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Properties

Label 841.2.d.j
Level 841841
Weight 22
Character orbit 841.d
Analytic conductor 6.7156.715
Analytic rank 00
Dimension 1212
Inner twists 66

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [841,2,Mod(190,841)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(841, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("841.190");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 841=292 841 = 29^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 841.d (of order 77, degree 66, 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: 6.715418809996.71541880999
Analytic rank: 00
Dimension: 1212
Relative dimension: 22 over Q(ζ7)\Q(\zeta_{7})
Coefficient field: 12.0.74049191673856.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x12+2x10+4x8+8x6+16x4+32x2+64 x^{12} + 2x^{10} + 4x^{8} + 8x^{6} + 16x^{4} + 32x^{2} + 64 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: SU(2)[C7]\mathrm{SU}(2)[C_{7}]

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,…,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β9−β2)q2+(β10−β3)q3+(−2β11+β4)q4−β2q5+(3β10+3β8+3β6+⋯+3)q6+2β3q7+⋯+(−2β7−4)q99+O(q100) q + (\beta_{9} - \beta_{2}) q^{2} + (\beta_{10} - \beta_{3}) q^{3} + ( - 2 \beta_{11} + \beta_{4}) q^{4} - \beta_{2} q^{5} + (3 \beta_{10} + 3 \beta_{8} + 3 \beta_{6} + \cdots + 3) q^{6} + 2 \beta_{3} q^{7}+ \cdots + ( - 2 \beta_{7} - 4) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+2q2−2q3−2q4+2q5+6q6+6q8−2q10−2q11+60q12+2q13−8q14+2q15−6q16−24q17+8q18−12q19+2q20+8q21−2q22+⋯−48q99+O(q100) 12 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 6 q^{6} + 6 q^{8} - 2 q^{10} - 2 q^{11} + 60 q^{12} + 2 q^{13} - 8 q^{14} + 2 q^{15} - 6 q^{16} - 24 q^{17} + 8 q^{18} - 12 q^{19} + 2 q^{20} + 8 q^{21} - 2 q^{22}+ \cdots - 48 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12+2x10+4x8+8x6+16x4+32x2+64 x^{12} + 2x^{10} + 4x^{8} + 8x^{6} + 16x^{4} + 32x^{2} + 64 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν2)/2 ( \nu^{2} ) / 2 Copy content Toggle raw display
β3\beta_{3}== (ν3)/2 ( \nu^{3} ) / 2 Copy content Toggle raw display
β4\beta_{4}== (ν4)/4 ( \nu^{4} ) / 4 Copy content Toggle raw display
β5\beta_{5}== (ν5)/4 ( \nu^{5} ) / 4 Copy content Toggle raw display
β6\beta_{6}== (ν6)/8 ( \nu^{6} ) / 8 Copy content Toggle raw display
β7\beta_{7}== (ν7)/8 ( \nu^{7} ) / 8 Copy content Toggle raw display
β8\beta_{8}== (ν8)/16 ( \nu^{8} ) / 16 Copy content Toggle raw display
β9\beta_{9}== (ν9)/16 ( \nu^{9} ) / 16 Copy content Toggle raw display
β10\beta_{10}== (ν10)/32 ( \nu^{10} ) / 32 Copy content Toggle raw display
β11\beta_{11}== (ν11)/32 ( \nu^{11} ) / 32 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== 2β2 2\beta_{2} Copy content Toggle raw display
ν3\nu^{3}== 2β3 2\beta_{3} Copy content Toggle raw display
ν4\nu^{4}== 4β4 4\beta_{4} Copy content Toggle raw display
ν5\nu^{5}== 4β5 4\beta_{5} Copy content Toggle raw display
ν6\nu^{6}== 8β6 8\beta_{6} Copy content Toggle raw display
ν7\nu^{7}== 8β7 8\beta_{7} Copy content Toggle raw display
ν8\nu^{8}== 16β8 16\beta_{8} Copy content Toggle raw display
ν9\nu^{9}== 16β9 16\beta_{9} Copy content Toggle raw display
ν10\nu^{10}== 32β10 32\beta_{10} Copy content Toggle raw display
ν11\nu^{11}== 32β11 32\beta_{11} Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/841Z)×\left(\mathbb{Z}/841\mathbb{Z}\right)^\times.

nn 22
χ(n)\chi(n) β4\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}
190.1
−0.314692 + 1.37876i
0.314692 − 1.37876i
−0.314692 − 1.37876i
0.314692 + 1.37876i
1.27416 − 0.613604i
−1.27416 + 0.613604i
0.881748 + 1.10568i
−0.881748 − 1.10568i
0.881748 − 1.10568i
−0.881748 + 1.10568i
1.27416 + 0.613604i
−1.27416 − 0.613604i
−0.373194 − 0.179721i −0.258258 + 0.323845i −1.14001 − 1.42952i 0.900969 + 0.433884i 0.154582 − 0.0744427i 1.76350 − 2.21135i 0.352871 + 1.54603i 0.629384 + 2.75751i −0.258258 − 0.323845i
190.2 2.17513 + 1.04749i 1.50524 − 1.88751i 2.38699 + 2.99318i 0.900969 + 0.433884i 5.25123 − 2.52886i −1.76350 + 2.21135i 0.982255 + 4.30354i −0.629384 − 2.75751i 1.50524 + 1.88751i
571.1 −0.373194 + 0.179721i −0.258258 − 0.323845i −1.14001 + 1.42952i 0.900969 − 0.433884i 0.154582 + 0.0744427i 1.76350 + 2.21135i 0.352871 − 1.54603i 0.629384 − 2.75751i −0.258258 + 0.323845i
571.2 2.17513 − 1.04749i 1.50524 + 1.88751i 2.38699 − 2.99318i 0.900969 − 0.433884i 5.25123 + 2.52886i −1.76350 − 2.21135i 0.982255 − 4.30354i −0.629384 + 2.75751i 1.50524 − 1.88751i
574.1 −1.50524 + 1.88751i −0.537213 + 2.35368i −0.851905 − 3.73244i −0.623490 + 0.781831i −3.63396 − 4.55685i 0.629384 − 2.75751i 3.97707 + 1.91526i −2.54832 − 1.22721i −0.537213 − 2.35368i
574.2 0.258258 − 0.323845i 0.0921712 − 0.403828i 0.406863 + 1.78258i −0.623490 + 0.781831i −0.106974 − 0.134141i −0.629384 + 2.75751i 1.42874 + 0.688047i 2.54832 + 1.22721i 0.0921712 + 0.403828i
605.1 −0.0921712 + 0.403828i 0.373194 − 0.179721i 1.64736 + 0.793325i 0.222521 − 0.974928i 0.0381786 + 0.167271i −2.54832 + 1.22721i −0.988722 + 1.23982i −1.76350 + 2.21135i 0.373194 + 0.179721i
605.2 0.537213 − 2.35368i −2.17513 + 1.04749i −3.44929 − 1.66109i 0.222521 − 0.974928i 1.29695 + 5.68230i 2.54832 − 1.22721i −2.75222 + 3.45117i 1.76350 − 2.21135i −2.17513 − 1.04749i
645.1 −0.0921712 − 0.403828i 0.373194 + 0.179721i 1.64736 − 0.793325i 0.222521 + 0.974928i 0.0381786 − 0.167271i −2.54832 − 1.22721i −0.988722 − 1.23982i −1.76350 − 2.21135i 0.373194 − 0.179721i
645.2 0.537213 + 2.35368i −2.17513 − 1.04749i −3.44929 + 1.66109i 0.222521 + 0.974928i 1.29695 − 5.68230i 2.54832 + 1.22721i −2.75222 − 3.45117i 1.76350 + 2.21135i −2.17513 + 1.04749i
778.1 −1.50524 − 1.88751i −0.537213 − 2.35368i −0.851905 + 3.73244i −0.623490 − 0.781831i −3.63396 + 4.55685i 0.629384 + 2.75751i 3.97707 − 1.91526i −2.54832 + 1.22721i −0.537213 + 2.35368i
778.2 0.258258 + 0.323845i 0.0921712 + 0.403828i 0.406863 − 1.78258i −0.623490 − 0.781831i −0.106974 + 0.134141i −0.629384 − 2.75751i 1.42874 − 0.688047i 2.54832 − 1.22721i 0.0921712 − 0.403828i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 190.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 5 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.d.j 12
29.b even 2 1 841.2.d.f 12
29.c odd 4 2 841.2.e.k 24
29.d even 7 1 29.2.a.a ✓ 2
29.d even 7 5 inner 841.2.d.j 12
29.e even 14 1 841.2.a.d 2
29.e even 14 5 841.2.d.f 12
29.f odd 28 2 841.2.b.a 4
29.f odd 28 10 841.2.e.k 24
87.h odd 14 1 7569.2.a.c 2
87.j odd 14 1 261.2.a.d 2
116.j odd 14 1 464.2.a.h 2
145.n even 14 1 725.2.a.b 2
145.p odd 28 2 725.2.b.b 4
203.n odd 14 1 1421.2.a.j 2
232.p odd 14 1 1856.2.a.w 2
232.s even 14 1 1856.2.a.r 2
319.m odd 14 1 3509.2.a.j 2
348.s even 14 1 4176.2.a.bq 2
377.w even 14 1 4901.2.a.g 2
435.w odd 14 1 6525.2.a.o 2
493.p even 14 1 8381.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a ✓ 2 29.d even 7 1
261.2.a.d 2 87.j odd 14 1
464.2.a.h 2 116.j odd 14 1
725.2.a.b 2 145.n even 14 1
725.2.b.b 4 145.p odd 28 2
841.2.a.d 2 29.e even 14 1
841.2.b.a 4 29.f odd 28 2
841.2.d.f 12 29.b even 2 1
841.2.d.f 12 29.e even 14 5
841.2.d.j 12 1.a even 1 1 trivial
841.2.d.j 12 29.d even 7 5 inner
841.2.e.k 24 29.c odd 4 2
841.2.e.k 24 29.f odd 28 10
1421.2.a.j 2 203.n odd 14 1
1856.2.a.r 2 232.s even 14 1
1856.2.a.w 2 232.p odd 14 1
3509.2.a.j 2 319.m odd 14 1
4176.2.a.bq 2 348.s even 14 1
4901.2.a.g 2 377.w even 14 1
6525.2.a.o 2 435.w odd 14 1
7569.2.a.c 2 87.h odd 14 1
8381.2.a.e 2 493.p even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T212−2T211+5T210−12T29+29T28−70T27+169T26+⋯+1 T_{2}^{12} - 2 T_{2}^{11} + 5 T_{2}^{10} - 12 T_{2}^{9} + 29 T_{2}^{8} - 70 T_{2}^{7} + 169 T_{2}^{6} + \cdots + 1 acting on S2new(841,[χ])S_{2}^{\mathrm{new}}(841, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12−2T11+⋯+1 T^{12} - 2 T^{11} + \cdots + 1 Copy content Toggle raw display
33 T12+2T11+⋯+1 T^{12} + 2 T^{11} + \cdots + 1 Copy content Toggle raw display
55 (T6−T5+T4+⋯+1)2 (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} Copy content Toggle raw display
77 T12+8T10+⋯+262144 T^{12} + 8 T^{10} + \cdots + 262144 Copy content Toggle raw display
1111 T12+2T11+⋯+1 T^{12} + 2 T^{11} + \cdots + 1 Copy content Toggle raw display
1313 T12−2T11+⋯+117649 T^{12} - 2 T^{11} + \cdots + 117649 Copy content Toggle raw display
1717 (T2+4T−4)6 (T^{2} + 4 T - 4)^{6} Copy content Toggle raw display
1919 (T6+6T5+⋯+46656)2 (T^{6} + 6 T^{5} + \cdots + 46656)^{2} Copy content Toggle raw display
2323 T12+⋯+481890304 T^{12} + \cdots + 481890304 Copy content Toggle raw display
2929 T12 T^{12} Copy content Toggle raw display
3131 T12+⋯+4750104241 T^{12} + \cdots + 4750104241 Copy content Toggle raw display
3737 (T6−4T5+⋯+4096)2 (T^{6} - 4 T^{5} + \cdots + 4096)^{2} Copy content Toggle raw display
4141 (T2−8T−56)6 (T^{2} - 8 T - 56)^{6} Copy content Toggle raw display
4343 T12+⋯+148035889 T^{12} + \cdots + 148035889 Copy content Toggle raw display
4747 T12+2T11+⋯+24137569 T^{12} + 2 T^{11} + \cdots + 24137569 Copy content Toggle raw display
5353 T12+⋯+128100283921 T^{12} + \cdots + 128100283921 Copy content Toggle raw display
5959 (T2−4T−28)6 (T^{2} - 4 T - 28)^{6} Copy content Toggle raw display
6161 T12−4T11+⋯+4096 T^{12} - 4 T^{11} + \cdots + 4096 Copy content Toggle raw display
6767 T12+⋯+1073741824 T^{12} + \cdots + 1073741824 Copy content Toggle raw display
7171 T12+⋯+481890304 T^{12} + \cdots + 481890304 Copy content Toggle raw display
7373 (T6+4T5+⋯+4096)2 (T^{6} + 4 T^{5} + \cdots + 4096)^{2} Copy content Toggle raw display
7979 T12−2T11+⋯+1 T^{12} - 2 T^{11} + \cdots + 1 Copy content Toggle raw display
8383 T12+⋯+481890304 T^{12} + \cdots + 481890304 Copy content Toggle raw display
8989 T12+⋯+30840979456 T^{12} + \cdots + 30840979456 Copy content Toggle raw display
9797 T12+⋯+30840979456 T^{12} + \cdots + 30840979456 Copy content Toggle raw display
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