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Properties

Label 38.6.c.b
Level 3838
Weight 66
Character orbit 38.c
Analytic conductor 6.0956.095
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] = [38,6,Mod(7,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 38=2⋅19 38 = 2 \cdot 19
Weight: k k == 6 6
Character orbit: [χ][\chi] == 38.c (of order 33, degree 22, 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: 6.094585152896.09458515289
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ3)\Q(\zeta_{3})
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: x8−2x7+386x6+3436x5+128708x4+568528x3+7340704x2−19430784x+211527936 x^{8} - 2x^{7} + 386x^{6} + 3436x^{5} + 128708x^{4} + 568528x^{3} + 7340704x^{2} - 19430784x + 211527936 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 22⋅3 2^{2}\cdot 3
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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+(4β2−4)q2+(−β3+3β2−3)q3−16β2q4+(β5+β4+9β2−9)q5+(−12β2−4β1)q6+(−β7−β6+β3+⋯+9)q7+⋯+(−364β7−1108β5+⋯−804β1)q99+O(q100) q + (4 \beta_{2} - 4) q^{2} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{3} - 16 \beta_{2} q^{4} + (\beta_{5} + \beta_{4} + 9 \beta_{2} - 9) q^{5} + ( - 12 \beta_{2} - 4 \beta_1) q^{6} + ( - \beta_{7} - \beta_{6} + \beta_{3} + \cdots + 9) q^{7}+ \cdots + ( - 364 \beta_{7} - 1108 \beta_{5} + \cdots - 804 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−16q2−14q3−64q4−36q5−56q6+76q7+512q8+156q9−144q10+144q11+448q12−674q13−152q14−20q15−1024q16+522q17−1248q18+⋯−138328q99+O(q100) 8 q - 16 q^{2} - 14 q^{3} - 64 q^{4} - 36 q^{5} - 56 q^{6} + 76 q^{7} + 512 q^{8} + 156 q^{9} - 144 q^{10} + 144 q^{11} + 448 q^{12} - 674 q^{13} - 152 q^{14} - 20 q^{15} - 1024 q^{16} + 522 q^{17} - 1248 q^{18}+ \cdots - 138328 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−2x7+386x6+3436x5+128708x4+568528x3+7340704x2−19430784x+211527936 x^{8} - 2x^{7} + 386x^{6} + 3436x^{5} + 128708x^{4} + 568528x^{3} + 7340704x^{2} - 19430784x + 211527936 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (390471533ν7+3142267604ν6+128794858006ν5+2734091869688ν4+⋯+21 ⁣⋯ ⁣64)/27 ⁣⋯ ⁣56 ( 390471533 \nu^{7} + 3142267604 \nu^{6} + 128794858006 \nu^{5} + 2734091869688 \nu^{4} + \cdots + 21\!\cdots\!64 ) / 27\!\cdots\!56 Copy content Toggle raw display
β3\beta_{3}== (−6473945ν7+36183422ν6−2297742050ν5−13997685940ν4+⋯+136296431422848)/44885936230176 ( - 6473945 \nu^{7} + 36183422 \nu^{6} - 2297742050 \nu^{5} - 13997685940 \nu^{4} + \cdots + 136296431422848 ) / 44885936230176 Copy content Toggle raw display
β4\beta_{4}== (−18756773ν7+22066664ν6−6657181370ν5−40555089316ν4+⋯+20 ⁣⋯ ⁣88)/22442968115088 ( - 18756773 \nu^{7} + 22066664 \nu^{6} - 6657181370 \nu^{5} - 40555089316 \nu^{4} + \cdots + 20\!\cdots\!88 ) / 22442968115088 Copy content Toggle raw display
β5\beta_{5}== (−14831978953ν7−117204668308ν6−4803969780662ν5+⋯−79 ⁣⋯ ⁣28)/13 ⁣⋯ ⁣28 ( - 14831978953 \nu^{7} - 117204668308 \nu^{6} - 4803969780662 \nu^{5} + \cdots - 79\!\cdots\!28 ) / 13\!\cdots\!28 Copy content Toggle raw display
β6\beta_{6}== (35647275847ν7−1347014053248ν6+26507480732490ν5+⋯−12 ⁣⋯ ⁣44)/90 ⁣⋯ ⁣52 ( 35647275847 \nu^{7} - 1347014053248 \nu^{6} + 26507480732490 \nu^{5} + \cdots - 12\!\cdots\!44 ) / 90\!\cdots\!52 Copy content Toggle raw display
β7\beta_{7}== (71425529135ν7+280447752060ν6+11494956177090ν5+561804417935112ν4+⋯+19 ⁣⋯ ⁣60)/90 ⁣⋯ ⁣52 ( 71425529135 \nu^{7} + 280447752060 \nu^{6} + 11494956177090 \nu^{5} + 561804417935112 \nu^{4} + \cdots + 19\!\cdots\!60 ) / 90\!\cdots\!52 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}== 2β5+2β4−8β3+188β2−188 2\beta_{5} + 2\beta_{4} - 8\beta_{3} + 188\beta_{2} - 188 Copy content Toggle raw display
ν3\nu^{3}== −2β7−2β6+28β4−326β3−326β1−1414 -2\beta_{7} - 2\beta_{6} + 28\beta_{4} - 326\beta_{3} - 326\beta _1 - 1414 Copy content Toggle raw display
ν4\nu^{4}== −4β7−820β5−60100β2−5044β1 -4\beta_{7} - 820\beta_{5} - 60100\beta_{2} - 5044\beta_1 Copy content Toggle raw display
ν5\nu^{5}== 772β6−15008β5−15008β4+130764β3−911516β2+911516 772\beta_{6} - 15008\beta_{5} - 15008\beta_{4} + 130764\beta_{3} - 911516\beta_{2} + 911516 Copy content Toggle raw display
ν6\nu^{6}== 5744β7+5744β6−351576β4+2507520β3+2507520β1+23936064 5744\beta_{7} + 5744\beta_{6} - 351576\beta_{4} + 2507520\beta_{3} + 2507520\beta _1 + 23936064 Copy content Toggle raw display
ν7\nu^{7}== 282648β7+7124496β5+455799624β2+56964328β1 282648\beta_{7} + 7124496\beta_{5} + 455799624\beta_{2} + 56964328\beta_1 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/38Z)×\left(\mathbb{Z}/38\mathbb{Z}\right)^\times.

nn 2121
χ(n)\chi(n) −β2-\beta_{2}

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}
7.1
10.6694 − 18.4800i
2.36148 − 4.09020i
−5.68703 + 9.85023i
−6.34386 + 10.9879i
10.6694 + 18.4800i
2.36148 + 4.09020i
−5.68703 − 9.85023i
−6.34386 − 10.9879i
−2.00000 − 3.46410i −12.1694 − 21.0781i −8.00000 + 13.8564i −28.6588 − 49.6385i −48.6777 + 84.3122i 33.7394 64.0000 −174.689 + 302.571i −114.635 + 198.554i
7.2 −2.00000 − 3.46410i −3.86148 − 6.68827i −8.00000 + 13.8564i 46.3693 + 80.3140i −15.4459 + 26.7531i 13.5472 64.0000 91.6780 − 158.791i 185.477 − 321.256i
7.3 −2.00000 − 3.46410i 4.18703 + 7.25215i −8.00000 + 13.8564i −12.5904 − 21.8073i 16.7481 − 29.0086i −187.086 64.0000 86.4375 − 149.714i −50.3618 + 87.2292i
7.4 −2.00000 − 3.46410i 4.84386 + 8.38982i −8.00000 + 13.8564i −23.1201 − 40.0451i 19.3755 − 33.5593i 177.800 64.0000 74.5740 − 129.166i −92.4803 + 160.181i
11.1 −2.00000 + 3.46410i −12.1694 + 21.0781i −8.00000 − 13.8564i −28.6588 + 49.6385i −48.6777 − 84.3122i 33.7394 64.0000 −174.689 − 302.571i −114.635 − 198.554i
11.2 −2.00000 + 3.46410i −3.86148 + 6.68827i −8.00000 − 13.8564i 46.3693 − 80.3140i −15.4459 − 26.7531i 13.5472 64.0000 91.6780 + 158.791i 185.477 + 321.256i
11.3 −2.00000 + 3.46410i 4.18703 − 7.25215i −8.00000 − 13.8564i −12.5904 + 21.8073i 16.7481 + 29.0086i −187.086 64.0000 86.4375 + 149.714i −50.3618 − 87.2292i
11.4 −2.00000 + 3.46410i 4.84386 − 8.38982i −8.00000 − 13.8564i −23.1201 + 40.0451i 19.3755 + 33.5593i 177.800 64.0000 74.5740 + 129.166i −92.4803 − 160.181i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.6.c.b ✓ 8
3.b odd 2 1 342.6.g.d 8
4.b odd 2 1 304.6.i.b 8
19.c even 3 1 inner 38.6.c.b ✓ 8
19.c even 3 1 722.6.a.j 4
19.d odd 6 1 722.6.a.g 4
57.h odd 6 1 342.6.g.d 8
76.g odd 6 1 304.6.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.c.b ✓ 8 1.a even 1 1 trivial
38.6.c.b ✓ 8 19.c even 3 1 inner
304.6.i.b 8 4.b odd 2 1
304.6.i.b 8 76.g odd 6 1
342.6.g.d 8 3.b odd 2 1
342.6.g.d 8 57.h odd 6 1
722.6.a.g 4 19.d odd 6 1
722.6.a.j 4 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T38+14T37+506T36−2752T35+91967T34−180832T33+⋯+232532001 T_{3}^{8} + 14 T_{3}^{7} + 506 T_{3}^{6} - 2752 T_{3}^{5} + 91967 T_{3}^{4} - 180832 T_{3}^{3} + \cdots + 232532001 acting on S6new(38,[χ])S_{6}^{\mathrm{new}}(38, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+4T+16)4 (T^{2} + 4 T + 16)^{4} Copy content Toggle raw display
33 T8+14T7+⋯+232532001 T^{8} + 14 T^{7} + \cdots + 232532001 Copy content Toggle raw display
55 T8+⋯+38307063132900 T^{8} + \cdots + 38307063132900 Copy content Toggle raw display
77 (T4−38T3+⋯−15204096)2 (T^{4} - 38 T^{3} + \cdots - 15204096)^{2} Copy content Toggle raw display
1111 (T4−72T3+⋯−423612144)2 (T^{4} - 72 T^{3} + \cdots - 423612144)^{2} Copy content Toggle raw display
1313 T8+⋯+10 ⁣⋯ ⁣44 T^{8} + \cdots + 10\!\cdots\!44 Copy content Toggle raw display
1717 T8+⋯+23 ⁣⋯ ⁣24 T^{8} + \cdots + 23\!\cdots\!24 Copy content Toggle raw display
1919 T8+⋯+37 ⁣⋯ ⁣01 T^{8} + \cdots + 37\!\cdots\!01 Copy content Toggle raw display
2323 T8+⋯+42 ⁣⋯ ⁣44 T^{8} + \cdots + 42\!\cdots\!44 Copy content Toggle raw display
2929 T8+⋯+41 ⁣⋯ ⁣00 T^{8} + \cdots + 41\!\cdots\!00 Copy content Toggle raw display
3131 (T4+⋯−9213535294816)2 (T^{4} + \cdots - 9213535294816)^{2} Copy content Toggle raw display
3737 (T4+⋯+27 ⁣⋯ ⁣80)2 (T^{4} + \cdots + 27\!\cdots\!80)^{2} Copy content Toggle raw display
4141 T8+⋯+51 ⁣⋯ ⁣25 T^{8} + \cdots + 51\!\cdots\!25 Copy content Toggle raw display
4343 T8+⋯+78 ⁣⋯ ⁣00 T^{8} + \cdots + 78\!\cdots\!00 Copy content Toggle raw display
4747 T8+⋯+88 ⁣⋯ ⁣00 T^{8} + \cdots + 88\!\cdots\!00 Copy content Toggle raw display
5353 T8+⋯+20 ⁣⋯ ⁣76 T^{8} + \cdots + 20\!\cdots\!76 Copy content Toggle raw display
5959 T8+⋯+59 ⁣⋯ ⁣25 T^{8} + \cdots + 59\!\cdots\!25 Copy content Toggle raw display
6161 T8+⋯+16 ⁣⋯ ⁣16 T^{8} + \cdots + 16\!\cdots\!16 Copy content Toggle raw display
6767 T8+⋯+52 ⁣⋯ ⁣25 T^{8} + \cdots + 52\!\cdots\!25 Copy content Toggle raw display
7171 T8+⋯+31 ⁣⋯ ⁣96 T^{8} + \cdots + 31\!\cdots\!96 Copy content Toggle raw display
7373 T8+⋯+28 ⁣⋯ ⁣09 T^{8} + \cdots + 28\!\cdots\!09 Copy content Toggle raw display
7979 T8+⋯+64 ⁣⋯ ⁣16 T^{8} + \cdots + 64\!\cdots\!16 Copy content Toggle raw display
8383 (T4+⋯−61 ⁣⋯ ⁣52)2 (T^{4} + \cdots - 61\!\cdots\!52)^{2} Copy content Toggle raw display
8989 T8+⋯+61 ⁣⋯ ⁣16 T^{8} + \cdots + 61\!\cdots\!16 Copy content Toggle raw display
9797 T8+⋯+53 ⁣⋯ ⁣25 T^{8} + \cdots + 53\!\cdots\!25 Copy content Toggle raw display
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