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Properties

Label 3528.3.f.b
Level $3528$
Weight $3$
Character orbit 3528.f
Analytic conductor $96.131$
Analytic rank $0$
Dimension $8$
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] = [3528,3,Mod(2449,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.2449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3528.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(96.1310372663\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.35911766016.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 5) q^{11} + ( - \beta_{7} + \beta_{5} + 2 \beta_1) q^{13} + (\beta_{7} + \beta_{6} + \cdots + 3 \beta_1) q^{17}+ \cdots + ( - 7 \beta_{7} - 4 \beta_{6} + \cdots + 57 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 44 q^{11} + 96 q^{23} - 84 q^{25} - 68 q^{29} + 236 q^{37} - 92 q^{43} + 20 q^{53} - 296 q^{65} - 44 q^{67} + 392 q^{71} - 328 q^{79} + 200 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4244 \nu^{7} - 873 \nu^{6} + 33756 \nu^{5} + 71462 \nu^{4} - 213594 \nu^{3} - 469674 \nu^{2} + \cdots + 1364084 ) / 329245 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\nu^{7} + 41\nu^{6} - 222\nu^{5} - 414\nu^{4} + 488\nu^{3} + 1608\nu^{2} - 207\nu + 3637 ) / 2045 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1281 \nu^{7} + 4752 \nu^{6} + 4396 \nu^{5} - 64593 \nu^{4} - 60254 \nu^{3} + 157791 \nu^{2} + \cdots - 327681 ) / 47035 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2179 \nu^{7} - 2932 \nu^{6} - 12056 \nu^{5} - 18227 \nu^{4} + 160624 \nu^{3} - 25261 \nu^{2} + \cdots - 458804 ) / 47035 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} + 5\nu^{6} - 86\nu^{5} - 202\nu^{4} + 472\nu^{3} + 1080\nu^{2} - 227\nu - 3151 ) / 161 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3669 \nu^{7} - 2692 \nu^{6} - 21336 \nu^{5} - 34317 \nu^{4} + 219344 \nu^{3} + 103859 \nu^{2} + \cdots - 924324 ) / 47035 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 36277 \nu^{7} + 27681 \nu^{6} + 246648 \nu^{5} + 464221 \nu^{4} - 1787752 \nu^{3} + \cdots + 8535392 ) / 329245 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{2} + 8\beta _1 + 8 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - 4\beta_{5} - \beta_{4} - 4\beta_{2} - 32\beta _1 + 32 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{7} + 18\beta_{6} + 9\beta_{5} - 21\beta_{2} + 2\beta _1 + 196 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{7} + 10\beta_{6} - 23\beta_{5} - 10\beta_{4} - 7\beta_{3} + 23\beta_{2} - 121\beta _1 - 121 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{7} + 33\beta_{6} - 141\beta_{5} + 33\beta_{4} + 49\beta_{3} - 141\beta_{2} - 785\beta _1 + 785 ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 182\beta_{7} + 150\beta_{6} + 75\beta_{5} + 245\beta_{2} - 16\beta _1 - 672 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -22\beta_{7} - 8\beta_{6} - 207\beta_{5} + 8\beta_{4} + 22\beta_{3} + 207\beta_{2} - 734\beta _1 - 734 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
−1.33172 − 1.34622i
1.83172 − 0.480194i
2.40015 + 0.808379i
−1.90015 + 1.67440i
−1.90015 − 1.67440i
2.40015 − 0.808379i
1.83172 + 0.480194i
−1.33172 + 1.34622i
0 0 0 − 7.85832i 0 0 0 0 0
2449.2 0 0 0 − 6.12627i 0 0 0 0 0
2449.3 0 0 0 − 5.40561i 0 0 0 0 0
2449.4 0 0 0 − 3.67356i 0 0 0 0 0
2449.5 0 0 0 3.67356i 0 0 0 0 0
2449.6 0 0 0 5.40561i 0 0 0 0 0
2449.7 0 0 0 6.12627i 0 0 0 0 0
2449.8 0 0 0 7.85832i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2449.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.3.f.b 8
3.b odd 2 1 1176.3.f.c 8
7.b odd 2 1 inner 3528.3.f.b 8
7.c even 3 1 504.3.by.c 8
7.d odd 6 1 504.3.by.c 8
12.b even 2 1 2352.3.f.g 8
21.c even 2 1 1176.3.f.c 8
21.g even 6 1 168.3.z.b ✓ 8
21.g even 6 1 1176.3.z.c 8
21.h odd 6 1 168.3.z.b ✓ 8
21.h odd 6 1 1176.3.z.c 8
28.f even 6 1 1008.3.cg.p 8
28.g odd 6 1 1008.3.cg.p 8
84.h odd 2 1 2352.3.f.g 8
84.j odd 6 1 336.3.bh.g 8
84.n even 6 1 336.3.bh.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.b ✓ 8 21.g even 6 1
168.3.z.b ✓ 8 21.h odd 6 1
336.3.bh.g 8 84.j odd 6 1
336.3.bh.g 8 84.n even 6 1
504.3.by.c 8 7.c even 3 1
504.3.by.c 8 7.d odd 6 1
1008.3.cg.p 8 28.f even 6 1
1008.3.cg.p 8 28.g odd 6 1
1176.3.f.c 8 3.b odd 2 1
1176.3.f.c 8 21.c even 2 1
1176.3.z.c 8 21.g even 6 1
1176.3.z.c 8 21.h odd 6 1
2352.3.f.g 8 12.b even 2 1
2352.3.f.g 8 84.h odd 2 1
3528.3.f.b 8 1.a even 1 1 trivial
3528.3.f.b 8 7.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 \(S_{3}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5}^{8} + 142T_{5}^{6} + 6953T_{5}^{4} + 138152T_{5}^{2} + 913936 \) Copy content Toggle raw display
\( T_{11}^{4} + 22T_{11}^{3} - 43T_{11}^{2} - 1324T_{11} + 4228 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 142 T^{6} + \cdots + 913936 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 22 T^{3} + \cdots + 4228)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 262 T^{6} + \cdots + 2408704 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 3470623744 \) Copy content Toggle raw display
$19$ \( T^{8} + 1678 T^{6} + \cdots + 17272336 \) Copy content Toggle raw display
$23$ \( (T^{4} - 48 T^{3} + \cdots + 150400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 34 T^{3} + \cdots + 224128)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 25912950625 \) Copy content Toggle raw display
$37$ \( (T^{4} - 118 T^{3} + \cdots - 130652)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 580010189056 \) Copy content Toggle raw display
$43$ \( (T^{4} + 46 T^{3} + \cdots + 1658308)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 52408029184 \) Copy content Toggle raw display
$53$ \( (T^{4} - 10 T^{3} + \cdots + 1221256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 1048985640000 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 43785853599744 \) Copy content Toggle raw display
$67$ \( (T^{4} + 22 T^{3} + \cdots + 308848)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 196 T^{3} + \cdots - 13209344)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 622497709609216 \) Copy content Toggle raw display
$79$ \( (T^{4} + 164 T^{3} + \cdots - 163415975)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 839297841424 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 723343446016 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
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