Properties

Label 637.2.h.a
Level $637$
Weight $2$
Character orbit 637.h
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.h (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: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} -3 \zeta_{6} q^{3} - q^{4} + 3 \zeta_{6} q^{5} -3 \zeta_{6} q^{6} -3 q^{8} + ( -6 + 6 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + q^{2} -3 \zeta_{6} q^{3} - q^{4} + 3 \zeta_{6} q^{5} -3 \zeta_{6} q^{6} -3 q^{8} + ( -6 + 6 \zeta_{6} ) q^{9} + 3 \zeta_{6} q^{10} + 3 \zeta_{6} q^{11} + 3 \zeta_{6} q^{12} + ( -1 + 4 \zeta_{6} ) q^{13} + ( 9 - 9 \zeta_{6} ) q^{15} - q^{16} + 2 q^{17} + ( -6 + 6 \zeta_{6} ) q^{18} + ( -1 + \zeta_{6} ) q^{19} -3 \zeta_{6} q^{20} + 3 \zeta_{6} q^{22} + 9 \zeta_{6} q^{24} + ( -4 + 4 \zeta_{6} ) q^{25} + ( -1 + 4 \zeta_{6} ) q^{26} + 9 q^{27} + ( -7 + 7 \zeta_{6} ) q^{29} + ( 9 - 9 \zeta_{6} ) q^{30} + ( 3 - 3 \zeta_{6} ) q^{31} + 5 q^{32} + ( 9 - 9 \zeta_{6} ) q^{33} + 2 q^{34} + ( 6 - 6 \zeta_{6} ) q^{36} + 2 q^{37} + ( -1 + \zeta_{6} ) q^{38} + ( 12 - 9 \zeta_{6} ) q^{39} -9 \zeta_{6} q^{40} + ( 3 - 3 \zeta_{6} ) q^{41} + 7 \zeta_{6} q^{43} -3 \zeta_{6} q^{44} -18 q^{45} + \zeta_{6} q^{47} + 3 \zeta_{6} q^{48} + ( -4 + 4 \zeta_{6} ) q^{50} -6 \zeta_{6} q^{51} + ( 1 - 4 \zeta_{6} ) q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} + 9 q^{54} + ( -9 + 9 \zeta_{6} ) q^{55} + 3 q^{57} + ( -7 + 7 \zeta_{6} ) q^{58} + 4 q^{59} + ( -9 + 9 \zeta_{6} ) q^{60} + ( -13 + 13 \zeta_{6} ) q^{61} + ( 3 - 3 \zeta_{6} ) q^{62} + 7 q^{64} + ( -12 + 9 \zeta_{6} ) q^{65} + ( 9 - 9 \zeta_{6} ) q^{66} + 3 \zeta_{6} q^{67} -2 q^{68} -13 \zeta_{6} q^{71} + ( 18 - 18 \zeta_{6} ) q^{72} + ( -13 + 13 \zeta_{6} ) q^{73} + 2 q^{74} + 12 q^{75} + ( 1 - \zeta_{6} ) q^{76} + ( 12 - 9 \zeta_{6} ) q^{78} + 3 \zeta_{6} q^{79} -3 \zeta_{6} q^{80} -9 \zeta_{6} q^{81} + ( 3 - 3 \zeta_{6} ) q^{82} + 6 \zeta_{6} q^{85} + 7 \zeta_{6} q^{86} + 21 q^{87} -9 \zeta_{6} q^{88} -6 q^{89} -18 q^{90} -9 q^{93} + \zeta_{6} q^{94} -3 q^{95} -15 \zeta_{6} q^{96} -5 \zeta_{6} q^{97} -18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 3q^{3} - 2q^{4} + 3q^{5} - 3q^{6} - 6q^{8} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 3q^{3} - 2q^{4} + 3q^{5} - 3q^{6} - 6q^{8} - 6q^{9} + 3q^{10} + 3q^{11} + 3q^{12} + 2q^{13} + 9q^{15} - 2q^{16} + 4q^{17} - 6q^{18} - q^{19} - 3q^{20} + 3q^{22} + 9q^{24} - 4q^{25} + 2q^{26} + 18q^{27} - 7q^{29} + 9q^{30} + 3q^{31} + 10q^{32} + 9q^{33} + 4q^{34} + 6q^{36} + 4q^{37} - q^{38} + 15q^{39} - 9q^{40} + 3q^{41} + 7q^{43} - 3q^{44} - 36q^{45} + q^{47} + 3q^{48} - 4q^{50} - 6q^{51} - 2q^{52} - 3q^{53} + 18q^{54} - 9q^{55} + 6q^{57} - 7q^{58} + 8q^{59} - 9q^{60} - 13q^{61} + 3q^{62} + 14q^{64} - 15q^{65} + 9q^{66} + 3q^{67} - 4q^{68} - 13q^{71} + 18q^{72} - 13q^{73} + 4q^{74} + 24q^{75} + q^{76} + 15q^{78} + 3q^{79} - 3q^{80} - 9q^{81} + 3q^{82} + 6q^{85} + 7q^{86} + 42q^{87} - 9q^{88} - 12q^{89} - 36q^{90} - 18q^{93} + q^{94} - 6q^{95} - 15q^{96} - 5q^{97} - 36q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 −1.50000 2.59808i −1.00000 1.50000 + 2.59808i −1.50000 2.59808i 0 −3.00000 −3.00000 + 5.19615i 1.50000 + 2.59808i
471.1 1.00000 −1.50000 + 2.59808i −1.00000 1.50000 2.59808i −1.50000 + 2.59808i 0 −3.00000 −3.00000 5.19615i 1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

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

Hecke characteristic polynomials

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