Properties

Label 60.4.d.a
Level $60$
Weight $4$
Character orbit 60.d
Analytic conductor $3.540$
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] = [60,4,Mod(49,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 60.d (of order \(2\), degree \(1\), 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: \(3.54011460034\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + (5 i - 10) q^{5} + 22 i q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} + (5 i - 10) q^{5} + 22 i q^{7} - 9 q^{9} - 14 q^{11} + 30 i q^{13} + ( - 30 i - 15) q^{15} + 62 i q^{17} + 120 q^{19} - 66 q^{21} - 188 i q^{23} + ( - 100 i + 75) q^{25} - 27 i q^{27} - 96 q^{29} + 184 q^{31} - 42 i q^{33} + ( - 220 i - 110) q^{35} + 406 i q^{37} - 90 q^{39} + 130 q^{41} - 148 i q^{43} + ( - 45 i + 90) q^{45} + 448 i q^{47} - 141 q^{49} - 186 q^{51} + 414 i q^{53} + ( - 70 i + 140) q^{55} + 360 i q^{57} - 266 q^{59} - 838 q^{61} - 198 i q^{63} + ( - 300 i - 150) q^{65} + 248 i q^{67} + 564 q^{69} + 1020 q^{71} - 484 i q^{73} + (225 i + 300) q^{75} - 308 i q^{77} + 48 q^{79} + 81 q^{81} - 548 i q^{83} + ( - 620 i - 310) q^{85} - 288 i q^{87} + 650 q^{89} - 660 q^{91} + 552 i q^{93} + (600 i - 1200) q^{95} - 1816 i q^{97} + 126 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 20 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 20 q^{5} - 18 q^{9} - 28 q^{11} - 30 q^{15} + 240 q^{19} - 132 q^{21} + 150 q^{25} - 192 q^{29} + 368 q^{31} - 220 q^{35} - 180 q^{39} + 260 q^{41} + 180 q^{45} - 282 q^{49} - 372 q^{51} + 280 q^{55} - 532 q^{59} - 1676 q^{61} - 300 q^{65} + 1128 q^{69} + 2040 q^{71} + 600 q^{75} + 96 q^{79} + 162 q^{81} - 620 q^{85} + 1300 q^{89} - 1320 q^{91} - 2400 q^{95} + 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 3.00000i 0 −10.0000 5.00000i 0 22.0000i 0 −9.00000 0
49.2 0 3.00000i 0 −10.0000 + 5.00000i 0 22.0000i 0 −9.00000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.4.d.a 2
3.b odd 2 1 180.4.d.b 2
4.b odd 2 1 240.4.f.a 2
5.b even 2 1 inner 60.4.d.a 2
5.c odd 4 1 300.4.a.d 1
5.c odd 4 1 300.4.a.f 1
8.b even 2 1 960.4.f.j 2
8.d odd 2 1 960.4.f.i 2
12.b even 2 1 720.4.f.h 2
15.d odd 2 1 180.4.d.b 2
15.e even 4 1 900.4.a.d 1
15.e even 4 1 900.4.a.o 1
20.d odd 2 1 240.4.f.a 2
20.e even 4 1 1200.4.a.q 1
20.e even 4 1 1200.4.a.w 1
40.e odd 2 1 960.4.f.i 2
40.f even 2 1 960.4.f.j 2
60.h even 2 1 720.4.f.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.d.a 2 1.a even 1 1 trivial
60.4.d.a 2 5.b even 2 1 inner
180.4.d.b 2 3.b odd 2 1
180.4.d.b 2 15.d odd 2 1
240.4.f.a 2 4.b odd 2 1
240.4.f.a 2 20.d odd 2 1
300.4.a.d 1 5.c odd 4 1
300.4.a.f 1 5.c odd 4 1
720.4.f.h 2 12.b even 2 1
720.4.f.h 2 60.h even 2 1
900.4.a.d 1 15.e even 4 1
900.4.a.o 1 15.e even 4 1
960.4.f.i 2 8.d odd 2 1
960.4.f.i 2 40.e odd 2 1
960.4.f.j 2 8.b even 2 1
960.4.f.j 2 40.f even 2 1
1200.4.a.q 1 20.e even 4 1
1200.4.a.w 1 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 20T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 484 \) Copy content Toggle raw display
$11$ \( (T + 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 900 \) Copy content Toggle raw display
$17$ \( T^{2} + 3844 \) Copy content Toggle raw display
$19$ \( (T - 120)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 35344 \) Copy content Toggle raw display
$29$ \( (T + 96)^{2} \) Copy content Toggle raw display
$31$ \( (T - 184)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 164836 \) Copy content Toggle raw display
$41$ \( (T - 130)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 21904 \) Copy content Toggle raw display
$47$ \( T^{2} + 200704 \) Copy content Toggle raw display
$53$ \( T^{2} + 171396 \) Copy content Toggle raw display
$59$ \( (T + 266)^{2} \) Copy content Toggle raw display
$61$ \( (T + 838)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 61504 \) Copy content Toggle raw display
$71$ \( (T - 1020)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 234256 \) Copy content Toggle raw display
$79$ \( (T - 48)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 300304 \) Copy content Toggle raw display
$89$ \( (T - 650)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 3297856 \) Copy content Toggle raw display
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