Properties

Label 450.3.g.e
Level $450$
Weight $3$
Character orbit 450.g
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(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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{7} + (2 i - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{7} + (2 i - 2) q^{8} - 12 q^{11} + (12 i - 12) q^{13} - 6 i q^{14} - 4 q^{16} + ( - 12 i - 12) q^{17} + 20 i q^{19} + ( - 12 i - 12) q^{22} + (3 i - 3) q^{23} - 24 q^{26} + ( - 6 i + 6) q^{28} + 30 i q^{29} - 8 q^{31} + ( - 4 i - 4) q^{32} - 24 i q^{34} + ( - 48 i - 48) q^{37} + (20 i - 20) q^{38} + 48 q^{41} + (27 i - 27) q^{43} - 24 i q^{44} - 6 q^{46} + ( - 27 i - 27) q^{47} - 31 i q^{49} + ( - 24 i - 24) q^{52} + ( - 12 i + 12) q^{53} + 12 q^{56} + (30 i - 30) q^{58} + 60 i q^{59} + 32 q^{61} + ( - 8 i - 8) q^{62} - 8 i q^{64} + ( - 3 i - 3) q^{67} + ( - 24 i + 24) q^{68} + 48 q^{71} + (12 i - 12) q^{73} - 96 i q^{74} - 40 q^{76} + (36 i + 36) q^{77} + 40 i q^{79} + (48 i + 48) q^{82} + (93 i - 93) q^{83} - 54 q^{86} + ( - 24 i + 24) q^{88} - 30 i q^{89} + 72 q^{91} + ( - 6 i - 6) q^{92} - 54 i q^{94} + (12 i + 12) q^{97} + ( - 31 i + 31) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 6 q^{7} - 4 q^{8} - 24 q^{11} - 24 q^{13} - 8 q^{16} - 24 q^{17} - 24 q^{22} - 6 q^{23} - 48 q^{26} + 12 q^{28} - 16 q^{31} - 8 q^{32} - 96 q^{37} - 40 q^{38} + 96 q^{41} - 54 q^{43} - 12 q^{46} - 54 q^{47} - 48 q^{52} + 24 q^{53} + 24 q^{56} - 60 q^{58} + 64 q^{61} - 16 q^{62} - 6 q^{67} + 48 q^{68} + 96 q^{71} - 24 q^{73} - 80 q^{76} + 72 q^{77} + 96 q^{82} - 186 q^{83} - 108 q^{86} + 48 q^{88} + 144 q^{91} - 12 q^{92} + 24 q^{97} + 62 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
1.00000i
1.00000i
1.00000 + 1.00000i 0 2.00000i 0 0 −3.00000 3.00000i −2.00000 + 2.00000i 0 0
343.1 1.00000 1.00000i 0 2.00000i 0 0 −3.00000 + 3.00000i −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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.g.e 2
3.b odd 2 1 50.3.c.a 2
5.b even 2 1 450.3.g.c 2
5.c odd 4 1 450.3.g.c 2
5.c odd 4 1 inner 450.3.g.e 2
12.b even 2 1 400.3.p.a 2
15.d odd 2 1 50.3.c.b yes 2
15.e even 4 1 50.3.c.a 2
15.e even 4 1 50.3.c.b yes 2
60.h even 2 1 400.3.p.g 2
60.l odd 4 1 400.3.p.a 2
60.l odd 4 1 400.3.p.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.3.c.a 2 3.b odd 2 1
50.3.c.a 2 15.e even 4 1
50.3.c.b yes 2 15.d odd 2 1
50.3.c.b yes 2 15.e even 4 1
400.3.p.a 2 12.b even 2 1
400.3.p.a 2 60.l odd 4 1
400.3.p.g 2 60.h even 2 1
400.3.p.g 2 60.l odd 4 1
450.3.g.c 2 5.b even 2 1
450.3.g.c 2 5.c odd 4 1
450.3.g.e 2 1.a even 1 1 trivial
450.3.g.e 2 5.c odd 4 1 inner

Hecke kernels

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

\( T_{7}^{2} + 6T_{7} + 18 \) Copy content Toggle raw display
\( T_{11} + 12 \) Copy content Toggle raw display
\( T_{17}^{2} + 24T_{17} + 288 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$11$ \( (T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 24T + 288 \) Copy content Toggle raw display
$17$ \( T^{2} + 24T + 288 \) Copy content Toggle raw display
$19$ \( T^{2} + 400 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$29$ \( T^{2} + 900 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 96T + 4608 \) Copy content Toggle raw display
$41$ \( (T - 48)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 54T + 1458 \) Copy content Toggle raw display
$47$ \( T^{2} + 54T + 1458 \) Copy content Toggle raw display
$53$ \( T^{2} - 24T + 288 \) Copy content Toggle raw display
$59$ \( T^{2} + 3600 \) Copy content Toggle raw display
$61$ \( (T - 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$71$ \( (T - 48)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 24T + 288 \) Copy content Toggle raw display
$79$ \( T^{2} + 1600 \) Copy content Toggle raw display
$83$ \( T^{2} + 186T + 17298 \) Copy content Toggle raw display
$89$ \( T^{2} + 900 \) Copy content Toggle raw display
$97$ \( T^{2} - 24T + 288 \) Copy content Toggle raw display
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