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Properties

Label 900.5.l.k
Level $900$
Weight $5$
Character orbit 900.l
Analytic conductor $93.033$
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] = [900,5,Mod(757,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.757");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 900.l (of order \(4\), 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: \(93.0329667755\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} + 28x^{5} + 97x^{4} - 168x^{3} + 288x^{2} + 864x + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{2} - 17 \beta_1 + 18) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{2} - 17 \beta_1 + 18) q^{7} + ( - 3 \beta_{7} - 3 \beta_{5} + \cdots - 33) q^{11}+ \cdots + ( - 70 \beta_{5} - 186 \beta_{4} + \cdots + 1985) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 140 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 140 q^{7} - 288 q^{11} - 300 q^{13} - 1020 q^{17} + 1320 q^{23} + 1472 q^{31} + 300 q^{37} + 3480 q^{41} + 6360 q^{43} + 4800 q^{47} + 3900 q^{53} - 11544 q^{61} + 920 q^{67} + 3600 q^{71} - 2960 q^{73} + 19800 q^{77} + 12720 q^{83} + 32400 q^{91} + 15600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} + 28x^{5} + 97x^{4} - 168x^{3} + 288x^{2} + 864x + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 343\nu^{7} - 2023\nu^{6} + 4880\nu^{5} + 5404\nu^{4} + 17563\nu^{3} - 175455\nu^{2} + 180684\nu + 145584 ) / 486540 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 331\nu^{7} - 2731\nu^{6} + 12650\nu^{5} - 15812\nu^{4} + 15391\nu^{3} - 43155\nu^{2} + 579618\nu - 160812 ) / 40545 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 805 \nu^{7} + 2272 \nu^{6} - 1316 \nu^{5} - 33580 \nu^{4} - 105637 \nu^{3} + 85764 \nu^{2} + \cdots - 1032840 ) / 64872 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19477 \nu^{7} + 87082 \nu^{6} - 220340 \nu^{5} - 367996 \nu^{4} - 1944877 \nu^{3} + \cdots - 10153296 ) / 973080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 634\nu^{7} - 4411\nu^{6} + 13697\nu^{5} - 8147\nu^{4} + 31120\nu^{3} - 215496\nu^{2} + 303399\nu - 460809 ) / 24327 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -839\nu^{7} + 3234\nu^{6} - 5960\nu^{5} - 27972\nu^{4} - 55399\nu^{3} + 80710\nu^{2} - 211512\nu - 579672 ) / 27030 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16072 \nu^{7} + 87637 \nu^{6} - 179600 \nu^{5} - 568036 \nu^{4} - 107452 \nu^{3} + \cdots - 19941876 ) / 486540 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{4} + 2\beta_{3} - \beta_{2} - 10\beta _1 + 10 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{2} - 85\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} + 25\beta_{6} - 22\beta_{4} - 22\beta_{3} - 286\beta _1 - 286 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -21\beta_{7} + 32\beta_{6} - 21\beta_{5} - 46\beta_{3} + 32\beta_{2} + 21\beta _1 - 1184 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -224\beta_{5} + 350\beta_{4} - 350\beta_{3} + 497\beta_{2} + 6302\beta _1 - 6078 ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 413\beta_{7} - 727\beta_{6} - 413\beta_{5} + 962\beta_{4} + 727\beta_{2} + 21505\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4920\beta_{7} - 9841\beta_{6} + 6502\beta_{4} + 6502\beta_{3} + 124670\beta _1 + 124670 ) / 20 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
3.17086 + 3.17086i
−1.84806 − 1.84806i
−0.946115 − 0.946115i
1.62332 + 1.62332i
3.17086 − 3.17086i
−1.84806 + 1.84806i
−0.946115 + 0.946115i
1.62332 − 1.62332i
0 0 0 0 0 −21.1834 − 21.1834i 0 0 0
757.2 0 0 0 0 0 9.71439 + 9.71439i 0 0 0
757.3 0 0 0 0 0 29.2390 + 29.2390i 0 0 0
757.4 0 0 0 0 0 52.2300 + 52.2300i 0 0 0
793.1 0 0 0 0 0 −21.1834 + 21.1834i 0 0 0
793.2 0 0 0 0 0 9.71439 − 9.71439i 0 0 0
793.3 0 0 0 0 0 29.2390 − 29.2390i 0 0 0
793.4 0 0 0 0 0 52.2300 − 52.2300i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.5.l.k 8
3.b odd 2 1 300.5.k.d 8
5.b even 2 1 180.5.l.b 8
5.c odd 4 1 180.5.l.b 8
5.c odd 4 1 inner 900.5.l.k 8
15.d odd 2 1 60.5.k.a ✓ 8
15.e even 4 1 60.5.k.a ✓ 8
15.e even 4 1 300.5.k.d 8
60.h even 2 1 240.5.bg.d 8
60.l odd 4 1 240.5.bg.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.k.a ✓ 8 15.d odd 2 1
60.5.k.a ✓ 8 15.e even 4 1
180.5.l.b 8 5.b even 2 1
180.5.l.b 8 5.c odd 4 1
240.5.bg.d 8 60.h even 2 1
240.5.bg.d 8 60.l odd 4 1
300.5.k.d 8 3.b odd 2 1
300.5.k.d 8 15.e even 4 1
900.5.l.k 8 1.a even 1 1 trivial
900.5.l.k 8 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 140 T_{7}^{7} + 9800 T_{7}^{6} - 245480 T_{7}^{5} + 3113188 T_{7}^{4} - 69856960 T_{7}^{3} + \cdots + 1580189787136 \) acting on \(S_{5}^{\mathrm{new}}(900, [\chi])\). 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
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 1580189787136 \) Copy content Toggle raw display
$11$ \( (T^{4} + 144 T^{3} + \cdots + 311125216)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 69\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 1071194996864)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 3221711600000)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 162789583002624)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 301327179200000)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
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