Properties

Label 1368.5.o.a
Level $1368$
Weight $5$
Character orbit 1368.o
Analytic conductor $141.410$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1368,5,Mod(721,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.721");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1368.o (of order \(2\), degree \(1\), 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: \(141.410109499\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + 37911701888064 x^{8} + 915101881952256 x^{6} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{50} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - \beta_{5} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + ( - \beta_{5} + 2) q^{7} + ( - \beta_{6} - \beta_{3} + 1) q^{11} + \beta_{13} q^{13} + (\beta_{7} + \beta_{3} - 11) q^{17} + (\beta_{16} - \beta_{3} - 30) q^{19} + ( - \beta_{9} - 3 \beta_{3} - 29) q^{23} + ( - \beta_{8} + \beta_{6} + \beta_{4} + 3 \beta_{3} + 71) q^{25} + (\beta_{19} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{10} - \beta_1) q^{29} + (\beta_{19} - 2 \beta_{17} - \beta_{16} + \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + \beta_{11} + \beta_{10} + \cdots + 20 \beta_1) q^{31}+ \cdots + (\beta_{19} + 3 \beta_{18} + 16 \beta_{17} + 7 \beta_{16} - 7 \beta_{15} - 7 \beta_{14} + \cdots - 130 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 32 q^{7} + 24 q^{11} - 216 q^{17} - 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} - 1256 q^{43} + 3768 q^{47} - 2740 q^{49} - 10128 q^{55} + 352 q^{61} + 1352 q^{73} - 9288 q^{77} + 16104 q^{83} + 10232 q^{85} - 14232 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + 37911701888064 x^{8} + 915101881952256 x^{6} + \cdots + 34\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 70\!\cdots\!53 \nu^{18} + \cdots + 76\!\cdots\!72 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70\!\cdots\!53 \nu^{18} + \cdots + 70\!\cdots\!72 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!71 \nu^{18} + \cdots + 58\!\cdots\!08 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64\!\cdots\!81 \nu^{18} + \cdots - 93\!\cdots\!96 ) / 30\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!99 \nu^{18} + \cdots - 39\!\cdots\!40 ) / 89\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!25 \nu^{18} + \cdots + 38\!\cdots\!16 ) / 28\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 67\!\cdots\!57 \nu^{18} + \cdots - 32\!\cdots\!60 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!95 \nu^{18} + \cdots - 24\!\cdots\!56 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 70\!\cdots\!53 \nu^{19} + \cdots - 70\!\cdots\!44 \nu ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 47\!\cdots\!39 \nu^{19} + \cdots - 33\!\cdots\!68 \nu ) / 32\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 64\!\cdots\!81 \nu^{19} + \cdots + 94\!\cdots\!08 \nu ) / 30\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!19 \nu^{19} + \cdots + 19\!\cdots\!12 \nu ) / 51\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!93 \nu^{19} + \cdots + 18\!\cdots\!40 \nu ) / 48\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20\!\cdots\!75 \nu^{19} + \cdots + 33\!\cdots\!00 ) / 34\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 20\!\cdots\!75 \nu^{19} + \cdots + 33\!\cdots\!00 ) / 34\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 37\!\cdots\!53 \nu^{19} + \cdots - 60\!\cdots\!24 \nu ) / 26\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!41 \nu^{19} + \cdots + 15\!\cdots\!88 \nu ) / 58\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 15\!\cdots\!19 \nu^{19} + \cdots - 24\!\cdots\!12 \nu ) / 51\!\cdots\!88 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} - \beta_{15} + \beta_{12} - 175\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{9} + 10 \beta_{7} - 22 \beta_{6} + 46 \beta_{5} - 6 \beta_{4} + 277 \beta_{3} - 235 \beta_{2} + 17500 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{19} + 28 \beta_{18} - 241 \beta_{17} - 291 \beta_{16} + 291 \beta_{15} - 182 \beta_{14} + 112 \beta_{13} - 333 \beta_{12} - 42 \beta_{11} - 56 \beta_{10} + 35457 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 234 \beta_{16} + 234 \beta_{15} - 898 \beta_{9} + 146 \beta_{8} - 3504 \beta_{7} + 7428 \beta_{6} - 15102 \beta_{5} + 2540 \beta_{4} - 65087 \beta_{3} + 53061 \beta_{2} - 3550438 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7980 \beta_{19} - 11004 \beta_{18} + 51293 \beta_{17} + 76025 \beta_{16} - 76025 \beta_{15} + 80286 \beta_{14} - 50292 \beta_{13} + 89693 \beta_{12} + 16776 \beta_{11} + 19352 \beta_{10} - 7612981 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 19822 \beta_{16} + 19822 \beta_{15} + 282914 \beta_{9} - 51140 \beta_{8} + 1007638 \beta_{7} - 1946034 \beta_{6} + 3990734 \beta_{5} - 848226 \beta_{4} + 14782537 \beta_{3} + \cdots + 763911296 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3084394 \beta_{19} + 3297452 \beta_{18} - 10737017 \beta_{17} - 19356063 \beta_{16} + 19356063 \beta_{15} - 26691478 \beta_{14} + 15782024 \beta_{13} - 22579949 \beta_{12} + \cdots + 1682376793 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20655942 \beta_{16} - 20655942 \beta_{15} - 78374274 \beta_{9} + 12438198 \beta_{8} - 269994396 \beta_{7} + 473039808 \beta_{6} - 987799614 \beta_{5} + \cdots - 169264445898 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1002420040 \beta_{19} - 904682252 \beta_{18} + 2263300229 \beta_{17} + 4867692981 \beta_{16} - 4867692981 \beta_{15} + 7971477598 \beta_{14} - 4357837964 \beta_{13} + \cdots - 378339049933 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8124839966 \beta_{16} + 8124839966 \beta_{15} + 20445235362 \beta_{9} - 2525244936 \beta_{8} + 69769361570 \beta_{7} - 112419964174 \beta_{6} + \cdots + 38178353707492 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 296492631078 \beta_{19} + 239102205084 \beta_{18} - 483370031553 \beta_{17} - 1214351170587 \beta_{16} + 1214351170587 \beta_{15} + \cdots + 86120707333457 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2588688905206 \beta_{16} - 2588688905206 \beta_{15} - 5162871005442 \beta_{9} + 434472103386 \beta_{8} - 17650004264584 \beta_{7} + \cdots - 87\!\cdots\!06 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 82864254888292 \beta_{19} - 62001344303324 \beta_{18} + 104708928870253 \beta_{17} + 301217752722353 \beta_{16} - 301217752722353 \beta_{15} + \cdots - 19\!\cdots\!17 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 753426470023374 \beta_{16} + 753426470023374 \beta_{15} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 22\!\cdots\!26 \beta_{19} + \cdots + 45\!\cdots\!89 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 20\!\cdots\!02 \beta_{16} + \cdots - 46\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 58\!\cdots\!48 \beta_{19} + \cdots - 10\!\cdots\!09 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
6.40102i
6.40102i
9.35107i
9.35107i
13.8615i
13.8615i
1.51452i
1.51452i
15.6226i
15.6226i
6.69459i
6.69459i
11.4302i
11.4302i
1.48358i
1.48358i
5.59979i
5.59979i
14.8343i
14.8343i
0 0 0 −47.0652 0 32.2009 0 0 0
721.2 0 0 0 −47.0652 0 32.2009 0 0 0
721.3 0 0 0 −21.0112 0 −9.13929 0 0 0
721.4 0 0 0 −21.0112 0 −9.13929 0 0 0
721.5 0 0 0 −16.6305 0 −34.1005 0 0 0
721.6 0 0 0 −16.6305 0 −34.1005 0 0 0
721.7 0 0 0 −15.8064 0 −74.7269 0 0 0
721.8 0 0 0 −15.8064 0 −74.7269 0 0 0
721.9 0 0 0 −6.30407 0 −28.3216 0 0 0
721.10 0 0 0 −6.30407 0 −28.3216 0 0 0
721.11 0 0 0 −2.80752 0 75.3813 0 0 0
721.12 0 0 0 −2.80752 0 75.3813 0 0 0
721.13 0 0 0 9.10929 0 75.9168 0 0 0
721.14 0 0 0 9.10929 0 75.9168 0 0 0
721.15 0 0 0 22.0416 0 25.0930 0 0 0
721.16 0 0 0 22.0416 0 25.0930 0 0 0
721.17 0 0 0 33.0135 0 −43.4389 0 0 0
721.18 0 0 0 33.0135 0 −43.4389 0 0 0
721.19 0 0 0 45.4605 0 −2.86483 0 0 0
721.20 0 0 0 45.4605 0 −2.86483 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.5.o.a 20
3.b odd 2 1 152.5.e.a 20
12.b even 2 1 304.5.e.f 20
19.b odd 2 1 inner 1368.5.o.a 20
57.d even 2 1 152.5.e.a 20
228.b odd 2 1 304.5.e.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.5.e.a 20 3.b odd 2 1
152.5.e.a 20 57.d even 2 1
304.5.e.f 20 12.b even 2 1
304.5.e.f 20 228.b odd 2 1
1368.5.o.a 20 1.a even 1 1 trivial
1368.5.o.a 20 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 3478 T_{5}^{8} - 6348 T_{5}^{7} + 3312217 T_{5}^{6} + 21809988 T_{5}^{5} - 1014011772 T_{5}^{4} - 10460647104 T_{5}^{3} + 49729423104 T_{5}^{2} + \cdots + 1386394827776 \) acting on \(S_{5}^{\mathrm{new}}(1368, [\chi])\). 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
$5$ \( (T^{10} - 3478 T^{8} + \cdots + 1386394827776)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} - 16 T^{9} + \cdots + 379548178285972)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 12 T^{9} + \cdots - 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 269268 T^{18} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{10} + 108 T^{9} + \cdots - 60\!\cdots\!04)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + 596 T^{19} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{10} + 288 T^{9} + \cdots - 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + 6992228 T^{18} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{20} + 10576576 T^{18} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{20} + 24738912 T^{18} + \cdots + 92\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + 38072064 T^{18} + \cdots + 92\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{10} + 628 T^{9} + \cdots - 33\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 1884 T^{9} + \cdots + 62\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 112629156 T^{18} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{20} + 151851092 T^{18} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{10} - 176 T^{9} + \cdots + 33\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + 250778052 T^{18} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{20} + 308190992 T^{18} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{10} - 676 T^{9} + \cdots - 80\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + 562672512 T^{18} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{10} - 8052 T^{9} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 529225648 T^{18} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{20} + 1129821904 T^{18} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
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