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Properties

Label 888.2.c.a.443.13
Level $888$
Weight $2$
Character 888.443
Analytic conductor $7.091$
Analytic rank $0$
Dimension $16$
CM discriminant -111
Inner twists $8$

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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] = [888,2,Mod(443,888)] 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("888.443"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{8} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 443.13
Root \(1.19106 + 0.762485i\) of defining polynomial
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.a.443.14

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.19106 - 0.762485i) q^{2} -1.73205 q^{3} +(0.837235 - 1.81633i) q^{4} -2.96213i q^{5} +(-2.06297 + 1.32066i) q^{6} -2.65929i q^{7} +(-0.387726 - 2.80173i) q^{8} +3.00000 q^{9} +(-2.25858 - 3.52806i) q^{10} +(-1.45013 + 3.14597i) q^{12} +(-2.02766 - 3.16736i) q^{14} +5.13056i q^{15} +(-2.59808 - 3.04138i) q^{16} +1.03899 q^{17} +(3.57317 - 2.28745i) q^{18} +(-5.38019 - 2.48000i) q^{20} +4.60602i q^{21} +8.24478i q^{23} +(0.671561 + 4.85273i) q^{24} -3.77420 q^{25} -5.19615 q^{27} +(-4.83013 - 2.22645i) q^{28} -2.08062i q^{29} +(3.91197 + 6.11079i) q^{30} +(-5.41346 - 1.64147i) q^{32} +(1.23750 - 0.792216i) q^{34} -7.87714 q^{35} +(2.51170 - 5.44898i) q^{36} -6.08276i q^{37} +(-8.29907 + 1.14849i) q^{40} +(3.51202 + 5.48603i) q^{42} -8.88638i q^{45} +(6.28651 + 9.82000i) q^{46} +(4.50000 + 5.26783i) q^{48} -0.0717968 q^{49} +(-4.49529 + 2.87777i) q^{50} -1.79959 q^{51} +(-6.18891 + 3.96199i) q^{54} +(-7.45059 + 1.03107i) q^{56} +(-1.58644 - 2.47814i) q^{58} -15.3317 q^{59} +(9.31876 + 4.29548i) q^{60} -7.97786i q^{63} +(-7.69934 + 2.17260i) q^{64} +11.2726 q^{67} +(0.869881 - 1.88715i) q^{68} -14.2804i q^{69} +(-9.38213 + 6.00620i) q^{70} +(-1.16318 - 8.40518i) q^{72} +17.0731 q^{73} +(-4.63801 - 7.24492i) q^{74} +6.53711 q^{75} +(-9.00896 + 7.69583i) q^{80} +9.00000 q^{81} +(8.36603 + 3.85632i) q^{84} -3.07763i q^{85} +3.60374i q^{87} -13.1533 q^{89} +(-6.77573 - 10.5842i) q^{90} +(14.9752 + 6.90281i) q^{92} +(9.37639 + 2.84310i) q^{96} +(-0.0855141 + 0.0547439i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{9} - 80 q^{25} - 8 q^{28} - 24 q^{30} - 40 q^{34} - 56 q^{40} + 72 q^{48} - 112 q^{49} + 88 q^{58} + 104 q^{70} + 144 q^{81} + 120 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19106 − 0.762485i 0.842205 − 0.539158i
\(3\) −1.73205 −1.00000
\(4\) 0.837235 − 1.81633i 0.418617 − 0.908163i
\(5\) − 2.96213i − 1.32470i −0.749193 − 0.662352i \(-0.769558\pi\)
0.749193 − 0.662352i \(-0.230442\pi\)
\(6\) −2.06297 + 1.32066i −0.842205 + 0.539158i
\(7\) − 2.65929i − 1.00512i −0.864544 − 0.502558i \(-0.832392\pi\)
0.864544 − 0.502558i \(-0.167608\pi\)
\(8\) −0.387726 − 2.80173i −0.137082 − 0.990560i
\(9\) 3.00000 1.00000
\(10\) −2.25858 − 3.52806i −0.714225 − 1.11567i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.45013 + 3.14597i −0.418617 + 0.908163i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −2.02766 − 3.16736i −0.541916 − 0.846513i
\(15\) 5.13056i 1.32470i
\(16\) −2.59808 − 3.04138i −0.649519 − 0.760345i
\(17\) 1.03899 0.251993 0.125996 − 0.992031i \(-0.459787\pi\)
0.125996 + 0.992031i \(0.459787\pi\)
\(18\) 3.57317 − 2.28745i 0.842205 − 0.539158i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −5.38019 − 2.48000i −1.20305 − 0.554544i
\(21\) 4.60602i 1.00512i
\(22\) 0 0
\(23\) 8.24478i 1.71915i 0.511006 + 0.859577i \(0.329273\pi\)
−0.511006 + 0.859577i \(0.670727\pi\)
\(24\) 0.671561 + 4.85273i 0.137082 + 0.990560i
\(25\) −3.77420 −0.754841
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) −4.83013 − 2.22645i −0.912808 − 0.420759i
\(29\) − 2.08062i − 0.386361i −0.981163 − 0.193181i \(-0.938120\pi\)
0.981163 − 0.193181i \(-0.0618803\pi\)
\(30\) 3.91197 + 6.11079i 0.714225 + 1.11567i
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.41346 − 1.64147i −0.956974 − 0.290173i
\(33\) 0 0
\(34\) 1.23750 − 0.792216i 0.212230 − 0.135864i
\(35\) −7.87714 −1.33148
\(36\) 2.51170 − 5.44898i 0.418617 − 0.908163i
\(37\) − 6.08276i − 1.00000i
\(38\) 0 0
\(39\) 0 0
\(40\) −8.29907 + 1.14849i −1.31220 + 0.181593i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 3.51202 + 5.48603i 0.541916 + 0.846513i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) − 8.88638i − 1.32470i
\(46\) 6.28651 + 9.82000i 0.926896 + 1.44788i
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 4.50000 + 5.26783i 0.649519 + 0.760345i
\(49\) −0.0717968 −0.0102567
\(50\) −4.49529 + 2.87777i −0.635730 + 0.406978i
\(51\) −1.79959 −0.251993
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −6.18891 + 3.96199i −0.842205 + 0.539158i
\(55\) 0 0
\(56\) −7.45059 + 1.03107i −0.995627 + 0.137783i
\(57\) 0 0
\(58\) −1.58644 − 2.47814i −0.208310 − 0.325395i
\(59\) −15.3317 −1.99601 −0.998007 − 0.0630973i \(-0.979902\pi\)
−0.998007 + 0.0630973i \(0.979902\pi\)
\(60\) 9.31876 + 4.29548i 1.20305 + 0.554544i
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) − 7.97786i − 1.00512i
\(64\) −7.69934 + 2.17260i −0.962417 + 0.271575i
\(65\) 0 0
\(66\) 0 0
\(67\) 11.2726 1.37717 0.688584 − 0.725156i \(-0.258232\pi\)
0.688584 + 0.725156i \(0.258232\pi\)
\(68\) 0.869881 − 1.88715i 0.105489 − 0.228851i
\(69\) − 14.2804i − 1.71915i
\(70\) −9.38213 + 6.00620i −1.12138 + 0.717878i
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.16318 − 8.40518i −0.137082 − 0.990560i
\(73\) 17.0731 1.99826 0.999130 − 0.0416990i \(-0.0132771\pi\)
0.999130 + 0.0416990i \(0.0132771\pi\)
\(74\) −4.63801 − 7.24492i −0.539158 − 0.842205i
\(75\) 6.53711 0.754841
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −9.00896 + 7.69583i −1.00723 + 0.860420i
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 8.36603 + 3.85632i 0.912808 + 0.420759i
\(85\) − 3.07763i − 0.333816i
\(86\) 0 0
\(87\) 3.60374i 0.386361i
\(88\) 0 0
\(89\) −13.1533 −1.39425 −0.697123 − 0.716952i \(-0.745537\pi\)
−0.697123 + 0.716952i \(0.745537\pi\)
\(90\) −6.77573 − 10.5842i −0.714225 − 1.11567i
\(91\) 0 0
\(92\) 14.9752 + 6.90281i 1.56127 + 0.719668i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 9.37639 + 2.84310i 0.956974 + 0.290173i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.0855141 + 0.0547439i −0.00863822 + 0.00552997i
\(99\) 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 888.2.c.a.443.13 yes 16
3.2 odd 2 inner 888.2.c.a.443.4 yes 16
8.3 odd 2 inner 888.2.c.a.443.14 yes 16
24.11 even 2 inner 888.2.c.a.443.3 ✓ 16
37.36 even 2 inner 888.2.c.a.443.4 yes 16
111.110 odd 2 CM 888.2.c.a.443.13 yes 16
296.147 odd 2 inner 888.2.c.a.443.3 ✓ 16
888.443 even 2 inner 888.2.c.a.443.14 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.a.443.3 ✓ 16 24.11 even 2 inner
888.2.c.a.443.3 ✓ 16 296.147 odd 2 inner
888.2.c.a.443.4 yes 16 3.2 odd 2 inner
888.2.c.a.443.4 yes 16 37.36 even 2 inner
888.2.c.a.443.13 yes 16 1.1 even 1 trivial
888.2.c.a.443.13 yes 16 111.110 odd 2 CM
888.2.c.a.443.14 yes 16 8.3 odd 2 inner
888.2.c.a.443.14 yes 16 888.443 even 2 inner