Properties

Label 225.10.a.a
Level $225$
Weight $10$
Character orbit 225.a
Self dual yes
Analytic conductor $115.883$
Analytic rank $0$
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] = [225,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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)
 
\(f(q)\) \(=\) \( q - 36 q^{2} + 784 q^{4} + 4480 q^{7} - 9792 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 36 q^{2} + 784 q^{4} + 4480 q^{7} - 9792 q^{8} - 1476 q^{11} + 151522 q^{13} - 161280 q^{14} - 48896 q^{16} + 108162 q^{17} + 593084 q^{19} + 53136 q^{22} - 969480 q^{23} - 5454792 q^{26} + 3512320 q^{28} + 6642522 q^{29} + 7070600 q^{31} + 6773760 q^{32} - 3893832 q^{34} + 7472410 q^{37} - 21351024 q^{38} + 4350150 q^{41} + 4358716 q^{43} - 1157184 q^{44} + 34901280 q^{46} + 28309248 q^{47} - 20283207 q^{49} + 118793248 q^{52} + 16111710 q^{53} - 43868160 q^{56} - 239130792 q^{58} + 86075964 q^{59} + 32213918 q^{61} - 254541600 q^{62} - 218820608 q^{64} - 99531452 q^{67} + 84799008 q^{68} + 44170488 q^{71} + 23560630 q^{73} - 269006760 q^{74} + 464977856 q^{76} - 6612480 q^{77} - 401754760 q^{79} - 156605400 q^{82} - 744528708 q^{83} - 156913776 q^{86} + 14452992 q^{88} - 769871034 q^{89} + 678818560 q^{91} - 760072320 q^{92} - 1019132928 q^{94} - 907130882 q^{97} + 730195452 q^{98}+O(q^{100}) \) 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
−36.0000 0 784.000 0 0 4480.00 −9792.00 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 225.10.a.a 1
3.b odd 2 1 75.10.a.d 1
5.b even 2 1 9.10.a.c 1
5.c odd 4 2 225.10.b.a 2
15.d odd 2 1 3.10.a.a 1
15.e even 4 2 75.10.b.a 2
20.d odd 2 1 144.10.a.l 1
45.h odd 6 2 81.10.c.e 2
45.j even 6 2 81.10.c.a 2
60.h even 2 1 48.10.a.e 1
105.g even 2 1 147.10.a.a 1
120.i odd 2 1 192.10.a.m 1
120.m even 2 1 192.10.a.f 1
165.d even 2 1 363.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.10.a.a 1 15.d odd 2 1
9.10.a.c 1 5.b even 2 1
48.10.a.e 1 60.h even 2 1
75.10.a.d 1 3.b odd 2 1
75.10.b.a 2 15.e even 4 2
81.10.c.a 2 45.j even 6 2
81.10.c.e 2 45.h odd 6 2
144.10.a.l 1 20.d odd 2 1
147.10.a.a 1 105.g even 2 1
192.10.a.f 1 120.m even 2 1
192.10.a.m 1 120.i odd 2 1
225.10.a.a 1 1.a even 1 1 trivial
225.10.b.a 2 5.c odd 4 2
363.10.a.b 1 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 36 \) Copy content Toggle raw display
\( T_{7} - 4480 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 36 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 4480 \) Copy content Toggle raw display
$11$ \( T + 1476 \) Copy content Toggle raw display
$13$ \( T - 151522 \) Copy content Toggle raw display
$17$ \( T - 108162 \) Copy content Toggle raw display
$19$ \( T - 593084 \) Copy content Toggle raw display
$23$ \( T + 969480 \) Copy content Toggle raw display
$29$ \( T - 6642522 \) Copy content Toggle raw display
$31$ \( T - 7070600 \) Copy content Toggle raw display
$37$ \( T - 7472410 \) Copy content Toggle raw display
$41$ \( T - 4350150 \) Copy content Toggle raw display
$43$ \( T - 4358716 \) Copy content Toggle raw display
$47$ \( T - 28309248 \) Copy content Toggle raw display
$53$ \( T - 16111710 \) Copy content Toggle raw display
$59$ \( T - 86075964 \) Copy content Toggle raw display
$61$ \( T - 32213918 \) Copy content Toggle raw display
$67$ \( T + 99531452 \) Copy content Toggle raw display
$71$ \( T - 44170488 \) Copy content Toggle raw display
$73$ \( T - 23560630 \) Copy content Toggle raw display
$79$ \( T + 401754760 \) Copy content Toggle raw display
$83$ \( T + 744528708 \) Copy content Toggle raw display
$89$ \( T + 769871034 \) Copy content Toggle raw display
$97$ \( T + 907130882 \) Copy content Toggle raw display
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