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(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,0,-4,-2,0,4,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 475.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 888.475
Dual form 888.2.r.a.43.1

$q$-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+(-1.00000 - 1.00000i) q^{2} -1.00000i q^{3} +2.00000i q^{4} +(-2.00000 - 2.00000i) q^{5} +(-1.00000 + 1.00000i) q^{6} +2.00000i q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} +4.00000i q^{10} -4.00000i q^{11} +2.00000 q^{12} +(-1.00000 - 1.00000i) q^{13} +(2.00000 - 2.00000i) q^{14} +(-2.00000 + 2.00000i) q^{15} -4.00000 q^{16} +(-2.00000 + 2.00000i) q^{17} +(1.00000 + 1.00000i) q^{18} +(1.00000 - 1.00000i) q^{19} +(4.00000 - 4.00000i) q^{20} +2.00000 q^{21} +(-4.00000 + 4.00000i) q^{22} +(-2.00000 - 2.00000i) q^{23} +(-2.00000 - 2.00000i) q^{24} +3.00000i q^{25} +2.00000i q^{26} +1.00000i q^{27} -4.00000 q^{28} +(-2.00000 + 2.00000i) q^{29} +4.00000 q^{30} +(-7.00000 + 7.00000i) q^{31} +(4.00000 + 4.00000i) q^{32} -4.00000 q^{33} +4.00000 q^{34} +(4.00000 - 4.00000i) q^{35} -2.00000i q^{36} +(-1.00000 + 6.00000i) q^{37} -2.00000 q^{38} +(-1.00000 + 1.00000i) q^{39} -8.00000 q^{40} +2.00000i q^{41} +(-2.00000 - 2.00000i) q^{42} +(3.00000 - 3.00000i) q^{43} +8.00000 q^{44} +(2.00000 + 2.00000i) q^{45} +4.00000i q^{46} +4.00000i q^{47} +4.00000i q^{48} +3.00000 q^{49} +(3.00000 - 3.00000i) q^{50} +(2.00000 + 2.00000i) q^{51} +(2.00000 - 2.00000i) q^{52} +2.00000i q^{53} +(1.00000 - 1.00000i) q^{54} +(-8.00000 + 8.00000i) q^{55} +(4.00000 + 4.00000i) q^{56} +(-1.00000 - 1.00000i) q^{57} +4.00000 q^{58} +(-8.00000 + 8.00000i) q^{59} +(-4.00000 - 4.00000i) q^{60} +(-1.00000 + 1.00000i) q^{61} +14.0000 q^{62} -2.00000i q^{63} -8.00000i q^{64} +4.00000i q^{65} +(4.00000 + 4.00000i) q^{66} +2.00000i q^{67} +(-4.00000 - 4.00000i) q^{68} +(-2.00000 + 2.00000i) q^{69} -8.00000 q^{70} +(-2.00000 + 2.00000i) q^{72} -6.00000i q^{73} +(7.00000 - 5.00000i) q^{74} +3.00000 q^{75} +(2.00000 + 2.00000i) q^{76} +8.00000 q^{77} +2.00000 q^{78} +(-3.00000 - 3.00000i) q^{79} +(8.00000 + 8.00000i) q^{80} +1.00000 q^{81} +(2.00000 - 2.00000i) q^{82} +4.00000i q^{84} +8.00000 q^{85} -6.00000 q^{86} +(2.00000 + 2.00000i) q^{87} +(-8.00000 - 8.00000i) q^{88} -4.00000i q^{90} +(2.00000 - 2.00000i) q^{91} +(4.00000 - 4.00000i) q^{92} +(7.00000 + 7.00000i) q^{93} +(4.00000 - 4.00000i) q^{94} -4.00000 q^{95} +(4.00000 - 4.00000i) q^{96} +(3.00000 - 3.00000i) q^{97} +(-3.00000 - 3.00000i) q^{98} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{5} - 2 q^{6} + 4 q^{8} - 2 q^{9} + 4 q^{12} - 2 q^{13} + 4 q^{14} - 4 q^{15} - 8 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} + 8 q^{20} + 4 q^{21} - 8 q^{22} - 4 q^{23} - 4 q^{24} - 8 q^{28}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.00000i −0.707107 − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) 2.00000i 1.00000i
\(5\) −2.00000 − 2.00000i −0.894427 − 0.894427i 0.100509 − 0.994936i \(-0.467953\pi\)
−0.994936 + 0.100509i \(0.967953\pi\)
\(6\) −1.00000 + 1.00000i −0.408248 + 0.408248i
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 2.00000 − 2.00000i 0.707107 − 0.707107i
\(9\) −1.00000 −0.333333
\(10\) 4.00000i 1.26491i
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) 2.00000 0.577350
\(13\) −1.00000 − 1.00000i −0.277350 − 0.277350i 0.554700 − 0.832050i \(-0.312833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 − 2.00000i 0.534522 − 0.534522i
\(15\) −2.00000 + 2.00000i −0.516398 + 0.516398i
\(16\) −4.00000 −1.00000
\(17\) −2.00000 + 2.00000i −0.485071 + 0.485071i −0.906747 − 0.421676i \(-0.861442\pi\)
0.421676 + 0.906747i \(0.361442\pi\)
\(18\) 1.00000 + 1.00000i 0.235702 + 0.235702i
\(19\) 1.00000 − 1.00000i 0.229416 − 0.229416i −0.583033 − 0.812449i \(-0.698134\pi\)
0.812449 + 0.583033i \(0.198134\pi\)
\(20\) 4.00000 − 4.00000i 0.894427 − 0.894427i
\(21\) 2.00000 0.436436
\(22\) −4.00000 + 4.00000i −0.852803 + 0.852803i
\(23\) −2.00000 − 2.00000i −0.417029 − 0.417029i 0.467150 − 0.884178i \(-0.345281\pi\)
−0.884178 + 0.467150i \(0.845281\pi\)
\(24\) −2.00000 − 2.00000i −0.408248 − 0.408248i
\(25\) 3.00000i 0.600000i
\(26\) 2.00000i 0.392232i
\(27\) 1.00000i 0.192450i
\(28\) −4.00000 −0.755929
\(29\) −2.00000 + 2.00000i −0.371391 + 0.371391i −0.867984 − 0.496593i \(-0.834584\pi\)
0.496593 + 0.867984i \(0.334584\pi\)
\(30\) 4.00000 0.730297
\(31\) −7.00000 + 7.00000i −1.25724 + 1.25724i −0.304830 + 0.952407i \(0.598600\pi\)
−0.952407 + 0.304830i \(0.901400\pi\)
\(32\) 4.00000 + 4.00000i 0.707107 + 0.707107i
\(33\) −4.00000 −0.696311
\(34\) 4.00000 0.685994
\(35\) 4.00000 − 4.00000i 0.676123 − 0.676123i
\(36\) − 2.00000i − 0.333333i
\(37\) −1.00000 + 6.00000i −0.164399 + 0.986394i
\(38\) −2.00000 −0.324443
\(39\) −1.00000 + 1.00000i −0.160128 + 0.160128i
\(40\) −8.00000 −1.26491
\(41\) 2.00000i 0.312348i 0.987730 + 0.156174i \(0.0499160\pi\)
−0.987730 + 0.156174i \(0.950084\pi\)
\(42\) −2.00000 − 2.00000i −0.308607 − 0.308607i
\(43\) 3.00000 − 3.00000i 0.457496 − 0.457496i −0.440337 − 0.897833i \(-0.645141\pi\)
0.897833 + 0.440337i \(0.145141\pi\)
\(44\) 8.00000 1.20605
\(45\) 2.00000 + 2.00000i 0.298142 + 0.298142i
\(46\) 4.00000i 0.589768i
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 4.00000i 0.577350i
\(49\) 3.00000 0.428571
\(50\) 3.00000 − 3.00000i 0.424264 − 0.424264i
\(51\) 2.00000 + 2.00000i 0.280056 + 0.280056i
\(52\) 2.00000 − 2.00000i 0.277350 − 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 1.00000 − 1.00000i 0.136083 − 0.136083i
\(55\) −8.00000 + 8.00000i −1.07872 + 1.07872i
\(56\) 4.00000 + 4.00000i 0.534522 + 0.534522i
\(57\) −1.00000 − 1.00000i −0.132453 − 0.132453i
\(58\) 4.00000 0.525226
\(59\) −8.00000 + 8.00000i −1.04151 + 1.04151i −0.0424110 + 0.999100i \(0.513504\pi\)
−0.999100 + 0.0424110i \(0.986496\pi\)
\(60\) −4.00000 − 4.00000i −0.516398 − 0.516398i
\(61\) −1.00000 + 1.00000i −0.128037 + 0.128037i −0.768221 − 0.640184i \(-0.778858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 14.0000 1.77800
\(63\) − 2.00000i − 0.251976i
\(64\) − 8.00000i − 1.00000i
\(65\) 4.00000i 0.496139i
\(66\) 4.00000 + 4.00000i 0.492366 + 0.492366i
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) −4.00000 − 4.00000i −0.485071 − 0.485071i
\(69\) −2.00000 + 2.00000i −0.240772 + 0.240772i
\(70\) −8.00000 −0.956183
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −2.00000 + 2.00000i −0.235702 + 0.235702i
\(73\) − 6.00000i − 0.702247i −0.936329 − 0.351123i \(-0.885800\pi\)
0.936329 − 0.351123i \(-0.114200\pi\)
\(74\) 7.00000 − 5.00000i 0.813733 − 0.581238i
\(75\) 3.00000 0.346410
\(76\) 2.00000 + 2.00000i 0.229416 + 0.229416i
\(77\) 8.00000 0.911685
\(78\) 2.00000 0.226455
\(79\) −3.00000 − 3.00000i −0.337526 − 0.337526i 0.517909 − 0.855436i \(-0.326710\pi\)
−0.855436 + 0.517909i \(0.826710\pi\)
\(80\) 8.00000 + 8.00000i 0.894427 + 0.894427i
\(81\) 1.00000 0.111111
\(82\) 2.00000 − 2.00000i 0.220863 − 0.220863i
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 4.00000i 0.436436i
\(85\) 8.00000 0.867722
\(86\) −6.00000 −0.646997
\(87\) 2.00000 + 2.00000i 0.214423 + 0.214423i
\(88\) −8.00000 − 8.00000i −0.852803 − 0.852803i
\(89\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) − 4.00000i − 0.421637i
\(91\) 2.00000 − 2.00000i 0.209657 − 0.209657i
\(92\) 4.00000 − 4.00000i 0.417029 − 0.417029i
\(93\) 7.00000 + 7.00000i 0.725866 + 0.725866i
\(94\) 4.00000 − 4.00000i 0.412568 − 0.412568i
\(95\) −4.00000 −0.410391
\(96\) 4.00000 − 4.00000i 0.408248 − 0.408248i
\(97\) 3.00000 − 3.00000i 0.304604 − 0.304604i −0.538208 − 0.842812i \(-0.680899\pi\)
0.842812 + 0.538208i \(0.180899\pi\)
\(98\) −3.00000 − 3.00000i −0.303046 − 0.303046i
\(99\) 4.00000i 0.402015i
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.r.a.475.1 yes 2
8.3 odd 2 888.2.r.c.475.1 yes 2
37.6 odd 4 888.2.r.c.43.1 yes 2
296.43 even 4 inner 888.2.r.a.43.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.a.43.1 ✓ 2 296.43 even 4 inner
888.2.r.a.475.1 yes 2 1.1 even 1 trivial
888.2.r.c.43.1 yes 2 37.6 odd 4
888.2.r.c.475.1 yes 2 8.3 odd 2