Properties

Label 1050.3.f.b
Level $1050$
Weight $3$
Character orbit 1050.f
Analytic conductor $28.610$
Analytic rank $0$
Dimension $8$
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.3
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + 2 q^{4} - \beta_{3} q^{6} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{7} + 2 \beta_1 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + 2 q^{4} - \beta_{3} q^{6} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{7} + 2 \beta_1 q^{8} - 3 q^{9} + ( - \beta_{2} - 6 \beta_1 - 2) q^{11} + 2 \beta_{5} q^{12} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{13} + (2 \beta_{7} - 2 \beta_{5} - \beta_{2} + 4) q^{14} + 4 q^{16} + (2 \beta_{7} - 6 \beta_{6}) q^{17} - 3 \beta_1 q^{18} + ( - 2 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 7 \beta_{3}) q^{19} + ( - 3 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{21} + ( - 2 \beta_{4} - 2 \beta_1 - 12) q^{22} + ( - 2 \beta_{4} - 4 \beta_{2} + 8 \beta_1) q^{23} - 2 \beta_{3} q^{24} + ( - 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 2 \beta_{3}) q^{26} - 3 \beta_{5} q^{27} + ( - 4 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 4 \beta_1) q^{28} + ( - 4 \beta_{4} - 14 \beta_1 - 18) q^{29} + ( - 2 \beta_{7} + 2 \beta_{6} + 14 \beta_{5} - \beta_{3}) q^{31} + 4 \beta_1 q^{32} + (3 \beta_{7} - 2 \beta_{5} + 6 \beta_{3}) q^{33} + (6 \beta_{7} - 4 \beta_{6}) q^{34} - 6 q^{36} + ( - 8 \beta_{4} - 4 \beta_{2} - 10 \beta_1 + 6) q^{37} + (2 \beta_{7} + 4 \beta_{6} + 14 \beta_{5} - 6 \beta_{3}) q^{38} + ( - 2 \beta_{4} - \beta_{2} + 6 \beta_1 - 6) q^{39} + ( - 2 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 4 \beta_{3}) q^{41} + (3 \beta_{7} + 4 \beta_{5} + 2 \beta_{2} + 6) q^{42} + (4 \beta_{4} - 8 \beta_{2} - 4 \beta_1 + 8) q^{43} + ( - 2 \beta_{2} - 12 \beta_1 - 4) q^{44} + ( - 8 \beta_{4} - 2 \beta_{2} + 16) q^{46} + (8 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} - 14 \beta_{3}) q^{47} + 4 \beta_{5} q^{48} + (2 \beta_{7} + 12 \beta_{5} - 8 \beta_{2} - 3) q^{49} + (6 \beta_{4} + 2 \beta_{2}) q^{51} + ( - 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + 4 \beta_{3}) q^{52} + (14 \beta_{4} + 2 \beta_{2} + 11 \beta_1 - 16) q^{53} + 3 \beta_{3} q^{54} + (4 \beta_{7} - 4 \beta_{5} - 2 \beta_{2} + 8) q^{56} + (2 \beta_{4} - 2 \beta_{2} - 21 \beta_1 - 18) q^{57} + ( - 4 \beta_{2} - 18 \beta_1 - 28) q^{58} + (8 \beta_{7} + 24 \beta_{5} + 14 \beta_{3}) q^{59} + ( - 10 \beta_{7} - 16 \beta_{6} - 28 \beta_{5} + 16 \beta_{3}) q^{61} + ( - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 14 \beta_{3}) q^{62} + (6 \beta_{6} + 3 \beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{63} + 8 q^{64} + ( - 6 \beta_{6} - 12 \beta_{5} + 2 \beta_{3}) q^{66} + (6 \beta_{4} + 4 \beta_{2} - 36 \beta_1 + 24) q^{67} + (4 \beta_{7} - 12 \beta_{6}) q^{68} + (12 \beta_{7} - 6 \beta_{6} - 8 \beta_{3}) q^{69} + ( - 4 \beta_{4} + 15 \beta_{2} + 14 \beta_1 + 22) q^{71} - 6 \beta_1 q^{72} + (9 \beta_{7} + 14 \beta_{6} + 38 \beta_{5} - 2 \beta_{3}) q^{73} + ( - 8 \beta_{4} - 8 \beta_{2} + 6 \beta_1 - 20) q^{74} + ( - 4 \beta_{7} - 4 \beta_{6} + 12 \beta_{5} - 14 \beta_{3}) q^{76} + ( - 12 \beta_{7} + 10 \beta_{6} + 12 \beta_{5} - 2 \beta_{4} - 12 \beta_{3} + \cdots - 24) q^{77}+ \cdots + (3 \beta_{2} + 18 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} - 24 q^{9} - 16 q^{11} + 32 q^{14} + 32 q^{16} - 96 q^{22} - 144 q^{29} - 48 q^{36} + 48 q^{37} - 48 q^{39} + 48 q^{42} + 64 q^{43} - 32 q^{44} + 128 q^{46} - 24 q^{49} - 128 q^{53} + 64 q^{56} - 144 q^{57} - 224 q^{58} + 64 q^{64} + 192 q^{67} + 176 q^{71} - 160 q^{74} - 192 q^{77} + 96 q^{78} - 288 q^{79} + 72 q^{81} - 64 q^{86} - 192 q^{88} + 64 q^{91} - 336 q^{93} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -10\nu^{7} + 49\nu^{5} - 133\nu^{3} + 801\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} + \nu^{5} + 5\nu^{3} - 63\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 2\nu^{2} + 18 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} - 56\nu^{2} + 225 ) / 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 22 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} - 7\nu^{5} + 19\nu^{3} - 81\nu ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 2\beta_{5} + \beta_{4} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{6} + \beta_{5} + 4\beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} + 19\beta_{3} + 5\beta_{2} + 19\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{6} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{7} + 13\beta_{3} + 29\beta_{2} - 13\beta_1 ) / 4 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
−1.72286 0.178197i
1.01575 + 1.40294i
−1.72286 + 0.178197i
1.01575 1.40294i
1.72286 + 0.178197i
−1.01575 1.40294i
1.72286 0.178197i
−1.01575 + 1.40294i
−1.41421 1.73205i 2.00000 0 2.44949i −6.70141 + 2.02265i −2.82843 −3.00000 0
601.2 −1.41421 1.73205i 2.00000 0 2.44949i 1.04456 6.92163i −2.82843 −3.00000 0
601.3 −1.41421 1.73205i 2.00000 0 2.44949i −6.70141 2.02265i −2.82843 −3.00000 0
601.4 −1.41421 1.73205i 2.00000 0 2.44949i 1.04456 + 6.92163i −2.82843 −3.00000 0
601.5 1.41421 1.73205i 2.00000 0 2.44949i −1.04456 + 6.92163i 2.82843 −3.00000 0
601.6 1.41421 1.73205i 2.00000 0 2.44949i 6.70141 2.02265i 2.82843 −3.00000 0
601.7 1.41421 1.73205i 2.00000 0 2.44949i −1.04456 6.92163i 2.82843 −3.00000 0
601.8 1.41421 1.73205i 2.00000 0 2.44949i 6.70141 + 2.02265i 2.82843 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.f.b 8
5.b even 2 1 210.3.f.a 8
5.c odd 4 2 1050.3.h.b 16
7.b odd 2 1 inner 1050.3.f.b 8
15.d odd 2 1 630.3.f.c 8
20.d odd 2 1 1680.3.s.a 8
35.c odd 2 1 210.3.f.a 8
35.f even 4 2 1050.3.h.b 16
105.g even 2 1 630.3.f.c 8
140.c even 2 1 1680.3.s.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.f.a 8 5.b even 2 1
210.3.f.a 8 35.c odd 2 1
630.3.f.c 8 15.d odd 2 1
630.3.f.c 8 105.g even 2 1
1050.3.f.b 8 1.a even 1 1 trivial
1050.3.f.b 8 7.b odd 2 1 inner
1050.3.h.b 16 5.c odd 4 2
1050.3.h.b 16 35.f even 4 2
1680.3.s.a 8 20.d odd 2 1
1680.3.s.a 8 140.c even 2 1

Hecke kernels

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

\( T_{11}^{4} + 8T_{11}^{3} - 180T_{11}^{2} - 784T_{11} + 964 \) Copy content Toggle raw display
\( T_{23}^{4} - 1336T_{23}^{2} - 15360T_{23} + 54544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{6} - 2842 T^{4} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{3} - 180 T^{2} - 784 T + 964)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 264 T^{6} + \cdots + 14714896 \) Copy content Toggle raw display
$17$ \( (T^{4} + 440 T^{2} + 19600)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 1848 T^{6} + \cdots + 2869102096 \) Copy content Toggle raw display
$23$ \( (T^{4} - 1336 T^{2} - 15360 T + 54544)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 72 T^{3} + 680 T^{2} + \cdots - 281456)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 2616 T^{6} + \cdots + 79120688656 \) Copy content Toggle raw display
$37$ \( (T^{4} - 24 T^{3} - 3064 T^{2} + \cdots - 883184)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 2016 T^{6} + \cdots + 7126061056 \) Copy content Toggle raw display
$43$ \( (T^{4} - 32 T^{3} - 4000 T^{2} + \cdots + 2654464)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 8016 T^{6} + \cdots + 1081300500736 \) Copy content Toggle raw display
$53$ \( (T^{4} + 64 T^{3} - 5068 T^{2} + \cdots + 3722116)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 14176 T^{6} + \cdots + 9016903929856 \) Copy content Toggle raw display
$61$ \( T^{8} + 24672 T^{6} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{4} - 96 T^{3} - 3768 T^{2} + \cdots - 7371504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 88 T^{3} - 11860 T^{2} + \cdots + 25706884)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 24584 T^{6} + \cdots + 483086161936 \) Copy content Toggle raw display
$79$ \( (T^{4} + 144 T^{3} + 2912 T^{2} + \cdots - 2641664)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 177346111242496 \) Copy content Toggle raw display
$89$ \( T^{8} + 43344 T^{6} + \cdots + 86\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{8} + 35144 T^{6} + \cdots + 76019496215056 \) Copy content Toggle raw display
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