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Properties

Label 1520.2.d.k
Level $1520$
Weight $2$
Character orbit 1520.d
Analytic conductor $12.137$
Analytic rank $0$
Dimension $12$
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(609,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 9 x^{10} - 8 x^{9} - 11 x^{8} + 60 x^{7} - 126 x^{6} + 180 x^{5} - 99 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{11} q^{3} - \beta_{6} q^{5} + (\beta_{11} + \beta_{5}) q^{7} + ( - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{3} - \beta_{6} q^{5} + (\beta_{11} + \beta_{5}) q^{7} + ( - \beta_{3} - 1) q^{9} + (\beta_{10} + \beta_{9}) q^{11} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{4}) q^{13}+ \cdots + ( - \beta_{10} - 5 \beta_{9} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{5} - 16 q^{9} + 4 q^{11} + 12 q^{15} + 12 q^{19} + 36 q^{21} + 18 q^{25} + 4 q^{29} - 16 q^{31} - 6 q^{35} - 36 q^{39} + 8 q^{41} - 2 q^{45} - 4 q^{49} + 68 q^{51} + 18 q^{55} - 4 q^{59} + 20 q^{61} + 20 q^{65} - 36 q^{69} + 16 q^{71} + 16 q^{75} - 40 q^{79} + 12 q^{81} + 6 q^{85} - 64 q^{89} - 20 q^{91} + 6 q^{95} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 9 x^{10} - 8 x^{9} - 11 x^{8} + 60 x^{7} - 126 x^{6} + 180 x^{5} - 99 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3 \nu^{11} - 25 \nu^{10} + 82 \nu^{9} - 90 \nu^{8} - 85 \nu^{7} + 377 \nu^{6} - 891 \nu^{5} + \cdots - 8343 ) / 648 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} - 4 \nu^{9} - 6 \nu^{8} + 23 \nu^{7} - 46 \nu^{6} + 59 \nu^{5} - 36 \nu^{4} + \cdots - 54 ) / 108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7 \nu^{11} + 2 \nu^{10} - 30 \nu^{9} + 79 \nu^{8} - 47 \nu^{7} - 45 \nu^{6} + 351 \nu^{5} + \cdots + 4617 ) / 486 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4 \nu^{11} - 13 \nu^{10} + 24 \nu^{9} - 5 \nu^{8} - 68 \nu^{7} + 207 \nu^{6} - 324 \nu^{5} + \cdots - 1701 ) / 243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7 \nu^{11} - 31 \nu^{10} + 48 \nu^{9} - 2 \nu^{8} - 107 \nu^{7} + 345 \nu^{6} - 549 \nu^{5} + \cdots - 1215 ) / 486 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41 \nu^{11} + 11 \nu^{10} + 126 \nu^{9} - 338 \nu^{8} + 217 \nu^{7} - 3 \nu^{6} - 945 \nu^{5} + \cdots - 19683 ) / 1944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25 \nu^{11} + 88 \nu^{10} - 132 \nu^{9} - 34 \nu^{8} + 479 \nu^{7} - 1242 \nu^{6} + 1827 \nu^{5} + \cdots + 6318 ) / 972 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29 \nu^{11} + 44 \nu^{10} + 54 \nu^{9} - 254 \nu^{8} + 301 \nu^{7} - 354 \nu^{6} - 45 \nu^{5} + \cdots - 12150 ) / 972 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 61 \nu^{11} + 205 \nu^{10} - 186 \nu^{9} - 178 \nu^{8} + 1037 \nu^{7} - 2457 \nu^{6} + 3339 \nu^{5} + \cdots + 5103 ) / 1944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 55 \nu^{11} - 253 \nu^{10} + 438 \nu^{9} - 62 \nu^{8} - 1151 \nu^{7} + 3393 \nu^{6} - 5697 \nu^{5} + \cdots - 26487 ) / 1944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49 \nu^{11} - 181 \nu^{10} + 264 \nu^{9} + 22 \nu^{8} - 875 \nu^{7} + 2253 \nu^{6} - 3771 \nu^{5} + \cdots - 13851 ) / 972 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{10} + \beta_{9} - \beta_{8} - 4\beta_{7} - \beta_{6} + 2\beta_{5} - 2\beta_{3} + 2\beta _1 - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{9} + \beta_{8} - 4\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{11} - 3 \beta_{10} - 14 \beta_{9} + 12 \beta_{8} + 3 \beta_{7} - 13 \beta_{6} + 8 \beta_{5} + \cdots - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 10 \beta_{11} + 8 \beta_{10} - 6 \beta_{9} + 2 \beta_{8} + \beta_{7} + 9 \beta_{4} - 2 \beta_{3} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 16 \beta_{11} + 20 \beta_{10} + 24 \beta_{9} - 47 \beta_{8} - 9 \beta_{7} + 35 \beta_{6} + 28 \beta_{5} + \cdots - 58 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7 \beta_{11} + 20 \beta_{10} + 15 \beta_{9} - 15 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + \cdots - 33 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 24 \beta_{11} - 26 \beta_{10} + 113 \beta_{9} + 17 \beta_{8} - 128 \beta_{7} + 11 \beta_{6} + 78 \beta_{5} + \cdots - 156 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 12 \beta_{11} + 44 \beta_{10} + 64 \beta_{9} + 12 \beta_{8} - 101 \beta_{7} - 60 \beta_{6} + \cdots + 208 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 200 \beta_{11} - 45 \beta_{10} + 14 \beta_{9} + 128 \beta_{8} + 149 \beta_{7} - 87 \beta_{6} + 44 \beta_{5} + \cdots - 42 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
0.664288 + 1.59960i
1.35119 + 1.08364i
0.588529 − 1.62900i
1.62784 − 0.591735i
−0.505103 + 1.65677i
−1.72675 − 0.135465i
−1.72675 + 0.135465i
−0.505103 − 1.65677i
1.62784 + 0.591735i
0.588529 + 1.62900i
1.35119 − 1.08364i
0.664288 − 1.59960i
0 − 2.97875i 0 2.21811 − 0.282801i 0 4.30732i 0 −5.87292 0
609.2 0 − 2.84742i 0 0.691595 + 2.12643i 0 0.145034i 0 −5.10782 0
609.3 0 − 2.43031i 0 −2.06801 + 0.850485i 0 3.60737i 0 −2.90640 0
609.4 0 − 1.59277i 0 −1.59909 − 1.56299i 0 − 1.66290i 0 0.463073 0
609.5 0 − 0.664406i 0 1.55549 + 1.60638i 0 − 0.345799i 0 2.55856 0
609.6 0 − 0.366738i 0 2.20191 + 0.389375i 0 − 3.08675i 0 2.86550 0
609.7 0 0.366738i 0 2.20191 − 0.389375i 0 3.08675i 0 2.86550 0
609.8 0 0.664406i 0 1.55549 − 1.60638i 0 0.345799i 0 2.55856 0
609.9 0 1.59277i 0 −1.59909 + 1.56299i 0 1.66290i 0 0.463073 0
609.10 0 2.43031i 0 −2.06801 − 0.850485i 0 − 3.60737i 0 −2.90640 0
609.11 0 2.84742i 0 0.691595 − 2.12643i 0 − 0.145034i 0 −5.10782 0
609.12 0 2.97875i 0 2.21811 + 0.282801i 0 − 4.30732i 0 −5.87292 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 609.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.d.k 12
4.b odd 2 1 760.2.d.e ✓ 12
5.b even 2 1 inner 1520.2.d.k 12
5.c odd 4 1 7600.2.a.cg 6
5.c odd 4 1 7600.2.a.cn 6
20.d odd 2 1 760.2.d.e ✓ 12
20.e even 4 1 3800.2.a.z 6
20.e even 4 1 3800.2.a.be 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.e ✓ 12 4.b odd 2 1
760.2.d.e ✓ 12 20.d odd 2 1
1520.2.d.k 12 1.a even 1 1 trivial
1520.2.d.k 12 5.b even 2 1 inner
3800.2.a.z 6 20.e even 4 1
3800.2.a.be 6 20.e even 4 1
7600.2.a.cg 6 5.c odd 4 1
7600.2.a.cn 6 5.c odd 4 1

Hecke kernels

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

\( T_{3}^{12} + 26T_{3}^{10} + 245T_{3}^{8} + 996T_{3}^{6} + 1588T_{3}^{4} + 672T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{12} + 44T_{7}^{10} + 662T_{7}^{8} + 3892T_{7}^{6} + 6897T_{7}^{4} + 904T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 26 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 44 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{6} - 2 T^{5} + \cdots - 2584)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 142 T^{10} + \cdots + 87909376 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 251666496 \) Copy content Toggle raw display
$19$ \( (T - 1)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 126 T^{10} + \cdots + 73984 \) Copy content Toggle raw display
$29$ \( (T^{6} - 2 T^{5} + \cdots + 10064)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots - 2304)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 152571904 \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} + \cdots - 4832)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 250 T^{10} + \cdots + 21455424 \) Copy content Toggle raw display
$47$ \( T^{12} + 94 T^{10} + \cdots + 1327104 \) Copy content Toggle raw display
$53$ \( T^{12} + 334 T^{10} + \cdots + 7573504 \) Copy content Toggle raw display
$59$ \( (T^{6} + 2 T^{5} + \cdots - 374496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 10 T^{5} + \cdots + 40256)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 434 T^{10} + \cdots + 7573504 \) Copy content Toggle raw display
$71$ \( (T^{6} - 8 T^{5} - 62 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 328 T^{10} + \cdots + 10291264 \) Copy content Toggle raw display
$79$ \( (T^{6} + 20 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 232 T^{10} + \cdots + 38539264 \) Copy content Toggle raw display
$89$ \( (T^{6} + 32 T^{5} + \cdots + 69504)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 38805848064 \) Copy content Toggle raw display
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