Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3744,2,Mod(1585,3744)] 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("3744.1585"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 936)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1585.14
Root \(-0.752864 - 0.902863i\) of defining polynomial
Character \(\chi\) \(=\) 3744.1585
Dual form 3744.2.m.h.1585.16

$q$-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)
 
\(f(q)\) \(=\) \(q+2.68999 q^{5} -4.15163i q^{7} +4.35250 q^{11} +(3.53159 - 0.726543i) q^{13} +5.87130 q^{17} +5.71423 q^{19} -3.62866 q^{23} +2.23607 q^{25} +3.08672i q^{29} -9.28334i q^{31} -11.1679i q^{35} -2.69790 q^{37} +11.1074i q^{41} +3.80423i q^{43} +4.91034i q^{47} -10.2361 q^{49} -1.17902i q^{53} +11.7082 q^{55} +2.29753 q^{59} +7.05342i q^{61} +(9.49996 - 1.95440i) q^{65} -10.0795 q^{67} +2.08191i q^{71} +13.4350i q^{73} -18.0700i q^{77} -10.9443 q^{79} -9.73249 q^{83} +15.7938 q^{85} -12.1877i q^{89} +(-3.01634 - 14.6619i) q^{91} +15.3713 q^{95} +5.13170i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 128 q^{49} + 80 q^{55} - 32 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.68999 1.20300 0.601501 − 0.798872i \(-0.294570\pi\)
0.601501 + 0.798872i \(0.294570\pi\)
\(6\) 0 0
\(7\) − 4.15163i − 1.56917i −0.620021 − 0.784585i \(-0.712876\pi\)
0.620021 − 0.784585i \(-0.287124\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.35250 1.31233 0.656164 − 0.754618i \(-0.272178\pi\)
0.656164 + 0.754618i \(0.272178\pi\)
\(12\) 0 0
\(13\) 3.53159 − 0.726543i 0.979487 − 0.201507i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.87130 1.42400 0.711999 − 0.702180i \(-0.247790\pi\)
0.711999 + 0.702180i \(0.247790\pi\)
\(18\) 0 0
\(19\) 5.71423 1.31094 0.655468 − 0.755223i \(-0.272472\pi\)
0.655468 + 0.755223i \(0.272472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.62866 −0.756628 −0.378314 − 0.925677i \(-0.623496\pi\)
−0.378314 + 0.925677i \(0.623496\pi\)
\(24\) 0 0
\(25\) 2.23607 0.447214
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.08672i 0.573190i 0.958052 + 0.286595i \(0.0925234\pi\)
−0.958052 + 0.286595i \(0.907477\pi\)
\(30\) 0 0
\(31\) − 9.28334i − 1.66734i −0.552266 − 0.833668i \(-0.686237\pi\)
0.552266 − 0.833668i \(-0.313763\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 11.1679i − 1.88771i
\(36\) 0 0
\(37\) −2.69790 −0.443531 −0.221766 − 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.1074i 1.73468i 0.497715 + 0.867340i \(0.334172\pi\)
−0.497715 + 0.867340i \(0.665828\pi\)
\(42\) 0 0
\(43\) 3.80423i 0.580139i 0.957006 + 0.290070i \(0.0936784\pi\)
−0.957006 + 0.290070i \(0.906322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.91034i 0.716247i 0.933674 + 0.358123i \(0.116583\pi\)
−0.933674 + 0.358123i \(0.883417\pi\)
\(48\) 0 0
\(49\) −10.2361 −1.46230
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.17902i − 0.161951i −0.996716 − 0.0809757i \(-0.974196\pi\)
0.996716 − 0.0809757i \(-0.0258036\pi\)
\(54\) 0 0
\(55\) 11.7082 1.57873
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.29753 0.299113 0.149556 − 0.988753i \(-0.452215\pi\)
0.149556 + 0.988753i \(0.452215\pi\)
\(60\) 0 0
\(61\) 7.05342i 0.903098i 0.892246 + 0.451549i \(0.149128\pi\)
−0.892246 + 0.451549i \(0.850872\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.49996 − 1.95440i 1.17832 − 0.242413i
\(66\) 0 0
\(67\) −10.0795 −1.23141 −0.615705 − 0.787977i \(-0.711129\pi\)
−0.615705 + 0.787977i \(0.711129\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.08191i 0.247078i 0.992340 + 0.123539i \(0.0394244\pi\)
−0.992340 + 0.123539i \(0.960576\pi\)
\(72\) 0 0
\(73\) 13.4350i 1.57244i 0.617944 + 0.786222i \(0.287966\pi\)
−0.617944 + 0.786222i \(0.712034\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 18.0700i − 2.05927i
\(78\) 0 0
\(79\) −10.9443 −1.23133 −0.615663 − 0.788009i \(-0.711112\pi\)
−0.615663 + 0.788009i \(0.711112\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.73249 −1.06828 −0.534140 − 0.845396i \(-0.679364\pi\)
−0.534140 + 0.845396i \(0.679364\pi\)
\(84\) 0 0
\(85\) 15.7938 1.71307
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 12.1877i − 1.29190i −0.763381 − 0.645949i \(-0.776462\pi\)
0.763381 − 0.645949i \(-0.223538\pi\)
\(90\) 0 0
\(91\) −3.01634 − 14.6619i −0.316198 − 1.53698i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.3713 1.57706
\(96\) 0 0
\(97\) 5.13170i 0.521045i 0.965468 + 0.260523i \(0.0838949\pi\)
−0.965468 + 0.260523i \(0.916105\pi\)
\(98\) 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 3744.2.m.h.1585.14 16
3.2 odd 2 inner 3744.2.m.h.1585.2 16
4.3 odd 2 936.2.m.h.181.6 yes 16
8.3 odd 2 936.2.m.h.181.10 yes 16
8.5 even 2 inner 3744.2.m.h.1585.1 16
12.11 even 2 936.2.m.h.181.12 yes 16
13.12 even 2 inner 3744.2.m.h.1585.3 16
24.5 odd 2 inner 3744.2.m.h.1585.13 16
24.11 even 2 936.2.m.h.181.8 yes 16
39.38 odd 2 inner 3744.2.m.h.1585.15 16
52.51 odd 2 936.2.m.h.181.11 yes 16
104.51 odd 2 936.2.m.h.181.7 yes 16
104.77 even 2 inner 3744.2.m.h.1585.16 16
156.155 even 2 936.2.m.h.181.5 ✓ 16
312.77 odd 2 inner 3744.2.m.h.1585.4 16
312.155 even 2 936.2.m.h.181.9 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
936.2.m.h.181.5 ✓ 16 156.155 even 2
936.2.m.h.181.6 yes 16 4.3 odd 2
936.2.m.h.181.7 yes 16 104.51 odd 2
936.2.m.h.181.8 yes 16 24.11 even 2
936.2.m.h.181.9 yes 16 312.155 even 2
936.2.m.h.181.10 yes 16 8.3 odd 2
936.2.m.h.181.11 yes 16 52.51 odd 2
936.2.m.h.181.12 yes 16 12.11 even 2
3744.2.m.h.1585.1 16 8.5 even 2 inner
3744.2.m.h.1585.2 16 3.2 odd 2 inner
3744.2.m.h.1585.3 16 13.12 even 2 inner
3744.2.m.h.1585.4 16 312.77 odd 2 inner
3744.2.m.h.1585.13 16 24.5 odd 2 inner
3744.2.m.h.1585.14 16 1.1 even 1 trivial
3744.2.m.h.1585.15 16 39.38 odd 2 inner
3744.2.m.h.1585.16 16 104.77 even 2 inner