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(7,171)] 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("171.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-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: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 171.49
Dual form 171.2.h.b.7.1

$q$-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} -1.73205i q^{3} -1.00000 q^{4} +(0.500000 + 0.866025i) q^{5} +1.73205i q^{6} +(-1.50000 - 2.59808i) q^{7} +3.00000 q^{8} -3.00000 q^{9} +(-0.500000 - 0.866025i) q^{10} +(-1.50000 - 2.59808i) q^{11} +1.73205i q^{12} -6.00000 q^{13} +(1.50000 + 2.59808i) q^{14} +(1.50000 - 0.866025i) q^{15} -1.00000 q^{16} +(-1.50000 + 2.59808i) q^{17} +3.00000 q^{18} +(-4.00000 - 1.73205i) q^{19} +(-0.500000 - 0.866025i) q^{20} +(-4.50000 + 2.59808i) q^{21} +(1.50000 + 2.59808i) q^{22} +8.00000 q^{23} -5.19615i q^{24} +(2.00000 - 3.46410i) q^{25} +6.00000 q^{26} +5.19615i q^{27} +(1.50000 + 2.59808i) q^{28} +(2.50000 - 4.33013i) q^{29} +(-1.50000 + 0.866025i) q^{30} +(3.50000 - 6.06218i) q^{31} -5.00000 q^{32} +(-4.50000 + 2.59808i) q^{33} +(1.50000 - 2.59808i) q^{34} +(1.50000 - 2.59808i) q^{35} +3.00000 q^{36} +2.00000 q^{37} +(4.00000 + 1.73205i) q^{38} +10.3923i q^{39} +(1.50000 + 2.59808i) q^{40} +(0.500000 + 0.866025i) q^{41} +(4.50000 - 2.59808i) q^{42} +8.00000 q^{43} +(1.50000 + 2.59808i) q^{44} +(-1.50000 - 2.59808i) q^{45} -8.00000 q^{46} +(-4.50000 + 7.79423i) q^{47} +1.73205i q^{48} +(-1.00000 + 1.73205i) q^{49} +(-2.00000 + 3.46410i) q^{50} +(4.50000 + 2.59808i) q^{51} +6.00000 q^{52} +(-1.50000 - 2.59808i) q^{53} -5.19615i q^{54} +(1.50000 - 2.59808i) q^{55} +(-4.50000 - 7.79423i) q^{56} +(-3.00000 + 6.92820i) q^{57} +(-2.50000 + 4.33013i) q^{58} +(-1.50000 - 2.59808i) q^{59} +(-1.50000 + 0.866025i) q^{60} +(-3.50000 + 6.06218i) q^{61} +(-3.50000 + 6.06218i) q^{62} +(4.50000 + 7.79423i) q^{63} +7.00000 q^{64} +(-3.00000 - 5.19615i) q^{65} +(4.50000 - 2.59808i) q^{66} -4.00000 q^{67} +(1.50000 - 2.59808i) q^{68} -13.8564i q^{69} +(-1.50000 + 2.59808i) q^{70} +(7.50000 - 12.9904i) q^{71} -9.00000 q^{72} +(2.50000 - 4.33013i) q^{73} -2.00000 q^{74} +(-6.00000 - 3.46410i) q^{75} +(4.00000 + 1.73205i) q^{76} +(-4.50000 + 7.79423i) q^{77} -10.3923i q^{78} -12.0000 q^{79} +(-0.500000 - 0.866025i) q^{80} +9.00000 q^{81} +(-0.500000 - 0.866025i) q^{82} +(0.500000 + 0.866025i) q^{83} +(4.50000 - 2.59808i) q^{84} -3.00000 q^{85} -8.00000 q^{86} +(-7.50000 - 4.33013i) q^{87} +(-4.50000 - 7.79423i) q^{88} +(0.500000 + 0.866025i) q^{89} +(1.50000 + 2.59808i) q^{90} +(9.00000 + 15.5885i) q^{91} -8.00000 q^{92} +(-10.5000 - 6.06218i) q^{93} +(4.50000 - 7.79423i) q^{94} +(-0.500000 - 4.33013i) q^{95} +8.66025i q^{96} -2.00000 q^{97} +(1.00000 - 1.73205i) q^{98} +(4.50000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{4} + q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{9} - q^{10} - 3 q^{11} - 12 q^{13} + 3 q^{14} + 3 q^{15} - 2 q^{16} - 3 q^{17} + 6 q^{18} - 8 q^{19} - q^{20} - 9 q^{21} + 3 q^{22} + 16 q^{23}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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 \(0.615027\pi\)
\(3\) − 1.73205i − 1.00000i
\(4\) −1.00000 −0.500000
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i 0.955901 − 0.293691i \(-0.0948835\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 1.73205i 0.707107i
\(7\) −1.50000 − 2.59808i −0.566947 − 0.981981i −0.996866 − 0.0791130i \(-0.974791\pi\)
0.429919 − 0.902867i \(-0.358542\pi\)
\(8\) 3.00000 1.06066
\(9\) −3.00000 −1.00000
\(10\) −0.500000 − 0.866025i −0.158114 − 0.273861i
\(11\) −1.50000 − 2.59808i −0.452267 − 0.783349i 0.546259 − 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 1.73205i 0.500000i
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.50000 + 2.59808i 0.400892 + 0.694365i
\(15\) 1.50000 − 0.866025i 0.387298 − 0.223607i
\(16\) −1.00000 −0.250000
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 − 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 3.00000 0.707107
\(19\) −4.00000 − 1.73205i −0.917663 − 0.397360i
\(20\) −0.500000 − 0.866025i −0.111803 − 0.193649i
\(21\) −4.50000 + 2.59808i −0.981981 + 0.566947i
\(22\) 1.50000 + 2.59808i 0.319801 + 0.553912i
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) − 5.19615i − 1.06066i
\(25\) 2.00000 − 3.46410i 0.400000 − 0.692820i
\(26\) 6.00000 1.17670
\(27\) 5.19615i 1.00000i
\(28\) 1.50000 + 2.59808i 0.283473 + 0.490990i
\(29\) 2.50000 − 4.33013i 0.464238 − 0.804084i −0.534928 − 0.844897i \(-0.679661\pi\)
0.999167 + 0.0408130i \(0.0129948\pi\)
\(30\) −1.50000 + 0.866025i −0.273861 + 0.158114i
\(31\) 3.50000 − 6.06218i 0.628619 − 1.08880i −0.359211 − 0.933257i \(-0.616954\pi\)
0.987829 − 0.155543i \(-0.0497126\pi\)
\(32\) −5.00000 −0.883883
\(33\) −4.50000 + 2.59808i −0.783349 + 0.452267i
\(34\) 1.50000 − 2.59808i 0.257248 − 0.445566i
\(35\) 1.50000 − 2.59808i 0.253546 − 0.439155i
\(36\) 3.00000 0.500000
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 + 1.73205i 0.648886 + 0.280976i
\(39\) 10.3923i 1.66410i
\(40\) 1.50000 + 2.59808i 0.237171 + 0.410792i
\(41\) 0.500000 + 0.866025i 0.0780869 + 0.135250i 0.902424 − 0.430848i \(-0.141786\pi\)
−0.824338 + 0.566099i \(0.808452\pi\)
\(42\) 4.50000 − 2.59808i 0.694365 − 0.400892i
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 1.50000 + 2.59808i 0.226134 + 0.391675i
\(45\) −1.50000 − 2.59808i −0.223607 − 0.387298i
\(46\) −8.00000 −1.17954
\(47\) −4.50000 + 7.79423i −0.656392 + 1.13691i 0.325150 + 0.945662i \(0.394585\pi\)
−0.981543 + 0.191243i \(0.938748\pi\)
\(48\) 1.73205i 0.250000i
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) −2.00000 + 3.46410i −0.282843 + 0.489898i
\(51\) 4.50000 + 2.59808i 0.630126 + 0.363803i
\(52\) 6.00000 0.832050
\(53\) −1.50000 − 2.59808i −0.206041 − 0.356873i 0.744423 − 0.667708i \(-0.232725\pi\)
−0.950464 + 0.310835i \(0.899391\pi\)
\(54\) − 5.19615i − 0.707107i
\(55\) 1.50000 − 2.59808i 0.202260 − 0.350325i
\(56\) −4.50000 − 7.79423i −0.601338 − 1.04155i
\(57\) −3.00000 + 6.92820i −0.397360 + 0.917663i
\(58\) −2.50000 + 4.33013i −0.328266 + 0.568574i
\(59\) −1.50000 − 2.59808i −0.195283 − 0.338241i 0.751710 − 0.659494i \(-0.229229\pi\)
−0.946993 + 0.321253i \(0.895896\pi\)
\(60\) −1.50000 + 0.866025i −0.193649 + 0.111803i
\(61\) −3.50000 + 6.06218i −0.448129 + 0.776182i −0.998264 − 0.0588933i \(-0.981243\pi\)
0.550135 + 0.835076i \(0.314576\pi\)
\(62\) −3.50000 + 6.06218i −0.444500 + 0.769897i
\(63\) 4.50000 + 7.79423i 0.566947 + 0.981981i
\(64\) 7.00000 0.875000
\(65\) −3.00000 − 5.19615i −0.372104 − 0.644503i
\(66\) 4.50000 − 2.59808i 0.553912 − 0.319801i
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 1.50000 − 2.59808i 0.181902 − 0.315063i
\(69\) − 13.8564i − 1.66812i
\(70\) −1.50000 + 2.59808i −0.179284 + 0.310530i
\(71\) 7.50000 − 12.9904i 0.890086 − 1.54167i 0.0503155 − 0.998733i \(-0.483977\pi\)
0.839771 − 0.542941i \(-0.182689\pi\)
\(72\) −9.00000 −1.06066
\(73\) 2.50000 − 4.33013i 0.292603 − 0.506803i −0.681822 − 0.731519i \(-0.738812\pi\)
0.974424 + 0.224716i \(0.0721453\pi\)
\(74\) −2.00000 −0.232495
\(75\) −6.00000 − 3.46410i −0.692820 − 0.400000i
\(76\) 4.00000 + 1.73205i 0.458831 + 0.198680i
\(77\) −4.50000 + 7.79423i −0.512823 + 0.888235i
\(78\) − 10.3923i − 1.17670i
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −0.500000 − 0.866025i −0.0559017 − 0.0968246i
\(81\) 9.00000 1.00000
\(82\) −0.500000 − 0.866025i −0.0552158 − 0.0956365i
\(83\) 0.500000 + 0.866025i 0.0548821 + 0.0950586i 0.892161 − 0.451717i \(-0.149188\pi\)
−0.837279 + 0.546776i \(0.815855\pi\)
\(84\) 4.50000 − 2.59808i 0.490990 − 0.283473i
\(85\) −3.00000 −0.325396
\(86\) −8.00000 −0.862662
\(87\) −7.50000 − 4.33013i −0.804084 − 0.464238i
\(88\) −4.50000 − 7.79423i −0.479702 − 0.830868i
\(89\) 0.500000 + 0.866025i 0.0529999 + 0.0917985i 0.891308 − 0.453398i \(-0.149788\pi\)
−0.838308 + 0.545197i \(0.816455\pi\)
\(90\) 1.50000 + 2.59808i 0.158114 + 0.273861i
\(91\) 9.00000 + 15.5885i 0.943456 + 1.63411i
\(92\) −8.00000 −0.834058
\(93\) −10.5000 − 6.06218i −1.08880 − 0.628619i
\(94\) 4.50000 − 7.79423i 0.464140 − 0.803913i
\(95\) −0.500000 − 4.33013i −0.0512989 − 0.444262i
\(96\) 8.66025i 0.883883i
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 − 1.73205i 0.101015 − 0.174964i
\(99\) 4.50000 + 7.79423i 0.452267 + 0.783349i
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.h.b.49.1 yes 2
3.2 odd 2 513.2.h.a.334.1 2
9.2 odd 6 513.2.g.b.505.1 2
9.7 even 3 171.2.g.b.106.1 ✓ 2
19.7 even 3 171.2.g.b.121.1 yes 2
57.26 odd 6 513.2.g.b.64.1 2
171.7 even 3 inner 171.2.h.b.7.1 yes 2
171.83 odd 6 513.2.h.a.235.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.b.106.1 ✓ 2 9.7 even 3
171.2.g.b.121.1 yes 2 19.7 even 3
171.2.h.b.7.1 yes 2 171.7 even 3 inner
171.2.h.b.49.1 yes 2 1.1 even 1 trivial
513.2.g.b.64.1 2 57.26 odd 6
513.2.g.b.505.1 2 9.2 odd 6
513.2.h.a.235.1 2 171.83 odd 6
513.2.h.a.334.1 2 3.2 odd 2