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Properties

Label 576.6.k.a.433.9
Level $576$
Weight $6$
Character 576.433
Analytic conductor $92.381$
Analytic rank $0$
Dimension $18$
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] = [576,6,Mod(145,576)] 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("576.145"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 576.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0,0,2,0,0,0,0,0,-606] 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: \(92.3810802123\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 8 x^{17} - 3867 x^{16} + 20528 x^{15} + 5993890 x^{14} - 12125584 x^{13} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{72}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 433.9
Root \(-33.1823 + 1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 576.433
Dual form 576.6.k.a.145.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(67.3647 + 67.3647i) q^{5} +148.379i q^{7} +(-256.205 - 256.205i) q^{11} +(-218.586 + 218.586i) q^{13} +463.168 q^{17} +(-920.791 + 920.791i) q^{19} -1053.65i q^{23} +5951.00i q^{25} +(-1290.79 + 1290.79i) q^{29} -10036.9 q^{31} +(-9995.50 + 9995.50i) q^{35} +(-9409.85 - 9409.85i) q^{37} +368.682i q^{41} +(9168.57 + 9168.57i) q^{43} -7638.65 q^{47} -5209.29 q^{49} +(1242.05 + 1242.05i) q^{53} -34518.4i q^{55} +(16255.3 + 16255.3i) q^{59} +(-1394.11 + 1394.11i) q^{61} -29449.9 q^{65} +(17400.9 - 17400.9i) q^{67} -67414.6i q^{71} +19543.2i q^{73} +(38015.5 - 38015.5i) q^{77} +43996.5 q^{79} +(73286.1 - 73286.1i) q^{83} +(31201.2 + 31201.2i) q^{85} +88671.1i q^{89} +(-32433.5 - 32433.5i) q^{91} -124058. q^{95} -73610.3 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{5} - 606 q^{11} - 2 q^{13} + 4 q^{17} + 2362 q^{19} - 4070 q^{29} + 11536 q^{31} + 8636 q^{35} - 10650 q^{37} + 15382 q^{43} + 44176 q^{47} - 14410 q^{49} - 24726 q^{53} - 29734 q^{59} - 48082 q^{61}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0 0
\(4\) 0 0
\(5\) 67.3647 + 67.3647i 1.20506 + 1.20506i 0.972609 + 0.232447i \(0.0746733\pi\)
0.232447 + 0.972609i \(0.425327\pi\)
\(6\) 0 0
\(7\) 148.379i 1.14453i 0.820069 + 0.572265i \(0.193935\pi\)
−0.820069 + 0.572265i \(0.806065\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −256.205 − 256.205i −0.638420 − 0.638420i 0.311745 − 0.950166i \(-0.399086\pi\)
−0.950166 + 0.311745i \(0.899086\pi\)
\(12\) 0 0
\(13\) −218.586 + 218.586i −0.358727 + 0.358727i −0.863343 − 0.504617i \(-0.831634\pi\)
0.504617 + 0.863343i \(0.331634\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 463.168 0.388701 0.194351 − 0.980932i \(-0.437740\pi\)
0.194351 + 0.980932i \(0.437740\pi\)
\(18\) 0 0
\(19\) −920.791 + 920.791i −0.585163 + 0.585163i −0.936318 − 0.351154i \(-0.885789\pi\)
0.351154 + 0.936318i \(0.385789\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 1053.65i − 0.415315i −0.978202 − 0.207657i \(-0.933416\pi\)
0.978202 − 0.207657i \(-0.0665839\pi\)
\(24\) 0 0
\(25\) 5951.00i 1.90432i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1290.79 + 1290.79i −0.285010 + 0.285010i −0.835103 − 0.550093i \(-0.814592\pi\)
0.550093 + 0.835103i \(0.314592\pi\)
\(30\) 0 0
\(31\) −10036.9 −1.87584 −0.937921 − 0.346848i \(-0.887252\pi\)
−0.937921 + 0.346848i \(0.887252\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9995.50 + 9995.50i −1.37922 + 1.37922i
\(36\) 0 0
\(37\) −9409.85 − 9409.85i −1.13000 − 1.13000i −0.990176 − 0.139824i \(-0.955346\pi\)
−0.139824 − 0.990176i \(-0.544654\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 368.682i 0.0342525i 0.999853 + 0.0171263i \(0.00545173\pi\)
−0.999853 + 0.0171263i \(0.994548\pi\)
\(42\) 0 0
\(43\) 9168.57 + 9168.57i 0.756189 + 0.756189i 0.975627 − 0.219438i \(-0.0704222\pi\)
−0.219438 + 0.975627i \(0.570422\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7638.65 −0.504396 −0.252198 − 0.967676i \(-0.581153\pi\)
−0.252198 + 0.967676i \(0.581153\pi\)
\(48\) 0 0
\(49\) −5209.29 −0.309948
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1242.05 + 1242.05i 0.0607363 + 0.0607363i 0.736823 − 0.676086i \(-0.236325\pi\)
−0.676086 + 0.736823i \(0.736325\pi\)
\(54\) 0 0
\(55\) − 34518.4i − 1.53866i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 16255.3 + 16255.3i 0.607945 + 0.607945i 0.942409 − 0.334464i \(-0.108555\pi\)
−0.334464 + 0.942409i \(0.608555\pi\)
\(60\) 0 0
\(61\) −1394.11 + 1394.11i −0.0479704 + 0.0479704i −0.730685 − 0.682715i \(-0.760799\pi\)
0.682715 + 0.730685i \(0.260799\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −29449.9 −0.864572
\(66\) 0 0
\(67\) 17400.9 − 17400.9i 0.473569 − 0.473569i −0.429498 − 0.903068i \(-0.641310\pi\)
0.903068 + 0.429498i \(0.141310\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 67414.6i − 1.58711i −0.608495 − 0.793557i \(-0.708227\pi\)
0.608495 − 0.793557i \(-0.291773\pi\)
\(72\) 0 0
\(73\) 19543.2i 0.429228i 0.976699 + 0.214614i \(0.0688494\pi\)
−0.976699 + 0.214614i \(0.931151\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 38015.5 − 38015.5i 0.730691 − 0.730691i
\(78\) 0 0
\(79\) 43996.5 0.793140 0.396570 − 0.918004i \(-0.370200\pi\)
0.396570 + 0.918004i \(0.370200\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 73286.1 − 73286.1i 1.16769 − 1.16769i 0.184937 − 0.982750i \(-0.440792\pi\)
0.982750 − 0.184937i \(-0.0592080\pi\)
\(84\) 0 0
\(85\) 31201.2 + 31201.2i 0.468407 + 0.468407i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 88671.1i 1.18661i 0.804979 + 0.593304i \(0.202177\pi\)
−0.804979 + 0.593304i \(0.797823\pi\)
\(90\) 0 0
\(91\) −32433.5 − 32433.5i −0.410573 − 0.410573i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −124058. −1.41031
\(96\) 0 0
\(97\) −73610.3 −0.794345 −0.397173 − 0.917744i \(-0.630009\pi\)
−0.397173 + 0.917744i \(0.630009\pi\)
\(98\) 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 576.6.k.a.433.9 18
3.2 odd 2 64.6.e.a.49.5 18
4.3 odd 2 144.6.k.a.37.9 18
12.11 even 2 16.6.e.a.5.1 ✓ 18
16.3 odd 4 144.6.k.a.109.9 18
16.13 even 4 inner 576.6.k.a.145.9 18
24.5 odd 2 128.6.e.a.97.5 18
24.11 even 2 128.6.e.b.97.5 18
48.5 odd 4 128.6.e.a.33.5 18
48.11 even 4 128.6.e.b.33.5 18
48.29 odd 4 64.6.e.a.17.5 18
48.35 even 4 16.6.e.a.13.1 yes 18
96.29 odd 8 1024.6.a.l.1.10 18
96.35 even 8 1024.6.a.k.1.9 18
96.77 odd 8 1024.6.a.l.1.9 18
96.83 even 8 1024.6.a.k.1.10 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.6.e.a.5.1 ✓ 18 12.11 even 2
16.6.e.a.13.1 yes 18 48.35 even 4
64.6.e.a.17.5 18 48.29 odd 4
64.6.e.a.49.5 18 3.2 odd 2
128.6.e.a.33.5 18 48.5 odd 4
128.6.e.a.97.5 18 24.5 odd 2
128.6.e.b.33.5 18 48.11 even 4
128.6.e.b.97.5 18 24.11 even 2
144.6.k.a.37.9 18 4.3 odd 2
144.6.k.a.109.9 18 16.3 odd 4
576.6.k.a.145.9 18 16.13 even 4 inner
576.6.k.a.433.9 18 1.1 even 1 trivial
1024.6.a.k.1.9 18 96.35 even 8
1024.6.a.k.1.10 18 96.83 even 8
1024.6.a.l.1.9 18 96.77 odd 8
1024.6.a.l.1.10 18 96.29 odd 8