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Properties

Label 5175.2.a.cj
Level $5175$
Weight $2$
Character orbit 5175.a
Self dual yes
Analytic conductor $41.323$
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] = [5175,2,Mod(1,5175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5175.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: \(41.3225830460\)
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: \( x^{10} - 25x^{8} - 6x^{7} + 229x^{6} + 100x^{5} - 917x^{4} - 510x^{3} + 1466x^{2} + 736x - 642 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1035)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{7} - \beta_{6}) q^{4} + \beta_{2} q^{7} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{7} - \beta_{6}) q^{4} + \beta_{2} q^{7} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{8}+ \cdots + ( - \beta_{7} - 3 \beta_{6} - \beta_{5} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 8 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} + 8 q^{4} + 12 q^{8} + 20 q^{16} + 30 q^{17} - 4 q^{19} - 10 q^{23} - 10 q^{31} + 48 q^{32} + 20 q^{34} + 24 q^{38} - 4 q^{46} + 24 q^{47} + 12 q^{49} + 30 q^{53} + 20 q^{61} + 32 q^{62} + 28 q^{64} + 44 q^{68} - 16 q^{76} + 44 q^{77} - 4 q^{79} + 18 q^{83} - 16 q^{91} - 8 q^{92} - 32 q^{94} + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 25x^{8} - 6x^{7} + 229x^{6} + 100x^{5} - 917x^{4} - 510x^{3} + 1466x^{2} + 736x - 642 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3 \nu^{9} + 26 \nu^{8} - 68 \nu^{7} - 520 \nu^{6} + 547 \nu^{5} + 3294 \nu^{4} - 1844 \nu^{3} + \cdots + 2812 ) / 524 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23 \nu^{9} - 19 \nu^{8} - 696 \nu^{7} + 642 \nu^{6} + 6901 \nu^{5} - 5793 \nu^{4} - 26626 \nu^{3} + \cdots - 20798 ) / 1048 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 79 \nu^{9} + 145 \nu^{8} + 1616 \nu^{7} - 2638 \nu^{6} - 11697 \nu^{5} + 16355 \nu^{4} + \cdots + 22978 ) / 1048 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 109 \nu^{9} - 147 \nu^{8} - 2296 \nu^{7} + 2678 \nu^{6} + 16643 \nu^{5} - 16165 \nu^{4} + \cdots - 27870 ) / 1048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 71 \nu^{9} - 127 \nu^{8} - 1522 \nu^{7} + 2278 \nu^{6} + 11723 \nu^{5} - 13873 \nu^{4} - 37790 \nu^{3} + \cdots - 26634 ) / 524 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 95 \nu^{9} + 181 \nu^{8} + 2066 \nu^{7} - 3358 \nu^{6} - 16099 \nu^{5} + 20795 \nu^{4} + \cdots + 35054 ) / 524 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 139 \nu^{9} - 280 \nu^{8} - 2976 \nu^{7} + 5076 \nu^{6} + 22899 \nu^{5} - 30516 \nu^{4} - 73736 \nu^{3} + \cdots - 47172 ) / 524 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 83 \nu^{9} - 154 \nu^{8} - 1794 \nu^{7} + 2818 \nu^{6} + 13911 \nu^{5} - 17334 \nu^{4} - 45166 \nu^{3} + \cdots - 30844 ) / 262 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 181 \nu^{9} + 309 \nu^{8} + 3928 \nu^{7} - 5656 \nu^{6} - 30295 \nu^{5} + 34573 \nu^{4} + \cdots + 58894 ) / 524 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{4} + \beta_{3} + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + 7\beta_{8} + 2\beta_{7} + 10\beta_{6} - 6\beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{9} + 5\beta_{8} + \beta_{7} - 2\beta_{5} + 5\beta_{4} + 5\beta_{3} + \beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{9} + 27 \beta_{8} + 12 \beta_{7} + 40 \beta_{6} - 22 \beta_{5} + 6 \beta_{4} - 5 \beta_{3} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 49\beta_{9} + 47\beta_{8} + 15\beta_{7} - 30\beta_{5} + 47\beta_{4} + 39\beta_{3} + 4\beta_{2} + 15\beta _1 + 245 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 103 \beta_{9} + 223 \beta_{8} + 118 \beta_{7} + 303 \beta_{6} - 179 \beta_{5} + 69 \beta_{4} + \cdots + 367 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 480 \beta_{9} + 440 \beta_{8} + 173 \beta_{7} + 2 \beta_{6} - 344 \beta_{5} + 440 \beta_{4} + \cdots + 1941 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1184 \beta_{9} + 1916 \beta_{8} + 1104 \beta_{7} + 2263 \beta_{6} - 1545 \beta_{5} + 778 \beta_{4} + \cdots + 3463 \) 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
2.54115
−1.88216
−1.50036
2.78369
0.515417
3.11999
−2.61509
−1.77800
1.56388
−2.74851
−2.03496 0 2.14106 0 0 −2.85626 −0.287051 0 0
1.2 −2.03496 0 2.14106 0 0 2.85626 −0.287051 0 0
1.3 −0.558450 0 −1.68813 0 0 −1.31549 2.05964 0 0
1.4 −0.558450 0 −1.68813 0 0 1.31549 2.05964 0 0
1.5 0.449855 0 −1.79763 0 0 −3.10616 −1.70838 0 0
1.6 0.449855 0 −1.79763 0 0 3.10616 −1.70838 0 0
1.7 1.45526 0 0.117782 0 0 −2.91412 −2.73912 0 0
1.8 1.45526 0 0.117782 0 0 2.91412 −2.73912 0 0
1.9 2.68829 0 5.22692 0 0 −3.60151 8.67492 0 0
1.10 2.68829 0 5.22692 0 0 3.60151 8.67492 0 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
\(3\) \(1\)
\(5\) \(-1\)
\(23\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5175.2.a.cj 10
3.b odd 2 1 5175.2.a.ci 10
5.b even 2 1 5175.2.a.ci 10
5.c odd 4 2 1035.2.b.g ✓ 20
15.d odd 2 1 inner 5175.2.a.cj 10
15.e even 4 2 1035.2.b.g ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1035.2.b.g ✓ 20 5.c odd 4 2
1035.2.b.g ✓ 20 15.e even 4 2
5175.2.a.ci 10 3.b odd 2 1
5175.2.a.ci 10 5.b even 2 1
5175.2.a.cj 10 1.a even 1 1 trivial
5175.2.a.cj 10 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5175))\):

\( T_{2}^{5} - 2T_{2}^{4} - 5T_{2}^{3} + 8T_{2}^{2} + 2T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{10} - 41T_{7}^{8} + 639T_{7}^{6} - 4639T_{7}^{4} + 14988T_{7}^{2} - 15004 \) Copy content Toggle raw display
\( T_{11}^{10} - 60T_{11}^{8} + 1228T_{11}^{6} - 11276T_{11}^{4} + 47104T_{11}^{2} - 71424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} - 2 T^{4} - 5 T^{3} + \cdots - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 41 T^{8} + \cdots - 15004 \) Copy content Toggle raw display
$11$ \( T^{10} - 60 T^{8} + \cdots - 71424 \) Copy content Toggle raw display
$13$ \( T^{10} - 68 T^{8} + \cdots - 1984 \) Copy content Toggle raw display
$17$ \( (T^{5} - 15 T^{4} + \cdots + 178)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 2 T^{4} + \cdots - 524)^{2} \) Copy content Toggle raw display
$23$ \( (T + 1)^{10} \) Copy content Toggle raw display
$29$ \( T^{10} - 157 T^{8} + \cdots - 23571904 \) Copy content Toggle raw display
$31$ \( (T^{5} + 5 T^{4} - 17 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 209 T^{8} + \cdots - 11011696 \) Copy content Toggle raw display
$41$ \( T^{10} - 289 T^{8} + \cdots - 7936 \) Copy content Toggle raw display
$43$ \( T^{10} - 244 T^{8} + \cdots - 44046784 \) Copy content Toggle raw display
$47$ \( (T^{5} - 12 T^{4} + \cdots - 1864)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} - 15 T^{4} + \cdots + 7286)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 397 T^{8} + \cdots - 3017664 \) Copy content Toggle raw display
$61$ \( (T^{5} - 10 T^{4} + \cdots - 24)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} - 129 T^{8} + \cdots - 124 \) Copy content Toggle raw display
$71$ \( T^{10} - 361 T^{8} + \cdots - 26927344 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 843406336 \) Copy content Toggle raw display
$79$ \( (T^{5} + 2 T^{4} + \cdots + 7232)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 9 T^{4} + \cdots - 60408)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 1487912704 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 524164864 \) Copy content Toggle raw display
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