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Properties

Label 255.2.b.c
Level $255$
Weight $2$
Character orbit 255.b
Analytic conductor $2.036$
Analytic rank $0$
Dimension $10$
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] = [255,2,Mod(154,255)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(255, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("255.154");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 255 = 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 255.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: \(2.03618525154\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 20x^{8} + 140x^{6} + 393x^{4} + 352x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 1) q^{4}+ \cdots - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 1) q^{4}+ \cdots + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 20 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 20 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{9} - 2 q^{10} - 8 q^{11} - 22 q^{14} + 40 q^{16} + 12 q^{19} - 14 q^{20} + 4 q^{21} - 18 q^{24} - 4 q^{25} + 28 q^{26} - 32 q^{29} - 14 q^{30} + 16 q^{31} - 4 q^{34} - 12 q^{35} + 20 q^{36} + 8 q^{39} - 50 q^{40} + 48 q^{41} + 66 q^{44} - 4 q^{45} + 12 q^{46} - 14 q^{49} - 44 q^{50} + 10 q^{51} - 4 q^{54} + 18 q^{55} + 24 q^{56} - 8 q^{59} + 10 q^{60} + 28 q^{61} - 38 q^{64} - 8 q^{65} - 14 q^{66} - 20 q^{69} - 54 q^{70} + 4 q^{71} + 74 q^{74} + 8 q^{75} - 6 q^{76} - 8 q^{79} - 42 q^{80} + 10 q^{81} + 54 q^{84} - 20 q^{89} + 2 q^{90} + 40 q^{91} + 94 q^{94} - 20 q^{95} + 40 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 20x^{8} + 140x^{6} + 393x^{4} + 352x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{8} - 44\nu^{6} - 172\nu^{4} - 115\nu^{2} - 56 ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7 \nu^{9} - 12 \nu^{8} + 116 \nu^{7} - 176 \nu^{6} + 628 \nu^{5} - 848 \nu^{4} + 1375 \nu^{3} + \cdots - 704 ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{9} + 12 \nu^{8} + 116 \nu^{7} + 176 \nu^{6} + 628 \nu^{5} + 848 \nu^{4} + 1375 \nu^{3} + \cdots + 704 ) / 320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{8} - 44\nu^{6} - 192\nu^{4} - 235\nu^{2} - 56 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{9} - 116\nu^{7} - 628\nu^{5} - 1215\nu^{3} - 584\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} - 28\nu^{7} - 244\nu^{5} - 745\nu^{3} - 472\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11 \nu^{9} + 16 \nu^{8} - 148 \nu^{7} + 288 \nu^{6} - 484 \nu^{5} + 1664 \nu^{4} + 125 \nu^{3} + \cdots + 1312 ) / 320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11 \nu^{9} + 16 \nu^{8} + 148 \nu^{7} + 288 \nu^{6} + 484 \nu^{5} + 1664 \nu^{4} - 125 \nu^{3} + \cdots + 1312 ) / 320 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 6\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{9} + 4\beta_{8} + 3\beta_{7} - 16\beta_{6} - 8\beta_{4} - 8\beta_{3} + 34\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{9} + 3\beta_{8} + 48\beta_{5} + 34\beta_{4} - 34\beta_{3} - 58\beta_{2} - 121 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 54\beta_{9} - 54\beta_{8} - 44\beta_{7} + 168\beta_{6} + 58\beta_{4} + 58\beta_{3} - 193\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -44\beta_{9} - 44\beta_{8} - 341\beta_{5} - 193\beta_{4} + 193\beta_{3} + 417\beta_{2} + 839 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -536\beta_{9} + 536\beta_{8} + 460\beta_{7} - 1545\beta_{6} - 417\beta_{4} - 417\beta_{3} + 1106\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/255\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(86\) \(241\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
154.1
− 2.75765i
− 2.42928i
− 2.25682i
− 1.07666i
− 0.491470i
0.491470i
1.07666i
2.25682i
2.42928i
2.75765i
− 2.75765i 1.00000i −5.60465 −1.43779 + 1.71253i 2.75765 0.846996i 9.94037i −1.00000 4.72257 + 3.96492i
154.2 − 2.42928i − 1.00000i −3.90138 2.20351 + 0.380215i −2.42928 − 4.33066i 4.61898i −1.00000 0.923647 − 5.35292i
154.3 − 2.25682i 1.00000i −3.09322 1.63701 − 1.52322i 2.25682 − 1.16360i 2.46720i −1.00000 −3.43762 − 3.69444i
154.4 − 1.07666i − 1.00000i 0.840795 0.751636 − 2.10595i −1.07666 1.76413i − 3.05858i −1.00000 −2.26740 − 0.809259i
154.5 − 0.491470i 1.00000i 1.75846 −1.15437 − 1.91506i 0.491470 − 4.24993i − 1.84717i −1.00000 −0.941193 + 0.567337i
154.6 0.491470i − 1.00000i 1.75846 −1.15437 + 1.91506i 0.491470 4.24993i 1.84717i −1.00000 −0.941193 − 0.567337i
154.7 1.07666i 1.00000i 0.840795 0.751636 + 2.10595i −1.07666 − 1.76413i 3.05858i −1.00000 −2.26740 + 0.809259i
154.8 2.25682i − 1.00000i −3.09322 1.63701 + 1.52322i 2.25682 1.16360i − 2.46720i −1.00000 −3.43762 + 3.69444i
154.9 2.42928i 1.00000i −3.90138 2.20351 − 0.380215i −2.42928 4.33066i − 4.61898i −1.00000 0.923647 + 5.35292i
154.10 2.75765i − 1.00000i −5.60465 −1.43779 − 1.71253i 2.75765 − 0.846996i − 9.94037i −1.00000 4.72257 − 3.96492i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 154.10
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 255.2.b.c ✓ 10
3.b odd 2 1 765.2.b.d 10
4.b odd 2 1 4080.2.m.s 10
5.b even 2 1 inner 255.2.b.c ✓ 10
5.c odd 4 1 1275.2.a.u 5
5.c odd 4 1 1275.2.a.v 5
15.d odd 2 1 765.2.b.d 10
15.e even 4 1 3825.2.a.bk 5
15.e even 4 1 3825.2.a.br 5
20.d odd 2 1 4080.2.m.s 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.b.c ✓ 10 1.a even 1 1 trivial
255.2.b.c ✓ 10 5.b even 2 1 inner
765.2.b.d 10 3.b odd 2 1
765.2.b.d 10 15.d odd 2 1
1275.2.a.u 5 5.c odd 4 1
1275.2.a.v 5 5.c odd 4 1
3825.2.a.bk 5 15.e even 4 1
3825.2.a.br 5 15.e even 4 1
4080.2.m.s 10 4.b odd 2 1
4080.2.m.s 10 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 20T_{2}^{8} + 140T_{2}^{6} + 393T_{2}^{4} + 352T_{2}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(255, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 20 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 42 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{5} + 4 T^{4} - 14 T^{3} + \cdots + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 46 T^{8} + \cdots + 20736 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$19$ \( (T^{5} - 6 T^{4} + \cdots - 900)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 74 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{5} + 16 T^{4} + \cdots - 5120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} + \cdots - 4608)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 174 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( (T^{5} - 24 T^{4} + \cdots + 324)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 210 T^{8} + \cdots + 46656 \) Copy content Toggle raw display
$47$ \( T^{10} + 162 T^{8} + \cdots + 719104 \) Copy content Toggle raw display
$53$ \( T^{10} + 54 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( (T^{5} + 4 T^{4} + \cdots + 2560)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 14 T^{4} + \cdots + 3008)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 187580416 \) Copy content Toggle raw display
$71$ \( (T^{5} - 2 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 158 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( (T^{5} + 4 T^{4} + \cdots - 40640)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11781799936 \) Copy content Toggle raw display
$89$ \( (T^{5} + 10 T^{4} + \cdots + 162560)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 748 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
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