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 = [4,0,-4,-8,0,8,4] 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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 420.239
Dual form 420.2.l.a.239.2

$q$-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+1.41421i q^{2} +(-1.00000 - 1.41421i) q^{3} -2.00000 q^{4} +(2.12132 - 0.707107i) q^{5} +(2.00000 - 1.41421i) q^{6} +1.00000 q^{7} -2.82843i q^{8} +(-1.00000 + 2.82843i) q^{9} +(1.00000 + 3.00000i) q^{10} -4.24264 q^{11} +(2.00000 + 2.82843i) q^{12} -6.00000i q^{13} +1.41421i q^{14} +(-3.12132 - 2.29289i) q^{15} +4.00000 q^{16} +4.24264 q^{17} +(-4.00000 - 1.41421i) q^{18} -6.00000i q^{19} +(-4.24264 + 1.41421i) q^{20} +(-1.00000 - 1.41421i) q^{21} -6.00000i q^{22} -1.41421i q^{23} +(-4.00000 + 2.82843i) q^{24} +(4.00000 - 3.00000i) q^{25} +8.48528 q^{26} +(5.00000 - 1.41421i) q^{27} -2.00000 q^{28} +2.82843i q^{29} +(3.24264 - 4.41421i) q^{30} +5.65685i q^{32} +(4.24264 + 6.00000i) q^{33} +6.00000i q^{34} +(2.12132 - 0.707107i) q^{35} +(2.00000 - 5.65685i) q^{36} -6.00000i q^{37} +8.48528 q^{38} +(-8.48528 + 6.00000i) q^{39} +(-2.00000 - 6.00000i) q^{40} -1.41421i q^{41} +(2.00000 - 1.41421i) q^{42} +8.00000 q^{43} +8.48528 q^{44} +(-0.121320 + 6.70711i) q^{45} +2.00000 q^{46} +2.82843i q^{47} +(-4.00000 - 5.65685i) q^{48} +1.00000 q^{49} +(4.24264 + 5.65685i) q^{50} +(-4.24264 - 6.00000i) q^{51} +12.0000i q^{52} -8.48528 q^{53} +(2.00000 + 7.07107i) q^{54} +(-9.00000 + 3.00000i) q^{55} -2.82843i q^{56} +(-8.48528 + 6.00000i) q^{57} -4.00000 q^{58} +(6.24264 + 4.58579i) q^{60} -10.0000 q^{61} +(-1.00000 + 2.82843i) q^{63} -8.00000 q^{64} +(-4.24264 - 12.7279i) q^{65} +(-8.48528 + 6.00000i) q^{66} -4.00000 q^{67} -8.48528 q^{68} +(-2.00000 + 1.41421i) q^{69} +(1.00000 + 3.00000i) q^{70} +12.7279 q^{71} +(8.00000 + 2.82843i) q^{72} +6.00000i q^{73} +8.48528 q^{74} +(-8.24264 - 2.65685i) q^{75} +12.0000i q^{76} -4.24264 q^{77} +(-8.48528 - 12.0000i) q^{78} +(8.48528 - 2.82843i) q^{80} +(-7.00000 - 5.65685i) q^{81} +2.00000 q^{82} +2.82843i q^{83} +(2.00000 + 2.82843i) q^{84} +(9.00000 - 3.00000i) q^{85} +11.3137i q^{86} +(4.00000 - 2.82843i) q^{87} +12.0000i q^{88} +7.07107i q^{89} +(-9.48528 - 0.171573i) q^{90} -6.00000i q^{91} +2.82843i q^{92} -4.00000 q^{94} +(-4.24264 - 12.7279i) q^{95} +(8.00000 - 5.65685i) q^{96} +6.00000i q^{97} +1.41421i q^{98} +(4.24264 - 12.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{4} + 8 q^{6} + 4 q^{7} - 4 q^{9} + 4 q^{10} + 8 q^{12} - 4 q^{15} + 16 q^{16} - 16 q^{18} - 4 q^{21} - 16 q^{24} + 16 q^{25} + 20 q^{27} - 8 q^{28} - 4 q^{30} + 8 q^{36} - 8 q^{40}+ \cdots + 32 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) −1.00000 − 1.41421i −0.577350 − 0.816497i
\(4\) −2.00000 −1.00000
\(5\) 2.12132 − 0.707107i 0.948683 − 0.316228i
\(6\) 2.00000 − 1.41421i 0.816497 − 0.577350i
\(7\) 1.00000 0.377964
\(8\) − 2.82843i − 1.00000i
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 1.00000 + 3.00000i 0.316228 + 0.948683i
\(11\) −4.24264 −1.27920 −0.639602 − 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 2.00000 + 2.82843i 0.577350 + 0.816497i
\(13\) − 6.00000i − 1.66410i −0.554700 − 0.832050i \(-0.687167\pi\)
0.554700 − 0.832050i \(-0.312833\pi\)
\(14\) 1.41421i 0.377964i
\(15\) −3.12132 − 2.29289i −0.805921 − 0.592022i
\(16\) 4.00000 1.00000
\(17\) 4.24264 1.02899 0.514496 − 0.857493i \(-0.327979\pi\)
0.514496 + 0.857493i \(0.327979\pi\)
\(18\) −4.00000 − 1.41421i −0.942809 − 0.333333i
\(19\) − 6.00000i − 1.37649i −0.725476 − 0.688247i \(-0.758380\pi\)
0.725476 − 0.688247i \(-0.241620\pi\)
\(20\) −4.24264 + 1.41421i −0.948683 + 0.316228i
\(21\) −1.00000 − 1.41421i −0.218218 − 0.308607i
\(22\) − 6.00000i − 1.27920i
\(23\) − 1.41421i − 0.294884i −0.989071 − 0.147442i \(-0.952896\pi\)
0.989071 − 0.147442i \(-0.0471040\pi\)
\(24\) −4.00000 + 2.82843i −0.816497 + 0.577350i
\(25\) 4.00000 − 3.00000i 0.800000 − 0.600000i
\(26\) 8.48528 1.66410
\(27\) 5.00000 − 1.41421i 0.962250 − 0.272166i
\(28\) −2.00000 −0.377964
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 3.24264 − 4.41421i 0.592022 − 0.805921i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 4.24264 + 6.00000i 0.738549 + 1.04447i
\(34\) 6.00000i 1.02899i
\(35\) 2.12132 − 0.707107i 0.358569 − 0.119523i
\(36\) 2.00000 − 5.65685i 0.333333 − 0.942809i
\(37\) − 6.00000i − 0.986394i −0.869918 − 0.493197i \(-0.835828\pi\)
0.869918 − 0.493197i \(-0.164172\pi\)
\(38\) 8.48528 1.37649
\(39\) −8.48528 + 6.00000i −1.35873 + 0.960769i
\(40\) −2.00000 − 6.00000i −0.316228 − 0.948683i
\(41\) − 1.41421i − 0.220863i −0.993884 − 0.110432i \(-0.964777\pi\)
0.993884 − 0.110432i \(-0.0352233\pi\)
\(42\) 2.00000 − 1.41421i 0.308607 − 0.218218i
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 8.48528 1.27920
\(45\) −0.121320 + 6.70711i −0.0180854 + 0.999836i
\(46\) 2.00000 0.294884
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) −4.00000 − 5.65685i −0.577350 − 0.816497i
\(49\) 1.00000 0.142857
\(50\) 4.24264 + 5.65685i 0.600000 + 0.800000i
\(51\) −4.24264 − 6.00000i −0.594089 − 0.840168i
\(52\) 12.0000i 1.66410i
\(53\) −8.48528 −1.16554 −0.582772 − 0.812636i \(-0.698032\pi\)
−0.582772 + 0.812636i \(0.698032\pi\)
\(54\) 2.00000 + 7.07107i 0.272166 + 0.962250i
\(55\) −9.00000 + 3.00000i −1.21356 + 0.404520i
\(56\) − 2.82843i − 0.377964i
\(57\) −8.48528 + 6.00000i −1.12390 + 0.794719i
\(58\) −4.00000 −0.525226
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 6.24264 + 4.58579i 0.805921 + 0.592022i
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.00000 + 2.82843i −0.125988 + 0.356348i
\(64\) −8.00000 −1.00000
\(65\) −4.24264 − 12.7279i −0.526235 − 1.57870i
\(66\) −8.48528 + 6.00000i −1.04447 + 0.738549i
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −8.48528 −1.02899
\(69\) −2.00000 + 1.41421i −0.240772 + 0.170251i
\(70\) 1.00000 + 3.00000i 0.119523 + 0.358569i
\(71\) 12.7279 1.51053 0.755263 − 0.655422i \(-0.227509\pi\)
0.755263 + 0.655422i \(0.227509\pi\)
\(72\) 8.00000 + 2.82843i 0.942809 + 0.333333i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 8.48528 0.986394
\(75\) −8.24264 − 2.65685i −0.951778 − 0.306787i
\(76\) 12.0000i 1.37649i
\(77\) −4.24264 −0.483494
\(78\) −8.48528 − 12.0000i −0.960769 − 1.35873i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.48528 − 2.82843i 0.948683 − 0.316228i
\(81\) −7.00000 − 5.65685i −0.777778 − 0.628539i
\(82\) 2.00000 0.220863
\(83\) 2.82843i 0.310460i 0.987878 + 0.155230i \(0.0496119\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 2.00000 + 2.82843i 0.218218 + 0.308607i
\(85\) 9.00000 − 3.00000i 0.976187 − 0.325396i
\(86\) 11.3137i 1.21999i
\(87\) 4.00000 − 2.82843i 0.428845 − 0.303239i
\(88\) 12.0000i 1.27920i
\(89\) 7.07107i 0.749532i 0.927119 + 0.374766i \(0.122277\pi\)
−0.927119 + 0.374766i \(0.877723\pi\)
\(90\) −9.48528 − 0.171573i −0.999836 − 0.0180854i
\(91\) − 6.00000i − 0.628971i
\(92\) 2.82843i 0.294884i
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) −4.24264 − 12.7279i −0.435286 − 1.30586i
\(96\) 8.00000 − 5.65685i 0.816497 − 0.577350i
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 1.41421i 0.142857i
\(99\) 4.24264 − 12.0000i 0.426401 − 1.20605i
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.a.239.4 yes 4
3.2 odd 2 inner 420.2.l.a.239.1 ✓ 4
4.3 odd 2 420.2.l.b.239.2 yes 4
5.4 even 2 420.2.l.b.239.1 yes 4
12.11 even 2 420.2.l.b.239.3 yes 4
15.14 odd 2 420.2.l.b.239.4 yes 4
20.19 odd 2 inner 420.2.l.a.239.3 yes 4
60.59 even 2 inner 420.2.l.a.239.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.l.a.239.1 ✓ 4 3.2 odd 2 inner
420.2.l.a.239.2 yes 4 60.59 even 2 inner
420.2.l.a.239.3 yes 4 20.19 odd 2 inner
420.2.l.a.239.4 yes 4 1.1 even 1 trivial
420.2.l.b.239.1 yes 4 5.4 even 2
420.2.l.b.239.2 yes 4 4.3 odd 2
420.2.l.b.239.3 yes 4 12.11 even 2
420.2.l.b.239.4 yes 4 15.14 odd 2