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Properties

Label 99.9.c.b
Level 9999
Weight 99
Character orbit 99.c
Analytic conductor 40.33040.330
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] = [99,9,Mod(10,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.10");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: N N == 99=32⋅11 99 = 3^{2} \cdot 11
Weight: k k == 9 9
Character orbit: [χ][\chi] == 99.c (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: 40.330482396140.3304823961
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+1374x4+436560x2+40320000 x^{6} + 1374x^{4} + 436560x^{2} + 40320000 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 27⋅5 2^{7}\cdot 5
Twist minimal: no (minimal twist has level 11)
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+(−2β4+3β3−203)q4+(3β4+5β3+73)q5+(2β5+β2−13β1)q7+(−2β5+3β2−227β1)q8+⋯+(−64232β5+⋯+778197β1)q98+O(q100) q + \beta_1 q^{2} + ( - 2 \beta_{4} + 3 \beta_{3} - 203) q^{4} + (3 \beta_{4} + 5 \beta_{3} + 73) q^{5} + (2 \beta_{5} + \beta_{2} - 13 \beta_1) q^{7} + ( - 2 \beta_{5} + 3 \beta_{2} - 227 \beta_1) q^{8}+ \cdots + ( - 64232 \beta_{5} + \cdots + 778197 \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 6q−1212q4+448q5+32318q11+33768q14+312264q16+894868q20−550440q22−683084q23−141498q25+657432q26+942684q31−1345128q34−3804816q37+⋯−222185616q97+O(q100) 6 q - 1212 q^{4} + 448 q^{5} + 32318 q^{11} + 33768 q^{14} + 312264 q^{16} + 894868 q^{20} - 550440 q^{22} - 683084 q^{23} - 141498 q^{25} + 657432 q^{26} + 942684 q^{31} - 1345128 q^{34} - 3804816 q^{37}+ \cdots - 222185616 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6+1374x4+436560x2+40320000 x^{6} + 1374x^{4} + 436560x^{2} + 40320000 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (−ν5−1134ν3−171240ν)/120 ( -\nu^{5} - 1134\nu^{3} - 171240\nu ) / 120 Copy content Toggle raw display
β3\beta_{3}== (−ν4−1134ν2−181080)/120 ( -\nu^{4} - 1134\nu^{2} - 181080 ) / 120 Copy content Toggle raw display
β4\beta_{4}== (−ν4−1174ν2−199440)/80 ( -\nu^{4} - 1174\nu^{2} - 199440 ) / 80 Copy content Toggle raw display
β5\beta_{5}== (−ν5−1174ν3−200800ν)/80 ( -\nu^{5} - 1174\nu^{3} - 200800\nu ) / 80 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== −2β4+3β3−459 -2\beta_{4} + 3\beta_{3} - 459 Copy content Toggle raw display
ν3\nu^{3}== −2β5+3β2−739β1 -2\beta_{5} + 3\beta_{2} - 739\beta_1 Copy content Toggle raw display
ν4\nu^{4}== 2268β4−3522β3+339426 2268\beta_{4} - 3522\beta_{3} + 339426 Copy content Toggle raw display
ν5\nu^{5}== 2268β5−3522β2+666786β1 2268\beta_{5} - 3522\beta_{2} + 666786\beta_1 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/99Z)×\left(\mathbb{Z}/99\mathbb{Z}\right)^\times.

nn 4646 5656
χ(n)\chi(n) −1-1 11

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}
10.1
− 31.0620i
− 14.5774i
− 14.0233i
14.0233i
14.5774i
31.0620i
− 31.0620i 0 −708.845 −583.097 0 1704.90i 14066.3i 0 18112.1i
10.2 − 14.5774i 0 43.4987 870.143 0 − 4079.11i − 4365.92i 0 − 12684.4i
10.3 − 14.0233i 0 59.3467 −63.0464 0 1667.89i − 4422.21i 0 884.120i
10.4 14.0233i 0 59.3467 −63.0464 0 − 1667.89i 4422.21i 0 − 884.120i
10.5 14.5774i 0 43.4987 870.143 0 4079.11i 4365.92i 0 12684.4i
10.6 31.0620i 0 −708.845 −583.097 0 − 1704.90i − 14066.3i 0 − 18112.1i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.9.c.b 6
3.b odd 2 1 11.9.b.b ✓ 6
11.b odd 2 1 inner 99.9.c.b 6
12.b even 2 1 176.9.h.c 6
33.d even 2 1 11.9.b.b ✓ 6
132.d odd 2 1 176.9.h.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.9.b.b ✓ 6 3.b odd 2 1
11.9.b.b ✓ 6 33.d even 2 1
99.9.c.b 6 1.a even 1 1 trivial
99.9.c.b 6 11.b odd 2 1 inner
176.9.h.c 6 12.b even 2 1
176.9.h.c 6 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T26+1374T24+436560T22+40320000 T_{2}^{6} + 1374T_{2}^{4} + 436560T_{2}^{2} + 40320000 acting on S9new(99,[χ])S_{9}^{\mathrm{new}}(99, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6+1374T4+⋯+40320000 T^{6} + 1374 T^{4} + \cdots + 40320000 Copy content Toggle raw display
33 T6 T^{6} Copy content Toggle raw display
55 (T3−224T2+⋯−31988350)2 (T^{3} - 224 T^{2} + \cdots - 31988350)^{2} Copy content Toggle raw display
77 T6+⋯+13 ⁣⋯ ⁣00 T^{6} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
1111 T6+⋯+98 ⁣⋯ ⁣41 T^{6} + \cdots + 98\!\cdots\!41 Copy content Toggle raw display
1313 T6+⋯+77 ⁣⋯ ⁣00 T^{6} + \cdots + 77\!\cdots\!00 Copy content Toggle raw display
1717 T6+⋯+30 ⁣⋯ ⁣00 T^{6} + \cdots + 30\!\cdots\!00 Copy content Toggle raw display
1919 T6+⋯+88 ⁣⋯ ⁣00 T^{6} + \cdots + 88\!\cdots\!00 Copy content Toggle raw display
2323 (T3+⋯−26 ⁣⋯ ⁣50)2 (T^{3} + \cdots - 26\!\cdots\!50)^{2} Copy content Toggle raw display
2929 T6+⋯+20 ⁣⋯ ⁣00 T^{6} + \cdots + 20\!\cdots\!00 Copy content Toggle raw display
3131 (T3+⋯+11 ⁣⋯ ⁣58)2 (T^{3} + \cdots + 11\!\cdots\!58)^{2} Copy content Toggle raw display
3737 (T3+⋯−12 ⁣⋯ ⁣50)2 (T^{3} + \cdots - 12\!\cdots\!50)^{2} Copy content Toggle raw display
4141 T6+⋯+29 ⁣⋯ ⁣00 T^{6} + \cdots + 29\!\cdots\!00 Copy content Toggle raw display
4343 T6+⋯+67 ⁣⋯ ⁣00 T^{6} + \cdots + 67\!\cdots\!00 Copy content Toggle raw display
4747 (T3+⋯−86 ⁣⋯ ⁣00)2 (T^{3} + \cdots - 86\!\cdots\!00)^{2} Copy content Toggle raw display
5353 (T3+⋯−20 ⁣⋯ ⁣00)2 (T^{3} + \cdots - 20\!\cdots\!00)^{2} Copy content Toggle raw display
5959 (T3+⋯+48 ⁣⋯ ⁣18)2 (T^{3} + \cdots + 48\!\cdots\!18)^{2} Copy content Toggle raw display
6161 T6+⋯+83 ⁣⋯ ⁣00 T^{6} + \cdots + 83\!\cdots\!00 Copy content Toggle raw display
6767 (T3+⋯+61 ⁣⋯ ⁣50)2 (T^{3} + \cdots + 61\!\cdots\!50)^{2} Copy content Toggle raw display
7171 (T3+⋯−10 ⁣⋯ ⁣42)2 (T^{3} + \cdots - 10\!\cdots\!42)^{2} Copy content Toggle raw display
7373 T6+⋯+10 ⁣⋯ ⁣00 T^{6} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
7979 T6+⋯+18 ⁣⋯ ⁣00 T^{6} + \cdots + 18\!\cdots\!00 Copy content Toggle raw display
8383 T6+⋯+50 ⁣⋯ ⁣00 T^{6} + \cdots + 50\!\cdots\!00 Copy content Toggle raw display
8989 (T3+⋯+62 ⁣⋯ ⁣02)2 (T^{3} + \cdots + 62\!\cdots\!02)^{2} Copy content Toggle raw display
9797 (T3+⋯−22 ⁣⋯ ⁣50)2 (T^{3} + \cdots - 22\!\cdots\!50)^{2} Copy content Toggle raw display
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