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Properties

Label 152.4.v.a
Level 152152
Weight 44
Character orbit 152.v
Analytic conductor 8.9688.968
Analytic rank 00
Dimension 1212
CM discriminant -8
Inner twists 44

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [152,4,Mod(3,152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 13])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 152=23⋅19 152 = 2^{3} \cdot 19
Weight: k k == 4 4
Character orbit: [χ][\chi] == 152.v (of order 1818, degree 66, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 8.968290320878.96829032087
Analytic rank: 00
Dimension: 1212
Relative dimension: 22 over Q(ζ18)\Q(\zeta_{18})
Coefficient field: 12.0.101559956668416.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x12−8x6+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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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−2β11q2+(−5β10+β9+⋯−β1)q3−8β4q4+(−10β11+4β8+⋯−4)q6+(16β9−16β3)q8+(10β11−27β10+⋯−23)q9+⋯+(−485β11+485β9+⋯+950)q99+O(q100) q - 2 \beta_{11} q^{2} + ( - 5 \beta_{10} + \beta_{9} + \cdots - \beta_1) q^{3} - 8 \beta_{4} q^{4} + ( - 10 \beta_{11} + 4 \beta_{8} + \cdots - 4) q^{6} + (16 \beta_{9} - 16 \beta_{3}) q^{8} + (10 \beta_{11} - 27 \beta_{10} + \cdots - 23) q^{9}+ \cdots + ( - 485 \beta_{11} + 485 \beta_{9} + \cdots + 950) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+30q3−24q6−138q9−1200q22+192q24−1710q27+30q33+1104q36+1080q38−1566q41−864q44+3840q48−2058q49+2718q51+648q54+⋯+9942q99+O(q100) 12 q + 30 q^{3} - 24 q^{6} - 138 q^{9} - 1200 q^{22} + 192 q^{24} - 1710 q^{27} + 30 q^{33} + 1104 q^{36} + 1080 q^{38} - 1566 q^{41} - 864 q^{44} + 3840 q^{48} - 2058 q^{49} + 2718 q^{51} + 648 q^{54}+ \cdots + 9942 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12−8x6+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

Display νj\nu^j in terms of βi\beta_i

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

Display βi\beta_i in terms of νj\nu^j

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.81808 + 2.16670i −1.45741 − 4.00419i −1.38919 − 7.87846i 0 11.3256 + 4.12217i 0 19.5959 + 11.3137i 6.77366 − 5.68378i 0
3.2 1.81808 − 2.16670i −2.93952 − 8.07626i −1.38919 − 7.87846i 0 −22.8431 − 8.31421i 0 −19.5959 − 11.3137i −35.9020 + 30.1254i 0
51.1 −1.81808 − 2.16670i −1.45741 + 4.00419i −1.38919 + 7.87846i 0 11.3256 − 4.12217i 0 19.5959 − 11.3137i 6.77366 + 5.68378i 0
51.2 1.81808 + 2.16670i −2.93952 + 8.07626i −1.38919 + 7.87846i 0 −22.8431 + 8.31421i 0 −19.5959 + 11.3137i −35.9020 − 30.1254i 0
59.1 −2.78546 − 0.491151i 6.01452 − 7.16782i 7.51754 + 2.73616i 0 −20.2737 + 17.0116i 0 −19.5959 − 11.3137i −10.5148 − 59.6321i 0
59.2 2.78546 + 0.491151i 6.64593 − 7.92031i 7.51754 + 2.73616i 0 22.4020 − 18.7975i 0 19.5959 + 11.3137i −13.8744 − 78.6858i 0
67.1 −2.78546 + 0.491151i 6.01452 + 7.16782i 7.51754 − 2.73616i 0 −20.2737 − 17.0116i 0 −19.5959 + 11.3137i −10.5148 + 59.6321i 0
67.2 2.78546 − 0.491151i 6.64593 + 7.92031i 7.51754 − 2.73616i 0 22.4020 + 18.7975i 0 19.5959 − 11.3137i −13.8744 + 78.6858i 0
91.1 −0.967379 + 2.65785i 5.98571 − 1.05544i −6.12836 − 5.14230i 0 −2.98524 + 16.9302i 0 19.5959 − 11.3137i 9.34311 − 3.40062i 0
91.2 0.967379 − 2.65785i 0.750768 − 0.132381i −6.12836 − 5.14230i 0 0.374429 − 2.12349i 0 −19.5959 + 11.3137i −24.8256 + 9.03577i 0
147.1 −0.967379 − 2.65785i 5.98571 + 1.05544i −6.12836 + 5.14230i 0 −2.98524 − 16.9302i 0 19.5959 + 11.3137i 9.34311 + 3.40062i 0
147.2 0.967379 + 2.65785i 0.750768 + 0.132381i −6.12836 + 5.14230i 0 0.374429 + 2.12349i 0 −19.5959 − 11.3137i −24.8256 − 9.03577i 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 152.4.v.a ✓ 12
8.d odd 2 1 CM 152.4.v.a ✓ 12
19.f odd 18 1 inner 152.4.v.a ✓ 12
152.v even 18 1 inner 152.4.v.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.v.a ✓ 12 1.a even 1 1 trivial
152.4.v.a ✓ 12 8.d odd 2 1 CM
152.4.v.a ✓ 12 19.f odd 18 1 inner
152.4.v.a ✓ 12 152.v even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T312−30T311+519T310−6000T39+58026T38−484080T37+⋯+269517889 T_{3}^{12} - 30 T_{3}^{11} + 519 T_{3}^{10} - 6000 T_{3}^{9} + 58026 T_{3}^{8} - 484080 T_{3}^{7} + \cdots + 269517889 acting on S4new(152,[χ])S_{4}^{\mathrm{new}}(152, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12−512T6+262144 T^{12} - 512 T^{6} + 262144 Copy content Toggle raw display
33 T12+⋯+269517889 T^{12} + \cdots + 269517889 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+⋯+73 ⁣⋯ ⁣81 T^{12} + \cdots + 73\!\cdots\!81 Copy content Toggle raw display
1313 T12 T^{12} Copy content Toggle raw display
1717 T12+⋯+85 ⁣⋯ ⁣61 T^{12} + \cdots + 85\!\cdots\!61 Copy content Toggle raw display
1919 T12+⋯+10 ⁣⋯ ⁣41 T^{12} + \cdots + 10\!\cdots\!41 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+⋯+72 ⁣⋯ ⁣89 T^{12} + \cdots + 72\!\cdots\!89 Copy content Toggle raw display
4343 T12+⋯+13 ⁣⋯ ⁣21 T^{12} + \cdots + 13\!\cdots\!21 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+⋯+24 ⁣⋯ ⁣89 T^{12} + \cdots + 24\!\cdots\!89 Copy content Toggle raw display
6161 T12 T^{12} Copy content Toggle raw display
6767 T12+⋯+54 ⁣⋯ ⁣09 T^{12} + \cdots + 54\!\cdots\!09 Copy content Toggle raw display
7171 T12 T^{12} Copy content Toggle raw display
7373 T12+⋯+30 ⁣⋯ ⁣21 T^{12} + \cdots + 30\!\cdots\!21 Copy content Toggle raw display
7979 T12 T^{12} Copy content Toggle raw display
8383 T12+⋯+29 ⁣⋯ ⁣81 T^{12} + \cdots + 29\!\cdots\!81 Copy content Toggle raw display
8989 T12+⋯+17 ⁣⋯ ⁣49 T^{12} + \cdots + 17\!\cdots\!49 Copy content Toggle raw display
9797 T12+⋯+51 ⁣⋯ ⁣89 T^{12} + \cdots + 51\!\cdots\!89 Copy content Toggle raw display
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