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Properties

Label 1520.2.q.n
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(881,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1500534351369.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 13x^{6} - 18x^{5} + 147x^{4} - 156x^{3} + 369x^{2} + 180x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} + ( - \beta_{3} + 1) q^{7} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} + ( - \beta_{3} + 1) q^{7} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 4) q^{9}+ \cdots + (3 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9} + 8 q^{11} - q^{13} + q^{15} - 7 q^{17} + 19 q^{19} - 24 q^{21} + 5 q^{23} - 4 q^{25} + 28 q^{27} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 22 q^{37} - 38 q^{39} - 13 q^{41} + 7 q^{43} + 26 q^{45} + 10 q^{47} - 2 q^{49} + 16 q^{51} - 4 q^{55} - 25 q^{57} - 8 q^{59} - 11 q^{61} - 24 q^{63} + 2 q^{65} - 20 q^{67} - 26 q^{69} - 5 q^{71} + 12 q^{73} - 2 q^{75} + 18 q^{77} - 22 q^{79} - 40 q^{81} - 26 q^{83} - 7 q^{85} + 12 q^{87} + 9 q^{89} + 18 q^{91} - 34 q^{93} - 17 q^{95} - 41 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 13x^{6} - 18x^{5} + 147x^{4} - 156x^{3} + 369x^{2} + 180x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 645 \nu^{7} + 7549 \nu^{6} - 4949 \nu^{5} + 69950 \nu^{4} - 179383 \nu^{3} + 607596 \nu^{2} + \cdots - 596568 ) / 285804 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -172\nu^{7} - 104\nu^{6} - 1849\nu^{5} + 129\nu^{4} - 25077\nu^{3} - 7869\nu^{2} - 4644\nu - 38412 ) / 71451 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1067 \nu^{7} + 1755 \nu^{6} - 13455 \nu^{5} + 26602 \nu^{4} - 157365 \nu^{3} + 266760 \nu^{2} + \cdots + 112320 ) / 285804 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1147 \nu^{7} - 6417 \nu^{6} - 14315 \nu^{5} - 68606 \nu^{4} - 59729 \nu^{3} - 689580 \nu^{2} + \cdots - 1458636 ) / 285804 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2467 \nu^{7} - 1647 \nu^{6} + 44383 \nu^{5} - 43530 \nu^{4} + 465241 \nu^{3} - 536148 \nu^{2} + \cdots - 581748 ) / 285804 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3941 \nu^{7} - 1125 \nu^{6} + 40381 \nu^{5} - 72422 \nu^{4} + 418435 \nu^{3} - 456804 \nu^{2} + \cdots - 548340 ) / 285804 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} + 7\beta_{4} + 2\beta_{3} - 2\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 2\beta_{5} - 10\beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} - 10\beta_{6} - 67\beta_{4} - 11\beta_{3} + 11\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{7} + 29\beta_{6} - 29\beta_{5} + 98\beta_{4} + 122\beta_{3} - 32\beta_{2} - 90\beta _1 - 98 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{7} + 109\beta_{5} - 236\beta_{3} + 119\beta_{2} + 697 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -87\beta_{7} - 365\beta_{6} - 1424\beta_{4} - 226\beta_{3} + 226\beta_{2} + 945\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−1.72425 − 2.98649i
−0.282064 − 0.488549i
1.08493 + 1.87916i
1.42138 + 2.46190i
−1.72425 + 2.98649i
−0.282064 + 0.488549i
1.08493 − 1.87916i
1.42138 − 2.46190i
0 −1.72425 − 2.98649i 0 −0.500000 − 0.866025i 0 4.44850 0 −4.44607 + 7.70083i 0
881.2 0 −0.282064 − 0.488549i 0 −0.500000 − 0.866025i 0 1.56413 0 1.34088 − 2.32247i 0
881.3 0 1.08493 + 1.87916i 0 −0.500000 − 0.866025i 0 −1.16987 0 −0.854162 + 1.47945i 0
881.4 0 1.42138 + 2.46190i 0 −0.500000 − 0.866025i 0 −1.84276 0 −2.54064 + 4.40052i 0
961.1 0 −1.72425 + 2.98649i 0 −0.500000 + 0.866025i 0 4.44850 0 −4.44607 − 7.70083i 0
961.2 0 −0.282064 + 0.488549i 0 −0.500000 + 0.866025i 0 1.56413 0 1.34088 + 2.32247i 0
961.3 0 1.08493 − 1.87916i 0 −0.500000 + 0.866025i 0 −1.16987 0 −0.854162 − 1.47945i 0
961.4 0 1.42138 − 2.46190i 0 −0.500000 + 0.866025i 0 −1.84276 0 −2.54064 − 4.40052i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.4
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.n 8
4.b odd 2 1 760.2.q.d ✓ 8
19.c even 3 1 inner 1520.2.q.n 8
76.g odd 6 1 760.2.q.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.q.d ✓ 8 4.b odd 2 1
760.2.q.d ✓ 8 76.g odd 6 1
1520.2.q.n 8 1.a even 1 1 trivial
1520.2.q.n 8 19.c even 3 1 inner

Hecke kernels

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

\( T_{3}^{8} - T_{3}^{7} + 13T_{3}^{6} - 18T_{3}^{5} + 147T_{3}^{4} - 156T_{3}^{3} + 369T_{3}^{2} + 180T_{3} + 144 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} - 9T_{7}^{2} + 8T_{7} + 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 3 T^{3} - 9 T^{2} + \cdots + 15)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} - 27 T^{2} + \cdots - 97)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} + \cdots + 68644 \) Copy content Toggle raw display
$17$ \( T^{8} + 7 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{8} - 19 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$29$ \( T^{8} + 33 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$31$ \( (T^{4} - 5 T^{3} + \cdots + 160)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 11 T^{3} + 33 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 13 T^{7} + \cdots + 1089 \) Copy content Toggle raw display
$43$ \( T^{8} - 7 T^{7} + \cdots + 357604 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 379456 \) Copy content Toggle raw display
$53$ \( T^{8} + 117 T^{6} + \cdots + 793881 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{8} + 11 T^{7} + \cdots + 38539264 \) Copy content Toggle raw display
$67$ \( T^{8} + 20 T^{7} + \cdots + 36192256 \) Copy content Toggle raw display
$71$ \( T^{8} + 5 T^{7} + \cdots + 381924 \) Copy content Toggle raw display
$73$ \( T^{8} - 12 T^{7} + \cdots + 337750884 \) Copy content Toggle raw display
$79$ \( T^{8} + 22 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$83$ \( (T^{4} + 13 T^{3} + \cdots - 3430)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 9 T^{7} + \cdots + 20575296 \) Copy content Toggle raw display
$97$ \( T^{8} + 41 T^{7} + \cdots + 93354244 \) Copy content Toggle raw display
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