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Properties

Label 234.2.a.c.1.1
Level $234$
Weight $2$
Character 234.1
Self dual yes
Analytic conductor $1.868$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [234,2,Mod(1,234)] 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("234.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 234.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} -1.00000 q^{25} +1.00000 q^{26} +4.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -8.00000 q^{35} -2.00000 q^{37} -8.00000 q^{38} -2.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -8.00000 q^{47} +9.00000 q^{49} -1.00000 q^{50} +1.00000 q^{52} +10.0000 q^{53} -8.00000 q^{55} +4.00000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -16.0000 q^{67} -2.00000 q^{68} -8.00000 q^{70} +8.00000 q^{71} +2.00000 q^{73} -2.00000 q^{74} -8.00000 q^{76} +16.0000 q^{77} +8.00000 q^{79} -2.00000 q^{80} +10.0000 q^{82} -12.0000 q^{83} +4.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} -14.0000 q^{89} +4.00000 q^{91} -8.00000 q^{94} +16.0000 q^{95} +10.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −16.0000 −1.95471 −0.977356 − 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 16.0000 1.64157
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 9.00000 0.909137
\(99\) 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 234.2.a.c.1.1 1
3.2 odd 2 78.2.a.a.1.1 ✓ 1
4.3 odd 2 1872.2.a.c.1.1 1
5.2 odd 4 5850.2.e.bb.5149.2 2
5.3 odd 4 5850.2.e.bb.5149.1 2
5.4 even 2 5850.2.a.d.1.1 1
8.3 odd 2 7488.2.a.bk.1.1 1
8.5 even 2 7488.2.a.bz.1.1 1
9.2 odd 6 2106.2.e.q.1405.1 2
9.4 even 3 2106.2.e.j.703.1 2
9.5 odd 6 2106.2.e.q.703.1 2
9.7 even 3 2106.2.e.j.1405.1 2
12.11 even 2 624.2.a.h.1.1 1
13.5 odd 4 3042.2.b.g.1351.1 2
13.8 odd 4 3042.2.b.g.1351.2 2
13.12 even 2 3042.2.a.f.1.1 1
15.2 even 4 1950.2.e.i.1249.1 2
15.8 even 4 1950.2.e.i.1249.2 2
15.14 odd 2 1950.2.a.w.1.1 1
21.20 even 2 3822.2.a.j.1.1 1
24.5 odd 2 2496.2.a.t.1.1 1
24.11 even 2 2496.2.a.b.1.1 1
33.32 even 2 9438.2.a.t.1.1 1
39.2 even 12 1014.2.i.d.823.1 4
39.5 even 4 1014.2.b.b.337.2 2
39.8 even 4 1014.2.b.b.337.1 2
39.11 even 12 1014.2.i.d.823.2 4
39.17 odd 6 1014.2.e.c.991.1 2
39.20 even 12 1014.2.i.d.361.1 4
39.23 odd 6 1014.2.e.c.529.1 2
39.29 odd 6 1014.2.e.f.529.1 2
39.32 even 12 1014.2.i.d.361.2 4
39.35 odd 6 1014.2.e.f.991.1 2
39.38 odd 2 1014.2.a.d.1.1 1
156.155 even 2 8112.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.a.a.1.1 ✓ 1 3.2 odd 2
234.2.a.c.1.1 1 1.1 even 1 trivial
624.2.a.h.1.1 1 12.11 even 2
1014.2.a.d.1.1 1 39.38 odd 2
1014.2.b.b.337.1 2 39.8 even 4
1014.2.b.b.337.2 2 39.5 even 4
1014.2.e.c.529.1 2 39.23 odd 6
1014.2.e.c.991.1 2 39.17 odd 6
1014.2.e.f.529.1 2 39.29 odd 6
1014.2.e.f.991.1 2 39.35 odd 6
1014.2.i.d.361.1 4 39.20 even 12
1014.2.i.d.361.2 4 39.32 even 12
1014.2.i.d.823.1 4 39.2 even 12
1014.2.i.d.823.2 4 39.11 even 12
1872.2.a.c.1.1 1 4.3 odd 2
1950.2.a.w.1.1 1 15.14 odd 2
1950.2.e.i.1249.1 2 15.2 even 4
1950.2.e.i.1249.2 2 15.8 even 4
2106.2.e.j.703.1 2 9.4 even 3
2106.2.e.j.1405.1 2 9.7 even 3
2106.2.e.q.703.1 2 9.5 odd 6
2106.2.e.q.1405.1 2 9.2 odd 6
2496.2.a.b.1.1 1 24.11 even 2
2496.2.a.t.1.1 1 24.5 odd 2
3042.2.a.f.1.1 1 13.12 even 2
3042.2.b.g.1351.1 2 13.5 odd 4
3042.2.b.g.1351.2 2 13.8 odd 4
3822.2.a.j.1.1 1 21.20 even 2
5850.2.a.d.1.1 1 5.4 even 2
5850.2.e.bb.5149.1 2 5.3 odd 4
5850.2.e.bb.5149.2 2 5.2 odd 4
7488.2.a.bk.1.1 1 8.3 odd 2
7488.2.a.bz.1.1 1 8.5 even 2
8112.2.a.v.1.1 1 156.155 even 2
9438.2.a.t.1.1 1 33.32 even 2