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Properties

Label 441.2.bb.c.298.4
Level $441$
Weight $2$
Character 441.298
Analytic conductor $3.521$
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] = [441,2,Mod(37,441)] 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("441.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 32])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.bb (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 147)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label 298.4
Character \(\chi\) \(=\) 441.298
Dual form 441.2.bb.c.37.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.84692 + 1.25921i) q^{2} +(1.09483 + 2.78958i) q^{4} +(1.49226 - 0.460300i) q^{5} +(1.29066 + 2.30959i) q^{7} +(-0.495786 + 2.17218i) q^{8} +(3.33570 + 1.02893i) q^{10} +(0.278287 - 3.71348i) q^{11} +(-0.768961 + 0.370312i) q^{13} +(-0.524503 + 5.89084i) q^{14} +(0.742617 - 0.689048i) q^{16} +(-6.99311 + 1.05404i) q^{17} +(0.365872 + 0.633708i) q^{19} +(2.91781 + 3.65882i) q^{20} +(5.19003 - 6.50809i) q^{22} +(-1.11780 - 0.168481i) q^{23} +(-2.11624 + 1.44283i) q^{25} +(-1.88651 - 0.284346i) q^{26} +(-5.02973 + 6.12901i) q^{28} +(2.81777 + 3.53337i) q^{29} +(-3.36687 + 5.83159i) q^{31} +(6.64552 - 1.00165i) q^{32} +(-14.2430 - 6.85906i) q^{34} +(2.98910 + 2.85241i) q^{35} +(3.19665 - 8.14493i) q^{37} +(-0.122235 + 1.63112i) q^{38} +(0.260015 + 3.46966i) q^{40} +(2.11766 - 9.27809i) q^{41} +(-1.48956 - 6.52618i) q^{43} +(10.6637 - 3.28933i) q^{44} +(-1.85233 - 1.71871i) q^{46} +(-2.66472 - 1.81678i) q^{47} +(-3.66838 + 5.96179i) q^{49} -5.72535 q^{50} +(-1.87490 - 1.73965i) q^{52} +(2.22020 + 5.65698i) q^{53} +(-1.29404 - 5.66957i) q^{55} +(-5.65673 + 1.65849i) q^{56} +(0.754945 + 10.0740i) q^{58} +(-4.93430 - 1.52203i) q^{59} +(2.49712 - 6.36255i) q^{61} +(-13.5615 + 6.53089i) q^{62} +(11.7096 + 5.63905i) q^{64} +(-0.977033 + 0.906554i) q^{65} +(7.43668 - 12.8807i) q^{67} +(-10.5966 - 18.3539i) q^{68} +(1.92887 + 9.03208i) q^{70} +(-6.24439 + 7.83022i) q^{71} +(-3.44149 + 2.34637i) q^{73} +(16.1602 - 11.0178i) q^{74} +(-1.36721 + 1.71443i) q^{76} +(8.93578 - 4.15012i) q^{77} +(-1.41313 - 2.44762i) q^{79} +(0.791006 - 1.37006i) q^{80} +(15.5942 - 14.4693i) q^{82} +(-13.8048 - 6.64804i) q^{83} +(-9.95034 + 4.79183i) q^{85} +(5.46674 - 13.9290i) q^{86} +(7.92838 + 2.44558i) q^{88} +(0.515731 + 6.88196i) q^{89} +(-1.84774 - 1.29803i) q^{91} +(-0.753807 - 3.30265i) q^{92} +(-2.63383 - 6.71090i) q^{94} +(0.837671 + 0.777245i) q^{95} +7.03718 q^{97} +(-14.2824 + 6.39171i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + q^{2} + 3 q^{4} - 6 q^{8} + 30 q^{10} + 9 q^{11} + 42 q^{14} + 29 q^{16} + 5 q^{17} - 26 q^{19} + 5 q^{20} + q^{22} + 4 q^{23} - 56 q^{25} + 62 q^{26} + 7 q^{28} - 12 q^{29} - 36 q^{31} + 14 q^{32}+ \cdots - 119 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{5}{21}\right)\) \(1\)

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 0.998135 − 0.0610429i \(-0.0194426\pi\)
0.307836 + 0.951439i \(0.400395\pi\)
\(3\) 0 0
\(4\) 1.09483 + 2.78958i 0.547415 + 1.39479i
\(5\) 1.49226 − 0.460300i 0.667358 − 0.205853i 0.0574782 − 0.998347i \(-0.481694\pi\)
0.609880 + 0.792494i \(0.291218\pi\)
\(6\) 0 0
\(7\) 1.29066 + 2.30959i 0.487824 + 0.872942i
\(8\) −0.495786 + 2.17218i −0.175287 + 0.767981i
\(9\) 0 0
\(10\) 3.33570 + 1.02893i 1.05484 + 0.325375i
\(11\) 0.278287 − 3.71348i 0.0839067 − 1.11966i −0.783278 − 0.621671i \(-0.786454\pi\)
0.867185 − 0.497986i \(-0.165927\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) 0.742617 − 0.689048i 0.185654 − 0.172262i
\(17\) −6.99311 + 1.05404i −1.69608 + 0.255643i −0.924690 − 0.380722i \(-0.875675\pi\)
−0.771388 + 0.636365i \(0.780437\pi\)
\(18\) 0 0
\(19\) 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 0
\(22\) 5.19003 − 6.50809i 1.10652 − 1.38753i
\(23\) −1.11780 − 0.168481i −0.233077 − 0.0351307i 0.0314652 − 0.999505i \(-0.489983\pi\)
−0.264542 + 0.964374i \(0.585221\pi\)
\(24\) 0 0
\(25\) −2.11624 + 1.44283i −0.423248 + 0.288565i
\(26\) −1.88651 − 0.284346i −0.369976 − 0.0557648i
\(27\) 0 0
\(28\) −5.02973 + 6.12901i −0.950529 + 1.15827i
\(29\) 2.81777 + 3.53337i 0.523247 + 0.656131i 0.971295 − 0.237879i \(-0.0764520\pi\)
−0.448048 + 0.894010i \(0.647881\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 0 0
\(34\) −14.2430 − 6.85906i −2.44265 − 1.17632i
\(35\) 2.98910 + 2.85241i 0.505251 + 0.482144i
\(36\) 0 0
\(37\) 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 0
\(40\) 0.260015 + 3.46966i 0.0411120 + 0.548601i
\(41\) 2.11766 − 9.27809i 0.330723 − 1.44899i −0.487010 − 0.873396i \(-0.661913\pi\)
0.817733 − 0.575597i \(-0.195230\pi\)
\(42\) 0 0
\(43\) −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 0
\(46\) −1.85233 − 1.71871i −0.273111 − 0.253410i
\(47\) −2.66472 − 1.81678i −0.388690 − 0.265004i 0.353170 − 0.935559i \(-0.385104\pi\)
−0.741860 + 0.670555i \(0.766056\pi\)
\(48\) 0 0
\(49\) −3.66838 + 5.96179i −0.524055 + 0.851685i
\(50\) −5.72535 −0.809687
\(51\) 0 0
\(52\) −1.87490 − 1.73965i −0.260002 − 0.241246i
\(53\) 2.22020 + 5.65698i 0.304968 + 0.777046i 0.998401 + 0.0565251i \(0.0180021\pi\)
−0.693433 + 0.720521i \(0.743903\pi\)
\(54\) 0 0
\(55\) −1.29404 − 5.66957i −0.174489 − 0.764484i
\(56\) −5.65673 + 1.65849i −0.755912 + 0.221625i
\(57\) 0 0
\(58\) 0.754945 + 10.0740i 0.0991291 + 1.32279i
\(59\) −4.93430 − 1.52203i −0.642392 − 0.198152i −0.0435908 − 0.999049i \(-0.513880\pi\)
−0.598801 + 0.800898i \(0.704356\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 0 0
\(64\) 11.7096 + 5.63905i 1.46370 + 0.704881i
\(65\) −0.977033 + 0.906554i −0.121186 + 0.112444i
\(66\) 0 0
\(67\) 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 0
\(70\) 1.92887 + 9.03208i 0.230543 + 1.07954i
\(71\) −6.24439 + 7.83022i −0.741073 + 0.929276i −0.999323 − 0.0368025i \(-0.988283\pi\)
0.258250 + 0.966078i \(0.416854\pi\)
\(72\) 0 0
\(73\) −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
\(75\) 0 0
\(76\) −1.36721 + 1.71443i −0.156830 + 0.196659i
\(77\) 8.93578 − 4.15012i 1.01833 − 0.472950i
\(78\) 0 0
\(79\) −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 0
\(82\) 15.5942 − 14.4693i 1.72209 − 1.59787i
\(83\) −13.8048 − 6.64804i −1.51527 − 0.729717i −0.522831 − 0.852436i \(-0.675124\pi\)
−0.992441 + 0.122719i \(0.960838\pi\)
\(84\) 0 0
\(85\) −9.95034 + 4.79183i −1.07927 + 0.519747i
\(86\) 5.46674 − 13.9290i 0.589493 − 1.50200i
\(87\) 0 0
\(88\) 7.92838 + 2.44558i 0.845168 + 0.260700i
\(89\) 0.515731 + 6.88196i 0.0546674 + 0.729486i 0.955438 + 0.295191i \(0.0953834\pi\)
−0.900771 + 0.434295i \(0.856998\pi\)
\(90\) 0 0
\(91\) −1.84774 − 1.29803i −0.193695 − 0.136071i
\(92\) −0.753807 − 3.30265i −0.0785898 − 0.344325i
\(93\) 0 0
\(94\) −2.63383 − 6.71090i −0.271659 − 0.692176i
\(95\) 0.837671 + 0.777245i 0.0859432 + 0.0797436i
\(96\) 0 0
\(97\) 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
\(99\) 0 0
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 441.2.bb.c.298.4 48
3.2 odd 2 147.2.m.a.4.1 ✓ 48
49.37 even 21 inner 441.2.bb.c.37.4 48
147.74 odd 42 7203.2.a.i.1.4 24
147.86 odd 42 147.2.m.a.37.1 yes 48
147.122 even 42 7203.2.a.k.1.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.m.a.4.1 ✓ 48 3.2 odd 2
147.2.m.a.37.1 yes 48 147.86 odd 42
441.2.bb.c.37.4 48 49.37 even 21 inner
441.2.bb.c.298.4 48 1.1 even 1 trivial
7203.2.a.i.1.4 24 147.74 odd 42
7203.2.a.k.1.4 24 147.122 even 42