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Properties

Label 300.11.k.c
Level $300$
Weight $11$
Character orbit 300.k
Analytic conductor $190.607$
Analytic rank $0$
Dimension $16$
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] = [300,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.157");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
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: \(190.607175802\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 63831600 x^{13} + 120528248672 x^{12} - 17600989215600 x^{11} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{12}\cdot 5^{16} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 81 \beta_{3} q^{3} + ( - \beta_{8} - 143 \beta_{2}) q^{7} - 19683 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 \beta_{3} q^{3} + ( - \beta_{8} - 143 \beta_{2}) q^{7} - 19683 \beta_1 q^{9} + ( - \beta_{5} - \beta_{4} - 20694) q^{11} + (\beta_{15} + \beta_{12} - 8265 \beta_{3}) q^{13} + ( - 4 \beta_{14} + \cdots - 121158 \beta_{2}) q^{17}+ \cdots + (19683 \beta_{7} + \cdots + 407320002 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 331104 q^{11} + 555984 q^{21} - 140804816 q^{31} + 29553600 q^{41} + 471062304 q^{51} + 3576862832 q^{61} + 1853192640 q^{71} - 6198727824 q^{81} + 7033272240 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 63831600 x^{13} + 120528248672 x^{12} - 17600989215600 x^{11} + \cdots + 38\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29\!\cdots\!05 \nu^{15} + \cdots - 50\!\cdots\!00 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 41\!\cdots\!25 \nu^{15} + \cdots - 17\!\cdots\!00 ) / 69\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55\!\cdots\!67 \nu^{15} + \cdots + 88\!\cdots\!20 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97\!\cdots\!13 \nu^{15} + \cdots + 21\!\cdots\!00 ) / 99\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 55\!\cdots\!67 \nu^{15} + \cdots - 71\!\cdots\!80 ) / 34\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!29 \nu^{15} + \cdots - 13\!\cdots\!40 ) / 37\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 74\!\cdots\!47 \nu^{15} + \cdots - 80\!\cdots\!20 ) / 17\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27\!\cdots\!25 \nu^{15} + \cdots - 30\!\cdots\!88 ) / 17\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!25 \nu^{15} + \cdots - 35\!\cdots\!12 ) / 56\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!87 \nu^{15} + \cdots + 77\!\cdots\!20 ) / 10\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37\!\cdots\!33 \nu^{15} + \cdots + 71\!\cdots\!80 ) / 17\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!45 \nu^{15} + \cdots - 53\!\cdots\!32 ) / 18\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 78\!\cdots\!58 \nu^{15} + \cdots - 75\!\cdots\!68 ) / 56\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 96\!\cdots\!75 \nu^{15} + \cdots - 23\!\cdots\!12 ) / 18\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 99\!\cdots\!92 \nu^{15} + \cdots - 49\!\cdots\!32 ) / 18\!\cdots\!18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - 10\beta_{7} + 26\beta_{6} - 26\beta_{5} - 10\beta_{4} - 3600\beta_{2} ) / 7200 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} + \beta_{14} - 3 \beta_{13} + 30 \beta_{12} + 180 \beta_{11} - 3 \beta_{9} + \cdots + 330885600 \beta_1 ) / 2400 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4860 \beta_{15} + 3240 \beta_{13} - 188460 \beta_{12} - 12503 \beta_{11} - 12503 \beta_{10} + \cdots + 172345320000 ) / 14400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6253 \beta_{15} + 6253 \beta_{14} - 784041 \beta_{13} + 7407600 \beta_{12} + 19497390 \beta_{10} + \cdots - 36158474601600 ) / 1200 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 877370400 \beta_{14} + 18225566993 \beta_{11} - 18225566993 \beta_{10} - 1462598100 \beta_{9} + \cdots + 39\!\cdots\!00 ) / 7200 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 109353214359 \beta_{15} + 109353214359 \beta_{14} + 1189699824123 \beta_{13} + \cdots - 36\!\cdots\!00 \beta_1 ) / 4800 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 148044363536475 \beta_{15} - 398778651424125 \beta_{13} + \cdots - 68\!\cdots\!00 ) / 3600 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 27\!\cdots\!99 \beta_{15} + \cdots + 48\!\cdots\!00 ) / 2400 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20\!\cdots\!60 \beta_{14} + \cdots - 86\!\cdots\!00 ) / 14400 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 54\!\cdots\!21 \beta_{15} + \cdots + 67\!\cdots\!00 \beta_1 ) / 1200 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 33\!\cdots\!00 \beta_{15} + \cdots + 13\!\cdots\!00 ) / 7200 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 16\!\cdots\!65 \beta_{15} + \cdots - 15\!\cdots\!00 ) / 960 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 56\!\cdots\!95 \beta_{14} + \cdots + 19\!\cdots\!00 ) / 3600 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 13\!\cdots\!13 \beta_{15} + \cdots - 10\!\cdots\!00 \beta_1 ) / 2400 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 14\!\cdots\!08 \beta_{15} + \cdots - 46\!\cdots\!00 ) / 2880 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−381.708 − 381.708i
297.652 + 297.652i
−97.2366 − 97.2366i
178.843 + 178.843i
180.068 + 180.068i
−96.0119 − 96.0119i
298.877 + 298.877i
−380.484 − 380.484i
−381.708 + 381.708i
297.652 − 297.652i
−97.2366 + 97.2366i
178.843 − 178.843i
180.068 − 180.068i
−96.0119 + 96.0119i
298.877 − 298.877i
−380.484 + 380.484i
0 −99.2043 + 99.2043i 0 0 0 −17583.1 − 17583.1i 0 − 19683.0i 0
157.2 0 −99.2043 + 99.2043i 0 0 0 −7200.85 − 7200.85i 0 − 19683.0i 0
157.3 0 −99.2043 + 99.2043i 0 0 0 6761.61 + 6761.61i 0 − 19683.0i 0
157.4 0 −99.2043 + 99.2043i 0 0 0 17321.8 + 17321.8i 0 − 19683.0i 0
157.5 0 99.2043 − 99.2043i 0 0 0 −17321.8 − 17321.8i 0 − 19683.0i 0
157.6 0 99.2043 − 99.2043i 0 0 0 −6761.61 − 6761.61i 0 − 19683.0i 0
157.7 0 99.2043 − 99.2043i 0 0 0 7200.85 + 7200.85i 0 − 19683.0i 0
157.8 0 99.2043 − 99.2043i 0 0 0 17583.1 + 17583.1i 0 − 19683.0i 0
193.1 0 −99.2043 − 99.2043i 0 0 0 −17583.1 + 17583.1i 0 19683.0i 0
193.2 0 −99.2043 − 99.2043i 0 0 0 −7200.85 + 7200.85i 0 19683.0i 0
193.3 0 −99.2043 − 99.2043i 0 0 0 6761.61 − 6761.61i 0 19683.0i 0
193.4 0 −99.2043 − 99.2043i 0 0 0 17321.8 − 17321.8i 0 19683.0i 0
193.5 0 99.2043 + 99.2043i 0 0 0 −17321.8 + 17321.8i 0 19683.0i 0
193.6 0 99.2043 + 99.2043i 0 0 0 −6761.61 + 6761.61i 0 19683.0i 0
193.7 0 99.2043 + 99.2043i 0 0 0 7200.85 − 7200.85i 0 19683.0i 0
193.8 0 99.2043 + 99.2043i 0 0 0 17583.1 − 17583.1i 0 19683.0i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.11.k.c ✓ 16
5.b even 2 1 inner 300.11.k.c ✓ 16
5.c odd 4 2 inner 300.11.k.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.11.k.c ✓ 16 1.a even 1 1 trivial
300.11.k.c ✓ 16 5.b even 2 1 inner
300.11.k.c ✓ 16 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + \cdots + 12\!\cdots\!61 \) acting on \(S_{11}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 387420489)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 29\!\cdots\!04)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 89\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 27\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 46\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 42\!\cdots\!99)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 84\!\cdots\!00)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 94\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 58\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 26\!\cdots\!96)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 29\!\cdots\!41)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 79\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 18\!\cdots\!00)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 66\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 26\!\cdots\!61 \) Copy content Toggle raw display
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