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: \(1\)
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: \( x^{4} - 2x^{3} - 45829x^{2} + 45830x + 94023351 \) 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 1.4
Root \(46.8997\) 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)
 
\(f(q)\) \(=\) \(q+2138.03 q^{5} +6324.93 q^{7} -72731.0 q^{11} +145456. q^{13} -198122. q^{17} -235718. q^{19} -1.02028e6 q^{23} +2.61804e6 q^{25} -5.88023e6 q^{29} -9.92408e6 q^{31} +1.35229e7 q^{35} -1.75951e7 q^{37} -4.41988e6 q^{41} -3.21267e7 q^{43} -4.32490e7 q^{47} -348915. q^{49} -7.26048e6 q^{53} -1.55501e8 q^{55} -1.86313e7 q^{59} +1.48600e8 q^{61} +3.10989e8 q^{65} +2.02616e8 q^{67} +2.35986e8 q^{71} -1.13179e8 q^{73} -4.60018e8 q^{77} +2.89403e8 q^{79} -3.59791e8 q^{83} -4.23589e8 q^{85} +4.34227e8 q^{89} +9.19998e8 q^{91} -5.03972e8 q^{95} +4.30724e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 83840 q^{11} + 167656 q^{13} - 1181952 q^{23} + 3017132 q^{25} + 15366784 q^{35} - 20265992 q^{37} - 52473088 q^{47} - 567324 q^{49} - 38492416 q^{59} + 172363992 q^{61} + 196785152 q^{71} - 135464424 q^{73}+ \cdots - 4974152 q^{97}+O(q^{100}) \) 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
\(5\) 2138.03 1.52985 0.764924 − 0.644120i \(-0.222776\pi\)
0.764924 + 0.644120i \(0.222776\pi\)
\(6\) 0 0
\(7\) 6324.93 0.995667 0.497834 − 0.867273i \(-0.334129\pi\)
0.497834 + 0.867273i \(0.334129\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −72731.0 −1.49779 −0.748897 − 0.662686i \(-0.769416\pi\)
−0.748897 + 0.662686i \(0.769416\pi\)
\(12\) 0 0
\(13\) 145456. 1.41249 0.706247 − 0.707966i \(-0.250387\pi\)
0.706247 + 0.707966i \(0.250387\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −198122. −0.575323 −0.287661 − 0.957732i \(-0.592878\pi\)
−0.287661 + 0.957732i \(0.592878\pi\)
\(18\) 0 0
\(19\) −235718. −0.414956 −0.207478 − 0.978240i \(-0.566525\pi\)
−0.207478 + 0.978240i \(0.566525\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.02028e6 −0.760230 −0.380115 − 0.924939i \(-0.624116\pi\)
−0.380115 + 0.924939i \(0.624116\pi\)
\(24\) 0 0
\(25\) 2.61804e6 1.34044
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.88023e6 −1.54384 −0.771922 − 0.635718i \(-0.780704\pi\)
−0.771922 + 0.635718i \(0.780704\pi\)
\(30\) 0 0
\(31\) −9.92408e6 −1.93002 −0.965012 − 0.262206i \(-0.915550\pi\)
−0.965012 + 0.262206i \(0.915550\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.35229e7 1.52322
\(36\) 0 0
\(37\) −1.75951e7 −1.54342 −0.771709 − 0.635976i \(-0.780598\pi\)
−0.771709 + 0.635976i \(0.780598\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.41988e6 −0.244277 −0.122139 − 0.992513i \(-0.538975\pi\)
−0.122139 + 0.992513i \(0.538975\pi\)
\(42\) 0 0
\(43\) −3.21267e7 −1.43304 −0.716520 − 0.697567i \(-0.754266\pi\)
−0.716520 + 0.697567i \(0.754266\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.32490e7 −1.29281 −0.646406 − 0.762993i \(-0.723729\pi\)
−0.646406 + 0.762993i \(0.723729\pi\)
\(48\) 0 0
\(49\) −348915. −0.00864644
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.26048e6 −0.126393 −0.0631966 − 0.998001i \(-0.520130\pi\)
−0.0631966 + 0.998001i \(0.520130\pi\)
\(54\) 0 0
\(55\) −1.55501e8 −2.29140
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.86313e7 −0.200174 −0.100087 − 0.994979i \(-0.531912\pi\)
−0.100087 + 0.994979i \(0.531912\pi\)
\(60\) 0 0
\(61\) 1.48600e8 1.37415 0.687077 − 0.726585i \(-0.258894\pi\)
0.687077 + 0.726585i \(0.258894\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.10989e8 2.16090
\(66\) 0 0
\(67\) 2.02616e8 1.22839 0.614196 − 0.789154i \(-0.289480\pi\)
0.614196 + 0.789154i \(0.289480\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.35986e8 1.10211 0.551053 − 0.834470i \(-0.314226\pi\)
0.551053 + 0.834470i \(0.314226\pi\)
\(72\) 0 0
\(73\) −1.13179e8 −0.466459 −0.233230 − 0.972422i \(-0.574929\pi\)
−0.233230 + 0.972422i \(0.574929\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.60018e8 −1.49131
\(78\) 0 0
\(79\) 2.89403e8 0.835952 0.417976 − 0.908458i \(-0.362740\pi\)
0.417976 + 0.908458i \(0.362740\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.59791e8 −0.832146 −0.416073 − 0.909331i \(-0.636594\pi\)
−0.416073 + 0.909331i \(0.636594\pi\)
\(84\) 0 0
\(85\) −4.23589e8 −0.880157
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.34227e8 0.733604 0.366802 − 0.930299i \(-0.380453\pi\)
0.366802 + 0.930299i \(0.380453\pi\)
\(90\) 0 0
\(91\) 9.19998e8 1.40637
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.03972e8 −0.634819
\(96\) 0 0
\(97\) 4.30724e7 0.0493999 0.0247000 − 0.999695i \(-0.492137\pi\)
0.0247000 + 0.999695i \(0.492137\pi\)
\(98\) 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.q.1.4 yes 4
3.2 odd 2 288.10.a.r.1.1 yes 4
4.3 odd 2 288.10.a.r.1.4 yes 4
12.11 even 2 inner 288.10.a.q.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.10.a.q.1.1 ✓ 4 12.11 even 2 inner
288.10.a.q.1.4 yes 4 1.1 even 1 trivial
288.10.a.r.1.1 yes 4 3.2 odd 2
288.10.a.r.1.4 yes 4 4.3 odd 2