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Properties

Label 2520.2.bi.s
Level $2520$
Weight $2$
Character orbit 2520.bi
Analytic conductor $20.122$
Analytic rank $0$
Dimension $10$
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] = [2520,2,Mod(361,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2520.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.bi (of order \(3\), 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: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 29x^{8} + 247x^{6} + 855x^{4} + 1212x^{2} + 588 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + \beta_{6} q^{7} - \beta_{3} q^{11} + (\beta_{9} - \beta_{6} + \beta_{4}) q^{13} + (\beta_{9} - \beta_{8} + \cdots + \beta_{3}) q^{17}+ \cdots + (\beta_{8} - 2 \beta_{7} + \beta_{5} + \cdots + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5 q^{5} - q^{7} + 2 q^{11} + 6 q^{13} - 2 q^{17} + q^{19} - 8 q^{23} - 5 q^{25} + 7 q^{31} + q^{35} - 11 q^{37} - 20 q^{41} + 6 q^{43} - 23 q^{49} + 14 q^{53} + 4 q^{55} - 4 q^{59} - 6 q^{61} + 3 q^{65} - 7 q^{67} + 32 q^{71} + 3 q^{73} - 8 q^{77} - 19 q^{79} - 28 q^{83} - 4 q^{85} + 18 q^{89} - 21 q^{91} - q^{95} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 29x^{8} + 247x^{6} + 855x^{4} + 1212x^{2} + 588 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19\nu^{9} + 467\nu^{7} + 2607\nu^{5} + 4121\nu^{3} + 642\nu - 812 ) / 1624 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10 \nu^{9} - 28 \nu^{8} - 171 \nu^{7} - 763 \nu^{6} + 316 \nu^{5} - 5530 \nu^{4} + 4701 \nu^{3} + \cdots - 8190 ) / 812 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19 \nu^{9} + 84 \nu^{8} - 467 \nu^{7} + 2086 \nu^{6} - 2607 \nu^{5} + 12124 \nu^{4} - 4121 \nu^{3} + \cdots + 10360 ) / 1624 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{8} - 89\nu^{6} - 781\nu^{4} - 2583\nu^{2} - 2400 ) / 58 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53 \nu^{9} - 14 \nu^{8} - 1495 \nu^{7} - 686 \nu^{6} - 11845 \nu^{5} - 9870 \nu^{4} - 34381 \nu^{3} + \cdots - 35560 ) / 1624 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32 \nu^{9} - 49 \nu^{8} - 872 \nu^{7} - 1386 \nu^{6} - 6378 \nu^{5} - 10997 \nu^{4} - 16300 \nu^{3} + \cdots - 25396 ) / 812 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{8} - 347\nu^{6} - 2437\nu^{4} - 6089\nu^{2} - 4368 ) / 58 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53 \nu^{9} - 14 \nu^{8} + 1495 \nu^{7} - 686 \nu^{6} + 11845 \nu^{5} - 9870 \nu^{4} + 34381 \nu^{3} + \cdots - 35560 ) / 1624 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32 \nu^{9} + 49 \nu^{8} - 872 \nu^{7} + 1386 \nu^{6} - 6378 \nu^{5} + 10997 \nu^{4} - 16300 \nu^{3} + \cdots + 25396 ) / 812 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{7} - \beta_{6} + 2\beta_{3} + 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{5} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 19 \beta_{9} - 3 \beta_{8} - 10 \beta_{7} + 7 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 20 \beta_{3} + \cdots - 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -40\beta_{9} - 18\beta_{8} - 19\beta_{7} + 40\beta_{6} - 18\beta_{5} - 5\beta_{4} + 76 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 337 \beta_{9} + 78 \beta_{8} + 148 \beta_{7} - 67 \beta_{6} - 78 \beta_{5} + 54 \beta_{4} + 296 \beta_{3} + \cdots + 388 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 716\beta_{9} + 309\beta_{8} + 335\beta_{7} - 716\beta_{6} + 309\beta_{5} + 116\beta_{4} - 1226 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5893 \beta_{9} - 1485 \beta_{8} - 2470 \beta_{7} + 901 \beta_{6} + 1485 \beta_{5} - 927 \beta_{4} + \cdots - 6784 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -12550\beta_{9} - 5342\beta_{8} - 5853\beta_{7} + 12550\beta_{6} - 5342\beta_{5} - 2159\beta_{4} + 20952 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 102691 \beta_{9} + 26448 \beta_{8} + 42538 \beta_{7} - 14437 \beta_{6} - 26448 \beta_{5} + 16026 \beta_{4} + \cdots + 118372 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
1.05780i
2.19149i
− 2.03852i
− 4.17259i
1.22977i
− 1.05780i
− 2.19149i
2.03852i
4.17259i
− 1.22977i
0 0 0 0.500000 + 0.866025i 0 −2.62478 − 0.332488i 0 0 0
361.2 0 0 0 0.500000 + 0.866025i 0 −0.780339 − 2.52806i 0 0 0
361.3 0 0 0 0.500000 + 0.866025i 0 0.194868 + 2.63857i 0 0 0
361.4 0 0 0 0.500000 + 0.866025i 0 0.835066 − 2.51051i 0 0 0
361.5 0 0 0 0.500000 + 0.866025i 0 1.87518 + 1.86646i 0 0 0
1801.1 0 0 0 0.500000 − 0.866025i 0 −2.62478 + 0.332488i 0 0 0
1801.2 0 0 0 0.500000 − 0.866025i 0 −0.780339 + 2.52806i 0 0 0
1801.3 0 0 0 0.500000 − 0.866025i 0 0.194868 − 2.63857i 0 0 0
1801.4 0 0 0 0.500000 − 0.866025i 0 0.835066 + 2.51051i 0 0 0
1801.5 0 0 0 0.500000 − 0.866025i 0 1.87518 − 1.86646i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.2.bi.s yes 10
3.b odd 2 1 2520.2.bi.r ✓ 10
7.c even 3 1 inner 2520.2.bi.s yes 10
21.h odd 6 1 2520.2.bi.r ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.2.bi.r ✓ 10 3.b odd 2 1
2520.2.bi.r ✓ 10 21.h odd 6 1
2520.2.bi.s yes 10 1.a even 1 1 trivial
2520.2.bi.s yes 10 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2520, [\chi])\):

\( T_{11}^{10} - 2 T_{11}^{9} + 45 T_{11}^{8} - 150 T_{11}^{7} + 1695 T_{11}^{6} - 4554 T_{11}^{5} + \cdots + 448900 \) Copy content Toggle raw display
\( T_{13}^{5} - 3T_{13}^{4} - 42T_{13}^{3} + 44T_{13}^{2} + 525T_{13} + 471 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + \cdots + 448900 \) Copy content Toggle raw display
$13$ \( (T^{5} - 3 T^{4} + \cdots + 471)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 2 T^{9} + \cdots + 473344 \) Copy content Toggle raw display
$19$ \( T^{10} - T^{9} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( T^{10} + 8 T^{9} + \cdots + 9604 \) Copy content Toggle raw display
$29$ \( (T^{5} - 90 T^{3} + \cdots + 384)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 7 T^{9} + \cdots + 20611600 \) Copy content Toggle raw display
$37$ \( T^{10} + 11 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$41$ \( (T^{5} + 10 T^{4} + \cdots - 498)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 3 T^{4} + \cdots - 14424)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 111 T^{8} + \cdots + 46656 \) Copy content Toggle raw display
$53$ \( T^{10} - 14 T^{9} + \cdots + 112896 \) Copy content Toggle raw display
$59$ \( T^{10} + 4 T^{9} + \cdots + 8809024 \) Copy content Toggle raw display
$61$ \( T^{10} + 6 T^{9} + \cdots + 451584 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 117852736 \) Copy content Toggle raw display
$71$ \( (T^{5} - 16 T^{4} + \cdots + 216)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 1056770064 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 1256560704 \) Copy content Toggle raw display
$83$ \( (T^{5} + 14 T^{4} + \cdots + 186128)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 18 T^{9} + \cdots + 331776 \) Copy content Toggle raw display
$97$ \( (T^{5} - 24 T^{4} + \cdots - 50496)^{2} \) Copy content Toggle raw display
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