Properties

Label 1530.2.f.b
Level $1530$
Weight $2$
Character orbit 1530.f
Analytic conductor $12.217$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1530,2,Mod(1189,1530)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1530, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1530.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.f (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: \(12.2171115093\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( - i - 2) q^{5} + 2 q^{7} - i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - q^{4} + ( - i - 2) q^{5} + 2 q^{7} - i q^{8} + ( - 2 i + 1) q^{10} - i q^{13} + 2 i q^{14} + q^{16} + (4 i - 1) q^{17} - 5 q^{19} + (i + 2) q^{20} + 4 q^{23} + (4 i + 3) q^{25} + q^{26} - 2 q^{28} - 9 i q^{29} - 5 i q^{31} + i q^{32} + ( - i - 4) q^{34} + ( - 2 i - 4) q^{35} + 2 q^{37} - 5 i q^{38} + (2 i - 1) q^{40} - 10 i q^{41} - 6 i q^{43} + 4 i q^{46} - 7 i q^{47} - 3 q^{49} + (3 i - 4) q^{50} + i q^{52} + i q^{53} - 2 i q^{56} + 9 q^{58} + 5 q^{59} - 5 i q^{61} + 5 q^{62} - q^{64} + (2 i - 1) q^{65} + 2 i q^{67} + ( - 4 i + 1) q^{68} + ( - 4 i + 2) q^{70} + 5 i q^{71} + 11 q^{73} + 2 i q^{74} + 5 q^{76} - 16 i q^{79} + ( - i - 2) q^{80} + 10 q^{82} + 6 i q^{83} + ( - 7 i + 6) q^{85} + 6 q^{86} - 5 q^{89} - 2 i q^{91} - 4 q^{92} + 7 q^{94} + (5 i + 10) q^{95} + 7 q^{97} - 3 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 4 q^{5} + 4 q^{7} + 2 q^{10} + 2 q^{16} - 2 q^{17} - 10 q^{19} + 4 q^{20} + 8 q^{23} + 6 q^{25} + 2 q^{26} - 4 q^{28} - 8 q^{34} - 8 q^{35} + 4 q^{37} - 2 q^{40} - 6 q^{49} - 8 q^{50} + 18 q^{58} + 10 q^{59} + 10 q^{62} - 2 q^{64} - 2 q^{65} + 2 q^{68} + 4 q^{70} + 22 q^{73} + 10 q^{76} - 4 q^{80} + 20 q^{82} + 12 q^{85} + 12 q^{86} - 10 q^{89} - 8 q^{92} + 14 q^{94} + 20 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(1261\) \(1361\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 2.00000 1.00000i 0 1.00000 + 2.00000i
1189.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 2.00000 1.00000i 0 1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.f.b 2
3.b odd 2 1 170.2.d.b yes 2
5.b even 2 1 1530.2.f.e 2
12.b even 2 1 1360.2.o.a 2
15.d odd 2 1 170.2.d.a 2
15.e even 4 1 850.2.b.c 2
15.e even 4 1 850.2.b.i 2
17.b even 2 1 1530.2.f.e 2
51.c odd 2 1 170.2.d.a 2
60.h even 2 1 1360.2.o.b 2
85.c even 2 1 inner 1530.2.f.b 2
204.h even 2 1 1360.2.o.b 2
255.h odd 2 1 170.2.d.b yes 2
255.o even 4 1 850.2.b.c 2
255.o even 4 1 850.2.b.i 2
1020.b even 2 1 1360.2.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.d.a 2 15.d odd 2 1
170.2.d.a 2 51.c odd 2 1
170.2.d.b yes 2 3.b odd 2 1
170.2.d.b yes 2 255.h odd 2 1
850.2.b.c 2 15.e even 4 1
850.2.b.c 2 255.o even 4 1
850.2.b.i 2 15.e even 4 1
850.2.b.i 2 255.o even 4 1
1360.2.o.a 2 12.b even 2 1
1360.2.o.a 2 1020.b even 2 1
1360.2.o.b 2 60.h even 2 1
1360.2.o.b 2 204.h even 2 1
1530.2.f.b 2 1.a even 1 1 trivial
1530.2.f.b 2 85.c even 2 1 inner
1530.2.f.e 2 5.b even 2 1
1530.2.f.e 2 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 17 \) Copy content Toggle raw display
$19$ \( (T + 5)^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 81 \) Copy content Toggle raw display
$31$ \( T^{2} + 25 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 49 \) Copy content Toggle raw display
$53$ \( T^{2} + 1 \) Copy content Toggle raw display
$59$ \( (T - 5)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 25 \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 256 \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T + 5)^{2} \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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