Properties

Label 2352.2.q.j
Level $2352$
Weight $2$
Character orbit 2352.q
Analytic conductor $18.781$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} + 2 q^{13} - 2 q^{15} + ( - 6 \zeta_{6} + 6) q^{17} + 4 \zeta_{6} q^{19} - 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + q^{27} + 6 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + 10 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{39} + 10 q^{41} - 12 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 8 \zeta_{6} q^{47} + 6 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{53} - 4 q^{57} + (4 \zeta_{6} - 4) q^{59} - 10 \zeta_{6} q^{61} + 4 \zeta_{6} q^{65} + ( - 12 \zeta_{6} + 12) q^{67} + 4 q^{69} - 4 q^{71} + ( - 2 \zeta_{6} + 2) q^{73} + \zeta_{6} q^{75} + 8 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 4 q^{83} + 12 q^{85} + (6 \zeta_{6} - 6) q^{87} + 6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{93} + (8 \zeta_{6} - 8) q^{95} - 10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{5} - q^{9} + 4 q^{13} - 4 q^{15} + 6 q^{17} + 4 q^{19} - 4 q^{23} + q^{25} + 2 q^{27} + 12 q^{29} + 8 q^{31} + 10 q^{37} - 2 q^{39} + 20 q^{41} - 24 q^{43} + 2 q^{45} + 8 q^{47} + 6 q^{51} - 6 q^{53} - 8 q^{57} - 4 q^{59} - 10 q^{61} + 4 q^{65} + 12 q^{67} + 8 q^{69} - 8 q^{71} + 2 q^{73} + q^{75} + 8 q^{79} - q^{81} + 8 q^{83} + 24 q^{85} - 6 q^{87} + 6 q^{89} + 8 q^{93} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i 0 1.00000 1.73205i 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 −0.500000 + 0.866025i 0 1.00000 + 1.73205i 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 2352.2.q.j 2
4.b odd 2 1 1176.2.q.j 2
7.b odd 2 1 2352.2.q.o 2
7.c even 3 1 2352.2.a.q 1
7.c even 3 1 inner 2352.2.q.j 2
7.d odd 6 1 336.2.a.c 1
7.d odd 6 1 2352.2.q.o 2
12.b even 2 1 3528.2.s.h 2
21.g even 6 1 1008.2.a.e 1
21.h odd 6 1 7056.2.a.br 1
28.d even 2 1 1176.2.q.b 2
28.f even 6 1 168.2.a.b 1
28.f even 6 1 1176.2.q.b 2
28.g odd 6 1 1176.2.a.a 1
28.g odd 6 1 1176.2.q.j 2
35.i odd 6 1 8400.2.a.bx 1
56.j odd 6 1 1344.2.a.n 1
56.k odd 6 1 9408.2.a.cy 1
56.m even 6 1 1344.2.a.c 1
56.p even 6 1 9408.2.a.bc 1
84.h odd 2 1 3528.2.s.v 2
84.j odd 6 1 504.2.a.b 1
84.j odd 6 1 3528.2.s.v 2
84.n even 6 1 3528.2.a.w 1
84.n even 6 1 3528.2.s.h 2
112.v even 12 2 5376.2.c.bd 2
112.x odd 12 2 5376.2.c.f 2
140.s even 6 1 4200.2.a.i 1
140.x odd 12 2 4200.2.t.m 2
168.ba even 6 1 4032.2.a.bj 1
168.be odd 6 1 4032.2.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.b 1 28.f even 6 1
336.2.a.c 1 7.d odd 6 1
504.2.a.b 1 84.j odd 6 1
1008.2.a.e 1 21.g even 6 1
1176.2.a.a 1 28.g odd 6 1
1176.2.q.b 2 28.d even 2 1
1176.2.q.b 2 28.f even 6 1
1176.2.q.j 2 4.b odd 2 1
1176.2.q.j 2 28.g odd 6 1
1344.2.a.c 1 56.m even 6 1
1344.2.a.n 1 56.j odd 6 1
2352.2.a.q 1 7.c even 3 1
2352.2.q.j 2 1.a even 1 1 trivial
2352.2.q.j 2 7.c even 3 1 inner
2352.2.q.o 2 7.b odd 2 1
2352.2.q.o 2 7.d odd 6 1
3528.2.a.w 1 84.n even 6 1
3528.2.s.h 2 12.b even 2 1
3528.2.s.h 2 84.n even 6 1
3528.2.s.v 2 84.h odd 2 1
3528.2.s.v 2 84.j odd 6 1
4032.2.a.be 1 168.be odd 6 1
4032.2.a.bj 1 168.ba even 6 1
4200.2.a.i 1 140.s even 6 1
4200.2.t.m 2 140.x odd 12 2
5376.2.c.f 2 112.x odd 12 2
5376.2.c.bd 2 112.v even 12 2
7056.2.a.br 1 21.h odd 6 1
8400.2.a.bx 1 35.i odd 6 1
9408.2.a.bc 1 56.p even 6 1
9408.2.a.cy 1 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{2} - 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( (T + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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