Properties

Label 46.14.a.a
Level 4646
Weight 1414
Character orbit 46.a
Self dual yes
Analytic conductor 49.32649.326
Analytic rank 11
Dimension 55
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] = [46,14,Mod(1,46)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(46, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("46.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: N N == 46=223 46 = 2 \cdot 23
Weight: k k == 14 14
Character orbit: [χ][\chi] == 46.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: 49.326227317949.3262273179
Analytic rank: 11
Dimension: 55
Coefficient field: Q[x]/(x5)\mathbb{Q}[x]/(x^{5} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x5x43257825x3+1372618617x2+279354108456x130815431589168 x^{5} - x^{4} - 3257825x^{3} + 1372618617x^{2} + 279354108456x - 130815431589168 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 2959 2^{9}\cdot 59
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,β2,β3,β41,\beta_1,\beta_2,\beta_3,\beta_4 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q64q2+(β1+138)q3+4096q4+(β4+7β114360)q5+(64β18832)q6+(2β4+3β3++6272)q7262144q8+(19β414β3++704547)q9++(98867847β4+2970699615747)q99+O(q100) q - 64 q^{2} + ( - \beta_1 + 138) q^{3} + 4096 q^{4} + ( - \beta_{4} + 7 \beta_1 - 14360) q^{5} + (64 \beta_1 - 8832) q^{6} + ( - 2 \beta_{4} + 3 \beta_{3} + \cdots + 6272) q^{7} - 262144 q^{8} + (19 \beta_{4} - 14 \beta_{3} + \cdots + 704547) q^{9}+ \cdots + ( - 98867847 \beta_{4} + \cdots - 2970699615747) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 5q320q2+690q3+20480q471798q544160q6+31366q71310720q8+3522701q9+4595072q107592076q11+2826240q12+28315858q132007424q14+14853301668582q99+O(q100) 5 q - 320 q^{2} + 690 q^{3} + 20480 q^{4} - 71798 q^{5} - 44160 q^{6} + 31366 q^{7} - 1310720 q^{8} + 3522701 q^{9} + 4595072 q^{10} - 7592076 q^{11} + 2826240 q^{12} + 28315858 q^{13} - 2007424 q^{14}+ \cdots - 14853301668582 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x5x43257825x3+1372618617x2+279354108456x130815431589168 x^{5} - x^{4} - 3257825x^{3} + 1372618617x^{2} + 279354108456x - 130815431589168 : Copy content Toggle raw display

β1\beta_{1}== (12481ν412829265ν3+32956482977ν2+17271508397517ν3307289103907668)/4598290128456 ( -12481\nu^{4} - 12829265\nu^{3} + 32956482977\nu^{2} + 17271508397517\nu - 3307289103907668 ) / 4598290128456 Copy content Toggle raw display
β2\beta_{2}== (259893ν4582275233ν3659872651633ν2+66 ⁣ ⁣20)/16860397137672 ( 259893 \nu^{4} - 582275233 \nu^{3} - 659872651633 \nu^{2} + \cdots - 66\!\cdots\!20 ) / 16860397137672 Copy content Toggle raw display
β3\beta_{3}== (3998666ν41423823965ν3+12709676152996ν2+16 ⁣ ⁣06)/12645297853254 ( - 3998666 \nu^{4} - 1423823965 \nu^{3} + 12709676152996 \nu^{2} + \cdots - 16\!\cdots\!06 ) / 12645297853254 Copy content Toggle raw display
β4\beta_{4}== (23195195ν48660318164ν3+72151790122993ν2+78 ⁣ ⁣20)/25290595706508 ( - 23195195 \nu^{4} - 8660318164 \nu^{3} + 72151790122993 \nu^{2} + \cdots - 78\!\cdots\!20 ) / 25290595706508 Copy content Toggle raw display
ν\nu== (11β428β3+13β2381β1+385)/1888 ( 11\beta_{4} - 28\beta_{3} + 13\beta_{2} - 381\beta _1 + 385 ) / 1888 Copy content Toggle raw display
ν2\nu^{2}== (5815β4+20144β3+15655β2293031β1+1230145423)/944 ( -5815\beta_{4} + 20144\beta_{3} + 15655\beta_{2} - 293031\beta _1 + 1230145423 ) / 944 Copy content Toggle raw display
ν3\nu^{3}== (38119673β496700036β3+8974447β21563865983β11551179111813)/1888 ( 38119673\beta_{4} - 96700036\beta_{3} + 8974447\beta_{2} - 1563865983\beta _1 - 1551179111813 ) / 1888 Copy content Toggle raw display
ν4\nu^{4}== (6833834341β4+20879125655β3+11429988907β2145354014783β1+948897413820376)/236 ( - 6833834341 \beta_{4} + 20879125655 \beta_{3} + 11429988907 \beta_{2} - 145354014783 \beta _1 + 948897413820376 ) / 236 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
422.240
344.754
−1965.85
−303.975
1503.84
−64.0000 −1710.21 4096.00 −66208.2 109454. −138269. −262144. 1.33050e6 4.23733e6
1.2 −64.0000 −1136.86 4096.00 57154.9 72759.2 −17496.6 −262144. −301866. −3.65791e6
1.3 −64.0000 −115.417 4096.00 8488.20 7386.68 418547. −262144. −1.58100e6 −543245.
1.4 −64.0000 1281.56 4096.00 −32225.1 −82019.7 32545.6 −262144. 48065.7 2.06240e6
1.5 −64.0000 2370.93 4096.00 −39007.8 −151740. −263961. −262144. 4.02700e6 2.49650e6
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
2323 +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 46.14.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.14.a.a 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T35690T345509108T33+924234162T32+6086476932219T3+681843043869048 T_{3}^{5} - 690T_{3}^{4} - 5509108T_{3}^{3} + 924234162T_{3}^{2} + 6086476932219T_{3} + 681843043869048 acting on S14new(Γ0(46))S_{14}^{\mathrm{new}}(\Gamma_0(46)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T+64)5 (T + 64)^{5} Copy content Toggle raw display
33 T5++681843043869048 T^{5} + \cdots + 681843043869048 Copy content Toggle raw display
55 T5++40 ⁣ ⁣00 T^{5} + \cdots + 40\!\cdots\!00 Copy content Toggle raw display
77 T5++86 ⁣ ⁣24 T^{5} + \cdots + 86\!\cdots\!24 Copy content Toggle raw display
1111 T5+25 ⁣ ⁣40 T^{5} + \cdots - 25\!\cdots\!40 Copy content Toggle raw display
1313 T5+14 ⁣ ⁣74 T^{5} + \cdots - 14\!\cdots\!74 Copy content Toggle raw display
1717 T5++60 ⁣ ⁣20 T^{5} + \cdots + 60\!\cdots\!20 Copy content Toggle raw display
1919 T5+34 ⁣ ⁣12 T^{5} + \cdots - 34\!\cdots\!12 Copy content Toggle raw display
2323 (T+148035889)5 (T + 148035889)^{5} Copy content Toggle raw display
2929 T5++10 ⁣ ⁣30 T^{5} + \cdots + 10\!\cdots\!30 Copy content Toggle raw display
3131 T5+15 ⁣ ⁣00 T^{5} + \cdots - 15\!\cdots\!00 Copy content Toggle raw display
3737 T5+58 ⁣ ⁣28 T^{5} + \cdots - 58\!\cdots\!28 Copy content Toggle raw display
4141 T5+52 ⁣ ⁣18 T^{5} + \cdots - 52\!\cdots\!18 Copy content Toggle raw display
4343 T5++16 ⁣ ⁣00 T^{5} + \cdots + 16\!\cdots\!00 Copy content Toggle raw display
4747 T5++13 ⁣ ⁣72 T^{5} + \cdots + 13\!\cdots\!72 Copy content Toggle raw display
5353 T5+27 ⁣ ⁣08 T^{5} + \cdots - 27\!\cdots\!08 Copy content Toggle raw display
5959 T5+15 ⁣ ⁣40 T^{5} + \cdots - 15\!\cdots\!40 Copy content Toggle raw display
6161 T5+20 ⁣ ⁣52 T^{5} + \cdots - 20\!\cdots\!52 Copy content Toggle raw display
6767 T5+32 ⁣ ⁣80 T^{5} + \cdots - 32\!\cdots\!80 Copy content Toggle raw display
7171 T5+92 ⁣ ⁣36 T^{5} + \cdots - 92\!\cdots\!36 Copy content Toggle raw display
7373 T5++63 ⁣ ⁣46 T^{5} + \cdots + 63\!\cdots\!46 Copy content Toggle raw display
7979 T5++51 ⁣ ⁣16 T^{5} + \cdots + 51\!\cdots\!16 Copy content Toggle raw display
8383 T5++86 ⁣ ⁣76 T^{5} + \cdots + 86\!\cdots\!76 Copy content Toggle raw display
8989 T5++94 ⁣ ⁣04 T^{5} + \cdots + 94\!\cdots\!04 Copy content Toggle raw display
9797 T5++32 ⁣ ⁣88 T^{5} + \cdots + 32\!\cdots\!88 Copy content Toggle raw display
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