Properties

Label 837.2.h.a
Level $837$
Weight $2$
Character orbit 837.h
Analytic conductor $6.683$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 837 = 3^{3} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 837.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.68347864918\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 q^{4} + ( -5 + 5 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 q^{4} + ( -5 + 5 \zeta_{6} ) q^{7} -5 \zeta_{6} q^{13} + 4 q^{16} + ( 7 - 7 \zeta_{6} ) q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 10 - 10 \zeta_{6} ) q^{28} + ( 1 - 6 \zeta_{6} ) q^{31} + ( -11 + 11 \zeta_{6} ) q^{37} + ( 13 - 13 \zeta_{6} ) q^{43} -18 \zeta_{6} q^{49} + 10 \zeta_{6} q^{52} -13 q^{61} -8 q^{64} -11 \zeta_{6} q^{67} + 10 \zeta_{6} q^{73} + ( -14 + 14 \zeta_{6} ) q^{76} + ( 13 - 13 \zeta_{6} ) q^{79} + 25 q^{91} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} - 5q^{7} + O(q^{10}) \) \( 2q - 4q^{4} - 5q^{7} - 5q^{13} + 8q^{16} + 7q^{19} + 5q^{25} + 10q^{28} - 4q^{31} - 11q^{37} + 13q^{43} - 18q^{49} + 10q^{52} - 26q^{61} - 16q^{64} - 11q^{67} + 10q^{73} - 14q^{76} + 13q^{79} + 50q^{91} + 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/837\mathbb{Z}\right)^\times\).

\(n\) \(218\) \(406\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
676.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −2.00000 0 0 −2.50000 4.33013i 0 0 0
811.1 0 0 −2.00000 0 0 −2.50000 + 4.33013i 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
31.c even 3 1 inner
93.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 837.2.h.a 2
3.b odd 2 1 CM 837.2.h.a 2
31.c even 3 1 inner 837.2.h.a 2
93.h odd 6 1 inner 837.2.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
837.2.h.a 2 1.a even 1 1 trivial
837.2.h.a 2 3.b odd 2 1 CM
837.2.h.a 2 31.c even 3 1 inner
837.2.h.a 2 93.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(837, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + 5 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 49 - 7 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 31 + 4 T + T^{2} \)
$37$ \( 121 + 11 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 169 - 13 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 13 + T )^{2} \)
$67$ \( 121 + 11 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 169 - 13 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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