Properties

Label 3648.2.a.br
Level $3648$
Weight $2$
Character orbit 3648.a
Self dual yes
Analytic conductor $29.129$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.1294266574\)
Analytic rank: \(0\)
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 456)
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.

\(f(q)\) \(=\) \( q + q^{3} - \beta q^{5} - \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta q^{5} - \beta q^{7} + q^{9} + ( - \beta + 4) q^{11} + 2 \beta q^{13} - \beta q^{15} + ( - 3 \beta + 2) q^{17} + q^{19} - \beta q^{21} + (2 \beta - 6) q^{23} + (\beta - 1) q^{25} + q^{27} + ( - 4 \beta + 2) q^{29} + 2 q^{31} + ( - \beta + 4) q^{33} + (\beta + 4) q^{35} + 8 q^{37} + 2 \beta q^{39} + (2 \beta - 2) q^{41} + \beta q^{43} - \beta q^{45} + ( - 3 \beta + 2) q^{47} + (\beta - 3) q^{49} + ( - 3 \beta + 2) q^{51} + (4 \beta + 2) q^{53} + ( - 3 \beta + 4) q^{55} + q^{57} + 12 q^{59} + (3 \beta - 2) q^{61} - \beta q^{63} + ( - 2 \beta - 8) q^{65} - 4 \beta q^{67} + (2 \beta - 6) q^{69} + (7 \beta - 6) q^{73} + (\beta - 1) q^{75} + ( - 3 \beta + 4) q^{77} + (6 \beta - 2) q^{79} + q^{81} + 4 q^{83} + (\beta + 12) q^{85} + ( - 4 \beta + 2) q^{87} - 6 q^{89} + ( - 2 \beta - 8) q^{91} + 2 q^{93} - \beta q^{95} + ( - 4 \beta - 2) q^{97} + ( - \beta + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{5} - q^{7} + 2 q^{9} + 7 q^{11} + 2 q^{13} - q^{15} + q^{17} + 2 q^{19} - q^{21} - 10 q^{23} - q^{25} + 2 q^{27} + 4 q^{31} + 7 q^{33} + 9 q^{35} + 16 q^{37} + 2 q^{39} - 2 q^{41} + q^{43} - q^{45} + q^{47} - 5 q^{49} + q^{51} + 8 q^{53} + 5 q^{55} + 2 q^{57} + 24 q^{59} - q^{61} - q^{63} - 18 q^{65} - 4 q^{67} - 10 q^{69} - 5 q^{73} - q^{75} + 5 q^{77} + 2 q^{79} + 2 q^{81} + 8 q^{83} + 25 q^{85} - 12 q^{89} - 18 q^{91} + 4 q^{93} - q^{95} - 8 q^{97} + 7 q^{99}+O(q^{100}) \) 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
0 1.00000 0 −2.56155 0 −2.56155 0 1.00000 0
1.2 0 1.00000 0 1.56155 0 1.56155 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3648.2.a.br 2
4.b odd 2 1 3648.2.a.bl 2
8.b even 2 1 912.2.a.m 2
8.d odd 2 1 456.2.a.f 2
24.f even 2 1 1368.2.a.k 2
24.h odd 2 1 2736.2.a.z 2
152.b even 2 1 8664.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.f 2 8.d odd 2 1
912.2.a.m 2 8.b even 2 1
1368.2.a.k 2 24.f even 2 1
2736.2.a.z 2 24.h odd 2 1
3648.2.a.bl 2 4.b odd 2 1
3648.2.a.br 2 1.a even 1 1 trivial
8664.2.a.r 2 152.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\):

\( T_{5}^{2} + T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 8 \) Copy content Toggle raw display
\( T_{23}^{2} + 10T_{23} + 8 \) Copy content Toggle raw display
\( T_{31} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$17$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 68 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 5T - 202 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 152 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
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