↑

Properties

Label 350.3.i.a
Level $350$
Weight $3$
Character orbit 350.i
Analytic conductor $9.537$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,3,Mod(199,350)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.199");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.i (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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{6} - 2 \beta_{5} + \cdots + \beta_1) q^{3}+ \cdots + 6 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{6} - 2 \beta_{5} + \cdots + \beta_1) q^{3}+ \cdots + (54 \beta_{7} - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 36 q^{11} + 72 q^{14} - 16 q^{16} - 12 q^{19} + 108 q^{21} - 48 q^{24} + 48 q^{26} - 96 q^{29} - 84 q^{31} - 24 q^{39} - 72 q^{44} + 72 q^{46} + 40 q^{49} + 108 q^{51} + 72 q^{54} + 96 q^{56} + 156 q^{59} - 84 q^{61} - 64 q^{64} - 288 q^{66} - 48 q^{71} - 192 q^{74} - 220 q^{79} + 36 q^{81} - 24 q^{84} + 48 q^{86} + 756 q^{89} + 48 q^{91} + 24 q^{94} - 96 q^{96} - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.258819 + 0.965926i
0.965926 − 0.258819i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
−1.22474 + 0.707107i −2.09077 + 3.62132i 1.00000 − 1.73205i 0 − 5.91359i −6.63103 − 2.24264i 2.82843i −4.24264 − 7.34847i 0
199.2 −1.22474 + 0.707107i −0.358719 + 0.621320i 1.00000 − 1.73205i 0 − 1.01461i −3.16693 − 6.24264i 2.82843i 4.24264 + 7.34847i 0
199.3 1.22474 − 0.707107i 0.358719 − 0.621320i 1.00000 − 1.73205i 0 − 1.01461i 3.16693 + 6.24264i − 2.82843i 4.24264 + 7.34847i 0
199.4 1.22474 − 0.707107i 2.09077 − 3.62132i 1.00000 − 1.73205i 0 − 5.91359i 6.63103 + 2.24264i − 2.82843i −4.24264 − 7.34847i 0
299.1 −1.22474 − 0.707107i −2.09077 − 3.62132i 1.00000 + 1.73205i 0 5.91359i −6.63103 + 2.24264i − 2.82843i −4.24264 + 7.34847i 0
299.2 −1.22474 − 0.707107i −0.358719 − 0.621320i 1.00000 + 1.73205i 0 1.01461i −3.16693 + 6.24264i − 2.82843i 4.24264 − 7.34847i 0
299.3 1.22474 + 0.707107i 0.358719 + 0.621320i 1.00000 + 1.73205i 0 1.01461i 3.16693 − 6.24264i 2.82843i 4.24264 − 7.34847i 0
299.4 1.22474 + 0.707107i 2.09077 + 3.62132i 1.00000 + 1.73205i 0 5.91359i 6.63103 − 2.24264i 2.82843i −4.24264 + 7.34847i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.i.a 8
5.b even 2 1 inner 350.3.i.a 8
5.c odd 4 1 14.3.d.a ✓ 4
5.c odd 4 1 350.3.k.a 4
7.d odd 6 1 inner 350.3.i.a 8
15.e even 4 1 126.3.n.c 4
20.e even 4 1 112.3.s.b 4
35.f even 4 1 98.3.d.a 4
35.i odd 6 1 inner 350.3.i.a 8
35.k even 12 1 14.3.d.a ✓ 4
35.k even 12 1 98.3.b.b 4
35.k even 12 1 350.3.k.a 4
35.l odd 12 1 98.3.b.b 4
35.l odd 12 1 98.3.d.a 4
40.i odd 4 1 448.3.s.d 4
40.k even 4 1 448.3.s.c 4
60.l odd 4 1 1008.3.cg.l 4
105.k odd 4 1 882.3.n.b 4
105.w odd 12 1 126.3.n.c 4
105.w odd 12 1 882.3.c.f 4
105.x even 12 1 882.3.c.f 4
105.x even 12 1 882.3.n.b 4
140.j odd 4 1 784.3.s.c 4
140.w even 12 1 784.3.c.e 4
140.w even 12 1 784.3.s.c 4
140.x odd 12 1 112.3.s.b 4
140.x odd 12 1 784.3.c.e 4
280.bp odd 12 1 448.3.s.c 4
280.bv even 12 1 448.3.s.d 4
420.br even 12 1 1008.3.cg.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a ✓ 4 5.c odd 4 1
14.3.d.a ✓ 4 35.k even 12 1
98.3.b.b 4 35.k even 12 1
98.3.b.b 4 35.l odd 12 1
98.3.d.a 4 35.f even 4 1
98.3.d.a 4 35.l odd 12 1
112.3.s.b 4 20.e even 4 1
112.3.s.b 4 140.x odd 12 1
126.3.n.c 4 15.e even 4 1
126.3.n.c 4 105.w odd 12 1
350.3.i.a 8 1.a even 1 1 trivial
350.3.i.a 8 5.b even 2 1 inner
350.3.i.a 8 7.d odd 6 1 inner
350.3.i.a 8 35.i odd 6 1 inner
350.3.k.a 4 5.c odd 4 1
350.3.k.a 4 35.k even 12 1
448.3.s.c 4 40.k even 4 1
448.3.s.c 4 280.bp odd 12 1
448.3.s.d 4 40.i odd 4 1
448.3.s.d 4 280.bv even 12 1
784.3.c.e 4 140.w even 12 1
784.3.c.e 4 140.x odd 12 1
784.3.s.c 4 140.j odd 4 1
784.3.s.c 4 140.w even 12 1
882.3.c.f 4 105.w odd 12 1
882.3.c.f 4 105.x even 12 1
882.3.n.b 4 105.k odd 4 1
882.3.n.b 4 105.x even 12 1
1008.3.cg.l 4 60.l odd 4 1
1008.3.cg.l 4 420.br even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 18T_{3}^{6} + 315T_{3}^{4} + 162T_{3}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 18 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 20 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} - 18 T^{3} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 264 T^{2} + 7056)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 198 T^{6} + \cdots + 6765201 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} + 9 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 774 T^{6} + \cdots + 15752961 \) Copy content Toggle raw display
$29$ \( (T^{2} + 24 T + 72)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 42 T^{3} + \cdots + 1447209)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 1330863361 \) Copy content Toggle raw display
$41$ \( (T^{4} + 1224 T^{2} + 345744)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 152 T^{2} + 4624)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 40135886713521 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 2311278643521 \) Copy content Toggle raw display
$59$ \( (T^{4} - 78 T^{3} + \cdots + 10517049)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 42 T^{3} + \cdots + 35964009)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 106042233977761 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T - 1764)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( (T^{4} + 110 T^{3} + \cdots + 6630625)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 27936 T^{2} + 189778176)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 378 T^{3} + \cdots + 71419401)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 11016 T^{2} + 6780816)^{2} \) Copy content Toggle raw display
show more
show less