Properties

Label 4950.2.d.j
Level $4950$
Weight $2$
Character orbit 4950.d
Analytic conductor $39.526$
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] = [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: 8.0.1027968876544.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} - 16x^{5} - 27x^{4} - 16x^{3} + 214x^{2} - 488x + 908 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_{5} - \beta_{4}) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - \beta_{5} - \beta_{4}) q^{7} - q^{8} + ( - \beta_{6} + \beta_1) q^{11} + \beta_{7} q^{13} + (\beta_{5} + \beta_{4}) q^{14} + q^{16} - \beta_{3} q^{17} + ( - \beta_{7} - \beta_{6} + \beta_{4}) q^{19} + (\beta_{6} - \beta_1) q^{22} + ( - \beta_{7} + \beta_{6}) q^{23} - \beta_{7} q^{26} + ( - \beta_{5} - \beta_{4}) q^{28} + ( - 2 \beta_{2} + \beta_1 + 2) q^{29} + (\beta_{2} - 2 \beta_1 - 2) q^{31} - q^{32} + \beta_{3} q^{34} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{37} + (\beta_{7} + \beta_{6} - \beta_{4}) q^{38} + ( - \beta_{3} - 2 \beta_{2} - 2) q^{41} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4}) q^{43} + ( - \beta_{6} + \beta_1) q^{44} + (\beta_{7} - \beta_{6}) q^{46} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4}) q^{47} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{49} + \beta_{7} q^{52} + ( - \beta_{6} - \beta_{5} + \beta_{4}) q^{53} + (\beta_{5} + \beta_{4}) q^{56} + (2 \beta_{2} - \beta_1 - 2) q^{58} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5}) q^{59} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{61} + ( - \beta_{2} + 2 \beta_1 + 2) q^{62} + q^{64} + (3 \beta_{2} - \beta_1) q^{67} - \beta_{3} q^{68} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4}) q^{71} + ( - \beta_{6} + \beta_{5} + 3 \beta_{4}) q^{73} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{74} + ( - \beta_{7} - \beta_{6} + \beta_{4}) q^{76} + (2 \beta_{7} - 4 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{77} + ( - 4 \beta_{6} + \beta_{5} - \beta_{4}) q^{79} + (\beta_{3} + 2 \beta_{2} + 2) q^{82} + (\beta_{2} - 2 \beta_1) q^{83} + (\beta_{7} - 2 \beta_{6} + \beta_{4}) q^{86} + (\beta_{6} - \beta_1) q^{88} + ( - \beta_{7} - 3 \beta_{6} + \beta_{5} - 4 \beta_{4}) q^{89} + ( - 2 \beta_{3} + \beta_{2} - 5 \beta_1 - 2) q^{91} + ( - \beta_{7} + \beta_{6}) q^{92} + (2 \beta_{7} - \beta_{6} - \beta_{4}) q^{94} + (\beta_{3} - 2 \beta_{2} - 3 \beta_1 - 6) q^{97} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 3) q^{98}+O(q^{100}) \) 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 \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 4 q^{17} + 16 q^{29} - 16 q^{31} - 8 q^{32} + 4 q^{34} + 12 q^{37} - 20 q^{41} - 28 q^{49} - 16 q^{58} + 16 q^{62} + 8 q^{64} - 4 q^{68} - 12 q^{74} - 12 q^{77} + 20 q^{82} - 24 q^{91} - 44 q^{97} + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2x^{6} - 16x^{5} - 27x^{4} - 16x^{3} + 214x^{2} - 488x + 908 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 621 \nu^{7} - 5603 \nu^{6} - 6004 \nu^{5} - 1274 \nu^{4} + 100105 \nu^{3} + 170934 \nu^{2} - 67448 \nu - 1673544 ) / 831215 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6057 \nu^{7} - 9964 \nu^{6} - 37812 \nu^{5} + 102228 \nu^{4} + 394735 \nu^{3} + 658862 \nu^{2} + 307416 \nu + 79928 ) / 1662430 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 64\nu^{7} + 250\nu^{6} + 160\nu^{5} - 399\nu^{4} - 5648\nu^{3} - 15530\nu^{2} + 6872\nu + 14503 ) / 15113 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1079 \nu^{7} + 2798 \nu^{6} + 10254 \nu^{5} + 4844 \nu^{4} - 23435 \nu^{3} - 120614 \nu^{2} - 14492 \nu - 486816 ) / 151130 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2745 \nu^{7} + 4583 \nu^{6} + 14419 \nu^{5} - 9793 \nu^{4} - 87338 \nu^{3} - 134775 \nu^{2} + 334416 \nu - 1053375 ) / 166243 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3202 \nu^{7} + 7785 \nu^{6} + 23118 \nu^{5} + 6013 \nu^{4} - 93664 \nu^{3} - 235751 \nu^{2} + 30220 \nu - 1246171 ) / 166243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 46813 \nu^{7} + 106826 \nu^{6} + 321058 \nu^{5} - 73892 \nu^{4} - 1135095 \nu^{3} - 3579578 \nu^{2} + 2262756 \nu - 18262052 ) / 1662430 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} - \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{2} + 2\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{7} + 6\beta_{6} + 4\beta_{5} + 8\beta_{4} + 3\beta_{3} + 8\beta_{2} + 4\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} - 13\beta_{6} + 11\beta_{5} + 37\beta_{4} - 20\beta_{3} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60\beta_{7} + 24\beta_{6} - 56\beta_{5} - 140\beta_{4} - 2\beta_{3} + 46\beta_{2} - 69\beta _1 - 106 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -7\beta_{7} + 96\beta_{6} - 14\beta_{5} - 123\beta_{4} + 70\beta_{3} + 28\beta_{2} + 352\beta _1 + 798 \) 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
−0.625703 + 2.96730i
2.26020 + 0.0814034i
0.625703 + 1.55309i
−2.26020 1.33281i
−2.26020 + 1.33281i
0.625703 1.55309i
2.26020 0.0814034i
−0.625703 2.96730i
−1.00000 0 1.00000 0 0 5.04973i −1.00000 0 0
4751.2 −1.00000 0 1.00000 0 0 3.35921i −1.00000 0 0
4751.3 −1.00000 0 1.00000 0 0 2.22130i −1.00000 0 0
4751.4 −1.00000 0 1.00000 0 0 0.530782i −1.00000 0 0
4751.5 −1.00000 0 1.00000 0 0 0.530782i −1.00000 0 0
4751.6 −1.00000 0 1.00000 0 0 2.22130i −1.00000 0 0
4751.7 −1.00000 0 1.00000 0 0 3.35921i −1.00000 0 0
4751.8 −1.00000 0 1.00000 0 0 5.04973i −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
33.d even 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.j yes 8
3.b odd 2 1 4950.2.d.k yes 8
5.b even 2 1 4950.2.d.l yes 8
5.c odd 4 2 4950.2.f.g 16
11.b odd 2 1 4950.2.d.k yes 8
15.d odd 2 1 4950.2.d.i 8
15.e even 4 2 4950.2.f.h 16
33.d even 2 1 inner 4950.2.d.j yes 8
55.d odd 2 1 4950.2.d.i 8
55.e even 4 2 4950.2.f.h 16
165.d even 2 1 4950.2.d.l yes 8
165.l odd 4 2 4950.2.f.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4950.2.d.i 8 15.d odd 2 1
4950.2.d.i 8 55.d odd 2 1
4950.2.d.j yes 8 1.a even 1 1 trivial
4950.2.d.j yes 8 33.d even 2 1 inner
4950.2.d.k yes 8 3.b odd 2 1
4950.2.d.k yes 8 11.b odd 2 1
4950.2.d.l yes 8 5.b even 2 1
4950.2.d.l yes 8 165.d even 2 1
4950.2.f.g 16 5.c odd 4 2
4950.2.f.g 16 165.l odd 4 2
4950.2.f.h 16 15.e even 4 2
4950.2.f.h 16 55.e even 4 2

Hecke kernels

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

\( T_{7}^{8} + 42T_{7}^{6} + 481T_{7}^{4} + 1552T_{7}^{2} + 400 \) Copy content Toggle raw display
\( T_{17}^{2} + T_{17} - 22 \) Copy content Toggle raw display
\( T_{29}^{4} - 8T_{29}^{3} - 43T_{29}^{2} + 236T_{29} - 220 \) 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
$7$ \( T^{8} + 42 T^{6} + 481 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{8} - 114 T^{4} + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 56 T^{6} + 973 T^{4} + \cdots + 2500 \) Copy content Toggle raw display
$17$ \( (T^{2} + T - 22)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 98 T^{6} + 2490 T^{4} + \cdots + 7921 \) Copy content Toggle raw display
$23$ \( T^{8} + 62 T^{6} + 970 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 43 T^{2} + 236 T - 220)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} - 25 T^{2} - 164 T + 398)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} - 53 T^{2} + 8 T + 160)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 10 T^{3} - 75 T^{2} - 856 T - 1772)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 124 T^{6} + 697 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{8} + 214 T^{6} + 13357 T^{4} + \cdots + 1716100 \) Copy content Toggle raw display
$53$ \( T^{8} + 76 T^{6} + 1716 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$59$ \( T^{8} + 278 T^{6} + 20473 T^{4} + \cdots + 48400 \) Copy content Toggle raw display
$61$ \( T^{8} + 208 T^{6} + 10616 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$67$ \( (T^{4} - 146 T^{2} + 968)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 494 T^{6} + 81034 T^{4} + \cdots + 4950625 \) Copy content Toggle raw display
$73$ \( T^{8} + 116 T^{6} + 3284 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$79$ \( T^{8} + 346 T^{6} + \cdots + 32672656 \) Copy content Toggle raw display
$83$ \( (T^{4} - 49 T^{2} + 578)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 472 T^{6} + 54709 T^{4} + \cdots + 1562500 \) Copy content Toggle raw display
$97$ \( (T^{4} + 22 T^{3} - 66 T^{2} - 3570 T - 15043)^{2} \) Copy content Toggle raw display
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