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Properties

Label 350.4.c.i
Level $350$
Weight $4$
Character orbit 350.c
Analytic conductor $20.651$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [350,4,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, 4, 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, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,4,0,0,52,0,-70] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 i q^{2} + i q^{3} - 4 q^{4} + 2 q^{6} + 7 i q^{7} + 8 i q^{8} + 26 q^{9} - 35 q^{11} - 4 i q^{12} - 58 i q^{13} + 14 q^{14} + 16 q^{16} + 107 i q^{17} - 52 i q^{18} - 23 q^{19} - 7 q^{21} + 70 i q^{22} + \cdots - 910 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 70 q^{11} + 28 q^{14} + 32 q^{16} - 46 q^{19} - 14 q^{21} - 16 q^{24} - 232 q^{26} + 348 q^{29} + 152 q^{31} + 428 q^{34} - 208 q^{36} + 116 q^{39} + 862 q^{41} + 280 q^{44}+ \cdots - 1820 q^{99}+O(q^{100}) \) 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
1.00000i
− 1.00000i
− 2.00000i 1.00000i −4.00000 0 2.00000 7.00000i 8.00000i 26.0000 0
99.2 2.00000i − 1.00000i −4.00000 0 2.00000 − 7.00000i − 8.00000i 26.0000 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 350.4.c.i 2
5.b even 2 1 inner 350.4.c.i 2
5.c odd 4 1 350.4.a.d ✓ 1
5.c odd 4 1 350.4.a.s yes 1
35.f even 4 1 2450.4.a.l 1
35.f even 4 1 2450.4.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.4.a.d ✓ 1 5.c odd 4 1
350.4.a.s yes 1 5.c odd 4 1
350.4.c.i 2 1.a even 1 1 trivial
350.4.c.i 2 5.b even 2 1 inner
2450.4.a.l 1 35.f even 4 1
2450.4.a.bd 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} + 35 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T + 35)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3364 \) Copy content Toggle raw display
$17$ \( T^{2} + 11449 \) Copy content Toggle raw display
$19$ \( (T + 23)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 40000 \) Copy content Toggle raw display
$29$ \( (T - 174)^{2} \) Copy content Toggle raw display
$31$ \( (T - 76)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 33856 \) Copy content Toggle raw display
$41$ \( (T - 431)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 20736 \) Copy content Toggle raw display
$47$ \( T^{2} + 276676 \) Copy content Toggle raw display
$53$ \( T^{2} + 11664 \) Copy content Toggle raw display
$59$ \( (T + 76)^{2} \) Copy content Toggle raw display
$61$ \( (T - 118)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 471969 \) Copy content Toggle raw display
$71$ \( (T - 530)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 89401 \) Copy content Toggle raw display
$79$ \( (T + 402)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 804609 \) Copy content Toggle raw display
$89$ \( (T - 799)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2280100 \) Copy content Toggle raw display
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