Properties

Label 1008.3.bk.b
Level $1008$
Weight $3$
Character orbit 1008.bk
Analytic conductor $27.466$
Analytic rank $0$
Dimension $20$
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] = [1008,3,Mod(143,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.143");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.bk (of order \(6\), degree \(2\), 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: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 44 x^{18} + 1263 x^{16} - 20896 x^{14} + 250941 x^{12} - 2032238 x^{10} + 12100168 x^{8} - 44511576 x^{6} + 104461920 x^{4} - 736992 x^{2} + \cdots + 5184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + \beta_{9} q^{7} + ( - \beta_{18} + \beta_{17} - \beta_1) q^{11} + (\beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 1) q^{13} + ( - \beta_{18} + \beta_{17} - \beta_{15} - \beta_{14} + \beta_{3} - \beta_{2} - \beta_1) q^{17} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} + 2 \beta_{5} + 3) q^{19} + ( - \beta_{18} - \beta_{15} - 3 \beta_{2} - \beta_1) q^{23} + ( - \beta_{10} - 2 \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} + \cdots + 2) q^{25}+ \cdots + (\beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - 7 \beta_{8} - 7 \beta_{7} - 2 \beta_{6} + 20 \beta_{5} + \cdots + 11) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 10 q^{7} + 14 q^{19} - 14 q^{25} + 22 q^{31} - 10 q^{37} + 158 q^{49} - 240 q^{55} - 180 q^{61} + 318 q^{67} - 246 q^{73} + 66 q^{79} + 144 q^{85} - 42 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 44 x^{18} + 1263 x^{16} - 20896 x^{14} + 250941 x^{12} - 2032238 x^{10} + 12100168 x^{8} - 44511576 x^{6} + 104461920 x^{4} - 736992 x^{2} + \cdots + 5184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 86\!\cdots\!23 \nu^{19} + \cdots + 15\!\cdots\!44 \nu ) / 31\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26\!\cdots\!43 \nu^{19} + \cdots + 19\!\cdots\!72 \nu ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!43 \nu^{19} + \cdots + 38\!\cdots\!40 \nu ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 70\!\cdots\!11 \nu^{18} + \cdots + 70\!\cdots\!36 ) / 51\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!43 \nu^{18} + \cdots - 19\!\cdots\!36 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35\!\cdots\!87 \nu^{18} + \cdots + 48\!\cdots\!72 ) / 15\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!35 \nu^{18} + \cdots - 30\!\cdots\!48 ) / 34\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!79 \nu^{18} + \cdots + 28\!\cdots\!08 ) / 51\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74\!\cdots\!33 \nu^{18} + \cdots - 81\!\cdots\!56 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 63\!\cdots\!17 \nu^{18} + \cdots + 24\!\cdots\!64 ) / 74\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!03 \nu^{18} + \cdots - 17\!\cdots\!60 ) / 77\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 64\!\cdots\!14 \nu^{18} + \cdots + 42\!\cdots\!84 ) / 38\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3413825939 \nu^{19} + 150096723988 \nu^{17} - 4307260216977 \nu^{15} + 71217972964760 \nu^{13} - 854994701116371 \nu^{11} + \cdots + 7522682195808 \nu ) / 49473480381696 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 32\!\cdots\!87 \nu^{19} + \cdots - 47\!\cdots\!56 \nu ) / 31\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 53\!\cdots\!53 \nu^{19} + \cdots + 39\!\cdots\!68 \nu ) / 51\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17\!\cdots\!01 \nu^{19} + \cdots + 48\!\cdots\!56 \nu ) / 15\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 89\!\cdots\!31 \nu^{19} + \cdots - 13\!\cdots\!04 \nu ) / 77\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 12\!\cdots\!21 \nu^{19} + \cdots - 14\!\cdots\!00 \nu ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 66\!\cdots\!71 \nu^{19} + \cdots - 13\!\cdots\!88 \nu ) / 15\!\cdots\!32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + 28\beta_{5} + 29 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{19} - \beta_{17} - 2\beta_{15} - \beta_{14} + 7\beta_{13} - 14\beta_{3} + 26\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{12} + 8 \beta_{11} - 13 \beta_{10} - 38 \beta_{9} - 25 \beta_{8} - 38 \beta_{7} + 23 \beta_{6} + 380 \beta_{5} + 23 \beta_{4} + 34 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 56 \beta_{19} + 29 \beta_{18} - 58 \beta_{17} - 56 \beta_{16} - 25 \beta_{15} - 50 \beta_{14} + 195 \beta_{13} - 464 \beta_{3} + 232 \beta_{2} + 224 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 106 \beta_{12} + 106 \beta_{11} + 543 \beta_{10} - 543 \beta_{9} + 183 \beta_{8} - 183 \beta_{7} + 487 \beta_{4} - 5787 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 579 \beta_{18} - 663 \beta_{17} - 1242 \beta_{16} + 515 \beta_{15} - 515 \beta_{14} - 4240 \beta_{3} - 2998 \beta_{2} + 4928 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4544 \beta_{12} - 2272 \beta_{11} + 14054 \beta_{10} + 2857 \beta_{9} + 14054 \beta_{8} + 11197 \beta_{7} - 9959 \beta_{6} - 109748 \beta_{5} - 114877 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 25700 \beta_{19} - 2254 \beta_{18} + 13977 \beta_{17} + 20362 \beta_{15} + 10181 \beta_{14} - 90447 \beta_{13} + 81212 \beta_{3} - 136724 \beta_{2} - 2254 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 46062 \beta_{12} - 92124 \beta_{11} + 48967 \beta_{10} + 274910 \beta_{9} + 225943 \beta_{8} + 274910 \beta_{7} - 200443 \beta_{6} - 2081146 \beta_{5} - 200443 \beta_{4} - 228848 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 518510 \beta_{19} - 284935 \beta_{18} + 569870 \beta_{17} + 518510 \beta_{16} + 200083 \beta_{15} + 400166 \beta_{14} - 1830949 \beta_{13} + 3175264 \beta_{3} - 1587632 \beta_{2} - 2115884 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 918676 \beta_{12} - 918676 \beta_{11} - 4516477 \beta_{10} + 4516477 \beta_{9} - 896305 \beta_{8} + 896305 \beta_{7} - 4004075 \beta_{4} + 37805475 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 4634703 \beta_{18} + 5723201 \beta_{17} + 10357904 \beta_{16} - 3940381 \beta_{15} + 3940381 \beta_{14} + 31304308 \beta_{3} + 20946404 \beta_{2} - 41283054 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 36477332 \beta_{12} + 18238666 \beta_{11} - 106966710 \beta_{10} - 17061183 \beta_{9} - 106966710 \beta_{8} - 89905527 \beta_{7} + 79699207 \beta_{6} + 796396438 \beta_{5} + \cdots + 831696287 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 206082066 \beta_{19} + 22291884 \beta_{18} - 114186975 \beta_{17} - 155639926 \beta_{15} - 77819963 \beta_{14} + 729964469 \beta_{13} - 619451176 \beta_{3} + \cdots + 22291884 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 361721992 \beta_{12} + 723443984 \beta_{11} - 331645897 \beta_{10} - 2117986262 \beta_{9} - 1786340365 \beta_{8} - 2117986262 \beta_{7} + 1583647967 \beta_{6} + \cdots + 1756264270 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 4093432316 \beta_{19} + 2271238521 \beta_{18} - 4542477042 \beta_{17} - 4093432316 \beta_{16} - 1539988037 \beta_{15} - 3079976074 \beta_{14} + 14507869791 \beta_{13} + \cdots + 16779108312 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 7173408390 \beta_{12} + 7173408390 \beta_{11} + 35463416791 \beta_{10} - 35463416791 \beta_{9} + 6515319991 \beta_{8} - 6515319991 \beta_{7} + \cdots - 290693850567 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 36139136767 \beta_{18} - 45112782391 \beta_{17} - 81251919158 \beta_{16} + 30511502371 \beta_{15} - 30511502371 \beta_{14} - 243431145344 \beta_{3} + \cdots + 324197751812 \beta_1 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1 + \beta_{5}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
−3.85738 2.22706i
−2.62188 1.51374i
−2.40156 1.38654i
−2.33957 1.35075i
−0.0727418 0.0419975i
0.0727418 + 0.0419975i
2.33957 + 1.35075i
2.40156 + 1.38654i
2.62188 + 1.51374i
3.85738 + 2.22706i
−3.85738 + 2.22706i
−2.62188 + 1.51374i
−2.40156 + 1.38654i
−2.33957 + 1.35075i
−0.0727418 + 0.0419975i
0.0727418 0.0419975i
2.33957 1.35075i
2.40156 1.38654i
2.62188 1.51374i
3.85738 2.22706i
0 0 0 −3.85738 6.68117i 0 6.70362 + 2.01532i 0 0 0
143.2 0 0 0 −2.62188 4.54123i 0 1.49953 6.83750i 0 0 0
143.3 0 0 0 −2.40156 4.15963i 0 −6.15580 3.33258i 0 0 0
143.4 0 0 0 −2.33957 4.05226i 0 5.56798 + 4.24236i 0 0 0
143.5 0 0 0 −0.0727418 0.125993i 0 −5.11533 + 4.77843i 0 0 0
143.6 0 0 0 0.0727418 + 0.125993i 0 −5.11533 + 4.77843i 0 0 0
143.7 0 0 0 2.33957 + 4.05226i 0 5.56798 + 4.24236i 0 0 0
143.8 0 0 0 2.40156 + 4.15963i 0 −6.15580 3.33258i 0 0 0
143.9 0 0 0 2.62188 + 4.54123i 0 1.49953 6.83750i 0 0 0
143.10 0 0 0 3.85738 + 6.68117i 0 6.70362 + 2.01532i 0 0 0
719.1 0 0 0 −3.85738 + 6.68117i 0 6.70362 2.01532i 0 0 0
719.2 0 0 0 −2.62188 + 4.54123i 0 1.49953 + 6.83750i 0 0 0
719.3 0 0 0 −2.40156 + 4.15963i 0 −6.15580 + 3.33258i 0 0 0
719.4 0 0 0 −2.33957 + 4.05226i 0 5.56798 4.24236i 0 0 0
719.5 0 0 0 −0.0727418 + 0.125993i 0 −5.11533 4.77843i 0 0 0
719.6 0 0 0 0.0727418 0.125993i 0 −5.11533 4.77843i 0 0 0
719.7 0 0 0 2.33957 4.05226i 0 5.56798 4.24236i 0 0 0
719.8 0 0 0 2.40156 4.15963i 0 −6.15580 + 3.33258i 0 0 0
719.9 0 0 0 2.62188 4.54123i 0 1.49953 + 6.83750i 0 0 0
719.10 0 0 0 3.85738 6.68117i 0 6.70362 2.01532i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.bk.b yes 20
3.b odd 2 1 inner 1008.3.bk.b yes 20
4.b odd 2 1 1008.3.bk.a 20
7.d odd 6 1 1008.3.bk.a 20
12.b even 2 1 1008.3.bk.a 20
21.g even 6 1 1008.3.bk.a 20
28.f even 6 1 inner 1008.3.bk.b yes 20
84.j odd 6 1 inner 1008.3.bk.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.3.bk.a 20 4.b odd 2 1
1008.3.bk.a 20 7.d odd 6 1
1008.3.bk.a 20 12.b even 2 1
1008.3.bk.a 20 21.g even 6 1
1008.3.bk.b yes 20 1.a even 1 1 trivial
1008.3.bk.b yes 20 3.b odd 2 1 inner
1008.3.bk.b yes 20 28.f even 6 1 inner
1008.3.bk.b yes 20 84.j odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{20} + 132 T_{5}^{18} + 11367 T_{5}^{16} + 564192 T_{5}^{14} + 20326221 T_{5}^{12} + 493833834 T_{5}^{10} + 8821022472 T_{5}^{8} + 97346816712 T_{5}^{6} + 685374657120 T_{5}^{4} + \cdots + 306110016 \) Copy content Toggle raw display
\( T_{19}^{10} - 7 T_{19}^{9} + 516 T_{19}^{8} + 1411 T_{19}^{7} + 179126 T_{19}^{6} + 179871 T_{19}^{5} + 21935993 T_{19}^{4} + 20933374 T_{19}^{3} + 2088347646 T_{19}^{2} - 1037079460 T_{19} + 520296100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 132 T^{18} + \cdots + 306110016 \) Copy content Toggle raw display
$7$ \( (T^{10} - 5 T^{9} - 27 T^{8} + \cdots + 282475249)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 504 T^{18} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + 1053 T^{8} + \cdots + 60279187500)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + 1020 T^{18} + \cdots + 20061226008576 \) Copy content Toggle raw display
$19$ \( (T^{10} - 7 T^{9} + 516 T^{8} + \cdots + 520296100)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 1692 T^{18} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{10} + 4590 T^{8} + \cdots + 487629476352)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 11 T^{9} + \cdots + 4929914156281)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 5 T^{9} + \cdots + 3070464171076)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 11682 T^{8} + \cdots - 125506023630336)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 10893 T^{8} + \cdots + 671491870022700)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} - 9090 T^{18} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{20} - 23742 T^{18} + \cdots + 63\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{20} - 20682 T^{18} + \cdots + 44\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{10} + 90 T^{9} + \cdots + 37\!\cdots\!52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 159 T^{9} + \cdots + 40\!\cdots\!08)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 19014 T^{8} + \cdots - 31\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 123 T^{9} + \cdots + 46\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 33 T^{9} + \cdots + 86\!\cdots\!43)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 35928 T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 61920 T^{18} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{10} + 28938 T^{8} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
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