Properties

Label 33.2.d.a.32.1
Level $33$
Weight $2$
Character 33.32
Analytic conductor $0.264$
Analytic rank $0$
Dimension $2$
CM discriminant -11
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] = [33,2,Mod(32,33)] 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("33.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(33, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.263506326670\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 32.1
Root \(0.500000 - 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 33.32
Dual form 33.2.d.a.32.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 1.65831i) q^{3} -2.00000 q^{4} +3.31662i q^{5} +(-2.50000 - 1.65831i) q^{9} -3.31662i q^{11} +(-1.00000 + 3.31662i) q^{12} +(5.50000 + 1.65831i) q^{15} +4.00000 q^{16} -6.63325i q^{20} +3.31662i q^{23} -6.00000 q^{25} +(-4.00000 + 3.31662i) q^{27} +5.00000 q^{31} +(-5.50000 - 1.65831i) q^{33} +(5.00000 + 3.31662i) q^{36} -7.00000 q^{37} +6.63325i q^{44} +(5.50000 - 8.29156i) q^{45} -6.63325i q^{47} +(2.00000 - 6.63325i) q^{48} +7.00000 q^{49} +13.2665i q^{53} +11.0000 q^{55} +3.31662i q^{59} +(-11.0000 - 3.31662i) q^{60} -8.00000 q^{64} -13.0000 q^{67} +(5.50000 + 1.65831i) q^{69} -16.5831i q^{71} +(-3.00000 + 9.94987i) q^{75} +13.2665i q^{80} +(3.50000 + 8.29156i) q^{81} -16.5831i q^{89} -6.63325i q^{92} +(2.50000 - 8.29156i) q^{93} +17.0000 q^{97} +(-5.50000 + 8.29156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 4 q^{4} - 5 q^{9} - 2 q^{12} + 11 q^{15} + 8 q^{16} - 12 q^{25} - 8 q^{27} + 10 q^{31} - 11 q^{33} + 10 q^{36} - 14 q^{37} + 11 q^{45} + 4 q^{48} + 14 q^{49} + 22 q^{55} - 22 q^{60} - 16 q^{64}+ \cdots - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1\) \(-1\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0.500000 − 1.65831i 0.288675 − 0.957427i
\(4\) −2.00000 −1.00000
\(5\) 3.31662i 1.48324i 0.670820 + 0.741620i \(0.265942\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −2.50000 − 1.65831i −0.833333 − 0.552771i
\(10\) 0 0
\(11\) − 3.31662i − 1.00000i
\(12\) −1.00000 + 3.31662i −0.288675 + 0.957427i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 5.50000 + 1.65831i 1.42009 + 0.428174i
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) − 6.63325i − 1.48324i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.31662i 0.691564i 0.938315 + 0.345782i \(0.112386\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) −6.00000 −1.20000
\(26\) 0 0
\(27\) −4.00000 + 3.31662i −0.769800 + 0.638285i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) −5.50000 − 1.65831i −0.957427 − 0.288675i
\(34\) 0 0
\(35\) 0 0
\(36\) 5.00000 + 3.31662i 0.833333 + 0.552771i
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 6.63325i 1.00000i
\(45\) 5.50000 − 8.29156i 0.819892 − 1.23603i
\(46\) 0 0
\(47\) − 6.63325i − 0.967559i −0.875190 − 0.483779i \(-0.839264\pi\)
0.875190 − 0.483779i \(-0.160736\pi\)
\(48\) 2.00000 − 6.63325i 0.288675 − 0.957427i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.2665i 1.82229i 0.412082 + 0.911147i \(0.364802\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 11.0000 1.48324
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.31662i 0.431788i 0.976417 + 0.215894i \(0.0692665\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) −11.0000 − 3.31662i −1.42009 − 0.428174i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 − 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) 5.50000 + 1.65831i 0.662122 + 0.199637i
\(70\) 0 0
\(71\) − 16.5831i − 1.96805i −0.178017 − 0.984027i \(-0.556968\pi\)
0.178017 − 0.984027i \(-0.443032\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −3.00000 + 9.94987i −0.346410 + 1.14891i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 13.2665i 1.48324i
\(81\) 3.50000 + 8.29156i 0.388889 + 0.921285i
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 16.5831i − 1.75781i −0.476999 − 0.878904i \(-0.658275\pi\)
0.476999 − 0.878904i \(-0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 6.63325i − 0.691564i
\(93\) 2.50000 − 8.29156i 0.259238 − 0.859795i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 − 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) −5.50000 + 8.29156i −0.552771 + 0.833333i
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 33.2.d.a.32.1 ✓ 2
3.2 odd 2 inner 33.2.d.a.32.2 yes 2
4.3 odd 2 528.2.b.a.65.2 2
5.2 odd 4 825.2.d.a.824.1 4
5.3 odd 4 825.2.d.a.824.4 4
5.4 even 2 825.2.f.a.626.2 2
8.3 odd 2 2112.2.b.f.65.1 2
8.5 even 2 2112.2.b.e.65.2 2
9.2 odd 6 891.2.g.a.296.2 4
9.4 even 3 891.2.g.a.593.2 4
9.5 odd 6 891.2.g.a.593.1 4
9.7 even 3 891.2.g.a.296.1 4
11.2 odd 10 363.2.f.c.161.1 8
11.3 even 5 363.2.f.c.233.2 8
11.4 even 5 363.2.f.c.215.1 8
11.5 even 5 363.2.f.c.239.2 8
11.6 odd 10 363.2.f.c.239.2 8
11.7 odd 10 363.2.f.c.215.1 8
11.8 odd 10 363.2.f.c.233.2 8
11.9 even 5 363.2.f.c.161.1 8
11.10 odd 2 CM 33.2.d.a.32.1 ✓ 2
12.11 even 2 528.2.b.a.65.1 2
15.2 even 4 825.2.d.a.824.3 4
15.8 even 4 825.2.d.a.824.2 4
15.14 odd 2 825.2.f.a.626.1 2
24.5 odd 2 2112.2.b.e.65.1 2
24.11 even 2 2112.2.b.f.65.2 2
33.2 even 10 363.2.f.c.161.2 8
33.5 odd 10 363.2.f.c.239.1 8
33.8 even 10 363.2.f.c.233.1 8
33.14 odd 10 363.2.f.c.233.1 8
33.17 even 10 363.2.f.c.239.1 8
33.20 odd 10 363.2.f.c.161.2 8
33.26 odd 10 363.2.f.c.215.2 8
33.29 even 10 363.2.f.c.215.2 8
33.32 even 2 inner 33.2.d.a.32.2 yes 2
44.43 even 2 528.2.b.a.65.2 2
55.32 even 4 825.2.d.a.824.1 4
55.43 even 4 825.2.d.a.824.4 4
55.54 odd 2 825.2.f.a.626.2 2
88.21 odd 2 2112.2.b.e.65.2 2
88.43 even 2 2112.2.b.f.65.1 2
99.32 even 6 891.2.g.a.593.1 4
99.43 odd 6 891.2.g.a.296.1 4
99.65 even 6 891.2.g.a.296.2 4
99.76 odd 6 891.2.g.a.593.2 4
132.131 odd 2 528.2.b.a.65.1 2
165.32 odd 4 825.2.d.a.824.3 4
165.98 odd 4 825.2.d.a.824.2 4
165.164 even 2 825.2.f.a.626.1 2
264.131 odd 2 2112.2.b.f.65.2 2
264.197 even 2 2112.2.b.e.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.d.a.32.1 ✓ 2 1.1 even 1 trivial
33.2.d.a.32.1 ✓ 2 11.10 odd 2 CM
33.2.d.a.32.2 yes 2 3.2 odd 2 inner
33.2.d.a.32.2 yes 2 33.32 even 2 inner
363.2.f.c.161.1 8 11.2 odd 10
363.2.f.c.161.1 8 11.9 even 5
363.2.f.c.161.2 8 33.2 even 10
363.2.f.c.161.2 8 33.20 odd 10
363.2.f.c.215.1 8 11.4 even 5
363.2.f.c.215.1 8 11.7 odd 10
363.2.f.c.215.2 8 33.26 odd 10
363.2.f.c.215.2 8 33.29 even 10
363.2.f.c.233.1 8 33.8 even 10
363.2.f.c.233.1 8 33.14 odd 10
363.2.f.c.233.2 8 11.3 even 5
363.2.f.c.233.2 8 11.8 odd 10
363.2.f.c.239.1 8 33.5 odd 10
363.2.f.c.239.1 8 33.17 even 10
363.2.f.c.239.2 8 11.5 even 5
363.2.f.c.239.2 8 11.6 odd 10
528.2.b.a.65.1 2 12.11 even 2
528.2.b.a.65.1 2 132.131 odd 2
528.2.b.a.65.2 2 4.3 odd 2
528.2.b.a.65.2 2 44.43 even 2
825.2.d.a.824.1 4 5.2 odd 4
825.2.d.a.824.1 4 55.32 even 4
825.2.d.a.824.2 4 15.8 even 4
825.2.d.a.824.2 4 165.98 odd 4
825.2.d.a.824.3 4 15.2 even 4
825.2.d.a.824.3 4 165.32 odd 4
825.2.d.a.824.4 4 5.3 odd 4
825.2.d.a.824.4 4 55.43 even 4
825.2.f.a.626.1 2 15.14 odd 2
825.2.f.a.626.1 2 165.164 even 2
825.2.f.a.626.2 2 5.4 even 2
825.2.f.a.626.2 2 55.54 odd 2
891.2.g.a.296.1 4 9.7 even 3
891.2.g.a.296.1 4 99.43 odd 6
891.2.g.a.296.2 4 9.2 odd 6
891.2.g.a.296.2 4 99.65 even 6
891.2.g.a.593.1 4 9.5 odd 6
891.2.g.a.593.1 4 99.32 even 6
891.2.g.a.593.2 4 9.4 even 3
891.2.g.a.593.2 4 99.76 odd 6
2112.2.b.e.65.1 2 24.5 odd 2
2112.2.b.e.65.1 2 264.197 even 2
2112.2.b.e.65.2 2 8.5 even 2
2112.2.b.e.65.2 2 88.21 odd 2
2112.2.b.f.65.1 2 8.3 odd 2
2112.2.b.f.65.1 2 88.43 even 2
2112.2.b.f.65.2 2 24.11 even 2
2112.2.b.f.65.2 2 264.131 odd 2