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Properties

Label 4598.2.a.by
Level 45984598
Weight 22
Character orbit 4598.a
Self dual yes
Analytic conductor 36.71536.715
Analytic rank 00
Dimension 88
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4598,2,Mod(1,4598)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4598, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4598.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 4598=2⋅112⋅19 4598 = 2 \cdot 11^{2} \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 4598.a (trivial)

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: yes
Analytic conductor: 36.715214849436.7152148494
Analytic rank: 00
Dimension: 88
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: x8−16x6−4x5+75x4+32x3−90x2−28x−2 x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: −1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

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−q2+(β6+1)q3+q4+(β6+β5+⋯−β2)q5+(−β6−1)q6+(β3+β2+β1−1)q7−q8+(β7+β6−β4+⋯+3)q9+⋯+(3β6−β5−β4+⋯−4)q98+O(q100) q - q^{2} + (\beta_{6} + 1) q^{3} + q^{4} + (\beta_{6} + \beta_{5} + \cdots - \beta_{2}) q^{5} + ( - \beta_{6} - 1) q^{6} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots + 3) q^{9}+ \cdots + (3 \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 4) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−8q2+8q3+8q4−8q6−4q7−8q8+22q9+8q12+12q13+4q14+4q15+8q16+4q17−22q18+8q19+20q21+14q23−8q24+36q25+⋯−34q98+O(q100) 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25}+ \cdots - 34 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−16x6−4x5+75x4+32x3−90x2−28x−2 x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν7−ν6−15ν5+11ν4+64ν3−24ν2−66ν−2)/8 ( \nu^{7} - \nu^{6} - 15\nu^{5} + 11\nu^{4} + 64\nu^{3} - 24\nu^{2} - 66\nu - 2 ) / 8 Copy content Toggle raw display
β3\beta_{3}== (ν6−2ν5−13ν4+20ν3+44ν2−40ν−18)/4 ( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 20\nu^{3} + 44\nu^{2} - 40\nu - 18 ) / 4 Copy content Toggle raw display
β4\beta_{4}== (ν7+ν6−19ν5−15ν4+108ν3+56ν2−166ν−18)/4 ( \nu^{7} + \nu^{6} - 19\nu^{5} - 15\nu^{4} + 108\nu^{3} + 56\nu^{2} - 166\nu - 18 ) / 4 Copy content Toggle raw display
β5\beta_{5}== (3ν7−2ν6−45ν5+14ν4+196ν3+8ν2−218ν−32)/4 ( 3\nu^{7} - 2\nu^{6} - 45\nu^{5} + 14\nu^{4} + 196\nu^{3} + 8\nu^{2} - 218\nu - 32 ) / 4 Copy content Toggle raw display
β6\beta_{6}== (5ν7+3ν6−87ν5−53ν4+448ν3+232ν2−610ν−98)/8 ( 5\nu^{7} + 3\nu^{6} - 87\nu^{5} - 53\nu^{4} + 448\nu^{3} + 232\nu^{2} - 610\nu - 98 ) / 8 Copy content Toggle raw display
β7\beta_{7}== (7ν7−3ν6−109ν5+13ν4+496ν3+72ν2−582ν−78)/8 ( 7\nu^{7} - 3\nu^{6} - 109\nu^{5} + 13\nu^{4} + 496\nu^{3} + 72\nu^{2} - 582\nu - 78 ) / 8 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}== −β7+β5+β3+β2+3 -\beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 3 Copy content Toggle raw display
ν3\nu^{3}== −2β7+2β5+β4+5β1+1 -2\beta_{7} + 2\beta_{5} + \beta_{4} + 5\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== −9β7+β6+8β5+6β3+10β2+18 -9\beta_{7} + \beta_{6} + 8\beta_{5} + 6\beta_{3} + 10\beta_{2} + 18 Copy content Toggle raw display
ν5\nu^{5}== −25β7+3β6+24β5+8β4−2β3+30β1+12 -25\beta_{7} + 3\beta_{6} + 24\beta_{5} + 8\beta_{4} - 2\beta_{3} + 30\beta _1 + 12 Copy content Toggle raw display
ν6\nu^{6}== −83β7+19β6+68β5−4β4+34β3+86β2+124 -83\beta_{7} + 19\beta_{6} + 68\beta_{5} - 4\beta_{4} + 34\beta_{3} + 86\beta_{2} + 124 Copy content Toggle raw display
ν7\nu^{7}== −255β7+53β6+236β5+52β4−38β3+8β2+196β1+116 -255\beta_{7} + 53\beta_{6} + 236\beta_{5} + 52\beta_{4} - 38\beta_{3} + 8\beta_{2} + 196\beta _1 + 116 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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}
1.1
−1.93433
−0.115899
2.55131
3.02984
−1.84109
1.24609
−0.182716
−2.75320
−1.00000 −2.48823 1.00000 3.25915 2.48823 −4.58972 −1.00000 3.19127 −3.25915
1.2 −1.00000 −2.11129 1.00000 −2.75070 2.11129 −3.66381 −1.00000 1.45756 2.75070
1.3 −1.00000 −0.103249 1.00000 −3.91798 0.103249 4.45623 −1.00000 −2.98934 3.91798
1.4 −1.00000 1.55594 1.00000 2.24033 −1.55594 2.27752 −1.00000 −0.579065 −2.24033
1.5 −1.00000 1.67352 1.00000 −0.566489 −1.67352 −3.40033 −1.00000 −0.199325 0.566489
1.6 −1.00000 2.79123 1.00000 4.02325 −2.79123 −0.274917 −1.00000 4.79095 −4.02325
1.7 −1.00000 3.30349 1.00000 1.88260 −3.30349 −2.41197 −1.00000 7.91304 −1.88260
1.8 −1.00000 3.37860 1.00000 −4.17017 −3.37860 3.60700 −1.00000 8.41492 4.17017
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
1111 +1 +1
1919 −1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4598.2.a.by ✓ 8
11.b odd 2 1 4598.2.a.cb yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.by ✓ 8 1.a even 1 1 trivial
4598.2.a.cb yes 8 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(4598))S_{2}^{\mathrm{new}}(\Gamma_0(4598)):

T38−8T37+9T36+72T35−183T34−80T33+557T32−368T3−44 T_{3}^{8} - 8T_{3}^{7} + 9T_{3}^{6} + 72T_{3}^{5} - 183T_{3}^{4} - 80T_{3}^{3} + 557T_{3}^{2} - 368T_{3} - 44 Copy content Toggle raw display
T58−38T56+12T55+470T54−288T53−1984T52+1536T5+1408 T_{5}^{8} - 38T_{5}^{6} + 12T_{5}^{5} + 470T_{5}^{4} - 288T_{5}^{3} - 1984T_{5}^{2} + 1536T_{5} + 1408 Copy content Toggle raw display
T78+4T77−37T76−152T75+387T74+1716T73−867T72−5408T7−1388 T_{7}^{8} + 4T_{7}^{7} - 37T_{7}^{6} - 152T_{7}^{5} + 387T_{7}^{4} + 1716T_{7}^{3} - 867T_{7}^{2} - 5408T_{7} - 1388 Copy content Toggle raw display
T138−12T137+34T136+84T135−451T134+72T133+1088T132−96T13−368 T_{13}^{8} - 12T_{13}^{7} + 34T_{13}^{6} + 84T_{13}^{5} - 451T_{13}^{4} + 72T_{13}^{3} + 1088T_{13}^{2} - 96T_{13} - 368 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T+1)8 (T + 1)^{8} Copy content Toggle raw display
33 T8−8T7+⋯−44 T^{8} - 8 T^{7} + \cdots - 44 Copy content Toggle raw display
55 T8−38T6+⋯+1408 T^{8} - 38 T^{6} + \cdots + 1408 Copy content Toggle raw display
77 T8+4T7+⋯−1388 T^{8} + 4 T^{7} + \cdots - 1388 Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 T8−12T7+⋯−368 T^{8} - 12 T^{7} + \cdots - 368 Copy content Toggle raw display
1717 T8−4T7+⋯−92 T^{8} - 4 T^{7} + \cdots - 92 Copy content Toggle raw display
1919 (T−1)8 (T - 1)^{8} Copy content Toggle raw display
2323 T8−14T7+⋯+35296 T^{8} - 14 T^{7} + \cdots + 35296 Copy content Toggle raw display
2929 T8−2T7+⋯+964 T^{8} - 2 T^{7} + \cdots + 964 Copy content Toggle raw display
3131 T8−132T6+⋯−9344 T^{8} - 132 T^{6} + \cdots - 9344 Copy content Toggle raw display
3737 T8−24T7+⋯+29569 T^{8} - 24 T^{7} + \cdots + 29569 Copy content Toggle raw display
4141 T8+8T7+⋯−3872 T^{8} + 8 T^{7} + \cdots - 3872 Copy content Toggle raw display
4343 T8+8T7+⋯−67328 T^{8} + 8 T^{7} + \cdots - 67328 Copy content Toggle raw display
4747 T8+16T7+⋯−2659283 T^{8} + 16 T^{7} + \cdots - 2659283 Copy content Toggle raw display
5353 T8−36T7+⋯+2770816 T^{8} - 36 T^{7} + \cdots + 2770816 Copy content Toggle raw display
5959 T8+24T7+⋯−20032604 T^{8} + 24 T^{7} + \cdots - 20032604 Copy content Toggle raw display
6161 T8+12T7+⋯+16192 T^{8} + 12 T^{7} + \cdots + 16192 Copy content Toggle raw display
6767 T8−16T7+⋯+457168 T^{8} - 16 T^{7} + \cdots + 457168 Copy content Toggle raw display
7171 T8−4T7+⋯+3806464 T^{8} - 4 T^{7} + \cdots + 3806464 Copy content Toggle raw display
7373 T8−20T7+⋯−37382204 T^{8} - 20 T^{7} + \cdots - 37382204 Copy content Toggle raw display
7979 T8−12T7+⋯+51808 T^{8} - 12 T^{7} + \cdots + 51808 Copy content Toggle raw display
8383 T8+20T7+⋯−251648 T^{8} + 20 T^{7} + \cdots - 251648 Copy content Toggle raw display
8989 T8−8T7+⋯−320672 T^{8} - 8 T^{7} + \cdots - 320672 Copy content Toggle raw display
9797 T8−4T7+⋯−1921664 T^{8} - 4 T^{7} + \cdots - 1921664 Copy content Toggle raw display
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