Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,4,Mod(599,1150)] 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("1150.599"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-48,0,-20,0,0,-122,0,-98] 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: no
Analytic conductor: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 219x^{10} + 17685x^{8} + 640366x^{6} + 10000368x^{4} + 54897345x^{2} + 95531076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.1
Root \(-8.71212i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.s.599.12

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -9.71212i q^{3} -4.00000 q^{4} -19.4242 q^{6} +33.1288i q^{7} +8.00000i q^{8} -67.3252 q^{9} +1.59705 q^{11} +38.8485i q^{12} +15.1238i q^{13} +66.2575 q^{14} +16.0000 q^{16} -63.9661i q^{17} +134.650i q^{18} +110.153 q^{19} +321.750 q^{21} -3.19409i q^{22} -23.0000i q^{23} +77.6969 q^{24} +30.2477 q^{26} +391.643i q^{27} -132.515i q^{28} -191.979 q^{29} +292.477 q^{31} -32.0000i q^{32} -15.5107i q^{33} -127.932 q^{34} +269.301 q^{36} -62.7459i q^{37} -220.306i q^{38} +146.885 q^{39} +296.314 q^{41} -643.501i q^{42} +90.3963i q^{43} -6.38819 q^{44} -46.0000 q^{46} +142.749i q^{47} -155.394i q^{48} -754.515 q^{49} -621.246 q^{51} -60.4954i q^{52} -589.530i q^{53} +783.286 q^{54} -265.030 q^{56} -1069.82i q^{57} +383.959i q^{58} +175.690 q^{59} -809.615 q^{61} -584.955i q^{62} -2230.40i q^{63} -64.0000 q^{64} -31.0214 q^{66} -431.173i q^{67} +255.864i q^{68} -223.379 q^{69} +453.657 q^{71} -538.602i q^{72} -111.788i q^{73} -125.492 q^{74} -440.612 q^{76} +52.9082i q^{77} -293.769i q^{78} +1175.57 q^{79} +1985.90 q^{81} -592.628i q^{82} +95.0748i q^{83} -1287.00 q^{84} +180.793 q^{86} +1864.52i q^{87} +12.7764i q^{88} -836.699 q^{89} -501.034 q^{91} +92.0000i q^{92} -2840.57i q^{93} +285.499 q^{94} -310.788 q^{96} +599.979i q^{97} +1509.03i q^{98} -107.522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 48 q^{4} - 20 q^{6} - 122 q^{9} - 98 q^{11} + 168 q^{14} + 192 q^{16} + 458 q^{19} + 184 q^{21} + 80 q^{24} - 64 q^{26} + 364 q^{29} + 228 q^{31} + 700 q^{34} + 488 q^{36} + 286 q^{39} + 486 q^{41}+ \cdots + 284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) − 9.71212i − 1.86910i −0.355835 − 0.934549i \(-0.615803\pi\)
0.355835 − 0.934549i \(-0.384197\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −19.4242 −1.32165
\(7\) 33.1288i 1.78878i 0.447283 + 0.894392i \(0.352392\pi\)
−0.447283 + 0.894392i \(0.647608\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −67.3252 −2.49353
\(10\) 0 0
\(11\) 1.59705 0.0437753 0.0218876 − 0.999760i \(-0.493032\pi\)
0.0218876 + 0.999760i \(0.493032\pi\)
\(12\) 38.8485i 0.934549i
\(13\) 15.1238i 0.322661i 0.986900 + 0.161331i \(0.0515786\pi\)
−0.986900 + 0.161331i \(0.948421\pi\)
\(14\) 66.2575 1.26486
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 63.9661i − 0.912591i −0.889828 − 0.456296i \(-0.849176\pi\)
0.889828 − 0.456296i \(-0.150824\pi\)
\(18\) 134.650i 1.76319i
\(19\) 110.153 1.33004 0.665022 − 0.746824i \(-0.268422\pi\)
0.665022 + 0.746824i \(0.268422\pi\)
\(20\) 0 0
\(21\) 321.750 3.34341
\(22\) − 3.19409i − 0.0309538i
\(23\) − 23.0000i − 0.208514i
\(24\) 77.6969 0.660826
\(25\) 0 0
\(26\) 30.2477 0.228156
\(27\) 391.643i 2.79155i
\(28\) − 132.515i − 0.894392i
\(29\) −191.979 −1.22930 −0.614649 − 0.788801i \(-0.710702\pi\)
−0.614649 + 0.788801i \(0.710702\pi\)
\(30\) 0 0
\(31\) 292.477 1.69453 0.847266 − 0.531169i \(-0.178247\pi\)
0.847266 + 0.531169i \(0.178247\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 15.5107i − 0.0818202i
\(34\) −127.932 −0.645299
\(35\) 0 0
\(36\) 269.301 1.24676
\(37\) − 62.7459i − 0.278794i −0.990237 − 0.139397i \(-0.955484\pi\)
0.990237 − 0.139397i \(-0.0445163\pi\)
\(38\) − 220.306i − 0.940483i
\(39\) 146.885 0.603086
\(40\) 0 0
\(41\) 296.314 1.12869 0.564347 − 0.825538i \(-0.309128\pi\)
0.564347 + 0.825538i \(0.309128\pi\)
\(42\) − 643.501i − 2.36415i
\(43\) 90.3963i 0.320589i 0.987069 + 0.160294i \(0.0512443\pi\)
−0.987069 + 0.160294i \(0.948756\pi\)
\(44\) −6.38819 −0.0218876
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 142.749i 0.443024i 0.975158 + 0.221512i \(0.0710992\pi\)
−0.975158 + 0.221512i \(0.928901\pi\)
\(48\) − 155.394i − 0.467274i
\(49\) −754.515 −2.19975
\(50\) 0 0
\(51\) −621.246 −1.70572
\(52\) − 60.4954i − 0.161331i
\(53\) − 589.530i − 1.52789i −0.645281 − 0.763945i \(-0.723260\pi\)
0.645281 − 0.763945i \(-0.276740\pi\)
\(54\) 783.286 1.97392
\(55\) 0 0
\(56\) −265.030 −0.632431
\(57\) − 1069.82i − 2.48598i
\(58\) 383.959i 0.869245i
\(59\) 175.690 0.387675 0.193838 − 0.981034i \(-0.437907\pi\)
0.193838 + 0.981034i \(0.437907\pi\)
\(60\) 0 0
\(61\) −809.615 −1.69935 −0.849677 − 0.527303i \(-0.823203\pi\)
−0.849677 + 0.527303i \(0.823203\pi\)
\(62\) − 584.955i − 1.19821i
\(63\) − 2230.40i − 4.46038i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −31.0214 −0.0578556
\(67\) − 431.173i − 0.786212i −0.919493 − 0.393106i \(-0.871401\pi\)
0.919493 − 0.393106i \(-0.128599\pi\)
\(68\) 255.864i 0.456296i
\(69\) −223.379 −0.389734
\(70\) 0 0
\(71\) 453.657 0.758298 0.379149 − 0.925336i \(-0.376217\pi\)
0.379149 + 0.925336i \(0.376217\pi\)
\(72\) − 538.602i − 0.881595i
\(73\) − 111.788i − 0.179230i −0.995976 − 0.0896149i \(-0.971436\pi\)
0.995976 − 0.0896149i \(-0.0285636\pi\)
\(74\) −125.492 −0.197137
\(75\) 0 0
\(76\) −440.612 −0.665022
\(77\) 52.9082i 0.0783045i
\(78\) − 293.769i − 0.426446i
\(79\) 1175.57 1.67420 0.837102 − 0.547047i \(-0.184248\pi\)
0.837102 + 0.547047i \(0.184248\pi\)
\(80\) 0 0
\(81\) 1985.90 2.72415
\(82\) − 592.628i − 0.798107i
\(83\) 95.0748i 0.125733i 0.998022 + 0.0628663i \(0.0200242\pi\)
−0.998022 + 0.0628663i \(0.979976\pi\)
\(84\) −1287.00 −1.67171
\(85\) 0 0
\(86\) 180.793 0.226690
\(87\) 1864.52i 2.29768i
\(88\) 12.7764i 0.0154769i
\(89\) −836.699 −0.996516 −0.498258 − 0.867029i \(-0.666027\pi\)
−0.498258 + 0.867029i \(0.666027\pi\)
\(90\) 0 0
\(91\) −501.034 −0.577172
\(92\) 92.0000i 0.104257i
\(93\) − 2840.57i − 3.16724i
\(94\) 285.499 0.313265
\(95\) 0 0
\(96\) −310.788 −0.330413
\(97\) 599.979i 0.628027i 0.949419 + 0.314013i \(0.101674\pi\)
−0.949419 + 0.314013i \(0.898326\pi\)
\(98\) 1509.03i 1.55546i
\(99\) −107.522 −0.109155
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 1150.4.b.s.599.1 12
5.2 odd 4 1150.4.a.x.1.1 yes 6
5.3 odd 4 1150.4.a.w.1.6 ✓ 6
5.4 even 2 inner 1150.4.b.s.599.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.6 ✓ 6 5.3 odd 4
1150.4.a.x.1.1 yes 6 5.2 odd 4
1150.4.b.s.599.1 12 1.1 even 1 trivial
1150.4.b.s.599.12 12 5.4 even 2 inner