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Properties

Label 3528.2.s.q.361.1
Level $3528$
Weight $2$
Character 3528.361
Analytic conductor $28.171$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(361,3528)] 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("3528.361"); 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, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,1,0,0,0,0,0,3,0,12,0,0,0,5,0,1,0,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3528.361
Dual form 3528.2.s.q.3313.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{5} +(1.50000 - 2.59808i) q^{11} +6.00000 q^{13} +(2.50000 - 4.33013i) q^{17} +(0.500000 + 0.866025i) q^{19} +(-3.50000 - 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} -2.00000 q^{29} +(-2.50000 + 4.33013i) q^{31} +(-1.50000 - 2.59808i) q^{37} -2.00000 q^{41} -4.00000 q^{43} +(-2.50000 - 4.33013i) q^{47} +(-0.500000 + 0.866025i) q^{53} +3.00000 q^{55} +(-7.50000 + 12.9904i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(3.00000 + 5.19615i) q^{65} +(4.50000 - 7.79423i) q^{67} +(3.50000 - 6.06218i) q^{73} +(-0.500000 - 0.866025i) q^{79} +12.0000 q^{83} +5.00000 q^{85} +(-3.50000 - 6.06218i) q^{89} +(-0.500000 + 0.866025i) q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + 3 q^{11} + 12 q^{13} + 5 q^{17} + q^{19} - 7 q^{23} + 4 q^{25} - 4 q^{29} - 5 q^{31} - 3 q^{37} - 4 q^{41} - 8 q^{43} - 5 q^{47} - q^{53} + 6 q^{55} - 15 q^{59} - 5 q^{61} + 6 q^{65} + 9 q^{67}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i 0.955901 − 0.293691i \(-0.0948835\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.50000 − 2.59808i 0.452267 − 0.783349i −0.546259 − 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.50000 − 4.33013i 0.606339 − 1.05021i −0.385499 − 0.922708i \(-0.625971\pi\)
0.991838 − 0.127502i \(-0.0406959\pi\)
\(18\) 0 0
\(19\) 0.500000 + 0.866025i 0.114708 + 0.198680i 0.917663 − 0.397360i \(-0.130073\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.50000 − 6.06218i −0.729800 − 1.26405i −0.956967 − 0.290196i \(-0.906280\pi\)
0.227167 − 0.973856i \(-0.427054\pi\)
\(24\) 0 0
\(25\) 2.00000 − 3.46410i 0.400000 − 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −2.50000 + 4.33013i −0.449013 + 0.777714i −0.998322 − 0.0579057i \(-0.981558\pi\)
0.549309 + 0.835619i \(0.314891\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.50000 − 2.59808i −0.246598 − 0.427121i 0.715981 − 0.698119i \(-0.245980\pi\)
−0.962580 + 0.270998i \(0.912646\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.50000 − 4.33013i −0.364662 − 0.631614i 0.624059 − 0.781377i \(-0.285482\pi\)
−0.988722 + 0.149763i \(0.952149\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.500000 + 0.866025i −0.0686803 + 0.118958i −0.898321 − 0.439340i \(-0.855212\pi\)
0.829640 + 0.558298i \(0.188546\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.50000 + 12.9904i −0.976417 + 1.69120i −0.301239 + 0.953549i \(0.597400\pi\)
−0.675178 + 0.737655i \(0.735933\pi\)
\(60\) 0 0
\(61\) −2.50000 − 4.33013i −0.320092 − 0.554416i 0.660415 − 0.750901i \(-0.270381\pi\)
−0.980507 + 0.196485i \(0.937047\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 4.50000 − 7.79423i 0.549762 − 0.952217i −0.448528 − 0.893769i \(-0.648052\pi\)
0.998290 − 0.0584478i \(-0.0186151\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 3.50000 − 6.06218i 0.409644 − 0.709524i −0.585206 − 0.810885i \(-0.698986\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.500000 − 0.866025i −0.0562544 − 0.0974355i 0.836527 − 0.547926i \(-0.184582\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.50000 − 6.06218i −0.370999 − 0.642590i 0.618720 − 0.785611i \(-0.287651\pi\)
−0.989720 + 0.143022i \(0.954318\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.500000 + 0.866025i −0.0512989 + 0.0888523i
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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.s.q.361.1 2
3.2 odd 2 392.2.i.b.361.1 2
7.2 even 3 inner 3528.2.s.q.3313.1 2
7.3 odd 6 3528.2.a.p.1.1 1
7.4 even 3 3528.2.a.j.1.1 1
7.5 odd 6 504.2.s.c.289.1 2
7.6 odd 2 504.2.s.c.361.1 2
12.11 even 2 784.2.i.h.753.1 2
21.2 odd 6 392.2.i.b.177.1 2
21.5 even 6 56.2.i.b.9.1 ✓ 2
21.11 odd 6 392.2.a.e.1.1 1
21.17 even 6 392.2.a.c.1.1 1
21.20 even 2 56.2.i.b.25.1 yes 2
28.3 even 6 7056.2.a.bj.1.1 1
28.11 odd 6 7056.2.a.u.1.1 1
28.19 even 6 1008.2.s.g.289.1 2
28.27 even 2 1008.2.s.g.865.1 2
84.11 even 6 784.2.a.c.1.1 1
84.23 even 6 784.2.i.h.177.1 2
84.47 odd 6 112.2.i.a.65.1 2
84.59 odd 6 784.2.a.h.1.1 1
84.83 odd 2 112.2.i.a.81.1 2
105.47 odd 12 1400.2.bh.a.849.2 4
105.59 even 6 9800.2.a.be.1.1 1
105.62 odd 4 1400.2.bh.a.249.1 4
105.68 odd 12 1400.2.bh.a.849.1 4
105.74 odd 6 9800.2.a.s.1.1 1
105.83 odd 4 1400.2.bh.a.249.2 4
105.89 even 6 1400.2.q.d.401.1 2
105.104 even 2 1400.2.q.d.1201.1 2
168.5 even 6 448.2.i.b.65.1 2
168.11 even 6 3136.2.a.t.1.1 1
168.53 odd 6 3136.2.a.i.1.1 1
168.59 odd 6 3136.2.a.j.1.1 1
168.83 odd 2 448.2.i.d.193.1 2
168.101 even 6 3136.2.a.u.1.1 1
168.125 even 2 448.2.i.b.193.1 2
168.131 odd 6 448.2.i.d.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.i.b.9.1 ✓ 2 21.5 even 6
56.2.i.b.25.1 yes 2 21.20 even 2
112.2.i.a.65.1 2 84.47 odd 6
112.2.i.a.81.1 2 84.83 odd 2
392.2.a.c.1.1 1 21.17 even 6
392.2.a.e.1.1 1 21.11 odd 6
392.2.i.b.177.1 2 21.2 odd 6
392.2.i.b.361.1 2 3.2 odd 2
448.2.i.b.65.1 2 168.5 even 6
448.2.i.b.193.1 2 168.125 even 2
448.2.i.d.65.1 2 168.131 odd 6
448.2.i.d.193.1 2 168.83 odd 2
504.2.s.c.289.1 2 7.5 odd 6
504.2.s.c.361.1 2 7.6 odd 2
784.2.a.c.1.1 1 84.11 even 6
784.2.a.h.1.1 1 84.59 odd 6
784.2.i.h.177.1 2 84.23 even 6
784.2.i.h.753.1 2 12.11 even 2
1008.2.s.g.289.1 2 28.19 even 6
1008.2.s.g.865.1 2 28.27 even 2
1400.2.q.d.401.1 2 105.89 even 6
1400.2.q.d.1201.1 2 105.104 even 2
1400.2.bh.a.249.1 4 105.62 odd 4
1400.2.bh.a.249.2 4 105.83 odd 4
1400.2.bh.a.849.1 4 105.68 odd 12
1400.2.bh.a.849.2 4 105.47 odd 12
3136.2.a.i.1.1 1 168.53 odd 6
3136.2.a.j.1.1 1 168.59 odd 6
3136.2.a.t.1.1 1 168.11 even 6
3136.2.a.u.1.1 1 168.101 even 6
3528.2.a.j.1.1 1 7.4 even 3
3528.2.a.p.1.1 1 7.3 odd 6
3528.2.s.q.361.1 2 1.1 even 1 trivial
3528.2.s.q.3313.1 2 7.2 even 3 inner
7056.2.a.u.1.1 1 28.11 odd 6
7056.2.a.bj.1.1 1 28.3 even 6
9800.2.a.s.1.1 1 105.74 odd 6
9800.2.a.be.1.1 1 105.59 even 6