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Properties

Label 900.2.h.b
Level $900$
Weight $2$
Character orbit 900.h
Analytic conductor $7.187$
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] = [900,2,Mod(899,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.h (of order \(2\), degree \(1\), 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 180)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + (\beta_{5} + \beta_{3}) q^{4} + ( - \beta_{6} - 2 \beta_{4}) q^{7} + (\beta_{7} - \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{2} + (\beta_{5} + \beta_{3}) q^{4} + ( - \beta_{6} - 2 \beta_{4}) q^{7} + (\beta_{7} - \beta_{5}) q^{8} + ( - \beta_{2} - \beta_1) q^{11} + (2 \beta_{6} - \beta_{2} + \beta_1) q^{13} + (3 \beta_{6} + 2 \beta_{4} + \cdots + \beta_1) q^{14}+ \cdots + ( - 8 \beta_{5} - 7 \beta_{3} - 15) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} - 4 q^{8} + 4 q^{16} + 32 q^{17} + 36 q^{32} - 16 q^{34} + 32 q^{38} - 40 q^{46} + 56 q^{49} + 64 q^{53} + 16 q^{61} + 40 q^{62} - 52 q^{64} - 16 q^{68} + 48 q^{76} - 16 q^{77} + 8 q^{92} - 48 q^{94} - 92 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu^{7} - 10\nu^{6} + 35\nu^{5} - 60\nu^{4} + 86\nu^{3} - 72\nu^{2} + 38\nu - 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3\nu^{7} + 11\nu^{6} - 38\nu^{5} + 70\nu^{4} - 101\nu^{3} + 89\nu^{2} - 48\nu + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 78\nu - 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 116\nu^{4} - 172\nu^{3} + 158\nu^{2} - 94\nu + 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 9\nu^{7} - 31\nu^{6} + 108\nu^{5} - 190\nu^{4} + 275\nu^{3} - 235\nu^{2} + 130\nu - 31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 166\nu - 42 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -14\nu^{7} + 50\nu^{6} - 174\nu^{5} + 315\nu^{4} - 460\nu^{3} + 406\nu^{2} - 232\nu + 58 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - \beta_{5} + 3\beta_{3} + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 2\beta_{3} - \beta_{2} - \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 4\beta_{6} + 5\beta_{5} - 5\beta_{3} - 6\beta _1 - 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{6} + \beta_{5} + \beta_{4} - 5\beta_{3} + 3\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{7} + 2\beta_{6} - 23\beta_{5} + 10\beta_{4} - \beta_{3} + 10\beta_{2} + 30\beta _1 + 45 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} - 20\beta_{6} - 22\beta_{5} - 5\beta_{4} + 42\beta_{3} - 26\beta_{2} + 19\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 65\beta_{7} - 54\beta_{6} + 71\beta_{5} - 70\beta_{4} + 93\beta_{3} - 98\beta_{2} - 98\beta _1 - 157 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
899.1
0.500000 + 0.0297061i
0.500000 + 1.44392i
0.500000 − 0.0297061i
0.500000 − 1.44392i
0.500000 + 2.10607i
0.500000 + 0.691860i
0.500000 − 2.10607i
0.500000 − 0.691860i
−1.20711 − 0.736813i 0 0.914214 + 1.77882i 0 0 −5.03127 0.207107 − 2.82083i 0 0
899.2 −1.20711 − 0.736813i 0 0.914214 + 1.77882i 0 0 5.03127 0.207107 − 2.82083i 0 0
899.3 −1.20711 + 0.736813i 0 0.914214 − 1.77882i 0 0 −5.03127 0.207107 + 2.82083i 0 0
899.4 −1.20711 + 0.736813i 0 0.914214 − 1.77882i 0 0 5.03127 0.207107 + 2.82083i 0 0
899.5 0.207107 − 1.39897i 0 −1.91421 − 0.579471i 0 0 −1.63899 −1.20711 + 2.55791i 0 0
899.6 0.207107 − 1.39897i 0 −1.91421 − 0.579471i 0 0 1.63899 −1.20711 + 2.55791i 0 0
899.7 0.207107 + 1.39897i 0 −1.91421 + 0.579471i 0 0 −1.63899 −1.20711 − 2.55791i 0 0
899.8 0.207107 + 1.39897i 0 −1.91421 + 0.579471i 0 0 1.63899 −1.20711 − 2.55791i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 899.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.h.b 8
3.b odd 2 1 900.2.h.c 8
4.b odd 2 1 inner 900.2.h.b 8
5.b even 2 1 900.2.h.c 8
5.c odd 4 1 180.2.e.a ✓ 8
5.c odd 4 1 900.2.e.d 8
12.b even 2 1 900.2.h.c 8
15.d odd 2 1 inner 900.2.h.b 8
15.e even 4 1 180.2.e.a ✓ 8
15.e even 4 1 900.2.e.d 8
20.d odd 2 1 900.2.h.c 8
20.e even 4 1 180.2.e.a ✓ 8
20.e even 4 1 900.2.e.d 8
40.i odd 4 1 2880.2.h.e 8
40.k even 4 1 2880.2.h.e 8
60.h even 2 1 inner 900.2.h.b 8
60.l odd 4 1 180.2.e.a ✓ 8
60.l odd 4 1 900.2.e.d 8
120.q odd 4 1 2880.2.h.e 8
120.w even 4 1 2880.2.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.e.a ✓ 8 5.c odd 4 1
180.2.e.a ✓ 8 15.e even 4 1
180.2.e.a ✓ 8 20.e even 4 1
180.2.e.a ✓ 8 60.l odd 4 1
900.2.e.d 8 5.c odd 4 1
900.2.e.d 8 15.e even 4 1
900.2.e.d 8 20.e even 4 1
900.2.e.d 8 60.l odd 4 1
900.2.h.b 8 1.a even 1 1 trivial
900.2.h.b 8 4.b odd 2 1 inner
900.2.h.b 8 15.d odd 2 1 inner
900.2.h.b 8 60.h even 2 1 inner
900.2.h.c 8 3.b odd 2 1
900.2.h.c 8 5.b even 2 1
900.2.h.c 8 12.b even 2 1
900.2.h.c 8 20.d odd 2 1
2880.2.h.e 8 40.i odd 4 1
2880.2.h.e 8 40.k even 4 1
2880.2.h.e 8 120.q odd 4 1
2880.2.h.e 8 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} - 28T_{7}^{2} + 68 \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 3 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 28 T^{2} + 68)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 12 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T - 4)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 40 T^{2} + 272)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 72 T^{2} + 784)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 40 T^{2} + 272)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 108 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 112 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 16 T + 56)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 180 T^{2} + 5508)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 160 T^{2} + 4352)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 224 T^{2} + 4352)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 88 T^{2} + 784)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 232 T^{2} + 13328)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 112 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 36 T^{2} + 196)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 216 T^{2} + 1296)^{2} \) Copy content Toggle raw display
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