Properties

Label 450.3.g.f
Level $450$
Weight $3$
Character orbit 450.g
Analytic conductor $12.262$
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] = [450,3,Mod(307,450)]
 
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([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.307");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.g (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: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + 2 \beta_{2} q^{4} + \beta_1 q^{7} + ( - 2 \beta_{2} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + 2 \beta_{2} q^{4} + \beta_1 q^{7} + ( - 2 \beta_{2} + 2) q^{8} + (6 \beta_{3} - 6 \beta_1) q^{11} - 9 \beta_{3} q^{13} + ( - \beta_{3} - \beta_1) q^{14} - 4 q^{16} + ( - 6 \beta_{2} - 6) q^{17} + 17 \beta_{2} q^{19} + 12 \beta_1 q^{22} + ( - 30 \beta_{2} + 30) q^{23} + (9 \beta_{3} - 9 \beta_1) q^{26} + 2 \beta_{3} q^{28} + ( - 18 \beta_{3} - 18 \beta_1) q^{29} + 31 q^{31} + (4 \beta_{2} + 4) q^{32} + 12 \beta_{2} q^{34} - 16 \beta_1 q^{37} + ( - 17 \beta_{2} + 17) q^{38} + (30 \beta_{3} - 30 \beta_1) q^{41} - 17 \beta_{3} q^{43} + ( - 12 \beta_{3} - 12 \beta_1) q^{44} - 60 q^{46} - 46 \beta_{2} q^{49} + 18 \beta_1 q^{52} + ( - 42 \beta_{2} + 42) q^{53} + ( - 2 \beta_{3} + 2 \beta_1) q^{56} + 36 \beta_{3} q^{58} + ( - 12 \beta_{3} - 12 \beta_1) q^{59} + 47 q^{61} + ( - 31 \beta_{2} - 31) q^{62} - 8 \beta_{2} q^{64} - 57 \beta_1 q^{67} + ( - 12 \beta_{2} + 12) q^{68} + (18 \beta_{3} - 18 \beta_1) q^{71} + 40 \beta_{3} q^{73} + (16 \beta_{3} + 16 \beta_1) q^{74} - 34 q^{76} + ( - 18 \beta_{2} - 18) q^{77} - 38 \beta_{2} q^{79} + 60 \beta_1 q^{82} + (24 \beta_{2} - 24) q^{83} + (17 \beta_{3} - 17 \beta_1) q^{86} + 24 \beta_{3} q^{88} + (36 \beta_{3} + 36 \beta_1) q^{89} + 27 q^{91} + (60 \beta_{2} + 60) q^{92} - 79 \beta_1 q^{97} + (46 \beta_{2} - 46) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 8 q^{8} - 16 q^{16} - 24 q^{17} + 120 q^{23} + 124 q^{31} + 16 q^{32} + 68 q^{38} - 240 q^{46} + 168 q^{53} + 188 q^{61} - 124 q^{62} + 48 q^{68} - 136 q^{76} - 72 q^{77} - 96 q^{83} + 108 q^{91} + 240 q^{92} - 184 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−1.00000 1.00000i 0 2.00000i 0 0 −1.22474 1.22474i 2.00000 2.00000i 0 0
307.2 −1.00000 1.00000i 0 2.00000i 0 0 1.22474 + 1.22474i 2.00000 2.00000i 0 0
343.1 −1.00000 + 1.00000i 0 2.00000i 0 0 −1.22474 + 1.22474i 2.00000 + 2.00000i 0 0
343.2 −1.00000 + 1.00000i 0 2.00000i 0 0 1.22474 1.22474i 2.00000 + 2.00000i 0 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.c odd 4 1 inner
15.d odd 2 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 450.3.g.f 4
3.b odd 2 1 450.3.g.i yes 4
5.b even 2 1 450.3.g.i yes 4
5.c odd 4 1 inner 450.3.g.f 4
5.c odd 4 1 450.3.g.i yes 4
15.d odd 2 1 inner 450.3.g.f 4
15.e even 4 1 inner 450.3.g.f 4
15.e even 4 1 450.3.g.i yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.3.g.f 4 1.a even 1 1 trivial
450.3.g.f 4 5.c odd 4 1 inner
450.3.g.f 4 15.d odd 2 1 inner
450.3.g.f 4 15.e even 4 1 inner
450.3.g.i yes 4 3.b odd 2 1
450.3.g.i yes 4 5.b even 2 1
450.3.g.i yes 4 5.c odd 4 1
450.3.g.i yes 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 216 \) Copy content Toggle raw display
\( T_{17}^{2} + 12T_{17} + 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} - 216)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 59049 \) Copy content Toggle raw display
$17$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 289)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 60 T + 1800)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1944)^{2} \) Copy content Toggle raw display
$31$ \( (T - 31)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 589824 \) Copy content Toggle raw display
$41$ \( (T^{2} - 5400)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 751689 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 84 T + 3528)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 864)^{2} \) Copy content Toggle raw display
$61$ \( (T - 47)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 95004009 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1944)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 23040000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1444)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 48 T + 1152)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 7776)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 350550729 \) Copy content Toggle raw display
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