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Properties

Label 8670.2.a.bx.1.3
Level $8670$
Weight $2$
Character 8670.1
Self dual yes
Analytic conductor $69.230$
Analytic rank $0$
Dimension $4$
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 = [4,4,-4,4,-4,-4,-4,4,4,-4,0,-4,4] 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: \(4\)
Coefficient field: 4.4.4352.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.334904\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.473626 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.61706 q^{11} -1.00000 q^{12} -2.55765 q^{13} +0.473626 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +1.88118 q^{19} -1.00000 q^{20} -0.473626 q^{21} -3.61706 q^{22} -3.41421 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.55765 q^{26} -1.00000 q^{27} +0.473626 q^{28} -1.61706 q^{29} +1.00000 q^{30} -1.25559 q^{31} +1.00000 q^{32} +3.61706 q^{33} -0.473626 q^{35} +1.00000 q^{36} +0.118824 q^{37} +1.88118 q^{38} +2.55765 q^{39} -1.00000 q^{40} +5.47010 q^{41} -0.473626 q^{42} +3.88784 q^{43} -3.61706 q^{44} -1.00000 q^{45} -3.41421 q^{46} -0.168043 q^{47} -1.00000 q^{48} -6.77568 q^{49} +1.00000 q^{50} -2.55765 q^{52} +7.89450 q^{53} -1.00000 q^{54} +3.61706 q^{55} +0.473626 q^{56} -1.88118 q^{57} -1.61706 q^{58} +6.67647 q^{59} +1.00000 q^{60} -9.27391 q^{61} -1.25559 q^{62} +0.473626 q^{63} +1.00000 q^{64} +2.55765 q^{65} +3.61706 q^{66} -1.05941 q^{67} +3.41421 q^{69} -0.473626 q^{70} -2.13048 q^{71} +1.00000 q^{72} -1.52637 q^{73} +0.118824 q^{74} -1.00000 q^{75} +1.88118 q^{76} -1.71313 q^{77} +2.55765 q^{78} +13.3674 q^{79} -1.00000 q^{80} +1.00000 q^{81} +5.47010 q^{82} +0.237649 q^{83} -0.473626 q^{84} +3.88784 q^{86} +1.61706 q^{87} -3.61706 q^{88} +13.7757 q^{89} -1.00000 q^{90} -1.21137 q^{91} -3.41421 q^{92} +1.25559 q^{93} -0.168043 q^{94} -1.88118 q^{95} -1.00000 q^{96} -2.47363 q^{97} -6.77568 q^{98} -3.61706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{10} - 4 q^{12} + 4 q^{13} - 4 q^{14} + 4 q^{15} + 4 q^{16} + 4 q^{18} + 8 q^{19} - 4 q^{20} + 4 q^{21} - 8 q^{23}+ \cdots - 4 q^{98}+O(q^{100}) \) 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
\(7\) 0.473626 0.179014 0.0895069 − 0.995986i \(-0.471471\pi\)
0.0895069 + 0.995986i \(0.471471\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −3.61706 −1.09058 −0.545292 − 0.838246i \(-0.683581\pi\)
−0.545292 + 0.838246i \(0.683581\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.55765 −0.709364 −0.354682 − 0.934987i \(-0.615411\pi\)
−0.354682 + 0.934987i \(0.615411\pi\)
\(14\) 0.473626 0.126582
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) 1.88118 0.431571 0.215786 − 0.976441i \(-0.430769\pi\)
0.215786 + 0.976441i \(0.430769\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.473626 −0.103354
\(22\) −3.61706 −0.771160
\(23\) −3.41421 −0.711913 −0.355956 − 0.934503i \(-0.615845\pi\)
−0.355956 + 0.934503i \(0.615845\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.55765 −0.501596
\(27\) −1.00000 −0.192450
\(28\) 0.473626 0.0895069
\(29\) −1.61706 −0.300280 −0.150140 − 0.988665i \(-0.547973\pi\)
−0.150140 + 0.988665i \(0.547973\pi\)
\(30\) 1.00000 0.182574
\(31\) −1.25559 −0.225511 −0.112756 − 0.993623i \(-0.535968\pi\)
−0.112756 + 0.993623i \(0.535968\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.61706 0.629649
\(34\) 0 0
\(35\) −0.473626 −0.0800574
\(36\) 1.00000 0.166667
\(37\) 0.118824 0.0195346 0.00976730 − 0.999952i \(-0.496891\pi\)
0.00976730 + 0.999952i \(0.496891\pi\)
\(38\) 1.88118 0.305167
\(39\) 2.55765 0.409551
\(40\) −1.00000 −0.158114
\(41\) 5.47010 0.854285 0.427143 − 0.904184i \(-0.359520\pi\)
0.427143 + 0.904184i \(0.359520\pi\)
\(42\) −0.473626 −0.0730820
\(43\) 3.88784 0.592890 0.296445 − 0.955050i \(-0.404199\pi\)
0.296445 + 0.955050i \(0.404199\pi\)
\(44\) −3.61706 −0.545292
\(45\) −1.00000 −0.149071
\(46\) −3.41421 −0.503398
\(47\) −0.168043 −0.0245116 −0.0122558 − 0.999925i \(-0.503901\pi\)
−0.0122558 + 0.999925i \(0.503901\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.77568 −0.967954
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.55765 −0.354682
\(53\) 7.89450 1.08439 0.542197 − 0.840252i \(-0.317593\pi\)
0.542197 + 0.840252i \(0.317593\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.61706 0.487724
\(56\) 0.473626 0.0632909
\(57\) −1.88118 −0.249168
\(58\) −1.61706 −0.212330
\(59\) 6.67647 0.869203 0.434601 − 0.900623i \(-0.356889\pi\)
0.434601 + 0.900623i \(0.356889\pi\)
\(60\) 1.00000 0.129099
\(61\) −9.27391 −1.18740 −0.593701 − 0.804685i \(-0.702334\pi\)
−0.593701 + 0.804685i \(0.702334\pi\)
\(62\) −1.25559 −0.159461
\(63\) 0.473626 0.0596712
\(64\) 1.00000 0.125000
\(65\) 2.55765 0.317237
\(66\) 3.61706 0.445229
\(67\) −1.05941 −0.129428 −0.0647139 − 0.997904i \(-0.520613\pi\)
−0.0647139 + 0.997904i \(0.520613\pi\)
\(68\) 0 0
\(69\) 3.41421 0.411023
\(70\) −0.473626 −0.0566091
\(71\) −2.13048 −0.252841 −0.126421 − 0.991977i \(-0.540349\pi\)
−0.126421 + 0.991977i \(0.540349\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.52637 −0.178649 −0.0893243 − 0.996003i \(-0.528471\pi\)
−0.0893243 + 0.996003i \(0.528471\pi\)
\(74\) 0.118824 0.0138131
\(75\) −1.00000 −0.115470
\(76\) 1.88118 0.215786
\(77\) −1.71313 −0.195230
\(78\) 2.55765 0.289597
\(79\) 13.3674 1.50395 0.751973 − 0.659194i \(-0.229103\pi\)
0.751973 + 0.659194i \(0.229103\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 5.47010 0.604071
\(83\) 0.237649 0.0260853 0.0130427 − 0.999915i \(-0.495848\pi\)
0.0130427 + 0.999915i \(0.495848\pi\)
\(84\) −0.473626 −0.0516768
\(85\) 0 0
\(86\) 3.88784 0.419236
\(87\) 1.61706 0.173367
\(88\) −3.61706 −0.385580
\(89\) 13.7757 1.46022 0.730110 − 0.683330i \(-0.239469\pi\)
0.730110 + 0.683330i \(0.239469\pi\)
\(90\) −1.00000 −0.105409
\(91\) −1.21137 −0.126986
\(92\) −3.41421 −0.355956
\(93\) 1.25559 0.130199
\(94\) −0.168043 −0.0173323
\(95\) −1.88118 −0.193005
\(96\) −1.00000 −0.102062
\(97\) −2.47363 −0.251159 −0.125579 − 0.992084i \(-0.540079\pi\)
−0.125579 + 0.992084i \(0.540079\pi\)
\(98\) −6.77568 −0.684447
\(99\) −3.61706 −0.363528
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.bx.1.3 4
17.8 even 8 510.2.p.c.421.4 yes 8
17.15 even 8 510.2.p.c.361.4 ✓ 8
17.16 even 2 8670.2.a.ca.1.2 4
51.8 odd 8 1530.2.q.j.1441.4 8
51.32 odd 8 1530.2.q.j.361.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.p.c.361.4 ✓ 8 17.15 even 8
510.2.p.c.421.4 yes 8 17.8 even 8
1530.2.q.j.361.4 8 51.32 odd 8
1530.2.q.j.1441.4 8 51.8 odd 8
8670.2.a.bx.1.3 4 1.1 even 1 trivial
8670.2.a.ca.1.2 4 17.16 even 2