↑

Properties

Label 450.4.f.d
Level $450$
Weight $4$
Character orbit 450.f
Analytic conductor $26.551$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,4,Mod(107,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.f (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: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2}) q^{2} - 4 \beta_1 q^{4} + (\beta_{4} - 6 \beta_1 - 6) q^{7} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2}) q^{2} - 4 \beta_1 q^{4} + (\beta_{4} - 6 \beta_1 - 6) q^{7} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{8} - 3 \beta_{3} q^{11} + ( - 3 \beta_{5} + 24 \beta_1 - 24) q^{13} + ( - \beta_{7} - \beta_{6} + 12 \beta_{2}) q^{14} - 16 q^{16} + ( - 3 \beta_{7} - 6 \beta_{3} - 6 \beta_{2}) q^{17} + ( - 6 \beta_{5} - 6 \beta_{4} - 37 \beta_1) q^{19} + ( - 6 \beta_1 - 6) q^{22} + (9 \beta_{6} - 24 \beta_{3} + 24 \beta_{2}) q^{23} + ( - 3 \beta_{7} + 3 \beta_{6} + 48 \beta_{3}) q^{26} + ( - 4 \beta_{5} + 24 \beta_1 - 24) q^{28} + ( - 6 \beta_{7} - 6 \beta_{6} + 111 \beta_{2}) q^{29} + (18 \beta_{5} - 18 \beta_{4} + 19) q^{31} + (16 \beta_{3} + 16 \beta_{2}) q^{32} + ( - 6 \beta_{5} - 6 \beta_{4} - 24 \beta_1) q^{34} + (14 \beta_{4} + 84 \beta_1 + 84) q^{37} + (12 \beta_{6} - 37 \beta_{3} + 37 \beta_{2}) q^{38} + ( - 18 \beta_{7} + 18 \beta_{6} - 42 \beta_{3}) q^{41} + ( - 11 \beta_{5} + 270 \beta_1 - 270) q^{43} + 12 \beta_{2} q^{44} + (18 \beta_{5} - 18 \beta_{4} - 96) q^{46} + ( - 27 \beta_{7} + 90 \beta_{3} + 90 \beta_{2}) q^{47} + ( - 12 \beta_{5} - 12 \beta_{4} - 196 \beta_1) q^{49} + ( - 12 \beta_{4} + 96 \beta_1 + 96) q^{52} + ( - 12 \beta_{6} + \cdots + 246 \beta_{2}) q^{53}+ \cdots + (24 \beta_{6} - 196 \beta_{3} + 196 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{7} - 192 q^{13} - 128 q^{16} - 48 q^{22} - 192 q^{28} + 152 q^{31} + 672 q^{37} - 2160 q^{43} - 768 q^{46} + 768 q^{52} - 1776 q^{58} + 1000 q^{61} + 816 q^{67} - 3936 q^{73} - 1184 q^{76} - 672 q^{82} - 192 q^{88} + 4104 q^{91} + 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 5\zeta_{24}^{5} + 5\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 10\zeta_{24}^{7} - 5\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -5\zeta_{24}^{6} + 10\zeta_{24}^{4} + 10\zeta_{24}^{2} - 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -5\zeta_{24}^{6} - 10\zeta_{24}^{4} + 10\zeta_{24}^{2} + 5 \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( 2\beta_{4} + 5\beta_{3} + 5\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 10\beta_1 ) / 20 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 10 ) / 20 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{4} - 5\beta_{3} - 5\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( 2\beta_{5} - 5\beta_{3} + 5\beta_{2} ) / 20 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−1.41421 + 1.41421i 0 − 4.00000i 0 0 −12.1237 − 12.1237i 5.65685 + 5.65685i 0 0
107.2 −1.41421 + 1.41421i 0 − 4.00000i 0 0 0.123724 + 0.123724i 5.65685 + 5.65685i 0 0
107.3 1.41421 − 1.41421i 0 − 4.00000i 0 0 −12.1237 − 12.1237i −5.65685 − 5.65685i 0 0
107.4 1.41421 − 1.41421i 0 − 4.00000i 0 0 0.123724 + 0.123724i −5.65685 − 5.65685i 0 0
143.1 −1.41421 − 1.41421i 0 4.00000i 0 0 −12.1237 + 12.1237i 5.65685 − 5.65685i 0 0
143.2 −1.41421 − 1.41421i 0 4.00000i 0 0 0.123724 − 0.123724i 5.65685 − 5.65685i 0 0
143.3 1.41421 + 1.41421i 0 4.00000i 0 0 −12.1237 + 12.1237i −5.65685 + 5.65685i 0 0
143.4 1.41421 + 1.41421i 0 4.00000i 0 0 0.123724 − 0.123724i −5.65685 + 5.65685i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.4.f.d ✓ 8
3.b odd 2 1 inner 450.4.f.d ✓ 8
5.b even 2 1 450.4.f.f yes 8
5.c odd 4 1 inner 450.4.f.d ✓ 8
5.c odd 4 1 450.4.f.f yes 8
15.d odd 2 1 450.4.f.f yes 8
15.e even 4 1 inner 450.4.f.d ✓ 8
15.e even 4 1 450.4.f.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.4.f.d ✓ 8 1.a even 1 1 trivial
450.4.f.d ✓ 8 3.b odd 2 1 inner
450.4.f.d ✓ 8 5.c odd 4 1 inner
450.4.f.d ✓ 8 15.e even 4 1 inner
450.4.f.f yes 8 5.b even 2 1
450.4.f.f yes 8 5.c odd 4 1
450.4.f.f yes 8 15.d odd 2 1
450.4.f.f yes 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 24T_{7}^{3} + 288T_{7}^{2} - 72T_{7} + 9 \) acting on \(S_{4}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 24 T^{3} + 288 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 96 T^{3} + \cdots + 227529)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 2115384078096 \) Copy content Toggle raw display
$19$ \( (T^{4} + 13538 T^{2} + 16248961)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} - 70884 T^{2} + 191600964)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 38 T - 48239)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 336 T^{3} + \cdots + 345744)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 201456 T^{2} + 8774443584)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1080 T^{3} + \cdots + 18693725625)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{4} - 25956 T^{2} + 74338884)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 250 T - 524375)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 408 T^{3} + \cdots + 8698733289)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1196784 T^{2} + 211607360064)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1968 T^{3} + \cdots + 53744221584)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1257608 T^{2} + 203577830416)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1700496 T^{2} + 283185751104)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 96 T^{3} + \cdots + 10306919529)^{2} \) Copy content Toggle raw display
show more
show less