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Properties

Label 350.2.g.a
Level $350$
Weight $2$
Character orbit 350.g
Analytic conductor $2.795$
Analytic rank $0$
Dimension $8$
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(293,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(4))
 
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("350.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.g (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + \beta_{3} q^{4} + ( - \beta_{7} + \beta_{4}) q^{6} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + (\beta_{5} - \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + \beta_{3} q^{4} + ( - \beta_{7} + \beta_{4}) q^{6} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} - 8 q^{16} - 8 q^{18} + 16 q^{21} + 16 q^{22} + 8 q^{23} - 8 q^{28} + 8 q^{36} - 32 q^{37} - 32 q^{43} - 16 q^{46} - 32 q^{51} + 32 q^{53} - 8 q^{56} + 8 q^{57} + 16 q^{58} - 16 q^{67} - 16 q^{71} - 8 q^{72} + 16 q^{77} + 40 q^{78} + 24 q^{81} - 16 q^{88} + 16 q^{91} + 8 q^{92} + 16 q^{93} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{5} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{16}^{6} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display

Display \(\zeta_{16}^j\) in terms of \(\beta_i\)

\(\zeta_{16}\)\(=\) \( ( \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( \beta_{5} \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{16}^j\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.382683 + 0.923880i
−0.382683 − 0.923880i
−0.923880 + 0.382683i
0.923880 − 0.382683i
0.382683 − 0.923880i
−0.382683 + 0.923880i
−0.923880 − 0.382683i
0.923880 + 0.382683i
−0.707107 + 0.707107i −0.541196 + 0.541196i − 1.00000i 0 − 0.765367i −1.55487 − 2.14065i 0.707107 + 0.707107i 2.41421i 0
293.2 −0.707107 + 0.707107i 0.541196 − 0.541196i − 1.00000i 0 0.765367i 2.14065 + 1.55487i 0.707107 + 0.707107i 2.41421i 0
293.3 0.707107 − 0.707107i −1.30656 + 1.30656i − 1.00000i 0 1.84776i 0.941740 − 2.47247i −0.707107 − 0.707107i − 0.414214i 0
293.4 0.707107 − 0.707107i 1.30656 − 1.30656i − 1.00000i 0 − 1.84776i 2.47247 − 0.941740i −0.707107 − 0.707107i − 0.414214i 0
307.1 −0.707107 − 0.707107i −0.541196 − 0.541196i 1.00000i 0 0.765367i −1.55487 + 2.14065i 0.707107 − 0.707107i − 2.41421i 0
307.2 −0.707107 − 0.707107i 0.541196 + 0.541196i 1.00000i 0 − 0.765367i 2.14065 − 1.55487i 0.707107 − 0.707107i − 2.41421i 0
307.3 0.707107 + 0.707107i −1.30656 − 1.30656i 1.00000i 0 − 1.84776i 0.941740 + 2.47247i −0.707107 + 0.707107i 0.414214i 0
307.4 0.707107 + 0.707107i 1.30656 + 1.30656i 1.00000i 0 1.84776i 2.47247 + 0.941740i −0.707107 + 0.707107i 0.414214i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.g.a 8
5.b even 2 1 70.2.g.a ✓ 8
5.c odd 4 1 70.2.g.a ✓ 8
5.c odd 4 1 inner 350.2.g.a 8
7.b odd 2 1 inner 350.2.g.a 8
15.d odd 2 1 630.2.p.a 8
15.e even 4 1 630.2.p.a 8
20.d odd 2 1 560.2.bj.c 8
20.e even 4 1 560.2.bj.c 8
35.c odd 2 1 70.2.g.a ✓ 8
35.f even 4 1 70.2.g.a ✓ 8
35.f even 4 1 inner 350.2.g.a 8
35.i odd 6 2 490.2.l.a 16
35.j even 6 2 490.2.l.a 16
35.k even 12 2 490.2.l.a 16
35.l odd 12 2 490.2.l.a 16
105.g even 2 1 630.2.p.a 8
105.k odd 4 1 630.2.p.a 8
140.c even 2 1 560.2.bj.c 8
140.j odd 4 1 560.2.bj.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.g.a ✓ 8 5.b even 2 1
70.2.g.a ✓ 8 5.c odd 4 1
70.2.g.a ✓ 8 35.c odd 2 1
70.2.g.a ✓ 8 35.f even 4 1
350.2.g.a 8 1.a even 1 1 trivial
350.2.g.a 8 5.c odd 4 1 inner
350.2.g.a 8 7.b odd 2 1 inner
350.2.g.a 8 35.f even 4 1 inner
490.2.l.a 16 35.i odd 6 2
490.2.l.a 16 35.j even 6 2
490.2.l.a 16 35.k even 12 2
490.2.l.a 16 35.l odd 12 2
560.2.bj.c 8 20.d odd 2 1
560.2.bj.c 8 20.e even 4 1
560.2.bj.c 8 140.c even 2 1
560.2.bj.c 8 140.j odd 4 1
630.2.p.a 8 15.d odd 2 1
630.2.p.a 8 15.e even 4 1
630.2.p.a 8 105.g even 2 1
630.2.p.a 8 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 12T_{3}^{4} + 4 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 1548 T^{4} + 334084 \) Copy content Toggle raw display
$17$ \( T^{8} + 768 T^{4} + 16384 \) Copy content Toggle raw display
$19$ \( (T^{4} - 52 T^{2} + 98)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T + 32)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 192T^{4} + 1024 \) Copy content Toggle raw display
$53$ \( (T^{4} - 16 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 148 T^{2} + 4418)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T + 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 3264 T^{4} + 2458624 \) Copy content Toggle raw display
$79$ \( (T^{4} + 108 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 6732 T^{4} + 11303044 \) Copy content Toggle raw display
$89$ \( (T^{4} - 416 T^{2} + 36992)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 62208 T^{4} + 107495424 \) Copy content Toggle raw display
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