Properties

Label 416.4.i.d
Level 416416
Weight 44
Character orbit 416.i
Analytic conductor 24.54524.545
Analytic rank 00
Dimension 88
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,4,Mod(289,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 416=2513 416 = 2^{5} \cdot 13
Weight: k k == 4 4
Character orbit: [χ][\chi] == 416.i (of order 33, degree 22, 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: 24.544794562424.5447945624
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x8+)\mathbb{Q}[x]/(x^{8} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8+74x6+5367x4+8066x2+11881 x^{8} + 74x^{6} + 5367x^{4} + 8066x^{2} + 11881 Copy content Toggle raw display
Coefficient ring: Z[a1,,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 26 2^{6}
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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,,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+β5q3+(β2+2)q5+(β3+2β1)q7+(3β6+10β410)q9+(2β75β5)q11+(52β4+13)q13+(β79β5)q15++(61β7+101β5++101β1)q99+O(q100) q + \beta_{5} q^{3} + (\beta_{2} + 2) q^{5} + ( - \beta_{3} + 2 \beta_1) q^{7} + (3 \beta_{6} + 10 \beta_{4} - 10) q^{9} + (2 \beta_{7} - 5 \beta_{5}) q^{11} + ( - 52 \beta_{4} + 13) q^{13} + ( - \beta_{7} - 9 \beta_{5}) q^{15}+ \cdots + (61 \beta_{7} + 101 \beta_{5} + \cdots + 101 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+16q540q9104q13+180q17+696q21+152q25+4q29+636q33108q37716q41+1600q451096q49+4528q53104q57+580q61208q65+156q97+O(q100) 8 q + 16 q^{5} - 40 q^{9} - 104 q^{13} + 180 q^{17} + 696 q^{21} + 152 q^{25} + 4 q^{29} + 636 q^{33} - 108 q^{37} - 716 q^{41} + 1600 q^{45} - 1096 q^{49} + 4528 q^{53} - 104 q^{57} + 580 q^{61} - 208 q^{65}+ \cdots - 156 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+74x6+5367x4+8066x2+11881 x^{8} + 74x^{6} + 5367x^{4} + 8066x^{2} + 11881 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν6+190513)/16101 ( -\nu^{6} + 190513 ) / 16101 Copy content Toggle raw display
β3\beta_{3}== (ν7+367624ν)/16101 ( -\nu^{7} + 367624\nu ) / 16101 Copy content Toggle raw display
β4\beta_{4}== (74ν65367ν4397158ν211881)/585003 ( -74\nu^{6} - 5367\nu^{4} - 397158\nu^{2} - 11881 ) / 585003 Copy content Toggle raw display
β5\beta_{5}== (74ν75367ν5397158ν3596884ν)/585003 ( -74\nu^{7} - 5367\nu^{5} - 397158\nu^{3} - 596884\nu ) / 585003 Copy content Toggle raw display
β6\beta_{6}== (2629ν6+198579ν4+14109843ν2+21205514)/1755009 ( 2629\nu^{6} + 198579\nu^{4} + 14109843\nu^{2} + 21205514 ) / 1755009 Copy content Toggle raw display
β7\beta_{7}== (5071ν7+375690ν5+27216057ν3+40902686ν)/1755009 ( 5071\nu^{7} + 375690\nu^{5} + 27216057\nu^{3} + 40902686\nu ) / 1755009 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== 3β637β4+3β2 -3\beta_{6} - 37\beta_{4} + 3\beta_{2} Copy content Toggle raw display
ν3\nu^{3}== 3β770β5+3β370β1 -3\beta_{7} - 70\beta_{5} + 3\beta_{3} - 70\beta_1 Copy content Toggle raw display
ν4\nu^{4}== 222β6+2629β42629 222\beta_{6} + 2629\beta_{4} - 2629 Copy content Toggle raw display
ν5\nu^{5}== 222β7+5071β5 222\beta_{7} + 5071\beta_{5} Copy content Toggle raw display
ν6\nu^{6}== 16101β2+190513 -16101\beta_{2} + 190513 Copy content Toggle raw display
ν7\nu^{7}== 16101β3+367624β1 -16101\beta_{3} + 367624\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/416Z)×\left(\mathbb{Z}/416\mathbb{Z}\right)^\times.

nn 261261 287287 353353
χ(n)\chi(n) 11 11 1+β4-1 + \beta_{4}

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}
289.1
−4.25724 + 7.37376i
−0.613091 + 1.06190i
0.613091 1.06190i
4.25724 7.37376i
−4.25724 7.37376i
−0.613091 1.06190i
0.613091 + 1.06190i
4.25724 + 7.37376i
0 −4.25724 7.37376i 0 −9.83216 0 −12.0572 + 20.8837i 0 −22.7482 + 39.4011i 0
289.2 0 −0.613091 1.06190i 0 13.8322 0 12.7720 22.1218i 0 12.7482 22.0806i 0
289.3 0 0.613091 + 1.06190i 0 13.8322 0 −12.7720 + 22.1218i 0 12.7482 22.0806i 0
289.4 0 4.25724 + 7.37376i 0 −9.83216 0 12.0572 20.8837i 0 −22.7482 + 39.4011i 0
321.1 0 −4.25724 + 7.37376i 0 −9.83216 0 −12.0572 20.8837i 0 −22.7482 39.4011i 0
321.2 0 −0.613091 + 1.06190i 0 13.8322 0 12.7720 + 22.1218i 0 12.7482 + 22.0806i 0
321.3 0 0.613091 1.06190i 0 13.8322 0 −12.7720 22.1218i 0 12.7482 + 22.0806i 0
321.4 0 4.25724 7.37376i 0 −9.83216 0 12.0572 + 20.8837i 0 −22.7482 39.4011i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.c even 3 1 inner
52.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.4.i.d 8
4.b odd 2 1 inner 416.4.i.d 8
13.c even 3 1 inner 416.4.i.d 8
52.j odd 6 1 inner 416.4.i.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.4.i.d 8 1.a even 1 1 trivial
416.4.i.d 8 4.b odd 2 1 inner
416.4.i.d 8 13.c even 3 1 inner
416.4.i.d 8 52.j odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T38+74T36+5367T34+8066T32+11881 T_{3}^{8} + 74T_{3}^{6} + 5367T_{3}^{4} + 8066T_{3}^{2} + 11881 acting on S4new(416,[χ])S_{4}^{\mathrm{new}}(416, [\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+74T6++11881 T^{8} + 74 T^{6} + \cdots + 11881 Copy content Toggle raw display
55 (T24T136)4 (T^{2} - 4 T - 136)^{4} Copy content Toggle raw display
77 T8++143966366041 T^{8} + \cdots + 143966366041 Copy content Toggle raw display
1111 T8++9691696696201 T^{8} + \cdots + 9691696696201 Copy content Toggle raw display
1313 (T2+26T+2197)4 (T^{2} + 26 T + 2197)^{4} Copy content Toggle raw display
1717 (T490T3++3553225)2 (T^{4} - 90 T^{3} + \cdots + 3553225)^{2} Copy content Toggle raw display
1919 T8++1548343801 T^{8} + \cdots + 1548343801 Copy content Toggle raw display
2323 T8++16 ⁣ ⁣25 T^{8} + \cdots + 16\!\cdots\!25 Copy content Toggle raw display
2929 (T42T3++312481)2 (T^{4} - 2 T^{3} + \cdots + 312481)^{2} Copy content Toggle raw display
3131 (T490624T2+206377984)2 (T^{4} - 90624 T^{2} + 206377984)^{2} Copy content Toggle raw display
3737 (T4+54T3++346921)2 (T^{4} + 54 T^{3} + \cdots + 346921)^{2} Copy content Toggle raw display
4141 (T4+358T3++3075700681)2 (T^{4} + 358 T^{3} + \cdots + 3075700681)^{2} Copy content Toggle raw display
4343 T8++14 ⁣ ⁣61 T^{8} + \cdots + 14\!\cdots\!61 Copy content Toggle raw display
4747 (T4259704T2+9804141904)2 (T^{4} - 259704 T^{2} + 9804141904)^{2} Copy content Toggle raw display
5353 (T21132T+296696)4 (T^{2} - 1132 T + 296696)^{4} Copy content Toggle raw display
5959 T8++2077994325625 T^{8} + \cdots + 2077994325625 Copy content Toggle raw display
6161 (T4290T3++174736540225)2 (T^{4} - 290 T^{3} + \cdots + 174736540225)^{2} Copy content Toggle raw display
6767 T8++1677100110841 T^{8} + \cdots + 1677100110841 Copy content Toggle raw display
7171 T8++90 ⁣ ⁣01 T^{8} + \cdots + 90\!\cdots\!01 Copy content Toggle raw display
7373 (T2564T511976)4 (T^{2} - 564 T - 511976)^{4} Copy content Toggle raw display
7979 (T4++2436698733904)2 (T^{4} + \cdots + 2436698733904)^{2} Copy content Toggle raw display
8383 (T4242656T2+9680483584)2 (T^{4} - 242656 T^{2} + 9680483584)^{2} Copy content Toggle raw display
8989 (T4++1158036406641)2 (T^{4} + \cdots + 1158036406641)^{2} Copy content Toggle raw display
9797 (T4+78T3++327091342561)2 (T^{4} + 78 T^{3} + \cdots + 327091342561)^{2} Copy content Toggle raw display
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