Properties

Label 350.2.j.e
Level $350$
Weight $2$
Character orbit 350.j
Analytic conductor $2.795$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,2,Mod(149,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.j (of order \(6\), degree \(2\), 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: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + 2 q^{6} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + 2 q^{6} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} + 2 \zeta_{12} q^{12} - 2 \zeta_{12}^{3} q^{13} + (3 \zeta_{12}^{2} - 2) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{17} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{18} + ( - 8 \zeta_{12}^{2} + 8) q^{19} + (4 \zeta_{12}^{2} + 2) q^{21} - 9 \zeta_{12} q^{23} + 2 \zeta_{12}^{2} q^{24} + ( - 2 \zeta_{12}^{2} + 2) q^{26} + 4 \zeta_{12}^{3} q^{27} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{28} + 6 q^{29} - 5 \zeta_{12}^{2} q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} - 3 q^{34} + q^{36} - 8 \zeta_{12} q^{37} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{38} - 4 \zeta_{12}^{2} q^{39} - 3 q^{41} + (4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{42} + 10 \zeta_{12}^{3} q^{43} - 9 \zeta_{12}^{2} q^{46} + 3 \zeta_{12} q^{47} + 2 \zeta_{12}^{3} q^{48} + (5 \zeta_{12}^{2} - 8) q^{49} + (6 \zeta_{12}^{2} - 6) q^{51} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{52} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{53} + (4 \zeta_{12}^{2} - 4) q^{54} + (\zeta_{12}^{2} - 3) q^{56} - 16 \zeta_{12}^{3} q^{57} + 6 \zeta_{12} q^{58} + 12 \zeta_{12}^{2} q^{59} + ( - 4 \zeta_{12}^{2} + 4) q^{61} - 5 \zeta_{12}^{3} q^{62} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{63} - q^{64} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{67} - 3 \zeta_{12} q^{68} - 18 q^{69} - 9 q^{71} + \zeta_{12} q^{72} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{73} - 8 \zeta_{12}^{2} q^{74} + 8 q^{76} - 4 \zeta_{12}^{3} q^{78} + ( - 5 \zeta_{12}^{2} + 5) q^{79} + 11 \zeta_{12}^{2} q^{81} - 3 \zeta_{12} q^{82} + 6 \zeta_{12}^{3} q^{83} + (6 \zeta_{12}^{2} - 4) q^{84} + (10 \zeta_{12}^{2} - 10) q^{86} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{87} + ( - 3 \zeta_{12}^{2} + 3) q^{89} + ( - 2 \zeta_{12}^{2} + 6) q^{91} - 9 \zeta_{12}^{3} q^{92} - 10 \zeta_{12} q^{93} + 3 \zeta_{12}^{2} q^{94} + (2 \zeta_{12}^{2} - 2) q^{96} + 5 \zeta_{12}^{3} q^{97} + (5 \zeta_{12}^{3} - 8 \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 8 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 8 q^{6} + 2 q^{9} - 2 q^{14} - 2 q^{16} + 16 q^{19} + 16 q^{21} + 4 q^{24} + 4 q^{26} + 24 q^{29} - 10 q^{31} - 12 q^{34} + 4 q^{36} - 8 q^{39} - 12 q^{41} - 18 q^{46} - 22 q^{49} - 12 q^{51} - 8 q^{54} - 10 q^{56} + 24 q^{59} + 8 q^{61} - 4 q^{64} - 72 q^{69} - 36 q^{71} - 16 q^{74} + 32 q^{76} + 10 q^{79} + 22 q^{81} - 4 q^{84} - 20 q^{86} + 6 q^{89} + 20 q^{91} + 6 q^{94} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −1.73205 + 1.00000i 0.500000 + 0.866025i 0 2.00000 −0.866025 2.50000i 1.00000i 0.500000 0.866025i 0
149.2 0.866025 + 0.500000i 1.73205 1.00000i 0.500000 + 0.866025i 0 2.00000 0.866025 + 2.50000i 1.00000i 0.500000 0.866025i 0
249.1 −0.866025 + 0.500000i −1.73205 1.00000i 0.500000 0.866025i 0 2.00000 −0.866025 + 2.50000i 1.00000i 0.500000 + 0.866025i 0
249.2 0.866025 0.500000i 1.73205 + 1.00000i 0.500000 0.866025i 0 2.00000 0.866025 2.50000i 1.00000i 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.j.e 4
5.b even 2 1 inner 350.2.j.e 4
5.c odd 4 1 350.2.e.a 2
5.c odd 4 1 350.2.e.k yes 2
7.c even 3 1 inner 350.2.j.e 4
7.c even 3 1 2450.2.c.o 2
7.d odd 6 1 2450.2.c.d 2
35.i odd 6 1 2450.2.c.d 2
35.j even 6 1 inner 350.2.j.e 4
35.j even 6 1 2450.2.c.o 2
35.k even 12 1 2450.2.a.o 1
35.k even 12 1 2450.2.a.u 1
35.l odd 12 1 350.2.e.a 2
35.l odd 12 1 350.2.e.k yes 2
35.l odd 12 1 2450.2.a.e 1
35.l odd 12 1 2450.2.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.e.a 2 5.c odd 4 1
350.2.e.a 2 35.l odd 12 1
350.2.e.k yes 2 5.c odd 4 1
350.2.e.k yes 2 35.l odd 12 1
350.2.j.e 4 1.a even 1 1 trivial
350.2.j.e 4 5.b even 2 1 inner
350.2.j.e 4 7.c even 3 1 inner
350.2.j.e 4 35.j even 6 1 inner
2450.2.a.e 1 35.l odd 12 1
2450.2.a.o 1 35.k even 12 1
2450.2.a.u 1 35.k even 12 1
2450.2.a.be 1 35.l odd 12 1
2450.2.c.d 2 7.d odd 6 1
2450.2.c.d 2 35.i odd 6 1
2450.2.c.o 2 7.c even 3 1
2450.2.c.o 2 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} - 4T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$41$ \( (T + 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 9)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
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