Properties

Label 350.6.a.n
Level $350$
Weight $6$
Character orbit 350.a
Self dual yes
Analytic conductor $56.134$
Analytic rank $0$
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] = [350,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(56.1343369345\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 23 q^{3} + 16 q^{4} + 92 q^{6} - 49 q^{7} + 64 q^{8} + 286 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 23 q^{3} + 16 q^{4} + 92 q^{6} - 49 q^{7} + 64 q^{8} + 286 q^{9} + 555 q^{11} + 368 q^{12} + 241 q^{13} - 196 q^{14} + 256 q^{16} + 1491 q^{17} + 1144 q^{18} - 2038 q^{19} - 1127 q^{21} + 2220 q^{22} + 1230 q^{23} + 1472 q^{24} + 964 q^{26} + 989 q^{27} - 784 q^{28} - 5001 q^{29} + 5696 q^{31} + 1024 q^{32} + 12765 q^{33} + 5964 q^{34} + 4576 q^{36} + 5602 q^{37} - 8152 q^{38} + 5543 q^{39} - 2424 q^{41} - 4508 q^{42} - 602 q^{43} + 8880 q^{44} + 4920 q^{46} + 23163 q^{47} + 5888 q^{48} + 2401 q^{49} + 34293 q^{51} + 3856 q^{52} + 25296 q^{53} + 3956 q^{54} - 3136 q^{56} - 46874 q^{57} - 20004 q^{58} + 5724 q^{59} - 36112 q^{61} + 22784 q^{62} - 14014 q^{63} + 4096 q^{64} + 51060 q^{66} - 66104 q^{67} + 23856 q^{68} + 28290 q^{69} + 16080 q^{71} + 18304 q^{72} + 80482 q^{73} + 22408 q^{74} - 32608 q^{76} - 27195 q^{77} + 22172 q^{78} - 64147 q^{79} - 46751 q^{81} - 9696 q^{82} + 106284 q^{83} - 18032 q^{84} - 2408 q^{86} - 115023 q^{87} + 35520 q^{88} - 71676 q^{89} - 11809 q^{91} + 19680 q^{92} + 131008 q^{93} + 92652 q^{94} + 23552 q^{96} - 151025 q^{97} + 9604 q^{98} + 158730 q^{99}+O(q^{100}) \) 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
4.00000 23.0000 16.0000 0 92.0000 −49.0000 64.0000 286.000 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 350.6.a.n 1
5.b even 2 1 70.6.a.a 1
5.c odd 4 2 350.6.c.h 2
15.d odd 2 1 630.6.a.j 1
20.d odd 2 1 560.6.a.i 1
35.c odd 2 1 490.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.a 1 5.b even 2 1
350.6.a.n 1 1.a even 1 1 trivial
350.6.c.h 2 5.c odd 4 2
490.6.a.i 1 35.c odd 2 1
560.6.a.i 1 20.d odd 2 1
630.6.a.j 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3} - 23 \) Copy content Toggle raw display
\( T_{13} - 241 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T - 23 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T - 555 \) Copy content Toggle raw display
$13$ \( T - 241 \) Copy content Toggle raw display
$17$ \( T - 1491 \) Copy content Toggle raw display
$19$ \( T + 2038 \) Copy content Toggle raw display
$23$ \( T - 1230 \) Copy content Toggle raw display
$29$ \( T + 5001 \) Copy content Toggle raw display
$31$ \( T - 5696 \) Copy content Toggle raw display
$37$ \( T - 5602 \) Copy content Toggle raw display
$41$ \( T + 2424 \) Copy content Toggle raw display
$43$ \( T + 602 \) Copy content Toggle raw display
$47$ \( T - 23163 \) Copy content Toggle raw display
$53$ \( T - 25296 \) Copy content Toggle raw display
$59$ \( T - 5724 \) Copy content Toggle raw display
$61$ \( T + 36112 \) Copy content Toggle raw display
$67$ \( T + 66104 \) Copy content Toggle raw display
$71$ \( T - 16080 \) Copy content Toggle raw display
$73$ \( T - 80482 \) Copy content Toggle raw display
$79$ \( T + 64147 \) Copy content Toggle raw display
$83$ \( T - 106284 \) Copy content Toggle raw display
$89$ \( T + 71676 \) Copy content Toggle raw display
$97$ \( T + 151025 \) Copy content Toggle raw display
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