Properties

Label 156.3.l.a
Level $156$
Weight $3$
Character orbit 156.l
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.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{3} + 4 \beta_{2} q^{4} - 4 \beta_{3} q^{5} + (\beta_{3} + 4 \beta_{2} + 4) q^{6} + ( - 3 \beta_{2} - 3) q^{7} + 4 \beta_{3} q^{8} + (2 \beta_{3} + 2 \beta_1 + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{3} + 4 \beta_{2} q^{4} - 4 \beta_{3} q^{5} + (\beta_{3} + 4 \beta_{2} + 4) q^{6} + ( - 3 \beta_{2} - 3) q^{7} + 4 \beta_{3} q^{8} + (2 \beta_{3} + 2 \beta_1 + 7) q^{9} + 16 q^{10} + 2 \beta_{3} q^{11} + (4 \beta_{3} + 4 \beta_1 - 4) q^{12} - 13 \beta_{2} q^{13} + ( - 3 \beta_{3} - 3 \beta_1) q^{14} + ( - 16 \beta_{2} + 4 \beta_1 + 16) q^{15} - 16 q^{16} + (10 \beta_{3} - 10 \beta_1) q^{17} + (8 \beta_{2} + 7 \beta_1 - 8) q^{18} + (17 \beta_{2} - 17) q^{19} + 16 \beta_1 q^{20} + ( - 3 \beta_{2} - 6 \beta_1 + 3) q^{21} - 8 q^{22} + (6 \beta_{3} + 6 \beta_1) q^{23} + (16 \beta_{2} - 4 \beta_1 - 16) q^{24} - 39 \beta_{2} q^{25} - 13 \beta_{3} q^{26} + ( - 5 \beta_{3} + 23 \beta_{2} + 5 \beta_1) q^{27} + ( - 12 \beta_{2} + 12) q^{28} + (8 \beta_{3} + 8 \beta_1) q^{29} + ( - 16 \beta_{3} + 16 \beta_{2} + 16 \beta_1) q^{30} + ( - 21 \beta_{2} + 21) q^{31} - 16 \beta_1 q^{32} + (8 \beta_{2} - 2 \beta_1 - 8) q^{33} + ( - 40 \beta_{2} - 40) q^{34} + (12 \beta_{3} - 12 \beta_1) q^{35} + (8 \beta_{3} + 28 \beta_{2} - 8 \beta_1) q^{36} + ( - 21 \beta_{2} + 21) q^{37} + (17 \beta_{3} - 17 \beta_1) q^{38} + ( - 13 \beta_{3} - 13 \beta_1 + 13) q^{39} + 64 \beta_{2} q^{40} - 28 \beta_{3} q^{41} + ( - 3 \beta_{3} - 24 \beta_{2} + 3 \beta_1) q^{42} + 14 q^{43} - 8 \beta_1 q^{44} + ( - 28 \beta_{3} + 32 \beta_{2} + 32) q^{45} + (24 \beta_{2} - 24) q^{46} + 22 \beta_{3} q^{47} + (16 \beta_{3} - 16 \beta_{2} - 16 \beta_1) q^{48} - 31 \beta_{2} q^{49} - 39 \beta_{3} q^{50} + ( - 10 \beta_{3} - 10 \beta_1 - 80) q^{51} + 52 q^{52} + (18 \beta_{3} + 18 \beta_1) q^{53} + (23 \beta_{3} + 20 \beta_{2} + 20) q^{54} + 32 \beta_{2} q^{55} + ( - 12 \beta_{3} + 12 \beta_1) q^{56} + (34 \beta_{3} - 17 \beta_{2} - 17) q^{57} + (32 \beta_{2} - 32) q^{58} - 26 \beta_{3} q^{59} + (16 \beta_{3} + 64 \beta_{2} + 64) q^{60} + 16 q^{61} + ( - 21 \beta_{3} + 21 \beta_1) q^{62} + ( - 12 \beta_{3} - 21 \beta_{2} - 21) q^{63} - 64 \beta_{2} q^{64} - 52 \beta_1 q^{65} + (8 \beta_{3} - 8 \beta_{2} - 8 \beta_1) q^{66} + ( - \beta_{2} + 1) q^{67} + ( - 40 \beta_{3} - 40 \beta_1) q^{68} + (6 \beta_{3} + 48 \beta_{2} - 6 \beta_1) q^{69} + ( - 48 \beta_{2} - 48) q^{70} - 18 \beta_1 q^{71} + (28 \beta_{3} - 32 \beta_{2} - 32) q^{72} + (7 \beta_{2} - 7) q^{73} + ( - 21 \beta_{3} + 21 \beta_1) q^{74} + ( - 39 \beta_{3} - 39 \beta_1 + 39) q^{75} + ( - 68 \beta_{2} - 68) q^{76} + ( - 6 \beta_{3} + 6 \beta_1) q^{77} + ( - 52 \beta_{2} + 13 \beta_1 + 52) q^{78} - 106 \beta_{2} q^{79} + 64 \beta_{3} q^{80} + (28 \beta_{3} + 28 \beta_1 + 17) q^{81} + 112 q^{82} + 70 \beta_1 q^{83} + ( - 24 \beta_{3} + 12 \beta_{2} + 12) q^{84} + (160 \beta_{2} - 160) q^{85} + 14 \beta_1 q^{86} + (8 \beta_{3} + 64 \beta_{2} - 8 \beta_1) q^{87} - 32 \beta_{2} q^{88} + 56 \beta_1 q^{89} + (32 \beta_{3} + 32 \beta_1 + 112) q^{90} + (39 \beta_{2} - 39) q^{91} + (24 \beta_{3} - 24 \beta_1) q^{92} + ( - 42 \beta_{3} + 21 \beta_{2} + 21) q^{93} - 88 q^{94} + (68 \beta_{3} + 68 \beta_1) q^{95} + ( - 16 \beta_{3} - 64 \beta_{2} - 64) q^{96} + ( - \beta_{2} - 1) q^{97} - 31 \beta_{3} q^{98} + (14 \beta_{3} - 16 \beta_{2} - 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{6} - 12 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{6} - 12 q^{7} + 28 q^{9} + 64 q^{10} - 16 q^{12} + 64 q^{15} - 64 q^{16} - 32 q^{18} - 68 q^{19} + 12 q^{21} - 32 q^{22} - 64 q^{24} + 48 q^{28} + 84 q^{31} - 32 q^{33} - 160 q^{34} + 84 q^{37} + 52 q^{39} + 56 q^{43} + 128 q^{45} - 96 q^{46} - 320 q^{51} + 208 q^{52} + 80 q^{54} - 68 q^{57} - 128 q^{58} + 256 q^{60} + 64 q^{61} - 84 q^{63} + 4 q^{67} - 192 q^{70} - 128 q^{72} - 28 q^{73} + 156 q^{75} - 272 q^{76} + 208 q^{78} + 68 q^{81} + 448 q^{82} + 48 q^{84} - 640 q^{85} + 448 q^{90} - 156 q^{91} + 84 q^{93} - 352 q^{94} - 256 q^{96} - 4 q^{97} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{8}^{3} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−1.41421 1.41421i −2.82843 + 1.00000i 4.00000i −5.65685 + 5.65685i 5.41421 + 2.58579i −3.00000 3.00000i 5.65685 5.65685i 7.00000 5.65685i 16.0000
47.2 1.41421 + 1.41421i 2.82843 + 1.00000i 4.00000i 5.65685 5.65685i 2.58579 + 5.41421i −3.00000 3.00000i −5.65685 + 5.65685i 7.00000 + 5.65685i 16.0000
83.1 −1.41421 + 1.41421i −2.82843 1.00000i 4.00000i −5.65685 5.65685i 5.41421 2.58579i −3.00000 + 3.00000i 5.65685 + 5.65685i 7.00000 + 5.65685i 16.0000
83.2 1.41421 1.41421i 2.82843 1.00000i 4.00000i 5.65685 + 5.65685i 2.58579 5.41421i −3.00000 + 3.00000i −5.65685 5.65685i 7.00000 5.65685i 16.0000
\(n\): 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
52.f even 4 1 inner
156.l odd 4 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(156, [\chi])\):

\( T_{5}^{4} + 4096 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} - 14T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 4096 \) Copy content Toggle raw display
$7$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 34 T + 578)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 42 T + 882)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 42 T + 882)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 9834496 \) Copy content Toggle raw display
$43$ \( (T - 14)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 3748096 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 7311616 \) Copy content Toggle raw display
$61$ \( (T - 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 1679616 \) Copy content Toggle raw display
$73$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 11236)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 384160000 \) Copy content Toggle raw display
$89$ \( T^{4} + 157351936 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
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