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Properties

Label 1083.2.a.l.1.1
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 1.1
Root \(2.51414\) 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)
 
\(f(q)\) \(=\) \(q-2.51414 q^{2} +1.00000 q^{3} +4.32088 q^{4} +3.32088 q^{5} -2.51414 q^{6} +2.32088 q^{7} -5.83502 q^{8} +1.00000 q^{9} -8.34916 q^{10} -1.70739 q^{11} +4.32088 q^{12} +4.02827 q^{13} -5.83502 q^{14} +3.32088 q^{15} +6.02827 q^{16} -2.51414 q^{18} +14.3492 q^{20} +2.32088 q^{21} +4.29261 q^{22} -2.34916 q^{23} -5.83502 q^{24} +6.02827 q^{25} -10.1276 q^{26} +1.00000 q^{27} +10.0283 q^{28} +6.64177 q^{29} -8.34916 q^{30} +6.70739 q^{31} -3.48586 q^{32} -1.70739 q^{33} +7.70739 q^{35} +4.32088 q^{36} -1.00000 q^{37} +4.02827 q^{39} -19.3774 q^{40} -6.64177 q^{41} -5.83502 q^{42} +0.707389 q^{43} -7.37743 q^{44} +3.32088 q^{45} +5.90611 q^{46} -6.00000 q^{47} +6.02827 q^{48} -1.61350 q^{49} -15.1559 q^{50} +17.4057 q^{52} -9.96265 q^{53} -2.51414 q^{54} -5.67004 q^{55} -13.5424 q^{56} -16.6983 q^{58} +1.70739 q^{59} +14.3492 q^{60} +3.38650 q^{61} -16.8633 q^{62} +2.32088 q^{63} -3.29261 q^{64} +13.3774 q^{65} +4.29261 q^{66} -8.37743 q^{67} -2.34916 q^{69} -19.3774 q^{70} -9.41478 q^{71} -5.83502 q^{72} +11.6418 q^{73} +2.51414 q^{74} +6.02827 q^{75} -3.96265 q^{77} -10.1276 q^{78} -3.34916 q^{79} +20.0192 q^{80} +1.00000 q^{81} +16.6983 q^{82} -10.0565 q^{83} +10.0283 q^{84} -1.77847 q^{86} +6.64177 q^{87} +9.96265 q^{88} -2.67912 q^{89} -8.34916 q^{90} +9.34916 q^{91} -10.1504 q^{92} +6.70739 q^{93} +15.0848 q^{94} -3.48586 q^{96} +17.7266 q^{97} +4.05655 q^{98} -1.70739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{5} - q^{6} - q^{7} - 3 q^{8} + 3 q^{9} - 4 q^{10} + 5 q^{12} - q^{13} - 3 q^{14} + 2 q^{15} + 5 q^{16} - q^{18} + 22 q^{20} - q^{21} + 18 q^{22} + 14 q^{23}+ \cdots - 14 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.51414 −1.77776 −0.888882 − 0.458137i \(-0.848517\pi\)
−0.888882 + 0.458137i \(0.848517\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.32088 2.16044
\(5\) 3.32088 1.48514 0.742572 − 0.669766i \(-0.233606\pi\)
0.742572 + 0.669766i \(0.233606\pi\)
\(6\) −2.51414 −1.02639
\(7\) 2.32088 0.877212 0.438606 − 0.898679i \(-0.355472\pi\)
0.438606 + 0.898679i \(0.355472\pi\)
\(8\) −5.83502 −2.06299
\(9\) 1.00000 0.333333
\(10\) −8.34916 −2.64024
\(11\) −1.70739 −0.514797 −0.257399 − 0.966305i \(-0.582865\pi\)
−0.257399 + 0.966305i \(0.582865\pi\)
\(12\) 4.32088 1.24733
\(13\) 4.02827 1.11724 0.558621 − 0.829423i \(-0.311331\pi\)
0.558621 + 0.829423i \(0.311331\pi\)
\(14\) −5.83502 −1.55948
\(15\) 3.32088 0.857449
\(16\) 6.02827 1.50707
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −2.51414 −0.592588
\(19\) 0 0
\(20\) 14.3492 3.20857
\(21\) 2.32088 0.506459
\(22\) 4.29261 0.915188
\(23\) −2.34916 −0.489833 −0.244917 − 0.969544i \(-0.578761\pi\)
−0.244917 + 0.969544i \(0.578761\pi\)
\(24\) −5.83502 −1.19107
\(25\) 6.02827 1.20565
\(26\) −10.1276 −1.98619
\(27\) 1.00000 0.192450
\(28\) 10.0283 1.89517
\(29\) 6.64177 1.23335 0.616673 − 0.787220i \(-0.288480\pi\)
0.616673 + 0.787220i \(0.288480\pi\)
\(30\) −8.34916 −1.52434
\(31\) 6.70739 1.20468 0.602341 − 0.798239i \(-0.294235\pi\)
0.602341 + 0.798239i \(0.294235\pi\)
\(32\) −3.48586 −0.616219
\(33\) −1.70739 −0.297218
\(34\) 0 0
\(35\) 7.70739 1.30279
\(36\) 4.32088 0.720147
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 4.02827 0.645040
\(40\) −19.3774 −3.06384
\(41\) −6.64177 −1.03727 −0.518635 − 0.854996i \(-0.673560\pi\)
−0.518635 + 0.854996i \(0.673560\pi\)
\(42\) −5.83502 −0.900363
\(43\) 0.707389 0.107876 0.0539379 − 0.998544i \(-0.482823\pi\)
0.0539379 + 0.998544i \(0.482823\pi\)
\(44\) −7.37743 −1.11219
\(45\) 3.32088 0.495048
\(46\) 5.90611 0.870808
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 6.02827 0.870106
\(49\) −1.61350 −0.230499
\(50\) −15.1559 −2.14337
\(51\) 0 0
\(52\) 17.4057 2.41374
\(53\) −9.96265 −1.36848 −0.684238 − 0.729259i \(-0.739865\pi\)
−0.684238 + 0.729259i \(0.739865\pi\)
\(54\) −2.51414 −0.342131
\(55\) −5.67004 −0.764548
\(56\) −13.5424 −1.80968
\(57\) 0 0
\(58\) −16.6983 −2.19260
\(59\) 1.70739 0.222283 0.111142 − 0.993805i \(-0.464549\pi\)
0.111142 + 0.993805i \(0.464549\pi\)
\(60\) 14.3492 1.85247
\(61\) 3.38650 0.433598 0.216799 − 0.976216i \(-0.430438\pi\)
0.216799 + 0.976216i \(0.430438\pi\)
\(62\) −16.8633 −2.14164
\(63\) 2.32088 0.292404
\(64\) −3.29261 −0.411576
\(65\) 13.3774 1.65927
\(66\) 4.29261 0.528384
\(67\) −8.37743 −1.02347 −0.511733 − 0.859144i \(-0.670996\pi\)
−0.511733 + 0.859144i \(0.670996\pi\)
\(68\) 0 0
\(69\) −2.34916 −0.282805
\(70\) −19.3774 −2.31605
\(71\) −9.41478 −1.11733 −0.558664 − 0.829394i \(-0.688686\pi\)
−0.558664 + 0.829394i \(0.688686\pi\)
\(72\) −5.83502 −0.687664
\(73\) 11.6418 1.36257 0.681283 − 0.732020i \(-0.261422\pi\)
0.681283 + 0.732020i \(0.261422\pi\)
\(74\) 2.51414 0.292262
\(75\) 6.02827 0.696085
\(76\) 0 0
\(77\) −3.96265 −0.451586
\(78\) −10.1276 −1.14673
\(79\) −3.34916 −0.376810 −0.188405 − 0.982091i \(-0.560332\pi\)
−0.188405 + 0.982091i \(0.560332\pi\)
\(80\) 20.0192 2.23821
\(81\) 1.00000 0.111111
\(82\) 16.6983 1.84402
\(83\) −10.0565 −1.10385 −0.551925 − 0.833894i \(-0.686106\pi\)
−0.551925 + 0.833894i \(0.686106\pi\)
\(84\) 10.0283 1.09417
\(85\) 0 0
\(86\) −1.77847 −0.191778
\(87\) 6.64177 0.712072
\(88\) 9.96265 1.06202
\(89\) −2.67912 −0.283986 −0.141993 − 0.989868i \(-0.545351\pi\)
−0.141993 + 0.989868i \(0.545351\pi\)
\(90\) −8.34916 −0.880079
\(91\) 9.34916 0.980058
\(92\) −10.1504 −1.05826
\(93\) 6.70739 0.695524
\(94\) 15.0848 1.55588
\(95\) 0 0
\(96\) −3.48586 −0.355774
\(97\) 17.7266 1.79986 0.899931 − 0.436032i \(-0.143616\pi\)
0.899931 + 0.436032i \(0.143616\pi\)
\(98\) 4.05655 0.409773
\(99\) −1.70739 −0.171599
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.1.1 3
3.2 odd 2 3249.2.a.y.1.3 3
19.7 even 3 57.2.e.b.49.3 yes 6
19.11 even 3 57.2.e.b.7.3 ✓ 6
19.18 odd 2 1083.2.a.o.1.3 3
57.11 odd 6 171.2.f.b.64.1 6
57.26 odd 6 171.2.f.b.163.1 6
57.56 even 2 3249.2.a.t.1.1 3
76.7 odd 6 912.2.q.l.49.1 6
76.11 odd 6 912.2.q.l.577.1 6
228.11 even 6 2736.2.s.z.577.3 6
228.83 even 6 2736.2.s.z.1873.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.3 ✓ 6 19.11 even 3
57.2.e.b.49.3 yes 6 19.7 even 3
171.2.f.b.64.1 6 57.11 odd 6
171.2.f.b.163.1 6 57.26 odd 6
912.2.q.l.49.1 6 76.7 odd 6
912.2.q.l.577.1 6 76.11 odd 6
1083.2.a.l.1.1 3 1.1 even 1 trivial
1083.2.a.o.1.3 3 19.18 odd 2
2736.2.s.z.577.3 6 228.11 even 6
2736.2.s.z.1873.3 6 228.83 even 6
3249.2.a.t.1.1 3 57.56 even 2
3249.2.a.y.1.3 3 3.2 odd 2