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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,6,Mod(145,288)] 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("288.145"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.1905401061\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-54 +2 \sqrt{73}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 8x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 145.2
Root \(2.38600 - 1.51888i\) of defining polynomial
Character \(\chi\) \(=\) 288.145
Dual form 288.6.d.b.145.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.38521i q^{5} -160.704 q^{7} -129.129i q^{11} +759.659i q^{13} -323.408 q^{17} -198.511i q^{19} -1193.15 q^{23} +3123.08 q^{25} -5987.24i q^{29} +4872.45 q^{31} +222.609i q^{35} -3698.56i q^{37} +10437.9 q^{41} +9873.11i q^{43} -6297.98 q^{47} +9018.79 q^{49} -21728.1i q^{53} -178.871 q^{55} +33513.4i q^{59} -48506.8i q^{61} +1052.29 q^{65} -33182.4i q^{67} +59464.1 q^{71} +51278.6 q^{73} +20751.6i q^{77} +73724.5 q^{79} -61628.0i q^{83} +447.988i q^{85} +106735. q^{89} -122080. i q^{91} -274.980 q^{95} +12562.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 96 q^{7} - 200 q^{17} + 2336 q^{23} + 1556 q^{25} + 12928 q^{31} + 4568 q^{41} - 54720 q^{47} + 9828 q^{49} - 85472 q^{55} + 19520 q^{65} + 206688 q^{71} + 39976 q^{73} + 247872 q^{79} + 84632 q^{89}+ \cdots - 99576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.38521i − 0.0247794i −0.999923 − 0.0123897i \(-0.996056\pi\)
0.999923 − 0.0123897i \(-0.00394386\pi\)
\(6\) 0 0
\(7\) −160.704 −1.23960 −0.619800 − 0.784760i \(-0.712786\pi\)
−0.619800 + 0.784760i \(0.712786\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 129.129i − 0.321768i −0.986973 − 0.160884i \(-0.948566\pi\)
0.986973 − 0.160884i \(-0.0514345\pi\)
\(12\) 0 0
\(13\) 759.659i 1.24670i 0.781945 + 0.623348i \(0.214228\pi\)
−0.781945 + 0.623348i \(0.785772\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −323.408 −0.271412 −0.135706 − 0.990749i \(-0.543330\pi\)
−0.135706 + 0.990749i \(0.543330\pi\)
\(18\) 0 0
\(19\) − 198.511i − 0.126154i −0.998009 − 0.0630771i \(-0.979909\pi\)
0.998009 − 0.0630771i \(-0.0200914\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1193.15 −0.470302 −0.235151 − 0.971959i \(-0.575558\pi\)
−0.235151 + 0.971959i \(0.575558\pi\)
\(24\) 0 0
\(25\) 3123.08 0.999386
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 5987.24i − 1.32200i −0.750386 − 0.661000i \(-0.770132\pi\)
0.750386 − 0.661000i \(-0.229868\pi\)
\(30\) 0 0
\(31\) 4872.45 0.910632 0.455316 − 0.890330i \(-0.349526\pi\)
0.455316 + 0.890330i \(0.349526\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 222.609i 0.0307165i
\(36\) 0 0
\(37\) − 3698.56i − 0.444149i −0.975030 − 0.222074i \(-0.928717\pi\)
0.975030 − 0.222074i \(-0.0712828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10437.9 0.969734 0.484867 − 0.874588i \(-0.338868\pi\)
0.484867 + 0.874588i \(0.338868\pi\)
\(42\) 0 0
\(43\) 9873.11i 0.814297i 0.913362 + 0.407148i \(0.133477\pi\)
−0.913362 + 0.407148i \(0.866523\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6297.98 −0.415869 −0.207935 − 0.978143i \(-0.566674\pi\)
−0.207935 + 0.978143i \(0.566674\pi\)
\(48\) 0 0
\(49\) 9018.79 0.536609
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 21728.1i − 1.06251i −0.847212 − 0.531255i \(-0.821721\pi\)
0.847212 − 0.531255i \(-0.178279\pi\)
\(54\) 0 0
\(55\) −178.871 −0.00797320
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 33513.4i 1.25340i 0.779261 + 0.626699i \(0.215594\pi\)
−0.779261 + 0.626699i \(0.784406\pi\)
\(60\) 0 0
\(61\) − 48506.8i − 1.66908i −0.550944 − 0.834542i \(-0.685732\pi\)
0.550944 − 0.834542i \(-0.314268\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1052.29 0.0308923
\(66\) 0 0
\(67\) − 33182.4i − 0.903068i −0.892254 − 0.451534i \(-0.850877\pi\)
0.892254 − 0.451534i \(-0.149123\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 59464.1 1.39994 0.699970 − 0.714173i \(-0.253197\pi\)
0.699970 + 0.714173i \(0.253197\pi\)
\(72\) 0 0
\(73\) 51278.6 1.12624 0.563118 − 0.826377i \(-0.309602\pi\)
0.563118 + 0.826377i \(0.309602\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 20751.6i 0.398863i
\(78\) 0 0
\(79\) 73724.5 1.32906 0.664530 − 0.747262i \(-0.268632\pi\)
0.664530 + 0.747262i \(0.268632\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 61628.0i − 0.981935i −0.871178 − 0.490967i \(-0.836643\pi\)
0.871178 − 0.490967i \(-0.163357\pi\)
\(84\) 0 0
\(85\) 447.988i 0.00672541i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 106735. 1.42834 0.714169 − 0.699974i \(-0.246805\pi\)
0.714169 + 0.699974i \(0.246805\pi\)
\(90\) 0 0
\(91\) − 122080.i − 1.54540i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −274.980 −0.00312602
\(96\) 0 0
\(97\) 12562.9 0.135569 0.0677846 − 0.997700i \(-0.478407\pi\)
0.0677846 + 0.997700i \(0.478407\pi\)
\(98\) 0 0
\(99\) 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 288.6.d.b.145.2 4
3.2 odd 2 32.6.b.a.17.4 4
4.3 odd 2 72.6.d.b.37.4 4
8.3 odd 2 72.6.d.b.37.3 4
8.5 even 2 inner 288.6.d.b.145.3 4
12.11 even 2 8.6.b.a.5.1 ✓ 4
15.2 even 4 800.6.f.a.49.7 8
15.8 even 4 800.6.f.a.49.2 8
15.14 odd 2 800.6.d.a.401.1 4
24.5 odd 2 32.6.b.a.17.1 4
24.11 even 2 8.6.b.a.5.2 yes 4
48.5 odd 4 256.6.a.n.1.4 4
48.11 even 4 256.6.a.k.1.1 4
48.29 odd 4 256.6.a.n.1.1 4
48.35 even 4 256.6.a.k.1.4 4
60.23 odd 4 200.6.f.a.149.4 8
60.47 odd 4 200.6.f.a.149.5 8
60.59 even 2 200.6.d.a.101.4 4
120.29 odd 2 800.6.d.a.401.4 4
120.53 even 4 800.6.f.a.49.8 8
120.59 even 2 200.6.d.a.101.3 4
120.77 even 4 800.6.f.a.49.1 8
120.83 odd 4 200.6.f.a.149.6 8
120.107 odd 4 200.6.f.a.149.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.6.b.a.5.1 ✓ 4 12.11 even 2
8.6.b.a.5.2 yes 4 24.11 even 2
32.6.b.a.17.1 4 24.5 odd 2
32.6.b.a.17.4 4 3.2 odd 2
72.6.d.b.37.3 4 8.3 odd 2
72.6.d.b.37.4 4 4.3 odd 2
200.6.d.a.101.3 4 120.59 even 2
200.6.d.a.101.4 4 60.59 even 2
200.6.f.a.149.3 8 120.107 odd 4
200.6.f.a.149.4 8 60.23 odd 4
200.6.f.a.149.5 8 60.47 odd 4
200.6.f.a.149.6 8 120.83 odd 4
256.6.a.k.1.1 4 48.11 even 4
256.6.a.k.1.4 4 48.35 even 4
256.6.a.n.1.1 4 48.29 odd 4
256.6.a.n.1.4 4 48.5 odd 4
288.6.d.b.145.2 4 1.1 even 1 trivial
288.6.d.b.145.3 4 8.5 even 2 inner
800.6.d.a.401.1 4 15.14 odd 2
800.6.d.a.401.4 4 120.29 odd 2
800.6.f.a.49.1 8 120.77 even 4
800.6.f.a.49.2 8 15.8 even 4
800.6.f.a.49.7 8 15.2 even 4
800.6.f.a.49.8 8 120.53 even 4