Properties

Label 1575.4.a.bo
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $5$
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] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.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: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.78066700.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 18x^{3} + 7x^{2} + 30x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 5 \)
Twist minimal: no (minimal twist has level 105)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1 + 5) q^{4} + 7 q^{7} + (\beta_{2} + 5 \beta_1 - 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1 + 5) q^{4} + 7 q^{7} + (\beta_{2} + 5 \beta_1 - 6) q^{8} + ( - \beta_{4} - 8 \beta_1 - 15) q^{11} + (\beta_{4} - \beta_{2} - 6 \beta_1 - 1) q^{13} + 7 \beta_1 q^{14} + ( - 2 \beta_{4} - 5 \beta_{3} + \cdots + 29) q^{16}+ \cdots + 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8} - 66 q^{11} - 2 q^{13} - 7 q^{14} + 155 q^{16} - 108 q^{17} + 174 q^{19} - 506 q^{22} + 116 q^{23} - 446 q^{26} + 189 q^{28} - 370 q^{29} + 342 q^{31} + 55 q^{32} + 112 q^{34} - 408 q^{37} + 34 q^{38} - 802 q^{41} - 584 q^{43} - 290 q^{44} + 640 q^{46} - 716 q^{47} + 245 q^{49} + 338 q^{52} - 98 q^{53} - 231 q^{56} + 482 q^{58} - 704 q^{59} + 650 q^{61} - 2070 q^{62} + 75 q^{64} + 180 q^{67} - 4520 q^{68} - 1470 q^{71} - 534 q^{73} + 1312 q^{74} + 4370 q^{76} - 462 q^{77} - 820 q^{79} - 1338 q^{82} - 1520 q^{83} - 832 q^{86} - 3258 q^{88} - 286 q^{89} - 14 q^{91} - 1288 q^{92} + 2540 q^{94} + 278 q^{97} - 49 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 18x^{3} + 7x^{2} + 30x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{4} + 3\nu^{3} + 27\nu^{2} - 20\nu + 10 ) / 15 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{4} + 24\nu^{3} - 84\nu^{2} - 320\nu + 70 ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{4} - 12\nu^{3} - 138\nu^{2} + 140\nu + 155 ) / 15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -32\nu^{4} + 18\nu^{3} + 582\nu^{2} + 100\nu - 785 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + 4\beta_{3} + \beta_{2} + 2\beta _1 + 5 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} - 18\beta _1 + 70 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{4} + 23\beta_{3} + 12\beta_{2} + 4\beta _1 + 70 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 38\beta_{4} + 2\beta_{3} + 53\beta_{2} - 644\beta _1 + 2150 ) / 20 \) Copy content Toggle raw display

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
−3.71490
4.40248
0.329739
1.35311
−1.37042
−5.18660 0 18.9008 0 0 7.00000 −56.5383 0 0
1.2 −3.33774 0 3.14050 0 0 7.00000 16.2197 0 0
1.3 0.428319 0 −7.81654 0 0 7.00000 −6.77452 0 0
1.4 2.20666 0 −3.13065 0 0 7.00000 −24.5616 0 0
1.5 4.88936 0 15.9059 0 0 7.00000 38.6546 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

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 1575.4.a.bo 5
3.b odd 2 1 525.4.a.x 5
5.b even 2 1 1575.4.a.bp 5
5.c odd 4 2 315.4.d.b 10
15.d odd 2 1 525.4.a.w 5
15.e even 4 2 105.4.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.b 10 15.e even 4 2
315.4.d.b 10 5.c odd 4 2
525.4.a.w 5 15.d odd 2 1
525.4.a.x 5 3.b odd 2 1
1575.4.a.bo 5 1.a even 1 1 trivial
1575.4.a.bp 5 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{5} + T_{2}^{4} - 33T_{2}^{3} - 17T_{2}^{2} + 200T_{2} - 80 \) Copy content Toggle raw display
\( T_{11}^{5} + 66T_{11}^{4} - 2100T_{11}^{3} - 140456T_{11}^{2} + 1472448T_{11} + 55852416 \) Copy content Toggle raw display
\( T_{13}^{5} + 2T_{13}^{4} - 4152T_{13}^{3} + 15504T_{13}^{2} + 4336080T_{13} - 41380960 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} + \cdots - 80 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( (T - 7)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 66 T^{4} + \cdots + 55852416 \) Copy content Toggle raw display
$13$ \( T^{5} + 2 T^{4} + \cdots - 41380960 \) Copy content Toggle raw display
$17$ \( T^{5} + 108 T^{4} + \cdots + 232018688 \) Copy content Toggle raw display
$19$ \( T^{5} - 174 T^{4} + \cdots - 784374624 \) Copy content Toggle raw display
$23$ \( T^{5} - 116 T^{4} + \cdots - 488160000 \) Copy content Toggle raw display
$29$ \( T^{5} + 370 T^{4} + \cdots - 1150048 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 52737095200 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 315202167808 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 531402107648 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 132088069120 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 65728742400 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 462468251232 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 33724261457920 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 1105143174112 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 1579424171008 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 237519904000 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 1941655936032 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 43229481181184 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 128527499264 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 1125486676224 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 19868737339616 \) Copy content Toggle raw display
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