Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9200,2,Mod(1,9200)] 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("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,0,0,-4,0,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 92)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -4.00000 q^{7} +6.00000 q^{9} -2.00000 q^{11} +5.00000 q^{13} -4.00000 q^{17} +2.00000 q^{19} +12.0000 q^{21} +1.00000 q^{23} -9.00000 q^{27} -7.00000 q^{29} +3.00000 q^{31} +6.00000 q^{33} -2.00000 q^{37} -15.0000 q^{39} -9.00000 q^{41} -8.00000 q^{43} +9.00000 q^{47} +9.00000 q^{49} +12.0000 q^{51} -2.00000 q^{53} -6.00000 q^{57} -2.00000 q^{61} -24.0000 q^{63} +14.0000 q^{67} -3.00000 q^{69} +3.00000 q^{71} +3.00000 q^{73} +8.00000 q^{77} +6.00000 q^{79} +9.00000 q^{81} +8.00000 q^{83} +21.0000 q^{87} +12.0000 q^{89} -20.0000 q^{91} -9.00000 q^{93} -12.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 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
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 12.0000 2.61861
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −7.00000 −1.29987 −0.649934 − 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 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
\(39\) −15.0000 −2.40192
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −24.0000 −3.02372
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 − 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 3.00000 0.351123 0.175562 − 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 21.0000 2.25144
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) 0 0
\(93\) −9.00000 −0.933257
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
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 9200.2.a.b.1.1 1
4.3 odd 2 2300.2.a.h.1.1 1
5.4 even 2 368.2.a.g.1.1 1
15.14 odd 2 3312.2.a.q.1.1 1
20.3 even 4 2300.2.c.b.1749.2 2
20.7 even 4 2300.2.c.b.1749.1 2
20.19 odd 2 92.2.a.a.1.1 ✓ 1
40.19 odd 2 1472.2.a.n.1.1 1
40.29 even 2 1472.2.a.b.1.1 1
60.59 even 2 828.2.a.c.1.1 1
115.114 odd 2 8464.2.a.s.1.1 1
140.139 even 2 4508.2.a.d.1.1 1
460.459 even 2 2116.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.a.a.1.1 ✓ 1 20.19 odd 2
368.2.a.g.1.1 1 5.4 even 2
828.2.a.c.1.1 1 60.59 even 2
1472.2.a.b.1.1 1 40.29 even 2
1472.2.a.n.1.1 1 40.19 odd 2
2116.2.a.a.1.1 1 460.459 even 2
2300.2.a.h.1.1 1 4.3 odd 2
2300.2.c.b.1749.1 2 20.7 even 4
2300.2.c.b.1749.2 2 20.3 even 4
3312.2.a.q.1.1 1 15.14 odd 2
4508.2.a.d.1.1 1 140.139 even 2
8464.2.a.s.1.1 1 115.114 odd 2
9200.2.a.b.1.1 1 1.1 even 1 trivial