Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1083,2,Mod(1,1083)] 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("1083.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1083, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1083 = 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1083.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.64779853890\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1083.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.87939 q^{2} +1.00000 q^{3} +1.53209 q^{4} +0.879385 q^{5} -1.87939 q^{6} -2.87939 q^{7} +0.879385 q^{8} +1.00000 q^{9} -1.65270 q^{10} -1.83750 q^{11} +1.53209 q^{12} +2.75877 q^{13} +5.41147 q^{14} +0.879385 q^{15} -4.71688 q^{16} -7.10607 q^{17} -1.87939 q^{18} +1.34730 q^{20} -2.87939 q^{21} +3.45336 q^{22} +6.59627 q^{23} +0.879385 q^{24} -4.22668 q^{25} -5.18479 q^{26} +1.00000 q^{27} -4.41147 q^{28} +3.12836 q^{29} -1.65270 q^{30} -7.65270 q^{31} +7.10607 q^{32} -1.83750 q^{33} +13.3550 q^{34} -2.53209 q^{35} +1.53209 q^{36} -2.83750 q^{37} +2.75877 q^{39} +0.773318 q^{40} -3.98545 q^{41} +5.41147 q^{42} -11.4534 q^{43} -2.81521 q^{44} +0.879385 q^{45} -12.3969 q^{46} +2.20708 q^{47} -4.71688 q^{48} +1.29086 q^{49} +7.94356 q^{50} -7.10607 q^{51} +4.22668 q^{52} -2.70233 q^{53} -1.87939 q^{54} -1.61587 q^{55} -2.53209 q^{56} -5.87939 q^{58} +8.41147 q^{59} +1.34730 q^{60} +0.615867 q^{61} +14.3824 q^{62} -2.87939 q^{63} -3.92127 q^{64} +2.42602 q^{65} +3.45336 q^{66} +3.67499 q^{67} -10.8871 q^{68} +6.59627 q^{69} +4.75877 q^{70} +7.45336 q^{71} +0.879385 q^{72} -10.0077 q^{73} +5.33275 q^{74} -4.22668 q^{75} +5.29086 q^{77} -5.18479 q^{78} -1.61081 q^{79} -4.14796 q^{80} +1.00000 q^{81} +7.49020 q^{82} +0.985452 q^{83} -4.41147 q^{84} -6.24897 q^{85} +21.5253 q^{86} +3.12836 q^{87} -1.61587 q^{88} -17.0574 q^{89} -1.65270 q^{90} -7.94356 q^{91} +10.1061 q^{92} -7.65270 q^{93} -4.14796 q^{94} +7.10607 q^{96} +5.90167 q^{97} -2.42602 q^{98} -1.83750 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 6 q^{10} - 3 q^{11} - 3 q^{13} + 6 q^{14} - 3 q^{15} - 6 q^{16} - 9 q^{17} + 3 q^{20} - 3 q^{21} - 3 q^{22} + 6 q^{23} - 3 q^{24} - 6 q^{25} - 12 q^{26}+ \cdots - 3 q^{99}+O(q^{100}) \) 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 \(0.731340\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.53209 0.766044
\(5\) 0.879385 0.393273 0.196637 − 0.980476i \(-0.436998\pi\)
0.196637 + 0.980476i \(0.436998\pi\)
\(6\) −1.87939 −0.767256
\(7\) −2.87939 −1.08831 −0.544153 − 0.838986i \(-0.683149\pi\)
−0.544153 + 0.838986i \(0.683149\pi\)
\(8\) 0.879385 0.310910
\(9\) 1.00000 0.333333
\(10\) −1.65270 −0.522631
\(11\) −1.83750 −0.554026 −0.277013 − 0.960866i \(-0.589345\pi\)
−0.277013 + 0.960866i \(0.589345\pi\)
\(12\) 1.53209 0.442276
\(13\) 2.75877 0.765145 0.382573 − 0.923925i \(-0.375038\pi\)
0.382573 + 0.923925i \(0.375038\pi\)
\(14\) 5.41147 1.44628
\(15\) 0.879385 0.227056
\(16\) −4.71688 −1.17922
\(17\) −7.10607 −1.72347 −0.861737 − 0.507355i \(-0.830623\pi\)
−0.861737 + 0.507355i \(0.830623\pi\)
\(18\) −1.87939 −0.442975
\(19\) 0 0
\(20\) 1.34730 0.301265
\(21\) −2.87939 −0.628333
\(22\) 3.45336 0.736260
\(23\) 6.59627 1.37542 0.687708 − 0.725987i \(-0.258617\pi\)
0.687708 + 0.725987i \(0.258617\pi\)
\(24\) 0.879385 0.179504
\(25\) −4.22668 −0.845336
\(26\) −5.18479 −1.01682
\(27\) 1.00000 0.192450
\(28\) −4.41147 −0.833690
\(29\) 3.12836 0.580921 0.290461 − 0.956887i \(-0.406192\pi\)
0.290461 + 0.956887i \(0.406192\pi\)
\(30\) −1.65270 −0.301741
\(31\) −7.65270 −1.37447 −0.687233 − 0.726437i \(-0.741175\pi\)
−0.687233 + 0.726437i \(0.741175\pi\)
\(32\) 7.10607 1.25619
\(33\) −1.83750 −0.319867
\(34\) 13.3550 2.29037
\(35\) −2.53209 −0.428001
\(36\) 1.53209 0.255348
\(37\) −2.83750 −0.466481 −0.233241 − 0.972419i \(-0.574933\pi\)
−0.233241 + 0.972419i \(0.574933\pi\)
\(38\) 0 0
\(39\) 2.75877 0.441757
\(40\) 0.773318 0.122272
\(41\) −3.98545 −0.622423 −0.311212 − 0.950341i \(-0.600735\pi\)
−0.311212 + 0.950341i \(0.600735\pi\)
\(42\) 5.41147 0.835009
\(43\) −11.4534 −1.74662 −0.873311 − 0.487164i \(-0.838032\pi\)
−0.873311 + 0.487164i \(0.838032\pi\)
\(44\) −2.81521 −0.424408
\(45\) 0.879385 0.131091
\(46\) −12.3969 −1.82783
\(47\) 2.20708 0.321936 0.160968 − 0.986960i \(-0.448538\pi\)
0.160968 + 0.986960i \(0.448538\pi\)
\(48\) −4.71688 −0.680823
\(49\) 1.29086 0.184408
\(50\) 7.94356 1.12339
\(51\) −7.10607 −0.995048
\(52\) 4.22668 0.586135
\(53\) −2.70233 −0.371194 −0.185597 − 0.982626i \(-0.559422\pi\)
−0.185597 + 0.982626i \(0.559422\pi\)
\(54\) −1.87939 −0.255752
\(55\) −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 \(0.315564\pi\)
\(60\) 1.34730 0.173935
\(61\) 0.615867 0.0788537 0.0394268 − 0.999222i \(-0.487447\pi\)
0.0394268 + 0.999222i \(0.487447\pi\)
\(62\) 14.3824 1.82656
\(63\) −2.87939 −0.362768
\(64\) −3.92127 −0.490159
\(65\) 2.42602 0.300911
\(66\) 3.45336 0.425080
\(67\) 3.67499 0.448972 0.224486 − 0.974477i \(-0.427930\pi\)
0.224486 + 0.974477i \(0.427930\pi\)
\(68\) −10.8871 −1.32026
\(69\) 6.59627 0.794097
\(70\) 4.75877 0.568782
\(71\) 7.45336 0.884551 0.442276 − 0.896879i \(-0.354171\pi\)
0.442276 + 0.896879i \(0.354171\pi\)
\(72\) 0.879385 0.103637
\(73\) −10.0077 −1.17132 −0.585659 − 0.810558i \(-0.699164\pi\)
−0.585659 + 0.810558i \(0.699164\pi\)
\(74\) 5.33275 0.619919
\(75\) −4.22668 −0.488055
\(76\) 0 0
\(77\) 5.29086 0.602949
\(78\) −5.18479 −0.587062
\(79\) −1.61081 −0.181231 −0.0906154 − 0.995886i \(-0.528883\pi\)
−0.0906154 + 0.995886i \(0.528883\pi\)
\(80\) −4.14796 −0.463756
\(81\) 1.00000 0.111111
\(82\) 7.49020 0.827154
\(83\) 0.985452 0.108167 0.0540837 − 0.998536i \(-0.482776\pi\)
0.0540837 + 0.998536i \(0.482776\pi\)
\(84\) −4.41147 −0.481331
\(85\) −6.24897 −0.677796
\(86\) 21.5253 2.32113
\(87\) 3.12836 0.335395
\(88\) −1.61587 −0.172252
\(89\) −17.0574 −1.80808 −0.904039 − 0.427450i \(-0.859412\pi\)
−0.904039 + 0.427450i \(0.859412\pi\)
\(90\) −1.65270 −0.174210
\(91\) −7.94356 −0.832712
\(92\) 10.1061 1.05363
\(93\) −7.65270 −0.793548
\(94\) −4.14796 −0.427829
\(95\) 0 0
\(96\) 7.10607 0.725260
\(97\) 5.90167 0.599224 0.299612 − 0.954061i \(-0.403143\pi\)
0.299612 + 0.954061i \(0.403143\pi\)
\(98\) −2.42602 −0.245065
\(99\) −1.83750 −0.184675
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 1083.2.a.n.1.1 3
3.2 odd 2 3249.2.a.x.1.3 3
19.3 odd 18 57.2.i.a.28.1 ✓ 6
19.13 odd 18 57.2.i.a.55.1 yes 6
19.18 odd 2 1083.2.a.m.1.3 3
57.32 even 18 171.2.u.a.55.1 6
57.41 even 18 171.2.u.a.28.1 6
57.56 even 2 3249.2.a.w.1.1 3
76.3 even 18 912.2.bo.b.769.1 6
76.51 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 19.3 odd 18
57.2.i.a.55.1 yes 6 19.13 odd 18
171.2.u.a.28.1 6 57.41 even 18
171.2.u.a.55.1 6 57.32 even 18
912.2.bo.b.625.1 6 76.51 even 18
912.2.bo.b.769.1 6 76.3 even 18
1083.2.a.m.1.3 3 19.18 odd 2
1083.2.a.n.1.1 3 1.1 even 1 trivial
3249.2.a.w.1.1 3 57.56 even 2
3249.2.a.x.1.3 3 3.2 odd 2