Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2304,2,Mod(1151,2304)] 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("2304.1151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.3975326257\)
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: \(\Z[a_1, \ldots, a_{17}]\)
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 1151.3
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1151
Dual form 2304.2.f.c.1151.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{5} -4.00000i q^{7} +5.65685i q^{11} +4.00000i q^{13} +4.24264i q^{17} -5.65685 q^{23} -3.00000 q^{25} +1.41421 q^{29} +4.00000i q^{31} -5.65685i q^{35} +6.00000i q^{37} +9.89949i q^{41} -8.00000 q^{43} +5.65685 q^{47} -9.00000 q^{49} +4.24264 q^{53} +8.00000i q^{55} -11.3137i q^{59} -2.00000i q^{61} +5.65685i q^{65} +8.00000 q^{67} +5.65685 q^{71} +22.6274 q^{77} -4.00000i q^{79} +5.65685i q^{83} +6.00000i q^{85} +4.24264i q^{89} +16.0000 q^{91} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{25} - 32 q^{43} - 36 q^{49} + 32 q^{67} + 64 q^{91} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) 1.41421 0.632456 0.316228 − 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 − 0.755929i \(-0.727186\pi\)
0.654654 − 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685i 1.70561i 0.522233 + 0.852803i \(0.325099\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 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
\(17\) 4.24264i 1.02899i 0.857493 + 0.514496i \(0.172021\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.65685 −1.17954 −0.589768 − 0.807573i \(-0.700781\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41421 0.262613 0.131306 − 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 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
\(35\) − 5.65685i − 0.956183i
\(36\) 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 \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.65685 0.825137 0.412568 − 0.910927i \(-0.364632\pi\)
0.412568 + 0.910927i \(0.364632\pi\)
\(48\) 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
\(55\) 8.00000i 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 11.3137i − 1.47292i −0.676481 − 0.736460i \(-0.736496\pi\)
0.676481 − 0.736460i \(-0.263504\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 0.256074i −0.991769 − 0.128037i \(-0.959132\pi\)
0.991769 − 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.65685i 0.701646i
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 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 \(-0.927756\pi\)
0.974355 − 0.225018i \(-0.0722440\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.65685i 0.620920i 0.950586 + 0.310460i \(0.100483\pi\)
−0.950586 + 0.310460i \(0.899517\pi\)
\(84\) 0 0
\(85\) 6.00000i 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.24264i 0.449719i 0.974391 + 0.224860i \(0.0721923\pi\)
−0.974391 + 0.224860i \(0.927808\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 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 2304.2.f.c.1151.3 4
3.2 odd 2 inner 2304.2.f.c.1151.1 4
4.3 odd 2 2304.2.f.e.1151.4 4
8.3 odd 2 inner 2304.2.f.c.1151.2 4
8.5 even 2 2304.2.f.e.1151.1 4
12.11 even 2 2304.2.f.e.1151.2 4
16.3 odd 4 576.2.c.c.575.1 4
16.5 even 4 288.2.c.a.287.4 yes 4
16.11 odd 4 288.2.c.a.287.3 yes 4
16.13 even 4 576.2.c.c.575.2 4
24.5 odd 2 2304.2.f.e.1151.3 4
24.11 even 2 inner 2304.2.f.c.1151.4 4
48.5 odd 4 288.2.c.a.287.2 yes 4
48.11 even 4 288.2.c.a.287.1 ✓ 4
48.29 odd 4 576.2.c.c.575.4 4
48.35 even 4 576.2.c.c.575.3 4
80.27 even 4 7200.2.o.n.7199.3 4
80.37 odd 4 7200.2.o.a.7199.1 4
80.43 even 4 7200.2.o.a.7199.4 4
80.53 odd 4 7200.2.o.n.7199.2 4
80.59 odd 4 7200.2.h.d.1151.4 4
80.69 even 4 7200.2.h.d.1151.1 4
144.5 odd 12 2592.2.s.d.1727.1 8
144.11 even 12 2592.2.s.d.863.3 8
144.43 odd 12 2592.2.s.d.863.1 8
144.59 even 12 2592.2.s.d.1727.2 8
144.85 even 12 2592.2.s.d.1727.3 8
144.101 odd 12 2592.2.s.d.863.4 8
144.133 even 12 2592.2.s.d.863.2 8
144.139 odd 12 2592.2.s.d.1727.4 8
240.53 even 4 7200.2.o.n.7199.4 4
240.59 even 4 7200.2.h.d.1151.3 4
240.107 odd 4 7200.2.o.n.7199.1 4
240.149 odd 4 7200.2.h.d.1151.2 4
240.197 even 4 7200.2.o.a.7199.3 4
240.203 odd 4 7200.2.o.a.7199.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.c.a.287.1 ✓ 4 48.11 even 4
288.2.c.a.287.2 yes 4 48.5 odd 4
288.2.c.a.287.3 yes 4 16.11 odd 4
288.2.c.a.287.4 yes 4 16.5 even 4
576.2.c.c.575.1 4 16.3 odd 4
576.2.c.c.575.2 4 16.13 even 4
576.2.c.c.575.3 4 48.35 even 4
576.2.c.c.575.4 4 48.29 odd 4
2304.2.f.c.1151.1 4 3.2 odd 2 inner
2304.2.f.c.1151.2 4 8.3 odd 2 inner
2304.2.f.c.1151.3 4 1.1 even 1 trivial
2304.2.f.c.1151.4 4 24.11 even 2 inner
2304.2.f.e.1151.1 4 8.5 even 2
2304.2.f.e.1151.2 4 12.11 even 2
2304.2.f.e.1151.3 4 24.5 odd 2
2304.2.f.e.1151.4 4 4.3 odd 2
2592.2.s.d.863.1 8 144.43 odd 12
2592.2.s.d.863.2 8 144.133 even 12
2592.2.s.d.863.3 8 144.11 even 12
2592.2.s.d.863.4 8 144.101 odd 12
2592.2.s.d.1727.1 8 144.5 odd 12
2592.2.s.d.1727.2 8 144.59 even 12
2592.2.s.d.1727.3 8 144.85 even 12
2592.2.s.d.1727.4 8 144.139 odd 12
7200.2.h.d.1151.1 4 80.69 even 4
7200.2.h.d.1151.2 4 240.149 odd 4
7200.2.h.d.1151.3 4 240.59 even 4
7200.2.h.d.1151.4 4 80.59 odd 4
7200.2.o.a.7199.1 4 80.37 odd 4
7200.2.o.a.7199.2 4 240.203 odd 4
7200.2.o.a.7199.3 4 240.197 even 4
7200.2.o.a.7199.4 4 80.43 even 4
7200.2.o.n.7199.1 4 240.107 odd 4
7200.2.o.n.7199.2 4 80.53 odd 4
7200.2.o.n.7199.3 4 80.27 even 4
7200.2.o.n.7199.4 4 240.53 even 4