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Properties

Label 1080.2.k.c
Level $1080$
Weight $2$
Character orbit 1080.k
Analytic conductor $8.624$
Analytic rank $0$
Dimension $16$
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] = [1080,2,Mod(541,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.k (of order \(2\), degree \(1\), 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: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 2 x^{13} + 2 x^{12} - 4 x^{11} + 6 x^{10} - 16 x^{9} + 8 x^{8} - 32 x^{7} + 24 x^{6} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{8} q^{5} - \beta_{11} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{8} q^{5} - \beta_{11} q^{7} + \beta_{3} q^{8} - \beta_{10} q^{10} + (\beta_{14} - \beta_{10} + \cdots - \beta_1) q^{11}+ \cdots + ( - 3 \beta_{14} - \beta_{13} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{4} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{4} + 6 q^{8} + 2 q^{10} - 10 q^{14} - 6 q^{16} - 24 q^{17} + 20 q^{22} - 16 q^{25} + 14 q^{26} + 20 q^{28} - 8 q^{31} + 10 q^{32} - 6 q^{34} - 26 q^{38} + 4 q^{40} + 40 q^{41} - 40 q^{44} - 6 q^{46} + 8 q^{47} + 24 q^{49} + 4 q^{52} + 48 q^{56} - 38 q^{62} - 14 q^{64} - 50 q^{68} + 6 q^{70} - 8 q^{73} + 62 q^{74} - 28 q^{76} + 40 q^{79} + 8 q^{80} + 32 q^{82} - 64 q^{86} + 32 q^{88} + 56 q^{89} - 66 q^{92} + 6 q^{94} - 8 q^{95} + 8 q^{97} + 66 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 2 x^{13} + 2 x^{12} - 4 x^{11} + 6 x^{10} - 16 x^{9} + 8 x^{8} - 32 x^{7} + 24 x^{6} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} - 10 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} - 14 \nu^{11} - 8 \nu^{10} - 42 \nu^{9} - 44 \nu^{8} + \cdots + 128 ) / 512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 8 \nu^{12} + 6 \nu^{11} - 16 \nu^{10} - 6 \nu^{9} - 28 \nu^{8} + \cdots + 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} + 14 \nu^{14} - 19 \nu^{13} + 36 \nu^{12} - 46 \nu^{11} + 120 \nu^{10} - 106 \nu^{9} + \cdots + 128 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} - 3 \nu^{13} + 20 \nu^{12} - 30 \nu^{11} + 56 \nu^{10} - 106 \nu^{9} + \cdots + 1152 ) / 512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{15} + 6 \nu^{14} - 7 \nu^{13} + 20 \nu^{12} - 22 \nu^{11} + 40 \nu^{10} - 34 \nu^{9} + \cdots - 384 ) / 512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + \nu^{13} - 4 \nu^{12} - 2 \nu^{11} + 8 \nu^{10} - 10 \nu^{9} + 4 \nu^{8} + \cdots + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{15} - 2 \nu^{14} + 7 \nu^{13} - 8 \nu^{12} + 14 \nu^{11} - 8 \nu^{10} + 2 \nu^{9} - 28 \nu^{8} + \cdots + 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{15} + 2 \nu^{14} + \nu^{13} + 10 \nu^{11} + 14 \nu^{9} - 4 \nu^{8} + 32 \nu^{7} - 64 \nu^{6} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} + \nu^{13} + 2 \nu^{12} - 6 \nu^{11} + 8 \nu^{10} - 10 \nu^{9} + 16 \nu^{8} - 24 \nu^{7} + \cdots + 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} + \nu^{13} - 2 \nu^{12} + 2 \nu^{11} - 4 \nu^{10} + 6 \nu^{9} - 16 \nu^{8} + 8 \nu^{7} + \cdots + 64 \nu ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 7 \nu^{15} + 22 \nu^{14} - 27 \nu^{13} + 44 \nu^{12} - 78 \nu^{11} + 120 \nu^{10} - 154 \nu^{9} + \cdots + 128 ) / 512 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7 \nu^{15} + 6 \nu^{14} - 11 \nu^{13} + 20 \nu^{12} - 30 \nu^{11} + 32 \nu^{10} - 42 \nu^{9} + \cdots - 640 ) / 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{11} + 3 \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} - 3 \beta_{5} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{15} - \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{9} - 9 \beta_{8} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - \beta_{15} - \beta_{14} - \beta_{11} - 7 \beta_{10} + 4 \beta_{9} + \beta_{8} + \beta_{7} - 3 \beta_{6} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( \beta_{15} - 3 \beta_{14} + 5 \beta_{11} + 3 \beta_{10} + 8 \beta_{9} + 11 \beta_{8} - 5 \beta_{7} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - \beta_{15} - \beta_{14} - 4 \beta_{13} - 12 \beta_{12} - 5 \beta_{11} + 13 \beta_{10} + 5 \beta_{8} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3 \beta_{15} - 15 \beta_{14} - 8 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} - \beta_{10} - 20 \beta_{9} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11 \beta_{15} - 5 \beta_{14} - 4 \beta_{13} + 20 \beta_{12} + 7 \beta_{11} + 25 \beta_{10} + 25 \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 21 \beta_{15} + 29 \beta_{14} + 36 \beta_{13} - 16 \beta_{12} - 7 \beta_{11} + 47 \beta_{10} + 8 \beta_{9} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 21 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} + 12 \beta_{12} - 17 \beta_{11} + \beta_{10} - 32 \beta_{9} + \cdots - 17 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
−1.23284 − 0.692893i
−1.23284 + 0.692893i
−1.19981 − 0.748637i
−1.19981 + 0.748637i
−0.712695 − 1.22150i
−0.712695 + 1.22150i
−0.334028 − 1.37420i
−0.334028 + 1.37420i
0.138827 − 1.40738i
0.138827 + 1.40738i
0.807363 − 1.16111i
0.807363 + 1.16111i
1.12266 − 0.860014i
1.12266 + 0.860014i
1.41052 − 0.102150i
1.41052 + 0.102150i
−1.23284 − 0.692893i 0 1.03980 + 1.70846i 1.00000i 0 −0.176600 −0.0981293 − 2.82672i 0 0.692893 − 1.23284i
541.2 −1.23284 + 0.692893i 0 1.03980 − 1.70846i − 1.00000i 0 −0.176600 −0.0981293 + 2.82672i 0 0.692893 + 1.23284i
541.3 −1.19981 − 0.748637i 0 0.879085 + 1.79644i − 1.00000i 0 2.18630 0.290149 − 2.81351i 0 −0.748637 + 1.19981i
541.4 −1.19981 + 0.748637i 0 0.879085 − 1.79644i 1.00000i 0 2.18630 0.290149 + 2.81351i 0 −0.748637 − 1.19981i
541.5 −0.712695 − 1.22150i 0 −0.984132 + 1.74112i 1.00000i 0 3.54638 2.82816 − 0.0387664i 0 1.22150 − 0.712695i
541.6 −0.712695 + 1.22150i 0 −0.984132 − 1.74112i − 1.00000i 0 3.54638 2.82816 + 0.0387664i 0 1.22150 + 0.712695i
541.7 −0.334028 − 1.37420i 0 −1.77685 + 0.918043i − 1.00000i 0 −2.97489 1.85509 + 2.13509i 0 −1.37420 + 0.334028i
541.8 −0.334028 + 1.37420i 0 −1.77685 − 0.918043i 1.00000i 0 −2.97489 1.85509 − 2.13509i 0 −1.37420 − 0.334028i
541.9 0.138827 − 1.40738i 0 −1.96145 − 0.390767i 1.00000i 0 −0.468215 −0.822262 + 2.70627i 0 1.40738 + 0.138827i
541.10 0.138827 + 1.40738i 0 −1.96145 + 0.390767i − 1.00000i 0 −0.468215 −0.822262 − 2.70627i 0 1.40738 − 0.138827i
541.11 0.807363 − 1.16111i 0 −0.696330 − 1.87487i − 1.00000i 0 −0.855386 −2.73911 − 0.705185i 0 −1.16111 − 0.807363i
541.12 0.807363 + 1.16111i 0 −0.696330 + 1.87487i 1.00000i 0 −0.855386 −2.73911 + 0.705185i 0 −1.16111 + 0.807363i
541.13 1.12266 − 0.860014i 0 0.520752 − 1.93101i 1.00000i 0 −5.10207 −1.07607 − 2.61574i 0 0.860014 + 1.12266i
541.14 1.12266 + 0.860014i 0 0.520752 + 1.93101i − 1.00000i 0 −5.10207 −1.07607 + 2.61574i 0 0.860014 − 1.12266i
541.15 1.41052 − 0.102150i 0 1.97913 − 0.288170i 1.00000i 0 3.84449 2.76217 − 0.608638i 0 0.102150 + 1.41052i
541.16 1.41052 + 0.102150i 0 1.97913 + 0.288170i − 1.00000i 0 3.84449 2.76217 + 0.608638i 0 0.102150 − 1.41052i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.k.c yes 16
3.b odd 2 1 1080.2.k.b ✓ 16
4.b odd 2 1 4320.2.k.b 16
8.b even 2 1 inner 1080.2.k.c yes 16
8.d odd 2 1 4320.2.k.b 16
12.b even 2 1 4320.2.k.c 16
24.f even 2 1 4320.2.k.c 16
24.h odd 2 1 1080.2.k.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.k.b ✓ 16 3.b odd 2 1
1080.2.k.b ✓ 16 24.h odd 2 1
1080.2.k.c yes 16 1.a even 1 1 trivial
1080.2.k.c yes 16 8.b even 2 1 inner
4320.2.k.b 16 4.b odd 2 1
4320.2.k.b 16 8.d odd 2 1
4320.2.k.c 16 12.b even 2 1
4320.2.k.c 16 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\):

\( T_{7}^{8} - 34T_{7}^{6} + 16T_{7}^{5} + 289T_{7}^{4} - 96T_{7}^{3} - 540T_{7}^{2} - 272T_{7} - 32 \) Copy content Toggle raw display
\( T_{17}^{8} + 12T_{17}^{7} + 8T_{17}^{6} - 404T_{17}^{5} - 1893T_{17}^{4} - 3252T_{17}^{3} - 2472T_{17}^{2} - 792T_{17} - 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 34 T^{6} + \cdots - 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 96 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{16} + 116 T^{14} + \cdots + 2259009 \) Copy content Toggle raw display
$17$ \( (T^{8} + 12 T^{7} + \cdots - 72)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 132 T^{14} + \cdots + 430336 \) Copy content Toggle raw display
$23$ \( (T^{8} - 78 T^{6} + \cdots + 8464)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 252 T^{14} + \cdots + 19289664 \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + \cdots + 65392)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 669988086784 \) Copy content Toggle raw display
$41$ \( (T^{8} - 20 T^{7} + \cdots - 256)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 855445504 \) Copy content Toggle raw display
$47$ \( (T^{8} - 4 T^{7} + \cdots - 118072)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 433532698624 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1011167002624 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10751788536064 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 151613669376 \) Copy content Toggle raw display
$71$ \( (T^{8} - 352 T^{6} + \cdots + 30652544)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 4 T^{7} + \cdots + 317952)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 20 T^{7} + \cdots - 4755497)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 190626545664 \) Copy content Toggle raw display
$89$ \( (T^{8} - 28 T^{7} + \cdots + 10877824)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 4 T^{7} + \cdots - 13677056)^{2} \) Copy content Toggle raw display
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