Properties

Label 39.6.j.b
Level 3939
Weight 66
Character orbit 39.j
Analytic conductor 6.2556.255
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] = [39,6,Mod(4,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 39=313 39 = 3 \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 39.j (of order 66, 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: 6.254968972716.25496897271
Analytic rank: 00
Dimension: 1212
Relative dimension: 66 over Q(ζ6)\Q(\zeta_{6})
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+310x10+35389x8+1834428x6+43601364x4+433521504x2+1332542016 x^{12} + 310x^{10} + 35389x^{8} + 1834428x^{6} + 43601364x^{4} + 433521504x^{2} + 1332542016 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 33132 3^{3}\cdot 13^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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+(β2+β1)q2+9β3q3+(β520β3+20)q4+(β7β6+20β310)q59β2q6+(β11+β5+β4++23)q7++(81β11+81β10+1782)q99+O(q100) q + ( - \beta_{2} + \beta_1) q^{2} + 9 \beta_{3} q^{3} + ( - \beta_{5} - 20 \beta_{3} + 20) q^{4} + ( - \beta_{7} - \beta_{6} + 20 \beta_{3} - 10) q^{5} - 9 \beta_{2} q^{6} + ( - \beta_{11} + \beta_{5} + \beta_{4} + \cdots + 23) q^{7}+ \cdots + (81 \beta_{11} + 81 \beta_{10} + \cdots - 1782) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q+54q3+118q4+414q7486q956q10+360q11+2124q12+986q133204q141620q151706q16+528q175880q19+6678q20+322q22+6216q23+402804q98+O(q100) 12 q + 54 q^{3} + 118 q^{4} + 414 q^{7} - 486 q^{9} - 56 q^{10} + 360 q^{11} + 2124 q^{12} + 986 q^{13} - 3204 q^{14} - 1620 q^{15} - 1706 q^{16} + 528 q^{17} - 5880 q^{19} + 6678 q^{20} + 322 q^{22} + 6216 q^{23}+ \cdots - 402804 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12+310x10+35389x8+1834428x6+43601364x4+433521504x2+1332542016 x^{12} + 310x^{10} + 35389x^{8} + 1834428x^{6} + 43601364x^{4} + 433521504x^{2} + 1332542016 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν6+155ν4+5682ν2+2496ν+36504)/4992 ( \nu^{6} + 155\nu^{4} + 5682\nu^{2} + 2496\nu + 36504 ) / 4992 Copy content Toggle raw display
β3\beta_{3}== (ν11+310ν9+35389ν7+1797924ν5+37943244ν3+226105776ν+91113984)/182227968 ( \nu^{11} + 310\nu^{9} + 35389\nu^{7} + 1797924\nu^{5} + 37943244\nu^{3} + 226105776\nu + 91113984 ) / 182227968 Copy content Toggle raw display
β4\beta_{4}== ν252 -\nu^{2} - 52 Copy content Toggle raw display
β5\beta_{5}== (ν11310ν934687ν71689114ν533954480ν3+1752192ν2++91113984)/3504384 ( - \nu^{11} - 310 \nu^{9} - 34687 \nu^{7} - 1689114 \nu^{5} - 33954480 \nu^{3} + 1752192 \nu^{2} + \cdots + 91113984 ) / 3504384 Copy content Toggle raw display
β6\beta_{6}== (181ν11+2028ν10+54082ν9+555672ν8+5715889ν7++93107102400)/668169216 ( 181 \nu^{11} + 2028 \nu^{10} + 54082 \nu^{9} + 555672 \nu^{8} + 5715889 \nu^{7} + \cdots + 93107102400 ) / 668169216 Copy content Toggle raw display
β7\beta_{7}== (181ν112028ν10+54082ν9555672ν8+5715889ν7+93107102400)/668169216 ( 181 \nu^{11} - 2028 \nu^{10} + 54082 \nu^{9} - 555672 \nu^{8} + 5715889 \nu^{7} + \cdots - 93107102400 ) / 668169216 Copy content Toggle raw display
β8\beta_{8}== (604ν11+1521ν10+172030ν9+416754ν8+17408188ν7++345324846672)/1002253824 ( 604 \nu^{11} + 1521 \nu^{10} + 172030 \nu^{9} + 416754 \nu^{8} + 17408188 \nu^{7} + \cdots + 345324846672 ) / 1002253824 Copy content Toggle raw display
β9\beta_{9}== (604ν11+1521ν10172030ν9+416754ν817408188ν7++346327100496)/1002253824 ( - 604 \nu^{11} + 1521 \nu^{10} - 172030 \nu^{9} + 416754 \nu^{8} - 17408188 \nu^{7} + \cdots + 346327100496 ) / 1002253824 Copy content Toggle raw display
β10\beta_{10}== (1138ν117605ν10328444ν92284542ν833403846ν7+796094198640)/1002253824 ( - 1138 \nu^{11} - 7605 \nu^{10} - 328444 \nu^{9} - 2284542 \nu^{8} - 33403846 \nu^{7} + \cdots - 796094198640 ) / 1002253824 Copy content Toggle raw display
β11\beta_{11}== (1138ν11+7605ν10328444ν9+2284542ν833403846ν7++796094198640)/1002253824 ( - 1138 \nu^{11} + 7605 \nu^{10} - 328444 \nu^{9} + 2284542 \nu^{8} - 33403846 \nu^{7} + \cdots + 796094198640 ) / 1002253824 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β452 -\beta_{4} - 52 Copy content Toggle raw display
ν3\nu^{3}== β11+β10β9+β8+3β7+3β6+2β5+β4+2β381β1 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 3\beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} + 2\beta_{3} - 81\beta_1 Copy content Toggle raw display
ν4\nu^{4}== 4β9+4β8+2β72β6+113β480β2+40β1+4258 4\beta_{9} + 4\beta_{8} + 2\beta_{7} - 2\beta_{6} + 113\beta_{4} - 80\beta_{2} + 40\beta _1 + 4258 Copy content Toggle raw display
ν5\nu^{5}== 119β11119β10+107β9107β8417β7417β6++2112 - 119 \beta_{11} - 119 \beta_{10} + 107 \beta_{9} - 107 \beta_{8} - 417 \beta_{7} - 417 \beta_{6} + \cdots + 2112 Copy content Toggle raw display
ν6\nu^{6}== 620β9620β8310β7+310β611833β4+17392β2+401030 - 620 \beta_{9} - 620 \beta_{8} - 310 \beta_{7} + 310 \beta_{6} - 11833 \beta_{4} + 17392 \beta_{2} + \cdots - 401030 Copy content Toggle raw display
ν7\nu^{7}== 12763β11+12763β1010903β9+10903β8+47589β7+457152 12763 \beta_{11} + 12763 \beta_{10} - 10903 \beta_{9} + 10903 \beta_{8} + 47589 \beta_{7} + \cdots - 457152 Copy content Toggle raw display
ν8\nu^{8}== 2496β112496β10+75868β9+75868β8+44174β7++39868894 2496 \beta_{11} - 2496 \beta_{10} + 75868 \beta_{9} + 75868 \beta_{8} + 44174 \beta_{7} + \cdots + 39868894 Copy content Toggle raw display
ν9\nu^{9}== 1338611β111338611β10+1148447β91148447β85131389β7++69466176 - 1338611 \beta_{11} - 1338611 \beta_{10} + 1148447 \beta_{9} - 1148447 \beta_{8} - 5131389 \beta_{7} + \cdots + 69466176 Copy content Toggle raw display
ν10\nu^{10}== 683904β11+683904β108649788β98649788β86199390β7+4072808942 - 683904 \beta_{11} + 683904 \beta_{10} - 8649788 \beta_{9} - 8649788 \beta_{8} - 6199390 \beta_{7} + \cdots - 4072808942 Copy content Toggle raw display
ν11\nu^{11}== 139309315β11+139309315β10124606927β9+124606927β8+9244691904 139309315 \beta_{11} + 139309315 \beta_{10} - 124606927 \beta_{9} + 124606927 \beta_{8} + \cdots - 9244691904 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/39Z)×\left(\mathbb{Z}/39\mathbb{Z}\right)^\times.

nn 1414 2828
χ(n)\chi(n) 11 1β31 - \beta_{3}

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
9.68706i
7.60446i
2.32967i
3.82029i
5.30461i
10.4963i
9.68706i
7.60446i
2.32967i
3.82029i
5.30461i
10.4963i
−8.38924 4.84353i 4.50000 7.79423i 30.9196 + 53.5543i 92.4599i −75.5032 + 43.5918i 209.382 120.887i 289.054i −40.5000 70.1481i −447.832 + 775.669i
4.2 −6.58566 3.80223i 4.50000 7.79423i 12.9139 + 22.3675i 60.8553i −59.2709 + 34.2201i −28.1411 + 16.2473i 46.9362i −40.5000 70.1481i 231.386 400.772i
4.3 −2.01755 1.16483i 4.50000 7.79423i −13.2863 23.0126i 3.62321i −18.1580 + 10.4835i −67.5002 + 38.9713i 136.455i −40.5000 70.1481i −4.22043 + 7.31001i
4.4 3.30847 + 1.91015i 4.50000 7.79423i −8.70269 15.0735i 26.1960i 29.7762 17.1913i 210.299 121.416i 188.743i −40.5000 70.1481i −50.0383 + 86.6688i
4.5 4.59393 + 2.65231i 4.50000 7.79423i −1.93054 3.34380i 98.3488i 41.3454 23.8708i −133.821 + 77.2615i 190.229i −40.5000 70.1481i 260.851 451.808i
4.6 9.09005 + 5.24814i 4.50000 7.79423i 39.0860 + 67.6990i 3.45756i 81.8105 47.2333i 16.7808 9.68841i 484.636i −40.5000 70.1481i −18.1458 + 31.4294i
10.1 −8.38924 + 4.84353i 4.50000 + 7.79423i 30.9196 53.5543i 92.4599i −75.5032 43.5918i 209.382 + 120.887i 289.054i −40.5000 + 70.1481i −447.832 775.669i
10.2 −6.58566 + 3.80223i 4.50000 + 7.79423i 12.9139 22.3675i 60.8553i −59.2709 34.2201i −28.1411 16.2473i 46.9362i −40.5000 + 70.1481i 231.386 + 400.772i
10.3 −2.01755 + 1.16483i 4.50000 + 7.79423i −13.2863 + 23.0126i 3.62321i −18.1580 10.4835i −67.5002 38.9713i 136.455i −40.5000 + 70.1481i −4.22043 7.31001i
10.4 3.30847 1.91015i 4.50000 + 7.79423i −8.70269 + 15.0735i 26.1960i 29.7762 + 17.1913i 210.299 + 121.416i 188.743i −40.5000 + 70.1481i −50.0383 86.6688i
10.5 4.59393 2.65231i 4.50000 + 7.79423i −1.93054 + 3.34380i 98.3488i 41.3454 + 23.8708i −133.821 77.2615i 190.229i −40.5000 + 70.1481i 260.851 + 451.808i
10.6 9.09005 5.24814i 4.50000 + 7.79423i 39.0860 67.6990i 3.45756i 81.8105 + 47.2333i 16.7808 + 9.68841i 484.636i −40.5000 + 70.1481i −18.1458 31.4294i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.6.j.b 12
3.b odd 2 1 117.6.q.d 12
13.e even 6 1 inner 39.6.j.b 12
13.f odd 12 2 507.6.a.m 12
39.h odd 6 1 117.6.q.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.6.j.b 12 1.a even 1 1 trivial
39.6.j.b 12 13.e even 6 1 inner
117.6.q.d 12 3.b odd 2 1
117.6.q.d 12 39.h odd 6 1
507.6.a.m 12 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T212155T210+18343T28+7488T27807702T26+26627004T24++1332542016 T_{2}^{12} - 155 T_{2}^{10} + 18343 T_{2}^{8} + 7488 T_{2}^{7} - 807702 T_{2}^{6} + 26627004 T_{2}^{4} + \cdots + 1332542016 acting on S6new(39,[χ])S_{6}^{\mathrm{new}}(39, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12++1332542016 T^{12} + \cdots + 1332542016 Copy content Toggle raw display
33 (T29T+81)6 (T^{2} - 9 T + 81)^{6} Copy content Toggle raw display
55 T12++32 ⁣ ⁣56 T^{12} + \cdots + 32\!\cdots\!56 Copy content Toggle raw display
77 T12++19 ⁣ ⁣56 T^{12} + \cdots + 19\!\cdots\!56 Copy content Toggle raw display
1111 T12++15 ⁣ ⁣84 T^{12} + \cdots + 15\!\cdots\!84 Copy content Toggle raw display
1313 T12++26 ⁣ ⁣49 T^{12} + \cdots + 26\!\cdots\!49 Copy content Toggle raw display
1717 T12++30 ⁣ ⁣56 T^{12} + \cdots + 30\!\cdots\!56 Copy content Toggle raw display
1919 T12++47 ⁣ ⁣04 T^{12} + \cdots + 47\!\cdots\!04 Copy content Toggle raw display
2323 T12++10 ⁣ ⁣84 T^{12} + \cdots + 10\!\cdots\!84 Copy content Toggle raw display
2929 T12++62 ⁣ ⁣44 T^{12} + \cdots + 62\!\cdots\!44 Copy content Toggle raw display
3131 T12++75 ⁣ ⁣84 T^{12} + \cdots + 75\!\cdots\!84 Copy content Toggle raw display
3737 T12++96 ⁣ ⁣76 T^{12} + \cdots + 96\!\cdots\!76 Copy content Toggle raw display
4141 T12++62 ⁣ ⁣64 T^{12} + \cdots + 62\!\cdots\!64 Copy content Toggle raw display
4343 T12++36 ⁣ ⁣24 T^{12} + \cdots + 36\!\cdots\!24 Copy content Toggle raw display
4747 T12++18 ⁣ ⁣36 T^{12} + \cdots + 18\!\cdots\!36 Copy content Toggle raw display
5353 (T6++31 ⁣ ⁣72)2 (T^{6} + \cdots + 31\!\cdots\!72)^{2} Copy content Toggle raw display
5959 T12++56 ⁣ ⁣96 T^{12} + \cdots + 56\!\cdots\!96 Copy content Toggle raw display
6161 T12++22 ⁣ ⁣49 T^{12} + \cdots + 22\!\cdots\!49 Copy content Toggle raw display
6767 T12++20 ⁣ ⁣56 T^{12} + \cdots + 20\!\cdots\!56 Copy content Toggle raw display
7171 T12++19 ⁣ ⁣04 T^{12} + \cdots + 19\!\cdots\!04 Copy content Toggle raw display
7373 T12++12 ⁣ ⁣89 T^{12} + \cdots + 12\!\cdots\!89 Copy content Toggle raw display
7979 (T6++42 ⁣ ⁣36)2 (T^{6} + \cdots + 42\!\cdots\!36)^{2} Copy content Toggle raw display
8383 T12++81 ⁣ ⁣24 T^{12} + \cdots + 81\!\cdots\!24 Copy content Toggle raw display
8989 T12++10 ⁣ ⁣44 T^{12} + \cdots + 10\!\cdots\!44 Copy content Toggle raw display
9797 T12++12 ⁣ ⁣96 T^{12} + \cdots + 12\!\cdots\!96 Copy content Toggle raw display
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