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Properties

Label 575.4.a.o
Level $575$
Weight $4$
Character orbit 575.a
Self dual yes
Analytic conductor $33.926$
Analytic rank $0$
Dimension $13$
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] = [575,4,Mod(1,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.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: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 3 x^{12} - 85 x^{11} + 222 x^{10} + 2763 x^{9} - 6071 x^{8} - 43370 x^{7} + 78313 x^{6} + \cdots - 1836192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 6) q^{4} + ( - \beta_{12} + \beta_{5} + \beta_{3} + \cdots + 5) q^{6} + ( - \beta_{9} - \beta_{3}) q^{7} + (\beta_{11} - \beta_{10} - \beta_{8} + \cdots - 5) q^{8}+ \cdots + (3 \beta_{12} + 23 \beta_{11} + \cdots + 338) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 3 q^{2} + 3 q^{3} + 75 q^{4} + 62 q^{6} + q^{7} - 78 q^{8} + 108 q^{9} + 118 q^{11} - 27 q^{12} + 83 q^{13} + 40 q^{14} + 411 q^{16} + 71 q^{17} - 217 q^{18} + 360 q^{19} + 434 q^{21} + 169 q^{22}+ \cdots + 4599 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 3 x^{12} - 85 x^{11} + 222 x^{10} + 2763 x^{9} - 6071 x^{8} - 43370 x^{7} + 78313 x^{6} + \cdots - 1836192 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 927008643605 \nu^{12} - 23068503726904 \nu^{11} + 154680408089001 \nu^{10} + \cdots - 77\!\cdots\!84 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9926266331284 \nu^{12} - 62494768112453 \nu^{11} - 667811947939288 \nu^{10} + \cdots - 14\!\cdots\!32 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14311662374174 \nu^{12} - 49227503230541 \nu^{11} + \cdots - 51\!\cdots\!20 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17619277385029 \nu^{12} - 88862232750707 \nu^{11} + \cdots - 37\!\cdots\!36 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19025189846323 \nu^{12} - 31779431612917 \nu^{11} + \cdots - 19\!\cdots\!80 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1208155986455 \nu^{12} - 4444771924598 \nu^{11} - 83992250465313 \nu^{10} + \cdots - 13\!\cdots\!88 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23474222377227 \nu^{12} - 151030031338655 \nu^{11} + \cdots - 99\!\cdots\!32 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27296051144703 \nu^{12} + 34539730556514 \nu^{11} + \cdots + 39\!\cdots\!96 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1898111792200 \nu^{12} - 1120188199631 \nu^{11} - 162760931475086 \nu^{10} + \cdots + 93\!\cdots\!44 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 39234183388288 \nu^{12} - 148180680340021 \nu^{11} + \cdots - 12\!\cdots\!24 ) / 15\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} + 21\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{5} - 3\beta_{4} - 3\beta_{3} + 35\beta_{2} + 6\beta _1 + 303 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{12} - 42 \beta_{11} + 36 \beta_{10} - 3 \beta_{9} + 39 \beta_{8} - 33 \beta_{7} - 4 \beta_{6} + \cdots + 240 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 49 \beta_{12} + 65 \beta_{11} - 47 \beta_{10} + 90 \beta_{9} + 2 \beta_{8} - 23 \beta_{7} + \cdots + 7751 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 83 \beta_{12} - 1408 \beta_{11} + 1124 \beta_{10} - 152 \beta_{9} + 1292 \beta_{8} - 1014 \beta_{7} + \cdots + 8634 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1867 \beta_{12} + 2662 \beta_{11} - 1623 \beta_{10} + 3140 \beta_{9} + 191 \beta_{8} - 1291 \beta_{7} + \cdots + 213017 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2083 \beta_{12} - 44125 \beta_{11} + 34223 \beta_{10} - 5470 \beta_{9} + 40967 \beta_{8} - 30904 \beta_{7} + \cdots + 284573 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 66211 \beta_{12} + 91633 \beta_{11} - 49378 \beta_{10} + 100919 \beta_{9} + 12460 \beta_{8} + \cdots + 6070526 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 27721 \beta_{12} - 1347093 \beta_{11} + 1039808 \beta_{10} - 173863 \beta_{9} + 1280490 \beta_{8} + \cdots + 9063348 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2283005 \beta_{12} + 2904754 \beta_{11} - 1391809 \beta_{10} + 3133647 \beta_{9} + 642873 \beta_{8} + \cdots + 176616341 \) 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
5.57433
5.38033
4.46434
2.94429
2.58481
1.65311
1.00574
−1.74638
−2.46740
−2.75027
−3.96763
−4.24401
−5.43126
−5.57433 −8.43510 23.0731 0 47.0200 −2.78544 −84.0227 44.1509 0
1.2 −5.38033 6.50247 20.9480 0 −34.9855 18.6903 −69.6645 15.2821 0
1.3 −4.46434 −2.57761 11.9303 0 11.5073 −20.3094 −17.5462 −20.3559 0
1.4 −2.94429 −9.02583 0.668861 0 26.5747 −5.11159 21.5850 54.4656 0
1.5 −2.58481 6.24621 −1.31876 0 −16.1453 29.3198 24.0872 12.0152 0
1.6 −1.65311 0.577468 −5.26724 0 −0.954616 6.70797 21.9322 −26.6665 0
1.7 −1.00574 1.93572 −6.98848 0 −1.94684 −26.1202 15.0746 −23.2530 0
1.8 1.74638 7.96785 −4.95017 0 13.9149 −27.5580 −22.6159 36.4866 0
1.9 2.46740 −5.30990 −1.91192 0 −13.1017 −35.3432 −24.4567 1.19507 0
1.10 2.75027 −2.01254 −0.436007 0 −5.53504 28.8523 −23.2013 −22.9497 0
1.11 3.96763 −5.18197 7.74209 0 −20.5601 14.2471 −1.02330 −0.147229 0
1.12 4.24401 8.98231 10.0116 0 38.1210 26.5161 8.53740 53.6819 0
1.13 5.43126 3.33092 21.4986 0 18.0911 −6.10564 73.3142 −15.9050 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(23\) \( +1 \)

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 575.4.a.o ✓ 13
5.b even 2 1 575.4.a.p yes 13
5.c odd 4 2 575.4.b.l 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.4.a.o ✓ 13 1.a even 1 1 trivial
575.4.a.p yes 13 5.b even 2 1
575.4.b.l 26 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(575))\):

\( T_{2}^{13} + 3 T_{2}^{12} - 85 T_{2}^{11} - 222 T_{2}^{10} + 2763 T_{2}^{9} + 6071 T_{2}^{8} + \cdots + 1836192 \) Copy content Toggle raw display
\( T_{3}^{13} - 3 T_{3}^{12} - 225 T_{3}^{11} + 636 T_{3}^{10} + 18634 T_{3}^{9} - 47017 T_{3}^{8} + \cdots - 117619600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 3 T^{12} + \cdots + 1836192 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots - 117619600 \) Copy content Toggle raw display
$5$ \( T^{13} \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots - 33\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots - 66\!\cdots\!14 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T + 23)^{13} \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 46\!\cdots\!35 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 83\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 20\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 65\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 59\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 34\!\cdots\!85 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 21\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 88\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
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