Properties

Label 450.3.d.f
Level $450$
Weight $3$
Character orbit 450.d
Analytic conductor $12.262$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,3,Mod(251,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.251");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.d (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - 2 q^{4} + 4 q^{7} + 2 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 2 q^{4} + 4 q^{7} + 2 \beta q^{8} + 12 \beta q^{11} - 8 q^{13} - 4 \beta q^{14} + 4 q^{16} + 9 \beta q^{17} - 16 q^{19} + 24 q^{22} + 12 \beta q^{23} + 8 \beta q^{26} - 8 q^{28} + 3 \beta q^{29} + 44 q^{31} - 4 \beta q^{32} + 18 q^{34} + 34 q^{37} + 16 \beta q^{38} + 33 \beta q^{41} + 40 q^{43} - 24 \beta q^{44} + 24 q^{46} + 60 \beta q^{47} - 33 q^{49} + 16 q^{52} - 27 \beta q^{53} + 8 \beta q^{56} + 6 q^{58} + 24 \beta q^{59} + 50 q^{61} - 44 \beta q^{62} - 8 q^{64} - 8 q^{67} - 18 \beta q^{68} - 36 \beta q^{71} + 16 q^{73} - 34 \beta q^{74} + 32 q^{76} + 48 \beta q^{77} - 76 q^{79} + 66 q^{82} - 84 \beta q^{83} - 40 \beta q^{86} - 48 q^{88} + 9 \beta q^{89} - 32 q^{91} - 24 \beta q^{92} + 120 q^{94} - 176 q^{97} + 33 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 8 q^{7} - 16 q^{13} + 8 q^{16} - 32 q^{19} + 48 q^{22} - 16 q^{28} + 88 q^{31} + 36 q^{34} + 68 q^{37} + 80 q^{43} + 48 q^{46} - 66 q^{49} + 32 q^{52} + 12 q^{58} + 100 q^{61} - 16 q^{64} - 16 q^{67} + 32 q^{73} + 64 q^{76} - 152 q^{79} + 132 q^{82} - 96 q^{88} - 64 q^{91} + 240 q^{94} - 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.41421i
1.41421i
1.41421i 0 −2.00000 0 0 4.00000 2.82843i 0 0
251.2 1.41421i 0 −2.00000 0 0 4.00000 2.82843i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.d.f 2
3.b odd 2 1 inner 450.3.d.f 2
4.b odd 2 1 3600.3.l.d 2
5.b even 2 1 18.3.b.a 2
5.c odd 4 2 450.3.b.b 4
12.b even 2 1 3600.3.l.d 2
15.d odd 2 1 18.3.b.a 2
15.e even 4 2 450.3.b.b 4
20.d odd 2 1 144.3.e.b 2
20.e even 4 2 3600.3.c.b 4
35.c odd 2 1 882.3.b.a 2
35.i odd 6 2 882.3.s.d 4
35.j even 6 2 882.3.s.b 4
40.e odd 2 1 576.3.e.f 2
40.f even 2 1 576.3.e.c 2
45.h odd 6 2 162.3.d.b 4
45.j even 6 2 162.3.d.b 4
55.d odd 2 1 2178.3.c.d 2
60.h even 2 1 144.3.e.b 2
60.l odd 4 2 3600.3.c.b 4
65.d even 2 1 3042.3.c.e 2
65.g odd 4 2 3042.3.d.a 4
80.k odd 4 2 2304.3.h.c 4
80.q even 4 2 2304.3.h.f 4
105.g even 2 1 882.3.b.a 2
105.o odd 6 2 882.3.s.b 4
105.p even 6 2 882.3.s.d 4
120.i odd 2 1 576.3.e.c 2
120.m even 2 1 576.3.e.f 2
165.d even 2 1 2178.3.c.d 2
180.n even 6 2 1296.3.q.f 4
180.p odd 6 2 1296.3.q.f 4
195.e odd 2 1 3042.3.c.e 2
195.n even 4 2 3042.3.d.a 4
240.t even 4 2 2304.3.h.c 4
240.bm odd 4 2 2304.3.h.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 5.b even 2 1
18.3.b.a 2 15.d odd 2 1
144.3.e.b 2 20.d odd 2 1
144.3.e.b 2 60.h even 2 1
162.3.d.b 4 45.h odd 6 2
162.3.d.b 4 45.j even 6 2
450.3.b.b 4 5.c odd 4 2
450.3.b.b 4 15.e even 4 2
450.3.d.f 2 1.a even 1 1 trivial
450.3.d.f 2 3.b odd 2 1 inner
576.3.e.c 2 40.f even 2 1
576.3.e.c 2 120.i odd 2 1
576.3.e.f 2 40.e odd 2 1
576.3.e.f 2 120.m even 2 1
882.3.b.a 2 35.c odd 2 1
882.3.b.a 2 105.g even 2 1
882.3.s.b 4 35.j even 6 2
882.3.s.b 4 105.o odd 6 2
882.3.s.d 4 35.i odd 6 2
882.3.s.d 4 105.p even 6 2
1296.3.q.f 4 180.n even 6 2
1296.3.q.f 4 180.p odd 6 2
2178.3.c.d 2 55.d odd 2 1
2178.3.c.d 2 165.d even 2 1
2304.3.h.c 4 80.k odd 4 2
2304.3.h.c 4 240.t even 4 2
2304.3.h.f 4 80.q even 4 2
2304.3.h.f 4 240.bm odd 4 2
3042.3.c.e 2 65.d even 2 1
3042.3.c.e 2 195.e odd 2 1
3042.3.d.a 4 65.g odd 4 2
3042.3.d.a 4 195.n even 4 2
3600.3.c.b 4 20.e even 4 2
3600.3.c.b 4 60.l odd 4 2
3600.3.l.d 2 4.b odd 2 1
3600.3.l.d 2 12.b even 2 1

Hecke kernels

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

\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 288 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 288 \) Copy content Toggle raw display
$13$ \( (T + 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 162 \) Copy content Toggle raw display
$19$ \( (T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 288 \) Copy content Toggle raw display
$29$ \( T^{2} + 18 \) Copy content Toggle raw display
$31$ \( (T - 44)^{2} \) Copy content Toggle raw display
$37$ \( (T - 34)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2178 \) Copy content Toggle raw display
$43$ \( (T - 40)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 7200 \) Copy content Toggle raw display
$53$ \( T^{2} + 1458 \) Copy content Toggle raw display
$59$ \( T^{2} + 1152 \) Copy content Toggle raw display
$61$ \( (T - 50)^{2} \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 2592 \) Copy content Toggle raw display
$73$ \( (T - 16)^{2} \) Copy content Toggle raw display
$79$ \( (T + 76)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14112 \) Copy content Toggle raw display
$89$ \( T^{2} + 162 \) Copy content Toggle raw display
$97$ \( (T + 176)^{2} \) Copy content Toggle raw display
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