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Properties

Label 171.2.a.c.1.1
Level $171$
Weight $2$
Character 171.1
Self dual yes
Analytic conductor $1.365$
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] = [171,2,Mod(1,171)] 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("171.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,0,2,-1] 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: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 171.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} -2.00000 q^{10} +3.00000 q^{11} -6.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} -2.00000 q^{20} +6.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} -12.0000 q^{26} +6.00000 q^{28} +10.0000 q^{29} +2.00000 q^{31} -8.00000 q^{32} -6.00000 q^{34} -3.00000 q^{35} +8.00000 q^{37} -2.00000 q^{38} +8.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} -8.00000 q^{46} -3.00000 q^{47} +2.00000 q^{49} -8.00000 q^{50} -12.0000 q^{52} +6.00000 q^{53} -3.00000 q^{55} +20.0000 q^{58} +7.00000 q^{61} +4.00000 q^{62} -8.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} -6.00000 q^{70} -12.0000 q^{71} -11.0000 q^{73} +16.0000 q^{74} -2.00000 q^{76} +9.00000 q^{77} +4.00000 q^{80} +16.0000 q^{82} -4.00000 q^{83} +3.00000 q^{85} -2.00000 q^{86} -10.0000 q^{89} -18.0000 q^{91} -8.00000 q^{92} -6.00000 q^{94} +1.00000 q^{95} -2.00000 q^{97} +4.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −12.0000 −2.35339
\(27\) 0 0
\(28\) 6.00000 1.13389
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 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
\(46\) −8.00000 −1.17954
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) −8.00000 −1.13137
\(51\) 0 0
\(52\) −12.0000 −1.66410
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 20.0000 2.62613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 16.0000 1.85996
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 16.0000 1.76690
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 4.00000 0.404061
\(99\) 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 171.2.a.c.1.1 1
3.2 odd 2 57.2.a.b.1.1 ✓ 1
4.3 odd 2 2736.2.a.h.1.1 1
5.4 even 2 4275.2.a.a.1.1 1
7.6 odd 2 8379.2.a.q.1.1 1
12.11 even 2 912.2.a.d.1.1 1
15.2 even 4 1425.2.c.a.799.1 2
15.8 even 4 1425.2.c.a.799.2 2
15.14 odd 2 1425.2.a.i.1.1 1
19.18 odd 2 3249.2.a.a.1.1 1
21.20 even 2 2793.2.a.a.1.1 1
24.5 odd 2 3648.2.a.h.1.1 1
24.11 even 2 3648.2.a.y.1.1 1
33.32 even 2 6897.2.a.g.1.1 1
39.38 odd 2 9633.2.a.p.1.1 1
57.56 even 2 1083.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 ✓ 1 3.2 odd 2
171.2.a.c.1.1 1 1.1 even 1 trivial
912.2.a.d.1.1 1 12.11 even 2
1083.2.a.d.1.1 1 57.56 even 2
1425.2.a.i.1.1 1 15.14 odd 2
1425.2.c.a.799.1 2 15.2 even 4
1425.2.c.a.799.2 2 15.8 even 4
2736.2.a.h.1.1 1 4.3 odd 2
2793.2.a.a.1.1 1 21.20 even 2
3249.2.a.a.1.1 1 19.18 odd 2
3648.2.a.h.1.1 1 24.5 odd 2
3648.2.a.y.1.1 1 24.11 even 2
4275.2.a.a.1.1 1 5.4 even 2
6897.2.a.g.1.1 1 33.32 even 2
8379.2.a.q.1.1 1 7.6 odd 2
9633.2.a.p.1.1 1 39.38 odd 2