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Properties

Label 21.10.e.a
Level 2121
Weight 1010
Character orbit 21.e
Analytic conductor 10.81610.816
Analytic rank 00
Dimension 1010
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,10,Mod(4,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: N N == 21=3⋅7 21 = 3 \cdot 7
Weight: k k == 10 10
Character orbit: [χ][\chi] == 21.e (of order 33, degree 22, 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: 10.815752559410.8157525594
Analytic rank: 00
Dimension: 1010
Relative dimension: 55 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x10−⋯ )\mathbb{Q}[x]/(x^{10} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x10−2x9+1600x8−7420x7+2144441x6−9353044x5+682842856x4+⋯+6442540462656 x^{10} - 2 x^{9} + 1600 x^{8} - 7420 x^{7} + 2144441 x^{6} - 9353044 x^{5} + 682842856 x^{4} + \cdots + 6442540462656 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 24⋅33⋅74 2^{4}\cdot 3^{3}\cdot 7^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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,…,β91,\beta_1,\ldots,\beta_{9} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β3+7β2+β1)q2+(−81β2+81)q3+(−β7+177β2+⋯−177)q4+(−β9+β8+⋯−9β1)q5+(81β3+567)q6+⋯+(118098β6+137781β5+⋯+53564004)q99+O(q100) q + (\beta_{3} + 7 \beta_{2} + \beta_1) q^{2} + ( - 81 \beta_{2} + 81) q^{3} + ( - \beta_{7} + 177 \beta_{2} + \cdots - 177) q^{4} + ( - \beta_{9} + \beta_{8} + \cdots - 9 \beta_1) q^{5} + (81 \beta_{3} + 567) q^{6}+ \cdots + (118098 \beta_{6} + 137781 \beta_{5} + \cdots + 53564004) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 10q+33q2+405q3−853q4−165q5+5346q6−1981q7−82158q8−32805q9+28697q10−40227q11+69093q12+462460q13−331968q14−26730q15−618577q16+⋯+527858694q99+O(q100) 10 q + 33 q^{2} + 405 q^{3} - 853 q^{4} - 165 q^{5} + 5346 q^{6} - 1981 q^{7} - 82158 q^{8} - 32805 q^{9} + 28697 q^{10} - 40227 q^{11} + 69093 q^{12} + 462460 q^{13} - 331968 q^{14} - 26730 q^{15} - 618577 q^{16}+ \cdots + 527858694 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x10−2x9+1600x8−7420x7+2144441x6−9353044x5+682842856x4+⋯+6442540462656 x^{10} - 2 x^{9} + 1600 x^{8} - 7420 x^{7} + 2144441 x^{6} - 9353044 x^{5} + 682842856 x^{4} + \cdots + 6442540462656 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (−33 ⁣⋯ ⁣15ν9+⋯+32 ⁣⋯ ⁣48)/34 ⁣⋯ ⁣56 ( - 33\!\cdots\!15 \nu^{9} + \cdots + 32\!\cdots\!48 ) / 34\!\cdots\!56 Copy content Toggle raw display
β3\beta_{3}== (−35 ⁣⋯ ⁣89ν9+⋯−22 ⁣⋯ ⁣80)/36 ⁣⋯ ⁣07 ( - 35\!\cdots\!89 \nu^{9} + \cdots - 22\!\cdots\!80 ) / 36\!\cdots\!07 Copy content Toggle raw display
β4\beta_{4}== (67 ⁣⋯ ⁣40ν9+⋯−20 ⁣⋯ ⁣64)/33 ⁣⋯ ⁣37 ( 67\!\cdots\!40 \nu^{9} + \cdots - 20\!\cdots\!64 ) / 33\!\cdots\!37 Copy content Toggle raw display
β5\beta_{5}== (−21 ⁣⋯ ⁣71ν9+⋯+14 ⁣⋯ ⁣40)/13 ⁣⋯ ⁣28 ( - 21\!\cdots\!71 \nu^{9} + \cdots + 14\!\cdots\!40 ) / 13\!\cdots\!28 Copy content Toggle raw display
β6\beta_{6}== (24 ⁣⋯ ⁣03ν9+⋯−51 ⁣⋯ ⁣80)/10 ⁣⋯ ⁣96 ( 24\!\cdots\!03 \nu^{9} + \cdots - 51\!\cdots\!80 ) / 10\!\cdots\!96 Copy content Toggle raw display
β7\beta_{7}== (−25 ⁣⋯ ⁣38ν9+⋯−16 ⁣⋯ ⁣24)/42 ⁣⋯ ⁣57 ( - 25\!\cdots\!38 \nu^{9} + \cdots - 16\!\cdots\!24 ) / 42\!\cdots\!57 Copy content Toggle raw display
β8\beta_{8}== (49 ⁣⋯ ⁣27ν9+⋯+56 ⁣⋯ ⁣72)/47 ⁣⋯ ⁣84 ( 49\!\cdots\!27 \nu^{9} + \cdots + 56\!\cdots\!72 ) / 47\!\cdots\!84 Copy content Toggle raw display
β9\beta_{9}== (−13 ⁣⋯ ⁣04ν9+⋯+49 ⁣⋯ ⁣64)/11 ⁣⋯ ⁣96 ( - 13\!\cdots\!04 \nu^{9} + \cdots + 49\!\cdots\!64 ) / 11\!\cdots\!96 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β7−β4−2β3−640β2−2β1 \beta_{7} - \beta_{4} - 2\beta_{3} - 640\beta_{2} - 2\beta_1 Copy content Toggle raw display
ν3\nu^{3}== 10β6+8β5−2β4+1067β3+1696 10\beta_{6} + 8\beta_{5} - 2\beta_{4} + 1067\beta_{3} + 1696 Copy content Toggle raw display
ν4\nu^{4}== −34β9−184β8−1367β7−34β6+685752β2+2444β1−685752 -34\beta_{9} - 184\beta_{8} - 1367\beta_{7} - 34\beta_{6} + 685752\beta_{2} + 2444\beta _1 - 685752 Copy content Toggle raw display
ν5\nu^{5}== 15892β9+12400β8−620β7−12400β5+620β4+⋯−1295269β1 15892 \beta_{9} + 12400 \beta_{8} - 620 \beta_{7} - 12400 \beta_{5} + 620 \beta_{4} + \cdots - 1295269 \beta_1 Copy content Toggle raw display
ν6\nu^{6}== 75540β6+311312β5+1758973β4+3606598β3+834505616 75540\beta_{6} + 311312\beta_{5} + 1758973\beta_{4} + 3606598\beta_{3} + 834505616 Copy content Toggle raw display
ν7\nu^{7}== −21259086β9−16836984β8−1034394β7−21259086β6+3143952144β2+⋯−3143952144 - 21259086 \beta_{9} - 16836984 \beta_{8} - 1034394 \beta_{7} - 21259086 \beta_{6} + 3143952144 \beta_{2} + \cdots - 3143952144 Copy content Toggle raw display
ν8\nu^{8}== 122929302β9+432605448β8+2241938803β7−432605448β5+⋯−5656546160β1 122929302 \beta_{9} + 432605448 \beta_{8} + 2241938803 \beta_{7} - 432605448 \beta_{5} + \cdots - 5656546160 \beta_1 Copy content Toggle raw display
ν9\nu^{9}== 27428612344β6+22065406496β5+3286684624β4+2062615185737β3+4699650485848 27428612344\beta_{6} + 22065406496\beta_{5} + 3286684624\beta_{4} + 2062615185737\beta_{3} + 4699650485848 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/21Z)×\left(\mathbb{Z}/21\mathbb{Z}\right)^\times.

nn 88 1010
χ(n)\chi(n) 11 −β2-\beta_{2}

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}
4.1
17.6118 − 30.5045i
7.71783 − 13.3677i
3.35159 − 5.80513i
−9.66321 + 16.7372i
−18.0180 + 31.2081i
17.6118 + 30.5045i
7.71783 + 13.3677i
3.35159 + 5.80513i
−9.66321 − 16.7372i
−18.0180 − 31.2081i
−14.1118 − 24.4423i 40.5000 − 70.1481i −142.286 + 246.446i −101.336 − 175.518i −2286.11 −4617.22 − 4362.90i −6418.86 −3280.50 − 5681.99i −2860.06 + 4953.76i
4.2 −4.21783 − 7.30550i 40.5000 − 70.1481i 220.420 − 381.778i 1069.58 + 1852.56i −683.289 6190.35 + 1425.89i −8037.83 −3280.50 − 5681.99i 9022.59 − 15627.6i
4.3 0.148406 + 0.257047i 40.5000 − 70.1481i 255.956 − 443.329i −734.607 − 1272.38i 24.0418 −2964.10 + 5618.52i 303.910 −3280.50 − 5681.99i 218.040 − 377.657i
4.4 13.1632 + 22.7993i 40.5000 − 70.1481i −90.5399 + 156.820i −337.364 − 584.331i 2132.44 5108.86 − 3775.33i 8711.94 −3280.50 − 5681.99i 8881.57 − 15383.3i
4.5 21.5180 + 37.2703i 40.5000 − 70.1481i −670.050 + 1160.56i 21.2298 + 36.7711i 3485.92 −4708.39 + 4264.34i −35638.2 −3280.50 − 5681.99i −913.647 + 1582.48i
16.1 −14.1118 + 24.4423i 40.5000 + 70.1481i −142.286 − 246.446i −101.336 + 175.518i −2286.11 −4617.22 + 4362.90i −6418.86 −3280.50 + 5681.99i −2860.06 − 4953.76i
16.2 −4.21783 + 7.30550i 40.5000 + 70.1481i 220.420 + 381.778i 1069.58 − 1852.56i −683.289 6190.35 − 1425.89i −8037.83 −3280.50 + 5681.99i 9022.59 + 15627.6i
16.3 0.148406 − 0.257047i 40.5000 + 70.1481i 255.956 + 443.329i −734.607 + 1272.38i 24.0418 −2964.10 − 5618.52i 303.910 −3280.50 + 5681.99i 218.040 + 377.657i
16.4 13.1632 − 22.7993i 40.5000 + 70.1481i −90.5399 − 156.820i −337.364 + 584.331i 2132.44 5108.86 + 3775.33i 8711.94 −3280.50 + 5681.99i 8881.57 + 15383.3i
16.5 21.5180 − 37.2703i 40.5000 + 70.1481i −670.050 − 1160.56i 21.2298 − 36.7711i 3485.92 −4708.39 − 4264.34i −35638.2 −3280.50 + 5681.99i −913.647 − 1582.48i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
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 21.10.e.a ✓ 10
3.b odd 2 1 63.10.e.a 10
7.c even 3 1 inner 21.10.e.a ✓ 10
7.c even 3 1 147.10.a.i 5
7.d odd 6 1 147.10.a.j 5
21.h odd 6 1 63.10.e.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.10.e.a ✓ 10 1.a even 1 1 trivial
21.10.e.a ✓ 10 7.c even 3 1 inner
63.10.e.a 10 3.b odd 2 1
63.10.e.a 10 21.h odd 6 1
147.10.a.i 5 7.c even 3 1
147.10.a.j 5 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T210−33T29+2251T28−12390T27+1925068T26−12244560T25+⋯+6410244096 T_{2}^{10} - 33 T_{2}^{9} + 2251 T_{2}^{8} - 12390 T_{2}^{7} + 1925068 T_{2}^{6} - 12244560 T_{2}^{5} + \cdots + 6410244096 acting on S10new(21,[χ])S_{10}^{\mathrm{new}}(21, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10+⋯+6410244096 T^{10} + \cdots + 6410244096 Copy content Toggle raw display
33 (T2−81T+6561)5 (T^{2} - 81 T + 6561)^{5} Copy content Toggle raw display
55 T10+⋯+33 ⁣⋯ ⁣00 T^{10} + \cdots + 33\!\cdots\!00 Copy content Toggle raw display
77 T10+⋯+10 ⁣⋯ ⁣07 T^{10} + \cdots + 10\!\cdots\!07 Copy content Toggle raw display
1111 T10+⋯+36 ⁣⋯ ⁣00 T^{10} + \cdots + 36\!\cdots\!00 Copy content Toggle raw display
1313 (T5+⋯+34 ⁣⋯ ⁣32)2 (T^{5} + \cdots + 34\!\cdots\!32)^{2} Copy content Toggle raw display
1717 T10+⋯+26 ⁣⋯ ⁣76 T^{10} + \cdots + 26\!\cdots\!76 Copy content Toggle raw display
1919 T10+⋯+21 ⁣⋯ ⁣36 T^{10} + \cdots + 21\!\cdots\!36 Copy content Toggle raw display
2323 T10+⋯+10 ⁣⋯ ⁣04 T^{10} + \cdots + 10\!\cdots\!04 Copy content Toggle raw display
2929 (T5+⋯+13 ⁣⋯ ⁣16)2 (T^{5} + \cdots + 13\!\cdots\!16)^{2} Copy content Toggle raw display
3131 T10+⋯+12 ⁣⋯ ⁣41 T^{10} + \cdots + 12\!\cdots\!41 Copy content Toggle raw display
3737 T10+⋯+76 ⁣⋯ ⁣44 T^{10} + \cdots + 76\!\cdots\!44 Copy content Toggle raw display
4141 (T5+⋯−31 ⁣⋯ ⁣00)2 (T^{5} + \cdots - 31\!\cdots\!00)^{2} Copy content Toggle raw display
4343 (T5+⋯+94 ⁣⋯ ⁣76)2 (T^{5} + \cdots + 94\!\cdots\!76)^{2} Copy content Toggle raw display
4747 T10+⋯+12 ⁣⋯ ⁣00 T^{10} + \cdots + 12\!\cdots\!00 Copy content Toggle raw display
5353 T10+⋯+61 ⁣⋯ ⁣64 T^{10} + \cdots + 61\!\cdots\!64 Copy content Toggle raw display
5959 T10+⋯+25 ⁣⋯ ⁣00 T^{10} + \cdots + 25\!\cdots\!00 Copy content Toggle raw display
6161 T10+⋯+16 ⁣⋯ ⁣84 T^{10} + \cdots + 16\!\cdots\!84 Copy content Toggle raw display
6767 T10+⋯+43 ⁣⋯ ⁣00 T^{10} + \cdots + 43\!\cdots\!00 Copy content Toggle raw display
7171 (T5+⋯−12 ⁣⋯ ⁣64)2 (T^{5} + \cdots - 12\!\cdots\!64)^{2} Copy content Toggle raw display
7373 T10+⋯+62 ⁣⋯ ⁣00 T^{10} + \cdots + 62\!\cdots\!00 Copy content Toggle raw display
7979 T10+⋯+70 ⁣⋯ ⁣89 T^{10} + \cdots + 70\!\cdots\!89 Copy content Toggle raw display
8383 (T5+⋯−25 ⁣⋯ ⁣08)2 (T^{5} + \cdots - 25\!\cdots\!08)^{2} Copy content Toggle raw display
8989 T10+⋯+26 ⁣⋯ ⁣04 T^{10} + \cdots + 26\!\cdots\!04 Copy content Toggle raw display
9797 (T5+⋯−37 ⁣⋯ ⁣36)2 (T^{5} + \cdots - 37\!\cdots\!36)^{2} Copy content Toggle raw display
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