Properties

Label 225.4.a.b
Level $225$
Weight $4$
Character orbit 225.a
Self dual yes
Analytic conductor $13.275$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,4,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

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: yes
Analytic conductor: \(13.2754297513\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 8 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 8 q^{4} - 6 q^{7} - 32 q^{11} + 38 q^{13} + 24 q^{14} - 64 q^{16} + 26 q^{17} + 100 q^{19} + 128 q^{22} - 78 q^{23} - 152 q^{26} - 48 q^{28} + 50 q^{29} - 108 q^{31} + 256 q^{32} - 104 q^{34} - 266 q^{37} - 400 q^{38} - 22 q^{41} - 442 q^{43} - 256 q^{44} + 312 q^{46} - 514 q^{47} - 307 q^{49} + 304 q^{52} + 2 q^{53} - 200 q^{58} - 500 q^{59} - 518 q^{61} + 432 q^{62} - 512 q^{64} - 126 q^{67} + 208 q^{68} - 412 q^{71} + 878 q^{73} + 1064 q^{74} + 800 q^{76} + 192 q^{77} + 600 q^{79} + 88 q^{82} + 282 q^{83} + 1768 q^{86} + 150 q^{89} - 228 q^{91} - 624 q^{92} + 2056 q^{94} - 386 q^{97} + 1228 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−4.00000 0 8.00000 0 0 −6.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.a.b 1
3.b odd 2 1 25.4.a.c 1
5.b even 2 1 45.4.a.d 1
5.c odd 4 2 225.4.b.c 2
12.b even 2 1 400.4.a.m 1
15.d odd 2 1 5.4.a.a 1
15.e even 4 2 25.4.b.a 2
20.d odd 2 1 720.4.a.u 1
21.c even 2 1 1225.4.a.k 1
24.f even 2 1 1600.4.a.s 1
24.h odd 2 1 1600.4.a.bi 1
35.c odd 2 1 2205.4.a.q 1
45.h odd 6 2 405.4.e.l 2
45.j even 6 2 405.4.e.c 2
60.h even 2 1 80.4.a.d 1
60.l odd 4 2 400.4.c.k 2
105.g even 2 1 245.4.a.a 1
105.o odd 6 2 245.4.e.f 2
105.p even 6 2 245.4.e.g 2
120.i odd 2 1 320.4.a.g 1
120.m even 2 1 320.4.a.h 1
165.d even 2 1 605.4.a.d 1
195.e odd 2 1 845.4.a.b 1
240.t even 4 2 1280.4.d.l 2
240.bm odd 4 2 1280.4.d.e 2
255.h odd 2 1 1445.4.a.a 1
285.b even 2 1 1805.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 15.d odd 2 1
25.4.a.c 1 3.b odd 2 1
25.4.b.a 2 15.e even 4 2
45.4.a.d 1 5.b even 2 1
80.4.a.d 1 60.h even 2 1
225.4.a.b 1 1.a even 1 1 trivial
225.4.b.c 2 5.c odd 4 2
245.4.a.a 1 105.g even 2 1
245.4.e.f 2 105.o odd 6 2
245.4.e.g 2 105.p even 6 2
320.4.a.g 1 120.i odd 2 1
320.4.a.h 1 120.m even 2 1
400.4.a.m 1 12.b even 2 1
400.4.c.k 2 60.l odd 4 2
405.4.e.c 2 45.j even 6 2
405.4.e.l 2 45.h odd 6 2
605.4.a.d 1 165.d even 2 1
720.4.a.u 1 20.d odd 2 1
845.4.a.b 1 195.e odd 2 1
1225.4.a.k 1 21.c even 2 1
1280.4.d.e 2 240.bm odd 4 2
1280.4.d.l 2 240.t even 4 2
1445.4.a.a 1 255.h odd 2 1
1600.4.a.s 1 24.f even 2 1
1600.4.a.bi 1 24.h odd 2 1
1805.4.a.h 1 285.b even 2 1
2205.4.a.q 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 4 \) Copy content Toggle raw display
\( T_{7} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 6 \) Copy content Toggle raw display
$11$ \( T + 32 \) Copy content Toggle raw display
$13$ \( T - 38 \) Copy content Toggle raw display
$17$ \( T - 26 \) Copy content Toggle raw display
$19$ \( T - 100 \) Copy content Toggle raw display
$23$ \( T + 78 \) Copy content Toggle raw display
$29$ \( T - 50 \) Copy content Toggle raw display
$31$ \( T + 108 \) Copy content Toggle raw display
$37$ \( T + 266 \) Copy content Toggle raw display
$41$ \( T + 22 \) Copy content Toggle raw display
$43$ \( T + 442 \) Copy content Toggle raw display
$47$ \( T + 514 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T + 500 \) Copy content Toggle raw display
$61$ \( T + 518 \) Copy content Toggle raw display
$67$ \( T + 126 \) Copy content Toggle raw display
$71$ \( T + 412 \) Copy content Toggle raw display
$73$ \( T - 878 \) Copy content Toggle raw display
$79$ \( T - 600 \) Copy content Toggle raw display
$83$ \( T - 282 \) Copy content Toggle raw display
$89$ \( T - 150 \) Copy content Toggle raw display
$97$ \( T + 386 \) Copy content Toggle raw display
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