Properties

Label 684.5.h.b
Level $684$
Weight $5$
Character orbit 684.h
Self dual yes
Analytic conductor $70.705$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 684.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7050547493\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 3\sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 17 + 3 \beta ) q^{5} + ( 39 + 5 \beta ) q^{7} +O(q^{10})\) \( q + ( 17 + 3 \beta ) q^{5} + ( 39 + 5 \beta ) q^{7} + ( -119 - 5 \beta ) q^{11} + ( -159 + 35 \beta ) q^{17} + 361 q^{19} + 158 q^{23} + ( 816 + 93 \beta ) q^{25} + ( 2583 + 187 \beta ) q^{35} + ( -1721 + 85 \beta ) q^{43} + ( 441 - 325 \beta ) q^{47} + ( 2320 + 365 \beta ) q^{49} + ( -3943 - 427 \beta ) q^{55} + ( -1841 - 515 \beta ) q^{61} + ( 4879 - 275 \beta ) q^{73} + ( -7841 - 765 \beta ) q^{77} + 5678 q^{83} + ( 10737 + 13 \beta ) q^{85} + ( 6137 + 1083 \beta ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 31q^{5} + 73q^{7} + O(q^{10}) \) \( 2q + 31q^{5} + 73q^{7} - 233q^{11} - 353q^{17} + 722q^{19} + 316q^{23} + 1539q^{25} + 4979q^{35} - 3527q^{43} + 1207q^{47} + 4275q^{49} - 7459q^{55} - 3167q^{61} + 10033q^{73} - 14917q^{77} + 11356q^{83} + 21461q^{85} + 11191q^{95} + O(q^{100}) \)

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
−3.27492
4.27492
0 0 0 −18.4743 0 −20.1238 0 0 0
37.2 0 0 0 49.4743 0 93.1238 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 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.5.h.b 2
3.b odd 2 1 76.5.c.a 2
12.b even 2 1 304.5.e.b 2
19.b odd 2 1 CM 684.5.h.b 2
57.d even 2 1 76.5.c.a 2
228.b odd 2 1 304.5.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.c.a 2 3.b odd 2 1
76.5.c.a 2 57.d even 2 1
304.5.e.b 2 12.b even 2 1
304.5.e.b 2 228.b odd 2 1
684.5.h.b 2 1.a even 1 1 trivial
684.5.h.b 2 19.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 31 T_{5} - 914 \) acting on \(S_{5}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -914 - 31 T + T^{2} \)
$7$ \( -1874 - 73 T + T^{2} \)
$11$ \( 10366 + 233 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -125954 + 353 T + T^{2} \)
$19$ \( ( -361 + T )^{2} \)
$23$ \( ( -158 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 2183326 + 3527 T + T^{2} \)
$47$ \( -13182194 - 1207 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( -31507634 + 3167 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 15466366 - 10033 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -5678 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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