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Properties

Label 980.2.k.k.687.13
Level $980$
Weight $2$
Character 980.687
Analytic conductor $7.825$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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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(687,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("980.687"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,-2,0,0,8] 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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 687.13
Character \(\chi\) \(=\) 980.687
Dual form 980.2.k.k.883.13

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.566415 - 1.29583i) q^{2} +(-0.792756 - 0.792756i) q^{3} +(-1.35835 - 1.46795i) q^{4} +(0.427372 + 2.19485i) q^{5} +(-1.47630 + 0.578247i) q^{6} +(-2.67161 + 0.928715i) q^{8} -1.74308i q^{9} +(3.08622 + 0.689394i) q^{10} -2.98364i q^{11} +(-0.0868916 + 2.24057i) q^{12} +(4.05418 - 4.05418i) q^{13} +(1.40118 - 2.07878i) q^{15} +(-0.309783 + 3.98799i) q^{16} +(1.68722 + 1.68722i) q^{17} +(-2.25873 - 0.987305i) q^{18} -2.51388 q^{19} +(2.64142 - 3.60873i) q^{20} +(-3.86629 - 1.68998i) q^{22} +(-4.24161 - 4.24161i) q^{23} +(2.85418 + 1.38169i) q^{24} +(-4.63471 + 1.87603i) q^{25} +(-2.95718 - 7.54987i) q^{26} +(-3.76010 + 3.76010i) q^{27} +2.55433i q^{29} +(-1.90010 - 2.99314i) q^{30} -3.60553i q^{31} +(4.99228 + 2.66028i) q^{32} +(-2.36530 + 2.36530i) q^{33} +(3.14202 - 1.23068i) q^{34} +(-2.55876 + 2.36770i) q^{36} +(-3.37486 - 3.37486i) q^{37} +(-1.42390 + 3.25756i) q^{38} -6.42795 q^{39} +(-3.18016 - 5.46686i) q^{40} -7.93727 q^{41} +(-7.62646 - 7.62646i) q^{43} +(-4.37985 + 4.05282i) q^{44} +(3.82579 - 0.744942i) q^{45} +(-7.89891 + 3.09389i) q^{46} +(-2.09272 + 2.09272i) q^{47} +(3.40708 - 2.91592i) q^{48} +(-0.194151 + 7.06840i) q^{50} -2.67511i q^{51} +(-11.4583 - 0.444366i) q^{52} +(1.80994 - 1.80994i) q^{53} +(2.74267 + 7.00223i) q^{54} +(6.54863 - 1.27512i) q^{55} +(1.99289 + 1.99289i) q^{57} +(3.30997 + 1.44681i) q^{58} +2.09389 q^{59} +(-4.95484 + 0.766842i) q^{60} -1.90096 q^{61} +(-4.67216 - 2.04223i) q^{62} +(6.27498 - 4.96233i) q^{64} +(10.6309 + 7.16566i) q^{65} +(1.72528 + 4.40476i) q^{66} +(2.19990 - 2.19990i) q^{67} +(0.184931 - 4.76859i) q^{68} +6.72512i q^{69} -8.09721i q^{71} +(1.61882 + 4.65682i) q^{72} +(6.89558 - 6.89558i) q^{73} +(-6.28481 + 2.46167i) q^{74} +(5.16142 + 2.18696i) q^{75} +(3.41472 + 3.69026i) q^{76} +(-3.64089 + 8.32952i) q^{78} +8.06116 q^{79} +(-8.88541 + 1.02443i) q^{80} +0.732452 q^{81} +(-4.49579 + 10.2853i) q^{82} +(5.99790 + 5.99790i) q^{83} +(-2.98212 + 4.42426i) q^{85} +(-14.2023 + 5.56285i) q^{86} +(2.02496 - 2.02496i) q^{87} +(2.77095 + 7.97111i) q^{88} +2.05525i q^{89} +(1.20167 - 5.37951i) q^{90} +(-0.464910 + 11.9881i) q^{92} +(-2.85831 + 2.85831i) q^{93} +(1.52646 + 3.89715i) q^{94} +(-1.07436 - 5.51758i) q^{95} +(-1.84871 - 6.06661i) q^{96} +(6.63160 + 6.63160i) q^{97} -5.20071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 2 q^{2} + 8 q^{5} - 8 q^{6} - 2 q^{8} - 2 q^{10} - 10 q^{12} + 28 q^{16} - 4 q^{17} + 20 q^{18} - 28 q^{20} - 8 q^{22} + 16 q^{25} + 4 q^{26} + 32 q^{30} + 38 q^{32} + 64 q^{33} + 8 q^{36} + 4 q^{37}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.566415 − 1.29583i 0.400516 − 0.916290i
\(3\) −0.792756 − 0.792756i −0.457698 − 0.457698i 0.440201 − 0.897899i \(-0.354907\pi\)
−0.897899 + 0.440201i \(0.854907\pi\)
\(4\) −1.35835 − 1.46795i −0.679174 − 0.733977i
\(5\) 0.427372 + 2.19485i 0.191126 + 0.981565i
\(6\) −1.47630 + 0.578247i −0.602699 + 0.236068i
\(7\) 0 0
\(8\) −2.67161 + 0.928715i −0.944556 + 0.328350i
\(9\) − 1.74308i − 0.581026i
\(10\) 3.08622 + 0.689394i 0.975948 + 0.218005i
\(11\) − 2.98364i − 0.899601i −0.893129 − 0.449800i \(-0.851495\pi\)
0.893129 − 0.449800i \(-0.148505\pi\)
\(12\) −0.0868916 + 2.24057i −0.0250834 + 0.646796i
\(13\) 4.05418 − 4.05418i 1.12443 − 1.12443i 0.133359 − 0.991068i \(-0.457424\pi\)
0.991068 − 0.133359i \(-0.0425763\pi\)
\(14\) 0 0
\(15\) 1.40118 − 2.07878i 0.361782 − 0.536738i
\(16\) −0.309783 + 3.98799i −0.0774456 + 0.996997i
\(17\) 1.68722 + 1.68722i 0.409211 + 0.409211i 0.881463 − 0.472252i \(-0.156559\pi\)
−0.472252 + 0.881463i \(0.656559\pi\)
\(18\) −2.25873 − 0.987305i −0.532388 − 0.232710i
\(19\) −2.51388 −0.576723 −0.288362 − 0.957522i \(-0.593111\pi\)
−0.288362 + 0.957522i \(0.593111\pi\)
\(20\) 2.64142 − 3.60873i 0.590639 − 0.806936i
\(21\) 0 0
\(22\) −3.86629 − 1.68998i −0.824295 − 0.360304i
\(23\) −4.24161 − 4.24161i −0.884436 − 0.884436i 0.109545 − 0.993982i \(-0.465060\pi\)
−0.993982 + 0.109545i \(0.965060\pi\)
\(24\) 2.85418 + 1.38169i 0.582606 + 0.282036i
\(25\) −4.63471 + 1.87603i −0.926941 + 0.375206i
\(26\) −2.95718 − 7.54987i −0.579950 − 1.48065i
\(27\) −3.76010 + 3.76010i −0.723632 + 0.723632i
\(28\) 0 0
\(29\) 2.55433i 0.474327i 0.971470 + 0.237163i \(0.0762177\pi\)
−0.971470 + 0.237163i \(0.923782\pi\)
\(30\) −1.90010 − 2.99314i −0.346908 − 0.546470i
\(31\) − 3.60553i − 0.647573i −0.946130 − 0.323787i \(-0.895044\pi\)
0.946130 − 0.323787i \(-0.104956\pi\)
\(32\) 4.99228 + 2.66028i 0.882520 + 0.470276i
\(33\) −2.36530 + 2.36530i −0.411745 + 0.411745i
\(34\) 3.14202 − 1.23068i 0.538851 − 0.211060i
\(35\) 0 0
\(36\) −2.55876 + 2.36770i −0.426460 + 0.394617i
\(37\) −3.37486 − 3.37486i −0.554823 − 0.554823i 0.373006 − 0.927829i \(-0.378327\pi\)
−0.927829 + 0.373006i \(0.878327\pi\)
\(38\) −1.42390 + 3.25756i −0.230987 + 0.528446i
\(39\) −6.42795 −1.02930
\(40\) −3.18016 − 5.46686i −0.502827 − 0.864387i
\(41\) −7.93727 −1.23959 −0.619797 − 0.784763i \(-0.712785\pi\)
−0.619797 + 0.784763i \(0.712785\pi\)
\(42\) 0 0
\(43\) −7.62646 − 7.62646i −1.16302 − 1.16302i −0.983810 − 0.179215i \(-0.942644\pi\)
−0.179215 − 0.983810i \(-0.557356\pi\)
\(44\) −4.37985 + 4.05282i −0.660287 + 0.610985i
\(45\) 3.82579 − 0.744942i 0.570315 − 0.111049i
\(46\) −7.89891 + 3.09389i −1.16463 + 0.456169i
\(47\) −2.09272 + 2.09272i −0.305254 + 0.305254i −0.843065 − 0.537811i \(-0.819251\pi\)
0.537811 + 0.843065i \(0.319251\pi\)
\(48\) 3.40708 − 2.91592i 0.491770 − 0.420876i
\(49\) 0 0
\(50\) −0.194151 + 7.06840i −0.0274571 + 0.999623i
\(51\) − 2.67511i − 0.374590i
\(52\) −11.4583 − 0.444366i −1.58899 − 0.0616225i
\(53\) 1.80994 − 1.80994i 0.248615 − 0.248615i −0.571787 − 0.820402i \(-0.693750\pi\)
0.820402 + 0.571787i \(0.193750\pi\)
\(54\) 2.74267 + 7.00223i 0.373230 + 0.952882i
\(55\) 6.54863 − 1.27512i 0.883017 − 0.171938i
\(56\) 0 0
\(57\) 1.99289 + 1.99289i 0.263965 + 0.263965i
\(58\) 3.30997 + 1.44681i 0.434621 + 0.189975i
\(59\) 2.09389 0.272601 0.136301 − 0.990668i \(-0.456479\pi\)
0.136301 + 0.990668i \(0.456479\pi\)
\(60\) −4.95484 + 0.766842i −0.639667 + 0.0989988i
\(61\) −1.90096 −0.243393 −0.121696 − 0.992567i \(-0.538833\pi\)
−0.121696 + 0.992567i \(0.538833\pi\)
\(62\) −4.67216 − 2.04223i −0.593365 − 0.259363i
\(63\) 0 0
\(64\) 6.27498 − 4.96233i 0.784372 − 0.620291i
\(65\) 10.6309 + 7.16566i 1.31861 + 0.888791i
\(66\) 1.72528 + 4.40476i 0.212367 + 0.542188i
\(67\) 2.19990 − 2.19990i 0.268761 − 0.268761i −0.559840 − 0.828601i \(-0.689137\pi\)
0.828601 + 0.559840i \(0.189137\pi\)
\(68\) 0.184931 − 4.76859i 0.0224262 − 0.578277i
\(69\) 6.72512i 0.809609i
\(70\) 0 0
\(71\) − 8.09721i − 0.960962i −0.877005 − 0.480481i \(-0.840462\pi\)
0.877005 − 0.480481i \(-0.159538\pi\)
\(72\) 1.61882 + 4.65682i 0.190780 + 0.548811i
\(73\) 6.89558 − 6.89558i 0.807067 − 0.807067i −0.177122 − 0.984189i \(-0.556679\pi\)
0.984189 + 0.177122i \(0.0566788\pi\)
\(74\) −6.28481 + 2.46167i −0.730595 + 0.286163i
\(75\) 5.16142 + 2.18696i 0.595990 + 0.252528i
\(76\) 3.41472 + 3.69026i 0.391695 + 0.423302i
\(77\) 0 0
\(78\) −3.64089 + 8.32952i −0.412249 + 0.943133i
\(79\) 8.06116 0.906952 0.453476 − 0.891269i \(-0.350184\pi\)
0.453476 + 0.891269i \(0.350184\pi\)
\(80\) −8.88541 + 1.02443i −0.993419 + 0.114535i
\(81\) 0.732452 0.0813835
\(82\) −4.49579 + 10.2853i −0.496477 + 1.13583i
\(83\) 5.99790 + 5.99790i 0.658356 + 0.658356i 0.954991 − 0.296635i \(-0.0958646\pi\)
−0.296635 + 0.954991i \(0.595865\pi\)
\(84\) 0 0
\(85\) −2.98212 + 4.42426i −0.323456 + 0.479878i
\(86\) −14.2023 + 5.56285i −1.53148 + 0.599858i
\(87\) 2.02496 − 2.02496i 0.217098 − 0.217098i
\(88\) 2.77095 + 7.97111i 0.295384 + 0.849723i
\(89\) 2.05525i 0.217856i 0.994050 + 0.108928i \(0.0347418\pi\)
−0.994050 + 0.108928i \(0.965258\pi\)
\(90\) 1.20167 − 5.37951i 0.126667 − 0.567051i
\(91\) 0 0
\(92\) −0.464910 + 11.9881i −0.0484702 + 1.24984i
\(93\) −2.85831 + 2.85831i −0.296393 + 0.296393i
\(94\) 1.52646 + 3.89715i 0.157442 + 0.401960i
\(95\) −1.07436 − 5.51758i −0.110227 − 0.566092i
\(96\) −1.84871 − 6.06661i −0.188683 − 0.619171i
\(97\) 6.63160 + 6.63160i 0.673337 + 0.673337i 0.958484 − 0.285147i \(-0.0920425\pi\)
−0.285147 + 0.958484i \(0.592042\pi\)
\(98\) 0 0
\(99\) −5.20071 −0.522691
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 980.2.k.k.687.13 36
4.3 odd 2 inner 980.2.k.k.687.3 36
5.3 odd 4 inner 980.2.k.k.883.3 36
7.2 even 3 140.2.w.b.67.2 yes 72
7.3 odd 6 980.2.x.m.667.13 72
7.4 even 3 140.2.w.b.107.13 yes 72
7.5 odd 6 980.2.x.m.67.2 72
7.6 odd 2 980.2.k.j.687.13 36
20.3 even 4 inner 980.2.k.k.883.13 36
28.3 even 6 980.2.x.m.667.10 72
28.11 odd 6 140.2.w.b.107.10 yes 72
28.19 even 6 980.2.x.m.67.14 72
28.23 odd 6 140.2.w.b.67.14 yes 72
28.27 even 2 980.2.k.j.687.3 36
35.2 odd 12 700.2.be.e.543.9 72
35.3 even 12 980.2.x.m.863.14 72
35.4 even 6 700.2.be.e.107.6 72
35.9 even 6 700.2.be.e.207.17 72
35.13 even 4 980.2.k.j.883.3 36
35.18 odd 12 140.2.w.b.23.14 yes 72
35.23 odd 12 140.2.w.b.123.10 yes 72
35.32 odd 12 700.2.be.e.443.5 72
35.33 even 12 980.2.x.m.263.10 72
140.3 odd 12 980.2.x.m.863.2 72
140.23 even 12 140.2.w.b.123.13 yes 72
140.39 odd 6 700.2.be.e.107.9 72
140.67 even 12 700.2.be.e.443.17 72
140.79 odd 6 700.2.be.e.207.5 72
140.83 odd 4 980.2.k.j.883.13 36
140.103 odd 12 980.2.x.m.263.13 72
140.107 even 12 700.2.be.e.543.6 72
140.123 even 12 140.2.w.b.23.2 ✓ 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.w.b.23.2 ✓ 72 140.123 even 12
140.2.w.b.23.14 yes 72 35.18 odd 12
140.2.w.b.67.2 yes 72 7.2 even 3
140.2.w.b.67.14 yes 72 28.23 odd 6
140.2.w.b.107.10 yes 72 28.11 odd 6
140.2.w.b.107.13 yes 72 7.4 even 3
140.2.w.b.123.10 yes 72 35.23 odd 12
140.2.w.b.123.13 yes 72 140.23 even 12
700.2.be.e.107.6 72 35.4 even 6
700.2.be.e.107.9 72 140.39 odd 6
700.2.be.e.207.5 72 140.79 odd 6
700.2.be.e.207.17 72 35.9 even 6
700.2.be.e.443.5 72 35.32 odd 12
700.2.be.e.443.17 72 140.67 even 12
700.2.be.e.543.6 72 140.107 even 12
700.2.be.e.543.9 72 35.2 odd 12
980.2.k.j.687.3 36 28.27 even 2
980.2.k.j.687.13 36 7.6 odd 2
980.2.k.j.883.3 36 35.13 even 4
980.2.k.j.883.13 36 140.83 odd 4
980.2.k.k.687.3 36 4.3 odd 2 inner
980.2.k.k.687.13 36 1.1 even 1 trivial
980.2.k.k.883.3 36 5.3 odd 4 inner
980.2.k.k.883.13 36 20.3 even 4 inner
980.2.x.m.67.2 72 7.5 odd 6
980.2.x.m.67.14 72 28.19 even 6
980.2.x.m.263.10 72 35.33 even 12
980.2.x.m.263.13 72 140.103 odd 12
980.2.x.m.667.10 72 28.3 even 6
980.2.x.m.667.13 72 7.3 odd 6
980.2.x.m.863.2 72 140.3 odd 12
980.2.x.m.863.14 72 35.3 even 12