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: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} - 2 q^{4} + (\beta_{2} - 2) q^{6} - \beta_{3} q^{7} - 2 \beta_1 q^{8} + (2 \beta_{2} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} - 2 q^{4} + (\beta_{2} - 2) q^{6} - \beta_{3} q^{7} - 2 \beta_1 q^{8} + (2 \beta_{2} + 5) q^{9} + ( - 2 \beta_{2} + 5 \beta_1) q^{11} + ( - 2 \beta_{3} - 2 \beta_1) q^{12} + ( - 4 \beta_{3} - 10) q^{13} - \beta_{2} q^{14} + 4 q^{16} + (5 \beta_{2} - 2 \beta_1) q^{17} + ( - 4 \beta_{3} + 5 \beta_1) q^{18} - 16 q^{19} + ( - \beta_{2} - 7) q^{21} + (4 \beta_{3} - 10) q^{22} + ( - 10 \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{2} + 4) q^{24} + ( - 4 \beta_{2} - 10 \beta_1) q^{26} + (\beta_{3} + 19 \beta_1) q^{27} + 2 \beta_{3} q^{28} + (2 \beta_{2} - 20 \beta_1) q^{29} + (10 \beta_{3} - 32) q^{31} + 4 \beta_1 q^{32} + (4 \beta_{3} + 5 \beta_{2} - 14 \beta_1 - 10) q^{33} + ( - 10 \beta_{3} + 4) q^{34} + ( - 4 \beta_{2} - 10) q^{36} - 20 q^{37} - 16 \beta_1 q^{38} + ( - 10 \beta_{3} - 4 \beta_{2} - 10 \beta_1 - 28) q^{39} + (9 \beta_{2} + 30 \beta_1) q^{41} + (2 \beta_{3} - 7 \beta_1) q^{42} + ( - 12 \beta_{3} - 20) q^{43} + (4 \beta_{2} - 10 \beta_1) q^{44} + (20 \beta_{3} - 2) q^{46} - 6 \beta_1 q^{47} + (4 \beta_{3} + 4 \beta_1) q^{48} + 7 q^{49} + ( - 10 \beta_{3} - 2 \beta_{2} + 35 \beta_1 + 4) q^{51} + (8 \beta_{3} + 20) q^{52} - 36 \beta_1 q^{53} + (\beta_{2} - 38) q^{54} + 2 \beta_{2} q^{56} + ( - 16 \beta_{3} - 16 \beta_1) q^{57} + ( - 4 \beta_{3} + 40) q^{58} + (8 \beta_{2} - 20 \beta_1) q^{59} + ( - 20 \beta_{3} - 14) q^{61} + (10 \beta_{2} - 32 \beta_1) q^{62} + ( - 5 \beta_{3} - 14 \beta_1) q^{63} - 8 q^{64} + ( - 10 \beta_{3} + 4 \beta_{2} - 10 \beta_1 + 28) q^{66} + (4 \beta_{3} - 60) q^{67} + ( - 10 \beta_{2} + 4 \beta_1) q^{68} + (20 \beta_{3} + \beta_{2} - 70 \beta_1 - 2) q^{69} + ( - 14 \beta_{2} - 25 \beta_1) q^{71} + (8 \beta_{3} - 10 \beta_1) q^{72} + (16 \beta_{3} - 30) q^{73} - 20 \beta_1 q^{74} + 32 q^{76} + ( - 5 \beta_{2} + 14 \beta_1) q^{77} + (8 \beta_{3} - 10 \beta_{2} - 28 \beta_1 + 20) q^{78} + ( - 20 \beta_{3} - 32) q^{79} + (20 \beta_{2} - 31) q^{81} + ( - 18 \beta_{3} - 60) q^{82} + (20 \beta_{2} - 50 \beta_1) q^{83} + (2 \beta_{2} + 14) q^{84} + ( - 12 \beta_{2} - 20 \beta_1) q^{86} + ( - 4 \beta_{3} - 20 \beta_{2} + 14 \beta_1 + 40) q^{87} + ( - 8 \beta_{3} + 20) q^{88} + (3 \beta_{2} - 30 \beta_1) q^{89} + (10 \beta_{3} + 28) q^{91} + (20 \beta_{2} - 2 \beta_1) q^{92} + ( - 32 \beta_{3} + 10 \beta_{2} - 32 \beta_1 + 70) q^{93} + 12 q^{94} + (4 \beta_{2} - 8) q^{96} + ( - 8 \beta_{3} + 90) q^{97} + 7 \beta_1 q^{98} + ( - 20 \beta_{3} - 10 \beta_{2} + 25 \beta_1 + 56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 8 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 8 q^{6} + 20 q^{9} - 40 q^{13} + 16 q^{16} - 64 q^{19} - 28 q^{21} - 40 q^{22} + 16 q^{24} - 128 q^{31} - 40 q^{33} + 16 q^{34} - 40 q^{36} - 80 q^{37} - 112 q^{39} - 80 q^{43} - 8 q^{46} + 28 q^{49} + 16 q^{51} + 80 q^{52} - 152 q^{54} + 160 q^{58} - 56 q^{61} - 32 q^{64} + 112 q^{66} - 240 q^{67} - 8 q^{69} - 120 q^{73} + 128 q^{76} + 80 q^{78} - 128 q^{79} - 124 q^{81} - 240 q^{82} + 56 q^{84} + 160 q^{87} + 80 q^{88} + 112 q^{91} + 280 q^{93} + 48 q^{94} - 32 q^{96} + 360 q^{97} + 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display

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 \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{4} + 212T_{11}^{2} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} + 20T_{13} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

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