↑

Properties

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

Related objects

Downloads

Learn more

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: yes
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
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 1024 q^{4} + 11904 q^{9} + 4400 q^{11} - 704 q^{14} + 32768 q^{16} + 34384 q^{21} - 129648 q^{29} - 380928 q^{36} + 456336 q^{39} - 140800 q^{44} + 330816 q^{46} + 442384 q^{49} + 318512 q^{51} + 22528 q^{56} - 1048576 q^{64} + 832496 q^{71} + 1651072 q^{74} - 3169552 q^{79} + 5997616 q^{81} - 1100288 q^{84} - 160960 q^{86} - 4121376 q^{91} + 4000464 q^{99}+O(q^{100}) \) 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 −46.4127 −32.0000 0 262.550i −342.731 + 13.5908i 181.019i 1425.14 0
349.2 5.65685i −46.4127 −32.0000 0 − 262.550i −342.731 − 13.5908i − 181.019i 1425.14 0
349.3 − 5.65685i −21.5881 −32.0000 0 122.121i 59.6086 + 337.781i 181.019i −262.954 0
349.4 5.65685i −21.5881 −32.0000 0 − 122.121i 59.6086 − 337.781i − 181.019i −262.954 0
349.5 − 5.65685i −50.6007 −32.0000 0 286.241i −216.460 − 266.072i 181.019i 1831.43 0
349.6 5.65685i −50.6007 −32.0000 0 − 286.241i −216.460 + 266.072i − 181.019i 1831.43 0
349.7 − 5.65685i −1.01458 −32.0000 0 5.73933i −188.001 − 286.888i 181.019i −727.971 0
349.8 5.65685i −1.01458 −32.0000 0 − 5.73933i −188.001 + 286.888i − 181.019i −727.971 0
349.9 − 5.65685i 46.4127 −32.0000 0 − 262.550i 342.731 + 13.5908i 181.019i 1425.14 0
349.10 5.65685i 46.4127 −32.0000 0 262.550i 342.731 − 13.5908i − 181.019i 1425.14 0
349.11 − 5.65685i 1.01458 −32.0000 0 − 5.73933i 188.001 − 286.888i 181.019i −727.971 0
349.12 5.65685i 1.01458 −32.0000 0 5.73933i 188.001 + 286.888i − 181.019i −727.971 0
349.13 − 5.65685i 23.6280 −32.0000 0 − 133.660i 184.853 + 288.926i 181.019i −170.719 0
349.14 5.65685i 23.6280 −32.0000 0 133.660i 184.853 − 288.926i − 181.019i −170.719 0
349.15 − 5.65685i −23.6280 −32.0000 0 133.660i −184.853 + 288.926i 181.019i −170.719 0
349.16 5.65685i −23.6280 −32.0000 0 − 133.660i −184.853 − 288.926i − 181.019i −170.719 0
349.17 − 5.65685i −35.6764 −32.0000 0 201.816i 247.077 − 237.912i 181.019i 543.809 0
349.18 5.65685i −35.6764 −32.0000 0 − 201.816i 247.077 + 237.912i − 181.019i 543.809 0
349.19 − 5.65685i 4.00374 −32.0000 0 − 22.6486i 340.543 + 40.9842i 181.019i −712.970 0
349.20 5.65685i 4.00374 −32.0000 0 22.6486i 340.543 − 40.9842i − 181.019i −712.970 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.b 32
5.b even 2 1 inner 350.7.d.b 32
5.c odd 4 1 350.7.b.b ✓ 16
5.c odd 4 1 350.7.b.d yes 16
7.b odd 2 1 inner 350.7.d.b 32
35.c odd 2 1 inner 350.7.d.b 32
35.f even 4 1 350.7.b.b ✓ 16
35.f even 4 1 350.7.b.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.7.b.b ✓ 16 5.c odd 4 1
350.7.b.b ✓ 16 35.f even 4 1
350.7.b.d yes 16 5.c odd 4 1
350.7.b.d yes 16 35.f even 4 1
350.7.d.b 32 1.a even 1 1 trivial
350.7.d.b 32 5.b even 2 1 inner
350.7.d.b 32 7.b odd 2 1 inner
350.7.d.b 32 35.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 8808 T_{3}^{14} + 30534946 T_{3}^{12} - 52758144084 T_{3}^{10} + 47329974329769 T_{3}^{8} + \cdots + 53\!\cdots\!36 \) acting on \(S_{7}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display