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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3249,2,Mod(1,3249)] 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("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3249.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,3,0,-3,-3,0,6,3,0,3,6,0,-6,9,0,0,-3,0,3,-6,0,-6,12,0, -3,-9,0,24,9,0,-15,3,0,6,0,0,-9,6,0,-21,12,0,9,3,0,-12,9,0,-6,18,0,6,-3, 0,-12,15,0,-9,3,0,-3,15,0,-6,3,0,-3,9,0,-6,3,0,0,0,0,9,-3,0,21,15,0,-6, 18,0,-6,0,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.9433956167\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 3249.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.87939 q^{2} +1.53209 q^{4} -0.879385 q^{5} -2.87939 q^{7} +0.879385 q^{8} +1.65270 q^{10} +1.83750 q^{11} -2.75877 q^{13} +5.41147 q^{14} -4.71688 q^{16} +7.10607 q^{17} -1.34730 q^{20} -3.45336 q^{22} -6.59627 q^{23} -4.22668 q^{25} +5.18479 q^{26} -4.41147 q^{28} +3.12836 q^{29} +7.65270 q^{31} +7.10607 q^{32} -13.3550 q^{34} +2.53209 q^{35} +2.83750 q^{37} -0.773318 q^{40} -3.98545 q^{41} -11.4534 q^{43} +2.81521 q^{44} +12.3969 q^{46} -2.20708 q^{47} +1.29086 q^{49} +7.94356 q^{50} -4.22668 q^{52} -2.70233 q^{53} -1.61587 q^{55} -2.53209 q^{56} -5.87939 q^{58} +8.41147 q^{59} +0.615867 q^{61} -14.3824 q^{62} -3.92127 q^{64} +2.42602 q^{65} -3.67499 q^{67} +10.8871 q^{68} -4.75877 q^{70} +7.45336 q^{71} -10.0077 q^{73} -5.33275 q^{74} -5.29086 q^{77} +1.61081 q^{79} +4.14796 q^{80} +7.49020 q^{82} -0.985452 q^{83} -6.24897 q^{85} +21.5253 q^{86} +1.61587 q^{88} -17.0574 q^{89} +7.94356 q^{91} -10.1061 q^{92} +4.14796 q^{94} -5.90167 q^{97} -2.42602 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7} - 3 q^{8} + 6 q^{10} + 3 q^{11} + 3 q^{13} + 6 q^{14} - 6 q^{16} + 9 q^{17} - 3 q^{20} + 3 q^{22} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} - 9 q^{29} + 24 q^{31} + 9 q^{32} - 15 q^{34}+ \cdots - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87939 −1.32893 −0.664463 − 0.747321i \(-0.731340\pi\)
−0.664463 + 0.747321i \(0.731340\pi\)
\(3\) 0 0
\(4\) 1.53209 0.766044
\(5\) −0.879385 −0.393273 −0.196637 − 0.980476i \(-0.563002\pi\)
−0.196637 + 0.980476i \(0.563002\pi\)
\(6\) 0 0
\(7\) −2.87939 −1.08831 −0.544153 − 0.838986i \(-0.683149\pi\)
−0.544153 + 0.838986i \(0.683149\pi\)
\(8\) 0.879385 0.310910
\(9\) 0 0
\(10\) 1.65270 0.522631
\(11\) 1.83750 0.554026 0.277013 − 0.960866i \(-0.410655\pi\)
0.277013 + 0.960866i \(0.410655\pi\)
\(12\) 0 0
\(13\) −2.75877 −0.765145 −0.382573 − 0.923925i \(-0.624962\pi\)
−0.382573 + 0.923925i \(0.624962\pi\)
\(14\) 5.41147 1.44628
\(15\) 0 0
\(16\) −4.71688 −1.17922
\(17\) 7.10607 1.72347 0.861737 − 0.507355i \(-0.169377\pi\)
0.861737 + 0.507355i \(0.169377\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) −1.34730 −0.301265
\(21\) 0 0
\(22\) −3.45336 −0.736260
\(23\) −6.59627 −1.37542 −0.687708 − 0.725987i \(-0.741383\pi\)
−0.687708 + 0.725987i \(0.741383\pi\)
\(24\) 0 0
\(25\) −4.22668 −0.845336
\(26\) 5.18479 1.01682
\(27\) 0 0
\(28\) −4.41147 −0.833690
\(29\) 3.12836 0.580921 0.290461 − 0.956887i \(-0.406192\pi\)
0.290461 + 0.956887i \(0.406192\pi\)
\(30\) 0 0
\(31\) 7.65270 1.37447 0.687233 − 0.726437i \(-0.258825\pi\)
0.687233 + 0.726437i \(0.258825\pi\)
\(32\) 7.10607 1.25619
\(33\) 0 0
\(34\) −13.3550 −2.29037
\(35\) 2.53209 0.428001
\(36\) 0 0
\(37\) 2.83750 0.466481 0.233241 − 0.972419i \(-0.425067\pi\)
0.233241 + 0.972419i \(0.425067\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.773318 −0.122272
\(41\) −3.98545 −0.622423 −0.311212 − 0.950341i \(-0.600735\pi\)
−0.311212 + 0.950341i \(0.600735\pi\)
\(42\) 0 0
\(43\) −11.4534 −1.74662 −0.873311 − 0.487164i \(-0.838032\pi\)
−0.873311 + 0.487164i \(0.838032\pi\)
\(44\) 2.81521 0.424408
\(45\) 0 0
\(46\) 12.3969 1.82783
\(47\) −2.20708 −0.321936 −0.160968 − 0.986960i \(-0.551462\pi\)
−0.160968 + 0.986960i \(0.551462\pi\)
\(48\) 0 0
\(49\) 1.29086 0.184408
\(50\) 7.94356 1.12339
\(51\) 0 0
\(52\) −4.22668 −0.586135
\(53\) −2.70233 −0.371194 −0.185597 − 0.982626i \(-0.559422\pi\)
−0.185597 + 0.982626i \(0.559422\pi\)
\(54\) 0 0
\(55\) −1.61587 −0.217883
\(56\) −2.53209 −0.338365
\(57\) 0 0
\(58\) −5.87939 −0.772001
\(59\) 8.41147 1.09508 0.547540 − 0.836779i \(-0.315564\pi\)
0.547540 + 0.836779i \(0.315564\pi\)
\(60\) 0 0
\(61\) 0.615867 0.0788537 0.0394268 − 0.999222i \(-0.487447\pi\)
0.0394268 + 0.999222i \(0.487447\pi\)
\(62\) −14.3824 −1.82656
\(63\) 0 0
\(64\) −3.92127 −0.490159
\(65\) 2.42602 0.300911
\(66\) 0 0
\(67\) −3.67499 −0.448972 −0.224486 − 0.974477i \(-0.572070\pi\)
−0.224486 + 0.974477i \(0.572070\pi\)
\(68\) 10.8871 1.32026
\(69\) 0 0
\(70\) −4.75877 −0.568782
\(71\) 7.45336 0.884551 0.442276 − 0.896879i \(-0.354171\pi\)
0.442276 + 0.896879i \(0.354171\pi\)
\(72\) 0 0
\(73\) −10.0077 −1.17132 −0.585659 − 0.810558i \(-0.699164\pi\)
−0.585659 + 0.810558i \(0.699164\pi\)
\(74\) −5.33275 −0.619919
\(75\) 0 0
\(76\) 0 0
\(77\) −5.29086 −0.602949
\(78\) 0 0
\(79\) 1.61081 0.181231 0.0906154 − 0.995886i \(-0.471117\pi\)
0.0906154 + 0.995886i \(0.471117\pi\)
\(80\) 4.14796 0.463756
\(81\) 0 0
\(82\) 7.49020 0.827154
\(83\) −0.985452 −0.108167 −0.0540837 − 0.998536i \(-0.517224\pi\)
−0.0540837 + 0.998536i \(0.517224\pi\)
\(84\) 0 0
\(85\) −6.24897 −0.677796
\(86\) 21.5253 2.32113
\(87\) 0 0
\(88\) 1.61587 0.172252
\(89\) −17.0574 −1.80808 −0.904039 − 0.427450i \(-0.859412\pi\)
−0.904039 + 0.427450i \(0.859412\pi\)
\(90\) 0 0
\(91\) 7.94356 0.832712
\(92\) −10.1061 −1.05363
\(93\) 0 0
\(94\) 4.14796 0.427829
\(95\) 0 0
\(96\) 0 0
\(97\) −5.90167 −0.599224 −0.299612 − 0.954061i \(-0.596857\pi\)
−0.299612 + 0.954061i \(0.596857\pi\)
\(98\) −2.42602 −0.245065
\(99\) 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 3249.2.a.w.1.1 3
3.2 odd 2 1083.2.a.m.1.3 3
19.6 even 9 171.2.u.a.55.1 6
19.16 even 9 171.2.u.a.28.1 6
19.18 odd 2 3249.2.a.x.1.3 3
57.35 odd 18 57.2.i.a.28.1 ✓ 6
57.44 odd 18 57.2.i.a.55.1 yes 6
57.56 even 2 1083.2.a.n.1.1 3
228.35 even 18 912.2.bo.b.769.1 6
228.215 even 18 912.2.bo.b.625.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.i.a.28.1 ✓ 6 57.35 odd 18
57.2.i.a.55.1 yes 6 57.44 odd 18
171.2.u.a.28.1 6 19.16 even 9
171.2.u.a.55.1 6 19.6 even 9
912.2.bo.b.625.1 6 228.215 even 18
912.2.bo.b.769.1 6 228.35 even 18
1083.2.a.m.1.3 3 3.2 odd 2
1083.2.a.n.1.1 3 57.56 even 2
3249.2.a.w.1.1 3 1.1 even 1 trivial
3249.2.a.x.1.3 3 19.18 odd 2