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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5328,2,Mod(2737,5328)] 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("5328.2737"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5328.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,4,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0, 0,0,0,0,0,0,-6,0,0,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(42.5442941969\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.49179812660224.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 6x^{7} + 53x^{6} - 46x^{5} + 18x^{4} + 12x^{3} + 196x^{2} - 112x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 296)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2737.1
Root \(0.279838 + 0.279838i\) of defining polynomial
Character \(\chi\) \(=\) 5328.2737
Dual form 5328.2.h.q.2737.10

$q$-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)
 
\(f(q)\) \(=\) \(q-3.19874i q^{5} +3.28371 q^{7} +1.70530 q^{11} +1.45481i q^{13} +4.77715i q^{17} +0.209741i q^{19} -8.73294i q^{23} -5.23196 q^{25} -2.65144i q^{29} +9.47491i q^{31} -10.5037i q^{35} +(3.65355 + 4.86329i) q^{37} +11.3457 q^{41} -8.67366i q^{43} +12.6681 q^{47} +3.78272 q^{49} +5.11378 q^{53} -5.45481i q^{55} +1.65780i q^{59} -1.90947i q^{61} +4.65355 q^{65} -1.90947 q^{67} -9.51127 q^{71} +6.10278 q^{73} +5.59969 q^{77} +3.62473i q^{79} -3.20221 q^{83} +15.2809 q^{85} -10.1727i q^{89} +4.77715i q^{91} +0.670907 q^{95} +10.1905i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{7} + 2 q^{11} + 4 q^{25} - 6 q^{37} + 26 q^{41} + 8 q^{47} + 14 q^{49} + 20 q^{53} + 4 q^{65} + 2 q^{67} - 4 q^{71} - 14 q^{73} - 36 q^{83} + 8 q^{85} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1333\) \(1999\) \(2369\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) − 3.19874i − 1.43052i −0.698858 − 0.715261i \(-0.746308\pi\)
0.698858 − 0.715261i \(-0.253692\pi\)
\(6\) 0 0
\(7\) 3.28371 1.24112 0.620562 − 0.784157i \(-0.286904\pi\)
0.620562 + 0.784157i \(0.286904\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.70530 0.514166 0.257083 − 0.966389i \(-0.417239\pi\)
0.257083 + 0.966389i \(0.417239\pi\)
\(12\) 0 0
\(13\) 1.45481i 0.403490i 0.979438 + 0.201745i \(0.0646613\pi\)
−0.979438 + 0.201745i \(0.935339\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.77715i 1.15863i 0.815104 + 0.579315i \(0.196680\pi\)
−0.815104 + 0.579315i \(0.803320\pi\)
\(18\) 0 0
\(19\) 0.209741i 0.0481179i 0.999711 + 0.0240589i \(0.00765894\pi\)
−0.999711 + 0.0240589i \(0.992341\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 8.73294i − 1.82094i −0.413571 − 0.910472i \(-0.635719\pi\)
0.413571 − 0.910472i \(-0.364281\pi\)
\(24\) 0 0
\(25\) −5.23196 −1.04639
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.65144i − 0.492360i −0.969224 − 0.246180i \(-0.920825\pi\)
0.969224 − 0.246180i \(-0.0791754\pi\)
\(30\) 0 0
\(31\) 9.47491i 1.70174i 0.525373 + 0.850872i \(0.323926\pi\)
−0.525373 + 0.850872i \(0.676074\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 10.5037i − 1.77545i
\(36\) 0 0
\(37\) 3.65355 + 4.86329i 0.600640 + 0.799520i
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.3457 1.77191 0.885953 − 0.463774i \(-0.153505\pi\)
0.885953 + 0.463774i \(0.153505\pi\)
\(42\) 0 0
\(43\) − 8.67366i − 1.32272i −0.750069 − 0.661360i \(-0.769980\pi\)
0.750069 − 0.661360i \(-0.230020\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.6681 1.84783 0.923915 − 0.382598i \(-0.124970\pi\)
0.923915 + 0.382598i \(0.124970\pi\)
\(48\) 0 0
\(49\) 3.78272 0.540389
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.11378 0.702432 0.351216 − 0.936295i \(-0.385768\pi\)
0.351216 + 0.936295i \(0.385768\pi\)
\(54\) 0 0
\(55\) − 5.45481i − 0.735526i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.65780i 0.215827i 0.994160 + 0.107914i \(0.0344170\pi\)
−0.994160 + 0.107914i \(0.965583\pi\)
\(60\) 0 0
\(61\) − 1.90947i − 0.244482i −0.992500 − 0.122241i \(-0.960992\pi\)
0.992500 − 0.122241i \(-0.0390081\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.65355 0.577202
\(66\) 0 0
\(67\) −1.90947 −0.233278 −0.116639 − 0.993174i \(-0.537212\pi\)
−0.116639 + 0.993174i \(0.537212\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.51127 −1.12878 −0.564390 − 0.825508i \(-0.690889\pi\)
−0.564390 + 0.825508i \(0.690889\pi\)
\(72\) 0 0
\(73\) 6.10278 0.714277 0.357138 − 0.934051i \(-0.383752\pi\)
0.357138 + 0.934051i \(0.383752\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.59969 0.638144
\(78\) 0 0
\(79\) 3.62473i 0.407814i 0.978990 + 0.203907i \(0.0653640\pi\)
−0.978990 + 0.203907i \(0.934636\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.20221 −0.351488 −0.175744 − 0.984436i \(-0.556233\pi\)
−0.175744 + 0.984436i \(0.556233\pi\)
\(84\) 0 0
\(85\) 15.2809 1.65744
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 10.1727i − 1.07830i −0.842209 − 0.539151i \(-0.818745\pi\)
0.842209 − 0.539151i \(-0.181255\pi\)
\(90\) 0 0
\(91\) 4.77715i 0.500782i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.670907 0.0688337
\(96\) 0 0
\(97\) 10.1905i 1.03469i 0.855778 + 0.517344i \(0.173079\pi\)
−0.855778 + 0.517344i \(0.826921\pi\)
\(98\) 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 5328.2.h.q.2737.1 10
3.2 odd 2 592.2.g.d.369.2 10
4.3 odd 2 2664.2.h.c.73.1 10
12.11 even 2 296.2.g.a.73.10 yes 10
24.5 odd 2 2368.2.g.p.961.9 10
24.11 even 2 2368.2.g.o.961.1 10
37.36 even 2 inner 5328.2.h.q.2737.10 10
111.110 odd 2 592.2.g.d.369.1 10
148.147 odd 2 2664.2.h.c.73.10 10
444.443 even 2 296.2.g.a.73.9 ✓ 10
888.221 odd 2 2368.2.g.p.961.10 10
888.443 even 2 2368.2.g.o.961.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.g.a.73.9 ✓ 10 444.443 even 2
296.2.g.a.73.10 yes 10 12.11 even 2
592.2.g.d.369.1 10 111.110 odd 2
592.2.g.d.369.2 10 3.2 odd 2
2368.2.g.o.961.1 10 24.11 even 2
2368.2.g.o.961.2 10 888.443 even 2
2368.2.g.p.961.9 10 24.5 odd 2
2368.2.g.p.961.10 10 888.221 odd 2
2664.2.h.c.73.1 10 4.3 odd 2
2664.2.h.c.73.10 10 148.147 odd 2
5328.2.h.q.2737.1 10 1.1 even 1 trivial
5328.2.h.q.2737.10 10 37.36 even 2 inner