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Properties

Label 576.4.c.e.575.2
Level $576$
Weight $4$
Character 576.575
Analytic conductor $33.985$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [576,4,Mod(575,576)] 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("576.575"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 576.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,108] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.9851001633\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 13x^{2} - 12x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.2
Root \(0.500000 + 0.522278i\) of defining polynomial
Character \(\chi\) \(=\) 576.575
Dual form 576.4.c.e.575.3

$q$-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)
 
\(f(q)\) \(=\) \(q-9.89949i q^{5} +30.9839i q^{7} -43.8178 q^{11} +28.0000 q^{13} -49.4975i q^{17} -43.8178 q^{23} +27.0000 q^{25} -137.179i q^{29} -216.887i q^{31} +306.725 q^{35} -266.000 q^{37} -227.688i q^{41} +61.9677i q^{43} +306.725 q^{47} -617.000 q^{49} -120.208i q^{53} +433.774i q^{55} -613.449 q^{59} -350.000 q^{61} -277.186i q^{65} -433.774i q^{67} -657.267 q^{71} -112.000 q^{73} -1357.65i q^{77} +216.887i q^{79} -306.725 q^{83} -490.000 q^{85} -1296.83i q^{89} +867.548i q^{91} -616.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 112 q^{13} + 108 q^{25} - 1064 q^{37} - 2468 q^{49} - 1400 q^{61} - 448 q^{73} - 1960 q^{85} - 2464 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(5\) − 9.89949i − 0.885438i −0.896660 − 0.442719i \(-0.854014\pi\)
0.896660 − 0.442719i \(-0.145986\pi\)
\(6\) 0 0
\(7\) 30.9839i 1.67297i 0.547989 + 0.836486i \(0.315394\pi\)
−0.547989 + 0.836486i \(0.684606\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −43.8178 −1.20105 −0.600526 − 0.799605i \(-0.705042\pi\)
−0.600526 + 0.799605i \(0.705042\pi\)
\(12\) 0 0
\(13\) 28.0000 0.597369 0.298685 − 0.954352i \(-0.403452\pi\)
0.298685 + 0.954352i \(0.403452\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 49.4975i − 0.706171i −0.935591 − 0.353085i \(-0.885133\pi\)
0.935591 − 0.353085i \(-0.114867\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −43.8178 −0.397245 −0.198623 − 0.980076i \(-0.563647\pi\)
−0.198623 + 0.980076i \(0.563647\pi\)
\(24\) 0 0
\(25\) 27.0000 0.216000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 137.179i − 0.878395i −0.898391 − 0.439197i \(-0.855263\pi\)
0.898391 − 0.439197i \(-0.144737\pi\)
\(30\) 0 0
\(31\) − 216.887i − 1.25658i −0.777978 − 0.628291i \(-0.783755\pi\)
0.777978 − 0.628291i \(-0.216245\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 306.725 1.48131
\(36\) 0 0
\(37\) −266.000 −1.18190 −0.590948 − 0.806710i \(-0.701246\pi\)
−0.590948 + 0.806710i \(0.701246\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 227.688i − 0.867291i −0.901083 − 0.433646i \(-0.857227\pi\)
0.901083 − 0.433646i \(-0.142773\pi\)
\(42\) 0 0
\(43\) 61.9677i 0.219767i 0.993944 + 0.109884i \(0.0350478\pi\)
−0.993944 + 0.109884i \(0.964952\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 306.725 0.951923 0.475962 − 0.879466i \(-0.342100\pi\)
0.475962 + 0.879466i \(0.342100\pi\)
\(48\) 0 0
\(49\) −617.000 −1.79883
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 120.208i − 0.311545i −0.987793 − 0.155772i \(-0.950213\pi\)
0.987793 − 0.155772i \(-0.0497866\pi\)
\(54\) 0 0
\(55\) 433.774i 1.06346i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −613.449 −1.35363 −0.676816 − 0.736152i \(-0.736641\pi\)
−0.676816 + 0.736152i \(0.736641\pi\)
\(60\) 0 0
\(61\) −350.000 −0.734638 −0.367319 − 0.930095i \(-0.619724\pi\)
−0.367319 + 0.930095i \(0.619724\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 277.186i − 0.528933i
\(66\) 0 0
\(67\) − 433.774i − 0.790954i −0.918476 − 0.395477i \(-0.870579\pi\)
0.918476 − 0.395477i \(-0.129421\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −657.267 −1.09864 −0.549319 − 0.835613i \(-0.685113\pi\)
−0.549319 + 0.835613i \(0.685113\pi\)
\(72\) 0 0
\(73\) −112.000 −0.179570 −0.0897850 − 0.995961i \(-0.528618\pi\)
−0.0897850 + 0.995961i \(0.528618\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1357.65i − 2.00932i
\(78\) 0 0
\(79\) 216.887i 0.308882i 0.988002 + 0.154441i \(0.0493577\pi\)
−0.988002 + 0.154441i \(0.950642\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −306.725 −0.405631 −0.202816 − 0.979217i \(-0.565009\pi\)
−0.202816 + 0.979217i \(0.565009\pi\)
\(84\) 0 0
\(85\) −490.000 −0.625270
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1296.83i − 1.54454i −0.635294 − 0.772270i \(-0.719121\pi\)
0.635294 − 0.772270i \(-0.280879\pi\)
\(90\) 0 0
\(91\) 867.548i 0.999382i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −616.000 −0.644797 −0.322399 − 0.946604i \(-0.604489\pi\)
−0.322399 + 0.946604i \(0.604489\pi\)
\(98\) 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 576.4.c.e.575.2 4
3.2 odd 2 inner 576.4.c.e.575.4 4
4.3 odd 2 inner 576.4.c.e.575.1 4
8.3 odd 2 36.4.b.b.35.4 yes 4
8.5 even 2 36.4.b.b.35.2 yes 4
12.11 even 2 inner 576.4.c.e.575.3 4
16.3 odd 4 2304.4.f.g.1151.4 8
16.5 even 4 2304.4.f.g.1151.5 8
16.11 odd 4 2304.4.f.g.1151.7 8
16.13 even 4 2304.4.f.g.1151.2 8
24.5 odd 2 36.4.b.b.35.3 yes 4
24.11 even 2 36.4.b.b.35.1 ✓ 4
48.5 odd 4 2304.4.f.g.1151.1 8
48.11 even 4 2304.4.f.g.1151.3 8
48.29 odd 4 2304.4.f.g.1151.6 8
48.35 even 4 2304.4.f.g.1151.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.4.b.b.35.1 ✓ 4 24.11 even 2
36.4.b.b.35.2 yes 4 8.5 even 2
36.4.b.b.35.3 yes 4 24.5 odd 2
36.4.b.b.35.4 yes 4 8.3 odd 2
576.4.c.e.575.1 4 4.3 odd 2 inner
576.4.c.e.575.2 4 1.1 even 1 trivial
576.4.c.e.575.3 4 12.11 even 2 inner
576.4.c.e.575.4 4 3.2 odd 2 inner
2304.4.f.g.1151.1 8 48.5 odd 4
2304.4.f.g.1151.2 8 16.13 even 4
2304.4.f.g.1151.3 8 48.11 even 4
2304.4.f.g.1151.4 8 16.3 odd 4
2304.4.f.g.1151.5 8 16.5 even 4
2304.4.f.g.1151.6 8 48.29 odd 4
2304.4.f.g.1151.7 8 16.11 odd 4
2304.4.f.g.1151.8 8 48.35 even 4