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Properties

Label 156.2.u.b
Level $156$
Weight $2$
Character orbit 156.u
Analytic conductor $1.246$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,2,Mod(41,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.u (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 111 x^{12} - 216 x^{11} + 432 x^{10} - 864 x^{9} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{15} - \beta_{13} - \beta_{9}) q^{3} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{3}) q^{5}+ \cdots + ( - \beta_{15} + \beta_{10} + \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} - \beta_{13} - \beta_{9}) q^{3} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{3}) q^{5}+ \cdots + (2 \beta_{15} - \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} + 6 q^{9} + 12 q^{15} + 8 q^{19} - 6 q^{21} - 36 q^{27} - 32 q^{31} - 30 q^{33} - 20 q^{37} + 24 q^{39} - 30 q^{45} + 12 q^{49} + 6 q^{57} + 6 q^{63} - 64 q^{67} + 78 q^{69} - 4 q^{73} + 60 q^{75} + 96 q^{79} + 6 q^{81} + 96 q^{85} + 56 q^{91} + 18 q^{93} + 28 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 111 x^{12} - 216 x^{11} + 432 x^{10} - 864 x^{9} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{15} + 13 \nu^{14} - 66 \nu^{13} + 192 \nu^{12} - 405 \nu^{11} + 741 \nu^{10} + \cdots + 45927 ) / 11664 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{15} - 15 \nu^{14} - 18 \nu^{13} + 252 \nu^{12} - 597 \nu^{11} + 1017 \nu^{10} - 1503 \nu^{9} + \cdots + 115911 ) / 17496 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{15} - 15 \nu^{14} - 18 \nu^{13} + 252 \nu^{12} - 597 \nu^{11} + 1017 \nu^{10} - 1503 \nu^{9} + \cdots + 115911 ) / 17496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17 \nu^{15} - 45 \nu^{14} - 30 \nu^{13} + 144 \nu^{12} - 273 \nu^{11} + 63 \nu^{10} - 243 \nu^{9} + \cdots - 19683 ) / 34992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7 \nu^{15} + 51 \nu^{14} - 138 \nu^{13} + 288 \nu^{12} - 561 \nu^{11} + 1107 \nu^{10} + \cdots + 6561 ) / 17496 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 6 \nu^{14} + 21 \nu^{13} - 54 \nu^{12} + 111 \nu^{11} - 216 \nu^{10} + 432 \nu^{9} + \cdots - 13122 ) / 2187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19 \nu^{15} - 129 \nu^{14} + 354 \nu^{13} - 720 \nu^{12} + 1245 \nu^{11} - 2529 \nu^{10} + \cdots - 37179 ) / 34992 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7 \nu^{15} - 39 \nu^{14} + 134 \nu^{13} - 312 \nu^{12} + 585 \nu^{11} - 1107 \nu^{10} + 2283 \nu^{9} + \cdots - 29889 ) / 11664 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17 \nu^{15} + 42 \nu^{14} - 24 \nu^{13} - 72 \nu^{12} + 165 \nu^{11} - 72 \nu^{10} + 27 \nu^{9} + \cdots - 61236 ) / 17496 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{15} + 30 \nu^{13} - 99 \nu^{12} + 201 \nu^{11} - 252 \nu^{10} + 621 \nu^{9} - 1620 \nu^{8} + \cdots - 19683 ) / 4374 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 41 \nu^{15} + 183 \nu^{14} - 510 \nu^{13} + 1008 \nu^{12} - 1743 \nu^{11} + 3591 \nu^{10} + \cdots + 50301 ) / 34992 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26 \nu^{15} - 63 \nu^{14} + 54 \nu^{13} + 72 \nu^{12} - 318 \nu^{11} + 171 \nu^{10} + 90 \nu^{9} + \cdots + 85293 ) / 17496 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13 \nu^{15} - 83 \nu^{14} + 342 \nu^{13} - 888 \nu^{12} + 1803 \nu^{11} - 3363 \nu^{10} + \cdots - 164025 ) / 11664 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 9 \nu^{15} - 49 \nu^{14} + 154 \nu^{13} - 348 \nu^{12} + 651 \nu^{11} - 1245 \nu^{10} + 2577 \nu^{9} + \cdots - 31347 ) / 5832 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + 2 \beta_{10} + \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + 3 \beta_{5} + 2 \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} + 4\beta_{14} + \beta_{13} + \beta_{11} - \beta_{8} - 6\beta_{7} + \beta_{6} + 3\beta_{5} + 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{13} - 3\beta_{12} - 3\beta_{10} - 6\beta_{9} + 6\beta_{8} + 6\beta_{6} - 3\beta_{2} - 3\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6 \beta_{15} - 3 \beta_{14} - 3 \beta_{13} + 9 \beta_{12} - 3 \beta_{11} + 3 \beta_{9} + 6 \beta_{8} + \cdots + 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9\beta_{15} + 3\beta_{14} + 9\beta_{13} + 18\beta_{12} - 18\beta_{9} + 9\beta_{8} - 9\beta_{5} - 3\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 24 \beta_{15} + 3 \beta_{14} - 6 \beta_{13} - 9 \beta_{12} - 6 \beta_{11} + 9 \beta_{10} - 45 \beta_{9} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 18 \beta_{15} + 36 \beta_{14} + 36 \beta_{13} + 27 \beta_{12} + 18 \beta_{11} - 9 \beta_{10} + \cdots - 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 45 \beta_{15} - 54 \beta_{14} + 54 \beta_{11} - 81 \beta_{10} + 18 \beta_{9} - 63 \beta_{8} + 45 \beta_{7} + \cdots - 72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 180 \beta_{15} - 126 \beta_{14} - 162 \beta_{13} + 108 \beta_{12} - 180 \beta_{10} + 126 \beta_{9} + \cdots - 198 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 36 \beta_{15} + 198 \beta_{14} + 36 \beta_{13} + 270 \beta_{12} + 36 \beta_{11} - 216 \beta_{10} + \cdots - 117 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 54 \beta_{15} + 162 \beta_{14} - 270 \beta_{13} - 540 \beta_{12} - 162 \beta_{11} - 216 \beta_{10} + \cdots - 270 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 486 \beta_{15} + 162 \beta_{14} + 162 \beta_{13} - 324 \beta_{11} - 162 \beta_{10} + 324 \beta_{9} + \cdots - 324 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 837 \beta_{15} - 621 \beta_{14} + 1134 \beta_{13} + 648 \beta_{12} + 162 \beta_{11} - 270 \beta_{10} + \cdots - 1593 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{9} + \beta_{15}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
1.64067 + 0.555155i
0.614744 + 1.61929i
0.339557 − 1.69844i
−1.09497 − 1.34203i
−1.42240 − 0.988319i
1.06168 − 1.36851i
0.144709 + 1.72600i
1.71601 − 0.235189i
1.64067 − 0.555155i
0.614744 − 1.61929i
0.339557 + 1.69844i
−1.09497 + 1.34203i
−1.42240 + 0.988319i
1.06168 + 1.36851i
0.144709 − 1.72600i
1.71601 + 0.235189i
0 −1.69844 − 0.339557i 0 1.67398 + 1.67398i 0 1.53475 − 0.411235i 0 2.76940 + 1.15343i 0
41.2 0 −1.34203 + 1.09497i 0 −2.65779 − 2.65779i 0 −3.76680 + 1.00931i 0 0.602073 − 2.93896i 0
41.3 0 0.555155 − 1.64067i 0 −1.67398 − 1.67398i 0 1.53475 − 0.411235i 0 −2.38361 − 1.82166i 0
41.4 0 1.61929 − 0.614744i 0 2.65779 + 2.65779i 0 −3.76680 + 1.00931i 0 2.24418 − 1.99089i 0
89.1 0 −1.72600 + 0.144709i 0 −0.744930 + 0.744930i 0 0.929711 + 3.46973i 0 2.95812 − 0.499533i 0
89.2 0 0.235189 + 1.71601i 0 2.75301 − 2.75301i 0 0.302340 + 1.12835i 0 −2.88937 + 0.807173i 0
89.3 0 0.988319 − 1.42240i 0 0.744930 − 0.744930i 0 0.929711 + 3.46973i 0 −1.04645 − 2.81157i 0
89.4 0 1.36851 + 1.06168i 0 −2.75301 + 2.75301i 0 0.302340 + 1.12835i 0 0.745654 + 2.90586i 0
137.1 0 −1.69844 + 0.339557i 0 1.67398 − 1.67398i 0 1.53475 + 0.411235i 0 2.76940 − 1.15343i 0
137.2 0 −1.34203 − 1.09497i 0 −2.65779 + 2.65779i 0 −3.76680 − 1.00931i 0 0.602073 + 2.93896i 0
137.3 0 0.555155 + 1.64067i 0 −1.67398 + 1.67398i 0 1.53475 + 0.411235i 0 −2.38361 + 1.82166i 0
137.4 0 1.61929 + 0.614744i 0 2.65779 − 2.65779i 0 −3.76680 − 1.00931i 0 2.24418 + 1.99089i 0
149.1 0 −1.72600 − 0.144709i 0 −0.744930 − 0.744930i 0 0.929711 − 3.46973i 0 2.95812 + 0.499533i 0
149.2 0 0.235189 − 1.71601i 0 2.75301 + 2.75301i 0 0.302340 − 1.12835i 0 −2.88937 − 0.807173i 0
149.3 0 0.988319 + 1.42240i 0 0.744930 + 0.744930i 0 0.929711 − 3.46973i 0 −1.04645 + 2.81157i 0
149.4 0 1.36851 − 1.06168i 0 −2.75301 − 2.75301i 0 0.302340 − 1.12835i 0 0.745654 − 2.90586i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.u.b ✓ 16
3.b odd 2 1 inner 156.2.u.b ✓ 16
4.b odd 2 1 624.2.cn.e 16
12.b even 2 1 624.2.cn.e 16
13.f odd 12 1 inner 156.2.u.b ✓ 16
39.k even 12 1 inner 156.2.u.b ✓ 16
52.l even 12 1 624.2.cn.e 16
156.v odd 12 1 624.2.cn.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.u.b ✓ 16 1.a even 1 1 trivial
156.2.u.b ✓ 16 3.b odd 2 1 inner
156.2.u.b ✓ 16 13.f odd 12 1 inner
156.2.u.b ✓ 16 39.k even 12 1 inner
624.2.cn.e 16 4.b odd 2 1
624.2.cn.e 16 12.b even 2 1
624.2.cn.e 16 52.l even 12 1
624.2.cn.e 16 156.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 462T_{5}^{12} + 59913T_{5}^{8} + 1513512T_{5}^{4} + 1774224 \) acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 3 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 462 T^{12} + \cdots + 1774224 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + \cdots + 676)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 66 T^{14} + \cdots + 1774224 \) Copy content Toggle raw display
$13$ \( (T^{8} - 23 T^{6} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 3167100729 \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{7} + 17 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 454201344 \) Copy content Toggle raw display
$29$ \( T^{16} - 120 T^{14} + \cdots + 110889 \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 10 T^{7} + \cdots + 169)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 639253018089 \) Copy content Toggle raw display
$43$ \( (T^{8} - 27 T^{6} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 588644941824 \) Copy content Toggle raw display
$53$ \( (T^{8} + 198 T^{6} + \cdots + 1363968)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 393049974438144 \) Copy content Toggle raw display
$61$ \( (T^{8} + 114 T^{6} + \cdots + 31329)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 32 T^{7} + \cdots + 21641104)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{8} + 2 T^{7} + \cdots + 111556)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + \cdots - 192)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 138522354280704 \) Copy content Toggle raw display
$89$ \( T^{16} + 66 T^{14} + \cdots + 1774224 \) Copy content Toggle raw display
$97$ \( (T^{8} - 14 T^{7} + \cdots + 9897316)^{2} \) Copy content Toggle raw display
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