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Properties

Label 3969.2.a.l.1.2
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
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] = [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 = [3,-3,0,3,3,0,0,-6,0,0,-6] 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: \(3\)
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 63)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) 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)
 
\(f(q)\) \(=\) \(q-1.34730 q^{2} -0.184793 q^{4} +2.53209 q^{5} +2.94356 q^{8} -3.41147 q^{10} -0.467911 q^{11} -5.82295 q^{13} -3.59627 q^{16} +3.87939 q^{17} +2.18479 q^{19} -0.467911 q^{20} +0.630415 q^{22} +0.106067 q^{23} +1.41147 q^{25} +7.84524 q^{26} -8.78106 q^{29} +7.68004 q^{31} -1.04189 q^{32} -5.22668 q^{34} -7.68004 q^{37} -2.94356 q^{38} +7.45336 q^{40} -2.22668 q^{41} +1.22668 q^{43} +0.0864665 q^{44} -0.142903 q^{46} -5.33275 q^{47} -1.90167 q^{50} +1.07604 q^{52} +0.716881 q^{53} -1.18479 q^{55} +11.8307 q^{58} +0.736482 q^{59} -0.958111 q^{61} -10.3473 q^{62} +8.59627 q^{64} -14.7442 q^{65} -9.63816 q^{67} -0.716881 q^{68} -13.2344 q^{71} +10.2686 q^{73} +10.3473 q^{74} -0.403733 q^{76} -12.6382 q^{79} -9.10607 q^{80} +3.00000 q^{82} -2.73143 q^{83} +9.82295 q^{85} -1.65270 q^{86} -1.37733 q^{88} -8.11381 q^{89} -0.0196004 q^{92} +7.18479 q^{94} +5.53209 q^{95} +13.6040 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8} - 6 q^{11} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} + 9 q^{22} - 12 q^{23} - 6 q^{25} - 3 q^{26} - 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} + 6 q^{38}+ \cdots + 3 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34730 −0.952682 −0.476341 − 0.879261i \(-0.658037\pi\)
−0.476341 + 0.879261i \(0.658037\pi\)
\(3\) 0 0
\(4\) −0.184793 −0.0923963
\(5\) 2.53209 1.13238 0.566192 − 0.824273i \(-0.308416\pi\)
0.566192 + 0.824273i \(0.308416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.94356 1.04071
\(9\) 0 0
\(10\) −3.41147 −1.07880
\(11\) −0.467911 −0.141081 −0.0705403 − 0.997509i \(-0.522472\pi\)
−0.0705403 + 0.997509i \(0.522472\pi\)
\(12\) 0 0
\(13\) −5.82295 −1.61500 −0.807498 − 0.589871i \(-0.799179\pi\)
−0.807498 + 0.589871i \(0.799179\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.59627 −0.899067
\(17\) 3.87939 0.940889 0.470445 − 0.882430i \(-0.344094\pi\)
0.470445 + 0.882430i \(0.344094\pi\)
\(18\) 0 0
\(19\) 2.18479 0.501226 0.250613 − 0.968087i \(-0.419368\pi\)
0.250613 + 0.968087i \(0.419368\pi\)
\(20\) −0.467911 −0.104628
\(21\) 0 0
\(22\) 0.630415 0.134405
\(23\) 0.106067 0.0221165 0.0110582 − 0.999939i \(-0.496480\pi\)
0.0110582 + 0.999939i \(0.496480\pi\)
\(24\) 0 0
\(25\) 1.41147 0.282295
\(26\) 7.84524 1.53858
\(27\) 0 0
\(28\) 0 0
\(29\) −8.78106 −1.63060 −0.815301 − 0.579038i \(-0.803428\pi\)
−0.815301 + 0.579038i \(0.803428\pi\)
\(30\) 0 0
\(31\) 7.68004 1.37938 0.689688 − 0.724106i \(-0.257748\pi\)
0.689688 + 0.724106i \(0.257748\pi\)
\(32\) −1.04189 −0.184182
\(33\) 0 0
\(34\) −5.22668 −0.896368
\(35\) 0 0
\(36\) 0 0
\(37\) −7.68004 −1.26259 −0.631296 − 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) −2.94356 −0.477509
\(39\) 0 0
\(40\) 7.45336 1.17848
\(41\) −2.22668 −0.347749 −0.173875 − 0.984768i \(-0.555629\pi\)
−0.173875 + 0.984768i \(0.555629\pi\)
\(42\) 0 0
\(43\) 1.22668 0.187067 0.0935336 − 0.995616i \(-0.470184\pi\)
0.0935336 + 0.995616i \(0.470184\pi\)
\(44\) 0.0864665 0.0130353
\(45\) 0 0
\(46\) −0.142903 −0.0210700
\(47\) −5.33275 −0.777861 −0.388931 − 0.921267i \(-0.627155\pi\)
−0.388931 + 0.921267i \(0.627155\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.90167 −0.268937
\(51\) 0 0
\(52\) 1.07604 0.149220
\(53\) 0.716881 0.0984712 0.0492356 − 0.998787i \(-0.484321\pi\)
0.0492356 + 0.998787i \(0.484321\pi\)
\(54\) 0 0
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) 0 0
\(58\) 11.8307 1.55345
\(59\) 0.736482 0.0958818 0.0479409 − 0.998850i \(-0.484734\pi\)
0.0479409 + 0.998850i \(0.484734\pi\)
\(60\) 0 0
\(61\) −0.958111 −0.122674 −0.0613368 − 0.998117i \(-0.519536\pi\)
−0.0613368 + 0.998117i \(0.519536\pi\)
\(62\) −10.3473 −1.31411
\(63\) 0 0
\(64\) 8.59627 1.07453
\(65\) −14.7442 −1.82880
\(66\) 0 0
\(67\) −9.63816 −1.17749 −0.588744 − 0.808320i \(-0.700377\pi\)
−0.588744 + 0.808320i \(0.700377\pi\)
\(68\) −0.716881 −0.0869346
\(69\) 0 0
\(70\) 0 0
\(71\) −13.2344 −1.57064 −0.785318 − 0.619092i \(-0.787501\pi\)
−0.785318 + 0.619092i \(0.787501\pi\)
\(72\) 0 0
\(73\) 10.2686 1.20185 0.600923 − 0.799307i \(-0.294800\pi\)
0.600923 + 0.799307i \(0.294800\pi\)
\(74\) 10.3473 1.20285
\(75\) 0 0
\(76\) −0.403733 −0.0463114
\(77\) 0 0
\(78\) 0 0
\(79\) −12.6382 −1.42190 −0.710952 − 0.703241i \(-0.751736\pi\)
−0.710952 + 0.703241i \(0.751736\pi\)
\(80\) −9.10607 −1.01809
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) −2.73143 −0.299813 −0.149907 − 0.988700i \(-0.547897\pi\)
−0.149907 + 0.988700i \(0.547897\pi\)
\(84\) 0 0
\(85\) 9.82295 1.06545
\(86\) −1.65270 −0.178216
\(87\) 0 0
\(88\) −1.37733 −0.146823
\(89\) −8.11381 −0.860062 −0.430031 − 0.902814i \(-0.641497\pi\)
−0.430031 + 0.902814i \(0.641497\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.0196004 −0.00204348
\(93\) 0 0
\(94\) 7.18479 0.741055
\(95\) 5.53209 0.567580
\(96\) 0 0
\(97\) 13.6040 1.38128 0.690639 − 0.723200i \(-0.257329\pi\)
0.690639 + 0.723200i \(0.257329\pi\)
\(98\) 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.l.1.2 3
3.2 odd 2 3969.2.a.q.1.2 3
7.6 odd 2 567.2.a.c.1.2 3
9.2 odd 6 441.2.f.c.148.2 6
9.4 even 3 1323.2.f.d.883.2 6
9.5 odd 6 441.2.f.c.295.2 6
9.7 even 3 1323.2.f.d.442.2 6
21.20 even 2 567.2.a.h.1.2 3
28.27 even 2 9072.2.a.bs.1.1 3
63.2 odd 6 441.2.g.b.67.2 6
63.4 even 3 1323.2.g.e.667.2 6
63.5 even 6 441.2.h.d.214.2 6
63.11 odd 6 441.2.h.e.373.2 6
63.13 odd 6 189.2.f.b.127.2 6
63.16 even 3 1323.2.g.e.361.2 6
63.20 even 6 63.2.f.a.22.2 ✓ 6
63.23 odd 6 441.2.h.e.214.2 6
63.25 even 3 1323.2.h.b.226.2 6
63.31 odd 6 1323.2.g.d.667.2 6
63.32 odd 6 441.2.g.b.79.2 6
63.34 odd 6 189.2.f.b.64.2 6
63.38 even 6 441.2.h.d.373.2 6
63.40 odd 6 1323.2.h.c.802.2 6
63.41 even 6 63.2.f.a.43.2 yes 6
63.47 even 6 441.2.g.c.67.2 6
63.52 odd 6 1323.2.h.c.226.2 6
63.58 even 3 1323.2.h.b.802.2 6
63.59 even 6 441.2.g.c.79.2 6
63.61 odd 6 1323.2.g.d.361.2 6
84.83 odd 2 9072.2.a.ca.1.3 3
252.83 odd 6 1008.2.r.h.337.1 6
252.139 even 6 3024.2.r.k.2017.3 6
252.167 odd 6 1008.2.r.h.673.1 6
252.223 even 6 3024.2.r.k.1009.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 63.20 even 6
63.2.f.a.43.2 yes 6 63.41 even 6
189.2.f.b.64.2 6 63.34 odd 6
189.2.f.b.127.2 6 63.13 odd 6
441.2.f.c.148.2 6 9.2 odd 6
441.2.f.c.295.2 6 9.5 odd 6
441.2.g.b.67.2 6 63.2 odd 6
441.2.g.b.79.2 6 63.32 odd 6
441.2.g.c.67.2 6 63.47 even 6
441.2.g.c.79.2 6 63.59 even 6
441.2.h.d.214.2 6 63.5 even 6
441.2.h.d.373.2 6 63.38 even 6
441.2.h.e.214.2 6 63.23 odd 6
441.2.h.e.373.2 6 63.11 odd 6
567.2.a.c.1.2 3 7.6 odd 2
567.2.a.h.1.2 3 21.20 even 2
1008.2.r.h.337.1 6 252.83 odd 6
1008.2.r.h.673.1 6 252.167 odd 6
1323.2.f.d.442.2 6 9.7 even 3
1323.2.f.d.883.2 6 9.4 even 3
1323.2.g.d.361.2 6 63.61 odd 6
1323.2.g.d.667.2 6 63.31 odd 6
1323.2.g.e.361.2 6 63.16 even 3
1323.2.g.e.667.2 6 63.4 even 3
1323.2.h.b.226.2 6 63.25 even 3
1323.2.h.b.802.2 6 63.58 even 3
1323.2.h.c.226.2 6 63.52 odd 6
1323.2.h.c.802.2 6 63.40 odd 6
3024.2.r.k.1009.3 6 252.223 even 6
3024.2.r.k.2017.3 6 252.139 even 6
3969.2.a.l.1.2 3 1.1 even 1 trivial
3969.2.a.q.1.2 3 3.2 odd 2
9072.2.a.bs.1.1 3 28.27 even 2
9072.2.a.ca.1.3 3 84.83 odd 2