Properties

Label 1530.2.q.a
Level $1530$
Weight $2$
Character orbit 1530.q
Analytic conductor $12.217$
Analytic rank $0$
Dimension $4$
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(361,1530)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1530, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1530.361");
 
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.q (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: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
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 a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{8}^{2} q^{2} - q^{4} - \zeta_{8}^{3} q^{5} + ( - 2 \zeta_{8}^{2} - 2) q^{7} + \zeta_{8}^{2} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{2} q^{2} - q^{4} - \zeta_{8}^{3} q^{5} + ( - 2 \zeta_{8}^{2} - 2) q^{7} + \zeta_{8}^{2} q^{8} - \zeta_{8} q^{10} + (2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{11} + ( - \zeta_{8}^{3} + \zeta_{8} + 4) q^{13} + (2 \zeta_{8}^{2} - 2) q^{14} + q^{16} + (\zeta_{8}^{2} - 4) q^{17} + (2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 2 \zeta_{8}) q^{19} + \zeta_{8}^{3} q^{20} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{22} + (3 \zeta_{8}^{2} + 2 \zeta_{8} + 3) q^{23} - \zeta_{8}^{2} q^{25} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} - \zeta_{8}) q^{26} + (2 \zeta_{8}^{2} + 2) q^{28} - 8 \zeta_{8}^{3} q^{29} + ( - 6 \zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{31} - \zeta_{8}^{2} q^{32} + (4 \zeta_{8}^{2} + 1) q^{34} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{35} + ( - 2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4) q^{37} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 4) q^{38} + \zeta_{8} q^{40} + (2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{41} + ( - 5 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 5 \zeta_{8}) q^{43} + ( - 2 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{44} + ( - 2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{46} - 8 q^{47} + \zeta_{8}^{2} q^{49} - q^{50} + (\zeta_{8}^{3} - \zeta_{8} - 4) q^{52} + (4 \zeta_{8}^{3} + 6 \zeta_{8}^{2} + 4 \zeta_{8}) q^{53} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} + 2) q^{55} + ( - 2 \zeta_{8}^{2} + 2) q^{56} - 8 \zeta_{8} q^{58} + ( - 5 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 5 \zeta_{8}) q^{59} + (2 \zeta_{8}^{2} - 10 \zeta_{8} + 2) q^{61} + ( - \zeta_{8}^{2} - 6 \zeta_{8} - 1) q^{62} - q^{64} + ( - 4 \zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{65} + ( - \zeta_{8}^{3} + \zeta_{8} - 6) q^{67} + ( - \zeta_{8}^{2} + 4) q^{68} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{70} + ( - 4 \zeta_{8}^{2} + 4) q^{71} - 10 \zeta_{8}^{3} q^{73} + (4 \zeta_{8}^{2} - 2 \zeta_{8} + 4) q^{74} + ( - 2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{76} + ( - 4 \zeta_{8}^{3} - 8 \zeta_{8}^{2} - 4 \zeta_{8}) q^{77} + (5 \zeta_{8}^{2} - 10 \zeta_{8} + 5) q^{79} - \zeta_{8}^{3} q^{80} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{82} + (4 \zeta_{8}^{3} + \zeta_{8}) q^{85} + (5 \zeta_{8}^{3} - 5 \zeta_{8} - 2) q^{86} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{88} + ( - 10 \zeta_{8}^{3} + 10 \zeta_{8} + 2) q^{89} + ( - 8 \zeta_{8}^{2} - 4 \zeta_{8} - 8) q^{91} + ( - 3 \zeta_{8}^{2} - 2 \zeta_{8} - 3) q^{92} + 8 \zeta_{8}^{2} q^{94} + (2 \zeta_{8}^{2} - 4 \zeta_{8} + 2) q^{95} + ( - 10 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4) q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 8 q^{7} + 8 q^{11} + 16 q^{13} - 8 q^{14} + 4 q^{16} - 16 q^{17} + 8 q^{22} + 12 q^{23} + 8 q^{28} + 4 q^{31} + 4 q^{34} - 16 q^{37} - 16 q^{38} + 8 q^{41} - 8 q^{44} + 12 q^{46} - 32 q^{47} - 4 q^{50} - 16 q^{52} + 8 q^{55} + 8 q^{56} + 8 q^{61} - 4 q^{62} - 4 q^{64} + 4 q^{65} - 24 q^{67} + 16 q^{68} + 16 q^{71} + 16 q^{74} + 20 q^{79} + 8 q^{82} - 8 q^{86} - 8 q^{88} + 8 q^{89} - 32 q^{91} - 12 q^{92} + 8 q^{95} - 16 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(307\) \(1261\) \(1361\)
\(\chi(n)\) \(1\) \(-\zeta_{8}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
1.00000i 0 −1.00000 −0.707107 + 0.707107i 0 −2.00000 2.00000i 1.00000i 0 0.707107 + 0.707107i
361.2 1.00000i 0 −1.00000 0.707107 0.707107i 0 −2.00000 2.00000i 1.00000i 0 −0.707107 0.707107i
1441.1 1.00000i 0 −1.00000 −0.707107 0.707107i 0 −2.00000 + 2.00000i 1.00000i 0 0.707107 0.707107i
1441.2 1.00000i 0 −1.00000 0.707107 + 0.707107i 0 −2.00000 + 2.00000i 1.00000i 0 −0.707107 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.q.a 4
3.b odd 2 1 510.2.p.a 4
17.c even 4 1 inner 1530.2.q.a 4
51.f odd 4 1 510.2.p.a 4
51.g odd 8 1 8670.2.a.bi 2
51.g odd 8 1 8670.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.p.a 4 3.b odd 2 1
510.2.p.a 4 51.f odd 4 1
1530.2.q.a 4 1.a even 1 1 trivial
1530.2.q.a 4 17.c even 4 1 inner
8670.2.a.bi 2 51.g odd 8 1
8670.2.a.bj 2 51.g odd 8 1

Hecke kernels

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

\( T_{7}^{2} + 4T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{4} - 8T_{11}^{3} + 32T_{11}^{2} - 32T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 8 T^{3} + 32 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} - 8 T + 14)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8 T + 17)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$29$ \( T^{4} + 4096 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + 8 T^{2} + 136 T + 1156 \) Copy content Toggle raw display
$37$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} + 32 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 108T^{2} + 2116 \) Copy content Toggle raw display
$47$ \( (T + 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 108T^{2} + 2116 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T + 34)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T + 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 10000 \) Copy content Toggle raw display
$79$ \( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 4 T - 196)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
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