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 $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1500,2,Mod(301,1500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1500, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1500.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

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: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.26265625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) Copy content Toggle raw display
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

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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + (\beta_{7} - \beta_{6} - \beta_{2} - 1) q^{7} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + (\beta_{7} - \beta_{6} - \beta_{2} - 1) q^{7} - \beta_1 q^{9} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{11} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{13} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_1) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{19} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{21} + (\beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} + 1) q^{23} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{27} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{29} + (\beta_{7} - \beta_{6} - 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - \beta_1) q^{31} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{33} + ( - 3 \beta_{5} + 2 \beta_1) q^{37} + (\beta_{7} - \beta_{5} + 2 \beta_{4}) q^{39} + ( - 4 \beta_{5} - 2 \beta_{3} + \beta_1 + 2) q^{41} + ( - 2 \beta_{7} + \beta_{6} - \beta_{3} + 2 \beta_{2} + 2) q^{43} + (2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{47} + ( - 3 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + \cdots - 2) q^{49}+ \cdots + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} - 8 q^{7} - 2 q^{9} + 8 q^{11} - 3 q^{17} + 5 q^{19} + 7 q^{21} + 7 q^{23} - 2 q^{27} - 3 q^{29} - 3 q^{31} - 7 q^{33} + q^{37} + 10 q^{41} + 12 q^{43} + 33 q^{47} - 8 q^{49} - 8 q^{51} + 19 q^{53} - 10 q^{57} - 38 q^{59} + 46 q^{61} - 3 q^{63} + 8 q^{67} + 2 q^{69} - 25 q^{71} + 26 q^{73} - 23 q^{77} - 16 q^{79} - 2 q^{81} - 8 q^{83} - 3 q^{87} - 30 q^{89} + 25 q^{91} + 22 q^{93} + 14 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{3} - 2\nu^{2} + 12\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 15\nu^{6} - 4\nu^{5} - 8\nu^{4} + 3\nu^{3} + 2\nu^{2} + 92\nu - 96 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 13\nu^{6} - 6\nu^{5} - 8\nu^{4} + 3\nu^{3} + 4\nu^{2} + 96\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{7} - 21\nu^{6} - 4\nu^{5} - 4\nu^{4} + 5\nu^{3} + 10\nu^{2} + 120\nu - 144 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} - 31\nu^{6} - 4\nu^{5} - 8\nu^{4} + 11\nu^{3} + 18\nu^{2} + 180\nu - 224 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} - 4\beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - 3\beta_{2} + 2\beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 8\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 11\beta_{6} - 9\beta_{5} + 3\beta_{4} - 11\beta_{3} + 6\beta_{2} + 6\beta _1 + 19 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} - 7\beta_{6} + 7\beta_{5} - 9\beta_{4} - 7\beta_{3} + 7\beta_{2} + 12\beta _1 - 12 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{7} + 3\beta_{6} - 3\beta_{5} + 6\beta_{4} - 7\beta_{3} + 7\beta_{2} + 57\beta _1 + 3 ) / 5 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 301.2
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 \( T_{7}^{4} + 4T_{7}^{3} - 4T_{7}^{2} - 11T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} - 4 T^{2} - 11 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + 23 T^{6} + 39 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{6} + 5 T^{5} + 310 T^{4} + \cdots + 2025 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + 13 T^{6} + T^{5} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + 5 T^{6} + 45 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{8} - 7 T^{7} + 18 T^{6} + \cdots + 29241 \) Copy content Toggle raw display
$29$ \( T^{8} + 3 T^{7} + 8 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + 58 T^{6} + \cdots + 962361 \) Copy content Toggle raw display
$37$ \( T^{8} - T^{7} + 57 T^{6} + \cdots + 1042441 \) Copy content Toggle raw display
$41$ \( T^{8} - 10 T^{7} + 155 T^{6} + \cdots + 7317025 \) Copy content Toggle raw display
$43$ \( (T^{4} - 6 T^{3} - 19 T^{2} + 64 T + 131)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 33 T^{7} + 568 T^{6} + \cdots + 16801801 \) Copy content Toggle raw display
$53$ \( T^{8} - 19 T^{7} + 217 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$59$ \( T^{8} + 38 T^{7} + 773 T^{6} + \cdots + 13697401 \) Copy content Toggle raw display
$61$ \( T^{8} - 46 T^{7} + 1037 T^{6} + \cdots + 62552281 \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + 183 T^{6} + \cdots + 408321 \) Copy content Toggle raw display
$71$ \( T^{8} + 25 T^{7} + 275 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$73$ \( T^{8} - 26 T^{7} + 267 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$79$ \( T^{8} + 16 T^{7} + 117 T^{6} + \cdots + 408321 \) Copy content Toggle raw display
$83$ \( T^{8} + 8 T^{7} - 17 T^{6} + \cdots + 46908801 \) Copy content Toggle raw display
$89$ \( T^{8} + 30 T^{7} + 625 T^{6} + \cdots + 97515625 \) Copy content Toggle raw display
$97$ \( T^{8} - 14 T^{7} + 87 T^{6} + \cdots + 64304361 \) Copy content Toggle raw display
show more
show less