↑

Properties

Label 4896.2.c.m.577.1
Level $4896$
Weight $2$
Character 4896.577
Analytic conductor $39.095$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4896,2,Mod(577,4896)] 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("4896.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4896, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4896.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,-4,0,0,0,0,0,0,0,-12,0,0,0,0, 0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(35)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.0947568296\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2 + \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) 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 544)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.1
Root \(-1.84776i\) of defining polynomial
Character \(\chi\) \(=\) 4896.577
Dual form 4896.2.c.m.577.4

$q$-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)
 
\(f(q)\) \(=\) \(q-3.69552i q^{5} +1.08239i q^{7} +2.61313i q^{11} +4.82843 q^{13} +(-3.82843 - 1.53073i) q^{17} +5.65685 q^{19} -1.08239i q^{23} -8.65685 q^{25} +6.75699i q^{29} +8.47343i q^{31} +4.00000 q^{35} -3.69552i q^{37} -3.06147i q^{41} +9.65685 q^{43} +9.65685 q^{47} +5.82843 q^{49} +2.00000 q^{53} +9.65685 q^{55} +1.65685 q^{59} +8.02509i q^{61} -17.8435i q^{65} -13.6569 q^{67} +10.6382i q^{71} +10.4525i q^{73} -2.82843 q^{77} -3.24718i q^{79} +1.65685 q^{83} +(-5.65685 + 14.1480i) q^{85} +6.48528 q^{89} +5.22625i q^{91} -20.9050i q^{95} -13.5140i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{13} - 4 q^{17} - 12 q^{25} + 16 q^{35} + 16 q^{43} + 16 q^{47} + 12 q^{49} + 8 q^{53} + 16 q^{55} - 16 q^{59} - 32 q^{67} - 16 q^{83} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4896\mathbb{Z}\right)^\times\).

\(n\) \(613\) \(2143\) \(3809\) \(4321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(-0.690417\pi\)
0.563167 − 0.826343i \(-0.309583\pi\)
\(6\) 0 0
\(7\) 1.08239i 0.409106i 0.978856 + 0.204553i \(0.0655740\pi\)
−0.978856 + 0.204553i \(0.934426\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.61313i 0.787887i 0.919135 + 0.393944i \(0.128889\pi\)
−0.919135 + 0.393944i \(0.871111\pi\)
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 − 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.82843 − 1.53073i −0.928530 − 0.371257i
\(18\) 0 0
\(19\) 5.65685 1.29777 0.648886 − 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 1.08239i − 0.225694i −0.993612 − 0.112847i \(-0.964003\pi\)
0.993612 − 0.112847i \(-0.0359971\pi\)
\(24\) 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
\(31\) 8.47343i 1.52187i 0.648827 + 0.760936i \(0.275260\pi\)
−0.648827 + 0.760936i \(0.724740\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) − 3.69552i − 0.607539i −0.952745 − 0.303770i \(-0.901755\pi\)
0.952745 − 0.303770i \(-0.0982453\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.06147i − 0.478121i −0.971005 − 0.239060i \(-0.923161\pi\)
0.971005 − 0.239060i \(-0.0768394\pi\)
\(42\) 0 0
\(43\) 9.65685 1.47266 0.736328 − 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.65685 1.40860 0.704298 − 0.709904i \(-0.251262\pi\)
0.704298 + 0.709904i \(0.251262\pi\)
\(48\) 0 0
\(49\) 5.82843 0.832632
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 9.65685 1.30213
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.65685 0.215704 0.107852 − 0.994167i \(-0.465603\pi\)
0.107852 + 0.994167i \(0.465603\pi\)
\(60\) 0 0
\(61\) 8.02509i 1.02751i 0.857938 + 0.513754i \(0.171745\pi\)
−0.857938 + 0.513754i \(0.828255\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 17.8435i − 2.21322i
\(66\) 0 0
\(67\) −13.6569 −1.66845 −0.834225 − 0.551424i \(-0.814085\pi\)
−0.834225 + 0.551424i \(0.814085\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.6382i 1.26252i 0.775570 + 0.631262i \(0.217463\pi\)
−0.775570 + 0.631262i \(0.782537\pi\)
\(72\) 0 0
\(73\) 10.4525i 1.22337i 0.791100 + 0.611687i \(0.209509\pi\)
−0.791100 + 0.611687i \(0.790491\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) − 3.24718i − 0.365336i −0.983175 − 0.182668i \(-0.941527\pi\)
0.983175 − 0.182668i \(-0.0584733\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.65685 0.181863 0.0909317 − 0.995857i \(-0.471016\pi\)
0.0909317 + 0.995857i \(0.471016\pi\)
\(84\) 0 0
\(85\) −5.65685 + 14.1480i −0.613572 + 1.53457i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.48528 0.687438 0.343719 − 0.939072i \(-0.388313\pi\)
0.343719 + 0.939072i \(0.388313\pi\)
\(90\) 0 0
\(91\) 5.22625i 0.547860i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 20.9050i − 2.14481i
\(96\) 0 0
\(97\) − 13.5140i − 1.37214i −0.727537 − 0.686068i \(-0.759335\pi\)
0.727537 − 0.686068i \(-0.240665\pi\)
\(98\) 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 4896.2.c.m.577.1 4
3.2 odd 2 544.2.b.e.33.1 yes 4
4.3 odd 2 4896.2.c.l.577.1 4
12.11 even 2 544.2.b.d.33.4 yes 4
17.16 even 2 inner 4896.2.c.m.577.4 4
24.5 odd 2 1088.2.b.m.577.4 4
24.11 even 2 1088.2.b.l.577.1 4
51.38 odd 4 9248.2.a.bd.1.1 4
51.47 odd 4 9248.2.a.bd.1.4 4
51.50 odd 2 544.2.b.e.33.4 yes 4
68.67 odd 2 4896.2.c.l.577.4 4
204.47 even 4 9248.2.a.be.1.1 4
204.191 even 4 9248.2.a.be.1.4 4
204.203 even 2 544.2.b.d.33.1 ✓ 4
408.101 odd 2 1088.2.b.m.577.1 4
408.203 even 2 1088.2.b.l.577.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
544.2.b.d.33.1 ✓ 4 204.203 even 2
544.2.b.d.33.4 yes 4 12.11 even 2
544.2.b.e.33.1 yes 4 3.2 odd 2
544.2.b.e.33.4 yes 4 51.50 odd 2
1088.2.b.l.577.1 4 24.11 even 2
1088.2.b.l.577.4 4 408.203 even 2
1088.2.b.m.577.1 4 408.101 odd 2
1088.2.b.m.577.4 4 24.5 odd 2
4896.2.c.l.577.1 4 4.3 odd 2
4896.2.c.l.577.4 4 68.67 odd 2
4896.2.c.m.577.1 4 1.1 even 1 trivial
4896.2.c.m.577.4 4 17.16 even 2 inner
9248.2.a.bd.1.1 4 51.38 odd 4
9248.2.a.bd.1.4 4 51.47 odd 4
9248.2.a.be.1.1 4 204.47 even 4
9248.2.a.be.1.4 4 204.191 even 4