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Properties

Label 1680.2.q.c.559.1
Level $1680$
Weight $2$
Character 1680.559
Analytic conductor $13.415$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1680,2,Mod(559,1680)] 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("1680.559"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.q (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,0,0,0,-32,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 559.1
Character \(\chi\) \(=\) 1680.559
Dual form 1680.2.q.c.559.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-1.58345 - 1.57882i) q^{5} +(2.37980 - 1.15610i) q^{7} -1.00000 q^{9} +1.26306i q^{11} -6.82818 q^{13} +(-1.57882 + 1.58345i) q^{15} +1.90384 q^{17} -2.53661 q^{19} +(-1.15610 - 2.37980i) q^{21} -5.50258 q^{23} +(0.0146493 + 4.99998i) q^{25} +1.00000i q^{27} +6.49989 q^{29} -5.78049 q^{31} +1.26306 q^{33} +(-5.59357 - 1.92664i) q^{35} -6.24970i q^{37} +6.82818i q^{39} +11.5678i q^{41} -7.55118 q^{43} +(1.58345 + 1.57882i) q^{45} +0.158389i q^{47} +(4.32685 - 5.50258i) q^{49} -1.90384i q^{51} +0.469217i q^{53} +(1.99415 - 2.00000i) q^{55} +2.53661i q^{57} +6.99413 q^{59} -3.90063i q^{61} +(-2.37980 + 1.15610i) q^{63} +(10.8121 + 10.7805i) q^{65} -10.7551 q^{67} +5.50258i q^{69} +3.31768i q^{71} -0.691529 q^{73} +(4.99998 - 0.0146493i) q^{75} +(1.46023 + 3.00583i) q^{77} +10.1922i q^{79} +1.00000 q^{81} +(-3.01465 - 3.00583i) q^{85} -6.49989i q^{87} -4.26668i q^{89} +(-16.2497 + 7.89407i) q^{91} +5.78049i q^{93} +(4.01661 + 4.00486i) q^{95} -2.82332 q^{97} -1.26306i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{9} + 8 q^{21} - 32 q^{25} + 16 q^{49} + 48 q^{65} + 32 q^{81} - 64 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\) \(-1\)

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 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) −1.58345 − 1.57882i −0.708142 − 0.706070i
\(6\) 0 0
\(7\) 2.37980 − 1.15610i 0.899478 − 0.436966i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.26306i 0.380828i 0.981704 + 0.190414i \(0.0609830\pi\)
−0.981704 + 0.190414i \(0.939017\pi\)
\(12\) 0 0
\(13\) −6.82818 −1.89380 −0.946898 − 0.321535i \(-0.895801\pi\)
−0.946898 + 0.321535i \(0.895801\pi\)
\(14\) 0 0
\(15\) −1.57882 + 1.58345i −0.407650 + 0.408846i
\(16\) 0 0
\(17\) 1.90384 0.461750 0.230875 − 0.972983i \(-0.425841\pi\)
0.230875 + 0.972983i \(0.425841\pi\)
\(18\) 0 0
\(19\) −2.53661 −0.581939 −0.290969 − 0.956732i \(-0.593978\pi\)
−0.290969 + 0.956732i \(0.593978\pi\)
\(20\) 0 0
\(21\) −1.15610 − 2.37980i −0.252282 − 0.519314i
\(22\) 0 0
\(23\) −5.50258 −1.14737 −0.573683 − 0.819077i \(-0.694486\pi\)
−0.573683 + 0.819077i \(0.694486\pi\)
\(24\) 0 0
\(25\) 0.0146493 + 4.99998i 0.00292985 + 0.999996i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 6.49989 1.20700 0.603500 − 0.797363i \(-0.293772\pi\)
0.603500 + 0.797363i \(0.293772\pi\)
\(30\) 0 0
\(31\) −5.78049 −1.03821 −0.519103 − 0.854712i \(-0.673734\pi\)
−0.519103 + 0.854712i \(0.673734\pi\)
\(32\) 0 0
\(33\) 1.26306 0.219871
\(34\) 0 0
\(35\) −5.59357 − 1.92664i −0.945487 − 0.325661i
\(36\) 0 0
\(37\) − 6.24970i − 1.02745i −0.857956 − 0.513723i \(-0.828266\pi\)
0.857956 − 0.513723i \(-0.171734\pi\)
\(38\) 0 0
\(39\) 6.82818i 1.09338i
\(40\) 0 0
\(41\) 11.5678i 1.80659i 0.429025 + 0.903293i \(0.358857\pi\)
−0.429025 + 0.903293i \(0.641143\pi\)
\(42\) 0 0
\(43\) −7.55118 −1.15154 −0.575772 − 0.817610i \(-0.695299\pi\)
−0.575772 + 0.817610i \(0.695299\pi\)
\(44\) 0 0
\(45\) 1.58345 + 1.57882i 0.236047 + 0.235357i
\(46\) 0 0
\(47\) 0.158389i 0.0231034i 0.999933 + 0.0115517i \(0.00367711\pi\)
−0.999933 + 0.0115517i \(0.996323\pi\)
\(48\) 0 0
\(49\) 4.32685 − 5.50258i 0.618122 − 0.786082i
\(50\) 0 0
\(51\) − 1.90384i − 0.266592i
\(52\) 0 0
\(53\) 0.469217i 0.0644520i 0.999481 + 0.0322260i \(0.0102596\pi\)
−0.999481 + 0.0322260i \(0.989740\pi\)
\(54\) 0 0
\(55\) 1.99415 − 2.00000i 0.268891 − 0.269680i
\(56\) 0 0
\(57\) 2.53661i 0.335983i
\(58\) 0 0
\(59\) 6.99413 0.910558 0.455279 − 0.890349i \(-0.349540\pi\)
0.455279 + 0.890349i \(0.349540\pi\)
\(60\) 0 0
\(61\) − 3.90063i − 0.499424i −0.968320 − 0.249712i \(-0.919664\pi\)
0.968320 − 0.249712i \(-0.0803359\pi\)
\(62\) 0 0
\(63\) −2.37980 + 1.15610i −0.299826 + 0.145655i
\(64\) 0 0
\(65\) 10.8121 + 10.7805i 1.34108 + 1.33715i
\(66\) 0 0
\(67\) −10.7551 −1.31394 −0.656971 − 0.753916i \(-0.728163\pi\)
−0.656971 + 0.753916i \(0.728163\pi\)
\(68\) 0 0
\(69\) 5.50258i 0.662432i
\(70\) 0 0
\(71\) 3.31768i 0.393736i 0.980430 + 0.196868i \(0.0630771\pi\)
−0.980430 + 0.196868i \(0.936923\pi\)
\(72\) 0 0
\(73\) −0.691529 −0.0809374 −0.0404687 − 0.999181i \(-0.512885\pi\)
−0.0404687 + 0.999181i \(0.512885\pi\)
\(74\) 0 0
\(75\) 4.99998 − 0.0146493i 0.577348 − 0.00169155i
\(76\) 0 0
\(77\) 1.46023 + 3.00583i 0.166409 + 0.342546i
\(78\) 0 0
\(79\) 10.1922i 1.14672i 0.819304 + 0.573359i \(0.194360\pi\)
−0.819304 + 0.573359i \(0.805640\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −3.01465 − 3.00583i −0.326985 − 0.326028i
\(86\) 0 0
\(87\) − 6.49989i − 0.696862i
\(88\) 0 0
\(89\) − 4.26668i − 0.452267i −0.974096 − 0.226133i \(-0.927391\pi\)
0.974096 − 0.226133i \(-0.0726085\pi\)
\(90\) 0 0
\(91\) −16.2497 + 7.89407i −1.70343 + 0.827523i
\(92\) 0 0
\(93\) 5.78049i 0.599409i
\(94\) 0 0
\(95\) 4.01661 + 4.00486i 0.412095 + 0.410890i
\(96\) 0 0
\(97\) −2.82332 −0.286665 −0.143332 − 0.989675i \(-0.545782\pi\)
−0.143332 + 0.989675i \(0.545782\pi\)
\(98\) 0 0
\(99\) − 1.26306i − 0.126943i
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 1680.2.q.c.559.1 ✓ 32
4.3 odd 2 inner 1680.2.q.c.559.7 yes 32
5.4 even 2 inner 1680.2.q.c.559.8 yes 32
7.6 odd 2 inner 1680.2.q.c.559.4 yes 32
20.19 odd 2 inner 1680.2.q.c.559.2 yes 32
28.27 even 2 inner 1680.2.q.c.559.6 yes 32
35.34 odd 2 inner 1680.2.q.c.559.5 yes 32
140.139 even 2 inner 1680.2.q.c.559.3 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1680.2.q.c.559.1 ✓ 32 1.1 even 1 trivial
1680.2.q.c.559.2 yes 32 20.19 odd 2 inner
1680.2.q.c.559.3 yes 32 140.139 even 2 inner
1680.2.q.c.559.4 yes 32 7.6 odd 2 inner
1680.2.q.c.559.5 yes 32 35.34 odd 2 inner
1680.2.q.c.559.6 yes 32 28.27 even 2 inner
1680.2.q.c.559.7 yes 32 4.3 odd 2 inner
1680.2.q.c.559.8 yes 32 5.4 even 2 inner