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Properties

Label 46.11.b.a
Level 4646
Weight 1111
Character orbit 46.b
Analytic conductor 29.22629.226
Analytic rank 00
Dimension 2020
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [46,11,Mod(45,46)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(46, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("46.45");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: N N == 46=2⋅23 46 = 2 \cdot 23
Weight: k k == 11 11
Character orbit: [χ][\chi] == 46.b (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: 29.226433623029.2264336230
Analytic rank: 00
Dimension: 2020
Coefficient field: Q[x]/(x20−⋯ )\mathbb{Q}[x]/(x^{20} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x20−4x19+120620418x18+14359423740x17+⋯+24 ⁣⋯ ⁣12 x^{20} - 4 x^{19} + 120620418 x^{18} + 14359423740 x^{17} + \cdots + 24\!\cdots\!12 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a23]\Z[a_1, \ldots, a_{23}]
Coefficient ring index: multiple of 262⋅34⋅72 2^{62}\cdot 3^{4}\cdot 7^{2}
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,…,β191,\beta_1,\ldots,\beta_{19} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−β1q2+(β3+β1−6)q3+512q4−β2q5+(−β4+4β3+⋯−607)q6+β5q7−512β1q8+(−β7+2β4+⋯+18136)q9+⋯+(1177β17+⋯+858930β2)q99+O(q100) q - \beta_1 q^{2} + (\beta_{3} + \beta_1 - 6) q^{3} + 512 q^{4} - \beta_{2} q^{5} + ( - \beta_{4} + 4 \beta_{3} + \cdots - 607) q^{6} + \beta_{5} q^{7} - 512 \beta_1 q^{8} + ( - \beta_{7} + 2 \beta_{4} + \cdots + 18136) q^{9}+ \cdots + (1177 \beta_{17} + \cdots + 858930 \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 20q−124q3+10240q4−12160q6+362880q9−63488q12+1231780q13+5242880q16+6493952q18+2731876q23−6225920q24−47471412q25−13969408q26−60370732q27+⋯+20634273536q98+O(q100) 20 q - 124 q^{3} + 10240 q^{4} - 12160 q^{6} + 362880 q^{9} - 63488 q^{12} + 1231780 q^{13} + 5242880 q^{16} + 6493952 q^{18} + 2731876 q^{23} - 6225920 q^{24} - 47471412 q^{25} - 13969408 q^{26} - 60370732 q^{27}+ \cdots + 20634273536 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x20−4x19+120620418x18+14359423740x17+⋯+24 ⁣⋯ ⁣12 x^{20} - 4 x^{19} + 120620418 x^{18} + 14359423740 x^{17} + \cdots + 24\!\cdots\!12 : Copy content Toggle raw display

β1\beta_{1}== (26 ⁣⋯ ⁣22ν19+⋯+10 ⁣⋯ ⁣64)/51 ⁣⋯ ⁣00 ( 26\!\cdots\!22 \nu^{19} + \cdots + 10\!\cdots\!64 ) / 51\!\cdots\!00 Copy content Toggle raw display
β2\beta_{2}== (−98 ⁣⋯ ⁣75ν19+⋯+57 ⁣⋯ ⁣72)/29 ⁣⋯ ⁣00 ( - 98\!\cdots\!75 \nu^{19} + \cdots + 57\!\cdots\!72 ) / 29\!\cdots\!00 Copy content Toggle raw display
β3\beta_{3}== (−56 ⁣⋯ ⁣77ν19+⋯−22 ⁣⋯ ⁣64)/14 ⁣⋯ ⁣00 ( - 56\!\cdots\!77 \nu^{19} + \cdots - 22\!\cdots\!64 ) / 14\!\cdots\!00 Copy content Toggle raw display
β4\beta_{4}== (−51 ⁣⋯ ⁣47ν19+⋯−45 ⁣⋯ ⁣28)/37 ⁣⋯ ⁣00 ( - 51\!\cdots\!47 \nu^{19} + \cdots - 45\!\cdots\!28 ) / 37\!\cdots\!00 Copy content Toggle raw display
β5\beta_{5}== (−10 ⁣⋯ ⁣43ν19+⋯+18 ⁣⋯ ⁣52)/10 ⁣⋯ ⁣00 ( - 10\!\cdots\!43 \nu^{19} + \cdots + 18\!\cdots\!52 ) / 10\!\cdots\!00 Copy content Toggle raw display
β6\beta_{6}== (−19 ⁣⋯ ⁣93ν19+⋯+20 ⁣⋯ ⁣36)/14 ⁣⋯ ⁣00 ( - 19\!\cdots\!93 \nu^{19} + \cdots + 20\!\cdots\!36 ) / 14\!\cdots\!00 Copy content Toggle raw display
β7\beta_{7}== (43 ⁣⋯ ⁣91ν19+⋯+85 ⁣⋯ ⁣40)/14 ⁣⋯ ⁣00 ( 43\!\cdots\!91 \nu^{19} + \cdots + 85\!\cdots\!40 ) / 14\!\cdots\!00 Copy content Toggle raw display
β8\beta_{8}== (92 ⁣⋯ ⁣99ν19+⋯−16 ⁣⋯ ⁣24)/10 ⁣⋯ ⁣00 ( 92\!\cdots\!99 \nu^{19} + \cdots - 16\!\cdots\!24 ) / 10\!\cdots\!00 Copy content Toggle raw display
β9\beta_{9}== (11 ⁣⋯ ⁣53ν19+⋯+76 ⁣⋯ ⁣28)/10 ⁣⋯ ⁣00 ( 11\!\cdots\!53 \nu^{19} + \cdots + 76\!\cdots\!28 ) / 10\!\cdots\!00 Copy content Toggle raw display
β10\beta_{10}== (21 ⁣⋯ ⁣94ν19+⋯+50 ⁣⋯ ⁣80)/14 ⁣⋯ ⁣00 ( 21\!\cdots\!94 \nu^{19} + \cdots + 50\!\cdots\!80 ) / 14\!\cdots\!00 Copy content Toggle raw display
β11\beta_{11}== (77 ⁣⋯ ⁣43ν19+⋯−12 ⁣⋯ ⁣36)/50 ⁣⋯ ⁣00 ( 77\!\cdots\!43 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 50\!\cdots\!00 Copy content Toggle raw display
β12\beta_{12}== (−57 ⁣⋯ ⁣69ν19+⋯+79 ⁣⋯ ⁣36)/25 ⁣⋯ ⁣00 ( - 57\!\cdots\!69 \nu^{19} + \cdots + 79\!\cdots\!36 ) / 25\!\cdots\!00 Copy content Toggle raw display
β13\beta_{13}== (−35 ⁣⋯ ⁣69ν19+⋯+45 ⁣⋯ ⁣16)/12 ⁣⋯ ⁣00 ( - 35\!\cdots\!69 \nu^{19} + \cdots + 45\!\cdots\!16 ) / 12\!\cdots\!00 Copy content Toggle raw display
β14\beta_{14}== (25 ⁣⋯ ⁣97ν19+⋯+62 ⁣⋯ ⁣64)/74 ⁣⋯ ⁣00 ( 25\!\cdots\!97 \nu^{19} + \cdots + 62\!\cdots\!64 ) / 74\!\cdots\!00 Copy content Toggle raw display
β15\beta_{15}== (56 ⁣⋯ ⁣95ν19+⋯−56 ⁣⋯ ⁣08)/10 ⁣⋯ ⁣00 ( 56\!\cdots\!95 \nu^{19} + \cdots - 56\!\cdots\!08 ) / 10\!\cdots\!00 Copy content Toggle raw display
β16\beta_{16}== (86 ⁣⋯ ⁣35ν19+⋯+20 ⁣⋯ ⁣32)/14 ⁣⋯ ⁣00 ( 86\!\cdots\!35 \nu^{19} + \cdots + 20\!\cdots\!32 ) / 14\!\cdots\!00 Copy content Toggle raw display
β17\beta_{17}== (81 ⁣⋯ ⁣39ν19+⋯−13 ⁣⋯ ⁣76)/10 ⁣⋯ ⁣00 ( 81\!\cdots\!39 \nu^{19} + \cdots - 13\!\cdots\!76 ) / 10\!\cdots\!00 Copy content Toggle raw display
β18\beta_{18}== (13 ⁣⋯ ⁣15ν19+⋯+41 ⁣⋯ ⁣76)/10 ⁣⋯ ⁣00 ( 13\!\cdots\!15 \nu^{19} + \cdots + 41\!\cdots\!76 ) / 10\!\cdots\!00 Copy content Toggle raw display
β19\beta_{19}== (14 ⁣⋯ ⁣87ν19+⋯+39 ⁣⋯ ⁣96)/10 ⁣⋯ ⁣00 ( 14\!\cdots\!87 \nu^{19} + \cdots + 39\!\cdots\!96 ) / 10\!\cdots\!00 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== −β3+β2−β1 -\beta_{3} + \beta_{2} - \beta_1 Copy content Toggle raw display
ν2\nu^{2}== 2β18+β17+β16+4β14+3β13+2β12+3β11+⋯−12060212 2 \beta_{18} + \beta_{17} + \beta_{16} + 4 \beta_{14} + 3 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \cdots - 12060212 Copy content Toggle raw display
ν3\nu^{3}== −81β19−81β18−17443β17+8550β16+15981β15+⋯−2218701337 - 81 \beta_{19} - 81 \beta_{18} - 17443 \beta_{17} + 8550 \beta_{16} + 15981 \beta_{15} + \cdots - 2218701337 Copy content Toggle raw display
ν4\nu^{4}== −3050809β19−61737913β18−33686228β17−2022431β16+⋯+249555470342379 - 3050809 \beta_{19} - 61737913 \beta_{18} - 33686228 \beta_{17} - 2022431 \beta_{16} + \cdots + 249555470342379 Copy content Toggle raw display
ν5\nu^{5}== 3293454927β19+5049468135β18+671098803882β17−335747076570β16+⋯+88 ⁣⋯ ⁣17 3293454927 \beta_{19} + 5049468135 \beta_{18} + 671098803882 \beta_{17} - 335747076570 \beta_{16} + \cdots + 88\!\cdots\!17 Copy content Toggle raw display
ν6\nu^{6}== 157215249582401β19+⋯−59 ⁣⋯ ⁣67 157215249582401 \beta_{19} + \cdots - 59\!\cdots\!67 Copy content Toggle raw display
ν7\nu^{7}== −11 ⁣⋯ ⁣81β19+⋯−29 ⁣⋯ ⁣29 - 11\!\cdots\!81 \beta_{19} + \cdots - 29\!\cdots\!29 Copy content Toggle raw display
ν8\nu^{8}== −60 ⁣⋯ ⁣45β19+⋯+15 ⁣⋯ ⁣83 - 60\!\cdots\!45 \beta_{19} + \cdots + 15\!\cdots\!83 Copy content Toggle raw display
ν9\nu^{9}== 40 ⁣⋯ ⁣15β19+⋯+87 ⁣⋯ ⁣69 40\!\cdots\!15 \beta_{19} + \cdots + 87\!\cdots\!69 Copy content Toggle raw display
ν10\nu^{10}== 21 ⁣⋯ ⁣17β19+⋯−41 ⁣⋯ ⁣55 21\!\cdots\!17 \beta_{19} + \cdots - 41\!\cdots\!55 Copy content Toggle raw display
ν11\nu^{11}== −13 ⁣⋯ ⁣61β19+⋯−23 ⁣⋯ ⁣61 - 13\!\cdots\!61 \beta_{19} + \cdots - 23\!\cdots\!61 Copy content Toggle raw display
ν12\nu^{12}== −70 ⁣⋯ ⁣45β19+⋯+11 ⁣⋯ ⁣71 - 70\!\cdots\!45 \beta_{19} + \cdots + 11\!\cdots\!71 Copy content Toggle raw display
ν13\nu^{13}== 47 ⁣⋯ ⁣99β19+⋯+56 ⁣⋯ ⁣25 47\!\cdots\!99 \beta_{19} + \cdots + 56\!\cdots\!25 Copy content Toggle raw display
ν14\nu^{14}== 23 ⁣⋯ ⁣45β19+⋯−34 ⁣⋯ ⁣11 23\!\cdots\!45 \beta_{19} + \cdots - 34\!\cdots\!11 Copy content Toggle raw display
ν15\nu^{15}== −16 ⁣⋯ ⁣41β19+⋯−10 ⁣⋯ ⁣65 - 16\!\cdots\!41 \beta_{19} + \cdots - 10\!\cdots\!65 Copy content Toggle raw display
ν16\nu^{16}== −74 ⁣⋯ ⁣89β19+⋯+10 ⁣⋯ ⁣99 - 74\!\cdots\!89 \beta_{19} + \cdots + 10\!\cdots\!99 Copy content Toggle raw display
ν17\nu^{17}== 54 ⁣⋯ ⁣03β19+⋯+11 ⁣⋯ ⁣69 54\!\cdots\!03 \beta_{19} + \cdots + 11\!\cdots\!69 Copy content Toggle raw display
ν18\nu^{18}== 23 ⁣⋯ ⁣65β19+⋯−31 ⁣⋯ ⁣19 23\!\cdots\!65 \beta_{19} + \cdots - 31\!\cdots\!19 Copy content Toggle raw display
ν19\nu^{19}== −18 ⁣⋯ ⁣69β19+⋯+33 ⁣⋯ ⁣15 - 18\!\cdots\!69 \beta_{19} + \cdots + 33\!\cdots\!15 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/46Z)×\left(\mathbb{Z}/46\mathbb{Z}\right)^\times.

nn 55
χ(n)\chi(n) −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}
45.1
350.088 + 2602.68i
350.088 − 2602.68i
121.392 + 3722.70i
121.392 − 3722.70i
−35.9811 + 5626.40i
−35.9811 − 5626.40i
−197.103 + 1180.32i
−197.103 − 1180.32i
−371.746 + 1225.89i
−371.746 − 1225.89i
458.105 + 4984.71i
458.105 − 4984.71i
203.515 + 1077.37i
203.515 − 1077.37i
84.1332 + 4281.32i
84.1332 − 4281.32i
−230.180 + 1828.23i
−230.180 − 1828.23i
−380.223 + 4304.29i
−380.223 − 4304.29i
−22.6274 −356.088 512.000 − 2602.68i 8057.35 − 4637.41i −11585.2 67749.5 58892.0i
45.2 −22.6274 −356.088 512.000 2602.68i 8057.35 4637.41i −11585.2 67749.5 − 58892.0i
45.3 −22.6274 −127.392 512.000 − 3722.70i 2882.54 − 27422.0i −11585.2 −42820.4 84235.2i
45.4 −22.6274 −127.392 512.000 3722.70i 2882.54 27422.0i −11585.2 −42820.4 − 84235.2i
45.5 −22.6274 29.9811 512.000 − 5626.40i −678.395 23833.7i −11585.2 −58150.1 127311.i
45.6 −22.6274 29.9811 512.000 5626.40i −678.395 − 23833.7i −11585.2 −58150.1 − 127311.i
45.7 −22.6274 191.103 512.000 − 1180.32i −4324.16 − 17666.5i −11585.2 −22528.8 26707.5i
45.8 −22.6274 191.103 512.000 1180.32i −4324.16 17666.5i −11585.2 −22528.8 − 26707.5i
45.9 −22.6274 365.746 512.000 − 1225.89i −8275.89 7881.79i −11585.2 74721.1 27738.6i
45.10 −22.6274 365.746 512.000 1225.89i −8275.89 − 7881.79i −11585.2 74721.1 − 27738.6i
45.11 22.6274 −464.105 512.000 − 4984.71i −10501.5 − 20466.0i 11585.2 156344. − 112791.i
45.12 22.6274 −464.105 512.000 4984.71i −10501.5 20466.0i 11585.2 156344. 112791.i
45.13 22.6274 −209.515 512.000 − 1077.37i −4740.79 6799.51i 11585.2 −15152.3 − 24378.1i
45.14 22.6274 −209.515 512.000 1077.37i −4740.79 − 6799.51i 11585.2 −15152.3 24378.1i
45.15 22.6274 −90.1332 512.000 − 4281.32i −2039.48 14153.3i 11585.2 −50925.0 − 96875.3i
45.16 22.6274 −90.1332 512.000 4281.32i −2039.48 − 14153.3i 11585.2 −50925.0 96875.3i
45.17 22.6274 224.180 512.000 − 1828.23i 5072.62 − 19992.6i 11585.2 −8792.19 − 41368.1i
45.18 22.6274 224.180 512.000 1828.23i 5072.62 19992.6i 11585.2 −8792.19 41368.1i
45.19 22.6274 374.223 512.000 − 4304.29i 8467.70 13952.6i 11585.2 80993.8 − 97395.0i
45.20 22.6274 374.223 512.000 4304.29i 8467.70 − 13952.6i 11585.2 80993.8 97395.0i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 46.11.b.a ✓ 20
23.b odd 2 1 inner 46.11.b.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.11.b.a ✓ 20 1.a even 1 1 trivial
46.11.b.a ✓ 20 23.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace S11new(46,[χ])S_{11}^{\mathrm{new}}(46, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2−512)10 (T^{2} - 512)^{10} Copy content Toggle raw display
33 (T10+⋯−69 ⁣⋯ ⁣76)2 (T^{10} + \cdots - 69\!\cdots\!76)^{2} Copy content Toggle raw display
55 T20+⋯+20 ⁣⋯ ⁣00 T^{20} + \cdots + 20\!\cdots\!00 Copy content Toggle raw display
77 T20+⋯+53 ⁣⋯ ⁣84 T^{20} + \cdots + 53\!\cdots\!84 Copy content Toggle raw display
1111 T20+⋯+13 ⁣⋯ ⁣84 T^{20} + \cdots + 13\!\cdots\!84 Copy content Toggle raw display
1313 (T10+⋯−55 ⁣⋯ ⁣12)2 (T^{10} + \cdots - 55\!\cdots\!12)^{2} Copy content Toggle raw display
1717 T20+⋯+39 ⁣⋯ ⁣00 T^{20} + \cdots + 39\!\cdots\!00 Copy content Toggle raw display
1919 T20+⋯+76 ⁣⋯ ⁣16 T^{20} + \cdots + 76\!\cdots\!16 Copy content Toggle raw display
2323 T20+⋯+14 ⁣⋯ ⁣01 T^{20} + \cdots + 14\!\cdots\!01 Copy content Toggle raw display
2929 (T10+⋯−50 ⁣⋯ ⁣36)2 (T^{10} + \cdots - 50\!\cdots\!36)^{2} Copy content Toggle raw display
3131 (T10+⋯−15 ⁣⋯ ⁣48)2 (T^{10} + \cdots - 15\!\cdots\!48)^{2} Copy content Toggle raw display
3737 T20+⋯+27 ⁣⋯ ⁣24 T^{20} + \cdots + 27\!\cdots\!24 Copy content Toggle raw display
4141 (T10+⋯+18 ⁣⋯ ⁣28)2 (T^{10} + \cdots + 18\!\cdots\!28)^{2} Copy content Toggle raw display
4343 T20+⋯+18 ⁣⋯ ⁣16 T^{20} + \cdots + 18\!\cdots\!16 Copy content Toggle raw display
4747 (T10+⋯−47 ⁣⋯ ⁣84)2 (T^{10} + \cdots - 47\!\cdots\!84)^{2} Copy content Toggle raw display
5353 T20+⋯+40 ⁣⋯ ⁣76 T^{20} + \cdots + 40\!\cdots\!76 Copy content Toggle raw display
5959 (T10+⋯+19 ⁣⋯ ⁣44)2 (T^{10} + \cdots + 19\!\cdots\!44)^{2} Copy content Toggle raw display
6161 T20+⋯+50 ⁣⋯ ⁣64 T^{20} + \cdots + 50\!\cdots\!64 Copy content Toggle raw display
6767 T20+⋯+70 ⁣⋯ ⁣84 T^{20} + \cdots + 70\!\cdots\!84 Copy content Toggle raw display
7171 (T10+⋯−82 ⁣⋯ ⁣52)2 (T^{10} + \cdots - 82\!\cdots\!52)^{2} Copy content Toggle raw display
7373 (T10+⋯−13 ⁣⋯ ⁣16)2 (T^{10} + \cdots - 13\!\cdots\!16)^{2} Copy content Toggle raw display
7979 T20+⋯+38 ⁣⋯ ⁣44 T^{20} + \cdots + 38\!\cdots\!44 Copy content Toggle raw display
8383 T20+⋯+53 ⁣⋯ ⁣00 T^{20} + \cdots + 53\!\cdots\!00 Copy content Toggle raw display
8989 T20+⋯+10 ⁣⋯ ⁣00 T^{20} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
9797 T20+⋯+23 ⁣⋯ ⁣24 T^{20} + \cdots + 23\!\cdots\!24 Copy content Toggle raw display
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