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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(14,171)] 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("171.14"); 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([15, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.x (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 110.15
Character \(\chi\) \(=\) 171.110
Dual form 171.2.x.a.14.15

$q$-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+(1.30761 - 1.09721i) q^{2} +(0.387860 + 1.68807i) q^{3} +(0.158664 - 0.899827i) q^{4} +(1.05557 - 2.90015i) q^{5} +(2.35933 + 1.78176i) q^{6} +(-0.421554 - 0.730152i) q^{7} +(0.927128 + 1.60583i) q^{8} +(-2.69913 + 1.30947i) q^{9} +(-1.80181 - 4.95044i) q^{10} +5.25332i q^{11} +(1.58051 - 0.0811718i) q^{12} +(-2.06528 - 5.67432i) q^{13} +(-1.35236 - 0.492218i) q^{14} +(5.30506 + 0.657017i) q^{15} +(4.69147 + 1.70755i) 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} +(1.06904 - 0.994807i) q^{21} +(5.76401 + 6.86928i) q^{22} +(-5.64507 - 0.995378i) q^{23} +(-2.35115 + 2.18789i) q^{24} +(-3.46643 - 2.90868i) q^{25} +(-8.92651 - 5.15373i) q^{26} +(-3.25735 - 4.04842i) q^{27} +(-0.723896 + 0.263476i) q^{28} +(-0.889964 + 5.04724i) q^{29} +(7.65781 - 4.96165i) q^{30} +4.15289i q^{31} +(4.52328 - 1.64634i) q^{32} +(-8.86796 + 2.03755i) q^{33} +(-1.55555 - 4.27385i) q^{34} +(-2.56253 + 0.451843i) q^{35} +(0.750038 + 2.63651i) q^{36} -4.72641i q^{37} +(-3.00129 + 6.80828i) q^{38} +(8.77759 - 5.68718i) q^{39} +(5.63581 - 0.993745i) q^{40} +(-2.20913 + 1.85368i) q^{41} +(0.306371 - 2.47378i) q^{42} +(-0.757486 - 4.29592i) q^{43} +(4.72708 + 0.833512i) q^{44} +(0.948531 + 9.21012i) q^{45} +(-8.47367 + 4.89227i) q^{46} +(5.55641 + 0.979745i) q^{47} +(-1.06283 + 8.58179i) q^{48} +(3.14459 - 5.44658i) q^{49} -7.72417 q^{50} +(4.58000 + 0.567221i) q^{51} +(-5.43359 + 0.958089i) q^{52} +(9.57018 + 8.03034i) q^{53} +(-8.70130 - 1.71974i) q^{54} +(15.2354 + 5.54525i) q^{55} +(0.781668 - 1.35389i) q^{56} +(-4.76669 - 5.85480i) q^{57} +(4.37417 + 7.57628i) q^{58} +(-0.518580 - 2.94101i) q^{59} +(1.43292 - 4.66939i) q^{60} +(6.29738 - 2.29206i) q^{61} +(4.55661 + 5.43035i) q^{62} +(2.09394 + 1.41877i) q^{63} +(-0.884270 + 1.53160i) q^{64} -18.6365 q^{65} +(-9.36017 + 12.3943i) q^{66} +(-1.11799 + 1.33237i) q^{67} +(-2.10838 - 1.21727i) q^{68} +(-0.509232 - 9.91531i) q^{69} +(-2.85501 + 3.40247i) q^{70} +(5.22326 - 4.38284i) q^{71} +(-4.60522 - 3.12031i) q^{72} +(-0.376772 - 2.13678i) q^{73} +(-5.18587 - 6.18028i) q^{74} +(3.56556 - 6.97973i) q^{75} +(1.11187 + 3.82441i) q^{76} +(3.83573 - 2.21456i) q^{77} +(5.23759 - 17.0675i) q^{78} +(1.27977 - 3.51615i) q^{79} +(9.90433 - 11.8035i) q^{80} +(5.57060 - 7.06883i) q^{81} +(-0.854794 + 4.84778i) q^{82} +(7.69241 - 4.44121i) q^{83} +(-0.725535 - 1.11979i) q^{84} +(-6.29940 - 5.28582i) q^{85} +(-5.70403 - 4.78625i) q^{86} +(-8.86525 + 0.455303i) q^{87} +(-8.43596 + 4.87050i) q^{88} +(-1.29317 + 7.33393i) q^{89} +(11.3458 + 11.0025i) q^{90} +(-3.27249 + 3.90000i) q^{91} +(-1.79133 + 4.92165i) q^{92} +(-7.01036 + 1.61074i) q^{93} +(8.34058 - 4.81544i) q^{94} +(1.45540 + 13.3738i) q^{95} +(4.53352 + 6.99704i) q^{96} +(1.81856 + 2.16727i) q^{97} +(-1.86417 - 10.5723i) q^{98} +(-6.87905 - 14.1794i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 9 q^{2} - 3 q^{4} - 9 q^{5} + 3 q^{7} - 24 q^{9} - 12 q^{10} - 9 q^{12} - 6 q^{13} - 9 q^{14} - 36 q^{15} - 9 q^{16} + 27 q^{17} + 36 q^{18} - 15 q^{19} - 18 q^{20} + 3 q^{21} + 30 q^{22} - 45 q^{23}+ \cdots + 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30761 − 1.09721i 0.924617 − 0.775846i −0.0502260 − 0.998738i \(-0.515994\pi\)
0.974843 + 0.222892i \(0.0715497\pi\)
\(3\) 0.387860 + 1.68807i 0.223931 + 0.974605i
\(4\) 0.158664 − 0.899827i 0.0793318 − 0.449913i
\(5\) 1.05557 − 2.90015i 0.472065 − 1.29699i −0.444025 − 0.896014i \(-0.646450\pi\)
0.916090 − 0.400973i \(-0.131328\pi\)
\(6\) 2.35933 + 1.78176i 0.963194 + 0.727401i
\(7\) −0.421554 − 0.730152i −0.159332 − 0.275972i 0.775296 − 0.631598i \(-0.217601\pi\)
−0.934628 + 0.355627i \(0.884267\pi\)
\(8\) 0.927128 + 1.60583i 0.327789 + 0.567748i
\(9\) −2.69913 + 1.30947i −0.899710 + 0.436488i
\(10\) −1.80181 − 4.95044i −0.569783 − 1.56547i
\(11\) 5.25332i 1.58394i 0.610562 + 0.791969i \(0.290944\pi\)
−0.610562 + 0.791969i \(0.709056\pi\)
\(12\) 1.58051 − 0.0811718i 0.456253 − 0.0234323i
\(13\) −2.06528 − 5.67432i −0.572807 − 1.57377i −0.800048 − 0.599936i \(-0.795193\pi\)
0.227241 − 0.973838i \(-0.427029\pi\)
\(14\) −1.35236 − 0.492218i −0.361433 − 0.131551i
\(15\) 5.30506 + 0.657017i 1.36976 + 0.169641i
\(16\) 4.69147 + 1.70755i 1.17287 + 0.426888i
\(17\) 0.911301 − 2.50378i 0.221023 − 0.607256i −0.778776 − 0.627302i \(-0.784159\pi\)
0.999799 + 0.0200463i \(0.00638136\pi\)
\(18\) −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
\(21\) 1.06904 − 0.994807i 0.233284 − 0.217085i
\(22\) 5.76401 + 6.86928i 1.22889 + 1.46454i
\(23\) −5.64507 − 0.995378i −1.17708 − 0.207551i −0.449311 − 0.893375i \(-0.648331\pi\)
−0.727767 + 0.685825i \(0.759442\pi\)
\(24\) −2.35115 + 2.18789i −0.479927 + 0.446601i
\(25\) −3.46643 − 2.90868i −0.693287 − 0.581737i
\(26\) −8.92651 − 5.15373i −1.75063 − 1.01073i
\(27\) −3.25735 − 4.04842i −0.626877 − 0.779118i
\(28\) −0.723896 + 0.263476i −0.136803 + 0.0497924i
\(29\) −0.889964 + 5.04724i −0.165262 + 0.937249i 0.783531 + 0.621352i \(0.213416\pi\)
−0.948794 + 0.315896i \(0.897695\pi\)
\(30\) 7.65781 − 4.96165i 1.39812 − 0.905870i
\(31\) 4.15289i 0.745882i 0.927855 + 0.372941i \(0.121651\pi\)
−0.927855 + 0.372941i \(0.878349\pi\)
\(32\) 4.52328 − 1.64634i 0.799610 − 0.291034i
\(33\) −8.86796 + 2.03755i −1.54371 + 0.354693i
\(34\) −1.55555 − 4.27385i −0.266775 − 0.732959i
\(35\) −2.56253 + 0.451843i −0.433147 + 0.0763755i
\(36\) 0.750038 + 2.63651i 0.125006 + 0.439419i
\(37\) − 4.72641i − 0.777017i −0.921445 − 0.388508i \(-0.872990\pi\)
0.921445 − 0.388508i \(-0.127010\pi\)
\(38\) −3.00129 + 6.80828i −0.486873 + 1.10445i
\(39\) 8.77759 − 5.68718i 1.40554 − 0.910677i
\(40\) 5.63581 − 0.993745i 0.891099 − 0.157125i
\(41\) −2.20913 + 1.85368i −0.345009 + 0.289497i −0.798782 − 0.601620i \(-0.794522\pi\)
0.453773 + 0.891117i \(0.350078\pi\)
\(42\) 0.306371 − 2.47378i 0.0472741 − 0.381712i
\(43\) −0.757486 − 4.29592i −0.115516 − 0.655121i −0.986494 − 0.163800i \(-0.947625\pi\)
0.870978 − 0.491322i \(-0.163486\pi\)
\(44\) 4.72708 + 0.833512i 0.712634 + 0.125657i
\(45\) 0.948531 + 9.21012i 0.141399 + 1.37296i
\(46\) −8.47367 + 4.89227i −1.24937 + 0.721326i
\(47\) 5.55641 + 0.979745i 0.810485 + 0.142910i 0.563508 − 0.826111i \(-0.309451\pi\)
0.246978 + 0.969021i \(0.420563\pi\)
\(48\) −1.06283 + 8.58179i −0.153407 + 1.23868i
\(49\) 3.14459 − 5.44658i 0.449226 − 0.778083i
\(50\) −7.72417 −1.09236
\(51\) 4.58000 + 0.567221i 0.641328 + 0.0794268i
\(52\) −5.43359 + 0.958089i −0.753504 + 0.132863i
\(53\) 9.57018 + 8.03034i 1.31457 + 1.10305i 0.987427 + 0.158075i \(0.0505289\pi\)
0.327139 + 0.944976i \(0.393916\pi\)
\(54\) −8.70130 − 1.71974i −1.18410 − 0.234027i
\(55\) 15.2354 + 5.54525i 2.05435 + 0.747721i
\(56\) 0.781668 − 1.35389i 0.104455 − 0.180921i
\(57\) −4.76669 − 5.85480i −0.631363 − 0.775487i
\(58\) 4.37417 + 7.57628i 0.574356 + 0.994814i
\(59\) −0.518580 − 2.94101i −0.0675133 − 0.382887i −0.999777 − 0.0211081i \(-0.993281\pi\)
0.932264 − 0.361779i \(-0.117831\pi\)
\(60\) 1.43292 − 4.66939i 0.184989 − 0.602815i
\(61\) 6.29738 − 2.29206i 0.806297 − 0.293468i 0.0942040 − 0.995553i \(-0.469969\pi\)
0.712094 + 0.702085i \(0.247747\pi\)
\(62\) 4.55661 + 5.43035i 0.578689 + 0.689655i
\(63\) 2.09394 + 1.41877i 0.263811 + 0.178748i
\(64\) −0.884270 + 1.53160i −0.110534 + 0.191450i
\(65\) −18.6365 −2.31157
\(66\) −9.36017 + 12.3943i −1.15216 + 1.52564i
\(67\) −1.11799 + 1.33237i −0.136584 + 0.162774i −0.830001 − 0.557762i \(-0.811660\pi\)
0.693417 + 0.720537i \(0.256105\pi\)
\(68\) −2.10838 − 1.21727i −0.255678 − 0.147616i
\(69\) −0.509232 − 9.91531i −0.0613044 − 1.19366i
\(70\) −2.85501 + 3.40247i −0.341239 + 0.406673i
\(71\) 5.22326 − 4.38284i 0.619887 − 0.520147i −0.277881 − 0.960615i \(-0.589632\pi\)
0.897768 + 0.440469i \(0.145188\pi\)
\(72\) −4.60522 − 3.12031i −0.542730 − 0.367732i
\(73\) −0.376772 − 2.13678i −0.0440978 − 0.250091i 0.954788 − 0.297288i \(-0.0960822\pi\)
−0.998886 + 0.0471971i \(0.984971\pi\)
\(74\) −5.18587 − 6.18028i −0.602845 − 0.718443i
\(75\) 3.56556 − 6.97973i 0.411715 − 0.805949i
\(76\) 1.11187 + 3.82441i 0.127540 + 0.438690i
\(77\) 3.83573 − 2.21456i 0.437122 − 0.252372i
\(78\) 5.23759 − 17.0675i 0.593040 − 1.93251i
\(79\) 1.27977 − 3.51615i 0.143986 − 0.395598i −0.846646 − 0.532156i \(-0.821382\pi\)
0.990632 + 0.136558i \(0.0436041\pi\)
\(80\) 9.90433 − 11.8035i 1.10734 − 1.31967i
\(81\) 5.57060 − 7.06883i 0.618956 − 0.785426i
\(82\) −0.854794 + 4.84778i −0.0943962 + 0.535347i
\(83\) 7.69241 − 4.44121i 0.844351 − 0.487486i −0.0143896 − 0.999896i \(-0.504581\pi\)
0.858741 + 0.512410i \(0.171247\pi\)
\(84\) −0.725535 − 1.11979i −0.0791624 − 0.122179i
\(85\) −6.29940 − 5.28582i −0.683266 − 0.573328i
\(86\) −5.70403 − 4.78625i −0.615081 − 0.516114i
\(87\) −8.86525 + 0.455303i −0.950455 + 0.0488136i
\(88\) −8.43596 + 4.87050i −0.899276 + 0.519197i
\(89\) −1.29317 + 7.33393i −0.137076 + 0.777395i 0.836316 + 0.548248i \(0.184705\pi\)
−0.973392 + 0.229147i \(0.926406\pi\)
\(90\) 11.3458 + 11.0025i 1.19595 + 1.15976i
\(91\) −3.27249 + 3.90000i −0.343050 + 0.408831i
\(92\) −1.79133 + 4.92165i −0.186760 + 0.513118i
\(93\) −7.01036 + 1.61074i −0.726940 + 0.167026i
\(94\) 8.34058 − 4.81544i 0.860265 − 0.496674i
\(95\) 1.45540 + 13.3738i 0.149321 + 1.37212i
\(96\) 4.53352 + 6.99704i 0.462701 + 0.714132i
\(97\) 1.81856 + 2.16727i 0.184647 + 0.220053i 0.850425 − 0.526096i \(-0.176345\pi\)
−0.665778 + 0.746150i \(0.731900\pi\)
\(98\) −1.86417 − 10.5723i −0.188310 − 1.06796i
\(99\) −6.87905 − 14.1794i −0.691370 − 1.42508i
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.x.a.110.15 yes 108
3.2 odd 2 513.2.bo.a.224.4 108
9.4 even 3 513.2.cd.a.395.15 108
9.5 odd 6 171.2.bd.a.167.4 yes 108
19.14 odd 18 171.2.bd.a.128.4 yes 108
57.14 even 18 513.2.cd.a.413.15 108
171.14 even 18 inner 171.2.x.a.14.15 ✓ 108
171.166 odd 18 513.2.bo.a.71.4 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.14.15 ✓ 108 171.14 even 18 inner
171.2.x.a.110.15 yes 108 1.1 even 1 trivial
171.2.bd.a.128.4 yes 108 19.14 odd 18
171.2.bd.a.167.4 yes 108 9.5 odd 6
513.2.bo.a.71.4 108 171.166 odd 18
513.2.bo.a.224.4 108 3.2 odd 2
513.2.cd.a.395.15 108 9.4 even 3
513.2.cd.a.413.15 108 57.14 even 18