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Properties

Label 3969.2.a.p.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 = [3,1,0,3,-5,0,0,6,0,0,2] 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: 3.3.321.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) 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 \(0.239123\) 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)
 
\(f(q)\) \(=\) \(q+0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -0.942820 q^{8} +0.282630 q^{10} +3.70370 q^{11} -1.00000 q^{13} +3.66019 q^{16} -6.94282 q^{17} -1.94282 q^{19} -2.29630 q^{20} +0.885640 q^{22} +5.60301 q^{23} -3.60301 q^{25} -0.239123 q^{26} -0.239123 q^{29} -1.66019 q^{31} +2.76088 q^{32} -1.66019 q^{34} -9.54583 q^{37} -0.464574 q^{38} -1.11436 q^{40} -10.1819 q^{41} +2.22545 q^{43} -7.19562 q^{44} +1.33981 q^{46} +5.82846 q^{47} -0.861564 q^{50} +1.94282 q^{52} +11.6030 q^{53} +4.37756 q^{55} -0.0571799 q^{58} +2.60301 q^{59} +7.60301 q^{61} -0.396990 q^{62} -6.66019 q^{64} -1.18194 q^{65} +3.50808 q^{67} +13.4887 q^{68} -8.60301 q^{71} -15.1488 q^{73} -2.28263 q^{74} +3.77455 q^{76} +7.37756 q^{79} +4.32614 q^{80} -2.43474 q^{82} -6.94282 q^{83} -8.20602 q^{85} +0.532157 q^{86} -3.49192 q^{88} +2.74720 q^{89} -10.8856 q^{92} +1.39372 q^{94} -2.29630 q^{95} -7.16827 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8} + 2 q^{11} - 3 q^{13} + 3 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} - 15 q^{22} + 6 q^{25} - q^{26} - q^{29} + 3 q^{31} + 8 q^{32} + 3 q^{34} - 3 q^{37} + 8 q^{38}+ \cdots - 3 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.239123 0.169086 0.0845428 − 0.996420i \(-0.473057\pi\)
0.0845428 + 0.996420i \(0.473057\pi\)
\(3\) 0 0
\(4\) −1.94282 −0.971410
\(5\) 1.18194 0.528581 0.264291 − 0.964443i \(-0.414862\pi\)
0.264291 + 0.964443i \(0.414862\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −0.942820 −0.333337
\(9\) 0 0
\(10\) 0.282630 0.0893755
\(11\) 3.70370 1.11671 0.558353 − 0.829603i \(-0.311433\pi\)
0.558353 + 0.829603i \(0.311433\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.66019 0.915047
\(17\) −6.94282 −1.68388 −0.841941 − 0.539570i \(-0.818587\pi\)
−0.841941 + 0.539570i \(0.818587\pi\)
\(18\) 0 0
\(19\) −1.94282 −0.445713 −0.222857 − 0.974851i \(-0.571538\pi\)
−0.222857 + 0.974851i \(0.571538\pi\)
\(20\) −2.29630 −0.513469
\(21\) 0 0
\(22\) 0.885640 0.188819
\(23\) 5.60301 1.16831 0.584154 − 0.811643i \(-0.301426\pi\)
0.584154 + 0.811643i \(0.301426\pi\)
\(24\) 0 0
\(25\) −3.60301 −0.720602
\(26\) −0.239123 −0.0468959
\(27\) 0 0
\(28\) 0 0
\(29\) −0.239123 −0.0444041 −0.0222020 − 0.999754i \(-0.507068\pi\)
−0.0222020 + 0.999754i \(0.507068\pi\)
\(30\) 0 0
\(31\) −1.66019 −0.298179 −0.149089 − 0.988824i \(-0.547634\pi\)
−0.149089 + 0.988824i \(0.547634\pi\)
\(32\) 2.76088 0.488059
\(33\) 0 0
\(34\) −1.66019 −0.284720
\(35\) 0 0
\(36\) 0 0
\(37\) −9.54583 −1.56932 −0.784662 − 0.619923i \(-0.787164\pi\)
−0.784662 + 0.619923i \(0.787164\pi\)
\(38\) −0.464574 −0.0753638
\(39\) 0 0
\(40\) −1.11436 −0.176196
\(41\) −10.1819 −1.59015 −0.795076 − 0.606510i \(-0.792569\pi\)
−0.795076 + 0.606510i \(0.792569\pi\)
\(42\) 0 0
\(43\) 2.22545 0.339378 0.169689 − 0.985498i \(-0.445724\pi\)
0.169689 + 0.985498i \(0.445724\pi\)
\(44\) −7.19562 −1.08478
\(45\) 0 0
\(46\) 1.33981 0.197544
\(47\) 5.82846 0.850168 0.425084 − 0.905154i \(-0.360245\pi\)
0.425084 + 0.905154i \(0.360245\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −0.861564 −0.121843
\(51\) 0 0
\(52\) 1.94282 0.269421
\(53\) 11.6030 1.59380 0.796898 − 0.604114i \(-0.206473\pi\)
0.796898 + 0.604114i \(0.206473\pi\)
\(54\) 0 0
\(55\) 4.37756 0.590270
\(56\) 0 0
\(57\) 0 0
\(58\) −0.0571799 −0.00750809
\(59\) 2.60301 0.338883 0.169442 − 0.985540i \(-0.445804\pi\)
0.169442 + 0.985540i \(0.445804\pi\)
\(60\) 0 0
\(61\) 7.60301 0.973466 0.486733 − 0.873551i \(-0.338189\pi\)
0.486733 + 0.873551i \(0.338189\pi\)
\(62\) −0.396990 −0.0504178
\(63\) 0 0
\(64\) −6.66019 −0.832524
\(65\) −1.18194 −0.146602
\(66\) 0 0
\(67\) 3.50808 0.428580 0.214290 − 0.976770i \(-0.431256\pi\)
0.214290 + 0.976770i \(0.431256\pi\)
\(68\) 13.4887 1.63574
\(69\) 0 0
\(70\) 0 0
\(71\) −8.60301 −1.02099 −0.510495 − 0.859881i \(-0.670538\pi\)
−0.510495 + 0.859881i \(0.670538\pi\)
\(72\) 0 0
\(73\) −15.1488 −1.77304 −0.886519 − 0.462693i \(-0.846883\pi\)
−0.886519 + 0.462693i \(0.846883\pi\)
\(74\) −2.28263 −0.265350
\(75\) 0 0
\(76\) 3.77455 0.432971
\(77\) 0 0
\(78\) 0 0
\(79\) 7.37756 0.830040 0.415020 − 0.909812i \(-0.363775\pi\)
0.415020 + 0.909812i \(0.363775\pi\)
\(80\) 4.32614 0.483677
\(81\) 0 0
\(82\) −2.43474 −0.268872
\(83\) −6.94282 −0.762074 −0.381037 − 0.924560i \(-0.624433\pi\)
−0.381037 + 0.924560i \(0.624433\pi\)
\(84\) 0 0
\(85\) −8.20602 −0.890068
\(86\) 0.532157 0.0573840
\(87\) 0 0
\(88\) −3.49192 −0.372240
\(89\) 2.74720 0.291203 0.145602 − 0.989343i \(-0.453488\pi\)
0.145602 + 0.989343i \(0.453488\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −10.8856 −1.13491
\(93\) 0 0
\(94\) 1.39372 0.143751
\(95\) −2.29630 −0.235596
\(96\) 0 0
\(97\) −7.16827 −0.727828 −0.363914 − 0.931433i \(-0.618560\pi\)
−0.363914 + 0.931433i \(0.618560\pi\)
\(98\) 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.p.1.2 3
3.2 odd 2 3969.2.a.m.1.2 3
7.6 odd 2 567.2.a.g.1.2 3
9.2 odd 6 441.2.f.d.148.2 6
9.4 even 3 1323.2.f.c.883.2 6
9.5 odd 6 441.2.f.d.295.2 6
9.7 even 3 1323.2.f.c.442.2 6
21.20 even 2 567.2.a.d.1.2 3
28.27 even 2 9072.2.a.cd.1.1 3
63.2 odd 6 441.2.g.d.67.2 6
63.4 even 3 1323.2.g.b.667.2 6
63.5 even 6 441.2.h.c.214.2 6
63.11 odd 6 441.2.h.b.373.2 6
63.13 odd 6 189.2.f.a.127.2 6
63.16 even 3 1323.2.g.b.361.2 6
63.20 even 6 63.2.f.b.22.2 ✓ 6
63.23 odd 6 441.2.h.b.214.2 6
63.25 even 3 1323.2.h.e.226.2 6
63.31 odd 6 1323.2.g.c.667.2 6
63.32 odd 6 441.2.g.d.79.2 6
63.34 odd 6 189.2.f.a.64.2 6
63.38 even 6 441.2.h.c.373.2 6
63.40 odd 6 1323.2.h.d.802.2 6
63.41 even 6 63.2.f.b.43.2 yes 6
63.47 even 6 441.2.g.e.67.2 6
63.52 odd 6 1323.2.h.d.226.2 6
63.58 even 3 1323.2.h.e.802.2 6
63.59 even 6 441.2.g.e.79.2 6
63.61 odd 6 1323.2.g.c.361.2 6
84.83 odd 2 9072.2.a.bq.1.3 3
252.83 odd 6 1008.2.r.k.337.2 6
252.139 even 6 3024.2.r.g.2017.3 6
252.167 odd 6 1008.2.r.k.673.2 6
252.223 even 6 3024.2.r.g.1009.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 63.20 even 6
63.2.f.b.43.2 yes 6 63.41 even 6
189.2.f.a.64.2 6 63.34 odd 6
189.2.f.a.127.2 6 63.13 odd 6
441.2.f.d.148.2 6 9.2 odd 6
441.2.f.d.295.2 6 9.5 odd 6
441.2.g.d.67.2 6 63.2 odd 6
441.2.g.d.79.2 6 63.32 odd 6
441.2.g.e.67.2 6 63.47 even 6
441.2.g.e.79.2 6 63.59 even 6
441.2.h.b.214.2 6 63.23 odd 6
441.2.h.b.373.2 6 63.11 odd 6
441.2.h.c.214.2 6 63.5 even 6
441.2.h.c.373.2 6 63.38 even 6
567.2.a.d.1.2 3 21.20 even 2
567.2.a.g.1.2 3 7.6 odd 2
1008.2.r.k.337.2 6 252.83 odd 6
1008.2.r.k.673.2 6 252.167 odd 6
1323.2.f.c.442.2 6 9.7 even 3
1323.2.f.c.883.2 6 9.4 even 3
1323.2.g.b.361.2 6 63.16 even 3
1323.2.g.b.667.2 6 63.4 even 3
1323.2.g.c.361.2 6 63.61 odd 6
1323.2.g.c.667.2 6 63.31 odd 6
1323.2.h.d.226.2 6 63.52 odd 6
1323.2.h.d.802.2 6 63.40 odd 6
1323.2.h.e.226.2 6 63.25 even 3
1323.2.h.e.802.2 6 63.58 even 3
3024.2.r.g.1009.3 6 252.223 even 6
3024.2.r.g.2017.3 6 252.139 even 6
3969.2.a.m.1.2 3 3.2 odd 2
3969.2.a.p.1.2 3 1.1 even 1 trivial
9072.2.a.bq.1.3 3 84.83 odd 2
9072.2.a.cd.1.1 3 28.27 even 2