Properties

Label 5994.2.a.ba
Level 59945994
Weight 22
Character orbit 5994.a
Self dual yes
Analytic conductor 47.86247.862
Analytic rank 00
Dimension 1010
CM no
Inner twists 11

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5994,2,Mod(1,5994)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5994, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5994.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 5994=23437 5994 = 2 \cdot 3^{4} \cdot 37
Weight: k k == 2 2
Character orbit: [χ][\chi] == 5994.a (trivial)

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: yes
Analytic conductor: 47.862330971647.8623309716
Analytic rank: 00
Dimension: 1010
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: x102x926x8+49x7+236x6420x5860x4+1461x3+993x21638x+99 x^{10} - 2x^{9} - 26x^{8} + 49x^{7} + 236x^{6} - 420x^{5} - 860x^{4} + 1461x^{3} + 993x^{2} - 1638x + 99 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 32 3^{2}
Twist minimal: no (minimal twist has level 666)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(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,,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) == qq2+q4+β6q5β9q7q8β6q10+(β5+β2)q11+(β9β6β4+1)q13+β9q14+q16+(β7+β4+β3+1)q17++(2β9+β6+2β4+3)q98+O(q100) q - q^{2} + q^{4} + \beta_{6} q^{5} - \beta_{9} q^{7} - q^{8} - \beta_{6} q^{10} + (\beta_{5} + \beta_{2}) q^{11} + ( - \beta_{9} - \beta_{6} - \beta_{4} + 1) q^{13} + \beta_{9} q^{14} + q^{16} + (\beta_{7} + \beta_{4} + \beta_{3} + \cdots - 1) q^{17}+ \cdots + (2 \beta_{9} + \beta_{6} + 2 \beta_{4} + \cdots - 3) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q10q2+10q4q5+q710q8+q103q11+12q13q14+10q1612q17+24q19q20+3q223q23+21q2512q26+q284q29+35q98+O(q100) 10 q - 10 q^{2} + 10 q^{4} - q^{5} + q^{7} - 10 q^{8} + q^{10} - 3 q^{11} + 12 q^{13} - q^{14} + 10 q^{16} - 12 q^{17} + 24 q^{19} - q^{20} + 3 q^{22} - 3 q^{23} + 21 q^{25} - 12 q^{26} + q^{28} - 4 q^{29}+ \cdots - 35 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x102x926x8+49x7+236x6420x5860x4+1461x3+993x21638x+99 x^{10} - 2x^{9} - 26x^{8} + 49x^{7} + 236x^{6} - 420x^{5} - 860x^{4} + 1461x^{3} + 993x^{2} - 1638x + 99 : Copy content Toggle raw display

β1\beta_{1}== (12122ν9+3878ν8319687ν7111802ν6+2840680ν5+934458ν4+997761)/272073 ( 12122 \nu^{9} + 3878 \nu^{8} - 319687 \nu^{7} - 111802 \nu^{6} + 2840680 \nu^{5} + 934458 \nu^{4} + \cdots - 997761 ) / 272073 Copy content Toggle raw display
β2\beta_{2}== (26903ν94417ν8+686777ν7+167612ν65914326ν51567558ν4++163557)/272073 ( - 26903 \nu^{9} - 4417 \nu^{8} + 686777 \nu^{7} + 167612 \nu^{6} - 5914326 \nu^{5} - 1567558 \nu^{4} + \cdots + 163557 ) / 272073 Copy content Toggle raw display
β3\beta_{3}== (41434ν9836ν8+1069090ν7+114325ν69388827ν5++2748360)/272073 ( - 41434 \nu^{9} - 836 \nu^{8} + 1069090 \nu^{7} + 114325 \nu^{6} - 9388827 \nu^{5} + \cdots + 2748360 ) / 272073 Copy content Toggle raw display
β4\beta_{4}== (47656ν9+1827ν8+1222032ν7+118233ν610625383ν5++3520176)/272073 ( - 47656 \nu^{9} + 1827 \nu^{8} + 1222032 \nu^{7} + 118233 \nu^{6} - 10625383 \nu^{5} + \cdots + 3520176 ) / 272073 Copy content Toggle raw display
β5\beta_{5}== (60657ν95654ν8+1574170ν7+307810ν613868663ν5++3609258)/272073 ( - 60657 \nu^{9} - 5654 \nu^{8} + 1574170 \nu^{7} + 307810 \nu^{6} - 13868663 \nu^{5} + \cdots + 3609258 ) / 272073 Copy content Toggle raw display
β6\beta_{6}== (21496ν92373ν8+554947ν7+118506ν64849246ν51242004ν4++672259)/90691 ( - 21496 \nu^{9} - 2373 \nu^{8} + 554947 \nu^{7} + 118506 \nu^{6} - 4849246 \nu^{5} - 1242004 \nu^{4} + \cdots + 672259 ) / 90691 Copy content Toggle raw display
β7\beta_{7}== (113902ν912498ν8+2951775ν7+629301ν625978321ν5++3991668)/272073 ( - 113902 \nu^{9} - 12498 \nu^{8} + 2951775 \nu^{7} + 629301 \nu^{6} - 25978321 \nu^{5} + \cdots + 3991668 ) / 272073 Copy content Toggle raw display
β8\beta_{8}== (53070ν93960ν8+1379192ν7+236054ν612163529ν5++2694093)/90691 ( - 53070 \nu^{9} - 3960 \nu^{8} + 1379192 \nu^{7} + 236054 \nu^{6} - 12163529 \nu^{5} + \cdots + 2694093 ) / 90691 Copy content Toggle raw display
β9\beta_{9}== (163381ν9+810ν84214799ν7530595ν6+36878434ν5+6849019ν4+7890852)/272073 ( 163381 \nu^{9} + 810 \nu^{8} - 4214799 \nu^{7} - 530595 \nu^{6} + 36878434 \nu^{5} + 6849019 \nu^{4} + \cdots - 7890852 ) / 272073 Copy content Toggle raw display
ν\nu== (β7β62β5+β42β11)/3 ( \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta _1 - 1 ) / 3 Copy content Toggle raw display
ν2\nu^{2}== (β8β5β4β3+β2+3β1+17)/3 ( \beta_{8} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 3\beta _1 + 17 ) / 3 Copy content Toggle raw display
ν3\nu^{3}== (3β9+2β8+7β7β619β5+11β4+β34β211β16)/3 ( 3\beta_{9} + 2\beta_{8} + 7\beta_{7} - \beta_{6} - 19\beta_{5} + 11\beta_{4} + \beta_{3} - 4\beta_{2} - 11\beta _1 - 6 ) / 3 Copy content Toggle raw display
ν4\nu^{4}== (6β9+14β8β7+4β612β511β3+8β2+35β1+137)/3 ( 6\beta_{9} + 14\beta_{8} - \beta_{7} + 4\beta_{6} - 12\beta_{5} - 11\beta_{3} + 8\beta_{2} + 35\beta _1 + 137 ) / 3 Copy content Toggle raw display
ν5\nu^{5}== (42β9+33β8+50β7+34β6190β5+116β4+6β3+35)/3 ( 42 \beta_{9} + 33 \beta_{8} + 50 \beta_{7} + 34 \beta_{6} - 190 \beta_{5} + 116 \beta_{4} + 6 \beta_{3} + \cdots - 35 ) / 3 Copy content Toggle raw display
ν6\nu^{6}== (93β9+158β8+66β6152β5+67β4125β3+53β2++1237)/3 ( 93 \beta_{9} + 158 \beta_{8} + 66 \beta_{6} - 152 \beta_{5} + 67 \beta_{4} - 125 \beta_{3} + 53 \beta_{2} + \cdots + 1237 ) / 3 Copy content Toggle raw display
ν7\nu^{7}== (480β9+439β8+386β7+568β61961β5+1195β4+168)/3 ( 480 \beta_{9} + 439 \beta_{8} + 386 \beta_{7} + 568 \beta_{6} - 1961 \beta_{5} + 1195 \beta_{4} + \cdots - 168 ) / 3 Copy content Toggle raw display
ν8\nu^{8}== (1092β9+1711β8+121β7+812β61935β5+1146β4++11752)/3 ( 1092 \beta_{9} + 1711 \beta_{8} + 121 \beta_{7} + 812 \beta_{6} - 1935 \beta_{5} + 1146 \beta_{4} + \cdots + 11752 ) / 3 Copy content Toggle raw display
ν9\nu^{9}== (5214β9+5400β8+3229β7+7028β620582β5+12289β4+163)/3 ( 5214 \beta_{9} + 5400 \beta_{8} + 3229 \beta_{7} + 7028 \beta_{6} - 20582 \beta_{5} + 12289 \beta_{4} + \cdots - 163 ) / 3 Copy content Toggle raw display

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}
1.1
0.0630659
−3.14737
2.06486
3.32874
2.05114
3.11329
−2.79044
−1.50682
1.15778
−2.33425
−1.00000 0 1.00000 −3.69049 0 0.539128 −1.00000 0 3.69049
1.2 −1.00000 0 1.00000 −3.49123 0 2.86107 −1.00000 0 3.49123
1.3 −1.00000 0 1.00000 −3.28277 0 −3.13291 −1.00000 0 3.28277
1.4 −1.00000 0 1.00000 −1.43355 0 0.697985 −1.00000 0 1.43355
1.5 −1.00000 0 1.00000 −0.0738674 0 4.67803 −1.00000 0 0.0738674
1.6 −1.00000 0 1.00000 0.363318 0 −5.03822 −1.00000 0 −0.363318
1.7 −1.00000 0 1.00000 0.949474 0 −4.04317 −1.00000 0 −0.949474
1.8 −1.00000 0 1.00000 2.84378 0 −0.483762 −1.00000 0 −2.84378
1.9 −1.00000 0 1.00000 3.33861 0 0.195962 −1.00000 0 −3.33861
1.10 −1.00000 0 1.00000 3.47672 0 4.72589 −1.00000 0 −3.47672
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
33 1 -1
3737 +1 +1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5994.2.a.ba 10
3.b odd 2 1 5994.2.a.bb 10
9.c even 3 2 1998.2.e.e 20
9.d odd 6 2 666.2.e.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.e.e 20 9.d odd 6 2
1998.2.e.e 20 9.c even 3 2
5994.2.a.ba 10 1.a even 1 1 trivial
5994.2.a.bb 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(5994))S_{2}^{\mathrm{new}}(\Gamma_0(5994)):

T510+T5935T5826T57+424T56+196T551958T54+51 T_{5}^{10} + T_{5}^{9} - 35 T_{5}^{8} - 26 T_{5}^{7} + 424 T_{5}^{6} + 196 T_{5}^{5} - 1958 T_{5}^{4} + \cdots - 51 Copy content Toggle raw display
T1110+3T11967T118204T117+1466T116+4343T115+5039 T_{11}^{10} + 3 T_{11}^{9} - 67 T_{11}^{8} - 204 T_{11}^{7} + 1466 T_{11}^{6} + 4343 T_{11}^{5} + \cdots - 5039 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T+1)10 (T + 1)^{10} Copy content Toggle raw display
33 T10 T^{10} Copy content Toggle raw display
55 T10+T9+51 T^{10} + T^{9} + \cdots - 51 Copy content Toggle raw display
77 T10T9++144 T^{10} - T^{9} + \cdots + 144 Copy content Toggle raw display
1111 T10+3T9+5039 T^{10} + 3 T^{9} + \cdots - 5039 Copy content Toggle raw display
1313 T1012T9++169 T^{10} - 12 T^{9} + \cdots + 169 Copy content Toggle raw display
1717 T10+12T9++769872 T^{10} + 12 T^{9} + \cdots + 769872 Copy content Toggle raw display
1919 T1024T9++7248 T^{10} - 24 T^{9} + \cdots + 7248 Copy content Toggle raw display
2323 T10+3T9+40937 T^{10} + 3 T^{9} + \cdots - 40937 Copy content Toggle raw display
2929 T10+4T9+254741 T^{10} + 4 T^{9} + \cdots - 254741 Copy content Toggle raw display
3131 T1011T9++2288361 T^{10} - 11 T^{9} + \cdots + 2288361 Copy content Toggle raw display
3737 (T+1)10 (T + 1)^{10} Copy content Toggle raw display
4141 T103T9+107747 T^{10} - 3 T^{9} + \cdots - 107747 Copy content Toggle raw display
4343 T106T9++340464 T^{10} - 6 T^{9} + \cdots + 340464 Copy content Toggle raw display
4747 T10+8T9+10100112 T^{10} + 8 T^{9} + \cdots - 10100112 Copy content Toggle raw display
5353 T1010T9++1011376 T^{10} - 10 T^{9} + \cdots + 1011376 Copy content Toggle raw display
5959 T10+10T9++14068272 T^{10} + 10 T^{9} + \cdots + 14068272 Copy content Toggle raw display
6161 T102T9+1169519 T^{10} - 2 T^{9} + \cdots - 1169519 Copy content Toggle raw display
6767 T107T9++1275319 T^{10} - 7 T^{9} + \cdots + 1275319 Copy content Toggle raw display
7171 T10+15T9++11950032 T^{10} + 15 T^{9} + \cdots + 11950032 Copy content Toggle raw display
7373 T103T9+15936 T^{10} - 3 T^{9} + \cdots - 15936 Copy content Toggle raw display
7979 T10+118547631 T^{10} + \cdots - 118547631 Copy content Toggle raw display
8383 T1023T9+69498213 T^{10} - 23 T^{9} + \cdots - 69498213 Copy content Toggle raw display
8989 T10+21T9+3312 T^{10} + 21 T^{9} + \cdots - 3312 Copy content Toggle raw display
9797 T10++18102071296 T^{10} + \cdots + 18102071296 Copy content Toggle raw display
show more
show less