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Properties

Label 155.2.b.b
Level $155$
Weight $2$
Character orbit 155.b
Analytic conductor $1.238$
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] = [155,2,Mod(94,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("155.94");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.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: \(1.23768123133\)
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: \( x^{10} + 16x^{8} + 88x^{6} + 183x^{4} + 92x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{5} - \beta_{3} - 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{9} + \beta_{7} - \beta_{5} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{9} + \beta_{6} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{5} - \beta_{3} - 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{9} + \beta_{7} - \beta_{5} + \cdots + 1) q^{6}+ \cdots + ( - 4 \beta_{9} + 2 \beta_{7} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 12 q^{4} + 8 q^{6} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 12 q^{4} + 8 q^{6} - 22 q^{9} - 7 q^{10} + 16 q^{11} + 6 q^{14} - 10 q^{15} + 8 q^{16} + 4 q^{19} - 21 q^{20} + 20 q^{21} + 8 q^{24} - 4 q^{25} + 28 q^{26} - 32 q^{29} - 42 q^{30} - 10 q^{31} + 28 q^{34} - 4 q^{35} + 44 q^{36} + 16 q^{39} - 2 q^{40} + 36 q^{41} - 52 q^{44} + 8 q^{46} - 22 q^{49} - 29 q^{50} + 20 q^{51} - 8 q^{55} + 12 q^{56} - 12 q^{59} - 28 q^{60} + 70 q^{64} + 2 q^{65} - 29 q^{70} + 12 q^{71} - 28 q^{74} + 42 q^{75} - 30 q^{76} - 28 q^{79} + 25 q^{80} - 14 q^{81} + 40 q^{84} + 22 q^{85} + 36 q^{86} - 48 q^{89} - 37 q^{90} - 4 q^{91} + 48 q^{94} + 12 q^{95} - 60 q^{96} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 16x^{8} + 88x^{6} + 183x^{4} + 92x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 11\nu^{5} + \nu^{4} + 34\nu^{3} + 5\nu^{2} + 22\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 11\nu^{5} + \nu^{4} - 34\nu^{3} + 5\nu^{2} - 22\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 11\nu^{5} + \nu^{4} + 34\nu^{3} + 7\nu^{2} + 22\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 10\nu^{5} - 9\nu^{4} - 27\nu^{3} - 20\nu^{2} - 12\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 10\nu^{5} + 9\nu^{4} - 27\nu^{3} + 20\nu^{2} - 12\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 14\nu^{7} + 68\nu^{5} + 127\nu^{3} + 54\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{8} - \nu^{7} + 11\nu^{6} - 10\nu^{5} + 35\nu^{4} - 27\nu^{3} + 27\nu^{2} - 12\nu + 2 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + \beta_{4} + 6\beta_{3} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - 7\beta_{2} + 25\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} - \beta_{6} + 25\beta_{5} - 9\beta_{4} - 34\beta_{3} - 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{7} - 11\beta_{6} + 10\beta_{4} - 10\beta_{3} + 43\beta_{2} - 127\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{9} - 12\beta_{7} + 10\beta_{6} - 127\beta_{5} + 64\beta_{4} + 191\beta_{3} + 423 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4\beta_{8} + 86\beta_{7} + 86\beta_{6} - 72\beta_{4} + 72\beta_{3} - 253\beta_{2} + 659\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(96\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1
− 2.40287i
− 2.32578i
− 2.02998i
− 0.805123i
− 0.218967i
0.218967i
0.805123i
2.02998i
2.32578i
2.40287i
− 2.40287i − 2.55247i −3.77377 2.23377 + 0.101341i −6.13324 1.03502i 4.26212i −3.51510 0.243510 − 5.36745i
94.2 − 2.32578i 1.74020i −3.40926 1.10691 − 1.94287i 4.04732 − 3.02582i 3.27764i −0.0282830 −4.51870 − 2.57443i
94.3 − 2.02998i 3.01138i −2.12080 −1.81150 + 1.31090i 6.11304 3.60704i 0.245225i −6.06843 2.66110 + 3.67730i
94.4 − 0.805123i 0.681921i 1.35178 −1.41046 − 1.73511i 0.549030 − 0.986124i − 2.69859i 2.53498 −1.39698 + 1.13559i
94.5 − 0.218967i − 2.63119i 1.95205 −0.118717 − 2.23291i −0.576144 4.66797i − 0.865369i −3.92317 −0.488934 + 0.0259950i
94.6 0.218967i 2.63119i 1.95205 −0.118717 + 2.23291i −0.576144 − 4.66797i 0.865369i −3.92317 −0.488934 − 0.0259950i
94.7 0.805123i − 0.681921i 1.35178 −1.41046 + 1.73511i 0.549030 0.986124i 2.69859i 2.53498 −1.39698 − 1.13559i
94.8 2.02998i − 3.01138i −2.12080 −1.81150 − 1.31090i 6.11304 − 3.60704i − 0.245225i −6.06843 2.66110 − 3.67730i
94.9 2.32578i − 1.74020i −3.40926 1.10691 + 1.94287i 4.04732 3.02582i − 3.27764i −0.0282830 −4.51870 + 2.57443i
94.10 2.40287i 2.55247i −3.77377 2.23377 − 0.101341i −6.13324 − 1.03502i − 4.26212i −3.51510 0.243510 + 5.36745i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.10
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 155.2.b.b ✓ 10
3.b odd 2 1 1395.2.c.e 10
4.b odd 2 1 2480.2.d.g 10
5.b even 2 1 inner 155.2.b.b ✓ 10
5.c odd 4 2 775.2.a.l 10
15.d odd 2 1 1395.2.c.e 10
15.e even 4 2 6975.2.a.ch 10
20.d odd 2 1 2480.2.d.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.b.b ✓ 10 1.a even 1 1 trivial
155.2.b.b ✓ 10 5.b even 2 1 inner
775.2.a.l 10 5.c odd 4 2
1395.2.c.e 10 3.b odd 2 1
1395.2.c.e 10 15.d odd 2 1
2480.2.d.g 10 4.b odd 2 1
2480.2.d.g 10 20.d odd 2 1
6975.2.a.ch 10 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 16T_{2}^{8} + 88T_{2}^{6} + 183T_{2}^{4} + 92T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 16 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{10} + 26 T^{8} + \cdots + 576 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{8} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 46 T^{8} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( (T^{5} - 8 T^{4} + \cdots - 288)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 56 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{10} + 58 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( (T^{5} - 2 T^{4} + \cdots - 2404)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 108 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( (T^{5} + 16 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{10} \) Copy content Toggle raw display
$37$ \( T^{10} + 206 T^{8} + \cdots + 7311616 \) Copy content Toggle raw display
$41$ \( (T^{5} - 18 T^{4} + \cdots + 438)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 599662144 \) Copy content Toggle raw display
$47$ \( T^{10} + 156 T^{8} + \cdots + 173056 \) Copy content Toggle raw display
$53$ \( T^{10} + 298 T^{8} + \cdots + 84934656 \) Copy content Toggle raw display
$59$ \( (T^{5} + 6 T^{4} + \cdots - 41204)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 124 T^{3} + \cdots - 3488)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 192 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$71$ \( (T^{5} - 6 T^{4} + \cdots + 5944)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 338 T^{8} + \cdots + 8856576 \) Copy content Toggle raw display
$79$ \( (T^{5} + 14 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 490 T^{8} + \cdots + 760384 \) Copy content Toggle raw display
$89$ \( (T^{5} + 24 T^{4} + \cdots - 2704)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 478 T^{8} + \cdots + 8667136 \) Copy content Toggle raw display
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