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Properties

Label 840.2.bz.a.19.13
Level $840$
Weight $2$
Character 840.19
Analytic conductor $6.707$
Analytic rank $0$
Dimension $96$
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] = [840,2,Mod(19,840)] 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("840.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bz (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 19.13
Character \(\chi\) \(=\) 840.19
Dual form 840.2.bz.a.619.13

$q$-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+(-1.06845 + 0.926509i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.283163 - 1.97985i) q^{4} +(1.15715 + 1.91338i) q^{5} +(1.33660 + 0.462049i) q^{6} +(-1.30898 + 2.29925i) q^{7} +(1.53181 + 2.37772i) q^{8} +(-0.500000 + 0.866025i) q^{9} +(-3.00911 - 0.972235i) q^{10} +(-0.215464 - 0.373194i) q^{11} +(-1.85618 + 0.744700i) q^{12} +4.07715i q^{13} +(-0.731697 - 3.66942i) q^{14} +(1.07846 - 1.95881i) q^{15} +(-3.83964 - 1.12124i) q^{16} +(-0.866070 - 1.50008i) q^{17} +(-0.268156 - 1.38856i) q^{18} +(-3.86480 - 2.23134i) q^{19} +(4.11586 - 1.74919i) q^{20} +(2.64570 - 0.0160136i) q^{21} +(0.575979 + 0.199109i) q^{22} +(0.440788 - 0.763466i) q^{23} +(1.29327 - 2.51544i) q^{24} +(-2.32201 + 4.42812i) q^{25} +(-3.77751 - 4.35622i) q^{26} +1.00000 q^{27} +(4.18153 + 3.24266i) q^{28} -6.46005i q^{29} +(0.662577 + 3.09209i) q^{30} +(-1.15678 - 2.00361i) q^{31} +(5.14129 - 2.35947i) q^{32} +(-0.215464 + 0.373194i) q^{33} +(2.31519 + 0.800334i) q^{34} +(-5.91402 + 0.156000i) q^{35} +(1.57302 + 1.23515i) q^{36} +(-3.81436 + 6.60666i) q^{37} +(6.19669 - 1.19670i) q^{38} +(3.53091 - 2.03857i) q^{39} +(-2.77695 + 5.68230i) q^{40} +6.70434i q^{41} +(-2.81196 + 2.46838i) q^{42} +4.01301i q^{43} +(-0.799881 + 0.320912i) q^{44} +(-2.23561 + 0.0454322i) q^{45} +(0.236400 + 1.22412i) q^{46} +(-9.14574 - 5.28030i) q^{47} +(0.948794 + 3.88584i) q^{48} +(-3.57313 - 6.01937i) q^{49} +(-1.62174 - 6.88258i) q^{50} +(-0.866070 + 1.50008i) q^{51} +(8.07215 + 1.15450i) q^{52} +(1.73662 + 3.00792i) q^{53} +(-1.06845 + 0.926509i) q^{54} +(0.464737 - 0.844104i) q^{55} +(-7.47210 + 0.409610i) q^{56} +4.46268i q^{57} +(5.98530 + 6.90223i) q^{58} +(-0.708208 + 0.408884i) q^{59} +(-3.57277 - 2.68985i) q^{60} +(-6.34043 + 10.9819i) q^{61} +(3.09233 + 1.06898i) q^{62} +(-1.33672 - 2.28324i) q^{63} +(-3.30714 + 7.28442i) q^{64} +(-7.80112 + 4.71787i) q^{65} +(-0.115556 - 0.598367i) q^{66} +(-7.57067 + 4.37093i) q^{67} +(-3.21517 + 1.28993i) q^{68} -0.881575 q^{69} +(6.17429 - 5.64607i) q^{70} -6.08373i q^{71} +(-2.82507 + 0.137721i) q^{72} +(-3.99832 - 6.92529i) q^{73} +(-2.04569 - 10.5929i) q^{74} +(4.99587 - 0.203137i) q^{75} +(-5.51210 + 7.01990i) q^{76} +(1.14011 - 0.00690071i) q^{77} +(-1.88384 + 5.44953i) q^{78} +(9.91239 + 5.72292i) q^{79} +(-2.29767 - 8.64411i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-6.21163 - 7.16324i) q^{82} -10.1969 q^{83} +(0.717461 - 5.24264i) q^{84} +(1.86804 - 3.39293i) q^{85} +(-3.71808 - 4.28769i) q^{86} +(-5.59457 + 3.23003i) q^{87} +(0.557304 - 1.08397i) q^{88} +(1.68998 + 0.975709i) q^{89} +(2.34654 - 2.11985i) q^{90} +(-9.37439 - 5.33692i) q^{91} +(-1.38674 - 1.08888i) q^{92} +(-1.15678 + 2.00361i) q^{93} +(14.6640 - 2.83189i) q^{94} +(-0.202750 - 9.97680i) q^{95} +(-4.61401 - 3.27276i) q^{96} +4.18222 q^{97} +(9.39470 + 3.12085i) q^{98} +0.430927 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 48 q^{3} - 48 q^{9} + 13 q^{10} + 14 q^{14} + 4 q^{16} - 22 q^{20} + 96 q^{27} + 4 q^{28} - 5 q^{30} + 30 q^{32} - 8 q^{35} + 12 q^{38} - 23 q^{40} + 2 q^{42} - 16 q^{44} - 22 q^{46} - 8 q^{48} + 12 q^{50}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06845 + 0.926509i −0.755507 + 0.655141i
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) 0.283163 − 1.97985i 0.141582 − 0.989927i
\(5\) 1.15715 + 1.91338i 0.517493 + 0.855688i
\(6\) 1.33660 + 0.462049i 0.545666 + 0.188631i
\(7\) −1.30898 + 2.29925i −0.494749 + 0.869036i
\(8\) 1.53181 + 2.37772i 0.541575 + 0.840652i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) −3.00911 − 0.972235i −0.951565 − 0.307448i
\(11\) −0.215464 − 0.373194i −0.0649647 − 0.112522i 0.831714 − 0.555205i \(-0.187360\pi\)
−0.896678 + 0.442683i \(0.854027\pi\)
\(12\) −1.85618 + 0.744700i −0.535834 + 0.214976i
\(13\) 4.07715i 1.13080i 0.824818 + 0.565399i \(0.191278\pi\)
−0.824818 + 0.565399i \(0.808722\pi\)
\(14\) −0.731697 − 3.66942i −0.195554 − 0.980693i
\(15\) 1.07846 − 1.95881i 0.278457 − 0.505762i
\(16\) −3.83964 − 1.12124i −0.959909 − 0.280311i
\(17\) −0.866070 − 1.50008i −0.210053 − 0.363822i 0.741678 − 0.670756i \(-0.234030\pi\)
−0.951731 + 0.306934i \(0.900697\pi\)
\(18\) −0.268156 − 1.38856i −0.0632050 − 0.327286i
\(19\) −3.86480 − 2.23134i −0.886645 − 0.511905i −0.0138015 − 0.999905i \(-0.504393\pi\)
−0.872844 + 0.488000i \(0.837727\pi\)
\(20\) 4.11586 − 1.74919i 0.920335 − 0.391130i
\(21\) 2.64570 − 0.0160136i 0.577340 − 0.00349446i
\(22\) 0.575979 + 0.199109i 0.122799 + 0.0424503i
\(23\) 0.440788 − 0.763466i 0.0919106 − 0.159194i −0.816404 − 0.577481i \(-0.804036\pi\)
0.908315 + 0.418287i \(0.137369\pi\)
\(24\) 1.29327 − 2.51544i 0.263987 − 0.513463i
\(25\) −2.32201 + 4.42812i −0.464403 + 0.885624i
\(26\) −3.77751 − 4.35622i −0.740831 − 0.854325i
\(27\) 1.00000 0.192450
\(28\) 4.18153 + 3.24266i 0.790234 + 0.612805i
\(29\) − 6.46005i − 1.19960i −0.800149 − 0.599801i \(-0.795246\pi\)
0.800149 − 0.599801i \(-0.204754\pi\)
\(30\) 0.662577 + 3.09209i 0.120969 + 0.564535i
\(31\) −1.15678 − 2.00361i −0.207765 − 0.359859i 0.743245 − 0.669019i \(-0.233286\pi\)
−0.951010 + 0.309160i \(0.899952\pi\)
\(32\) 5.14129 − 2.35947i 0.908861 − 0.417099i
\(33\) −0.215464 + 0.373194i −0.0375074 + 0.0649647i
\(34\) 2.31519 + 0.800334i 0.397051 + 0.137256i
\(35\) −5.91402 + 0.156000i −0.999652 + 0.0263689i
\(36\) 1.57302 + 1.23515i 0.262170 + 0.205859i
\(37\) −3.81436 + 6.60666i −0.627077 + 1.08613i 0.361059 + 0.932543i \(0.382415\pi\)
−0.988135 + 0.153586i \(0.950918\pi\)
\(38\) 6.19669 − 1.19670i 1.00524 − 0.194130i
\(39\) 3.53091 − 2.03857i 0.565399 − 0.326433i
\(40\) −2.77695 + 5.68230i −0.439075 + 0.898451i
\(41\) 6.70434i 1.04704i 0.852013 + 0.523521i \(0.175382\pi\)
−0.852013 + 0.523521i \(0.824618\pi\)
\(42\) −2.81196 + 2.46838i −0.433895 + 0.380879i
\(43\) 4.01301i 0.611978i 0.952035 + 0.305989i \(0.0989870\pi\)
−0.952035 + 0.305989i \(0.901013\pi\)
\(44\) −0.799881 + 0.320912i −0.120587 + 0.0483793i
\(45\) −2.23561 + 0.0454322i −0.333265 + 0.00677264i
\(46\) 0.236400 + 1.22412i 0.0348552 + 0.180486i
\(47\) −9.14574 − 5.28030i −1.33404 − 0.770211i −0.348127 − 0.937447i \(-0.613182\pi\)
−0.985917 + 0.167237i \(0.946516\pi\)
\(48\) 0.948794 + 3.88584i 0.136947 + 0.560873i
\(49\) −3.57313 − 6.01937i −0.510447 − 0.859909i
\(50\) −1.62174 − 6.88258i −0.229349 − 0.973344i
\(51\) −0.866070 + 1.50008i −0.121274 + 0.210053i
\(52\) 8.07215 + 1.15450i 1.11941 + 0.160100i
\(53\) 1.73662 + 3.00792i 0.238543 + 0.413169i 0.960296 − 0.278981i \(-0.0899968\pi\)
−0.721753 + 0.692150i \(0.756663\pi\)
\(54\) −1.06845 + 0.926509i −0.145397 + 0.126082i
\(55\) 0.464737 − 0.844104i 0.0626651 − 0.113819i
\(56\) −7.47210 + 0.409610i −0.998501 + 0.0547364i
\(57\) 4.46268i 0.591097i
\(58\) 5.98530 + 6.90223i 0.785908 + 0.906307i
\(59\) −0.708208 + 0.408884i −0.0922008 + 0.0532322i −0.545391 − 0.838181i \(-0.683619\pi\)
0.453191 + 0.891414i \(0.350286\pi\)
\(60\) −3.57277 − 2.68985i −0.461243 − 0.347258i
\(61\) −6.34043 + 10.9819i −0.811808 + 1.40609i 0.0997889 + 0.995009i \(0.468183\pi\)
−0.911597 + 0.411085i \(0.865150\pi\)
\(62\) 3.09233 + 1.06898i 0.392726 + 0.135761i
\(63\) −1.33672 − 2.28324i −0.168411 − 0.287661i
\(64\) −3.30714 + 7.28442i −0.413392 + 0.910553i
\(65\) −7.80112 + 4.71787i −0.967609 + 0.585179i
\(66\) −0.115556 − 0.598367i −0.0142239 − 0.0736539i
\(67\) −7.57067 + 4.37093i −0.924904 + 0.533994i −0.885196 − 0.465218i \(-0.845976\pi\)
−0.0397079 + 0.999211i \(0.512643\pi\)
\(68\) −3.21517 + 1.28993i −0.389897 + 0.156426i
\(69\) −0.881575 −0.106129
\(70\) 6.17429 − 5.64607i 0.737969 − 0.674835i
\(71\) − 6.08373i − 0.722006i −0.932565 − 0.361003i \(-0.882434\pi\)
0.932565 − 0.361003i \(-0.117566\pi\)
\(72\) −2.82507 + 0.137721i −0.332938 + 0.0162306i
\(73\) −3.99832 − 6.92529i −0.467967 − 0.810543i 0.531363 − 0.847145i \(-0.321680\pi\)
−0.999330 + 0.0366012i \(0.988347\pi\)
\(74\) −2.04569 − 10.5929i −0.237806 − 1.23140i
\(75\) 4.99587 − 0.203137i 0.576874 − 0.0234563i
\(76\) −5.51210 + 7.01990i −0.632281 + 0.805237i
\(77\) 1.14011 − 0.00690071i 0.129927 − 0.000786409i
\(78\) −1.88384 + 5.44953i −0.213303 + 0.617038i
\(79\) 9.91239 + 5.72292i 1.11523 + 0.643879i 0.940179 − 0.340681i \(-0.110657\pi\)
0.175052 + 0.984559i \(0.443991\pi\)
\(80\) −2.29767 − 8.64411i −0.256888 − 0.966441i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −6.21163 − 7.16324i −0.685960 − 0.791047i
\(83\) −10.1969 −1.11926 −0.559630 − 0.828743i \(-0.689057\pi\)
−0.559630 + 0.828743i \(0.689057\pi\)
\(84\) 0.717461 − 5.24264i 0.0782814 − 0.572019i
\(85\) 1.86804 − 3.39293i 0.202617 − 0.368015i
\(86\) −3.71808 − 4.28769i −0.400931 − 0.462353i
\(87\) −5.59457 + 3.23003i −0.599801 + 0.346295i
\(88\) 0.557304 − 1.08397i 0.0594088 − 0.115552i
\(89\) 1.68998 + 0.975709i 0.179137 + 0.103425i 0.586887 − 0.809669i \(-0.300353\pi\)
−0.407750 + 0.913094i \(0.633686\pi\)
\(90\) 2.34654 − 2.11985i 0.247347 − 0.223452i
\(91\) −9.37439 − 5.33692i −0.982703 − 0.559461i
\(92\) −1.38674 − 1.08888i −0.144577 − 0.113524i
\(93\) −1.15678 + 2.00361i −0.119953 + 0.207765i
\(94\) 14.6640 − 2.83189i 1.51248 − 0.292087i
\(95\) −0.202750 − 9.97680i −0.0208017 − 1.02360i
\(96\) −4.61401 − 3.27276i −0.470915 − 0.334024i
\(97\) 4.18222 0.424640 0.212320 − 0.977200i \(-0.431898\pi\)
0.212320 + 0.977200i \(0.431898\pi\)
\(98\) 9.39470 + 3.12085i 0.949008 + 0.315253i
\(99\) 0.430927 0.0433098
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 840.2.bz.a.19.13 ✓ 96
5.4 even 2 840.2.bz.b.19.36 yes 96
7.3 odd 6 840.2.bz.b.619.21 yes 96
8.3 odd 2 inner 840.2.bz.a.19.28 yes 96
35.24 odd 6 inner 840.2.bz.a.619.28 yes 96
40.19 odd 2 840.2.bz.b.19.21 yes 96
56.3 even 6 840.2.bz.b.619.36 yes 96
280.59 even 6 inner 840.2.bz.a.619.13 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bz.a.19.13 ✓ 96 1.1 even 1 trivial
840.2.bz.a.19.28 yes 96 8.3 odd 2 inner
840.2.bz.a.619.13 yes 96 280.59 even 6 inner
840.2.bz.a.619.28 yes 96 35.24 odd 6 inner
840.2.bz.b.19.21 yes 96 40.19 odd 2
840.2.bz.b.19.36 yes 96 5.4 even 2
840.2.bz.b.619.21 yes 96 7.3 odd 6
840.2.bz.b.619.36 yes 96 56.3 even 6