Properties

Label 684.4.a.g
Level $684$
Weight $4$
Character orbit 684.a
Self dual yes
Analytic conductor $40.357$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 684.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3573064439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta q^{5} + ( -17 + 4 \beta ) q^{7} +O(q^{10})\) \( q + 5 \beta q^{5} + ( -17 + 4 \beta ) q^{7} + ( 38 - 5 \beta ) q^{11} + ( -23 + 11 \beta ) q^{13} + ( -3 + 6 \beta ) q^{17} -19 q^{19} + ( -27 + 59 \beta ) q^{23} + ( 75 + 25 \beta ) q^{25} + ( -45 - 65 \beta ) q^{29} + ( -84 + 80 \beta ) q^{31} + ( 160 - 65 \beta ) q^{35} + ( 186 + 8 \beta ) q^{37} + ( 56 + 30 \beta ) q^{41} + ( 136 - 117 \beta ) q^{43} + ( 216 + 23 \beta ) q^{47} + ( 74 - 120 \beta ) q^{49} + ( 199 - 123 \beta ) q^{53} + ( -200 + 165 \beta ) q^{55} + ( 419 + 35 \beta ) q^{59} + ( 310 - 175 \beta ) q^{61} + ( 440 - 60 \beta ) q^{65} + ( 353 - 61 \beta ) q^{67} + ( 846 + 20 \beta ) q^{71} + ( -463 - 64 \beta ) q^{73} + ( -806 + 217 \beta ) q^{77} + ( -642 + 10 \beta ) q^{79} + ( 102 - 114 \beta ) q^{83} + ( 240 + 15 \beta ) q^{85} + ( 484 - 80 \beta ) q^{89} + ( 743 - 235 \beta ) q^{91} -95 \beta q^{95} + ( 126 + 458 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 5q^{5} - 30q^{7} + O(q^{10}) \) \( 2q + 5q^{5} - 30q^{7} + 71q^{11} - 35q^{13} - 38q^{19} + 5q^{23} + 175q^{25} - 155q^{29} - 88q^{31} + 255q^{35} + 380q^{37} + 142q^{41} + 155q^{43} + 455q^{47} + 28q^{49} + 275q^{53} - 235q^{55} + 873q^{59} + 445q^{61} + 820q^{65} + 645q^{67} + 1712q^{71} - 990q^{73} - 1395q^{77} - 1274q^{79} + 90q^{83} + 495q^{85} + 888q^{89} + 1251q^{91} - 95q^{95} + 710q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.37228
3.37228
0 0 0 −11.8614 0 −26.4891 0 0 0
1.2 0 0 0 16.8614 0 −3.51087 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.4.a.g 2
3.b odd 2 1 76.4.a.a 2
12.b even 2 1 304.4.a.f 2
15.d odd 2 1 1900.4.a.b 2
15.e even 4 2 1900.4.c.b 4
24.f even 2 1 1216.4.a.h 2
24.h odd 2 1 1216.4.a.o 2
57.d even 2 1 1444.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.a 2 3.b odd 2 1
304.4.a.f 2 12.b even 2 1
684.4.a.g 2 1.a even 1 1 trivial
1216.4.a.h 2 24.f even 2 1
1216.4.a.o 2 24.h odd 2 1
1444.4.a.d 2 57.d even 2 1
1900.4.a.b 2 15.d odd 2 1
1900.4.c.b 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 5 T_{5} - 200 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(684))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -200 - 5 T + T^{2} \)
$7$ \( 93 + 30 T + T^{2} \)
$11$ \( 1054 - 71 T + T^{2} \)
$13$ \( -692 + 35 T + T^{2} \)
$17$ \( -297 + T^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( -28712 - 5 T + T^{2} \)
$29$ \( -28850 + 155 T + T^{2} \)
$31$ \( -50864 + 88 T + T^{2} \)
$37$ \( 35572 - 380 T + T^{2} \)
$41$ \( -2384 - 142 T + T^{2} \)
$43$ \( -106928 - 155 T + T^{2} \)
$47$ \( 47392 - 455 T + T^{2} \)
$53$ \( -105908 - 275 T + T^{2} \)
$59$ \( 180426 - 873 T + T^{2} \)
$61$ \( -203150 - 445 T + T^{2} \)
$67$ \( 73308 - 645 T + T^{2} \)
$71$ \( 729436 - 1712 T + T^{2} \)
$73$ \( 211233 + 990 T + T^{2} \)
$79$ \( 404944 + 1274 T + T^{2} \)
$83$ \( -105192 - 90 T + T^{2} \)
$89$ \( 144336 - 888 T + T^{2} \)
$97$ \( -1604528 - 710 T + T^{2} \)
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