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Properties

Label 1859.4.a.k
Level 18591859
Weight 44
Character orbit 1859.a
Self dual yes
Analytic conductor 109.685109.685
Analytic rank 00
Dimension 1818
CM no
Inner twists 11

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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] = [1859,4,Mod(1,1859)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1859, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) 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("1859.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 1859=11⋅132 1859 = 11 \cdot 13^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1859.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,2,0,76,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 109.684550701109.684550701
Analytic rank: 00
Dimension: 1818
Coefficient field: Q[x]/(x18−⋯ )\mathbb{Q}[x]/(x^{18} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x18−2x17−108x16+212x15+4721x14−8963x13−107626x12+194656x11+⋯+9847296 x^{18} - 2 x^{17} - 108 x^{16} + 212 x^{15} + 4721 x^{14} - 8963 x^{13} - 107626 x^{12} + 194656 x^{11} + \cdots + 9847296 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a17]\Z[a_1, \ldots, a_{17}]
Coefficient ring index: 25⋅32 2^{5}\cdot 3^{2}
Twist minimal: no (minimal twist has level 143)
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

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,…,β171,\beta_1,\ldots,\beta_{17} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2−β5q3+(β2+4)q4+(β11+1)q5+(β8−β5+3)q6+(β14−β5+2)q7+(β3+4β1−1)q8+(β6+β1+10)q9+⋯+(11β6+11β1+110)q99+O(q100) q + \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{11} + 1) q^{5} + (\beta_{8} - \beta_{5} + 3) q^{6} + (\beta_{14} - \beta_{5} + 2) q^{7} + (\beta_{3} + 4 \beta_1 - 1) q^{8} + (\beta_{6} + \beta_1 + 10) q^{9}+ \cdots + (11 \beta_{6} + 11 \beta_1 + 110) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 18q+2q2+76q4+20q5+49q6+28q7−12q8+180q9+56q10+198q11+54q12+4q14+60q15+364q16−138q17+298q18+24q19+160q20+⋯+1980q99+O(q100) 18 q + 2 q^{2} + 76 q^{4} + 20 q^{5} + 49 q^{6} + 28 q^{7} - 12 q^{8} + 180 q^{9} + 56 q^{10} + 198 q^{11} + 54 q^{12} + 4 q^{14} + 60 q^{15} + 364 q^{16} - 138 q^{17} + 298 q^{18} + 24 q^{19} + 160 q^{20}+ \cdots + 1980 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x18−2x17−108x16+212x15+4721x14−8963x13−107626x12+194656x11+⋯+9847296 x^{18} - 2 x^{17} - 108 x^{16} + 212 x^{15} + 4721 x^{14} - 8963 x^{13} - 107626 x^{12} + 194656 x^{11} + \cdots + 9847296 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2−12 \nu^{2} - 12 Copy content Toggle raw display
β3\beta_{3}== ν3−20ν+1 \nu^{3} - 20\nu + 1 Copy content Toggle raw display
β4\beta_{4}== (−87 ⁣⋯ ⁣43ν17+⋯+69 ⁣⋯ ⁣48)/38 ⁣⋯ ⁣12 ( - 87\!\cdots\!43 \nu^{17} + \cdots + 69\!\cdots\!48 ) / 38\!\cdots\!12 Copy content Toggle raw display
β5\beta_{5}== (19 ⁣⋯ ⁣81ν17+⋯+17 ⁣⋯ ⁣96)/77 ⁣⋯ ⁣24 ( 19\!\cdots\!81 \nu^{17} + \cdots + 17\!\cdots\!96 ) / 77\!\cdots\!24 Copy content Toggle raw display
β6\beta_{6}== (21 ⁣⋯ ⁣85ν17+⋯−26 ⁣⋯ ⁣64)/77 ⁣⋯ ⁣24 ( 21\!\cdots\!85 \nu^{17} + \cdots - 26\!\cdots\!64 ) / 77\!\cdots\!24 Copy content Toggle raw display
β7\beta_{7}== (13 ⁣⋯ ⁣15ν17+⋯+43 ⁣⋯ ⁣04)/38 ⁣⋯ ⁣12 ( 13\!\cdots\!15 \nu^{17} + \cdots + 43\!\cdots\!04 ) / 38\!\cdots\!12 Copy content Toggle raw display
β8\beta_{8}== (18 ⁣⋯ ⁣99ν17+⋯+10 ⁣⋯ ⁣00)/38 ⁣⋯ ⁣12 ( 18\!\cdots\!99 \nu^{17} + \cdots + 10\!\cdots\!00 ) / 38\!\cdots\!12 Copy content Toggle raw display
β9\beta_{9}== (47 ⁣⋯ ⁣53ν17+⋯−37 ⁣⋯ ⁣80)/96 ⁣⋯ ⁣28 ( 47\!\cdots\!53 \nu^{17} + \cdots - 37\!\cdots\!80 ) / 96\!\cdots\!28 Copy content Toggle raw display
β10\beta_{10}== (11 ⁣⋯ ⁣97ν17+⋯+16 ⁣⋯ ⁣20)/19 ⁣⋯ ⁣56 ( 11\!\cdots\!97 \nu^{17} + \cdots + 16\!\cdots\!20 ) / 19\!\cdots\!56 Copy content Toggle raw display
β11\beta_{11}== (55 ⁣⋯ ⁣81ν17+⋯+18 ⁣⋯ ⁣60)/77 ⁣⋯ ⁣24 ( 55\!\cdots\!81 \nu^{17} + \cdots + 18\!\cdots\!60 ) / 77\!\cdots\!24 Copy content Toggle raw display
β12\beta_{12}== (−57 ⁣⋯ ⁣87ν17+⋯+75 ⁣⋯ ⁣60)/77 ⁣⋯ ⁣24 ( - 57\!\cdots\!87 \nu^{17} + \cdots + 75\!\cdots\!60 ) / 77\!\cdots\!24 Copy content Toggle raw display
β13\beta_{13}== (−19 ⁣⋯ ⁣00ν17+⋯−24 ⁣⋯ ⁣40)/24 ⁣⋯ ⁣32 ( - 19\!\cdots\!00 \nu^{17} + \cdots - 24\!\cdots\!40 ) / 24\!\cdots\!32 Copy content Toggle raw display
β14\beta_{14}== (12 ⁣⋯ ⁣95ν17+⋯+58 ⁣⋯ ⁣40)/14 ⁣⋯ ⁣12 ( 12\!\cdots\!95 \nu^{17} + \cdots + 58\!\cdots\!40 ) / 14\!\cdots\!12 Copy content Toggle raw display
β15\beta_{15}== (20 ⁣⋯ ⁣57ν17+⋯−96 ⁣⋯ ⁣72)/14 ⁣⋯ ⁣12 ( 20\!\cdots\!57 \nu^{17} + \cdots - 96\!\cdots\!72 ) / 14\!\cdots\!12 Copy content Toggle raw display
β16\beta_{16}== (14 ⁣⋯ ⁣05ν17+⋯+58 ⁣⋯ ⁣56)/96 ⁣⋯ ⁣28 ( 14\!\cdots\!05 \nu^{17} + \cdots + 58\!\cdots\!56 ) / 96\!\cdots\!28 Copy content Toggle raw display
β17\beta_{17}== (−63 ⁣⋯ ⁣33ν17+⋯−28 ⁣⋯ ⁣44)/38 ⁣⋯ ⁣12 ( - 63\!\cdots\!33 \nu^{17} + \cdots - 28\!\cdots\!44 ) / 38\!\cdots\!12 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}== β2+12 \beta_{2} + 12 Copy content Toggle raw display
ν3\nu^{3}== β3+20β1−1 \beta_{3} + 20\beta _1 - 1 Copy content Toggle raw display
ν4\nu^{4}== β17+2β11+β10+β8−β7−β6−β5+β4+⋯+243 \beta_{17} + 2 \beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 243 Copy content Toggle raw display
ν5\nu^{5}== β17+β16−3β13+β12−5β11−2β10+2β9+⋯−54 \beta_{17} + \beta_{16} - 3 \beta_{13} + \beta_{12} - 5 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \cdots - 54 Copy content Toggle raw display
ν6\nu^{6}== 43β17+5β16−9β15−2β14+7β13−3β12+110β11+⋯+5694 43 \beta_{17} + 5 \beta_{16} - 9 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 3 \beta_{12} + 110 \beta_{11} + \cdots + 5694 Copy content Toggle raw display
ν7\nu^{7}== 58β17+42β16+11β15+41β14−146β13+34β12+⋯−2429 58 \beta_{17} + 42 \beta_{16} + 11 \beta_{15} + 41 \beta_{14} - 146 \beta_{13} + 34 \beta_{12} + \cdots - 2429 Copy content Toggle raw display
ν8\nu^{8}== 1392β17+243β16−505β15−164β14+397β13−177β12+⋯+142456 1392 \beta_{17} + 243 \beta_{16} - 505 \beta_{15} - 164 \beta_{14} + 397 \beta_{13} - 177 \beta_{12} + \cdots + 142456 Copy content Toggle raw display
ν9\nu^{9}== 2365β17+1423β16+839β15+2748β14−5235β13+811β12+⋯−95658 2365 \beta_{17} + 1423 \beta_{16} + 839 \beta_{15} + 2748 \beta_{14} - 5235 \beta_{13} + 811 \beta_{12} + \cdots - 95658 Copy content Toggle raw display
ν10\nu^{10}== 40683β17+8386β16−20201β15−9310β14+15992β13+⋯+3703901 40683 \beta_{17} + 8386 \beta_{16} - 20201 \beta_{15} - 9310 \beta_{14} + 15992 \beta_{13} + \cdots + 3703901 Copy content Toggle raw display
ν11\nu^{11}== 84881β17+45180β16+42082β15+125248β14−168488β13+⋯−3497365 84881 \beta_{17} + 45180 \beta_{16} + 42082 \beta_{15} + 125248 \beta_{14} - 168488 \beta_{13} + \cdots - 3497365 Copy content Toggle raw display
ν12\nu^{12}== 1132592β17+253357β16−712064β15−429353β14+565631β13+⋯+98915292 1132592 \beta_{17} + 253357 \beta_{16} - 712064 \beta_{15} - 429353 \beta_{14} + 565631 \beta_{13} + \cdots + 98915292 Copy content Toggle raw display
ν13\nu^{13}== 2858244β17+1392248β16+1758968β15+4867680β14−5175540β13+⋯−122138637 2858244 \beta_{17} + 1392248 \beta_{16} + 1758968 \beta_{15} + 4867680 \beta_{14} - 5175540 \beta_{13} + \cdots - 122138637 Copy content Toggle raw display
ν14\nu^{14}== 30685237β17+7156458β16−23629696β15−17447418β14+18790318β13+⋯+2695761351 30685237 \beta_{17} + 7156458 \beta_{16} - 23629696 \beta_{15} - 17447418 \beta_{14} + 18790318 \beta_{13} + \cdots + 2695761351 Copy content Toggle raw display
ν15\nu^{15}== 92570193β17+42169053β16+66545956β15+173975032β14+⋯−4134600078 92570193 \beta_{17} + 42169053 \beta_{16} + 66545956 \beta_{15} + 173975032 \beta_{14} + \cdots - 4134600078 Copy content Toggle raw display
ν16\nu^{16}== 817043035β17+194236989β16−758941021β15−653984098β14+⋯+74662407118 817043035 \beta_{17} + 194236989 \beta_{16} - 758941021 \beta_{15} - 653984098 \beta_{14} + \cdots + 74662407118 Copy content Toggle raw display
ν17\nu^{17}== 2919002722β17+1262961878β16+2368607115β15+5916221793β14+⋯−136855788777 2919002722 \beta_{17} + 1262961878 \beta_{16} + 2368607115 \beta_{15} + 5916221793 \beta_{14} + \cdots - 136855788777 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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}
1.1
−5.51522
−4.98751
−4.27424
−3.98165
−2.40428
−2.37174
−2.32803
−0.901974
0.370518
0.468651
1.26989
2.18512
2.22320
3.37885
3.97618
4.83029
4.83238
5.22956
−5.51522 3.00574 22.4176 18.8802 −16.5773 −0.324979 −79.5165 −17.9655 −104.128
1.2 −4.98751 −4.47034 16.8752 −20.1605 22.2959 23.8481 −44.2654 −7.01607 100.551
1.3 −4.27424 4.33951 10.2691 0.295728 −18.5481 11.5737 −9.69883 −8.16862 −1.26401
1.4 −3.98165 −8.48796 7.85352 −2.25841 33.7961 −28.5605 0.583227 45.0454 8.99219
1.5 −2.40428 7.21858 −2.21943 −9.84845 −17.3555 −8.26104 24.5704 25.1078 23.6784
1.6 −2.37174 −6.00967 −2.37485 22.0017 14.2534 21.9243 24.6064 9.11608 −52.1824
1.7 −2.32803 −4.42417 −2.58028 −12.1881 10.2996 −11.2388 24.6312 −7.42671 28.3743
1.8 −0.901974 8.17246 −7.18644 11.3696 −7.37135 8.81530 13.6978 39.7891 −10.2551
1.9 0.370518 −0.548979 −7.86272 −11.6192 −0.203406 19.9902 −5.87742 −26.6986 −4.30511
1.10 0.468651 −6.65460 −7.78037 3.45502 −3.11869 11.5900 −7.39549 17.2837 1.61920
1.11 1.26989 3.96847 −6.38737 −4.99906 5.03953 −32.3730 −18.2704 −11.2513 −6.34827
1.12 2.18512 −9.62593 −3.22525 10.6961 −21.0338 −26.7047 −24.5285 65.6585 23.3723
1.13 2.22320 2.89827 −3.05739 19.1319 6.44342 4.29594 −24.5828 −18.6000 42.5339
1.14 3.37885 7.92479 3.41661 −15.9579 26.7767 29.7230 −15.4866 35.8023 −53.9193
1.15 3.97618 −1.00306 7.81003 −0.345299 −3.98837 −17.7538 −0.755338 −25.9939 −1.37297
1.16 4.83029 −7.08819 15.3317 −9.07266 −34.2380 −2.15508 35.4141 23.2424 −43.8235
1.17 4.83238 9.74664 15.3519 8.44122 47.0995 −12.5767 35.5272 67.9971 40.7912
1.18 5.22956 1.03843 19.3483 12.1782 5.43054 36.1879 59.3469 −25.9217 63.6866
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
1111 −1 -1
1313 −1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.4.a.k 18
13.b even 2 1 1859.4.a.j 18
13.d odd 4 2 143.4.b.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.b.a ✓ 36 13.d odd 4 2
1859.4.a.j 18 13.b even 2 1
1859.4.a.k 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T218−2T217−108T216+212T215+4721T214−8963T213+⋯+9847296 T_{2}^{18} - 2 T_{2}^{17} - 108 T_{2}^{16} + 212 T_{2}^{15} + 4721 T_{2}^{14} - 8963 T_{2}^{13} + \cdots + 9847296 acting on S4new(Γ0(1859))S_{4}^{\mathrm{new}}(\Gamma_0(1859)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T18−2T17+⋯+9847296 T^{18} - 2 T^{17} + \cdots + 9847296 Copy content Toggle raw display
33 T18+⋯−179055689728 T^{18} + \cdots - 179055689728 Copy content Toggle raw display
55 T18+⋯−16 ⁣⋯ ⁣84 T^{18} + \cdots - 16\!\cdots\!84 Copy content Toggle raw display
77 T18+⋯−20 ⁣⋯ ⁣40 T^{18} + \cdots - 20\!\cdots\!40 Copy content Toggle raw display
1111 (T−11)18 (T - 11)^{18} Copy content Toggle raw display
1313 T18 T^{18} Copy content Toggle raw display
1717 T18+⋯−14 ⁣⋯ ⁣00 T^{18} + \cdots - 14\!\cdots\!00 Copy content Toggle raw display
1919 T18+⋯+34 ⁣⋯ ⁣04 T^{18} + \cdots + 34\!\cdots\!04 Copy content Toggle raw display
2323 T18+⋯−69 ⁣⋯ ⁣76 T^{18} + \cdots - 69\!\cdots\!76 Copy content Toggle raw display
2929 T18+⋯−19 ⁣⋯ ⁣96 T^{18} + \cdots - 19\!\cdots\!96 Copy content Toggle raw display
3131 T18+⋯−76 ⁣⋯ ⁣12 T^{18} + \cdots - 76\!\cdots\!12 Copy content Toggle raw display
3737 T18+⋯+66 ⁣⋯ ⁣92 T^{18} + \cdots + 66\!\cdots\!92 Copy content Toggle raw display
4141 T18+⋯+16 ⁣⋯ ⁣24 T^{18} + \cdots + 16\!\cdots\!24 Copy content Toggle raw display
4343 T18+⋯+26 ⁣⋯ ⁣04 T^{18} + \cdots + 26\!\cdots\!04 Copy content Toggle raw display
4747 T18+⋯+10 ⁣⋯ ⁣64 T^{18} + \cdots + 10\!\cdots\!64 Copy content Toggle raw display
5353 T18+⋯+26 ⁣⋯ ⁣24 T^{18} + \cdots + 26\!\cdots\!24 Copy content Toggle raw display
5959 T18+⋯−14 ⁣⋯ ⁣36 T^{18} + \cdots - 14\!\cdots\!36 Copy content Toggle raw display
6161 T18+⋯−22 ⁣⋯ ⁣56 T^{18} + \cdots - 22\!\cdots\!56 Copy content Toggle raw display
6767 T18+⋯+46 ⁣⋯ ⁣36 T^{18} + \cdots + 46\!\cdots\!36 Copy content Toggle raw display
7171 T18+⋯−39 ⁣⋯ ⁣52 T^{18} + \cdots - 39\!\cdots\!52 Copy content Toggle raw display
7373 T18+⋯−26 ⁣⋯ ⁣84 T^{18} + \cdots - 26\!\cdots\!84 Copy content Toggle raw display
7979 T18+⋯+38 ⁣⋯ ⁣00 T^{18} + \cdots + 38\!\cdots\!00 Copy content Toggle raw display
8383 T18+⋯−82 ⁣⋯ ⁣76 T^{18} + \cdots - 82\!\cdots\!76 Copy content Toggle raw display
8989 T18+⋯−26 ⁣⋯ ⁣00 T^{18} + \cdots - 26\!\cdots\!00 Copy content Toggle raw display
9797 T18+⋯+46 ⁣⋯ ⁣68 T^{18} + \cdots + 46\!\cdots\!68 Copy content Toggle raw display
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