Properties

Label 1089.1.s.b
Level $1089$
Weight $1$
Character orbit 1089.s
Analytic conductor $0.543$
Analytic rank $0$
Dimension $16$
Projective image $S_{4}$
CM/RM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,1,Mod(40,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.40");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.s (of order \(30\), degree \(8\), not 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: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{30})\)
Coefficient field: 16.0.26873856000000000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.107811.1

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{7} q^{3} + \beta_{2} q^{4} + (\beta_{15} + \beta_{3}) q^{5} - \beta_{8} q^{6} + \beta_{6} q^{7} + ( - \beta_{15} + \beta_{9} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{7} q^{3} + \beta_{2} q^{4} + (\beta_{15} + \beta_{3}) q^{5} - \beta_{8} q^{6} + \beta_{6} q^{7} + ( - \beta_{15} + \beta_{9} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{9} + \beta_{13} q^{10} - \beta_{9} q^{12} + 2 \beta_{7} q^{14} + ( - \beta_{11} + \beta_{2}) q^{15} - \beta_{3} q^{16} + ( - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{8} + \beta_{6} - \beta_1) q^{18} + ( - \beta_{15} + \beta_{9} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{20} - \beta_{13} q^{21} - \beta_{15} q^{27} + \beta_{8} q^{28} - \beta_{6} q^{29} + \beta_{14} q^{30} + (\beta_{15} - \beta_{9} - \beta_{7} + \beta_{3} + \beta_{2} + 1) q^{31} - \beta_{4} q^{32} + (\beta_{10} - \beta_1) q^{35} + (\beta_{15} + \beta_{11} - \beta_{5} + 1) q^{36} - \beta_{11} q^{37} + (2 \beta_{15} - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{42}+ \cdots + \beta_{13} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9} - 8 q^{12} + 4 q^{14} + 2 q^{15} - 2 q^{16} + 2 q^{20} + 4 q^{27} + 2 q^{31} + 4 q^{36} + 4 q^{37} - 4 q^{42} - 16 q^{45} - 2 q^{47} - 4 q^{48} - 2 q^{49} - 4 q^{53} - 4 q^{58} + 2 q^{59} + 4 q^{60} - 4 q^{64} + 8 q^{67} - 4 q^{70} + 4 q^{71} - 4 q^{80} + 2 q^{81} - 2 q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{10} ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{12} ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{13} ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -\nu^{13} + 32\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{14} - 32\nu^{4} ) / 128 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{14} + 2\beta_{12} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{5} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{7} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{9} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32\beta_{10} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 64\beta_{11} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 64\beta_{12} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 128\beta_{15} + 128\beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 128\beta_{13} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-\beta_{11}\) \(-1 + \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−1.40647 0.147826i
1.40647 + 0.147826i
−1.05097 0.946294i
1.05097 + 0.946294i
−0.294032 1.38331i
0.294032 + 1.38331i
−0.575212 1.29195i
0.575212 + 1.29195i
−0.294032 + 1.38331i
0.294032 1.38331i
−1.05097 + 0.946294i
1.05097 0.946294i
−0.575212 + 1.29195i
0.575212 1.29195i
−1.40647 + 0.147826i
1.40647 0.147826i
−1.40647 0.147826i −0.669131 0.743145i 0.978148 + 0.207912i 0.104528 + 0.994522i 0.831254 + 1.14412i −1.05097 0.946294i 0 −0.104528 + 0.994522i 1.41421i
40.2 1.40647 + 0.147826i −0.669131 0.743145i 0.978148 + 0.207912i 0.104528 + 0.994522i −0.831254 1.14412i 1.05097 + 0.946294i 0 −0.104528 + 0.994522i 1.41421i
94.1 −1.05097 0.946294i −0.913545 + 0.406737i 0.104528 + 0.994522i −0.669131 0.743145i 1.34500 + 0.437016i −0.575212 + 1.29195i 0 0.669131 0.743145i 1.41421i
94.2 1.05097 + 0.946294i −0.913545 + 0.406737i 0.104528 + 0.994522i −0.669131 0.743145i −1.34500 0.437016i 0.575212 1.29195i 0 0.669131 0.743145i 1.41421i
112.1 −0.294032 1.38331i 0.104528 + 0.994522i −0.913545 + 0.406737i 0.978148 + 0.207912i 1.34500 0.437016i 1.40647 + 0.147826i 0 −0.978148 + 0.207912i 1.41421i
112.2 0.294032 + 1.38331i 0.104528 + 0.994522i −0.913545 + 0.406737i 0.978148 + 0.207912i −1.34500 + 0.437016i −1.40647 0.147826i 0 −0.978148 + 0.207912i 1.41421i
403.1 −0.575212 1.29195i 0.978148 0.207912i −0.669131 + 0.743145i −0.913545 0.406737i −0.831254 1.14412i 0.294032 1.38331i 0 0.913545 0.406737i 1.41421i
403.2 0.575212 + 1.29195i 0.978148 0.207912i −0.669131 + 0.743145i −0.913545 0.406737i 0.831254 + 1.14412i −0.294032 + 1.38331i 0 0.913545 0.406737i 1.41421i
457.1 −0.294032 + 1.38331i 0.104528 0.994522i −0.913545 0.406737i 0.978148 0.207912i 1.34500 + 0.437016i 1.40647 0.147826i 0 −0.978148 0.207912i 1.41421i
457.2 0.294032 1.38331i 0.104528 0.994522i −0.913545 0.406737i 0.978148 0.207912i −1.34500 0.437016i −1.40647 + 0.147826i 0 −0.978148 0.207912i 1.41421i
475.1 −1.05097 + 0.946294i −0.913545 0.406737i 0.104528 0.994522i −0.669131 + 0.743145i 1.34500 0.437016i −0.575212 1.29195i 0 0.669131 + 0.743145i 1.41421i
475.2 1.05097 0.946294i −0.913545 0.406737i 0.104528 0.994522i −0.669131 + 0.743145i −1.34500 + 0.437016i 0.575212 + 1.29195i 0 0.669131 + 0.743145i 1.41421i
481.1 −0.575212 + 1.29195i 0.978148 + 0.207912i −0.669131 0.743145i −0.913545 + 0.406737i −0.831254 + 1.14412i 0.294032 + 1.38331i 0 0.913545 + 0.406737i 1.41421i
481.2 0.575212 1.29195i 0.978148 + 0.207912i −0.669131 0.743145i −0.913545 + 0.406737i 0.831254 1.14412i −0.294032 1.38331i 0 0.913545 + 0.406737i 1.41421i
844.1 −1.40647 + 0.147826i −0.669131 + 0.743145i 0.978148 0.207912i 0.104528 0.994522i 0.831254 1.14412i −1.05097 + 0.946294i 0 −0.104528 0.994522i 1.41421i
844.2 1.40647 0.147826i −0.669131 + 0.743145i 0.978148 0.207912i 0.104528 0.994522i −0.831254 + 1.14412i 1.05097 0.946294i 0 −0.104528 0.994522i 1.41421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
99.h odd 6 1 inner
99.m even 15 3 inner
99.o odd 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.s.b 16
3.b odd 2 1 3267.1.w.b 16
9.c even 3 1 inner 1089.1.s.b 16
9.d odd 6 1 3267.1.w.b 16
11.b odd 2 1 inner 1089.1.s.b 16
11.c even 5 1 1089.1.h.a 4
11.c even 5 3 inner 1089.1.s.b 16
11.d odd 10 1 1089.1.h.a 4
11.d odd 10 3 inner 1089.1.s.b 16
33.d even 2 1 3267.1.w.b 16
33.f even 10 1 3267.1.h.a 4
33.f even 10 3 3267.1.w.b 16
33.h odd 10 1 3267.1.h.a 4
33.h odd 10 3 3267.1.w.b 16
99.g even 6 1 3267.1.w.b 16
99.h odd 6 1 inner 1089.1.s.b 16
99.m even 15 1 1089.1.h.a 4
99.m even 15 3 inner 1089.1.s.b 16
99.n odd 30 1 3267.1.h.a 4
99.n odd 30 3 3267.1.w.b 16
99.o odd 30 1 1089.1.h.a 4
99.o odd 30 3 inner 1089.1.s.b 16
99.p even 30 1 3267.1.h.a 4
99.p even 30 3 3267.1.w.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.h.a 4 11.c even 5 1
1089.1.h.a 4 11.d odd 10 1
1089.1.h.a 4 99.m even 15 1
1089.1.h.a 4 99.o odd 30 1
1089.1.s.b 16 1.a even 1 1 trivial
1089.1.s.b 16 9.c even 3 1 inner
1089.1.s.b 16 11.b odd 2 1 inner
1089.1.s.b 16 11.c even 5 3 inner
1089.1.s.b 16 11.d odd 10 3 inner
1089.1.s.b 16 99.h odd 6 1 inner
1089.1.s.b 16 99.m even 15 3 inner
1089.1.s.b 16 99.o odd 30 3 inner
3267.1.h.a 4 33.f even 10 1
3267.1.h.a 4 33.h odd 10 1
3267.1.h.a 4 99.n odd 30 1
3267.1.h.a 4 99.p even 30 1
3267.1.w.b 16 3.b odd 2 1
3267.1.w.b 16 9.d odd 6 1
3267.1.w.b 16 33.d even 2 1
3267.1.w.b 16 33.f even 10 3
3267.1.w.b 16 33.h odd 10 3
3267.1.w.b 16 99.g even 6 1
3267.1.w.b 16 99.n odd 30 3
3267.1.w.b 16 99.p even 30 3

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 2T_{2}^{14} - 8T_{2}^{10} - 16T_{2}^{8} - 32T_{2}^{6} + 128T_{2}^{2} + 256 \) acting on \(S_{1}^{\mathrm{new}}(1089, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( (T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( (T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1)^{2} \) Copy content Toggle raw display
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