Properties

Label 385.2.o.a
Level $385$
Weight $2$
Character orbit 385.o
Analytic conductor $3.074$
Analytic rank $0$
Dimension $16$
CM discriminant -55
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [385,2,Mod(54,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.54");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.o (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: \(3.07424047782\)
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: \( x^{16} + 3x^{12} - 7x^{8} + 48x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{14} + \beta_{10}) q^{2} + ( - \beta_{7} + 2 \beta_{5} + \beta_1) q^{4} + (\beta_{7} + \beta_{6} - \beta_1) q^{5} - \beta_{13} q^{7} + ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12}) q^{8} + (3 \beta_{5} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} + \beta_{10}) q^{2} + ( - \beta_{7} + 2 \beta_{5} + \beta_1) q^{4} + (\beta_{7} + \beta_{6} - \beta_1) q^{5} - \beta_{13} q^{7} + ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12}) q^{8} + (3 \beta_{5} + 3) q^{9} + (\beta_{15} - \beta_{14} + \beta_{10} + \beta_{4}) q^{10} + ( - \beta_{9} + \beta_{6} + \beta_1) q^{11} + (\beta_{14} - \beta_{12} - 2 \beta_{10} - 2 \beta_{8} - \beta_{4} + \beta_{2}) q^{13} + ( - \beta_{11} + \beta_{9} + 2 \beta_{6} - \beta_{3} - \beta_1) q^{14} + (\beta_{11} - 2 \beta_{9} - 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{3} - 2) q^{16} + (\beta_{14} - \beta_{13} + \beta_{12} - \beta_{4} - \beta_{2}) q^{17} + 3 \beta_{10} q^{18} + ( - \beta_{11} + 2 \beta_{9} - 2 \beta_{7} + 5 \beta_{5} - \beta_{3} + 2) q^{20} + ( - \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{4} + \beta_{2}) q^{22} - 5 \beta_{5} q^{25} + (2 \beta_{11} + 3 \beta_{9} + 2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_1 + 1) q^{26} + ( - \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 3 \beta_{10} + 2 \beta_{4} + \cdots - \beta_{2}) q^{28}+ \cdots + ( - 3 \beta_{9} - 3 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 24 q^{9} + 8 q^{14} - 20 q^{16} + 40 q^{25} - 96 q^{36} - 44 q^{44} - 112 q^{56} + 48 q^{59} + 80 q^{64} + 120 q^{70} + 64 q^{71} - 120 q^{80} - 72 q^{81} + 68 q^{86} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 3x^{12} - 7x^{8} + 48x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 35\nu^{11} + 217\nu^{7} - 176\nu^{3} ) / 896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{12} - 7\nu^{8} + 91\nu^{4} + 256 ) / 112 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{13} + 7\nu^{9} - 91\nu^{5} - 256\nu ) / 224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{12} + 7\nu^{8} + 21\nu^{4} - 256 ) / 112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{14} + 3\nu^{10} - 7\nu^{6} + 48\nu^{2} ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{14} - 21\nu^{10} + 49\nu^{6} + 768\nu^{2} ) / 448 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} - 12\nu^{13} - 21\nu^{11} + 28\nu^{9} + 49\nu^{7} + 84\nu^{5} + 216\nu^{3} - 1024\nu ) / 448 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{14} - 7\nu^{10} - 21\nu^{6} + 256\nu^{2} ) / 112 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7\nu^{15} - 24\nu^{13} + 21\nu^{11} + 56\nu^{9} - 49\nu^{7} + 168\nu^{5} + 336\nu^{3} - 1152\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\nu^{12} + 7\nu^{8} - 91\nu^{4} + 624 ) / 112 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7\nu^{15} + 24\nu^{13} + 21\nu^{11} - 56\nu^{9} - 49\nu^{7} - 168\nu^{5} + 336\nu^{3} + 256\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -13\nu^{13} - 7\nu^{9} + 91\nu^{5} - 624\nu ) / 224 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -9\nu^{15} - 24\nu^{13} + 21\nu^{11} + 56\nu^{9} - 49\nu^{7} + 168\nu^{5} - 768\nu^{3} - 2048\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2\nu^{15} + 3\nu^{13} - 7\nu^{9} - 21\nu^{5} + 26\nu^{3} + 144\nu ) / 112 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{14} - 3\beta_{12} + \beta_{10} - 2\beta_{8} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{15} - 6\beta_{14} + 5\beta_{12} + 3\beta_{10} + \beta_{8} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{14} + \beta_{12} + 2\beta_{10} + 3\beta_{8} - 14\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{9} + 4\beta_{7} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5\beta_{14} - 4\beta_{12} - 8\beta_{10} + 9\beta_{8} + 21\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{11} + 13\beta_{5} + 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 13\beta_{14} - 42\beta_{13} - 26\beta_{12} + 39\beta_{10} + 13\beta_{8} ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -7\beta_{9} + 12\beta_{6} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -49\beta_{15} - 3\beta_{14} + 55\beta_{12} + 61\beta_{10} - 52\beta_{8} + 49\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7\beta_{11} + 7\beta_{3} - 55 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 110\beta_{14} - 98\beta_{13} + 165\beta_{12} - 55\beta_{10} + 110\beta_{8} - 98\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 28\beta_{7} + 28\beta_{6} - 69\beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 483\beta_{15} + 134\beta_{14} - 177\beta_{12} + 129\beta_{10} + 43\beta_{8} ) / 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(276\)
\(\chi(n)\) \(-1\) \(-1\) \(1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1
−0.964601 + 1.03419i
0.318275 + 1.37793i
−1.37793 + 0.318275i
1.03419 + 0.964601i
−1.03419 0.964601i
1.37793 0.318275i
−0.318275 1.37793i
0.964601 1.03419i
−0.964601 1.03419i
0.318275 1.37793i
−1.37793 0.318275i
1.03419 0.964601i
−1.03419 + 0.964601i
1.37793 + 0.318275i
−0.318275 + 1.37793i
0.964601 + 1.03419i
−1.37793 + 2.38665i 0 −2.79740 4.84524i 1.93649 + 1.11803i 0 −2.61087 + 0.428223i 9.90680 1.50000 2.59808i −5.33671 + 3.08115i
54.2 −1.03419 + 1.79127i 0 −1.13909 1.97296i 1.93649 + 1.11803i 0 0.428223 2.61087i 0.575379 1.50000 2.59808i −4.00539 + 2.31251i
54.3 −0.964601 + 1.67074i 0 −0.860910 1.49114i −1.93649 1.11803i 0 2.61087 + 0.428223i −0.536664 1.50000 2.59808i 3.73588 2.15691i
54.4 −0.318275 + 0.551268i 0 0.797402 + 1.38114i −1.93649 1.11803i 0 0.428223 + 2.61087i −2.28827 1.50000 2.59808i 1.23267 0.711685i
54.5 0.318275 0.551268i 0 0.797402 + 1.38114i −1.93649 1.11803i 0 −0.428223 2.61087i 2.28827 1.50000 2.59808i −1.23267 + 0.711685i
54.6 0.964601 1.67074i 0 −0.860910 1.49114i −1.93649 1.11803i 0 −2.61087 0.428223i 0.536664 1.50000 2.59808i −3.73588 + 2.15691i
54.7 1.03419 1.79127i 0 −1.13909 1.97296i 1.93649 + 1.11803i 0 −0.428223 + 2.61087i −0.575379 1.50000 2.59808i 4.00539 2.31251i
54.8 1.37793 2.38665i 0 −2.79740 4.84524i 1.93649 + 1.11803i 0 2.61087 0.428223i −9.90680 1.50000 2.59808i 5.33671 3.08115i
164.1 −1.37793 2.38665i 0 −2.79740 + 4.84524i 1.93649 1.11803i 0 −2.61087 0.428223i 9.90680 1.50000 + 2.59808i −5.33671 3.08115i
164.2 −1.03419 1.79127i 0 −1.13909 + 1.97296i 1.93649 1.11803i 0 0.428223 + 2.61087i 0.575379 1.50000 + 2.59808i −4.00539 2.31251i
164.3 −0.964601 1.67074i 0 −0.860910 + 1.49114i −1.93649 + 1.11803i 0 2.61087 0.428223i −0.536664 1.50000 + 2.59808i 3.73588 + 2.15691i
164.4 −0.318275 0.551268i 0 0.797402 1.38114i −1.93649 + 1.11803i 0 0.428223 2.61087i −2.28827 1.50000 + 2.59808i 1.23267 + 0.711685i
164.5 0.318275 + 0.551268i 0 0.797402 1.38114i −1.93649 + 1.11803i 0 −0.428223 + 2.61087i 2.28827 1.50000 + 2.59808i −1.23267 0.711685i
164.6 0.964601 + 1.67074i 0 −0.860910 + 1.49114i −1.93649 + 1.11803i 0 −2.61087 + 0.428223i 0.536664 1.50000 + 2.59808i −3.73588 2.15691i
164.7 1.03419 + 1.79127i 0 −1.13909 + 1.97296i 1.93649 1.11803i 0 −0.428223 2.61087i −0.575379 1.50000 + 2.59808i 4.00539 + 2.31251i
164.8 1.37793 + 2.38665i 0 −2.79740 + 4.84524i 1.93649 1.11803i 0 2.61087 + 0.428223i −9.90680 1.50000 + 2.59808i 5.33671 + 3.08115i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 54.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
35.i odd 6 1 inner
77.i even 6 1 inner
385.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.o.a 16
5.b even 2 1 inner 385.2.o.a 16
7.d odd 6 1 inner 385.2.o.a 16
11.b odd 2 1 inner 385.2.o.a 16
35.i odd 6 1 inner 385.2.o.a 16
55.d odd 2 1 CM 385.2.o.a 16
77.i even 6 1 inner 385.2.o.a 16
385.o even 6 1 inner 385.2.o.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.o.a 16 1.a even 1 1 trivial
385.2.o.a 16 5.b even 2 1 inner
385.2.o.a 16 7.d odd 6 1 inner
385.2.o.a 16 11.b odd 2 1 inner
385.2.o.a 16 35.i odd 6 1 inner
385.2.o.a 16 55.d odd 2 1 CM
385.2.o.a 16 77.i even 6 1 inner
385.2.o.a 16 385.o even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 16T_{2}^{14} + 173T_{2}^{12} + 1024T_{2}^{10} + 4408T_{2}^{8} + 11048T_{2}^{6} + 19037T_{2}^{4} + 7448T_{2}^{2} + 2401 \) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 16 T^{14} + 173 T^{12} + \cdots + 2401 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} - 78 T^{4} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 11 T^{2} + 121)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 104 T^{6} + 3218 T^{4} + 26728 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 68 T^{6} + 3644 T^{4} + \cdots + 960400)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} - 106 T^{6} + 11067 T^{4} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{8} - 344 T^{6} + 33458 T^{4} + \cdots + 2798929)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{4} - 12 T^{3} + 5 T^{2} + 516 T + 1849)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 149)^{8} \) Copy content Toggle raw display
$73$ \( T^{16} - 584 T^{14} + \cdots + 64\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} + 664 T^{6} + 124178 T^{4} + \cdots + 45064369)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 354 T^{6} + 117747 T^{4} + \cdots + 57289761)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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