Properties

Label 350.2.e.h
Level $350$
Weight $2$
Character orbit 350.e
Analytic conductor $2.795$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,2,Mod(51,350)]
 
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([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.e (of order \(3\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - 2 q^{6} + (2 \zeta_{6} + 1) q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - 2 q^{6} + (2 \zeta_{6} + 1) q^{7} - q^{8} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} - 2 \zeta_{6} q^{12} - 5 q^{13} + (3 \zeta_{6} - 2) q^{14} - \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + ( - \zeta_{6} + 1) q^{18} + \zeta_{6} q^{19} + (2 \zeta_{6} - 6) q^{21} - 3 q^{22} + 3 \zeta_{6} q^{23} + ( - 2 \zeta_{6} + 2) q^{24} - 5 \zeta_{6} q^{26} - 4 q^{27} + (\zeta_{6} - 3) q^{28} - 6 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + ( - \zeta_{6} + 1) q^{32} - 6 \zeta_{6} q^{33} + 6 q^{34} + q^{36} + 11 \zeta_{6} q^{37} + (\zeta_{6} - 1) q^{38} + ( - 10 \zeta_{6} + 10) q^{39} + 3 q^{41} + ( - 4 \zeta_{6} - 2) q^{42} + 10 q^{43} - 3 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{46} + 3 \zeta_{6} q^{47} + 2 q^{48} + (8 \zeta_{6} - 3) q^{49} + 12 \zeta_{6} q^{51} + ( - 5 \zeta_{6} + 5) q^{52} + ( - 3 \zeta_{6} + 3) q^{53} - 4 \zeta_{6} q^{54} + ( - 2 \zeta_{6} - 1) q^{56} - 2 q^{57} - 6 \zeta_{6} q^{58} + 4 \zeta_{6} q^{61} + 4 q^{62} + ( - 3 \zeta_{6} + 2) q^{63} + q^{64} + ( - 6 \zeta_{6} + 6) q^{66} + (4 \zeta_{6} - 4) q^{67} + 6 \zeta_{6} q^{68} - 6 q^{69} + 12 q^{71} + \zeta_{6} q^{72} + (4 \zeta_{6} - 4) q^{73} + (11 \zeta_{6} - 11) q^{74} - q^{76} + (3 \zeta_{6} - 9) q^{77} + 10 q^{78} + 10 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 3 \zeta_{6} q^{82} + 12 q^{83} + ( - 6 \zeta_{6} + 4) q^{84} + 10 \zeta_{6} q^{86} + ( - 12 \zeta_{6} + 12) q^{87} + ( - 3 \zeta_{6} + 3) q^{88} - 6 \zeta_{6} q^{89} + ( - 10 \zeta_{6} - 5) q^{91} - 3 q^{92} + 8 \zeta_{6} q^{93} + (3 \zeta_{6} - 3) q^{94} + 2 \zeta_{6} q^{96} - 14 q^{97} + (5 \zeta_{6} - 8) q^{98} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - 4 q^{6} + 4 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - 4 q^{6} + 4 q^{7} - 2 q^{8} - q^{9} - 3 q^{11} - 2 q^{12} - 10 q^{13} - q^{14} - q^{16} + 6 q^{17} + q^{18} + q^{19} - 10 q^{21} - 6 q^{22} + 3 q^{23} + 2 q^{24} - 5 q^{26} - 8 q^{27} - 5 q^{28} - 12 q^{29} + 4 q^{31} + q^{32} - 6 q^{33} + 12 q^{34} + 2 q^{36} + 11 q^{37} - q^{38} + 10 q^{39} + 6 q^{41} - 8 q^{42} + 20 q^{43} - 3 q^{44} - 3 q^{46} + 3 q^{47} + 4 q^{48} + 2 q^{49} + 12 q^{51} + 5 q^{52} + 3 q^{53} - 4 q^{54} - 4 q^{56} - 4 q^{57} - 6 q^{58} + 4 q^{61} + 8 q^{62} + q^{63} + 2 q^{64} + 6 q^{66} - 4 q^{67} + 6 q^{68} - 12 q^{69} + 24 q^{71} + q^{72} - 4 q^{73} - 11 q^{74} - 2 q^{76} - 15 q^{77} + 20 q^{78} + 10 q^{79} + 11 q^{81} + 3 q^{82} + 24 q^{83} + 2 q^{84} + 10 q^{86} + 12 q^{87} + 3 q^{88} - 6 q^{89} - 20 q^{91} - 6 q^{92} + 8 q^{93} - 3 q^{94} + 2 q^{96} - 28 q^{97} - 11 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i −1.00000 1.73205i −0.500000 0.866025i 0 −2.00000 2.00000 1.73205i −1.00000 −0.500000 + 0.866025i 0
151.1 0.500000 + 0.866025i −1.00000 + 1.73205i −0.500000 + 0.866025i 0 −2.00000 2.00000 + 1.73205i −1.00000 −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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.e.h 2
5.b even 2 1 70.2.e.b 2
5.c odd 4 2 350.2.j.a 4
7.c even 3 1 inner 350.2.e.h 2
7.c even 3 1 2450.2.a.p 1
7.d odd 6 1 2450.2.a.f 1
15.d odd 2 1 630.2.k.e 2
20.d odd 2 1 560.2.q.d 2
35.c odd 2 1 490.2.e.a 2
35.i odd 6 1 490.2.a.j 1
35.i odd 6 1 490.2.e.a 2
35.j even 6 1 70.2.e.b 2
35.j even 6 1 490.2.a.g 1
35.k even 12 2 2450.2.c.p 2
35.l odd 12 2 350.2.j.a 4
35.l odd 12 2 2450.2.c.f 2
105.o odd 6 1 630.2.k.e 2
105.o odd 6 1 4410.2.a.m 1
105.p even 6 1 4410.2.a.c 1
140.p odd 6 1 560.2.q.d 2
140.p odd 6 1 3920.2.a.be 1
140.s even 6 1 3920.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 5.b even 2 1
70.2.e.b 2 35.j even 6 1
350.2.e.h 2 1.a even 1 1 trivial
350.2.e.h 2 7.c even 3 1 inner
350.2.j.a 4 5.c odd 4 2
350.2.j.a 4 35.l odd 12 2
490.2.a.g 1 35.j even 6 1
490.2.a.j 1 35.i odd 6 1
490.2.e.a 2 35.c odd 2 1
490.2.e.a 2 35.i odd 6 1
560.2.q.d 2 20.d odd 2 1
560.2.q.d 2 140.p odd 6 1
630.2.k.e 2 15.d odd 2 1
630.2.k.e 2 105.o odd 6 1
2450.2.a.f 1 7.d odd 6 1
2450.2.a.p 1 7.c even 3 1
2450.2.c.f 2 35.l odd 12 2
2450.2.c.p 2 35.k even 12 2
3920.2.a.g 1 140.s even 6 1
3920.2.a.be 1 140.p odd 6 1
4410.2.a.c 1 105.p even 6 1
4410.2.a.m 1 105.o odd 6 1

Hecke kernels

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

\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13} + 5 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
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