Properties

Label 900.2.k.g
Level $900$
Weight $2$
Character orbit 900.k
Analytic conductor $7.187$
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] = [900,2,Mod(307,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.k (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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 300)
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 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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{6} + \beta_{2} - 2) q^{4} + ( - 2 \beta_{7} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{7} + (2 \beta_{5} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{6} + \beta_{2} - 2) q^{4} + ( - 2 \beta_{7} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{7} + (2 \beta_{5} + 2 \beta_1) q^{8} + 2 \beta_{3} q^{11} + ( - \beta_{7} + 3 \beta_{5} + \beta_{4} - \beta_1) q^{13} + (2 \beta_{6} + \beta_{3} + \beta_{2} + 4) q^{14} + (2 \beta_{6} - 2 \beta_{2}) q^{16} - 6 \beta_1 q^{17} + (3 \beta_{6} - 3 \beta_{3} + 3 \beta_{2} - 2) q^{19} + (2 \beta_{7} + 2 \beta_{5}) q^{22} + ( - 2 \beta_{7} - 2 \beta_{4} - 4 \beta_1) q^{23} + (\beta_{6} + 5 \beta_{3} - 4 \beta_{2} + 2) q^{26} + (\beta_{7} + 3 \beta_{5} - 3 \beta_{4} - 4 \beta_1) q^{28} + ( - 2 \beta_{6} - 4 \beta_{3} + 2 \beta_{2} - 2) q^{29} + (\beta_{6} - 2 \beta_{3} - \beta_{2} + 1) q^{31} + ( - 4 \beta_{5} + 4 \beta_{4} - 4 \beta_1) q^{32} + ( - 6 \beta_{6} + 6 \beta_{3}) q^{34} + ( - 4 \beta_{7} - 4 \beta_{4} - 4 \beta_1) q^{37} + ( - 3 \beta_{7} + 3 \beta_{5} + 2 \beta_{4} - 6 \beta_1) q^{38} + ( - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{41} + (2 \beta_{7} + 2 \beta_{4} + \beta_1) q^{43} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{44} + ( - 4 \beta_{6} + 6 \beta_{3} + 2 \beta_{2} - 4) q^{46} + ( - 4 \beta_{7} + 2 \beta_{5} + 4 \beta_{4} - 4 \beta_1) q^{47} + (4 \beta_{6} - 6 \beta_{3} - 4 \beta_{2} + 4) q^{49} + (5 \beta_{7} - 3 \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{52} + (2 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{53} + ( - 8 \beta_{6} + 6 \beta_{3} - 6) q^{56} + ( - 4 \beta_{7} - 2 \beta_{4} + 4 \beta_1) q^{58} + ( - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2} - 8) q^{59} + 3 q^{61} + ( - 2 \beta_{7} - 4 \beta_{5} + \beta_{4} - 2 \beta_1) q^{62} + 8 q^{64} + ( - 2 \beta_{7} + 7 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{67} + (6 \beta_{7} + 6 \beta_{5} - 6 \beta_{4} + 12 \beta_1) q^{68} + ( - 2 \beta_{6} - 6 \beta_{3} + 2 \beta_{2} - 2) q^{71} - 4 \beta_{5} q^{73} + ( - 4 \beta_{6} + 8 \beta_{3} + 4 \beta_{2} - 8) q^{74} + ( - \beta_{6} + 12 \beta_{3} - 5 \beta_{2} + 4) q^{76} + ( - 4 \beta_{7} - 4 \beta_{4} - 2 \beta_1) q^{77} + ( - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2} - 4) q^{79} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_1) q^{82} + (4 \beta_{7} + 4 \beta_{4} + 6 \beta_1) q^{83} + (\beta_{6} - 3 \beta_{3} - 2 \beta_{2} + 4) q^{86} + ( - 4 \beta_{5} + 4 \beta_1) q^{88} + (4 \beta_{6} - 4 \beta_{2} + 4) q^{89} + ( - 7 \beta_{6} - 2 \beta_{3} + 7 \beta_{2} - 7) q^{91} + (6 \beta_{7} + 10 \beta_{5} - 2 \beta_{4} + 8 \beta_1) q^{92} + (4 \beta_{6} + 10 \beta_{3} - 6 \beta_{2} + 8) q^{94} + ( - \beta_{7} - \beta_{4} - 8 \beta_1) q^{97} + ( - 6 \beta_{7} - 14 \beta_{5} + 4 \beta_{4} - 8 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 28 q^{14} - 16 q^{16} - 16 q^{19} - 4 q^{26} + 24 q^{34} + 16 q^{41} - 8 q^{46} - 16 q^{56} - 64 q^{59} + 24 q^{61} + 64 q^{64} - 32 q^{74} + 16 q^{76} - 32 q^{79} + 20 q^{86} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{4} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( -\beta_{6} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{3}\)

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
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
−0.707107 1.22474i 0 −1.00000 + 1.73205i 0 0 −3.15660 + 3.15660i 2.82843 0 0
307.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 0 0 1.74238 1.74238i 2.82843 0 0
307.3 0.707107 1.22474i 0 −1.00000 1.73205i 0 0 −1.74238 + 1.74238i −2.82843 0 0
307.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0 0 3.15660 3.15660i −2.82843 0 0
343.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i 0 0 1.74238 + 1.74238i 2.82843 0 0
343.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 0 0 −3.15660 3.15660i 2.82843 0 0
343.3 0.707107 1.22474i 0 −1.00000 1.73205i 0 0 3.15660 + 3.15660i −2.82843 0 0
343.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0 0 −1.74238 1.74238i −2.82843 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.k.g 8
3.b odd 2 1 300.2.j.a 8
4.b odd 2 1 900.2.k.l 8
5.b even 2 1 inner 900.2.k.g 8
5.c odd 4 2 900.2.k.l 8
12.b even 2 1 300.2.j.c yes 8
15.d odd 2 1 300.2.j.a 8
15.e even 4 2 300.2.j.c yes 8
20.d odd 2 1 900.2.k.l 8
20.e even 4 2 inner 900.2.k.g 8
60.h even 2 1 300.2.j.c yes 8
60.l odd 4 2 300.2.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.j.a 8 3.b odd 2 1
300.2.j.a 8 15.d odd 2 1
300.2.j.a 8 60.l odd 4 2
300.2.j.c yes 8 12.b even 2 1
300.2.j.c yes 8 15.e even 4 2
300.2.j.c yes 8 60.h even 2 1
900.2.k.g 8 1.a even 1 1 trivial
900.2.k.g 8 5.b even 2 1 inner
900.2.k.g 8 20.e even 4 2 inner
900.2.k.l 8 4.b odd 2 1
900.2.k.l 8 5.c odd 4 2
900.2.k.l 8 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{8} + 434T_{7}^{4} + 14641 \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{8} + 1106T_{13}^{4} + 28561 \) Copy content Toggle raw display
\( T_{17}^{4} + 1296 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 23 \) Copy content Toggle raw display
\( T_{23}^{8} + 3104T_{23}^{4} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 434 T^{4} + 14641 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 1106 T^{4} + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 23)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 3104T^{4} + 256 \) Copy content Toggle raw display
$29$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 14 T^{2} + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 14336 T^{4} + \cdots + 1048576 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 8)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 434 T^{4} + 14641 \) Copy content Toggle raw display
$47$ \( T^{8} + 27936 T^{4} + 20736 \) Copy content Toggle raw display
$53$ \( T^{8} + 896T^{4} + 4096 \) Copy content Toggle raw display
$59$ \( (T^{2} + 16 T + 52)^{4} \) Copy content Toggle raw display
$61$ \( (T - 3)^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 25074 T^{4} + \cdots + 22667121 \) Copy content Toggle raw display
$71$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 4)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 27936 T^{4} + 20736 \) Copy content Toggle raw display
$89$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 10514 T^{4} + \cdots + 13845841 \) Copy content Toggle raw display
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