Properties

Label 950.2.b.h
Level $950$
Weight $2$
Character orbit 950.b
Analytic conductor $7.586$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(799,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.b (of order \(2\), degree \(1\), 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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.63107136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 13x^{4} + 42x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{3} + \beta_1) q^{3} - q^{4} + ( - \beta_{2} + 1) q^{6} + ( - \beta_{5} + \beta_{3}) q^{7} - \beta_{3} q^{8} + (\beta_{4} + 2 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{3} + \beta_1) q^{3} - q^{4} + ( - \beta_{2} + 1) q^{6} + ( - \beta_{5} + \beta_{3}) q^{7} - \beta_{3} q^{8} + (\beta_{4} + 2 \beta_{2} - 2) q^{9} + 2 \beta_{2} q^{11} + (\beta_{3} - \beta_1) q^{12} + (\beta_{5} + 2 \beta_{3} - \beta_1) q^{13} + ( - \beta_{4} - 1) q^{14} + q^{16} + (\beta_{5} + 4 \beta_{3} - \beta_1) q^{17} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{18} - q^{19} + (\beta_{2} + 2) q^{21} + 2 \beta_1 q^{22} + ( - 2 \beta_{5} + \beta_{3} + \beta_1) q^{23} + (\beta_{2} - 1) q^{24} + (\beta_{4} + \beta_{2} - 2) q^{26} + (2 \beta_{5} + 6 \beta_{3} - 3 \beta_1) q^{27} + (\beta_{5} - \beta_{3}) q^{28} + ( - \beta_{4} - \beta_{2} - 2) q^{29} + (2 \beta_{2} + 2) q^{31} + \beta_{3} q^{32} + (2 \beta_{5} + 8 \beta_{3} - 2 \beta_1) q^{33} + (\beta_{4} + \beta_{2} - 4) q^{34} + ( - \beta_{4} - 2 \beta_{2} + 2) q^{36} + ( - \beta_{3} + 2 \beta_1) q^{37} - \beta_{3} q^{38} + ( - \beta_{4} - 5 \beta_{2} + 5) q^{39} + (2 \beta_{3} + \beta_1) q^{42} + (4 \beta_{3} + 2 \beta_1) q^{43} - 2 \beta_{2} q^{44} + ( - 2 \beta_{4} - \beta_{2} - 1) q^{46} + ( - 2 \beta_{5} + \beta_{3} + 2 \beta_1) q^{47} + ( - \beta_{3} + \beta_1) q^{48} + ( - 3 \beta_{4} - 3 \beta_{2} - 3) q^{49} + ( - \beta_{4} - 7 \beta_{2} + 7) q^{51} + ( - \beta_{5} - 2 \beta_{3} + \beta_1) q^{52} + ( - \beta_{5} + 5 \beta_{3} - 4 \beta_1) q^{53} + (2 \beta_{4} + 3 \beta_{2} - 6) q^{54} + (\beta_{4} + 1) q^{56} + (\beta_{3} - \beta_1) q^{57} + (\beta_{5} - 2 \beta_{3} - \beta_1) q^{58} + (4 \beta_{4} + \beta_{2} - 1) q^{59} + 2 \beta_{4} q^{61} + (2 \beta_{3} + 2 \beta_1) q^{62} + ( - 2 \beta_{5} + 5 \beta_{3} + \beta_1) q^{63} - q^{64} + (2 \beta_{4} + 2 \beta_{2} - 8) q^{66} + ( - \beta_{5} + \beta_{3} + 2 \beta_1) q^{67} + ( - \beta_{5} - 4 \beta_{3} + \beta_1) q^{68} + (\beta_{4} + 4 \beta_{2} - 1) q^{69} + ( - 2 \beta_{4} + 4 \beta_{2} - 4) q^{71} + (\beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{72} + ( - 2 \beta_{5} + 5 \beta_{3} - \beta_1) q^{73} + ( - 2 \beta_{2} + 1) q^{74} + q^{76} + ( - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{77} + (\beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{78} + (4 \beta_{2} - 8) q^{79} + ( - 7 \beta_{2} + 10) q^{81} + (2 \beta_{5} + 2 \beta_{3} - 4 \beta_1) q^{83} + ( - \beta_{2} - 2) q^{84} + ( - 2 \beta_{2} - 4) q^{86} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{87} - 2 \beta_1 q^{88} + (4 \beta_{2} - 6) q^{89} + (\beta_{4} + 2 \beta_{2} + 6) q^{91} + (2 \beta_{5} - \beta_{3} - \beta_1) q^{92} + (2 \beta_{5} + 6 \beta_{3}) q^{93} + ( - 2 \beta_{4} - 2 \beta_{2} - 1) q^{94} + ( - \beta_{2} + 1) q^{96} + (2 \beta_{5} - 8 \beta_{3} - 6 \beta_1) q^{97} + (3 \beta_{5} - 3 \beta_{3} - 3 \beta_1) q^{98} + ( - 2 \beta_{4} - 8 \beta_{2} + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 4 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 4 q^{6} - 10 q^{9} + 4 q^{11} - 4 q^{14} + 6 q^{16} - 6 q^{19} + 14 q^{21} - 4 q^{24} - 12 q^{26} - 12 q^{29} + 16 q^{31} - 24 q^{34} + 10 q^{36} + 22 q^{39} - 4 q^{44} - 4 q^{46} - 18 q^{49} + 30 q^{51} - 34 q^{54} + 4 q^{56} - 12 q^{59} - 4 q^{61} - 6 q^{64} - 48 q^{66} - 12 q^{71} + 2 q^{74} + 6 q^{76} - 40 q^{79} + 46 q^{81} - 14 q^{84} - 28 q^{86} - 28 q^{89} + 38 q^{91} - 6 q^{94} + 4 q^{96} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 13x^{4} + 42x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 7\nu^{2} + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + 21\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 19\nu^{3} - 75\nu ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{4} + 3\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{5} + 19\beta_{3} + 39\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
2.77339i
0.480031i
2.25342i
2.25342i
0.480031i
2.77339i
1.00000i 1.77339i −1.00000 0 −1.77339 2.69168i 1.00000i −0.144903 0
799.2 1.00000i 0.519969i −1.00000 0 0.519969 4.76957i 1.00000i 2.72963 0
799.3 1.00000i 3.25342i −1.00000 0 3.25342 0.0778929i 1.00000i −7.58473 0
799.4 1.00000i 3.25342i −1.00000 0 3.25342 0.0778929i 1.00000i −7.58473 0
799.5 1.00000i 0.519969i −1.00000 0 0.519969 4.76957i 1.00000i 2.72963 0
799.6 1.00000i 1.77339i −1.00000 0 −1.77339 2.69168i 1.00000i −0.144903 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.b.h 6
5.b even 2 1 inner 950.2.b.h 6
5.c odd 4 1 950.2.a.j 3
5.c odd 4 1 950.2.a.l yes 3
15.e even 4 1 8550.2.a.ci 3
15.e even 4 1 8550.2.a.cp 3
20.e even 4 1 7600.2.a.bk 3
20.e even 4 1 7600.2.a.bz 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 5.c odd 4 1
950.2.a.l yes 3 5.c odd 4 1
950.2.b.h 6 1.a even 1 1 trivial
950.2.b.h 6 5.b even 2 1 inner
7600.2.a.bk 3 20.e even 4 1
7600.2.a.bz 3 20.e even 4 1
8550.2.a.ci 3 15.e even 4 1
8550.2.a.cp 3 15.e even 4 1

Hecke kernels

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

\( T_{3}^{6} + 14T_{3}^{4} + 37T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 30T_{7}^{4} + 165T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 24T_{11} + 24 \) Copy content Toggle raw display
\( T_{13}^{6} + 42T_{13}^{4} + 429T_{13}^{2} + 1225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 14 T^{4} + 37 T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 30 T^{4} + 165 T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{3} - 2 T^{2} - 24 T + 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 42 T^{4} + 429 T^{2} + \cdots + 1225 \) Copy content Toggle raw display
$17$ \( T^{6} + 78 T^{4} + 1305 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 106 T^{4} + 2157 T^{2} + \cdots + 12321 \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 3 T - 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} - 4 T + 56)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 51 T^{4} + 627 T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 116 T^{4} + 2272 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$47$ \( T^{6} + 123 T^{4} + 2979 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$53$ \( T^{6} + 310 T^{4} + 28365 T^{2} + \cdots + 751689 \) Copy content Toggle raw display
$59$ \( (T^{3} + 6 T^{2} - 201 T - 1431)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 2 T^{2} - 56 T - 120)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 62 T^{4} + 1129 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$71$ \( (T^{3} + 6 T^{2} - 192 T - 1512)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 198 T^{4} + 8337 T^{2} + \cdots + 100489 \) Copy content Toggle raw display
$79$ \( (T^{3} + 20 T^{2} + 32 T - 320)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 232 T^{4} + 10320 T^{2} + \cdots + 28224 \) Copy content Toggle raw display
$89$ \( (T^{3} + 14 T^{2} - 36 T - 312)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 696 T^{4} + 138576 T^{2} + \cdots + 5953600 \) Copy content Toggle raw display
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