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Properties

Label 144.6.s.b
Level $144$
Weight $6$
Character orbit 144.s
Analytic conductor $23.095$
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] = [144,6,Mod(47,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.47");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 144.s (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: \(23.0952700531\)
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: \( x^{20} - 19 x^{18} - 44 x^{16} - 222597 x^{14} + 1089207 x^{12} + 5219984088 x^{10} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{25} \)
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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{3} + ( - \beta_{4} - 6 \beta_1 - 6) q^{5} + (\beta_{15} + \beta_{14}) q^{7} + ( - \beta_{10} + 8 \beta_1 + 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{3} + ( - \beta_{4} - 6 \beta_1 - 6) q^{5} + (\beta_{15} + \beta_{14}) q^{7} + ( - \beta_{10} + 8 \beta_1 + 20) q^{9} + (\beta_{18} + \beta_{16} + \cdots + \beta_{6}) q^{11}+ \cdots + ( - 51 \beta_{19} + \cdots - 714 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 174 q^{5} + 480 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 174 q^{5} + 480 q^{9} + 362 q^{13} + 2034 q^{21} + 2944 q^{25} + 9438 q^{29} + 11646 q^{33} + 9712 q^{37} + 24426 q^{41} - 1602 q^{45} + 36248 q^{49} + 61020 q^{57} + 49546 q^{61} - 11418 q^{65} - 13482 q^{69} - 137656 q^{73} + 80082 q^{77} - 52344 q^{81} + 17700 q^{85} - 80694 q^{93} + 41774 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 19 x^{18} - 44 x^{16} - 222597 x^{14} + 1089207 x^{12} + 5219984088 x^{10} + \cdots + 12\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2470379843 \nu^{18} - 329095547674 \nu^{16} + 40326693725770 \nu^{14} + \cdots - 79\!\cdots\!61 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 65498215817 \nu^{18} - 11380527291022 \nu^{16} + 355246482469534 \nu^{14} + \cdots - 87\!\cdots\!63 ) / 44\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29348934955 \nu^{18} - 3189859116982 \nu^{16} + 280234625950726 \nu^{14} + \cdots - 74\!\cdots\!07 ) / 28\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1699276160119 \nu^{18} + 106907950452478 \nu^{16} + \cdots + 39\!\cdots\!99 ) / 89\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1739213709427 \nu^{18} - 237002256117818 \nu^{16} + \cdots - 58\!\cdots\!65 ) / 89\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 364431320813 \nu^{19} - 23132357245370 \nu^{17} + \cdots + 23\!\cdots\!71 \nu ) / 93\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19271608 \nu^{18} - 1795087303 \nu^{16} - 11914264793 \nu^{14} + 5728691484405 \nu^{12} + \cdots - 37\!\cdots\!27 ) / 38\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 550997696237 \nu^{19} - 26677118378426 \nu^{17} + \cdots + 16\!\cdots\!75 \nu ) / 93\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3795983913565 \nu^{18} - 932467222947230 \nu^{16} + \cdots - 21\!\cdots\!55 ) / 44\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1446663674543 \nu^{18} - 79119358383614 \nu^{16} + \cdots + 52\!\cdots\!93 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 624954604655 \nu^{19} - 133293886316386 \nu^{17} + \cdots - 23\!\cdots\!41 \nu ) / 46\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 246939629327 \nu^{19} + 17281925330050 \nu^{17} + \cdots + 71\!\cdots\!49 \nu ) / 84\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17865320209403 \nu^{18} + \cdots + 16\!\cdots\!27 ) / 59\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16481414129111 \nu^{19} + \cdots - 58\!\cdots\!09 \nu ) / 48\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 77155849015 \nu^{19} + 12879001354030 \nu^{17} - 267374564048446 \nu^{15} + \cdots + 14\!\cdots\!91 \nu ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11500253594497 \nu^{19} + \cdots + 42\!\cdots\!89 \nu ) / 17\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 25599337664965 \nu^{19} + \cdots - 83\!\cdots\!79 \nu ) / 32\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 279793254401285 \nu^{19} + \cdots - 28\!\cdots\!07 \nu ) / 28\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 12355270295605 \nu^{19} + 613248307456714 \nu^{17} + \cdots - 23\!\cdots\!67 \nu ) / 74\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + \beta_{11} - 2\beta_{8} - \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{2} - 84\beta _1 + 59 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9 \beta_{19} + 18 \beta_{18} - 27 \beta_{16} + 18 \beta_{15} - 63 \beta_{14} - \beta_{12} + \cdots + 55 \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{13} - 27 \beta_{10} - 9 \beta_{9} + 72 \beta_{7} + 4 \beta_{5} + 200 \beta_{4} + \cdots - 1272 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1269 \beta_{19} - 306 \beta_{18} - 549 \beta_{17} + 693 \beta_{16} - 5643 \beta_{15} + \cdots - 6244 \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 61 \beta_{13} + 5319 \beta_{10} + 2358 \beta_{9} - 5688 \beta_{7} - 1194 \beta_{5} + 18785 \beta_{4} + \cdots + 918865 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 69705 \beta_{19} - 193995 \beta_{18} - 3789 \beta_{17} - 121833 \beta_{16} + 631323 \beta_{15} + \cdots + 3037510 \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 47601 \beta_{13} + 131463 \beta_{10} - 288099 \beta_{9} + 69489 \beta_{7} + 58823 \beta_{5} + \cdots - 58870571 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3169278 \beta_{19} - 7023447 \beta_{18} + 1745307 \beta_{17} - 537165 \beta_{16} + \cdots - 257412682 \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 17224994 \beta_{13} + 58447602 \beta_{10} + 8595972 \beta_{9} + 9112266 \beta_{7} + 11403248 \beta_{5} + \cdots - 23793265464 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 186798258 \beta_{19} + 992435472 \beta_{18} - 71851230 \beta_{17} + 796441932 \beta_{16} + \cdots + 21146328187 \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 419827466 \beta_{13} + 1379474334 \beta_{10} - 1496231604 \beta_{9} - 130991238 \beta_{7} + \cdots - 1159977157399 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 32687152578 \beta_{19} - 88947061956 \beta_{18} - 25553895570 \beta_{17} + \cdots + 976090099013 \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 139174990758 \beta_{13} + 228058898598 \beta_{10} + 61096513284 \beta_{9} - 161390789838 \beta_{7} + \cdots + 43550290309055 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4757519930265 \beta_{19} + 4934755705806 \beta_{18} - 1058776370334 \beta_{17} + \cdots + 3077015301793 \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 3767200865431 \beta_{13} - 19631433928905 \beta_{10} - 11464035135093 \beta_{9} + 10264162569282 \beta_{7} + \cdots - 93\!\cdots\!08 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 340391371963257 \beta_{19} + 476269027800606 \beta_{18} - 221325292039779 \beta_{17} + \cdots - 87\!\cdots\!26 \beta_{6} ) / 27 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 295029025005319 \beta_{13} + \cdots + 24\!\cdots\!13 ) / 9 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 18\!\cdots\!85 \beta_{19} + \cdots - 95\!\cdots\!80 \beta_{6} ) / 27 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1 - \beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−8.44629 − 3.10810i
−5.37530 − 7.21846i
−8.79856 + 1.89349i
−1.18210 − 8.92203i
−5.31078 + 7.26606i
5.31078 − 7.26606i
1.18210 + 8.92203i
8.79856 − 1.89349i
5.37530 + 7.21846i
8.44629 + 3.10810i
−8.44629 + 3.10810i
−5.37530 + 7.21846i
−8.79856 − 1.89349i
−1.18210 + 8.92203i
−5.31078 − 7.26606i
5.31078 + 7.26606i
1.18210 − 8.92203i
8.79856 + 1.89349i
5.37530 − 7.21846i
8.44629 − 3.10810i
0 −15.3611 − 2.65255i 0 63.8385 − 36.8572i 0 20.9224 + 12.0795i 0 228.928 + 81.4923i 0
47.2 0 −14.3143 + 6.17254i 0 −37.6433 + 21.7334i 0 −213.857 − 123.470i 0 166.800 − 176.711i 0
47.3 0 −11.5580 − 10.4600i 0 −80.5546 + 46.5082i 0 89.6151 + 51.7393i 0 24.1765 + 241.794i 0
47.4 0 −9.49985 + 12.3593i 0 −14.3487 + 8.28421i 0 149.607 + 86.3756i 0 −62.5056 − 234.823i 0
47.5 0 −1.67357 − 15.4984i 0 25.2081 − 14.5539i 0 −5.79097 − 3.34342i 0 −237.398 + 51.8751i 0
47.6 0 1.67357 + 15.4984i 0 25.2081 − 14.5539i 0 5.79097 + 3.34342i 0 −237.398 + 51.8751i 0
47.7 0 9.49985 − 12.3593i 0 −14.3487 + 8.28421i 0 −149.607 − 86.3756i 0 −62.5056 − 234.823i 0
47.8 0 11.5580 + 10.4600i 0 −80.5546 + 46.5082i 0 −89.6151 − 51.7393i 0 24.1765 + 241.794i 0
47.9 0 14.3143 − 6.17254i 0 −37.6433 + 21.7334i 0 213.857 + 123.470i 0 166.800 − 176.711i 0
47.10 0 15.3611 + 2.65255i 0 63.8385 − 36.8572i 0 −20.9224 − 12.0795i 0 228.928 + 81.4923i 0
95.1 0 −15.3611 + 2.65255i 0 63.8385 + 36.8572i 0 20.9224 − 12.0795i 0 228.928 − 81.4923i 0
95.2 0 −14.3143 − 6.17254i 0 −37.6433 − 21.7334i 0 −213.857 + 123.470i 0 166.800 + 176.711i 0
95.3 0 −11.5580 + 10.4600i 0 −80.5546 − 46.5082i 0 89.6151 − 51.7393i 0 24.1765 − 241.794i 0
95.4 0 −9.49985 − 12.3593i 0 −14.3487 − 8.28421i 0 149.607 − 86.3756i 0 −62.5056 + 234.823i 0
95.5 0 −1.67357 + 15.4984i 0 25.2081 + 14.5539i 0 −5.79097 + 3.34342i 0 −237.398 − 51.8751i 0
95.6 0 1.67357 − 15.4984i 0 25.2081 + 14.5539i 0 5.79097 − 3.34342i 0 −237.398 − 51.8751i 0
95.7 0 9.49985 + 12.3593i 0 −14.3487 − 8.28421i 0 −149.607 + 86.3756i 0 −62.5056 + 234.823i 0
95.8 0 11.5580 − 10.4600i 0 −80.5546 − 46.5082i 0 −89.6151 + 51.7393i 0 24.1765 − 241.794i 0
95.9 0 14.3143 + 6.17254i 0 −37.6433 − 21.7334i 0 213.857 − 123.470i 0 166.800 + 176.711i 0
95.10 0 15.3611 − 2.65255i 0 63.8385 + 36.8572i 0 −20.9224 + 12.0795i 0 228.928 − 81.4923i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.6.s.b ✓ 20
3.b odd 2 1 432.6.s.c 20
4.b odd 2 1 inner 144.6.s.b ✓ 20
9.c even 3 1 432.6.s.c 20
9.d odd 6 1 inner 144.6.s.b ✓ 20
12.b even 2 1 432.6.s.c 20
36.f odd 6 1 432.6.s.c 20
36.h even 6 1 inner 144.6.s.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.6.s.b ✓ 20 1.a even 1 1 trivial
144.6.s.b ✓ 20 4.b odd 2 1 inner
144.6.s.b ✓ 20 9.d odd 6 1 inner
144.6.s.b ✓ 20 36.h even 6 1 inner
432.6.s.c 20 3.b odd 2 1
432.6.s.c 20 9.c even 3 1
432.6.s.c 20 12.b even 2 1
432.6.s.c 20 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(144, [\chi])\):

\( T_{5}^{10} + 87 T_{5}^{9} - 4764 T_{5}^{8} - 633969 T_{5}^{7} + 33760512 T_{5}^{6} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
\( T_{7}^{20} - 102159 T_{7}^{18} + 7580300922 T_{7}^{16} - 249295276800999 T_{7}^{14} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 20\!\cdots\!48)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 77\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 48\!\cdots\!92)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 11\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 21\!\cdots\!84)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 69\!\cdots\!47)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 84\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 65\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 51\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 21\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 10\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 17\!\cdots\!16)^{4} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 13\!\cdots\!61)^{2} \) Copy content Toggle raw display
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