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Properties

Label 234.2.a.e.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 = [1,1,0,1,3] 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 26)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} -6.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{19} +3.00000 q^{20} -6.00000 q^{22} +4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} +2.00000 q^{38} +3.00000 q^{40} -1.00000 q^{43} -6.00000 q^{44} -3.00000 q^{47} -6.00000 q^{49} +4.00000 q^{50} +1.00000 q^{52} -18.0000 q^{55} -1.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +14.0000 q^{67} +3.00000 q^{68} -3.00000 q^{70} +3.00000 q^{71} +2.00000 q^{73} -7.00000 q^{74} +2.00000 q^{76} +6.00000 q^{77} +8.00000 q^{79} +3.00000 q^{80} -12.0000 q^{83} +9.00000 q^{85} -1.00000 q^{86} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} -3.00000 q^{94} +6.00000 q^{95} -10.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −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
\(34\) 3.00000 0.514496
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) 3.00000 0.356034 0.178017 − 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −6.00000 −0.606092
\(99\) 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.e.1.1 1
3.2 odd 2 26.2.a.a.1.1 ✓ 1
4.3 odd 2 1872.2.a.q.1.1 1
5.2 odd 4 5850.2.e.a.5149.2 2
5.3 odd 4 5850.2.e.a.5149.1 2
5.4 even 2 5850.2.a.p.1.1 1
8.3 odd 2 7488.2.a.h.1.1 1
8.5 even 2 7488.2.a.g.1.1 1
9.2 odd 6 2106.2.e.ba.1405.1 2
9.4 even 3 2106.2.e.b.703.1 2
9.5 odd 6 2106.2.e.ba.703.1 2
9.7 even 3 2106.2.e.b.1405.1 2
12.11 even 2 208.2.a.a.1.1 1
13.5 odd 4 3042.2.b.a.1351.1 2
13.8 odd 4 3042.2.b.a.1351.2 2
13.12 even 2 3042.2.a.a.1.1 1
15.2 even 4 650.2.b.d.599.1 2
15.8 even 4 650.2.b.d.599.2 2
15.14 odd 2 650.2.a.j.1.1 1
21.2 odd 6 1274.2.f.p.1145.1 2
21.5 even 6 1274.2.f.r.1145.1 2
21.11 odd 6 1274.2.f.p.79.1 2
21.17 even 6 1274.2.f.r.79.1 2
21.20 even 2 1274.2.a.d.1.1 1
24.5 odd 2 832.2.a.d.1.1 1
24.11 even 2 832.2.a.i.1.1 1
33.32 even 2 3146.2.a.n.1.1 1
39.2 even 12 338.2.e.a.147.1 4
39.5 even 4 338.2.b.c.337.2 2
39.8 even 4 338.2.b.c.337.1 2
39.11 even 12 338.2.e.a.147.2 4
39.17 odd 6 338.2.c.a.315.1 2
39.20 even 12 338.2.e.a.23.1 4
39.23 odd 6 338.2.c.a.191.1 2
39.29 odd 6 338.2.c.d.191.1 2
39.32 even 12 338.2.e.a.23.2 4
39.35 odd 6 338.2.c.d.315.1 2
39.38 odd 2 338.2.a.f.1.1 1
48.5 odd 4 3328.2.b.m.1665.1 2
48.11 even 4 3328.2.b.j.1665.2 2
48.29 odd 4 3328.2.b.m.1665.2 2
48.35 even 4 3328.2.b.j.1665.1 2
51.50 odd 2 7514.2.a.c.1.1 1
57.56 even 2 9386.2.a.j.1.1 1
60.59 even 2 5200.2.a.x.1.1 1
156.47 odd 4 2704.2.f.d.337.1 2
156.83 odd 4 2704.2.f.d.337.2 2
156.155 even 2 2704.2.a.f.1.1 1
195.194 odd 2 8450.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.a.1.1 ✓ 1 3.2 odd 2
208.2.a.a.1.1 1 12.11 even 2
234.2.a.e.1.1 1 1.1 even 1 trivial
338.2.a.f.1.1 1 39.38 odd 2
338.2.b.c.337.1 2 39.8 even 4
338.2.b.c.337.2 2 39.5 even 4
338.2.c.a.191.1 2 39.23 odd 6
338.2.c.a.315.1 2 39.17 odd 6
338.2.c.d.191.1 2 39.29 odd 6
338.2.c.d.315.1 2 39.35 odd 6
338.2.e.a.23.1 4 39.20 even 12
338.2.e.a.23.2 4 39.32 even 12
338.2.e.a.147.1 4 39.2 even 12
338.2.e.a.147.2 4 39.11 even 12
650.2.a.j.1.1 1 15.14 odd 2
650.2.b.d.599.1 2 15.2 even 4
650.2.b.d.599.2 2 15.8 even 4
832.2.a.d.1.1 1 24.5 odd 2
832.2.a.i.1.1 1 24.11 even 2
1274.2.a.d.1.1 1 21.20 even 2
1274.2.f.p.79.1 2 21.11 odd 6
1274.2.f.p.1145.1 2 21.2 odd 6
1274.2.f.r.79.1 2 21.17 even 6
1274.2.f.r.1145.1 2 21.5 even 6
1872.2.a.q.1.1 1 4.3 odd 2
2106.2.e.b.703.1 2 9.4 even 3
2106.2.e.b.1405.1 2 9.7 even 3
2106.2.e.ba.703.1 2 9.5 odd 6
2106.2.e.ba.1405.1 2 9.2 odd 6
2704.2.a.f.1.1 1 156.155 even 2
2704.2.f.d.337.1 2 156.47 odd 4
2704.2.f.d.337.2 2 156.83 odd 4
3042.2.a.a.1.1 1 13.12 even 2
3042.2.b.a.1351.1 2 13.5 odd 4
3042.2.b.a.1351.2 2 13.8 odd 4
3146.2.a.n.1.1 1 33.32 even 2
3328.2.b.j.1665.1 2 48.35 even 4
3328.2.b.j.1665.2 2 48.11 even 4
3328.2.b.m.1665.1 2 48.5 odd 4
3328.2.b.m.1665.2 2 48.29 odd 4
5200.2.a.x.1.1 1 60.59 even 2
5850.2.a.p.1.1 1 5.4 even 2
5850.2.e.a.5149.1 2 5.3 odd 4
5850.2.e.a.5149.2 2 5.2 odd 4
7488.2.a.g.1.1 1 8.5 even 2
7488.2.a.h.1.1 1 8.3 odd 2
7514.2.a.c.1.1 1 51.50 odd 2
8450.2.a.c.1.1 1 195.194 odd 2
9386.2.a.j.1.1 1 57.56 even 2