Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1152,4,Mod(1,1152)] 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("1152.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,4,0,-8,0,0,0,92,0,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.9702003266\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1152.1

$q$-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)
 
\(f(q)\) \(=\) \(q+15.8564 q^{5} -17.8564 q^{7} +52.9282 q^{11} -8.43078 q^{13} -129.138 q^{17} -50.4974 q^{19} -128.708 q^{23} +126.426 q^{25} +111.282 q^{29} -302.851 q^{31} -283.138 q^{35} +182.995 q^{37} +94.5744 q^{41} -184.641 q^{43} -296.841 q^{47} -24.1487 q^{49} -102.995 q^{53} +839.251 q^{55} -93.3693 q^{59} +338.974 q^{61} -133.682 q^{65} -489.041 q^{67} +86.9845 q^{71} -154.267 q^{73} -945.108 q^{77} -449.415 q^{79} +383.933 q^{83} -2047.67 q^{85} +517.672 q^{89} +150.543 q^{91} -800.708 q^{95} +1739.39 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 8 q^{7} + 92 q^{11} - 100 q^{13} - 92 q^{17} - 4 q^{19} - 8 q^{23} + 142 q^{25} + 84 q^{29} - 384 q^{31} - 400 q^{35} + 172 q^{37} + 300 q^{41} - 300 q^{43} + 16 q^{47} - 270 q^{49} - 12 q^{53}+ \cdots + 2204 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 15.8564 1.41824 0.709120 − 0.705088i \(-0.249092\pi\)
0.709120 + 0.705088i \(0.249092\pi\)
\(6\) 0 0
\(7\) −17.8564 −0.964155 −0.482078 − 0.876128i \(-0.660118\pi\)
−0.482078 + 0.876128i \(0.660118\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 52.9282 1.45077 0.725384 − 0.688344i \(-0.241662\pi\)
0.725384 + 0.688344i \(0.241662\pi\)
\(12\) 0 0
\(13\) −8.43078 −0.179868 −0.0899338 − 0.995948i \(-0.528666\pi\)
−0.0899338 + 0.995948i \(0.528666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −129.138 −1.84239 −0.921196 − 0.389098i \(-0.872787\pi\)
−0.921196 + 0.389098i \(0.872787\pi\)
\(18\) 0 0
\(19\) −50.4974 −0.609732 −0.304866 − 0.952395i \(-0.598612\pi\)
−0.304866 + 0.952395i \(0.598612\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −128.708 −1.16684 −0.583422 − 0.812169i \(-0.698287\pi\)
−0.583422 + 0.812169i \(0.698287\pi\)
\(24\) 0 0
\(25\) 126.426 1.01141
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 111.282 0.712571 0.356285 − 0.934377i \(-0.384043\pi\)
0.356285 + 0.934377i \(0.384043\pi\)
\(30\) 0 0
\(31\) −302.851 −1.75464 −0.877318 − 0.479910i \(-0.840669\pi\)
−0.877318 + 0.479910i \(0.840669\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −283.138 −1.36740
\(36\) 0 0
\(37\) 182.995 0.813086 0.406543 − 0.913632i \(-0.366734\pi\)
0.406543 + 0.913632i \(0.366734\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 94.5744 0.360245 0.180122 − 0.983644i \(-0.442351\pi\)
0.180122 + 0.983644i \(0.442351\pi\)
\(42\) 0 0
\(43\) −184.641 −0.654825 −0.327413 − 0.944881i \(-0.606177\pi\)
−0.327413 + 0.944881i \(0.606177\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −296.841 −0.921249 −0.460624 − 0.887595i \(-0.652375\pi\)
−0.460624 + 0.887595i \(0.652375\pi\)
\(48\) 0 0
\(49\) −24.1487 −0.0704045
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −102.995 −0.266933 −0.133466 − 0.991053i \(-0.542611\pi\)
−0.133466 + 0.991053i \(0.542611\pi\)
\(54\) 0 0
\(55\) 839.251 2.05754
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −93.3693 −0.206028 −0.103014 − 0.994680i \(-0.532849\pi\)
−0.103014 + 0.994680i \(0.532849\pi\)
\(60\) 0 0
\(61\) 338.974 0.711495 0.355748 − 0.934582i \(-0.384226\pi\)
0.355748 + 0.934582i \(0.384226\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −133.682 −0.255095
\(66\) 0 0
\(67\) −489.041 −0.891729 −0.445865 − 0.895100i \(-0.647104\pi\)
−0.445865 + 0.895100i \(0.647104\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 86.9845 0.145397 0.0726983 − 0.997354i \(-0.476839\pi\)
0.0726983 + 0.997354i \(0.476839\pi\)
\(72\) 0 0
\(73\) −154.267 −0.247336 −0.123668 − 0.992324i \(-0.539466\pi\)
−0.123668 + 0.992324i \(0.539466\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −945.108 −1.39877
\(78\) 0 0
\(79\) −449.415 −0.640040 −0.320020 − 0.947411i \(-0.603690\pi\)
−0.320020 + 0.947411i \(0.603690\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 383.933 0.507737 0.253868 − 0.967239i \(-0.418297\pi\)
0.253868 + 0.967239i \(0.418297\pi\)
\(84\) 0 0
\(85\) −2047.67 −2.61295
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 517.672 0.616551 0.308276 − 0.951297i \(-0.400248\pi\)
0.308276 + 0.951297i \(0.400248\pi\)
\(90\) 0 0
\(91\) 150.543 0.173420
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −800.708 −0.864746
\(96\) 0 0
\(97\) 1739.39 1.82071 0.910355 − 0.413829i \(-0.135809\pi\)
0.910355 + 0.413829i \(0.135809\pi\)
\(98\) 0 0
\(99\) 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 1152.4.a.s.1.2 2
3.2 odd 2 128.4.a.e.1.2 ✓ 2
4.3 odd 2 1152.4.a.t.1.2 2
8.3 odd 2 1152.4.a.r.1.1 2
8.5 even 2 1152.4.a.q.1.1 2
12.11 even 2 128.4.a.g.1.1 yes 2
24.5 odd 2 128.4.a.h.1.1 yes 2
24.11 even 2 128.4.a.f.1.2 yes 2
48.5 odd 4 256.4.b.i.129.2 4
48.11 even 4 256.4.b.h.129.3 4
48.29 odd 4 256.4.b.i.129.3 4
48.35 even 4 256.4.b.h.129.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.a.e.1.2 ✓ 2 3.2 odd 2
128.4.a.f.1.2 yes 2 24.11 even 2
128.4.a.g.1.1 yes 2 12.11 even 2
128.4.a.h.1.1 yes 2 24.5 odd 2
256.4.b.h.129.2 4 48.35 even 4
256.4.b.h.129.3 4 48.11 even 4
256.4.b.i.129.2 4 48.5 odd 4
256.4.b.i.129.3 4 48.29 odd 4
1152.4.a.q.1.1 2 8.5 even 2
1152.4.a.r.1.1 2 8.3 odd 2
1152.4.a.s.1.2 2 1.1 even 1 trivial
1152.4.a.t.1.2 2 4.3 odd 2