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 = [4,2] 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.b.31.2

$q$-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+(1.36603 + 0.366025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(1.73205 + 1.00000i) 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} +(1.00000 + 1.00000i) q^{10} +(-1.73205 + 1.00000i) q^{11} -2.00000i q^{12} +0.267949i q^{13} +(-0.267949 + 3.73205i) q^{14} -1.00000i q^{15} +(2.00000 + 3.46410i) q^{16} +(6.00000 - 3.46410i) q^{17} +(-1.00000 + 1.00000i) 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} +(0.732051 - 2.73205i) q^{24} +(0.500000 + 0.866025i) q^{25} +(-0.0980762 + 0.366025i) q^{26} +1.00000 q^{27} +(-1.73205 + 5.00000i) q^{28} +2.53590 q^{29} +(0.366025 - 1.36603i) q^{30} +(-3.23205 - 5.59808i) q^{31} +(1.46410 + 5.46410i) 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} +(2.46410 - 2.46410i) q^{38} +(0.232051 - 0.133975i) q^{39} +(0.732051 + 2.73205i) q^{40} -6.00000i q^{41} +(3.36603 - 1.63397i) q^{42} +9.19615i q^{43} -4.00000 q^{44} +(-0.866025 + 0.500000i) q^{45} +(-8.00000 - 8.00000i) 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} +(-0.267949 + 0.464102i) q^{52} +(-1.26795 - 2.19615i) q^{53} +(1.36603 + 0.366025i) q^{54} -2.00000 q^{55} +(-4.19615 + 6.19615i) q^{56} -2.46410 q^{57} +(3.46410 + 0.928203i) q^{58} +(1.46410 + 2.53590i) q^{59} +(1.00000 - 1.73205i) 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} +(2.00000 + 2.00000i) q^{66} +(-1.03590 + 0.598076i) q^{67} +13.8564 q^{68} +8.00000i q^{69} +(-2.09808 + 3.09808i) q^{70} -15.4641i q^{71} +(-2.73205 + 0.732051i) q^{72} +(2.76795 - 1.59808i) q^{73} +(-2.46410 + 2.46410i) 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} +4.00000i q^{80} +(-0.500000 - 0.866025i) q^{81} +(2.19615 - 8.19615i) q^{82} +2.00000 q^{83} +(5.19615 - 1.00000i) q^{84} +6.92820 q^{85} +(-3.36603 + 12.5622i) q^{86} +(-1.26795 - 2.19615i) q^{87} +(-5.46410 - 1.46410i) 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} +(0.535898 - 0.535898i) q^{94} +(2.13397 - 1.23205i) q^{95} +(4.00000 - 4.00000i) q^{96} +14.9282i q^{97} +(-9.83013 + 1.16987i) q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 2 q^{6} + 2 q^{7} + 8 q^{8} - 2 q^{9} + 4 q^{10} - 8 q^{14} + 8 q^{16} + 24 q^{17} - 4 q^{18} - 2 q^{19} + 4 q^{20} + 8 q^{21} - 4 q^{22} - 4 q^{24} + 2 q^{25} + 10 q^{26} + 4 q^{27}+ \cdots - 22 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36603 + 0.366025i 0.965926 + 0.258819i
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) 1.73205 + 1.00000i 0.866025 + 0.500000i
\(5\) 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
\(10\) 1.00000 + 1.00000i 0.316228 + 0.316228i
\(11\) −1.73205 + 1.00000i −0.522233 + 0.301511i −0.737848 − 0.674967i \(-0.764158\pi\)
0.215615 + 0.976478i \(0.430824\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) 0.267949i 0.0743157i 0.999309 + 0.0371579i \(0.0118304\pi\)
−0.999309 + 0.0371579i \(0.988170\pi\)
\(14\) −0.267949 + 3.73205i −0.0716124 + 0.997433i
\(15\) − 1.00000i − 0.258199i
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 6.00000 − 3.46410i 1.45521 − 0.840168i 0.456444 − 0.889752i \(-0.349123\pi\)
0.998770 + 0.0495842i \(0.0157896\pi\)
\(18\) −1.00000 + 1.00000i −0.235702 + 0.235702i
\(19\) 1.23205 − 2.13397i 0.282652 − 0.489567i −0.689385 − 0.724395i \(-0.742119\pi\)
0.972037 + 0.234828i \(0.0754526\pi\)
\(20\) 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 \(-0.647321\pi\)
−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
\(26\) −0.0980762 + 0.366025i −0.0192343 + 0.0717835i
\(27\) 1.00000 0.192450
\(28\) −1.73205 + 5.00000i −0.327327 + 0.944911i
\(29\) 2.53590 0.470905 0.235452 − 0.971886i \(-0.424343\pi\)
0.235452 + 0.971886i \(0.424343\pi\)
\(30\) 0.366025 − 1.36603i 0.0668268 − 0.249401i
\(31\) −3.23205 − 5.59808i −0.580493 − 1.00544i −0.995421 − 0.0955896i \(-0.969526\pi\)
0.414927 − 0.909855i \(-0.363807\pi\)
\(32\) 1.46410 + 5.46410i 0.258819 + 0.965926i
\(33\) 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 −0.949349 − 0.314225i \(-0.898256\pi\)
0.746801 + 0.665048i \(0.231589\pi\)
\(38\) 2.46410 − 2.46410i 0.399730 − 0.399730i
\(39\) 0.232051 − 0.133975i 0.0371579 − 0.0214531i
\(40\) 0.732051 + 2.73205i 0.115747 + 0.431975i
\(41\) − 6.00000i − 0.937043i −0.883452 − 0.468521i \(-0.844787\pi\)
0.883452 − 0.468521i \(-0.155213\pi\)
\(42\) 3.36603 − 1.63397i 0.519389 − 0.252128i
\(43\) 9.19615i 1.40240i 0.712965 + 0.701200i \(0.247352\pi\)
−0.712965 + 0.701200i \(0.752648\pi\)
\(44\) −4.00000 −0.603023
\(45\) −0.866025 + 0.500000i −0.129099 + 0.0745356i
\(46\) −8.00000 − 8.00000i −1.17954 − 1.17954i
\(47\) 0.267949 − 0.464102i 0.0390844 − 0.0676962i −0.845821 − 0.533466i \(-0.820889\pi\)
0.884906 + 0.465770i \(0.154223\pi\)
\(48\) 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
\(52\) −0.267949 + 0.464102i −0.0371579 + 0.0643593i
\(53\) −1.26795 − 2.19615i −0.174166 − 0.301665i 0.765706 − 0.643191i \(-0.222390\pi\)
−0.939872 + 0.341526i \(0.889056\pi\)
\(54\) 1.36603 + 0.366025i 0.185893 + 0.0498097i
\(55\) −2.00000 −0.269680
\(56\) −4.19615 + 6.19615i −0.560734 + 0.827996i
\(57\) −2.46410 −0.326378
\(58\) 3.46410 + 0.928203i 0.454859 + 0.121879i
\(59\) 1.46410 + 2.53590i 0.190610 + 0.330146i 0.945452 − 0.325760i \(-0.105620\pi\)
−0.754843 + 0.655906i \(0.772287\pi\)
\(60\) 1.00000 − 1.73205i 0.129099 − 0.223607i
\(61\) −9.46410 − 5.46410i −1.21175 − 0.699607i −0.248613 − 0.968603i \(-0.579975\pi\)
−0.963141 + 0.268996i \(0.913308\pi\)
\(62\) −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
\(66\) 2.00000 + 2.00000i 0.246183 + 0.246183i
\(67\) −1.03590 + 0.598076i −0.126555 + 0.0730666i −0.561941 − 0.827177i \(-0.689945\pi\)
0.435386 + 0.900244i \(0.356612\pi\)
\(68\) 13.8564 1.68034
\(69\) 8.00000i 0.963087i
\(70\) −2.09808 + 3.09808i −0.250768 + 0.370291i
\(71\) − 15.4641i − 1.83525i −0.397446 − 0.917626i \(-0.630103\pi\)
0.397446 − 0.917626i \(-0.369897\pi\)
\(72\) −2.73205 + 0.732051i −0.321975 + 0.0862730i
\(73\) 2.76795 − 1.59808i 0.323964 − 0.187041i −0.329194 − 0.944262i \(-0.606777\pi\)
0.653158 + 0.757222i \(0.273444\pi\)
\(74\) −2.46410 + 2.46410i −0.286446 + 0.286446i
\(75\) 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 \(-0.410511\pi\)
−0.693300 + 0.720649i \(0.743844\pi\)
\(80\) 4.00000i 0.447214i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 2.19615 − 8.19615i 0.242524 − 0.905114i
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 5.19615 − 1.00000i 0.566947 − 0.109109i
\(85\) 6.92820 0.751469
\(86\) −3.36603 + 12.5622i −0.362968 + 1.35461i
\(87\) −1.26795 − 2.19615i −0.135938 − 0.235452i
\(88\) −5.46410 − 1.46410i −0.582475 − 0.156074i
\(89\) −14.1962 − 8.19615i −1.50479 − 0.868790i −0.999985 − 0.00555677i \(-0.998231\pi\)
−0.504805 − 0.863234i \(-0.668435\pi\)
\(90\) −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
\(94\) 0.535898 − 0.535898i 0.0552737 − 0.0552737i
\(95\) 2.13397 − 1.23205i 0.218941 − 0.126406i
\(96\) 4.00000 − 4.00000i 0.408248 − 0.408248i
\(97\) 14.9282i 1.51573i 0.652412 + 0.757865i \(0.273757\pi\)
−0.652412 + 0.757865i \(0.726243\pi\)
\(98\) −9.83013 + 1.16987i −0.992993 + 0.118175i
\(99\) − 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.b.271.2 yes 4
4.3 odd 2 420.2.bi.a.271.1 yes 4
7.3 odd 6 420.2.bi.a.31.2 ✓ 4
28.3 even 6 inner 420.2.bi.b.31.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bi.a.31.2 ✓ 4 7.3 odd 6
420.2.bi.a.271.1 yes 4 4.3 odd 2
420.2.bi.b.31.2 yes 4 28.3 even 6 inner
420.2.bi.b.271.2 yes 4 1.1 even 1 trivial