↑

Properties

Label 354.4.c.a
Level $354$
Weight $4$
Character orbit 354.c
Analytic conductor $20.887$
Analytic rank $0$
Dimension $30$
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] = [354,4,Mod(353,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.353");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 354.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: \(20.8866761420\)
Analytic rank: \(0\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 60 q^{2} + 5 q^{3} + 120 q^{4} - 10 q^{6} + 6 q^{7} - 240 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 60 q^{2} + 5 q^{3} + 120 q^{4} - 10 q^{6} + 6 q^{7} - 240 q^{8} + 27 q^{9} - 60 q^{11} + 20 q^{12} - 12 q^{14} + 20 q^{15} + 480 q^{16} - 54 q^{18} + 90 q^{19} + 132 q^{21} + 120 q^{22} + 24 q^{23} - 40 q^{24} - 1080 q^{25} - 55 q^{27} + 24 q^{28} - 40 q^{30} - 960 q^{32} + 336 q^{33} + 108 q^{36} - 180 q^{38} + 652 q^{39} - 264 q^{42} - 240 q^{44} - 878 q^{45} - 48 q^{46} + 792 q^{47} + 80 q^{48} + 2016 q^{49} + 2160 q^{50} + 650 q^{51} + 110 q^{54} - 48 q^{56} + 846 q^{57} - 480 q^{59} + 80 q^{60} + 887 q^{63} + 1920 q^{64} - 1416 q^{65} - 672 q^{66} - 590 q^{69} - 216 q^{72} - 952 q^{75} + 360 q^{76} + 864 q^{77} - 1304 q^{78} + 738 q^{79} - 1217 q^{81} + 876 q^{83} + 528 q^{84} + 1176 q^{85} + 534 q^{87} + 480 q^{88} - 300 q^{89} + 1756 q^{90} + 96 q^{92} + 1684 q^{93} - 1584 q^{94} - 160 q^{96} - 4032 q^{98} + 730 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1 −2.00000 −5.11454 − 0.917350i 4.00000 − 12.8770i 10.2291 + 1.83470i −4.12434 −8.00000 25.3169 + 9.38364i 25.7541i
353.2 −2.00000 −5.11454 + 0.917350i 4.00000 12.8770i 10.2291 − 1.83470i −4.12434 −8.00000 25.3169 − 9.38364i − 25.7541i
353.3 −2.00000 −4.86384 − 1.82840i 4.00000 − 6.14441i 9.72768 + 3.65680i −15.5071 −8.00000 20.3139 + 17.7861i 12.2888i
353.4 −2.00000 −4.86384 + 1.82840i 4.00000 6.14441i 9.72768 − 3.65680i −15.5071 −8.00000 20.3139 − 17.7861i − 12.2888i
353.5 −2.00000 −4.66792 − 2.28265i 4.00000 − 8.40010i 9.33585 + 4.56530i 31.0296 −8.00000 16.5790 + 21.3105i 16.8002i
353.6 −2.00000 −4.66792 + 2.28265i 4.00000 8.40010i 9.33585 − 4.56530i 31.0296 −8.00000 16.5790 − 21.3105i − 16.8002i
353.7 −2.00000 −3.62921 − 3.71872i 4.00000 18.5626i 7.25841 + 7.43744i 9.26632 −8.00000 −0.657741 + 26.9920i − 37.1253i
353.8 −2.00000 −3.62921 + 3.71872i 4.00000 − 18.5626i 7.25841 − 7.43744i 9.26632 −8.00000 −0.657741 − 26.9920i 37.1253i
353.9 −2.00000 −3.04040 − 4.21378i 4.00000 12.5447i 6.08081 + 8.42756i −33.4405 −8.00000 −8.51188 + 25.6232i − 25.0894i
353.10 −2.00000 −3.04040 + 4.21378i 4.00000 − 12.5447i 6.08081 − 8.42756i −33.4405 −8.00000 −8.51188 − 25.6232i 25.0894i
353.11 −2.00000 −1.90975 − 4.83248i 4.00000 5.55280i 3.81949 + 9.66496i 16.8053 −8.00000 −19.7057 + 18.4576i − 11.1056i
353.12 −2.00000 −1.90975 + 4.83248i 4.00000 − 5.55280i 3.81949 − 9.66496i 16.8053 −8.00000 −19.7057 − 18.4576i 11.1056i
353.13 −2.00000 −1.46648 − 4.98492i 4.00000 − 18.9931i 2.93296 + 9.96984i −10.1890 −8.00000 −22.6989 + 14.6206i 37.9861i
353.14 −2.00000 −1.46648 + 4.98492i 4.00000 18.9931i 2.93296 − 9.96984i −10.1890 −8.00000 −22.6989 − 14.6206i − 37.9861i
353.15 −2.00000 1.05107 − 5.08874i 4.00000 6.49284i −2.10214 + 10.1775i −27.4926 −8.00000 −24.7905 − 10.6972i − 12.9857i
353.16 −2.00000 1.05107 + 5.08874i 4.00000 − 6.49284i −2.10214 − 10.1775i −27.4926 −8.00000 −24.7905 + 10.6972i 12.9857i
353.17 −2.00000 1.53016 − 4.96574i 4.00000 − 1.00990i −3.06032 + 9.93149i 8.17095 −8.00000 −22.3172 − 15.1967i 2.01980i
353.18 −2.00000 1.53016 + 4.96574i 4.00000 1.00990i −3.06032 − 9.93149i 8.17095 −8.00000 −22.3172 + 15.1967i − 2.01980i
353.19 −2.00000 2.99142 − 4.24869i 4.00000 19.3201i −5.98284 + 8.49739i 6.85976 −8.00000 −9.10280 − 25.4193i − 38.6403i
353.20 −2.00000 2.99142 + 4.24869i 4.00000 − 19.3201i −5.98284 − 8.49739i 6.85976 −8.00000 −9.10280 + 25.4193i 38.6403i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.30
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.4.c.a ✓ 30
3.b odd 2 1 354.4.c.b yes 30
59.b odd 2 1 354.4.c.b yes 30
177.d even 2 1 inner 354.4.c.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.4.c.a ✓ 30 1.a even 1 1 trivial
354.4.c.a ✓ 30 177.d even 2 1 inner
354.4.c.b yes 30 3.b odd 2 1
354.4.c.b yes 30 59.b odd 2 1