Properties

Label 128.12.a.e
Level 128128
Weight 1212
Character orbit 128.a
Self dual yes
Analytic conductor 98.34898.348
Analytic rank 00
Dimension 66
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] = [128,12,Mod(1,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: N N == 128=27 128 = 2^{7}
Weight: k k == 12 12
Character orbit: [χ][\chi] == 128.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: 98.347927111698.3479271116
Analytic rank: 00
Dimension: 66
Coefficient field: Q[x]/(x6)\mathbb{Q}[x]/(x^{6} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x62x54442x4+153566x31333532x24433532x+49754286 x^{6} - 2x^{5} - 4442x^{4} + 153566x^{3} - 1333532x^{2} - 4433532x + 49754286 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 24332 2^{43}\cdot 3^{2}
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,,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+(β13)q3+(β24β1299)q5+(β3+β2++8233)q7+(β4β3+4β2++52271)q9+(β5+β4+114826)q11++(35428β5+63532β4++40042305135)q99+O(q100) q + ( - \beta_1 - 3) q^{3} + (\beta_{2} - 4 \beta_1 - 299) q^{5} + ( - \beta_{3} + \beta_{2} + \cdots + 8233) q^{7} + (\beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots + 52271) q^{9} + ( - \beta_{5} + \beta_{4} + \cdots - 114826) q^{11}+ \cdots + (35428 \beta_{5} + 63532 \beta_{4} + \cdots + 40042305135) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q20q31804q5+49368q7+313814q9688460q112290348q13+4828264q15+4127636q17+9936364q19+20325616q21+9921320q23+51633002q25132503384q27++240482467988q99+O(q100) 6 q - 20 q^{3} - 1804 q^{5} + 49368 q^{7} + 313814 q^{9} - 688460 q^{11} - 2290348 q^{13} + 4828264 q^{15} + 4127636 q^{17} + 9936364 q^{19} + 20325616 q^{21} + 9921320 q^{23} + 51633002 q^{25} - 132503384 q^{27}+ \cdots + 240482467988 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x62x54442x4+153566x31333532x24433532x+49754286 x^{6} - 2x^{5} - 4442x^{4} + 153566x^{3} - 1333532x^{2} - 4433532x + 49754286 : Copy content Toggle raw display

β1\beta_{1}== (22274044ν5418462072ν4+87404493092ν31674466085912ν2++36980862273615)/112429558305 ( - 22274044 \nu^{5} - 418462072 \nu^{4} + 87404493092 \nu^{3} - 1674466085912 \nu^{2} + \cdots + 36980862273615 ) / 112429558305 Copy content Toggle raw display
β2\beta_{2}== (124903048ν5+4134146512ν4460640866040ν3+1939774343120ν2+90301388773827)/112429558305 ( 124903048 \nu^{5} + 4134146512 \nu^{4} - 460640866040 \nu^{3} + 1939774343120 \nu^{2} + \cdots - 90301388773827 ) / 112429558305 Copy content Toggle raw display
β3\beta_{3}== (1172219372ν5+17119693592ν44885995122548ν3+98676597031288ν2+26 ⁣ ⁣09)/37476519435 ( 1172219372 \nu^{5} + 17119693592 \nu^{4} - 4885995122548 \nu^{3} + 98676597031288 \nu^{2} + \cdots - 26\!\cdots\!09 ) / 37476519435 Copy content Toggle raw display
β4\beta_{4}== (3310701220ν590663757192ν4+12727119015164ν3120373678691624ν2++38 ⁣ ⁣73)/37476519435 ( - 3310701220 \nu^{5} - 90663757192 \nu^{4} + 12727119015164 \nu^{3} - 120373678691624 \nu^{2} + \cdots + 38\!\cdots\!73 ) / 37476519435 Copy content Toggle raw display
β5\beta_{5}== (39454874912ν5+547867641152ν4165635503038688ν3+20 ⁣ ⁣39)/112429558305 ( 39454874912 \nu^{5} + 547867641152 \nu^{4} - 165635503038688 \nu^{3} + \cdots - 20\!\cdots\!39 ) / 112429558305 Copy content Toggle raw display
ν\nu== (β4+β3+56β2+26β1+4106)/12288 ( \beta_{4} + \beta_{3} + 56\beta_{2} + 26\beta _1 + 4106 ) / 12288 Copy content Toggle raw display
ν2\nu^{2}== (37β593β4551β34974β27876β1+36406075)/24576 ( 37\beta_{5} - 93\beta_{4} - 551\beta_{3} - 4974\beta_{2} - 7876\beta _1 + 36406075 ) / 24576 Copy content Toggle raw display
ν3\nu^{3}== (3561β5+20459β4+55565β3+1049350β2773528β13555021155)/49152 ( -3561\beta_{5} + 20459\beta_{4} + 55565\beta_{3} + 1049350\beta_{2} - 773528\beta _1 - 3555021155 ) / 49152 Copy content Toggle raw display
ν4\nu^{4}== (367135β51449757β45846283β374555306β244318904β1+357482673941)/49152 ( 367135 \beta_{5} - 1449757 \beta_{4} - 5846283 \beta_{3} - 74555306 \beta_{2} - 44318904 \beta _1 + 357482673941 ) / 49152 Copy content Toggle raw display
ν5\nu^{5}== (26433905β5+122446027β4+411662141β3+6319119158β2+26054324125227)/49152 ( - 26433905 \beta_{5} + 122446027 \beta_{4} + 411662141 \beta_{3} + 6319119158 \beta_{2} + \cdots - 26054324125227 ) / 49152 Copy content Toggle raw display

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
23.8940
6.34629
18.7879
−80.0003
−5.65189
38.6241
0 −795.850 0 −3925.84 0 13521.5 0 456230. 0
1.2 0 −219.634 0 4589.61 0 25262.2 0 −128908. 0
1.3 0 −181.938 0 4531.11 0 −62513.8 0 −144046. 0
1.4 0 28.4295 0 −13262.1 0 35677.5 0 −176339. 0
1.5 0 551.275 0 −3654.34 0 −40487.8 0 126757. 0
1.6 0 597.717 0 9917.53 0 77908.4 0 180119. 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +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 128.12.a.e 6
4.b odd 2 1 128.12.a.g yes 6
8.b even 2 1 128.12.a.h yes 6
8.d odd 2 1 128.12.a.f yes 6
16.e even 4 2 256.12.b.p 12
16.f odd 4 2 256.12.b.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.12.a.e 6 1.a even 1 1 trivial
128.12.a.f yes 6 8.d odd 2 1
128.12.a.g yes 6 4.b odd 2 1
128.12.a.h yes 6 8.b even 2 1
256.12.b.p 12 16.e even 4 2
256.12.b.q 12 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S12new(Γ0(128))S_{12}^{\mathrm{new}}(\Gamma_0(128)):

T36+20T35688148T34+32764128T33+81557676144T32+8149590626112T3297910794836160 T_{3}^{6} + 20T_{3}^{5} - 688148T_{3}^{4} + 32764128T_{3}^{3} + 81557676144T_{3}^{2} + 8149590626112T_{3} - 297910794836160 Copy content Toggle raw display
T56+1804T55170673668T54+115715684000T53+39 ⁣ ⁣00 T_{5}^{6} + 1804 T_{5}^{5} - 170673668 T_{5}^{4} + 115715684000 T_{5}^{3} + \cdots - 39\!\cdots\!00 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6 T^{6} Copy content Toggle raw display
33 T6+297910794836160 T^{6} + \cdots - 297910794836160 Copy content Toggle raw display
55 T6+39 ⁣ ⁣00 T^{6} + \cdots - 39\!\cdots\!00 Copy content Toggle raw display
77 T6++24 ⁣ ⁣40 T^{6} + \cdots + 24\!\cdots\!40 Copy content Toggle raw display
1111 T6+55 ⁣ ⁣28 T^{6} + \cdots - 55\!\cdots\!28 Copy content Toggle raw display
1313 T6+31 ⁣ ⁣40 T^{6} + \cdots - 31\!\cdots\!40 Copy content Toggle raw display
1717 T6+49 ⁣ ⁣40 T^{6} + \cdots - 49\!\cdots\!40 Copy content Toggle raw display
1919 T6++21 ⁣ ⁣40 T^{6} + \cdots + 21\!\cdots\!40 Copy content Toggle raw display
2323 T6+21 ⁣ ⁣96 T^{6} + \cdots - 21\!\cdots\!96 Copy content Toggle raw display
2929 T6+11 ⁣ ⁣28 T^{6} + \cdots - 11\!\cdots\!28 Copy content Toggle raw display
3131 T6++20 ⁣ ⁣48 T^{6} + \cdots + 20\!\cdots\!48 Copy content Toggle raw display
3737 T6++15 ⁣ ⁣12 T^{6} + \cdots + 15\!\cdots\!12 Copy content Toggle raw display
4141 T6++75 ⁣ ⁣92 T^{6} + \cdots + 75\!\cdots\!92 Copy content Toggle raw display
4343 T6+91 ⁣ ⁣00 T^{6} + \cdots - 91\!\cdots\!00 Copy content Toggle raw display
4747 T6++16 ⁣ ⁣00 T^{6} + \cdots + 16\!\cdots\!00 Copy content Toggle raw display
5353 T6+19 ⁣ ⁣40 T^{6} + \cdots - 19\!\cdots\!40 Copy content Toggle raw display
5959 T6+14 ⁣ ⁣80 T^{6} + \cdots - 14\!\cdots\!80 Copy content Toggle raw display
6161 T6++71 ⁣ ⁣20 T^{6} + \cdots + 71\!\cdots\!20 Copy content Toggle raw display
6767 T6+22 ⁣ ⁣76 T^{6} + \cdots - 22\!\cdots\!76 Copy content Toggle raw display
7171 T6++10 ⁣ ⁣20 T^{6} + \cdots + 10\!\cdots\!20 Copy content Toggle raw display
7373 T6+29 ⁣ ⁣80 T^{6} + \cdots - 29\!\cdots\!80 Copy content Toggle raw display
7979 T6++13 ⁣ ⁣60 T^{6} + \cdots + 13\!\cdots\!60 Copy content Toggle raw display
8383 T6+31 ⁣ ⁣72 T^{6} + \cdots - 31\!\cdots\!72 Copy content Toggle raw display
8989 T6+60 ⁣ ⁣84 T^{6} + \cdots - 60\!\cdots\!84 Copy content Toggle raw display
9797 T6+23 ⁣ ⁣84 T^{6} + \cdots - 23\!\cdots\!84 Copy content Toggle raw display
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