Properties

Label 1638.2.a.y
Level $1638$
Weight $2$
Character orbit 1638.a
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \( x^{2} - x - 14 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} + q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta q^{5} + q^{7} + q^{8} + \beta q^{10} + (\beta - 2) q^{11} - q^{13} + q^{14} + q^{16} + ( - \beta + 4) q^{17} + ( - \beta + 2) q^{19} + \beta q^{20} + (\beta - 2) q^{22} + ( - \beta + 2) q^{23} + (\beta + 9) q^{25} - q^{26} + q^{28} + ( - \beta - 4) q^{29} + 8 q^{31} + q^{32} + ( - \beta + 4) q^{34} + \beta q^{35} - \beta q^{37} + ( - \beta + 2) q^{38} + \beta q^{40} + ( - 2 \beta + 2) q^{41} + ( - \beta + 2) q^{43} + (\beta - 2) q^{44} + ( - \beta + 2) q^{46} + q^{49} + (\beta + 9) q^{50} - q^{52} - 10 q^{53} + ( - \beta + 14) q^{55} + q^{56} + ( - \beta - 4) q^{58} - 8 q^{59} + ( - \beta - 8) q^{61} + 8 q^{62} + q^{64} - \beta q^{65} + (2 \beta + 4) q^{67} + ( - \beta + 4) q^{68} + \beta q^{70} - 3 \beta q^{73} - \beta q^{74} + ( - \beta + 2) q^{76} + (\beta - 2) q^{77} + (2 \beta + 4) q^{79} + \beta q^{80} + ( - 2 \beta + 2) q^{82} + (2 \beta - 4) q^{83} + (3 \beta - 14) q^{85} + ( - \beta + 2) q^{86} + (\beta - 2) q^{88} + 14 q^{89} - q^{91} + ( - \beta + 2) q^{92} + (\beta - 14) q^{95} + (4 \beta - 2) q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} - 3 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 7 q^{17} + 3 q^{19} + q^{20} - 3 q^{22} + 3 q^{23} + 19 q^{25} - 2 q^{26} + 2 q^{28} - 9 q^{29} + 16 q^{31} + 2 q^{32} + 7 q^{34} + q^{35} - q^{37} + 3 q^{38} + q^{40} + 2 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 2 q^{49} + 19 q^{50} - 2 q^{52} - 20 q^{53} + 27 q^{55} + 2 q^{56} - 9 q^{58} - 16 q^{59} - 17 q^{61} + 16 q^{62} + 2 q^{64} - q^{65} + 10 q^{67} + 7 q^{68} + q^{70} - 3 q^{73} - q^{74} + 3 q^{76} - 3 q^{77} + 10 q^{79} + q^{80} + 2 q^{82} - 6 q^{83} - 25 q^{85} + 3 q^{86} - 3 q^{88} + 28 q^{89} - 2 q^{91} + 3 q^{92} - 27 q^{95} + 2 q^{98}+O(q^{100}) \) 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
−3.27492
4.27492
1.00000 0 1.00000 −3.27492 0 1.00000 1.00000 0 −3.27492
1.2 1.00000 0 1.00000 4.27492 0 1.00000 1.00000 0 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 1638.2.a.y 2
3.b odd 2 1 546.2.a.h 2
12.b even 2 1 4368.2.a.bh 2
21.c even 2 1 3822.2.a.bm 2
39.d 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 3.b odd 2 1
1638.2.a.y 2 1.a even 1 1 trivial
3822.2.a.bm 2 21.c even 2 1
4368.2.a.bh 2 12.b even 2 1
7098.2.a.bu 2 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1638))\):

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