Properties

Label 950.2.b.g
Level $950$
Weight $2$
Character orbit 950.b
Analytic conductor $7.586$
Analytic rank $0$
Dimension $6$
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(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: 6.0.4227136.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 22x^{2} + 9 \) 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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} - q^{4} + ( - \beta_{5} + \beta_{2} + 1) q^{6} + (\beta_{4} - 2 \beta_{3} - \beta_1) q^{7} - \beta_{4} q^{8} + ( - \beta_{5} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} - q^{4} + ( - \beta_{5} + \beta_{2} + 1) q^{6} + (\beta_{4} - 2 \beta_{3} - \beta_1) q^{7} - \beta_{4} q^{8} + ( - \beta_{5} - 4) q^{9} + (2 \beta_{5} + 2 \beta_{2}) q^{11} + (\beta_{4} + \beta_{3} - \beta_1) q^{12} + ( - \beta_{3} + 2 \beta_1) q^{13} + (\beta_{5} + 2 \beta_{2} - 1) q^{14} + q^{16} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_1) q^{17} + ( - 4 \beta_{4} - \beta_1) q^{18} + q^{19} + ( - 5 \beta_{5} - 3 \beta_{2} - 2) q^{21} + (2 \beta_{3} + 2 \beta_1) q^{22} + ( - 5 \beta_{4} + \beta_{3} + \beta_1) q^{23} + (\beta_{5} - \beta_{2} - 1) q^{24} + ( - 2 \beta_{5} + \beta_{2}) q^{26} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_1) q^{27} + ( - \beta_{4} + 2 \beta_{3} + \beta_1) q^{28} + ( - 2 \beta_{5} + \beta_{2} - 4) q^{29} + (2 \beta_{5} - 2 \beta_{2} - 2) q^{31} + \beta_{4} q^{32} + ( - 6 \beta_{3} - 4 \beta_1) q^{33} + ( - 2 \beta_{5} - \beta_{2} + 2) q^{34} + (\beta_{5} + 4) q^{36} + ( - 5 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{37} + \beta_{4} q^{38} + ( - 2 \beta_{5} + 3 \beta_{2} - 9) q^{39} + (4 \beta_{5} - 4) q^{41} + ( - 2 \beta_{4} - 3 \beta_{3} - 5 \beta_1) q^{42} + (2 \beta_{3} - 2 \beta_1) q^{43} + ( - 2 \beta_{5} - 2 \beta_{2}) q^{44} + ( - \beta_{5} - \beta_{2} + 5) q^{46} + (3 \beta_{4} + 2 \beta_{3} + 4 \beta_1) q^{47} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{48} + (2 \beta_{5} - 3 \beta_{2} - 9) q^{49} + (4 \beta_{5} + 3 \beta_{2} - 5) q^{51} + (\beta_{3} - 2 \beta_1) q^{52} + (3 \beta_{4} + 2 \beta_{3} + \beta_1) q^{53} + (\beta_{5} - 3 \beta_{2} + 2) q^{54} + ( - \beta_{5} - 2 \beta_{2} + 1) q^{56} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{57} + ( - 4 \beta_{4} + \beta_{3} - 2 \beta_1) q^{58} + (3 \beta_{5} + \beta_{2} + 1) q^{59} + ( - 2 \beta_{5} - 4 \beta_{2} + 8) q^{61} + ( - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{62} + ( - \beta_{4} + 9 \beta_{3} + \beta_1) q^{63} - q^{64} + (4 \beta_{5} + 6 \beta_{2}) q^{66} + ( - \beta_{4} - 4 \beta_{3} + 3 \beta_1) q^{67} + (2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{68} + (7 \beta_{5} - 2 \beta_{2} - 5) q^{69} - 2 \beta_{5} q^{71} + (4 \beta_{4} + \beta_1) q^{72} + (3 \beta_{4} + 3 \beta_{3} + 3 \beta_1) q^{73} + (2 \beta_{5} + 2 \beta_{2} + 5) q^{74} - q^{76} + ( - 18 \beta_{4} - 6 \beta_{3} + 4 \beta_1) q^{77} + ( - 9 \beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{78} + (4 \beta_{2} - 8) q^{79} + (5 \beta_{5} - \beta_{2} - 2) q^{81} + ( - 4 \beta_{4} + 4 \beta_1) q^{82} + ( - 2 \beta_{4} - 4 \beta_{3} - 6 \beta_1) q^{83} + (5 \beta_{5} + 3 \beta_{2} + 2) q^{84} + (2 \beta_{5} - 2 \beta_{2}) q^{86} + ( - 5 \beta_{4} + 7 \beta_{3} - 6 \beta_1) q^{87} + ( - 2 \beta_{3} - 2 \beta_1) q^{88} + 10 q^{89} - 7 \beta_{5} q^{91} + (5 \beta_{4} - \beta_{3} - \beta_1) q^{92} + (14 \beta_{4} + 2 \beta_1) q^{93} + ( - 4 \beta_{5} - 2 \beta_{2} - 3) q^{94} + ( - \beta_{5} + \beta_{2} + 1) q^{96} - 6 \beta_{3} q^{97} + ( - 9 \beta_{4} - 3 \beta_{3} + 2 \beta_1) q^{98} + ( - 6 \beta_{5} - 8 \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 4 q^{6} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 4 q^{6} - 26 q^{9} + 4 q^{11} - 4 q^{14} + 6 q^{16} + 6 q^{19} - 22 q^{21} - 4 q^{24} - 4 q^{26} - 28 q^{29} - 8 q^{31} + 8 q^{34} + 26 q^{36} - 58 q^{39} - 16 q^{41} - 4 q^{44} + 28 q^{46} - 50 q^{49} - 22 q^{51} + 14 q^{54} + 4 q^{56} + 12 q^{59} + 44 q^{61} - 6 q^{64} + 8 q^{66} - 16 q^{69} - 4 q^{71} + 34 q^{74} - 6 q^{76} - 48 q^{79} - 2 q^{81} + 22 q^{84} + 4 q^{86} + 60 q^{89} - 14 q^{91} - 26 q^{94} + 4 q^{96} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 22x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 6\nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 5\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{4} - 6\beta_{3} + 17\beta_1 \) 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
2.19869i
1.91223i
0.713538i
0.713538i
1.91223i
2.19869i
1.00000i 3.03293i −1.00000 0 −3.03293 2.46980i 1.00000i −6.19869 0
799.2 1.00000i 2.25561i −1.00000 0 2.25561 4.22547i 1.00000i −2.08777 0
799.3 1.00000i 2.77733i −1.00000 0 2.77733 4.69527i 1.00000i −4.71354 0
799.4 1.00000i 2.77733i −1.00000 0 2.77733 4.69527i 1.00000i −4.71354 0
799.5 1.00000i 2.25561i −1.00000 0 2.25561 4.22547i 1.00000i −2.08777 0
799.6 1.00000i 3.03293i −1.00000 0 −3.03293 2.46980i 1.00000i −6.19869 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.g 6
5.b even 2 1 inner 950.2.b.g 6
5.c odd 4 1 950.2.a.k 3
5.c odd 4 1 950.2.a.m yes 3
15.e even 4 1 8550.2.a.cj 3
15.e even 4 1 8550.2.a.co 3
20.e even 4 1 7600.2.a.bm 3
20.e even 4 1 7600.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.k 3 5.c odd 4 1
950.2.a.m yes 3 5.c odd 4 1
950.2.b.g 6 1.a even 1 1 trivial
950.2.b.g 6 5.b even 2 1 inner
7600.2.a.bm 3 20.e even 4 1
7600.2.a.cb 3 20.e even 4 1
8550.2.a.cj 3 15.e even 4 1
8550.2.a.co 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 \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{6} + 22T_{3}^{4} + 157T_{3}^{2} + 361 \) Copy content Toggle raw display
\( T_{7}^{6} + 46T_{7}^{4} + 637T_{7}^{2} + 2401 \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 32T_{11} + 24 \) Copy content Toggle raw display
\( T_{13}^{6} + 50T_{13}^{4} + 445T_{13}^{2} + 441 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 22 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 46 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{3} - 2 T^{2} - 32 T + 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 50 T^{4} + \cdots + 441 \) Copy content Toggle raw display
$17$ \( T^{6} + 46 T^{4} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 82 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$29$ \( (T^{3} + 14 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 4 T^{2} + \cdots - 152)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 163 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( (T^{3} + 8 T^{2} - 48 T - 64)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 84 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$47$ \( T^{6} + 219 T^{4} + \cdots + 275625 \) Copy content Toggle raw display
$53$ \( T^{6} + 78 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$59$ \( (T^{3} - 6 T^{2} + \cdots + 175)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 22 T^{2} + \cdots + 536)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 262 T^{4} + \cdots + 219961 \) Copy content Toggle raw display
$71$ \( (T^{3} + 2 T^{2} - 16 T - 24)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 198 T^{4} + \cdots + 59049 \) Copy content Toggle raw display
$79$ \( (T^{3} + 24 T^{2} + \cdots - 320)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 472 T^{4} + \cdots + 2383936 \) Copy content Toggle raw display
$89$ \( (T - 10)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 360 T^{4} + \cdots + 419904 \) Copy content Toggle raw display
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