Properties

Label 637.2.g.a
Level $637$
Weight $2$
Character orbit 637.g
Analytic conductor $5.086$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 3.00000 0.500000 + 0.866025i 1.50000 + 2.59808i −1.50000 + 2.59808i 0 −3.00000 6.00000 −3.00000
373.1 −0.500000 0.866025i 3.00000 0.500000 0.866025i 1.50000 2.59808i −1.50000 2.59808i 0 −3.00000 6.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.g.a 2
7.b odd 2 1 91.2.g.a 2
7.c even 3 1 637.2.f.a 2
7.c even 3 1 637.2.h.a 2
7.d odd 6 1 91.2.h.a yes 2
7.d odd 6 1 637.2.f.b 2
13.c even 3 1 637.2.h.a 2
21.c even 2 1 819.2.n.c 2
21.g even 6 1 819.2.s.a 2
91.g even 3 1 inner 637.2.g.a 2
91.g even 3 1 8281.2.a.j 1
91.h even 3 1 637.2.f.a 2
91.l odd 6 1 1183.2.e.c 2
91.m odd 6 1 91.2.g.a 2
91.m odd 6 1 8281.2.a.i 1
91.n odd 6 1 91.2.h.a yes 2
91.n odd 6 1 1183.2.e.a 2
91.p odd 6 1 8281.2.a.c 1
91.t odd 6 1 1183.2.e.c 2
91.u even 6 1 8281.2.a.g 1
91.v odd 6 1 637.2.f.b 2
91.v odd 6 1 1183.2.e.a 2
273.bf even 6 1 819.2.n.c 2
273.bn even 6 1 819.2.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 7.b odd 2 1
91.2.g.a 2 91.m odd 6 1
91.2.h.a yes 2 7.d odd 6 1
91.2.h.a yes 2 91.n odd 6 1
637.2.f.a 2 7.c even 3 1
637.2.f.a 2 91.h even 3 1
637.2.f.b 2 7.d odd 6 1
637.2.f.b 2 91.v odd 6 1
637.2.g.a 2 1.a even 1 1 trivial
637.2.g.a 2 91.g even 3 1 inner
637.2.h.a 2 7.c even 3 1
637.2.h.a 2 13.c even 3 1
819.2.n.c 2 21.c even 2 1
819.2.n.c 2 273.bf even 6 1
819.2.s.a 2 21.g even 6 1
819.2.s.a 2 273.bn even 6 1
1183.2.e.a 2 91.n odd 6 1
1183.2.e.a 2 91.v odd 6 1
1183.2.e.c 2 91.l odd 6 1
1183.2.e.c 2 91.t odd 6 1
8281.2.a.c 1 91.p odd 6 1
8281.2.a.g 1 91.u even 6 1
8281.2.a.i 1 91.m odd 6 1
8281.2.a.j 1 91.g even 3 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{3} - 3 \)
\( T_{5}^{2} - 3 T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 13 - 2 T + T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( 49 + 7 T + T^{2} \)
$31$ \( 9 - 3 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( 9 - 3 T + T^{2} \)
$43$ \( 49 - 7 T + T^{2} \)
$47$ \( 1 - T + T^{2} \)
$53$ \( 9 + 3 T + T^{2} \)
$59$ \( 16 + 4 T + T^{2} \)
$61$ \( ( -13 + T )^{2} \)
$67$ \( ( 3 + T )^{2} \)
$71$ \( 169 + 13 T + T^{2} \)
$73$ \( 169 + 13 T + T^{2} \)
$79$ \( 9 - 3 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( 25 + 5 T + T^{2} \)
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