Properties

Label 39.4.j.a
Level $39$
Weight $4$
Character orbit 39.j
Analytic conductor $2.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.30107449022\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{3} - 8 \zeta_{6} q^{4} + ( - 6 \zeta_{6} + 3) q^{5} + ( - 6 \zeta_{6} + 12) q^{7} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{3} - 8 \zeta_{6} q^{4} + ( - 6 \zeta_{6} + 3) q^{5} + ( - 6 \zeta_{6} + 12) q^{7} - 9 \zeta_{6} q^{9} + (30 \zeta_{6} + 30) q^{11} - 24 q^{12} + ( - 39 \zeta_{6} - 13) q^{13} + ( - 9 \zeta_{6} - 9) q^{15} + (64 \zeta_{6} - 64) q^{16} + 117 \zeta_{6} q^{17} + (14 \zeta_{6} - 28) q^{19} + (24 \zeta_{6} - 48) q^{20} + ( - 36 \zeta_{6} + 18) q^{21} + (18 \zeta_{6} - 18) q^{23} + 98 q^{25} - 27 q^{27} + ( - 48 \zeta_{6} - 48) q^{28} + ( - 99 \zeta_{6} + 99) q^{29} + ( - 224 \zeta_{6} + 112) q^{31} + ( - 90 \zeta_{6} + 180) q^{33} - 54 \zeta_{6} q^{35} + (72 \zeta_{6} - 72) q^{36} + (65 \zeta_{6} + 65) q^{37} + (39 \zeta_{6} - 156) q^{39} + ( - 21 \zeta_{6} - 21) q^{41} + 82 \zeta_{6} q^{43} + ( - 480 \zeta_{6} + 240) q^{44} + (27 \zeta_{6} - 54) q^{45} + (84 \zeta_{6} - 42) q^{47} + 192 \zeta_{6} q^{48} + (235 \zeta_{6} - 235) q^{49} + 351 q^{51} + (416 \zeta_{6} - 312) q^{52} - 261 q^{53} + ( - 270 \zeta_{6} + 270) q^{55} + (84 \zeta_{6} - 42) q^{57} + (456 \zeta_{6} - 912) q^{59} + (144 \zeta_{6} - 72) q^{60} + 719 \zeta_{6} q^{61} + ( - 54 \zeta_{6} - 54) q^{63} + 512 q^{64} + (195 \zeta_{6} - 273) q^{65} + ( - 406 \zeta_{6} - 406) q^{67} + ( - 936 \zeta_{6} + 936) q^{68} + 54 \zeta_{6} q^{69} + (270 \zeta_{6} - 540) q^{71} + ( - 790 \zeta_{6} + 395) q^{73} + ( - 294 \zeta_{6} + 294) q^{75} + (112 \zeta_{6} + 112) q^{76} + 540 q^{77} - 440 q^{79} + (192 \zeta_{6} + 192) q^{80} + (81 \zeta_{6} - 81) q^{81} + ( - 1380 \zeta_{6} + 690) q^{83} + (144 \zeta_{6} - 288) q^{84} + ( - 351 \zeta_{6} + 702) q^{85} - 297 \zeta_{6} q^{87} + (876 \zeta_{6} + 876) q^{89} + ( - 156 \zeta_{6} - 390) q^{91} + 144 q^{92} + ( - 336 \zeta_{6} - 336) q^{93} + 126 \zeta_{6} q^{95} + (668 \zeta_{6} - 1336) q^{97} + ( - 540 \zeta_{6} + 270) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 8 q^{4} + 18 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 8 q^{4} + 18 q^{7} - 9 q^{9} + 90 q^{11} - 48 q^{12} - 65 q^{13} - 27 q^{15} - 64 q^{16} + 117 q^{17} - 42 q^{19} - 72 q^{20} - 18 q^{23} + 196 q^{25} - 54 q^{27} - 144 q^{28} + 99 q^{29} + 270 q^{33} - 54 q^{35} - 72 q^{36} + 195 q^{37} - 273 q^{39} - 63 q^{41} + 82 q^{43} - 81 q^{45} + 192 q^{48} - 235 q^{49} + 702 q^{51} - 208 q^{52} - 522 q^{53} + 270 q^{55} - 1368 q^{59} + 719 q^{61} - 162 q^{63} + 1024 q^{64} - 351 q^{65} - 1218 q^{67} + 936 q^{68} + 54 q^{69} - 810 q^{71} + 294 q^{75} + 336 q^{76} + 1080 q^{77} - 880 q^{79} + 576 q^{80} - 81 q^{81} - 432 q^{84} + 1053 q^{85} - 297 q^{87} + 2628 q^{89} - 936 q^{91} + 288 q^{92} - 1008 q^{93} + 126 q^{95} - 2004 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 2.59808i −4.00000 6.92820i 5.19615i 0 9.00000 5.19615i 0 −4.50000 7.79423i 0
10.1 0 1.50000 + 2.59808i −4.00000 + 6.92820i 5.19615i 0 9.00000 + 5.19615i 0 −4.50000 + 7.79423i 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
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.4.j.a 2
3.b odd 2 1 117.4.q.b 2
4.b odd 2 1 624.4.bv.a 2
13.c even 3 1 507.4.b.a 2
13.e even 6 1 inner 39.4.j.a 2
13.e even 6 1 507.4.b.a 2
13.f odd 12 2 507.4.a.g 2
39.h odd 6 1 117.4.q.b 2
39.k even 12 2 1521.4.a.m 2
52.i odd 6 1 624.4.bv.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.a 2 1.a even 1 1 trivial
39.4.j.a 2 13.e even 6 1 inner
117.4.q.b 2 3.b odd 2 1
117.4.q.b 2 39.h odd 6 1
507.4.a.g 2 13.f odd 12 2
507.4.b.a 2 13.c even 3 1
507.4.b.a 2 13.e even 6 1
624.4.bv.a 2 4.b odd 2 1
624.4.bv.a 2 52.i odd 6 1
1521.4.a.m 2 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 27 \) Copy content Toggle raw display
$7$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$11$ \( T^{2} - 90T + 2700 \) Copy content Toggle raw display
$13$ \( T^{2} + 65T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 117T + 13689 \) Copy content Toggle raw display
$19$ \( T^{2} + 42T + 588 \) Copy content Toggle raw display
$23$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$29$ \( T^{2} - 99T + 9801 \) Copy content Toggle raw display
$31$ \( T^{2} + 37632 \) Copy content Toggle raw display
$37$ \( T^{2} - 195T + 12675 \) Copy content Toggle raw display
$41$ \( T^{2} + 63T + 1323 \) Copy content Toggle raw display
$43$ \( T^{2} - 82T + 6724 \) Copy content Toggle raw display
$47$ \( T^{2} + 5292 \) Copy content Toggle raw display
$53$ \( (T + 261)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1368 T + 623808 \) Copy content Toggle raw display
$61$ \( T^{2} - 719T + 516961 \) Copy content Toggle raw display
$67$ \( T^{2} + 1218 T + 494508 \) Copy content Toggle raw display
$71$ \( T^{2} + 810T + 218700 \) Copy content Toggle raw display
$73$ \( T^{2} + 468075 \) Copy content Toggle raw display
$79$ \( (T + 440)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1428300 \) Copy content Toggle raw display
$89$ \( T^{2} - 2628 T + 2302128 \) Copy content Toggle raw display
$97$ \( T^{2} + 2004 T + 1338672 \) Copy content Toggle raw display
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