Properties

Label 1520.2.q.o
Level $1520$
Weight $2$
Character orbit 1520.q
Analytic conductor $12.137$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.4601315889.1
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_1) q^{3} - \beta_{5} q^{5} + (\beta_{6} - \beta_{2} + 1) q^{7} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_1) q^{3} - \beta_{5} q^{5} + (\beta_{6} - \beta_{2} + 1) q^{7} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{9} + (\beta_{6} - 2 \beta_{3} - \beta_{2}) q^{11} + (\beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_1 - 2) q^{13} + ( - \beta_{5} + \beta_{3} + \beta_1 + 1) q^{15} + (\beta_{7} + 2 \beta_1) q^{17} + ( - \beta_{4} - 2 \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{21} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_1) q^{23} + (\beta_{5} - 1) q^{25} + (\beta_{6} - 3 \beta_{4} - \beta_{3} - \beta_{2} - 4) q^{27} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1) q^{29} + (\beta_{6} + 3 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{31} + ( - 3 \beta_{7} - 6 \beta_{5} + 2 \beta_1) q^{33} + ( - \beta_{5} + \beta_{2}) q^{35} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 1) q^{37} + ( - 2 \beta_{4} - 3 \beta_{3} - 5) q^{39} + (2 \beta_{7} + 3 \beta_{5} - 3 \beta_{2} - 2 \beta_1) q^{41} + ( - \beta_{7} - \beta_{5} + \beta_{2} + 4 \beta_1) q^{43} + (\beta_{4} + 2 \beta_{3} + 1) q^{45} + ( - 2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 3) q^{47} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{49} + ( - \beta_{7} + \beta_{6} - 6 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 6) q^{51} + (3 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{53} + (\beta_{2} - 2 \beta_1) q^{55} + ( - 4 \beta_{7} - 4 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{57} - 5 \beta_1 q^{59} + ( - 4 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{4} + 1) q^{61} + (2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_1 - 1) q^{63} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 2) q^{65} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 2) q^{67} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 3) q^{69} + (3 \beta_{7} + 6 \beta_{5} - \beta_1) q^{71} + ( - \beta_{7} + 4 \beta_{5} - \beta_{2} + 3 \beta_1) q^{73} + ( - \beta_{3} - 1) q^{75} + (2 \beta_{6} + \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 3) q^{77} + (2 \beta_{7} + 5 \beta_{5} - \beta_{2} - \beta_1) q^{79} + ( - 2 \beta_{7} - 4 \beta_{5} - 3 \beta_{2} + 2 \beta_1) q^{81} + (2 \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{83} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{85} + (2 \beta_{6} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3) q^{87} + ( - 2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 3) q^{89} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} - 2 \beta_1 + 1) q^{91} + (3 \beta_{7} + 4 \beta_{5} + 3 \beta_{2} - 5 \beta_1) q^{93} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{95} + (\beta_{5} - 4 \beta_{2} - 5 \beta_1) q^{97} + ( - 5 \beta_{7} - 12 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} + 5 \beta_1 + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{3} - 4 q^{5} + 8 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{3} - 4 q^{5} + 8 q^{7} - q^{9} + 4 q^{11} - 7 q^{13} + 3 q^{15} + q^{17} - 5 q^{19} + 4 q^{21} + 2 q^{23} - 4 q^{25} - 24 q^{27} + q^{29} - 19 q^{33} - 4 q^{35} - 4 q^{37} - 30 q^{39} + 8 q^{41} + q^{43} + 2 q^{45} - 12 q^{47} - 20 q^{49} + 22 q^{51} + 5 q^{53} - 2 q^{55} + 7 q^{57} - 5 q^{59} - 3 q^{63} + 14 q^{65} + 4 q^{67} - 18 q^{69} + 20 q^{71} + 20 q^{73} - 6 q^{75} + 28 q^{77} + 17 q^{79} - 12 q^{81} - 2 q^{83} + q^{85} + 32 q^{87} - 11 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{95} - q^{97} + 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{5} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} - 12\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(-1 + \beta_{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
1.07988 + 1.87040i
0.689667 + 1.19454i
−0.245959 0.426014i
−1.02359 1.77290i
1.07988 1.87040i
0.689667 1.19454i
−0.245959 + 0.426014i
−1.02359 + 1.77290i
0 −0.579878 1.00438i 0 −0.500000 0.866025i 0 2.43525 0 0.827483 1.43324i 0
881.2 0 −0.189667 0.328513i 0 −0.500000 0.866025i 0 −1.89307 0 1.42805 2.47346i 0
881.3 0 0.745959 + 1.29204i 0 −0.500000 0.866025i 0 2.84864 0 0.387090 0.670459i 0
881.4 0 1.52359 + 2.63893i 0 −0.500000 0.866025i 0 0.609175 0 −3.14263 + 5.44319i 0
961.1 0 −0.579878 + 1.00438i 0 −0.500000 + 0.866025i 0 2.43525 0 0.827483 + 1.43324i 0
961.2 0 −0.189667 + 0.328513i 0 −0.500000 + 0.866025i 0 −1.89307 0 1.42805 + 2.47346i 0
961.3 0 0.745959 1.29204i 0 −0.500000 + 0.866025i 0 2.84864 0 0.387090 + 0.670459i 0
961.4 0 1.52359 2.63893i 0 −0.500000 + 0.866025i 0 0.609175 0 −3.14263 5.44319i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.q.o 8
4.b odd 2 1 95.2.e.c 8
12.b even 2 1 855.2.k.h 8
19.c even 3 1 inner 1520.2.q.o 8
20.d odd 2 1 475.2.e.e 8
20.e even 4 2 475.2.j.c 16
76.f even 6 1 1805.2.a.i 4
76.g odd 6 1 95.2.e.c 8
76.g odd 6 1 1805.2.a.o 4
228.m even 6 1 855.2.k.h 8
380.p odd 6 1 475.2.e.e 8
380.p odd 6 1 9025.2.a.bg 4
380.s even 6 1 9025.2.a.bp 4
380.v even 12 2 475.2.j.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 4.b odd 2 1
95.2.e.c 8 76.g odd 6 1
475.2.e.e 8 20.d odd 2 1
475.2.e.e 8 380.p odd 6 1
475.2.j.c 16 20.e even 4 2
475.2.j.c 16 380.v even 12 2
855.2.k.h 8 12.b even 2 1
855.2.k.h 8 228.m even 6 1
1520.2.q.o 8 1.a even 1 1 trivial
1520.2.q.o 8 19.c even 3 1 inner
1805.2.a.i 4 76.f even 6 1
1805.2.a.o 4 76.g odd 6 1
9025.2.a.bg 4 380.p odd 6 1
9025.2.a.bp 4 380.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{8} - 3T_{3}^{7} + 11T_{3}^{6} - 4T_{3}^{5} + 17T_{3}^{4} + 2T_{3}^{3} + 29T_{3}^{2} + 10T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} - T_{7}^{2} + 15T_{7} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + 11 T^{6} - 4 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} - T^{2} + 15 T - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 25 T^{2} + 19 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 7 T^{7} + 42 T^{6} + 95 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - T^{7} + 31 T^{6} + 64 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + 31 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + 21 T^{6} + 48 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} + 64 T^{6} + 451 T^{5} + \cdots + 19881 \) Copy content Toggle raw display
$31$ \( (T^{4} - 67 T^{2} + 5 T + 1063)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 31 T^{2} - 123 T - 118)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} + 151 T^{6} + \cdots + 5008644 \) Copy content Toggle raw display
$43$ \( T^{8} - T^{7} + 99 T^{6} + \cdots + 630436 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + 199 T^{6} + \cdots + 5363856 \) Copy content Toggle raw display
$53$ \( T^{8} - 5 T^{7} + 111 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$59$ \( T^{8} + 5 T^{7} + 150 T^{6} + \cdots + 3515625 \) Copy content Toggle raw display
$61$ \( T^{8} + 130 T^{6} + 176 T^{5} + \cdots + 9296401 \) Copy content Toggle raw display
$67$ \( T^{8} - 4 T^{7} + 52 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + 309 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + 305 T^{6} + \cdots + 2979076 \) Copy content Toggle raw display
$79$ \( T^{8} - 17 T^{7} + 217 T^{6} + \cdots + 33856 \) Copy content Toggle raw display
$83$ \( (T^{4} + T^{3} - 62 T^{2} - 55 T + 366)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 11 T^{7} + 211 T^{6} + \cdots + 14561856 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + 267 T^{6} + \cdots + 55383364 \) Copy content Toggle raw display
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