Properties

Label 1050.2.d.e
Level $1050$
Weight $2$
Character orbit 1050.d
Analytic conductor $8.384$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{3} q^{3} + q^{4} + \beta_{3} q^{6} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} -3 \beta_{2} q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{3} q^{3} + q^{4} + \beta_{3} q^{6} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} -3 \beta_{2} q^{9} + \beta_{3} q^{12} + ( -\beta_{1} + \beta_{3} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{14} + q^{16} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{17} -3 \beta_{2} q^{18} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{21} -6 q^{23} + \beta_{3} q^{24} + ( -\beta_{1} + \beta_{3} ) q^{26} + 3 \beta_{1} q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{28} -6 \beta_{2} q^{29} + q^{32} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{34} -3 \beta_{2} q^{36} -2 \beta_{2} q^{37} + ( -\beta_{1} - \beta_{3} ) q^{38} + ( 3 - 3 \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{42} -4 \beta_{2} q^{43} -6 q^{46} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{47} + \beta_{3} q^{48} + ( 5 + 2 \beta_{1} + 2 \beta_{3} ) q^{49} + ( 6 + 6 \beta_{2} ) q^{51} + ( -\beta_{1} + \beta_{3} ) q^{52} -6 q^{53} + 3 \beta_{1} q^{54} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{56} + ( 3 + 3 \beta_{2} ) q^{57} -6 \beta_{2} q^{58} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{59} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{61} + ( -3 + 3 \beta_{1} + 3 \beta_{3} ) q^{63} + q^{64} + 8 \beta_{2} q^{67} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{68} -6 \beta_{3} q^{69} -3 \beta_{2} q^{72} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{73} -2 \beta_{2} q^{74} + ( -\beta_{1} - \beta_{3} ) q^{76} + ( 3 - 3 \beta_{2} ) q^{78} + 10 q^{79} -9 q^{81} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{82} + ( \beta_{1} + \beta_{3} ) q^{83} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{84} -4 \beta_{2} q^{86} + 6 \beta_{1} q^{87} + ( 6 + \beta_{1} + \beta_{3} ) q^{91} -6 q^{92} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{94} + \beta_{3} q^{96} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{97} + ( 5 + 2 \beta_{1} + 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{8} + 4q^{16} + 12q^{21} - 24q^{23} + 4q^{32} + 12q^{39} + 12q^{42} - 24q^{46} + 20q^{49} + 24q^{51} - 24q^{53} + 12q^{57} - 12q^{63} + 4q^{64} + 12q^{78} + 40q^{79} - 36q^{81} + 12q^{84} + 24q^{91} - 24q^{92} + 20q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1049.1
1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
1.00000 −1.22474 1.22474i 1.00000 0 −1.22474 1.22474i −2.44949 + 1.00000i 1.00000 3.00000i 0
1049.2 1.00000 −1.22474 + 1.22474i 1.00000 0 −1.22474 + 1.22474i −2.44949 1.00000i 1.00000 3.00000i 0
1049.3 1.00000 1.22474 1.22474i 1.00000 0 1.22474 1.22474i 2.44949 1.00000i 1.00000 3.00000i 0
1049.4 1.00000 1.22474 + 1.22474i 1.00000 0 1.22474 + 1.22474i 2.44949 + 1.00000i 1.00000 3.00000i 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
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.d.e 4
3.b odd 2 1 1050.2.d.b 4
5.b even 2 1 1050.2.d.b 4
5.c odd 4 1 42.2.d.a 4
5.c odd 4 1 1050.2.b.b 4
7.b odd 2 1 inner 1050.2.d.e 4
15.d odd 2 1 inner 1050.2.d.e 4
15.e even 4 1 42.2.d.a 4
15.e even 4 1 1050.2.b.b 4
20.e even 4 1 336.2.k.b 4
21.c even 2 1 1050.2.d.b 4
35.c odd 2 1 1050.2.d.b 4
35.f even 4 1 42.2.d.a 4
35.f even 4 1 1050.2.b.b 4
35.k even 12 2 294.2.f.b 8
35.l odd 12 2 294.2.f.b 8
40.i odd 4 1 1344.2.k.c 4
40.k even 4 1 1344.2.k.d 4
45.k odd 12 2 1134.2.m.g 8
45.l even 12 2 1134.2.m.g 8
60.l odd 4 1 336.2.k.b 4
105.g even 2 1 inner 1050.2.d.e 4
105.k odd 4 1 42.2.d.a 4
105.k odd 4 1 1050.2.b.b 4
105.w odd 12 2 294.2.f.b 8
105.x even 12 2 294.2.f.b 8
120.q odd 4 1 1344.2.k.d 4
120.w even 4 1 1344.2.k.c 4
140.j odd 4 1 336.2.k.b 4
280.s even 4 1 1344.2.k.c 4
280.y odd 4 1 1344.2.k.d 4
315.cb even 12 2 1134.2.m.g 8
315.cf odd 12 2 1134.2.m.g 8
420.w even 4 1 336.2.k.b 4
840.bm even 4 1 1344.2.k.d 4
840.bp odd 4 1 1344.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 5.c odd 4 1
42.2.d.a 4 15.e even 4 1
42.2.d.a 4 35.f even 4 1
42.2.d.a 4 105.k odd 4 1
294.2.f.b 8 35.k even 12 2
294.2.f.b 8 35.l odd 12 2
294.2.f.b 8 105.w odd 12 2
294.2.f.b 8 105.x even 12 2
336.2.k.b 4 20.e even 4 1
336.2.k.b 4 60.l odd 4 1
336.2.k.b 4 140.j odd 4 1
336.2.k.b 4 420.w even 4 1
1050.2.b.b 4 5.c odd 4 1
1050.2.b.b 4 15.e even 4 1
1050.2.b.b 4 35.f even 4 1
1050.2.b.b 4 105.k odd 4 1
1050.2.d.b 4 3.b odd 2 1
1050.2.d.b 4 5.b even 2 1
1050.2.d.b 4 21.c even 2 1
1050.2.d.b 4 35.c odd 2 1
1050.2.d.e 4 1.a even 1 1 trivial
1050.2.d.e 4 7.b odd 2 1 inner
1050.2.d.e 4 15.d odd 2 1 inner
1050.2.d.e 4 105.g even 2 1 inner
1134.2.m.g 8 45.k odd 12 2
1134.2.m.g 8 45.l even 12 2
1134.2.m.g 8 315.cb even 12 2
1134.2.m.g 8 315.cf odd 12 2
1344.2.k.c 4 40.i odd 4 1
1344.2.k.c 4 120.w even 4 1
1344.2.k.c 4 280.s even 4 1
1344.2.k.c 4 840.bp odd 4 1
1344.2.k.d 4 40.k even 4 1
1344.2.k.d 4 120.q odd 4 1
1344.2.k.d 4 280.y odd 4 1
1344.2.k.d 4 840.bm even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11} \)
\( T_{13}^{2} - 6 \)
\( T_{23} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{4} \)
$3$ \( 1 + 9 T^{4} \)
$5$ 1
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( ( 1 + 20 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 6 T + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{4} \)
$59$ \( ( 1 - 32 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 28 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 + 50 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 + 170 T^{2} + 9409 T^{4} )^{2} \)
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