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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 167.11
Character \(\chi\) \(=\) 171.167
Dual form 171.2.bd.a.128.11

$q$-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+(0.114947 + 0.651899i) q^{2} +(-1.71194 - 0.263172i) q^{3} +(1.46763 - 0.534172i) q^{4} +(-2.11716 + 0.373313i) q^{5} +(-0.0252213 - 1.14626i) q^{6} +4.63689 q^{7} +(1.17888 + 2.04188i) q^{8} +(2.86148 + 0.901071i) q^{9} +(-0.486725 - 1.33727i) q^{10} +(2.06942 + 1.19478i) q^{11} +(-2.65307 + 0.528232i) q^{12} +(-1.12514 + 1.34089i) q^{13} +(0.532998 + 3.02278i) q^{14} +(3.72270 - 0.0819109i) q^{15} +(1.19725 - 1.00461i) q^{16} +(1.52972 - 4.20288i) q^{17} +(-0.258487 + 1.96897i) q^{18} +(-3.09784 + 3.06650i) q^{19} +(-2.90779 + 1.67881i) q^{20} +(-7.93807 - 1.22030i) q^{21} +(-0.541001 + 1.48639i) q^{22} +(-1.97169 - 5.41717i) q^{23} +(-1.48081 - 3.80583i) q^{24} +(-0.355445 + 0.129372i) q^{25} +(-1.00346 - 0.579348i) q^{26} +(-4.66155 - 2.29564i) q^{27} +(6.80521 - 2.47690i) q^{28} +(5.01985 + 4.21215i) q^{29} +(0.481313 + 2.41741i) q^{30} +(-2.62988 + 1.51836i) q^{31} +(4.40483 + 3.69609i) q^{32} +(-3.22829 - 2.59000i) q^{33} +(2.91569 + 0.514115i) q^{34} +(-9.81705 + 1.73101i) q^{35} +(4.68091 - 0.206088i) q^{36} -6.49789i q^{37} +(-2.35514 - 1.66699i) q^{38} +(2.27906 - 1.99942i) q^{39} +(-3.25815 - 3.88291i) q^{40} +(-0.835095 - 0.303950i) q^{41} +(-0.116948 - 5.31509i) q^{42} +(-10.1747 - 3.70329i) q^{43} +(3.67535 + 0.648063i) q^{44} +(-6.39460 - 0.839486i) q^{45} +(3.30481 - 1.90803i) q^{46} +(-5.36369 + 6.39219i) q^{47} +(-2.31400 + 1.40475i) q^{48} +14.5007 q^{49} +(-0.125195 - 0.216844i) q^{50} +(-3.72488 + 6.79250i) q^{51} +(-0.935021 + 2.56895i) q^{52} +(-5.26690 - 4.41945i) q^{53} +(0.960694 - 3.30274i) q^{54} +(-4.82732 - 1.75700i) q^{55} +(5.46634 + 9.46798i) q^{56} +(6.11033 - 4.43440i) q^{57} +(-2.16888 + 3.75661i) q^{58} +(4.79130 - 4.02038i) q^{59} +(5.41978 - 2.10878i) q^{60} +(0.265379 - 1.50504i) q^{61} +(-1.29212 - 1.53989i) q^{62} +(13.2684 + 4.17816i) q^{63} +(-0.340260 + 0.589347i) q^{64} +(1.88154 - 3.25892i) q^{65} +(1.31734 - 2.40223i) q^{66} +(-2.99917 - 0.528834i) q^{67} -6.98539i q^{68} +(1.94976 + 9.79276i) q^{69} +(-2.25689 - 6.20075i) q^{70} +(-7.57102 + 6.35284i) q^{71} +(1.53347 + 6.90506i) q^{72} +(1.89766 + 10.7622i) q^{73} +(4.23597 - 0.746916i) q^{74} +(0.642548 - 0.127933i) q^{75} +(-2.90843 + 6.15525i) q^{76} +(9.59565 + 5.54005i) q^{77} +(1.56539 + 1.25589i) q^{78} +(-1.38770 - 1.65380i) q^{79} +(-2.15973 + 2.57387i) q^{80} +(7.37614 + 5.15679i) q^{81} +(0.102153 - 0.579336i) q^{82} +2.14060i q^{83} +(-12.3020 + 2.44935i) q^{84} +(-1.66969 + 9.46926i) q^{85} +(1.24462 - 7.05857i) q^{86} +(-7.48516 - 8.53204i) q^{87} +5.63401i q^{88} +(2.28391 - 12.9527i) q^{89} +(-0.187783 - 4.26513i) q^{90} +(-5.21716 + 6.21757i) q^{91} +(-5.78740 - 6.89715i) q^{92} +(4.90179 - 1.90724i) q^{93} +(-4.78361 - 2.76182i) q^{94} +(5.41386 - 7.64874i) q^{95} +(-6.56809 - 7.48671i) q^{96} +(1.81563 - 0.320145i) q^{97} +(1.66682 + 9.45300i) q^{98} +(4.84502 + 5.28353i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.114947 + 0.651899i 0.0812801 + 0.460962i 0.998097 + 0.0616556i \(0.0196380\pi\)
−0.916817 + 0.399307i \(0.869251\pi\)
\(3\) −1.71194 − 0.263172i −0.988389 − 0.151943i
\(4\) 1.46763 − 0.534172i 0.733813 − 0.267086i
\(5\) −2.11716 + 0.373313i −0.946824 + 0.166951i −0.625680 − 0.780080i \(-0.715178\pi\)
−0.321144 + 0.947030i \(0.604067\pi\)
\(6\) −0.0252213 − 1.14626i −0.0102966 − 0.467960i
\(7\) 4.63689 1.75258 0.876289 − 0.481786i \(-0.160012\pi\)
0.876289 + 0.481786i \(0.160012\pi\)
\(8\) 1.17888 + 2.04188i 0.416798 + 0.721915i
\(9\) 2.86148 + 0.901071i 0.953827 + 0.300357i
\(10\) −0.486725 − 1.33727i −0.153916 − 0.422880i
\(11\) 2.06942 + 1.19478i 0.623953 + 0.360239i 0.778406 − 0.627761i \(-0.216028\pi\)
−0.154454 + 0.988000i \(0.549362\pi\)
\(12\) −2.65307 + 0.528232i −0.765875 + 0.152488i
\(13\) −1.12514 + 1.34089i −0.312059 + 0.371897i −0.899163 − 0.437614i \(-0.855823\pi\)
0.587104 + 0.809511i \(0.300268\pi\)
\(14\) 0.532998 + 3.02278i 0.142450 + 0.807872i
\(15\) 3.72270 − 0.0819109i 0.961198 − 0.0211493i
\(16\) 1.19725 − 1.00461i 0.299312 − 0.251152i
\(17\) 1.52972 − 4.20288i 0.371013 − 1.01935i −0.603958 − 0.797016i \(-0.706411\pi\)
0.974971 − 0.222333i \(-0.0713672\pi\)
\(18\) −0.258487 + 1.96897i −0.0609261 + 0.464091i
\(19\) −3.09784 + 3.06650i −0.710693 + 0.703503i
\(20\) −2.90779 + 1.67881i −0.650202 + 0.375394i
\(21\) −7.93807 − 1.22030i −1.73223 − 0.266291i
\(22\) −0.541001 + 1.48639i −0.115342 + 0.316899i
\(23\) −1.97169 − 5.41717i −0.411125 − 1.12956i −0.956593 − 0.291426i \(-0.905870\pi\)
0.545468 − 0.838132i \(-0.316352\pi\)
\(24\) −1.48081 − 3.80583i −0.302269 − 0.776862i
\(25\) −0.355445 + 0.129372i −0.0710891 + 0.0258743i
\(26\) −1.00346 − 0.579348i −0.196795 − 0.113619i
\(27\) −4.66155 − 2.29564i −0.897115 − 0.441797i
\(28\) 6.80521 − 2.47690i 1.28606 − 0.468089i
\(29\) 5.01985 + 4.21215i 0.932162 + 0.782177i 0.976204 − 0.216853i \(-0.0695791\pi\)
−0.0440421 + 0.999030i \(0.514024\pi\)
\(30\) 0.481313 + 2.41741i 0.0878753 + 0.441357i
\(31\) −2.62988 + 1.51836i −0.472341 + 0.272706i −0.717219 − 0.696848i \(-0.754585\pi\)
0.244878 + 0.969554i \(0.421252\pi\)
\(32\) 4.40483 + 3.69609i 0.778671 + 0.653382i
\(33\) −3.22829 − 2.59000i −0.561973 − 0.450862i
\(34\) 2.91569 + 0.514115i 0.500037 + 0.0881701i
\(35\) −9.81705 + 1.73101i −1.65938 + 0.292594i
\(36\) 4.68091 − 0.206088i 0.780152 − 0.0343481i
\(37\) − 6.49789i − 1.06825i −0.845407 − 0.534123i \(-0.820642\pi\)
0.845407 − 0.534123i \(-0.179358\pi\)
\(38\) −2.35514 − 1.66699i −0.382053 − 0.270422i
\(39\) 2.27906 − 1.99942i 0.364942 − 0.320164i
\(40\) −3.25815 − 3.88291i −0.515158 − 0.613942i
\(41\) −0.835095 − 0.303950i −0.130420 − 0.0474690i 0.275986 − 0.961162i \(-0.410996\pi\)
−0.406406 + 0.913693i \(0.633218\pi\)
\(42\) −0.116948 − 5.31509i −0.0180455 − 0.820137i
\(43\) −10.1747 − 3.70329i −1.55163 − 0.564747i −0.582831 − 0.812594i \(-0.698055\pi\)
−0.968799 + 0.247847i \(0.920277\pi\)
\(44\) 3.67535 + 0.648063i 0.554080 + 0.0976992i
\(45\) −6.39460 − 0.839486i −0.953251 − 0.125143i
\(46\) 3.30481 − 1.90803i 0.487267 − 0.281324i
\(47\) −5.36369 + 6.39219i −0.782374 + 0.932397i −0.999038 − 0.0438491i \(-0.986038\pi\)
0.216664 + 0.976246i \(0.430482\pi\)
\(48\) −2.31400 + 1.40475i −0.333997 + 0.202758i
\(49\) 14.5007 2.07153
\(50\) −0.125195 − 0.216844i −0.0177052 − 0.0306663i
\(51\) −3.72488 + 6.79250i −0.521587 + 0.951141i
\(52\) −0.935021 + 2.56895i −0.129664 + 0.356249i
\(53\) −5.26690 − 4.41945i −0.723464 − 0.607059i 0.204877 − 0.978788i \(-0.434321\pi\)
−0.928341 + 0.371729i \(0.878765\pi\)
\(54\) 0.960694 − 3.30274i 0.130734 − 0.449446i
\(55\) −4.82732 − 1.75700i −0.650916 − 0.236914i
\(56\) 5.46634 + 9.46798i 0.730470 + 1.26521i
\(57\) 6.11033 − 4.43440i 0.809333 − 0.587350i
\(58\) −2.16888 + 3.75661i −0.284788 + 0.493267i
\(59\) 4.79130 − 4.02038i 0.623774 − 0.523409i −0.275213 − 0.961383i \(-0.588748\pi\)
0.898987 + 0.437974i \(0.144304\pi\)
\(60\) 5.41978 − 2.10878i 0.699691 − 0.272242i
\(61\) 0.265379 − 1.50504i 0.0339783 − 0.192700i −0.963094 − 0.269165i \(-0.913252\pi\)
0.997072 + 0.0764649i \(0.0243633\pi\)
\(62\) −1.29212 − 1.53989i −0.164099 − 0.195566i
\(63\) 13.2684 + 4.17816i 1.67166 + 0.526399i
\(64\) −0.340260 + 0.589347i −0.0425324 + 0.0736683i
\(65\) 1.88154 − 3.25892i 0.233376 − 0.404219i
\(66\) 1.31734 − 2.40223i 0.162153 − 0.295694i
\(67\) −2.99917 − 0.528834i −0.366407 − 0.0646074i −0.0125864 − 0.999921i \(-0.504006\pi\)
−0.353820 + 0.935313i \(0.615118\pi\)
\(68\) − 6.98539i − 0.847104i
\(69\) 1.94976 + 9.79276i 0.234724 + 1.17891i
\(70\) −2.25689 − 6.20075i −0.269750 − 0.741131i
\(71\) −7.57102 + 6.35284i −0.898515 + 0.753943i −0.969899 − 0.243506i \(-0.921703\pi\)
0.0713849 + 0.997449i \(0.477258\pi\)
\(72\) 1.53347 + 6.90506i 0.180721 + 0.813770i
\(73\) 1.89766 + 10.7622i 0.222104 + 1.25962i 0.868144 + 0.496312i \(0.165313\pi\)
−0.646040 + 0.763303i \(0.723576\pi\)
\(74\) 4.23597 − 0.746916i 0.492421 − 0.0868272i
\(75\) 0.642548 − 0.127933i 0.0741951 − 0.0147724i
\(76\) −2.90843 + 6.15525i −0.333620 + 0.706055i
\(77\) 9.59565 + 5.54005i 1.09353 + 0.631348i
\(78\) 1.56539 + 1.25589i 0.177246 + 0.142202i
\(79\) −1.38770 − 1.65380i −0.156129 − 0.186067i 0.682310 − 0.731063i \(-0.260976\pi\)
−0.838438 + 0.544996i \(0.816531\pi\)
\(80\) −2.15973 + 2.57387i −0.241465 + 0.287767i
\(81\) 7.37614 + 5.15679i 0.819571 + 0.572977i
\(82\) 0.102153 − 0.579336i 0.0112809 − 0.0639770i
\(83\) 2.14060i 0.234962i 0.993075 + 0.117481i \(0.0374819\pi\)
−0.993075 + 0.117481i \(0.962518\pi\)
\(84\) −12.3020 + 2.44935i −1.34226 + 0.267246i
\(85\) −1.66969 + 9.46926i −0.181103 + 1.02708i
\(86\) 1.24462 − 7.05857i 0.134211 − 0.761146i
\(87\) −7.48516 − 8.53204i −0.802493 − 0.914731i
\(88\) 5.63401i 0.600588i
\(89\) 2.28391 − 12.9527i 0.242094 − 1.37298i −0.585052 − 0.810996i \(-0.698926\pi\)
0.827146 − 0.561987i \(-0.189963\pi\)
\(90\) −0.187783 − 4.26513i −0.0197941 − 0.449584i
\(91\) −5.21716 + 6.21757i −0.546907 + 0.651779i
\(92\) −5.78740 − 6.89715i −0.603378 − 0.719078i
\(93\) 4.90179 − 1.90724i 0.508292 − 0.197771i
\(94\) −4.78361 − 2.76182i −0.493391 − 0.284860i
\(95\) 5.41386 − 7.64874i 0.555451 − 0.784744i
\(96\) −6.56809 − 7.48671i −0.670353 − 0.764109i
\(97\) 1.81563 − 0.320145i 0.184350 − 0.0325058i −0.0807110 − 0.996738i \(-0.525719\pi\)
0.265061 + 0.964232i \(0.414608\pi\)
\(98\) 1.66682 + 9.45300i 0.168374 + 0.954897i
\(99\) 4.84502 + 5.28353i 0.486943 + 0.531015i
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.167.11 yes 108
3.2 odd 2 513.2.cd.a.395.8 108
9.2 odd 6 171.2.x.a.110.8 yes 108
9.7 even 3 513.2.bo.a.224.11 108
19.14 odd 18 171.2.x.a.14.8 ✓ 108
57.14 even 18 513.2.bo.a.71.11 108
171.52 odd 18 513.2.cd.a.413.8 108
171.128 even 18 inner 171.2.bd.a.128.11 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.14.8 ✓ 108 19.14 odd 18
171.2.x.a.110.8 yes 108 9.2 odd 6
171.2.bd.a.128.11 yes 108 171.128 even 18 inner
171.2.bd.a.167.11 yes 108 1.1 even 1 trivial
513.2.bo.a.71.11 108 57.14 even 18
513.2.bo.a.224.11 108 9.7 even 3
513.2.cd.a.395.8 108 3.2 odd 2
513.2.cd.a.413.8 108 171.52 odd 18