Properties

Label 350.10.c.j
Level $350$
Weight $10$
Character orbit 350.c
Analytic conductor $180.263$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [350,10,Mod(99,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.99"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-1024,0,448] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(180.262542657\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{2305})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1153x^{2} + 331776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 14)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 16 \beta_1 q^{2} + (\beta_{2} - 7 \beta_1) q^{3} - 256 q^{4} + ( - 16 \beta_{3} + 112) q^{6} - 2401 \beta_1 q^{7} - 4096 \beta_1 q^{8} + (14 \beta_{3} - 37991) q^{9} + (210 \beta_{3} + 22470) q^{11}+ \cdots + ( - 7663530 \beta_{3} - 684240270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1024 q^{4} + 448 q^{6} - 151964 q^{9} + 89880 q^{11} + 153664 q^{14} + 262144 q^{16} - 1017548 q^{19} - 67228 q^{21} - 114688 q^{24} - 3209024 q^{26} - 4012656 q^{29} + 4377464 q^{31} - 27853056 q^{34}+ \cdots - 2736961080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 1153x^{2} + 331776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 577\nu ) / 576 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{3} + 8645\nu ) / 576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 10\nu^{2} + 5765 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 5\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 5765 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -577\beta_{2} + 8645\beta_1 ) / 10 \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
23.5052i
24.5052i
24.5052i
23.5052i
16.0000i 233.052i −256.000 0 −3728.83 2401.00i 4096.00i −34630.3 0
99.2 16.0000i 247.052i −256.000 0 3952.83 2401.00i 4096.00i −41351.7 0
99.3 16.0000i 247.052i −256.000 0 3952.83 2401.00i 4096.00i −41351.7 0
99.4 16.0000i 233.052i −256.000 0 −3728.83 2401.00i 4096.00i −34630.3 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
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 350.10.c.j 4
5.b even 2 1 inner 350.10.c.j 4
5.c odd 4 1 14.10.a.c 2
5.c odd 4 1 350.10.a.j 2
15.e even 4 1 126.10.a.o 2
20.e even 4 1 112.10.a.c 2
35.f even 4 1 98.10.a.e 2
35.k even 12 2 98.10.c.h 4
35.l odd 12 2 98.10.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.c 2 5.c odd 4 1
98.10.a.e 2 35.f even 4 1
98.10.c.h 4 35.k even 12 2
98.10.c.j 4 35.l odd 12 2
112.10.a.c 2 20.e even 4 1
126.10.a.o 2 15.e even 4 1
350.10.a.j 2 5.c odd 4 1
350.10.c.j 4 1.a even 1 1 trivial
350.10.c.j 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 115348T_{3}^{2} + 3314995776 \) acting on \(S_{10}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 3314995776 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 44940 T - 2036361600)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 57\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{2} + 508774 T - 352577596856)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots - 31904129519604)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 2367849772544)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 527816477266884)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{2} + \cdots + 15\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 68\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 50\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
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