LMFDB - Newform orbit 3960.2.d.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(3169,3960)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3960.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3960.d (of order $2$, degree $1$, 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:	$31.6207592004$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 24x^{12} + 226x^{10} + 1052x^{8} + 2497x^{6} + 2788x^{4} + 1156x^{2} + 64 $ x^14 + 24x^12 + 226x^10 + 1052x^8 + 2497x^6 + 2788x^4 + 1156x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 basis $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{10} q^{5} + \beta_{5} q^{7}+O(q^{10}) $ q + b10 * q^5 + b5 * q^7	$ q + \beta_{10} q^{5} + \beta_{5} q^{7} + q^{11} + ( - \beta_{5} + \beta_1) q^{13} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{17} + ( - \beta_{11} - \beta_{4}) q^{19} + ( - \beta_{2} - \beta_1) q^{23} + (\beta_{12} - \beta_{11}) q^{25} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_{7}) q^{29}+ \cdots + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{97}+O(q^{100}) $ q + b10 * q^5 + b5 * q^7 + q^11 + (-b5 + b1) * q^13 + (-b5 + b3 - b2) * q^17 + (-b11 - b4) * q^19 + (-b2 - b1) * q^23 + (b12 - b11) * q^25 + (-b13 + b12 - b11 - b10 + b9 - b7) * q^29 + (-b13 + b12 - b10 + b9 - b7 - b4) * q^31 + (b12 - b7 - b5 - b4 + b1 - 1) * q^35 + (b13 + b12 - b6 - b5 + 2b3 + b1) q^37 + (-b11 - b10 + b9 - b7 + 2) * q^41 + (b6 - b2) * q^43 + (b6 + b5 - b2) * q^47 + (-b11 + 2b10 - 2b9 + b8 + b7 - b4) * q^49 + (-b13 - b12 - b3 - b2) * q^53 + b10 * q^55 + (b13 - b12 + b11 - b8 - b7 + b4 + 1) * q^59 + (b13 - b12 + 2b11 - 2b8 + b7 + b4) * q^61 + (-b12 + b11 - b10 - b9 - b8 + b6 + 2b5 + b4 - b3 - b2 - b1) q^65 + (b10 + b9 + b3 + b2) * q^67 + (b11 + b8 - 2b7 - 1) q^71 + (-2b10 - 2b9 + b6 + 2b5 - 2b3 + b2 + b1) * q^73 + b5 * q^77 + (-b13 + b12 - b11 + 2b8 - b7 - 2b4 - 2) * q^79 + (b13 + b12 + b5 - b2) * q^83 + (b13 - b12 + b11 + b5 + 2b4 - 2b3 + b2) * q^85 + (b13 - b12 + 2b11 + b10 - b9 - b7 + b4 - 4) q^89 + (-3b10 + 3b9 - 2b8 + b7 + b4 + 4) q^91 + (b13 - b12 - b8 - b6 - b5 - b2 - 1) * q^95 + (-b13 - b12 - 2b10 - 2b9 + b6 - b5 + 2b3 - 2b2 - b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q+O(q^{10}) $ 14 * q	$ 14 q + 14 q^{11} - 2 q^{25} - 12 q^{35} + 32 q^{41} - 6 q^{49} + 24 q^{59} + 4 q^{61} + 4 q^{65} - 8 q^{71} - 32 q^{79} + 4 q^{85} - 48 q^{89} + 56 q^{91} - 8 q^{95}+O(q^{100}) $ 14 * q + 14 * q^11 - 2 * q^25 - 12 * q^35 + 32 * q^41 - 6 * q^49 + 24 * q^59 + 4 * q^61 + 4 * q^65 - 8 * q^71 - 32 * q^79 + 4 * q^85 - 48 * q^89 + 56 * q^91 - 8 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{14} + 24x^{12} + 226x^{10} + 1052x^{8} + 2497x^{6} + 2788x^{4} + 1156x^{2} + 64 $ x^14 + 24*x^12 + 226*x^10 + 1052*x^8 + 2497*x^6 + 2788*x^4 + 1156*x^2 + 64 :

$\beta_{1}$	$=$	$ 2\nu $ 2*v
$\beta_{2}$	$=$	$ \nu^{3} + 5\nu $ v^3 + 5*v
$\beta_{3}$	$=$	$ ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 34\nu ) / 4 $ (v^7 + 12v^5 + 41v^3 + 34*v) / 4
$\beta_{4}$	$=$	$ ( \nu^{8} + 12\nu^{6} + 41\nu^{4} + 34\nu^{2} ) / 4 $ (v^8 + 12v^6 + 41v^4 + 34*v^2) / 4
$\beta_{5}$	$=$	$ ( \nu^{9} + 16\nu^{7} + 85\nu^{5} + 162\nu^{3} + 72\nu ) / 8 $ (v^9 + 16v^7 + 85v^5 + 162v^3 + 72v) / 8
$\beta_{6}$	$=$	$ ( \nu^{9} + 16\nu^{7} + 93\nu^{5} + 234\nu^{3} + 200\nu ) / 8 $ (v^9 + 16v^7 + 93v^5 + 234v^3 + 200v) / 8
$\beta_{7}$	$=$	$ ( \nu^{10} + 17\nu^{8} + 101\nu^{6} + 247\nu^{4} + 226\nu^{2} + 48 ) / 8 $ (v^10 + 17v^8 + 101v^6 + 247v^4 + 226v^2 + 48) / 8
$\beta_{8}$	$=$	$ ( -\nu^{12} - 19\nu^{10} - 129\nu^{8} - 361\nu^{6} - 314\nu^{4} + 88\nu^{2} + 16 ) / 16 $ (-v^12 - 19v^10 - 129v^8 - 361v^6 - 314v^4 + 88*v^2 + 16) / 16
$\beta_{9}$	$=$	$ ( \nu^{11} + \nu^{10} + 19 \nu^{9} + 17 \nu^{8} + 131 \nu^{7} + 101 \nu^{6} + 393 \nu^{5} + \cdots + 164 \nu ) / 16 $ (v^11 + v^10 + 19v^9 + 17v^8 + 131v^7 + 101v^6 + 393v^5 + 239v^4 + 476v^3 + 170v^2 + 164*v) / 16
$\beta_{10}$	$=$	$ ( \nu^{11} - \nu^{10} + 19 \nu^{9} - 17 \nu^{8} + 131 \nu^{7} - 101 \nu^{6} + 393 \nu^{5} + \cdots + 164 \nu ) / 16 $ (v^11 - v^10 + 19v^9 - 17v^8 + 131v^7 - 101v^6 + 393v^5 - 239v^4 + 476v^3 - 170v^2 + 164*v) / 16
$\beta_{11}$	$=$	$ ( \nu^{12} + 21\nu^{10} + 167\nu^{8} + 627\nu^{6} + 1132\nu^{4} + 868\nu^{2} + 128 ) / 16 $ (v^12 + 21v^10 + 167v^8 + 627v^6 + 1132v^4 + 868*v^2 + 128) / 16
$\beta_{12}$	$=$	$ ( \nu^{13} + \nu^{12} + 22 \nu^{11} + 21 \nu^{10} + 186 \nu^{9} + 167 \nu^{8} + 752 \nu^{7} + 619 \nu^{6} + \cdots + 48 ) / 16 $ (v^13 + v^12 + 22v^11 + 21v^10 + 186v^9 + 167v^8 + 752v^7 + 619v^6 + 1461v^5 + 1052v^4 + 1178v^3 + 668v^2 + 240*v + 48) / 16
$\beta_{13}$	$=$	$ ( \nu^{13} - \nu^{12} + 22 \nu^{11} - 21 \nu^{10} + 186 \nu^{9} - 167 \nu^{8} + 752 \nu^{7} - 619 \nu^{6} + \cdots - 48 ) / 16 $ (v^13 - v^12 + 22v^11 - 21v^10 + 186v^9 - 167v^8 + 752v^7 - 619v^6 + 1461v^5 - 1052v^4 + 1178v^3 - 668v^2 + 240*v - 48) / 16

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{13} - \beta_{12} + \beta_{11} - \beta_{8} + \beta_{7} + \beta_{4} - 7 ) / 2 $ (b13 - b12 + b11 - b8 + b7 + b4 - 7) / 2
$\nu^{3}$	$=$	$ ( 2\beta_{2} - 5\beta_1 ) / 2 $ (2b2 - 5b1) / 2
$\nu^{4}$	$=$	$ ( -7\beta_{13} + 7\beta_{12} - 7\beta_{11} + 2\beta_{10} - 2\beta_{9} + 7\beta_{8} - 5\beta_{7} - 7\beta_{4} + 37 ) / 2 $ (-7b13 + 7b12 - 7b11 + 2b10 - 2b9 + 7b8 - 5b7 - 7b4 + 37) / 2
$\nu^{5}$	$=$	$ ( 2\beta_{6} - 2\beta_{5} - 18\beta_{2} + 29\beta_1 ) / 2 $ (2b6 - 2b5 - 18b2 + 29b1) / 2
$\nu^{6}$	$=$	$ ( 47 \beta_{13} - 47 \beta_{12} + 49 \beta_{11} - 20 \beta_{10} + 20 \beta_{9} - 45 \beta_{8} + 25 \beta_{7} + \cdots - 215 ) / 2 $ (47b13 - 47b12 + 49b11 - 20b10 + 20b9 - 45b8 + 25b7 + 45b4 - 215) / 2
$\nu^{7}$	$=$	$ ( -24\beta_{6} + 24\beta_{5} + 8\beta_{3} + 134\beta_{2} - 177\beta_1 ) / 2 $ (-24b6 + 24b5 + 8b3 + 134b2 - 177*b1) / 2
$\nu^{8}$	$=$	$ ( - 311 \beta_{13} + 311 \beta_{12} - 335 \beta_{11} + 158 \beta_{10} - 158 \beta_{9} + 287 \beta_{8} + \cdots + 1301 ) / 2 $ (-311b13 + 311b12 - 335b11 + 158b10 - 158b9 + 287b8 - 129b7 - 279b4 + 1301) / 2
$\nu^{9}$	$=$	$ ( 214\beta_{6} - 198\beta_{5} - 128\beta_{3} - 938\beta_{2} + 1105\beta_1 ) / 2 $ (214b6 - 198b5 - 128b3 - 938b2 + 1105*b1) / 2
$\nu^{10}$	$=$	$ ( 2043 \beta_{13} - 2043 \beta_{12} + 2249 \beta_{11} - 1160 \beta_{10} + 1160 \beta_{9} - 1837 \beta_{8} + \cdots - 8055 ) / 2 $ (2043b13 - 2043b12 + 2249b11 - 1160b10 + 1160b9 - 1837b8 + 693b7 + 1701b4 - 8055) / 2
$\nu^{11}$	$=$	$ ( 16\beta_{10} + 16\beta_{9} - 1708\beta_{6} + 1404\beta_{5} + 1384\beta_{3} + 6390\beta_{2} - 6989\beta_1 ) / 2 $ (16b10 + 16b9 - 1708b6 + 1404b5 + 1384b3 + 6390b2 - 6989*b1) / 2
$\nu^{12}$	$=$	$ ( - 13379 \beta_{13} + 13379 \beta_{12} - 14919 \beta_{11} + 8250 \beta_{10} - 8250 \beta_{9} + \cdots + 50629 ) / 2 $ (-13379b13 + 13379b12 - 14919b11 + 8250b10 - 8250b9 + 11807b8 - 3893b7 - 10287b4 + 50629) / 2
$\nu^{13}$	$=$	$ ( 16 \beta_{13} + 16 \beta_{12} - 352 \beta_{10} - 352 \beta_{9} + 12898 \beta_{6} - 9186 \beta_{5} + \cdots + 44613 \beta_1 ) / 2 $ (16b13 + 16b12 - 352b10 - 352b9 + 12898b6 - 9186b5 - 12656b3 - 42938b2 + 44613*b1) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$.

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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} $

3169.1

		1.31120i
	−	1.31120i
	−	2.47637i
		2.47637i
	−	1.91990i
		1.91990i
		2.21557i
	−	2.21557i
	−	0.254885i
		0.254885i
		0.876610i
	−	0.876610i
	−	2.59233i
		2.59233i

−2.15086

−

0.611385i

−

4.56390i

3169.2

−2.15086

0.611385i

4.56390i

3169.3

−1.40211

−

1.74186i

0.169515i

3169.4

−1.40211

1.74186i

−

0.169515i

3169.5

−1.26135

−

1.84634i

−

3.12979i

3169.6

−1.26135

1.84634i

3.12979i

3169.7

0.101079

−

2.23378i

3.61424i

3169.8

0.101079

2.23378i

−

3.61424i

3169.9

0.628932

−

2.14580i

−

1.96993i

3169.10

0.628932

2.14580i

1.96993i

3169.11

1.85471

−

1.24902i

−

1.00883i

3169.12

1.85471

1.24902i

1.00883i

3169.13

2.22960

−

0.169899i

−

1.83994i

3169.14

2.22960

0.169899i

1.83994i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3960.2.d.j	yes	14
3.b	odd	2	1		3960.2.d.i	✓	14
5.b	even	2	1	inner	3960.2.d.j	yes	14
15.d	odd	2	1		3960.2.d.i	✓	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3960.2.d.i	✓	14	3.b	odd	2	1
3960.2.d.i	✓	14	15.d	odd	2	1
3960.2.d.j	yes	14	1.a	even	1	1	trivial
3960.2.d.j	yes	14	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3960, [\chi])$:

$ T_{7}^{14} + 52T_{7}^{12} + 988T_{7}^{10} + 8608T_{7}^{8} + 35312T_{7}^{6} + 63808T_{7}^{4} + 37440T_{7}^{2} + 1024 $

$ T_{29}^{7} - 116T_{29}^{5} - 80T_{29}^{4} + 2528T_{29}^{3} + 6016T_{29}^{2} + 3456T_{29} + 512 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{14} $ T^14
$3$	$ T^{14} $ T^14
$5$	$ T^{14} + T^{12} + \cdots + 78125 $ T^14 + T^12 - 8T^11 - 5T^10 - 32T^9 + 3T^8 + 208T^7 + 15T^6 - 800T^5 - 625T^4 - 5000T^3 + 3125T^2 + 78125
$7$	$ T^{14} + 52 T^{12} + \cdots + 1024 $ T^14 + 52T^12 + 988T^10 + 8608T^8 + 35312T^6 + 63808T^4 + 37440T^2 + 1024
$11$	$ (T - 1)^{14} $ (T - 1)^14
$13$	$ T^{14} + 100 T^{12} + \cdots + 82944 $ T^14 + 100T^12 + 3228T^10 + 42016T^8 + 230128T^6 + 475200T^4 + 350784T^2 + 82944
$17$	$ T^{14} + 92 T^{12} + \cdots + 4096 $ T^14 + 92T^12 + 2200T^10 + 14848T^8 + 41232T^6 + 49984T^4 + 24576T^2 + 4096
$19$	$ (T^{7} - 68 T^{5} + \cdots + 256)^{2} $ (T^7 - 68T^5 - 112T^4 + 784T^3 + 1792T^2 + 1216*T + 256)^2
$23$	$ T^{14} + 148 T^{12} + \cdots + 9437184 $ T^14 + 148T^12 + 8292T^10 + 217664T^8 + 2667520T^6 + 13221888T^4 + 24182784T^2 + 9437184
$29$	$ (T^{7} - 116 T^{5} + \cdots + 512)^{2} $ (T^7 - 116T^5 - 80T^4 + 2528T^3 + 6016T^2 + 3456*T + 512)^2
$31$	$ (T^{7} - 128 T^{5} + \cdots + 65536)^{2} $ (T^7 - 128T^5 + 32T^4 + 4432T^3 - 4736T^2 - 41600*T + 65536)^2
$37$	$ T^{14} + \cdots + 17994612736 $ T^14 + 360T^12 + 50320T^10 + 3395328T^8 + 111007744T^6 + 1556512768T^4 + 9111142400T^2 + 17994612736
$41$	$ (T^{7} - 16 T^{6} + \cdots - 6144)^{2} $ (T^7 - 16T^6 - 24T^5 + 1248T^4 - 3424T^3 - 9984T^2 + 19200T - 6144)^2
$43$	$ T^{14} + \cdots + 764411904 $ T^14 + 192T^12 + 14536T^10 + 560480T^8 + 11740432T^6 + 129700224T^4 + 644661504T^2 + 764411904
$47$	$ T^{14} + \cdots + 220463104 $ T^14 + 244T^12 + 20836T^10 + 745408T^8 + 11093504T^6 + 74579968T^4 + 218775552T^2 + 220463104
$53$	$ T^{14} + \cdots + 19323224064 $ T^14 + 360T^12 + 50740T^10 + 3542288T^8 + 127000576T^6 + 2181765120T^4 + 14787919872T^2 + 19323224064
$59$	$ (T^{7} - 12 T^{6} + \cdots + 52736)^{2} $ (T^7 - 12T^6 - 128T^5 + 1824T^4 - 176T^3 - 40000T^2 + 46208T + 52736)^2
$61$	$ (T^{7} - 2 T^{6} + \cdots + 1157056)^{2} $ (T^7 - 2T^6 - 230T^5 + 508T^4 + 16352T^3 - 44480T^2 - 358752T + 1157056)^2
$67$	$ T^{14} + 136 T^{12} + \cdots + 16777216 $ T^14 + 136T^12 + 6672T^10 + 150528T^8 + 1691648T^6 + 9437184T^4 + 23330816T^2 + 16777216
$71$	$ (T^{7} + 4 T^{6} + \cdots - 128768)^{2} $ (T^7 + 4T^6 - 242T^5 - 1088T^4 + 14176T^3 + 74240T^2 + 24160T - 128768)^2
$73$	$ T^{14} + \cdots + 212572635136 $ T^14 + 624T^12 + 152008T^10 + 18054944T^8 + 1057750288T^6 + 27146211712T^4 + 248252780800T^2 + 212572635136
$79$	$ (T^{7} + 16 T^{6} + \cdots + 7424)^{2} $ (T^7 + 16T^6 - 182T^5 - 3264T^4 + 1728T^3 + 80128T^2 - 117280T + 7424)^2
$83$	$ T^{14} + \cdots + 140847087616 $ T^14 + 440T^12 + 70840T^10 + 5551584T^8 + 227852688T^6 + 4767187456T^4 + 44632936448T^2 + 140847087616
$89$	$ (T^{7} + 24 T^{6} + \cdots - 65536)^{2} $ (T^7 + 24T^6 + 40T^5 - 1760T^4 - 3568T^3 + 28672T^2 + 53760T - 65536)^2
$97$	$ T^{14} + 904 T^{12} + \cdots + 16777216 $ T^14 + 904T^12 + 290320T^10 + 38941696T^8 + 1929826304T^6 + 19278069760T^4 + 9036890112T^2 + 16777216
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{10} q^{5} + \beta_{5} q^{7}+O(q^{10}) \) q + b10 * q^5 + b5 * q^7	\( q + \beta_{10} q^{5} + \beta_{5} q^{7} + q^{11} + ( - \beta_{5} + \beta_1) q^{13} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{17} + ( - \beta_{11} - \beta_{4}) q^{19} + ( - \beta_{2} - \beta_1) q^{23} + (\beta_{12} - \beta_{11}) q^{25} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_{7}) q^{29}+ \cdots + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{97}+O(q^{100}) \) q + b10 * q^5 + b5 * q^7 + q^11 + (-b5 + b1) * q^13 + (-b5 + b3 - b2) * q^17 + (-b11 - b4) * q^19 + (-b2 - b1) * q^23 + (b12 - b11) * q^25 + (-b13 + b12 - b11 - b10 + b9 - b7) * q^29 + (-b13 + b12 - b10 + b9 - b7 - b4) * q^31 + (b12 - b7 - b5 - b4 + b1 - 1) * q^35 + (b13 + b12 - b6 - b5 + 2b3 + b1) q^37 + (-b11 - b10 + b9 - b7 + 2) * q^41 + (b6 - b2) * q^43 + (b6 + b5 - b2) * q^47 + (-b11 + 2b10 - 2b9 + b8 + b7 - b4) * q^49 + (-b13 - b12 - b3 - b2) * q^53 + b10 * q^55 + (b13 - b12 + b11 - b8 - b7 + b4 + 1) * q^59 + (b13 - b12 + 2b11 - 2b8 + b7 + b4) * q^61 + (-b12 + b11 - b10 - b9 - b8 + b6 + 2b5 + b4 - b3 - b2 - b1) q^65 + (b10 + b9 + b3 + b2) * q^67 + (b11 + b8 - 2b7 - 1) q^71 + (-2b10 - 2b9 + b6 + 2b5 - 2b3 + b2 + b1) * q^73 + b5 * q^77 + (-b13 + b12 - b11 + 2b8 - b7 - 2b4 - 2) * q^79 + (b13 + b12 + b5 - b2) * q^83 + (b13 - b12 + b11 + b5 + 2b4 - 2b3 + b2) * q^85 + (b13 - b12 + 2b11 + b10 - b9 - b7 + b4 - 4) q^89 + (-3b10 + 3b9 - 2b8 + b7 + b4 + 4) q^91 + (b13 - b12 - b8 - b6 - b5 - b2 - 1) * q^95 + (-b13 - b12 - 2b10 - 2b9 + b6 - b5 + 2b3 - 2b2 - b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q+O(q^{10}) \) 14 * q	\( 14 q + 14 q^{11} - 2 q^{25} - 12 q^{35} + 32 q^{41} - 6 q^{49} + 24 q^{59} + 4 q^{61} + 4 q^{65} - 8 q^{71} - 32 q^{79} + 4 q^{85} - 48 q^{89} + 56 q^{91} - 8 q^{95}+O(q^{100}) \) 14 * q + 14 * q^11 - 2 * q^25 - 12 * q^35 + 32 * q^41 - 6 * q^49 + 24 * q^59 + 4 * q^61 + 4 * q^65 - 8 * q^71 - 32 * q^79 + 4 * q^85 - 48 * q^89 + 56 * q^91 - 8 * q^95

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