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Properties

Label 228.5.l.a
Level $228$
Weight $5$
Character orbit 228.l
Analytic conductor $23.568$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [228,5,Mod(145,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.145");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 228.l (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: \(23.5683515831\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 2358 x^{12} + 15572 x^{11} + 4050518 x^{10} + 21628620 x^{9} + 2974230644 x^{8} + \cdots + 96\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{3} - 6) q^{3} + (4 \beta_{3} - \beta_{2} + \beta_1) q^{5} + (\beta_{4} + 8) q^{7} + (27 \beta_{3} + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_{3} - 6) q^{3} + (4 \beta_{3} - \beta_{2} + \beta_1) q^{5} + (\beta_{4} + 8) q^{7} + (27 \beta_{3} + 27) q^{9} + ( - \beta_{9} - \beta_{2} - 19) q^{11} + (\beta_{9} - \beta_{8} - 3 \beta_{3} + \cdots + 3) q^{13}+ \cdots + (27 \beta_{10} - 513 \beta_{3} + \cdots - 513) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 63 q^{3} - 30 q^{5} + 106 q^{7} + 189 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 63 q^{3} - 30 q^{5} + 106 q^{7} + 189 q^{9} - 264 q^{11} + 57 q^{13} + 270 q^{15} + 282 q^{17} + 2 q^{19} - 477 q^{21} + 96 q^{23} - 465 q^{25} - 630 q^{29} + 1188 q^{33} + 1434 q^{35} - 342 q^{39} - 228 q^{41} - 2093 q^{43} - 1620 q^{45} - 4710 q^{47} + 4440 q^{49} - 2538 q^{51} + 2364 q^{53} + 6368 q^{55} + 1143 q^{57} + 11838 q^{59} - 1661 q^{61} + 1431 q^{63} - 20319 q^{67} + 624 q^{71} + 5851 q^{73} - 1080 q^{77} - 13299 q^{79} - 5103 q^{81} + 12252 q^{83} - 5740 q^{85} + 3780 q^{87} - 20010 q^{89} + 15951 q^{91} + 855 q^{93} + 7770 q^{95} + 44904 q^{97} - 3564 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 2358 x^{12} + 15572 x^{11} + 4050518 x^{10} + 21628620 x^{9} + 2974230644 x^{8} + \cdots + 96\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45\!\cdots\!87 \nu^{13} + \cdots - 87\!\cdots\!04 ) / 76\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 58\!\cdots\!89 \nu^{13} + \cdots - 48\!\cdots\!52 ) / 49\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!15 \nu^{13} + \cdots - 47\!\cdots\!36 ) / 56\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 93\!\cdots\!77 \nu^{13} + \cdots + 81\!\cdots\!56 ) / 25\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 46\!\cdots\!61 \nu^{13} + \cdots + 38\!\cdots\!96 ) / 93\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!07 \nu^{13} + \cdots + 49\!\cdots\!16 ) / 60\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!53 \nu^{13} + \cdots - 10\!\cdots\!04 ) / 12\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\!\cdots\!03 \nu^{13} + \cdots + 31\!\cdots\!52 ) / 28\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!57 \nu^{13} + \cdots + 10\!\cdots\!28 ) / 20\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32\!\cdots\!35 \nu^{13} + \cdots - 68\!\cdots\!48 ) / 20\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!65 \nu^{13} + \cdots + 18\!\cdots\!08 ) / 36\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!29 \nu^{13} + \cdots - 95\!\cdots\!32 ) / 30\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \cdots + 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -24\beta_{11} - 5\beta_{9} - 24\beta_{8} + 32\beta_{6} - 21\beta_{5} + 76\beta_{4} - 1089\beta_{2} - 4019 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2092 \beta_{13} + 3312 \beta_{12} - 3934 \beta_{11} + 4477 \beta_{10} + 311 \beta_{9} + 3623 \beta_{8} + \cdots - 744202 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 56301 \beta_{13} + 84193 \beta_{12} - 127944 \beta_{11} + 186485 \beta_{10} + 58541 \beta_{9} + \cdots - 1344205 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 931094 \beta_{11} + 1100938 \beta_{9} + 931094 \beta_{8} - 5391724 \beta_{6} + 3548710 \beta_{5} + \cdots + 1037678792 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 116993148 \beta_{13} - 173762754 \beta_{12} + 315808710 \beta_{11} - 348376152 \beta_{10} + \cdots + 27754913184 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6008535928 \beta_{13} - 9046820708 \beta_{12} + 11087930954 \beta_{11} - 15047909474 \beta_{10} + \cdots + 67911729458 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 117209306898 \beta_{11} - 64369155584 \beta_{9} - 117209306898 \beta_{8} + 331479668078 \beta_{6} + \cdots - 54989934315002 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10386279836428 \beta_{13} + 15589901162184 \beta_{12} - 23612628413176 \beta_{11} + \cdots - 27\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 411517283403300 \beta_{13} + 611678113682392 \beta_{12} - 810884972900448 \beta_{11} + \cdots - 59\!\cdots\!52 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 75\!\cdots\!48 \beta_{11} + \cdots + 47\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 74\!\cdots\!48 \beta_{13} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/228\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\) \(115\)
\(\chi(n)\) \(1\) \(1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
21.1442 − 36.6229i
9.98915 − 17.3017i
9.71311 − 16.8236i
−0.566103 + 0.980520i
−9.25681 + 16.0333i
−14.8161 + 25.6623i
−15.2075 + 26.3401i
21.1442 + 36.6229i
9.98915 + 17.3017i
9.71311 + 16.8236i
−0.566103 − 0.980520i
−9.25681 − 16.0333i
−14.8161 − 25.6623i
−15.2075 − 26.3401i
0 −4.50000 + 2.59808i 0 −23.1442 − 40.0870i 0 −21.4281 0 13.5000 − 23.3827i 0
145.2 0 −4.50000 + 2.59808i 0 −11.9892 − 20.7658i 0 −2.95794 0 13.5000 − 23.3827i 0
145.3 0 −4.50000 + 2.59808i 0 −11.7131 − 20.2877i 0 78.7599 0 13.5000 − 23.3827i 0
145.4 0 −4.50000 + 2.59808i 0 −1.43390 − 2.48358i 0 −73.3640 0 13.5000 − 23.3827i 0
145.5 0 −4.50000 + 2.59808i 0 7.25681 + 12.5692i 0 −14.1648 0 13.5000 − 23.3827i 0
145.6 0 −4.50000 + 2.59808i 0 12.8161 + 22.1982i 0 82.2037 0 13.5000 − 23.3827i 0
145.7 0 −4.50000 + 2.59808i 0 13.2075 + 22.8760i 0 3.95127 0 13.5000 − 23.3827i 0
217.1 0 −4.50000 − 2.59808i 0 −23.1442 + 40.0870i 0 −21.4281 0 13.5000 + 23.3827i 0
217.2 0 −4.50000 − 2.59808i 0 −11.9892 + 20.7658i 0 −2.95794 0 13.5000 + 23.3827i 0
217.3 0 −4.50000 − 2.59808i 0 −11.7131 + 20.2877i 0 78.7599 0 13.5000 + 23.3827i 0
217.4 0 −4.50000 − 2.59808i 0 −1.43390 + 2.48358i 0 −73.3640 0 13.5000 + 23.3827i 0
217.5 0 −4.50000 − 2.59808i 0 7.25681 − 12.5692i 0 −14.1648 0 13.5000 + 23.3827i 0
217.6 0 −4.50000 − 2.59808i 0 12.8161 − 22.1982i 0 82.2037 0 13.5000 + 23.3827i 0
217.7 0 −4.50000 − 2.59808i 0 13.2075 − 22.8760i 0 3.95127 0 13.5000 + 23.3827i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.5.l.a ✓ 14
3.b odd 2 1 684.5.y.e 14
19.d odd 6 1 inner 228.5.l.a ✓ 14
57.f even 6 1 684.5.y.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.5.l.a ✓ 14 1.a even 1 1 trivial
228.5.l.a ✓ 14 19.d odd 6 1 inner
684.5.y.e 14 3.b odd 2 1
684.5.y.e 14 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 30 T_{5}^{13} + 2870 T_{5}^{12} + 18756 T_{5}^{11} + 3847014 T_{5}^{10} + \cdots + 53\!\cdots\!84 \) acting on \(S_{5}^{\mathrm{new}}(228, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 27)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 53\!\cdots\!84 \) Copy content Toggle raw display
$7$ \( (T^{7} - 53 T^{6} + \cdots - 1684994347)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots + 29667711140352)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 98\!\cdots\!27 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 63\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 58\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 27\!\cdots\!83 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 35\!\cdots\!07 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 50\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 57\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 25\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 82\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 75\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 19\!\cdots\!63 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 35\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 22\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 87\!\cdots\!83 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 61\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 49\!\cdots\!68 \) Copy content Toggle raw display
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