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,-6,0,44,-33,0,83,-87,0,-54,-85,0,0,-158,0,216,-178] 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: \(0\)
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: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) 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 1.6
Root \(-0.588238\) 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)
 
\(f(q)\) \(=\) \(q+0.213700 q^{2} -7.95433 q^{4} +15.3391 q^{5} +32.3928 q^{7} -3.40944 q^{8} +3.27797 q^{10} +29.5925 q^{11} +6.92233 q^{14} +62.9061 q^{16} +78.1958 q^{17} -10.6600 q^{19} -122.013 q^{20} +6.32392 q^{22} +26.8789 q^{23} +110.289 q^{25} -257.663 q^{28} +190.785 q^{29} +128.108 q^{31} +40.7185 q^{32} +16.7104 q^{34} +496.877 q^{35} +379.934 q^{37} -2.27805 q^{38} -52.2979 q^{40} -464.631 q^{41} +322.758 q^{43} -235.389 q^{44} +5.74401 q^{46} -248.529 q^{47} +706.292 q^{49} +23.5688 q^{50} -740.167 q^{53} +453.924 q^{55} -110.441 q^{56} +40.7706 q^{58} -340.673 q^{59} -590.834 q^{61} +27.3767 q^{62} -494.547 q^{64} +340.777 q^{67} -621.995 q^{68} +106.183 q^{70} -36.2243 q^{71} +164.572 q^{73} +81.1919 q^{74} +84.7935 q^{76} +958.584 q^{77} -327.242 q^{79} +964.925 q^{80} -99.2916 q^{82} -1404.48 q^{83} +1199.46 q^{85} +68.9734 q^{86} -100.894 q^{88} +736.986 q^{89} -213.803 q^{92} -53.1107 q^{94} -163.516 q^{95} -1494.87 q^{97} +150.935 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29}+ \cdots - 1457 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.213700 0.0755543 0.0377772 − 0.999286i \(-0.487972\pi\)
0.0377772 + 0.999286i \(0.487972\pi\)
\(3\) 0 0
\(4\) −7.95433 −0.994292
\(5\) 15.3391 1.37197 0.685987 − 0.727614i \(-0.259371\pi\)
0.685987 + 0.727614i \(0.259371\pi\)
\(6\) 0 0
\(7\) 32.3928 1.74905 0.874523 − 0.484984i \(-0.161175\pi\)
0.874523 + 0.484984i \(0.161175\pi\)
\(8\) −3.40944 −0.150677
\(9\) 0 0
\(10\) 3.27797 0.103659
\(11\) 29.5925 0.811135 0.405567 − 0.914065i \(-0.367074\pi\)
0.405567 + 0.914065i \(0.367074\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 6.92233 0.132148
\(15\) 0 0
\(16\) 62.9061 0.982907
\(17\) 78.1958 1.11560 0.557802 − 0.829974i \(-0.311645\pi\)
0.557802 + 0.829974i \(0.311645\pi\)
\(18\) 0 0
\(19\) −10.6600 −0.128715 −0.0643574 − 0.997927i \(-0.520500\pi\)
−0.0643574 + 0.997927i \(0.520500\pi\)
\(20\) −122.013 −1.36414
\(21\) 0 0
\(22\) 6.32392 0.0612847
\(23\) 26.8789 0.243680 0.121840 − 0.992550i \(-0.461121\pi\)
0.121840 + 0.992550i \(0.461121\pi\)
\(24\) 0 0
\(25\) 110.289 0.882314
\(26\) 0 0
\(27\) 0 0
\(28\) −257.663 −1.73906
\(29\) 190.785 1.22165 0.610824 − 0.791766i \(-0.290838\pi\)
0.610824 + 0.791766i \(0.290838\pi\)
\(30\) 0 0
\(31\) 128.108 0.742222 0.371111 − 0.928589i \(-0.378977\pi\)
0.371111 + 0.928589i \(0.378977\pi\)
\(32\) 40.7185 0.224940
\(33\) 0 0
\(34\) 16.7104 0.0842887
\(35\) 496.877 2.39965
\(36\) 0 0
\(37\) 379.934 1.68813 0.844065 − 0.536241i \(-0.180156\pi\)
0.844065 + 0.536241i \(0.180156\pi\)
\(38\) −2.27805 −0.00972496
\(39\) 0 0
\(40\) −52.2979 −0.206725
\(41\) −464.631 −1.76983 −0.884917 − 0.465749i \(-0.845785\pi\)
−0.884917 + 0.465749i \(0.845785\pi\)
\(42\) 0 0
\(43\) 322.758 1.14466 0.572328 − 0.820025i \(-0.306041\pi\)
0.572328 + 0.820025i \(0.306041\pi\)
\(44\) −235.389 −0.806504
\(45\) 0 0
\(46\) 5.74401 0.0184110
\(47\) −248.529 −0.771314 −0.385657 − 0.922642i \(-0.626025\pi\)
−0.385657 + 0.922642i \(0.626025\pi\)
\(48\) 0 0
\(49\) 706.292 2.05916
\(50\) 23.5688 0.0666626
\(51\) 0 0
\(52\) 0 0
\(53\) −740.167 −1.91830 −0.959148 − 0.282904i \(-0.908702\pi\)
−0.959148 + 0.282904i \(0.908702\pi\)
\(54\) 0 0
\(55\) 453.924 1.11286
\(56\) −110.441 −0.263542
\(57\) 0 0
\(58\) 40.7706 0.0923008
\(59\) −340.673 −0.751727 −0.375864 − 0.926675i \(-0.622654\pi\)
−0.375864 + 0.926675i \(0.622654\pi\)
\(60\) 0 0
\(61\) −590.834 −1.24014 −0.620070 − 0.784547i \(-0.712896\pi\)
−0.620070 + 0.784547i \(0.712896\pi\)
\(62\) 27.3767 0.0560781
\(63\) 0 0
\(64\) −494.547 −0.965912
\(65\) 0 0
\(66\) 0 0
\(67\) 340.777 0.621382 0.310691 − 0.950511i \(-0.399440\pi\)
0.310691 + 0.950511i \(0.399440\pi\)
\(68\) −621.995 −1.10924
\(69\) 0 0
\(70\) 106.183 0.181304
\(71\) −36.2243 −0.0605498 −0.0302749 − 0.999542i \(-0.509638\pi\)
−0.0302749 + 0.999542i \(0.509638\pi\)
\(72\) 0 0
\(73\) 164.572 0.263859 0.131929 − 0.991259i \(-0.457883\pi\)
0.131929 + 0.991259i \(0.457883\pi\)
\(74\) 81.1919 0.127546
\(75\) 0 0
\(76\) 84.7935 0.127980
\(77\) 958.584 1.41871
\(78\) 0 0
\(79\) −327.242 −0.466046 −0.233023 − 0.972471i \(-0.574862\pi\)
−0.233023 + 0.972471i \(0.574862\pi\)
\(80\) 964.925 1.34852
\(81\) 0 0
\(82\) −99.2916 −0.133719
\(83\) −1404.48 −1.85737 −0.928684 − 0.370871i \(-0.879059\pi\)
−0.928684 + 0.370871i \(0.879059\pi\)
\(84\) 0 0
\(85\) 1199.46 1.53058
\(86\) 68.9734 0.0864837
\(87\) 0 0
\(88\) −100.894 −0.122220
\(89\) 736.986 0.877756 0.438878 − 0.898547i \(-0.355376\pi\)
0.438878 + 0.898547i \(0.355376\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −213.803 −0.242289
\(93\) 0 0
\(94\) −53.1107 −0.0582761
\(95\) −163.516 −0.176593
\(96\) 0 0
\(97\) −1494.87 −1.56476 −0.782378 − 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(98\) 150.935 0.155578
\(99\) 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.bf.1.6 9
3.2 odd 2 507.4.a.p.1.4 yes 9
13.12 even 2 1521.4.a.bi.1.4 9
39.5 even 4 507.4.b.k.337.10 18
39.8 even 4 507.4.b.k.337.9 18
39.38 odd 2 507.4.a.o.1.6 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.6 ✓ 9 39.38 odd 2
507.4.a.p.1.4 yes 9 3.2 odd 2
507.4.b.k.337.9 18 39.8 even 4
507.4.b.k.337.10 18 39.5 even 4
1521.4.a.bf.1.6 9 1.1 even 1 trivial
1521.4.a.bi.1.4 9 13.12 even 2