Properties

Label 450.2.p.e
Level $450$
Weight $2$
Character orbit 450.p
Analytic conductor $3.593$
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] = [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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} + (2 \zeta_{24}^{7} - \zeta_{24}^{3}) q^{3} - \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{6} - \zeta_{24}^{2}) q^{6} + (\zeta_{24}^{7} - 2 \zeta_{24}^{3} - 3 \zeta_{24}) q^{7} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{8} - 3 \zeta_{24}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} + (2 \zeta_{24}^{7} - \zeta_{24}^{3}) q^{3} - \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{6} - \zeta_{24}^{2}) q^{6} + (\zeta_{24}^{7} - 2 \zeta_{24}^{3} - 3 \zeta_{24}) q^{7} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{8} - 3 \zeta_{24}^{6} q^{9} + ( - 2 \zeta_{24}^{4} + 4) q^{11} + ( - \zeta_{24}^{5} + 2 \zeta_{24}) q^{12} + ( - 3 \zeta_{24}^{7} - \zeta_{24}^{5} + 3 \zeta_{24}^{3} + 2 \zeta_{24}) q^{13} + ( - 2 \zeta_{24}^{6} - 3 \zeta_{24}^{4} + \zeta_{24}^{2} + 3) q^{14} + \zeta_{24}^{4} q^{16} + 3 \zeta_{24} q^{18} + ( - 2 \zeta_{24}^{6} + 6 \zeta_{24}^{4} - 3) q^{19} + ( - 3 \zeta_{24}^{6} - 3 \zeta_{24}^{4} + 3 \zeta_{24}^{2} + 6) q^{21} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{3}) q^{22} + ( - 3 \zeta_{24}^{7} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{3}) q^{23} + (2 \zeta_{24}^{4} - 1) q^{24} + (3 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 1) q^{26} + (3 \zeta_{24}^{5} + 3 \zeta_{24}) q^{27} + (\zeta_{24}^{5} + 3 \zeta_{24}^{3} + \zeta_{24}) q^{28} + ( - \zeta_{24}^{6} - 9 \zeta_{24}^{4} - \zeta_{24}^{2}) q^{29} + (6 \zeta_{24}^{6} - 5 \zeta_{24}^{4} - 3 \zeta_{24}^{2} + 5) q^{31} + (\zeta_{24}^{7} - \zeta_{24}^{3}) q^{32} + 6 \zeta_{24}^{7} q^{33} + (3 \zeta_{24}^{4} - 3) q^{36} + ( - 6 \zeta_{24}^{7} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{3} + 3 \zeta_{24}) q^{37} + (3 \zeta_{24}^{7} - 6 \zeta_{24}^{3} + 2 \zeta_{24}) q^{38} + (6 \zeta_{24}^{6} + 3 \zeta_{24}^{4} - 3 \zeta_{24}^{2} - 3) q^{39} + ( - \zeta_{24}^{4} - 1) q^{41} + (3 \zeta_{24}^{7} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{3}) q^{42} + ( - 3 \zeta_{24}^{7} + 4 \zeta_{24}^{5} - 2 \zeta_{24}) q^{43} + (2 \zeta_{24}^{6} - 4 \zeta_{24}^{2}) q^{44} + ( - 3 \zeta_{24}^{6} + 6 \zeta_{24}^{2} - 3) q^{46} - 6 \zeta_{24}^{7} q^{47} + (\zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{48} + (6 \zeta_{24}^{4} + 5 \zeta_{24}^{2} + 6) q^{49} + (\zeta_{24}^{7} - 2 \zeta_{24}^{3} - 3 \zeta_{24}) q^{52} + ( - 6 \zeta_{24}^{7} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{3} + 3 \zeta_{24}) q^{53} + (3 \zeta_{24}^{4} - 6) q^{54} + (3 \zeta_{24}^{6} + \zeta_{24}^{4} - 3 \zeta_{24}^{2} - 2) q^{56} + (2 \zeta_{24}^{5} - 9 \zeta_{24}^{3} + 2 \zeta_{24}) q^{57} + ( - 9 \zeta_{24}^{7} - \zeta_{24}^{5} + 9 \zeta_{24}^{3} + 2 \zeta_{24}) q^{58} + (4 \zeta_{24}^{6} - 9 \zeta_{24}^{4} - 2 \zeta_{24}^{2} + 9) q^{59} + 4 \zeta_{24}^{4} q^{61} + ( - 3 \zeta_{24}^{5} + 5 \zeta_{24}^{3} - 3 \zeta_{24}) q^{62} + (9 \zeta_{24}^{7} + 6 \zeta_{24}^{5} - 3 \zeta_{24}) q^{63} - \zeta_{24}^{6} q^{64} - 6 \zeta_{24}^{2} q^{66} + (2 \zeta_{24}^{7} - 9 \zeta_{24}^{5} + 2 \zeta_{24}^{3}) q^{67} + ( - 3 \zeta_{24}^{4} + 9 \zeta_{24}^{2} - 3) q^{69} + ( - 9 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 1) q^{71} - 3 \zeta_{24}^{3} q^{72} + ( - 3 \zeta_{24}^{5} - 3 \zeta_{24}) q^{73} + (3 \zeta_{24}^{6} + 3 \zeta_{24}^{4} + 3 \zeta_{24}^{2}) q^{74} + ( - 6 \zeta_{24}^{6} + 2 \zeta_{24}^{4} + 3 \zeta_{24}^{2} - 2) q^{76} + (6 \zeta_{24}^{7} + 6 \zeta_{24}^{5} - 6 \zeta_{24}^{3} - 12 \zeta_{24}) q^{77} + ( - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{3} - 3 \zeta_{24}) q^{78} + (10 \zeta_{24}^{6} - 10 \zeta_{24}^{2}) q^{79} - 9 q^{81} + ( - 2 \zeta_{24}^{7} + \zeta_{24}^{3}) q^{82} + (3 \zeta_{24}^{7} - 6 \zeta_{24}^{3} - 12 \zeta_{24}) q^{83} + (3 \zeta_{24}^{6} - 6 \zeta_{24}^{2} - 3) q^{84} + ( - 2 \zeta_{24}^{4} + 3 \zeta_{24}^{2} - 2) q^{86} + ( - 9 \zeta_{24}^{7} + 18 \zeta_{24}^{3} + 3 \zeta_{24}) q^{87} + ( - 4 \zeta_{24}^{5} + 2 \zeta_{24}) q^{88} + ( - 5 \zeta_{24}^{6} + 10 \zeta_{24}^{2}) q^{89} + (6 \zeta_{24}^{6} - 12 \zeta_{24}^{2} - 12) q^{91} + ( - 3 \zeta_{24}^{7} + 6 \zeta_{24}^{5} - 3 \zeta_{24}) q^{92} + (5 \zeta_{24}^{7} - 9 \zeta_{24}^{5} + 5 \zeta_{24}^{3}) q^{93} + 6 \zeta_{24}^{2} q^{94} + ( - 2 \zeta_{24}^{6} + \zeta_{24}^{2}) q^{96} + (5 \zeta_{24}^{7} - 10 \zeta_{24}^{3} + 6 \zeta_{24}) q^{97} + (12 \zeta_{24}^{7} + 5 \zeta_{24}^{5} - 6 \zeta_{24}^{3} - 5 \zeta_{24}) q^{98} + ( - 6 \zeta_{24}^{6} - 6 \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 \( 8 q + 24 q^{11} + 12 q^{14} + 4 q^{16} + 36 q^{21} - 36 q^{29} + 20 q^{31} - 12 q^{36} - 12 q^{39} - 12 q^{41} - 24 q^{46} + 72 q^{49} - 36 q^{54} - 12 q^{56} + 36 q^{59} + 16 q^{61} - 36 q^{69} + 12 q^{74} - 8 q^{76} - 72 q^{81} - 24 q^{84} - 24 q^{86} - 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 −1.22474 1.22474i 0.866025 + 0.500000i 0 0.866025 + 1.50000i −0.328169 + 1.22474i −0.707107 0.707107i 3.00000i 0
257.2 0.965926 + 0.258819i 1.22474 + 1.22474i 0.866025 + 0.500000i 0 0.866025 + 1.50000i 0.328169 1.22474i 0.707107 + 0.707107i 3.00000i 0
293.1 −0.258819 + 0.965926i −1.22474 + 1.22474i −0.866025 0.500000i 0 −0.866025 1.50000i −4.57081 1.22474i 0.707107 0.707107i 3.00000i 0
293.2 0.258819 0.965926i 1.22474 1.22474i −0.866025 0.500000i 0 −0.866025 1.50000i 4.57081 + 1.22474i −0.707107 + 0.707107i 3.00000i 0
407.1 −0.258819 0.965926i −1.22474 1.22474i −0.866025 + 0.500000i 0 −0.866025 + 1.50000i −4.57081 + 1.22474i 0.707107 + 0.707107i 3.00000i 0
407.2 0.258819 + 0.965926i 1.22474 + 1.22474i −0.866025 + 0.500000i 0 −0.866025 + 1.50000i 4.57081 1.22474i −0.707107 0.707107i 3.00000i 0
443.1 −0.965926 + 0.258819i −1.22474 + 1.22474i 0.866025 0.500000i 0 0.866025 1.50000i −0.328169 1.22474i −0.707107 + 0.707107i 3.00000i 0
443.2 0.965926 0.258819i 1.22474 1.22474i 0.866025 0.500000i 0 0.866025 1.50000i 0.328169 + 1.22474i 0.707107 0.707107i 3.00000i 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
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.e yes 8
3.b odd 2 1 1350.2.q.a 8
5.b even 2 1 inner 450.2.p.e yes 8
5.c odd 4 2 450.2.p.c 8
9.c even 3 1 1350.2.q.d 8
9.d odd 6 1 450.2.p.c 8
15.d odd 2 1 1350.2.q.a 8
15.e even 4 2 1350.2.q.d 8
45.h odd 6 1 450.2.p.c 8
45.j even 6 1 1350.2.q.d 8
45.k odd 12 2 1350.2.q.a 8
45.l even 12 2 inner 450.2.p.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.p.c 8 5.c odd 4 2
450.2.p.c 8 9.d odd 6 1
450.2.p.c 8 45.h odd 6 1
450.2.p.e yes 8 1.a even 1 1 trivial
450.2.p.e yes 8 5.b even 2 1 inner
450.2.p.e yes 8 45.l even 12 2 inner
1350.2.q.a 8 3.b odd 2 1
1350.2.q.a 8 15.d odd 2 1
1350.2.q.a 8 45.k odd 12 2
1350.2.q.d 8 9.c even 3 1
1350.2.q.d 8 15.e even 4 2
1350.2.q.d 8 45.j even 6 1

Hecke kernels

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

\( T_{7}^{8} - 36T_{7}^{6} + 396T_{7}^{4} + 1296T_{7}^{2} + 1296 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 36 T^{6} + 396 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} - 36 T^{6} + 396 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 62 T^{2} + 529)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 108 T^{6} + 3564 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$29$ \( (T^{4} + 18 T^{3} + 246 T^{2} + 1404 T + 6084)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 4536 T^{4} + 104976 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} - 72 T^{6} + 1719 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$47$ \( T^{8} - 1296 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$53$ \( T^{8} + 4536 T^{4} + 104976 \) Copy content Toggle raw display
$59$ \( (T^{4} - 18 T^{3} + 255 T^{2} - 1242 T + 4761)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 216 T^{6} + \cdots + 22667121 \) Copy content Toggle raw display
$71$ \( (T^{4} + 168 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 100 T^{2} + 10000)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 432 T^{6} + \cdots + 187388721 \) Copy content Toggle raw display
$89$ \( (T^{2} - 75)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 360 T^{6} + 41679 T^{4} + \cdots + 2313441 \) Copy content Toggle raw display
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