Properties

Label 392.2.i.a
Level 392
Weight 2
Character orbit 392.i
Analytic conductor 3.130
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{5} -\zeta_{6} q^{9} -8 q^{15} + ( 2 - 2 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} -8 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} -4 q^{27} + 2 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{37} -2 q^{41} + 8 q^{43} + ( 4 - 4 \zeta_{6} ) q^{45} + 4 \zeta_{6} q^{47} + 4 \zeta_{6} q^{51} + ( 10 - 10 \zeta_{6} ) q^{53} -4 q^{57} + ( -6 + 6 \zeta_{6} ) q^{59} -4 \zeta_{6} q^{61} + ( 12 - 12 \zeta_{6} ) q^{67} + 16 q^{69} + ( 14 - 14 \zeta_{6} ) q^{73} -22 \zeta_{6} q^{75} + 8 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} + 6 q^{83} + 8 q^{85} + ( -4 + 4 \zeta_{6} ) q^{87} -10 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + ( -8 + 8 \zeta_{6} ) q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} - q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} - q^{9} - 16q^{15} + 2q^{17} + 2q^{19} - 8q^{23} - 11q^{25} - 8q^{27} + 4q^{29} - 4q^{31} + 6q^{37} - 4q^{41} + 16q^{43} + 4q^{45} + 4q^{47} + 4q^{51} + 10q^{53} - 8q^{57} - 6q^{59} - 4q^{61} + 12q^{67} + 32q^{69} + 14q^{73} - 22q^{75} + 8q^{79} + 11q^{81} + 12q^{83} + 16q^{85} - 4q^{87} - 10q^{89} - 8q^{93} - 8q^{95} - 4q^{97} + O(q^{100}) \)

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 −1.00000 1.73205i 0 2.00000 3.46410i 0 0 0 −0.500000 + 0.866025i 0
361.1 0 −1.00000 + 1.73205i 0 2.00000 + 3.46410i 0 0 0 −0.500000 0.866025i 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 392.2.i.a 2
3.b odd 2 1 3528.2.s.a 2
4.b odd 2 1 784.2.i.j 2
7.b odd 2 1 392.2.i.e 2
7.c even 3 1 56.2.a.b 1
7.c even 3 1 inner 392.2.i.a 2
7.d odd 6 1 392.2.a.b 1
7.d odd 6 1 392.2.i.e 2
21.c even 2 1 3528.2.s.ba 2
21.g even 6 1 3528.2.a.b 1
21.g even 6 1 3528.2.s.ba 2
21.h odd 6 1 504.2.a.h 1
21.h odd 6 1 3528.2.s.a 2
28.d even 2 1 784.2.i.b 2
28.f even 6 1 784.2.a.i 1
28.f even 6 1 784.2.i.b 2
28.g odd 6 1 112.2.a.a 1
28.g odd 6 1 784.2.i.j 2
35.i odd 6 1 9800.2.a.bj 1
35.j even 6 1 1400.2.a.a 1
35.l odd 12 2 1400.2.g.b 2
56.j odd 6 1 3136.2.a.w 1
56.k odd 6 1 448.2.a.h 1
56.m even 6 1 3136.2.a.c 1
56.p even 6 1 448.2.a.c 1
77.h odd 6 1 6776.2.a.h 1
84.j odd 6 1 7056.2.a.c 1
84.n even 6 1 1008.2.a.m 1
91.r even 6 1 9464.2.a.h 1
112.u odd 12 2 1792.2.b.h 2
112.w even 12 2 1792.2.b.a 2
140.p odd 6 1 2800.2.a.bd 1
140.w even 12 2 2800.2.g.g 2
168.s odd 6 1 4032.2.a.d 1
168.v even 6 1 4032.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 7.c even 3 1
112.2.a.a 1 28.g odd 6 1
392.2.a.b 1 7.d odd 6 1
392.2.i.a 2 1.a even 1 1 trivial
392.2.i.a 2 7.c even 3 1 inner
392.2.i.e 2 7.b odd 2 1
392.2.i.e 2 7.d odd 6 1
448.2.a.c 1 56.p even 6 1
448.2.a.h 1 56.k odd 6 1
504.2.a.h 1 21.h odd 6 1
784.2.a.i 1 28.f even 6 1
784.2.i.b 2 28.d even 2 1
784.2.i.b 2 28.f even 6 1
784.2.i.j 2 4.b odd 2 1
784.2.i.j 2 28.g odd 6 1
1008.2.a.m 1 84.n even 6 1
1400.2.a.a 1 35.j even 6 1
1400.2.g.b 2 35.l odd 12 2
1792.2.b.a 2 112.w even 12 2
1792.2.b.h 2 112.u odd 12 2
2800.2.a.bd 1 140.p odd 6 1
2800.2.g.g 2 140.w even 12 2
3136.2.a.c 1 56.m even 6 1
3136.2.a.w 1 56.j odd 6 1
3528.2.a.b 1 21.g even 6 1
3528.2.s.a 2 3.b odd 2 1
3528.2.s.a 2 21.h odd 6 1
3528.2.s.ba 2 21.c even 2 1
3528.2.s.ba 2 21.g even 6 1
4032.2.a.a 1 168.v even 6 1
4032.2.a.d 1 168.s odd 6 1
6776.2.a.h 1 77.h odd 6 1
7056.2.a.c 1 84.j odd 6 1
9464.2.a.h 1 91.r even 6 1
9800.2.a.bj 1 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{2} + 2 T_{3} + 4 \)
\( T_{5}^{2} - 4 T_{5} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 4 T - 31 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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