Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,4,Mod(99,350)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//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=2527 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

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: 20.650668502020.6506685020
Analytic rank: 00
Dimension: 22
Coefficient field: Q(1)\Q(\sqrt{-1})
comment: defining polynomial
 
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: no (minimal twist has level 14)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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+8iq34q416q6+7iq78iq837q928q1132iq12+18iq1314q14+16q1674iq1774iq1880q1956q2156iq22++1036q99+O(q100) q + 2 i q^{2} + 8 i q^{3} - 4 q^{4} - 16 q^{6} + 7 i q^{7} - 8 i q^{8} - 37 q^{9} - 28 q^{11} - 32 i q^{12} + 18 i q^{13} - 14 q^{14} + 16 q^{16} - 74 i q^{17} - 74 i q^{18} - 80 q^{19} - 56 q^{21} - 56 i q^{22} + \cdots + 1036 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q8q432q674q956q1128q14+32q16160q19112q21+128q2472q26380q29+144q31+296q34+296q36288q39+324q41+224q44++2072q99+O(q100) 2 q - 8 q^{4} - 32 q^{6} - 74 q^{9} - 56 q^{11} - 28 q^{14} + 32 q^{16} - 160 q^{19} - 112 q^{21} + 128 q^{24} - 72 q^{26} - 380 q^{29} + 144 q^{31} + 296 q^{34} + 296 q^{36} - 288 q^{39} + 324 q^{41} + 224 q^{44}+ \cdots + 2072 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.

comment: embeddings in the coefficient field
 
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 8.00000i −4.00000 0 −16.0000 7.00000i 8.00000i −37.0000 0
99.2 2.00000i 8.00000i −4.00000 0 −16.0000 7.00000i 8.00000i −37.0000 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.b 2
5.b even 2 1 inner 350.4.c.b 2
5.c odd 4 1 14.4.a.a 1
5.c odd 4 1 350.4.a.l 1
15.e even 4 1 126.4.a.h 1
20.e even 4 1 112.4.a.a 1
35.f even 4 1 98.4.a.a 1
35.f even 4 1 2450.4.a.bo 1
35.k even 12 2 98.4.c.f 2
35.l odd 12 2 98.4.c.d 2
40.i odd 4 1 448.4.a.b 1
40.k even 4 1 448.4.a.o 1
55.e even 4 1 1694.4.a.g 1
60.l odd 4 1 1008.4.a.s 1
65.h odd 4 1 2366.4.a.h 1
105.k odd 4 1 882.4.a.i 1
105.w odd 12 2 882.4.g.k 2
105.x even 12 2 882.4.g.b 2
140.j odd 4 1 784.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 5.c odd 4 1
98.4.a.a 1 35.f even 4 1
98.4.c.d 2 35.l odd 12 2
98.4.c.f 2 35.k even 12 2
112.4.a.a 1 20.e even 4 1
126.4.a.h 1 15.e even 4 1
350.4.a.l 1 5.c odd 4 1
350.4.c.b 2 1.a even 1 1 trivial
350.4.c.b 2 5.b even 2 1 inner
448.4.a.b 1 40.i odd 4 1
448.4.a.o 1 40.k even 4 1
784.4.a.s 1 140.j odd 4 1
882.4.a.i 1 105.k odd 4 1
882.4.g.b 2 105.x even 12 2
882.4.g.k 2 105.w odd 12 2
1008.4.a.s 1 60.l odd 4 1
1694.4.a.g 1 55.e even 4 1
2366.4.a.h 1 65.h odd 4 1
2450.4.a.bo 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+64 T_{3}^{2} + 64 Copy content Toggle raw display
T11+28 T_{11} + 28 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+64 T^{2} + 64 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+28)2 (T + 28)^{2} Copy content Toggle raw display
1313 T2+324 T^{2} + 324 Copy content Toggle raw display
1717 T2+5476 T^{2} + 5476 Copy content Toggle raw display
1919 (T+80)2 (T + 80)^{2} Copy content Toggle raw display
2323 T2+12544 T^{2} + 12544 Copy content Toggle raw display
2929 (T+190)2 (T + 190)^{2} Copy content Toggle raw display
3131 (T72)2 (T - 72)^{2} Copy content Toggle raw display
3737 T2+119716 T^{2} + 119716 Copy content Toggle raw display
4141 (T162)2 (T - 162)^{2} Copy content Toggle raw display
4343 T2+169744 T^{2} + 169744 Copy content Toggle raw display
4747 T2+576 T^{2} + 576 Copy content Toggle raw display
5353 T2+101124 T^{2} + 101124 Copy content Toggle raw display
5959 (T200)2 (T - 200)^{2} Copy content Toggle raw display
6161 (T+198)2 (T + 198)^{2} Copy content Toggle raw display
6767 T2+512656 T^{2} + 512656 Copy content Toggle raw display
7171 (T392)2 (T - 392)^{2} Copy content Toggle raw display
7373 T2+289444 T^{2} + 289444 Copy content Toggle raw display
7979 (T+240)2 (T + 240)^{2} Copy content Toggle raw display
8383 T2+1149184 T^{2} + 1149184 Copy content Toggle raw display
8989 (T+810)2 (T + 810)^{2} Copy content Toggle raw display
9797 T2+1833316 T^{2} + 1833316 Copy content Toggle raw display
show more
show less