Properties

Label 671.2.bl.a
Level $671$
Weight $2$
Character orbit 671.bl
Analytic conductor $5.358$
Analytic rank $0$
Dimension $480$
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] = [671,2,Mod(25,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([24, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bl (of order \(15\), degree \(8\), 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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{2} - 2 q^{3} + 50 q^{4} - 6 q^{5} + q^{6} - 6 q^{7} - 114 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{2} - 2 q^{3} + 50 q^{4} - 6 q^{5} + q^{6} - 6 q^{7} - 114 q^{9} - 28 q^{10} - 13 q^{11} - 72 q^{12} - 7 q^{13} - 17 q^{14} + 13 q^{15} + 38 q^{16} - 2 q^{17} + 17 q^{18} - 16 q^{19} + 8 q^{20} + q^{21} - 47 q^{22} - 46 q^{23} + 56 q^{24} - 214 q^{25} - 43 q^{26} + 19 q^{27} - 2 q^{28} - 28 q^{29} + 51 q^{30} - 10 q^{31} + 2 q^{32} + 5 q^{33} - 20 q^{34} - 9 q^{35} - 209 q^{36} - 6 q^{37} - 68 q^{38} - 62 q^{39} + 20 q^{40} - 122 q^{41} + 99 q^{42} + 9 q^{43} + 59 q^{44} - 31 q^{45} + 17 q^{46} + 11 q^{47} + 59 q^{48} + 58 q^{49} - 5 q^{50} + 27 q^{51} - 21 q^{52} - 24 q^{54} + 165 q^{55} + 31 q^{56} - 16 q^{57} - 42 q^{58} - q^{59} + 30 q^{60} - 16 q^{61} + 85 q^{62} + 18 q^{63} - 144 q^{64} - 37 q^{65} + 132 q^{66} + 4 q^{67} - 32 q^{68} - 79 q^{69} - 201 q^{70} + 13 q^{71} + 149 q^{72} + 23 q^{73} + 119 q^{74} - 46 q^{75} - 66 q^{76} - 19 q^{77} + 71 q^{78} + 5 q^{79} + 11 q^{80} - 74 q^{81} + 55 q^{82} - 2 q^{83} + 24 q^{84} - 38 q^{85} - 188 q^{86} - 73 q^{87} + 27 q^{88} + 24 q^{89} + 125 q^{90} - 21 q^{91} + 122 q^{92} - 15 q^{93} + 94 q^{94} + 80 q^{95} + 111 q^{96} - 98 q^{97} - 147 q^{98} + 159 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.285584 + 2.71715i −0.403470 + 0.293138i −5.34504 1.13612i −1.62469 2.81404i −0.681275 1.18000i −0.0580480 + 0.552290i 2.92494 9.00203i −0.850193 + 2.61662i 8.11015 3.61087i
25.2 −0.281438 + 2.67771i −1.08221 + 0.786270i −5.13461 1.09140i 1.29436 + 2.24190i −1.80083 3.11912i −0.0373564 + 0.355422i 2.70348 8.32047i −0.374098 + 1.15136i −6.36743 + 2.83496i
25.3 −0.278667 + 2.65134i 1.36335 0.990534i −4.99567 1.06186i −0.0346755 0.0600598i 2.24632 + 3.89075i 0.0117499 0.111793i 2.55985 7.87840i −0.0494776 + 0.152276i 0.168902 0.0752001i
25.4 −0.264807 + 2.51947i 1.12049 0.814085i −4.32131 0.918523i 1.77724 + 3.07827i 1.75435 + 3.03862i −0.466240 + 4.43597i 1.89281 5.82546i −0.334284 + 1.02882i −8.22622 + 3.66255i
25.5 −0.259044 + 2.46464i 2.52872 1.83722i −4.05106 0.861079i −1.07529 1.86245i 3.87305 + 6.70831i 0.0234834 0.223430i 1.64003 5.04750i 2.09199 6.43848i 4.86882 2.16774i
25.6 −0.257677 + 2.45163i −2.00543 + 1.45703i −3.98780 0.847633i 0.396793 + 0.687265i −3.05534 5.29201i 0.406827 3.87070i 1.58211 4.86923i 0.971759 2.99077i −1.78716 + 0.795697i
25.7 −0.233053 + 2.21735i −2.47998 + 1.80181i −2.90603 0.617697i −1.93612 3.35346i −3.41728 5.91890i 0.0513781 0.488830i 0.668963 2.05886i 1.97673 6.08374i 7.88702 3.51153i
25.8 −0.230219 + 2.19039i −0.668805 + 0.485916i −2.78852 0.592718i −0.140529 0.243404i −0.910374 1.57681i −0.0854680 + 0.813173i 0.579063 1.78217i −0.715864 + 2.20320i 0.565503 0.251778i
25.9 −0.220803 + 2.10080i 1.10241 0.800945i −2.40832 0.511903i −0.922080 1.59709i 1.43921 + 2.49279i 0.240891 2.29192i 0.301652 0.928389i −0.353265 + 1.08724i 3.55877 1.58446i
25.10 −0.219890 + 2.09211i −1.49834 + 1.08861i −2.37228 0.504243i −0.827887 1.43394i −1.94802 3.37407i −0.393963 + 3.74831i 0.276454 0.850839i 0.132907 0.409047i 3.18201 1.41672i
25.11 −0.216483 + 2.05970i 0.649806 0.472112i −2.23921 0.475960i 1.63567 + 2.83307i 0.831737 + 1.44061i 0.530941 5.05157i 0.185110 0.569710i −0.727693 + 2.23961i −6.18937 + 2.75568i
25.12 −0.202793 + 1.92945i 1.75434 1.27461i −1.72536 0.366736i 0.246757 + 0.427396i 2.10352 + 3.64340i −0.474463 + 4.51421i −0.141544 + 0.435629i 0.526052 1.61902i −0.874681 + 0.389433i
25.13 −0.201519 + 1.91733i −2.58944 + 1.88134i −1.67924 0.356933i 1.74045 + 3.01455i −3.08532 5.34392i −0.219352 + 2.08699i −0.168744 + 0.519340i 2.23871 6.89003i −6.13062 + 2.72953i
25.14 −0.199584 + 1.89891i 2.76603 2.00964i −1.60974 0.342160i 1.60160 + 2.77406i 3.26407 + 5.65354i 0.186263 1.77217i −0.209047 + 0.643381i 2.68523 8.26429i −5.58735 + 2.48765i
25.15 −0.190665 + 1.81405i 0.321613 0.233666i −1.29815 0.275930i −1.21559 2.10547i 0.362562 + 0.627976i 0.219407 2.08752i −0.379261 + 1.16724i −0.878216 + 2.70287i 4.05121 1.80371i
25.16 −0.174592 + 1.66113i −1.40510 + 1.02086i −0.772583 0.164218i 0.317456 + 0.549850i −1.45047 2.51229i −0.373806 + 3.55653i −0.624617 + 1.92237i 0.00508971 0.0156645i −0.968800 + 0.431337i
25.17 −0.161327 + 1.53493i 0.869183 0.631498i −0.373680 0.0794282i 1.42302 + 2.46474i 0.829081 + 1.43601i −0.00370149 + 0.0352173i −0.771661 + 2.37493i −0.570362 + 1.75540i −4.01277 + 1.78660i
25.18 −0.129542 + 1.23251i −0.658544 + 0.478460i 0.454007 + 0.0965023i 1.33569 + 2.31348i −0.504396 0.873639i 0.0276602 0.263170i −0.943678 + 2.90434i −0.722295 + 2.22300i −3.02441 + 1.34655i
25.19 −0.125849 + 1.19737i 1.72539 1.25357i 0.538429 + 0.114447i −0.362674 0.628170i 1.28385 + 2.22369i −0.0567740 + 0.540169i −0.948890 + 2.92038i 0.478476 1.47260i 0.797796 0.355202i
25.20 −0.115962 + 1.10330i −1.60803 + 1.16830i 0.752464 + 0.159941i −0.568079 0.983942i −1.10252 1.90962i 0.388961 3.70072i −0.949356 + 2.92182i 0.293774 0.904143i 1.15146 0.512664i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.bl even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.bl.a 480
11.c even 5 1 671.2.bp.a yes 480
61.i even 15 1 671.2.bp.a yes 480
671.bl even 15 1 inner 671.2.bl.a 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.bl.a 480 1.a even 1 1 trivial
671.2.bl.a 480 671.bl even 15 1 inner
671.2.bp.a yes 480 11.c even 5 1
671.2.bp.a yes 480 61.i even 15 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).