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Properties

Label 1008.2.bt.d.17.3
Level $1008$
Weight $2$
Character 1008.17
Analytic conductor $8.049$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [1008,2,Mod(17,1008)] 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("1008.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bt (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.3
Root \(0.587308 + 2.01725i\) of defining polynomial
Character \(\chi\) \(=\) 1008.17
Dual form 1008.2.bt.d.593.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.587308 - 1.01725i) q^{5} +(-2.35282 + 1.21006i) q^{7} +(1.44725 + 0.835571i) q^{11} -1.14525i q^{13} +(2.07609 - 3.59589i) q^{17} +(-1.83453 + 1.05917i) q^{19} +(4.22379 - 2.43860i) q^{23} +(1.81014 - 3.13525i) q^{25} -8.32591i q^{29} +(-7.18124 - 4.14609i) q^{31} +(2.61276 + 1.68272i) q^{35} +(-2.19996 - 3.81045i) q^{37} +2.67577 q^{41} -4.08536 q^{43} +(-1.75282 - 3.03597i) q^{47} +(4.07150 - 5.69411i) q^{49} +(1.59764 + 0.922398i) q^{53} -1.96295i q^{55} +(2.98667 - 5.17306i) q^{59} +(-12.8011 + 7.39071i) q^{61} +(-1.16500 + 0.672614i) q^{65} +(4.22435 - 7.31679i) q^{67} -5.48320i q^{71} +(0.846715 + 0.488851i) q^{73} +(-4.41621 - 0.214682i) q^{77} +(5.56097 + 9.63187i) q^{79} -10.2258 q^{83} -4.87721 q^{85} +(6.29987 + 10.9117i) q^{89} +(1.38582 + 2.69456i) q^{91} +(2.15487 + 1.24411i) q^{95} +9.52049i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7} - 12 q^{19} + 12 q^{25} - 24 q^{31} + 4 q^{37} - 8 q^{43} + 32 q^{49} + 28 q^{67} - 60 q^{73} + 32 q^{79} - 32 q^{85} + 84 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

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
\(5\) −0.587308 − 1.01725i −0.262652 − 0.454926i 0.704294 − 0.709909i \(-0.251264\pi\)
−0.966946 + 0.254982i \(0.917930\pi\)
\(6\) 0 0
\(7\) −2.35282 + 1.21006i −0.889281 + 0.457361i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.44725 + 0.835571i 0.436363 + 0.251934i 0.702054 − 0.712124i \(-0.252267\pi\)
−0.265691 + 0.964058i \(0.585600\pi\)
\(12\) 0 0
\(13\) − 1.14525i − 0.317635i −0.987308 − 0.158818i \(-0.949232\pi\)
0.987308 − 0.158818i \(-0.0507682\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.07609 − 3.59589i 0.503525 − 0.872131i −0.496466 − 0.868056i \(-0.665369\pi\)
0.999992 − 0.00407535i \(-0.00129723\pi\)
\(18\) 0 0
\(19\) −1.83453 + 1.05917i −0.420870 + 0.242989i −0.695449 − 0.718575i \(-0.744795\pi\)
0.274579 + 0.961564i \(0.411461\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.22379 − 2.43860i 0.880720 − 0.508484i 0.00982439 − 0.999952i \(-0.496873\pi\)
0.870896 + 0.491468i \(0.163539\pi\)
\(24\) 0 0
\(25\) 1.81014 − 3.13525i 0.362028 − 0.627051i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.32591i − 1.54608i −0.634355 − 0.773042i \(-0.718734\pi\)
0.634355 − 0.773042i \(-0.281266\pi\)
\(30\) 0 0
\(31\) −7.18124 − 4.14609i −1.28979 − 0.744660i −0.311173 − 0.950353i \(-0.600722\pi\)
−0.978617 + 0.205693i \(0.934055\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.61276 + 1.68272i 0.441637 + 0.284431i
\(36\) 0 0
\(37\) −2.19996 − 3.81045i −0.361672 − 0.626434i 0.626564 − 0.779370i \(-0.284461\pi\)
−0.988236 + 0.152936i \(0.951127\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.67577 0.417885 0.208942 − 0.977928i \(-0.432998\pi\)
0.208942 + 0.977928i \(0.432998\pi\)
\(42\) 0 0
\(43\) −4.08536 −0.623011 −0.311505 − 0.950244i \(-0.600833\pi\)
−0.311505 + 0.950244i \(0.600833\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.75282 − 3.03597i −0.255674 − 0.442841i 0.709404 − 0.704802i \(-0.248964\pi\)
−0.965078 + 0.261961i \(0.915631\pi\)
\(48\) 0 0
\(49\) 4.07150 − 5.69411i 0.581643 − 0.813444i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.59764 + 0.922398i 0.219453 + 0.126701i 0.605697 − 0.795695i \(-0.292894\pi\)
−0.386244 + 0.922397i \(0.626228\pi\)
\(54\) 0 0
\(55\) − 1.96295i − 0.264684i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.98667 − 5.17306i 0.388831 − 0.673475i −0.603462 − 0.797392i \(-0.706212\pi\)
0.992293 + 0.123917i \(0.0395457\pi\)
\(60\) 0 0
\(61\) −12.8011 + 7.39071i −1.63901 + 0.946284i −0.657838 + 0.753160i \(0.728529\pi\)
−0.981174 + 0.193124i \(0.938138\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.16500 + 0.672614i −0.144501 + 0.0834275i
\(66\) 0 0
\(67\) 4.22435 − 7.31679i 0.516087 − 0.893889i −0.483739 − 0.875212i \(-0.660721\pi\)
0.999826 − 0.0186762i \(-0.00594518\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 5.48320i − 0.650736i −0.945587 − 0.325368i \(-0.894512\pi\)
0.945587 − 0.325368i \(-0.105488\pi\)
\(72\) 0 0
\(73\) 0.846715 + 0.488851i 0.0991004 + 0.0572157i 0.548731 − 0.835999i \(-0.315111\pi\)
−0.449631 + 0.893215i \(0.648444\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.41621 − 0.214682i −0.503274 − 0.0244652i
\(78\) 0 0
\(79\) 5.56097 + 9.63187i 0.625657 + 1.08367i 0.988413 + 0.151786i \(0.0485025\pi\)
−0.362756 + 0.931884i \(0.618164\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.2258 −1.12242 −0.561212 − 0.827672i \(-0.689665\pi\)
−0.561212 + 0.827672i \(0.689665\pi\)
\(84\) 0 0
\(85\) −4.87721 −0.529007
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.29987 + 10.9117i 0.667785 + 1.15664i 0.978522 + 0.206142i \(0.0660909\pi\)
−0.310737 + 0.950496i \(0.600576\pi\)
\(90\) 0 0
\(91\) 1.38582 + 2.69456i 0.145274 + 0.282467i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.15487 + 1.24411i 0.221085 + 0.127643i
\(96\) 0 0
\(97\) 9.52049i 0.966660i 0.875438 + 0.483330i \(0.160573\pi\)
−0.875438 + 0.483330i \(0.839427\pi\)
\(98\) 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 1008.2.bt.d.17.3 16
3.2 odd 2 inner 1008.2.bt.d.17.6 16
4.3 odd 2 504.2.bl.a.17.3 ✓ 16
7.3 odd 6 7056.2.k.h.881.5 16
7.4 even 3 7056.2.k.h.881.11 16
7.5 odd 6 inner 1008.2.bt.d.593.6 16
12.11 even 2 504.2.bl.a.17.6 yes 16
21.5 even 6 inner 1008.2.bt.d.593.3 16
21.11 odd 6 7056.2.k.h.881.6 16
21.17 even 6 7056.2.k.h.881.12 16
28.3 even 6 3528.2.k.b.881.6 16
28.11 odd 6 3528.2.k.b.881.12 16
28.19 even 6 504.2.bl.a.89.6 yes 16
28.23 odd 6 3528.2.bl.a.1097.3 16
28.27 even 2 3528.2.bl.a.521.6 16
84.11 even 6 3528.2.k.b.881.5 16
84.23 even 6 3528.2.bl.a.1097.6 16
84.47 odd 6 504.2.bl.a.89.3 yes 16
84.59 odd 6 3528.2.k.b.881.11 16
84.83 odd 2 3528.2.bl.a.521.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bl.a.17.3 ✓ 16 4.3 odd 2
504.2.bl.a.17.6 yes 16 12.11 even 2
504.2.bl.a.89.3 yes 16 84.47 odd 6
504.2.bl.a.89.6 yes 16 28.19 even 6
1008.2.bt.d.17.3 16 1.1 even 1 trivial
1008.2.bt.d.17.6 16 3.2 odd 2 inner
1008.2.bt.d.593.3 16 21.5 even 6 inner
1008.2.bt.d.593.6 16 7.5 odd 6 inner
3528.2.k.b.881.5 16 84.11 even 6
3528.2.k.b.881.6 16 28.3 even 6
3528.2.k.b.881.11 16 84.59 odd 6
3528.2.k.b.881.12 16 28.11 odd 6
3528.2.bl.a.521.3 16 84.83 odd 2
3528.2.bl.a.521.6 16 28.27 even 2
3528.2.bl.a.1097.3 16 28.23 odd 6
3528.2.bl.a.1097.6 16 84.23 even 6
7056.2.k.h.881.5 16 7.3 odd 6
7056.2.k.h.881.6 16 21.11 odd 6
7056.2.k.h.881.11 16 7.4 even 3
7056.2.k.h.881.12 16 21.17 even 6