↑

Properties

Label 8670.2.a.cl.1.3
Level $8670$
Weight $2$
Character 8670.1
Self dual yes
Analytic conductor $69.230$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.3
Root \(1.81937\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.15876 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.27618 q^{11} -1.00000 q^{12} +2.78328 q^{13} -1.15876 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +4.10800 q^{19} +1.00000 q^{20} +1.15876 q^{21} -5.27618 q^{22} +6.69627 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.78328 q^{26} -1.00000 q^{27} -1.15876 q^{28} -0.726891 q^{29} -1.00000 q^{30} +2.12151 q^{31} +1.00000 q^{32} +5.27618 q^{33} -1.15876 q^{35} +1.00000 q^{36} -9.50510 q^{37} +4.10800 q^{38} -2.78328 q^{39} +1.00000 q^{40} +2.47997 q^{41} +1.15876 q^{42} +10.8964 q^{43} -5.27618 q^{44} +1.00000 q^{45} +6.69627 q^{46} +0.233879 q^{47} -1.00000 q^{48} -5.65728 q^{49} +1.00000 q^{50} +2.78328 q^{52} -4.86498 q^{53} -1.00000 q^{54} -5.27618 q^{55} -1.15876 q^{56} -4.10800 q^{57} -0.726891 q^{58} +4.74220 q^{59} -1.00000 q^{60} +5.13432 q^{61} +2.12151 q^{62} -1.15876 q^{63} +1.00000 q^{64} +2.78328 q^{65} +5.27618 q^{66} -12.3690 q^{67} -6.69627 q^{69} -1.15876 q^{70} -2.66415 q^{71} +1.00000 q^{72} -11.2421 q^{73} -9.50510 q^{74} -1.00000 q^{75} +4.10800 q^{76} +6.11381 q^{77} -2.78328 q^{78} -0.128561 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.47997 q^{82} -2.53209 q^{83} +1.15876 q^{84} +10.8964 q^{86} +0.726891 q^{87} -5.27618 q^{88} -11.1237 q^{89} +1.00000 q^{90} -3.22515 q^{91} +6.69627 q^{92} -2.12151 q^{93} +0.233879 q^{94} +4.10800 q^{95} -1.00000 q^{96} -5.26158 q^{97} -5.65728 q^{98} -5.27618 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{13} - 8 q^{15} + 8 q^{16} + 8 q^{18} + 16 q^{19} + 8 q^{20} + 8 q^{22} - 8 q^{24} + 8 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(7\) −1.15876 −0.437969 −0.218985 − 0.975728i \(-0.570274\pi\)
−0.218985 + 0.975728i \(0.570274\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −5.27618 −1.59083 −0.795413 − 0.606067i \(-0.792746\pi\)
−0.795413 + 0.606067i \(0.792746\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.78328 0.771944 0.385972 − 0.922510i \(-0.373866\pi\)
0.385972 + 0.922510i \(0.373866\pi\)
\(14\) −1.15876 −0.309691
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) 4.10800 0.942439 0.471220 − 0.882016i \(-0.343814\pi\)
0.471220 + 0.882016i \(0.343814\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.15876 0.252862
\(22\) −5.27618 −1.12488
\(23\) 6.69627 1.39627 0.698135 − 0.715966i \(-0.254014\pi\)
0.698135 + 0.715966i \(0.254014\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.78328 0.545847
\(27\) −1.00000 −0.192450
\(28\) −1.15876 −0.218985
\(29\) −0.726891 −0.134980 −0.0674901 − 0.997720i \(-0.521499\pi\)
−0.0674901 + 0.997720i \(0.521499\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.12151 0.381034 0.190517 − 0.981684i \(-0.438984\pi\)
0.190517 + 0.981684i \(0.438984\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.27618 0.918464
\(34\) 0 0
\(35\) −1.15876 −0.195866
\(36\) 1.00000 0.166667
\(37\) −9.50510 −1.56263 −0.781315 − 0.624137i \(-0.785451\pi\)
−0.781315 + 0.624137i \(0.785451\pi\)
\(38\) 4.10800 0.666405
\(39\) −2.78328 −0.445682
\(40\) 1.00000 0.158114
\(41\) 2.47997 0.387307 0.193653 − 0.981070i \(-0.437966\pi\)
0.193653 + 0.981070i \(0.437966\pi\)
\(42\) 1.15876 0.178800
\(43\) 10.8964 1.66169 0.830844 − 0.556506i \(-0.187858\pi\)
0.830844 + 0.556506i \(0.187858\pi\)
\(44\) −5.27618 −0.795413
\(45\) 1.00000 0.149071
\(46\) 6.69627 0.987311
\(47\) 0.233879 0.0341148 0.0170574 − 0.999855i \(-0.494570\pi\)
0.0170574 + 0.999855i \(0.494570\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.65728 −0.808183
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.78328 0.385972
\(53\) −4.86498 −0.668257 −0.334128 − 0.942528i \(-0.608442\pi\)
−0.334128 + 0.942528i \(0.608442\pi\)
\(54\) −1.00000 −0.136083
\(55\) −5.27618 −0.711439
\(56\) −1.15876 −0.154845
\(57\) −4.10800 −0.544117
\(58\) −0.726891 −0.0954454
\(59\) 4.74220 0.617381 0.308691 − 0.951162i \(-0.400109\pi\)
0.308691 + 0.951162i \(0.400109\pi\)
\(60\) −1.00000 −0.129099
\(61\) 5.13432 0.657382 0.328691 − 0.944438i \(-0.393393\pi\)
0.328691 + 0.944438i \(0.393393\pi\)
\(62\) 2.12151 0.269432
\(63\) −1.15876 −0.145990
\(64\) 1.00000 0.125000
\(65\) 2.78328 0.345224
\(66\) 5.27618 0.649452
\(67\) −12.3690 −1.51111 −0.755557 − 0.655082i \(-0.772634\pi\)
−0.755557 + 0.655082i \(0.772634\pi\)
\(68\) 0 0
\(69\) −6.69627 −0.806136
\(70\) −1.15876 −0.138498
\(71\) −2.66415 −0.316177 −0.158088 − 0.987425i \(-0.550533\pi\)
−0.158088 + 0.987425i \(0.550533\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.2421 −1.31579 −0.657896 − 0.753109i \(-0.728553\pi\)
−0.657896 + 0.753109i \(0.728553\pi\)
\(74\) −9.50510 −1.10495
\(75\) −1.00000 −0.115470
\(76\) 4.10800 0.471220
\(77\) 6.11381 0.696733
\(78\) −2.78328 −0.315145
\(79\) −0.128561 −0.0144642 −0.00723212 − 0.999974i \(-0.502302\pi\)
−0.00723212 + 0.999974i \(0.502302\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.47997 0.273867
\(83\) −2.53209 −0.277933 −0.138967 − 0.990297i \(-0.544378\pi\)
−0.138967 + 0.990297i \(0.544378\pi\)
\(84\) 1.15876 0.126431
\(85\) 0 0
\(86\) 10.8964 1.17499
\(87\) 0.726891 0.0779309
\(88\) −5.27618 −0.562442
\(89\) −11.1237 −1.17911 −0.589553 − 0.807730i \(-0.700696\pi\)
−0.589553 + 0.807730i \(0.700696\pi\)
\(90\) 1.00000 0.105409
\(91\) −3.22515 −0.338088
\(92\) 6.69627 0.698135
\(93\) −2.12151 −0.219990
\(94\) 0.233879 0.0241228
\(95\) 4.10800 0.421472
\(96\) −1.00000 −0.102062
\(97\) −5.26158 −0.534232 −0.267116 − 0.963664i \(-0.586071\pi\)
−0.267116 + 0.963664i \(0.586071\pi\)
\(98\) −5.65728 −0.571472
\(99\) −5.27618 −0.530276
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.1.3 8
17.5 odd 16 510.2.u.d.331.4 yes 16
17.7 odd 16 510.2.u.d.151.4 ✓ 16
17.16 even 2 8670.2.a.cm.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.u.d.151.4 ✓ 16 17.7 odd 16
510.2.u.d.331.4 yes 16 17.5 odd 16
8670.2.a.cl.1.3 8 1.1 even 1 trivial
8670.2.a.cm.1.6 8 17.16 even 2