Properties

Label 3762.2.a.y
Level $3762$
Weight $2$
Character orbit 3762.a
Self dual yes
Analytic conductor $30.040$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.0397212404\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 2 q^{5} + (\beta + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - 2 q^{5} + (\beta + 1) q^{7} + q^{8} - 2 q^{10} - q^{11} + ( - \beta - 1) q^{13} + (\beta + 1) q^{14} + q^{16} + ( - \beta - 1) q^{17} - q^{19} - 2 q^{20} - q^{22} + ( - \beta - 3) q^{23} - q^{25} + ( - \beta - 1) q^{26} + (\beta + 1) q^{28} + ( - 3 \beta + 1) q^{29} + 2 q^{31} + q^{32} + ( - \beta - 1) q^{34} + ( - 2 \beta - 2) q^{35} + (2 \beta - 2) q^{37} - q^{38} - 2 q^{40} - 2 q^{41} - q^{44} + ( - \beta - 3) q^{46} - 8 q^{47} + (3 \beta - 2) q^{49} - q^{50} + ( - \beta - 1) q^{52} + (5 \beta - 5) q^{53} + 2 q^{55} + (\beta + 1) q^{56} + ( - 3 \beta + 1) q^{58} + ( - 3 \beta + 3) q^{59} - 4 \beta q^{61} + 2 q^{62} + q^{64} + (2 \beta + 2) q^{65} + (3 \beta - 3) q^{67} + ( - \beta - 1) q^{68} + ( - 2 \beta - 2) q^{70} - 6 q^{71} + (3 \beta - 5) q^{73} + (2 \beta - 2) q^{74} - q^{76} + ( - \beta - 1) q^{77} - 4 q^{79} - 2 q^{80} - 2 q^{82} + (4 \beta - 8) q^{83} + (2 \beta + 2) q^{85} - q^{88} + (2 \beta + 4) q^{89} + ( - 3 \beta - 5) q^{91} + ( - \beta - 3) q^{92} - 8 q^{94} + 2 q^{95} + (2 \beta - 4) q^{97} + (3 \beta - 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 7 q^{23} - 2 q^{25} - 3 q^{26} + 3 q^{28} - q^{29} + 4 q^{31} + 2 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 7 q^{46} - 16 q^{47} - q^{49} - 2 q^{50} - 3 q^{52} - 5 q^{53} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 3 q^{67} - 3 q^{68} - 6 q^{70} - 12 q^{71} - 7 q^{73} - 2 q^{74} - 2 q^{76} - 3 q^{77} - 8 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 6 q^{85} - 2 q^{88} + 10 q^{89} - 13 q^{91} - 7 q^{92} - 16 q^{94} + 4 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
1.00000 0 1.00000 −2.00000 0 −0.561553 1.00000 0 −2.00000
1.2 1.00000 0 1.00000 −2.00000 0 3.56155 1.00000 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

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 3762.2.a.y 2
3.b odd 2 1 418.2.a.e 2
12.b even 2 1 3344.2.a.k 2
33.d even 2 1 4598.2.a.bj 2
57.d even 2 1 7942.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.e 2 3.b odd 2 1
3344.2.a.k 2 12.b even 2 1
3762.2.a.y 2 1.a even 1 1 trivial
4598.2.a.bj 2 33.d even 2 1
7942.2.a.x 2 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3762))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) 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
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 5T - 100 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 32 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
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