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Properties

Label 980.2.k.j.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.j.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 0.897899 − 0.440201i \(-0.145093\pi\)
−0.440201 + 0.897899i \(0.645093\pi\)
\(4\) −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 \(0.542576\pi\)
−0.991068 + 0.133359i \(0.957424\pi\)
\(14\) 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 0.472252 − 0.881463i \(-0.343441\pi\)
−0.881463 + 0.472252i \(0.843441\pi\)
\(18\) −2.25873 − 0.987305i −0.532388 − 0.232710i
\(19\) 2.51388 0.576723 0.288362 − 0.957522i \(-0.406889\pi\)
0.288362 + 0.957522i \(0.406889\pi\)
\(20\) −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 \(0.104956\pi\)
−0.946130 + 0.323787i \(0.895044\pi\)
\(32\) 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 \(-0.287215\pi\)
0.619797 + 0.784763i \(0.287215\pi\)
\(42\) 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 −0.537811 − 0.843065i \(-0.680749\pi\)
0.843065 + 0.537811i \(0.180749\pi\)
\(48\) −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 \(-0.543521\pi\)
−0.136301 + 0.990668i \(0.543521\pi\)
\(60\) −4.95484 + 0.766842i −0.639667 + 0.0989988i
\(61\) 1.90096 0.243393 0.121696 − 0.992567i \(-0.461167\pi\)
0.121696 + 0.992567i \(0.461167\pi\)
\(62\) 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 −0.984189 − 0.177122i \(-0.943321\pi\)
0.177122 + 0.984189i \(0.443321\pi\)
\(74\) −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 0.296635 − 0.954991i \(-0.404135\pi\)
−0.954991 + 0.296635i \(0.904135\pi\)
\(84\) 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 \(-0.965258\pi\)
0.994050 − 0.108928i \(-0.0347418\pi\)
\(90\) −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 0.285147 − 0.958484i \(-0.407958\pi\)
−0.958484 + 0.285147i \(0.907958\pi\)
\(98\) 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.j.687.13 36
4.3 odd 2 inner 980.2.k.j.687.3 36
5.3 odd 4 inner 980.2.k.j.883.3 36
7.2 even 3 980.2.x.m.67.2 72
7.3 odd 6 140.2.w.b.107.13 yes 72
7.4 even 3 980.2.x.m.667.13 72
7.5 odd 6 140.2.w.b.67.2 yes 72
7.6 odd 2 980.2.k.k.687.13 36
20.3 even 4 inner 980.2.k.j.883.13 36
28.3 even 6 140.2.w.b.107.10 yes 72
28.11 odd 6 980.2.x.m.667.10 72
28.19 even 6 140.2.w.b.67.14 yes 72
28.23 odd 6 980.2.x.m.67.14 72
28.27 even 2 980.2.k.k.687.3 36
35.3 even 12 140.2.w.b.23.14 yes 72
35.12 even 12 700.2.be.e.543.9 72
35.13 even 4 980.2.k.k.883.3 36
35.17 even 12 700.2.be.e.443.5 72
35.18 odd 12 980.2.x.m.863.14 72
35.19 odd 6 700.2.be.e.207.17 72
35.23 odd 12 980.2.x.m.263.10 72
35.24 odd 6 700.2.be.e.107.6 72
35.33 even 12 140.2.w.b.123.10 yes 72
140.3 odd 12 140.2.w.b.23.2 ✓ 72
140.19 even 6 700.2.be.e.207.5 72
140.23 even 12 980.2.x.m.263.13 72
140.47 odd 12 700.2.be.e.543.6 72
140.59 even 6 700.2.be.e.107.9 72
140.83 odd 4 980.2.k.k.883.13 36
140.87 odd 12 700.2.be.e.443.17 72
140.103 odd 12 140.2.w.b.123.13 yes 72
140.123 even 12 980.2.x.m.863.2 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.w.b.23.2 ✓ 72 140.3 odd 12
140.2.w.b.23.14 yes 72 35.3 even 12
140.2.w.b.67.2 yes 72 7.5 odd 6
140.2.w.b.67.14 yes 72 28.19 even 6
140.2.w.b.107.10 yes 72 28.3 even 6
140.2.w.b.107.13 yes 72 7.3 odd 6
140.2.w.b.123.10 yes 72 35.33 even 12
140.2.w.b.123.13 yes 72 140.103 odd 12
700.2.be.e.107.6 72 35.24 odd 6
700.2.be.e.107.9 72 140.59 even 6
700.2.be.e.207.5 72 140.19 even 6
700.2.be.e.207.17 72 35.19 odd 6
700.2.be.e.443.5 72 35.17 even 12
700.2.be.e.443.17 72 140.87 odd 12
700.2.be.e.543.6 72 140.47 odd 12
700.2.be.e.543.9 72 35.12 even 12
980.2.k.j.687.3 36 4.3 odd 2 inner
980.2.k.j.687.13 36 1.1 even 1 trivial
980.2.k.j.883.3 36 5.3 odd 4 inner
980.2.k.j.883.13 36 20.3 even 4 inner
980.2.k.k.687.3 36 28.27 even 2
980.2.k.k.687.13 36 7.6 odd 2
980.2.k.k.883.3 36 35.13 even 4
980.2.k.k.883.13 36 140.83 odd 4
980.2.x.m.67.2 72 7.2 even 3
980.2.x.m.67.14 72 28.23 odd 6
980.2.x.m.263.10 72 35.23 odd 12
980.2.x.m.263.13 72 140.23 even 12
980.2.x.m.667.10 72 28.11 odd 6
980.2.x.m.667.13 72 7.4 even 3
980.2.x.m.863.2 72 140.123 even 12
980.2.x.m.863.14 72 35.18 odd 12