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Properties

Label 490.2.l.a.117.3
Level $490$
Weight $2$
Character 490.117
Analytic conductor $3.913$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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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] = [490,2,Mod(117,490)] 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("490.117"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(490, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.l (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0] 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: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 117.3
Root \(0.608761 + 0.793353i\) of defining polynomial
Character \(\chi\) \(=\) 490.117
Dual form 490.2.l.a.423.3

$q$-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+(0.965926 - 0.258819i) q^{2} +(-0.478235 + 1.78480i) q^{3} +(0.866025 - 0.500000i) q^{4} +(2.01088 + 0.977945i) q^{5} +1.84776i q^{6} +(0.707107 - 0.707107i) q^{8} +(-0.358719 - 0.207107i) q^{9} +(2.19547 + 0.424170i) q^{10} +(-1.41421 - 2.44949i) q^{11} +(0.478235 + 1.78480i) q^{12} +(4.23671 + 4.23671i) q^{13} +(-2.70711 + 3.12132i) q^{15} +(0.500000 - 0.866025i) q^{16} +(-5.04817 - 1.35265i) q^{17} +(-0.400100 - 0.107206i) q^{18} +(0.699709 - 1.21193i) q^{19} +(2.23044 - 0.158513i) q^{20} +(-2.00000 - 2.00000i) q^{22} +(0.151613 + 0.565826i) q^{23} +(0.923880 + 1.60021i) q^{24} +(3.08725 + 3.93305i) q^{25} +(5.18889 + 2.99581i) q^{26} +(-3.37849 + 3.37849i) q^{27} -0.828427i q^{29} +(-1.80701 + 3.71561i) q^{30} +(-1.32565 + 0.765367i) q^{31} +(0.258819 - 0.965926i) q^{32} +(5.04817 - 1.35265i) q^{33} -5.22625 q^{34} -0.414214 q^{36} +(-3.53225 + 0.946464i) q^{37} +(0.362196 - 1.35173i) q^{38} +(-9.58783 + 5.53553i) q^{39} +(2.11342 - 0.730392i) q^{40} -3.69552i q^{41} +(4.00000 - 4.00000i) q^{43} +(-2.44949 - 1.41421i) q^{44} +(-0.518801 - 0.767274i) q^{45} +(0.292893 + 0.507306i) q^{46} +(-0.396183 - 1.47858i) q^{47} +(1.30656 + 1.30656i) q^{48} +(4.00000 + 3.00000i) q^{50} +(4.82843 - 8.36308i) q^{51} +(5.78746 + 1.55075i) q^{52} +(11.2597 + 3.01702i) q^{53} +(-2.38896 + 4.13779i) q^{54} +(-0.448342 - 6.30864i) q^{55} +(1.82843 + 1.82843i) q^{57} +(-0.214413 - 0.800199i) q^{58} +(-4.61940 - 8.00103i) q^{59} +(-0.783763 + 4.05670i) q^{60} +(-5.57717 - 3.21998i) q^{61} +(-1.08239 + 1.08239i) q^{62} -1.00000i q^{64} +(4.37623 + 12.6628i) q^{65} +(4.52607 - 2.61313i) q^{66} +(3.83788 - 14.3232i) q^{67} +(-5.04817 + 1.35265i) q^{68} -1.08239 q^{69} -0.585786 q^{71} +(-0.400100 + 0.107206i) q^{72} +(-1.51676 + 5.66062i) q^{73} +(-3.16693 + 1.82843i) q^{74} +(-8.49614 + 3.62919i) q^{75} -1.39942i q^{76} +(-7.82843 + 7.82843i) q^{78} +(-4.39167 - 2.53553i) q^{79} +(1.85236 - 1.25250i) q^{80} +(-5.03553 - 8.72180i) q^{81} +(-0.956470 - 3.56960i) q^{82} +(-5.31911 - 5.31911i) q^{83} +(-8.82843 - 7.65685i) q^{85} +(2.82843 - 4.89898i) q^{86} +(1.47858 + 0.396183i) q^{87} +(-2.73205 - 0.732051i) q^{88} +(5.67459 - 9.82868i) q^{89} +(-0.699709 - 0.606854i) q^{90} +(0.414214 + 0.414214i) q^{92} +(-0.732051 - 2.73205i) q^{93} +(-0.765367 - 1.32565i) q^{94} +(2.59223 - 1.75277i) q^{95} +(1.60021 + 0.923880i) q^{96} +(4.59220 - 4.59220i) q^{97} +1.17157i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{15} + 8 q^{16} - 8 q^{18} - 32 q^{22} + 8 q^{23} - 16 q^{30} + 16 q^{36} - 32 q^{37} + 64 q^{43} + 16 q^{46} + 64 q^{50} + 32 q^{51} + 32 q^{53} - 16 q^{57} + 16 q^{58} + 8 q^{60} - 8 q^{65}+ \cdots - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.965926 − 0.258819i 0.683013 − 0.183013i
\(3\) −0.478235 + 1.78480i −0.276109 + 1.03045i 0.678985 + 0.734152i \(0.262420\pi\)
−0.955094 + 0.296302i \(0.904247\pi\)
\(4\) 0.866025 − 0.500000i 0.433013 − 0.250000i
\(5\) 2.01088 + 0.977945i 0.899291 + 0.437350i
\(6\) 1.84776i 0.754344i
\(7\) 0 0
\(8\) 0.707107 − 0.707107i 0.250000 − 0.250000i
\(9\) −0.358719 − 0.207107i −0.119573 − 0.0690356i
\(10\) 2.19547 + 0.424170i 0.694268 + 0.134134i
\(11\) −1.41421 − 2.44949i −0.426401 − 0.738549i 0.570149 − 0.821541i \(-0.306886\pi\)
−0.996550 + 0.0829925i \(0.973552\pi\)
\(12\) 0.478235 + 1.78480i 0.138055 + 0.515227i
\(13\) 4.23671 + 4.23671i 1.17505 + 1.17505i 0.980989 + 0.194064i \(0.0621670\pi\)
0.194064 + 0.980989i \(0.437833\pi\)
\(14\) 0 0
\(15\) −2.70711 + 3.12132i −0.698972 + 0.805921i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) −5.04817 − 1.35265i −1.22436 − 0.328067i −0.411980 − 0.911193i \(-0.635163\pi\)
−0.812382 + 0.583126i \(0.801829\pi\)
\(18\) −0.400100 − 0.107206i −0.0943044 − 0.0252688i
\(19\) 0.699709 − 1.21193i 0.160524 − 0.278036i −0.774533 − 0.632534i \(-0.782015\pi\)
0.935057 + 0.354498i \(0.115348\pi\)
\(20\) 2.23044 − 0.158513i 0.498742 − 0.0354445i
\(21\) 0 0
\(22\) −2.00000 − 2.00000i −0.426401 − 0.426401i
\(23\) 0.151613 + 0.565826i 0.0316134 + 0.117983i 0.979929 − 0.199344i \(-0.0638813\pi\)
−0.948316 + 0.317327i \(0.897215\pi\)
\(24\) 0.923880 + 1.60021i 0.188586 + 0.326641i
\(25\) 3.08725 + 3.93305i 0.617449 + 0.786611i
\(26\) 5.18889 + 2.99581i 1.01763 + 0.587527i
\(27\) −3.37849 + 3.37849i −0.650191 + 0.650191i
\(28\) 0 0
\(29\) − 0.828427i − 0.153835i −0.997037 − 0.0769175i \(-0.975492\pi\)
0.997037 − 0.0769175i \(-0.0245078\pi\)
\(30\) −1.80701 + 3.71561i −0.329913 + 0.678375i
\(31\) −1.32565 + 0.765367i −0.238095 + 0.137464i −0.614301 − 0.789072i \(-0.710562\pi\)
0.376206 + 0.926536i \(0.377228\pi\)
\(32\) 0.258819 − 0.965926i 0.0457532 − 0.170753i
\(33\) 5.04817 − 1.35265i 0.878774 − 0.235467i
\(34\) −5.22625 −0.896295
\(35\) 0 0
\(36\) −0.414214 −0.0690356
\(37\) −3.53225 + 0.946464i −0.580698 + 0.155598i −0.537199 − 0.843456i \(-0.680517\pi\)
−0.0434997 + 0.999053i \(0.513851\pi\)
\(38\) 0.362196 − 1.35173i 0.0587559 − 0.219280i
\(39\) −9.58783 + 5.53553i −1.53528 + 0.886395i
\(40\) 2.11342 − 0.730392i 0.334160 − 0.115485i
\(41\) − 3.69552i − 0.577143i −0.957458 − 0.288571i \(-0.906820\pi\)
0.957458 − 0.288571i \(-0.0931803\pi\)
\(42\) 0 0
\(43\) 4.00000 − 4.00000i 0.609994 − 0.609994i −0.332950 − 0.942944i \(-0.608044\pi\)
0.942944 + 0.332950i \(0.108044\pi\)
\(44\) −2.44949 − 1.41421i −0.369274 − 0.213201i
\(45\) −0.518801 − 0.767274i −0.0773383 − 0.114378i
\(46\) 0.292893 + 0.507306i 0.0431847 + 0.0747982i
\(47\) −0.396183 − 1.47858i −0.0577892 − 0.215672i 0.930993 − 0.365037i \(-0.118944\pi\)
−0.988782 + 0.149365i \(0.952277\pi\)
\(48\) 1.30656 + 1.30656i 0.188586 + 0.188586i
\(49\) 0 0
\(50\) 4.00000 + 3.00000i 0.565685 + 0.424264i
\(51\) 4.82843 − 8.36308i 0.676115 − 1.17107i
\(52\) 5.78746 + 1.55075i 0.802576 + 0.215050i
\(53\) 11.2597 + 3.01702i 1.54663 + 0.414419i 0.928402 − 0.371578i \(-0.121183\pi\)
0.618231 + 0.785997i \(0.287850\pi\)
\(54\) −2.38896 + 4.13779i −0.325096 + 0.563082i
\(55\) −0.448342 − 6.30864i −0.0604544 − 0.850657i
\(56\) 0 0
\(57\) 1.82843 + 1.82843i 0.242181 + 0.242181i
\(58\) −0.214413 − 0.800199i −0.0281538 − 0.105071i
\(59\) −4.61940 − 8.00103i −0.601394 − 1.04165i −0.992610 − 0.121347i \(-0.961279\pi\)
0.391216 − 0.920299i \(-0.372055\pi\)
\(60\) −0.783763 + 4.05670i −0.101183 + 0.523717i
\(61\) −5.57717 − 3.21998i −0.714083 − 0.412276i 0.0984878 − 0.995138i \(-0.468599\pi\)
−0.812571 + 0.582862i \(0.801933\pi\)
\(62\) −1.08239 + 1.08239i −0.137464 + 0.137464i
\(63\) 0 0
\(64\) − 1.00000i − 0.125000i
\(65\) 4.37623 + 12.6628i 0.542805 + 1.57062i
\(66\) 4.52607 − 2.61313i 0.557120 − 0.321654i
\(67\) 3.83788 − 14.3232i 0.468872 − 1.74985i −0.174852 − 0.984595i \(-0.555945\pi\)
0.643724 − 0.765258i \(-0.277389\pi\)
\(68\) −5.04817 + 1.35265i −0.612181 + 0.164033i
\(69\) −1.08239 −0.130305
\(70\) 0 0
\(71\) −0.585786 −0.0695201 −0.0347600 − 0.999396i \(-0.511067\pi\)
−0.0347600 + 0.999396i \(0.511067\pi\)
\(72\) −0.400100 + 0.107206i −0.0471522 + 0.0126344i
\(73\) −1.51676 + 5.66062i −0.177523 + 0.662525i 0.818585 + 0.574385i \(0.194759\pi\)
−0.996108 + 0.0881398i \(0.971908\pi\)
\(74\) −3.16693 + 1.82843i −0.368148 + 0.212550i
\(75\) −8.49614 + 3.62919i −0.981049 + 0.419062i
\(76\) − 1.39942i − 0.160524i
\(77\) 0 0
\(78\) −7.82843 + 7.82843i −0.886395 + 0.886395i
\(79\) −4.39167 − 2.53553i −0.494102 − 0.285270i 0.232173 − 0.972675i \(-0.425417\pi\)
−0.726275 + 0.687405i \(0.758750\pi\)
\(80\) 1.85236 − 1.25250i 0.207101 − 0.140033i
\(81\) −5.03553 − 8.72180i −0.559504 − 0.969089i
\(82\) −0.956470 − 3.56960i −0.105624 − 0.394196i
\(83\) −5.31911 − 5.31911i −0.583848 − 0.583848i 0.352111 − 0.935958i \(-0.385464\pi\)
−0.935958 + 0.352111i \(0.885464\pi\)
\(84\) 0 0
\(85\) −8.82843 − 7.65685i −0.957577 − 0.830502i
\(86\) 2.82843 − 4.89898i 0.304997 − 0.528271i
\(87\) 1.47858 + 0.396183i 0.158520 + 0.0424753i
\(88\) −2.73205 − 0.732051i −0.291238 − 0.0780369i
\(89\) 5.67459 − 9.82868i 0.601506 − 1.04184i −0.391088 − 0.920353i \(-0.627901\pi\)
0.992593 − 0.121485i \(-0.0387656\pi\)
\(90\) −0.699709 − 0.606854i −0.0737558 − 0.0639680i
\(91\) 0 0
\(92\) 0.414214 + 0.414214i 0.0431847 + 0.0431847i
\(93\) −0.732051 − 2.73205i −0.0759101 − 0.283300i
\(94\) −0.765367 − 1.32565i −0.0789416 − 0.136731i
\(95\) 2.59223 − 1.75277i 0.265957 − 0.179830i
\(96\) 1.60021 + 0.923880i 0.163320 + 0.0942931i
\(97\) 4.59220 − 4.59220i 0.466267 − 0.466267i −0.434436 − 0.900703i \(-0.643052\pi\)
0.900703 + 0.434436i \(0.143052\pi\)
\(98\) 0 0
\(99\) 1.17157i 0.117748i
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 490.2.l.a.117.3 16
5.3 odd 4 inner 490.2.l.a.313.1 16
7.2 even 3 70.2.g.a.27.1 yes 8
7.3 odd 6 inner 490.2.l.a.227.1 16
7.4 even 3 inner 490.2.l.a.227.2 16
7.5 odd 6 70.2.g.a.27.2 yes 8
7.6 odd 2 inner 490.2.l.a.117.4 16
21.2 odd 6 630.2.p.a.307.4 8
21.5 even 6 630.2.p.a.307.3 8
28.19 even 6 560.2.bj.c.97.1 8
28.23 odd 6 560.2.bj.c.97.4 8
35.2 odd 12 350.2.g.a.293.3 8
35.3 even 12 inner 490.2.l.a.423.3 16
35.9 even 6 350.2.g.a.307.4 8
35.12 even 12 350.2.g.a.293.4 8
35.13 even 4 inner 490.2.l.a.313.2 16
35.18 odd 12 inner 490.2.l.a.423.4 16
35.19 odd 6 350.2.g.a.307.3 8
35.23 odd 12 70.2.g.a.13.2 yes 8
35.33 even 12 70.2.g.a.13.1 ✓ 8
105.23 even 12 630.2.p.a.433.3 8
105.68 odd 12 630.2.p.a.433.4 8
140.23 even 12 560.2.bj.c.433.1 8
140.103 odd 12 560.2.bj.c.433.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.g.a.13.1 ✓ 8 35.33 even 12
70.2.g.a.13.2 yes 8 35.23 odd 12
70.2.g.a.27.1 yes 8 7.2 even 3
70.2.g.a.27.2 yes 8 7.5 odd 6
350.2.g.a.293.3 8 35.2 odd 12
350.2.g.a.293.4 8 35.12 even 12
350.2.g.a.307.3 8 35.19 odd 6
350.2.g.a.307.4 8 35.9 even 6
490.2.l.a.117.3 16 1.1 even 1 trivial
490.2.l.a.117.4 16 7.6 odd 2 inner
490.2.l.a.227.1 16 7.3 odd 6 inner
490.2.l.a.227.2 16 7.4 even 3 inner
490.2.l.a.313.1 16 5.3 odd 4 inner
490.2.l.a.313.2 16 35.13 even 4 inner
490.2.l.a.423.3 16 35.3 even 12 inner
490.2.l.a.423.4 16 35.18 odd 12 inner
560.2.bj.c.97.1 8 28.19 even 6
560.2.bj.c.97.4 8 28.23 odd 6
560.2.bj.c.433.1 8 140.23 even 12
560.2.bj.c.433.4 8 140.103 odd 12
630.2.p.a.307.3 8 21.5 even 6
630.2.p.a.307.4 8 21.2 odd 6
630.2.p.a.433.3 8 105.23 even 12
630.2.p.a.433.4 8 105.68 odd 12