Properties

Label 684.3.h.b
Level $684$
Weight $3$
Character orbit 684.h
Analytic conductor $18.638$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
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: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 q^{7} +O(q^{10})\) \( q -2 q^{7} + ( -8 + 16 \zeta_{6} ) q^{13} + ( -5 - 16 \zeta_{6} ) q^{19} -25 q^{25} + ( -24 + 48 \zeta_{6} ) q^{31} + ( -40 + 80 \zeta_{6} ) q^{37} -22 q^{43} -45 q^{49} -74 q^{61} + ( -32 + 64 \zeta_{6} ) q^{67} -46 q^{73} + ( -40 + 80 \zeta_{6} ) q^{79} + ( 16 - 32 \zeta_{6} ) q^{91} + ( -112 + 224 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} + O(q^{10}) \) \( 2 q - 4 q^{7} - 26 q^{19} - 50 q^{25} - 44 q^{43} - 90 q^{49} - 148 q^{61} - 92 q^{73} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −2.00000 0 0 0
37.2 0 0 0 0 0 −2.00000 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 \(\Q(\sqrt{-3}) \)
19.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.h.b 2
3.b odd 2 1 CM 684.3.h.b 2
4.b odd 2 1 2736.3.o.g 2
12.b even 2 1 2736.3.o.g 2
19.b odd 2 1 inner 684.3.h.b 2
57.d even 2 1 inner 684.3.h.b 2
76.d even 2 1 2736.3.o.g 2
228.b odd 2 1 2736.3.o.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.h.b 2 1.a even 1 1 trivial
684.3.h.b 2 3.b odd 2 1 CM
684.3.h.b 2 19.b odd 2 1 inner
684.3.h.b 2 57.d even 2 1 inner
2736.3.o.g 2 4.b odd 2 1
2736.3.o.g 2 12.b even 2 1
2736.3.o.g 2 76.d even 2 1
2736.3.o.g 2 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 192 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 361 + 26 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1728 + T^{2} \)
$37$ \( 4800 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 22 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 74 + T )^{2} \)
$67$ \( 3072 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 46 + T )^{2} \)
$79$ \( 4800 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 37632 + T^{2} \)
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