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Properties

Label 2816.2.e.o
Level 28162816
Weight 22
Character orbit 2816.e
Analytic conductor 22.48622.486
Analytic rank 00
Dimension 1616
Inner twists 88

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2816,2,Mod(2815,2816)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2816.2815"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 2816=28⋅11 2816 = 2^{8} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2816.e (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0, 0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(89)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 22.485873209222.4858732092
Analytic rank: 00
Dimension: 1616
Coefficient field: 16.0.1622647227216566419456.8
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x16−4x14+8x12−44x10+161x8−88x6+32x4−32x2+16 x^{16} - 4x^{14} + 8x^{12} - 44x^{10} + 161x^{8} - 88x^{6} + 32x^{4} - 32x^{2} + 16 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 226 2^{26}
Twist minimal: no (minimal twist has level 88)
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) == q+β2q3+β8q5+β6q7+β1q9+(−β9−β7)q11+β3q13+β4q15+β15q17−β7q19+β3q21+⋯+(β13−3β9+⋯+β2)q99+O(q100) q + \beta_{2} q^{3} + \beta_{8} q^{5} + \beta_{6} q^{7} + \beta_1 q^{9} + ( - \beta_{9} - \beta_{7}) q^{11} + \beta_{3} q^{13} + \beta_{4} q^{15} + \beta_{15} q^{17} - \beta_{7} q^{19} + \beta_{3} q^{21}+ \cdots + (\beta_{13} - 3 \beta_{9} + \cdots + \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q+32q25+16q33+16q49−16q81+80q89−16q97+O(q100) 16 q + 32 q^{25} + 16 q^{33} + 16 q^{49} - 16 q^{81} + 80 q^{89} - 16 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16−4x14+8x12−44x10+161x8−88x6+32x4−32x2+16 x^{16} - 4x^{14} + 8x^{12} - 44x^{10} + 161x^{8} - 88x^{6} + 32x^{4} - 32x^{2} + 16 : Copy content Toggle raw display

β1\beta_{1}== (29ν14−56ν12+324ν10−908ν8+485ν6−188ν4+180ν2+69664)/24660 ( 29\nu^{14} - 56\nu^{12} + 324\nu^{10} - 908\nu^{8} + 485\nu^{6} - 188\nu^{4} + 180\nu^{2} + 69664 ) / 24660 Copy content Toggle raw display
β2\beta_{2}== (699ν14−2134ν12+2972ν10−25892ν8+82779ν6+34934ν4−6508ν2−9680)/49320 ( 699\nu^{14} - 2134\nu^{12} + 2972\nu^{10} - 25892\nu^{8} + 82779\nu^{6} + 34934\nu^{4} - 6508\nu^{2} - 9680 ) / 49320 Copy content Toggle raw display
β3\beta_{3}== (857ν15−2184ν13+2498ν11−29932ν9+89333ν7+96240ν5+⋯−28036ν)/24660 ( 857 \nu^{15} - 2184 \nu^{13} + 2498 \nu^{11} - 29932 \nu^{9} + 89333 \nu^{7} + 96240 \nu^{5} + \cdots - 28036 \nu ) / 24660 Copy content Toggle raw display
β4\beta_{4}== (2223ν14−8072ν12+15388ν10−96436ν8+326295ν6−97556ν4+⋯−58072)/49320 ( 2223 \nu^{14} - 8072 \nu^{12} + 15388 \nu^{10} - 96436 \nu^{8} + 326295 \nu^{6} - 97556 \nu^{4} + \cdots - 58072 ) / 49320 Copy content Toggle raw display
β5\beta_{5}== (2875ν15−10342ν13+18912ν11−119836ν9+415771ν7−106762ν5+⋯−33376ν)/49320 ( 2875 \nu^{15} - 10342 \nu^{13} + 18912 \nu^{11} - 119836 \nu^{9} + 415771 \nu^{7} - 106762 \nu^{5} + \cdots - 33376 \nu ) / 49320 Copy content Toggle raw display
β6\beta_{6}== (−885ν15+3712ν13−7757ν11+40616ν9−150771ν7+106522ν5+⋯+67946ν)/12330 ( - 885 \nu^{15} + 3712 \nu^{13} - 7757 \nu^{11} + 40616 \nu^{9} - 150771 \nu^{7} + 106522 \nu^{5} + \cdots + 67946 \nu ) / 12330 Copy content Toggle raw display
β7\beta_{7}== (−3715ν15+13438ν13−22968ν11+147724ν9−527779ν7+⋯+74944ν)/49320 ( - 3715 \nu^{15} + 13438 \nu^{13} - 22968 \nu^{11} + 147724 \nu^{9} - 527779 \nu^{7} + \cdots + 74944 \nu ) / 49320 Copy content Toggle raw display
β8\beta_{8}== (−13ν14+53ν12−104ν10+564ν8−2113ν6+1157ν4+196ν2+240)/180 ( -13\nu^{14} + 53\nu^{12} - 104\nu^{10} + 564\nu^{8} - 2113\nu^{6} + 1157\nu^{4} + 196\nu^{2} + 240 ) / 180 Copy content Toggle raw display
β9\beta_{9}== (3959ν14−12114ν12+17852ν10−148372ν8+480839ν6+203994ν4+⋯−56080)/49320 ( 3959 \nu^{14} - 12114 \nu^{12} + 17852 \nu^{10} - 148372 \nu^{8} + 480839 \nu^{6} + 203994 \nu^{4} + \cdots - 56080 ) / 49320 Copy content Toggle raw display
β10\beta_{10}== (3247ν15−13498ν13+28482ν11−149284ν9+551755ν7+⋯−248548ν)/24660 ( 3247 \nu^{15} - 13498 \nu^{13} + 28482 \nu^{11} - 149284 \nu^{9} + 551755 \nu^{7} + \cdots - 248548 \nu ) / 24660 Copy content Toggle raw display
β11\beta_{11}== (4203ν15−10790ν13+12736ν11−149020ν9+433899ν7+⋯−135856ν)/24660 ( 4203 \nu^{15} - 10790 \nu^{13} + 12736 \nu^{11} - 149020 \nu^{9} + 433899 \nu^{7} + \cdots - 135856 \nu ) / 24660 Copy content Toggle raw display
β12\beta_{12}== (111ν14−460ν12+900ν10−4836ν8+18183ν6−9960ν4−1692ν2−2064)/548 ( 111\nu^{14} - 460\nu^{12} + 900\nu^{10} - 4836\nu^{8} + 18183\nu^{6} - 9960\nu^{4} - 1692\nu^{2} - 2064 ) / 548 Copy content Toggle raw display
β13\beta_{13}== (14121ν15−51314ν13+87376ν11−564772ν9+2020185ν7+⋯−286864ν)/49320 ( 14121 \nu^{15} - 51314 \nu^{13} + 87376 \nu^{11} - 564772 \nu^{9} + 2020185 \nu^{7} + \cdots - 286864 \nu ) / 49320 Copy content Toggle raw display
β14\beta_{14}== (12847ν14−46728ν12+88732ν10−544724ν8+1891735ν6−565524ν4+⋯−336488)/49320 ( 12847 \nu^{14} - 46728 \nu^{12} + 88732 \nu^{10} - 544724 \nu^{8} + 1891735 \nu^{6} - 565524 \nu^{4} + \cdots - 336488 ) / 49320 Copy content Toggle raw display
β15\beta_{15}== (16591ν15−60222ν13+112024ν11−693356ν9+2419999ν7+⋯−193808ν)/49320 ( 16591 \nu^{15} - 60222 \nu^{13} + 112024 \nu^{11} - 693356 \nu^{9} + 2419999 \nu^{7} + \cdots - 193808 \nu ) / 49320 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β15+β13−β11−β10+5β7+β6+β5+2β3)/16 ( \beta_{15} + \beta_{13} - \beta_{11} - \beta_{10} + 5\beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{3} ) / 16 Copy content Toggle raw display
ν2\nu^{2}== (β14+β12−β9+4β8−3β4+3β2−β1+4)/8 ( \beta_{14} + \beta_{12} - \beta_{9} + 4\beta_{8} - 3\beta_{4} + 3\beta_{2} - \beta _1 + 4 ) / 8 Copy content Toggle raw display
ν3\nu^{3}== (β15−β13−β11−7β10−β7−13β6−3β5+6β3)/16 ( \beta_{15} - \beta_{13} - \beta_{11} - 7\beta_{10} - \beta_{7} - 13\beta_{6} - 3\beta_{5} + 6\beta_{3} ) / 16 Copy content Toggle raw display
ν4\nu^{4}== (2β12+3β9+6β8−15β2)/4 ( 2\beta_{12} + 3\beta_{9} + 6\beta_{8} - 15\beta_{2} ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (7β15+7β13+β11−3β10+27β7−5β6−41β5−2β3)/16 ( 7\beta_{15} + 7\beta_{13} + \beta_{11} - 3\beta_{10} + 27\beta_{7} - 5\beta_{6} - 41\beta_{5} - 2\beta_{3} ) / 16 Copy content Toggle raw display
ν6\nu^{6}== (3β14−7β12+15β9−20β8−17β4−85β2−35β1+100)/8 ( 3\beta_{14} - 7\beta_{12} + 15\beta_{9} - 20\beta_{8} - 17\beta_{4} - 85\beta_{2} - 35\beta _1 + 100 ) / 8 Copy content Toggle raw display
ν7\nu^{7}== (5β15+27β13−31β11+31β10+103β7+57β6+⋯+150β3)/16 ( 5 \beta_{15} + 27 \beta_{13} - 31 \beta_{11} + 31 \beta_{10} + 103 \beta_{7} + 57 \beta_{6} + \cdots + 150 \beta_{3} ) / 16 Copy content Toggle raw display
ν8\nu^{8}== (15β14−87β4+12β1−34)/4 ( 15\beta_{14} - 87\beta_{4} + 12\beta _1 - 34 ) / 4 Copy content Toggle raw display
ν9\nu^{9}== (23β15−133β13−89β11−89β10−509β7−163β6+⋯+430β3)/16 ( 23 \beta_{15} - 133 \beta_{13} - 89 \beta_{11} - 89 \beta_{10} - 509 \beta_{7} - 163 \beta_{6} + \cdots + 430 \beta_{3} ) / 16 Copy content Toggle raw display
ν10\nu^{10}== (17β14+41β12+187β9+116β8−99β4−1089β2+451β1−1276)/8 ( 17\beta_{14} + 41\beta_{12} + 187\beta_{9} + 116\beta_{8} - 99\beta_{4} - 1089\beta_{2} + 451\beta _1 - 1276 ) / 8 Copy content Toggle raw display
ν11\nu^{11}== (263β15−263β13+61β11+355β10−1007β7+⋯−294β3)/16 ( 263 \beta_{15} - 263 \beta_{13} + 61 \beta_{11} + 355 \beta_{10} - 1007 \beta_{7} + \cdots - 294 \beta_{3} ) / 16 Copy content Toggle raw display
ν12\nu^{12}== (−350β12+261β9−990β8−1521β2)/4 ( -350\beta_{12} + 261\beta_{9} - 990\beta_{8} - 1521\beta_{2} ) / 4 Copy content Toggle raw display
ν13\nu^{13}== (−7β15−7β13−577β11+3363β10−27β7+⋯+2786β3)/16 ( - 7 \beta_{15} - 7 \beta_{13} - 577 \beta_{11} + 3363 \beta_{10} - 27 \beta_{7} + \cdots + 2786 \beta_{3} ) / 16 Copy content Toggle raw display
ν14\nu^{14}== (693β14−1673β12−1287β9−4732β8−4039β4+7501β2+⋯−8788)/8 ( 693 \beta_{14} - 1673 \beta_{12} - 1287 \beta_{9} - 4732 \beta_{8} - 4039 \beta_{4} + 7501 \beta_{2} + \cdots - 8788 ) / 8 Copy content Toggle raw display
ν15\nu^{15}== (−1277β15−7443β13−1649β11+1649β10−28495β7+⋯+7962β3)/16 ( - 1277 \beta_{15} - 7443 \beta_{13} - 1649 \beta_{11} + 1649 \beta_{10} - 28495 \beta_{7} + \cdots + 7962 \beta_{3} ) / 16 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/2816Z)×\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times.

nn 10251025 15411541 20472047
χ(n)\chi(n) −1-1 11 −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}
2815.1
−1.00237 + 1.55225i
1.00237 − 1.55225i
1.80639 + 0.388825i
−1.80639 − 0.388825i
−0.415194 − 0.642962i
0.415194 + 0.642962i
−0.748229 + 0.161057i
0.748229 − 0.161057i
−0.415194 + 0.642962i
0.415194 − 0.642962i
−0.748229 − 0.161057i
0.748229 + 0.161057i
−1.00237 − 1.55225i
1.00237 + 1.55225i
1.80639 − 0.388825i
−1.80639 + 0.388825i
0 − 2.41421i 0 −2.64575 0 −2.27411 0 −2.82843 0
2815.2 0 − 2.41421i 0 −2.64575 0 2.27411 0 −2.82843 0
2815.3 0 − 2.41421i 0 2.64575 0 −2.27411 0 −2.82843 0
2815.4 0 − 2.41421i 0 2.64575 0 2.27411 0 −2.82843 0
2815.5 0 − 0.414214i 0 −2.64575 0 −3.29066 0 2.82843 0
2815.6 0 − 0.414214i 0 −2.64575 0 3.29066 0 2.82843 0
2815.7 0 − 0.414214i 0 2.64575 0 −3.29066 0 2.82843 0
2815.8 0 − 0.414214i 0 2.64575 0 3.29066 0 2.82843 0
2815.9 0 0.414214i 0 −2.64575 0 −3.29066 0 2.82843 0
2815.10 0 0.414214i 0 −2.64575 0 3.29066 0 2.82843 0
2815.11 0 0.414214i 0 2.64575 0 −3.29066 0 2.82843 0
2815.12 0 0.414214i 0 2.64575 0 3.29066 0 2.82843 0
2815.13 0 2.41421i 0 −2.64575 0 −2.27411 0 −2.82843 0
2815.14 0 2.41421i 0 −2.64575 0 2.27411 0 −2.82843 0
2815.15 0 2.41421i 0 2.64575 0 −2.27411 0 −2.82843 0
2815.16 0 2.41421i 0 2.64575 0 2.27411 0 −2.82843 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2815.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner
88.b odd 2 1 inner
88.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2816.2.e.o 16
4.b odd 2 1 inner 2816.2.e.o 16
8.b even 2 1 inner 2816.2.e.o 16
8.d odd 2 1 inner 2816.2.e.o 16
11.b odd 2 1 inner 2816.2.e.o 16
16.e even 4 1 88.2.g.b ✓ 8
16.e even 4 1 352.2.g.b 8
16.f odd 4 1 88.2.g.b ✓ 8
16.f odd 4 1 352.2.g.b 8
44.c even 2 1 inner 2816.2.e.o 16
48.i odd 4 1 792.2.h.g 8
48.i odd 4 1 3168.2.h.g 8
48.k even 4 1 792.2.h.g 8
48.k even 4 1 3168.2.h.g 8
88.b odd 2 1 inner 2816.2.e.o 16
88.g even 2 1 inner 2816.2.e.o 16
176.i even 4 1 88.2.g.b ✓ 8
176.i even 4 1 352.2.g.b 8
176.l odd 4 1 88.2.g.b ✓ 8
176.l odd 4 1 352.2.g.b 8
176.u odd 20 4 968.2.k.g 32
176.v odd 20 4 968.2.k.g 32
176.w even 20 4 968.2.k.g 32
176.x even 20 4 968.2.k.g 32
528.s odd 4 1 792.2.h.g 8
528.s odd 4 1 3168.2.h.g 8
528.x even 4 1 792.2.h.g 8
528.x even 4 1 3168.2.h.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.g.b ✓ 8 16.e even 4 1
88.2.g.b ✓ 8 16.f odd 4 1
88.2.g.b ✓ 8 176.i even 4 1
88.2.g.b ✓ 8 176.l odd 4 1
352.2.g.b 8 16.e even 4 1
352.2.g.b 8 16.f odd 4 1
352.2.g.b 8 176.i even 4 1
352.2.g.b 8 176.l odd 4 1
792.2.h.g 8 48.i odd 4 1
792.2.h.g 8 48.k even 4 1
792.2.h.g 8 528.s odd 4 1
792.2.h.g 8 528.x even 4 1
968.2.k.g 32 176.u odd 20 4
968.2.k.g 32 176.v odd 20 4
968.2.k.g 32 176.w even 20 4
968.2.k.g 32 176.x even 20 4
2816.2.e.o 16 1.a even 1 1 trivial
2816.2.e.o 16 4.b odd 2 1 inner
2816.2.e.o 16 8.b even 2 1 inner
2816.2.e.o 16 8.d odd 2 1 inner
2816.2.e.o 16 11.b odd 2 1 inner
2816.2.e.o 16 44.c even 2 1 inner
2816.2.e.o 16 88.b odd 2 1 inner
2816.2.e.o 16 88.g even 2 1 inner
3168.2.h.g 8 48.i odd 4 1
3168.2.h.g 8 48.k even 4 1
3168.2.h.g 8 528.s odd 4 1
3168.2.h.g 8 528.x even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(2816,[χ])S_{2}^{\mathrm{new}}(2816, [\chi]):

T34+6T32+1 T_{3}^{4} + 6T_{3}^{2} + 1 Copy content Toggle raw display
T52−7 T_{5}^{2} - 7 Copy content Toggle raw display
T74−16T72+56 T_{7}^{4} - 16T_{7}^{2} + 56 Copy content Toggle raw display
T134+32T132+56 T_{13}^{4} + 32T_{13}^{2} + 56 Copy content Toggle raw display
T194−16T192+56 T_{19}^{4} - 16T_{19}^{2} + 56 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16 T^{16} Copy content Toggle raw display
33 (T4+6T2+1)4 (T^{4} + 6 T^{2} + 1)^{4} Copy content Toggle raw display
55 (T2−7)8 (T^{2} - 7)^{8} Copy content Toggle raw display
77 (T4−16T2+56)4 (T^{4} - 16 T^{2} + 56)^{4} Copy content Toggle raw display
1111 (T8−20T6+⋯+14641)2 (T^{8} - 20 T^{6} + \cdots + 14641)^{2} Copy content Toggle raw display
1313 (T4+32T2+56)4 (T^{4} + 32 T^{2} + 56)^{4} Copy content Toggle raw display
1717 (T4+32T2+56)4 (T^{4} + 32 T^{2} + 56)^{4} Copy content Toggle raw display
1919 (T4−16T2+56)4 (T^{4} - 16 T^{2} + 56)^{4} Copy content Toggle raw display
2323 (T4+42T2+49)4 (T^{4} + 42 T^{2} + 49)^{4} Copy content Toggle raw display
2929 (T4+64T2+896)4 (T^{4} + 64 T^{2} + 896)^{4} Copy content Toggle raw display
3131 (T4+42T2+49)4 (T^{4} + 42 T^{2} + 49)^{4} Copy content Toggle raw display
3737 (T4−126T2+2401)4 (T^{4} - 126 T^{2} + 2401)^{4} Copy content Toggle raw display
4141 (T4+128T2+2744)4 (T^{4} + 128 T^{2} + 2744)^{4} Copy content Toggle raw display
4343 (T4−128T2+3584)4 (T^{4} - 128 T^{2} + 3584)^{4} Copy content Toggle raw display
4747 T16 T^{16} Copy content Toggle raw display
5353 (T2−56)8 (T^{2} - 56)^{8} Copy content Toggle raw display
5959 (T4+38T2+289)4 (T^{4} + 38 T^{2} + 289)^{4} Copy content Toggle raw display
6161 (T4+64T2+896)4 (T^{4} + 64 T^{2} + 896)^{4} Copy content Toggle raw display
6767 (T4+102T2+2401)4 (T^{4} + 102 T^{2} + 2401)^{4} Copy content Toggle raw display
7171 (T4+42T2+49)4 (T^{4} + 42 T^{2} + 49)^{4} Copy content Toggle raw display
7373 (T4+64T2+56)4 (T^{4} + 64 T^{2} + 56)^{4} Copy content Toggle raw display
7979 (T4−128T2+3584)4 (T^{4} - 128 T^{2} + 3584)^{4} Copy content Toggle raw display
8383 (T4−64T2+896)4 (T^{4} - 64 T^{2} + 896)^{4} Copy content Toggle raw display
8989 (T2−10T−7)8 (T^{2} - 10 T - 7)^{8} Copy content Toggle raw display
9797 (T2+2T−71)8 (T^{2} + 2 T - 71)^{8} Copy content Toggle raw display
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