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Properties

Label 805.2.bp.a
Level $805$
Weight $2$
Character orbit 805.bp
Analytic conductor $6.428$
Analytic rank $0$
Dimension $640$
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] = [805,2,Mod(61,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 55, 51]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.bp (of order \(66\), degree \(20\), 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: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(32\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 2 q^{2} + 34 q^{4} - 32 q^{5} + 3 q^{7} + 12 q^{8} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 2 q^{2} + 34 q^{4} - 32 q^{5} + 3 q^{7} + 12 q^{8} - 34 q^{9} + 2 q^{10} + 12 q^{11} - 12 q^{14} + 24 q^{16} + q^{17} + 18 q^{18} - 4 q^{19} + 68 q^{20} - 40 q^{21} - 16 q^{23} - 18 q^{24} + 32 q^{25} - 12 q^{26} + 2 q^{28} - 30 q^{29} - 6 q^{30} + 3 q^{31} + 348 q^{32} + 10 q^{33} - 16 q^{34} - 8 q^{35} - 54 q^{36} + 39 q^{37} + 77 q^{38} - 210 q^{39} - 16 q^{40} - 44 q^{41} - 50 q^{42} - 88 q^{43} - 37 q^{44} - 318 q^{45} + 26 q^{46} - 36 q^{47} - 352 q^{48} + 19 q^{49} - 18 q^{50} + 14 q^{51} + 33 q^{53} + 100 q^{54} - 125 q^{56} + 15 q^{58} - 27 q^{59} - 10 q^{61} + 120 q^{63} - 124 q^{64} - 2 q^{66} - 72 q^{68} + 150 q^{69} - 6 q^{70} + 22 q^{71} - 106 q^{72} - 18 q^{73} - 35 q^{74} + 104 q^{76} - 345 q^{77} + 140 q^{78} - 22 q^{79} - 46 q^{80} + 24 q^{81} + 102 q^{82} - 22 q^{83} + 208 q^{84} + 2 q^{85} - 170 q^{86} - 354 q^{87} - 6 q^{88} + 18 q^{89} + 36 q^{90} - 68 q^{91} + 156 q^{92} - 10 q^{93} + 29 q^{94} - 29 q^{95} - 60 q^{96} + 36 q^{97} - 7 q^{98} + 264 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −2.13127 − 1.67605i −0.673782 − 0.479798i 1.26165 + 5.20060i −0.981929 + 0.189251i 0.631846 + 2.15187i 2.63770 + 0.206239i 3.77487 − 8.26581i −0.757428 − 2.18844i 2.40995 + 1.24242i
61.2 −2.09465 − 1.64725i 2.37586 + 1.69184i 1.20260 + 4.95720i −0.981929 + 0.189251i −2.18971 − 7.45745i 0.874116 − 2.49718i 3.43274 − 7.51666i 1.80118 + 5.20416i 2.36854 + 1.22107i
61.3 −2.02980 − 1.59625i 0.247273 + 0.176082i 1.10055 + 4.53651i −0.981929 + 0.189251i −0.220843 − 0.752121i −1.74315 + 1.99034i 2.86210 − 6.26713i −0.951065 − 2.74792i 2.29521 + 1.18326i
61.4 −1.96348 − 1.54410i −2.31461 − 1.64823i 0.999493 + 4.11997i −0.981929 + 0.189251i 1.99967 + 6.81024i −1.81646 − 1.92366i 2.32382 − 5.08845i 1.65957 + 4.79502i 2.22022 + 1.14460i
61.5 −1.69521 − 1.33313i 1.30191 + 0.927089i 0.624984 + 2.57622i −0.981929 + 0.189251i −0.971088 − 3.30722i −1.16417 + 2.37586i 0.583175 − 1.27697i −0.145716 − 0.421020i 1.91687 + 0.988214i
61.6 −1.67046 − 1.31366i −2.50228 − 1.78186i 0.593202 + 2.44521i −0.981929 + 0.189251i 1.83918 + 6.26368i 1.97477 + 1.76077i 0.455651 − 0.997736i 2.10515 + 6.08243i 1.88888 + 0.973787i
61.7 −1.57513 − 1.23870i 1.47624 + 1.05123i 0.475150 + 1.95860i −0.981929 + 0.189251i −1.02312 − 3.48444i −1.57538 − 2.12560i 0.0128260 − 0.0280850i 0.0930087 + 0.268731i 1.78109 + 0.918216i
61.8 −1.46567 − 1.15261i −0.501468 − 0.357094i 0.348143 + 1.43507i −0.981929 + 0.189251i 0.323394 + 1.10138i 1.25432 − 2.32952i −0.405343 + 0.887578i −0.857250 − 2.47686i 1.65731 + 0.854404i
61.9 −1.19735 − 0.941606i −1.25146 − 0.891161i 0.0755063 + 0.311241i −0.981929 + 0.189251i 0.659313 + 2.24541i −2.54222 − 0.732900i −1.06290 + 2.32742i −0.209218 − 0.604497i 1.35391 + 0.697990i
61.10 −1.03106 − 0.810838i −0.834207 − 0.594036i −0.0658822 − 0.271570i −0.981929 + 0.189251i 0.378454 + 1.28890i 0.227272 + 2.63597i −1.24207 + 2.71975i −0.638182 − 1.84391i 1.16588 + 0.601055i
61.11 −0.931535 − 0.732567i 2.11885 + 1.50883i −0.140415 − 0.578799i −0.981929 + 0.189251i −0.868468 − 2.95773i 0.501814 + 2.59773i −1.27781 + 2.79801i 1.23177 + 3.55898i 1.05334 + 0.543035i
61.12 −0.518407 − 0.407679i 2.70767 + 1.92813i −0.368975 − 1.52094i −0.981929 + 0.189251i −0.617619 − 2.10342i 1.65397 − 2.06504i −0.976712 + 2.13870i 2.63263 + 7.60648i 0.586192 + 0.302203i
61.13 −0.513227 − 0.403606i −1.86758 − 1.32990i −0.371014 − 1.52934i −0.981929 + 0.189251i 0.421738 + 1.43631i 2.32561 − 1.26157i −0.969300 + 2.12247i 0.738025 + 2.13238i 0.580336 + 0.299184i
61.14 −0.414454 − 0.325930i 0.474594 + 0.337957i −0.405976 − 1.67346i −0.981929 + 0.189251i −0.0865470 − 0.294752i −2.42640 − 1.05480i −0.815235 + 1.78511i −0.870179 − 2.51422i 0.468647 + 0.241604i
61.15 −0.252578 − 0.198629i −0.210353 − 0.149792i −0.447176 − 1.84328i −0.981929 + 0.189251i 0.0233775 + 0.0796164i 1.51448 + 2.16942i −0.520149 + 1.13897i −0.959393 − 2.77198i 0.285604 + 0.147239i
61.16 −0.250471 − 0.196973i 1.94666 + 1.38621i −0.447580 − 1.84495i −0.981929 + 0.189251i −0.214536 − 0.730644i −2.64404 + 0.0952085i −0.516038 + 1.12997i 0.886703 + 2.56196i 0.283222 + 0.146011i
61.17 −0.0971952 − 0.0764352i −1.97832 − 1.40875i −0.467913 − 1.92876i −0.981929 + 0.189251i 0.0846048 + 0.288137i −2.37952 + 1.15667i −0.204678 + 0.448183i 0.947953 + 2.73893i 0.109904 + 0.0566596i
61.18 −0.0760207 − 0.0597833i −2.35795 − 1.67909i −0.469313 − 1.93453i −0.981929 + 0.189251i 0.0788714 + 0.268611i 2.64569 − 0.0172925i −0.160327 + 0.351066i 1.75938 + 5.08339i 0.0859610 + 0.0443160i
61.19 0.0980737 + 0.0771260i 0.640829 + 0.456332i −0.467848 − 1.92850i −0.981929 + 0.189251i 0.0276534 + 0.0941788i 1.68803 − 2.03729i 0.206514 − 0.452203i −0.778781 − 2.25014i −0.110898 − 0.0571717i
61.20 0.521747 + 0.410306i 0.485924 + 0.346025i −0.367649 − 1.51547i −0.981929 + 0.189251i 0.111553 + 0.379915i 2.13135 + 1.56760i 0.981455 − 2.14909i −0.864815 − 2.49872i −0.589969 − 0.304150i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
161.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.bp.a ✓ 640
7.d odd 6 1 805.2.bp.b yes 640
23.d odd 22 1 805.2.bp.b yes 640
161.o even 66 1 inner 805.2.bp.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.bp.a ✓ 640 1.a even 1 1 trivial
805.2.bp.a ✓ 640 161.o even 66 1 inner
805.2.bp.b yes 640 7.d odd 6 1
805.2.bp.b yes 640 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{640} + 2 T_{2}^{639} - 47 T_{2}^{638} - 106 T_{2}^{637} + 980 T_{2}^{636} + 2496 T_{2}^{635} + \cdots + 19\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\). Copy content Toggle raw display