↑

Properties

Label 663.2.bq.b
Level $663$
Weight $2$
Character orbit 663.bq
Analytic conductor $5.294$
Analytic rank $0$
Dimension $148$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [663,2,Mod(137,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 11, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.bq (of order \(12\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(148\)
Relative dimension: \(37\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 148 q + 2 q^{5} - 6 q^{6} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 148 q + 2 q^{5} - 6 q^{6} + 6 q^{7} + 28 q^{11} - 4 q^{13} + 20 q^{15} + 60 q^{16} + 74 q^{17} - 32 q^{18} - 4 q^{19} + 16 q^{20} + 16 q^{21} - 8 q^{22} - 112 q^{24} - 28 q^{26} + 12 q^{27} - 20 q^{28} + 8 q^{30} - 10 q^{31} - 74 q^{33} - 18 q^{36} - 2 q^{37} + 18 q^{39} - 24 q^{40} - 18 q^{41} + 20 q^{42} - 42 q^{43} - 28 q^{44} + 8 q^{45} - 40 q^{46} - 32 q^{47} + 32 q^{48} - 30 q^{49} + 104 q^{50} - 100 q^{52} + 6 q^{54} + 8 q^{55} - 80 q^{56} - 4 q^{57} - 12 q^{58} + 48 q^{59} + 48 q^{60} - 46 q^{63} - 50 q^{65} + 8 q^{67} - 28 q^{69} + 28 q^{70} + 8 q^{71} - 172 q^{72} + 30 q^{73} + 60 q^{75} + 20 q^{76} - 48 q^{77} + 170 q^{78} + 16 q^{79} - 28 q^{80} + 60 q^{82} + 16 q^{83} - 38 q^{84} - 2 q^{85} - 48 q^{86} + 20 q^{87} + 72 q^{88} + 24 q^{89} + 56 q^{90} + 6 q^{91} - 14 q^{93} - 80 q^{94} + 18 q^{95} - 106 q^{96} - 14 q^{97} + 160 q^{98} + 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 −0.726491 − 2.71130i −0.326797 + 1.70094i −5.09132 + 2.93947i −1.71229 + 1.71229i 4.84918 − 0.349675i −2.80509 − 0.751620i 7.69898 + 7.69898i −2.78641 − 1.11172i 5.88651 + 3.39858i
137.2 −0.714371 − 2.66607i −1.36767 − 1.06277i −4.86555 + 2.80913i 0.723971 − 0.723971i −1.85640 + 4.40551i 0.728576 + 0.195221i 7.06174 + 7.06174i 0.741029 + 2.90704i −2.44734 − 1.41297i
137.3 −0.611008 − 2.28031i −0.287801 − 1.70797i −3.09444 + 1.78658i 0.247415 − 0.247415i −3.71886 + 1.69986i 1.80643 + 0.484032i 2.62607 + 2.62607i −2.83434 + 0.983114i −0.715355 − 0.413010i
137.4 −0.592465 − 2.21111i 0.577085 + 1.63309i −2.80594 + 1.62001i −0.784907 + 0.784907i 3.26903 − 2.24355i 3.31161 + 0.887342i 2.00715 + 2.00715i −2.33394 + 1.88486i 2.20054 + 1.27048i
137.5 −0.586013 − 2.18703i 1.39644 + 1.02468i −2.70764 + 1.56325i 1.96512 − 1.96512i 1.42266 − 3.65452i 1.71356 + 0.459148i 1.80356 + 1.80356i 0.900081 + 2.86179i −5.44937 − 3.14620i
137.6 −0.523420 − 1.95343i −1.47276 + 0.911582i −1.80987 + 1.04493i −0.176769 + 0.176769i 2.55158 + 2.39979i −3.72390 − 0.997815i 0.128498 + 0.128498i 1.33804 − 2.68508i 0.437830 + 0.252781i
137.7 −0.461814 − 1.72351i −1.72680 + 0.134707i −1.02518 + 0.591886i −0.885045 + 0.885045i 1.02963 + 2.91396i 0.544863 + 0.145996i −1.02984 − 1.02984i 2.96371 − 0.465224i 1.93411 + 1.11666i
137.8 −0.457331 − 1.70678i 0.750643 − 1.56094i −0.971910 + 0.561132i −1.80087 + 1.80087i −3.00748 − 0.567319i 4.77550 + 1.27959i −1.09669 − 1.09669i −1.87307 − 2.34342i 3.89728 + 2.25010i
137.9 −0.444639 − 1.65941i 1.70287 − 0.316614i −0.823902 + 0.475680i 2.30143 − 2.30143i −1.28255 − 2.68498i 1.74581 + 0.467788i −1.27386 − 1.27386i 2.79951 − 1.07830i −4.84234 − 2.79573i
137.10 −0.414132 − 1.54556i 1.20452 − 1.24464i −0.485205 + 0.280133i −0.299161 + 0.299161i −2.42250 − 1.34622i −2.40527 − 0.644491i −1.62896 − 1.62896i −0.0982512 − 2.99839i 0.586263 + 0.338479i
137.11 −0.412602 − 1.53985i −0.703642 − 1.58268i −0.468847 + 0.270689i 2.89886 − 2.89886i −2.14677 + 1.73652i −4.99367 − 1.33805i −1.64423 − 1.64423i −2.00978 + 2.22729i −5.65988 − 3.26773i
137.12 −0.369840 − 1.38026i −1.16209 − 1.28435i −0.0362936 + 0.0209541i −1.45476 + 1.45476i −1.34296 + 2.07899i −0.570094 − 0.152756i −1.97850 − 1.97850i −0.299115 + 2.98505i 2.54597 + 1.46992i
137.13 −0.289390 − 1.08002i −1.47932 + 0.900894i 0.649360 − 0.374908i 2.49821 − 2.49821i 1.40108 + 1.33698i 4.40191 + 1.17949i −2.17408 − 2.17408i 1.37678 − 2.66542i −3.42106 − 1.97515i
137.14 −0.247731 − 0.924544i 1.64689 + 0.536425i 0.938639 − 0.541923i −2.56486 + 2.56486i 0.0879633 − 1.65551i 1.61538 + 0.432840i −2.08719 − 2.08719i 2.42450 + 1.76687i 3.00672 + 1.73593i
137.15 −0.188543 − 0.703652i 0.394281 + 1.68658i 1.27247 − 0.734663i 0.851330 − 0.851330i 1.11242 − 0.595429i 0.898202 + 0.240673i −1.78708 − 1.78708i −2.68909 + 1.32997i −0.759552 − 0.438528i
137.16 −0.161743 − 0.603633i 1.71475 − 0.244227i 1.39384 − 0.804733i −0.246023 + 0.246023i −0.424771 − 0.995575i −4.23357 − 1.13438i −1.59499 − 1.59499i 2.88071 − 0.837573i 0.188300 + 0.108715i
137.17 −0.0967418 − 0.361045i −1.54182 + 0.789164i 1.61106 − 0.930144i −2.50331 + 2.50331i 0.434082 + 0.480323i 1.38640 + 0.371484i −1.02029 − 1.02029i 1.75444 − 2.43350i 1.14598 + 0.661633i
137.18 −0.0650176 − 0.242649i −0.258689 − 1.71262i 1.67740 − 0.968447i 1.16231 − 1.16231i −0.398747 + 0.174121i 0.694384 + 0.186060i −0.699316 − 0.699316i −2.86616 + 0.886075i −0.357604 − 0.206463i
137.19 0.0355121 + 0.132533i 1.72302 + 0.176643i 1.71575 − 0.990587i 1.21178 − 1.21178i 0.0377770 + 0.234630i 1.05906 + 0.283775i 0.386256 + 0.386256i 2.93759 + 0.608718i 0.203633 + 0.117567i
137.20 0.118257 + 0.441340i 0.520308 + 1.65205i 1.55125 − 0.895617i −1.73824 + 1.73824i −0.667586 + 0.424999i −1.33154 − 0.356786i 1.22488 + 1.22488i −2.45856 + 1.71915i −0.972711 − 0.561595i
See next 80 embeddings (of 148 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.37
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.bq.b yes 148
3.b odd 2 1 663.2.bq.a ✓ 148
13.f odd 12 1 663.2.bq.a ✓ 148
39.k even 12 1 inner 663.2.bq.b yes 148
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.bq.a ✓ 148 3.b odd 2 1
663.2.bq.a ✓ 148 13.f odd 12 1
663.2.bq.b yes 148 1.a even 1 1 trivial
663.2.bq.b yes 148 39.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{148} - 271 T_{2}^{144} - 52 T_{2}^{141} + 41538 T_{2}^{140} + 436 T_{2}^{139} + \cdots + 26384154624 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display