Properties

Label 1050.3.l.e
Level $1050$
Weight $3$
Character orbit 1050.l
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.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.1
Defining polynomial: \(x^{8} + 23 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{2} -\beta_{6} q^{3} + 2 \beta_{3} q^{4} + ( -\beta_{1} - \beta_{6} ) q^{6} -\beta_{2} q^{7} + ( -2 + 2 \beta_{3} ) q^{8} -3 \beta_{3} q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{2} -\beta_{6} q^{3} + 2 \beta_{3} q^{4} + ( -\beta_{1} - \beta_{6} ) q^{6} -\beta_{2} q^{7} + ( -2 + 2 \beta_{3} ) q^{8} -3 \beta_{3} q^{9} + ( -4 + 3 \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{11} -2 \beta_{1} q^{12} + ( 5 - 5 \beta_{3} + 5 \beta_{6} + \beta_{7} ) q^{13} + ( -\beta_{2} - \beta_{7} ) q^{14} -4 q^{16} + ( -5 - \beta_{1} + 9 \beta_{2} - 5 \beta_{3} ) q^{17} + ( 3 - 3 \beta_{3} ) q^{18} + ( -2 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} - \beta_{5} + 2 \beta_{6} + 7 \beta_{7} ) q^{19} + \beta_{4} q^{21} + ( -4 + 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + \beta_{5} ) q^{22} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} - 8 \beta_{6} + \beta_{7} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{24} + ( 10 + 5 \beta_{1} - \beta_{2} + 5 \beta_{6} + \beta_{7} ) q^{26} + 3 \beta_{1} q^{27} -2 \beta_{7} q^{28} + ( 4 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} + \beta_{5} - 4 \beta_{6} + 9 \beta_{7} ) q^{29} + ( 12 - 6 \beta_{1} - 2 \beta_{2} - 6 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} ) q^{31} + ( -4 - 4 \beta_{3} ) q^{32} + ( -9 + 9 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{33} + ( -\beta_{1} + 9 \beta_{2} - 10 \beta_{3} + \beta_{6} + 9 \beta_{7} ) q^{34} + 6 q^{36} + ( 14 - 15 \beta_{1} + 4 \beta_{2} + 14 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{37} + ( -3 + 3 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} + 14 \beta_{7} ) q^{38} + ( 5 \beta_{1} + 15 \beta_{3} + \beta_{5} - 5 \beta_{6} ) q^{39} + ( -20 - 10 \beta_{1} + 2 \beta_{2} - 10 \beta_{6} - 2 \beta_{7} ) q^{41} + ( \beta_{4} - \beta_{5} ) q^{42} + ( -8 + 8 \beta_{3} + \beta_{4} + \beta_{5} - 13 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 6 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{44} + ( -2 - 8 \beta_{1} - \beta_{2} + 2 \beta_{4} - 8 \beta_{6} + \beta_{7} ) q^{46} + ( 8 + 13 \beta_{1} + 7 \beta_{2} + 8 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{47} + 4 \beta_{6} q^{48} + 7 \beta_{3} q^{49} + ( 3 + 5 \beta_{1} - 9 \beta_{4} + 5 \beta_{6} ) q^{51} + ( 10 + 10 \beta_{1} - 2 \beta_{2} + 10 \beta_{3} ) q^{52} + ( 10 - 10 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 12 \beta_{6} - 4 \beta_{7} ) q^{53} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{54} + ( 2 \beta_{2} - 2 \beta_{7} ) q^{56} + ( 6 - 3 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} ) q^{57} + ( 4 - 4 \beta_{3} + \beta_{4} + \beta_{5} - 8 \beta_{6} + 18 \beta_{7} ) q^{58} + ( -20 \beta_{1} + 5 \beta_{2} + 15 \beta_{3} - 11 \beta_{5} + 20 \beta_{6} + 5 \beta_{7} ) q^{59} + ( -16 + 16 \beta_{1} - 16 \beta_{2} - 8 \beta_{4} + 16 \beta_{6} + 16 \beta_{7} ) q^{61} + ( 12 - 12 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{62} + 3 \beta_{7} q^{63} -8 \beta_{3} q^{64} + ( -18 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} - 3 \beta_{7} ) q^{66} + ( -38 - 7 \beta_{1} + 4 \beta_{2} - 38 \beta_{3} - \beta_{4} + \beta_{5} ) q^{67} + ( 10 - 10 \beta_{3} + 2 \beta_{6} + 18 \beta_{7} ) q^{68} + ( -\beta_{1} + 3 \beta_{2} - 24 \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{69} + ( 30 - \beta_{1} - 19 \beta_{2} + \beta_{4} - \beta_{6} + 19 \beta_{7} ) q^{71} + ( 6 + 6 \beta_{3} ) q^{72} + ( 3 - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 19 \beta_{6} - 27 \beta_{7} ) q^{73} + ( -15 \beta_{1} + 4 \beta_{2} + 28 \beta_{3} - 4 \beta_{5} + 15 \beta_{6} + 4 \beta_{7} ) q^{74} + ( -6 + 4 \beta_{1} - 14 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} + 14 \beta_{7} ) q^{76} + ( -7 + 7 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{77} + ( -15 + 15 \beta_{3} + \beta_{4} + \beta_{5} - 10 \beta_{6} ) q^{78} + ( 6 \beta_{1} + 19 \beta_{2} - 29 \beta_{3} - 6 \beta_{5} - 6 \beta_{6} + 19 \beta_{7} ) q^{79} -9 q^{81} + ( -20 - 20 \beta_{1} + 4 \beta_{2} - 20 \beta_{3} ) q^{82} + ( 8 - 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{83} -2 \beta_{5} q^{84} + ( -16 - 13 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 13 \beta_{6} + 2 \beta_{7} ) q^{86} + ( -12 + 4 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{87} + ( 8 - 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} ) q^{88} + ( 29 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 5 \beta_{5} - 29 \beta_{6} + 2 \beta_{7} ) q^{89} + ( 7 - 5 \beta_{2} - 5 \beta_{4} + 5 \beta_{7} ) q^{91} + ( -2 - 16 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{92} + ( 18 - 18 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 12 \beta_{6} - 18 \beta_{7} ) q^{93} + ( 13 \beta_{1} + 7 \beta_{2} + 16 \beta_{3} + 18 \beta_{5} - 13 \beta_{6} + 7 \beta_{7} ) q^{94} + ( 4 \beta_{1} + 4 \beta_{6} ) q^{96} + ( 34 + 42 \beta_{1} + 12 \beta_{2} + 34 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -7 + 7 \beta_{3} ) q^{98} + ( -9 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} - 3 \beta_{5} + 9 \beta_{6} - 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{8} + O(q^{10}) \) \( 8 q + 8 q^{2} - 16 q^{8} - 32 q^{11} + 40 q^{13} - 32 q^{16} - 40 q^{17} + 24 q^{18} - 32 q^{22} - 8 q^{23} + 80 q^{26} + 96 q^{31} - 32 q^{32} - 72 q^{33} + 48 q^{36} + 112 q^{37} - 24 q^{38} - 160 q^{41} - 64 q^{43} - 16 q^{46} + 64 q^{47} + 24 q^{51} + 80 q^{52} + 80 q^{53} + 48 q^{57} + 32 q^{58} - 128 q^{61} + 96 q^{62} - 144 q^{66} - 304 q^{67} + 80 q^{68} + 240 q^{71} + 48 q^{72} + 24 q^{73} - 48 q^{76} - 56 q^{77} - 120 q^{78} - 72 q^{81} - 160 q^{82} + 64 q^{83} - 128 q^{86} - 96 q^{87} + 64 q^{88} + 56 q^{91} - 16 q^{92} + 144 q^{93} + 272 q^{97} - 56 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 23 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 19 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 29 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 24 \nu^{2} \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{4} + 23 \)\()/5\)
\(\beta_{5}\)\(=\)\( -\nu^{6} - 22 \nu^{2} \)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{7} - 91 \nu^{3} \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( 6 \nu^{7} + 139 \nu^{3} \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 5 \beta_{3}\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} + 3 \beta_{6}\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{4} - 23\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-19 \beta_{2} + 29 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-12 \beta_{5} - 55 \beta_{3}\)
\(\nu^{7}\)\(=\)\((\)\(-91 \beta_{7} - 139 \beta_{6}\)\()/2\)

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.323042 0.323042i
−1.54779 + 1.54779i
1.54779 1.54779i
−0.323042 + 0.323042i
0.323042 + 0.323042i
−1.54779 1.54779i
1.54779 + 1.54779i
−0.323042 0.323042i
1.00000 1.00000i −1.22474 1.22474i 2.00000i 0 −2.44949 −1.87083 + 1.87083i −2.00000 2.00000i 3.00000i 0
43.2 1.00000 1.00000i −1.22474 1.22474i 2.00000i 0 −2.44949 1.87083 1.87083i −2.00000 2.00000i 3.00000i 0
43.3 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 0 2.44949 −1.87083 + 1.87083i −2.00000 2.00000i 3.00000i 0
43.4 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 0 2.44949 1.87083 1.87083i −2.00000 2.00000i 3.00000i 0
757.1 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 −1.87083 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.2 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.3 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 0 2.44949 −1.87083 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.4 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 0 2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.l.e yes 8
5.b even 2 1 1050.3.l.a 8
5.c odd 4 1 1050.3.l.a 8
5.c odd 4 1 inner 1050.3.l.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.l.a 8 5.b even 2 1
1050.3.l.a 8 5.c odd 4 1
1050.3.l.e yes 8 1.a even 1 1 trivial
1050.3.l.e yes 8 5.c odd 4 1 inner

Hecke kernels

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

\( T_{11}^{4} + 16 T_{11}^{3} - 82 T_{11}^{2} - 160 T_{11} + 625 \)
\(T_{13}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 - 2 T + T^{2} )^{4} \)
$3$ \( ( 9 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 49 + T^{4} )^{2} \)
$11$ \( ( 625 - 160 T - 82 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$13$ \( 1157776 - 1377280 T + 819200 T^{2} + 131040 T^{3} + 20648 T^{4} - 6720 T^{5} + 800 T^{6} - 40 T^{7} + T^{8} \)
$17$ \( 69482851216 - 5482796800 T + 216320000 T^{2} + 6928160 T^{3} + 178408 T^{4} - 12800 T^{5} + 800 T^{6} + 40 T^{7} + T^{8} \)
$19$ \( 145676568976 + 1253503040 T^{2} + 3066648 T^{4} + 2960 T^{6} + T^{8} \)
$23$ \( 244140625 - 71875000 T + 10580000 T^{2} + 2037000 T^{3} + 189650 T^{4} + 840 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$29$ \( 1070879337889 + 5038353292 T^{2} + 8103750 T^{4} + 5068 T^{6} + T^{8} \)
$31$ \( ( -379760 + 66624 T - 1192 T^{2} - 48 T^{3} + T^{4} )^{2} \)
$37$ \( 1163755500625 - 111519444400 T + 5343298688 T^{2} - 75130608 T^{3} + 2352914 T^{4} - 152880 T^{5} + 6272 T^{6} - 112 T^{7} + T^{8} \)
$41$ \( ( -68864 - 20480 T + 1088 T^{2} + 80 T^{3} + T^{4} )^{2} \)
$43$ \( 223532401 + 560243872 T + 702075392 T^{2} + 15421056 T^{3} + 178898 T^{4} + 12768 T^{5} + 2048 T^{6} + 64 T^{7} + T^{8} \)
$47$ \( 6469392250000 + 1371251720000 T + 145325187200 T^{2} - 4007848320 T^{3} + 54758696 T^{4} - 44016 T^{5} + 2048 T^{6} - 64 T^{7} + T^{8} \)
$53$ \( 383805030400 - 45358776320 T + 2680291328 T^{2} - 65961984 T^{3} + 1289216 T^{4} - 55296 T^{5} + 3200 T^{6} - 80 T^{7} + T^{8} \)
$59$ \( 125230513134736 + 247059022784 T^{2} + 143675736 T^{4} + 22064 T^{6} + T^{8} \)
$61$ \( ( -22016000 - 1085440 T - 11392 T^{2} + 64 T^{3} + T^{4} )^{2} \)
$67$ \( 38759303135809 + 4837615596880 T + 301895580800 T^{2} + 11104159152 T^{3} + 267307682 T^{4} + 4307664 T^{5} + 46208 T^{6} + 304 T^{7} + T^{8} \)
$71$ \( ( 17125945 + 495336 T - 4762 T^{2} - 120 T^{3} + T^{4} )^{2} \)
$73$ \( 165058256250000 + 6382638000000 T + 123405120000 T^{2} - 5951340000 T^{3} + 133065000 T^{4} - 194400 T^{5} + 288 T^{6} - 24 T^{7} + T^{8} \)
$79$ \( 7023613249 + 689858784972 T^{2} + 241774310 T^{4} + 27468 T^{6} + T^{8} \)
$83$ \( 66382491904 - 8442609664 T + 536870912 T^{2} - 16751616 T^{3} + 515360 T^{4} - 32256 T^{5} + 2048 T^{6} - 64 T^{7} + T^{8} \)
$89$ \( 362929170490000 + 466959816000 T^{2} + 169779416 T^{4} + 22608 T^{6} + T^{8} \)
$97$ \( 392594596000000 - 29109143680000 T + 1079156787200 T^{2} - 4014311680 T^{3} + 40504096 T^{4} - 1723712 T^{5} + 36992 T^{6} - 272 T^{7} + T^{8} \)
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