Properties

Label 225.5.g.c
Level $225$
Weight $5$
Character orbit 225.g
Analytic conductor $23.258$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,5,Mod(82,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.82");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), degree \(2\), 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: \(2\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (5 i + 5) q^{2} + 34 i q^{4} + ( - 40 i - 40) q^{7} + (90 i - 90) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 i + 5) q^{2} + 34 i q^{4} + ( - 40 i - 40) q^{7} + (90 i - 90) q^{8} - 100 q^{11} + (205 i - 205) q^{13} - 400 i q^{14} - 356 q^{16} + ( - 235 i - 235) q^{17} + 72 i q^{19} + ( - 500 i - 500) q^{22} + (340 i - 340) q^{23} - 2050 q^{26} + ( - 1360 i + 1360) q^{28} + 450 i q^{29} + 428 q^{31} + ( - 340 i - 340) q^{32} - 2350 i q^{34} + (755 i + 755) q^{37} + (360 i - 360) q^{38} + 950 q^{41} + ( - 1220 i + 1220) q^{43} - 3400 i q^{44} - 3400 q^{46} + (320 i + 320) q^{47} + 799 i q^{49} + ( - 6970 i - 6970) q^{52} + (505 i - 505) q^{53} + 7200 q^{56} + (2250 i - 2250) q^{58} + 6300 i q^{59} - 3808 q^{61} + (2140 i + 2140) q^{62} + 2296 i q^{64} + ( - 340 i - 340) q^{67} + ( - 7990 i + 7990) q^{68} - 3400 q^{71} + (415 i - 415) q^{73} + 7550 i q^{74} - 2448 q^{76} + (4000 i + 4000) q^{77} - 6732 i q^{79} + (4750 i + 4750) q^{82} + ( - 680 i + 680) q^{83} + 12200 q^{86} + ( - 9000 i + 9000) q^{88} + 2250 i q^{89} + 16400 q^{91} + ( - 11560 i - 11560) q^{92} + 3200 i q^{94} + ( - 1615 i - 1615) q^{97} + (3995 i - 3995) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{2} - 80 q^{7} - 180 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{2} - 80 q^{7} - 180 q^{8} - 200 q^{11} - 410 q^{13} - 712 q^{16} - 470 q^{17} - 1000 q^{22} - 680 q^{23} - 4100 q^{26} + 2720 q^{28} + 856 q^{31} - 680 q^{32} + 1510 q^{37} - 720 q^{38} + 1900 q^{41} + 2440 q^{43} - 6800 q^{46} + 640 q^{47} - 13940 q^{52} - 1010 q^{53} + 14400 q^{56} - 4500 q^{58} - 7616 q^{61} + 4280 q^{62} - 680 q^{67} + 15980 q^{68} - 6800 q^{71} - 830 q^{73} - 4896 q^{76} + 8000 q^{77} + 9500 q^{82} + 1360 q^{83} + 24400 q^{86} + 18000 q^{88} + 32800 q^{91} - 23120 q^{92} - 3230 q^{97} - 7990 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
1.00000i
1.00000i
5.00000 + 5.00000i 0 34.0000i 0 0 −40.0000 40.0000i −90.0000 + 90.0000i 0 0
118.1 5.00000 5.00000i 0 34.0000i 0 0 −40.0000 + 40.0000i −90.0000 90.0000i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.5.g.c 2
3.b odd 2 1 225.5.g.a 2
5.b even 2 1 45.5.g.a 2
5.c odd 4 1 45.5.g.a 2
5.c odd 4 1 inner 225.5.g.c 2
15.d odd 2 1 45.5.g.c yes 2
15.e even 4 1 45.5.g.c yes 2
15.e even 4 1 225.5.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.g.a 2 5.b even 2 1
45.5.g.a 2 5.c odd 4 1
45.5.g.c yes 2 15.d odd 2 1
45.5.g.c yes 2 15.e even 4 1
225.5.g.a 2 3.b odd 2 1
225.5.g.a 2 15.e even 4 1
225.5.g.c 2 1.a even 1 1 trivial
225.5.g.c 2 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 10T_{2} + 50 \) acting on \(S_{5}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 80T + 3200 \) Copy content Toggle raw display
$11$ \( (T + 100)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 410T + 84050 \) Copy content Toggle raw display
$17$ \( T^{2} + 470T + 110450 \) Copy content Toggle raw display
$19$ \( T^{2} + 5184 \) Copy content Toggle raw display
$23$ \( T^{2} + 680T + 231200 \) Copy content Toggle raw display
$29$ \( T^{2} + 202500 \) Copy content Toggle raw display
$31$ \( (T - 428)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 1510 T + 1140050 \) Copy content Toggle raw display
$41$ \( (T - 950)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 2440 T + 2976800 \) Copy content Toggle raw display
$47$ \( T^{2} - 640T + 204800 \) Copy content Toggle raw display
$53$ \( T^{2} + 1010 T + 510050 \) Copy content Toggle raw display
$59$ \( T^{2} + 39690000 \) Copy content Toggle raw display
$61$ \( (T + 3808)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 680T + 231200 \) Copy content Toggle raw display
$71$ \( (T + 3400)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 830T + 344450 \) Copy content Toggle raw display
$79$ \( T^{2} + 45319824 \) Copy content Toggle raw display
$83$ \( T^{2} - 1360 T + 924800 \) Copy content Toggle raw display
$89$ \( T^{2} + 5062500 \) Copy content Toggle raw display
$97$ \( T^{2} + 3230 T + 5216450 \) Copy content Toggle raw display
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