↑

Properties

Label 1320.2.bw.d
Level $1320$
Weight $2$
Character orbit 1320.bw
Analytic conductor $10.540$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1320,2,Mod(361,1320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1320, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1320.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.bw (of order \(5\), degree \(4\), 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: \(10.5402530668\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 23 x^{10} - 116 x^{9} + 340 x^{8} - 1281 x^{7} + 3863 x^{6} - 8123 x^{5} + \cdots + 48605 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4} + \beta_{3} - 1) q^{3} + \beta_{6} q^{5} + ( - \beta_{9} - \beta_{7}) q^{7} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{4} + \beta_{3} - 1) q^{3} + \beta_{6} q^{5} + ( - \beta_{9} - \beta_{7}) q^{7} - \beta_{3} q^{9} + (\beta_{11} - 2 \beta_{6} + \cdots - \beta_1) q^{11}+ \cdots + ( - \beta_{8} + \beta_{5} + \beta_{4} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} + 3 q^{5} + 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{3} + 3 q^{5} + 4 q^{7} - 3 q^{9} - 6 q^{11} - 13 q^{13} + 3 q^{15} + 2 q^{17} + 20 q^{19} - 6 q^{21} - 28 q^{23} - 3 q^{25} - 3 q^{27} - 12 q^{29} - 5 q^{31} - 6 q^{33} + q^{35} - 9 q^{37} - 13 q^{39} + 12 q^{41} + 16 q^{43} - 12 q^{45} + 13 q^{47} + 5 q^{49} + 2 q^{51} - 23 q^{53} - 4 q^{55} - 20 q^{57} - 13 q^{59} - 11 q^{61} + 4 q^{63} + 8 q^{65} - 10 q^{67} + 12 q^{69} + 25 q^{71} - 19 q^{73} - 3 q^{75} - 13 q^{77} + 14 q^{79} - 3 q^{81} - 33 q^{83} - 2 q^{85} + 18 q^{87} + 26 q^{89} - 2 q^{91} - 5 q^{93} - 20 q^{95} - 9 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 23 x^{10} - 116 x^{9} + 340 x^{8} - 1281 x^{7} + 3863 x^{6} - 8123 x^{5} + \cdots + 48605 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3243343421746 \nu^{11} - 1429663057448 \nu^{10} + 14014928374804 \nu^{9} + \cdots + 14\!\cdots\!80 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3798351690541 \nu^{11} + 21431890554381 \nu^{10} - 74483140494988 \nu^{9} + \cdots + 14\!\cdots\!05 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9239847942383 \nu^{11} + 17052179430690 \nu^{10} - 158166746004238 \nu^{9} + \cdots + 16\!\cdots\!40 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10036835482758 \nu^{11} + 12878948387455 \nu^{10} + 125543945896646 \nu^{9} + \cdots + 18\!\cdots\!05 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10209433762729 \nu^{11} - 3025721566645 \nu^{10} + 210681744287436 \nu^{9} + \cdots - 25\!\cdots\!85 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10667364396459 \nu^{11} + 54349756670571 \nu^{10} - 205378674438272 \nu^{9} + \cdots + 44\!\cdots\!15 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 173187173134 \nu^{11} + 284182512903 \nu^{10} - 3916507590794 \nu^{9} + \cdots + 82\!\cdots\!45 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66191816836426 \nu^{11} + 78640875403007 \nu^{10} + \cdots + 27\!\cdots\!05 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27602579721542 \nu^{11} - 24135232255331 \nu^{10} + 607294180804822 \nu^{9} + \cdots - 49\!\cdots\!45 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 92374526807454 \nu^{11} - 73424522833793 \nu^{10} + \cdots - 10\!\cdots\!45 ) / 32\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} + 3\beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{11} - \beta_{10} + \beta_{8} + 2 \beta_{7} + 10 \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{11} + 9 \beta_{10} + 10 \beta_{9} - 14 \beta_{8} - 5 \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 37 \beta_{11} + 18 \beta_{10} - 29 \beta_{9} + 46 \beta_{8} - 9 \beta_{7} - 103 \beta_{6} - \beta_{5} + \cdots - 107 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 59 \beta_{11} - 114 \beta_{10} - 59 \beta_{9} + 153 \beta_{8} + 155 \beta_{7} + 198 \beta_{6} + \cdots - 115 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 465 \beta_{11} + 25 \beta_{10} + 608 \beta_{9} - 491 \beta_{8} - 111 \beta_{7} + 1134 \beta_{6} + \cdots + 1070 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 187 \beta_{11} + 1767 \beta_{10} - 70 \beta_{9} - 839 \beta_{8} - 1800 \beta_{7} - 3543 \beta_{6} + \cdots + 551 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6081 \beta_{11} - 1846 \beta_{10} - 7295 \beta_{9} + 6681 \beta_{8} + 5760 \beta_{7} - 10381 \beta_{6} + \cdots - 11514 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2203 \beta_{11} - 19522 \beta_{10} + 15191 \beta_{9} + 5046 \beta_{8} + 16241 \beta_{7} + 58477 \beta_{6} + \cdots - 6332 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 75261 \beta_{11} + 54146 \beta_{10} + 77691 \beta_{9} - 81257 \beta_{8} - 101405 \beta_{7} + \cdots + 142425 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(881\) \(991\) \(1057\) \(1201\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.09280 + 3.05269i
0.0530924 − 1.56220i
2.34872 − 0.539435i
−1.09280 − 3.05269i
0.0530924 + 1.56220i
2.34872 + 0.539435i
1.87886 − 0.592611i
−0.641604 + 3.51866i
−1.04628 − 2.33826i
1.87886 + 0.592611i
−0.641604 − 3.51866i
−1.04628 + 2.33826i
0 −0.809017 + 0.587785i 0 −0.309017 − 0.951057i 0 −1.76818 − 1.28466i 0 0.309017 − 0.951057i 0
361.2 0 −0.809017 + 0.587785i 0 −0.309017 − 0.951057i 0 0.0859053 + 0.0624139i 0 0.309017 − 0.951057i 0
361.3 0 −0.809017 + 0.587785i 0 −0.309017 − 0.951057i 0 3.80031 + 2.76109i 0 0.309017 − 0.951057i 0
841.1 0 −0.809017 − 0.587785i 0 −0.309017 + 0.951057i 0 −1.76818 + 1.28466i 0 0.309017 + 0.951057i 0
841.2 0 −0.809017 − 0.587785i 0 −0.309017 + 0.951057i 0 0.0859053 − 0.0624139i 0 0.309017 + 0.951057i 0
841.3 0 −0.809017 − 0.587785i 0 −0.309017 + 0.951057i 0 3.80031 − 2.76109i 0 0.309017 + 0.951057i 0
961.1 0 0.309017 − 0.951057i 0 0.809017 − 0.587785i 0 −1.16120 − 3.57381i 0 −0.809017 − 0.587785i 0
961.2 0 0.309017 − 0.951057i 0 0.809017 − 0.587785i 0 0.396533 + 1.22040i 0 −0.809017 − 0.587785i 0
961.3 0 0.309017 − 0.951057i 0 0.809017 − 0.587785i 0 0.646634 + 1.99013i 0 −0.809017 − 0.587785i 0
1081.1 0 0.309017 + 0.951057i 0 0.809017 + 0.587785i 0 −1.16120 + 3.57381i 0 −0.809017 + 0.587785i 0
1081.2 0 0.309017 + 0.951057i 0 0.809017 + 0.587785i 0 0.396533 − 1.22040i 0 −0.809017 + 0.587785i 0
1081.3 0 0.309017 + 0.951057i 0 0.809017 + 0.587785i 0 0.646634 − 1.99013i 0 −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1320.2.bw.d ✓ 12
11.c even 5 1 inner 1320.2.bw.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.bw.d ✓ 12 1.a even 1 1 trivial
1320.2.bw.d ✓ 12 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 4 T_{7}^{11} + 16 T_{7}^{10} - 46 T_{7}^{9} + 292 T_{7}^{8} + 214 T_{7}^{7} + 1251 T_{7}^{6} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(1320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} - 4 T^{11} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 182925625 \) Copy content Toggle raw display
$17$ \( T^{12} - 2 T^{11} + \cdots + 3748096 \) Copy content Toggle raw display
$19$ \( T^{12} - 20 T^{11} + \cdots + 326041 \) Copy content Toggle raw display
$23$ \( (T^{6} + 14 T^{5} + \cdots + 5651)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 12 T^{11} + \cdots + 234256 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 171610000 \) Copy content Toggle raw display
$37$ \( T^{12} + 9 T^{11} + \cdots + 203401 \) Copy content Toggle raw display
$41$ \( T^{12} - 12 T^{11} + \cdots + 3258025 \) Copy content Toggle raw display
$43$ \( (T^{6} - 8 T^{5} + \cdots - 81436)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 37330490521 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 26586933025 \) Copy content Toggle raw display
$59$ \( T^{12} + 13 T^{11} + \cdots + 5313025 \) Copy content Toggle raw display
$61$ \( T^{12} + 11 T^{11} + \cdots + 68624656 \) Copy content Toggle raw display
$67$ \( (T^{6} + 5 T^{5} + \cdots - 188116)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 23016006250000 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2239560976 \) Copy content Toggle raw display
$79$ \( T^{12} - 14 T^{11} + \cdots + 2085136 \) Copy content Toggle raw display
$83$ \( T^{12} + 33 T^{11} + \cdots + 23658496 \) Copy content Toggle raw display
$89$ \( (T^{6} - 13 T^{5} + \cdots + 183995)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 5026810000 \) Copy content Toggle raw display
show more
show less