↑

Properties

Label 300.8.d.h
Level $300$
Weight $8$
Character orbit 300.d
Analytic conductor $93.716$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [300,8,Mod(49,300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-4374,0,6276] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(93.7155076452\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 1145x^{4} + 319204x^{2} + 15920100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 27 \beta_1 q^{3} + ( - \beta_{2} + 117 \beta_1) q^{7} - 729 q^{9} + ( - \beta_{4} - \beta_{3} + 1046) q^{11} + (\beta_{5} + 6 \beta_{2} - 1195 \beta_1) q^{13} + (\beta_{5} - 7 \beta_{2} + 1742 \beta_1) q^{17}+ \cdots + (729 \beta_{4} + 729 \beta_{3} - 762534) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4374 q^{9} + 6276 q^{11} - 42390 q^{19} + 18954 q^{21} - 53196 q^{29} - 286926 q^{31} - 193590 q^{39} - 962640 q^{41} - 3382476 q^{49} + 282204 q^{51} - 3981516 q^{59} + 215442 q^{61} - 2183436 q^{69}+ \cdots - 4575204 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 1145x^{4} + 319204x^{2} + 15920100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 5135\nu^{3} + 2605474\nu ) / 18202380 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 5135\nu^{3} + 57212614\nu ) / 910119 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -30\nu^{4} - 17190\nu^{2} + 165320 ) / 2281 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 220\nu^{4} + 217300\nu^{2} + 33610920 ) / 2281 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -416\nu^{5} - 402600\nu^{3} - 73211704\nu ) / 43339 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 20\beta_1 ) / 60 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{4} + 22\beta_{3} - 45800 ) / 120 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} - 1166\beta_{2} + 547480\beta_1 ) / 120 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -1719\beta_{4} - 21730\beta_{3} + 26904680 ) / 120 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -15405\beta_{5} + 776462\beta_{2} - 522805240\beta_1 ) / 120 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
18.3302i
7.97258i
− 27.3028i
27.3028i
− 7.97258i
− 18.3302i
0 − 27.0000i 0 0 0 − 1002.81i 0 −729.000 0
49.2 0 − 27.0000i 0 0 0 − 381.355i 0 −729.000 0
49.3 0 − 27.0000i 0 0 0 1735.17i 0 −729.000 0
49.4 0 27.0000i 0 0 0 − 1735.17i 0 −729.000 0
49.5 0 27.0000i 0 0 0 381.355i 0 −729.000 0
49.6 0 27.0000i 0 0 0 1002.81i 0 −729.000 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 49.6
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.h 6
5.b even 2 1 inner 300.8.d.h 6
5.c odd 4 1 300.8.a.m ✓ 3
5.c odd 4 1 300.8.a.n yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.8.a.m ✓ 3 5.c odd 4 1
300.8.a.n yes 3 5.c odd 4 1
300.8.d.h 6 1.a even 1 1 trivial
300.8.d.h 6 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 \(S_{8}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{6} + 4161867T_{7}^{4} + 3611876246163T_{7}^{2} + 440331532447984969 \) Copy content Toggle raw display
\( T_{11}^{3} - 3138T_{11}^{2} - 65608452T_{11} + 287673763464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 44\!\cdots\!69 \) Copy content Toggle raw display
$11$ \( (T^{3} - 3138 T^{2} + \cdots + 287673763464)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 1414307134375)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots - 773565070226904)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 15\!\cdots\!39)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 56\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots - 12\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 86\!\cdots\!43)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 27\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 56\!\cdots\!88)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
show more
show less