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Properties

Label 1575.4.a.bn
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: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{4} + 7 q^{7} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \cdots - 10) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{4} + 7 q^{7} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \cdots - 10) q^{8}+ \cdots + (49 \beta_1 - 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{2} + 18 q^{4} + 35 q^{7} - 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{2} + 18 q^{4} + 35 q^{7} - 42 q^{8} - 42 q^{11} + 34 q^{13} - 28 q^{14} + 74 q^{16} - 238 q^{17} - 36 q^{19} + 358 q^{22} - 152 q^{23} + 310 q^{26} + 126 q^{28} + 44 q^{29} + 60 q^{31} - 710 q^{32} - 482 q^{34} + 312 q^{37} - 280 q^{38} + 426 q^{41} + 304 q^{43} - 712 q^{44} + 88 q^{46} - 370 q^{47} + 245 q^{49} - 1156 q^{52} - 976 q^{53} - 294 q^{56} - 2722 q^{58} + 432 q^{59} - 442 q^{61} + 956 q^{62} + 1362 q^{64} - 804 q^{67} + 420 q^{68} - 440 q^{71} + 564 q^{73} + 1512 q^{74} - 1336 q^{76} - 294 q^{77} + 1790 q^{79} - 3480 q^{82} - 1656 q^{83} - 1216 q^{86} - 1092 q^{88} - 746 q^{89} + 238 q^{91} - 572 q^{92} - 826 q^{94} + 518 q^{97} - 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 17\nu^{2} + 41\nu + 2 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 25\nu^{2} + 49\nu + 90 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - \nu^{3} - 25\nu^{2} + 11\nu + 74 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} - 4\beta_{3} + 19\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{4} - 29\beta_{3} + 25\beta_{2} + 33\beta _1 + 209 \) 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
−4.31366
−1.85474
−0.555276
2.67516
5.04851
−5.31366 0 20.2350 0 0 7.00000 −65.0123 0 0
1.2 −2.85474 0 0.149548 0 0 7.00000 22.4110 0 0
1.3 −1.55528 0 −5.58112 0 0 7.00000 21.1224 0 0
1.4 1.67516 0 −5.19383 0 0 7.00000 −22.1018 0 0
1.5 4.04851 0 8.39045 0 0 7.00000 1.58074 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.bn 5
3.b odd 2 1 175.4.a.j 5
5.b even 2 1 1575.4.a.bq 5
5.c odd 4 2 315.4.d.c 10
15.d odd 2 1 175.4.a.i 5
15.e even 4 2 35.4.b.a ✓ 10
21.c even 2 1 1225.4.a.bh 5
60.l odd 4 2 560.4.g.f 10
105.g even 2 1 1225.4.a.be 5
105.k odd 4 2 245.4.b.d 10
105.w odd 12 4 245.4.j.f 20
105.x even 12 4 245.4.j.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.b.a ✓ 10 15.e even 4 2
175.4.a.i 5 15.d odd 2 1
175.4.a.j 5 3.b odd 2 1
245.4.b.d 10 105.k odd 4 2
245.4.j.e 20 105.x even 12 4
245.4.j.f 20 105.w odd 12 4
315.4.d.c 10 5.c odd 4 2
560.4.g.f 10 60.l odd 4 2
1225.4.a.be 5 105.g even 2 1
1225.4.a.bh 5 21.c even 2 1
1575.4.a.bn 5 1.a even 1 1 trivial
1575.4.a.bq 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{5} + 4T_{2}^{4} - 21T_{2}^{3} - 70T_{2}^{2} + 54T_{2} + 160 \) Copy content Toggle raw display
\( T_{11}^{5} + 42T_{11}^{4} - 2271T_{11}^{3} - 151744T_{11}^{2} - 2556120T_{11} - 11139072 \) Copy content Toggle raw display
\( T_{13}^{5} - 34T_{13}^{4} - 7833T_{13}^{3} + 110628T_{13}^{2} + 12637182T_{13} - 51332920 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 4 T^{4} + \cdots + 160 \) 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
$11$ \( T^{5} + 42 T^{4} + \cdots - 11139072 \) Copy content Toggle raw display
$13$ \( T^{5} - 34 T^{4} + \cdots - 51332920 \) Copy content Toggle raw display
$17$ \( T^{5} + 238 T^{4} + \cdots - 359983792 \) Copy content Toggle raw display
$19$ \( T^{5} + 36 T^{4} + \cdots - 20133120 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 8730849408 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 126081243400 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots - 34433580800 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 43388412544 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 109849343072 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 49627325440 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 32437777344 \) Copy content Toggle raw display
$53$ \( T^{5} + 976 T^{4} + \cdots + 744413184 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 5009454255200 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 650238065792 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 15885557727232 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 7767441797120 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 9409174483008 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 201540544400 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 2803375323136 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 68855772276480 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 12547593502192 \) Copy content Toggle raw display
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