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Properties

Label 300.5.f.a
Level $300$
Weight $5$
Character orbit 300.f
Analytic conductor $31.011$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,5,Mod(199,300)]
 
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([1, 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("300.199");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 300.f (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: \(31.0109889252\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.592240896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 12)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{5}) q^{2} - \beta_{4} q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{4} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 5) q^{6} + ( - 12 \beta_{7} + 4 \beta_{6} + \cdots + 4 \beta_{4}) q^{7}+ \cdots + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{5}) q^{2} - \beta_{4} q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{4} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 5) q^{6} + ( - 12 \beta_{7} + 4 \beta_{6} + \cdots + 4 \beta_{4}) q^{7}+ \cdots + (486 \beta_{3} - 162 \beta_{2} + \cdots + 270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{4} - 36 q^{6} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{4} - 36 q^{6} + 216 q^{9} - 1200 q^{14} + 224 q^{16} - 288 q^{21} - 2592 q^{24} + 3384 q^{26} - 1776 q^{29} + 968 q^{34} + 1080 q^{36} + 1104 q^{41} + 7392 q^{44} - 768 q^{46} + 1144 q^{49} - 972 q^{54} + 3456 q^{56} + 8464 q^{61} + 29440 q^{64} + 9648 q^{66} + 19584 q^{69} + 8232 q^{74} - 3744 q^{76} + 5832 q^{81} + 21024 q^{84} - 39120 q^{86} + 50160 q^{89} + 10464 q^{94} - 16704 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 16\nu^{4} - 76\nu^{2} + 171 ) / 18 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{6} + 40\nu^{4} - 460\nu^{2} + 531 ) / 90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{6} + 40\nu^{4} - 190\nu^{2} + 81 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 16\nu^{5} - 76\nu^{3} + 207\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} - 40\nu^{5} + 190\nu^{3} - 81\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 28\nu^{3} - 57\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 14\nu^{5} - 80\nu^{3} + 126\nu ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} - 2\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 10 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} - 5\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{3} - 19\beta_{2} + 21\beta _1 - 91 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -57\beta_{7} - 60\beta_{5} + 40\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{3} + 10\beta _1 - 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -291\beta_{7} + 291\beta_{6} + 360\beta_{5} + 434\beta_{4} ) / 12 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.12824 + 0.651388i
−1.12824 − 0.651388i
−1.99426 + 1.15139i
−1.99426 − 1.15139i
1.99426 + 1.15139i
1.99426 − 1.15139i
1.12824 + 0.651388i
1.12824 − 0.651388i
−3.98852 − 0.302776i 5.19615 15.8167 + 2.41526i 0 −20.7250 − 1.57327i 43.0318 −62.3538 − 14.4222i 27.0000 0
199.2 −3.98852 + 0.302776i 5.19615 15.8167 − 2.41526i 0 −20.7250 + 1.57327i 43.0318 −62.3538 + 14.4222i 27.0000 0
199.3 −2.25647 − 3.30278i −5.19615 −5.81665 + 14.9053i 0 11.7250 + 17.1617i 56.8882 62.3538 − 14.4222i 27.0000 0
199.4 −2.25647 + 3.30278i −5.19615 −5.81665 − 14.9053i 0 11.7250 − 17.1617i 56.8882 62.3538 + 14.4222i 27.0000 0
199.5 2.25647 − 3.30278i 5.19615 −5.81665 − 14.9053i 0 11.7250 − 17.1617i −56.8882 −62.3538 − 14.4222i 27.0000 0
199.6 2.25647 + 3.30278i 5.19615 −5.81665 + 14.9053i 0 11.7250 + 17.1617i −56.8882 −62.3538 + 14.4222i 27.0000 0
199.7 3.98852 − 0.302776i −5.19615 15.8167 − 2.41526i 0 −20.7250 + 1.57327i −43.0318 62.3538 − 14.4222i 27.0000 0
199.8 3.98852 + 0.302776i −5.19615 15.8167 + 2.41526i 0 −20.7250 − 1.57327i −43.0318 62.3538 + 14.4222i 27.0000 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.5.f.a 8
4.b odd 2 1 inner 300.5.f.a 8
5.b even 2 1 inner 300.5.f.a 8
5.c odd 4 1 12.5.d.a ✓ 4
5.c odd 4 1 300.5.c.a 4
15.e even 4 1 36.5.d.b 4
20.d odd 2 1 inner 300.5.f.a 8
20.e even 4 1 12.5.d.a ✓ 4
20.e even 4 1 300.5.c.a 4
40.i odd 4 1 192.5.g.d 4
40.k even 4 1 192.5.g.d 4
60.l odd 4 1 36.5.d.b 4
80.i odd 4 1 768.5.b.g 8
80.j even 4 1 768.5.b.g 8
80.s even 4 1 768.5.b.g 8
80.t odd 4 1 768.5.b.g 8
120.q odd 4 1 576.5.g.m 4
120.w even 4 1 576.5.g.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.5.d.a ✓ 4 5.c odd 4 1
12.5.d.a ✓ 4 20.e even 4 1
36.5.d.b 4 15.e even 4 1
36.5.d.b 4 60.l odd 4 1
192.5.g.d 4 40.i odd 4 1
192.5.g.d 4 40.k even 4 1
300.5.c.a 4 5.c odd 4 1
300.5.c.a 4 20.e even 4 1
300.5.f.a 8 1.a even 1 1 trivial
300.5.f.a 8 4.b odd 2 1 inner
300.5.f.a 8 5.b even 2 1 inner
300.5.f.a 8 20.d odd 2 1 inner
576.5.g.m 4 120.q odd 4 1
576.5.g.m 4 120.w even 4 1
768.5.b.g 8 80.i odd 4 1
768.5.b.g 8 80.j even 4 1
768.5.b.g 8 80.s even 4 1
768.5.b.g 8 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 5088T_{7}^{2} + 5992704 \) acting on \(S_{5}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 20 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 5088 T^{2} + 5992704)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 22368 T^{2} + 77158656)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 70856 T^{2} + 599074576)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 51656 T^{2} + 367565584)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 325728 T^{2} + 283855104)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 463872 T^{2} + 44930433024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 444 T + 8516)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1604064 T^{2} + 310721515776)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 1198140403216)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 276 T - 144028)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 4698628487424)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 1723191791616)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 1939992122896)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 60549065443584)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2116 T - 10861436)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 3924392696064)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 10870367256576)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 28\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 783891587748096)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 87\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12540 T + 7350788)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 910698888572944)^{2} \) Copy content Toggle raw display
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