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Properties

Label 45.10.b.d
Level $45$
Weight $10$
Character orbit 45.b
Analytic conductor $23.177$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,10,Mod(19,45)]
 
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([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 45.b (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: \(23.1766126274\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 774x^{6} - 2822x^{5} + 257730x^{4} + 2421892x^{3} - 2488179x^{2} - 34986488x + 335502416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8}\cdot 5^{7} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 18) q^{4} + ( - \beta_{4} - \beta_{3} + 24 \beta_1) q^{5} - \beta_{5} q^{7} + (8 \beta_{4} + 220 \beta_1) q^{8} + (\beta_{6} + \beta_{5} + 60 \beta_{2} - 12835) q^{10}+ \cdots + (910560 \beta_{4} - 53014073 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 144 q^{4} - 102680 q^{10} - 1009088 q^{16} - 2079872 q^{19} - 595720 q^{25} + 12077344 q^{31} + 25286480 q^{34} + 18248480 q^{40} + 3119840 q^{46} - 174619144 q^{49} - 105991200 q^{55} - 165845744 q^{61}+ \cdots - 4117219360 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 774x^{6} - 2822x^{5} + 257730x^{4} + 2421892x^{3} - 2488179x^{2} - 34986488x + 335502416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6904938953 \nu^{7} - 37144894182 \nu^{6} - 4851062706478 \nu^{5} - 6257780345790 \nu^{4} + \cdots + 60\!\cdots\!44 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 116957544 \nu^{7} + 4833928896 \nu^{6} + 7776782184 \nu^{5} - 1371549880020 \nu^{4} + \cdots + 85\!\cdots\!78 ) / 227580881007625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 668560664711 \nu^{7} - 2645074001034 \nu^{6} - 530608174016386 \nu^{5} + \cdots - 23\!\cdots\!72 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 331927634177 \nu^{7} - 2247832279038 \nu^{6} - 234962282948902 \nu^{5} + \cdots + 35\!\cdots\!96 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9891204792 \nu^{7} - 237035160948 \nu^{6} - 5741279386692 \nu^{5} + 138230829015540 \nu^{4} + \cdots + 75\!\cdots\!16 ) / 227580881007625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25452206052 \nu^{7} + 794327480988 \nu^{6} + 13593374795352 \nu^{5} + \cdots - 28\!\cdots\!71 ) / 227580881007625 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 644689654177 \nu^{7} - 1707245576362 \nu^{6} + 657301176385502 \nu^{5} + \cdots + 37\!\cdots\!04 ) / 18\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 8\beta_{4} - 84\beta_{3} - 60\beta_{2} + 1775\beta _1 + 1800 ) / 7200 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 10\beta_{6} + 30\beta_{5} - 44\beta_{4} - 48\beta_{3} - 595\beta_{2} + 1810\beta _1 + 279360 ) / 1440 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 319 \beta_{7} + 300 \beta_{6} + 1800 \beta_{5} - 18328 \beta_{4} - 15996 \beta_{3} + \cdots + 11806200 ) / 7200 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 826 \beta_{7} + 6070 \beta_{6} + 24330 \beta_{5} - 60892 \beta_{4} - 36624 \beta_{3} + \cdots + 36397440 ) / 1440 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 30689 \beta_{7} + 456500 \beta_{6} + 1989000 \beta_{5} - 13178452 \beta_{4} + 1437276 \beta_{3} + \cdots + 2081314800 ) / 7200 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 221484 \beta_{7} + 2967010 \beta_{6} + 12410670 \beta_{5} - 48906792 \beta_{4} + 10041216 \beta_{3} + \cdots - 34516536840 ) / 1440 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 117363141 \beta_{7} + 276526600 \beta_{6} + 1170804600 \beta_{5} - 6107526168 \beta_{4} + \cdots - 4421476909800 ) / 7200 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−16.5231 − 7.54427i
23.3674 − 7.54427i
−8.93777 − 3.05515i
3.09348 − 3.05515i
−8.93777 + 3.05515i
3.09348 + 3.05515i
−16.5231 + 7.54427i
23.3674 + 7.54427i
− 30.1771i 0 −398.657 −755.297 − 1175.86i 0 − 10271.7i − 3420.35i 0 −35484.1 + 22792.7i
19.2 − 30.1771i 0 −398.657 755.297 − 1175.86i 0 10271.7i − 3420.35i 0 −35484.1 − 22792.7i
19.3 − 12.2206i 0 362.657 −1143.76 + 803.080i 0 4342.19i − 10688.8i 0 9814.11 + 13977.4i
19.4 − 12.2206i 0 362.657 1143.76 + 803.080i 0 − 4342.19i − 10688.8i 0 9814.11 − 13977.4i
19.5 12.2206i 0 362.657 −1143.76 − 803.080i 0 − 4342.19i 10688.8i 0 9814.11 − 13977.4i
19.6 12.2206i 0 362.657 1143.76 − 803.080i 0 4342.19i 10688.8i 0 9814.11 + 13977.4i
19.7 30.1771i 0 −398.657 −755.297 + 1175.86i 0 10271.7i 3420.35i 0 −35484.1 − 22792.7i
19.8 30.1771i 0 −398.657 755.297 + 1175.86i 0 − 10271.7i 3420.35i 0 −35484.1 + 22792.7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.10.b.d ✓ 8
3.b odd 2 1 inner 45.10.b.d ✓ 8
5.b even 2 1 inner 45.10.b.d ✓ 8
5.c odd 4 2 225.10.a.x 8
15.d odd 2 1 inner 45.10.b.d ✓ 8
15.e even 4 2 225.10.a.x 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.10.b.d ✓ 8 1.a even 1 1 trivial
45.10.b.d ✓ 8 3.b odd 2 1 inner
45.10.b.d ✓ 8 5.b even 2 1 inner
45.10.b.d ✓ 8 15.d odd 2 1 inner
225.10.a.x 8 5.c odd 4 2
225.10.a.x 8 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 1060T_{2}^{2} + 136000 \) acting on \(S_{10}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1060 T^{2} + 136000)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 519968 T - 179554976144)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 39095996869376)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 709545978526076)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 13\!\cdots\!16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
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