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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 = [2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,-12] 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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4607.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 4608.4607
Dual form 4608.2.c.e.4607.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.82843i q^{5} -1.41421i q^{7} +2.00000 q^{13} +1.41421i q^{17} -2.82843i q^{19} -6.00000 q^{23} -3.00000 q^{25} -5.65685i q^{29} -9.89949i q^{31} +4.00000 q^{35} -2.00000 q^{37} -4.24264i q^{41} -8.48528i q^{43} -10.0000 q^{47} +5.00000 q^{49} +5.65685i q^{53} +12.0000 q^{59} +10.0000 q^{61} +5.65685i q^{65} -5.65685i q^{67} -2.00000 q^{71} -10.0000 q^{73} +12.7279i q^{79} +4.00000 q^{83} -4.00000 q^{85} -9.89949i q^{89} -2.82843i q^{91} +8.00000 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{13} - 12 q^{23} - 6 q^{25} + 8 q^{35} - 4 q^{37} - 20 q^{47} + 10 q^{49} + 24 q^{59} + 20 q^{61} - 4 q^{71} - 20 q^{73} + 8 q^{83} - 8 q^{85} + 16 q^{95} + 24 q^{97}+O(q^{100}) \) 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
\(5\) 2.82843i 1.26491i 0.774597 + 0.632456i \(0.217953\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 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
\(19\) − 2.82843i − 0.648886i −0.945905 − 0.324443i \(-0.894823\pi\)
0.945905 − 0.324443i \(-0.105177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 5.65685i − 1.05045i −0.850963 − 0.525226i \(-0.823981\pi\)
0.850963 − 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) − 9.89949i − 1.77800i −0.457905 − 0.889001i \(-0.651400\pi\)
0.457905 − 0.889001i \(-0.348600\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 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
\(43\) − 8.48528i − 1.29399i −0.762493 − 0.646997i \(-0.776025\pi\)
0.762493 − 0.646997i \(-0.223975\pi\)
\(44\) 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
\(53\) 5.65685i 0.777029i 0.921443 + 0.388514i \(0.127012\pi\)
−0.921443 + 0.388514i \(0.872988\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.65685i 0.701646i
\(66\) 0 0
\(67\) − 5.65685i − 0.691095i −0.938401 − 0.345547i \(-0.887693\pi\)
0.938401 − 0.345547i \(-0.112307\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.7279i 1.43200i 0.698099 + 0.716002i \(0.254030\pi\)
−0.698099 + 0.716002i \(0.745970\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 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
\(91\) − 2.82843i − 0.296500i
\(92\) 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.c.e.4607.2 yes 2
3.2 odd 2 4608.2.c.f.4607.1 yes 2
4.3 odd 2 4608.2.c.f.4607.2 yes 2
8.3 odd 2 4608.2.c.d.4607.1 yes 2
8.5 even 2 4608.2.c.c.4607.1 ✓ 2
12.11 even 2 inner 4608.2.c.e.4607.1 yes 2
16.3 odd 4 4608.2.f.j.2303.3 4
16.5 even 4 4608.2.f.l.2303.2 4
16.11 odd 4 4608.2.f.j.2303.1 4
16.13 even 4 4608.2.f.l.2303.4 4
24.5 odd 2 4608.2.c.d.4607.2 yes 2
24.11 even 2 4608.2.c.c.4607.2 yes 2
48.5 odd 4 4608.2.f.j.2303.4 4
48.11 even 4 4608.2.f.l.2303.3 4
48.29 odd 4 4608.2.f.j.2303.2 4
48.35 even 4 4608.2.f.l.2303.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.c.4607.1 ✓ 2 8.5 even 2
4608.2.c.c.4607.2 yes 2 24.11 even 2
4608.2.c.d.4607.1 yes 2 8.3 odd 2
4608.2.c.d.4607.2 yes 2 24.5 odd 2
4608.2.c.e.4607.1 yes 2 12.11 even 2 inner
4608.2.c.e.4607.2 yes 2 1.1 even 1 trivial
4608.2.c.f.4607.1 yes 2 3.2 odd 2
4608.2.c.f.4607.2 yes 2 4.3 odd 2
4608.2.f.j.2303.1 4 16.11 odd 4
4608.2.f.j.2303.2 4 48.29 odd 4
4608.2.f.j.2303.3 4 16.3 odd 4
4608.2.f.j.2303.4 4 48.5 odd 4
4608.2.f.l.2303.1 4 48.35 even 4
4608.2.f.l.2303.2 4 16.5 even 4
4608.2.f.l.2303.3 4 48.11 even 4
4608.2.f.l.2303.4 4 16.13 even 4