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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4608,2,Mod(2305,4608)] 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("4608.2305"); 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([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,-16,0,-4] 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2305.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2305
Dual form 4608.2.d.o.2305.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.585786i q^{5} +3.41421 q^{7} -2.00000i q^{11} +2.82843i q^{13} -3.65685 q^{17} +5.65685i q^{19} -1.17157 q^{23} +4.65685 q^{25} -0.585786i q^{29} -4.58579 q^{31} +2.00000i q^{35} +9.65685i q^{37} -11.6569 q^{41} +1.65685i q^{43} +12.4853 q^{47} +4.65685 q^{49} +11.8995i q^{53} +1.17157 q^{55} +4.00000i q^{59} -9.65685i q^{61} -1.65685 q^{65} +8.00000i q^{67} -9.17157 q^{71} +1.65685 q^{73} -6.82843i q^{77} -5.75736 q^{79} +9.31371i q^{83} -2.14214i q^{85} +2.00000 q^{89} +9.65685i q^{91} -3.31371 q^{95} +13.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} + 8 q^{17} - 16 q^{23} - 4 q^{25} - 24 q^{31} - 24 q^{41} + 16 q^{47} - 4 q^{49} + 16 q^{55} + 16 q^{65} - 48 q^{71} - 16 q^{73} - 40 q^{79} + 8 q^{89} + 32 q^{95} + 8 q^{97}+O(q^{100}) \) 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
\(5\) 0.585786i 0.261972i 0.991384 + 0.130986i \(0.0418142\pi\)
−0.991384 + 0.130986i \(0.958186\pi\)
\(6\) 0 0
\(7\) 3.41421 1.29045 0.645226 − 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i 0.919866 + 0.392232i \(0.128297\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.65685 −0.886917 −0.443459 − 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) 5.65685i 1.29777i 0.760886 + 0.648886i \(0.224765\pi\)
−0.760886 + 0.648886i \(0.775235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.17157 −0.244290 −0.122145 − 0.992512i \(-0.538977\pi\)
−0.122145 + 0.992512i \(0.538977\pi\)
\(24\) 0 0
\(25\) 4.65685 0.931371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.585786i − 0.108778i −0.998520 − 0.0543889i \(-0.982679\pi\)
0.998520 − 0.0543889i \(-0.0173211\pi\)
\(30\) 0 0
\(31\) −4.58579 −0.823632 −0.411816 − 0.911267i \(-0.635105\pi\)
−0.411816 + 0.911267i \(0.635105\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) 9.65685i 1.58758i 0.608194 + 0.793789i \(0.291894\pi\)
−0.608194 + 0.793789i \(0.708106\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.6569 −1.82049 −0.910247 − 0.414065i \(-0.864109\pi\)
−0.910247 + 0.414065i \(0.864109\pi\)
\(42\) 0 0
\(43\) 1.65685i 0.252668i 0.991988 + 0.126334i \(0.0403211\pi\)
−0.991988 + 0.126334i \(0.959679\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.4853 1.82117 0.910583 − 0.413327i \(-0.135633\pi\)
0.910583 + 0.413327i \(0.135633\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.8995i 1.63452i 0.576268 + 0.817261i \(0.304508\pi\)
−0.576268 + 0.817261i \(0.695492\pi\)
\(54\) 0 0
\(55\) 1.17157 0.157975
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) − 9.65685i − 1.23643i −0.786008 − 0.618217i \(-0.787855\pi\)
0.786008 − 0.618217i \(-0.212145\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.65685 −0.205507
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.17157 −1.08847 −0.544233 − 0.838934i \(-0.683179\pi\)
−0.544233 + 0.838934i \(0.683179\pi\)
\(72\) 0 0
\(73\) 1.65685 0.193920 0.0969601 − 0.995288i \(-0.469088\pi\)
0.0969601 + 0.995288i \(0.469088\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6.82843i − 0.778171i
\(78\) 0 0
\(79\) −5.75736 −0.647754 −0.323877 − 0.946099i \(-0.604986\pi\)
−0.323877 + 0.946099i \(0.604986\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.31371i 1.02231i 0.859488 + 0.511156i \(0.170783\pi\)
−0.859488 + 0.511156i \(0.829217\pi\)
\(84\) 0 0
\(85\) − 2.14214i − 0.232347i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 9.65685i 1.01231i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.31371 −0.339979
\(96\) 0 0
\(97\) 13.3137 1.35180 0.675901 − 0.736992i \(-0.263755\pi\)
0.675901 + 0.736992i \(0.263755\pi\)
\(98\) 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.d.o.2305.3 4
3.2 odd 2 1536.2.d.f.769.2 4
4.3 odd 2 4608.2.d.c.2305.3 4
8.3 odd 2 4608.2.d.c.2305.2 4
8.5 even 2 inner 4608.2.d.o.2305.2 4
12.11 even 2 1536.2.d.a.769.4 4
16.3 odd 4 4608.2.a.r.1.1 2
16.5 even 4 4608.2.a.a.1.2 2
16.11 odd 4 4608.2.a.e.1.2 2
16.13 even 4 4608.2.a.n.1.1 2
24.5 odd 2 1536.2.d.f.769.3 4
24.11 even 2 1536.2.d.a.769.1 4
48.5 odd 4 1536.2.a.e.1.1 yes 2
48.11 even 4 1536.2.a.l.1.1 yes 2
48.29 odd 4 1536.2.a.g.1.2 yes 2
48.35 even 4 1536.2.a.b.1.2 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.2 ✓ 2 48.35 even 4
1536.2.a.e.1.1 yes 2 48.5 odd 4
1536.2.a.g.1.2 yes 2 48.29 odd 4
1536.2.a.l.1.1 yes 2 48.11 even 4
1536.2.d.a.769.1 4 24.11 even 2
1536.2.d.a.769.4 4 12.11 even 2
1536.2.d.f.769.2 4 3.2 odd 2
1536.2.d.f.769.3 4 24.5 odd 2
4608.2.a.a.1.2 2 16.5 even 4
4608.2.a.e.1.2 2 16.11 odd 4
4608.2.a.n.1.1 2 16.13 even 4
4608.2.a.r.1.1 2 16.3 odd 4
4608.2.d.c.2305.2 4 8.3 odd 2
4608.2.d.c.2305.3 4 4.3 odd 2
4608.2.d.o.2305.2 4 8.5 even 2 inner
4608.2.d.o.2305.3 4 1.1 even 1 trivial