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Properties

Label 450.2.j.d
Level $450$
Weight $2$
Character orbit 450.j
Analytic conductor $3.593$
Analytic rank $0$
Dimension $4$
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(49,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.j (of order \(6\), degree \(2\), 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: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{2} + 1) q^{6} + 4 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{2} + 1) q^{6} + 4 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + (3 \zeta_{12}^{2} - 3) q^{11} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{12} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{13} + 4 \zeta_{12}^{2} q^{14} + (\zeta_{12}^{2} - 1) q^{16} - 3 \zeta_{12}^{3} q^{17} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{18} + 4 q^{19} + ( - 8 \zeta_{12}^{2} + 4) q^{21} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{22} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{23} + ( - \zeta_{12}^{2} + 2) q^{24} + 4 q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + 4 \zeta_{12}^{3} q^{28} + (6 \zeta_{12}^{2} - 6) q^{29} - 8 \zeta_{12}^{2} q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{33} + ( - 3 \zeta_{12}^{2} + 3) q^{34} - 3 q^{36} + 8 \zeta_{12}^{3} q^{37} + 4 \zeta_{12} q^{38} + ( - 4 \zeta_{12}^{2} - 4) q^{39} + 6 \zeta_{12}^{2} q^{41} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{42} - \zeta_{12} q^{43} - 3 q^{44} + 6 q^{46} - 12 \zeta_{12} q^{47} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{48} + 9 \zeta_{12}^{2} q^{49} + (3 \zeta_{12}^{2} - 6) q^{51} + 4 \zeta_{12} q^{52} + (3 \zeta_{12}^{2} + 3) q^{54} + (4 \zeta_{12}^{2} - 4) q^{56} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{57} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{58} - 9 \zeta_{12}^{2} q^{59} + (8 \zeta_{12}^{2} - 8) q^{61} - 8 \zeta_{12}^{3} q^{62} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{63} - q^{64} + (3 \zeta_{12}^{2} + 3) q^{66} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{67} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{68} + ( - 6 \zeta_{12}^{2} - 6) q^{69} - 6 q^{71} - 3 \zeta_{12} q^{72} - 14 \zeta_{12}^{3} q^{73} + (8 \zeta_{12}^{2} - 8) q^{74} + 4 \zeta_{12}^{2} q^{76} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{77} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{78} + ( - 8 \zeta_{12}^{2} + 8) q^{79} - 9 \zeta_{12}^{2} q^{81} + 6 \zeta_{12}^{3} q^{82} - 9 \zeta_{12} q^{83} + ( - 4 \zeta_{12}^{2} + 8) q^{84} - \zeta_{12}^{2} q^{86} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12}) q^{87} - 3 \zeta_{12} q^{88} + 9 q^{89} + 16 q^{91} + 6 \zeta_{12} q^{92} + (16 \zeta_{12}^{3} - 8 \zeta_{12}) q^{93} - 12 \zeta_{12}^{2} q^{94} + (\zeta_{12}^{2} + 1) q^{96} + 7 \zeta_{12} q^{97} + 9 \zeta_{12}^{3} q^{98} - 9 \zeta_{12}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 6 q^{9} - 6 q^{11} + 8 q^{14} - 2 q^{16} + 16 q^{19} + 6 q^{24} + 16 q^{26} - 12 q^{29} - 16 q^{31} + 6 q^{34} - 12 q^{36} - 24 q^{39} + 12 q^{41} - 12 q^{44} + 24 q^{46} + 18 q^{49} - 18 q^{51} + 18 q^{54} - 8 q^{56} - 18 q^{59} - 16 q^{61} - 4 q^{64} + 18 q^{66} - 36 q^{69} - 24 q^{71} - 16 q^{74} + 8 q^{76} + 16 q^{79} - 18 q^{81} + 24 q^{84} - 2 q^{86} + 36 q^{89} + 64 q^{91} - 24 q^{94} + 6 q^{96} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0.866025 − 1.50000i 0.500000 − 0.866025i 0 1.73205i −3.46410 + 2.00000i 1.00000i −1.50000 − 2.59808i 0
49.2 0.866025 − 0.500000i −0.866025 + 1.50000i 0.500000 − 0.866025i 0 1.73205i 3.46410 − 2.00000i − 1.00000i −1.50000 − 2.59808i 0
349.1 −0.866025 − 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0 − 1.73205i −3.46410 − 2.00000i − 1.00000i −1.50000 + 2.59808i 0
349.2 0.866025 + 0.500000i −0.866025 − 1.50000i 0.500000 + 0.866025i 0 − 1.73205i 3.46410 + 2.00000i 1.00000i −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.j.d 4
3.b odd 2 1 1350.2.j.d 4
5.b even 2 1 inner 450.2.j.d 4
5.c odd 4 1 450.2.e.a ✓ 2
5.c odd 4 1 450.2.e.h yes 2
9.c even 3 1 inner 450.2.j.d 4
9.c even 3 1 4050.2.c.q 2
9.d odd 6 1 1350.2.j.d 4
9.d odd 6 1 4050.2.c.e 2
15.d odd 2 1 1350.2.j.d 4
15.e even 4 1 1350.2.e.e 2
15.e even 4 1 1350.2.e.f 2
45.h odd 6 1 1350.2.j.d 4
45.h odd 6 1 4050.2.c.e 2
45.j even 6 1 inner 450.2.j.d 4
45.j even 6 1 4050.2.c.q 2
45.k odd 12 1 450.2.e.a ✓ 2
45.k odd 12 1 450.2.e.h yes 2
45.k odd 12 1 4050.2.a.b 1
45.k odd 12 1 4050.2.a.bj 1
45.l even 12 1 1350.2.e.e 2
45.l even 12 1 1350.2.e.f 2
45.l even 12 1 4050.2.a.p 1
45.l even 12 1 4050.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.a ✓ 2 5.c odd 4 1
450.2.e.a ✓ 2 45.k odd 12 1
450.2.e.h yes 2 5.c odd 4 1
450.2.e.h yes 2 45.k odd 12 1
450.2.j.d 4 1.a even 1 1 trivial
450.2.j.d 4 5.b even 2 1 inner
450.2.j.d 4 9.c even 3 1 inner
450.2.j.d 4 45.j even 6 1 inner
1350.2.e.e 2 15.e even 4 1
1350.2.e.e 2 45.l even 12 1
1350.2.e.f 2 15.e even 4 1
1350.2.e.f 2 45.l even 12 1
1350.2.j.d 4 3.b odd 2 1
1350.2.j.d 4 9.d odd 6 1
1350.2.j.d 4 15.d odd 2 1
1350.2.j.d 4 45.h odd 6 1
4050.2.a.b 1 45.k odd 12 1
4050.2.a.p 1 45.l even 12 1
4050.2.a.t 1 45.l even 12 1
4050.2.a.bj 1 45.k odd 12 1
4050.2.c.e 2 9.d odd 6 1
4050.2.c.e 2 45.h odd 6 1
4050.2.c.q 2 9.c even 3 1
4050.2.c.q 2 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 \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} - 16T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$89$ \( (T - 9)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
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