Properties

Label 637.2.e.c
Level $637$
Weight $2$
Character orbit 637.e
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.e (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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} -\zeta_{6} q^{9} -4 \zeta_{6} q^{12} + q^{13} + 6 q^{15} -4 \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + 7 \zeta_{6} q^{19} + 6 q^{20} -3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 4 q^{27} -9 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} -2 q^{36} -2 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{39} -6 q^{41} - q^{43} + ( 3 - 3 \zeta_{6} ) q^{45} -3 \zeta_{6} q^{47} -8 q^{48} -12 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{52} + ( 9 - 9 \zeta_{6} ) q^{53} + 14 q^{57} + ( 12 - 12 \zeta_{6} ) q^{60} + 10 \zeta_{6} q^{61} -8 q^{64} + 3 \zeta_{6} q^{65} + ( -14 + 14 \zeta_{6} ) q^{67} -12 \zeta_{6} q^{68} -6 q^{69} -6 q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + 8 \zeta_{6} q^{75} + 14 q^{76} + \zeta_{6} q^{79} + ( 12 - 12 \zeta_{6} ) q^{80} + ( 11 - 11 \zeta_{6} ) q^{81} + 3 q^{83} + 18 q^{85} + ( -18 + 18 \zeta_{6} ) q^{87} -15 \zeta_{6} q^{89} -6 q^{92} + 10 \zeta_{6} q^{93} + ( -21 + 21 \zeta_{6} ) q^{95} - q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{4} + 3q^{5} - q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{4} + 3q^{5} - q^{9} - 4q^{12} + 2q^{13} + 12q^{15} - 4q^{16} + 6q^{17} + 7q^{19} + 12q^{20} - 3q^{23} - 4q^{25} + 8q^{27} - 18q^{29} - 5q^{31} - 4q^{36} - 2q^{37} + 2q^{39} - 12q^{41} - 2q^{43} + 3q^{45} - 3q^{47} - 16q^{48} - 12q^{51} + 2q^{52} + 9q^{53} + 28q^{57} + 12q^{60} + 10q^{61} - 16q^{64} + 3q^{65} - 14q^{67} - 12q^{68} - 12q^{69} - 12q^{71} - 11q^{73} + 8q^{75} + 28q^{76} + q^{79} + 12q^{80} + 11q^{81} + 6q^{83} + 36q^{85} - 18q^{87} - 15q^{89} - 12q^{92} + 10q^{93} - 21q^{95} - 2q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 + 1.73205i 1.00000 + 1.73205i 1.50000 2.59808i 0 0 0 −0.500000 + 0.866025i 0
508.1 0 1.00000 1.73205i 1.00000 1.73205i 1.50000 + 2.59808i 0 0 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c 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.e.c 2
7.b odd 2 1 637.2.e.b 2
7.c even 3 1 91.2.a.b 1
7.c even 3 1 inner 637.2.e.c 2
7.d odd 6 1 637.2.a.b 1
7.d odd 6 1 637.2.e.b 2
21.g even 6 1 5733.2.a.f 1
21.h odd 6 1 819.2.a.c 1
28.g odd 6 1 1456.2.a.k 1
35.j even 6 1 2275.2.a.d 1
56.k odd 6 1 5824.2.a.f 1
56.p even 6 1 5824.2.a.bd 1
91.r even 6 1 1183.2.a.a 1
91.s odd 6 1 8281.2.a.h 1
91.z odd 12 2 1183.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.b 1 7.c even 3 1
637.2.a.b 1 7.d odd 6 1
637.2.e.b 2 7.b odd 2 1
637.2.e.b 2 7.d odd 6 1
637.2.e.c 2 1.a even 1 1 trivial
637.2.e.c 2 7.c even 3 1 inner
819.2.a.c 1 21.h odd 6 1
1183.2.a.a 1 91.r even 6 1
1183.2.c.a 2 91.z odd 12 2
1456.2.a.k 1 28.g odd 6 1
2275.2.a.d 1 35.j even 6 1
5733.2.a.f 1 21.g even 6 1
5824.2.a.f 1 56.k odd 6 1
5824.2.a.bd 1 56.p even 6 1
8281.2.a.h 1 91.s odd 6 1

Hecke kernels

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

\( T_{2} \)
\( T_{3}^{2} - 2 T_{3} + 4 \)
\( T_{5}^{2} - 3 T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 - 2 T + T^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 49 - 7 T + T^{2} \)
$23$ \( 9 + 3 T + T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( 9 + 3 T + T^{2} \)
$53$ \( 81 - 9 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 100 - 10 T + T^{2} \)
$67$ \( 196 + 14 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 121 + 11 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( ( -3 + T )^{2} \)
$89$ \( 225 + 15 T + T^{2} \)
$97$ \( ( 1 + T )^{2} \)
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