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Properties

Label 45.4.f.a
Level $45$
Weight $4$
Character orbit 45.f
Analytic conductor $2.655$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,4,Mod(8,45)]
 
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([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.f (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: \(2.65508595026\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{5} - 6 \beta_1) q^{4} + (\beta_{11} + \beta_{10} + \beta_{9}) q^{5} + ( - \beta_{4} + 2 \beta_1 + 2) q^{7} + ( - 3 \beta_{10} - 7 \beta_{9} + \cdots + 2 \beta_{7}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{5} - 6 \beta_1) q^{4} + (\beta_{11} + \beta_{10} + \beta_{9}) q^{5} + ( - \beta_{4} + 2 \beta_1 + 2) q^{7} + ( - 3 \beta_{10} - 7 \beta_{9} + \cdots + 2 \beta_{7}) q^{8}+ \cdots + ( - 48 \beta_{10} + \cdots + 32 \beta_{7}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{10} - 56\nu^{8} + 634\nu^{6} + 3248\nu^{4} + 83197\nu^{2} + 418024 ) / 182400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -71\nu^{10} + 2538\nu^{8} - 4502\nu^{6} - 356014\nu^{4} - 1896111\nu^{2} - 1578712 ) / 3620640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1029\nu^{10} - 22972\nu^{8} + 158738\nu^{6} - 2560724\nu^{4} + 22729229\nu^{2} + 78379688 ) / 9051600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5723\nu^{10} - 110024\nu^{8} + 167406\nu^{6} - 2434608\nu^{4} + 44277523\nu^{2} + 559571096 ) / 18103200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1265\nu^{10} - 2936\nu^{8} - 23478\nu^{6} - 1362480\nu^{4} - 16244903\nu^{2} - 51726904 ) / 2896512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 113621 \nu^{11} + 12179272 \nu^{9} - 76074438 \nu^{7} - 253832976 \nu^{5} + \cdots - 93476563288 \nu ) / 25778956800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28451 \nu^{11} + 1391000 \nu^{9} - 20705978 \nu^{7} + 10214960 \nu^{5} - 1219736029 \nu^{3} - 1295931240 \nu ) / 2577895680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -6773\nu^{11} + 146104\nu^{9} - 577306\nu^{7} + 5223568\nu^{5} - 79774373\nu^{3} - 554840616\nu ) / 226131200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1093919 \nu^{11} + 7804552 \nu^{9} - 15466318 \nu^{7} + 1380835984 \nu^{5} + \cdots + 28260515592 \nu ) / 25778956800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -1095\nu^{11} + 7908\nu^{9} + 18534\nu^{7} + 745404\nu^{5} + 8928141\nu^{3} - 11971272\nu ) / 21482464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -9087\nu^{11} + 123320\nu^{9} - 524974\nu^{7} + 11892160\nu^{5} - 5216047\nu^{3} - 669132280\nu ) / 135678720 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -6\beta_{11} - \beta_{10} + 12\beta_{9} - \beta_{8} + 6\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{5} - 3\beta_{4} + 11\beta_{3} - 20\beta_{2} - 64\beta _1 + 64 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{11} + 53\beta_{10} + 66\beta_{9} - 137\beta_{8} + 6\beta_{7} + 120\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{5} - 35\beta_{4} + 5\beta_{3} - 268\beta_{2} + 1136 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{11} - 615\beta_{10} + 1704\beta_{9} - 1259\beta_{8} - 348\beta_{7} + 1518\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 676\beta_{5} - 1157\beta_{4} + 909\beta_{3} - 3212\beta_{2} + 5824\beta _1 + 25216 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5940\beta_{11} - 2133\beta_{10} + 21438\beta_{9} - 12439\beta_{8} - 6918\beta_{7} + 24264\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 252\beta_{5} - 4255\beta_{4} + 5385\beta_{3} - 14588\beta_{2} - 20480\beta _1 + 129968 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -88602\beta_{11} + 24551\beta_{10} + 292920\beta_{9} - 220021\beta_{8} - 38724\beta_{7} + 359538\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 30812\beta_{5} - 127323\beta_{4} + 201011\beta_{3} - 706676\beta_{2} - 856384\beta _1 + 4399744 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 776748 \beta_{11} - 305227 \beta_{10} + 4045218 \beta_{9} - 3762665 \beta_{8} + \cdots + 4953432 \beta_{6} ) / 24 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−3.78139 − 0.0336790i
−0.347140 + 2.27426i
2.02004 + 2.30794i
−2.02004 − 2.30794i
0.347140 − 2.27426i
3.78139 + 0.0336790i
−3.78139 + 0.0336790i
−0.347140 − 2.27426i
2.02004 − 2.30794i
−2.02004 + 2.30794i
0.347140 + 2.27426i
3.78139 − 0.0336790i
−3.74771 − 3.74771i 0 20.0907i −0.572391 + 11.1657i 0 1.80948 − 1.80948i 45.3126 − 45.3126i 0 43.9909 − 39.7006i
8.2 −2.62140 − 2.62140i 0 5.74350i −8.38994 − 7.38978i 0 −10.8652 + 10.8652i −5.91519 + 5.91519i 0 2.62183 + 41.3650i
8.3 −0.287902 − 0.287902i 0 − 7.83422i −9.93887 − 5.12043i 0 15.0557 − 15.0557i −4.55871 + 4.55871i 0 1.38724 + 4.33560i
8.4 0.287902 + 0.287902i 0 − 7.83422i 9.93887 + 5.12043i 0 15.0557 − 15.0557i 4.55871 − 4.55871i 0 1.38724 + 4.33560i
8.5 2.62140 + 2.62140i 0 5.74350i 8.38994 + 7.38978i 0 −10.8652 + 10.8652i 5.91519 − 5.91519i 0 2.62183 + 41.3650i
8.6 3.74771 + 3.74771i 0 20.0907i 0.572391 − 11.1657i 0 1.80948 − 1.80948i −45.3126 + 45.3126i 0 43.9909 − 39.7006i
17.1 −3.74771 + 3.74771i 0 − 20.0907i −0.572391 − 11.1657i 0 1.80948 + 1.80948i 45.3126 + 45.3126i 0 43.9909 + 39.7006i
17.2 −2.62140 + 2.62140i 0 − 5.74350i −8.38994 + 7.38978i 0 −10.8652 − 10.8652i −5.91519 − 5.91519i 0 2.62183 − 41.3650i
17.3 −0.287902 + 0.287902i 0 7.83422i −9.93887 + 5.12043i 0 15.0557 + 15.0557i −4.55871 − 4.55871i 0 1.38724 − 4.33560i
17.4 0.287902 − 0.287902i 0 7.83422i 9.93887 − 5.12043i 0 15.0557 + 15.0557i 4.55871 + 4.55871i 0 1.38724 − 4.33560i
17.5 2.62140 − 2.62140i 0 − 5.74350i 8.38994 − 7.38978i 0 −10.8652 − 10.8652i 5.91519 + 5.91519i 0 2.62183 − 41.3650i
17.6 3.74771 − 3.74771i 0 − 20.0907i 0.572391 + 11.1657i 0 1.80948 + 1.80948i −45.3126 − 45.3126i 0 43.9909 + 39.7006i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.6
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 45.4.f.a ✓ 12
3.b odd 2 1 inner 45.4.f.a ✓ 12
4.b odd 2 1 720.4.w.d 12
5.b even 2 1 225.4.f.c 12
5.c odd 4 1 inner 45.4.f.a ✓ 12
5.c odd 4 1 225.4.f.c 12
12.b even 2 1 720.4.w.d 12
15.d odd 2 1 225.4.f.c 12
15.e even 4 1 inner 45.4.f.a ✓ 12
15.e even 4 1 225.4.f.c 12
20.e even 4 1 720.4.w.d 12
60.l odd 4 1 720.4.w.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.f.a ✓ 12 1.a even 1 1 trivial
45.4.f.a ✓ 12 3.b odd 2 1 inner
45.4.f.a ✓ 12 5.c odd 4 1 inner
45.4.f.a ✓ 12 15.e even 4 1 inner
225.4.f.c 12 5.b even 2 1
225.4.f.c 12 5.c odd 4 1
225.4.f.c 12 15.d odd 2 1
225.4.f.c 12 15.e even 4 1
720.4.w.d 12 4.b odd 2 1
720.4.w.d 12 12.b even 2 1
720.4.w.d 12 20.e even 4 1
720.4.w.d 12 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 978 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 12 T^{5} + \cdots + 700928)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 4344 T^{4} + \cdots + 2390999552)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 54 T^{5} + \cdots + 40033352)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + 26640 T^{4} + \cdots + 604661760000)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 3276052522632)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 312 T^{2} + \cdots + 333184)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 414 T^{5} + \cdots + 431026413512)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 12363096915968)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 19\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 24 T^{2} + \cdots - 70019072)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 25\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 826210599600128)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 35\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
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