Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [99,2,Mod(34,99)] 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("99.34"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 99.67
Dual form 99.2.e.c.34.1

$q$-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 + 1.73205i) q^{2} +1.73205i q^{3} +(-1.00000 + 1.73205i) q^{4} +(1.00000 - 1.73205i) q^{5} +(-3.00000 + 1.73205i) q^{6} +(-2.00000 - 3.46410i) q^{7} -3.00000 q^{9} +4.00000 q^{10} +(-0.500000 - 0.866025i) q^{11} +(-3.00000 - 1.73205i) q^{12} +(-2.00000 + 3.46410i) q^{13} +(4.00000 - 6.92820i) q^{14} +(3.00000 + 1.73205i) q^{15} +(2.00000 + 3.46410i) q^{16} +4.00000 q^{17} +(-3.00000 - 5.19615i) q^{18} -6.00000 q^{19} +(2.00000 + 3.46410i) q^{20} +(6.00000 - 3.46410i) q^{21} +(1.00000 - 1.73205i) q^{22} +(0.500000 - 0.866025i) q^{23} +(0.500000 + 0.866025i) q^{25} -8.00000 q^{26} -5.19615i q^{27} +8.00000 q^{28} +6.92820i q^{30} +(-0.500000 + 0.866025i) q^{31} +(-4.00000 + 6.92820i) q^{32} +(1.50000 - 0.866025i) q^{33} +(4.00000 + 6.92820i) q^{34} -8.00000 q^{35} +(3.00000 - 5.19615i) q^{36} +3.00000 q^{37} +(-6.00000 - 10.3923i) q^{38} +(-6.00000 - 3.46410i) q^{39} +(1.00000 - 1.73205i) q^{41} +(12.0000 + 6.92820i) q^{42} +(-6.00000 - 10.3923i) q^{43} +2.00000 q^{44} +(-3.00000 + 5.19615i) q^{45} +2.00000 q^{46} +(3.50000 + 6.06218i) q^{47} +(-6.00000 + 3.46410i) q^{48} +(-4.50000 + 7.79423i) q^{49} +(-1.00000 + 1.73205i) q^{50} +6.92820i q^{51} +(-4.00000 - 6.92820i) q^{52} +3.00000 q^{53} +(9.00000 - 5.19615i) q^{54} -2.00000 q^{55} -10.3923i q^{57} +(-5.50000 + 9.52628i) q^{59} +(-6.00000 + 3.46410i) q^{60} -2.00000 q^{62} +(6.00000 + 10.3923i) q^{63} -8.00000 q^{64} +(4.00000 + 6.92820i) q^{65} +(3.00000 + 1.73205i) q^{66} +(2.00000 - 3.46410i) q^{67} +(-4.00000 + 6.92820i) q^{68} +(1.50000 + 0.866025i) q^{69} +(-8.00000 - 13.8564i) q^{70} +15.0000 q^{71} -8.00000 q^{73} +(3.00000 + 5.19615i) q^{74} +(-1.50000 + 0.866025i) q^{75} +(6.00000 - 10.3923i) q^{76} +(-2.00000 + 3.46410i) q^{77} -13.8564i q^{78} +(5.00000 + 8.66025i) q^{79} +8.00000 q^{80} +9.00000 q^{81} +4.00000 q^{82} +(-6.00000 - 10.3923i) q^{83} +13.8564i q^{84} +(4.00000 - 6.92820i) q^{85} +(12.0000 - 20.7846i) q^{86} +3.00000 q^{89} -12.0000 q^{90} +16.0000 q^{91} +(1.00000 + 1.73205i) q^{92} +(-1.50000 - 0.866025i) q^{93} +(-7.00000 + 12.1244i) q^{94} +(-6.00000 + 10.3923i) q^{95} +(-12.0000 - 6.92820i) q^{96} +(-8.50000 - 14.7224i) q^{97} -18.0000 q^{98} +(1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{6} - 4 q^{7} - 6 q^{9} + 8 q^{10} - q^{11} - 6 q^{12} - 4 q^{13} + 8 q^{14} + 6 q^{15} + 4 q^{16} + 8 q^{17} - 6 q^{18} - 12 q^{19} + 4 q^{20} + 12 q^{21} + 2 q^{22}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 + 1.73205i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) 1.73205i 1.00000i
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) 1.00000 − 1.73205i 0.447214 − 0.774597i −0.550990 − 0.834512i \(-0.685750\pi\)
0.998203 + 0.0599153i \(0.0190830\pi\)
\(6\) −3.00000 + 1.73205i −1.22474 + 0.707107i
\(7\) −2.00000 − 3.46410i −0.755929 − 1.30931i −0.944911 − 0.327327i \(-0.893852\pi\)
0.188982 − 0.981981i \(-0.439481\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 4.00000 1.26491
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i
\(12\) −3.00000 − 1.73205i −0.866025 − 0.500000i
\(13\) −2.00000 + 3.46410i −0.554700 + 0.960769i 0.443227 + 0.896410i \(0.353834\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 4.00000 − 6.92820i 1.06904 − 1.85164i
\(15\) 3.00000 + 1.73205i 0.774597 + 0.447214i
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −3.00000 − 5.19615i −0.707107 − 1.22474i
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 2.00000 + 3.46410i 0.447214 + 0.774597i
\(21\) 6.00000 − 3.46410i 1.30931 − 0.755929i
\(22\) 1.00000 − 1.73205i 0.213201 − 0.369274i
\(23\) 0.500000 − 0.866025i 0.104257 − 0.180579i −0.809177 − 0.587565i \(-0.800087\pi\)
0.913434 + 0.406986i \(0.133420\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) −8.00000 −1.56893
\(27\) − 5.19615i − 1.00000i
\(28\) 8.00000 1.51186
\(29\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 6.92820i 1.26491i
\(31\) −0.500000 + 0.866025i −0.0898027 + 0.155543i −0.907428 − 0.420208i \(-0.861957\pi\)
0.817625 + 0.575751i \(0.195290\pi\)
\(32\) −4.00000 + 6.92820i −0.707107 + 1.22474i
\(33\) 1.50000 − 0.866025i 0.261116 − 0.150756i
\(34\) 4.00000 + 6.92820i 0.685994 + 1.18818i
\(35\) −8.00000 −1.35225
\(36\) 3.00000 − 5.19615i 0.500000 − 0.866025i
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −6.00000 − 10.3923i −0.973329 − 1.68585i
\(39\) −6.00000 − 3.46410i −0.960769 − 0.554700i
\(40\) 0 0
\(41\) 1.00000 − 1.73205i 0.156174 − 0.270501i −0.777312 − 0.629115i \(-0.783417\pi\)
0.933486 + 0.358614i \(0.116751\pi\)
\(42\) 12.0000 + 6.92820i 1.85164 + 1.06904i
\(43\) −6.00000 − 10.3923i −0.914991 − 1.58481i −0.806914 − 0.590669i \(-0.798864\pi\)
−0.108078 − 0.994142i \(-0.534469\pi\)
\(44\) 2.00000 0.301511
\(45\) −3.00000 + 5.19615i −0.447214 + 0.774597i
\(46\) 2.00000 0.294884
\(47\) 3.50000 + 6.06218i 0.510527 + 0.884260i 0.999926 + 0.0121990i \(0.00388317\pi\)
−0.489398 + 0.872060i \(0.662783\pi\)
\(48\) −6.00000 + 3.46410i −0.866025 + 0.500000i
\(49\) −4.50000 + 7.79423i −0.642857 + 1.11346i
\(50\) −1.00000 + 1.73205i −0.141421 + 0.244949i
\(51\) 6.92820i 0.970143i
\(52\) −4.00000 − 6.92820i −0.554700 − 0.960769i
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 9.00000 − 5.19615i 1.22474 − 0.707107i
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) − 10.3923i − 1.37649i
\(58\) 0 0
\(59\) −5.50000 + 9.52628i −0.716039 + 1.24022i 0.246518 + 0.969138i \(0.420713\pi\)
−0.962557 + 0.271078i \(0.912620\pi\)
\(60\) −6.00000 + 3.46410i −0.774597 + 0.447214i
\(61\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) −2.00000 −0.254000
\(63\) 6.00000 + 10.3923i 0.755929 + 1.30931i
\(64\) −8.00000 −1.00000
\(65\) 4.00000 + 6.92820i 0.496139 + 0.859338i
\(66\) 3.00000 + 1.73205i 0.369274 + 0.213201i
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) −4.00000 + 6.92820i −0.485071 + 0.840168i
\(69\) 1.50000 + 0.866025i 0.180579 + 0.104257i
\(70\) −8.00000 − 13.8564i −0.956183 − 1.65616i
\(71\) 15.0000 1.78017 0.890086 − 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 3.00000 + 5.19615i 0.348743 + 0.604040i
\(75\) −1.50000 + 0.866025i −0.173205 + 0.100000i
\(76\) 6.00000 − 10.3923i 0.688247 − 1.19208i
\(77\) −2.00000 + 3.46410i −0.227921 + 0.394771i
\(78\) − 13.8564i − 1.56893i
\(79\) 5.00000 + 8.66025i 0.562544 + 0.974355i 0.997274 + 0.0737937i \(0.0235106\pi\)
−0.434730 + 0.900561i \(0.643156\pi\)
\(80\) 8.00000 0.894427
\(81\) 9.00000 1.00000
\(82\) 4.00000 0.441726
\(83\) −6.00000 − 10.3923i −0.658586 − 1.14070i −0.980982 − 0.194099i \(-0.937822\pi\)
0.322396 − 0.946605i \(-0.395512\pi\)
\(84\) 13.8564i 1.51186i
\(85\) 4.00000 − 6.92820i 0.433861 − 0.751469i
\(86\) 12.0000 − 20.7846i 1.29399 − 2.24126i
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) −12.0000 −1.26491
\(91\) 16.0000 1.67726
\(92\) 1.00000 + 1.73205i 0.104257 + 0.180579i
\(93\) −1.50000 − 0.866025i −0.155543 − 0.0898027i
\(94\) −7.00000 + 12.1244i −0.721995 + 1.25053i
\(95\) −6.00000 + 10.3923i −0.615587 + 1.06623i
\(96\) −12.0000 − 6.92820i −1.22474 − 0.707107i
\(97\) −8.50000 − 14.7224i −0.863044 − 1.49484i −0.868976 − 0.494854i \(-0.835222\pi\)
0.00593185 − 0.999982i \(-0.498112\pi\)
\(98\) −18.0000 −1.81827
\(99\) 1.50000 + 2.59808i 0.150756 + 0.261116i
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 99.2.e.c.67.1 yes 2
3.2 odd 2 297.2.e.a.199.1 2
9.2 odd 6 297.2.e.a.100.1 2
9.4 even 3 891.2.a.a.1.1 1
9.5 odd 6 891.2.a.h.1.1 1
9.7 even 3 inner 99.2.e.c.34.1 ✓ 2
11.10 odd 2 1089.2.e.a.364.1 2
99.32 even 6 9801.2.a.a.1.1 1
99.43 odd 6 1089.2.e.a.727.1 2
99.76 odd 6 9801.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.e.c.34.1 ✓ 2 9.7 even 3 inner
99.2.e.c.67.1 yes 2 1.1 even 1 trivial
297.2.e.a.100.1 2 9.2 odd 6
297.2.e.a.199.1 2 3.2 odd 2
891.2.a.a.1.1 1 9.4 even 3
891.2.a.h.1.1 1 9.5 odd 6
1089.2.e.a.364.1 2 11.10 odd 2
1089.2.e.a.727.1 2 99.43 odd 6
9801.2.a.a.1.1 1 99.32 even 6
9801.2.a.l.1.1 1 99.76 odd 6