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Properties

Label 350.9.b.a
Level $350$
Weight $9$
Character orbit 350.b
Analytic conductor $142.583$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.251");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(142.582513521\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.3520512.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 120x^{2} + 3438 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + ( - \beta_{2} + 2 \beta_1) q^{3} + 128 q^{4} + ( - 14 \beta_{2} - 6 \beta_1) q^{6} + ( - 147 \beta_{3} + 49 \beta_1 + 1519) q^{7} - 256 \beta_{3} q^{8} + (1062 \beta_{3} - 3423) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + ( - \beta_{2} + 2 \beta_1) q^{3} + 128 q^{4} + ( - 14 \beta_{2} - 6 \beta_1) q^{6} + ( - 147 \beta_{3} + 49 \beta_1 + 1519) q^{7} - 256 \beta_{3} q^{8} + (1062 \beta_{3} - 3423) q^{9} + (1308 \beta_{3} - 3390) q^{11} + ( - 128 \beta_{2} + 256 \beta_1) q^{12} + (360 \beta_{2} - 229 \beta_1) q^{13} + ( - 3038 \beta_{3} - 98 \beta_{2} + \cdots + 9408) q^{14}+ \cdots + ( - 8077464 \beta_{3} + 56055042) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 512 q^{4} + 6076 q^{7} - 13692 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 512 q^{4} + 6076 q^{7} - 13692 q^{9} - 13560 q^{11} + 37632 q^{14} + 65536 q^{16} - 271872 q^{18} - 413952 q^{21} - 334848 q^{22} + 894072 q^{23} + 777728 q^{28} + 317064 q^{29} - 1752576 q^{36} + 2495096 q^{37} + 10228992 q^{39} + 1881600 q^{42} - 9186568 q^{43} - 1735680 q^{44} + 3059712 q^{46} + 931588 q^{49} - 324096 q^{51} + 38727288 q^{53} + 4816896 q^{56} + 30690816 q^{57} - 34661376 q^{58} - 40780740 q^{63} + 8388608 q^{64} + 12320248 q^{67} + 62168712 q^{71} - 34799616 q^{72} - 22957056 q^{74} - 45208968 q^{77} + 116728320 q^{78} + 24889736 q^{79} - 70788348 q^{81} - 52985856 q^{84} + 74680320 q^{86} - 42860544 q^{88} + 38158848 q^{91} + 114441216 q^{92} + 408466944 q^{93} + 228652032 q^{98} + 224220168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 120x^{2} + 3438 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 156\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 108\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{2} + 240 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + 3\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{3} - 240 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 39\beta_{2} - 81\beta_1 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
− 6.87547i
6.87547i
− 8.52807i
8.52807i
−11.3137 − 63.0589i 128.000 0 713.430i 687.442 − 2300.48i −1448.15 2584.58 0
251.2 −11.3137 63.0589i 128.000 0 − 713.430i 687.442 + 2300.48i −1448.15 2584.58 0
251.3 11.3137 − 126.458i 128.000 0 − 1430.71i 2350.56 − 489.571i 1448.15 −9430.58 0
251.4 11.3137 126.458i 128.000 0 1430.71i 2350.56 + 489.571i 1448.15 −9430.58 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.9.b.a 4
5.b even 2 1 14.9.b.a ✓ 4
5.c odd 4 2 350.9.d.a 8
7.b odd 2 1 inner 350.9.b.a 4
15.d odd 2 1 126.9.c.a 4
20.d odd 2 1 112.9.c.c 4
35.c odd 2 1 14.9.b.a ✓ 4
35.f even 4 2 350.9.d.a 8
35.i odd 6 2 98.9.d.a 8
35.j even 6 2 98.9.d.a 8
105.g even 2 1 126.9.c.a 4
140.c even 2 1 112.9.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.9.b.a ✓ 4 5.b even 2 1
14.9.b.a ✓ 4 35.c odd 2 1
98.9.d.a 8 35.i odd 6 2
98.9.d.a 8 35.j even 6 2
112.9.c.c 4 20.d odd 2 1
112.9.c.c 4 140.c even 2 1
126.9.c.a 4 15.d odd 2 1
126.9.c.a 4 105.g even 2 1
350.9.b.a 4 1.a even 1 1 trivial
350.9.b.a 4 7.b odd 2 1 inner
350.9.d.a 8 5.c odd 4 2
350.9.d.a 8 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} + 19968T_{3}^{2} + 63589248 \) Copy content Toggle raw display
\( T_{23}^{2} - 447036T_{23} + 45389086596 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 19968 T^{2} + 63589248 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6780 T - 43255548)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 202796647224192 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 23\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T^{2} - 447036 T + 45389086596)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 158532 T - 580343359356)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( (T^{2} - 1247548 T + 131756883844)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 54\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 2551346607364)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{2} + \cdots + 90844298538372)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 38\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots - 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 599142107080188)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 50828841800444)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 97\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 95\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 34\!\cdots\!12 \) Copy content Toggle raw display
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