Properties

Label 39.4.e.b
Level $39$
Weight $4$
Character orbit 39.e
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.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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (3 \zeta_{6} - 3) q^{3} + 7 \zeta_{6} q^{4} + 7 q^{5} + 3 \zeta_{6} q^{6} + 10 \zeta_{6} q^{7} + 15 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (3 \zeta_{6} - 3) q^{3} + 7 \zeta_{6} q^{4} + 7 q^{5} + 3 \zeta_{6} q^{6} + 10 \zeta_{6} q^{7} + 15 q^{8} - 9 \zeta_{6} q^{9} + ( - 7 \zeta_{6} + 7) q^{10} + ( - 22 \zeta_{6} + 22) q^{11} - 21 q^{12} + ( - 13 \zeta_{6} - 39) q^{13} + 10 q^{14} + (21 \zeta_{6} - 21) q^{15} + (41 \zeta_{6} - 41) q^{16} - 37 \zeta_{6} q^{17} - 9 q^{18} - 30 \zeta_{6} q^{19} + 49 \zeta_{6} q^{20} - 30 q^{21} - 22 \zeta_{6} q^{22} + ( - 162 \zeta_{6} + 162) q^{23} + (45 \zeta_{6} - 45) q^{24} - 76 q^{25} + (39 \zeta_{6} - 52) q^{26} + 27 q^{27} + (70 \zeta_{6} - 70) q^{28} + ( - 113 \zeta_{6} + 113) q^{29} + 21 \zeta_{6} q^{30} + 196 q^{31} + 161 \zeta_{6} q^{32} + 66 \zeta_{6} q^{33} - 37 q^{34} + 70 \zeta_{6} q^{35} + ( - 63 \zeta_{6} + 63) q^{36} + (13 \zeta_{6} - 13) q^{37} - 30 q^{38} + ( - 117 \zeta_{6} + 156) q^{39} + 105 q^{40} + (285 \zeta_{6} - 285) q^{41} + (30 \zeta_{6} - 30) q^{42} + 246 \zeta_{6} q^{43} + 154 q^{44} - 63 \zeta_{6} q^{45} - 162 \zeta_{6} q^{46} - 462 q^{47} - 123 \zeta_{6} q^{48} + ( - 243 \zeta_{6} + 243) q^{49} + (76 \zeta_{6} - 76) q^{50} + 111 q^{51} + ( - 364 \zeta_{6} + 91) q^{52} - 537 q^{53} + ( - 27 \zeta_{6} + 27) q^{54} + ( - 154 \zeta_{6} + 154) q^{55} + 150 \zeta_{6} q^{56} + 90 q^{57} - 113 \zeta_{6} q^{58} - 576 \zeta_{6} q^{59} - 147 q^{60} + 635 \zeta_{6} q^{61} + ( - 196 \zeta_{6} + 196) q^{62} + ( - 90 \zeta_{6} + 90) q^{63} - 167 q^{64} + ( - 91 \zeta_{6} - 273) q^{65} + 66 q^{66} + (202 \zeta_{6} - 202) q^{67} + ( - 259 \zeta_{6} + 259) q^{68} + 486 \zeta_{6} q^{69} + 70 q^{70} + 1086 \zeta_{6} q^{71} - 135 \zeta_{6} q^{72} - 805 q^{73} + 13 \zeta_{6} q^{74} + ( - 228 \zeta_{6} + 228) q^{75} + ( - 210 \zeta_{6} + 210) q^{76} + 220 q^{77} + ( - 156 \zeta_{6} + 39) q^{78} + 884 q^{79} + (287 \zeta_{6} - 287) q^{80} + (81 \zeta_{6} - 81) q^{81} + 285 \zeta_{6} q^{82} + 518 q^{83} - 210 \zeta_{6} q^{84} - 259 \zeta_{6} q^{85} + 246 q^{86} + 339 \zeta_{6} q^{87} + ( - 330 \zeta_{6} + 330) q^{88} + (194 \zeta_{6} - 194) q^{89} - 63 q^{90} + ( - 520 \zeta_{6} + 130) q^{91} + 1134 q^{92} + (588 \zeta_{6} - 588) q^{93} + (462 \zeta_{6} - 462) q^{94} - 210 \zeta_{6} q^{95} - 483 q^{96} + 1202 \zeta_{6} q^{97} - 243 \zeta_{6} q^{98} - 198 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 3 q^{3} + 7 q^{4} + 14 q^{5} + 3 q^{6} + 10 q^{7} + 30 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 3 q^{3} + 7 q^{4} + 14 q^{5} + 3 q^{6} + 10 q^{7} + 30 q^{8} - 9 q^{9} + 7 q^{10} + 22 q^{11} - 42 q^{12} - 91 q^{13} + 20 q^{14} - 21 q^{15} - 41 q^{16} - 37 q^{17} - 18 q^{18} - 30 q^{19} + 49 q^{20} - 60 q^{21} - 22 q^{22} + 162 q^{23} - 45 q^{24} - 152 q^{25} - 65 q^{26} + 54 q^{27} - 70 q^{28} + 113 q^{29} + 21 q^{30} + 392 q^{31} + 161 q^{32} + 66 q^{33} - 74 q^{34} + 70 q^{35} + 63 q^{36} - 13 q^{37} - 60 q^{38} + 195 q^{39} + 210 q^{40} - 285 q^{41} - 30 q^{42} + 246 q^{43} + 308 q^{44} - 63 q^{45} - 162 q^{46} - 924 q^{47} - 123 q^{48} + 243 q^{49} - 76 q^{50} + 222 q^{51} - 182 q^{52} - 1074 q^{53} + 27 q^{54} + 154 q^{55} + 150 q^{56} + 180 q^{57} - 113 q^{58} - 576 q^{59} - 294 q^{60} + 635 q^{61} + 196 q^{62} + 90 q^{63} - 334 q^{64} - 637 q^{65} + 132 q^{66} - 202 q^{67} + 259 q^{68} + 486 q^{69} + 140 q^{70} + 1086 q^{71} - 135 q^{72} - 1610 q^{73} + 13 q^{74} + 228 q^{75} + 210 q^{76} + 440 q^{77} - 78 q^{78} + 1768 q^{79} - 287 q^{80} - 81 q^{81} + 285 q^{82} + 1036 q^{83} - 210 q^{84} - 259 q^{85} + 492 q^{86} + 339 q^{87} + 330 q^{88} - 194 q^{89} - 126 q^{90} - 260 q^{91} + 2268 q^{92} - 588 q^{93} - 462 q^{94} - 210 q^{95} - 966 q^{96} + 1202 q^{97} - 243 q^{98} - 396 q^{99}+O(q^{100}) \) 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
0.500000 + 0.866025i −1.50000 2.59808i 3.50000 6.06218i 7.00000 1.50000 2.59808i 5.00000 8.66025i 15.0000 −4.50000 + 7.79423i 3.50000 + 6.06218i
22.1 0.500000 0.866025i −1.50000 + 2.59808i 3.50000 + 6.06218i 7.00000 1.50000 + 2.59808i 5.00000 + 8.66025i 15.0000 −4.50000 7.79423i 3.50000 6.06218i
\(n\): 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.b 2
3.b odd 2 1 117.4.g.a 2
4.b odd 2 1 624.4.q.c 2
13.c even 3 1 inner 39.4.e.b 2
13.c even 3 1 507.4.a.b 1
13.e even 6 1 507.4.a.d 1
13.f odd 12 2 507.4.b.d 2
39.h odd 6 1 1521.4.a.e 1
39.i odd 6 1 117.4.g.a 2
39.i odd 6 1 1521.4.a.h 1
52.j odd 6 1 624.4.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.b 2 1.a even 1 1 trivial
39.4.e.b 2 13.c even 3 1 inner
117.4.g.a 2 3.b odd 2 1
117.4.g.a 2 39.i odd 6 1
507.4.a.b 1 13.c even 3 1
507.4.a.d 1 13.e even 6 1
507.4.b.d 2 13.f odd 12 2
624.4.q.c 2 4.b odd 2 1
624.4.q.c 2 52.j odd 6 1
1521.4.a.e 1 39.h odd 6 1
1521.4.a.h 1 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( (T - 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$11$ \( T^{2} - 22T + 484 \) Copy content Toggle raw display
$13$ \( T^{2} + 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 37T + 1369 \) Copy content Toggle raw display
$19$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$23$ \( T^{2} - 162T + 26244 \) Copy content Toggle raw display
$29$ \( T^{2} - 113T + 12769 \) Copy content Toggle raw display
$31$ \( (T - 196)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$41$ \( T^{2} + 285T + 81225 \) Copy content Toggle raw display
$43$ \( T^{2} - 246T + 60516 \) Copy content Toggle raw display
$47$ \( (T + 462)^{2} \) Copy content Toggle raw display
$53$ \( (T + 537)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 576T + 331776 \) Copy content Toggle raw display
$61$ \( T^{2} - 635T + 403225 \) Copy content Toggle raw display
$67$ \( T^{2} + 202T + 40804 \) Copy content Toggle raw display
$71$ \( T^{2} - 1086 T + 1179396 \) Copy content Toggle raw display
$73$ \( (T + 805)^{2} \) Copy content Toggle raw display
$79$ \( (T - 884)^{2} \) Copy content Toggle raw display
$83$ \( (T - 518)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 194T + 37636 \) Copy content Toggle raw display
$97$ \( T^{2} - 1202 T + 1444804 \) Copy content Toggle raw display
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