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Properties

Label 1260.2.c.e
Level $1260$
Weight $2$
Character orbit 1260.c
Analytic conductor $10.061$
Analytic rank $0$
Dimension $16$
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] = [1260,2,Mod(811,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.811");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.c (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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 420)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{8} q^{4} - \beta_{3} q^{5} - \beta_{15} q^{7} + ( - \beta_{7} - \beta_{4} + \cdots + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + \beta_{8} q^{4} - \beta_{3} q^{5} - \beta_{15} q^{7} + ( - \beta_{7} - \beta_{4} + \cdots + \beta_{2}) q^{8}+ \cdots + ( - \beta_{14} - \beta_{13} + 2 \beta_{12} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 2 q^{4} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} - 2 q^{4} + 4 q^{7} - 2 q^{8} - 10 q^{14} + 6 q^{16} + 24 q^{19} - 12 q^{22} - 16 q^{25} - 12 q^{26} - 22 q^{28} - 16 q^{29} - 8 q^{31} + 18 q^{32} - 24 q^{34} + 24 q^{37} + 28 q^{38} - 12 q^{40} + 8 q^{44} - 20 q^{46} + 16 q^{47} - 16 q^{49} + 2 q^{50} + 20 q^{52} + 32 q^{53} + 2 q^{56} - 32 q^{58} + 8 q^{59} + 16 q^{62} - 2 q^{64} + 8 q^{65} + 4 q^{68} - 20 q^{70} + 4 q^{74} - 16 q^{76} + 8 q^{77} - 16 q^{80} + 4 q^{82} + 8 q^{83} - 64 q^{86} - 52 q^{88} - 16 q^{91} - 64 q^{92} - 16 q^{94} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3 \nu^{15} + \nu^{13} - 26 \nu^{12} - 3 \nu^{11} + 19 \nu^{9} + 26 \nu^{8} - 104 \nu^{7} + 56 \nu^{6} + \cdots - 1152 ) / 512 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{14} - 5 \nu^{12} + 2 \nu^{11} + 7 \nu^{10} + 17 \nu^{8} - 18 \nu^{7} + 20 \nu^{6} + \cdots - 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} + 3 \nu^{11} + 2 \nu^{10} - 7 \nu^{9} + 12 \nu^{8} + \cdots - 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7 \nu^{15} + 4 \nu^{14} + 3 \nu^{13} + 14 \nu^{12} + 31 \nu^{11} + 12 \nu^{10} + 25 \nu^{9} + \cdots - 384 ) / 512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5 \nu^{15} - 16 \nu^{14} + 15 \nu^{13} - 38 \nu^{12} + 3 \nu^{11} + 32 \nu^{10} - 3 \nu^{9} + \cdots - 2176 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{15} + 12 \nu^{14} - 7 \nu^{13} + 10 \nu^{12} - 19 \nu^{11} - 28 \nu^{10} - 5 \nu^{9} + \cdots + 1408 ) / 512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{15} + \nu^{14} - \nu^{13} + \nu^{12} + \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 5 \nu^{8} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{15} + 6 \nu^{14} - 3 \nu^{13} + 12 \nu^{12} + \nu^{11} - 10 \nu^{10} + 7 \nu^{9} + \cdots + 704 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17 \nu^{15} + 4 \nu^{14} + 11 \nu^{13} - 34 \nu^{12} - 57 \nu^{11} - 20 \nu^{10} - 79 \nu^{9} + \cdots + 128 ) / 512 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17 \nu^{15} - 28 \nu^{14} + 27 \nu^{13} - 66 \nu^{12} - 9 \nu^{11} + 44 \nu^{10} - 31 \nu^{9} + \cdots - 3456 ) / 512 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15 \nu^{15} - 36 \nu^{14} + 5 \nu^{13} - 62 \nu^{12} + 9 \nu^{11} + 84 \nu^{10} - 33 \nu^{9} + \cdots - 4224 ) / 512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3 \nu^{15} + 5 \nu^{13} - 2 \nu^{12} - 7 \nu^{11} - 17 \nu^{9} + 18 \nu^{8} - 20 \nu^{7} + \cdots + 384 \nu ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11 \nu^{15} - 10 \nu^{14} + 9 \nu^{13} - 32 \nu^{12} - 15 \nu^{11} + 18 \nu^{10} - 37 \nu^{9} + \cdots - 1408 ) / 256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21 \nu^{15} + 40 \nu^{14} - 23 \nu^{13} + 78 \nu^{12} - 11 \nu^{11} - 104 \nu^{10} + 11 \nu^{9} + \cdots + 5504 ) / 512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + 2\beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} - 2\beta_{9} + 2\beta_{6} - 2\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} - 4 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{12} + 2 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{15} + 2\beta_{14} - 3\beta_{10} - \beta_{9} + 5\beta_{7} - \beta_{6} + \beta_{4} - 4\beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8 \beta_{14} + 4 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 4 \beta_{7} + \cdots - 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5 \beta_{15} + 6 \beta_{13} + 2 \beta_{12} - 6 \beta_{11} + \beta_{10} + 9 \beta_{9} + 14 \beta_{8} + \cdots + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6 \beta_{15} + 2 \beta_{14} - 6 \beta_{12} + 4 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 7 \beta_{15} + 6 \beta_{14} - 4 \beta_{13} + 12 \beta_{11} - 7 \beta_{10} + 3 \beta_{9} - 36 \beta_{8} + \cdots + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} + 20 \beta_{12} - 18 \beta_{11} + 12 \beta_{10} + 2 \beta_{9} + \cdots + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 5 \beta_{15} + 12 \beta_{14} + 34 \beta_{13} + 2 \beta_{12} - 50 \beta_{11} - 35 \beta_{10} + \cdots - 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2 \beta_{15} - 18 \beta_{14} + 16 \beta_{13} - 6 \beta_{12} + 8 \beta_{11} - 6 \beta_{10} + 30 \beta_{9} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 73 \beta_{15} - 118 \beta_{14} + 8 \beta_{13} + 48 \beta_{12} - 40 \beta_{11} + 77 \beta_{10} + \cdots - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 56 \beta_{15} - 128 \beta_{14} + 32 \beta_{13} + 132 \beta_{12} + 10 \beta_{11} + 52 \beta_{10} + \cdots + 24 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
811.1
1.40936 − 0.117062i
1.40936 + 0.117062i
1.10145 − 0.887017i
1.10145 + 0.887017i
1.07312 − 0.921096i
1.07312 + 0.921096i
0.309204 − 1.38000i
0.309204 + 1.38000i
−0.102186 − 1.41052i
−0.102186 + 1.41052i
−0.449546 − 1.34086i
−0.449546 + 1.34086i
−0.947441 − 1.04993i
−0.947441 + 1.04993i
−1.39396 − 0.238466i
−1.39396 + 0.238466i
−1.40936 − 0.117062i 0 1.97259 + 0.329965i − 1.00000i 0 −0.776136 − 2.52935i −2.74147 − 0.695955i 0 −0.117062 + 1.40936i
811.2 −1.40936 + 0.117062i 0 1.97259 − 0.329965i 1.00000i 0 −0.776136 + 2.52935i −2.74147 + 0.695955i 0 −0.117062 − 1.40936i
811.3 −1.10145 − 0.887017i 0 0.426402 + 1.95402i − 1.00000i 0 −0.391948 + 2.61656i 1.26358 − 2.53049i 0 −0.887017 + 1.10145i
811.4 −1.10145 + 0.887017i 0 0.426402 − 1.95402i 1.00000i 0 −0.391948 − 2.61656i 1.26358 + 2.53049i 0 −0.887017 − 1.10145i
811.5 −1.07312 − 0.921096i 0 0.303166 + 1.97689i 1.00000i 0 1.82575 − 1.91485i 1.49557 − 2.40068i 0 0.921096 − 1.07312i
811.6 −1.07312 + 0.921096i 0 0.303166 − 1.97689i − 1.00000i 0 1.82575 + 1.91485i 1.49557 + 2.40068i 0 0.921096 + 1.07312i
811.7 −0.309204 − 1.38000i 0 −1.80879 + 0.853401i − 1.00000i 0 2.64459 + 0.0785232i 1.73698 + 2.23224i 0 −1.38000 + 0.309204i
811.8 −0.309204 + 1.38000i 0 −1.80879 − 0.853401i 1.00000i 0 2.64459 − 0.0785232i 1.73698 − 2.23224i 0 −1.38000 − 0.309204i
811.9 0.102186 − 1.41052i 0 −1.97912 − 0.288270i 1.00000i 0 0.178143 − 2.63975i −0.608847 + 2.76212i 0 1.41052 + 0.102186i
811.10 0.102186 + 1.41052i 0 −1.97912 + 0.288270i − 1.00000i 0 0.178143 + 2.63975i −0.608847 − 2.76212i 0 1.41052 − 0.102186i
811.11 0.449546 − 1.34086i 0 −1.59582 − 1.20556i 1.00000i 0 −1.40015 + 2.24490i −2.33388 + 1.59781i 0 1.34086 + 0.449546i
811.12 0.449546 + 1.34086i 0 −1.59582 + 1.20556i − 1.00000i 0 −1.40015 − 2.24490i −2.33388 − 1.59781i 0 1.34086 − 0.449546i
811.13 0.947441 − 1.04993i 0 −0.204711 − 1.98950i − 1.00000i 0 2.29670 − 1.31346i −2.28279 − 1.67000i 0 −1.04993 − 0.947441i
811.14 0.947441 + 1.04993i 0 −0.204711 + 1.98950i 1.00000i 0 2.29670 + 1.31346i −2.28279 + 1.67000i 0 −1.04993 + 0.947441i
811.15 1.39396 − 0.238466i 0 1.88627 − 0.664826i − 1.00000i 0 −2.37694 − 1.16196i 2.47085 − 1.37655i 0 −0.238466 − 1.39396i
811.16 1.39396 + 0.238466i 0 1.88627 + 0.664826i 1.00000i 0 −2.37694 + 1.16196i 2.47085 + 1.37655i 0 −0.238466 + 1.39396i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 811.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.c.e 16
3.b odd 2 1 420.2.c.b yes 16
4.b odd 2 1 1260.2.c.d 16
7.b odd 2 1 1260.2.c.d 16
12.b even 2 1 420.2.c.a ✓ 16
21.c even 2 1 420.2.c.a ✓ 16
28.d even 2 1 inner 1260.2.c.e 16
84.h odd 2 1 420.2.c.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.c.a ✓ 16 12.b even 2 1
420.2.c.a ✓ 16 21.c even 2 1
420.2.c.b yes 16 3.b odd 2 1
420.2.c.b yes 16 84.h odd 2 1
1260.2.c.d 16 4.b odd 2 1
1260.2.c.d 16 7.b odd 2 1
1260.2.c.e 16 1.a even 1 1 trivial
1260.2.c.e 16 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{16} + 108 T_{11}^{14} + 4292 T_{11}^{12} + 75904 T_{11}^{10} + 549248 T_{11}^{8} + 947712 T_{11}^{6} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{19}^{8} - 12T_{19}^{7} + 6T_{19}^{6} + 288T_{19}^{5} - 352T_{19}^{4} - 2400T_{19}^{3} + 1696T_{19}^{2} + 6784T_{19} + 1408 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 108 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( T^{16} + 108 T^{14} + \cdots + 1982464 \) Copy content Toggle raw display
$17$ \( T^{16} + 104 T^{14} + \cdots + 589824 \) Copy content Toggle raw display
$19$ \( (T^{8} - 12 T^{7} + \cdots + 1408)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 110166016 \) Copy content Toggle raw display
$29$ \( (T^{8} + 8 T^{7} + \cdots - 475392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + \cdots + 435072)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 12 T^{7} + \cdots - 843008)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 186292371456 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 32480690176 \) Copy content Toggle raw display
$47$ \( (T^{8} - 8 T^{7} + \cdots + 1243904)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 16 T^{7} + \cdots + 311424)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 4 T^{7} + \cdots - 323072)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 334139490304 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1665379926016 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 57232008953856 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 34021064704 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 57415827456 \) Copy content Toggle raw display
$83$ \( (T^{8} - 4 T^{7} + \cdots - 30015488)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 278378643456 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 970650566656 \) Copy content Toggle raw display
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