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Properties

Label 42.5.b.a
Level 4242
Weight 55
Character orbit 42.b
Analytic conductor 4.3424.342
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [42,5,Mod(29,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.29");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: N N == 42=2⋅3⋅7 42 = 2 \cdot 3 \cdot 7
Weight: k k == 5 5
Character orbit: [χ][\chi] == 42.b (of order 22, degree 11, 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: 4.341538449524.34153844952
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: x8−44x6+646x4−3060x2+6561 x^{8} - 44x^{6} + 646x^{4} - 3060x^{2} + 6561 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 27⋅33⋅72 2^{7}\cdot 3^{3}\cdot 7^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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+β1q2+(−β3+2)q3−8q4+(−β6−β3−4β1)q5+(β4+β1−4)q6−β2q7−8β1q8+(−β6+β5+β4+⋯+20)q9+⋯+(−48β7−78β6+⋯+1388)q99+O(q100) q + \beta_1 q^{2} + ( - \beta_{3} + 2) q^{3} - 8 q^{4} + ( - \beta_{6} - \beta_{3} - 4 \beta_1) q^{5} + (\beta_{4} + \beta_1 - 4) q^{6} - \beta_{2} q^{7} - 8 \beta_1 q^{8} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 20) q^{9}+ \cdots + ( - 48 \beta_{7} - 78 \beta_{6} + \cdots + 1388) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q+12q3−64q4−32q6+152q9+256q10−96q12−520q13−616q15+512q16+768q18+1336q19−196q21−1024q22+256q24−3672q25+36q27+⋯+10976q99+O(q100) 8 q + 12 q^{3} - 64 q^{4} - 32 q^{6} + 152 q^{9} + 256 q^{10} - 96 q^{12} - 520 q^{13} - 616 q^{15} + 512 q^{16} + 768 q^{18} + 1336 q^{19} - 196 q^{21} - 1024 q^{22} + 256 q^{24} - 3672 q^{25} + 36 q^{27}+ \cdots + 10976 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−44x6+646x4−3060x2+6561 x^{8} - 44x^{6} + 646x^{4} - 3060x^{2} + 6561 : Copy content Toggle raw display

β1\beta_{1}== (14ν7−535ν5+6452ν3−14571ν)/7047 ( 14\nu^{7} - 535\nu^{5} + 6452\nu^{3} - 14571\nu ) / 7047 Copy content Toggle raw display
β2\beta_{2}== (7ν6−224ν4+1225ν2+6678)/522 ( 7\nu^{6} - 224\nu^{4} + 1225\nu^{2} + 6678 ) / 522 Copy content Toggle raw display
β3\beta_{3}== (−56ν7−297ν6+2140ν5+11853ν4−25808ν3−145935ν2+114660ν+386127)/28188 ( -56\nu^{7} - 297\nu^{6} + 2140\nu^{5} + 11853\nu^{4} - 25808\nu^{3} - 145935\nu^{2} + 114660\nu + 386127 ) / 28188 Copy content Toggle raw display
β4\beta_{4}== (10ν7−54ν6−494ν5+1728ν4+8188ν3−18846ν2−44748ν+51840)/2349 ( 10\nu^{7} - 54\nu^{6} - 494\nu^{5} + 1728\nu^{4} + 8188\nu^{3} - 18846\nu^{2} - 44748\nu + 51840 ) / 2349 Copy content Toggle raw display
β5\beta_{5}== (−112ν7+108ν6+4280ν5−1107ν4−42220ν3−4590ν2−5580ν−16767)/14094 ( -112\nu^{7} + 108\nu^{6} + 4280\nu^{5} - 1107\nu^{4} - 42220\nu^{3} - 4590\nu^{2} - 5580\nu - 16767 ) / 14094 Copy content Toggle raw display
β6\beta_{6}== (172ν7−351ν6−7244ν5+22977ν4+91348ν3−362097ν2−262908ν+1067499)/28188 ( 172\nu^{7} - 351\nu^{6} - 7244\nu^{5} + 22977\nu^{4} + 91348\nu^{3} - 362097\nu^{2} - 262908\nu + 1067499 ) / 28188 Copy content Toggle raw display
β7\beta_{7}== (472ν7+567ν6−22064ν5−25191ν4+308800ν3+352917ν2+⋯−1101033)/28188 ( 472 \nu^{7} + 567 \nu^{6} - 22064 \nu^{5} - 25191 \nu^{4} + 308800 \nu^{3} + 352917 \nu^{2} + \cdots - 1101033 ) / 28188 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (7β7−14β6+7β5−14β4+77β3+9β2+119β1−49)/252 ( 7\beta_{7} - 14\beta_{6} + 7\beta_{5} - 14\beta_{4} + 77\beta_{3} + 9\beta_{2} + 119\beta _1 - 49 ) / 252 Copy content Toggle raw display
ν2\nu^{2}== (2β7+2β6+5β5−7β4−14β3−27β2−2β1+700)/63 ( 2\beta_{7} + 2\beta_{6} + 5\beta_{5} - 7\beta_{4} - 14\beta_{3} - 27\beta_{2} - 2\beta _1 + 700 ) / 63 Copy content Toggle raw display
ν3\nu^{3}== (73β7−326β6+361β5−56β4+1127β3+171β2+3077β1−763)/252 ( 73\beta_{7} - 326\beta_{6} + 361\beta_{5} - 56\beta_{4} + 1127\beta_{3} + 171\beta_{2} + 3077\beta _1 - 763 ) / 252 Copy content Toggle raw display
ν4\nu^{4}== (38β7+164β6+158β5−196β4−266β3−540β2−38β1+10339)/63 ( 38\beta_{7} + 164\beta_{6} + 158\beta_{5} - 196\beta_{4} - 266\beta_{3} - 540\beta_{2} - 38\beta _1 + 10339 ) / 63 Copy content Toggle raw display
ν5\nu^{5}== (−485β7−6806β6+7723β5−434β4+17129β3+2709β2+⋯−11725)/252 ( - 485 \beta_{7} - 6806 \beta_{6} + 7723 \beta_{5} - 434 \beta_{4} + 17129 \beta_{3} + 2709 \beta_{2} + \cdots - 11725 ) / 252 Copy content Toggle raw display
ν6\nu^{6}== (866β7+4898β6+4181β5−5047β4−6062β3−7857β2+⋯+148246)/63 ( 866 \beta_{7} + 4898 \beta_{6} + 4181 \beta_{5} - 5047 \beta_{4} - 6062 \beta_{3} - 7857 \beta_{2} + \cdots + 148246 ) / 63 Copy content Toggle raw display
ν7\nu^{7}== (−6413β7−17774β6+19435β5−764β4+30761β3+4869β2+⋯−21061)/36 ( - 6413 \beta_{7} - 17774 \beta_{6} + 19435 \beta_{5} - 764 \beta_{4} + 30761 \beta_{3} + 4869 \beta_{2} + \cdots - 21061 ) / 36 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/42Z)×\left(\mathbb{Z}/42\mathbb{Z}\right)^\times.

nn 2929 3131
χ(n)\chi(n) −1-1 11

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}
29.1
4.40877 − 0.707107i
1.88752 − 0.707107i
−1.88752 − 0.707107i
−4.40877 − 0.707107i
4.40877 + 0.707107i
1.88752 + 0.707107i
−1.88752 + 0.707107i
−4.40877 + 0.707107i
− 2.82843i −8.64042 − 2.51857i −8.00000 34.6139i −7.12360 + 24.4388i 18.5203 22.6274i 68.3136 + 43.5230i 97.9029
29.2 − 2.82843i −0.952172 − 8.94949i −8.00000 − 10.3985i −25.3130 + 2.69315i −18.5203 22.6274i −79.1867 + 17.0429i −29.4113
29.3 − 2.82843i 6.59792 + 6.12106i −8.00000 47.9925i 17.3130 − 18.6617i −18.5203 22.6274i 6.06518 + 80.7726i 135.743
29.4 − 2.82843i 8.99466 − 0.309853i −8.00000 − 26.9531i −0.876398 − 25.4408i 18.5203 22.6274i 80.8080 − 5.57406i −76.2349
29.5 2.82843i −8.64042 + 2.51857i −8.00000 − 34.6139i −7.12360 − 24.4388i 18.5203 − 22.6274i 68.3136 − 43.5230i 97.9029
29.6 2.82843i −0.952172 + 8.94949i −8.00000 10.3985i −25.3130 − 2.69315i −18.5203 − 22.6274i −79.1867 − 17.0429i −29.4113
29.7 2.82843i 6.59792 − 6.12106i −8.00000 − 47.9925i 17.3130 + 18.6617i −18.5203 − 22.6274i 6.06518 − 80.7726i 135.743
29.8 2.82843i 8.99466 + 0.309853i −8.00000 26.9531i −0.876398 + 25.4408i 18.5203 − 22.6274i 80.8080 + 5.57406i −76.2349
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.5.b.a ✓ 8
3.b odd 2 1 inner 42.5.b.a ✓ 8
4.b odd 2 1 336.5.d.a 8
7.b odd 2 1 294.5.b.d 8
12.b even 2 1 336.5.d.a 8
21.c even 2 1 294.5.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.5.b.a ✓ 8 1.a even 1 1 trivial
42.5.b.a ✓ 8 3.b odd 2 1 inner
294.5.b.d 8 7.b odd 2 1
294.5.b.d 8 21.c even 2 1
336.5.d.a 8 4.b odd 2 1
336.5.d.a 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace S5new(42,[χ])S_{5}^{\mathrm{new}}(42, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+8)4 (T^{2} + 8)^{4} Copy content Toggle raw display
33 T8−12T7+⋯+43046721 T^{8} - 12 T^{7} + \cdots + 43046721 Copy content Toggle raw display
55 T8+⋯+216772185744 T^{8} + \cdots + 216772185744 Copy content Toggle raw display
77 (T2−343)4 (T^{2} - 343)^{4} Copy content Toggle raw display
1111 T8+⋯+17 ⁣⋯ ⁣16 T^{8} + \cdots + 17\!\cdots\!16 Copy content Toggle raw display
1313 (T4+260T3+⋯−628257852)2 (T^{4} + 260 T^{3} + \cdots - 628257852)^{2} Copy content Toggle raw display
1717 T8+⋯+26 ⁣⋯ ⁣56 T^{8} + \cdots + 26\!\cdots\!56 Copy content Toggle raw display
1919 (T4−668T3+⋯−3943244156)2 (T^{4} - 668 T^{3} + \cdots - 3943244156)^{2} Copy content Toggle raw display
2323 T8+⋯+55 ⁣⋯ ⁣04 T^{8} + \cdots + 55\!\cdots\!04 Copy content Toggle raw display
2929 T8+⋯+24 ⁣⋯ ⁣76 T^{8} + \cdots + 24\!\cdots\!76 Copy content Toggle raw display
3131 (T4−2440T3+⋯−407095948224)2 (T^{4} - 2440 T^{3} + \cdots - 407095948224)^{2} Copy content Toggle raw display
3737 (T4+⋯+4270226378896)2 (T^{4} + \cdots + 4270226378896)^{2} Copy content Toggle raw display
4141 T8+⋯+10 ⁣⋯ ⁣64 T^{8} + \cdots + 10\!\cdots\!64 Copy content Toggle raw display
4343 (T4+⋯+3649641123088)2 (T^{4} + \cdots + 3649641123088)^{2} Copy content Toggle raw display
4747 T8+⋯+71 ⁣⋯ ⁣64 T^{8} + \cdots + 71\!\cdots\!64 Copy content Toggle raw display
5353 T8+⋯+21 ⁣⋯ ⁣36 T^{8} + \cdots + 21\!\cdots\!36 Copy content Toggle raw display
5959 T8+⋯+91 ⁣⋯ ⁣96 T^{8} + \cdots + 91\!\cdots\!96 Copy content Toggle raw display
6161 (T4+⋯+43195048412292)2 (T^{4} + \cdots + 43195048412292)^{2} Copy content Toggle raw display
6767 (T4+⋯−26829784150256)2 (T^{4} + \cdots - 26829784150256)^{2} Copy content Toggle raw display
7171 T8+⋯+16 ⁣⋯ ⁣96 T^{8} + \cdots + 16\!\cdots\!96 Copy content Toggle raw display
7373 (T4+⋯−479820270553328)2 (T^{4} + \cdots - 479820270553328)^{2} Copy content Toggle raw display
7979 (T4+⋯+343544312164608)2 (T^{4} + \cdots + 343544312164608)^{2} Copy content Toggle raw display
8383 T8+⋯+36 ⁣⋯ ⁣76 T^{8} + \cdots + 36\!\cdots\!76 Copy content Toggle raw display
8989 T8+⋯+17 ⁣⋯ ⁣64 T^{8} + \cdots + 17\!\cdots\!64 Copy content Toggle raw display
9797 (T4+⋯−47 ⁣⋯ ⁣88)2 (T^{4} + \cdots - 47\!\cdots\!88)^{2} Copy content Toggle raw display
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