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Properties

Label 2888.2.a.y.1.4
Level $2888$
Weight $2$
Character 2888.1
Self dual yes
Analytic conductor $23.061$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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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] = [2888,2,Mod(1,2888)] 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("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2888.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,3,0,3,0,9,0,6,0,3,0,-6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.0607961037\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 12x^{7} + 35x^{6} + 45x^{5} - 117x^{4} - 55x^{3} + 96x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.0745540\) of defining polynomial
Character \(\chi\) \(=\) 2888.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.0745540 q^{3} -0.900387 q^{5} -4.87539 q^{7} -2.99444 q^{9} -1.50202 q^{11} +4.14084 q^{13} +0.0671274 q^{15} -3.76137 q^{17} +0.363480 q^{21} +5.75304 q^{23} -4.18930 q^{25} +0.446910 q^{27} -7.40665 q^{29} +3.76276 q^{31} +0.111982 q^{33} +4.38974 q^{35} -3.40626 q^{37} -0.308716 q^{39} -2.19857 q^{41} +1.90770 q^{43} +2.69616 q^{45} -1.53171 q^{47} +16.7695 q^{49} +0.280425 q^{51} -9.93607 q^{53} +1.35240 q^{55} +7.80483 q^{59} -3.39158 q^{61} +14.5991 q^{63} -3.72836 q^{65} -2.56669 q^{67} -0.428912 q^{69} +7.24656 q^{71} +8.24601 q^{73} +0.312329 q^{75} +7.32295 q^{77} +16.0443 q^{79} +8.95001 q^{81} +8.36430 q^{83} +3.38669 q^{85} +0.552195 q^{87} -12.3930 q^{89} -20.1882 q^{91} -0.280528 q^{93} +11.0167 q^{97} +4.49772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9} + 3 q^{11} - 6 q^{13} - 3 q^{17} + 15 q^{21} + 24 q^{23} + 30 q^{25} + 12 q^{27} - 15 q^{29} + 6 q^{31} - 18 q^{33} + 15 q^{35} + 24 q^{37} + 6 q^{39} - 12 q^{41}+ \cdots + 48 q^{99}+O(q^{100}) \) 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
\(3\) −0.0745540 −0.0430438 −0.0215219 − 0.999768i \(-0.506851\pi\)
−0.0215219 + 0.999768i \(0.506851\pi\)
\(4\) 0 0
\(5\) −0.900387 −0.402665 −0.201333 − 0.979523i \(-0.564527\pi\)
−0.201333 + 0.979523i \(0.564527\pi\)
\(6\) 0 0
\(7\) −4.87539 −1.84273 −0.921363 − 0.388704i \(-0.872923\pi\)
−0.921363 + 0.388704i \(0.872923\pi\)
\(8\) 0 0
\(9\) −2.99444 −0.998147
\(10\) 0 0
\(11\) −1.50202 −0.452877 −0.226438 − 0.974025i \(-0.572708\pi\)
−0.226438 + 0.974025i \(0.572708\pi\)
\(12\) 0 0
\(13\) 4.14084 1.14846 0.574231 − 0.818693i \(-0.305301\pi\)
0.574231 + 0.818693i \(0.305301\pi\)
\(14\) 0 0
\(15\) 0.0671274 0.0173322
\(16\) 0 0
\(17\) −3.76137 −0.912266 −0.456133 − 0.889912i \(-0.650766\pi\)
−0.456133 + 0.889912i \(0.650766\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 0.363480 0.0793178
\(22\) 0 0
\(23\) 5.75304 1.19959 0.599796 − 0.800153i \(-0.295248\pi\)
0.599796 + 0.800153i \(0.295248\pi\)
\(24\) 0 0
\(25\) −4.18930 −0.837861
\(26\) 0 0
\(27\) 0.446910 0.0860078
\(28\) 0 0
\(29\) −7.40665 −1.37538 −0.687690 − 0.726004i \(-0.741375\pi\)
−0.687690 + 0.726004i \(0.741375\pi\)
\(30\) 0 0
\(31\) 3.76276 0.675811 0.337905 − 0.941180i \(-0.390282\pi\)
0.337905 + 0.941180i \(0.390282\pi\)
\(32\) 0 0
\(33\) 0.111982 0.0194935
\(34\) 0 0
\(35\) 4.38974 0.742001
\(36\) 0 0
\(37\) −3.40626 −0.559986 −0.279993 − 0.960002i \(-0.590332\pi\)
−0.279993 + 0.960002i \(0.590332\pi\)
\(38\) 0 0
\(39\) −0.308716 −0.0494342
\(40\) 0 0
\(41\) −2.19857 −0.343359 −0.171679 − 0.985153i \(-0.554919\pi\)
−0.171679 + 0.985153i \(0.554919\pi\)
\(42\) 0 0
\(43\) 1.90770 0.290921 0.145461 − 0.989364i \(-0.453534\pi\)
0.145461 + 0.989364i \(0.453534\pi\)
\(44\) 0 0
\(45\) 2.69616 0.401919
\(46\) 0 0
\(47\) −1.53171 −0.223423 −0.111711 − 0.993741i \(-0.535633\pi\)
−0.111711 + 0.993741i \(0.535633\pi\)
\(48\) 0 0
\(49\) 16.7695 2.39564
\(50\) 0 0
\(51\) 0.280425 0.0392674
\(52\) 0 0
\(53\) −9.93607 −1.36482 −0.682412 − 0.730968i \(-0.739069\pi\)
−0.682412 + 0.730968i \(0.739069\pi\)
\(54\) 0 0
\(55\) 1.35240 0.182358
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.80483 1.01610 0.508051 − 0.861327i \(-0.330366\pi\)
0.508051 + 0.861327i \(0.330366\pi\)
\(60\) 0 0
\(61\) −3.39158 −0.434247 −0.217124 − 0.976144i \(-0.569667\pi\)
−0.217124 + 0.976144i \(0.569667\pi\)
\(62\) 0 0
\(63\) 14.5991 1.83931
\(64\) 0 0
\(65\) −3.72836 −0.462446
\(66\) 0 0
\(67\) −2.56669 −0.313571 −0.156785 − 0.987633i \(-0.550113\pi\)
−0.156785 + 0.987633i \(0.550113\pi\)
\(68\) 0 0
\(69\) −0.428912 −0.0516350
\(70\) 0 0
\(71\) 7.24656 0.860008 0.430004 − 0.902827i \(-0.358512\pi\)
0.430004 + 0.902827i \(0.358512\pi\)
\(72\) 0 0
\(73\) 8.24601 0.965122 0.482561 − 0.875862i \(-0.339707\pi\)
0.482561 + 0.875862i \(0.339707\pi\)
\(74\) 0 0
\(75\) 0.312329 0.0360647
\(76\) 0 0
\(77\) 7.32295 0.834527
\(78\) 0 0
\(79\) 16.0443 1.80513 0.902563 − 0.430557i \(-0.141683\pi\)
0.902563 + 0.430557i \(0.141683\pi\)
\(80\) 0 0
\(81\) 8.95001 0.994445
\(82\) 0 0
\(83\) 8.36430 0.918101 0.459051 − 0.888410i \(-0.348190\pi\)
0.459051 + 0.888410i \(0.348190\pi\)
\(84\) 0 0
\(85\) 3.38669 0.367338
\(86\) 0 0
\(87\) 0.552195 0.0592015
\(88\) 0 0
\(89\) −12.3930 −1.31366 −0.656829 − 0.754039i \(-0.728103\pi\)
−0.656829 + 0.754039i \(0.728103\pi\)
\(90\) 0 0
\(91\) −20.1882 −2.11630
\(92\) 0 0
\(93\) −0.280528 −0.0290894
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0167 1.11858 0.559289 − 0.828973i \(-0.311074\pi\)
0.559289 + 0.828973i \(0.311074\pi\)
\(98\) 0 0
\(99\) 4.49772 0.452038
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 2888.2.a.y.1.4 9
4.3 odd 2 5776.2.a.cd.1.6 9
19.6 even 9 152.2.q.c.17.2 yes 18
19.16 even 9 152.2.q.c.9.2 ✓ 18
19.18 odd 2 2888.2.a.x.1.6 9
76.35 odd 18 304.2.u.f.161.2 18
76.63 odd 18 304.2.u.f.17.2 18
76.75 even 2 5776.2.a.ce.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.q.c.9.2 ✓ 18 19.16 even 9
152.2.q.c.17.2 yes 18 19.6 even 9
304.2.u.f.17.2 18 76.63 odd 18
304.2.u.f.161.2 18 76.35 odd 18
2888.2.a.x.1.6 9 19.18 odd 2
2888.2.a.y.1.4 9 1.1 even 1 trivial
5776.2.a.cd.1.6 9 4.3 odd 2
5776.2.a.ce.1.4 9 76.75 even 2