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Properties

Label 144.2.l.a
Level $144$
Weight $2$
Character orbit 144.l
Analytic conductor $1.150$
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] = [144,2,Mod(35,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.l (of order \(4\), degree \(2\), 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: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} - \beta_{7} q^{4} + \beta_{6} q^{5} + ( - \beta_{9} - \beta_{4}) q^{7} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} - \beta_{7} q^{4} + \beta_{6} q^{5} + ( - \beta_{9} - \beta_{4}) q^{7} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_{2}) q^{8}+ \cdots + ( - 4 \beta_{14} + \beta_{11} + \cdots - 2 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{10} - 16 q^{16} + 16 q^{19} - 40 q^{22} - 24 q^{28} + 24 q^{34} + 72 q^{40} - 32 q^{43} + 40 q^{46} + 16 q^{49} + 24 q^{52} - 64 q^{55} + 24 q^{58} - 32 q^{61} - 48 q^{64} - 16 q^{67} - 72 q^{70} + 80 q^{82} - 32 q^{85} + 48 q^{88} + 48 q^{91} + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} + 21\nu^{12} - 18\nu^{10} + 18\nu^{8} - 189\nu^{6} + 81\nu^{4} + 72\nu^{2} + 1072 ) / 480 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{15} - 6\nu^{11} - 12\nu^{9} - 81\nu^{7} - 276\nu^{5} + 336\nu^{3} + 64\nu ) / 1536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{14} + 2\nu^{12} - 6\nu^{10} + 16\nu^{8} + 27\nu^{6} + 62\nu^{4} - 256\nu^{2} - 96 ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{14} + 24\nu^{12} + 18\nu^{10} + 372\nu^{8} - 141\nu^{6} - 876\nu^{4} - 1872\nu^{2} + 1088 ) / 1920 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\nu^{14} - 108\nu^{12} + 114\nu^{10} - 324\nu^{8} + 747\nu^{6} - 528\nu^{4} + 3024\nu^{2} - 6016 ) / 1920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\nu^{15} - 66\nu^{13} + 138\nu^{11} - 168\nu^{9} + 339\nu^{7} - 486\nu^{5} + 1008\nu^{3} - 1952\nu ) / 1920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{14} - \nu^{12} + 2\nu^{10} - 10\nu^{8} + 13\nu^{6} - 13\nu^{4} + 56\nu^{2} - 80 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27\nu^{15} - 72\nu^{13} + 66\nu^{11} - 236\nu^{9} + 1003\nu^{7} - 812\nu^{5} + 1936\nu^{3} - 2624\nu ) / 2560 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 89\nu^{14} - 204\nu^{12} + 102\nu^{10} - 732\nu^{8} + 1401\nu^{6} - 984\nu^{4} + 4752\nu^{2} - 9728 ) / 1920 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\nu^{14} - 24\nu^{12} + 18\nu^{10} - 108\nu^{8} + 219\nu^{6} - 156\nu^{4} + 864\nu^{2} - 1472 ) / 192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -39\nu^{15} + 64\nu^{13} - 42\nu^{11} + 172\nu^{9} - 471\nu^{7} + 564\nu^{5} - 1552\nu^{3} + 4288\nu ) / 2560 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} + \nu^{13} - 2\nu^{11} + 10\nu^{9} - 13\nu^{7} + 13\nu^{5} - 56\nu^{3} + 80\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -71\nu^{15} + 216\nu^{13} - 138\nu^{11} + 828\nu^{9} - 2199\nu^{7} + 1356\nu^{5} - 5328\nu^{3} + 14912\nu ) / 3840 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 89\nu^{15} - 204\nu^{13} + 102\nu^{11} - 732\nu^{9} + 1401\nu^{7} - 984\nu^{5} + 4752\nu^{3} - 9728\nu ) / 3840 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 149\nu^{15} - 264\nu^{13} + 222\nu^{11} - 1332\nu^{9} + 2181\nu^{7} - 1764\nu^{5} + 11952\nu^{3} - 18368\nu ) / 3840 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} + \beta_{11} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{15} - \beta_{14} + \beta_{13} + 2\beta_{12} + \beta_{11} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{9} - \beta_{7} + 3\beta_{5} - 3\beta_{4} + \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{15} + \beta_{14} + \beta_{13} + 2\beta_{12} + 3\beta_{11} - \beta_{8} - 8\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{10} - 5\beta_{9} - \beta_{7} - 3\beta_{5} + 4\beta_{3} - 3\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{14} - \beta_{13} + \beta_{11} + 7\beta_{8} - 2\beta_{6} - 6\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4\beta_{10} - 6\beta_{9} - 9\beta_{7} + 15\beta_{5} + 5\beta_{4} + 9\beta_{3} + 7\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8\beta_{15} - 13\beta_{14} + 7\beta_{13} + 20\beta_{12} - 21\beta_{11} + 9\beta_{8} + 2\beta_{6} - 10\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -15\beta_{10} - 17\beta_{9} + 19\beta_{7} + 45\beta_{5} - 4\beta_{4} - 4\beta_{3} - 11\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6\beta_{15} - 29\beta_{14} + 5\beta_{13} - 10\beta_{12} - 15\beta_{11} - 3\beta_{8} + 28\beta_{6} - 28\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 24\beta_{10} - 42\beta_{9} + 23\beta_{7} - 13\beta_{5} + 5\beta_{4} + 25\beta_{3} + 7\beta _1 - 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 6\beta_{15} - 91\beta_{14} - 19\beta_{13} - 66\beta_{12} - 37\beta_{11} + 3\beta_{8} - 20\beta_{6} - 20\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -27\beta_{10} + 27\beta_{9} + 15\beta_{7} + 69\beta_{5} + 24\beta_{4} + 84\beta_{3} + 69\beta _1 + 41 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 12 \beta_{15} + 25 \beta_{14} + 91 \beta_{13} - 60 \beta_{12} - 167 \beta_{11} + 19 \beta_{8} + \cdots - 90 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{5}\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
0.944649 − 1.05244i
−1.36166 + 0.381939i
−1.40927 + 0.118126i
0.517174 + 1.31626i
−0.517174 − 1.31626i
1.40927 − 0.118126i
1.36166 − 0.381939i
−0.944649 + 1.05244i
0.944649 + 1.05244i
−1.36166 − 0.381939i
−1.40927 − 0.118126i
0.517174 − 1.31626i
−0.517174 + 1.31626i
1.40927 + 0.118126i
1.36166 + 0.381939i
−0.944649 − 1.05244i
−1.32068 + 0.505776i 0 1.48838 − 1.33594i −2.10489 − 2.10489i 0 −4.40731 −1.28999 + 2.51713i 0 3.84448 + 1.71528i
35.2 −1.12697 + 0.854358i 0 0.540143 − 1.92568i 0.763878 + 0.763878i 0 1.33620 1.03649 + 2.63167i 0 −1.51350 − 0.208245i
35.3 −0.957325 − 1.04093i 0 −0.167056 + 1.99301i 0.236253 + 0.236253i 0 3.27830 2.23450 − 1.73407i 0 0.0197510 − 0.472092i
35.4 −0.263185 + 1.38951i 0 −1.86147 − 0.731395i 2.63251 + 2.63251i 0 −0.207188 1.50619 − 2.39403i 0 −4.35074 + 2.96506i
35.5 0.263185 − 1.38951i 0 −1.86147 − 0.731395i −2.63251 − 2.63251i 0 −0.207188 −1.50619 + 2.39403i 0 −4.35074 + 2.96506i
35.6 0.957325 + 1.04093i 0 −0.167056 + 1.99301i −0.236253 − 0.236253i 0 3.27830 −2.23450 + 1.73407i 0 0.0197510 − 0.472092i
35.7 1.12697 − 0.854358i 0 0.540143 − 1.92568i −0.763878 − 0.763878i 0 1.33620 −1.03649 − 2.63167i 0 −1.51350 − 0.208245i
35.8 1.32068 − 0.505776i 0 1.48838 − 1.33594i 2.10489 + 2.10489i 0 −4.40731 1.28999 − 2.51713i 0 3.84448 + 1.71528i
107.1 −1.32068 − 0.505776i 0 1.48838 + 1.33594i −2.10489 + 2.10489i 0 −4.40731 −1.28999 − 2.51713i 0 3.84448 − 1.71528i
107.2 −1.12697 − 0.854358i 0 0.540143 + 1.92568i 0.763878 − 0.763878i 0 1.33620 1.03649 − 2.63167i 0 −1.51350 + 0.208245i
107.3 −0.957325 + 1.04093i 0 −0.167056 − 1.99301i 0.236253 − 0.236253i 0 3.27830 2.23450 + 1.73407i 0 0.0197510 + 0.472092i
107.4 −0.263185 − 1.38951i 0 −1.86147 + 0.731395i 2.63251 − 2.63251i 0 −0.207188 1.50619 + 2.39403i 0 −4.35074 − 2.96506i
107.5 0.263185 + 1.38951i 0 −1.86147 + 0.731395i −2.63251 + 2.63251i 0 −0.207188 −1.50619 − 2.39403i 0 −4.35074 − 2.96506i
107.6 0.957325 − 1.04093i 0 −0.167056 − 1.99301i −0.236253 + 0.236253i 0 3.27830 −2.23450 − 1.73407i 0 0.0197510 + 0.472092i
107.7 1.12697 + 0.854358i 0 0.540143 + 1.92568i −0.763878 + 0.763878i 0 1.33620 −1.03649 + 2.63167i 0 −1.51350 + 0.208245i
107.8 1.32068 + 0.505776i 0 1.48838 + 1.33594i 2.10489 − 2.10489i 0 −4.40731 1.28999 + 2.51713i 0 3.84448 − 1.71528i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.l.a ✓ 16
3.b odd 2 1 inner 144.2.l.a ✓ 16
4.b odd 2 1 576.2.l.a 16
8.b even 2 1 1152.2.l.a 16
8.d odd 2 1 1152.2.l.b 16
12.b even 2 1 576.2.l.a 16
16.e even 4 1 576.2.l.a 16
16.e even 4 1 1152.2.l.b 16
16.f odd 4 1 inner 144.2.l.a ✓ 16
16.f odd 4 1 1152.2.l.a 16
24.f even 2 1 1152.2.l.b 16
24.h odd 2 1 1152.2.l.a 16
48.i odd 4 1 576.2.l.a 16
48.i odd 4 1 1152.2.l.b 16
48.k even 4 1 inner 144.2.l.a ✓ 16
48.k even 4 1 1152.2.l.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.l.a ✓ 16 1.a even 1 1 trivial
144.2.l.a ✓ 16 3.b odd 2 1 inner
144.2.l.a ✓ 16 16.f odd 4 1 inner
144.2.l.a ✓ 16 48.k even 4 1 inner
576.2.l.a 16 4.b odd 2 1
576.2.l.a 16 12.b even 2 1
576.2.l.a 16 16.e even 4 1
576.2.l.a 16 48.i odd 4 1
1152.2.l.a 16 8.b even 2 1
1152.2.l.a 16 16.f odd 4 1
1152.2.l.a 16 24.h odd 2 1
1152.2.l.a 16 48.k even 4 1
1152.2.l.b 16 8.d odd 2 1
1152.2.l.b 16 16.e even 4 1
1152.2.l.b 16 24.f even 2 1
1152.2.l.b 16 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 272 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 16 T + 4)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + 960 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{8} + 64 T^{5} + \cdots + 400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 72 T^{6} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 96 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 16062013696 \) Copy content Toggle raw display
$31$ \( (T^{8} + 192 T^{6} + \cdots + 1648656)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 512 T^{5} + \cdots + 35344)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 168 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 16 T^{7} + \cdots + 4129024)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 160 T^{6} + \cdots + 665856)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 22663495936 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2186423566336 \) Copy content Toggle raw display
$61$ \( (T^{8} + 16 T^{7} + \cdots + 258064)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 8 T^{7} + \cdots + 7573504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 192 T^{6} + \cdots + 73984)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 176 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 224 T^{6} + \cdots + 3825936)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 102776124276736 \) Copy content Toggle raw display
$89$ \( (T^{8} - 200 T^{6} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 72 T^{2} + \cdots - 176)^{4} \) Copy content Toggle raw display
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