Properties

Label 1045.2.b.a
Level $1045$
Weight $2$
Character orbit 1045.b
Analytic conductor $8.344$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 2.00000i −3.82843 −1.00000 2.00000i 4.82843 4.82843i 4.41421i −1.00000 −4.82843 + 2.41421i
419.2 0.414214i 2.00000i 1.82843 −1.00000 + 2.00000i −0.828427 0.828427i 1.58579i −1.00000 0.828427 + 0.414214i
419.3 0.414214i 2.00000i 1.82843 −1.00000 2.00000i −0.828427 0.828427i 1.58579i −1.00000 0.828427 0.414214i
419.4 2.41421i 2.00000i −3.82843 −1.00000 + 2.00000i 4.82843 4.82843i 4.41421i −1.00000 −4.82843 2.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.a 4
5.b even 2 1 inner 1045.2.b.a 4
5.c odd 4 1 5225.2.a.d 2
5.c odd 4 1 5225.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.a 4 1.a even 1 1 trivial
1045.2.b.a 4 5.b even 2 1 inner
5225.2.a.d 2 5.c odd 4 1
5225.2.a.g 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1045, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 68)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 144T^{2} + 3136 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 24 T + 136)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$89$ \( (T^{2} - 20 T + 68)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
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