Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,2,Mod(1,63)] 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("63.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 63.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{7} -1.73205 q^{8} -6.00000 q^{10} +3.46410 q^{11} +2.00000 q^{13} +1.73205 q^{14} -5.00000 q^{16} +3.46410 q^{17} -4.00000 q^{19} -3.46410 q^{20} +6.00000 q^{22} -3.46410 q^{23} +7.00000 q^{25} +3.46410 q^{26} +1.00000 q^{28} -4.00000 q^{31} -5.19615 q^{32} +6.00000 q^{34} -3.46410 q^{35} +2.00000 q^{37} -6.92820 q^{38} +6.00000 q^{40} +10.3923 q^{41} -4.00000 q^{43} +3.46410 q^{44} -6.00000 q^{46} +6.92820 q^{47} +1.00000 q^{49} +12.1244 q^{50} +2.00000 q^{52} -6.92820 q^{53} -12.0000 q^{55} -1.73205 q^{56} -6.92820 q^{59} -10.0000 q^{61} -6.92820 q^{62} +1.00000 q^{64} -6.92820 q^{65} -4.00000 q^{67} +3.46410 q^{68} -6.00000 q^{70} -10.3923 q^{71} +14.0000 q^{73} +3.46410 q^{74} -4.00000 q^{76} +3.46410 q^{77} +8.00000 q^{79} +17.3205 q^{80} +18.0000 q^{82} -12.0000 q^{85} -6.92820 q^{86} -6.00000 q^{88} -3.46410 q^{89} +2.00000 q^{91} -3.46410 q^{92} +12.0000 q^{94} +13.8564 q^{95} +14.0000 q^{97} +1.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{7} - 12 q^{10} + 4 q^{13} - 10 q^{16} - 8 q^{19} + 12 q^{22} + 14 q^{25} + 2 q^{28} - 8 q^{31} + 12 q^{34} + 4 q^{37} + 12 q^{40} - 8 q^{43} - 12 q^{46} + 2 q^{49} + 4 q^{52} - 24 q^{55}+ \cdots + 28 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 − 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.46410 −1.54919 −0.774597 − 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) −6.00000 −1.89737
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.73205 0.462910
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 3.46410 0.840168 0.420084 − 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −3.46410 −0.774597
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −3.46410 −0.722315 −0.361158 − 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 3.46410 0.679366
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −6.92820 −1.12390
\(39\) 0 0
\(40\) 6.00000 0.948683
\(41\) 10.3923 1.62301 0.811503 − 0.584349i \(-0.198650\pi\)
0.811503 + 0.584349i \(0.198650\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.46410 0.522233
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 6.92820 1.01058 0.505291 − 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 12.1244 1.71464
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.92820 −0.951662 −0.475831 − 0.879537i \(-0.657853\pi\)
−0.475831 + 0.879537i \(0.657853\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) −1.73205 −0.231455
\(57\) 0 0
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 − 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −6.92820 −0.879883
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.92820 −0.859338
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) −10.3923 −1.23334 −0.616670 − 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 3.46410 0.402694
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 3.46410 0.394771
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 17.3205 1.93649
\(81\) 0 0
\(82\) 18.0000 1.98777
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −6.92820 −0.747087
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) −3.46410 −0.367194 −0.183597 − 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −3.46410 −0.361158
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 13.8564 1.42164
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 1.73205 0.174964
\(99\) 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 63.2.a.b.1.2 yes 2
3.2 odd 2 inner 63.2.a.b.1.1 ✓ 2
4.3 odd 2 1008.2.a.n.1.1 2
5.2 odd 4 1575.2.d.i.1324.4 4
5.3 odd 4 1575.2.d.i.1324.1 4
5.4 even 2 1575.2.a.q.1.1 2
7.2 even 3 441.2.e.j.361.1 4
7.3 odd 6 441.2.e.i.226.1 4
7.4 even 3 441.2.e.j.226.1 4
7.5 odd 6 441.2.e.i.361.1 4
7.6 odd 2 441.2.a.g.1.2 2
8.3 odd 2 4032.2.a.bq.1.2 2
8.5 even 2 4032.2.a.bt.1.2 2
9.2 odd 6 567.2.f.j.190.2 4
9.4 even 3 567.2.f.j.379.1 4
9.5 odd 6 567.2.f.j.379.2 4
9.7 even 3 567.2.f.j.190.1 4
11.10 odd 2 7623.2.a.bi.1.1 2
12.11 even 2 1008.2.a.n.1.2 2
15.2 even 4 1575.2.d.i.1324.2 4
15.8 even 4 1575.2.d.i.1324.3 4
15.14 odd 2 1575.2.a.q.1.2 2
21.2 odd 6 441.2.e.j.361.2 4
21.5 even 6 441.2.e.i.361.2 4
21.11 odd 6 441.2.e.j.226.2 4
21.17 even 6 441.2.e.i.226.2 4
21.20 even 2 441.2.a.g.1.1 2
24.5 odd 2 4032.2.a.bt.1.1 2
24.11 even 2 4032.2.a.bq.1.1 2
28.27 even 2 7056.2.a.cm.1.2 2
33.32 even 2 7623.2.a.bi.1.2 2
84.83 odd 2 7056.2.a.cm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.a.b.1.1 ✓ 2 3.2 odd 2 inner
63.2.a.b.1.2 yes 2 1.1 even 1 trivial
441.2.a.g.1.1 2 21.20 even 2
441.2.a.g.1.2 2 7.6 odd 2
441.2.e.i.226.1 4 7.3 odd 6
441.2.e.i.226.2 4 21.17 even 6
441.2.e.i.361.1 4 7.5 odd 6
441.2.e.i.361.2 4 21.5 even 6
441.2.e.j.226.1 4 7.4 even 3
441.2.e.j.226.2 4 21.11 odd 6
441.2.e.j.361.1 4 7.2 even 3
441.2.e.j.361.2 4 21.2 odd 6
567.2.f.j.190.1 4 9.7 even 3
567.2.f.j.190.2 4 9.2 odd 6
567.2.f.j.379.1 4 9.4 even 3
567.2.f.j.379.2 4 9.5 odd 6
1008.2.a.n.1.1 2 4.3 odd 2
1008.2.a.n.1.2 2 12.11 even 2
1575.2.a.q.1.1 2 5.4 even 2
1575.2.a.q.1.2 2 15.14 odd 2
1575.2.d.i.1324.1 4 5.3 odd 4
1575.2.d.i.1324.2 4 15.2 even 4
1575.2.d.i.1324.3 4 15.8 even 4
1575.2.d.i.1324.4 4 5.2 odd 4
4032.2.a.bq.1.1 2 24.11 even 2
4032.2.a.bq.1.2 2 8.3 odd 2
4032.2.a.bt.1.1 2 24.5 odd 2
4032.2.a.bt.1.2 2 8.5 even 2
7056.2.a.cm.1.1 2 84.83 odd 2
7056.2.a.cm.1.2 2 28.27 even 2
7623.2.a.bi.1.1 2 11.10 odd 2
7623.2.a.bi.1.2 2 33.32 even 2