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Properties

Label 663.2.j.b
Level $663$
Weight $2$
Character orbit 663.j
Analytic conductor $5.294$
Analytic rank $0$
Dimension $40$
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] = [663,2,Mod(157,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.j (of order \(4\), degree \(2\), 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: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 48 q^{4} - 4 q^{5} - 4 q^{6} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 48 q^{4} - 4 q^{5} - 4 q^{6} + 8 q^{7} + 4 q^{10} - 4 q^{11} + 40 q^{13} - 32 q^{14} + 88 q^{16} - 4 q^{17} + 60 q^{20} + 8 q^{22} + 20 q^{23} + 4 q^{24} - 16 q^{28} - 36 q^{29} - 48 q^{30} + 4 q^{31} - 24 q^{34} - 8 q^{35} - 12 q^{37} + 16 q^{38} - 16 q^{40} + 4 q^{41} + 44 q^{44} + 4 q^{45} - 8 q^{47} + 16 q^{48} + 96 q^{50} + 4 q^{51} - 48 q^{52} - 4 q^{54} - 72 q^{55} + 36 q^{56} + 8 q^{57} + 80 q^{58} - 36 q^{61} - 40 q^{62} + 8 q^{63} - 144 q^{64} - 4 q^{65} + 64 q^{67} - 52 q^{68} + 48 q^{69} - 20 q^{71} + 28 q^{73} + 48 q^{74} + 16 q^{75} - 4 q^{78} - 24 q^{79} - 220 q^{80} - 40 q^{81} + 68 q^{82} - 88 q^{84} - 12 q^{85} + 32 q^{86} - 4 q^{88} + 8 q^{89} + 4 q^{90} + 8 q^{91} - 44 q^{92} + 96 q^{95} + 16 q^{96} - 32 q^{97} - 16 q^{98} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 − 2.77487i −0.707107 + 0.707107i −5.69991 −2.16859 + 2.16859i 1.96213 + 1.96213i −2.75967 − 2.75967i 10.2668i − 1.00000i 6.01756 + 6.01756i
157.2 − 2.71965i 0.707107 − 0.707107i −5.39650 −1.81457 + 1.81457i −1.92308 − 1.92308i 2.78375 + 2.78375i 9.23730i − 1.00000i 4.93500 + 4.93500i
157.3 − 2.46160i 0.707107 − 0.707107i −4.05949 0.753427 − 0.753427i −1.74062 − 1.74062i −0.604967 − 0.604967i 5.06966i − 1.00000i −1.85464 − 1.85464i
157.4 − 1.95310i −0.707107 + 0.707107i −1.81460 −0.478184 + 0.478184i 1.38105 + 1.38105i −1.38709 − 1.38709i − 0.362113i − 1.00000i 0.933940 + 0.933940i
157.5 − 1.64448i 0.707107 − 0.707107i −0.704321 2.26180 − 2.26180i −1.16282 − 1.16282i −1.77533 − 1.77533i − 2.13072i − 1.00000i −3.71949 − 3.71949i
157.6 − 1.41186i −0.707107 + 0.707107i 0.00664320 2.11664 − 2.11664i 0.998338 + 0.998338i 1.31644 + 1.31644i − 2.83311i − 1.00000i −2.98841 − 2.98841i
157.7 − 0.918771i 0.707107 − 0.707107i 1.15586 −0.0765262 + 0.0765262i −0.649669 − 0.649669i 1.88436 + 1.88436i − 2.89951i − 1.00000i 0.0703101 + 0.0703101i
157.8 − 0.717200i −0.707107 + 0.707107i 1.48562 −2.40275 + 2.40275i 0.507137 + 0.507137i 1.97254 + 1.97254i − 2.49989i − 1.00000i 1.72325 + 1.72325i
157.9 − 0.595701i 0.707107 − 0.707107i 1.64514 −1.78479 + 1.78479i −0.421224 − 0.421224i −2.20758 − 2.20758i − 2.17141i − 1.00000i 1.06320 + 1.06320i
157.10 − 0.469240i −0.707107 + 0.707107i 1.77981 −0.169004 + 0.169004i 0.331803 + 0.331803i −1.41774 − 1.41774i − 1.77364i − 1.00000i 0.0793034 + 0.0793034i
157.11 − 0.0907040i 0.707107 − 0.707107i 1.99177 2.06692 − 2.06692i −0.0641374 − 0.0641374i 0.803125 + 0.803125i − 0.362070i − 1.00000i −0.187478 − 0.187478i
157.12 0.656776i 0.707107 − 0.707107i 1.56865 −2.07358 + 2.07358i 0.464411 + 0.464411i −1.72443 − 1.72443i 2.34380i − 1.00000i −1.36188 − 1.36188i
157.13 0.713253i −0.707107 + 0.707107i 1.49127 1.57785 − 1.57785i −0.504346 − 0.504346i 3.28083 + 3.28083i 2.49016i − 1.00000i 1.12541 + 1.12541i
157.14 1.17287i −0.707107 + 0.707107i 0.624386 2.72927 − 2.72927i −0.829341 − 0.829341i −3.25290 − 3.25290i 3.07805i − 1.00000i 3.20107 + 3.20107i
157.15 1.50350i 0.707107 − 0.707107i −0.260500 1.66625 − 1.66625i 1.06313 + 1.06313i 0.205896 + 0.205896i 2.61533i − 1.00000i 2.50520 + 2.50520i
157.16 1.90856i −0.707107 + 0.707107i −1.64259 −2.34301 + 2.34301i −1.34955 − 1.34955i 3.06815 + 3.06815i 0.682132i − 1.00000i −4.47178 − 4.47178i
157.17 2.15604i 0.707107 − 0.707107i −2.64850 −0.516807 + 0.516807i 1.52455 + 1.52455i 3.14656 + 3.14656i − 1.39818i − 1.00000i −1.11426 − 1.11426i
157.18 2.46509i −0.707107 + 0.707107i −4.07669 −0.953525 + 0.953525i −1.74308 − 1.74308i −0.373947 − 0.373947i − 5.11924i − 1.00000i −2.35053 − 2.35053i
157.19 2.48072i −0.707107 + 0.707107i −4.15395 2.50551 − 2.50551i −1.75413 − 1.75413i 1.55339 + 1.55339i − 5.34334i − 1.00000i 6.21546 + 6.21546i
157.20 2.70039i 0.707107 − 0.707107i −5.29211 −2.89634 + 2.89634i 1.90946 + 1.90946i −0.511388 − 0.511388i − 8.88997i − 1.00000i −7.82124 − 7.82124i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.j.b ✓ 40
17.c even 4 1 inner 663.2.j.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.j.b ✓ 40 1.a even 1 1 trivial
663.2.j.b ✓ 40 17.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 64 T_{2}^{38} + 1874 T_{2}^{36} + 33280 T_{2}^{34} + 400649 T_{2}^{32} + 3462596 T_{2}^{30} + \cdots + 6241 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display