Properties

Label 39.4.b.a
Level $39$
Weight $4$
Character orbit 39.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.30107449022\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.5054412.1
Defining polynomial: \( x^{4} + 29x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{3} - 7) q^{4} + \beta_{2} q^{5} - 3 \beta_1 q^{6} + 6 \beta_1 q^{7} + (2 \beta_{2} - 9 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{3} - 7) q^{4} + \beta_{2} q^{5} - 3 \beta_1 q^{6} + 6 \beta_1 q^{7} + (2 \beta_{2} - 9 \beta_1) q^{8} + 9 q^{9} + ( - 2 \beta_{3} + 6) q^{10} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - 3 \beta_{3} + 21) q^{12} + (2 \beta_{3} - 3 \beta_{2} - 19) q^{13} + (6 \beta_{3} - 90) q^{14} - 3 \beta_{2} q^{15} + ( - 5 \beta_{3} + 91) q^{16} + 54 q^{17} + 9 \beta_1 q^{18} + ( - 6 \beta_{2} + 6 \beta_1) q^{19} + (4 \beta_{2} + 26 \beta_1) q^{20} - 18 \beta_1 q^{21} + (4 \beta_{3} - 36) q^{22} + ( - 12 \beta_{3} + 36) q^{23} + ( - 6 \beta_{2} + 27 \beta_1) q^{24} + ( - 8 \beta_{3} - 7) q^{25} + (6 \beta_{3} + 4 \beta_{2} - 39 \beta_1 - 18) q^{26} - 27 q^{27} + (12 \beta_{2} - 102 \beta_1) q^{28} + (12 \beta_{3} - 18) q^{29} + (6 \beta_{3} - 18) q^{30} + ( - 6 \beta_{2} - 18 \beta_1) q^{31} + (6 \beta_{2} + 69 \beta_1) q^{32} + (3 \beta_{2} - 6 \beta_1) q^{33} + 54 \beta_1 q^{34} + ( - 12 \beta_{3} + 36) q^{35} + (9 \beta_{3} - 63) q^{36} + (24 \beta_{2} + 48 \beta_1) q^{37} + (18 \beta_{3} - 126) q^{38} + ( - 6 \beta_{3} + 9 \beta_{2} + 57) q^{39} + (2 \beta_{3} - 318) q^{40} + ( - 13 \beta_{2} - 4 \beta_1) q^{41} + ( - 18 \beta_{3} + 270) q^{42} + (12 \beta_{3} + 260) q^{43} - 60 \beta_1 q^{44} + 9 \beta_{2} q^{45} + ( - 24 \beta_{2} + 156 \beta_1) q^{46} + ( - 21 \beta_{2} - 122 \beta_1) q^{47} + (15 \beta_{3} - 273) q^{48} + (36 \beta_{3} - 197) q^{49} + ( - 16 \beta_{2} + 73 \beta_1) q^{50} - 162 q^{51} + ( - 31 \beta_{3} - 12 \beta_{2} - 78 \beta_1 + 457) q^{52} + ( - 36 \beta_{3} - 198) q^{53} - 27 \beta_1 q^{54} + (4 \beta_{3} + 144) q^{55} + ( - 78 \beta_{3} + 882) q^{56} + (18 \beta_{2} - 18 \beta_1) q^{57} + (24 \beta_{2} - 138 \beta_1) q^{58} + ( - 15 \beta_{2} + 34 \beta_1) q^{59} + ( - 12 \beta_{2} - 78 \beta_1) q^{60} + ( - 8 \beta_{3} - 566) q^{61} + ( - 6 \beta_{3} + 234) q^{62} + 54 \beta_1 q^{63} + (17 \beta_{3} - 271) q^{64} + (24 \beta_{3} + 3 \beta_{2} + 52 \beta_1 + 396) q^{65} + ( - 12 \beta_{3} + 108) q^{66} + ( - 12 \beta_{2} + 150 \beta_1) q^{67} + (54 \beta_{3} - 378) q^{68} + (36 \beta_{3} - 108) q^{69} + ( - 24 \beta_{2} + 156 \beta_1) q^{70} + ( - 23 \beta_{2} + 6 \beta_1) q^{71} + (18 \beta_{2} - 81 \beta_1) q^{72} + 54 \beta_{2} q^{73} - 576 q^{74} + (24 \beta_{3} + 21) q^{75} + ( - 12 \beta_{2} - 258 \beta_1) q^{76} + (24 \beta_{3} - 216) q^{77} + ( - 18 \beta_{3} - 12 \beta_{2} + 117 \beta_1 + 54) q^{78} + ( - 32 \beta_{3} + 88) q^{79} + (36 \beta_{2} - 130 \beta_1) q^{80} + 81 q^{81} + (22 \beta_{3} - 18) q^{82} + (25 \beta_{2} + 138 \beta_1) q^{83} + ( - 36 \beta_{2} + 306 \beta_1) q^{84} + 54 \beta_{2} q^{85} + (24 \beta_{2} + 140 \beta_1) q^{86} + ( - 36 \beta_{3} + 54) q^{87} + ( - 28 \beta_{3} + 612) q^{88} + (15 \beta_{2} + 4 \beta_1) q^{89} + ( - 18 \beta_{3} + 54) q^{90} + (36 \beta_{3} + 24 \beta_{2} - 234 \beta_1 - 108) q^{91} + (108 \beta_{3} - 2196) q^{92} + (18 \beta_{2} + 54 \beta_1) q^{93} + ( - 80 \beta_{3} + 1704) q^{94} + (36 \beta_{3} + 828) q^{95} + ( - 18 \beta_{2} - 207 \beta_1) q^{96} + ( - 54 \beta_{2} - 336 \beta_1) q^{97} + (72 \beta_{2} - 557 \beta_1) q^{98} + ( - 9 \beta_{2} + 18 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 26 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 26 q^{4} + 36 q^{9} + 20 q^{10} + 78 q^{12} - 72 q^{13} - 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} - 44 q^{25} - 60 q^{26} - 108 q^{27} - 48 q^{29} - 60 q^{30} + 120 q^{35} - 234 q^{36} - 468 q^{38} + 216 q^{39} - 1268 q^{40} + 1044 q^{42} + 1064 q^{43} - 1062 q^{48} - 716 q^{49} - 648 q^{51} + 1766 q^{52} - 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} - 1050 q^{64} + 1632 q^{65} + 408 q^{66} - 1404 q^{68} - 360 q^{69} - 2304 q^{74} + 132 q^{75} - 816 q^{77} + 180 q^{78} + 288 q^{79} + 324 q^{81} - 28 q^{82} + 144 q^{87} + 2392 q^{88} + 180 q^{90} - 360 q^{91} - 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 29x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 25\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 25\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
5.21898i
1.32750i
1.32750i
5.21898i
5.21898i −3.00000 −19.2377 5.83936i 15.6569i 31.3139i 58.6495i 9.00000 30.4755
25.2 1.32750i −3.00000 6.23774 15.4241i 3.98251i 7.96501i 18.9006i 9.00000 −20.4755
25.3 1.32750i −3.00000 6.23774 15.4241i 3.98251i 7.96501i 18.9006i 9.00000 −20.4755
25.4 5.21898i −3.00000 −19.2377 5.83936i 15.6569i 31.3139i 58.6495i 9.00000 30.4755
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 29T_{2}^{2} + 48 \) acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 29T^{2} + 48 \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 272T^{2} + 8112 \) Copy content Toggle raw display
$7$ \( T^{4} + 1044 T^{2} + 62208 \) Copy content Toggle raw display
$11$ \( T^{4} + 428 T^{2} + 43200 \) Copy content Toggle raw display
$13$ \( T^{4} + 72 T^{3} + 3094 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T - 54)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 11556 T^{2} + \cdots + 31492800 \) Copy content Toggle raw display
$23$ \( (T^{2} - 60 T - 22464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 24 T - 23220)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 17028 T^{2} + \cdots + 47044800 \) Copy content Toggle raw display
$37$ \( T^{4} + 200448 T^{2} + \cdots + 2293235712 \) Copy content Toggle raw display
$41$ \( T^{4} + 45392 T^{2} + \cdots + 128314800 \) Copy content Toggle raw display
$43$ \( (T^{2} - 532 T + 47392)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 500348 T^{2} + \cdots + 62387841792 \) Copy content Toggle raw display
$53$ \( (T^{2} + 432 T - 163620)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 104924 T^{2} + \cdots + 2436066048 \) Copy content Toggle raw display
$61$ \( (T^{2} + 1140 T + 314516)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 727668 T^{2} + \cdots + 143327232 \) Copy content Toggle raw display
$71$ \( T^{4} + 147692 T^{2} + \cdots + 3298756800 \) Copy content Toggle raw display
$73$ \( T^{4} + 793152 T^{2} + \cdots + 68976790272 \) Copy content Toggle raw display
$79$ \( (T^{2} - 144 T - 160960)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 653276 T^{2} + \cdots + 106682723328 \) Copy content Toggle raw display
$89$ \( T^{4} + 60464 T^{2} + \cdots + 249304368 \) Copy content Toggle raw display
$97$ \( T^{4} + 3704256 T^{2} + \cdots + 3383532000000 \) Copy content Toggle raw display
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