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(106,171)] 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("171.106"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 171.121
Dual form 171.2.g.b.106.1

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

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i −0.633316 − 0.773893i \(-0.718307\pi\)
0.986869 + 0.161521i \(0.0516399\pi\)
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 1.50000 − 0.866025i 0.612372 − 0.353553i
\(7\) −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
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) −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
\(13\) 3.00000 + 5.19615i 0.832050 + 1.44115i 0.896410 + 0.443227i \(0.146166\pi\)
−0.0643593 + 0.997927i \(0.520500\pi\)
\(14\) −3.00000 −0.801784
\(15\) −1.50000 − 0.866025i −0.387298 − 0.223607i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) −1.50000 − 2.59808i −0.363803 − 0.630126i 0.624780 − 0.780801i \(-0.285189\pi\)
−0.988583 + 0.150675i \(0.951855\pi\)
\(18\) 3.00000 0.707107
\(19\) −4.00000 + 1.73205i −0.917663 + 0.397360i
\(20\) −0.500000 − 0.866025i −0.111803 − 0.193649i
\(21\) − 5.19615i − 1.13389i
\(22\) −3.00000 −0.639602
\(23\) −4.00000 − 6.92820i −0.834058 − 1.44463i −0.894795 − 0.446476i \(-0.852679\pi\)
0.0607377 − 0.998154i \(-0.480655\pi\)
\(24\) 4.50000 + 2.59808i 0.918559 + 0.530330i
\(25\) −4.00000 −0.800000
\(26\) 6.00000 1.17670
\(27\) 5.19615i 1.00000i
\(28\) 1.50000 − 2.59808i 0.283473 − 0.490990i
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) −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 \(-0.0497126\pi\)
\(32\) 2.50000 + 4.33013i 0.441942 + 0.765466i
\(33\) − 5.19615i − 0.904534i
\(34\) −3.00000 −0.514496
\(35\) 1.50000 + 2.59808i 0.253546 + 0.439155i
\(36\) −1.50000 + 2.59808i −0.250000 + 0.433013i
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −0.500000 + 4.33013i −0.0811107 + 0.702439i
\(39\) 10.3923i 1.66410i
\(40\) −3.00000 −0.474342
\(41\) −1.00000 −0.156174 −0.0780869 − 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) −4.50000 − 2.59808i −0.694365 − 0.400892i
\(43\) −4.00000 + 6.92820i −0.609994 + 1.05654i 0.381246 + 0.924473i \(0.375495\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 1.50000 − 2.59808i 0.226134 − 0.391675i
\(45\) −1.50000 − 2.59808i −0.223607 − 0.387298i
\(46\) −8.00000 −1.17954
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.50000 − 0.866025i 0.216506 − 0.125000i
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) −2.00000 + 3.46410i −0.282843 + 0.489898i
\(51\) − 5.19615i − 0.727607i
\(52\) −3.00000 + 5.19615i −0.416025 + 0.720577i
\(53\) −1.50000 + 2.59808i −0.206041 + 0.356873i −0.950464 − 0.310835i \(-0.899391\pi\)
0.744423 + 0.667708i \(0.232725\pi\)
\(54\) 4.50000 + 2.59808i 0.612372 + 0.353553i
\(55\) 1.50000 + 2.59808i 0.202260 + 0.350325i
\(56\) −4.50000 − 7.79423i −0.601338 − 1.04155i
\(57\) −7.50000 − 0.866025i −0.993399 − 0.114708i
\(58\) −2.50000 + 4.33013i −0.328266 + 0.568574i
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) − 1.73205i − 0.223607i
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) −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
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 − 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 1.50000 − 2.59808i 0.181902 − 0.315063i
\(69\) − 13.8564i − 1.66812i
\(70\) 3.00000 0.358569
\(71\) 7.50000 + 12.9904i 0.890086 + 1.54167i 0.839771 + 0.542941i \(0.182689\pi\)
0.0503155 + 0.998733i \(0.483977\pi\)
\(72\) 4.50000 + 7.79423i 0.530330 + 0.918559i
\(73\) 2.50000 + 4.33013i 0.292603 + 0.506803i 0.974424 − 0.224716i \(-0.0721453\pi\)
−0.681822 + 0.731519i \(0.738812\pi\)
\(74\) 1.00000 − 1.73205i 0.116248 − 0.201347i
\(75\) −6.00000 − 3.46410i −0.692820 − 0.400000i
\(76\) −3.50000 − 2.59808i −0.401478 − 0.298020i
\(77\) −4.50000 + 7.79423i −0.512823 + 0.888235i
\(78\) 9.00000 + 5.19615i 1.01905 + 0.588348i
\(79\) 6.00000 − 10.3923i 0.675053 − 1.16923i −0.301401 − 0.953498i \(-0.597454\pi\)
0.976453 − 0.215728i \(-0.0692125\pi\)
\(80\) −0.500000 + 0.866025i −0.0559017 + 0.0968246i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) −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
\(85\) 1.50000 + 2.59808i 0.162698 + 0.281801i
\(86\) 4.00000 + 6.92820i 0.431331 + 0.747087i
\(87\) −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 −0.838308 − 0.545197i \(-0.816455\pi\)
0.891308 + 0.453398i \(0.149788\pi\)
\(90\) −3.00000 −0.316228
\(91\) 9.00000 − 15.5885i 0.943456 − 1.63411i
\(92\) 4.00000 − 6.92820i 0.417029 − 0.722315i
\(93\) 10.5000 − 6.06218i 1.08880 − 0.628619i
\(94\) 4.50000 − 7.79423i 0.464140 − 0.803913i
\(95\) 4.00000 − 1.73205i 0.410391 − 0.177705i
\(96\) 8.66025i 0.883883i
\(97\) 1.00000 − 1.73205i 0.101535 − 0.175863i −0.810782 − 0.585348i \(-0.800958\pi\)
0.912317 + 0.409484i \(0.134291\pi\)
\(98\) 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.g.b.121.1 yes 2
3.2 odd 2 513.2.g.b.64.1 2
9.2 odd 6 513.2.h.a.235.1 2
9.7 even 3 171.2.h.b.7.1 yes 2
19.11 even 3 171.2.h.b.49.1 yes 2
57.11 odd 6 513.2.h.a.334.1 2
171.11 odd 6 513.2.g.b.505.1 2
171.106 even 3 inner 171.2.g.b.106.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.b.106.1 ✓ 2 171.106 even 3 inner
171.2.g.b.121.1 yes 2 1.1 even 1 trivial
171.2.h.b.7.1 yes 2 9.7 even 3
171.2.h.b.49.1 yes 2 19.11 even 3
513.2.g.b.64.1 2 3.2 odd 2
513.2.g.b.505.1 2 171.11 odd 6
513.2.h.a.235.1 2 9.2 odd 6
513.2.h.a.334.1 2 57.11 odd 6