Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.2123079070\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 6664.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.11491 q^{3} +3.47283 q^{5} +6.70265 q^{9} +4.22982 q^{11} +1.28415 q^{13} +10.8176 q^{15} +1.00000 q^{17} -2.94567 q^{19} -0.715853 q^{23} +7.06058 q^{25} +11.5334 q^{27} -6.94567 q^{29} -8.98680 q^{31} +13.1755 q^{33} -10.9457 q^{37} +4.00000 q^{39} -2.06058 q^{41} -0.399055 q^{43} +23.2772 q^{45} +12.4596 q^{47} +3.11491 q^{51} +6.64832 q^{53} +14.6894 q^{55} -9.17548 q^{57} +1.43171 q^{59} +6.88509 q^{61} +4.45963 q^{65} +10.4185 q^{67} -2.22982 q^{69} -13.9736 q^{71} +14.5202 q^{73} +21.9930 q^{75} +4.00000 q^{79} +15.8176 q^{81} +1.05433 q^{83} +3.47283 q^{85} -21.6351 q^{87} -4.60719 q^{89} -27.9930 q^{93} -10.2298 q^{95} -10.2709 q^{97} +28.3510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 5 q^{5} + 2 q^{9} + 2 q^{13} + 8 q^{15} + 3 q^{17} + 2 q^{19} - 4 q^{23} + 4 q^{25} + 12 q^{27} - 10 q^{29} - 7 q^{31} + 16 q^{33} - 22 q^{37} + 12 q^{39} + 11 q^{41} + 7 q^{43} + 20 q^{45}+ \cdots + 38 q^{99}+O(q^{100}) \) 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 0
\(3\) 3.11491 1.79839 0.899196 − 0.437545i \(-0.144152\pi\)
0.899196 + 0.437545i \(0.144152\pi\)
\(4\) 0 0
\(5\) 3.47283 1.55310 0.776549 − 0.630057i \(-0.216968\pi\)
0.776549 + 0.630057i \(0.216968\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 6.70265 2.23422
\(10\) 0 0
\(11\) 4.22982 1.27534 0.637669 − 0.770311i \(-0.279899\pi\)
0.637669 + 0.770311i \(0.279899\pi\)
\(12\) 0 0
\(13\) 1.28415 0.356158 0.178079 − 0.984016i \(-0.443012\pi\)
0.178079 + 0.984016i \(0.443012\pi\)
\(14\) 0 0
\(15\) 10.8176 2.79308
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −2.94567 −0.675783 −0.337891 − 0.941185i \(-0.609714\pi\)
−0.337891 + 0.941185i \(0.609714\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.715853 −0.149266 −0.0746328 − 0.997211i \(-0.523778\pi\)
−0.0746328 + 0.997211i \(0.523778\pi\)
\(24\) 0 0
\(25\) 7.06058 1.41212
\(26\) 0 0
\(27\) 11.5334 2.21961
\(28\) 0 0
\(29\) −6.94567 −1.28978 −0.644889 − 0.764276i \(-0.723096\pi\)
−0.644889 + 0.764276i \(0.723096\pi\)
\(30\) 0 0
\(31\) −8.98680 −1.61408 −0.807038 − 0.590499i \(-0.798931\pi\)
−0.807038 + 0.590499i \(0.798931\pi\)
\(32\) 0 0
\(33\) 13.1755 2.29356
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.9457 −1.79946 −0.899728 − 0.436450i \(-0.856235\pi\)
−0.899728 + 0.436450i \(0.856235\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.06058 −0.321808 −0.160904 − 0.986970i \(-0.551441\pi\)
−0.160904 + 0.986970i \(0.551441\pi\)
\(42\) 0 0
\(43\) −0.399055 −0.0608553 −0.0304276 − 0.999537i \(-0.509687\pi\)
−0.0304276 + 0.999537i \(0.509687\pi\)
\(44\) 0 0
\(45\) 23.2772 3.46996
\(46\) 0 0
\(47\) 12.4596 1.81742 0.908712 − 0.417424i \(-0.137067\pi\)
0.908712 + 0.417424i \(0.137067\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.11491 0.436174
\(52\) 0 0
\(53\) 6.64832 0.913217 0.456608 − 0.889668i \(-0.349064\pi\)
0.456608 + 0.889668i \(0.349064\pi\)
\(54\) 0 0
\(55\) 14.6894 1.98072
\(56\) 0 0
\(57\) −9.17548 −1.21532
\(58\) 0 0
\(59\) 1.43171 0.186392 0.0931961 − 0.995648i \(-0.470292\pi\)
0.0931961 + 0.995648i \(0.470292\pi\)
\(60\) 0 0
\(61\) 6.88509 0.881546 0.440773 − 0.897619i \(-0.354704\pi\)
0.440773 + 0.897619i \(0.354704\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.45963 0.553149
\(66\) 0 0
\(67\) 10.4185 1.27282 0.636411 − 0.771350i \(-0.280418\pi\)
0.636411 + 0.771350i \(0.280418\pi\)
\(68\) 0 0
\(69\) −2.22982 −0.268438
\(70\) 0 0
\(71\) −13.9736 −1.65836 −0.829180 − 0.558981i \(-0.811192\pi\)
−0.829180 + 0.558981i \(0.811192\pi\)
\(72\) 0 0
\(73\) 14.5202 1.69946 0.849731 − 0.527217i \(-0.176764\pi\)
0.849731 + 0.527217i \(0.176764\pi\)
\(74\) 0 0
\(75\) 21.9930 2.53954
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 15.8176 1.75751
\(82\) 0 0
\(83\) 1.05433 0.115728 0.0578640 − 0.998324i \(-0.481571\pi\)
0.0578640 + 0.998324i \(0.481571\pi\)
\(84\) 0 0
\(85\) 3.47283 0.376682
\(86\) 0 0
\(87\) −21.6351 −2.31953
\(88\) 0 0
\(89\) −4.60719 −0.488361 −0.244180 − 0.969730i \(-0.578519\pi\)
−0.244180 + 0.969730i \(0.578519\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −27.9930 −2.90274
\(94\) 0 0
\(95\) −10.2298 −1.04956
\(96\) 0 0
\(97\) −10.2709 −1.04286 −0.521428 − 0.853295i \(-0.674601\pi\)
−0.521428 + 0.853295i \(0.674601\pi\)
\(98\) 0 0
\(99\) 28.3510 2.84938
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 6664.2.a.m.1.3 3
7.6 odd 2 952.2.a.c.1.1 ✓ 3
21.20 even 2 8568.2.a.be.1.3 3
28.27 even 2 1904.2.a.o.1.3 3
56.13 odd 2 7616.2.a.bh.1.3 3
56.27 even 2 7616.2.a.bb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.c.1.1 ✓ 3 7.6 odd 2
1904.2.a.o.1.3 3 28.27 even 2
6664.2.a.m.1.3 3 1.1 even 1 trivial
7616.2.a.bb.1.1 3 56.27 even 2
7616.2.a.bh.1.3 3 56.13 odd 2
8568.2.a.be.1.3 3 21.20 even 2