Properties

Label 644.2.a.c
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: 5.5.8580816.1
Defining polynomial: \( x^{5} - x^{4} - 12x^{3} + 10x^{2} + 20x + 6 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{4} + 1) q^{5} + q^{7} + (\beta_{4} + \beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{4} + 1) q^{5} + q^{7} + (\beta_{4} + \beta_{3} + \beta_{2} + 2) q^{9} + ( - \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{3} + 1) q^{13} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{15} + ( - \beta_{3} + \beta_1 + 1) q^{17} + (2 \beta_{2} - 2 \beta_1 + 2) q^{19} + \beta_1 q^{21} + q^{23} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 + 3) q^{25} + ( - \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 1) q^{27} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{29} + ( - \beta_{4} - \beta_{3}) q^{31} + (\beta_{4} + \beta_{3} - \beta_1) q^{33} + (\beta_{4} + 1) q^{35} + ( - 2 \beta_{2} + 4) q^{37} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{39} + (\beta_{3} - 2 \beta_1 + 3) q^{41} + (2 \beta_{3} + 2 \beta_{2} + 2) q^{43} + (3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 7) q^{45} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 4) q^{47} + q^{49} + (\beta_{4} + 3 \beta_{2} - 2 \beta_1 + 6) q^{51} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 2) q^{53} + (3 \beta_{4} - \beta_{3} + 3 \beta_1 - 4) q^{55} + ( - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_1 - 6) q^{57} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{59} + ( - \beta_{4} - 2 \beta_{2} - 3) q^{61} + (\beta_{4} + \beta_{3} + \beta_{2} + 2) q^{63} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{65} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{67} + \beta_1 q^{69} + (\beta_{3} - 2 \beta_1 - 3) q^{71} + ( - 2 \beta_{4} - \beta_{3} + 2 \beta_1 + 3) q^{73} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 10) q^{75} + ( - \beta_{4} - \beta_{2}) q^{77} + (\beta_{4} - 3 \beta_{2}) q^{79} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 7) q^{81} + (2 \beta_{4} + 4 \beta_{2} - 2) q^{83} + ( - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 2) q^{85} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 7) q^{87} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 5) q^{89} + ( - \beta_{3} + 1) q^{91} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 3) q^{93} + (2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{95} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 7) q^{97} + (\beta_{4} - \beta_{2} + 4 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} + 4 q^{5} + 5 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{3} + 4 q^{5} + 5 q^{7} + 10 q^{9} + 2 q^{11} + 3 q^{13} - 6 q^{15} + 4 q^{17} + 6 q^{19} + q^{21} + 5 q^{23} + 15 q^{25} + q^{27} + 5 q^{29} - q^{31} + 4 q^{35} + 22 q^{37} - q^{39} + 15 q^{41} + 12 q^{43} + 36 q^{45} - 21 q^{47} + 5 q^{49} + 24 q^{51} + 2 q^{53} - 22 q^{55} - 34 q^{57} - 12 q^{61} + 10 q^{63} - 2 q^{65} + 22 q^{67} + q^{69} - 15 q^{71} + 17 q^{73} - 47 q^{75} + 2 q^{77} + 2 q^{79} + 37 q^{81} - 16 q^{83} - 8 q^{85} - 33 q^{87} - 24 q^{89} + 3 q^{91} + 5 q^{93} - 8 q^{95} + 38 q^{97} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 12x^{3} + 10x^{2} + 20x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + \nu^{3} + 11\nu^{2} - 11\nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + 2\nu^{3} + 12\nu^{2} - 20\nu - 15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{4} - 3\nu^{3} - 22\nu^{2} + 31\nu + 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - 2\beta_{2} + 9\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{4} + 11\beta_{3} + 8\beta_{2} - 2\beta _1 + 43 \) 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
−3.18838
−0.604308
−0.465082
2.16481
3.09295
0 −3.18838 0 3.43522 0 1.00000 0 7.16574 0
1.2 0 −0.604308 0 −3.83889 0 1.00000 0 −2.63481 0
1.3 0 −0.465082 0 3.21918 0 1.00000 0 −2.78370 0
1.4 0 2.16481 0 −0.502631 0 1.00000 0 1.68642 0
1.5 0 3.09295 0 1.68712 0 1.00000 0 6.56635 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.c 5
3.b odd 2 1 5796.2.a.s 5
4.b odd 2 1 2576.2.a.bc 5
7.b odd 2 1 4508.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.a.c 5 1.a even 1 1 trivial
2576.2.a.bc 5 4.b odd 2 1
4508.2.a.g 5 7.b odd 2 1
5796.2.a.s 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - T_{3}^{4} - 12T_{3}^{3} + 10T_{3}^{2} + 20T_{3} + 6 \) 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
$3$ \( T^{5} - T^{4} - 12 T^{3} + 10 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$5$ \( T^{5} - 4 T^{4} - 12 T^{3} + 62 T^{2} + \cdots - 36 \) Copy content Toggle raw display
$7$ \( (T - 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 2 T^{4} - 24 T^{3} + 12 T^{2} + \cdots - 72 \) Copy content Toggle raw display
$13$ \( T^{5} - 3 T^{4} - 28 T^{3} + 64 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{5} - 4 T^{4} - 36 T^{3} + 74 T^{2} + \cdots + 96 \) Copy content Toggle raw display
$19$ \( T^{5} - 6 T^{4} - 88 T^{3} + \cdots - 3968 \) Copy content Toggle raw display
$23$ \( (T - 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 5 T^{4} - 72 T^{3} + 256 T^{2} + \cdots + 516 \) Copy content Toggle raw display
$31$ \( T^{5} + T^{4} - 54 T^{3} + 2 T^{2} + \cdots - 738 \) Copy content Toggle raw display
$37$ \( T^{5} - 22 T^{4} + 112 T^{3} + \cdots + 9824 \) Copy content Toggle raw display
$41$ \( T^{5} - 15 T^{4} + 12 T^{3} + \cdots + 432 \) Copy content Toggle raw display
$43$ \( T^{5} - 12 T^{4} - 88 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$47$ \( T^{5} + 21 T^{4} + 54 T^{3} + \cdots - 20898 \) Copy content Toggle raw display
$53$ \( T^{5} - 2 T^{4} - 264 T^{3} + \cdots - 27936 \) Copy content Toggle raw display
$59$ \( T^{5} - 144 T^{3} + 174 T^{2} + \cdots - 432 \) Copy content Toggle raw display
$61$ \( T^{5} + 12 T^{4} - 16 T^{3} + \cdots + 108 \) Copy content Toggle raw display
$67$ \( T^{5} - 22 T^{4} + 40 T^{3} + \cdots - 10216 \) Copy content Toggle raw display
$71$ \( T^{5} + 15 T^{4} + 12 T^{3} + \cdots - 1152 \) Copy content Toggle raw display
$73$ \( T^{5} - 17 T^{4} - 72 T^{3} + \cdots + 3504 \) Copy content Toggle raw display
$79$ \( T^{5} - 2 T^{4} - 240 T^{3} + \cdots - 6984 \) Copy content Toggle raw display
$83$ \( T^{5} + 16 T^{4} - 192 T^{3} + \cdots + 66432 \) Copy content Toggle raw display
$89$ \( T^{5} + 24 T^{4} - 2394 T^{2} + \cdots + 52488 \) Copy content Toggle raw display
$97$ \( T^{5} - 38 T^{4} + 396 T^{3} + \cdots + 173424 \) Copy content Toggle raw display
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