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(1,1150)] 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("1150.1"); 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([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,12,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.8521965066\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 109x^{4} + 94x^{3} + 2808x^{2} + 81x - 9774 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-8.71212\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -9.71212 q^{3} +4.00000 q^{4} -19.4242 q^{6} -33.1288 q^{7} +8.00000 q^{8} +67.3252 q^{9} +1.59705 q^{11} -38.8485 q^{12} +15.1238 q^{13} -66.2575 q^{14} +16.0000 q^{16} +63.9661 q^{17} +134.650 q^{18} -110.153 q^{19} +321.750 q^{21} +3.19409 q^{22} -23.0000 q^{23} -77.6969 q^{24} +30.2477 q^{26} -391.643 q^{27} -132.515 q^{28} +191.979 q^{29} +292.477 q^{31} +32.0000 q^{32} -15.5107 q^{33} +127.932 q^{34} +269.301 q^{36} +62.7459 q^{37} -220.306 q^{38} -146.885 q^{39} +296.314 q^{41} +643.501 q^{42} +90.3963 q^{43} +6.38819 q^{44} -46.0000 q^{46} -142.749 q^{47} -155.394 q^{48} +754.515 q^{49} -621.246 q^{51} +60.4954 q^{52} -589.530 q^{53} -783.286 q^{54} -265.030 q^{56} +1069.82 q^{57} +383.959 q^{58} -175.690 q^{59} -809.615 q^{61} +584.955 q^{62} -2230.40 q^{63} +64.0000 q^{64} -31.0214 q^{66} +431.173 q^{67} +255.864 q^{68} +223.379 q^{69} +453.657 q^{71} +538.602 q^{72} -111.788 q^{73} +125.492 q^{74} -440.612 q^{76} -52.9082 q^{77} -293.769 q^{78} -1175.57 q^{79} +1985.90 q^{81} +592.628 q^{82} +95.0748 q^{83} +1287.00 q^{84} +180.793 q^{86} -1864.52 q^{87} +12.7764 q^{88} +836.699 q^{89} -501.034 q^{91} -92.0000 q^{92} -2840.57 q^{93} -285.499 q^{94} -310.788 q^{96} -599.979 q^{97} +1509.03 q^{98} +107.522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} - 5 q^{3} + 24 q^{4} - 10 q^{6} - 42 q^{7} + 48 q^{8} + 61 q^{9} - 49 q^{11} - 20 q^{12} - 16 q^{13} - 84 q^{14} + 96 q^{16} - 175 q^{17} + 122 q^{18} - 229 q^{19} + 92 q^{21} - 98 q^{22}+ \cdots - 142 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −9.71212 −1.86910 −0.934549 − 0.355835i \(-0.884197\pi\)
−0.934549 + 0.355835i \(0.884197\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −19.4242 −1.32165
\(7\) −33.1288 −1.78878 −0.894392 − 0.447283i \(-0.852392\pi\)
−0.894392 + 0.447283i \(0.852392\pi\)
\(8\) 8.00000 0.353553
\(9\) 67.3252 2.49353
\(10\) 0 0
\(11\) 1.59705 0.0437753 0.0218876 − 0.999760i \(-0.493032\pi\)
0.0218876 + 0.999760i \(0.493032\pi\)
\(12\) −38.8485 −0.934549
\(13\) 15.1238 0.322661 0.161331 − 0.986900i \(-0.448421\pi\)
0.161331 + 0.986900i \(0.448421\pi\)
\(14\) −66.2575 −1.26486
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 63.9661 0.912591 0.456296 − 0.889828i \(-0.349176\pi\)
0.456296 + 0.889828i \(0.349176\pi\)
\(18\) 134.650 1.76319
\(19\) −110.153 −1.33004 −0.665022 − 0.746824i \(-0.731578\pi\)
−0.665022 + 0.746824i \(0.731578\pi\)
\(20\) 0 0
\(21\) 321.750 3.34341
\(22\) 3.19409 0.0309538
\(23\) −23.0000 −0.208514
\(24\) −77.6969 −0.660826
\(25\) 0 0
\(26\) 30.2477 0.228156
\(27\) −391.643 −2.79155
\(28\) −132.515 −0.894392
\(29\) 191.979 1.22930 0.614649 − 0.788801i \(-0.289298\pi\)
0.614649 + 0.788801i \(0.289298\pi\)
\(30\) 0 0
\(31\) 292.477 1.69453 0.847266 − 0.531169i \(-0.178247\pi\)
0.847266 + 0.531169i \(0.178247\pi\)
\(32\) 32.0000 0.176777
\(33\) −15.5107 −0.0818202
\(34\) 127.932 0.645299
\(35\) 0 0
\(36\) 269.301 1.24676
\(37\) 62.7459 0.278794 0.139397 − 0.990237i \(-0.455484\pi\)
0.139397 + 0.990237i \(0.455484\pi\)
\(38\) −220.306 −0.940483
\(39\) −146.885 −0.603086
\(40\) 0 0
\(41\) 296.314 1.12869 0.564347 − 0.825538i \(-0.309128\pi\)
0.564347 + 0.825538i \(0.309128\pi\)
\(42\) 643.501 2.36415
\(43\) 90.3963 0.320589 0.160294 − 0.987069i \(-0.448756\pi\)
0.160294 + 0.987069i \(0.448756\pi\)
\(44\) 6.38819 0.0218876
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) −142.749 −0.443024 −0.221512 − 0.975158i \(-0.571099\pi\)
−0.221512 + 0.975158i \(0.571099\pi\)
\(48\) −155.394 −0.467274
\(49\) 754.515 2.19975
\(50\) 0 0
\(51\) −621.246 −1.70572
\(52\) 60.4954 0.161331
\(53\) −589.530 −1.52789 −0.763945 − 0.645281i \(-0.776740\pi\)
−0.763945 + 0.645281i \(0.776740\pi\)
\(54\) −783.286 −1.97392
\(55\) 0 0
\(56\) −265.030 −0.632431
\(57\) 1069.82 2.48598
\(58\) 383.959 0.869245
\(59\) −175.690 −0.387675 −0.193838 − 0.981034i \(-0.562093\pi\)
−0.193838 + 0.981034i \(0.562093\pi\)
\(60\) 0 0
\(61\) −809.615 −1.69935 −0.849677 − 0.527303i \(-0.823203\pi\)
−0.849677 + 0.527303i \(0.823203\pi\)
\(62\) 584.955 1.19821
\(63\) −2230.40 −4.46038
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −31.0214 −0.0578556
\(67\) 431.173 0.786212 0.393106 − 0.919493i \(-0.371401\pi\)
0.393106 + 0.919493i \(0.371401\pi\)
\(68\) 255.864 0.456296
\(69\) 223.379 0.389734
\(70\) 0 0
\(71\) 453.657 0.758298 0.379149 − 0.925336i \(-0.376217\pi\)
0.379149 + 0.925336i \(0.376217\pi\)
\(72\) 538.602 0.881595
\(73\) −111.788 −0.179230 −0.0896149 − 0.995976i \(-0.528564\pi\)
−0.0896149 + 0.995976i \(0.528564\pi\)
\(74\) 125.492 0.197137
\(75\) 0 0
\(76\) −440.612 −0.665022
\(77\) −52.9082 −0.0783045
\(78\) −293.769 −0.426446
\(79\) −1175.57 −1.67420 −0.837102 − 0.547047i \(-0.815752\pi\)
−0.837102 + 0.547047i \(0.815752\pi\)
\(80\) 0 0
\(81\) 1985.90 2.72415
\(82\) 592.628 0.798107
\(83\) 95.0748 0.125733 0.0628663 − 0.998022i \(-0.479976\pi\)
0.0628663 + 0.998022i \(0.479976\pi\)
\(84\) 1287.00 1.67171
\(85\) 0 0
\(86\) 180.793 0.226690
\(87\) −1864.52 −2.29768
\(88\) 12.7764 0.0154769
\(89\) 836.699 0.996516 0.498258 − 0.867029i \(-0.333973\pi\)
0.498258 + 0.867029i \(0.333973\pi\)
\(90\) 0 0
\(91\) −501.034 −0.577172
\(92\) −92.0000 −0.104257
\(93\) −2840.57 −3.16724
\(94\) −285.499 −0.313265
\(95\) 0 0
\(96\) −310.788 −0.330413
\(97\) −599.979 −0.628027 −0.314013 − 0.949419i \(-0.601674\pi\)
−0.314013 + 0.949419i \(0.601674\pi\)
\(98\) 1509.03 1.55546
\(99\) 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.a.x.1.1 yes 6
5.2 odd 4 1150.4.b.s.599.12 12
5.3 odd 4 1150.4.b.s.599.1 12
5.4 even 2 1150.4.a.w.1.6 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.6 ✓ 6 5.4 even 2
1150.4.a.x.1.1 yes 6 1.1 even 1 trivial
1150.4.b.s.599.1 12 5.3 odd 4
1150.4.b.s.599.12 12 5.2 odd 4