Properties

Label 350.2.j.a
Level $350$
Weight $2$
Character orbit 350.j
Analytic conductor $2.795$
Analytic rank $0$
Dimension $4$
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: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} - 2 q^{6} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) 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} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} - 3 \zeta_{12}^{2} q^{11} - 2 \zeta_{12} q^{12} + 5 \zeta_{12}^{3} q^{13} + (3 \zeta_{12}^{2} - 1) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{17} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{18} + (\zeta_{12}^{2} - 1) q^{19} + ( - 2 \zeta_{12}^{2} - 4) q^{21} - 3 \zeta_{12}^{3} q^{22} - 3 \zeta_{12} q^{23} - 2 \zeta_{12}^{2} q^{24} + (5 \zeta_{12}^{2} - 5) q^{26} - 4 \zeta_{12}^{3} q^{27} + (3 \zeta_{12}^{3} - \zeta_{12}) q^{28} + 6 q^{29} + 4 \zeta_{12}^{2} q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + 6 \zeta_{12} q^{33} - 6 q^{34} + q^{36} + 11 \zeta_{12} q^{37} + (\zeta_{12}^{3} - \zeta_{12}) q^{38} - 10 \zeta_{12}^{2} q^{39} + 3 q^{41} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{42} - 10 \zeta_{12}^{3} q^{43} + ( - 3 \zeta_{12}^{2} + 3) q^{44} - 3 \zeta_{12}^{2} q^{46} + 3 \zeta_{12} q^{47} - 2 \zeta_{12}^{3} q^{48} + (8 \zeta_{12}^{2} - 5) q^{49} + ( - 12 \zeta_{12}^{2} + 12) q^{51} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{52} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{53} + ( - 4 \zeta_{12}^{2} + 4) q^{54} + (2 \zeta_{12}^{2} - 3) q^{56} - 2 \zeta_{12}^{3} q^{57} + 6 \zeta_{12} q^{58} + ( - 4 \zeta_{12}^{2} + 4) q^{61} + 4 \zeta_{12}^{3} q^{62} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{63} - q^{64} + 6 \zeta_{12}^{2} q^{66} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{67} - 6 \zeta_{12} q^{68} + 6 q^{69} + 12 q^{71} + \zeta_{12} q^{72} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{73} + 11 \zeta_{12}^{2} q^{74} - q^{76} + ( - 9 \zeta_{12}^{3} + 3 \zeta_{12}) q^{77} - 10 \zeta_{12}^{3} q^{78} + (10 \zeta_{12}^{2} - 10) q^{79} + 11 \zeta_{12}^{2} q^{81} + 3 \zeta_{12} q^{82} - 12 \zeta_{12}^{3} q^{83} + ( - 6 \zeta_{12}^{2} + 2) q^{84} + ( - 10 \zeta_{12}^{2} + 10) q^{86} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{87} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{88} + ( - 6 \zeta_{12}^{2} + 6) q^{89} + (10 \zeta_{12}^{2} - 15) q^{91} - 3 \zeta_{12}^{3} q^{92} - 8 \zeta_{12} q^{93} + 3 \zeta_{12}^{2} q^{94} + ( - 2 \zeta_{12}^{2} + 2) q^{96} - 14 \zeta_{12}^{3} q^{97} + (8 \zeta_{12}^{3} - 5 \zeta_{12}) q^{98} - 3 q^{99} +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 \( 4 q + 2 q^{4} - 8 q^{6} + 2 q^{9} - 6 q^{11} + 2 q^{14} - 2 q^{16} - 2 q^{19} - 20 q^{21} - 4 q^{24} - 10 q^{26} + 24 q^{29} + 8 q^{31} - 24 q^{34} + 4 q^{36} - 20 q^{39} + 12 q^{41} + 6 q^{44} - 6 q^{46} - 4 q^{49} + 24 q^{51} + 8 q^{54} - 8 q^{56} + 8 q^{61} - 4 q^{64} + 12 q^{66} + 24 q^{69} + 48 q^{71} + 22 q^{74} - 4 q^{76} - 20 q^{79} + 22 q^{81} - 4 q^{84} + 20 q^{86} + 12 q^{89} - 40 q^{91} + 6 q^{94} + 4 q^{96} - 12 q^{99}+O(q^{100}) \) 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 −1.73205 2.00000i 1.00000i 0.500000 0.866025i 0
149.2 0.866025 + 0.500000i −1.73205 + 1.00000i 0.500000 + 0.866025i 0 −2.00000 1.73205 + 2.00000i 1.00000i 0.500000 0.866025i 0
249.1 −0.866025 + 0.500000i 1.73205 + 1.00000i 0.500000 0.866025i 0 −2.00000 −1.73205 + 2.00000i 1.00000i 0.500000 + 0.866025i 0
249.2 0.866025 0.500000i −1.73205 1.00000i 0.500000 0.866025i 0 −2.00000 1.73205 2.00000i 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.a 4
5.b even 2 1 inner 350.2.j.a 4
5.c odd 4 1 70.2.e.b 2
5.c odd 4 1 350.2.e.h 2
7.c even 3 1 inner 350.2.j.a 4
7.c even 3 1 2450.2.c.f 2
7.d odd 6 1 2450.2.c.p 2
15.e even 4 1 630.2.k.e 2
20.e even 4 1 560.2.q.d 2
35.f even 4 1 490.2.e.a 2
35.i odd 6 1 2450.2.c.p 2
35.j even 6 1 inner 350.2.j.a 4
35.j even 6 1 2450.2.c.f 2
35.k even 12 1 490.2.a.j 1
35.k even 12 1 490.2.e.a 2
35.k even 12 1 2450.2.a.f 1
35.l odd 12 1 70.2.e.b 2
35.l odd 12 1 350.2.e.h 2
35.l odd 12 1 490.2.a.g 1
35.l odd 12 1 2450.2.a.p 1
105.w odd 12 1 4410.2.a.c 1
105.x even 12 1 630.2.k.e 2
105.x even 12 1 4410.2.a.m 1
140.w even 12 1 560.2.q.d 2
140.w even 12 1 3920.2.a.be 1
140.x odd 12 1 3920.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 5.c odd 4 1
70.2.e.b 2 35.l odd 12 1
350.2.e.h 2 5.c odd 4 1
350.2.e.h 2 35.l odd 12 1
350.2.j.a 4 1.a even 1 1 trivial
350.2.j.a 4 5.b even 2 1 inner
350.2.j.a 4 7.c even 3 1 inner
350.2.j.a 4 35.j even 6 1 inner
490.2.a.g 1 35.l odd 12 1
490.2.a.j 1 35.k even 12 1
490.2.e.a 2 35.f even 4 1
490.2.e.a 2 35.k even 12 1
560.2.q.d 2 20.e even 4 1
560.2.q.d 2 140.w even 12 1
630.2.k.e 2 15.e even 4 1
630.2.k.e 2 105.x even 12 1
2450.2.a.f 1 35.k even 12 1
2450.2.a.p 1 35.l odd 12 1
2450.2.c.f 2 7.c even 3 1
2450.2.c.f 2 35.j even 6 1
2450.2.c.p 2 7.d odd 6 1
2450.2.c.p 2 35.i odd 6 1
3920.2.a.g 1 140.x odd 12 1
3920.2.a.be 1 140.w even 12 1
4410.2.a.c 1 105.w odd 12 1
4410.2.a.m 1 105.x even 12 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
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 25 \) 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
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 121 T^{2} + 14641 \) 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
$53$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T - 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
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