Properties

Label 300.5.k.d
Level $300$
Weight $5$
Character orbit 300.k
Analytic conductor $31.011$
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] = [300,5,Mod(157,300)]
 
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, 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("300.157");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 300.k (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: \(31.0109889252\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{6} + 4 \beta_{2} + 17 \beta_1 + 17) q^{7} - 27 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{6} + 4 \beta_{2} + 17 \beta_1 + 17) q^{7} - 27 \beta_1 q^{9} + ( - 3 \beta_{7} - 3 \beta_{6} + \beta_{4} + 39) q^{11} + (5 \beta_{7} - 6 \beta_{5} - 6 \beta_{4} + 8 \beta_{3} + 40 \beta_1 - 40) q^{13} + (3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 12 \beta_{2} + 126 \beta_1 + 126) q^{17} + (\beta_{7} - \beta_{6} + 12 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} + 26 \beta_1) q^{19} + ( - 9 \beta_{4} + 16 \beta_{3} + 16 \beta_{2} + 99) q^{21} + ( - 8 \beta_{7} + 11 \beta_{5} + 11 \beta_{4} + 20 \beta_{3} + 161 \beta_1 - 161) q^{23} - 27 \beta_{2} q^{27} + ( - 6 \beta_{7} + 6 \beta_{6} - 19 \beta_{5} - 96 \beta_{3} + 96 \beta_{2} + 471 \beta_1) q^{29} + (14 \beta_{7} + 14 \beta_{6} + 42 \beta_{4} + 80 \beta_{3} + 80 \beta_{2} + \cdots + 170) q^{31}+ \cdots + ( - 81 \beta_{7} + 81 \beta_{6} + 27 \beta_{5} - 1053 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 140 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 140 q^{7} + 288 q^{11} - 300 q^{13} + 1020 q^{17} + 792 q^{21} - 1320 q^{23} + 1472 q^{31} + 180 q^{33} + 300 q^{37} - 3480 q^{41} + 6360 q^{43} - 4800 q^{47} + 2232 q^{51} - 3900 q^{53} - 360 q^{57} - 11544 q^{61} + 3780 q^{63} + 920 q^{67} - 3600 q^{71} - 2960 q^{73} - 19800 q^{77} - 5832 q^{81} - 12720 q^{83} + 19620 q^{87} + 32400 q^{91} + 14760 q^{93} + 15600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(\beta_{2}\)\(=\) \( ( 331 \nu^{7} - 2731 \nu^{6} + 12650 \nu^{5} - 15812 \nu^{4} + 15391 \nu^{3} - 43155 \nu^{2} + 579618 \nu - 160812 ) / 162180 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 839 \nu^{7} + 3234 \nu^{6} - 5960 \nu^{5} - 27972 \nu^{4} - 55399 \nu^{3} + 80710 \nu^{2} - 211512 \nu - 579672 ) / 108120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 805 \nu^{7} - 2272 \nu^{6} + 1316 \nu^{5} + 33580 \nu^{4} + 105637 \nu^{3} - 85764 \nu^{2} - 328788 \nu + 1032840 ) / 64872 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19477 \nu^{7} - 87082 \nu^{6} + 220340 \nu^{5} + 367996 \nu^{4} + 1944877 \nu^{3} - 3558870 \nu^{2} + 13561236 \nu + 10153296 ) / 973080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 634 \nu^{7} + 4411 \nu^{6} - 13697 \nu^{5} + 8147 \nu^{4} - 31120 \nu^{3} + 215496 \nu^{2} - 303399 \nu + 485136 ) / 24327 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3283 \nu^{7} - 17932 \nu^{6} + 36896 \nu^{5} + 114688 \nu^{4} + 25003 \nu^{3} - 865116 \nu^{2} + 1334448 \nu + 4114800 ) / 97308 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - 2\beta_{2} + 5\beta _1 + 5 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} + 4\beta_{3} - 4\beta_{2} + 86\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} + 11\beta_{5} + 11\beta_{4} + 50\beta_{3} + 147\beta _1 - 147 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21\beta_{7} + 21\beta_{6} + 46\beta_{4} + 128\beta_{3} + 128\beta_{2} - 1226 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 112\beta_{6} - 175\beta_{5} + 175\beta_{4} + 994\beta_{2} - 3151\beta _1 - 3151 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -413\beta_{7} + 413\beta_{6} - 962\beta_{5} - 2908\beta_{3} + 2908\beta_{2} - 21918\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2460\beta_{7} - 3251\beta_{5} - 3251\beta_{4} - 19682\beta_{3} - 64795\beta _1 + 64795 ) / 10 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
3.17086 + 3.17086i
−0.946115 0.946115i
−1.84806 1.84806i
1.62332 + 1.62332i
3.17086 3.17086i
−0.946115 + 0.946115i
−1.84806 + 1.84806i
1.62332 1.62332i
0 −3.67423 + 3.67423i 0 0 0 −21.1834 21.1834i 0 27.0000i 0
157.2 0 −3.67423 + 3.67423i 0 0 0 29.2390 + 29.2390i 0 27.0000i 0
157.3 0 3.67423 3.67423i 0 0 0 9.71439 + 9.71439i 0 27.0000i 0
157.4 0 3.67423 3.67423i 0 0 0 52.2300 + 52.2300i 0 27.0000i 0
193.1 0 −3.67423 3.67423i 0 0 0 −21.1834 + 21.1834i 0 27.0000i 0
193.2 0 −3.67423 3.67423i 0 0 0 29.2390 29.2390i 0 27.0000i 0
193.3 0 3.67423 + 3.67423i 0 0 0 9.71439 9.71439i 0 27.0000i 0
193.4 0 3.67423 + 3.67423i 0 0 0 52.2300 52.2300i 0 27.0000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.4
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 300.5.k.d 8
3.b odd 2 1 900.5.l.k 8
5.b even 2 1 60.5.k.a 8
5.c odd 4 1 60.5.k.a 8
5.c odd 4 1 inner 300.5.k.d 8
15.d odd 2 1 180.5.l.b 8
15.e even 4 1 180.5.l.b 8
15.e even 4 1 900.5.l.k 8
20.d odd 2 1 240.5.bg.d 8
20.e even 4 1 240.5.bg.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.k.a 8 5.b even 2 1
60.5.k.a 8 5.c odd 4 1
180.5.l.b 8 15.d odd 2 1
180.5.l.b 8 15.e even 4 1
240.5.bg.d 8 20.d odd 2 1
240.5.bg.d 8 20.e even 4 1
300.5.k.d 8 1.a even 1 1 trivial
300.5.k.d 8 5.c odd 4 1 inner
900.5.l.k 8 3.b odd 2 1
900.5.l.k 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 140 T_{7}^{7} + 9800 T_{7}^{6} - 245480 T_{7}^{5} + 3113188 T_{7}^{4} - 69856960 T_{7}^{3} + 9400947200 T_{7}^{2} - 172367518720 T_{7} + 1580189787136 \) acting on \(S_{5}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 140 T^{7} + \cdots + 1580189787136 \) Copy content Toggle raw display
$11$ \( (T^{4} - 144 T^{3} - 30374 T^{2} + \cdots + 311125216)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 300 T^{7} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} - 1020 T^{7} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{8} + 235000 T^{6} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + 1320 T^{7} + \cdots + 69\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + 3665716 T^{6} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{4} - 736 T^{3} + \cdots - 1071194996864)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 300 T^{7} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + 1740 T^{3} + \cdots - 3221711600000)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 6360 T^{7} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + 4800 T^{7} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + 3900 T^{7} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + 34454956 T^{6} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + 5772 T^{3} + \cdots - 162789583002624)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 920 T^{7} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1800 T^{3} + \cdots - 301327179200000)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 2960 T^{7} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{8} + 156443664 T^{6} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + 12720 T^{7} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + 112438600 T^{6} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} - 15600 T^{7} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
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