Properties

Label 350.4.a.k
Level $350$
Weight $4$
Character orbit 350.a
Self dual yes
Analytic conductor $20.651$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(20.6506685020\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 10 q^{3} + 4 q^{4} - 20 q^{6} - 7 q^{7} + 8 q^{8} + 73 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 10 q^{3} + 4 q^{4} - 20 q^{6} - 7 q^{7} + 8 q^{8} + 73 q^{9} + 9 q^{11} - 40 q^{12} + 52 q^{13} - 14 q^{14} + 16 q^{16} - 96 q^{17} + 146 q^{18} - 10 q^{19} + 70 q^{21} + 18 q^{22} - 75 q^{23} - 80 q^{24} + 104 q^{26} - 460 q^{27} - 28 q^{28} + 189 q^{29} - 232 q^{31} + 32 q^{32} - 90 q^{33} - 192 q^{34} + 292 q^{36} - 305 q^{37} - 20 q^{38} - 520 q^{39} - 438 q^{41} + 140 q^{42} - 353 q^{43} + 36 q^{44} - 150 q^{46} + 486 q^{47} - 160 q^{48} + 49 q^{49} + 960 q^{51} + 208 q^{52} + 354 q^{53} - 920 q^{54} - 56 q^{56} + 100 q^{57} + 378 q^{58} - 672 q^{59} + 206 q^{61} - 464 q^{62} - 511 q^{63} + 64 q^{64} - 180 q^{66} - 599 q^{67} - 384 q^{68} + 750 q^{69} - 471 q^{71} + 584 q^{72} - 614 q^{73} - 610 q^{74} - 40 q^{76} - 63 q^{77} - 1040 q^{78} + 743 q^{79} + 2629 q^{81} - 876 q^{82} - 996 q^{83} + 280 q^{84} - 706 q^{86} - 1890 q^{87} + 72 q^{88} + 180 q^{89} - 364 q^{91} - 300 q^{92} + 2320 q^{93} + 972 q^{94} - 320 q^{96} + 184 q^{97} + 98 q^{98} + 657 q^{99}+O(q^{100}) \) 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
2.00000 −10.0000 4.00000 0 −20.0000 −7.00000 8.00000 73.0000 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 350.4.a.k yes 1
5.b even 2 1 350.4.a.j 1
5.c odd 4 2 350.4.c.a 2
7.b odd 2 1 2450.4.a.bp 1
35.c odd 2 1 2450.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.4.a.j 1 5.b even 2 1
350.4.a.k yes 1 1.a even 1 1 trivial
350.4.c.a 2 5.c odd 4 2
2450.4.a.a 1 35.c odd 2 1
2450.4.a.bp 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3} + 10 \) Copy content Toggle raw display
\( T_{11} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 10 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T - 9 \) Copy content Toggle raw display
$13$ \( T - 52 \) Copy content Toggle raw display
$17$ \( T + 96 \) Copy content Toggle raw display
$19$ \( T + 10 \) Copy content Toggle raw display
$23$ \( T + 75 \) Copy content Toggle raw display
$29$ \( T - 189 \) Copy content Toggle raw display
$31$ \( T + 232 \) Copy content Toggle raw display
$37$ \( T + 305 \) Copy content Toggle raw display
$41$ \( T + 438 \) Copy content Toggle raw display
$43$ \( T + 353 \) Copy content Toggle raw display
$47$ \( T - 486 \) Copy content Toggle raw display
$53$ \( T - 354 \) Copy content Toggle raw display
$59$ \( T + 672 \) Copy content Toggle raw display
$61$ \( T - 206 \) Copy content Toggle raw display
$67$ \( T + 599 \) Copy content Toggle raw display
$71$ \( T + 471 \) Copy content Toggle raw display
$73$ \( T + 614 \) Copy content Toggle raw display
$79$ \( T - 743 \) Copy content Toggle raw display
$83$ \( T + 996 \) Copy content Toggle raw display
$89$ \( T - 180 \) Copy content Toggle raw display
$97$ \( T - 184 \) Copy content Toggle raw display
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