Properties

Label 36.2.e.a
Level $36$
Weight $2$
Character orbit 36.e
Analytic conductor $0.287$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.287461447277\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 1) q^{3} + (3 \zeta_{6} - 3) q^{5} + \zeta_{6} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 1) q^{3} + (3 \zeta_{6} - 3) q^{5} + \zeta_{6} q^{7} - 3 q^{9} - 3 \zeta_{6} q^{11} + ( - \zeta_{6} + 1) q^{13} + (3 \zeta_{6} + 3) q^{15} + 6 q^{17} - 4 q^{19} + ( - \zeta_{6} + 2) q^{21} + ( - 3 \zeta_{6} + 3) q^{23} - 4 \zeta_{6} q^{25} + (6 \zeta_{6} - 3) q^{27} - 3 \zeta_{6} q^{29} + (5 \zeta_{6} - 5) q^{31} + (3 \zeta_{6} - 6) q^{33} - 3 q^{35} + 2 q^{37} + ( - \zeta_{6} - 1) q^{39} + (3 \zeta_{6} - 3) q^{41} + \zeta_{6} q^{43} + ( - 9 \zeta_{6} + 9) q^{45} + 9 \zeta_{6} q^{47} + ( - 6 \zeta_{6} + 6) q^{49} + ( - 12 \zeta_{6} + 6) q^{51} - 6 q^{53} + 9 q^{55} + (8 \zeta_{6} - 4) q^{57} + ( - 3 \zeta_{6} + 3) q^{59} + 13 \zeta_{6} q^{61} - 3 \zeta_{6} q^{63} + 3 \zeta_{6} q^{65} + ( - 7 \zeta_{6} + 7) q^{67} + ( - 3 \zeta_{6} - 3) q^{69} - 12 q^{71} - 10 q^{73} + (4 \zeta_{6} - 8) q^{75} + ( - 3 \zeta_{6} + 3) q^{77} - 11 \zeta_{6} q^{79} + 9 q^{81} + 9 \zeta_{6} q^{83} + (18 \zeta_{6} - 18) q^{85} + (3 \zeta_{6} - 6) q^{87} + 6 q^{89} + q^{91} + (5 \zeta_{6} + 5) q^{93} + ( - 12 \zeta_{6} + 12) q^{95} - 11 \zeta_{6} q^{97} + 9 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} + q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} + q^{7} - 6 q^{9} - 3 q^{11} + q^{13} + 9 q^{15} + 12 q^{17} - 8 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{29} - 5 q^{31} - 9 q^{33} - 6 q^{35} + 4 q^{37} - 3 q^{39} - 3 q^{41} + q^{43} + 9 q^{45} + 9 q^{47} + 6 q^{49} - 12 q^{53} + 18 q^{55} + 3 q^{59} + 13 q^{61} - 3 q^{63} + 3 q^{65} + 7 q^{67} - 9 q^{69} - 24 q^{71} - 20 q^{73} - 12 q^{75} + 3 q^{77} - 11 q^{79} + 18 q^{81} + 9 q^{83} - 18 q^{85} - 9 q^{87} + 12 q^{89} + 2 q^{91} + 15 q^{93} + 12 q^{95} - 11 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 −1.50000 + 2.59808i 0 0.500000 + 0.866025i 0 −3.00000 0
25.1 0 1.73205i 0 −1.50000 2.59808i 0 0.500000 0.866025i 0 −3.00000 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.2.e.a 2
3.b odd 2 1 108.2.e.a 2
4.b odd 2 1 144.2.i.a 2
5.b even 2 1 900.2.i.b 2
5.c odd 4 2 900.2.s.b 4
7.b odd 2 1 1764.2.j.b 2
7.c even 3 1 1764.2.i.a 2
7.c even 3 1 1764.2.l.c 2
7.d odd 6 1 1764.2.i.c 2
7.d odd 6 1 1764.2.l.a 2
8.b even 2 1 576.2.i.f 2
8.d odd 2 1 576.2.i.e 2
9.c even 3 1 inner 36.2.e.a 2
9.c even 3 1 324.2.a.c 1
9.d odd 6 1 108.2.e.a 2
9.d odd 6 1 324.2.a.a 1
12.b even 2 1 432.2.i.c 2
15.d odd 2 1 2700.2.i.b 2
15.e even 4 2 2700.2.s.b 4
21.c even 2 1 5292.2.j.a 2
21.g even 6 1 5292.2.i.a 2
21.g even 6 1 5292.2.l.c 2
21.h odd 6 1 5292.2.i.c 2
21.h odd 6 1 5292.2.l.a 2
24.f even 2 1 1728.2.i.c 2
24.h odd 2 1 1728.2.i.d 2
36.f odd 6 1 144.2.i.a 2
36.f odd 6 1 1296.2.a.k 1
36.h even 6 1 432.2.i.c 2
36.h even 6 1 1296.2.a.b 1
45.h odd 6 1 2700.2.i.b 2
45.h odd 6 1 8100.2.a.g 1
45.j even 6 1 900.2.i.b 2
45.j even 6 1 8100.2.a.j 1
45.k odd 12 2 900.2.s.b 4
45.k odd 12 2 8100.2.d.h 2
45.l even 12 2 2700.2.s.b 4
45.l even 12 2 8100.2.d.c 2
63.g even 3 1 1764.2.i.a 2
63.h even 3 1 1764.2.l.c 2
63.i even 6 1 5292.2.l.c 2
63.j odd 6 1 5292.2.l.a 2
63.k odd 6 1 1764.2.i.c 2
63.l odd 6 1 1764.2.j.b 2
63.n odd 6 1 5292.2.i.c 2
63.o even 6 1 5292.2.j.a 2
63.s even 6 1 5292.2.i.a 2
63.t odd 6 1 1764.2.l.a 2
72.j odd 6 1 1728.2.i.d 2
72.j odd 6 1 5184.2.a.ba 1
72.l even 6 1 1728.2.i.c 2
72.l even 6 1 5184.2.a.bb 1
72.n even 6 1 576.2.i.f 2
72.n even 6 1 5184.2.a.e 1
72.p odd 6 1 576.2.i.e 2
72.p odd 6 1 5184.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 1.a even 1 1 trivial
36.2.e.a 2 9.c even 3 1 inner
108.2.e.a 2 3.b odd 2 1
108.2.e.a 2 9.d odd 6 1
144.2.i.a 2 4.b odd 2 1
144.2.i.a 2 36.f odd 6 1
324.2.a.a 1 9.d odd 6 1
324.2.a.c 1 9.c even 3 1
432.2.i.c 2 12.b even 2 1
432.2.i.c 2 36.h even 6 1
576.2.i.e 2 8.d odd 2 1
576.2.i.e 2 72.p odd 6 1
576.2.i.f 2 8.b even 2 1
576.2.i.f 2 72.n even 6 1
900.2.i.b 2 5.b even 2 1
900.2.i.b 2 45.j even 6 1
900.2.s.b 4 5.c odd 4 2
900.2.s.b 4 45.k odd 12 2
1296.2.a.b 1 36.h even 6 1
1296.2.a.k 1 36.f odd 6 1
1728.2.i.c 2 24.f even 2 1
1728.2.i.c 2 72.l even 6 1
1728.2.i.d 2 24.h odd 2 1
1728.2.i.d 2 72.j odd 6 1
1764.2.i.a 2 7.c even 3 1
1764.2.i.a 2 63.g even 3 1
1764.2.i.c 2 7.d odd 6 1
1764.2.i.c 2 63.k odd 6 1
1764.2.j.b 2 7.b odd 2 1
1764.2.j.b 2 63.l odd 6 1
1764.2.l.a 2 7.d odd 6 1
1764.2.l.a 2 63.t odd 6 1
1764.2.l.c 2 7.c even 3 1
1764.2.l.c 2 63.h even 3 1
2700.2.i.b 2 15.d odd 2 1
2700.2.i.b 2 45.h odd 6 1
2700.2.s.b 4 15.e even 4 2
2700.2.s.b 4 45.l even 12 2
5184.2.a.e 1 72.n even 6 1
5184.2.a.f 1 72.p odd 6 1
5184.2.a.ba 1 72.j odd 6 1
5184.2.a.bb 1 72.l even 6 1
5292.2.i.a 2 21.g even 6 1
5292.2.i.a 2 63.s even 6 1
5292.2.i.c 2 21.h odd 6 1
5292.2.i.c 2 63.n odd 6 1
5292.2.j.a 2 21.c even 2 1
5292.2.j.a 2 63.o even 6 1
5292.2.l.a 2 21.h odd 6 1
5292.2.l.a 2 63.j odd 6 1
5292.2.l.c 2 21.g even 6 1
5292.2.l.c 2 63.i even 6 1
8100.2.a.g 1 45.h odd 6 1
8100.2.a.j 1 45.j even 6 1
8100.2.d.c 2 45.l even 12 2
8100.2.d.h 2 45.k odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
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