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Properties

Label 1001.2.i.e
Level $1001$
Weight $2$
Character orbit 1001.i
Analytic conductor $7.993$
Analytic rank $0$
Dimension $50$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1001,2,Mod(144,1001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1001.144");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1001.i (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.99302524233\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(25\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 2 q^{3} - 30 q^{4} + 3 q^{5} + 4 q^{6} - q^{7} + 6 q^{8} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - 2 q^{3} - 30 q^{4} + 3 q^{5} + 4 q^{6} - q^{7} + 6 q^{8} - 37 q^{9} - 9 q^{10} + 25 q^{11} - q^{12} + 50 q^{13} - 14 q^{14} + 44 q^{15} - 48 q^{16} - 7 q^{17} + 4 q^{18} + 5 q^{19} - 56 q^{20} + 8 q^{21} - 7 q^{23} + 4 q^{24} - 58 q^{25} + 22 q^{27} - 38 q^{28} + 28 q^{29} - 29 q^{30} - 4 q^{31} + 4 q^{32} + 2 q^{33} + 40 q^{34} - 2 q^{35} + 124 q^{36} - 43 q^{37} + 12 q^{38} - 2 q^{39} - q^{40} - 24 q^{41} + 2 q^{42} + 44 q^{43} + 30 q^{44} - 10 q^{45} - 17 q^{46} + 10 q^{47} + 28 q^{48} - 21 q^{49} - 72 q^{50} + 9 q^{51} - 30 q^{52} - 22 q^{53} - 4 q^{54} + 6 q^{55} - 12 q^{56} + 48 q^{57} - 89 q^{58} + 32 q^{59} - 85 q^{60} - 34 q^{61} + 52 q^{62} - 15 q^{63} + 74 q^{64} + 3 q^{65} + 2 q^{66} - 46 q^{67} + 8 q^{68} + 92 q^{69} + 2 q^{70} - 22 q^{71} - 18 q^{72} + 21 q^{73} + 23 q^{74} - 28 q^{75} + 36 q^{76} + q^{77} + 4 q^{78} - 19 q^{79} + 101 q^{80} - 89 q^{81} + 36 q^{82} + 8 q^{83} + 33 q^{84} + 128 q^{85} + 17 q^{86} + 11 q^{87} + 3 q^{88} + 7 q^{89} + 240 q^{90} - q^{91} - 72 q^{92} - 79 q^{93} + 38 q^{94} - 35 q^{95} + 33 q^{96} - 12 q^{97} - 54 q^{98} - 74 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
144.1 −1.33197 − 2.30704i 1.49979 − 2.59771i −2.54830 + 4.41378i 0.314861 + 0.545355i −7.99069 2.62853 + 0.301351i 8.24916 −2.99872 − 5.19393i 0.838772 − 1.45280i
144.2 −1.32587 − 2.29648i −0.177231 + 0.306973i −2.51588 + 4.35764i 1.82388 + 3.15906i 0.939944 −0.396934 + 2.61581i 8.03947 1.43718 + 2.48927i 4.83648 − 8.37703i
144.3 −1.29311 − 2.23974i −1.12306 + 1.94520i −2.34429 + 4.06043i −0.969541 − 1.67929i 5.80899 0.615999 − 2.57304i 6.95330 −1.02253 − 1.77108i −2.50746 + 4.34304i
144.4 −1.15903 − 2.00750i −1.60335 + 2.77708i −1.68671 + 2.92146i −1.10286 − 1.91021i 7.43331 −0.570065 + 2.58361i 3.18366 −3.64144 − 6.30715i −2.55650 + 4.42800i
144.5 −1.05802 − 1.83255i 0.964193 − 1.67003i −1.23882 + 2.14570i 2.12791 + 3.68565i −4.08055 −0.522865 − 2.59357i 1.01070 −0.359338 − 0.622392i 4.50275 − 7.79900i
144.6 −0.968874 − 1.67814i 0.313422 − 0.542862i −0.877434 + 1.51976i −1.84550 − 3.19649i −1.21466 −2.36366 + 1.18874i −0.475002 1.30353 + 2.25779i −3.57611 + 6.19400i
144.7 −0.745367 − 1.29101i −0.802405 + 1.38981i −0.111144 + 0.192506i 0.689297 + 1.19390i 2.39234 −2.28967 − 1.32568i −2.65010 0.212293 + 0.367702i 1.02756 − 1.77978i
144.8 −0.720625 − 1.24816i 0.301718 − 0.522592i −0.0386012 + 0.0668593i −0.325194 − 0.563253i −0.869703 1.86700 − 1.87465i −2.77123 1.31793 + 2.28273i −0.468686 + 0.811789i
144.9 −0.451047 − 0.781236i −0.797248 + 1.38087i 0.593113 − 1.02730i −0.211103 − 0.365642i 1.43839 1.06669 + 2.42119i −2.87428 0.228792 + 0.396279i −0.190435 + 0.329843i
144.10 −0.418855 − 0.725479i 1.43997 − 2.49410i 0.649120 − 1.12431i −1.48281 − 2.56830i −2.41256 2.59664 − 0.507406i −2.76297 −2.64704 − 4.58481i −1.24217 + 2.15150i
144.11 −0.197595 − 0.342245i −1.39646 + 2.41874i 0.921912 − 1.59680i −1.97536 − 3.42143i 1.10374 1.82002 − 1.92029i −1.51904 −2.40022 − 4.15730i −0.780644 + 1.35211i
144.12 −0.108156 − 0.187332i 0.296654 − 0.513821i 0.976605 − 1.69153i 1.11289 + 1.92758i −0.128340 −1.68455 + 2.04017i −0.855127 1.32399 + 2.29322i 0.240731 − 0.416959i
144.13 0.00412290 + 0.00714108i 1.41159 − 2.44495i 0.999966 − 1.73199i 1.25144 + 2.16756i 0.0232795 −0.322273 − 2.62605i 0.0329827 −2.48520 − 4.30449i −0.0103191 + 0.0178733i
144.14 0.161955 + 0.280514i −0.113624 + 0.196803i 0.947541 − 1.64119i −0.314070 − 0.543986i −0.0736079 −2.14433 − 1.54979i 1.26166 1.47418 + 2.55335i 0.101731 − 0.176202i
144.15 0.311019 + 0.538701i −1.62910 + 2.82168i 0.806534 − 1.39696i 0.245613 + 0.425415i −2.02672 −2.62311 − 0.345363i 2.24747 −3.80791 − 6.59549i −0.152781 + 0.264624i
144.16 0.353640 + 0.612523i −1.28055 + 2.21798i 0.749877 − 1.29882i 0.816459 + 1.41415i −1.81142 2.16866 + 1.51556i 2.47531 −1.77962 − 3.08239i −0.577466 + 1.00020i
144.17 0.564239 + 0.977291i −0.0142628 + 0.0247039i 0.363269 − 0.629200i 1.95373 + 3.38395i −0.0321906 1.57976 + 2.12235i 3.07684 1.49959 + 2.59737i −2.20474 + 3.81872i
144.18 0.621113 + 1.07580i 0.794203 − 1.37560i 0.228438 − 0.395666i −2.04151 − 3.53600i 1.97316 −0.808776 − 2.51910i 3.05199 0.238485 + 0.413068i 2.53601 − 4.39250i
144.19 0.715204 + 1.23877i 1.23486 − 2.13884i −0.0230324 + 0.0398932i 0.313136 + 0.542367i 3.53271 −2.64557 + 0.0308780i 2.79492 −1.54977 − 2.68428i −0.447912 + 0.775806i
144.20 0.884556 + 1.53210i 0.593055 − 1.02720i −0.564880 + 0.978400i −0.880157 − 1.52448i 2.09836 0.0376759 + 2.64548i 1.53955 0.796572 + 1.37970i 1.55710 − 2.69697i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 144.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1001.2.i.e ✓ 50
7.c even 3 1 inner 1001.2.i.e ✓ 50
7.c even 3 1 7007.2.a.bg 25
7.d odd 6 1 7007.2.a.bf 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1001.2.i.e ✓ 50 1.a even 1 1 trivial
1001.2.i.e ✓ 50 7.c even 3 1 inner
7007.2.a.bf 25 7.d odd 6 1
7007.2.a.bg 25 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} + 40 T_{2}^{48} - 2 T_{2}^{47} + 907 T_{2}^{46} - 76 T_{2}^{45} + 14083 T_{2}^{44} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(1001, [\chi])\). Copy content Toggle raw display