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Properties

Label 968.6.a.q
Level 968968
Weight 66
Character orbit 968.a
Self dual yes
Analytic conductor 155.252155.252
Analytic rank 00
Dimension 1616
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] = [968,6,Mod(1,968)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(968, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("968.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: N N == 968=23⋅112 968 = 2^{3} \cdot 11^{2}
Weight: k k == 6 6
Character orbit: [χ][\chi] == 968.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,12,0,81,0,47] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 155.251537579155.251537579
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: x16−4x15−2635x14+10644x13+2721739x12−11107836x11+⋯+53 ⁣⋯ ⁣20 x^{16} - 4 x^{15} - 2635 x^{14} + 10644 x^{13} + 2721739 x^{12} - 11107836 x^{11} + \cdots + 53\!\cdots\!20 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 226⋅118 2^{26}\cdot 11^{8}
Twist minimal: no (minimal twist has level 88)
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,…,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+(β1+1)q3+(−β5+5)q5+(β6−β2+2)q7+(β3+2β2+β1+90)q9+(−β9+β6+β5+⋯−52)q13+(−β10−β9−2β6+⋯−47)q15+⋯+(−20β15+3β14+⋯+11731)q97+O(q100) q + (\beta_1 + 1) q^{3} + ( - \beta_{5} + 5) q^{5} + (\beta_{6} - \beta_{2} + 2) q^{7} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 90) q^{9} + ( - \beta_{9} + \beta_{6} + \beta_{5} + \cdots - 52) q^{13} + ( - \beta_{10} - \beta_{9} - 2 \beta_{6} + \cdots - 47) q^{15}+ \cdots + ( - 20 \beta_{15} + 3 \beta_{14} + \cdots + 11731) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q+12q3+81q5+47q7+1416q9−859q13−738q15−1226q17−616q19−1141q21+2258q23+10307q25+564q27−1613q29+18511q31+23544q35+⋯+171314q97+O(q100) 16 q + 12 q^{3} + 81 q^{5} + 47 q^{7} + 1416 q^{9} - 859 q^{13} - 738 q^{15} - 1226 q^{17} - 616 q^{19} - 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} - 1613 q^{29} + 18511 q^{31} + 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16−4x15−2635x14+10644x13+2721739x12−11107836x11+⋯+53 ⁣⋯ ⁣20 x^{16} - 4 x^{15} - 2635 x^{14} + 10644 x^{13} + 2721739 x^{12} - 11107836 x^{11} + \cdots + 53\!\cdots\!20 : Copy content Toggle raw display

β1\beta_{1}== (−20 ⁣⋯ ⁣29ν15+⋯−15 ⁣⋯ ⁣60)/30 ⁣⋯ ⁣00 ( - 20\!\cdots\!29 \nu^{15} + \cdots - 15\!\cdots\!60 ) / 30\!\cdots\!00 Copy content Toggle raw display
β2\beta_{2}== (20 ⁣⋯ ⁣29ν15+⋯+14 ⁣⋯ ⁣60)/28 ⁣⋯ ⁣00 ( 20\!\cdots\!29 \nu^{15} + \cdots + 14\!\cdots\!60 ) / 28\!\cdots\!00 Copy content Toggle raw display
β3\beta_{3}== (−97 ⁣⋯ ⁣01ν15+⋯−27 ⁣⋯ ⁣40)/77 ⁣⋯ ⁣00 ( - 97\!\cdots\!01 \nu^{15} + \cdots - 27\!\cdots\!40 ) / 77\!\cdots\!00 Copy content Toggle raw display
β4\beta_{4}== (20 ⁣⋯ ⁣16ν15+⋯+63 ⁣⋯ ⁣40)/15 ⁣⋯ ⁣00 ( 20\!\cdots\!16 \nu^{15} + \cdots + 63\!\cdots\!40 ) / 15\!\cdots\!00 Copy content Toggle raw display
β5\beta_{5}== (24 ⁣⋯ ⁣32ν15+⋯+42 ⁣⋯ ⁣00)/10 ⁣⋯ ⁣00 ( 24\!\cdots\!32 \nu^{15} + \cdots + 42\!\cdots\!00 ) / 10\!\cdots\!00 Copy content Toggle raw display
β6\beta_{6}== (−30 ⁣⋯ ⁣49ν15+⋯−30 ⁣⋯ ⁣20)/27 ⁣⋯ ⁣00 ( - 30\!\cdots\!49 \nu^{15} + \cdots - 30\!\cdots\!20 ) / 27\!\cdots\!00 Copy content Toggle raw display
β7\beta_{7}== (−33 ⁣⋯ ⁣45ν15+⋯−14 ⁣⋯ ⁣20)/27 ⁣⋯ ⁣00 ( - 33\!\cdots\!45 \nu^{15} + \cdots - 14\!\cdots\!20 ) / 27\!\cdots\!00 Copy content Toggle raw display
β8\beta_{8}== (36 ⁣⋯ ⁣57ν15+⋯+15 ⁣⋯ ⁣00)/13 ⁣⋯ ⁣00 ( 36\!\cdots\!57 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 13\!\cdots\!00 Copy content Toggle raw display
β9\beta_{9}== (−42 ⁣⋯ ⁣97ν15+⋯−38 ⁣⋯ ⁣20)/13 ⁣⋯ ⁣00 ( - 42\!\cdots\!97 \nu^{15} + \cdots - 38\!\cdots\!20 ) / 13\!\cdots\!00 Copy content Toggle raw display
β10\beta_{10}== (43 ⁣⋯ ⁣39ν15+⋯+49 ⁣⋯ ⁣00)/13 ⁣⋯ ⁣00 ( 43\!\cdots\!39 \nu^{15} + \cdots + 49\!\cdots\!00 ) / 13\!\cdots\!00 Copy content Toggle raw display
β11\beta_{11}== (87 ⁣⋯ ⁣13ν15+⋯+10 ⁣⋯ ⁣00)/16 ⁣⋯ ⁣00 ( 87\!\cdots\!13 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 16\!\cdots\!00 Copy content Toggle raw display
β12\beta_{12}== (10 ⁣⋯ ⁣71ν15+⋯+18 ⁣⋯ ⁣60)/13 ⁣⋯ ⁣00 ( 10\!\cdots\!71 \nu^{15} + \cdots + 18\!\cdots\!60 ) / 13\!\cdots\!00 Copy content Toggle raw display
β13\beta_{13}== (−13 ⁣⋯ ⁣74ν15+⋯−65 ⁣⋯ ⁣80)/13 ⁣⋯ ⁣00 ( - 13\!\cdots\!74 \nu^{15} + \cdots - 65\!\cdots\!80 ) / 13\!\cdots\!00 Copy content Toggle raw display
β14\beta_{14}== (−16 ⁣⋯ ⁣91ν15+⋯−22 ⁣⋯ ⁣20)/16 ⁣⋯ ⁣00 ( - 16\!\cdots\!91 \nu^{15} + \cdots - 22\!\cdots\!20 ) / 16\!\cdots\!00 Copy content Toggle raw display
β15\beta_{15}== (−37 ⁣⋯ ⁣59ν15+⋯−35 ⁣⋯ ⁣20)/27 ⁣⋯ ⁣00 ( - 37\!\cdots\!59 \nu^{15} + \cdots - 35\!\cdots\!20 ) / 27\!\cdots\!00 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β2+11β1+6)/11 ( \beta_{2} + 11\beta _1 + 6 ) / 11 Copy content Toggle raw display
ν2\nu^{2}== (−2β4+11β3+23β2+β1+3649)/11 ( -2\beta_{4} + 11\beta_{3} + 23\beta_{2} + \beta _1 + 3649 ) / 11 Copy content Toggle raw display
ν3\nu^{3}== (14β15−14β14−16β13+3β12−11β10−11β9+⋯+1613)/11 ( 14 \beta_{15} - 14 \beta_{14} - 16 \beta_{13} + 3 \beta_{12} - 11 \beta_{10} - 11 \beta_{9} + \cdots + 1613 ) / 11 Copy content Toggle raw display
ν4\nu^{4}== (86β15−50β14−112β13−118β12+481β11−22β10+⋯+2070581)/11 ( 86 \beta_{15} - 50 \beta_{14} - 112 \beta_{13} - 118 \beta_{12} + 481 \beta_{11} - 22 \beta_{10} + \cdots + 2070581 ) / 11 Copy content Toggle raw display
ν5\nu^{5}== (16360β15−14289β14−14919β13+3286β12+3541β11+⋯+896230)/11 ( 16360 \beta_{15} - 14289 \beta_{14} - 14919 \beta_{13} + 3286 \beta_{12} + 3541 \beta_{11} + \cdots + 896230 ) / 11 Copy content Toggle raw display
ν6\nu^{6}== (80100β15−35489β14−109901β13−77894β12+540559β11+⋯+1305606147)/11 ( 80100 \beta_{15} - 35489 \beta_{14} - 109901 \beta_{13} - 77894 \beta_{12} + 540559 \beta_{11} + \cdots + 1305606147 ) / 11 Copy content Toggle raw display
ν7\nu^{7}== (14233192β15−11923064β14−11762716β13+2752214β12+⋯+1173802456)/11 ( 14233192 \beta_{15} - 11923064 \beta_{14} - 11762716 \beta_{13} + 2752214 \beta_{12} + \cdots + 1173802456 ) / 11 Copy content Toggle raw display
ν8\nu^{8}== (57895680β15−23701634β14−87244208β13−42550746β12+⋯+855837391975)/11 ( 57895680 \beta_{15} - 23701634 \beta_{14} - 87244208 \beta_{13} - 42550746 \beta_{12} + \cdots + 855837391975 ) / 11 Copy content Toggle raw display
ν9\nu^{9}== (11233153028β15−9268631135β14−8820173389β13+2102923450β12+⋯+1389135395578)/11 ( 11233153028 \beta_{15} - 9268631135 \beta_{14} - 8820173389 \beta_{13} + 2102923450 \beta_{12} + \cdots + 1389135395578 ) / 11 Copy content Toggle raw display
ν10\nu^{10}== (39771429124β15−17454103611β14−65800025875β13−21605764158β12+⋯+573922922803927)/11 ( 39771429124 \beta_{15} - 17454103611 \beta_{14} - 65800025875 \beta_{13} - 21605764158 \beta_{12} + \cdots + 573922922803927 ) / 11 Copy content Toggle raw display
ν11\nu^{11}== (8486951484528β15−6950351199814β14−6480825394054β13+1531120812430β12+⋯+14 ⁣⋯ ⁣64)/11 ( 8486951484528 \beta_{15} - 6950351199814 \beta_{14} - 6480825394054 \beta_{13} + 1531120812430 \beta_{12} + \cdots + 14\!\cdots\!64 ) / 11 Copy content Toggle raw display
ν12\nu^{12}== (27666473523240β15−14098244508668β14−48969656825462β13+⋯+39 ⁣⋯ ⁣07)/11 ( 27666473523240 \beta_{15} - 14098244508668 \beta_{14} - 48969656825462 \beta_{13} + \cdots + 39\!\cdots\!07 ) / 11 Copy content Toggle raw display
ν13\nu^{13}== (62 ⁣⋯ ⁣24β15+⋯+13 ⁣⋯ ⁣22)/11 ( 62\!\cdots\!24 \beta_{15} + \cdots + 13\!\cdots\!22 ) / 11 Copy content Toggle raw display
ν14\nu^{14}== (19 ⁣⋯ ⁣56β15+⋯+26 ⁣⋯ ⁣27)/11 ( 19\!\cdots\!56 \beta_{15} + \cdots + 26\!\cdots\!27 ) / 11 Copy content Toggle raw display
ν15\nu^{15}== (45 ⁣⋯ ⁣40β15+⋯+11 ⁣⋯ ⁣92)/11 ( 45\!\cdots\!40 \beta_{15} + \cdots + 11\!\cdots\!92 ) / 11 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
−26.5066
−23.8164
−25.8955
−21.8511
−10.4080
−6.91330
−4.22794
−5.39811
3.21970
6.63850
10.0842
13.3325
19.5696
22.3288
26.8952
26.9484
0 −27.1247 0 88.7213 0 184.080 0 492.748 0
1.2 0 −24.4344 0 −57.4032 0 −156.812 0 354.040 0
1.3 0 −24.2774 0 −51.2310 0 25.3165 0 346.393 0
1.4 0 −20.2331 0 59.3479 0 −110.064 0 166.379 0
1.5 0 −11.0260 0 −30.6908 0 62.4872 0 −121.427 0
1.6 0 −5.29527 0 6.33197 0 −24.6791 0 −214.960 0
1.7 0 −4.84597 0 11.7645 0 −55.0471 0 −219.517 0
1.8 0 −3.78008 0 89.9154 0 257.151 0 −228.711 0
1.9 0 4.83773 0 −82.0306 0 11.3366 0 −219.596 0
1.10 0 6.02047 0 76.6139 0 −112.762 0 −206.754 0
1.11 0 11.7022 0 −24.8620 0 −76.4943 0 −106.058 0
1.12 0 12.7145 0 −55.8973 0 194.771 0 −81.3426 0
1.13 0 21.1877 0 85.4976 0 −215.502 0 205.918 0
1.14 0 21.7107 0 −83.2917 0 −240.887 0 228.355 0
1.15 0 26.2772 0 56.0244 0 77.2326 0 447.491 0
1.16 0 28.5665 0 −7.81027 0 226.871 0 573.042 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 −1 -1
1111 +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 968.6.a.q 16
11.b odd 2 1 968.6.a.p 16
11.c even 5 2 88.6.i.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.6.i.b ✓ 32 11.c even 5 2
968.6.a.p 16 11.b odd 2 1
968.6.a.q 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S6new(Γ0(968))S_{6}^{\mathrm{new}}(\Gamma_0(968)):

T316−12T315−2580T314+29404T313+2591498T312+⋯+52 ⁣⋯ ⁣25 T_{3}^{16} - 12 T_{3}^{15} - 2580 T_{3}^{14} + 29404 T_{3}^{13} + 2591498 T_{3}^{12} + \cdots + 52\!\cdots\!25 Copy content Toggle raw display
T716−47T715−180613T714+4592698T713+12459543981T712+⋯+30 ⁣⋯ ⁣84 T_{7}^{16} - 47 T_{7}^{15} - 180613 T_{7}^{14} + 4592698 T_{7}^{13} + 12459543981 T_{7}^{12} + \cdots + 30\!\cdots\!84 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+⋯+52 ⁣⋯ ⁣25 T^{16} + \cdots + 52\!\cdots\!25 Copy content Toggle raw display
55 T16+⋯+86 ⁣⋯ ⁣24 T^{16} + \cdots + 86\!\cdots\!24 Copy content Toggle raw display
77 T16+⋯+30 ⁣⋯ ⁣84 T^{16} + \cdots + 30\!\cdots\!84 Copy content Toggle raw display
1111 T16 T^{16} Copy content Toggle raw display
1313 T16+⋯+36 ⁣⋯ ⁣00 T^{16} + \cdots + 36\!\cdots\!00 Copy content Toggle raw display
1717 T16+⋯+30 ⁣⋯ ⁣05 T^{16} + \cdots + 30\!\cdots\!05 Copy content Toggle raw display
1919 T16+⋯+19 ⁣⋯ ⁣95 T^{16} + \cdots + 19\!\cdots\!95 Copy content Toggle raw display
2323 T16+⋯−34 ⁣⋯ ⁣96 T^{16} + \cdots - 34\!\cdots\!96 Copy content Toggle raw display
2929 T16+⋯+46 ⁣⋯ ⁣00 T^{16} + \cdots + 46\!\cdots\!00 Copy content Toggle raw display
3131 T16+⋯+37 ⁣⋯ ⁣80 T^{16} + \cdots + 37\!\cdots\!80 Copy content Toggle raw display
3737 T16+⋯−16 ⁣⋯ ⁣00 T^{16} + \cdots - 16\!\cdots\!00 Copy content Toggle raw display
4141 T16+⋯+23 ⁣⋯ ⁣25 T^{16} + \cdots + 23\!\cdots\!25 Copy content Toggle raw display
4343 T16+⋯−75 ⁣⋯ ⁣80 T^{16} + \cdots - 75\!\cdots\!80 Copy content Toggle raw display
4747 T16+⋯+55 ⁣⋯ ⁣96 T^{16} + \cdots + 55\!\cdots\!96 Copy content Toggle raw display
5353 T16+⋯−91 ⁣⋯ ⁣80 T^{16} + \cdots - 91\!\cdots\!80 Copy content Toggle raw display
5959 T16+⋯+32 ⁣⋯ ⁣31 T^{16} + \cdots + 32\!\cdots\!31 Copy content Toggle raw display
6161 T16+⋯+99 ⁣⋯ ⁣80 T^{16} + \cdots + 99\!\cdots\!80 Copy content Toggle raw display
6767 T16+⋯+15 ⁣⋯ ⁣00 T^{16} + \cdots + 15\!\cdots\!00 Copy content Toggle raw display
7171 T16+⋯−24 ⁣⋯ ⁣00 T^{16} + \cdots - 24\!\cdots\!00 Copy content Toggle raw display
7373 T16+⋯+24 ⁣⋯ ⁣45 T^{16} + \cdots + 24\!\cdots\!45 Copy content Toggle raw display
7979 T16+⋯−30 ⁣⋯ ⁣20 T^{16} + \cdots - 30\!\cdots\!20 Copy content Toggle raw display
8383 T16+⋯−62 ⁣⋯ ⁣25 T^{16} + \cdots - 62\!\cdots\!25 Copy content Toggle raw display
8989 T16+⋯−34 ⁣⋯ ⁣20 T^{16} + \cdots - 34\!\cdots\!20 Copy content Toggle raw display
9797 T16+⋯+53 ⁣⋯ ⁣05 T^{16} + \cdots + 53\!\cdots\!05 Copy content Toggle raw display
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