Properties

Label 900.4.d.h
Level $900$
Weight $4$
Character orbit 900.d
Analytic conductor $53.102$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,4,Mod(649,900)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.d (of order \(2\), degree \(1\), not 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: \(53.1017190052\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 60)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 14 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 14 \beta q^{7} + 24 q^{11} - 35 \beta q^{13} + 51 \beta q^{17} - 20 q^{19} + 36 \beta q^{23} + 306 q^{29} - 136 q^{31} + 107 \beta q^{37} + 150 q^{41} - 146 \beta q^{43} - 36 \beta q^{47} - 441 q^{49} + 207 \beta q^{53} - 744 q^{59} - 418 q^{61} - 94 \beta q^{67} - 480 q^{71} + 217 \beta q^{73} + 336 \beta q^{77} - 1352 q^{79} + 306 \beta q^{83} - 30 q^{89} + 1960 q^{91} + 143 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 48 q^{11} - 40 q^{19} + 612 q^{29} - 272 q^{31} + 300 q^{41} - 882 q^{49} - 1488 q^{59} - 836 q^{61} - 960 q^{71} - 2704 q^{79} - 60 q^{89} + 3920 q^{91}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 0 0 28.0000i 0 0 0
649.2 0 0 0 0 0 28.0000i 0 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.4.d.h 2
3.b odd 2 1 300.4.d.b 2
5.b even 2 1 inner 900.4.d.h 2
5.c odd 4 1 180.4.a.d 1
5.c odd 4 1 900.4.a.q 1
12.b even 2 1 1200.4.f.n 2
15.d odd 2 1 300.4.d.b 2
15.e even 4 1 60.4.a.a 1
15.e even 4 1 300.4.a.i 1
20.e even 4 1 720.4.a.bb 1
45.k odd 12 2 1620.4.i.f 2
45.l even 12 2 1620.4.i.l 2
60.h even 2 1 1200.4.f.n 2
60.l odd 4 1 240.4.a.i 1
60.l odd 4 1 1200.4.a.a 1
120.q odd 4 1 960.4.a.r 1
120.w even 4 1 960.4.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.a 1 15.e even 4 1
180.4.a.d 1 5.c odd 4 1
240.4.a.i 1 60.l odd 4 1
300.4.a.i 1 15.e even 4 1
300.4.d.b 2 3.b odd 2 1
300.4.d.b 2 15.d odd 2 1
720.4.a.bb 1 20.e even 4 1
900.4.a.q 1 5.c odd 4 1
900.4.d.h 2 1.a even 1 1 trivial
900.4.d.h 2 5.b even 2 1 inner
960.4.a.r 1 120.q odd 4 1
960.4.a.bc 1 120.w even 4 1
1200.4.a.a 1 60.l odd 4 1
1200.4.f.n 2 12.b even 2 1
1200.4.f.n 2 60.h even 2 1
1620.4.i.f 2 45.k odd 12 2
1620.4.i.l 2 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 784 \) Copy content Toggle raw display
\( T_{11} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 784 \) Copy content Toggle raw display
$11$ \( (T - 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4900 \) Copy content Toggle raw display
$17$ \( T^{2} + 10404 \) Copy content Toggle raw display
$19$ \( (T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5184 \) Copy content Toggle raw display
$29$ \( (T - 306)^{2} \) Copy content Toggle raw display
$31$ \( (T + 136)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 45796 \) Copy content Toggle raw display
$41$ \( (T - 150)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 85264 \) Copy content Toggle raw display
$47$ \( T^{2} + 5184 \) Copy content Toggle raw display
$53$ \( T^{2} + 171396 \) Copy content Toggle raw display
$59$ \( (T + 744)^{2} \) Copy content Toggle raw display
$61$ \( (T + 418)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 35344 \) Copy content Toggle raw display
$71$ \( (T + 480)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 188356 \) Copy content Toggle raw display
$79$ \( (T + 1352)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 374544 \) Copy content Toggle raw display
$89$ \( (T + 30)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 81796 \) Copy content Toggle raw display
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