Properties

Label 684.3.h.e
Level $684$
Weight $3$
Character orbit 684.h
Analytic conductor $18.638$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(6\)
Coefficient field: 6.0.219615408.1
Defining polynomial: \(x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} -\beta_{2} q^{7} +O(q^{10})\) \( q + \beta_{3} q^{5} -\beta_{2} q^{7} + ( 4 + \beta_{2} ) q^{11} + ( -\beta_{1} + \beta_{4} ) q^{13} + ( 8 - \beta_{2} + 2 \beta_{3} ) q^{17} + ( -1 - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + ( -6 + 3 \beta_{2} + \beta_{3} ) q^{23} + ( 5 - 3 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{4} ) q^{29} + ( -\beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{31} + ( -2 + \beta_{2} + 4 \beta_{3} ) q^{35} + ( 3 \beta_{1} + \beta_{4} ) q^{37} + ( 3 \beta_{4} - 4 \beta_{5} ) q^{41} + ( -36 + \beta_{2} + 2 \beta_{3} ) q^{43} + ( -4 - 4 \beta_{2} + 5 \beta_{3} ) q^{47} + ( -7 + 5 \beta_{2} - 2 \beta_{3} ) q^{49} + ( -2 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{53} + ( 2 - \beta_{2} ) q^{55} + ( 4 \beta_{1} - 5 \beta_{4} - 2 \beta_{5} ) q^{59} + ( 32 + 7 \beta_{2} + 4 \beta_{3} ) q^{61} + ( -6 \beta_{1} + 9 \beta_{4} - 4 \beta_{5} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{67} + ( -4 \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{71} + ( 12 + \beta_{2} + 14 \beta_{3} ) q^{73} + ( -42 - 9 \beta_{2} + 2 \beta_{3} ) q^{77} + ( 3 \beta_{1} - 8 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 66 + 4 \beta_{3} ) q^{83} + ( 58 - 5 \beta_{2} + 16 \beta_{3} ) q^{85} + ( 3 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 11 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -60 - 2 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{95} + ( 7 \beta_{4} - 2 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5} - 2 q^{7} + O(q^{10}) \) \( 6 q + 2 q^{5} - 2 q^{7} + 26 q^{11} + 50 q^{17} - 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 210 q^{43} - 22 q^{47} - 36 q^{49} + 10 q^{55} + 214 q^{61} + 102 q^{73} - 266 q^{77} + 404 q^{83} + 370 q^{85} - 358 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 32 \nu^{3} + 43 \nu^{2} + 16 \nu - 15 \)\()/19\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} + 9 \nu^{2} - 8 \nu - 7 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} - 13 \nu^{2} + 12 \nu - 17 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{5} - 20 \nu^{4} + 128 \nu^{3} - 172 \nu^{2} + 392 \nu - 168 \)\()/19\)
\(\beta_{5}\)\(=\)\((\)\( 8 \nu^{5} - 20 \nu^{4} + 204 \nu^{3} - 286 \nu^{2} + 1114 \nu - 510 \)\()/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 4 \beta_{1} + 12\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - 12 \beta_{3} - 12 \beta_{2} + 4 \beta_{1} - 132\)\()/24\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{5} - 7 \beta_{4} - 9 \beta_{3} - 9 \beta_{2} - 16 \beta_{1} - 102\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(12 \beta_{5} - 29 \beta_{4} + 72 \beta_{3} + 120 \beta_{2} - 68 \beta_{1} + 1044\)\()/24\)
\(\nu^{5}\)\(=\)\((\)\(-66 \beta_{5} + 181 \beta_{4} + 210 \beta_{3} + 330 \beta_{2} + 232 \beta_{1} + 2952\)\()/24\)

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 0.460304i
0.500000 + 0.460304i
0.500000 + 3.19918i
0.500000 3.19918i
0.500000 + 2.79345i
0.500000 2.79345i
0 0 0 −5.83539 0 5.24085 0 0 0
37.2 0 0 0 −5.83539 0 5.24085 0 0 0
37.3 0 0 0 −0.556406 0 −9.52587 0 0 0
37.4 0 0 0 −0.556406 0 −9.52587 0 0 0
37.5 0 0 0 7.39180 0 3.28502 0 0 0
37.6 0 0 0 7.39180 0 3.28502 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
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.e 6
3.b odd 2 1 228.3.h.a 6
4.b odd 2 1 2736.3.o.m 6
12.b even 2 1 912.3.o.c 6
19.b odd 2 1 inner 684.3.h.e 6
57.d even 2 1 228.3.h.a 6
76.d even 2 1 2736.3.o.m 6
228.b odd 2 1 912.3.o.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.h.a 6 3.b odd 2 1
228.3.h.a 6 57.d even 2 1
684.3.h.e 6 1.a even 1 1 trivial
684.3.h.e 6 19.b odd 2 1 inner
912.3.o.c 6 12.b even 2 1
912.3.o.c 6 228.b odd 2 1
2736.3.o.m 6 4.b odd 2 1
2736.3.o.m 6 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - T_{5}^{2} - 44 T_{5} - 24 \) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( ( -24 - 44 T - T^{2} + T^{3} )^{2} \)
$7$ \( ( 164 - 64 T + T^{2} + T^{3} )^{2} \)
$11$ \( ( 12 - 8 T - 13 T^{2} + T^{3} )^{2} \)
$13$ \( 2495232 + 98064 T^{2} + 792 T^{4} + T^{6} \)
$17$ \( ( 108 - 32 T - 25 T^{2} + T^{3} )^{2} \)
$19$ \( 47045881 + 1303210 T + 146927 T^{2} + 7372 T^{3} + 407 T^{4} + 10 T^{5} + T^{6} \)
$23$ \( ( -5136 - 560 T + 14 T^{2} + T^{3} )^{2} \)
$29$ \( 322486272 + 2011392 T^{2} + 3168 T^{4} + T^{6} \)
$31$ \( 62208 + 1009296 T^{2} + 3192 T^{4} + T^{6} \)
$37$ \( 832267008 + 8883216 T^{2} + 5976 T^{4} + T^{6} \)
$41$ \( 29104386048 + 35322624 T^{2} + 10896 T^{4} + T^{6} \)
$43$ \( ( 35764 + 3432 T + 105 T^{2} + T^{3} )^{2} \)
$47$ \( ( -25272 - 2084 T + 11 T^{2} + T^{3} )^{2} \)
$53$ \( 63700992 + 4349952 T^{2} + 4272 T^{4} + T^{6} \)
$59$ \( 207873257472 + 115418880 T^{2} + 19536 T^{4} + T^{6} \)
$61$ \( ( 104276 - 64 T - 107 T^{2} + T^{3} )^{2} \)
$67$ \( 511377408 + 4886784 T^{2} + 4320 T^{4} + T^{6} \)
$71$ \( 48612814848 + 41654016 T^{2} + 11472 T^{4} + T^{6} \)
$73$ \( ( 115492 - 7896 T - 51 T^{2} + T^{3} )^{2} \)
$79$ \( 60723965952 + 68395536 T^{2} + 19848 T^{4} + T^{6} \)
$83$ \( ( -259992 + 12892 T - 202 T^{2} + T^{3} )^{2} \)
$89$ \( 956510208 + 3278592 T^{2} + 3408 T^{4} + T^{6} \)
$97$ \( 168210432 + 16512768 T^{2} + 8400 T^{4} + T^{6} \)
show more
show less