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 = [4,0,0,0,0,0,0,0,0,0,83840] 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: \(0\)
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: \( x^{4} - 2x^{3} - 45829x^{2} + 45830x + 94023351 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-208.492\) 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)
 
\(f(q)\) \(=\) \(q-918.506 q^{5} -6357.58 q^{7} -30811.0 q^{11} -61628.0 q^{13} -205280. q^{17} -155714. q^{19} -429306. q^{23} -1.10947e6 q^{25} -5.02406e6 q^{29} -1.80416e6 q^{31} +5.83948e6 q^{35} +7.46208e6 q^{37} -1.52344e7 q^{41} -3.56357e7 q^{43} -1.70124e7 q^{47} +65252.9 q^{49} +5.94555e7 q^{53} +2.83001e7 q^{55} +614954. q^{59} -6.24183e7 q^{61} +5.66056e7 q^{65} -1.38482e8 q^{67} +1.37593e8 q^{71} +4.54470e7 q^{73} +1.95883e8 q^{77} +3.81318e8 q^{79} -2.48669e7 q^{83} +1.88551e8 q^{85} -7.76045e8 q^{89} +3.91805e8 q^{91} +1.43024e8 q^{95} -4.55595e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 83840 q^{11} + 167656 q^{13} + 1181952 q^{23} + 3017132 q^{25} - 15366784 q^{35} - 20265992 q^{37} + 52473088 q^{47} - 567324 q^{49} + 38492416 q^{59} + 172363992 q^{61} - 196785152 q^{71} - 135464424 q^{73}+ \cdots - 4974152 q^{97}+O(q^{100}) \) 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
\(5\) −918.506 −0.657229 −0.328615 − 0.944464i \(-0.606582\pi\)
−0.328615 + 0.944464i \(0.606582\pi\)
\(6\) 0 0
\(7\) −6357.58 −1.00081 −0.500404 − 0.865792i \(-0.666815\pi\)
−0.500404 + 0.865792i \(0.666815\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −30811.0 −0.634510 −0.317255 − 0.948340i \(-0.602761\pi\)
−0.317255 + 0.948340i \(0.602761\pi\)
\(12\) 0 0
\(13\) −61628.0 −0.598457 −0.299228 − 0.954182i \(-0.596729\pi\)
−0.299228 + 0.954182i \(0.596729\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −205280. −0.596109 −0.298055 − 0.954549i \(-0.596338\pi\)
−0.298055 + 0.954549i \(0.596338\pi\)
\(18\) 0 0
\(19\) −155714. −0.274118 −0.137059 − 0.990563i \(-0.543765\pi\)
−0.137059 + 0.990563i \(0.543765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −429306. −0.319883 −0.159942 − 0.987126i \(-0.551131\pi\)
−0.159942 + 0.987126i \(0.551131\pi\)
\(24\) 0 0
\(25\) −1.10947e6 −0.568050
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.02406e6 −1.31906 −0.659529 − 0.751679i \(-0.729244\pi\)
−0.659529 + 0.751679i \(0.729244\pi\)
\(30\) 0 0
\(31\) −1.80416e6 −0.350870 −0.175435 − 0.984491i \(-0.556133\pi\)
−0.175435 + 0.984491i \(0.556133\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.83948e6 0.657760
\(36\) 0 0
\(37\) 7.46208e6 0.654564 0.327282 − 0.944927i \(-0.393867\pi\)
0.327282 + 0.944927i \(0.393867\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.52344e7 −0.841974 −0.420987 − 0.907067i \(-0.638316\pi\)
−0.420987 + 0.907067i \(0.638316\pi\)
\(42\) 0 0
\(43\) −3.56357e7 −1.58956 −0.794781 − 0.606896i \(-0.792415\pi\)
−0.794781 + 0.606896i \(0.792415\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.70124e7 −0.508541 −0.254271 − 0.967133i \(-0.581835\pi\)
−0.254271 + 0.967133i \(0.581835\pi\)
\(48\) 0 0
\(49\) 65252.9 0.00161703
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.94555e7 1.03502 0.517512 − 0.855676i \(-0.326858\pi\)
0.517512 + 0.855676i \(0.326858\pi\)
\(54\) 0 0
\(55\) 2.83001e7 0.417018
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 614954. 0.00660707 0.00330353 − 0.999995i \(-0.498948\pi\)
0.00330353 + 0.999995i \(0.498948\pi\)
\(60\) 0 0
\(61\) −6.24183e7 −0.577201 −0.288601 − 0.957450i \(-0.593190\pi\)
−0.288601 + 0.957450i \(0.593190\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.66056e7 0.393323
\(66\) 0 0
\(67\) −1.38482e8 −0.839571 −0.419786 − 0.907623i \(-0.637895\pi\)
−0.419786 + 0.907623i \(0.637895\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.37593e8 0.642592 0.321296 − 0.946979i \(-0.395882\pi\)
0.321296 + 0.946979i \(0.395882\pi\)
\(72\) 0 0
\(73\) 4.54470e7 0.187306 0.0936532 − 0.995605i \(-0.470146\pi\)
0.0936532 + 0.995605i \(0.470146\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.95883e8 0.635023
\(78\) 0 0
\(79\) 3.81318e8 1.10145 0.550726 − 0.834686i \(-0.314351\pi\)
0.550726 + 0.834686i \(0.314351\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.48669e7 −0.0575135 −0.0287567 − 0.999586i \(-0.509155\pi\)
−0.0287567 + 0.999586i \(0.509155\pi\)
\(84\) 0 0
\(85\) 1.88551e8 0.391780
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.76045e8 −1.31109 −0.655544 − 0.755157i \(-0.727560\pi\)
−0.655544 + 0.755157i \(0.727560\pi\)
\(90\) 0 0
\(91\) 3.91805e8 0.598940
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.43024e8 0.180158
\(96\) 0 0
\(97\) −4.55595e7 −0.0522524 −0.0261262 − 0.999659i \(-0.508317\pi\)
−0.0261262 + 0.999659i \(0.508317\pi\)
\(98\) 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.r.1.2 yes 4
3.2 odd 2 288.10.a.q.1.3 yes 4
4.3 odd 2 288.10.a.q.1.2 ✓ 4
12.11 even 2 inner 288.10.a.r.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.10.a.q.1.2 ✓ 4 4.3 odd 2
288.10.a.q.1.3 yes 4 3.2 odd 2
288.10.a.r.1.2 yes 4 1.1 even 1 trivial
288.10.a.r.1.3 yes 4 12.11 even 2 inner