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Properties

Label 156.3.x.b
Level $156$
Weight $3$
Character orbit 156.x
Analytic conductor $4.251$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,3,Mod(37,156)]
 
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, 0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 156.x (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: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 63 x^{10} - 186 x^{9} + 291 x^{8} + 21300 x^{7} + 268814 x^{6} + 1894608 x^{5} + \cdots + 53608789 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{3} + (\beta_{8} + \beta_{4} - \beta_{3} - 1) q^{5} + (\beta_{10} - 2 \beta_{8} + \beta_{7} + \cdots + 1) q^{7}+ \cdots + ( - 3 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{3} + (\beta_{8} + \beta_{4} - \beta_{3} - 1) q^{5} + (\beta_{10} - 2 \beta_{8} + \beta_{7} + \cdots + 1) q^{7}+ \cdots + (3 \beta_{10} - 3 \beta_{7} - 3 \beta_{6} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} - 18 q^{9} + 6 q^{11} - 6 q^{13} + 24 q^{15} + 42 q^{17} + 36 q^{19} + 6 q^{21} + 30 q^{23} - 120 q^{29} - 114 q^{31} + 42 q^{33} - 66 q^{35} + 36 q^{37} - 42 q^{39} + 234 q^{41} - 36 q^{43} + 24 q^{47} + 78 q^{49} - 96 q^{53} - 234 q^{55} + 48 q^{57} - 84 q^{59} - 42 q^{61} + 18 q^{63} - 510 q^{65} + 474 q^{67} - 54 q^{69} + 294 q^{71} + 330 q^{73} - 198 q^{75} + 48 q^{79} - 54 q^{81} - 270 q^{85} - 24 q^{87} + 468 q^{89} - 774 q^{91} + 162 q^{93} - 186 q^{95} + 78 q^{97} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 63 x^{10} - 186 x^{9} + 291 x^{8} + 21300 x^{7} + 268814 x^{6} + 1894608 x^{5} + \cdots + 53608789 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17\!\cdots\!02 \nu^{11} + \cdots - 26\!\cdots\!27 ) / 76\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38\!\cdots\!11 \nu^{11} + \cdots + 44\!\cdots\!40 ) / 76\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 109818360934983 \nu^{11} - 323217558029922 \nu^{10} + \cdots + 20\!\cdots\!43 ) / 20\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 504293089500 \nu^{11} - 1630928179701 \nu^{10} - 26855005709828 \nu^{9} + \cdots + 75\!\cdots\!63 ) / 85\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 95\!\cdots\!65 \nu^{11} + \cdots - 28\!\cdots\!67 ) / 15\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 86\!\cdots\!04 \nu^{11} + \cdots - 14\!\cdots\!85 ) / 76\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 60\!\cdots\!11 \nu^{11} + \cdots + 11\!\cdots\!03 ) / 38\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 323217558029922 \nu^{11} - 974798726086891 \nu^{10} + \cdots + 56\!\cdots\!59 ) / 20\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 58\!\cdots\!23 \nu^{11} + \cdots - 12\!\cdots\!15 ) / 15\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42\!\cdots\!00 \nu^{11} + \cdots - 70\!\cdots\!31 ) / 76\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 54\!\cdots\!64 \nu^{11} + \cdots - 75\!\cdots\!02 ) / 76\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{6} + \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} - 3\beta_{7} + 2\beta_{6} - \beta_{5} + 33\beta_{4} + 10\beta_{3} - \beta_{2} + 31\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{11} + 3 \beta_{10} + \beta_{9} - 24 \beta_{8} - 54 \beta_{7} + 27 \beta_{6} + 3 \beta_{5} + \cdots + 105 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 24 \beta_{11} + 59 \beta_{10} + 57 \beta_{9} - 471 \beta_{8} - 311 \beta_{7} + 320 \beta_{6} + \cdots + 1351 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 411 \beta_{11} + 321 \beta_{10} + 106 \beta_{9} - 4860 \beta_{8} - 2888 \beta_{7} + 3022 \beta_{6} + \cdots + 1190 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4458 \beta_{11} + 3470 \beta_{10} + 948 \beta_{9} - 40908 \beta_{8} - 24468 \beta_{7} + 10448 \beta_{6} + \cdots - 46357 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 33994 \beta_{11} + 11970 \beta_{10} - 7804 \beta_{9} - 335550 \beta_{8} - 160038 \beta_{7} + \cdots - 1131234 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 237629 \beta_{11} + 8572 \beta_{10} - 173815 \beta_{9} - 2365680 \beta_{8} - 57159 \beta_{7} + \cdots - 15359442 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 699114 \beta_{11} - 1482615 \beta_{10} - 3516249 \beta_{9} - 9170154 \beta_{8} + 12953250 \beta_{7} + \cdots - 186046947 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7772580 \beta_{11} - 27621195 \beta_{10} - 42211521 \beta_{9} + 78782229 \beta_{8} + \cdots - 1929448405 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 227619155 \beta_{11} - 397918719 \beta_{10} - 454460936 \beta_{9} + 2394309516 \beta_{8} + \cdots - 17698542500 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−5.44627 + 0.459324i
−3.14938 − 0.156126i
9.46168 − 3.53525i
1.28509 − 5.79602i
−0.611299 + 1.28140i
−1.53982 + 4.74667i
−5.44627 − 0.459324i
−3.14938 + 0.156126i
9.46168 + 3.53525i
1.28509 + 5.79602i
−0.611299 − 1.28140i
−1.53982 − 4.74667i
0 0.866025 + 1.50000i 0 −4.62092 − 4.62092i 0 −2.83806 − 10.5918i 0 −1.50000 + 2.59808i 0
37.2 0 0.866025 + 1.50000i 0 −2.93948 − 2.93948i 0 3.10624 + 11.5927i 0 −1.50000 + 2.59808i 0
37.3 0 0.866025 + 1.50000i 0 6.29245 + 6.29245i 0 −1.13421 − 4.23291i 0 −1.50000 + 2.59808i 0
85.1 0 −0.866025 + 1.50000i 0 −5.87695 − 5.87695i 0 11.7827 + 3.15717i 0 −1.50000 − 2.59808i 0
85.2 0 −0.866025 + 1.50000i 0 −0.695927 − 0.695927i 0 −12.8099 − 3.43240i 0 −1.50000 − 2.59808i 0
85.3 0 −0.866025 + 1.50000i 0 1.84083 + 1.84083i 0 1.89319 + 0.507277i 0 −1.50000 − 2.59808i 0
97.1 0 0.866025 − 1.50000i 0 −4.62092 + 4.62092i 0 −2.83806 + 10.5918i 0 −1.50000 − 2.59808i 0
97.2 0 0.866025 − 1.50000i 0 −2.93948 + 2.93948i 0 3.10624 − 11.5927i 0 −1.50000 − 2.59808i 0
97.3 0 0.866025 − 1.50000i 0 6.29245 − 6.29245i 0 −1.13421 + 4.23291i 0 −1.50000 − 2.59808i 0
145.1 0 −0.866025 − 1.50000i 0 −5.87695 + 5.87695i 0 11.7827 − 3.15717i 0 −1.50000 + 2.59808i 0
145.2 0 −0.866025 − 1.50000i 0 −0.695927 + 0.695927i 0 −12.8099 + 3.43240i 0 −1.50000 + 2.59808i 0
145.3 0 −0.866025 − 1.50000i 0 1.84083 − 1.84083i 0 1.89319 − 0.507277i 0 −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.x.b ✓ 12
3.b odd 2 1 468.3.cd.d 12
13.f odd 12 1 inner 156.3.x.b ✓ 12
39.k even 12 1 468.3.cd.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.x.b ✓ 12 1.a even 1 1 trivial
156.3.x.b ✓ 12 13.f odd 12 1 inner
468.3.cd.d 12 3.b odd 2 1
468.3.cd.d 12 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 12 T_{5}^{11} + 72 T_{5}^{10} + 192 T_{5}^{9} + 6318 T_{5}^{8} + 75312 T_{5}^{7} + \cdots + 26501904 \) acting on \(S_{3}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 12 T^{11} + \cdots + 26501904 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 33437048164 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 1234392105024 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16332840000 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 7597233790864 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 235433671447104 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 90\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{6} + 48 T^{5} + \cdots - 565036200)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 60\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 820139167877184 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( (T^{6} - 24 T^{5} + \cdots - 25596936)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
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