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Properties

Label 3.25.b.b
Level 33
Weight 2525
Character orbit 3.b
Analytic conductor 10.94910.949
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,25,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: N N == 3 3
Weight: k k == 25 25
Character orbit: [χ][\chi] == 3.b (of order 22, degree 11, 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: 10.949014567710.9490145677
Analytic rank: 00
Dimension: 66
Coefficient field: Q[x]/(x6+⋯ )\mathbb{Q}[x]/(x^{6} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x6+253102x4+17425276096x2+250659115499520 x^{6} + 253102x^{4} + 17425276096x^{2} + 250659115499520 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 213⋅324⋅52 2^{13}\cdot 3^{24}\cdot 5^{2}
Twist minimal: yes
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 a basis 1,β1,…,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+(β2+16β1−102807)q3+(β5−8β3+⋯−10557800)q4+(−4β5+β4+⋯−16866β1)q5+(−515β5−3β4+⋯−445648392)q6+⋯+(−33 ⁣⋯ ⁣72β5+⋯−30 ⁣⋯ ⁣20)q99+O(q100) q + \beta_1 q^{2} + (\beta_{2} + 16 \beta_1 - 102807) q^{3} + (\beta_{5} - 8 \beta_{3} + \cdots - 10557800) q^{4} + ( - 4 \beta_{5} + \beta_{4} + \cdots - 16866 \beta_1) q^{5} + ( - 515 \beta_{5} - 3 \beta_{4} + \cdots - 445648392) q^{6}+ \cdots + ( - 33\!\cdots\!72 \beta_{5} + \cdots - 30\!\cdots\!20) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 6q−616842q3−63346800q4−2673890352q6+1988064876q7−791186949882q9+2770823311200q10−18492861842352q12−73542438063924q13+227151045057600q15−433335898904448q16+⋯−18 ⁣⋯ ⁣20q99+O(q100) 6 q - 616842 q^{3} - 63346800 q^{4} - 2673890352 q^{6} + 1988064876 q^{7} - 791186949882 q^{9} + 2770823311200 q^{10} - 18492861842352 q^{12} - 73542438063924 q^{13} + 227151045057600 q^{15} - 433335898904448 q^{16}+ \cdots - 18\!\cdots\!20 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6+253102x4+17425276096x2+250659115499520 x^{6} + 253102x^{4} + 17425276096x^{2} + 250659115499520 : Copy content Toggle raw display

β1\beta_{1}== 18ν 18\nu Copy content Toggle raw display
β2\beta_{2}== (927ν5−22464ν4+158192658ν3−8746814592ν2+4583765471040ν−519220416540672)/1041302528 ( 927 \nu^{5} - 22464 \nu^{4} + 158192658 \nu^{3} - 8746814592 \nu^{2} + 4583765471040 \nu - 519220416540672 ) / 1041302528 Copy content Toggle raw display
β3\beta_{3}== (−927ν5+162432ν4−158192658ν3+11341271808ν2+⋯−624717182509056)/520651264 ( - 927 \nu^{5} + 162432 \nu^{4} - 158192658 \nu^{3} + 11341271808 \nu^{2} + \cdots - 624717182509056 ) / 520651264 Copy content Toggle raw display
β4\beta_{4}== (129501ν5+252288ν4+36839351862ν3+46328530176ν2+⋯+14 ⁣⋯ ⁣32)/520651264 ( 129501 \nu^{5} + 252288 \nu^{4} + 36839351862 \nu^{3} + 46328530176 \nu^{2} + \cdots + 14\!\cdots\!32 ) / 520651264 Copy content Toggle raw display
β5\beta_{5}== (927ν5+2217024ν4+158192658ν3+370146519936ν2+⋯+96 ⁣⋯ ⁣28)/1041302528 ( 927 \nu^{5} + 2217024 \nu^{4} + 158192658 \nu^{3} + 370146519936 \nu^{2} + \cdots + 96\!\cdots\!28 ) / 1041302528 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β1)/18 ( \beta_1 ) / 18 Copy content Toggle raw display
ν2\nu^{2}== (β5−8β3−17β2+8β1−27335016)/324 ( \beta_{5} - 8\beta_{3} - 17\beta_{2} + 8\beta _1 - 27335016 ) / 324 Copy content Toggle raw display
ν3\nu^{3}== (−736β5+103β4+3393β3−21256β2−18690797β1)/2916 ( -736\beta_{5} + 103\beta_{4} + 3393\beta_{3} - 21256\beta_{2} - 18690797\beta_1 ) / 2916 Copy content Toggle raw display
ν4\nu^{4}== (−250237β5+18272248β3+36794733β2−18272248β1+42588282369096)/4374 ( -250237\beta_{5} + 18272248\beta_{3} + 36794733\beta_{2} - 18272248\beta _1 + 42588282369096 ) / 4374 Copy content Toggle raw display
ν5\nu^{5}== (103238176β5−8788481β4−481592855β3+3026837432β2+1194462231243β1)/1458 ( 103238176\beta_{5} - 8788481\beta_{4} - 481592855\beta_{3} + 3026837432\beta_{2} + 1194462231243\beta_1 ) / 1458 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/3Z)×\left(\mathbb{Z}/3\mathbb{Z}\right)^\times.

nn 22
χ(n)\chi(n) −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}
2.1
− 380.197i
− 298.476i
− 139.516i
139.516i
298.476i
380.197i
− 6843.54i 162008. − 506145.i −3.00569e7 3.79015e8i −3.46383e9 − 1.10871e9i −1.14171e10 9.08798e10i −2.29937e11 − 1.63999e11i 2.59380e12
2.2 − 5372.56i −24323.9 + 530884.i −1.20872e7 − 5.28531e7i 2.85221e9 + 1.30681e8i 2.07202e10 − 2.51975e10i −2.81246e11 − 2.58263e10i −2.83957e11
2.3 − 2511.29i −446105. − 288825.i 1.04706e7 − 3.68111e8i −7.25325e8 + 1.12030e9i −8.30906e9 − 6.84273e10i 1.15589e11 + 2.57693e11i −9.24434e11
2.4 2511.29i −446105. + 288825.i 1.04706e7 3.68111e8i −7.25325e8 − 1.12030e9i −8.30906e9 6.84273e10i 1.15589e11 − 2.57693e11i −9.24434e11
2.5 5372.56i −24323.9 − 530884.i −1.20872e7 5.28531e7i 2.85221e9 − 1.30681e8i 2.07202e10 2.51975e10i −2.81246e11 + 2.58263e10i −2.83957e11
2.6 6843.54i 162008. + 506145.i −3.00569e7 − 3.79015e8i −3.46383e9 + 1.10871e9i −1.14171e10 − 9.08798e10i −2.29937e11 + 1.63999e11i 2.59380e12
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.25.b.b ✓ 6
3.b odd 2 1 inner 3.25.b.b ✓ 6
4.b odd 2 1 48.25.e.b 6
12.b even 2 1 48.25.e.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.25.b.b ✓ 6 1.a even 1 1 trivial
3.25.b.b ✓ 6 3.b odd 2 1 inner
48.25.e.b 6 4.b odd 2 1
48.25.e.b 6 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T26+82005048T24+1829235783453696T22+8525473984011546132480 T_{2}^{6} + 82005048T_{2}^{4} + 1829235783453696T_{2}^{2} + 8525473984011546132480 acting on S25new(3,[χ])S_{25}^{\mathrm{new}}(3, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6+⋯+85 ⁣⋯ ⁣80 T^{6} + \cdots + 85\!\cdots\!80 Copy content Toggle raw display
33 T6+⋯+22 ⁣⋯ ⁣41 T^{6} + \cdots + 22\!\cdots\!41 Copy content Toggle raw display
55 T6+⋯+54 ⁣⋯ ⁣00 T^{6} + \cdots + 54\!\cdots\!00 Copy content Toggle raw display
77 (T3+⋯−19 ⁣⋯ ⁣00)2 (T^{3} + \cdots - 19\!\cdots\!00)^{2} Copy content Toggle raw display
1111 T6+⋯+21 ⁣⋯ ⁣00 T^{6} + \cdots + 21\!\cdots\!00 Copy content Toggle raw display
1313 (T3+⋯−73 ⁣⋯ ⁣00)2 (T^{3} + \cdots - 73\!\cdots\!00)^{2} Copy content Toggle raw display
1717 T6+⋯+54 ⁣⋯ ⁣20 T^{6} + \cdots + 54\!\cdots\!20 Copy content Toggle raw display
1919 (T3+⋯+24 ⁣⋯ ⁣72)2 (T^{3} + \cdots + 24\!\cdots\!72)^{2} Copy content Toggle raw display
2323 T6+⋯+10 ⁣⋯ ⁣80 T^{6} + \cdots + 10\!\cdots\!80 Copy content Toggle raw display
2929 T6+⋯+72 ⁣⋯ ⁣00 T^{6} + \cdots + 72\!\cdots\!00 Copy content Toggle raw display
3131 (T3+⋯+26 ⁣⋯ ⁣72)2 (T^{3} + \cdots + 26\!\cdots\!72)^{2} Copy content Toggle raw display
3737 (T3+⋯−85 ⁣⋯ ⁣00)2 (T^{3} + \cdots - 85\!\cdots\!00)^{2} Copy content Toggle raw display
4141 T6+⋯+12 ⁣⋯ ⁣00 T^{6} + \cdots + 12\!\cdots\!00 Copy content Toggle raw display
4343 (T3+⋯−31 ⁣⋯ ⁣00)2 (T^{3} + \cdots - 31\!\cdots\!00)^{2} Copy content Toggle raw display
4747 T6+⋯+15 ⁣⋯ ⁣20 T^{6} + \cdots + 15\!\cdots\!20 Copy content Toggle raw display
5353 T6+⋯+53 ⁣⋯ ⁣20 T^{6} + \cdots + 53\!\cdots\!20 Copy content Toggle raw display
5959 T6+⋯+77 ⁣⋯ ⁣00 T^{6} + \cdots + 77\!\cdots\!00 Copy content Toggle raw display
6161 (T3+⋯−18 ⁣⋯ ⁣88)2 (T^{3} + \cdots - 18\!\cdots\!88)^{2} Copy content Toggle raw display
6767 (T3+⋯−20 ⁣⋯ ⁣00)2 (T^{3} + \cdots - 20\!\cdots\!00)^{2} Copy content Toggle raw display
7171 T6+⋯+12 ⁣⋯ ⁣00 T^{6} + \cdots + 12\!\cdots\!00 Copy content Toggle raw display
7373 (T3+⋯+14 ⁣⋯ ⁣00)2 (T^{3} + \cdots + 14\!\cdots\!00)^{2} Copy content Toggle raw display
7979 (T3+⋯+25 ⁣⋯ ⁣72)2 (T^{3} + \cdots + 25\!\cdots\!72)^{2} Copy content Toggle raw display
8383 T6+⋯+29 ⁣⋯ ⁣20 T^{6} + \cdots + 29\!\cdots\!20 Copy content Toggle raw display
8989 T6+⋯+54 ⁣⋯ ⁣00 T^{6} + \cdots + 54\!\cdots\!00 Copy content Toggle raw display
9797 (T3+⋯−61 ⁣⋯ ⁣00)2 (T^{3} + \cdots - 61\!\cdots\!00)^{2} Copy content Toggle raw display
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