Properties

Label 392.2.i.f
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} -6 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} -6 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} -2 q^{13} -3 q^{15} + ( 3 - 3 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + 3 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} -9 q^{27} -6 q^{29} + ( -1 + \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + 5 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{39} + 10 q^{41} -4 q^{43} + ( -6 + 6 \zeta_{6} ) q^{45} + \zeta_{6} q^{47} -9 \zeta_{6} q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} - q^{55} + 15 q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + 3 \zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + ( -11 + 11 \zeta_{6} ) q^{67} + 9 q^{69} + 16 q^{71} + ( 7 - 7 \zeta_{6} ) q^{73} -12 \zeta_{6} q^{75} + 11 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 4 q^{83} -3 q^{85} + ( -18 + 18 \zeta_{6} ) q^{87} -9 \zeta_{6} q^{89} + 3 \zeta_{6} q^{93} + ( 5 - 5 \zeta_{6} ) q^{95} -6 q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} - q^{5} - 6q^{9} + O(q^{10}) \) \( 2q + 3q^{3} - q^{5} - 6q^{9} + q^{11} - 4q^{13} - 6q^{15} + 3q^{17} + 5q^{19} + 3q^{23} + 4q^{25} - 18q^{27} - 12q^{29} - q^{31} - 3q^{33} + 5q^{37} - 6q^{39} + 20q^{41} - 8q^{43} - 6q^{45} + q^{47} - 9q^{51} + 9q^{53} - 2q^{55} + 30q^{57} + 3q^{59} + 3q^{61} + 2q^{65} - 11q^{67} + 18q^{69} + 32q^{71} + 7q^{73} - 12q^{75} + 11q^{79} - 9q^{81} + 8q^{83} - 6q^{85} - 18q^{87} - 9q^{89} + 3q^{93} + 5q^{95} - 12q^{97} - 12q^{99} + O(q^{100}) \)

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 1.50000 + 2.59808i 0 −0.500000 + 0.866025i 0 0 0 −3.00000 + 5.19615i 0
361.1 0 1.50000 2.59808i 0 −0.500000 0.866025i 0 0 0 −3.00000 5.19615i 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.f 2
3.b odd 2 1 3528.2.s.o 2
4.b odd 2 1 784.2.i.a 2
7.b odd 2 1 56.2.i.a 2
7.c even 3 1 392.2.a.a 1
7.c even 3 1 inner 392.2.i.f 2
7.d odd 6 1 56.2.i.a 2
7.d odd 6 1 392.2.a.f 1
21.c even 2 1 504.2.s.e 2
21.g even 6 1 504.2.s.e 2
21.g even 6 1 3528.2.a.r 1
21.h odd 6 1 3528.2.a.k 1
21.h odd 6 1 3528.2.s.o 2
28.d even 2 1 112.2.i.c 2
28.f even 6 1 112.2.i.c 2
28.f even 6 1 784.2.a.a 1
28.g odd 6 1 784.2.a.j 1
28.g odd 6 1 784.2.i.a 2
35.c odd 2 1 1400.2.q.g 2
35.f even 4 2 1400.2.bh.f 4
35.i odd 6 1 1400.2.q.g 2
35.i odd 6 1 9800.2.a.b 1
35.j even 6 1 9800.2.a.bp 1
35.k even 12 2 1400.2.bh.f 4
56.e even 2 1 448.2.i.a 2
56.h odd 2 1 448.2.i.f 2
56.j odd 6 1 448.2.i.f 2
56.j odd 6 1 3136.2.a.b 1
56.k odd 6 1 3136.2.a.a 1
56.m even 6 1 448.2.i.a 2
56.m even 6 1 3136.2.a.bc 1
56.p even 6 1 3136.2.a.bb 1
84.h odd 2 1 1008.2.s.e 2
84.j odd 6 1 1008.2.s.e 2
84.j odd 6 1 7056.2.a.bi 1
84.n even 6 1 7056.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 7.b odd 2 1
56.2.i.a 2 7.d odd 6 1
112.2.i.c 2 28.d even 2 1
112.2.i.c 2 28.f even 6 1
392.2.a.a 1 7.c even 3 1
392.2.a.f 1 7.d odd 6 1
392.2.i.f 2 1.a even 1 1 trivial
392.2.i.f 2 7.c even 3 1 inner
448.2.i.a 2 56.e even 2 1
448.2.i.a 2 56.m even 6 1
448.2.i.f 2 56.h odd 2 1
448.2.i.f 2 56.j odd 6 1
504.2.s.e 2 21.c even 2 1
504.2.s.e 2 21.g even 6 1
784.2.a.a 1 28.f even 6 1
784.2.a.j 1 28.g odd 6 1
784.2.i.a 2 4.b odd 2 1
784.2.i.a 2 28.g odd 6 1
1008.2.s.e 2 84.h odd 2 1
1008.2.s.e 2 84.j odd 6 1
1400.2.q.g 2 35.c odd 2 1
1400.2.q.g 2 35.i odd 6 1
1400.2.bh.f 4 35.f even 4 2
1400.2.bh.f 4 35.k even 12 2
3136.2.a.a 1 56.k odd 6 1
3136.2.a.b 1 56.j odd 6 1
3136.2.a.bb 1 56.p even 6 1
3136.2.a.bc 1 56.m even 6 1
3528.2.a.k 1 21.h odd 6 1
3528.2.a.r 1 21.g even 6 1
3528.2.s.o 2 3.b odd 2 1
3528.2.s.o 2 21.h odd 6 1
7056.2.a.s 1 84.n even 6 1
7056.2.a.bi 1 84.j odd 6 1
9800.2.a.b 1 35.i odd 6 1
9800.2.a.bp 1 35.j even 6 1

Hecke kernels

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

\( T_{3}^{2} - 3 T_{3} + 9 \)
\( T_{5}^{2} + T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} ) \)
$5$ \( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \)
$37$ \( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \)
$79$ \( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 6 T + 97 T^{2} )^{2} \)
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