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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 = [2,0,0,0,0,0,0,0,0,0,8,0,-12,0,0,0,0,0,0,0,0,0,-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: \(\Z[a_1, \ldots, a_{19}]\)
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.g.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)
 
\(f(q)\) \(=\) \(q+4.24264i q^{7} +4.00000 q^{11} -6.00000 q^{13} -4.24264i q^{17} -2.82843i q^{19} -6.00000 q^{23} +5.00000 q^{25} -8.48528i q^{29} -4.24264i q^{31} -6.00000 q^{37} +1.41421i q^{41} -2.82843i q^{43} +6.00000 q^{47} -11.0000 q^{49} -8.48528i q^{53} -4.00000 q^{59} +6.00000 q^{61} -11.3137i q^{67} +6.00000 q^{71} +6.00000 q^{73} +16.9706i q^{77} -4.24264i q^{79} +16.0000 q^{83} +12.7279i q^{89} -25.4558i q^{91} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{11} - 12 q^{13} - 12 q^{23} + 10 q^{25} - 12 q^{37} + 12 q^{47} - 22 q^{49} - 8 q^{59} + 12 q^{61} + 12 q^{71} + 12 q^{73} + 32 q^{83} - 24 q^{97}+O(q^{100}) \) 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
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 4.24264i 1.60357i 0.597614 + 0.801784i \(0.296115\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−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 \(-0.827979\pi\)
0.857493 − 0.514496i \(-0.172021\pi\)
\(18\) 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
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.48528i − 1.57568i −0.615882 − 0.787839i \(-0.711200\pi\)
0.615882 − 0.787839i \(-0.288800\pi\)
\(30\) 0 0
\(31\) − 4.24264i − 0.762001i −0.924575 − 0.381000i \(-0.875580\pi\)
0.924575 − 0.381000i \(-0.124420\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.41421i 0.220863i 0.993884 + 0.110432i \(0.0352233\pi\)
−0.993884 + 0.110432i \(0.964777\pi\)
\(42\) 0 0
\(43\) − 2.82843i − 0.431331i −0.976467 − 0.215666i \(-0.930808\pi\)
0.976467 − 0.215666i \(-0.0691921\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 8.48528i − 1.16554i −0.812636 − 0.582772i \(-0.801968\pi\)
0.812636 − 0.582772i \(-0.198032\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.3137i − 1.38219i −0.722764 − 0.691095i \(-0.757129\pi\)
0.722764 − 0.691095i \(-0.242871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.9706i 1.93398i
\(78\) 0 0
\(79\) − 4.24264i − 0.477334i −0.971101 − 0.238667i \(-0.923290\pi\)
0.971101 − 0.238667i \(-0.0767105\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.7279i 1.34916i 0.738203 + 0.674579i \(0.235675\pi\)
−0.738203 + 0.674579i \(0.764325\pi\)
\(90\) 0 0
\(91\) − 25.4558i − 2.66850i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 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.g.4607.2 yes 2
3.2 odd 2 4608.2.c.a.4607.2 yes 2
4.3 odd 2 4608.2.c.a.4607.1 ✓ 2
8.3 odd 2 4608.2.c.h.4607.1 yes 2
8.5 even 2 4608.2.c.b.4607.2 yes 2
12.11 even 2 inner 4608.2.c.g.4607.1 yes 2
16.3 odd 4 4608.2.f.i.2303.4 4
16.5 even 4 4608.2.f.k.2303.2 4
16.11 odd 4 4608.2.f.i.2303.3 4
16.13 even 4 4608.2.f.k.2303.1 4
24.5 odd 2 4608.2.c.h.4607.2 yes 2
24.11 even 2 4608.2.c.b.4607.1 yes 2
48.5 odd 4 4608.2.f.i.2303.1 4
48.11 even 4 4608.2.f.k.2303.4 4
48.29 odd 4 4608.2.f.i.2303.2 4
48.35 even 4 4608.2.f.k.2303.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.a.4607.1 ✓ 2 4.3 odd 2
4608.2.c.a.4607.2 yes 2 3.2 odd 2
4608.2.c.b.4607.1 yes 2 24.11 even 2
4608.2.c.b.4607.2 yes 2 8.5 even 2
4608.2.c.g.4607.1 yes 2 12.11 even 2 inner
4608.2.c.g.4607.2 yes 2 1.1 even 1 trivial
4608.2.c.h.4607.1 yes 2 8.3 odd 2
4608.2.c.h.4607.2 yes 2 24.5 odd 2
4608.2.f.i.2303.1 4 48.5 odd 4
4608.2.f.i.2303.2 4 48.29 odd 4
4608.2.f.i.2303.3 4 16.11 odd 4
4608.2.f.i.2303.4 4 16.3 odd 4
4608.2.f.k.2303.1 4 16.13 even 4
4608.2.f.k.2303.2 4 16.5 even 4
4608.2.f.k.2303.3 4 48.35 even 4
4608.2.f.k.2303.4 4 48.11 even 4