Properties

Label 3724.1.bb.b
Level $3724$
Weight $1$
Character orbit 3724.bb
Analytic conductor $1.859$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3724.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.85851810705\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.283024.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{4} -\zeta_{12}^{4} q^{6} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{4} -\zeta_{12}^{4} q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{3} q^{11} + \zeta_{12}^{3} q^{12} -\zeta_{12}^{4} q^{13} -\zeta_{12}^{2} q^{16} + \zeta_{12}^{2} q^{17} -\zeta_{12}^{5} q^{19} -\zeta_{12}^{2} q^{22} + \zeta_{12} q^{23} -\zeta_{12}^{2} q^{24} -\zeta_{12}^{4} q^{25} + \zeta_{12}^{3} q^{26} -\zeta_{12}^{3} q^{27} + \zeta_{12}^{4} q^{29} -\zeta_{12}^{3} q^{31} + \zeta_{12} q^{32} -\zeta_{12}^{2} q^{33} -\zeta_{12} q^{34} + q^{37} + \zeta_{12}^{4} q^{38} + \zeta_{12}^{3} q^{39} -\zeta_{12}^{2} q^{41} + \zeta_{12}^{5} q^{43} + \zeta_{12} q^{44} - q^{46} -\zeta_{12} q^{47} + \zeta_{12} q^{48} + \zeta_{12}^{3} q^{50} -\zeta_{12} q^{51} -\zeta_{12}^{2} q^{52} + \zeta_{12}^{2} q^{54} + \zeta_{12}^{4} q^{57} -\zeta_{12}^{3} q^{58} -\zeta_{12}^{5} q^{59} + \zeta_{12}^{4} q^{61} + \zeta_{12}^{2} q^{62} - q^{64} + \zeta_{12} q^{66} + q^{68} - q^{69} -\zeta_{12}^{5} q^{71} + \zeta_{12}^{2} q^{73} + \zeta_{12}^{5} q^{74} + \zeta_{12}^{3} q^{75} -\zeta_{12}^{3} q^{76} -\zeta_{12}^{2} q^{78} + \zeta_{12}^{2} q^{81} + \zeta_{12} q^{82} -\zeta_{12}^{4} q^{86} -\zeta_{12}^{3} q^{87} - q^{88} -\zeta_{12}^{4} q^{89} -\zeta_{12}^{5} q^{92} + \zeta_{12}^{2} q^{93} + q^{94} - q^{96} + \zeta_{12}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 2q^{6} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{6} + 2q^{13} - 2q^{16} + 2q^{17} - 2q^{22} - 2q^{24} + 2q^{25} - 2q^{29} - 2q^{33} + 4q^{37} - 2q^{38} - 2q^{41} - 4q^{46} - 2q^{52} + 2q^{54} - 2q^{57} - 2q^{61} + 2q^{62} - 4q^{64} + 4q^{68} - 4q^{69} + 2q^{73} - 2q^{78} + 2q^{81} + 2q^{86} - 4q^{88} + 2q^{89} + 2q^{93} + 4q^{94} - 4q^{96} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times\).

\(n\) \(1863\) \(3041\) \(3137\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2059.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 0 1.00000i 0 0
2059.2 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 0 1.00000i 0 0
3431.1 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.00000i 0 0
3431.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.00000i 0 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
4.b odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3724.1.bb.b 4
4.b odd 2 1 inner 3724.1.bb.b 4
7.b odd 2 1 3724.1.bb.a 4
7.c even 3 1 532.1.n.a 4
7.c even 3 1 532.1.bk.a yes 4
7.d odd 6 1 3724.1.n.a 4
7.d odd 6 1 3724.1.bk.a 4
19.c even 3 1 inner 3724.1.bb.b 4
28.d even 2 1 3724.1.bb.a 4
28.f even 6 1 3724.1.n.a 4
28.f even 6 1 3724.1.bk.a 4
28.g odd 6 1 532.1.n.a 4
28.g odd 6 1 532.1.bk.a yes 4
76.g odd 6 1 inner 3724.1.bb.b 4
133.g even 3 1 532.1.bk.a yes 4
133.h even 3 1 532.1.n.a 4
133.k odd 6 1 3724.1.bk.a 4
133.m odd 6 1 3724.1.bb.a 4
133.t odd 6 1 3724.1.n.a 4
532.n odd 6 1 532.1.bk.a yes 4
532.r even 6 1 3724.1.n.a 4
532.u even 6 1 3724.1.bb.a 4
532.bk odd 6 1 532.1.n.a 4
532.bn even 6 1 3724.1.bk.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.1.n.a 4 7.c even 3 1
532.1.n.a 4 28.g odd 6 1
532.1.n.a 4 133.h even 3 1
532.1.n.a 4 532.bk odd 6 1
532.1.bk.a yes 4 7.c even 3 1
532.1.bk.a yes 4 28.g odd 6 1
532.1.bk.a yes 4 133.g even 3 1
532.1.bk.a yes 4 532.n odd 6 1
3724.1.n.a 4 7.d odd 6 1
3724.1.n.a 4 28.f even 6 1
3724.1.n.a 4 133.t odd 6 1
3724.1.n.a 4 532.r even 6 1
3724.1.bb.a 4 7.b odd 2 1
3724.1.bb.a 4 28.d even 2 1
3724.1.bb.a 4 133.m odd 6 1
3724.1.bb.a 4 532.u even 6 1
3724.1.bb.b 4 1.a even 1 1 trivial
3724.1.bb.b 4 4.b odd 2 1 inner
3724.1.bb.b 4 19.c even 3 1 inner
3724.1.bb.b 4 76.g odd 6 1 inner
3724.1.bk.a 4 7.d odd 6 1
3724.1.bk.a 4 28.f even 6 1
3724.1.bk.a 4 133.k odd 6 1
3724.1.bk.a 4 532.bn even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} - T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3724, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 1 - T + T^{2} )^{2} \)
$17$ \( ( 1 - T + T^{2} )^{2} \)
$19$ \( 1 - T^{2} + T^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 1 + T + T^{2} )^{2} \)
$31$ \( ( 1 + T^{2} )^{2} \)
$37$ \( ( -1 + T )^{4} \)
$41$ \( ( 1 + T + T^{2} )^{2} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( 1 - T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( 1 - T^{2} + T^{4} \)
$61$ \( ( 1 + T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( 1 - T^{2} + T^{4} \)
$73$ \( ( 1 - T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 1 - T + T^{2} )^{2} \)
$97$ \( ( 1 - T + T^{2} )^{2} \)
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