Properties

Label 637.2.a.l
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.746052.1
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_{4} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{4} - \beta_{2} q^{5} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - \beta_{4} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{4} - \beta_{2} q^{5} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{8} + (\beta_{2} + 1) q^{9} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{10} + ( - \beta_{3} + 2) q^{11} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{12} + q^{13} + (2 \beta_{4} + \beta_{2} + 2 \beta_1) q^{15} + (\beta_{3} - 2 \beta_{2} - 3 \beta_1 + 2) q^{16} + (\beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{17} + ( - \beta_{3} + \beta_{2} + 2) q^{18} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{19} + (3 \beta_{3} - \beta_1 + 1) q^{20} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{22} + ( - \beta_{4} + 2) q^{23} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{24} + (2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 1) q^{25} + ( - \beta_1 + 1) q^{26} + ( - \beta_{2} - 2 \beta_1) q^{27} + (\beta_{4} + 2 \beta_1 - 1) q^{29} + (4 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{30} + ( - \beta_{4} - 2 \beta_{2} - 2) q^{31} + (4 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{32} + ( - 3 \beta_{4} - 2 \beta_1 + 2) q^{33} + (3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{34} + ( - \beta_{4} - 2 \beta_{3} + 1) q^{36} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{37} + (4 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{38} - \beta_{4} q^{39} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 6) q^{40} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{41} + (\beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{43} + ( - 3 \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{44} + ( - 2 \beta_{4} + 2 \beta_{3} - 6) q^{45} + ( - \beta_{4} - \beta_{3} - 3 \beta_1 + 1) q^{46} + (4 \beta_{4} - \beta_{3}) q^{47} + (3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 5) q^{48} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{50} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2) q^{51} + ( - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{52} + (2 \beta_{4} + 2 \beta_{3} + 4 \beta_1 + 3) q^{53} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{54} + (2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 2) q^{55} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{57} + (3 \beta_{4} - \beta_{3} + 2 \beta_1 - 6) q^{58} + ( - \beta_{3} + 2) q^{59} + (2 \beta_{4} - \beta_{3} + 5 \beta_1 - 7) q^{60} + (2 \beta_{4} - 4 \beta_{3} - 3) q^{61} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 5) q^{62} + (3 \beta_{4} + 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 3) q^{64} - \beta_{2} q^{65} + ( - 5 \beta_{4} - \beta_{3} - 5 \beta_1 + 5) q^{66} + (\beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 4) q^{67} + (2 \beta_{4} - 3 \beta_{3} + \beta_{2} + 7 \beta_1 - 8) q^{68} + ( - 2 \beta_{4} + \beta_{2} + 4) q^{69} + (\beta_{4} - \beta_{3} + 4 \beta_1 + 2) q^{71} + ( - 3 \beta_{4} - \beta_{3} - 4) q^{72} + ( - 2 \beta_{3} + \beta_{2}) q^{73} + ( - \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + \beta_1 - 7) q^{74} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 - 4) q^{75} + (2 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{76} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{78} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{79} + (2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 11) q^{80} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 5) q^{81} + ( - 3 \beta_{4} - \beta_{2} - 8 \beta_1 + 8) q^{82} + (\beta_{4} + 3 \beta_{2} + 2 \beta_1 + 2) q^{83} + ( - 4 \beta_{4} + \beta_{2} - 4) q^{85} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{86} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{87} + ( - 2 \beta_{4} - 2 \beta_{3} - 1) q^{88} + ( - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{89} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{90} + ( - 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 5) q^{92} + (6 \beta_{4} + 3 \beta_{2} + 4 \beta_1 + 4) q^{93} + (3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 5 \beta_1 + 4) q^{94} + ( - 6 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{95} + (\beta_{4} - \beta_{3} + \beta_{2} + 9 \beta_1 - 9) q^{96} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{97} + ( - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26} - 3 q^{29} - 13 q^{30} - 6 q^{31} + 22 q^{32} + 8 q^{33} - 22 q^{34} + 7 q^{36} + 4 q^{37} - 10 q^{38} + 28 q^{40} + 14 q^{41} + 2 q^{43} - 32 q^{45} + 3 q^{46} + q^{47} - 23 q^{48} + 9 q^{50} - 8 q^{51} + 8 q^{52} + 17 q^{53} + 23 q^{54} - 16 q^{57} - 27 q^{58} + 11 q^{59} - 29 q^{60} - 11 q^{61} - 23 q^{62} + 9 q^{64} + 2 q^{65} + 21 q^{66} + 13 q^{67} - 32 q^{68} + 18 q^{69} + 15 q^{71} - 19 q^{72} - 33 q^{74} - 20 q^{75} + 8 q^{76} - 5 q^{78} + 2 q^{79} + 55 q^{80} - 19 q^{81} + 34 q^{82} + 6 q^{83} - 22 q^{85} + 28 q^{86} - 8 q^{87} - 3 q^{88} - 4 q^{89} - 34 q^{90} + 21 q^{92} + 18 q^{93} + 20 q^{94} - 12 q^{95} - 37 q^{96} - 12 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 7x^{3} + 8x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 4\nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{4} + 2\beta_{3} + \beta_{2} + 6\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{4} + 9\beta_{3} + 2\beta_{2} + 13\beta _1 + 19 \) 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
3.00852
1.19566
−0.265608
−1.21332
−1.72525
−2.00852 1.75906 2.03417 0.905722 −3.53311 0 −0.0686323 0.0942784 −1.81916
1.2 −0.195656 −0.259788 −1.96172 3.93251 0.0508292 0 0.775135 −2.93251 −0.769420
1.3 1.26561 −2.62728 −0.398235 −2.90260 −3.32511 0 −3.03523 3.90260 −3.67356
1.4 2.21332 2.47443 2.89879 −2.12280 5.47671 0 1.98932 3.12280 −4.69843
1.5 2.72525 −1.34642 5.42699 2.18716 −3.66932 0 9.33940 −1.18716 5.96057
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(13\) \(-1\)

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 637.2.a.l 5
3.b odd 2 1 5733.2.a.bl 5
7.b odd 2 1 637.2.a.k 5
7.c even 3 2 91.2.e.c 10
7.d odd 6 2 637.2.e.m 10
13.b even 2 1 8281.2.a.bw 5
21.c even 2 1 5733.2.a.bm 5
21.h odd 6 2 819.2.j.h 10
28.g odd 6 2 1456.2.r.p 10
91.b odd 2 1 8281.2.a.bx 5
91.r even 6 2 1183.2.e.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 7.c even 3 2
637.2.a.k 5 7.b odd 2 1
637.2.a.l 5 1.a even 1 1 trivial
637.2.e.m 10 7.d odd 6 2
819.2.j.h 10 21.h odd 6 2
1183.2.e.f 10 91.r even 6 2
1456.2.r.p 10 28.g odd 6 2
5733.2.a.bl 5 3.b odd 2 1
5733.2.a.bm 5 21.c even 2 1
8281.2.a.bw 5 13.b even 2 1
8281.2.a.bx 5 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2}^{5} - 4T_{2}^{4} - T_{2}^{3} + 17T_{2}^{2} - 12T_{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{5} - 9T_{3}^{3} + 16T_{3} + 4 \) Copy content Toggle raw display
\( T_{17}^{5} + 5T_{17}^{4} - 22T_{17}^{3} - 106T_{17}^{2} + 93T_{17} + 429 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 4 T^{4} - T^{3} + 17 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{5} - 9 T^{3} + 16 T + 4 \) Copy content Toggle raw display
$5$ \( T^{5} - 2 T^{4} - 15 T^{3} + 20 T^{2} + \cdots - 48 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 11 T^{4} + 36 T^{3} - 22 T^{2} + \cdots + 33 \) Copy content Toggle raw display
$13$ \( (T - 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + 5 T^{4} - 22 T^{3} - 106 T^{2} + \cdots + 429 \) Copy content Toggle raw display
$19$ \( T^{5} - 9 T^{4} - 14 T^{3} + 176 T^{2} + \cdots - 223 \) Copy content Toggle raw display
$23$ \( T^{5} - 10 T^{4} + 31 T^{3} - 26 T^{2} + \cdots + 12 \) Copy content Toggle raw display
$29$ \( T^{5} + 3 T^{4} - 25 T^{3} - 19 T^{2} + \cdots - 108 \) Copy content Toggle raw display
$31$ \( T^{5} + 6 T^{4} - 61 T^{3} - 102 T^{2} + \cdots - 356 \) Copy content Toggle raw display
$37$ \( T^{5} - 4 T^{4} - 111 T^{3} + \cdots - 7036 \) Copy content Toggle raw display
$41$ \( T^{5} - 14 T^{4} - 28 T^{3} + \cdots - 1584 \) Copy content Toggle raw display
$43$ \( T^{5} - 2 T^{4} - 72 T^{3} + 308 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{5} - T^{4} - 124 T^{3} + 26 T^{2} + \cdots - 5169 \) Copy content Toggle raw display
$53$ \( T^{5} - 17 T^{4} - 74 T^{3} + \cdots + 19959 \) Copy content Toggle raw display
$59$ \( T^{5} - 11 T^{4} + 36 T^{3} - 22 T^{2} + \cdots + 33 \) Copy content Toggle raw display
$61$ \( T^{5} + 11 T^{4} - 122 T^{3} + \cdots - 8461 \) Copy content Toggle raw display
$67$ \( T^{5} - 13 T^{4} - 162 T^{3} + \cdots - 22699 \) Copy content Toggle raw display
$71$ \( T^{5} - 15 T^{4} - 25 T^{3} + \cdots - 6336 \) Copy content Toggle raw display
$73$ \( T^{5} - 75 T^{3} + 42 T^{2} + \cdots - 712 \) Copy content Toggle raw display
$79$ \( T^{5} - 2 T^{4} - 137 T^{3} + \cdots - 1000 \) Copy content Toggle raw display
$83$ \( T^{5} - 6 T^{4} - 124 T^{3} + \cdots - 7488 \) Copy content Toggle raw display
$89$ \( T^{5} + 4 T^{4} - 155 T^{3} + \cdots + 7692 \) Copy content Toggle raw display
$97$ \( T^{5} + 12 T^{4} - 16 T^{3} + \cdots - 2384 \) Copy content Toggle raw display
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