Properties

Label 684.2.f.b
Level $684$
Weight $2$
Character orbit 684.f
Analytic conductor $5.462$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14453810176.1
Defining polynomial: \(x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{2} + \beta_{6} ) q^{5} + ( \beta_{5} - \beta_{6} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{2} + \beta_{6} ) q^{5} + ( \beta_{5} - \beta_{6} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{10} + ( \beta_{2} + \beta_{5} ) q^{11} + ( -\beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{13} + ( 2 \beta_{4} - \beta_{7} ) q^{14} + ( -\beta_{2} + \beta_{5} + \beta_{6} ) q^{16} + q^{17} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{19} + ( -3 - \beta_{5} - \beta_{6} ) q^{20} + ( \beta_{3} + 2 \beta_{4} ) q^{22} + ( -\beta_{2} - \beta_{6} ) q^{23} + ( -\beta_{2} + \beta_{6} ) q^{25} + ( 2 - \beta_{2} + \beta_{5} + 3 \beta_{6} ) q^{26} + ( 3 - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{28} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{29} + ( \beta_{1} - \beta_{7} ) q^{31} + ( -\beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{32} + \beta_{1} q^{34} + ( \beta_{2} + 3 \beta_{5} - 2 \beta_{6} ) q^{35} + ( 3 \beta_{1} + \beta_{7} ) q^{37} + ( -2 - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{38} + ( -3 \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{40} + ( 3 \beta_{1} + \beta_{7} ) q^{41} + ( -3 \beta_{2} - \beta_{5} - 2 \beta_{6} ) q^{43} + ( -1 - 2 \beta_{2} - \beta_{5} + 3 \beta_{6} ) q^{44} + ( -\beta_{3} - \beta_{7} ) q^{46} + ( -3 \beta_{2} - 3 \beta_{5} ) q^{47} + ( -4 + 4 \beta_{2} - 4 \beta_{6} ) q^{49} + ( -\beta_{3} + \beta_{7} ) q^{50} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 3 \beta_{7} ) q^{52} + ( 4 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{53} + ( -\beta_{2} + \beta_{5} - 2 \beta_{6} ) q^{55} + ( 3 \beta_{1} - \beta_{3} - 4 \beta_{4} + 2 \beta_{7} ) q^{56} + ( -4 + \beta_{2} + \beta_{5} + 3 \beta_{6} ) q^{58} + ( 3 \beta_{1} + \beta_{4} - 3 \beta_{7} ) q^{59} + ( 7 - 3 \beta_{2} + 3 \beta_{6} ) q^{61} + ( 2 + 2 \beta_{2} ) q^{62} + ( -2 - 3 \beta_{2} - 3 \beta_{5} + \beta_{6} ) q^{64} + ( -6 \beta_{1} - 2 \beta_{7} ) q^{65} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{7} ) q^{67} + ( -1 + \beta_{2} ) q^{68} + ( \beta_{3} + 6 \beta_{4} - 2 \beta_{7} ) q^{70} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{71} + ( 3 + 2 \beta_{2} - 2 \beta_{6} ) q^{73} + ( -6 + 2 \beta_{2} ) q^{74} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{76} + ( -5 + 3 \beta_{2} - 3 \beta_{6} ) q^{77} + ( 3 \beta_{1} + 4 \beta_{4} - 3 \beta_{7} ) q^{79} + ( 6 + 2 \beta_{5} - 2 \beta_{6} ) q^{80} + ( -6 + 2 \beta_{2} ) q^{82} + ( 2 \beta_{2} - 2 \beta_{5} + 4 \beta_{6} ) q^{83} + ( 1 - \beta_{2} + \beta_{6} ) q^{85} + ( -3 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{86} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{7} ) q^{88} + ( -5 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{7} ) q^{89} + ( -\beta_{1} - 7 \beta_{4} + \beta_{7} ) q^{91} + ( 4 + \beta_{2} - \beta_{5} - \beta_{6} ) q^{92} + ( -3 \beta_{3} - 6 \beta_{4} ) q^{94} + ( -\beta_{2} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{95} + ( 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{97} + ( -4 \beta_{1} + 4 \beta_{3} - 4 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 6q^{4} + 4q^{5} + O(q^{10}) \) \( 8q - 6q^{4} + 4q^{5} - 6q^{16} + 8q^{17} - 20q^{20} - 4q^{25} + 6q^{26} + 22q^{28} - 18q^{38} - 16q^{44} - 16q^{49} - 38q^{58} + 44q^{61} + 20q^{62} - 18q^{64} - 6q^{68} + 32q^{73} - 44q^{74} - 16q^{76} - 28q^{77} + 48q^{80} - 44q^{82} + 4q^{85} + 38q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + \nu \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} - 2 \nu^{3} - 4 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + \nu^{4} - 2 \nu^{2} - 4 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 3 \nu^{4} + 6 \nu^{2} + 8 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} + 6 \nu^{3} + 8 \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{6} + \beta_{5} - \beta_{2}\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 2 \beta_{4} - \beta_{3}\)
\(\nu^{6}\)\(=\)\(\beta_{6} - 3 \beta_{5} - 3 \beta_{2} - 2\)
\(\nu^{7}\)\(=\)\(\beta_{7} - 6 \beta_{4} - 3 \beta_{3} - 2 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
−1.06789 0.927153i
−1.06789 + 0.927153i
−0.331077 1.37491i
−0.331077 + 1.37491i
0.331077 1.37491i
0.331077 + 1.37491i
1.06789 0.927153i
1.06789 + 0.927153i
−1.06789 0.927153i 0 0.280776 + 1.98019i −1.56155 0 0.868210i 1.53610 2.37495i 0 1.66757 + 1.44780i
379.2 −1.06789 + 0.927153i 0 0.280776 1.98019i −1.56155 0 0.868210i 1.53610 + 2.37495i 0 1.66757 1.44780i
379.3 −0.331077 1.37491i 0 −1.78078 + 0.910404i 2.56155 0 4.15286i 1.84130 + 2.14700i 0 −0.848071 3.52191i
379.4 −0.331077 + 1.37491i 0 −1.78078 0.910404i 2.56155 0 4.15286i 1.84130 2.14700i 0 −0.848071 + 3.52191i
379.5 0.331077 1.37491i 0 −1.78078 0.910404i 2.56155 0 4.15286i −1.84130 + 2.14700i 0 0.848071 3.52191i
379.6 0.331077 + 1.37491i 0 −1.78078 + 0.910404i 2.56155 0 4.15286i −1.84130 2.14700i 0 0.848071 + 3.52191i
379.7 1.06789 0.927153i 0 0.280776 1.98019i −1.56155 0 0.868210i −1.53610 2.37495i 0 −1.66757 + 1.44780i
379.8 1.06789 + 0.927153i 0 0.280776 + 1.98019i −1.56155 0 0.868210i −1.53610 + 2.37495i 0 −1.66757 1.44780i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.f.b 8
3.b odd 2 1 76.2.d.a 8
4.b odd 2 1 inner 684.2.f.b 8
12.b even 2 1 76.2.d.a 8
19.b odd 2 1 inner 684.2.f.b 8
24.f even 2 1 1216.2.h.d 8
24.h odd 2 1 1216.2.h.d 8
57.d even 2 1 76.2.d.a 8
76.d even 2 1 inner 684.2.f.b 8
228.b odd 2 1 76.2.d.a 8
456.l odd 2 1 1216.2.h.d 8
456.p even 2 1 1216.2.h.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.d.a 8 3.b odd 2 1
76.2.d.a 8 12.b even 2 1
76.2.d.a 8 57.d even 2 1
76.2.d.a 8 228.b odd 2 1
684.2.f.b 8 1.a even 1 1 trivial
684.2.f.b 8 4.b odd 2 1 inner
684.2.f.b 8 19.b odd 2 1 inner
684.2.f.b 8 76.d even 2 1 inner
1216.2.h.d 8 24.f even 2 1
1216.2.h.d 8 24.h odd 2 1
1216.2.h.d 8 456.l odd 2 1
1216.2.h.d 8 456.p even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{2} - T_{5} - 4 \)
\( T_{31}^{4} - 20 T_{31}^{2} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 12 T^{2} + 6 T^{4} + 3 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( -4 - T + T^{2} )^{4} \)
$7$ \( ( 13 + 18 T^{2} + T^{4} )^{2} \)
$11$ \( ( 52 + 15 T^{2} + T^{4} )^{2} \)
$13$ \( ( 416 + 41 T^{2} + T^{4} )^{2} \)
$17$ \( ( -1 + T )^{8} \)
$19$ \( 130321 - 5776 T^{2} + 718 T^{4} - 16 T^{6} + T^{8} \)
$23$ \( ( 52 + 19 T^{2} + T^{4} )^{2} \)
$29$ \( ( 104 + 73 T^{2} + T^{4} )^{2} \)
$31$ \( ( 32 - 20 T^{2} + T^{4} )^{2} \)
$37$ \( ( 416 + 44 T^{2} + T^{4} )^{2} \)
$41$ \( ( 416 + 44 T^{2} + T^{4} )^{2} \)
$43$ \( ( 208 + 123 T^{2} + T^{4} )^{2} \)
$47$ \( ( 4212 + 135 T^{2} + T^{4} )^{2} \)
$53$ \( ( 104 + 97 T^{2} + T^{4} )^{2} \)
$59$ \( ( 5408 - 175 T^{2} + T^{4} )^{2} \)
$61$ \( ( -8 - 11 T + T^{2} )^{4} \)
$67$ \( ( 8 - 95 T^{2} + T^{4} )^{2} \)
$71$ \( ( 512 - 124 T^{2} + T^{4} )^{2} \)
$73$ \( ( -1 - 8 T + T^{2} )^{4} \)
$79$ \( ( 11552 - 244 T^{2} + T^{4} )^{2} \)
$83$ \( ( 13312 + 236 T^{2} + T^{4} )^{2} \)
$89$ \( ( 6656 + 232 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1664 + 292 T^{2} + T^{4} )^{2} \)
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