Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4232,2,Mod(1,4232)] 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("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4232 = 2^{3} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4232.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(33.7926901354\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 4232.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.56155 q^{3} -2.00000 q^{5} -0.561553 q^{9} +3.12311 q^{11} +0.438447 q^{13} -3.12311 q^{15} -5.12311 q^{17} +3.12311 q^{19} -1.00000 q^{25} -5.56155 q^{27} +3.56155 q^{29} -2.43845 q^{31} +4.87689 q^{33} -8.24621 q^{37} +0.684658 q^{39} -9.80776 q^{41} +8.00000 q^{43} +1.12311 q^{45} -0.684658 q^{47} -7.00000 q^{49} -8.00000 q^{51} -2.00000 q^{53} -6.24621 q^{55} +4.87689 q^{57} +10.2462 q^{59} +4.24621 q^{61} -0.876894 q^{65} -3.12311 q^{67} +13.5616 q^{71} -14.6847 q^{73} -1.56155 q^{75} -3.12311 q^{79} -7.00000 q^{81} -14.2462 q^{83} +10.2462 q^{85} +5.56155 q^{87} -11.3693 q^{89} -3.80776 q^{93} -6.24621 q^{95} -11.3693 q^{97} -1.75379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 4 q^{5} + 3 q^{9} - 2 q^{11} + 5 q^{13} + 2 q^{15} - 2 q^{17} - 2 q^{19} - 2 q^{25} - 7 q^{27} + 3 q^{29} - 9 q^{31} + 18 q^{33} - 11 q^{39} + q^{41} + 16 q^{43} - 6 q^{45} + 11 q^{47} - 14 q^{49}+ \cdots - 20 q^{99}+O(q^{100}) \) 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
\(3\) 1.56155 0.901563 0.450781 − 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 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
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 3.12311 0.941652 0.470826 − 0.882226i \(-0.343956\pi\)
0.470826 + 0.882226i \(0.343956\pi\)
\(12\) 0 0
\(13\) 0.438447 0.121603 0.0608017 − 0.998150i \(-0.480634\pi\)
0.0608017 + 0.998150i \(0.480634\pi\)
\(14\) 0 0
\(15\) −3.12311 −0.806382
\(16\) 0 0
\(17\) −5.12311 −1.24254 −0.621268 − 0.783598i \(-0.713382\pi\)
−0.621268 + 0.783598i \(0.713382\pi\)
\(18\) 0 0
\(19\) 3.12311 0.716490 0.358245 − 0.933628i \(-0.383375\pi\)
0.358245 + 0.933628i \(0.383375\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) 3.56155 0.661364 0.330682 − 0.943742i \(-0.392721\pi\)
0.330682 + 0.943742i \(0.392721\pi\)
\(30\) 0 0
\(31\) −2.43845 −0.437958 −0.218979 − 0.975730i \(-0.570273\pi\)
−0.218979 + 0.975730i \(0.570273\pi\)
\(32\) 0 0
\(33\) 4.87689 0.848958
\(34\) 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
\(39\) 0.684658 0.109633
\(40\) 0 0
\(41\) −9.80776 −1.53172 −0.765858 − 0.643010i \(-0.777685\pi\)
−0.765858 + 0.643010i \(0.777685\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 1.12311 0.167423
\(46\) 0 0
\(47\) −0.684658 −0.0998677 −0.0499338 − 0.998753i \(-0.515901\pi\)
−0.0499338 + 0.998753i \(0.515901\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −6.24621 −0.842239
\(56\) 0 0
\(57\) 4.87689 0.645960
\(58\) 0 0
\(59\) 10.2462 1.33394 0.666972 − 0.745083i \(-0.267590\pi\)
0.666972 + 0.745083i \(0.267590\pi\)
\(60\) 0 0
\(61\) 4.24621 0.543672 0.271836 − 0.962344i \(-0.412369\pi\)
0.271836 + 0.962344i \(0.412369\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.876894 −0.108765
\(66\) 0 0
\(67\) −3.12311 −0.381548 −0.190774 − 0.981634i \(-0.561100\pi\)
−0.190774 + 0.981634i \(0.561100\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.5616 1.60946 0.804730 − 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(72\) 0 0
\(73\) −14.6847 −1.71871 −0.859355 − 0.511380i \(-0.829134\pi\)
−0.859355 + 0.511380i \(0.829134\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.12311 −0.351377 −0.175688 − 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −14.2462 −1.56372 −0.781862 − 0.623451i \(-0.785730\pi\)
−0.781862 + 0.623451i \(0.785730\pi\)
\(84\) 0 0
\(85\) 10.2462 1.11136
\(86\) 0 0
\(87\) 5.56155 0.596261
\(88\) 0 0
\(89\) −11.3693 −1.20515 −0.602573 − 0.798064i \(-0.705858\pi\)
−0.602573 + 0.798064i \(0.705858\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.80776 −0.394847
\(94\) 0 0
\(95\) −6.24621 −0.640848
\(96\) 0 0
\(97\) −11.3693 −1.15438 −0.577190 − 0.816610i \(-0.695851\pi\)
−0.577190 + 0.816610i \(0.695851\pi\)
\(98\) 0 0
\(99\) −1.75379 −0.176262
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 4232.2.a.o.1.2 2
4.3 odd 2 8464.2.a.bd.1.1 2
23.22 odd 2 184.2.a.e.1.2 ✓ 2
69.68 even 2 1656.2.a.j.1.2 2
92.91 even 2 368.2.a.i.1.1 2
115.22 even 4 4600.2.e.o.4049.2 4
115.68 even 4 4600.2.e.o.4049.3 4
115.114 odd 2 4600.2.a.s.1.1 2
161.160 even 2 9016.2.a.w.1.1 2
184.45 odd 2 1472.2.a.u.1.1 2
184.91 even 2 1472.2.a.p.1.2 2
276.275 odd 2 3312.2.a.t.1.1 2
460.459 even 2 9200.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.e.1.2 ✓ 2 23.22 odd 2
368.2.a.i.1.1 2 92.91 even 2
1472.2.a.p.1.2 2 184.91 even 2
1472.2.a.u.1.1 2 184.45 odd 2
1656.2.a.j.1.2 2 69.68 even 2
3312.2.a.t.1.1 2 276.275 odd 2
4232.2.a.o.1.2 2 1.1 even 1 trivial
4600.2.a.s.1.1 2 115.114 odd 2
4600.2.e.o.4049.2 4 115.22 even 4
4600.2.e.o.4049.3 4 115.68 even 4
8464.2.a.bd.1.1 2 4.3 odd 2
9016.2.a.w.1.1 2 161.160 even 2
9200.2.a.br.1.2 2 460.459 even 2