Properties

Label 510.2.c.a
Level $510$
Weight $2$
Character orbit 510.c
Analytic conductor $4.072$
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] = [510,2,Mod(271,510)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(510, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.c (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: \(4.07237050309\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + i q^{3} + q^{4} - i q^{5} - i q^{6} - q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + i q^{3} + q^{4} - i q^{5} - i q^{6} - q^{8} - q^{9} + i q^{10} + 4 i q^{11} + i q^{12} + q^{15} + q^{16} + (4 i - 1) q^{17} + q^{18} + 8 q^{19} - i q^{20} - 4 i q^{22} - 2 i q^{23} - i q^{24} - q^{25} - i q^{27} + 6 i q^{29} - q^{30} + 2 i q^{31} - q^{32} - 4 q^{33} + ( - 4 i + 1) q^{34} - q^{36} + 10 i q^{37} - 8 q^{38} + i q^{40} + 2 i q^{41} + 6 q^{43} + 4 i q^{44} + i q^{45} + 2 i q^{46} - 8 q^{47} + i q^{48} + 7 q^{49} + q^{50} + ( - i - 4) q^{51} - 6 q^{53} + i q^{54} + 4 q^{55} + 8 i q^{57} - 6 i q^{58} + 6 q^{59} + q^{60} + 2 i q^{61} - 2 i q^{62} + q^{64} + 4 q^{66} - 10 q^{67} + (4 i - 1) q^{68} + 2 q^{69} - 4 i q^{71} + q^{72} + 2 i q^{73} - 10 i q^{74} - i q^{75} + 8 q^{76} + 2 i q^{79} - i q^{80} + q^{81} - 2 i q^{82} + 12 q^{83} + (i + 4) q^{85} - 6 q^{86} - 6 q^{87} - 4 i q^{88} + 14 q^{89} - i q^{90} - 2 i q^{92} - 2 q^{93} + 8 q^{94} - 8 i q^{95} - i q^{96} - 14 i q^{97} - 7 q^{98} - 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} + 16 q^{19} - 2 q^{25} - 2 q^{30} - 2 q^{32} - 8 q^{33} + 2 q^{34} - 2 q^{36} - 16 q^{38} + 12 q^{43} - 16 q^{47} + 14 q^{49} + 2 q^{50} - 8 q^{51} - 12 q^{53} + 8 q^{55} + 12 q^{59} + 2 q^{60} + 2 q^{64} + 8 q^{66} - 20 q^{67} - 2 q^{68} + 4 q^{69} + 2 q^{72} + 16 q^{76} + 2 q^{81} + 24 q^{83} + 8 q^{85} - 12 q^{86} - 12 q^{87} + 28 q^{89} - 4 q^{93} + 16 q^{94} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(307\) \(341\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000 1.00000i 1.00000i 0 −1.00000 −1.00000 1.00000i
271.2 −1.00000 1.00000i 1.00000 1.00000i 1.00000i 0 −1.00000 −1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 510.2.c.a 2
3.b odd 2 1 1530.2.c.e 2
4.b odd 2 1 4080.2.h.h 2
5.b even 2 1 2550.2.c.h 2
5.c odd 4 1 2550.2.f.d 2
5.c odd 4 1 2550.2.f.k 2
17.b even 2 1 inner 510.2.c.a 2
17.c even 4 1 8670.2.a.s 1
17.c even 4 1 8670.2.a.w 1
51.c odd 2 1 1530.2.c.e 2
68.d odd 2 1 4080.2.h.h 2
85.c even 2 1 2550.2.c.h 2
85.g odd 4 1 2550.2.f.d 2
85.g odd 4 1 2550.2.f.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.c.a 2 1.a even 1 1 trivial
510.2.c.a 2 17.b even 2 1 inner
1530.2.c.e 2 3.b odd 2 1
1530.2.c.e 2 51.c odd 2 1
2550.2.c.h 2 5.b even 2 1
2550.2.c.h 2 85.c even 2 1
2550.2.f.d 2 5.c odd 4 1
2550.2.f.d 2 85.g odd 4 1
2550.2.f.k 2 5.c odd 4 1
2550.2.f.k 2 85.g odd 4 1
4080.2.h.h 2 4.b odd 2 1
4080.2.h.h 2 68.d odd 2 1
8670.2.a.s 1 17.c even 4 1
8670.2.a.w 1 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 17 \) Copy content Toggle raw display
$19$ \( (T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( (T + 10)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 4 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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