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Properties

Label 3528.2.bl.a.521.4
Level $3528$
Weight $2$
Character 3528.521
Analytic conductor $28.171$
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] = [3528,2,Mod(521,3528)] 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("3528.521"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.bl (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.1712218331\)
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: \(\Z[a_1, \ldots, a_{19}]\)
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 521.4
Root \(0.144868 + 1.25092i\) of defining polynomial
Character \(\chi\) \(=\) 3528.521
Dual form 3528.2.bl.a.1097.4

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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
\(5\) −0.144868 − 0.250919i −0.0647871 − 0.112215i 0.831812 − 0.555057i \(-0.187304\pi\)
−0.896600 + 0.442842i \(0.853970\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.23077 − 3.01999i −1.57714 − 0.910560i −0.995256 − 0.0972858i \(-0.968984\pi\)
−0.581880 − 0.813275i \(-0.697683\pi\)
\(12\) 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 −0.458856 − 0.888511i \(-0.651741\pi\)
0.998901 − 0.0468746i \(-0.0149261\pi\)
\(18\) 0 0
\(19\) −3.51739 + 2.03076i −0.806944 + 0.465889i −0.845893 − 0.533352i \(-0.820932\pi\)
0.0389497 + 0.999241i \(0.487599\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.11743 − 0.645146i 0.232999 − 0.134522i −0.378956 − 0.925415i \(-0.623717\pi\)
0.611955 + 0.790893i \(0.290383\pi\)
\(24\) 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 \(0.0111714\pi\)
−0.999384 + 0.0350888i \(0.988829\pi\)
\(30\) 0 0
\(31\) 3.09749 + 1.78834i 0.556326 + 0.321195i 0.751670 − 0.659540i \(-0.229249\pi\)
−0.195343 + 0.980735i \(0.562582\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.01555 + 1.75898i 0.166955 + 0.289174i 0.937348 − 0.348395i \(-0.113273\pi\)
−0.770393 + 0.637569i \(0.779940\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.50384 0.859556 0.429778 − 0.902935i \(-0.358592\pi\)
0.429778 + 0.902935i \(0.358592\pi\)
\(42\) 0 0
\(43\) −6.45419 −0.984254 −0.492127 − 0.870523i \(-0.663780\pi\)
−0.492127 + 0.870523i \(0.663780\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.38833 + 9.33287i 0.785969 + 1.36134i 0.928418 + 0.371536i \(0.121169\pi\)
−0.142449 + 0.989802i \(0.545498\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.77422 + 5.64315i 1.34259 + 0.775146i 0.987187 − 0.159567i \(-0.0510097\pi\)
0.355405 + 0.934713i \(0.384343\pi\)
\(54\) 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 −0.809998 − 0.586433i \(-0.800532\pi\)
0.912865 + 0.408262i \(0.133865\pi\)
\(60\) 0 0
\(61\) −9.54984 + 5.51360i −1.22273 + 0.705945i −0.965500 − 0.260405i \(-0.916144\pi\)
−0.257233 + 0.966350i \(0.582811\pi\)
\(62\) 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 \(-0.913663\pi\)
0.713749 + 0.700402i \(0.246996\pi\)
\(68\) 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 \(-0.0628695\pi\)
0.320340 + 0.947303i \(0.396203\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.01355 + 10.4158i 0.676577 + 1.17187i 0.976005 + 0.217747i \(0.0698707\pi\)
−0.299428 + 0.954119i \(0.596796\pi\)
\(80\) 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 \(-0.0513271\pi\)
−0.632555 + 0.774516i \(0.717994\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 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 3528.2.bl.a.521.4 16
3.2 odd 2 inner 3528.2.bl.a.521.5 16
7.2 even 3 504.2.bl.a.89.4 yes 16
7.3 odd 6 3528.2.k.b.881.8 16
7.4 even 3 3528.2.k.b.881.10 16
7.5 odd 6 inner 3528.2.bl.a.1097.5 16
7.6 odd 2 504.2.bl.a.17.5 yes 16
21.2 odd 6 504.2.bl.a.89.5 yes 16
21.5 even 6 inner 3528.2.bl.a.1097.4 16
21.11 odd 6 3528.2.k.b.881.7 16
21.17 even 6 3528.2.k.b.881.9 16
21.20 even 2 504.2.bl.a.17.4 ✓ 16
28.3 even 6 7056.2.k.h.881.7 16
28.11 odd 6 7056.2.k.h.881.9 16
28.23 odd 6 1008.2.bt.d.593.4 16
28.27 even 2 1008.2.bt.d.17.5 16
84.11 even 6 7056.2.k.h.881.8 16
84.23 even 6 1008.2.bt.d.593.5 16
84.59 odd 6 7056.2.k.h.881.10 16
84.83 odd 2 1008.2.bt.d.17.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bl.a.17.4 ✓ 16 21.20 even 2
504.2.bl.a.17.5 yes 16 7.6 odd 2
504.2.bl.a.89.4 yes 16 7.2 even 3
504.2.bl.a.89.5 yes 16 21.2 odd 6
1008.2.bt.d.17.4 16 84.83 odd 2
1008.2.bt.d.17.5 16 28.27 even 2
1008.2.bt.d.593.4 16 28.23 odd 6
1008.2.bt.d.593.5 16 84.23 even 6
3528.2.k.b.881.7 16 21.11 odd 6
3528.2.k.b.881.8 16 7.3 odd 6
3528.2.k.b.881.9 16 21.17 even 6
3528.2.k.b.881.10 16 7.4 even 3
3528.2.bl.a.521.4 16 1.1 even 1 trivial
3528.2.bl.a.521.5 16 3.2 odd 2 inner
3528.2.bl.a.1097.4 16 21.5 even 6 inner
3528.2.bl.a.1097.5 16 7.5 odd 6 inner
7056.2.k.h.881.7 16 28.3 even 6
7056.2.k.h.881.8 16 84.11 even 6
7056.2.k.h.881.9 16 28.11 odd 6
7056.2.k.h.881.10 16 84.59 odd 6