↑

Properties

Label 63.4.c.a.62.2
Level $63$
Weight $4$
Character 63.62
Analytic conductor $3.717$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71712033036\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 62.2
Root \(2.57794i\) of defining polynomial
Character \(\chi\) \(=\) 63.62
Dual form 63.4.c.a.62.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.66471i q^{2} +5.22876 q^{4} +18.5203 q^{7} -22.0220i q^{8} -29.3750i q^{11} -30.8308i q^{14} +5.16995 q^{16} -48.9007 q^{22} +181.692i q^{23} -125.000 q^{25} +96.8379 q^{28} +304.463i q^{29} -184.782i q^{32} -10.5830 q^{37} -534.442 q^{43} -153.595i q^{44} +302.464 q^{46} +343.000 q^{49} +208.088i q^{50} -768.909i q^{53} -407.853i q^{56} +506.840 q^{58} -266.248 q^{64} +740.000 q^{67} +1178.70i q^{71} +17.6176i q^{74} -544.032i q^{77} -1384.00 q^{79} +889.688i q^{86} -646.895 q^{88} +950.024i q^{92} -570.994i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{4} + 444 q^{16} - 820 q^{22} - 500 q^{25} + 980 q^{28} - 748 q^{46} + 1372 q^{49} + 260 q^{58} - 3552 q^{64} + 2960 q^{67} - 5536 q^{79} + 8260 q^{88}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.66471i − 0.588562i −0.955719 − 0.294281i \(-0.904920\pi\)
0.955719 − 0.294281i \(-0.0950802\pi\)
\(3\) 0 0
\(4\) 5.22876 0.653595
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 18.5203 1.00000
\(8\) − 22.0220i − 0.973243i
\(9\) 0 0
\(10\) 0 0
\(11\) − 29.3750i − 0.805172i −0.915382 − 0.402586i \(-0.868111\pi\)
0.915382 − 0.402586i \(-0.131889\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) − 30.8308i − 0.588562i
\(15\) 0 0
\(16\) 5.16995 0.0807804
\(17\) 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
\(22\) −48.9007 −0.473894
\(23\) 181.692i 1.64719i 0.567176 + 0.823597i \(0.308036\pi\)
−0.567176 + 0.823597i \(0.691964\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 96.8379 0.653595
\(29\) 304.463i 1.94956i 0.223165 + 0.974781i \(0.428361\pi\)
−0.223165 + 0.974781i \(0.571639\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 184.782i − 1.02079i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.5830 −0.0470226 −0.0235113 − 0.999724i \(-0.507485\pi\)
−0.0235113 + 0.999724i \(0.507485\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −534.442 −1.89539 −0.947693 − 0.319183i \(-0.896592\pi\)
−0.947693 + 0.319183i \(0.896592\pi\)
\(44\) − 153.595i − 0.526256i
\(45\) 0 0
\(46\) 302.464 0.969476
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 343.000 1.00000
\(50\) 208.088i 0.588562i
\(51\) 0 0
\(52\) 0 0
\(53\) − 768.909i − 1.99279i −0.0848489 − 0.996394i \(-0.527041\pi\)
0.0848489 − 0.996394i \(-0.472959\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 407.853i − 0.973243i
\(57\) 0 0
\(58\) 506.840 1.14744
\(59\) 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
\(64\) −266.248 −0.520016
\(65\) 0 0
\(66\) 0 0
\(67\) 740.000 1.34933 0.674667 − 0.738122i \(-0.264287\pi\)
0.674667 + 0.738122i \(0.264287\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1178.70i 1.97022i 0.171931 + 0.985109i \(0.445000\pi\)
−0.171931 + 0.985109i \(0.555000\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 17.6176i 0.0276757i
\(75\) 0 0
\(76\) 0 0
\(77\) − 544.032i − 0.805172i
\(78\) 0 0
\(79\) −1384.00 −1.97104 −0.985520 − 0.169559i \(-0.945766\pi\)
−0.985520 + 0.169559i \(0.945766\pi\)
\(80\) 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
\(86\) 889.688i 1.11555i
\(87\) 0 0
\(88\) −646.895 −0.783628
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 950.024i 1.07660i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 570.994i − 0.588562i
\(99\) 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 63.4.c.a.62.2 ✓ 4
3.2 odd 2 inner 63.4.c.a.62.3 yes 4
4.3 odd 2 1008.4.k.a.881.2 4
7.2 even 3 441.4.p.b.80.2 8
7.3 odd 6 441.4.p.b.215.3 8
7.4 even 3 441.4.p.b.215.3 8
7.5 odd 6 441.4.p.b.80.2 8
7.6 odd 2 CM 63.4.c.a.62.2 ✓ 4
12.11 even 2 1008.4.k.a.881.1 4
21.2 odd 6 441.4.p.b.80.3 8
21.5 even 6 441.4.p.b.80.3 8
21.11 odd 6 441.4.p.b.215.2 8
21.17 even 6 441.4.p.b.215.2 8
21.20 even 2 inner 63.4.c.a.62.3 yes 4
28.27 even 2 1008.4.k.a.881.2 4
84.83 odd 2 1008.4.k.a.881.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.c.a.62.2 ✓ 4 1.1 even 1 trivial
63.4.c.a.62.2 ✓ 4 7.6 odd 2 CM
63.4.c.a.62.3 yes 4 3.2 odd 2 inner
63.4.c.a.62.3 yes 4 21.20 even 2 inner
441.4.p.b.80.2 8 7.2 even 3
441.4.p.b.80.2 8 7.5 odd 6
441.4.p.b.80.3 8 21.2 odd 6
441.4.p.b.80.3 8 21.5 even 6
441.4.p.b.215.2 8 21.11 odd 6
441.4.p.b.215.2 8 21.17 even 6
441.4.p.b.215.3 8 7.3 odd 6
441.4.p.b.215.3 8 7.4 even 3
1008.4.k.a.881.1 4 12.11 even 2
1008.4.k.a.881.1 4 84.83 odd 2
1008.4.k.a.881.2 4 4.3 odd 2
1008.4.k.a.881.2 4 28.27 even 2