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Properties

Label 950.2.e.l
Level $950$
Weight $2$
Character orbit 950.e
Analytic conductor $7.586$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.39075800976.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 12x^{6} - 13x^{5} + 125x^{4} - 116x^{3} + 232x^{2} + 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_1 q^{3} + ( - \beta_{5} - 1) q^{4} + (\beta_{4} - \beta_1) q^{6} + (\beta_{7} + \beta_{6} + \cdots + \beta_{2}) q^{7}+ \cdots + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_1 q^{3} + ( - \beta_{5} - 1) q^{4} + (\beta_{4} - \beta_1) q^{6} + (\beta_{7} + \beta_{6} + \cdots + \beta_{2}) q^{7}+ \cdots + (\beta_{7} - \beta_{6} - 16 \beta_{5} + \cdots - 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + q^{3} - 4 q^{4} + q^{6} + 12 q^{7} + 8 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + q^{3} - 4 q^{4} + q^{6} + 12 q^{7} + 8 q^{8} - 11 q^{9} + 10 q^{11} - 2 q^{12} + 9 q^{13} - 6 q^{14} - 4 q^{16} + 5 q^{17} + 22 q^{18} - q^{21} - 5 q^{22} + 6 q^{23} + q^{24} - 18 q^{26} + 16 q^{27} - 6 q^{28} + 17 q^{29} + 22 q^{31} - 4 q^{32} - 4 q^{33} + 5 q^{34} - 11 q^{36} - 8 q^{37} - 36 q^{39} + 7 q^{41} - q^{42} - 13 q^{43} - 5 q^{44} - 12 q^{46} + 14 q^{47} + q^{48} + 44 q^{49} - 9 q^{51} + 9 q^{52} - 14 q^{53} - 8 q^{54} + 12 q^{56} - 48 q^{57} - 34 q^{58} + 14 q^{59} - 9 q^{61} - 11 q^{62} - 45 q^{63} + 8 q^{64} - 4 q^{66} + 6 q^{67} - 10 q^{68} + 54 q^{69} + 14 q^{71} - 11 q^{72} - 11 q^{73} + 4 q^{74} - 10 q^{77} + 18 q^{78} - 17 q^{79} - 36 q^{81} + 7 q^{82} - 46 q^{83} + 2 q^{84} - 13 q^{86} - 2 q^{87} + 10 q^{88} + 14 q^{89} - 25 q^{91} + 6 q^{92} + 13 q^{93} - 28 q^{94} - 2 q^{96} - 17 q^{97} - 22 q^{98} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 12x^{6} - 13x^{5} + 125x^{4} - 116x^{3} + 232x^{2} + 96x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} + 403\nu^{6} - 150\nu^{5} + 4817\nu^{4} - 2663\nu^{3} + 45286\nu^{2} - 24992\nu + 58672 ) / 15888 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -93\nu^{7} + 85\nu^{6} - 2301\nu^{5} + 2933\nu^{4} - 20057\nu^{3} + 31681\nu^{2} - 116300\nu + 82432 ) / 39720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -154\nu^{7} - 55\nu^{6} - 1568\nu^{5} - 126\nu^{4} - 19421\nu^{3} - 4592\nu^{2} - 2240\nu - 17824 ) / 39720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 557\nu^{7} - 865\nu^{6} + 6574\nu^{5} - 10377\nu^{4} + 69373\nu^{3} - 103454\nu^{2} + 120040\nu - 30448 ) / 79440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -456\nu^{7} + 310\nu^{6} - 5997\nu^{5} + 2636\nu^{4} - 61934\nu^{3} + 20857\nu^{2} - 121580\nu - 124136 ) / 39720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 944\nu^{7} - 845\nu^{6} + 10063\nu^{5} - 11264\nu^{4} + 104841\nu^{3} - 84843\nu^{2} + 76320\nu + 145024 ) / 39720 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - 5\beta_{5} - \beta_{2} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - \beta_{6} - 10\beta_{4} + \beta_{3} - 2\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} - 2\beta_{6} + 46\beta_{5} + 10\beta_{4} + 10\beta_{3} + 10\beta_{2} - 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{7} - 13\beta_{6} - 25\beta_{5} + 88\beta_{4} - 24\beta_{3} + 11\beta_{2} - 75\beta _1 - 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{7} + 123\beta_{6} - 154\beta_{4} - 123\beta_{3} + 22\beta_{2} + 409 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 92\beta_{7} + 246\beta_{6} + 366\beta_{5} + 154\beta_{4} + 154\beta_{3} + 154\beta_{2} + 725\beta _1 - 246 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5}\)

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
−1.63248 + 2.82754i
−0.236942 + 0.410396i
0.851703 − 1.47519i
1.51772 − 2.62877i
−1.63248 − 2.82754i
−0.236942 − 0.410396i
0.851703 + 1.47519i
1.51772 + 2.62877i
−0.500000 + 0.866025i −1.63248 + 2.82754i −0.500000 − 0.866025i 0 −1.63248 − 2.82754i 2.62013 1.00000 −3.82998 − 6.63372i 0
201.2 −0.500000 + 0.866025i −0.236942 + 0.410396i −0.500000 − 0.866025i 0 −0.236942 − 0.410396i 2.19155 1.00000 1.38772 + 2.40360i 0
201.3 −0.500000 + 0.866025i 0.851703 − 1.47519i −0.500000 − 0.866025i 0 0.851703 + 1.47519i −3.74324 1.00000 0.0492032 + 0.0852224i 0
201.4 −0.500000 + 0.866025i 1.51772 − 2.62877i −0.500000 − 0.866025i 0 1.51772 + 2.62877i 4.93155 1.00000 −3.10694 − 5.38138i 0
501.1 −0.500000 − 0.866025i −1.63248 − 2.82754i −0.500000 + 0.866025i 0 −1.63248 + 2.82754i 2.62013 1.00000 −3.82998 + 6.63372i 0
501.2 −0.500000 − 0.866025i −0.236942 − 0.410396i −0.500000 + 0.866025i 0 −0.236942 + 0.410396i 2.19155 1.00000 1.38772 − 2.40360i 0
501.3 −0.500000 − 0.866025i 0.851703 + 1.47519i −0.500000 + 0.866025i 0 0.851703 − 1.47519i −3.74324 1.00000 0.0492032 − 0.0852224i 0
501.4 −0.500000 − 0.866025i 1.51772 + 2.62877i −0.500000 + 0.866025i 0 1.51772 − 2.62877i 4.93155 1.00000 −3.10694 + 5.38138i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.4
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.l ✓ 8
5.b even 2 1 950.2.e.m yes 8
5.c odd 4 2 950.2.j.i 16
19.c even 3 1 inner 950.2.e.l ✓ 8
95.i even 6 1 950.2.e.m yes 8
95.m odd 12 2 950.2.j.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.l ✓ 8 1.a even 1 1 trivial
950.2.e.l ✓ 8 19.c even 3 1 inner
950.2.e.m yes 8 5.b even 2 1
950.2.e.m yes 8 95.i even 6 1
950.2.j.i 16 5.c odd 4 2
950.2.j.i 16 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 \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{8} - T_{3}^{7} + 12T_{3}^{6} - 13T_{3}^{5} + 125T_{3}^{4} - 116T_{3}^{3} + 232T_{3}^{2} + 96T_{3} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} - 7T_{7}^{2} + 82T_{7} - 106 \) Copy content Toggle raw display
\( T_{11}^{4} - 5T_{11}^{3} - 20T_{11}^{2} + 123T_{11} - 117 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{3} + \cdots - 106)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 5 T^{3} + \cdots - 117)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 9 T^{7} + \cdots + 64516 \) Copy content Toggle raw display
$17$ \( T^{8} - 5 T^{7} + \cdots + 50625 \) Copy content Toggle raw display
$19$ \( T^{8} + 35 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( (T^{4} - 3 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 17 T^{7} + \cdots + 202500 \) Copy content Toggle raw display
$31$ \( (T^{4} - 11 T^{3} + \cdots + 1118)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots + 3096)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 7 T^{7} + \cdots + 5948721 \) Copy content Toggle raw display
$43$ \( T^{8} + 13 T^{7} + \cdots + 29584 \) Copy content Toggle raw display
$47$ \( T^{8} - 14 T^{7} + \cdots + 54756 \) Copy content Toggle raw display
$53$ \( T^{8} + 14 T^{7} + \cdots + 54756 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} + \cdots + 363609 \) Copy content Toggle raw display
$61$ \( T^{8} + 9 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( T^{8} - 6 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + \cdots + 2862864 \) Copy content Toggle raw display
$73$ \( T^{8} + 11 T^{7} + \cdots + 6718464 \) Copy content Toggle raw display
$79$ \( T^{8} + 17 T^{7} + \cdots + 7022500 \) Copy content Toggle raw display
$83$ \( (T^{4} + 23 T^{3} + \cdots - 1872)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 14 T^{7} + \cdots + 8555625 \) Copy content Toggle raw display
$97$ \( T^{8} + 17 T^{7} + \cdots + 81 \) Copy content Toggle raw display
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