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Properties

Label 1150.4.b.p
Level $1150$
Weight $4$
Character orbit 1150.b
Analytic conductor $67.852$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,4,Mod(599,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.599");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 77x^{8} + 1924x^{6} + 16594x^{4} + 33128x^{2} + 10201 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{5} q^{2} + (\beta_{8} - 2 \beta_{5}) q^{3} - 4 q^{4} + ( - 2 \beta_{3} - 4) q^{6} + (\beta_{9} - \beta_{7} - 4 \beta_{5}) q^{7} + 8 \beta_{5} q^{8} + (\beta_{6} - 7 \beta_{3} + 2 \beta_{2} + 1) q^{9}+ \cdots + ( - 49 \beta_{6} - 5 \beta_{4} + \cdots + 342) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 40 q^{4} - 48 q^{6} - 22 q^{9} - 108 q^{11} - 96 q^{14} + 160 q^{16} + 100 q^{19} - 316 q^{21} + 192 q^{24} - 144 q^{26} + 208 q^{29} - 684 q^{31} + 528 q^{34} + 88 q^{36} - 386 q^{39} + 4 q^{41}+ \cdots + 3480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 77x^{8} + 1924x^{6} + 16594x^{4} + 33128x^{2} + 10201 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} - 78\nu^{4} - 480\nu^{2} + 202 ) / 303 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} - 103\nu^{6} - 3342\nu^{4} - 35257\nu^{2} - 42925 ) / 6363 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 39\nu^{4} + 63\nu^{2} + 4949 ) / 303 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 77\nu^{7} + 2025\nu^{5} + 20533\nu^{3} + 57368\nu ) / 30603 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{8} - 940\nu^{6} - 21522\nu^{4} - 158965\nu^{2} - 165640 ) / 6363 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\nu^{9} + 1333\nu^{7} + 36339\nu^{5} + 352768\nu^{3} + 938290\nu ) / 30603 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 365\nu^{9} + 27095\nu^{7} + 638529\nu^{5} + 4895714\nu^{3} + 6719126\nu ) / 642663 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1196\nu^{9} + 91688\nu^{7} + 2253129\nu^{5} + 18248099\nu^{3} + 24492803\nu ) / 642663 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} - \beta_{2} - 32 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} + 5\beta_{8} + 2\beta_{7} - 5\beta_{5} - 14\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{6} - 64\beta_{4} - 26\beta_{3} + 51\beta_{2} + 888 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 154\beta_{9} - 406\beta_{8} - 164\beta_{7} + 910\beta_{5} + 849\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -39\beta_{6} + 1008\beta_{4} + 507\beta_{3} - 1026\beta_{2} - 13375 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5046\beta_{9} + 13434\beta_{8} + 6042\beta_{7} - 42786\beta_{5} - 26593\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1350\beta_{6} - 64274\beta_{4} - 30276\beta_{3} + 76171\beta_{2} + 829928 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 79412\beta_{9} - 208799\beta_{8} - 107633\beta_{7} + 859154\beta_{5} + 422996\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
− 3.55548i
1.50693i
5.76842i
− 5.33268i
0.612813i
− 0.612813i
5.33268i
− 5.76842i
− 1.50693i
3.55548i
− 2.00000i − 10.0533i −4.00000 0 −20.1065 0.670353i 8.00000i −74.0682 0
599.2 − 2.00000i − 5.31348i −4.00000 0 −10.6270 − 34.1697i 8.00000i −1.23302 0
599.3 − 2.00000i − 1.80744i −4.00000 0 −3.61488 9.19471i 8.00000i 23.7332 0
599.4 − 2.00000i 2.43578i −4.00000 0 4.87156 3.42292i 8.00000i 21.0670 0
599.5 − 2.00000i 2.73841i −4.00000 0 5.47681 − 3.11829i 8.00000i 19.5011 0
599.6 2.00000i − 2.73841i −4.00000 0 5.47681 3.11829i − 8.00000i 19.5011 0
599.7 2.00000i − 2.43578i −4.00000 0 4.87156 − 3.42292i − 8.00000i 21.0670 0
599.8 2.00000i 1.80744i −4.00000 0 −3.61488 − 9.19471i − 8.00000i 23.7332 0
599.9 2.00000i 5.31348i −4.00000 0 −10.6270 34.1697i − 8.00000i −1.23302 0
599.10 2.00000i 10.0533i −4.00000 0 −20.1065 − 0.670353i − 8.00000i −74.0682 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 599.10
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 1150.4.b.p 10
5.b even 2 1 inner 1150.4.b.p 10
5.c odd 4 1 1150.4.a.s ✓ 5
5.c odd 4 1 1150.4.a.t yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1150.4.a.s ✓ 5 5.c odd 4 1
1150.4.a.t yes 5 5.c odd 4 1
1150.4.b.p 10 1.a even 1 1 trivial
1150.4.b.p 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{10} + 146T_{3}^{8} + 5101T_{3}^{6} + 59221T_{3}^{4} + 270956T_{3}^{2} + 414736 \) Copy content Toggle raw display
\( T_{7}^{10} + 1274T_{7}^{8} + 126241T_{7}^{6} + 2315464T_{7}^{4} + 12260816T_{7}^{2} + 5053504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 146 T^{8} + \cdots + 414736 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 1274 T^{8} + \cdots + 5053504 \) Copy content Toggle raw display
$11$ \( (T^{5} + 54 T^{4} + \cdots + 65678040)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 96\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} - 50 T^{4} + \cdots - 2606683072)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{5} \) Copy content Toggle raw display
$29$ \( (T^{5} - 104 T^{4} + \cdots - 24501628185)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 342 T^{4} + \cdots + 11316658220)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{5} - 2 T^{4} + \cdots + 235392970113)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{5} - 277 T^{4} + \cdots + 433939150092)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 19500464743200)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{5} - 208 T^{4} + \cdots + 569270032548)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 69\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 85004646248488)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 710453302020000)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
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