Properties

Label 102.14.a.c
Level 102102
Weight 1414
Character orbit 102.a
Self dual yes
Analytic conductor 109.376109.376
Analytic rank 11
Dimension 44
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] = [102,14,Mod(1,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: N N == 102=2317 102 = 2 \cdot 3 \cdot 17
Weight: k k == 14 14
Character orbit: [χ][\chi] == 102.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: 109.375547531109.375547531
Analytic rank: 11
Dimension: 44
Coefficient field: Q[x]/(x4)\mathbb{Q}[x]/(x^{4} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x42x313078169x25525405910x+22928880943080 x^{4} - 2x^{3} - 13078169x^{2} - 5525405910x + 22928880943080 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 253 2^{5}\cdot 3
Twist minimal: yes
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,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+64q2729q3+4096q4+(β33β29189)q546656q6+(7β3+4β2++44047)q7+262144q8+531441q9+(64β3192β2588096)q10++(140300424β3+521040699630)q99+O(q100) q + 64 q^{2} - 729 q^{3} + 4096 q^{4} + (\beta_{3} - 3 \beta_{2} - 9189) q^{5} - 46656 q^{6} + (7 \beta_{3} + 4 \beta_{2} + \cdots + 44047) q^{7} + 262144 q^{8} + 531441 q^{9} + (64 \beta_{3} - 192 \beta_{2} - 588096) q^{10}+ \cdots + ( - 140300424 \beta_{3} + \cdots - 521040699630) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+256q22916q3+16384q436750q5186624q6+176212q7+1048576q8+2125764q92352000q103922006q1111943936q1210375302q13+11277568q14+2084314790646q99+O(q100) 4 q + 256 q^{2} - 2916 q^{3} + 16384 q^{4} - 36750 q^{5} - 186624 q^{6} + 176212 q^{7} + 1048576 q^{8} + 2125764 q^{9} - 2352000 q^{10} - 3922006 q^{11} - 11943936 q^{12} - 10375302 q^{13} + 11277568 q^{14}+ \cdots - 2084314790646 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x42x313078169x25525405910x+22928880943080 x^{4} - 2x^{3} - 13078169x^{2} - 5525405910x + 22928880943080 : Copy content Toggle raw display

β1\beta_{1}== (324ν3+451039ν2+3228342050ν1602037001250)/139744605 ( -324\nu^{3} + 451039\nu^{2} + 3228342050\nu - 1602037001250 ) / 139744605 Copy content Toggle raw display
β2\beta_{2}== (2ν3+38293ν21607610ν258737722380)/19963515 ( 2\nu^{3} + 38293\nu^{2} - 1607610\nu - 258737722380 ) / 19963515 Copy content Toggle raw display
β3\beta_{3}== (62ν3143818ν2308030704ν+682444569135)/27948921 ( 62\nu^{3} - 143818\nu^{2} - 308030704\nu + 682444569135 ) / 27948921 Copy content Toggle raw display
ν\nu== (β3+β2+β1+7)/12 ( \beta_{3} + \beta_{2} + \beta _1 + 7 ) / 12 Copy content Toggle raw display
ν2\nu^{2}== (223β3+2693β297β1+39235811)/6 ( -223\beta_{3} + 2693\beta_{2} - 97\beta _1 + 39235811 ) / 6 Copy content Toggle raw display
ν3\nu^{3}== (4671572β3+8730923β2+2259113β1+24987525146)/6 ( 4671572\beta_{3} + 8730923\beta_{2} + 2259113\beta _1 + 24987525146 ) / 6 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
−2909.34
3583.54
−1856.70
1184.50
64.0000 −729.000 4096.00 −54016.2 −46656.0 −335501. 262144. 531441. −3.45704e6
1.2 64.0000 −729.000 4096.00 −36242.2 −46656.0 157321. 262144. 531441. −2.31950e6
1.3 64.0000 −729.000 4096.00 24273.0 −46656.0 561533. 262144. 531441. 1.55347e6
1.4 64.0000 −729.000 4096.00 29235.4 −46656.0 −207141. 262144. 531441. 1.87107e6
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
33 +1 +1
1717 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 102.14.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.14.a.c 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T54+36750T532162286135T5240701412457500T5+1389219666096442500 T_{5}^{4} + 36750T_{5}^{3} - 2162286135T_{5}^{2} - 40701412457500T_{5} + 1389219666096442500 acting on S14new(Γ0(102))S_{14}^{\mathrm{new}}(\Gamma_0(102)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T64)4 (T - 64)^{4} Copy content Toggle raw display
33 (T+729)4 (T + 729)^{4} Copy content Toggle raw display
55 T4++13 ⁣ ⁣00 T^{4} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
77 T4++61 ⁣ ⁣44 T^{4} + \cdots + 61\!\cdots\!44 Copy content Toggle raw display
1111 T4+77 ⁣ ⁣00 T^{4} + \cdots - 77\!\cdots\!00 Copy content Toggle raw display
1313 T4++97 ⁣ ⁣56 T^{4} + \cdots + 97\!\cdots\!56 Copy content Toggle raw display
1717 (T24137569)4 (T - 24137569)^{4} Copy content Toggle raw display
1919 T4++63 ⁣ ⁣64 T^{4} + \cdots + 63\!\cdots\!64 Copy content Toggle raw display
2323 T4++13 ⁣ ⁣96 T^{4} + \cdots + 13\!\cdots\!96 Copy content Toggle raw display
2929 T4+10 ⁣ ⁣00 T^{4} + \cdots - 10\!\cdots\!00 Copy content Toggle raw display
3131 T4+14 ⁣ ⁣00 T^{4} + \cdots - 14\!\cdots\!00 Copy content Toggle raw display
3737 T4+37 ⁣ ⁣64 T^{4} + \cdots - 37\!\cdots\!64 Copy content Toggle raw display
4141 T4+21 ⁣ ⁣04 T^{4} + \cdots - 21\!\cdots\!04 Copy content Toggle raw display
4343 T4+32 ⁣ ⁣84 T^{4} + \cdots - 32\!\cdots\!84 Copy content Toggle raw display
4747 T4+79 ⁣ ⁣36 T^{4} + \cdots - 79\!\cdots\!36 Copy content Toggle raw display
5353 T4+12 ⁣ ⁣76 T^{4} + \cdots - 12\!\cdots\!76 Copy content Toggle raw display
5959 T4++96 ⁣ ⁣80 T^{4} + \cdots + 96\!\cdots\!80 Copy content Toggle raw display
6161 T4++87 ⁣ ⁣80 T^{4} + \cdots + 87\!\cdots\!80 Copy content Toggle raw display
6767 T4+97 ⁣ ⁣80 T^{4} + \cdots - 97\!\cdots\!80 Copy content Toggle raw display
7171 T4++32 ⁣ ⁣36 T^{4} + \cdots + 32\!\cdots\!36 Copy content Toggle raw display
7373 T4++59 ⁣ ⁣44 T^{4} + \cdots + 59\!\cdots\!44 Copy content Toggle raw display
7979 T4++17 ⁣ ⁣00 T^{4} + \cdots + 17\!\cdots\!00 Copy content Toggle raw display
8383 T4++62 ⁣ ⁣24 T^{4} + \cdots + 62\!\cdots\!24 Copy content Toggle raw display
8989 T4+67 ⁣ ⁣84 T^{4} + \cdots - 67\!\cdots\!84 Copy content Toggle raw display
9797 T4+13 ⁣ ⁣80 T^{4} + \cdots - 13\!\cdots\!80 Copy content Toggle raw display
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