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Properties

Label 8670.2.a.cl.1.4
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 1.4
Root \(0.189865\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.14570 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.90321 q^{11} -1.00000 q^{12} -4.02662 q^{13} -1.14570 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +6.07525 q^{19} +1.00000 q^{20} +1.14570 q^{21} -4.90321 q^{22} -5.08643 q^{23} -1.00000 q^{24} +1.00000 q^{25} -4.02662 q^{26} -1.00000 q^{27} -1.14570 q^{28} -2.29110 q^{29} -1.00000 q^{30} +1.73077 q^{31} +1.00000 q^{32} +4.90321 q^{33} -1.14570 q^{35} +1.00000 q^{36} +7.06097 q^{37} +6.07525 q^{38} +4.02662 q^{39} +1.00000 q^{40} -0.765973 q^{41} +1.14570 q^{42} -6.30839 q^{43} -4.90321 q^{44} +1.00000 q^{45} -5.08643 q^{46} +12.4649 q^{47} -1.00000 q^{48} -5.68736 q^{49} +1.00000 q^{50} -4.02662 q^{52} +5.78505 q^{53} -1.00000 q^{54} -4.90321 q^{55} -1.14570 q^{56} -6.07525 q^{57} -2.29110 q^{58} -4.60207 q^{59} -1.00000 q^{60} +1.27471 q^{61} +1.73077 q^{62} -1.14570 q^{63} +1.00000 q^{64} -4.02662 q^{65} +4.90321 q^{66} +3.86625 q^{67} +5.08643 q^{69} -1.14570 q^{70} +8.45962 q^{71} +1.00000 q^{72} -10.7122 q^{73} +7.06097 q^{74} -1.00000 q^{75} +6.07525 q^{76} +5.61763 q^{77} +4.02662 q^{78} +3.11474 q^{79} +1.00000 q^{80} +1.00000 q^{81} -0.765973 q^{82} +17.3584 q^{83} +1.14570 q^{84} -6.30839 q^{86} +2.29110 q^{87} -4.90321 q^{88} -0.747481 q^{89} +1.00000 q^{90} +4.61332 q^{91} -5.08643 q^{92} -1.73077 q^{93} +12.4649 q^{94} +6.07525 q^{95} -1.00000 q^{96} +8.11757 q^{97} -5.68736 q^{98} -4.90321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{13} - 8 q^{15} + 8 q^{16} + 8 q^{18} + 16 q^{19} + 8 q^{20} + 8 q^{22} - 8 q^{24} + 8 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(7\) −1.14570 −0.433035 −0.216518 − 0.976279i \(-0.569470\pi\)
−0.216518 + 0.976279i \(0.569470\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.90321 −1.47837 −0.739187 − 0.673500i \(-0.764790\pi\)
−0.739187 + 0.673500i \(0.764790\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.02662 −1.11678 −0.558392 − 0.829577i \(-0.688582\pi\)
−0.558392 + 0.829577i \(0.688582\pi\)
\(14\) −1.14570 −0.306202
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) 6.07525 1.39376 0.696879 − 0.717189i \(-0.254572\pi\)
0.696879 + 0.717189i \(0.254572\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.14570 0.250013
\(22\) −4.90321 −1.04537
\(23\) −5.08643 −1.06059 −0.530297 − 0.847812i \(-0.677920\pi\)
−0.530297 + 0.847812i \(0.677920\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −4.02662 −0.789686
\(27\) −1.00000 −0.192450
\(28\) −1.14570 −0.216518
\(29\) −2.29110 −0.425446 −0.212723 − 0.977113i \(-0.568233\pi\)
−0.212723 + 0.977113i \(0.568233\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.73077 0.310856 0.155428 − 0.987847i \(-0.450324\pi\)
0.155428 + 0.987847i \(0.450324\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.90321 0.853540
\(34\) 0 0
\(35\) −1.14570 −0.193659
\(36\) 1.00000 0.166667
\(37\) 7.06097 1.16082 0.580408 − 0.814326i \(-0.302893\pi\)
0.580408 + 0.814326i \(0.302893\pi\)
\(38\) 6.07525 0.985535
\(39\) 4.02662 0.644776
\(40\) 1.00000 0.158114
\(41\) −0.765973 −0.119625 −0.0598124 − 0.998210i \(-0.519050\pi\)
−0.0598124 + 0.998210i \(0.519050\pi\)
\(42\) 1.14570 0.176786
\(43\) −6.30839 −0.962021 −0.481010 − 0.876715i \(-0.659730\pi\)
−0.481010 + 0.876715i \(0.659730\pi\)
\(44\) −4.90321 −0.739187
\(45\) 1.00000 0.149071
\(46\) −5.08643 −0.749953
\(47\) 12.4649 1.81820 0.909098 − 0.416583i \(-0.136772\pi\)
0.909098 + 0.416583i \(0.136772\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.68736 −0.812480
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.02662 −0.558392
\(53\) 5.78505 0.794638 0.397319 − 0.917681i \(-0.369941\pi\)
0.397319 + 0.917681i \(0.369941\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.90321 −0.661149
\(56\) −1.14570 −0.153101
\(57\) −6.07525 −0.804686
\(58\) −2.29110 −0.300836
\(59\) −4.60207 −0.599139 −0.299569 − 0.954075i \(-0.596843\pi\)
−0.299569 + 0.954075i \(0.596843\pi\)
\(60\) −1.00000 −0.129099
\(61\) 1.27471 0.163210 0.0816048 − 0.996665i \(-0.473995\pi\)
0.0816048 + 0.996665i \(0.473995\pi\)
\(62\) 1.73077 0.219809
\(63\) −1.14570 −0.144345
\(64\) 1.00000 0.125000
\(65\) −4.02662 −0.499441
\(66\) 4.90321 0.603544
\(67\) 3.86625 0.472338 0.236169 − 0.971712i \(-0.424108\pi\)
0.236169 + 0.971712i \(0.424108\pi\)
\(68\) 0 0
\(69\) 5.08643 0.612334
\(70\) −1.14570 −0.136938
\(71\) 8.45962 1.00397 0.501986 − 0.864876i \(-0.332603\pi\)
0.501986 + 0.864876i \(0.332603\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.7122 −1.25377 −0.626887 − 0.779110i \(-0.715671\pi\)
−0.626887 + 0.779110i \(0.715671\pi\)
\(74\) 7.06097 0.820821
\(75\) −1.00000 −0.115470
\(76\) 6.07525 0.696879
\(77\) 5.61763 0.640188
\(78\) 4.02662 0.455925
\(79\) 3.11474 0.350435 0.175218 − 0.984530i \(-0.443937\pi\)
0.175218 + 0.984530i \(0.443937\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −0.765973 −0.0845876
\(83\) 17.3584 1.90534 0.952668 − 0.304013i \(-0.0983265\pi\)
0.952668 + 0.304013i \(0.0983265\pi\)
\(84\) 1.14570 0.125007
\(85\) 0 0
\(86\) −6.30839 −0.680252
\(87\) 2.29110 0.245631
\(88\) −4.90321 −0.522684
\(89\) −0.747481 −0.0792328 −0.0396164 − 0.999215i \(-0.512614\pi\)
−0.0396164 + 0.999215i \(0.512614\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.61332 0.483607
\(92\) −5.08643 −0.530297
\(93\) −1.73077 −0.179473
\(94\) 12.4649 1.28566
\(95\) 6.07525 0.623307
\(96\) −1.00000 −0.102062
\(97\) 8.11757 0.824215 0.412107 − 0.911135i \(-0.364793\pi\)
0.412107 + 0.911135i \(0.364793\pi\)
\(98\) −5.68736 −0.574510
\(99\) −4.90321 −0.492791
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.1.4 8
17.3 odd 16 510.2.u.d.451.1 yes 16
17.6 odd 16 510.2.u.d.121.1 ✓ 16
17.16 even 2 8670.2.a.cm.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.u.d.121.1 ✓ 16 17.6 odd 16
510.2.u.d.451.1 yes 16 17.3 odd 16
8670.2.a.cl.1.4 8 1.1 even 1 trivial
8670.2.a.cm.1.5 8 17.16 even 2