Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.e (of order \(3\), 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{2} + (3 \zeta_{6} - 3) q^{3} - \zeta_{6} q^{4} - 9 q^{5} - 9 \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} - 21 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{2} + (3 \zeta_{6} - 3) q^{3} - \zeta_{6} q^{4} - 9 q^{5} - 9 \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} - 21 q^{8} - 9 \zeta_{6} q^{9} + ( - 27 \zeta_{6} + 27) q^{10} + (30 \zeta_{6} - 30) q^{11} + 3 q^{12} + (39 \zeta_{6} + 13) q^{13} + 6 q^{14} + ( - 27 \zeta_{6} + 27) q^{15} + ( - 71 \zeta_{6} + 71) q^{16} + 111 \zeta_{6} q^{17} + 27 q^{18} + 46 \zeta_{6} q^{19} + 9 \zeta_{6} q^{20} + 6 q^{21} - 90 \zeta_{6} q^{22} + ( - 6 \zeta_{6} + 6) q^{23} + ( - 63 \zeta_{6} + 63) q^{24} - 44 q^{25} + (39 \zeta_{6} - 156) q^{26} + 27 q^{27} + (2 \zeta_{6} - 2) q^{28} + ( - 105 \zeta_{6} + 105) q^{29} + 81 \zeta_{6} q^{30} - 100 q^{31} + 45 \zeta_{6} q^{32} - 90 \zeta_{6} q^{33} - 333 q^{34} + 18 \zeta_{6} q^{35} + (9 \zeta_{6} - 9) q^{36} + (17 \zeta_{6} - 17) q^{37} - 138 q^{38} + (39 \zeta_{6} - 156) q^{39} + 189 q^{40} + ( - 231 \zeta_{6} + 231) q^{41} + (18 \zeta_{6} - 18) q^{42} + 514 \zeta_{6} q^{43} + 30 q^{44} + 81 \zeta_{6} q^{45} + 18 \zeta_{6} q^{46} - 162 q^{47} + 213 \zeta_{6} q^{48} + ( - 339 \zeta_{6} + 339) q^{49} + ( - 132 \zeta_{6} + 132) q^{50} - 333 q^{51} + ( - 52 \zeta_{6} + 39) q^{52} + 639 q^{53} + (81 \zeta_{6} - 81) q^{54} + ( - 270 \zeta_{6} + 270) q^{55} + 42 \zeta_{6} q^{56} - 138 q^{57} + 315 \zeta_{6} q^{58} - 600 \zeta_{6} q^{59} - 27 q^{60} - 233 \zeta_{6} q^{61} + ( - 300 \zeta_{6} + 300) q^{62} + (18 \zeta_{6} - 18) q^{63} + 433 q^{64} + ( - 351 \zeta_{6} - 117) q^{65} + 270 q^{66} + (926 \zeta_{6} - 926) q^{67} + ( - 111 \zeta_{6} + 111) q^{68} + 18 \zeta_{6} q^{69} - 54 q^{70} + 930 \zeta_{6} q^{71} + 189 \zeta_{6} q^{72} - 253 q^{73} - 51 \zeta_{6} q^{74} + ( - 132 \zeta_{6} + 132) q^{75} + ( - 46 \zeta_{6} + 46) q^{76} + 60 q^{77} + ( - 468 \zeta_{6} + 351) q^{78} - 1324 q^{79} + (639 \zeta_{6} - 639) q^{80} + (81 \zeta_{6} - 81) q^{81} + 693 \zeta_{6} q^{82} + 810 q^{83} - 6 \zeta_{6} q^{84} - 999 \zeta_{6} q^{85} - 1542 q^{86} + 315 \zeta_{6} q^{87} + ( - 630 \zeta_{6} + 630) q^{88} + (498 \zeta_{6} - 498) q^{89} - 243 q^{90} + ( - 104 \zeta_{6} + 78) q^{91} - 6 q^{92} + ( - 300 \zeta_{6} + 300) q^{93} + ( - 486 \zeta_{6} + 486) q^{94} - 414 \zeta_{6} q^{95} - 135 q^{96} - 1358 \zeta_{6} q^{97} + 1017 \zeta_{6} q^{98} + 270 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 3 q^{3} - q^{4} - 18 q^{5} - 9 q^{6} - 2 q^{7} - 42 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 3 q^{3} - q^{4} - 18 q^{5} - 9 q^{6} - 2 q^{7} - 42 q^{8} - 9 q^{9} + 27 q^{10} - 30 q^{11} + 6 q^{12} + 65 q^{13} + 12 q^{14} + 27 q^{15} + 71 q^{16} + 111 q^{17} + 54 q^{18} + 46 q^{19} + 9 q^{20} + 12 q^{21} - 90 q^{22} + 6 q^{23} + 63 q^{24} - 88 q^{25} - 273 q^{26} + 54 q^{27} - 2 q^{28} + 105 q^{29} + 81 q^{30} - 200 q^{31} + 45 q^{32} - 90 q^{33} - 666 q^{34} + 18 q^{35} - 9 q^{36} - 17 q^{37} - 276 q^{38} - 273 q^{39} + 378 q^{40} + 231 q^{41} - 18 q^{42} + 514 q^{43} + 60 q^{44} + 81 q^{45} + 18 q^{46} - 324 q^{47} + 213 q^{48} + 339 q^{49} + 132 q^{50} - 666 q^{51} + 26 q^{52} + 1278 q^{53} - 81 q^{54} + 270 q^{55} + 42 q^{56} - 276 q^{57} + 315 q^{58} - 600 q^{59} - 54 q^{60} - 233 q^{61} + 300 q^{62} - 18 q^{63} + 866 q^{64} - 585 q^{65} + 540 q^{66} - 926 q^{67} + 111 q^{68} + 18 q^{69} - 108 q^{70} + 930 q^{71} + 189 q^{72} - 506 q^{73} - 51 q^{74} + 132 q^{75} + 46 q^{76} + 120 q^{77} + 234 q^{78} - 2648 q^{79} - 639 q^{80} - 81 q^{81} + 693 q^{82} + 1620 q^{83} - 6 q^{84} - 999 q^{85} - 3084 q^{86} + 315 q^{87} + 630 q^{88} - 498 q^{89} - 486 q^{90} + 52 q^{91} - 12 q^{92} + 300 q^{93} + 486 q^{94} - 414 q^{95} - 270 q^{96} - 1358 q^{97} + 1017 q^{98} + 540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.50000 2.59808i −1.50000 2.59808i −0.500000 + 0.866025i −9.00000 −4.50000 + 7.79423i −1.00000 + 1.73205i −21.0000 −4.50000 + 7.79423i 13.5000 + 23.3827i
22.1 −1.50000 + 2.59808i −1.50000 + 2.59808i −0.500000 0.866025i −9.00000 −4.50000 7.79423i −1.00000 1.73205i −21.0000 −4.50000 7.79423i 13.5000 23.3827i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( (T + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$13$ \( T^{2} - 65T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 111T + 12321 \) Copy content Toggle raw display
$19$ \( T^{2} - 46T + 2116 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 105T + 11025 \) Copy content Toggle raw display
$31$ \( (T + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$41$ \( T^{2} - 231T + 53361 \) Copy content Toggle raw display
$43$ \( T^{2} - 514T + 264196 \) Copy content Toggle raw display
$47$ \( (T + 162)^{2} \) Copy content Toggle raw display
$53$ \( (T - 639)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 600T + 360000 \) Copy content Toggle raw display
$61$ \( T^{2} + 233T + 54289 \) Copy content Toggle raw display
$67$ \( T^{2} + 926T + 857476 \) Copy content Toggle raw display
$71$ \( T^{2} - 930T + 864900 \) Copy content Toggle raw display
$73$ \( (T + 253)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1324)^{2} \) Copy content Toggle raw display
$83$ \( (T - 810)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 498T + 248004 \) Copy content Toggle raw display
$97$ \( T^{2} + 1358 T + 1844164 \) Copy content Toggle raw display
show more
show less