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Properties

Label 228.3.r.a
Level $228$
Weight $3$
Character orbit 228.r
Analytic conductor $6.213$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [228,3,Mod(13,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.r (of order \(18\), degree \(6\), 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: \(6.21255002741\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 186 x^{16} + 736 x^{15} + 15318 x^{14} - 30747 x^{13} - 692832 x^{12} + \cdots + 86407195071 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - 2 \beta_{3}) q^{3} + (\beta_{10} + \beta_{9} - \beta_{8} - 1) q^{5} + (\beta_{16} - \beta_{14} + \cdots - \beta_{5}) q^{7}+ \cdots + (3 \beta_{9} + 3 \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - 2 \beta_{3}) q^{3} + (\beta_{10} + \beta_{9} - \beta_{8} - 1) q^{5} + (\beta_{16} - \beta_{14} + \cdots - \beta_{5}) q^{7}+ \cdots + (3 \beta_{16} - 3 \beta_{15} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 9 q^{7} + 15 q^{11} - 12 q^{13} - 18 q^{15} + 18 q^{17} - 42 q^{19} + 45 q^{21} + 69 q^{23} + 90 q^{25} - 81 q^{27} + 102 q^{29} + 9 q^{33} + 30 q^{35} - 18 q^{39} - 18 q^{41} - 78 q^{43} - 9 q^{45} - 33 q^{47} - 78 q^{49} - 63 q^{51} - 132 q^{53} - 231 q^{55} - 99 q^{57} + 348 q^{59} + 117 q^{61} + 63 q^{63} + 261 q^{65} + 30 q^{67} - 108 q^{69} + 75 q^{73} - 180 q^{77} - 384 q^{79} + 45 q^{83} + 27 q^{85} - 9 q^{87} - 357 q^{89} - 738 q^{91} - 54 q^{93} - 330 q^{95} + 477 q^{97} + 45 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 186 x^{16} + 736 x^{15} + 15318 x^{14} - 30747 x^{13} - 692832 x^{12} + \cdots + 86407195071 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 67\!\cdots\!09 \nu^{17} + \cdots + 17\!\cdots\!76 ) / 10\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!58 \nu^{17} + \cdots - 51\!\cdots\!16 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21\!\cdots\!23 \nu^{17} + \cdots - 15\!\cdots\!77 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!99 \nu^{17} + \cdots + 16\!\cdots\!96 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32\!\cdots\!79 \nu^{17} + \cdots + 24\!\cdots\!29 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34\!\cdots\!93 \nu^{17} + \cdots - 14\!\cdots\!26 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 36\!\cdots\!96 \nu^{17} + \cdots + 10\!\cdots\!47 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!33 \nu^{17} + \cdots + 11\!\cdots\!04 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 34\!\cdots\!36 \nu^{17} + \cdots - 11\!\cdots\!10 ) / 16\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 31\!\cdots\!73 \nu^{17} + \cdots - 17\!\cdots\!93 ) / 10\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!06 \nu^{17} + \cdots - 26\!\cdots\!30 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 45\!\cdots\!88 \nu^{17} + \cdots - 27\!\cdots\!28 ) / 10\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 45\!\cdots\!08 \nu^{17} + \cdots - 86\!\cdots\!32 ) / 10\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\!\cdots\!12 \nu^{17} + \cdots + 58\!\cdots\!78 ) / 33\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10\!\cdots\!72 \nu^{17} + \cdots + 31\!\cdots\!46 ) / 10\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 14\!\cdots\!67 \nu^{17} + \cdots - 46\!\cdots\!64 ) / 10\!\cdots\!13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{14} - \beta_{12} + \beta_{11} - 6 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + \beta_{6} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} - 8 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 40 \beta_{9} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{17} + 19 \beta_{16} - 18 \beta_{15} - 56 \beta_{14} + 23 \beta_{13} - 80 \beta_{12} + \cdots + 888 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 95 \beta_{17} + \beta_{16} - 65 \beta_{15} - 98 \beta_{14} + 291 \beta_{13} - 754 \beta_{12} + \cdots + 5763 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 901 \beta_{17} + 1090 \beta_{16} - 3068 \beta_{15} - 3064 \beta_{14} + 2783 \beta_{13} - 6361 \beta_{12} + \cdots + 59680 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12894 \beta_{17} + 1079 \beta_{16} - 29477 \beta_{15} - 15655 \beta_{14} + 28435 \beta_{13} + \cdots + 512984 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 119582 \beta_{17} + 39051 \beta_{16} - 384802 \beta_{15} - 191652 \beta_{14} + 271307 \beta_{13} + \cdots + 4836175 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1391943 \beta_{17} - 36348 \beta_{16} - 3895755 \beta_{15} - 1333900 \beta_{14} + 2661126 \beta_{13} + \cdots + 43983958 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 13442058 \beta_{17} - 363362 \beta_{16} - 41807834 \beta_{15} - 12616484 \beta_{14} + 25239608 \beta_{13} + \cdots + 405084504 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 141093579 \beta_{17} - 23772043 \beta_{16} - 418920216 \beta_{15} - 95944459 \beta_{14} + \cdots + 3704518849 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1380308360 \beta_{17} - 298681087 \beta_{16} - 4234581109 \beta_{15} - 811828890 \beta_{14} + \cdots + 33846212862 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13843490466 \beta_{17} - 4057126396 \beta_{16} - 41838157803 \beta_{15} - 6180295990 \beta_{14} + \cdots + 308593238563 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 135179065699 \beta_{17} - 46191268543 \beta_{16} - 412636433144 \beta_{15} - 48079393729 \beta_{14} + \cdots + 2807044934499 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1326583248668 \beta_{17} - 521882126992 \beta_{16} - 4030151620474 \beta_{15} - 344076478256 \beta_{14} + \cdots + 25497520572857 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 12871020359946 \beta_{17} - 5595318560018 \beta_{16} - 39219479635990 \beta_{15} + \cdots + 231120788616736 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 124751293467922 \beta_{17} - 59131700705958 \beta_{16} - 379556786694809 \beta_{15} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(77\) \(97\) \(115\)
\(\chi(n)\) \(1\) \(-\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
5.31416 − 0.984808i
−2.01589 − 0.984808i
−4.00401 − 0.984808i
−5.23052 − 0.342020i
−2.35255 − 0.342020i
9.17547 − 0.342020i
3.81126 − 0.642788i
2.83307 − 0.642788i
−4.53099 − 0.642788i
−5.23052 + 0.342020i
−2.35255 + 0.342020i
9.17547 + 0.342020i
5.31416 + 0.984808i
−2.01589 + 0.984808i
−4.00401 + 0.984808i
3.81126 + 0.642788i
2.83307 + 0.642788i
−4.53099 + 0.642788i
0 0.592396 − 1.62760i 0 −1.21899 − 6.91325i 0 −2.39816 + 4.15373i 0 −2.29813 − 1.92836i 0
13.2 0 0.592396 − 1.62760i 0 0.0538569 + 0.305438i 0 3.68992 − 6.39114i 0 −2.29813 − 1.92836i 0
13.3 0 0.592396 − 1.62760i 0 0.399090 + 2.26335i 0 −5.19210 + 8.99298i 0 −2.29813 − 1.92836i 0
97.1 0 1.11334 − 1.32683i 0 −5.47175 + 1.99155i 0 4.45513 + 7.71652i 0 −0.520945 − 2.95442i 0
97.2 0 1.11334 − 1.32683i 0 −2.76735 + 1.00723i 0 −2.75609 − 4.77368i 0 −0.520945 − 2.95442i 0
97.3 0 1.11334 − 1.32683i 0 8.06545 − 2.93558i 0 1.15212 + 1.99553i 0 −0.520945 − 2.95442i 0
109.1 0 −1.70574 − 0.300767i 0 −2.06673 − 1.73419i 0 3.45786 + 5.98918i 0 2.81908 + 1.02606i 0
109.2 0 −1.70574 − 0.300767i 0 −1.31739 − 1.10542i 0 −0.385791 − 0.668210i 0 2.81908 + 1.02606i 0
109.3 0 −1.70574 − 0.300767i 0 4.32381 + 3.62811i 0 −6.52290 − 11.2980i 0 2.81908 + 1.02606i 0
181.1 0 1.11334 + 1.32683i 0 −5.47175 − 1.99155i 0 4.45513 − 7.71652i 0 −0.520945 + 2.95442i 0
181.2 0 1.11334 + 1.32683i 0 −2.76735 − 1.00723i 0 −2.75609 + 4.77368i 0 −0.520945 + 2.95442i 0
181.3 0 1.11334 + 1.32683i 0 8.06545 + 2.93558i 0 1.15212 − 1.99553i 0 −0.520945 + 2.95442i 0
193.1 0 0.592396 + 1.62760i 0 −1.21899 + 6.91325i 0 −2.39816 − 4.15373i 0 −2.29813 + 1.92836i 0
193.2 0 0.592396 + 1.62760i 0 0.0538569 − 0.305438i 0 3.68992 + 6.39114i 0 −2.29813 + 1.92836i 0
193.3 0 0.592396 + 1.62760i 0 0.399090 − 2.26335i 0 −5.19210 − 8.99298i 0 −2.29813 + 1.92836i 0
205.1 0 −1.70574 + 0.300767i 0 −2.06673 + 1.73419i 0 3.45786 − 5.98918i 0 2.81908 − 1.02606i 0
205.2 0 −1.70574 + 0.300767i 0 −1.31739 + 1.10542i 0 −0.385791 + 0.668210i 0 2.81908 − 1.02606i 0
205.3 0 −1.70574 + 0.300767i 0 4.32381 − 3.62811i 0 −6.52290 + 11.2980i 0 2.81908 − 1.02606i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.3.r.a ✓ 18
19.f odd 18 1 inner 228.3.r.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.r.a ✓ 18 1.a even 1 1 trivial
228.3.r.a ✓ 18 19.f odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 45 T_{5}^{16} - 196 T_{5}^{15} + 18873 T_{5}^{13} + 148251 T_{5}^{12} + 14139 T_{5}^{11} + \cdots + 371988369 \) acting on \(S_{3}^{\mathrm{new}}(228, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{6} + 9 T^{3} + 27)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 371988369 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 8385269607289 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 132896467093329 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 18\!\cdots\!03 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 14\!\cdots\!63 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 34\!\cdots\!43 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 23\!\cdots\!63 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 29\!\cdots\!23 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 48\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 27\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 66\!\cdots\!83 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 56\!\cdots\!83 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 10\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 56\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 28\!\cdots\!43 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 12\!\cdots\!83 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 28\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 37\!\cdots\!87 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 17\!\cdots\!27 \) Copy content Toggle raw display
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