Properties

Label 950.2.e.j
Level $950$
Weight $2$
Character orbit 950.e
Analytic conductor $7.586$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(201,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} - \beta_{3} q^{3} + \beta_1 q^{4} + ( - \beta_{3} + \beta_{2}) q^{6} + q^{7} - q^{8} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} - \beta_{3} q^{3} + \beta_1 q^{4} + ( - \beta_{3} + \beta_{2}) q^{6} + q^{7} - q^{8} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{9} + ( - \beta_{2} - 1) q^{11} + \beta_{2} q^{12} + ( - \beta_{3} + \beta_{2} + 3 \beta_1) q^{13} + (\beta_1 + 1) q^{14} + ( - \beta_1 - 1) q^{16} + ( - \beta_{3} + 2 \beta_1 + 2) q^{17} + ( - \beta_{2} - 2) q^{18} + (3 \beta_1 - 2) q^{19} - \beta_{3} q^{21} + ( - \beta_{3} - \beta_1 - 1) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{23} + \beta_{3} q^{24} + (\beta_{2} - 3) q^{26} + 5 q^{27} + \beta_1 q^{28} + (\beta_{3} - \beta_{2} + 4 \beta_1) q^{29} + ( - \beta_{2} - 2) q^{31} - \beta_1 q^{32} + (2 \beta_{3} + 5 \beta_1 + 5) q^{33} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{34} + ( - \beta_{3} - 2 \beta_1 - 2) q^{36} + ( - 2 \beta_{2} + 2) q^{37} + ( - 2 \beta_1 - 5) q^{38} + (2 \beta_{2} - 5) q^{39} + (3 \beta_{3} + 3 \beta_1 + 3) q^{41} + ( - \beta_{3} + \beta_{2}) q^{42} + (\beta_{3} + 3 \beta_1 + 3) q^{43} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{44} + (2 \beta_{2} - 1) q^{46} + (4 \beta_{3} - 4 \beta_{2} + \beta_1) q^{47} + (\beta_{3} - \beta_{2}) q^{48} - 6 q^{49} + ( - \beta_{3} + \beta_{2} + 5 \beta_1) q^{51} + (\beta_{3} - 3 \beta_1 - 3) q^{52} + (2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{53} + (5 \beta_1 + 5) q^{54} - q^{56} + (2 \beta_{3} + 3 \beta_{2}) q^{57} + ( - \beta_{2} - 4) q^{58} + ( - 2 \beta_{3} + \beta_1 + 1) q^{59} + (3 \beta_{3} - 3 \beta_{2} - 7 \beta_1) q^{61} + ( - \beta_{3} - 2 \beta_1 - 2) q^{62} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{63} + q^{64} + (2 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{66} + ( - 2 \beta_{3} + 2 \beta_{2} - 10 \beta_1) q^{67} + (\beta_{2} - 2) q^{68} + ( - \beta_{2} - 10) q^{69} + (2 \beta_{3} - \beta_1 - 1) q^{71} + ( - \beta_{3} + \beta_{2} - 2 \beta_1) q^{72} + \beta_{3} q^{73} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{74} + ( - 5 \beta_1 - 3) q^{76} + ( - \beta_{2} - 1) q^{77} + (2 \beta_{3} - 5 \beta_1 - 5) q^{78} + ( - 5 \beta_{3} - \beta_1 - 1) q^{79} + ( - 2 \beta_{3} + 6 \beta_1 + 6) q^{81} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{82} + ( - \beta_{2} - 1) q^{83} + \beta_{2} q^{84} + (\beta_{3} - \beta_{2} + 3 \beta_1) q^{86} + (5 \beta_{2} + 5) q^{87} + (\beta_{2} + 1) q^{88} + (2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{89} + ( - \beta_{3} + \beta_{2} + 3 \beta_1) q^{91} + (2 \beta_{3} - \beta_1 - 1) q^{92} + (3 \beta_{3} + 5 \beta_1 + 5) q^{93} + ( - 4 \beta_{2} - 1) q^{94} - \beta_{2} q^{96} + ( - 3 \beta_{3} - 4 \beta_1 - 4) q^{97} + ( - 6 \beta_1 - 6) q^{98} + ( - 4 \beta_{3} + 4 \beta_{2} - 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{6} + 4 q^{7} - 4 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{6} + 4 q^{7} - 4 q^{8} - 5 q^{9} - 6 q^{11} + 2 q^{12} - 5 q^{13} + 2 q^{14} - 2 q^{16} + 3 q^{17} - 10 q^{18} - 14 q^{19} - q^{21} - 3 q^{22} + q^{24} - 10 q^{26} + 20 q^{27} - 2 q^{28} - 9 q^{29} - 10 q^{31} + 2 q^{32} + 12 q^{33} - 3 q^{34} - 5 q^{36} + 4 q^{37} - 16 q^{38} - 16 q^{39} + 9 q^{41} + q^{42} + 7 q^{43} + 3 q^{44} - 6 q^{47} - q^{48} - 24 q^{49} - 9 q^{51} - 5 q^{52} + 10 q^{54} - 4 q^{56} + 8 q^{57} - 18 q^{58} + 11 q^{61} - 5 q^{62} - 5 q^{63} + 4 q^{64} - 12 q^{66} + 22 q^{67} - 6 q^{68} - 42 q^{69} + 5 q^{72} + q^{73} + 2 q^{74} - 2 q^{76} - 6 q^{77} - 8 q^{78} - 7 q^{79} + 10 q^{81} - 9 q^{82} - 6 q^{83} + 2 q^{84} - 7 q^{86} + 30 q^{87} + 6 q^{88} - 5 q^{91} + 13 q^{93} - 12 q^{94} - 2 q^{96} - 11 q^{97} - 12 q^{98} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 3\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} + \nu^{2} + \nu - 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 5\beta _1 + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} - \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
1.39564 0.228425i
−0.895644 + 1.09445i
1.39564 + 0.228425i
−0.895644 1.09445i
0.500000 0.866025i −1.39564 + 2.41733i −0.500000 0.866025i 0 1.39564 + 2.41733i 1.00000 −1.00000 −2.39564 4.14938i 0
201.2 0.500000 0.866025i 0.895644 1.55130i −0.500000 0.866025i 0 −0.895644 1.55130i 1.00000 −1.00000 −0.104356 0.180750i 0
501.1 0.500000 + 0.866025i −1.39564 2.41733i −0.500000 + 0.866025i 0 1.39564 2.41733i 1.00000 −1.00000 −2.39564 + 4.14938i 0
501.2 0.500000 + 0.866025i 0.895644 + 1.55130i −0.500000 + 0.866025i 0 −0.895644 + 1.55130i 1.00000 −1.00000 −0.104356 + 0.180750i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.e.j yes 4
5.b even 2 1 950.2.e.i 4
5.c odd 4 2 950.2.j.h 8
19.c even 3 1 inner 950.2.e.j yes 4
95.i even 6 1 950.2.e.i 4
95.m odd 12 2 950.2.j.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.i 4 5.b even 2 1
950.2.e.i 4 95.i even 6 1
950.2.e.j yes 4 1.a even 1 1 trivial
950.2.e.j yes 4 19.c even 3 1 inner
950.2.j.h 8 5.c odd 4 2
950.2.j.h 8 95.m odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{4} + T_{3}^{3} + 6T_{3}^{2} - 5T_{3} + 25 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + 6 T^{2} - 5 T + 25 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} + 24 T^{2} + 5 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} + 12 T^{2} + 9 T + 9 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$29$ \( T^{4} + 9 T^{3} + 66 T^{2} + 135 T + 225 \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T - 20)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 9 T^{3} + 108 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$43$ \( T^{4} - 7 T^{3} + 42 T^{2} - 49 T + 49 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + 111 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$53$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$59$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$61$ \( T^{4} - 11 T^{3} + 138 T^{2} + \cdots + 289 \) Copy content Toggle raw display
$67$ \( T^{4} - 22 T^{3} + 384 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$71$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$73$ \( T^{4} - T^{3} + 6 T^{2} + 5 T + 25 \) Copy content Toggle raw display
$79$ \( T^{4} + 7 T^{3} + 168 T^{2} + \cdots + 14161 \) Copy content Toggle raw display
$83$ \( (T^{2} + 3 T - 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$97$ \( T^{4} + 11 T^{3} + 138 T^{2} + \cdots + 289 \) Copy content Toggle raw display
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