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Properties

Label 450.2.e.n
Level $450$
Weight $2$
Character orbit 450.e
Analytic conductor $3.593$
Analytic rank $0$
Dimension $4$
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] = [450,2,Mod(151,450)]
 
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([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.e (of order \(3\), 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: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} - \beta_{2} - \beta_1) q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{2} - \beta_1 + 1) q^{6} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{7} - q^{8} + ( - \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{3} - \beta_{2} - \beta_1) q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{2} - \beta_1 + 1) q^{6} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{7} - q^{8} + ( - \beta_{2} + 2 \beta_1 + 1) q^{9} + (\beta_{3} + \beta_{2} + \beta_1) q^{11} + ( - \beta_{3} + 1) q^{12} + (2 \beta_{3} - \beta_1) q^{13} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{14} - \beta_{2} q^{16} + (2 \beta_{3} - 4 \beta_1 - 1) q^{17} + (2 \beta_{3} + 1) q^{18} + (\beta_{3} - 2 \beta_1 - 3) q^{19} + ( - 2 \beta_{3} + 3 \beta_1 - 4) q^{21} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{22} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{23} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{24} + (\beta_{3} - 2 \beta_1) q^{26} + ( - \beta_{3} - 5) q^{27} + (\beta_{3} - 2 \beta_1 + 2) q^{28} + 6 \beta_{2} q^{29} + ( - 2 \beta_{3} + 4 \beta_{2} + \cdots - 4) q^{31}+ \cdots + (\beta_{3} + 8 \beta_{2} + 2 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} - 4 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} - 4 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{11} + 4 q^{12} + 4 q^{14} - 2 q^{16} - 4 q^{17} + 4 q^{18} - 12 q^{19} - 16 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 20 q^{27} + 8 q^{28} + 12 q^{29} - 8 q^{31} + 2 q^{32} - 10 q^{33} - 2 q^{34} + 2 q^{36} + 32 q^{37} - 6 q^{38} + 2 q^{41} - 8 q^{42} - 10 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{47} - 2 q^{48} - 6 q^{49} + 26 q^{51} + 24 q^{53} - 10 q^{54} + 4 q^{56} + 18 q^{57} - 12 q^{58} - 2 q^{59} - 4 q^{61} - 16 q^{62} - 8 q^{63} + 4 q^{64} + 4 q^{66} - 14 q^{67} + 2 q^{68} + 8 q^{69} - 2 q^{72} + 20 q^{73} + 16 q^{74} + 6 q^{76} - 8 q^{77} + 12 q^{78} + 14 q^{81} + 4 q^{82} + 8 q^{83} + 8 q^{84} + 10 q^{86} + 12 q^{87} - 2 q^{88} + 32 q^{89} - 24 q^{91} - 4 q^{92} + 16 q^{93} - 4 q^{94} - 4 q^{96} - 26 q^{97} - 12 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

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

Display \(\nu^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
1.22474 − 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 − 0.707107i
0.500000 − 0.866025i −1.72474 + 0.158919i −0.500000 − 0.866025i 0 −0.724745 + 1.57313i 0.224745 − 0.389270i −1.00000 2.94949 − 0.548188i 0
151.2 0.500000 − 0.866025i 0.724745 + 1.57313i −0.500000 − 0.866025i 0 1.72474 + 0.158919i −2.22474 + 3.85337i −1.00000 −1.94949 + 2.28024i 0
301.1 0.500000 + 0.866025i −1.72474 − 0.158919i −0.500000 + 0.866025i 0 −0.724745 − 1.57313i 0.224745 + 0.389270i −1.00000 2.94949 + 0.548188i 0
301.2 0.500000 + 0.866025i 0.724745 − 1.57313i −0.500000 + 0.866025i 0 1.72474 − 0.158919i −2.22474 − 3.85337i −1.00000 −1.94949 − 2.28024i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.e.n 4
3.b odd 2 1 1350.2.e.j 4
5.b even 2 1 450.2.e.k 4
5.c odd 4 2 90.2.i.b ✓ 8
9.c even 3 1 inner 450.2.e.n 4
9.c even 3 1 4050.2.a.bq 2
9.d odd 6 1 1350.2.e.j 4
9.d odd 6 1 4050.2.a.bz 2
15.d odd 2 1 1350.2.e.m 4
15.e even 4 2 270.2.i.b 8
20.e even 4 2 720.2.by.c 8
45.h odd 6 1 1350.2.e.m 4
45.h odd 6 1 4050.2.a.bm 2
45.j even 6 1 450.2.e.k 4
45.j even 6 1 4050.2.a.bs 2
45.k odd 12 2 90.2.i.b ✓ 8
45.k odd 12 2 810.2.c.f 4
45.l even 12 2 270.2.i.b 8
45.l even 12 2 810.2.c.e 4
60.l odd 4 2 2160.2.by.d 8
180.v odd 12 2 2160.2.by.d 8
180.x even 12 2 720.2.by.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.i.b ✓ 8 5.c odd 4 2
90.2.i.b ✓ 8 45.k odd 12 2
270.2.i.b 8 15.e even 4 2
270.2.i.b 8 45.l even 12 2
450.2.e.k 4 5.b even 2 1
450.2.e.k 4 45.j even 6 1
450.2.e.n 4 1.a even 1 1 trivial
450.2.e.n 4 9.c even 3 1 inner
720.2.by.c 8 20.e even 4 2
720.2.by.c 8 180.x even 12 2
810.2.c.e 4 45.l even 12 2
810.2.c.f 4 45.k odd 12 2
1350.2.e.j 4 3.b odd 2 1
1350.2.e.j 4 9.d odd 6 1
1350.2.e.m 4 15.d odd 2 1
1350.2.e.m 4 45.h odd 6 1
2160.2.by.d 8 60.l odd 4 2
2160.2.by.d 8 180.v odd 12 2
4050.2.a.bm 2 45.h odd 6 1
4050.2.a.bq 2 9.c even 3 1
4050.2.a.bs 2 45.j even 6 1
4050.2.a.bz 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} + 4T_{7}^{3} + 18T_{7}^{2} - 8T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} + 9T_{11}^{2} + 10T_{11} + 25 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} - 23 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$13$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T - 23)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$37$ \( (T - 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 30)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + \cdots + 22201 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$67$ \( T^{4} + 14 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 10 T - 71)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 54T^{2} + 2916 \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 40)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
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