LMFDB - Newform orbit 4100.2.d.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4100,2,Mod(1149,4100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4100.1149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4100 = 2^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4100.d (of order $2$, degree $1$, not 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:	$32.7386648287$
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} + 27x^{12} + 281x^{10} + 1405x^{8} + 3397x^{6} + 3488x^{4} + 1281x^{2} + 121 $ x^14 + 27x^12 + 281x^10 + 1405x^8 + 3397x^6 + 3488x^4 + 1281x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
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_1 q^{3} + \beta_{11} q^{7} + ( - \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) $ q + b1 * q^3 + b11 * q^7 + (-b3 + b2 - 1) * q^9	$ q + \beta_1 q^{3} + \beta_{11} q^{7} + ( - \beta_{3} + \beta_{2} - 1) q^{9} + (\beta_{8} - \beta_{6} + \cdots - \beta_{2}) q^{11}+ \cdots + (\beta_{10} + 2 \beta_{6} - 2 \beta_{3} + \cdots - 6) q^{99}+O(q^{100}) $ q + b1 * q^3 + b11 * q^7 + (-b3 + b2 - 1) * q^9 + (b8 - b6 + b3 - b2) * q^11 + (b5 - b4) * q^13 + (b13 - b12 + b11 - b7 + b4) * q^17 + (-b8 - b3 - b2) * q^19 + (-b9 + 2b6 + b2 - 3) q^21 + (b12 + b7 - b4 + b1) * q^23 + (-b7 + b5 + 2b4 - b1) q^27 + (2b10 + b9 + b8 + b3 - b2) q^29 + (-b10 + b9) * q^31 + (-2b12 + 2b11 + b7 - 2b5 - b4 + 2b1) * q^33 + (-2b13 + b12 - b11 + 4b4 + 2b1) q^37 + (-b6 - 2b3 - b2 - 1) q^39 - q^41 + (2b13 + b7 - b5 - 3b4 + b1) * q^43 + (-b13 - b12 - b11 + b7 + 2b4 + 2b1) * q^47 + (-b10 - 2b9 - b8 + b6 - 2b3 + b2 - 3) * q^49 + (2b10 + b8 + 3b3 - 4) * q^51 + (-2b13 + 2b12 + b5 - b4 + b1) * q^53 + (b13 + 2b12 - b5 + 3b4 + b1) * q^57 + (-3b10 - b8 + b6 - 2b3 - 2) * q^59 + (-b10 + 2b9 - 2b8 + 2b6 - 3b3 + b2 - 1) * q^61 + (-2b13 - 2b11 - 2b7 + 3b5 + 3b4 - 2b1) * q^63 + (b13 - 4b12 + 2b11 + b7 - 2b5 + 5b4 + b1) * q^67 + (-2b10 - b9 + 2b6 - 3b3 + 2b2 - 4) * q^69 + (b10 - 2b6 + 2b3 - b2 - 3) * q^71 + (b12 - 2b7 - b5 - 2b4 + b1) * q^73 + (b12 + 2b7 - 2b5 + 3b4 + 4b1) * q^77 + (b10 - b9 - 3b6 - b2 - 1) q^79 + (b10 + b8 - 2b6 + b3 + b2 - 2) q^81 + (2b13 - b12 + b7 - 2b5) * q^83 + (-b13 + 4b12 - 3b11 + 2b7 + b5 + 2b1) * q^87 + (3b9 + b8 - 2b3) * q^89 + (-b10 - 3b9 + b8 + 2b3 - 2b2 - 3) q^91 + (-3b12 + 3b11 - b7 - b5 - b4 - b1) * q^93 + (2b12 - 3b11 + 3b5 + 6b4 - b1) * q^97 + (b10 + 2b6 - 2b3 + b2 - 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 12 q^{9}+O(q^{10}) $ 14 * q - 12 * q^9	$ 14 q - 12 q^{9} - 6 q^{11} - 6 q^{19} - 26 q^{21} + 2 q^{29} - 8 q^{31} - 26 q^{39} - 14 q^{41} - 34 q^{49} - 42 q^{51} - 40 q^{59} - 20 q^{61} - 50 q^{69} - 46 q^{71} - 22 q^{79} - 26 q^{81} - 16 q^{89} - 38 q^{91} - 72 q^{99}+O(q^{100}) $ 14 * q - 12 * q^9 - 6 * q^11 - 6 * q^19 - 26 * q^21 + 2 * q^29 - 8 * q^31 - 26 * q^39 - 14 * q^41 - 34 * q^49 - 42 * q^51 - 40 * q^59 - 20 * q^61 - 50 * q^69 - 46 * q^71 - 22 * q^79 - 26 * q^81 - 16 * q^89 - 38 * q^91 - 72 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{14} + 27x^{12} + 281x^{10} + 1405x^{8} + 3397x^{6} + 3488x^{4} + 1281x^{2} + 121 $ x^14 + 27*x^12 + 281*x^10 + 1405*x^8 + 3397*x^6 + 3488*x^4 + 1281*x^2 + 121 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{12} + \nu^{10} + 273\nu^{8} + 2300\nu^{6} + 6117\nu^{4} + 4783\nu^{2} + 3602 ) / 873 $ (-v^12 + v^10 + 273v^8 + 2300v^6 + 6117v^4 + 4783v^2 + 3602) / 873
$\beta_{3}$	$=$	$ ( -\nu^{12} + \nu^{10} + 273\nu^{8} + 2300\nu^{6} + 6117\nu^{4} + 3910\nu^{2} + 110 ) / 873 $ (-v^12 + v^10 + 273v^8 + 2300v^6 + 6117v^4 + 3910v^2 + 110) / 873
$\beta_{4}$	$=$	$ ( -10\nu^{13} - 281\nu^{11} - 2799\nu^{9} - 11047\nu^{7} - 8670\nu^{5} + 32407\nu^{3} + 30200\nu ) / 9603 $ (-10v^13 - 281v^11 - 2799v^9 - 11047v^7 - 8670v^5 + 32407v^3 + 30200*v) / 9603
$\beta_{5}$	$=$	$ ( -31\nu^{13} - 551\nu^{11} - 2595\nu^{9} + 3206\nu^{7} + 49947\nu^{5} + 117427\nu^{3} + 119228\nu ) / 9603 $ (-31v^13 - 551v^11 - 2595v^9 + 3206v^7 + 49947v^5 + 117427v^3 + 119228*v) / 9603
$\beta_{6}$	$=$	$ ( -8\nu^{12} - 186\nu^{10} - 1599\nu^{8} - 6238\nu^{6} - 10913\nu^{4} - 7714\nu^{2} - 1642 ) / 291 $ (-8v^12 - 186v^10 - 1599v^8 - 6238v^6 - 10913v^4 - 7714v^2 - 1642) / 291
$\beta_{7}$	$=$	$ ( -17\nu^{13} - 371\nu^{11} - 2731\nu^{9} - 6296\nu^{7} + 10869\nu^{5} + 57546\nu^{3} + 37469\nu ) / 3201 $ (-17v^13 - 371v^11 - 2731v^9 - 6296v^7 + 10869v^5 + 57546v^3 + 37469*v) / 3201
$\beta_{8}$	$=$	$ ( -3\nu^{12} - 94\nu^{10} - 1121\nu^{8} - 6195\nu^{6} - 15114\nu^{4} - 11744\nu^{2} - 1513 ) / 291 $ (-3v^12 - 94v^10 - 1121v^8 - 6195v^6 - 15114v^4 - 11744v^2 - 1513) / 291
$\beta_{9}$	$=$	$ ( -3\nu^{12} - 94\nu^{10} - 1121\nu^{8} - 6292\nu^{6} - 16375\nu^{4} - 15915\nu^{2} - 3356 ) / 291 $ (-3v^12 - 94v^10 - 1121v^8 - 6292v^6 - 16375v^4 - 15915v^2 - 3356) / 291
$\beta_{10}$	$=$	$ ( -37\nu^{12} - 836\nu^{10} - 6777\nu^{8} - 23443\nu^{6} - 31497\nu^{4} - 11888\nu^{2} + 578 ) / 873 $ (-37v^12 - 836v^10 - 6777v^8 - 23443v^6 - 31497v^4 - 11888v^2 + 578) / 873
$\beta_{11}$	$=$	$ ( -109\nu^{13} - 3383\nu^{11} - 39792\nu^{9} - 218683\nu^{7} - 549045\nu^{5} - 492788\nu^{3} - 70945\nu ) / 9603 $ (-109v^13 - 3383v^11 - 39792v^9 - 218683v^7 - 549045v^5 - 492788v^3 - 70945*v) / 9603
$\beta_{12}$	$=$	$ ( -181\nu^{13} - 4766\nu^{11} - 47781\nu^{9} - 226519\nu^{7} - 509037\nu^{5} - 468803\nu^{3} - 122389\nu ) / 9603 $ (-181v^13 - 4766v^11 - 47781v^9 - 226519v^7 - 509037v^5 - 468803v^3 - 122389*v) / 9603
$\beta_{13}$	$=$	$ ( 482\nu^{13} + 12904\nu^{11} + 132351\nu^{9} + 643220\nu^{7} + 1458219\nu^{5} + 1249741\nu^{3} + 272900\nu ) / 9603 $ (482v^13 + 12904v^11 + 132351v^9 + 643220v^7 + 1458219v^5 + 1249741v^3 + 272900*v) / 9603

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{3} + \beta_{2} - 4 $ -b3 + b2 - 4
$\nu^{3}$	$=$	$ -\beta_{7} + \beta_{5} + 2\beta_{4} - 7\beta_1 $ -b7 + b5 + 2b4 - 7b1
$\nu^{4}$	$=$	$ \beta_{10} + \beta_{8} - 2\beta_{6} + 10\beta_{3} - 8\beta_{2} + 25 $ b10 + b8 - 2b6 + 10b3 - 8*b2 + 25
$\nu^{5}$	$=$	$ \beta_{13} + \beta_{12} + 2\beta_{11} + 12\beta_{7} - 9\beta_{5} - 25\beta_{4} + 49\beta_1 $ b13 + b12 + 2b11 + 12b7 - 9b5 - 25b4 + 49*b1
$\nu^{6}$	$=$	$ -13\beta_{10} - 3\beta_{9} - 10\beta_{8} + 26\beta_{6} - 87\beta_{3} + 61\beta_{2} - 172 $ -13b10 - 3b9 - 10b8 + 26b6 - 87b3 + 61b2 - 172
$\nu^{7}$	$=$	$ -16\beta_{13} - 19\beta_{12} - 29\beta_{11} - 116\beta_{7} + 74\beta_{5} + 251\beta_{4} - 352\beta_1 $ -16b13 - 19b12 - 29b11 - 116b7 + 74b5 + 251b4 - 352*b1
$\nu^{8}$	$=$	$ 135\beta_{10} + 48\beta_{9} + 81\beta_{8} - 267\beta_{6} + 726\beta_{3} - 474\beta_{2} + 1243 $ 135b10 + 48b9 + 81b8 - 267b6 + 726b3 - 474b2 + 1243
$\nu^{9}$	$=$	$ 186\beta_{13} + 243\beta_{12} + 312\beta_{11} + 1047\beta_{7} - 606\beta_{5} - 2295\beta_{4} + 2602\beta_1 $ 186b13 + 243b12 + 312b11 + 1047b7 - 606b5 - 2295b4 + 2602*b1
$\nu^{10}$	$=$	$ -1290\beta_{10} - 555\beta_{9} - 618\beta_{8} + 2520\beta_{6} - 5956\beta_{3} + 3763\beta_{2} - 9316 $ -1290b10 - 555b9 - 618b8 + 2520b6 - 5956b3 + 3763b2 - 9316
$\nu^{11}$	$=$	$ -1902\beta_{13} - 2634\beta_{12} - 3015\beta_{11} - 9148\beta_{7} + 4993\beta_{5} + 20039\beta_{4} - 19723\beta_1 $ -1902b13 - 2634b12 - 3015b11 - 9148b7 + 4993b5 + 20039b4 - 19723*b1
$\nu^{12}$	$=$	$ 11782\beta_{10} + 5649\beta_{9} + 4612\beta_{8} - 22805\beta_{6} + 48529\beta_{3} - 30365\beta_{2} + 71818 $ 11782b10 + 5649b9 + 4612b8 - 22805b6 + 48529b3 - 30365b2 + 71818
$\nu^{13}$	$=$	$ 18193 \beta_{13} + 26122 \beta_{12} + 27695 \beta_{11} + 78504 \beta_{7} - 41388 \beta_{5} + \cdots + 152623 \beta_1 $ 18193b13 + 26122b12 + 27695b11 + 78504b7 - 41388b5 - 170809b4 + 152623*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$.

$n$	$1477$	$2051$	$3901$
$\chi(n)$	$-1$	$1$	$1$

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} $

1149.1

	−	2.88961i
	−	2.56508i
	−	2.32767i
	−	2.18506i
	−	1.12349i
	−	0.687910i
	−	0.377545i
		0.377545i
		0.687910i
		1.12349i
		2.18506i
		2.32767i
		2.56508i
		2.88961i

−

2.88961i

−

4.40706i

−5.34984

1149.2

−

2.56508i

2.92431i

−3.57966

1149.3

−

2.32767i

−

2.51583i

−2.41803

1149.4

−

2.18506i

1.94794i

−1.77449

1149.5

−

1.12349i

−

2.98128i

1.73778

1149.6

−

0.687910i

−

4.33809i

2.52678

1149.7

−

0.377545i

0.441953i

2.85746

1149.8

0.377545i

−

0.441953i

2.85746

1149.9

0.687910i

4.33809i

2.52678

1149.10

1.12349i

2.98128i

1.73778

1149.11

2.18506i

−

1.94794i

−1.77449

1149.12

2.32767i

2.51583i

−2.41803

1149.13

2.56508i

−

2.92431i

−3.57966

1149.14

2.88961i

4.40706i

−5.34984

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	4100.2.d.f		14
5.b	even	2	1	inner	4100.2.d.f		14
5.c	odd	4	1		4100.2.a.h	✓	7
5.c	odd	4	1		4100.2.a.i	yes	7

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4100.2.a.h	✓	7	5.c	odd	4	1
4100.2.a.i	yes	7	5.c	odd	4	1
4100.2.d.f		14	1.a	even	1	1	trivial
4100.2.d.f		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 $S_{2}^{\mathrm{new}}(4100, [\chi])$:

$ T_{3}^{14} + 27T_{3}^{12} + 281T_{3}^{10} + 1405T_{3}^{8} + 3397T_{3}^{6} + 3488T_{3}^{4} + 1281T_{3}^{2} + 121 $

$ T_{7}^{14} + 66T_{7}^{12} + 1709T_{7}^{10} + 22171T_{7}^{8} + 152626T_{7}^{6} + 533126T_{7}^{4} + 765681T_{7}^{2} + 130321 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{14} $ T^14
$3$	$ T^{14} + 27 T^{12} + \cdots + 121 $ T^14 + 27T^12 + 281T^10 + 1405T^8 + 3397T^6 + 3488T^4 + 1281T^2 + 121
$5$	$ T^{14} $ T^14
$7$	$ T^{14} + 66 T^{12} + \cdots + 130321 $ T^14 + 66T^12 + 1709T^10 + 22171T^8 + 152626T^6 + 533126T^4 + 765681T^2 + 130321
$11$	$ (T^{7} + 3 T^{6} + \cdots + 192)^{2} $ (T^7 + 3T^6 - 36T^5 - 65T^4 + 393T^3 + 189T^2 - 736T + 192)^2
$13$	$ T^{14} + 87 T^{12} + \cdots + 326041 $ T^14 + 87T^12 + 2893T^10 + 46621T^8 + 374989T^6 + 1350760T^4 + 1428489T^2 + 326041
$17$	$ T^{14} + 137 T^{12} + \cdots + 51984 $ T^14 + 137T^12 + 6100T^10 + 95227T^8 + 436627T^6 + 714137T^4 + 401080T^2 + 51984
$19$	$ (T^{7} + 3 T^{6} + \cdots - 14227)^{2} $ (T^7 + 3T^6 - 73T^5 - 216T^4 + 1317T^3 + 3224T^2 - 6457T - 14227)^2
$23$	$ T^{14} + 102 T^{12} + \cdots + 729 $ T^14 + 102T^12 + 1781T^10 + 11568T^8 + 31264T^6 + 31140T^4 + 11016T^2 + 729
$29$	$ (T^{7} - T^{6} + \cdots + 19707)^{2} $ (T^7 - T^6 - 125T^5 + 55T^4 + 4547T^3 + 332T^2 - 41857*T + 19707)^2
$31$	$ (T^{7} + 4 T^{6} - 71 T^{5} + \cdots - 81)^{2} $ (T^7 + 4T^6 - 71T^5 - 89T^4 + 1524T^3 - 1944T^2 - 945T - 81)^2
$37$	$ T^{14} + \cdots + 4555035081 $ T^14 + 350T^12 + 48361T^10 + 3325024T^8 + 117273860T^6 + 1940652472T^4 + 11033017128T^2 + 4555035081
$41$	$ (T + 1)^{14} $ (T + 1)^14
$43$	$ T^{14} + 294 T^{12} + \cdots + 1008016 $ T^14 + 294T^12 + 25997T^10 + 721816T^8 + 3405640T^6 + 5783993T^4 + 4082940T^2 + 1008016
$47$	$ T^{14} + 310 T^{12} + \cdots + 4782969 $ T^14 + 310T^12 + 31361T^10 + 1229785T^8 + 20504224T^6 + 127856394T^4 + 100567008T^2 + 4782969
$53$	$ T^{14} + \cdots + 609151761 $ T^14 + 432T^12 + 63542T^10 + 3726012T^8 + 77965901T^6 + 586274109T^4 + 1184221798T^2 + 609151761
$59$	$ (T^{7} + 20 T^{6} + \cdots + 421761)^{2} $ (T^7 + 20T^6 - 82T^5 - 4006T^4 - 20226T^3 + 73683T^2 + 615482T + 421761)^2
$61$	$ (T^{7} + 10 T^{6} + \cdots + 124757)^{2} $ (T^7 + 10T^6 - 171T^5 - 1540T^4 + 6498T^3 + 34068T^2 - 145736T + 124757)^2
$67$	$ T^{14} + \cdots + 12372335361 $ T^14 + 696T^12 + 183586T^10 + 22451092T^8 + 1211546925T^6 + 20019691525T^4 + 38619384210T^2 + 12372335361
$71$	$ (T^{7} + 23 T^{6} + \cdots - 34413)^{2} $ (T^7 + 23T^6 + 52T^5 - 2900T^4 - 33385T^3 - 145768T^2 - 240058T - 34413)^2
$73$	$ T^{14} + \cdots + 5547904449201 $ T^14 + 673T^12 + 177358T^10 + 23581948T^8 + 1682938863T^6 + 62700721380T^4 + 1072433057652T^2 + 5547904449201
$79$	$ (T^{7} + 11 T^{6} + \cdots + 325089)^{2} $ (T^7 + 11T^6 - 218T^5 - 2518T^4 + 7429T^3 + 141881T^2 + 400452T + 325089)^2
$83$	$ T^{14} + 424 T^{12} + \cdots + 12027024 $ T^14 + 424T^12 + 51816T^10 + 2456153T^8 + 42378210T^6 + 158986753T^4 + 104856136T^2 + 12027024
$89$	$ (T^{7} + 8 T^{6} + \cdots - 938961)^{2} $ (T^7 + 8T^6 - 306T^5 - 3158T^4 + 7839T^3 + 132995T^2 + 79138T - 938961)^2
$97$	$ T^{14} + \cdots + 25528689729 $ T^14 + 929T^12 + 327186T^10 + 54092118T^8 + 4171098995T^6 + 121950385825T^4 + 111374071110T^2 + 25528689729
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{11} q^{7} + ( - \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \) q + b1 * q^3 + b11 * q^7 + (-b3 + b2 - 1) * q^9	\( q + \beta_1 q^{3} + \beta_{11} q^{7} + ( - \beta_{3} + \beta_{2} - 1) q^{9} + (\beta_{8} - \beta_{6} + \cdots - \beta_{2}) q^{11}+ \cdots + (\beta_{10} + 2 \beta_{6} - 2 \beta_{3} + \cdots - 6) q^{99}+O(q^{100}) \) q + b1 * q^3 + b11 * q^7 + (-b3 + b2 - 1) * q^9 + (b8 - b6 + b3 - b2) * q^11 + (b5 - b4) * q^13 + (b13 - b12 + b11 - b7 + b4) * q^17 + (-b8 - b3 - b2) * q^19 + (-b9 + 2b6 + b2 - 3) q^21 + (b12 + b7 - b4 + b1) * q^23 + (-b7 + b5 + 2b4 - b1) q^27 + (2b10 + b9 + b8 + b3 - b2) q^29 + (-b10 + b9) * q^31 + (-2b12 + 2b11 + b7 - 2b5 - b4 + 2b1) * q^33 + (-2b13 + b12 - b11 + 4b4 + 2b1) q^37 + (-b6 - 2b3 - b2 - 1) q^39 - q^41 + (2b13 + b7 - b5 - 3b4 + b1) * q^43 + (-b13 - b12 - b11 + b7 + 2b4 + 2b1) * q^47 + (-b10 - 2b9 - b8 + b6 - 2b3 + b2 - 3) * q^49 + (2b10 + b8 + 3b3 - 4) * q^51 + (-2b13 + 2b12 + b5 - b4 + b1) * q^53 + (b13 + 2b12 - b5 + 3b4 + b1) * q^57 + (-3b10 - b8 + b6 - 2b3 - 2) * q^59 + (-b10 + 2b9 - 2b8 + 2b6 - 3b3 + b2 - 1) * q^61 + (-2b13 - 2b11 - 2b7 + 3b5 + 3b4 - 2b1) * q^63 + (b13 - 4b12 + 2b11 + b7 - 2b5 + 5b4 + b1) * q^67 + (-2b10 - b9 + 2b6 - 3b3 + 2b2 - 4) * q^69 + (b10 - 2b6 + 2b3 - b2 - 3) * q^71 + (b12 - 2b7 - b5 - 2b4 + b1) * q^73 + (b12 + 2b7 - 2b5 + 3b4 + 4b1) * q^77 + (b10 - b9 - 3b6 - b2 - 1) q^79 + (b10 + b8 - 2b6 + b3 + b2 - 2) q^81 + (2b13 - b12 + b7 - 2b5) * q^83 + (-b13 + 4b12 - 3b11 + 2b7 + b5 + 2b1) * q^87 + (3b9 + b8 - 2b3) * q^89 + (-b10 - 3b9 + b8 + 2b3 - 2b2 - 3) q^91 + (-3b12 + 3b11 - b7 - b5 - b4 - b1) * q^93 + (2b12 - 3b11 + 3b5 + 6b4 - b1) * q^97 + (b10 + 2b6 - 2b3 + b2 - 6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 12 q^{9}+O(q^{10}) \) 14 * q - 12 * q^9	\( 14 q - 12 q^{9} - 6 q^{11} - 6 q^{19} - 26 q^{21} + 2 q^{29} - 8 q^{31} - 26 q^{39} - 14 q^{41} - 34 q^{49} - 42 q^{51} - 40 q^{59} - 20 q^{61} - 50 q^{69} - 46 q^{71} - 22 q^{79} - 26 q^{81} - 16 q^{89} - 38 q^{91} - 72 q^{99}+O(q^{100}) \) 14 * q - 12 * q^9 - 6 * q^11 - 6 * q^19 - 26 * q^21 + 2 * q^29 - 8 * q^31 - 26 * q^39 - 14 * q^41 - 34 * q^49 - 42 * q^51 - 40 * q^59 - 20 * q^61 - 50 * q^69 - 46 * q^71 - 22 * q^79 - 26 * q^81 - 16 * q^89 - 38 * q^91 - 72 * q^99

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