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Properties

Label 8670.2.a.cl.1.7
Level $8670$
Weight $2$
Character 8670.1
Self dual yes
Analytic conductor $69.230$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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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] = [8670,2,Mod(1,8670)] 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("8670.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8670, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8,-8,8,8,-8,0,8,8,8,8,-8,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.2302985525\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.75178704896.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} - 8x^{5} + 72x^{4} + 48x^{3} - 104x^{2} - 72x + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(3.46685\) of defining polynomial
Character \(\chi\) \(=\) 8670.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.48865 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.32453 q^{11} -1.00000 q^{12} +4.58999 q^{13} +3.48865 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +5.23817 q^{19} +1.00000 q^{20} -3.48865 q^{21} +3.32453 q^{22} +2.85383 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.58999 q^{26} -1.00000 q^{27} +3.48865 q^{28} -4.08431 q^{29} -1.00000 q^{30} -6.29394 q^{31} +1.00000 q^{32} -3.32453 q^{33} +3.48865 q^{35} +1.00000 q^{36} +8.93863 q^{37} +5.23817 q^{38} -4.58999 q^{39} +1.00000 q^{40} +10.4223 q^{41} -3.48865 q^{42} +2.30859 q^{43} +3.32453 q^{44} +1.00000 q^{45} +2.85383 q^{46} -2.23232 q^{47} -1.00000 q^{48} +5.17066 q^{49} +1.00000 q^{50} +4.58999 q^{52} -5.09848 q^{53} -1.00000 q^{54} +3.32453 q^{55} +3.48865 q^{56} -5.23817 q^{57} -4.08431 q^{58} +10.6028 q^{59} -1.00000 q^{60} -7.38458 q^{61} -6.29394 q^{62} +3.48865 q^{63} +1.00000 q^{64} +4.58999 q^{65} -3.32453 q^{66} -2.27383 q^{67} -2.85383 q^{69} +3.48865 q^{70} -1.25654 q^{71} +1.00000 q^{72} -13.2416 q^{73} +8.93863 q^{74} -1.00000 q^{75} +5.23817 q^{76} +11.5981 q^{77} -4.58999 q^{78} -10.6617 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.4223 q^{82} -3.85763 q^{83} -3.48865 q^{84} +2.30859 q^{86} +4.08431 q^{87} +3.32453 q^{88} +7.99627 q^{89} +1.00000 q^{90} +16.0129 q^{91} +2.85383 q^{92} +6.29394 q^{93} -2.23232 q^{94} +5.23817 q^{95} -1.00000 q^{96} -6.79053 q^{97} +5.17066 q^{98} +3.32453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{13} - 8 q^{15} + 8 q^{16} + 8 q^{18} + 16 q^{19} + 8 q^{20} + 8 q^{22} - 8 q^{24} + 8 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 3.48865 1.31858 0.659292 − 0.751887i \(-0.270856\pi\)
0.659292 + 0.751887i \(0.270856\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.32453 1.00238 0.501191 − 0.865337i \(-0.332895\pi\)
0.501191 + 0.865337i \(0.332895\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.58999 1.27303 0.636517 − 0.771263i \(-0.280374\pi\)
0.636517 + 0.771263i \(0.280374\pi\)
\(14\) 3.48865 0.932380
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) 5.23817 1.20172 0.600860 − 0.799354i \(-0.294825\pi\)
0.600860 + 0.799354i \(0.294825\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.48865 −0.761285
\(22\) 3.32453 0.708791
\(23\) 2.85383 0.595065 0.297532 − 0.954712i \(-0.403836\pi\)
0.297532 + 0.954712i \(0.403836\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 4.58999 0.900171
\(27\) −1.00000 −0.192450
\(28\) 3.48865 0.659292
\(29\) −4.08431 −0.758437 −0.379219 − 0.925307i \(-0.623807\pi\)
−0.379219 + 0.925307i \(0.623807\pi\)
\(30\) −1.00000 −0.182574
\(31\) −6.29394 −1.13042 −0.565212 − 0.824945i \(-0.691206\pi\)
−0.565212 + 0.824945i \(0.691206\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.32453 −0.578726
\(34\) 0 0
\(35\) 3.48865 0.589689
\(36\) 1.00000 0.166667
\(37\) 8.93863 1.46950 0.734751 − 0.678337i \(-0.237299\pi\)
0.734751 + 0.678337i \(0.237299\pi\)
\(38\) 5.23817 0.849744
\(39\) −4.58999 −0.734987
\(40\) 1.00000 0.158114
\(41\) 10.4223 1.62770 0.813848 − 0.581078i \(-0.197369\pi\)
0.813848 + 0.581078i \(0.197369\pi\)
\(42\) −3.48865 −0.538310
\(43\) 2.30859 0.352057 0.176029 − 0.984385i \(-0.443675\pi\)
0.176029 + 0.984385i \(0.443675\pi\)
\(44\) 3.32453 0.501191
\(45\) 1.00000 0.149071
\(46\) 2.85383 0.420774
\(47\) −2.23232 −0.325618 −0.162809 − 0.986658i \(-0.552055\pi\)
−0.162809 + 0.986658i \(0.552055\pi\)
\(48\) −1.00000 −0.144338
\(49\) 5.17066 0.738666
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 4.58999 0.636517
\(53\) −5.09848 −0.700330 −0.350165 − 0.936688i \(-0.613874\pi\)
−0.350165 + 0.936688i \(0.613874\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.32453 0.448279
\(56\) 3.48865 0.466190
\(57\) −5.23817 −0.693813
\(58\) −4.08431 −0.536296
\(59\) 10.6028 1.38037 0.690185 − 0.723633i \(-0.257529\pi\)
0.690185 + 0.723633i \(0.257529\pi\)
\(60\) −1.00000 −0.129099
\(61\) −7.38458 −0.945499 −0.472750 − 0.881197i \(-0.656738\pi\)
−0.472750 + 0.881197i \(0.656738\pi\)
\(62\) −6.29394 −0.799331
\(63\) 3.48865 0.439528
\(64\) 1.00000 0.125000
\(65\) 4.58999 0.569318
\(66\) −3.32453 −0.409221
\(67\) −2.27383 −0.277793 −0.138896 − 0.990307i \(-0.544356\pi\)
−0.138896 + 0.990307i \(0.544356\pi\)
\(68\) 0 0
\(69\) −2.85383 −0.343561
\(70\) 3.48865 0.416973
\(71\) −1.25654 −0.149124 −0.0745622 − 0.997216i \(-0.523756\pi\)
−0.0745622 + 0.997216i \(0.523756\pi\)
\(72\) 1.00000 0.117851
\(73\) −13.2416 −1.54982 −0.774908 − 0.632073i \(-0.782204\pi\)
−0.774908 + 0.632073i \(0.782204\pi\)
\(74\) 8.93863 1.03909
\(75\) −1.00000 −0.115470
\(76\) 5.23817 0.600860
\(77\) 11.5981 1.32173
\(78\) −4.58999 −0.519714
\(79\) −10.6617 −1.19954 −0.599768 − 0.800174i \(-0.704741\pi\)
−0.599768 + 0.800174i \(0.704741\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.4223 1.15095
\(83\) −3.85763 −0.423430 −0.211715 − 0.977331i \(-0.567905\pi\)
−0.211715 + 0.977331i \(0.567905\pi\)
\(84\) −3.48865 −0.380643
\(85\) 0 0
\(86\) 2.30859 0.248942
\(87\) 4.08431 0.437884
\(88\) 3.32453 0.354396
\(89\) 7.99627 0.847602 0.423801 − 0.905755i \(-0.360696\pi\)
0.423801 + 0.905755i \(0.360696\pi\)
\(90\) 1.00000 0.105409
\(91\) 16.0129 1.67860
\(92\) 2.85383 0.297532
\(93\) 6.29394 0.652651
\(94\) −2.23232 −0.230247
\(95\) 5.23817 0.537425
\(96\) −1.00000 −0.102062
\(97\) −6.79053 −0.689474 −0.344737 − 0.938699i \(-0.612032\pi\)
−0.344737 + 0.938699i \(0.612032\pi\)
\(98\) 5.17066 0.522316
\(99\) 3.32453 0.334127
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 8670.2.a.cl.1.7 8
17.11 odd 16 510.2.u.d.121.4 ✓ 16
17.14 odd 16 510.2.u.d.451.4 yes 16
17.16 even 2 8670.2.a.cm.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.u.d.121.4 ✓ 16 17.11 odd 16
510.2.u.d.451.4 yes 16 17.14 odd 16
8670.2.a.cl.1.7 8 1.1 even 1 trivial
8670.2.a.cm.1.2 8 17.16 even 2