Properties

Label 684.3.y.g
Level $684$
Weight $3$
Character orbit 684.y
Analytic conductor $18.638$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 34 x^{6} + 921 x^{4} - 7990 x^{2} + 55225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{5} ) q^{5} + ( -1 + \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{5} ) q^{5} + ( -1 + \beta_{3} ) q^{7} -\beta_{4} q^{11} + ( -5 + 5 \beta_{2} - 4 \beta_{3} + 2 \beta_{6} ) q^{13} + ( -\beta_{1} + \beta_{5} + \beta_{7} ) q^{17} + ( 5 + 2 \beta_{2} - 7 \beta_{3} + 8 \beta_{6} ) q^{19} + \beta_{5} q^{23} + ( -9 - 9 \beta_{2} + 6 \beta_{6} ) q^{25} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{29} + ( -9 - 18 \beta_{2} - 5 \beta_{3} + 10 \beta_{6} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{5} - \beta_{7} ) q^{35} + ( -1 - 2 \beta_{2} - 6 \beta_{3} + 12 \beta_{6} ) q^{37} + ( -2 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{41} + ( 5 \beta_{2} - 11 \beta_{3} + 11 \beta_{6} ) q^{43} + ( -4 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} ) q^{47} + ( -42 - 2 \beta_{3} ) q^{49} + ( -4 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{53} + ( -2 \beta_{2} - 28 \beta_{3} + 28 \beta_{6} ) q^{55} + ( 5 \beta_{1} + 4 \beta_{4} + 5 \beta_{5} + 2 \beta_{7} ) q^{59} + ( 31 + 31 \beta_{2} + 34 \beta_{6} ) q^{61} + ( -3 \beta_{1} + 2 \beta_{4} + 6 \beta_{5} + 4 \beta_{7} ) q^{65} + ( -19 + 19 \beta_{2} - 38 \beta_{3} + 19 \beta_{6} ) q^{67} + ( -3 \beta_{1} - 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} ) q^{71} + ( -25 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{73} -5 \beta_{1} q^{77} + ( -66 - 33 \beta_{2} + \beta_{3} + \beta_{6} ) q^{79} + ( 17 \beta_{1} - 3 \beta_{4} ) q^{83} + ( 32 + 32 \beta_{2} + 22 \beta_{6} ) q^{85} + ( -4 \beta_{1} - 5 \beta_{4} + 2 \beta_{5} + 5 \beta_{7} ) q^{89} + ( -7 + 7 \beta_{2} - 6 \beta_{3} + 3 \beta_{6} ) q^{91} + ( 4 \beta_{1} + 8 \beta_{4} - 10 \beta_{5} + 7 \beta_{7} ) q^{95} + ( 36 + 18 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} - 60q^{13} + 32q^{19} - 36q^{25} - 20q^{43} - 336q^{49} + 8q^{55} + 124q^{61} - 228q^{67} + 100q^{73} - 396q^{79} + 128q^{85} - 84q^{91} + 216q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 34 x^{6} + 921 x^{4} - 7990 x^{2} + 55225\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 36 \nu^{7} - 5219 \nu^{5} + 105301 \nu^{3} - 1514105 \nu \)\()/649305\)
\(\beta_{2}\)\(=\)\((\)\( -34 \nu^{6} + 921 \nu^{4} - 31314 \nu^{2} + 55225 \)\()/216435\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 7667 \)\()/2763\)
\(\beta_{4}\)\(=\)\((\)\( 48 \nu^{7} - 2149 \nu^{5} + 58637 \nu^{3} - 888535 \nu \)\()/129861\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 34 \nu^{5} + 686 \nu^{3} - 3995 \nu \)\()/2115\)
\(\beta_{6}\)\(=\)\((\)\( -343 \nu^{6} + 15657 \nu^{4} - 315903 \nu^{2} + 2740570 \)\()/649305\)
\(\beta_{7}\)\(=\)\((\)\( 95 \nu^{7} - 2149 \nu^{5} + 58637 \nu^{3} + 250980 \nu \)\()/129861\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\(3 \beta_{6} - 3 \beta_{3} - 17 \beta_{2}\)
\(\nu^{3}\)\(=\)\((\)\(34 \beta_{7} - 70 \beta_{5} + 17 \beta_{4} + 35 \beta_{1}\)\()/6\)
\(\nu^{4}\)\(=\)\(102 \beta_{6} - 343 \beta_{2} - 343\)
\(\nu^{5}\)\(=\)\((\)\(343 \beta_{7} - 955 \beta_{5} + 686 \beta_{4} - 955 \beta_{1}\)\()/6\)
\(\nu^{6}\)\(=\)\(2763 \beta_{3} - 7667\)
\(\nu^{7}\)\(=\)\((\)\(-7667 \beta_{7} + 24245 \beta_{5} + 7667 \beta_{4} - 48490 \beta_{1}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(1 + \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−4.27333 2.46721i
2.69047 + 1.55335i
−2.69047 1.55335i
4.27333 + 2.46721i
−4.27333 + 2.46721i
2.69047 1.55335i
−2.69047 + 1.55335i
4.27333 2.46721i
0 0 0 −3.48916 6.04340i 0 −3.44949 0 0 0
145.2 0 0 0 −2.19676 3.80490i 0 1.44949 0 0 0
145.3 0 0 0 2.19676 + 3.80490i 0 1.44949 0 0 0
145.4 0 0 0 3.48916 + 6.04340i 0 −3.44949 0 0 0
217.1 0 0 0 −3.48916 + 6.04340i 0 −3.44949 0 0 0
217.2 0 0 0 −2.19676 + 3.80490i 0 1.44949 0 0 0
217.3 0 0 0 2.19676 3.80490i 0 1.44949 0 0 0
217.4 0 0 0 3.48916 6.04340i 0 −3.44949 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.y.g 8
3.b odd 2 1 inner 684.3.y.g 8
19.d odd 6 1 inner 684.3.y.g 8
57.f even 6 1 inner 684.3.y.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.y.g 8 1.a even 1 1 trivial
684.3.y.g 8 3.b odd 2 1 inner
684.3.y.g 8 19.d odd 6 1 inner
684.3.y.g 8 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{8} + 68 T_{5}^{6} + 3684 T_{5}^{4} + 63920 T_{5}^{2} + 883600 \)
\( T_{7}^{2} + 2 T_{7} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 883600 + 63920 T^{2} + 3684 T^{4} + 68 T^{6} + T^{8} \)
$7$ \( ( -5 + 2 T + T^{2} )^{4} \)
$11$ \( ( 23500 - 332 T^{2} + T^{4} )^{2} \)
$13$ \( ( 9 + 90 T + 303 T^{2} + 30 T^{3} + T^{4} )^{2} \)
$17$ \( 14137600 + 1473920 T^{2} + 149904 T^{4} + 392 T^{6} + T^{8} \)
$19$ \( ( 130321 - 5776 T + 570 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$23$ \( 883600 + 63920 T^{2} + 3684 T^{4} + 68 T^{6} + T^{8} \)
$29$ \( 1145145600 - 39795840 T^{2} + 1349136 T^{4} - 1176 T^{6} + T^{8} \)
$31$ \( ( 42849 + 1386 T^{2} + T^{4} )^{2} \)
$37$ \( ( 416025 + 1302 T^{2} + T^{4} )^{2} \)
$41$ \( 23745739161600 - 23858012160 T^{2} + 19097856 T^{4} - 4896 T^{6} + T^{8} \)
$43$ \( ( 491401 - 7010 T + 801 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$47$ \( 926521753600 + 6037176320 T^{2} + 38375424 T^{4} + 6272 T^{6} + T^{8} \)
$53$ \( 44732250000 - 393390000 T^{2} + 3248100 T^{4} - 1860 T^{6} + T^{8} \)
$59$ \( 27957656250000 - 46762650000 T^{2} + 72928836 T^{4} - 8844 T^{6} + T^{8} \)
$61$ \( ( 35700625 + 370450 T + 9819 T^{2} - 62 T^{3} + T^{4} )^{2} \)
$67$ \( ( 29322225 - 617310 T - 1083 T^{2} + 114 T^{3} + T^{4} )^{2} \)
$71$ \( 7513300281600 - 29011167360 T^{2} + 109280016 T^{4} - 10584 T^{6} + T^{8} \)
$73$ \( ( 361201 - 30050 T + 1899 T^{2} - 50 T^{3} + T^{4} )^{2} \)
$79$ \( ( 10556001 + 643302 T + 16317 T^{2} + 198 T^{3} + T^{4} )^{2} \)
$83$ \( ( 18954160 - 23048 T^{2} + T^{4} )^{2} \)
$89$ \( 7447769396211600 - 2198590518960 T^{2} + 562726116 T^{4} - 25476 T^{6} + T^{8} \)
$97$ \( ( 972 - 54 T + T^{2} )^{4} \)
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