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Properties

Label 287.2.h.c
Level $287$
Weight $2$
Character orbit 287.h
Analytic conductor $2.292$
Analytic rank $0$
Dimension $40$
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] = [287,2,Mod(57,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 287.h (of order \(5\), degree \(4\), 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: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} - 3 q^{4} - 4 q^{5} - 19 q^{6} - 10 q^{7} + 16 q^{8} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} - 3 q^{4} - 4 q^{5} - 19 q^{6} - 10 q^{7} + 16 q^{8} + 60 q^{9} + 5 q^{10} - 10 q^{11} - 12 q^{12} - 17 q^{13} + 2 q^{14} - 3 q^{15} - 23 q^{16} - 8 q^{17} - 2 q^{18} + 23 q^{19} - 13 q^{22} - q^{23} + 46 q^{24} - 34 q^{25} + 3 q^{26} - 18 q^{28} - 18 q^{29} - 19 q^{30} - 3 q^{31} - 26 q^{32} - 6 q^{33} - 44 q^{34} + q^{35} - 38 q^{36} - 5 q^{37} + 28 q^{38} + 17 q^{39} + 14 q^{40} - 11 q^{41} - 24 q^{42} + 13 q^{43} + 66 q^{44} + 43 q^{45} - 20 q^{46} - 27 q^{47} + 39 q^{48} - 10 q^{49} + 106 q^{50} - 18 q^{51} - 30 q^{52} - 30 q^{53} - 109 q^{54} + 118 q^{55} + 16 q^{56} - 40 q^{57} - 23 q^{58} - 37 q^{59} + 96 q^{60} - 41 q^{61} - 13 q^{62} - 30 q^{63} + 10 q^{64} + 6 q^{65} - 30 q^{66} - 6 q^{67} - 26 q^{68} - 31 q^{69} + 5 q^{70} - 31 q^{71} + 107 q^{72} - 46 q^{73} + 75 q^{74} - 61 q^{75} + 43 q^{76} - 10 q^{77} + 34 q^{78} + 76 q^{79} + 64 q^{80} + 16 q^{81} - 16 q^{82} - 52 q^{83} - 7 q^{84} + 86 q^{85} - 17 q^{86} - 20 q^{87} - 52 q^{88} - 16 q^{89} + 6 q^{90} + 18 q^{91} + 97 q^{92} + 32 q^{93} - 5 q^{94} - 102 q^{95} + 38 q^{96} - 18 q^{97} - 3 q^{98} - 71 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 −0.794058 + 2.44386i 3.29652 −3.72389 − 2.70556i 0.799736 + 0.581042i −2.61763 + 8.05622i 0.309017 + 0.951057i 5.41126 − 3.93151i 7.86702 −2.05502 + 1.49306i
57.2 −0.662436 + 2.03877i 0.259063 −2.09972 − 1.52553i 0.0549948 + 0.0399561i −0.171613 + 0.528169i 0.309017 + 0.951057i 1.03258 − 0.750212i −2.93289 −0.117892 + 0.0856533i
57.3 −0.508764 + 1.56581i −0.349538 −0.574899 − 0.417688i −3.08337 − 2.24020i 0.177832 − 0.547311i 0.309017 + 0.951057i −1.71741 + 1.24777i −2.87782 5.07644 − 3.68825i
57.4 −0.279797 + 0.861126i 2.59818 0.954783 + 0.693690i −1.43627 − 1.04351i −0.726962 + 2.23736i 0.309017 + 0.951057i −2.32953 + 1.69251i 3.75053 1.30046 − 0.944836i
57.5 −0.149394 + 0.459786i −1.52804 1.42895 + 1.03819i 1.42271 + 1.03366i 0.228280 − 0.702574i 0.309017 + 0.951057i −1.47306 + 1.07024i −0.665079 −0.687806 + 0.499720i
57.6 0.223825 − 0.688863i 1.30681 1.19360 + 0.867201i 1.13385 + 0.823787i 0.292498 − 0.900216i 0.309017 + 0.951057i 2.03650 − 1.47961i −1.29224 0.821260 − 0.596680i
57.7 0.228466 − 0.703146i −2.48892 1.17582 + 0.854281i −2.55629 − 1.85725i −0.568634 + 1.75007i 0.309017 + 0.951057i 2.06558 − 1.50073i 3.19473 −1.88994 + 1.37312i
57.8 0.450739 − 1.38723i 0.560317 −0.103210 − 0.0749862i 0.738519 + 0.536566i 0.252556 − 0.777289i 0.309017 + 0.951057i 2.20955 − 1.60533i −2.68604 1.07722 − 0.782646i
57.9 0.571535 − 1.75900i −3.10319 −1.14941 − 0.835096i 2.72110 + 1.97700i −1.77358 + 5.45852i 0.309017 + 0.951057i 0.866730 − 0.629716i 6.62977 5.03276 − 3.65651i
57.10 0.728900 − 2.24332i −0.551198 −2.88317 − 2.09475i −1.91302 − 1.38989i −0.401768 + 1.23652i 0.309017 + 0.951057i −2.98417 + 2.16813i −2.69618 −4.51238 + 3.27844i
78.1 −2.17748 + 1.58203i 2.12926 1.62055 − 4.98755i −1.11426 + 3.42935i −4.63642 + 3.36856i −0.809017 − 0.587785i 2.69829 + 8.30447i 1.53376 −2.99905 − 9.23012i
78.2 −1.91136 + 1.38868i −1.05524 1.10682 − 3.40644i 0.886655 − 2.72884i 2.01694 − 1.46539i −0.809017 − 0.587785i 1.15479 + 3.55408i −1.88648 2.09479 + 6.44709i
78.3 −1.52332 + 1.10676i 2.82445 0.477560 − 1.46978i 0.756805 − 2.32921i −4.30254 + 3.12598i −0.809017 − 0.587785i −0.264503 − 0.814055i 4.97750 1.42501 + 4.38572i
78.4 −1.49937 + 1.08936i −1.69208 0.443381 − 1.36459i −0.353412 + 1.08769i 2.53705 − 1.84327i −0.809017 − 0.587785i −0.323689 − 0.996211i −0.136877 −0.654987 − 2.01584i
78.5 −0.224492 + 0.163103i −2.77195 −0.594240 + 1.82888i −1.20463 + 3.70748i 0.622280 − 0.452113i −0.809017 − 0.587785i −0.336391 − 1.03531i 4.68369 −0.334271 − 1.02878i
78.6 0.539909 − 0.392267i −3.00502 −0.480406 + 1.47854i 0.818883 − 2.52026i −1.62244 + 1.17877i −0.809017 − 0.587785i 0.733059 + 2.25612i 6.03012 −0.546493 − 1.68193i
78.7 0.635695 − 0.461859i 2.52804 −0.427240 + 1.31491i 0.910194 − 2.80129i 1.60706 − 1.16760i −0.809017 − 0.587785i 0.821337 + 2.52781i 3.39096 −0.715196 − 2.20115i
78.8 0.991525 − 0.720385i 2.24376 −0.153867 + 0.473553i −1.02540 + 3.15585i 2.22474 − 1.61637i −0.809017 − 0.587785i 0.946036 + 2.91160i 2.03444 1.25672 + 3.86779i
78.9 1.84039 − 1.33712i 1.18259 0.981102 − 3.01952i 0.156878 − 0.482822i 2.17642 − 1.58126i −0.809017 − 0.587785i −0.825922 − 2.54193i −1.60148 −0.356874 − 1.09834i
78.10 2.01949 − 1.46724i −2.38381 1.30749 − 4.02404i 0.286326 − 0.881220i −4.81408 + 3.49763i −0.809017 − 0.587785i −1.72104 − 5.29681i 2.68257 −0.714733 − 2.19972i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.h.c ✓ 40
41.d even 5 1 inner 287.2.h.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.h.c ✓ 40 1.a even 1 1 trivial
287.2.h.c ✓ 40 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 3 T_{2}^{39} + 16 T_{2}^{38} + 31 T_{2}^{37} + 150 T_{2}^{36} + 270 T_{2}^{35} + \cdots + 92416 \) acting on \(S_{2}^{\mathrm{new}}(287, [\chi])\). Copy content Toggle raw display