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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 110.8
Character \(\chi\) \(=\) 171.110
Dual form 171.2.x.a.14.8

$q$-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+(-0.507087 + 0.425497i) q^{2} +(-1.51869 + 0.832819i) q^{3} +(-0.271206 + 1.53809i) q^{4} +(-0.735283 + 2.02017i) q^{5} +(0.415745 - 1.06851i) q^{6} +(-2.31844 - 4.01566i) q^{7} +(-1.17888 - 2.04188i) q^{8} +(1.61282 - 2.52958i) q^{9} +(-0.486725 - 1.33727i) q^{10} +2.38956i q^{11} +(-0.869071 - 2.56174i) q^{12} +(-0.598676 - 1.64485i) q^{13} +(2.88431 + 1.04980i) q^{14} +(-0.565774 - 3.68037i) q^{15} +(-1.46864 - 0.534541i) 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} +(6.86531 + 4.16769i) q^{21} +(-1.01675 - 1.21171i) q^{22} +(-5.67725 - 1.00105i) q^{23} +(3.49087 + 2.11919i) q^{24} +(0.289762 + 0.243139i) q^{25} +(1.00346 + 0.579348i) q^{26} +(-0.342690 + 5.18484i) q^{27} +(6.80521 - 2.47690i) q^{28} +(-1.13791 + 6.45339i) q^{29} +(1.85288 + 1.62553i) q^{30} -3.03673i q^{31} +(5.40332 - 1.96665i) q^{32} +(-1.99007 - 3.62899i) q^{33} +(-1.01261 - 2.78212i) q^{34} +(9.81705 - 1.73101i) q^{35} +(3.45331 + 3.16670i) q^{36} -6.49789i q^{37} +(0.266090 - 2.87310i) q^{38} +(2.27906 + 1.99942i) q^{39} +(4.99177 - 0.880184i) q^{40} +(-0.680776 + 0.571239i) q^{41} +(-5.25465 + 0.807785i) q^{42} +(1.88021 + 10.6632i) q^{43} +(-3.67535 - 0.648063i) q^{44} +(3.92432 + 5.11815i) q^{45} +(3.30481 - 1.90803i) q^{46} +(-8.21764 - 1.44899i) q^{47} +(2.67558 - 0.411311i) q^{48} +(-7.25036 + 12.5580i) q^{49} -0.250389 q^{50} +(-1.17707 - 7.65685i) q^{51} +(2.69229 - 0.474723i) q^{52} +(5.26690 + 4.41945i) q^{53} +(-2.03236 - 2.77498i) q^{54} +(-4.82732 - 1.75700i) q^{55} +(-5.46634 + 9.46798i) q^{56} +(2.15081 - 7.23699i) q^{57} +(-2.16888 - 3.75661i) q^{58} +(-1.08610 - 6.15958i) q^{59} +(5.81417 + 0.127930i) q^{60} +(-1.43609 + 0.522694i) q^{61} +(1.29212 + 1.53989i) q^{62} +(-13.8972 - 0.611858i) q^{63} +(-0.340260 + 0.589347i) q^{64} +3.76308 q^{65} +(2.55326 + 0.993448i) q^{66} +(1.95757 - 2.33294i) q^{67} +(-6.04953 - 3.49270i) q^{68} +(9.45566 - 3.20784i) q^{69} +(-4.24156 + 5.05490i) q^{70} +(7.57102 - 6.35284i) q^{71} +(-7.06645 - 0.311118i) q^{72} +(1.89766 + 10.7622i) q^{73} +(2.76483 + 3.29500i) q^{74} +(-0.642548 - 0.127933i) q^{75} +(-3.87639 - 5.59640i) q^{76} +(9.59565 - 5.54005i) q^{77} +(-2.00643 - 0.0441477i) q^{78} +(-0.738381 + 2.02868i) q^{79} +(2.15973 - 2.57387i) q^{80} +(-3.79760 - 8.15955i) q^{81} +(0.102153 - 0.579336i) q^{82} +(-1.85382 + 1.07030i) q^{83} +(-8.27219 + 9.42914i) q^{84} +(-7.36577 - 6.18062i) q^{85} +(-5.49060 - 4.60716i) q^{86} +(-3.64638 - 10.7484i) q^{87} +(4.87920 - 2.81701i) q^{88} +(-2.28391 + 12.9527i) q^{89} +(-4.16773 - 0.925563i) q^{90} +(-5.21716 + 6.21757i) q^{91} +(3.07941 - 8.46061i) q^{92} +(2.52904 + 4.61184i) q^{93} +(4.78361 - 2.76182i) q^{94} +(-3.91707 - 8.51291i) q^{95} +(-6.56809 + 7.48671i) q^{96} +(-1.18507 - 1.41231i) q^{97} +(-1.66682 - 9.45300i) q^{98} +(6.04459 + 3.85394i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.507087 + 0.425497i −0.358565 + 0.300872i −0.804218 − 0.594334i \(-0.797416\pi\)
0.445653 + 0.895206i \(0.352971\pi\)
\(3\) −1.51869 + 0.832819i −0.876815 + 0.480828i
\(4\) −0.271206 + 1.53809i −0.135603 + 0.769044i
\(5\) −0.735283 + 2.02017i −0.328829 + 0.903449i 0.659580 + 0.751634i \(0.270734\pi\)
−0.988409 + 0.151815i \(0.951488\pi\)
\(6\) 0.415745 − 1.06851i 0.169727 − 0.436217i
\(7\) −2.31844 − 4.01566i −0.876289 − 1.51778i −0.855383 − 0.517996i \(-0.826678\pi\)
−0.0209060 − 0.999781i \(-0.506655\pi\)
\(8\) −1.17888 − 2.04188i −0.416798 − 0.721915i
\(9\) 1.61282 − 2.52958i 0.537608 − 0.843195i
\(10\) −0.486725 − 1.33727i −0.153916 − 0.422880i
\(11\) 2.38956i 0.720479i 0.932860 + 0.360239i \(0.117305\pi\)
−0.932860 + 0.360239i \(0.882695\pi\)
\(12\) −0.869071 − 2.56174i −0.250879 − 0.739511i
\(13\) −0.598676 − 1.64485i −0.166043 − 0.456199i 0.828567 − 0.559890i \(-0.189157\pi\)
−0.994610 + 0.103691i \(0.966935\pi\)
\(14\) 2.88431 + 1.04980i 0.770863 + 0.280571i
\(15\) −0.565774 − 3.68037i −0.146082 − 0.950268i
\(16\) −1.46864 − 0.534541i −0.367160 − 0.133635i
\(17\) −1.52972 + 4.20288i −0.371013 + 1.01935i 0.603958 + 0.797016i \(0.293589\pi\)
−0.974971 + 0.222333i \(0.928633\pi\)
\(18\) 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
\(21\) 6.86531 + 4.16769i 1.49813 + 0.909465i
\(22\) −1.01675 − 1.21171i −0.216772 − 0.258338i
\(23\) −5.67725 − 1.00105i −1.18379 − 0.208734i −0.453109 − 0.891455i \(-0.649685\pi\)
−0.730679 + 0.682721i \(0.760796\pi\)
\(24\) 3.49087 + 2.11919i 0.712571 + 0.432577i
\(25\) 0.289762 + 0.243139i 0.0579523 + 0.0486278i
\(26\) 1.00346 + 0.579348i 0.196795 + 0.113619i
\(27\) −0.342690 + 5.18484i −0.0659506 + 0.997823i
\(28\) 6.80521 − 2.47690i 1.28606 − 0.468089i
\(29\) −1.13791 + 6.45339i −0.211304 + 1.19836i 0.675902 + 0.736991i \(0.263754\pi\)
−0.887206 + 0.461373i \(0.847357\pi\)
\(30\) 1.85288 + 1.62553i 0.338289 + 0.296781i
\(31\) − 3.03673i − 0.545412i −0.962097 − 0.272706i \(-0.912081\pi\)
0.962097 − 0.272706i \(-0.0879186\pi\)
\(32\) 5.40332 − 1.96665i 0.955181 − 0.347658i
\(33\) −1.99007 − 3.62899i −0.346427 − 0.631726i
\(34\) −1.01261 − 2.78212i −0.173661 − 0.477130i
\(35\) 9.81705 − 1.73101i 1.65938 − 0.292594i
\(36\) 3.45331 + 3.16670i 0.575552 + 0.527784i
\(37\) − 6.49789i − 1.06825i −0.845407 − 0.534123i \(-0.820642\pi\)
0.845407 − 0.534123i \(-0.179358\pi\)
\(38\) 0.266090 − 2.87310i 0.0431654 − 0.466079i
\(39\) 2.27906 + 1.99942i 0.364942 + 0.320164i
\(40\) 4.99177 − 0.880184i 0.789268 − 0.139169i
\(41\) −0.680776 + 0.571239i −0.106319 + 0.0892125i −0.694398 − 0.719591i \(-0.744329\pi\)
0.588078 + 0.808804i \(0.299885\pi\)
\(42\) −5.25465 + 0.807785i −0.810811 + 0.124644i
\(43\) 1.88021 + 10.6632i 0.286730 + 1.62612i 0.699041 + 0.715082i \(0.253610\pi\)
−0.412311 + 0.911043i \(0.635278\pi\)
\(44\) −3.67535 − 0.648063i −0.554080 − 0.0976992i
\(45\) 3.92432 + 5.11815i 0.585003 + 0.762968i
\(46\) 3.30481 − 1.90803i 0.487267 − 0.281324i
\(47\) −8.21764 − 1.44899i −1.19867 − 0.211357i −0.461545 − 0.887117i \(-0.652705\pi\)
−0.737122 + 0.675760i \(0.763816\pi\)
\(48\) 2.67558 − 0.411311i 0.386187 − 0.0593676i
\(49\) −7.25036 + 12.5580i −1.03577 + 1.79400i
\(50\) −0.250389 −0.0354104
\(51\) −1.17707 − 7.65685i −0.164823 − 1.07217i
\(52\) 2.69229 − 0.474723i 0.373353 − 0.0658322i
\(53\) 5.26690 + 4.41945i 0.723464 + 0.607059i 0.928341 − 0.371729i \(-0.121235\pi\)
−0.204877 + 0.978788i \(0.565679\pi\)
\(54\) −2.03236 − 2.77498i −0.276569 − 0.377627i
\(55\) −4.82732 − 1.75700i −0.650916 − 0.236914i
\(56\) −5.46634 + 9.46798i −0.730470 + 1.26521i
\(57\) 2.15081 − 7.23699i 0.284882 − 0.958563i
\(58\) −2.16888 − 3.75661i −0.284788 − 0.493267i
\(59\) −1.08610 − 6.15958i −0.141398 − 0.801909i −0.970189 − 0.242350i \(-0.922082\pi\)
0.828791 − 0.559559i \(-0.189029\pi\)
\(60\) 5.81417 + 0.127930i 0.750606 + 0.0165156i
\(61\) −1.43609 + 0.522694i −0.183873 + 0.0669241i −0.432316 − 0.901722i \(-0.642303\pi\)
0.248443 + 0.968647i \(0.420081\pi\)
\(62\) 1.29212 + 1.53989i 0.164099 + 0.195566i
\(63\) −13.8972 − 0.611858i −1.75088 − 0.0770869i
\(64\) −0.340260 + 0.589347i −0.0425324 + 0.0736683i
\(65\) 3.76308 0.466752
\(66\) 2.55326 + 0.993448i 0.314285 + 0.122285i
\(67\) 1.95757 − 2.33294i 0.239155 − 0.285014i −0.633095 − 0.774074i \(-0.718216\pi\)
0.872250 + 0.489060i \(0.162660\pi\)
\(68\) −6.04953 − 3.49270i −0.733613 − 0.423552i
\(69\) 9.45566 − 3.20784i 1.13833 − 0.386178i
\(70\) −4.24156 + 5.05490i −0.506964 + 0.604176i
\(71\) 7.57102 − 6.35284i 0.898515 − 0.753943i −0.0713849 − 0.997449i \(-0.522742\pi\)
0.969899 + 0.243506i \(0.0782974\pi\)
\(72\) −7.06645 − 0.311118i −0.832789 − 0.0366655i
\(73\) 1.89766 + 10.7622i 0.222104 + 1.25962i 0.868144 + 0.496312i \(0.165313\pi\)
−0.646040 + 0.763303i \(0.723576\pi\)
\(74\) 2.76483 + 3.29500i 0.321405 + 0.383036i
\(75\) −0.642548 − 0.127933i −0.0741951 − 0.0147724i
\(76\) −3.87639 − 5.59640i −0.444652 − 0.641951i
\(77\) 9.59565 − 5.54005i 1.09353 − 0.631348i
\(78\) −2.00643 − 0.0441477i −0.227184 − 0.00499874i
\(79\) −0.738381 + 2.02868i −0.0830743 + 0.228245i −0.974274 − 0.225366i \(-0.927642\pi\)
0.891200 + 0.453611i \(0.149864\pi\)
\(80\) 2.15973 − 2.57387i 0.241465 − 0.287767i
\(81\) −3.79760 − 8.15955i −0.421955 − 0.906617i
\(82\) 0.102153 − 0.579336i 0.0112809 − 0.0639770i
\(83\) −1.85382 + 1.07030i −0.203483 + 0.117481i −0.598279 − 0.801288i \(-0.704149\pi\)
0.394796 + 0.918769i \(0.370815\pi\)
\(84\) −8.27219 + 9.42914i −0.902570 + 1.02880i
\(85\) −7.36577 − 6.18062i −0.798930 − 0.670382i
\(86\) −5.49060 − 4.60716i −0.592066 − 0.496802i
\(87\) −3.64638 − 10.7484i −0.390933 − 1.15234i
\(88\) 4.87920 − 2.81701i 0.520124 − 0.300294i
\(89\) −2.28391 + 12.9527i −0.242094 + 1.37298i 0.585052 + 0.810996i \(0.301074\pi\)
−0.827146 + 0.561987i \(0.810037\pi\)
\(90\) −4.16773 − 0.925563i −0.439317 − 0.0975629i
\(91\) −5.21716 + 6.21757i −0.546907 + 0.651779i
\(92\) 3.07941 − 8.46061i 0.321051 − 0.882080i
\(93\) 2.52904 + 4.61184i 0.262250 + 0.478225i
\(94\) 4.78361 − 2.76182i 0.493391 − 0.284860i
\(95\) −3.91707 − 8.51291i −0.401883 − 0.873406i
\(96\) −6.56809 + 7.48671i −0.670353 + 0.764109i
\(97\) −1.18507 − 1.41231i −0.120326 − 0.143399i 0.702519 − 0.711665i \(-0.252059\pi\)
−0.822845 + 0.568267i \(0.807614\pi\)
\(98\) −1.66682 − 9.45300i −0.168374 − 0.954897i
\(99\) 6.04459 + 3.85394i 0.607504 + 0.387335i
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.110.8 yes 108
3.2 odd 2 513.2.bo.a.224.11 108
9.4 even 3 513.2.cd.a.395.8 108
9.5 odd 6 171.2.bd.a.167.11 yes 108
19.14 odd 18 171.2.bd.a.128.11 yes 108
57.14 even 18 513.2.cd.a.413.8 108
171.14 even 18 inner 171.2.x.a.14.8 ✓ 108
171.166 odd 18 513.2.bo.a.71.11 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.14.8 ✓ 108 171.14 even 18 inner
171.2.x.a.110.8 yes 108 1.1 even 1 trivial
171.2.bd.a.128.11 yes 108 19.14 odd 18
171.2.bd.a.167.11 yes 108 9.5 odd 6
513.2.bo.a.71.11 108 171.166 odd 18
513.2.bo.a.224.11 108 3.2 odd 2
513.2.cd.a.395.8 108 9.4 even 3
513.2.cd.a.413.8 108 57.14 even 18