Properties

Label 960.4.b.b
Level 960960
Weight 44
Character orbit 960.b
Analytic conductor 56.64256.642
Analytic rank 00
Dimension 1616
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [960,4,Mod(671,960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 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("960.671"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 960=2635 960 = 2^{6} \cdot 3 \cdot 5
Weight: k k == 4 4
Character orbit: [χ][\chi] == 960.b (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,80] 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: no
Analytic conductor: 56.641833605556.6418336055
Analytic rank: 00
Dimension: 1616
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+3472x12+2676096x8+573099300x4+31755240000 x^{16} + 3472x^{12} + 2676096x^{8} + 573099300x^{4} + 31755240000 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 22334 2^{23}\cdot 3^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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,,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) == qβ4q3+5q5+(β7β4β1)q7+(β92)q9+(β8β7β3)q11β14q135β4q15+(β14+2β12β10+1)q17++(9β154β13++60β1)q99+O(q100) q - \beta_{4} q^{3} + 5 q^{5} + (\beta_{7} - \beta_{4} - \beta_1) q^{7} + (\beta_{9} - 2) q^{9} + ( - \beta_{8} - \beta_{7} - \beta_{3}) q^{11} - \beta_{14} q^{13} - 5 \beta_{4} q^{15} + (\beta_{14} + 2 \beta_{12} - \beta_{10} + \cdots - 1) q^{17}+ \cdots + ( - 9 \beta_{15} - 4 \beta_{13} + \cdots + 60 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q+80q532q9248q21+400q25+192q29520q33160q45+576q49+960q532056q571992q69+784q73+4992q773760q814032q93+1520q97+O(q100) 16 q + 80 q^{5} - 32 q^{9} - 248 q^{21} + 400 q^{25} + 192 q^{29} - 520 q^{33} - 160 q^{45} + 576 q^{49} + 960 q^{53} - 2056 q^{57} - 1992 q^{69} + 784 q^{73} + 4992 q^{77} - 3760 q^{81} - 4032 q^{93} + 1520 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16+3472x12+2676096x8+573099300x4+31755240000 x^{16} + 3472x^{12} + 2676096x^{8} + 573099300x^{4} + 31755240000 : Copy content Toggle raw display

β1\beta_{1}== (9491ν1433546752ν1027708714936ν68590329053100ν2)/14972808163500 ( -9491\nu^{14} - 33546752\nu^{10} - 27708714936\nu^{6} - 8590329053100\nu^{2} ) / 14972808163500 Copy content Toggle raw display
β2\beta_{2}== (4231ν12+13092190ν8+6863423634ν4+625475850885)/13611643785 ( 4231\nu^{12} + 13092190\nu^{8} + 6863423634\nu^{4} + 625475850885 ) / 13611643785 Copy content Toggle raw display
β3\beta_{3}== (28473ν15+350020ν14100640256ν11+621724840ν10++179673697962000ν)/179673697962000 ( - 28473 \nu^{15} + 350020 \nu^{14} - 100640256 \nu^{11} + 621724840 \nu^{10} + \cdots + 179673697962000 \nu ) / 179673697962000 Copy content Toggle raw display
β4\beta_{4}== (28473ν15+1672642ν14100640256ν11+5623847224ν10++179673697962000ν)/359347395924000 ( - 28473 \nu^{15} + 1672642 \nu^{14} - 100640256 \nu^{11} + 5623847224 \nu^{10} + \cdots + 179673697962000 \nu ) / 359347395924000 Copy content Toggle raw display
β5\beta_{5}== (9491ν156020080ν12+33546752ν1120452693360ν8+17 ⁣ ⁣00)/29945616327000 ( 9491 \nu^{15} - 6020080 \nu^{12} + 33546752 \nu^{11} - 20452693360 \nu^{8} + \cdots - 17\!\cdots\!00 ) / 29945616327000 Copy content Toggle raw display
β6\beta_{6}== (9491ν153010040ν1233546752ν1110226346680ν8+854413921053000)/14972808163500 ( - 9491 \nu^{15} - 3010040 \nu^{12} - 33546752 \nu^{11} - 10226346680 \nu^{8} + \cdots - 854413921053000 ) / 14972808163500 Copy content Toggle raw display
β7\beta_{7}== (9491ν15+1739994ν1433546752ν11+5643774168ν10++59891232654000ν)/119782465308000 ( - 9491 \nu^{15} + 1739994 \nu^{14} - 33546752 \nu^{11} + 5643774168 \nu^{10} + \cdots + 59891232654000 \nu ) / 119782465308000 Copy content Toggle raw display
β8\beta_{8}== (28473ν15+1672642ν14+100640256ν11+5623847224ν10+179673697962000ν)/119782465308000 ( 28473 \nu^{15} + 1672642 \nu^{14} + 100640256 \nu^{11} + 5623847224 \nu^{10} + \cdots - 179673697962000 \nu ) / 119782465308000 Copy content Toggle raw display
β9\beta_{9}== (836321ν15+9030120ν131782000ν12+2811923612ν11+35 ⁣ ⁣00)/179673697962000 ( 836321 \nu^{15} + 9030120 \nu^{13} - 1782000 \nu^{12} + 2811923612 \nu^{11} + \cdots - 35\!\cdots\!00 ) / 179673697962000 Copy content Toggle raw display
β10\beta_{10}== (836321ν159030120ν13+34338480ν122811923612ν11++66 ⁣ ⁣00)/179673697962000 ( - 836321 \nu^{15} - 9030120 \nu^{13} + 34338480 \nu^{12} - 2811923612 \nu^{11} + \cdots + 66\!\cdots\!00 ) / 179673697962000 Copy content Toggle raw display
β11\beta_{11}== (8826043ν155359602ν14+75804960ν1330043408096ν11++31 ⁣ ⁣00ν)/10 ⁣ ⁣00 ( - 8826043 \nu^{15} - 5359602 \nu^{14} + 75804960 \nu^{13} - 30043408096 \nu^{11} + \cdots + 31\!\cdots\!00 \nu ) / 10\!\cdots\!00 Copy content Toggle raw display
β12\beta_{12}== (4455731ν1537902480ν1383773800ν1215172664432ν11+12 ⁣ ⁣00)/539021093886000 ( - 4455731 \nu^{15} - 37902480 \nu^{13} - 83773800 \nu^{12} - 15172664432 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 539021093886000 Copy content Toggle raw display
β13\beta_{13}== (1701115ν1556946ν14+18060240ν135724487480ν11++51 ⁣ ⁣00ν)/179673697962000 ( - 1701115 \nu^{15} - 56946 \nu^{14} + 18060240 \nu^{13} - 5724487480 \nu^{11} + \cdots + 51\!\cdots\!00 \nu ) / 179673697962000 Copy content Toggle raw display
β14\beta_{14}== (6244177ν1585255500ν1320426779144ν11276078217800ν9+15 ⁣ ⁣00ν)/539021093886000 ( - 6244177 \nu^{15} - 85255500 \nu^{13} - 20426779144 \nu^{11} - 276078217800 \nu^{9} + \cdots - 15\!\cdots\!00 \nu ) / 539021093886000 Copy content Toggle raw display
β15\beta_{15}== (3564376ν15+85419ν14+58165140ν1311387166772ν11++70 ⁣ ⁣00ν)/269510546943000 ( - 3564376 \nu^{15} + 85419 \nu^{14} + 58165140 \nu^{13} - 11387166772 \nu^{11} + \cdots + 70\!\cdots\!00 \nu ) / 269510546943000 Copy content Toggle raw display
ν\nu== (2β8β6+β5+6β4)/12 ( -2\beta_{8} - \beta_{6} + \beta_{5} + 6\beta_{4} ) / 12 Copy content Toggle raw display
ν2\nu^{2}== (2β86β7+6β375β1)/12 ( 2\beta_{8} - 6\beta_{7} + 6\beta_{3} - 75\beta_1 ) / 12 Copy content Toggle raw display
ν3\nu^{3}== (12β13+18β12+18β1112β10+12β9+74β8+9)/12 ( - 12 \beta_{13} + 18 \beta_{12} + 18 \beta_{11} - 12 \beta_{10} + 12 \beta_{9} + 74 \beta_{8} + \cdots - 9 ) / 12 Copy content Toggle raw display
ν4\nu^{4}== (174β10174β943β686β5+18β25208)/6 ( -174\beta_{10} - 174\beta_{9} - 43\beta_{6} - 86\beta_{5} + 18\beta_{2} - 5208 ) / 6 Copy content Toggle raw display
ν5\nu^{5}== (36β1572β14858β131116β12+1116β11+894β10++558)/12 ( 36 \beta_{15} - 72 \beta_{14} - 858 \beta_{13} - 1116 \beta_{12} + 1116 \beta_{11} + 894 \beta_{10} + \cdots + 558 ) / 12 Copy content Toggle raw display
ν6\nu^{6}== (2633β8+6009β75220β44344β3+28020β1)/3 ( -2633\beta_{8} + 6009\beta_{7} - 5220\beta_{4} - 4344\beta_{3} + 28020\beta_1 ) / 3 Copy content Toggle raw display
ν7\nu^{7}== (1665β153330β14+24261β1329394β1229394β11++14697)/6 ( - 1665 \beta_{15} - 3330 \beta_{14} + 24261 \beta_{13} - 29394 \beta_{12} - 29394 \beta_{11} + \cdots + 14697 ) / 6 Copy content Toggle raw display
ν8\nu^{8}== (212937β10+212937β9+36209β6+72418β553874β2+5026944)/3 ( 212937\beta_{10} + 212937\beta_{9} + 36209\beta_{6} + 72418\beta_{5} - 53874\beta_{2} + 5026944 ) / 3 Copy content Toggle raw display
ν9\nu^{9}== (107748β15+215496β14+1275864β13+1493118β121493118β11+746559)/6 ( - 107748 \beta_{15} + 215496 \beta_{14} + 1275864 \beta_{13} + 1493118 \beta_{12} - 1493118 \beta_{11} + \cdots - 746559 ) / 6 Copy content Toggle raw display
ν10\nu^{10}== (7202168β816563174β7+17223930β4+10472874β362245920β1)/3 ( 7202168\beta_{8} - 16563174\beta_{7} + 17223930\beta_{4} + 10472874\beta_{3} - 62245920\beta_1 ) / 3 Copy content Toggle raw display
ν11\nu^{11}== (6090300β15+12180600β1465205726β13+75017844β12+75017844β11+37508922)/6 ( 6090300 \beta_{15} + 12180600 \beta_{14} - 65205726 \beta_{13} + 75017844 \beta_{12} + 75017844 \beta_{11} + \cdots - 37508922 ) / 6 Copy content Toggle raw display
ν12\nu^{12}== (517772112β10517772112β977166509β6154333018β5+11774469009)/3 ( - 517772112 \beta_{10} - 517772112 \beta_{9} - 77166509 \beta_{6} - 154333018 \beta_{5} + \cdots - 11774469009 ) / 3 Copy content Toggle raw display
ν13\nu^{13}== (161756739β15323513478β141645330287β131876829814β12++938414907)/3 ( 161756739 \beta_{15} - 323513478 \beta_{14} - 1645330287 \beta_{13} - 1876829814 \beta_{12} + \cdots + 938414907 ) / 3 Copy content Toggle raw display
ν14\nu^{14}== (18222259178β8+42358499214β745639807840β425692734454β3+150447844695β1)/3 ( - 18222259178 \beta_{8} + 42358499214 \beta_{7} - 45639807840 \beta_{4} - 25692734454 \beta_{3} + 150447844695 \beta_1 ) / 3 Copy content Toggle raw display
ν15\nu^{15}== (8332882380β1516665764760β14+82538255208β1393743968122β12++46871984061)/3 ( - 8332882380 \beta_{15} - 16665764760 \beta_{14} + 82538255208 \beta_{13} - 93743968122 \beta_{12} + \cdots + 46871984061 ) / 3 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/960Z)×\left(\mathbb{Z}/960\mathbb{Z}\right)^\times.

nn 511511 577577 641641 901901
χ(n)\chi(n) 1-1 11 1-1 1-1

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}
671.1
4.99314 + 4.99314i
4.99314 4.99314i
3.61508 3.61508i
3.61508 + 3.61508i
2.71061 2.71061i
2.71061 + 2.71061i
2.15693 + 2.15693i
2.15693 2.15693i
−2.15693 2.15693i
−2.15693 + 2.15693i
−2.71061 + 2.71061i
−2.71061 2.71061i
−3.61508 + 3.61508i
−3.61508 3.61508i
−4.99314 4.99314i
−4.99314 + 4.99314i
0 −4.99314 1.43824i 0 5.00000 0 16.1856i 0 22.8629 + 14.3627i 0
671.2 0 −4.99314 + 1.43824i 0 5.00000 0 16.1856i 0 22.8629 14.3627i 0
671.3 0 −3.61508 3.73245i 0 5.00000 0 29.1389i 0 −0.862344 + 26.9862i 0
671.4 0 −3.61508 + 3.73245i 0 5.00000 0 29.1389i 0 −0.862344 26.9862i 0
671.5 0 −2.71061 4.43313i 0 5.00000 0 7.10557i 0 −12.3052 + 24.0329i 0
671.6 0 −2.71061 + 4.43313i 0 5.00000 0 7.10557i 0 −12.3052 24.0329i 0
671.7 0 −2.15693 4.72733i 0 5.00000 0 8.15228i 0 −17.6953 + 20.3930i 0
671.8 0 −2.15693 + 4.72733i 0 5.00000 0 8.15228i 0 −17.6953 20.3930i 0
671.9 0 2.15693 4.72733i 0 5.00000 0 8.15228i 0 −17.6953 20.3930i 0
671.10 0 2.15693 + 4.72733i 0 5.00000 0 8.15228i 0 −17.6953 + 20.3930i 0
671.11 0 2.71061 4.43313i 0 5.00000 0 7.10557i 0 −12.3052 24.0329i 0
671.12 0 2.71061 + 4.43313i 0 5.00000 0 7.10557i 0 −12.3052 + 24.0329i 0
671.13 0 3.61508 3.73245i 0 5.00000 0 29.1389i 0 −0.862344 26.9862i 0
671.14 0 3.61508 + 3.73245i 0 5.00000 0 29.1389i 0 −0.862344 + 26.9862i 0
671.15 0 4.99314 1.43824i 0 5.00000 0 16.1856i 0 22.8629 14.3627i 0
671.16 0 4.99314 + 1.43824i 0 5.00000 0 16.1856i 0 22.8629 + 14.3627i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 671.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.4.b.b yes 16
3.b odd 2 1 960.4.b.a 16
4.b odd 2 1 inner 960.4.b.b yes 16
8.b even 2 1 960.4.b.a 16
8.d odd 2 1 960.4.b.a 16
12.b even 2 1 960.4.b.a 16
24.f even 2 1 inner 960.4.b.b yes 16
24.h odd 2 1 inner 960.4.b.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.4.b.a 16 3.b odd 2 1
960.4.b.a 16 8.b even 2 1
960.4.b.a 16 8.d odd 2 1
960.4.b.a 16 12.b even 2 1
960.4.b.b yes 16 1.a even 1 1 trivial
960.4.b.b yes 16 4.b odd 2 1 inner
960.4.b.b yes 16 24.f even 2 1 inner
960.4.b.b yes 16 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(960,[χ])S_{4}^{\mathrm{new}}(960, [\chi]):

T78+1228T76+355728T74+29741776T72+746382400 T_{7}^{8} + 1228T_{7}^{6} + 355728T_{7}^{4} + 29741776T_{7}^{2} + 746382400 Copy content Toggle raw display
T29448T29337488T292+1914544T29+39321360 T_{29}^{4} - 48T_{29}^{3} - 37488T_{29}^{2} + 1914544T_{29} + 39321360 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++282429536481 T^{16} + \cdots + 282429536481 Copy content Toggle raw display
55 (T5)16 (T - 5)^{16} Copy content Toggle raw display
77 (T8+1228T6++746382400)2 (T^{8} + 1228 T^{6} + \cdots + 746382400)^{2} Copy content Toggle raw display
1111 (T8+5328T6++20666937600)2 (T^{8} + 5328 T^{6} + \cdots + 20666937600)^{2} Copy content Toggle raw display
1313 (T8++5700677875200)2 (T^{8} + \cdots + 5700677875200)^{2} Copy content Toggle raw display
1717 (T8++430656937920000)2 (T^{8} + \cdots + 430656937920000)^{2} Copy content Toggle raw display
1919 (T8++420208715520000)2 (T^{8} + \cdots + 420208715520000)^{2} Copy content Toggle raw display
2323 (T8++26 ⁣ ⁣00)2 (T^{8} + \cdots + 26\!\cdots\!00)^{2} Copy content Toggle raw display
2929 (T448T3++39321360)4 (T^{4} - 48 T^{3} + \cdots + 39321360)^{4} Copy content Toggle raw display
3131 (T8++21 ⁣ ⁣00)2 (T^{8} + \cdots + 21\!\cdots\!00)^{2} Copy content Toggle raw display
3737 (T8++35 ⁣ ⁣00)2 (T^{8} + \cdots + 35\!\cdots\!00)^{2} Copy content Toggle raw display
4141 (T8++11 ⁣ ⁣00)2 (T^{8} + \cdots + 11\!\cdots\!00)^{2} Copy content Toggle raw display
4343 (T8++769198518000000)2 (T^{8} + \cdots + 769198518000000)^{2} Copy content Toggle raw display
4747 (T8++16 ⁣ ⁣00)2 (T^{8} + \cdots + 16\!\cdots\!00)^{2} Copy content Toggle raw display
5353 (T4240T3+1424598000)4 (T^{4} - 240 T^{3} + \cdots - 1424598000)^{4} Copy content Toggle raw display
5959 (T8++71 ⁣ ⁣00)2 (T^{8} + \cdots + 71\!\cdots\!00)^{2} Copy content Toggle raw display
6161 (T8++13 ⁣ ⁣00)2 (T^{8} + \cdots + 13\!\cdots\!00)^{2} Copy content Toggle raw display
6767 (T8++63 ⁣ ⁣00)2 (T^{8} + \cdots + 63\!\cdots\!00)^{2} Copy content Toggle raw display
7171 (T8++14 ⁣ ⁣00)2 (T^{8} + \cdots + 14\!\cdots\!00)^{2} Copy content Toggle raw display
7373 (T4196T3+801702800)4 (T^{4} - 196 T^{3} + \cdots - 801702800)^{4} Copy content Toggle raw display
7979 (T8++56 ⁣ ⁣00)2 (T^{8} + \cdots + 56\!\cdots\!00)^{2} Copy content Toggle raw display
8383 (T8++19 ⁣ ⁣00)2 (T^{8} + \cdots + 19\!\cdots\!00)^{2} Copy content Toggle raw display
8989 (T8++16 ⁣ ⁣00)2 (T^{8} + \cdots + 16\!\cdots\!00)^{2} Copy content Toggle raw display
9797 (T4380T3++191786509040)4 (T^{4} - 380 T^{3} + \cdots + 191786509040)^{4} Copy content Toggle raw display
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