LMFDB - Newform orbit 99.4.f.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [99,4,Mod(37,99)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("99.37"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$99 = 3^{2} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	99.f (of order $5$ , degree $4$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.84118909057$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + x^{2} - x + 1$ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a primitive root of unity $\zeta_{10}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots - 2) q^{2} - 12 \zeta_{10}^{3} q^{4} + (11 \zeta_{10}^{2} - 12 \zeta_{10} + 11) q^{5} + (19 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{7} + (16 \zeta_{10}^{3} + \cdots + 16 \zeta_{10}) q^{8} + \cdots + ( - 744 \zeta_{10}^{3} + 744 \zeta_{10}^{2} - 948) q^{98} +O(q^{100})$ q + (2z^3 + 2z^2 - 2z - 2) q^2 - 12z^3 q^4 + (11z^2 - 12z + 11) * q^5 + (19z^3 - 6z + 6) * q^7 + (16z^3 - 8z^2 + 16z) q^8 + (-26z^3 + 26z^2 - 42) * q^10 + (10z^3 + 6z^2 + 35z - 20) q^11 + (33z^3 - 11z^2 + 11z - 33) q^13 + (-88z^3 + 74z^2 - 88z) q^14 + 16z q^16 + (-57z^2 - 21z - 57) * q^17 + (-45z^3 - 38z^2 - 45z) q^19 + (12z^3 - 144z^2 + 144z - 12) q^20 + (60z^3 - 184z^2 + 12z - 54) q^22 + (44z^3 - 44z^2 + 15) * q^23 + (-143z^3 + 140z^2 - 143z) q^25 + (-154z^3 - 88z + 88) * q^26 + (-72z^2 + 300z - 72) * q^28 + (162z^3 + 192z - 192) * q^29 + (44z^3 + 33z^2 - 33z - 44) q^31 + (192z^3 - 192z^2 - 96) * q^32 + (-198z^3 + 198z^2 + 384) * q^34 + (-85z^3 + 366z^2 - 366z + 85) q^35 + (267z^3 + 22z - 22) * q^37 + (242z^2 + 104z + 242) * q^38 + (168z^3 - 104z + 104) * q^40 + (34z^3 - 61z^2 + 34z) q^41 + (84z^3 - 84z^2 - 75) * q^43 + (-180z^3 + 420z^2 - 300z + 492) q^44 + (-146z^3 + 294z^2 - 294z + 146) q^46 + (-55z^3 - 145z^2 - 55z) q^47 + (264z^2 - 318z + 264) * q^49 + (-274z^2 + 852z - 274) * q^50 + (264z^3 + 132z^2 + 264z) q^52 + (-98z^3 + 241z^2 - 241z + 98) q^53 + (369z^3 - 520z^2 + 571z - 276) q^55 + (352z^3 - 352z^2 - 56) * q^56 + (-264z^3 - 828z^2 - 264z) q^58 + (-33z^3 - 451z + 451) * q^59 + (279z^2 + 438z + 279) * q^61 + (-396z^3 - 22z + 22) * q^62 + (-832z^3 + 832z^2 - 832z + 832) q^64 + (99z^3 - 99z^2 - 209) * q^65 + (561z^3 - 561z^2 - 243) * q^67 + (936z^3 - 252z^2 + 252z - 936) q^68 + (-954z^3 + 902z - 902) * q^70 + (-275z^2 + 708z - 275) * q^71 + (-28z^3 + 318z - 318) * q^73 + (-1024z^3 + 402z^2 - 1024z) q^74 + (540z^3 - 540z^2 - 996) * q^76 + (249z^3 - 779z^2 + 745z - 839) q^77 + (65z^3 + 637z^2 - 637z - 65) q^79 + (176z^3 - 192z^2 + 176z) q^80 + (176z^2 - 258z + 176) * q^82 + (466z^2 - 431z + 466) * q^83 + (-174z^3 - 375z^2 - 174z) q^85 + (-486z^3 + 354z^2 - 354z + 486) q^86 + (168z^3 + 48z^2 + 192z - 688) q^88 + (154z^3 - 154z^2 + 1279) * q^89 + (-352z^3 - 143z^2 - 352z) q^91 + (-180z^3 + 528z - 528) * q^92 + (690z^2 - 70z + 690) * q^94 + (-412z^3 - 373z + 373) * q^95 + (-609z^3 - 282z^2 + 282z + 609) q^97 + (-744z^3 + 744z^2 - 948) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 10 q^{2} - 12 q^{4} + 21 q^{5} + 37 q^{7} + 40 q^{8} - 220 q^{10} - 41 q^{11} - 77 q^{13} - 250 q^{14} + 16 q^{16} - 192 q^{17} - 52 q^{19} + 252 q^{20} + 40 q^{22} + 148 q^{23} - 426 q^{25} + 110 q^{26}+ \cdots - 5280 q^{98}+O(q^{100})$ 4 * q - 10 * q^2 - 12 * q^4 + 21 * q^5 + 37 * q^7 + 40 * q^8 - 220 * q^10 - 41 * q^11 - 77 * q^13 - 250 * q^14 + 16 * q^16 - 192 * q^17 - 52 * q^19 + 252 * q^20 + 40 * q^22 + 148 * q^23 - 426 * q^25 + 110 * q^26 + 84 * q^28 - 414 * q^29 - 198 * q^31 + 1140 * q^34 - 477 * q^35 + 201 * q^37 + 830 * q^38 + 480 * q^40 + 129 * q^41 - 132 * q^43 + 1068 * q^44 - 150 * q^46 + 35 * q^47 + 474 * q^49 + 30 * q^50 + 396 * q^52 - 188 * q^53 + 356 * q^55 + 480 * q^56 + 300 * q^58 + 1320 * q^59 + 1275 * q^61 - 330 * q^62 + 832 * q^64 - 638 * q^65 + 150 * q^67 - 2304 * q^68 - 3660 * q^70 - 117 * q^71 - 982 * q^73 - 2450 * q^74 - 2904 * q^76 - 1583 * q^77 - 1469 * q^79 + 544 * q^80 + 270 * q^82 + 967 * q^83 + 27 * q^85 + 750 * q^86 - 2440 * q^88 + 5424 * q^89 - 561 * q^91 - 1764 * q^92 + 2000 * q^94 + 707 * q^95 + 2391 * q^97 - 5280 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$ .

$n$	$46$	$56$
$\chi(n)$	$-\zeta_{10}^{3}$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ 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

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

37.1

0.809017	−	0.587785i
−0.309017	−	0.951057i
−0.309017	+	0.951057i
0.809017	+	0.587785i

−3.61803

−

2.62866i

3.70820

11.4127i

4.69098

−

3.40820i

−4.72542

−

14.5434i

5.52786

−

17.0130i

−25.9311

64.1

−1.38197

4.25325i

−9.70820

−

7.05342i

5.80902

17.8783i

23.2254

16.8743i

14.4721

−

10.5146i

−84.0689

82.1

−1.38197

−

4.25325i

−9.70820

7.05342i

5.80902

−

17.8783i

23.2254

−

16.8743i

14.4721

10.5146i

−84.0689

91.1

−3.61803

2.62866i

3.70820

−

11.4127i

4.69098

3.40820i

−4.72542

14.5434i

5.52786

17.0130i

−25.9311

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.4.f.a		4
3.b	odd	2	1		33.4.e.a	✓	4
11.c	even	5	1	inner	99.4.f.a		4
11.c	even	5	1		1089.4.a.p		2
11.d	odd	10	1		1089.4.a.q		2
33.f	even	10	1		363.4.a.o		2
33.h	odd	10	1		33.4.e.a	✓	4
33.h	odd	10	1		363.4.a.n		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.4.e.a	✓	4	3.b	odd	2	1
33.4.e.a	✓	4	33.h	odd	10	1
99.4.f.a		4	1.a	even	1	1	trivial
99.4.f.a		4	11.c	even	5	1	inner
363.4.a.n		2	33.h	odd	10	1
363.4.a.o		2	33.f	even	10	1
1089.4.a.p		2	11.c	even	5	1
1089.4.a.q		2	11.d	odd	10	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{4} + 10T_{2}^{3} + 60T_{2}^{2} + 200T_{2} + 400$ T2^4 + 10*T2^3 + 60*T2^2 + 200*T2 + 400 acting on $S_{4}^{\mathrm{new}}(99, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 10 T^{3} + \cdots + 400$ T^4 + 10T^3 + 60T^2 + 200*T + 400
$3$	$T^{4}$ T^4
$5$	$T^{4} - 21 T^{3} + \cdots + 11881$ T^4 - 21T^3 + 496T^2 - 3706*T + 11881
$7$	$T^{4} - 37 T^{3} + \cdots + 192721$ T^4 - 37T^3 + 619T^2 - 3073*T + 192721
$11$	$T^{4} + 41 T^{3} + \cdots + 1771561$ T^4 + 41T^3 + 1881T^2 + 54571*T + 1771561
$13$	$T^{4} + 77 T^{3} + \cdots + 14641$ T^4 + 77T^3 + 2299T^2 + 3993*T + 14641
$17$	$T^{4} + 192 T^{3} + \cdots + 2595321$ T^4 + 192T^3 + 14634T^2 + 149823*T + 2595321
$19$	$T^{4} + 52 T^{3} + \cdots + 1274641$ T^4 + 52T^3 + 11254T^2 - 195317*T + 1274641
$23$	$(T^{2} - 74 T - 1051)^{2}$ (T^2 - 74*T - 1051)^2
$29$	$T^{4} + \cdots + 1740892176$ T^4 + 414T^3 + 142596T^2 + 22781304*T + 1740892176
$31$	$T^{4} + 198 T^{3} + \cdots + 54479161$ T^4 + 198T^3 + 22264T^2 + 1380247*T + 54479161
$37$	$T^{4} + \cdots + 4216034761$ T^4 - 201T^3 + 67351T^2 - 20193541*T + 4216034761
$41$	$T^{4} - 129 T^{3} + \cdots + 241081$ T^4 - 129T^3 + 6271T^2 + 20131*T + 241081
$43$	$(T^{2} + 66 T - 7731)^{2}$ (T^2 + 66*T - 7731)^2
$47$	$T^{4} - 35 T^{3} + \cdots + 674700625$ T^4 - 35T^3 + 41100T^2 - 8052250*T + 674700625
$53$	$T^{4} + 188 T^{3} + \cdots + 10042561$ T^4 + 188T^3 + 105414T^2 - 1670063*T + 10042561
$59$	$T^{4} + \cdots + 47173668025$ T^4 - 1320T^3 + 799810T^2 - 203077325*T + 47173668025
$61$	$T^{4} + \cdots + 55792802025$ T^4 - 1275T^3 + 625410T^2 + 28344600*T + 55792802025
$67$	$(T^{2} - 75 T - 391995)^{2}$ (T^2 - 75*T - 391995)^2
$71$	$T^{4} + \cdots + 53332821721$ T^4 + 117T^3 + 609064T^2 - 290521262*T + 53332821721
$73$	$T^{4} + \cdots + 8360542096$ T^4 + 982T^3 + 414184T^2 + 55593088*T + 8360542096
$79$	$T^{4} + \cdots + 285379255681$ T^4 + 1469T^3 + 1929811T^2 + 1090320569*T + 285379255681
$83$	$T^{4} + \cdots + 53935882081$ T^4 - 967T^3 + 853539T^2 - 316544483*T + 53935882081
$89$	$(T^{2} - 2712 T + 1809091)^{2}$ (T^2 - 2712*T + 1809091)^2
$97$	$T^{4} + \cdots + 932420053161$ T^4 - 2391T^3 + 3003786T^2 - 1993037616*T + 932420053161
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