Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1521,4,Mod(1,1521)] 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("1521.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1521, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-8,0,32,-41,0,1,-111,0,198,-37,0,0,-98,0,32,134] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
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: \( x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.37739\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.37739 q^{2} +3.40677 q^{4} +15.7127 q^{5} -17.1681 q^{7} +15.5131 q^{8} -53.0679 q^{10} -52.8187 q^{11} +57.9835 q^{14} -79.6481 q^{16} +71.0654 q^{17} +92.6916 q^{19} +53.5295 q^{20} +178.390 q^{22} -190.712 q^{23} +121.888 q^{25} -58.4878 q^{28} +128.204 q^{29} +3.29674 q^{31} +144.898 q^{32} -240.016 q^{34} -269.757 q^{35} +241.546 q^{37} -313.056 q^{38} +243.753 q^{40} +97.1824 q^{41} +376.151 q^{43} -179.941 q^{44} +644.109 q^{46} -577.354 q^{47} -48.2555 q^{49} -411.664 q^{50} +307.686 q^{53} -829.924 q^{55} -266.331 q^{56} -432.995 q^{58} -349.914 q^{59} +127.467 q^{61} -11.1344 q^{62} +147.809 q^{64} -903.564 q^{67} +242.104 q^{68} +911.076 q^{70} -826.106 q^{71} -131.760 q^{73} -815.796 q^{74} +315.779 q^{76} +906.799 q^{77} -556.244 q^{79} -1251.48 q^{80} -328.223 q^{82} +254.664 q^{83} +1116.63 q^{85} -1270.41 q^{86} -819.384 q^{88} -183.410 q^{89} -649.712 q^{92} +1949.95 q^{94} +1456.43 q^{95} -780.498 q^{97} +162.978 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8} + 198 q^{10} - 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} - 72 q^{19} - 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} + 132 q^{28} + 547 q^{29}+ \cdots - 5593 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.37739 −1.19409 −0.597044 − 0.802208i \(-0.703658\pi\)
−0.597044 + 0.802208i \(0.703658\pi\)
\(3\) 0 0
\(4\) 3.40677 0.425846
\(5\) 15.7127 1.40538 0.702692 − 0.711494i \(-0.251981\pi\)
0.702692 + 0.711494i \(0.251981\pi\)
\(6\) 0 0
\(7\) −17.1681 −0.926992 −0.463496 − 0.886099i \(-0.653405\pi\)
−0.463496 + 0.886099i \(0.653405\pi\)
\(8\) 15.5131 0.685590
\(9\) 0 0
\(10\) −53.0679 −1.67815
\(11\) −52.8187 −1.44777 −0.723884 − 0.689922i \(-0.757645\pi\)
−0.723884 + 0.689922i \(0.757645\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 57.9835 1.10691
\(15\) 0 0
\(16\) −79.6481 −1.24450
\(17\) 71.0654 1.01388 0.506938 − 0.861982i \(-0.330777\pi\)
0.506938 + 0.861982i \(0.330777\pi\)
\(18\) 0 0
\(19\) 92.6916 1.11921 0.559603 − 0.828761i \(-0.310954\pi\)
0.559603 + 0.828761i \(0.310954\pi\)
\(20\) 53.5295 0.598478
\(21\) 0 0
\(22\) 178.390 1.72876
\(23\) −190.712 −1.72896 −0.864482 − 0.502663i \(-0.832354\pi\)
−0.864482 + 0.502663i \(0.832354\pi\)
\(24\) 0 0
\(25\) 121.888 0.975106
\(26\) 0 0
\(27\) 0 0
\(28\) −58.4878 −0.394756
\(29\) 128.204 0.820927 0.410463 − 0.911877i \(-0.365367\pi\)
0.410463 + 0.911877i \(0.365367\pi\)
\(30\) 0 0
\(31\) 3.29674 0.0191004 0.00955020 − 0.999954i \(-0.496960\pi\)
0.00955020 + 0.999954i \(0.496960\pi\)
\(32\) 144.898 0.800454
\(33\) 0 0
\(34\) −240.016 −1.21066
\(35\) −269.757 −1.30278
\(36\) 0 0
\(37\) 241.546 1.07324 0.536621 − 0.843823i \(-0.319700\pi\)
0.536621 + 0.843823i \(0.319700\pi\)
\(38\) −313.056 −1.33643
\(39\) 0 0
\(40\) 243.753 0.963518
\(41\) 97.1824 0.370179 0.185090 − 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(42\) 0 0
\(43\) 376.151 1.33401 0.667006 − 0.745052i \(-0.267575\pi\)
0.667006 + 0.745052i \(0.267575\pi\)
\(44\) −179.941 −0.616527
\(45\) 0 0
\(46\) 644.109 2.06454
\(47\) −577.354 −1.79182 −0.895912 − 0.444231i \(-0.853477\pi\)
−0.895912 + 0.444231i \(0.853477\pi\)
\(48\) 0 0
\(49\) −48.2555 −0.140686
\(50\) −411.664 −1.16436
\(51\) 0 0
\(52\) 0 0
\(53\) 307.686 0.797433 0.398716 − 0.917074i \(-0.369456\pi\)
0.398716 + 0.917074i \(0.369456\pi\)
\(54\) 0 0
\(55\) −829.924 −2.03467
\(56\) −266.331 −0.635536
\(57\) 0 0
\(58\) −432.995 −0.980259
\(59\) −349.914 −0.772117 −0.386059 − 0.922474i \(-0.626164\pi\)
−0.386059 + 0.922474i \(0.626164\pi\)
\(60\) 0 0
\(61\) 127.467 0.267548 0.133774 − 0.991012i \(-0.457290\pi\)
0.133774 + 0.991012i \(0.457290\pi\)
\(62\) −11.1344 −0.0228076
\(63\) 0 0
\(64\) 147.809 0.288689
\(65\) 0 0
\(66\) 0 0
\(67\) −903.564 −1.64758 −0.823791 − 0.566894i \(-0.808145\pi\)
−0.823791 + 0.566894i \(0.808145\pi\)
\(68\) 242.104 0.431755
\(69\) 0 0
\(70\) 911.076 1.55563
\(71\) −826.106 −1.38086 −0.690428 − 0.723401i \(-0.742578\pi\)
−0.690428 + 0.723401i \(0.742578\pi\)
\(72\) 0 0
\(73\) −131.760 −0.211252 −0.105626 − 0.994406i \(-0.533685\pi\)
−0.105626 + 0.994406i \(0.533685\pi\)
\(74\) −815.796 −1.28155
\(75\) 0 0
\(76\) 315.779 0.476610
\(77\) 906.799 1.34207
\(78\) 0 0
\(79\) −556.244 −0.792182 −0.396091 − 0.918211i \(-0.629633\pi\)
−0.396091 + 0.918211i \(0.629633\pi\)
\(80\) −1251.48 −1.74900
\(81\) 0 0
\(82\) −328.223 −0.442026
\(83\) 254.664 0.336783 0.168391 − 0.985720i \(-0.446143\pi\)
0.168391 + 0.985720i \(0.446143\pi\)
\(84\) 0 0
\(85\) 1116.63 1.42489
\(86\) −1270.41 −1.59293
\(87\) 0 0
\(88\) −819.384 −0.992576
\(89\) −183.410 −0.218443 −0.109222 − 0.994017i \(-0.534836\pi\)
−0.109222 + 0.994017i \(0.534836\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −649.712 −0.736273
\(93\) 0 0
\(94\) 1949.95 2.13960
\(95\) 1456.43 1.57292
\(96\) 0 0
\(97\) −780.498 −0.816986 −0.408493 − 0.912762i \(-0.633946\pi\)
−0.408493 + 0.912762i \(0.633946\pi\)
\(98\) 162.978 0.167992
\(99\) 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 1521.4.a.be.1.3 9
3.2 odd 2 507.4.a.q.1.7 yes 9
13.12 even 2 1521.4.a.bj.1.7 9
39.5 even 4 507.4.b.j.337.4 18
39.8 even 4 507.4.b.j.337.15 18
39.38 odd 2 507.4.a.n.1.3 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.n.1.3 ✓ 9 39.38 odd 2
507.4.a.q.1.7 yes 9 3.2 odd 2
507.4.b.j.337.4 18 39.5 even 4
507.4.b.j.337.15 18 39.8 even 4
1521.4.a.be.1.3 9 1.1 even 1 trivial
1521.4.a.bj.1.7 9 13.12 even 2