LMFDB - Embedded newform 6042.2.a.be.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 36 x^{10} + 105 x^{9} + 465 x^{8} - 1287 x^{7} - 2668 x^{6} + 6443 x^{5} + \cdots + 1496 $ x^12 - 3x^11 - 36x^10 + 105x^9 + 465x^8 - 1287x^7 - 2668x^6 + 6443x^5 + 7140x^4 - 10858x^3 - 10086x^2 + 2072*x + 1496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.89280$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.89280 q^{5} +1.00000 q^{6} -4.43415 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.89280 q^{5} +1.00000 q^{6} -4.43415 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.89280 q^{10} +2.55061 q^{11} -1.00000 q^{12} +1.38802 q^{13} +4.43415 q^{14} -2.89280 q^{15} +1.00000 q^{16} -0.833577 q^{17} -1.00000 q^{18} -1.00000 q^{19} +2.89280 q^{20} +4.43415 q^{21} -2.55061 q^{22} +1.26007 q^{23} +1.00000 q^{24} +3.36829 q^{25} -1.38802 q^{26} -1.00000 q^{27} -4.43415 q^{28} +1.25574 q^{29} +2.89280 q^{30} -5.61422 q^{31} -1.00000 q^{32} -2.55061 q^{33} +0.833577 q^{34} -12.8271 q^{35} +1.00000 q^{36} -6.56122 q^{37} +1.00000 q^{38} -1.38802 q^{39} -2.89280 q^{40} +12.1665 q^{41} -4.43415 q^{42} -4.05933 q^{43} +2.55061 q^{44} +2.89280 q^{45} -1.26007 q^{46} -9.52989 q^{47} -1.00000 q^{48} +12.6617 q^{49} -3.36829 q^{50} +0.833577 q^{51} +1.38802 q^{52} -1.00000 q^{53} +1.00000 q^{54} +7.37839 q^{55} +4.43415 q^{56} +1.00000 q^{57} -1.25574 q^{58} +4.34318 q^{59} -2.89280 q^{60} -11.4325 q^{61} +5.61422 q^{62} -4.43415 q^{63} +1.00000 q^{64} +4.01525 q^{65} +2.55061 q^{66} -7.92927 q^{67} -0.833577 q^{68} -1.26007 q^{69} +12.8271 q^{70} +4.38742 q^{71} -1.00000 q^{72} +7.83474 q^{73} +6.56122 q^{74} -3.36829 q^{75} -1.00000 q^{76} -11.3098 q^{77} +1.38802 q^{78} -11.4441 q^{79} +2.89280 q^{80} +1.00000 q^{81} -12.1665 q^{82} +4.76641 q^{83} +4.43415 q^{84} -2.41137 q^{85} +4.05933 q^{86} -1.25574 q^{87} -2.55061 q^{88} +13.7811 q^{89} -2.89280 q^{90} -6.15467 q^{91} +1.26007 q^{92} +5.61422 q^{93} +9.52989 q^{94} -2.89280 q^{95} +1.00000 q^{96} -5.66278 q^{97} -12.6617 q^{98} +2.55061 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} + 12 q^{16} - 4 q^{17} - 12 q^{18} - 12 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} - 18 q^{23} + 12 q^{24} + 21 q^{25} - 4 q^{26} - 12 q^{27} - q^{28} + 10 q^{29} - 3 q^{30} + 14 q^{31} - 12 q^{32} + 7 q^{33} + 4 q^{34} - 19 q^{35} + 12 q^{36} + 14 q^{37} + 12 q^{38} - 4 q^{39} + 3 q^{40} - 9 q^{41} - q^{42} - 4 q^{43} - 7 q^{44} - 3 q^{45} + 18 q^{46} - 14 q^{47} - 12 q^{48} + 37 q^{49} - 21 q^{50} + 4 q^{51} + 4 q^{52} - 12 q^{53} + 12 q^{54} - 4 q^{55} + q^{56} + 12 q^{57} - 10 q^{58} - 18 q^{59} + 3 q^{60} - 7 q^{61} - 14 q^{62} - q^{63} + 12 q^{64} + 3 q^{65} - 7 q^{66} + 8 q^{67} - 4 q^{68} + 18 q^{69} + 19 q^{70} - q^{71} - 12 q^{72} - 31 q^{73} - 14 q^{74} - 21 q^{75} - 12 q^{76} - 25 q^{77} + 4 q^{78} - 3 q^{80} + 12 q^{81} + 9 q^{82} - 48 q^{83} + q^{84} + 4 q^{85} + 4 q^{86} - 10 q^{87} + 7 q^{88} + 13 q^{89} + 3 q^{90} + 9 q^{91} - 18 q^{92} - 14 q^{93} + 14 q^{94} + 3 q^{95} + 12 q^{96} + 25 q^{97} - 37 q^{98} - 7 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 7 * q^11 - 12 * q^12 + 4 * q^13 + q^14 + 3 * q^15 + 12 * q^16 - 4 * q^17 - 12 * q^18 - 12 * q^19 - 3 * q^20 + q^21 + 7 * q^22 - 18 * q^23 + 12 * q^24 + 21 * q^25 - 4 * q^26 - 12 * q^27 - q^28 + 10 * q^29 - 3 * q^30 + 14 * q^31 - 12 * q^32 + 7 * q^33 + 4 * q^34 - 19 * q^35 + 12 * q^36 + 14 * q^37 + 12 * q^38 - 4 * q^39 + 3 * q^40 - 9 * q^41 - q^42 - 4 * q^43 - 7 * q^44 - 3 * q^45 + 18 * q^46 - 14 * q^47 - 12 * q^48 + 37 * q^49 - 21 * q^50 + 4 * q^51 + 4 * q^52 - 12 * q^53 + 12 * q^54 - 4 * q^55 + q^56 + 12 * q^57 - 10 * q^58 - 18 * q^59 + 3 * q^60 - 7 * q^61 - 14 * q^62 - q^63 + 12 * q^64 + 3 * q^65 - 7 * q^66 + 8 * q^67 - 4 * q^68 + 18 * q^69 + 19 * q^70 - q^71 - 12 * q^72 - 31 * q^73 - 14 * q^74 - 21 * q^75 - 12 * q^76 - 25 * q^77 + 4 * q^78 - 3 * q^80 + 12 * q^81 + 9 * q^82 - 48 * q^83 + q^84 + 4 * q^85 + 4 * q^86 - 10 * q^87 + 7 * q^88 + 13 * q^89 + 3 * q^90 + 9 * q^91 - 18 * q^92 - 14 * q^93 + 14 * q^94 + 3 * q^95 + 12 * q^96 + 25 * q^97 - 37 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.89280	1.29370	0.646850	−	0.762618i	$-0.276086\pi$
$5$	2.89280	1.29370	0.646850	+	0.762618i	$0.276086\pi$
$6$	1.00000	0.408248
$7$	−4.43415	−1.67595	−0.837976	−	0.545708i	$-0.816261\pi$
$7$	−4.43415	−1.67595	−0.837976	+	0.545708i	$0.816261\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.89280	−0.914784
$11$	2.55061	0.769036	0.384518	−	0.923117i	$-0.374368\pi$
$11$	2.55061	0.769036	0.384518	+	0.923117i	$0.374368\pi$
$12$	−1.00000	−0.288675
$13$	1.38802	0.384967	0.192483	−	0.981300i	$-0.438346\pi$
$13$	1.38802	0.384967	0.192483	+	0.981300i	$0.438346\pi$
$14$	4.43415	1.18508
$15$	−2.89280	−0.746918
$16$	1.00000	0.250000
$17$	−0.833577	−0.202172	−0.101086	−	0.994878i	$-0.532232\pi$
$17$	−0.833577	−0.202172	−0.101086	+	0.994878i	$0.532232\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.89280	0.646850
$21$	4.43415	0.967611
$22$	−2.55061	−0.543791
$23$	1.26007	0.262743	0.131371	−	0.991333i	$-0.458062\pi$
$23$	1.26007	0.262743	0.131371	+	0.991333i	$0.458062\pi$
$24$	1.00000	0.204124
$25$	3.36829	0.673658
$26$	−1.38802	−0.272212
$27$	−1.00000	−0.192450
$28$	−4.43415	−0.837976
$29$	1.25574	0.233186	0.116593	−	0.993180i	$-0.462803\pi$
$29$	1.25574	0.233186	0.116593	+	0.993180i	$0.462803\pi$
$30$	2.89280	0.528151
$31$	−5.61422	−1.00834	−0.504172	−	0.863603i	$-0.668202\pi$
$31$	−5.61422	−1.00834	−0.504172	+	0.863603i	$0.668202\pi$
$32$	−1.00000	−0.176777
$33$	−2.55061	−0.444003
$34$	0.833577	0.142957
$35$	−12.8271	−2.16818
$36$	1.00000	0.166667
$37$	−6.56122	−1.07866	−0.539329	−	0.842095i	$-0.681322\pi$
$37$	−6.56122	−1.07866	−0.539329	+	0.842095i	$0.681322\pi$
$38$	1.00000	0.162221
$39$	−1.38802	−0.222261
$40$	−2.89280	−0.457392
$41$	12.1665	1.90008	0.950041	−	0.312124i	$-0.101041\pi$
$41$	12.1665	1.90008	0.950041	+	0.312124i	$0.101041\pi$
$42$	−4.43415	−0.684204
$43$	−4.05933	−0.619042	−0.309521	−	0.950893i	$-0.600169\pi$
$43$	−4.05933	−0.619042	−0.309521	+	0.950893i	$0.600169\pi$
$44$	2.55061	0.384518
$45$	2.89280	0.431233
$46$	−1.26007	−0.185787
$47$	−9.52989	−1.39008	−0.695039	−	0.718972i	$-0.744613\pi$
$47$	−9.52989	−1.39008	−0.695039	+	0.718972i	$0.744613\pi$
$48$	−1.00000	−0.144338
$49$	12.6617	1.80881
$50$	−3.36829	−0.476348
$51$	0.833577	0.116724
$52$	1.38802	0.192483
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	1.00000	0.136083
$55$	7.37839	0.994902
$56$	4.43415	0.592538
$57$	1.00000	0.132453
$58$	−1.25574	−0.164887
$59$	4.34318	0.565434	0.282717	−	0.959203i	$-0.408764\pi$
$59$	4.34318	0.565434	0.282717	+	0.959203i	$0.408764\pi$
$60$	−2.89280	−0.373459
$61$	−11.4325	−1.46378	−0.731891	−	0.681422i	$-0.761362\pi$
$61$	−11.4325	−1.46378	−0.731891	+	0.681422i	$0.761362\pi$
$62$	5.61422	0.713007
$63$	−4.43415	−0.558650
$64$	1.00000	0.125000
$65$	4.01525	0.498031
$66$	2.55061	0.313958
$67$	−7.92927	−0.968714	−0.484357	−	0.874870i	$-0.660946\pi$
$67$	−7.92927	−0.968714	−0.484357	+	0.874870i	$0.660946\pi$
$68$	−0.833577	−0.101086
$69$	−1.26007	−0.151694
$70$	12.8271	1.53313
$71$	4.38742	0.520691	0.260345	−	0.965516i	$-0.416164\pi$
$71$	4.38742	0.520691	0.260345	+	0.965516i	$0.416164\pi$
$72$	−1.00000	−0.117851
$73$	7.83474	0.916987	0.458494	−	0.888698i	$-0.348389\pi$
$73$	7.83474	0.916987	0.458494	+	0.888698i	$0.348389\pi$
$74$	6.56122	0.762726
$75$	−3.36829	−0.388937
$76$	−1.00000	−0.114708
$77$	−11.3098	−1.28887
$78$	1.38802	0.157162
$79$	−11.4441	−1.28756	−0.643781	−	0.765209i	$-0.722635\pi$
$79$	−11.4441	−1.28756	−0.643781	+	0.765209i	$0.722635\pi$
$80$	2.89280	0.323425
$81$	1.00000	0.111111
$82$	−12.1665	−1.34356
$83$	4.76641	0.523181	0.261591	−	0.965179i	$-0.415753\pi$
$83$	4.76641	0.523181	0.261591	+	0.965179i	$0.415753\pi$
$84$	4.43415	0.483805
$85$	−2.41137	−0.261550
$86$	4.05933	0.437729
$87$	−1.25574	−0.134630
$88$	−2.55061	−0.271895
$89$	13.7811	1.46079	0.730397	−	0.683023i	$-0.239335\pi$
$89$	13.7811	1.46079	0.730397	+	0.683023i	$0.239335\pi$
$90$	−2.89280	−0.304928
$91$	−6.15467	−0.645185
$92$	1.26007	0.131371
$93$	5.61422	0.582167
$94$	9.52989	0.982933
$95$	−2.89280	−0.296795
$96$	1.00000	0.102062
$97$	−5.66278	−0.574968	−0.287484	−	0.957785i	$-0.592819\pi$
$97$	−5.66278	−0.574968	−0.287484	+	0.957785i	$0.592819\pi$
$98$	−12.6617	−1.27902
$99$	2.55061	0.256345
$100$	3.36829	0.336829
$101$	11.6482	1.15904	0.579520	−	0.814958i	$-0.303240\pi$
$101$	11.6482	1.15904	0.579520	+	0.814958i	$0.303240\pi$
$102$	−0.833577	−0.0825364
$103$	15.1470	1.49247	0.746237	−	0.665680i	$-0.231858\pi$
$103$	15.1470	1.49247	0.746237	+	0.665680i	$0.231858\pi$
$104$	−1.38802	−0.136106
$105$	12.8271	1.25180
$106$	1.00000	0.0971286
$107$	−18.5935	−1.79750	−0.898752	−	0.438457i	$-0.855525\pi$
$107$	−18.5935	−1.79750	−0.898752	+	0.438457i	$0.855525\pi$
$108$	−1.00000	−0.0962250
$109$	1.79861	0.172276	0.0861379	−	0.996283i	$-0.472547\pi$
$109$	1.79861	0.172276	0.0861379	+	0.996283i	$0.472547\pi$
$110$	−7.37839	−0.703502
$111$	6.56122	0.622763
$112$	−4.43415	−0.418988
$113$	3.64902	0.343271	0.171635	−	0.985161i	$-0.445095\pi$
$113$	3.64902	0.343271	0.171635	+	0.985161i	$0.445095\pi$
$114$	−1.00000	−0.0936586
$115$	3.64513	0.339910
$116$	1.25574	0.116593
$117$	1.38802	0.128322
$118$	−4.34318	−0.399823
$119$	3.69621	0.338831
$120$	2.89280	0.264075
$121$	−4.49441	−0.408583
$122$	11.4325	1.03505
$123$	−12.1665	−1.09701
$124$	−5.61422	−0.504172
$125$	−4.72021	−0.422189
$126$	4.43415	0.395026
$127$	−10.9397	−0.970741	−0.485370	−	0.874309i	$-0.661315\pi$
$127$	−10.9397	−0.970741	−0.485370	+	0.874309i	$0.661315\pi$
$128$	−1.00000	−0.0883883
$129$	4.05933	0.357404
$130$	−4.01525	−0.352161
$131$	3.12574	0.273097	0.136549	−	0.990633i	$-0.456399\pi$
$131$	3.12574	0.273097	0.136549	+	0.990633i	$0.456399\pi$
$132$	−2.55061	−0.222002
$133$	4.43415	0.384490
$134$	7.92927	0.684985
$135$	−2.89280	−0.248973
$136$	0.833577	0.0714786
$137$	−12.8454	−1.09746	−0.548730	−	0.836000i	$-0.684888\pi$
$137$	−12.8454	−1.09746	−0.548730	+	0.836000i	$0.684888\pi$
$138$	1.26007	0.107264
$139$	−13.4067	−1.13714	−0.568572	−	0.822633i	$-0.692504\pi$
$139$	−13.4067	−1.13714	−0.568572	+	0.822633i	$0.692504\pi$
$140$	−12.8271	−1.08409
$141$	9.52989	0.802562
$142$	−4.38742	−0.368184
$143$	3.54028	0.296053
$144$	1.00000	0.0833333
$145$	3.63262	0.301672
$146$	−7.83474	−0.648408
$147$	−12.6617	−1.04432
$148$	−6.56122	−0.539329
$149$	−19.7899	−1.62125	−0.810627	−	0.585563i	$-0.800874\pi$
$149$	−19.7899	−1.62125	−0.810627	+	0.585563i	$0.800874\pi$
$150$	3.36829	0.275020
$151$	20.1482	1.63964	0.819820	−	0.572621i	$-0.194073\pi$
$151$	20.1482	1.63964	0.819820	+	0.572621i	$0.194073\pi$
$152$	1.00000	0.0811107
$153$	−0.833577	−0.0673907
$154$	11.3098	0.911367
$155$	−16.2408	−1.30449
$156$	−1.38802	−0.111130
$157$	0.585212	0.0467050	0.0233525	−	0.999727i	$-0.492566\pi$
$157$	0.585212	0.0467050	0.0233525	+	0.999727i	$0.492566\pi$
$158$	11.4441	0.910444
$159$	1.00000	0.0793052
$160$	−2.89280	−0.228696
$161$	−5.58734	−0.440344
$162$	−1.00000	−0.0785674
$163$	−21.8455	−1.71107	−0.855535	−	0.517746i	$-0.826771\pi$
$163$	−21.8455	−1.71107	−0.855535	+	0.517746i	$0.826771\pi$
$164$	12.1665	0.950041
$165$	−7.37839	−0.574407
$166$	−4.76641	−0.369945
$167$	13.4068	1.03745	0.518724	−	0.854942i	$-0.326407\pi$
$167$	13.4068	1.03745	0.518724	+	0.854942i	$0.326407\pi$
$168$	−4.43415	−0.342102
$169$	−11.0734	−0.851801
$170$	2.41137	0.184944
$171$	−1.00000	−0.0764719
$172$	−4.05933	−0.309521
$173$	5.23740	0.398192	0.199096	−	0.979980i	$-0.436199\pi$
$173$	5.23740	0.398192	0.199096	+	0.979980i	$0.436199\pi$
$174$	1.25574	0.0951977
$175$	−14.9355	−1.12902
$176$	2.55061	0.192259
$177$	−4.34318	−0.326454
$178$	−13.7811	−1.03294
$179$	−11.5623	−0.864208	−0.432104	−	0.901824i	$-0.642229\pi$
$179$	−11.5623	−0.864208	−0.432104	+	0.901824i	$0.642229\pi$
$180$	2.89280	0.215617
$181$	9.43359	0.701193	0.350597	−	0.936527i	$-0.385979\pi$
$181$	9.43359	0.701193	0.350597	+	0.936527i	$0.385979\pi$
$182$	6.15467	0.456215
$183$	11.4325	0.845115
$184$	−1.26007	−0.0928935
$185$	−18.9803	−1.39546
$186$	−5.61422	−0.411655
$187$	−2.12613	−0.155478
$188$	−9.52989	−0.695039
$189$	4.43415	0.322537
$190$	2.89280	0.209866
$191$	−19.7591	−1.42972	−0.714859	−	0.699268i	$-0.753509\pi$
$191$	−19.7591	−1.42972	−0.714859	+	0.699268i	$0.753509\pi$
$192$	−1.00000	−0.0721688
$193$	−16.2667	−1.17090	−0.585451	−	0.810708i	$-0.699082\pi$
$193$	−16.2667	−1.17090	−0.585451	+	0.810708i	$0.699082\pi$
$194$	5.66278	0.406564
$195$	−4.01525	−0.287538
$196$	12.6617	0.904406
$197$	4.34202	0.309356	0.154678	−	0.987965i	$-0.450566\pi$
$197$	4.34202	0.309356	0.154678	+	0.987965i	$0.450566\pi$
$198$	−2.55061	−0.181264
$199$	9.35771	0.663350	0.331675	−	0.943394i	$-0.392386\pi$
$199$	9.35771	0.663350	0.331675	+	0.943394i	$0.392386\pi$
$200$	−3.36829	−0.238174
$201$	7.92927	0.559288
$202$	−11.6482	−0.819565
$203$	−5.56816	−0.390808
$204$	0.833577	0.0583621
$205$	35.1951	2.45814
$206$	−15.1470	−1.05534
$207$	1.26007	0.0875808
$208$	1.38802	0.0962416
$209$	−2.55061	−0.176429
$210$	−12.8271	−0.885155
$211$	12.6719	0.872368	0.436184	−	0.899858i	$-0.356330\pi$
$211$	12.6719	0.872368	0.436184	+	0.899858i	$0.356330\pi$
$212$	−1.00000	−0.0686803
$213$	−4.38742	−0.300621
$214$	18.5935	1.27103
$215$	−11.7428	−0.800855
$216$	1.00000	0.0680414
$217$	24.8943	1.68993
$218$	−1.79861	−0.121817
$219$	−7.83474	−0.529423
$220$	7.37839	0.497451
$221$	−1.15702	−0.0778295
$222$	−6.56122	−0.440360
$223$	13.6729	0.915608	0.457804	−	0.889053i	$-0.348636\pi$
$223$	13.6729	0.915608	0.457804	+	0.889053i	$0.348636\pi$
$224$	4.43415	0.296269
$225$	3.36829	0.224553
$226$	−3.64902	−0.242729
$227$	5.42303	0.359939	0.179969	−	0.983672i	$-0.442400\pi$
$227$	5.42303	0.359939	0.179969	+	0.983672i	$0.442400\pi$
$228$	1.00000	0.0662266
$229$	11.3460	0.749768	0.374884	−	0.927072i	$-0.377683\pi$
$229$	11.3460	0.749768	0.374884	+	0.927072i	$0.377683\pi$
$230$	−3.64513	−0.240353
$231$	11.3098	0.744128
$232$	−1.25574	−0.0824437
$233$	−16.7811	−1.09936	−0.549682	−	0.835374i	$-0.685251\pi$
$233$	−16.7811	−1.09936	−0.549682	+	0.835374i	$0.685251\pi$
$234$	−1.38802	−0.0907375
$235$	−27.5681	−1.79834
$236$	4.34318	0.282717
$237$	11.4441	0.743375
$238$	−3.69621	−0.239589
$239$	17.5565	1.13563	0.567817	−	0.823155i	$-0.307788\pi$
$239$	17.5565	1.13563	0.567817	+	0.823155i	$0.307788\pi$
$240$	−2.89280	−0.186729
$241$	−13.3030	−0.856920	−0.428460	−	0.903561i	$-0.640944\pi$
$241$	−13.3030	−0.856920	−0.428460	+	0.903561i	$0.640944\pi$
$242$	4.49441	0.288912
$243$	−1.00000	−0.0641500
$244$	−11.4325	−0.731891
$245$	36.6277	2.34006
$246$	12.1665	0.775705
$247$	−1.38802	−0.0883174
$248$	5.61422	0.356503
$249$	−4.76641	−0.302059
$250$	4.72021	0.298532
$251$	−30.0251	−1.89517	−0.947583	−	0.319511i	$-0.896481\pi$
$251$	−30.0251	−1.89517	−0.947583	+	0.319511i	$0.896481\pi$
$252$	−4.43415	−0.279325
$253$	3.21394	0.202059
$254$	10.9397	0.686417
$255$	2.41137	0.151006
$256$	1.00000	0.0625000
$257$	−14.9943	−0.935318	−0.467659	−	0.883909i	$-0.654902\pi$
$257$	−14.9943	−0.935318	−0.467659	+	0.883909i	$0.654902\pi$
$258$	−4.05933	−0.252723
$259$	29.0934	1.80778
$260$	4.01525	0.249015
$261$	1.25574	0.0777286
$262$	−3.12574	−0.193109
$263$	26.3825	1.62682	0.813408	−	0.581694i	$-0.197610\pi$
$263$	26.3825	1.62682	0.813408	+	0.581694i	$0.197610\pi$
$264$	2.55061	0.156979
$265$	−2.89280	−0.177703
$266$	−4.43415	−0.271875
$267$	−13.7811	−0.843390
$268$	−7.92927	−0.484357
$269$	8.20107	0.500028	0.250014	−	0.968242i	$-0.419565\pi$
$269$	8.20107	0.500028	0.250014	+	0.968242i	$0.419565\pi$
$270$	2.89280	0.176050
$271$	−2.32053	−0.140962	−0.0704811	−	0.997513i	$-0.522453\pi$
$271$	−2.32053	−0.140962	−0.0704811	+	0.997513i	$0.522453\pi$
$272$	−0.833577	−0.0505430
$273$	6.15467	0.372498
$274$	12.8454	0.776021
$275$	8.59118	0.518067
$276$	−1.26007	−0.0758472
$277$	7.56983	0.454827	0.227413	−	0.973798i	$-0.426973\pi$
$277$	7.56983	0.454827	0.227413	+	0.973798i	$0.426973\pi$
$278$	13.4067	0.804083
$279$	−5.61422	−0.336115
$280$	12.8271	0.766566
$281$	−22.1411	−1.32083	−0.660413	−	0.750902i	$-0.729619\pi$
$281$	−22.1411	−1.32083	−0.660413	+	0.750902i	$0.729619\pi$
$282$	−9.52989	−0.567497
$283$	28.5111	1.69481	0.847406	−	0.530946i	$-0.178163\pi$
$283$	28.5111	1.69481	0.847406	+	0.530946i	$0.178163\pi$
$284$	4.38742	0.260345
$285$	2.89280	0.171355
$286$	−3.54028	−0.209341
$287$	−53.9479	−3.18445
$288$	−1.00000	−0.0589256
$289$	−16.3051	−0.959126
$290$	−3.63262	−0.213315
$291$	5.66278	0.331958
$292$	7.83474	0.458494
$293$	−17.3041	−1.01091	−0.505457	−	0.862852i	$-0.668676\pi$
$293$	−17.3041	−1.01091	−0.505457	+	0.862852i	$0.668676\pi$
$294$	12.6617	0.738445
$295$	12.5640	0.731502
$296$	6.56122	0.381363
$297$	−2.55061	−0.148001
$298$	19.7899	1.14640
$299$	1.74900	0.101147
$300$	−3.36829	−0.194468
$301$	17.9997	1.03749
$302$	−20.1482	−1.15940
$303$	−11.6482	−0.669172
$304$	−1.00000	−0.0573539
$305$	−33.0719	−1.89369
$306$	0.833577	0.0476524
$307$	−13.9626	−0.796889	−0.398444	−	0.917192i	$-0.630450\pi$
$307$	−13.9626	−0.796889	−0.398444	+	0.917192i	$0.630450\pi$
$308$	−11.3098	−0.644434
$309$	−15.1470	−0.861681
$310$	16.2408	0.922416
$311$	26.2304	1.48739	0.743695	−	0.668519i	$-0.233071\pi$
$311$	26.2304	1.48739	0.743695	+	0.668519i	$0.233071\pi$
$312$	1.38802	0.0785810
$313$	−10.3564	−0.585376	−0.292688	−	0.956208i	$-0.594550\pi$
$313$	−10.3564	−0.585376	−0.292688	+	0.956208i	$0.594550\pi$
$314$	−0.585212	−0.0330254
$315$	−12.8271	−0.722726
$316$	−11.4441	−0.643781
$317$	−6.44040	−0.361729	−0.180864	−	0.983508i	$-0.557890\pi$
$317$	−6.44040	−0.361729	−0.180864	+	0.983508i	$0.557890\pi$
$318$	−1.00000	−0.0560772
$319$	3.20291	0.179328
$320$	2.89280	0.161712
$321$	18.5935	1.03779
$322$	5.58734	0.311370
$323$	0.833577	0.0463815
$324$	1.00000	0.0555556
$325$	4.67524	0.259336
$326$	21.8455	1.20991
$327$	−1.79861	−0.0994635
$328$	−12.1665	−0.671781
$329$	42.2570	2.32970
$330$	7.37839	0.406167
$331$	20.7036	1.13797	0.568987	−	0.822346i	$-0.307335\pi$
$331$	20.7036	1.13797	0.568987	+	0.822346i	$0.307335\pi$
$332$	4.76641	0.261591
$333$	−6.56122	−0.359553
$334$	−13.4068	−0.733586
$335$	−22.9378	−1.25323
$336$	4.43415	0.241903
$337$	17.6542	0.961687	0.480843	−	0.876806i	$-0.340331\pi$
$337$	17.6542	0.961687	0.480843	+	0.876806i	$0.340331\pi$
$338$	11.0734	0.602314
$339$	−3.64902	−0.198187
$340$	−2.41137	−0.130775
$341$	−14.3197	−0.775453
$342$	1.00000	0.0540738
$343$	−25.1048	−1.35553
$344$	4.05933	0.218865
$345$	−3.64513	−0.196247
$346$	−5.23740	−0.281564
$347$	−8.40797	−0.451363	−0.225682	−	0.974201i	$-0.572461\pi$
$347$	−8.40797	−0.451363	−0.225682	+	0.974201i	$0.572461\pi$
$348$	−1.25574	−0.0673150
$349$	−31.1774	−1.66889	−0.834443	−	0.551095i	$-0.814210\pi$
$349$	−31.1774	−1.66889	−0.834443	+	0.551095i	$0.814210\pi$
$350$	14.9355	0.798336
$351$	−1.38802	−0.0740868
$352$	−2.55061	−0.135948
$353$	12.7492	0.678573	0.339286	−	0.940683i	$-0.389814\pi$
$353$	12.7492	0.678573	0.339286	+	0.940683i	$0.389814\pi$
$354$	4.34318	0.230838
$355$	12.6919	0.673617
$356$	13.7811	0.730397
$357$	−3.69621	−0.195624
$358$	11.5623	0.611088
$359$	−37.2835	−1.96775	−0.983875	−	0.178859i	$-0.942759\pi$
$359$	−37.2835	−1.96775	−0.983875	+	0.178859i	$0.942759\pi$
$360$	−2.89280	−0.152464
$361$	1.00000	0.0526316
$362$	−9.43359	−0.495818
$363$	4.49441	0.235896
$364$	−6.15467	−0.322593
$365$	22.6643	1.18631
$366$	−11.4325	−0.597587
$367$	−22.5155	−1.17530	−0.587650	−	0.809116i	$-0.699947\pi$
$367$	−22.5155	−1.17530	−0.587650	+	0.809116i	$0.699947\pi$
$368$	1.26007	0.0656856
$369$	12.1665	0.633361
$370$	18.9803	0.986738
$371$	4.43415	0.230210
$372$	5.61422	0.291084
$373$	−11.6486	−0.603142	−0.301571	−	0.953444i	$-0.597511\pi$
$373$	−11.6486	−0.603142	−0.301571	+	0.953444i	$0.597511\pi$
$374$	2.12613	0.109939
$375$	4.72021	0.243751
$376$	9.52989	0.491467
$377$	1.74299	0.0897688
$378$	−4.43415	−0.228068
$379$	−28.0449	−1.44057	−0.720286	−	0.693677i	$-0.755990\pi$
$379$	−28.0449	−1.44057	−0.720286	+	0.693677i	$0.755990\pi$
$380$	−2.89280	−0.148397
$381$	10.9397	0.560457
$382$	19.7591	1.01096
$383$	−4.53275	−0.231613	−0.115806	−	0.993272i	$-0.536945\pi$
$383$	−4.53275	−0.231613	−0.115806	+	0.993272i	$0.536945\pi$
$384$	1.00000	0.0510310
$385$	−32.7169	−1.66741
$386$	16.2667	0.827953
$387$	−4.05933	−0.206347
$388$	−5.66278	−0.287484
$389$	29.9810	1.52010	0.760048	−	0.649867i	$-0.225176\pi$
$389$	29.9810	1.52010	0.760048	+	0.649867i	$0.225176\pi$
$390$	4.01525	0.203320
$391$	−1.05036	−0.0531192
$392$	−12.6617	−0.639512
$393$	−3.12574	−0.157673
$394$	−4.34202	−0.218748
$395$	−33.1055	−1.66572
$396$	2.55061	0.128173
$397$	20.1615	1.01188	0.505939	−	0.862569i	$-0.331146\pi$
$397$	20.1615	1.01188	0.505939	+	0.862569i	$0.331146\pi$
$398$	−9.35771	−0.469059
$399$	−4.43415	−0.221985
$400$	3.36829	0.168414
$401$	−12.3324	−0.615852	−0.307926	−	0.951410i	$-0.599635\pi$
$401$	−12.3324	−0.615852	−0.307926	+	0.951410i	$0.599635\pi$
$402$	−7.92927	−0.395476
$403$	−7.79263	−0.388179
$404$	11.6482	0.579520
$405$	2.89280	0.143744
$406$	5.56816	0.276343
$407$	−16.7351	−0.829527
$408$	−0.833577	−0.0412682
$409$	−35.8668	−1.77350	−0.886749	−	0.462251i	$-0.847042\pi$
$409$	−35.8668	−1.77350	−0.886749	+	0.462251i	$0.847042\pi$
$410$	−35.1951	−1.73816
$411$	12.8454	0.633619
$412$	15.1470	0.746237
$413$	−19.2583	−0.947641
$414$	−1.26007	−0.0619290
$415$	13.7883	0.676839
$416$	−1.38802	−0.0680531
$417$	13.4067	0.656531
$418$	2.55061	0.124754
$419$	−21.5076	−1.05071	−0.525357	−	0.850882i	$-0.676068\pi$
$419$	−21.5076	−1.05071	−0.525357	+	0.850882i	$0.676068\pi$
$420$	12.8271	0.625899
$421$	0.870578	0.0424294	0.0212147	−	0.999775i	$-0.493247\pi$
$421$	0.870578	0.0424294	0.0212147	+	0.999775i	$0.493247\pi$
$422$	−12.6719	−0.616857
$423$	−9.52989	−0.463359
$424$	1.00000	0.0485643
$425$	−2.80773	−0.136195
$426$	4.38742	0.212571
$427$	50.6935	2.45323
$428$	−18.5935	−0.898752
$429$	−3.54028	−0.170926
$430$	11.7428	0.566290
$431$	−30.9457	−1.49060	−0.745300	−	0.666729i	$-0.767694\pi$
$431$	−30.9457	−1.49060	−0.745300	+	0.666729i	$0.767694\pi$
$432$	−1.00000	−0.0481125
$433$	8.66839	0.416576	0.208288	−	0.978068i	$-0.433211\pi$
$433$	8.66839	0.416576	0.208288	+	0.978068i	$0.433211\pi$
$434$	−24.8943	−1.19496
$435$	−3.63262	−0.174171
$436$	1.79861	0.0861379
$437$	−1.26007	−0.0602773
$438$	7.83474	0.374359
$439$	−5.15221	−0.245901	−0.122951	−	0.992413i	$-0.539236\pi$
$439$	−5.15221	−0.245901	−0.122951	+	0.992413i	$0.539236\pi$
$440$	−7.37839	−0.351751
$441$	12.6617	0.602938
$442$	1.15702	0.0550338
$443$	5.48007	0.260366	0.130183	−	0.991490i	$-0.458444\pi$
$443$	5.48007	0.260366	0.130183	+	0.991490i	$0.458444\pi$
$444$	6.56122	0.311382
$445$	39.8660	1.88983
$446$	−13.6729	−0.647433
$447$	19.7899	0.936032
$448$	−4.43415	−0.209494
$449$	20.6561	0.974821	0.487410	−	0.873173i	$-0.337942\pi$
$449$	20.6561	0.974821	0.487410	+	0.873173i	$0.337942\pi$
$450$	−3.36829	−0.158783
$451$	31.0318	1.46123
$452$	3.64902	0.171635
$453$	−20.1482	−0.946647
$454$	−5.42303	−0.254515
$455$	−17.8042	−0.834676
$456$	−1.00000	−0.0468293
$457$	−37.1398	−1.73733	−0.868663	−	0.495403i	$-0.835020\pi$
$457$	−37.1398	−1.73733	−0.868663	+	0.495403i	$0.835020\pi$
$458$	−11.3460	−0.530166
$459$	0.833577	0.0389080
$460$	3.64513	0.169955
$461$	−30.3791	−1.41490	−0.707448	−	0.706766i	$-0.750153\pi$
$461$	−30.3791	−1.41490	−0.707448	+	0.706766i	$0.750153\pi$
$462$	−11.3098	−0.526178
$463$	34.6250	1.60916	0.804580	−	0.593844i	$-0.202391\pi$
$463$	34.6250	1.60916	0.804580	+	0.593844i	$0.202391\pi$
$464$	1.25574	0.0582965
$465$	16.2408	0.753150
$466$	16.7811	0.777368
$467$	10.2845	0.475909	0.237955	−	0.971276i	$-0.423523\pi$
$467$	10.2845	0.475909	0.237955	+	0.971276i	$0.423523\pi$
$468$	1.38802	0.0641611
$469$	35.1596	1.62352
$470$	27.5681	1.27162
$471$	−0.585212	−0.0269651
$472$	−4.34318	−0.199911
$473$	−10.3538	−0.476066
$474$	−11.4441	−0.525645
$475$	−3.36829	−0.154548
$476$	3.69621	0.169415
$477$	−1.00000	−0.0457869
$478$	−17.5565	−0.803015
$479$	−32.0832	−1.46592	−0.732960	−	0.680272i	$-0.761862\pi$
$479$	−32.0832	−1.46592	−0.732960	+	0.680272i	$0.761862\pi$
$480$	2.89280	0.132038
$481$	−9.10708	−0.415247
$482$	13.3030	0.605934
$483$	5.58734	0.254233
$484$	−4.49441	−0.204292
$485$	−16.3813	−0.743836
$486$	1.00000	0.0453609
$487$	19.7931	0.896913	0.448456	−	0.893805i	$-0.351974\pi$
$487$	19.7931	0.896913	0.448456	+	0.893805i	$0.351974\pi$
$488$	11.4325	0.517525
$489$	21.8455	0.987886
$490$	−36.6277	−1.65467
$491$	−2.26270	−0.102114	−0.0510571	−	0.998696i	$-0.516259\pi$
$491$	−2.26270	−0.102114	−0.0510571	+	0.998696i	$0.516259\pi$
$492$	−12.1665	−0.548507
$493$	−1.04676	−0.0471437
$494$	1.38802	0.0624498
$495$	7.37839	0.331634
$496$	−5.61422	−0.252086
$497$	−19.4545	−0.872652
$498$	4.76641	0.213588
$499$	8.60450	0.385190	0.192595	−	0.981278i	$-0.438310\pi$
$499$	8.60450	0.385190	0.192595	+	0.981278i	$0.438310\pi$
$500$	−4.72021	−0.211094
$501$	−13.4068	−0.598970
$502$	30.0251	1.34008
$503$	−42.9936	−1.91699	−0.958494	−	0.285111i	$-0.907969\pi$
$503$	−42.9936	−1.91699	−0.958494	+	0.285111i	$0.907969\pi$
$504$	4.43415	0.197513
$505$	33.6959	1.49945
$506$	−3.21394	−0.142877
$507$	11.0734	0.491787
$508$	−10.9397	−0.485370
$509$	8.60981	0.381623	0.190812	−	0.981627i	$-0.438888\pi$
$509$	8.60981	0.381623	0.190812	+	0.981627i	$0.438888\pi$
$510$	−2.41137	−0.106777
$511$	−34.7404	−1.53683
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	14.9943	0.661370
$515$	43.8171	1.93081
$516$	4.05933	0.178702
$517$	−24.3070	−1.06902
$518$	−29.0934	−1.27829
$519$	−5.23740	−0.229896
$520$	−4.01525	−0.176081
$521$	−7.50369	−0.328743	−0.164371	−	0.986399i	$-0.552560\pi$
$521$	−7.50369	−0.328743	−0.164371	+	0.986399i	$0.552560\pi$
$522$	−1.25574	−0.0549624
$523$	1.91238	0.0836224	0.0418112	−	0.999126i	$-0.486687\pi$
$523$	1.91238	0.0836224	0.0418112	+	0.999126i	$0.486687\pi$
$524$	3.12574	0.136549
$525$	14.9355	0.651839
$526$	−26.3825	−1.15033
$527$	4.67988	0.203859
$528$	−2.55061	−0.111001
$529$	−21.4122	−0.930966
$530$	2.89280	0.125655
$531$	4.34318	0.188478
$532$	4.43415	0.192245
$533$	16.8873	0.731468
$534$	13.7811	0.596366
$535$	−53.7874	−2.32543
$536$	7.92927	0.342492
$537$	11.5623	0.498951
$538$	−8.20107	−0.353573
$539$	32.2950	1.39104
$540$	−2.89280	−0.124486
$541$	−5.73367	−0.246510	−0.123255	−	0.992375i	$-0.539333\pi$
$541$	−5.73367	−0.246510	−0.123255	+	0.992375i	$0.539333\pi$
$542$	2.32053	0.0996753
$543$	−9.43359	−0.404834
$544$	0.833577	0.0357393
$545$	5.20302	0.222873
$546$	−6.15467	−0.263396
$547$	18.3378	0.784068	0.392034	−	0.919951i	$-0.371772\pi$
$547$	18.3378	0.784068	0.392034	+	0.919951i	$0.371772\pi$
$548$	−12.8454	−0.548730
$549$	−11.4325	−0.487927
$550$	−8.59118	−0.366329
$551$	−1.25574	−0.0534965
$552$	1.26007	0.0536321
$553$	50.7449	2.15789
$554$	−7.56983	−0.321611
$555$	18.9803	0.805668
$556$	−13.4067	−0.568572
$557$	3.78108	0.160210	0.0801048	−	0.996786i	$-0.474475\pi$
$557$	3.78108	0.160210	0.0801048	+	0.996786i	$0.474475\pi$
$558$	5.61422	0.237669
$559$	−5.63442	−0.238311
$560$	−12.8271	−0.542044
$561$	2.12613	0.0897651
$562$	22.1411	0.933965
$563$	−28.0386	−1.18168	−0.590842	−	0.806787i	$-0.701204\pi$
$563$	−28.0386	−1.18168	−0.590842	+	0.806787i	$0.701204\pi$
$564$	9.52989	0.401281
$565$	10.5559	0.444089
$566$	−28.5111	−1.19841
$567$	−4.43415	−0.186217
$568$	−4.38742	−0.184092
$569$	9.31294	0.390419	0.195209	−	0.980762i	$-0.437461\pi$
$569$	9.31294	0.390419	0.195209	+	0.980762i	$0.437461\pi$
$570$	−2.89280	−0.121166
$571$	−32.2212	−1.34841	−0.674207	−	0.738543i	$-0.735514\pi$
$571$	−32.2212	−1.34841	−0.674207	+	0.738543i	$0.735514\pi$
$572$	3.54028	0.148027
$573$	19.7591	0.825448
$574$	53.9479	2.25174
$575$	4.24428	0.176999
$576$	1.00000	0.0416667
$577$	20.0203	0.833457	0.416728	−	0.909031i	$-0.363177\pi$
$577$	20.0203	0.833457	0.416728	+	0.909031i	$0.363177\pi$
$578$	16.3051	0.678205
$579$	16.2667	0.676020
$580$	3.63262	0.150836
$581$	−21.1350	−0.876826
$582$	−5.66278	−0.234730
$583$	−2.55061	−0.105635
$584$	−7.83474	−0.324204
$585$	4.01525	0.166010
$586$	17.3041	0.714824
$587$	−39.7285	−1.63977	−0.819885	−	0.572528i	$-0.805963\pi$
$587$	−39.7285	−1.63977	−0.819885	+	0.572528i	$0.805963\pi$
$588$	−12.6617	−0.522159
$589$	5.61422	0.231330
$590$	−12.5640	−0.517250
$591$	−4.34202	−0.178607
$592$	−6.56122	−0.269664
$593$	−39.1353	−1.60709	−0.803547	−	0.595242i	$-0.797056\pi$
$593$	−39.1353	−1.60709	−0.803547	+	0.595242i	$0.797056\pi$
$594$	2.55061	0.104653
$595$	10.6924	0.438345
$596$	−19.7899	−0.810627
$597$	−9.35771	−0.382985
$598$	−1.74900	−0.0715218
$599$	7.81482	0.319305	0.159653	−	0.987173i	$-0.448963\pi$
$599$	7.81482	0.319305	0.159653	+	0.987173i	$0.448963\pi$
$600$	3.36829	0.137510
$601$	−0.980464	−0.0399940	−0.0199970	−	0.999800i	$-0.506366\pi$
$601$	−0.980464	−0.0399940	−0.0199970	+	0.999800i	$0.506366\pi$
$602$	−17.9997	−0.733613
$603$	−7.92927	−0.322905
$604$	20.1482	0.819820
$605$	−13.0014	−0.528584
$606$	11.6482	0.473176
$607$	−32.8897	−1.33495	−0.667475	−	0.744632i	$-0.732625\pi$
$607$	−32.8897	−1.33495	−0.667475	+	0.744632i	$0.732625\pi$
$608$	1.00000	0.0405554
$609$	5.56816	0.225633
$610$	33.0719	1.33904
$611$	−13.2276	−0.535133
$612$	−0.833577	−0.0336954
$613$	−0.792129	−0.0319938	−0.0159969	−	0.999872i	$-0.505092\pi$
$613$	−0.792129	−0.0319938	−0.0159969	+	0.999872i	$0.505092\pi$
$614$	13.9626	0.563486
$615$	−35.1951	−1.41921
$616$	11.3098	0.455683
$617$	−27.4464	−1.10495	−0.552476	−	0.833529i	$-0.686317\pi$
$617$	−27.4464	−1.10495	−0.552476	+	0.833529i	$0.686317\pi$
$618$	15.1470	0.609300
$619$	−12.1298	−0.487537	−0.243769	−	0.969833i	$-0.578384\pi$
$619$	−12.1298	−0.487537	−0.243769	+	0.969833i	$0.578384\pi$
$620$	−16.2408	−0.652247
$621$	−1.26007	−0.0505648
$622$	−26.2304	−1.05174
$623$	−61.1075	−2.44822
$624$	−1.38802	−0.0555651
$625$	−30.4961	−1.21984
$626$	10.3564	0.413923
$627$	2.55061	0.101861
$628$	0.585212	0.0233525
$629$	5.46928	0.218074
$630$	12.8271	0.511044
$631$	−25.4732	−1.01407	−0.507035	−	0.861925i	$-0.669259\pi$
$631$	−25.4732	−1.01407	−0.507035	+	0.861925i	$0.669259\pi$
$632$	11.4441	0.455222
$633$	−12.6719	−0.503662
$634$	6.44040	0.255781
$635$	−31.6463	−1.25585
$636$	1.00000	0.0396526
$637$	17.5746	0.696332
$638$	−3.20291	−0.126804
$639$	4.38742	0.173564
$640$	−2.89280	−0.114348
$641$	34.5409	1.36428	0.682141	−	0.731220i	$-0.261049\pi$
$641$	34.5409	1.36428	0.682141	+	0.731220i	$0.261049\pi$
$642$	−18.5935	−0.733828
$643$	−30.4163	−1.19950	−0.599751	−	0.800187i	$-0.704734\pi$
$643$	−30.4163	−1.19950	−0.599751	+	0.800187i	$0.704734\pi$
$644$	−5.58734	−0.220172
$645$	11.7428	0.462374
$646$	−0.833577	−0.0327966
$647$	1.40469	0.0552240	0.0276120	−	0.999619i	$-0.491210\pi$
$647$	1.40469	0.0552240	0.0276120	+	0.999619i	$0.491210\pi$
$648$	−1.00000	−0.0392837
$649$	11.0777	0.434840
$650$	−4.67524	−0.183378
$651$	−24.8943	−0.975684
$652$	−21.8455	−0.855535
$653$	28.4211	1.11220	0.556101	−	0.831115i	$-0.312297\pi$
$653$	28.4211	1.11220	0.556101	+	0.831115i	$0.312297\pi$
$654$	1.79861	0.0703313
$655$	9.04214	0.353306
$656$	12.1665	0.475021
$657$	7.83474	0.305662
$658$	−42.2570	−1.64735
$659$	−23.4991	−0.915395	−0.457698	−	0.889108i	$-0.651326\pi$
$659$	−23.4991	−0.915395	−0.457698	+	0.889108i	$0.651326\pi$
$660$	−7.37839	−0.287203
$661$	23.4619	0.912562	0.456281	−	0.889836i	$-0.349181\pi$
$661$	23.4619	0.912562	0.456281	+	0.889836i	$0.349181\pi$
$662$	−20.7036	−0.804669
$663$	1.15702	0.0449349
$664$	−4.76641	−0.184972
$665$	12.8271	0.497414
$666$	6.56122	0.254242
$667$	1.58232	0.0612679
$668$	13.4068	0.518724
$669$	−13.6729	−0.528627
$670$	22.9378	0.886164
$671$	−29.1598	−1.12570
$672$	−4.43415	−0.171051
$673$	35.3649	1.36322	0.681609	−	0.731716i	$-0.261280\pi$
$673$	35.3649	1.36322	0.681609	+	0.731716i	$0.261280\pi$
$674$	−17.6542	−0.680015
$675$	−3.36829	−0.129646
$676$	−11.0734	−0.425900
$677$	−16.0544	−0.617022	−0.308511	−	0.951221i	$-0.599831\pi$
$677$	−16.0544	−0.617022	−0.308511	+	0.951221i	$0.599831\pi$
$678$	3.64902	0.140140
$679$	25.1096	0.963618
$680$	2.41137	0.0924719
$681$	−5.42303	−0.207811
$682$	14.3197	0.548328
$683$	−39.6307	−1.51642	−0.758212	−	0.652008i	$-0.773927\pi$
$683$	−39.6307	−1.51642	−0.758212	+	0.652008i	$0.773927\pi$
$684$	−1.00000	−0.0382360
$685$	−37.1593	−1.41978
$686$	25.1048	0.958505
$687$	−11.3460	−0.432879
$688$	−4.05933	−0.154761
$689$	−1.38802	−0.0528792
$690$	3.64513	0.138768
$691$	−5.96218	−0.226812	−0.113406	−	0.993549i	$-0.536176\pi$
$691$	−5.96218	−0.226812	−0.113406	+	0.993549i	$0.536176\pi$
$692$	5.23740	0.199096
$693$	−11.3098	−0.429623
$694$	8.40797	0.319162
$695$	−38.7830	−1.47112
$696$	1.25574	0.0475989
$697$	−10.1417	−0.384144
$698$	31.1774	1.18008
$699$	16.7811	0.634718
$700$	−14.9355	−0.564509
$701$	−30.2634	−1.14303	−0.571516	−	0.820591i	$-0.693644\pi$
$701$	−30.2634	−1.14303	−0.571516	+	0.820591i	$0.693644\pi$
$702$	1.38802	0.0523873
$703$	6.56122	0.247461
$704$	2.55061	0.0961295
$705$	27.5681	1.03827
$706$	−12.7492	−0.479823
$707$	−51.6499	−1.94250
$708$	−4.34318	−0.163227
$709$	39.4640	1.48210	0.741050	−	0.671449i	$-0.234328\pi$
$709$	39.4640	1.48210	0.741050	+	0.671449i	$0.234328\pi$
$710$	−12.6919	−0.476319
$711$	−11.4441	−0.429188
$712$	−13.7811	−0.516468
$713$	−7.07430	−0.264935
$714$	3.69621	0.138327
$715$	10.2413	0.383004
$716$	−11.5623	−0.432104
$717$	−17.5565	−0.655659
$718$	37.2835	1.39141
$719$	41.7993	1.55885	0.779426	−	0.626494i	$-0.215511\pi$
$719$	41.7993	1.55885	0.779426	+	0.626494i	$0.215511\pi$
$720$	2.89280	0.107808
$721$	−67.1639	−2.50132
$722$	−1.00000	−0.0372161
$723$	13.3030	0.494743
$724$	9.43359	0.350597
$725$	4.22971	0.157088
$726$	−4.49441	−0.166803
$727$	−33.9611	−1.25955	−0.629773	−	0.776779i	$-0.716852\pi$
$727$	−33.9611	−1.25955	−0.629773	+	0.776779i	$0.716852\pi$
$728$	6.15467	0.228107
$729$	1.00000	0.0370370
$730$	−22.6643	−0.838845
$731$	3.38377	0.125153
$732$	11.4325	0.422558
$733$	7.11285	0.262719	0.131360	−	0.991335i	$-0.458066\pi$
$733$	7.11285	0.262719	0.131360	+	0.991335i	$0.458066\pi$
$734$	22.5155	0.831062
$735$	−36.6277	−1.35103
$736$	−1.26007	−0.0464468
$737$	−20.2244	−0.744977
$738$	−12.1665	−0.447854
$739$	−41.2387	−1.51699	−0.758495	−	0.651679i	$-0.774065\pi$
$739$	−41.2387	−1.51699	−0.758495	+	0.651679i	$0.774065\pi$
$740$	−18.9803	−0.697729
$741$	1.38802	0.0509901
$742$	−4.43415	−0.162783
$743$	−22.7810	−0.835753	−0.417877	−	0.908504i	$-0.637226\pi$
$743$	−22.7810	−0.835753	−0.417877	+	0.908504i	$0.637226\pi$
$744$	−5.61422	−0.205827
$745$	−57.2483	−2.09742
$746$	11.6486	0.426486
$747$	4.76641	0.174394
$748$	−2.12613	−0.0777389
$749$	82.4465	3.01253
$750$	−4.72021	−0.172358
$751$	28.4933	1.03973	0.519867	−	0.854247i	$-0.325981\pi$
$751$	28.4933	1.03973	0.519867	+	0.854247i	$0.325981\pi$
$752$	−9.52989	−0.347519
$753$	30.0251	1.09417
$754$	−1.74299	−0.0634761
$755$	58.2848	2.12120
$756$	4.43415	0.161268
$757$	−35.4717	−1.28924	−0.644621	−	0.764502i	$-0.722985\pi$
$757$	−35.4717	−1.28924	−0.644621	+	0.764502i	$0.722985\pi$
$758$	28.0449	1.01864
$759$	−3.21394	−0.116659
$760$	2.89280	0.104933
$761$	43.6789	1.58336	0.791679	−	0.610937i	$-0.209207\pi$
$761$	43.6789	1.58336	0.791679	+	0.610937i	$0.209207\pi$
$762$	−10.9397	−0.396303
$763$	−7.97531	−0.288726
$764$	−19.7591	−0.714859
$765$	−2.41137	−0.0871833
$766$	4.53275	0.163775
$767$	6.02841	0.217673
$768$	−1.00000	−0.0360844
$769$	1.90532	0.0687078	0.0343539	−	0.999410i	$-0.489063\pi$
$769$	1.90532	0.0687078	0.0343539	+	0.999410i	$0.489063\pi$
$770$	32.7169	1.17903
$771$	14.9943	0.540006
$772$	−16.2667	−0.585451
$773$	7.28153	0.261898	0.130949	−	0.991389i	$-0.458198\pi$
$773$	7.28153	0.261898	0.130949	+	0.991389i	$0.458198\pi$
$774$	4.05933	0.145910
$775$	−18.9103	−0.679279
$776$	5.66278	0.203282
$777$	−29.0934	−1.04372
$778$	−29.9810	−1.07487
$779$	−12.1665	−0.435909
$780$	−4.01525	−0.143769
$781$	11.1906	0.400430
$782$	1.05036	0.0375610
$783$	−1.25574	−0.0448766
$784$	12.6617	0.452203
$785$	1.69290	0.0604222
$786$	3.12574	0.111491
$787$	−15.1015	−0.538309	−0.269155	−	0.963097i	$-0.586744\pi$
$787$	−15.1015	−0.538309	−0.269155	+	0.963097i	$0.586744\pi$
$788$	4.34202	0.154678
$789$	−26.3825	−0.939243
$790$	33.1055	1.17784
$791$	−16.1803	−0.575305
$792$	−2.55061	−0.0906318
$793$	−15.8685	−0.563507
$794$	−20.1615	−0.715506
$795$	2.89280	0.102597
$796$	9.35771	0.331675
$797$	21.2932	0.754244	0.377122	−	0.926163i	$-0.376914\pi$
$797$	21.2932	0.754244	0.377122	+	0.926163i	$0.376914\pi$
$798$	4.43415	0.156967
$799$	7.94390	0.281035
$800$	−3.36829	−0.119087
$801$	13.7811	0.486931
$802$	12.3324	0.435473
$803$	19.9833	0.705197
$804$	7.92927	0.279644
$805$	−16.1630	−0.569672
$806$	7.79263	0.274484
$807$	−8.20107	−0.288691
$808$	−11.6482	−0.409783
$809$	−16.8992	−0.594143	−0.297072	−	0.954855i	$-0.596010\pi$
$809$	−16.8992	−0.594143	−0.297072	+	0.954855i	$0.596010\pi$
$810$	−2.89280	−0.101643
$811$	−6.12427	−0.215052	−0.107526	−	0.994202i	$-0.534293\pi$
$811$	−6.12427	−0.215052	−0.107526	+	0.994202i	$0.534293\pi$
$812$	−5.56816	−0.195404
$813$	2.32053	0.0813845
$814$	16.7351	0.586564
$815$	−63.1946	−2.21361
$816$	0.833577	0.0291810
$817$	4.05933	0.142018
$818$	35.8668	1.25405
$819$	−6.15467	−0.215062
$820$	35.1951	1.22907
$821$	36.6883	1.28043	0.640215	−	0.768196i	$-0.278845\pi$
$821$	36.6883	1.28043	0.640215	+	0.768196i	$0.278845\pi$
$822$	−12.8454	−0.448036
$823$	−17.2712	−0.602036	−0.301018	−	0.953619i	$-0.597326\pi$
$823$	−17.2712	−0.602036	−0.301018	+	0.953619i	$0.597326\pi$
$824$	−15.1470	−0.527670
$825$	−8.59118	−0.299106
$826$	19.2583	0.670083
$827$	−30.2628	−1.05234	−0.526170	−	0.850379i	$-0.676372\pi$
$827$	−30.2628	−1.05234	−0.526170	+	0.850379i	$0.676372\pi$
$828$	1.26007	0.0437904
$829$	−30.1771	−1.04809	−0.524047	−	0.851689i	$-0.675578\pi$
$829$	−30.1771	−1.04809	−0.524047	+	0.851689i	$0.675578\pi$
$830$	−13.7883	−0.478597
$831$	−7.56983	−0.262594
$832$	1.38802	0.0481208
$833$	−10.5545	−0.365692
$834$	−13.4067	−0.464237
$835$	38.7831	1.34214
$836$	−2.55061	−0.0882145
$837$	5.61422	0.194056
$838$	21.5076	0.742967
$839$	−25.9193	−0.894835	−0.447418	−	0.894325i	$-0.647656\pi$
$839$	−25.9193	−0.894835	−0.447418	+	0.894325i	$0.647656\pi$
$840$	−12.8271	−0.442577
$841$	−27.4231	−0.945624
$842$	−0.870578	−0.0300021
$843$	22.1411	0.762580
$844$	12.6719	0.436184
$845$	−32.0332	−1.10197
$846$	9.52989	0.327644
$847$	19.9289	0.684765
$848$	−1.00000	−0.0343401
$849$	−28.5111	−0.978500
$850$	2.80773	0.0963043
$851$	−8.26759	−0.283409
$852$	−4.38742	−0.150310
$853$	19.7487	0.676184	0.338092	−	0.941113i	$-0.390219\pi$
$853$	19.7487	0.676184	0.338092	+	0.941113i	$0.390219\pi$
$854$	−50.6935	−1.73469
$855$	−2.89280	−0.0989317
$856$	18.5935	0.635514
$857$	−33.9875	−1.16099	−0.580496	−	0.814263i	$-0.697141\pi$
$857$	−33.9875	−1.16099	−0.580496	+	0.814263i	$0.697141\pi$
$858$	3.54028	0.120863
$859$	−17.0189	−0.580677	−0.290339	−	0.956924i	$-0.593768\pi$
$859$	−17.0189	−0.580677	−0.290339	+	0.956924i	$0.593768\pi$
$860$	−11.7428	−0.400427
$861$	53.9479	1.83854
$862$	30.9457	1.05401
$863$	29.6765	1.01020	0.505100	−	0.863061i	$-0.331456\pi$
$863$	29.6765	1.01020	0.505100	+	0.863061i	$0.331456\pi$
$864$	1.00000	0.0340207
$865$	15.1508	0.515141
$866$	−8.66839	−0.294564
$867$	16.3051	0.553752
$868$	24.8943	0.844967
$869$	−29.1894	−0.990183
$870$	3.63262	0.123157
$871$	−11.0060	−0.372923
$872$	−1.79861	−0.0609087
$873$	−5.66278	−0.191656
$874$	1.26007	0.0426225
$875$	20.9301	0.707567
$876$	−7.83474	−0.264711
$877$	−14.2333	−0.480625	−0.240312	−	0.970696i	$-0.577250\pi$
$877$	−14.2333	−0.480625	−0.240312	+	0.970696i	$0.577250\pi$
$878$	5.15221	0.173879
$879$	17.3041	0.583651
$880$	7.37839	0.248725
$881$	0.153708	0.00517856	0.00258928	−	0.999997i	$-0.499176\pi$
$881$	0.153708	0.00517856	0.00258928	+	0.999997i	$0.499176\pi$
$882$	−12.6617	−0.426341
$883$	43.2642	1.45596	0.727978	−	0.685601i	$-0.240460\pi$
$883$	43.2642	1.45596	0.727978	+	0.685601i	$0.240460\pi$
$884$	−1.15702	−0.0389147
$885$	−12.5640	−0.422333
$886$	−5.48007	−0.184106
$887$	3.47419	0.116652	0.0583260	−	0.998298i	$-0.481424\pi$
$887$	3.47419	0.116652	0.0583260	+	0.998298i	$0.481424\pi$
$888$	−6.56122	−0.220180
$889$	48.5082	1.62691
$890$	−39.8660	−1.33631
$891$	2.55061	0.0854485
$892$	13.6729	0.457804
$893$	9.52989	0.318906
$894$	−19.7899	−0.661874
$895$	−33.4475	−1.11803
$896$	4.43415	0.148135
$897$	−1.74900	−0.0583973
$898$	−20.6561	−0.689302
$899$	−7.05003	−0.235132
$900$	3.36829	0.112276
$901$	0.833577	0.0277705
$902$	−31.0318	−1.03325
$903$	−17.9997	−0.598992
$904$	−3.64902	−0.121364
$905$	27.2895	0.907133
$906$	20.1482	0.669380
$907$	25.1706	0.835775	0.417887	−	0.908499i	$-0.362771\pi$
$907$	25.1706	0.835775	0.417887	+	0.908499i	$0.362771\pi$
$908$	5.42303	0.179969
$909$	11.6482	0.386347
$910$	17.8042	0.590205
$911$	−14.0987	−0.467112	−0.233556	−	0.972343i	$-0.575036\pi$
$911$	−14.0987	−0.467112	−0.233556	+	0.972343i	$0.575036\pi$
$912$	1.00000	0.0331133
$913$	12.1572	0.402345
$914$	37.1398	1.22848
$915$	33.0719	1.09332
$916$	11.3460	0.374884
$917$	−13.8600	−0.457697
$918$	−0.833577	−0.0275121
$919$	53.4407	1.76284	0.881422	−	0.472329i	$-0.156587\pi$
$919$	53.4407	1.76284	0.881422	+	0.472329i	$0.156587\pi$
$920$	−3.64513	−0.120176
$921$	13.9626	0.460084
$922$	30.3791	1.00048
$923$	6.08981	0.200448
$924$	11.3098	0.372064
$925$	−22.1001	−0.726646
$926$	−34.6250	−1.13785
$927$	15.1470	0.497492
$928$	−1.25574	−0.0412218
$929$	30.4780	0.999951	0.499975	−	0.866040i	$-0.333342\pi$
$929$	30.4780	0.999951	0.499975	+	0.866040i	$0.333342\pi$
$930$	−16.2408	−0.532557
$931$	−12.6617	−0.414970
$932$	−16.7811	−0.549682
$933$	−26.2304	−0.858745
$934$	−10.2845	−0.336519
$935$	−6.15046	−0.201141
$936$	−1.38802	−0.0453687
$937$	9.06231	0.296053	0.148026	−	0.988983i	$-0.452708\pi$
$937$	9.06231	0.296053	0.148026	+	0.988983i	$0.452708\pi$
$938$	−35.1596	−1.14800
$939$	10.3564	0.337967
$940$	−27.5681	−0.899171
$941$	22.0517	0.718865	0.359432	−	0.933171i	$-0.382970\pi$
$941$	22.0517	0.718865	0.359432	+	0.933171i	$0.382970\pi$
$942$	0.585212	0.0190672
$943$	15.3306	0.499233
$944$	4.34318	0.141359
$945$	12.8271	0.417266
$946$	10.3538	0.336630
$947$	5.64289	0.183369	0.0916847	−	0.995788i	$-0.470775\pi$
$947$	5.64289	0.183369	0.0916847	+	0.995788i	$0.470775\pi$
$948$	11.4441	0.371687
$949$	10.8748	0.353009
$950$	3.36829	0.109282
$951$	6.44040	0.208844
$952$	−3.69621	−0.119795
$953$	31.5878	1.02323	0.511615	−	0.859215i	$-0.329047\pi$
$953$	31.5878	1.02323	0.511615	+	0.859215i	$0.329047\pi$
$954$	1.00000	0.0323762
$955$	−57.1591	−1.84963
$956$	17.5565	0.567817
$957$	−3.20291	−0.103535
$958$	32.0832	1.03656
$959$	56.9586	1.83929
$960$	−2.89280	−0.0933647
$961$	0.519463	0.0167569
$962$	9.10708	0.293624
$963$	−18.5935	−0.599168
$964$	−13.3030	−0.428460
$965$	−47.0563	−1.51479
$966$	−5.58734	−0.179770
$967$	−23.8768	−0.767826	−0.383913	−	0.923369i	$-0.625424\pi$
$967$	−23.8768	−0.767826	−0.383913	+	0.923369i	$0.625424\pi$
$968$	4.49441	0.144456
$969$	−0.833577	−0.0267783
$970$	16.3813	0.525971
$971$	−5.01705	−0.161005	−0.0805025	−	0.996754i	$-0.525652\pi$
$971$	−5.01705	−0.161005	−0.0805025	+	0.996754i	$0.525652\pi$
$972$	−1.00000	−0.0320750
$973$	59.4475	1.90580
$974$	−19.7931	−0.634213
$975$	−4.67524	−0.149728
$976$	−11.4325	−0.365946
$977$	60.4627	1.93437	0.967187	−	0.254064i	$-0.0817675\pi$
$977$	60.4627	1.93437	0.967187	+	0.254064i	$0.0817675\pi$
$978$	−21.8455	−0.698541
$979$	35.1501	1.12340
$980$	36.6277	1.17003
$981$	1.79861	0.0574253
$982$	2.26270	0.0722057
$983$	51.2576	1.63486	0.817432	−	0.576026i	$-0.195397\pi$
$983$	51.2576	1.63486	0.817432	+	0.576026i	$0.195397\pi$
$984$	12.1665	0.387853
$985$	12.5606	0.400213
$986$	1.04676	0.0333356
$987$	−42.2570	−1.34505
$988$	−1.38802	−0.0441587
$989$	−5.11504	−0.162649
$990$	−7.37839	−0.234501
$991$	41.9449	1.33242	0.666212	−	0.745762i	$-0.267914\pi$
$991$	41.9449	1.33242	0.666212	+	0.745762i	$0.267914\pi$
$992$	5.61422	0.178252
$993$	−20.7036	−0.657010
$994$	19.4545	0.617058
$995$	27.0700	0.858176
$996$	−4.76641	−0.151029
$997$	16.4564	0.521178	0.260589	−	0.965450i	$-0.416083\pi$
$997$	16.4564	0.521178	0.260589	+	0.965450i	$0.416083\pi$
$998$	−8.60450	−0.272371
$999$	6.56122	0.207588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.be.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.be.1.10	✓	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.89280 q^{5} +1.00000 q^{6} -4.43415 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.89280 q^{5} +1.00000 q^{6} -4.43415 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.89280 q^{10} +2.55061 q^{11} -1.00000 q^{12} +1.38802 q^{13} +4.43415 q^{14} -2.89280 q^{15} +1.00000 q^{16} -0.833577 q^{17} -1.00000 q^{18} -1.00000 q^{19} +2.89280 q^{20} +4.43415 q^{21} -2.55061 q^{22} +1.26007 q^{23} +1.00000 q^{24} +3.36829 q^{25} -1.38802 q^{26} -1.00000 q^{27} -4.43415 q^{28} +1.25574 q^{29} +2.89280 q^{30} -5.61422 q^{31} -1.00000 q^{32} -2.55061 q^{33} +0.833577 q^{34} -12.8271 q^{35} +1.00000 q^{36} -6.56122 q^{37} +1.00000 q^{38} -1.38802 q^{39} -2.89280 q^{40} +12.1665 q^{41} -4.43415 q^{42} -4.05933 q^{43} +2.55061 q^{44} +2.89280 q^{45} -1.26007 q^{46} -9.52989 q^{47} -1.00000 q^{48} +12.6617 q^{49} -3.36829 q^{50} +0.833577 q^{51} +1.38802 q^{52} -1.00000 q^{53} +1.00000 q^{54} +7.37839 q^{55} +4.43415 q^{56} +1.00000 q^{57} -1.25574 q^{58} +4.34318 q^{59} -2.89280 q^{60} -11.4325 q^{61} +5.61422 q^{62} -4.43415 q^{63} +1.00000 q^{64} +4.01525 q^{65} +2.55061 q^{66} -7.92927 q^{67} -0.833577 q^{68} -1.26007 q^{69} +12.8271 q^{70} +4.38742 q^{71} -1.00000 q^{72} +7.83474 q^{73} +6.56122 q^{74} -3.36829 q^{75} -1.00000 q^{76} -11.3098 q^{77} +1.38802 q^{78} -11.4441 q^{79} +2.89280 q^{80} +1.00000 q^{81} -12.1665 q^{82} +4.76641 q^{83} +4.43415 q^{84} -2.41137 q^{85} +4.05933 q^{86} -1.25574 q^{87} -2.55061 q^{88} +13.7811 q^{89} -2.89280 q^{90} -6.15467 q^{91} +1.26007 q^{92} +5.61422 q^{93} +9.52989 q^{94} -2.89280 q^{95} +1.00000 q^{96} -5.66278 q^{97} -12.6617 q^{98} +2.55061 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} + 12 q^{16} - 4 q^{17} - 12 q^{18} - 12 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} - 18 q^{23} + 12 q^{24} + 21 q^{25} - 4 q^{26} - 12 q^{27} - q^{28} + 10 q^{29} - 3 q^{30} + 14 q^{31} - 12 q^{32} + 7 q^{33} + 4 q^{34} - 19 q^{35} + 12 q^{36} + 14 q^{37} + 12 q^{38} - 4 q^{39} + 3 q^{40} - 9 q^{41} - q^{42} - 4 q^{43} - 7 q^{44} - 3 q^{45} + 18 q^{46} - 14 q^{47} - 12 q^{48} + 37 q^{49} - 21 q^{50} + 4 q^{51} + 4 q^{52} - 12 q^{53} + 12 q^{54} - 4 q^{55} + q^{56} + 12 q^{57} - 10 q^{58} - 18 q^{59} + 3 q^{60} - 7 q^{61} - 14 q^{62} - q^{63} + 12 q^{64} + 3 q^{65} - 7 q^{66} + 8 q^{67} - 4 q^{68} + 18 q^{69} + 19 q^{70} - q^{71} - 12 q^{72} - 31 q^{73} - 14 q^{74} - 21 q^{75} - 12 q^{76} - 25 q^{77} + 4 q^{78} - 3 q^{80} + 12 q^{81} + 9 q^{82} - 48 q^{83} + q^{84} + 4 q^{85} + 4 q^{86} - 10 q^{87} + 7 q^{88} + 13 q^{89} + 3 q^{90} + 9 q^{91} - 18 q^{92} - 14 q^{93} + 14 q^{94} + 3 q^{95} + 12 q^{96} + 25 q^{97} - 37 q^{98} - 7 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 7 * q^11 - 12 * q^12 + 4 * q^13 + q^14 + 3 * q^15 + 12 * q^16 - 4 * q^17 - 12 * q^18 - 12 * q^19 - 3 * q^20 + q^21 + 7 * q^22 - 18 * q^23 + 12 * q^24 + 21 * q^25 - 4 * q^26 - 12 * q^27 - q^28 + 10 * q^29 - 3 * q^30 + 14 * q^31 - 12 * q^32 + 7 * q^33 + 4 * q^34 - 19 * q^35 + 12 * q^36 + 14 * q^37 + 12 * q^38 - 4 * q^39 + 3 * q^40 - 9 * q^41 - q^42 - 4 * q^43 - 7 * q^44 - 3 * q^45 + 18 * q^46 - 14 * q^47 - 12 * q^48 + 37 * q^49 - 21 * q^50 + 4 * q^51 + 4 * q^52 - 12 * q^53 + 12 * q^54 - 4 * q^55 + q^56 + 12 * q^57 - 10 * q^58 - 18 * q^59 + 3 * q^60 - 7 * q^61 - 14 * q^62 - q^63 + 12 * q^64 + 3 * q^65 - 7 * q^66 + 8 * q^67 - 4 * q^68 + 18 * q^69 + 19 * q^70 - q^71 - 12 * q^72 - 31 * q^73 - 14 * q^74 - 21 * q^75 - 12 * q^76 - 25 * q^77 + 4 * q^78 - 3 * q^80 + 12 * q^81 + 9 * q^82 - 48 * q^83 + q^84 + 4 * q^85 + 4 * q^86 - 10 * q^87 + 7 * q^88 + 13 * q^89 + 3 * q^90 + 9 * q^91 - 18 * q^92 - 14 * q^93 + 14 * q^94 + 3 * q^95 + 12 * q^96 + 25 * q^97 - 37 * q^98 - 7 * q^99

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