↑

Properties

Label 1200.2.a.k.1.1
Level $1200$
Weight $2$
Character 1200.1
Self dual yes
Analytic conductor $9.582$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1200,2,Mod(1,1200)] 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("1200.1"); 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([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,0,0,-4,0,1,0,0,0,-2] 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: yes
Analytic conductor: \(9.58204824255\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -4.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} -6.00000 q^{17} +4.00000 q^{19} -4.00000 q^{21} +1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} -2.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} +9.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} +4.00000 q^{57} -10.0000 q^{61} -4.00000 q^{63} -4.00000 q^{67} -2.00000 q^{73} -8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -6.00000 q^{87} +18.0000 q^{89} +8.00000 q^{91} -8.00000 q^{93} -2.00000 q^{97} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 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
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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.a.k.1.1 1
3.2 odd 2 3600.2.a.f.1.1 1
4.3 odd 2 150.2.a.b.1.1 1
5.2 odd 4 1200.2.f.e.49.1 2
5.3 odd 4 1200.2.f.e.49.2 2
5.4 even 2 240.2.a.b.1.1 1
8.3 odd 2 4800.2.a.cq.1.1 1
8.5 even 2 4800.2.a.d.1.1 1
12.11 even 2 450.2.a.d.1.1 1
15.2 even 4 3600.2.f.i.2449.1 2
15.8 even 4 3600.2.f.i.2449.2 2
15.14 odd 2 720.2.a.j.1.1 1
20.3 even 4 150.2.c.a.49.1 2
20.7 even 4 150.2.c.a.49.2 2
20.19 odd 2 30.2.a.a.1.1 ✓ 1
28.27 even 2 7350.2.a.ct.1.1 1
40.3 even 4 4800.2.f.p.3649.2 2
40.13 odd 4 4800.2.f.w.3649.1 2
40.19 odd 2 960.2.a.e.1.1 1
40.27 even 4 4800.2.f.p.3649.1 2
40.29 even 2 960.2.a.p.1.1 1
40.37 odd 4 4800.2.f.w.3649.2 2
60.23 odd 4 450.2.c.b.199.2 2
60.47 odd 4 450.2.c.b.199.1 2
60.59 even 2 90.2.a.c.1.1 1
80.19 odd 4 3840.2.k.y.1921.2 2
80.29 even 4 3840.2.k.f.1921.1 2
80.59 odd 4 3840.2.k.y.1921.1 2
80.69 even 4 3840.2.k.f.1921.2 2
120.29 odd 2 2880.2.a.q.1.1 1
120.59 even 2 2880.2.a.a.1.1 1
140.19 even 6 1470.2.i.q.361.1 2
140.39 odd 6 1470.2.i.o.961.1 2
140.59 even 6 1470.2.i.q.961.1 2
140.79 odd 6 1470.2.i.o.361.1 2
140.139 even 2 1470.2.a.d.1.1 1
180.59 even 6 810.2.e.b.541.1 2
180.79 odd 6 810.2.e.l.271.1 2
180.119 even 6 810.2.e.b.271.1 2
180.139 odd 6 810.2.e.l.541.1 2
220.219 even 2 3630.2.a.w.1.1 1
260.99 even 4 5070.2.b.k.1351.1 2
260.239 even 4 5070.2.b.k.1351.2 2
260.259 odd 2 5070.2.a.w.1.1 1
340.339 odd 2 8670.2.a.g.1.1 1
420.419 odd 2 4410.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 ✓ 1 20.19 odd 2
90.2.a.c.1.1 1 60.59 even 2
150.2.a.b.1.1 1 4.3 odd 2
150.2.c.a.49.1 2 20.3 even 4
150.2.c.a.49.2 2 20.7 even 4
240.2.a.b.1.1 1 5.4 even 2
450.2.a.d.1.1 1 12.11 even 2
450.2.c.b.199.1 2 60.47 odd 4
450.2.c.b.199.2 2 60.23 odd 4
720.2.a.j.1.1 1 15.14 odd 2
810.2.e.b.271.1 2 180.119 even 6
810.2.e.b.541.1 2 180.59 even 6
810.2.e.l.271.1 2 180.79 odd 6
810.2.e.l.541.1 2 180.139 odd 6
960.2.a.e.1.1 1 40.19 odd 2
960.2.a.p.1.1 1 40.29 even 2
1200.2.a.k.1.1 1 1.1 even 1 trivial
1200.2.f.e.49.1 2 5.2 odd 4
1200.2.f.e.49.2 2 5.3 odd 4
1470.2.a.d.1.1 1 140.139 even 2
1470.2.i.o.361.1 2 140.79 odd 6
1470.2.i.o.961.1 2 140.39 odd 6
1470.2.i.q.361.1 2 140.19 even 6
1470.2.i.q.961.1 2 140.59 even 6
2880.2.a.a.1.1 1 120.59 even 2
2880.2.a.q.1.1 1 120.29 odd 2
3600.2.a.f.1.1 1 3.2 odd 2
3600.2.f.i.2449.1 2 15.2 even 4
3600.2.f.i.2449.2 2 15.8 even 4
3630.2.a.w.1.1 1 220.219 even 2
3840.2.k.f.1921.1 2 80.29 even 4
3840.2.k.f.1921.2 2 80.69 even 4
3840.2.k.y.1921.1 2 80.59 odd 4
3840.2.k.y.1921.2 2 80.19 odd 4
4410.2.a.z.1.1 1 420.419 odd 2
4800.2.a.d.1.1 1 8.5 even 2
4800.2.a.cq.1.1 1 8.3 odd 2
4800.2.f.p.3649.1 2 40.27 even 4
4800.2.f.p.3649.2 2 40.3 even 4
4800.2.f.w.3649.1 2 40.13 odd 4
4800.2.f.w.3649.2 2 40.37 odd 4
5070.2.a.w.1.1 1 260.259 odd 2
5070.2.b.k.1351.1 2 260.99 even 4
5070.2.b.k.1351.2 2 260.239 even 4
7350.2.a.ct.1.1 1 28.27 even 2
8670.2.a.g.1.1 1 340.339 odd 2