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Properties

Label 671.2.bo.a
Level $671$
Weight $2$
Character orbit 671.bo
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(16,671)]
 
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([12, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bo (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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{2} - 2 q^{3} + 50 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} - 10 q^{8} - 114 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{2} - 2 q^{3} + 50 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} - 10 q^{8} - 114 q^{9} - 8 q^{10} - 18 q^{11} + 33 q^{12} - 7 q^{13} + 13 q^{14} - 12 q^{15} + 78 q^{16} - 12 q^{17} - 8 q^{18} - 16 q^{19} - 52 q^{20} - 14 q^{21} + 3 q^{22} - 26 q^{23} + 36 q^{24} + 51 q^{25} + 42 q^{26} - 26 q^{27} - 52 q^{28} - 8 q^{29} + 16 q^{30} - 15 q^{31} - 23 q^{32} - 10 q^{33} + 30 q^{34} - 84 q^{35} + 26 q^{36} - 6 q^{37} - 38 q^{38} + 53 q^{39} + 40 q^{40} + 33 q^{41} + 29 q^{42} - 76 q^{43} + 4 q^{44} - q^{45} - 28 q^{46} - 49 q^{47} - 21 q^{48} + 68 q^{49} + 10 q^{50} - 48 q^{51} - 86 q^{52} + 15 q^{53} + 136 q^{54} + 10 q^{55} - 34 q^{56} - 46 q^{57} + 98 q^{58} + 4 q^{59} - 45 q^{60} + 14 q^{61} - 30 q^{62} + 103 q^{63} - 124 q^{64} - 32 q^{65} + 82 q^{66} + 4 q^{67} + 23 q^{68} + 36 q^{69} + 4 q^{70} + 33 q^{71} - 61 q^{72} + 33 q^{73} + 84 q^{74} - 81 q^{75} - 36 q^{76} + 121 q^{77} - 9 q^{78} - 80 q^{79} - 49 q^{80} - 104 q^{81} - 65 q^{82} + 38 q^{83} - 11 q^{84} - 93 q^{85} - 98 q^{86} + 2 q^{87} + 52 q^{88} - 16 q^{89} - 270 q^{90} + 59 q^{91} + 72 q^{92} + 15 q^{93} + 4 q^{94} - 330 q^{95} - 109 q^{96} - 23 q^{97} - 47 q^{98} + 94 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.291878 − 2.77703i −0.717005 + 2.20672i −5.67041 + 1.20528i 0.453028 + 0.0962941i 6.33739 + 1.34705i 1.96421 + 0.417507i 3.27642 + 10.0838i −1.92844 − 1.40110i 0.135183 − 1.28618i
16.2 −0.286555 − 2.72639i 0.569725 − 1.75343i −5.39478 + 1.14670i 0.0387212 + 0.00823044i −4.94379 − 1.05084i −2.62129 − 0.557172i 2.97796 + 9.16521i −0.322888 − 0.234592i 0.0113436 − 0.107927i
16.3 −0.264872 − 2.52009i 0.00192934 − 0.00593790i −4.32438 + 0.919176i −4.03276 − 0.857189i −0.0154750 − 0.00328932i 4.85771 + 1.03254i 1.89573 + 5.83447i 2.42702 + 1.76333i −1.09203 + 10.3899i
16.4 −0.257849 − 2.45327i 0.296341 − 0.912045i −3.99576 + 0.849325i 2.49094 + 0.529465i −2.31391 − 0.491836i 1.88631 + 0.400947i 1.58937 + 4.89158i 1.68304 + 1.22280i 0.656635 − 6.24747i
16.5 −0.256882 − 2.44407i −0.696487 + 2.14357i −3.95120 + 0.839854i 3.14962 + 0.669472i 5.41795 + 1.15162i −4.25555 − 0.904545i 1.54882 + 4.76677i −1.68274 − 1.22258i 0.827157 − 7.86987i
16.6 −0.252656 − 2.40386i −0.0802338 + 0.246934i −3.75840 + 0.798873i 2.17761 + 0.462865i 0.613866 + 0.130481i −0.547109 − 0.116292i 1.37611 + 4.23523i 2.37251 + 1.72373i 0.562476 − 5.35160i
16.7 −0.251744 − 2.39518i −0.434628 + 1.33765i −3.71722 + 0.790119i −3.20140 − 0.680478i 3.31332 + 0.704267i −2.98490 − 0.634461i 1.33981 + 4.12350i 0.826655 + 0.600600i −0.823936 + 7.83922i
16.8 −0.244617 − 2.32738i 0.821511 − 2.52835i −3.40056 + 0.722812i −2.84180 − 0.604043i −6.08538 − 1.29349i −1.40696 − 0.299059i 1.06777 + 3.28626i −3.29062 − 2.39078i −0.710683 + 6.76170i
16.9 −0.230388 − 2.19200i 0.864314 − 2.66009i −2.79548 + 0.594197i 2.39029 + 0.508072i −6.03003 − 1.28172i 3.78799 + 0.805163i 0.584332 + 1.79839i −3.90197 − 2.83495i 0.562998 − 5.35657i
16.10 −0.220377 − 2.09675i 0.280558 − 0.863469i −2.39150 + 0.508329i −1.15357 − 0.245198i −1.87231 − 0.397971i −0.463869 − 0.0985984i 0.289869 + 0.892125i 1.76019 + 1.27885i −0.259899 + 2.47278i
16.11 −0.215195 − 2.04744i −0.621413 + 1.91251i −2.18941 + 0.465374i −1.64819 − 0.350333i 4.04948 + 0.860744i −0.0274548 − 0.00583569i 0.151617 + 0.466630i −0.844501 − 0.613566i −0.362606 + 3.44996i
16.12 −0.198811 − 1.89156i −0.586899 + 1.80629i −1.58217 + 0.336301i 2.82418 + 0.600299i 3.53339 + 0.751044i 3.66803 + 0.779664i −0.224800 − 0.691864i −0.491182 − 0.356865i 0.574022 − 5.46145i
16.13 −0.191354 − 1.82061i 1.04429 − 3.21399i −1.32172 + 0.280941i 2.18932 + 0.465354i −6.05127 − 1.28624i −2.96533 − 0.630300i −0.366996 − 1.12950i −6.81216 − 4.94933i 0.428295 − 4.07495i
16.14 −0.184476 − 1.75517i −0.941366 + 2.89723i −1.09029 + 0.231748i −1.83535 − 0.390115i 5.25878 + 1.11779i 2.80280 + 0.595753i −0.482840 − 1.48603i −5.08070 − 3.69135i −0.346141 + 3.29331i
16.15 −0.165121 − 1.57102i −0.227252 + 0.699410i −0.484557 + 0.102996i −1.08369 − 0.230346i 1.13631 + 0.241531i −2.76074 − 0.586814i −0.734475 − 2.26048i 1.98952 + 1.44547i −0.182938 + 1.74054i
16.16 −0.153909 − 1.46434i −1.03247 + 3.17760i −0.164315 + 0.0349261i 1.62503 + 0.345410i 4.81201 + 1.02282i −2.52407 − 0.536509i −0.833565 − 2.56545i −6.60413 − 4.79818i 0.255693 − 2.43275i
16.17 −0.152164 − 1.44775i 0.233842 − 0.719692i −0.116521 + 0.0247673i −0.866362 − 0.184151i −1.07751 − 0.229033i 4.25982 + 0.905453i −0.846098 − 2.60402i 1.96378 + 1.42677i −0.134774 + 1.28229i
16.18 −0.151974 − 1.44594i 0.480498 − 1.47882i −0.111352 + 0.0236685i 1.55918 + 0.331414i −2.21131 − 0.470028i −4.52491 − 0.961798i −0.847417 − 2.60808i 0.471021 + 0.342217i 0.242249 − 2.30485i
16.19 −0.151234 − 1.43889i 0.0840938 − 0.258814i −0.0912417 + 0.0193940i −0.0300022 − 0.00637716i −0.385123 − 0.0818605i 0.0224112 + 0.00476365i −0.852478 − 2.62366i 2.36714 + 1.71983i −0.00463871 + 0.0441343i
16.20 −0.139041 − 1.32289i −0.409404 + 1.26002i 0.225600 − 0.0479528i 3.78591 + 0.804720i 1.72378 + 0.366401i 2.34108 + 0.497611i −0.916896 − 2.82192i 1.00702 + 0.731643i 0.538156 − 5.12021i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.60
Significant digits:
Format:

Inner twists

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

Hecke kernels

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