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Properties

Label 3969.2.a.u.1.3
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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] = [3969,2,Mod(1,3969)] 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("3969.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3969, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,0,0,0,0,-8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 567)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.18890\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.456850 q^{2} -1.79129 q^{4} -4.37780 q^{5} -1.73205 q^{8} -2.00000 q^{10} +2.64575 q^{11} -4.00000 q^{13} +2.79129 q^{16} +3.46410 q^{17} +3.58258 q^{19} +7.84190 q^{20} +1.20871 q^{22} +3.46410 q^{23} +14.1652 q^{25} -1.82740 q^{26} -1.82740 q^{29} -9.16515 q^{31} +4.73930 q^{32} +1.58258 q^{34} +3.00000 q^{37} +1.63670 q^{38} +7.58258 q^{40} +4.37780 q^{41} -8.58258 q^{43} -4.73930 q^{44} +1.58258 q^{46} +2.74110 q^{47} +6.47135 q^{50} +7.16515 q^{52} +8.66025 q^{53} -11.5826 q^{55} -0.834849 q^{58} +3.46410 q^{59} -2.41742 q^{61} -4.18710 q^{62} -3.41742 q^{64} +17.5112 q^{65} -0.582576 q^{67} -6.20520 q^{68} -11.4014 q^{71} +3.16515 q^{73} +1.37055 q^{74} -6.41742 q^{76} -8.58258 q^{79} -12.2197 q^{80} +2.00000 q^{82} +6.20520 q^{83} -15.1652 q^{85} -3.92095 q^{86} -4.58258 q^{88} +8.75560 q^{89} -6.20520 q^{92} +1.25227 q^{94} -15.6838 q^{95} -7.58258 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 8 q^{10} - 16 q^{13} + 2 q^{16} - 4 q^{19} + 14 q^{22} + 20 q^{25} - 12 q^{34} + 12 q^{37} + 12 q^{40} - 16 q^{43} - 12 q^{46} - 8 q^{52} - 28 q^{55} - 40 q^{58} - 28 q^{61} - 32 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.456850 0.323042 0.161521 − 0.986869i \(-0.448360\pi\)
0.161521 + 0.986869i \(0.448360\pi\)
\(3\) 0 0
\(4\) −1.79129 −0.895644
\(5\) −4.37780 −1.95781 −0.978906 − 0.204310i \(-0.934505\pi\)
−0.978906 + 0.204310i \(0.934505\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 2.64575 0.797724 0.398862 − 0.917011i \(-0.369405\pi\)
0.398862 + 0.917011i \(0.369405\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.79129 0.697822
\(17\) 3.46410 0.840168 0.420084 − 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 3.58258 0.821899 0.410950 − 0.911658i \(-0.365197\pi\)
0.410950 + 0.911658i \(0.365197\pi\)
\(20\) 7.84190 1.75350
\(21\) 0 0
\(22\) 1.20871 0.257698
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 14.1652 2.83303
\(26\) −1.82740 −0.358383
\(27\) 0 0
\(28\) 0 0
\(29\) −1.82740 −0.339340 −0.169670 − 0.985501i \(-0.554270\pi\)
−0.169670 + 0.985501i \(0.554270\pi\)
\(30\) 0 0
\(31\) −9.16515 −1.64611 −0.823055 − 0.567962i \(-0.807732\pi\)
−0.823055 + 0.567962i \(0.807732\pi\)
\(32\) 4.73930 0.837798
\(33\) 0 0
\(34\) 1.58258 0.271409
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 1.63670 0.265508
\(39\) 0 0
\(40\) 7.58258 1.19891
\(41\) 4.37780 0.683698 0.341849 − 0.939755i \(-0.388947\pi\)
0.341849 + 0.939755i \(0.388947\pi\)
\(42\) 0 0
\(43\) −8.58258 −1.30883 −0.654415 − 0.756135i \(-0.727085\pi\)
−0.654415 + 0.756135i \(0.727085\pi\)
\(44\) −4.73930 −0.714477
\(45\) 0 0
\(46\) 1.58258 0.233338
\(47\) 2.74110 0.399831 0.199915 − 0.979813i \(-0.435933\pi\)
0.199915 + 0.979813i \(0.435933\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 6.47135 0.915188
\(51\) 0 0
\(52\) 7.16515 0.993628
\(53\) 8.66025 1.18958 0.594789 − 0.803882i \(-0.297236\pi\)
0.594789 + 0.803882i \(0.297236\pi\)
\(54\) 0 0
\(55\) −11.5826 −1.56179
\(56\) 0 0
\(57\) 0 0
\(58\) −0.834849 −0.109621
\(59\) 3.46410 0.450988 0.225494 − 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 0 0
\(61\) −2.41742 −0.309519 −0.154760 − 0.987952i \(-0.549460\pi\)
−0.154760 + 0.987952i \(0.549460\pi\)
\(62\) −4.18710 −0.531762
\(63\) 0 0
\(64\) −3.41742 −0.427178
\(65\) 17.5112 2.17200
\(66\) 0 0
\(67\) −0.582576 −0.0711729 −0.0355865 − 0.999367i \(-0.511330\pi\)
−0.0355865 + 0.999367i \(0.511330\pi\)
\(68\) −6.20520 −0.752491
\(69\) 0 0
\(70\) 0 0
\(71\) −11.4014 −1.35309 −0.676546 − 0.736400i \(-0.736524\pi\)
−0.676546 + 0.736400i \(0.736524\pi\)
\(72\) 0 0
\(73\) 3.16515 0.370453 0.185226 − 0.982696i \(-0.440698\pi\)
0.185226 + 0.982696i \(0.440698\pi\)
\(74\) 1.37055 0.159323
\(75\) 0 0
\(76\) −6.41742 −0.736129
\(77\) 0 0
\(78\) 0 0
\(79\) −8.58258 −0.965615 −0.482808 − 0.875726i \(-0.660383\pi\)
−0.482808 + 0.875726i \(0.660383\pi\)
\(80\) −12.2197 −1.36620
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) 6.20520 0.681110 0.340555 − 0.940225i \(-0.389385\pi\)
0.340555 + 0.940225i \(0.389385\pi\)
\(84\) 0 0
\(85\) −15.1652 −1.64489
\(86\) −3.92095 −0.422807
\(87\) 0 0
\(88\) −4.58258 −0.488504
\(89\) 8.75560 0.928092 0.464046 − 0.885811i \(-0.346397\pi\)
0.464046 + 0.885811i \(0.346397\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.20520 −0.646937
\(93\) 0 0
\(94\) 1.25227 0.129162
\(95\) −15.6838 −1.60912
\(96\) 0 0
\(97\) −7.58258 −0.769894 −0.384947 − 0.922939i \(-0.625780\pi\)
−0.384947 + 0.922939i \(0.625780\pi\)
\(98\) 0 0
\(99\) 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 3969.2.a.u.1.3 4
3.2 odd 2 inner 3969.2.a.u.1.2 4
7.6 odd 2 567.2.a.i.1.3 yes 4
21.20 even 2 567.2.a.i.1.2 ✓ 4
28.27 even 2 9072.2.a.ci.1.4 4
63.13 odd 6 567.2.f.n.379.2 8
63.20 even 6 567.2.f.n.190.3 8
63.34 odd 6 567.2.f.n.190.2 8
63.41 even 6 567.2.f.n.379.3 8
84.83 odd 2 9072.2.a.ci.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.i.1.2 ✓ 4 21.20 even 2
567.2.a.i.1.3 yes 4 7.6 odd 2
567.2.f.n.190.2 8 63.34 odd 6
567.2.f.n.190.3 8 63.20 even 6
567.2.f.n.379.2 8 63.13 odd 6
567.2.f.n.379.3 8 63.41 even 6
3969.2.a.u.1.2 4 3.2 odd 2 inner
3969.2.a.u.1.3 4 1.1 even 1 trivial
9072.2.a.ci.1.1 4 84.83 odd 2
9072.2.a.ci.1.4 4 28.27 even 2