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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 128.11
Character \(\chi\) \(=\) 171.128
Dual form 171.2.bd.a.167.11

$q$-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 \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{18}\right)\)

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 −0.916817 − 0.399307i \(-0.869251\pi\)
0.998097 − 0.0616556i \(-0.0196380\pi\)
\(3\) −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 −0.321144 − 0.947030i \(-0.604067\pi\)
−0.625680 + 0.780080i \(0.715178\pi\)
\(6\) −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 −0.154454 − 0.988000i \(-0.549362\pi\)
0.778406 + 0.627761i \(0.216028\pi\)
\(12\) −2.65307 − 0.528232i −0.765875 − 0.152488i
\(13\) −1.12514 − 1.34089i −0.312059 − 0.371897i 0.587104 − 0.809511i \(-0.300268\pi\)
−0.899163 + 0.437614i \(0.855823\pi\)
\(14\) 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 0.974971 + 0.222333i \(0.0713672\pi\)
−0.603958 + 0.797016i \(0.706411\pi\)
\(18\) −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 0.545468 + 0.838132i \(0.316352\pi\)
−0.956593 + 0.291426i \(0.905870\pi\)
\(24\) −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 −0.0440421 − 0.999030i \(-0.514024\pi\)
0.976204 + 0.216853i \(0.0695791\pi\)
\(30\) 0.481313 − 2.41741i 0.0878753 − 0.441357i
\(31\) −2.62988 − 1.51836i −0.472341 − 0.272706i 0.244878 − 0.969554i \(-0.421252\pi\)
−0.717219 + 0.696848i \(0.754585\pi\)
\(32\) 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 \(0.179358\pi\)
−0.845407 + 0.534123i \(0.820642\pi\)
\(38\) −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 −0.406406 − 0.913693i \(-0.633218\pi\)
0.275986 + 0.961162i \(0.410996\pi\)
\(42\) −0.116948 + 5.31509i −0.0180455 + 0.820137i
\(43\) −10.1747 + 3.70329i −1.55163 + 0.564747i −0.968799 − 0.247847i \(-0.920277\pi\)
−0.582831 + 0.812594i \(0.698055\pi\)
\(44\) 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 0.216664 − 0.976246i \(-0.430482\pi\)
−0.999038 + 0.0438491i \(0.986038\pi\)
\(48\) −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 −0.928341 − 0.371729i \(-0.878765\pi\)
0.204877 + 0.978788i \(0.434321\pi\)
\(54\) 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 0.898987 − 0.437974i \(-0.144304\pi\)
−0.275213 + 0.961383i \(0.588748\pi\)
\(60\) 5.41978 + 2.10878i 0.699691 + 0.272242i
\(61\) 0.265379 + 1.50504i 0.0339783 + 0.192700i 0.997072 − 0.0764649i \(-0.0243633\pi\)
−0.963094 + 0.269165i \(0.913252\pi\)
\(62\) −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 −0.353820 − 0.935313i \(-0.615118\pi\)
−0.0125864 + 0.999921i \(0.504006\pi\)
\(68\) 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 0.0713849 − 0.997449i \(-0.477258\pi\)
−0.969899 + 0.243506i \(0.921703\pi\)
\(72\) 1.53347 − 6.90506i 0.180721 − 0.813770i
\(73\) 1.89766 − 10.7622i 0.222104 − 1.25962i −0.646040 − 0.763303i \(-0.723576\pi\)
0.868144 − 0.496312i \(-0.165313\pi\)
\(74\) 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 −0.838438 − 0.544996i \(-0.816531\pi\)
0.682310 + 0.731063i \(0.260976\pi\)
\(80\) −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 \(-0.962518\pi\)
0.993075 − 0.117481i \(-0.0374819\pi\)
\(84\) −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 0.827146 + 0.561987i \(0.189963\pi\)
−0.585052 + 0.810996i \(0.698926\pi\)
\(90\) −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 0.265061 − 0.964232i \(-0.414608\pi\)
−0.0807110 + 0.996738i \(0.525719\pi\)
\(98\) 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.128.11 yes 108
3.2 odd 2 513.2.cd.a.413.8 108
9.4 even 3 513.2.bo.a.71.11 108
9.5 odd 6 171.2.x.a.14.8 ✓ 108
19.15 odd 18 171.2.x.a.110.8 yes 108
57.53 even 18 513.2.bo.a.224.11 108
171.148 odd 18 513.2.cd.a.395.8 108
171.167 even 18 inner 171.2.bd.a.167.11 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.14.8 ✓ 108 9.5 odd 6
171.2.x.a.110.8 yes 108 19.15 odd 18
171.2.bd.a.128.11 yes 108 1.1 even 1 trivial
171.2.bd.a.167.11 yes 108 171.167 even 18 inner
513.2.bo.a.71.11 108 9.4 even 3
513.2.bo.a.224.11 108 57.53 even 18
513.2.cd.a.395.8 108 171.148 odd 18
513.2.cd.a.413.8 108 3.2 odd 2