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Properties

Label 1680.2.q.c.559.17
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 559.17
Character \(\chi\) \(=\) 1680.559
Dual form 1680.2.q.c.559.19

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-0.890213 - 2.05122i) q^{5} +(-0.425671 + 2.61128i) q^{7} -1.00000 q^{9} +2.24665i q^{11} +2.86877 q^{13} +(-2.05122 + 0.890213i) q^{15} -0.466226 q^{17} +4.40993 q^{19} +(2.61128 + 0.425671i) q^{21} -2.22309 q^{23} +(-3.41504 + 3.65205i) q^{25} +1.00000i q^{27} +0.668748 q^{29} +9.53655 q^{31} +2.24665 q^{33} +(5.73527 - 1.45145i) q^{35} +6.08295i q^{37} -2.86877i q^{39} -7.46155i q^{41} +3.53577 q^{43} +(0.890213 + 2.05122i) q^{45} +8.72140i q^{47} +(-6.63761 - 2.22309i) q^{49} +0.466226i q^{51} +3.45359i q^{53} +(4.60839 - 2.00000i) q^{55} -4.40993i q^{57} +8.26044 q^{59} -7.17688i q^{61} +(0.425671 - 2.61128i) q^{63} +(-2.55382 - 5.88450i) q^{65} +13.4434 q^{67} +2.22309i q^{69} -0.252645i q^{71} +14.5429 q^{73} +(3.65205 + 3.41504i) q^{75} +(-5.86665 - 0.956334i) q^{77} -9.20165i q^{79} +1.00000 q^{81} +(0.415041 + 0.956334i) q^{85} -0.668748i q^{87} -13.9638i q^{89} +(-1.22115 + 7.49118i) q^{91} -9.53655i q^{93} +(-3.92578 - 9.04575i) q^{95} -6.17698 q^{97} -2.24665i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 32 q^{9} + 8 q^{21} - 32 q^{25} + 16 q^{49} + 48 q^{65} + 32 q^{81} - 64 q^{85}+O(q^{100}) \) 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
\(5\) −0.890213 − 2.05122i −0.398115 − 0.917335i
\(6\) 0 0
\(7\) −0.425671 + 2.61128i −0.160888 + 0.986973i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.24665i 0.677391i 0.940896 + 0.338696i \(0.109986\pi\)
−0.940896 + 0.338696i \(0.890014\pi\)
\(12\) 0 0
\(13\) 2.86877 0.795654 0.397827 − 0.917460i \(-0.369764\pi\)
0.397827 + 0.917460i \(0.369764\pi\)
\(14\) 0 0
\(15\) −2.05122 + 0.890213i −0.529624 + 0.229852i
\(16\) 0 0
\(17\) −0.466226 −0.113076 −0.0565382 − 0.998400i \(-0.518006\pi\)
−0.0565382 + 0.998400i \(0.518006\pi\)
\(18\) 0 0
\(19\) 4.40993 1.01171 0.505854 − 0.862619i \(-0.331178\pi\)
0.505854 + 0.862619i \(0.331178\pi\)
\(20\) 0 0
\(21\) 2.61128 + 0.425671i 0.569829 + 0.0928890i
\(22\) 0 0
\(23\) −2.22309 −0.463547 −0.231774 − 0.972770i \(-0.574453\pi\)
−0.231774 + 0.972770i \(0.574453\pi\)
\(24\) 0 0
\(25\) −3.41504 + 3.65205i −0.683008 + 0.730411i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0.668748 0.124183 0.0620917 − 0.998070i \(-0.480223\pi\)
0.0620917 + 0.998070i \(0.480223\pi\)
\(30\) 0 0
\(31\) 9.53655 1.71281 0.856407 − 0.516301i \(-0.172691\pi\)
0.856407 + 0.516301i \(0.172691\pi\)
\(32\) 0 0
\(33\) 2.24665 0.391092
\(34\) 0 0
\(35\) 5.73527 − 1.45145i 0.969437 − 0.245340i
\(36\) 0 0
\(37\) 6.08295i 1.00003i 0.866016 + 0.500016i \(0.166672\pi\)
−0.866016 + 0.500016i \(0.833328\pi\)
\(38\) 0 0
\(39\) − 2.86877i − 0.459371i
\(40\) 0 0
\(41\) − 7.46155i − 1.16530i −0.812724 − 0.582649i \(-0.802016\pi\)
0.812724 − 0.582649i \(-0.197984\pi\)
\(42\) 0 0
\(43\) 3.53577 0.539200 0.269600 − 0.962972i \(-0.413108\pi\)
0.269600 + 0.962972i \(0.413108\pi\)
\(44\) 0 0
\(45\) 0.890213 + 2.05122i 0.132705 + 0.305778i
\(46\) 0 0
\(47\) 8.72140i 1.27215i 0.771629 + 0.636073i \(0.219442\pi\)
−0.771629 + 0.636073i \(0.780558\pi\)
\(48\) 0 0
\(49\) −6.63761 − 2.22309i −0.948230 − 0.317585i
\(50\) 0 0
\(51\) 0.466226i 0.0652847i
\(52\) 0 0
\(53\) 3.45359i 0.474388i 0.971462 + 0.237194i \(0.0762277\pi\)
−0.971462 + 0.237194i \(0.923772\pi\)
\(54\) 0 0
\(55\) 4.60839 − 2.00000i 0.621395 − 0.269680i
\(56\) 0 0
\(57\) − 4.40993i − 0.584109i
\(58\) 0 0
\(59\) 8.26044 1.07542 0.537709 − 0.843131i \(-0.319290\pi\)
0.537709 + 0.843131i \(0.319290\pi\)
\(60\) 0 0
\(61\) − 7.17688i − 0.918906i −0.888202 − 0.459453i \(-0.848045\pi\)
0.888202 − 0.459453i \(-0.151955\pi\)
\(62\) 0 0
\(63\) 0.425671 − 2.61128i 0.0536295 − 0.328991i
\(64\) 0 0
\(65\) −2.55382 − 5.88450i −0.316762 − 0.729882i
\(66\) 0 0
\(67\) 13.4434 1.64237 0.821183 − 0.570664i \(-0.193314\pi\)
0.821183 + 0.570664i \(0.193314\pi\)
\(68\) 0 0
\(69\) 2.22309i 0.267629i
\(70\) 0 0
\(71\) − 0.252645i − 0.0299835i −0.999888 − 0.0149917i \(-0.995228\pi\)
0.999888 − 0.0149917i \(-0.00477220\pi\)
\(72\) 0 0
\(73\) 14.5429 1.70212 0.851060 − 0.525068i \(-0.175960\pi\)
0.851060 + 0.525068i \(0.175960\pi\)
\(74\) 0 0
\(75\) 3.65205 + 3.41504i 0.421703 + 0.394335i
\(76\) 0 0
\(77\) −5.86665 − 0.956334i −0.668567 − 0.108984i
\(78\) 0 0
\(79\) − 9.20165i − 1.03527i −0.855603 − 0.517633i \(-0.826813\pi\)
0.855603 − 0.517633i \(-0.173187\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0.415041 + 0.956334i 0.0450175 + 0.103729i
\(86\) 0 0
\(87\) − 0.668748i − 0.0716974i
\(88\) 0 0
\(89\) − 13.9638i − 1.48016i −0.672521 − 0.740078i \(-0.734789\pi\)
0.672521 − 0.740078i \(-0.265211\pi\)
\(90\) 0 0
\(91\) −1.22115 + 7.49118i −0.128012 + 0.785289i
\(92\) 0 0
\(93\) − 9.53655i − 0.988894i
\(94\) 0 0
\(95\) −3.92578 − 9.04575i −0.402776 − 0.928075i
\(96\) 0 0
\(97\) −6.17698 −0.627177 −0.313589 − 0.949559i \(-0.601531\pi\)
−0.313589 + 0.949559i \(0.601531\pi\)
\(98\) 0 0
\(99\) − 2.24665i − 0.225797i
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.559.17 yes 32
4.3 odd 2 inner 1680.2.q.c.559.11 yes 32
5.4 even 2 inner 1680.2.q.c.559.12 yes 32
7.6 odd 2 inner 1680.2.q.c.559.20 yes 32
20.19 odd 2 inner 1680.2.q.c.559.18 yes 32
28.27 even 2 inner 1680.2.q.c.559.10 yes 32
35.34 odd 2 inner 1680.2.q.c.559.9 ✓ 32
140.139 even 2 inner 1680.2.q.c.559.19 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1680.2.q.c.559.9 ✓ 32 35.34 odd 2 inner
1680.2.q.c.559.10 yes 32 28.27 even 2 inner
1680.2.q.c.559.11 yes 32 4.3 odd 2 inner
1680.2.q.c.559.12 yes 32 5.4 even 2 inner
1680.2.q.c.559.17 yes 32 1.1 even 1 trivial
1680.2.q.c.559.18 yes 32 20.19 odd 2 inner
1680.2.q.c.559.19 yes 32 140.139 even 2 inner
1680.2.q.c.559.20 yes 32 7.6 odd 2 inner