Properties

Label 288.10.a.p.1.1
Level $288$
Weight $10$
Character 288.1
Self dual yes
Analytic conductor $148.330$
Analytic rank $1$
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] = [288,10,Mod(1,288)] 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("288.1"); S:= CuspForms(chi, 10); 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, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 288.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4288,0,0,0,0,0,0] 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: yes
Analytic conductor: \(148.330320815\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 555x^{2} + 14000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{5}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.14680\) of defining polynomial
Character \(\chi\) \(=\) 288.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2276.85 q^{5} -11327.3 q^{7} +63096.7 q^{11} -64234.0 q^{13} +325090. q^{17} -715036. q^{19} +1.76406e6 q^{23} +3.23092e6 q^{25} -2.50198e6 q^{29} -1.43574e6 q^{31} +2.57906e7 q^{35} +1.87493e7 q^{37} +6.36737e6 q^{41} +4.30368e7 q^{43} -2.70080e7 q^{47} +8.79549e7 q^{49} -2.33610e7 q^{53} -1.43662e8 q^{55} -1.37018e8 q^{59} +9.86600e7 q^{61} +1.46251e8 q^{65} +1.40808e8 q^{67} -1.11322e8 q^{71} -3.87755e8 q^{73} -7.14717e8 q^{77} -1.55264e8 q^{79} -7.34374e7 q^{83} -7.40180e8 q^{85} +1.10191e9 q^{89} +7.27600e8 q^{91} +1.62803e9 q^{95} -2.94108e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4288 q^{5} + 128616 q^{13} - 386432 q^{17} + 2590892 q^{25} + 1245376 q^{29} + 53020792 q^{37} + 16360832 q^{41} + 141308132 q^{49} - 143107904 q^{53} + 71932760 q^{61} + 326656128 q^{65} - 254795496 q^{73}+ \cdots - 2467261512 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2276.85 −1.62918 −0.814591 − 0.580036i \(-0.803038\pi\)
−0.814591 + 0.580036i \(0.803038\pi\)
\(6\) 0 0
\(7\) −11327.3 −1.78314 −0.891572 − 0.452879i \(-0.850397\pi\)
−0.891572 + 0.452879i \(0.850397\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 63096.7 1.29939 0.649695 − 0.760195i \(-0.274897\pi\)
0.649695 + 0.760195i \(0.274897\pi\)
\(12\) 0 0
\(13\) −64234.0 −0.623764 −0.311882 − 0.950121i \(-0.600959\pi\)
−0.311882 + 0.950121i \(0.600959\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 325090. 0.944024 0.472012 − 0.881592i \(-0.343528\pi\)
0.472012 + 0.881592i \(0.343528\pi\)
\(18\) 0 0
\(19\) −715036. −1.25874 −0.629371 − 0.777105i \(-0.716687\pi\)
−0.629371 + 0.777105i \(0.716687\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.76406e6 1.31443 0.657215 − 0.753703i \(-0.271734\pi\)
0.657215 + 0.753703i \(0.271734\pi\)
\(24\) 0 0
\(25\) 3.23092e6 1.65423
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.50198e6 −0.656891 −0.328445 − 0.944523i \(-0.606525\pi\)
−0.328445 + 0.944523i \(0.606525\pi\)
\(30\) 0 0
\(31\) −1.43574e6 −0.279222 −0.139611 − 0.990206i \(-0.544585\pi\)
−0.139611 + 0.990206i \(0.544585\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.57906e7 2.90507
\(36\) 0 0
\(37\) 1.87493e7 1.64467 0.822333 − 0.569007i \(-0.192672\pi\)
0.822333 + 0.569007i \(0.192672\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.36737e6 0.351911 0.175956 − 0.984398i \(-0.443698\pi\)
0.175956 + 0.984398i \(0.443698\pi\)
\(42\) 0 0
\(43\) 4.30368e7 1.91969 0.959846 − 0.280528i \(-0.0905096\pi\)
0.959846 + 0.280528i \(0.0905096\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.70080e7 −0.807333 −0.403666 − 0.914906i \(-0.632264\pi\)
−0.403666 + 0.914906i \(0.632264\pi\)
\(48\) 0 0
\(49\) 8.79549e7 2.17960
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.33610e7 −0.406677 −0.203339 − 0.979108i \(-0.565179\pi\)
−0.203339 + 0.979108i \(0.565179\pi\)
\(54\) 0 0
\(55\) −1.43662e8 −2.11694
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.37018e8 −1.47213 −0.736063 − 0.676913i \(-0.763317\pi\)
−0.736063 + 0.676913i \(0.763317\pi\)
\(60\) 0 0
\(61\) 9.86600e7 0.912340 0.456170 − 0.889893i \(-0.349221\pi\)
0.456170 + 0.889893i \(0.349221\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.46251e8 1.01622
\(66\) 0 0
\(67\) 1.40808e8 0.853669 0.426834 − 0.904330i \(-0.359629\pi\)
0.426834 + 0.904330i \(0.359629\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.11322e8 −0.519897 −0.259949 − 0.965622i \(-0.583706\pi\)
−0.259949 + 0.965622i \(0.583706\pi\)
\(72\) 0 0
\(73\) −3.87755e8 −1.59810 −0.799052 − 0.601262i \(-0.794665\pi\)
−0.799052 + 0.601262i \(0.794665\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.14717e8 −2.31700
\(78\) 0 0
\(79\) −1.55264e8 −0.448487 −0.224243 − 0.974533i \(-0.571991\pi\)
−0.224243 + 0.974533i \(0.571991\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.34374e7 −0.169850 −0.0849250 − 0.996387i \(-0.527065\pi\)
−0.0849250 + 0.996387i \(0.527065\pi\)
\(84\) 0 0
\(85\) −7.40180e8 −1.53799
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.10191e9 1.86162 0.930808 − 0.365509i \(-0.119105\pi\)
0.930808 + 0.365509i \(0.119105\pi\)
\(90\) 0 0
\(91\) 7.27600e8 1.11226
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.62803e9 2.05072
\(96\) 0 0
\(97\) −2.94108e8 −0.337314 −0.168657 − 0.985675i \(-0.553943\pi\)
−0.168657 + 0.985675i \(0.553943\pi\)
\(98\) 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 288.10.a.p.1.1 ✓ 4
3.2 odd 2 288.10.a.s.1.3 yes 4
4.3 odd 2 inner 288.10.a.p.1.2 yes 4
12.11 even 2 288.10.a.s.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.10.a.p.1.1 ✓ 4 1.1 even 1 trivial
288.10.a.p.1.2 yes 4 4.3 odd 2 inner
288.10.a.s.1.3 yes 4 3.2 odd 2
288.10.a.s.1.4 yes 4 12.11 even 2