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Properties

Label 420.2.l.c.239.6
Level $420$
Weight $2$
Character 420.239
Analytic conductor $3.354$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [420,2,Mod(239,420)] 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("420.239"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-2,2,-8,4,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.386672896.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.6
Root \(0.621372 - 1.27039i\) of defining polynomial
Character \(\chi\) \(=\) 420.239
Dual form 420.2.l.c.239.5

$q$-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+(0.621372 + 1.27039i) q^{2} +(-1.72779 - 0.121372i) q^{3} +(-1.22779 + 1.57877i) q^{4} +(-1.00000 - 2.00000i) q^{5} +(-0.919412 - 2.27039i) q^{6} -1.00000 q^{7} +(-2.76858 - 0.578773i) q^{8} +(2.97054 + 0.419412i) q^{9} +(1.91941 - 2.51314i) q^{10} +3.45559 q^{11} +(2.31299 - 2.57877i) q^{12} -4.83882i q^{13} +(-0.621372 - 1.27039i) q^{14} +(1.48505 + 3.57696i) q^{15} +(-0.985049 - 3.87681i) q^{16} +5.94108 q^{17} +(1.31299 + 4.03436i) q^{18} -1.08157i q^{19} +(4.38534 + 0.876813i) q^{20} +(1.72779 + 0.121372i) q^{21} +(2.14721 + 4.38995i) q^{22} -0.596080i q^{23} +(4.71328 + 1.33603i) q^{24} +(-3.00000 + 4.00000i) q^{25} +(6.14721 - 3.00671i) q^{26} +(-5.08157 - 1.08520i) q^{27} +(1.22779 - 1.57877i) q^{28} -4.83882i q^{29} +(-3.62137 + 4.10922i) q^{30} -9.56706i q^{31} +(4.31299 - 3.66034i) q^{32} +(-5.97054 - 0.419412i) q^{33} +(3.69162 + 7.54750i) q^{34} +(1.00000 + 2.00000i) q^{35} +(-4.30936 + 4.17485i) q^{36} -2.91117i q^{37} +(1.37402 - 0.672057i) q^{38} +(-0.587299 + 8.36049i) q^{39} +(1.61103 + 6.11593i) q^{40} +6.91117i q^{41} +(0.919412 + 2.27039i) q^{42} -7.39666 q^{43} +(-4.24274 + 5.45559i) q^{44} +(-2.13171 - 6.36049i) q^{45} +(0.757255 - 0.370388i) q^{46} +0.242745i q^{47} +(1.23142 + 6.81789i) q^{48} +1.00000 q^{49} +(-6.94569 - 1.32569i) q^{50} +(-10.2649 - 0.721082i) q^{51} +(7.63941 + 5.94108i) q^{52} +11.8223 q^{53} +(-1.77892 - 7.12990i) q^{54} +(-3.45559 - 6.91117i) q^{55} +(2.76858 + 0.578773i) q^{56} +(-0.131272 + 1.86873i) q^{57} +(6.14721 - 3.00671i) q^{58} +3.25197 q^{59} +(-7.47054 - 2.04721i) q^{60} -12.6486 q^{61} +(12.1539 - 5.94470i) q^{62} +(-2.97054 - 0.419412i) q^{63} +(7.33004 + 3.20476i) q^{64} +(-9.67765 + 4.83882i) q^{65} +(-3.17711 - 7.84554i) q^{66} +5.71901 q^{67} +(-7.29441 + 9.37961i) q^{68} +(-0.0723476 + 1.02990i) q^{69} +(-1.91941 + 2.51314i) q^{70} -8.00000 q^{71} +(-7.98142 - 2.88044i) q^{72} -8.00000i q^{73} +(3.69833 - 1.80892i) q^{74} +(5.66887 - 6.54706i) q^{75} +(1.70755 + 1.32794i) q^{76} -3.45559 q^{77} +(-10.9860 + 4.44887i) q^{78} +0.949416i q^{79} +(-6.76858 + 5.84691i) q^{80} +(8.64819 + 2.49176i) q^{81} +(-8.77990 + 4.29441i) q^{82} -16.9637i q^{83} +(-2.31299 + 2.57877i) q^{84} +(-5.94108 - 11.8822i) q^{85} +(-4.59608 - 9.39666i) q^{86} +(-0.587299 + 8.36049i) q^{87} +(-9.56706 - 2.00000i) q^{88} +5.23352i q^{89} +(6.75573 - 6.66034i) q^{90} +4.83882i q^{91} +(0.941075 + 0.731863i) q^{92} +(-1.16118 + 16.5299i) q^{93} +(-0.308381 + 0.150835i) q^{94} +(-2.16314 + 1.08157i) q^{95} +(-7.89622 + 5.80084i) q^{96} +3.30587i q^{97} +(0.621372 + 1.27039i) q^{98} +(10.2649 + 1.44932i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 2 q^{4} - 8 q^{5} + 4 q^{6} - 8 q^{7} + 6 q^{8} + 2 q^{9} + 4 q^{10} + 4 q^{11} + 14 q^{12} + 10 q^{15} - 6 q^{16} + 4 q^{17} + 6 q^{18} + 6 q^{20} + 2 q^{21} - 6 q^{22} + 6 q^{24} - 24 q^{25}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.621372 + 1.27039i 0.439377 + 0.898303i
\(3\) −1.72779 − 0.121372i −0.997542 − 0.0700743i
\(4\) −1.22779 + 1.57877i −0.613897 + 0.789387i
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) −0.919412 − 2.27039i −0.375348 − 0.926884i
\(7\) −1.00000 −0.377964
\(8\) −2.76858 − 0.578773i −0.978840 − 0.204627i
\(9\) 2.97054 + 0.419412i 0.990179 + 0.139804i
\(10\) 1.91941 − 2.51314i 0.606971 − 0.794724i
\(11\) 3.45559 1.04190 0.520949 − 0.853588i \(-0.325578\pi\)
0.520949 + 0.853588i \(0.325578\pi\)
\(12\) 2.31299 − 2.57877i 0.667703 − 0.744428i
\(13\) − 4.83882i − 1.34205i −0.741435 − 0.671024i \(-0.765855\pi\)
0.741435 − 0.671024i \(-0.234145\pi\)
\(14\) −0.621372 − 1.27039i −0.166069 − 0.339527i
\(15\) 1.48505 + 3.57696i 0.383438 + 0.923567i
\(16\) −0.985049 − 3.87681i −0.246262 − 0.969203i
\(17\) 5.94108 1.44092 0.720461 − 0.693495i \(-0.243930\pi\)
0.720461 + 0.693495i \(0.243930\pi\)
\(18\) 1.31299 + 4.03436i 0.309475 + 0.950908i
\(19\) − 1.08157i − 0.248129i −0.992274 − 0.124064i \(-0.960407\pi\)
0.992274 − 0.124064i \(-0.0395930\pi\)
\(20\) 4.38534 + 0.876813i 0.980592 + 0.196061i
\(21\) 1.72779 + 0.121372i 0.377035 + 0.0264856i
\(22\) 2.14721 + 4.38995i 0.457786 + 0.935940i
\(23\) − 0.596080i − 0.124291i −0.998067 − 0.0621456i \(-0.980206\pi\)
0.998067 − 0.0621456i \(-0.0197943\pi\)
\(24\) 4.71328 + 1.33603i 0.962095 + 0.272716i
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 6.14721 − 3.00671i 1.20557 − 0.589665i
\(27\) −5.08157 − 1.08520i −0.977948 − 0.208847i
\(28\) 1.22779 − 1.57877i 0.232031 − 0.298360i
\(29\) − 4.83882i − 0.898547i −0.893394 − 0.449274i \(-0.851683\pi\)
0.893394 − 0.449274i \(-0.148317\pi\)
\(30\) −3.62137 + 4.10922i −0.661169 + 0.750237i
\(31\) − 9.56706i − 1.71829i −0.511729 − 0.859147i \(-0.670995\pi\)
0.511729 − 0.859147i \(-0.329005\pi\)
\(32\) 4.31299 − 3.66034i 0.762436 − 0.647063i
\(33\) −5.97054 − 0.419412i −1.03934 − 0.0730103i
\(34\) 3.69162 + 7.54750i 0.633107 + 1.29438i
\(35\) 1.00000 + 2.00000i 0.169031 + 0.338062i
\(36\) −4.30936 + 4.17485i −0.718227 + 0.695809i
\(37\) − 2.91117i − 0.478594i −0.970946 − 0.239297i \(-0.923083\pi\)
0.970946 − 0.239297i \(-0.0769169\pi\)
\(38\) 1.37402 − 0.672057i 0.222895 − 0.109022i
\(39\) −0.587299 + 8.36049i −0.0940431 + 1.33875i
\(40\) 1.61103 + 6.11593i 0.254726 + 0.967013i
\(41\) 6.91117i 1.07934i 0.841875 + 0.539672i \(0.181452\pi\)
−0.841875 + 0.539672i \(0.818548\pi\)
\(42\) 0.919412 + 2.27039i 0.141868 + 0.350329i
\(43\) −7.39666 −1.12798 −0.563990 − 0.825782i \(-0.690734\pi\)
−0.563990 + 0.825782i \(0.690734\pi\)
\(44\) −4.24274 + 5.45559i −0.639618 + 0.822461i
\(45\) −2.13171 − 6.36049i −0.317777 − 0.948165i
\(46\) 0.757255 − 0.370388i 0.111651 − 0.0546107i
\(47\) 0.242745i 0.0354079i 0.999843 + 0.0177040i \(0.00563564\pi\)
−0.999843 + 0.0177040i \(0.994364\pi\)
\(48\) 1.23142 + 6.81789i 0.177741 + 0.984077i
\(49\) 1.00000 0.142857
\(50\) −6.94569 − 1.32569i −0.982268 − 0.187481i
\(51\) −10.2649 − 0.721082i −1.43738 − 0.100972i
\(52\) 7.63941 + 5.94108i 1.05939 + 0.823879i
\(53\) 11.8223 1.62392 0.811962 − 0.583710i \(-0.198400\pi\)
0.811962 + 0.583710i \(0.198400\pi\)
\(54\) −1.77892 − 7.12990i −0.242080 − 0.970256i
\(55\) −3.45559 − 6.91117i −0.465951 − 0.931902i
\(56\) 2.76858 + 0.578773i 0.369967 + 0.0773418i
\(57\) −0.131272 + 1.86873i −0.0173875 + 0.247519i
\(58\) 6.14721 − 3.00671i 0.807168 − 0.394801i
\(59\) 3.25197 0.423370 0.211685 − 0.977338i \(-0.432105\pi\)
0.211685 + 0.977338i \(0.432105\pi\)
\(60\) −7.47054 − 2.04721i −0.964442 − 0.264294i
\(61\) −12.6486 −1.61949 −0.809745 − 0.586781i \(-0.800395\pi\)
−0.809745 + 0.586781i \(0.800395\pi\)
\(62\) 12.1539 − 5.94470i 1.54355 − 0.754978i
\(63\) −2.97054 − 0.419412i −0.374253 − 0.0528410i
\(64\) 7.33004 + 3.20476i 0.916255 + 0.400595i
\(65\) −9.67765 + 4.83882i −1.20036 + 0.600182i
\(66\) −3.17711 − 7.84554i −0.391075 − 0.965719i
\(67\) 5.71901 0.698689 0.349344 − 0.936994i \(-0.386404\pi\)
0.349344 + 0.936994i \(0.386404\pi\)
\(68\) −7.29441 + 9.37961i −0.884577 + 1.13744i
\(69\) −0.0723476 + 1.02990i −0.00870963 + 0.123986i
\(70\) −1.91941 + 2.51314i −0.229414 + 0.300377i
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −7.98142 − 2.88044i −0.940619 − 0.339463i
\(73\) − 8.00000i − 0.936329i −0.883641 − 0.468165i \(-0.844915\pi\)
0.883641 − 0.468165i \(-0.155085\pi\)
\(74\) 3.69833 − 1.80892i 0.429922 − 0.210283i
\(75\) 5.66887 − 6.54706i 0.654585 − 0.755989i
\(76\) 1.70755 + 1.32794i 0.195870 + 0.152326i
\(77\) −3.45559 −0.393801
\(78\) −10.9860 + 4.44887i −1.24392 + 0.503736i
\(79\) 0.949416i 0.106818i 0.998573 + 0.0534088i \(0.0170086\pi\)
−0.998573 + 0.0534088i \(0.982991\pi\)
\(80\) −6.76858 + 5.84691i −0.756750 + 0.653704i
\(81\) 8.64819 + 2.49176i 0.960910 + 0.276862i
\(82\) −8.77990 + 4.29441i −0.969578 + 0.474238i
\(83\) − 16.9637i − 1.86201i −0.365006 − 0.931005i \(-0.618933\pi\)
0.365006 − 0.931005i \(-0.381067\pi\)
\(84\) −2.31299 + 2.57877i −0.252368 + 0.281367i
\(85\) −5.94108 − 11.8822i −0.644400 − 1.28880i
\(86\) −4.59608 − 9.39666i −0.495608 − 1.01327i
\(87\) −0.587299 + 8.36049i −0.0629651 + 0.896338i
\(88\) −9.56706 − 2.00000i −1.01985 − 0.213201i
\(89\) 5.23352i 0.554752i 0.960761 + 0.277376i \(0.0894648\pi\)
−0.960761 + 0.277376i \(0.910535\pi\)
\(90\) 6.75573 − 6.66034i 0.712116 − 0.702062i
\(91\) 4.83882i 0.507247i
\(92\) 0.941075 + 0.731863i 0.0981139 + 0.0763020i
\(93\) −1.16118 + 16.5299i −0.120408 + 1.71407i
\(94\) −0.308381 + 0.150835i −0.0318070 + 0.0155574i
\(95\) −2.16314 + 1.08157i −0.221933 + 0.110967i
\(96\) −7.89622 + 5.80084i −0.805905 + 0.592045i
\(97\) 3.30587i 0.335660i 0.985816 + 0.167830i \(0.0536760\pi\)
−0.985816 + 0.167830i \(0.946324\pi\)
\(98\) 0.621372 + 1.27039i 0.0627681 + 0.128329i
\(99\) 10.2649 + 1.44932i 1.03167 + 0.145662i
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 420.2.l.c.239.6 yes 8
3.2 odd 2 420.2.l.d.239.3 yes 8
4.3 odd 2 420.2.l.e.239.5 yes 8
5.4 even 2 420.2.l.f.239.3 yes 8
12.11 even 2 420.2.l.f.239.4 yes 8
15.14 odd 2 420.2.l.e.239.6 yes 8
20.19 odd 2 420.2.l.d.239.4 yes 8
60.59 even 2 inner 420.2.l.c.239.5 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.l.c.239.5 ✓ 8 60.59 even 2 inner
420.2.l.c.239.6 yes 8 1.1 even 1 trivial
420.2.l.d.239.3 yes 8 3.2 odd 2
420.2.l.d.239.4 yes 8 20.19 odd 2
420.2.l.e.239.5 yes 8 4.3 odd 2
420.2.l.e.239.6 yes 8 15.14 odd 2
420.2.l.f.239.3 yes 8 5.4 even 2
420.2.l.f.239.4 yes 8 12.11 even 2