Properties

Label 156.3.n.d
Level $156$
Weight $3$
Character orbit 156.n
Analytic conductor $4.251$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{142})\)
Defining polynomial: \( x^{4} + 142x^{2} + 20164 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + ( - \beta_{2} - 2) q^{3} + 4 \beta_{2} q^{4} + ( - 2 \beta_{2} - 1) q^{5} + (4 \beta_{2} + 2) q^{6} + (2 \beta_{2} + \beta_1 + 2) q^{7} + 8 q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + ( - \beta_{2} - 2) q^{3} + 4 \beta_{2} q^{4} + ( - 2 \beta_{2} - 1) q^{5} + (4 \beta_{2} + 2) q^{6} + (2 \beta_{2} + \beta_1 + 2) q^{7} + 8 q^{8} + (3 \beta_{2} + 3) q^{9} + (2 \beta_{2} - 2) q^{10} + ( - \beta_{3} + 6 \beta_{2} - \beta_1) q^{11} + ( - 4 \beta_{2} + 4) q^{12} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 - 6) q^{13} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{14} + 3 \beta_{2} q^{15} + ( - 16 \beta_{2} - 16) q^{16} + (13 \beta_{2} - \beta_1 + 13) q^{17} - 6 \beta_{2} q^{18} + (8 \beta_{2} + \beta_1 + 8) q^{19} + (4 \beta_{2} + 8) q^{20} + ( - \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{21} + (2 \beta_{3} + 12) q^{22} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 + 8) q^{23} + ( - 8 \beta_{2} - 16) q^{24} + 22 q^{25} + (2 \beta_{3} + 12 \beta_{2} + 6) q^{26} + ( - 6 \beta_{2} - 3) q^{27} + (4 \beta_{3} - 8) q^{28} + ( - 2 \beta_{3} - 7 \beta_{2} - 2 \beta_1) q^{29} + 6 q^{30} + ( - 2 \beta_{3} - 32) q^{31} + 32 \beta_{2} q^{32} + (2 \beta_{3} - 6 \beta_{2} + \beta_1 + 6) q^{33} + (2 \beta_{3} - 26 \beta_{2} + 2 \beta_1) q^{34} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{35} - 12 q^{36} + ( - \beta_{3} - 7 \beta_{2} + \beta_1 - 14) q^{37} + ( - 2 \beta_{3} - 16 \beta_{2} - 2 \beta_1) q^{38} + (2 \beta_{3} + 9 \beta_{2} + \beta_1 + 9) q^{39} + ( - 16 \beta_{2} - 8) q^{40} + ( - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 6) q^{41} + (4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{42} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{43} + ( - 24 \beta_{2} + 4 \beta_1 - 24) q^{44} + ( - 3 \beta_{2} + 3) q^{45} + ( - 2 \beta_{3} - 16 \beta_{2} - 4 \beta_1 - 8) q^{46} + (\beta_{3} + 12) q^{47} + (32 \beta_{2} + 16) q^{48} + (4 \beta_{3} + 97 \beta_{2} + 4 \beta_1) q^{49} + ( - 44 \beta_{2} - 44) q^{50} + (\beta_{3} - 26 \beta_{2} + 2 \beta_1 - 13) q^{51} + ( - 12 \beta_{2} + 4 \beta_1 + 12) q^{52} + (2 \beta_{3} - 25) q^{53} + (6 \beta_{2} - 6) q^{54} + (\beta_{3} + 6 \beta_{2} - \beta_1 + 12) q^{55} + (16 \beta_{2} + 8 \beta_1 + 16) q^{56} + ( - \beta_{3} - 16 \beta_{2} - 2 \beta_1 - 8) q^{57} + (4 \beta_{3} - 14) q^{58} + (56 \beta_{2} + 56) q^{59} + ( - 12 \beta_{2} - 12) q^{60} + ( - 27 \beta_{2} + 5 \beta_1 - 27) q^{61} + (64 \beta_{2} - 4 \beta_1 + 64) q^{62} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{63} + 64 q^{64} + (\beta_{3} + 9 \beta_{2} - \beta_1) q^{65} + ( - 2 \beta_{3} - 12 \beta_{2} + 2 \beta_1 - 24) q^{66} + ( - 7 \beta_{3} - 20 \beta_{2} - 7 \beta_1) q^{67} + ( - 4 \beta_{3} - 52) q^{68} + ( - 12 \beta_{2} - 3 \beta_1 - 12) q^{69} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 8) q^{70} + ( - 88 \beta_{2} - 3 \beta_1 - 88) q^{71} + (24 \beta_{2} + 24) q^{72} + (2 \beta_{3} - 98 \beta_{2} + 4 \beta_1 - 49) q^{73} + ( - 2 \beta_{3} + 28 \beta_{2} - 4 \beta_1 + 14) q^{74} + ( - 22 \beta_{2} - 44) q^{75} + (4 \beta_{3} - 32) q^{76} + (4 \beta_{3} + 130) q^{77} + ( - 2 \beta_{3} - 18 \beta_{2} + 2 \beta_1) q^{78} + (2 \beta_{3} + 8 \beta_{2} + 4 \beta_1 + 4) q^{79} + (16 \beta_{2} - 16) q^{80} + 9 \beta_{2} q^{81} + ( - 6 \beta_{3} + 12 \beta_{2} - 12 \beta_1 + 6) q^{82} + (9 \beta_{3} - 26) q^{83} + ( - 4 \beta_{3} + 8 \beta_{2} + 4 \beta_1 + 16) q^{84} + (2 \beta_{3} - 13 \beta_{2} + \beta_1 + 13) q^{85} + ( - 2 \beta_{3} + 8 \beta_{2} + 2 \beta_1 + 16) q^{86} + (4 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 7) q^{87} + ( - 8 \beta_{3} + 48 \beta_{2} - 8 \beta_1) q^{88} + (2 \beta_{3} - 48 \beta_{2} - 2 \beta_1 - 96) q^{89} + ( - 6 \beta_{2} - 12) q^{90} + ( - 5 \beta_{3} - 12 \beta_{2} - 6 \beta_1 + 136) q^{91} + (8 \beta_{3} + 16 \beta_{2} + 4 \beta_1 - 16) q^{92} + (2 \beta_{3} + 32 \beta_{2} - 2 \beta_1 + 64) q^{93} + ( - 24 \beta_{2} + 2 \beta_1 - 24) q^{94} + ( - 2 \beta_{3} - 8 \beta_{2} - \beta_1 + 8) q^{95} + ( - 32 \beta_{2} + 32) q^{96} + ( - 8 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 2) q^{97} + ( - 8 \beta_{3} + 194) q^{98} + ( - 3 \beta_{3} - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} + 4 q^{7} + 32 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} + 4 q^{7} + 32 q^{8} + 6 q^{9} - 12 q^{10} - 12 q^{11} + 24 q^{12} - 18 q^{13} + 8 q^{14} - 6 q^{15} - 32 q^{16} + 26 q^{17} + 12 q^{18} + 16 q^{19} + 24 q^{20} + 48 q^{22} + 24 q^{23} - 48 q^{24} + 88 q^{25} - 32 q^{28} + 14 q^{29} + 24 q^{30} - 128 q^{31} - 64 q^{32} + 36 q^{33} + 52 q^{34} + 12 q^{35} - 48 q^{36} - 42 q^{37} + 32 q^{38} + 18 q^{39} - 18 q^{41} - 24 q^{42} - 24 q^{43} - 48 q^{44} + 18 q^{45} + 48 q^{47} - 194 q^{49} - 88 q^{50} + 72 q^{52} - 100 q^{53} - 36 q^{54} + 36 q^{55} + 32 q^{56} - 56 q^{58} + 112 q^{59} - 24 q^{60} - 54 q^{61} + 128 q^{62} - 12 q^{63} + 256 q^{64} - 18 q^{65} - 72 q^{66} + 40 q^{67} - 208 q^{68} - 24 q^{69} - 24 q^{70} - 176 q^{71} + 48 q^{72} - 132 q^{75} - 128 q^{76} + 520 q^{77} + 36 q^{78} - 96 q^{80} - 18 q^{81} - 104 q^{83} + 48 q^{84} + 78 q^{85} + 48 q^{86} - 42 q^{87} - 96 q^{88} - 288 q^{89} - 36 q^{90} + 568 q^{91} - 96 q^{92} + 192 q^{93} - 48 q^{94} + 48 q^{95} + 192 q^{96} - 12 q^{97} + 776 q^{98} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 142x^{2} + 20164 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 142 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 142 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 142\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 142\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−5.95819 10.3199i
5.95819 + 10.3199i
−5.95819 + 10.3199i
5.95819 10.3199i
−1.00000 1.73205i −1.50000 0.866025i −2.00000 + 3.46410i 1.73205i 3.46410i −4.95819 8.58783i 8.00000 1.50000 + 2.59808i −3.00000 + 1.73205i
43.2 −1.00000 1.73205i −1.50000 0.866025i −2.00000 + 3.46410i 1.73205i 3.46410i 6.95819 + 12.0519i 8.00000 1.50000 + 2.59808i −3.00000 + 1.73205i
127.1 −1.00000 + 1.73205i −1.50000 + 0.866025i −2.00000 3.46410i 1.73205i 3.46410i −4.95819 + 8.58783i 8.00000 1.50000 2.59808i −3.00000 1.73205i
127.2 −1.00000 + 1.73205i −1.50000 + 0.866025i −2.00000 3.46410i 1.73205i 3.46410i 6.95819 12.0519i 8.00000 1.50000 2.59808i −3.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.n.d yes 4
4.b odd 2 1 156.3.n.c 4
13.e even 6 1 156.3.n.c 4
52.i odd 6 1 inner 156.3.n.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.n.c 4 4.b odd 2 1
156.3.n.c 4 13.e even 6 1
156.3.n.d yes 4 1.a even 1 1 trivial
156.3.n.d yes 4 52.i odd 6 1 inner

Hecke kernels

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

\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} + 154T_{7}^{2} + 552T_{7} + 19044 \) Copy content Toggle raw display
\( T_{11}^{4} + 12T_{11}^{3} + 250T_{11}^{2} - 1272T_{11} + 11236 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 154 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + 250 T^{2} + \cdots + 11236 \) Copy content Toggle raw display
$13$ \( T^{4} + 18 T^{3} + 277 T^{2} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{4} - 26 T^{3} + 649 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{4} - 16 T^{3} + 334 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$23$ \( T^{4} - 24 T^{3} - 186 T^{2} + \cdots + 142884 \) Copy content Toggle raw display
$29$ \( T^{4} - 14 T^{3} + 715 T^{2} + \cdots + 269361 \) Copy content Toggle raw display
$31$ \( (T^{2} + 64 T + 456)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 42 T^{3} + 309 T^{2} + \cdots + 77841 \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} - 3699 T^{2} + \cdots + 14493249 \) Copy content Toggle raw display
$43$ \( T^{4} + 24 T^{3} - 186 T^{2} + \cdots + 142884 \) Copy content Toggle raw display
$47$ \( (T^{2} - 24 T + 2)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 50 T + 57)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 56 T + 3136)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 54 T^{3} + 5737 T^{2} + \cdots + 7958041 \) Copy content Toggle raw display
$67$ \( T^{4} - 40 T^{3} + 8158 T^{2} + \cdots + 43007364 \) Copy content Toggle raw display
$71$ \( T^{4} + 176 T^{3} + \cdots + 41809156 \) Copy content Toggle raw display
$73$ \( T^{4} + 17814 T^{2} + \cdots + 30239001 \) Copy content Toggle raw display
$79$ \( T^{4} + 3504 T^{2} + \cdots + 2742336 \) Copy content Toggle raw display
$83$ \( (T^{2} + 52 T - 10826)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 288 T^{3} + \cdots + 27123264 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} - 6756 T^{2} + \cdots + 46294416 \) Copy content Toggle raw display
show more
show less