↑

Properties

Label 5775.2.a.cn
Level $5775$
Weight $2$
Character orbit 5775.a
Self dual yes
Analytic conductor $46.114$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5775,2,Mod(1,5775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("5775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.a (trivial)

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: yes
Analytic conductor: \(46.1136071673\)
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: \( x^{10} - 3x^{9} - 12x^{8} + 38x^{7} + 42x^{6} - 152x^{5} - 31x^{4} + 197x^{3} - 16x^{2} - 58x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8} + q^{9} + q^{11} + ( - \beta_{2} - 1) q^{12} - \beta_{7} q^{13} - \beta_1 q^{14} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{16}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 10 q^{3} + 13 q^{4} + 3 q^{6} + 10 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} - 10 q^{3} + 13 q^{4} + 3 q^{6} + 10 q^{7} - 9 q^{8} + 10 q^{9} + 10 q^{11} - 13 q^{12} - 3 q^{14} + 19 q^{16} - 3 q^{17} - 3 q^{18} + 17 q^{19} - 10 q^{21} - 3 q^{22} - 9 q^{23} + 9 q^{24} + 14 q^{26} - 10 q^{27} + 13 q^{28} + 5 q^{29} + 6 q^{31} - 31 q^{32} - 10 q^{33} + 16 q^{34} + 13 q^{36} + 14 q^{38} + 26 q^{41} + 3 q^{42} + 3 q^{43} + 13 q^{44} + 20 q^{46} + 6 q^{47} - 19 q^{48} + 10 q^{49} + 3 q^{51} - 19 q^{53} + 3 q^{54} - 9 q^{56} - 17 q^{57} + 26 q^{58} + 7 q^{59} + 39 q^{61} - 30 q^{62} + 10 q^{63} + 13 q^{64} + 3 q^{66} - 2 q^{67} - 8 q^{68} + 9 q^{69} - 9 q^{72} - 8 q^{73} + 4 q^{74} + 42 q^{76} + 10 q^{77} - 14 q^{78} + 26 q^{79} + 10 q^{81} + 12 q^{82} + 3 q^{83} - 13 q^{84} + 2 q^{86} - 5 q^{87} - 9 q^{88} + 5 q^{89} - 64 q^{92} - 6 q^{93} + 48 q^{94} + 31 q^{96} + 35 q^{97} - 3 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 12x^{8} + 38x^{7} + 42x^{6} - 152x^{5} - 31x^{4} + 197x^{3} - 16x^{2} - 58x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{9} + 9\nu^{8} + 20\nu^{7} - 106\nu^{6} - 58\nu^{5} + 391\nu^{4} + 36\nu^{3} - 448\nu^{2} + 20\nu + 48 ) / 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} - 4\nu^{8} - 49\nu^{7} + 64\nu^{6} + 239\nu^{5} - 273\nu^{4} - 358\nu^{3} + 254\nu^{2} + 122\nu - 15 ) / 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{9} + 4\nu^{8} + 49\nu^{7} - 64\nu^{6} - 239\nu^{5} + 292\nu^{4} + 339\nu^{3} - 368\nu^{2} - 46\nu + 72 ) / 19 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{9} - 4\nu^{8} - 49\nu^{7} + 64\nu^{6} + 258\nu^{5} - 311\nu^{4} - 491\nu^{3} + 463\nu^{2} + 293\nu - 110 ) / 19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2\nu^{9} - 9\nu^{8} - 20\nu^{7} + 125\nu^{6} + 20\nu^{5} - 543\nu^{4} + 211\nu^{3} + 733\nu^{2} - 267\nu - 162 ) / 19 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8\nu^{9} - 17\nu^{8} - 99\nu^{7} + 196\nu^{6} + 384\nu^{5} - 690\nu^{4} - 467\nu^{3} + 709\nu^{2} + 110\nu - 97 ) / 19 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 2\beta_{6} + \beta_{5} + 9\beta_{3} + 10\beta_{2} + 28\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} + 2\beta_{7} + 12\beta_{6} + 10\beta_{5} + \beta_{4} + 13\beta_{3} + 48\beta_{2} + 12\beta _1 + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} + 2 \beta_{8} + 12 \beta_{7} + 27 \beta_{6} + 13 \beta_{5} + 3 \beta_{4} + 70 \beta_{3} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{9} + 14 \beta_{8} + 28 \beta_{7} + 109 \beta_{6} + 79 \beta_{5} + 19 \beta_{4} + 125 \beta_{3} + \cdots + 511 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 19 \beta_{9} + 30 \beta_{8} + 111 \beta_{7} + 262 \beta_{6} + 122 \beta_{5} + 53 \beta_{4} + 526 \beta_{3} + \cdots + 331 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.74150
2.55795
1.93467
1.40171
0.792050
0.0342944
−0.568210
−1.28094
−2.28323
−2.32980
−2.74150 −1.00000 5.51584 0 2.74150 1.00000 −9.63869 1.00000 0
1.2 −2.55795 −1.00000 4.54312 0 2.55795 1.00000 −6.50518 1.00000 0
1.3 −1.93467 −1.00000 1.74296 0 1.93467 1.00000 0.497289 1.00000 0
1.4 −1.40171 −1.00000 −0.0352144 0 1.40171 1.00000 2.85278 1.00000 0
1.5 −0.792050 −1.00000 −1.37266 0 0.792050 1.00000 2.67131 1.00000 0
1.6 −0.0342944 −1.00000 −1.99882 0 0.0342944 1.00000 0.137137 1.00000 0
1.7 0.568210 −1.00000 −1.67714 0 −0.568210 1.00000 −2.08939 1.00000 0
1.8 1.28094 −1.00000 −0.359182 0 −1.28094 1.00000 −3.02198 1.00000 0
1.9 2.28323 −1.00000 3.21315 0 −2.28323 1.00000 2.76990 1.00000 0
1.10 2.32980 −1.00000 3.42795 0 −2.32980 1.00000 3.32682 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5775.2.a.cn 10
5.b even 2 1 5775.2.a.co 10
5.c odd 4 2 1155.2.c.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.f ✓ 20 5.c odd 4 2
5775.2.a.cn 10 1.a even 1 1 trivial
5775.2.a.co 10 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5775))\):

\( T_{2}^{10} + 3T_{2}^{9} - 12T_{2}^{8} - 38T_{2}^{7} + 42T_{2}^{6} + 152T_{2}^{5} - 31T_{2}^{4} - 197T_{2}^{3} - 16T_{2}^{2} + 58T_{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{10} - 76 T_{13}^{8} + 46 T_{13}^{7} + 1813 T_{13}^{6} - 1168 T_{13}^{5} - 16330 T_{13}^{4} + \cdots + 7648 \) Copy content Toggle raw display
\( T_{17}^{10} + 3 T_{17}^{9} - 91 T_{17}^{8} - 247 T_{17}^{7} + 2706 T_{17}^{6} + 6048 T_{17}^{5} + \cdots + 13184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 3 T^{9} + \cdots + 2 \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( (T - 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 76 T^{8} + \cdots + 7648 \) Copy content Toggle raw display
$17$ \( T^{10} + 3 T^{9} + \cdots + 13184 \) Copy content Toggle raw display
$19$ \( T^{10} - 17 T^{9} + \cdots - 1905888 \) Copy content Toggle raw display
$23$ \( T^{10} + 9 T^{9} + \cdots - 1886976 \) Copy content Toggle raw display
$29$ \( T^{10} - 5 T^{9} + \cdots - 2190256 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 117243392 \) Copy content Toggle raw display
$37$ \( T^{10} - 232 T^{8} + \cdots - 7297792 \) Copy content Toggle raw display
$41$ \( T^{10} - 26 T^{9} + \cdots - 6464512 \) Copy content Toggle raw display
$43$ \( T^{10} - 3 T^{9} + \cdots + 10135552 \) Copy content Toggle raw display
$47$ \( T^{10} - 6 T^{9} + \cdots + 152064 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 125348736 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 104226256 \) Copy content Toggle raw display
$61$ \( T^{10} - 39 T^{9} + \cdots - 5992448 \) Copy content Toggle raw display
$67$ \( T^{10} + 2 T^{9} + \cdots - 1974784 \) Copy content Toggle raw display
$71$ \( T^{10} - 434 T^{8} + \cdots - 851968 \) Copy content Toggle raw display
$73$ \( T^{10} + 8 T^{9} + \cdots + 58368 \) Copy content Toggle raw display
$79$ \( T^{10} - 26 T^{9} + \cdots + 40370176 \) Copy content Toggle raw display
$83$ \( T^{10} - 3 T^{9} + \cdots + 56626688 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 928874496 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 210510464 \) Copy content Toggle raw display
show more
show less