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Properties

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

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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: \( x^{10} - 3x^{9} - 12x^{8} + 38x^{7} + 42x^{6} - 152x^{5} - 31x^{4} + 205x^{3} - 48x^{2} - 34x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} - q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8} + q^{9} - q^{11} + ( - \beta_{2} - 1) q^{12} + (\beta_{8} - \beta_{5} - \beta_{3}) q^{13} + \beta_1 q^{14} + (\beta_{9} + \beta_{8} + \cdots + 2 \beta_1) q^{16}+ \cdots - q^{99}+O(q^{100}) \) 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 \( 10 q - 3 q^{2} - 10 q^{3} + 13 q^{4} + 3 q^{6} - 10 q^{7} - 9 q^{8} + 10 q^{9} - 10 q^{11} - 13 q^{12} + 3 q^{14} + 19 q^{16} - 13 q^{17} - 3 q^{18} - q^{19} + 10 q^{21} + 3 q^{22} - 13 q^{23} + 9 q^{24} + 10 q^{26} - 10 q^{27} - 13 q^{28} + 19 q^{29} + 10 q^{31} - 31 q^{32} + 10 q^{33} + 16 q^{34} + 13 q^{36} - 14 q^{38} + 6 q^{41} - 3 q^{42} + 13 q^{43} - 13 q^{44} + 4 q^{46} - 34 q^{47} - 19 q^{48} + 10 q^{49} + 13 q^{51} + 16 q^{52} - 11 q^{53} + 3 q^{54} + 9 q^{56} + q^{57} + 38 q^{58} + 11 q^{59} + 17 q^{61} + 6 q^{62} - 10 q^{63} + 13 q^{64} - 3 q^{66} - 6 q^{67} - 36 q^{68} + 13 q^{69} + 36 q^{71} - 9 q^{72} + 8 q^{73} + 36 q^{74} + 22 q^{76} + 10 q^{77} - 10 q^{78} - 2 q^{79} + 10 q^{81} - 4 q^{82} - 51 q^{83} + 13 q^{84} + 26 q^{86} - 19 q^{87} + 9 q^{88} - 3 q^{89} - 8 q^{92} - 10 q^{93} - 8 q^{94} + 31 q^{96} + 3 q^{97} - 3 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 12x^{8} + 38x^{7} + 42x^{6} - 152x^{5} - 31x^{4} + 205x^{3} - 48x^{2} - 34x + 10 \) : 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
\(\beta_{4}\)\(=\) \( \nu^{7} - 2\nu^{6} - 10\nu^{5} + 19\nu^{4} + 24\nu^{3} - 42\nu^{2} - 7\nu + 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 4\nu^{7} - 6\nu^{6} + 38\nu^{5} - 13\nu^{4} - 82\nu^{3} + 72\nu^{2} + 8\nu - 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{9} - 4\nu^{8} - 7\nu^{7} + 40\nu^{6} - 3\nu^{5} - 100\nu^{4} + 47\nu^{3} + 44\nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{9} - 4\nu^{8} - 7\nu^{7} + 41\nu^{6} - 4\nu^{5} - 110\nu^{4} + 56\nu^{3} + 67\nu^{2} - 20\nu - 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\nu^{9} + 3\nu^{8} + 11\nu^{7} - 34\nu^{6} - 36\nu^{5} + 114\nu^{4} + 44\nu^{3} - 122\nu^{2} - 24\nu + 21 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\nu^{8} - 9\nu^{7} - 10\nu^{6} + 87\nu^{5} - 45\nu^{4} - 197\nu^{3} + 185\nu^{2} + 39\nu - 30 \) 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
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} + 7\beta_{2} + 2\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - 2\beta_{5} + \beta_{4} + 9\beta_{3} + 10\beta_{2} + 29\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{9} + 10\beta_{8} + \beta_{7} + 9\beta_{6} - 12\beta_{5} + 11\beta_{4} + 48\beta_{2} + 22\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} - 25 \beta_{5} + 14 \beta_{4} + 66 \beta_{3} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 93 \beta_{9} + 77 \beta_{8} + 14 \beta_{7} + 63 \beta_{6} - 108 \beta_{5} + 97 \beta_{4} + 4 \beta_{3} + \cdots + 438 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 126 \beta_{9} + 15 \beta_{8} + 30 \beta_{7} - 14 \beta_{6} - 233 \beta_{5} + 149 \beta_{4} + 458 \beta_{3} + \cdots + 111 \) 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
2.74619
2.51841
2.02181
1.41336
0.445112
0.332438
−0.437361
−1.76011
−1.76958
−2.51027
−2.74619 −1.00000 5.54154 0 2.74619 −1.00000 −9.72574 1.00000 0
1.2 −2.51841 −1.00000 4.34238 0 2.51841 −1.00000 −5.89907 1.00000 0
1.3 −2.02181 −1.00000 2.08773 0 2.02181 −1.00000 −0.177380 1.00000 0
1.4 −1.41336 −1.00000 −0.00241967 0 1.41336 −1.00000 2.83014 1.00000 0
1.5 −0.445112 −1.00000 −1.80187 0 0.445112 −1.00000 1.69226 1.00000 0
1.6 −0.332438 −1.00000 −1.88948 0 0.332438 −1.00000 1.29301 1.00000 0
1.7 0.437361 −1.00000 −1.80872 0 −0.437361 −1.00000 −1.66578 1.00000 0
1.8 1.76011 −1.00000 1.09799 0 −1.76011 −1.00000 −1.58764 1.00000 0
1.9 1.76958 −1.00000 1.13141 0 −1.76958 −1.00000 −1.53703 1.00000 0
1.10 2.51027 −1.00000 4.30144 0 −2.51027 −1.00000 5.77723 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.cm 10
5.b even 2 1 5775.2.a.cp 10
5.c odd 4 2 1155.2.c.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.e ✓ 20 5.c odd 4 2
5775.2.a.cm 10 1.a even 1 1 trivial
5775.2.a.cp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5775))\):

\( T_{2}^{10} + 3T_{2}^{9} - 12T_{2}^{8} - 38T_{2}^{7} + 42T_{2}^{6} + 152T_{2}^{5} - 31T_{2}^{4} - 205T_{2}^{3} - 48T_{2}^{2} + 34T_{2} + 10 \) Copy content Toggle raw display
\( T_{13}^{10} - 92 T_{13}^{8} - 50 T_{13}^{7} + 2749 T_{13}^{6} + 2312 T_{13}^{5} - 29290 T_{13}^{4} + \cdots - 1600 \) Copy content Toggle raw display
\( T_{17}^{10} + 13 T_{17}^{9} - 23 T_{17}^{8} - 837 T_{17}^{7} - 1614 T_{17}^{6} + 12384 T_{17}^{5} + \cdots - 151168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 3 T^{9} + \cdots + 10 \) 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
$13$ \( T^{10} - 92 T^{8} + \cdots - 1600 \) Copy content Toggle raw display
$17$ \( T^{10} + 13 T^{9} + \cdots - 151168 \) Copy content Toggle raw display
$19$ \( T^{10} + T^{9} + \cdots + 277120 \) Copy content Toggle raw display
$23$ \( T^{10} + 13 T^{9} + \cdots + 2694400 \) Copy content Toggle raw display
$29$ \( T^{10} - 19 T^{9} + \cdots + 169600 \) Copy content Toggle raw display
$31$ \( T^{10} - 10 T^{9} + \cdots + 6656 \) Copy content Toggle raw display
$37$ \( T^{10} - 160 T^{8} + \cdots - 54016 \) Copy content Toggle raw display
$41$ \( T^{10} - 6 T^{9} + \cdots + 23487488 \) Copy content Toggle raw display
$43$ \( T^{10} - 13 T^{9} + \cdots - 12904448 \) Copy content Toggle raw display
$47$ \( T^{10} + 34 T^{9} + \cdots + 22309888 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 126732416 \) Copy content Toggle raw display
$59$ \( T^{10} - 11 T^{9} + \cdots + 842144 \) Copy content Toggle raw display
$61$ \( T^{10} - 17 T^{9} + \cdots - 6926336 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 354232960 \) Copy content Toggle raw display
$71$ \( T^{10} - 36 T^{9} + \cdots + 4096000 \) Copy content Toggle raw display
$73$ \( T^{10} - 8 T^{9} + \cdots - 13293824 \) Copy content Toggle raw display
$79$ \( T^{10} + 2 T^{9} + \cdots + 4341760 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 2321474560 \) Copy content Toggle raw display
$89$ \( T^{10} + 3 T^{9} + \cdots + 14123008 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 2222343296 \) Copy content Toggle raw display
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