Properties

Label 900.3.c.o
Level $900$
Weight $3$
Character orbit 900.c
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(451,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.451");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.c (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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.15012375625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{6} + 28x^{4} + 112x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 180)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + ( - \beta_{7} - \beta_{3}) q^{7} + (\beta_{6} + \beta_{5} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + ( - \beta_{7} - \beta_{3}) q^{7} + (\beta_{6} + \beta_{5} - 2 \beta_1) q^{8} + ( - \beta_{6} - 2 \beta_{5}) q^{11} + (\beta_{7} + \beta_{4} - 2 \beta_{3} - 1) q^{13} + ( - 2 \beta_{6} + 2 \beta_{2}) q^{14} + (\beta_{4} - 2 \beta_{3} - 1) q^{16} + (3 \beta_{6} - 2 \beta_{2} - 4 \beta_1) q^{17} + (\beta_{7} - \beta_{4} - 4 \beta_{3} + 1) q^{19} + ( - \beta_{7} - 2 \beta_{4} - \beta_{3}) q^{22} + (\beta_{6} + \beta_{2} + 5 \beta_1) q^{23} + ( - 4 \beta_{5} + 4 \beta_{2} - 4 \beta_1) q^{26} + (4 \beta_{7} + 2 \beta_{3} + 12) q^{28} + (2 \beta_{6} - \beta_{2} - 3 \beta_1) q^{29} + (3 \beta_{7} - \beta_{4} - 2 \beta_{3} + 1) q^{31} + ( - \beta_{6} - 3 \beta_{5} + 6 \beta_{2} - 4 \beta_1) q^{32} + ( - 5 \beta_{7} - 7 \beta_{3} - 14) q^{34} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3} - 35) q^{37} + ( - 4 \beta_{6} - 4 \beta_{5} - 8 \beta_{2} + 4 \beta_1) q^{38} + (4 \beta_{6} - 8 \beta_1) q^{41} + ( - 2 \beta_{7} + 2 \beta_{4} + 8 \beta_{3} - 2) q^{43} + ( - 4 \beta_{6} + 2 \beta_{5} - 10 \beta_{2} + 6 \beta_1) q^{44} + (4 \beta_{3} - 24) q^{46} + ( - 3 \beta_{6} - 8 \beta_{5} + \beta_{2} + 5 \beta_1) q^{47} + ( - 4 \beta_{7} - 4 \beta_{4} + 8 \beta_{3} - 23) q^{49} + ( - 4 \beta_{4} - 8 \beta_{3} - 28) q^{52} + (4 \beta_{6} + 7 \beta_{2} - 15 \beta_1) q^{53} + (6 \beta_{6} - 2 \beta_{5} - 8 \beta_{2} + 12 \beta_1) q^{56} + ( - 3 \beta_{7} - 5 \beta_{3} - 10) q^{58} + (9 \beta_{6} + 6 \beta_{5} + 6 \beta_{2} + 30 \beta_1) q^{59} + ( - 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{3}) q^{61} + ( - 4 \beta_{5} - 12 \beta_{2} + 4 \beta_1) q^{62} + (4 \beta_{7} - 3 \beta_{4} - 6 \beta_{3} - 21) q^{64} + ( - 6 \beta_{7} - 4 \beta_{4} - 26 \beta_{3} + 4) q^{67} + ( - 12 \beta_{6} - 2 \beta_{5} + 10 \beta_{2} - 14 \beta_1) q^{68} + ( - 4 \beta_{6} - 4 \beta_{5} - 2 \beta_{2} - 10 \beta_1) q^{71} + ( - 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{3} + 28) q^{73} + (4 \beta_{5} - 4 \beta_{2} - 32 \beta_1) q^{74} + ( - 8 \beta_{7} - 4 \beta_{4} + 4 \beta_{3} + 44) q^{76} + ( - 6 \beta_{6} + 16 \beta_{2} - 4 \beta_1) q^{77} + ( - 7 \beta_{7} - 3 \beta_{4} - 22 \beta_{3} + 3) q^{79} + ( - 4 \beta_{7} - 12 \beta_{3} - 24) q^{82} + (6 \beta_{6} - 4 \beta_{5} + 8 \beta_{2} + 40 \beta_1) q^{83} + (8 \beta_{6} + 8 \beta_{5} + 16 \beta_{2} - 8 \beta_1) q^{86} + ( - 4 \beta_{7} + 2 \beta_{4} + 12 \beta_{3} + 78) q^{88} + (4 \beta_{6} + 26 \beta_{2} - 34 \beta_1) q^{89} + ( - 14 \beta_{7} + 2 \beta_{4} - 4 \beta_{3} - 2) q^{91} + (4 \beta_{6} + 4 \beta_{5} - 24 \beta_1) q^{92} + ( - 4 \beta_{7} - 8 \beta_{4} - 24) q^{94} + ( - 6 \beta_{7} - 6 \beta_{4} + 12 \beta_{3} - 36) q^{97} + (16 \beta_{5} - 16 \beta_{2} - 11 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 14 q^{4} - 16 q^{13} - 14 q^{16} + 4 q^{22} + 92 q^{28} - 116 q^{34} - 272 q^{37} - 184 q^{46} - 152 q^{49} - 232 q^{52} - 84 q^{58} + 16 q^{61} - 182 q^{64} + 240 q^{73} + 384 q^{76} - 208 q^{82} + 652 q^{88} - 168 q^{94} - 240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{6} + 28x^{4} + 112x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 4\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 2\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 13\nu^{5} - 28\nu^{3} - 144\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + 13\nu^{5} + 60\nu^{3} + 208\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 3\nu^{4} + 12\nu^{2} + 48 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 2\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} - 3\beta_{5} + 6\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{7} - 3\beta_{4} - 6\beta_{3} - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 7\beta_{5} - 26\beta_{2} - 12\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−1.46040 1.36646i
−1.46040 + 1.36646i
−0.342371 1.97048i
−0.342371 + 1.97048i
0.342371 1.97048i
0.342371 + 1.97048i
1.46040 1.36646i
1.46040 + 1.36646i
−1.46040 1.36646i 0 0.265564 + 3.99117i 0 0 1.87135i 5.06596 6.19161i 0 0
451.2 −1.46040 + 1.36646i 0 0.265564 3.99117i 0 0 1.87135i 5.06596 + 6.19161i 0 0
451.3 −0.342371 1.97048i 0 −3.76556 + 1.34927i 0 0 11.5108i 3.94792 + 6.95801i 0 0
451.4 −0.342371 + 1.97048i 0 −3.76556 1.34927i 0 0 11.5108i 3.94792 6.95801i 0 0
451.5 0.342371 1.97048i 0 −3.76556 1.34927i 0 0 11.5108i −3.94792 + 6.95801i 0 0
451.6 0.342371 + 1.97048i 0 −3.76556 + 1.34927i 0 0 11.5108i −3.94792 6.95801i 0 0
451.7 1.46040 1.36646i 0 0.265564 3.99117i 0 0 1.87135i −5.06596 6.19161i 0 0
451.8 1.46040 + 1.36646i 0 0.265564 + 3.99117i 0 0 1.87135i −5.06596 + 6.19161i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.o 8
3.b odd 2 1 inner 900.3.c.o 8
4.b odd 2 1 inner 900.3.c.o 8
5.b even 2 1 180.3.c.c 8
5.c odd 4 2 900.3.f.i 16
12.b even 2 1 inner 900.3.c.o 8
15.d odd 2 1 180.3.c.c 8
15.e even 4 2 900.3.f.i 16
20.d odd 2 1 180.3.c.c 8
20.e even 4 2 900.3.f.i 16
40.e odd 2 1 2880.3.e.i 8
40.f even 2 1 2880.3.e.i 8
60.h even 2 1 180.3.c.c 8
60.l odd 4 2 900.3.f.i 16
120.i odd 2 1 2880.3.e.i 8
120.m even 2 1 2880.3.e.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.c.c 8 5.b even 2 1
180.3.c.c 8 15.d odd 2 1
180.3.c.c 8 20.d odd 2 1
180.3.c.c 8 60.h even 2 1
900.3.c.o 8 1.a even 1 1 trivial
900.3.c.o 8 3.b odd 2 1 inner
900.3.c.o 8 4.b odd 2 1 inner
900.3.c.o 8 12.b even 2 1 inner
900.3.f.i 16 5.c odd 4 2
900.3.f.i 16 15.e even 4 2
900.3.f.i 16 20.e even 4 2
900.3.f.i 16 60.l odd 4 2
2880.3.e.i 8 40.e odd 2 1
2880.3.e.i 8 40.f even 2 1
2880.3.e.i 8 120.i odd 2 1
2880.3.e.i 8 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} + 136T_{7}^{2} + 464 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 256 \) Copy content Toggle raw display
\( T_{17}^{4} - 1096T_{17}^{2} + 150544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 7 T^{6} + 28 T^{4} + 112 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 136 T^{2} + 464)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 328 T^{2} + 22736)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 256)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1096 T^{2} + 150544)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 944 T^{2} + 118784)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 368 T^{2} + 29696)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 456 T^{2} + 35344)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1840 T^{2} + 742400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 68 T + 896)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 832)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3776 T^{2} + 1900544)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 5552 T^{2} + 7602176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 3624 T^{2} + 21904)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16488 T^{2} + 51452496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 1036)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 21664 T^{2} + 75732224)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2848 T^{2} + 363776)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 60 T - 140)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16816 T^{2} + 65598464)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 24608 T^{2} + 69852416)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 28704 T^{2} + \cdots + 160985344)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 60 T - 8460)^{4} \) Copy content Toggle raw display
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