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Properties

Label 350.7.d.c
Level $350$
Weight $7$
Character orbit 350.d
Analytic conductor $80.519$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,7,Mod(349,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.349");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 350.d (of order \(2\), degree \(1\), not 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: \(80.5189292669\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 1024 q^{4} + 11904 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 1024 q^{4} + 11904 q^{9} + 4400 q^{11} + 8416 q^{14} + 32768 q^{16} - 81416 q^{21} + 24912 q^{29} - 380928 q^{36} - 385344 q^{39} - 140800 q^{44} - 20544 q^{46} - 650696 q^{49} - 1567648 q^{51} - 269312 q^{56} - 1048576 q^{64} - 1097104 q^{71} - 4531328 q^{74} + 10673648 q^{79} + 7488496 q^{81} + 2605312 q^{84} + 6288320 q^{86} + 992784 q^{91} - 9330096 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1 − 5.65685i −49.5013 −32.0000 0 280.022i 330.321 + 92.3978i 181.019i 1721.38 0
349.2 5.65685i −49.5013 −32.0000 0 − 280.022i 330.321 − 92.3978i − 181.019i 1721.38 0
349.3 − 5.65685i 12.9369 −32.0000 0 − 73.1823i −211.214 + 270.255i 181.019i −561.636 0
349.4 5.65685i 12.9369 −32.0000 0 73.1823i −211.214 − 270.255i − 181.019i −561.636 0
349.5 − 5.65685i 49.5013 −32.0000 0 − 280.022i −330.321 + 92.3978i 181.019i 1721.38 0
349.6 5.65685i 49.5013 −32.0000 0 280.022i −330.321 − 92.3978i − 181.019i 1721.38 0
349.7 − 5.65685i 25.0969 −32.0000 0 − 141.969i −34.0693 + 341.304i 181.019i −99.1463 0
349.8 5.65685i 25.0969 −32.0000 0 141.969i −34.0693 − 341.304i − 181.019i −99.1463 0
349.9 − 5.65685i 5.30248 −32.0000 0 − 29.9954i 318.063 + 128.394i 181.019i −700.884 0
349.10 5.65685i 5.30248 −32.0000 0 29.9954i 318.063 − 128.394i − 181.019i −700.884 0
349.11 − 5.65685i −27.7162 −32.0000 0 156.787i 109.424 − 325.078i 181.019i 39.1898 0
349.12 5.65685i −27.7162 −32.0000 0 − 156.787i 109.424 + 325.078i − 181.019i 39.1898 0
349.13 − 5.65685i 42.0947 −32.0000 0 − 238.124i 33.5913 − 341.351i 181.019i 1042.97 0
349.14 5.65685i 42.0947 −32.0000 0 238.124i 33.5913 + 341.351i − 181.019i 1042.97 0
349.15 − 5.65685i 20.8810 −32.0000 0 − 118.121i −323.590 − 113.749i 181.019i −292.985 0
349.16 5.65685i 20.8810 −32.0000 0 118.121i −323.590 + 113.749i − 181.019i −292.985 0
349.17 − 5.65685i −20.8810 −32.0000 0 118.121i 323.590 − 113.749i 181.019i −292.985 0
349.18 5.65685i −20.8810 −32.0000 0 − 118.121i 323.590 + 113.749i − 181.019i −292.985 0
349.19 − 5.65685i 27.7162 −32.0000 0 − 156.787i −109.424 − 325.078i 181.019i 39.1898 0
349.20 5.65685i 27.7162 −32.0000 0 156.787i −109.424 + 325.078i − 181.019i 39.1898 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.7.d.c 32
5.b even 2 1 inner 350.7.d.c 32
5.c odd 4 1 70.7.b.a ✓ 16
5.c odd 4 1 350.7.b.c 16
7.b odd 2 1 inner 350.7.d.c 32
15.e even 4 1 630.7.f.a 16
20.e even 4 1 560.7.f.b 16
35.c odd 2 1 inner 350.7.d.c 32
35.f even 4 1 70.7.b.a ✓ 16
35.f even 4 1 350.7.b.c 16
105.k odd 4 1 630.7.f.a 16
140.j odd 4 1 560.7.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.7.b.a ✓ 16 5.c odd 4 1
70.7.b.a ✓ 16 35.f even 4 1
350.7.b.c 16 5.c odd 4 1
350.7.b.c 16 35.f even 4 1
350.7.d.c 32 1.a even 1 1 trivial
350.7.d.c 32 5.b even 2 1 inner
350.7.d.c 32 7.b odd 2 1 inner
350.7.d.c 32 35.c odd 2 1 inner
560.7.f.b 16 20.e even 4 1
560.7.f.b 16 140.j odd 4 1
630.7.f.a 16 15.e even 4 1
630.7.f.a 16 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 8808 T_{3}^{14} + 30348586 T_{3}^{12} - 52122411564 T_{3}^{10} + 47554824291849 T_{3}^{8} + \cdots + 11\!\cdots\!36 \) acting on \(S_{7}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display