Properties

Label 36.3.g.a
Level $36$
Weight $3$
Character orbit 36.g
Analytic conductor $0.981$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.980928951697\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1 + 1) q^{3} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} + 7 \beta_1 - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1 + 1) q^{3} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} + 7 \beta_1 - 7) q^{9} + ( - 2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 7) q^{11} + ( - 3 \beta_{3} + 4 \beta_1) q^{13} + (3 \beta_{3} - 12 \beta_1 - 6) q^{15} + (2 \beta_{3} - \beta_{2} + 14 \beta_1 - 8) q^{17} + (3 \beta_{2} + 2) q^{19} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 25) q^{21} + ( - \beta_{3} + 2 \beta_{2} - 16 \beta_1 + 34) q^{23} + (9 \beta_{3} - 9 \beta_{2} - 11 \beta_1 + 2) q^{25} + ( - 6 \beta_{2} + 24 \beta_1 - 15) q^{27} + (7 \beta_{3} + 7 \beta_{2} - 14 \beta_1 - 7) q^{29} + (9 \beta_{3} - 8 \beta_1) q^{31} + ( - 12 \beta_{3} + 3 \beta_{2} - 27 \beta_1 + 9) q^{33} + ( - 10 \beta_{3} + 5 \beta_{2} + 56 \beta_1 - 23) q^{35} + ( - 12 \beta_{2} - 22) q^{37} + (\beta_{3} - 4 \beta_{2} - 24 \beta_1 + 24) q^{39} + ( - 8 \beta_{3} + 16 \beta_{2} + 7 \beta_1 + 2) q^{41} + (23 \beta_1 - 23) q^{43} + ( - 15 \beta_{3} + 12 \beta_{2} + 30 \beta_1 - 30) q^{45} + ( - 3 \beta_{3} - 3 \beta_{2} - 12 \beta_1 - 15) q^{47} + (3 \beta_{3} - 27 \beta_1) q^{49} + (9 \beta_{3} - 15 \beta_{2} + 15 \beta_1 - 24) q^{51} + (16 \beta_{3} - 8 \beta_{2} - 32 \beta_1 + 8) q^{53} + (9 \beta_{2} + 27) q^{55} + ( - \beta_{3} + 3 \beta_{2} + 25 \beta_1 + 2) q^{57} + (8 \beta_{3} - 16 \beta_{2} - 25 \beta_1 + 34) q^{59} + ( - 9 \beta_{3} + 9 \beta_{2} - 16 \beta_1 + 25) q^{61} + (23 \beta_{3} - \beta_{2} - 40 \beta_1 + 49) q^{63} + ( - 7 \beta_{3} - 7 \beta_{2} + 32 \beta_1 + 25) q^{65} + ( - 18 \beta_{3} + 67 \beta_1) q^{67} + (15 \beta_{3} + 18 \beta_{2} - 24 \beta_1 + 42) q^{69} + ( - 4 \beta_{3} + 2 \beta_{2} - 28 \beta_1 + 16) q^{71} + (9 \beta_{2} + 26) q^{73} + (9 \beta_{3} + 2 \beta_{2} - 11 \beta_1 - 70) q^{75} + (17 \beta_{3} - 34 \beta_{2} + 38 \beta_1 - 110) q^{77} + ( - 15 \beta_{3} + 15 \beta_{2} - 34 \beta_1 + 49) q^{79} + (15 \beta_{3} - 30 \beta_{2} - 39 \beta_1 - 15) q^{81} + ( - 3 \beta_{3} - 3 \beta_{2} - 12 \beta_1 - 15) q^{83} + ( - 18 \beta_{3} + 18 \beta_1) q^{85} + ( - 21 \beta_{3} + 21 \beta_{2} + 126 \beta_1 - 63) q^{87} + ( - 16 \beta_{3} + 8 \beta_{2} - 112 \beta_1 + 64) q^{89} + (9 \beta_{2} - 71) q^{91} + (\beta_{3} + 8 \beta_{2} + 72 \beta_1 - 72) q^{93} + ( - 4 \beta_{3} + 8 \beta_{2} + 26 \beta_1 - 44) q^{95} + ( - 6 \beta_{3} + 6 \beta_{2} + 101 \beta_1 - 95) q^{97} + ( - 33 \beta_{3} + 30 \beta_{2} - 78 \beta_1 + 105) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + 9 q^{5} - q^{7} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} + 9 q^{5} - q^{7} - 15 q^{9} - 36 q^{11} + 5 q^{13} - 45 q^{15} + 2 q^{19} + 99 q^{21} + 99 q^{23} + 13 q^{25} - 63 q^{29} - 7 q^{31} - 36 q^{33} - 64 q^{37} + 57 q^{39} - 18 q^{41} - 46 q^{43} - 99 q^{45} - 81 q^{47} - 51 q^{49} - 27 q^{51} + 90 q^{55} + 51 q^{57} + 126 q^{59} + 41 q^{61} + 141 q^{63} + 171 q^{65} + 116 q^{67} + 99 q^{69} + 86 q^{73} - 297 q^{75} - 279 q^{77} + 83 q^{79} - 63 q^{81} - 81 q^{83} + 18 q^{85} - 63 q^{87} - 302 q^{91} - 159 q^{93} - 144 q^{95} - 196 q^{97} + 171 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 −0.686141 + 2.92048i 0 6.55842 + 3.78651i 0 −4.55842 7.89542i 0 −8.05842 4.00772i 0
5.2 0 2.18614 2.05446i 0 −2.05842 1.18843i 0 4.05842 + 7.02939i 0 0.558422 8.98266i 0
29.1 0 −0.686141 2.92048i 0 6.55842 3.78651i 0 −4.55842 + 7.89542i 0 −8.05842 + 4.00772i 0
29.2 0 2.18614 + 2.05446i 0 −2.05842 + 1.18843i 0 4.05842 7.02939i 0 0.558422 + 8.98266i 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
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.3.g.a 4
3.b odd 2 1 108.3.g.a 4
4.b odd 2 1 144.3.q.b 4
5.b even 2 1 900.3.p.a 4
5.c odd 4 2 900.3.u.a 8
8.b even 2 1 576.3.q.d 4
8.d odd 2 1 576.3.q.g 4
9.c even 3 1 108.3.g.a 4
9.c even 3 1 324.3.c.b 4
9.d odd 6 1 inner 36.3.g.a 4
9.d odd 6 1 324.3.c.b 4
12.b even 2 1 432.3.q.b 4
15.d odd 2 1 2700.3.p.b 4
15.e even 4 2 2700.3.u.b 8
24.f even 2 1 1728.3.q.h 4
24.h odd 2 1 1728.3.q.g 4
36.f odd 6 1 432.3.q.b 4
36.f odd 6 1 1296.3.e.e 4
36.h even 6 1 144.3.q.b 4
36.h even 6 1 1296.3.e.e 4
45.h odd 6 1 900.3.p.a 4
45.j even 6 1 2700.3.p.b 4
45.k odd 12 2 2700.3.u.b 8
45.l even 12 2 900.3.u.a 8
72.j odd 6 1 576.3.q.d 4
72.l even 6 1 576.3.q.g 4
72.n even 6 1 1728.3.q.g 4
72.p odd 6 1 1728.3.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 1.a even 1 1 trivial
36.3.g.a 4 9.d odd 6 1 inner
108.3.g.a 4 3.b odd 2 1
108.3.g.a 4 9.c even 3 1
144.3.q.b 4 4.b odd 2 1
144.3.q.b 4 36.h even 6 1
324.3.c.b 4 9.c even 3 1
324.3.c.b 4 9.d odd 6 1
432.3.q.b 4 12.b even 2 1
432.3.q.b 4 36.f odd 6 1
576.3.q.d 4 8.b even 2 1
576.3.q.d 4 72.j odd 6 1
576.3.q.g 4 8.d odd 2 1
576.3.q.g 4 72.l even 6 1
900.3.p.a 4 5.b even 2 1
900.3.p.a 4 45.h odd 6 1
900.3.u.a 8 5.c odd 4 2
900.3.u.a 8 45.l even 12 2
1296.3.e.e 4 36.f odd 6 1
1296.3.e.e 4 36.h even 6 1
1728.3.q.g 4 24.h odd 2 1
1728.3.q.g 4 72.n even 6 1
1728.3.q.h 4 24.f even 2 1
1728.3.q.h 4 72.p odd 6 1
2700.3.p.b 4 15.d odd 2 1
2700.3.p.b 4 45.j even 6 1
2700.3.u.b 8 15.e even 4 2
2700.3.u.b 8 45.k odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + 12 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} - 9 T^{3} + 9 T^{2} + 162 T + 324 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + 75 T^{2} - 74 T + 5476 \) Copy content Toggle raw display
$11$ \( T^{4} + 36 T^{3} + 441 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} - 5 T^{3} + 93 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$17$ \( T^{4} + 387 T^{2} + 20736 \) Copy content Toggle raw display
$19$ \( (T^{2} - T - 74)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 99 T^{3} + 4059 T^{2} + \cdots + 627264 \) Copy content Toggle raw display
$29$ \( T^{4} + 63 T^{3} + 441 T^{2} + \cdots + 777924 \) Copy content Toggle raw display
$31$ \( T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336 \) Copy content Toggle raw display
$37$ \( (T^{2} + 32 T - 932)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} - 1449 T^{2} + \cdots + 2424249 \) Copy content Toggle raw display
$43$ \( (T^{2} + 23 T + 529)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 81 T^{3} + 2511 T^{2} + \cdots + 104976 \) Copy content Toggle raw display
$53$ \( T^{4} + 4032 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$59$ \( T^{4} - 126 T^{3} + 5031 T^{2} + \cdots + 68121 \) Copy content Toggle raw display
$61$ \( T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504 \) Copy content Toggle raw display
$67$ \( T^{4} - 116 T^{3} + 12765 T^{2} + \cdots + 477481 \) Copy content Toggle raw display
$71$ \( T^{4} + 1548 T^{2} + 331776 \) Copy content Toggle raw display
$73$ \( (T^{2} - 43 T - 206)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 83 T^{3} + 7023 T^{2} + \cdots + 17956 \) Copy content Toggle raw display
$83$ \( T^{4} + 81 T^{3} + 2511 T^{2} + \cdots + 104976 \) Copy content Toggle raw display
$89$ \( T^{4} + 24768 T^{2} + \cdots + 84934656 \) Copy content Toggle raw display
$97$ \( T^{4} + 196 T^{3} + \cdots + 86620249 \) Copy content Toggle raw display
show more
show less