Properties

Label 39.2.k.a
Level $39$
Weight $2$
Character orbit 39.k
Analytic conductor $0.311$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.311416567883\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}) q^{13} + 4 \zeta_{12}^{2} q^{16} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 5 \zeta_{12} + 5) q^{19} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 4) q^{21} - 5 \zeta_{12}^{3} q^{25} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} - 6) q^{28} + (\zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} - 1) q^{31} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} + (7 \zeta_{12}^{3} + 7 \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{37} + ( - 2 \zeta_{12}^{2} - 5) q^{39} + ( - \zeta_{12}^{2} - 1) q^{43} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + ( - 7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 7 \zeta_{12} + 6) q^{49} + ( - 2 \zeta_{12}^{2} + 8) q^{52} + (\zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12} - 1) q^{57} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{61} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 3 \zeta_{12} + 9) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 9 \zeta_{12} + 7) q^{67} + ( - 9 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 9) q^{73} + (5 \zeta_{12}^{2} - 10) q^{75} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 10 \zeta_{12} - 4) q^{76} + (7 \zeta_{12}^{3} - 14 \zeta_{12}) q^{79} - 9 \zeta_{12}^{2} q^{81} + (2 \zeta_{12}^{3} + 10 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{84} + (9 \zeta_{12}^{3} + 11 \zeta_{12}^{2} - 10 \zeta_{12} - 5) q^{91} + (11 \zeta_{12}^{3} - 11 \zeta_{12}^{2} - 4 \zeta_{12} + 7) q^{93} + ( - 11 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{7} - 6 q^{9} + 8 q^{16} + 14 q^{19} + 18 q^{21} - 20 q^{28} - 14 q^{31} - 2 q^{37} - 24 q^{39} - 6 q^{43} + 18 q^{49} + 28 q^{52} + 12 q^{57} + 24 q^{63} + 32 q^{67} - 34 q^{73} - 30 q^{75} - 28 q^{76} - 18 q^{81} - 12 q^{84} + 2 q^{91} + 6 q^{93} - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 1.50000i 1.73205 + 1.00000i 0 0 −4.23205 + 1.13397i 0 −1.50000 + 2.59808i 0
11.1 0 0.866025 + 1.50000i −1.73205 1.00000i 0 0 −0.767949 2.86603i 0 −1.50000 + 2.59808i 0
20.1 0 −0.866025 + 1.50000i 1.73205 1.00000i 0 0 −4.23205 1.13397i 0 −1.50000 2.59808i 0
32.1 0 0.866025 1.50000i −1.73205 + 1.00000i 0 0 −0.767949 + 2.86603i 0 −1.50000 2.59808i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.k.a 4
3.b odd 2 1 CM 39.2.k.a 4
4.b odd 2 1 624.2.cn.b 4
5.b even 2 1 975.2.bo.c 4
5.c odd 4 1 975.2.bp.a 4
5.c odd 4 1 975.2.bp.d 4
12.b even 2 1 624.2.cn.b 4
13.b even 2 1 507.2.k.c 4
13.c even 3 1 507.2.f.c 4
13.c even 3 1 507.2.k.b 4
13.d odd 4 1 507.2.k.a 4
13.d odd 4 1 507.2.k.b 4
13.e even 6 1 507.2.f.b 4
13.e even 6 1 507.2.k.a 4
13.f odd 12 1 inner 39.2.k.a 4
13.f odd 12 1 507.2.f.b 4
13.f odd 12 1 507.2.f.c 4
13.f odd 12 1 507.2.k.c 4
15.d odd 2 1 975.2.bo.c 4
15.e even 4 1 975.2.bp.a 4
15.e even 4 1 975.2.bp.d 4
39.d odd 2 1 507.2.k.c 4
39.f even 4 1 507.2.k.a 4
39.f even 4 1 507.2.k.b 4
39.h odd 6 1 507.2.f.b 4
39.h odd 6 1 507.2.k.a 4
39.i odd 6 1 507.2.f.c 4
39.i odd 6 1 507.2.k.b 4
39.k even 12 1 inner 39.2.k.a 4
39.k even 12 1 507.2.f.b 4
39.k even 12 1 507.2.f.c 4
39.k even 12 1 507.2.k.c 4
52.l even 12 1 624.2.cn.b 4
65.o even 12 1 975.2.bp.a 4
65.s odd 12 1 975.2.bo.c 4
65.t even 12 1 975.2.bp.d 4
156.v odd 12 1 624.2.cn.b 4
195.bc odd 12 1 975.2.bp.d 4
195.bh even 12 1 975.2.bo.c 4
195.bn odd 12 1 975.2.bp.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.a 4 1.a even 1 1 trivial
39.2.k.a 4 3.b odd 2 1 CM
39.2.k.a 4 13.f odd 12 1 inner
39.2.k.a 4 39.k even 12 1 inner
507.2.f.b 4 13.e even 6 1
507.2.f.b 4 13.f odd 12 1
507.2.f.b 4 39.h odd 6 1
507.2.f.b 4 39.k even 12 1
507.2.f.c 4 13.c even 3 1
507.2.f.c 4 13.f odd 12 1
507.2.f.c 4 39.i odd 6 1
507.2.f.c 4 39.k even 12 1
507.2.k.a 4 13.d odd 4 1
507.2.k.a 4 13.e even 6 1
507.2.k.a 4 39.f even 4 1
507.2.k.a 4 39.h odd 6 1
507.2.k.b 4 13.c even 3 1
507.2.k.b 4 13.d odd 4 1
507.2.k.b 4 39.f even 4 1
507.2.k.b 4 39.i odd 6 1
507.2.k.c 4 13.b even 2 1
507.2.k.c 4 13.f odd 12 1
507.2.k.c 4 39.d odd 2 1
507.2.k.c 4 39.k even 12 1
624.2.cn.b 4 4.b odd 2 1
624.2.cn.b 4 12.b even 2 1
624.2.cn.b 4 52.l even 12 1
624.2.cn.b 4 156.v odd 12 1
975.2.bo.c 4 5.b even 2 1
975.2.bo.c 4 15.d odd 2 1
975.2.bo.c 4 65.s odd 12 1
975.2.bo.c 4 195.bh even 12 1
975.2.bp.a 4 5.c odd 4 1
975.2.bp.a 4 15.e even 4 1
975.2.bp.a 4 65.o even 12 1
975.2.bp.a 4 195.bn odd 12 1
975.2.bp.d 4 5.c odd 4 1
975.2.bp.d 4 15.e even 4 1
975.2.bp.d 4 65.t even 12 1
975.2.bp.d 4 195.bc odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + 41 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 14 T^{3} + 50 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + 122 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$67$ \( T^{4} - 32 T^{3} + 281 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 34 T^{3} + 578 T^{2} + \cdots + 20449 \) Copy content Toggle raw display
$79$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 38 T^{3} + 557 T^{2} + \cdots + 28561 \) Copy content Toggle raw display
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