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 = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,12] 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: \(4\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2303.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2303
Dual form 4608.2.f.l.2303.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82843 q^{5} -1.41421i q^{7} +2.00000i q^{13} -1.41421i q^{17} -2.82843 q^{19} +6.00000 q^{23} +3.00000 q^{25} -5.65685 q^{29} +9.89949i q^{31} +4.00000i q^{35} +2.00000i q^{37} -4.24264i q^{41} +8.48528 q^{43} -10.0000 q^{47} +5.00000 q^{49} -5.65685 q^{53} -12.0000i q^{59} +10.0000i q^{61} -5.65685i q^{65} -5.65685 q^{67} +2.00000 q^{71} +10.0000 q^{73} -12.7279i q^{79} +4.00000i q^{83} +4.00000i q^{85} -9.89949i q^{89} +2.82843 q^{91} +8.00000 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 24 q^{23} + 12 q^{25} - 40 q^{47} + 20 q^{49} + 8 q^{71} + 40 q^{73} + 32 q^{95} + 48 q^{97}+O(q^{100}) \) 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
\(5\) −2.82843 −1.26491 −0.632456 − 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) − 1.41421i − 0.534522i −0.963624 − 0.267261i \(-0.913881\pi\)
0.963624 − 0.267261i \(-0.0861187\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.41421i − 0.342997i −0.985184 − 0.171499i \(-0.945139\pi\)
0.985184 − 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 − 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.65685 −1.05045 −0.525226 − 0.850963i \(-0.676019\pi\)
−0.525226 + 0.850963i \(0.676019\pi\)
\(30\) 0 0
\(31\) 9.89949i 1.77800i 0.457905 + 0.889001i \(0.348600\pi\)
−0.457905 + 0.889001i \(0.651400\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000i 0.676123i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.24264i − 0.662589i −0.943527 − 0.331295i \(-0.892515\pi\)
0.943527 − 0.331295i \(-0.107485\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 − 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.0000 −1.45865 −0.729325 − 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.65685 −0.777029 −0.388514 − 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 − 0.781133i \(-0.714642\pi\)
0.624364 − 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) 10.0000i 1.28037i 0.768221 + 0.640184i \(0.221142\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 5.65685i − 0.701646i
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 − 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 12.7279i − 1.43200i −0.698099 − 0.716002i \(-0.745970\pi\)
0.698099 − 0.716002i \(-0.254030\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 4.00000i 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 9.89949i − 1.04934i −0.851304 − 0.524672i \(-0.824188\pi\)
0.851304 − 0.524672i \(-0.175812\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 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.l.2303.1 4
3.2 odd 2 4608.2.f.j.2303.3 4
4.3 odd 2 4608.2.f.j.2303.2 4
8.3 odd 2 4608.2.f.j.2303.4 4
8.5 even 2 inner 4608.2.f.l.2303.3 4
12.11 even 2 inner 4608.2.f.l.2303.4 4
16.3 odd 4 4608.2.c.d.4607.2 yes 2
16.5 even 4 4608.2.c.e.4607.1 yes 2
16.11 odd 4 4608.2.c.f.4607.1 yes 2
16.13 even 4 4608.2.c.c.4607.2 yes 2
24.5 odd 2 4608.2.f.j.2303.1 4
24.11 even 2 inner 4608.2.f.l.2303.2 4
48.5 odd 4 4608.2.c.f.4607.2 yes 2
48.11 even 4 4608.2.c.e.4607.2 yes 2
48.29 odd 4 4608.2.c.d.4607.1 yes 2
48.35 even 4 4608.2.c.c.4607.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.c.4607.1 ✓ 2 48.35 even 4
4608.2.c.c.4607.2 yes 2 16.13 even 4
4608.2.c.d.4607.1 yes 2 48.29 odd 4
4608.2.c.d.4607.2 yes 2 16.3 odd 4
4608.2.c.e.4607.1 yes 2 16.5 even 4
4608.2.c.e.4607.2 yes 2 48.11 even 4
4608.2.c.f.4607.1 yes 2 16.11 odd 4
4608.2.c.f.4607.2 yes 2 48.5 odd 4
4608.2.f.j.2303.1 4 24.5 odd 2
4608.2.f.j.2303.2 4 4.3 odd 2
4608.2.f.j.2303.3 4 3.2 odd 2
4608.2.f.j.2303.4 4 8.3 odd 2
4608.2.f.l.2303.1 4 1.1 even 1 trivial
4608.2.f.l.2303.2 4 24.11 even 2 inner
4608.2.f.l.2303.3 4 8.5 even 2 inner
4608.2.f.l.2303.4 4 12.11 even 2 inner