Properties

Label 1216.2.n.f
Level 12161216
Weight 22
Character orbit 1216.n
Analytic conductor 9.7109.710
Analytic rank 00
Dimension 1616
Inner twists 44

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(255,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("1216.255");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1216=2619 1216 = 2^{6} \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1216.n (of order 66, degree 22, 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: 9.709808885799.70980888579
Analytic rank: 00
Dimension: 1616
Relative dimension: 88 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x163x15+6x149x13+12x129x11+3x10+6x910x8++256 x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + \cdots + 256 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 26 2^{6}
Twist minimal: no (minimal twist has level 76)
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,,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+β7q3+(β10β5)q5β9q7+(β12β5)q9+(β15β9)q11+(β12+β5β4++2)q13++(β14β13+β2)q99+O(q100) q + \beta_{7} q^{3} + (\beta_{10} - \beta_{5}) q^{5} - \beta_{9} q^{7} + ( - \beta_{12} - \beta_{5}) q^{9} + (\beta_{15} - \beta_{9}) q^{11} + ( - \beta_{12} + \beta_{5} - \beta_{4} + \cdots + 2) q^{13}+ \cdots + ( - \beta_{14} - \beta_{13} + \cdots - \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q+2q54q9+18q13+2q172q25+6q29+18q3348q4124q45+16q496q5326q57+26q61+16q7380q77+12q8114q85+18q89+12q97+O(q100) 16 q + 2 q^{5} - 4 q^{9} + 18 q^{13} + 2 q^{17} - 2 q^{25} + 6 q^{29} + 18 q^{33} - 48 q^{41} - 24 q^{45} + 16 q^{49} - 6 q^{53} - 26 q^{57} + 26 q^{61} + 16 q^{73} - 80 q^{77} + 12 q^{81} - 14 q^{85} + 18 q^{89}+ \cdots - 12 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x163x15+6x149x13+12x129x11+3x10+6x910x8++256 x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + \cdots + 256 : Copy content Toggle raw display

β1\beta_{1}== (26ν15397ν14+335ν13666ν12+419ν11428ν10+5056)/40768 ( - 26 \nu^{15} - 397 \nu^{14} + 335 \nu^{13} - 666 \nu^{12} + 419 \nu^{11} - 428 \nu^{10} + \cdots - 5056 ) / 40768 Copy content Toggle raw display
β2\beta_{2}== (79ν15+341ν14+1114ν13629ν12+1716ν11965ν10+32128)/81536 ( - 79 \nu^{15} + 341 \nu^{14} + 1114 \nu^{13} - 629 \nu^{12} + 1716 \nu^{11} - 965 \nu^{10} + \cdots - 32128 ) / 81536 Copy content Toggle raw display
β3\beta_{3}== (79ν15341ν141114ν13+629ν121716ν11+965ν10++32128)/81536 ( 79 \nu^{15} - 341 \nu^{14} - 1114 \nu^{13} + 629 \nu^{12} - 1716 \nu^{11} + 965 \nu^{10} + \cdots + 32128 ) / 81536 Copy content Toggle raw display
β4\beta_{4}== (13ν15+34ν1417ν13+165ν12269ν11+179ν10+62ν9++13504)/5824 ( 13 \nu^{15} + 34 \nu^{14} - 17 \nu^{13} + 165 \nu^{12} - 269 \nu^{11} + 179 \nu^{10} + 62 \nu^{9} + \cdots + 13504 ) / 5824 Copy content Toggle raw display
β5\beta_{5}== (230ν15+97ν14+949ν13290ν12367ν11+1404ν10++52928)/81536 ( 230 \nu^{15} + 97 \nu^{14} + 949 \nu^{13} - 290 \nu^{12} - 367 \nu^{11} + 1404 \nu^{10} + \cdots + 52928 ) / 81536 Copy content Toggle raw display
β6\beta_{6}== (334ν151881ν14785ν131002ν12+671ν115272ν10+179776)/81536 ( - 334 \nu^{15} - 1881 \nu^{14} - 785 \nu^{13} - 1002 \nu^{12} + 671 \nu^{11} - 5272 \nu^{10} + \cdots - 179776 ) / 81536 Copy content Toggle raw display
β7\beta_{7}== (885ν15431ν14+114ν13+4223ν124660ν11+5903ν10++273536)/163072 ( 885 \nu^{15} - 431 \nu^{14} + 114 \nu^{13} + 4223 \nu^{12} - 4660 \nu^{11} + 5903 \nu^{10} + \cdots + 273536 ) / 163072 Copy content Toggle raw display
β8\beta_{8}== (1077ν15+13ν14+1334ν134609ν12+2272ν112425ν10+183168)/163072 ( 1077 \nu^{15} + 13 \nu^{14} + 1334 \nu^{13} - 4609 \nu^{12} + 2272 \nu^{11} - 2425 \nu^{10} + \cdots - 183168 ) / 163072 Copy content Toggle raw display
β9\beta_{9}== (1261ν15+7769ν146660ν13+15817ν1217286ν11+13101ν10++869632)/163072 ( - 1261 \nu^{15} + 7769 \nu^{14} - 6660 \nu^{13} + 15817 \nu^{12} - 17286 \nu^{11} + 13101 \nu^{10} + \cdots + 869632 ) / 163072 Copy content Toggle raw display
β10\beta_{10}== (1277ν156409ν14+12364ν1320081ν12+17782ν1113893ν10+548096)/163072 ( 1277 \nu^{15} - 6409 \nu^{14} + 12364 \nu^{13} - 20081 \nu^{12} + 17782 \nu^{11} - 13893 \nu^{10} + \cdots - 548096 ) / 163072 Copy content Toggle raw display
β11\beta_{11}== (1346ν152357ν14+3471ν136350ν12+7867ν113172ν10+249664)/81536 ( 1346 \nu^{15} - 2357 \nu^{14} + 3471 \nu^{13} - 6350 \nu^{12} + 7867 \nu^{11} - 3172 \nu^{10} + \cdots - 249664 ) / 81536 Copy content Toggle raw display
β12\beta_{12}== (1481ν153622ν14+6053ν138885ν12+8867ν113803ν10+227392)/81536 ( 1481 \nu^{15} - 3622 \nu^{14} + 6053 \nu^{13} - 8885 \nu^{12} + 8867 \nu^{11} - 3803 \nu^{10} + \cdots - 227392 ) / 81536 Copy content Toggle raw display
β13\beta_{13}== (415ν15+887ν141364ν13+2381ν122428ν11+771ν10++80320)/20384 ( - 415 \nu^{15} + 887 \nu^{14} - 1364 \nu^{13} + 2381 \nu^{12} - 2428 \nu^{11} + 771 \nu^{10} + \cdots + 80320 ) / 20384 Copy content Toggle raw display
β14\beta_{14}== (17ν15+33ν1446ν13+67ν1274ν11+63ν105ν9++2304)/832 ( - 17 \nu^{15} + 33 \nu^{14} - 46 \nu^{13} + 67 \nu^{12} - 74 \nu^{11} + 63 \nu^{10} - 5 \nu^{9} + \cdots + 2304 ) / 832 Copy content Toggle raw display
β15\beta_{15}== (2119ν15+4811ν149276ν13+10107ν128898ν11+1623ν10++166528)/81536 ( - 2119 \nu^{15} + 4811 \nu^{14} - 9276 \nu^{13} + 10107 \nu^{12} - 8898 \nu^{11} + 1623 \nu^{10} + \cdots + 166528 ) / 81536 Copy content Toggle raw display
ν\nu== (β3+β2)/2 ( \beta_{3} + \beta_{2} ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β13+β12+β5β4+β3β1)/2 ( \beta_{13} + \beta_{12} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (β14+2β122β112β102β7+2β5β4+β1)/2 ( -\beta_{14} + 2\beta_{12} - 2\beta_{11} - 2\beta_{10} - 2\beta_{7} + 2\beta_{5} - \beta_{4} + \beta_1 ) / 2 Copy content Toggle raw display
ν4\nu^{4}== (β15β13β12β10+2β9+β8+2β6+2β5++1)/2 ( - \beta_{15} - \beta_{13} - \beta_{12} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 1 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (β14+2β12+2β11+4β9+2β62β5β4+4)/2 ( \beta_{14} + 2 \beta_{12} + 2 \beta_{11} + 4 \beta_{9} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \cdots - 4 ) / 2 Copy content Toggle raw display
ν6\nu^{6}== (β15+4β124β11β10+3β8+4β7β54β4++1)/2 ( \beta_{15} + 4 \beta_{12} - 4 \beta_{11} - \beta_{10} + 3 \beta_{8} + 4 \beta_{7} - \beta_{5} - 4 \beta_{4} + \cdots + 1 ) / 2 Copy content Toggle raw display
ν7\nu^{7}== (4β154β132β12+4β102β8+6β7+2β5+2)/2 ( 4 \beta_{15} - 4 \beta_{13} - 2 \beta_{12} + 4 \beta_{10} - 2 \beta_{8} + 6 \beta_{7} + 2 \beta_{5} + \cdots - 2 ) / 2 Copy content Toggle raw display
ν8\nu^{8}== (4β143β139β12+6β106β86β69β5+6)/2 ( - 4 \beta_{14} - 3 \beta_{13} - 9 \beta_{12} + 6 \beta_{10} - 6 \beta_{8} - 6 \beta_{6} - 9 \beta_{5} + \cdots - 6 ) / 2 Copy content Toggle raw display
ν9\nu^{9}== (4β155β14+8β134β12+6β11+2β106β9+16)/2 ( 4 \beta_{15} - 5 \beta_{14} + 8 \beta_{13} - 4 \beta_{12} + 6 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} + \cdots - 16 ) / 2 Copy content Toggle raw display
ν10\nu^{10}== (5β15+24β145β13+3β12+8β1111β1012β9+13)/2 ( - 5 \beta_{15} + 24 \beta_{14} - 5 \beta_{13} + 3 \beta_{12} + 8 \beta_{11} - 11 \beta_{10} - 12 \beta_{9} + \cdots - 13 ) / 2 Copy content Toggle raw display
ν11\nu^{11}== (5β14+4β12+22β1122β108β922β84β6++66)/2 ( 5 \beta_{14} + 4 \beta_{12} + 22 \beta_{11} - 22 \beta_{10} - 8 \beta_{9} - 22 \beta_{8} - 4 \beta_{6} + \cdots + 66 ) / 2 Copy content Toggle raw display
ν12\nu^{12}== (7β1524β146β1233β1010β939β833β5+39)/2 ( - 7 \beta_{15} - 24 \beta_{14} - 6 \beta_{12} - 33 \beta_{10} - 10 \beta_{9} - 39 \beta_{8} - 33 \beta_{5} + \cdots - 39 ) / 2 Copy content Toggle raw display
ν13\nu^{13}== (24β15+24β1318β12+26β102β918β8+64)/2 ( - 24 \beta_{15} + 24 \beta_{13} - 18 \beta_{12} + 26 \beta_{10} - 2 \beta_{9} - 18 \beta_{8} + \cdots - 64 ) / 2 Copy content Toggle raw display
ν14\nu^{14}== (24β147β13+29β1252β1120β10+20β8++20)/2 ( - 24 \beta_{14} - 7 \beta_{13} + 29 \beta_{12} - 52 \beta_{11} - 20 \beta_{10} + 20 \beta_{8} + \cdots + 20 ) / 2 Copy content Toggle raw display
ν15\nu^{15}== (20β1541β1440β1374β12+42β1126β10++124)/2 ( - 20 \beta_{15} - 41 \beta_{14} - 40 \beta_{13} - 74 \beta_{12} + 42 \beta_{11} - 26 \beta_{10} + \cdots + 124 ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1216Z)×\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times.

nn 191191 705705 837837
χ(n)\chi(n) 1-1 β1\beta_{1} 11

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}
255.1
−0.327894 1.37568i
1.34543 + 0.435684i
−0.835469 + 1.14105i
1.16486 0.801943i
−0.112075 + 1.40977i
0.570443 1.29406i
1.05003 + 0.947334i
−1.35532 + 0.403874i
−0.327894 + 1.37568i
1.34543 0.435684i
−0.835469 1.14105i
1.16486 + 0.801943i
−0.112075 1.40977i
0.570443 + 1.29406i
1.05003 0.947334i
−1.35532 0.403874i
0 −1.42689 + 2.47144i 0 0.139977 0.242447i 0 1.55280i 0 −2.57201 4.45486i 0
255.2 0 −0.982349 + 1.70148i 0 0.349646 0.605604i 0 3.80025i 0 −0.430019 0.744815i 0
255.3 0 −0.637123 + 1.10353i 0 1.60333 2.77705i 0 1.25044i 0 0.688149 + 1.19191i 0
255.4 0 −0.305055 + 0.528371i 0 −1.59295 + 2.75907i 0 2.36291i 0 1.31388 + 2.27571i 0
255.5 0 0.305055 0.528371i 0 −1.59295 + 2.75907i 0 2.36291i 0 1.31388 + 2.27571i 0
255.6 0 0.637123 1.10353i 0 1.60333 2.77705i 0 1.25044i 0 0.688149 + 1.19191i 0
255.7 0 0.982349 1.70148i 0 0.349646 0.605604i 0 3.80025i 0 −0.430019 0.744815i 0
255.8 0 1.42689 2.47144i 0 0.139977 0.242447i 0 1.55280i 0 −2.57201 4.45486i 0
639.1 0 −1.42689 2.47144i 0 0.139977 + 0.242447i 0 1.55280i 0 −2.57201 + 4.45486i 0
639.2 0 −0.982349 1.70148i 0 0.349646 + 0.605604i 0 3.80025i 0 −0.430019 + 0.744815i 0
639.3 0 −0.637123 1.10353i 0 1.60333 + 2.77705i 0 1.25044i 0 0.688149 1.19191i 0
639.4 0 −0.305055 0.528371i 0 −1.59295 2.75907i 0 2.36291i 0 1.31388 2.27571i 0
639.5 0 0.305055 + 0.528371i 0 −1.59295 2.75907i 0 2.36291i 0 1.31388 2.27571i 0
639.6 0 0.637123 + 1.10353i 0 1.60333 + 2.77705i 0 1.25044i 0 0.688149 1.19191i 0
639.7 0 0.982349 + 1.70148i 0 0.349646 + 0.605604i 0 3.80025i 0 −0.430019 + 0.744815i 0
639.8 0 1.42689 + 2.47144i 0 0.139977 + 0.242447i 0 1.55280i 0 −2.57201 + 4.45486i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.n.f 16
4.b odd 2 1 inner 1216.2.n.f 16
8.b even 2 1 76.2.f.a 16
8.d odd 2 1 76.2.f.a 16
19.d odd 6 1 inner 1216.2.n.f 16
24.f even 2 1 684.2.r.a 16
24.h odd 2 1 684.2.r.a 16
76.f even 6 1 inner 1216.2.n.f 16
152.l odd 6 1 76.2.f.a 16
152.o even 6 1 76.2.f.a 16
456.s odd 6 1 684.2.r.a 16
456.v even 6 1 684.2.r.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.f.a 16 8.b even 2 1
76.2.f.a 16 8.d odd 2 1
76.2.f.a 16 152.l odd 6 1
76.2.f.a 16 152.o even 6 1
684.2.r.a 16 24.f even 2 1
684.2.r.a 16 24.h odd 2 1
684.2.r.a 16 456.s odd 6 1
684.2.r.a 16 456.v even 6 1
1216.2.n.f 16 1.a even 1 1 trivial
1216.2.n.f 16 4.b odd 2 1 inner
1216.2.n.f 16 19.d odd 6 1 inner
1216.2.n.f 16 76.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T316+14T314+140T312+644T310+2137T38+3388T36+3836T34+1330T32+361 T_{3}^{16} + 14T_{3}^{14} + 140T_{3}^{12} + 644T_{3}^{10} + 2137T_{3}^{8} + 3388T_{3}^{6} + 3836T_{3}^{4} + 1330T_{3}^{2} + 361 acting on S2new(1216,[χ])S_{2}^{\mathrm{new}}(1216, [\chi]). 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+14T14++361 T^{16} + 14 T^{14} + \cdots + 361 Copy content Toggle raw display
55 (T8T7+11T6++4)2 (T^{8} - T^{7} + 11 T^{6} + \cdots + 4)^{2} Copy content Toggle raw display
77 (T8+24T6++304)2 (T^{8} + 24 T^{6} + \cdots + 304)^{2} Copy content Toggle raw display
1111 (T8+45T6++3724)2 (T^{8} + 45 T^{6} + \cdots + 3724)^{2} Copy content Toggle raw display
1313 (T89T7++3136)2 (T^{8} - 9 T^{7} + \cdots + 3136)^{2} Copy content Toggle raw display
1717 (T8T7+19T6++784)2 (T^{8} - T^{7} + 19 T^{6} + \cdots + 784)^{2} Copy content Toggle raw display
1919 T16++16983563041 T^{16} + \cdots + 16983563041 Copy content Toggle raw display
2323 T16++757115775376 T^{16} + \cdots + 757115775376 Copy content Toggle raw display
2929 (T83T7++98596)2 (T^{8} - 3 T^{7} + \cdots + 98596)^{2} Copy content Toggle raw display
3131 (T8124T6++7600)2 (T^{8} - 124 T^{6} + \cdots + 7600)^{2} Copy content Toggle raw display
3737 (T8+124T6++784)2 (T^{8} + 124 T^{6} + \cdots + 784)^{2} Copy content Toggle raw display
4141 (T8+24T7++841)2 (T^{8} + 24 T^{7} + \cdots + 841)^{2} Copy content Toggle raw display
4343 T16105T14++1478656 T^{16} - 105 T^{14} + \cdots + 1478656 Copy content Toggle raw display
4747 T1669T14++37896336 T^{16} - 69 T^{14} + \cdots + 37896336 Copy content Toggle raw display
5353 (T8+3T7++802816)2 (T^{8} + 3 T^{7} + \cdots + 802816)^{2} Copy content Toggle raw display
5959 T16++1300683225625 T^{16} + \cdots + 1300683225625 Copy content Toggle raw display
6161 (T813T7++209764)2 (T^{8} - 13 T^{7} + \cdots + 209764)^{2} Copy content Toggle raw display
6767 T16+106T14++361 T^{16} + 106 T^{14} + \cdots + 361 Copy content Toggle raw display
7171 T16++3550253056 T^{16} + \cdots + 3550253056 Copy content Toggle raw display
7373 (T88T7++1190281)2 (T^{8} - 8 T^{7} + \cdots + 1190281)^{2} Copy content Toggle raw display
7979 T16++136386521399296 T^{16} + \cdots + 136386521399296 Copy content Toggle raw display
8383 (T8+181T6++255664)2 (T^{8} + 181 T^{6} + \cdots + 255664)^{2} Copy content Toggle raw display
8989 (T89T7++250000)2 (T^{8} - 9 T^{7} + \cdots + 250000)^{2} Copy content Toggle raw display
9797 (T8+6T7++24649)2 (T^{8} + 6 T^{7} + \cdots + 24649)^{2} Copy content Toggle raw display
show more
show less