Properties

Label 156.3.g.b
Level $156$
Weight $3$
Character orbit 156.g
Analytic conductor $4.251$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 156.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2) q^{3} + (\beta_{2} - 2 \beta_1) q^{5} - 7 \beta_{2} q^{7} + ( - 5 \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2) q^{3} + (\beta_{2} - 2 \beta_1) q^{5} - 7 \beta_{2} q^{7} + ( - 5 \beta_{3} + 1) q^{9} + ( - 6 \beta_{2} + 12 \beta_1) q^{11} - 13 \beta_{2} q^{13} + ( - 8 \beta_{2} + 5 \beta_1) q^{15} + (14 \beta_{3} + 7) q^{17} - 30 \beta_{2} q^{19} + (14 \beta_{2} + 7 \beta_1) q^{21} + ( - 16 \beta_{3} - 8) q^{23} - 14 q^{25} + (16 \beta_{3} + 13) q^{27} + ( - 20 \beta_{3} - 10) q^{29} - 10 \beta_{2} q^{31} + (48 \beta_{2} - 30 \beta_1) q^{33} + (14 \beta_{3} + 7) q^{35} + 13 \beta_{2} q^{37} + (26 \beta_{2} + 13 \beta_1) q^{39} + (14 \beta_{2} - 28 \beta_1) q^{41} + 55 q^{43} + (31 \beta_{2} - 7 \beta_1) q^{45} + (15 \beta_{2} - 30 \beta_1) q^{47} + ( - 35 \beta_{3} - 56) q^{51} + (24 \beta_{3} + 12) q^{53} - 66 q^{55} + (60 \beta_{2} + 30 \beta_1) q^{57} + ( - 28 \beta_{2} + 56 \beta_1) q^{59} + 14 q^{61} + ( - 7 \beta_{2} - 35 \beta_1) q^{63} + (26 \beta_{3} + 13) q^{65} - 16 \beta_{2} q^{67} + (40 \beta_{3} + 64) q^{69} + (9 \beta_{2} - 18 \beta_1) q^{71} + 4 \beta_{2} q^{73} + ( - 14 \beta_{3} + 28) q^{75} + ( - 84 \beta_{3} - 42) q^{77} + 54 q^{79} + ( - 35 \beta_{3} - 74) q^{81} + (10 \beta_{2} - 20 \beta_1) q^{83} - 77 \beta_{2} q^{85} + (50 \beta_{3} + 80) q^{87} + ( - 6 \beta_{2} + 12 \beta_1) q^{89} - 91 q^{91} + (20 \beta_{2} + 10 \beta_1) q^{93} + (60 \beta_{3} + 30) q^{95} + 78 \beta_{2} q^{97} + ( - 186 \beta_{2} + 42 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{3} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{3} + 14 q^{9} - 56 q^{25} + 20 q^{27} + 220 q^{43} - 154 q^{51} - 264 q^{55} + 56 q^{61} + 176 q^{69} + 140 q^{75} + 216 q^{79} - 226 q^{81} + 220 q^{87} - 364 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

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

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
1.65831 0.500000i
−1.65831 + 0.500000i
1.65831 + 0.500000i
−1.65831 0.500000i
0 −2.50000 1.65831i 0 −3.31662 0 7.00000i 0 3.50000 + 8.29156i 0
77.2 0 −2.50000 1.65831i 0 3.31662 0 7.00000i 0 3.50000 + 8.29156i 0
77.3 0 −2.50000 + 1.65831i 0 −3.31662 0 7.00000i 0 3.50000 8.29156i 0
77.4 0 −2.50000 + 1.65831i 0 3.31662 0 7.00000i 0 3.50000 8.29156i 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
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.g.b 4
3.b odd 2 1 inner 156.3.g.b 4
4.b odd 2 1 624.3.l.h 4
12.b even 2 1 624.3.l.h 4
13.b even 2 1 inner 156.3.g.b 4
39.d odd 2 1 inner 156.3.g.b 4
52.b odd 2 1 624.3.l.h 4
156.h even 2 1 624.3.l.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.g.b 4 1.a even 1 1 trivial
156.3.g.b 4 3.b odd 2 1 inner
156.3.g.b 4 13.b even 2 1 inner
156.3.g.b 4 39.d odd 2 1 inner
624.3.l.h 4 4.b odd 2 1
624.3.l.h 4 12.b even 2 1
624.3.l.h 4 52.b odd 2 1
624.3.l.h 4 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 11 \) acting on \(S_{3}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 5 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 11)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 396)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 539)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 900)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 704)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1100)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2156)^{2} \) Copy content Toggle raw display
$43$ \( (T - 55)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2475)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1584)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8624)^{2} \) Copy content Toggle raw display
$61$ \( (T - 14)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 891)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T - 54)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1100)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 396)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 6084)^{2} \) Copy content Toggle raw display
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