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Properties

Label 1014.2.g.b
Level 10141014
Weight 22
Character orbit 1014.g
Analytic conductor 8.0978.097
Analytic rank 00
Dimension 1212
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,2,Mod(239,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1014=2⋅3⋅132 1014 = 2 \cdot 3 \cdot 13^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1014.g (of order 44, 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: 8.096830764968.09683076496
Analytic rank: 00
Dimension: 1212
Relative dimension: 66 over Q(i)\Q(i)
Coefficient field: 12.0.58498535041007616.52
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x12−12x9+72x6−324x3+729 x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

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,…,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−β6q2+β7q3−β8q4+(−β9+β5)q5+β9q6+(−β11−β8+β1+1)q7+β2q8+(−β11+β5+β4)q9+⋯+(3β8−3β7+3β3+⋯−3)q99+O(q100) q - \beta_{6} q^{2} + \beta_{7} q^{3} - \beta_{8} q^{4} + ( - \beta_{9} + \beta_{5}) q^{5} + \beta_{9} q^{6} + ( - \beta_{11} - \beta_{8} + \beta_1 + 1) q^{7} + \beta_{2} q^{8} + ( - \beta_{11} + \beta_{5} + \beta_{4}) q^{9}+ \cdots + (3 \beta_{8} - 3 \beta_{7} + 3 \beta_{3} + \cdots - 3) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+12q7−12q16+12q19−36q27−12q28−12q31−36q33−12q37+36q42−36q45+36q54+36q57+36q63+12q67+12q73+12q76+72q79+⋯−36q99+O(q100) 12 q + 12 q^{7} - 12 q^{16} + 12 q^{19} - 36 q^{27} - 12 q^{28} - 12 q^{31} - 36 q^{33} - 12 q^{37} + 36 q^{42} - 36 q^{45} + 36 q^{54} + 36 q^{57} + 36 q^{63} + 12 q^{67} + 12 q^{73} + 12 q^{76} + 72 q^{79}+ \cdots - 36 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12−12x9+72x6−324x3+729 x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν9−3ν6+18ν3−81)/81 ( \nu^{9} - 3\nu^{6} + 18\nu^{3} - 81 ) / 81 Copy content Toggle raw display
β3\beta_{3}== (ν10−3ν7+18ν4−81ν)/81 ( \nu^{10} - 3\nu^{7} + 18\nu^{4} - 81\nu ) / 81 Copy content Toggle raw display
β4\beta_{4}== (−ν11+3ν8−45ν5+162ν2)/243 ( -\nu^{11} + 3\nu^{8} - 45\nu^{5} + 162\nu^{2} ) / 243 Copy content Toggle raw display
β5\beta_{5}== (−ν11+12ν8−72ν5+324ν2)/243 ( -\nu^{11} + 12\nu^{8} - 72\nu^{5} + 324\nu^{2} ) / 243 Copy content Toggle raw display
β6\beta_{6}== (−2ν9+15ν6−63ν3+243)/81 ( -2\nu^{9} + 15\nu^{6} - 63\nu^{3} + 243 ) / 81 Copy content Toggle raw display
β7\beta_{7}== (−2ν10+15ν7−63ν4+243ν)/81 ( -2\nu^{10} + 15\nu^{7} - 63\nu^{4} + 243\nu ) / 81 Copy content Toggle raw display
β8\beta_{8}== (2ν9−15ν6+90ν3−324)/81 ( 2\nu^{9} - 15\nu^{6} + 90\nu^{3} - 324 ) / 81 Copy content Toggle raw display
β9\beta_{9}== (2ν10−15ν7+90ν4−324ν)/81 ( 2\nu^{10} - 15\nu^{7} + 90\nu^{4} - 324\nu ) / 81 Copy content Toggle raw display
β10\beta_{10}== (−ν11+6ν8−27ν5+135ν2)/81 ( -\nu^{11} + 6\nu^{8} - 27\nu^{5} + 135\nu^{2} ) / 81 Copy content Toggle raw display
β11\beta_{11}== (4ν11−30ν8+153ν5−486ν2)/243 ( 4\nu^{11} - 30\nu^{8} + 153\nu^{5} - 486\nu^{2} ) / 243 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}== β11+β10+β5 \beta_{11} + \beta_{10} + \beta_{5} Copy content Toggle raw display
ν3\nu^{3}== 3β8+3β6+3 3\beta_{8} + 3\beta_{6} + 3 Copy content Toggle raw display
ν4\nu^{4}== 3β9+3β7+3β1 3\beta_{9} + 3\beta_{7} + 3\beta_1 Copy content Toggle raw display
ν5\nu^{5}== 3β11+6β10−6β4 3\beta_{11} + 6\beta_{10} - 6\beta_{4} Copy content Toggle raw display
ν6\nu^{6}== 9β8+18β6+18β2 9\beta_{8} + 18\beta_{6} + 18\beta_{2} Copy content Toggle raw display
ν7\nu^{7}== 9β9+18β7+18β3 9\beta_{9} + 18\beta_{7} + 18\beta_{3} Copy content Toggle raw display
ν8\nu^{8}== −9β11+9β5−45β4 -9\beta_{11} + 9\beta_{5} - 45\beta_{4} Copy content Toggle raw display
ν9\nu^{9}== −27β8+135β2+27 -27\beta_{8} + 135\beta_{2} + 27 Copy content Toggle raw display
ν10\nu^{10}== −27β9+135β3+27β1 -27\beta_{9} + 135\beta_{3} + 27\beta_1 Copy content Toggle raw display
ν11\nu^{11}== −108β10+189β5−108β4 -108\beta_{10} + 189\beta_{5} - 108\beta_{4} 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/1014Z)×\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times.

nn 677677 847847
χ(n)\chi(n) −1-1 β8\beta_{8}

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}
239.1
−0.779723 + 1.54662i
−0.949550 − 1.44857i
1.72927 − 0.0980500i
1.54662 − 0.779723i
−1.44857 − 0.949550i
−0.0980500 + 1.72927i
−0.779723 − 1.54662i
−0.949550 + 1.44857i
1.72927 + 0.0980500i
1.54662 + 0.779723i
−1.44857 + 0.949550i
−0.0980500 − 1.72927i
−0.707107 − 0.707107i −1.64497 + 0.542278i 1.00000i −2.32634 − 2.32634i 1.54662 + 0.779723i 1.76690 + 1.76690i 0.707107 − 0.707107i 2.41187 − 1.78406i 3.28995i
239.2 −0.707107 − 0.707107i 0.352860 − 1.69573i 1.00000i 0.499019 + 0.499019i −1.44857 + 0.949550i −1.39812 − 1.39812i 0.707107 − 0.707107i −2.75098 − 1.19671i − 0.705720i
239.3 −0.707107 − 0.707107i 1.29211 + 1.15345i 1.00000i 1.82732 + 1.82732i −0.0980500 − 1.72927i 2.63122 + 2.63122i 0.707107 − 0.707107i 0.339111 + 2.98077i − 2.58423i
239.4 0.707107 + 0.707107i −1.64497 − 0.542278i 1.00000i 2.32634 + 2.32634i −0.779723 − 1.54662i 1.76690 + 1.76690i −0.707107 + 0.707107i 2.41187 + 1.78406i 3.28995i
239.5 0.707107 + 0.707107i 0.352860 + 1.69573i 1.00000i −0.499019 − 0.499019i −0.949550 + 1.44857i −1.39812 − 1.39812i −0.707107 + 0.707107i −2.75098 + 1.19671i − 0.705720i
239.6 0.707107 + 0.707107i 1.29211 − 1.15345i 1.00000i −1.82732 − 1.82732i 1.72927 + 0.0980500i 2.63122 + 2.63122i −0.707107 + 0.707107i 0.339111 − 2.98077i − 2.58423i
437.1 −0.707107 + 0.707107i −1.64497 − 0.542278i − 1.00000i −2.32634 + 2.32634i 1.54662 − 0.779723i 1.76690 − 1.76690i 0.707107 + 0.707107i 2.41187 + 1.78406i − 3.28995i
437.2 −0.707107 + 0.707107i 0.352860 + 1.69573i − 1.00000i 0.499019 − 0.499019i −1.44857 − 0.949550i −1.39812 + 1.39812i 0.707107 + 0.707107i −2.75098 + 1.19671i 0.705720i
437.3 −0.707107 + 0.707107i 1.29211 − 1.15345i − 1.00000i 1.82732 − 1.82732i −0.0980500 + 1.72927i 2.63122 − 2.63122i 0.707107 + 0.707107i 0.339111 − 2.98077i 2.58423i
437.4 0.707107 − 0.707107i −1.64497 + 0.542278i − 1.00000i 2.32634 − 2.32634i −0.779723 + 1.54662i 1.76690 − 1.76690i −0.707107 − 0.707107i 2.41187 − 1.78406i − 3.28995i
437.5 0.707107 − 0.707107i 0.352860 − 1.69573i − 1.00000i −0.499019 + 0.499019i −0.949550 − 1.44857i −1.39812 + 1.39812i −0.707107 − 0.707107i −2.75098 − 1.19671i 0.705720i
437.6 0.707107 − 0.707107i 1.29211 + 1.15345i − 1.00000i −1.82732 + 1.82732i 1.72927 − 0.0980500i 2.63122 − 2.63122i −0.707107 − 0.707107i 0.339111 + 2.98077i 2.58423i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.g.b 12
3.b odd 2 1 inner 1014.2.g.b 12
13.b even 2 1 78.2.g.a ✓ 12
13.d odd 4 1 78.2.g.a ✓ 12
13.d odd 4 1 inner 1014.2.g.b 12
39.d odd 2 1 78.2.g.a ✓ 12
39.f even 4 1 78.2.g.a ✓ 12
39.f even 4 1 inner 1014.2.g.b 12
52.b odd 2 1 624.2.bf.f 12
52.f even 4 1 624.2.bf.f 12
156.h even 2 1 624.2.bf.f 12
156.l odd 4 1 624.2.bf.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.g.a ✓ 12 13.b even 2 1
78.2.g.a ✓ 12 13.d odd 4 1
78.2.g.a ✓ 12 39.d odd 2 1
78.2.g.a ✓ 12 39.f even 4 1
624.2.bf.f 12 52.b odd 2 1
624.2.bf.f 12 52.f even 4 1
624.2.bf.f 12 156.h even 2 1
624.2.bf.f 12 156.l odd 4 1
1014.2.g.b 12 1.a even 1 1 trivial
1014.2.g.b 12 3.b odd 2 1 inner
1014.2.g.b 12 13.d odd 4 1 inner
1014.2.g.b 12 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1014,[χ])S_{2}^{\mathrm{new}}(1014, [\chi]):

T512+162T58+5265T54+1296 T_{5}^{12} + 162T_{5}^{8} + 5265T_{5}^{4} + 1296 Copy content Toggle raw display
T76−6T75+18T74−8T73+9T72−78T7+338 T_{7}^{6} - 6T_{7}^{5} + 18T_{7}^{4} - 8T_{7}^{3} + 9T_{7}^{2} - 78T_{7} + 338 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4+1)3 (T^{4} + 1)^{3} Copy content Toggle raw display
33 (T6+6T3+27)2 (T^{6} + 6 T^{3} + 27)^{2} Copy content Toggle raw display
55 T12+162T8+⋯+1296 T^{12} + 162 T^{8} + \cdots + 1296 Copy content Toggle raw display
77 (T6−6T5+⋯+338)2 (T^{6} - 6 T^{5} + \cdots + 338)^{2} Copy content Toggle raw display
1111 T12+648T8+⋯+331776 T^{12} + 648 T^{8} + \cdots + 331776 Copy content Toggle raw display
1313 T12 T^{12} Copy content Toggle raw display
1717 (T6−54T4+⋯−1152)2 (T^{6} - 54 T^{4} + \cdots - 1152)^{2} Copy content Toggle raw display
1919 (T6−6T5+⋯+1568)2 (T^{6} - 6 T^{5} + \cdots + 1568)^{2} Copy content Toggle raw display
2323 (T6−36T4+⋯−288)2 (T^{6} - 36 T^{4} + \cdots - 288)^{2} Copy content Toggle raw display
2929 (T6+36T4+⋯+288)2 (T^{6} + 36 T^{4} + \cdots + 288)^{2} Copy content Toggle raw display
3131 (T6+6T5+⋯+512)2 (T^{6} + 6 T^{5} + \cdots + 512)^{2} Copy content Toggle raw display
3737 (T6+6T5+⋯+4802)2 (T^{6} + 6 T^{5} + \cdots + 4802)^{2} Copy content Toggle raw display
4141 T12+18576T8+⋯+5308416 T^{12} + 18576 T^{8} + \cdots + 5308416 Copy content Toggle raw display
4343 (T6+162T4+⋯+324)2 (T^{6} + 162 T^{4} + \cdots + 324)^{2} Copy content Toggle raw display
4747 T12+15714T8+⋯+1679616 T^{12} + 15714 T^{8} + \cdots + 1679616 Copy content Toggle raw display
5353 (T6+324T4+⋯+1152)2 (T^{6} + 324 T^{4} + \cdots + 1152)^{2} Copy content Toggle raw display
5959 T12+2592T8+⋯+5308416 T^{12} + 2592 T^{8} + \cdots + 5308416 Copy content Toggle raw display
6161 (T3−72T+192)4 (T^{3} - 72 T + 192)^{4} Copy content Toggle raw display
6767 (T6−6T5+⋯+392)2 (T^{6} - 6 T^{5} + \cdots + 392)^{2} Copy content Toggle raw display
7171 T12+13986T8+⋯+20736 T^{12} + 13986 T^{8} + \cdots + 20736 Copy content Toggle raw display
7373 (T6−6T5+⋯+1568)2 (T^{6} - 6 T^{5} + \cdots + 1568)^{2} Copy content Toggle raw display
7979 (T3−18T2+⋯+12)4 (T^{3} - 18 T^{2} + \cdots + 12)^{4} Copy content Toggle raw display
8383 T12+⋯+21743271936 T^{12} + \cdots + 21743271936 Copy content Toggle raw display
8989 T12+⋯+27710263296 T^{12} + \cdots + 27710263296 Copy content Toggle raw display
9797 (T6−30T5+⋯+392)2 (T^{6} - 30 T^{5} + \cdots + 392)^{2} Copy content Toggle raw display
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