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Properties

Label 176.4.m.e
Level 176176
Weight 44
Character orbit 176.m
Analytic conductor 10.38410.384
Analytic rank 00
Dimension 1616
Inner twists 22

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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] = [176,4,Mod(49,176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(176, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 4])) 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("176.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 176=24⋅11 176 = 2^{4} \cdot 11
Weight: k k == 4 4
Character orbit: [χ][\chi] == 176.m (of order 55, degree 44, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 10.384336161010.3843361610
Analytic rank: 00
Dimension: 1616
Relative dimension: 44 over Q(ζ5)\Q(\zeta_{5})
Coefficient field: Q[x]/(x16−⋯ )\mathbb{Q}[x]/(x^{16} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x16−4x15+60x14−83x13+1685x12−14618x11+106543x10−521269x9+⋯+2025 x^{16} - 4 x^{15} + 60 x^{14} - 83 x^{13} + 1685 x^{12} - 14618 x^{11} + 106543 x^{10} - 521269 x^{9} + \cdots + 2025 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 212 2^{12}
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: SU(2)[C5]\mathrm{SU}(2)[C_{5}]

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,…,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β6+β3)q3+(β13+2β5−β3−1)q5+(β14+β12+⋯+β2)q7+(β13−β10+⋯−β2)q9+(−β15−β14−β13+⋯−5)q11+⋯+(−10β15−19β14+⋯+495)q99+O(q100) q + (\beta_{6} + \beta_{3}) q^{3} + (\beta_{13} + 2 \beta_{5} - \beta_{3} - 1) q^{5} + (\beta_{14} + \beta_{12} + \cdots + \beta_{2}) q^{7} + (\beta_{13} - \beta_{10} + \cdots - \beta_{2}) q^{9} + ( - \beta_{15} - \beta_{14} - \beta_{13} + \cdots - 5) q^{11}+ \cdots + ( - 10 \beta_{15} - 19 \beta_{14} + \cdots + 495) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q+q3−q5+13q7+7q9−83q11+69q13−93q15−217q17−126q19+34q21+92q23+307q25−158q27−553q29−205q31−198q33−7q35+⋯+3265q99+O(q100) 16 q + q^{3} - q^{5} + 13 q^{7} + 7 q^{9} - 83 q^{11} + 69 q^{13} - 93 q^{15} - 217 q^{17} - 126 q^{19} + 34 q^{21} + 92 q^{23} + 307 q^{25} - 158 q^{27} - 553 q^{29} - 205 q^{31} - 198 q^{33} - 7 q^{35}+ \cdots + 3265 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16−4x15+60x14−83x13+1685x12−14618x11+106543x10−521269x9+⋯+2025 x^{16} - 4 x^{15} + 60 x^{14} - 83 x^{13} + 1685 x^{12} - 14618 x^{11} + 106543 x^{10} - 521269 x^{9} + \cdots + 2025 : Copy content Toggle raw display

β1\beta_{1}== (−11 ⁣⋯ ⁣53ν15+⋯+79 ⁣⋯ ⁣70)/36 ⁣⋯ ⁣15 ( - 11\!\cdots\!53 \nu^{15} + \cdots + 79\!\cdots\!70 ) / 36\!\cdots\!15 Copy content Toggle raw display
β2\beta_{2}== (40 ⁣⋯ ⁣92ν15+⋯−98 ⁣⋯ ⁣00)/11 ⁣⋯ ⁣55 ( 40\!\cdots\!92 \nu^{15} + \cdots - 98\!\cdots\!00 ) / 11\!\cdots\!55 Copy content Toggle raw display
β3\beta_{3}== (−26 ⁣⋯ ⁣21ν15+⋯−57 ⁣⋯ ⁣45)/35 ⁣⋯ ⁣95 ( - 26\!\cdots\!21 \nu^{15} + \cdots - 57\!\cdots\!45 ) / 35\!\cdots\!95 Copy content Toggle raw display
β4\beta_{4}== (−12 ⁣⋯ ⁣87ν15+⋯−17 ⁣⋯ ⁣20)/10 ⁣⋯ ⁣85 ( - 12\!\cdots\!87 \nu^{15} + \cdots - 17\!\cdots\!20 ) / 10\!\cdots\!85 Copy content Toggle raw display
β5\beta_{5}== (40 ⁣⋯ ⁣92ν15+⋯+35 ⁣⋯ ⁣00)/35 ⁣⋯ ⁣95 ( 40\!\cdots\!92 \nu^{15} + \cdots + 35\!\cdots\!00 ) / 35\!\cdots\!95 Copy content Toggle raw display
β6\beta_{6}== (22 ⁣⋯ ⁣07ν15+⋯+52 ⁣⋯ ⁣50)/36 ⁣⋯ ⁣15 ( 22\!\cdots\!07 \nu^{15} + \cdots + 52\!\cdots\!50 ) / 36\!\cdots\!15 Copy content Toggle raw display
β7\beta_{7}== (13 ⁣⋯ ⁣13ν15+⋯+66 ⁣⋯ ⁣00)/22 ⁣⋯ ⁣69 ( 13\!\cdots\!13 \nu^{15} + \cdots + 66\!\cdots\!00 ) / 22\!\cdots\!69 Copy content Toggle raw display
β8\beta_{8}== (20 ⁣⋯ ⁣46ν15+⋯+49 ⁣⋯ ⁣30)/33 ⁣⋯ ⁣35 ( 20\!\cdots\!46 \nu^{15} + \cdots + 49\!\cdots\!30 ) / 33\!\cdots\!35 Copy content Toggle raw display
β9\beta_{9}== (−69 ⁣⋯ ⁣07ν15+⋯−34 ⁣⋯ ⁣30)/11 ⁣⋯ ⁣45 ( - 69\!\cdots\!07 \nu^{15} + \cdots - 34\!\cdots\!30 ) / 11\!\cdots\!45 Copy content Toggle raw display
β10\beta_{10}== (−72 ⁣⋯ ⁣30ν15+⋯−10 ⁣⋯ ⁣70)/11 ⁣⋯ ⁣45 ( - 72\!\cdots\!30 \nu^{15} + \cdots - 10\!\cdots\!70 ) / 11\!\cdots\!45 Copy content Toggle raw display
β11\beta_{11}== (92 ⁣⋯ ⁣29ν15+⋯+80 ⁣⋯ ⁣25)/11 ⁣⋯ ⁣45 ( 92\!\cdots\!29 \nu^{15} + \cdots + 80\!\cdots\!25 ) / 11\!\cdots\!45 Copy content Toggle raw display
β12\beta_{12}== (63 ⁣⋯ ⁣81ν15+⋯+30 ⁣⋯ ⁣90)/66 ⁣⋯ ⁣07 ( 63\!\cdots\!81 \nu^{15} + \cdots + 30\!\cdots\!90 ) / 66\!\cdots\!07 Copy content Toggle raw display
β13\beta_{13}== (12 ⁣⋯ ⁣14ν15+⋯+10 ⁣⋯ ⁣00)/11 ⁣⋯ ⁣45 ( 12\!\cdots\!14 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 11\!\cdots\!45 Copy content Toggle raw display
β14\beta_{14}== (−36 ⁣⋯ ⁣53ν15+⋯−17 ⁣⋯ ⁣45)/33 ⁣⋯ ⁣35 ( - 36\!\cdots\!53 \nu^{15} + \cdots - 17\!\cdots\!45 ) / 33\!\cdots\!35 Copy content Toggle raw display
β15\beta_{15}== (−16 ⁣⋯ ⁣65ν15+⋯−37 ⁣⋯ ⁣40)/11 ⁣⋯ ⁣45 ( - 16\!\cdots\!65 \nu^{15} + \cdots - 37\!\cdots\!40 ) / 11\!\cdots\!45 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β14+β12−β11−β9+β7−β5+2β4−β3−β2+1)/4 ( \beta_{14} + \beta_{12} - \beta_{11} - \beta_{9} + \beta_{7} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} + 1 ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (−β15−3β14+β13−β12−2β11+β10+β9+⋯−8)/4 ( - \beta_{15} - 3 \beta_{14} + \beta_{13} - \beta_{12} - 2 \beta_{11} + \beta_{10} + \beta_{9} + \cdots - 8 ) / 4 Copy content Toggle raw display
ν3\nu^{3}== (3β14+4β13−7β12+3β11−38β10+4β9+β8+⋯−3)/4 ( 3 \beta_{14} + 4 \beta_{13} - 7 \beta_{12} + 3 \beta_{11} - 38 \beta_{10} + 4 \beta_{9} + \beta_{8} + \cdots - 3 ) / 4 Copy content Toggle raw display
ν4\nu^{4}== (145β15+205β13+619β11−23β10+182β9+292β8+⋯+449)/4 ( 145 \beta_{15} + 205 \beta_{13} + 619 \beta_{11} - 23 \beta_{10} + 182 \beta_{9} + 292 \beta_{8} + \cdots + 449 ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (542β15−116β14−116β13+542β12−2087β9−1545β8+⋯+17046)/4 ( 542 \beta_{15} - 116 \beta_{14} - 116 \beta_{13} + 542 \beta_{12} - 2087 \beta_{9} - 1545 \beta_{8} + \cdots + 17046 ) / 4 Copy content Toggle raw display
ν6\nu^{6}== (−2907β15+9622β14−2907β13+8170β12−36405β11+⋯−138061)/4 ( - 2907 \beta_{15} + 9622 \beta_{14} - 2907 \beta_{13} + 8170 \beta_{12} - 36405 \beta_{11} + \cdots - 138061 ) / 4 Copy content Toggle raw display
ν7\nu^{7}== (−118634β15−75235β14−78418β13−80969β12−79299β11+⋯−464527)/4 ( - 118634 \beta_{15} - 75235 \beta_{14} - 78418 \beta_{13} - 80969 \beta_{12} - 79299 \beta_{11} + \cdots - 464527 ) / 4 Copy content Toggle raw display
ν8\nu^{8}== (72825β14−10492β13−251551β12+808915β11−486800β10+⋯−808915)/4 ( 72825 \beta_{14} - 10492 \beta_{13} - 251551 \beta_{12} + 808915 \beta_{11} - 486800 \beta_{10} + \cdots - 808915 ) / 4 Copy content Toggle raw display
ν9\nu^{9}== (6738409β15+6316675β13+16661425β11−2549724β10+3766951β9+⋯+23661739)/4 ( 6738409 \beta_{15} + 6316675 \beta_{13} + 16661425 \beta_{11} - 2549724 \beta_{10} + 3766951 \beta_{9} + \cdots + 23661739 ) / 4 Copy content Toggle raw display
ν10\nu^{10}== (9413058β15+51998β14+51998β13+9413058β12−22301985β9+⋯+241753968)/2 ( 9413058 \beta_{15} + 51998 \beta_{14} + 51998 \beta_{13} + 9413058 \beta_{12} - 22301985 \beta_{9} + \cdots + 241753968 ) / 2 Copy content Toggle raw display
ν11\nu^{11}== (−169942957β15+192426688β14−169942957β13+226267472β12+⋯−2419533373)/4 ( - 169942957 \beta_{15} + 192426688 \beta_{14} - 169942957 \beta_{13} + 226267472 \beta_{12} + \cdots - 2419533373 ) / 4 Copy content Toggle raw display
ν12\nu^{12}== (−2921919832β15−1350049445β14−1817938598β13−1609241489β12+⋯−20162136479)/4 ( - 2921919832 \beta_{15} - 1350049445 \beta_{14} - 1817938598 \beta_{13} - 1609241489 \beta_{12} + \cdots - 20162136479 ) / 4 Copy content Toggle raw display
ν13\nu^{13}== (−1325892545β14+1292850551β13−5588325403β12+14594975574β11+⋯−14594975574)/2 ( - 1325892545 \beta_{14} + 1292850551 \beta_{13} - 5588325403 \beta_{12} + 14594975574 \beta_{11} + \cdots - 14594975574 ) / 2 Copy content Toggle raw display
ν14\nu^{14}== (90392633315β15+79843906181β13+243403074732β11−44007542631β10+⋯+531038420813)/2 ( 90392633315 \beta_{15} + 79843906181 \beta_{13} + 243403074732 \beta_{11} - 44007542631 \beta_{10} + \cdots + 531038420813 ) / 2 Copy content Toggle raw display
ν15\nu^{15}== (726568911906β15+213910212068β14+213910212068β13+726568911906β12+⋯+11522693609186)/4 ( 726568911906 \beta_{15} + 213910212068 \beta_{14} + 213910212068 \beta_{13} + 726568911906 \beta_{12} + \cdots + 11522693609186 ) / 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/176Z)×\left(\mathbb{Z}/176\mathbb{Z}\right)^\times.

nn 111111 133133 145145
χ(n)\chi(n) 11 11 −1−β3+β4+β5-1 - \beta_{3} + \beta_{4} + \beta_{5}

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}
49.1
−5.58272 + 4.05608i
2.60776 − 1.89465i
3.98665 − 2.89647i
−0.0116883 + 0.00849208i
2.43785 + 7.50293i
−1.72876 − 5.32058i
0.432930 + 1.33242i
−0.142016 − 0.437082i
−5.58272 − 4.05608i
2.60776 + 1.89465i
3.98665 + 2.89647i
−0.0116883 − 0.00849208i
2.43785 − 7.50293i
−1.72876 + 5.32058i
0.432930 − 1.33242i
−0.142016 + 0.437082i
0 −5.04830 − 3.66781i 0 2.49068 − 7.66551i 0 −20.6812 + 15.0258i 0 3.68910 + 11.3539i 0
49.2 0 −4.57022 − 3.32046i 0 −3.76204 + 11.5784i 0 27.3815 − 19.8938i 0 1.51800 + 4.67192i 0
49.3 0 1.87500 + 1.36227i 0 1.88546 − 5.80286i 0 3.69974 − 2.68802i 0 −6.68360 − 20.5700i 0
49.4 0 6.31647 + 4.58918i 0 −3.65918 + 11.2618i 0 −7.70898 + 5.60090i 0 10.4937 + 32.2964i 0
81.1 0 −1.71205 + 5.26914i 0 5.02952 − 3.65416i 0 3.12006 + 9.60257i 0 −2.98931 − 2.17186i 0
81.2 0 −0.320520 + 0.986460i 0 −7.72334 + 5.61133i 0 −5.56367 − 17.1232i 0 20.9731 + 15.2378i 0
81.3 0 1.76631 − 5.43615i 0 16.1524 − 11.7354i 0 −1.37031 − 4.21739i 0 −4.58836 − 3.33364i 0
81.4 0 2.19331 − 6.75031i 0 −10.9135 + 7.92910i 0 7.62293 + 23.4610i 0 −18.9126 − 13.7408i 0
97.1 0 −5.04830 + 3.66781i 0 2.49068 + 7.66551i 0 −20.6812 − 15.0258i 0 3.68910 − 11.3539i 0
97.2 0 −4.57022 + 3.32046i 0 −3.76204 − 11.5784i 0 27.3815 + 19.8938i 0 1.51800 − 4.67192i 0
97.3 0 1.87500 − 1.36227i 0 1.88546 + 5.80286i 0 3.69974 + 2.68802i 0 −6.68360 + 20.5700i 0
97.4 0 6.31647 − 4.58918i 0 −3.65918 − 11.2618i 0 −7.70898 − 5.60090i 0 10.4937 − 32.2964i 0
113.1 0 −1.71205 − 5.26914i 0 5.02952 + 3.65416i 0 3.12006 − 9.60257i 0 −2.98931 + 2.17186i 0
113.2 0 −0.320520 − 0.986460i 0 −7.72334 − 5.61133i 0 −5.56367 + 17.1232i 0 20.9731 − 15.2378i 0
113.3 0 1.76631 + 5.43615i 0 16.1524 + 11.7354i 0 −1.37031 + 4.21739i 0 −4.58836 + 3.33364i 0
113.4 0 2.19331 + 6.75031i 0 −10.9135 − 7.92910i 0 7.62293 − 23.4610i 0 −18.9126 + 13.7408i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.4.m.e 16
4.b odd 2 1 88.4.i.a ✓ 16
11.c even 5 1 inner 176.4.m.e 16
11.c even 5 1 1936.4.a.bw 8
11.d odd 10 1 1936.4.a.bv 8
44.g even 10 1 968.4.a.o 8
44.h odd 10 1 88.4.i.a ✓ 16
44.h odd 10 1 968.4.a.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.4.i.a ✓ 16 4.b odd 2 1
88.4.i.a ✓ 16 44.h odd 10 1
176.4.m.e 16 1.a even 1 1 trivial
176.4.m.e 16 11.c even 5 1 inner
968.4.a.n 8 44.h odd 10 1
968.4.a.o 8 44.g even 10 1
1936.4.a.bv 8 11.d odd 10 1
1936.4.a.bw 8 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T316−T315+51T314+164T313+2923T312+5816T311+⋯+22114366681 T_{3}^{16} - T_{3}^{15} + 51 T_{3}^{14} + 164 T_{3}^{13} + 2923 T_{3}^{12} + 5816 T_{3}^{11} + \cdots + 22114366681 acting on S4new(176,[χ])S_{4}^{\mathrm{new}}(176, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16 T^{16} Copy content Toggle raw display
33 T16+⋯+22114366681 T^{16} + \cdots + 22114366681 Copy content Toggle raw display
55 T16+⋯+12 ⁣⋯ ⁣96 T^{16} + \cdots + 12\!\cdots\!96 Copy content Toggle raw display
77 T16+⋯+56 ⁣⋯ ⁣36 T^{16} + \cdots + 56\!\cdots\!36 Copy content Toggle raw display
1111 T16+⋯+98 ⁣⋯ ⁣41 T^{16} + \cdots + 98\!\cdots\!41 Copy content Toggle raw display
1313 T16+⋯+10 ⁣⋯ ⁣00 T^{16} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
1717 T16+⋯+79 ⁣⋯ ⁣21 T^{16} + \cdots + 79\!\cdots\!21 Copy content Toggle raw display
1919 T16+⋯+49 ⁣⋯ ⁣81 T^{16} + \cdots + 49\!\cdots\!81 Copy content Toggle raw display
2323 (T8+⋯+47 ⁣⋯ ⁣56)2 (T^{8} + \cdots + 47\!\cdots\!56)^{2} Copy content Toggle raw display
2929 T16+⋯+21 ⁣⋯ ⁣96 T^{16} + \cdots + 21\!\cdots\!96 Copy content Toggle raw display
3131 T16+⋯+19 ⁣⋯ ⁣96 T^{16} + \cdots + 19\!\cdots\!96 Copy content Toggle raw display
3737 T16+⋯+96 ⁣⋯ ⁣76 T^{16} + \cdots + 96\!\cdots\!76 Copy content Toggle raw display
4141 T16+⋯+22 ⁣⋯ ⁣21 T^{16} + \cdots + 22\!\cdots\!21 Copy content Toggle raw display
4343 (T8+⋯+25 ⁣⋯ ⁣84)2 (T^{8} + \cdots + 25\!\cdots\!84)^{2} Copy content Toggle raw display
4747 T16+⋯+28 ⁣⋯ ⁣76 T^{16} + \cdots + 28\!\cdots\!76 Copy content Toggle raw display
5353 T16+⋯+19 ⁣⋯ ⁣56 T^{16} + \cdots + 19\!\cdots\!56 Copy content Toggle raw display
5959 T16+⋯+48 ⁣⋯ ⁣61 T^{16} + \cdots + 48\!\cdots\!61 Copy content Toggle raw display
6161 T16+⋯+43 ⁣⋯ ⁣96 T^{16} + \cdots + 43\!\cdots\!96 Copy content Toggle raw display
6767 (T8+⋯−18 ⁣⋯ ⁣20)2 (T^{8} + \cdots - 18\!\cdots\!20)^{2} Copy content Toggle raw display
7171 T16+⋯+99 ⁣⋯ ⁣76 T^{16} + \cdots + 99\!\cdots\!76 Copy content Toggle raw display
7373 T16+⋯+79 ⁣⋯ ⁣61 T^{16} + \cdots + 79\!\cdots\!61 Copy content Toggle raw display
7979 T16+⋯+23 ⁣⋯ ⁣96 T^{16} + \cdots + 23\!\cdots\!96 Copy content Toggle raw display
8383 T16+⋯+12 ⁣⋯ ⁣81 T^{16} + \cdots + 12\!\cdots\!81 Copy content Toggle raw display
8989 (T8+⋯+15 ⁣⋯ ⁣20)2 (T^{8} + \cdots + 15\!\cdots\!20)^{2} Copy content Toggle raw display
9797 T16+⋯+40 ⁣⋯ ⁣21 T^{16} + \cdots + 40\!\cdots\!21 Copy content Toggle raw display
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