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(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 31.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 420.31
Dual form 420.2.bi.a.271.1

$q$-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} +(0.366025 + 1.36603i) q^{6} +(-0.500000 + 2.59808i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.366025 + 1.36603i) q^{10} +(1.73205 + 1.00000i) q^{11} +(-1.73205 - 1.00000i) q^{12} -0.267949i q^{13} +(-2.09808 - 3.09808i) q^{14} -1.00000i q^{15} -4.00000 q^{16} +(6.00000 + 3.46410i) q^{17} +(1.36603 + 0.366025i) q^{18} +(-1.23205 - 2.13397i) q^{19} +(-1.00000 - 1.73205i) q^{20} +(2.00000 + 1.73205i) q^{21} +(-2.73205 + 0.732051i) q^{22} +(6.92820 - 4.00000i) q^{23} +(2.73205 - 0.732051i) q^{24} +(0.500000 - 0.866025i) q^{25} +(0.267949 + 0.267949i) q^{26} -1.00000 q^{27} +(5.19615 + 1.00000i) q^{28} +2.53590 q^{29} +(1.00000 + 1.00000i) q^{30} +(3.23205 - 5.59808i) q^{31} +(4.00000 - 4.00000i) q^{32} +(1.73205 - 1.00000i) q^{33} +(-9.46410 + 2.53590i) q^{34} +(0.866025 + 2.50000i) q^{35} +(-1.73205 + 1.00000i) q^{36} +(-1.23205 - 2.13397i) q^{37} +(3.36603 + 0.901924i) q^{38} +(-0.232051 - 0.133975i) q^{39} +(2.73205 + 0.732051i) q^{40} +6.00000i q^{41} +(-3.73205 + 0.267949i) q^{42} +9.19615i q^{43} +(2.00000 - 3.46410i) q^{44} +(-0.866025 - 0.500000i) q^{45} +(-2.92820 + 10.9282i) q^{46} +(-0.267949 - 0.464102i) q^{47} +(-2.00000 + 3.46410i) q^{48} +(-6.50000 - 2.59808i) q^{49} +(0.366025 + 1.36603i) q^{50} +(6.00000 - 3.46410i) q^{51} -0.535898 q^{52} +(-1.26795 + 2.19615i) q^{53} +(1.00000 - 1.00000i) q^{54} +2.00000 q^{55} +(-6.19615 + 4.19615i) q^{56} -2.46410 q^{57} +(-2.53590 + 2.53590i) q^{58} +(-1.46410 + 2.53590i) q^{59} -2.00000 q^{60} +(-9.46410 + 5.46410i) q^{61} +(2.36603 + 8.83013i) q^{62} +(2.50000 - 0.866025i) q^{63} +8.00000i q^{64} +(-0.133975 - 0.232051i) q^{65} +(-0.732051 + 2.73205i) q^{66} +(1.03590 + 0.598076i) q^{67} +(6.92820 - 12.0000i) q^{68} -8.00000i q^{69} +(-3.36603 - 1.63397i) q^{70} -15.4641i q^{71} +(0.732051 - 2.73205i) q^{72} +(2.76795 + 1.59808i) q^{73} +(3.36603 + 0.901924i) q^{74} +(-0.500000 - 0.866025i) q^{75} +(-4.26795 + 2.46410i) q^{76} +(-3.46410 + 4.00000i) q^{77} +(0.366025 - 0.0980762i) q^{78} +(3.69615 - 2.13397i) 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} +(-9.19615 - 9.19615i) q^{86} +(1.26795 - 2.19615i) q^{87} +(1.46410 + 5.46410i) q^{88} +(-14.1962 + 8.19615i) q^{89} +(1.36603 - 0.366025i) q^{90} +(0.696152 + 0.133975i) q^{91} +(-8.00000 - 13.8564i) q^{92} +(-3.23205 - 5.59808i) q^{93} +(0.732051 + 0.196152i) q^{94} +(-2.13397 - 1.23205i) q^{95} +(-1.46410 - 5.46410i) q^{96} -14.9282i q^{97} +(9.09808 - 3.90192i) 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 \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

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

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
\(6\) 0.366025 + 1.36603i 0.149429 + 0.557678i
\(7\) −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
\(10\) −0.366025 + 1.36603i −0.115747 + 0.431975i
\(11\) 1.73205 + 1.00000i 0.522233 + 0.301511i 0.737848 − 0.674967i \(-0.235842\pi\)
−0.215615 + 0.976478i \(0.569176\pi\)
\(12\) −1.73205 − 1.00000i −0.500000 − 0.288675i
\(13\) − 0.267949i − 0.0743157i −0.999309 − 0.0371579i \(-0.988170\pi\)
0.999309 − 0.0371579i \(-0.0118304\pi\)
\(14\) −2.09808 − 3.09808i −0.560734 − 0.827996i
\(15\) − 1.00000i − 0.258199i
\(16\) −4.00000 −1.00000
\(17\) 6.00000 + 3.46410i 1.45521 + 0.840168i 0.998770 − 0.0495842i \(-0.0157896\pi\)
0.456444 + 0.889752i \(0.349123\pi\)
\(18\) 1.36603 + 0.366025i 0.321975 + 0.0862730i
\(19\) −1.23205 − 2.13397i −0.282652 − 0.489567i 0.689385 − 0.724395i \(-0.257881\pi\)
−0.972037 + 0.234828i \(0.924547\pi\)
\(20\) −1.00000 − 1.73205i −0.223607 − 0.387298i
\(21\) 2.00000 + 1.73205i 0.436436 + 0.377964i
\(22\) −2.73205 + 0.732051i −0.582475 + 0.156074i
\(23\) 6.92820 − 4.00000i 1.44463 − 0.834058i 0.446476 − 0.894795i \(-0.352679\pi\)
0.998154 + 0.0607377i \(0.0193453\pi\)
\(24\) 2.73205 − 0.732051i 0.557678 − 0.149429i
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 0.267949 + 0.267949i 0.0525492 + 0.0525492i
\(27\) −1.00000 −0.192450
\(28\) 5.19615 + 1.00000i 0.981981 + 0.188982i
\(29\) 2.53590 0.470905 0.235452 − 0.971886i \(-0.424343\pi\)
0.235452 + 0.971886i \(0.424343\pi\)
\(30\) 1.00000 + 1.00000i 0.182574 + 0.182574i
\(31\) 3.23205 − 5.59808i 0.580493 − 1.00544i −0.414927 − 0.909855i \(-0.636193\pi\)
0.995421 − 0.0955896i \(-0.0304737\pi\)
\(32\) 4.00000 − 4.00000i 0.707107 − 0.707107i
\(33\) 1.73205 − 1.00000i 0.301511 − 0.174078i
\(34\) −9.46410 + 2.53590i −1.62308 + 0.434903i
\(35\) 0.866025 + 2.50000i 0.146385 + 0.422577i
\(36\) −1.73205 + 1.00000i −0.288675 + 0.166667i
\(37\) −1.23205 − 2.13397i −0.202548 − 0.350823i 0.746801 − 0.665048i \(-0.231589\pi\)
−0.949349 + 0.314225i \(0.898256\pi\)
\(38\) 3.36603 + 0.901924i 0.546041 + 0.146311i
\(39\) −0.232051 − 0.133975i −0.0371579 − 0.0214531i
\(40\) 2.73205 + 0.732051i 0.431975 + 0.115747i
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) −3.73205 + 0.267949i −0.575868 + 0.0413455i
\(43\) 9.19615i 1.40240i 0.712965 + 0.701200i \(0.247352\pi\)
−0.712965 + 0.701200i \(0.752648\pi\)
\(44\) 2.00000 − 3.46410i 0.301511 − 0.522233i
\(45\) −0.866025 − 0.500000i −0.129099 − 0.0745356i
\(46\) −2.92820 + 10.9282i −0.431740 + 1.61128i
\(47\) −0.267949 − 0.464102i −0.0390844 − 0.0676962i 0.845821 − 0.533466i \(-0.179111\pi\)
−0.884906 + 0.465770i \(0.845777\pi\)
\(48\) −2.00000 + 3.46410i −0.288675 + 0.500000i
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) 0.366025 + 1.36603i 0.0517638 + 0.193185i
\(51\) 6.00000 − 3.46410i 0.840168 − 0.485071i
\(52\) −0.535898 −0.0743157
\(53\) −1.26795 + 2.19615i −0.174166 + 0.301665i −0.939872 − 0.341526i \(-0.889056\pi\)
0.765706 + 0.643191i \(0.222390\pi\)
\(54\) 1.00000 − 1.00000i 0.136083 − 0.136083i
\(55\) 2.00000 0.269680
\(56\) −6.19615 + 4.19615i −0.827996 + 0.560734i
\(57\) −2.46410 −0.326378
\(58\) −2.53590 + 2.53590i −0.332980 + 0.332980i
\(59\) −1.46410 + 2.53590i −0.190610 + 0.330146i −0.945452 − 0.325760i \(-0.894380\pi\)
0.754843 + 0.655906i \(0.227713\pi\)
\(60\) −2.00000 −0.258199
\(61\) −9.46410 + 5.46410i −1.21175 + 0.699607i −0.963141 − 0.268996i \(-0.913308\pi\)
−0.248613 + 0.968603i \(0.579975\pi\)
\(62\) 2.36603 + 8.83013i 0.300486 + 1.12143i
\(63\) 2.50000 − 0.866025i 0.314970 − 0.109109i
\(64\) 8.00000i 1.00000i
\(65\) −0.133975 − 0.232051i −0.0166175 − 0.0287824i
\(66\) −0.732051 + 2.73205i −0.0901092 + 0.336292i
\(67\) 1.03590 + 0.598076i 0.126555 + 0.0730666i 0.561941 − 0.827177i \(-0.310055\pi\)
−0.435386 + 0.900244i \(0.643388\pi\)
\(68\) 6.92820 − 12.0000i 0.840168 − 1.45521i
\(69\) − 8.00000i − 0.963087i
\(70\) −3.36603 − 1.63397i −0.402317 − 0.195297i
\(71\) − 15.4641i − 1.83525i −0.397446 − 0.917626i \(-0.630103\pi\)
0.397446 − 0.917626i \(-0.369897\pi\)
\(72\) 0.732051 − 2.73205i 0.0862730 − 0.321975i
\(73\) 2.76795 + 1.59808i 0.323964 + 0.187041i 0.653158 − 0.757222i \(-0.273444\pi\)
−0.329194 + 0.944262i \(0.606777\pi\)
\(74\) 3.36603 + 0.901924i 0.391293 + 0.104847i
\(75\) −0.500000 − 0.866025i −0.0577350 − 0.100000i
\(76\) −4.26795 + 2.46410i −0.489567 + 0.282652i
\(77\) −3.46410 + 4.00000i −0.394771 + 0.455842i
\(78\) 0.366025 − 0.0980762i 0.0414442 − 0.0111049i
\(79\) 3.69615 − 2.13397i 0.415850 − 0.240091i −0.277450 − 0.960740i \(-0.589489\pi\)
0.693300 + 0.720649i \(0.256156\pi\)
\(80\) −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
\(86\) −9.19615 − 9.19615i −0.991647 − 0.991647i
\(87\) 1.26795 − 2.19615i 0.135938 − 0.235452i
\(88\) 1.46410 + 5.46410i 0.156074 + 0.582475i
\(89\) −14.1962 + 8.19615i −1.50479 + 0.868790i −0.504805 + 0.863234i \(0.668435\pi\)
−0.999985 + 0.00555677i \(0.998231\pi\)
\(90\) 1.36603 − 0.366025i 0.143992 − 0.0385825i
\(91\) 0.696152 + 0.133975i 0.0729766 + 0.0140444i
\(92\) −8.00000 − 13.8564i −0.834058 − 1.44463i
\(93\) −3.23205 − 5.59808i −0.335148 − 0.580493i
\(94\) 0.732051 + 0.196152i 0.0755053 + 0.0202316i
\(95\) −2.13397 − 1.23205i −0.218941 − 0.126406i
\(96\) −1.46410 − 5.46410i −0.149429 − 0.557678i
\(97\) − 14.9282i − 1.51573i −0.652412 − 0.757865i \(-0.726243\pi\)
0.652412 − 0.757865i \(-0.273757\pi\)
\(98\) 9.09808 − 3.90192i 0.919044 − 0.394154i
\(99\) − 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.31.2 ✓ 4
4.3 odd 2 420.2.bi.b.31.2 yes 4
7.5 odd 6 420.2.bi.b.271.2 yes 4
28.19 even 6 inner 420.2.bi.a.271.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bi.a.31.2 ✓ 4 1.1 even 1 trivial
420.2.bi.a.271.1 yes 4 28.19 even 6 inner
420.2.bi.b.31.2 yes 4 4.3 odd 2
420.2.bi.b.271.2 yes 4 7.5 odd 6