Properties

Label 684.3.h.c
Level $684$
Weight $3$
Character orbit 684.h
Analytic conductor $18.638$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(\Q(\sqrt{-29}) \)
Defining polynomial: \(x^{2} + 29\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-29}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{5} - q^{7} +O(q^{10})\) \( q + 4 q^{5} - q^{7} -14 q^{11} -\beta q^{13} -23 q^{17} + ( 10 - \beta ) q^{19} + q^{23} -9 q^{25} -3 \beta q^{29} + 2 \beta q^{31} -4 q^{35} -2 \beta q^{37} + 2 \beta q^{41} + 68 q^{43} -26 q^{47} -48 q^{49} -5 \beta q^{53} -56 q^{55} + \beta q^{59} -40 q^{61} -4 \beta q^{65} + \beta q^{67} -2 \beta q^{71} -7 q^{73} + 14 q^{77} -6 \beta q^{79} -32 q^{83} -92 q^{85} -8 \beta q^{89} + \beta q^{91} + ( 40 - 4 \beta ) q^{95} + 6 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + 8q^{5} - 2q^{7} - 28q^{11} - 46q^{17} + 20q^{19} + 2q^{23} - 18q^{25} - 8q^{35} + 136q^{43} - 52q^{47} - 96q^{49} - 112q^{55} - 80q^{61} - 14q^{73} + 28q^{77} - 64q^{83} - 184q^{85} + 80q^{95} + O(q^{100}) \)

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
5.38516i
5.38516i
0 0 0 4.00000 0 −1.00000 0 0 0
37.2 0 0 0 4.00000 0 −1.00000 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
19.b odd 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.c 2
3.b odd 2 1 76.3.c.a 2
4.b odd 2 1 2736.3.o.i 2
12.b even 2 1 304.3.e.b 2
15.d odd 2 1 1900.3.e.b 2
15.e even 4 2 1900.3.g.b 4
19.b odd 2 1 inner 684.3.h.c 2
24.f even 2 1 1216.3.e.l 2
24.h odd 2 1 1216.3.e.k 2
57.d even 2 1 76.3.c.a 2
76.d even 2 1 2736.3.o.i 2
228.b odd 2 1 304.3.e.b 2
285.b even 2 1 1900.3.e.b 2
285.j odd 4 2 1900.3.g.b 4
456.l odd 2 1 1216.3.e.l 2
456.p even 2 1 1216.3.e.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.a 2 3.b odd 2 1
76.3.c.a 2 57.d even 2 1
304.3.e.b 2 12.b even 2 1
304.3.e.b 2 228.b odd 2 1
684.3.h.c 2 1.a even 1 1 trivial
684.3.h.c 2 19.b odd 2 1 inner
1216.3.e.k 2 24.h odd 2 1
1216.3.e.k 2 456.p even 2 1
1216.3.e.l 2 24.f even 2 1
1216.3.e.l 2 456.l odd 2 1
1900.3.e.b 2 15.d odd 2 1
1900.3.e.b 2 285.b even 2 1
1900.3.g.b 4 15.e even 4 2
1900.3.g.b 4 285.j odd 4 2
2736.3.o.i 2 4.b odd 2 1
2736.3.o.i 2 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 4 \) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -4 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 14 + T )^{2} \)
$13$ \( 261 + T^{2} \)
$17$ \( ( 23 + T )^{2} \)
$19$ \( 361 - 20 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 2349 + T^{2} \)
$31$ \( 1044 + T^{2} \)
$37$ \( 1044 + T^{2} \)
$41$ \( 1044 + T^{2} \)
$43$ \( ( -68 + T )^{2} \)
$47$ \( ( 26 + T )^{2} \)
$53$ \( 6525 + T^{2} \)
$59$ \( 261 + T^{2} \)
$61$ \( ( 40 + T )^{2} \)
$67$ \( 261 + T^{2} \)
$71$ \( 1044 + T^{2} \)
$73$ \( ( 7 + T )^{2} \)
$79$ \( 9396 + T^{2} \)
$83$ \( ( 32 + T )^{2} \)
$89$ \( 16704 + T^{2} \)
$97$ \( 9396 + T^{2} \)
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