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Properties

Label 63.2.c.a.62.3
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 62.3
Root \(1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 63.62
Dual form 63.2.c.a.62.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.16372i q^{2} +0.645751 q^{4} -2.64575 q^{7} +3.07892i q^{8} -6.57008i q^{11} -3.07892i q^{14} -2.29150 q^{16} +7.64575 q^{22} +1.91520i q^{23} -5.00000 q^{25} -1.70850 q^{28} +8.89753i q^{29} +3.49117i q^{32} +10.5830 q^{37} -5.29150 q^{43} -4.24264i q^{44} -2.22876 q^{46} +7.00000 q^{49} -5.81861i q^{50} +0.412247i q^{53} -8.14605i q^{56} -10.3542 q^{58} -8.64575 q^{64} -4.00000 q^{67} -15.0554i q^{71} +12.3157i q^{74} +17.3828i q^{77} +8.00000 q^{79} -6.15784i q^{86} +20.2288 q^{88} +1.23674i q^{92} +8.14605i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 12 q^{16} + 20 q^{22} - 20 q^{25} - 28 q^{28} + 44 q^{46} + 28 q^{49} - 52 q^{58} - 24 q^{64} - 16 q^{67} + 32 q^{79} + 28 q^{88}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.16372i 0.822876i 0.911438 + 0.411438i \(0.134973\pi\)
−0.911438 + 0.411438i \(0.865027\pi\)
\(3\) 0 0
\(4\) 0.645751 0.322876
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.64575 −1.00000
\(8\) 3.07892i 1.08856i
\(9\) 0 0
\(10\) 0 0
\(11\) − 6.57008i − 1.98096i −0.137675 − 0.990478i \(-0.543963\pi\)
0.137675 − 0.990478i \(-0.456037\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) − 3.07892i − 0.822876i
\(15\) 0 0
\(16\) −2.29150 −0.572876
\(17\) 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
\(22\) 7.64575 1.63008
\(23\) 1.91520i 0.399346i 0.979863 + 0.199673i \(0.0639880\pi\)
−0.979863 + 0.199673i \(0.936012\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −1.70850 −0.322876
\(29\) 8.89753i 1.65223i 0.563502 + 0.826115i \(0.309454\pi\)
−0.563502 + 0.826115i \(0.690546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 3.49117i 0.617157i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.5830 1.73984 0.869918 − 0.493197i \(-0.164172\pi\)
0.869918 + 0.493197i \(0.164172\pi\)
\(38\) 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 \(-0.632197\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) − 4.24264i − 0.639602i
\(45\) 0 0
\(46\) −2.22876 −0.328612
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) − 5.81861i − 0.822876i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.412247i 0.0566265i 0.999599 + 0.0283132i \(0.00901359\pi\)
−0.999599 + 0.0283132i \(0.990986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 8.14605i − 1.08856i
\(57\) 0 0
\(58\) −10.3542 −1.35958
\(59\) 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
\(64\) −8.64575 −1.08072
\(65\) 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
\(71\) − 15.0554i − 1.78674i −0.449319 − 0.893372i \(-0.648333\pi\)
0.449319 − 0.893372i \(-0.351667\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 12.3157i 1.43167i
\(75\) 0 0
\(76\) 0 0
\(77\) 17.3828i 1.98096i
\(78\) 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
\(86\) − 6.15784i − 0.664017i
\(87\) 0 0
\(88\) 20.2288 2.15639
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.23674i 0.128939i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 8.14605i 0.822876i
\(99\) 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.62.3 yes 4
3.2 odd 2 inner 63.2.c.a.62.2 ✓ 4
4.3 odd 2 1008.2.k.a.881.4 4
5.2 odd 4 1575.2.g.d.1574.3 8
5.3 odd 4 1575.2.g.d.1574.6 8
5.4 even 2 1575.2.b.a.251.2 4
7.2 even 3 441.2.p.b.80.3 8
7.3 odd 6 441.2.p.b.215.2 8
7.4 even 3 441.2.p.b.215.2 8
7.5 odd 6 441.2.p.b.80.3 8
7.6 odd 2 CM 63.2.c.a.62.3 yes 4
8.3 odd 2 4032.2.k.b.3905.3 4
8.5 even 2 4032.2.k.c.3905.2 4
9.2 odd 6 567.2.o.f.377.2 8
9.4 even 3 567.2.o.f.188.2 8
9.5 odd 6 567.2.o.f.188.3 8
9.7 even 3 567.2.o.f.377.3 8
12.11 even 2 1008.2.k.a.881.3 4
15.2 even 4 1575.2.g.d.1574.5 8
15.8 even 4 1575.2.g.d.1574.4 8
15.14 odd 2 1575.2.b.a.251.3 4
21.2 odd 6 441.2.p.b.80.2 8
21.5 even 6 441.2.p.b.80.2 8
21.11 odd 6 441.2.p.b.215.3 8
21.17 even 6 441.2.p.b.215.3 8
21.20 even 2 inner 63.2.c.a.62.2 ✓ 4
24.5 odd 2 4032.2.k.c.3905.1 4
24.11 even 2 4032.2.k.b.3905.4 4
28.27 even 2 1008.2.k.a.881.4 4
35.13 even 4 1575.2.g.d.1574.6 8
35.27 even 4 1575.2.g.d.1574.3 8
35.34 odd 2 1575.2.b.a.251.2 4
56.13 odd 2 4032.2.k.c.3905.2 4
56.27 even 2 4032.2.k.b.3905.3 4
63.13 odd 6 567.2.o.f.188.2 8
63.20 even 6 567.2.o.f.377.2 8
63.34 odd 6 567.2.o.f.377.3 8
63.41 even 6 567.2.o.f.188.3 8
84.83 odd 2 1008.2.k.a.881.3 4
105.62 odd 4 1575.2.g.d.1574.5 8
105.83 odd 4 1575.2.g.d.1574.4 8
105.104 even 2 1575.2.b.a.251.3 4
168.83 odd 2 4032.2.k.b.3905.4 4
168.125 even 2 4032.2.k.c.3905.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.c.a.62.2 ✓ 4 3.2 odd 2 inner
63.2.c.a.62.2 ✓ 4 21.20 even 2 inner
63.2.c.a.62.3 yes 4 1.1 even 1 trivial
63.2.c.a.62.3 yes 4 7.6 odd 2 CM
441.2.p.b.80.2 8 21.2 odd 6
441.2.p.b.80.2 8 21.5 even 6
441.2.p.b.80.3 8 7.2 even 3
441.2.p.b.80.3 8 7.5 odd 6
441.2.p.b.215.2 8 7.3 odd 6
441.2.p.b.215.2 8 7.4 even 3
441.2.p.b.215.3 8 21.11 odd 6
441.2.p.b.215.3 8 21.17 even 6
567.2.o.f.188.2 8 9.4 even 3
567.2.o.f.188.2 8 63.13 odd 6
567.2.o.f.188.3 8 9.5 odd 6
567.2.o.f.188.3 8 63.41 even 6
567.2.o.f.377.2 8 9.2 odd 6
567.2.o.f.377.2 8 63.20 even 6
567.2.o.f.377.3 8 9.7 even 3
567.2.o.f.377.3 8 63.34 odd 6
1008.2.k.a.881.3 4 12.11 even 2
1008.2.k.a.881.3 4 84.83 odd 2
1008.2.k.a.881.4 4 4.3 odd 2
1008.2.k.a.881.4 4 28.27 even 2
1575.2.b.a.251.2 4 5.4 even 2
1575.2.b.a.251.2 4 35.34 odd 2
1575.2.b.a.251.3 4 15.14 odd 2
1575.2.b.a.251.3 4 105.104 even 2
1575.2.g.d.1574.3 8 5.2 odd 4
1575.2.g.d.1574.3 8 35.27 even 4
1575.2.g.d.1574.4 8 15.8 even 4
1575.2.g.d.1574.4 8 105.83 odd 4
1575.2.g.d.1574.5 8 15.2 even 4
1575.2.g.d.1574.5 8 105.62 odd 4
1575.2.g.d.1574.6 8 5.3 odd 4
1575.2.g.d.1574.6 8 35.13 even 4
4032.2.k.b.3905.3 4 8.3 odd 2
4032.2.k.b.3905.3 4 56.27 even 2
4032.2.k.b.3905.4 4 24.11 even 2
4032.2.k.b.3905.4 4 168.83 odd 2
4032.2.k.c.3905.1 4 24.5 odd 2
4032.2.k.c.3905.1 4 168.125 even 2
4032.2.k.c.3905.2 4 8.5 even 2
4032.2.k.c.3905.2 4 56.13 odd 2