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Properties

Label 8712.2.a.ch.1.5
Level $8712$
Weight $2$
Character 8712.1
Self dual yes
Analytic conductor $69.566$
Analytic rank $1$
Dimension $6$
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] = [8712,2,Mod(1,8712)] 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("8712.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8712, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8712.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-5,0,-1,0,0,0,0,0,-4,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.5656702409\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.62158000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 14x^{4} + 22x^{3} + 38x^{2} - 60x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 792)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.17002\) of defining polynomial
Character \(\chi\) \(=\) 8712.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.893131 q^{5} +2.25714 q^{7} -2.48389 q^{13} +0.588971 q^{17} -2.56922 q^{19} -0.671062 q^{23} -4.20232 q^{25} +3.49691 q^{29} -6.99216 q^{31} +2.01592 q^{35} +6.99816 q^{37} -9.21137 q^{41} +2.14385 q^{43} +4.33931 q^{47} -1.90534 q^{49} -7.15219 q^{53} -13.7394 q^{59} -4.49525 q^{61} -2.21844 q^{65} +3.49896 q^{67} -0.442063 q^{71} +6.38715 q^{73} +4.79928 q^{79} -7.10614 q^{83} +0.526029 q^{85} -11.4193 q^{89} -5.60648 q^{91} -2.29465 q^{95} +7.10193 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{5} - q^{7} - 4 q^{13} - 6 q^{17} + 14 q^{19} - 6 q^{23} + 7 q^{25} - 7 q^{29} - 7 q^{31} + 2 q^{35} + 2 q^{37} + 6 q^{41} + 12 q^{43} - 26 q^{47} + 9 q^{49} - 9 q^{53} - 17 q^{59} - 20 q^{61}+ \cdots + 25 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.893131 0.399420 0.199710 − 0.979855i \(-0.436000\pi\)
0.199710 + 0.979855i \(0.436000\pi\)
\(6\) 0 0
\(7\) 2.25714 0.853117 0.426559 − 0.904460i \(-0.359726\pi\)
0.426559 + 0.904460i \(0.359726\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.48389 −0.688908 −0.344454 − 0.938803i \(-0.611936\pi\)
−0.344454 + 0.938803i \(0.611936\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.588971 0.142847 0.0714233 − 0.997446i \(-0.477246\pi\)
0.0714233 + 0.997446i \(0.477246\pi\)
\(18\) 0 0
\(19\) −2.56922 −0.589419 −0.294710 − 0.955587i \(-0.595223\pi\)
−0.294710 + 0.955587i \(0.595223\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.671062 −0.139926 −0.0699631 − 0.997550i \(-0.522288\pi\)
−0.0699631 + 0.997550i \(0.522288\pi\)
\(24\) 0 0
\(25\) −4.20232 −0.840463
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.49691 0.649361 0.324680 − 0.945824i \(-0.394743\pi\)
0.324680 + 0.945824i \(0.394743\pi\)
\(30\) 0 0
\(31\) −6.99216 −1.25583 −0.627915 − 0.778282i \(-0.716091\pi\)
−0.627915 + 0.778282i \(0.716091\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.01592 0.340752
\(36\) 0 0
\(37\) 6.99816 1.15049 0.575246 − 0.817981i \(-0.304906\pi\)
0.575246 + 0.817981i \(0.304906\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.21137 −1.43857 −0.719287 − 0.694713i \(-0.755532\pi\)
−0.719287 + 0.694713i \(0.755532\pi\)
\(42\) 0 0
\(43\) 2.14385 0.326935 0.163467 − 0.986549i \(-0.447732\pi\)
0.163467 + 0.986549i \(0.447732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.33931 0.632953 0.316477 − 0.948600i \(-0.397500\pi\)
0.316477 + 0.948600i \(0.397500\pi\)
\(48\) 0 0
\(49\) −1.90534 −0.272191
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.15219 −0.982428 −0.491214 − 0.871039i \(-0.663447\pi\)
−0.491214 + 0.871039i \(0.663447\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.7394 −1.78872 −0.894358 − 0.447351i \(-0.852367\pi\)
−0.894358 + 0.447351i \(0.852367\pi\)
\(60\) 0 0
\(61\) −4.49525 −0.575557 −0.287779 − 0.957697i \(-0.592917\pi\)
−0.287779 + 0.957697i \(0.592917\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.21844 −0.275164
\(66\) 0 0
\(67\) 3.49896 0.427466 0.213733 − 0.976892i \(-0.431438\pi\)
0.213733 + 0.976892i \(0.431438\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.442063 −0.0524632 −0.0262316 − 0.999656i \(-0.508351\pi\)
−0.0262316 + 0.999656i \(0.508351\pi\)
\(72\) 0 0
\(73\) 6.38715 0.747559 0.373780 − 0.927518i \(-0.378062\pi\)
0.373780 + 0.927518i \(0.378062\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.79928 0.539962 0.269981 − 0.962866i \(-0.412983\pi\)
0.269981 + 0.962866i \(0.412983\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.10614 −0.780000 −0.390000 − 0.920815i \(-0.627525\pi\)
−0.390000 + 0.920815i \(0.627525\pi\)
\(84\) 0 0
\(85\) 0.526029 0.0570558
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.4193 −1.21045 −0.605223 − 0.796056i \(-0.706916\pi\)
−0.605223 + 0.796056i \(0.706916\pi\)
\(90\) 0 0
\(91\) −5.60648 −0.587719
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.29465 −0.235426
\(96\) 0 0
\(97\) 7.10193 0.721091 0.360546 − 0.932742i \(-0.382590\pi\)
0.360546 + 0.932742i \(0.382590\pi\)
\(98\) 0 0
\(99\) 0 0
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 8712.2.a.ch.1.5 6
3.2 odd 2 8712.2.a.cj.1.2 6
11.5 even 5 792.2.r.i.289.2 yes 12
11.9 even 5 792.2.r.i.433.2 yes 12
11.10 odd 2 8712.2.a.ci.1.5 6
33.5 odd 10 792.2.r.h.289.2 ✓ 12
33.20 odd 10 792.2.r.h.433.2 yes 12
33.32 even 2 8712.2.a.ck.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
792.2.r.h.289.2 ✓ 12 33.5 odd 10
792.2.r.h.433.2 yes 12 33.20 odd 10
792.2.r.i.289.2 yes 12 11.5 even 5
792.2.r.i.433.2 yes 12 11.9 even 5
8712.2.a.ch.1.5 6 1.1 even 1 trivial
8712.2.a.ci.1.5 6 11.10 odd 2
8712.2.a.cj.1.2 6 3.2 odd 2
8712.2.a.ck.1.2 6 33.32 even 2