Properties

Label 1530.2.n.q
Level $1530$
Weight $2$
Character orbit 1530.n
Analytic conductor $12.217$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1530,2,Mod(829,1530)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1530, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1530.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.n (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.110166016.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 510)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{7} - q^{8} + (\beta_{5} + \beta_{3} - \beta_1) q^{10} + (2 \beta_{5} + 2 \beta_{2} + 2) q^{11} + ( - \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4}) q^{13} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{14} + q^{16} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{17} + (2 \beta_{3} - 2 \beta_{2}) q^{19} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{20} + ( - 2 \beta_{5} - 2 \beta_{2} - 2) q^{22} + ( - \beta_{7} - 2 \beta_{5} - 3 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{23} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + 2) q^{25} + (\beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4}) q^{26} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{28} + ( - 2 \beta_{6} - 2) q^{29} + ( - \beta_{7} - 2 \beta_{5} - 5 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{31} - q^{32} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 1) q^{34} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{35} + (2 \beta_{7} - 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{37} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{38} + (\beta_{5} + \beta_{3} - \beta_1) q^{40} + ( - \beta_{4} + 3 \beta_{2} + 1) q^{41} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{43} + (2 \beta_{5} + 2 \beta_{2} + 2) q^{44} + (\beta_{7} + 2 \beta_{5} + 3 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{46} + (3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2}) q^{47} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2}) q^{49} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 2) q^{50} + ( - \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4}) q^{52} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_1 + 4) q^{53} + (2 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{55} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{56} + (2 \beta_{6} + 2) q^{58} + (\beta_{7} - 3 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{59} + (\beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{61} + (\beta_{7} + 2 \beta_{5} + 5 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{62} + q^{64} + ( - 2 \beta_{7} + \beta_{6} - 5 \beta_{5} + 7 \beta_{3} - 4 \beta_{2} - 2) q^{65} + (\beta_{7} - 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2}) q^{67} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{68} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 2) q^{70} + (3 \beta_{6} - 4 \beta_{5} + \beta_{3} + 7) q^{71} + (\beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{3} + 6 \beta_{2} + \beta_1 - 4) q^{73} + ( - 2 \beta_{7} + 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{74} + (2 \beta_{3} - 2 \beta_{2}) q^{76} + (6 \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 6) q^{77} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{79} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{80} + (\beta_{4} - 3 \beta_{2} - 1) q^{82} + (2 \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{83} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_1 - 7) q^{85} + ( - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 5) q^{86} + ( - 2 \beta_{5} - 2 \beta_{2} - 2) q^{88} + (2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{89} + ( - 2 \beta_{5} - 6 \beta_{4} + 4) q^{91} + ( - \beta_{7} - 2 \beta_{5} - 3 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{92} + ( - 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2}) q^{94} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{95} + (\beta_{7} - 5 \beta_{5} - 5 \beta_{4} - \beta_{3} + 6 \beta_{2} + \beta_1) q^{97} + (2 \beta_{7} - 3 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} + 4 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} + 4 q^{7} - 8 q^{8} + 16 q^{11} - 4 q^{14} + 8 q^{16} - 4 q^{17} - 16 q^{22} + 16 q^{23} + 16 q^{25} + 4 q^{28} - 8 q^{29} + 16 q^{31} - 8 q^{32} + 4 q^{34} - 16 q^{35} + 4 q^{41} + 24 q^{43} + 16 q^{44} - 16 q^{46} - 16 q^{50} + 16 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 16 q^{61} - 16 q^{62} + 8 q^{64} - 20 q^{65} - 4 q^{68} + 16 q^{70} + 44 q^{71} - 20 q^{73} + 48 q^{77} + 16 q^{79} - 4 q^{82} - 16 q^{83} - 56 q^{85} - 24 q^{86} - 16 q^{88} - 16 q^{89} + 8 q^{91} + 16 q^{92} + 16 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} + 9\nu^{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{5} + 10\nu^{4} + 9\nu^{3} + 18\nu^{2} + 9\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - \nu^{5} + 10\nu^{4} - 9\nu^{3} + 18\nu^{2} - 9\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + \nu^{5} + 28\nu^{4} + 9\nu^{3} + 40\nu^{2} + 13\nu + 13 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} + 19\nu^{5} + 29\nu^{3} + 9\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + \nu^{5} - 28\nu^{4} + 9\nu^{3} - 40\nu^{2} + 13\nu - 13 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 29\nu^{5} + 47\nu^{3} + 14\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{4} - 3\beta_{3} - 3\beta_{2} + 2\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 2\beta_{6} + 3\beta_{5} - 2\beta_{4} - 3\beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{6} - 9\beta_{4} + 27\beta_{3} + 27\beta_{2} - 14\beta _1 + 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 36\beta_{7} + 27\beta_{6} - 54\beta_{5} + 27\beta_{4} + 41\beta_{3} - 41\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -36\beta_{6} + 36\beta_{4} - 106\beta_{3} - 106\beta_{2} + 52\beta _1 - 151 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -284\beta_{7} - 203\beta_{6} + 428\beta_{5} - 203\beta_{4} - 307\beta_{3} + 307\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(1261\) \(1361\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
2.77462i
1.22833i
0.814115i
0.360409i
2.77462i
1.22833i
0.814115i
0.360409i
−1.00000 0 1.00000 −2.21680 + 0.292893i 0 2.56350 + 2.56350i −1.00000 0 2.21680 0.292893i
829.2 −1.00000 0 1.00000 −1.44423 + 1.70711i 0 1.55123 + 1.55123i −1.00000 0 1.44423 1.70711i
829.3 −1.00000 0 1.00000 1.44423 + 1.70711i 0 −3.37966 3.37966i −1.00000 0 −1.44423 1.70711i
829.4 −1.00000 0 1.00000 2.21680 + 0.292893i 0 1.26493 + 1.26493i −1.00000 0 −2.21680 0.292893i
1279.1 −1.00000 0 1.00000 −2.21680 0.292893i 0 2.56350 2.56350i −1.00000 0 2.21680 + 0.292893i
1279.2 −1.00000 0 1.00000 −1.44423 1.70711i 0 1.55123 1.55123i −1.00000 0 1.44423 + 1.70711i
1279.3 −1.00000 0 1.00000 1.44423 1.70711i 0 −3.37966 + 3.37966i −1.00000 0 −1.44423 + 1.70711i
1279.4 −1.00000 0 1.00000 2.21680 0.292893i 0 1.26493 1.26493i −1.00000 0 −2.21680 + 0.292893i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.n.q 8
3.b odd 2 1 510.2.m.b yes 8
5.b even 2 1 1530.2.n.r 8
15.d odd 2 1 510.2.m.a 8
17.c even 4 1 1530.2.n.r 8
51.f odd 4 1 510.2.m.a 8
85.j even 4 1 inner 1530.2.n.q 8
255.i odd 4 1 510.2.m.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.m.a 8 15.d odd 2 1
510.2.m.a 8 51.f odd 4 1
510.2.m.b yes 8 3.b odd 2 1
510.2.m.b yes 8 255.i odd 4 1
1530.2.n.q 8 1.a even 1 1 trivial
1530.2.n.q 8 85.j even 4 1 inner
1530.2.n.r 8 5.b even 2 1
1530.2.n.r 8 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\):

\( T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} - 32T_{7}^{5} + 460T_{7}^{4} - 2144T_{7}^{3} + 5408T_{7}^{2} - 7072T_{7} + 4624 \) Copy content Toggle raw display
\( T_{11}^{4} - 8T_{11}^{3} + 32T_{11}^{2} - 32T_{11} + 16 \) Copy content Toggle raw display
\( T_{23}^{8} - 16T_{23}^{7} + 128T_{23}^{6} - 432T_{23}^{5} + 840T_{23}^{4} - 1856T_{23}^{3} + 15488T_{23}^{2} - 51392T_{23} + 85264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{6} + 34 T^{4} - 200 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + 8 T^{6} - 32 T^{5} + \cdots + 4624 \) Copy content Toggle raw display
$11$ \( (T^{4} - 8 T^{3} + 32 T^{2} - 32 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 76 T^{6} + 1708 T^{4} + \cdots + 35344 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} - 24 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 85264 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$31$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 85264 \) Copy content Toggle raw display
$37$ \( T^{8} + 512 T^{5} + 12032 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} + 8 T^{6} + 64 T^{5} + \cdots + 784 \) Copy content Toggle raw display
$43$ \( (T^{4} - 12 T^{3} - 2 T^{2} + 168 T + 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 264 T^{6} + 22448 T^{4} + \cdots + 9535744 \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} - 96 T^{2} + 768 T - 1168)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 284 T^{6} + 23116 T^{4} + \cdots + 1336336 \) Copy content Toggle raw display
$61$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$67$ \( T^{8} + 148 T^{6} + 4876 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$71$ \( T^{8} - 44 T^{7} + 968 T^{6} + \cdots + 795664 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} + 200 T^{6} + \cdots + 59043856 \) Copy content Toggle raw display
$79$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$83$ \( (T^{4} + 8 T^{3} - 96 T^{2} - 960 T - 1600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 8 T^{3} - 96 T^{2} - 768 T - 1168)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 20 T^{7} + 200 T^{6} + \cdots + 85264 \) Copy content Toggle raw display
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