Properties

Label 152.2.v.a
Level 152152
Weight 22
Character orbit 152.v
Analytic conductor 1.2141.214
Analytic rank 00
Dimension 1212
CM discriminant -8
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [152,2,Mod(3,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 152=2319 152 = 2^{3} \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 152.v (of order 1818, degree 66, 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: 1.213726110721.21372611072
Analytic rank: 00
Dimension: 1212
Relative dimension: 22 over Q(ζ18)\Q(\zeta_{18})
Coefficient field: 12.0.101559956668416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x128x6+64 x^{12} - 8x^{6} + 64 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: U(1)[D18]\mathrm{U}(1)[D_{18}]

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) == qβ11q2+(β10β9β6++β1)q32β4q4+(β112β82β6++2)q6+(2β92β3)q8++(5β115β9+2β8++2)q99+O(q100) q - \beta_{11} q^{2} + (\beta_{10} - \beta_{9} - \beta_{6} + \cdots + \beta_1) q^{3} - 2 \beta_{4} q^{4} + (\beta_{11} - 2 \beta_{8} - 2 \beta_{6} + \cdots + 2) q^{6} + (2 \beta_{9} - 2 \beta_{3}) q^{8}+ \cdots + (5 \beta_{11} - 5 \beta_{9} + 2 \beta_{8} + \cdots + 2) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q6q3+12q6+6q9+24q2224q2490q27+30q3312q36+36q38+18q4172q4448q4842q49+90q5136q5418q59+48q64+30q99+O(q100) 12 q - 6 q^{3} + 12 q^{6} + 6 q^{9} + 24 q^{22} - 24 q^{24} - 90 q^{27} + 30 q^{33} - 12 q^{36} + 36 q^{38} + 18 q^{41} - 72 q^{44} - 48 q^{48} - 42 q^{49} + 90 q^{51} - 36 q^{54} - 18 q^{59} + 48 q^{64}+ \cdots - 30 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x128x6+64 x^{12} - 8x^{6} + 64 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν2)/2 ( \nu^{2} ) / 2 Copy content Toggle raw display
β3\beta_{3}== (ν3)/2 ( \nu^{3} ) / 2 Copy content Toggle raw display
β4\beta_{4}== (ν4)/4 ( \nu^{4} ) / 4 Copy content Toggle raw display
β5\beta_{5}== (ν5)/4 ( \nu^{5} ) / 4 Copy content Toggle raw display
β6\beta_{6}== (ν6)/8 ( \nu^{6} ) / 8 Copy content Toggle raw display
β7\beta_{7}== (ν7)/8 ( \nu^{7} ) / 8 Copy content Toggle raw display
β8\beta_{8}== (ν8)/16 ( \nu^{8} ) / 16 Copy content Toggle raw display
β9\beta_{9}== (ν9)/16 ( \nu^{9} ) / 16 Copy content Toggle raw display
β10\beta_{10}== (ν10)/32 ( \nu^{10} ) / 32 Copy content Toggle raw display
β11\beta_{11}== (ν11)/32 ( \nu^{11} ) / 32 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== 2β2 2\beta_{2} Copy content Toggle raw display
ν3\nu^{3}== 2β3 2\beta_{3} Copy content Toggle raw display
ν4\nu^{4}== 4β4 4\beta_{4} Copy content Toggle raw display
ν5\nu^{5}== 4β5 4\beta_{5} Copy content Toggle raw display
ν6\nu^{6}== 8β6 8\beta_{6} Copy content Toggle raw display
ν7\nu^{7}== 8β7 8\beta_{7} Copy content Toggle raw display
ν8\nu^{8}== 16β8 16\beta_{8} Copy content Toggle raw display
ν9\nu^{9}== 16β9 16\beta_{9} Copy content Toggle raw display
ν10\nu^{10}== 32β10 32\beta_{10} Copy content Toggle raw display
ν11\nu^{11}== 32β11 32\beta_{11} Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/152Z)×\left(\mathbb{Z}/152\mathbb{Z}\right)^\times.

nn 3939 7777 9797
χ(n)\chi(n) 1-1 1-1 β4-\beta_{4}

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}
3.1
0.483690 1.32893i
−0.483690 + 1.32893i
0.483690 + 1.32893i
−0.483690 1.32893i
−0.909039 1.08335i
0.909039 + 1.08335i
−0.909039 + 1.08335i
0.909039 1.08335i
−1.39273 0.245576i
1.39273 + 0.245576i
−1.39273 + 0.245576i
1.39273 0.245576i
−0.909039 + 1.08335i −0.301363 0.827987i −0.347296 1.96962i 0 1.17095 + 0.426191i 0 2.44949 + 1.41421i 1.70339 1.42931i 0
3.2 0.909039 1.08335i 1.18075 + 3.24408i −0.347296 1.96962i 0 4.58782 + 1.66983i 0 −2.44949 1.41421i −6.83175 + 5.73252i 0
51.1 −0.909039 1.08335i −0.301363 + 0.827987i −0.347296 + 1.96962i 0 1.17095 0.426191i 0 2.44949 1.41421i 1.70339 + 1.42931i 0
51.2 0.909039 + 1.08335i 1.18075 3.24408i −0.347296 + 1.96962i 0 4.58782 1.66983i 0 −2.44949 + 1.41421i −6.83175 5.73252i 0
59.1 −1.39273 0.245576i −0.950339 + 1.13257i 1.87939 + 0.684040i 0 1.60169 1.34398i 0 −2.44949 1.41421i 0.141375 + 0.801775i 0
59.2 1.39273 + 0.245576i −1.58175 + 1.88506i 1.87939 + 0.684040i 0 −2.66587 + 2.23693i 0 2.44949 + 1.41421i −0.530560 3.00895i 0
67.1 −1.39273 + 0.245576i −0.950339 1.13257i 1.87939 0.684040i 0 1.60169 + 1.34398i 0 −2.44949 + 1.41421i 0.141375 0.801775i 0
67.2 1.39273 0.245576i −1.58175 1.88506i 1.87939 0.684040i 0 −2.66587 2.23693i 0 2.44949 1.41421i −0.530560 + 3.00895i 0
91.1 −0.483690 + 1.32893i −3.29112 + 0.580314i −1.53209 1.28558i 0 0.820687 4.65435i 0 2.44949 1.41421i 7.67564 2.79370i 0
91.2 0.483690 1.32893i 1.94383 0.342749i −1.53209 1.28558i 0 0.484720 2.74898i 0 −2.44949 + 1.41421i 0.841902 0.306427i 0
147.1 −0.483690 1.32893i −3.29112 0.580314i −1.53209 + 1.28558i 0 0.820687 + 4.65435i 0 2.44949 + 1.41421i 7.67564 + 2.79370i 0
147.2 0.483690 + 1.32893i 1.94383 + 0.342749i −1.53209 + 1.28558i 0 0.484720 + 2.74898i 0 −2.44949 1.41421i 0.841902 + 0.306427i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by Q(2)\Q(\sqrt{-2})
19.f odd 18 1 inner
152.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.2.v.a 12
4.b odd 2 1 608.2.bh.a 12
8.b even 2 1 608.2.bh.a 12
8.d odd 2 1 CM 152.2.v.a 12
19.f odd 18 1 inner 152.2.v.a 12
76.k even 18 1 608.2.bh.a 12
152.s odd 18 1 608.2.bh.a 12
152.v even 18 1 inner 152.2.v.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.v.a 12 1.a even 1 1 trivial
152.2.v.a 12 8.d odd 2 1 CM
152.2.v.a 12 19.f odd 18 1 inner
152.2.v.a 12 152.v even 18 1 inner
608.2.bh.a 12 4.b odd 2 1
608.2.bh.a 12 8.b even 2 1
608.2.bh.a 12 76.k even 18 1
608.2.bh.a 12 152.s odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T312+6T311+15T310+48T39+138T38+48T37932T36++5329 T_{3}^{12} + 6 T_{3}^{11} + 15 T_{3}^{10} + 48 T_{3}^{9} + 138 T_{3}^{8} + 48 T_{3}^{7} - 932 T_{3}^{6} + \cdots + 5329 acting on S2new(152,[χ])S_{2}^{\mathrm{new}}(152, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T128T6+64 T^{12} - 8T^{6} + 64 Copy content Toggle raw display
33 T12+6T11++5329 T^{12} + 6 T^{11} + \cdots + 5329 Copy content Toggle raw display
55 T12 T^{12} Copy content Toggle raw display
77 T12 T^{12} Copy content Toggle raw display
1111 T12+66T10++13461561 T^{12} + 66 T^{10} + \cdots + 13461561 Copy content Toggle raw display
1313 T12 T^{12} Copy content Toggle raw display
1717 T12+486T9++11390625 T^{12} + 486 T^{9} + \cdots + 11390625 Copy content Toggle raw display
1919 T12+106T9++47045881 T^{12} + 106 T^{9} + \cdots + 47045881 Copy content Toggle raw display
2323 T12 T^{12} Copy content Toggle raw display
2929 T12 T^{12} Copy content Toggle raw display
3131 T12 T^{12} Copy content Toggle raw display
3737 T12 T^{12} Copy content Toggle raw display
4141 T12++41437894969 T^{12} + \cdots + 41437894969 Copy content Toggle raw display
4343 T12++594823321 T^{12} + \cdots + 594823321 Copy content Toggle raw display
4747 T12 T^{12} Copy content Toggle raw display
5353 T12 T^{12} Copy content Toggle raw display
5959 T12++260443853569 T^{12} + \cdots + 260443853569 Copy content Toggle raw display
6161 T12 T^{12} Copy content Toggle raw display
6767 T12++87534914769 T^{12} + \cdots + 87534914769 Copy content Toggle raw display
7171 T12 T^{12} Copy content Toggle raw display
7373 T12++964620586801 T^{12} + \cdots + 964620586801 Copy content Toggle raw display
7979 T12 T^{12} Copy content Toggle raw display
8383 T12++11478765321 T^{12} + \cdots + 11478765321 Copy content Toggle raw display
8989 T12++168425239515625 T^{12} + \cdots + 168425239515625 Copy content Toggle raw display
9797 T12++7482560872329 T^{12} + \cdots + 7482560872329 Copy content Toggle raw display
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