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Properties

Label 575.3.d.e
Level $575$
Weight $3$
Character orbit 575.d
Analytic conductor $15.668$
Analytic rank $0$
Dimension $6$
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] = [575,3,Mod(551,575)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.551");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 575.d (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: \(15.6676152007\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 34x^{4} + 50x^{3} + 690x^{2} - 600x + 4725 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 115)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_{2} q^{3} - 3 q^{4} - \beta_{2} q^{6} - \beta_{3} q^{7} - 7 q^{8} + (\beta_{2} + \beta_1 + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_{2} q^{3} - 3 q^{4} - \beta_{2} q^{6} - \beta_{3} q^{7} - 7 q^{8} + (\beta_{2} + \beta_1 + 8) q^{9} + \beta_{5} q^{11} + 3 \beta_{2} q^{12} + (\beta_{2} - \beta_1 + 1) q^{13} - \beta_{3} q^{14} + 5 q^{16} + (\beta_{5} + \beta_{4}) q^{17} + (\beta_{2} + \beta_1 + 8) q^{18} + ( - \beta_{4} - \beta_{3}) q^{19} + (\beta_{5} + 3 \beta_{4} + 3 \beta_{3}) q^{21} + \beta_{5} q^{22} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{23}+ \cdots + (2 \beta_{5} + 5 \beta_{4} - 17 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 2 q^{3} - 18 q^{4} - 2 q^{6} - 42 q^{8} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} - 2 q^{3} - 18 q^{4} - 2 q^{6} - 42 q^{8} + 48 q^{9} + 6 q^{12} + 10 q^{13} + 30 q^{16} + 48 q^{18} + 10 q^{23} + 14 q^{24} + 10 q^{26} - 56 q^{27} + 72 q^{29} - 6 q^{31} + 198 q^{32} - 144 q^{36} - 148 q^{39} + 142 q^{41} + 10 q^{46} - 112 q^{47} - 10 q^{48} - 304 q^{49} - 30 q^{52} - 56 q^{54} + 72 q^{58} + 236 q^{59} - 6 q^{62} + 78 q^{64} + 156 q^{69} - 218 q^{71} - 336 q^{72} - 184 q^{77} - 148 q^{78} + 354 q^{81} + 142 q^{82} + 584 q^{87} - 30 q^{92} + 176 q^{93} - 112 q^{94} - 66 q^{96} - 304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 34x^{4} + 50x^{3} + 690x^{2} - 600x + 4725 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 62\nu^{5} - 5599\nu^{4} + 5812\nu^{3} + 221635\nu^{2} - 193350\nu - 3002445 ) / 249285 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -272\nu^{5} + 439\nu^{4} + 6668\nu^{3} - 7360\nu^{2} + 44100\nu + 88575 ) / 249285 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -718\nu^{5} + 3725\nu^{4} + 30799\nu^{3} - 22361\nu^{2} - 712095\nu + 312630 ) / 249285 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 544\nu^{5} - 878\nu^{4} - 13336\nu^{3} + 14720\nu^{2} + 410370\nu - 177150 ) / 83095 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1751\nu^{5} - 290\nu^{4} + 80318\nu^{3} - 97237\nu^{2} - 1648065\nu + 741585 ) / 249285 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 6\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} + 4\beta_{3} + 3\beta_{2} + 3\beta _1 + 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24\beta_{5} + 55\beta_{4} + 42\beta_{3} + 66\beta_{2} + 6\beta _1 + 42 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -64\beta_{5} + 20\beta_{4} + 188\beta_{3} + 33\beta_{2} - 12\beta _1 - 159 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 490\beta_{5} + 1575\beta_{4} + 1420\beta_{3} - 2964\beta_{2} - 54\beta _1 + 522 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1
5.33482 + 2.23607i
5.33482 − 2.23607i
0.396209 − 2.23607i
0.396209 + 2.23607i
−4.73103 − 2.23607i
−4.73103 + 2.23607i
1.00000 −5.33482 −3.00000 0 −5.33482 − 13.3268i −7.00000 19.4603 0
551.2 1.00000 −5.33482 −3.00000 0 −5.33482 13.3268i −7.00000 19.4603 0
551.3 1.00000 −0.396209 −3.00000 0 −0.396209 − 8.16531i −7.00000 −8.84302 0
551.4 1.00000 −0.396209 −3.00000 0 −0.396209 8.16531i −7.00000 −8.84302 0
551.5 1.00000 4.73103 −3.00000 0 4.73103 − 7.39757i −7.00000 13.3827 0
551.6 1.00000 4.73103 −3.00000 0 4.73103 7.39757i −7.00000 13.3827 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.3.d.e 6
5.b even 2 1 115.3.d.a ✓ 6
5.c odd 4 2 575.3.c.c 12
15.d odd 2 1 1035.3.g.a 6
20.d odd 2 1 1840.3.k.a 6
23.b odd 2 1 inner 575.3.d.e 6
115.c odd 2 1 115.3.d.a ✓ 6
115.e even 4 2 575.3.c.c 12
345.h even 2 1 1035.3.g.a 6
460.g even 2 1 1840.3.k.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.d.a ✓ 6 5.b even 2 1
115.3.d.a ✓ 6 115.c odd 2 1
575.3.c.c 12 5.c odd 4 2
575.3.c.c 12 115.e even 4 2
575.3.d.e 6 1.a even 1 1 trivial
575.3.d.e 6 23.b odd 2 1 inner
1035.3.g.a 6 15.d odd 2 1
1035.3.g.a 6 345.h even 2 1
1840.3.k.a 6 20.d odd 2 1
1840.3.k.a 6 460.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{3}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} + T^{2} - 25 T - 10)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 299 T^{4} + \cdots + 648000 \) Copy content Toggle raw display
$11$ \( T^{6} + 731 T^{4} + \cdots + 8632980 \) Copy content Toggle raw display
$13$ \( (T^{3} - 5 T^{2} - 257 T + 54)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 971 T^{4} + \cdots + 524880 \) Copy content Toggle raw display
$19$ \( T^{6} + 779 T^{4} + \cdots + 714420 \) Copy content Toggle raw display
$23$ \( T^{6} - 10 T^{5} + \cdots + 148035889 \) Copy content Toggle raw display
$29$ \( (T^{3} - 36 T^{2} + \cdots + 8872)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 3 T^{2} + \cdots - 30050)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 1856 T^{4} + \cdots + 123405120 \) Copy content Toggle raw display
$41$ \( (T^{3} - 71 T^{2} + \cdots + 8462)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 2411208000 \) Copy content Toggle raw display
$47$ \( (T^{3} + 56 T^{2} + \cdots - 16376)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1620)^{3} \) Copy content Toggle raw display
$59$ \( (T^{3} - 118 T^{2} + \cdots + 118680)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 18819 T^{4} + \cdots + 649116180 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 5202247680 \) Copy content Toggle raw display
$71$ \( (T^{3} + 109 T^{2} + \cdots - 33718)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 912 T + 344)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 70054917120 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 29786849280 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 39983258880 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 25687244880 \) Copy content Toggle raw display
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