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Properties

Label 231.2.a.a.1.1
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
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] = [231,2,Mod(1,231)] 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("231.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 231.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} -5.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -6.00000 q^{39} -6.00000 q^{40} +10.0000 q^{41} +1.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} -2.00000 q^{45} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +3.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -12.0000 q^{65} -1.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} +2.00000 q^{70} +3.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +6.00000 q^{78} +16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +4.00000 q^{83} +1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -3.00000 q^{88} +18.0000 q^{89} +2.00000 q^{90} +6.00000 q^{91} -8.00000 q^{93} +8.00000 q^{94} -8.00000 q^{95} +5.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.00000 −0.365148
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −5.00000 −0.883883
\(33\) 1.00000 0.174078
\(34\) −2.00000 −0.342997
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −6.00000 −0.960769
\(40\) −6.00000 −0.948683
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) 3.00000 0.400892
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −2.00000 −0.258199
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −8.00000 −1.01600
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) −12.0000 −1.48842
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) −1.00000 −0.113961
\(78\) 6.00000 0.679366
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) −3.00000 −0.319801
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 2.00000 0.210819
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) −8.00000 −0.820783
\(96\) 5.00000 0.510310
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
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 231.2.a.a.1.1 ✓ 1
3.2 odd 2 693.2.a.d.1.1 1
4.3 odd 2 3696.2.a.t.1.1 1
5.4 even 2 5775.2.a.t.1.1 1
7.6 odd 2 1617.2.a.e.1.1 1
11.10 odd 2 2541.2.a.h.1.1 1
21.20 even 2 4851.2.a.p.1.1 1
33.32 even 2 7623.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
693.2.a.d.1.1 1 3.2 odd 2
1617.2.a.e.1.1 1 7.6 odd 2
2541.2.a.h.1.1 1 11.10 odd 2
3696.2.a.t.1.1 1 4.3 odd 2
4851.2.a.p.1.1 1 21.20 even 2
5775.2.a.t.1.1 1 5.4 even 2
7623.2.a.f.1.1 1 33.32 even 2