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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1200,2,Mod(1199,1200)] 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("1200.1199"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1199.3
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1199
Dual form 1200.2.o.i.1199.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} +3.46410 q^{7} +3.00000 q^{9} -2.00000i q^{13} +3.46410i q^{19} +6.00000 q^{21} +5.19615 q^{27} -10.3923i q^{31} +10.0000i q^{37} -3.46410i q^{39} -10.3923 q^{43} +5.00000 q^{49} +6.00000i q^{57} +14.0000 q^{61} +10.3923 q^{63} -3.46410 q^{67} +10.0000i q^{73} -17.3205i q^{79} +9.00000 q^{81} -6.92820i q^{91} -18.0000i q^{93} +14.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} + 24 q^{21} + 20 q^{49} + 56 q^{61} + 36 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.73205 1.00000
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 − 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) − 10.3923i − 1.86651i −0.359211 − 0.933257i \(-0.616954\pi\)
0.359211 − 0.933257i \(-0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) − 3.46410i − 0.554700i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −10.3923 −1.58481 −0.792406 − 0.609994i \(-0.791172\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 10.3923 1.30931
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.46410 −0.423207 −0.211604 − 0.977356i \(-0.567869\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 17.3205i − 1.94871i −0.225018 − 0.974355i \(-0.572244\pi\)
0.225018 − 0.974355i \(-0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) − 6.92820i − 0.726273i
\(92\) 0 0
\(93\) − 18.0000i − 1.86651i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 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 1200.2.o.i.1199.3 4
3.2 odd 2 CM 1200.2.o.i.1199.3 4
4.3 odd 2 inner 1200.2.o.i.1199.1 4
5.2 odd 4 48.2.c.a.47.1 ✓ 2
5.3 odd 4 1200.2.h.e.1151.2 2
5.4 even 2 inner 1200.2.o.i.1199.2 4
12.11 even 2 inner 1200.2.o.i.1199.1 4
15.2 even 4 48.2.c.a.47.1 ✓ 2
15.8 even 4 1200.2.h.e.1151.2 2
15.14 odd 2 inner 1200.2.o.i.1199.2 4
20.3 even 4 1200.2.h.e.1151.1 2
20.7 even 4 48.2.c.a.47.2 yes 2
20.19 odd 2 inner 1200.2.o.i.1199.4 4
35.27 even 4 2352.2.h.c.2255.2 2
40.27 even 4 192.2.c.a.191.1 2
40.37 odd 4 192.2.c.a.191.2 2
45.2 even 12 1296.2.s.e.863.1 2
45.7 odd 12 1296.2.s.e.863.1 2
45.22 odd 12 1296.2.s.b.431.1 2
45.32 even 12 1296.2.s.b.431.1 2
60.23 odd 4 1200.2.h.e.1151.1 2
60.47 odd 4 48.2.c.a.47.2 yes 2
60.59 even 2 inner 1200.2.o.i.1199.4 4
80.27 even 4 768.2.f.d.383.4 4
80.37 odd 4 768.2.f.d.383.1 4
80.67 even 4 768.2.f.d.383.2 4
80.77 odd 4 768.2.f.d.383.3 4
105.62 odd 4 2352.2.h.c.2255.2 2
120.77 even 4 192.2.c.a.191.2 2
120.107 odd 4 192.2.c.a.191.1 2
140.27 odd 4 2352.2.h.c.2255.1 2
180.7 even 12 1296.2.s.b.863.1 2
180.47 odd 12 1296.2.s.b.863.1 2
180.67 even 12 1296.2.s.e.431.1 2
180.167 odd 12 1296.2.s.e.431.1 2
240.77 even 4 768.2.f.d.383.3 4
240.107 odd 4 768.2.f.d.383.4 4
240.197 even 4 768.2.f.d.383.1 4
240.227 odd 4 768.2.f.d.383.2 4
420.167 even 4 2352.2.h.c.2255.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.c.a.47.1 ✓ 2 5.2 odd 4
48.2.c.a.47.1 ✓ 2 15.2 even 4
48.2.c.a.47.2 yes 2 20.7 even 4
48.2.c.a.47.2 yes 2 60.47 odd 4
192.2.c.a.191.1 2 40.27 even 4
192.2.c.a.191.1 2 120.107 odd 4
192.2.c.a.191.2 2 40.37 odd 4
192.2.c.a.191.2 2 120.77 even 4
768.2.f.d.383.1 4 80.37 odd 4
768.2.f.d.383.1 4 240.197 even 4
768.2.f.d.383.2 4 80.67 even 4
768.2.f.d.383.2 4 240.227 odd 4
768.2.f.d.383.3 4 80.77 odd 4
768.2.f.d.383.3 4 240.77 even 4
768.2.f.d.383.4 4 80.27 even 4
768.2.f.d.383.4 4 240.107 odd 4
1200.2.h.e.1151.1 2 20.3 even 4
1200.2.h.e.1151.1 2 60.23 odd 4
1200.2.h.e.1151.2 2 5.3 odd 4
1200.2.h.e.1151.2 2 15.8 even 4
1200.2.o.i.1199.1 4 4.3 odd 2 inner
1200.2.o.i.1199.1 4 12.11 even 2 inner
1200.2.o.i.1199.2 4 5.4 even 2 inner
1200.2.o.i.1199.2 4 15.14 odd 2 inner
1200.2.o.i.1199.3 4 1.1 even 1 trivial
1200.2.o.i.1199.3 4 3.2 odd 2 CM
1200.2.o.i.1199.4 4 20.19 odd 2 inner
1200.2.o.i.1199.4 4 60.59 even 2 inner
1296.2.s.b.431.1 2 45.22 odd 12
1296.2.s.b.431.1 2 45.32 even 12
1296.2.s.b.863.1 2 180.7 even 12
1296.2.s.b.863.1 2 180.47 odd 12
1296.2.s.e.431.1 2 180.67 even 12
1296.2.s.e.431.1 2 180.167 odd 12
1296.2.s.e.863.1 2 45.2 even 12
1296.2.s.e.863.1 2 45.7 odd 12
2352.2.h.c.2255.1 2 140.27 odd 4
2352.2.h.c.2255.1 2 420.167 even 4
2352.2.h.c.2255.2 2 35.27 even 4
2352.2.h.c.2255.2 2 105.62 odd 4