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Properties

Label 197.3.c.a
Level $197$
Weight $3$
Character orbit 197.c
Analytic conductor $5.368$
Analytic rank $0$
Dimension $64$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,3,Mod(14,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.14");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 197.c (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: \(5.36786120790\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 2 q^{2} - 2 q^{3} - 10 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 2 q^{2} - 2 q^{3} - 10 q^{5} - 6 q^{8} + 10 q^{11} + 68 q^{12} + 2 q^{13} - 2 q^{14} - 280 q^{16} - 30 q^{17} - 68 q^{18} + 114 q^{20} - 20 q^{21} + 48 q^{23} + 60 q^{24} + 22 q^{27} + 392 q^{28} + 56 q^{29} - 70 q^{30} - 64 q^{31} - 156 q^{32} - 52 q^{34} - 92 q^{35} - 36 q^{36} - 160 q^{37} + 40 q^{38} + 52 q^{40} + 80 q^{42} + 140 q^{44} + 324 q^{45} + 214 q^{46} + 216 q^{48} - 836 q^{49} + 126 q^{50} + 124 q^{51} - 10 q^{52} - 16 q^{53} + 400 q^{54} - 40 q^{56} + 68 q^{57} - 266 q^{58} - 364 q^{59} - 68 q^{60} - 168 q^{61} - 12 q^{63} + 238 q^{66} - 112 q^{67} - 148 q^{68} - 48 q^{69} + 48 q^{70} + 150 q^{71} - 474 q^{72} + 18 q^{73} + 236 q^{74} + 64 q^{75} - 260 q^{76} - 388 q^{77} - 782 q^{78} + 202 q^{79} + 796 q^{80} - 516 q^{81} + 72 q^{82} - 294 q^{84} - 696 q^{85} + 618 q^{86} + 464 q^{87} - 244 q^{88} - 536 q^{89} + 1484 q^{90} + 212 q^{91} + 132 q^{94} + 370 q^{95} - 530 q^{98} - 66 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −2.75292 + 2.75292i −0.348601 + 0.348601i − 11.1571i −4.86430 + 4.86430i − 1.91934i 12.7739i 19.7030 + 19.7030i 8.75696i − 26.7821i
14.2 −2.60079 + 2.60079i −0.624484 + 0.624484i − 9.52825i 1.64362 − 1.64362i − 3.24831i − 8.36970i 14.3778 + 14.3778i 8.22004i 8.54942i
14.3 −2.37240 + 2.37240i 3.08237 − 3.08237i − 7.25653i 3.27580 − 3.27580i 14.6252i − 0.819881i 7.72579 + 7.72579i − 10.0021i 15.5430i
14.4 −2.36282 + 2.36282i −3.56182 + 3.56182i − 7.16585i −0.0568632 + 0.0568632i − 16.8319i 1.89117i 7.48035 + 7.48035i − 16.3731i − 0.268715i
14.5 −2.07857 + 2.07857i 3.03733 − 3.03733i − 4.64088i −3.00380 + 3.00380i 12.6266i 7.49199i 1.33210 + 1.33210i − 9.45073i − 12.4872i
14.6 −1.90042 + 1.90042i 0.406457 − 0.406457i − 3.22320i 1.91596 − 1.91596i 1.54488i − 0.163555i −1.47625 − 1.47625i 8.66959i 7.28227i
14.7 −1.85403 + 1.85403i 0.722049 − 0.722049i − 2.87488i −6.14279 + 6.14279i 2.67741i − 9.45582i −2.08602 − 2.08602i 7.95729i − 22.7779i
14.8 −1.84369 + 1.84369i −1.89732 + 1.89732i − 2.79835i 6.24444 − 6.24444i − 6.99613i 9.93768i −2.21546 − 2.21546i 1.80035i 23.0256i
14.9 −1.68863 + 1.68863i −3.21724 + 3.21724i − 1.70294i −3.76601 + 3.76601i − 10.8655i − 2.48441i −3.87889 − 3.87889i − 11.7013i − 12.7188i
14.10 −1.11355 + 1.11355i 2.19442 − 2.19442i 1.52002i −1.87378 + 1.87378i 4.88717i 5.07096i −6.14681 − 6.14681i − 0.630916i − 4.17310i
14.11 −0.904445 + 0.904445i −2.55489 + 2.55489i 2.36396i 3.86642 − 3.86642i − 4.62152i − 11.7409i −5.75585 − 5.75585i − 4.05494i 6.99393i
14.12 −0.892438 + 0.892438i −0.994798 + 0.994798i 2.40711i −0.178552 + 0.178552i − 1.77559i 2.79633i −5.71795 − 5.71795i 7.02075i − 0.318693i
14.13 −0.770755 + 0.770755i 4.07862 − 4.07862i 2.81187i −1.79164 + 1.79164i 6.28722i − 12.4564i −5.25028 − 5.25028i − 24.2702i − 2.76183i
14.14 −0.694722 + 0.694722i 2.63608 − 2.63608i 3.03472i 5.73458 − 5.73458i 3.66268i − 1.32953i −4.88717 − 4.88717i − 4.89782i 7.96787i
14.15 −0.573175 + 0.573175i −2.28798 + 2.28798i 3.34294i −5.47154 + 5.47154i − 2.62283i 8.43976i −4.20879 − 4.20879i − 1.46972i − 6.27231i
14.16 −0.0624262 + 0.0624262i 1.58284 − 1.58284i 3.99221i 3.24675 − 3.24675i 0.197621i 10.7744i −0.498923 − 0.498923i 3.98923i 0.405365i
14.17 −0.00732944 + 0.00732944i 0.226886 − 0.226886i 3.99989i −2.04970 + 2.04970i 0.00332589i − 5.87781i −0.0586348 − 0.0586348i 8.89705i − 0.0300462i
14.18 0.351579 − 0.351579i −3.57720 + 3.57720i 3.75279i 0.963817 − 0.963817i 2.51533i 6.75956i 2.72571 + 2.72571i − 16.5927i − 0.677714i
14.19 0.667408 − 0.667408i 2.53367 − 2.53367i 3.10913i −6.79914 + 6.79914i − 3.38198i 5.78196i 4.74469 + 4.74469i − 3.83893i 9.07559i
14.20 0.711462 − 0.711462i −2.03514 + 2.03514i 2.98765i 3.89762 − 3.89762i 2.89584i 0.0611598i 4.97144 + 4.97144i 0.716450i − 5.54602i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.32
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.3.c.a ✓ 64
197.c odd 4 1 inner 197.3.c.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.3.c.a ✓ 64 1.a even 1 1 trivial
197.3.c.a ✓ 64 197.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(197, [\chi])\).