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Properties

Label 55.3.i.d
Level 5555
Weight 33
Character orbit 55.i
Analytic conductor 1.4991.499
Analytic rank 00
Dimension 1212
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,3,Mod(6,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.6");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 55=5⋅11 55 = 5 \cdot 11
Weight: k k == 3 3
Character orbit: [χ][\chi] == 55.i (of order 1010, degree 44, 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: 1.498641453981.49864145398
Analytic rank: 00
Dimension: 1212
Relative dimension: 33 over Q(ζ10)\Q(\zeta_{10})
Coefficient field: Q[x]/(x12+⋯ )\mathbb{Q}[x]/(x^{12} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x12+25x10+235x8+1025x6+2090x4+1880x2+605 x^{12} + 25x^{10} + 235x^{8} + 1025x^{6} + 2090x^{4} + 1880x^{2} + 605 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C10]\mathrm{SU}(2)[C_{10}]

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,…,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(−β10−β4−β3+⋯+1)q2+(β11−β10−β9+⋯−1)q3+(−β11−3β10+⋯+2)q4+(2β10−β3−2)q5+⋯+(32β11−2β10+⋯+76)q99+O(q100) q + ( - \beta_{10} - \beta_{4} - \beta_{3} + \cdots + 1) q^{2} + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots - 1) q^{3} + ( - \beta_{11} - 3 \beta_{10} + \cdots + 2) q^{4} + (2 \beta_{10} - \beta_{3} - 2) q^{5}+ \cdots + (32 \beta_{11} - 2 \beta_{10} + \cdots + 76) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+5q2−13q3+13q4−15q5+20q6−5q7−35q8+4q9+12q11−2q12+70q13−60q14+35q15−43q16−15q17−30q18−80q19+20q20+⋯+669q99+O(q100) 12 q + 5 q^{2} - 13 q^{3} + 13 q^{4} - 15 q^{5} + 20 q^{6} - 5 q^{7} - 35 q^{8} + 4 q^{9} + 12 q^{11} - 2 q^{12} + 70 q^{13} - 60 q^{14} + 35 q^{15} - 43 q^{16} - 15 q^{17} - 30 q^{18} - 80 q^{19} + 20 q^{20}+ \cdots + 669 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12+25x10+235x8+1025x6+2090x4+1880x2+605 x^{12} + 25x^{10} + 235x^{8} + 1025x^{6} + 2090x^{4} + 1880x^{2} + 605 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν11+33ν10−74ν9+616ν8−1613ν7+3344ν6−10536ν5+⋯−14069)/3058 ( \nu^{11} + 33 \nu^{10} - 74 \nu^{9} + 616 \nu^{8} - 1613 \nu^{7} + 3344 \nu^{6} - 10536 \nu^{5} + \cdots - 14069 ) / 3058 Copy content Toggle raw display
β3\beta_{3}== (−ν11+77ν10+74ν9+1947ν8+1613ν7+17996ν6+10536ν5+⋯+57893)/3058 ( - \nu^{11} + 77 \nu^{10} + 74 \nu^{9} + 1947 \nu^{8} + 1613 \nu^{7} + 17996 \nu^{6} + 10536 \nu^{5} + \cdots + 57893 ) / 3058 Copy content Toggle raw display
β4\beta_{4}== (−ν11−77ν10+74ν9−1947ν8+1613ν7−17996ν6+10536ν5+⋯−57893)/3058 ( - \nu^{11} - 77 \nu^{10} + 74 \nu^{9} - 1947 \nu^{8} + 1613 \nu^{7} - 17996 \nu^{6} + 10536 \nu^{5} + \cdots - 57893 ) / 3058 Copy content Toggle raw display
β5\beta_{5}== (48ν11+77ν10+1035ν9+1947ν8+8200ν7+17996ν6+29422ν5+⋯+76241)/3058 ( 48 \nu^{11} + 77 \nu^{10} + 1035 \nu^{9} + 1947 \nu^{8} + 8200 \nu^{7} + 17996 \nu^{6} + 29422 \nu^{5} + \cdots + 76241 ) / 3058 Copy content Toggle raw display
β6\beta_{6}== (48ν11−77ν10+1035ν9−1947ν8+8200ν7−17996ν6+29422ν5+⋯−76241)/3058 ( 48 \nu^{11} - 77 \nu^{10} + 1035 \nu^{9} - 1947 \nu^{8} + 8200 \nu^{7} - 17996 \nu^{6} + 29422 \nu^{5} + \cdots - 76241 ) / 3058 Copy content Toggle raw display
β7\beta_{7}== (7ν11+9ν10+177ν9+168ν8+1636ν7+1051ν6+6590ν5+⋯+55)/278 ( 7 \nu^{11} + 9 \nu^{10} + 177 \nu^{9} + 168 \nu^{8} + 1636 \nu^{7} + 1051 \nu^{6} + 6590 \nu^{5} + \cdots + 55 ) / 278 Copy content Toggle raw display
β8\beta_{8}== (7ν11−9ν10+177ν9−168ν8+1636ν7−1051ν6+6590ν5+⋯−55)/278 ( 7 \nu^{11} - 9 \nu^{10} + 177 \nu^{9} - 168 \nu^{8} + 1636 \nu^{7} - 1051 \nu^{6} + 6590 \nu^{5} + \cdots - 55 ) / 278 Copy content Toggle raw display
β9\beta_{9}== (50ν10+1165ν8+9978ν6+37957ν4+60559ν2+139ν+30484)/278 ( 50\nu^{10} + 1165\nu^{8} + 9978\nu^{6} + 37957\nu^{4} + 60559\nu^{2} + 139\nu + 30484 ) / 278 Copy content Toggle raw display
β10\beta_{10}== (−125ν11+77ν10−2982ν9+1947ν8−26196ν7+17996ν6+⋯+59422)/3058 ( - 125 \nu^{11} + 77 \nu^{10} - 2982 \nu^{9} + 1947 \nu^{8} - 26196 \nu^{7} + 17996 \nu^{6} + \cdots + 59422 ) / 3058 Copy content Toggle raw display
β11\beta_{11}== (483ν11+330ν10+11657ν9+7689ν8+103154ν7+65549ν6+⋯+189574)/3058 ( 483 \nu^{11} + 330 \nu^{10} + 11657 \nu^{9} + 7689 \nu^{8} + 103154 \nu^{7} + 65549 \nu^{6} + \cdots + 189574 ) / 3058 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}== −β9−β8+β7−β6+β5−2β4+β3−β2+β1−2 -\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 2 Copy content Toggle raw display
ν3\nu^{3}== 2β10+β8+β7+β6+β5+β4−β3−6β1−1 2\beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 6\beta _1 - 1 Copy content Toggle raw display
ν4\nu^{4}== 9β9+9β8−9β7+8β6−8β5+19β4−10β3+9β2−9β1+6 9\beta_{9} + 9\beta_{8} - 9\beta_{7} + 8\beta_{6} - 8\beta_{5} + 19\beta_{4} - 10\beta_{3} + 9\beta_{2} - 9\beta _1 + 6 Copy content Toggle raw display
ν5\nu^{5}== 2β11−12β10−β9−11β8−11β7−8β6−8β5+⋯+6 2 \beta_{11} - 12 \beta_{10} - \beta_{9} - 11 \beta_{8} - 11 \beta_{7} - 8 \beta_{6} - 8 \beta_{5} + \cdots + 6 Copy content Toggle raw display
ν6\nu^{6}== −72β9−73β8+73β7−59β6+59β5−160β4+82β3+⋯−16 - 72 \beta_{9} - 73 \beta_{8} + 73 \beta_{7} - 59 \beta_{6} + 59 \beta_{5} - 160 \beta_{4} + 82 \beta_{3} + \cdots - 16 Copy content Toggle raw display
ν7\nu^{7}== −32β11+40β10+16β9+97β8+97β7+57β6+57β5+⋯−20 - 32 \beta_{11} + 40 \beta_{10} + 16 \beta_{9} + 97 \beta_{8} + 97 \beta_{7} + 57 \beta_{6} + 57 \beta_{5} + \cdots - 20 Copy content Toggle raw display
ν8\nu^{8}== 566β9+586β8−586β7+437β6−437β5+1311β4+⋯−34 566 \beta_{9} + 586 \beta_{8} - 586 \beta_{7} + 437 \beta_{6} - 437 \beta_{5} + 1311 \beta_{4} + \cdots - 34 Copy content Toggle raw display
ν9\nu^{9}== 364β11+106β10−182β9−801β8−801β7−404β6+⋯−53 364 \beta_{11} + 106 \beta_{10} - 182 \beta_{9} - 801 \beta_{8} - 801 \beta_{7} - 404 \beta_{6} + \cdots - 53 Copy content Toggle raw display
ν10\nu^{10}== −4435β9−4707β8+4707β7−3270β6+3270β5−10618β4+⋯+1243 - 4435 \beta_{9} - 4707 \beta_{8} + 4707 \beta_{7} - 3270 \beta_{6} + 3270 \beta_{5} - 10618 \beta_{4} + \cdots + 1243 Copy content Toggle raw display
ν11\nu^{11}== −3608β11−3776β10+1804β9+6437β8+6437β7+2903β6+⋯+1888 - 3608 \beta_{11} - 3776 \beta_{10} + 1804 \beta_{9} + 6437 \beta_{8} + 6437 \beta_{7} + 2903 \beta_{6} + \cdots + 1888 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/55Z)×\left(\mathbb{Z}/55\mathbb{Z}\right)^\times.

nn 1212 4646
χ(n)\chi(n) 11 1+β3−β4−β101 + \beta_{3} - \beta_{4} - \beta_{10}

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}
6.1
1.48398i
− 0.914937i
− 2.47115i
− 2.81976i
− 0.987790i
2.63198i
− 1.48398i
0.914937i
2.47115i
2.81976i
0.987790i
− 2.63198i
−3.40164 + 1.10526i −3.07249 − 2.23230i 7.11349 − 5.16825i −0.690983 + 2.12663i 12.9188 + 4.19757i 4.78108 + 6.58059i −10.0760 + 13.8684i 1.67591 + 5.15793i − 7.99774i
6.2 0.289909 − 0.0941972i −4.17687 − 3.03467i −3.16089 + 2.29652i −0.690983 + 2.12663i −1.49677 − 0.486330i −5.94293 − 8.17974i −1.41674 + 1.94998i 5.45583 + 16.7913i 0.681617i
6.3 2.68468 − 0.872305i 0.0862408 + 0.0626576i 3.21052 − 2.33258i −0.690983 + 2.12663i 0.286186 + 0.0929874i −1.76520 − 2.42959i −0.0523905 + 0.0721094i −2.77764 − 8.54870i 6.31206i
41.1 0.0936958 + 0.128961i −0.602908 − 1.85556i 1.22822 − 3.78006i −1.80902 − 1.31433i 0.182805 − 0.251610i 0.0125693 + 0.00408401i 1.20897 − 0.392819i 4.20154 − 3.05260i − 0.356440i
41.2 0.759198 + 1.04495i 1.55259 + 4.77837i 0.720536 − 2.21758i −1.80902 − 1.31433i −3.81442 + 5.25010i −3.31915 − 1.07846i 7.77792 − 2.52720i −13.1411 + 9.54757i − 2.88816i
41.3 2.07416 + 2.85483i −0.286558 − 0.881935i −2.61187 + 8.03851i −1.80902 − 1.31433i 1.92341 − 2.64735i 3.73363 + 1.21313i −14.9418 + 4.85489i 6.58546 − 4.78462i − 7.89056i
46.1 −3.40164 − 1.10526i −3.07249 + 2.23230i 7.11349 + 5.16825i −0.690983 − 2.12663i 12.9188 − 4.19757i 4.78108 − 6.58059i −10.0760 − 13.8684i 1.67591 − 5.15793i 7.99774i
46.2 0.289909 + 0.0941972i −4.17687 + 3.03467i −3.16089 − 2.29652i −0.690983 − 2.12663i −1.49677 + 0.486330i −5.94293 + 8.17974i −1.41674 − 1.94998i 5.45583 − 16.7913i − 0.681617i
46.3 2.68468 + 0.872305i 0.0862408 − 0.0626576i 3.21052 + 2.33258i −0.690983 − 2.12663i 0.286186 − 0.0929874i −1.76520 + 2.42959i −0.0523905 − 0.0721094i −2.77764 + 8.54870i − 6.31206i
51.1 0.0936958 − 0.128961i −0.602908 + 1.85556i 1.22822 + 3.78006i −1.80902 + 1.31433i 0.182805 + 0.251610i 0.0125693 − 0.00408401i 1.20897 + 0.392819i 4.20154 + 3.05260i 0.356440i
51.2 0.759198 − 1.04495i 1.55259 − 4.77837i 0.720536 + 2.21758i −1.80902 + 1.31433i −3.81442 − 5.25010i −3.31915 + 1.07846i 7.77792 + 2.52720i −13.1411 − 9.54757i 2.88816i
51.3 2.07416 − 2.85483i −0.286558 + 0.881935i −2.61187 − 8.03851i −1.80902 + 1.31433i 1.92341 + 2.64735i 3.73363 − 1.21313i −14.9418 − 4.85489i 6.58546 + 4.78462i 7.89056i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.3.i.d ✓ 12
5.b even 2 1 275.3.x.f 12
5.c odd 4 2 275.3.q.f 24
11.c even 5 1 605.3.c.d 12
11.d odd 10 1 inner 55.3.i.d ✓ 12
11.d odd 10 1 605.3.c.d 12
55.h odd 10 1 275.3.x.f 12
55.l even 20 2 275.3.q.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.3.i.d ✓ 12 1.a even 1 1 trivial
55.3.i.d ✓ 12 11.d odd 10 1 inner
275.3.q.f 24 5.c odd 4 2
275.3.q.f 24 55.l even 20 2
275.3.x.f 12 5.b even 2 1
275.3.x.f 12 55.h odd 10 1
605.3.c.d 12 11.c even 5 1
605.3.c.d 12 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T212−5T211+80T29−215T28−300T27+2425T26−4660T25+⋯+5 T_{2}^{12} - 5 T_{2}^{11} + 80 T_{2}^{9} - 215 T_{2}^{8} - 300 T_{2}^{7} + 2425 T_{2}^{6} - 4660 T_{2}^{5} + \cdots + 5 acting on S3new(55,[χ])S_{3}^{\mathrm{new}}(55, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12−5T11+⋯+5 T^{12} - 5 T^{11} + \cdots + 5 Copy content Toggle raw display
33 T12+13T11+⋯+361 T^{12} + 13 T^{11} + \cdots + 361 Copy content Toggle raw display
55 (T4+5T3+15T2+⋯+25)3 (T^{4} + 5 T^{3} + 15 T^{2} + \cdots + 25)^{3} Copy content Toggle raw display
77 T12+5T11+⋯+2000 T^{12} + 5 T^{11} + \cdots + 2000 Copy content Toggle raw display
1111 T12+⋯+3138428376721 T^{12} + \cdots + 3138428376721 Copy content Toggle raw display
1313 T12+⋯+2507904080 T^{12} + \cdots + 2507904080 Copy content Toggle raw display
1717 T12+⋯+605990405 T^{12} + \cdots + 605990405 Copy content Toggle raw display
1919 T12+⋯+373311161071805 T^{12} + \cdots + 373311161071805 Copy content Toggle raw display
2323 (T6−21T5+⋯+8843956)2 (T^{6} - 21 T^{5} + \cdots + 8843956)^{2} Copy content Toggle raw display
2929 T12+⋯+13 ⁣⋯ ⁣80 T^{12} + \cdots + 13\!\cdots\!80 Copy content Toggle raw display
3131 T12+⋯+4020067216 T^{12} + \cdots + 4020067216 Copy content Toggle raw display
3737 T12+⋯+10 ⁣⋯ ⁣76 T^{12} + \cdots + 10\!\cdots\!76 Copy content Toggle raw display
4141 T12+⋯+25 ⁣⋯ ⁣05 T^{12} + \cdots + 25\!\cdots\!05 Copy content Toggle raw display
4343 T12+⋯+10 ⁣⋯ ⁣25 T^{12} + \cdots + 10\!\cdots\!25 Copy content Toggle raw display
4747 T12+⋯+32774983803136 T^{12} + \cdots + 32774983803136 Copy content Toggle raw display
5353 T12+⋯+17 ⁣⋯ ⁣96 T^{12} + \cdots + 17\!\cdots\!96 Copy content Toggle raw display
5959 T12+⋯+36 ⁣⋯ ⁣01 T^{12} + \cdots + 36\!\cdots\!01 Copy content Toggle raw display
6161 T12+⋯+33 ⁣⋯ ⁣80 T^{12} + \cdots + 33\!\cdots\!80 Copy content Toggle raw display
6767 (T6+106T5+⋯−31173869579)2 (T^{6} + 106 T^{5} + \cdots - 31173869579)^{2} Copy content Toggle raw display
7171 T12+⋯+10 ⁣⋯ ⁣96 T^{12} + \cdots + 10\!\cdots\!96 Copy content Toggle raw display
7373 T12+⋯+29 ⁣⋯ ⁣05 T^{12} + \cdots + 29\!\cdots\!05 Copy content Toggle raw display
7979 T12+⋯+12 ⁣⋯ ⁣80 T^{12} + \cdots + 12\!\cdots\!80 Copy content Toggle raw display
8383 T12+⋯+20 ⁣⋯ ⁣05 T^{12} + \cdots + 20\!\cdots\!05 Copy content Toggle raw display
8989 (T6+57T5+⋯−494725464851)2 (T^{6} + 57 T^{5} + \cdots - 494725464851)^{2} Copy content Toggle raw display
9797 T12+⋯+24 ⁣⋯ ⁣41 T^{12} + \cdots + 24\!\cdots\!41 Copy content Toggle raw display
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