Properties

Label 1216.2.n.e
Level 12161216
Weight 22
Character orbit 1216.n
Analytic conductor 9.7109.710
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1216,2,Mod(255,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 9.709808885799.70980888579
Analytic rank: 00
Dimension: 66
Relative dimension: 33 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: 6.0.31726512.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x6x5+10x4+3x3+84x227x+9 x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q3+(β3+β11)q5+(β5β4)q7+(2β5+β4++β1)q9+(β5+β4+2β3+1)q11+(2β34)q13++(3β49β3+18)q99+O(q100) q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_{5} - \beta_{4}) q^{7} + ( - 2 \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{9} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 1) q^{11} + ( - 2 \beta_{3} - 4) q^{13}+ \cdots + ( - 3 \beta_{4} - 9 \beta_{3} + \cdots - 18) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+q32q510q918q1318q152q17+17q196q215q2526q27+12q294q31+3q336q35+3q41+18q4312q45+18q47+84q99+O(q100) 6 q + q^{3} - 2 q^{5} - 10 q^{9} - 18 q^{13} - 18 q^{15} - 2 q^{17} + 17 q^{19} - 6 q^{21} - 5 q^{25} - 26 q^{27} + 12 q^{29} - 4 q^{31} + 3 q^{33} - 6 q^{35} + 3 q^{41} + 18 q^{43} - 12 q^{45} + 18 q^{47}+ \cdots - 84 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6x5+10x4+3x3+84x227x+9 x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν5+10ν4100ν3+84ν227ν+270)/813 ( -\nu^{5} + 10\nu^{4} - 100\nu^{3} + 84\nu^{2} - 27\nu + 270 ) / 813 Copy content Toggle raw display
β3\beta_{3}== (30ν529ν4+290ν3+190ν2+2436ν783)/813 ( 30\nu^{5} - 29\nu^{4} + 290\nu^{3} + 190\nu^{2} + 2436\nu - 783 ) / 813 Copy content Toggle raw display
β4\beta_{4}== (161ν516ν41466ν31923ν214916ν5310)/2439 ( -161\nu^{5} - 16\nu^{4} - 1466\nu^{3} - 1923\nu^{2} - 14916\nu - 5310 ) / 2439 Copy content Toggle raw display
β5\beta_{5}== (191ν5284ν4+2027ν3597ν2+15726ν10107)/2439 ( 191\nu^{5} - 284\nu^{4} + 2027\nu^{3} - 597\nu^{2} + 15726\nu - 10107 ) / 2439 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== 2β5+β4+6β3β2+β1 -2\beta_{5} + \beta_{4} + 6\beta_{3} - \beta_{2} + \beta_1 Copy content Toggle raw display
ν3\nu^{3}== β5β410β23 -\beta_{5} - \beta_{4} - 10\beta_{2} - 3 Copy content Toggle raw display
ν4\nu^{4}== 10β520β457β316β157 10\beta_{5} - 20\beta_{4} - 57\beta_{3} - 16\beta _1 - 57 Copy content Toggle raw display
ν5\nu^{5}== 32β516β466β3+103β2103β1 32\beta_{5} - 16\beta_{4} - 66\beta_{3} + 103\beta_{2} - 103\beta_1 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 β3-\beta_{3} 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.35887 + 2.35363i
0.162698 0.281802i
1.69617 2.93786i
−1.35887 2.35363i
0.162698 + 0.281802i
1.69617 + 2.93786i
0 −1.35887 + 2.35363i 0 −1.85887 + 3.21966i 0 2.36936i 0 −2.19306 3.79849i 0
255.2 0 0.162698 0.281802i 0 −0.337302 + 0.584224i 0 3.59084i 0 1.44706 + 2.50638i 0
255.3 0 1.69617 2.93786i 0 1.19617 2.07183i 0 1.22147i 0 −4.25400 7.36814i 0
639.1 0 −1.35887 2.35363i 0 −1.85887 3.21966i 0 2.36936i 0 −2.19306 + 3.79849i 0
639.2 0 0.162698 + 0.281802i 0 −0.337302 0.584224i 0 3.59084i 0 1.44706 2.50638i 0
639.3 0 1.69617 + 2.93786i 0 1.19617 + 2.07183i 0 1.22147i 0 −4.25400 + 7.36814i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.e 6
4.b odd 2 1 1216.2.n.d 6
8.b even 2 1 304.2.n.d 6
8.d odd 2 1 304.2.n.e yes 6
19.d odd 6 1 1216.2.n.d 6
24.f even 2 1 2736.2.bm.m 6
24.h odd 2 1 2736.2.bm.l 6
76.f even 6 1 inner 1216.2.n.e 6
152.l odd 6 1 304.2.n.e yes 6
152.o even 6 1 304.2.n.d 6
456.s odd 6 1 2736.2.bm.l 6
456.v even 6 1 2736.2.bm.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.d 6 8.b even 2 1
304.2.n.d 6 152.o even 6 1
304.2.n.e yes 6 8.d odd 2 1
304.2.n.e yes 6 152.l odd 6 1
1216.2.n.d 6 4.b odd 2 1
1216.2.n.d 6 19.d odd 6 1
1216.2.n.e 6 1.a even 1 1 trivial
1216.2.n.e 6 76.f even 6 1 inner
2736.2.bm.l 6 24.h odd 2 1
2736.2.bm.l 6 456.s odd 6 1
2736.2.bm.m 6 24.f even 2 1
2736.2.bm.m 6 456.v even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T36T35+10T34+3T33+84T3227T3+9 T_{3}^{6} - T_{3}^{5} + 10T_{3}^{4} + 3T_{3}^{3} + 84T_{3}^{2} - 27T_{3} + 9 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 T6 T^{6} Copy content Toggle raw display
33 T6T5+10T4++9 T^{6} - T^{5} + 10 T^{4} + \cdots + 9 Copy content Toggle raw display
55 T6+2T5++36 T^{6} + 2 T^{5} + \cdots + 36 Copy content Toggle raw display
77 T6+20T4++108 T^{6} + 20 T^{4} + \cdots + 108 Copy content Toggle raw display
1111 T6+29T4++507 T^{6} + 29 T^{4} + \cdots + 507 Copy content Toggle raw display
1313 (T2+6T+12)3 (T^{2} + 6 T + 12)^{3} Copy content Toggle raw display
1717 T6+2T5++144 T^{6} + 2 T^{5} + \cdots + 144 Copy content Toggle raw display
1919 T617T5++6859 T^{6} - 17 T^{5} + \cdots + 6859 Copy content Toggle raw display
2323 T610T4++108 T^{6} - 10 T^{4} + \cdots + 108 Copy content Toggle raw display
2929 T612T5++2700 T^{6} - 12 T^{5} + \cdots + 2700 Copy content Toggle raw display
3131 (T3+2T254T+66)2 (T^{3} + 2 T^{2} - 54 T + 66)^{2} Copy content Toggle raw display
3737 T6+72T4++2700 T^{6} + 72 T^{4} + \cdots + 2700 Copy content Toggle raw display
4141 T63T5++54675 T^{6} - 3 T^{5} + \cdots + 54675 Copy content Toggle raw display
4343 (T26T+12)3 (T^{2} - 6 T + 12)^{3} Copy content Toggle raw display
4747 T618T5++12 T^{6} - 18 T^{5} + \cdots + 12 Copy content Toggle raw display
5353 (T212T+48)3 (T^{2} - 12 T + 48)^{3} Copy content Toggle raw display
5959 T627T5++263169 T^{6} - 27 T^{5} + \cdots + 263169 Copy content Toggle raw display
6161 T610T5++36 T^{6} - 10 T^{5} + \cdots + 36 Copy content Toggle raw display
6767 T6+11T5++245025 T^{6} + 11 T^{5} + \cdots + 245025 Copy content Toggle raw display
7171 (T2+6T+36)3 (T^{2} + 6 T + 36)^{3} Copy content Toggle raw display
7373 T6+5T5++289 T^{6} + 5 T^{5} + \cdots + 289 Copy content Toggle raw display
7979 T616T5++2304 T^{6} - 16 T^{5} + \cdots + 2304 Copy content Toggle raw display
8383 T6+113T4++3 T^{6} + 113 T^{4} + \cdots + 3 Copy content Toggle raw display
8989 T6+24T5++97200 T^{6} + 24 T^{5} + \cdots + 97200 Copy content Toggle raw display
9797 T621T5++151875 T^{6} - 21 T^{5} + \cdots + 151875 Copy content Toggle raw display
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