Properties

Label 4950.2.d.g
Level $4950$
Weight $2$
Character orbit 4950.d
Analytic conductor $39.526$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4950,2,Mod(4751,4950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4950.4751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4950.d (of order \(2\), degree \(1\), 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: \(39.5259490005\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.231260962816.24
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} - 4x^{5} + 41x^{4} - 122x^{3} + 293x^{2} - 244x + 236 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 990)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - \beta_{3} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - \beta_{3} q^{7} - q^{8} + ( - \beta_{3} - \beta_1) q^{11} + ( - \beta_{4} - \beta_1) q^{13} + \beta_{3} q^{14} + q^{16} - 2 q^{17} + (\beta_{7} - \beta_{6} - \beta_{2} + 1) q^{19} + (\beta_{3} + \beta_1) q^{22} - 2 \beta_{2} q^{23} + (\beta_{4} + \beta_1) q^{26} - \beta_{3} q^{28} + (\beta_{5} + \beta_1) q^{29} + ( - 2 \beta_{7} - 2 \beta_{6} - 2) q^{31} - q^{32} + 2 q^{34} + ( - \beta_{4} + \beta_1) q^{37} + ( - \beta_{7} + \beta_{6} + \beta_{2} - 1) q^{38} + ( - 2 \beta_{5} - \beta_{4} - \beta_1) q^{41} + ( - 3 \beta_{4} + \beta_{3} - 3 \beta_1) q^{43} + ( - \beta_{3} - \beta_1) q^{44} + 2 \beta_{2} q^{46} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} - 1) q^{47} + 5 q^{49} + ( - \beta_{4} - \beta_1) q^{52} + ( - \beta_{7} + \beta_{6} + \beta_{2} - 1) q^{53} + \beta_{3} q^{56} + ( - \beta_{5} - \beta_1) q^{58} + ( - \beta_{4} + 2 \beta_{3} - \beta_1) q^{59} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} - 1) q^{61} + (2 \beta_{7} + 2 \beta_{6} + 2) q^{62} + q^{64} + ( - 2 \beta_{4} + 2 \beta_1) q^{67} - 2 q^{68} + (2 \beta_{4} - 8 \beta_{3} + 2 \beta_1) q^{73} + (\beta_{4} - \beta_1) q^{74} + (\beta_{7} - \beta_{6} - \beta_{2} + 1) q^{76} + (\beta_{7} - 2) q^{77} + (\beta_{7} - \beta_{6} - \beta_{2} + 1) q^{79} + (2 \beta_{5} + \beta_{4} + \beta_1) q^{82} + (3 \beta_{7} + 3 \beta_{6} - 5) q^{83} + (3 \beta_{4} - \beta_{3} + 3 \beta_1) q^{86} + (\beta_{3} + \beta_1) q^{88} + (2 \beta_{4} + 3 \beta_{3} + 2 \beta_1) q^{89} + (\beta_{7} + \beta_{6} - 1) q^{91} - 2 \beta_{2} q^{92} + (\beta_{7} - \beta_{6} - 2 \beta_{2} + 1) q^{94} + ( - 2 \beta_{5} - 2 \beta_1) q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 16 q^{17} - 16 q^{31} - 8 q^{32} + 16 q^{34} + 40 q^{49} + 16 q^{62} + 8 q^{64} - 16 q^{68} - 20 q^{77} - 40 q^{83} - 8 q^{91} - 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + x^{6} - 4x^{5} + 41x^{4} - 122x^{3} + 293x^{2} - 244x + 236 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3828 \nu^{7} - 2122 \nu^{6} + 11870 \nu^{5} + 9627 \nu^{4} - 111066 \nu^{3} + 128270 \nu^{2} + 171251 \nu - 1190947 ) / 331315 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 363\nu^{7} - 463\nu^{6} + 70\nu^{5} - 2972\nu^{4} + 7211\nu^{3} - 27175\nu^{2} + 94354\nu - 51328 ) / 30820 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -233\nu^{7} + 633\nu^{6} - 50\nu^{5} - 348\nu^{4} - 11341\nu^{3} + 32225\nu^{2} - 59234\nu + 41368 ) / 19780 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5801 \nu^{7} + 2004 \nu^{6} - 7775 \nu^{5} - 29389 \nu^{4} + 182332 \nu^{3} - 257910 \nu^{2} + 609883 \nu + 869599 ) / 331315 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7686 \nu^{7} + 1971 \nu^{6} + 18640 \nu^{5} + 71779 \nu^{4} - 342442 \nu^{3} + 205615 \nu^{2} - 648543 \nu - 208599 ) / 331315 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -733\nu^{7} + 2633\nu^{6} - 1330\nu^{5} + 992\nu^{4} - 32221\nu^{3} + 146485\nu^{2} - 264054\nu + 291028 ) / 30820 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1437\nu^{7} - 2597\nu^{6} - 2270\nu^{5} - 4888\nu^{4} + 61149\nu^{3} - 153425\nu^{2} + 286406\nu - 131872 ) / 30820 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} + 3\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - \beta_{5} - 2\beta_{4} - 8\beta_{3} + 4\beta_{2} + \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} - 5\beta_{5} + 6\beta_{4} - 4\beta_{3} - 12\beta_{2} + 9\beta _1 + 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 19\beta_{6} + 19\beta_{5} + 14\beta_{4} - 48\beta_{3} + 5\beta _1 - 73 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -81\beta_{7} - 41\beta_{6} - 49\beta_{5} - 6\beta_{4} - 92\beta_{3} + 20\beta_{2} - 39\beta _1 + 111 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 39\beta_{7} - 29\beta_{6} + 67\beta_{5} + 294\beta_{4} + 184\beta_{3} - 180\beta_{2} + 157\beta _1 - 409 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -225\beta_{7} + 111\beta_{6} + 15\beta_{5} - 298\beta_{4} - 660\beta_{3} + 644\beta_{2} - 635\beta _1 - 1049 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2377\) \(4501\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4751.1
2.20793 1.10418i
−2.15569 + 1.81129i
0.594137 + 1.81129i
0.353624 1.10418i
2.20793 + 1.10418i
−2.15569 1.81129i
0.594137 1.81129i
0.353624 + 1.10418i
−1.00000 0 1.00000 0 0 1.41421i −1.00000 0 0
4751.2 −1.00000 0 1.00000 0 0 1.41421i −1.00000 0 0
4751.3 −1.00000 0 1.00000 0 0 1.41421i −1.00000 0 0
4751.4 −1.00000 0 1.00000 0 0 1.41421i −1.00000 0 0
4751.5 −1.00000 0 1.00000 0 0 1.41421i −1.00000 0 0
4751.6 −1.00000 0 1.00000 0 0 1.41421i −1.00000 0 0
4751.7 −1.00000 0 1.00000 0 0 1.41421i −1.00000 0 0
4751.8 −1.00000 0 1.00000 0 0 1.41421i −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4751.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4950.2.d.g 8
3.b odd 2 1 4950.2.d.n 8
5.b even 2 1 4950.2.d.n 8
5.c odd 4 2 990.2.f.c 16
11.b odd 2 1 4950.2.d.n 8
15.d odd 2 1 inner 4950.2.d.g 8
15.e even 4 2 990.2.f.c 16
33.d even 2 1 inner 4950.2.d.g 8
55.d odd 2 1 inner 4950.2.d.g 8
55.e even 4 2 990.2.f.c 16
165.d even 2 1 4950.2.d.n 8
165.l odd 4 2 990.2.f.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.f.c 16 5.c odd 4 2
990.2.f.c 16 15.e even 4 2
990.2.f.c 16 55.e even 4 2
990.2.f.c 16 165.l odd 4 2
4950.2.d.g 8 1.a even 1 1 trivial
4950.2.d.g 8 15.d odd 2 1 inner
4950.2.d.g 8 33.d even 2 1 inner
4950.2.d.g 8 55.d odd 2 1 inner
4950.2.d.n 8 3.b odd 2 1
4950.2.d.n 8 5.b even 2 1
4950.2.d.n 8 11.b odd 2 1
4950.2.d.n 8 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4950, [\chi])\):

\( T_{7}^{2} + 2 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{29}^{4} - 44T_{29}^{2} + 416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} - 2 T^{6} - 182 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 18 T^{2} + 64)^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 82 T^{2} + 1664)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 88 T^{2} + 1664)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 44 T^{2} + 416)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 46 T^{2} + 104)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 158 T^{2} + 104)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 154 T^{2} + 5776)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 116 T^{2} + 1664)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 82 T^{2} + 1664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 26 T^{2} + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 116 T^{2} + 1664)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 184 T^{2} + 1664)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 264 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 82 T^{2} + 1664)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T - 128)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 132 T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 176 T^{2} + 6656)^{2} \) Copy content Toggle raw display
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