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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 = [2,0] 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 99.67
Dual form 99.2.e.b.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)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(1.50000 - 2.59808i) q^{5} +(2.00000 + 3.46410i) q^{7} +(1.50000 - 2.59808i) q^{9} +(-0.500000 - 0.866025i) q^{11} +3.46410i q^{12} +(-1.00000 + 1.73205i) q^{13} +5.19615i q^{15} +(-2.00000 - 3.46410i) q^{16} -6.00000 q^{17} +2.00000 q^{19} +(-3.00000 - 5.19615i) q^{20} +(-6.00000 - 3.46410i) q^{21} +(-1.50000 + 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} +5.19615i q^{27} +8.00000 q^{28} +(3.00000 + 5.19615i) q^{29} +(-4.00000 + 6.92820i) q^{31} +(1.50000 + 0.866025i) q^{33} +12.0000 q^{35} +(-3.00000 - 5.19615i) q^{36} +2.00000 q^{37} -3.46410i q^{39} +(-4.00000 - 6.92820i) q^{43} -2.00000 q^{44} +(-4.50000 - 7.79423i) q^{45} +(-1.50000 - 2.59808i) q^{47} +(6.00000 + 3.46410i) q^{48} +(-4.50000 + 7.79423i) q^{49} +(9.00000 - 5.19615i) q^{51} +(2.00000 + 3.46410i) q^{52} +3.00000 q^{53} -3.00000 q^{55} +(-3.00000 + 1.73205i) q^{57} +(9.00000 + 5.19615i) q^{60} +(-4.00000 - 6.92820i) q^{61} +12.0000 q^{63} -8.00000 q^{64} +(3.00000 + 5.19615i) q^{65} +(6.50000 - 11.2583i) q^{67} +(-6.00000 + 10.3923i) q^{68} -5.19615i q^{69} +2.00000 q^{73} +(6.00000 + 3.46410i) q^{75} +(2.00000 - 3.46410i) q^{76} +(2.00000 - 3.46410i) q^{77} +(-1.00000 - 1.73205i) q^{79} -12.0000 q^{80} +(-4.50000 - 7.79423i) q^{81} +(9.00000 + 15.5885i) q^{83} +(-12.0000 + 6.92820i) q^{84} +(-9.00000 + 15.5885i) q^{85} +(-9.00000 - 5.19615i) q^{87} +3.00000 q^{89} -8.00000 q^{91} +(3.00000 + 5.19615i) q^{92} -13.8564i q^{93} +(3.00000 - 5.19615i) q^{95} +(-1.00000 - 1.73205i) q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 2 q^{4} + 3 q^{5} + 4 q^{7} + 3 q^{9} - q^{11} - 2 q^{13} - 4 q^{16} - 12 q^{17} + 4 q^{19} - 6 q^{20} - 12 q^{21} - 3 q^{23} - 4 q^{25} + 16 q^{28} + 6 q^{29} - 8 q^{31} + 3 q^{33} + 24 q^{35}+ \cdots - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) −1.50000 + 0.866025i −0.866025 + 0.500000i
\(4\) 1.00000 − 1.73205i 0.500000 − 0.866025i
\(5\) 1.50000 − 2.59808i 0.670820 − 1.16190i −0.306851 − 0.951757i \(-0.599275\pi\)
0.977672 − 0.210138i \(-0.0673912\pi\)
\(6\) 0 0
\(7\) 2.00000 + 3.46410i 0.755929 + 1.30931i 0.944911 + 0.327327i \(0.106148\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.50000 − 2.59808i 0.500000 − 0.866025i
\(10\) 0 0
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i
\(12\) 3.46410i 1.00000i
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 − 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 5.19615i 1.34164i
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −3.00000 − 5.19615i −0.670820 − 1.16190i
\(21\) −6.00000 − 3.46410i −1.30931 − 0.755929i
\(22\) 0 0
\(23\) −1.50000 + 2.59808i −0.312772 + 0.541736i −0.978961 − 0.204046i \(-0.934591\pi\)
0.666190 + 0.745782i \(0.267924\pi\)
\(24\) 0 0
\(25\) −2.00000 − 3.46410i −0.400000 − 0.692820i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 8.00000 1.51186
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 0 0
\(33\) 1.50000 + 0.866025i 0.261116 + 0.150756i
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) −3.00000 − 5.19615i −0.500000 − 0.866025i
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) − 3.46410i − 0.554700i
\(40\) 0 0
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) −4.00000 − 6.92820i −0.609994 − 1.05654i −0.991241 − 0.132068i \(-0.957838\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) −2.00000 −0.301511
\(45\) −4.50000 − 7.79423i −0.670820 − 1.16190i
\(46\) 0 0
\(47\) −1.50000 − 2.59808i −0.218797 − 0.378968i 0.735643 − 0.677369i \(-0.236880\pi\)
−0.954441 + 0.298401i \(0.903547\pi\)
\(48\) 6.00000 + 3.46410i 0.866025 + 0.500000i
\(49\) −4.50000 + 7.79423i −0.642857 + 1.11346i
\(50\) 0 0
\(51\) 9.00000 − 5.19615i 1.26025 − 0.727607i
\(52\) 2.00000 + 3.46410i 0.277350 + 0.480384i
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −3.00000 + 1.73205i −0.397360 + 0.229416i
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 9.00000 + 5.19615i 1.16190 + 0.670820i
\(61\) −4.00000 − 6.92820i −0.512148 − 0.887066i −0.999901 − 0.0140840i \(-0.995517\pi\)
0.487753 − 0.872982i \(-0.337817\pi\)
\(62\) 0 0
\(63\) 12.0000 1.51186
\(64\) −8.00000 −1.00000
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 6.50000 − 11.2583i 0.794101 − 1.37542i −0.129307 − 0.991605i \(-0.541275\pi\)
0.923408 − 0.383819i \(-0.125391\pi\)
\(68\) −6.00000 + 10.3923i −0.727607 + 1.26025i
\(69\) − 5.19615i − 0.625543i
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 6.00000 + 3.46410i 0.692820 + 0.400000i
\(76\) 2.00000 − 3.46410i 0.229416 − 0.397360i
\(77\) 2.00000 − 3.46410i 0.227921 − 0.394771i
\(78\) 0 0
\(79\) −1.00000 − 1.73205i −0.112509 − 0.194871i 0.804272 − 0.594261i \(-0.202555\pi\)
−0.916781 + 0.399390i \(0.869222\pi\)
\(80\) −12.0000 −1.34164
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) 9.00000 + 15.5885i 0.987878 + 1.71106i 0.628372 + 0.777913i \(0.283721\pi\)
0.359506 + 0.933143i \(0.382945\pi\)
\(84\) −12.0000 + 6.92820i −1.30931 + 0.755929i
\(85\) −9.00000 + 15.5885i −0.976187 + 1.69081i
\(86\) 0 0
\(87\) −9.00000 − 5.19615i −0.964901 − 0.557086i
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 3.00000 + 5.19615i 0.312772 + 0.541736i
\(93\) − 13.8564i − 1.43684i
\(94\) 0 0
\(95\) 3.00000 − 5.19615i 0.307794 − 0.533114i
\(96\) 0 0
\(97\) −1.00000 − 1.73205i −0.101535 − 0.175863i 0.810782 − 0.585348i \(-0.199042\pi\)
−0.912317 + 0.409484i \(0.865709\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
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.b.67.1 yes 2
3.2 odd 2 297.2.e.b.199.1 2
9.2 odd 6 297.2.e.b.100.1 2
9.4 even 3 891.2.a.d.1.1 1
9.5 odd 6 891.2.a.e.1.1 1
9.7 even 3 inner 99.2.e.b.34.1 ✓ 2
11.10 odd 2 1089.2.e.b.364.1 2
99.32 even 6 9801.2.a.g.1.1 1
99.43 odd 6 1089.2.e.b.727.1 2
99.76 odd 6 9801.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.e.b.34.1 ✓ 2 9.7 even 3 inner
99.2.e.b.67.1 yes 2 1.1 even 1 trivial
297.2.e.b.100.1 2 9.2 odd 6
297.2.e.b.199.1 2 3.2 odd 2
891.2.a.d.1.1 1 9.4 even 3
891.2.a.e.1.1 1 9.5 odd 6
1089.2.e.b.364.1 2 11.10 odd 2
1089.2.e.b.727.1 2 99.43 odd 6
9801.2.a.f.1.1 1 99.76 odd 6
9801.2.a.g.1.1 1 99.32 even 6