \beta - 3) q^{23} - q^{24} + 3 \beta q^{25} - q^{26} - q^{27} + (3 \beta - 1) q^{29} + (\beta + 1) q^{30} + ( - 4 \beta + 4) q^{31} + q^{32} + (\beta - 1) q^{33} + (3 \beta - 1) q^{34} + q^{36} + ( - \beta - 5) q^{37} + (3 \beta - 3) q^{38} + q^{39} + ( - \beta - 1) q^{40} + (2 \beta - 4) q^{41} + (\beta - 9) q^{43} + ( - \beta + 1) q^{44} + ( - \beta - 1) q^{45} + ( - \beta - 3) q^{46} - q^{48} + 3 \beta q^{50} + ( - 3 \beta + 1) q^{51} - q^{52} + (4 \beta + 2) q^{53} - q^{54} + (\beta + 3) q^{55} + ( - 3 \beta + 3) q^{57} + (3 \beta - 1) q^{58} + ( - 4 \beta + 8) q^{59} + (\beta + 1) q^{60} + ( - 3 \beta + 1) q^{61} + ( - 4 \beta + 4) q^{62} + q^{64} + (\beta + 1) q^{65} + (\beta - 1) q^{66} + ( - 2 \beta - 2) q^{67} + (3 \beta - 1) q^{68} + (\beta + 3) q^{69} + 8 q^{71} + q^{72} + (\beta + 1) q^{73} + ( - \beta - 5) q^{74} - 3 \beta q^{75} + (3 \beta - 3) q^{76} + q^{78} + ( - 2 \beta - 6) q^{79} + ( - \beta - 1) q^{80} + q^{81} + (2 \beta - 4) q^{82} + (2 \beta - 14) q^{83} + ( - 5 \beta - 11) q^{85} + (\beta - 9) q^{86} + ( - 3 \beta + 1) q^{87} + ( - \beta + 1) q^{88} - 10 q^{89} + ( - \beta - 1) q^{90} + ( - \beta - 3) q^{92} + (4 \beta - 4) q^{93} + ( - 3 \beta - 9) q^{95} - q^{96} + ( - 8 \beta + 6) q^{97} + ( - \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 2 q^{13} + 3 q^{15} + 2 q^{16} + q^{17} + 2 q^{18} - 3 q^{19} - 3 q^{20} + q^{22} - 7 q^{23} - 2 q^{24} + 3 q^{25} - 2 q^{26} - 2 q^{27} + q^{29} + 3 q^{30} + 4 q^{31} + 2 q^{32} - q^{33} + q^{34} + 2 q^{36} - 11 q^{37} - 3 q^{38} + 2 q^{39} - 3 q^{40} - 6 q^{41} - 17 q^{43} + q^{44} - 3 q^{45} - 7 q^{46} - 2 q^{48} + 3 q^{50} - q^{51} - 2 q^{52} + 8 q^{53} - 2 q^{54} + 7 q^{55} + 3 q^{57} + q^{58} + 12 q^{59} + 3 q^{60} - q^{61} + 4 q^{62} + 2 q^{64} + 3 q^{65} - q^{66} - 6 q^{67} + q^{68} + 7 q^{69} + 16 q^{71} + 2 q^{72} + 3 q^{73} - 11 q^{74} - 3 q^{75} - 3 q^{76} + 2 q^{78} - 14 q^{79} - 3 q^{80} + 2 q^{81} - 6 q^{82} - 26 q^{83} - 27 q^{85} - 17 q^{86} - q^{87} + q^{88} - 20 q^{89} - 3 q^{90} - 7 q^{92} - 4 q^{93} - 21 q^{95} - 2 q^{96} + 4 q^{97} + q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
1.00000 −1.00000 1.00000 −3.56155 −1.00000 0 1.00000 1.00000 −3.56155
1.2 1.00000 −1.00000 1.00000 0.561553 −1.00000 0 1.00000 1.00000 0.561553
\(n\): 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.bo 2
7.b odd 2 1 546.2.a.j 2
21.c even 2 1 1638.2.a.u 2
28.d even 2 1 4368.2.a.be 2
91.b odd 2 1 7098.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.j 2 7.b odd 2 1
1638.2.a.u 2 21.c even 2 1
3822.2.a.bo 2 1.a even 1 1 trivial
4368.2.a.be 2 28.d even 2 1
7098.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\):

\( T_{5}^{2} + 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - T_{17} - 38 \) Copy content Toggle raw display
\( T_{29}^{2} - T_{29} - 38 \) 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
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} + 17T + 68 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T - 32 \) Copy content Toggle raw display
$61$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 26T + 152 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 268 \) Copy content Toggle raw display
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