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Properties

Label 147.2.m.a.4.1
Level $147$
Weight $2$
Character 147.4
Analytic conductor $1.174$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [147,2,Mod(4,147)] 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("147.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.m (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label 4.1
Character \(\chi\) \(=\) 147.4
Dual form 147.2.m.a.37.1

$q$-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+(-1.84692 - 1.25921i) q^{2} +(0.733052 + 0.680173i) q^{3} +(1.09483 + 2.78958i) q^{4} +(-1.49226 + 0.460300i) q^{5} +(-0.497409 - 2.17929i) q^{6} +(1.29066 + 2.30959i) q^{7} +(0.495786 - 2.17218i) q^{8} +(0.0747301 + 0.997204i) q^{9} +(3.33570 + 1.02893i) q^{10} +(-0.278287 + 3.71348i) q^{11} +(-1.09483 + 2.78958i) q^{12} +(-0.768961 + 0.370312i) q^{13} +(0.524503 - 5.89084i) q^{14} +(-1.40699 - 0.677569i) q^{15} +(0.742617 - 0.689048i) q^{16} +(6.99311 - 1.05404i) q^{17} +(1.11767 - 1.93586i) q^{18} +(0.365872 + 0.633708i) q^{19} +(-2.91781 - 3.65882i) q^{20} +(-0.624796 + 2.57092i) q^{21} +(5.19003 - 6.50809i) q^{22} +(1.11780 + 0.168481i) q^{23} +(1.84089 - 1.25510i) q^{24} +(-2.11624 + 1.44283i) q^{25} +(1.88651 + 0.284346i) q^{26} +(-0.623490 + 0.781831i) q^{27} +(-5.02973 + 6.12901i) q^{28} +(-2.81777 - 3.53337i) q^{29} +(1.74539 + 3.02311i) q^{30} +(-3.36687 + 5.83159i) q^{31} +(-6.64552 + 1.00165i) q^{32} +(-2.72981 + 2.53289i) q^{33} +(-14.2430 - 6.85906i) q^{34} +(-2.98910 - 2.85241i) q^{35} +(-2.69997 + 1.30023i) q^{36} +(3.19665 - 8.14493i) q^{37} +(0.122235 - 1.63112i) q^{38} +(-0.815564 - 0.251568i) q^{39} +(0.260015 + 3.46966i) q^{40} +(-2.11766 + 9.27809i) q^{41} +(4.39128 - 3.96154i) q^{42} +(-1.48956 - 6.52618i) q^{43} +(-10.6637 + 3.28933i) q^{44} +(-0.570530 - 1.45369i) q^{45} +(-1.85233 - 1.71871i) q^{46} +(2.66472 + 1.81678i) q^{47} +1.01305 q^{48} +(-3.66838 + 5.96179i) q^{49} +5.72535 q^{50} +(5.84324 + 3.98385i) q^{51} +(-1.87490 - 1.73965i) q^{52} +(-2.22020 - 5.65698i) q^{53} +(2.13603 - 0.658877i) q^{54} +(-1.29404 - 5.66957i) q^{55} +(5.65673 - 1.65849i) q^{56} +(-0.162828 + 0.713397i) q^{57} +(0.754945 + 10.0740i) q^{58} +(4.93430 + 1.52203i) q^{59} +(0.349723 - 4.66673i) q^{60} +(2.49712 - 6.36255i) q^{61} +(13.5615 - 6.53089i) q^{62} +(-2.20668 + 1.45965i) q^{63} +(11.7096 + 5.63905i) q^{64} +(0.977033 - 0.906554i) q^{65} +(8.23119 - 1.24065i) q^{66} +(7.43668 - 12.8807i) q^{67} +(10.5966 + 18.3539i) q^{68} +(0.704807 + 0.883800i) q^{69} +(1.92887 + 9.03208i) q^{70} +(6.24439 - 7.83022i) q^{71} +(2.20315 + 0.332072i) q^{72} +(-3.44149 + 2.34637i) q^{73} +(-16.1602 + 11.0178i) q^{74} +(-2.53268 - 0.381741i) q^{75} +(-1.36721 + 1.71443i) q^{76} +(-8.93578 + 4.15012i) q^{77} +(1.18951 + 1.49159i) q^{78} +(-1.41313 - 2.44762i) q^{79} +(-0.791006 + 1.37006i) q^{80} +(-0.988831 + 0.149042i) q^{81} +(15.5942 - 14.4693i) q^{82} +(13.8048 + 6.64804i) q^{83} +(-7.85584 + 1.07180i) q^{84} +(-9.95034 + 4.79183i) q^{85} +(-5.46674 + 13.9290i) q^{86} +(0.337732 - 4.50672i) q^{87} +(7.92838 + 2.44558i) q^{88} +(-0.515731 - 6.88196i) q^{89} +(-0.776773 + 3.40326i) q^{90} +(-1.84774 - 1.29803i) q^{91} +(0.753807 + 3.30265i) q^{92} +(-6.43458 + 1.98480i) q^{93} +(-2.63383 - 6.71090i) q^{94} +(-0.837671 - 0.777245i) q^{95} +(-5.55281 - 3.78584i) q^{96} +7.03718 q^{97} +(14.2824 - 6.39171i) q^{98} -3.72389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - q^{2} - 4 q^{3} + 3 q^{4} - 2 q^{6} + 6 q^{8} + 4 q^{9} + 30 q^{10} - 9 q^{11} - 3 q^{12} - 42 q^{14} - 7 q^{15} + 29 q^{16} - 5 q^{17} + 6 q^{18} - 26 q^{19} - 5 q^{20} + 7 q^{21} + q^{22} - 4 q^{23}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{21}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.84692 − 1.25921i −1.30597 − 0.890396i −0.307836 − 0.951439i \(-0.599605\pi\)
−0.998135 + 0.0610429i \(0.980557\pi\)
\(3\) 0.733052 + 0.680173i 0.423228 + 0.392698i
\(4\) 1.09483 + 2.78958i 0.547415 + 1.39479i
\(5\) −1.49226 + 0.460300i −0.667358 + 0.205853i −0.609880 − 0.792494i \(-0.708782\pi\)
−0.0574782 + 0.998347i \(0.518306\pi\)
\(6\) −0.497409 − 2.17929i −0.203067 − 0.889693i
\(7\) 1.29066 + 2.30959i 0.487824 + 0.872942i
\(8\) 0.495786 − 2.17218i 0.175287 − 0.767981i
\(9\) 0.0747301 + 0.997204i 0.0249100 + 0.332401i
\(10\) 3.33570 + 1.02893i 1.05484 + 0.325375i
\(11\) −0.278287 + 3.71348i −0.0839067 + 1.11966i 0.783278 + 0.621671i \(0.213546\pi\)
−0.867185 + 0.497986i \(0.834073\pi\)
\(12\) −1.09483 + 2.78958i −0.316050 + 0.805283i
\(13\) −0.768961 + 0.370312i −0.213271 + 0.102706i −0.537470 − 0.843283i \(-0.680620\pi\)
0.324198 + 0.945989i \(0.394905\pi\)
\(14\) 0.524503 − 5.89084i 0.140179 − 1.57439i
\(15\) −1.40699 − 0.677569i −0.363282 − 0.174947i
\(16\) 0.742617 − 0.689048i 0.185654 − 0.172262i
\(17\) 6.99311 − 1.05404i 1.69608 − 0.255643i 0.771388 − 0.636365i \(-0.219563\pi\)
0.924690 + 0.380722i \(0.124325\pi\)
\(18\) 1.11767 − 1.93586i 0.263437 − 0.456286i
\(19\) 0.365872 + 0.633708i 0.0839367 + 0.145383i 0.904938 − 0.425544i \(-0.139917\pi\)
−0.821001 + 0.570927i \(0.806584\pi\)
\(20\) −2.91781 − 3.65882i −0.652443 − 0.818138i
\(21\) −0.624796 + 2.57092i −0.136342 + 0.561021i
\(22\) 5.19003 − 6.50809i 1.10652 − 1.38753i
\(23\) 1.11780 + 0.168481i 0.233077 + 0.0351307i 0.264542 − 0.964374i \(-0.414779\pi\)
−0.0314652 + 0.999505i \(0.510017\pi\)
\(24\) 1.84089 − 1.25510i 0.375771 − 0.256196i
\(25\) −2.11624 + 1.44283i −0.423248 + 0.288565i
\(26\) 1.88651 + 0.284346i 0.369976 + 0.0557648i
\(27\) −0.623490 + 0.781831i −0.119991 + 0.150464i
\(28\) −5.02973 + 6.12901i −0.950529 + 1.15827i
\(29\) −2.81777 − 3.53337i −0.523247 − 0.656131i 0.448048 − 0.894010i \(-0.352119\pi\)
−0.971295 + 0.237879i \(0.923548\pi\)
\(30\) 1.74539 + 3.02311i 0.318664 + 0.551942i
\(31\) −3.36687 + 5.83159i −0.604707 + 1.04738i 0.387390 + 0.921916i \(0.373377\pi\)
−0.992098 + 0.125468i \(0.959957\pi\)
\(32\) −6.64552 + 1.00165i −1.17477 + 0.177069i
\(33\) −2.72981 + 2.53289i −0.475199 + 0.440920i
\(34\) −14.2430 − 6.85906i −2.44265 − 1.17632i
\(35\) −2.98910 − 2.85241i −0.505251 − 0.482144i
\(36\) −2.69997 + 1.30023i −0.449994 + 0.216706i
\(37\) 3.19665 − 8.14493i 0.525526 − 1.33902i −0.383985 − 0.923339i \(-0.625449\pi\)
0.909511 − 0.415679i \(-0.136456\pi\)
\(38\) 0.122235 − 1.63112i 0.0198292 − 0.264602i
\(39\) −0.815564 − 0.251568i −0.130595 − 0.0402832i
\(40\) 0.260015 + 3.46966i 0.0411120 + 0.548601i
\(41\) −2.11766 + 9.27809i −0.330723 + 1.44899i 0.487010 + 0.873396i \(0.338087\pi\)
−0.817733 + 0.575597i \(0.804770\pi\)
\(42\) 4.39128 − 3.96154i 0.677589 − 0.611279i
\(43\) −1.48956 − 6.52618i −0.227155 − 0.995233i −0.951947 − 0.306264i \(-0.900921\pi\)
0.724791 − 0.688969i \(-0.241936\pi\)
\(44\) −10.6637 + 3.28933i −1.60762 + 0.495885i
\(45\) −0.570530 − 1.45369i −0.0850496 − 0.216703i
\(46\) −1.85233 − 1.71871i −0.273111 − 0.253410i
\(47\) 2.66472 + 1.81678i 0.388690 + 0.265004i 0.741860 − 0.670555i \(-0.233944\pi\)
−0.353170 + 0.935559i \(0.614896\pi\)
\(48\) 1.01305 0.146221
\(49\) −3.66838 + 5.96179i −0.524055 + 0.851685i
\(50\) 5.72535 0.809687
\(51\) 5.84324 + 3.98385i 0.818217 + 0.557851i
\(52\) −1.87490 − 1.73965i −0.260002 − 0.241246i
\(53\) −2.22020 − 5.65698i −0.304968 − 0.777046i −0.998401 − 0.0565251i \(-0.981998\pi\)
0.693433 − 0.720521i \(-0.256097\pi\)
\(54\) 2.13603 − 0.658877i 0.290677 − 0.0896619i
\(55\) −1.29404 − 5.66957i −0.174489 − 0.764484i
\(56\) 5.65673 − 1.65849i 0.755912 − 0.221625i
\(57\) −0.162828 + 0.713397i −0.0215671 + 0.0944917i
\(58\) 0.754945 + 10.0740i 0.0991291 + 1.32279i
\(59\) 4.93430 + 1.52203i 0.642392 + 0.198152i 0.598801 − 0.800898i \(-0.295644\pi\)
0.0435908 + 0.999049i \(0.486120\pi\)
\(60\) 0.349723 − 4.66673i 0.0451490 − 0.602472i
\(61\) 2.49712 − 6.36255i 0.319723 − 0.814641i −0.677167 − 0.735830i \(-0.736792\pi\)
0.996890 − 0.0788112i \(-0.0251124\pi\)
\(62\) 13.5615 − 6.53089i 1.72232 − 0.829424i
\(63\) −2.20668 + 1.45965i −0.278015 + 0.183898i
\(64\) 11.7096 + 5.63905i 1.46370 + 0.704881i
\(65\) 0.977033 − 0.906554i 0.121186 − 0.112444i
\(66\) 8.23119 − 1.24065i 1.01319 − 0.152714i
\(67\) 7.43668 − 12.8807i 0.908536 − 1.57363i 0.0924360 − 0.995719i \(-0.470535\pi\)
0.816100 − 0.577911i \(-0.196132\pi\)
\(68\) 10.5966 + 18.3539i 1.28503 + 2.22573i
\(69\) 0.704807 + 0.883800i 0.0848488 + 0.106397i
\(70\) 1.92887 + 9.03208i 0.230543 + 1.07954i
\(71\) 6.24439 − 7.83022i 0.741073 − 0.929276i −0.258250 − 0.966078i \(-0.583146\pi\)
0.999323 + 0.0368025i \(0.0117172\pi\)
\(72\) 2.20315 + 0.332072i 0.259644 + 0.0391351i
\(73\) −3.44149 + 2.34637i −0.402796 + 0.274622i −0.747727 − 0.664006i \(-0.768855\pi\)
0.344931 + 0.938628i \(0.387902\pi\)
\(74\) −16.1602 + 11.0178i −1.87858 + 1.28079i
\(75\) −2.53268 − 0.381741i −0.292449 − 0.0440796i
\(76\) −1.36721 + 1.71443i −0.156830 + 0.196659i
\(77\) −8.93578 + 4.15012i −1.01833 + 0.472950i
\(78\) 1.18951 + 1.49159i 0.134685 + 0.168890i
\(79\) −1.41313 − 2.44762i −0.158990 − 0.275379i 0.775515 − 0.631329i \(-0.217490\pi\)
−0.934505 + 0.355951i \(0.884157\pi\)
\(80\) −0.791006 + 1.37006i −0.0884372 + 0.153178i
\(81\) −0.988831 + 0.149042i −0.109870 + 0.0165603i
\(82\) 15.5942 − 14.4693i 1.72209 − 1.59787i
\(83\) 13.8048 + 6.64804i 1.51527 + 0.729717i 0.992441 − 0.122719i \(-0.0391615\pi\)
0.522831 + 0.852436i \(0.324876\pi\)
\(84\) −7.85584 + 1.07180i −0.857142 + 0.116943i
\(85\) −9.95034 + 4.79183i −1.07927 + 0.519747i
\(86\) −5.46674 + 13.9290i −0.589493 + 1.50200i
\(87\) 0.337732 − 4.50672i 0.0362086 − 0.483171i
\(88\) 7.92838 + 2.44558i 0.845168 + 0.260700i
\(89\) −0.515731 − 6.88196i −0.0546674 − 0.729486i −0.955438 − 0.295191i \(-0.904617\pi\)
0.900771 − 0.434295i \(-0.143002\pi\)
\(90\) −0.776773 + 3.40326i −0.0818790 + 0.358735i
\(91\) −1.84774 − 1.29803i −0.193695 − 0.136071i
\(92\) 0.753807 + 3.30265i 0.0785898 + 0.344325i
\(93\) −6.43458 + 1.98480i −0.667234 + 0.205815i
\(94\) −2.63383 − 6.71090i −0.271659 − 0.692176i
\(95\) −0.837671 − 0.777245i −0.0859432 − 0.0797436i
\(96\) −5.55281 − 3.78584i −0.566731 − 0.386391i
\(97\) 7.03718 0.714518 0.357259 − 0.934005i \(-0.383711\pi\)
0.357259 + 0.934005i \(0.383711\pi\)
\(98\) 14.2824 − 6.39171i 1.44274 − 0.645660i
\(99\) −3.72389 −0.374266
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 147.2.m.a.4.1 ✓ 48
3.2 odd 2 441.2.bb.c.298.4 48
49.24 odd 42 7203.2.a.k.1.4 24
49.25 even 21 7203.2.a.i.1.4 24
49.37 even 21 inner 147.2.m.a.37.1 yes 48
147.86 odd 42 441.2.bb.c.37.4 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.m.a.4.1 ✓ 48 1.1 even 1 trivial
147.2.m.a.37.1 yes 48 49.37 even 21 inner
441.2.bb.c.37.4 48 147.86 odd 42
441.2.bb.c.298.4 48 3.2 odd 2
7203.2.a.i.1.4 24 49.25 even 21
7203.2.a.k.1.4 24 49.24 odd 42