Properties

Label 1575.4.a.bc
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 13x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} - 7 q^{7} + ( - \beta_{2} + 2 \beta_1 - 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} - 7 q^{7} + ( - \beta_{2} + 2 \beta_1 - 8) q^{8} + (4 \beta_{2} - 8 \beta_1 + 26) q^{11} + (\beta_{2} - 5 \beta_1 + 36) q^{13} + 7 \beta_1 q^{14} + ( - 10 \beta_{2} + 2 \beta_1 - 27) q^{16} + (5 \beta_{2} + 3 \beta_1 - 50) q^{17} + ( - \beta_{2} - 7 \beta_1 - 44) q^{19} + (8 \beta_{2} - 34 \beta_1 + 76) q^{22} + ( - \beta_{2} + 23 \beta_1 + 52) q^{23} + (5 \beta_{2} - 35 \beta_1 + 46) q^{26} + ( - 7 \beta_{2} - 7 \beta_1 - 7) q^{28} + (6 \beta_{2} + 34 \beta_1 - 66) q^{29} + (7 \beta_{2} - 47 \beta_1 - 104) q^{31} + (6 \beta_{2} + 49 \beta_1 + 36) q^{32} + ( - 3 \beta_{2} + 27 \beta_1 - 22) q^{34} + (32 \beta_{2} + 32 \beta_1 + 84) q^{37} + (7 \beta_{2} + 55 \beta_1 + 62) q^{38} + ( - 43 \beta_{2} + 61 \beta_1 + 34) q^{41} + ( - 12 \beta_{2} - 108 \beta_1 - 4) q^{43} + (2 \beta_{2} - 10 \beta_1 + 106) q^{44} + ( - 23 \beta_{2} - 71 \beta_1 - 208) q^{46} + (50 \beta_{2} - 82 \beta_1 + 52) q^{47} + 49 q^{49} + (27 \beta_{2} + 9 \beta_1 + 32) q^{52} + (27 \beta_{2} + 105 \beta_1 - 144) q^{53} + (7 \beta_{2} - 14 \beta_1 + 56) q^{56} + ( - 34 \beta_{2} + 8 \beta_1 - 300) q^{58} + (4 \beta_{2} - 112 \beta_1 + 332) q^{59} + (58 \beta_{2} + 106 \beta_1 - 306) q^{61} + (47 \beta_{2} + 123 \beta_1 + 430) q^{62} + (31 \beta_{2} - 125 \beta_1 - 219) q^{64} + (66 \beta_{2} - 90 \beta_1 + 20) q^{67} + ( - 67 \beta_{2} - 17 \beta_1 + 154) q^{68} + ( - 32 \beta_{2} - 136 \beta_1 + 374) q^{71} + ( - 65 \beta_{2} + 133 \beta_1 + 452) q^{73} + ( - 32 \beta_{2} - 244 \beta_1 - 256) q^{74} + ( - 47 \beta_{2} - 89 \beta_1 - 136) q^{76} + ( - 28 \beta_{2} + 56 \beta_1 - 182) q^{77} + (30 \beta_{2} - 198 \beta_1 - 464) q^{79} + ( - 61 \beta_{2} + 77 \beta_1 - 592) q^{82} + ( - 56 \beta_{2} - 252 \beta_1 + 356) q^{83} + (108 \beta_{2} + 160 \beta_1 + 960) q^{86} + ( - 54 \beta_{2} + 168 \beta_1 - 516) q^{88} + (43 \beta_{2} + 31 \beta_1 + 650) q^{89} + ( - 7 \beta_{2} + 35 \beta_1 - 252) q^{91} + (79 \beta_{2} + 187 \beta_1 + 200) q^{92} + (82 \beta_{2} - 170 \beta_1 + 788) q^{94} + ( - 27 \beta_{2} + 183 \beta_1 + 556) q^{97} - 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{4} - 21 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{4} - 21 q^{7} - 21 q^{8} + 66 q^{11} + 102 q^{13} + 7 q^{14} - 69 q^{16} - 152 q^{17} - 138 q^{19} + 186 q^{22} + 180 q^{23} + 98 q^{26} - 21 q^{28} - 170 q^{29} - 366 q^{31} + 151 q^{32} - 36 q^{34} + 252 q^{37} + 234 q^{38} + 206 q^{41} - 108 q^{43} + 306 q^{44} - 672 q^{46} + 24 q^{47} + 147 q^{49} + 78 q^{52} - 354 q^{53} + 147 q^{56} - 858 q^{58} + 880 q^{59} - 870 q^{61} + 1366 q^{62} - 813 q^{64} - 96 q^{67} + 512 q^{68} + 1018 q^{71} + 1554 q^{73} - 980 q^{74} - 450 q^{76} - 462 q^{77} - 1620 q^{79} - 1638 q^{82} + 872 q^{83} + 2932 q^{86} - 1326 q^{88} + 1938 q^{89} - 714 q^{91} + 708 q^{92} + 2112 q^{94} + 1878 q^{97} - 49 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 13x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 9 \) 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
4.10645
0.0765073
−3.18296
−4.10645 0 8.86293 0 0 −7.00000 −3.54358 0 0
1.2 −0.0765073 0 −7.99415 0 0 −7.00000 1.22367 0 0
1.3 3.18296 0 2.13122 0 0 −7.00000 −18.6801 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

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.bc 3
3.b odd 2 1 525.4.a.r 3
5.b even 2 1 1575.4.a.bd 3
5.c odd 4 2 315.4.d.a 6
15.d odd 2 1 525.4.a.q 3
15.e even 4 2 105.4.d.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.a 6 15.e even 4 2
315.4.d.a 6 5.c odd 4 2
525.4.a.q 3 15.d odd 2 1
525.4.a.r 3 3.b odd 2 1
1575.4.a.bc 3 1.a even 1 1 trivial
1575.4.a.bd 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{3} + T_{2}^{2} - 13T_{2} - 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 66T_{11}^{2} - 276T_{11} + 6120 \) Copy content Toggle raw display
\( T_{13}^{3} - 102T_{13}^{2} + 3084T_{13} - 28696 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 13T - 1 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 66 T^{2} + \cdots + 6120 \) Copy content Toggle raw display
$13$ \( T^{3} - 102 T^{2} + \cdots - 28696 \) Copy content Toggle raw display
$17$ \( T^{3} + 152 T^{2} + \cdots + 68272 \) Copy content Toggle raw display
$19$ \( T^{3} + 138 T^{2} + \cdots + 70632 \) Copy content Toggle raw display
$23$ \( T^{3} - 180 T^{2} + \cdots + 228816 \) Copy content Toggle raw display
$29$ \( T^{3} + 170 T^{2} + \cdots - 1680584 \) Copy content Toggle raw display
$31$ \( T^{3} + 366 T^{2} + \cdots - 3510632 \) Copy content Toggle raw display
$37$ \( T^{3} - 252 T^{2} + \cdots + 8221888 \) Copy content Toggle raw display
$41$ \( T^{3} - 206 T^{2} + \cdots + 18222200 \) Copy content Toggle raw display
$43$ \( T^{3} + 108 T^{2} + \cdots + 13701184 \) Copy content Toggle raw display
$47$ \( T^{3} - 24 T^{2} + \cdots - 20893824 \) Copy content Toggle raw display
$53$ \( T^{3} + 354 T^{2} + \cdots - 53538840 \) Copy content Toggle raw display
$59$ \( T^{3} - 880 T^{2} + \cdots + 22878976 \) Copy content Toggle raw display
$61$ \( T^{3} + 870 T^{2} + \cdots - 112468792 \) Copy content Toggle raw display
$67$ \( T^{3} + 96 T^{2} + \cdots - 35189888 \) Copy content Toggle raw display
$71$ \( T^{3} - 1018 T^{2} + \cdots + 133243912 \) Copy content Toggle raw display
$73$ \( T^{3} - 1554 T^{2} + \cdots + 199646136 \) Copy content Toggle raw display
$79$ \( T^{3} + 1620 T^{2} + \cdots - 258624448 \) Copy content Toggle raw display
$83$ \( T^{3} - 872 T^{2} + \cdots + 688370432 \) Copy content Toggle raw display
$89$ \( T^{3} - 1938 T^{2} + \cdots - 181473048 \) Copy content Toggle raw display
$97$ \( T^{3} - 1878 T^{2} + \cdots + 140508392 \) Copy content Toggle raw display
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