↑

Properties

Label 300.4.i.e
Level $300$
Weight $4$
Character orbit 300.i
Analytic conductor $17.701$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,4,Mod(257,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.257");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 300.i (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: \(17.7005730017\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{5} - \beta_{4}) q^{7} + (\beta_{7} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{5} - \beta_{4}) q^{7} + (\beta_{7} + 3 \beta_1) q^{9} + 2 \beta_{6} q^{11} + ( - 8 \beta_{3} - 8 \beta_{2}) q^{13} + ( - 6 \beta_{5} + 6 \beta_{4}) q^{17} - 28 \beta_1 q^{19} + (\beta_{6} + 24) q^{21} + ( - 21 \beta_{3} + 21 \beta_{2}) q^{23} + ( - 27 \beta_{5} - 6 \beta_{4}) q^{27} + 2 \beta_{7} q^{29} - 56 q^{31} + ( - 54 \beta_{3} - 6 \beta_{2}) q^{33} + ( - 28 \beta_{5} - 28 \beta_{4}) q^{37} + ( - 8 \beta_{7} + 192 \beta_1) q^{39} - 14 \beta_{6} q^{41} + ( - 63 \beta_{3} - 63 \beta_{2}) q^{43} + ( - 57 \beta_{5} + 57 \beta_{4}) q^{47} - 295 \beta_1 q^{49} + ( - 6 \beta_{6} + 180) q^{51} + ( - 84 \beta_{3} + 84 \beta_{2}) q^{53} + 28 \beta_{4} q^{57} - 14 \beta_{7} q^{59} - 406 q^{61} + ( - 27 \beta_{3} + 21 \beta_{2}) q^{63} + ( - 91 \beta_{5} - 91 \beta_{4}) q^{67} + (21 \beta_{7} + 630 \beta_1) q^{69} + 28 \beta_{6} q^{71} + ( - 44 \beta_{3} - 44 \beta_{2}) q^{73} + (48 \beta_{5} - 48 \beta_{4}) q^{77} - 944 \beta_1 q^{79} + (6 \beta_{6} + 711) q^{81} + (63 \beta_{3} - 63 \beta_{2}) q^{83} + ( - 54 \beta_{5} - 6 \beta_{4}) q^{87} + 28 \beta_{7} q^{89} - 384 q^{91} - 56 \beta_{2} q^{93} + (244 \beta_{5} + 244 \beta_{4}) q^{97} + ( - 6 \beta_{7} + 1440 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 192 q^{21} - 448 q^{31} + 1440 q^{51} - 3248 q^{61} + 5688 q^{81} - 3072 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 5\nu^{4} - 20\nu^{3} - 59\nu^{2} + 70\nu - 164 ) / 33 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 600\nu^{7} - 1255\nu^{6} - 3399\nu^{5} + 4368\nu^{4} + 24349\nu^{3} - 3663\nu^{2} + 16716\nu + 41137 ) / 5577 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 600\nu^{7} - 2945\nu^{6} + 1671\nu^{5} + 10452\nu^{4} + 3731\nu^{3} - 55377\nu^{2} + 79584\nu - 78853 ) / 5577 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -764\nu^{7} + 1660\nu^{6} + 3696\nu^{5} - 5785\nu^{4} - 27274\nu^{3} + 11022\nu^{2} - 51847\nu - 54079 ) / 5577 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -764\nu^{7} + 3688\nu^{6} - 2388\nu^{5} - 10855\nu^{4} - 6994\nu^{3} + 59694\nu^{2} - 111673\nu + 123371 ) / 5577 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -32\nu^{7} + 112\nu^{6} - 8\nu^{5} - 260\nu^{4} - 208\nu^{3} + 628\nu^{2} - 5272\nu + 2520 ) / 169 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2400\nu^{7} + 8400\nu^{6} + 3456\nu^{5} - 29640\nu^{4} - 56160\nu^{3} + 118080\nu^{2} - 237216\nu + 97740 ) / 1859 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - 6\beta_{3} - 6\beta_{2} + 12 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 12\beta_{5} + 12\beta_{4} - 6\beta_{3} - 6\beta_{2} - 72\beta _1 + 36 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 9\beta_{6} - 36\beta_{5} - 27\beta_{3} - 27\beta_{2} - 108\beta _1 + 48 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 6\beta_{6} - 28\beta_{5} + 4\beta_{4} + 8\beta_{3} - 40\beta_{2} - 216\beta _1 - 44 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40\beta_{7} + 60\beta_{6} - 360\beta_{5} - 180\beta_{4} + 129\beta_{3} - 231\beta_{2} - 1440\beta _1 - 408 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 127\beta_{7} + 135\beta_{6} - 576\beta_{5} - 864\beta_{4} + 684\beta_{3} - 756\beta_{2} - 3132\beta _1 - 4644 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 617 \beta_{7} + 126 \beta_{6} - 1245 \beta_{5} - 2841 \beta_{4} + 3213 \beta_{3} - 567 \beta_{2} + \cdots - 14772 ) / 24 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.606711 − 1.22474i
0.393289 + 1.22474i
2.84278 − 1.22474i
−1.84278 + 1.22474i
0.606711 + 1.22474i
0.393289 − 1.22474i
2.84278 + 1.22474i
−1.84278 − 1.22474i
0 −5.18810 − 0.289123i 0 0 0 −4.89898 − 4.89898i 0 26.8328 + 3.00000i 0
257.2 0 −0.289123 − 5.18810i 0 0 0 4.89898 + 4.89898i 0 −26.8328 + 3.00000i 0
257.3 0 0.289123 + 5.18810i 0 0 0 −4.89898 − 4.89898i 0 −26.8328 + 3.00000i 0
257.4 0 5.18810 + 0.289123i 0 0 0 4.89898 + 4.89898i 0 26.8328 + 3.00000i 0
293.1 0 −5.18810 + 0.289123i 0 0 0 −4.89898 + 4.89898i 0 26.8328 − 3.00000i 0
293.2 0 −0.289123 + 5.18810i 0 0 0 4.89898 − 4.89898i 0 −26.8328 − 3.00000i 0
293.3 0 0.289123 − 5.18810i 0 0 0 −4.89898 + 4.89898i 0 −26.8328 − 3.00000i 0
293.4 0 5.18810 − 0.289123i 0 0 0 4.89898 − 4.89898i 0 26.8328 − 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.4.i.e ✓ 8
3.b odd 2 1 inner 300.4.i.e ✓ 8
5.b even 2 1 inner 300.4.i.e ✓ 8
5.c odd 4 2 inner 300.4.i.e ✓ 8
15.d odd 2 1 inner 300.4.i.e ✓ 8
15.e even 4 2 inner 300.4.i.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.4.i.e ✓ 8 1.a even 1 1 trivial
300.4.i.e ✓ 8 3.b odd 2 1 inner
300.4.i.e ✓ 8 5.b even 2 1 inner
300.4.i.e ✓ 8 5.c odd 4 2 inner
300.4.i.e ✓ 8 15.d odd 2 1 inner
300.4.i.e ✓ 8 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 2304 \) acting on \(S_{4}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 1422 T^{4} + 531441 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2880)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 9437184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 4665600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 784)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 700131600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2880)^{4} \) Copy content Toggle raw display
$31$ \( (T + 56)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1416167424)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 141120)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 36294822144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 38001603600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 179233689600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 141120)^{4} \) Copy content Toggle raw display
$61$ \( (T + 406)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 157996710144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 564480)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8635613184)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 891136)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 56710659600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 564480)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 8166609321984)^{2} \) Copy content Toggle raw display
show more
show less