Properties

Label 1045.2.a.e
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1045,2,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

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: yes
Analytic conductor: \(8.34436701122\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.36497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_1 - 1) q^{3} + \beta_{3} q^{4} - q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{4} - \beta_{2}) q^{8} + (\beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_1 - 1) q^{3} + \beta_{3} q^{4} - q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{4} - \beta_{2}) q^{8} + (\beta_{2} - \beta_1) q^{9} - \beta_{4} q^{10} + q^{11} + (\beta_{2} + 1) q^{12} + ( - \beta_{4} - 2 \beta_{3} - \beta_1) q^{13} + ( - \beta_{4} - 2 \beta_1 + 1) q^{14} + ( - \beta_1 + 1) q^{15} + (\beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{16} + ( - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 2) q^{17} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{18} + q^{19} - \beta_{3} q^{20} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2) q^{21} + \beta_{4} q^{22} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{23} + (2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{24} + q^{25} + ( - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{26} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{27} + ( - \beta_{4} + \beta_{3} - 2) q^{28} + (3 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{29} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{30} + (\beta_{3} + 3 \beta_{2} + 3 \beta_1 - 2) q^{31} + ( - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{32} + (\beta_1 - 1) q^{33} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{34} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{35} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{36} + (4 \beta_{3} - 4 \beta_1 + 3) q^{37} + \beta_{4} q^{38} + (\beta_{4} - 2 \beta_{2} + \beta_1 - 3) q^{39} + (\beta_{4} + \beta_{2}) q^{40} + ( - 5 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 5) q^{41} + (\beta_{4} - \beta_{2} + 2 \beta_1 - 4) q^{42} + ( - 3 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - \beta_1) q^{43} + \beta_{3} q^{44} + ( - \beta_{2} + \beta_1) q^{45} + (4 \beta_{4} + 2 \beta_{3} + 5 \beta_1) q^{46} + (3 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{47} + ( - \beta_{4} - 2 \beta_{2} - 3 \beta_1 + 1) q^{48} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{49} + \beta_{4} q^{50} + (3 \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{51} + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_1 - 5) q^{52} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 2) q^{53} + (3 \beta_{4} + \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 1) q^{54} - q^{55} + (\beta_{4} - \beta_{3} - \beta_{2} + 4 \beta_1 - 4) q^{56} + (\beta_1 - 1) q^{57} + ( - 6 \beta_{4} + 5 \beta_{3} - 2 \beta_1 + 2) q^{58} + (3 \beta_{4} - 2 \beta_{2} - \beta_1 - 4) q^{59} + ( - \beta_{2} - 1) q^{60} + ( - 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 7) q^{61} + ( - 4 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 6 \beta_1 - 3) q^{62} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{63} + ( - 3 \beta_{4} - \beta_{3} - 4 \beta_1 + 1) q^{64} + (\beta_{4} + 2 \beta_{3} + \beta_1) q^{65} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{66} + (4 \beta_{4} + \beta_{3} + 3 \beta_{2} + 7 \beta_1 - 1) q^{67} + ( - \beta_{4} - 5 \beta_{3} + 2 \beta_1 + 1) q^{68} + (4 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + 3) q^{69} + (\beta_{4} + 2 \beta_1 - 1) q^{70} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 5) q^{71} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - 5) q^{72} + (2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2}) q^{73} + (7 \beta_{4} + 4 \beta_1 + 4) q^{74} + (\beta_1 - 1) q^{75} + \beta_{3} q^{76} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{77} + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{78} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 3) q^{79} + ( - \beta_{4} + 2 \beta_{3} - \beta_1 + 2) q^{80} + (3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{81} + ( - 3 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 5) q^{82} + (\beta_{4} - 6 \beta_{3} + 7 \beta_1 - 8) q^{83} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 4) q^{84} + (3 \beta_{4} - 2 \beta_{3} + 3 \beta_1 + 2) q^{85} + (3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 5) q^{86} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 7 \beta_1 + 7) q^{87} + ( - \beta_{4} - \beta_{2}) q^{88} + (3 \beta_{4} + 4 \beta_{3} - 3 \beta_1 - 1) q^{89} + (\beta_{4} + \beta_{3} - \beta_{2} - 1) q^{90} + (4 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{91} + (4 \beta_{4} - \beta_{2} - \beta_1 + 5) q^{92} + ( - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 6) q^{93} + ( - \beta_{4} + \beta_{3} - 7 \beta_{2} - 6 \beta_1 + 2) q^{94} - q^{95} + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{96} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3) q^{97} + ( - 3 \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{98} + (\beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9} + q^{10} + 5 q^{11} + 3 q^{12} - 3 q^{13} + 2 q^{14} + 3 q^{15} - 11 q^{16} - 11 q^{17} + 3 q^{18} + 5 q^{19} - q^{20} - 3 q^{21} - q^{22} + 5 q^{24} + 5 q^{25} - 8 q^{26} - 8 q^{28} - 15 q^{29} + 4 q^{30} - 9 q^{31} - 3 q^{33} - 14 q^{34} - 3 q^{35} - 7 q^{36} + 11 q^{37} - q^{38} - 10 q^{39} - 3 q^{40} - 23 q^{41} - 15 q^{42} + 9 q^{43} + q^{44} + 4 q^{45} + 8 q^{46} - 6 q^{47} + 4 q^{48} - 12 q^{49} - q^{50} - 27 q^{52} - 13 q^{53} + 9 q^{54} - 5 q^{55} - 12 q^{56} - 3 q^{57} + 17 q^{58} - 21 q^{59} - 3 q^{60} - 31 q^{61} - 18 q^{62} + 10 q^{63} - q^{64} + 3 q^{65} - 4 q^{66} + 5 q^{68} - 2 q^{70} - 28 q^{71} - 20 q^{72} - 14 q^{73} + 21 q^{74} - 3 q^{75} + q^{76} + 3 q^{77} + 13 q^{78} + 3 q^{79} + 11 q^{80} - 3 q^{81} - 18 q^{82} - 33 q^{83} + 13 q^{84} + 11 q^{85} - 20 q^{86} + 22 q^{87} + 3 q^{88} - 10 q^{89} - 3 q^{90} + 14 q^{91} + 21 q^{92} + 30 q^{93} + 14 q^{94} - 5 q^{95} - 3 q^{96} - 10 q^{97} + 8 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{4} + 2\beta_{3} + 5\beta_{2} + 5\beta _1 + 7 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.31801
−0.506287
1.33419
−1.55629
0.410375
−1.88661 1.31801 1.55930 −1.00000 −2.48658 0.927281 0.831437 −1.26284 1.88661
1.2 −1.46888 −1.50629 0.157597 −1.00000 2.21255 −2.37015 2.70626 −0.731099 1.46888
1.3 −0.584664 0.334185 −1.65817 −1.00000 −0.195386 1.85355 2.13880 −2.88832 0.584664
1.4 0.913732 −2.55629 −1.16509 −1.00000 −2.33576 3.50085 −2.89205 3.53460 −0.913732
1.5 2.02642 −0.589625 2.10637 −1.00000 −1.19483 −0.911544 0.215549 −2.65234 −2.02642
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.a.e 5
3.b odd 2 1 9405.2.a.u 5
5.b even 2 1 5225.2.a.i 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.e 5 1.a even 1 1 trivial
5225.2.a.i 5 5.b even 2 1
9405.2.a.u 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + T_{2}^{4} - 5T_{2}^{3} - 5T_{2}^{2} + 4T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} - 5 T^{3} - 5 T^{2} + 4 T + 3 \) Copy content Toggle raw display
$3$ \( T^{5} + 3 T^{4} - T^{3} - 6 T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 3 T^{4} - 7 T^{3} + 18 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 3 T^{4} - 33 T^{3} - 11 T^{2} + \cdots - 299 \) Copy content Toggle raw display
$17$ \( T^{5} + 11 T^{4} + 7 T^{3} - 199 T^{2} + \cdots - 69 \) Copy content Toggle raw display
$19$ \( (T - 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} - 74 T^{3} - 37 T^{2} + \cdots + 1821 \) Copy content Toggle raw display
$29$ \( T^{5} + 15 T^{4} - 29 T^{3} + \cdots - 1047 \) Copy content Toggle raw display
$31$ \( T^{5} + 9 T^{4} - 57 T^{3} + \cdots - 3025 \) Copy content Toggle raw display
$37$ \( T^{5} - 11 T^{4} - 54 T^{3} + 874 T^{2} + \cdots - 79 \) Copy content Toggle raw display
$41$ \( T^{5} + 23 T^{4} + 104 T^{3} + \cdots - 22467 \) Copy content Toggle raw display
$43$ \( T^{5} - 9 T^{4} - 135 T^{3} + \cdots - 2783 \) Copy content Toggle raw display
$47$ \( T^{5} + 6 T^{4} - 134 T^{3} + \cdots + 21621 \) Copy content Toggle raw display
$53$ \( T^{5} + 13 T^{4} + 37 T^{3} - 9 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$59$ \( T^{5} + 21 T^{4} + 73 T^{3} + \cdots - 6849 \) Copy content Toggle raw display
$61$ \( T^{5} + 31 T^{4} + 269 T^{3} + \cdots + 3307 \) Copy content Toggle raw display
$67$ \( T^{5} - 195 T^{3} + 236 T^{2} + \cdots + 1031 \) Copy content Toggle raw display
$71$ \( T^{5} + 28 T^{4} + 262 T^{3} + \cdots - 141 \) Copy content Toggle raw display
$73$ \( T^{5} + 14 T^{4} - 223 T^{3} + \cdots + 115699 \) Copy content Toggle raw display
$79$ \( T^{5} - 3 T^{4} - 154 T^{3} + \cdots + 6125 \) Copy content Toggle raw display
$83$ \( T^{5} + 33 T^{4} + 201 T^{3} + \cdots - 77949 \) Copy content Toggle raw display
$89$ \( T^{5} + 10 T^{4} - 149 T^{3} + \cdots + 6909 \) Copy content Toggle raw display
$97$ \( T^{5} + 10 T^{4} - 31 T^{3} + \cdots - 1597 \) Copy content Toggle raw display
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