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Properties

Label 1680.2.k.g.209.4
Level $1680$
Weight $2$
Character 1680.209
Analytic conductor $13.415$
Analytic rank $0$
Dimension $8$
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(209,1680)] 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("1680.209"); 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([0, 0, 1, 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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(i, \sqrt{2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.4
Root \(-1.14412 - 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 1680.209
Dual form 1680.2.k.g.209.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.41421 + 1.00000i) q^{3} +2.23607 q^{5} +(-1.41421 - 2.23607i) q^{7} +(1.00000 - 2.82843i) q^{9} +2.82843i q^{11} -2.82843 q^{13} +(-3.16228 + 2.23607i) q^{15} +4.00000i q^{17} -6.32456i q^{19} +(4.23607 + 1.74806i) q^{21} +6.32456 q^{23} +5.00000 q^{25} +(1.41421 + 5.00000i) q^{27} -5.65685i q^{29} +6.32456i q^{31} +(-2.82843 - 4.00000i) q^{33} +(-3.16228 - 5.00000i) q^{35} -8.94427i q^{37} +(4.00000 - 2.82843i) q^{39} +4.47214 q^{41} -4.47214i q^{43} +(2.23607 - 6.32456i) q^{45} -6.00000i q^{47} +(-3.00000 + 6.32456i) q^{49} +(-4.00000 - 5.65685i) q^{51} -6.32456 q^{53} +6.32456i q^{55} +(6.32456 + 8.94427i) q^{57} +8.94427 q^{59} +(-7.73877 + 1.76393i) q^{63} -6.32456 q^{65} +4.47214i q^{67} +(-8.94427 + 6.32456i) q^{69} -14.1421i q^{71} +8.48528 q^{73} +(-7.07107 + 5.00000i) q^{75} +(6.32456 - 4.00000i) q^{77} +12.0000 q^{79} +(-7.00000 - 5.65685i) q^{81} -10.0000i q^{83} +8.94427i q^{85} +(5.65685 + 8.00000i) q^{87} -4.47214 q^{89} +(4.00000 + 6.32456i) q^{91} +(-6.32456 - 8.94427i) q^{93} -14.1421i q^{95} +8.48528 q^{97} +(8.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} + 16 q^{21} + 40 q^{25} + 32 q^{39} - 24 q^{49} - 32 q^{51} + 96 q^{79} - 56 q^{81} + 32 q^{91} + 64 q^{99}+O(q^{100}) \) 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
\(3\) −1.41421 + 1.00000i −0.816497 + 0.577350i
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) −1.41421 − 2.23607i −0.534522 − 0.845154i
\(8\) 0 0
\(9\) 1.00000 − 2.82843i 0.333333 − 0.942809i
\(10\) 0 0
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) −2.82843 −0.784465 −0.392232 − 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) 0 0
\(15\) −3.16228 + 2.23607i −0.816497 + 0.577350i
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) − 6.32456i − 1.45095i −0.688247 − 0.725476i \(-0.741620\pi\)
0.688247 − 0.725476i \(-0.258380\pi\)
\(20\) 0 0
\(21\) 4.23607 + 1.74806i 0.924386 + 0.381459i
\(22\) 0 0
\(23\) 6.32456 1.31876 0.659380 − 0.751809i \(-0.270819\pi\)
0.659380 + 0.751809i \(0.270819\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 1.41421 + 5.00000i 0.272166 + 0.962250i
\(28\) 0 0
\(29\) − 5.65685i − 1.05045i −0.850963 − 0.525226i \(-0.823981\pi\)
0.850963 − 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) 6.32456i 1.13592i 0.823055 + 0.567962i \(0.192268\pi\)
−0.823055 + 0.567962i \(0.807732\pi\)
\(32\) 0 0
\(33\) −2.82843 − 4.00000i −0.492366 − 0.696311i
\(34\) 0 0
\(35\) −3.16228 − 5.00000i −0.534522 − 0.845154i
\(36\) 0 0
\(37\) − 8.94427i − 1.47043i −0.677834 − 0.735215i \(-0.737081\pi\)
0.677834 − 0.735215i \(-0.262919\pi\)
\(38\) 0 0
\(39\) 4.00000 − 2.82843i 0.640513 − 0.452911i
\(40\) 0 0
\(41\) 4.47214 0.698430 0.349215 − 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) − 4.47214i − 0.681994i −0.940064 − 0.340997i \(-0.889235\pi\)
0.940064 − 0.340997i \(-0.110765\pi\)
\(44\) 0 0
\(45\) 2.23607 − 6.32456i 0.333333 − 0.942809i
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 − 0.437595i \(-0.855830\pi\)
0.899172 − 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) −3.00000 + 6.32456i −0.428571 + 0.903508i
\(50\) 0 0
\(51\) −4.00000 − 5.65685i −0.560112 − 0.792118i
\(52\) 0 0
\(53\) −6.32456 −0.868744 −0.434372 − 0.900733i \(-0.643030\pi\)
−0.434372 + 0.900733i \(0.643030\pi\)
\(54\) 0 0
\(55\) 6.32456i 0.852803i
\(56\) 0 0
\(57\) 6.32456 + 8.94427i 0.837708 + 1.18470i
\(58\) 0 0
\(59\) 8.94427 1.16445 0.582223 − 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −7.73877 + 1.76393i −0.974993 + 0.222235i
\(64\) 0 0
\(65\) −6.32456 −0.784465
\(66\) 0 0
\(67\) 4.47214i 0.546358i 0.961963 + 0.273179i \(0.0880752\pi\)
−0.961963 + 0.273179i \(0.911925\pi\)
\(68\) 0 0
\(69\) −8.94427 + 6.32456i −1.07676 + 0.761387i
\(70\) 0 0
\(71\) − 14.1421i − 1.67836i −0.543852 − 0.839181i \(-0.683035\pi\)
0.543852 − 0.839181i \(-0.316965\pi\)
\(72\) 0 0
\(73\) 8.48528 0.993127 0.496564 − 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 0 0
\(75\) −7.07107 + 5.00000i −0.816497 + 0.577350i
\(76\) 0 0
\(77\) 6.32456 − 4.00000i 0.720750 − 0.455842i
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −7.00000 − 5.65685i −0.777778 − 0.628539i
\(82\) 0 0
\(83\) − 10.0000i − 1.09764i −0.835940 − 0.548821i \(-0.815077\pi\)
0.835940 − 0.548821i \(-0.184923\pi\)
\(84\) 0 0
\(85\) 8.94427i 0.970143i
\(86\) 0 0
\(87\) 5.65685 + 8.00000i 0.606478 + 0.857690i
\(88\) 0 0
\(89\) −4.47214 −0.474045 −0.237023 − 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) 0 0
\(91\) 4.00000 + 6.32456i 0.419314 + 0.662994i
\(92\) 0 0
\(93\) −6.32456 − 8.94427i −0.655826 − 0.927478i
\(94\) 0 0
\(95\) − 14.1421i − 1.45095i
\(96\) 0 0
\(97\) 8.48528 0.861550 0.430775 − 0.902459i \(-0.358240\pi\)
0.430775 + 0.902459i \(0.358240\pi\)
\(98\) 0 0
\(99\) 8.00000 + 2.82843i 0.804030 + 0.284268i
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.k.g.209.4 8
3.2 odd 2 inner 1680.2.k.g.209.1 8
4.3 odd 2 420.2.f.b.209.6 yes 8
5.4 even 2 inner 1680.2.k.g.209.6 8
7.6 odd 2 inner 1680.2.k.g.209.5 8
12.11 even 2 420.2.f.b.209.7 yes 8
15.14 odd 2 inner 1680.2.k.g.209.7 8
20.3 even 4 2100.2.d.i.1301.4 4
20.7 even 4 2100.2.d.f.1301.1 4
20.19 odd 2 420.2.f.b.209.4 yes 8
21.20 even 2 inner 1680.2.k.g.209.8 8
28.27 even 2 420.2.f.b.209.3 yes 8
35.34 odd 2 inner 1680.2.k.g.209.3 8
60.23 odd 4 2100.2.d.f.1301.4 4
60.47 odd 4 2100.2.d.i.1301.1 4
60.59 even 2 420.2.f.b.209.1 ✓ 8
84.83 odd 2 420.2.f.b.209.2 yes 8
105.104 even 2 inner 1680.2.k.g.209.2 8
140.27 odd 4 2100.2.d.i.1301.3 4
140.83 odd 4 2100.2.d.f.1301.2 4
140.139 even 2 420.2.f.b.209.5 yes 8
420.83 even 4 2100.2.d.i.1301.2 4
420.167 even 4 2100.2.d.f.1301.3 4
420.419 odd 2 420.2.f.b.209.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.f.b.209.1 ✓ 8 60.59 even 2
420.2.f.b.209.2 yes 8 84.83 odd 2
420.2.f.b.209.3 yes 8 28.27 even 2
420.2.f.b.209.4 yes 8 20.19 odd 2
420.2.f.b.209.5 yes 8 140.139 even 2
420.2.f.b.209.6 yes 8 4.3 odd 2
420.2.f.b.209.7 yes 8 12.11 even 2
420.2.f.b.209.8 yes 8 420.419 odd 2
1680.2.k.g.209.1 8 3.2 odd 2 inner
1680.2.k.g.209.2 8 105.104 even 2 inner
1680.2.k.g.209.3 8 35.34 odd 2 inner
1680.2.k.g.209.4 8 1.1 even 1 trivial
1680.2.k.g.209.5 8 7.6 odd 2 inner
1680.2.k.g.209.6 8 5.4 even 2 inner
1680.2.k.g.209.7 8 15.14 odd 2 inner
1680.2.k.g.209.8 8 21.20 even 2 inner
2100.2.d.f.1301.1 4 20.7 even 4
2100.2.d.f.1301.2 4 140.83 odd 4
2100.2.d.f.1301.3 4 420.167 even 4
2100.2.d.f.1301.4 4 60.23 odd 4
2100.2.d.i.1301.1 4 60.47 odd 4
2100.2.d.i.1301.2 4 420.83 even 4
2100.2.d.i.1301.3 4 140.27 odd 4
2100.2.d.i.1301.4 4 20.3 even 4