Properties

Label 225.8.a.i
Level $225$
Weight $8$
Character orbit 225.a
Self dual yes
Analytic conductor $70.287$
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] = [225,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(70.2866307339\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 6 q^{2} - 92 q^{4} + 64 q^{7} - 1320 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 q^{2} - 92 q^{4} + 64 q^{7} - 1320 q^{8} + 948 q^{11} + 5098 q^{13} + 384 q^{14} + 3856 q^{16} + 28386 q^{17} - 8620 q^{19} + 5688 q^{22} - 15288 q^{23} + 30588 q^{26} - 5888 q^{28} - 36510 q^{29} - 276808 q^{31} + 192096 q^{32} + 170316 q^{34} - 268526 q^{37} - 51720 q^{38} + 629718 q^{41} - 685772 q^{43} - 87216 q^{44} - 91728 q^{46} + 583296 q^{47} - 819447 q^{49} - 469016 q^{52} - 428058 q^{53} - 84480 q^{56} - 219060 q^{58} - 1306380 q^{59} + 300662 q^{61} - 1660848 q^{62} + 659008 q^{64} + 507244 q^{67} - 2611512 q^{68} - 5560632 q^{71} - 1369082 q^{73} - 1611156 q^{74} + 793040 q^{76} + 60672 q^{77} - 6913720 q^{79} + 3778308 q^{82} - 4376748 q^{83} - 4114632 q^{86} - 1251360 q^{88} + 8528310 q^{89} + 326272 q^{91} + 1406496 q^{92} + 3499776 q^{94} + 8826814 q^{97} - 4916682 q^{98}+O(q^{100}) \) 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
6.00000 0 −92.0000 0 0 64.0000 −1320.00 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 225.8.a.i 1
3.b odd 2 1 75.8.a.a 1
5.b even 2 1 9.8.a.a 1
5.c odd 4 2 225.8.b.f 2
15.d odd 2 1 3.8.a.a 1
15.e even 4 2 75.8.b.c 2
20.d odd 2 1 144.8.a.b 1
35.c odd 2 1 441.8.a.a 1
40.e odd 2 1 576.8.a.x 1
40.f even 2 1 576.8.a.w 1
45.h odd 6 2 81.8.c.a 2
45.j even 6 2 81.8.c.c 2
60.h even 2 1 48.8.a.g 1
105.g even 2 1 147.8.a.b 1
105.o odd 6 2 147.8.e.b 2
105.p even 6 2 147.8.e.a 2
120.i odd 2 1 192.8.a.i 1
120.m even 2 1 192.8.a.a 1
165.d even 2 1 363.8.a.b 1
195.e odd 2 1 507.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.8.a.a 1 15.d odd 2 1
9.8.a.a 1 5.b even 2 1
48.8.a.g 1 60.h even 2 1
75.8.a.a 1 3.b odd 2 1
75.8.b.c 2 15.e even 4 2
81.8.c.a 2 45.h odd 6 2
81.8.c.c 2 45.j even 6 2
144.8.a.b 1 20.d odd 2 1
147.8.a.b 1 105.g even 2 1
147.8.e.a 2 105.p even 6 2
147.8.e.b 2 105.o odd 6 2
192.8.a.a 1 120.m even 2 1
192.8.a.i 1 120.i odd 2 1
225.8.a.i 1 1.a even 1 1 trivial
225.8.b.f 2 5.c odd 4 2
363.8.a.b 1 165.d even 2 1
441.8.a.a 1 35.c odd 2 1
507.8.a.a 1 195.e odd 2 1
576.8.a.w 1 40.f even 2 1
576.8.a.x 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} - 6 \) Copy content Toggle raw display
\( T_{7} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 6 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 64 \) Copy content Toggle raw display
$11$ \( T - 948 \) Copy content Toggle raw display
$13$ \( T - 5098 \) Copy content Toggle raw display
$17$ \( T - 28386 \) Copy content Toggle raw display
$19$ \( T + 8620 \) Copy content Toggle raw display
$23$ \( T + 15288 \) Copy content Toggle raw display
$29$ \( T + 36510 \) Copy content Toggle raw display
$31$ \( T + 276808 \) Copy content Toggle raw display
$37$ \( T + 268526 \) Copy content Toggle raw display
$41$ \( T - 629718 \) Copy content Toggle raw display
$43$ \( T + 685772 \) Copy content Toggle raw display
$47$ \( T - 583296 \) Copy content Toggle raw display
$53$ \( T + 428058 \) Copy content Toggle raw display
$59$ \( T + 1306380 \) Copy content Toggle raw display
$61$ \( T - 300662 \) Copy content Toggle raw display
$67$ \( T - 507244 \) Copy content Toggle raw display
$71$ \( T + 5560632 \) Copy content Toggle raw display
$73$ \( T + 1369082 \) Copy content Toggle raw display
$79$ \( T + 6913720 \) Copy content Toggle raw display
$83$ \( T + 4376748 \) Copy content Toggle raw display
$89$ \( T - 8528310 \) Copy content Toggle raw display
$97$ \( T - 8826814 \) Copy content Toggle raw display
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