Properties

Label 1575.4.a.t
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
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 \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 11 q^{4} + 7 q^{7} + 3 \beta q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 11 q^{4} + 7 q^{7} + 3 \beta q^{8} + 10 \beta q^{11} - 82 q^{13} + 7 \beta q^{14} - 31 q^{16} - 18 \beta q^{17} - 20 q^{19} + 190 q^{22} - 30 \beta q^{23} - 82 \beta q^{26} + 77 q^{28} - 56 \beta q^{29} + 156 q^{31} - 55 \beta q^{32} - 342 q^{34} - 186 q^{37} - 20 \beta q^{38} + 38 \beta q^{41} - 164 q^{43} + 110 \beta q^{44} - 570 q^{46} + 108 \beta q^{47} + 49 q^{49} - 902 q^{52} - 36 \beta q^{53} + 21 \beta q^{56} - 1064 q^{58} - 36 \beta q^{59} + 790 q^{61} + 156 \beta q^{62} - 797 q^{64} + 44 q^{67} - 198 \beta q^{68} - 102 \beta q^{71} - 126 q^{73} - 186 \beta q^{74} - 220 q^{76} + 70 \beta q^{77} - 712 q^{79} + 722 q^{82} - 336 \beta q^{83} - 164 \beta q^{86} + 570 q^{88} + 334 \beta q^{89} - 574 q^{91} - 330 \beta q^{92} + 2052 q^{94} - 798 q^{97} + 49 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 22 q^{4} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 22 q^{4} + 14 q^{7} - 164 q^{13} - 62 q^{16} - 40 q^{19} + 380 q^{22} + 154 q^{28} + 312 q^{31} - 684 q^{34} - 372 q^{37} - 328 q^{43} - 1140 q^{46} + 98 q^{49} - 1804 q^{52} - 2128 q^{58} + 1580 q^{61} - 1594 q^{64} + 88 q^{67} - 252 q^{73} - 440 q^{76} - 1424 q^{79} + 1444 q^{82} + 1140 q^{88} - 1148 q^{91} + 4104 q^{94} - 1596 q^{97}+O(q^{100}) \) 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
−4.35890
4.35890
−4.35890 0 11.0000 0 0 7.00000 −13.0767 0 0
1.2 4.35890 0 11.0000 0 0 7.00000 13.0767 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.4.a.t 2
3.b odd 2 1 inner 1575.4.a.t 2
5.b even 2 1 63.4.a.d 2
15.d odd 2 1 63.4.a.d 2
20.d odd 2 1 1008.4.a.be 2
35.c odd 2 1 441.4.a.q 2
35.i odd 6 2 441.4.e.s 4
35.j even 6 2 441.4.e.r 4
60.h even 2 1 1008.4.a.be 2
105.g even 2 1 441.4.a.q 2
105.o odd 6 2 441.4.e.r 4
105.p even 6 2 441.4.e.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 5.b even 2 1
63.4.a.d 2 15.d odd 2 1
441.4.a.q 2 35.c odd 2 1
441.4.a.q 2 105.g even 2 1
441.4.e.r 4 35.j even 6 2
441.4.e.r 4 105.o odd 6 2
441.4.e.s 4 35.i odd 6 2
441.4.e.s 4 105.p even 6 2
1008.4.a.be 2 20.d odd 2 1
1008.4.a.be 2 60.h even 2 1
1575.4.a.t 2 1.a even 1 1 trivial
1575.4.a.t 2 3.b 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} - 19 \) Copy content Toggle raw display
\( T_{11}^{2} - 1900 \) Copy content Toggle raw display
\( T_{13} + 82 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 19 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 1900 \) Copy content Toggle raw display
$13$ \( (T + 82)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6156 \) Copy content Toggle raw display
$19$ \( (T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 17100 \) Copy content Toggle raw display
$29$ \( T^{2} - 59584 \) Copy content Toggle raw display
$31$ \( (T - 156)^{2} \) Copy content Toggle raw display
$37$ \( (T + 186)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 27436 \) Copy content Toggle raw display
$43$ \( (T + 164)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 221616 \) Copy content Toggle raw display
$53$ \( T^{2} - 24624 \) Copy content Toggle raw display
$59$ \( T^{2} - 24624 \) Copy content Toggle raw display
$61$ \( (T - 790)^{2} \) Copy content Toggle raw display
$67$ \( (T - 44)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 197676 \) Copy content Toggle raw display
$73$ \( (T + 126)^{2} \) Copy content Toggle raw display
$79$ \( (T + 712)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 2145024 \) Copy content Toggle raw display
$89$ \( T^{2} - 2119564 \) Copy content Toggle raw display
$97$ \( (T + 798)^{2} \) Copy content Toggle raw display
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