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Properties

Label 712.2.k.c
Level $712$
Weight $2$
Character orbit 712.k
Analytic conductor $5.685$
Analytic rank $0$
Dimension $168$
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] = [712,2,Mod(301,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 712.k (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: \(5.68534862392\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(84\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 4 q^{4} - 2 q^{6} - 12 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q - 4 q^{4} - 2 q^{6} - 12 q^{7} + 12 q^{8} + 10 q^{12} - 4 q^{14} - 28 q^{15} - 36 q^{16} - 12 q^{22} + 24 q^{23} + 4 q^{24} + 112 q^{25} + 18 q^{28} - 8 q^{30} + 16 q^{31} - 20 q^{32} + 48 q^{33} - 8 q^{38} - 88 q^{39} + 8 q^{41} + 28 q^{44} - 18 q^{46} + 20 q^{48} - 52 q^{50} + 56 q^{54} + 4 q^{56} + 16 q^{57} - 12 q^{58} - 52 q^{60} - 34 q^{62} - 20 q^{63} + 44 q^{64} - 120 q^{65} + 6 q^{66} - 62 q^{70} + 24 q^{73} + 54 q^{74} + 54 q^{76} + 136 q^{78} - 152 q^{81} - 26 q^{82} - 50 q^{86} + 176 q^{87} - 20 q^{88} + 16 q^{89} - 16 q^{90} - 108 q^{92} - 100 q^{95} + 2 q^{96} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1 −1.41388 − 0.0305234i 0.381010 − 0.381010i 1.99814 + 0.0863130i 2.70646 −0.550334 + 0.527074i −1.81614 − 1.81614i −2.82250 − 0.183026i 2.70966i −3.82662 − 0.0826102i
301.2 −1.41388 + 0.0305234i −0.381010 + 0.381010i 1.99814 − 0.0863130i −2.70646 0.527074 − 0.550334i −1.81614 − 1.81614i −2.82250 + 0.183026i 2.70966i 3.82662 − 0.0826102i
301.3 −1.41290 − 0.0610428i 2.08354 − 2.08354i 1.99255 + 0.172494i 0.675004 −3.07101 + 2.81664i 3.04231 + 3.04231i −2.80473 − 0.365347i − 5.68230i −0.953710 − 0.0412041i
301.4 −1.41290 + 0.0610428i −2.08354 + 2.08354i 1.99255 − 0.172494i −0.675004 2.81664 − 3.07101i 3.04231 + 3.04231i −2.80473 + 0.365347i − 5.68230i 0.953710 − 0.0412041i
301.5 −1.40476 − 0.163207i −1.91833 + 1.91833i 1.94673 + 0.458536i 2.08238 3.00789 − 2.38171i −1.99052 − 1.99052i −2.65986 − 0.961855i − 4.35997i −2.92525 − 0.339860i
301.6 −1.40476 + 0.163207i 1.91833 − 1.91833i 1.94673 − 0.458536i −2.08238 −2.38171 + 3.00789i −1.99052 − 1.99052i −2.65986 + 0.961855i − 4.35997i 2.92525 − 0.339860i
301.7 −1.35163 − 0.416049i 0.108019 − 0.108019i 1.65381 + 1.12469i 0.729379 −0.190942 + 0.101060i 1.31154 + 1.31154i −1.76741 − 2.20823i 2.97666i −0.985850 − 0.303457i
301.8 −1.35163 + 0.416049i −0.108019 + 0.108019i 1.65381 − 1.12469i −0.729379 0.101060 − 0.190942i 1.31154 + 1.31154i −1.76741 + 2.20823i 2.97666i 0.985850 − 0.303457i
301.9 −1.32919 − 0.482961i 1.17935 − 1.17935i 1.53350 + 1.28389i −4.24029 −2.13716 + 0.997999i 1.85123 + 1.85123i −1.41824 − 2.44716i 0.218281i 5.63616 + 2.04789i
301.10 −1.32919 + 0.482961i −1.17935 + 1.17935i 1.53350 − 1.28389i 4.24029 0.997999 − 2.13716i 1.85123 + 1.85123i −1.41824 + 2.44716i 0.218281i −5.63616 + 2.04789i
301.11 −1.20390 − 0.742033i −0.875054 + 0.875054i 0.898773 + 1.78667i 2.77380 1.70280 − 0.404163i 1.90084 + 1.90084i 0.243735 − 2.81791i 1.46856i −3.33940 − 2.05826i
301.12 −1.20390 + 0.742033i 0.875054 − 0.875054i 0.898773 − 1.78667i −2.77380 −0.404163 + 1.70280i 1.90084 + 1.90084i 0.243735 + 2.81791i 1.46856i 3.33940 − 2.05826i
301.13 −1.17359 − 0.789107i −0.720582 + 0.720582i 0.754619 + 1.85217i −0.0644480 1.41428 − 0.277050i −1.31622 − 1.31622i 0.575952 − 2.76917i 1.96152i 0.0756354 + 0.0508564i
301.14 −1.17359 + 0.789107i 0.720582 − 0.720582i 0.754619 − 1.85217i 0.0644480 −0.277050 + 1.41428i −1.31622 − 1.31622i 0.575952 + 2.76917i 1.96152i −0.0756354 + 0.0508564i
301.15 −1.15796 − 0.811872i 2.00510 − 2.00510i 0.681729 + 1.88022i 2.28952 −3.94969 + 0.693934i −1.77700 − 1.77700i 0.737088 − 2.73070i − 5.04081i −2.65117 − 1.85880i
301.16 −1.15796 + 0.811872i −2.00510 + 2.00510i 0.681729 − 1.88022i −2.28952 0.693934 − 3.94969i −1.77700 − 1.77700i 0.737088 + 2.73070i − 5.04081i 2.65117 − 1.85880i
301.17 −1.13487 − 0.843848i 0.841874 − 0.841874i 0.575839 + 1.91531i −2.45801 −1.66583 + 0.245000i −3.49231 − 3.49231i 0.962730 − 2.65954i 1.58250i 2.78951 + 2.07418i
301.18 −1.13487 + 0.843848i −0.841874 + 0.841874i 0.575839 − 1.91531i 2.45801 0.245000 − 1.66583i −3.49231 − 3.49231i 0.962730 + 2.65954i 1.58250i −2.78951 + 2.07418i
301.19 −1.05793 − 0.938496i −2.22254 + 2.22254i 0.238452 + 1.98573i −3.34009 4.43716 − 0.265459i −1.86634 − 1.86634i 1.61134 − 2.32456i − 6.87941i 3.53360 + 3.13466i
301.20 −1.05793 + 0.938496i 2.22254 − 2.22254i 0.238452 − 1.98573i 3.34009 −0.265459 + 4.43716i −1.86634 − 1.86634i 1.61134 + 2.32456i − 6.87941i −3.53360 + 3.13466i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.84
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
89.c even 4 1 inner
712.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.2.k.c ✓ 168
8.b even 2 1 inner 712.2.k.c ✓ 168
89.c even 4 1 inner 712.2.k.c ✓ 168
712.k even 4 1 inner 712.2.k.c ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.k.c ✓ 168 1.a even 1 1 trivial
712.2.k.c ✓ 168 8.b even 2 1 inner
712.2.k.c ✓ 168 89.c even 4 1 inner
712.2.k.c ✓ 168 712.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{168} + 1190 T_{3}^{164} + 664347 T_{3}^{160} + 231459176 T_{3}^{156} + 56476741949 T_{3}^{152} + \cdots + 78\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(712, [\chi])\). Copy content Toggle raw display