↑

Properties

Label 525.3.o.l
Level $525$
Weight $3$
Character orbit 525.o
Analytic conductor $14.305$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,3,Mod(376,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.376");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.o (of order \(6\), degree \(2\), 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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.523596960000.16
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{5} + 1) q^{3} + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots - 1) q^{4}+ \cdots - 3 \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{5} + 1) q^{3} + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots - 1) q^{4}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{6} + \cdots + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 12 q^{3} - 6 q^{4} + 16 q^{7} + 32 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 12 q^{3} - 6 q^{4} + 16 q^{7} + 32 q^{8} + 12 q^{9} + 20 q^{11} - 18 q^{12} - 16 q^{14} - 2 q^{16} + 18 q^{17} + 6 q^{18} + 48 q^{21} + 16 q^{22} - 62 q^{23} + 48 q^{24} + 120 q^{26} + 120 q^{28} - 100 q^{29} - 126 q^{31} - 36 q^{32} + 60 q^{33} - 36 q^{36} + 80 q^{37} - 114 q^{38} - 12 q^{39} - 90 q^{42} - 352 q^{43} - 18 q^{44} - 82 q^{46} + 72 q^{47} + 38 q^{49} + 18 q^{51} + 48 q^{52} + 76 q^{53} + 18 q^{54} + 196 q^{56} + 40 q^{58} - 54 q^{59} - 396 q^{61} + 96 q^{63} - 4 q^{64} + 24 q^{66} - 184 q^{67} + 312 q^{68} + 164 q^{71} + 48 q^{72} - 348 q^{73} - 140 q^{74} - 152 q^{77} + 240 q^{78} - 206 q^{79} - 36 q^{81} - 204 q^{82} + 132 q^{84} + 178 q^{86} - 150 q^{87} - 124 q^{88} + 282 q^{89} - 114 q^{91} + 288 q^{92} - 126 q^{93} + 30 q^{94} - 108 q^{96} + 592 q^{98} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{7} - 144\nu^{6} + 411\nu^{5} - 1614\nu^{4} + 1477\nu^{3} - 9240\nu^{2} + 13280\nu - 8440 ) / 3090 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{7} - 38\nu^{6} + 87\nu^{5} + 132\nu^{4} - 161\nu^{3} + 480\nu^{2} + 300\nu + 12250 ) / 3090 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 87\nu^{7} - 24\nu^{6} + 841\nu^{5} + 1276\nu^{4} + 10117\nu^{3} + 4640\nu^{2} + 2900\nu + 13700 ) / 21630 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 137\nu^{7} - 361\nu^{6} + 1805\nu^{5} - 1115\nu^{4} + 11191\nu^{3} - 16967\nu^{2} + 21390\nu - 10830 ) / 21630 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -69\nu^{7} + 154\nu^{6} - 1079\nu^{5} + 636\nu^{4} - 7761\nu^{3} + 4766\nu^{2} - 21870\nu + 3590 ) / 3090 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 598\nu^{7} - 1781\nu^{6} + 8184\nu^{5} - 6851\nu^{4} + 45838\nu^{3} - 67845\nu^{2} + 82420\nu + 40300 ) / 21630 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 5\beta_{5} + \beta_{4} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} + 9\beta_{4} + \beta_{3} + 2\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{7} - 2\beta_{6} + 43\beta_{5} + 13\beta_{3} + 2\beta_{2} - 17\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -21\beta_{7} - 26\beta_{6} + 59\beta_{5} - 95\beta_{4} + 13\beta_{3} - 13\beta_{2} - 95\beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34\beta_{7} - 34\beta_{6} - 34\beta_{5} - 243\beta_{4} - 121\beta_{3} - 68\beta_{2} + 407 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 466\beta_{7} + 155\beta_{6} - 1060\beta_{5} - 466\beta_{3} - 155\beta_{2} + 1081\beta _1 - 155 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
376.1
1.76021 − 3.04878i
0.836732 − 1.44926i
−0.336732 + 0.583237i
−1.26021 + 2.18275i
1.76021 + 3.04878i
0.836732 + 1.44926i
−0.336732 − 0.583237i
−1.26021 − 2.18275i
−1.76021 + 3.04878i 1.50000 − 0.866025i −4.19671 − 7.26891i 0 6.09756i −0.244004 + 6.99575i 15.4667 1.50000 − 2.59808i 0
376.2 −0.836732 + 1.44926i 1.50000 − 0.866025i 0.599760 + 1.03881i 0 2.89852i −4.76104 + 5.13152i −8.70121 1.50000 − 2.59808i 0
376.3 0.336732 − 0.583237i 1.50000 − 0.866025i 1.77322 + 3.07131i 0 − 1.16647i 6.82455 − 1.55742i 5.08226 1.50000 − 2.59808i 0
376.4 1.26021 − 2.18275i 1.50000 − 0.866025i −1.17628 − 2.03737i 0 − 4.36551i 6.18050 + 3.28656i 4.15226 1.50000 − 2.59808i 0
451.1 −1.76021 − 3.04878i 1.50000 + 0.866025i −4.19671 + 7.26891i 0 − 6.09756i −0.244004 − 6.99575i 15.4667 1.50000 + 2.59808i 0
451.2 −0.836732 − 1.44926i 1.50000 + 0.866025i 0.599760 − 1.03881i 0 − 2.89852i −4.76104 − 5.13152i −8.70121 1.50000 + 2.59808i 0
451.3 0.336732 + 0.583237i 1.50000 + 0.866025i 1.77322 − 3.07131i 0 1.16647i 6.82455 + 1.55742i 5.08226 1.50000 + 2.59808i 0
451.4 1.26021 + 2.18275i 1.50000 + 0.866025i −1.17628 + 2.03737i 0 4.36551i 6.18050 − 3.28656i 4.15226 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 376.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.o.l 8
5.b even 2 1 105.3.n.a ✓ 8
5.c odd 4 2 525.3.s.h 16
7.d odd 6 1 inner 525.3.o.l 8
15.d odd 2 1 315.3.w.a 8
35.i odd 6 1 105.3.n.a ✓ 8
35.i odd 6 1 735.3.h.a 8
35.j even 6 1 735.3.h.a 8
35.k even 12 2 525.3.s.h 16
105.p even 6 1 315.3.w.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.a ✓ 8 5.b even 2 1
105.3.n.a ✓ 8 35.i odd 6 1
315.3.w.a 8 15.d odd 2 1
315.3.w.a 8 105.p even 6 1
525.3.o.l 8 1.a even 1 1 trivial
525.3.o.l 8 7.d odd 6 1 inner
525.3.s.h 16 5.c odd 4 2
525.3.s.h 16 35.k even 12 2
735.3.h.a 8 35.i odd 6 1
735.3.h.a 8 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{8} + 2T_{2}^{7} + 13T_{2}^{6} + 2T_{2}^{5} + 91T_{2}^{4} + 50T_{2}^{3} + 190T_{2}^{2} - 100T_{2} + 100 \) Copy content Toggle raw display
\( T_{11}^{8} - 20T_{11}^{7} + 337T_{11}^{6} - 1880T_{11}^{5} + 10183T_{11}^{4} + 18970T_{11}^{3} + 96982T_{11}^{2} - 4340T_{11} + 196 \) Copy content Toggle raw display
\( T_{13}^{8} + 1164T_{13}^{6} + 420414T_{13}^{4} + 47028060T_{13}^{2} + 1230957225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 16 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} - 20 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 1230957225 \) Copy content Toggle raw display
$17$ \( T^{8} - 18 T^{7} + \cdots + 138297600 \) Copy content Toggle raw display
$19$ \( T^{8} - 846 T^{6} + \cdots + 9054081 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 7138560100 \) Copy content Toggle raw display
$29$ \( (T^{4} + 50 T^{3} + \cdots - 1825400)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 385089749136 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 5596891350625 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13887679716 \) Copy content Toggle raw display
$43$ \( (T^{4} + 176 T^{3} + \cdots + 762376)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 72 T^{7} + \cdots + 26010000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 98219560000 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 2582886122496 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 84471609600 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 273278017600 \) Copy content Toggle raw display
$71$ \( (T^{4} - 82 T^{3} + \cdots + 22760224)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 4446784387600 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 111959592561216 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 580473805464576 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 2211287961600 \) Copy content Toggle raw display
show more
show less