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Properties

Label 45.9.g.b
Level $45$
Weight $9$
Character orbit 45.g
Analytic conductor $18.332$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.28");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(18.3320374528\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3006 x^{14} + 3660359 x^{12} + 2360769624 x^{10} + 888292333775 x^{8} + 201214811046486 x^{6} + \cdots + 60\!\cdots\!84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{12}\cdot 5^{19} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{6} - 158 \beta_{2}) q^{4} + (\beta_{9} - 7 \beta_{4} - 4 \beta_{3}) q^{5} + (\beta_{11} + 3 \beta_{6} + 265 \beta_{2} + \cdots - 264) q^{7}+ \cdots + (\beta_{12} + 2 \beta_{9} + \cdots - 174 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{6} - 158 \beta_{2}) q^{4} + (\beta_{9} - 7 \beta_{4} - 4 \beta_{3}) q^{5} + (\beta_{11} + 3 \beta_{6} + 265 \beta_{2} + \cdots - 264) q^{7}+ \cdots + (17100 \beta_{15} + \cdots + 10050 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4220 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4220 q^{7} - 47000 q^{10} - 37940 q^{13} - 508916 q^{16} + 844700 q^{22} - 1664300 q^{25} + 6009380 q^{28} - 944752 q^{31} + 10616140 q^{37} - 17493600 q^{40} + 4050760 q^{43} - 34233160 q^{46} + 7180240 q^{52} + 17430500 q^{55} - 27842100 q^{58} + 32032232 q^{61} + 75463480 q^{67} + 272773500 q^{70} - 198258320 q^{73} - 196046088 q^{76} + 172139600 q^{82} + 183614500 q^{85} - 624395100 q^{88} - 477271600 q^{91} + 662476480 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 3006 x^{14} + 3660359 x^{12} + 2360769624 x^{10} + 888292333775 x^{8} + 201214811046486 x^{6} + \cdots + 60\!\cdots\!84 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 99\!\cdots\!58 \nu^{14} + \cdots + 36\!\cdots\!32 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 85\!\cdots\!37 \nu^{15} + \cdots + 32\!\cdots\!48 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 42\!\cdots\!37 \nu^{15} + \cdots + 35\!\cdots\!64 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42\!\cdots\!37 \nu^{15} + \cdots - 35\!\cdots\!64 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!33 \nu^{15} + \cdots - 36\!\cdots\!36 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!34 \nu^{15} + \cdots - 24\!\cdots\!36 \nu ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16\!\cdots\!66 \nu^{15} + \cdots - 33\!\cdots\!40 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!15 \nu^{15} + \cdots + 40\!\cdots\!96 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33\!\cdots\!31 \nu^{15} + \cdots + 28\!\cdots\!72 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 96\!\cdots\!44 \nu^{15} + \cdots - 32\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 96\!\cdots\!44 \nu^{15} + \cdots + 32\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!76 \nu^{15} + \cdots - 12\!\cdots\!92 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 90\!\cdots\!59 \nu^{15} + \cdots + 18\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!97 \nu^{15} + \cdots + 32\!\cdots\!40 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 79\!\cdots\!93 \nu^{15} + \cdots - 78\!\cdots\!52 ) / 55\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{15} - 5 \beta_{14} - 5 \beta_{12} - 12 \beta_{9} - 24 \beta_{8} + 6 \beta_{5} + \cdots + 2250 \beta_{2} ) / 2250 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 12 \beta_{15} - 10 \beta_{14} - 2 \beta_{13} + 10 \beta_{12} + 163 \beta_{11} - 161 \beta_{10} + \cdots - 845241 ) / 2250 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1399 \beta_{15} + 2485 \beta_{14} + 12 \beta_{13} + 2485 \beta_{12} - 483 \beta_{11} - 489 \beta_{10} + \cdots - 489 ) / 2250 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1716 \beta_{15} + 660 \beta_{14} + 90 \beta_{13} - 660 \beta_{12} - 10235 \beta_{11} + 10145 \beta_{10} + \cdots + 32145445 ) / 150 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1031457 \beta_{15} - 1672155 \beta_{14} - 13420 \beta_{13} - 1672155 \beta_{12} + 757655 \beta_{11} + \cdots + 764365 ) / 2250 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 33703422 \beta_{15} - 9934010 \beta_{14} - 769822 \beta_{13} + 9934010 \beta_{12} + \cdots - 347245653051 ) / 2250 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 812907399 \beta_{15} + 1287049185 \beta_{14} + 10589852 \beta_{13} + 1287049185 \beta_{12} + \cdots - 885199469 ) / 2250 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2513011832 \beta_{15} + 674088520 \beta_{14} + 30891690 \beta_{13} - 674088520 \beta_{12} + \cdots + 18334095524645 ) / 150 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 664730210237 \beta_{15} - 1044899305055 \beta_{14} - 8087500140 \beta_{13} - 1044899305055 \beta_{12} + \cdots + 932675669205 ) / 2250 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 39170782881282 \beta_{15} - 10146762661510 \beta_{14} - 312966859742 \beta_{13} + \cdots - 22\!\cdots\!11 ) / 2250 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 553584944897299 \beta_{15} + 867860493952885 \beta_{14} + 6590583625692 \beta_{13} + \cdots - 938559340898949 ) / 2250 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 26\!\cdots\!48 \beta_{15} + 664701587703180 \beta_{14} + 15600768084650 \beta_{13} + \cdots + 12\!\cdots\!25 ) / 150 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 46\!\cdots\!17 \beta_{15} + \cdots + 91\!\cdots\!45 ) / 2250 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 37\!\cdots\!42 \beta_{15} + \cdots - 15\!\cdots\!71 ) / 2250 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 39\!\cdots\!99 \beta_{15} + \cdots - 87\!\cdots\!29 ) / 2250 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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
− 17.4315i
14.3888i
13.5567i
29.6766i
− 27.6766i
− 11.5567i
− 12.3888i
19.4315i
17.4315i
− 14.3888i
− 13.5567i
− 29.6766i
27.6766i
11.5567i
12.3888i
− 19.4315i
−21.5835 + 21.5835i 0 − 675.695i 47.4620 + 623.195i 0 −2752.21 + 2752.21i 9058.50 + 9058.50i 0 −14475.1 − 12426.3i
28.2 −15.1900 + 15.1900i 0 − 205.471i −115.601 − 614.216i 0 605.930 − 605.930i −767.539 − 767.539i 0 11085.9 + 7573.95i
28.3 −10.7931 + 10.7931i 0 23.0166i 489.683 + 388.375i 0 2354.79 − 2354.79i −3011.46 − 3011.46i 0 −9477.00 + 1093.43i
28.4 −3.79807 + 3.79807i 0 227.149i −563.744 + 269.849i 0 −1263.51 + 1263.51i −1835.03 − 1835.03i 0 1116.23 − 3166.04i
28.5 3.79807 − 3.79807i 0 227.149i 563.744 − 269.849i 0 −1263.51 + 1263.51i 1835.03 + 1835.03i 0 1116.23 − 3166.04i
28.6 10.7931 − 10.7931i 0 23.0166i −489.683 − 388.375i 0 2354.79 − 2354.79i 3011.46 + 3011.46i 0 −9477.00 + 1093.43i
28.7 15.1900 − 15.1900i 0 − 205.471i 115.601 + 614.216i 0 605.930 − 605.930i 767.539 + 767.539i 0 11085.9 + 7573.95i
28.8 21.5835 − 21.5835i 0 − 675.695i −47.4620 − 623.195i 0 −2752.21 + 2752.21i −9058.50 − 9058.50i 0 −14475.1 − 12426.3i
37.1 −21.5835 − 21.5835i 0 675.695i 47.4620 − 623.195i 0 −2752.21 − 2752.21i 9058.50 − 9058.50i 0 −14475.1 + 12426.3i
37.2 −15.1900 − 15.1900i 0 205.471i −115.601 + 614.216i 0 605.930 + 605.930i −767.539 + 767.539i 0 11085.9 − 7573.95i
37.3 −10.7931 − 10.7931i 0 − 23.0166i 489.683 − 388.375i 0 2354.79 + 2354.79i −3011.46 + 3011.46i 0 −9477.00 − 1093.43i
37.4 −3.79807 − 3.79807i 0 − 227.149i −563.744 − 269.849i 0 −1263.51 − 1263.51i −1835.03 + 1835.03i 0 1116.23 + 3166.04i
37.5 3.79807 + 3.79807i 0 − 227.149i 563.744 + 269.849i 0 −1263.51 − 1263.51i 1835.03 − 1835.03i 0 1116.23 + 3166.04i
37.6 10.7931 + 10.7931i 0 − 23.0166i −489.683 + 388.375i 0 2354.79 + 2354.79i 3011.46 − 3011.46i 0 −9477.00 − 1093.43i
37.7 15.1900 + 15.1900i 0 205.471i 115.601 − 614.216i 0 605.930 + 605.930i 767.539 − 767.539i 0 11085.9 − 7573.95i
37.8 21.5835 + 21.5835i 0 675.695i −47.4620 + 623.195i 0 −2752.21 − 2752.21i −9058.50 + 9058.50i 0 −14475.1 + 12426.3i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.9.g.b ✓ 16
3.b odd 2 1 inner 45.9.g.b ✓ 16
5.c odd 4 1 inner 45.9.g.b ✓ 16
15.e even 4 1 inner 45.9.g.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.9.g.b ✓ 16 1.a even 1 1 trivial
45.9.g.b ✓ 16 3.b odd 2 1 inner
45.9.g.b ✓ 16 5.c odd 4 1 inner
45.9.g.b ✓ 16 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 1136125T_{2}^{12} + 244480702500T_{2}^{8} + 10236982600000000T_{2}^{4} + 8352100000000000000 \) acting on \(S_{9}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 81\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 45\!\cdots\!44)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 27\!\cdots\!16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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