LMFDB - Embedded newform 4010.2.a.k.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 7 x^{14} - 7 x^{13} + 133 x^{12} - 99 x^{11} - 941 x^{10} + 1290 x^{9} + 3031 x^{8} + \cdots - 512 $ x^15 - 7x^14 - 7x^13 + 133x^12 - 99x^11 - 941x^10 + 1290x^9 + 3031x^8 - 5452x^7 - 4098x^6 + 9986x^5 + 850x^4 - 7216x^3 + 1688x^2 + 1441x - 512
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.25248$ of defining polynomial
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.71977 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.71977 q^{6} -0.594829 q^{7} -1.00000 q^{8} +4.39715 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.71977 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.71977 q^{6} -0.594829 q^{7} -1.00000 q^{8} +4.39715 q^{9} +1.00000 q^{10} -5.15188 q^{11} +2.71977 q^{12} +0.430765 q^{13} +0.594829 q^{14} -2.71977 q^{15} +1.00000 q^{16} +3.55242 q^{17} -4.39715 q^{18} -5.11480 q^{19} -1.00000 q^{20} -1.61780 q^{21} +5.15188 q^{22} +6.74147 q^{23} -2.71977 q^{24} +1.00000 q^{25} -0.430765 q^{26} +3.79991 q^{27} -0.594829 q^{28} -7.79165 q^{29} +2.71977 q^{30} -8.50979 q^{31} -1.00000 q^{32} -14.0119 q^{33} -3.55242 q^{34} +0.594829 q^{35} +4.39715 q^{36} -0.601851 q^{37} +5.11480 q^{38} +1.17158 q^{39} +1.00000 q^{40} +4.94186 q^{41} +1.61780 q^{42} +2.44358 q^{43} -5.15188 q^{44} -4.39715 q^{45} -6.74147 q^{46} -2.58807 q^{47} +2.71977 q^{48} -6.64618 q^{49} -1.00000 q^{50} +9.66177 q^{51} +0.430765 q^{52} +3.18711 q^{53} -3.79991 q^{54} +5.15188 q^{55} +0.594829 q^{56} -13.9111 q^{57} +7.79165 q^{58} -14.3834 q^{59} -2.71977 q^{60} -2.26604 q^{61} +8.50979 q^{62} -2.61555 q^{63} +1.00000 q^{64} -0.430765 q^{65} +14.0119 q^{66} +2.61956 q^{67} +3.55242 q^{68} +18.3352 q^{69} -0.594829 q^{70} +6.99728 q^{71} -4.39715 q^{72} +5.20733 q^{73} +0.601851 q^{74} +2.71977 q^{75} -5.11480 q^{76} +3.06449 q^{77} -1.17158 q^{78} -17.2296 q^{79} -1.00000 q^{80} -2.85655 q^{81} -4.94186 q^{82} -6.99067 q^{83} -1.61780 q^{84} -3.55242 q^{85} -2.44358 q^{86} -21.1915 q^{87} +5.15188 q^{88} -12.7136 q^{89} +4.39715 q^{90} -0.256231 q^{91} +6.74147 q^{92} -23.1447 q^{93} +2.58807 q^{94} +5.11480 q^{95} -2.71977 q^{96} -1.67993 q^{97} +6.64618 q^{98} -22.6536 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{10} - 2 q^{11} - 6 q^{12} - 13 q^{13} + 5 q^{14} + 6 q^{15} + 15 q^{16} + 11 q^{17} - 19 q^{18} - 15 q^{19} - 15 q^{20} - 2 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 15 q^{25} + 13 q^{26} - 12 q^{27} - 5 q^{28} + 28 q^{29} - 6 q^{30} - 12 q^{31} - 15 q^{32} - 22 q^{33} - 11 q^{34} + 5 q^{35} + 19 q^{36} - 23 q^{37} + 15 q^{38} - 2 q^{39} + 15 q^{40} + 24 q^{41} + 2 q^{42} - 24 q^{43} - 2 q^{44} - 19 q^{45} + 3 q^{46} - 3 q^{47} - 6 q^{48} + 20 q^{49} - 15 q^{50} - 5 q^{51} - 13 q^{52} + 10 q^{53} + 12 q^{54} + 2 q^{55} + 5 q^{56} - 11 q^{57} - 28 q^{58} + 2 q^{59} + 6 q^{60} + 15 q^{61} + 12 q^{62} - 2 q^{63} + 15 q^{64} + 13 q^{65} + 22 q^{66} - 48 q^{67} + 11 q^{68} + 21 q^{69} - 5 q^{70} + 15 q^{71} - 19 q^{72} - 47 q^{73} + 23 q^{74} - 6 q^{75} - 15 q^{76} + 7 q^{77} + 2 q^{78} - 34 q^{79} - 15 q^{80} + 43 q^{81} - 24 q^{82} - 32 q^{83} - 2 q^{84} - 11 q^{85} + 24 q^{86} + 14 q^{87} + 2 q^{88} + 25 q^{89} + 19 q^{90} - 32 q^{91} - 3 q^{92} - 42 q^{93} + 3 q^{94} + 15 q^{95} + 6 q^{96} - 34 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^10 - 2 * q^11 - 6 * q^12 - 13 * q^13 + 5 * q^14 + 6 * q^15 + 15 * q^16 + 11 * q^17 - 19 * q^18 - 15 * q^19 - 15 * q^20 - 2 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 15 * q^25 + 13 * q^26 - 12 * q^27 - 5 * q^28 + 28 * q^29 - 6 * q^30 - 12 * q^31 - 15 * q^32 - 22 * q^33 - 11 * q^34 + 5 * q^35 + 19 * q^36 - 23 * q^37 + 15 * q^38 - 2 * q^39 + 15 * q^40 + 24 * q^41 + 2 * q^42 - 24 * q^43 - 2 * q^44 - 19 * q^45 + 3 * q^46 - 3 * q^47 - 6 * q^48 + 20 * q^49 - 15 * q^50 - 5 * q^51 - 13 * q^52 + 10 * q^53 + 12 * q^54 + 2 * q^55 + 5 * q^56 - 11 * q^57 - 28 * q^58 + 2 * q^59 + 6 * q^60 + 15 * q^61 + 12 * q^62 - 2 * q^63 + 15 * q^64 + 13 * q^65 + 22 * q^66 - 48 * q^67 + 11 * q^68 + 21 * q^69 - 5 * q^70 + 15 * q^71 - 19 * q^72 - 47 * q^73 + 23 * q^74 - 6 * q^75 - 15 * q^76 + 7 * q^77 + 2 * q^78 - 34 * q^79 - 15 * q^80 + 43 * q^81 - 24 * q^82 - 32 * q^83 - 2 * q^84 - 11 * q^85 + 24 * q^86 + 14 * q^87 + 2 * q^88 + 25 * q^89 + 19 * q^90 - 32 * q^91 - 3 * q^92 - 42 * q^93 + 3 * q^94 + 15 * q^95 + 6 * q^96 - 34 * q^97 - 20 * q^98 + 4 * 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
$3$	2.71977	1.57026	0.785130	−	0.619331i	$-0.212596\pi$
$3$	2.71977	1.57026	0.785130	+	0.619331i	$0.212596\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.71977	−1.11034
$7$	−0.594829	−0.224824	−0.112412	−	0.993662i	$-0.535858\pi$
$7$	−0.594829	−0.224824	−0.112412	+	0.993662i	$0.535858\pi$
$8$	−1.00000	−0.353553
$9$	4.39715	1.46572
$10$	1.00000	0.316228
$11$	−5.15188	−1.55335	−0.776675	−	0.629902i	$-0.783095\pi$
$11$	−5.15188	−1.55335	−0.776675	+	0.629902i	$0.783095\pi$
$12$	2.71977	0.785130
$13$	0.430765	0.119473	0.0597363	−	0.998214i	$-0.480974\pi$
$13$	0.430765	0.119473	0.0597363	+	0.998214i	$0.480974\pi$
$14$	0.594829	0.158975
$15$	−2.71977	−0.702241
$16$	1.00000	0.250000
$17$	3.55242	0.861589	0.430795	−	0.902450i	$-0.358233\pi$
$17$	3.55242	0.861589	0.430795	+	0.902450i	$0.358233\pi$
$18$	−4.39715	−1.03642
$19$	−5.11480	−1.17342	−0.586708	−	0.809798i	$-0.699576\pi$
$19$	−5.11480	−1.17342	−0.586708	+	0.809798i	$0.699576\pi$
$20$	−1.00000	−0.223607
$21$	−1.61780	−0.353033
$22$	5.15188	1.09838
$23$	6.74147	1.40569	0.702847	−	0.711341i	$-0.251912\pi$
$23$	6.74147	1.40569	0.702847	+	0.711341i	$0.251912\pi$
$24$	−2.71977	−0.555171
$25$	1.00000	0.200000
$26$	−0.430765	−0.0844799
$27$	3.79991	0.731294
$28$	−0.594829	−0.112412
$29$	−7.79165	−1.44687	−0.723436	−	0.690391i	$-0.757439\pi$
$29$	−7.79165	−1.44687	−0.723436	+	0.690391i	$0.757439\pi$
$30$	2.71977	0.496560
$31$	−8.50979	−1.52840	−0.764202	−	0.644977i	$-0.776867\pi$
$31$	−8.50979	−1.52840	−0.764202	+	0.644977i	$0.776867\pi$
$32$	−1.00000	−0.176777
$33$	−14.0119	−2.43916
$34$	−3.55242	−0.609236
$35$	0.594829	0.100544
$36$	4.39715	0.732858
$37$	−0.601851	−0.0989437	−0.0494719	−	0.998776i	$-0.515754\pi$
$37$	−0.601851	−0.0989437	−0.0494719	+	0.998776i	$0.515754\pi$
$38$	5.11480	0.829731
$39$	1.17158	0.187603
$40$	1.00000	0.158114
$41$	4.94186	0.771789	0.385895	−	0.922543i	$-0.373893\pi$
$41$	4.94186	0.771789	0.385895	+	0.922543i	$0.373893\pi$
$42$	1.61780	0.249632
$43$	2.44358	0.372642	0.186321	−	0.982489i	$-0.440344\pi$
$43$	2.44358	0.372642	0.186321	+	0.982489i	$0.440344\pi$
$44$	−5.15188	−0.776675
$45$	−4.39715	−0.655488
$46$	−6.74147	−0.993976
$47$	−2.58807	−0.377509	−0.188755	−	0.982024i	$-0.560445\pi$
$47$	−2.58807	−0.377509	−0.188755	+	0.982024i	$0.560445\pi$
$48$	2.71977	0.392565
$49$	−6.64618	−0.949454
$50$	−1.00000	−0.141421
$51$	9.66177	1.35292
$52$	0.430765	0.0597363
$53$	3.18711	0.437783	0.218891	−	0.975749i	$-0.429756\pi$
$53$	3.18711	0.437783	0.218891	+	0.975749i	$0.429756\pi$
$54$	−3.79991	−0.517103
$55$	5.15188	0.694679
$56$	0.594829	0.0794874
$57$	−13.9111	−1.84257
$58$	7.79165	1.02309
$59$	−14.3834	−1.87255	−0.936277	−	0.351261i	$-0.885753\pi$
$59$	−14.3834	−1.87255	−0.936277	+	0.351261i	$0.885753\pi$
$60$	−2.71977	−0.351121
$61$	−2.26604	−0.290137	−0.145068	−	0.989422i	$-0.546340\pi$
$61$	−2.26604	−0.290137	−0.145068	+	0.989422i	$0.546340\pi$
$62$	8.50979	1.08074
$63$	−2.61555	−0.329528
$64$	1.00000	0.125000
$65$	−0.430765	−0.0534298
$66$	14.0119	1.72475
$67$	2.61956	0.320031	0.160015	−	0.987115i	$-0.448846\pi$
$67$	2.61956	0.320031	0.160015	+	0.987115i	$0.448846\pi$
$68$	3.55242	0.430795
$69$	18.3352	2.20730
$70$	−0.594829	−0.0710957
$71$	6.99728	0.830424	0.415212	−	0.909725i	$-0.363707\pi$
$71$	6.99728	0.830424	0.415212	+	0.909725i	$0.363707\pi$
$72$	−4.39715	−0.518209
$73$	5.20733	0.609472	0.304736	−	0.952437i	$-0.401432\pi$
$73$	5.20733	0.609472	0.304736	+	0.952437i	$0.401432\pi$
$74$	0.601851	0.0699638
$75$	2.71977	0.314052
$76$	−5.11480	−0.586708
$77$	3.06449	0.349231
$78$	−1.17158	−0.132655
$79$	−17.2296	−1.93848	−0.969242	−	0.246112i	$-0.920847\pi$
$79$	−17.2296	−1.93848	−0.969242	+	0.246112i	$0.920847\pi$
$80$	−1.00000	−0.111803
$81$	−2.85655	−0.317394
$82$	−4.94186	−0.545738
$83$	−6.99067	−0.767325	−0.383663	−	0.923473i	$-0.625337\pi$
$83$	−6.99067	−0.767325	−0.383663	+	0.923473i	$0.625337\pi$
$84$	−1.61780	−0.176516
$85$	−3.55242	−0.385314
$86$	−2.44358	−0.263498
$87$	−21.1915	−2.27197
$88$	5.15188	0.549192
$89$	−12.7136	−1.34764	−0.673822	−	0.738894i	$-0.735349\pi$
$89$	−12.7136	−1.34764	−0.673822	+	0.738894i	$0.735349\pi$
$90$	4.39715	0.463500
$91$	−0.256231	−0.0268603
$92$	6.74147	0.702847
$93$	−23.1447	−2.39999
$94$	2.58807	0.266939
$95$	5.11480	0.524768
$96$	−2.71977	−0.277585
$97$	−1.67993	−0.170571	−0.0852857	−	0.996357i	$-0.527180\pi$
$97$	−1.67993	−0.170571	−0.0852857	+	0.996357i	$0.527180\pi$
$98$	6.64618	0.671365
$99$	−22.6536	−2.27677
$100$	1.00000	0.100000
$101$	12.8386	1.27749	0.638743	−	0.769420i	$-0.279455\pi$
$101$	12.8386	1.27749	0.638743	+	0.769420i	$0.279455\pi$
$102$	−9.66177	−0.956658
$103$	3.27639	0.322833	0.161416	−	0.986886i	$-0.448394\pi$
$103$	3.27639	0.322833	0.161416	+	0.986886i	$0.448394\pi$
$104$	−0.430765	−0.0422399
$105$	1.61780	0.157881
$106$	−3.18711	−0.309559
$107$	1.95485	0.188983	0.0944915	−	0.995526i	$-0.469877\pi$
$107$	1.95485	0.188983	0.0944915	+	0.995526i	$0.469877\pi$
$108$	3.79991	0.365647
$109$	0.904117	0.0865987	0.0432993	−	0.999062i	$-0.486213\pi$
$109$	0.904117	0.0865987	0.0432993	+	0.999062i	$0.486213\pi$
$110$	−5.15188	−0.491212
$111$	−1.63690	−0.155367
$112$	−0.594829	−0.0562061
$113$	−12.5979	−1.18511	−0.592555	−	0.805530i	$-0.701881\pi$
$113$	−12.5979	−1.18511	−0.592555	+	0.805530i	$0.701881\pi$
$114$	13.9111	1.30289
$115$	−6.74147	−0.628646
$116$	−7.79165	−0.723436
$117$	1.89413	0.175113
$118$	14.3834	1.32410
$119$	−2.11309	−0.193706
$120$	2.71977	0.248280
$121$	15.5418	1.41289
$122$	2.26604	0.205158
$123$	13.4407	1.21191
$124$	−8.50979	−0.764202
$125$	−1.00000	−0.0894427
$126$	2.61555	0.233012
$127$	−15.4067	−1.36712	−0.683560	−	0.729895i	$-0.739569\pi$
$127$	−15.4067	−1.36712	−0.683560	+	0.729895i	$0.739569\pi$
$128$	−1.00000	−0.0883883
$129$	6.64597	0.585145
$130$	0.430765	0.0377806
$131$	4.16970	0.364308	0.182154	−	0.983270i	$-0.441693\pi$
$131$	4.16970	0.364308	0.182154	+	0.983270i	$0.441693\pi$
$132$	−14.0119	−1.21958
$133$	3.04243	0.263813
$134$	−2.61956	−0.226296
$135$	−3.79991	−0.327045
$136$	−3.55242	−0.304618
$137$	15.4928	1.32364	0.661819	−	0.749664i	$-0.269785\pi$
$137$	15.4928	1.32364	0.661819	+	0.749664i	$0.269785\pi$
$138$	−18.3352	−1.56080
$139$	−6.32300	−0.536309	−0.268155	−	0.963376i	$-0.586414\pi$
$139$	−6.32300	−0.536309	−0.268155	+	0.963376i	$0.586414\pi$
$140$	0.594829	0.0502722
$141$	−7.03896	−0.592787
$142$	−6.99728	−0.587199
$143$	−2.21925	−0.185583
$144$	4.39715	0.366429
$145$	7.79165	0.647061
$146$	−5.20733	−0.430962
$147$	−18.0761	−1.49089
$148$	−0.601851	−0.0494719
$149$	21.4347	1.75600	0.877998	−	0.478665i	$-0.158879\pi$
$149$	21.4347	1.75600	0.877998	+	0.478665i	$0.158879\pi$
$150$	−2.71977	−0.222068
$151$	−1.44587	−0.117663	−0.0588316	−	0.998268i	$-0.518738\pi$
$151$	−1.44587	−0.117663	−0.0588316	+	0.998268i	$0.518738\pi$
$152$	5.11480	0.414865
$153$	15.6205	1.26284
$154$	−3.06449	−0.246943
$155$	8.50979	0.683523
$156$	1.17158	0.0938015
$157$	−17.1502	−1.36873	−0.684367	−	0.729138i	$-0.739921\pi$
$157$	−17.1502	−1.36873	−0.684367	+	0.729138i	$0.739921\pi$
$158$	17.2296	1.37071
$159$	8.66819	0.687432
$160$	1.00000	0.0790569
$161$	−4.01002	−0.316034
$162$	2.85655	0.224432
$163$	−12.1007	−0.947800	−0.473900	−	0.880579i	$-0.657154\pi$
$163$	−12.1007	−0.947800	−0.473900	+	0.880579i	$0.657154\pi$
$164$	4.94186	0.385895
$165$	14.0119	1.09083
$166$	6.99067	0.542581
$167$	−20.9442	−1.62071	−0.810357	−	0.585936i	$-0.800727\pi$
$167$	−20.9442	−1.62071	−0.810357	+	0.585936i	$0.800727\pi$
$168$	1.61780	0.124816
$169$	−12.8144	−0.985726
$170$	3.55242	0.272458
$171$	−22.4905	−1.71989
$172$	2.44358	0.186321
$173$	4.62138	0.351357	0.175678	−	0.984448i	$-0.443788\pi$
$173$	4.62138	0.351357	0.175678	+	0.984448i	$0.443788\pi$
$174$	21.1915	1.60652
$175$	−0.594829	−0.0449649
$176$	−5.15188	−0.388337
$177$	−39.1194	−2.94040
$178$	12.7136	0.952928
$179$	−13.5752	−1.01465	−0.507327	−	0.861753i	$-0.669366\pi$
$179$	−13.5752	−1.01465	−0.507327	+	0.861753i	$0.669366\pi$
$180$	−4.39715	−0.327744
$181$	1.90393	0.141518	0.0707589	−	0.997493i	$-0.477458\pi$
$181$	1.90393	0.141518	0.0707589	+	0.997493i	$0.477458\pi$
$182$	0.256231	0.0189931
$183$	−6.16311	−0.455590
$184$	−6.74147	−0.496988
$185$	0.601851	0.0442490
$186$	23.1447	1.69705
$187$	−18.3016	−1.33835
$188$	−2.58807	−0.188755
$189$	−2.26030	−0.164413
$190$	−5.11480	−0.371067
$191$	−11.4968	−0.831877	−0.415939	−	0.909393i	$-0.636547\pi$
$191$	−11.4968	−0.831877	−0.415939	+	0.909393i	$0.636547\pi$
$192$	2.71977	0.196282
$193$	−21.6763	−1.56029	−0.780147	−	0.625596i	$-0.784856\pi$
$193$	−21.6763	−1.56029	−0.780147	+	0.625596i	$0.784856\pi$
$194$	1.67993	0.120612
$195$	−1.17158	−0.0838986
$196$	−6.64618	−0.474727
$197$	−9.16597	−0.653048	−0.326524	−	0.945189i	$-0.605877\pi$
$197$	−9.16597	−0.653048	−0.326524	+	0.945189i	$0.605877\pi$
$198$	22.6536	1.60992
$199$	−20.8817	−1.48027	−0.740133	−	0.672461i	$-0.765237\pi$
$199$	−20.8817	−1.48027	−0.740133	+	0.672461i	$0.765237\pi$
$200$	−1.00000	−0.0707107
$201$	7.12461	0.502531
$202$	−12.8386	−0.903319
$203$	4.63470	0.325292
$204$	9.66177	0.676459
$205$	−4.94186	−0.345155
$206$	−3.27639	−0.228277
$207$	29.6432	2.06035
$208$	0.430765	0.0298682
$209$	26.3508	1.82273
$210$	−1.61780	−0.111639
$211$	19.4256	1.33731	0.668657	−	0.743571i	$-0.266869\pi$
$211$	19.4256	1.33731	0.668657	+	0.743571i	$0.266869\pi$
$212$	3.18711	0.218891
$213$	19.0310	1.30398
$214$	−1.95485	−0.133631
$215$	−2.44358	−0.166651
$216$	−3.79991	−0.258551
$217$	5.06187	0.343622
$218$	−0.904117	−0.0612345
$219$	14.1627	0.957029
$220$	5.15188	0.347339
$221$	1.53026	0.102936
$222$	1.63690	0.109861
$223$	24.0478	1.61036	0.805181	−	0.593030i	$-0.202068\pi$
$223$	24.0478	1.61036	0.805181	+	0.593030i	$0.202068\pi$
$224$	0.594829	0.0397437
$225$	4.39715	0.293143
$226$	12.5979	0.837999
$227$	4.82553	0.320281	0.160141	−	0.987094i	$-0.448805\pi$
$227$	4.82553	0.320281	0.160141	+	0.987094i	$0.448805\pi$
$228$	−13.9111	−0.921284
$229$	11.8566	0.783508	0.391754	−	0.920070i	$-0.371868\pi$
$229$	11.8566	0.783508	0.391754	+	0.920070i	$0.371868\pi$
$230$	6.74147	0.444520
$231$	8.33470	0.548383
$232$	7.79165	0.511547
$233$	23.3533	1.52992	0.764961	−	0.644076i	$-0.222758\pi$
$233$	23.3533	1.52992	0.764961	+	0.644076i	$0.222758\pi$
$234$	−1.89413	−0.123823
$235$	2.58807	0.168827
$236$	−14.3834	−0.936277
$237$	−46.8606	−3.04392
$238$	2.11309	0.136971
$239$	17.3614	1.12301	0.561506	−	0.827473i	$-0.310222\pi$
$239$	17.3614	1.12301	0.561506	+	0.827473i	$0.310222\pi$
$240$	−2.71977	−0.175560
$241$	10.5726	0.681039	0.340519	−	0.940237i	$-0.389397\pi$
$241$	10.5726	0.681039	0.340519	+	0.940237i	$0.389397\pi$
$242$	−15.5418	−0.999067
$243$	−19.1689	−1.22968
$244$	−2.26604	−0.145068
$245$	6.64618	0.424609
$246$	−13.4407	−0.856950
$247$	−2.20328	−0.140191
$248$	8.50979	0.540372
$249$	−19.0130	−1.20490
$250$	1.00000	0.0632456
$251$	−11.0779	−0.699232	−0.349616	−	0.936893i	$-0.613688\pi$
$251$	−11.0779	−0.699232	−0.349616	+	0.936893i	$0.613688\pi$
$252$	−2.61555	−0.164764
$253$	−34.7312	−2.18353
$254$	15.4067	0.966699
$255$	−9.66177	−0.605044
$256$	1.00000	0.0625000
$257$	23.2093	1.44775	0.723877	−	0.689929i	$-0.242358\pi$
$257$	23.2093	1.44775	0.723877	+	0.689929i	$0.242358\pi$
$258$	−6.64597	−0.413760
$259$	0.357999	0.0222450
$260$	−0.430765	−0.0267149
$261$	−34.2610	−2.12070
$262$	−4.16970	−0.257605
$263$	21.4733	1.32410	0.662049	−	0.749461i	$-0.269687\pi$
$263$	21.4733	1.32410	0.662049	+	0.749461i	$0.269687\pi$
$264$	14.0119	0.862374
$265$	−3.18711	−0.195782
$266$	−3.04243	−0.186544
$267$	−34.5782	−2.11615
$268$	2.61956	0.160015
$269$	13.3091	0.811469	0.405735	−	0.913991i	$-0.367016\pi$
$269$	13.3091	0.811469	0.405735	+	0.913991i	$0.367016\pi$
$270$	3.79991	0.231255
$271$	−27.6321	−1.67853	−0.839266	−	0.543721i	$-0.817015\pi$
$271$	−27.6321	−1.67853	−0.839266	+	0.543721i	$0.817015\pi$
$272$	3.55242	0.215397
$273$	−0.696890	−0.0421777
$274$	−15.4928	−0.935954
$275$	−5.15188	−0.310670
$276$	18.3352	1.10365
$277$	11.5551	0.694278	0.347139	−	0.937814i	$-0.387153\pi$
$277$	11.5551	0.694278	0.347139	+	0.937814i	$0.387153\pi$
$278$	6.32300	0.379228
$279$	−37.4188	−2.24021
$280$	−0.594829	−0.0355478
$281$	−12.2034	−0.727991	−0.363996	−	0.931401i	$-0.618588\pi$
$281$	−12.2034	−0.727991	−0.363996	+	0.931401i	$0.618588\pi$
$282$	7.03896	0.419164
$283$	−21.0410	−1.25076	−0.625378	−	0.780322i	$-0.715055\pi$
$283$	−21.0410	−1.25076	−0.625378	+	0.780322i	$0.715055\pi$
$284$	6.99728	0.415212
$285$	13.9111	0.824022
$286$	2.21925	0.131227
$287$	−2.93956	−0.173517
$288$	−4.39715	−0.259104
$289$	−4.38028	−0.257664
$290$	−7.79165	−0.457541
$291$	−4.56903	−0.267842
$292$	5.20733	0.304736
$293$	6.73977	0.393742	0.196871	−	0.980429i	$-0.436922\pi$
$293$	6.73977	0.393742	0.196871	+	0.980429i	$0.436922\pi$
$294$	18.0761	1.05422
$295$	14.3834	0.837432
$296$	0.601851	0.0349819
$297$	−19.5767	−1.13595
$298$	−21.4347	−1.24168
$299$	2.90399	0.167942
$300$	2.71977	0.157026
$301$	−1.45351	−0.0837790
$302$	1.44587	0.0832005
$303$	34.9180	2.00599
$304$	−5.11480	−0.293354
$305$	2.26604	0.129753
$306$	−15.6205	−0.892966
$307$	21.8952	1.24962	0.624811	−	0.780776i	$-0.285176\pi$
$307$	21.8952	1.24962	0.624811	+	0.780776i	$0.285176\pi$
$308$	3.06449	0.174615
$309$	8.91103	0.506931
$310$	−8.50979	−0.483324
$311$	19.9821	1.13308	0.566541	−	0.824033i	$-0.308281\pi$
$311$	19.9821	1.13308	0.566541	+	0.824033i	$0.308281\pi$
$312$	−1.17158	−0.0663277
$313$	6.20088	0.350494	0.175247	−	0.984524i	$-0.443928\pi$
$313$	6.20088	0.350494	0.175247	+	0.984524i	$0.443928\pi$
$314$	17.1502	0.967841
$315$	2.61555	0.147370
$316$	−17.2296	−0.969242
$317$	−10.6814	−0.599929	−0.299965	−	0.953950i	$-0.596975\pi$
$317$	−10.6814	−0.599929	−0.299965	+	0.953950i	$0.596975\pi$
$318$	−8.66819	−0.486088
$319$	40.1416	2.24750
$320$	−1.00000	−0.0559017
$321$	5.31675	0.296752
$322$	4.01002	0.223470
$323$	−18.1700	−1.01100
$324$	−2.85655	−0.158697
$325$	0.430765	0.0238945
$326$	12.1007	0.670196
$327$	2.45899	0.135982
$328$	−4.94186	−0.272869
$329$	1.53946	0.0848732
$330$	−14.0119	−0.771331
$331$	−13.2248	−0.726898	−0.363449	−	0.931614i	$-0.618401\pi$
$331$	−13.2248	−0.726898	−0.363449	+	0.931614i	$0.618401\pi$
$332$	−6.99067	−0.383663
$333$	−2.64643	−0.145023
$334$	20.9442	1.14602
$335$	−2.61956	−0.143122
$336$	−1.61780	−0.0882581
$337$	10.7103	0.583427	0.291713	−	0.956506i	$-0.405775\pi$
$337$	10.7103	0.583427	0.291713	+	0.956506i	$0.405775\pi$
$338$	12.8144	0.697014
$339$	−34.2633	−1.86093
$340$	−3.55242	−0.192657
$341$	43.8414	2.37415
$342$	22.4905	1.21615
$343$	8.11715	0.438285
$344$	−2.44358	−0.131749
$345$	−18.3352	−0.987137
$346$	−4.62138	−0.248447
$347$	26.7432	1.43565	0.717825	−	0.696223i	$-0.245138\pi$
$347$	26.7432	1.43565	0.717825	+	0.696223i	$0.245138\pi$
$348$	−21.1915	−1.13598
$349$	19.1024	1.02253	0.511264	−	0.859424i	$-0.329177\pi$
$349$	19.1024	1.02253	0.511264	+	0.859424i	$0.329177\pi$
$350$	0.594829	0.0317950
$351$	1.63687	0.0873696
$352$	5.15188	0.274596
$353$	−2.45986	−0.130925	−0.0654625	−	0.997855i	$-0.520852\pi$
$353$	−2.45986	−0.130925	−0.0654625	+	0.997855i	$0.520852\pi$
$354$	39.1194	2.07917
$355$	−6.99728	−0.371377
$356$	−12.7136	−0.673822
$357$	−5.74711	−0.304169
$358$	13.5752	0.717469
$359$	3.91065	0.206396	0.103198	−	0.994661i	$-0.467092\pi$
$359$	3.91065	0.206396	0.103198	+	0.994661i	$0.467092\pi$
$360$	4.39715	0.231750
$361$	7.16122	0.376906
$362$	−1.90393	−0.100068
$363$	42.2702	2.21861
$364$	−0.256231	−0.0134302
$365$	−5.20733	−0.272564
$366$	6.16311	0.322151
$367$	29.4789	1.53879	0.769393	−	0.638776i	$-0.220559\pi$
$367$	29.4789	1.53879	0.769393	+	0.638776i	$0.220559\pi$
$368$	6.74147	0.351424
$369$	21.7301	1.13122
$370$	−0.601851	−0.0312888
$371$	−1.89578	−0.0984242
$372$	−23.1447	−1.20000
$373$	25.0946	1.29935	0.649676	−	0.760211i	$-0.274905\pi$
$373$	25.0946	1.29935	0.649676	+	0.760211i	$0.274905\pi$
$374$	18.3016	0.946356
$375$	−2.71977	−0.140448
$376$	2.58807	0.133470
$377$	−3.35637	−0.172862
$378$	2.26030	0.116257
$379$	−7.83169	−0.402287	−0.201143	−	0.979562i	$-0.564466\pi$
$379$	−7.83169	−0.402287	−0.201143	+	0.979562i	$0.564466\pi$
$380$	5.11480	0.262384
$381$	−41.9025	−2.14673
$382$	11.4968	0.588226
$383$	10.0381	0.512925	0.256463	−	0.966554i	$-0.417443\pi$
$383$	10.0381	0.512925	0.256463	+	0.966554i	$0.417443\pi$
$384$	−2.71977	−0.138793
$385$	−3.06449	−0.156181
$386$	21.6763	1.10329
$387$	10.7448	0.546188
$388$	−1.67993	−0.0852857
$389$	−8.12206	−0.411805	−0.205902	−	0.978573i	$-0.566013\pi$
$389$	−8.12206	−0.411805	−0.205902	+	0.978573i	$0.566013\pi$
$390$	1.17158	0.0593253
$391$	23.9486	1.21113
$392$	6.64618	0.335683
$393$	11.3406	0.572058
$394$	9.16597	0.461775
$395$	17.2296	0.866916
$396$	−22.6536	−1.13838
$397$	−26.4713	−1.32856	−0.664279	−	0.747485i	$-0.731261\pi$
$397$	−26.4713	−1.32856	−0.664279	+	0.747485i	$0.731261\pi$
$398$	20.8817	1.04671
$399$	8.27472	0.414254
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−7.12461	−0.355343
$403$	−3.66572	−0.182602
$404$	12.8386	0.638743
$405$	2.85655	0.141943
$406$	−4.63470	−0.230016
$407$	3.10066	0.153694
$408$	−9.66177	−0.478329
$409$	−3.19920	−0.158190	−0.0790952	−	0.996867i	$-0.525203\pi$
$409$	−3.19920	−0.158190	−0.0790952	+	0.996867i	$0.525203\pi$
$410$	4.94186	0.244061
$411$	42.1368	2.07846
$412$	3.27639	0.161416
$413$	8.55565	0.420996
$414$	−29.6432	−1.45689
$415$	6.99067	0.343158
$416$	−0.430765	−0.0211200
$417$	−17.1971	−0.842145
$418$	−26.3508	−1.28886
$419$	9.80519	0.479015	0.239507	−	0.970895i	$-0.423014\pi$
$419$	9.80519	0.479015	0.239507	+	0.970895i	$0.423014\pi$
$420$	1.61780	0.0789405
$421$	−34.3451	−1.67388	−0.836939	−	0.547296i	$-0.815657\pi$
$421$	−34.3451	−1.67388	−0.836939	+	0.547296i	$0.815657\pi$
$422$	−19.4256	−0.945624
$423$	−11.3801	−0.553321
$424$	−3.18711	−0.154780
$425$	3.55242	0.172318
$426$	−19.0310	−0.922054
$427$	1.34791	0.0652298
$428$	1.95485	0.0944915
$429$	−6.03584	−0.291413
$430$	2.44358	0.117840
$431$	22.5844	1.08785	0.543927	−	0.839133i	$-0.316937\pi$
$431$	22.5844	1.08785	0.543927	+	0.839133i	$0.316937\pi$
$432$	3.79991	0.182823
$433$	−26.8669	−1.29114	−0.645570	−	0.763701i	$-0.723380\pi$
$433$	−26.8669	−1.29114	−0.645570	+	0.763701i	$0.723380\pi$
$434$	−5.06187	−0.242978
$435$	21.1915	1.01605
$436$	0.904117	0.0432993
$437$	−34.4813	−1.64946
$438$	−14.1627	−0.676722
$439$	2.51933	0.120241	0.0601205	−	0.998191i	$-0.480852\pi$
$439$	2.51933	0.120241	0.0601205	+	0.998191i	$0.480852\pi$
$440$	−5.15188	−0.245606
$441$	−29.2242	−1.39163
$442$	−1.53026	−0.0727870
$443$	−15.6162	−0.741950	−0.370975	−	0.928643i	$-0.620976\pi$
$443$	−15.6162	−0.741950	−0.370975	+	0.928643i	$0.620976\pi$
$444$	−1.63690	−0.0776837
$445$	12.7136	0.602685
$446$	−24.0478	−1.13870
$447$	58.2973	2.75737
$448$	−0.594829	−0.0281030
$449$	−3.09920	−0.146260	−0.0731301	−	0.997322i	$-0.523299\pi$
$449$	−3.09920	−0.146260	−0.0731301	+	0.997322i	$0.523299\pi$
$450$	−4.39715	−0.207283
$451$	−25.4599	−1.19886
$452$	−12.5979	−0.592555
$453$	−3.93243	−0.184762
$454$	−4.82553	−0.226473
$455$	0.256231	0.0120123
$456$	13.9111	0.651446
$457$	−2.80648	−0.131282	−0.0656409	−	0.997843i	$-0.520909\pi$
$457$	−2.80648	−0.131282	−0.0656409	+	0.997843i	$0.520909\pi$
$458$	−11.8566	−0.554024
$459$	13.4989	0.630075
$460$	−6.74147	−0.314323
$461$	10.1931	0.474742	0.237371	−	0.971419i	$-0.423714\pi$
$461$	10.1931	0.474742	0.237371	+	0.971419i	$0.423714\pi$
$462$	−8.33470	−0.387765
$463$	−1.35804	−0.0631136	−0.0315568	−	0.999502i	$-0.510047\pi$
$463$	−1.35804	−0.0631136	−0.0315568	+	0.999502i	$0.510047\pi$
$464$	−7.79165	−0.361718
$465$	23.1447	1.07331
$466$	−23.3533	−1.08182
$467$	−3.75856	−0.173925	−0.0869627	−	0.996212i	$-0.527716\pi$
$467$	−3.75856	−0.173925	−0.0869627	+	0.996212i	$0.527716\pi$
$468$	1.89413	0.0875564
$469$	−1.55819	−0.0719507
$470$	−2.58807	−0.119379
$471$	−46.6445	−2.14927
$472$	14.3834	0.662048
$473$	−12.5890	−0.578844
$474$	46.8606	2.15238
$475$	−5.11480	−0.234683
$476$	−2.11309	−0.0968531
$477$	14.0142	0.641665
$478$	−17.3614	−0.794090
$479$	−11.0719	−0.505887	−0.252943	−	0.967481i	$-0.581399\pi$
$479$	−11.0719	−0.505887	−0.252943	+	0.967481i	$0.581399\pi$
$480$	2.71977	0.124140
$481$	−0.259256	−0.0118211
$482$	−10.5726	−0.481567
$483$	−10.9063	−0.496256
$484$	15.5418	0.706447
$485$	1.67993	0.0762819
$486$	19.1689	0.869519
$487$	−16.7406	−0.758587	−0.379294	−	0.925276i	$-0.623833\pi$
$487$	−16.7406	−0.758587	−0.379294	+	0.925276i	$0.623833\pi$
$488$	2.26604	0.102579
$489$	−32.9111	−1.48829
$490$	−6.64618	−0.300244
$491$	−21.5509	−0.972576	−0.486288	−	0.873798i	$-0.661649\pi$
$491$	−21.5509	−0.972576	−0.486288	+	0.873798i	$0.661649\pi$
$492$	13.4407	0.605955
$493$	−27.6792	−1.24661
$494$	2.20328	0.0991301
$495$	22.6536	1.01820
$496$	−8.50979	−0.382101
$497$	−4.16219	−0.186700
$498$	19.0130	0.851993
$499$	−16.3321	−0.731125	−0.365563	−	0.930787i	$-0.619123\pi$
$499$	−16.3321	−0.731125	−0.365563	+	0.930787i	$0.619123\pi$
$500$	−1.00000	−0.0447214
$501$	−56.9635	−2.54494
$502$	11.0779	0.494431
$503$	36.2525	1.61642	0.808208	−	0.588896i	$-0.200437\pi$
$503$	36.2525	1.61642	0.808208	+	0.588896i	$0.200437\pi$
$504$	2.61555	0.116506
$505$	−12.8386	−0.571309
$506$	34.7312	1.54399
$507$	−34.8523	−1.54785
$508$	−15.4067	−0.683560
$509$	38.3403	1.69940	0.849702	−	0.527263i	$-0.176782\pi$
$509$	38.3403	1.69940	0.849702	+	0.527263i	$0.176782\pi$
$510$	9.66177	0.427831
$511$	−3.09747	−0.137024
$512$	−1.00000	−0.0441942
$513$	−19.4358	−0.858112
$514$	−23.2093	−1.02372
$515$	−3.27639	−0.144375
$516$	6.64597	0.292573
$517$	13.3334	0.586403
$518$	−0.357999	−0.0157296
$519$	12.5691	0.551722
$520$	0.430765	0.0188903
$521$	−30.3563	−1.32993	−0.664966	−	0.746873i	$-0.731554\pi$
$521$	−30.3563	−1.32993	−0.664966	+	0.746873i	$0.731554\pi$
$522$	34.2610	1.49956
$523$	34.4281	1.50544	0.752718	−	0.658343i	$-0.228743\pi$
$523$	34.4281	1.50544	0.752718	+	0.658343i	$0.228743\pi$
$524$	4.16970	0.182154
$525$	−1.61780	−0.0706065
$526$	−21.4733	−0.936278
$527$	−30.2304	−1.31686
$528$	−14.0119	−0.609790
$529$	22.4474	0.975976
$530$	3.18711	0.138439
$531$	−63.2458	−2.74463
$532$	3.04243	0.131906
$533$	2.12878	0.0922077
$534$	34.5782	1.49634
$535$	−1.95485	−0.0845157
$536$	−2.61956	−0.113148
$537$	−36.9213	−1.59327
$538$	−13.3091	−0.573795
$539$	34.2403	1.47483
$540$	−3.79991	−0.163522
$541$	−38.4208	−1.65184	−0.825920	−	0.563788i	$-0.809344\pi$
$541$	−38.4208	−1.65184	−0.825920	+	0.563788i	$0.809344\pi$
$542$	27.6321	1.18690
$543$	5.17824	0.222220
$544$	−3.55242	−0.152309
$545$	−0.904117	−0.0387281
$546$	0.696890	0.0298242
$547$	5.93390	0.253715	0.126858	−	0.991921i	$-0.459511\pi$
$547$	5.93390	0.253715	0.126858	+	0.991921i	$0.459511\pi$
$548$	15.4928	0.661819
$549$	−9.96412	−0.425258
$550$	5.15188	0.219677
$551$	39.8528	1.69778
$552$	−18.3352	−0.780400
$553$	10.2487	0.435818
$554$	−11.5551	−0.490928
$555$	1.63690	0.0694824
$556$	−6.32300	−0.268155
$557$	−12.8056	−0.542592	−0.271296	−	0.962496i	$-0.587452\pi$
$557$	−12.8056	−0.542592	−0.271296	+	0.962496i	$0.587452\pi$
$558$	37.4188	1.58406
$559$	1.05261	0.0445206
$560$	0.594829	0.0251361
$561$	−49.7763	−2.10156
$562$	12.2034	0.514768
$563$	−26.7424	−1.12706	−0.563529	−	0.826096i	$-0.690557\pi$
$563$	−26.7424	−1.12706	−0.563529	+	0.826096i	$0.690557\pi$
$564$	−7.03896	−0.296394
$565$	12.5979	0.529997
$566$	21.0410	0.884418
$567$	1.69916	0.0713579
$568$	−6.99728	−0.293599
$569$	−16.0171	−0.671473	−0.335737	−	0.941956i	$-0.608985\pi$
$569$	−16.0171	−0.671473	−0.335737	+	0.941956i	$0.608985\pi$
$570$	−13.9111	−0.582671
$571$	−2.12746	−0.0890314	−0.0445157	−	0.999009i	$-0.514174\pi$
$571$	−2.12746	−0.0890314	−0.0445157	+	0.999009i	$0.514174\pi$
$572$	−2.21925	−0.0927913
$573$	−31.2686	−1.30626
$574$	2.93956	0.122695
$575$	6.74147	0.281139
$576$	4.39715	0.183214
$577$	13.4658	0.560589	0.280294	−	0.959914i	$-0.409568\pi$
$577$	13.4658	0.560589	0.280294	+	0.959914i	$0.409568\pi$
$578$	4.38028	0.182196
$579$	−58.9545	−2.45007
$580$	7.79165	0.323531
$581$	4.15825	0.172513
$582$	4.56903	0.189393
$583$	−16.4196	−0.680029
$584$	−5.20733	−0.215481
$585$	−1.89413	−0.0783128
$586$	−6.73977	−0.278417
$587$	−42.7090	−1.76279	−0.881394	−	0.472382i	$-0.843394\pi$
$587$	−42.7090	−1.76279	−0.881394	+	0.472382i	$0.843394\pi$
$588$	−18.0761	−0.745445
$589$	43.5259	1.79345
$590$	−14.3834	−0.592154
$591$	−24.9293	−1.02546
$592$	−0.601851	−0.0247359
$593$	3.95945	0.162595	0.0812976	−	0.996690i	$-0.474094\pi$
$593$	3.95945	0.162595	0.0812976	+	0.996690i	$0.474094\pi$
$594$	19.5767	0.803241
$595$	2.11309	0.0866281
$596$	21.4347	0.877998
$597$	−56.7934	−2.32440
$598$	−2.90399	−0.118753
$599$	5.57202	0.227666	0.113833	−	0.993500i	$-0.463687\pi$
$599$	5.57202	0.227666	0.113833	+	0.993500i	$0.463687\pi$
$600$	−2.71977	−0.111034
$601$	−34.0271	−1.38799	−0.693997	−	0.719977i	$-0.744152\pi$
$601$	−34.0271	−1.38799	−0.693997	+	0.719977i	$0.744152\pi$
$602$	1.45351	0.0592407
$603$	11.5186	0.469074
$604$	−1.44587	−0.0588316
$605$	−15.5418	−0.631865
$606$	−34.9180	−1.41845
$607$	22.1670	0.899733	0.449866	−	0.893096i	$-0.351472\pi$
$607$	22.1670	0.899733	0.449866	+	0.893096i	$0.351472\pi$
$608$	5.11480	0.207433
$609$	12.6053	0.510793
$610$	−2.26604	−0.0917494
$611$	−1.11485	−0.0451020
$612$	15.6205	0.631422
$613$	−19.0501	−0.769424	−0.384712	−	0.923037i	$-0.625699\pi$
$613$	−19.0501	−0.769424	−0.384712	+	0.923037i	$0.625699\pi$
$614$	−21.8952	−0.883617
$615$	−13.4407	−0.541983
$616$	−3.06449	−0.123472
$617$	26.3274	1.05990	0.529951	−	0.848028i	$-0.322210\pi$
$617$	26.3274	1.05990	0.529951	+	0.848028i	$0.322210\pi$
$618$	−8.91103	−0.358454
$619$	35.2029	1.41492	0.707461	−	0.706752i	$-0.249840\pi$
$619$	35.2029	1.41492	0.707461	+	0.706752i	$0.249840\pi$
$620$	8.50979	0.341762
$621$	25.6170	1.02798
$622$	−19.9821	−0.801211
$623$	7.56245	0.302983
$624$	1.17158	0.0469008
$625$	1.00000	0.0400000
$626$	−6.20088	−0.247837
$627$	71.6682	2.86215
$628$	−17.1502	−0.684367
$629$	−2.13803	−0.0852489
$630$	−2.61555	−0.104206
$631$	19.0064	0.756635	0.378317	−	0.925676i	$-0.376503\pi$
$631$	19.0064	0.756635	0.378317	+	0.925676i	$0.376503\pi$
$632$	17.2296	0.685357
$633$	52.8332	2.09993
$634$	10.6814	0.424214
$635$	15.4067	0.611394
$636$	8.66819	0.343716
$637$	−2.86294	−0.113434
$638$	−40.1416	−1.58922
$639$	30.7681	1.21717
$640$	1.00000	0.0395285
$641$	−22.7832	−0.899883	−0.449942	−	0.893058i	$-0.648555\pi$
$641$	−22.7832	−0.899883	−0.449942	+	0.893058i	$0.648555\pi$
$642$	−5.31675	−0.209836
$643$	3.83924	0.151405	0.0757024	−	0.997130i	$-0.475880\pi$
$643$	3.83924	0.151405	0.0757024	+	0.997130i	$0.475880\pi$
$644$	−4.01002	−0.158017
$645$	−6.64597	−0.261685
$646$	18.1700	0.714887
$647$	1.36012	0.0534719	0.0267359	−	0.999643i	$-0.491489\pi$
$647$	1.36012	0.0534719	0.0267359	+	0.999643i	$0.491489\pi$
$648$	2.85655	0.112216
$649$	74.1013	2.90873
$650$	−0.430765	−0.0168960
$651$	13.7671	0.539576
$652$	−12.1007	−0.473900
$653$	−3.84236	−0.150363	−0.0751815	−	0.997170i	$-0.523954\pi$
$653$	−3.84236	−0.150363	−0.0751815	+	0.997170i	$0.523954\pi$
$654$	−2.45899	−0.0961541
$655$	−4.16970	−0.162923
$656$	4.94186	0.192947
$657$	22.8974	0.893312
$658$	−1.53946	−0.0600144
$659$	−49.9855	−1.94716	−0.973580	−	0.228348i	$-0.926668\pi$
$659$	−49.9855	−1.94716	−0.973580	+	0.228348i	$0.926668\pi$
$660$	14.0119	0.545413
$661$	41.3643	1.60889	0.804443	−	0.594029i	$-0.202464\pi$
$661$	41.3643	1.60889	0.804443	+	0.594029i	$0.202464\pi$
$662$	13.2248	0.513995
$663$	4.16195	0.161637
$664$	6.99067	0.271290
$665$	−3.04243	−0.117981
$666$	2.64643	0.102547
$667$	−52.5272	−2.03386
$668$	−20.9442	−0.810357
$669$	65.4045	2.52869
$670$	2.61956	0.101203
$671$	11.6744	0.450684
$672$	1.61780	0.0624079
$673$	19.3094	0.744323	0.372162	−	0.928168i	$-0.378617\pi$
$673$	19.3094	0.744323	0.372162	+	0.928168i	$0.378617\pi$
$674$	−10.7103	−0.412545
$675$	3.79991	0.146259
$676$	−12.8144	−0.492863
$677$	−50.0450	−1.92338	−0.961692	−	0.274133i	$-0.911609\pi$
$677$	−50.0450	−1.92338	−0.961692	+	0.274133i	$0.911609\pi$
$678$	34.2633	1.31588
$679$	0.999274	0.0383486
$680$	3.55242	0.136229
$681$	13.1243	0.502925
$682$	−43.8414	−1.67877
$683$	−9.28981	−0.355465	−0.177732	−	0.984079i	$-0.556876\pi$
$683$	−9.28981	−0.355465	−0.177732	+	0.984079i	$0.556876\pi$
$684$	−22.4905	−0.859947
$685$	−15.4928	−0.591949
$686$	−8.11715	−0.309914
$687$	32.2473	1.23031
$688$	2.44358	0.0931606
$689$	1.37289	0.0523030
$690$	18.3352	0.698011
$691$	−23.1226	−0.879624	−0.439812	−	0.898090i	$-0.644955\pi$
$691$	−23.1226	−0.879624	−0.439812	+	0.898090i	$0.644955\pi$
$692$	4.62138	0.175678
$693$	13.4750	0.511873
$694$	−26.7432	−1.01516
$695$	6.32300	0.239845
$696$	21.1915	0.803261
$697$	17.5556	0.664966
$698$	−19.1024	−0.723036
$699$	63.5155	2.40238
$700$	−0.594829	−0.0224824
$701$	40.1712	1.51725	0.758623	−	0.651530i	$-0.225872\pi$
$701$	40.1712	1.51725	0.758623	+	0.651530i	$0.225872\pi$
$702$	−1.63687	−0.0617796
$703$	3.07835	0.116102
$704$	−5.15188	−0.194169
$705$	7.03896	0.265102
$706$	2.45986	0.0925779
$707$	−7.63676	−0.287210
$708$	−39.1194	−1.47020
$709$	−14.6968	−0.551949	−0.275974	−	0.961165i	$-0.589000\pi$
$709$	−14.6968	−0.551949	−0.275974	+	0.961165i	$0.589000\pi$
$710$	6.99728	0.262603
$711$	−75.7611	−2.84126
$712$	12.7136	0.476464
$713$	−57.3685	−2.14847
$714$	5.74711	0.215080
$715$	2.21925	0.0829951
$716$	−13.5752	−0.507327
$717$	47.2189	1.76342
$718$	−3.91065	−0.145944
$719$	43.7984	1.63341	0.816703	−	0.577059i	$-0.195800\pi$
$719$	43.7984	1.63341	0.816703	+	0.577059i	$0.195800\pi$
$720$	−4.39715	−0.163872
$721$	−1.94889	−0.0725806
$722$	−7.16122	−0.266513
$723$	28.7549	1.06941
$724$	1.90393	0.0707589
$725$	−7.79165	−0.289375
$726$	−42.2702	−1.56879
$727$	39.0805	1.44942	0.724708	−	0.689056i	$-0.241975\pi$
$727$	39.0805	1.44942	0.724708	+	0.689056i	$0.241975\pi$
$728$	0.256231	0.00949657
$729$	−43.5653	−1.61353
$730$	5.20733	0.192732
$731$	8.68063	0.321065
$732$	−6.16311	−0.227795
$733$	−6.76627	−0.249918	−0.124959	−	0.992162i	$-0.539880\pi$
$733$	−6.76627	−0.249918	−0.124959	+	0.992162i	$0.539880\pi$
$734$	−29.4789	−1.08809
$735$	18.0761	0.666746
$736$	−6.74147	−0.248494
$737$	−13.4957	−0.497119
$738$	−21.7301	−0.799896
$739$	−6.77570	−0.249248	−0.124624	−	0.992204i	$-0.539773\pi$
$739$	−6.77570	−0.249248	−0.124624	+	0.992204i	$0.539773\pi$
$740$	0.601851	0.0221245
$741$	−5.99240	−0.220136
$742$	1.89578	0.0695964
$743$	8.62411	0.316388	0.158194	−	0.987408i	$-0.449433\pi$
$743$	8.62411	0.316388	0.158194	+	0.987408i	$0.449433\pi$
$744$	23.1447	0.848525
$745$	−21.4347	−0.785305
$746$	−25.0946	−0.918780
$747$	−30.7390	−1.12468
$748$	−18.3016	−0.669175
$749$	−1.16280	−0.0424880
$750$	2.71977	0.0993119
$751$	36.6638	1.33788	0.668940	−	0.743316i	$-0.266748\pi$
$751$	36.6638	1.33788	0.668940	+	0.743316i	$0.266748\pi$
$752$	−2.58807	−0.0943773
$753$	−30.1294	−1.09797
$754$	3.35637	0.122232
$755$	1.44587	0.0526206
$756$	−2.26030	−0.0822063
$757$	−38.8656	−1.41259	−0.706296	−	0.707916i	$-0.749635\pi$
$757$	−38.8656	−1.41259	−0.706296	+	0.707916i	$0.749635\pi$
$758$	7.83169	0.284460
$759$	−94.4609	−3.42871
$760$	−5.11480	−0.185533
$761$	37.6874	1.36617	0.683083	−	0.730341i	$-0.260639\pi$
$761$	37.6874	1.36617	0.683083	+	0.730341i	$0.260639\pi$
$762$	41.9025	1.51797
$763$	−0.537795	−0.0194695
$764$	−11.4968	−0.415939
$765$	−15.6205	−0.564761
$766$	−10.0381	−0.362693
$767$	−6.19585	−0.223719
$768$	2.71977	0.0981412
$769$	27.4527	0.989968	0.494984	−	0.868902i	$-0.335174\pi$
$769$	27.4527	0.989968	0.494984	+	0.868902i	$0.335174\pi$
$770$	3.06449	0.110436
$771$	63.1238	2.27335
$772$	−21.6763	−0.780147
$773$	−14.4256	−0.518854	−0.259427	−	0.965763i	$-0.583534\pi$
$773$	−14.4256	−0.518854	−0.259427	+	0.965763i	$0.583534\pi$
$774$	−10.7448	−0.386213
$775$	−8.50979	−0.305681
$776$	1.67993	0.0603061
$777$	0.973674	0.0349304
$778$	8.12206	0.291190
$779$	−25.2767	−0.905631
$780$	−1.17158	−0.0419493
$781$	−36.0491	−1.28994
$782$	−23.9486	−0.856399
$783$	−29.6076	−1.05809
$784$	−6.64618	−0.237364
$785$	17.1502	0.612116
$786$	−11.3406	−0.404506
$787$	−44.7132	−1.59385	−0.796927	−	0.604076i	$-0.793542\pi$
$787$	−44.7132	−1.59385	−0.796927	+	0.604076i	$0.793542\pi$
$788$	−9.16597	−0.326524
$789$	58.4023	2.07918
$790$	−17.2296	−0.613002
$791$	7.49359	0.266441
$792$	22.6536	0.804959
$793$	−0.976131	−0.0346634
$794$	26.4713	0.939433
$795$	−8.66819	−0.307429
$796$	−20.8817	−0.740133
$797$	20.3035	0.719186	0.359593	−	0.933109i	$-0.382915\pi$
$797$	20.3035	0.719186	0.359593	+	0.933109i	$0.382915\pi$
$798$	−8.27472	−0.292922
$799$	−9.19393	−0.325258
$800$	−1.00000	−0.0353553
$801$	−55.9038	−1.97526
$802$	1.00000	0.0353112
$803$	−26.8275	−0.946722
$804$	7.12461	0.251266
$805$	4.01002	0.141335
$806$	3.66572	0.129119
$807$	36.1976	1.27422
$808$	−12.8386	−0.451660
$809$	32.4567	1.14112	0.570559	−	0.821257i	$-0.306727\pi$
$809$	32.4567	1.14112	0.570559	+	0.821257i	$0.306727\pi$
$810$	−2.85655	−0.100369
$811$	31.4619	1.10478	0.552388	−	0.833587i	$-0.313717\pi$
$811$	31.4619	1.10478	0.552388	+	0.833587i	$0.313717\pi$
$812$	4.63470	0.162646
$813$	−75.1530	−2.63573
$814$	−3.10066	−0.108678
$815$	12.1007	0.423869
$816$	9.66177	0.338230
$817$	−12.4984	−0.437265
$818$	3.19920	0.111858
$819$	−1.12669	−0.0393696
$820$	−4.94186	−0.172577
$821$	27.9797	0.976500	0.488250	−	0.872704i	$-0.337635\pi$
$821$	27.9797	0.976500	0.488250	+	0.872704i	$0.337635\pi$
$822$	−42.1368	−1.46969
$823$	45.4505	1.58431	0.792153	−	0.610323i	$-0.208960\pi$
$823$	45.4505	1.58431	0.792153	+	0.610323i	$0.208960\pi$
$824$	−3.27639	−0.114139
$825$	−14.0119	−0.487832
$826$	−8.55565	−0.297689
$827$	−42.2763	−1.47009	−0.735045	−	0.678019i	$-0.762839\pi$
$827$	−42.2763	−1.47009	−0.735045	+	0.678019i	$0.762839\pi$
$828$	29.6432	1.03017
$829$	−39.1815	−1.36083	−0.680415	−	0.732827i	$-0.738200\pi$
$829$	−39.1815	−1.36083	−0.680415	+	0.732827i	$0.738200\pi$
$830$	−6.99067	−0.242650
$831$	31.4272	1.09020
$832$	0.430765	0.0149341
$833$	−23.6100	−0.818040
$834$	17.1971	0.595486
$835$	20.9442	0.724805
$836$	26.3508	0.911363
$837$	−32.3365	−1.11771
$838$	−9.80519	−0.338715
$839$	7.17337	0.247652	0.123826	−	0.992304i	$-0.460483\pi$
$839$	7.17337	0.247652	0.123826	+	0.992304i	$0.460483\pi$
$840$	−1.61780	−0.0558193
$841$	31.7098	1.09344
$842$	34.3451	1.18361
$843$	−33.1903	−1.14314
$844$	19.4256	0.668657
$845$	12.8144	0.440830
$846$	11.3801	0.391257
$847$	−9.24473	−0.317653
$848$	3.18711	0.109446
$849$	−57.2266	−1.96401
$850$	−3.55242	−0.121847
$851$	−4.05736	−0.139085
$852$	19.0310	0.651991
$853$	−21.4921	−0.735876	−0.367938	−	0.929850i	$-0.619936\pi$
$853$	−21.4921	−0.735876	−0.367938	+	0.929850i	$0.619936\pi$
$854$	−1.34791	−0.0461245
$855$	22.4905	0.769160
$856$	−1.95485	−0.0668156
$857$	27.8792	0.952335	0.476168	−	0.879355i	$-0.342026\pi$
$857$	27.8792	0.952335	0.476168	+	0.879355i	$0.342026\pi$
$858$	6.03584	0.206060
$859$	−41.6685	−1.42171	−0.710856	−	0.703337i	$-0.751692\pi$
$859$	−41.6685	−1.42171	−0.710856	+	0.703337i	$0.751692\pi$
$860$	−2.44358	−0.0833254
$861$	−7.99494	−0.272467
$862$	−22.5844	−0.769228
$863$	12.9026	0.439211	0.219605	−	0.975589i	$-0.429523\pi$
$863$	12.9026	0.439211	0.219605	+	0.975589i	$0.429523\pi$
$864$	−3.79991	−0.129276
$865$	−4.62138	−0.157132
$866$	26.8669	0.912975
$867$	−11.9134	−0.404599
$868$	5.06187	0.171811
$869$	88.7648	3.01114
$870$	−21.1915	−0.718459
$871$	1.12842	0.0382349
$872$	−0.904117	−0.0306173
$873$	−7.38692	−0.250009
$874$	34.4813	1.16635
$875$	0.594829	0.0201089
$876$	14.1627	0.478514
$877$	−13.9215	−0.470095	−0.235047	−	0.971984i	$-0.575525\pi$
$877$	−13.9215	−0.470095	−0.235047	+	0.971984i	$0.575525\pi$
$878$	−2.51933	−0.0850232
$879$	18.3306	0.618276
$880$	5.15188	0.173670
$881$	15.0713	0.507765	0.253882	−	0.967235i	$-0.418292\pi$
$881$	15.0713	0.507765	0.253882	+	0.967235i	$0.418292\pi$
$882$	29.2242	0.984030
$883$	28.6998	0.965825	0.482912	−	0.875669i	$-0.339579\pi$
$883$	28.6998	0.965825	0.482912	+	0.875669i	$0.339579\pi$
$884$	1.53026	0.0514682
$885$	39.1194	1.31499
$886$	15.6162	0.524638
$887$	−49.9217	−1.67621	−0.838103	−	0.545512i	$-0.816335\pi$
$887$	−49.9217	−1.67621	−0.838103	+	0.545512i	$0.816335\pi$
$888$	1.63690	0.0549307
$889$	9.16433	0.307362
$890$	−12.7136	−0.426163
$891$	14.7166	0.493024
$892$	24.0478	0.805181
$893$	13.2375	0.442975
$894$	−58.2973	−1.94975
$895$	13.5752	0.453767
$896$	0.594829	0.0198718
$897$	7.89818	0.263712
$898$	3.09920	0.103422
$899$	66.3053	2.21141
$900$	4.39715	0.146572
$901$	11.3220	0.377189
$902$	25.4599	0.847721
$903$	−3.95322	−0.131555
$904$	12.5979	0.418999
$905$	−1.90393	−0.0632887
$906$	3.93243	0.130646
$907$	−34.9319	−1.15989	−0.579947	−	0.814654i	$-0.696927\pi$
$907$	−34.9319	−1.15989	−0.579947	+	0.814654i	$0.696927\pi$
$908$	4.82553	0.160141
$909$	56.4531	1.87243
$910$	−0.256231	−0.00849399
$911$	26.7853	0.887437	0.443718	−	0.896166i	$-0.353659\pi$
$911$	26.7853	0.887437	0.443718	+	0.896166i	$0.353659\pi$
$912$	−13.9111	−0.460642
$913$	36.0150	1.19192
$914$	2.80648	0.0928303
$915$	6.16311	0.203746
$916$	11.8566	0.391754
$917$	−2.48026	−0.0819053
$918$	−13.4989	−0.445530
$919$	−14.8975	−0.491424	−0.245712	−	0.969343i	$-0.579022\pi$
$919$	−14.8975	−0.491424	−0.245712	+	0.969343i	$0.579022\pi$
$920$	6.74147	0.222260
$921$	59.5498	1.96223
$922$	−10.1931	−0.335693
$923$	3.01418	0.0992130
$924$	8.33470	0.274191
$925$	−0.601851	−0.0197887
$926$	1.35804	0.0446280
$927$	14.4068	0.473181
$928$	7.79165	0.255773
$929$	9.69067	0.317940	0.158970	−	0.987283i	$-0.449183\pi$
$929$	9.69067	0.317940	0.158970	+	0.987283i	$0.449183\pi$
$930$	−23.1447	−0.758944
$931$	33.9939	1.11411
$932$	23.3533	0.764961
$933$	54.3468	1.77923
$934$	3.75856	0.122984
$935$	18.3016	0.598528
$936$	−1.89413	−0.0619117
$937$	36.7285	1.19987	0.599933	−	0.800050i	$-0.295194\pi$
$937$	36.7285	1.19987	0.599933	+	0.800050i	$0.295194\pi$
$938$	1.55819	0.0508768
$939$	16.8650	0.550367
$940$	2.58807	0.0844136
$941$	−18.2824	−0.595990	−0.297995	−	0.954567i	$-0.596318\pi$
$941$	−18.2824	−0.595990	−0.297995	+	0.954567i	$0.596318\pi$
$942$	46.6445	1.51976
$943$	33.3154	1.08490
$944$	−14.3834	−0.468139
$945$	2.26030	0.0735276
$946$	12.5890	0.409304
$947$	17.8109	0.578778	0.289389	−	0.957212i	$-0.406548\pi$
$947$	17.8109	0.578778	0.289389	+	0.957212i	$0.406548\pi$
$948$	−46.8606	−1.52196
$949$	2.24313	0.0728152
$950$	5.11480	0.165946
$951$	−29.0510	−0.942045
$952$	2.11309	0.0684855
$953$	15.7490	0.510162	0.255081	−	0.966920i	$-0.417898\pi$
$953$	15.7490	0.510162	0.255081	+	0.966920i	$0.417898\pi$
$954$	−14.0142	−0.453725
$955$	11.4968	0.372027
$956$	17.3614	0.561506
$957$	109.176	3.52916
$958$	11.0719	0.357716
$959$	−9.21556	−0.297586
$960$	−2.71977	−0.0877802
$961$	41.4166	1.33602
$962$	0.259256	0.00835876
$963$	8.59578	0.276995
$964$	10.5726	0.340519
$965$	21.6763	0.697785
$966$	10.9063	0.350906
$967$	−7.50734	−0.241420	−0.120710	−	0.992688i	$-0.538517\pi$
$967$	−7.50734	−0.241420	−0.120710	+	0.992688i	$0.538517\pi$
$968$	−15.5418	−0.499533
$969$	−49.4181	−1.58754
$970$	−1.67993	−0.0539394
$971$	43.3159	1.39007	0.695037	−	0.718974i	$-0.255388\pi$
$971$	43.3159	1.39007	0.695037	+	0.718974i	$0.255388\pi$
$972$	−19.1689	−0.614842
$973$	3.76110	0.120575
$974$	16.7406	0.536402
$975$	1.17158	0.0375206
$976$	−2.26604	−0.0725342
$977$	−49.8233	−1.59399	−0.796994	−	0.603987i	$-0.793578\pi$
$977$	−49.8233	−1.59399	−0.796994	+	0.603987i	$0.793578\pi$
$978$	32.9111	1.05238
$979$	65.4992	2.09336
$980$	6.64618	0.212304
$981$	3.97553	0.126929
$982$	21.5509	0.687715
$983$	−3.02102	−0.0963555	−0.0481777	−	0.998839i	$-0.515341\pi$
$983$	−3.02102	−0.0963555	−0.0481777	+	0.998839i	$0.515341\pi$
$984$	−13.4407	−0.428475
$985$	9.16597	0.292052
$986$	27.6792	0.881487
$987$	4.18698	0.133273
$988$	−2.20328	−0.0700956
$989$	16.4733	0.523821
$990$	−22.6536	−0.719977
$991$	−35.2163	−1.11868	−0.559342	−	0.828937i	$-0.688946\pi$
$991$	−35.2163	−1.11868	−0.559342	+	0.828937i	$0.688946\pi$
$992$	8.50979	0.270186
$993$	−35.9683	−1.14142
$994$	4.16219	0.132017
$995$	20.8817	0.661995
$996$	−19.0130	−0.602450
$997$	52.2631	1.65519	0.827594	−	0.561327i	$-0.189709\pi$
$997$	52.2631	1.65519	0.827594	+	0.561327i	$0.189709\pi$
$998$	16.3321	0.516984
$999$	−2.28698	−0.0723569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.k.1.14	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.k.1.14	✓	15	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:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.71977 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.71977 q^{6} -0.594829 q^{7} -1.00000 q^{8} +4.39715 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.71977 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.71977 q^{6} -0.594829 q^{7} -1.00000 q^{8} +4.39715 q^{9} +1.00000 q^{10} -5.15188 q^{11} +2.71977 q^{12} +0.430765 q^{13} +0.594829 q^{14} -2.71977 q^{15} +1.00000 q^{16} +3.55242 q^{17} -4.39715 q^{18} -5.11480 q^{19} -1.00000 q^{20} -1.61780 q^{21} +5.15188 q^{22} +6.74147 q^{23} -2.71977 q^{24} +1.00000 q^{25} -0.430765 q^{26} +3.79991 q^{27} -0.594829 q^{28} -7.79165 q^{29} +2.71977 q^{30} -8.50979 q^{31} -1.00000 q^{32} -14.0119 q^{33} -3.55242 q^{34} +0.594829 q^{35} +4.39715 q^{36} -0.601851 q^{37} +5.11480 q^{38} +1.17158 q^{39} +1.00000 q^{40} +4.94186 q^{41} +1.61780 q^{42} +2.44358 q^{43} -5.15188 q^{44} -4.39715 q^{45} -6.74147 q^{46} -2.58807 q^{47} +2.71977 q^{48} -6.64618 q^{49} -1.00000 q^{50} +9.66177 q^{51} +0.430765 q^{52} +3.18711 q^{53} -3.79991 q^{54} +5.15188 q^{55} +0.594829 q^{56} -13.9111 q^{57} +7.79165 q^{58} -14.3834 q^{59} -2.71977 q^{60} -2.26604 q^{61} +8.50979 q^{62} -2.61555 q^{63} +1.00000 q^{64} -0.430765 q^{65} +14.0119 q^{66} +2.61956 q^{67} +3.55242 q^{68} +18.3352 q^{69} -0.594829 q^{70} +6.99728 q^{71} -4.39715 q^{72} +5.20733 q^{73} +0.601851 q^{74} +2.71977 q^{75} -5.11480 q^{76} +3.06449 q^{77} -1.17158 q^{78} -17.2296 q^{79} -1.00000 q^{80} -2.85655 q^{81} -4.94186 q^{82} -6.99067 q^{83} -1.61780 q^{84} -3.55242 q^{85} -2.44358 q^{86} -21.1915 q^{87} +5.15188 q^{88} -12.7136 q^{89} +4.39715 q^{90} -0.256231 q^{91} +6.74147 q^{92} -23.1447 q^{93} +2.58807 q^{94} +5.11480 q^{95} -2.71977 q^{96} -1.67993 q^{97} +6.64618 q^{98} -22.6536 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{10} - 2 q^{11} - 6 q^{12} - 13 q^{13} + 5 q^{14} + 6 q^{15} + 15 q^{16} + 11 q^{17} - 19 q^{18} - 15 q^{19} - 15 q^{20} - 2 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 15 q^{25} + 13 q^{26} - 12 q^{27} - 5 q^{28} + 28 q^{29} - 6 q^{30} - 12 q^{31} - 15 q^{32} - 22 q^{33} - 11 q^{34} + 5 q^{35} + 19 q^{36} - 23 q^{37} + 15 q^{38} - 2 q^{39} + 15 q^{40} + 24 q^{41} + 2 q^{42} - 24 q^{43} - 2 q^{44} - 19 q^{45} + 3 q^{46} - 3 q^{47} - 6 q^{48} + 20 q^{49} - 15 q^{50} - 5 q^{51} - 13 q^{52} + 10 q^{53} + 12 q^{54} + 2 q^{55} + 5 q^{56} - 11 q^{57} - 28 q^{58} + 2 q^{59} + 6 q^{60} + 15 q^{61} + 12 q^{62} - 2 q^{63} + 15 q^{64} + 13 q^{65} + 22 q^{66} - 48 q^{67} + 11 q^{68} + 21 q^{69} - 5 q^{70} + 15 q^{71} - 19 q^{72} - 47 q^{73} + 23 q^{74} - 6 q^{75} - 15 q^{76} + 7 q^{77} + 2 q^{78} - 34 q^{79} - 15 q^{80} + 43 q^{81} - 24 q^{82} - 32 q^{83} - 2 q^{84} - 11 q^{85} + 24 q^{86} + 14 q^{87} + 2 q^{88} + 25 q^{89} + 19 q^{90} - 32 q^{91} - 3 q^{92} - 42 q^{93} + 3 q^{94} + 15 q^{95} + 6 q^{96} - 34 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^10 - 2 * q^11 - 6 * q^12 - 13 * q^13 + 5 * q^14 + 6 * q^15 + 15 * q^16 + 11 * q^17 - 19 * q^18 - 15 * q^19 - 15 * q^20 - 2 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 15 * q^25 + 13 * q^26 - 12 * q^27 - 5 * q^28 + 28 * q^29 - 6 * q^30 - 12 * q^31 - 15 * q^32 - 22 * q^33 - 11 * q^34 + 5 * q^35 + 19 * q^36 - 23 * q^37 + 15 * q^38 - 2 * q^39 + 15 * q^40 + 24 * q^41 + 2 * q^42 - 24 * q^43 - 2 * q^44 - 19 * q^45 + 3 * q^46 - 3 * q^47 - 6 * q^48 + 20 * q^49 - 15 * q^50 - 5 * q^51 - 13 * q^52 + 10 * q^53 + 12 * q^54 + 2 * q^55 + 5 * q^56 - 11 * q^57 - 28 * q^58 + 2 * q^59 + 6 * q^60 + 15 * q^61 + 12 * q^62 - 2 * q^63 + 15 * q^64 + 13 * q^65 + 22 * q^66 - 48 * q^67 + 11 * q^68 + 21 * q^69 - 5 * q^70 + 15 * q^71 - 19 * q^72 - 47 * q^73 + 23 * q^74 - 6 * q^75 - 15 * q^76 + 7 * q^77 + 2 * q^78 - 34 * q^79 - 15 * q^80 + 43 * q^81 - 24 * q^82 - 32 * q^83 - 2 * q^84 - 11 * q^85 + 24 * q^86 + 14 * q^87 + 2 * q^88 + 25 * q^89 + 19 * q^90 - 32 * q^91 - 3 * q^92 - 42 * q^93 + 3 * q^94 + 15 * q^95 + 6 * q^96 - 34 * q^97 - 20 * q^98 + 4 * q^99

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