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Properties

Label 1320.2.bw.e
Level $1320$
Weight $2$
Character orbit 1320.bw
Analytic conductor $10.540$
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] = [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: \( x^{12} - 2 x^{11} + 7 x^{10} - 6 x^{9} + 130 x^{8} - 768 x^{7} + 3132 x^{6} - 7488 x^{5} + 18450 x^{4} + \cdots + 9801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} - \beta_{2} q^{5} + (\beta_{9} + \beta_{2} - 1) q^{7} + ( - \beta_{8} + \beta_{6} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} - \beta_{2} q^{5} + (\beta_{9} + \beta_{2} - 1) q^{7} + ( - \beta_{8} + \beta_{6} + \beta_{2} - 1) q^{9} + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots + 2) q^{11}+ \cdots + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 3 q^{5} - 5 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 3 q^{5} - 5 q^{7} - 3 q^{9} + 6 q^{11} + 12 q^{13} + 3 q^{15} - 13 q^{17} + 20 q^{23} - 3 q^{25} + 3 q^{27} - 3 q^{29} - 3 q^{31} - q^{33} + 5 q^{35} + 5 q^{37} - 12 q^{39} + q^{41} + 6 q^{43} + 12 q^{45} - 10 q^{47} - 12 q^{51} - 3 q^{53} + q^{55} + 5 q^{57} + 23 q^{59} - 7 q^{61} - 5 q^{63} - 28 q^{65} + 44 q^{67} + 5 q^{69} - 9 q^{71} - 27 q^{73} + 3 q^{75} + 65 q^{77} - q^{79} - 3 q^{81} - 21 q^{83} + 12 q^{85} + 28 q^{87} + 32 q^{89} - 19 q^{91} + 3 q^{93} + 5 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 7 x^{10} - 6 x^{9} + 130 x^{8} - 768 x^{7} + 3132 x^{6} - 7488 x^{5} + 18450 x^{4} + \cdots + 9801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 44186172435 \nu^{11} + 394896426073 \nu^{10} - 1366423697738 \nu^{9} + \cdots + 11\!\cdots\!53 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 51122652869 \nu^{11} - 69267011678 \nu^{10} + 396495450808 \nu^{9} + \cdots - 363280094747460 ) / 53\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 211719510031 \nu^{11} + 832151086151 \nu^{10} - 765215188390 \nu^{9} + \cdots + 10\!\cdots\!99 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 306524081203 \nu^{11} + 1057120490693 \nu^{10} - 1152337134346 \nu^{9} + \cdots - 433068676035435 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 107400746229 \nu^{11} - 503230975653 \nu^{10} + 957494685578 \nu^{9} + \cdots - 39\!\cdots\!93 ) / 35\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 773725403149 \nu^{11} - 3001813342491 \nu^{10} + 2225051889174 \nu^{9} + \cdots - 38\!\cdots\!39 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1041059210099 \nu^{11} - 1870398910167 \nu^{10} + 6455263384542 \nu^{9} + \cdots - 80\!\cdots\!51 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1574255653957 \nu^{11} + 3941109503319 \nu^{10} - 12707090119206 \nu^{9} + \cdots + 15\!\cdots\!23 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 77918734737 \nu^{11} - 190818329135 \nu^{10} + 330403926238 \nu^{9} + \cdots + 171834534760185 ) / 368112747652848 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1326153586469 \nu^{11} - 596987156904 \nu^{10} + 6495544266480 \nu^{9} + \cdots + 90\!\cdots\!62 ) / 53\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{7} - 2\beta_{6} - 2\beta_{4} - 7\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} - 6\beta_{8} + 12\beta_{6} + 12\beta_{5} - \beta_{3} + 12\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{11} + 2 \beta_{10} - 9 \beta_{9} - 3 \beta_{8} + 9 \beta_{7} - 58 \beta_{6} - 38 \beta_{5} + \cdots - 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{11} - 18 \beta_{10} + 36 \beta_{8} + 18 \beta_{7} - 36 \beta_{6} + 38 \beta_{5} - 38 \beta_{4} + \cdots + 311 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 122 \beta_{11} + 86 \beta_{9} + 570 \beta_{8} - 122 \beta_{7} + 192 \beta_{6} + 192 \beta_{4} + \cdots - 942 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 281 \beta_{10} + 281 \beta_{9} - 283 \beta_{7} - 2550 \beta_{6} - 890 \beta_{5} - 1726 \beta_{4} + \cdots + 2550 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1447 \beta_{11} - 613 \beta_{10} - 1447 \beta_{9} - 6678 \beta_{8} + 17818 \beta_{6} + 10264 \beta_{5} + \cdots - 5231 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1376 \beta_{11} + 1376 \beta_{10} - 2779 \beta_{9} + 4095 \beta_{8} + 2779 \beta_{7} - 43474 \beta_{6} + \cdots - 43158 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10884 \beta_{11} - 18460 \beta_{10} - 27048 \beta_{8} + 18460 \beta_{7} + 27048 \beta_{6} + \cdots + 339833 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 60796 \beta_{11} + 23720 \beta_{9} + 294276 \beta_{8} - 60796 \beta_{7} + 330640 \beta_{6} + \cdots - 1194416 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(661\) \(881\) \(991\) \(1057\) \(1201\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(\beta_{8}\)

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
1.66571 + 1.21021i
−3.31518 − 2.40862i
2.14947 + 1.56168i
1.66571 − 1.21021i
−3.31518 + 2.40862i
2.14947 − 1.56168i
0.176245 + 0.542427i
1.03688 + 3.19120i
−0.713130 − 2.19479i
0.176245 − 0.542427i
1.03688 − 3.19120i
−0.713130 + 2.19479i
0 0.809017 − 0.587785i 0 0.309017 + 0.951057i 0 −2.45962 − 1.78702i 0 0.309017 − 0.951057i 0
361.2 0 0.809017 − 0.587785i 0 0.309017 + 0.951057i 0 −2.14647 − 1.55950i 0 0.309017 − 0.951057i 0
361.3 0 0.809017 − 0.587785i 0 0.309017 + 0.951057i 0 2.79708 + 2.03220i 0 0.309017 − 0.951057i 0
841.1 0 0.809017 + 0.587785i 0 0.309017 − 0.951057i 0 −2.45962 + 1.78702i 0 0.309017 + 0.951057i 0
841.2 0 0.809017 + 0.587785i 0 0.309017 − 0.951057i 0 −2.14647 + 1.55950i 0 0.309017 + 0.951057i 0
841.3 0 0.809017 + 0.587785i 0 0.309017 − 0.951057i 0 2.79708 − 2.03220i 0 0.309017 + 0.951057i 0
961.1 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 −1.25654 − 3.86723i 0 −0.809017 − 0.587785i 0
961.2 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 −0.232215 − 0.714683i 0 −0.809017 − 0.587785i 0
961.3 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 0.797771 + 2.45529i 0 −0.809017 − 0.587785i 0
1081.1 0 −0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0 −1.25654 + 3.86723i 0 −0.809017 + 0.587785i 0
1081.2 0 −0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0 −0.232215 + 0.714683i 0 −0.809017 + 0.587785i 0
1081.3 0 −0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0 0.797771 − 2.45529i 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.e ✓ 12
11.c even 5 1 inner 1320.2.bw.e ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.bw.e ✓ 12 1.a even 1 1 trivial
1320.2.bw.e ✓ 12 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 5 T_{7}^{11} + 23 T_{7}^{10} + 45 T_{7}^{9} + 134 T_{7}^{8} + 625 T_{7}^{7} + 3427 T_{7}^{6} + \cdots + 48400 \) 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
$7$ \( T^{12} + 5 T^{11} + \cdots + 48400 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} - 12 T^{11} + \cdots + 32041 \) Copy content Toggle raw display
$17$ \( T^{12} + 13 T^{11} + \cdots + 249001 \) Copy content Toggle raw display
$19$ \( T^{12} + 47 T^{10} + \cdots + 48400 \) Copy content Toggle raw display
$23$ \( (T^{6} - 10 T^{5} + \cdots + 12725)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + \cdots + 151321 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 238362721 \) Copy content Toggle raw display
$37$ \( T^{12} - 5 T^{11} + \cdots + 15912121 \) Copy content Toggle raw display
$41$ \( T^{12} - T^{11} + \cdots + 10471696 \) Copy content Toggle raw display
$43$ \( (T^{6} - 3 T^{5} + \cdots - 11231)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 10 T^{11} + \cdots + 36060025 \) Copy content Toggle raw display
$53$ \( T^{12} + 3 T^{11} + \cdots + 90706576 \) Copy content Toggle raw display
$59$ \( T^{12} - 23 T^{11} + \cdots + 12257001 \) Copy content Toggle raw display
$61$ \( T^{12} + 7 T^{11} + \cdots + 19360000 \) Copy content Toggle raw display
$67$ \( (T^{6} - 22 T^{5} + \cdots - 44541)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 9 T^{11} + \cdots + 65286400 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 56539328400 \) Copy content Toggle raw display
$79$ \( T^{12} + T^{11} + \cdots + 5948721 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2695686400 \) Copy content Toggle raw display
$89$ \( (T^{6} - 16 T^{5} + \cdots - 11484)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 13778064400 \) Copy content Toggle raw display
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