Properties

Label 3168.2.h.i
Level 31683168
Weight 22
Character orbit 3168.h
Analytic conductor 25.29725.297
Analytic rank 00
Dimension 1616
Inner twists 88

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3168,2,Mod(2287,3168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3168.2287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3168=253211 3168 = 2^{5} \cdot 3^{2} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3168.h (of order 22, degree 11, not 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: 25.296607360325.2966073603
Analytic rank: 00
Dimension: 1616
Coefficient field: 16.0.3342602057661458415616.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x1620x14+164x12666x10+1300x8924x6+273x4+404x2+64 x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 Copy content Toggle raw display
Coefficient ring: Z[a1,,a29]\Z[a_1, \ldots, a_{29}]
Coefficient ring index: 226 2^{26}
Twist minimal: no (minimal twist has level 792)
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,,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β4q5β14q7+(β13β7)q11+(β14β11)q13β7q17β8q19+β5q23+q25+β2q29+β12q31++(3β31)q97+O(q100) q + \beta_{4} q^{5} - \beta_{14} q^{7} + ( - \beta_{13} - \beta_{7}) q^{11} + (\beta_{14} - \beta_{11}) q^{13} - \beta_{7} q^{17} - \beta_{8} q^{19} + \beta_{5} q^{23} + q^{25} + \beta_{2} q^{29} + \beta_{12} q^{31}+ \cdots + ( - 3 \beta_{3} - 1) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q+16q2532q49+32q6764q9116q97+O(q100) 16 q + 16 q^{25} - 32 q^{49} + 32 q^{67} - 64 q^{91} - 16 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x1620x14+164x12666x10+1300x8924x6+273x4+404x2+64 x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 : Copy content Toggle raw display

β1\beta_{1}== (1017ν1418889ν12+141809ν10504639ν8+779299ν6250803ν4++1865592)/347684 ( 1017 \nu^{14} - 18889 \nu^{12} + 141809 \nu^{10} - 504639 \nu^{8} + 779299 \nu^{6} - 250803 \nu^{4} + \cdots + 1865592 ) / 347684 Copy content Toggle raw display
β2\beta_{2}== (2447ν14+49511ν12412949ν10+1685873ν83080369ν6+360024)/347684 ( - 2447 \nu^{14} + 49511 \nu^{12} - 412949 \nu^{10} + 1685873 \nu^{8} - 3080369 \nu^{6} + \cdots - 360024 ) / 347684 Copy content Toggle raw display
β3\beta_{3}== (800ν1415844ν12+126372ν10485558ν8+824496ν6258554ν4++310223)/86921 ( 800 \nu^{14} - 15844 \nu^{12} + 126372 \nu^{10} - 485558 \nu^{8} + 824496 \nu^{6} - 258554 \nu^{4} + \cdots + 310223 ) / 86921 Copy content Toggle raw display
β4\beta_{4}== (6891ν15135786ν13+1092346ν114305788ν9+7949022ν7++3564012ν)/695368 ( 6891 \nu^{15} - 135786 \nu^{13} + 1092346 \nu^{11} - 4305788 \nu^{9} + 7949022 \nu^{7} + \cdots + 3564012 \nu ) / 695368 Copy content Toggle raw display
β5\beta_{5}== (9337ν15192870ν13+1637078ν116938020ν9+14314690ν7++5304612ν)/695368 ( 9337 \nu^{15} - 192870 \nu^{13} + 1637078 \nu^{11} - 6938020 \nu^{9} + 14314690 \nu^{7} + \cdots + 5304612 \nu ) / 695368 Copy content Toggle raw display
β6\beta_{6}== (3522ν1568884ν13+545053ν112077259ν9+3556830ν7++1812748ν)/173842 ( 3522 \nu^{15} - 68884 \nu^{13} + 545053 \nu^{11} - 2077259 \nu^{9} + 3556830 \nu^{7} + \cdots + 1812748 \nu ) / 173842 Copy content Toggle raw display
β7\beta_{7}== (13987ν14+283685ν122375450ν10+10006389ν821090917ν6+3647880)/173842 ( - 13987 \nu^{14} + 283685 \nu^{12} - 2375450 \nu^{10} + 10006389 \nu^{8} - 21090917 \nu^{6} + \cdots - 3647880 ) / 173842 Copy content Toggle raw display
β8\beta_{8}== (11595ν15231812ν13+1903314ν117753568ν9+15174808ν7++6116828ν)/347684 ( 11595 \nu^{15} - 231812 \nu^{13} + 1903314 \nu^{11} - 7753568 \nu^{9} + 15174808 \nu^{7} + \cdots + 6116828 \nu ) / 347684 Copy content Toggle raw display
β9\beta_{9}== (25585ν14518343ν12+4328246ν1018126731ν8+37755575ν6++6522868)/173842 ( 25585 \nu^{14} - 518343 \nu^{12} + 4328246 \nu^{10} - 18126731 \nu^{8} + 37755575 \nu^{6} + \cdots + 6522868 ) / 173842 Copy content Toggle raw display
β10\beta_{10}== (38555ν14+783139ν126561887ν10+27628481ν858067193ν6+10043576)/173842 ( - 38555 \nu^{14} + 783139 \nu^{12} - 6561887 \nu^{10} + 27628481 \nu^{8} - 58067193 \nu^{6} + \cdots - 10043576 ) / 173842 Copy content Toggle raw display
β11\beta_{11}== (56101ν15+1136722ν139499560ν11+39874286ν983528306ν7+14653004ν)/347684 ( - 56101 \nu^{15} + 1136722 \nu^{13} - 9499560 \nu^{11} + 39874286 \nu^{9} - 83528306 \nu^{7} + \cdots - 14653004 \nu ) / 347684 Copy content Toggle raw display
β12\beta_{12}== (141343ν142858839ν12+23840107ν1099747945ν8+207692533ν6++35860056)/347684 ( 141343 \nu^{14} - 2858839 \nu^{12} + 23840107 \nu^{10} - 99747945 \nu^{8} + 207692533 \nu^{6} + \cdots + 35860056 ) / 347684 Copy content Toggle raw display
β13\beta_{13}== (155031ν1523196ν143141792ν13+469316ν12+26258800ν11+5749976)/695368 ( 155031 \nu^{15} - 23196 \nu^{14} - 3141792 \nu^{13} + 469316 \nu^{12} + 26258800 \nu^{11} + \cdots - 5749976 ) / 695368 Copy content Toggle raw display
β14\beta_{14}== (204751ν15+4145926ν1334610826ν11+144971388ν9302181482ν7+51983964ν)/695368 ( - 204751 \nu^{15} + 4145926 \nu^{13} - 34610826 \nu^{11} + 144971388 \nu^{9} - 302181482 \nu^{7} + \cdots - 51983964 \nu ) / 695368 Copy content Toggle raw display
β15\beta_{15}== (101947ν152063569ν13+17216979ν1172051334ν9+149956769ν7++25677692ν)/86921 ( 101947 \nu^{15} - 2063569 \nu^{13} + 17216979 \nu^{11} - 72051334 \nu^{9} + 149956769 \nu^{7} + \cdots + 25677692 \nu ) / 86921 Copy content Toggle raw display
ν\nu== (β14+2β13+β11+β9+β7+β4)/4 ( \beta_{14} + 2\beta_{13} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_{4} ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (β10+β9β72β1+10)/4 ( \beta_{10} + \beta_{9} - \beta_{7} - 2\beta _1 + 10 ) / 4 Copy content Toggle raw display
ν3\nu^{3}== (β15+8β14+14β13+12β11+7β93β8++16β4)/8 ( \beta_{15} + 8 \beta_{14} + 14 \beta_{13} + 12 \beta_{11} + 7 \beta_{9} - 3 \beta_{8} + \cdots + 16 \beta_{4} ) / 8 Copy content Toggle raw display
ν4\nu^{4}== (2β12+7β10+9β913β7+2β38β1+36)/4 ( -2\beta_{12} + 7\beta_{10} + 9\beta_{9} - 13\beta_{7} + 2\beta_{3} - 8\beta _1 + 36 ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (5β15+32β14+42β13+36β11+21β925β8++122β4)/8 ( 5 \beta_{15} + 32 \beta_{14} + 42 \beta_{13} + 36 \beta_{11} + 21 \beta_{9} - 25 \beta_{8} + \cdots + 122 \beta_{4} ) / 8 Copy content Toggle raw display
ν6\nu^{6}== (20β12+42β10+71β987β7+9β3+4β220β1+79)/4 ( -20\beta_{12} + 42\beta_{10} + 71\beta_{9} - 87\beta_{7} + 9\beta_{3} + 4\beta_{2} - 20\beta _1 + 79 ) / 4 Copy content Toggle raw display
ν7\nu^{7}== (11β15+56β14+22β13+8β11+11β9175β8++754β4)/8 ( 11 \beta_{15} + 56 \beta_{14} + 22 \beta_{13} + 8 \beta_{11} + 11 \beta_{9} - 175 \beta_{8} + \cdots + 754 \beta_{4} ) / 8 Copy content Toggle raw display
ν8\nu^{8}== (142β12+213β10+465β9453β7+6β3+16β2+48β1264)/4 ( -142\beta_{12} + 213\beta_{10} + 465\beta_{9} - 453\beta_{7} + 6\beta_{3} + 16\beta_{2} + 48\beta _1 - 264 ) / 4 Copy content Toggle raw display
ν9\nu^{9}== (81β15608β14998β13860β11499β9++3908β4)/8 ( - 81 \beta_{15} - 608 \beta_{14} - 998 \beta_{13} - 860 \beta_{11} - 499 \beta_{9} + \cdots + 3908 \beta_{4} ) / 8 Copy content Toggle raw display
ν10\nu^{10}== (810β12+883β10+2511β91923β7340β396β2+4950)/4 ( - 810 \beta_{12} + 883 \beta_{10} + 2511 \beta_{9} - 1923 \beta_{7} - 340 \beta_{3} - 96 \beta_{2} + \cdots - 4950 ) / 4 Copy content Toggle raw display
ν11\nu^{11}== (1357β158816β1410474β138260β115237β9++16454β4)/8 ( - 1357 \beta_{15} - 8816 \beta_{14} - 10474 \beta_{13} - 8260 \beta_{11} - 5237 \beta_{9} + \cdots + 16454 \beta_{4} ) / 8 Copy content Toggle raw display
ν12\nu^{12}== (3584β12+2632β10+10561β95981β74199β32024β2+41469)/4 ( - 3584 \beta_{12} + 2632 \beta_{10} + 10561 \beta_{9} - 5981 \beta_{7} - 4199 \beta_{3} - 2024 \beta_{2} + \cdots - 41469 ) / 4 Copy content Toggle raw display
ν13\nu^{13}== (12063β1574328β1474454β1354976β1137227β9++46594β4)/8 ( - 12063 \beta_{15} - 74328 \beta_{14} - 74454 \beta_{13} - 54976 \beta_{11} - 37227 \beta_{9} + \cdots + 46594 \beta_{4} ) / 8 Copy content Toggle raw display
ν14\nu^{14}== (9250β12+1295β10+24871β94567β734006β3+266984)/4 ( - 9250 \beta_{12} + 1295 \beta_{10} + 24871 \beta_{9} - 4567 \beta_{7} - 34006 \beta_{3} + \cdots - 266984 ) / 4 Copy content Toggle raw display
ν15\nu^{15}== (82603β15491936β14431322β13298844β11215661β9+36304β4)/8 ( - 82603 \beta_{15} - 491936 \beta_{14} - 431322 \beta_{13} - 298844 \beta_{11} - 215661 \beta_{9} + \cdots - 36304 \beta_{4} ) / 8 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3168Z)×\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times.

nn 353353 991991 11891189 17291729
χ(n)\chi(n) 11 1-1 1-1 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}
2287.1
2.34517 0.500000i
0.0131233 0.500000i
0.946412 0.500000i
−2.14576 0.500000i
2.14576 0.500000i
−0.946412 0.500000i
−0.0131233 0.500000i
−2.34517 0.500000i
2.34517 + 0.500000i
0.0131233 + 0.500000i
0.946412 + 0.500000i
−2.14576 + 0.500000i
2.14576 + 0.500000i
−0.946412 + 0.500000i
−0.0131233 + 0.500000i
−2.34517 + 0.500000i
0 0 0 2.00000i 0 −3.02045 0 0 0
2287.2 0 0 0 2.00000i 0 −3.02045 0 0 0
2287.3 0 0 0 2.00000i 0 −0.936426 0 0 0
2287.4 0 0 0 2.00000i 0 −0.936426 0 0 0
2287.5 0 0 0 2.00000i 0 0.936426 0 0 0
2287.6 0 0 0 2.00000i 0 0.936426 0 0 0
2287.7 0 0 0 2.00000i 0 3.02045 0 0 0
2287.8 0 0 0 2.00000i 0 3.02045 0 0 0
2287.9 0 0 0 2.00000i 0 −3.02045 0 0 0
2287.10 0 0 0 2.00000i 0 −3.02045 0 0 0
2287.11 0 0 0 2.00000i 0 −0.936426 0 0 0
2287.12 0 0 0 2.00000i 0 −0.936426 0 0 0
2287.13 0 0 0 2.00000i 0 0.936426 0 0 0
2287.14 0 0 0 2.00000i 0 0.936426 0 0 0
2287.15 0 0 0 2.00000i 0 3.02045 0 0 0
2287.16 0 0 0 2.00000i 0 3.02045 0 0 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2287.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
11.b odd 2 1 inner
24.f even 2 1 inner
33.d even 2 1 inner
88.g even 2 1 inner
264.p odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3168.2.h.i 16
3.b odd 2 1 inner 3168.2.h.i 16
4.b odd 2 1 792.2.h.i 16
8.b even 2 1 792.2.h.i 16
8.d odd 2 1 inner 3168.2.h.i 16
11.b odd 2 1 inner 3168.2.h.i 16
12.b even 2 1 792.2.h.i 16
24.f even 2 1 inner 3168.2.h.i 16
24.h odd 2 1 792.2.h.i 16
33.d even 2 1 inner 3168.2.h.i 16
44.c even 2 1 792.2.h.i 16
88.b odd 2 1 792.2.h.i 16
88.g even 2 1 inner 3168.2.h.i 16
132.d odd 2 1 792.2.h.i 16
264.m even 2 1 792.2.h.i 16
264.p odd 2 1 inner 3168.2.h.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
792.2.h.i 16 4.b odd 2 1
792.2.h.i 16 8.b even 2 1
792.2.h.i 16 12.b even 2 1
792.2.h.i 16 24.h odd 2 1
792.2.h.i 16 44.c even 2 1
792.2.h.i 16 88.b odd 2 1
792.2.h.i 16 132.d odd 2 1
792.2.h.i 16 264.m even 2 1
3168.2.h.i 16 1.a even 1 1 trivial
3168.2.h.i 16 3.b odd 2 1 inner
3168.2.h.i 16 8.d odd 2 1 inner
3168.2.h.i 16 11.b odd 2 1 inner
3168.2.h.i 16 24.f even 2 1 inner
3168.2.h.i 16 33.d even 2 1 inner
3168.2.h.i 16 88.g even 2 1 inner
3168.2.h.i 16 264.p odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(3168,[χ])S_{2}^{\mathrm{new}}(3168, [\chi]):

T52+4 T_{5}^{2} + 4 Copy content Toggle raw display
T7410T72+8 T_{7}^{4} - 10T_{7}^{2} + 8 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16 T^{16} Copy content Toggle raw display
33 T16 T^{16} Copy content Toggle raw display
55 (T2+4)8 (T^{2} + 4)^{8} Copy content Toggle raw display
77 (T410T2+8)4 (T^{4} - 10 T^{2} + 8)^{4} Copy content Toggle raw display
1111 (T816T6++14641)2 (T^{8} - 16 T^{6} + \cdots + 14641)^{2} Copy content Toggle raw display
1313 (T420T2+32)4 (T^{4} - 20 T^{2} + 32)^{4} Copy content Toggle raw display
1717 (T4+14T2+32)4 (T^{4} + 14 T^{2} + 32)^{4} Copy content Toggle raw display
1919 (T4+58T2+416)4 (T^{4} + 58 T^{2} + 416)^{4} Copy content Toggle raw display
2323 (T4+36T2+256)4 (T^{4} + 36 T^{2} + 256)^{4} Copy content Toggle raw display
2929 (T446T2+104)4 (T^{4} - 46 T^{2} + 104)^{4} Copy content Toggle raw display
3131 (T4+72T2+208)4 (T^{4} + 72 T^{2} + 208)^{4} Copy content Toggle raw display
3737 (T4+60T2+832)4 (T^{4} + 60 T^{2} + 832)^{4} Copy content Toggle raw display
4141 (T4+62T2+128)4 (T^{4} + 62 T^{2} + 128)^{4} Copy content Toggle raw display
4343 (T4+122T2+1664)4 (T^{4} + 122 T^{2} + 1664)^{4} Copy content Toggle raw display
4747 (T2+36)8 (T^{2} + 36)^{8} Copy content Toggle raw display
5353 (T2+68)8 (T^{2} + 68)^{8} Copy content Toggle raw display
5959 (T4236T2+13312)4 (T^{4} - 236 T^{2} + 13312)^{4} Copy content Toggle raw display
6161 (T4148T2+5408)4 (T^{4} - 148 T^{2} + 5408)^{4} Copy content Toggle raw display
6767 (T24T64)8 (T^{2} - 4 T - 64)^{8} Copy content Toggle raw display
7171 (T4+168T2+2704)4 (T^{4} + 168 T^{2} + 2704)^{4} Copy content Toggle raw display
7373 (T4+184T2+1664)4 (T^{4} + 184 T^{2} + 1664)^{4} Copy content Toggle raw display
7979 (T4170T2+2312)4 (T^{4} - 170 T^{2} + 2312)^{4} Copy content Toggle raw display
8383 (T4+28T2+128)4 (T^{4} + 28 T^{2} + 128)^{4} Copy content Toggle raw display
8989 (T4288T2+3328)4 (T^{4} - 288 T^{2} + 3328)^{4} Copy content Toggle raw display
9797 (T2+2T152)8 (T^{2} + 2 T - 152)^{8} Copy content Toggle raw display
show more
show less