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Properties

Label 1088.2.b.m.577.1
Level $1088$
Weight $2$
Character 1088.577
Analytic conductor $8.688$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1088,2,Mod(577,1088)] 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("1088.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1088, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,-8,0,16,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.68772373992\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 \(-0.765367i\) of defining polynomial
Character \(\chi\) \(=\) 1088.577
Dual form 1088.2.b.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)
 
\(f(q)\) \(=\) \(q-2.61313i q^{3} +3.69552i q^{5} -1.08239i q^{7} -3.82843 q^{9} -2.61313i q^{11} -4.82843 q^{13} +9.65685 q^{15} +(3.82843 - 1.53073i) q^{17} -5.65685 q^{19} -2.82843 q^{21} -1.08239i q^{23} -8.65685 q^{25} +2.16478i q^{27} -6.75699i q^{29} -8.47343i q^{31} -6.82843 q^{33} +4.00000 q^{35} -3.69552i q^{37} +12.6173i q^{39} -3.06147i q^{41} -9.65685 q^{43} -14.1480i q^{45} -9.65685 q^{47} +5.82843 q^{49} +(-4.00000 - 10.0042i) q^{51} +2.00000 q^{53} +9.65685 q^{55} +14.7821i q^{57} +1.65685 q^{59} +8.02509i q^{61} +4.14386i q^{63} -17.8435i q^{65} +13.6569 q^{67} -2.82843 q^{69} +10.6382i q^{71} -10.4525i q^{73} +22.6215i q^{75} -2.82843 q^{77} +3.24718i q^{79} -5.82843 q^{81} +1.65685 q^{83} +(5.65685 + 14.1480i) q^{85} -17.6569 q^{87} -6.48528 q^{89} +5.22625i q^{91} -22.1421 q^{93} -20.9050i q^{95} +13.5140i q^{97} +10.0042i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 8 q^{13} + 16 q^{15} + 4 q^{17} - 12 q^{25} - 16 q^{33} + 16 q^{35} - 16 q^{43} - 16 q^{47} + 12 q^{49} - 16 q^{51} + 8 q^{53} + 16 q^{55} - 16 q^{59} + 32 q^{67} - 12 q^{81} - 16 q^{83}+ \cdots - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1088\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.61313i − 1.50869i −0.656479 − 0.754344i \(-0.727955\pi\)
0.656479 − 0.754344i \(-0.272045\pi\)
\(4\) 0 0
\(5\) 3.69552i 1.65269i 0.563167 + 0.826343i \(0.309583\pi\)
−0.563167 + 0.826343i \(0.690417\pi\)
\(6\) 0 0
\(7\) − 1.08239i − 0.409106i −0.978856 − 0.204553i \(-0.934426\pi\)
0.978856 − 0.204553i \(-0.0655740\pi\)
\(8\) 0 0
\(9\) −3.82843 −1.27614
\(10\) 0 0
\(11\) − 2.61313i − 0.787887i −0.919135 − 0.393944i \(-0.871111\pi\)
0.919135 − 0.393944i \(-0.128889\pi\)
\(12\) 0 0
\(13\) −4.82843 −1.33916 −0.669582 − 0.742738i \(-0.733527\pi\)
−0.669582 + 0.742738i \(0.733527\pi\)
\(14\) 0 0
\(15\) 9.65685 2.49339
\(16\) 0 0
\(17\) 3.82843 − 1.53073i 0.928530 − 0.371257i
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 − 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 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
\(27\) 2.16478i 0.416613i
\(28\) 0 0
\(29\) − 6.75699i − 1.25474i −0.778721 − 0.627370i \(-0.784131\pi\)
0.778721 − 0.627370i \(-0.215869\pi\)
\(30\) 0 0
\(31\) − 8.47343i − 1.52187i −0.648827 − 0.760936i \(-0.724740\pi\)
0.648827 − 0.760936i \(-0.275260\pi\)
\(32\) 0 0
\(33\) −6.82843 −1.18868
\(34\) 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
\(39\) 12.6173i 2.02038i
\(40\) 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 \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) − 14.1480i − 2.10906i
\(46\) 0 0
\(47\) −9.65685 −1.40860 −0.704298 − 0.709904i \(-0.748738\pi\)
−0.704298 + 0.709904i \(0.748738\pi\)
\(48\) 0 0
\(49\) 5.82843 0.832632
\(50\) 0 0
\(51\) −4.00000 − 10.0042i −0.560112 − 1.40086i
\(52\) 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
\(57\) 14.7821i 1.95793i
\(58\) 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
\(63\) 4.14386i 0.522077i
\(64\) 0 0
\(65\) − 17.8435i − 2.21322i
\(66\) 0 0
\(67\) 13.6569 1.66845 0.834225 − 0.551424i \(-0.185915\pi\)
0.834225 + 0.551424i \(0.185915\pi\)
\(68\) 0 0
\(69\) −2.82843 −0.340503
\(70\) 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 \(-0.790491\pi\)
0.791100 − 0.611687i \(-0.209509\pi\)
\(74\) 0 0
\(75\) 22.6215i 2.61210i
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) 3.24718i 0.365336i 0.983175 + 0.182668i \(0.0584733\pi\)
−0.983175 + 0.182668i \(0.941527\pi\)
\(80\) 0 0
\(81\) −5.82843 −0.647603
\(82\) 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
\(87\) −17.6569 −1.89301
\(88\) 0 0
\(89\) −6.48528 −0.687438 −0.343719 − 0.939072i \(-0.611687\pi\)
−0.343719 + 0.939072i \(0.611687\pi\)
\(90\) 0 0
\(91\) 5.22625i 0.547860i
\(92\) 0 0
\(93\) −22.1421 −2.29603
\(94\) 0 0
\(95\) − 20.9050i − 2.14481i
\(96\) 0 0
\(97\) 13.5140i 1.37214i 0.727537 + 0.686068i \(0.240665\pi\)
−0.727537 + 0.686068i \(0.759335\pi\)
\(98\) 0 0
\(99\) 10.0042i 1.00546i
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 1088.2.b.m.577.1 4
4.3 odd 2 1088.2.b.l.577.4 4
8.3 odd 2 544.2.b.d.33.1 ✓ 4
8.5 even 2 544.2.b.e.33.4 yes 4
17.16 even 2 inner 1088.2.b.m.577.4 4
24.5 odd 2 4896.2.c.m.577.4 4
24.11 even 2 4896.2.c.l.577.4 4
68.67 odd 2 1088.2.b.l.577.1 4
136.13 even 4 9248.2.a.bd.1.1 4
136.21 even 4 9248.2.a.bd.1.4 4
136.67 odd 2 544.2.b.d.33.4 yes 4
136.101 even 2 544.2.b.e.33.1 yes 4
136.115 odd 4 9248.2.a.be.1.4 4
136.123 odd 4 9248.2.a.be.1.1 4
408.101 odd 2 4896.2.c.m.577.1 4
408.203 even 2 4896.2.c.l.577.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
544.2.b.d.33.1 ✓ 4 8.3 odd 2
544.2.b.d.33.4 yes 4 136.67 odd 2
544.2.b.e.33.1 yes 4 136.101 even 2
544.2.b.e.33.4 yes 4 8.5 even 2
1088.2.b.l.577.1 4 68.67 odd 2
1088.2.b.l.577.4 4 4.3 odd 2
1088.2.b.m.577.1 4 1.1 even 1 trivial
1088.2.b.m.577.4 4 17.16 even 2 inner
4896.2.c.l.577.1 4 408.203 even 2
4896.2.c.l.577.4 4 24.11 even 2
4896.2.c.m.577.1 4 408.101 odd 2
4896.2.c.m.577.4 4 24.5 odd 2
9248.2.a.bd.1.1 4 136.13 even 4
9248.2.a.bd.1.4 4 136.21 even 4
9248.2.a.be.1.1 4 136.123 odd 4
9248.2.a.be.1.4 4 136.115 odd 4