Properties

Label 1500.2.m.a
Level $1500$
Weight $2$
Character orbit 1500.m
Analytic conductor $11.978$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1500.m (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.26265625.1
Defining polynomial: \(x^{8} - 3 x^{7} + 2 x^{6} + x^{4} + 8 x^{2} - 24 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{6} q^{3} + ( -1 - \beta_{2} - \beta_{6} + \beta_{7} ) q^{7} -\beta_{1} q^{9} +O(q^{10})\) \( q -\beta_{6} q^{3} + ( -1 - \beta_{2} - \beta_{6} + \beta_{7} ) q^{7} -\beta_{1} q^{9} + ( 2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{11} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{17} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{19} + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{21} + ( 1 + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{23} + ( -1 + \beta_{1} + \beta_{3} + \beta_{6} ) q^{27} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{29} + ( -\beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} ) q^{31} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{33} + ( 2 \beta_{1} - 3 \beta_{5} ) q^{37} + ( 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{39} + ( 2 + \beta_{1} - 2 \beta_{3} - 4 \beta_{5} ) q^{41} + ( 2 + 2 \beta_{2} - \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{43} + ( 4 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{49} + ( -1 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{51} + ( 5 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{57} + ( -8 + 4 \beta_{1} - \beta_{2} + 8 \beta_{3} - \beta_{5} ) q^{59} + ( 9 - \beta_{1} - \beta_{3} - 2 \beta_{5} - 9 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{63} + ( -3 \beta_{1} - 2 \beta_{2} + 9 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{67} + ( \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{69} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{71} + ( 7 - 4 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -5 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{77} + ( -3 + 7 \beta_{1} + 4 \beta_{2} - \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{79} -\beta_{3} q^{81} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 6 \beta_{7} ) q^{83} + ( -2 \beta_{1} + \beta_{5} ) q^{87} + ( -2 - 4 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{89} + ( 4 + \beta_{1} - 4 \beta_{3} - \beta_{5} ) q^{91} + ( 2 + 3 \beta_{1} + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{93} + ( 6 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} ) q^{97} + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{3} - 8q^{7} - 2q^{9} + O(q^{10}) \) \( 8q - 2q^{3} - 8q^{7} - 2q^{9} + 8q^{11} - 3q^{17} + 5q^{19} + 7q^{21} + 7q^{23} - 2q^{27} - 3q^{29} - 3q^{31} - 7q^{33} + q^{37} + 10q^{41} + 12q^{43} + 33q^{47} - 8q^{49} - 8q^{51} + 19q^{53} - 10q^{57} - 38q^{59} + 46q^{61} - 3q^{63} + 8q^{67} + 2q^{69} - 25q^{71} + 26q^{73} - 23q^{77} - 16q^{79} - 2q^{81} - 8q^{83} - 3q^{87} - 30q^{89} + 25q^{91} + 22q^{93} + 14q^{97} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 2 x^{6} + x^{4} + 8 x^{2} - 24 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + \nu^{3} + 2 \nu^{2} + 4 \nu - 8 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + \nu^{3} - 2 \nu^{2} + 12 \nu - 12 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{7} + 9 \nu^{6} + 2 \nu^{5} + 4 \nu^{4} + \nu^{3} - 4 \nu^{2} - 60 \nu + 64 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 11 \nu^{7} - 15 \nu^{6} - 4 \nu^{5} - 8 \nu^{4} + 3 \nu^{3} + 2 \nu^{2} + 92 \nu - 96 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( 11 \nu^{7} - 13 \nu^{6} - 6 \nu^{5} - 8 \nu^{4} + 3 \nu^{3} + 4 \nu^{2} + 96 \nu - 88 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 13 \nu^{7} - 21 \nu^{6} - 4 \nu^{5} - 4 \nu^{4} + 5 \nu^{3} + 10 \nu^{2} + 120 \nu - 144 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 19 \nu^{7} - 31 \nu^{6} - 4 \nu^{5} - 8 \nu^{4} + 11 \nu^{3} + 18 \nu^{2} + 180 \nu - 224 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_{1} + 4\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_{1} + 3\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 3\)\()/5\)
\(\nu^{4}\)\(=\)\(-\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} + 2 \beta_{1} + 1\)
\(\nu^{5}\)\(=\)\((\)\(6 \beta_{7} - 11 \beta_{6} - 9 \beta_{5} + 3 \beta_{4} - 11 \beta_{3} + 6 \beta_{2} + 6 \beta_{1} + 19\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 12 \beta_{1} - 12\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(-8 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 57 \beta_{1} + 3\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(877\) \(1001\)
\(\chi(n)\) \(1\) \(-\beta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
1.40799 + 0.132563i
−1.21700 0.720348i
−0.0272949 + 1.41395i
1.33631 0.462894i
−0.0272949 1.41395i
1.33631 + 0.462894i
1.40799 0.132563i
−1.21700 + 0.720348i
0 0.309017 0.951057i 0 0 0 −1.50430 0 −0.809017 0.587785i 0
301.2 0 0.309017 0.951057i 0 0 0 1.74037 0 −0.809017 0.587785i 0
601.1 0 −0.809017 0.587785i 0 0 0 −4.32440 0 0.309017 + 0.951057i 0
601.2 0 −0.809017 0.587785i 0 0 0 0.0883282 0 0.309017 + 0.951057i 0
901.1 0 −0.809017 + 0.587785i 0 0 0 −4.32440 0 0.309017 0.951057i 0
901.2 0 −0.809017 + 0.587785i 0 0 0 0.0883282 0 0.309017 0.951057i 0
1201.1 0 0.309017 + 0.951057i 0 0 0 −1.50430 0 −0.809017 + 0.587785i 0
1201.2 0 0.309017 + 0.951057i 0 0 0 1.74037 0 −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1201.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1500.2.m.a 8
5.b even 2 1 300.2.m.b 8
5.c odd 4 2 1500.2.o.b 16
15.d odd 2 1 900.2.n.b 8
25.d even 5 1 inner 1500.2.m.a 8
25.d even 5 1 7500.2.a.f 4
25.e even 10 1 300.2.m.b 8
25.e even 10 1 7500.2.a.e 4
25.f odd 20 2 1500.2.o.b 16
25.f odd 20 2 7500.2.d.c 8
75.h odd 10 1 900.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.b 8 5.b even 2 1
300.2.m.b 8 25.e even 10 1
900.2.n.b 8 15.d odd 2 1
900.2.n.b 8 75.h odd 10 1
1500.2.m.a 8 1.a even 1 1 trivial
1500.2.m.a 8 25.d even 5 1 inner
1500.2.o.b 16 5.c odd 4 2
1500.2.o.b 16 25.f odd 20 2
7500.2.a.e 4 25.e even 10 1
7500.2.a.f 4 25.d even 5 1
7500.2.d.c 8 25.f odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 4 T_{7}^{3} - 4 T_{7}^{2} - 11 T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1500, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 1 - 11 T - 4 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$11$ \( 361 - 323 T + 1033 T^{2} + 149 T^{3} + 100 T^{4} + 39 T^{5} + 23 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( 2025 + 2025 T + 2925 T^{2} + 975 T^{3} + 310 T^{4} + 5 T^{5} - 5 T^{6} + T^{8} \)
$17$ \( 81 + 108 T + 333 T^{2} + 21 T^{3} - 20 T^{4} + T^{5} + 13 T^{6} + 3 T^{7} + T^{8} \)
$19$ \( 25 - 25 T + 225 T^{2} + 225 T^{3} + 160 T^{4} + 45 T^{5} + 5 T^{6} - 5 T^{7} + T^{8} \)
$23$ \( 29241 + 513 T + 12798 T^{2} + 6171 T^{3} + 1705 T^{4} + 61 T^{5} + 18 T^{6} - 7 T^{7} + T^{8} \)
$29$ \( 1 + 3 T + 58 T^{2} - 129 T^{3} + 105 T^{4} + 21 T^{5} + 8 T^{6} + 3 T^{7} + T^{8} \)
$31$ \( 962361 - 55917 T + 78228 T^{2} - 21699 T^{3} + 3625 T^{4} + 511 T^{5} + 58 T^{6} + 3 T^{7} + T^{8} \)
$37$ \( 1042441 + 57176 T + 12657 T^{2} + 343 T^{3} + 1550 T^{4} - 113 T^{5} + 57 T^{6} - T^{7} + T^{8} \)
$41$ \( 7317025 + 1704150 T + 228475 T^{2} - 5800 T^{3} + 4085 T^{4} - 520 T^{5} + 155 T^{6} - 10 T^{7} + T^{8} \)
$43$ \( ( 131 + 64 T - 19 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$47$ \( 16801801 - 4947493 T + 1291508 T^{2} - 275851 T^{3} + 48275 T^{4} - 6151 T^{5} + 568 T^{6} - 33 T^{7} + T^{8} \)
$53$ \( 9801 + 9504 T + 5787 T^{2} + 2547 T^{3} + 5080 T^{4} - 1347 T^{5} + 217 T^{6} - 19 T^{7} + T^{8} \)
$59$ \( 13697401 + 8874998 T + 2975713 T^{2} + 622126 T^{3} + 94675 T^{4} + 10026 T^{5} + 773 T^{6} + 38 T^{7} + T^{8} \)
$61$ \( 62552281 - 11594594 T + 3590217 T^{2} - 782922 T^{3} + 126055 T^{4} - 13818 T^{5} + 1037 T^{6} - 46 T^{7} + T^{8} \)
$67$ \( 408321 + 1755972 T + 2888388 T^{2} + 13704 T^{3} + 34630 T^{4} - 2416 T^{5} + 183 T^{6} - 8 T^{7} + T^{8} \)
$71$ \( 25 - 650 T + 44875 T^{2} + 10175 T^{3} + 4060 T^{4} + 1395 T^{5} + 275 T^{6} + 25 T^{7} + T^{8} \)
$73$ \( 121 + 506 T + 3182 T^{2} + 3618 T^{3} + 1600 T^{4} - 228 T^{5} + 267 T^{6} - 26 T^{7} + T^{8} \)
$79$ \( 408321 + 300969 T + 629937 T^{2} - 22443 T^{3} + 2350 T^{4} - 247 T^{5} + 117 T^{6} + 16 T^{7} + T^{8} \)
$83$ \( 46908801 + 14876028 T + 9296163 T^{2} + 531546 T^{3} + 43855 T^{4} - 1054 T^{5} - 17 T^{6} + 8 T^{7} + T^{8} \)
$89$ \( 97515625 - 11109375 T + 359375 T^{2} + 106875 T^{3} + 57750 T^{4} + 7125 T^{5} + 625 T^{6} + 30 T^{7} + T^{8} \)
$97$ \( 64304361 - 13423806 T + 5936247 T^{2} - 188028 T^{3} + 16705 T^{4} + 908 T^{5} + 87 T^{6} - 14 T^{7} + T^{8} \)
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