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(2,171)] 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("171.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.bd (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 167.4
Character \(\chi\) \(=\) 171.167
Dual form 171.2.bd.a.128.4

$q$-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+(-0.296410 - 1.68103i) q^{2} +(-0.212883 - 1.71892i) q^{3} +(-0.858604 + 0.312506i) q^{4} +(3.03939 - 0.535926i) q^{5} +(-2.82645 + 0.867368i) q^{6} +0.843107 q^{7} +(-0.927128 - 1.60583i) q^{8} +(-2.90936 + 0.731858i) q^{9} +(-1.80181 - 4.95044i) q^{10} +(4.54951 + 2.62666i) q^{11} +(0.719956 + 1.40934i) q^{12} +(-3.88147 + 4.62575i) q^{13} +(-0.249906 - 1.41729i) q^{14} +(-1.56825 - 5.11037i) q^{15} +(-3.82452 + 3.20915i) q^{16} +(-0.911301 + 2.50378i) q^{17} +(2.09264 + 4.67378i) q^{18} +(-3.91070 + 1.92521i) q^{19} +(-2.44215 + 1.40998i) q^{20} +(-0.179483 - 1.44923i) q^{21} +(3.06697 - 8.42642i) q^{22} +(-1.96051 - 5.38646i) q^{23} +(-2.56293 + 1.93551i) q^{24} +(4.25221 - 1.54768i) q^{25} +(8.92651 + 5.15373i) q^{26} +(1.87736 + 4.84515i) q^{27} +(-0.723896 + 0.263476i) q^{28} +(3.92605 + 3.29435i) q^{29} +(-8.12583 + 4.15104i) q^{30} +(3.59651 - 2.07645i) q^{31} +(3.68741 + 3.09410i) q^{32} +(3.54650 - 8.37941i) q^{33} +(4.47904 + 0.789775i) q^{34} +(2.56253 - 0.451843i) q^{35} +(2.26928 - 1.53757i) q^{36} -4.72641i q^{37} +(4.39550 + 6.00333i) q^{38} +(8.77759 + 5.68718i) q^{39} +(-3.67851 - 4.38388i) q^{40} +(-2.70990 - 0.986324i) q^{41} +(-2.38300 + 0.731284i) q^{42} +(4.09912 + 1.49196i) q^{43} +(-4.72708 - 0.833512i) q^{44} +(-8.45046 + 3.78361i) q^{45} +(-8.47367 + 4.89227i) q^{46} +(3.62669 - 4.32212i) q^{47} +(6.33045 + 5.89086i) q^{48} -6.28917 q^{49} +(-3.86209 - 6.68933i) q^{50} +(4.49779 + 1.03344i) q^{51} +(1.88707 - 5.18467i) q^{52} +(-9.57018 - 8.03034i) q^{53} +(7.58836 - 4.59204i) q^{54} +(15.2354 + 5.54525i) q^{55} +(-0.781668 - 1.35389i) q^{56} +(4.14181 + 6.31232i) q^{57} +(4.37417 - 7.57628i) q^{58} +(2.28770 - 1.91961i) q^{59} +(2.94353 + 3.89770i) q^{60} +(-1.16371 + 6.59973i) q^{61} +(-4.55661 - 5.43035i) q^{62} +(-2.45290 + 0.617035i) q^{63} +(-0.884270 + 1.53160i) q^{64} +(-9.31823 + 16.1396i) q^{65} +(-15.1372 - 3.47802i) q^{66} +(1.71286 + 0.302023i) q^{67} -2.43454i q^{68} +(-8.84153 + 4.51665i) q^{69} +(-1.51912 - 4.17375i) q^{70} +(-5.22326 + 4.38284i) q^{71} +(3.87259 + 3.99342i) q^{72} +(-0.376772 - 2.13678i) q^{73} +(-7.94521 + 1.40096i) q^{74} +(-3.56556 - 6.97973i) q^{75} +(2.75610 - 2.87511i) q^{76} +(3.83573 + 2.21456i) q^{77} +(6.95853 - 16.4411i) q^{78} +(2.40519 + 2.86639i) q^{79} +(-9.90433 + 11.8035i) q^{80} +(7.92877 - 4.25848i) q^{81} +(-0.854794 + 4.84778i) q^{82} -8.88243i q^{83} +(0.607000 + 1.18823i) q^{84} +(-1.42796 + 8.09835i) q^{85} +(1.29300 - 7.33296i) q^{86} +(4.82693 - 7.44988i) q^{87} -9.74101i q^{88} +(1.29317 - 7.33393i) q^{89} +(8.86514 + 13.0839i) q^{90} +(-3.27249 + 3.90000i) q^{91} +(3.36661 + 4.01217i) q^{92} +(-4.33488 - 5.74007i) q^{93} +(-8.34058 - 4.81544i) q^{94} +(-10.8544 + 7.94732i) q^{95} +(4.53352 - 6.99704i) q^{96} +(-2.78619 + 0.491281i) q^{97} +(1.86417 + 10.5723i) q^{98} +(-15.1585 - 4.31231i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 9 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 9 q^{6} - 6 q^{7} + 3 q^{9} - 12 q^{10} - 9 q^{11} - 9 q^{12} - 6 q^{13} - 9 q^{14} - 9 q^{15} + 9 q^{16} - 27 q^{17} - 36 q^{18} - 15 q^{19} - 18 q^{20}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.296410 − 1.68103i −0.209594 − 1.18866i −0.890045 − 0.455872i \(-0.849327\pi\)
0.680452 − 0.732793i \(-0.261784\pi\)
\(3\) −0.212883 − 1.71892i −0.122908 − 0.992418i
\(4\) −0.858604 + 0.312506i −0.429302 + 0.156253i
\(5\) 3.03939 − 0.535926i 1.35926 − 0.239674i 0.553959 − 0.832544i \(-0.313117\pi\)
0.805298 + 0.592871i \(0.202005\pi\)
\(6\) −2.82645 + 0.867368i −1.15389 + 0.354101i
\(7\) 0.843107 0.318665 0.159332 − 0.987225i \(-0.449066\pi\)
0.159332 + 0.987225i \(0.449066\pi\)
\(8\) −0.927128 − 1.60583i −0.327789 − 0.567748i
\(9\) −2.90936 + 0.731858i −0.969787 + 0.243953i
\(10\) −1.80181 − 4.95044i −0.569783 − 1.56547i
\(11\) 4.54951 + 2.62666i 1.37173 + 0.791969i 0.991146 − 0.132778i \(-0.0423897\pi\)
0.380584 + 0.924746i \(0.375723\pi\)
\(12\) 0.719956 + 1.40934i 0.207833 + 0.406842i
\(13\) −3.88147 + 4.62575i −1.07652 + 1.28295i −0.119535 + 0.992830i \(0.538140\pi\)
−0.956990 + 0.290122i \(0.906304\pi\)
\(14\) −0.249906 − 1.41729i −0.0667901 − 0.378785i
\(15\) −1.56825 − 5.11037i −0.404920 − 1.31949i
\(16\) −3.82452 + 3.20915i −0.956129 + 0.802288i
\(17\) −0.911301 + 2.50378i −0.221023 + 0.607256i −0.999799 − 0.0200463i \(-0.993619\pi\)
0.778776 + 0.627302i \(0.215841\pi\)
\(18\) 2.09264 + 4.67378i 0.493239 + 1.10162i
\(19\) −3.91070 + 1.92521i −0.897176 + 0.441674i
\(20\) −2.44215 + 1.40998i −0.546082 + 0.315281i
\(21\) −0.179483 − 1.44923i −0.0391665 − 0.316248i
\(22\) 3.06697 − 8.42642i 0.653879 − 1.79652i
\(23\) −1.96051 − 5.38646i −0.408795 − 1.12315i −0.957825 − 0.287352i \(-0.907225\pi\)
0.549030 − 0.835803i \(-0.314997\pi\)
\(24\) −2.56293 + 1.93551i −0.523155 + 0.395085i
\(25\) 4.25221 − 1.54768i 0.850442 − 0.309536i
\(26\) 8.92651 + 5.15373i 1.75063 + 1.01073i
\(27\) 1.87736 + 4.84515i 0.361298 + 0.932450i
\(28\) −0.723896 + 0.263476i −0.136803 + 0.0497924i
\(29\) 3.92605 + 3.29435i 0.729050 + 0.611746i 0.929872 − 0.367882i \(-0.119917\pi\)
−0.200822 + 0.979628i \(0.564361\pi\)
\(30\) −8.12583 + 4.15104i −1.48357 + 0.757872i
\(31\) 3.59651 − 2.07645i 0.645953 − 0.372941i −0.140951 − 0.990017i \(-0.545016\pi\)
0.786904 + 0.617076i \(0.211683\pi\)
\(32\) 3.68741 + 3.09410i 0.651848 + 0.546965i
\(33\) 3.54650 − 8.37941i 0.617367 − 1.45867i
\(34\) 4.47904 + 0.789775i 0.768149 + 0.135445i
\(35\) 2.56253 − 0.451843i 0.433147 − 0.0763755i
\(36\) 2.26928 − 1.53757i 0.378213 − 0.256262i
\(37\) − 4.72641i − 0.777017i −0.921445 − 0.388508i \(-0.872990\pi\)
0.921445 − 0.388508i \(-0.127010\pi\)
\(38\) 4.39550 + 6.00333i 0.713045 + 0.973869i
\(39\) 8.77759 + 5.68718i 1.40554 + 0.910677i
\(40\) −3.67851 − 4.38388i −0.581624 − 0.693152i
\(41\) −2.70990 − 0.986324i −0.423216 − 0.154038i 0.121627 − 0.992576i \(-0.461189\pi\)
−0.544843 + 0.838538i \(0.683411\pi\)
\(42\) −2.38300 + 0.731284i −0.367704 + 0.112840i
\(43\) 4.09912 + 1.49196i 0.625110 + 0.227521i 0.635101 − 0.772429i \(-0.280958\pi\)
−0.00999175 + 0.999950i \(0.503181\pi\)
\(44\) −4.72708 − 0.833512i −0.712634 − 0.125657i
\(45\) −8.45046 + 3.78361i −1.25972 + 0.564027i
\(46\) −8.47367 + 4.89227i −1.24937 + 0.721326i
\(47\) 3.62669 − 4.32212i 0.529007 − 0.630446i −0.433679 − 0.901067i \(-0.642785\pi\)
0.962686 + 0.270622i \(0.0872293\pi\)
\(48\) 6.33045 + 5.89086i 0.913721 + 0.850272i
\(49\) −6.28917 −0.898453
\(50\) −3.86209 − 6.68933i −0.546181 − 0.946014i
\(51\) 4.49779 + 1.03344i 0.629817 + 0.144710i
\(52\) 1.88707 − 5.18467i 0.261689 − 0.718985i
\(53\) −9.57018 − 8.03034i −1.31457 − 1.10305i −0.987427 − 0.158075i \(-0.949471\pi\)
−0.327139 − 0.944976i \(-0.606084\pi\)
\(54\) 7.58836 − 4.59204i 1.03265 − 0.624898i
\(55\) 15.2354 + 5.54525i 2.05435 + 0.747721i
\(56\) −0.781668 − 1.35389i −0.104455 − 0.180921i
\(57\) 4.14181 + 6.31232i 0.548596 + 0.836088i
\(58\) 4.37417 − 7.57628i 0.574356 − 0.994814i
\(59\) 2.28770 − 1.91961i 0.297833 − 0.249912i −0.481608 − 0.876387i \(-0.659947\pi\)
0.779442 + 0.626475i \(0.215503\pi\)
\(60\) 2.94353 + 3.89770i 0.380008 + 0.503191i
\(61\) −1.16371 + 6.59973i −0.148998 + 0.845008i 0.815072 + 0.579359i \(0.196697\pi\)
−0.964070 + 0.265649i \(0.914414\pi\)
\(62\) −4.55661 − 5.43035i −0.578689 − 0.689655i
\(63\) −2.45290 + 0.617035i −0.309037 + 0.0777391i
\(64\) −0.884270 + 1.53160i −0.110534 + 0.191450i
\(65\) −9.31823 + 16.1396i −1.15578 + 2.00188i
\(66\) −15.1372 − 3.47802i −1.86326 − 0.428115i
\(67\) 1.71286 + 0.302023i 0.209259 + 0.0368980i 0.277295 − 0.960785i \(-0.410562\pi\)
−0.0680360 + 0.997683i \(0.521673\pi\)
\(68\) − 2.43454i − 0.295232i
\(69\) −8.84153 + 4.51665i −1.06439 + 0.543741i
\(70\) −1.51912 − 4.17375i −0.181570 − 0.498859i
\(71\) −5.22326 + 4.38284i −0.619887 + 0.520147i −0.897768 − 0.440469i \(-0.854812\pi\)
0.277881 + 0.960615i \(0.410368\pi\)
\(72\) 3.87259 + 3.99342i 0.456389 + 0.470629i
\(73\) −0.376772 − 2.13678i −0.0440978 − 0.250091i 0.954788 − 0.297288i \(-0.0960822\pi\)
−0.998886 + 0.0471971i \(0.984971\pi\)
\(74\) −7.94521 + 1.40096i −0.923612 + 0.162858i
\(75\) −3.56556 − 6.97973i −0.411715 − 0.805949i
\(76\) 2.75610 − 2.87511i 0.316146 − 0.329798i
\(77\) 3.83573 + 2.21456i 0.437122 + 0.252372i
\(78\) 6.95853 − 16.4411i 0.787898 − 1.86159i
\(79\) 2.40519 + 2.86639i 0.270605 + 0.322494i 0.884184 − 0.467139i \(-0.154715\pi\)
−0.613579 + 0.789633i \(0.710271\pi\)
\(80\) −9.90433 + 11.8035i −1.10734 + 1.31967i
\(81\) 7.92877 − 4.25848i 0.880974 − 0.473165i
\(82\) −0.854794 + 4.84778i −0.0943962 + 0.535347i
\(83\) − 8.88243i − 0.974973i −0.873131 − 0.487486i \(-0.837914\pi\)
0.873131 − 0.487486i \(-0.162086\pi\)
\(84\) 0.607000 + 1.18823i 0.0662291 + 0.129646i
\(85\) −1.42796 + 8.09835i −0.154884 + 0.878390i
\(86\) 1.29300 − 7.33296i 0.139428 − 0.790733i
\(87\) 4.82693 − 7.44988i 0.517501 − 0.798711i
\(88\) − 9.74101i − 1.03839i
\(89\) 1.29317 − 7.33393i 0.137076 − 0.777395i −0.836316 − 0.548248i \(-0.815295\pi\)
0.973392 − 0.229147i \(-0.0735937\pi\)
\(90\) 8.86514 + 13.0839i 0.934468 + 1.37917i
\(91\) −3.27249 + 3.90000i −0.343050 + 0.408831i
\(92\) 3.36661 + 4.01217i 0.350993 + 0.418297i
\(93\) −4.33488 − 5.74007i −0.449506 − 0.595218i
\(94\) −8.34058 − 4.81544i −0.860265 − 0.496674i
\(95\) −10.8544 + 7.94732i −1.11363 + 0.815378i
\(96\) 4.53352 − 6.99704i 0.462701 − 0.714132i
\(97\) −2.78619 + 0.491281i −0.282895 + 0.0498820i −0.313295 − 0.949656i \(-0.601433\pi\)
0.0304001 + 0.999538i \(0.490322\pi\)
\(98\) 1.86417 + 10.5723i 0.188310 + 1.06796i
\(99\) −15.1585 − 4.31231i −1.52349 − 0.433404i
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.bd.a.167.4 yes 108
3.2 odd 2 513.2.cd.a.395.15 108
9.2 odd 6 171.2.x.a.110.15 yes 108
9.7 even 3 513.2.bo.a.224.4 108
19.14 odd 18 171.2.x.a.14.15 ✓ 108
57.14 even 18 513.2.bo.a.71.4 108
171.52 odd 18 513.2.cd.a.413.15 108
171.128 even 18 inner 171.2.bd.a.128.4 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.14.15 ✓ 108 19.14 odd 18
171.2.x.a.110.15 yes 108 9.2 odd 6
171.2.bd.a.128.4 yes 108 171.128 even 18 inner
171.2.bd.a.167.4 yes 108 1.1 even 1 trivial
513.2.bo.a.71.4 108 57.14 even 18
513.2.bo.a.224.4 108 9.7 even 3
513.2.cd.a.395.15 108 3.2 odd 2
513.2.cd.a.413.15 108 171.52 odd 18