Properties

Label 1050.3.e.a
Level $1050$
Weight $3$
Character orbit 1050.e
Analytic conductor $28.610$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} -2 q^{4} + ( -2 + \beta_{2} ) q^{6} -\beta_{3} q^{7} -2 \beta_{1} q^{8} + ( 5 + 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} -2 q^{4} + ( -2 + \beta_{2} ) q^{6} -\beta_{3} q^{7} -2 \beta_{1} q^{8} + ( 5 + 2 \beta_{2} ) q^{9} + ( 5 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{12} + ( -10 - 4 \beta_{3} ) q^{13} -\beta_{2} q^{14} + 4 q^{16} + ( -2 \beta_{1} + 5 \beta_{2} ) q^{17} + ( 5 \beta_{1} - 4 \beta_{3} ) q^{18} -16 q^{19} + ( -7 - \beta_{2} ) q^{21} + ( -10 + 4 \beta_{3} ) q^{22} + ( \beta_{1} - 10 \beta_{2} ) q^{23} + ( 4 - 2 \beta_{2} ) q^{24} + ( -10 \beta_{1} - 4 \beta_{2} ) q^{26} + ( 19 \beta_{1} + \beta_{3} ) q^{27} + 2 \beta_{3} q^{28} + ( -20 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -32 + 10 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( -10 - 14 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{33} + ( 4 - 10 \beta_{3} ) q^{34} + ( -10 - 4 \beta_{2} ) q^{36} -20 q^{37} -16 \beta_{1} q^{38} + ( -28 - 10 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} ) q^{39} + ( 30 \beta_{1} + 9 \beta_{2} ) q^{41} + ( -7 \beta_{1} + 2 \beta_{3} ) q^{42} + ( -20 - 12 \beta_{3} ) q^{43} + ( -10 \beta_{1} + 4 \beta_{2} ) q^{44} + ( -2 + 20 \beta_{3} ) q^{46} -6 \beta_{1} q^{47} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{48} + 7 q^{49} + ( 4 + 35 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} ) q^{51} + ( 20 + 8 \beta_{3} ) q^{52} -36 \beta_{1} q^{53} + ( -38 + \beta_{2} ) q^{54} + 2 \beta_{2} q^{56} + ( -16 \beta_{1} - 16 \beta_{3} ) q^{57} + ( 40 - 4 \beta_{3} ) q^{58} + ( -20 \beta_{1} + 8 \beta_{2} ) q^{59} + ( -14 - 20 \beta_{3} ) q^{61} + ( -32 \beta_{1} + 10 \beta_{2} ) q^{62} + ( -14 \beta_{1} - 5 \beta_{3} ) q^{63} -8 q^{64} + ( 28 - 10 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} ) q^{66} + ( -60 + 4 \beta_{3} ) q^{67} + ( 4 \beta_{1} - 10 \beta_{2} ) q^{68} + ( -2 - 70 \beta_{1} + \beta_{2} + 20 \beta_{3} ) q^{69} + ( -25 \beta_{1} - 14 \beta_{2} ) q^{71} + ( -10 \beta_{1} + 8 \beta_{3} ) q^{72} + ( -30 + 16 \beta_{3} ) q^{73} -20 \beta_{1} q^{74} + 32 q^{76} + ( 14 \beta_{1} - 5 \beta_{2} ) q^{77} + ( 20 - 28 \beta_{1} - 10 \beta_{2} + 8 \beta_{3} ) q^{78} + ( -32 - 20 \beta_{3} ) q^{79} + ( -31 + 20 \beta_{2} ) q^{81} + ( -60 - 18 \beta_{3} ) q^{82} + ( -50 \beta_{1} + 20 \beta_{2} ) q^{83} + ( 14 + 2 \beta_{2} ) q^{84} + ( -20 \beta_{1} - 12 \beta_{2} ) q^{86} + ( 40 + 14 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} ) q^{87} + ( 20 - 8 \beta_{3} ) q^{88} + ( -30 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 28 + 10 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 20 \beta_{2} ) q^{92} + ( 70 - 32 \beta_{1} + 10 \beta_{2} - 32 \beta_{3} ) q^{93} + 12 q^{94} + ( -8 + 4 \beta_{2} ) q^{96} + ( 90 - 8 \beta_{3} ) q^{97} + 7 \beta_{1} q^{98} + ( 56 + 25 \beta_{1} - 10 \beta_{2} - 20 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 8 q^{6} + 20 q^{9} + O(q^{10}) \) \( 4 q - 8 q^{4} - 8 q^{6} + 20 q^{9} - 40 q^{13} + 16 q^{16} - 64 q^{19} - 28 q^{21} - 40 q^{22} + 16 q^{24} - 128 q^{31} - 40 q^{33} + 16 q^{34} - 40 q^{36} - 80 q^{37} - 112 q^{39} - 80 q^{43} - 8 q^{46} + 28 q^{49} + 16 q^{51} + 80 q^{52} - 152 q^{54} + 160 q^{58} - 56 q^{61} - 32 q^{64} + 112 q^{66} - 240 q^{67} - 8 q^{69} - 120 q^{73} + 128 q^{76} + 80 q^{78} - 128 q^{79} - 124 q^{81} - 240 q^{82} + 56 q^{84} + 160 q^{87} + 80 q^{88} + 112 q^{91} + 280 q^{93} + 48 q^{94} - 32 q^{96} + 360 q^{97} + 224 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} + 11 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
2.57794i
1.16372i
2.57794i
1.16372i
1.41421i −2.64575 1.41421i −2.00000 0 −2.00000 + 3.74166i 2.64575 2.82843i 5.00000 + 7.48331i 0
701.2 1.41421i 2.64575 1.41421i −2.00000 0 −2.00000 3.74166i −2.64575 2.82843i 5.00000 7.48331i 0
701.3 1.41421i −2.64575 + 1.41421i −2.00000 0 −2.00000 3.74166i 2.64575 2.82843i 5.00000 7.48331i 0
701.4 1.41421i 2.64575 + 1.41421i −2.00000 0 −2.00000 + 3.74166i −2.64575 2.82843i 5.00000 + 7.48331i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.e.a 4
3.b odd 2 1 inner 1050.3.e.a 4
5.b even 2 1 42.3.b.a 4
5.c odd 4 2 1050.3.c.a 8
15.d odd 2 1 42.3.b.a 4
15.e even 4 2 1050.3.c.a 8
20.d odd 2 1 336.3.d.b 4
35.c odd 2 1 294.3.b.h 4
35.i odd 6 2 294.3.h.g 8
35.j even 6 2 294.3.h.d 8
40.e odd 2 1 1344.3.d.e 4
40.f even 2 1 1344.3.d.c 4
45.h odd 6 2 1134.3.q.a 8
45.j even 6 2 1134.3.q.a 8
60.h even 2 1 336.3.d.b 4
105.g even 2 1 294.3.b.h 4
105.o odd 6 2 294.3.h.d 8
105.p even 6 2 294.3.h.g 8
120.i odd 2 1 1344.3.d.c 4
120.m even 2 1 1344.3.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a 4 5.b even 2 1
42.3.b.a 4 15.d odd 2 1
294.3.b.h 4 35.c odd 2 1
294.3.b.h 4 105.g even 2 1
294.3.h.d 8 35.j even 6 2
294.3.h.d 8 105.o odd 6 2
294.3.h.g 8 35.i odd 6 2
294.3.h.g 8 105.p even 6 2
336.3.d.b 4 20.d odd 2 1
336.3.d.b 4 60.h even 2 1
1050.3.c.a 8 5.c odd 4 2
1050.3.c.a 8 15.e even 4 2
1050.3.e.a 4 1.a even 1 1 trivial
1050.3.e.a 4 3.b odd 2 1 inner
1134.3.q.a 8 45.h odd 6 2
1134.3.q.a 8 45.j even 6 2
1344.3.d.c 4 40.f even 2 1
1344.3.d.c 4 120.i odd 2 1
1344.3.d.e 4 40.e odd 2 1
1344.3.d.e 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{4} + 212 T_{11}^{2} + 36 \)
\( T_{13}^{2} + 20 T_{13} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( 81 - 10 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 36 + 212 T^{2} + T^{4} \)
$13$ \( ( -12 + 20 T + T^{2} )^{2} \)
$17$ \( 116964 + 716 T^{2} + T^{4} \)
$19$ \( ( 16 + T )^{4} \)
$23$ \( 1954404 + 2804 T^{2} + T^{4} \)
$29$ \( 553536 + 1712 T^{2} + T^{4} \)
$31$ \( ( 324 + 64 T + T^{2} )^{2} \)
$37$ \( ( 20 + T )^{4} \)
$41$ \( 443556 + 5868 T^{2} + T^{4} \)
$43$ \( ( -608 + 40 T + T^{2} )^{2} \)
$47$ \( ( 72 + T^{2} )^{2} \)
$53$ \( ( 2592 + T^{2} )^{2} \)
$59$ \( 9216 + 3392 T^{2} + T^{4} \)
$61$ \( ( -2604 + 28 T + T^{2} )^{2} \)
$67$ \( ( 3488 + 120 T + T^{2} )^{2} \)
$71$ \( 2232036 + 7988 T^{2} + T^{4} \)
$73$ \( ( -892 + 60 T + T^{2} )^{2} \)
$79$ \( ( -1776 + 64 T + T^{2} )^{2} \)
$83$ \( 360000 + 21200 T^{2} + T^{4} \)
$89$ \( 2802276 + 3852 T^{2} + T^{4} \)
$97$ \( ( 7652 - 180 T + T^{2} )^{2} \)
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