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Properties

Label 671.2.bp.a
Level $671$
Weight $2$
Character orbit 671.bp
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(137,671)]
 
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, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bp (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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{2} - 12 q^{3} + 50 q^{4} - 6 q^{5} - 14 q^{6} - q^{7} + 436 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{2} - 12 q^{3} + 50 q^{4} - 6 q^{5} - 14 q^{6} - q^{7} + 436 q^{9} - 28 q^{10} - 13 q^{11} - 72 q^{12} - 7 q^{13} + 13 q^{14} - 32 q^{15} + 28 q^{16} - 32 q^{17} - 68 q^{18} + 29 q^{19} - 22 q^{20} + q^{21} - 7 q^{22} - 46 q^{23} - 54 q^{24} + 46 q^{25} + 2 q^{26} - 36 q^{27} + 8 q^{28} + 2 q^{29} - 24 q^{30} + 2 q^{32} - 20 q^{33} - 20 q^{34} - 49 q^{35} + 6 q^{36} + 4 q^{37} + 27 q^{38} - 37 q^{39} - 55 q^{40} + 33 q^{41} - q^{42} + 9 q^{43} - 76 q^{44} - 31 q^{45} - 8 q^{46} - 4 q^{47} - 76 q^{48} - 207 q^{49} - 5 q^{50} - 28 q^{51} + 39 q^{52} - 15 q^{53} - 24 q^{54} + 30 q^{55} + 31 q^{56} + 29 q^{57} + 88 q^{58} - 26 q^{59} + 30 q^{60} + 19 q^{61} + 85 q^{62} - 27 q^{63} - 44 q^{64} - 37 q^{65} + 22 q^{66} + 4 q^{67} - 187 q^{68} + 91 q^{69} + 139 q^{70} + 13 q^{71} + 149 q^{72} + 48 q^{73} - 151 q^{74} + 89 q^{75} - 66 q^{76} - 4 q^{77} + 71 q^{78} + 40 q^{79} - 139 q^{80} + 336 q^{81} + 25 q^{82} - 27 q^{83} + 79 q^{84} + 7 q^{85} + 117 q^{86} - 73 q^{87} + 152 q^{88} + 24 q^{89} - 220 q^{90} - 81 q^{91} + 37 q^{92} - 45 q^{93} - 36 q^{94} + 35 q^{95} - 44 q^{96} + 32 q^{97} - 147 q^{98} - 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 −2.57449 + 1.14624i 0.529240 3.97588 − 4.41566i 1.19623 − 0.532596i −1.36252 + 0.606635i −1.38263 + 2.39478i −3.43277 + 10.5650i −2.71990 −2.46920 + 2.74232i
137.2 −2.43783 + 1.08539i −2.39080 3.42669 − 3.80572i −2.00788 + 0.893966i 5.82838 − 2.59496i −0.136549 + 0.236510i −2.57374 + 7.92116i 2.71594 3.92457 − 4.35868i
137.3 −2.35494 + 1.04849i 3.34787 3.10816 − 3.45196i 0.0183215 − 0.00815726i −7.88404 + 3.51020i −1.21247 + 2.10006i −2.10702 + 6.48474i 8.20825 −0.0345933 + 0.0384197i
137.4 −2.21929 + 0.988093i −3.12919 2.61068 − 2.89945i 1.80666 − 0.804378i 6.94460 − 3.09194i −0.128366 + 0.222336i −1.42753 + 4.39348i 6.79186 −3.21472 + 3.57030i
137.5 −2.17011 + 0.966197i 0.867144 2.43760 − 2.70723i 1.49944 − 0.667593i −1.88180 + 0.837832i 1.81831 − 3.14940i −1.20602 + 3.71175i −2.24806 −2.60893 + 2.89751i
137.6 −2.14577 + 0.955358i 0.623440 2.35336 − 2.61367i −3.77163 + 1.67924i −1.33776 + 0.595608i −1.91296 + 3.31334i −1.10111 + 3.38887i −2.61132 6.48878 − 7.20652i
137.7 −2.09682 + 0.933564i −1.39103 2.18685 − 2.42874i 2.66607 − 1.18701i 2.91674 − 1.29862i 0.00498853 − 0.00864039i −0.899498 + 2.76837i −1.06503 −4.48212 + 4.97790i
137.8 −2.08703 + 0.929204i 2.16609 2.15400 − 2.39226i −1.33270 + 0.593358i −4.52069 + 2.01274i 0.762723 − 1.32108i −0.860639 + 2.64877i 1.69194 2.23004 − 2.47671i
137.9 −2.06451 + 0.919179i 1.96864 2.07905 − 2.30902i 3.94607 − 1.75690i −4.06427 + 1.80953i 0.857271 − 1.48484i −0.773131 + 2.37945i 0.875529 −6.53179 + 7.25429i
137.10 −2.00024 + 0.890563i −0.229366 1.86958 − 2.07638i −0.920880 + 0.410002i 0.458785 − 0.204264i −0.400791 + 0.694190i −0.537254 + 1.65350i −2.94739 1.47685 − 1.64020i
137.11 −1.99641 + 0.888861i −1.51549 1.85734 − 2.06278i −2.27536 + 1.01305i 3.02555 − 1.34706i 1.72939 − 2.99539i −0.523869 + 1.61230i −0.703282 3.64209 − 4.04495i
137.12 −1.49033 + 0.663538i 1.79780 0.442541 − 0.491491i −1.22283 + 0.544439i −2.67931 + 1.19291i −1.13085 + 1.95869i 0.674833 − 2.07692i 0.232067 1.46116 − 1.62279i
137.13 −1.45198 + 0.646464i −2.30627 0.352077 − 0.391021i 2.10166 − 0.935721i 3.34866 − 1.49092i 2.53048 − 4.38292i 0.723870 − 2.22784i 2.31886 −2.44667 + 2.71730i
137.14 −1.41158 + 0.628476i −2.57569 0.259313 − 0.287996i −0.484850 + 0.215869i 3.63579 − 1.61876i −1.23772 + 2.14379i 0.769923 − 2.36958i 3.63417 0.548735 − 0.609432i
137.15 −1.38776 + 0.617870i −2.32214 0.205851 − 0.228621i −0.499363 + 0.222331i 3.22257 − 1.43478i −2.30818 + 3.99788i 0.794437 − 2.44502i 2.39234 0.555624 − 0.617083i
137.16 −1.34365 + 0.598232i −0.290905 0.109257 − 0.121342i 0.756690 − 0.336900i 0.390875 − 0.174029i 1.27131 − 2.20198i 0.834798 − 2.56924i −2.91537 −0.815183 + 0.905353i
137.17 −1.33715 + 0.595335i 3.06873 0.0952715 − 0.105810i 0.742728 − 0.330684i −4.10334 + 1.82692i 1.44889 − 2.50954i 0.840209 − 2.58590i 6.41711 −0.796268 + 0.884345i
137.18 −1.26633 + 0.563806i 1.92528 −0.0525500 + 0.0583627i 1.36638 − 0.608352i −2.43804 + 1.08548i −1.74446 + 3.02150i 0.890340 − 2.74018i 0.706700 −1.38729 + 1.54075i
137.19 −1.22445 + 0.545161i −2.78243 −0.136178 + 0.151241i −3.78811 + 1.68658i 3.40696 − 1.51687i 0.359396 − 0.622493i 0.912662 − 2.80889i 4.74193 3.71891 − 4.13027i
137.20 −1.08503 + 0.483086i 2.37673 −0.394344 + 0.437964i −3.17227 + 1.41238i −2.57882 + 1.14816i 1.55889 − 2.70008i 0.950348 − 2.92487i 2.64883 2.75970 − 3.06496i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.60
Significant digits:
Format:

Inner twists

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

Hecke kernels

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