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Properties

Label 1575.4.a.bt
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $10$
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] = [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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 67x^{8} + 1523x^{6} - 13569x^{4} + 36944x^{2} - 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 315)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} - 7 q^{7} + (\beta_{3} + 6 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} - 7 q^{7} + (\beta_{3} + 6 \beta_1) q^{8} + ( - \beta_{8} - \beta_{3} - 5 \beta_1) q^{11} + (\beta_{4} - 2 \beta_{2} - 10) q^{13} - 7 \beta_1 q^{14} + (\beta_{5} - \beta_{4} + 5 \beta_{2} + 29) q^{16} + ( - \beta_{9} + \beta_{8} - \beta_{3}) q^{17} + (\beta_{6} - \beta_{5} - 4 \beta_{2} + 6) q^{19} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 60) q^{22}+ \cdots + 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 54 q^{4} - 70 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 54 q^{4} - 70 q^{7} - 104 q^{13} + 310 q^{16} + 36 q^{19} - 644 q^{22} - 378 q^{28} - 24 q^{31} + 116 q^{34} - 732 q^{37} - 212 q^{43} - 184 q^{46} + 490 q^{49} - 2692 q^{52} - 3196 q^{58} + 1024 q^{61} + 1590 q^{64} - 2388 q^{67} - 3432 q^{73} - 4184 q^{76} - 776 q^{79} - 1860 q^{82} - 10164 q^{88} + 728 q^{91} - 4028 q^{94} - 4024 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 67x^{8} + 1523x^{6} - 13569x^{4} + 36944x^{2} - 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 22\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 48\nu^{6} + 541\nu^{4} + 210\nu^{2} - 5952 ) / 112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 48\nu^{6} + 653\nu^{4} - 3038\nu^{2} + 5248 ) / 112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - 62\nu^{6} + 1241\nu^{4} - 8540\nu^{2} + 8160 ) / 112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - 69\nu^{7} + 1591\nu^{5} - 13699\nu^{3} + 31568\nu ) / 224 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{9} + 207\nu^{7} - 4885\nu^{5} + 45353\nu^{3} - 127520\nu ) / 224 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2\nu^{9} - 131\nu^{7} + 2888\nu^{5} - 24815\nu^{3} + 65432\nu ) / 112 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 22\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{4} + 29\beta_{2} + 277 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{8} - 6\beta_{7} + 38\beta_{3} + 543\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{6} + 50\beta_{5} - 42\beta_{4} + 825\beta_{2} + 6733 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16\beta_{9} - 84\beta_{8} - 316\beta_{7} + 1227\beta_{3} + 14360\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -384\beta_{6} + 1859\beta_{5} - 1363\beta_{4} + 23701\beta_{2} + 176549 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1104\beta_{9} - 2614\beta_{8} - 12034\beta_{7} + 37904\beta_{3} + 396737\beta_1 \) 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
−5.44617
−4.31226
−3.73445
−2.18874
−0.0833494
0.0833494
2.18874
3.73445
4.31226
5.44617
−5.44617 0 21.6607 0 0 −7.00000 −74.3987 0 0
1.2 −4.31226 0 10.5956 0 0 −7.00000 −11.1930 0 0
1.3 −3.73445 0 5.94609 0 0 −7.00000 7.67022 0 0
1.4 −2.18874 0 −3.20940 0 0 −7.00000 24.5345 0 0
1.5 −0.0833494 0 −7.99305 0 0 −7.00000 1.33301 0 0
1.6 0.0833494 0 −7.99305 0 0 −7.00000 −1.33301 0 0
1.7 2.18874 0 −3.20940 0 0 −7.00000 −24.5345 0 0
1.8 3.73445 0 5.94609 0 0 −7.00000 −7.67022 0 0
1.9 4.31226 0 10.5956 0 0 −7.00000 11.1930 0 0
1.10 5.44617 0 21.6607 0 0 −7.00000 74.3987 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bt 10
3.b odd 2 1 inner 1575.4.a.bt 10
5.b even 2 1 1575.4.a.bu 10
5.c odd 4 2 315.4.d.d ✓ 20
15.d odd 2 1 1575.4.a.bu 10
15.e even 4 2 315.4.d.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.d.d ✓ 20 5.c odd 4 2
315.4.d.d ✓ 20 15.e even 4 2
1575.4.a.bt 10 1.a even 1 1 trivial
1575.4.a.bt 10 3.b odd 2 1 inner
1575.4.a.bu 10 5.b even 2 1
1575.4.a.bu 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{10} - 67T_{2}^{8} + 1523T_{2}^{6} - 13569T_{2}^{4} + 36944T_{2}^{2} - 256 \) Copy content Toggle raw display
\( T_{11}^{10} - 8988T_{11}^{8} + 28913136T_{11}^{6} - 40462757440T_{11}^{4} + 22772208205824T_{11}^{2} - 2775688151040000 \) Copy content Toggle raw display
\( T_{13}^{5} + 52T_{13}^{4} - 6384T_{13}^{3} - 366408T_{13}^{2} + 4517184T_{13} + 317871232 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 67 T^{8} + \cdots - 256 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T + 7)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + 52 T^{4} + \cdots + 317871232)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 48\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{5} - 18 T^{4} + \cdots - 1947671040)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + 12 T^{4} + \cdots + 76972365952)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 366 T^{4} + \cdots - 131111593984)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{5} + 106 T^{4} + \cdots - 58598844416)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 24\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 21\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 21\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 1370493512000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 20083004612608)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 90\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 15592344641664)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 8769624080384)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 203983416135296)^{2} \) Copy content Toggle raw display
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