↑

Properties

Label 420.2.c.b.391.7
Level $420$
Weight $2$
Character 420.391
Analytic conductor $3.354$
Analytic rank $0$
Dimension $16$
Inner twists $2$

Related objects

Downloads

Learn more

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(391,420)] 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("420.391"); 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, 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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,2,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 391.7
Root \(-0.102186 - 1.41052i\) of defining polynomial
Character \(\chi\) \(=\) 420.391
Dual form 420.2.c.b.391.8

$q$-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+(-0.102186 - 1.41052i) q^{2} +1.00000 q^{3} +(-1.97912 + 0.288270i) q^{4} +1.00000i q^{5} +(-0.102186 - 1.41052i) q^{6} +(0.178143 + 2.63975i) q^{7} +(0.608847 + 2.76212i) q^{8} +1.00000 q^{9} +(1.41052 - 0.102186i) q^{10} +5.22855i q^{11} +(-1.97912 + 0.288270i) q^{12} +4.52534i q^{13} +(3.70520 - 0.521019i) q^{14} +1.00000i q^{15} +(3.83380 - 1.14104i) q^{16} -6.70156i q^{17} +(-0.102186 - 1.41052i) q^{18} -2.81981 q^{19} +(-0.288270 - 1.97912i) q^{20} +(0.178143 + 2.63975i) q^{21} +(7.37496 - 0.534284i) q^{22} -0.858617i q^{23} +(0.608847 + 2.76212i) q^{24} -1.00000 q^{25} +(6.38308 - 0.462426i) q^{26} +1.00000 q^{27} +(-1.11353 - 5.17301i) q^{28} +6.47333 q^{29} +(1.41052 - 0.102186i) q^{30} -2.60723 q^{31} +(-2.00121 - 5.29104i) q^{32} +5.22855i q^{33} +(-9.45266 + 0.684805i) q^{34} +(-2.63975 + 0.178143i) q^{35} +(-1.97912 + 0.288270i) q^{36} +2.13976 q^{37} +(0.288144 + 3.97738i) q^{38} +4.52534i q^{39} +(-2.76212 + 0.608847i) q^{40} -8.71476i q^{41} +(3.70520 - 0.521019i) q^{42} +7.42042i q^{43} +(-1.50723 - 10.3479i) q^{44} +1.00000i q^{45} +(-1.21109 + 0.0877385i) q^{46} +9.82671 q^{47} +(3.83380 - 1.14104i) q^{48} +(-6.93653 + 0.940508i) q^{49} +(0.102186 + 1.41052i) q^{50} -6.70156i q^{51} +(-1.30452 - 8.95618i) q^{52} +3.69301 q^{53} +(-0.102186 - 1.41052i) q^{54} -5.22855 q^{55} +(-7.18284 + 2.09926i) q^{56} -2.81981 q^{57} +(-0.661483 - 9.13075i) q^{58} -4.27962 q^{59} +(-0.288270 - 1.97912i) q^{60} -10.7054i q^{61} +(0.266423 + 3.67755i) q^{62} +(0.178143 + 2.63975i) q^{63} +(-7.25861 + 3.36342i) q^{64} -4.52534 q^{65} +(7.37496 - 0.534284i) q^{66} -4.52269i q^{67} +(1.93186 + 13.2632i) q^{68} -0.858617i q^{69} +(0.521019 + 3.70520i) q^{70} +7.23513i q^{71} +(0.608847 + 2.76212i) q^{72} +9.24697i q^{73} +(-0.218653 - 3.01816i) q^{74} -1.00000 q^{75} +(5.58072 - 0.812865i) q^{76} +(-13.8020 + 0.931432i) q^{77} +(6.38308 - 0.462426i) q^{78} -2.68314i q^{79} +(1.14104 + 3.83380i) q^{80} +1.00000 q^{81} +(-12.2923 + 0.890525i) q^{82} +16.2812 q^{83} +(-1.11353 - 5.17301i) q^{84} +6.70156 q^{85} +(10.4666 - 0.758262i) q^{86} +6.47333 q^{87} +(-14.4419 + 3.18339i) q^{88} -8.53516i q^{89} +(1.41052 - 0.102186i) q^{90} +(-11.9458 + 0.806161i) q^{91} +(0.247513 + 1.69930i) q^{92} -2.60723 q^{93} +(-1.00415 - 13.8607i) q^{94} -2.81981i q^{95} +(-2.00121 - 5.29104i) q^{96} +10.5209i q^{97} +(2.03542 + 9.68799i) q^{98} +5.22855i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 16 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{7} + 2 q^{8} + 16 q^{9} - 2 q^{12} + 10 q^{14} + 6 q^{16} + 2 q^{18} + 24 q^{19} + 4 q^{21} - 12 q^{22} + 2 q^{24} - 16 q^{25} + 12 q^{26} + 16 q^{27}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.102186 − 1.41052i −0.0722563 − 0.997386i
\(3\) 1.00000 0.577350
\(4\) −1.97912 + 0.288270i −0.989558 + 0.144135i
\(5\) 1.00000i 0.447214i
\(6\) −0.102186 − 1.41052i −0.0417172 − 0.575841i
\(7\) 0.178143 + 2.63975i 0.0673319 + 0.997731i
\(8\) 0.608847 + 2.76212i 0.215260 + 0.976557i
\(9\) 1.00000 0.333333
\(10\) 1.41052 − 0.102186i 0.446045 − 0.0323140i
\(11\) 5.22855i 1.57647i 0.615376 + 0.788234i \(0.289004\pi\)
−0.615376 + 0.788234i \(0.710996\pi\)
\(12\) −1.97912 + 0.288270i −0.571322 + 0.0832163i
\(13\) 4.52534i 1.25510i 0.778574 + 0.627552i \(0.215943\pi\)
−0.778574 + 0.627552i \(0.784057\pi\)
\(14\) 3.70520 − 0.521019i 0.990258 − 0.139248i
\(15\) 1.00000i 0.258199i
\(16\) 3.83380 − 1.14104i 0.958450 − 0.285260i
\(17\) − 6.70156i − 1.62537i −0.582705 − 0.812684i \(-0.698006\pi\)
0.582705 − 0.812684i \(-0.301994\pi\)
\(18\) −0.102186 − 1.41052i −0.0240854 − 0.332462i
\(19\) −2.81981 −0.646908 −0.323454 − 0.946244i \(-0.604844\pi\)
−0.323454 + 0.946244i \(0.604844\pi\)
\(20\) −0.288270 − 1.97912i −0.0644591 − 0.442544i
\(21\) 0.178143 + 2.63975i 0.0388741 + 0.576040i
\(22\) 7.37496 − 0.534284i 1.57235 − 0.113910i
\(23\) − 0.858617i − 0.179034i −0.995985 − 0.0895170i \(-0.971468\pi\)
0.995985 − 0.0895170i \(-0.0285323\pi\)
\(24\) 0.608847 + 2.76212i 0.124280 + 0.563815i
\(25\) −1.00000 −0.200000
\(26\) 6.38308 − 0.462426i 1.25182 − 0.0906893i
\(27\) 1.00000 0.192450
\(28\) −1.11353 − 5.17301i −0.210437 − 0.977607i
\(29\) 6.47333 1.20207 0.601034 − 0.799224i \(-0.294756\pi\)
0.601034 + 0.799224i \(0.294756\pi\)
\(30\) 1.41052 − 0.102186i 0.257524 − 0.0186565i
\(31\) −2.60723 −0.468273 −0.234137 − 0.972204i \(-0.575226\pi\)
−0.234137 + 0.972204i \(0.575226\pi\)
\(32\) −2.00121 − 5.29104i −0.353768 − 0.935333i
\(33\) 5.22855i 0.910174i
\(34\) −9.45266 + 0.684805i −1.62112 + 0.117443i
\(35\) −2.63975 + 0.178143i −0.446199 + 0.0301117i
\(36\) −1.97912 + 0.288270i −0.329853 + 0.0480450i
\(37\) 2.13976 0.351774 0.175887 − 0.984410i \(-0.443721\pi\)
0.175887 + 0.984410i \(0.443721\pi\)
\(38\) 0.288144 + 3.97738i 0.0467432 + 0.645217i
\(39\) 4.52534i 0.724635i
\(40\) −2.76212 + 0.608847i −0.436729 + 0.0962672i
\(41\) − 8.71476i − 1.36102i −0.732740 − 0.680508i \(-0.761759\pi\)
0.732740 − 0.680508i \(-0.238241\pi\)
\(42\) 3.70520 − 0.521019i 0.571725 − 0.0803950i
\(43\) 7.42042i 1.13160i 0.824541 + 0.565802i \(0.191433\pi\)
−0.824541 + 0.565802i \(0.808567\pi\)
\(44\) −1.50723 − 10.3479i −0.227224 − 1.56001i
\(45\) 1.00000i 0.149071i
\(46\) −1.21109 + 0.0877385i −0.178566 + 0.0129363i
\(47\) 9.82671 1.43337 0.716686 − 0.697396i \(-0.245658\pi\)
0.716686 + 0.697396i \(0.245658\pi\)
\(48\) 3.83380 − 1.14104i 0.553362 − 0.164695i
\(49\) −6.93653 + 0.940508i −0.990933 + 0.134358i
\(50\) 0.102186 + 1.41052i 0.0144513 + 0.199477i
\(51\) − 6.70156i − 0.938406i
\(52\) −1.30452 − 8.95618i −0.180904 − 1.24200i
\(53\) 3.69301 0.507274 0.253637 − 0.967299i \(-0.418373\pi\)
0.253637 + 0.967299i \(0.418373\pi\)
\(54\) −0.102186 − 1.41052i −0.0139057 − 0.191947i
\(55\) −5.22855 −0.705018
\(56\) −7.18284 + 2.09926i −0.959847 + 0.280525i
\(57\) −2.81981 −0.373493
\(58\) −0.661483 − 9.13075i −0.0868570 − 1.19893i
\(59\) −4.27962 −0.557159 −0.278579 − 0.960413i \(-0.589864\pi\)
−0.278579 + 0.960413i \(0.589864\pi\)
\(60\) −0.288270 − 1.97912i −0.0372155 − 0.255503i
\(61\) − 10.7054i − 1.37069i −0.728221 − 0.685343i \(-0.759653\pi\)
0.728221 − 0.685343i \(-0.240347\pi\)
\(62\) 0.266423 + 3.67755i 0.0338357 + 0.467049i
\(63\) 0.178143 + 2.63975i 0.0224440 + 0.332577i
\(64\) −7.25861 + 3.36342i −0.907326 + 0.420427i
\(65\) −4.52534 −0.561300
\(66\) 7.37496 − 0.534284i 0.907795 − 0.0657658i
\(67\) − 4.52269i − 0.552534i −0.961081 − 0.276267i \(-0.910903\pi\)
0.961081 − 0.276267i \(-0.0890975\pi\)
\(68\) 1.93186 + 13.2632i 0.234272 + 1.60840i
\(69\) − 0.858617i − 0.103365i
\(70\) 0.521019 + 3.70520i 0.0622737 + 0.442857i
\(71\) 7.23513i 0.858652i 0.903150 + 0.429326i \(0.141249\pi\)
−0.903150 + 0.429326i \(0.858751\pi\)
\(72\) 0.608847 + 2.76212i 0.0717533 + 0.325519i
\(73\) 9.24697i 1.08228i 0.840934 + 0.541138i \(0.182006\pi\)
−0.840934 + 0.541138i \(0.817994\pi\)
\(74\) −0.218653 − 3.01816i −0.0254179 − 0.350854i
\(75\) −1.00000 −0.115470
\(76\) 5.58072 − 0.812865i 0.640153 − 0.0932420i
\(77\) −13.8020 + 0.931432i −1.57289 + 0.106147i
\(78\) 6.38308 − 0.462426i 0.722741 − 0.0523595i
\(79\) − 2.68314i − 0.301877i −0.988543 − 0.150938i \(-0.951770\pi\)
0.988543 − 0.150938i \(-0.0482295\pi\)
\(80\) 1.14104 + 3.83380i 0.127572 + 0.428632i
\(81\) 1.00000 0.111111
\(82\) −12.2923 + 0.890525i −1.35746 + 0.0983421i
\(83\) 16.2812 1.78710 0.893548 − 0.448967i \(-0.148208\pi\)
0.893548 + 0.448967i \(0.148208\pi\)
\(84\) −1.11353 − 5.17301i −0.121496 − 0.564422i
\(85\) 6.70156 0.726886
\(86\) 10.4666 − 0.758262i 1.12865 − 0.0817655i
\(87\) 6.47333 0.694014
\(88\) −14.4419 + 3.18339i −1.53951 + 0.339350i
\(89\) − 8.53516i − 0.904725i −0.891834 − 0.452362i \(-0.850581\pi\)
0.891834 − 0.452362i \(-0.149419\pi\)
\(90\) 1.41052 − 0.102186i 0.148682 − 0.0107713i
\(91\) −11.9458 + 0.806161i −1.25226 + 0.0845086i
\(92\) 0.247513 + 1.69930i 0.0258051 + 0.177165i
\(93\) −2.60723 −0.270358
\(94\) −1.00415 − 13.8607i −0.103570 − 1.42963i
\(95\) − 2.81981i − 0.289306i
\(96\) −2.00121 − 5.29104i −0.204248 − 0.540015i
\(97\) 10.5209i 1.06824i 0.845410 + 0.534118i \(0.179356\pi\)
−0.845410 + 0.534118i \(0.820644\pi\)
\(98\) 2.03542 + 9.68799i 0.205608 + 0.978634i
\(99\) 5.22855i 0.525489i
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.c.b.391.7 yes 16
3.2 odd 2 1260.2.c.e.811.10 16
4.3 odd 2 420.2.c.a.391.8 yes 16
7.6 odd 2 420.2.c.a.391.7 ✓ 16
12.11 even 2 1260.2.c.d.811.9 16
21.20 even 2 1260.2.c.d.811.10 16
28.27 even 2 inner 420.2.c.b.391.8 yes 16
84.83 odd 2 1260.2.c.e.811.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.c.a.391.7 ✓ 16 7.6 odd 2
420.2.c.a.391.8 yes 16 4.3 odd 2
420.2.c.b.391.7 yes 16 1.1 even 1 trivial
420.2.c.b.391.8 yes 16 28.27 even 2 inner
1260.2.c.d.811.9 16 12.11 even 2
1260.2.c.d.811.10 16 21.20 even 2
1260.2.c.e.811.9 16 84.83 odd 2
1260.2.c.e.811.10 16 3.2 odd 2