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Properties

Label 1680.2.k.f.209.1
Level $1680$
Weight $2$
Character 1680.209
Analytic conductor $13.415$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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 = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,8] 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: 8.0.10070523904.11
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.1
Root \(-1.68014 + 0.420861i\) of defining polynomial
Character \(\chi\) \(=\) 1680.209
Dual form 1680.2.k.f.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)
 
\(f(q)\) \(=\) \(q+(-1.68014 - 0.420861i) q^{3} +(-1.08495 - 1.95522i) q^{5} +(0.595188 - 2.57794i) q^{7} +(2.64575 + 1.41421i) q^{9} +2.82843i q^{11} -3.36028 q^{13} +(1.00000 + 3.74166i) q^{15} +4.75216i q^{17} +5.59388i q^{19} +(-2.08495 + 4.08080i) q^{21} +7.29150 q^{23} +(-2.64575 + 4.24264i) q^{25} +(-3.85005 - 3.48957i) q^{27} +0.500983i q^{29} -3.06871i q^{31} +(1.19038 - 4.75216i) q^{33} +(-5.68618 + 1.63322i) q^{35} -3.32941i q^{37} +(5.64575 + 1.41421i) q^{39} +4.33981 q^{41} +10.3117i q^{43} +(-0.105422 - 6.70738i) q^{45} -7.82087i q^{47} +(-6.29150 - 3.06871i) q^{49} +(2.00000 - 7.98430i) q^{51} +8.58301 q^{53} +(5.53019 - 3.06871i) q^{55} +(2.35425 - 9.39851i) q^{57} +2.16991 q^{59} +2.52517i q^{61} +(5.22047 - 5.97885i) q^{63} +(3.64575 + 6.57008i) q^{65} -10.3117i q^{67} +(-12.2508 - 3.06871i) q^{69} +9.81076i q^{71} -5.53019 q^{73} +(6.23080 - 6.01474i) q^{75} +(7.29150 + 1.68345i) q^{77} -3.29150 q^{79} +(5.00000 + 7.48331i) q^{81} +6.97915i q^{83} +(9.29150 - 5.15587i) q^{85} +(0.210845 - 0.841723i) q^{87} +15.8219 q^{89} +(-2.00000 + 8.66259i) q^{91} +(-1.29150 + 5.15587i) q^{93} +(10.9373 - 6.06910i) q^{95} +14.6315 q^{97} +(-4.00000 + 7.48331i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{15} - 8 q^{21} + 16 q^{23} + 16 q^{35} + 24 q^{39} - 8 q^{49} + 16 q^{51} - 16 q^{53} + 40 q^{57} + 8 q^{63} + 8 q^{65} + 16 q^{77} + 16 q^{79} + 40 q^{81} + 32 q^{85} - 16 q^{91} + 32 q^{93}+ \cdots - 32 q^{99}+O(q^{100}) \) 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
\(3\) −1.68014 − 0.420861i −0.970030 − 0.242984i
\(4\) 0 0
\(5\) −1.08495 − 1.95522i −0.485206 − 0.874400i
\(6\) 0 0
\(7\) 0.595188 − 2.57794i 0.224960 − 0.974368i
\(8\) 0 0
\(9\) 2.64575 + 1.41421i 0.881917 + 0.471405i
\(10\) 0 0
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) −3.36028 −0.931975 −0.465987 − 0.884791i \(-0.654301\pi\)
−0.465987 + 0.884791i \(0.654301\pi\)
\(14\) 0 0
\(15\) 1.00000 + 3.74166i 0.258199 + 0.966092i
\(16\) 0 0
\(17\) 4.75216i 1.15257i 0.817250 + 0.576284i \(0.195498\pi\)
−0.817250 + 0.576284i \(0.804502\pi\)
\(18\) 0 0
\(19\) 5.59388i 1.28332i 0.766987 + 0.641662i \(0.221755\pi\)
−0.766987 + 0.641662i \(0.778245\pi\)
\(20\) 0 0
\(21\) −2.08495 + 4.08080i −0.454974 + 0.890505i
\(22\) 0 0
\(23\) 7.29150 1.52038 0.760192 − 0.649699i \(-0.225105\pi\)
0.760192 + 0.649699i \(0.225105\pi\)
\(24\) 0 0
\(25\) −2.64575 + 4.24264i −0.529150 + 0.848528i
\(26\) 0 0
\(27\) −3.85005 − 3.48957i −0.740942 − 0.671569i
\(28\) 0 0
\(29\) 0.500983i 0.0930303i 0.998918 + 0.0465151i \(0.0148116\pi\)
−0.998918 + 0.0465151i \(0.985188\pi\)
\(30\) 0 0
\(31\) − 3.06871i − 0.551157i −0.961279 − 0.275578i \(-0.911131\pi\)
0.961279 − 0.275578i \(-0.0888694\pi\)
\(32\) 0 0
\(33\) 1.19038 − 4.75216i 0.207218 − 0.827245i
\(34\) 0 0
\(35\) −5.68618 + 1.63322i −0.961139 + 0.276064i
\(36\) 0 0
\(37\) − 3.32941i − 0.547352i −0.961822 − 0.273676i \(-0.911760\pi\)
0.961822 − 0.273676i \(-0.0882396\pi\)
\(38\) 0 0
\(39\) 5.64575 + 1.41421i 0.904044 + 0.226455i
\(40\) 0 0
\(41\) 4.33981 0.677765 0.338883 − 0.940829i \(-0.389951\pi\)
0.338883 + 0.940829i \(0.389951\pi\)
\(42\) 0 0
\(43\) 10.3117i 1.57253i 0.617892 + 0.786263i \(0.287987\pi\)
−0.617892 + 0.786263i \(0.712013\pi\)
\(44\) 0 0
\(45\) −0.105422 − 6.70738i −0.0157154 − 0.999877i
\(46\) 0 0
\(47\) − 7.82087i − 1.14079i −0.821370 − 0.570396i \(-0.806790\pi\)
0.821370 − 0.570396i \(-0.193210\pi\)
\(48\) 0 0
\(49\) −6.29150 − 3.06871i −0.898786 − 0.438387i
\(50\) 0 0
\(51\) 2.00000 − 7.98430i 0.280056 − 1.11803i
\(52\) 0 0
\(53\) 8.58301 1.17897 0.589483 − 0.807781i \(-0.299331\pi\)
0.589483 + 0.807781i \(0.299331\pi\)
\(54\) 0 0
\(55\) 5.53019 − 3.06871i 0.745691 − 0.413785i
\(56\) 0 0
\(57\) 2.35425 − 9.39851i 0.311828 − 1.24486i
\(58\) 0 0
\(59\) 2.16991 0.282498 0.141249 − 0.989974i \(-0.454888\pi\)
0.141249 + 0.989974i \(0.454888\pi\)
\(60\) 0 0
\(61\) 2.52517i 0.323315i 0.986847 + 0.161657i \(0.0516839\pi\)
−0.986847 + 0.161657i \(0.948316\pi\)
\(62\) 0 0
\(63\) 5.22047 − 5.97885i 0.657717 − 0.753265i
\(64\) 0 0
\(65\) 3.64575 + 6.57008i 0.452200 + 0.814919i
\(66\) 0 0
\(67\) − 10.3117i − 1.25978i −0.776684 − 0.629890i \(-0.783100\pi\)
0.776684 − 0.629890i \(-0.216900\pi\)
\(68\) 0 0
\(69\) −12.2508 − 3.06871i −1.47482 − 0.369430i
\(70\) 0 0
\(71\) 9.81076i 1.16432i 0.813073 + 0.582161i \(0.197793\pi\)
−0.813073 + 0.582161i \(0.802207\pi\)
\(72\) 0 0
\(73\) −5.53019 −0.647260 −0.323630 − 0.946184i \(-0.604903\pi\)
−0.323630 + 0.946184i \(0.604903\pi\)
\(74\) 0 0
\(75\) 6.23080 − 6.01474i 0.719471 − 0.694523i
\(76\) 0 0
\(77\) 7.29150 + 1.68345i 0.830944 + 0.191846i
\(78\) 0 0
\(79\) −3.29150 −0.370323 −0.185161 − 0.982708i \(-0.559281\pi\)
−0.185161 + 0.982708i \(0.559281\pi\)
\(80\) 0 0
\(81\) 5.00000 + 7.48331i 0.555556 + 0.831479i
\(82\) 0 0
\(83\) 6.97915i 0.766061i 0.923736 + 0.383030i \(0.125120\pi\)
−0.923736 + 0.383030i \(0.874880\pi\)
\(84\) 0 0
\(85\) 9.29150 − 5.15587i 1.00780 − 0.559233i
\(86\) 0 0
\(87\) 0.210845 − 0.841723i 0.0226049 − 0.0902422i
\(88\) 0 0
\(89\) 15.8219 1.67712 0.838558 − 0.544812i \(-0.183399\pi\)
0.838558 + 0.544812i \(0.183399\pi\)
\(90\) 0 0
\(91\) −2.00000 + 8.66259i −0.209657 + 0.908086i
\(92\) 0 0
\(93\) −1.29150 + 5.15587i −0.133923 + 0.534639i
\(94\) 0 0
\(95\) 10.9373 − 6.06910i 1.12214 − 0.622677i
\(96\) 0 0
\(97\) 14.6315 1.48560 0.742802 − 0.669511i \(-0.233496\pi\)
0.742802 + 0.669511i \(0.233496\pi\)
\(98\) 0 0
\(99\) −4.00000 + 7.48331i −0.402015 + 0.752101i
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.f.209.1 8
3.2 odd 2 1680.2.k.e.209.2 8
4.3 odd 2 210.2.d.b.209.8 yes 8
5.4 even 2 1680.2.k.e.209.8 8
7.6 odd 2 inner 1680.2.k.f.209.8 8
12.11 even 2 210.2.d.a.209.7 yes 8
15.14 odd 2 inner 1680.2.k.f.209.7 8
20.3 even 4 1050.2.b.f.251.4 16
20.7 even 4 1050.2.b.f.251.13 16
20.19 odd 2 210.2.d.a.209.1 ✓ 8
21.20 even 2 1680.2.k.e.209.7 8
28.27 even 2 210.2.d.b.209.1 yes 8
35.34 odd 2 1680.2.k.e.209.1 8
60.23 odd 4 1050.2.b.f.251.14 16
60.47 odd 4 1050.2.b.f.251.3 16
60.59 even 2 210.2.d.b.209.2 yes 8
84.83 odd 2 210.2.d.a.209.2 yes 8
105.104 even 2 inner 1680.2.k.f.209.2 8
140.27 odd 4 1050.2.b.f.251.12 16
140.83 odd 4 1050.2.b.f.251.5 16
140.139 even 2 210.2.d.a.209.8 yes 8
420.83 even 4 1050.2.b.f.251.11 16
420.167 even 4 1050.2.b.f.251.6 16
420.419 odd 2 210.2.d.b.209.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.d.a.209.1 ✓ 8 20.19 odd 2
210.2.d.a.209.2 yes 8 84.83 odd 2
210.2.d.a.209.7 yes 8 12.11 even 2
210.2.d.a.209.8 yes 8 140.139 even 2
210.2.d.b.209.1 yes 8 28.27 even 2
210.2.d.b.209.2 yes 8 60.59 even 2
210.2.d.b.209.7 yes 8 420.419 odd 2
210.2.d.b.209.8 yes 8 4.3 odd 2
1050.2.b.f.251.3 16 60.47 odd 4
1050.2.b.f.251.4 16 20.3 even 4
1050.2.b.f.251.5 16 140.83 odd 4
1050.2.b.f.251.6 16 420.167 even 4
1050.2.b.f.251.11 16 420.83 even 4
1050.2.b.f.251.12 16 140.27 odd 4
1050.2.b.f.251.13 16 20.7 even 4
1050.2.b.f.251.14 16 60.23 odd 4
1680.2.k.e.209.1 8 35.34 odd 2
1680.2.k.e.209.2 8 3.2 odd 2
1680.2.k.e.209.7 8 21.20 even 2
1680.2.k.e.209.8 8 5.4 even 2
1680.2.k.f.209.1 8 1.1 even 1 trivial
1680.2.k.f.209.2 8 105.104 even 2 inner
1680.2.k.f.209.7 8 15.14 odd 2 inner
1680.2.k.f.209.8 8 7.6 odd 2 inner