Properties

Label 644.2.a.d
Level $644$
Weight $2$
Character orbit 644.a
Self dual yes
Analytic conductor $5.142$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.6963152.1
Defining polynomial: \( x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + (\beta_{4} - \beta_1 + 1) q^{11} + ( - \beta_{4} + \beta_{2} + 2) q^{13} + ( - \beta_{4} - 2 \beta_{2} + \beta_1 + 1) q^{15} + (\beta_{4} - \beta_{2} + \beta_1 + 1) q^{17} + ( - 2 \beta_{3} + 2) q^{19} + (\beta_1 - 1) q^{21} - q^{23} + ( - \beta_{4} - \beta_1 + 4) q^{25} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{27} + (2 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{29} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{31} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{33} + ( - \beta_{3} + \beta_{2} - 1) q^{35} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{37} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 2) q^{39} + (\beta_{4} - \beta_{2} + 2 \beta_1) q^{41} + (2 \beta_{3} + 2 \beta_1 - 2) q^{43} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{45} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{47} + q^{49} + (\beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{51} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{53} + (\beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 2) q^{55} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 4) q^{57} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{59} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{61} + ( - \beta_{2} + \beta_1 - 2) q^{63} + ( - 2 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{65} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{67} + (\beta_1 - 1) q^{69} + ( - \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 2) q^{71} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - \beta_{4} + 3 \beta_{3} - 4 \beta_1 + 9) q^{75} + ( - \beta_{4} + \beta_1 - 1) q^{77} + (\beta_{4} + 2 \beta_{2} - 3 \beta_1 - 1) q^{79} + (\beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 5 \beta_1 - 2) q^{81} + (2 \beta_{3} + 2 \beta_{2} - 6) q^{83} + (3 \beta_{4} - 2 \beta_{3} + \beta_1 + 3) q^{85} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{87} + (\beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{89} + (\beta_{4} - \beta_{2} - 2) q^{91} + ( - 6 \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 5) q^{93} + (2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 4) q^{95} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{97} + (\beta_{4} - 2 \beta_{3} + 6 \beta_{2} - 5 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9} + 2 q^{11} + 13 q^{13} + 4 q^{15} + 4 q^{17} + 12 q^{19} - 3 q^{21} - 5 q^{23} + 19 q^{25} + 15 q^{27} + 13 q^{29} - 3 q^{31} + 24 q^{33} - 2 q^{35} - 4 q^{37} + 3 q^{39} + q^{41} - 8 q^{43} - 16 q^{45} + 5 q^{47} + 5 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 12 q^{59} + 20 q^{61} - 10 q^{63} - 12 q^{65} - 12 q^{67} - 3 q^{69} + 9 q^{71} - 9 q^{73} + 35 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} - 28 q^{83} + 16 q^{85} - 15 q^{87} + 32 q^{89} - 13 q^{91} - 15 q^{93} - 36 q^{95} + 4 q^{97} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 7\nu - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 9\nu + 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 7\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 7\beta_{2} + 12\beta _1 + 29 \) 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
3.11181
2.76321
−0.435854
−1.18229
−2.25688
0 −2.11181 0 1.77860 0 −1.00000 0 1.45975 0
1.2 0 −1.76321 0 −3.11657 0 −1.00000 0 0.108911 0
1.3 0 1.43585 0 2.34236 0 −1.00000 0 −0.938323 0
1.4 0 2.18229 0 4.04332 0 −1.00000 0 1.76237 0
1.5 0 3.25688 0 −3.04771 0 −1.00000 0 7.60729 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(23\) \(1\)

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 644.2.a.d 5
3.b odd 2 1 5796.2.a.t 5
4.b odd 2 1 2576.2.a.bb 5
7.b odd 2 1 4508.2.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.a.d 5 1.a even 1 1 trivial
2576.2.a.bb 5 4.b odd 2 1
4508.2.a.f 5 7.b odd 2 1
5796.2.a.t 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 3T_{3}^{4} - 8T_{3}^{3} + 22T_{3}^{2} + 16T_{3} - 38 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(644))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 3 T^{4} - 8 T^{3} + 22 T^{2} + \cdots - 38 \) Copy content Toggle raw display
$5$ \( T^{5} - 2 T^{4} - 20 T^{3} + 34 T^{2} + \cdots - 160 \) Copy content Toggle raw display
$7$ \( (T + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 2 T^{4} - 48 T^{3} + 68 T^{2} + \cdots - 360 \) Copy content Toggle raw display
$13$ \( T^{5} - 13 T^{4} + 20 T^{3} + \cdots + 1424 \) Copy content Toggle raw display
$17$ \( T^{5} - 4 T^{4} - 48 T^{3} + 70 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{5} - 12 T^{4} + 320 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$23$ \( (T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 13 T^{4} + 304 T^{2} + \cdots - 796 \) Copy content Toggle raw display
$31$ \( T^{5} + 3 T^{4} - 142 T^{3} + \cdots + 20810 \) Copy content Toggle raw display
$37$ \( T^{5} + 4 T^{4} - 88 T^{3} + \cdots + 1152 \) Copy content Toggle raw display
$41$ \( T^{5} - T^{4} - 84 T^{3} + 96 T^{2} + \cdots - 2032 \) Copy content Toggle raw display
$43$ \( T^{5} + 8 T^{4} - 88 T^{3} + \cdots - 1984 \) Copy content Toggle raw display
$47$ \( T^{5} - 5 T^{4} - 130 T^{3} + \cdots - 1198 \) Copy content Toggle raw display
$53$ \( T^{5} + 8 T^{4} - 144 T^{3} + \cdots + 28224 \) Copy content Toggle raw display
$59$ \( T^{5} - 12 T^{4} - 44 T^{3} + \cdots - 216 \) Copy content Toggle raw display
$61$ \( T^{5} - 20 T^{4} + 60 T^{3} + \cdots + 292 \) Copy content Toggle raw display
$67$ \( T^{5} + 12 T^{4} - 96 T^{3} + \cdots - 9008 \) Copy content Toggle raw display
$71$ \( T^{5} - 9 T^{4} - 180 T^{3} + \cdots - 19840 \) Copy content Toggle raw display
$73$ \( T^{5} + 9 T^{4} - 112 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$79$ \( T^{5} + 8 T^{4} - 160 T^{3} + \cdots - 6320 \) Copy content Toggle raw display
$83$ \( T^{5} + 28 T^{4} + 160 T^{3} + \cdots - 4096 \) Copy content Toggle raw display
$89$ \( T^{5} - 32 T^{4} + 148 T^{3} + \cdots + 200 \) Copy content Toggle raw display
$97$ \( T^{5} - 4 T^{4} - 196 T^{3} + \cdots + 27700 \) Copy content Toggle raw display
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