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Properties

Label 2352.2.k.i.881.16
Level $2352$
Weight $2$
Character 2352.881
Analytic conductor $18.781$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [2352,2,Mod(881,2352)] 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("2352.881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,-4,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,36,0,0,0, 0,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.7808145554\)
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: \( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.16
Root \(1.73018 - 0.0805675i\) of defining polynomial
Character \(\chi\) \(=\) 2352.881
Dual form 2352.2.k.i.881.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.53866 + 0.795315i) q^{3} -3.80034 q^{5} +(1.73495 + 2.44744i) q^{9} -0.357425i q^{11} +4.04570i q^{13} +(-5.84742 - 3.02246i) q^{15} -0.103938 q^{17} -2.45507i q^{19} -1.33007i q^{23} +9.44255 q^{25} +(0.723015 + 5.14560i) q^{27} -4.97265i q^{29} +7.88669i q^{31} +(0.284265 - 0.549955i) q^{33} -10.9124 q^{37} +(-3.21760 + 6.22495i) q^{39} -6.15464 q^{41} -0.502751 q^{43} +(-6.59339 - 9.30108i) q^{45} -11.4516 q^{47} +(-0.159925 - 0.0826633i) q^{51} -5.86753i q^{53} +1.35833i q^{55} +(1.95255 - 3.77751i) q^{57} -7.54728 q^{59} -9.47414i q^{61} -15.3750i q^{65} -2.68750 q^{67} +(1.05783 - 2.04653i) q^{69} +5.78975i q^{71} -0.235473i q^{73} +(14.5289 + 7.50980i) q^{75} -3.22495 q^{79} +(-2.97990 + 8.49236i) q^{81} -9.07747 q^{83} +0.394999 q^{85} +(3.95482 - 7.65122i) q^{87} +6.82427 q^{89} +(-6.27240 + 12.1349i) q^{93} +9.33007i q^{95} -5.14243i q^{97} +(0.874774 - 0.620114i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{9} - 8 q^{15} + 36 q^{25} + 4 q^{37} - 44 q^{39} - 20 q^{43} + 12 q^{51} - 8 q^{57} + 28 q^{67} + 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.53866 + 0.795315i 0.888346 + 0.459175i
\(4\) 0 0
\(5\) −3.80034 −1.69956 −0.849781 − 0.527136i \(-0.823266\pi\)
−0.849781 + 0.527136i \(0.823266\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.73495 + 2.44744i 0.578317 + 0.815812i
\(10\) 0 0
\(11\) − 0.357425i − 0.107768i −0.998547 − 0.0538838i \(-0.982840\pi\)
0.998547 − 0.0538838i \(-0.0171600\pi\)
\(12\) 0 0
\(13\) 4.04570i 1.12207i 0.827791 + 0.561037i \(0.189597\pi\)
−0.827791 + 0.561037i \(0.810403\pi\)
\(14\) 0 0
\(15\) −5.84742 − 3.02246i −1.50980 − 0.780396i
\(16\) 0 0
\(17\) −0.103938 −0.0252086 −0.0126043 − 0.999921i \(-0.504012\pi\)
−0.0126043 + 0.999921i \(0.504012\pi\)
\(18\) 0 0
\(19\) − 2.45507i − 0.563231i −0.959527 − 0.281615i \(-0.909130\pi\)
0.959527 − 0.281615i \(-0.0908702\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 1.33007i − 0.277340i −0.990339 − 0.138670i \(-0.955717\pi\)
0.990339 − 0.138670i \(-0.0442827\pi\)
\(24\) 0 0
\(25\) 9.44255 1.88851
\(26\) 0 0
\(27\) 0.723015 + 5.14560i 0.139144 + 0.990272i
\(28\) 0 0
\(29\) − 4.97265i − 0.923398i −0.887037 − 0.461699i \(-0.847240\pi\)
0.887037 − 0.461699i \(-0.152760\pi\)
\(30\) 0 0
\(31\) 7.88669i 1.41649i 0.705966 + 0.708246i \(0.250513\pi\)
−0.705966 + 0.708246i \(0.749487\pi\)
\(32\) 0 0
\(33\) 0.284265 − 0.549955i 0.0494842 − 0.0957348i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.9124 −1.79400 −0.896998 − 0.442035i \(-0.854257\pi\)
−0.896998 + 0.442035i \(0.854257\pi\)
\(38\) 0 0
\(39\) −3.21760 + 6.22495i −0.515228 + 0.996790i
\(40\) 0 0
\(41\) −6.15464 −0.961193 −0.480597 − 0.876942i \(-0.659580\pi\)
−0.480597 + 0.876942i \(0.659580\pi\)
\(42\) 0 0
\(43\) −0.502751 −0.0766688 −0.0383344 − 0.999265i \(-0.512205\pi\)
−0.0383344 + 0.999265i \(0.512205\pi\)
\(44\) 0 0
\(45\) −6.59339 − 9.30108i −0.982885 − 1.38652i
\(46\) 0 0
\(47\) −11.4516 −1.67038 −0.835190 − 0.549961i \(-0.814643\pi\)
−0.835190 + 0.549961i \(0.814643\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.159925 − 0.0826633i −0.0223940 − 0.0115752i
\(52\) 0 0
\(53\) − 5.86753i − 0.805967i −0.915207 − 0.402983i \(-0.867973\pi\)
0.915207 − 0.402983i \(-0.132027\pi\)
\(54\) 0 0
\(55\) 1.35833i 0.183158i
\(56\) 0 0
\(57\) 1.95255 − 3.77751i 0.258622 − 0.500344i
\(58\) 0 0
\(59\) −7.54728 −0.982572 −0.491286 − 0.870998i \(-0.663473\pi\)
−0.491286 + 0.870998i \(0.663473\pi\)
\(60\) 0 0
\(61\) − 9.47414i − 1.21304i −0.795068 − 0.606520i \(-0.792565\pi\)
0.795068 − 0.606520i \(-0.207435\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 15.3750i − 1.90703i
\(66\) 0 0
\(67\) −2.68750 −0.328330 −0.164165 − 0.986433i \(-0.552493\pi\)
−0.164165 + 0.986433i \(0.552493\pi\)
\(68\) 0 0
\(69\) 1.05783 − 2.04653i 0.127348 − 0.246374i
\(70\) 0 0
\(71\) 5.78975i 0.687117i 0.939131 + 0.343558i \(0.111632\pi\)
−0.939131 + 0.343558i \(0.888368\pi\)
\(72\) 0 0
\(73\) − 0.235473i − 0.0275600i −0.999905 − 0.0137800i \(-0.995614\pi\)
0.999905 − 0.0137800i \(-0.00438645\pi\)
\(74\) 0 0
\(75\) 14.5289 + 7.50980i 1.67765 + 0.867157i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.22495 −0.362835 −0.181418 − 0.983406i \(-0.558069\pi\)
−0.181418 + 0.983406i \(0.558069\pi\)
\(80\) 0 0
\(81\) −2.97990 + 8.49236i −0.331100 + 0.943596i
\(82\) 0 0
\(83\) −9.07747 −0.996382 −0.498191 − 0.867067i \(-0.666002\pi\)
−0.498191 + 0.867067i \(0.666002\pi\)
\(84\) 0 0
\(85\) 0.394999 0.0428436
\(86\) 0 0
\(87\) 3.95482 − 7.65122i 0.424001 − 0.820297i
\(88\) 0 0
\(89\) 6.82427 0.723371 0.361685 − 0.932300i \(-0.382201\pi\)
0.361685 + 0.932300i \(0.382201\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.27240 + 12.1349i −0.650418 + 1.25833i
\(94\) 0 0
\(95\) 9.33007i 0.957245i
\(96\) 0 0
\(97\) − 5.14243i − 0.522134i −0.965321 − 0.261067i \(-0.915926\pi\)
0.965321 − 0.261067i \(-0.0840744\pi\)
\(98\) 0 0
\(99\) 0.874774 − 0.620114i 0.0879181 − 0.0623238i
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 2352.2.k.i.881.16 16
3.2 odd 2 inner 2352.2.k.i.881.2 16
4.3 odd 2 1176.2.k.a.881.1 16
7.4 even 3 336.2.bc.f.257.5 16
7.5 odd 6 336.2.bc.f.17.7 16
7.6 odd 2 inner 2352.2.k.i.881.1 16
12.11 even 2 1176.2.k.a.881.15 16
21.5 even 6 336.2.bc.f.17.5 16
21.11 odd 6 336.2.bc.f.257.7 16
21.20 even 2 inner 2352.2.k.i.881.15 16
28.3 even 6 1176.2.u.b.1097.5 16
28.11 odd 6 168.2.u.a.89.4 yes 16
28.19 even 6 168.2.u.a.17.2 ✓ 16
28.23 odd 6 1176.2.u.b.521.7 16
28.27 even 2 1176.2.k.a.881.16 16
84.11 even 6 168.2.u.a.89.2 yes 16
84.23 even 6 1176.2.u.b.521.5 16
84.47 odd 6 168.2.u.a.17.4 yes 16
84.59 odd 6 1176.2.u.b.1097.7 16
84.83 odd 2 1176.2.k.a.881.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.u.a.17.2 ✓ 16 28.19 even 6
168.2.u.a.17.4 yes 16 84.47 odd 6
168.2.u.a.89.2 yes 16 84.11 even 6
168.2.u.a.89.4 yes 16 28.11 odd 6
336.2.bc.f.17.5 16 21.5 even 6
336.2.bc.f.17.7 16 7.5 odd 6
336.2.bc.f.257.5 16 7.4 even 3
336.2.bc.f.257.7 16 21.11 odd 6
1176.2.k.a.881.1 16 4.3 odd 2
1176.2.k.a.881.2 16 84.83 odd 2
1176.2.k.a.881.15 16 12.11 even 2
1176.2.k.a.881.16 16 28.27 even 2
1176.2.u.b.521.5 16 84.23 even 6
1176.2.u.b.521.7 16 28.23 odd 6
1176.2.u.b.1097.5 16 28.3 even 6
1176.2.u.b.1097.7 16 84.59 odd 6
2352.2.k.i.881.1 16 7.6 odd 2 inner
2352.2.k.i.881.2 16 3.2 odd 2 inner
2352.2.k.i.881.15 16 21.20 even 2 inner
2352.2.k.i.881.16 16 1.1 even 1 trivial