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Properties

Label 450.3.i.c
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 \beta_{2} q^{3} + 2 \beta_{2} q^{4} - 3 \beta_{3} q^{6} + (4 \beta_{3} - 7 \beta_{2} - 2 \beta_1 + 7) q^{7} - 2 \beta_{3} q^{8} + (9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 \beta_{2} q^{3} + 2 \beta_{2} q^{4} - 3 \beta_{3} q^{6} + (4 \beta_{3} - 7 \beta_{2} - 2 \beta_1 + 7) q^{7} - 2 \beta_{3} q^{8} + (9 \beta_{2} - 9) q^{9} + ( - 2 \beta_{2} - 7 \beta_1 - 2) q^{11} + (6 \beta_{2} - 6) q^{12} + ( - 7 \beta_{3} - 6 \beta_{2} - 7 \beta_1) q^{13} + (7 \beta_{3} - 4 \beta_{2} - 7 \beta_1 + 8) q^{14} + (4 \beta_{2} - 4) q^{16} + ( - 17 \beta_{3} - 4 \beta_{2} + 2) q^{17} + ( - 9 \beta_{3} + 9 \beta_1) q^{18} + (\beta_{3} - 2 \beta_1 + 18) q^{19} + (6 \beta_{3} - 12 \beta_1 + 21) q^{21} + (2 \beta_{3} + 14 \beta_{2} + 2 \beta_1) q^{22} + (2 \beta_{3} + 7 \beta_{2} - 2 \beta_1 - 14) q^{23} + ( - 6 \beta_{3} + 6 \beta_1) q^{24} + (6 \beta_{3} + 28 \beta_{2} - 14) q^{26} - 27 q^{27} + (4 \beta_{3} - 8 \beta_1 + 14) q^{28} + (3 \beta_{2} - 6 \beta_1 + 3) q^{29} + (13 \beta_{3} - 22 \beta_{2} + 13 \beta_1) q^{31} + ( - 4 \beta_{3} + 4 \beta_1) q^{32} + ( - 21 \beta_{3} - 12 \beta_{2} + 6) q^{33} + (4 \beta_{3} + 34 \beta_{2} + \cdots - 34) q^{34}+ \cdots + ( - 63 \beta_{3} - 18 \beta_{2} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 4 q^{4} + 14 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 4 q^{4} + 14 q^{7} - 18 q^{9} - 12 q^{11} - 12 q^{12} - 12 q^{13} + 24 q^{14} - 8 q^{16} + 72 q^{19} + 84 q^{21} + 28 q^{22} - 42 q^{23} - 108 q^{27} + 56 q^{28} + 18 q^{29} - 44 q^{31} - 68 q^{34} - 72 q^{36} + 104 q^{37} + 12 q^{38} + 36 q^{39} + 18 q^{41} + 72 q^{42} + 8 q^{43} + 16 q^{46} - 174 q^{47} - 48 q^{48} - 48 q^{49} + 36 q^{51} + 24 q^{52} + 48 q^{56} + 108 q^{57} + 24 q^{58} - 96 q^{59} - 10 q^{61} + 126 q^{63} - 32 q^{64} - 84 q^{66} - 14 q^{67} + 24 q^{68} - 126 q^{69} - 64 q^{73} - 36 q^{74} + 72 q^{76} + 84 q^{77} - 252 q^{78} + 144 q^{79} - 162 q^{81} - 32 q^{82} + 54 q^{83} + 84 q^{84} - 192 q^{86} - 56 q^{88} + 168 q^{91} - 84 q^{92} + 132 q^{93} - 40 q^{94} + 40 q^{97} + 108 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

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

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 1.50000 + 2.59808i 1.00000 + 1.73205i 0 − 4.24264i 1.05051 − 1.81954i − 2.82843i −4.50000 + 7.79423i 0
101.2 1.22474 + 0.707107i 1.50000 + 2.59808i 1.00000 + 1.73205i 0 4.24264i 5.94949 − 10.3048i 2.82843i −4.50000 + 7.79423i 0
401.1 −1.22474 + 0.707107i 1.50000 − 2.59808i 1.00000 − 1.73205i 0 4.24264i 1.05051 + 1.81954i 2.82843i −4.50000 − 7.79423i 0
401.2 1.22474 − 0.707107i 1.50000 − 2.59808i 1.00000 − 1.73205i 0 − 4.24264i 5.94949 + 10.3048i − 2.82843i −4.50000 − 7.79423i 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.c 4
3.b odd 2 1 1350.3.i.c 4
5.b even 2 1 450.3.i.a 4
5.c odd 4 2 90.3.j.a ✓ 8
9.c even 3 1 1350.3.i.c 4
9.d odd 6 1 inner 450.3.i.c 4
15.d odd 2 1 1350.3.i.a 4
15.e even 4 2 270.3.j.a 8
45.h odd 6 1 450.3.i.a 4
45.j even 6 1 1350.3.i.a 4
45.k odd 12 2 270.3.j.a 8
45.k odd 12 2 810.3.b.a 8
45.l even 12 2 90.3.j.a ✓ 8
45.l even 12 2 810.3.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.j.a ✓ 8 5.c odd 4 2
90.3.j.a ✓ 8 45.l even 12 2
270.3.j.a 8 15.e even 4 2
270.3.j.a 8 45.k odd 12 2
450.3.i.a 4 5.b even 2 1
450.3.i.a 4 45.h odd 6 1
450.3.i.c 4 1.a even 1 1 trivial
450.3.i.c 4 9.d odd 6 1 inner
810.3.b.a 8 45.k odd 12 2
810.3.b.a 8 45.l even 12 2
1350.3.i.a 4 15.d odd 2 1
1350.3.i.a 4 45.j even 6 1
1350.3.i.c 4 3.b odd 2 1
1350.3.i.c 4 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 14T_{7}^{3} + 171T_{7}^{2} - 350T_{7} + 625 \) 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
$3$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 14 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + \cdots + 7396 \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} + \cdots + 66564 \) Copy content Toggle raw display
$17$ \( T^{4} + 1180 T^{2} + 320356 \) Copy content Toggle raw display
$19$ \( (T^{2} - 36 T + 318)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 42 T^{3} + \cdots + 19321 \) Copy content Toggle raw display
$29$ \( T^{4} - 18 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{4} + 44 T^{3} + \cdots + 280900 \) Copy content Toggle raw display
$37$ \( (T^{2} - 52 T + 622)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + \cdots + 2310400 \) Copy content Toggle raw display
$47$ \( T^{4} + 174 T^{3} + \cdots + 5396329 \) Copy content Toggle raw display
$53$ \( T^{4} + 2832 T^{2} + 14400 \) Copy content Toggle raw display
$59$ \( T^{4} + 96 T^{3} + \cdots + 21529600 \) Copy content Toggle raw display
$61$ \( T^{4} + 10 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{4} + 14 T^{3} + \cdots + 5527201 \) Copy content Toggle raw display
$71$ \( T^{4} + 21084 T^{2} + 92968164 \) Copy content Toggle raw display
$73$ \( (T^{2} + 32 T + 232)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 144 T^{3} + \cdots + 46656 \) Copy content Toggle raw display
$83$ \( T^{4} - 54 T^{3} + \cdots + 44521 \) Copy content Toggle raw display
$89$ \( T^{4} + 28774 T^{2} + 125731369 \) Copy content Toggle raw display
$97$ \( T^{4} - 40 T^{3} + \cdots + 9339136 \) Copy content Toggle raw display
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