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Properties

Label 2166.4.a.bk
Level $2166$
Weight $4$
Character orbit 2166.a
Self dual yes
Analytic conductor $127.798$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2166,4,Mod(1,2166)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2166, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2166.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

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: yes
Analytic conductor: \(127.798137072\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3 x^{8} - 753 x^{7} + 41 x^{6} + 190713 x^{5} + 502293 x^{4} - 15827924 x^{3} - 81474654 x^{2} + \cdots + 33478707 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 19 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + \beta_1 q^{5} - 6 q^{6} + ( - \beta_{7} - \beta_{6} + 3) q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + \beta_1 q^{5} - 6 q^{6} + ( - \beta_{7} - \beta_{6} + 3) q^{7} - 8 q^{8} + 9 q^{9} - 2 \beta_1 q^{10} + (\beta_{8} + 3 \beta_{7} + 2 \beta_{5} + \cdots - 6) q^{11}+ \cdots + (9 \beta_{8} + 27 \beta_{7} + \cdots - 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 18 q^{2} + 27 q^{3} + 36 q^{4} - 3 q^{5} - 54 q^{6} + 30 q^{7} - 72 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 18 q^{2} + 27 q^{3} + 36 q^{4} - 3 q^{5} - 54 q^{6} + 30 q^{7} - 72 q^{8} + 81 q^{9} + 6 q^{10} - 57 q^{11} + 108 q^{12} - 93 q^{13} - 60 q^{14} - 9 q^{15} + 144 q^{16} - 147 q^{17} - 162 q^{18} - 12 q^{20} + 90 q^{21} + 114 q^{22} - 24 q^{23} - 216 q^{24} + 390 q^{25} + 186 q^{26} + 243 q^{27} + 120 q^{28} - 225 q^{29} + 18 q^{30} - 429 q^{31} - 288 q^{32} - 171 q^{33} + 294 q^{34} - 438 q^{35} + 324 q^{36} - 159 q^{37} - 279 q^{39} + 24 q^{40} - 567 q^{41} - 180 q^{42} + 1230 q^{43} - 228 q^{44} - 27 q^{45} + 48 q^{46} - 342 q^{47} + 432 q^{48} + 249 q^{49} - 780 q^{50} - 441 q^{51} - 372 q^{52} + 285 q^{53} - 486 q^{54} + 174 q^{55} - 240 q^{56} + 450 q^{58} + 171 q^{59} - 36 q^{60} - 816 q^{61} + 858 q^{62} + 270 q^{63} + 576 q^{64} - 1209 q^{65} + 342 q^{66} - 2388 q^{67} - 588 q^{68} - 72 q^{69} + 876 q^{70} - 2022 q^{71} - 648 q^{72} + 645 q^{73} + 318 q^{74} + 1170 q^{75} - 3843 q^{77} + 558 q^{78} - 3516 q^{79} - 48 q^{80} + 729 q^{81} + 1134 q^{82} - 156 q^{83} + 360 q^{84} - 291 q^{85} - 2460 q^{86} - 675 q^{87} + 456 q^{88} - 1959 q^{89} + 54 q^{90} - 501 q^{91} - 96 q^{92} - 1287 q^{93} + 684 q^{94} - 864 q^{96} + 2457 q^{97} - 498 q^{98} - 513 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3 x^{8} - 753 x^{7} + 41 x^{6} + 190713 x^{5} + 502293 x^{4} - 15827924 x^{3} - 81474654 x^{2} + \cdots + 33478707 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2829731826091 \nu^{8} + 48686189331726 \nu^{7} + \cdots + 13\!\cdots\!91 ) / 13\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 624093487200499 \nu^{8} + \cdots - 32\!\cdots\!03 ) / 14\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6505394672963 \nu^{8} + 155329279979678 \nu^{7} + \cdots - 13\!\cdots\!07 ) / 13\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!89 \nu^{8} + \cdots - 21\!\cdots\!53 ) / 14\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13355138808331 \nu^{8} + 247139231452430 \nu^{7} + \cdots - 28\!\cdots\!57 ) / 13\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18882011094212 \nu^{8} + 334694474924855 \nu^{7} + \cdots - 30\!\cdots\!80 ) / 66\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 89\!\cdots\!63 \nu^{8} + \cdots - 90\!\cdots\!35 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35\!\cdots\!97 \nu^{8} + \cdots + 58\!\cdots\!88 ) / 47\!\cdots\!26 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5\beta_{5} - 2\beta_{3} - 19\beta_1 ) / 19 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 38\beta_{8} + 19\beta_{7} + 57\beta_{6} + 174\beta_{5} + 133\beta_{4} + 71\beta_{3} - 114\beta _1 + 3211 ) / 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 76 \beta_{8} - 323 \beta_{7} - 418 \beta_{6} + 2973 \beta_{5} + 190 \beta_{4} + 289 \beta_{3} + \cdots + 12597 ) / 19 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6954 \beta_{8} - 5358 \beta_{7} + 7049 \beta_{6} + 84315 \beta_{5} + 35625 \beta_{4} + 23844 \beta_{3} + \cdots + 838679 ) / 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 52478 \beta_{8} - 262599 \beta_{7} - 209665 \beta_{6} + 1499697 \beta_{5} + 193097 \beta_{4} + \cdots + 6115074 ) / 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 749284 \beta_{8} - 5177386 \beta_{7} - 752381 \beta_{6} + 35332872 \beta_{5} + 10113035 \beta_{4} + \cdots + 237128094 ) / 19 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 27831846 \beta_{8} - 138780978 \beta_{7} - 91140644 \beta_{6} + 654036480 \beta_{5} + \cdots + 2478317839 ) / 19 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 178774990 \beta_{8} - 2783757906 \beta_{7} - 1052433560 \beta_{6} + 14100106599 \beta_{5} + \cdots + 71727137424 ) / 19 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
19.4354
15.1929
16.6546
0.303078
−3.68876
−2.09936
−13.9157
−14.1094
−14.7729
−2.00000 3.00000 4.00000 −19.0881 −6.00000 18.3084 −8.00000 9.00000 38.1762
1.2 −2.00000 3.00000 4.00000 −17.0723 −6.00000 −8.15489 −8.00000 9.00000 34.1447
1.3 −2.00000 3.00000 4.00000 −15.1225 −6.00000 4.48182 −8.00000 9.00000 30.2450
1.4 −2.00000 3.00000 4.00000 1.22901 −6.00000 24.0257 −8.00000 9.00000 −2.45802
1.5 −2.00000 3.00000 4.00000 1.80938 −6.00000 −16.5374 −8.00000 9.00000 −3.61875
1.6 −2.00000 3.00000 4.00000 2.44666 −6.00000 12.4652 −8.00000 9.00000 −4.89332
1.7 −2.00000 3.00000 4.00000 12.0363 −6.00000 33.8548 −8.00000 9.00000 −24.0726
1.8 −2.00000 3.00000 4.00000 14.4567 −6.00000 −15.8300 −8.00000 9.00000 −28.9133
1.9 −2.00000 3.00000 4.00000 16.3050 −6.00000 −22.6136 −8.00000 9.00000 −32.6099
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2166.4.a.bk 9
19.b odd 2 1 2166.4.a.bl 9
19.e even 9 2 114.4.i.c ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.i.c ✓ 18 19.e even 9 2
2166.4.a.bk 9 1.a even 1 1 trivial
2166.4.a.bl 9 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{9} + 3 T_{5}^{8} - 753 T_{5}^{7} - 179 T_{5}^{6} + 188415 T_{5}^{5} - 472023 T_{5}^{4} + \cdots + 76070664 \) Copy content Toggle raw display
\( T_{13}^{9} + 93 T_{13}^{8} - 12459 T_{13}^{7} - 1206157 T_{13}^{6} + 56436993 T_{13}^{5} + \cdots + 83\!\cdots\!87 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{9} \) Copy content Toggle raw display
$3$ \( (T - 3)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 3 T^{8} + \cdots + 76070664 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 40163896491 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 20463570638712 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 83\!\cdots\!87 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 12\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 34\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 15\!\cdots\!91 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 18\!\cdots\!13 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 12\!\cdots\!03 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 49\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 29\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 17\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 11\!\cdots\!67 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 16\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 53\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 72\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 68\!\cdots\!33 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 96\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 19\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 16\!\cdots\!11 \) Copy content Toggle raw display
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