LMFDB - Embedded newform 6042.2.a.be.1.6

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			1.6
Root			$0.427666$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.427666 q^{5} +1.00000 q^{6} -1.56626 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.427666 q^{5} +1.00000 q^{6} -1.56626 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.427666 q^{10} +4.81756 q^{11} -1.00000 q^{12} -0.388531 q^{13} +1.56626 q^{14} +0.427666 q^{15} +1.00000 q^{16} +4.71032 q^{17} -1.00000 q^{18} -1.00000 q^{19} -0.427666 q^{20} +1.56626 q^{21} -4.81756 q^{22} -4.41251 q^{23} +1.00000 q^{24} -4.81710 q^{25} +0.388531 q^{26} -1.00000 q^{27} -1.56626 q^{28} +4.21904 q^{29} -0.427666 q^{30} -5.84517 q^{31} -1.00000 q^{32} -4.81756 q^{33} -4.71032 q^{34} +0.669836 q^{35} +1.00000 q^{36} +4.83968 q^{37} +1.00000 q^{38} +0.388531 q^{39} +0.427666 q^{40} -5.41921 q^{41} -1.56626 q^{42} -1.29954 q^{43} +4.81756 q^{44} -0.427666 q^{45} +4.41251 q^{46} -12.1622 q^{47} -1.00000 q^{48} -4.54683 q^{49} +4.81710 q^{50} -4.71032 q^{51} -0.388531 q^{52} -1.00000 q^{53} +1.00000 q^{54} -2.06031 q^{55} +1.56626 q^{56} +1.00000 q^{57} -4.21904 q^{58} +0.782618 q^{59} +0.427666 q^{60} +7.68687 q^{61} +5.84517 q^{62} -1.56626 q^{63} +1.00000 q^{64} +0.166162 q^{65} +4.81756 q^{66} -1.73804 q^{67} +4.71032 q^{68} +4.41251 q^{69} -0.669836 q^{70} +5.66582 q^{71} -1.00000 q^{72} +0.430955 q^{73} -4.83968 q^{74} +4.81710 q^{75} -1.00000 q^{76} -7.54555 q^{77} -0.388531 q^{78} +13.8973 q^{79} -0.427666 q^{80} +1.00000 q^{81} +5.41921 q^{82} -6.44884 q^{83} +1.56626 q^{84} -2.01444 q^{85} +1.29954 q^{86} -4.21904 q^{87} -4.81756 q^{88} +6.44903 q^{89} +0.427666 q^{90} +0.608541 q^{91} -4.41251 q^{92} +5.84517 q^{93} +12.1622 q^{94} +0.427666 q^{95} +1.00000 q^{96} -2.09682 q^{97} +4.54683 q^{98} +4.81756 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.427666	−0.191258	−0.0956290	−	0.995417i	$-0.530486\pi$
$5$	−0.427666	−0.191258	−0.0956290	+	0.995417i	$0.530486\pi$
$6$	1.00000	0.408248
$7$	−1.56626	−0.591990	−0.295995	−	0.955189i	$-0.595651\pi$
$7$	−1.56626	−0.591990	−0.295995	+	0.955189i	$0.595651\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.427666	0.135240
$11$	4.81756	1.45255	0.726274	−	0.687405i	$-0.241250\pi$
$11$	4.81756	1.45255	0.726274	+	0.687405i	$0.241250\pi$
$12$	−1.00000	−0.288675
$13$	−0.388531	−0.107759	−0.0538796	−	0.998547i	$-0.517159\pi$
$13$	−0.388531	−0.107759	−0.0538796	+	0.998547i	$0.517159\pi$
$14$	1.56626	0.418600
$15$	0.427666	0.110423
$16$	1.00000	0.250000
$17$	4.71032	1.14242	0.571210	−	0.820804i	$-0.306474\pi$
$17$	4.71032	1.14242	0.571210	+	0.820804i	$0.306474\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.427666	−0.0956290
$21$	1.56626	0.341786
$22$	−4.81756	−1.02711
$23$	−4.41251	−0.920073	−0.460036	−	0.887900i	$-0.652164\pi$
$23$	−4.41251	−0.920073	−0.460036	+	0.887900i	$0.652164\pi$
$24$	1.00000	0.204124
$25$	−4.81710	−0.963420
$26$	0.388531	0.0761972
$27$	−1.00000	−0.192450
$28$	−1.56626	−0.295995
$29$	4.21904	0.783456	0.391728	−	0.920081i	$-0.371877\pi$
$29$	4.21904	0.783456	0.391728	+	0.920081i	$0.371877\pi$
$30$	−0.427666	−0.0780808
$31$	−5.84517	−1.04982	−0.524911	−	0.851157i	$-0.675902\pi$
$31$	−5.84517	−1.04982	−0.524911	+	0.851157i	$0.675902\pi$
$32$	−1.00000	−0.176777
$33$	−4.81756	−0.838630
$34$	−4.71032	−0.807813
$35$	0.669836	0.113223
$36$	1.00000	0.166667
$37$	4.83968	0.795638	0.397819	−	0.917464i	$-0.369767\pi$
$37$	4.83968	0.795638	0.397819	+	0.917464i	$0.369767\pi$
$38$	1.00000	0.162221
$39$	0.388531	0.0622148
$40$	0.427666	0.0676199
$41$	−5.41921	−0.846338	−0.423169	−	0.906051i	$-0.639082\pi$
$41$	−5.41921	−0.846338	−0.423169	+	0.906051i	$0.639082\pi$
$42$	−1.56626	−0.241679
$43$	−1.29954	−0.198177	−0.0990886	−	0.995079i	$-0.531593\pi$
$43$	−1.29954	−0.198177	−0.0990886	+	0.995079i	$0.531593\pi$
$44$	4.81756	0.726274
$45$	−0.427666	−0.0637527
$46$	4.41251	0.650590
$47$	−12.1622	−1.77404	−0.887022	−	0.461727i	$-0.847230\pi$
$47$	−12.1622	−1.77404	−0.887022	+	0.461727i	$0.847230\pi$
$48$	−1.00000	−0.144338
$49$	−4.54683	−0.649547
$50$	4.81710	0.681241
$51$	−4.71032	−0.659577
$52$	−0.388531	−0.0538796
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	1.00000	0.136083
$55$	−2.06031	−0.277812
$56$	1.56626	0.209300
$57$	1.00000	0.132453
$58$	−4.21904	−0.553987
$59$	0.782618	0.101888	0.0509441	−	0.998702i	$-0.483777\pi$
$59$	0.782618	0.101888	0.0509441	+	0.998702i	$0.483777\pi$
$60$	0.427666	0.0552114
$61$	7.68687	0.984203	0.492102	−	0.870538i	$-0.336229\pi$
$61$	7.68687	0.984203	0.492102	+	0.870538i	$0.336229\pi$
$62$	5.84517	0.742337
$63$	−1.56626	−0.197330
$64$	1.00000	0.125000
$65$	0.166162	0.0206098
$66$	4.81756	0.593001
$67$	−1.73804	−0.212335	−0.106168	−	0.994348i	$-0.533858\pi$
$67$	−1.73804	−0.212335	−0.106168	+	0.994348i	$0.533858\pi$
$68$	4.71032	0.571210
$69$	4.41251	0.531204
$70$	−0.669836	−0.0800607
$71$	5.66582	0.672410	0.336205	−	0.941789i	$-0.390857\pi$
$71$	5.66582	0.672410	0.336205	+	0.941789i	$0.390857\pi$
$72$	−1.00000	−0.117851
$73$	0.430955	0.0504395	0.0252198	−	0.999682i	$-0.491971\pi$
$73$	0.430955	0.0504395	0.0252198	+	0.999682i	$0.491971\pi$
$74$	−4.83968	−0.562601
$75$	4.81710	0.556231
$76$	−1.00000	−0.114708
$77$	−7.54555	−0.859895
$78$	−0.388531	−0.0439925
$79$	13.8973	1.56357	0.781785	−	0.623548i	$-0.214310\pi$
$79$	13.8973	1.56357	0.781785	+	0.623548i	$0.214310\pi$
$80$	−0.427666	−0.0478145
$81$	1.00000	0.111111
$82$	5.41921	0.598451
$83$	−6.44884	−0.707852	−0.353926	−	0.935273i	$-0.615154\pi$
$83$	−6.44884	−0.707852	−0.353926	+	0.935273i	$0.615154\pi$
$84$	1.56626	0.170893
$85$	−2.01444	−0.218497
$86$	1.29954	0.140132
$87$	−4.21904	−0.452328
$88$	−4.81756	−0.513554
$89$	6.44903	0.683596	0.341798	−	0.939774i	$-0.388964\pi$
$89$	6.44903	0.683596	0.341798	+	0.939774i	$0.388964\pi$
$90$	0.427666	0.0450799
$91$	0.608541	0.0637924
$92$	−4.41251	−0.460036
$93$	5.84517	0.606115
$94$	12.1622	1.25444
$95$	0.427666	0.0438776
$96$	1.00000	0.102062
$97$	−2.09682	−0.212900	−0.106450	−	0.994318i	$-0.533948\pi$
$97$	−2.09682	−0.212900	−0.106450	+	0.994318i	$0.533948\pi$
$98$	4.54683	0.459299
$99$	4.81756	0.484183
$100$	−4.81710	−0.481710
$101$	10.5144	1.04623	0.523113	−	0.852263i	$-0.324771\pi$
$101$	10.5144	1.04623	0.523113	+	0.852263i	$0.324771\pi$
$102$	4.71032	0.466391
$103$	−13.3352	−1.31395	−0.656977	−	0.753910i	$-0.728165\pi$
$103$	−13.3352	−1.31395	−0.656977	+	0.753910i	$0.728165\pi$
$104$	0.388531	0.0380986
$105$	−0.669836	−0.0653693
$106$	1.00000	0.0971286
$107$	7.20210	0.696253	0.348127	−	0.937448i	$-0.386818\pi$
$107$	7.20210	0.696253	0.348127	+	0.937448i	$0.386818\pi$
$108$	−1.00000	−0.0962250
$109$	11.7162	1.12221	0.561103	−	0.827746i	$-0.310377\pi$
$109$	11.7162	1.12221	0.561103	+	0.827746i	$0.310377\pi$
$110$	2.06031	0.196442
$111$	−4.83968	−0.459362
$112$	−1.56626	−0.147998
$113$	−14.8757	−1.39939	−0.699695	−	0.714442i	$-0.746681\pi$
$113$	−14.8757	−1.39939	−0.699695	+	0.714442i	$0.746681\pi$
$114$	−1.00000	−0.0936586
$115$	1.88708	0.175971
$116$	4.21904	0.391728
$117$	−0.388531	−0.0359197
$118$	−0.782618	−0.0720458
$119$	−7.37758	−0.676302
$120$	−0.427666	−0.0390404
$121$	12.2089	1.10990
$122$	−7.68687	−0.695937
$123$	5.41921	0.488633
$124$	−5.84517	−0.524911
$125$	4.19844	0.375520
$126$	1.56626	0.139533
$127$	16.0592	1.42502	0.712511	−	0.701661i	$-0.247558\pi$
$127$	16.0592	1.42502	0.712511	+	0.701661i	$0.247558\pi$
$128$	−1.00000	−0.0883883
$129$	1.29954	0.114418
$130$	−0.166162	−0.0145733
$131$	−3.20412	−0.279945	−0.139973	−	0.990155i	$-0.544701\pi$
$131$	−3.20412	−0.279945	−0.139973	+	0.990155i	$0.544701\pi$
$132$	−4.81756	−0.419315
$133$	1.56626	0.135812
$134$	1.73804	0.150144
$135$	0.427666	0.0368076
$136$	−4.71032	−0.403907
$137$	−16.5623	−1.41501	−0.707505	−	0.706709i	$-0.750179\pi$
$137$	−16.5623	−1.41501	−0.707505	+	0.706709i	$0.750179\pi$
$138$	−4.41251	−0.375618
$139$	6.08399	0.516037	0.258019	−	0.966140i	$-0.416930\pi$
$139$	6.08399	0.516037	0.258019	+	0.966140i	$0.416930\pi$
$140$	0.669836	0.0566114
$141$	12.1622	1.02424
$142$	−5.66582	−0.475465
$143$	−1.87177	−0.156525
$144$	1.00000	0.0833333
$145$	−1.80434	−0.149842
$146$	−0.430955	−0.0356661
$147$	4.54683	0.375016
$148$	4.83968	0.397819
$149$	−3.28066	−0.268762	−0.134381	−	0.990930i	$-0.542905\pi$
$149$	−3.28066	−0.268762	−0.134381	+	0.990930i	$0.542905\pi$
$150$	−4.81710	−0.393315
$151$	−16.3697	−1.33214	−0.666072	−	0.745887i	$-0.732026\pi$
$151$	−16.3697	−1.33214	−0.666072	+	0.745887i	$0.732026\pi$
$152$	1.00000	0.0811107
$153$	4.71032	0.380807
$154$	7.54555	0.608038
$155$	2.49978	0.200787
$156$	0.388531	0.0311074
$157$	0.158093	0.0126172	0.00630861	−	0.999980i	$-0.497992\pi$
$157$	0.158093	0.0126172	0.00630861	+	0.999980i	$0.497992\pi$
$158$	−13.8973	−1.10561
$159$	1.00000	0.0793052
$160$	0.427666	0.0338100
$161$	6.91114	0.544674
$162$	−1.00000	−0.0785674
$163$	22.9015	1.79379	0.896893	−	0.442247i	$-0.145819\pi$
$163$	22.9015	1.79379	0.896893	+	0.442247i	$0.145819\pi$
$164$	−5.41921	−0.423169
$165$	2.06031	0.160395
$166$	6.44884	0.500527
$167$	−13.4022	−1.03709	−0.518547	−	0.855049i	$-0.673527\pi$
$167$	−13.4022	−1.03709	−0.518547	+	0.855049i	$0.673527\pi$
$168$	−1.56626	−0.120840
$169$	−12.8490	−0.988388
$170$	2.01444	0.154501
$171$	−1.00000	−0.0764719
$172$	−1.29954	−0.0990886
$173$	3.51675	0.267373	0.133687	−	0.991024i	$-0.457318\pi$
$173$	3.51675	0.267373	0.133687	+	0.991024i	$0.457318\pi$
$174$	4.21904	0.319844
$175$	7.54483	0.570336
$176$	4.81756	0.363137
$177$	−0.782618	−0.0588252
$178$	−6.44903	−0.483375
$179$	−18.8844	−1.41148	−0.705742	−	0.708469i	$-0.749386\pi$
$179$	−18.8844	−1.41148	−0.705742	+	0.708469i	$0.749386\pi$
$180$	−0.427666	−0.0318763
$181$	11.6974	0.869463	0.434731	−	0.900560i	$-0.356843\pi$
$181$	11.6974	0.869463	0.434731	+	0.900560i	$0.356843\pi$
$182$	−0.608541	−0.0451080
$183$	−7.68687	−0.568230
$184$	4.41251	0.325295
$185$	−2.06976	−0.152172
$186$	−5.84517	−0.428588
$187$	22.6922	1.65942
$188$	−12.1622	−0.887022
$189$	1.56626	0.113929
$190$	−0.427666	−0.0310261
$191$	−11.7659	−0.851352	−0.425676	−	0.904876i	$-0.639964\pi$
$191$	−11.7659	−0.851352	−0.425676	+	0.904876i	$0.639964\pi$
$192$	−1.00000	−0.0721688
$193$	5.56038	0.400245	0.200123	−	0.979771i	$-0.435866\pi$
$193$	5.56038	0.400245	0.200123	+	0.979771i	$0.435866\pi$
$194$	2.09682	0.150543
$195$	−0.166162	−0.0118991
$196$	−4.54683	−0.324774
$197$	−18.9492	−1.35007	−0.675037	−	0.737784i	$-0.735872\pi$
$197$	−18.9492	−1.35007	−0.675037	+	0.737784i	$0.735872\pi$
$198$	−4.81756	−0.342369
$199$	−25.2727	−1.79153	−0.895767	−	0.444523i	$-0.853373\pi$
$199$	−25.2727	−1.79153	−0.895767	+	0.444523i	$0.853373\pi$
$200$	4.81710	0.340621
$201$	1.73804	0.122592
$202$	−10.5144	−0.739793
$203$	−6.60811	−0.463798
$204$	−4.71032	−0.329788
$205$	2.31761	0.161869
$206$	13.3352	0.929106
$207$	−4.41251	−0.306691
$208$	−0.388531	−0.0269398
$209$	−4.81756	−0.333238
$210$	0.669836	0.0462231
$211$	−5.36682	−0.369467	−0.184734	−	0.982789i	$-0.559142\pi$
$211$	−5.36682	−0.369467	−0.184734	+	0.982789i	$0.559142\pi$
$212$	−1.00000	−0.0686803
$213$	−5.66582	−0.388216
$214$	−7.20210	−0.492325
$215$	0.555767	0.0379030
$216$	1.00000	0.0680414
$217$	9.15504	0.621485
$218$	−11.7162	−0.793520
$219$	−0.430955	−0.0291213
$220$	−2.06031	−0.138906
$221$	−1.83011	−0.123106
$222$	4.83968	0.324818
$223$	19.7809	1.32463	0.662314	−	0.749227i	$-0.269575\pi$
$223$	19.7809	1.32463	0.662314	+	0.749227i	$0.269575\pi$
$224$	1.56626	0.104650
$225$	−4.81710	−0.321140
$226$	14.8757	0.989518
$227$	4.85901	0.322504	0.161252	−	0.986913i	$-0.448447\pi$
$227$	4.85901	0.322504	0.161252	+	0.986913i	$0.448447\pi$
$228$	1.00000	0.0662266
$229$	−20.1052	−1.32859	−0.664294	−	0.747472i	$-0.731268\pi$
$229$	−20.1052	−1.32859	−0.664294	+	0.747472i	$0.731268\pi$
$230$	−1.88708	−0.124430
$231$	7.54555	0.496461
$232$	−4.21904	−0.276993
$233$	2.78758	0.182620	0.0913101	−	0.995823i	$-0.470895\pi$
$233$	2.78758	0.182620	0.0913101	+	0.995823i	$0.470895\pi$
$234$	0.388531	0.0253991
$235$	5.20137	0.339300
$236$	0.782618	0.0509441
$237$	−13.8973	−0.902727
$238$	7.37758	0.478218
$239$	17.0415	1.10232	0.551162	−	0.834398i	$-0.314184\pi$
$239$	17.0415	1.10232	0.551162	+	0.834398i	$0.314184\pi$
$240$	0.427666	0.0276057
$241$	9.85330	0.634707	0.317353	−	0.948307i	$-0.397206\pi$
$241$	9.85330	0.634707	0.317353	+	0.948307i	$0.397206\pi$
$242$	−12.2089	−0.784817
$243$	−1.00000	−0.0641500
$244$	7.68687	0.492102
$245$	1.94453	0.124231
$246$	−5.41921	−0.345516
$247$	0.388531	0.0247217
$248$	5.84517	0.371168
$249$	6.44884	0.408678
$250$	−4.19844	−0.265533
$251$	−14.9682	−0.944786	−0.472393	−	0.881388i	$-0.656610\pi$
$251$	−14.9682	−0.944786	−0.472393	+	0.881388i	$0.656610\pi$
$252$	−1.56626	−0.0986651
$253$	−21.2576	−1.33645
$254$	−16.0592	−1.00764
$255$	2.01444	0.126149
$256$	1.00000	0.0625000
$257$	−31.3096	−1.95304	−0.976519	−	0.215429i	$-0.930885\pi$
$257$	−31.3096	−1.95304	−0.976519	+	0.215429i	$0.930885\pi$
$258$	−1.29954	−0.0809055
$259$	−7.58019	−0.471010
$260$	0.166162	0.0103049
$261$	4.21904	0.261152
$262$	3.20412	0.197951
$263$	−27.7682	−1.71226	−0.856129	−	0.516762i	$-0.827137\pi$
$263$	−27.7682	−1.71226	−0.856129	+	0.516762i	$0.827137\pi$
$264$	4.81756	0.296500
$265$	0.427666	0.0262713
$266$	−1.56626	−0.0960335
$267$	−6.44903	−0.394674
$268$	−1.73804	−0.106168
$269$	−24.8474	−1.51497	−0.757486	−	0.652851i	$-0.773573\pi$
$269$	−24.8474	−1.51497	−0.757486	+	0.652851i	$0.773573\pi$
$270$	−0.427666	−0.0260269
$271$	−18.2597	−1.10920	−0.554599	−	0.832118i	$-0.687128\pi$
$271$	−18.2597	−1.10920	−0.554599	+	0.832118i	$0.687128\pi$
$272$	4.71032	0.285605
$273$	−0.608541	−0.0368306
$274$	16.5623	1.00056
$275$	−23.2067	−1.39942
$276$	4.41251	0.265602
$277$	14.3516	0.862305	0.431152	−	0.902279i	$-0.358107\pi$
$277$	14.3516	0.862305	0.431152	+	0.902279i	$0.358107\pi$
$278$	−6.08399	−0.364893
$279$	−5.84517	−0.349941
$280$	−0.669836	−0.0400303
$281$	12.1036	0.722041	0.361020	−	0.932558i	$-0.382428\pi$
$281$	12.1036	0.722041	0.361020	+	0.932558i	$0.382428\pi$
$282$	−12.1622	−0.724250
$283$	−22.1941	−1.31930	−0.659651	−	0.751572i	$-0.729296\pi$
$283$	−22.1941	−1.31930	−0.659651	+	0.751572i	$0.729296\pi$
$284$	5.66582	0.336205
$285$	−0.427666	−0.0253327
$286$	1.87177	0.110680
$287$	8.48788	0.501024
$288$	−1.00000	−0.0589256
$289$	5.18711	0.305124
$290$	1.80434	0.105954
$291$	2.09682	0.122918
$292$	0.430955	0.0252198
$293$	15.7437	0.919759	0.459880	−	0.887981i	$-0.347893\pi$
$293$	15.7437	0.919759	0.459880	+	0.887981i	$0.347893\pi$
$294$	−4.54683	−0.265177
$295$	−0.334699	−0.0194869
$296$	−4.83968	−0.281300
$297$	−4.81756	−0.279543
$298$	3.28066	0.190043
$299$	1.71440	0.0991463
$300$	4.81710	0.278116
$301$	2.03541	0.117319
$302$	16.3697	0.941968
$303$	−10.5144	−0.604039
$304$	−1.00000	−0.0573539
$305$	−3.28741	−0.188237
$306$	−4.71032	−0.269271
$307$	6.20366	0.354062	0.177031	−	0.984205i	$-0.443351\pi$
$307$	6.20366	0.354062	0.177031	+	0.984205i	$0.443351\pi$
$308$	−7.54555	−0.429947
$309$	13.3352	0.758612
$310$	−2.49978	−0.141978
$311$	13.6845	0.775978	0.387989	−	0.921664i	$-0.373170\pi$
$311$	13.6845	0.775978	0.387989	+	0.921664i	$0.373170\pi$
$312$	−0.388531	−0.0219962
$313$	−32.5278	−1.83858	−0.919289	−	0.393583i	$-0.871235\pi$
$313$	−32.5278	−1.83858	−0.919289	+	0.393583i	$0.871235\pi$
$314$	−0.158093	−0.00892172
$315$	0.669836	0.0377410
$316$	13.8973	0.781785
$317$	19.9115	1.11834	0.559171	−	0.829052i	$-0.311119\pi$
$317$	19.9115	1.11834	0.559171	+	0.829052i	$0.311119\pi$
$318$	−1.00000	−0.0560772
$319$	20.3255	1.13801
$320$	−0.427666	−0.0239073
$321$	−7.20210	−0.401982
$322$	−6.91114	−0.385143
$323$	−4.71032	−0.262089
$324$	1.00000	0.0555556
$325$	1.87159	0.103817
$326$	−22.9015	−1.26840
$327$	−11.7162	−0.647906
$328$	5.41921	0.299226
$329$	19.0492	1.05022
$330$	−2.06031	−0.113416
$331$	−8.28706	−0.455498	−0.227749	−	0.973720i	$-0.573137\pi$
$331$	−8.28706	−0.455498	−0.227749	+	0.973720i	$0.573137\pi$
$332$	−6.44884	−0.353926
$333$	4.83968	0.265213
$334$	13.4022	0.733336
$335$	0.743300	0.0406108
$336$	1.56626	0.0854464
$337$	15.5498	0.847049	0.423525	−	0.905885i	$-0.360793\pi$
$337$	15.5498	0.847049	0.423525	+	0.905885i	$0.360793\pi$
$338$	12.8490	0.698896
$339$	14.8757	0.807938
$340$	−2.01444	−0.109249
$341$	−28.1594	−1.52492
$342$	1.00000	0.0540738
$343$	18.0853	0.976516
$344$	1.29954	0.0700662
$345$	−1.88708	−0.101597
$346$	−3.51675	−0.189062
$347$	11.5934	0.622366	0.311183	−	0.950350i	$-0.399275\pi$
$347$	11.5934	0.622366	0.311183	+	0.950350i	$0.399275\pi$
$348$	−4.21904	−0.226164
$349$	17.3982	0.931305	0.465653	−	0.884968i	$-0.345820\pi$
$349$	17.3982	0.931305	0.465653	+	0.884968i	$0.345820\pi$
$350$	−7.54483	−0.403288
$351$	0.388531	0.0207383
$352$	−4.81756	−0.256777
$353$	−29.0994	−1.54881	−0.774403	−	0.632692i	$-0.781950\pi$
$353$	−29.0994	−1.54881	−0.774403	+	0.632692i	$0.781950\pi$
$354$	0.782618	0.0415957
$355$	−2.42308	−0.128604
$356$	6.44903	0.341798
$357$	7.37758	0.390463
$358$	18.8844	0.998069
$359$	−9.34900	−0.493421	−0.246711	−	0.969089i	$-0.579350\pi$
$359$	−9.34900	−0.493421	−0.246711	+	0.969089i	$0.579350\pi$
$360$	0.427666	0.0225400
$361$	1.00000	0.0526316
$362$	−11.6974	−0.614803
$363$	−12.2089	−0.640800
$364$	0.608541	0.0318962
$365$	−0.184305	−0.00964696
$366$	7.68687	0.401799
$367$	35.8981	1.87386	0.936932	−	0.349513i	$-0.113653\pi$
$367$	35.8981	1.87386	0.936932	+	0.349513i	$0.113653\pi$
$368$	−4.41251	−0.230018
$369$	−5.41921	−0.282113
$370$	2.06976	0.107602
$371$	1.56626	0.0813161
$372$	5.84517	0.303058
$373$	−20.1554	−1.04360	−0.521802	−	0.853066i	$-0.674740\pi$
$373$	−20.1554	−1.04360	−0.521802	+	0.853066i	$0.674740\pi$
$374$	−22.6922	−1.17339
$375$	−4.19844	−0.216806
$376$	12.1622	0.627219
$377$	−1.63923	−0.0844246
$378$	−1.56626	−0.0805597
$379$	−13.9274	−0.715405	−0.357702	−	0.933836i	$-0.616440\pi$
$379$	−13.9274	−0.715405	−0.357702	+	0.933836i	$0.616440\pi$
$380$	0.427666	0.0219388
$381$	−16.0592	−0.822737
$382$	11.7659	0.601997
$383$	4.26634	0.218000	0.109000	−	0.994042i	$-0.465235\pi$
$383$	4.26634	0.218000	0.109000	+	0.994042i	$0.465235\pi$
$384$	1.00000	0.0510310
$385$	3.22697	0.164462
$386$	−5.56038	−0.283016
$387$	−1.29954	−0.0660591
$388$	−2.09682	−0.106450
$389$	26.3513	1.33606	0.668031	−	0.744133i	$-0.267137\pi$
$389$	26.3513	1.33606	0.668031	+	0.744133i	$0.267137\pi$
$390$	0.166162	0.00841392
$391$	−20.7844	−1.05111
$392$	4.54683	0.229650
$393$	3.20412	0.161626
$394$	18.9492	0.954646
$395$	−5.94341	−0.299045
$396$	4.81756	0.242091
$397$	−30.7695	−1.54428	−0.772138	−	0.635455i	$-0.780812\pi$
$397$	−30.7695	−1.54428	−0.772138	+	0.635455i	$0.780812\pi$
$398$	25.2727	1.26681
$399$	−1.56626	−0.0784110
$400$	−4.81710	−0.240855
$401$	−22.7538	−1.13627	−0.568134	−	0.822936i	$-0.692335\pi$
$401$	−22.7538	−1.13627	−0.568134	+	0.822936i	$0.692335\pi$
$402$	−1.73804	−0.0866855
$403$	2.27103	0.113128
$404$	10.5144	0.523113
$405$	−0.427666	−0.0212509
$406$	6.60811	0.327955
$407$	23.3154	1.15570
$408$	4.71032	0.233196
$409$	35.3716	1.74901	0.874506	−	0.485015i	$-0.161186\pi$
$409$	35.3716	1.74901	0.874506	+	0.485015i	$0.161186\pi$
$410$	−2.31761	−0.114459
$411$	16.5623	0.816956
$412$	−13.3352	−0.656977
$413$	−1.22578	−0.0603168
$414$	4.41251	0.216863
$415$	2.75795	0.135382
$416$	0.388531	0.0190493
$417$	−6.08399	−0.297934
$418$	4.81756	0.235635
$419$	14.0641	0.687074	0.343537	−	0.939139i	$-0.388375\pi$
$419$	14.0641	0.687074	0.343537	+	0.939139i	$0.388375\pi$
$420$	−0.669836	−0.0326846
$421$	−17.3122	−0.843746	−0.421873	−	0.906655i	$-0.638627\pi$
$421$	−17.3122	−0.843746	−0.421873	+	0.906655i	$0.638627\pi$
$422$	5.36682	0.261253
$423$	−12.1622	−0.591348
$424$	1.00000	0.0485643
$425$	−22.6901	−1.10063
$426$	5.66582	0.274510
$427$	−12.0396	−0.582639
$428$	7.20210	0.348127
$429$	1.87177	0.0903700
$430$	−0.555767	−0.0268015
$431$	2.44035	0.117548	0.0587739	−	0.998271i	$-0.481281\pi$
$431$	2.44035	0.117548	0.0587739	+	0.998271i	$0.481281\pi$
$432$	−1.00000	−0.0481125
$433$	37.4176	1.79817	0.899087	−	0.437770i	$-0.144232\pi$
$433$	37.4176	1.79817	0.899087	+	0.437770i	$0.144232\pi$
$434$	−9.15504	−0.439456
$435$	1.80434	0.0865114
$436$	11.7162	0.561103
$437$	4.41251	0.211079
$438$	0.430955	0.0205918
$439$	−27.5258	−1.31374	−0.656868	−	0.754005i	$-0.728119\pi$
$439$	−27.5258	−1.31374	−0.656868	+	0.754005i	$0.728119\pi$
$440$	2.06031	0.0982212
$441$	−4.54683	−0.216516
$442$	1.83011	0.0870493
$443$	−30.1858	−1.43417	−0.717086	−	0.696984i	$-0.754525\pi$
$443$	−30.1858	−1.43417	−0.717086	+	0.696984i	$0.754525\pi$
$444$	−4.83968	−0.229681
$445$	−2.75803	−0.130743
$446$	−19.7809	−0.936653
$447$	3.28066	0.155170
$448$	−1.56626	−0.0739988
$449$	28.8307	1.36061	0.680303	−	0.732931i	$-0.261848\pi$
$449$	28.8307	1.36061	0.680303	+	0.732931i	$0.261848\pi$
$450$	4.81710	0.227080
$451$	−26.1073	−1.22935
$452$	−14.8757	−0.699695
$453$	16.3697	0.769114
$454$	−4.85901	−0.228044
$455$	−0.260252	−0.0122008
$456$	−1.00000	−0.0468293
$457$	−29.7394	−1.39115	−0.695574	−	0.718454i	$-0.744850\pi$
$457$	−29.7394	−1.39115	−0.695574	+	0.718454i	$0.744850\pi$
$458$	20.1052	0.939453
$459$	−4.71032	−0.219859
$460$	1.88708	0.0879856
$461$	−20.3174	−0.946276	−0.473138	−	0.880988i	$-0.656879\pi$
$461$	−20.3174	−0.946276	−0.473138	+	0.880988i	$0.656879\pi$
$462$	−7.54555	−0.351051
$463$	−33.1623	−1.54118	−0.770591	−	0.637330i	$-0.780039\pi$
$463$	−33.1623	−1.54118	−0.770591	+	0.637330i	$0.780039\pi$
$464$	4.21904	0.195864
$465$	−2.49978	−0.115924
$466$	−2.78758	−0.129132
$467$	−9.10992	−0.421557	−0.210778	−	0.977534i	$-0.567600\pi$
$467$	−9.10992	−0.421557	−0.210778	+	0.977534i	$0.567600\pi$
$468$	−0.388531	−0.0179599
$469$	2.72222	0.125700
$470$	−5.20137	−0.239921
$471$	−0.158093	−0.00728456
$472$	−0.782618	−0.0360229
$473$	−6.26059	−0.287862
$474$	13.8973	0.638325
$475$	4.81710	0.221024
$476$	−7.37758	−0.338151
$477$	−1.00000	−0.0457869
$478$	−17.0415	−0.779461
$479$	−33.7857	−1.54371	−0.771855	−	0.635799i	$-0.780671\pi$
$479$	−33.7857	−1.54371	−0.771855	+	0.635799i	$0.780671\pi$
$480$	−0.427666	−0.0195202
$481$	−1.88037	−0.0857373
$482$	−9.85330	−0.448805
$483$	−6.91114	−0.314468
$484$	12.2089	0.554949
$485$	0.896738	0.0407188
$486$	1.00000	0.0453609
$487$	25.3375	1.14815	0.574075	−	0.818802i	$-0.305362\pi$
$487$	25.3375	1.14815	0.574075	+	0.818802i	$0.305362\pi$
$488$	−7.68687	−0.347968
$489$	−22.9015	−1.03564
$490$	−1.94453	−0.0878447
$491$	−24.0912	−1.08722	−0.543610	−	0.839338i	$-0.682943\pi$
$491$	−24.0912	−1.08722	−0.543610	+	0.839338i	$0.682943\pi$
$492$	5.41921	0.244317
$493$	19.8730	0.895036
$494$	−0.388531	−0.0174808
$495$	−2.06031	−0.0926039
$496$	−5.84517	−0.262456
$497$	−8.87415	−0.398060
$498$	−6.44884	−0.288979
$499$	−23.7957	−1.06524	−0.532621	−	0.846354i	$-0.678793\pi$
$499$	−23.7957	−1.06524	−0.532621	+	0.846354i	$0.678793\pi$
$500$	4.19844	0.187760
$501$	13.4022	0.598766
$502$	14.9682	0.668065
$503$	3.15195	0.140539	0.0702693	−	0.997528i	$-0.477614\pi$
$503$	3.15195	0.140539	0.0702693	+	0.997528i	$0.477614\pi$
$504$	1.56626	0.0697667
$505$	−4.49667	−0.200099
$506$	21.2576	0.945013
$507$	12.8490	0.570646
$508$	16.0592	0.712511
$509$	14.5245	0.643786	0.321893	−	0.946776i	$-0.395681\pi$
$509$	14.5245	0.643786	0.321893	+	0.946776i	$0.395681\pi$
$510$	−2.01444	−0.0892010
$511$	−0.674988	−0.0298597
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	31.3096	1.38101
$515$	5.70300	0.251304
$516$	1.29954	0.0572088
$517$	−58.5923	−2.57689
$518$	7.58019	0.333054
$519$	−3.51675	−0.154368
$520$	−0.166162	−0.00728667
$521$	−10.1718	−0.445634	−0.222817	−	0.974860i	$-0.571525\pi$
$521$	−10.1718	−0.445634	−0.222817	+	0.974860i	$0.571525\pi$
$522$	−4.21904	−0.184662
$523$	−13.1376	−0.574467	−0.287234	−	0.957861i	$-0.592736\pi$
$523$	−13.1376	−0.574467	−0.287234	+	0.957861i	$0.592736\pi$
$524$	−3.20412	−0.139973
$525$	−7.54483	−0.329283
$526$	27.7682	1.21075
$527$	−27.5326	−1.19934
$528$	−4.81756	−0.209657
$529$	−3.52972	−0.153466
$530$	−0.427666	−0.0185766
$531$	0.782618	0.0339627
$532$	1.56626	0.0679059
$533$	2.10553	0.0912007
$534$	6.44903	0.279077
$535$	−3.08009	−0.133164
$536$	1.73804	0.0750719
$537$	18.8844	0.814920
$538$	24.8474	1.07125
$539$	−21.9046	−0.943499
$540$	0.427666	0.0184038
$541$	36.7326	1.57926	0.789630	−	0.613584i	$-0.210273\pi$
$541$	36.7326	1.57926	0.789630	+	0.613584i	$0.210273\pi$
$542$	18.2597	0.784321
$543$	−11.6974	−0.501985
$544$	−4.71032	−0.201953
$545$	−5.01061	−0.214631
$546$	0.608541	0.0260431
$547$	36.0512	1.54144	0.770719	−	0.637175i	$-0.219897\pi$
$547$	36.0512	1.54144	0.770719	+	0.637175i	$0.219897\pi$
$548$	−16.5623	−0.707505
$549$	7.68687	0.328068
$550$	23.2067	0.989536
$551$	−4.21904	−0.179737
$552$	−4.41251	−0.187809
$553$	−21.7668	−0.925618
$554$	−14.3516	−0.609741
$555$	2.06976	0.0878566
$556$	6.08399	0.258019
$557$	21.0757	0.893004	0.446502	−	0.894783i	$-0.352670\pi$
$557$	21.0757	0.893004	0.446502	+	0.894783i	$0.352670\pi$
$558$	5.84517	0.247446
$559$	0.504910	0.0213554
$560$	0.669836	0.0283057
$561$	−22.6922	−0.958067
$562$	−12.1036	−0.510560
$563$	−18.1589	−0.765307	−0.382654	−	0.923892i	$-0.624990\pi$
$563$	−18.1589	−0.765307	−0.382654	+	0.923892i	$0.624990\pi$
$564$	12.1622	0.512122
$565$	6.36184	0.267645
$566$	22.1941	0.932887
$567$	−1.56626	−0.0657767
$568$	−5.66582	−0.237733
$569$	−34.6164	−1.45120	−0.725598	−	0.688119i	$-0.758437\pi$
$569$	−34.6164	−1.45120	−0.725598	+	0.688119i	$0.758437\pi$
$570$	0.427666	0.0179130
$571$	−10.8694	−0.454871	−0.227435	−	0.973793i	$-0.573034\pi$
$571$	−10.8694	−0.454871	−0.227435	+	0.973793i	$0.573034\pi$
$572$	−1.87177	−0.0782627
$573$	11.7659	0.491528
$574$	−8.48788	−0.354277
$575$	21.2555	0.886417
$576$	1.00000	0.0416667
$577$	−41.3833	−1.72281	−0.861405	−	0.507918i	$-0.830415\pi$
$577$	−41.3833	−1.72281	−0.861405	+	0.507918i	$0.830415\pi$
$578$	−5.18711	−0.215755
$579$	−5.56038	−0.231082
$580$	−1.80434	−0.0749211
$581$	10.1006	0.419041
$582$	−2.09682	−0.0869160
$583$	−4.81756	−0.199523
$584$	−0.430955	−0.0178331
$585$	0.166162	0.00686994
$586$	−15.7437	−0.650368
$587$	−45.2616	−1.86815	−0.934074	−	0.357080i	$-0.883772\pi$
$587$	−45.2616	−1.86815	−0.934074	+	0.357080i	$0.883772\pi$
$588$	4.54683	0.187508
$589$	5.84517	0.240846
$590$	0.334699	0.0137793
$591$	18.9492	0.779466
$592$	4.83968	0.198909
$593$	−3.02620	−0.124271	−0.0621355	−	0.998068i	$-0.519791\pi$
$593$	−3.02620	−0.124271	−0.0621355	+	0.998068i	$0.519791\pi$
$594$	4.81756	0.197667
$595$	3.15514	0.129348
$596$	−3.28066	−0.134381
$597$	25.2727	1.03434
$598$	−1.71440	−0.0701070
$599$	19.7579	0.807284	0.403642	−	0.914917i	$-0.367744\pi$
$599$	19.7579	0.807284	0.403642	+	0.914917i	$0.367744\pi$
$600$	−4.81710	−0.196657
$601$	−31.6123	−1.28949	−0.644746	−	0.764397i	$-0.723037\pi$
$601$	−31.6123	−1.28949	−0.644746	+	0.764397i	$0.723037\pi$
$602$	−2.03541	−0.0829571
$603$	−1.73804	−0.0707784
$604$	−16.3697	−0.666072
$605$	−5.22132	−0.212277
$606$	10.5144	0.427120
$607$	−30.5914	−1.24167	−0.620833	−	0.783943i	$-0.713206\pi$
$607$	−30.5914	−1.24167	−0.620833	+	0.783943i	$0.713206\pi$
$608$	1.00000	0.0405554
$609$	6.60811	0.267774
$610$	3.28741	0.133103
$611$	4.72541	0.191170
$612$	4.71032	0.190403
$613$	6.83835	0.276198	0.138099	−	0.990418i	$-0.455901\pi$
$613$	6.83835	0.276198	0.138099	+	0.990418i	$0.455901\pi$
$614$	−6.20366	−0.250359
$615$	−2.31761	−0.0934550
$616$	7.54555	0.304019
$617$	28.3842	1.14270	0.571352	−	0.820705i	$-0.306419\pi$
$617$	28.3842	1.14270	0.571352	+	0.820705i	$0.306419\pi$
$618$	−13.3352	−0.536420
$619$	34.5278	1.38779	0.693894	−	0.720077i	$-0.255893\pi$
$619$	34.5278	1.38779	0.693894	+	0.720077i	$0.255893\pi$
$620$	2.49978	0.100394
$621$	4.41251	0.177068
$622$	−13.6845	−0.548699
$623$	−10.1008	−0.404682
$624$	0.388531	0.0155537
$625$	22.2900	0.891599
$626$	32.5278	1.30007
$627$	4.81756	0.192395
$628$	0.158093	0.00630861
$629$	22.7964	0.908953
$630$	−0.669836	−0.0266869
$631$	15.6077	0.621334	0.310667	−	0.950519i	$-0.399448\pi$
$631$	15.6077	0.621334	0.310667	+	0.950519i	$0.399448\pi$
$632$	−13.8973	−0.552805
$633$	5.36682	0.213312
$634$	−19.9115	−0.790788
$635$	−6.86796	−0.272547
$636$	1.00000	0.0396526
$637$	1.76659	0.0699947
$638$	−20.3255	−0.804693
$639$	5.66582	0.224137
$640$	0.427666	0.0169050
$641$	−41.0101	−1.61980	−0.809901	−	0.586566i	$-0.800479\pi$
$641$	−41.0101	−1.61980	−0.809901	+	0.586566i	$0.800479\pi$
$642$	7.20210	0.284244
$643$	2.85313	0.112517	0.0562583	−	0.998416i	$-0.482083\pi$
$643$	2.85313	0.112517	0.0562583	+	0.998416i	$0.482083\pi$
$644$	6.91114	0.272337
$645$	−0.555767	−0.0218833
$646$	4.71032	0.185325
$647$	−47.4929	−1.86714	−0.933570	−	0.358395i	$-0.883324\pi$
$647$	−47.4929	−1.86714	−0.933570	+	0.358395i	$0.883324\pi$
$648$	−1.00000	−0.0392837
$649$	3.77031	0.147998
$650$	−1.87159	−0.0734100
$651$	−9.15504	−0.358814
$652$	22.9015	0.896893
$653$	−13.7303	−0.537309	−0.268655	−	0.963237i	$-0.586579\pi$
$653$	−13.7303	−0.537309	−0.268655	+	0.963237i	$0.586579\pi$
$654$	11.7162	0.458139
$655$	1.37029	0.0535417
$656$	−5.41921	−0.211584
$657$	0.430955	0.0168132
$658$	−19.0492	−0.742615
$659$	−19.9785	−0.778250	−0.389125	−	0.921185i	$-0.627223\pi$
$659$	−19.9785	−0.778250	−0.389125	+	0.921185i	$0.627223\pi$
$660$	2.06031	0.0801973
$661$	−26.3123	−1.02343	−0.511715	−	0.859156i	$-0.670989\pi$
$661$	−26.3123	−1.02343	−0.511715	+	0.859156i	$0.670989\pi$
$662$	8.28706	0.322086
$663$	1.83011	0.0710754
$664$	6.44884	0.250263
$665$	−0.669836	−0.0259751
$666$	−4.83968	−0.187534
$667$	−18.6166	−0.720836
$668$	−13.4022	−0.518547
$669$	−19.7809	−0.764774
$670$	−0.743300	−0.0287162
$671$	37.0320	1.42960
$672$	−1.56626	−0.0604198
$673$	−26.3113	−1.01423	−0.507113	−	0.861879i	$-0.669287\pi$
$673$	−26.3113	−1.01423	−0.507113	+	0.861879i	$0.669287\pi$
$674$	−15.5498	−0.598954
$675$	4.81710	0.185410
$676$	−12.8490	−0.494194
$677$	26.8523	1.03202	0.516008	−	0.856584i	$-0.327417\pi$
$677$	26.8523	1.03202	0.516008	+	0.856584i	$0.327417\pi$
$678$	−14.8757	−0.571299
$679$	3.28416	0.126035
$680$	2.01444	0.0772504
$681$	−4.85901	−0.186198
$682$	28.1594	1.07828
$683$	−45.4541	−1.73925	−0.869626	−	0.493711i	$-0.835640\pi$
$683$	−45.4541	−1.73925	−0.869626	+	0.493711i	$0.835640\pi$
$684$	−1.00000	−0.0382360
$685$	7.08311	0.270632
$686$	−18.0853	−0.690501
$687$	20.1052	0.767060
$688$	−1.29954	−0.0495443
$689$	0.388531	0.0148019
$690$	1.88708	0.0718400
$691$	50.9942	1.93991	0.969956	−	0.243280i	$-0.0782234\pi$
$691$	50.9942	1.93991	0.969956	+	0.243280i	$0.0782234\pi$
$692$	3.51675	0.133687
$693$	−7.54555	−0.286632
$694$	−11.5934	−0.440079
$695$	−2.60191	−0.0986962
$696$	4.21904	0.159922
$697$	−25.5262	−0.966874
$698$	−17.3982	−0.658532
$699$	−2.78758	−0.105436
$700$	7.54483	0.285168
$701$	−13.5188	−0.510598	−0.255299	−	0.966862i	$-0.582174\pi$
$701$	−13.5188	−0.510598	−0.255299	+	0.966862i	$0.582174\pi$
$702$	−0.388531	−0.0146642
$703$	−4.83968	−0.182532
$704$	4.81756	0.181569
$705$	−5.20137	−0.195895
$706$	29.0994	1.09517
$707$	−16.4683	−0.619356
$708$	−0.782618	−0.0294126
$709$	42.2626	1.58721	0.793603	−	0.608436i	$-0.208203\pi$
$709$	42.2626	1.58721	0.793603	+	0.608436i	$0.208203\pi$
$710$	2.42308	0.0909366
$711$	13.8973	0.521190
$712$	−6.44903	−0.241688
$713$	25.7919	0.965913
$714$	−7.37758	−0.276099
$715$	0.800493	0.0299368
$716$	−18.8844	−0.705742
$717$	−17.0415	−0.636427
$718$	9.34900	0.348902
$719$	−22.4613	−0.837666	−0.418833	−	0.908063i	$-0.637561\pi$
$719$	−22.4613	−0.837666	−0.418833	+	0.908063i	$0.637561\pi$
$720$	−0.427666	−0.0159382
$721$	20.8864	0.777849
$722$	−1.00000	−0.0372161
$723$	−9.85330	−0.366448
$724$	11.6974	0.434731
$725$	−20.3235	−0.754797
$726$	12.2089	0.453114
$727$	−44.3321	−1.64419	−0.822094	−	0.569352i	$-0.807194\pi$
$727$	−44.3321	−1.64419	−0.822094	+	0.569352i	$0.807194\pi$
$728$	−0.608541	−0.0225540
$729$	1.00000	0.0370370
$730$	0.184305	0.00682143
$731$	−6.12123	−0.226402
$732$	−7.68687	−0.284115
$733$	35.5377	1.31261	0.656307	−	0.754494i	$-0.272117\pi$
$733$	35.5377	1.31261	0.656307	+	0.754494i	$0.272117\pi$
$734$	−35.8981	−1.32502
$735$	−1.94453	−0.0717249
$736$	4.41251	0.162647
$737$	−8.37311	−0.308427
$738$	5.41921	0.199484
$739$	−23.8987	−0.879130	−0.439565	−	0.898211i	$-0.644867\pi$
$739$	−23.8987	−0.879130	−0.439565	+	0.898211i	$0.644867\pi$
$740$	−2.06976	−0.0760861
$741$	−0.388531	−0.0142731
$742$	−1.56626	−0.0574992
$743$	2.88824	0.105959	0.0529796	−	0.998596i	$-0.483128\pi$
$743$	2.88824	0.105959	0.0529796	+	0.998596i	$0.483128\pi$
$744$	−5.84517	−0.214294
$745$	1.40303	0.0514029
$746$	20.1554	0.737940
$747$	−6.44884	−0.235951
$748$	22.6922	0.829711
$749$	−11.2804	−0.412175
$750$	4.19844	0.153305
$751$	−4.00165	−0.146022	−0.0730111	−	0.997331i	$-0.523261\pi$
$751$	−4.00165	−0.146022	−0.0730111	+	0.997331i	$0.523261\pi$
$752$	−12.1622	−0.443511
$753$	14.9682	0.545473
$754$	1.63923	0.0596972
$755$	7.00075	0.254783
$756$	1.56626	0.0569643
$757$	1.09654	0.0398546	0.0199273	−	0.999801i	$-0.493657\pi$
$757$	1.09654	0.0398546	0.0199273	+	0.999801i	$0.493657\pi$
$758$	13.9274	0.505868
$759$	21.2576	0.771600
$760$	−0.427666	−0.0155131
$761$	−5.82146	−0.211028	−0.105514	−	0.994418i	$-0.533649\pi$
$761$	−5.82146	−0.211028	−0.105514	+	0.994418i	$0.533649\pi$
$762$	16.0592	0.581763
$763$	−18.3506	−0.664335
$764$	−11.7659	−0.425676
$765$	−2.01444	−0.0728324
$766$	−4.26634	−0.154149
$767$	−0.304072	−0.0109794
$768$	−1.00000	−0.0360844
$769$	39.7643	1.43394	0.716969	−	0.697105i	$-0.245529\pi$
$769$	39.7643	1.43394	0.716969	+	0.697105i	$0.245529\pi$
$770$	−3.22697	−0.116292
$771$	31.3096	1.12759
$772$	5.56038	0.200123
$773$	−42.1916	−1.51753	−0.758763	−	0.651367i	$-0.774196\pi$
$773$	−42.1916	−1.51753	−0.758763	+	0.651367i	$0.774196\pi$
$774$	1.29954	0.0467108
$775$	28.1568	1.01142
$776$	2.09682	0.0752714
$777$	7.58019	0.271938
$778$	−26.3513	−0.944739
$779$	5.41921	0.194163
$780$	−0.166162	−0.00594954
$781$	27.2954	0.976708
$782$	20.7844	0.743247
$783$	−4.21904	−0.150776
$784$	−4.54683	−0.162387
$785$	−0.0676111	−0.00241314
$786$	−3.20412	−0.114287
$787$	−40.2467	−1.43464	−0.717320	−	0.696743i	$-0.754632\pi$
$787$	−40.2467	−1.43464	−0.717320	+	0.696743i	$0.754632\pi$
$788$	−18.9492	−0.675037
$789$	27.7682	0.988573
$790$	5.94341	0.211457
$791$	23.2992	0.828425
$792$	−4.81756	−0.171185
$793$	−2.98659	−0.106057
$794$	30.7695	1.09197
$795$	−0.427666	−0.0151677
$796$	−25.2727	−0.895767
$797$	11.2826	0.399650	0.199825	−	0.979832i	$-0.435963\pi$
$797$	11.2826	0.399650	0.199825	+	0.979832i	$0.435963\pi$
$798$	1.56626	0.0554450
$799$	−57.2880	−2.02670
$800$	4.81710	0.170310
$801$	6.44903	0.227865
$802$	22.7538	0.803463
$803$	2.07615	0.0732659
$804$	1.73804	0.0612959
$805$	−2.95566	−0.104173
$806$	−2.27103	−0.0799936
$807$	24.8474	0.874670
$808$	−10.5144	−0.369897
$809$	48.6665	1.71102	0.855511	−	0.517785i	$-0.173243\pi$
$809$	48.6665	1.71102	0.855511	+	0.517785i	$0.173243\pi$
$810$	0.427666	0.0150266
$811$	−27.5782	−0.968402	−0.484201	−	0.874957i	$-0.660890\pi$
$811$	−27.5782	−0.968402	−0.484201	+	0.874957i	$0.660890\pi$
$812$	−6.60811	−0.231899
$813$	18.2597	0.640395
$814$	−23.3154	−0.817205
$815$	−9.79420	−0.343076
$816$	−4.71032	−0.164894
$817$	1.29954	0.0454650
$818$	−35.3716	−1.23674
$819$	0.608541	0.0212641
$820$	2.31761	0.0809344
$821$	46.7885	1.63293	0.816465	−	0.577394i	$-0.195930\pi$
$821$	46.7885	1.63293	0.816465	+	0.577394i	$0.195930\pi$
$822$	−16.5623	−0.577675
$823$	−35.1781	−1.22623	−0.613115	−	0.789994i	$-0.710084\pi$
$823$	−35.1781	−1.22623	−0.613115	+	0.789994i	$0.710084\pi$
$824$	13.3352	0.464553
$825$	23.2067	0.807953
$826$	1.22578	0.0426504
$827$	−22.0985	−0.768440	−0.384220	−	0.923242i	$-0.625530\pi$
$827$	−22.0985	−0.768440	−0.384220	+	0.923242i	$0.625530\pi$
$828$	−4.41251	−0.153345
$829$	−32.0670	−1.11373	−0.556866	−	0.830602i	$-0.687996\pi$
$829$	−32.0670	−1.11373	−0.556866	+	0.830602i	$0.687996\pi$
$830$	−2.75795	−0.0957298
$831$	−14.3516	−0.497852
$832$	−0.388531	−0.0134699
$833$	−21.4170	−0.742056
$834$	6.08399	0.210671
$835$	5.73167	0.198352
$836$	−4.81756	−0.166619
$837$	5.84517	0.202038
$838$	−14.0641	−0.485835
$839$	38.3593	1.32431	0.662155	−	0.749367i	$-0.269642\pi$
$839$	38.3593	1.32431	0.662155	+	0.749367i	$0.269642\pi$
$840$	0.669836	0.0231115
$841$	−11.1997	−0.386197
$842$	17.3122	0.596619
$843$	−12.1036	−0.416870
$844$	−5.36682	−0.184734
$845$	5.49510	0.189037
$846$	12.1622	0.418146
$847$	−19.1223	−0.657049
$848$	−1.00000	−0.0343401
$849$	22.1941	0.761699
$850$	22.6901	0.778264
$851$	−21.3551	−0.732045
$852$	−5.66582	−0.194108
$853$	−2.76253	−0.0945871	−0.0472936	−	0.998881i	$-0.515060\pi$
$853$	−2.76253	−0.0945871	−0.0472936	+	0.998881i	$0.515060\pi$
$854$	12.0396	0.411988
$855$	0.427666	0.0146259
$856$	−7.20210	−0.246163
$857$	24.7242	0.844562	0.422281	−	0.906465i	$-0.361230\pi$
$857$	24.7242	0.844562	0.422281	+	0.906465i	$0.361230\pi$
$858$	−1.87177	−0.0639013
$859$	38.2591	1.30538	0.652692	−	0.757623i	$-0.273640\pi$
$859$	38.2591	1.30538	0.652692	+	0.757623i	$0.273640\pi$
$860$	0.555767	0.0189515
$861$	−8.48788	−0.289266
$862$	−2.44035	−0.0831188
$863$	17.0677	0.580992	0.290496	−	0.956876i	$-0.406180\pi$
$863$	17.0677	0.580992	0.290496	+	0.956876i	$0.406180\pi$
$864$	1.00000	0.0340207
$865$	−1.50399	−0.0511373
$866$	−37.4176	−1.27150
$867$	−5.18711	−0.176164
$868$	9.15504	0.310742
$869$	66.9511	2.27116
$870$	−1.80434	−0.0611728
$871$	0.675283	0.0228811
$872$	−11.7162	−0.396760
$873$	−2.09682	−0.0709666
$874$	−4.41251	−0.149256
$875$	−6.57584	−0.222304
$876$	−0.430955	−0.0145606
$877$	−54.0592	−1.82545	−0.912724	−	0.408576i	$-0.866025\pi$
$877$	−54.0592	−1.82545	−0.912724	+	0.408576i	$0.866025\pi$
$878$	27.5258	0.928952
$879$	−15.7437	−0.531023
$880$	−2.06031	−0.0694529
$881$	44.2094	1.48945	0.744725	−	0.667371i	$-0.232580\pi$
$881$	44.2094	1.48945	0.744725	+	0.667371i	$0.232580\pi$
$882$	4.54683	0.153100
$883$	6.72692	0.226379	0.113189	−	0.993573i	$-0.463893\pi$
$883$	6.72692	0.226379	0.113189	+	0.993573i	$0.463893\pi$
$884$	−1.83011	−0.0615531
$885$	0.334699	0.0112508
$886$	30.1858	1.01411
$887$	−36.5800	−1.22824	−0.614118	−	0.789214i	$-0.710488\pi$
$887$	−36.5800	−1.22824	−0.614118	+	0.789214i	$0.710488\pi$
$888$	4.83968	0.162409
$889$	−25.1528	−0.843599
$890$	2.75803	0.0924493
$891$	4.81756	0.161394
$892$	19.7809	0.662314
$893$	12.1622	0.406994
$894$	−3.28066	−0.109722
$895$	8.07620	0.269958
$896$	1.56626	0.0523250
$897$	−1.71440	−0.0572421
$898$	−28.8307	−0.962093
$899$	−24.6610	−0.822490
$900$	−4.81710	−0.160570
$901$	−4.71032	−0.156924
$902$	26.1073	0.869280
$903$	−2.03541	−0.0677342
$904$	14.8757	0.494759
$905$	−5.00259	−0.166292
$906$	−16.3697	−0.543846
$907$	−9.73639	−0.323291	−0.161646	−	0.986849i	$-0.551680\pi$
$907$	−9.73639	−0.323291	−0.161646	+	0.986849i	$0.551680\pi$
$908$	4.85901	0.161252
$909$	10.5144	0.348742
$910$	0.260252	0.00862727
$911$	24.3211	0.805793	0.402897	−	0.915245i	$-0.368003\pi$
$911$	24.3211	0.805793	0.402897	+	0.915245i	$0.368003\pi$
$912$	1.00000	0.0331133
$913$	−31.0677	−1.02819
$914$	29.7394	0.983691
$915$	3.28741	0.108679
$916$	−20.1052	−0.664294
$917$	5.01848	0.165725
$918$	4.71032	0.155464
$919$	50.7554	1.67427	0.837134	−	0.546998i	$-0.184230\pi$
$919$	50.7554	1.67427	0.837134	+	0.546998i	$0.184230\pi$
$920$	−1.88708	−0.0622152
$921$	−6.20366	−0.204418
$922$	20.3174	0.669118
$923$	−2.20135	−0.0724583
$924$	7.54555	0.248230
$925$	−23.3132	−0.766534
$926$	33.1623	1.08978
$927$	−13.3352	−0.437985
$928$	−4.21904	−0.138497
$929$	−21.4279	−0.703028	−0.351514	−	0.936183i	$-0.614333\pi$
$929$	−21.4279	−0.703028	−0.351514	+	0.936183i	$0.614333\pi$
$930$	2.49978	0.0819710
$931$	4.54683	0.149016
$932$	2.78758	0.0913101
$933$	−13.6845	−0.448011
$934$	9.10992	0.298086
$935$	−9.70470	−0.317378
$936$	0.388531	0.0126995
$937$	−54.9331	−1.79459	−0.897293	−	0.441436i	$-0.854469\pi$
$937$	−54.9331	−1.79459	−0.897293	+	0.441436i	$0.854469\pi$
$938$	−2.72222	−0.0888836
$939$	32.5278	1.06150
$940$	5.20137	0.169650
$941$	35.7889	1.16669	0.583343	−	0.812226i	$-0.301744\pi$
$941$	35.7889	1.16669	0.583343	+	0.812226i	$0.301744\pi$
$942$	0.158093	0.00515096
$943$	23.9123	0.778692
$944$	0.782618	0.0254720
$945$	−0.669836	−0.0217898
$946$	6.26059	0.203549
$947$	−19.6900	−0.639838	−0.319919	−	0.947445i	$-0.603656\pi$
$947$	−19.6900	−0.639838	−0.319919	+	0.947445i	$0.603656\pi$
$948$	−13.8973	−0.451364
$949$	−0.167440	−0.00543532
$950$	−4.81710	−0.156287
$951$	−19.9115	−0.645675
$952$	7.37758	0.239109
$953$	−31.6404	−1.02493	−0.512467	−	0.858707i	$-0.671268\pi$
$953$	−31.6404	−1.02493	−0.512467	+	0.858707i	$0.671268\pi$
$954$	1.00000	0.0323762
$955$	5.03188	0.162828
$956$	17.0415	0.551162
$957$	−20.3255	−0.657029
$958$	33.7857	1.09157
$959$	25.9408	0.837672
$960$	0.427666	0.0138029
$961$	3.16597	0.102128
$962$	1.88037	0.0606254
$963$	7.20210	0.232084
$964$	9.85330	0.317353
$965$	−2.37799	−0.0765501
$966$	6.91114	0.222362
$967$	39.8795	1.28244	0.641220	−	0.767357i	$-0.278429\pi$
$967$	39.8795	1.28244	0.641220	+	0.767357i	$0.278429\pi$
$968$	−12.2089	−0.392408
$969$	4.71032	0.151317
$970$	−0.896738	−0.0287925
$971$	16.7511	0.537567	0.268784	−	0.963201i	$-0.413378\pi$
$971$	16.7511	0.537567	0.268784	+	0.963201i	$0.413378\pi$
$972$	−1.00000	−0.0320750
$973$	−9.52910	−0.305489
$974$	−25.3375	−0.811865
$975$	−1.87159	−0.0599390
$976$	7.68687	0.246051
$977$	22.1244	0.707823	0.353912	−	0.935279i	$-0.384851\pi$
$977$	22.1244	0.707823	0.353912	+	0.935279i	$0.384851\pi$
$978$	22.9015	0.732310
$979$	31.0686	0.992956
$980$	1.94453	0.0621156
$981$	11.7162	0.374069
$982$	24.0912	0.768781
$983$	−12.4986	−0.398645	−0.199322	−	0.979934i	$-0.563874\pi$
$983$	−12.4986	−0.398645	−0.199322	+	0.979934i	$0.563874\pi$
$984$	−5.41921	−0.172758
$985$	8.10392	0.258212
$986$	−19.8730	−0.632886
$987$	−19.0492	−0.606343
$988$	0.388531	0.0123608
$989$	5.73422	0.182337
$990$	2.06031	0.0654808
$991$	−30.1720	−0.958444	−0.479222	−	0.877694i	$-0.659081\pi$
$991$	−30.1720	−0.958444	−0.479222	+	0.877694i	$0.659081\pi$
$992$	5.84517	0.185584
$993$	8.28706	0.262982
$994$	8.87415	0.281471
$995$	10.8083	0.342645
$996$	6.44884	0.204339
$997$	12.9835	0.411192	0.205596	−	0.978637i	$-0.434087\pi$
$997$	12.9835	0.411192	0.205596	+	0.978637i	$0.434087\pi$
$998$	23.7957	0.753240
$999$	−4.83968	−0.153121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.be.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.be.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.427666 q^{5} +1.00000 q^{6} -1.56626 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.427666 q^{5} +1.00000 q^{6} -1.56626 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.427666 q^{10} +4.81756 q^{11} -1.00000 q^{12} -0.388531 q^{13} +1.56626 q^{14} +0.427666 q^{15} +1.00000 q^{16} +4.71032 q^{17} -1.00000 q^{18} -1.00000 q^{19} -0.427666 q^{20} +1.56626 q^{21} -4.81756 q^{22} -4.41251 q^{23} +1.00000 q^{24} -4.81710 q^{25} +0.388531 q^{26} -1.00000 q^{27} -1.56626 q^{28} +4.21904 q^{29} -0.427666 q^{30} -5.84517 q^{31} -1.00000 q^{32} -4.81756 q^{33} -4.71032 q^{34} +0.669836 q^{35} +1.00000 q^{36} +4.83968 q^{37} +1.00000 q^{38} +0.388531 q^{39} +0.427666 q^{40} -5.41921 q^{41} -1.56626 q^{42} -1.29954 q^{43} +4.81756 q^{44} -0.427666 q^{45} +4.41251 q^{46} -12.1622 q^{47} -1.00000 q^{48} -4.54683 q^{49} +4.81710 q^{50} -4.71032 q^{51} -0.388531 q^{52} -1.00000 q^{53} +1.00000 q^{54} -2.06031 q^{55} +1.56626 q^{56} +1.00000 q^{57} -4.21904 q^{58} +0.782618 q^{59} +0.427666 q^{60} +7.68687 q^{61} +5.84517 q^{62} -1.56626 q^{63} +1.00000 q^{64} +0.166162 q^{65} +4.81756 q^{66} -1.73804 q^{67} +4.71032 q^{68} +4.41251 q^{69} -0.669836 q^{70} +5.66582 q^{71} -1.00000 q^{72} +0.430955 q^{73} -4.83968 q^{74} +4.81710 q^{75} -1.00000 q^{76} -7.54555 q^{77} -0.388531 q^{78} +13.8973 q^{79} -0.427666 q^{80} +1.00000 q^{81} +5.41921 q^{82} -6.44884 q^{83} +1.56626 q^{84} -2.01444 q^{85} +1.29954 q^{86} -4.21904 q^{87} -4.81756 q^{88} +6.44903 q^{89} +0.427666 q^{90} +0.608541 q^{91} -4.41251 q^{92} +5.84517 q^{93} +12.1622 q^{94} +0.427666 q^{95} +1.00000 q^{96} -2.09682 q^{97} +4.54683 q^{98} +4.81756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 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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