Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7200,2,Mod(7199,7200)] 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("7200.7199"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0] 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: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 7199.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 7200.7199
Dual form 7200.2.o.n.7199.3

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{7} +5.65685 q^{11} +4.00000i q^{13} -4.24264 q^{17} -5.65685i q^{23} -1.41421i q^{29} +4.00000i q^{31} +6.00000i q^{37} +9.89949i q^{41} +8.00000 q^{43} +5.65685i q^{47} +9.00000 q^{49} +4.24264 q^{53} +11.3137 q^{59} -2.00000 q^{61} -8.00000 q^{67} +5.65685 q^{71} +22.6274 q^{77} +4.00000i q^{79} -5.65685i q^{83} -4.24264i q^{89} +16.0000i q^{91} +8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{7} + 32 q^{43} + 36 q^{49} - 8 q^{61} - 32 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685 1.70561 0.852803 − 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.24264 −1.02899 −0.514496 − 0.857493i \(-0.672021\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 5.65685i − 1.17954i −0.807573 − 0.589768i \(-0.799219\pi\)
0.807573 − 0.589768i \(-0.200781\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.41421i − 0.262613i −0.991342 − 0.131306i \(-0.958083\pi\)
0.991342 − 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.89949i 1.54604i 0.634381 + 0.773021i \(0.281255\pi\)
−0.634381 + 0.773021i \(0.718745\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.65685i 0.825137i 0.910927 + 0.412568i \(0.135368\pi\)
−0.910927 + 0.412568i \(0.864632\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.24264 0.582772 0.291386 − 0.956606i \(-0.405884\pi\)
0.291386 + 0.956606i \(0.405884\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 − 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 − 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 22.6274 2.57863
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 5.65685i − 0.620920i −0.950586 − 0.310460i \(-0.899517\pi\)
0.950586 − 0.310460i \(-0.100483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.24264i − 0.449719i −0.974391 − 0.224860i \(-0.927808\pi\)
0.974391 − 0.224860i \(-0.0721923\pi\)
\(90\) 0 0
\(91\) 16.0000i 1.67726i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 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 7200.2.o.n.7199.4 4
3.2 odd 2 inner 7200.2.o.n.7199.2 4
4.3 odd 2 7200.2.o.a.7199.2 4
5.2 odd 4 288.2.c.a.287.2 yes 4
5.3 odd 4 7200.2.h.d.1151.2 4
5.4 even 2 7200.2.o.a.7199.3 4
12.11 even 2 7200.2.o.a.7199.4 4
15.2 even 4 288.2.c.a.287.4 yes 4
15.8 even 4 7200.2.h.d.1151.1 4
15.14 odd 2 7200.2.o.a.7199.1 4
20.3 even 4 7200.2.h.d.1151.3 4
20.7 even 4 288.2.c.a.287.1 ✓ 4
20.19 odd 2 inner 7200.2.o.n.7199.1 4
40.27 even 4 576.2.c.c.575.3 4
40.37 odd 4 576.2.c.c.575.4 4
45.2 even 12 2592.2.s.d.863.2 8
45.7 odd 12 2592.2.s.d.863.4 8
45.22 odd 12 2592.2.s.d.1727.1 8
45.32 even 12 2592.2.s.d.1727.3 8
60.23 odd 4 7200.2.h.d.1151.4 4
60.47 odd 4 288.2.c.a.287.3 yes 4
60.59 even 2 inner 7200.2.o.n.7199.3 4
80.27 even 4 2304.2.f.c.1151.4 4
80.37 odd 4 2304.2.f.e.1151.3 4
80.67 even 4 2304.2.f.e.1151.2 4
80.77 odd 4 2304.2.f.c.1151.1 4
120.77 even 4 576.2.c.c.575.2 4
120.107 odd 4 576.2.c.c.575.1 4
180.7 even 12 2592.2.s.d.863.3 8
180.47 odd 12 2592.2.s.d.863.1 8
180.67 even 12 2592.2.s.d.1727.2 8
180.167 odd 12 2592.2.s.d.1727.4 8
240.77 even 4 2304.2.f.c.1151.3 4
240.107 odd 4 2304.2.f.c.1151.2 4
240.197 even 4 2304.2.f.e.1151.1 4
240.227 odd 4 2304.2.f.e.1151.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.c.a.287.1 ✓ 4 20.7 even 4
288.2.c.a.287.2 yes 4 5.2 odd 4
288.2.c.a.287.3 yes 4 60.47 odd 4
288.2.c.a.287.4 yes 4 15.2 even 4
576.2.c.c.575.1 4 120.107 odd 4
576.2.c.c.575.2 4 120.77 even 4
576.2.c.c.575.3 4 40.27 even 4
576.2.c.c.575.4 4 40.37 odd 4
2304.2.f.c.1151.1 4 80.77 odd 4
2304.2.f.c.1151.2 4 240.107 odd 4
2304.2.f.c.1151.3 4 240.77 even 4
2304.2.f.c.1151.4 4 80.27 even 4
2304.2.f.e.1151.1 4 240.197 even 4
2304.2.f.e.1151.2 4 80.67 even 4
2304.2.f.e.1151.3 4 80.37 odd 4
2304.2.f.e.1151.4 4 240.227 odd 4
2592.2.s.d.863.1 8 180.47 odd 12
2592.2.s.d.863.2 8 45.2 even 12
2592.2.s.d.863.3 8 180.7 even 12
2592.2.s.d.863.4 8 45.7 odd 12
2592.2.s.d.1727.1 8 45.22 odd 12
2592.2.s.d.1727.2 8 180.67 even 12
2592.2.s.d.1727.3 8 45.32 even 12
2592.2.s.d.1727.4 8 180.167 odd 12
7200.2.h.d.1151.1 4 15.8 even 4
7200.2.h.d.1151.2 4 5.3 odd 4
7200.2.h.d.1151.3 4 20.3 even 4
7200.2.h.d.1151.4 4 60.23 odd 4
7200.2.o.a.7199.1 4 15.14 odd 2
7200.2.o.a.7199.2 4 4.3 odd 2
7200.2.o.a.7199.3 4 5.4 even 2
7200.2.o.a.7199.4 4 12.11 even 2
7200.2.o.n.7199.1 4 20.19 odd 2 inner
7200.2.o.n.7199.2 4 3.2 odd 2 inner
7200.2.o.n.7199.3 4 60.59 even 2 inner
7200.2.o.n.7199.4 4 1.1 even 1 trivial