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Properties

Label 576.2.bb.b.337.1
Level $576$
Weight $2$
Character 576.337
Analytic conductor $4.599$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [576,2,Mod(49,576)] 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("576.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.bb (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 337.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 576.337
Dual form 576.2.bb.b.241.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} +(0.500000 + 1.86603i) q^{5} +(3.86603 - 2.23205i) q^{7} -3.00000 q^{9} +(-1.86603 - 0.500000i) q^{11} +(2.23205 - 0.598076i) q^{13} +(3.23205 - 0.866025i) q^{15} +4.00000 q^{17} +(3.00000 + 3.00000i) q^{19} +(-3.86603 - 6.69615i) q^{21} +(-5.59808 - 3.23205i) q^{23} +(1.09808 - 0.633975i) q^{25} +5.19615i q^{27} +(0.232051 - 0.866025i) q^{29} +(4.59808 - 7.96410i) q^{31} +(-0.866025 + 3.23205i) q^{33} +(6.09808 + 6.09808i) q^{35} +(-4.26795 + 4.26795i) q^{37} +(-1.03590 - 3.86603i) q^{39} +(-0.696152 - 0.401924i) q^{41} +(-6.33013 - 1.69615i) q^{43} +(-1.50000 - 5.59808i) q^{45} +(0.598076 + 1.03590i) q^{47} +(6.46410 - 11.1962i) q^{49} -6.92820i q^{51} +(5.73205 - 5.73205i) q^{53} -3.73205i q^{55} +(5.19615 - 5.19615i) q^{57} +(-0.401924 - 1.50000i) q^{59} +(0.571797 - 2.13397i) q^{61} +(-11.5981 + 6.69615i) q^{63} +(2.23205 + 3.86603i) q^{65} +(-8.33013 + 2.23205i) q^{67} +(-5.59808 + 9.69615i) q^{69} +2.92820i q^{71} +7.46410i q^{73} +(-1.09808 - 1.90192i) q^{75} +(-8.33013 + 2.23205i) q^{77} +(0.866025 + 1.50000i) q^{79} +9.00000 q^{81} +(-3.79423 + 14.1603i) q^{83} +(2.00000 + 7.46410i) q^{85} +(-1.50000 - 0.401924i) q^{87} +15.8564i q^{89} +(7.29423 - 7.29423i) q^{91} +(-13.7942 - 7.96410i) q^{93} +(-4.09808 + 7.09808i) q^{95} +(-0.500000 - 0.866025i) q^{97} +(5.59808 + 1.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 12 q^{7} - 12 q^{9} - 4 q^{11} + 2 q^{13} + 6 q^{15} + 16 q^{17} + 12 q^{19} - 12 q^{21} - 12 q^{23} - 6 q^{25} - 6 q^{29} + 8 q^{31} + 14 q^{35} - 24 q^{37} - 18 q^{39} + 18 q^{41} - 8 q^{43}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) − 1.73205i − 1.00000i
\(4\) 0 0
\(5\) 0.500000 + 1.86603i 0.223607 + 0.834512i 0.982958 + 0.183831i \(0.0588499\pi\)
−0.759351 + 0.650681i \(0.774483\pi\)
\(6\) 0 0
\(7\) 3.86603 − 2.23205i 1.46122 − 0.843636i 0.462152 − 0.886801i \(-0.347077\pi\)
0.999068 + 0.0431647i \(0.0137440\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −1.86603 − 0.500000i −0.562628 − 0.150756i −0.0337145 − 0.999432i \(-0.510734\pi\)
−0.528913 + 0.848676i \(0.677400\pi\)
\(12\) 0 0
\(13\) 2.23205 − 0.598076i 0.619060 − 0.165876i 0.0643593 − 0.997927i \(-0.479500\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 3.23205 − 0.866025i 0.834512 − 0.223607i
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 3.00000 + 3.00000i 0.688247 + 0.688247i 0.961844 − 0.273597i \(-0.0882135\pi\)
−0.273597 + 0.961844i \(0.588214\pi\)
\(20\) 0 0
\(21\) −3.86603 − 6.69615i −0.843636 − 1.46122i
\(22\) 0 0
\(23\) −5.59808 − 3.23205i −1.16728 − 0.673929i −0.214242 − 0.976781i \(-0.568728\pi\)
−0.953038 + 0.302851i \(0.902061\pi\)
\(24\) 0 0
\(25\) 1.09808 − 0.633975i 0.219615 − 0.126795i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 0.232051 − 0.866025i 0.0430908 − 0.160817i −0.941028 − 0.338329i \(-0.890138\pi\)
0.984119 + 0.177512i \(0.0568049\pi\)
\(30\) 0 0
\(31\) 4.59808 − 7.96410i 0.825839 − 1.43039i −0.0754376 − 0.997151i \(-0.524035\pi\)
0.901277 − 0.433244i \(-0.142631\pi\)
\(32\) 0 0
\(33\) −0.866025 + 3.23205i −0.150756 + 0.562628i
\(34\) 0 0
\(35\) 6.09808 + 6.09808i 1.03076 + 1.03076i
\(36\) 0 0
\(37\) −4.26795 + 4.26795i −0.701647 + 0.701647i −0.964764 − 0.263117i \(-0.915249\pi\)
0.263117 + 0.964764i \(0.415249\pi\)
\(38\) 0 0
\(39\) −1.03590 − 3.86603i −0.165876 − 0.619060i
\(40\) 0 0
\(41\) −0.696152 − 0.401924i −0.108721 − 0.0627700i 0.444654 − 0.895703i \(-0.353327\pi\)
−0.553374 + 0.832933i \(0.686660\pi\)
\(42\) 0 0
\(43\) −6.33013 − 1.69615i −0.965335 − 0.258661i −0.258478 − 0.966017i \(-0.583221\pi\)
−0.706857 + 0.707356i \(0.749888\pi\)
\(44\) 0 0
\(45\) −1.50000 − 5.59808i −0.223607 − 0.834512i
\(46\) 0 0
\(47\) 0.598076 + 1.03590i 0.0872384 + 0.151101i 0.906343 − 0.422543i \(-0.138862\pi\)
−0.819104 + 0.573644i \(0.805529\pi\)
\(48\) 0 0
\(49\) 6.46410 − 11.1962i 0.923443 − 1.59945i
\(50\) 0 0
\(51\) − 6.92820i − 0.970143i
\(52\) 0 0
\(53\) 5.73205 − 5.73205i 0.787358 − 0.787358i −0.193703 − 0.981060i \(-0.562050\pi\)
0.981060 + 0.193703i \(0.0620497\pi\)
\(54\) 0 0
\(55\) − 3.73205i − 0.503230i
\(56\) 0 0
\(57\) 5.19615 − 5.19615i 0.688247 − 0.688247i
\(58\) 0 0
\(59\) −0.401924 − 1.50000i −0.0523260 − 0.195283i 0.934815 − 0.355135i \(-0.115565\pi\)
−0.987141 + 0.159852i \(0.948898\pi\)
\(60\) 0 0
\(61\) 0.571797 − 2.13397i 0.0732111 − 0.273227i −0.919611 − 0.392831i \(-0.871496\pi\)
0.992822 + 0.119604i \(0.0381624\pi\)
\(62\) 0 0
\(63\) −11.5981 + 6.69615i −1.46122 + 0.843636i
\(64\) 0 0
\(65\) 2.23205 + 3.86603i 0.276852 + 0.479521i
\(66\) 0 0
\(67\) −8.33013 + 2.23205i −1.01769 + 0.272688i −0.728838 − 0.684686i \(-0.759939\pi\)
−0.288849 + 0.957375i \(0.593273\pi\)
\(68\) 0 0
\(69\) −5.59808 + 9.69615i −0.673929 + 1.16728i
\(70\) 0 0
\(71\) 2.92820i 0.347514i 0.984789 + 0.173757i \(0.0555907\pi\)
−0.984789 + 0.173757i \(0.944409\pi\)
\(72\) 0 0
\(73\) 7.46410i 0.873607i 0.899557 + 0.436804i \(0.143889\pi\)
−0.899557 + 0.436804i \(0.856111\pi\)
\(74\) 0 0
\(75\) −1.09808 − 1.90192i −0.126795 − 0.219615i
\(76\) 0 0
\(77\) −8.33013 + 2.23205i −0.949306 + 0.254366i
\(78\) 0 0
\(79\) 0.866025 + 1.50000i 0.0974355 + 0.168763i 0.910622 − 0.413239i \(-0.135603\pi\)
−0.813187 + 0.582003i \(0.802269\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −3.79423 + 14.1603i −0.416471 + 1.55429i 0.365401 + 0.930850i \(0.380932\pi\)
−0.781872 + 0.623440i \(0.785735\pi\)
\(84\) 0 0
\(85\) 2.00000 + 7.46410i 0.216930 + 0.809595i
\(86\) 0 0
\(87\) −1.50000 − 0.401924i −0.160817 − 0.0430908i
\(88\) 0 0
\(89\) 15.8564i 1.68078i 0.541985 + 0.840388i \(0.317673\pi\)
−0.541985 + 0.840388i \(0.682327\pi\)
\(90\) 0 0
\(91\) 7.29423 − 7.29423i 0.764643 − 0.764643i
\(92\) 0 0
\(93\) −13.7942 − 7.96410i −1.43039 − 0.825839i
\(94\) 0 0
\(95\) −4.09808 + 7.09808i −0.420454 + 0.728247i
\(96\) 0 0
\(97\) −0.500000 − 0.866025i −0.0507673 − 0.0879316i 0.839525 − 0.543321i \(-0.182833\pi\)
−0.890292 + 0.455389i \(0.849500\pi\)
\(98\) 0 0
\(99\) 5.59808 + 1.50000i 0.562628 + 0.150756i
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 576.2.bb.b.337.1 4
3.2 odd 2 1728.2.bc.b.145.1 4
4.3 odd 2 144.2.x.d.13.1 yes 4
9.2 odd 6 1728.2.bc.c.721.1 4
9.7 even 3 576.2.bb.a.529.1 4
12.11 even 2 432.2.y.a.253.1 4
16.5 even 4 576.2.bb.a.49.1 4
16.11 odd 4 144.2.x.a.85.1 yes 4
36.7 odd 6 144.2.x.a.61.1 ✓ 4
36.11 even 6 432.2.y.d.397.1 4
48.5 odd 4 1728.2.bc.c.1009.1 4
48.11 even 4 432.2.y.d.37.1 4
144.11 even 12 432.2.y.a.181.1 4
144.43 odd 12 144.2.x.d.133.1 yes 4
144.101 odd 12 1728.2.bc.b.1585.1 4
144.133 even 12 inner 576.2.bb.b.241.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.x.a.61.1 ✓ 4 36.7 odd 6
144.2.x.a.85.1 yes 4 16.11 odd 4
144.2.x.d.13.1 yes 4 4.3 odd 2
144.2.x.d.133.1 yes 4 144.43 odd 12
432.2.y.a.181.1 4 144.11 even 12
432.2.y.a.253.1 4 12.11 even 2
432.2.y.d.37.1 4 48.11 even 4
432.2.y.d.397.1 4 36.11 even 6
576.2.bb.a.49.1 4 16.5 even 4
576.2.bb.a.529.1 4 9.7 even 3
576.2.bb.b.241.1 4 144.133 even 12 inner
576.2.bb.b.337.1 4 1.1 even 1 trivial
1728.2.bc.b.145.1 4 3.2 odd 2
1728.2.bc.b.1585.1 4 144.101 odd 12
1728.2.bc.c.721.1 4 9.2 odd 6
1728.2.bc.c.1009.1 4 48.5 odd 4