↑

Properties

Label 462.4.c.a
Level $462$
Weight $4$
Character orbit 462.c
Analytic conductor $27.259$
Analytic rank $0$
Dimension $36$
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] = [462,4,Mod(197,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.197");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 462.c (of order \(2\), degree \(1\), 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: \(27.2588824227\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 72 q^{2} - 2 q^{3} + 144 q^{4} + 4 q^{6} - 288 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 72 q^{2} - 2 q^{3} + 144 q^{4} + 4 q^{6} - 288 q^{8} - 54 q^{9} + 36 q^{11} - 8 q^{12} - 26 q^{15} + 576 q^{16} - 252 q^{17} + 108 q^{18} - 72 q^{22} + 16 q^{24} - 840 q^{25} + 532 q^{27} - 216 q^{29} + 52 q^{30} - 48 q^{31} - 1152 q^{32} - 10 q^{33} + 504 q^{34} - 252 q^{35} - 216 q^{36} + 132 q^{37} + 60 q^{39} + 1116 q^{41} + 144 q^{44} - 558 q^{45} - 32 q^{48} - 1764 q^{49} + 1680 q^{50} + 208 q^{51} - 1064 q^{54} - 1224 q^{55} + 1700 q^{57} + 432 q^{58} - 104 q^{60} + 96 q^{62} + 140 q^{63} + 2304 q^{64} + 1200 q^{65} + 20 q^{66} - 636 q^{67} - 1008 q^{68} - 1094 q^{69} + 504 q^{70} + 432 q^{72} - 264 q^{74} + 2360 q^{75} + 84 q^{77} - 120 q^{78} - 1274 q^{81} - 2232 q^{82} - 3348 q^{83} - 3296 q^{87} - 288 q^{88} + 1116 q^{90} + 1008 q^{91} + 3930 q^{93} + 816 q^{95} + 64 q^{96} - 348 q^{97} + 3528 q^{98} - 4018 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −2.00000 −5.04478 − 1.24508i 4.00000 3.59753i 10.0896 + 2.49016i 7.00000i −8.00000 23.8995 + 12.5623i − 7.19506i
197.2 −2.00000 −5.04478 + 1.24508i 4.00000 − 3.59753i 10.0896 − 2.49016i − 7.00000i −8.00000 23.8995 − 12.5623i 7.19506i
197.3 −2.00000 −4.97602 − 1.49641i 4.00000 − 17.3267i 9.95204 + 2.99282i − 7.00000i −8.00000 22.5215 + 14.8923i 34.6534i
197.4 −2.00000 −4.97602 + 1.49641i 4.00000 17.3267i 9.95204 − 2.99282i 7.00000i −8.00000 22.5215 − 14.8923i − 34.6534i
197.5 −2.00000 −4.62593 − 2.36659i 4.00000 16.9069i 9.25186 + 4.73319i − 7.00000i −8.00000 15.7985 + 21.8954i − 33.8139i
197.6 −2.00000 −4.62593 + 2.36659i 4.00000 − 16.9069i 9.25186 − 4.73319i 7.00000i −8.00000 15.7985 − 21.8954i 33.8139i
197.7 −2.00000 −3.76335 − 3.58290i 4.00000 − 1.65729i 7.52671 + 7.16580i 7.00000i −8.00000 1.32565 + 26.9674i 3.31459i
197.8 −2.00000 −3.76335 + 3.58290i 4.00000 1.65729i 7.52671 − 7.16580i − 7.00000i −8.00000 1.32565 − 26.9674i − 3.31459i
197.9 −2.00000 −3.10473 − 4.16661i 4.00000 7.04512i 6.20947 + 8.33322i − 7.00000i −8.00000 −7.72126 + 25.8724i − 14.0902i
197.10 −2.00000 −3.10473 + 4.16661i 4.00000 − 7.04512i 6.20947 − 8.33322i 7.00000i −8.00000 −7.72126 − 25.8724i 14.0902i
197.11 −2.00000 −2.97423 − 4.26075i 4.00000 − 11.0756i 5.94845 + 8.52150i − 7.00000i −8.00000 −9.30797 + 25.3449i 22.1511i
197.12 −2.00000 −2.97423 + 4.26075i 4.00000 11.0756i 5.94845 − 8.52150i 7.00000i −8.00000 −9.30797 − 25.3449i − 22.1511i
197.13 −2.00000 −1.99170 − 4.79928i 4.00000 − 20.9899i 3.98341 + 9.59857i 7.00000i −8.00000 −19.0662 + 19.1175i 41.9797i
197.14 −2.00000 −1.99170 + 4.79928i 4.00000 20.9899i 3.98341 − 9.59857i − 7.00000i −8.00000 −19.0662 − 19.1175i − 41.9797i
197.15 −2.00000 −1.29651 − 5.03181i 4.00000 − 6.24924i 2.59302 + 10.0636i − 7.00000i −8.00000 −23.6381 + 13.0476i 12.4985i
197.16 −2.00000 −1.29651 + 5.03181i 4.00000 6.24924i 2.59302 − 10.0636i 7.00000i −8.00000 −23.6381 − 13.0476i − 12.4985i
197.17 −2.00000 −1.02171 − 5.09471i 4.00000 11.3919i 2.04342 + 10.1894i 7.00000i −8.00000 −24.9122 + 10.4107i − 22.7839i
197.18 −2.00000 −1.02171 + 5.09471i 4.00000 − 11.3919i 2.04342 − 10.1894i − 7.00000i −8.00000 −24.9122 − 10.4107i 22.7839i
197.19 −2.00000 −0.879414 − 5.12119i 4.00000 19.6175i 1.75883 + 10.2424i 7.00000i −8.00000 −25.4533 + 9.00730i − 39.2350i
197.20 −2.00000 −0.879414 + 5.12119i 4.00000 − 19.6175i 1.75883 − 10.2424i − 7.00000i −8.00000 −25.4533 − 9.00730i 39.2350i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.4.c.a ✓ 36
3.b odd 2 1 462.4.c.b yes 36
11.b odd 2 1 462.4.c.b yes 36
33.d even 2 1 inner 462.4.c.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.c.a ✓ 36 1.a even 1 1 trivial
462.4.c.a ✓ 36 33.d even 2 1 inner
462.4.c.b yes 36 3.b odd 2 1
462.4.c.b yes 36 11.b odd 2 1