Properties

Label 520.2.d.c
Level $520$
Weight $2$
Character orbit 520.d
Analytic conductor $4.152$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(209,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 2x^{8} + 16x^{7} - 15x^{6} - 40x^{5} - 75x^{4} + 400x^{3} + 250x^{2} - 2500x + 3125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} - \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{8} - \beta_{6} - \beta_{5} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{3} - \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{8} - \beta_{6} - \beta_{5} - 4) q^{9} + (\beta_{8} + \beta_{7} + \beta_{2} + \cdots + 3) q^{11}+ \cdots + ( - 3 \beta_{8} - 5 \beta_{7} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{5} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{5} - 30 q^{9} + 16 q^{11} + 4 q^{15} - 36 q^{19} + 4 q^{21} + 12 q^{25} - 4 q^{29} - 8 q^{31} + 18 q^{35} - 4 q^{39} - 24 q^{41} + 36 q^{45} - 38 q^{49} - 4 q^{51} + 32 q^{55} - 28 q^{59} + 20 q^{61} + 4 q^{65} - 72 q^{69} + 36 q^{71} + 62 q^{75} - 16 q^{79} + 18 q^{81} + 8 q^{85} + 12 q^{89} + 52 q^{95} - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} + 2x^{8} + 16x^{7} - 15x^{6} - 40x^{5} - 75x^{4} + 400x^{3} + 250x^{2} - 2500x + 3125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 4\nu^{8} + 2\nu^{7} + 16\nu^{6} - 15\nu^{5} - 40\nu^{4} - 75\nu^{3} + 400\nu^{2} + 250\nu - 2500 ) / 625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4 \nu^{9} + 11 \nu^{8} + 22 \nu^{7} - 39 \nu^{6} - 100 \nu^{5} + 120 \nu^{4} + 700 \nu^{3} + \cdots + 3125 ) / 2500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} - 3\nu^{7} + 36\nu^{5} + 36\nu^{4} - 115\nu^{3} - 320\nu^{2} + 325\nu + 500 ) / 500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - 9\nu^{7} - 6\nu^{6} + 24\nu^{5} + 60\nu^{4} - 195\nu^{3} - 300\nu^{2} + 675\nu + 750 ) / 500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} + 6\nu^{8} + 5\nu^{7} - 32\nu^{6} - 8\nu^{5} + 80\nu^{4} + 235\nu^{3} - 450\nu^{2} - 1375\nu + 2500 ) / 500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8 \nu^{9} + 7 \nu^{8} + 34 \nu^{7} - 53 \nu^{6} - 80 \nu^{5} + 120 \nu^{4} + 900 \nu^{3} + \cdots + 4375 ) / 2500 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8 \nu^{9} + 27 \nu^{8} + 4 \nu^{7} - 113 \nu^{6} - 60 \nu^{5} + 320 \nu^{4} + 1700 \nu^{3} + \cdots + 14375 ) / 2500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18 \nu^{9} - 37 \nu^{8} - 74 \nu^{7} + 213 \nu^{6} + 300 \nu^{5} - 340 \nu^{4} - 2650 \nu^{3} + \cdots - 19375 ) / 2500 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{6} - \beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 3\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{3} + 3\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{8} + 7\beta_{7} - 2\beta_{6} + 5\beta_{5} + 3\beta_{4} + 12\beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{9} + 2 \beta_{8} + 10 \beta_{7} + 4 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} - 16 \beta_{3} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{9} + 6\beta_{8} - 26\beta_{6} - 6\beta_{5} + 16\beta_{4} + 80\beta_{3} - 16\beta_{2} - 24\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -20\beta_{9} - 42\beta_{8} - 2\beta_{7} - 28\beta_{5} + 60\beta_{4} + 40\beta_{3} + 2\beta_{2} + 73\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 113 \beta_{9} - 62 \beta_{8} - 73 \beta_{7} + 73 \beta_{6} - 62 \beta_{5} - 75 \beta_{4} + 316 \beta_{3} + \cdots - 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 123 \beta_{9} + 75 \beta_{8} - 183 \beta_{7} + 363 \beta_{6} - 91 \beta_{5} - 36 \beta_{4} - 81 \beta_{3} + \cdots + 849 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
1.64680 + 1.51263i
2.19360 + 0.433733i
−1.11501 1.93823i
−2.16128 0.573465i
1.43589 1.71412i
1.43589 + 1.71412i
−2.16128 + 0.573465i
−1.11501 + 1.93823i
2.19360 0.433733i
1.64680 1.51263i
0 3.08870i 0 −1.64680 1.51263i 0 3.54010i 0 −6.54010 0
209.2 0 3.05748i 0 −2.19360 0.433733i 0 3.34821i 0 −6.34821 0
209.3 0 2.57527i 0 1.11501 + 1.93823i 0 0.632021i 0 −3.63202 0
209.4 0 1.76881i 0 2.16128 + 0.573465i 0 2.87131i 0 −0.128689 0
209.5 0 1.16232i 0 −1.43589 + 1.71412i 0 4.64901i 0 1.64901 0
209.6 0 1.16232i 0 −1.43589 1.71412i 0 4.64901i 0 1.64901 0
209.7 0 1.76881i 0 2.16128 0.573465i 0 2.87131i 0 −0.128689 0
209.8 0 2.57527i 0 1.11501 1.93823i 0 0.632021i 0 −3.63202 0
209.9 0 3.05748i 0 −2.19360 + 0.433733i 0 3.34821i 0 −6.34821 0
209.10 0 3.08870i 0 −1.64680 + 1.51263i 0 3.54010i 0 −6.54010 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.10
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 520.2.d.c 10
3.b odd 2 1 4680.2.l.i 10
4.b odd 2 1 1040.2.d.f 10
5.b even 2 1 inner 520.2.d.c 10
5.c odd 4 1 2600.2.a.bb 5
5.c odd 4 1 2600.2.a.bc 5
15.d odd 2 1 4680.2.l.i 10
20.d odd 2 1 1040.2.d.f 10
20.e even 4 1 5200.2.a.cm 5
20.e even 4 1 5200.2.a.cn 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.d.c 10 1.a even 1 1 trivial
520.2.d.c 10 5.b even 2 1 inner
1040.2.d.f 10 4.b odd 2 1
1040.2.d.f 10 20.d odd 2 1
2600.2.a.bb 5 5.c odd 4 1
2600.2.a.bc 5 5.c odd 4 1
4680.2.l.i 10 3.b odd 2 1
4680.2.l.i 10 15.d odd 2 1
5200.2.a.cm 5 20.e even 4 1
5200.2.a.cn 5 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 30T_{3}^{8} + 333T_{3}^{6} + 1660T_{3}^{4} + 3556T_{3}^{2} + 2500 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 30 T^{8} + \cdots + 2500 \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 54 T^{8} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{5} - 8 T^{4} + \cdots - 1544)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{10} + 66 T^{8} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( (T^{5} + 18 T^{4} + \cdots - 5368)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 140 T^{8} + \cdots + 2611456 \) Copy content Toggle raw display
$29$ \( (T^{5} + 2 T^{4} + \cdots + 9232)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 4 T^{4} + \cdots + 1944)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 194 T^{8} + \cdots + 2755600 \) Copy content Toggle raw display
$41$ \( (T^{5} + 12 T^{4} + \cdots + 5120)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 178 T^{8} + \cdots + 25100100 \) Copy content Toggle raw display
$47$ \( T^{10} + 134 T^{8} + \cdots + 3600 \) Copy content Toggle raw display
$53$ \( T^{10} + 240 T^{8} + \cdots + 1982464 \) Copy content Toggle raw display
$59$ \( (T^{5} + 14 T^{4} + \cdots - 4008)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 10 T^{4} + \cdots + 1472)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1317980416 \) Copy content Toggle raw display
$71$ \( (T^{5} - 18 T^{4} + \cdots - 5770)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 352 T^{8} + \cdots + 85229824 \) Copy content Toggle raw display
$79$ \( (T^{5} + 8 T^{4} + \cdots + 14720)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 181494784 \) Copy content Toggle raw display
$89$ \( (T^{5} - 6 T^{4} + \cdots - 2208)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 324 T^{8} + \cdots + 25600 \) Copy content Toggle raw display
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