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Properties

Label 1008.2.bt.d.593.5
Level $1008$
Weight $2$
Character 1008.593
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 593.5
Root \(-0.144868 + 1.25092i\) of defining polynomial
Character \(\chi\) \(=\) 1008.593
Dual form 1008.2.bt.d.17.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.144868 - 0.250919i) q^{5} +(2.26907 - 1.36064i) q^{7} +(5.23077 - 3.01999i) q^{11} +5.46050i q^{13} +(-2.22666 - 3.85669i) q^{17} +(-3.51739 - 2.03076i) q^{19} +(-1.11743 - 0.645146i) q^{23} +(2.45803 + 4.25743i) q^{25} -0.377918i q^{29} +(3.09749 - 1.78834i) q^{31} +(-0.0126942 - 0.766467i) q^{35} +(1.01555 - 1.75898i) q^{37} -5.50384 q^{41} +6.45419 q^{43} +(5.38833 - 9.33287i) q^{47} +(3.29733 - 6.17475i) q^{49} +(9.77422 - 5.64315i) q^{53} -1.75000i q^{55} +(0.790140 + 1.36856i) q^{59} +(9.54984 + 5.51360i) q^{61} +(1.37015 + 0.791054i) q^{65} +(2.04381 + 3.53999i) q^{67} +0.410536i q^{71} +(-11.1149 + 6.41718i) q^{73} +(7.75986 - 13.9697i) q^{77} +(-6.01355 + 10.4158i) q^{79} -0.155917 q^{83} -1.29029 q^{85} +(-3.34409 + 5.79213i) q^{89} +(7.42976 + 12.3902i) q^{91} +(-1.01912 + 0.588387i) q^{95} -16.5090i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7} - 12 q^{19} + 12 q^{25} - 24 q^{31} + 4 q^{37} - 8 q^{43} + 32 q^{49} + 28 q^{67} - 60 q^{73} + 32 q^{79} - 32 q^{85} + 84 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(5\) 0.144868 − 0.250919i 0.0647871 − 0.112215i −0.831812 − 0.555057i \(-0.812696\pi\)
0.896600 + 0.442842i \(0.146030\pi\)
\(6\) 0 0
\(7\) 2.26907 − 1.36064i 0.857627 − 0.514273i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.23077 − 3.01999i 1.57714 − 0.910560i 0.581880 − 0.813275i \(-0.302317\pi\)
0.995256 − 0.0972858i \(-0.0310161\pi\)
\(12\) 0 0
\(13\) 5.46050i 1.51447i 0.653142 + 0.757235i \(0.273450\pi\)
−0.653142 + 0.757235i \(0.726550\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.22666 − 3.85669i −0.540045 − 0.935385i −0.998901 − 0.0468746i \(-0.985074\pi\)
0.458856 − 0.888511i \(-0.348259\pi\)
\(18\) 0 0
\(19\) −3.51739 − 2.03076i −0.806944 − 0.465889i 0.0389497 − 0.999241i \(-0.487599\pi\)
−0.845893 + 0.533352i \(0.820932\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.11743 − 0.645146i −0.232999 − 0.134522i 0.378956 − 0.925415i \(-0.376283\pi\)
−0.611955 + 0.790893i \(0.709617\pi\)
\(24\) 0 0
\(25\) 2.45803 + 4.25743i 0.491605 + 0.851485i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.377918i − 0.0701776i −0.999384 − 0.0350888i \(-0.988829\pi\)
0.999384 − 0.0350888i \(-0.0111714\pi\)
\(30\) 0 0
\(31\) 3.09749 − 1.78834i 0.556326 − 0.321195i −0.195343 − 0.980735i \(-0.562582\pi\)
0.751670 + 0.659540i \(0.229249\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.0126942 − 0.766467i −0.00214570 − 0.129557i
\(36\) 0 0
\(37\) 1.01555 − 1.75898i 0.166955 − 0.289174i −0.770393 − 0.637569i \(-0.779940\pi\)
0.937348 + 0.348395i \(0.113273\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.50384 −0.859556 −0.429778 − 0.902935i \(-0.641408\pi\)
−0.429778 + 0.902935i \(0.641408\pi\)
\(42\) 0 0
\(43\) 6.45419 0.984254 0.492127 − 0.870523i \(-0.336220\pi\)
0.492127 + 0.870523i \(0.336220\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.38833 − 9.33287i 0.785969 − 1.36134i −0.142449 − 0.989802i \(-0.545498\pi\)
0.928418 − 0.371536i \(-0.121169\pi\)
\(48\) 0 0
\(49\) 3.29733 − 6.17475i 0.471048 − 0.882108i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.77422 − 5.64315i 1.34259 − 0.775146i 0.355405 − 0.934713i \(-0.384343\pi\)
0.987187 + 0.159567i \(0.0510097\pi\)
\(54\) 0 0
\(55\) − 1.75000i − 0.235970i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.790140 + 1.36856i 0.102867 + 0.178172i 0.912865 − 0.408262i \(-0.133865\pi\)
−0.809998 + 0.586433i \(0.800532\pi\)
\(60\) 0 0
\(61\) 9.54984 + 5.51360i 1.22273 + 0.705945i 0.965500 − 0.260405i \(-0.0838560\pi\)
0.257233 + 0.966350i \(0.417189\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.37015 + 0.791054i 0.169946 + 0.0981182i
\(66\) 0 0
\(67\) 2.04381 + 3.53999i 0.249691 + 0.432478i 0.963440 − 0.267924i \(-0.0863375\pi\)
−0.713749 + 0.700402i \(0.753004\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.410536i 0.0487216i 0.999703 + 0.0243608i \(0.00775505\pi\)
−0.999703 + 0.0243608i \(0.992245\pi\)
\(72\) 0 0
\(73\) −11.1149 + 6.41718i −1.30090 + 0.751074i −0.980558 − 0.196229i \(-0.937131\pi\)
−0.320340 + 0.947303i \(0.603797\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.75986 − 13.9697i 0.884319 − 1.59200i
\(78\) 0 0
\(79\) −6.01355 + 10.4158i −0.676577 + 1.17187i 0.299428 + 0.954119i \(0.403204\pi\)
−0.976005 + 0.217747i \(0.930129\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.155917 −0.0171142 −0.00855708 − 0.999963i \(-0.502724\pi\)
−0.00855708 + 0.999963i \(0.502724\pi\)
\(84\) 0 0
\(85\) −1.29029 −0.139952
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.34409 + 5.79213i −0.354473 + 0.613965i −0.987028 − 0.160551i \(-0.948673\pi\)
0.632555 + 0.774516i \(0.282006\pi\)
\(90\) 0 0
\(91\) 7.42976 + 12.3902i 0.778851 + 1.29885i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.01912 + 0.588387i −0.104559 + 0.0603672i
\(96\) 0 0
\(97\) − 16.5090i − 1.67623i −0.545494 − 0.838115i \(-0.683658\pi\)
0.545494 − 0.838115i \(-0.316342\pi\)
\(98\) 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.593.5 16
3.2 odd 2 inner 1008.2.bt.d.593.4 16
4.3 odd 2 504.2.bl.a.89.5 yes 16
7.2 even 3 7056.2.k.h.881.8 16
7.3 odd 6 inner 1008.2.bt.d.17.4 16
7.5 odd 6 7056.2.k.h.881.10 16
12.11 even 2 504.2.bl.a.89.4 yes 16
21.2 odd 6 7056.2.k.h.881.9 16
21.5 even 6 7056.2.k.h.881.7 16
21.17 even 6 inner 1008.2.bt.d.17.5 16
28.3 even 6 504.2.bl.a.17.4 ✓ 16
28.11 odd 6 3528.2.bl.a.521.5 16
28.19 even 6 3528.2.k.b.881.9 16
28.23 odd 6 3528.2.k.b.881.7 16
28.27 even 2 3528.2.bl.a.1097.4 16
84.11 even 6 3528.2.bl.a.521.4 16
84.23 even 6 3528.2.k.b.881.10 16
84.47 odd 6 3528.2.k.b.881.8 16
84.59 odd 6 504.2.bl.a.17.5 yes 16
84.83 odd 2 3528.2.bl.a.1097.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bl.a.17.4 ✓ 16 28.3 even 6
504.2.bl.a.17.5 yes 16 84.59 odd 6
504.2.bl.a.89.4 yes 16 12.11 even 2
504.2.bl.a.89.5 yes 16 4.3 odd 2
1008.2.bt.d.17.4 16 7.3 odd 6 inner
1008.2.bt.d.17.5 16 21.17 even 6 inner
1008.2.bt.d.593.4 16 3.2 odd 2 inner
1008.2.bt.d.593.5 16 1.1 even 1 trivial
3528.2.k.b.881.7 16 28.23 odd 6
3528.2.k.b.881.8 16 84.47 odd 6
3528.2.k.b.881.9 16 28.19 even 6
3528.2.k.b.881.10 16 84.23 even 6
3528.2.bl.a.521.4 16 84.11 even 6
3528.2.bl.a.521.5 16 28.11 odd 6
3528.2.bl.a.1097.4 16 28.27 even 2
3528.2.bl.a.1097.5 16 84.83 odd 2
7056.2.k.h.881.7 16 21.5 even 6
7056.2.k.h.881.8 16 7.2 even 3
7056.2.k.h.881.9 16 21.2 odd 6
7056.2.k.h.881.10 16 7.5 odd 6