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Properties

Label 450.2.p.b
Level $450$
Weight $2$
Character orbit 450.p
Analytic conductor $3.593$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,2,Mod(257,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.p (of order \(12\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} + ( - \zeta_{24}^{5} + 2 \zeta_{24}) q^{3} - \zeta_{24}^{2} q^{4} + (2 \zeta_{24}^{4} - 1) q^{6} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{8} + ( - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} + ( - \zeta_{24}^{5} + 2 \zeta_{24}) q^{3} - \zeta_{24}^{2} q^{4} + (2 \zeta_{24}^{4} - 1) q^{6} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{8} + ( - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{2}) q^{9} + (\zeta_{24}^{4} - 2) q^{11} + (\zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{12} + ( - 4 \zeta_{24}^{5} + 8 \zeta_{24}) q^{13} + \zeta_{24}^{4} q^{16} + 3 \zeta_{24}^{3} q^{17} + 3 \zeta_{24}^{5} q^{18} + 4 \zeta_{24}^{6} q^{19} + ( - \zeta_{24}^{7} - \zeta_{24}^{3}) q^{22} - 6 \zeta_{24}^{5} q^{23} + ( - 2 \zeta_{24}^{6} + \zeta_{24}^{2}) q^{24} + (8 \zeta_{24}^{4} - 4) q^{26} + ( - 6 \zeta_{24}^{7} + 3 \zeta_{24}^{3}) q^{27} + (2 \zeta_{24}^{6} + 2 \zeta_{24}^{2}) q^{29} + (4 \zeta_{24}^{4} - 4) q^{31} + (\zeta_{24}^{7} - \zeta_{24}^{3}) q^{32} + (3 \zeta_{24}^{5} - 3 \zeta_{24}) q^{33} + (3 \zeta_{24}^{6} - 3 \zeta_{24}^{2}) q^{34} - 3 q^{36} + ( - 8 \zeta_{24}^{7} + 4 \zeta_{24}^{3}) q^{37} - 4 \zeta_{24} q^{38} + ( - 12 \zeta_{24}^{6} + 12 \zeta_{24}^{2}) q^{39} + ( - 4 \zeta_{24}^{4} - 4) q^{41} + ( - 10 \zeta_{24}^{5} + 5 \zeta_{24}) q^{43} + ( - \zeta_{24}^{6} + 2 \zeta_{24}^{2}) q^{44} + 6 q^{46} + 12 \zeta_{24}^{7} q^{47} + (\zeta_{24}^{5} + \zeta_{24}) q^{48} - 7 \zeta_{24}^{2} q^{49} + (3 \zeta_{24}^{4} + 3) q^{51} + (4 \zeta_{24}^{7} - 8 \zeta_{24}^{3}) q^{52} + (12 \zeta_{24}^{5} - 12 \zeta_{24}) q^{53} + (3 \zeta_{24}^{6} + 3 \zeta_{24}^{2}) q^{54} + (4 \zeta_{24}^{7} + 4 \zeta_{24}^{3}) q^{57} + (2 \zeta_{24}^{5} - 4 \zeta_{24}) q^{58} + (10 \zeta_{24}^{6} - 5 \zeta_{24}^{2}) q^{59} - 8 \zeta_{24}^{4} q^{61} - 4 \zeta_{24}^{3} q^{62} - \zeta_{24}^{6} q^{64} - 3 \zeta_{24}^{4} q^{66} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{3}) q^{67} - 3 \zeta_{24}^{5} q^{68} + ( - 6 \zeta_{24}^{6} - 6 \zeta_{24}^{2}) q^{69} + ( - 4 \zeta_{24}^{4} + 2) q^{71} - 3 \zeta_{24}^{7} q^{72} + (4 \zeta_{24}^{5} + 4 \zeta_{24}) q^{73} + (4 \zeta_{24}^{6} + 4 \zeta_{24}^{2}) q^{74} + ( - 4 \zeta_{24}^{4} + 4) q^{76} + 12 \zeta_{24}^{5} q^{78} + (4 \zeta_{24}^{6} - 4 \zeta_{24}^{2}) q^{79} + ( - 9 \zeta_{24}^{4} + 9) q^{81} + ( - 8 \zeta_{24}^{7} + 4 \zeta_{24}^{3}) q^{82} - 9 \zeta_{24} q^{83} + (5 \zeta_{24}^{4} + 5) q^{86} + 6 \zeta_{24}^{3} q^{87} + (2 \zeta_{24}^{5} - \zeta_{24}) q^{88} + (\zeta_{24}^{6} - 2 \zeta_{24}^{2}) q^{89} + 6 \zeta_{24}^{7} q^{92} + (8 \zeta_{24}^{5} - 4 \zeta_{24}) q^{93} - 12 \zeta_{24}^{2} q^{94} + (\zeta_{24}^{4} - 2) q^{96} + (9 \zeta_{24}^{7} - 18 \zeta_{24}^{3}) q^{97} + ( - 7 \zeta_{24}^{5} + 7 \zeta_{24}) q^{98} + (6 \zeta_{24}^{6} - 3 \zeta_{24}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{11} + 4 q^{16} - 16 q^{31} - 24 q^{36} - 48 q^{41} + 48 q^{46} + 36 q^{51} - 32 q^{61} - 12 q^{66} + 16 q^{76} + 36 q^{81} + 60 q^{86} - 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 - \zeta_{24}^{4}\) \(-\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 − 0.258819i −0.448288 − 1.67303i 0.866025 + 0.500000i 0 1.73205i 0 −0.707107 − 0.707107i −2.59808 + 1.50000i 0
257.2 0.965926 + 0.258819i 0.448288 + 1.67303i 0.866025 + 0.500000i 0 1.73205i 0 0.707107 + 0.707107i −2.59808 + 1.50000i 0
293.1 −0.258819 + 0.965926i 1.67303 − 0.448288i −0.866025 − 0.500000i 0 1.73205i 0 0.707107 − 0.707107i 2.59808 − 1.50000i 0
293.2 0.258819 − 0.965926i −1.67303 + 0.448288i −0.866025 − 0.500000i 0 1.73205i 0 −0.707107 + 0.707107i 2.59808 − 1.50000i 0
407.1 −0.258819 − 0.965926i 1.67303 + 0.448288i −0.866025 + 0.500000i 0 − 1.73205i 0 0.707107 + 0.707107i 2.59808 + 1.50000i 0
407.2 0.258819 + 0.965926i −1.67303 − 0.448288i −0.866025 + 0.500000i 0 − 1.73205i 0 −0.707107 − 0.707107i 2.59808 + 1.50000i 0
443.1 −0.965926 + 0.258819i −0.448288 + 1.67303i 0.866025 − 0.500000i 0 − 1.73205i 0 −0.707107 + 0.707107i −2.59808 − 1.50000i 0
443.2 0.965926 − 0.258819i 0.448288 − 1.67303i 0.866025 − 0.500000i 0 − 1.73205i 0 0.707107 − 0.707107i −2.59808 − 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.p.b ✓ 8
3.b odd 2 1 1350.2.q.e 8
5.b even 2 1 inner 450.2.p.b ✓ 8
5.c odd 4 2 inner 450.2.p.b ✓ 8
9.c even 3 1 1350.2.q.e 8
9.d odd 6 1 inner 450.2.p.b ✓ 8
15.d odd 2 1 1350.2.q.e 8
15.e even 4 2 1350.2.q.e 8
45.h odd 6 1 inner 450.2.p.b ✓ 8
45.j even 6 1 1350.2.q.e 8
45.k odd 12 2 1350.2.q.e 8
45.l even 12 2 inner 450.2.p.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.p.b ✓ 8 1.a even 1 1 trivial
450.2.p.b ✓ 8 5.b even 2 1 inner
450.2.p.b ✓ 8 5.c odd 4 2 inner
450.2.p.b ✓ 8 9.d odd 6 1 inner
450.2.p.b ✓ 8 45.h odd 6 1 inner
450.2.p.b ✓ 8 45.l even 12 2 inner
1350.2.q.e 8 3.b odd 2 1
1350.2.q.e 8 9.c even 3 1
1350.2.q.e 8 15.d odd 2 1
1350.2.q.e 8 15.e even 4 2
1350.2.q.e 8 45.j even 6 1
1350.2.q.e 8 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} - 2304 T^{4} + 5308416 \) Copy content Toggle raw display
$17$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 1296 T^{4} + 1679616 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 48)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} - 5625 T^{4} + 31640625 \) Copy content Toggle raw display
$47$ \( T^{8} - 20736 T^{4} + 429981696 \) Copy content Toggle raw display
$53$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 75 T^{2} + 5625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 6561 T^{4} + 43046721 \) Copy content Toggle raw display
$89$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 3486784401 \) Copy content Toggle raw display
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