Properties

Label 300.8.d.b
Level $300$
Weight $8$
Character orbit 300.d
Analytic conductor $93.716$
Analytic rank $0$
Dimension $2$
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] = [300,8,Mod(49,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.49");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 300.d (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: \(93.7155076452\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 27 i q^{3} + 1408 i q^{7} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 27 i q^{3} + 1408 i q^{7} - 729 q^{9} - 4044 q^{11} - 5890 i q^{13} - 31002 i q^{17} + 40300 q^{19} - 38016 q^{21} - 78912 i q^{23} - 19683 i q^{27} + 157194 q^{29} + 114824 q^{31} - 109188 i q^{33} + 471994 i q^{37} + 159030 q^{39} - 404310 q^{41} - 253852 i q^{43} - 437688 i q^{47} - 1158921 q^{49} + 837054 q^{51} + 334926 i q^{53} + 1088100 i q^{57} - 562596 q^{59} + 3246662 q^{61} - 1026432 i q^{63} - 3895148 i q^{67} + 2130624 q^{69} - 2345160 q^{71} + 5726954 i q^{73} - 5693952 i q^{77} + 5222008 q^{79} + 531441 q^{81} - 2928132 i q^{83} + 4244238 i q^{87} + 3160230 q^{89} + 8293120 q^{91} + 3100248 i q^{93} + 1898686 i q^{97} + 2948076 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1458 q^{9} - 8088 q^{11} + 80600 q^{19} - 76032 q^{21} + 314388 q^{29} + 229648 q^{31} + 318060 q^{39} - 808620 q^{41} - 2317842 q^{49} + 1674108 q^{51} - 1125192 q^{59} + 6493324 q^{61} + 4261248 q^{69} - 4690320 q^{71} + 10444016 q^{79} + 1062882 q^{81} + 6320460 q^{89} + 16586240 q^{91} + 5896152 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 27.0000i 0 0 0 1408.00i 0 −729.000 0
49.2 0 27.0000i 0 0 0 1408.00i 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
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 300.8.d.b 2
5.b even 2 1 inner 300.8.d.b 2
5.c odd 4 1 60.8.a.d 1
5.c odd 4 1 300.8.a.d 1
15.e even 4 1 180.8.a.a 1
20.e even 4 1 240.8.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.8.a.d 1 5.c odd 4 1
180.8.a.a 1 15.e even 4 1
240.8.a.g 1 20.e even 4 1
300.8.a.d 1 5.c odd 4 1
300.8.d.b 2 1.a even 1 1 trivial
300.8.d.b 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{2} + 1982464 \) Copy content Toggle raw display
\( T_{11} + 4044 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 729 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1982464 \) Copy content Toggle raw display
$11$ \( (T + 4044)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 34692100 \) Copy content Toggle raw display
$17$ \( T^{2} + 961124004 \) Copy content Toggle raw display
$19$ \( (T - 40300)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6227103744 \) Copy content Toggle raw display
$29$ \( (T - 157194)^{2} \) Copy content Toggle raw display
$31$ \( (T - 114824)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 222778336036 \) Copy content Toggle raw display
$41$ \( (T + 404310)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64440837904 \) Copy content Toggle raw display
$47$ \( T^{2} + 191570785344 \) Copy content Toggle raw display
$53$ \( T^{2} + 112175425476 \) Copy content Toggle raw display
$59$ \( (T + 562596)^{2} \) Copy content Toggle raw display
$61$ \( (T - 3246662)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 15172177941904 \) Copy content Toggle raw display
$71$ \( (T + 2345160)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 32798002118116 \) Copy content Toggle raw display
$79$ \( (T - 5222008)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8573957009424 \) Copy content Toggle raw display
$89$ \( (T - 3160230)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 3605008526596 \) Copy content Toggle raw display
show more
show less