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 = [8,-1] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.508277025.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.1
Root \(-0.734668 - 0.348716i\) of defining polynomial
Character \(\chi\) \(=\) 99.67
Dual form 99.2.e.e.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+(-1.23467 - 2.13851i) q^{2} +(1.66933 + 0.461883i) q^{3} +(-2.04881 + 3.54864i) q^{4} +(1.21814 - 2.10988i) q^{5} +(-1.07333 - 4.14015i) q^{6} +(-1.16933 - 2.02534i) q^{7} +5.17972 q^{8} +(2.57333 + 1.54207i) q^{9} -6.01598 q^{10} +(-0.500000 - 0.866025i) q^{11} +(-5.05919 + 4.97754i) q^{12} +(-2.35519 + 4.07931i) q^{13} +(-2.88747 + 5.00124i) q^{14} +(3.00799 - 2.95945i) q^{15} +(-2.29761 - 3.97958i) q^{16} -3.20799 q^{17} +(0.120523 - 7.40702i) q^{18} +7.77494 q^{19} +(4.99146 + 8.64547i) q^{20} +(-1.01653 - 3.92106i) q^{21} +(-1.23467 + 2.13851i) q^{22} +(-1.37948 + 2.38932i) q^{23} +(8.64666 + 2.39242i) q^{24} +(-0.467722 - 0.810117i) q^{25} +11.6315 q^{26} +(3.58348 + 3.76280i) q^{27} +9.58293 q^{28} +(1.18586 + 2.05397i) q^{29} +(-10.0427 - 2.77868i) q^{30} +(0.685860 - 1.18794i) q^{31} +(-0.493856 + 0.855383i) q^{32} +(-0.434663 - 1.67662i) q^{33} +(3.96080 + 6.86030i) q^{34} -5.69762 q^{35} +(-10.7445 + 5.97241i) q^{36} -8.47256 q^{37} +(-9.59946 - 16.6268i) q^{38} +(-5.81575 + 5.72189i) q^{39} +(6.30961 - 10.9286i) q^{40} +(-1.77332 + 3.07149i) q^{41} +(-7.13013 + 7.01505i) q^{42} +(3.73467 + 6.46863i) q^{43} +4.09762 q^{44} +(6.38825 - 3.55095i) q^{45} +6.81278 q^{46} +(-0.103993 - 0.180122i) q^{47} +(-1.99737 - 7.70446i) q^{48} +(0.765332 - 1.32559i) q^{49} +(-1.15496 + 2.00045i) q^{50} +(-5.35519 - 1.48171i) q^{51} +(-9.65066 - 16.7154i) q^{52} -9.11360 q^{53} +(3.62237 - 12.3091i) q^{54} -2.43628 q^{55} +(-6.05680 - 10.4907i) q^{56} +(12.9789 + 3.59111i) q^{57} +(2.92829 - 5.07194i) q^{58} +(0.120523 - 0.208751i) q^{59} +(4.33921 + 16.7376i) q^{60} +(-0.830670 - 1.43876i) q^{61} -3.38724 q^{62} +(0.114145 - 7.01505i) q^{63} -6.75145 q^{64} +(5.73789 + 9.93832i) q^{65} +(-3.04881 + 2.99960i) q^{66} +(3.84027 - 6.65155i) q^{67} +(6.57255 - 11.3840i) q^{68} +(-3.40639 + 3.35142i) q^{69} +(7.03467 + 12.1844i) q^{70} +1.07731 q^{71} +(13.3291 + 7.98749i) q^{72} -2.37495 q^{73} +(10.4608 + 18.1186i) q^{74} +(-0.406602 - 1.56839i) q^{75} +(-15.9294 + 27.5904i) q^{76} +(-1.16933 + 2.02534i) q^{77} +(19.4168 + 5.37239i) q^{78} +(-6.35680 - 11.0103i) q^{79} -11.1952 q^{80} +(4.24404 + 7.93651i) q^{81} +8.75786 q^{82} +(-5.25042 - 9.09399i) q^{83} +(15.9971 + 4.42619i) q^{84} +(-3.90777 + 6.76846i) q^{85} +(9.22215 - 15.9732i) q^{86} +(1.03090 + 3.97648i) q^{87} +(-2.58986 - 4.48577i) q^{88} -14.2933 q^{89} +(-15.4811 - 9.27707i) q^{90} +11.0160 q^{91} +(-5.65257 - 9.79053i) q^{92} +(1.69362 - 1.66628i) q^{93} +(-0.256795 + 0.444781i) q^{94} +(9.47095 - 16.4042i) q^{95} +(-1.21950 + 1.19981i) q^{96} +(-4.46694 - 7.73697i) q^{97} -3.77972 q^{98} +(0.0488078 - 2.99960i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 5 q^{3} - 11 q^{4} - 4 q^{5} + 17 q^{6} - q^{7} - 5 q^{9} + 2 q^{10} - 4 q^{11} - 2 q^{12} - 7 q^{13} - q^{14} - q^{15} - 17 q^{16} - 10 q^{17} - 2 q^{18} + 18 q^{19} + 10 q^{20} - 13 q^{21}+ \cdots - 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23467 − 2.13851i −0.873042 − 1.51215i −0.858834 − 0.512254i \(-0.828811\pi\)
−0.0142076 − 0.999899i \(-0.504523\pi\)
\(3\) 1.66933 + 0.461883i 0.963788 + 0.266668i
\(4\) −2.04881 + 3.54864i −1.02440 + 1.77432i
\(5\) 1.21814 − 2.10988i 0.544768 − 0.943566i −0.453853 − 0.891076i \(-0.649951\pi\)
0.998621 − 0.0524895i \(-0.0167156\pi\)
\(6\) −1.07333 − 4.14015i −0.438185 − 1.69021i
\(7\) −1.16933 − 2.02534i −0.441965 − 0.765506i 0.555870 − 0.831269i \(-0.312385\pi\)
−0.997835 + 0.0657628i \(0.979052\pi\)
\(8\) 5.17972 1.83131
\(9\) 2.57333 + 1.54207i 0.857776 + 0.514023i
\(10\) −6.01598 −1.90242
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i
\(12\) −5.05919 + 4.97754i −1.46046 + 1.43689i
\(13\) −2.35519 + 4.07931i −0.653212 + 1.13140i 0.329127 + 0.944286i \(0.393246\pi\)
−0.982339 + 0.187111i \(0.940088\pi\)
\(14\) −2.88747 + 5.00124i −0.771708 + 1.33664i
\(15\) 3.00799 − 2.95945i 0.776660 − 0.764126i
\(16\) −2.29761 − 3.97958i −0.574403 − 0.994895i
\(17\) −3.20799 −0.778051 −0.389026 − 0.921227i \(-0.627188\pi\)
−0.389026 + 0.921227i \(0.627188\pi\)
\(18\) 0.120523 − 7.40702i 0.0284075 − 1.74585i
\(19\) 7.77494 1.78369 0.891846 − 0.452338i \(-0.149410\pi\)
0.891846 + 0.452338i \(0.149410\pi\)
\(20\) 4.99146 + 8.64547i 1.11612 + 1.93318i
\(21\) −1.01653 − 3.92106i −0.221825 − 0.855644i
\(22\) −1.23467 + 2.13851i −0.263232 + 0.455931i
\(23\) −1.37948 + 2.38932i −0.287641 + 0.498209i −0.973246 − 0.229765i \(-0.926204\pi\)
0.685605 + 0.727973i \(0.259538\pi\)
\(24\) 8.64666 + 2.39242i 1.76499 + 0.488351i
\(25\) −0.467722 − 0.810117i −0.0935443 − 0.162023i
\(26\) 11.6315 2.28113
\(27\) 3.58348 + 3.76280i 0.689641 + 0.724151i
\(28\) 9.58293 1.81100
\(29\) 1.18586 + 2.05397i 0.220209 + 0.381413i 0.954871 − 0.297020i \(-0.0959928\pi\)
−0.734663 + 0.678433i \(0.762660\pi\)
\(30\) −10.0427 − 2.77868i −1.83353 − 0.507315i
\(31\) 0.685860 − 1.18794i 0.123184 − 0.213361i −0.797838 − 0.602872i \(-0.794023\pi\)
0.921022 + 0.389511i \(0.127356\pi\)
\(32\) −0.493856 + 0.855383i −0.0873022 + 0.151212i
\(33\) −0.434663 − 1.67662i −0.0756651 − 0.291863i
\(34\) 3.96080 + 6.86030i 0.679271 + 1.17653i
\(35\) −5.69762 −0.963074
\(36\) −10.7445 + 5.97241i −1.79075 + 0.995401i
\(37\) −8.47256 −1.39288 −0.696440 − 0.717615i \(-0.745234\pi\)
−0.696440 + 0.717615i \(0.745234\pi\)
\(38\) −9.59946 − 16.6268i −1.55724 − 2.69722i
\(39\) −5.81575 + 5.72189i −0.931266 + 0.916236i
\(40\) 6.30961 − 10.9286i 0.997637 − 1.72796i
\(41\) −1.77332 + 3.07149i −0.276947 + 0.479686i −0.970624 − 0.240600i \(-0.922656\pi\)
0.693678 + 0.720285i \(0.255989\pi\)
\(42\) −7.13013 + 7.01505i −1.10020 + 1.08245i
\(43\) 3.73467 + 6.46863i 0.569531 + 0.986457i 0.996612 + 0.0822439i \(0.0262087\pi\)
−0.427081 + 0.904213i \(0.640458\pi\)
\(44\) 4.09762 0.617739
\(45\) 6.38825 − 3.55095i 0.952304 − 0.529345i
\(46\) 6.81278 1.00449
\(47\) −0.103993 − 0.180122i −0.0151690 − 0.0262735i 0.858341 − 0.513079i \(-0.171495\pi\)
−0.873510 + 0.486806i \(0.838162\pi\)
\(48\) −1.99737 − 7.70446i −0.288296 − 1.11204i
\(49\) 0.765332 − 1.32559i 0.109333 − 0.189371i
\(50\) −1.15496 + 2.00045i −0.163336 + 0.282907i
\(51\) −5.35519 − 1.48171i −0.749877 − 0.207481i
\(52\) −9.65066 − 16.7154i −1.33831 − 2.31801i
\(53\) −9.11360 −1.25185 −0.625925 − 0.779884i \(-0.715278\pi\)
−0.625925 + 0.779884i \(0.715278\pi\)
\(54\) 3.62237 − 12.3091i 0.492942 − 1.67506i
\(55\) −2.43628 −0.328507
\(56\) −6.05680 − 10.4907i −0.809374 − 1.40188i
\(57\) 12.9789 + 3.59111i 1.71910 + 0.475654i
\(58\) 2.92829 − 5.07194i 0.384503 − 0.665978i
\(59\) 0.120523 − 0.208751i 0.0156907 − 0.0271771i −0.858073 − 0.513527i \(-0.828339\pi\)
0.873764 + 0.486350i \(0.161672\pi\)
\(60\) 4.33921 + 16.7376i 0.560189 + 2.16082i
\(61\) −0.830670 − 1.43876i −0.106356 − 0.184215i 0.807935 − 0.589271i \(-0.200585\pi\)
−0.914292 + 0.405057i \(0.867252\pi\)
\(62\) −3.38724 −0.430179
\(63\) 0.114145 − 7.01505i 0.0143809 − 0.883814i
\(64\) −6.75145 −0.843932
\(65\) 5.73789 + 9.93832i 0.711698 + 1.23270i
\(66\) −3.04881 + 2.99960i −0.375282 + 0.369226i
\(67\) 3.84027 − 6.65155i 0.469164 − 0.812616i −0.530214 − 0.847864i \(-0.677889\pi\)
0.999379 + 0.0352474i \(0.0112219\pi\)
\(68\) 6.57255 − 11.3840i 0.797039 − 1.38051i
\(69\) −3.40639 + 3.35142i −0.410081 + 0.403463i
\(70\) 7.03467 + 12.1844i 0.840804 + 1.45632i
\(71\) 1.07731 0.127854 0.0639268 − 0.997955i \(-0.479638\pi\)
0.0639268 + 0.997955i \(0.479638\pi\)
\(72\) 13.3291 + 7.98749i 1.57085 + 0.941334i
\(73\) −2.37495 −0.277966 −0.138983 − 0.990295i \(-0.544383\pi\)
−0.138983 + 0.990295i \(0.544383\pi\)
\(74\) 10.4608 + 18.1186i 1.21604 + 2.10625i
\(75\) −0.406602 − 1.56839i −0.0469504 − 0.181102i
\(76\) −15.9294 + 27.5904i −1.82722 + 3.16484i
\(77\) −1.16933 + 2.02534i −0.133258 + 0.230809i
\(78\) 19.4168 + 5.37239i 2.19852 + 0.608304i
\(79\) −6.35680 − 11.0103i −0.715196 − 1.23876i −0.962884 − 0.269916i \(-0.913004\pi\)
0.247687 − 0.968840i \(-0.420329\pi\)
\(80\) −11.1952 −1.25166
\(81\) 4.24404 + 7.93651i 0.471560 + 0.881834i
\(82\) 8.75786 0.967144
\(83\) −5.25042 − 9.09399i −0.576308 − 0.998195i −0.995898 − 0.0904812i \(-0.971159\pi\)
0.419590 − 0.907714i \(-0.362174\pi\)
\(84\) 15.9971 + 4.42619i 1.74542 + 0.482937i
\(85\) −3.90777 + 6.76846i −0.423857 + 0.734142i
\(86\) 9.22215 − 15.9732i 0.994450 − 1.72244i
\(87\) 1.03090 + 3.97648i 0.110524 + 0.426324i
\(88\) −2.58986 − 4.48577i −0.276080 − 0.478184i
\(89\) −14.2933 −1.51509 −0.757544 − 0.652784i \(-0.773601\pi\)
−0.757544 + 0.652784i \(0.773601\pi\)
\(90\) −15.4811 − 9.27707i −1.63185 − 0.977889i
\(91\) 11.0160 1.15479
\(92\) −5.65257 − 9.79053i −0.589321 − 1.02073i
\(93\) 1.69362 − 1.66628i 0.175620 − 0.172786i
\(94\) −0.256795 + 0.444781i −0.0264863 + 0.0458757i
\(95\) 9.47095 − 16.4042i 0.971699 − 1.68303i
\(96\) −1.21950 + 1.19981i −0.124464 + 0.122455i
\(97\) −4.46694 − 7.73697i −0.453549 − 0.785570i 0.545054 − 0.838401i \(-0.316509\pi\)
−0.998603 + 0.0528305i \(0.983176\pi\)
\(98\) −3.77972 −0.381810
\(99\) 0.0488078 − 2.99960i 0.00490536 − 0.301471i
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.e.67.1 yes 8
3.2 odd 2 297.2.e.e.199.4 8
9.2 odd 6 297.2.e.e.100.4 8
9.4 even 3 891.2.a.q.1.4 4
9.5 odd 6 891.2.a.p.1.1 4
9.7 even 3 inner 99.2.e.e.34.1 ✓ 8
11.10 odd 2 1089.2.e.i.364.4 8
99.32 even 6 9801.2.a.bl.1.4 4
99.43 odd 6 1089.2.e.i.727.4 8
99.76 odd 6 9801.2.a.bi.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.e.e.34.1 ✓ 8 9.7 even 3 inner
99.2.e.e.67.1 yes 8 1.1 even 1 trivial
297.2.e.e.100.4 8 9.2 odd 6
297.2.e.e.199.4 8 3.2 odd 2
891.2.a.p.1.1 4 9.5 odd 6
891.2.a.q.1.4 4 9.4 even 3
1089.2.e.i.364.4 8 11.10 odd 2
1089.2.e.i.727.4 8 99.43 odd 6
9801.2.a.bi.1.1 4 99.76 odd 6
9801.2.a.bl.1.4 4 99.32 even 6