Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [64,22,Mod(1,64)] 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("64.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,96764] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(178.865500344\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4963x + 96223 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(57.9766\) of defining polynomial
Character \(\chi\) \(=\) 64.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7983.67 q^{3} -3.09730e7 q^{5} -9.17385e7 q^{7} -1.03966e10 q^{9} +8.72158e10 q^{11} +2.36850e11 q^{13} +2.47278e11 q^{15} +7.42283e12 q^{17} +4.68680e9 q^{19} +7.32410e11 q^{21} -3.33703e14 q^{23} +4.82489e14 q^{25} +1.66515e14 q^{27} -3.23982e15 q^{29} -6.40473e15 q^{31} -6.96303e14 q^{33} +2.84142e15 q^{35} +1.61009e16 q^{37} -1.89093e15 q^{39} -5.77168e16 q^{41} -2.01468e17 q^{43} +3.22014e17 q^{45} -6.62056e17 q^{47} -5.50130e17 q^{49} -5.92615e16 q^{51} -4.62651e17 q^{53} -2.70133e18 q^{55} -3.74179e13 q^{57} +7.39502e18 q^{59} +5.50188e18 q^{61} +9.53770e17 q^{63} -7.33594e18 q^{65} +6.03520e18 q^{67} +2.66418e18 q^{69} +4.43147e19 q^{71} -2.48622e19 q^{73} -3.85204e18 q^{75} -8.00105e18 q^{77} -5.70948e19 q^{79} +1.07423e20 q^{81} -1.31820e20 q^{83} -2.29907e20 q^{85} +2.58657e19 q^{87} +3.97023e20 q^{89} -2.17282e19 q^{91} +5.11333e19 q^{93} -1.45164e17 q^{95} -9.80402e20 q^{97} -9.06749e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 96764 q^{3} + 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9} - 40335108684 q^{11} - 133734425946 q^{13} + 1223136458200 q^{15} + 7797732274422 q^{17} + 35788199781996 q^{19} - 198539224853088 q^{21}+ \cdots - 94\!\cdots\!60 q^{99}+O(q^{100}) \) 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
\(3\) −7983.67 −0.0780602 −0.0390301 − 0.999238i \(-0.512427\pi\)
−0.0390301 + 0.999238i \(0.512427\pi\)
\(4\) 0 0
\(5\) −3.09730e7 −1.41840 −0.709199 − 0.705008i \(-0.750943\pi\)
−0.709199 + 0.705008i \(0.750943\pi\)
\(6\) 0 0
\(7\) −9.17385e7 −0.122750 −0.0613751 − 0.998115i \(-0.519549\pi\)
−0.0613751 + 0.998115i \(0.519549\pi\)
\(8\) 0 0
\(9\) −1.03966e10 −0.993907
\(10\) 0 0
\(11\) 8.72158e10 1.01385 0.506923 − 0.861991i \(-0.330783\pi\)
0.506923 + 0.861991i \(0.330783\pi\)
\(12\) 0 0
\(13\) 2.36850e11 0.476505 0.238253 − 0.971203i \(-0.423425\pi\)
0.238253 + 0.971203i \(0.423425\pi\)
\(14\) 0 0
\(15\) 2.47278e11 0.110720
\(16\) 0 0
\(17\) 7.42283e12 0.893009 0.446505 − 0.894781i \(-0.352669\pi\)
0.446505 + 0.894781i \(0.352669\pi\)
\(18\) 0 0
\(19\) 4.68680e9 0.000175373 0 8.76867e−5 − 1.00000i \(-0.499972\pi\)
8.76867e−5 1.00000i \(0.499972\pi\)
\(20\) 0 0
\(21\) 7.32410e11 0.00958190
\(22\) 0 0
\(23\) −3.33703e14 −1.67965 −0.839823 − 0.542860i \(-0.817341\pi\)
−0.839823 + 0.542860i \(0.817341\pi\)
\(24\) 0 0
\(25\) 4.82489e14 1.01185
\(26\) 0 0
\(27\) 1.66515e14 0.155645
\(28\) 0 0
\(29\) −3.23982e15 −1.43002 −0.715010 − 0.699114i \(-0.753578\pi\)
−0.715010 + 0.699114i \(0.753578\pi\)
\(30\) 0 0
\(31\) −6.40473e15 −1.40347 −0.701735 − 0.712438i \(-0.747591\pi\)
−0.701735 + 0.712438i \(0.747591\pi\)
\(32\) 0 0
\(33\) −6.96303e14 −0.0791410
\(34\) 0 0
\(35\) 2.84142e15 0.174109
\(36\) 0 0
\(37\) 1.61009e16 0.550467 0.275234 − 0.961377i \(-0.411245\pi\)
0.275234 + 0.961377i \(0.411245\pi\)
\(38\) 0 0
\(39\) −1.89093e15 −0.0371961
\(40\) 0 0
\(41\) −5.77168e16 −0.671540 −0.335770 − 0.941944i \(-0.608997\pi\)
−0.335770 + 0.941944i \(0.608997\pi\)
\(42\) 0 0
\(43\) −2.01468e17 −1.42163 −0.710817 − 0.703377i \(-0.751675\pi\)
−0.710817 + 0.703377i \(0.751675\pi\)
\(44\) 0 0
\(45\) 3.22014e17 1.40976
\(46\) 0 0
\(47\) −6.62056e17 −1.83598 −0.917989 − 0.396607i \(-0.870188\pi\)
−0.917989 + 0.396607i \(0.870188\pi\)
\(48\) 0 0
\(49\) −5.50130e17 −0.984932
\(50\) 0 0
\(51\) −5.92615e16 −0.0697085
\(52\) 0 0
\(53\) −4.62651e17 −0.363376 −0.181688 − 0.983356i \(-0.558156\pi\)
−0.181688 + 0.983356i \(0.558156\pi\)
\(54\) 0 0
\(55\) −2.70133e18 −1.43804
\(56\) 0 0
\(57\) −3.74179e13 −1.36897e−5 0
\(58\) 0 0
\(59\) 7.39502e18 1.88362 0.941809 − 0.336148i \(-0.109124\pi\)
0.941809 + 0.336148i \(0.109124\pi\)
\(60\) 0 0
\(61\) 5.50188e18 0.987524 0.493762 − 0.869597i \(-0.335621\pi\)
0.493762 + 0.869597i \(0.335621\pi\)
\(62\) 0 0
\(63\) 9.53770e17 0.122002
\(64\) 0 0
\(65\) −7.33594e18 −0.675874
\(66\) 0 0
\(67\) 6.03520e18 0.404489 0.202244 − 0.979335i \(-0.435176\pi\)
0.202244 + 0.979335i \(0.435176\pi\)
\(68\) 0 0
\(69\) 2.66418e18 0.131113
\(70\) 0 0
\(71\) 4.43147e19 1.61561 0.807803 − 0.589453i \(-0.200657\pi\)
0.807803 + 0.589453i \(0.200657\pi\)
\(72\) 0 0
\(73\) −2.48622e19 −0.677095 −0.338548 − 0.940949i \(-0.609936\pi\)
−0.338548 + 0.940949i \(0.609936\pi\)
\(74\) 0 0
\(75\) −3.85204e18 −0.0789855
\(76\) 0 0
\(77\) −8.00105e18 −0.124450
\(78\) 0 0
\(79\) −5.70948e19 −0.678441 −0.339220 − 0.940707i \(-0.610163\pi\)
−0.339220 + 0.940707i \(0.610163\pi\)
\(80\) 0 0
\(81\) 1.07423e20 0.981757
\(82\) 0 0
\(83\) −1.31820e20 −0.932528 −0.466264 − 0.884646i \(-0.654400\pi\)
−0.466264 + 0.884646i \(0.654400\pi\)
\(84\) 0 0
\(85\) −2.29907e20 −1.26664
\(86\) 0 0
\(87\) 2.58657e19 0.111628
\(88\) 0 0
\(89\) 3.97023e20 1.34965 0.674823 − 0.737979i \(-0.264220\pi\)
0.674823 + 0.737979i \(0.264220\pi\)
\(90\) 0 0
\(91\) −2.17282e19 −0.0584911
\(92\) 0 0
\(93\) 5.11333e19 0.109555
\(94\) 0 0
\(95\) −1.45164e17 −0.000248749 0
\(96\) 0 0
\(97\) −9.80402e20 −1.34990 −0.674949 − 0.737864i \(-0.735835\pi\)
−0.674949 + 0.737864i \(0.735835\pi\)
\(98\) 0 0
\(99\) −9.06749e20 −1.00767
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 64.22.a.m.1.2 3
4.3 odd 2 64.22.a.l.1.2 3
8.3 odd 2 8.22.a.b.1.2 ✓ 3
8.5 even 2 16.22.a.f.1.2 3
24.11 even 2 72.22.a.f.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.2 ✓ 3 8.3 odd 2
16.22.a.f.1.2 3 8.5 even 2
64.22.a.l.1.2 3 4.3 odd 2
64.22.a.m.1.2 3 1.1 even 1 trivial
72.22.a.f.1.1 3 24.11 even 2