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Properties

Label 900.3.u.b
Level $900$
Weight $3$
Character orbit 900.u
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
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] = [900,3,Mod(149,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.u (of order \(6\), degree \(2\), 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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + (\beta_{6} + 4 \beta_1) q^{7} - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + (\beta_{6} + 4 \beta_1) q^{7} - 9 \beta_{2} q^{9} + ( - \beta_{7} - \beta_{5} - \beta_{2} + 1) q^{11} + ( - \beta_{4} + 10 \beta_{3} + 10 \beta_1) q^{13} + ( - \beta_{6} + \beta_{4} + \cdots - 2 \beta_1) q^{17}+ \cdots + ( - 18 \beta_{7} + 9 \beta_{5} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{9} + 12 q^{11} + 152 q^{19} + 48 q^{21} + 120 q^{29} - 8 q^{31} + 240 q^{39} - 180 q^{41} + 108 q^{49} - 36 q^{51} + 348 q^{59} + 88 q^{61} + 320 q^{79} - 324 q^{81} + 800 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 13\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} + 24\nu^{4} - 56\nu^{2} + 39 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -11\nu^{6} + 24\nu^{4} - 56\nu^{2} - 15 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{7} - 32\nu^{5} + 88\nu^{3} + 25\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + \beta_{4} - 18\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - \beta_{5} + 12\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{6} + 2\beta_{4} - 14\beta_{2} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 10\beta_{5} + 66\beta_{3} + 66\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{6} + 2\beta_{4} - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 13\beta_{5} + 174\beta_1 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.535233 + 0.309017i
1.40126 − 0.809017i
0.535233 − 0.309017i
−1.40126 + 0.809017i
−0.535233 − 0.309017i
1.40126 + 0.809017i
0.535233 + 0.309017i
−1.40126 − 0.809017i
0 −2.59808 + 1.50000i 0 0 0 −10.1723 + 5.87298i 0 4.50000 − 7.79423i 0
149.2 0 −2.59808 + 1.50000i 0 0 0 3.24410 − 1.87298i 0 4.50000 − 7.79423i 0
149.3 0 2.59808 − 1.50000i 0 0 0 −3.24410 + 1.87298i 0 4.50000 − 7.79423i 0
149.4 0 2.59808 − 1.50000i 0 0 0 10.1723 − 5.87298i 0 4.50000 − 7.79423i 0
749.1 0 −2.59808 − 1.50000i 0 0 0 −10.1723 − 5.87298i 0 4.50000 + 7.79423i 0
749.2 0 −2.59808 − 1.50000i 0 0 0 3.24410 + 1.87298i 0 4.50000 + 7.79423i 0
749.3 0 2.59808 + 1.50000i 0 0 0 −3.24410 − 1.87298i 0 4.50000 + 7.79423i 0
749.4 0 2.59808 + 1.50000i 0 0 0 10.1723 + 5.87298i 0 4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.u.b 8
3.b odd 2 1 2700.3.u.a 8
5.b even 2 1 inner 900.3.u.b 8
5.c odd 4 1 180.3.o.a ✓ 4
5.c odd 4 1 900.3.p.b 4
9.c even 3 1 2700.3.u.a 8
9.d odd 6 1 inner 900.3.u.b 8
15.d odd 2 1 2700.3.u.a 8
15.e even 4 1 540.3.o.a 4
15.e even 4 1 2700.3.p.a 4
20.e even 4 1 720.3.bs.a 4
45.h odd 6 1 inner 900.3.u.b 8
45.j even 6 1 2700.3.u.a 8
45.k odd 12 1 540.3.o.a 4
45.k odd 12 1 1620.3.g.a 4
45.k odd 12 1 2700.3.p.a 4
45.l even 12 1 180.3.o.a ✓ 4
45.l even 12 1 900.3.p.b 4
45.l even 12 1 1620.3.g.a 4
60.l odd 4 1 2160.3.bs.a 4
180.v odd 12 1 720.3.bs.a 4
180.x even 12 1 2160.3.bs.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.a ✓ 4 5.c odd 4 1
180.3.o.a ✓ 4 45.l even 12 1
540.3.o.a 4 15.e even 4 1
540.3.o.a 4 45.k odd 12 1
720.3.bs.a 4 20.e even 4 1
720.3.bs.a 4 180.v odd 12 1
900.3.p.b 4 5.c odd 4 1
900.3.p.b 4 45.l even 12 1
900.3.u.b 8 1.a even 1 1 trivial
900.3.u.b 8 5.b even 2 1 inner
900.3.u.b 8 9.d odd 6 1 inner
900.3.u.b 8 45.h odd 6 1 inner
1620.3.g.a 4 45.k odd 12 1
1620.3.g.a 4 45.l even 12 1
2160.3.bs.a 4 60.l odd 4 1
2160.3.bs.a 4 180.x even 12 1
2700.3.p.a 4 15.e even 4 1
2700.3.p.a 4 45.k odd 12 1
2700.3.u.a 8 3.b odd 2 1
2700.3.u.a 8 9.c even 3 1
2700.3.u.a 8 15.d odd 2 1
2700.3.u.a 8 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 152T_{7}^{6} + 21168T_{7}^{4} - 294272T_{7}^{2} + 3748096 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 152 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$11$ \( (T^{4} - 6 T^{3} + \cdots + 31329)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 320 T^{6} + \cdots + 2560000 \) Copy content Toggle raw display
$17$ \( (T^{4} - 366 T^{2} + 31329)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 38 T + 301)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 203928109056 \) Copy content Toggle raw display
$29$ \( (T^{4} - 60 T^{3} + \cdots + 176400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 8620096)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 196)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 45 T + 675)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 495504774241 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 116876510171136 \) Copy content Toggle raw display
$53$ \( (T^{4} - 1536 T^{2} + 166464)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 174 T^{3} + \cdots + 5489649)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 44 T^{3} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 54\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{4} + 11016 T^{2} + 5143824)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3818 T^{2} + 687241)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 160 T^{3} + \cdots + 34339600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8112 T^{2} + 65804544)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 9216 T^{2} + 1327104)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
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