Properties

Label 637.2.c.b
Level $637$
Weight $2$
Character orbit 637.c
Analytic conductor $5.086$
Analytic rank $0$
Dimension $2$
CM discriminant -91
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-13}) \)
Defining polynomial: \(x^{2} + 13\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} + \beta q^{5} -3 q^{9} +O(q^{10})\) \( q + 2 q^{4} + \beta q^{5} -3 q^{9} + \beta q^{13} + 4 q^{16} + \beta q^{19} + 2 \beta q^{20} + q^{23} -8 q^{25} -5 q^{29} -3 \beta q^{31} -6 q^{36} + 2 \beta q^{41} + 9 q^{43} -3 \beta q^{45} + \beta q^{47} + 2 \beta q^{52} + 11 q^{53} + 4 \beta q^{59} + 8 q^{64} -13 q^{65} -3 \beta q^{73} + 2 \beta q^{76} + 15 q^{79} + 4 \beta q^{80} + 9 q^{81} -5 \beta q^{83} + \beta q^{89} + 2 q^{92} -13 q^{95} -5 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} - 6q^{9} + O(q^{10}) \) \( 2q + 4q^{4} - 6q^{9} + 8q^{16} + 2q^{23} - 16q^{25} - 10q^{29} - 12q^{36} + 18q^{43} + 22q^{53} + 16q^{64} - 26q^{65} + 30q^{79} + 18q^{81} + 4q^{92} - 26q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
246.1
3.60555i
3.60555i
0 0 2.00000 3.60555i 0 0 0 −3.00000 0
246.2 0 0 2.00000 3.60555i 0 0 0 −3.00000 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
91.b odd 2 1 CM by \(\Q(\sqrt{-91}) \)
7.b odd 2 1 inner
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.c.b 2
7.b odd 2 1 inner 637.2.c.b 2
7.c even 3 2 637.2.r.b 4
7.d odd 6 2 637.2.r.b 4
13.b even 2 1 inner 637.2.c.b 2
13.d odd 4 2 8281.2.a.u 2
91.b odd 2 1 CM 637.2.c.b 2
91.i even 4 2 8281.2.a.u 2
91.r even 6 2 637.2.r.b 4
91.s odd 6 2 637.2.r.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.b 2 1.a even 1 1 trivial
637.2.c.b 2 7.b odd 2 1 inner
637.2.c.b 2 13.b even 2 1 inner
637.2.c.b 2 91.b odd 2 1 CM
637.2.r.b 4 7.c even 3 2
637.2.r.b 4 7.d odd 6 2
637.2.r.b 4 91.r even 6 2
637.2.r.b 4 91.s odd 6 2
8281.2.a.u 2 13.d odd 4 2
8281.2.a.u 2 91.i even 4 2

Hecke kernels

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

\( T_{2} \)
\( T_{3} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 13 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 13 + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 117 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( 52 + T^{2} \)
$43$ \( ( -9 + T )^{2} \)
$47$ \( 13 + T^{2} \)
$53$ \( ( -11 + T )^{2} \)
$59$ \( 208 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 117 + T^{2} \)
$79$ \( ( -15 + T )^{2} \)
$83$ \( 325 + T^{2} \)
$89$ \( 13 + T^{2} \)
$97$ \( 325 + T^{2} \)
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