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Properties

Label 287.2.h.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 10 q^{3} - 14 q^{4} - q^{5} + 9 q^{6} + 10 q^{7} + 3 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 10 q^{3} - 14 q^{4} - q^{5} + 9 q^{6} + 10 q^{7} + 3 q^{8} + 22 q^{9} + 10 q^{10} + 5 q^{11} - 17 q^{12} + 3 q^{13} - 8 q^{14} + 23 q^{15} - 18 q^{16} + 11 q^{17} - 38 q^{18} - 2 q^{19} + 31 q^{20} + 4 q^{22} + 2 q^{23} + 10 q^{24} - 21 q^{25} - 7 q^{26} - 52 q^{27} + 14 q^{28} - 11 q^{29} - 18 q^{30} - 3 q^{31} + 44 q^{32} - 51 q^{33} + 29 q^{34} - 9 q^{35} + 35 q^{36} + 11 q^{37} + 52 q^{38} - 5 q^{39} - 32 q^{40} + 29 q^{41} + 6 q^{42} - 32 q^{43} - 92 q^{44} - 56 q^{45} + 26 q^{46} + 29 q^{47} + 11 q^{48} - 10 q^{49} - 24 q^{50} - 4 q^{51} + 3 q^{52} + 30 q^{53} + 58 q^{54} - 100 q^{55} - 3 q^{56} - 49 q^{57} + 25 q^{58} + 5 q^{59} - 91 q^{60} + 22 q^{61} - 34 q^{62} + 13 q^{63} - 9 q^{64} + 21 q^{65} + 29 q^{66} + 9 q^{67} - 20 q^{68} + 30 q^{69} - 10 q^{70} + 34 q^{71} - 37 q^{72} - 20 q^{73} - 58 q^{74} + 41 q^{75} - 37 q^{76} - 5 q^{77} + 63 q^{78} + 66 q^{79} + 22 q^{80} + 96 q^{81} + 76 q^{82} - 22 q^{83} - 38 q^{84} - 26 q^{85} + 3 q^{86} + 49 q^{87} - 19 q^{89} - q^{90} + 22 q^{91} - 2 q^{92} - 39 q^{93} + 66 q^{94} + 71 q^{95} - 302 q^{96} + 47 q^{97} - 2 q^{98} + 28 q^{99}+O(q^{100}) \) 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 −0.837127 + 2.57641i 0.305172 −4.31909 − 3.13800i −1.36956 − 0.995044i −0.255467 + 0.786248i −0.309017 − 0.951057i 7.31716 − 5.31623i −2.90687 3.71014 − 2.69558i
57.2 −0.514559 + 1.58365i 1.87936 −0.625144 − 0.454194i 2.01594 + 1.46467i −0.967042 + 2.97625i −0.309017 − 0.951057i −1.65331 + 1.20120i 0.531995 −3.35684 + 2.43889i
57.3 −0.509577 + 1.56832i −1.29067 −0.581915 − 0.422786i 0.393282 + 0.285736i 0.657694 − 2.02417i −0.309017 − 0.951057i −1.70859 + 1.24136i −1.33418 −0.648532 + 0.471186i
57.4 −0.454415 + 1.39854i −3.02432 −0.131400 − 0.0954680i −2.14566 − 1.55892i 1.37430 − 4.22965i −0.309017 − 0.951057i −2.18612 + 1.58831i 6.14653 3.15524 − 2.29241i
57.5 0.0642451 − 0.197726i −0.765544 1.58307 + 1.15016i 3.13381 + 2.27685i −0.0491825 + 0.151368i −0.309017 − 0.951057i 0.665514 − 0.483524i −2.41394 0.651524 − 0.473360i
57.6 0.286127 − 0.880607i −2.35844 0.924433 + 0.671640i −0.331415 − 0.240787i −0.674812 + 2.07686i −0.309017 − 0.951057i 2.35413 − 1.71038i 2.56223 −0.306866 + 0.222951i
57.7 0.491871 − 1.51382i 0.558674 −0.431686 − 0.313638i −3.26045 − 2.36885i 0.274795 − 0.845733i −0.309017 − 0.951057i 1.88834 − 1.37196i −2.68788 −5.18974 + 3.77057i
57.8 0.567566 − 1.74679i 2.49604 −1.11111 − 0.807268i 0.226121 + 0.164287i 1.41667 − 4.36005i −0.309017 − 0.951057i 0.931061 − 0.676455i 3.23020 0.415314 − 0.301743i
57.9 0.718417 − 2.21106i 0.707185 −2.75463 − 2.00136i 3.19921 + 2.32437i 0.508054 − 1.56363i −0.309017 − 0.951057i −2.64242 + 1.91983i −2.49989 7.43769 − 5.40380i
57.10 0.805486 − 2.47903i −2.12549 −3.87876 − 2.81808i −0.434226 − 0.315483i −1.71205 + 5.26916i −0.309017 − 0.951057i −5.89282 + 4.28139i 1.51771 −1.13186 + 0.822342i
78.1 −2.21373 + 1.60837i −3.42753 1.69571 − 5.21887i −0.479452 + 1.47560i 7.58764 − 5.51274i 0.809017 + 0.587785i 2.94888 + 9.07572i 8.74800 −1.31193 − 4.03772i
78.2 −2.02082 + 1.46821i 0.931822 1.31004 − 4.03189i 0.525509 − 1.61735i −1.88305 + 1.36811i 0.809017 + 0.587785i 1.72855 + 5.31992i −2.13171 1.31266 + 4.03994i
78.3 −1.51768 + 1.10266i −0.230278 0.469455 − 1.44483i −1.06103 + 3.26553i 0.349487 − 0.253917i 0.809017 + 0.587785i −0.278726 − 0.857832i −2.94697 −1.99045 − 6.12596i
78.4 −1.06465 + 0.773512i −2.00647 −0.0828791 + 0.255076i 0.544444 − 1.67563i 2.13618 − 1.55203i 0.809017 + 0.587785i −0.922386 − 2.83881i 1.02590 0.716476 + 2.20509i
78.5 −0.740338 + 0.537887i 3.13658 −0.359256 + 1.10568i −0.751672 + 2.31341i −2.32213 + 1.68712i 0.809017 + 0.587785i −0.894326 − 2.75245i 6.83813 −0.687861 − 2.11702i
78.6 0.178644 − 0.129793i 1.96598 −0.602966 + 1.85574i 0.495330 − 1.52447i 0.351211 − 0.255170i 0.809017 + 0.587785i 0.269617 + 0.829796i 0.865075 −0.109377 − 0.336628i
78.7 0.667836 − 0.485211i −0.0680633 −0.407459 + 1.25403i −0.581628 + 1.79007i −0.0454551 + 0.0330250i 0.809017 + 0.587785i 0.846535 + 2.60537i −2.99537 0.480129 + 1.47768i
78.8 1.45435 − 1.05664i −0.241537 0.380592 − 1.17134i 1.25760 − 3.87050i −0.351279 + 0.255219i 0.809017 + 0.587785i 0.426843 + 1.31369i −2.94166 −2.26075 − 6.95788i
78.9 1.62919 − 1.18367i −2.63857 0.635133 − 1.95474i −1.15891 + 3.56676i −4.29872 + 3.12321i 0.809017 + 0.587785i −0.0344339 − 0.105977i 3.96205 2.33380 + 7.18269i
78.10 2.00917 − 1.45975i 1.19610 1.28786 − 3.96363i −0.717239 + 2.20744i 2.40317 − 1.74600i 0.809017 + 0.587785i −1.66350 − 5.11972i −1.56934 1.78124 + 5.48209i
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.d ✓ 40
41.d even 5 1 inner 287.2.h.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.h.d ✓ 40 1.a even 1 1 trivial
287.2.h.d ✓ 40 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 2 T_{2}^{39} + 19 T_{2}^{38} + 37 T_{2}^{37} + 228 T_{2}^{36} + 365 T_{2}^{35} + \cdots + 891136 \) acting on \(S_{2}^{\mathrm{new}}(287, [\chi])\). Copy content Toggle raw display