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Properties

Label 45.9.g.c
Level $45$
Weight $9$
Character orbit 45.g
Analytic conductor $18.332$
Analytic rank $0$
Dimension $16$
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] = [45,9,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: \( x^{16} - 4140 x^{13} + 1109893 x^{12} - 3063780 x^{11} + 8569800 x^{10} - 2336277960 x^{9} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{20}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{9} - 3 \beta_{3} + \cdots - 158 \beta_1) q^{4}+ \cdots + ( - \beta_{11} + 3 \beta_{10} + \cdots - 1091) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{9} - 3 \beta_{3} + \cdots - 158 \beta_1) q^{4}+ \cdots + ( - 18066 \beta_{14} - 6156 \beta_{13} + \cdots + 3127642) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 444 q^{5} + 4540 q^{7} - 17460 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 444 q^{5} + 4540 q^{7} - 17460 q^{8} - 25496 q^{10} + 23616 q^{11} + 133420 q^{13} - 471380 q^{16} - 573300 q^{17} + 863436 q^{20} - 234700 q^{22} - 651480 q^{23} + 1459756 q^{25} + 448848 q^{26} - 3567940 q^{28} + 1311776 q^{31} - 641460 q^{32} - 841080 q^{35} - 3607340 q^{37} - 8139840 q^{38} - 324552 q^{40} + 14740104 q^{41} - 4805480 q^{43} + 14024216 q^{46} - 26529600 q^{47} + 38452896 q^{50} - 15861080 q^{52} - 16612140 q^{53} - 7043284 q^{55} - 10752000 q^{56} + 63562980 q^{58} - 12550600 q^{61} + 35190840 q^{62} - 125689188 q^{65} + 46836760 q^{67} + 197811840 q^{68} + 754260 q^{70} + 85681968 q^{71} - 50835800 q^{73} + 101166648 q^{76} - 97175880 q^{77} - 339741204 q^{80} - 181542400 q^{82} + 208234800 q^{83} - 209242748 q^{85} + 187512576 q^{86} + 138207420 q^{88} + 38623856 q^{91} - 652331400 q^{92} + 74686896 q^{95} - 138370520 q^{97} + 50186520 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4140 x^{13} + 1109893 x^{12} - 3063780 x^{11} + 8569800 x^{10} - 2336277960 x^{9} + \cdots + 66\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 96\!\cdots\!78 \nu^{15} + \cdots + 52\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65\!\cdots\!74 \nu^{15} + \cdots - 16\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39\!\cdots\!03 \nu^{15} + \cdots - 29\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!51 \nu^{15} + \cdots - 89\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!41 \nu^{15} + \cdots - 59\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!99 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34\!\cdots\!22 \nu^{15} + \cdots + 89\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 63\!\cdots\!91 \nu^{15} + \cdots + 71\!\cdots\!00 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!91 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 48\!\cdots\!57 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!19 \nu^{15} + \cdots - 94\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 74\!\cdots\!32 \nu^{15} + \cdots - 22\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 78\!\cdots\!97 \nu^{15} + \cdots + 18\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 22\!\cdots\!11 \nu^{15} + \cdots - 33\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!02 \nu^{15} + \cdots + 32\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 53\beta_{2} ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{14} + 4 \beta_{12} + 23 \beta_{9} + 4 \beta_{8} + \beta_{7} + \beta_{6} + \cdots - 10934 \beta_1 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{14} + 2 \beta_{13} + \beta_{11} - \beta_{10} - 2 \beta_{8} - \beta_{7} + 23 \beta_{6} + \cdots + 777 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 252 \beta_{15} + 333 \beta_{14} + 387 \beta_{13} + 4492 \beta_{12} - 4159 \beta_{11} + 252 \beta_{10} + \cdots - 7495796 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 13815 \beta_{15} - 27477 \beta_{14} + 23787 \beta_{13} - 47574 \beta_{12} + 63126 \beta_{11} + \cdots + 25878825 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6274 \beta_{15} + 148605 \beta_{14} - 11011 \beta_{13} - 164692 \beta_{12} + 11011 \beta_{11} + \cdots + 206761164 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 62341218 \beta_{14} - 54319248 \beta_{13} + 16043940 \beta_{12} - 26251425 \beta_{11} + \cdots - 26709633549 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 63307764 \beta_{15} - 205029171 \beta_{14} - 274940109 \beta_{13} - 3918798500 \beta_{12} + \cdots + 4287413190448 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 193668291 \beta_{15} + 944521371 \beta_{14} - 505840601 \beta_{13} + 1011681202 \beta_{12} + \cdots - 948838803063 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 20529981126 \beta_{15} - 3350632870151 \beta_{14} + 118354382001 \beta_{13} + \cdots - 33\!\cdots\!20 \beta_1 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 55024501122810 \beta_{14} + 40353816653640 \beta_{13} - 29341368938340 \beta_{12} + \cdots + 23\!\cdots\!93 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2925541320320 \beta_{15} + 1815338707165 \beta_{14} + 3964540030435 \beta_{13} + \cdots - 98\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 10\!\cdots\!81 \beta_{15} + \cdots + 21\!\cdots\!13 ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 11\!\cdots\!66 \beta_{15} + \cdots + 21\!\cdots\!52 \beta_1 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 16\!\cdots\!06 \beta_{14} + \cdots - 71\!\cdots\!35 \) 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
16.9634 − 16.9634i
19.6124 − 19.6124i
9.06979 − 9.06979i
1.64191 − 1.64191i
0.963769 − 0.963769i
−8.66090 + 8.66090i
−20.7939 + 20.7939i
−18.7966 + 18.7966i
16.9634 + 16.9634i
19.6124 + 19.6124i
9.06979 + 9.06979i
1.64191 + 1.64191i
0.963769 + 0.963769i
−8.66090 − 8.66090i
−20.7939 − 20.7939i
−18.7966 − 18.7966i
−19.4129 + 19.4129i 0 − 497.724i 601.297 − 170.489i 0 1858.12 − 1858.12i 4692.57 + 4692.57i 0 −8363.24 + 14982.6i
28.2 −17.1630 + 17.1630i 0 − 333.134i −90.5008 + 618.413i 0 1268.31 − 1268.31i 1323.85 + 1323.85i 0 −9060.53 − 12167.1i
28.3 −6.62030 + 6.62030i 0 168.343i 372.705 − 501.713i 0 −216.619 + 216.619i −2809.28 − 2809.28i 0 854.067 + 5788.91i
28.4 −4.09140 + 4.09140i 0 222.521i 401.175 + 479.253i 0 −2635.24 + 2635.24i −1957.82 − 1957.82i 0 −3602.18 − 319.452i
28.5 −3.41326 + 3.41326i 0 232.699i −597.564 − 183.147i 0 2986.31 − 2986.31i −1668.06 − 1668.06i 0 2664.77 − 1414.51i
28.6 11.1104 − 11.1104i 0 9.11845i −542.419 − 310.494i 0 −1159.26 + 1159.26i 2945.57 + 2945.57i 0 −9476.20 + 2576.77i
28.7 18.3444 − 18.3444i 0 − 417.032i −510.688 + 360.309i 0 −1693.91 + 1693.91i −2954.03 − 2954.03i 0 −2758.61 + 15977.9i
28.8 21.2461 − 21.2461i 0 − 646.792i 587.994 + 211.868i 0 1862.28 − 1862.28i −8302.80 − 8302.80i 0 16993.9 − 7991.21i
37.1 −19.4129 − 19.4129i 0 497.724i 601.297 + 170.489i 0 1858.12 + 1858.12i 4692.57 − 4692.57i 0 −8363.24 − 14982.6i
37.2 −17.1630 − 17.1630i 0 333.134i −90.5008 − 618.413i 0 1268.31 + 1268.31i 1323.85 − 1323.85i 0 −9060.53 + 12167.1i
37.3 −6.62030 − 6.62030i 0 − 168.343i 372.705 + 501.713i 0 −216.619 − 216.619i −2809.28 + 2809.28i 0 854.067 − 5788.91i
37.4 −4.09140 − 4.09140i 0 − 222.521i 401.175 − 479.253i 0 −2635.24 − 2635.24i −1957.82 + 1957.82i 0 −3602.18 + 319.452i
37.5 −3.41326 − 3.41326i 0 − 232.699i −597.564 + 183.147i 0 2986.31 + 2986.31i −1668.06 + 1668.06i 0 2664.77 + 1414.51i
37.6 11.1104 + 11.1104i 0 − 9.11845i −542.419 + 310.494i 0 −1159.26 − 1159.26i 2945.57 − 2945.57i 0 −9476.20 − 2576.77i
37.7 18.3444 + 18.3444i 0 417.032i −510.688 − 360.309i 0 −1693.91 − 1693.91i −2954.03 + 2954.03i 0 −2758.61 − 15977.9i
37.8 21.2461 + 21.2461i 0 646.792i 587.994 − 211.868i 0 1862.28 + 1862.28i −8302.80 + 8302.80i 0 16993.9 + 7991.21i
\(n\): 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
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 45.9.g.c 16
3.b odd 2 1 15.9.f.a ✓ 16
5.c odd 4 1 inner 45.9.g.c 16
12.b even 2 1 240.9.bg.d 16
15.d odd 2 1 75.9.f.e 16
15.e even 4 1 15.9.f.a ✓ 16
15.e even 4 1 75.9.f.e 16
60.l odd 4 1 240.9.bg.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.f.a ✓ 16 3.b odd 2 1
15.9.f.a ✓ 16 15.e even 4 1
45.9.g.c 16 1.a even 1 1 trivial
45.9.g.c 16 5.c odd 4 1 inner
75.9.f.e 16 15.d odd 2 1
75.9.f.e 16 15.e even 4 1
240.9.bg.d 16 12.b even 2 1
240.9.bg.d 16 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 5820 T_{2}^{13} + 1126741 T_{2}^{12} + 3704100 T_{2}^{11} + 16936200 T_{2}^{10} + \cdots + 45\!\cdots\!56 \) acting on \(S_{9}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 45\!\cdots\!56 \) 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
$7$ \( T^{16} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 79\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 86\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 54\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 36\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 78\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
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