↑

Properties

Label 1150.4.b.r
Level $1150$
Weight $4$
Character orbit 1150.b
Analytic conductor $67.852$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \( x^{10} + 214x^{8} + 15751x^{6} + 460323x^{4} + 4609305x^{2} + 8503056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + ( - \beta_{2} + \beta_1) q^{3} - 4 q^{4} + ( - 2 \beta_{3} + 2) q^{6} + (\beta_{8} - \beta_{2}) q^{7} - 8 \beta_{2} q^{8} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} + ( - \beta_{2} + \beta_1) q^{3} - 4 q^{4} + ( - 2 \beta_{3} + 2) q^{6} + (\beta_{8} - \beta_{2}) q^{7} - 8 \beta_{2} q^{8} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - 16) q^{9} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 5) q^{11}+ \cdots + ( - 38 \beta_{6} - 5 \beta_{5} + \cdots - 457) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 40 q^{4} + 20 q^{6} - 168 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 40 q^{4} + 20 q^{6} - 168 q^{9} - 52 q^{11} + 12 q^{14} + 160 q^{16} - 148 q^{19} - 176 q^{21} - 80 q^{24} - 244 q^{26} - 74 q^{29} + 440 q^{31} - 924 q^{34} + 672 q^{36} - 390 q^{39} + 224 q^{41} + 208 q^{44} + 460 q^{46} - 364 q^{49} + 1126 q^{51} - 1472 q^{54} - 48 q^{56} + 362 q^{59} + 3148 q^{61} - 640 q^{64} + 1564 q^{66} - 230 q^{69} - 60 q^{71} - 880 q^{74} + 592 q^{76} - 1218 q^{79} + 2906 q^{81} + 704 q^{84} + 388 q^{86} + 3326 q^{89} + 722 q^{91} - 1480 q^{94} + 320 q^{96} - 4794 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 214x^{8} + 15751x^{6} + 460323x^{4} + 4609305x^{2} + 8503056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 1186\nu^{7} + 1168543\nu^{5} + 114771411\nu^{3} + 2260561041\nu ) / 3064538124 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} - 1186\nu^{6} - 117604\nu^{4} - 2320938\nu^{2} + 8748 ) / 3152817 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1009\nu^{8} + 145735\nu^{6} + 5161024\nu^{4} + 25556886\nu^{2} + 587055681 ) / 47292255 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1009\nu^{8} - 145735\nu^{6} - 5161024\nu^{4} + 21735369\nu^{2} + 1399219029 ) / 47292255 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3167\nu^{8} - 603245\nu^{6} - 37202327\nu^{4} - 774685323\nu^{2} - 2753079678 ) / 47292255 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12293\nu^{9} + 1968230\nu^{7} + 88943723\nu^{5} + 730801467\nu^{3} - 11731975443\nu ) / 1702521180 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -38569\nu^{9} - 7909030\nu^{7} - 539147659\nu^{5} - 13986322731\nu^{3} - 123029174781\nu ) / 5107563540 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -38569\nu^{9} - 7909030\nu^{7} - 539147659\nu^{5} - 13135062141\nu^{3} - 67697236431\nu ) / 2553781770 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{9} - 6\beta_{8} - 65\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{6} - 71\beta_{5} - 98\beta_{4} + 84\beta_{3} + 2793 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -321\beta_{9} + 645\beta_{8} + 3\beta_{7} + 3042\beta_{2} + 4804\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 972\beta_{6} + 5464\beta_{5} + 8335\beta_{4} - 12099\beta_{3} - 208509 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27921\beta_{9} - 59238\beta_{8} - 3531\beta_{7} - 456345\beta_{2} - 374465\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -94356\beta_{6} - 451358\beta_{5} - 681056\beta_{4} + 1317861\beta_{3} + 16311846 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -2326236\beta_{9} + 5174499\beta_{8} + 682137\beta_{7} + 51055488\beta_{2} + 30015592\beta_1 \) 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
− 9.27140i
− 3.88892i
− 1.53483i
6.21006i
8.48510i
− 8.48510i
− 6.21006i
1.53483i
3.88892i
9.27140i
− 2.00000i − 8.27140i −4.00000 0 −16.5428 − 22.8621i 8.00000i −41.4161 0
599.2 − 2.00000i − 2.88892i −4.00000 0 −5.77785 8.21839i 8.00000i 18.6541 0
599.3 − 2.00000i − 0.534830i −4.00000 0 −1.06966 28.9380i 8.00000i 26.7140 0
599.4 − 2.00000i 7.21006i −4.00000 0 14.4201 − 19.8879i 8.00000i −24.9850 0
599.5 − 2.00000i 9.48510i −4.00000 0 18.9702 8.59355i 8.00000i −62.9670 0
599.6 2.00000i − 9.48510i −4.00000 0 18.9702 − 8.59355i − 8.00000i −62.9670 0
599.7 2.00000i − 7.21006i −4.00000 0 14.4201 19.8879i − 8.00000i −24.9850 0
599.8 2.00000i 0.534830i −4.00000 0 −1.06966 − 28.9380i − 8.00000i 26.7140 0
599.9 2.00000i 2.88892i −4.00000 0 −5.77785 − 8.21839i − 8.00000i 18.6541 0
599.10 2.00000i 8.27140i −4.00000 0 −16.5428 22.8621i − 8.00000i −41.4161 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.r 10
5.b even 2 1 inner 1150.4.b.r 10
5.c odd 4 1 1150.4.a.q ✓ 5
5.c odd 4 1 1150.4.a.v yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1150.4.a.q ✓ 5 5.c odd 4 1
1150.4.a.v yes 5 5.c odd 4 1
1150.4.b.r 10 1.a even 1 1 trivial
1150.4.b.r 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 \(S_{4}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{10} + 219T_{3}^{8} + 16207T_{3}^{6} + 444682T_{3}^{4} + 2796369T_{3}^{2} + 763876 \) Copy content Toggle raw display
\( T_{7}^{10} + 1897T_{7}^{8} + 1228856T_{7}^{6} + 319822544T_{7}^{4} + 29343869248T_{7}^{2} + 863501845504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 219 T^{8} + \cdots + 763876 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 863501845504 \) Copy content Toggle raw display
$11$ \( (T^{5} + 26 T^{4} + \cdots + 3242920)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 27848266884 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + 74 T^{4} + \cdots + 112577688)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{5} \) Copy content Toggle raw display
$29$ \( (T^{5} + 37 T^{4} + \cdots + 20220755440)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 220 T^{4} + \cdots - 51046245360)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{5} - 112 T^{4} + \cdots + 4982446167)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 1123269020040)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 2333556256768)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 5642844385736)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 92\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 12948718884224)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 1214254682896)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 62\!\cdots\!16 \) Copy content Toggle raw display
show more
show less