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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1200,4,Mod(49,1200)] 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("1200.49"); S:= CuspForms(chi, 4); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-18,0,-104,0,0,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.b.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} -52.0000 q^{11} +22.0000i q^{13} +14.0000i q^{17} -20.0000 q^{19} -72.0000 q^{21} +168.000i q^{23} +27.0000i q^{27} -230.000 q^{29} +288.000 q^{31} +156.000i q^{33} +34.0000i q^{37} +66.0000 q^{39} +122.000 q^{41} +188.000i q^{43} +256.000i q^{47} -233.000 q^{49} +42.0000 q^{51} -338.000i q^{53} +60.0000i q^{57} +100.000 q^{59} +742.000 q^{61} +216.000i q^{63} -84.0000i q^{67} +504.000 q^{69} +328.000 q^{71} -38.0000i q^{73} +1248.00i q^{77} -240.000 q^{79} +81.0000 q^{81} -1212.00i q^{83} +690.000i q^{87} -330.000 q^{89} +528.000 q^{91} -864.000i q^{93} -866.000i q^{97} +468.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81}+ \cdots + 936 q^{99}+O(q^{100}) \) 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
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 24.0000i − 1.29588i −0.761692 − 0.647939i \(-0.775631\pi\)
0.761692 − 0.647939i \(-0.224369\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −52.0000 −1.42533 −0.712663 − 0.701506i \(-0.752511\pi\)
−0.712663 + 0.701506i \(0.752511\pi\)
\(12\) 0 0
\(13\) 22.0000i 0.469362i 0.972072 + 0.234681i \(0.0754045\pi\)
−0.972072 + 0.234681i \(0.924595\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14.0000i 0.199735i 0.995001 + 0.0998676i \(0.0318419\pi\)
−0.995001 + 0.0998676i \(0.968158\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 − 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) −72.0000 −0.748176
\(22\) 0 0
\(23\) 168.000i 1.52306i 0.648129 + 0.761531i \(0.275552\pi\)
−0.648129 + 0.761531i \(0.724448\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) −230.000 −1.47276 −0.736378 − 0.676570i \(-0.763465\pi\)
−0.736378 + 0.676570i \(0.763465\pi\)
\(30\) 0 0
\(31\) 288.000 1.66859 0.834296 − 0.551317i \(-0.185875\pi\)
0.834296 + 0.551317i \(0.185875\pi\)
\(32\) 0 0
\(33\) 156.000i 0.822913i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000i 0.151069i 0.997143 + 0.0755347i \(0.0240664\pi\)
−0.997143 + 0.0755347i \(0.975934\pi\)
\(38\) 0 0
\(39\) 66.0000 0.270986
\(40\) 0 0
\(41\) 122.000 0.464712 0.232356 − 0.972631i \(-0.425357\pi\)
0.232356 + 0.972631i \(0.425357\pi\)
\(42\) 0 0
\(43\) 188.000i 0.666738i 0.942796 + 0.333369i \(0.108185\pi\)
−0.942796 + 0.333369i \(0.891815\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 256.000i 0.794499i 0.917711 + 0.397249i \(0.130035\pi\)
−0.917711 + 0.397249i \(0.869965\pi\)
\(48\) 0 0
\(49\) −233.000 −0.679300
\(50\) 0 0
\(51\) 42.0000 0.115317
\(52\) 0 0
\(53\) − 338.000i − 0.875998i −0.898976 − 0.437999i \(-0.855687\pi\)
0.898976 − 0.437999i \(-0.144313\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 60.0000i 0.139424i
\(58\) 0 0
\(59\) 100.000 0.220659 0.110330 − 0.993895i \(-0.464809\pi\)
0.110330 + 0.993895i \(0.464809\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 − 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) 0 0
\(63\) 216.000i 0.431959i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 84.0000i − 0.153168i −0.997063 − 0.0765838i \(-0.975599\pi\)
0.997063 − 0.0765838i \(-0.0244013\pi\)
\(68\) 0 0
\(69\) 504.000 0.879340
\(70\) 0 0
\(71\) 328.000 0.548260 0.274130 − 0.961693i \(-0.411610\pi\)
0.274130 + 0.961693i \(0.411610\pi\)
\(72\) 0 0
\(73\) − 38.0000i − 0.0609255i −0.999536 − 0.0304628i \(-0.990302\pi\)
0.999536 − 0.0304628i \(-0.00969810\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1248.00i 1.84705i
\(78\) 0 0
\(79\) −240.000 −0.341799 −0.170899 − 0.985288i \(-0.554667\pi\)
−0.170899 + 0.985288i \(0.554667\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 1212.00i − 1.60282i −0.598114 − 0.801411i \(-0.704083\pi\)
0.598114 − 0.801411i \(-0.295917\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 690.000i 0.850296i
\(88\) 0 0
\(89\) −330.000 −0.393033 −0.196516 − 0.980501i \(-0.562963\pi\)
−0.196516 + 0.980501i \(0.562963\pi\)
\(90\) 0 0
\(91\) 528.000 0.608236
\(92\) 0 0
\(93\) − 864.000i − 0.963362i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 866.000i − 0.906484i −0.891387 − 0.453242i \(-0.850267\pi\)
0.891387 − 0.453242i \(-0.149733\pi\)
\(98\) 0 0
\(99\) 468.000 0.475109
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 1200.4.f.b.49.1 2
4.3 odd 2 75.4.b.b.49.1 2
5.2 odd 4 240.4.a.e.1.1 1
5.3 odd 4 1200.4.a.t.1.1 1
5.4 even 2 inner 1200.4.f.b.49.2 2
12.11 even 2 225.4.b.e.199.2 2
15.2 even 4 720.4.a.n.1.1 1
20.3 even 4 75.4.a.b.1.1 1
20.7 even 4 15.4.a.a.1.1 ✓ 1
20.19 odd 2 75.4.b.b.49.2 2
40.27 even 4 960.4.a.b.1.1 1
40.37 odd 4 960.4.a.ba.1.1 1
60.23 odd 4 225.4.a.f.1.1 1
60.47 odd 4 45.4.a.c.1.1 1
60.59 even 2 225.4.b.e.199.1 2
140.27 odd 4 735.4.a.e.1.1 1
180.7 even 12 405.4.e.g.271.1 2
180.47 odd 12 405.4.e.i.271.1 2
180.67 even 12 405.4.e.g.136.1 2
180.167 odd 12 405.4.e.i.136.1 2
220.87 odd 4 1815.4.a.e.1.1 1
420.167 even 4 2205.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 ✓ 1 20.7 even 4
45.4.a.c.1.1 1 60.47 odd 4
75.4.a.b.1.1 1 20.3 even 4
75.4.b.b.49.1 2 4.3 odd 2
75.4.b.b.49.2 2 20.19 odd 2
225.4.a.f.1.1 1 60.23 odd 4
225.4.b.e.199.1 2 60.59 even 2
225.4.b.e.199.2 2 12.11 even 2
240.4.a.e.1.1 1 5.2 odd 4
405.4.e.g.136.1 2 180.67 even 12
405.4.e.g.271.1 2 180.7 even 12
405.4.e.i.136.1 2 180.167 odd 12
405.4.e.i.271.1 2 180.47 odd 12
720.4.a.n.1.1 1 15.2 even 4
735.4.a.e.1.1 1 140.27 odd 4
960.4.a.b.1.1 1 40.27 even 4
960.4.a.ba.1.1 1 40.37 odd 4
1200.4.a.t.1.1 1 5.3 odd 4
1200.4.f.b.49.1 2 1.1 even 1 trivial
1200.4.f.b.49.2 2 5.4 even 2 inner
1815.4.a.e.1.1 1 220.87 odd 4
2205.4.a.l.1.1 1 420.167 even 4