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Properties

Label 1045.2.a.i
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 10\nu^{3} + 21\nu^{2} - 8\nu - 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 3\nu^{6} - 6\nu^{5} + 18\nu^{4} + 11\nu^{3} - 29\nu^{2} - 4\nu + 11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 2\nu^{6} + 9\nu^{5} - 13\nu^{4} - 24\nu^{3} + 19\nu^{2} + 12\nu - 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 8\beta_{2} + 9\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} + 9\beta_{3} + 19\beta_{2} + 31\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 2\beta_{6} + 3\beta_{5} + 14\beta_{4} + 24\beta_{3} + 61\beta_{2} + 71\beta _1 + 109 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{7} + 13\beta_{6} + 15\beta_{5} + 42\beta_{4} + 79\beta_{3} + 160\beta_{2} + 215\beta _1 + 268 \) Copy content Toggle raw display

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

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
−1.82030
−1.57959
−0.865980
−0.649219
0.714778
1.06639
2.25600
2.87791
−2.82030 0.978567 5.95409 1.00000 −2.75985 −4.31627 −11.1517 −2.04241 −2.82030
1.2 −2.57959 −2.53274 4.65426 1.00000 6.53343 2.87748 −6.84689 3.41479 −2.57959
1.3 −1.86598 2.01288 1.48188 1.00000 −3.75600 −2.04136 0.966800 1.05170 −1.86598
1.4 −1.64922 −2.98027 0.719922 1.00000 4.91512 −5.09280 2.11113 5.88203 −1.64922
1.5 −0.285222 −1.61587 −1.91865 1.00000 0.460881 −1.06724 1.11768 −0.388965 −0.285222
1.6 0.0663929 −0.255194 −1.99559 1.00000 −0.0169431 2.56056 −0.265279 −2.93488 0.0663929
1.7 1.25600 0.772194 −0.422456 1.00000 0.969878 −3.10169 −3.04261 −2.40372 1.25600
1.8 1.87791 −3.37956 1.52654 1.00000 −6.34651 −0.818685 −0.889109 8.42145 1.87791
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.i ✓ 8
3.b odd 2 1 9405.2.a.bf 8
5.b even 2 1 5225.2.a.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.i ✓ 8 1.a even 1 1 trivial
5225.2.a.o 8 5.b even 2 1
9405.2.a.bf 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 6T_{2}^{7} + 5T_{2}^{6} - 28T_{2}^{5} - 47T_{2}^{4} + 21T_{2}^{3} + 60T_{2}^{2} + 11T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 6 T^{7} + 5 T^{6} - 28 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 7 T^{7} + 7 T^{6} - 40 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 11 T^{7} + 23 T^{6} + \cdots + 896 \) Copy content Toggle raw display
$11$ \( (T - 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 17 T^{7} + 65 T^{6} + \cdots - 13392 \) Copy content Toggle raw display
$17$ \( T^{8} + 9 T^{7} - 47 T^{6} + \cdots + 8344 \) Copy content Toggle raw display
$19$ \( (T + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} - 48 T^{6} + \cdots + 1504 \) Copy content Toggle raw display
$29$ \( T^{8} + 3 T^{7} - 133 T^{6} + \cdots + 84152 \) Copy content Toggle raw display
$31$ \( T^{8} + T^{7} - 115 T^{6} + \cdots - 35840 \) Copy content Toggle raw display
$37$ \( T^{8} + 17 T^{7} - 28 T^{6} + \cdots - 280640 \) Copy content Toggle raw display
$41$ \( T^{8} + 5 T^{7} - 110 T^{6} + \cdots + 30440 \) Copy content Toggle raw display
$43$ \( T^{8} + 21 T^{7} + 101 T^{6} + \cdots - 18944 \) Copy content Toggle raw display
$47$ \( T^{8} + 8 T^{7} - 104 T^{6} + \cdots + 29984 \) Copy content Toggle raw display
$53$ \( T^{8} + 19 T^{7} + 59 T^{6} + \cdots + 10528 \) Copy content Toggle raw display
$59$ \( T^{8} + 33 T^{7} + 57 T^{6} + \cdots + 6694912 \) Copy content Toggle raw display
$61$ \( T^{8} + T^{7} - 369 T^{6} + \cdots + 25298072 \) Copy content Toggle raw display
$67$ \( T^{8} + 18 T^{7} - 43 T^{6} + \cdots + 56432 \) Copy content Toggle raw display
$71$ \( T^{8} + 18 T^{7} - 106 T^{6} + \cdots - 833024 \) Copy content Toggle raw display
$73$ \( T^{8} + 18 T^{7} + 67 T^{6} + \cdots - 22472 \) Copy content Toggle raw display
$79$ \( T^{8} + 5 T^{7} - 234 T^{6} + \cdots - 197248 \) Copy content Toggle raw display
$83$ \( T^{8} + 33 T^{7} + 197 T^{6} + \cdots + 2043584 \) Copy content Toggle raw display
$89$ \( T^{8} + 20 T^{7} + 23 T^{6} + \cdots + 216 \) Copy content Toggle raw display
$97$ \( T^{8} - 419 T^{6} + 556 T^{5} + \cdots + 9664 \) Copy content Toggle raw display
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