↑

Properties

Label 300.7.g.h
Level $300$
Weight $7$
Character orbit 300.g
Analytic conductor $69.016$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,7,Mod(101,300)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.101");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\), degree \(1\), 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: \(69.0162250860\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 202x^{6} + 620x^{5} + 12167x^{4} - 25372x^{3} - 177926x^{2} + 190716x + 977814 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{9}\cdot 5^{9} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 3) q^{3} + ( - \beta_{2} + 70) q^{7} + ( - \beta_{7} - 2 \beta_{3} + \cdots + 188) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 3) q^{3} + ( - \beta_{2} + 70) q^{7} + ( - \beta_{7} - 2 \beta_{3} + \cdots + 188) q^{9}+ \cdots + (333 \beta_{7} + 519 \beta_{6} + \cdots + 337578) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{3} + 560 q^{7} + 1492 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 20 q^{3} + 560 q^{7} + 1492 q^{9} + 6440 q^{13} - 15272 q^{19} - 868 q^{21} + 18620 q^{27} + 35032 q^{31} + 111120 q^{33} - 99880 q^{37} + 39608 q^{39} + 161000 q^{43} + 202560 q^{49} + 429120 q^{51} + 27160 q^{57} - 135608 q^{61} - 377240 q^{63} - 404920 q^{67} - 254940 q^{69} + 356960 q^{73} + 707704 q^{79} - 1198112 q^{81} + 1528440 q^{87} - 2004112 q^{91} + 467920 q^{93} + 1326320 q^{97} + 2650080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 202x^{6} + 620x^{5} + 12167x^{4} - 25372x^{3} - 177926x^{2} + 190716x + 977814 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12671 \nu^{7} - 1833194 \nu^{6} + 1945513 \nu^{5} + 346605310 \nu^{4} + 369000460 \nu^{3} + \cdots + 162559650069 ) / 281404449 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4237 \nu^{7} - 217442 \nu^{6} + 105199 \nu^{5} + 32090770 \nu^{4} - 70438880 \nu^{3} + \cdots - 30624072588 ) / 93801483 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23087 \nu^{7} - 111002 \nu^{6} + 5186194 \nu^{5} + 22636834 \nu^{4} - 334471142 \nu^{3} + \cdots + 11257731684 ) / 281404449 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12628 \nu^{7} - 445821 \nu^{6} - 2551343 \nu^{5} + 98340753 \nu^{4} + 90817366 \nu^{3} + \cdots + 54107138646 ) / 31267161 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 116552 \nu^{7} + 1018460 \nu^{6} - 29096440 \nu^{5} - 195438142 \nu^{4} + 2084250656 \nu^{3} + \cdots - 111179268045 ) / 93801483 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 274691 \nu^{7} - 1320718 \nu^{6} - 58719769 \nu^{5} + 196088852 \nu^{4} + 3420148484 \nu^{3} + \cdots + 22562586738 ) / 93801483 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1513381 \nu^{7} - 6801569 \nu^{6} - 275191817 \nu^{5} + 977705293 \nu^{4} + 14010431656 \nu^{3} + \cdots + 236183023836 ) / 281404449 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 66\beta_{7} - 3\beta_{6} + 431\beta_{5} + 6\beta_{4} + 10736\beta_{3} - 189\beta_{2} - 116\beta _1 + 21599 ) / 54000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 6 \beta_{7} - 237 \beta_{6} + 929 \beta_{5} + 294 \beta_{4} + 8444 \beta_{3} + 3069 \beta_{2} + \cdots + 1413281 ) / 27000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3714 \beta_{7} - 3507 \beta_{6} + 21589 \beta_{5} - 546 \beta_{4} + 436894 \beta_{3} + \cdots + 1899196 ) / 27000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1614 \beta_{7} - 13872 \beta_{6} + 33899 \beta_{5} + 18114 \beta_{4} + 261914 \beta_{3} + \cdots + 30415361 ) / 6750 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 504042 \beta_{7} - 672261 \beta_{6} + 2131477 \beta_{5} - 2778 \beta_{4} + 40995952 \beta_{3} + \cdots + 281191003 ) / 27000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 724038 \beta_{7} - 4390449 \beta_{6} + 8513803 \beta_{5} + 5224638 \beta_{4} + 55356868 \beta_{3} + \cdots + 5609235547 ) / 13500 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 33308346 \beta_{7} - 49648278 \beta_{6} + 106673876 \beta_{5} + 6049626 \beta_{4} + \cdots + 18240429779 ) / 13500 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
10.2694 − 0.707107i
10.2694 + 0.707107i
3.67163 − 0.707107i
3.67163 + 0.707107i
−9.26938 + 0.707107i
−9.26938 − 0.707107i
−2.67163 + 0.707107i
−2.67163 − 0.707107i
0 −25.0836 − 9.99060i 0 0 0 −134.460 0 529.376 + 501.201i 0
101.2 0 −25.0836 + 9.99060i 0 0 0 −134.460 0 529.376 − 501.201i 0
101.3 0 −11.2485 − 24.5453i 0 0 0 414.697 0 −475.944 + 552.194i 0
101.4 0 −11.2485 + 24.5453i 0 0 0 414.697 0 −475.944 − 552.194i 0
101.5 0 22.1779 − 15.3993i 0 0 0 −437.053 0 254.720 − 683.051i 0
101.6 0 22.1779 + 15.3993i 0 0 0 −437.053 0 254.720 + 683.051i 0
101.7 0 24.1542 − 12.0655i 0 0 0 436.815 0 437.848 − 582.864i 0
101.8 0 24.1542 + 12.0655i 0 0 0 436.815 0 437.848 + 582.864i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.7.g.h 8
3.b odd 2 1 inner 300.7.g.h 8
5.b even 2 1 60.7.g.a ✓ 8
5.c odd 4 2 300.7.b.e 16
15.d odd 2 1 60.7.g.a ✓ 8
15.e even 4 2 300.7.b.e 16
20.d odd 2 1 240.7.l.c 8
60.h even 2 1 240.7.l.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.g.a ✓ 8 5.b even 2 1
60.7.g.a ✓ 8 15.d odd 2 1
240.7.l.c 8 20.d odd 2 1
240.7.l.c 8 60.h even 2 1
300.7.b.e 16 5.c odd 4 2
300.7.b.e 16 15.e even 4 2
300.7.g.h 8 1.a even 1 1 trivial
300.7.g.h 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 280T_{7}^{3} - 246738T_{7}^{2} + 53487320T_{7} + 10645216936 \) acting on \(S_{7}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 280 T^{3} + \cdots + 10645216936)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} - 3220 T^{3} + \cdots - 261697247744)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 43\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 26\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 48\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 93\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 23\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 63\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
show more
show less