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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [420,2,Mod(31,420)] 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("420.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bi (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 420.271
Dual form 420.2.bi.a.31.1

$q$-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+(-1.00000 + 1.00000i) q^{2} +(0.500000 + 0.866025i) q^{3} -2.00000i q^{4} +(-0.866025 - 0.500000i) q^{5} +(-1.36603 - 0.366025i) q^{6} +(-0.500000 - 2.59808i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-1.73205 + 1.00000i) q^{11} +(1.73205 - 1.00000i) q^{12} -3.73205i q^{13} +(3.09808 + 2.09808i) q^{14} -1.00000i q^{15} -4.00000 q^{16} +(6.00000 - 3.46410i) q^{17} +(-0.366025 - 1.36603i) q^{18} +(2.23205 - 3.86603i) q^{19} +(-1.00000 + 1.73205i) q^{20} +(2.00000 - 1.73205i) q^{21} +(0.732051 - 2.73205i) q^{22} +(-6.92820 - 4.00000i) q^{23} +(-0.732051 + 2.73205i) q^{24} +(0.500000 + 0.866025i) q^{25} +(3.73205 + 3.73205i) q^{26} -1.00000 q^{27} +(-5.19615 + 1.00000i) q^{28} +9.46410 q^{29} +(1.00000 + 1.00000i) q^{30} +(-0.232051 - 0.401924i) q^{31} +(4.00000 - 4.00000i) q^{32} +(-1.73205 - 1.00000i) q^{33} +(-2.53590 + 9.46410i) q^{34} +(-0.866025 + 2.50000i) q^{35} +(1.73205 + 1.00000i) q^{36} +(2.23205 - 3.86603i) q^{37} +(1.63397 + 6.09808i) q^{38} +(3.23205 - 1.86603i) q^{39} +(-0.732051 - 2.73205i) q^{40} +6.00000i q^{41} +(-0.267949 + 3.73205i) q^{42} -1.19615i q^{43} +(2.00000 + 3.46410i) q^{44} +(0.866025 - 0.500000i) q^{45} +(10.9282 - 2.92820i) q^{46} +(-3.73205 + 6.46410i) q^{47} +(-2.00000 - 3.46410i) q^{48} +(-6.50000 + 2.59808i) q^{49} +(-1.36603 - 0.366025i) q^{50} +(6.00000 + 3.46410i) q^{51} -7.46410 q^{52} +(-4.73205 - 8.19615i) q^{53} +(1.00000 - 1.00000i) q^{54} +2.00000 q^{55} +(4.19615 - 6.19615i) q^{56} +4.46410 q^{57} +(-9.46410 + 9.46410i) q^{58} +(5.46410 + 9.46410i) q^{59} -2.00000 q^{60} +(-2.53590 - 1.46410i) q^{61} +(0.633975 + 0.169873i) q^{62} +(2.50000 + 0.866025i) q^{63} +8.00000i q^{64} +(-1.86603 + 3.23205i) q^{65} +(2.73205 - 0.732051i) q^{66} +(7.96410 - 4.59808i) q^{67} +(-6.92820 - 12.0000i) q^{68} -8.00000i q^{69} +(-1.63397 - 3.36603i) q^{70} -8.53590i q^{71} +(-2.73205 + 0.732051i) q^{72} +(6.23205 - 3.59808i) q^{73} +(1.63397 + 6.09808i) q^{74} +(-0.500000 + 0.866025i) q^{75} +(-7.73205 - 4.46410i) q^{76} +(3.46410 + 4.00000i) q^{77} +(-1.36603 + 5.09808i) q^{78} +(-6.69615 - 3.86603i) q^{79} +(3.46410 + 2.00000i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-6.00000 - 6.00000i) q^{82} -2.00000 q^{83} +(-3.46410 - 4.00000i) q^{84} -6.92820 q^{85} +(1.19615 + 1.19615i) q^{86} +(4.73205 + 8.19615i) q^{87} +(-5.46410 - 1.46410i) q^{88} +(-3.80385 - 2.19615i) q^{89} +(-0.366025 + 1.36603i) q^{90} +(-9.69615 + 1.86603i) q^{91} +(-8.00000 + 13.8564i) q^{92} +(0.232051 - 0.401924i) q^{93} +(-2.73205 - 10.1962i) q^{94} +(-3.86603 + 2.23205i) q^{95} +(5.46410 + 1.46410i) q^{96} -1.07180i q^{97} +(3.90192 - 9.09808i) q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 2 q^{3} - 2 q^{6} - 2 q^{7} + 8 q^{8} - 2 q^{9} + 2 q^{10} + 2 q^{14} - 16 q^{16} + 24 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{20} + 8 q^{21} - 4 q^{22} + 4 q^{24} + 2 q^{25} + 8 q^{26} - 4 q^{27}+ \cdots + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.00000i −0.707107 + 0.707107i
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) − 2.00000i − 1.00000i
\(5\) −0.866025 − 0.500000i −0.387298 − 0.223607i
\(6\) −1.36603 − 0.366025i −0.557678 − 0.149429i
\(7\) −0.500000 − 2.59808i −0.188982 − 0.981981i
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 1.36603 − 0.366025i 0.431975 − 0.115747i
\(11\) −1.73205 + 1.00000i −0.522233 + 0.301511i −0.737848 − 0.674967i \(-0.764158\pi\)
0.215615 + 0.976478i \(0.430824\pi\)
\(12\) 1.73205 − 1.00000i 0.500000 − 0.288675i
\(13\) − 3.73205i − 1.03508i −0.855658 − 0.517542i \(-0.826847\pi\)
0.855658 − 0.517542i \(-0.173153\pi\)
\(14\) 3.09808 + 2.09808i 0.827996 + 0.560734i
\(15\) − 1.00000i − 0.258199i
\(16\) −4.00000 −1.00000
\(17\) 6.00000 − 3.46410i 1.45521 − 0.840168i 0.456444 − 0.889752i \(-0.349123\pi\)
0.998770 + 0.0495842i \(0.0157896\pi\)
\(18\) −0.366025 − 1.36603i −0.0862730 − 0.321975i
\(19\) 2.23205 − 3.86603i 0.512068 − 0.886927i −0.487835 − 0.872936i \(-0.662213\pi\)
0.999902 − 0.0139909i \(-0.00445360\pi\)
\(20\) −1.00000 + 1.73205i −0.223607 + 0.387298i
\(21\) 2.00000 − 1.73205i 0.436436 − 0.377964i
\(22\) 0.732051 − 2.73205i 0.156074 − 0.582475i
\(23\) −6.92820 − 4.00000i −1.44463 − 0.834058i −0.446476 − 0.894795i \(-0.647321\pi\)
−0.998154 + 0.0607377i \(0.980655\pi\)
\(24\) −0.732051 + 2.73205i −0.149429 + 0.557678i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 3.73205 + 3.73205i 0.731915 + 0.731915i
\(27\) −1.00000 −0.192450
\(28\) −5.19615 + 1.00000i −0.981981 + 0.188982i
\(29\) 9.46410 1.75744 0.878720 − 0.477338i \(-0.158398\pi\)
0.878720 + 0.477338i \(0.158398\pi\)
\(30\) 1.00000 + 1.00000i 0.182574 + 0.182574i
\(31\) −0.232051 − 0.401924i −0.0416776 − 0.0721876i 0.844434 − 0.535659i \(-0.179937\pi\)
−0.886112 + 0.463472i \(0.846604\pi\)
\(32\) 4.00000 − 4.00000i 0.707107 − 0.707107i
\(33\) −1.73205 − 1.00000i −0.301511 − 0.174078i
\(34\) −2.53590 + 9.46410i −0.434903 + 1.62308i
\(35\) −0.866025 + 2.50000i −0.146385 + 0.422577i
\(36\) 1.73205 + 1.00000i 0.288675 + 0.166667i
\(37\) 2.23205 − 3.86603i 0.366947 − 0.635571i −0.622140 − 0.782906i \(-0.713736\pi\)
0.989087 + 0.147336i \(0.0470697\pi\)
\(38\) 1.63397 + 6.09808i 0.265066 + 0.989239i
\(39\) 3.23205 − 1.86603i 0.517542 − 0.298803i
\(40\) −0.732051 − 2.73205i −0.115747 − 0.431975i
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) −0.267949 + 3.73205i −0.0413455 + 0.575868i
\(43\) − 1.19615i − 0.182412i −0.995832 − 0.0912058i \(-0.970928\pi\)
0.995832 − 0.0912058i \(-0.0290721\pi\)
\(44\) 2.00000 + 3.46410i 0.301511 + 0.522233i
\(45\) 0.866025 − 0.500000i 0.129099 − 0.0745356i
\(46\) 10.9282 − 2.92820i 1.61128 − 0.431740i
\(47\) −3.73205 + 6.46410i −0.544376 + 0.942886i 0.454270 + 0.890864i \(0.349900\pi\)
−0.998646 + 0.0520223i \(0.983433\pi\)
\(48\) −2.00000 − 3.46410i −0.288675 − 0.500000i
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) −1.36603 − 0.366025i −0.193185 − 0.0517638i
\(51\) 6.00000 + 3.46410i 0.840168 + 0.485071i
\(52\) −7.46410 −1.03508
\(53\) −4.73205 − 8.19615i −0.649997 − 1.12583i −0.983123 − 0.182946i \(-0.941437\pi\)
0.333126 − 0.942882i \(-0.391897\pi\)
\(54\) 1.00000 − 1.00000i 0.136083 − 0.136083i
\(55\) 2.00000 0.269680
\(56\) 4.19615 − 6.19615i 0.560734 − 0.827996i
\(57\) 4.46410 0.591285
\(58\) −9.46410 + 9.46410i −1.24270 + 1.24270i
\(59\) 5.46410 + 9.46410i 0.711365 + 1.23212i 0.964345 + 0.264649i \(0.0852562\pi\)
−0.252979 + 0.967472i \(0.581410\pi\)
\(60\) −2.00000 −0.258199
\(61\) −2.53590 − 1.46410i −0.324689 − 0.187459i 0.328792 − 0.944402i \(-0.393359\pi\)
−0.653480 + 0.756943i \(0.726692\pi\)
\(62\) 0.633975 + 0.169873i 0.0805149 + 0.0215739i
\(63\) 2.50000 + 0.866025i 0.314970 + 0.109109i
\(64\) 8.00000i 1.00000i
\(65\) −1.86603 + 3.23205i −0.231452 + 0.400887i
\(66\) 2.73205 − 0.732051i 0.336292 − 0.0901092i
\(67\) 7.96410 − 4.59808i 0.972970 − 0.561744i 0.0728295 − 0.997344i \(-0.476797\pi\)
0.900140 + 0.435600i \(0.143464\pi\)
\(68\) −6.92820 − 12.0000i −0.840168 − 1.45521i
\(69\) − 8.00000i − 0.963087i
\(70\) −1.63397 − 3.36603i −0.195297 − 0.402317i
\(71\) − 8.53590i − 1.01302i −0.862233 − 0.506512i \(-0.830934\pi\)
0.862233 − 0.506512i \(-0.169066\pi\)
\(72\) −2.73205 + 0.732051i −0.321975 + 0.0862730i
\(73\) 6.23205 − 3.59808i 0.729406 − 0.421123i −0.0887986 − 0.996050i \(-0.528303\pi\)
0.818205 + 0.574927i \(0.194969\pi\)
\(74\) 1.63397 + 6.09808i 0.189946 + 0.708887i
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) −7.73205 − 4.46410i −0.886927 − 0.512068i
\(77\) 3.46410 + 4.00000i 0.394771 + 0.455842i
\(78\) −1.36603 + 5.09808i −0.154672 + 0.577243i
\(79\) −6.69615 − 3.86603i −0.753376 − 0.434962i 0.0735364 − 0.997293i \(-0.476571\pi\)
−0.826912 + 0.562331i \(0.809905\pi\)
\(80\) 3.46410 + 2.00000i 0.387298 + 0.223607i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −6.00000 − 6.00000i −0.662589 − 0.662589i
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) −3.46410 − 4.00000i −0.377964 − 0.436436i
\(85\) −6.92820 −0.751469
\(86\) 1.19615 + 1.19615i 0.128984 + 0.128984i
\(87\) 4.73205 + 8.19615i 0.507329 + 0.878720i
\(88\) −5.46410 − 1.46410i −0.582475 − 0.156074i
\(89\) −3.80385 − 2.19615i −0.403207 − 0.232792i 0.284660 − 0.958629i \(-0.408119\pi\)
−0.687867 + 0.725837i \(0.741453\pi\)
\(90\) −0.366025 + 1.36603i −0.0385825 + 0.143992i
\(91\) −9.69615 + 1.86603i −1.01643 + 0.195613i
\(92\) −8.00000 + 13.8564i −0.834058 + 1.44463i
\(93\) 0.232051 − 0.401924i 0.0240625 − 0.0416776i
\(94\) −2.73205 − 10.1962i −0.281790 − 1.05165i
\(95\) −3.86603 + 2.23205i −0.396646 + 0.229004i
\(96\) 5.46410 + 1.46410i 0.557678 + 0.149429i
\(97\) − 1.07180i − 0.108824i −0.998519 − 0.0544122i \(-0.982671\pi\)
0.998519 − 0.0544122i \(-0.0173285\pi\)
\(98\) 3.90192 − 9.09808i 0.394154 − 0.919044i
\(99\) − 2.00000i − 0.201008i
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.bi.a.271.2 yes 4
4.3 odd 2 420.2.bi.b.271.1 yes 4
7.3 odd 6 420.2.bi.b.31.1 yes 4
28.3 even 6 inner 420.2.bi.a.31.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bi.a.31.1 ✓ 4 28.3 even 6 inner
420.2.bi.a.271.2 yes 4 1.1 even 1 trivial
420.2.bi.b.31.1 yes 4 7.3 odd 6
420.2.bi.b.271.1 yes 4 4.3 odd 2