Properties

Label 45.5.c.a
Level $45$
Weight $5$
Character orbit 45.c
Analytic conductor $4.652$
Analytic rank $0$
Dimension $4$
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] = [45,5,Mod(26,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.26");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 45.c (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: \(4.65164833877\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 7) q^{4} + 5 \beta_{2} q^{5} + ( - 7 \beta_{3} - 12) q^{7} + ( - 27 \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 7) q^{4} + 5 \beta_{2} q^{5} + ( - 7 \beta_{3} - 12) q^{7} + ( - 27 \beta_{2} - \beta_1) q^{8} + ( - 5 \beta_{3} - 25) q^{10} + ( - 26 \beta_{2} + 11 \beta_1) q^{11} + (5 \beta_{3} + 202) q^{13} + ( - 138 \beta_{2} - 47 \beta_1) q^{14} + ( - 4 \beta_{3} + 41) q^{16} + (112 \beta_{2} + 44 \beta_1) q^{17} + (2 \beta_{3} - 440) q^{19} + ( - 35 \beta_{2} - 50 \beta_1) q^{20} + (15 \beta_{3} - 68) q^{22} + (294 \beta_{2} - 2 \beta_1) q^{23} - 125 q^{25} + (292 \beta_{2} + 227 \beta_1) q^{26} + (73 \beta_{3} + 1344) q^{28} + (350 \beta_{2} - 72 \beta_1) q^{29} + (114 \beta_{3} - 414) q^{31} + ( - 463 \beta_{2} + 5 \beta_1) q^{32} + ( - 156 \beta_{3} - 1352) q^{34} + ( - 60 \beta_{2} - 175 \beta_1) q^{35} + (137 \beta_{3} + 86) q^{37} + ( - 404 \beta_{2} - 430 \beta_1) q^{38} + (5 \beta_{3} + 675) q^{40} + ( - 616 \beta_{2} + 267 \beta_1) q^{41} + ( - 90 \beta_{3} + 1444) q^{43} + ( - 214 \beta_{2} + 183 \beta_1) q^{44} + ( - 292 \beta_{3} - 1434) q^{46} + (124 \beta_{2} + 422 \beta_1) q^{47} + (168 \beta_{3} + 2153) q^{49} + ( - 125 \beta_{2} - 125 \beta_1) q^{50} + ( - 439 \beta_{3} - 2314) q^{52} + (568 \beta_{2} - 962 \beta_1) q^{53} + ( - 55 \beta_{3} + 650) q^{55} + (450 \beta_{2} + 957 \beta_1) q^{56} + ( - 278 \beta_{3} - 454) q^{58} + ( - 1346 \beta_{2} + 331 \beta_1) q^{59} + (18 \beta_{3} - 5042) q^{61} + (1638 \beta_{2} + 156 \beta_1) q^{62} + (394 \beta_{3} + 2881) q^{64} + (1010 \beta_{2} + 125 \beta_1) q^{65} + (176 \beta_{3} - 396) q^{67} + ( - 2368 \beta_{2} - 1428 \beta_1) q^{68} + (235 \beta_{3} + 3450) q^{70} + ( - 388 \beta_{2} - 892 \beta_1) q^{71} + (394 \beta_{3} + 22) q^{73} + (2552 \beta_{2} + 771 \beta_1) q^{74} + (866 \beta_{3} + 2720) q^{76} + ( - 1074 \beta_{2} + 778 \beta_1) q^{77} + (134 \beta_{3} + 4434) q^{79} + (205 \beta_{2} - 100 \beta_1) q^{80} + (349 \beta_{3} - 1726) q^{82} + (744 \beta_{2} - 282 \beta_1) q^{83} + ( - 220 \beta_{3} - 2800) q^{85} + ( - 176 \beta_{2} + 994 \beta_1) q^{86} + (271 \beta_{3} - 3312) q^{88} + (3876 \beta_{2} + 291 \beta_1) q^{89} + ( - 1474 \beta_{3} - 5574) q^{91} + ( - 1986 \beta_{2} - 2926 \beta_1) q^{92} + ( - 546 \beta_{3} - 8216) q^{94} + ( - 2200 \beta_{2} + 50 \beta_1) q^{95} + (274 \beta_{3} + 6854) q^{97} + (5177 \beta_{2} + 2993 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{4} - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{4} - 48 q^{7} - 100 q^{10} + 808 q^{13} + 164 q^{16} - 1760 q^{19} - 272 q^{22} - 500 q^{25} + 5376 q^{28} - 1656 q^{31} - 5408 q^{34} + 344 q^{37} + 2700 q^{40} + 5776 q^{43} - 5736 q^{46} + 8612 q^{49} - 9256 q^{52} + 2600 q^{55} - 1816 q^{58} - 20168 q^{61} + 11524 q^{64} - 1584 q^{67} + 13800 q^{70} + 88 q^{73} + 10880 q^{76} + 17736 q^{79} - 6904 q^{82} - 11200 q^{85} - 13248 q^{88} - 22296 q^{91} - 32864 q^{94} + 27416 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 7\beta_1 ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
1.58114 0.707107i
−1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 + 0.707107i
6.47871i 0 −25.9737 11.1803i 0 −78.4078 64.6165i 0 −72.4342
26.2 2.00657i 0 11.9737 11.1803i 0 54.4078 56.1312i 0 22.4342
26.3 2.00657i 0 11.9737 11.1803i 0 54.4078 56.1312i 0 22.4342
26.4 6.47871i 0 −25.9737 11.1803i 0 −78.4078 64.6165i 0 −72.4342
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.5.c.a 4
3.b odd 2 1 inner 45.5.c.a 4
4.b odd 2 1 720.5.l.c 4
5.b even 2 1 225.5.c.b 4
5.c odd 4 2 225.5.d.b 8
12.b even 2 1 720.5.l.c 4
15.d odd 2 1 225.5.c.b 4
15.e even 4 2 225.5.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.c.a 4 1.a even 1 1 trivial
45.5.c.a 4 3.b odd 2 1 inner
225.5.c.b 4 5.b even 2 1
225.5.c.b 4 15.d odd 2 1
225.5.d.b 8 5.c odd 4 2
225.5.d.b 8 15.e even 4 2
720.5.l.c 4 4.b odd 2 1
720.5.l.c 4 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 46T^{2} + 169 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 24 T - 4266)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 11116 T^{2} + \cdots + 1444804 \) Copy content Toggle raw display
$13$ \( (T^{2} - 404 T + 38554)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 195136 T^{2} + \cdots + 776848384 \) Copy content Toggle raw display
$19$ \( (T^{2} + 880 T + 193240)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 864504 T^{2} + \cdots + 186717323664 \) Copy content Toggle raw display
$29$ \( T^{4} + 1411624 T^{2} + \cdots + 269556179344 \) Copy content Toggle raw display
$31$ \( (T^{2} + 828 T - 998244)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 172 T - 1681814)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6360964 T^{2} + \cdots + 377091790084 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2888 T + 1356136)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6564784 T^{2} + \cdots + 9788338191424 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 226348173496384 \) Copy content Toggle raw display
$59$ \( T^{4} + 22061356 T^{2} + \cdots + 50218227136324 \) Copy content Toggle raw display
$61$ \( (T^{2} + 10084 T + 25392604)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 792 T - 2631024)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 184124057069824 \) Copy content Toggle raw display
$73$ \( (T^{2} - 44 T - 13970756)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8868 T + 18044316)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 8398224 T^{2} + \cdots + 1785558717504 \) Copy content Toggle raw display
$89$ \( T^{4} + 153282276 T^{2} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{2} - 13708 T + 40220476)^{2} \) Copy content Toggle raw display
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