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Properties

Label 63.2.c.a.62.1
Level $63$
Weight $2$
Character 63.62
Analytic conductor $0.503$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $4$

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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] = [63,2,Mod(62,63)] 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("63.62"); 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([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 62.1
Root \(-2.57794i\) of defining polynomial
Character \(\chi\) \(=\) 63.62
Dual form 63.2.c.a.62.4

$q$-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)
 
\(f(q)\) \(=\) \(q-2.57794i q^{2} -4.64575 q^{4} +2.64575 q^{7} +6.82058i q^{8} +0.913230i q^{11} -6.82058i q^{14} +8.29150 q^{16} +2.35425 q^{22} +9.39851i q^{23} -5.00000 q^{25} -12.2915 q^{28} -6.06910i q^{29} -7.73381i q^{32} -10.5830 q^{37} +5.29150 q^{43} -4.24264i q^{44} +24.2288 q^{46} +7.00000 q^{49} +12.8897i q^{50} -14.5544i q^{53} +18.0455i q^{56} -15.6458 q^{58} -3.35425 q^{64} -4.00000 q^{67} -7.57205i q^{71} +27.2823i q^{74} +2.41618i q^{77} +8.00000 q^{79} -13.6412i q^{86} -6.22876 q^{88} -43.6631i q^{92} -18.0455i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 12 q^{16} + 20 q^{22} - 20 q^{25} - 28 q^{28} + 44 q^{46} + 28 q^{49} - 52 q^{58} - 24 q^{64} - 16 q^{67} + 32 q^{79} + 28 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.57794i − 1.82288i −0.411438 − 0.911438i \(-0.634973\pi\)
0.411438 − 0.911438i \(-0.365027\pi\)
\(3\) 0 0
\(4\) −4.64575 −2.32288
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) 6.82058i 2.41144i
\(9\) 0 0
\(10\) 0 0
\(11\) 0.913230i 0.275349i 0.990478 + 0.137675i \(0.0439628\pi\)
−0.990478 + 0.137675i \(0.956037\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) − 6.82058i − 1.82288i
\(15\) 0 0
\(16\) 8.29150 2.07288
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.35425 0.501928
\(23\) 9.39851i 1.95973i 0.199673 + 0.979863i \(0.436012\pi\)
−0.199673 + 0.979863i \(0.563988\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −12.2915 −2.32288
\(29\) − 6.06910i − 1.12700i −0.826115 − 0.563502i \(-0.809454\pi\)
0.826115 − 0.563502i \(-0.190546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 7.73381i − 1.36716i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.5830 −1.73984 −0.869918 − 0.493197i \(-0.835828\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 5.29150 0.806947 0.403473 − 0.914991i \(-0.367803\pi\)
0.403473 + 0.914991i \(0.367803\pi\)
\(44\) − 4.24264i − 0.639602i
\(45\) 0 0
\(46\) 24.2288 3.57234
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 12.8897i 1.82288i
\(51\) 0 0
\(52\) 0 0
\(53\) − 14.5544i − 1.99920i −0.0283132 − 0.999599i \(-0.509014\pi\)
0.0283132 − 0.999599i \(-0.490986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 18.0455i 2.41144i
\(57\) 0 0
\(58\) −15.6458 −2.05439
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −3.35425 −0.419281
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 7.57205i − 0.898637i −0.893372 − 0.449319i \(-0.851667\pi\)
0.893372 − 0.449319i \(-0.148333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 27.2823i 3.17150i
\(75\) 0 0
\(76\) 0 0
\(77\) 2.41618i 0.275349i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 13.6412i − 1.47096i
\(87\) 0 0
\(88\) −6.22876 −0.663988
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 43.6631i − 4.55220i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 18.0455i − 1.82288i
\(99\) 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.c.a.62.1 ✓ 4
3.2 odd 2 inner 63.2.c.a.62.4 yes 4
4.3 odd 2 1008.2.k.a.881.1 4
5.2 odd 4 1575.2.g.d.1574.8 8
5.3 odd 4 1575.2.g.d.1574.1 8
5.4 even 2 1575.2.b.a.251.4 4
7.2 even 3 441.2.p.b.80.1 8
7.3 odd 6 441.2.p.b.215.4 8
7.4 even 3 441.2.p.b.215.4 8
7.5 odd 6 441.2.p.b.80.1 8
7.6 odd 2 CM 63.2.c.a.62.1 ✓ 4
8.3 odd 2 4032.2.k.b.3905.2 4
8.5 even 2 4032.2.k.c.3905.3 4
9.2 odd 6 567.2.o.f.377.4 8
9.4 even 3 567.2.o.f.188.4 8
9.5 odd 6 567.2.o.f.188.1 8
9.7 even 3 567.2.o.f.377.1 8
12.11 even 2 1008.2.k.a.881.2 4
15.2 even 4 1575.2.g.d.1574.2 8
15.8 even 4 1575.2.g.d.1574.7 8
15.14 odd 2 1575.2.b.a.251.1 4
21.2 odd 6 441.2.p.b.80.4 8
21.5 even 6 441.2.p.b.80.4 8
21.11 odd 6 441.2.p.b.215.1 8
21.17 even 6 441.2.p.b.215.1 8
21.20 even 2 inner 63.2.c.a.62.4 yes 4
24.5 odd 2 4032.2.k.c.3905.4 4
24.11 even 2 4032.2.k.b.3905.1 4
28.27 even 2 1008.2.k.a.881.1 4
35.13 even 4 1575.2.g.d.1574.1 8
35.27 even 4 1575.2.g.d.1574.8 8
35.34 odd 2 1575.2.b.a.251.4 4
56.13 odd 2 4032.2.k.c.3905.3 4
56.27 even 2 4032.2.k.b.3905.2 4
63.13 odd 6 567.2.o.f.188.4 8
63.20 even 6 567.2.o.f.377.4 8
63.34 odd 6 567.2.o.f.377.1 8
63.41 even 6 567.2.o.f.188.1 8
84.83 odd 2 1008.2.k.a.881.2 4
105.62 odd 4 1575.2.g.d.1574.2 8
105.83 odd 4 1575.2.g.d.1574.7 8
105.104 even 2 1575.2.b.a.251.1 4
168.83 odd 2 4032.2.k.b.3905.1 4
168.125 even 2 4032.2.k.c.3905.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.c.a.62.1 ✓ 4 1.1 even 1 trivial
63.2.c.a.62.1 ✓ 4 7.6 odd 2 CM
63.2.c.a.62.4 yes 4 3.2 odd 2 inner
63.2.c.a.62.4 yes 4 21.20 even 2 inner
441.2.p.b.80.1 8 7.2 even 3
441.2.p.b.80.1 8 7.5 odd 6
441.2.p.b.80.4 8 21.2 odd 6
441.2.p.b.80.4 8 21.5 even 6
441.2.p.b.215.1 8 21.11 odd 6
441.2.p.b.215.1 8 21.17 even 6
441.2.p.b.215.4 8 7.3 odd 6
441.2.p.b.215.4 8 7.4 even 3
567.2.o.f.188.1 8 9.5 odd 6
567.2.o.f.188.1 8 63.41 even 6
567.2.o.f.188.4 8 9.4 even 3
567.2.o.f.188.4 8 63.13 odd 6
567.2.o.f.377.1 8 9.7 even 3
567.2.o.f.377.1 8 63.34 odd 6
567.2.o.f.377.4 8 9.2 odd 6
567.2.o.f.377.4 8 63.20 even 6
1008.2.k.a.881.1 4 4.3 odd 2
1008.2.k.a.881.1 4 28.27 even 2
1008.2.k.a.881.2 4 12.11 even 2
1008.2.k.a.881.2 4 84.83 odd 2
1575.2.b.a.251.1 4 15.14 odd 2
1575.2.b.a.251.1 4 105.104 even 2
1575.2.b.a.251.4 4 5.4 even 2
1575.2.b.a.251.4 4 35.34 odd 2
1575.2.g.d.1574.1 8 5.3 odd 4
1575.2.g.d.1574.1 8 35.13 even 4
1575.2.g.d.1574.2 8 15.2 even 4
1575.2.g.d.1574.2 8 105.62 odd 4
1575.2.g.d.1574.7 8 15.8 even 4
1575.2.g.d.1574.7 8 105.83 odd 4
1575.2.g.d.1574.8 8 5.2 odd 4
1575.2.g.d.1574.8 8 35.27 even 4
4032.2.k.b.3905.1 4 24.11 even 2
4032.2.k.b.3905.1 4 168.83 odd 2
4032.2.k.b.3905.2 4 8.3 odd 2
4032.2.k.b.3905.2 4 56.27 even 2
4032.2.k.c.3905.3 4 8.5 even 2
4032.2.k.c.3905.3 4 56.13 odd 2
4032.2.k.c.3905.4 4 24.5 odd 2
4032.2.k.c.3905.4 4 168.125 even 2