Properties

Label 1050.2.s.e
Level 1050
Weight 2
Character orbit 1050.s
Analytic conductor 8.384
Analytic rank 0
Dimension 8
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.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 32 \nu^{4} + 16 \nu^{2} + 45 \)\()/144\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu \)\()/432\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225 \)\()/144\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/48\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 13 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 80 \nu^{3} + 225 \nu \)\()/144\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 2 \beta_{4} - \beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-\beta_{7} + 15 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} - 13\)
\(\nu^{7}\)\(=\)\(48 \beta_{5} - 13 \beta_{1}\)

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.396143 + 1.68614i
−1.26217 1.18614i
1.26217 + 1.18614i
−0.396143 1.68614i
0.396143 1.68614i
−1.26217 + 1.18614i
1.26217 1.18614i
−0.396143 + 1.68614i
−0.866025 + 0.500000i −1.65831 0.500000i 0.500000 0.866025i 0 1.68614 0.396143i 0.866025 + 2.50000i 1.00000i 2.50000 + 1.65831i 0
101.2 −0.866025 + 0.500000i 1.65831 0.500000i 0.500000 0.866025i 0 −1.18614 + 1.26217i 0.866025 + 2.50000i 1.00000i 2.50000 1.65831i 0
101.3 0.866025 0.500000i −1.65831 + 0.500000i 0.500000 0.866025i 0 −1.18614 + 1.26217i −0.866025 2.50000i 1.00000i 2.50000 1.65831i 0
101.4 0.866025 0.500000i 1.65831 + 0.500000i 0.500000 0.866025i 0 1.68614 0.396143i −0.866025 2.50000i 1.00000i 2.50000 + 1.65831i 0
551.1 −0.866025 0.500000i −1.65831 + 0.500000i 0.500000 + 0.866025i 0 1.68614 + 0.396143i 0.866025 2.50000i 1.00000i 2.50000 1.65831i 0
551.2 −0.866025 0.500000i 1.65831 + 0.500000i 0.500000 + 0.866025i 0 −1.18614 1.26217i 0.866025 2.50000i 1.00000i 2.50000 + 1.65831i 0
551.3 0.866025 + 0.500000i −1.65831 0.500000i 0.500000 + 0.866025i 0 −1.18614 1.26217i −0.866025 + 2.50000i 1.00000i 2.50000 + 1.65831i 0
551.4 0.866025 + 0.500000i 1.65831 0.500000i 0.500000 + 0.866025i 0 1.68614 + 0.396143i −0.866025 + 2.50000i 1.00000i 2.50000 1.65831i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.g even 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.s.e 8
3.b odd 2 1 1050.2.s.d 8
5.b even 2 1 inner 1050.2.s.e 8
5.c odd 4 1 210.2.t.a 4
5.c odd 4 1 210.2.t.d yes 4
7.d odd 6 1 1050.2.s.d 8
15.d odd 2 1 1050.2.s.d 8
15.e even 4 1 210.2.t.b yes 4
15.e even 4 1 210.2.t.c yes 4
21.g even 6 1 inner 1050.2.s.e 8
35.i odd 6 1 1050.2.s.d 8
35.k even 12 1 210.2.t.b yes 4
35.k even 12 1 210.2.t.c yes 4
35.k even 12 1 1470.2.d.b 4
35.k even 12 1 1470.2.d.c 4
35.l odd 12 1 1470.2.d.a 4
35.l odd 12 1 1470.2.d.d 4
105.p even 6 1 inner 1050.2.s.e 8
105.w odd 12 1 210.2.t.a 4
105.w odd 12 1 210.2.t.d yes 4
105.w odd 12 1 1470.2.d.a 4
105.w odd 12 1 1470.2.d.d 4
105.x even 12 1 1470.2.d.b 4
105.x even 12 1 1470.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.a 4 5.c odd 4 1
210.2.t.a 4 105.w odd 12 1
210.2.t.b yes 4 15.e even 4 1
210.2.t.b yes 4 35.k even 12 1
210.2.t.c yes 4 15.e even 4 1
210.2.t.c yes 4 35.k even 12 1
210.2.t.d yes 4 5.c odd 4 1
210.2.t.d yes 4 105.w odd 12 1
1050.2.s.d 8 3.b odd 2 1
1050.2.s.d 8 7.d odd 6 1
1050.2.s.d 8 15.d odd 2 1
1050.2.s.d 8 35.i odd 6 1
1050.2.s.e 8 1.a even 1 1 trivial
1050.2.s.e 8 5.b even 2 1 inner
1050.2.s.e 8 21.g even 6 1 inner
1050.2.s.e 8 105.p even 6 1 inner
1470.2.d.a 4 35.l odd 12 1
1470.2.d.a 4 105.w odd 12 1
1470.2.d.b 4 35.k even 12 1
1470.2.d.b 4 105.x even 12 1
1470.2.d.c 4 35.k even 12 1
1470.2.d.c 4 105.x even 12 1
1470.2.d.d 4 35.l odd 12 1
1470.2.d.d 4 105.w odd 12 1

Hecke kernels

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

\( T_{11}^{4} + 9 T_{11}^{3} + 31 T_{11}^{2} + 36 T_{11} + 16 \)
\( T_{37}^{8} + 204 T_{37}^{6} + 32400 T_{37}^{4} + 1880064 T_{37}^{2} + 84934656 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 1 - 5 T^{2} + 9 T^{4} )^{2} \)
$5$ 1
$7$ \( ( 1 + 11 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 9 T + 53 T^{2} + 234 T^{3} + 852 T^{4} + 2574 T^{5} + 6413 T^{6} + 11979 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 10 T^{2} - 189 T^{4} + 2890 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{4}( 1 + T + 19 T^{2} )^{4} \)
$23$ \( 1 + 71 T^{2} + 2797 T^{4} + 84206 T^{6} + 2089006 T^{8} + 44544974 T^{10} + 782715277 T^{12} + 10510548119 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 - 47 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( 1 + 56 T^{2} + 802 T^{4} - 22624 T^{6} - 719789 T^{8} - 30972256 T^{10} + 1503077122 T^{12} + 143680678904 T^{14} + 3512479453921 T^{16} \)
$41$ \( ( 1 + 9 T + 94 T^{2} + 369 T^{3} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 49 T^{2} + 660 T^{4} + 90601 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( 1 - 112 T^{2} + 6178 T^{4} - 218176 T^{6} + 7936579 T^{8} - 481950784 T^{10} + 30146669218 T^{12} - 1207272116848 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 + 191 T^{2} + 21817 T^{4} + 1727786 T^{6} + 104667286 T^{8} + 4853350874 T^{10} + 172146623977 T^{12} + 4233392975639 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 + 9 T + 17 T^{2} - 486 T^{3} - 4164 T^{4} - 28674 T^{5} + 59177 T^{6} + 1848411 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( 1 - 205 T^{2} + 23209 T^{4} - 2016790 T^{6} + 145240510 T^{8} - 9053370310 T^{10} + 467687367289 T^{12} - 18543968344645 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - T - 83 T^{2} + 74 T^{3} + 736 T^{4} + 5846 T^{5} - 518003 T^{6} - 493039 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 190 T^{2} + 18051 T^{4} + 1308910 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 3 T - 163 T^{2} + 18 T^{3} + 20862 T^{4} + 1602 T^{5} - 1291123 T^{6} - 2114907 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 155 T^{2} + 12276 T^{4} - 1458395 T^{6} + 88529281 T^{8} )^{2} \)
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