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Properties

Label 45.13.g.a
Level $45$
Weight $13$
Character orbit 45.g
Analytic conductor $41.130$
Analytic rank $0$
Dimension $10$
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] = [45,13,Mod(28,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.28");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 45.g (of order \(4\), 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: \(41.1297217774\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 7950x^{8} + 16939113x^{6} + 4574579500x^{4} + 337520899536x^{2} + 6615595526400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2}\cdot 5^{9} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{8} + 8 \beta_{6} + \cdots + 1549 \beta_1) q^{4}+ \cdots + ( - 4 \beta_{9} + 31 \beta_{8} + \cdots + 47131) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{8} + 8 \beta_{6} + \cdots + 1549 \beta_1) q^{4}+ \cdots + ( - 18795772 \beta_{9} + \cdots + 148450654597) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 4250 q^{5} + 279598 q^{7} + 469980 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} + 4250 q^{5} + 279598 q^{7} + 469980 q^{8} + 1552150 q^{10} - 312620 q^{11} + 5290738 q^{13} + 30547960 q^{16} + 41269502 q^{17} - 334988100 q^{20} - 155490544 q^{22} + 510099842 q^{23} + 942201250 q^{25} - 1475846420 q^{26} - 3562106488 q^{28} + 3077089820 q^{31} + 4623883832 q^{32} - 9330787150 q^{35} - 2599618502 q^{37} + 15310240920 q^{38} + 15901243500 q^{40} - 7412079020 q^{41} - 5784410402 q^{43} - 7382547880 q^{46} - 16053249598 q^{47} + 68314688750 q^{50} + 96763417228 q^{52} - 101763514618 q^{53} - 84180068500 q^{55} + 172002747600 q^{56} + 135238672320 q^{58} + 7731718220 q^{61} - 193287375176 q^{62} + 338075024150 q^{65} - 80010636002 q^{67} - 204699541412 q^{68} + 376969924200 q^{70} + 46557252580 q^{71} - 448527032342 q^{73} + 305095930800 q^{76} + 425580405844 q^{77} - 873032236000 q^{80} - 671946416464 q^{82} + 91118376722 q^{83} + 543768569650 q^{85} - 414117747320 q^{86} - 1842434230560 q^{88} + 2737742572220 q^{91} - 906853941448 q^{92} + 1044695070000 q^{95} - 1409507601302 q^{97} + 1481746533298 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 7950x^{8} + 16939113x^{6} + 4574579500x^{4} + 337520899536x^{2} + 6615595526400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26465 \nu^{9} + 176059482 \nu^{7} + 167682655437 \nu^{5} - 471206316514012 \nu^{3} - 74\!\cdots\!92 \nu ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2320992487 \nu^{8} + 18354612359011 \nu^{6} + \cdots + 42\!\cdots\!28 ) / 38\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1460065153745 \nu^{9} + 56782570931856 \nu^{8} + \cdots - 65\!\cdots\!80 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1460065153745 \nu^{9} + 56782570931856 \nu^{8} + \cdots - 65\!\cdots\!80 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5235689538055 \nu^{9} + 55538770561995 \nu^{8} + \cdots + 59\!\cdots\!80 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5235689538055 \nu^{9} - 55538770561995 \nu^{8} + \cdots - 59\!\cdots\!80 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 376177622934695 \nu^{9} + \cdots + 16\!\cdots\!20 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 965372918064815 \nu^{9} + \cdots - 14\!\cdots\!76 \nu ) / 73\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 458260888172395 \nu^{9} + 686396955964080 \nu^{8} + \cdots + 11\!\cdots\!80 ) / 10\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 3 \beta_{9} - 8 \beta_{8} + \beta_{7} - 115 \beta_{6} - 115 \beta_{5} + 16 \beta_{4} - 18 \beta_{3} + \cdots - 2 ) / 1500 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{8} + 3 \beta_{7} - 1280 \beta_{6} + 1280 \beta_{5} - 272 \beta_{4} - 274 \beta_{3} + \cdots - 317606 ) / 200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 23829 \beta_{9} + 54169 \beta_{8} - 7943 \beta_{7} + 1387820 \beta_{6} + 1387820 \beta_{5} + \cdots + 15886 ) / 3000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 907 \beta_{9} - 907 \beta_{8} - 2721 \beta_{7} + 1004276 \beta_{6} - 1004276 \beta_{5} + \cdots + 234283294 ) / 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 91566861 \beta_{9} - 210463921 \beta_{8} + 30522287 \beta_{7} - 5260650380 \beta_{6} + \cdots - 61044574 ) / 3000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 21041599 \beta_{9} + 21041599 \beta_{8} + 63124797 \beta_{7} - 18616513460 \beta_{6} + \cdots - 4481712980374 ) / 200 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 351810148749 \beta_{9} + 824142222289 \beta_{8} - 117270049583 \beta_{7} + 19611758189420 \beta_{6} + \cdots + 234540099166 ) / 3000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 94863113647 \beta_{9} - 94863113647 \beta_{8} - 284589340941 \beta_{7} + 68767072032500 \beta_{6} + \cdots + 17\!\cdots\!62 ) / 200 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 13\!\cdots\!41 \beta_{9} + \cdots - 901852006264894 ) / 3000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
− 14.0132i
5.61354i
62.7587i
− 60.9123i
8.55327i
14.0132i
− 5.61354i
− 62.7587i
60.9123i
− 8.55327i
−77.8857 + 77.8857i 0 − 8036.37i 1256.84 − 15574.4i 0 107575. − 107575.i 306898. + 306898.i 0 1.11513e6 + 1.31091e6i
28.2 −34.6956 + 34.6956i 0 1688.43i −12911.0 + 8800.45i 0 25919.1 − 25919.1i −200694. − 200694.i 0 142616. − 753290.i
28.3 −3.05147 + 3.05147i 0 4077.38i 15614.1 + 583.350i 0 −139850. + 139850.i −24940.8 − 24940.8i 0 −49426.1 + 45865.9i
28.4 54.8437 − 54.8437i 0 − 1919.66i −15616.9 + 502.591i 0 78559.8 − 78559.8i 119359. + 119359.i 0 −828925. + 884053.i
28.5 61.7891 − 61.7891i 0 − 3539.78i 13781.9 − 7362.02i 0 67595.3 − 67595.3i 34368.1 + 34368.1i 0 396679. − 1.30646e6i
37.1 −77.8857 − 77.8857i 0 8036.37i 1256.84 + 15574.4i 0 107575. + 107575.i 306898. − 306898.i 0 1.11513e6 − 1.31091e6i
37.2 −34.6956 − 34.6956i 0 − 1688.43i −12911.0 − 8800.45i 0 25919.1 + 25919.1i −200694. + 200694.i 0 142616. + 753290.i
37.3 −3.05147 − 3.05147i 0 − 4077.38i 15614.1 − 583.350i 0 −139850. − 139850.i −24940.8 + 24940.8i 0 −49426.1 − 45865.9i
37.4 54.8437 + 54.8437i 0 1919.66i −15616.9 − 502.591i 0 78559.8 + 78559.8i 119359. − 119359.i 0 −828925. − 884053.i
37.5 61.7891 + 61.7891i 0 3539.78i 13781.9 + 7362.02i 0 67595.3 + 67595.3i 34368.1 − 34368.1i 0 396679. + 1.30646e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.13.g.a 10
3.b odd 2 1 5.13.c.a ✓ 10
5.c odd 4 1 inner 45.13.g.a 10
12.b even 2 1 80.13.p.c 10
15.d odd 2 1 25.13.c.b 10
15.e even 4 1 5.13.c.a ✓ 10
15.e even 4 1 25.13.c.b 10
60.l odd 4 1 80.13.p.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.13.c.a ✓ 10 3.b odd 2 1
5.13.c.a ✓ 10 15.e even 4 1
25.13.c.b 10 15.d odd 2 1
25.13.c.b 10 15.e even 4 1
45.13.g.a 10 1.a even 1 1 trivial
45.13.g.a 10 5.c odd 4 1 inner
80.13.p.c 10 12.b even 2 1
80.13.p.c 10 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 2 T_{2}^{9} + 2 T_{2}^{8} - 151200 T_{2}^{7} + 124044608 T_{2}^{6} - 2679039616 T_{2}^{5} + \cdots + 24\!\cdots\!32 \) acting on \(S_{13}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 33\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 12\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 29\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 82\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 91\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 61\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 16\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 48\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 32\!\cdots\!32 \) Copy content Toggle raw display
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