Properties

Label 392.6.i.e
Level $392$
Weight $6$
Character orbit 392.i
Analytic conductor $62.870$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 20 \zeta_{6} + 20) q^{3} - 74 \zeta_{6} q^{5} - 157 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 20 \zeta_{6} + 20) q^{3} - 74 \zeta_{6} q^{5} - 157 \zeta_{6} q^{9} + (124 \zeta_{6} - 124) q^{11} - 478 q^{13} - 1480 q^{15} + (1198 \zeta_{6} - 1198) q^{17} + 3044 \zeta_{6} q^{19} - 184 \zeta_{6} q^{23} + (2351 \zeta_{6} - 2351) q^{25} + 1720 q^{27} - 3282 q^{29} + (5728 \zeta_{6} - 5728) q^{31} + 2480 \zeta_{6} q^{33} - 10326 \zeta_{6} q^{37} + (9560 \zeta_{6} - 9560) q^{39} + 8886 q^{41} - 9188 q^{43} + (11618 \zeta_{6} - 11618) q^{45} + 23664 \zeta_{6} q^{47} + 23960 \zeta_{6} q^{51} + (11686 \zeta_{6} - 11686) q^{53} + 9176 q^{55} + 60880 q^{57} + ( - 16876 \zeta_{6} + 16876) q^{59} - 18482 \zeta_{6} q^{61} + 35372 \zeta_{6} q^{65} + ( - 15532 \zeta_{6} + 15532) q^{67} - 3680 q^{69} - 31960 q^{71} + (4886 \zeta_{6} - 4886) q^{73} + 47020 \zeta_{6} q^{75} - 44560 \zeta_{6} q^{79} + ( - 72551 \zeta_{6} + 72551) q^{81} - 67364 q^{83} + 88652 q^{85} + (65640 \zeta_{6} - 65640) q^{87} + 71994 \zeta_{6} q^{89} + 114560 \zeta_{6} q^{93} + ( - 225256 \zeta_{6} + 225256) q^{95} - 48866 q^{97} + 19468 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{3} - 74 q^{5} - 157 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{3} - 74 q^{5} - 157 q^{9} - 124 q^{11} - 956 q^{13} - 2960 q^{15} - 1198 q^{17} + 3044 q^{19} - 184 q^{23} - 2351 q^{25} + 3440 q^{27} - 6564 q^{29} - 5728 q^{31} + 2480 q^{33} - 10326 q^{37} - 9560 q^{39} + 17772 q^{41} - 18376 q^{43} - 11618 q^{45} + 23664 q^{47} + 23960 q^{51} - 11686 q^{53} + 18352 q^{55} + 121760 q^{57} + 16876 q^{59} - 18482 q^{61} + 35372 q^{65} + 15532 q^{67} - 7360 q^{69} - 63920 q^{71} - 4886 q^{73} + 47020 q^{75} - 44560 q^{79} + 72551 q^{81} - 134728 q^{83} + 177304 q^{85} - 65640 q^{87} + 71994 q^{89} + 114560 q^{93} + 225256 q^{95} - 97732 q^{97} + 38936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.500000 0.866025i
0.500000 + 0.866025i
0 10.0000 + 17.3205i 0 −37.0000 + 64.0859i 0 0 0 −78.5000 + 135.966i 0
361.1 0 10.0000 17.3205i 0 −37.0000 64.0859i 0 0 0 −78.5000 135.966i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.6.i.e 2
7.b odd 2 1 392.6.i.b 2
7.c even 3 1 392.6.a.b 1
7.c even 3 1 inner 392.6.i.e 2
7.d odd 6 1 8.6.a.a 1
7.d odd 6 1 392.6.i.b 2
21.g even 6 1 72.6.a.f 1
28.f even 6 1 16.6.a.a 1
28.g odd 6 1 784.6.a.l 1
35.i odd 6 1 200.6.a.a 1
35.k even 12 2 200.6.c.a 2
56.j odd 6 1 64.6.a.a 1
56.m even 6 1 64.6.a.g 1
77.i even 6 1 968.6.a.a 1
84.j odd 6 1 144.6.a.k 1
112.v even 12 2 256.6.b.d 2
112.x odd 12 2 256.6.b.f 2
140.s even 6 1 400.6.a.l 1
140.x odd 12 2 400.6.c.d 2
168.ba even 6 1 576.6.a.g 1
168.be odd 6 1 576.6.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 7.d odd 6 1
16.6.a.a 1 28.f even 6 1
64.6.a.a 1 56.j odd 6 1
64.6.a.g 1 56.m even 6 1
72.6.a.f 1 21.g even 6 1
144.6.a.k 1 84.j odd 6 1
200.6.a.a 1 35.i odd 6 1
200.6.c.a 2 35.k even 12 2
256.6.b.d 2 112.v even 12 2
256.6.b.f 2 112.x odd 12 2
392.6.a.b 1 7.c even 3 1
392.6.i.b 2 7.b odd 2 1
392.6.i.b 2 7.d odd 6 1
392.6.i.e 2 1.a even 1 1 trivial
392.6.i.e 2 7.c even 3 1 inner
400.6.a.l 1 140.s even 6 1
400.6.c.d 2 140.x odd 12 2
576.6.a.g 1 168.ba even 6 1
576.6.a.h 1 168.be odd 6 1
784.6.a.l 1 28.g odd 6 1
968.6.a.a 1 77.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 20T_{3} + 400 \) acting on \(S_{6}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 20T + 400 \) Copy content Toggle raw display
$5$ \( T^{2} + 74T + 5476 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 124T + 15376 \) Copy content Toggle raw display
$13$ \( (T + 478)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1198 T + 1435204 \) Copy content Toggle raw display
$19$ \( T^{2} - 3044 T + 9265936 \) Copy content Toggle raw display
$23$ \( T^{2} + 184T + 33856 \) Copy content Toggle raw display
$29$ \( (T + 3282)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5728 T + 32809984 \) Copy content Toggle raw display
$37$ \( T^{2} + 10326 T + 106626276 \) Copy content Toggle raw display
$41$ \( (T - 8886)^{2} \) Copy content Toggle raw display
$43$ \( (T + 9188)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 23664 T + 559984896 \) Copy content Toggle raw display
$53$ \( T^{2} + 11686 T + 136562596 \) Copy content Toggle raw display
$59$ \( T^{2} - 16876 T + 284799376 \) Copy content Toggle raw display
$61$ \( T^{2} + 18482 T + 341584324 \) Copy content Toggle raw display
$67$ \( T^{2} - 15532 T + 241243024 \) Copy content Toggle raw display
$71$ \( (T + 31960)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4886 T + 23872996 \) Copy content Toggle raw display
$79$ \( T^{2} + 44560 T + 1985593600 \) Copy content Toggle raw display
$83$ \( (T + 67364)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 71994 T + 5183136036 \) Copy content Toggle raw display
$97$ \( (T + 48866)^{2} \) Copy content Toggle raw display
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