Properties

Label 450.4.f.a
Level $450$
Weight $4$
Character orbit 450.f
Analytic conductor $26.551$
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] = [450,4,Mod(107,450)]
 
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([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.f (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: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2}) q^{2} + 4 \beta_1 q^{4} + (9 \beta_1 - 9) q^{7} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2}) q^{2} + 4 \beta_1 q^{4} + (9 \beta_1 - 9) q^{7} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{8} + 27 \beta_{2} q^{11} + (9 \beta_1 + 9) q^{13} - 18 \beta_{3} q^{14} - 16 q^{16} + ( - 39 \beta_{3} - 39 \beta_{2}) q^{17} - 38 \beta_1 q^{19} + (54 \beta_1 - 54) q^{22} + (21 \beta_{3} - 21 \beta_{2}) q^{23} + 18 \beta_{2} q^{26} + ( - 36 \beta_1 - 36) q^{28} - 54 \beta_{3} q^{29} - 236 q^{31} + ( - 16 \beta_{3} - 16 \beta_{2}) q^{32} - 156 \beta_1 q^{34} + (279 \beta_1 - 279) q^{37} + (38 \beta_{3} - 38 \beta_{2}) q^{38} + 243 \beta_{2} q^{41} + ( - 360 \beta_1 - 360) q^{43} - 108 \beta_{3} q^{44} + 84 q^{46} + ( - 180 \beta_{3} - 180 \beta_{2}) q^{47} + 181 \beta_1 q^{49} + (36 \beta_1 - 36) q^{52} + (249 \beta_{3} - 249 \beta_{2}) q^{53} - 72 \beta_{2} q^{56} + ( - 108 \beta_1 - 108) q^{58} + 243 \beta_{3} q^{59} + 110 q^{61} + ( - 236 \beta_{3} - 236 \beta_{2}) q^{62} - 64 \beta_1 q^{64} + (702 \beta_1 - 702) q^{67} + (156 \beta_{3} - 156 \beta_{2}) q^{68} + 54 \beta_{2} q^{71} + ( - 378 \beta_1 - 378) q^{73} - 558 \beta_{3} q^{74} + 152 q^{76} + ( - 243 \beta_{3} - 243 \beta_{2}) q^{77} + 272 \beta_1 q^{79} + (486 \beta_1 - 486) q^{82} + ( - 42 \beta_{3} + 42 \beta_{2}) q^{83} - 720 \beta_{2} q^{86} + ( - 216 \beta_1 - 216) q^{88} + 1053 \beta_{3} q^{89} - 162 q^{91} + (84 \beta_{3} + 84 \beta_{2}) q^{92} - 720 \beta_1 q^{94} + (684 \beta_1 - 684) q^{97} + ( - 181 \beta_{3} + 181 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{7} + 36 q^{13} - 64 q^{16} - 216 q^{22} - 144 q^{28} - 944 q^{31} - 1116 q^{37} - 1440 q^{43} + 336 q^{46} - 144 q^{52} - 432 q^{58} + 440 q^{61} - 2808 q^{67} - 1512 q^{73} + 608 q^{76} - 1944 q^{82} - 864 q^{88} - 648 q^{91} - 2736 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−1.41421 + 1.41421i 0 4.00000i 0 0 −9.00000 9.00000i 5.65685 + 5.65685i 0 0
107.2 1.41421 1.41421i 0 4.00000i 0 0 −9.00000 9.00000i −5.65685 5.65685i 0 0
143.1 −1.41421 1.41421i 0 4.00000i 0 0 −9.00000 + 9.00000i 5.65685 5.65685i 0 0
143.2 1.41421 + 1.41421i 0 4.00000i 0 0 −9.00000 + 9.00000i −5.65685 + 5.65685i 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
3.b odd 2 1 inner
5.c odd 4 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 450.4.f.a 4
3.b odd 2 1 inner 450.4.f.a 4
5.b even 2 1 450.4.f.c yes 4
5.c odd 4 1 inner 450.4.f.a 4
5.c odd 4 1 450.4.f.c yes 4
15.d odd 2 1 450.4.f.c yes 4
15.e even 4 1 inner 450.4.f.a 4
15.e even 4 1 450.4.f.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.4.f.a 4 1.a even 1 1 trivial
450.4.f.a 4 3.b odd 2 1 inner
450.4.f.a 4 5.c odd 4 1 inner
450.4.f.a 4 15.e even 4 1 inner
450.4.f.c yes 4 5.b even 2 1
450.4.f.c yes 4 5.c odd 4 1
450.4.f.c yes 4 15.d odd 2 1
450.4.f.c yes 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 18T_{7} + 162 \) acting on \(S_{4}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 18 T + 162)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1458)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18 T + 162)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 37015056 \) Copy content Toggle raw display
$19$ \( (T^{2} + 1444)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 3111696 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5832)^{2} \) Copy content Toggle raw display
$31$ \( (T + 236)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 558 T + 155682)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 118098)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 720 T + 259200)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 16796160000 \) Copy content Toggle raw display
$53$ \( T^{4} + 61505984016 \) Copy content Toggle raw display
$59$ \( (T^{2} - 118098)^{2} \) Copy content Toggle raw display
$61$ \( (T - 110)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1404 T + 985608)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 5832)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 756 T + 285768)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 73984)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 49787136 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2217618)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1368 T + 935712)^{2} \) Copy content Toggle raw display
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