Properties

Label 825.4.c.p
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
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] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not 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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 54x^{6} + 913x^{4} + 5292x^{2} + 8464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + ( - \beta_{5} + \beta_{3} - 7) q^{4} + ( - 3 \beta_{3} + 3) q^{6} + (\beta_{6} - \beta_{4} + \cdots - 2 \beta_1) q^{7}+ \cdots - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + ( - \beta_{5} + \beta_{3} - 7) q^{4} + ( - 3 \beta_{3} + 3) q^{6} + (\beta_{6} - \beta_{4} + \cdots - 2 \beta_1) q^{7}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9} - 88 q^{11} + 104 q^{14} + 132 q^{16} - 272 q^{19} + 204 q^{21} - 288 q^{24} - 640 q^{26} - 104 q^{29} + 984 q^{31} - 488 q^{34} + 468 q^{36} - 12 q^{39} + 536 q^{41} + 572 q^{44} + 736 q^{46} + 992 q^{49} + 444 q^{51} - 216 q^{54} - 1704 q^{56} + 2064 q^{59} + 232 q^{61} + 1836 q^{64} - 264 q^{66} + 384 q^{69} - 1840 q^{71} + 5712 q^{74} + 3144 q^{76} - 2304 q^{79} + 648 q^{81} + 120 q^{84} + 472 q^{86} + 2128 q^{89} + 5560 q^{91} + 2864 q^{94} + 1248 q^{96} + 792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 54x^{6} + 913x^{4} + 5292x^{2} + 8464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 54\nu^{5} + 821\nu^{3} + 2808\nu ) / 1656 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 27\nu^{2} + 92 ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 45\nu^{3} - 452\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{4} - 45\nu^{2} - 344 ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{7} - 610\nu^{5} - 8189\nu^{3} - 29696\nu ) / 1656 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 47\nu^{4} + 614\nu^{2} + 1840 ) / 36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - \beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - 2\beta_{4} - 13\beta_{2} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 27\beta_{5} + 45\beta_{3} + 286 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 45\beta_{6} + 54\beta_{4} + 585\beta_{2} + 493\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{7} - 655\beta_{5} - 1501\beta_{3} - 6686 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1609\beta_{6} - 1274\beta_{4} - 19261\beta_{2} - 12189\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
4.17080i
5.20196i
2.63835i
1.60719i
1.60719i
2.63835i
5.20196i
4.17080i
5.17080i 3.00000i −18.7372 0 15.5124 11.1745i 55.5199i −9.00000 0
199.2 4.20196i 3.00000i −9.65650 0 −12.6059 15.3793i 6.96057i −9.00000 0
199.3 3.63835i 3.00000i −5.23763 0 10.9151 20.8444i 10.0505i −9.00000 0
199.4 0.607192i 3.00000i 7.63132 0 −1.82158 8.95080i 9.49121i −9.00000 0
199.5 0.607192i 3.00000i 7.63132 0 −1.82158 8.95080i 9.49121i −9.00000 0
199.6 3.63835i 3.00000i −5.23763 0 10.9151 20.8444i 10.0505i −9.00000 0
199.7 4.20196i 3.00000i −9.65650 0 −12.6059 15.3793i 6.96057i −9.00000 0
199.8 5.17080i 3.00000i −18.7372 0 15.5124 11.1745i 55.5199i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.4.c.p 8
5.b even 2 1 inner 825.4.c.p 8
5.c odd 4 1 165.4.a.h 4
5.c odd 4 1 825.4.a.t 4
15.e even 4 1 495.4.a.m 4
15.e even 4 1 2475.4.a.be 4
55.e even 4 1 1815.4.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.h 4 5.c odd 4 1
495.4.a.m 4 15.e even 4 1
825.4.a.t 4 5.c odd 4 1
825.4.c.p 8 1.a even 1 1 trivial
825.4.c.p 8 5.b even 2 1 inner
1815.4.a.t 4 55.e even 4 1
2475.4.a.be 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{8} + 58T_{2}^{6} + 1081T_{2}^{4} + 6640T_{2}^{2} + 2304 \) Copy content Toggle raw display
\( T_{7}^{8} + 876T_{7}^{6} + 250320T_{7}^{4} + 27778816T_{7}^{2} + 1028100096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 58 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 1028100096 \) Copy content Toggle raw display
$11$ \( (T + 11)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 4643604736 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 565050932640000 \) Copy content Toggle raw display
$19$ \( (T^{4} + 136 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{4} + 52 T^{3} + \cdots + 315474624)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 492 T^{3} + \cdots - 903269376)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} - 268 T^{3} + \cdots - 6228069696)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} - 1032 T^{3} + \cdots - 2612441088)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 116 T^{3} + \cdots - 8546777488)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 82\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + 920 T^{3} + \cdots + 291456592896)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{4} + 1152 T^{3} + \cdots + 48148922944)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1064 T^{3} + \cdots + 479041129296)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
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