↑

Properties

Label 354.7.d.a
Level $354$
Weight $7$
Character orbit 354.d
Analytic conductor $81.439$
Analytic rank $0$
Dimension $60$
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] = [354,7,Mod(235,354)]
 
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([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.235");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 354.d (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: \(81.4391456014\)
Analytic rank: \(0\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 1920 q^{4} + 408 q^{7} + 14580 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 1920 q^{4} + 408 q^{7} + 14580 q^{9} + 4536 q^{15} + 61440 q^{16} - 15840 q^{17} - 5616 q^{19} - 17472 q^{22} + 226260 q^{25} - 34048 q^{26} - 13056 q^{28} - 75392 q^{29} + 278000 q^{35} - 466560 q^{36} + 67376 q^{41} + 209856 q^{46} + 269100 q^{49} - 206064 q^{51} + 490000 q^{53} - 373248 q^{57} - 863472 q^{59} - 145152 q^{60} - 155072 q^{62} + 99144 q^{63} - 1966080 q^{64} - 404352 q^{66} + 506880 q^{68} - 2041856 q^{71} - 2146176 q^{74} + 808704 q^{75} + 179712 q^{76} + 228096 q^{78} + 670248 q^{79} + 3542940 q^{81} + 873408 q^{85} + 1832576 q^{86} - 2568024 q^{87} + 559104 q^{88} + 1049472 q^{94} - 245856 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1 − 5.65685i −15.5885 −32.0000 −92.5158 88.1816i 505.471 181.019i 243.000 523.348i
235.2 5.65685i −15.5885 −32.0000 −92.5158 − 88.1816i 505.471 − 181.019i 243.000 − 523.348i
235.3 − 5.65685i 15.5885 −32.0000 117.083 − 88.1816i 626.230 181.019i 243.000 − 662.324i
235.4 5.65685i 15.5885 −32.0000 117.083 88.1816i 626.230 − 181.019i 243.000 662.324i
235.5 − 5.65685i 15.5885 −32.0000 −229.516 − 88.1816i −374.873 181.019i 243.000 1298.34i
235.6 5.65685i 15.5885 −32.0000 −229.516 88.1816i −374.873 − 181.019i 243.000 − 1298.34i
235.7 − 5.65685i −15.5885 −32.0000 133.407 88.1816i −65.3612 181.019i 243.000 − 754.665i
235.8 5.65685i −15.5885 −32.0000 133.407 − 88.1816i −65.3612 − 181.019i 243.000 754.665i
235.9 − 5.65685i −15.5885 −32.0000 204.428 88.1816i 398.882 181.019i 243.000 − 1156.42i
235.10 5.65685i −15.5885 −32.0000 204.428 − 88.1816i 398.882 − 181.019i 243.000 1156.42i
235.11 − 5.65685i −15.5885 −32.0000 −30.8626 88.1816i −9.02297 181.019i 243.000 174.585i
235.12 5.65685i −15.5885 −32.0000 −30.8626 − 88.1816i −9.02297 − 181.019i 243.000 − 174.585i
235.13 − 5.65685i −15.5885 −32.0000 151.165 88.1816i −215.009 181.019i 243.000 − 855.120i
235.14 5.65685i −15.5885 −32.0000 151.165 − 88.1816i −215.009 − 181.019i 243.000 855.120i
235.15 − 5.65685i −15.5885 −32.0000 65.2792 88.1816i −409.923 181.019i 243.000 − 369.275i
235.16 5.65685i −15.5885 −32.0000 65.2792 − 88.1816i −409.923 − 181.019i 243.000 369.275i
235.17 − 5.65685i 15.5885 −32.0000 −16.5609 − 88.1816i 303.476 181.019i 243.000 93.6824i
235.18 5.65685i 15.5885 −32.0000 −16.5609 88.1816i 303.476 − 181.019i 243.000 − 93.6824i
235.19 − 5.65685i −15.5885 −32.0000 −96.2981 88.1816i −484.887 181.019i 243.000 544.744i
235.20 5.65685i −15.5885 −32.0000 −96.2981 − 88.1816i −484.887 − 181.019i 243.000 − 544.744i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.7.d.a ✓ 60
59.b odd 2 1 inner 354.7.d.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.7.d.a ✓ 60 1.a even 1 1 trivial
354.7.d.a ✓ 60 59.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(354, [\chi])\).