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(4607,4608)] 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("4608.4607"); 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, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.c (of order \(2\), degree \(1\), 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] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == 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 4607.6
Root \(-0.382683 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 4608.4607
Dual form 4608.2.c.q.4607.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.53073i q^{5} +0.585786i q^{7} +5.22625 q^{11} -5.22625 q^{13} -1.41421i q^{17} -3.06147i q^{19} +0.828427 q^{23} +2.65685 q^{25} -5.86030i q^{29} -6.24264i q^{31} -0.896683 q^{35} -3.06147 q^{37} -3.75736i q^{41} -4.32957i q^{43} -8.82843 q^{47} +6.65685 q^{49} -5.86030i q^{53} +8.00000i q^{55} -4.32957 q^{59} +7.39104 q^{61} -8.00000i q^{65} +13.5140i q^{67} +10.4853 q^{71} +7.65685 q^{73} +3.06147i q^{77} -2.92893i q^{79} -9.55582 q^{83} +2.16478 q^{85} -7.07107i q^{89} -3.06147i q^{91} +4.68629 q^{95} +11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{23} - 24 q^{25} - 48 q^{47} + 8 q^{49} + 16 q^{71} + 16 q^{73} + 128 q^{95}+O(q^{100}) \) 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
\(5\) 1.53073i 0.684565i 0.939597 + 0.342282i \(0.111200\pi\)
−0.939597 + 0.342282i \(0.888800\pi\)
\(6\) 0 0
\(7\) 0.585786i 0.221406i 0.993854 + 0.110703i \(0.0353103\pi\)
−0.993854 + 0.110703i \(0.964690\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.22625 1.57577 0.787887 − 0.615820i \(-0.211175\pi\)
0.787887 + 0.615820i \(0.211175\pi\)
\(12\) 0 0
\(13\) −5.22625 −1.44950 −0.724751 − 0.689011i \(-0.758045\pi\)
−0.724751 + 0.689011i \(0.758045\pi\)
\(14\) 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
\(19\) − 3.06147i − 0.702349i −0.936310 − 0.351174i \(-0.885782\pi\)
0.936310 − 0.351174i \(-0.114218\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.828427 0.172739 0.0863695 − 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) 0 0
\(25\) 2.65685 0.531371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 5.86030i − 1.08823i −0.839010 − 0.544115i \(-0.816865\pi\)
0.839010 − 0.544115i \(-0.183135\pi\)
\(30\) 0 0
\(31\) − 6.24264i − 1.12121i −0.828083 − 0.560606i \(-0.810568\pi\)
0.828083 − 0.560606i \(-0.189432\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.896683 −0.151567
\(36\) 0 0
\(37\) −3.06147 −0.503302 −0.251651 − 0.967818i \(-0.580974\pi\)
−0.251651 + 0.967818i \(0.580974\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.75736i − 0.586801i −0.955990 − 0.293400i \(-0.905213\pi\)
0.955990 − 0.293400i \(-0.0947869\pi\)
\(42\) 0 0
\(43\) − 4.32957i − 0.660253i −0.943937 − 0.330127i \(-0.892909\pi\)
0.943937 − 0.330127i \(-0.107091\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.82843 −1.28776 −0.643879 − 0.765127i \(-0.722676\pi\)
−0.643879 + 0.765127i \(0.722676\pi\)
\(48\) 0 0
\(49\) 6.65685 0.950979
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.86030i − 0.804974i −0.915426 − 0.402487i \(-0.868146\pi\)
0.915426 − 0.402487i \(-0.131854\pi\)
\(54\) 0 0
\(55\) 8.00000i 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.32957 −0.563662 −0.281831 − 0.959464i \(-0.590942\pi\)
−0.281831 + 0.959464i \(0.590942\pi\)
\(60\) 0 0
\(61\) 7.39104 0.946325 0.473163 − 0.880975i \(-0.343112\pi\)
0.473163 + 0.880975i \(0.343112\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 8.00000i − 0.992278i
\(66\) 0 0
\(67\) 13.5140i 1.65099i 0.564406 + 0.825497i \(0.309105\pi\)
−0.564406 + 0.825497i \(0.690895\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.4853 1.24437 0.622187 − 0.782869i \(-0.286244\pi\)
0.622187 + 0.782869i \(0.286244\pi\)
\(72\) 0 0
\(73\) 7.65685 0.896167 0.448084 − 0.893992i \(-0.352107\pi\)
0.448084 + 0.893992i \(0.352107\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.06147i 0.348887i
\(78\) 0 0
\(79\) − 2.92893i − 0.329531i −0.986333 − 0.164765i \(-0.947313\pi\)
0.986333 − 0.164765i \(-0.0526867\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.55582 −1.04889 −0.524444 − 0.851445i \(-0.675727\pi\)
−0.524444 + 0.851445i \(0.675727\pi\)
\(84\) 0 0
\(85\) 2.16478 0.234804
\(86\) 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
\(91\) − 3.06147i − 0.320929i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.68629 0.480803
\(96\) 0 0
\(97\) 11.3137 1.14873 0.574367 − 0.818598i \(-0.305248\pi\)
0.574367 + 0.818598i \(0.305248\pi\)
\(98\) 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.c.q.4607.6 yes 8
3.2 odd 2 4608.2.c.r.4607.4 yes 8
4.3 odd 2 4608.2.c.r.4607.5 yes 8
8.3 odd 2 4608.2.c.r.4607.3 yes 8
8.5 even 2 inner 4608.2.c.q.4607.4 yes 8
12.11 even 2 inner 4608.2.c.q.4607.3 ✓ 8
16.3 odd 4 4608.2.f.n.2303.6 8
16.5 even 4 4608.2.f.o.2303.3 8
16.11 odd 4 4608.2.f.n.2303.4 8
16.13 even 4 4608.2.f.o.2303.5 8
24.5 odd 2 4608.2.c.r.4607.6 yes 8
24.11 even 2 inner 4608.2.c.q.4607.5 yes 8
48.5 odd 4 4608.2.f.n.2303.5 8
48.11 even 4 4608.2.f.o.2303.6 8
48.29 odd 4 4608.2.f.n.2303.3 8
48.35 even 4 4608.2.f.o.2303.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.q.4607.3 ✓ 8 12.11 even 2 inner
4608.2.c.q.4607.4 yes 8 8.5 even 2 inner
4608.2.c.q.4607.5 yes 8 24.11 even 2 inner
4608.2.c.q.4607.6 yes 8 1.1 even 1 trivial
4608.2.c.r.4607.3 yes 8 8.3 odd 2
4608.2.c.r.4607.4 yes 8 3.2 odd 2
4608.2.c.r.4607.5 yes 8 4.3 odd 2
4608.2.c.r.4607.6 yes 8 24.5 odd 2
4608.2.f.n.2303.3 8 48.29 odd 4
4608.2.f.n.2303.4 8 16.11 odd 4
4608.2.f.n.2303.5 8 48.5 odd 4
4608.2.f.n.2303.6 8 16.3 odd 4
4608.2.f.o.2303.3 8 16.5 even 4
4608.2.f.o.2303.4 8 48.35 even 4
4608.2.f.o.2303.5 8 16.13 even 4
4608.2.f.o.2303.6 8 48.11 even 4