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Properties

Label 1001.2.i.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 6 q^{2} + 2 q^{3} - 30 q^{4} + q^{5} + 4 q^{6} + q^{7} + 42 q^{8} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - 6 q^{2} + 2 q^{3} - 30 q^{4} + q^{5} + 4 q^{6} + q^{7} + 42 q^{8} - 37 q^{9} + 3 q^{10} - 25 q^{11} - 9 q^{12} - 50 q^{13} + 22 q^{14} - 32 q^{16} + q^{17} - 44 q^{18} - 5 q^{19} + 8 q^{20} + 2 q^{21} + 12 q^{22} - 15 q^{23} + 4 q^{24} - 50 q^{25} + 6 q^{26} - 34 q^{27} + 12 q^{28} + 48 q^{29} - q^{30} + 12 q^{31} - 48 q^{32} + 2 q^{33} - 16 q^{34} + 20 q^{35} + 60 q^{36} - 33 q^{37} - 16 q^{38} - 2 q^{39} + 21 q^{40} + 24 q^{41} - 42 q^{42} + 76 q^{43} - 30 q^{44} + 22 q^{45} - 39 q^{46} - 4 q^{47} + 164 q^{48} + 23 q^{49} + 32 q^{50} - 51 q^{51} + 30 q^{52} - 2 q^{53} - 10 q^{54} - 2 q^{55} - 72 q^{56} + 76 q^{57} - 17 q^{58} + 4 q^{59} + 33 q^{60} + 22 q^{61} + 84 q^{62} - 19 q^{63} + 82 q^{64} - q^{65} - 2 q^{66} - 24 q^{67} - 14 q^{68} - 60 q^{69} - 124 q^{70} + 18 q^{71} - 102 q^{72} - 11 q^{73} - 39 q^{74} + 16 q^{75} + 116 q^{76} + q^{77} - 4 q^{78} - 19 q^{79} + 33 q^{80} - 73 q^{81} + 32 q^{82} + 32 q^{83} - 109 q^{84} + 28 q^{85} - 27 q^{86} + 11 q^{87} - 21 q^{88} - 13 q^{89} + 80 q^{90} - q^{91} - 17 q^{93} + 56 q^{94} - 15 q^{95} - 55 q^{96} + 68 q^{97} - 22 q^{98} + 74 q^{99}+O(q^{100}) \) 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 −1.38692 − 2.40222i −0.563570 + 0.976132i −2.84711 + 4.93134i −1.24877 − 2.16293i 3.12651 1.34983 + 2.27551i 10.2472 0.864778 + 1.49784i −3.46389 + 5.99963i
144.2 −1.38050 − 2.39110i −1.49431 + 2.58822i −2.81159 + 4.86981i 1.63913 + 2.83906i 8.25161 −2.03317 − 1.69299i 10.0036 −2.96593 − 5.13714i 4.52565 − 7.83866i
144.3 −1.27968 − 2.21648i 1.33068 − 2.30480i −2.27518 + 3.94073i 1.23753 + 2.14346i −6.81138 −2.63170 + 0.272277i 6.52730 −2.04140 − 3.53582i 3.16729 − 5.48591i
144.4 −1.23499 − 2.13907i 0.565276 − 0.979087i −2.05041 + 3.55142i 0.362280 + 0.627487i −2.79245 1.50647 − 2.17498i 5.18901 0.860925 + 1.49117i 0.894826 − 1.54988i
144.5 −1.10707 − 1.91750i 1.56943 − 2.71833i −1.45121 + 2.51356i −2.00485 − 3.47250i −6.94987 −0.797633 + 2.52265i 1.99807 −3.42621 − 5.93438i −4.43902 + 7.68861i
144.6 −1.06972 − 1.85281i −1.15167 + 1.99475i −1.28861 + 2.23194i −1.42684 − 2.47136i 4.92786 −2.62366 − 0.341214i 1.23494 −1.15268 − 1.99650i −3.05265 + 5.28735i
144.7 −0.838641 − 1.45257i −1.35731 + 2.35094i −0.406638 + 0.704317i −0.392716 − 0.680205i 4.55320 2.59912 − 0.494548i −1.99047 −2.18460 − 3.78385i −0.658696 + 1.14089i
144.8 −0.820905 − 1.42185i −0.409713 + 0.709644i −0.347771 + 0.602357i 1.79433 + 3.10787i 1.34534 0.873238 + 2.49749i −2.14167 1.16427 + 2.01657i 2.94595 − 5.10254i
144.9 −0.738941 − 1.27988i 0.252914 − 0.438060i −0.0920676 + 0.159466i 0.747629 + 1.29493i −0.747555 −2.63790 + 0.203673i −2.68363 1.37207 + 2.37649i 1.10491 − 1.91376i
144.10 −0.702891 − 1.21744i 1.64875 − 2.85573i 0.0118876 − 0.0205898i 1.08426 + 1.87799i −4.63558 2.39918 − 1.11532i −2.84499 −3.93679 − 6.81871i 1.52423 − 2.64005i
144.11 −0.510452 − 0.884128i 0.692979 − 1.20028i 0.478878 − 0.829442i −1.61918 − 2.80451i −1.41493 2.64329 + 0.114204i −3.01958 0.539560 + 0.934545i −1.65303 + 2.86313i
144.12 −0.144247 − 0.249843i −1.56212 + 2.70567i 0.958386 − 1.65997i −1.43833 − 2.49126i 0.901324 −1.29360 + 2.30794i −1.12997 −3.38043 − 5.85507i −0.414950 + 0.718714i
144.13 −0.0543180 − 0.0940815i −1.02534 + 1.77594i 0.994099 − 1.72183i 0.123266 + 0.213504i 0.222777 −1.00041 − 2.44932i −0.433262 −0.602639 − 1.04380i 0.0133912 − 0.0231942i
144.14 −0.0242073 − 0.0419282i 0.929978 − 1.61077i 0.998828 − 1.73002i 1.10715 + 1.91765i −0.0900488 −0.821369 − 2.51503i −0.193545 −0.229717 − 0.397882i 0.0536024 − 0.0928420i
144.15 0.0550434 + 0.0953379i −0.555439 + 0.962048i 0.993940 − 1.72156i 0.906015 + 1.56926i −0.122293 1.45700 + 2.20843i 0.439013 0.882976 + 1.52936i −0.0997402 + 0.172755i
144.16 0.168532 + 0.291906i 0.926958 − 1.60554i 0.943194 − 1.63366i −1.52372 − 2.63917i 0.624888 0.675665 − 2.55802i 1.30996 −0.218502 − 0.378456i 0.513592 − 0.889567i
144.17 0.434546 + 0.752655i 1.23247 − 2.13471i 0.622340 − 1.07792i 2.15910 + 3.73967i 2.14227 −1.52058 + 2.16514i 2.81992 −1.53799 − 2.66387i −1.87646 + 3.25012i
144.18 0.608631 + 1.05418i −1.54163 + 2.67019i 0.259137 − 0.448838i 1.72234 + 2.98318i −3.75314 2.12081 − 1.58182i 3.06540 −3.25327 − 5.63483i −2.09653 + 3.63131i
144.19 0.677177 + 1.17291i 0.426504 − 0.738727i 0.0828624 − 0.143522i −0.558163 − 0.966767i 1.15527 −0.188015 + 2.63906i 2.93316 1.13619 + 1.96794i 0.755950 − 1.30934i
144.20 0.831424 + 1.44007i 1.70931 − 2.96061i −0.382533 + 0.662566i −0.618688 − 1.07160i 5.68465 −1.64713 − 2.07050i 2.05351 −4.34348 − 7.52313i 1.02878 − 1.78191i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 144.25
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} + 6 T_{2}^{49} + 58 T_{2}^{48} + 246 T_{2}^{47} + 1497 T_{2}^{46} + 5250 T_{2}^{45} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(1001, [\chi])\). Copy content Toggle raw display