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Properties

Label 20.18.c.a
Level 2020
Weight 1818
Character orbit 20.c
Analytic conductor 36.64436.644
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,18,Mod(9,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.9");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: N N == 20=22⋅5 20 = 2^{2} \cdot 5
Weight: k k == 18 18
Character orbit: [χ][\chi] == 20.c (of order 22, degree 11, 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: 36.644417468936.6444174689
Analytic rank: 00
Dimension: 88
Coefficient field: Q[x]/(x8−⋯ )\mathbb{Q}[x]/(x^{8} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−x7−10513788x6+47438777752x5−249513269598475x4+⋯+12 ⁣⋯ ⁣68 x^{8} - x^{7} - 10513788 x^{6} + 47438777752 x^{5} - 249513269598475 x^{4} + \cdots + 12\!\cdots\!68 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 235⋅34⋅512 2^{35}\cdot 3^{4}\cdot 5^{12}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,…,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q3+(β2−4β1+159600)q5+(β5+3β2+298β1)q7+(−β4−β3+⋯−47106031)q9+(β6+2β4+⋯−49525080)q11+⋯+(34305831β6+⋯−29 ⁣⋯ ⁣20)q99+O(q100) q + \beta_1 q^{3} + (\beta_{2} - 4 \beta_1 + 159600) q^{5} + (\beta_{5} + 3 \beta_{2} + 298 \beta_1) q^{7} + ( - \beta_{4} - \beta_{3} + \cdots - 47106031) q^{9} + (\beta_{6} + 2 \beta_{4} + \cdots - 49525080) q^{11}+ \cdots + (34305831 \beta_{6} + \cdots - 29\!\cdots\!20) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q+1276800q5−376848248q9−396200640q11+5677983200q15+11821646592q19−420670059472q21+1642212165000q25+1543712861232q29−13722543013312q31−13325691076800q35+⋯−23 ⁣⋯ ⁣60q99+O(q100) 8 q + 1276800 q^{5} - 376848248 q^{9} - 396200640 q^{11} + 5677983200 q^{15} + 11821646592 q^{19} - 420670059472 q^{21} + 1642212165000 q^{25} + 1543712861232 q^{29} - 13722543013312 q^{31} - 13325691076800 q^{35}+ \cdots - 23\!\cdots\!60 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−x7−10513788x6+47438777752x5−249513269598475x4+⋯+12 ⁣⋯ ⁣68 x^{8} - x^{7} - 10513788 x^{6} + 47438777752 x^{5} - 249513269598475 x^{4} + \cdots + 12\!\cdots\!68 : Copy content Toggle raw display

β1\beta_{1}== (16 ⁣⋯ ⁣39ν7+⋯−55 ⁣⋯ ⁣52)/27 ⁣⋯ ⁣50 ( 16\!\cdots\!39 \nu^{7} + \cdots - 55\!\cdots\!52 ) / 27\!\cdots\!50 Copy content Toggle raw display
β2\beta_{2}== (−34 ⁣⋯ ⁣39ν7+⋯+12 ⁣⋯ ⁣02)/12 ⁣⋯ ⁣50 ( - 34\!\cdots\!39 \nu^{7} + \cdots + 12\!\cdots\!02 ) / 12\!\cdots\!50 Copy content Toggle raw display
β3\beta_{3}== (45 ⁣⋯ ⁣69ν7+⋯−28 ⁣⋯ ⁣42)/12 ⁣⋯ ⁣50 ( 45\!\cdots\!69 \nu^{7} + \cdots - 28\!\cdots\!42 ) / 12\!\cdots\!50 Copy content Toggle raw display
β4\beta_{4}== (26 ⁣⋯ ⁣59ν7+⋯−91 ⁣⋯ ⁣77)/54 ⁣⋯ ⁣75 ( 26\!\cdots\!59 \nu^{7} + \cdots - 91\!\cdots\!77 ) / 54\!\cdots\!75 Copy content Toggle raw display
β5\beta_{5}== (−11 ⁣⋯ ⁣06ν7+⋯+36 ⁣⋯ ⁣33)/13 ⁣⋯ ⁣75 ( - 11\!\cdots\!06 \nu^{7} + \cdots + 36\!\cdots\!33 ) / 13\!\cdots\!75 Copy content Toggle raw display
β6\beta_{6}== (66 ⁣⋯ ⁣61ν7+⋯−14 ⁣⋯ ⁣98)/12 ⁣⋯ ⁣50 ( 66\!\cdots\!61 \nu^{7} + \cdots - 14\!\cdots\!98 ) / 12\!\cdots\!50 Copy content Toggle raw display
β7\beta_{7}== (−48 ⁣⋯ ⁣68ν7+⋯+96 ⁣⋯ ⁣99)/45 ⁣⋯ ⁣25 ( - 48\!\cdots\!68 \nu^{7} + \cdots + 96\!\cdots\!99 ) / 45\!\cdots\!25 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β4+6β2−524β1+625)/5000 ( \beta_{4} + 6\beta_{2} - 524\beta _1 + 625 ) / 5000 Copy content Toggle raw display
ν2\nu^{2}== (125β7−750β6−24875β5−10436β4+3250β3+⋯+105137885000)/40000 ( 125 \beta_{7} - 750 \beta_{6} - 24875 \beta_{5} - 10436 \beta_{4} + 3250 \beta_{3} + \cdots + 105137885000 ) / 40000 Copy content Toggle raw display
ν3\nu^{3}== (−315875β7+5651500β6−262683375β5−84814578β4+⋯−14 ⁣⋯ ⁣00)/80000 ( - 315875 \beta_{7} + 5651500 \beta_{6} - 262683375 \beta_{5} - 84814578 \beta_{4} + \cdots - 14\!\cdots\!00 ) / 80000 Copy content Toggle raw display
ν4\nu^{4}== (794969150β7+728628975β6+25977943650β5+8251976656β4+⋯+12 ⁣⋯ ⁣00)/8000 ( 794969150 \beta_{7} + 728628975 \beta_{6} + 25977943650 \beta_{5} + 8251976656 \beta_{4} + \cdots + 12\!\cdots\!00 ) / 8000 Copy content Toggle raw display
ν5\nu^{5}== (−76614562530125β7−186588028874250β6+⋯−29 ⁣⋯ ⁣00)/160000 ( - 76614562530125 \beta_{7} - 186588028874250 \beta_{6} + \cdots - 29\!\cdots\!00 ) / 160000 Copy content Toggle raw display
ν6\nu^{6}== (38 ⁣⋯ ⁣25β7+⋯−38 ⁣⋯ ⁣00)/160000 ( 38\!\cdots\!25 \beta_{7} + \cdots - 38\!\cdots\!00 ) / 160000 Copy content Toggle raw display
ν7\nu^{7}== (−15 ⁣⋯ ⁣00β7+⋯+72 ⁣⋯ ⁣00)/80000 ( - 15\!\cdots\!00 \beta_{7} + \cdots + 72\!\cdots\!00 ) / 80000 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/20Z)×\left(\mathbb{Z}/20\mathbb{Z}\right)^\times.

nn 1111 1717
χ(n)\chi(n) 11 −1-1

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
9.1
3569.15 + 2557.70i
−5032.46 + 787.915i
2565.99 − 1839.67i
−1102.18 + 4980.24i
−1102.18 − 4980.24i
2565.99 + 1839.67i
−5032.46 − 787.915i
3569.15 − 2557.70i
0 − 20057.4i 0 873406. − 10106.1i 0 − 2.05068e7i 0 −2.73159e8 0
9.2 0 − 14851.7i 0 −846917. + 213710.i 0 − 2.81651e6i 0 −9.14330e7 0
9.3 0 − 7565.06i 0 672773. + 557061.i 0 2.63351e7i 0 7.19100e7 0
9.4 0 − 4988.24i 0 −60861.6 − 871341.i 0 8.73679e6i 0 1.04258e8 0
9.5 0 4988.24i 0 −60861.6 + 871341.i 0 − 8.73679e6i 0 1.04258e8 0
9.6 0 7565.06i 0 672773. − 557061.i 0 − 2.63351e7i 0 7.19100e7 0
9.7 0 14851.7i 0 −846917. − 213710.i 0 2.81651e6i 0 −9.14330e7 0
9.8 0 20057.4i 0 873406. + 10106.1i 0 2.05068e7i 0 −2.73159e8 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
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 20.18.c.a ✓ 8
4.b odd 2 1 80.18.c.c 8
5.b even 2 1 inner 20.18.c.a ✓ 8
5.c odd 4 2 100.18.a.f 8
20.d odd 2 1 80.18.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.18.c.a ✓ 8 1.a even 1 1 trivial
20.18.c.a ✓ 8 5.b even 2 1 inner
80.18.c.c 8 4.b odd 2 1
80.18.c.c 8 20.d odd 2 1
100.18.a.f 8 5.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace S18new(20,[χ])S_{18}^{\mathrm{new}}(20, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T8+⋯+12 ⁣⋯ ⁣96 T^{8} + \cdots + 12\!\cdots\!96 Copy content Toggle raw display
55 T8+⋯+33 ⁣⋯ ⁣25 T^{8} + \cdots + 33\!\cdots\!25 Copy content Toggle raw display
77 T8+⋯+17 ⁣⋯ ⁣76 T^{8} + \cdots + 17\!\cdots\!76 Copy content Toggle raw display
1111 (T4+⋯+16 ⁣⋯ ⁣00)2 (T^{4} + \cdots + 16\!\cdots\!00)^{2} Copy content Toggle raw display
1313 T8+⋯+10 ⁣⋯ ⁣56 T^{8} + \cdots + 10\!\cdots\!56 Copy content Toggle raw display
1717 T8+⋯+15 ⁣⋯ ⁣96 T^{8} + \cdots + 15\!\cdots\!96 Copy content Toggle raw display
1919 (T4+⋯+31 ⁣⋯ ⁣76)2 (T^{4} + \cdots + 31\!\cdots\!76)^{2} Copy content Toggle raw display
2323 T8+⋯+76 ⁣⋯ ⁣36 T^{8} + \cdots + 76\!\cdots\!36 Copy content Toggle raw display
2929 (T4+⋯+11 ⁣⋯ ⁣56)2 (T^{4} + \cdots + 11\!\cdots\!56)^{2} Copy content Toggle raw display
3131 (T4+⋯−17 ⁣⋯ ⁣84)2 (T^{4} + \cdots - 17\!\cdots\!84)^{2} Copy content Toggle raw display
3737 T8+⋯+41 ⁣⋯ ⁣96 T^{8} + \cdots + 41\!\cdots\!96 Copy content Toggle raw display
4141 (T4+⋯+59 ⁣⋯ ⁣56)2 (T^{4} + \cdots + 59\!\cdots\!56)^{2} Copy content Toggle raw display
4343 T8+⋯+60 ⁣⋯ ⁣00 T^{8} + \cdots + 60\!\cdots\!00 Copy content Toggle raw display
4747 T8+⋯+86 ⁣⋯ ⁣96 T^{8} + \cdots + 86\!\cdots\!96 Copy content Toggle raw display
5353 T8+⋯+41 ⁣⋯ ⁣36 T^{8} + \cdots + 41\!\cdots\!36 Copy content Toggle raw display
5959 (T4+⋯+24 ⁣⋯ ⁣76)2 (T^{4} + \cdots + 24\!\cdots\!76)^{2} Copy content Toggle raw display
6161 (T4+⋯−16 ⁣⋯ ⁣44)2 (T^{4} + \cdots - 16\!\cdots\!44)^{2} Copy content Toggle raw display
6767 T8+⋯+11 ⁣⋯ ⁣56 T^{8} + \cdots + 11\!\cdots\!56 Copy content Toggle raw display
7171 (T4+⋯+84 ⁣⋯ ⁣56)2 (T^{4} + \cdots + 84\!\cdots\!56)^{2} Copy content Toggle raw display
7373 T8+⋯+12 ⁣⋯ ⁣36 T^{8} + \cdots + 12\!\cdots\!36 Copy content Toggle raw display
7979 (T4+⋯−34 ⁣⋯ ⁣24)2 (T^{4} + \cdots - 34\!\cdots\!24)^{2} Copy content Toggle raw display
8383 T8+⋯+66 ⁣⋯ ⁣36 T^{8} + \cdots + 66\!\cdots\!36 Copy content Toggle raw display
8989 (T4+⋯−18 ⁣⋯ ⁣44)2 (T^{4} + \cdots - 18\!\cdots\!44)^{2} Copy content Toggle raw display
9797 T8+⋯+70 ⁣⋯ ⁣76 T^{8} + \cdots + 70\!\cdots\!76 Copy content Toggle raw display
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