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 = [8,0,2,2,-8,-2,8] 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: \(8\)
Coefficient field: 8.0.386672896.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.4
Root \(-0.835949 - 1.14070i\) of defining polynomial
Character \(\chi\) \(=\) 420.239
Dual form 420.2.l.e.239.3

$q$-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+(-0.835949 + 1.14070i) q^{2} +(1.10238 - 1.33595i) q^{3} +(-0.602380 - 1.90713i) q^{4} +(-1.00000 - 2.00000i) q^{5} +(0.602380 + 2.37427i) q^{6} +1.00000 q^{7} +(2.67901 + 0.907128i) q^{8} +(-0.569517 - 2.94545i) q^{9} +(3.11734 + 0.531200i) q^{10} -2.20476 q^{11} +(-3.21188 - 1.29763i) q^{12} +1.89089i q^{13} +(-0.835949 + 1.14070i) q^{14} +(-3.77428 - 0.868811i) q^{15} +(-3.27428 + 2.29763i) q^{16} -1.13903 q^{17} +(3.83595 + 1.81259i) q^{18} -8.56279i q^{19} +(-3.21188 + 3.11189i) q^{20} +(1.10238 - 1.33595i) q^{21} +(1.84307 - 2.51496i) q^{22} -3.21899i q^{23} +(4.16517 - 2.57903i) q^{24} +(-3.00000 + 4.00000i) q^{25} +(-2.15693 - 1.58069i) q^{26} +(-4.56279 - 2.48615i) q^{27} +(-0.602380 - 1.90713i) q^{28} +1.89089i q^{29} +(4.14615 - 3.57903i) q^{30} -5.90658i q^{31} +(0.116226 - 5.65566i) q^{32} +(-2.43048 + 2.94545i) q^{33} +(0.952175 - 1.29929i) q^{34} +(-1.00000 - 2.00000i) q^{35} +(-5.27428 + 2.86042i) q^{36} -0.409519i q^{37} +(9.76755 + 7.15805i) q^{38} +(2.52613 + 2.08448i) q^{39} +(-0.864758 - 6.26516i) q^{40} +4.40952i q^{41} +(0.602380 + 2.37427i) q^{42} -0.934275 q^{43} +(1.32810 + 4.20476i) q^{44} +(-5.32137 + 4.08448i) q^{45} +(3.67190 + 2.69091i) q^{46} +2.67190i q^{47} +(-0.539980 + 6.90713i) q^{48} +1.00000 q^{49} +(-2.05494 - 6.76589i) q^{50} +(-1.25565 + 1.52169i) q^{51} +(3.60617 - 1.13903i) q^{52} +6.81904 q^{53} +(6.65021 - 3.12646i) q^{54} +(2.20476 + 4.40952i) q^{55} +(2.67901 + 0.907128i) q^{56} +(-11.4394 - 9.43945i) q^{57} +(-2.15693 - 1.58069i) q^{58} +13.5351 q^{59} +(0.616614 + 7.72139i) q^{60} +12.4694 q^{61} +(6.73762 + 4.93760i) q^{62} +(-0.569517 - 2.94545i) q^{63} +(6.35424 + 4.86042i) q^{64} +(3.78178 - 1.89089i) q^{65} +(-1.32810 - 5.23469i) q^{66} -10.8475 q^{67} +(0.686132 + 2.17229i) q^{68} +(-4.30041 - 3.54855i) q^{69} +(3.11734 + 0.531200i) q^{70} +8.00000 q^{71} +(1.14615 - 8.40752i) q^{72} -8.00000i q^{73} +(0.467138 + 0.342337i) q^{74} +(2.03665 + 8.41737i) q^{75} +(-16.3303 + 5.15805i) q^{76} -2.20476 q^{77} +(-4.48948 + 1.13903i) q^{78} +3.76609i q^{79} +(7.86954 + 4.25092i) q^{80} +(-8.35130 + 3.35496i) q^{81} +(-5.02993 - 3.68613i) q^{82} -6.84086i q^{83} +(-3.21188 - 1.29763i) q^{84} +(1.13903 + 2.27807i) q^{85} +(0.781006 - 1.06572i) q^{86} +(2.52613 + 2.08448i) q^{87} +(-5.90658 - 2.00000i) q^{88} +16.1913i q^{89} +(-0.210760 - 9.48449i) q^{90} +1.89089i q^{91} +(-6.13903 + 1.93906i) q^{92} +(-7.89089 - 6.51130i) q^{93} +(-3.04783 - 2.23357i) q^{94} +(-17.1256 + 8.56279i) q^{95} +(-7.42755 - 6.38996i) q^{96} +18.4917i q^{97} +(-0.835949 + 1.14070i) q^{98} +(1.25565 + 6.49400i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 2 q^{4} - 8 q^{5} - 2 q^{6} + 8 q^{7} + 6 q^{8} + 2 q^{9} - 4 q^{10} - 4 q^{11} - 10 q^{12} - 10 q^{15} - 6 q^{16} + 4 q^{17} + 24 q^{18} - 10 q^{20} + 2 q^{21} + 6 q^{22} - 18 q^{24} - 24 q^{25}+ \cdots - 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.835949 + 1.14070i −0.591105 + 0.806595i
\(3\) 1.10238 − 1.33595i 0.636459 − 0.771310i
\(4\) −0.602380 − 1.90713i −0.301190 − 0.953564i
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 0.602380 + 2.37427i 0.245921 + 0.969290i
\(7\) 1.00000 0.377964
\(8\) 2.67901 + 0.907128i 0.947175 + 0.320718i
\(9\) −0.569517 − 2.94545i −0.189839 − 0.981815i
\(10\) 3.11734 + 0.531200i 0.985790 + 0.167980i
\(11\) −2.20476 −0.664760 −0.332380 − 0.943146i \(-0.607852\pi\)
−0.332380 + 0.943146i \(0.607852\pi\)
\(12\) −3.21188 − 1.29763i −0.927189 − 0.374594i
\(13\) 1.89089i 0.524439i 0.965008 + 0.262219i \(0.0844544\pi\)
−0.965008 + 0.262219i \(0.915546\pi\)
\(14\) −0.835949 + 1.14070i −0.223417 + 0.304864i
\(15\) −3.77428 − 0.868811i −0.974514 − 0.224326i
\(16\) −3.27428 + 2.29763i −0.818569 + 0.574408i
\(17\) −1.13903 −0.276257 −0.138128 − 0.990414i \(-0.544109\pi\)
−0.138128 + 0.990414i \(0.544109\pi\)
\(18\) 3.83595 + 1.81259i 0.904142 + 0.427233i
\(19\) − 8.56279i − 1.96444i −0.187738 − 0.982219i \(-0.560115\pi\)
0.187738 − 0.982219i \(-0.439885\pi\)
\(20\) −3.21188 + 3.11189i −0.718197 + 0.695839i
\(21\) 1.10238 − 1.33595i 0.240559 − 0.291528i
\(22\) 1.84307 − 2.51496i 0.392943 − 0.536192i
\(23\) − 3.21899i − 0.671207i −0.942003 − 0.335603i \(-0.891060\pi\)
0.942003 − 0.335603i \(-0.108940\pi\)
\(24\) 4.16517 − 2.57903i 0.850211 − 0.526441i
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) −2.15693 − 1.58069i −0.423010 − 0.309998i
\(27\) −4.56279 − 2.48615i −0.878109 − 0.478461i
\(28\) −0.602380 − 1.90713i −0.113839 − 0.360413i
\(29\) 1.89089i 0.351130i 0.984468 + 0.175565i \(0.0561752\pi\)
−0.984468 + 0.175565i \(0.943825\pi\)
\(30\) 4.14615 − 3.57903i 0.756980 − 0.653438i
\(31\) − 5.90658i − 1.06085i −0.847731 − 0.530427i \(-0.822032\pi\)
0.847731 − 0.530427i \(-0.177968\pi\)
\(32\) 0.116226 − 5.65566i 0.0205460 − 0.999789i
\(33\) −2.43048 + 2.94545i −0.423093 + 0.512736i
\(34\) 0.952175 − 1.29929i 0.163297 − 0.222827i
\(35\) −1.00000 − 2.00000i −0.169031 − 0.338062i
\(36\) −5.27428 + 2.86042i −0.879046 + 0.476737i
\(37\) − 0.409519i − 0.0673246i −0.999433 − 0.0336623i \(-0.989283\pi\)
0.999433 − 0.0336623i \(-0.0107171\pi\)
\(38\) 9.76755 + 7.15805i 1.58451 + 1.16119i
\(39\) 2.52613 + 2.08448i 0.404505 + 0.333784i
\(40\) −0.864758 − 6.26516i −0.136730 − 0.990608i
\(41\) 4.40952i 0.688651i 0.938850 + 0.344326i \(0.111892\pi\)
−0.938850 + 0.344326i \(0.888108\pi\)
\(42\) 0.602380 + 2.37427i 0.0929492 + 0.366357i
\(43\) −0.934275 −0.142476 −0.0712378 − 0.997459i \(-0.522695\pi\)
−0.0712378 + 0.997459i \(0.522695\pi\)
\(44\) 1.32810 + 4.20476i 0.200219 + 0.633891i
\(45\) −5.32137 + 4.08448i −0.793264 + 0.608878i
\(46\) 3.67190 + 2.69091i 0.541392 + 0.396754i
\(47\) 2.67190i 0.389736i 0.980829 + 0.194868i \(0.0624279\pi\)
−0.980829 + 0.194868i \(0.937572\pi\)
\(48\) −0.539980 + 6.90713i −0.0779393 + 0.996958i
\(49\) 1.00000 0.142857
\(50\) −2.05494 − 6.76589i −0.290613 − 0.956841i
\(51\) −1.25565 + 1.52169i −0.175826 + 0.213079i
\(52\) 3.60617 − 1.13903i 0.500086 − 0.157956i
\(53\) 6.81904 0.936667 0.468334 − 0.883552i \(-0.344855\pi\)
0.468334 + 0.883552i \(0.344855\pi\)
\(54\) 6.65021 − 3.12646i 0.904978 − 0.425458i
\(55\) 2.20476 + 4.40952i 0.297290 + 0.594579i
\(56\) 2.67901 + 0.907128i 0.357998 + 0.121220i
\(57\) −11.4394 − 9.43945i −1.51519 − 1.25029i
\(58\) −2.15693 − 1.58069i −0.283219 − 0.207554i
\(59\) 13.5351 1.76212 0.881060 − 0.473005i \(-0.156831\pi\)
0.881060 + 0.473005i \(0.156831\pi\)
\(60\) 0.616614 + 7.72139i 0.0796046 + 0.996827i
\(61\) 12.4694 1.59654 0.798270 − 0.602300i \(-0.205749\pi\)
0.798270 + 0.602300i \(0.205749\pi\)
\(62\) 6.73762 + 4.93760i 0.855679 + 0.627076i
\(63\) −0.569517 − 2.94545i −0.0717524 − 0.371091i
\(64\) 6.35424 + 4.86042i 0.794280 + 0.607552i
\(65\) 3.78178 − 1.89089i 0.469072 − 0.234536i
\(66\) −1.32810 − 5.23469i −0.163478 − 0.644345i
\(67\) −10.8475 −1.32523 −0.662617 − 0.748958i \(-0.730554\pi\)
−0.662617 + 0.748958i \(0.730554\pi\)
\(68\) 0.686132 + 2.17229i 0.0832057 + 0.263428i
\(69\) −4.30041 − 3.54855i −0.517709 − 0.427196i
\(70\) 3.11734 + 0.531200i 0.372594 + 0.0634906i
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.14615 − 8.40752i 0.135075 − 0.990835i
\(73\) − 8.00000i − 0.936329i −0.883641 − 0.468165i \(-0.844915\pi\)
0.883641 − 0.468165i \(-0.155085\pi\)
\(74\) 0.467138 + 0.342337i 0.0543036 + 0.0397959i
\(75\) 2.03665 + 8.41737i 0.235173 + 0.971954i
\(76\) −16.3303 + 5.15805i −1.87322 + 0.591669i
\(77\) −2.20476 −0.251256
\(78\) −4.48948 + 1.13903i −0.508333 + 0.128970i
\(79\) 3.76609i 0.423718i 0.977300 + 0.211859i \(0.0679518\pi\)
−0.977300 + 0.211859i \(0.932048\pi\)
\(80\) 7.86954 + 4.25092i 0.879841 + 0.475268i
\(81\) −8.35130 + 3.35496i −0.927922 + 0.372774i
\(82\) −5.02993 − 3.68613i −0.555462 − 0.407065i
\(83\) − 6.84086i − 0.750882i −0.926846 − 0.375441i \(-0.877491\pi\)
0.926846 − 0.375441i \(-0.122509\pi\)
\(84\) −3.21188 − 1.29763i −0.350444 − 0.141583i
\(85\) 1.13903 + 2.27807i 0.123546 + 0.247091i
\(86\) 0.781006 − 1.06572i 0.0842180 − 0.114920i
\(87\) 2.52613 + 2.08448i 0.270830 + 0.223480i
\(88\) −5.90658 − 2.00000i −0.629644 − 0.213201i
\(89\) 16.1913i 1.71627i 0.513420 + 0.858137i \(0.328378\pi\)
−0.513420 + 0.858137i \(0.671622\pi\)
\(90\) −0.210760 − 9.48449i −0.0222160 − 0.999753i
\(91\) 1.89089i 0.198219i
\(92\) −6.13903 + 1.93906i −0.640039 + 0.202161i
\(93\) −7.89089 − 6.51130i −0.818247 − 0.675190i
\(94\) −3.04783 − 2.23357i −0.314359 − 0.230375i
\(95\) −17.1256 + 8.56279i −1.75705 + 0.878524i
\(96\) −7.42755 − 6.38996i −0.758071 − 0.652172i
\(97\) 18.4917i 1.87755i 0.344532 + 0.938774i \(0.388038\pi\)
−0.344532 + 0.938774i \(0.611962\pi\)
\(98\) −0.835949 + 1.14070i −0.0844436 + 0.115228i
\(99\) 1.25565 + 6.49400i 0.126197 + 0.652672i
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.e.239.4 yes 8
3.2 odd 2 420.2.l.f.239.5 yes 8
4.3 odd 2 420.2.l.c.239.3 ✓ 8
5.4 even 2 420.2.l.d.239.5 yes 8
12.11 even 2 420.2.l.d.239.6 yes 8
15.14 odd 2 420.2.l.c.239.4 yes 8
20.19 odd 2 420.2.l.f.239.6 yes 8
60.59 even 2 inner 420.2.l.e.239.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.l.c.239.3 ✓ 8 4.3 odd 2
420.2.l.c.239.4 yes 8 15.14 odd 2
420.2.l.d.239.5 yes 8 5.4 even 2
420.2.l.d.239.6 yes 8 12.11 even 2
420.2.l.e.239.3 yes 8 60.59 even 2 inner
420.2.l.e.239.4 yes 8 1.1 even 1 trivial
420.2.l.f.239.5 yes 8 3.2 odd 2
420.2.l.f.239.6 yes 8 20.19 odd 2