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Properties

Label 2816.2.g.c.1407.5
Level $2816$
Weight $2$
Character 2816.1407
Analytic conductor $22.486$
Analytic rank $0$
Dimension $8$
CM discriminant -11
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] = [2816,2,Mod(1407,2816)] 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("2816.1407"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2816 = 2^{8} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2816.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.4858732092\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(i, \sqrt{3}, \sqrt{11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 176)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1407.5
Root \(1.26217 + 1.18614i\) of defining polynomial
Character \(\chi\) \(=\) 2816.1407
Dual form 2816.2.g.c.1407.6

$q$-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)
 
\(f(q)\) \(=\) \(q+0.792287 q^{3} -1.37228i q^{5} -2.37228 q^{9} -3.31662 q^{11} -1.08724i q^{15} -6.13592i q^{23} +3.11684 q^{25} -4.25639 q^{27} +9.30506i q^{31} -2.62772 q^{33} +12.1168i q^{37} +3.25544i q^{45} +6.63325i q^{47} -7.00000 q^{49} -6.00000i q^{53} +4.55134i q^{55} -14.6487 q^{59} -16.2333 q^{67} -4.86141i q^{69} +10.8896i q^{71} +2.46943 q^{75} +3.74456 q^{81} -18.8614 q^{89} +7.37228i q^{93} -0.116844 q^{97} +7.86797 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9} - 44 q^{25} - 44 q^{33} - 56 q^{49} - 16 q^{81} - 36 q^{89} + 68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(1541\) \(2047\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0.792287 0.457427 0.228714 − 0.973494i \(-0.426548\pi\)
0.228714 + 0.973494i \(0.426548\pi\)
\(4\) 0 0
\(5\) − 1.37228i − 0.613703i −0.951757 − 0.306851i \(-0.900725\pi\)
0.951757 − 0.306851i \(-0.0992755\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2.37228 −0.790760
\(10\) 0 0
\(11\) −3.31662 −1.00000
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) − 1.08724i − 0.280724i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.13592i − 1.27943i −0.768613 − 0.639713i \(-0.779053\pi\)
0.768613 − 0.639713i \(-0.220947\pi\)
\(24\) 0 0
\(25\) 3.11684 0.623369
\(26\) 0 0
\(27\) −4.25639 −0.819142
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 9.30506i 1.67124i 0.549309 + 0.835619i \(0.314891\pi\)
−0.549309 + 0.835619i \(0.685109\pi\)
\(32\) 0 0
\(33\) −2.62772 −0.457427
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 12.1168i 1.99200i 0.0893706 + 0.995998i \(0.471514\pi\)
−0.0893706 + 0.995998i \(0.528486\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 3.25544i 0.485292i
\(46\) 0 0
\(47\) 6.63325i 0.967559i 0.875190 + 0.483779i \(0.160736\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 4.55134i 0.613703i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14.6487 −1.90710 −0.953549 − 0.301239i \(-0.902600\pi\)
−0.953549 + 0.301239i \(0.902600\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −16.2333 −1.98321 −0.991605 − 0.129307i \(-0.958725\pi\)
−0.991605 + 0.129307i \(0.958725\pi\)
\(68\) 0 0
\(69\) − 4.86141i − 0.585245i
\(70\) 0 0
\(71\) 10.8896i 1.29236i 0.763184 + 0.646181i \(0.223635\pi\)
−0.763184 + 0.646181i \(0.776365\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 2.46943 0.285146
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 3.74456 0.416063
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.8614 −1.99931 −0.999653 − 0.0263586i \(-0.991609\pi\)
−0.999653 + 0.0263586i \(0.991609\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.37228i 0.764470i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.116844 −0.0118637 −0.00593185 − 0.999982i \(-0.501888\pi\)
−0.00593185 + 0.999982i \(0.501888\pi\)
\(98\) 0 0
\(99\) 7.86797 0.790760
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 2816.2.g.c.1407.5 8
4.3 odd 2 inner 2816.2.g.c.1407.3 8
8.3 odd 2 inner 2816.2.g.c.1407.6 8
8.5 even 2 inner 2816.2.g.c.1407.4 8
11.10 odd 2 CM 2816.2.g.c.1407.5 8
16.3 odd 4 704.2.e.c.703.2 4
16.5 even 4 176.2.e.b.175.2 ✓ 4
16.11 odd 4 176.2.e.b.175.3 yes 4
16.13 even 4 704.2.e.c.703.3 4
44.43 even 2 inner 2816.2.g.c.1407.3 8
48.5 odd 4 1584.2.o.e.703.2 4
48.11 even 4 1584.2.o.e.703.1 4
88.21 odd 2 inner 2816.2.g.c.1407.4 8
88.43 even 2 inner 2816.2.g.c.1407.6 8
176.21 odd 4 176.2.e.b.175.2 ✓ 4
176.43 even 4 176.2.e.b.175.3 yes 4
176.109 odd 4 704.2.e.c.703.3 4
176.131 even 4 704.2.e.c.703.2 4
528.197 even 4 1584.2.o.e.703.2 4
528.395 odd 4 1584.2.o.e.703.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
176.2.e.b.175.2 ✓ 4 16.5 even 4
176.2.e.b.175.2 ✓ 4 176.21 odd 4
176.2.e.b.175.3 yes 4 16.11 odd 4
176.2.e.b.175.3 yes 4 176.43 even 4
704.2.e.c.703.2 4 16.3 odd 4
704.2.e.c.703.2 4 176.131 even 4
704.2.e.c.703.3 4 16.13 even 4
704.2.e.c.703.3 4 176.109 odd 4
1584.2.o.e.703.1 4 48.11 even 4
1584.2.o.e.703.1 4 528.395 odd 4
1584.2.o.e.703.2 4 48.5 odd 4
1584.2.o.e.703.2 4 528.197 even 4
2816.2.g.c.1407.3 8 4.3 odd 2 inner
2816.2.g.c.1407.3 8 44.43 even 2 inner
2816.2.g.c.1407.4 8 8.5 even 2 inner
2816.2.g.c.1407.4 8 88.21 odd 2 inner
2816.2.g.c.1407.5 8 1.1 even 1 trivial
2816.2.g.c.1407.5 8 11.10 odd 2 CM
2816.2.g.c.1407.6 8 8.3 odd 2 inner
2816.2.g.c.1407.6 8 88.43 even 2 inner