LMFDB - Embedded newform 402.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [402,2,Mod(1,402)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(402, 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("402.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 402 = 2 \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	402.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:	$3.20998616126$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	402.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{6} -3.70156 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{6} -3.70156 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.70156 q^{10} +4.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -3.70156 q^{14} -3.70156 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -7.40312 q^{19} +3.70156 q^{20} +3.70156 q^{21} +4.00000 q^{22} +7.70156 q^{23} -1.00000 q^{24} +8.70156 q^{25} +4.00000 q^{26} -1.00000 q^{27} -3.70156 q^{28} -4.00000 q^{29} -3.70156 q^{30} -0.298438 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -13.7016 q^{35} +1.00000 q^{36} +4.29844 q^{37} -7.40312 q^{38} -4.00000 q^{39} +3.70156 q^{40} -3.70156 q^{41} +3.70156 q^{42} -2.29844 q^{43} +4.00000 q^{44} +3.70156 q^{45} +7.70156 q^{46} +1.40312 q^{47} -1.00000 q^{48} +6.70156 q^{49} +8.70156 q^{50} +2.00000 q^{51} +4.00000 q^{52} -11.1047 q^{53} -1.00000 q^{54} +14.8062 q^{55} -3.70156 q^{56} +7.40312 q^{57} -4.00000 q^{58} -13.1047 q^{59} -3.70156 q^{60} -0.596876 q^{61} -0.298438 q^{62} -3.70156 q^{63} +1.00000 q^{64} +14.8062 q^{65} -4.00000 q^{66} -1.00000 q^{67} -2.00000 q^{68} -7.70156 q^{69} -13.7016 q^{70} -13.4031 q^{71} +1.00000 q^{72} -3.10469 q^{73} +4.29844 q^{74} -8.70156 q^{75} -7.40312 q^{76} -14.8062 q^{77} -4.00000 q^{78} -9.40312 q^{79} +3.70156 q^{80} +1.00000 q^{81} -3.70156 q^{82} +17.1047 q^{83} +3.70156 q^{84} -7.40312 q^{85} -2.29844 q^{86} +4.00000 q^{87} +4.00000 q^{88} -2.59688 q^{89} +3.70156 q^{90} -14.8062 q^{91} +7.70156 q^{92} +0.298438 q^{93} +1.40312 q^{94} -27.4031 q^{95} -1.00000 q^{96} -13.4031 q^{97} +6.70156 q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} + q^{10} + 8 q^{11} - 2 q^{12} + 8 q^{13} - q^{14} - q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 2 q^{19} + q^{20} + q^{21} + 8 q^{22} + 9 q^{23} - 2 q^{24} + 11 q^{25} + 8 q^{26} - 2 q^{27} - q^{28} - 8 q^{29} - q^{30} - 7 q^{31} + 2 q^{32} - 8 q^{33} - 4 q^{34} - 21 q^{35} + 2 q^{36} + 15 q^{37} - 2 q^{38} - 8 q^{39} + q^{40} - q^{41} + q^{42} - 11 q^{43} + 8 q^{44} + q^{45} + 9 q^{46} - 10 q^{47} - 2 q^{48} + 7 q^{49} + 11 q^{50} + 4 q^{51} + 8 q^{52} - 3 q^{53} - 2 q^{54} + 4 q^{55} - q^{56} + 2 q^{57} - 8 q^{58} - 7 q^{59} - q^{60} - 14 q^{61} - 7 q^{62} - q^{63} + 2 q^{64} + 4 q^{65} - 8 q^{66} - 2 q^{67} - 4 q^{68} - 9 q^{69} - 21 q^{70} - 14 q^{71} + 2 q^{72} + 13 q^{73} + 15 q^{74} - 11 q^{75} - 2 q^{76} - 4 q^{77} - 8 q^{78} - 6 q^{79} + q^{80} + 2 q^{81} - q^{82} + 15 q^{83} + q^{84} - 2 q^{85} - 11 q^{86} + 8 q^{87} + 8 q^{88} - 18 q^{89} + q^{90} - 4 q^{91} + 9 q^{92} + 7 q^{93} - 10 q^{94} - 42 q^{95} - 2 q^{96} - 14 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 + q^10 + 8 * q^11 - 2 * q^12 + 8 * q^13 - q^14 - q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 - 2 * q^19 + q^20 + q^21 + 8 * q^22 + 9 * q^23 - 2 * q^24 + 11 * q^25 + 8 * q^26 - 2 * q^27 - q^28 - 8 * q^29 - q^30 - 7 * q^31 + 2 * q^32 - 8 * q^33 - 4 * q^34 - 21 * q^35 + 2 * q^36 + 15 * q^37 - 2 * q^38 - 8 * q^39 + q^40 - q^41 + q^42 - 11 * q^43 + 8 * q^44 + q^45 + 9 * q^46 - 10 * q^47 - 2 * q^48 + 7 * q^49 + 11 * q^50 + 4 * q^51 + 8 * q^52 - 3 * q^53 - 2 * q^54 + 4 * q^55 - q^56 + 2 * q^57 - 8 * q^58 - 7 * q^59 - q^60 - 14 * q^61 - 7 * q^62 - q^63 + 2 * q^64 + 4 * q^65 - 8 * q^66 - 2 * q^67 - 4 * q^68 - 9 * q^69 - 21 * q^70 - 14 * q^71 + 2 * q^72 + 13 * q^73 + 15 * q^74 - 11 * q^75 - 2 * q^76 - 4 * q^77 - 8 * q^78 - 6 * q^79 + q^80 + 2 * q^81 - q^82 + 15 * q^83 + q^84 - 2 * q^85 - 11 * q^86 + 8 * q^87 + 8 * q^88 - 18 * q^89 + q^90 - 4 * q^91 + 9 * q^92 + 7 * q^93 - 10 * q^94 - 42 * q^95 - 2 * q^96 - 14 * q^97 + 7 * q^98 + 8 * 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
$5$	3.70156	1.65539	0.827694	−	0.561179i	$-0.189652\pi$
$5$	3.70156	1.65539	0.827694	+	0.561179i	$0.189652\pi$
$6$	−1.00000	−0.408248
$7$	−3.70156	−1.39906	−0.699529	−	0.714604i	$-0.746607\pi$
$7$	−3.70156	−1.39906	−0.699529	+	0.714604i	$0.746607\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.70156	1.17054
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−3.70156	−0.989284
$15$	−3.70156	−0.955739
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−7.40312	−1.69839	−0.849197	−	0.528077i	$-0.822913\pi$
$19$	−7.40312	−1.69839	−0.849197	+	0.528077i	$0.822913\pi$
$20$	3.70156	0.827694
$21$	3.70156	0.807747
$22$	4.00000	0.852803
$23$	7.70156	1.60589	0.802943	−	0.596055i	$-0.203266\pi$
$23$	7.70156	1.60589	0.802943	+	0.596055i	$0.203266\pi$
$24$	−1.00000	−0.204124
$25$	8.70156	1.74031
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	−3.70156	−0.699529
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−3.70156	−0.675810
$31$	−0.298438	−0.0536010	−0.0268005	−	0.999641i	$-0.508532\pi$
$31$	−0.298438	−0.0536010	−0.0268005	+	0.999641i	$0.508532\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	−2.00000	−0.342997
$35$	−13.7016	−2.31599
$36$	1.00000	0.166667
$37$	4.29844	0.706659	0.353329	−	0.935499i	$-0.385049\pi$
$37$	4.29844	0.706659	0.353329	+	0.935499i	$0.385049\pi$
$38$	−7.40312	−1.20095
$39$	−4.00000	−0.640513
$40$	3.70156	0.585268
$41$	−3.70156	−0.578087	−0.289043	−	0.957316i	$-0.593337\pi$
$41$	−3.70156	−0.578087	−0.289043	+	0.957316i	$0.593337\pi$
$42$	3.70156	0.571163
$43$	−2.29844	−0.350508	−0.175254	−	0.984523i	$-0.556075\pi$
$43$	−2.29844	−0.350508	−0.175254	+	0.984523i	$0.556075\pi$
$44$	4.00000	0.603023
$45$	3.70156	0.551796
$46$	7.70156	1.13553
$47$	1.40312	0.204667	0.102333	−	0.994750i	$-0.467369\pi$
$47$	1.40312	0.204667	0.102333	+	0.994750i	$0.467369\pi$
$48$	−1.00000	−0.144338
$49$	6.70156	0.957366
$50$	8.70156	1.23059
$51$	2.00000	0.280056
$52$	4.00000	0.554700
$53$	−11.1047	−1.52535	−0.762673	−	0.646784i	$-0.776113\pi$
$53$	−11.1047	−1.52535	−0.762673	+	0.646784i	$0.776113\pi$
$54$	−1.00000	−0.136083
$55$	14.8062	1.99647
$56$	−3.70156	−0.494642
$57$	7.40312	0.980568
$58$	−4.00000	−0.525226
$59$	−13.1047	−1.70608	−0.853042	−	0.521842i	$-0.825245\pi$
$59$	−13.1047	−1.70608	−0.853042	+	0.521842i	$0.825245\pi$
$60$	−3.70156	−0.477870
$61$	−0.596876	−0.0764221	−0.0382111	−	0.999270i	$-0.512166\pi$
$61$	−0.596876	−0.0764221	−0.0382111	+	0.999270i	$0.512166\pi$
$62$	−0.298438	−0.0379016
$63$	−3.70156	−0.466353
$64$	1.00000	0.125000
$65$	14.8062	1.83649
$66$	−4.00000	−0.492366
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	−2.00000	−0.242536
$69$	−7.70156	−0.927159
$70$	−13.7016	−1.63765
$71$	−13.4031	−1.59066	−0.795329	−	0.606178i	$-0.792702\pi$
$71$	−13.4031	−1.59066	−0.795329	+	0.606178i	$0.792702\pi$
$72$	1.00000	0.117851
$73$	−3.10469	−0.363376	−0.181688	−	0.983356i	$-0.558156\pi$
$73$	−3.10469	−0.363376	−0.181688	+	0.983356i	$0.558156\pi$
$74$	4.29844	0.499683
$75$	−8.70156	−1.00477
$76$	−7.40312	−0.849197
$77$	−14.8062	−1.68733
$78$	−4.00000	−0.452911
$79$	−9.40312	−1.05793	−0.528967	−	0.848642i	$-0.677421\pi$
$79$	−9.40312	−1.05793	−0.528967	+	0.848642i	$0.677421\pi$
$80$	3.70156	0.413847
$81$	1.00000	0.111111
$82$	−3.70156	−0.408769
$83$	17.1047	1.87748	0.938742	−	0.344622i	$-0.111993\pi$
$83$	17.1047	1.87748	0.938742	+	0.344622i	$0.111993\pi$
$84$	3.70156	0.403874
$85$	−7.40312	−0.802982
$86$	−2.29844	−0.247847
$87$	4.00000	0.428845
$88$	4.00000	0.426401
$89$	−2.59688	−0.275268	−0.137634	−	0.990483i	$-0.543950\pi$
$89$	−2.59688	−0.275268	−0.137634	+	0.990483i	$0.543950\pi$
$90$	3.70156	0.390179
$91$	−14.8062	−1.55212
$92$	7.70156	0.802943
$93$	0.298438	0.0309466
$94$	1.40312	0.144721
$95$	−27.4031	−2.81150
$96$	−1.00000	−0.102062
$97$	−13.4031	−1.36088	−0.680441	−	0.732803i	$-0.738212\pi$
$97$	−13.4031	−1.36088	−0.680441	+	0.732803i	$0.738212\pi$
$98$	6.70156	0.676960
$99$	4.00000	0.402015
$100$	8.70156	0.870156
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	2.00000	0.198030
$103$	18.8062	1.85303	0.926517	−	0.376252i	$-0.122787\pi$
$103$	18.8062	1.85303	0.926517	+	0.376252i	$0.122787\pi$
$104$	4.00000	0.392232
$105$	13.7016	1.33714
$106$	−11.1047	−1.07858
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−1.00000	−0.0962250
$109$	−3.40312	−0.325960	−0.162980	−	0.986629i	$-0.552111\pi$
$109$	−3.40312	−0.325960	−0.162980	+	0.986629i	$0.552111\pi$
$110$	14.8062	1.41172
$111$	−4.29844	−0.407990
$112$	−3.70156	−0.349765
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	7.40312	0.693366
$115$	28.5078	2.65837
$116$	−4.00000	−0.371391
$117$	4.00000	0.369800
$118$	−13.1047	−1.20638
$119$	7.40312	0.678643
$120$	−3.70156	−0.337905
$121$	5.00000	0.454545
$122$	−0.596876	−0.0540386
$123$	3.70156	0.333759
$124$	−0.298438	−0.0268005
$125$	13.7016	1.22550
$126$	−3.70156	−0.329761
$127$	10.8062	0.958899	0.479450	−	0.877569i	$-0.340836\pi$
$127$	10.8062	0.958899	0.479450	+	0.877569i	$0.340836\pi$
$128$	1.00000	0.0883883
$129$	2.29844	0.202366
$130$	14.8062	1.29859
$131$	−2.29844	−0.200815	−0.100408	−	0.994946i	$-0.532015\pi$
$131$	−2.29844	−0.200815	−0.100408	+	0.994946i	$0.532015\pi$
$132$	−4.00000	−0.348155
$133$	27.4031	2.37615
$134$	−1.00000	−0.0863868
$135$	−3.70156	−0.318580
$136$	−2.00000	−0.171499
$137$	15.1047	1.29048	0.645240	−	0.763980i	$-0.276757\pi$
$137$	15.1047	1.29048	0.645240	+	0.763980i	$0.276757\pi$
$138$	−7.70156	−0.655601
$139$	−1.70156	−0.144325	−0.0721623	−	0.997393i	$-0.522990\pi$
$139$	−1.70156	−0.144325	−0.0721623	+	0.997393i	$0.522990\pi$
$140$	−13.7016	−1.15799
$141$	−1.40312	−0.118164
$142$	−13.4031	−1.12477
$143$	16.0000	1.33799
$144$	1.00000	0.0833333
$145$	−14.8062	−1.22959
$146$	−3.10469	−0.256946
$147$	−6.70156	−0.552736
$148$	4.29844	0.353329
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	−8.70156	−0.710480
$151$	11.4031	0.927973	0.463987	−	0.885842i	$-0.346419\pi$
$151$	11.4031	0.927973	0.463987	+	0.885842i	$0.346419\pi$
$152$	−7.40312	−0.600473
$153$	−2.00000	−0.161690
$154$	−14.8062	−1.19312
$155$	−1.10469	−0.0887305
$156$	−4.00000	−0.320256
$157$	10.5078	0.838615	0.419307	−	0.907844i	$-0.362273\pi$
$157$	10.5078	0.838615	0.419307	+	0.907844i	$0.362273\pi$
$158$	−9.40312	−0.748072
$159$	11.1047	0.880659
$160$	3.70156	0.292634
$161$	−28.5078	−2.24673
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−3.70156	−0.289043
$165$	−14.8062	−1.15266
$166$	17.1047	1.32758
$167$	3.10469	0.240248	0.120124	−	0.992759i	$-0.461671\pi$
$167$	3.10469	0.240248	0.120124	+	0.992759i	$0.461671\pi$
$168$	3.70156	0.285582
$169$	3.00000	0.230769
$170$	−7.40312	−0.567794
$171$	−7.40312	−0.566131
$172$	−2.29844	−0.175254
$173$	−3.40312	−0.258735	−0.129367	−	0.991597i	$-0.541295\pi$
$173$	−3.40312	−0.258735	−0.129367	+	0.991597i	$0.541295\pi$
$174$	4.00000	0.303239
$175$	−32.2094	−2.43480
$176$	4.00000	0.301511
$177$	13.1047	0.985009
$178$	−2.59688	−0.194644
$179$	23.4031	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$179$	23.4031	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$180$	3.70156	0.275898
$181$	−3.10469	−0.230770	−0.115385	−	0.993321i	$-0.536810\pi$
$181$	−3.10469	−0.230770	−0.115385	+	0.993321i	$0.536810\pi$
$182$	−14.8062	−1.09751
$183$	0.596876	0.0441223
$184$	7.70156	0.567767
$185$	15.9109	1.16980
$186$	0.298438	0.0218825
$187$	−8.00000	−0.585018
$188$	1.40312	0.102333
$189$	3.70156	0.269249
$190$	−27.4031	−1.98803
$191$	10.8062	0.781913	0.390956	−	0.920409i	$-0.372144\pi$
$191$	10.8062	0.781913	0.390956	+	0.920409i	$0.372144\pi$
$192$	−1.00000	−0.0721688
$193$	−18.5078	−1.33222	−0.666111	−	0.745853i	$-0.732042\pi$
$193$	−18.5078	−1.33222	−0.666111	+	0.745853i	$0.732042\pi$
$194$	−13.4031	−0.962288
$195$	−14.8062	−1.06030
$196$	6.70156	0.478683
$197$	4.29844	0.306251	0.153126	−	0.988207i	$-0.451066\pi$
$197$	4.29844	0.306251	0.153126	+	0.988207i	$0.451066\pi$
$198$	4.00000	0.284268
$199$	6.80625	0.482482	0.241241	−	0.970465i	$-0.422446\pi$
$199$	6.80625	0.482482	0.241241	+	0.970465i	$0.422446\pi$
$200$	8.70156	0.615293
$201$	1.00000	0.0705346
$202$	−10.0000	−0.703598
$203$	14.8062	1.03919
$204$	2.00000	0.140028
$205$	−13.7016	−0.956959
$206$	18.8062	1.31029
$207$	7.70156	0.535296
$208$	4.00000	0.277350
$209$	−29.6125	−2.04834
$210$	13.7016	0.945498
$211$	22.2094	1.52896	0.764478	−	0.644650i	$-0.222997\pi$
$211$	22.2094	1.52896	0.764478	+	0.644650i	$0.222997\pi$
$212$	−11.1047	−0.762673
$213$	13.4031	0.918367
$214$	−8.00000	−0.546869
$215$	−8.50781	−0.580228
$216$	−1.00000	−0.0680414
$217$	1.10469	0.0749910
$218$	−3.40312	−0.230489
$219$	3.10469	0.209795
$220$	14.8062	0.998237
$221$	−8.00000	−0.538138
$222$	−4.29844	−0.288492
$223$	−27.4031	−1.83505	−0.917524	−	0.397679i	$-0.869816\pi$
$223$	−27.4031	−1.83505	−0.917524	+	0.397679i	$0.869816\pi$
$224$	−3.70156	−0.247321
$225$	8.70156	0.580104
$226$	6.00000	0.399114
$227$	−19.9109	−1.32154	−0.660768	−	0.750591i	$-0.729769\pi$
$227$	−19.9109	−1.32154	−0.660768	+	0.750591i	$0.729769\pi$
$228$	7.40312	0.490284
$229$	19.4031	1.28219	0.641097	−	0.767460i	$-0.278480\pi$
$229$	19.4031	1.28219	0.641097	+	0.767460i	$0.278480\pi$
$230$	28.5078	1.87975
$231$	14.8062	0.974180
$232$	−4.00000	−0.262613
$233$	29.9109	1.95953	0.979765	−	0.200150i	$-0.0641430\pi$
$233$	29.9109	1.95953	0.979765	+	0.200150i	$0.0641430\pi$
$234$	4.00000	0.261488
$235$	5.19375	0.338803
$236$	−13.1047	−0.853042
$237$	9.40312	0.610799
$238$	7.40312	0.479873
$239$	−11.4031	−0.737607	−0.368803	−	0.929507i	$-0.620232\pi$
$239$	−11.4031	−0.737607	−0.368803	+	0.929507i	$0.620232\pi$
$240$	−3.70156	−0.238935
$241$	30.5078	1.96518	0.982590	−	0.185785i	$-0.0594828\pi$
$241$	30.5078	1.96518	0.982590	+	0.185785i	$0.0594828\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	−0.596876	−0.0382111
$245$	24.8062	1.58481
$246$	3.70156	0.236003
$247$	−29.6125	−1.88420
$248$	−0.298438	−0.0189508
$249$	−17.1047	−1.08397
$250$	13.7016	0.866563
$251$	−26.8062	−1.69200	−0.845998	−	0.533187i	$-0.820994\pi$
$251$	−26.8062	−1.69200	−0.845998	+	0.533187i	$0.820994\pi$
$252$	−3.70156	−0.233176
$253$	30.8062	1.93677
$254$	10.8062	0.678044
$255$	7.40312	0.463602
$256$	1.00000	0.0625000
$257$	−1.40312	−0.0875245	−0.0437622	−	0.999042i	$-0.513934\pi$
$257$	−1.40312	−0.0875245	−0.0437622	+	0.999042i	$0.513934\pi$
$258$	2.29844	0.143094
$259$	−15.9109	−0.988657
$260$	14.8062	0.918245
$261$	−4.00000	−0.247594
$262$	−2.29844	−0.141998
$263$	−6.50781	−0.401289	−0.200644	−	0.979664i	$-0.564304\pi$
$263$	−6.50781	−0.401289	−0.200644	+	0.979664i	$0.564304\pi$
$264$	−4.00000	−0.246183
$265$	−41.1047	−2.52504
$266$	27.4031	1.68019
$267$	2.59688	0.158926
$268$	−1.00000	−0.0610847
$269$	26.2094	1.59801	0.799007	−	0.601322i	$-0.205359\pi$
$269$	26.2094	1.59801	0.799007	+	0.601322i	$0.205359\pi$
$270$	−3.70156	−0.225270
$271$	−14.5969	−0.886697	−0.443349	−	0.896349i	$-0.646210\pi$
$271$	−14.5969	−0.886697	−0.443349	+	0.896349i	$0.646210\pi$
$272$	−2.00000	−0.121268
$273$	14.8062	0.896115
$274$	15.1047	0.912507
$275$	34.8062	2.09890
$276$	−7.70156	−0.463580
$277$	−10.5078	−0.631353	−0.315677	−	0.948867i	$-0.602231\pi$
$277$	−10.5078	−0.631353	−0.315677	+	0.948867i	$0.602231\pi$
$278$	−1.70156	−0.102053
$279$	−0.298438	−0.0178670
$280$	−13.7016	−0.818825
$281$	−12.8062	−0.763957	−0.381978	−	0.924171i	$-0.624757\pi$
$281$	−12.8062	−0.763957	−0.381978	+	0.924171i	$0.624757\pi$
$282$	−1.40312	−0.0835548
$283$	8.59688	0.511031	0.255516	−	0.966805i	$-0.417755\pi$
$283$	8.59688	0.511031	0.255516	+	0.966805i	$0.417755\pi$
$284$	−13.4031	−0.795329
$285$	27.4031	1.62322
$286$	16.0000	0.946100
$287$	13.7016	0.808778
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−14.8062	−0.869453
$291$	13.4031	0.785705
$292$	−3.10469	−0.181688
$293$	−22.2094	−1.29749	−0.648743	−	0.761008i	$-0.724705\pi$
$293$	−22.2094	−1.29749	−0.648743	+	0.761008i	$0.724705\pi$
$294$	−6.70156	−0.390843
$295$	−48.5078	−2.82423
$296$	4.29844	0.249842
$297$	−4.00000	−0.232104
$298$	4.00000	0.231714
$299$	30.8062	1.78157
$300$	−8.70156	−0.502385
$301$	8.50781	0.490382
$302$	11.4031	0.656176
$303$	10.0000	0.574485
$304$	−7.40312	−0.424598
$305$	−2.20937	−0.126508
$306$	−2.00000	−0.114332
$307$	−26.8062	−1.52991	−0.764957	−	0.644082i	$-0.777240\pi$
$307$	−26.8062	−1.52991	−0.764957	+	0.644082i	$0.777240\pi$
$308$	−14.8062	−0.843664
$309$	−18.8062	−1.06985
$310$	−1.10469	−0.0627420
$311$	−11.4031	−0.646612	−0.323306	−	0.946294i	$-0.604794\pi$
$311$	−11.4031	−0.646612	−0.323306	+	0.946294i	$0.604794\pi$
$312$	−4.00000	−0.226455
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	10.5078	0.592990
$315$	−13.7016	−0.771996
$316$	−9.40312	−0.528967
$317$	18.2094	1.02274	0.511370	−	0.859361i	$-0.329138\pi$
$317$	18.2094	1.02274	0.511370	+	0.859361i	$0.329138\pi$
$318$	11.1047	0.622720
$319$	−16.0000	−0.895828
$320$	3.70156	0.206924
$321$	8.00000	0.446516
$322$	−28.5078	−1.58868
$323$	14.8062	0.823842
$324$	1.00000	0.0555556
$325$	34.8062	1.93070
$326$	12.0000	0.664619
$327$	3.40312	0.188193
$328$	−3.70156	−0.204385
$329$	−5.19375	−0.286341
$330$	−14.8062	−0.815057
$331$	31.9109	1.75398	0.876992	−	0.480505i	$-0.159547\pi$
$331$	31.9109	1.75398	0.876992	+	0.480505i	$0.159547\pi$
$332$	17.1047	0.938742
$333$	4.29844	0.235553
$334$	3.10469	0.169881
$335$	−3.70156	−0.202238
$336$	3.70156	0.201937
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	3.00000	0.163178
$339$	−6.00000	−0.325875
$340$	−7.40312	−0.401491
$341$	−1.19375	−0.0646453
$342$	−7.40312	−0.400315
$343$	1.10469	0.0596475
$344$	−2.29844	−0.123923
$345$	−28.5078	−1.53481
$346$	−3.40312	−0.182953
$347$	18.8062	1.00957	0.504786	−	0.863244i	$-0.331571\pi$
$347$	18.8062	1.00957	0.504786	+	0.863244i	$0.331571\pi$
$348$	4.00000	0.214423
$349$	2.50781	0.134240	0.0671200	−	0.997745i	$-0.478619\pi$
$349$	2.50781	0.134240	0.0671200	+	0.997745i	$0.478619\pi$
$350$	−32.2094	−1.72166
$351$	−4.00000	−0.213504
$352$	4.00000	0.213201
$353$	8.89531	0.473450	0.236725	−	0.971577i	$-0.423926\pi$
$353$	8.89531	0.473450	0.236725	+	0.971577i	$0.423926\pi$
$354$	13.1047	0.696506
$355$	−49.6125	−2.63316
$356$	−2.59688	−0.137634
$357$	−7.40312	−0.391815
$358$	23.4031	1.23689
$359$	−11.1047	−0.586083	−0.293041	−	0.956100i	$-0.594667\pi$
$359$	−11.1047	−0.586083	−0.293041	+	0.956100i	$0.594667\pi$
$360$	3.70156	0.195089
$361$	35.8062	1.88454
$362$	−3.10469	−0.163179
$363$	−5.00000	−0.262432
$364$	−14.8062	−0.776058
$365$	−11.4922	−0.601529
$366$	0.596876	0.0311992
$367$	−14.5969	−0.761951	−0.380975	−	0.924585i	$-0.624412\pi$
$367$	−14.5969	−0.761951	−0.380975	+	0.924585i	$0.624412\pi$
$368$	7.70156	0.401472
$369$	−3.70156	−0.192696
$370$	15.9109	0.827170
$371$	41.1047	2.13405
$372$	0.298438	0.0154733
$373$	10.2094	0.528621	0.264311	−	0.964438i	$-0.414856\pi$
$373$	10.2094	0.528621	0.264311	+	0.964438i	$0.414856\pi$
$374$	−8.00000	−0.413670
$375$	−13.7016	−0.707546
$376$	1.40312	0.0723606
$377$	−16.0000	−0.824042
$378$	3.70156	0.190388
$379$	21.1047	1.08407	0.542037	−	0.840354i	$-0.317653\pi$
$379$	21.1047	1.08407	0.542037	+	0.840354i	$0.317653\pi$
$380$	−27.4031	−1.40575
$381$	−10.8062	−0.553621
$382$	10.8062	0.552896
$383$	−2.80625	−0.143393	−0.0716963	−	0.997427i	$-0.522841\pi$
$383$	−2.80625	−0.143393	−0.0716963	+	0.997427i	$0.522841\pi$
$384$	−1.00000	−0.0510310
$385$	−54.8062	−2.79319
$386$	−18.5078	−0.942023
$387$	−2.29844	−0.116836
$388$	−13.4031	−0.680441
$389$	−5.19375	−0.263334	−0.131667	−	0.991294i	$-0.542033\pi$
$389$	−5.19375	−0.263334	−0.131667	+	0.991294i	$0.542033\pi$
$390$	−14.8062	−0.749744
$391$	−15.4031	−0.778969
$392$	6.70156	0.338480
$393$	2.29844	0.115941
$394$	4.29844	0.216552
$395$	−34.8062	−1.75129
$396$	4.00000	0.201008
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	6.80625	0.341166
$399$	−27.4031	−1.37187
$400$	8.70156	0.435078
$401$	−11.1047	−0.554542	−0.277271	−	0.960792i	$-0.589430\pi$
$401$	−11.1047	−0.554542	−0.277271	+	0.960792i	$0.589430\pi$
$402$	1.00000	0.0498755
$403$	−1.19375	−0.0594650
$404$	−10.0000	−0.497519
$405$	3.70156	0.183932
$406$	14.8062	0.734822
$407$	17.1938	0.852263
$408$	2.00000	0.0990148
$409$	10.5969	0.523982	0.261991	−	0.965070i	$-0.415621\pi$
$409$	10.5969	0.523982	0.261991	+	0.965070i	$0.415621\pi$
$410$	−13.7016	−0.676672
$411$	−15.1047	−0.745059
$412$	18.8062	0.926517
$413$	48.5078	2.38691
$414$	7.70156	0.378511
$415$	63.3141	3.10796
$416$	4.00000	0.196116
$417$	1.70156	0.0833259
$418$	−29.6125	−1.44839
$419$	9.10469	0.444793	0.222397	−	0.974956i	$-0.428612\pi$
$419$	9.10469	0.444793	0.222397	+	0.974956i	$0.428612\pi$
$420$	13.7016	0.668568
$421$	−12.2984	−0.599389	−0.299695	−	0.954035i	$-0.596885\pi$
$421$	−12.2984	−0.599389	−0.299695	+	0.954035i	$0.596885\pi$
$422$	22.2094	1.08114
$423$	1.40312	0.0682222
$424$	−11.1047	−0.539291
$425$	−17.4031	−0.844176
$426$	13.4031	0.649383
$427$	2.20937	0.106919
$428$	−8.00000	−0.386695
$429$	−16.0000	−0.772487
$430$	−8.50781	−0.410283
$431$	5.91093	0.284720	0.142360	−	0.989815i	$-0.454531\pi$
$431$	5.91093	0.284720	0.142360	+	0.989815i	$0.454531\pi$
$432$	−1.00000	−0.0481125
$433$	2.59688	0.124798	0.0623989	−	0.998051i	$-0.480125\pi$
$433$	2.59688	0.124798	0.0623989	+	0.998051i	$0.480125\pi$
$434$	1.10469	0.0530266
$435$	14.8062	0.709905
$436$	−3.40312	−0.162980
$437$	−57.0156	−2.72743
$438$	3.10469	0.148348
$439$	10.8062	0.515754	0.257877	−	0.966178i	$-0.416977\pi$
$439$	10.8062	0.515754	0.257877	+	0.966178i	$0.416977\pi$
$440$	14.8062	0.705860
$441$	6.70156	0.319122
$442$	−8.00000	−0.380521
$443$	−8.59688	−0.408450	−0.204225	−	0.978924i	$-0.565467\pi$
$443$	−8.59688	−0.408450	−0.204225	+	0.978924i	$0.565467\pi$
$444$	−4.29844	−0.203995
$445$	−9.61250	−0.455676
$446$	−27.4031	−1.29758
$447$	−4.00000	−0.189194
$448$	−3.70156	−0.174882
$449$	0.209373	0.00988091	0.00494045	−	0.999988i	$-0.498427\pi$
$449$	0.209373	0.00988091	0.00494045	+	0.999988i	$0.498427\pi$
$450$	8.70156	0.410196
$451$	−14.8062	−0.697199
$452$	6.00000	0.282216
$453$	−11.4031	−0.535766
$454$	−19.9109	−0.934466
$455$	−54.8062	−2.56936
$456$	7.40312	0.346683
$457$	−39.6125	−1.85299	−0.926497	−	0.376302i	$-0.877196\pi$
$457$	−39.6125	−1.85299	−0.926497	+	0.376302i	$0.877196\pi$
$458$	19.4031	0.906648
$459$	2.00000	0.0933520
$460$	28.5078	1.32918
$461$	−25.6125	−1.19289	−0.596446	−	0.802653i	$-0.703421\pi$
$461$	−25.6125	−1.19289	−0.596446	+	0.802653i	$0.703421\pi$
$462$	14.8062	0.688849
$463$	16.8953	0.785192	0.392596	−	0.919711i	$-0.371577\pi$
$463$	16.8953	0.785192	0.392596	+	0.919711i	$0.371577\pi$
$464$	−4.00000	−0.185695
$465$	1.10469	0.0512286
$466$	29.9109	1.38560
$467$	−17.6125	−0.815009	−0.407505	−	0.913203i	$-0.633601\pi$
$467$	−17.6125	−0.815009	−0.407505	+	0.913203i	$0.633601\pi$
$468$	4.00000	0.184900
$469$	3.70156	0.170922
$470$	5.19375	0.239570
$471$	−10.5078	−0.484174
$472$	−13.1047	−0.603192
$473$	−9.19375	−0.422729
$474$	9.40312	0.431900
$475$	−64.4187	−2.95573
$476$	7.40312	0.339322
$477$	−11.1047	−0.508449
$478$	−11.4031	−0.521567
$479$	−5.91093	−0.270078	−0.135039	−	0.990840i	$-0.543116\pi$
$479$	−5.91093	−0.270078	−0.135039	+	0.990840i	$0.543116\pi$
$480$	−3.70156	−0.168952
$481$	17.1938	0.783968
$482$	30.5078	1.38959
$483$	28.5078	1.29715
$484$	5.00000	0.227273
$485$	−49.6125	−2.25279
$486$	−1.00000	−0.0453609
$487$	42.5078	1.92621	0.963106	−	0.269121i	$-0.0867331\pi$
$487$	42.5078	1.92621	0.963106	+	0.269121i	$0.0867331\pi$
$488$	−0.596876	−0.0270193
$489$	−12.0000	−0.542659
$490$	24.8062	1.12063
$491$	1.10469	0.0498538	0.0249269	−	0.999689i	$-0.492065\pi$
$491$	1.10469	0.0498538	0.0249269	+	0.999689i	$0.492065\pi$
$492$	3.70156	0.166879
$493$	8.00000	0.360302
$494$	−29.6125	−1.33233
$495$	14.8062	0.665491
$496$	−0.298438	−0.0134003
$497$	49.6125	2.22542
$498$	−17.1047	−0.766479
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	13.7016	0.612752
$501$	−3.10469	−0.138707
$502$	−26.8062	−1.19642
$503$	−17.6125	−0.785302	−0.392651	−	0.919687i	$-0.628442\pi$
$503$	−17.6125	−0.785302	−0.392651	+	0.919687i	$0.628442\pi$
$504$	−3.70156	−0.164881
$505$	−37.0156	−1.64717
$506$	30.8062	1.36950
$507$	−3.00000	−0.133235
$508$	10.8062	0.479450
$509$	−27.4031	−1.21462	−0.607311	−	0.794464i	$-0.707752\pi$
$509$	−27.4031	−1.21462	−0.607311	+	0.794464i	$0.707752\pi$
$510$	7.40312	0.327816
$511$	11.4922	0.508385
$512$	1.00000	0.0441942
$513$	7.40312	0.326856
$514$	−1.40312	−0.0618892
$515$	69.6125	3.06749
$516$	2.29844	0.101183
$517$	5.61250	0.246837
$518$	−15.9109	−0.699086
$519$	3.40312	0.149381
$520$	14.8062	0.649297
$521$	44.8062	1.96300	0.981499	−	0.191469i	$-0.0613251\pi$
$521$	44.8062	1.96300	0.981499	+	0.191469i	$0.0613251\pi$
$522$	−4.00000	−0.175075
$523$	25.6125	1.11996	0.559978	−	0.828507i	$-0.310810\pi$
$523$	25.6125	1.11996	0.559978	+	0.828507i	$0.310810\pi$
$524$	−2.29844	−0.100408
$525$	32.2094	1.40573
$526$	−6.50781	−0.283754
$527$	0.596876	0.0260003
$528$	−4.00000	−0.174078
$529$	36.3141	1.57887
$530$	−41.1047	−1.78547
$531$	−13.1047	−0.568695
$532$	27.4031	1.18808
$533$	−14.8062	−0.641330
$534$	2.59688	0.112378
$535$	−29.6125	−1.28026
$536$	−1.00000	−0.0431934
$537$	−23.4031	−1.00992
$538$	26.2094	1.12997
$539$	26.8062	1.15463
$540$	−3.70156	−0.159290
$541$	12.5969	0.541582	0.270791	−	0.962638i	$-0.412715\pi$
$541$	12.5969	0.541582	0.270791	+	0.962638i	$0.412715\pi$
$542$	−14.5969	−0.626990
$543$	3.10469	0.133235
$544$	−2.00000	−0.0857493
$545$	−12.5969	−0.539591
$546$	14.8062	0.633649
$547$	36.5078	1.56096	0.780481	−	0.625180i	$-0.214974\pi$
$547$	36.5078	1.56096	0.780481	+	0.625180i	$0.214974\pi$
$548$	15.1047	0.645240
$549$	−0.596876	−0.0254740
$550$	34.8062	1.48414
$551$	29.6125	1.26153
$552$	−7.70156	−0.327800
$553$	34.8062	1.48011
$554$	−10.5078	−0.446434
$555$	−15.9109	−0.675382
$556$	−1.70156	−0.0721623
$557$	−6.20937	−0.263100	−0.131550	−	0.991310i	$-0.541995\pi$
$557$	−6.20937	−0.263100	−0.131550	+	0.991310i	$0.541995\pi$
$558$	−0.298438	−0.0126339
$559$	−9.19375	−0.388854
$560$	−13.7016	−0.578997
$561$	8.00000	0.337760
$562$	−12.8062	−0.540199
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	−1.40312	−0.0590822
$565$	22.2094	0.934355
$566$	8.59688	0.361354
$567$	−3.70156	−0.155451
$568$	−13.4031	−0.562383
$569$	−37.4031	−1.56802	−0.784010	−	0.620748i	$-0.786829\pi$
$569$	−37.4031	−1.56802	−0.784010	+	0.620748i	$0.786829\pi$
$570$	27.4031	1.14779
$571$	0.596876	0.0249785	0.0124892	−	0.999922i	$-0.496024\pi$
$571$	0.596876	0.0249785	0.0124892	+	0.999922i	$0.496024\pi$
$572$	16.0000	0.668994
$573$	−10.8062	−0.451438
$574$	13.7016	0.571892
$575$	67.0156	2.79474
$576$	1.00000	0.0416667
$577$	47.6125	1.98213	0.991067	−	0.133364i	$-0.0425780\pi$
$577$	47.6125	1.98213	0.991067	+	0.133364i	$0.0425780\pi$
$578$	−13.0000	−0.540729
$579$	18.5078	0.769158
$580$	−14.8062	−0.614796
$581$	−63.3141	−2.62671
$582$	13.4031	0.555577
$583$	−44.4187	−1.83964
$584$	−3.10469	−0.128473
$585$	14.8062	0.612163
$586$	−22.2094	−0.917461
$587$	25.6125	1.05714	0.528570	−	0.848889i	$-0.322728\pi$
$587$	25.6125	1.05714	0.528570	+	0.848889i	$0.322728\pi$
$588$	−6.70156	−0.276368
$589$	2.20937	0.0910356
$590$	−48.5078	−1.99703
$591$	−4.29844	−0.176814
$592$	4.29844	0.176665
$593$	−1.91093	−0.0784727	−0.0392363	−	0.999230i	$-0.512493\pi$
$593$	−1.91093	−0.0784727	−0.0392363	+	0.999230i	$0.512493\pi$
$594$	−4.00000	−0.164122
$595$	27.4031	1.12342
$596$	4.00000	0.163846
$597$	−6.80625	−0.278561
$598$	30.8062	1.25976
$599$	−30.8062	−1.25871	−0.629355	−	0.777118i	$-0.716681\pi$
$599$	−30.8062	−1.25871	−0.629355	+	0.777118i	$0.716681\pi$
$600$	−8.70156	−0.355240
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	8.50781	0.346752
$603$	−1.00000	−0.0407231
$604$	11.4031	0.463987
$605$	18.5078	0.752450
$606$	10.0000	0.406222
$607$	−24.5969	−0.998356	−0.499178	−	0.866499i	$-0.666365\pi$
$607$	−24.5969	−0.998356	−0.499178	+	0.866499i	$0.666365\pi$
$608$	−7.40312	−0.300236
$609$	−14.8062	−0.599979
$610$	−2.20937	−0.0894549
$611$	5.61250	0.227057
$612$	−2.00000	−0.0808452
$613$	43.1047	1.74098	0.870491	−	0.492184i	$-0.163801\pi$
$613$	43.1047	1.74098	0.870491	+	0.492184i	$0.163801\pi$
$614$	−26.8062	−1.08181
$615$	13.7016	0.552500
$616$	−14.8062	−0.596561
$617$	11.1938	0.450643	0.225322	−	0.974284i	$-0.427657\pi$
$617$	11.1938	0.450643	0.225322	+	0.974284i	$0.427657\pi$
$618$	−18.8062	−0.756498
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−1.10469	−0.0443653
$621$	−7.70156	−0.309053
$622$	−11.4031	−0.457224
$623$	9.61250	0.385117
$624$	−4.00000	−0.160128
$625$	7.20937	0.288375
$626$	−18.0000	−0.719425
$627$	29.6125	1.18261
$628$	10.5078	0.419307
$629$	−8.59688	−0.342780
$630$	−13.7016	−0.545883
$631$	−13.4031	−0.533570	−0.266785	−	0.963756i	$-0.585961\pi$
$631$	−13.4031	−0.533570	−0.266785	+	0.963756i	$0.585961\pi$
$632$	−9.40312	−0.374036
$633$	−22.2094	−0.882743
$634$	18.2094	0.723187
$635$	40.0000	1.58735
$636$	11.1047	0.440329
$637$	26.8062	1.06210
$638$	−16.0000	−0.633446
$639$	−13.4031	−0.530219
$640$	3.70156	0.146317
$641$	−13.3141	−0.525874	−0.262937	−	0.964813i	$-0.584691\pi$
$641$	−13.3141	−0.525874	−0.262937	+	0.964813i	$0.584691\pi$
$642$	8.00000	0.315735
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−28.5078	−1.12337
$645$	8.50781	0.334995
$646$	14.8062	0.582544
$647$	−29.0156	−1.14072	−0.570361	−	0.821394i	$-0.693197\pi$
$647$	−29.0156	−1.14072	−0.570361	+	0.821394i	$0.693197\pi$
$648$	1.00000	0.0392837
$649$	−52.4187	−2.05762
$650$	34.8062	1.36521
$651$	−1.10469	−0.0432961
$652$	12.0000	0.469956
$653$	−3.70156	−0.144853	−0.0724267	−	0.997374i	$-0.523074\pi$
$653$	−3.70156	−0.144853	−0.0724267	+	0.997374i	$0.523074\pi$
$654$	3.40312	0.133073
$655$	−8.50781	−0.332428
$656$	−3.70156	−0.144522
$657$	−3.10469	−0.121125
$658$	−5.19375	−0.202474
$659$	−13.1047	−0.510486	−0.255243	−	0.966877i	$-0.582155\pi$
$659$	−13.1047	−0.510486	−0.255243	+	0.966877i	$0.582155\pi$
$660$	−14.8062	−0.576332
$661$	−34.2094	−1.33059	−0.665295	−	0.746580i	$-0.731694\pi$
$661$	−34.2094	−1.33059	−0.665295	+	0.746580i	$0.731694\pi$
$662$	31.9109	1.24025
$663$	8.00000	0.310694
$664$	17.1047	0.663791
$665$	101.434	3.93346
$666$	4.29844	0.166561
$667$	−30.8062	−1.19282
$668$	3.10469	0.120124
$669$	27.4031	1.05947
$670$	−3.70156	−0.143004
$671$	−2.38750	−0.0921685
$672$	3.70156	0.142791
$673$	−12.2094	−0.470637	−0.235318	−	0.971918i	$-0.575613\pi$
$673$	−12.2094	−0.470637	−0.235318	+	0.971918i	$0.575613\pi$
$674$	2.00000	0.0770371
$675$	−8.70156	−0.334923
$676$	3.00000	0.115385
$677$	−3.70156	−0.142263	−0.0711313	−	0.997467i	$-0.522661\pi$
$677$	−3.70156	−0.142263	−0.0711313	+	0.997467i	$0.522661\pi$
$678$	−6.00000	−0.230429
$679$	49.6125	1.90395
$680$	−7.40312	−0.283897
$681$	19.9109	0.762989
$682$	−1.19375	−0.0457111
$683$	−6.20937	−0.237595	−0.118798	−	0.992919i	$-0.537904\pi$
$683$	−6.20937	−0.237595	−0.118798	+	0.992919i	$0.537904\pi$
$684$	−7.40312	−0.283066
$685$	55.9109	2.13625
$686$	1.10469	0.0421771
$687$	−19.4031	−0.740275
$688$	−2.29844	−0.0876271
$689$	−44.4187	−1.69222
$690$	−28.5078	−1.08527
$691$	47.4031	1.80330	0.901650	−	0.432467i	$-0.142357\pi$
$691$	47.4031	1.80330	0.901650	+	0.432467i	$0.142357\pi$
$692$	−3.40312	−0.129367
$693$	−14.8062	−0.562443
$694$	18.8062	0.713875
$695$	−6.29844	−0.238913
$696$	4.00000	0.151620
$697$	7.40312	0.280413
$698$	2.50781	0.0949220
$699$	−29.9109	−1.13134
$700$	−32.2094	−1.21740
$701$	−4.89531	−0.184893	−0.0924467	−	0.995718i	$-0.529469\pi$
$701$	−4.89531	−0.184893	−0.0924467	+	0.995718i	$0.529469\pi$
$702$	−4.00000	−0.150970
$703$	−31.8219	−1.20018
$704$	4.00000	0.150756
$705$	−5.19375	−0.195608
$706$	8.89531	0.334780
$707$	37.0156	1.39212
$708$	13.1047	0.492504
$709$	11.1047	0.417045	0.208523	−	0.978018i	$-0.433135\pi$
$709$	11.1047	0.417045	0.208523	+	0.978018i	$0.433135\pi$
$710$	−49.6125	−1.86192
$711$	−9.40312	−0.352645
$712$	−2.59688	−0.0973220
$713$	−2.29844	−0.0860772
$714$	−7.40312	−0.277055
$715$	59.2250	2.21489
$716$	23.4031	0.874616
$717$	11.4031	0.425857
$718$	−11.1047	−0.414423
$719$	−29.3141	−1.09323	−0.546615	−	0.837384i	$-0.684084\pi$
$719$	−29.3141	−1.09323	−0.546615	+	0.837384i	$0.684084\pi$
$720$	3.70156	0.137949
$721$	−69.6125	−2.59250
$722$	35.8062	1.33257
$723$	−30.5078	−1.13460
$724$	−3.10469	−0.115385
$725$	−34.8062	−1.29267
$726$	−5.00000	−0.185567
$727$	6.08907	0.225831	0.112915	−	0.993605i	$-0.463981\pi$
$727$	6.08907	0.225831	0.112915	+	0.993605i	$0.463981\pi$
$728$	−14.8062	−0.548756
$729$	1.00000	0.0370370
$730$	−11.4922	−0.425345
$731$	4.59688	0.170022
$732$	0.596876	0.0220612
$733$	−47.8219	−1.76634	−0.883171	−	0.469052i	$-0.844596\pi$
$733$	−47.8219	−1.76634	−0.883171	+	0.469052i	$0.844596\pi$
$734$	−14.5969	−0.538781
$735$	−24.8062	−0.914992
$736$	7.70156	0.283883
$737$	−4.00000	−0.147342
$738$	−3.70156	−0.136256
$739$	−24.5078	−0.901534	−0.450767	−	0.892642i	$-0.648850\pi$
$739$	−24.5078	−0.901534	−0.450767	+	0.892642i	$0.648850\pi$
$740$	15.9109	0.584898
$741$	29.6125	1.08784
$742$	41.1047	1.50900
$743$	13.3141	0.488445	0.244223	−	0.969719i	$-0.421467\pi$
$743$	13.3141	0.488445	0.244223	+	0.969719i	$0.421467\pi$
$744$	0.298438	0.0109413
$745$	14.8062	0.542459
$746$	10.2094	0.373792
$747$	17.1047	0.625828
$748$	−8.00000	−0.292509
$749$	29.6125	1.08202
$750$	−13.7016	−0.500310
$751$	−16.5969	−0.605629	−0.302814	−	0.953050i	$-0.597926\pi$
$751$	−16.5969	−0.605629	−0.302814	+	0.953050i	$0.597926\pi$
$752$	1.40312	0.0511667
$753$	26.8062	0.976874
$754$	−16.0000	−0.582686
$755$	42.2094	1.53616
$756$	3.70156	0.134625
$757$	−12.5969	−0.457841	−0.228921	−	0.973445i	$-0.573520\pi$
$757$	−12.5969	−0.457841	−0.228921	+	0.973445i	$0.573520\pi$
$758$	21.1047	0.766557
$759$	−30.8062	−1.11820
$760$	−27.4031	−0.994016
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−10.8062	−0.391469
$763$	12.5969	0.456038
$764$	10.8062	0.390956
$765$	−7.40312	−0.267661
$766$	−2.80625	−0.101394
$767$	−52.4187	−1.89273
$768$	−1.00000	−0.0360844
$769$	11.7906	0.425181	0.212590	−	0.977141i	$-0.431810\pi$
$769$	11.7906	0.425181	0.212590	+	0.977141i	$0.431810\pi$
$770$	−54.8062	−1.97508
$771$	1.40312	0.0505323
$772$	−18.5078	−0.666111
$773$	−48.0000	−1.72644	−0.863220	−	0.504828i	$-0.831556\pi$
$773$	−48.0000	−1.72644	−0.863220	+	0.504828i	$0.831556\pi$
$774$	−2.29844	−0.0826156
$775$	−2.59688	−0.0932825
$776$	−13.4031	−0.481144
$777$	15.9109	0.570802
$778$	−5.19375	−0.186205
$779$	27.4031	0.981819
$780$	−14.8062	−0.530149
$781$	−53.6125	−1.91841
$782$	−15.4031	−0.550815
$783$	4.00000	0.142948
$784$	6.70156	0.239342
$785$	38.8953	1.38823
$786$	2.29844	0.0819826
$787$	−15.4922	−0.552237	−0.276118	−	0.961124i	$-0.589048\pi$
$787$	−15.4922	−0.552237	−0.276118	+	0.961124i	$0.589048\pi$
$788$	4.29844	0.153126
$789$	6.50781	0.231684
$790$	−34.8062	−1.23835
$791$	−22.2094	−0.789674
$792$	4.00000	0.142134
$793$	−2.38750	−0.0847827
$794$	−6.00000	−0.212932
$795$	41.1047	1.45783
$796$	6.80625	0.241241
$797$	−16.0000	−0.566749	−0.283375	−	0.959009i	$-0.591454\pi$
$797$	−16.0000	−0.566749	−0.283375	+	0.959009i	$0.591454\pi$
$798$	−27.4031	−0.970060
$799$	−2.80625	−0.0992779
$800$	8.70156	0.307647
$801$	−2.59688	−0.0917561
$802$	−11.1047	−0.392120
$803$	−12.4187	−0.438248
$804$	1.00000	0.0352673
$805$	−105.523	−3.71921
$806$	−1.19375	−0.0420481
$807$	−26.2094	−0.922614
$808$	−10.0000	−0.351799
$809$	28.8953	1.01591	0.507953	−	0.861385i	$-0.330403\pi$
$809$	28.8953	1.01591	0.507953	+	0.861385i	$0.330403\pi$
$810$	3.70156	0.130060
$811$	41.7016	1.46434	0.732170	−	0.681122i	$-0.238508\pi$
$811$	41.7016	1.46434	0.732170	+	0.681122i	$0.238508\pi$
$812$	14.8062	0.519597
$813$	14.5969	0.511935
$814$	17.1938	0.602641
$815$	44.4187	1.55592
$816$	2.00000	0.0700140
$817$	17.0156	0.595301
$818$	10.5969	0.370511
$819$	−14.8062	−0.517372
$820$	−13.7016	−0.478479
$821$	28.0000	0.977207	0.488603	−	0.872506i	$-0.337507\pi$
$821$	28.0000	0.977207	0.488603	+	0.872506i	$0.337507\pi$
$822$	−15.1047	−0.526836
$823$	30.2094	1.05303	0.526516	−	0.850165i	$-0.323498\pi$
$823$	30.2094	1.05303	0.526516	+	0.850165i	$0.323498\pi$
$824$	18.8062	0.655147
$825$	−34.8062	−1.21180
$826$	48.5078	1.68780
$827$	−5.61250	−0.195166	−0.0975828	−	0.995227i	$-0.531111\pi$
$827$	−5.61250	−0.195166	−0.0975828	+	0.995227i	$0.531111\pi$
$828$	7.70156	0.267648
$829$	0.387503	0.0134585	0.00672927	−	0.999977i	$-0.497858\pi$
$829$	0.387503	0.0134585	0.00672927	+	0.999977i	$0.497858\pi$
$830$	63.3141	2.19766
$831$	10.5078	0.364512
$832$	4.00000	0.138675
$833$	−13.4031	−0.464391
$834$	1.70156	0.0589203
$835$	11.4922	0.397704
$836$	−29.6125	−1.02417
$837$	0.298438	0.0103155
$838$	9.10469	0.314516
$839$	19.0156	0.656492	0.328246	−	0.944592i	$-0.393542\pi$
$839$	19.0156	0.656492	0.328246	+	0.944592i	$0.393542\pi$
$840$	13.7016	0.472749
$841$	−13.0000	−0.448276
$842$	−12.2984	−0.423832
$843$	12.8062	0.441071
$844$	22.2094	0.764478
$845$	11.1047	0.382013
$846$	1.40312	0.0482404
$847$	−18.5078	−0.635936
$848$	−11.1047	−0.381336
$849$	−8.59688	−0.295044
$850$	−17.4031	−0.596922
$851$	33.1047	1.13481
$852$	13.4031	0.459183
$853$	23.6125	0.808476	0.404238	−	0.914654i	$-0.367537\pi$
$853$	23.6125	0.808476	0.404238	+	0.914654i	$0.367537\pi$
$854$	2.20937	0.0756032
$855$	−27.4031	−0.937167
$856$	−8.00000	−0.273434
$857$	15.7016	0.536355	0.268178	−	0.963369i	$-0.413579\pi$
$857$	15.7016	0.536355	0.268178	+	0.963369i	$0.413579\pi$
$858$	−16.0000	−0.546231
$859$	7.40312	0.252591	0.126296	−	0.991993i	$-0.459691\pi$
$859$	7.40312	0.252591	0.126296	+	0.991993i	$0.459691\pi$
$860$	−8.50781	−0.290114
$861$	−13.7016	−0.466948
$862$	5.91093	0.201327
$863$	−6.08907	−0.207274	−0.103637	−	0.994615i	$-0.533048\pi$
$863$	−6.08907	−0.207274	−0.103637	+	0.994615i	$0.533048\pi$
$864$	−1.00000	−0.0340207
$865$	−12.5969	−0.428307
$866$	2.59688	0.0882454
$867$	13.0000	0.441503
$868$	1.10469	0.0374955
$869$	−37.6125	−1.27592
$870$	14.8062	0.501979
$871$	−4.00000	−0.135535
$872$	−3.40312	−0.115244
$873$	−13.4031	−0.453627
$874$	−57.0156	−1.92858
$875$	−50.7172	−1.71455
$876$	3.10469	0.104898
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	10.8062	0.364693
$879$	22.2094	0.749104
$880$	14.8062	0.499119
$881$	−26.5969	−0.896072	−0.448036	−	0.894016i	$-0.647876\pi$
$881$	−26.5969	−0.896072	−0.448036	+	0.894016i	$0.647876\pi$
$882$	6.70156	0.225653
$883$	6.29844	0.211959	0.105980	−	0.994368i	$-0.466202\pi$
$883$	6.29844	0.211959	0.105980	+	0.994368i	$0.466202\pi$
$884$	−8.00000	−0.269069
$885$	48.5078	1.63057
$886$	−8.59688	−0.288818
$887$	41.3141	1.38719	0.693595	−	0.720365i	$-0.256026\pi$
$887$	41.3141	1.38719	0.693595	+	0.720365i	$0.256026\pi$
$888$	−4.29844	−0.144246
$889$	−40.0000	−1.34156
$890$	−9.61250	−0.322212
$891$	4.00000	0.134005
$892$	−27.4031	−0.917524
$893$	−10.3875	−0.347605
$894$	−4.00000	−0.133780
$895$	86.6281	2.89566
$896$	−3.70156	−0.123661
$897$	−30.8062	−1.02859
$898$	0.209373	0.00698686
$899$	1.19375	0.0398138
$900$	8.70156	0.290052
$901$	22.2094	0.739901
$902$	−14.8062	−0.492994
$903$	−8.50781	−0.283122
$904$	6.00000	0.199557
$905$	−11.4922	−0.382013
$906$	−11.4031	−0.378843
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−19.9109	−0.660768
$909$	−10.0000	−0.331679
$910$	−54.8062	−1.81681
$911$	−21.4031	−0.709117	−0.354559	−	0.935034i	$-0.615369\pi$
$911$	−21.4031	−0.709117	−0.354559	+	0.935034i	$0.615369\pi$
$912$	7.40312	0.245142
$913$	68.4187	2.26433
$914$	−39.6125	−1.31026
$915$	2.20937	0.0730396
$916$	19.4031	0.641097
$917$	8.50781	0.280953
$918$	2.00000	0.0660098
$919$	8.29844	0.273740	0.136870	−	0.990589i	$-0.456296\pi$
$919$	8.29844	0.273740	0.136870	+	0.990589i	$0.456296\pi$
$920$	28.5078	0.939875
$921$	26.8062	0.883296
$922$	−25.6125	−0.843503
$923$	−53.6125	−1.76468
$924$	14.8062	0.487090
$925$	37.4031	1.22981
$926$	16.8953	0.555214
$927$	18.8062	0.617678
$928$	−4.00000	−0.131306
$929$	4.80625	0.157688	0.0788439	−	0.996887i	$-0.474877\pi$
$929$	4.80625	0.157688	0.0788439	+	0.996887i	$0.474877\pi$
$930$	1.10469	0.0362241
$931$	−49.6125	−1.62598
$932$	29.9109	0.979765
$933$	11.4031	0.373322
$934$	−17.6125	−0.576299
$935$	−29.6125	−0.968432
$936$	4.00000	0.130744
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	3.70156	0.120860
$939$	18.0000	0.587408
$940$	5.19375	0.169401
$941$	19.6125	0.639349	0.319675	−	0.947527i	$-0.396426\pi$
$941$	19.6125	0.639349	0.319675	+	0.947527i	$0.396426\pi$
$942$	−10.5078	−0.342363
$943$	−28.5078	−0.928342
$944$	−13.1047	−0.426521
$945$	13.7016	0.445712
$946$	−9.19375	−0.298915
$947$	−11.3141	−0.367658	−0.183829	−	0.982958i	$-0.558849\pi$
$947$	−11.3141	−0.367658	−0.183829	+	0.982958i	$0.558849\pi$
$948$	9.40312	0.305399
$949$	−12.4187	−0.403130
$950$	−64.4187	−2.09002
$951$	−18.2094	−0.590479
$952$	7.40312	0.239937
$953$	33.4031	1.08203	0.541017	−	0.841012i	$-0.318040\pi$
$953$	33.4031	1.08203	0.541017	+	0.841012i	$0.318040\pi$
$954$	−11.1047	−0.359527
$955$	40.0000	1.29437
$956$	−11.4031	−0.368803
$957$	16.0000	0.517207
$958$	−5.91093	−0.190974
$959$	−55.9109	−1.80546
$960$	−3.70156	−0.119467
$961$	−30.9109	−0.997127
$962$	17.1938	0.554349
$963$	−8.00000	−0.257796
$964$	30.5078	0.982590
$965$	−68.5078	−2.20534
$966$	28.5078	0.917224
$967$	9.01562	0.289923	0.144961	−	0.989437i	$-0.453694\pi$
$967$	9.01562	0.289923	0.144961	+	0.989437i	$0.453694\pi$
$968$	5.00000	0.160706
$969$	−14.8062	−0.475645
$970$	−49.6125	−1.59296
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	−1.00000	−0.0320750
$973$	6.29844	0.201919
$974$	42.5078	1.36204
$975$	−34.8062	−1.11469
$976$	−0.596876	−0.0191055
$977$	43.6125	1.39529	0.697644	−	0.716445i	$-0.254232\pi$
$977$	43.6125	1.39529	0.697644	+	0.716445i	$0.254232\pi$
$978$	−12.0000	−0.383718
$979$	−10.3875	−0.331986
$980$	24.8062	0.792407
$981$	−3.40312	−0.108653
$982$	1.10469	0.0352520
$983$	−53.6125	−1.70997	−0.854987	−	0.518650i	$-0.826435\pi$
$983$	−53.6125	−1.70997	−0.854987	+	0.518650i	$0.826435\pi$
$984$	3.70156	0.118001
$985$	15.9109	0.506965
$986$	8.00000	0.254772
$987$	5.19375	0.165319
$988$	−29.6125	−0.942099
$989$	−17.7016	−0.562877
$990$	14.8062	0.470573
$991$	−7.70156	−0.244648	−0.122324	−	0.992490i	$-0.539035\pi$
$991$	−7.70156	−0.244648	−0.122324	+	0.992490i	$0.539035\pi$
$992$	−0.298438	−0.00947541
$993$	−31.9109	−1.01266
$994$	49.6125	1.57361
$995$	25.1938	0.798696
$996$	−17.1047	−0.541983
$997$	43.7016	1.38404	0.692021	−	0.721877i	$-0.256720\pi$
$997$	43.7016	1.38404	0.692021	+	0.721877i	$0.256720\pi$
$998$	0	0
$999$	−4.29844	−0.135997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.a.f.1.2	✓	2
3.2	odd	2		1206.2.a.h.1.1		2
4.3	odd	2		3216.2.a.p.1.2		2
12.11	even	2		9648.2.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.a.f.1.2	✓	2	1.1	even	1	trivial
1206.2.a.h.1.1		2	3.2	odd	2
3216.2.a.p.1.2		2	4.3	odd	2
9648.2.a.z.1.1		2	12.11	even	2

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:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	402.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{6} -3.70156 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{6} -3.70156 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.70156 q^{10} +4.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -3.70156 q^{14} -3.70156 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -7.40312 q^{19} +3.70156 q^{20} +3.70156 q^{21} +4.00000 q^{22} +7.70156 q^{23} -1.00000 q^{24} +8.70156 q^{25} +4.00000 q^{26} -1.00000 q^{27} -3.70156 q^{28} -4.00000 q^{29} -3.70156 q^{30} -0.298438 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -13.7016 q^{35} +1.00000 q^{36} +4.29844 q^{37} -7.40312 q^{38} -4.00000 q^{39} +3.70156 q^{40} -3.70156 q^{41} +3.70156 q^{42} -2.29844 q^{43} +4.00000 q^{44} +3.70156 q^{45} +7.70156 q^{46} +1.40312 q^{47} -1.00000 q^{48} +6.70156 q^{49} +8.70156 q^{50} +2.00000 q^{51} +4.00000 q^{52} -11.1047 q^{53} -1.00000 q^{54} +14.8062 q^{55} -3.70156 q^{56} +7.40312 q^{57} -4.00000 q^{58} -13.1047 q^{59} -3.70156 q^{60} -0.596876 q^{61} -0.298438 q^{62} -3.70156 q^{63} +1.00000 q^{64} +14.8062 q^{65} -4.00000 q^{66} -1.00000 q^{67} -2.00000 q^{68} -7.70156 q^{69} -13.7016 q^{70} -13.4031 q^{71} +1.00000 q^{72} -3.10469 q^{73} +4.29844 q^{74} -8.70156 q^{75} -7.40312 q^{76} -14.8062 q^{77} -4.00000 q^{78} -9.40312 q^{79} +3.70156 q^{80} +1.00000 q^{81} -3.70156 q^{82} +17.1047 q^{83} +3.70156 q^{84} -7.40312 q^{85} -2.29844 q^{86} +4.00000 q^{87} +4.00000 q^{88} -2.59688 q^{89} +3.70156 q^{90} -14.8062 q^{91} +7.70156 q^{92} +0.298438 q^{93} +1.40312 q^{94} -27.4031 q^{95} -1.00000 q^{96} -13.4031 q^{97} +6.70156 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} + q^{10} + 8 q^{11} - 2 q^{12} + 8 q^{13} - q^{14} - q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 2 q^{19} + q^{20} + q^{21} + 8 q^{22} + 9 q^{23} - 2 q^{24} + 11 q^{25} + 8 q^{26} - 2 q^{27} - q^{28} - 8 q^{29} - q^{30} - 7 q^{31} + 2 q^{32} - 8 q^{33} - 4 q^{34} - 21 q^{35} + 2 q^{36} + 15 q^{37} - 2 q^{38} - 8 q^{39} + q^{40} - q^{41} + q^{42} - 11 q^{43} + 8 q^{44} + q^{45} + 9 q^{46} - 10 q^{47} - 2 q^{48} + 7 q^{49} + 11 q^{50} + 4 q^{51} + 8 q^{52} - 3 q^{53} - 2 q^{54} + 4 q^{55} - q^{56} + 2 q^{57} - 8 q^{58} - 7 q^{59} - q^{60} - 14 q^{61} - 7 q^{62} - q^{63} + 2 q^{64} + 4 q^{65} - 8 q^{66} - 2 q^{67} - 4 q^{68} - 9 q^{69} - 21 q^{70} - 14 q^{71} + 2 q^{72} + 13 q^{73} + 15 q^{74} - 11 q^{75} - 2 q^{76} - 4 q^{77} - 8 q^{78} - 6 q^{79} + q^{80} + 2 q^{81} - q^{82} + 15 q^{83} + q^{84} - 2 q^{85} - 11 q^{86} + 8 q^{87} + 8 q^{88} - 18 q^{89} + q^{90} - 4 q^{91} + 9 q^{92} + 7 q^{93} - 10 q^{94} - 42 q^{95} - 2 q^{96} - 14 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 + q^10 + 8 * q^11 - 2 * q^12 + 8 * q^13 - q^14 - q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 - 2 * q^19 + q^20 + q^21 + 8 * q^22 + 9 * q^23 - 2 * q^24 + 11 * q^25 + 8 * q^26 - 2 * q^27 - q^28 - 8 * q^29 - q^30 - 7 * q^31 + 2 * q^32 - 8 * q^33 - 4 * q^34 - 21 * q^35 + 2 * q^36 + 15 * q^37 - 2 * q^38 - 8 * q^39 + q^40 - q^41 + q^42 - 11 * q^43 + 8 * q^44 + q^45 + 9 * q^46 - 10 * q^47 - 2 * q^48 + 7 * q^49 + 11 * q^50 + 4 * q^51 + 8 * q^52 - 3 * q^53 - 2 * q^54 + 4 * q^55 - q^56 + 2 * q^57 - 8 * q^58 - 7 * q^59 - q^60 - 14 * q^61 - 7 * q^62 - q^63 + 2 * q^64 + 4 * q^65 - 8 * q^66 - 2 * q^67 - 4 * q^68 - 9 * q^69 - 21 * q^70 - 14 * q^71 + 2 * q^72 + 13 * q^73 + 15 * q^74 - 11 * q^75 - 2 * q^76 - 4 * q^77 - 8 * q^78 - 6 * q^79 + q^80 + 2 * q^81 - q^82 + 15 * q^83 + q^84 - 2 * q^85 - 11 * q^86 + 8 * q^87 + 8 * q^88 - 18 * q^89 + q^90 - 4 * q^91 + 9 * q^92 + 7 * q^93 - 10 * q^94 - 42 * q^95 - 2 * q^96 - 14 * q^97 + 7 * q^98 + 8 * q^99

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