↑

Properties

Label 350.3.b.c
Level $350$
Weight $3$
Character orbit 350.b
Analytic conductor $9.537$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,3,Mod(251,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.251");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.b (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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.845277938384896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 50x^{6} - 52x^{5} + 315x^{4} + 48x^{3} - 334x^{2} - 28x + 98 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{5} q^{3} + 2 q^{4} + (\beta_{7} - \beta_{4}) q^{6} + ( - \beta_{6} - \beta_1) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{5} q^{3} + 2 q^{4} + (\beta_{7} - \beta_{4}) q^{6} + ( - \beta_{6} - \beta_1) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} - 4) q^{9} + (\beta_{3} - 7) q^{11} - 2 \beta_{5} q^{12} + ( - 3 \beta_{6} - 2 \beta_{5}) q^{13} + ( - \beta_{7} + \beta_{3}) q^{14} + 4 q^{16} + ( - \beta_{6} - 2 \beta_{5}) q^{17} + ( - \beta_{6} - \beta_{5} + \cdots - 2 \beta_1) q^{18}+ \cdots + ( - 12 \beta_{3} + 112) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} - 36 q^{9} - 60 q^{11} - 4 q^{14} + 32 q^{16} + 56 q^{21} + 92 q^{29} - 72 q^{36} - 204 q^{39} - 120 q^{44} + 104 q^{46} + 284 q^{49} - 212 q^{51} - 8 q^{56} + 64 q^{64} + 504 q^{71} + 112 q^{74} - 692 q^{79} - 136 q^{81} + 112 q^{84} - 280 q^{86} - 44 q^{91} + 944 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 50x^{6} - 52x^{5} + 315x^{4} + 48x^{3} - 334x^{2} - 28x + 98 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -7\nu^{7} - 80\nu^{6} - 352\nu^{5} - 3600\nu^{4} + 1928\nu^{3} - 23504\nu^{2} - 1216\nu + 25998 ) / 2163 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 248\nu^{7} - 35\nu^{6} + 12309\nu^{5} - 14450\nu^{4} + 75467\nu^{3} + 12583\nu^{2} - 125574\nu - 6020 ) / 45423 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -80\nu^{7} - 2\nu^{6} - 3964\nu^{5} + 4133\nu^{4} - 23168\nu^{3} - 3554\nu^{2} + 38780\nu + 686 ) / 2163 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -930\nu^{7} - 49\nu^{6} - 47060\nu^{5} + 45896\nu^{4} - 318150\nu^{3} - 31144\nu^{2} + 146092\nu + 17528 ) / 15141 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -439\nu^{7} - 29\nu^{6} - 22149\nu^{5} + 21355\nu^{4} - 147034\nu^{3} - 21251\nu^{2} + 67704\nu + 11389 ) / 6489 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 484\nu^{7} + 455\nu^{6} + 24456\nu^{5} - 2185\nu^{4} + 141526\nu^{3} + 164579\nu^{2} - 69510\nu - 83965 ) / 6489 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -163\nu^{7} - 156\nu^{6} - 8226\nu^{5} + 602\nu^{4} - 47143\nu^{3} - 56895\nu^{2} + 23214\nu + 28994 ) / 2163 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{5} - \beta_{3} - 7\beta_{2} + 2\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -25\beta_{7} - 27\beta_{6} + 10\beta_{5} - 13\beta_{4} + 3\beta_{3} + 20\beta_{2} - 3\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 76\beta_{7} + 42\beta_{6} - 60\beta_{5} + 28\beta_{4} + 49\beta_{3} + 336\beta_{2} - 72\beta _1 + 492 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1041\beta_{7} + 1192\beta_{6} - 286\beta_{5} + 471\beta_{4} - 190\beta_{3} - 1297\beta_{2} + 255\beta _1 - 1750 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4816 \beta_{7} - 3528 \beta_{6} + 3230 \beta_{5} - 2093 \beta_{4} - 1978 \beta_{3} - 13538 \beta_{2} + \cdots - 19162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 40008 \beta_{7} - 48744 \beta_{6} + 7808 \beta_{5} - 17832 \beta_{4} + 11144 \beta_{3} + \cdots + 107114 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.707107 − 2.75855i
0.707107 − 0.0750782i
0.707107 + 0.0750782i
0.707107 + 2.75855i
−0.707107 − 0.181255i
−0.707107 − 6.65972i
−0.707107 + 6.65972i
−0.707107 + 0.181255i
−1.41421 − 4.76222i 2.00000 0 6.73480i 6.84393 + 1.46990i −2.82843 −13.6788 0
251.2 −1.41421 − 2.07875i 2.00000 0 2.93980i −6.13682 + 3.36740i −2.82843 4.67878 0
251.3 −1.41421 2.07875i 2.00000 0 − 2.93980i −6.13682 − 3.36740i −2.82843 4.67878 0
251.4 −1.41421 4.76222i 2.00000 0 − 6.73480i 6.84393 − 1.46990i −2.82843 −13.6788 0
251.5 1.41421 − 4.76222i 2.00000 0 − 6.73480i −6.84393 + 1.46990i 2.82843 −13.6788 0
251.6 1.41421 − 2.07875i 2.00000 0 − 2.93980i 6.13682 + 3.36740i 2.82843 4.67878 0
251.7 1.41421 2.07875i 2.00000 0 2.93980i 6.13682 − 3.36740i 2.82843 4.67878 0
251.8 1.41421 4.76222i 2.00000 0 6.73480i −6.84393 − 1.46990i 2.82843 −13.6788 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.b.c 8
5.b even 2 1 inner 350.3.b.c 8
5.c odd 4 2 70.3.d.a ✓ 8
7.b odd 2 1 inner 350.3.b.c 8
15.e even 4 2 630.3.h.d 8
20.e even 4 2 560.3.p.g 8
35.c odd 2 1 inner 350.3.b.c 8
35.f even 4 2 70.3.d.a ✓ 8
35.k even 12 4 490.3.h.a 16
35.l odd 12 4 490.3.h.a 16
105.k odd 4 2 630.3.h.d 8
140.j odd 4 2 560.3.p.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.d.a ✓ 8 5.c odd 4 2
70.3.d.a ✓ 8 35.f even 4 2
350.3.b.c 8 1.a even 1 1 trivial
350.3.b.c 8 5.b even 2 1 inner
350.3.b.c 8 7.b odd 2 1 inner
350.3.b.c 8 35.c odd 2 1 inner
490.3.h.a 16 35.k even 12 4
490.3.h.a 16 35.l odd 12 4
560.3.p.g 8 20.e even 4 2
560.3.p.g 8 140.j odd 4 2
630.3.h.d 8 15.e even 4 2
630.3.h.d 8 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} + 27T_{3}^{2} + 98 \) Copy content Toggle raw display
\( T_{23}^{4} - 506T_{23}^{2} + 7056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 27 T^{2} + 98)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 142 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} + 15 T - 28)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 321 T^{2} + 21632)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 129 T^{2} + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1048 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 506 T^{2} + 7056)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 23 T - 626)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1096 T^{2} + 294912)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 1544 T^{2} + 331776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 10636 T^{2} + 28215072)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1562 T^{2} + 197136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 5227 T^{2} + 642978)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 7328 T^{2} + 2985984)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 10000 T^{2} + 11520000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4428 T^{2} + 4792608)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 338 T^{2} + 28224)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 126 T + 3632)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12100 T^{2} + 16866432)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 173 T + 5376)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1878 T^{2} + 137288)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5704 T^{2} + 8128512)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 15361 T^{2} + 23722272)^{2} \) Copy content Toggle raw display
show more
show less