Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2352,2,Mod(961,2352)] 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("2352.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,3,0,0,0,-1,0,3,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.7808145554\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1537.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1537
Dual form 2352.2.q.m.961.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(1.50000 + 2.59808i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(1.50000 - 2.59808i) q^{11} +4.00000 q^{13} -3.00000 q^{15} +(2.00000 + 3.46410i) q^{19} +(-2.00000 + 3.46410i) q^{25} +1.00000 q^{27} +9.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(1.50000 + 2.59808i) q^{33} +(-4.00000 - 6.92820i) q^{37} +(-2.00000 + 3.46410i) q^{39} +10.0000 q^{43} +(1.50000 - 2.59808i) q^{45} +(3.00000 + 5.19615i) q^{47} +(1.50000 - 2.59808i) q^{53} +9.00000 q^{55} -4.00000 q^{57} +(-1.50000 + 2.59808i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(6.00000 + 10.3923i) q^{65} +(-5.00000 + 8.66025i) q^{67} +6.00000 q^{71} +(1.00000 - 1.73205i) q^{73} +(-2.00000 - 3.46410i) q^{75} +(-0.500000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} -9.00000 q^{83} +(-4.50000 + 7.79423i) q^{87} +(3.00000 + 5.19615i) q^{89} +(0.500000 + 0.866025i) q^{93} +(-6.00000 + 10.3923i) q^{95} +1.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 3 q^{5} - q^{9} + 3 q^{11} + 8 q^{13} - 6 q^{15} + 4 q^{19} - 4 q^{25} + 2 q^{27} + 18 q^{29} + q^{31} + 3 q^{33} - 8 q^{37} - 4 q^{39} + 20 q^{43} + 3 q^{45} + 6 q^{47} + 3 q^{53} + 18 q^{55}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 1.50000 + 2.59808i 0.670820 + 1.16190i 0.977672 + 0.210138i \(0.0673912\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 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
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 − 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 0.500000 − 0.866025i 0.0898027 − 0.155543i −0.817625 − 0.575751i \(-0.804710\pi\)
0.907428 + 0.420208i \(0.138043\pi\)
\(32\) 0 0
\(33\) 1.50000 + 2.59808i 0.261116 + 0.452267i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 − 6.92820i −0.657596 − 1.13899i −0.981236 − 0.192809i \(-0.938240\pi\)
0.323640 − 0.946180i \(-0.395093\pi\)
\(38\) 0 0
\(39\) −2.00000 + 3.46410i −0.320256 + 0.554700i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 1.50000 − 2.59808i 0.223607 − 0.387298i
\(46\) 0 0
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 − 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.50000 − 2.59808i 0.206041 − 0.356873i −0.744423 − 0.667708i \(-0.767275\pi\)
0.950464 + 0.310835i \(0.100609\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −1.50000 + 2.59808i −0.195283 + 0.338241i −0.946993 − 0.321253i \(-0.895896\pi\)
0.751710 + 0.659494i \(0.229229\pi\)
\(60\) 0 0
\(61\) −5.00000 − 8.66025i −0.640184 − 1.10883i −0.985391 − 0.170305i \(-0.945525\pi\)
0.345207 − 0.938527i \(-0.387809\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 + 10.3923i 0.744208 + 1.28901i
\(66\) 0 0
\(67\) −5.00000 + 8.66025i −0.610847 + 1.05802i 0.380251 + 0.924883i \(0.375838\pi\)
−0.991098 + 0.133135i \(0.957496\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 1.00000 − 1.73205i 0.117041 − 0.202721i −0.801553 − 0.597924i \(-0.795992\pi\)
0.918594 + 0.395203i \(0.129326\pi\)
\(74\) 0 0
\(75\) −2.00000 − 3.46410i −0.230940 − 0.400000i
\(76\) 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
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.50000 + 7.79423i −0.482451 + 0.835629i
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 − 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.500000 + 0.866025i 0.0518476 + 0.0898027i
\(94\) 0 0
\(95\) −6.00000 + 10.3923i −0.615587 + 1.06623i
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
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 2352.2.q.m.1537.1 2
4.3 odd 2 294.2.e.f.67.1 2
7.2 even 3 inner 2352.2.q.m.961.1 2
7.3 odd 6 2352.2.a.m.1.1 1
7.4 even 3 2352.2.a.n.1.1 1
7.5 odd 6 336.2.q.d.289.1 2
7.6 odd 2 336.2.q.d.193.1 2
12.11 even 2 882.2.g.b.361.1 2
21.5 even 6 1008.2.s.n.289.1 2
21.11 odd 6 7056.2.a.bz.1.1 1
21.17 even 6 7056.2.a.g.1.1 1
21.20 even 2 1008.2.s.n.865.1 2
28.3 even 6 294.2.a.d.1.1 1
28.11 odd 6 294.2.a.a.1.1 1
28.19 even 6 42.2.e.b.37.1 yes 2
28.23 odd 6 294.2.e.f.79.1 2
28.27 even 2 42.2.e.b.25.1 ✓ 2
56.3 even 6 9408.2.a.d.1.1 1
56.5 odd 6 1344.2.q.j.961.1 2
56.11 odd 6 9408.2.a.db.1.1 1
56.13 odd 2 1344.2.q.j.193.1 2
56.19 even 6 1344.2.q.v.961.1 2
56.27 even 2 1344.2.q.v.193.1 2
56.45 odd 6 9408.2.a.bu.1.1 1
56.53 even 6 9408.2.a.bm.1.1 1
84.11 even 6 882.2.a.k.1.1 1
84.23 even 6 882.2.g.b.667.1 2
84.47 odd 6 126.2.g.b.37.1 2
84.59 odd 6 882.2.a.g.1.1 1
84.83 odd 2 126.2.g.b.109.1 2
140.19 even 6 1050.2.i.e.751.1 2
140.27 odd 4 1050.2.o.b.949.1 4
140.39 odd 6 7350.2.a.cw.1.1 1
140.47 odd 12 1050.2.o.b.499.2 4
140.59 even 6 7350.2.a.ce.1.1 1
140.83 odd 4 1050.2.o.b.949.2 4
140.103 odd 12 1050.2.o.b.499.1 4
140.139 even 2 1050.2.i.e.151.1 2
252.47 odd 6 1134.2.e.p.919.1 2
252.83 odd 6 1134.2.h.a.109.1 2
252.103 even 6 1134.2.h.p.541.1 2
252.131 odd 6 1134.2.h.a.541.1 2
252.139 even 6 1134.2.e.a.865.1 2
252.167 odd 6 1134.2.e.p.865.1 2
252.187 even 6 1134.2.e.a.919.1 2
252.223 even 6 1134.2.h.p.109.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.b.25.1 ✓ 2 28.27 even 2
42.2.e.b.37.1 yes 2 28.19 even 6
126.2.g.b.37.1 2 84.47 odd 6
126.2.g.b.109.1 2 84.83 odd 2
294.2.a.a.1.1 1 28.11 odd 6
294.2.a.d.1.1 1 28.3 even 6
294.2.e.f.67.1 2 4.3 odd 2
294.2.e.f.79.1 2 28.23 odd 6
336.2.q.d.193.1 2 7.6 odd 2
336.2.q.d.289.1 2 7.5 odd 6
882.2.a.g.1.1 1 84.59 odd 6
882.2.a.k.1.1 1 84.11 even 6
882.2.g.b.361.1 2 12.11 even 2
882.2.g.b.667.1 2 84.23 even 6
1008.2.s.n.289.1 2 21.5 even 6
1008.2.s.n.865.1 2 21.20 even 2
1050.2.i.e.151.1 2 140.139 even 2
1050.2.i.e.751.1 2 140.19 even 6
1050.2.o.b.499.1 4 140.103 odd 12
1050.2.o.b.499.2 4 140.47 odd 12
1050.2.o.b.949.1 4 140.27 odd 4
1050.2.o.b.949.2 4 140.83 odd 4
1134.2.e.a.865.1 2 252.139 even 6
1134.2.e.a.919.1 2 252.187 even 6
1134.2.e.p.865.1 2 252.167 odd 6
1134.2.e.p.919.1 2 252.47 odd 6
1134.2.h.a.109.1 2 252.83 odd 6
1134.2.h.a.541.1 2 252.131 odd 6
1134.2.h.p.109.1 2 252.223 even 6
1134.2.h.p.541.1 2 252.103 even 6
1344.2.q.j.193.1 2 56.13 odd 2
1344.2.q.j.961.1 2 56.5 odd 6
1344.2.q.v.193.1 2 56.27 even 2
1344.2.q.v.961.1 2 56.19 even 6
2352.2.a.m.1.1 1 7.3 odd 6
2352.2.a.n.1.1 1 7.4 even 3
2352.2.q.m.961.1 2 7.2 even 3 inner
2352.2.q.m.1537.1 2 1.1 even 1 trivial
7056.2.a.g.1.1 1 21.17 even 6
7056.2.a.bz.1.1 1 21.11 odd 6
7350.2.a.ce.1.1 1 140.59 even 6
7350.2.a.cw.1.1 1 140.39 odd 6
9408.2.a.d.1.1 1 56.3 even 6
9408.2.a.bm.1.1 1 56.53 even 6
9408.2.a.bu.1.1 1 56.45 odd 6
9408.2.a.db.1.1 1 56.11 odd 6