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Properties

Label 350.5.d.a
Level $350$
Weight $5$
Character orbit 350.d
Analytic conductor $36.179$
Analytic rank $0$
Dimension $8$
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] = [350,5,Mod(349,350)]
 
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([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.349");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 350.d (of order \(2\), degree \(1\), not 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: \(36.1794870793\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.6850489614336.26
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 60x^{6} + 1800x^{4} + 38340x^{2} + 408321 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{4} q^{3} - 8 q^{4} - \beta_{6} q^{6} + ( - \beta_{7} - 9 \beta_{5} + \cdots + 19 \beta_1) q^{7}+ \cdots + (6 \beta_{2} + 63) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{4} q^{3} - 8 q^{4} - \beta_{6} q^{6} + ( - \beta_{7} - 9 \beta_{5} + \cdots + 19 \beta_1) q^{7}+ \cdots + (2808 \beta_{2} + 7398) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4} + 504 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{4} + 504 q^{9} + 720 q^{11} + 576 q^{14} + 512 q^{16} + 1536 q^{21} - 2448 q^{29} - 4032 q^{36} + 1536 q^{39} - 5760 q^{44} + 6144 q^{46} + 3064 q^{49} - 22272 q^{51} - 4608 q^{56} - 4096 q^{64} - 43056 q^{71} + 6912 q^{74} - 25552 q^{79} - 59256 q^{81} - 12288 q^{84} - 23040 q^{86} + 11136 q^{91} + 59184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 60x^{6} + 1800x^{4} + 38340x^{2} + 408321 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -10\nu^{6} + 813\nu^{4} - 24390\nu^{2} - 191700 ) / 327807 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{6} - 60\nu^{4} - 1278\nu^{2} + 130680 ) / 50787 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 242\nu^{7} - 12390\nu^{5} + 371700\nu^{3} + 15128964\nu ) / 3605877 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -320\nu^{7} + 26016\nu^{5} - 999018\nu^{3} - 2856330\nu ) / 3605877 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} + 120\nu^{4} - 4878\nu^{2} - 38340 ) / 21087 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -724\nu^{7} + 44292\nu^{5} - 891684\nu^{3} - 41414868\nu ) / 3605877 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1124\nu^{7} + 76812\nu^{5} - 2741436\nu^{3} - 7123572\nu ) / 3605877 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{4} + 2\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -33\beta_{5} - 33\beta_{2} + 60\beta _1 + 60 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{7} + 48\beta_{6} - 66\beta_{4} + 126\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -495\beta_{5} + 1539\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -990\beta_{7} + 2529\beta_{6} + 3078\beta_{4} + 7038\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -80487\beta_{5} + 80487\beta_{2} + 223020\beta _1 - 223020 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -136242\beta_{7} + 55755\beta_{6} + 383994\beta_{4} + 160974\beta_{3} ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
1.31383 + 3.17185i
−6.80249 − 2.81768i
6.80249 + 2.81768i
−1.31383 − 3.17185i
1.31383 − 3.17185i
−6.80249 + 2.81768i
6.80249 − 2.81768i
−1.31383 + 3.17185i
− 2.82843i −12.6874 −8.00000 0 35.8854i −48.5729 + 6.45584i 22.6274i 79.9706 0
349.2 − 2.82843i −11.2707 −8.00000 0 31.8784i 20.6077 + 44.4558i 22.6274i 46.0294 0
349.3 − 2.82843i 11.2707 −8.00000 0 − 31.8784i −20.6077 + 44.4558i 22.6274i 46.0294 0
349.4 − 2.82843i 12.6874 −8.00000 0 − 35.8854i 48.5729 + 6.45584i 22.6274i 79.9706 0
349.5 2.82843i −12.6874 −8.00000 0 − 35.8854i −48.5729 − 6.45584i − 22.6274i 79.9706 0
349.6 2.82843i −11.2707 −8.00000 0 − 31.8784i 20.6077 − 44.4558i − 22.6274i 46.0294 0
349.7 2.82843i 11.2707 −8.00000 0 31.8784i −20.6077 − 44.4558i − 22.6274i 46.0294 0
349.8 2.82843i 12.6874 −8.00000 0 35.8854i 48.5729 − 6.45584i − 22.6274i 79.9706 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.8
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 350.5.d.a 8
5.b even 2 1 inner 350.5.d.a 8
5.c odd 4 1 14.5.b.a ✓ 4
5.c odd 4 1 350.5.b.a 4
7.b odd 2 1 inner 350.5.d.a 8
15.e even 4 1 126.5.c.a 4
20.e even 4 1 112.5.c.c 4
35.c odd 2 1 inner 350.5.d.a 8
35.f even 4 1 14.5.b.a ✓ 4
35.f even 4 1 350.5.b.a 4
35.k even 12 2 98.5.d.d 8
35.l odd 12 2 98.5.d.d 8
40.i odd 4 1 448.5.c.e 4
40.k even 4 1 448.5.c.f 4
60.l odd 4 1 1008.5.f.h 4
105.k odd 4 1 126.5.c.a 4
140.j odd 4 1 112.5.c.c 4
280.s even 4 1 448.5.c.e 4
280.y odd 4 1 448.5.c.f 4
420.w even 4 1 1008.5.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.5.b.a ✓ 4 5.c odd 4 1
14.5.b.a ✓ 4 35.f even 4 1
98.5.d.d 8 35.k even 12 2
98.5.d.d 8 35.l odd 12 2
112.5.c.c 4 20.e even 4 1
112.5.c.c 4 140.j odd 4 1
126.5.c.a 4 15.e even 4 1
126.5.c.a 4 105.k odd 4 1
350.5.b.a 4 5.c odd 4 1
350.5.b.a 4 35.f even 4 1
350.5.d.a 8 1.a even 1 1 trivial
350.5.d.a 8 5.b even 2 1 inner
350.5.d.a 8 7.b odd 2 1 inner
350.5.d.a 8 35.c odd 2 1 inner
448.5.c.e 4 40.i odd 4 1
448.5.c.e 4 280.s even 4 1
448.5.c.f 4 40.k even 4 1
448.5.c.f 4 280.y odd 4 1
1008.5.f.h 4 60.l odd 4 1
1008.5.f.h 4 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 288T_{3}^{2} + 20448 \) acting on \(S_{5}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 288 T^{2} + 20448)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{2} - 180 T - 2268)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2784 T^{2} + 1001952)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 116736 T^{2} + 2769149952)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 370848 T^{2} + 3940759008)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 225864 T^{2} + 1191906576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 612 T - 322236)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1990656 T^{2} + 137379151872)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2083976 T^{2} + 731647572496)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 2977977687552)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3773768 T^{2} + 34862864656)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 8659968 T^{2} + 299010613248)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 40888709447184)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 8158764780000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 1641231749088)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1464392 T^{2} + 353915528464)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 10764 T + 19321092)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 5086562360832)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 6388 T - 4378364)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 123161511902688)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 79956812938752)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 58\!\cdots\!52)^{2} \) Copy content Toggle raw display
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