Properties

Label 350.8.a.c
Level $350$
Weight $8$
Character orbit 350.a
Self dual yes
Analytic conductor $109.335$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,8,Mod(1,350)]
 
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([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

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: yes
Analytic conductor: \(109.334758919\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 8 q^{2} + 63 q^{3} + 64 q^{4} - 504 q^{6} - 343 q^{7} - 512 q^{8} + 1782 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 63 q^{3} + 64 q^{4} - 504 q^{6} - 343 q^{7} - 512 q^{8} + 1782 q^{9} - 2727 q^{11} + 4032 q^{12} - 5269 q^{13} + 2744 q^{14} + 4096 q^{16} + 17701 q^{17} - 14256 q^{18} + 712 q^{19} - 21609 q^{21} + 21816 q^{22} + 29330 q^{23} - 32256 q^{24} + 42152 q^{26} - 25515 q^{27} - 21952 q^{28} + 68491 q^{29} + 185026 q^{31} - 32768 q^{32} - 171801 q^{33} - 141608 q^{34} + 114048 q^{36} + 250046 q^{37} - 5696 q^{38} - 331947 q^{39} - 125814 q^{41} + 172872 q^{42} - 747476 q^{43} - 174528 q^{44} - 234640 q^{46} - 317317 q^{47} + 258048 q^{48} + 117649 q^{49} + 1115163 q^{51} - 337216 q^{52} - 1623246 q^{53} + 204120 q^{54} + 175616 q^{56} + 44856 q^{57} - 547928 q^{58} - 1519262 q^{59} - 3308640 q^{61} - 1480208 q^{62} - 611226 q^{63} + 262144 q^{64} + 1374408 q^{66} + 2272366 q^{67} + 1132864 q^{68} + 1847790 q^{69} - 4963104 q^{71} - 912384 q^{72} - 2351750 q^{73} - 2000368 q^{74} + 45568 q^{76} + 935361 q^{77} + 2655576 q^{78} - 2524249 q^{79} - 5504679 q^{81} + 1006512 q^{82} - 6051492 q^{83} - 1382976 q^{84} + 5979808 q^{86} + 4314933 q^{87} + 1396224 q^{88} + 8043880 q^{89} + 1807267 q^{91} + 1877120 q^{92} + 11656638 q^{93} + 2538536 q^{94} - 2064384 q^{96} + 2337645 q^{97} - 941192 q^{98} - 4859514 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−8.00000 63.0000 64.0000 0 −504.000 −343.000 −512.000 1782.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.8.a.c 1
5.b even 2 1 350.8.a.f 1
5.c odd 4 2 70.8.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.8.c.a 2 5.c odd 4 2
350.8.a.c 1 1.a even 1 1 trivial
350.8.a.f 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 63 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(350))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T - 63 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 343 \) Copy content Toggle raw display
$11$ \( T + 2727 \) Copy content Toggle raw display
$13$ \( T + 5269 \) Copy content Toggle raw display
$17$ \( T - 17701 \) Copy content Toggle raw display
$19$ \( T - 712 \) Copy content Toggle raw display
$23$ \( T - 29330 \) Copy content Toggle raw display
$29$ \( T - 68491 \) Copy content Toggle raw display
$31$ \( T - 185026 \) Copy content Toggle raw display
$37$ \( T - 250046 \) Copy content Toggle raw display
$41$ \( T + 125814 \) Copy content Toggle raw display
$43$ \( T + 747476 \) Copy content Toggle raw display
$47$ \( T + 317317 \) Copy content Toggle raw display
$53$ \( T + 1623246 \) Copy content Toggle raw display
$59$ \( T + 1519262 \) Copy content Toggle raw display
$61$ \( T + 3308640 \) Copy content Toggle raw display
$67$ \( T - 2272366 \) Copy content Toggle raw display
$71$ \( T + 4963104 \) Copy content Toggle raw display
$73$ \( T + 2351750 \) Copy content Toggle raw display
$79$ \( T + 2524249 \) Copy content Toggle raw display
$83$ \( T + 6051492 \) Copy content Toggle raw display
$89$ \( T - 8043880 \) Copy content Toggle raw display
$97$ \( T - 2337645 \) Copy content Toggle raw display
show more
show less