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Properties

Label 350.4.c.a
Level 350350
Weight 44
Character orbit 350.c
Analytic conductor 20.65120.651
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

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 N == 350=2⋅52⋅7 350 = 2 \cdot 5^{2} \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 350.c (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,-40,0,0,-146,0,18] 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: 20.650668502020.6506685020
Analytic rank: 00
Dimension: 22
Coefficient field: Q(−1)\Q(\sqrt{-1})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-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 qq-expansion are expressed in terms of i=−1i = \sqrt{-1}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+2iq2+10iq3−4q4−20q6−7iq7−8iq8−73q9+9q11−40iq12−52iq13+14q14+16q16−96iq17−146iq18+10q19+70q21+18iq22+⋯−657q99+O(q100) q + 2 i q^{2} + 10 i q^{3} - 4 q^{4} - 20 q^{6} - 7 i q^{7} - 8 i q^{8} - 73 q^{9} + 9 q^{11} - 40 i q^{12} - 52 i q^{13} + 14 q^{14} + 16 q^{16} - 96 i q^{17} - 146 i q^{18} + 10 q^{19} + 70 q^{21} + 18 i q^{22} + \cdots - 657 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q−8q4−40q6−146q9+18q11+28q14+32q16+20q19+140q21+160q24+208q26−378q29−464q31+384q34+584q36+1040q39−876q41+⋯−1314q99+O(q100) 2 q - 8 q^{4} - 40 q^{6} - 146 q^{9} + 18 q^{11} + 28 q^{14} + 32 q^{16} + 20 q^{19} + 140 q^{21} + 160 q^{24} + 208 q^{26} - 378 q^{29} - 464 q^{31} + 384 q^{34} + 584 q^{36} + 1040 q^{39} - 876 q^{41}+ \cdots - 1314 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

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

nn 101101 127127
χ(n)\chi(n) 11 −1-1

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
99.1
− 1.00000i
1.00000i
− 2.00000i − 10.0000i −4.00000 0 −20.0000 7.00000i 8.00000i −73.0000 0
99.2 2.00000i 10.0000i −4.00000 0 −20.0000 − 7.00000i − 8.00000i −73.0000 0
nn: 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.a 2
5.b even 2 1 inner 350.4.c.a 2
5.c odd 4 1 350.4.a.j ✓ 1
5.c odd 4 1 350.4.a.k yes 1
35.f even 4 1 2450.4.a.a 1
35.f even 4 1 2450.4.a.bp 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.4.a.j ✓ 1 5.c odd 4 1
350.4.a.k yes 1 5.c odd 4 1
350.4.c.a 2 1.a even 1 1 trivial
350.4.c.a 2 5.b even 2 1 inner
2450.4.a.a 1 35.f even 4 1
2450.4.a.bp 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 S4new(350,[χ])S_{4}^{\mathrm{new}}(350, [\chi]):

T32+100 T_{3}^{2} + 100 Copy content Toggle raw display
T11−9 T_{11} - 9 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+4 T^{2} + 4 Copy content Toggle raw display
33 T2+100 T^{2} + 100 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+49 T^{2} + 49 Copy content Toggle raw display
1111 (T−9)2 (T - 9)^{2} Copy content Toggle raw display
1313 T2+2704 T^{2} + 2704 Copy content Toggle raw display
1717 T2+9216 T^{2} + 9216 Copy content Toggle raw display
1919 (T−10)2 (T - 10)^{2} Copy content Toggle raw display
2323 T2+5625 T^{2} + 5625 Copy content Toggle raw display
2929 (T+189)2 (T + 189)^{2} Copy content Toggle raw display
3131 (T+232)2 (T + 232)^{2} Copy content Toggle raw display
3737 T2+93025 T^{2} + 93025 Copy content Toggle raw display
4141 (T+438)2 (T + 438)^{2} Copy content Toggle raw display
4343 T2+124609 T^{2} + 124609 Copy content Toggle raw display
4747 T2+236196 T^{2} + 236196 Copy content Toggle raw display
5353 T2+125316 T^{2} + 125316 Copy content Toggle raw display
5959 (T−672)2 (T - 672)^{2} Copy content Toggle raw display
6161 (T−206)2 (T - 206)^{2} Copy content Toggle raw display
6767 T2+358801 T^{2} + 358801 Copy content Toggle raw display
7171 (T+471)2 (T + 471)^{2} Copy content Toggle raw display
7373 T2+376996 T^{2} + 376996 Copy content Toggle raw display
7979 (T+743)2 (T + 743)^{2} Copy content Toggle raw display
8383 T2+992016 T^{2} + 992016 Copy content Toggle raw display
8989 (T+180)2 (T + 180)^{2} Copy content Toggle raw display
9797 T2+33856 T^{2} + 33856 Copy content Toggle raw display
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