Properties

Label 637.2.a.e
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−2.61803 2.23607 4.85410 −2.23607 −5.85410 0 −7.47214 2.00000 5.85410
1.2 −0.381966 −2.23607 −1.85410 2.23607 0.854102 0 1.47214 2.00000 −0.854102
\(n\): e.g. 2-40 or 990-1000
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.e 2
3.b odd 2 1 5733.2.a.w 2
7.b odd 2 1 637.2.a.f 2
7.c even 3 2 637.2.e.h 4
7.d odd 6 2 91.2.e.b 4
13.b even 2 1 8281.2.a.ba 2
21.c even 2 1 5733.2.a.v 2
21.g even 6 2 819.2.j.c 4
28.f even 6 2 1456.2.r.j 4
91.b odd 2 1 8281.2.a.z 2
91.s odd 6 2 1183.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 7.d odd 6 2
637.2.a.e 2 1.a even 1 1 trivial
637.2.a.f 2 7.b odd 2 1
637.2.e.h 4 7.c even 3 2
819.2.j.c 4 21.g even 6 2
1183.2.e.d 4 91.s odd 6 2
1456.2.r.j 4 28.f even 6 2
5733.2.a.v 2 21.c even 2 1
5733.2.a.w 2 3.b odd 2 1
8281.2.a.z 2 91.b odd 2 1
8281.2.a.ba 2 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{3}^{2} - 5 \)
\( T_{17}^{2} - 6 T_{17} - 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + T^{2} \)
$3$ \( -5 + T^{2} \)
$5$ \( -5 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -11 - 6 T + T^{2} \)
$19$ \( ( 3 + T )^{2} \)
$23$ \( 31 + 12 T + T^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( ( 5 + T )^{2} \)
$37$ \( -41 + 4 T + T^{2} \)
$41$ \( -20 + T^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( -11 - 6 T + T^{2} \)
$53$ \( -11 + 6 T + T^{2} \)
$59$ \( -11 + 6 T + T^{2} \)
$61$ \( ( 3 + T )^{2} \)
$67$ \( ( 3 + T )^{2} \)
$71$ \( -80 + T^{2} \)
$73$ \( -29 + 8 T + T^{2} \)
$79$ \( -29 - 8 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( -5 + T^{2} \)
$97$ \( -164 - 8 T + T^{2} \)
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