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Properties

Label 980.2.x.m
Level 980980
Weight 22
Character orbit 980.x
Analytic conductor 7.8257.825
Analytic rank 00
Dimension 7272
Inner twists 88

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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] = [980,2,Mod(67,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 8])) 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("980.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 980=22⋅5⋅72 980 = 2^{2} \cdot 5 \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 980.x (of order 1212, degree 44, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,8,16,0,-4,0,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 7.825339398097.82533939809
Analytic rank: 00
Dimension: 7272
Relative dimension: 1818 over Q(ζ12)\Q(\zeta_{12})
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: SU(2)[C12]\mathrm{SU}(2)[C_{12}]

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr⁡(f)(q)=\operatorname{Tr}(f)(q) = 72q+2q2+8q5+16q6−4q8−2q10−10q12−28q16−4q17−20q18+56q20−16q22−16q25+4q26−32q30−38q32+64q33+16q36−4q37+⋯+24q97+O(q100) 72 q + 2 q^{2} + 8 q^{5} + 16 q^{6} - 4 q^{8} - 2 q^{10} - 10 q^{12} - 28 q^{16} - 4 q^{17} - 20 q^{18} + 56 q^{20} - 16 q^{22} - 16 q^{25} + 4 q^{26} - 32 q^{30} - 38 q^{32} + 64 q^{33} + 16 q^{36} - 4 q^{37}+ \cdots + 24 q^{97}+O(q^{100}) Copy content Toggle raw display

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   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}
67.1 −1.41262 + 0.0671791i −2.38471 + 0.638980i 1.99097 − 0.189797i 0.525600 − 2.17342i 3.32575 − 1.06284i 0 −2.79973 + 0.401862i 2.68045 − 1.54756i −0.596463 + 3.10552i
67.2 −1.40543 + 0.157385i −1.08292 + 0.290169i 1.95046 − 0.442386i −1.68711 + 1.46754i 1.47630 − 0.578247i 0 −2.67161 + 0.928715i −1.50955 + 0.871538i 2.14014 − 2.32805i
67.3 −1.34754 + 0.429119i 2.71477 − 0.727420i 1.63171 − 1.15651i 2.01971 + 0.959579i −3.34610 + 2.14518i 0 −1.70252 + 2.25864i 4.24276 − 2.44956i −3.13340 − 0.426375i
67.4 −1.24267 − 0.675113i 0.402205 − 0.107770i 1.08845 + 1.67788i 2.22809 − 0.188711i −0.572564 − 0.137611i 0 −0.219816 − 2.81987i −2.44792 + 1.41331i −2.89618 − 1.26971i
67.5 −1.05431 + 0.942570i 1.20413 − 0.322645i 0.223125 − 1.98751i −2.09885 − 0.771256i −0.965404 + 1.47514i 0 1.63813 + 2.30576i −1.25225 + 0.722989i 2.93979 − 1.16517i
67.6 −0.809319 − 1.15974i 0.551941 − 0.147892i −0.690004 + 1.87720i −1.08510 − 1.95513i −0.618214 − 0.520418i 0 2.73551 − 0.719030i −2.31531 + 1.33674i −1.38926 + 2.84077i
67.7 −0.674418 − 1.24304i −2.47915 + 0.664287i −1.09032 + 1.67666i −1.93364 + 1.12296i 2.49773 + 2.63369i 0 2.81950 + 0.224544i 3.10685 − 1.79374i 2.69997 + 1.64626i
67.8 −0.667698 + 1.24667i −2.02821 + 0.543458i −1.10836 − 1.66480i 0.518800 + 2.17505i 0.676724 − 2.89137i 0 2.81549 − 0.270171i 1.22023 − 0.704499i −3.05797 − 0.805507i
67.9 −0.212826 − 1.39811i 0.807254 − 0.216303i −1.90941 + 0.595107i 0.780454 + 2.09545i −0.474220 − 1.08259i 0 1.23840 + 2.54291i −1.99320 + 1.15078i 2.76356 − 1.53712i
67.10 −0.0450897 + 1.41349i 2.02821 − 0.543458i −1.99593 − 0.127468i 0.518800 + 2.17505i 0.676724 + 2.89137i 0 0.270171 − 2.81549i 1.22023 − 0.704499i −3.09782 + 0.635249i
67.11 0.441772 + 1.34344i −1.20413 + 0.322645i −1.60968 + 1.18699i −2.09885 − 0.771256i −0.965404 − 1.47514i 0 −2.30576 − 1.63813i −1.25225 + 0.722989i 0.108927 − 3.16040i
67.12 0.883367 − 1.10438i −0.807254 + 0.216303i −0.439327 − 1.95115i 0.780454 + 2.09545i −0.474220 + 1.08259i 0 −2.54291 − 1.23840i −1.99320 + 1.15078i 3.00360 + 0.989126i
67.13 0.952442 + 1.04540i −2.71477 + 0.727420i −0.185707 + 1.99136i 2.01971 + 0.959579i −3.34610 − 2.14518i 0 −2.25864 + 1.70252i 4.24276 − 2.44956i 0.920513 + 3.02534i
67.14 1.13844 + 0.839014i 1.08292 − 0.290169i 0.592112 + 1.91034i −1.68711 + 1.46754i 1.47630 + 0.578247i 0 −0.928715 + 2.67161i −1.50955 + 0.871538i −3.15196 + 0.255205i
67.15 1.18977 + 0.764487i 2.38471 − 0.638980i 0.831118 + 1.81913i 0.525600 − 2.17342i 3.32575 + 1.06284i 0 −0.401862 + 2.79973i 2.68045 − 1.54756i 2.28689 − 2.18406i
67.16 1.20559 − 0.739299i 2.47915 − 0.664287i 0.906874 − 1.78258i −1.93364 + 1.12296i 2.49773 − 2.63369i 0 −0.224544 − 2.81950i 3.10685 − 1.79374i −1.50097 + 2.78336i
67.17 1.28076 − 0.599707i −0.551941 + 0.147892i 1.28070 − 1.53616i −1.08510 − 1.95513i −0.618214 + 0.520418i 0 0.719030 − 2.73551i −2.31531 + 1.33674i −2.56227 − 1.85332i
67.18 1.41374 + 0.0366689i −0.402205 + 0.107770i 1.99731 + 0.103680i 2.22809 − 0.188711i −0.572564 + 0.137611i 0 2.81987 + 0.219816i −2.44792 + 1.41331i 3.15686 − 0.185086i
263.1 −1.39811 + 0.212826i 0.216303 + 0.807254i 1.90941 − 0.595107i 1.42448 + 1.72362i −0.474220 − 1.08259i 0 −2.54291 + 1.23840i 1.99320 − 1.15078i −2.35841 − 2.10663i
263.2 −1.24304 + 0.674418i −0.664287 − 2.47915i 1.09032 − 1.67666i 1.93933 − 1.11310i 2.49773 + 2.63369i 0 −0.224544 + 2.81950i −3.10685 + 1.79374i −1.65998 + 2.69156i
See all 72 embeddings
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
20.e even 4 1 inner
28.g odd 6 1 inner
35.l odd 12 1 inner
140.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.x.m 72
4.b odd 2 1 inner 980.2.x.m 72
5.c odd 4 1 inner 980.2.x.m 72
7.b odd 2 1 140.2.w.b ✓ 72
7.c even 3 1 980.2.k.j 36
7.c even 3 1 inner 980.2.x.m 72
7.d odd 6 1 140.2.w.b ✓ 72
7.d odd 6 1 980.2.k.k 36
20.e even 4 1 inner 980.2.x.m 72
28.d even 2 1 140.2.w.b ✓ 72
28.f even 6 1 140.2.w.b ✓ 72
28.f even 6 1 980.2.k.k 36
28.g odd 6 1 980.2.k.j 36
28.g odd 6 1 inner 980.2.x.m 72
35.c odd 2 1 700.2.be.e 72
35.f even 4 1 140.2.w.b ✓ 72
35.f even 4 1 700.2.be.e 72
35.i odd 6 1 700.2.be.e 72
35.k even 12 1 140.2.w.b ✓ 72
35.k even 12 1 700.2.be.e 72
35.k even 12 1 980.2.k.k 36
35.l odd 12 1 980.2.k.j 36
35.l odd 12 1 inner 980.2.x.m 72
140.c even 2 1 700.2.be.e 72
140.j odd 4 1 140.2.w.b ✓ 72
140.j odd 4 1 700.2.be.e 72
140.s even 6 1 700.2.be.e 72
140.w even 12 1 980.2.k.j 36
140.w even 12 1 inner 980.2.x.m 72
140.x odd 12 1 140.2.w.b ✓ 72
140.x odd 12 1 700.2.be.e 72
140.x odd 12 1 980.2.k.k 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.w.b ✓ 72 7.b odd 2 1
140.2.w.b ✓ 72 7.d odd 6 1
140.2.w.b ✓ 72 28.d even 2 1
140.2.w.b ✓ 72 28.f even 6 1
140.2.w.b ✓ 72 35.f even 4 1
140.2.w.b ✓ 72 35.k even 12 1
140.2.w.b ✓ 72 140.j odd 4 1
140.2.w.b ✓ 72 140.x odd 12 1
700.2.be.e 72 35.c odd 2 1
700.2.be.e 72 35.f even 4 1
700.2.be.e 72 35.i odd 6 1
700.2.be.e 72 35.k even 12 1
700.2.be.e 72 140.c even 2 1
700.2.be.e 72 140.j odd 4 1
700.2.be.e 72 140.s even 6 1
700.2.be.e 72 140.x odd 12 1
980.2.k.j 36 7.c even 3 1
980.2.k.j 36 28.g odd 6 1
980.2.k.j 36 35.l odd 12 1
980.2.k.j 36 140.w even 12 1
980.2.k.k 36 7.d odd 6 1
980.2.k.k 36 28.f even 6 1
980.2.k.k 36 35.k even 12 1
980.2.k.k 36 140.x odd 12 1
980.2.x.m 72 1.a even 1 1 trivial
980.2.x.m 72 4.b odd 2 1 inner
980.2.x.m 72 5.c odd 4 1 inner
980.2.x.m 72 7.c even 3 1 inner
980.2.x.m 72 20.e even 4 1 inner
980.2.x.m 72 28.g odd 6 1 inner
980.2.x.m 72 35.l odd 12 1 inner
980.2.x.m 72 140.w even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(980,[χ])S_{2}^{\mathrm{new}}(980, [\chi]):

T372−167T368+17716T364−1150565T360+54628057T356+⋯+136048896 T_{3}^{72} - 167 T_{3}^{68} + 17716 T_{3}^{64} - 1150565 T_{3}^{60} + 54628057 T_{3}^{56} + \cdots + 136048896 Copy content Toggle raw display
T1136−86T1134+4450T1132−150556T1130+3773207T1128+⋯+84332160000 T_{11}^{36} - 86 T_{11}^{34} + 4450 T_{11}^{32} - 150556 T_{11}^{30} + 3773207 T_{11}^{28} + \cdots + 84332160000 Copy content Toggle raw display
T1318−36T1315+1009T1314−1068T1313+648T1312−5320T1311+⋯+12800 T_{13}^{18} - 36 T_{13}^{15} + 1009 T_{13}^{14} - 1068 T_{13}^{13} + 648 T_{13}^{12} - 5320 T_{13}^{11} + \cdots + 12800 Copy content Toggle raw display