Properties

Label 45.5.g.a
Level $45$
Weight $5$
Character orbit 45.g
Analytic conductor $4.652$
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] = [45,5,Mod(28,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.28");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(4.65164833877\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( - 5 i - 5) q^{2} + 34 i q^{4} + 25 q^{5} + (40 i + 40) q^{7} + ( - 90 i + 90) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 i - 5) q^{2} + 34 i q^{4} + 25 q^{5} + (40 i + 40) q^{7} + ( - 90 i + 90) q^{8} + ( - 125 i - 125) q^{10} - 100 q^{11} + ( - 205 i + 205) q^{13} - 400 i q^{14} - 356 q^{16} + (235 i + 235) q^{17} + 72 i q^{19} + 850 i q^{20} + (500 i + 500) q^{22} + ( - 340 i + 340) q^{23} + 625 q^{25} - 2050 q^{26} + (1360 i - 1360) q^{28} + 450 i q^{29} + 428 q^{31} + (340 i + 340) q^{32} - 2350 i q^{34} + (1000 i + 1000) q^{35} + ( - 755 i - 755) q^{37} + ( - 360 i + 360) q^{38} + ( - 2250 i + 2250) q^{40} + 950 q^{41} + (1220 i - 1220) q^{43} - 3400 i q^{44} - 3400 q^{46} + ( - 320 i - 320) q^{47} + 799 i q^{49} + ( - 3125 i - 3125) q^{50} + (6970 i + 6970) q^{52} + ( - 505 i + 505) q^{53} - 2500 q^{55} + 7200 q^{56} + ( - 2250 i + 2250) q^{58} + 6300 i q^{59} - 3808 q^{61} + ( - 2140 i - 2140) q^{62} + 2296 i q^{64} + ( - 5125 i + 5125) q^{65} + (340 i + 340) q^{67} + (7990 i - 7990) q^{68} - 10000 i q^{70} - 3400 q^{71} + ( - 415 i + 415) q^{73} + 7550 i q^{74} - 2448 q^{76} + ( - 4000 i - 4000) q^{77} - 6732 i q^{79} - 8900 q^{80} + ( - 4750 i - 4750) q^{82} + (680 i - 680) q^{83} + (5875 i + 5875) q^{85} + 12200 q^{86} + (9000 i - 9000) q^{88} + 2250 i q^{89} + 16400 q^{91} + (11560 i + 11560) q^{92} + 3200 i q^{94} + 1800 i q^{95} + (1615 i + 1615) q^{97} + ( - 3995 i + 3995) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{2} + 50 q^{5} + 80 q^{7} + 180 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{2} + 50 q^{5} + 80 q^{7} + 180 q^{8} - 250 q^{10} - 200 q^{11} + 410 q^{13} - 712 q^{16} + 470 q^{17} + 1000 q^{22} + 680 q^{23} + 1250 q^{25} - 4100 q^{26} - 2720 q^{28} + 856 q^{31} + 680 q^{32} + 2000 q^{35} - 1510 q^{37} + 720 q^{38} + 4500 q^{40} + 1900 q^{41} - 2440 q^{43} - 6800 q^{46} - 640 q^{47} - 6250 q^{50} + 13940 q^{52} + 1010 q^{53} - 5000 q^{55} + 14400 q^{56} + 4500 q^{58} - 7616 q^{61} - 4280 q^{62} + 10250 q^{65} + 680 q^{67} - 15980 q^{68} - 6800 q^{71} + 830 q^{73} - 4896 q^{76} - 8000 q^{77} - 17800 q^{80} - 9500 q^{82} - 1360 q^{83} + 11750 q^{85} + 24400 q^{86} - 18000 q^{88} + 32800 q^{91} + 23120 q^{92} + 3230 q^{97} + 7990 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
1.00000i
1.00000i
−5.00000 + 5.00000i 0 34.0000i 25.0000 0 40.0000 40.0000i 90.0000 + 90.0000i 0 −125.000 + 125.000i
37.1 −5.00000 5.00000i 0 34.0000i 25.0000 0 40.0000 + 40.0000i 90.0000 90.0000i 0 −125.000 125.000i
\(n\): 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 45.5.g.a 2
3.b odd 2 1 45.5.g.c yes 2
5.b even 2 1 225.5.g.c 2
5.c odd 4 1 inner 45.5.g.a 2
5.c odd 4 1 225.5.g.c 2
15.d odd 2 1 225.5.g.a 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 1.a even 1 1 trivial
45.5.g.a 2 5.c odd 4 1 inner
45.5.g.c yes 2 3.b odd 2 1
45.5.g.c yes 2 15.e even 4 1
225.5.g.a 2 15.d odd 2 1
225.5.g.a 2 15.e even 4 1
225.5.g.c 2 5.b even 2 1
225.5.g.c 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 10T_{2} + 50 \) acting on \(S_{5}^{\mathrm{new}}(45, [\chi])\). 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
$5$ \( (T - 25)^{2} \) 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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