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Properties

Label 1260.2.bm.d
Level $1260$
Weight $2$
Character orbit 1260.bm
Analytic conductor $10.061$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,2,Mod(109,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.109");
 
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.bm (of order \(6\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 4 x^{14} + 12 x^{13} + 162 x^{12} - 524 x^{11} - 88 x^{10} + 1492 x^{9} + \cdots + 1521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} + (\beta_{14} - \beta_{6} + \beta_{5}) q^{7} - \beta_{15} q^{11} + \beta_{6} q^{13} + (\beta_{12} + \beta_{10} + \beta_{2}) q^{17} + ( - \beta_{8} + 2 \beta_1) q^{19} + (\beta_{11} - \beta_{7} + \beta_{4}) q^{23}+ \cdots + ( - 5 \beta_{6} + 5 \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{19} - 8 q^{31} - 16 q^{49} + 24 q^{61} + 16 q^{79} - 80 q^{85} + 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 4 x^{14} + 12 x^{13} + 162 x^{12} - 524 x^{11} - 88 x^{10} + 1492 x^{9} + \cdots + 1521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1282138 \nu^{15} - 19217450 \nu^{14} + 57698230 \nu^{13} + 201080763 \nu^{12} + \cdots - 33231454884 ) / 3843901530 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 881253165 \nu^{15} + 7824103882 \nu^{14} - 27303457147 \nu^{13} - 96774671785 \nu^{12} + \cdots + 14207114782965 ) / 882816051390 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8517854635 \nu^{15} - 47901289016 \nu^{14} - 10808846529 \nu^{13} + 254360686162 \nu^{12} + \cdots - 26162339620443 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10028198071 \nu^{15} + 38930042921 \nu^{14} + 66475606146 \nu^{13} + \cdots + 17368954657899 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1683778105 \nu^{15} - 2653967414 \nu^{14} - 17809640997 \nu^{13} - 13603646729 \nu^{12} + \cdots - 1451250345714 ) / 378349736310 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2053516175 \nu^{15} + 8267524636 \nu^{14} + 7910711853 \nu^{13} - 29044383254 \nu^{12} + \cdots + 5795645027601 ) / 378349736310 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5036832092 \nu^{15} - 9832634070 \nu^{14} - 55413755011 \nu^{13} - 13428796079 \nu^{12} + \cdots + 8555250967614 ) / 882816051390 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15731411194 \nu^{15} + 64536858778 \nu^{14} + 74366831386 \nu^{13} + \cdots + 33091548064380 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19284019598 \nu^{15} + 17049266297 \nu^{14} + 261729007535 \nu^{13} + \cdots - 37242105966819 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20674411368 \nu^{15} + 71970333169 \nu^{14} + 126497443556 \nu^{13} + \cdots + 52612578521763 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21619765207 \nu^{15} + 91705215266 \nu^{14} + 116881780644 \nu^{13} + \cdots + 23157711300282 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26996336692 \nu^{15} - 78349820220 \nu^{14} - 212403246200 \nu^{13} + 137458795490 \nu^{12} + \cdots - 24377058701112 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 394610 \nu^{15} - 1397252 \nu^{14} - 2812046 \nu^{13} + 4855143 \nu^{12} + 71847798 \nu^{11} + \cdots - 209664117 ) / 32307210 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 49182588496 \nu^{15} + 144983220011 \nu^{14} + 358529986902 \nu^{13} + \cdots + 50559205112286 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30719491738 \nu^{15} + 97105774717 \nu^{14} + 197796899383 \nu^{13} + \cdots + 42196531034853 ) / 1324224077085 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - 2\beta_{13} - 3\beta_{11} + \beta_{8} - 2\beta_{6} + \beta_{5} - 3\beta_{4} - \beta_{3} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} + \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 12 \beta_{15} + 29 \beta_{14} - 2 \beta_{13} - 12 \beta_{12} - 21 \beta_{11} - 30 \beta_{10} + \cdots + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26 \beta_{15} - 30 \beta_{14} - 14 \beta_{13} + 18 \beta_{12} - 12 \beta_{11} + 30 \beta_{10} + \cdots - 45 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 80 \beta_{15} - 51 \beta_{14} + 90 \beta_{13} + 60 \beta_{12} + 165 \beta_{11} - 60 \beta_{10} + \cdots + 378 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 113 \beta_{15} + 620 \beta_{14} - 74 \beta_{13} - 57 \beta_{12} - 357 \beta_{11} - 207 \beta_{10} + \cdots - 252 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2380 \beta_{15} - 5865 \beta_{14} - 324 \beta_{13} - 252 \beta_{12} + 915 \beta_{11} + 1638 \beta_{10} + \cdots - 1050 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4604 \beta_{15} + 7656 \beta_{14} + 770 \beta_{13} + 2988 \beta_{12} + 768 \beta_{11} - 1188 \beta_{10} + \cdots + 3279 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 18888 \beta_{15} - 9587 \beta_{14} + 952 \beta_{13} - 23910 \beta_{12} - 11553 \beta_{11} - 7926 \beta_{10} + \cdots + 5226 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3865 \beta_{15} - 55682 \beta_{14} - 19685 \beta_{13} + 19185 \beta_{12} + 615 \beta_{11} + 22125 \beta_{10} + \cdots - 74475 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 210804 \beta_{15} + 570965 \beta_{14} + 189892 \beta_{13} + 165054 \beta_{12} + 190245 \beta_{11} + \cdots + 742668 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 386030 \beta_{15} - 640926 \beta_{14} - 211166 \beta_{13} - 693282 \beta_{12} - 501456 \beta_{11} + \cdots - 811167 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 993512 \beta_{15} - 202827 \beta_{14} - 512796 \beta_{13} + 5050266 \beta_{12} + 2078727 \beta_{11} + \cdots - 1963260 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1726925 \beta_{15} + 5874734 \beta_{14} + 4106908 \beta_{13} - 4201503 \beta_{12} + 1956759 \beta_{11} + \cdots + 15924624 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 20451160 \beta_{15} - 33583785 \beta_{14} - 34806450 \beta_{13} - 15889470 \beta_{12} - 49593435 \beta_{11} + \cdots - 134742744 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.45942 + 0.551253i
1.51479 − 1.11371i
0.752308 − 0.673492i
0.921698 + 0.861704i
2.22190 + 0.111032i
−2.89591 + 1.43281i
0.214591 − 0.363041i
−2.18880 + 2.65755i
1.45942 − 0.551253i
1.51479 + 1.11371i
0.752308 + 0.673492i
0.921698 − 0.861704i
2.22190 − 0.111032i
−2.89591 − 1.43281i
0.214591 + 0.363041i
−2.18880 − 2.65755i
0 0 0 −2.19944 + 0.403048i 0 2.42997 + 1.04655i 0 0 0
109.2 0 0 0 −1.70325 − 1.44877i 0 0.308646 − 2.62769i 0 0 0
109.3 0 0 0 −1.44877 + 1.70325i 0 −2.42997 − 1.04655i 0 0 0
109.4 0 0 0 −0.403048 − 2.19944i 0 −0.308646 + 2.62769i 0 0 0
109.5 0 0 0 0.403048 + 2.19944i 0 −0.308646 + 2.62769i 0 0 0
109.6 0 0 0 1.44877 − 1.70325i 0 −2.42997 − 1.04655i 0 0 0
109.7 0 0 0 1.70325 + 1.44877i 0 0.308646 − 2.62769i 0 0 0
109.8 0 0 0 2.19944 − 0.403048i 0 2.42997 + 1.04655i 0 0 0
289.1 0 0 0 −2.19944 − 0.403048i 0 2.42997 − 1.04655i 0 0 0
289.2 0 0 0 −1.70325 + 1.44877i 0 0.308646 + 2.62769i 0 0 0
289.3 0 0 0 −1.44877 − 1.70325i 0 −2.42997 + 1.04655i 0 0 0
289.4 0 0 0 −0.403048 + 2.19944i 0 −0.308646 − 2.62769i 0 0 0
289.5 0 0 0 0.403048 − 2.19944i 0 −0.308646 − 2.62769i 0 0 0
289.6 0 0 0 1.44877 + 1.70325i 0 −2.42997 + 1.04655i 0 0 0
289.7 0 0 0 1.70325 − 1.44877i 0 0.308646 + 2.62769i 0 0 0
289.8 0 0 0 2.19944 + 0.403048i 0 2.42997 − 1.04655i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.bm.d ✓ 16
3.b odd 2 1 inner 1260.2.bm.d ✓ 16
5.b even 2 1 inner 1260.2.bm.d ✓ 16
7.c even 3 1 inner 1260.2.bm.d ✓ 16
15.d odd 2 1 inner 1260.2.bm.d ✓ 16
21.h odd 6 1 inner 1260.2.bm.d ✓ 16
35.j even 6 1 inner 1260.2.bm.d ✓ 16
105.o odd 6 1 inner 1260.2.bm.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.bm.d ✓ 16 1.a even 1 1 trivial
1260.2.bm.d ✓ 16 3.b odd 2 1 inner
1260.2.bm.d ✓ 16 5.b even 2 1 inner
1260.2.bm.d ✓ 16 7.c even 3 1 inner
1260.2.bm.d ✓ 16 15.d odd 2 1 inner
1260.2.bm.d ✓ 16 21.h odd 6 1 inner
1260.2.bm.d ✓ 16 35.j even 6 1 inner
1260.2.bm.d ✓ 16 105.o odd 6 1 inner

Hecke kernels

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

\( T_{11}^{8} + 30T_{11}^{6} + 810T_{11}^{4} + 2700T_{11}^{2} + 8100 \) Copy content Toggle raw display
\( T_{17}^{8} - 50T_{17}^{6} + 2010T_{17}^{4} - 24500T_{17}^{2} + 240100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 10 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 30 T^{6} + \cdots + 8100)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8 T^{2} + 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} - 50 T^{6} + \cdots + 240100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 50 T^{6} + \cdots + 62500)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 50 T^{2} + 250)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} + \cdots + 3481)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 152 T^{6} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 30 T^{2} + 90)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{2} + 1)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 140 T^{6} + \cdots + 23425600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 120 T^{6} + \cdots + 2073600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 290 T^{6} + \cdots + 341880100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + 42 T^{2} + \cdots + 36)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 152 T^{6} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 210 T^{2} + 10890)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 128 T^{6} + \cdots + 13845841)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 210 T^{2} + 10890)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 110 T^{6} + \cdots + 8352100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 150)^{8} \) Copy content Toggle raw display
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