Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4608,2,Mod(2303,4608)] 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("4608.2303"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2303.8
Root \(-0.923880 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2303
Dual form 4608.2.f.n.2303.7

$q$-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)
 
\(f(q)\) \(=\) \(q+3.69552 q^{5} +3.41421i q^{7} +2.16478i q^{11} -2.16478i q^{13} +1.41421i q^{17} -7.39104 q^{19} -4.82843 q^{23} +8.65685 q^{25} -6.75699 q^{29} -2.24264i q^{31} +12.6173i q^{35} +7.39104i q^{37} +12.2426i q^{41} -10.4525 q^{43} +3.17157 q^{47} -4.65685 q^{49} +6.75699 q^{53} +8.00000i q^{55} +10.4525i q^{59} -3.06147i q^{61} -8.00000i q^{65} +11.7206 q^{67} -6.48528 q^{71} +3.65685 q^{73} -7.39104 q^{77} +17.0711i q^{79} -8.28772i q^{83} +5.22625i q^{85} -7.07107i q^{89} +7.39104 q^{91} -27.3137 q^{95} -11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{23} + 24 q^{25} + 48 q^{47} + 8 q^{49} + 16 q^{71} - 16 q^{73} - 128 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) 3.69552 1.65269 0.826343 − 0.563167i \(-0.190417\pi\)
0.826343 + 0.563167i \(0.190417\pi\)
\(6\) 0 0
\(7\) 3.41421i 1.29045i 0.763992 + 0.645226i \(0.223237\pi\)
−0.763992 + 0.645226i \(0.776763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.16478i 0.652707i 0.945248 + 0.326354i \(0.105820\pi\)
−0.945248 + 0.326354i \(0.894180\pi\)
\(12\) 0 0
\(13\) − 2.16478i − 0.600403i −0.953876 − 0.300202i \(-0.902946\pi\)
0.953876 − 0.300202i \(-0.0970540\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) −7.39104 −1.69562 −0.847810 − 0.530300i \(-0.822079\pi\)
−0.847810 + 0.530300i \(0.822079\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.82843 −1.00680 −0.503398 − 0.864054i \(-0.667917\pi\)
−0.503398 + 0.864054i \(0.667917\pi\)
\(24\) 0 0
\(25\) 8.65685 1.73137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.75699 −1.25474 −0.627370 − 0.778721i \(-0.715869\pi\)
−0.627370 + 0.778721i \(0.715869\pi\)
\(30\) 0 0
\(31\) − 2.24264i − 0.402790i −0.979510 − 0.201395i \(-0.935452\pi\)
0.979510 − 0.201395i \(-0.0645475\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.6173i 2.13271i
\(36\) 0 0
\(37\) 7.39104i 1.21508i 0.794290 + 0.607539i \(0.207843\pi\)
−0.794290 + 0.607539i \(0.792157\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.2426i 1.91198i 0.293400 + 0.955990i \(0.405213\pi\)
−0.293400 + 0.955990i \(0.594787\pi\)
\(42\) 0 0
\(43\) −10.4525 −1.59399 −0.796996 − 0.603985i \(-0.793579\pi\)
−0.796996 + 0.603985i \(0.793579\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.17157 0.462621 0.231311 − 0.972880i \(-0.425699\pi\)
0.231311 + 0.972880i \(0.425699\pi\)
\(48\) 0 0
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.75699 0.928143 0.464072 − 0.885798i \(-0.346388\pi\)
0.464072 + 0.885798i \(0.346388\pi\)
\(54\) 0 0
\(55\) 8.00000i 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.4525i 1.36080i 0.732841 + 0.680400i \(0.238194\pi\)
−0.732841 + 0.680400i \(0.761806\pi\)
\(60\) 0 0
\(61\) − 3.06147i − 0.391981i −0.980606 − 0.195990i \(-0.937208\pi\)
0.980606 − 0.195990i \(-0.0627921\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 8.00000i − 0.992278i
\(66\) 0 0
\(67\) 11.7206 1.43190 0.715950 − 0.698152i \(-0.245994\pi\)
0.715950 + 0.698152i \(0.245994\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.48528 −0.769661 −0.384831 − 0.922987i \(-0.625740\pi\)
−0.384831 + 0.922987i \(0.625740\pi\)
\(72\) 0 0
\(73\) 3.65685 0.428002 0.214001 − 0.976833i \(-0.431350\pi\)
0.214001 + 0.976833i \(0.431350\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.39104 −0.842287
\(78\) 0 0
\(79\) 17.0711i 1.92065i 0.278892 + 0.960323i \(0.410033\pi\)
−0.278892 + 0.960323i \(0.589967\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 8.28772i − 0.909695i −0.890569 − 0.454848i \(-0.849694\pi\)
0.890569 − 0.454848i \(-0.150306\pi\)
\(84\) 0 0
\(85\) 5.22625i 0.566867i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 7.07107i − 0.749532i −0.927119 − 0.374766i \(-0.877723\pi\)
0.927119 − 0.374766i \(-0.122277\pi\)
\(90\) 0 0
\(91\) 7.39104 0.774791
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −27.3137 −2.80233
\(96\) 0 0
\(97\) −11.3137 −1.14873 −0.574367 − 0.818598i \(-0.694752\pi\)
−0.574367 + 0.818598i \(0.694752\pi\)
\(98\) 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 4608.2.f.n.2303.8 8
3.2 odd 2 4608.2.f.o.2303.2 8
4.3 odd 2 4608.2.f.o.2303.7 8
8.3 odd 2 4608.2.f.o.2303.1 8
8.5 even 2 inner 4608.2.f.n.2303.2 8
12.11 even 2 inner 4608.2.f.n.2303.1 8
16.3 odd 4 4608.2.c.q.4607.2 yes 8
16.5 even 4 4608.2.c.r.4607.7 yes 8
16.11 odd 4 4608.2.c.q.4607.8 yes 8
16.13 even 4 4608.2.c.r.4607.1 yes 8
24.5 odd 2 4608.2.f.o.2303.8 8
24.11 even 2 inner 4608.2.f.n.2303.7 8
48.5 odd 4 4608.2.c.q.4607.1 ✓ 8
48.11 even 4 4608.2.c.r.4607.2 yes 8
48.29 odd 4 4608.2.c.q.4607.7 yes 8
48.35 even 4 4608.2.c.r.4607.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.q.4607.1 ✓ 8 48.5 odd 4
4608.2.c.q.4607.2 yes 8 16.3 odd 4
4608.2.c.q.4607.7 yes 8 48.29 odd 4
4608.2.c.q.4607.8 yes 8 16.11 odd 4
4608.2.c.r.4607.1 yes 8 16.13 even 4
4608.2.c.r.4607.2 yes 8 48.11 even 4
4608.2.c.r.4607.7 yes 8 16.5 even 4
4608.2.c.r.4607.8 yes 8 48.35 even 4
4608.2.f.n.2303.1 8 12.11 even 2 inner
4608.2.f.n.2303.2 8 8.5 even 2 inner
4608.2.f.n.2303.7 8 24.11 even 2 inner
4608.2.f.n.2303.8 8 1.1 even 1 trivial
4608.2.f.o.2303.1 8 8.3 odd 2
4608.2.f.o.2303.2 8 3.2 odd 2
4608.2.f.o.2303.7 8 4.3 odd 2
4608.2.f.o.2303.8 8 24.5 odd 2