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Properties

Label 671.2.cw.a
Level $671$
Weight $2$
Character orbit 671.cw
Analytic conductor $5.358$
Analytic rank $0$
Dimension $960$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [671,2,Mod(29,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([42, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.cw (of order \(60\), degree \(16\), 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: \(960\)
Relative dimension: \(60\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 960 q - 20 q^{2} - 18 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{8} + 200 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 960 q - 20 q^{2} - 18 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{8} + 200 q^{9} - 18 q^{11} - 28 q^{12} - 10 q^{13} - 18 q^{14} - 18 q^{15} - 138 q^{16} + 30 q^{17} - 20 q^{18} - 30 q^{19} + 40 q^{20} - 8 q^{22} - 40 q^{23} - 114 q^{25} - 64 q^{26} - 20 q^{28} - 10 q^{29} - 28 q^{31} - 84 q^{33} - 64 q^{34} - 30 q^{35} + 84 q^{36} - 36 q^{37} - 62 q^{38} + 60 q^{39} + 20 q^{40} - 18 q^{42} + 98 q^{44} + 36 q^{45} - 30 q^{46} - 22 q^{47} - 198 q^{48} - 60 q^{49} - 50 q^{50} + 20 q^{51} - 12 q^{53} + 124 q^{55} + 40 q^{56} - 130 q^{57} + 16 q^{58} - 8 q^{59} + 10 q^{61} + 300 q^{62} - 90 q^{63} - 24 q^{66} - 128 q^{67} - 160 q^{68} + 100 q^{69} - 64 q^{70} - 24 q^{71} + 190 q^{72} - 10 q^{73} + 40 q^{74} - 54 q^{75} - 100 q^{77} - 300 q^{78} - 20 q^{79} + 30 q^{80} - 280 q^{81} + 104 q^{82} + 20 q^{83} - 150 q^{84} - 50 q^{85} + 6 q^{86} - 32 q^{89} + 160 q^{90} + 24 q^{91} + 16 q^{92} - 150 q^{93} - 100 q^{94} - 40 q^{95} - 160 q^{96} - 66 q^{97} + 186 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.74544 − 2.15543i 0.610177 + 0.198259i −1.18352 + 5.56804i −0.795843 + 1.78749i −0.637692 − 1.66124i −0.0908057 − 1.73268i 9.12486 − 4.64935i −2.09404 − 1.52141i 5.24192 − 1.40457i
29.2 −1.70394 − 2.10419i 2.54109 + 0.825649i −1.10838 + 5.21451i 1.05320 − 2.36553i −2.59253 − 6.75378i 0.101671 + 1.94000i 8.03596 − 4.09452i 3.34838 + 2.43274i −6.77209 + 1.81458i
29.3 −1.67707 − 2.07101i −2.39664 − 0.778716i −1.06069 + 4.99013i −0.692437 + 1.55524i 2.40660 + 6.26942i 0.120837 + 2.30570i 7.36457 − 3.75244i 2.71044 + 1.96925i 4.38217 − 1.17420i
29.4 −1.55581 − 1.92126i −0.934233 − 0.303551i −0.854885 + 4.02192i 0.957802 − 2.15126i 0.870285 + 2.26717i 0.147564 + 2.81569i 4.65170 − 2.37016i −1.64640 − 1.19618i −5.62328 + 1.50675i
29.5 −1.55096 − 1.91527i 0.414186 + 0.134577i −0.846974 + 3.98470i 1.28492 − 2.88598i −0.384632 − 1.00200i −0.263501 − 5.02791i 4.55364 − 2.32020i −2.27361 − 1.65188i −7.52029 + 2.01506i
29.6 −1.48823 − 1.83782i 1.07670 + 0.349842i −0.746903 + 3.51390i −1.32632 + 2.97896i −0.959440 − 2.49943i 0.227498 + 4.34092i 3.35531 − 1.70962i −1.39015 − 1.01000i 7.44866 − 1.99586i
29.7 −1.48533 − 1.83423i −1.37447 − 0.446592i −0.742373 + 3.49259i 0.474604 − 1.06598i 1.22239 + 3.18443i 0.0303837 + 0.579755i 3.30295 − 1.68294i −0.737333 − 0.535704i −2.66020 + 0.712797i
29.8 −1.39939 − 1.72811i −1.55394 − 0.504904i −0.612226 + 2.88030i −0.615258 + 1.38189i 1.30204 + 3.39192i −0.153633 − 2.93149i 1.87162 − 0.953638i −0.267264 − 0.194179i 3.24904 − 0.870579i
29.9 −1.34807 − 1.66472i 2.20204 + 0.715487i −0.538198 + 2.53202i 0.0384973 − 0.0864663i −1.77741 − 4.63032i 0.0923272 + 1.76171i 1.12339 − 0.572398i 1.91002 + 1.38771i −0.195839 + 0.0524750i
29.10 −1.25960 − 1.55548i 3.00374 + 0.975975i −0.417094 + 1.96227i −1.41320 + 3.17410i −2.26541 − 5.90160i −0.165819 − 3.16402i 0.0108976 − 0.00555263i 5.64290 + 4.09980i 6.71732 − 1.79990i
29.11 −1.23296 − 1.52258i 1.18502 + 0.385035i −0.382238 + 1.79829i 0.517275 − 1.16182i −0.874834 − 2.27902i −0.174928 − 3.33783i −0.281989 + 0.143681i −1.17104 − 0.850810i −2.40675 + 0.644887i
29.12 −1.17161 − 1.44682i 0.837793 + 0.272215i −0.304792 + 1.43393i 1.26911 − 2.85047i −0.587722 − 1.53107i 0.136304 + 2.60083i −0.885853 + 0.451365i −1.79926 − 1.30724i −5.61103 + 1.50347i
29.13 −1.16231 − 1.43533i 0.218114 + 0.0708694i −0.293395 + 1.38031i −1.53621 + 3.45039i −0.151795 − 0.395438i −0.0831078 − 1.58579i −0.969027 + 0.493744i −2.38450 − 1.73244i 6.73802 − 1.80545i
29.14 −1.10563 − 1.36534i −3.15598 − 1.02544i −0.225915 + 1.06284i −0.0595229 + 0.133691i 2.08928 + 5.44277i 0.218878 + 4.17645i −1.42984 + 0.728538i 6.48166 + 4.70920i 0.248344 − 0.0665436i
29.15 −1.05343 − 1.30087i −2.77006 − 0.900046i −0.166740 + 0.784448i −0.230241 + 0.517129i 1.74720 + 4.55162i −0.170973 − 3.26235i −1.78681 + 0.910427i 4.43608 + 3.22300i 0.915260 − 0.245243i
29.16 −0.921245 − 1.13764i 0.00453379 + 0.00147312i −0.0297154 + 0.139800i −0.398069 + 0.894077i −0.00250085 − 0.00651494i 0.0357752 + 0.682632i −2.42222 + 1.23418i −2.42703 − 1.76334i 1.38386 − 0.370804i
29.17 −0.790977 − 0.976776i −0.838790 − 0.272539i 0.0873778 − 0.411080i −0.468948 + 1.05328i 0.397254 + 1.03488i 0.126834 + 2.42014i −2.71042 + 1.38103i −1.79776 − 1.30615i 1.39974 − 0.375060i
29.18 −0.745865 − 0.921067i 2.54314 + 0.826316i 0.123774 − 0.582311i −0.716561 + 1.60942i −1.13575 − 2.95872i 0.188971 + 3.60579i −2.74069 + 1.39645i 3.35771 + 2.43952i 2.01684 − 0.540412i
29.19 −0.725993 − 0.896527i 3.24653 + 1.05486i 0.139129 − 0.654550i 1.64571 − 3.69632i −1.41125 − 3.67642i −0.132251 − 2.52351i −2.74358 + 1.39793i 7.00015 + 5.08591i −4.50862 + 1.20808i
29.20 −0.690952 − 0.853255i 1.03873 + 0.337505i 0.165194 − 0.777178i 1.64813 − 3.70177i −0.429738 − 1.11951i 0.192880 + 3.68037i −2.73381 + 1.39294i −1.46199 − 1.06220i −4.29733 + 1.15147i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
61.h odd 12 1 inner
671.cw even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.cw.a ✓ 960
11.d odd 10 1 inner 671.2.cw.a ✓ 960
61.h odd 12 1 inner 671.2.cw.a ✓ 960
671.cw even 60 1 inner 671.2.cw.a ✓ 960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.cw.a ✓ 960 1.a even 1 1 trivial
671.2.cw.a ✓ 960 11.d odd 10 1 inner
671.2.cw.a ✓ 960 61.h odd 12 1 inner
671.2.cw.a ✓ 960 671.cw even 60 1 inner

Hecke kernels

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