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Properties

Label 450.3.i.b
Level $450$
Weight $3$
Character orbit 450.i
Analytic conductor $12.262$
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] = [450,3,Mod(101,450)]
 
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([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("450.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.i (of order \(6\), 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 \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{6} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} - 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{6} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} - 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots + 6) q^{11}+ \cdots + (9 \beta_{3} + 27 \beta_{2} + \cdots + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 12 q^{6} - 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 12 q^{6} - 2 q^{7} + 12 q^{9} + 18 q^{11} - 12 q^{12} + 10 q^{13} + 36 q^{14} - 8 q^{16} + 24 q^{18} - 40 q^{19} - 42 q^{21} + 12 q^{22} - 18 q^{23} - 8 q^{28} + 18 q^{29} + 38 q^{31} - 54 q^{33} + 24 q^{34} + 12 q^{36} - 128 q^{37} + 72 q^{38} - 102 q^{39} - 126 q^{41} + 48 q^{42} + 46 q^{43} - 24 q^{46} - 54 q^{47} - 24 q^{48} - 12 q^{49} - 72 q^{51} - 20 q^{52} + 36 q^{54} + 72 q^{56} - 144 q^{57} - 24 q^{58} + 126 q^{59} + 62 q^{61} - 222 q^{63} - 32 q^{64} - 72 q^{66} + 106 q^{67} - 72 q^{68} + 18 q^{69} + 96 q^{72} + 208 q^{73} + 72 q^{74} - 40 q^{76} + 90 q^{77} - 144 q^{78} + 14 q^{79} - 252 q^{81} - 144 q^{82} + 378 q^{83} - 144 q^{84} - 108 q^{86} + 54 q^{87} - 24 q^{88} + 412 q^{91} - 36 q^{92} - 222 q^{93} + 84 q^{94} + 48 q^{96} - 14 q^{97} + 126 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 − 0.707107i 2.44949 + 1.73205i 1.00000 + 1.73205i 0 −1.77526 − 3.85337i −4.17423 + 7.22999i − 2.82843i 3.00000 + 8.48528i 0
101.2 1.22474 + 0.707107i −2.44949 + 1.73205i 1.00000 + 1.73205i 0 −4.22474 + 0.389270i 3.17423 − 5.49794i 2.82843i 3.00000 − 8.48528i 0
401.1 −1.22474 + 0.707107i 2.44949 − 1.73205i 1.00000 − 1.73205i 0 −1.77526 + 3.85337i −4.17423 − 7.22999i 2.82843i 3.00000 − 8.48528i 0
401.2 1.22474 − 0.707107i −2.44949 − 1.73205i 1.00000 − 1.73205i 0 −4.22474 − 0.389270i 3.17423 + 5.49794i − 2.82843i 3.00000 + 8.48528i 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.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.i.b 4
3.b odd 2 1 1350.3.i.b 4
5.b even 2 1 18.3.d.a ✓ 4
5.c odd 4 2 450.3.k.a 8
9.c even 3 1 1350.3.i.b 4
9.d odd 6 1 inner 450.3.i.b 4
15.d odd 2 1 54.3.d.a 4
15.e even 4 2 1350.3.k.a 8
20.d odd 2 1 144.3.q.c 4
40.e odd 2 1 576.3.q.e 4
40.f even 2 1 576.3.q.f 4
45.h odd 6 1 18.3.d.a ✓ 4
45.h odd 6 1 162.3.b.a 4
45.j even 6 1 54.3.d.a 4
45.j even 6 1 162.3.b.a 4
45.k odd 12 2 1350.3.k.a 8
45.l even 12 2 450.3.k.a 8
60.h even 2 1 432.3.q.d 4
120.i odd 2 1 1728.3.q.d 4
120.m even 2 1 1728.3.q.c 4
180.n even 6 1 144.3.q.c 4
180.n even 6 1 1296.3.e.g 4
180.p odd 6 1 432.3.q.d 4
180.p odd 6 1 1296.3.e.g 4
360.z odd 6 1 1728.3.q.c 4
360.bd even 6 1 576.3.q.e 4
360.bh odd 6 1 576.3.q.f 4
360.bk even 6 1 1728.3.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a ✓ 4 5.b even 2 1
18.3.d.a ✓ 4 45.h odd 6 1
54.3.d.a 4 15.d odd 2 1
54.3.d.a 4 45.j even 6 1
144.3.q.c 4 20.d odd 2 1
144.3.q.c 4 180.n even 6 1
162.3.b.a 4 45.h odd 6 1
162.3.b.a 4 45.j even 6 1
432.3.q.d 4 60.h even 2 1
432.3.q.d 4 180.p odd 6 1
450.3.i.b 4 1.a even 1 1 trivial
450.3.i.b 4 9.d odd 6 1 inner
450.3.k.a 8 5.c odd 4 2
450.3.k.a 8 45.l even 12 2
576.3.q.e 4 40.e odd 2 1
576.3.q.e 4 360.bd even 6 1
576.3.q.f 4 40.f even 2 1
576.3.q.f 4 360.bh odd 6 1
1296.3.e.g 4 180.n even 6 1
1296.3.e.g 4 180.p odd 6 1
1350.3.i.b 4 3.b odd 2 1
1350.3.i.b 4 9.c even 3 1
1350.3.k.a 8 15.e even 4 2
1350.3.k.a 8 45.k odd 12 2
1728.3.q.c 4 120.m even 2 1
1728.3.q.c 4 360.z odd 6 1
1728.3.q.d 4 120.i odd 2 1
1728.3.q.d 4 360.bk even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 2T_{7}^{3} + 57T_{7}^{2} - 106T_{7} + 2809 \) acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - 6T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 2809 \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} - 10 T^{3} + \cdots + 36481 \) Copy content Toggle raw display
$17$ \( T^{4} + 360T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} - 18 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{4} - 38 T^{3} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( (T^{2} + 64 T + 808)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 126 T^{3} + \cdots + 455625 \) Copy content Toggle raw display
$43$ \( T^{4} - 46 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( T^{4} + 54 T^{3} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( T^{4} + 9000 T^{2} + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} - 126 T^{3} + \cdots + 2954961 \) Copy content Toggle raw display
$61$ \( T^{4} - 62 T^{3} + \cdots + 966289 \) Copy content Toggle raw display
$67$ \( T^{4} - 106 T^{3} + \cdots + 5396329 \) Copy content Toggle raw display
$71$ \( T^{4} + 7704 T^{2} + 2396304 \) Copy content Toggle raw display
$73$ \( (T^{2} - 104 T + 760)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} + \cdots + 1692601 \) Copy content Toggle raw display
$83$ \( T^{4} - 378 T^{3} + \cdots + 131262849 \) Copy content Toggle raw display
$89$ \( T^{4} + 22824 T^{2} + 36144144 \) Copy content Toggle raw display
$97$ \( T^{4} + 14 T^{3} + \cdots + 110986225 \) Copy content Toggle raw display
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