↑

Properties

Label 3312.2.a.t.1.2
Level $3312$
Weight $2$
Character 3312.1
Self dual yes
Analytic conductor $26.446$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3312,2,Mod(1,3312)] 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("3312.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4,0,0,0,0,0,2,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.4464531494\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3312.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{5} +5.12311 q^{11} +4.56155 q^{13} +3.12311 q^{17} -5.12311 q^{19} -1.00000 q^{23} -1.00000 q^{25} +0.561553 q^{29} +6.56155 q^{31} -8.24621 q^{37} -10.8078 q^{41} +8.00000 q^{43} +11.6847 q^{47} -7.00000 q^{49} -2.00000 q^{53} -10.2462 q^{55} -6.24621 q^{59} +12.2462 q^{61} -9.12311 q^{65} +5.12311 q^{67} +9.43845 q^{71} -2.31534 q^{73} +5.12311 q^{79} -2.24621 q^{83} -6.24621 q^{85} +13.3693 q^{89} +10.2462 q^{95} -13.3693 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 2 q^{11} + 5 q^{13} - 2 q^{17} - 2 q^{19} - 2 q^{23} - 2 q^{25} - 3 q^{29} + 9 q^{31} - q^{41} + 16 q^{43} + 11 q^{47} - 14 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} + 8 q^{61} - 10 q^{65}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.12311 1.54467 0.772337 − 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 0 0
\(13\) 4.56155 1.26515 0.632574 − 0.774500i \(-0.281999\pi\)
0.632574 + 0.774500i \(0.281999\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.12311 0.757464 0.378732 − 0.925506i \(-0.376360\pi\)
0.378732 + 0.925506i \(0.376360\pi\)
\(18\) 0 0
\(19\) −5.12311 −1.17532 −0.587661 − 0.809108i \(-0.699951\pi\)
−0.587661 + 0.809108i \(0.699951\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.561553 0.104278 0.0521389 − 0.998640i \(-0.483396\pi\)
0.0521389 + 0.998640i \(0.483396\pi\)
\(30\) 0 0
\(31\) 6.56155 1.17849 0.589245 − 0.807955i \(-0.299425\pi\)
0.589245 + 0.807955i \(0.299425\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.24621 −1.35567 −0.677834 − 0.735215i \(-0.737081\pi\)
−0.677834 + 0.735215i \(0.737081\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.8078 −1.68789 −0.843945 − 0.536430i \(-0.819772\pi\)
−0.843945 + 0.536430i \(0.819772\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.6847 1.70438 0.852191 − 0.523230i \(-0.175273\pi\)
0.852191 + 0.523230i \(0.175273\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −10.2462 −1.38160
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.24621 −0.813187 −0.406594 − 0.913609i \(-0.633284\pi\)
−0.406594 + 0.913609i \(0.633284\pi\)
\(60\) 0 0
\(61\) 12.2462 1.56797 0.783983 − 0.620782i \(-0.213185\pi\)
0.783983 + 0.620782i \(0.213185\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −9.12311 −1.13158
\(66\) 0 0
\(67\) 5.12311 0.625887 0.312943 − 0.949772i \(-0.398685\pi\)
0.312943 + 0.949772i \(0.398685\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.43845 1.12014 0.560069 − 0.828446i \(-0.310775\pi\)
0.560069 + 0.828446i \(0.310775\pi\)
\(72\) 0 0
\(73\) −2.31534 −0.270990 −0.135495 − 0.990778i \(-0.543263\pi\)
−0.135495 + 0.990778i \(0.543263\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.12311 0.576394 0.288197 − 0.957571i \(-0.406944\pi\)
0.288197 + 0.957571i \(0.406944\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.24621 −0.246554 −0.123277 − 0.992372i \(-0.539340\pi\)
−0.123277 + 0.992372i \(0.539340\pi\)
\(84\) 0 0
\(85\) −6.24621 −0.677497
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.3693 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.2462 1.05124
\(96\) 0 0
\(97\) −13.3693 −1.35745 −0.678724 − 0.734393i \(-0.737467\pi\)
−0.678724 + 0.734393i \(0.737467\pi\)
\(98\) 0 0
\(99\) 0 0
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 3312.2.a.t.1.2 2
3.2 odd 2 368.2.a.i.1.2 2
4.3 odd 2 1656.2.a.j.1.1 2
12.11 even 2 184.2.a.e.1.1 ✓ 2
15.14 odd 2 9200.2.a.br.1.1 2
24.5 odd 2 1472.2.a.p.1.1 2
24.11 even 2 1472.2.a.u.1.2 2
60.23 odd 4 4600.2.e.o.4049.1 4
60.47 odd 4 4600.2.e.o.4049.4 4
60.59 even 2 4600.2.a.s.1.2 2
69.68 even 2 8464.2.a.bd.1.2 2
84.83 odd 2 9016.2.a.w.1.2 2
276.275 odd 2 4232.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.e.1.1 ✓ 2 12.11 even 2
368.2.a.i.1.2 2 3.2 odd 2
1472.2.a.p.1.1 2 24.5 odd 2
1472.2.a.u.1.2 2 24.11 even 2
1656.2.a.j.1.1 2 4.3 odd 2
3312.2.a.t.1.2 2 1.1 even 1 trivial
4232.2.a.o.1.1 2 276.275 odd 2
4600.2.a.s.1.2 2 60.59 even 2
4600.2.e.o.4049.1 4 60.23 odd 4
4600.2.e.o.4049.4 4 60.47 odd 4
8464.2.a.bd.1.2 2 69.68 even 2
9016.2.a.w.1.2 2 84.83 odd 2
9200.2.a.br.1.1 2 15.14 odd 2