Properties

Label 450.2.s.b
Level $450$
Weight $2$
Character orbit 450.s
Analytic conductor $3.593$
Analytic rank $0$
Dimension $16$
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] = [450,2,Mod(17,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.s (of order \(20\), degree \(8\), 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: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-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 primitive root of unity \(\zeta_{40}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{40}^{11} q^{2} - \zeta_{40}^{2} q^{4} + (2 \zeta_{40}^{15} - 2 \zeta_{40}^{11} - \zeta_{40}^{3}) q^{5} + ( - 2 \zeta_{40}^{14} + 2 \zeta_{40}^{12} - 2 \zeta_{40}^{8} + 2 \zeta_{40}^{4} - 2) q^{7} - \zeta_{40}^{13} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{40}^{11} q^{2} - \zeta_{40}^{2} q^{4} + (2 \zeta_{40}^{15} - 2 \zeta_{40}^{11} - \zeta_{40}^{3}) q^{5} + ( - 2 \zeta_{40}^{14} + 2 \zeta_{40}^{12} - 2 \zeta_{40}^{8} + 2 \zeta_{40}^{4} - 2) q^{7} - \zeta_{40}^{13} q^{8} + ( - \zeta_{40}^{14} - 2 \zeta_{40}^{6} + 2 \zeta_{40}^{2}) q^{10} + ( - 2 \zeta_{40}^{15} + 2 \zeta_{40}^{13} + 2 \zeta_{40}^{11} - 2 \zeta_{40}^{7} + 2 \zeta_{40}^{3}) q^{11} + ( - 4 \zeta_{40}^{14} + \zeta_{40}^{12} + 2 \zeta_{40}^{10} + \zeta_{40}^{8} - 4 \zeta_{40}^{6} + \zeta_{40}^{4} + 3 \zeta_{40}^{2} + \cdots + 3) q^{13} + \cdots + ( - 4 \zeta_{40}^{15} + 4 \zeta_{40}^{11} - 4 \zeta_{40}^{7} + 8 \zeta_{40}^{3} - \zeta_{40}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} + 52 q^{13} + 4 q^{16} + 40 q^{19} - 8 q^{22} - 8 q^{28} - 16 q^{31} + 20 q^{34} + 52 q^{37} - 20 q^{40} + 40 q^{43} + 24 q^{46} - 52 q^{52} + 40 q^{55} + 36 q^{58} - 40 q^{61} - 64 q^{67} - 40 q^{70} - 84 q^{73} - 48 q^{76} + 40 q^{79} - 52 q^{82} - 100 q^{85} - 8 q^{88} - 112 q^{91} + 40 q^{94} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\zeta_{40}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.453990 + 0.891007i
0.453990 0.891007i
−0.453990 0.891007i
0.453990 + 0.891007i
−0.891007 + 0.453990i
0.891007 0.453990i
−0.891007 0.453990i
0.891007 + 0.453990i
0.156434 + 0.987688i
−0.156434 0.987688i
0.156434 0.987688i
−0.156434 + 0.987688i
0.987688 + 0.156434i
−0.987688 0.156434i
0.987688 0.156434i
−0.987688 + 0.156434i
−0.891007 0.453990i 0 0.587785 + 0.809017i 2.20854 0.349798i 0 2.52015 + 2.52015i −0.156434 0.987688i 0 −2.12663 0.690983i
17.2 0.891007 + 0.453990i 0 0.587785 + 0.809017i −2.20854 + 0.349798i 0 2.52015 + 2.52015i 0.156434 + 0.987688i 0 −2.12663 0.690983i
53.1 −0.891007 + 0.453990i 0 0.587785 0.809017i 2.20854 + 0.349798i 0 2.52015 2.52015i −0.156434 + 0.987688i 0 −2.12663 + 0.690983i
53.2 0.891007 0.453990i 0 0.587785 0.809017i −2.20854 0.349798i 0 2.52015 2.52015i 0.156434 0.987688i 0 −2.12663 + 0.690983i
197.1 −0.453990 0.891007i 0 −0.587785 + 0.809017i −0.349798 + 2.20854i 0 −1.28408 1.28408i 0.987688 + 0.156434i 0 2.12663 0.690983i
197.2 0.453990 + 0.891007i 0 −0.587785 + 0.809017i 0.349798 2.20854i 0 −1.28408 1.28408i −0.987688 0.156434i 0 2.12663 0.690983i
233.1 −0.453990 + 0.891007i 0 −0.587785 0.809017i −0.349798 2.20854i 0 −1.28408 + 1.28408i 0.987688 0.156434i 0 2.12663 + 0.690983i
233.2 0.453990 0.891007i 0 −0.587785 0.809017i 0.349798 + 2.20854i 0 −1.28408 + 1.28408i −0.987688 + 0.156434i 0 2.12663 + 0.690983i
287.1 −0.987688 + 0.156434i 0 0.951057 0.309017i 1.01515 + 1.99235i 0 −2.79360 2.79360i −0.891007 + 0.453990i 0 −1.31433 1.80902i
287.2 0.987688 0.156434i 0 0.951057 0.309017i −1.01515 1.99235i 0 −2.79360 2.79360i 0.891007 0.453990i 0 −1.31433 1.80902i
323.1 −0.987688 0.156434i 0 0.951057 + 0.309017i 1.01515 1.99235i 0 −2.79360 + 2.79360i −0.891007 0.453990i 0 −1.31433 + 1.80902i
323.2 0.987688 + 0.156434i 0 0.951057 + 0.309017i −1.01515 + 1.99235i 0 −2.79360 + 2.79360i 0.891007 + 0.453990i 0 −1.31433 + 1.80902i
377.1 −0.156434 + 0.987688i 0 −0.951057 0.309017i −1.99235 1.01515i 0 −0.442463 0.442463i 0.453990 0.891007i 0 1.31433 1.80902i
377.2 0.156434 0.987688i 0 −0.951057 0.309017i 1.99235 + 1.01515i 0 −0.442463 0.442463i −0.453990 + 0.891007i 0 1.31433 1.80902i
413.1 −0.156434 0.987688i 0 −0.951057 + 0.309017i −1.99235 + 1.01515i 0 −0.442463 + 0.442463i 0.453990 + 0.891007i 0 1.31433 + 1.80902i
413.2 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 1.99235 1.01515i 0 −0.442463 + 0.442463i −0.453990 0.891007i 0 1.31433 + 1.80902i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.s.b 16
3.b odd 2 1 inner 450.2.s.b 16
25.f odd 20 1 inner 450.2.s.b 16
75.l even 20 1 inner 450.2.s.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.s.b 16 1.a even 1 1 trivial
450.2.s.b 16 3.b odd 2 1 inner
450.2.s.b 16 25.f odd 20 1 inner
450.2.s.b 16 75.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} + 176T_{7}^{4} + 640T_{7}^{3} + 1152T_{7}^{2} + 768T_{7} + 256 \) acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 25 T^{12} + 625 T^{8} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + 8 T^{6} + 176 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 16 T^{14} + 192 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{8} - 26 T^{7} + 318 T^{6} + \cdots + 177241)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 120 T^{14} + 4724 T^{12} + \cdots + 2825761 \) Copy content Toggle raw display
$19$ \( (T^{8} - 20 T^{7} + 156 T^{6} + \cdots + 891136)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 80 T^{14} + 5584 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( T^{16} + 84 T^{14} + 3212 T^{12} + \cdots + 2825761 \) Copy content Toggle raw display
$31$ \( (T^{8} + 8 T^{7} + 108 T^{6} + 256 T^{5} + \cdots + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 26 T^{7} + 263 T^{6} + \cdots + 2621161)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 76 T^{14} + \cdots + 5076013506001 \) Copy content Toggle raw display
$43$ \( (T^{8} - 20 T^{7} + 200 T^{6} - 480 T^{5} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 80 T^{14} + 7280 T^{12} + \cdots + 40960000 \) Copy content Toggle raw display
$53$ \( T^{16} - 240 T^{14} + \cdots + 6870484987921 \) Copy content Toggle raw display
$59$ \( T^{16} + 256 T^{14} + \cdots + 4294967296 \) Copy content Toggle raw display
$61$ \( (T^{8} + 20 T^{7} + 340 T^{6} + \cdots + 483025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 32 T^{7} + 572 T^{6} + \cdots + 1597696)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 16 T^{14} + \cdots + 794123370496 \) Copy content Toggle raw display
$73$ \( (T^{8} + 42 T^{7} + 762 T^{6} + \cdots + 78961)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 20 T^{7} + 216 T^{6} + \cdots + 15241216)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 160 T^{14} + 17984 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( T^{16} - 244 T^{14} + \cdots + 30033479838961 \) Copy content Toggle raw display
$97$ \( (T^{8} - 6 T^{7} + 18 T^{6} - 270 T^{5} + \cdots + 6561)^{2} \) Copy content Toggle raw display
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