Properties

Label 225.5.d.a
Level $225$
Weight $5$
Character orbit 225.d
Analytic conductor $23.258$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,5,Mod(224,225)]
 
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([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.224");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(23.2582416939\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 9)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 2 q^{4} + 14 \beta_1 q^{7} - 14 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + 2 q^{4} + 14 \beta_1 q^{7} - 14 \beta_{3} q^{8} + 4 \beta_{2} q^{11} - 56 \beta_1 q^{13} + 28 \beta_{2} q^{14} - 284 q^{16} - 21 \beta_{3} q^{17} - 560 q^{19} + 36 \beta_1 q^{22} - 188 \beta_{3} q^{23} - 112 \beta_{2} q^{26} + 28 \beta_1 q^{28} + 233 \beta_{2} q^{29} - 364 q^{31} - 60 \beta_{3} q^{32} - 378 q^{34} + 413 \beta_1 q^{37} - 560 \beta_{3} q^{38} - 427 \beta_{2} q^{41} + 868 \beta_1 q^{43} + 8 \beta_{2} q^{44} - 3384 q^{46} - 308 \beta_{3} q^{47} + 1617 q^{49} - 112 \beta_1 q^{52} - 423 \beta_{3} q^{53} - 392 \beta_{2} q^{56} + 2097 \beta_1 q^{58} - 1064 \beta_{2} q^{59} + 2618 q^{61} - 364 \beta_{3} q^{62} + 3464 q^{64} + 1892 \beta_1 q^{67} - 42 \beta_{3} q^{68} - 2028 \beta_{2} q^{71} + 3304 \beta_1 q^{73} + 826 \beta_{2} q^{74} - 1120 q^{76} - 112 \beta_{3} q^{77} + 4276 q^{79} - 3843 \beta_1 q^{82} - 28 \beta_{3} q^{83} + 1736 \beta_{2} q^{86} - 504 \beta_1 q^{88} - 1029 \beta_{2} q^{89} + 3136 q^{91} - 376 \beta_{3} q^{92} - 5544 q^{94} + 2912 \beta_1 q^{97} + 1617 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 1136 q^{16} - 2240 q^{19} - 1456 q^{31} - 1512 q^{34} - 13536 q^{46} + 6468 q^{49} + 10472 q^{61} + 13856 q^{64} - 4480 q^{76} + 17104 q^{79} + 12544 q^{91} - 22176 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{8}^{3} + 3\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\zeta_{8}^{3} + 3\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
224.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−4.24264 0 2.00000 0 0 28.0000i 59.3970 0 0
224.2 −4.24264 0 2.00000 0 0 28.0000i 59.3970 0 0
224.3 4.24264 0 2.00000 0 0 28.0000i −59.3970 0 0
224.4 4.24264 0 2.00000 0 0 28.0000i −59.3970 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.5.d.a 4
3.b odd 2 1 inner 225.5.d.a 4
5.b even 2 1 inner 225.5.d.a 4
5.c odd 4 1 9.5.b.a 2
5.c odd 4 1 225.5.c.a 2
15.d odd 2 1 inner 225.5.d.a 4
15.e even 4 1 9.5.b.a 2
15.e even 4 1 225.5.c.a 2
20.e even 4 1 144.5.e.c 2
35.f even 4 1 441.5.b.a 2
40.i odd 4 1 576.5.e.d 2
40.k even 4 1 576.5.e.g 2
45.k odd 12 2 81.5.d.c 4
45.l even 12 2 81.5.d.c 4
60.l odd 4 1 144.5.e.c 2
105.k odd 4 1 441.5.b.a 2
120.q odd 4 1 576.5.e.g 2
120.w even 4 1 576.5.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.5.b.a 2 5.c odd 4 1
9.5.b.a 2 15.e even 4 1
81.5.d.c 4 45.k odd 12 2
81.5.d.c 4 45.l even 12 2
144.5.e.c 2 20.e even 4 1
144.5.e.c 2 60.l odd 4 1
225.5.c.a 2 5.c odd 4 1
225.5.c.a 2 15.e even 4 1
225.5.d.a 4 1.a even 1 1 trivial
225.5.d.a 4 3.b odd 2 1 inner
225.5.d.a 4 5.b even 2 1 inner
225.5.d.a 4 15.d odd 2 1 inner
441.5.b.a 2 35.f even 4 1
441.5.b.a 2 105.k odd 4 1
576.5.e.d 2 40.i odd 4 1
576.5.e.d 2 120.w even 4 1
576.5.e.g 2 40.k even 4 1
576.5.e.g 2 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 18 \) acting on \(S_{5}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 784)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12544)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 7938)^{2} \) Copy content Toggle raw display
$19$ \( (T + 560)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 636192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 977202)^{2} \) Copy content Toggle raw display
$31$ \( (T + 364)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 682276)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3281922)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3013696)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1707552)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 3220722)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 20377728)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2618)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 14318656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 74030112)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 43665664)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4276)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 14112)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 19059138)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 33918976)^{2} \) Copy content Toggle raw display
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