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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 = [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 4607.8
Root \(-0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 4608.4607
Dual form 4608.2.c.q.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)
 
\(f(q)\) \(=\) \(q+3.69552i q^{5} +3.41421i q^{7} +2.16478 q^{11} -2.16478 q^{13} +1.41421i q^{17} -7.39104i q^{19} -4.82843 q^{23} -8.65685 q^{25} +6.75699i q^{29} +2.24264i q^{31} -12.6173 q^{35} -7.39104 q^{37} -12.2426i q^{41} +10.4525i q^{43} -3.17157 q^{47} -4.65685 q^{49} +6.75699i q^{53} +8.00000i q^{55} +10.4525 q^{59} -3.06147 q^{61} -8.00000i q^{65} +11.7206i q^{67} -6.48528 q^{71} -3.65685 q^{73} +7.39104i q^{77} -17.0711i q^{79} +8.28772 q^{83} -5.22625 q^{85} +7.07107i q^{89} -7.39104i q^{91} +27.3137 q^{95} -11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{23} - 24 q^{25} - 48 q^{47} + 8 q^{49} + 16 q^{71} + 16 q^{73} + 128 q^{95}+O(q^{100}) \) 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
\(5\) 3.69552i 1.65269i 0.563167 + 0.826343i \(0.309583\pi\)
−0.563167 + 0.826343i \(0.690417\pi\)
\(6\) 0 0
\(7\) 3.41421i 1.29045i 0.763992 + 0.645226i \(0.223237\pi\)
−0.763992 + 0.645226i \(0.776763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.16478 0.652707 0.326354 − 0.945248i \(-0.394180\pi\)
0.326354 + 0.945248i \(0.394180\pi\)
\(12\) 0 0
\(13\) −2.16478 −0.600403 −0.300202 − 0.953876i \(-0.597054\pi\)
−0.300202 + 0.953876i \(0.597054\pi\)
\(14\) 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
\(19\) − 7.39104i − 1.69562i −0.530300 − 0.847810i \(-0.677921\pi\)
0.530300 − 0.847810i \(-0.322079\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.82843 −1.00680 −0.503398 − 0.864054i \(-0.667917\pi\)
−0.503398 + 0.864054i \(0.667917\pi\)
\(24\) 0 0
\(25\) −8.65685 −1.73137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.75699i 1.25474i 0.778721 + 0.627370i \(0.215869\pi\)
−0.778721 + 0.627370i \(0.784131\pi\)
\(30\) 0 0
\(31\) 2.24264i 0.402790i 0.979510 + 0.201395i \(0.0645475\pi\)
−0.979510 + 0.201395i \(0.935452\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −12.6173 −2.13271
\(36\) 0 0
\(37\) −7.39104 −1.21508 −0.607539 − 0.794290i \(-0.707843\pi\)
−0.607539 + 0.794290i \(0.707843\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 12.2426i − 1.91198i −0.293400 − 0.955990i \(-0.594787\pi\)
0.293400 − 0.955990i \(-0.405213\pi\)
\(42\) 0 0
\(43\) 10.4525i 1.59399i 0.603985 + 0.796996i \(0.293579\pi\)
−0.603985 + 0.796996i \(0.706421\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.17157 −0.462621 −0.231311 − 0.972880i \(-0.574301\pi\)
−0.231311 + 0.972880i \(0.574301\pi\)
\(48\) 0 0
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.75699i 0.928143i 0.885798 + 0.464072i \(0.153612\pi\)
−0.885798 + 0.464072i \(0.846388\pi\)
\(54\) 0 0
\(55\) 8.00000i 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.4525 1.36080 0.680400 − 0.732841i \(-0.261806\pi\)
0.680400 + 0.732841i \(0.261806\pi\)
\(60\) 0 0
\(61\) −3.06147 −0.391981 −0.195990 − 0.980606i \(-0.562792\pi\)
−0.195990 + 0.980606i \(0.562792\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 8.00000i − 0.992278i
\(66\) 0 0
\(67\) 11.7206i 1.43190i 0.698152 + 0.715950i \(0.254006\pi\)
−0.698152 + 0.715950i \(0.745994\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.48528 −0.769661 −0.384831 − 0.922987i \(-0.625740\pi\)
−0.384831 + 0.922987i \(0.625740\pi\)
\(72\) 0 0
\(73\) −3.65685 −0.428002 −0.214001 − 0.976833i \(-0.568650\pi\)
−0.214001 + 0.976833i \(0.568650\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.39104i 0.842287i
\(78\) 0 0
\(79\) − 17.0711i − 1.92065i −0.278892 − 0.960323i \(-0.589967\pi\)
0.278892 − 0.960323i \(-0.410033\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.28772 0.909695 0.454848 − 0.890569i \(-0.349694\pi\)
0.454848 + 0.890569i \(0.349694\pi\)
\(84\) 0 0
\(85\) −5.22625 −0.566867
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.07107i 0.749532i 0.927119 + 0.374766i \(0.122277\pi\)
−0.927119 + 0.374766i \(0.877723\pi\)
\(90\) 0 0
\(91\) − 7.39104i − 0.774791i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 27.3137 2.80233
\(96\) 0 0
\(97\) −11.3137 −1.14873 −0.574367 − 0.818598i \(-0.694752\pi\)
−0.574367 + 0.818598i \(0.694752\pi\)
\(98\) 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.4607.8 yes 8
3.2 odd 2 4608.2.c.r.4607.2 yes 8
4.3 odd 2 4608.2.c.r.4607.7 yes 8
8.3 odd 2 4608.2.c.r.4607.1 yes 8
8.5 even 2 inner 4608.2.c.q.4607.2 yes 8
12.11 even 2 inner 4608.2.c.q.4607.1 ✓ 8
16.3 odd 4 4608.2.f.n.2303.8 8
16.5 even 4 4608.2.f.o.2303.1 8
16.11 odd 4 4608.2.f.n.2303.2 8
16.13 even 4 4608.2.f.o.2303.7 8
24.5 odd 2 4608.2.c.r.4607.8 yes 8
24.11 even 2 inner 4608.2.c.q.4607.7 yes 8
48.5 odd 4 4608.2.f.n.2303.7 8
48.11 even 4 4608.2.f.o.2303.8 8
48.29 odd 4 4608.2.f.n.2303.1 8
48.35 even 4 4608.2.f.o.2303.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.q.4607.1 ✓ 8 12.11 even 2 inner
4608.2.c.q.4607.2 yes 8 8.5 even 2 inner
4608.2.c.q.4607.7 yes 8 24.11 even 2 inner
4608.2.c.q.4607.8 yes 8 1.1 even 1 trivial
4608.2.c.r.4607.1 yes 8 8.3 odd 2
4608.2.c.r.4607.2 yes 8 3.2 odd 2
4608.2.c.r.4607.7 yes 8 4.3 odd 2
4608.2.c.r.4607.8 yes 8 24.5 odd 2
4608.2.f.n.2303.1 8 48.29 odd 4
4608.2.f.n.2303.2 8 16.11 odd 4
4608.2.f.n.2303.7 8 48.5 odd 4
4608.2.f.n.2303.8 8 16.3 odd 4
4608.2.f.o.2303.1 8 16.5 even 4
4608.2.f.o.2303.2 8 48.35 even 4
4608.2.f.o.2303.7 8 16.13 even 4
4608.2.f.o.2303.8 8 48.11 even 4