Properties

Label 156.2.q.a
Level $156$
Weight $2$
Character orbit 156.q
Analytic conductor $1.246$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 1) q^{5} + ( - 2 \zeta_{6} + 4) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 1) q^{5} + ( - 2 \zeta_{6} + 4) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} + 2) q^{11} + ( - 3 \zeta_{6} - 1) q^{13} + (\zeta_{6} + 1) q^{15} - 3 \zeta_{6} q^{17} + (2 \zeta_{6} - 4) q^{19} + ( - 4 \zeta_{6} + 2) q^{21} + (6 \zeta_{6} - 6) q^{23} + 2 q^{25} - q^{27} + (9 \zeta_{6} - 9) q^{29} + ( - 2 \zeta_{6} + 4) q^{33} + 6 \zeta_{6} q^{35} + ( - 3 \zeta_{6} - 3) q^{37} + (\zeta_{6} - 4) q^{39} + ( - 5 \zeta_{6} - 5) q^{41} - 2 \zeta_{6} q^{43} + ( - \zeta_{6} + 2) q^{45} + ( - 4 \zeta_{6} + 2) q^{47} + ( - 5 \zeta_{6} + 5) q^{49} - 3 q^{51} - 9 q^{53} + (6 \zeta_{6} - 6) q^{55} + (4 \zeta_{6} - 2) q^{57} + ( - 8 \zeta_{6} + 16) q^{59} + 11 \zeta_{6} q^{61} + ( - 2 \zeta_{6} - 2) q^{63} + ( - 5 \zeta_{6} + 7) q^{65} + (6 \zeta_{6} + 6) q^{67} + 6 \zeta_{6} q^{69} + ( - 6 \zeta_{6} + 12) q^{71} + ( - 6 \zeta_{6} + 3) q^{73} + ( - 2 \zeta_{6} + 2) q^{75} + 12 q^{77} - 8 q^{79} + (\zeta_{6} - 1) q^{81} + (4 \zeta_{6} - 2) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + 9 \zeta_{6} q^{87} + ( - 4 \zeta_{6} - 4) q^{89} + ( - 4 \zeta_{6} - 10) q^{91} - 6 \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 8) q^{97} + ( - 4 \zeta_{6} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 6 q^{7} - q^{9} + 6 q^{11} - 5 q^{13} + 3 q^{15} - 3 q^{17} - 6 q^{19} - 6 q^{23} + 4 q^{25} - 2 q^{27} - 9 q^{29} + 6 q^{33} + 6 q^{35} - 9 q^{37} - 7 q^{39} - 15 q^{41} - 2 q^{43} + 3 q^{45} + 5 q^{49} - 6 q^{51} - 18 q^{53} - 6 q^{55} + 24 q^{59} + 11 q^{61} - 6 q^{63} + 9 q^{65} + 18 q^{67} + 6 q^{69} + 18 q^{71} + 2 q^{75} + 24 q^{77} - 16 q^{79} - q^{81} + 9 q^{85} + 9 q^{87} - 12 q^{89} - 24 q^{91} - 6 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 1.73205i 0 3.00000 + 1.73205i 0 −0.500000 + 0.866025i 0
121.1 0 0.500000 0.866025i 0 1.73205i 0 3.00000 1.73205i 0 −0.500000 0.866025i 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
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 156.2.q.a 2
3.b odd 2 1 468.2.t.c 2
4.b odd 2 1 624.2.bv.a 2
5.b even 2 1 3900.2.cd.a 2
5.c odd 4 2 3900.2.bw.e 4
12.b even 2 1 1872.2.by.b 2
13.b even 2 1 2028.2.q.a 2
13.c even 3 1 2028.2.b.b 2
13.c even 3 1 2028.2.q.a 2
13.d odd 4 2 2028.2.i.h 4
13.e even 6 1 inner 156.2.q.a 2
13.e even 6 1 2028.2.b.b 2
13.f odd 12 2 2028.2.a.h 2
13.f odd 12 2 2028.2.i.h 4
39.h odd 6 1 468.2.t.c 2
39.h odd 6 1 6084.2.b.c 2
39.i odd 6 1 6084.2.b.c 2
39.k even 12 2 6084.2.a.u 2
52.i odd 6 1 624.2.bv.a 2
52.l even 12 2 8112.2.a.bt 2
65.l even 6 1 3900.2.cd.a 2
65.r odd 12 2 3900.2.bw.e 4
156.r even 6 1 1872.2.by.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.q.a 2 1.a even 1 1 trivial
156.2.q.a 2 13.e even 6 1 inner
468.2.t.c 2 3.b odd 2 1
468.2.t.c 2 39.h odd 6 1
624.2.bv.a 2 4.b odd 2 1
624.2.bv.a 2 52.i odd 6 1
1872.2.by.b 2 12.b even 2 1
1872.2.by.b 2 156.r even 6 1
2028.2.a.h 2 13.f odd 12 2
2028.2.b.b 2 13.c even 3 1
2028.2.b.b 2 13.e even 6 1
2028.2.i.h 4 13.d odd 4 2
2028.2.i.h 4 13.f odd 12 2
2028.2.q.a 2 13.b even 2 1
2028.2.q.a 2 13.c even 3 1
3900.2.bw.e 4 5.c odd 4 2
3900.2.bw.e 4 65.r odd 12 2
3900.2.cd.a 2 5.b even 2 1
3900.2.cd.a 2 65.l even 6 1
6084.2.a.u 2 39.k even 12 2
6084.2.b.c 2 39.h odd 6 1
6084.2.b.c 2 39.i odd 6 1
8112.2.a.bt 2 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$41$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 24T + 192 \) Copy content Toggle raw display
$61$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$71$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 27 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
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