↑

Properties

Label 385.2.b.c
Level $385$
Weight $2$
Character orbit 385.b
Analytic conductor $3.074$
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] = [385,2,Mod(309,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.309");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.b (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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 14x^{10} + 71x^{8} + 156x^{6} + 135x^{4} + 26x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{10} + \beta_{2}) q^{3} + (\beta_{9} + \beta_{6} + \beta_{4} - 1) q^{4} + ( - \beta_{8} + \beta_{5} + 1) q^{5} + ( - \beta_{9} - \beta_{7} + \beta_{5} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{10} + \beta_{2}) q^{3} + (\beta_{9} + \beta_{6} + \beta_{4} - 1) q^{4} + ( - \beta_{8} + \beta_{5} + 1) q^{5} + ( - \beta_{9} - \beta_{7} + \beta_{5} + \cdots + 1) q^{6}+ \cdots + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{4} + 6 q^{5} - 4 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{4} + 6 q^{5} - 4 q^{6} - 20 q^{9} + 14 q^{10} - 12 q^{11} - 8 q^{15} - 12 q^{16} + 4 q^{19} - 6 q^{20} - 12 q^{21} - 44 q^{24} - 8 q^{25} + 16 q^{29} + 40 q^{30} + 8 q^{31} + 20 q^{34} + 2 q^{35} + 52 q^{36} + 16 q^{39} - 10 q^{40} + 8 q^{41} + 4 q^{44} - 2 q^{45} - 16 q^{46} - 12 q^{49} + 28 q^{50} - 8 q^{51} - 76 q^{54} - 6 q^{55} - 28 q^{59} + 16 q^{60} + 24 q^{61} + 44 q^{64} + 12 q^{65} + 4 q^{66} - 24 q^{69} + 6 q^{70} + 8 q^{71} + 16 q^{74} - 60 q^{75} - 16 q^{76} + 16 q^{79} + 22 q^{80} + 52 q^{81} + 24 q^{84} - 20 q^{85} + 24 q^{86} - 16 q^{89} - 46 q^{90} - 12 q^{91} + 20 q^{94} - 40 q^{95} - 32 q^{96} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 14x^{10} + 71x^{8} + 156x^{6} + 135x^{4} + 26x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 7\nu^{4} + 11\nu^{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} - 11\nu^{6} - 39\nu^{4} - 45\nu^{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + \nu^{10} + 13 \nu^{9} + 13 \nu^{8} + 60 \nu^{7} + 60 \nu^{6} + 118 \nu^{5} + 118 \nu^{4} + \cdots + 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{10} - 13\nu^{8} - 60\nu^{6} - 116\nu^{4} - 81\nu^{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - \nu^{10} + 13 \nu^{9} - 13 \nu^{8} + 60 \nu^{7} - 60 \nu^{6} + 118 \nu^{5} - 118 \nu^{4} + \cdots - 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} - 14\nu^{9} - 70\nu^{7} - 146\nu^{5} - 105\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{10} + 14\nu^{8} + 71\nu^{6} + 155\nu^{4} + 128\nu^{2} + 15 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} - 14\nu^{9} - 71\nu^{7} - 155\nu^{5} - 128\nu^{3} - 15\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \nu^{11} + 14\nu^{9} + 71\nu^{7} + 156\nu^{5} + 135\nu^{3} + 26\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{6} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{9} - \beta_{7} - 4\beta_{6} + \beta_{5} - 5\beta_{4} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 2\beta_{10} - 7\beta_{2} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24\beta_{9} + 7\beta_{7} + 17\beta_{6} - 7\beta_{5} + 24\beta_{4} + 2\beta_{3} - 59 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -9\beta_{11} - 20\beta_{10} + 2\beta_{8} + 40\beta_{2} - 76\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -114\beta_{9} - 38\beta_{7} - 76\beta_{6} + 38\beta_{5} - 116\beta_{4} - 22\beta_{3} + 273 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 62\beta_{11} + 144\beta_{10} - 22\beta_{8} - 2\beta_{7} - 2\beta_{5} - 214\beta_{2} + 349\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 541\beta_{9} + 190\beta_{7} + 349\beta_{6} - 190\beta_{5} + 567\beta_{4} + 166\beta_{3} - 1279 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -384\beta_{11} - 908\beta_{10} + 166\beta_{8} + 28\beta_{7} + 28\beta_{5} + 1113\beta_{2} - 1628\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(211\) \(232\) \(276\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
309.1
− 2.24274i
− 2.05338i
− 1.68955i
− 1.28581i
− 0.441021i
− 0.226645i
0.226645i
0.441021i
1.28581i
1.68955i
2.05338i
2.24274i
− 2.24274i 3.30979i −3.02989 1.71229 + 1.43807i 7.42300 1.00000i 2.30979i −7.95470 3.22522 − 3.84024i
309.2 − 2.05338i − 0.555739i −2.21636 1.42175 + 1.72587i −1.14114 − 1.00000i 0.444261i 2.69115 3.54386 − 2.91939i
309.3 − 1.68955i − 2.93524i −0.854580 −1.80653 + 1.31774i −4.95924 − 1.00000i − 1.93524i −5.61566 2.22639 + 3.05223i
309.4 − 1.28581i − 2.01740i 0.346698 −0.320060 − 2.21304i −2.59399 1.00000i − 3.01740i −1.06991 −2.84555 + 0.411535i
309.5 − 0.441021i − 0.678305i 1.80550 −0.164664 + 2.23000i −0.299147 1.00000i − 1.67831i 2.53990 0.983475 + 0.0726204i
309.6 − 0.226645i − 1.89494i 1.94863 2.15721 − 0.588587i −0.429477 − 1.00000i − 0.894936i −0.590784 −0.133400 − 0.488921i
309.7 0.226645i 1.89494i 1.94863 2.15721 + 0.588587i −0.429477 1.00000i 0.894936i −0.590784 −0.133400 + 0.488921i
309.8 0.441021i 0.678305i 1.80550 −0.164664 − 2.23000i −0.299147 − 1.00000i 1.67831i 2.53990 0.983475 − 0.0726204i
309.9 1.28581i 2.01740i 0.346698 −0.320060 + 2.21304i −2.59399 − 1.00000i 3.01740i −1.06991 −2.84555 − 0.411535i
309.10 1.68955i 2.93524i −0.854580 −1.80653 − 1.31774i −4.95924 1.00000i 1.93524i −5.61566 2.22639 − 3.05223i
309.11 2.05338i 0.555739i −2.21636 1.42175 − 1.72587i −1.14114 1.00000i − 0.444261i 2.69115 3.54386 + 2.91939i
309.12 2.24274i − 3.30979i −3.02989 1.71229 − 1.43807i 7.42300 − 1.00000i − 2.30979i −7.95470 3.22522 + 3.84024i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 309.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.b.c ✓ 12
5.b even 2 1 inner 385.2.b.c ✓ 12
5.c odd 4 1 1925.2.a.y 6
5.c odd 4 1 1925.2.a.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.b.c ✓ 12 1.a even 1 1 trivial
385.2.b.c ✓ 12 5.b even 2 1 inner
1925.2.a.y 6 5.c odd 4 1
1925.2.a.z 6 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\):

\( T_{2}^{12} + 14T_{2}^{10} + 71T_{2}^{8} + 156T_{2}^{6} + 135T_{2}^{4} + 26T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{12} + 80T_{13}^{10} + 1820T_{13}^{8} + 17496T_{13}^{6} + 78432T_{13}^{4} + 157508T_{13}^{2} + 114244 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 14 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + \cdots + 196 \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( (T + 1)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 80 T^{10} + \cdots + 114244 \) Copy content Toggle raw display
$17$ \( T^{12} + 84 T^{10} + \cdots + 1444 \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} - 38 T^{4} + \cdots + 72)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 120 T^{10} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( (T^{6} - 8 T^{5} - 48 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 4 T^{5} + \cdots + 794)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1447345936 \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} - 104 T^{4} + \cdots + 38)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 415996816 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 8793563076 \) Copy content Toggle raw display
$53$ \( T^{12} + 188 T^{10} + \cdots + 767376 \) Copy content Toggle raw display
$59$ \( (T^{6} + 14 T^{5} + \cdots - 69814)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 12 T^{5} + \cdots + 1966)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 4441422736 \) Copy content Toggle raw display
$71$ \( (T^{6} - 4 T^{5} + \cdots - 11728)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2492527343076 \) Copy content Toggle raw display
$79$ \( (T^{6} - 8 T^{5} + \cdots + 552316)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 256 T^{10} + \cdots + 1577536 \) Copy content Toggle raw display
$89$ \( (T^{6} + 8 T^{5} + \cdots - 1097208)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 55270129216 \) Copy content Toggle raw display
show more
show less