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Properties

Label 1960.2.g.e
Level $1960$
Weight $2$
Character orbit 1960.g
Analytic conductor $15.651$
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] = [1960,2,Mod(1569,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, 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("1960.1569");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.g (of order \(2\), degree \(1\), 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: \(15.6506787962\)
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: \( x^{12} + 25x^{10} + 230x^{8} + 950x^{6} + 1657x^{4} + 785x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 280)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{2} - 1) q^{9} + \beta_{3} q^{11} + ( - \beta_{11} + \beta_{7}) q^{13} + (\beta_{11} + \beta_{8} - \beta_{7} + \cdots + 1) q^{15}+ \cdots + ( - 2 \beta_{10} + \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 14 q^{9} + 2 q^{11} + 6 q^{15} - 10 q^{19} + 2 q^{25} + 6 q^{29} + 4 q^{31} - 20 q^{39} - 12 q^{41} - 8 q^{45} + 6 q^{55} - 48 q^{59} - 18 q^{61} + 26 q^{65} + 30 q^{69} + 8 q^{71} - 14 q^{75} + 44 q^{79} - 12 q^{81} - 22 q^{85} + 30 q^{89} + 26 q^{95} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 25x^{10} + 230x^{8} + 950x^{6} + 1657x^{4} + 785x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 19\nu^{6} + 115\nu^{4} + 233\nu^{2} + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 2 \nu^{9} + 20 \nu^{8} + 42 \nu^{7} + 126 \nu^{6} + 290 \nu^{5} + 244 \nu^{4} + \cdots + 292 \nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{10} + 2 \nu^{9} - 20 \nu^{8} + 42 \nu^{7} - 126 \nu^{6} + 290 \nu^{5} - 244 \nu^{4} + \cdots + 292 \nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 20\nu^{7} + 130\nu^{5} + 300\nu^{3} + 149\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} - 22\nu^{9} - 168\nu^{7} - 526\nu^{5} - 599\nu^{3} - 156\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} - 16 \nu^{9} - 42 \nu^{8} - 46 \nu^{7} - 290 \nu^{6} + 276 \nu^{5} - 702 \nu^{4} + \cdots - 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} + 16 \nu^{9} - 42 \nu^{8} + 46 \nu^{7} - 290 \nu^{6} - 276 \nu^{5} + \cdots - 372 \nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{10} - 62\nu^{8} - 424\nu^{6} - 1058\nu^{4} - 629\nu^{2} - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{11} - 24\nu^{9} - 208\nu^{7} - 786\nu^{5} - 1183\nu^{3} - 358\nu ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{7} + \beta_{6} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - 2\beta_{5} + 2\beta_{4} + \beta_{3} - 8\beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{11} + 2\beta_{9} - 2\beta_{8} + 16\beta_{7} - 10\beta_{6} + 2\beta_{5} + 2\beta_{4} + 39\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{10} - 12\beta_{9} - 12\beta_{8} + 30\beta_{5} - 30\beta_{4} - 14\beta_{3} + 67\beta_{2} - 190 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 159\beta_{11} - 30\beta_{9} + 30\beta_{8} - 189\beta_{7} + 91\beta_{6} - 22\beta_{5} - 22\beta_{4} - 276\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 38\beta_{10} + 113\beta_{9} + 113\beta_{8} - 340\beta_{5} + 340\beta_{4} + 159\beta_{3} - 586\beta_{2} + 1496 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1660 \beta_{11} + 340 \beta_{9} - 340 \beta_{8} + 2000 \beta_{7} - 812 \beta_{6} + 180 \beta_{5} + \cdots - 340 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 508 \beta_{10} - 992 \beta_{9} - 992 \beta_{8} + 3492 \beta_{5} - 3492 \beta_{4} - 1660 \beta_{3} + \cdots - 12416 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16573 \beta_{11} - 3492 \beta_{9} + 3492 \beta_{8} - 20081 \beta_{7} + 7237 \beta_{6} - 1316 \beta_{5} + \cdots + 3492 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1569.1
− 3.04302i
− 2.54431i
− 2.10821i
− 2.03837i
− 0.751428i
− 0.319986i
0.319986i
0.751428i
2.03837i
2.10821i
2.54431i
3.04302i
0 − 3.04302i 0 −1.75131 + 1.39029i 0 0 0 −6.25996 0
1569.2 0 − 2.54431i 0 1.65511 + 1.50353i 0 0 0 −3.47349 0
1569.3 0 − 2.10821i 0 2.19509 − 0.426103i 0 0 0 −1.44457 0
1569.4 0 − 2.03837i 0 0.240243 − 2.22312i 0 0 0 −1.15495 0
1569.5 0 − 0.751428i 0 −2.18976 − 0.452709i 0 0 0 2.43536 0
1569.6 0 − 0.319986i 0 −0.149376 + 2.23107i 0 0 0 2.89761 0
1569.7 0 0.319986i 0 −0.149376 − 2.23107i 0 0 0 2.89761 0
1569.8 0 0.751428i 0 −2.18976 + 0.452709i 0 0 0 2.43536 0
1569.9 0 2.03837i 0 0.240243 + 2.22312i 0 0 0 −1.15495 0
1569.10 0 2.10821i 0 2.19509 + 0.426103i 0 0 0 −1.44457 0
1569.11 0 2.54431i 0 1.65511 − 1.50353i 0 0 0 −3.47349 0
1569.12 0 3.04302i 0 −1.75131 − 1.39029i 0 0 0 −6.25996 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1569.12
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 1960.2.g.e 12
5.b even 2 1 inner 1960.2.g.e 12
5.c odd 4 1 9800.2.a.cw 6
5.c odd 4 1 9800.2.a.cy 6
7.b odd 2 1 1960.2.g.f 12
7.d odd 6 2 280.2.bg.a ✓ 24
28.f even 6 2 560.2.bw.f 24
35.c odd 2 1 1960.2.g.f 12
35.f even 4 1 9800.2.a.cv 6
35.f even 4 1 9800.2.a.cx 6
35.i odd 6 2 280.2.bg.a ✓ 24
35.k even 12 2 1400.2.q.n 12
35.k even 12 2 1400.2.q.o 12
140.s even 6 2 560.2.bw.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bg.a ✓ 24 7.d odd 6 2
280.2.bg.a ✓ 24 35.i odd 6 2
560.2.bw.f 24 28.f even 6 2
560.2.bw.f 24 140.s even 6 2
1400.2.q.n 12 35.k even 12 2
1400.2.q.o 12 35.k even 12 2
1960.2.g.e 12 1.a even 1 1 trivial
1960.2.g.e 12 5.b even 2 1 inner
1960.2.g.f 12 7.b odd 2 1
1960.2.g.f 12 35.c odd 2 1
9800.2.a.cv 6 35.f even 4 1
9800.2.a.cw 6 5.c odd 4 1
9800.2.a.cx 6 35.f even 4 1
9800.2.a.cy 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 \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3}^{12} + 25T_{3}^{10} + 230T_{3}^{8} + 950T_{3}^{6} + 1657T_{3}^{4} + 785T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{6} + 5T_{19}^{5} - 47T_{19}^{4} - 301T_{19}^{3} - 94T_{19}^{2} + 1300T_{19} + 568 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 25 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} - T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - T^{5} - 37 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 69 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{12} + 166 T^{10} + \cdots + 91240704 \) Copy content Toggle raw display
$19$ \( (T^{6} + 5 T^{5} + \cdots + 568)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 74 T^{10} + \cdots + 113569 \) Copy content Toggle raw display
$29$ \( (T^{6} - 3 T^{5} + \cdots - 7344)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 2 T^{5} + \cdots - 1968)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 231 T^{10} + \cdots + 11943936 \) Copy content Toggle raw display
$41$ \( (T^{6} + 6 T^{5} + \cdots + 2764)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 9165381696 \) Copy content Toggle raw display
$47$ \( T^{12} + 183 T^{10} + \cdots + 1937664 \) Copy content Toggle raw display
$53$ \( T^{12} + 179 T^{10} + \cdots + 118336 \) Copy content Toggle raw display
$59$ \( (T^{6} + 24 T^{5} + \cdots - 124704)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 9 T^{5} + \cdots - 105806)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 213 T^{10} + \cdots + 14760964 \) Copy content Toggle raw display
$71$ \( (T^{6} - 4 T^{5} + \cdots - 183296)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 32066781184 \) Copy content Toggle raw display
$79$ \( (T^{6} - 22 T^{5} + \cdots - 272)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 287 T^{10} + \cdots + 18496 \) Copy content Toggle raw display
$89$ \( (T^{6} - 15 T^{5} + \cdots + 115554)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 560 T^{10} + \cdots + 67108864 \) Copy content Toggle raw display
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