Properties

Label 39.2.f.a
Level $39$
Weight $2$
Character orbit 39.f
Analytic conductor $0.311$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.311416567883\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{3} - \zeta_{8}^{2} q^{4} - 2 \zeta_{8} q^{5} + (\zeta_{8}^{2} - \zeta_{8} - 1) q^{6} + (\zeta_{8}^{2} + 1) q^{7} - 3 \zeta_{8}^{3} q^{8} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{3} - \zeta_{8}^{2} q^{4} - 2 \zeta_{8} q^{5} + (\zeta_{8}^{2} - \zeta_{8} - 1) q^{6} + (\zeta_{8}^{2} + 1) q^{7} - 3 \zeta_{8}^{3} q^{8} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{9} - 2 \zeta_{8}^{2} q^{10} + 4 \zeta_{8}^{3} q^{11} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{12} + (3 \zeta_{8}^{2} - 2) q^{13} + (\zeta_{8}^{3} + \zeta_{8}) q^{14} + ( - 2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{15} + q^{16} + ( - 2 \zeta_{8}^{2} - \zeta_{8} + 2) q^{18} + ( - \zeta_{8}^{2} + 1) q^{19} + 2 \zeta_{8}^{3} q^{20} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{21} - 4 q^{22} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{23} + (3 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 3) q^{24} - \zeta_{8}^{2} q^{25} + (3 \zeta_{8}^{3} - 2 \zeta_{8}) q^{26} + (\zeta_{8}^{3} + \zeta_{8} + 5) q^{27} + ( - \zeta_{8}^{2} + 1) q^{28} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{29} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8}) q^{30} + (5 \zeta_{8}^{2} - 5) q^{31} - 5 \zeta_{8} q^{32} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 4) q^{33} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{35} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} - 2 \zeta_{8}) q^{36} + (\zeta_{8}^{2} + 1) q^{37} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{38} + (\zeta_{8}^{3} - 3 \zeta_{8}^{2} - 5 \zeta_{8} + 2) q^{39} - 6 q^{40} - 2 \zeta_{8} q^{41} + ( - \zeta_{8}^{3} - \zeta_{8} - 2) q^{42} - 6 \zeta_{8}^{2} q^{43} + 4 \zeta_{8} q^{44} + (4 \zeta_{8}^{2} + 2 \zeta_{8} - 4) q^{45} + (6 \zeta_{8}^{2} + 6) q^{46} + 4 \zeta_{8}^{3} q^{47} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{48} - 5 \zeta_{8}^{2} q^{49} - \zeta_{8}^{3} q^{50} + (2 \zeta_{8}^{2} + 3) q^{52} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{53} + (\zeta_{8}^{2} + 5 \zeta_{8} - 1) q^{54} + 8 q^{55} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{56} + (\zeta_{8}^{2} + 2 \zeta_{8} - 1) q^{57} + ( - 2 \zeta_{8}^{2} + 2) q^{58} + 4 \zeta_{8}^{3} q^{59} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2) q^{60} + 8 q^{61} + (5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{62} + ( - 4 \zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{63} - 7 \zeta_{8}^{2} q^{64} + ( - 6 \zeta_{8}^{3} + 4 \zeta_{8}) q^{65} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8} + 4) q^{66} + (5 \zeta_{8}^{2} - 5) q^{67} + (6 \zeta_{8}^{3} + 12 \zeta_{8}^{2} - 6 \zeta_{8}) q^{69} + ( - 2 \zeta_{8}^{2} + 2) q^{70} + 4 \zeta_{8} q^{71} + (3 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 6) q^{72} + (\zeta_{8}^{2} + 1) q^{73} + (\zeta_{8}^{3} + \zeta_{8}) q^{74} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{75} + ( - \zeta_{8}^{2} - 1) q^{76} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{77} + ( - 3 \zeta_{8}^{3} - 5 \zeta_{8}^{2} + 2 \zeta_{8} - 1) q^{78} - 10 q^{79} - 2 \zeta_{8} q^{80} + (4 \zeta_{8}^{3} + 4 \zeta_{8} - 7) q^{81} - 2 \zeta_{8}^{2} q^{82} - 8 \zeta_{8} q^{83} + (\zeta_{8}^{2} + 2 \zeta_{8} - 1) q^{84} - 6 \zeta_{8}^{3} q^{86} + (2 \zeta_{8}^{3} + 2 \zeta_{8} + 4) q^{87} + 12 \zeta_{8}^{2} q^{88} - 14 \zeta_{8}^{3} q^{89} + (4 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 4 \zeta_{8}) q^{90} + (\zeta_{8}^{2} - 5) q^{91} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{92} + ( - 5 \zeta_{8}^{2} - 10 \zeta_{8} + 5) q^{93} - 4 q^{94} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{95} + ( - 5 \zeta_{8}^{2} + 5 \zeta_{8} + 5) q^{96} + ( - 7 \zeta_{8}^{2} + 7) q^{97} - 5 \zeta_{8}^{3} q^{98} + ( - 4 \zeta_{8}^{3} + 8 \zeta_{8}^{2} + 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{6} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{6} + 4 q^{7} - 4 q^{9} - 8 q^{13} + 8 q^{15} + 4 q^{16} + 8 q^{18} + 4 q^{19} - 4 q^{21} - 16 q^{22} + 12 q^{24} + 20 q^{27} + 4 q^{28} - 20 q^{31} - 16 q^{33} + 4 q^{37} + 8 q^{39} - 24 q^{40} - 8 q^{42} - 16 q^{45} + 24 q^{46} - 4 q^{48} + 12 q^{52} - 4 q^{54} + 32 q^{55} - 4 q^{57} + 8 q^{58} - 8 q^{60} + 32 q^{61} - 4 q^{63} + 16 q^{66} - 20 q^{67} + 8 q^{70} - 24 q^{72} + 4 q^{73} - 4 q^{76} - 4 q^{78} - 40 q^{79} - 28 q^{81} - 4 q^{84} + 16 q^{87} - 20 q^{91} + 20 q^{93} - 16 q^{94} + 20 q^{96} + 28 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i −1.00000 1.41421i 1.00000i 1.41421 + 1.41421i −0.292893 + 1.70711i 1.00000 + 1.00000i −2.12132 + 2.12132i −1.00000 + 2.82843i 2.00000i
5.2 0.707107 + 0.707107i −1.00000 + 1.41421i 1.00000i −1.41421 1.41421i −1.70711 + 0.292893i 1.00000 + 1.00000i 2.12132 2.12132i −1.00000 2.82843i 2.00000i
8.1 −0.707107 + 0.707107i −1.00000 + 1.41421i 1.00000i 1.41421 1.41421i −0.292893 1.70711i 1.00000 1.00000i −2.12132 2.12132i −1.00000 2.82843i 2.00000i
8.2 0.707107 0.707107i −1.00000 1.41421i 1.00000i −1.41421 + 1.41421i −1.70711 0.292893i 1.00000 1.00000i 2.12132 + 2.12132i −1.00000 + 2.82843i 2.00000i
\(n\): 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 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.f.a 4
3.b odd 2 1 inner 39.2.f.a 4
4.b odd 2 1 624.2.bf.d 4
5.b even 2 1 975.2.o.j 4
5.c odd 4 1 975.2.n.c 4
5.c odd 4 1 975.2.n.d 4
12.b even 2 1 624.2.bf.d 4
13.b even 2 1 507.2.f.a 4
13.c even 3 2 507.2.k.j 8
13.d odd 4 1 inner 39.2.f.a 4
13.d odd 4 1 507.2.f.a 4
13.e even 6 2 507.2.k.i 8
13.f odd 12 2 507.2.k.i 8
13.f odd 12 2 507.2.k.j 8
15.d odd 2 1 975.2.o.j 4
15.e even 4 1 975.2.n.c 4
15.e even 4 1 975.2.n.d 4
39.d odd 2 1 507.2.f.a 4
39.f even 4 1 inner 39.2.f.a 4
39.f even 4 1 507.2.f.a 4
39.h odd 6 2 507.2.k.i 8
39.i odd 6 2 507.2.k.j 8
39.k even 12 2 507.2.k.i 8
39.k even 12 2 507.2.k.j 8
52.f even 4 1 624.2.bf.d 4
65.f even 4 1 975.2.n.d 4
65.g odd 4 1 975.2.o.j 4
65.k even 4 1 975.2.n.c 4
156.l odd 4 1 624.2.bf.d 4
195.j odd 4 1 975.2.n.c 4
195.n even 4 1 975.2.o.j 4
195.u odd 4 1 975.2.n.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.f.a 4 1.a even 1 1 trivial
39.2.f.a 4 3.b odd 2 1 inner
39.2.f.a 4 13.d odd 4 1 inner
39.2.f.a 4 39.f even 4 1 inner
507.2.f.a 4 13.b even 2 1
507.2.f.a 4 13.d odd 4 1
507.2.f.a 4 39.d odd 2 1
507.2.f.a 4 39.f even 4 1
507.2.k.i 8 13.e even 6 2
507.2.k.i 8 13.f odd 12 2
507.2.k.i 8 39.h odd 6 2
507.2.k.i 8 39.k even 12 2
507.2.k.j 8 13.c even 3 2
507.2.k.j 8 13.f odd 12 2
507.2.k.j 8 39.i odd 6 2
507.2.k.j 8 39.k even 12 2
624.2.bf.d 4 4.b odd 2 1
624.2.bf.d 4 12.b even 2 1
624.2.bf.d 4 52.f even 4 1
624.2.bf.d 4 156.l odd 4 1
975.2.n.c 4 5.c odd 4 1
975.2.n.c 4 15.e even 4 1
975.2.n.c 4 65.k even 4 1
975.2.n.c 4 195.j odd 4 1
975.2.n.d 4 5.c odd 4 1
975.2.n.d 4 15.e even 4 1
975.2.n.d 4 65.f even 4 1
975.2.n.d 4 195.u odd 4 1
975.2.o.j 4 5.b even 2 1
975.2.o.j 4 15.d odd 2 1
975.2.o.j 4 65.g odd 4 1
975.2.o.j 4 195.n even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(39, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 256 \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 256 \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 4096 \) Copy content Toggle raw display
$89$ \( T^{4} + 38416 \) Copy content Toggle raw display
$97$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
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