Properties

Label 3325.2.a.y
Level 33253325
Weight 22
Character orbit 3325.a
Self dual yes
Analytic conductor 26.55026.550
Analytic rank 00
Dimension 88
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3325,2,Mod(1,3325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3325.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3325=52719 3325 = 5^{2} \cdot 7 \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3325.a (trivial)

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: yes
Analytic conductor: 26.550258672126.5502586721
Analytic rank: 00
Dimension: 88
Coefficient field: Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x82x712x6+24x5+36x470x320x2+32x+3 x^{8} - 2x^{7} - 12x^{6} + 24x^{5} + 36x^{4} - 70x^{3} - 20x^{2} + 32x + 3 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 665)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(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 a basis 1,β1,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ1q2β2q3+(β3β2+1)q4+(β5+β2β1+2)q6q7+(β4+β22β1+1)q8+(β7β2)q9++(3β7+β6β3++5)q99+O(q100) q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_{2} - \beta_1 + 2) q^{6} - q^{7} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{8} + ( - \beta_{7} - \beta_{2}) q^{9}+ \cdots + ( - 3 \beta_{7} + \beta_{6} - \beta_{3} + \cdots + 5) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q2q2+3q3+12q4+9q68q7+7q9+16q11+19q122q13+2q14+24q16+7q17+5q18+8q193q21+10q227q23+q246q26++60q99+O(q100) 8 q - 2 q^{2} + 3 q^{3} + 12 q^{4} + 9 q^{6} - 8 q^{7} + 7 q^{9} + 16 q^{11} + 19 q^{12} - 2 q^{13} + 2 q^{14} + 24 q^{16} + 7 q^{17} + 5 q^{18} + 8 q^{19} - 3 q^{21} + 10 q^{22} - 7 q^{23} + q^{24} - 6 q^{26}+ \cdots + 60 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x82x712x6+24x5+36x470x320x2+32x+3 x^{8} - 2x^{7} - 12x^{6} + 24x^{5} + 36x^{4} - 70x^{3} - 20x^{2} + 32x + 3 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν611ν4+27ν2+4ν5)/4 ( \nu^{6} - 11\nu^{4} + 27\nu^{2} + 4\nu - 5 ) / 4 Copy content Toggle raw display
β3\beta_{3}== (ν611ν4+31ν2+4ν17)/4 ( \nu^{6} - 11\nu^{4} + 31\nu^{2} + 4\nu - 17 ) / 4 Copy content Toggle raw display
β4\beta_{4}== (ν611ν4+4ν3+27ν220ν1)/4 ( \nu^{6} - 11\nu^{4} + 4\nu^{3} + 27\nu^{2} - 20\nu - 1 ) / 4 Copy content Toggle raw display
β5\beta_{5}== (ν7ν611ν5+11ν4+27ν323ν25ν3)/4 ( \nu^{7} - \nu^{6} - 11\nu^{5} + 11\nu^{4} + 27\nu^{3} - 23\nu^{2} - 5\nu - 3 ) / 4 Copy content Toggle raw display
β6\beta_{6}== (ν7+13ν5+2ν447ν316ν2+43ν+10)/4 ( -\nu^{7} + 13\nu^{5} + 2\nu^{4} - 47\nu^{3} - 16\nu^{2} + 43\nu + 10 ) / 4 Copy content Toggle raw display
β7\beta_{7}== (ν7ν613ν5+13ν4+47ν339ν243ν+7)/4 ( \nu^{7} - \nu^{6} - 13\nu^{5} + 13\nu^{4} + 47\nu^{3} - 39\nu^{2} - 43\nu + 7 ) / 4 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3β2+3 \beta_{3} - \beta_{2} + 3 Copy content Toggle raw display
ν3\nu^{3}== β4β2+6β11 \beta_{4} - \beta_{2} + 6\beta _1 - 1 Copy content Toggle raw display
ν4\nu^{4}== β7+β6+7β36β2β1+18 \beta_{7} + \beta_{6} + 7\beta_{3} - 6\beta_{2} - \beta _1 + 18 Copy content Toggle raw display
ν5\nu^{5}== β7+β6+2β5+10β4β38β2+40β111 -\beta_{7} + \beta_{6} + 2\beta_{5} + 10\beta_{4} - \beta_{3} - 8\beta_{2} + 40\beta _1 - 11 Copy content Toggle raw display
ν6\nu^{6}== 11β7+11β6+50β335β215β1+122 11\beta_{7} + 11\beta_{6} + 50\beta_{3} - 35\beta_{2} - 15\beta _1 + 122 Copy content Toggle raw display
ν7\nu^{7}== 11β7+11β6+26β5+83β415β353β2+279β198 -11\beta_{7} + 11\beta_{6} + 26\beta_{5} + 83\beta_{4} - 15\beta_{3} - 53\beta_{2} + 279\beta _1 - 98 Copy content Toggle raw display

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}
1.1
2.66888
2.22346
1.84638
0.777584
−0.0903326
−0.730875
−1.92614
−2.76897
−2.66888 −0.321744 5.12292 0 0.858696 −1.00000 −8.33471 −2.89648 0
1.2 −2.22346 2.66100 2.94379 0 −5.91664 −1.00000 −2.09848 4.08093 0
1.3 −1.84638 −1.55242 1.40913 0 2.86637 −1.00000 1.09097 −0.589984 0
1.4 −0.777584 −2.65879 −1.39536 0 2.06743 −1.00000 2.64018 4.06915 0
1.5 0.0903326 1.28544 −1.99184 0 0.116117 −1.00000 −0.360593 −1.34766 0
1.6 0.730875 −0.878233 −1.46582 0 −0.641878 −1.00000 −2.53308 −2.22871 0
1.7 1.92614 3.21872 1.71000 0 6.19970 −1.00000 −0.558573 7.36017 0
1.8 2.76897 1.24603 5.66718 0 3.45021 −1.00000 10.1543 −1.44742 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 +1 +1
77 +1 +1
1919 1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3325.2.a.y 8
5.b even 2 1 665.2.a.k 8
15.d odd 2 1 5985.2.a.bq 8
35.c odd 2 1 4655.2.a.bi 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
665.2.a.k 8 5.b even 2 1
3325.2.a.y 8 1.a even 1 1 trivial
4655.2.a.bi 8 35.c odd 2 1
5985.2.a.bq 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(3325))S_{2}^{\mathrm{new}}(\Gamma_0(3325)):

T28+2T2712T2624T25+36T24+70T2320T2232T2+3 T_{2}^{8} + 2T_{2}^{7} - 12T_{2}^{6} - 24T_{2}^{5} + 36T_{2}^{4} + 70T_{2}^{3} - 20T_{2}^{2} - 32T_{2} + 3 Copy content Toggle raw display
T383T3711T36+31T35+32T3476T3332T32+48T3+16 T_{3}^{8} - 3T_{3}^{7} - 11T_{3}^{6} + 31T_{3}^{5} + 32T_{3}^{4} - 76T_{3}^{3} - 32T_{3}^{2} + 48T_{3} + 16 Copy content Toggle raw display
T11816T117+64T116+156T1151392T114+1408T113+3456T1123584T113072 T_{11}^{8} - 16T_{11}^{7} + 64T_{11}^{6} + 156T_{11}^{5} - 1392T_{11}^{4} + 1408T_{11}^{3} + 3456T_{11}^{2} - 3584T_{11} - 3072 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+2T7++3 T^{8} + 2 T^{7} + \cdots + 3 Copy content Toggle raw display
33 T83T7++16 T^{8} - 3 T^{7} + \cdots + 16 Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 (T+1)8 (T + 1)^{8} Copy content Toggle raw display
1111 T816T7+3072 T^{8} - 16 T^{7} + \cdots - 3072 Copy content Toggle raw display
1313 T8+2T7+6912 T^{8} + 2 T^{7} + \cdots - 6912 Copy content Toggle raw display
1717 T87T7++171936 T^{8} - 7 T^{7} + \cdots + 171936 Copy content Toggle raw display
1919 (T1)8 (T - 1)^{8} Copy content Toggle raw display
2323 T8+7T7+1294848 T^{8} + 7 T^{7} + \cdots - 1294848 Copy content Toggle raw display
2929 T86T7++35712 T^{8} - 6 T^{7} + \cdots + 35712 Copy content Toggle raw display
3131 T86T7+49152 T^{8} - 6 T^{7} + \cdots - 49152 Copy content Toggle raw display
3737 T8+3T7++5554096 T^{8} + 3 T^{7} + \cdots + 5554096 Copy content Toggle raw display
4141 T8+7T7+207456 T^{8} + 7 T^{7} + \cdots - 207456 Copy content Toggle raw display
4343 T8+23T7+72704 T^{8} + 23 T^{7} + \cdots - 72704 Copy content Toggle raw display
4747 T8144T6+98304 T^{8} - 144 T^{6} + \cdots - 98304 Copy content Toggle raw display
5353 T811T7+92016 T^{8} - 11 T^{7} + \cdots - 92016 Copy content Toggle raw display
5959 T84T7++13154304 T^{8} - 4 T^{7} + \cdots + 13154304 Copy content Toggle raw display
6161 T818T7++445312 T^{8} - 18 T^{7} + \cdots + 445312 Copy content Toggle raw display
6767 T8+4T7+5514688 T^{8} + 4 T^{7} + \cdots - 5514688 Copy content Toggle raw display
7171 T813T7+33408 T^{8} - 13 T^{7} + \cdots - 33408 Copy content Toggle raw display
7373 T8T7+32 T^{8} - T^{7} + \cdots - 32 Copy content Toggle raw display
7979 T810T7+1986048 T^{8} - 10 T^{7} + \cdots - 1986048 Copy content Toggle raw display
8383 T820T7+13879296 T^{8} - 20 T^{7} + \cdots - 13879296 Copy content Toggle raw display
8989 T8+2T7++17318016 T^{8} + 2 T^{7} + \cdots + 17318016 Copy content Toggle raw display
9797 T842T7++51008 T^{8} - 42 T^{7} + \cdots + 51008 Copy content Toggle raw display
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