Properties

Label 1045.2.a.c
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) 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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−0.414214 2.00000 −1.82843 1.00000 −0.828427 2.82843 1.58579 1.00000 −0.414214
1.2 2.41421 2.00000 3.82843 1.00000 4.82843 −2.82843 4.41421 1.00000 2.41421
\(n\): e.g. 2-40 or 990-1000
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.c 2
3.b odd 2 1 9405.2.a.o 2
5.b even 2 1 5225.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.c 2 1.a even 1 1 trivial
5225.2.a.e 2 5.b even 2 1
9405.2.a.o 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 8 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 24T + 136 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 124 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 184 \) Copy content Toggle raw display
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