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Properties

Label 1083.2.a.l.1.2
Level $1083$
Weight $2$
Character 1083.1
Self dual yes
Analytic conductor $8.648$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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,-1,3] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) 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 1.2
Root \(0.571993\) 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)
 
\(f(q)\) \(=\) \(q-0.571993 q^{2} +1.00000 q^{3} -1.67282 q^{4} -2.67282 q^{5} -0.571993 q^{6} -3.67282 q^{7} +2.10083 q^{8} +1.00000 q^{9} +1.52884 q^{10} -3.81681 q^{11} -1.67282 q^{12} +0.143987 q^{13} +2.10083 q^{14} -2.67282 q^{15} +2.14399 q^{16} -0.571993 q^{18} +4.47116 q^{20} -3.67282 q^{21} +2.18319 q^{22} +7.52884 q^{23} +2.10083 q^{24} +2.14399 q^{25} -0.0823593 q^{26} +1.00000 q^{27} +6.14399 q^{28} -5.34565 q^{29} +1.52884 q^{30} +8.81681 q^{31} -5.42801 q^{32} -3.81681 q^{33} +9.81681 q^{35} -1.67282 q^{36} -1.00000 q^{37} +0.143987 q^{39} -5.61515 q^{40} +5.34565 q^{41} +2.10083 q^{42} +2.81681 q^{43} +6.38485 q^{44} -2.67282 q^{45} -4.30644 q^{46} -6.00000 q^{47} +2.14399 q^{48} +6.48963 q^{49} -1.22635 q^{50} -0.240864 q^{52} +8.01847 q^{53} -0.571993 q^{54} +10.2017 q^{55} -7.71598 q^{56} +3.05767 q^{58} +3.81681 q^{59} +4.47116 q^{60} +11.4896 q^{61} -5.04316 q^{62} -3.67282 q^{63} -1.18319 q^{64} -0.384851 q^{65} +2.18319 q^{66} +5.38485 q^{67} +7.52884 q^{69} -5.61515 q^{70} -13.6336 q^{71} +2.10083 q^{72} -0.345647 q^{73} +0.571993 q^{74} +2.14399 q^{75} +14.0185 q^{77} -0.0823593 q^{78} +6.52884 q^{79} -5.73050 q^{80} +1.00000 q^{81} -3.05767 q^{82} -2.28797 q^{83} +6.14399 q^{84} -1.61120 q^{86} -5.34565 q^{87} -8.01847 q^{88} -8.67282 q^{89} +1.52884 q^{90} -0.528837 q^{91} -12.5944 q^{92} +8.81681 q^{93} +3.43196 q^{94} -5.42801 q^{96} -5.91369 q^{97} -3.71203 q^{98} -3.81681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{5} - q^{6} - q^{7} - 3 q^{8} + 3 q^{9} - 4 q^{10} + 5 q^{12} - q^{13} - 3 q^{14} + 2 q^{15} + 5 q^{16} - q^{18} + 22 q^{20} - q^{21} + 18 q^{22} + 14 q^{23}+ \cdots - 14 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.571993 −0.404460 −0.202230 − 0.979338i \(-0.564819\pi\)
−0.202230 + 0.979338i \(0.564819\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.67282 −0.836412
\(5\) −2.67282 −1.19532 −0.597662 − 0.801749i \(-0.703903\pi\)
−0.597662 + 0.801749i \(0.703903\pi\)
\(6\) −0.571993 −0.233515
\(7\) −3.67282 −1.38820 −0.694098 − 0.719880i \(-0.744197\pi\)
−0.694098 + 0.719880i \(0.744197\pi\)
\(8\) 2.10083 0.742756
\(9\) 1.00000 0.333333
\(10\) 1.52884 0.483461
\(11\) −3.81681 −1.15081 −0.575406 − 0.817868i \(-0.695156\pi\)
−0.575406 + 0.817868i \(0.695156\pi\)
\(12\) −1.67282 −0.482903
\(13\) 0.143987 0.0399347 0.0199673 − 0.999801i \(-0.493644\pi\)
0.0199673 + 0.999801i \(0.493644\pi\)
\(14\) 2.10083 0.561471
\(15\) −2.67282 −0.690120
\(16\) 2.14399 0.535997
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −0.571993 −0.134820
\(19\) 0 0
\(20\) 4.47116 0.999782
\(21\) −3.67282 −0.801476
\(22\) 2.18319 0.465458
\(23\) 7.52884 1.56987 0.784936 − 0.619577i \(-0.212696\pi\)
0.784936 + 0.619577i \(0.212696\pi\)
\(24\) 2.10083 0.428830
\(25\) 2.14399 0.428797
\(26\) −0.0823593 −0.0161520
\(27\) 1.00000 0.192450
\(28\) 6.14399 1.16110
\(29\) −5.34565 −0.992662 −0.496331 − 0.868133i \(-0.665320\pi\)
−0.496331 + 0.868133i \(0.665320\pi\)
\(30\) 1.52884 0.279126
\(31\) 8.81681 1.58355 0.791773 − 0.610816i \(-0.209158\pi\)
0.791773 + 0.610816i \(0.209158\pi\)
\(32\) −5.42801 −0.959545
\(33\) −3.81681 −0.664421
\(34\) 0 0
\(35\) 9.81681 1.65934
\(36\) −1.67282 −0.278804
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0.143987 0.0230563
\(40\) −5.61515 −0.887833
\(41\) 5.34565 0.834850 0.417425 − 0.908711i \(-0.362933\pi\)
0.417425 + 0.908711i \(0.362933\pi\)
\(42\) 2.10083 0.324165
\(43\) 2.81681 0.429560 0.214780 − 0.976663i \(-0.431097\pi\)
0.214780 + 0.976663i \(0.431097\pi\)
\(44\) 6.38485 0.962552
\(45\) −2.67282 −0.398441
\(46\) −4.30644 −0.634951
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 2.14399 0.309458
\(49\) 6.48963 0.927091
\(50\) −1.22635 −0.173431
\(51\) 0 0
\(52\) −0.240864 −0.0334018
\(53\) 8.01847 1.10142 0.550711 − 0.834696i \(-0.314357\pi\)
0.550711 + 0.834696i \(0.314357\pi\)
\(54\) −0.571993 −0.0778384
\(55\) 10.2017 1.37559
\(56\) −7.71598 −1.03109
\(57\) 0 0
\(58\) 3.05767 0.401492
\(59\) 3.81681 0.496906 0.248453 − 0.968644i \(-0.420078\pi\)
0.248453 + 0.968644i \(0.420078\pi\)
\(60\) 4.47116 0.577225
\(61\) 11.4896 1.47110 0.735548 − 0.677472i \(-0.236925\pi\)
0.735548 + 0.677472i \(0.236925\pi\)
\(62\) −5.04316 −0.640481
\(63\) −3.67282 −0.462732
\(64\) −1.18319 −0.147899
\(65\) −0.384851 −0.0477348
\(66\) 2.18319 0.268732
\(67\) 5.38485 0.657864 0.328932 − 0.944354i \(-0.393311\pi\)
0.328932 + 0.944354i \(0.393311\pi\)
\(68\) 0 0
\(69\) 7.52884 0.906365
\(70\) −5.61515 −0.671139
\(71\) −13.6336 −1.61801 −0.809007 − 0.587800i \(-0.799994\pi\)
−0.809007 + 0.587800i \(0.799994\pi\)
\(72\) 2.10083 0.247585
\(73\) −0.345647 −0.0404550 −0.0202275 − 0.999795i \(-0.506439\pi\)
−0.0202275 + 0.999795i \(0.506439\pi\)
\(74\) 0.571993 0.0664929
\(75\) 2.14399 0.247566
\(76\) 0 0
\(77\) 14.0185 1.59755
\(78\) −0.0823593 −0.00932536
\(79\) 6.52884 0.734552 0.367276 − 0.930112i \(-0.380291\pi\)
0.367276 + 0.930112i \(0.380291\pi\)
\(80\) −5.73050 −0.640689
\(81\) 1.00000 0.111111
\(82\) −3.05767 −0.337664
\(83\) −2.28797 −0.251138 −0.125569 − 0.992085i \(-0.540076\pi\)
−0.125569 + 0.992085i \(0.540076\pi\)
\(84\) 6.14399 0.670364
\(85\) 0 0
\(86\) −1.61120 −0.173740
\(87\) −5.34565 −0.573114
\(88\) −8.01847 −0.854772
\(89\) −8.67282 −0.919317 −0.459659 − 0.888096i \(-0.652028\pi\)
−0.459659 + 0.888096i \(0.652028\pi\)
\(90\) 1.52884 0.161154
\(91\) −0.528837 −0.0554372
\(92\) −12.5944 −1.31306
\(93\) 8.81681 0.914261
\(94\) 3.43196 0.353980
\(95\) 0 0
\(96\) −5.42801 −0.553994
\(97\) −5.91369 −0.600444 −0.300222 − 0.953869i \(-0.597061\pi\)
−0.300222 + 0.953869i \(0.597061\pi\)
\(98\) −3.71203 −0.374971
\(99\) −3.81681 −0.383604
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.l.1.2 3
3.2 odd 2 3249.2.a.y.1.2 3
19.7 even 3 57.2.e.b.49.2 yes 6
19.11 even 3 57.2.e.b.7.2 ✓ 6
19.18 odd 2 1083.2.a.o.1.2 3
57.11 odd 6 171.2.f.b.64.2 6
57.26 odd 6 171.2.f.b.163.2 6
57.56 even 2 3249.2.a.t.1.2 3
76.7 odd 6 912.2.q.l.49.3 6
76.11 odd 6 912.2.q.l.577.3 6
228.11 even 6 2736.2.s.z.577.1 6
228.83 even 6 2736.2.s.z.1873.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.2 ✓ 6 19.11 even 3
57.2.e.b.49.2 yes 6 19.7 even 3
171.2.f.b.64.2 6 57.11 odd 6
171.2.f.b.163.2 6 57.26 odd 6
912.2.q.l.49.3 6 76.7 odd 6
912.2.q.l.577.3 6 76.11 odd 6
1083.2.a.l.1.2 3 1.1 even 1 trivial
1083.2.a.o.1.2 3 19.18 odd 2
2736.2.s.z.577.1 6 228.11 even 6
2736.2.s.z.1873.1 6 228.83 even 6
3249.2.a.t.1.2 3 57.56 even 2
3249.2.a.y.1.2 3 3.2 odd 2