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Properties

Label 671.2.k.b
Level $671$
Weight $2$
Character orbit 671.k
Analytic conductor $5.358$
Analytic rank $0$
Dimension $236$
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(180,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.180");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.k (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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(59\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 236 q - 3 q^{2} - 6 q^{3} - 57 q^{4} - 16 q^{5} - 8 q^{6} + q^{7} - 8 q^{8} - 55 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 236 q - 3 q^{2} - 6 q^{3} - 57 q^{4} - 16 q^{5} - 8 q^{6} + q^{7} - 8 q^{8} - 55 q^{9} - 34 q^{10} - 20 q^{11} - 10 q^{12} + 5 q^{13} - 6 q^{14} - 9 q^{15} - 35 q^{16} - 2 q^{19} - 9 q^{20} - 62 q^{21} - 19 q^{22} + 7 q^{23} - 28 q^{24} + 212 q^{25} - 46 q^{26} - 37 q^{29} - 88 q^{30} + 7 q^{31} + 46 q^{32} + 9 q^{33} - 11 q^{34} - 38 q^{35} + 246 q^{36} - 10 q^{37} - 8 q^{38} + 34 q^{39} + 56 q^{40} - 100 q^{41} + 19 q^{42} - 18 q^{43} + 22 q^{44} - 30 q^{45} - 16 q^{46} - 21 q^{47} + 31 q^{48} - 4 q^{49} - 17 q^{50} + 12 q^{51} - 24 q^{52} - 10 q^{53} - 56 q^{54} - 99 q^{55} + 35 q^{56} + 4 q^{57} - 31 q^{58} + 10 q^{59} - 52 q^{60} + 18 q^{61} - 34 q^{62} + 89 q^{63} + 20 q^{65} + 8 q^{66} + 49 q^{67} + 57 q^{68} - 29 q^{69} + 40 q^{70} - 49 q^{71} - 58 q^{72} + 23 q^{73} - 27 q^{74} - 60 q^{75} + 31 q^{76} + 52 q^{77} + 67 q^{78} - 26 q^{79} - 53 q^{80} - 36 q^{81} + 14 q^{82} + 79 q^{83} + 122 q^{84} - 180 q^{85} + 25 q^{86} - 101 q^{87} - 41 q^{88} - 3 q^{89} - 12 q^{90} + 58 q^{91} - 30 q^{92} - 42 q^{93} - 123 q^{94} - 20 q^{95} + 19 q^{96} - 12 q^{97} + 73 q^{98} - 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
180.1 −2.27321 + 1.65158i 1.15847 + 0.841681i 1.82172 − 5.60667i 1.06077 −4.02356 1.18672 − 0.862203i 3.38216 + 10.4092i −0.293415 − 0.903039i −2.41135 + 1.75195i
180.2 −2.14057 + 1.55522i −2.46122 − 1.78818i 1.54531 − 4.75599i 1.95967 8.04942 2.75143 − 1.99903i 2.45349 + 7.55105i 1.93296 + 5.94904i −4.19482 + 3.04772i
180.3 −2.09812 + 1.52437i −1.02053 − 0.741455i 1.46035 − 4.49451i 3.58629 3.27144 −2.84046 + 2.06371i 2.18448 + 6.72315i −0.435334 − 1.33982i −7.52446 + 5.46684i
180.4 −2.02854 + 1.47382i −2.05621 − 1.49393i 1.32479 − 4.07730i −2.42605 6.37289 −2.36269 + 1.71660i 1.77214 + 5.45409i 1.06915 + 3.29050i 4.92135 − 3.57557i
180.5 −1.97331 + 1.43369i 0.757754 + 0.550541i 1.22044 − 3.75614i −1.94613 −2.28459 −0.0258442 + 0.0187769i 1.46937 + 4.52224i −0.655954 − 2.01882i 3.84031 − 2.79015i
180.6 −1.93108 + 1.40301i 2.68873 + 1.95347i 1.14259 − 3.51652i −1.56382 −7.93288 3.29063 − 2.39078i 1.25208 + 3.85349i 2.48614 + 7.65155i 3.01985 − 2.19405i
180.7 −1.90013 + 1.38053i −0.705925 − 0.512885i 1.08662 − 3.34427i −2.52692 2.04940 2.59047 − 1.88209i 1.10056 + 3.38718i −0.691771 − 2.12905i 4.80148 − 3.48848i
180.8 −1.88148 + 1.36698i 1.28115 + 0.930812i 1.05332 − 3.24178i 1.18055 −3.68287 −2.99571 + 2.17651i 1.01232 + 3.11560i −0.152110 − 0.468145i −2.22118 + 1.61378i
180.9 −1.72495 + 1.25325i 2.54897 + 1.85194i 0.786784 − 2.42147i 1.83166 −6.71779 −3.02373 + 2.19687i 0.359801 + 1.10735i 2.14054 + 6.58791i −3.15952 + 2.29553i
180.10 −1.71505 + 1.24606i −0.913512 − 0.663705i 0.770704 − 2.37198i 1.92220 2.39373 2.13928 − 1.55428i 0.323649 + 0.996089i −0.533051 − 1.64056i −3.29666 + 2.39517i
180.11 −1.67861 + 1.21958i 0.248337 + 0.180428i 0.712321 − 2.19230i −2.65958 −0.636909 −1.71408 + 1.24535i 0.195634 + 0.602101i −0.897934 − 2.76356i 4.46440 − 3.24358i
180.12 −1.63271 + 1.18623i 1.02186 + 0.742426i 0.640560 − 1.97144i 4.10270 −2.54910 2.28191 − 1.65791i 0.0454611 + 0.139915i −0.434046 − 1.33585i −6.69852 + 4.86676i
180.13 −1.55850 + 1.13232i −1.61326 − 1.17210i 0.528747 − 1.62732i 0.677114 3.84144 −2.03833 + 1.48093i −0.172003 − 0.529371i 0.301727 + 0.928619i −1.05528 + 0.766707i
180.14 −1.37425 + 0.998449i 2.02684 + 1.47258i 0.273621 − 0.842118i −3.34663 −4.25567 −0.0890797 + 0.0647202i −0.585042 − 1.80057i 1.01252 + 3.11620i 4.59910 − 3.34144i
180.15 −1.31336 + 0.954215i −2.17401 − 1.57951i 0.196365 − 0.604349i −4.11873 4.36246 3.39697 − 2.46804i −0.684540 − 2.10680i 1.30442 + 4.01458i 5.40939 − 3.93016i
180.16 −1.30708 + 0.949651i 0.404135 + 0.293621i 0.188594 − 0.580431i 0.612161 −0.807076 −0.828491 + 0.601934i −0.693822 − 2.13537i −0.849939 − 2.61584i −0.800145 + 0.581340i
180.17 −1.17061 + 0.850497i −1.60897 − 1.16898i 0.0289462 − 0.0890872i 1.59849 2.87769 −0.338026 + 0.245590i −0.852382 − 2.62336i 0.295204 + 0.908545i −1.87120 + 1.35951i
180.18 −1.12804 + 0.819569i 1.20125 + 0.872759i −0.0172526 + 0.0530981i −1.53027 −2.07034 1.09328 − 0.794316i −0.885802 − 2.72622i −0.245759 − 0.756369i 1.72620 − 1.25416i
180.19 −0.907535 + 0.659363i −1.15103 − 0.836272i −0.229174 + 0.705324i −3.63003 1.59601 −3.69859 + 2.68719i −0.950376 − 2.92496i −0.301533 − 0.928023i 3.29438 − 2.39351i
180.20 −0.814408 + 0.591702i 1.57851 + 1.14685i −0.304885 + 0.938338i −0.489483 −1.96414 3.36887 − 2.44763i −0.929069 − 2.85938i 0.249361 + 0.767455i 0.398639 − 0.289628i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 180.59
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.k even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.k.b ✓ 236
11.c even 5 1 671.2.l.b yes 236
61.e even 5 1 671.2.l.b yes 236
671.k even 5 1 inner 671.2.k.b ✓ 236
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.k.b ✓ 236 1.a even 1 1 trivial
671.2.k.b ✓ 236 671.k even 5 1 inner
671.2.l.b yes 236 11.c even 5 1
671.2.l.b yes 236 61.e even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{236} + 3 T_{2}^{235} + 92 T_{2}^{234} + 267 T_{2}^{233} + 4445 T_{2}^{232} + \cdots + 50\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display