Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1210,2,Mod(1,1210)] 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("1210.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1210.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,-1,1,1,-1,-3,1,-2,1,0,-1,0,-3,-1,1,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.66189864457\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1210.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} -2.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} +3.00000 q^{21} -1.00000 q^{24} +1.00000 q^{25} +5.00000 q^{27} -3.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} +8.00000 q^{34} -3.00000 q^{35} -2.00000 q^{36} +8.00000 q^{38} +1.00000 q^{40} +5.00000 q^{41} +3.00000 q^{42} +1.00000 q^{43} -2.00000 q^{45} -5.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} -8.00000 q^{51} +8.00000 q^{53} +5.00000 q^{54} -3.00000 q^{56} -8.00000 q^{57} +2.00000 q^{58} -10.0000 q^{59} -1.00000 q^{60} +7.00000 q^{61} +6.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -7.00000 q^{67} +8.00000 q^{68} -3.00000 q^{70} -14.0000 q^{71} -2.00000 q^{72} -16.0000 q^{73} -1.00000 q^{75} +8.00000 q^{76} +10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +5.00000 q^{82} -12.0000 q^{83} +3.00000 q^{84} +8.00000 q^{85} +1.00000 q^{86} -2.00000 q^{87} +9.00000 q^{89} -2.00000 q^{90} -6.00000 q^{93} -5.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +12.0000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 − 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −3.00000 −0.801784
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 − 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) −2.00000 −0.471405
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) −3.00000 −0.566947
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) −3.00000 −0.507093
\(36\) −2.00000 −0.333333
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 3.00000 0.462910
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −5.00000 −0.729325 −0.364662 − 0.931140i \(-0.618816\pi\)
−0.364662 + 0.931140i \(0.618816\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 − 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) −8.00000 −1.05963
\(58\) 2.00000 0.262613
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −1.00000 −0.129099
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 6.00000 0.762001
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) −14.0000 −1.66149 −0.830747 − 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) −2.00000 −0.235702
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 5.00000 0.552158
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 3.00000 0.327327
\(85\) 8.00000 0.867722
\(86\) 1.00000 0.107833
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 − 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) −5.00000 −0.515711
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 2.00000 0.202031
\(99\) 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 1210.2.a.j.1.1 yes 1
4.3 odd 2 9680.2.a.w.1.1 1
5.4 even 2 6050.2.a.l.1.1 1
11.10 odd 2 1210.2.a.c.1.1 ✓ 1
44.43 even 2 9680.2.a.t.1.1 1
55.54 odd 2 6050.2.a.bh.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1210.2.a.c.1.1 ✓ 1 11.10 odd 2
1210.2.a.j.1.1 yes 1 1.1 even 1 trivial
6050.2.a.l.1.1 1 5.4 even 2
6050.2.a.bh.1.1 1 55.54 odd 2
9680.2.a.t.1.1 1 44.43 even 2
9680.2.a.w.1.1 1 4.3 odd 2