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Properties

Label 210.2.d.a.209.1
Level $210$
Weight $2$
Character 210.209
Analytic conductor $1.677$
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] = [210,2,Mod(209,210)] 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("210.209"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.67685844245\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.1
Root \(-1.68014 + 0.420861i\) of defining polynomial
Character \(\chi\) \(=\) 210.209
Dual form 210.2.d.a.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-1.00000 q^{2} +(-1.68014 - 0.420861i) q^{3} +1.00000 q^{4} +(-1.08495 + 1.95522i) q^{5} +(1.68014 + 0.420861i) q^{6} +(0.595188 - 2.57794i) q^{7} -1.00000 q^{8} +(2.64575 + 1.41421i) q^{9} +(1.08495 - 1.95522i) q^{10} -2.82843i q^{11} +(-1.68014 - 0.420861i) q^{12} +3.36028 q^{13} +(-0.595188 + 2.57794i) q^{14} +(2.64575 - 2.82843i) q^{15} +1.00000 q^{16} -4.75216i q^{17} +(-2.64575 - 1.41421i) q^{18} -5.59388i q^{19} +(-1.08495 + 1.95522i) q^{20} +(-2.08495 + 4.08080i) q^{21} +2.82843i q^{22} +7.29150 q^{23} +(1.68014 + 0.420861i) q^{24} +(-2.64575 - 4.24264i) q^{25} -3.36028 q^{26} +(-3.85005 - 3.48957i) q^{27} +(0.595188 - 2.57794i) q^{28} +0.500983i q^{29} +(-2.64575 + 2.82843i) q^{30} +3.06871i q^{31} -1.00000 q^{32} +(-1.19038 + 4.75216i) q^{33} +4.75216i q^{34} +(4.39467 + 3.96066i) q^{35} +(2.64575 + 1.41421i) q^{36} +3.32941i q^{37} +5.59388i q^{38} +(-5.64575 - 1.41421i) q^{39} +(1.08495 - 1.95522i) q^{40} +4.33981 q^{41} +(2.08495 - 4.08080i) q^{42} +10.3117i q^{43} -2.82843i q^{44} +(-5.63561 + 3.63866i) q^{45} -7.29150 q^{46} -7.82087i q^{47} +(-1.68014 - 0.420861i) q^{48} +(-6.29150 - 3.06871i) q^{49} +(2.64575 + 4.24264i) q^{50} +(-2.00000 + 7.98430i) q^{51} +3.36028 q^{52} -8.58301 q^{53} +(3.85005 + 3.48957i) q^{54} +(5.53019 + 3.06871i) q^{55} +(-0.595188 + 2.57794i) q^{56} +(-2.35425 + 9.39851i) q^{57} -0.500983i q^{58} -2.16991 q^{59} +(2.64575 - 2.82843i) q^{60} +2.52517i q^{61} -3.06871i q^{62} +(5.22047 - 5.97885i) q^{63} +1.00000 q^{64} +(-3.64575 + 6.57008i) q^{65} +(1.19038 - 4.75216i) q^{66} -10.3117i q^{67} -4.75216i q^{68} +(-12.2508 - 3.06871i) q^{69} +(-4.39467 - 3.96066i) q^{70} -9.81076i q^{71} +(-2.64575 - 1.41421i) q^{72} +5.53019 q^{73} -3.32941i q^{74} +(2.65967 + 8.24173i) q^{75} -5.59388i q^{76} +(-7.29150 - 1.68345i) q^{77} +(5.64575 + 1.41421i) q^{78} +3.29150 q^{79} +(-1.08495 + 1.95522i) q^{80} +(5.00000 + 7.48331i) q^{81} -4.33981 q^{82} +6.97915i q^{83} +(-2.08495 + 4.08080i) q^{84} +(9.29150 + 5.15587i) q^{85} -10.3117i q^{86} +(0.210845 - 0.841723i) q^{87} +2.82843i q^{88} +15.8219 q^{89} +(5.63561 - 3.63866i) q^{90} +(2.00000 - 8.66259i) q^{91} +7.29150 q^{92} +(1.29150 - 5.15587i) q^{93} +7.82087i q^{94} +(10.9373 + 6.06910i) q^{95} +(1.68014 + 0.420861i) q^{96} -14.6315 q^{97} +(6.29150 + 3.06871i) q^{98} +(4.00000 - 7.48331i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 8 q^{21} + 16 q^{23} - 8 q^{32} + 16 q^{35} - 24 q^{39} + 8 q^{42} - 16 q^{46} - 8 q^{49} - 16 q^{51} + 16 q^{53} - 40 q^{57} + 8 q^{63} + 8 q^{64} - 8 q^{65}+ \cdots + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.68014 − 0.420861i −0.970030 − 0.242984i
\(4\) 1.00000 0.500000
\(5\) −1.08495 + 1.95522i −0.485206 + 0.874400i
\(6\) 1.68014 + 0.420861i 0.685915 + 0.171816i
\(7\) 0.595188 − 2.57794i 0.224960 − 0.974368i
\(8\) −1.00000 −0.353553
\(9\) 2.64575 + 1.41421i 0.881917 + 0.471405i
\(10\) 1.08495 − 1.95522i 0.343092 − 0.618294i
\(11\) − 2.82843i − 0.852803i −0.904534 − 0.426401i \(-0.859781\pi\)
0.904534 − 0.426401i \(-0.140219\pi\)
\(12\) −1.68014 − 0.420861i −0.485015 − 0.121492i
\(13\) 3.36028 0.931975 0.465987 − 0.884791i \(-0.345699\pi\)
0.465987 + 0.884791i \(0.345699\pi\)
\(14\) −0.595188 + 2.57794i −0.159071 + 0.688982i
\(15\) 2.64575 − 2.82843i 0.683130 − 0.730297i
\(16\) 1.00000 0.250000
\(17\) − 4.75216i − 1.15257i −0.817250 − 0.576284i \(-0.804502\pi\)
0.817250 − 0.576284i \(-0.195498\pi\)
\(18\) −2.64575 − 1.41421i −0.623610 − 0.333333i
\(19\) − 5.59388i − 1.28332i −0.766987 − 0.641662i \(-0.778245\pi\)
0.766987 − 0.641662i \(-0.221755\pi\)
\(20\) −1.08495 + 1.95522i −0.242603 + 0.437200i
\(21\) −2.08495 + 4.08080i −0.454974 + 0.890505i
\(22\) 2.82843i 0.603023i
\(23\) 7.29150 1.52038 0.760192 − 0.649699i \(-0.225105\pi\)
0.760192 + 0.649699i \(0.225105\pi\)
\(24\) 1.68014 + 0.420861i 0.342957 + 0.0859080i
\(25\) −2.64575 − 4.24264i −0.529150 − 0.848528i
\(26\) −3.36028 −0.659006
\(27\) −3.85005 − 3.48957i −0.740942 − 0.671569i
\(28\) 0.595188 − 2.57794i 0.112480 − 0.487184i
\(29\) 0.500983i 0.0930303i 0.998918 + 0.0465151i \(0.0148116\pi\)
−0.998918 + 0.0465151i \(0.985188\pi\)
\(30\) −2.64575 + 2.82843i −0.483046 + 0.516398i
\(31\) 3.06871i 0.551157i 0.961279 + 0.275578i \(0.0888694\pi\)
−0.961279 + 0.275578i \(0.911131\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.19038 + 4.75216i −0.207218 + 0.827245i
\(34\) 4.75216i 0.814988i
\(35\) 4.39467 + 3.96066i 0.742835 + 0.669474i
\(36\) 2.64575 + 1.41421i 0.440959 + 0.235702i
\(37\) 3.32941i 0.547352i 0.961822 + 0.273676i \(0.0882396\pi\)
−0.961822 + 0.273676i \(0.911760\pi\)
\(38\) 5.59388i 0.907447i
\(39\) −5.64575 − 1.41421i −0.904044 − 0.226455i
\(40\) 1.08495 − 1.95522i 0.171546 − 0.309147i
\(41\) 4.33981 0.677765 0.338883 − 0.940829i \(-0.389951\pi\)
0.338883 + 0.940829i \(0.389951\pi\)
\(42\) 2.08495 − 4.08080i 0.321715 − 0.629682i
\(43\) 10.3117i 1.57253i 0.617892 + 0.786263i \(0.287987\pi\)
−0.617892 + 0.786263i \(0.712013\pi\)
\(44\) − 2.82843i − 0.426401i
\(45\) −5.63561 + 3.63866i −0.840108 + 0.542420i
\(46\) −7.29150 −1.07507
\(47\) − 7.82087i − 1.14079i −0.821370 − 0.570396i \(-0.806790\pi\)
0.821370 − 0.570396i \(-0.193210\pi\)
\(48\) −1.68014 − 0.420861i −0.242508 − 0.0607461i
\(49\) −6.29150 − 3.06871i −0.898786 − 0.438387i
\(50\) 2.64575 + 4.24264i 0.374166 + 0.600000i
\(51\) −2.00000 + 7.98430i −0.280056 + 1.11803i
\(52\) 3.36028 0.465987
\(53\) −8.58301 −1.17897 −0.589483 − 0.807781i \(-0.700669\pi\)
−0.589483 + 0.807781i \(0.700669\pi\)
\(54\) 3.85005 + 3.48957i 0.523925 + 0.474871i
\(55\) 5.53019 + 3.06871i 0.745691 + 0.413785i
\(56\) −0.595188 + 2.57794i −0.0795353 + 0.344491i
\(57\) −2.35425 + 9.39851i −0.311828 + 1.24486i
\(58\) − 0.500983i − 0.0657823i
\(59\) −2.16991 −0.282498 −0.141249 − 0.989974i \(-0.545112\pi\)
−0.141249 + 0.989974i \(0.545112\pi\)
\(60\) 2.64575 − 2.82843i 0.341565 − 0.365148i
\(61\) 2.52517i 0.323315i 0.986847 + 0.161657i \(0.0516839\pi\)
−0.986847 + 0.161657i \(0.948316\pi\)
\(62\) − 3.06871i − 0.389727i
\(63\) 5.22047 − 5.97885i 0.657717 − 0.753265i
\(64\) 1.00000 0.125000
\(65\) −3.64575 + 6.57008i −0.452200 + 0.814919i
\(66\) 1.19038 − 4.75216i 0.146525 − 0.584950i
\(67\) − 10.3117i − 1.25978i −0.776684 − 0.629890i \(-0.783100\pi\)
0.776684 − 0.629890i \(-0.216900\pi\)
\(68\) − 4.75216i − 0.576284i
\(69\) −12.2508 − 3.06871i −1.47482 − 0.369430i
\(70\) −4.39467 − 3.96066i −0.525264 − 0.473390i
\(71\) − 9.81076i − 1.16432i −0.813073 − 0.582161i \(-0.802207\pi\)
0.813073 − 0.582161i \(-0.197793\pi\)
\(72\) −2.64575 − 1.41421i −0.311805 − 0.166667i
\(73\) 5.53019 0.647260 0.323630 − 0.946184i \(-0.395097\pi\)
0.323630 + 0.946184i \(0.395097\pi\)
\(74\) − 3.32941i − 0.387036i
\(75\) 2.65967 + 8.24173i 0.307113 + 0.951673i
\(76\) − 5.59388i − 0.641662i
\(77\) −7.29150 − 1.68345i −0.830944 − 0.191846i
\(78\) 5.64575 + 1.41421i 0.639255 + 0.160128i
\(79\) 3.29150 0.370323 0.185161 − 0.982708i \(-0.440719\pi\)
0.185161 + 0.982708i \(0.440719\pi\)
\(80\) −1.08495 + 1.95522i −0.121302 + 0.218600i
\(81\) 5.00000 + 7.48331i 0.555556 + 0.831479i
\(82\) −4.33981 −0.479252
\(83\) 6.97915i 0.766061i 0.923736 + 0.383030i \(0.125120\pi\)
−0.923736 + 0.383030i \(0.874880\pi\)
\(84\) −2.08495 + 4.08080i −0.227487 + 0.445252i
\(85\) 9.29150 + 5.15587i 1.00780 + 0.559233i
\(86\) − 10.3117i − 1.11194i
\(87\) 0.210845 − 0.841723i 0.0226049 − 0.0902422i
\(88\) 2.82843i 0.301511i
\(89\) 15.8219 1.67712 0.838558 − 0.544812i \(-0.183399\pi\)
0.838558 + 0.544812i \(0.183399\pi\)
\(90\) 5.63561 − 3.63866i 0.594046 − 0.383549i
\(91\) 2.00000 − 8.66259i 0.209657 − 0.908086i
\(92\) 7.29150 0.760192
\(93\) 1.29150 − 5.15587i 0.133923 − 0.534639i
\(94\) 7.82087i 0.806661i
\(95\) 10.9373 + 6.06910i 1.12214 + 0.622677i
\(96\) 1.68014 + 0.420861i 0.171479 + 0.0429540i
\(97\) −14.6315 −1.48560 −0.742802 − 0.669511i \(-0.766504\pi\)
−0.742802 + 0.669511i \(0.766504\pi\)
\(98\) 6.29150 + 3.06871i 0.635538 + 0.309987i
\(99\) 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 210.2.d.a.209.1 ✓ 8
3.2 odd 2 210.2.d.b.209.2 yes 8
4.3 odd 2 1680.2.k.e.209.8 8
5.2 odd 4 1050.2.b.f.251.4 16
5.3 odd 4 1050.2.b.f.251.13 16
5.4 even 2 210.2.d.b.209.8 yes 8
7.6 odd 2 inner 210.2.d.a.209.8 yes 8
12.11 even 2 1680.2.k.f.209.7 8
15.2 even 4 1050.2.b.f.251.14 16
15.8 even 4 1050.2.b.f.251.3 16
15.14 odd 2 inner 210.2.d.a.209.7 yes 8
20.19 odd 2 1680.2.k.f.209.1 8
21.20 even 2 210.2.d.b.209.7 yes 8
28.27 even 2 1680.2.k.e.209.1 8
35.13 even 4 1050.2.b.f.251.12 16
35.27 even 4 1050.2.b.f.251.5 16
35.34 odd 2 210.2.d.b.209.1 yes 8
60.59 even 2 1680.2.k.e.209.2 8
84.83 odd 2 1680.2.k.f.209.2 8
105.62 odd 4 1050.2.b.f.251.11 16
105.83 odd 4 1050.2.b.f.251.6 16
105.104 even 2 inner 210.2.d.a.209.2 yes 8
140.139 even 2 1680.2.k.f.209.8 8
420.419 odd 2 1680.2.k.e.209.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.d.a.209.1 ✓ 8 1.1 even 1 trivial
210.2.d.a.209.2 yes 8 105.104 even 2 inner
210.2.d.a.209.7 yes 8 15.14 odd 2 inner
210.2.d.a.209.8 yes 8 7.6 odd 2 inner
210.2.d.b.209.1 yes 8 35.34 odd 2
210.2.d.b.209.2 yes 8 3.2 odd 2
210.2.d.b.209.7 yes 8 21.20 even 2
210.2.d.b.209.8 yes 8 5.4 even 2
1050.2.b.f.251.3 16 15.8 even 4
1050.2.b.f.251.4 16 5.2 odd 4
1050.2.b.f.251.5 16 35.27 even 4
1050.2.b.f.251.6 16 105.83 odd 4
1050.2.b.f.251.11 16 105.62 odd 4
1050.2.b.f.251.12 16 35.13 even 4
1050.2.b.f.251.13 16 5.3 odd 4
1050.2.b.f.251.14 16 15.2 even 4
1680.2.k.e.209.1 8 28.27 even 2
1680.2.k.e.209.2 8 60.59 even 2
1680.2.k.e.209.7 8 420.419 odd 2
1680.2.k.e.209.8 8 4.3 odd 2
1680.2.k.f.209.1 8 20.19 odd 2
1680.2.k.f.209.2 8 84.83 odd 2
1680.2.k.f.209.7 8 12.11 even 2
1680.2.k.f.209.8 8 140.139 even 2