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Properties

Label 3249.2.a.y.1.3
Level $3249$
Weight $2$
Character 3249.1
Self dual yes
Analytic conductor $25.943$
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] = [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 = [3,1,0,5,-2,0,-1,3,0,-4,0,0,-1,3,0,5,0,0,0,-22,0,18,-14,0,5,21, 0,17,-4,0,15,17,0,0,-18,0,-3,0,0,-24,4,0,-3,-12,0,20,18,0,-2,23,0,5,6, 0,12,21,0,-8,0,0,13,23,0,-15,-6,0,9,0,0,-24,18,0,19,-1,0,0,-12,0,11,-10, 0,8,4,0,0,17,0,6,16,0,7] 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: 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 1.3
Root \(2.51414\) 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)
 
\(f(q)\) \(=\) \(q+2.51414 q^{2} +4.32088 q^{4} -3.32088 q^{5} +2.32088 q^{7} +5.83502 q^{8} -8.34916 q^{10} +1.70739 q^{11} +4.02827 q^{13} +5.83502 q^{14} +6.02827 q^{16} -14.3492 q^{20} +4.29261 q^{22} +2.34916 q^{23} +6.02827 q^{25} +10.1276 q^{26} +10.0283 q^{28} -6.64177 q^{29} +6.70739 q^{31} +3.48586 q^{32} -7.70739 q^{35} -1.00000 q^{37} -19.3774 q^{40} +6.64177 q^{41} +0.707389 q^{43} +7.37743 q^{44} +5.90611 q^{46} +6.00000 q^{47} -1.61350 q^{49} +15.1559 q^{50} +17.4057 q^{52} +9.96265 q^{53} -5.67004 q^{55} +13.5424 q^{56} -16.6983 q^{58} -1.70739 q^{59} +3.38650 q^{61} +16.8633 q^{62} -3.29261 q^{64} -13.3774 q^{65} -8.37743 q^{67} -19.3774 q^{70} +9.41478 q^{71} +11.6418 q^{73} -2.51414 q^{74} +3.96265 q^{77} -3.34916 q^{79} -20.0192 q^{80} +16.6983 q^{82} +10.0565 q^{83} +1.77847 q^{86} +9.96265 q^{88} +2.67912 q^{89} +9.34916 q^{91} +10.1504 q^{92} +15.0848 q^{94} +17.7266 q^{97} -4.05655 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} - 2 q^{5} - q^{7} + 3 q^{8} - 4 q^{10} - q^{13} + 3 q^{14} + 5 q^{16} - 22 q^{20} + 18 q^{22} - 14 q^{23} + 5 q^{25} + 21 q^{26} + 17 q^{28} - 4 q^{29} + 15 q^{31} + 17 q^{32} - 18 q^{35}+ \cdots + 14 q^{98}+O(q^{100}) \) 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 \(-0.151483\pi\)
0.888882 + 0.458137i \(0.151483\pi\)
\(3\) 0 0
\(4\) 4.32088 2.16044
\(5\) −3.32088 −1.48514 −0.742572 − 0.669766i \(-0.766394\pi\)
−0.742572 + 0.669766i \(0.766394\pi\)
\(6\) 0 0
\(7\) 2.32088 0.877212 0.438606 − 0.898679i \(-0.355472\pi\)
0.438606 + 0.898679i \(0.355472\pi\)
\(8\) 5.83502 2.06299
\(9\) 0 0
\(10\) −8.34916 −2.64024
\(11\) 1.70739 0.514797 0.257399 − 0.966305i \(-0.417135\pi\)
0.257399 + 0.966305i \(0.417135\pi\)
\(12\) 0 0
\(13\) 4.02827 1.11724 0.558621 − 0.829423i \(-0.311331\pi\)
0.558621 + 0.829423i \(0.311331\pi\)
\(14\) 5.83502 1.55948
\(15\) 0 0
\(16\) 6.02827 1.50707
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) −14.3492 −3.20857
\(21\) 0 0
\(22\) 4.29261 0.915188
\(23\) 2.34916 0.489833 0.244917 − 0.969544i \(-0.421239\pi\)
0.244917 + 0.969544i \(0.421239\pi\)
\(24\) 0 0
\(25\) 6.02827 1.20565
\(26\) 10.1276 1.98619
\(27\) 0 0
\(28\) 10.0283 1.89517
\(29\) −6.64177 −1.23335 −0.616673 − 0.787220i \(-0.711520\pi\)
−0.616673 + 0.787220i \(0.711520\pi\)
\(30\) 0 0
\(31\) 6.70739 1.20468 0.602341 − 0.798239i \(-0.294235\pi\)
0.602341 + 0.798239i \(0.294235\pi\)
\(32\) 3.48586 0.616219
\(33\) 0 0
\(34\) 0 0
\(35\) −7.70739 −1.30279
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −19.3774 −3.06384
\(41\) 6.64177 1.03727 0.518635 − 0.854996i \(-0.326440\pi\)
0.518635 + 0.854996i \(0.326440\pi\)
\(42\) 0 0
\(43\) 0.707389 0.107876 0.0539379 − 0.998544i \(-0.482823\pi\)
0.0539379 + 0.998544i \(0.482823\pi\)
\(44\) 7.37743 1.11219
\(45\) 0 0
\(46\) 5.90611 0.870808
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −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 \(-0.260135\pi\)
0.684238 + 0.729259i \(0.260135\pi\)
\(54\) 0 0
\(55\) −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 \(-0.535451\pi\)
−0.111142 + 0.993805i \(0.535451\pi\)
\(60\) 0 0
\(61\) 3.38650 0.433598 0.216799 − 0.976216i \(-0.430438\pi\)
0.216799 + 0.976216i \(0.430438\pi\)
\(62\) 16.8633 2.14164
\(63\) 0 0
\(64\) −3.29261 −0.411576
\(65\) −13.3774 −1.65927
\(66\) 0 0
\(67\) −8.37743 −1.02347 −0.511733 − 0.859144i \(-0.670996\pi\)
−0.511733 + 0.859144i \(0.670996\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −19.3774 −2.31605
\(71\) 9.41478 1.11733 0.558664 − 0.829394i \(-0.311314\pi\)
0.558664 + 0.829394i \(0.311314\pi\)
\(72\) 0 0
\(73\) 11.6418 1.36257 0.681283 − 0.732020i \(-0.261422\pi\)
0.681283 + 0.732020i \(0.261422\pi\)
\(74\) −2.51414 −0.292262
\(75\) 0 0
\(76\) 0 0
\(77\) 3.96265 0.451586
\(78\) 0 0
\(79\) −3.34916 −0.376810 −0.188405 − 0.982091i \(-0.560332\pi\)
−0.188405 + 0.982091i \(0.560332\pi\)
\(80\) −20.0192 −2.23821
\(81\) 0 0
\(82\) 16.6983 1.84402
\(83\) 10.0565 1.10385 0.551925 − 0.833894i \(-0.313894\pi\)
0.551925 + 0.833894i \(0.313894\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.77847 0.191778
\(87\) 0 0
\(88\) 9.96265 1.06202
\(89\) 2.67912 0.283986 0.141993 − 0.989868i \(-0.454649\pi\)
0.141993 + 0.989868i \(0.454649\pi\)
\(90\) 0 0
\(91\) 9.34916 0.980058
\(92\) 10.1504 1.05826
\(93\) 0 0
\(94\) 15.0848 1.55588
\(95\) 0 0
\(96\) 0 0
\(97\) 17.7266 1.79986 0.899931 − 0.436032i \(-0.143616\pi\)
0.899931 + 0.436032i \(0.143616\pi\)
\(98\) −4.05655 −0.409773
\(99\) 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.y.1.3 3
3.2 odd 2 1083.2.a.l.1.1 3
19.7 even 3 171.2.f.b.163.1 6
19.11 even 3 171.2.f.b.64.1 6
19.18 odd 2 3249.2.a.t.1.1 3
57.11 odd 6 57.2.e.b.7.3 ✓ 6
57.26 odd 6 57.2.e.b.49.3 yes 6
57.56 even 2 1083.2.a.o.1.3 3
76.7 odd 6 2736.2.s.z.1873.3 6
76.11 odd 6 2736.2.s.z.577.3 6
228.11 even 6 912.2.q.l.577.1 6
228.83 even 6 912.2.q.l.49.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.3 ✓ 6 57.11 odd 6
57.2.e.b.49.3 yes 6 57.26 odd 6
171.2.f.b.64.1 6 19.11 even 3
171.2.f.b.163.1 6 19.7 even 3
912.2.q.l.49.1 6 228.83 even 6
912.2.q.l.577.1 6 228.11 even 6
1083.2.a.l.1.1 3 3.2 odd 2
1083.2.a.o.1.3 3 57.56 even 2
2736.2.s.z.577.3 6 76.11 odd 6
2736.2.s.z.1873.3 6 76.7 odd 6
3249.2.a.t.1.1 3 19.18 odd 2
3249.2.a.y.1.3 3 1.1 even 1 trivial