Properties

Label 950.2.f.b
Level $950$
Weight $2$
Character orbit 950.f
Analytic conductor $7.586$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(493,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, 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("950.493");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.f (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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 17x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{5} q^{3} - \beta_{2} q^{4} + q^{6} + \beta_{3} q^{7} + \beta_{5} q^{8} - 2 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{5} q^{3} - \beta_{2} q^{4} + q^{6} + \beta_{3} q^{7} + \beta_{5} q^{8} - 2 \beta_{2} q^{9} + \beta_1 q^{12} + \beta_{5} q^{13} - \beta_{6} q^{14} - q^{16} + \beta_{3} q^{17} + 2 \beta_{5} q^{18} + ( - \beta_{6} + 2 \beta_{2}) q^{19} - \beta_{4} q^{21} + \beta_{7} q^{23} - \beta_{2} q^{24} - q^{26} + 5 \beta_1 q^{27} + \beta_{7} q^{28} + \beta_{6} q^{29} - \beta_1 q^{32} - \beta_{6} q^{34} - 2 q^{36} + 2 \beta_1 q^{37} + (\beta_{7} - 2 \beta_{5}) q^{38} - \beta_{2} q^{39} - 2 \beta_{4} q^{41} + \beta_{3} q^{42} + \beta_{4} q^{46} + 2 \beta_{3} q^{47} + \beta_{5} q^{48} + 8 \beta_{2} q^{49} - \beta_{4} q^{51} - \beta_1 q^{52} + 9 \beta_{5} q^{53} - 5 \beta_{2} q^{54} + \beta_{4} q^{56} + (\beta_{3} - 2 \beta_1) q^{57} - \beta_{7} q^{58} + 3 \beta_{6} q^{59} + 10 q^{61} + 2 \beta_{7} q^{63} + \beta_{2} q^{64} + 13 \beta_1 q^{67} + \beta_{7} q^{68} - \beta_{6} q^{69} + 2 \beta_{4} q^{71} - 2 \beta_1 q^{72} - \beta_{7} q^{73} - 2 \beta_{2} q^{74} + (\beta_{4} + 2) q^{76} + \beta_{5} q^{78} + 4 \beta_{6} q^{79} - q^{81} + 2 \beta_{3} q^{82} + 2 \beta_{7} q^{83} - \beta_{6} q^{84} - \beta_{3} q^{87} + 4 \beta_{6} q^{89} + \beta_{4} q^{91} - \beta_{3} q^{92} - 2 \beta_{6} q^{94} - q^{96} - 8 \beta_1 q^{97} - 8 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{6} - 8 q^{16} - 8 q^{26} - 16 q^{36} + 80 q^{61} + 16 q^{76} - 8 q^{81} - 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 17x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 5\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 33\nu^{2} ) / 112 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 61\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} + 17 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 13\nu^{3} ) / 448 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - \nu^{2} ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{7} - 251\nu^{3} ) / 448 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 11\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{4} - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{3} + 61\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -33\beta_{6} - 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 251\beta_{5} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
493.1
−1.01575 + 1.72286i
1.72286 1.01575i
−1.72286 + 1.01575i
1.01575 1.72286i
−1.01575 1.72286i
1.72286 + 1.01575i
−1.72286 1.01575i
1.01575 + 1.72286i
−0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 −2.73861 + 2.73861i 0.707107 0.707107i 2.00000i 0
493.2 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 2.73861 2.73861i 0.707107 0.707107i 2.00000i 0
493.3 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 −2.73861 + 2.73861i −0.707107 + 0.707107i 2.00000i 0
493.4 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 2.73861 2.73861i −0.707107 + 0.707107i 2.00000i 0
607.1 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 −2.73861 2.73861i 0.707107 + 0.707107i 2.00000i 0
607.2 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 2.73861 + 2.73861i 0.707107 + 0.707107i 2.00000i 0
607.3 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 −2.73861 2.73861i −0.707107 0.707107i 2.00000i 0
607.4 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 2.73861 + 2.73861i −0.707107 0.707107i 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 493.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
19.b odd 2 1 inner
95.d odd 2 1 inner
95.g even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.b 8
5.b even 2 1 inner 950.2.f.b 8
5.c odd 4 2 inner 950.2.f.b 8
19.b odd 2 1 inner 950.2.f.b 8
95.d odd 2 1 inner 950.2.f.b 8
95.g even 4 2 inner 950.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.f.b 8 1.a even 1 1 trivial
950.2.f.b 8 5.b even 2 1 inner
950.2.f.b 8 5.c odd 4 2 inner
950.2.f.b 8 19.b odd 2 1 inner
950.2.f.b 8 95.d odd 2 1 inner
950.2.f.b 8 95.g even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 225)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 22 T^{2} + 361)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 225)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 15)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 6561)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 135)^{4} \) Copy content Toggle raw display
$61$ \( (T - 10)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 28561)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 225)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 240)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 3600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 240)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
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