Properties

Label 39.4.j.b
Level $39$
Weight $4$
Character orbit 39.j
Analytic conductor $2.301$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-17})\)
Defining polynomial: \( x^{4} - 17x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 3 \beta_{2} + 3) q^{3} + 9 \beta_{2} q^{4} + (2 \beta_{3} - 6 \beta_{2} + 3) q^{5} + ( - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - 3 \beta_{3} + 11 \beta_{2} + 3 \beta_1 - 22) q^{7} + \beta_{3} q^{8} - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 3 \beta_{2} + 3) q^{3} + 9 \beta_{2} q^{4} + (2 \beta_{3} - 6 \beta_{2} + 3) q^{5} + ( - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - 3 \beta_{3} + 11 \beta_{2} + 3 \beta_1 - 22) q^{7} + \beta_{3} q^{8} - 9 \beta_{2} q^{9} + ( - 6 \beta_{3} + 34 \beta_{2} + 3 \beta_1 - 34) q^{10} + ( - 21 \beta_{2} + \beta_1 - 21) q^{11} + 27 q^{12} + (12 \beta_{3} + 12 \beta_{2} - 9 \beta_1 + 4) q^{13} + (11 \beta_{3} - 22 \beta_1 + 51) q^{14} + ( - 9 \beta_{2} + 6 \beta_1 - 9) q^{15} + ( - 55 \beta_{2} + 55) q^{16} + (\beta_{3} - 36 \beta_{2} + \beta_1) q^{17} - 9 \beta_{3} q^{18} + ( - 9 \beta_{3} - 37 \beta_{2} + 9 \beta_1 + 74) q^{19} + (18 \beta_{3} - 27 \beta_{2} - 18 \beta_1 + 54) q^{20} + ( - 9 \beta_{3} + 66 \beta_{2} - 33) q^{21} + ( - 21 \beta_{3} + 17 \beta_{2} - 21 \beta_1) q^{22} + (14 \beta_{3} + 69 \beta_{2} - 7 \beta_1 - 69) q^{23} + 3 \beta_1 q^{24} + ( - 12 \beta_{3} + 24 \beta_1 + 30) q^{25} + (12 \beta_{3} + 51 \beta_{2} + 4 \beta_1 - 204) q^{26} - 27 q^{27} + ( - 99 \beta_{2} + 27 \beta_1 - 99) q^{28} + (44 \beta_{3} + 3 \beta_{2} - 22 \beta_1 - 3) q^{29} + ( - 9 \beta_{3} + 102 \beta_{2} - 9 \beta_1) q^{30} + (42 \beta_{3} - 156 \beta_{2} + 78) q^{31} + ( - 63 \beta_{3} + 63 \beta_1) q^{32} + ( - 3 \beta_{3} + 63 \beta_{2} + 3 \beta_1 - 126) q^{33} + ( - 36 \beta_{3} + 34 \beta_{2} - 17) q^{34} + ( - 31 \beta_{3} + 201 \beta_{2} - 31 \beta_1) q^{35} + ( - 81 \beta_{2} + 81) q^{36} + (82 \beta_{2} + 45 \beta_1 + 82) q^{37} + ( - 37 \beta_{3} + 74 \beta_1 + 153) q^{38} + (27 \beta_{3} - 12 \beta_{2} + 9 \beta_1 + 48) q^{39} + ( - 3 \beta_{3} + 6 \beta_1 - 34) q^{40} + (30 \beta_{2} + \beta_1 + 30) q^{41} + (66 \beta_{3} - 153 \beta_{2} - 33 \beta_1 + 153) q^{42} + ( - 15 \beta_{3} + 235 \beta_{2} - 15 \beta_1) q^{43} + (9 \beta_{3} - 378 \beta_{2} + 189) q^{44} + ( - 18 \beta_{3} + 27 \beta_{2} + 18 \beta_1 - 54) q^{45} + (69 \beta_{3} + 119 \beta_{2} - 69 \beta_1 - 238) q^{46} + ( - 79 \beta_{3} - 222 \beta_{2} + 111) q^{47} - 165 \beta_{2} q^{48} + (132 \beta_{3} - 173 \beta_{2} - 66 \beta_1 + 173) q^{49} + (204 \beta_{2} + 30 \beta_1 + 204) q^{50} + ( - 3 \beta_{3} + 6 \beta_1 - 108) q^{51} + (27 \beta_{3} + 144 \beta_{2} - 108 \beta_1 - 108) q^{52} + (18 \beta_{3} - 36 \beta_1 - 567) q^{53} - 27 \beta_1 q^{54} + ( - 90 \beta_{3} + 223 \beta_{2} + 45 \beta_1 - 223) q^{55} + ( - 11 \beta_{3} + 51 \beta_{2} - 11 \beta_1) q^{56} + ( - 27 \beta_{3} - 222 \beta_{2} + 111) q^{57} + (3 \beta_{3} + 374 \beta_{2} - 3 \beta_1 - 748) q^{58} + ( - 8 \beta_{3} - 360 \beta_{2} + 8 \beta_1 + 720) q^{59} + (54 \beta_{3} - 162 \beta_{2} + 81) q^{60} + ( - 87 \beta_{3} - 80 \beta_{2} - 87 \beta_1) q^{61} + ( - 156 \beta_{3} + 714 \beta_{2} + 78 \beta_1 - 714) q^{62} + (99 \beta_{2} - 27 \beta_1 + 99) q^{63} + 631 q^{64} + (50 \beta_{3} - 366 \beta_{2} + 21 \beta_1 - 18) q^{65} + (63 \beta_{3} - 126 \beta_1 + 51) q^{66} + ( - 83 \beta_{2} + 21 \beta_1 - 83) q^{67} + (18 \beta_{3} - 324 \beta_{2} - 9 \beta_1 + 324) q^{68} + (21 \beta_{3} + 207 \beta_{2} + 21 \beta_1) q^{69} + (201 \beta_{3} - 1054 \beta_{2} + 527) q^{70} + ( - 17 \beta_{3} + 219 \beta_{2} + 17 \beta_1 - 438) q^{71} + ( - 9 \beta_{3} + 9 \beta_1) q^{72} + ( - 450 \beta_{2} + 225) q^{73} + (82 \beta_{3} + 765 \beta_{2} + 82 \beta_1) q^{74} + ( - 72 \beta_{3} - 90 \beta_{2} + 36 \beta_1 + 90) q^{75} + (333 \beta_{2} + 81 \beta_1 + 333) q^{76} + (74 \beta_{3} - 148 \beta_1 + 744) q^{77} + ( - 12 \beta_{3} + 612 \beta_{2} + 48 \beta_1 - 459) q^{78} + ( - 126 \beta_{3} + 252 \beta_1 + 2) q^{79} + ( - 165 \beta_{2} + 110 \beta_1 - 165) q^{80} + (81 \beta_{2} - 81) q^{81} + (30 \beta_{3} + 17 \beta_{2} + 30 \beta_1) q^{82} + ( - 241 \beta_{3} + 354 \beta_{2} - 177) q^{83} + ( - 81 \beta_{3} + 297 \beta_{2} + 81 \beta_1 - 594) q^{84} + ( - 81 \beta_{3} + 142 \beta_{2} + 81 \beta_1 - 284) q^{85} + (235 \beta_{3} - 510 \beta_{2} + 255) q^{86} + (66 \beta_{3} + 9 \beta_{2} + 66 \beta_1) q^{87} + ( - 42 \beta_{3} + 17 \beta_{2} + 21 \beta_1 - 17) q^{88} + ( - 42 \beta_{2} - 242 \beta_1 - 42) q^{89} + (27 \beta_{3} - 54 \beta_1 + 306) q^{90} + ( - 243 \beta_{3} + 524 \beta_{2} + 114 \beta_1 - 679) q^{91} + (63 \beta_{3} - 126 \beta_1 - 621) q^{92} + ( - 234 \beta_{2} + 126 \beta_1 - 234) q^{93} + ( - 222 \beta_{3} - 1343 \beta_{2} + 111 \beta_1 + 1343) q^{94} + (47 \beta_{3} - 27 \beta_{2} + 47 \beta_1) q^{95} - 189 \beta_{3} q^{96} + (402 \beta_{3} + 56 \beta_{2} - 402 \beta_1 - 112) q^{97} + ( - 173 \beta_{3} + 1122 \beta_{2} + 173 \beta_1 - 2244) q^{98} + ( - 9 \beta_{3} + 378 \beta_{2} - 189) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 18 q^{4} - 66 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 18 q^{4} - 66 q^{7} - 18 q^{9} - 68 q^{10} - 126 q^{11} + 108 q^{12} + 40 q^{13} + 204 q^{14} - 54 q^{15} + 110 q^{16} - 72 q^{17} + 222 q^{19} + 162 q^{20} + 34 q^{22} - 138 q^{23} + 120 q^{25} - 714 q^{26} - 108 q^{27} - 594 q^{28} - 6 q^{29} + 204 q^{30} - 378 q^{33} + 402 q^{35} + 162 q^{36} + 492 q^{37} + 612 q^{38} + 168 q^{39} - 136 q^{40} + 180 q^{41} + 306 q^{42} + 470 q^{43} - 162 q^{45} - 714 q^{46} - 330 q^{48} + 346 q^{49} + 1224 q^{50} - 432 q^{51} - 144 q^{52} - 2268 q^{53} - 446 q^{55} + 102 q^{56} - 2244 q^{58} + 2160 q^{59} - 160 q^{61} - 1428 q^{62} + 594 q^{63} + 2524 q^{64} - 804 q^{65} + 204 q^{66} - 498 q^{67} + 648 q^{68} + 414 q^{69} - 1314 q^{71} + 1530 q^{74} + 180 q^{75} + 1998 q^{76} + 2976 q^{77} - 612 q^{78} + 8 q^{79} - 990 q^{80} - 162 q^{81} + 34 q^{82} - 1782 q^{84} - 852 q^{85} + 18 q^{87} - 34 q^{88} - 252 q^{89} + 1224 q^{90} - 1668 q^{91} - 2484 q^{92} - 1404 q^{93} + 2686 q^{94} - 54 q^{95} - 336 q^{97} - 6732 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 17x^{2} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 17\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 17\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−3.57071 2.06155i
3.57071 + 2.06155i
−3.57071 + 2.06155i
3.57071 2.06155i
−3.57071 2.06155i 1.50000 2.59808i 4.50000 + 7.79423i 13.4424i −10.7121 + 6.18466i −27.2121 + 15.7109i 4.12311i −4.50000 7.79423i −27.7121 + 47.9988i
4.2 3.57071 + 2.06155i 1.50000 2.59808i 4.50000 + 7.79423i 3.05006i 10.7121 6.18466i −5.78786 + 3.34162i 4.12311i −4.50000 7.79423i −6.28786 + 10.8909i
10.1 −3.57071 + 2.06155i 1.50000 + 2.59808i 4.50000 7.79423i 13.4424i −10.7121 6.18466i −27.2121 15.7109i 4.12311i −4.50000 + 7.79423i −27.7121 47.9988i
10.2 3.57071 2.06155i 1.50000 + 2.59808i 4.50000 7.79423i 3.05006i 10.7121 + 6.18466i −5.78786 3.34162i 4.12311i −4.50000 + 7.79423i −6.28786 10.8909i
\(n\): 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.b 4
3.b odd 2 1 117.4.q.d 4
4.b odd 2 1 624.4.bv.c 4
13.c even 3 1 507.4.b.e 4
13.e even 6 1 inner 39.4.j.b 4
13.e even 6 1 507.4.b.e 4
13.f odd 12 2 507.4.a.k 4
39.h odd 6 1 117.4.q.d 4
39.k even 12 2 1521.4.a.z 4
52.i odd 6 1 624.4.bv.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.b 4 1.a even 1 1 trivial
39.4.j.b 4 13.e even 6 1 inner
117.4.q.d 4 3.b odd 2 1
117.4.q.d 4 39.h odd 6 1
507.4.a.k 4 13.f odd 12 2
507.4.b.e 4 13.c even 3 1
507.4.b.e 4 13.e even 6 1
624.4.bv.c 4 4.b odd 2 1
624.4.bv.c 4 52.i odd 6 1
1521.4.a.z 4 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 17T_{2}^{2} + 289 \) acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 17T^{2} + 289 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 190T^{2} + 1681 \) Copy content Toggle raw display
$7$ \( T^{4} + 66 T^{3} + 1662 T^{2} + \cdots + 44100 \) Copy content Toggle raw display
$11$ \( T^{4} + 126 T^{3} + 6598 T^{2} + \cdots + 1705636 \) Copy content Toggle raw display
$13$ \( T^{4} - 40 T^{3} + 663 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{4} + 72 T^{3} + 3939 T^{2} + \cdots + 1550025 \) Copy content Toggle raw display
$19$ \( T^{4} - 222 T^{3} + 19158 T^{2} + \cdots + 7452900 \) Copy content Toggle raw display
$23$ \( T^{4} + 138 T^{3} + 16782 T^{2} + \cdots + 5116644 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 24711 T^{2} + \cdots + 608855625 \) Copy content Toggle raw display
$31$ \( T^{4} + 96480 T^{2} + \cdots + 137733696 \) Copy content Toggle raw display
$37$ \( T^{4} - 492 T^{3} + \cdots + 203148009 \) Copy content Toggle raw display
$41$ \( T^{4} - 180 T^{3} + 13483 T^{2} + \cdots + 7198489 \) Copy content Toggle raw display
$43$ \( T^{4} - 470 T^{3} + \cdots + 1914062500 \) Copy content Toggle raw display
$47$ \( T^{4} + 286120 T^{2} + \cdots + 4779509956 \) Copy content Toggle raw display
$53$ \( (T^{2} + 1134 T + 304965)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 2160 T^{3} + \cdots + 150320594944 \) Copy content Toggle raw display
$61$ \( T^{4} + 160 T^{3} + \cdots + 144110585161 \) Copy content Toggle raw display
$67$ \( T^{4} + 498 T^{3} + \cdots + 173448900 \) Copy content Toggle raw display
$71$ \( T^{4} + 1314 T^{3} + \cdots + 19312660900 \) Copy content Toggle raw display
$73$ \( (T^{2} + 151875)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 809672)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 2162728 T^{2} + \cdots + 798145692100 \) Copy content Toggle raw display
$89$ \( T^{4} + 252 T^{3} + \cdots + 980686167616 \) Copy content Toggle raw display
$97$ \( T^{4} + 336 T^{3} + \cdots + 7495877379600 \) Copy content Toggle raw display
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