Properties

Label 3276.2.e.f
Level 32763276
Weight 22
Character orbit 3276.e
Analytic conductor 26.15926.159
Analytic rank 00
Dimension 88
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3276,2,Mod(2521,3276)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3276, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("3276.2521"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 3276=2232713 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3276.e (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 26.158991702226.1589917022
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.41589892096.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8+16x6+80x4+132x2+64 x^{8} + 16x^{6} + 80x^{4} + 132x^{2} + 64 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

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,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β4+β3)q5+β3q7+(β6+β4+β1)q11+(β7+β6+β3)q13+(2β5+β2)q17+(β6β3+2β1)q19++(3β6+2β4++4β1)q97+O(q100) q + (\beta_{4} + \beta_{3}) q^{5} + \beta_{3} q^{7} + (\beta_{6} + \beta_{4} + \beta_1) q^{11} + (\beta_{7} + \beta_{6} + \beta_{3}) q^{13} + (2 \beta_{5} + \beta_{2}) q^{17} + (\beta_{6} - \beta_{3} + 2 \beta_1) q^{19}+ \cdots + (3 \beta_{6} + 2 \beta_{4} + \cdots + 4 \beta_1) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+2q134q17+6q23+2q25+2q296q356q438q4922q5320q55+8q61+6q6526q7910q91+18q95+O(q100) 8 q + 2 q^{13} - 4 q^{17} + 6 q^{23} + 2 q^{25} + 2 q^{29} - 6 q^{35} - 6 q^{43} - 8 q^{49} - 22 q^{53} - 20 q^{55} + 8 q^{61} + 6 q^{65} - 26 q^{79} - 10 q^{91} + 18 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+16x6+80x4+132x2+64 x^{8} + 16x^{6} + 80x^{4} + 132x^{2} + 64 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν4+8ν2+8)/2 ( \nu^{4} + 8\nu^{2} + 8 ) / 2 Copy content Toggle raw display
β3\beta_{3}== (ν716ν572ν368ν)/16 ( -\nu^{7} - 16\nu^{5} - 72\nu^{3} - 68\nu ) / 16 Copy content Toggle raw display
β4\beta_{4}== (ν7+16ν5+80ν3+116ν)/16 ( \nu^{7} + 16\nu^{5} + 80\nu^{3} + 116\nu ) / 16 Copy content Toggle raw display
β5\beta_{5}== (ν614ν452ν236)/4 ( -\nu^{6} - 14\nu^{4} - 52\nu^{2} - 36 ) / 4 Copy content Toggle raw display
β6\beta_{6}== (3ν7+40ν5+144ν3+92ν)/16 ( 3\nu^{7} + 40\nu^{5} + 144\nu^{3} + 92\nu ) / 16 Copy content Toggle raw display
β7\beta_{7}== (ν6+14ν4+56ν2+52)/4 ( \nu^{6} + 14\nu^{4} + 56\nu^{2} + 52 ) / 4 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β7+β54 \beta_{7} + \beta_{5} - 4 Copy content Toggle raw display
ν3\nu^{3}== 2β4+2β36β1 2\beta_{4} + 2\beta_{3} - 6\beta_1 Copy content Toggle raw display
ν4\nu^{4}== 8β78β5+2β2+24 -8\beta_{7} - 8\beta_{5} + 2\beta_{2} + 24 Copy content Toggle raw display
ν5\nu^{5}== 2β618β424β3+40β1 -2\beta_{6} - 18\beta_{4} - 24\beta_{3} + 40\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 60β7+56β528β2164 60\beta_{7} + 56\beta_{5} - 28\beta_{2} - 164 Copy content Toggle raw display
ν7\nu^{7}== 32β6+144β4+224β3276β1 32\beta_{6} + 144\beta_{4} + 224\beta_{3} - 276\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3276Z)×\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times.

nn 16391639 20172017 23412341 25492549
χ(n)\chi(n) 11 1-1 11 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}
2521.1
1.29363i
0.927719i
2.77129i
2.40538i
2.40538i
2.77129i
0.927719i
1.29363i
0 0 0 2.79846i 0 1.00000i 0 0 0
2521.2 0 0 0 2.38393i 0 1.00000i 0 0 0
2521.3 0 0 0 2.32791i 0 1.00000i 0 0 0
2521.4 0 0 0 0.257562i 0 1.00000i 0 0 0
2521.5 0 0 0 0.257562i 0 1.00000i 0 0 0
2521.6 0 0 0 2.32791i 0 1.00000i 0 0 0
2521.7 0 0 0 2.38393i 0 1.00000i 0 0 0
2521.8 0 0 0 2.79846i 0 1.00000i 0 0 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2521.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3276.2.e.f 8
3.b odd 2 1 364.2.g.a 8
12.b even 2 1 1456.2.k.d 8
13.b even 2 1 inner 3276.2.e.f 8
21.c even 2 1 2548.2.g.g 8
21.g even 6 2 2548.2.y.e 16
21.h odd 6 2 2548.2.y.f 16
39.d odd 2 1 364.2.g.a 8
39.f even 4 1 4732.2.a.o 4
39.f even 4 1 4732.2.a.p 4
156.h even 2 1 1456.2.k.d 8
273.g even 2 1 2548.2.g.g 8
273.w odd 6 2 2548.2.y.f 16
273.ba even 6 2 2548.2.y.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.g.a 8 3.b odd 2 1
364.2.g.a 8 39.d odd 2 1
1456.2.k.d 8 12.b even 2 1
1456.2.k.d 8 156.h even 2 1
2548.2.g.g 8 21.c even 2 1
2548.2.g.g 8 273.g even 2 1
2548.2.y.e 16 21.g even 6 2
2548.2.y.e 16 273.ba even 6 2
2548.2.y.f 16 21.h odd 6 2
2548.2.y.f 16 273.w odd 6 2
3276.2.e.f 8 1.a even 1 1 trivial
3276.2.e.f 8 13.b even 2 1 inner
4732.2.a.o 4 39.f even 4 1
4732.2.a.p 4 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T58+19T56+119T54+249T52+16 T_{5}^{8} + 19T_{5}^{6} + 119T_{5}^{4} + 249T_{5}^{2} + 16 acting on S2new(3276,[χ])S_{2}^{\mathrm{new}}(3276, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 T8+19T6++16 T^{8} + 19 T^{6} + \cdots + 16 Copy content Toggle raw display
77 (T2+1)4 (T^{2} + 1)^{4} Copy content Toggle raw display
1111 T8+36T6++256 T^{8} + 36 T^{6} + \cdots + 256 Copy content Toggle raw display
1313 T82T7++28561 T^{8} - 2 T^{7} + \cdots + 28561 Copy content Toggle raw display
1717 (T4+2T352T2++44)2 (T^{4} + 2 T^{3} - 52 T^{2} + \cdots + 44)^{2} Copy content Toggle raw display
1919 T8+71T6++30976 T^{8} + 71 T^{6} + \cdots + 30976 Copy content Toggle raw display
2323 (T43T313T2+36)2 (T^{4} - 3 T^{3} - 13 T^{2} + \cdots - 36)^{2} Copy content Toggle raw display
2929 (T4T383T2++1182)2 (T^{4} - T^{3} - 83 T^{2} + \cdots + 1182)^{2} Copy content Toggle raw display
3131 T8+123T6++322624 T^{8} + 123 T^{6} + \cdots + 322624 Copy content Toggle raw display
3737 T8+212T6++20736 T^{8} + 212 T^{6} + \cdots + 20736 Copy content Toggle raw display
4141 T8+156T6++123904 T^{8} + 156 T^{6} + \cdots + 123904 Copy content Toggle raw display
4343 (T4+3T3++232)2 (T^{4} + 3 T^{3} + \cdots + 232)^{2} Copy content Toggle raw display
4747 T8+239T6++4032064 T^{8} + 239 T^{6} + \cdots + 4032064 Copy content Toggle raw display
5353 (T4+11T3+54)2 (T^{4} + 11 T^{3} + \cdots - 54)^{2} Copy content Toggle raw display
5959 T8+180T6++147456 T^{8} + 180 T^{6} + \cdots + 147456 Copy content Toggle raw display
6161 (T44T3112T2+16)2 (T^{4} - 4 T^{3} - 112 T^{2} + \cdots - 16)^{2} Copy content Toggle raw display
6767 T8+368T6++11943936 T^{8} + 368 T^{6} + \cdots + 11943936 Copy content Toggle raw display
7171 T8+408T6++2214144 T^{8} + 408 T^{6} + \cdots + 2214144 Copy content Toggle raw display
7373 T8+279T6++2421136 T^{8} + 279 T^{6} + \cdots + 2421136 Copy content Toggle raw display
7979 (T4+13T3+36)2 (T^{4} + 13 T^{3} + \cdots - 36)^{2} Copy content Toggle raw display
8383 T8+547T6++75481344 T^{8} + 547 T^{6} + \cdots + 75481344 Copy content Toggle raw display
8989 T8+439T6++57335184 T^{8} + 439 T^{6} + \cdots + 57335184 Copy content Toggle raw display
9797 T8+447T6++75134224 T^{8} + 447 T^{6} + \cdots + 75134224 Copy content Toggle raw display
show more
show less