LMFDB - Embedded newform 4730.2.a.be.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4730,2,Mod(1,4730)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4730, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4730.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4730 = 2 \cdot 5 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4730.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:	$37.7692401561$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 26 x^{10} + 79 x^{9} + 247 x^{8} - 766 x^{7} - 1023 x^{6} + 3281 x^{5} + 1634 x^{4} + \cdots + 392 $ x^12 - 3x^11 - 26x^10 + 79x^9 + 247x^8 - 766x^7 - 1023x^6 + 3281x^5 + 1634x^4 - 5557x^3 - 483x^2 + 1648*x + 392
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.342987$ of defining polynomial
Character	$\chi$	$=$	4730.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.342987 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.342987 q^{6} -3.87285 q^{7} +1.00000 q^{8} -2.88236 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.342987 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.342987 q^{6} -3.87285 q^{7} +1.00000 q^{8} -2.88236 q^{9} +1.00000 q^{10} -1.00000 q^{11} -0.342987 q^{12} +5.08439 q^{13} -3.87285 q^{14} -0.342987 q^{15} +1.00000 q^{16} -1.18024 q^{17} -2.88236 q^{18} -2.78382 q^{19} +1.00000 q^{20} +1.32834 q^{21} -1.00000 q^{22} -3.08439 q^{23} -0.342987 q^{24} +1.00000 q^{25} +5.08439 q^{26} +2.01757 q^{27} -3.87285 q^{28} +1.90897 q^{29} -0.342987 q^{30} +2.67757 q^{31} +1.00000 q^{32} +0.342987 q^{33} -1.18024 q^{34} -3.87285 q^{35} -2.88236 q^{36} +5.97555 q^{37} -2.78382 q^{38} -1.74388 q^{39} +1.00000 q^{40} +8.98944 q^{41} +1.32834 q^{42} +1.00000 q^{43} -1.00000 q^{44} -2.88236 q^{45} -3.08439 q^{46} +9.62693 q^{47} -0.342987 q^{48} +7.99899 q^{49} +1.00000 q^{50} +0.404809 q^{51} +5.08439 q^{52} -7.16428 q^{53} +2.01757 q^{54} -1.00000 q^{55} -3.87285 q^{56} +0.954814 q^{57} +1.90897 q^{58} +4.37336 q^{59} -0.342987 q^{60} +9.48586 q^{61} +2.67757 q^{62} +11.1630 q^{63} +1.00000 q^{64} +5.08439 q^{65} +0.342987 q^{66} +8.49930 q^{67} -1.18024 q^{68} +1.05791 q^{69} -3.87285 q^{70} -10.8233 q^{71} -2.88236 q^{72} -4.68226 q^{73} +5.97555 q^{74} -0.342987 q^{75} -2.78382 q^{76} +3.87285 q^{77} -1.74388 q^{78} -4.94508 q^{79} +1.00000 q^{80} +7.95508 q^{81} +8.98944 q^{82} -10.5687 q^{83} +1.32834 q^{84} -1.18024 q^{85} +1.00000 q^{86} -0.654754 q^{87} -1.00000 q^{88} +4.58827 q^{89} -2.88236 q^{90} -19.6911 q^{91} -3.08439 q^{92} -0.918373 q^{93} +9.62693 q^{94} -2.78382 q^{95} -0.342987 q^{96} +16.4470 q^{97} +7.99899 q^{98} +2.88236 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 12 q^{5} + 3 q^{6} + 8 q^{7} + 12 q^{8} + 25 q^{9}+O(q^{10}) $ 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 12 * q^5 + 3 * q^6 + 8 * q^7 + 12 * q^8 + 25 * q^9	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 12 q^{5} + 3 q^{6} + 8 q^{7} + 12 q^{8} + 25 q^{9} + 12 q^{10} - 12 q^{11} + 3 q^{12} + 16 q^{13} + 8 q^{14} + 3 q^{15} + 12 q^{16} + 18 q^{17} + 25 q^{18} - 4 q^{19} + 12 q^{20} + 4 q^{21} - 12 q^{22} + 8 q^{23} + 3 q^{24} + 12 q^{25} + 16 q^{26} + 6 q^{27} + 8 q^{28} + 20 q^{29} + 3 q^{30} + 5 q^{31} + 12 q^{32} - 3 q^{33} + 18 q^{34} + 8 q^{35} + 25 q^{36} + 19 q^{37} - 4 q^{38} + 6 q^{39} + 12 q^{40} + 16 q^{41} + 4 q^{42} + 12 q^{43} - 12 q^{44} + 25 q^{45} + 8 q^{46} - q^{47} + 3 q^{48} + 52 q^{49} + 12 q^{50} + q^{51} + 16 q^{52} + 11 q^{53} + 6 q^{54} - 12 q^{55} + 8 q^{56} + 9 q^{57} + 20 q^{58} - 11 q^{59} + 3 q^{60} + 18 q^{61} + 5 q^{62} + 15 q^{63} + 12 q^{64} + 16 q^{65} - 3 q^{66} - 10 q^{67} + 18 q^{68} + 8 q^{70} - 2 q^{71} + 25 q^{72} + 29 q^{73} + 19 q^{74} + 3 q^{75} - 4 q^{76} - 8 q^{77} + 6 q^{78} + 2 q^{79} + 12 q^{80} - 8 q^{81} + 16 q^{82} + 26 q^{83} + 4 q^{84} + 18 q^{85} + 12 q^{86} - 4 q^{87} - 12 q^{88} + 41 q^{89} + 25 q^{90} - 4 q^{91} + 8 q^{92} + 5 q^{93} - q^{94} - 4 q^{95} + 3 q^{96} - 7 q^{97} + 52 q^{98} - 25 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 12 * q^5 + 3 * q^6 + 8 * q^7 + 12 * q^8 + 25 * q^9 + 12 * q^10 - 12 * q^11 + 3 * q^12 + 16 * q^13 + 8 * q^14 + 3 * q^15 + 12 * q^16 + 18 * q^17 + 25 * q^18 - 4 * q^19 + 12 * q^20 + 4 * q^21 - 12 * q^22 + 8 * q^23 + 3 * q^24 + 12 * q^25 + 16 * q^26 + 6 * q^27 + 8 * q^28 + 20 * q^29 + 3 * q^30 + 5 * q^31 + 12 * q^32 - 3 * q^33 + 18 * q^34 + 8 * q^35 + 25 * q^36 + 19 * q^37 - 4 * q^38 + 6 * q^39 + 12 * q^40 + 16 * q^41 + 4 * q^42 + 12 * q^43 - 12 * q^44 + 25 * q^45 + 8 * q^46 - q^47 + 3 * q^48 + 52 * q^49 + 12 * q^50 + q^51 + 16 * q^52 + 11 * q^53 + 6 * q^54 - 12 * q^55 + 8 * q^56 + 9 * q^57 + 20 * q^58 - 11 * q^59 + 3 * q^60 + 18 * q^61 + 5 * q^62 + 15 * q^63 + 12 * q^64 + 16 * q^65 - 3 * q^66 - 10 * q^67 + 18 * q^68 + 8 * q^70 - 2 * q^71 + 25 * q^72 + 29 * q^73 + 19 * q^74 + 3 * q^75 - 4 * q^76 - 8 * q^77 + 6 * q^78 + 2 * q^79 + 12 * q^80 - 8 * q^81 + 16 * q^82 + 26 * q^83 + 4 * q^84 + 18 * q^85 + 12 * q^86 - 4 * q^87 - 12 * q^88 + 41 * q^89 + 25 * q^90 - 4 * q^91 + 8 * q^92 + 5 * q^93 - q^94 - 4 * q^95 + 3 * q^96 - 7 * q^97 + 52 * q^98 - 25 * 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
$3$	−0.342987	−0.198024	−0.0990119	−	0.995086i	$-0.531568\pi$
$3$	−0.342987	−0.198024	−0.0990119	+	0.995086i	$0.531568\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.342987	−0.140024
$7$	−3.87285	−1.46380	−0.731900	−	0.681412i	$-0.761366\pi$
$7$	−3.87285	−1.46380	−0.731900	+	0.681412i	$0.761366\pi$
$8$	1.00000	0.353553
$9$	−2.88236	−0.960787
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.342987	−0.0990119
$13$	5.08439	1.41016	0.705078	−	0.709130i	$-0.250912\pi$
$13$	5.08439	1.41016	0.705078	+	0.709130i	$0.250912\pi$
$14$	−3.87285	−1.03506
$15$	−0.342987	−0.0885589
$16$	1.00000	0.250000
$17$	−1.18024	−0.286251	−0.143126	−	0.989705i	$-0.545715\pi$
$17$	−1.18024	−0.286251	−0.143126	+	0.989705i	$0.545715\pi$
$18$	−2.88236	−0.679379
$19$	−2.78382	−0.638652	−0.319326	−	0.947645i	$-0.603456\pi$
$19$	−2.78382	−0.638652	−0.319326	+	0.947645i	$0.603456\pi$
$20$	1.00000	0.223607
$21$	1.32834	0.289867
$22$	−1.00000	−0.213201
$23$	−3.08439	−0.643139	−0.321570	−	0.946886i	$-0.604210\pi$
$23$	−3.08439	−0.643139	−0.321570	+	0.946886i	$0.604210\pi$
$24$	−0.342987	−0.0700120
$25$	1.00000	0.200000
$26$	5.08439	0.997130
$27$	2.01757	0.388282
$28$	−3.87285	−0.731900
$29$	1.90897	0.354488	0.177244	−	0.984167i	$-0.443282\pi$
$29$	1.90897	0.354488	0.177244	+	0.984167i	$0.443282\pi$
$30$	−0.342987	−0.0626206
$31$	2.67757	0.480906	0.240453	−	0.970661i	$-0.422704\pi$
$31$	2.67757	0.480906	0.240453	+	0.970661i	$0.422704\pi$
$32$	1.00000	0.176777
$33$	0.342987	0.0597064
$34$	−1.18024	−0.202410
$35$	−3.87285	−0.654632
$36$	−2.88236	−0.480393
$37$	5.97555	0.982374	0.491187	−	0.871054i	$-0.336563\pi$
$37$	5.97555	0.982374	0.491187	+	0.871054i	$0.336563\pi$
$38$	−2.78382	−0.451595
$39$	−1.74388	−0.279244
$40$	1.00000	0.158114
$41$	8.98944	1.40392	0.701958	−	0.712219i	$-0.252310\pi$
$41$	8.98944	1.40392	0.701958	+	0.712219i	$0.252310\pi$
$42$	1.32834	0.204967
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−1.00000	−0.150756
$45$	−2.88236	−0.429677
$46$	−3.08439	−0.454768
$47$	9.62693	1.40423	0.702116	−	0.712062i	$-0.252239\pi$
$47$	9.62693	1.40423	0.702116	+	0.712062i	$0.252239\pi$
$48$	−0.342987	−0.0495059
$49$	7.99899	1.14271
$50$	1.00000	0.141421
$51$	0.404809	0.0566845
$52$	5.08439	0.705078
$53$	−7.16428	−0.984090	−0.492045	−	0.870570i	$-0.663750\pi$
$53$	−7.16428	−0.984090	−0.492045	+	0.870570i	$0.663750\pi$
$54$	2.01757	0.274557
$55$	−1.00000	−0.134840
$56$	−3.87285	−0.517532
$57$	0.954814	0.126468
$58$	1.90897	0.250661
$59$	4.37336	0.569363	0.284682	−	0.958622i	$-0.408112\pi$
$59$	4.37336	0.569363	0.284682	+	0.958622i	$0.408112\pi$
$60$	−0.342987	−0.0442795
$61$	9.48586	1.21454	0.607270	−	0.794496i	$-0.292265\pi$
$61$	9.48586	1.21454	0.607270	+	0.794496i	$0.292265\pi$
$62$	2.67757	0.340052
$63$	11.1630	1.40640
$64$	1.00000	0.125000
$65$	5.08439	0.630641
$66$	0.342987	0.0422188
$67$	8.49930	1.03835	0.519177	−	0.854666i	$-0.326238\pi$
$67$	8.49930	1.03835	0.519177	+	0.854666i	$0.326238\pi$
$68$	−1.18024	−0.143126
$69$	1.05791	0.127357
$70$	−3.87285	−0.462894
$71$	−10.8233	−1.28449	−0.642247	−	0.766498i	$-0.721998\pi$
$71$	−10.8233	−1.28449	−0.642247	+	0.766498i	$0.721998\pi$
$72$	−2.88236	−0.339689
$73$	−4.68226	−0.548017	−0.274009	−	0.961727i	$-0.588350\pi$
$73$	−4.68226	−0.548017	−0.274009	+	0.961727i	$0.588350\pi$
$74$	5.97555	0.694643
$75$	−0.342987	−0.0396048
$76$	−2.78382	−0.319326
$77$	3.87285	0.441353
$78$	−1.74388	−0.197456
$79$	−4.94508	−0.556365	−0.278183	−	0.960528i	$-0.589732\pi$
$79$	−4.94508	−0.556365	−0.278183	+	0.960528i	$0.589732\pi$
$80$	1.00000	0.111803
$81$	7.95508	0.883897
$82$	8.98944	0.992718
$83$	−10.5687	−1.16007	−0.580035	−	0.814592i	$-0.696961\pi$
$83$	−10.5687	−1.16007	−0.580035	+	0.814592i	$0.696961\pi$
$84$	1.32834	0.144934
$85$	−1.18024	−0.128015
$86$	1.00000	0.107833
$87$	−0.654754	−0.0701970
$88$	−1.00000	−0.106600
$89$	4.58827	0.486356	0.243178	−	0.969982i	$-0.421810\pi$
$89$	4.58827	0.486356	0.243178	+	0.969982i	$0.421810\pi$
$90$	−2.88236	−0.303827
$91$	−19.6911	−2.06419
$92$	−3.08439	−0.321570
$93$	−0.918373	−0.0952308
$94$	9.62693	0.992942
$95$	−2.78382	−0.285614
$96$	−0.342987	−0.0350060
$97$	16.4470	1.66994	0.834969	−	0.550297i	$-0.185486\pi$
$97$	16.4470	1.66994	0.834969	+	0.550297i	$0.185486\pi$
$98$	7.99899	0.808020
$99$	2.88236	0.289688
$100$	1.00000	0.100000
$101$	18.4869	1.83951	0.919756	−	0.392490i	$-0.128386\pi$
$101$	18.4869	1.83951	0.919756	+	0.392490i	$0.128386\pi$
$102$	0.404809	0.0400820
$103$	3.66411	0.361036	0.180518	−	0.983572i	$-0.442223\pi$
$103$	3.66411	0.361036	0.180518	+	0.983572i	$0.442223\pi$
$104$	5.08439	0.498565
$105$	1.32834	0.129633
$106$	−7.16428	−0.695857
$107$	9.37944	0.906745	0.453372	−	0.891321i	$-0.350221\pi$
$107$	9.37944	0.906745	0.453372	+	0.891321i	$0.350221\pi$
$108$	2.01757	0.194141
$109$	0.816833	0.0782384	0.0391192	−	0.999235i	$-0.487545\pi$
$109$	0.816833	0.0782384	0.0391192	+	0.999235i	$0.487545\pi$
$110$	−1.00000	−0.0953463
$111$	−2.04954	−0.194533
$112$	−3.87285	−0.365950
$113$	−12.3567	−1.16242	−0.581212	−	0.813752i	$-0.697421\pi$
$113$	−12.3567	−1.16242	−0.581212	+	0.813752i	$0.697421\pi$
$114$	0.954814	0.0894265
$115$	−3.08439	−0.287621
$116$	1.90897	0.177244
$117$	−14.6550	−1.35486
$118$	4.37336	0.402601
$119$	4.57091	0.419015
$120$	−0.342987	−0.0313103
$121$	1.00000	0.0909091
$122$	9.48586	0.858809
$123$	−3.08326	−0.278009
$124$	2.67757	0.240453
$125$	1.00000	0.0894427
$126$	11.1630	0.994475
$127$	20.4964	1.81876	0.909382	−	0.415963i	$-0.136555\pi$
$127$	20.4964	1.81876	0.909382	+	0.415963i	$0.136555\pi$
$128$	1.00000	0.0883883
$129$	−0.342987	−0.0301983
$130$	5.08439	0.445930
$131$	7.50200	0.655453	0.327726	−	0.944773i	$-0.393718\pi$
$131$	7.50200	0.655453	0.327726	+	0.944773i	$0.393718\pi$
$132$	0.342987	0.0298532
$133$	10.7813	0.934859
$134$	8.49930	0.734228
$135$	2.01757	0.173645
$136$	−1.18024	−0.101205
$137$	0.602218	0.0514510	0.0257255	−	0.999669i	$-0.491810\pi$
$137$	0.602218	0.0514510	0.0257255	+	0.999669i	$0.491810\pi$
$138$	1.05791	0.0900549
$139$	12.3783	1.04991	0.524955	−	0.851130i	$-0.324082\pi$
$139$	12.3783	1.04991	0.524955	+	0.851130i	$0.324082\pi$
$140$	−3.87285	−0.327316
$141$	−3.30192	−0.278071
$142$	−10.8233	−0.908275
$143$	−5.08439	−0.425178
$144$	−2.88236	−0.240197
$145$	1.90897	0.158532
$146$	−4.68226	−0.387507
$147$	−2.74355	−0.226284
$148$	5.97555	0.491187
$149$	−9.45473	−0.774562	−0.387281	−	0.921962i	$-0.626586\pi$
$149$	−9.45473	−0.774562	−0.387281	+	0.921962i	$0.626586\pi$
$150$	−0.342987	−0.0280048
$151$	−11.5779	−0.942198	−0.471099	−	0.882080i	$-0.656142\pi$
$151$	−11.5779	−0.942198	−0.471099	+	0.882080i	$0.656142\pi$
$152$	−2.78382	−0.225797
$153$	3.40189	0.275026
$154$	3.87285	0.312083
$155$	2.67757	0.215068
$156$	−1.74388	−0.139622
$157$	−8.78374	−0.701019	−0.350510	−	0.936559i	$-0.613992\pi$
$157$	−8.78374	−0.701019	−0.350510	+	0.936559i	$0.613992\pi$
$158$	−4.94508	−0.393410
$159$	2.45726	0.194873
$160$	1.00000	0.0790569
$161$	11.9454	0.941427
$162$	7.95508	0.625010
$163$	8.63082	0.676018	0.338009	−	0.941143i	$-0.390247\pi$
$163$	8.63082	0.676018	0.338009	+	0.941143i	$0.390247\pi$
$164$	8.98944	0.701958
$165$	0.342987	0.0267015
$166$	−10.5687	−0.820293
$167$	−9.74131	−0.753805	−0.376903	−	0.926253i	$-0.623011\pi$
$167$	−9.74131	−0.753805	−0.376903	+	0.926253i	$0.623011\pi$
$168$	1.32834	0.102484
$169$	12.8510	0.988538
$170$	−1.18024	−0.0905206
$171$	8.02397	0.613608
$172$	1.00000	0.0762493
$173$	11.4785	0.872691	0.436346	−	0.899779i	$-0.356272\pi$
$173$	11.4785	0.872691	0.436346	+	0.899779i	$0.356272\pi$
$174$	−0.654754	−0.0496368
$175$	−3.87285	−0.292760
$176$	−1.00000	−0.0753778
$177$	−1.50001	−0.112747
$178$	4.58827	0.343905
$179$	−14.4576	−1.08061	−0.540305	−	0.841469i	$-0.681691\pi$
$179$	−14.4576	−1.08061	−0.540305	+	0.841469i	$0.681691\pi$
$180$	−2.88236	−0.214838
$181$	4.30048	0.319652	0.159826	−	0.987145i	$-0.448907\pi$
$181$	4.30048	0.319652	0.159826	+	0.987145i	$0.448907\pi$
$182$	−19.6911	−1.45960
$183$	−3.25353	−0.240508
$184$	−3.08439	−0.227384
$185$	5.97555	0.439331
$186$	−0.918373	−0.0673384
$187$	1.18024	0.0863080
$188$	9.62693	0.702116
$189$	−7.81377	−0.568368
$190$	−2.78382	−0.201959
$191$	−4.74647	−0.343443	−0.171721	−	0.985146i	$-0.554933\pi$
$191$	−4.74647	−0.343443	−0.171721	+	0.985146i	$0.554933\pi$
$192$	−0.342987	−0.0247530
$193$	−19.1907	−1.38138	−0.690690	−	0.723151i	$-0.742693\pi$
$193$	−19.1907	−1.38138	−0.690690	+	0.723151i	$0.742693\pi$
$194$	16.4470	1.18082
$195$	−1.74388	−0.124882
$196$	7.99899	0.571356
$197$	21.1138	1.50429	0.752147	−	0.658996i	$-0.229018\pi$
$197$	21.1138	1.50429	0.752147	+	0.658996i	$0.229018\pi$
$198$	2.88236	0.204840
$199$	0.404858	0.0286996	0.0143498	−	0.999897i	$-0.495432\pi$
$199$	0.404858	0.0286996	0.0143498	+	0.999897i	$0.495432\pi$
$200$	1.00000	0.0707107
$201$	−2.91515	−0.205619
$202$	18.4869	1.30073
$203$	−7.39317	−0.518899
$204$	0.404809	0.0283423
$205$	8.98944	0.627850
$206$	3.66411	0.255291
$207$	8.89031	0.617919
$208$	5.08439	0.352539
$209$	2.78382	0.192561
$210$	1.32834	0.0916641
$211$	−0.606821	−0.0417753	−0.0208876	−	0.999782i	$-0.506649\pi$
$211$	−0.606821	−0.0417753	−0.0208876	+	0.999782i	$0.506649\pi$
$212$	−7.16428	−0.492045
$213$	3.71227	0.254360
$214$	9.37944	0.641166
$215$	1.00000	0.0681994
$216$	2.01757	0.137279
$217$	−10.3698	−0.703950
$218$	0.816833	0.0553229
$219$	1.60596	0.108520
$220$	−1.00000	−0.0674200
$221$	−6.00082	−0.403659
$222$	−2.04954	−0.137556
$223$	−23.9828	−1.60601	−0.803003	−	0.595975i	$-0.796766\pi$
$223$	−23.9828	−1.60601	−0.803003	+	0.595975i	$0.796766\pi$
$224$	−3.87285	−0.258766
$225$	−2.88236	−0.192157
$226$	−12.3567	−0.821959
$227$	−0.142439	−0.00945401	−0.00472701	−	0.999989i	$-0.501505\pi$
$227$	−0.142439	−0.00945401	−0.00472701	+	0.999989i	$0.501505\pi$
$228$	0.954814	0.0632341
$229$	15.6290	1.03279	0.516397	−	0.856349i	$-0.327273\pi$
$229$	15.6290	1.03279	0.516397	+	0.856349i	$0.327273\pi$
$230$	−3.08439	−0.203378
$231$	−1.32834	−0.0873983
$232$	1.90897	0.125330
$233$	17.0333	1.11589	0.557946	−	0.829877i	$-0.311590\pi$
$233$	17.0333	1.11589	0.557946	+	0.829877i	$0.311590\pi$
$234$	−14.6550	−0.958029
$235$	9.62693	0.627992
$236$	4.37336	0.284682
$237$	1.69610	0.110174
$238$	4.57091	0.296288
$239$	−7.32611	−0.473887	−0.236943	−	0.971523i	$-0.576146\pi$
$239$	−7.32611	−0.473887	−0.236943	+	0.971523i	$0.576146\pi$
$240$	−0.342987	−0.0221397
$241$	9.79369	0.630867	0.315433	−	0.948948i	$-0.397850\pi$
$241$	9.79369	0.630867	0.315433	+	0.948948i	$0.397850\pi$
$242$	1.00000	0.0642824
$243$	−8.78121	−0.563315
$244$	9.48586	0.607270
$245$	7.99899	0.511037
$246$	−3.08326	−0.196582
$247$	−14.1540	−0.900598
$248$	2.67757	0.170026
$249$	3.62494	0.229721
$250$	1.00000	0.0632456
$251$	0.467866	0.0295315	0.0147657	−	0.999891i	$-0.495300\pi$
$251$	0.467866	0.0295315	0.0147657	+	0.999891i	$0.495300\pi$
$252$	11.1630	0.703200
$253$	3.08439	0.193914
$254$	20.4964	1.28606
$255$	0.404809	0.0253501
$256$	1.00000	0.0625000
$257$	16.3447	1.01956	0.509779	−	0.860306i	$-0.329727\pi$
$257$	16.3447	1.01956	0.509779	+	0.860306i	$0.329727\pi$
$258$	−0.342987	−0.0213535
$259$	−23.1424	−1.43800
$260$	5.08439	0.315320
$261$	−5.50235	−0.340587
$262$	7.50200	0.463475
$263$	19.5709	1.20680	0.603398	−	0.797440i	$-0.293813\pi$
$263$	19.5709	1.20680	0.603398	+	0.797440i	$0.293813\pi$
$264$	0.342987	0.0211094
$265$	−7.16428	−0.440098
$266$	10.7813	0.661045
$267$	−1.57372	−0.0963100
$268$	8.49930	0.519177
$269$	6.15925	0.375536	0.187768	−	0.982213i	$-0.439875\pi$
$269$	6.15925	0.375536	0.187768	+	0.982213i	$0.439875\pi$
$270$	2.01757	0.122786
$271$	3.09310	0.187893	0.0939464	−	0.995577i	$-0.470052\pi$
$271$	3.09310	0.187893	0.0939464	+	0.995577i	$0.470052\pi$
$272$	−1.18024	−0.0715628
$273$	6.75379	0.408758
$274$	0.602218	0.0363813
$275$	−1.00000	−0.0603023
$276$	1.05791	0.0636784
$277$	−13.6889	−0.822484	−0.411242	−	0.911526i	$-0.634905\pi$
$277$	−13.6889	−0.822484	−0.411242	+	0.911526i	$0.634905\pi$
$278$	12.3783	0.742399
$279$	−7.71772	−0.462048
$280$	−3.87285	−0.231447
$281$	−9.84644	−0.587390	−0.293695	−	0.955899i	$-0.594885\pi$
$281$	−9.84644	−0.587390	−0.293695	+	0.955899i	$0.594885\pi$
$282$	−3.30192	−0.196626
$283$	−8.73821	−0.519433	−0.259716	−	0.965685i	$-0.583629\pi$
$283$	−8.73821	−0.519433	−0.259716	+	0.965685i	$0.583629\pi$
$284$	−10.8233	−0.642247
$285$	0.954814	0.0565583
$286$	−5.08439	−0.300646
$287$	−34.8148	−2.05505
$288$	−2.88236	−0.169845
$289$	−15.6070	−0.918060
$290$	1.90897	0.112099
$291$	−5.64110	−0.330687
$292$	−4.68226	−0.274009
$293$	−8.02450	−0.468796	−0.234398	−	0.972141i	$-0.575312\pi$
$293$	−8.02450	−0.468796	−0.234398	+	0.972141i	$0.575312\pi$
$294$	−2.74355	−0.160007
$295$	4.37336	0.254627
$296$	5.97555	0.347322
$297$	−2.01757	−0.117072
$298$	−9.45473	−0.547698
$299$	−15.6822	−0.906926
$300$	−0.342987	−0.0198024
$301$	−3.87285	−0.223228
$302$	−11.5779	−0.666234
$303$	−6.34076	−0.364267
$304$	−2.78382	−0.159663
$305$	9.48586	0.543159
$306$	3.40189	0.194473
$307$	23.1615	1.32190	0.660948	−	0.750432i	$-0.270155\pi$
$307$	23.1615	1.32190	0.660948	+	0.750432i	$0.270155\pi$
$308$	3.87285	0.220676
$309$	−1.25674	−0.0714937
$310$	2.67757	0.152076
$311$	−26.5655	−1.50639	−0.753195	−	0.657798i	$-0.771488\pi$
$311$	−26.5655	−1.50639	−0.753195	+	0.657798i	$0.771488\pi$
$312$	−1.74388	−0.0987278
$313$	6.12755	0.346350	0.173175	−	0.984891i	$-0.444597\pi$
$313$	6.12755	0.346350	0.173175	+	0.984891i	$0.444597\pi$
$314$	−8.78374	−0.495695
$315$	11.1630	0.628961
$316$	−4.94508	−0.278183
$317$	14.7661	0.829347	0.414673	−	0.909970i	$-0.363896\pi$
$317$	14.7661	0.829347	0.414673	+	0.909970i	$0.363896\pi$
$318$	2.45726	0.137796
$319$	−1.90897	−0.106882
$320$	1.00000	0.0559017
$321$	−3.21703	−0.179557
$322$	11.9454	0.665690
$323$	3.28558	0.182815
$324$	7.95508	0.441949
$325$	5.08439	0.282031
$326$	8.63082	0.478017
$327$	−0.280163	−0.0154931
$328$	8.98944	0.496359
$329$	−37.2837	−2.05552
$330$	0.342987	0.0188808
$331$	33.5866	1.84609	0.923043	−	0.384697i	$-0.125694\pi$
$331$	33.5866	1.84609	0.923043	+	0.384697i	$0.125694\pi$
$332$	−10.5687	−0.580035
$333$	−17.2237	−0.943852
$334$	−9.74131	−0.533021
$335$	8.49930	0.464366
$336$	1.32834	0.0724668
$337$	−15.6502	−0.852522	−0.426261	−	0.904600i	$-0.640170\pi$
$337$	−15.6502	−0.852522	−0.426261	+	0.904600i	$0.640170\pi$
$338$	12.8510	0.699002
$339$	4.23821	0.230188
$340$	−1.18024	−0.0640077
$341$	−2.67757	−0.144999
$342$	8.02397	0.433886
$343$	−3.86893	−0.208902
$344$	1.00000	0.0539164
$345$	1.05791	0.0569557
$346$	11.4785	0.617086
$347$	−9.10091	−0.488563	−0.244281	−	0.969704i	$-0.578552\pi$
$347$	−9.10091	−0.488563	−0.244281	+	0.969704i	$0.578552\pi$
$348$	−0.654754	−0.0350985
$349$	−11.9911	−0.641872	−0.320936	−	0.947101i	$-0.603997\pi$
$349$	−11.9911	−0.641872	−0.320936	+	0.947101i	$0.603997\pi$
$350$	−3.87285	−0.207013
$351$	10.2581	0.547538
$352$	−1.00000	−0.0533002
$353$	16.3446	0.869933	0.434966	−	0.900447i	$-0.356760\pi$
$353$	16.3446	0.869933	0.434966	+	0.900447i	$0.356760\pi$
$354$	−1.50001	−0.0797245
$355$	−10.8233	−0.574443
$356$	4.58827	0.243178
$357$	−1.56776	−0.0829749
$358$	−14.4576	−0.764107
$359$	−35.5278	−1.87508	−0.937542	−	0.347872i	$-0.886904\pi$
$359$	−35.5278	−1.87508	−0.937542	+	0.347872i	$0.886904\pi$
$360$	−2.88236	−0.151914
$361$	−11.2504	−0.592124
$362$	4.30048	0.226028
$363$	−0.342987	−0.0180022
$364$	−19.6911	−1.03209
$365$	−4.68226	−0.245081
$366$	−3.25353	−0.170065
$367$	2.58400	0.134884	0.0674419	−	0.997723i	$-0.478516\pi$
$367$	2.58400	0.134884	0.0674419	+	0.997723i	$0.478516\pi$
$368$	−3.08439	−0.160785
$369$	−25.9108	−1.34886
$370$	5.97555	0.310654
$371$	27.7462	1.44051
$372$	−0.918373	−0.0476154
$373$	17.5403	0.908202	0.454101	−	0.890950i	$-0.349960\pi$
$373$	17.5403	0.908202	0.454101	+	0.890950i	$0.349960\pi$
$374$	1.18024	0.0610289
$375$	−0.342987	−0.0177118
$376$	9.62693	0.496471
$377$	9.70596	0.499882
$378$	−7.81377	−0.401897
$379$	12.8839	0.661801	0.330901	−	0.943666i	$-0.392648\pi$
$379$	12.8839	0.661801	0.330901	+	0.943666i	$0.392648\pi$
$380$	−2.78382	−0.142807
$381$	−7.03001	−0.360158
$382$	−4.74647	−0.242851
$383$	14.6945	0.750856	0.375428	−	0.926852i	$-0.377496\pi$
$383$	14.6945	0.750856	0.375428	+	0.926852i	$0.377496\pi$
$384$	−0.342987	−0.0175030
$385$	3.87285	0.197379
$386$	−19.1907	−0.976783
$387$	−2.88236	−0.146519
$388$	16.4470	0.834969
$389$	−12.3657	−0.626965	−0.313482	−	0.949594i	$-0.601496\pi$
$389$	−12.3657	−0.626965	−0.313482	+	0.949594i	$0.601496\pi$
$390$	−1.74388	−0.0883048
$391$	3.64033	0.184099
$392$	7.99899	0.404010
$393$	−2.57309	−0.129795
$394$	21.1138	1.06370
$395$	−4.94508	−0.248814
$396$	2.88236	0.144844
$397$	6.55812	0.329142	0.164571	−	0.986365i	$-0.447376\pi$
$397$	6.55812	0.329142	0.164571	+	0.986365i	$0.447376\pi$
$398$	0.404858	0.0202937
$399$	−3.69785	−0.185124
$400$	1.00000	0.0500000
$401$	−9.12400	−0.455631	−0.227815	−	0.973704i	$-0.573158\pi$
$401$	−9.12400	−0.455631	−0.227815	+	0.973704i	$0.573158\pi$
$402$	−2.91515	−0.145395
$403$	13.6138	0.678152
$404$	18.4869	0.919756
$405$	7.95508	0.395291
$406$	−7.39317	−0.366917
$407$	−5.97555	−0.296197
$408$	0.404809	0.0200410
$409$	5.90843	0.292153	0.146076	−	0.989273i	$-0.453335\pi$
$409$	5.90843	0.292153	0.146076	+	0.989273i	$0.453335\pi$
$410$	8.98944	0.443957
$411$	−0.206553	−0.0101885
$412$	3.66411	0.180518
$413$	−16.9374	−0.833434
$414$	8.89031	0.436935
$415$	−10.5687	−0.518799
$416$	5.08439	0.249283
$417$	−4.24558	−0.207907
$418$	2.78382	0.136161
$419$	−20.8979	−1.02093	−0.510465	−	0.859898i	$-0.670527\pi$
$419$	−20.8979	−1.02093	−0.510465	+	0.859898i	$0.670527\pi$
$420$	1.32834	0.0648163
$421$	1.44685	0.0705149	0.0352575	−	0.999378i	$-0.488775\pi$
$421$	1.44685	0.0705149	0.0352575	+	0.999378i	$0.488775\pi$
$422$	−0.606821	−0.0295396
$423$	−27.7483	−1.34917
$424$	−7.16428	−0.347928
$425$	−1.18024	−0.0572502
$426$	3.71227	0.179860
$427$	−36.7373	−1.77784
$428$	9.37944	0.453372
$429$	1.74388	0.0841953
$430$	1.00000	0.0482243
$431$	−12.0796	−0.581856	−0.290928	−	0.956745i	$-0.593964\pi$
$431$	−12.0796	−0.581856	−0.290928	+	0.956745i	$0.593964\pi$
$432$	2.01757	0.0970706
$433$	−27.3298	−1.31339	−0.656693	−	0.754158i	$-0.728045\pi$
$433$	−27.3298	−1.31339	−0.656693	+	0.754158i	$0.728045\pi$
$434$	−10.3698	−0.497768
$435$	−0.654754	−0.0313930
$436$	0.816833	0.0391192
$437$	8.58637	0.410742
$438$	1.60596	0.0767355
$439$	−6.28995	−0.300203	−0.150101	−	0.988671i	$-0.547960\pi$
$439$	−6.28995	−0.300203	−0.150101	+	0.988671i	$0.547960\pi$
$440$	−1.00000	−0.0476731
$441$	−23.0560	−1.09790
$442$	−6.00082	−0.285430
$443$	32.8673	1.56157	0.780785	−	0.624799i	$-0.214819\pi$
$443$	32.8673	1.56157	0.780785	+	0.624799i	$0.214819\pi$
$444$	−2.04954	−0.0972667
$445$	4.58827	0.217505
$446$	−23.9828	−1.13562
$447$	3.24285	0.153382
$448$	−3.87285	−0.182975
$449$	−38.0446	−1.79544	−0.897718	−	0.440570i	$-0.854776\pi$
$449$	−38.0446	−1.79544	−0.897718	+	0.440570i	$0.854776\pi$
$450$	−2.88236	−0.135876
$451$	−8.98944	−0.423296
$452$	−12.3567	−0.581212
$453$	3.97108	0.186578
$454$	−0.142439	−0.00668500
$455$	−19.6911	−0.923132
$456$	0.954814	0.0447133
$457$	10.3079	0.482184	0.241092	−	0.970502i	$-0.422494\pi$
$457$	10.3079	0.482184	0.241092	+	0.970502i	$0.422494\pi$
$458$	15.6290	0.730296
$459$	−2.38123	−0.111146
$460$	−3.08439	−0.143810
$461$	−25.5297	−1.18904	−0.594519	−	0.804082i	$-0.702657\pi$
$461$	−25.5297	−1.18904	−0.594519	+	0.804082i	$0.702657\pi$
$462$	−1.32834	−0.0617999
$463$	19.6089	0.911305	0.455652	−	0.890158i	$-0.349406\pi$
$463$	19.6089	0.911305	0.455652	+	0.890158i	$0.349406\pi$
$464$	1.90897	0.0886219
$465$	−0.918373	−0.0425885
$466$	17.0333	0.789054
$467$	16.8330	0.778938	0.389469	−	0.921040i	$-0.372659\pi$
$467$	16.8330	0.778938	0.389469	+	0.921040i	$0.372659\pi$
$468$	−14.6550	−0.677429
$469$	−32.9165	−1.51994
$470$	9.62693	0.444057
$471$	3.01271	0.138818
$472$	4.37336	0.201300
$473$	−1.00000	−0.0459800
$474$	1.69610	0.0779044
$475$	−2.78382	−0.127730
$476$	4.57091	0.209507
$477$	20.6500	0.945500
$478$	−7.32611	−0.335089
$479$	17.2296	0.787240	0.393620	−	0.919273i	$-0.371223\pi$
$479$	17.2296	0.787240	0.393620	+	0.919273i	$0.371223\pi$
$480$	−0.342987	−0.0156552
$481$	30.3820	1.38530
$482$	9.79369	0.446090
$483$	−4.09711	−0.186425
$484$	1.00000	0.0454545
$485$	16.4470	0.746819
$486$	−8.78121	−0.398324
$487$	−2.57189	−0.116543	−0.0582716	−	0.998301i	$-0.518559\pi$
$487$	−2.57189	−0.116543	−0.0582716	+	0.998301i	$0.518559\pi$
$488$	9.48586	0.429405
$489$	−2.96026	−0.133868
$490$	7.99899	0.361357
$491$	39.5104	1.78308	0.891539	−	0.452943i	$-0.149626\pi$
$491$	39.5104	1.78308	0.891539	+	0.452943i	$0.149626\pi$
$492$	−3.08326	−0.139004
$493$	−2.25305	−0.101472
$494$	−14.1540	−0.636819
$495$	2.88236	0.129552
$496$	2.67757	0.120226
$497$	41.9172	1.88024
$498$	3.62494	0.162437
$499$	−15.7140	−0.703457	−0.351728	−	0.936102i	$-0.614406\pi$
$499$	−15.7140	−0.703457	−0.351728	+	0.936102i	$0.614406\pi$
$500$	1.00000	0.0447214
$501$	3.34115	0.149271
$502$	0.467866	0.0208819
$503$	10.4555	0.466189	0.233095	−	0.972454i	$-0.425115\pi$
$503$	10.4555	0.466189	0.233095	+	0.972454i	$0.425115\pi$
$504$	11.1630	0.497237
$505$	18.4869	0.822655
$506$	3.08439	0.137118
$507$	−4.40773	−0.195754
$508$	20.4964	0.909382
$509$	24.0528	1.06612	0.533061	−	0.846076i	$-0.321041\pi$
$509$	24.0528	1.06612	0.533061	+	0.846076i	$0.321041\pi$
$510$	0.404809	0.0179252
$511$	18.1337	0.802188
$512$	1.00000	0.0441942
$513$	−5.61656	−0.247977
$514$	16.3447	0.720936
$515$	3.66411	0.161460
$516$	−0.342987	−0.0150992
$517$	−9.62693	−0.423392
$518$	−23.1424	−1.01682
$519$	−3.93697	−0.172814
$520$	5.08439	0.222965
$521$	31.6055	1.38466	0.692331	−	0.721580i	$-0.256584\pi$
$521$	31.6055	1.38466	0.692331	+	0.721580i	$0.256584\pi$
$522$	−5.50235	−0.240831
$523$	8.44666	0.369347	0.184673	−	0.982800i	$-0.440877\pi$
$523$	8.44666	0.369347	0.184673	+	0.982800i	$0.440877\pi$
$524$	7.50200	0.327726
$525$	1.32834	0.0579735
$526$	19.5709	0.853334
$527$	−3.16019	−0.137660
$528$	0.342987	0.0149266
$529$	−13.4866	−0.586372
$530$	−7.16428	−0.311197
$531$	−12.6056	−0.547037
$532$	10.7813	0.467429
$533$	45.7058	1.97974
$534$	−1.57372	−0.0681014
$535$	9.37944	0.405509
$536$	8.49930	0.367114
$537$	4.95877	0.213987
$538$	6.15925	0.265544
$539$	−7.99899	−0.344541
$540$	2.01757	0.0868226
$541$	−39.3323	−1.69103	−0.845513	−	0.533954i	$-0.820705\pi$
$541$	−39.3323	−1.69103	−0.845513	+	0.533954i	$0.820705\pi$
$542$	3.09310	0.132860
$543$	−1.47501	−0.0632987
$544$	−1.18024	−0.0506025
$545$	0.816833	0.0349893
$546$	6.75379	0.289036
$547$	2.16850	0.0927182	0.0463591	−	0.998925i	$-0.485238\pi$
$547$	2.16850	0.0927182	0.0463591	+	0.998925i	$0.485238\pi$
$548$	0.602218	0.0257255
$549$	−27.3417	−1.16691
$550$	−1.00000	−0.0426401
$551$	−5.31424	−0.226394
$552$	1.05791	0.0450274
$553$	19.1516	0.814408
$554$	−13.6889	−0.581584
$555$	−2.04954	−0.0869980
$556$	12.3783	0.524955
$557$	33.8051	1.43237	0.716184	−	0.697912i	$-0.245887\pi$
$557$	33.8051	1.43237	0.716184	+	0.697912i	$0.245887\pi$
$558$	−7.71772	−0.326717
$559$	5.08439	0.215047
$560$	−3.87285	−0.163658
$561$	−0.404809	−0.0170910
$562$	−9.84644	−0.415347
$563$	−11.3117	−0.476731	−0.238366	−	0.971176i	$-0.576612\pi$
$563$	−11.3117	−0.476731	−0.238366	+	0.971176i	$0.576612\pi$
$564$	−3.30192	−0.139036
$565$	−12.3567	−0.519852
$566$	−8.73821	−0.367294
$567$	−30.8088	−1.29385
$568$	−10.8233	−0.454137
$569$	−10.7096	−0.448969	−0.224485	−	0.974478i	$-0.572070\pi$
$569$	−10.7096	−0.448969	−0.224485	+	0.974478i	$0.572070\pi$
$570$	0.954814	0.0399928
$571$	3.34474	0.139973	0.0699865	−	0.997548i	$-0.477704\pi$
$571$	3.34474	0.139973	0.0699865	+	0.997548i	$0.477704\pi$
$572$	−5.08439	−0.212589
$573$	1.62798	0.0680098
$574$	−34.8148	−1.45314
$575$	−3.08439	−0.128628
$576$	−2.88236	−0.120098
$577$	5.16292	0.214935	0.107468	−	0.994209i	$-0.465726\pi$
$577$	5.16292	0.214935	0.107468	+	0.994209i	$0.465726\pi$
$578$	−15.6070	−0.649167
$579$	6.58218	0.273546
$580$	1.90897	0.0792658
$581$	40.9311	1.69811
$582$	−5.64110	−0.233831
$583$	7.16428	0.296714
$584$	−4.68226	−0.193753
$585$	−14.6550	−0.605911
$586$	−8.02450	−0.331489
$587$	22.9777	0.948393	0.474196	−	0.880419i	$-0.342739\pi$
$587$	22.9777	0.948393	0.474196	+	0.880419i	$0.342739\pi$
$588$	−2.74355	−0.113142
$589$	−7.45387	−0.307131
$590$	4.37336	0.180048
$591$	−7.24175	−0.297886
$592$	5.97555	0.245594
$593$	38.3036	1.57294	0.786470	−	0.617629i	$-0.211906\pi$
$593$	38.3036	1.57294	0.786470	+	0.617629i	$0.211906\pi$
$594$	−2.01757	−0.0827821
$595$	4.57091	0.187389
$596$	−9.45473	−0.387281
$597$	−0.138861	−0.00568321
$598$	−15.6822	−0.641294
$599$	−42.6395	−1.74220	−0.871101	−	0.491104i	$-0.836594\pi$
$599$	−42.6395	−1.74220	−0.871101	+	0.491104i	$0.836594\pi$
$600$	−0.342987	−0.0140024
$601$	−5.01872	−0.204718	−0.102359	−	0.994748i	$-0.532639\pi$
$601$	−5.01872	−0.204718	−0.102359	+	0.994748i	$0.532639\pi$
$602$	−3.87285	−0.157846
$603$	−24.4980	−0.997638
$604$	−11.5779	−0.471099
$605$	1.00000	0.0406558
$606$	−6.34076	−0.257576
$607$	16.6739	0.676774	0.338387	−	0.941007i	$-0.390119\pi$
$607$	16.6739	0.676774	0.338387	+	0.941007i	$0.390119\pi$
$608$	−2.78382	−0.112899
$609$	2.53576	0.102754
$610$	9.48586	0.384071
$611$	48.9471	1.98019
$612$	3.40189	0.137513
$613$	47.1966	1.90625	0.953126	−	0.302572i	$-0.0978453\pi$
$613$	47.1966	1.90625	0.953126	+	0.302572i	$0.0978453\pi$
$614$	23.1615	0.934721
$615$	−3.08326	−0.124329
$616$	3.87285	0.156042
$617$	−25.1261	−1.01154	−0.505770	−	0.862669i	$-0.668791\pi$
$617$	−25.1261	−1.01154	−0.505770	+	0.862669i	$0.668791\pi$
$618$	−1.25674	−0.0505537
$619$	42.6966	1.71612	0.858060	−	0.513549i	$-0.171670\pi$
$619$	42.6966	1.71612	0.858060	+	0.513549i	$0.171670\pi$
$620$	2.67757	0.107534
$621$	−6.22298	−0.249720
$622$	−26.5655	−1.06518
$623$	−17.7697	−0.711928
$624$	−1.74388	−0.0698111
$625$	1.00000	0.0400000
$626$	6.12755	0.244906
$627$	−0.954814	−0.0381316
$628$	−8.78374	−0.350510
$629$	−7.05260	−0.281206
$630$	11.1630	0.444743
$631$	−49.1073	−1.95493	−0.977464	−	0.211100i	$-0.932295\pi$
$631$	−49.1073	−1.95493	−0.977464	+	0.211100i	$0.932295\pi$
$632$	−4.94508	−0.196705
$633$	0.208132	0.00827250
$634$	14.7661	0.586437
$635$	20.4964	0.813376
$636$	2.45726	0.0974366
$637$	40.6699	1.61140
$638$	−1.90897	−0.0755770
$639$	31.1968	1.23412
$640$	1.00000	0.0395285
$641$	14.2281	0.561977	0.280988	−	0.959711i	$-0.409338\pi$
$641$	14.2281	0.561977	0.280988	+	0.959711i	$0.409338\pi$
$642$	−3.21703	−0.126966
$643$	−39.1359	−1.54337	−0.771685	−	0.636005i	$-0.780586\pi$
$643$	−39.1359	−1.54337	−0.771685	+	0.636005i	$0.780586\pi$
$644$	11.9454	0.470714
$645$	−0.342987	−0.0135051
$646$	3.28558	0.129270
$647$	−19.6826	−0.773805	−0.386902	−	0.922121i	$-0.626455\pi$
$647$	−19.6826	−0.773805	−0.386902	+	0.922121i	$0.626455\pi$
$648$	7.95508	0.312505
$649$	−4.37336	−0.171670
$650$	5.08439	0.199426
$651$	3.55672	0.139399
$652$	8.63082	0.338009
$653$	43.2651	1.69309	0.846547	−	0.532314i	$-0.178678\pi$
$653$	43.2651	1.69309	0.846547	+	0.532314i	$0.178678\pi$
$654$	−0.280163	−0.0109552
$655$	7.50200	0.293127
$656$	8.98944	0.350979
$657$	13.4960	0.526528
$658$	−37.2837	−1.45347
$659$	−37.6626	−1.46713	−0.733563	−	0.679622i	$-0.762144\pi$
$659$	−37.6626	−1.46713	−0.733563	+	0.679622i	$0.762144\pi$
$660$	0.342987	0.0133508
$661$	22.7164	0.883566	0.441783	−	0.897122i	$-0.354346\pi$
$661$	22.7164	0.883566	0.441783	+	0.897122i	$0.354346\pi$
$662$	33.5866	1.30538
$663$	2.05820	0.0799340
$664$	−10.5687	−0.410146
$665$	10.7813	0.418082
$666$	−17.2237	−0.667404
$667$	−5.88801	−0.227985
$668$	−9.74131	−0.376903
$669$	8.22579	0.318027
$670$	8.49930	0.328357
$671$	−9.48586	−0.366198
$672$	1.32834	0.0512418
$673$	−20.0130	−0.771446	−0.385723	−	0.922615i	$-0.626048\pi$
$673$	−20.0130	−0.771446	−0.385723	+	0.922615i	$0.626048\pi$
$674$	−15.6502	−0.602824
$675$	2.01757	0.0776565
$676$	12.8510	0.494269
$677$	14.0152	0.538650	0.269325	−	0.963049i	$-0.413200\pi$
$677$	14.0152	0.538650	0.269325	+	0.963049i	$0.413200\pi$
$678$	4.23821	0.162767
$679$	−63.6967	−2.44446
$680$	−1.18024	−0.0452603
$681$	0.0488548	0.00187212
$682$	−2.67757	−0.102529
$683$	21.4336	0.820135	0.410067	−	0.912055i	$-0.365505\pi$
$683$	21.4336	0.820135	0.410067	+	0.912055i	$0.365505\pi$
$684$	8.02397	0.306804
$685$	0.602218	0.0230096
$686$	−3.86893	−0.147716
$687$	−5.36055	−0.204518
$688$	1.00000	0.0381246
$689$	−36.4260	−1.38772
$690$	1.05791	0.0402738
$691$	−32.2678	−1.22752	−0.613762	−	0.789491i	$-0.710345\pi$
$691$	−32.2678	−1.22752	−0.613762	+	0.789491i	$0.710345\pi$
$692$	11.4785	0.436346
$693$	−11.1630	−0.424046
$694$	−9.10091	−0.345466
$695$	12.3783	0.469534
$696$	−0.654754	−0.0248184
$697$	−10.6097	−0.401872
$698$	−11.9911	−0.453872
$699$	−5.84222	−0.220973
$700$	−3.87285	−0.146380
$701$	−24.4051	−0.921769	−0.460884	−	0.887460i	$-0.652468\pi$
$701$	−24.4051	−0.921769	−0.460884	+	0.887460i	$0.652468\pi$
$702$	10.2581	0.387168
$703$	−16.6348	−0.627395
$704$	−1.00000	−0.0376889
$705$	−3.30192	−0.124357
$706$	16.3446	0.615135
$707$	−71.5969	−2.69268
$708$	−1.50001	−0.0563737
$709$	35.9324	1.34947	0.674734	−	0.738061i	$-0.264258\pi$
$709$	35.9324	1.34947	0.674734	+	0.738061i	$0.264258\pi$
$710$	−10.8233	−0.406193
$711$	14.2535	0.534548
$712$	4.58827	0.171953
$713$	−8.25866	−0.309289
$714$	−1.56776	−0.0586721
$715$	−5.08439	−0.190145
$716$	−14.4576	−0.540305
$717$	2.51276	0.0938409
$718$	−35.5278	−1.32588
$719$	−0.377559	−0.0140806	−0.00704028	−	0.999975i	$-0.502241\pi$
$719$	−0.377559	−0.0140806	−0.00704028	+	0.999975i	$0.502241\pi$
$720$	−2.88236	−0.107419
$721$	−14.1906	−0.528485
$722$	−11.2504	−0.418695
$723$	−3.35911	−0.124927
$724$	4.30048	0.159826
$725$	1.90897	0.0708975
$726$	−0.342987	−0.0127295
$727$	−45.9459	−1.70404	−0.852020	−	0.523509i	$-0.824623\pi$
$727$	−45.9459	−1.70404	−0.852020	+	0.523509i	$0.824623\pi$
$728$	−19.6911	−0.729800
$729$	−20.8534	−0.772348
$730$	−4.68226	−0.173298
$731$	−1.18024	−0.0436529
$732$	−3.25353	−0.120254
$733$	−21.4198	−0.791158	−0.395579	−	0.918432i	$-0.629456\pi$
$733$	−21.4198	−0.791158	−0.395579	+	0.918432i	$0.629456\pi$
$734$	2.58400	0.0953772
$735$	−2.74355	−0.101197
$736$	−3.08439	−0.113692
$737$	−8.49930	−0.313076
$738$	−25.9108	−0.953790
$739$	44.7591	1.64649	0.823245	−	0.567686i	$-0.192161\pi$
$739$	44.7591	1.64649	0.823245	+	0.567686i	$0.192161\pi$
$740$	5.97555	0.219666
$741$	4.85465	0.178340
$742$	27.7462	1.01860
$743$	−4.13324	−0.151634	−0.0758169	−	0.997122i	$-0.524156\pi$
$743$	−4.13324	−0.151634	−0.0758169	+	0.997122i	$0.524156\pi$
$744$	−0.918373	−0.0336692
$745$	−9.45473	−0.346395
$746$	17.5403	0.642196
$747$	30.4629	1.11458
$748$	1.18024	0.0431540
$749$	−36.3252	−1.32729
$750$	−0.342987	−0.0125241
$751$	−40.9647	−1.49482	−0.747412	−	0.664360i	$-0.768704\pi$
$751$	−40.9647	−1.49482	−0.747412	+	0.664360i	$0.768704\pi$
$752$	9.62693	0.351058
$753$	−0.160472	−0.00584793
$754$	9.70596	0.353470
$755$	−11.5779	−0.421364
$756$	−7.81377	−0.284184
$757$	−27.6766	−1.00592	−0.502962	−	0.864309i	$-0.667756\pi$
$757$	−27.6766	−1.00592	−0.502962	+	0.864309i	$0.667756\pi$
$758$	12.8839	0.467964
$759$	−1.05791	−0.0383995
$760$	−2.78382	−0.100980
$761$	50.2490	1.82153	0.910763	−	0.412930i	$-0.135495\pi$
$761$	50.2490	1.82153	0.910763	+	0.412930i	$0.135495\pi$
$762$	−7.03001	−0.254670
$763$	−3.16347	−0.114525
$764$	−4.74647	−0.171721
$765$	3.40189	0.122995
$766$	14.6945	0.530935
$767$	22.2359	0.802891
$768$	−0.342987	−0.0123765
$769$	22.3996	0.807749	0.403874	−	0.914814i	$-0.367663\pi$
$769$	22.3996	0.807749	0.403874	+	0.914814i	$0.367663\pi$
$770$	3.87285	0.139568
$771$	−5.60604	−0.201897
$772$	−19.1907	−0.690690
$773$	18.6315	0.670128	0.335064	−	0.942195i	$-0.391242\pi$
$773$	18.6315	0.670128	0.335064	+	0.942195i	$0.391242\pi$
$774$	−2.88236	−0.103604
$775$	2.67757	0.0961812
$776$	16.4470	0.590412
$777$	7.93755	0.284758
$778$	−12.3657	−0.443331
$779$	−25.0250	−0.896613
$780$	−1.74388	−0.0624409
$781$	10.8233	0.387290
$782$	3.64033	0.130178
$783$	3.85150	0.137641
$784$	7.99899	0.285678
$785$	−8.78374	−0.313505
$786$	−2.57309	−0.0917791
$787$	−25.2041	−0.898428	−0.449214	−	0.893424i	$-0.648296\pi$
$787$	−25.2041	−0.898428	−0.449214	+	0.893424i	$0.648296\pi$
$788$	21.1138	0.752147
$789$	−6.71259	−0.238974
$790$	−4.94508	−0.175938
$791$	47.8559	1.70156
$792$	2.88236	0.102420
$793$	48.2298	1.71269
$794$	6.55812	0.232739
$795$	2.45726	0.0871500
$796$	0.404858	0.0143498
$797$	−26.6689	−0.944660	−0.472330	−	0.881422i	$-0.656587\pi$
$797$	−26.6689	−0.944660	−0.472330	+	0.881422i	$0.656587\pi$
$798$	−3.69785	−0.130903
$799$	−11.3621	−0.401963
$800$	1.00000	0.0353553
$801$	−13.2250	−0.467284
$802$	−9.12400	−0.322179
$803$	4.68226	0.165233
$804$	−2.91515	−0.102809
$805$	11.9454	0.421019
$806$	13.6138	0.479526
$807$	−2.11255	−0.0743651
$808$	18.4869	0.650366
$809$	12.0456	0.423499	0.211750	−	0.977324i	$-0.432084\pi$
$809$	12.0456	0.423499	0.211750	+	0.977324i	$0.432084\pi$
$810$	7.95508	0.279513
$811$	28.4369	0.998554	0.499277	−	0.866442i	$-0.333599\pi$
$811$	28.4369	0.998554	0.499277	+	0.866442i	$0.333599\pi$
$812$	−7.39317	−0.259450
$813$	−1.06090	−0.0372072
$814$	−5.97555	−0.209443
$815$	8.63082	0.302324
$816$	0.404809	0.0141711
$817$	−2.78382	−0.0973935
$818$	5.90843	0.206583
$819$	56.7568	1.98324
$820$	8.98944	0.313925
$821$	−15.8847	−0.554382	−0.277191	−	0.960815i	$-0.589403\pi$
$821$	−15.8847	−0.554382	−0.277191	+	0.960815i	$0.589403\pi$
$822$	−0.206553	−0.00720437
$823$	16.8109	0.585990	0.292995	−	0.956114i	$-0.405348\pi$
$823$	16.8109	0.585990	0.292995	+	0.956114i	$0.405348\pi$
$824$	3.66411	0.127645
$825$	0.342987	0.0119413
$826$	−16.9374	−0.589327
$827$	25.1473	0.874457	0.437228	−	0.899350i	$-0.355960\pi$
$827$	25.1473	0.874457	0.437228	+	0.899350i	$0.355960\pi$
$828$	8.89031	0.308960
$829$	−35.8253	−1.24426	−0.622132	−	0.782912i	$-0.713733\pi$
$829$	−35.8253	−1.24426	−0.622132	+	0.782912i	$0.713733\pi$
$830$	−10.5687	−0.366846
$831$	4.69511	0.162871
$832$	5.08439	0.176269
$833$	−9.44075	−0.327103
$834$	−4.24558	−0.147013
$835$	−9.74131	−0.337112
$836$	2.78382	0.0962804
$837$	5.40220	0.186727
$838$	−20.8979	−0.721907
$839$	−22.1553	−0.764886	−0.382443	−	0.923979i	$-0.624917\pi$
$839$	−22.1553	−0.764886	−0.382443	+	0.923979i	$0.624917\pi$
$840$	1.32834	0.0458321
$841$	−25.3558	−0.874339
$842$	1.44685	0.0498616
$843$	3.37720	0.116317
$844$	−0.606821	−0.0208876
$845$	12.8510	0.442088
$846$	−27.7483	−0.954006
$847$	−3.87285	−0.133073
$848$	−7.16428	−0.246022
$849$	2.99710	0.102860
$850$	−1.18024	−0.0404820
$851$	−18.4309	−0.631803
$852$	3.71227	0.127180
$853$	−46.0026	−1.57510	−0.787550	−	0.616250i	$-0.788651\pi$
$853$	−46.0026	−1.57510	−0.787550	+	0.616250i	$0.788651\pi$
$854$	−36.7373	−1.25713
$855$	8.02397	0.274414
$856$	9.37944	0.320583
$857$	8.42733	0.287872	0.143936	−	0.989587i	$-0.454024\pi$
$857$	8.42733	0.287872	0.143936	+	0.989587i	$0.454024\pi$
$858$	1.74388	0.0595351
$859$	9.83910	0.335706	0.167853	−	0.985812i	$-0.446317\pi$
$859$	9.83910	0.335706	0.167853	+	0.985812i	$0.446317\pi$
$860$	1.00000	0.0340997
$861$	11.9410	0.406949
$862$	−12.0796	−0.411434
$863$	−43.7032	−1.48767	−0.743837	−	0.668361i	$-0.766996\pi$
$863$	−43.7032	−1.48767	−0.743837	+	0.668361i	$0.766996\pi$
$864$	2.01757	0.0686393
$865$	11.4785	0.390280
$866$	−27.3298	−0.928704
$867$	5.35301	0.181798
$868$	−10.3698	−0.351975
$869$	4.94508	0.167750
$870$	−0.654754	−0.0221982
$871$	43.2137	1.46424
$872$	0.816833	0.0276614
$873$	−47.4061	−1.60445
$874$	8.58637	0.290438
$875$	−3.87285	−0.130926
$876$	1.60596	0.0542602
$877$	−1.48322	−0.0500848	−0.0250424	−	0.999686i	$-0.507972\pi$
$877$	−1.48322	−0.0500848	−0.0250424	+	0.999686i	$0.507972\pi$
$878$	−6.28995	−0.212275
$879$	2.75230	0.0928328
$880$	−1.00000	−0.0337100
$881$	6.49472	0.218813	0.109406	−	0.993997i	$-0.465105\pi$
$881$	6.49472	0.218813	0.109406	+	0.993997i	$0.465105\pi$
$882$	−23.0560	−0.776334
$883$	27.3086	0.919007	0.459503	−	0.888176i	$-0.348027\pi$
$883$	27.3086	0.919007	0.459503	+	0.888176i	$0.348027\pi$
$884$	−6.00082	−0.201829
$885$	−1.50001	−0.0504222
$886$	32.8673	1.10420
$887$	3.86410	0.129744	0.0648719	−	0.997894i	$-0.479336\pi$
$887$	3.86410	0.129744	0.0648719	+	0.997894i	$0.479336\pi$
$888$	−2.04954	−0.0687780
$889$	−79.3796	−2.66231
$890$	4.58827	0.153799
$891$	−7.95508	−0.266505
$892$	−23.9828	−0.803003
$893$	−26.7996	−0.896816
$894$	3.24285	0.108457
$895$	−14.4576	−0.483264
$896$	−3.87285	−0.129383
$897$	5.37880	0.179593
$898$	−38.0446	−1.26957
$899$	5.11141	0.170475
$900$	−2.88236	−0.0960787
$901$	8.45560	0.281697
$902$	−8.98944	−0.299316
$903$	1.32834	0.0442044
$904$	−12.3567	−0.410979
$905$	4.30048	0.142953
$906$	3.97108	0.131930
$907$	5.73441	0.190408	0.0952040	−	0.995458i	$-0.469650\pi$
$907$	5.73441	0.190408	0.0952040	+	0.995458i	$0.469650\pi$
$908$	−0.142439	−0.00472701
$909$	−53.2858	−1.76738
$910$	−19.6911	−0.652753
$911$	29.0566	0.962689	0.481345	−	0.876531i	$-0.340149\pi$
$911$	29.0566	0.962689	0.481345	+	0.876531i	$0.340149\pi$
$912$	0.954814	0.0316171
$913$	10.5687	0.349774
$914$	10.3079	0.340955
$915$	−3.25353	−0.107558
$916$	15.6290	0.516397
$917$	−29.0541	−0.959452
$918$	−2.38123	−0.0785923
$919$	33.7416	1.11303	0.556516	−	0.830837i	$-0.312138\pi$
$919$	33.7416	1.11303	0.556516	+	0.830837i	$0.312138\pi$
$920$	−3.08439	−0.101689
$921$	−7.94409	−0.261767
$922$	−25.5297	−0.840776
$923$	−55.0301	−1.81134
$924$	−1.32834	−0.0436991
$925$	5.97555	0.196475
$926$	19.6089	0.644390
$927$	−10.5613	−0.346878
$928$	1.90897	0.0626651
$929$	−35.4076	−1.16169	−0.580843	−	0.814016i	$-0.697277\pi$
$929$	−35.4076	−1.16169	−0.580843	+	0.814016i	$0.697277\pi$
$930$	−0.918373	−0.0301146
$931$	−22.2677	−0.729795
$932$	17.0333	0.557946
$933$	9.11162	0.298301
$934$	16.8330	0.550792
$935$	1.18024	0.0385981
$936$	−14.6550	−0.479015
$937$	22.6892	0.741224	0.370612	−	0.928788i	$-0.379148\pi$
$937$	22.6892	0.741224	0.370612	+	0.928788i	$0.379148\pi$
$938$	−32.9165	−1.07476
$939$	−2.10167	−0.0685855
$940$	9.62693	0.313996
$941$	31.0603	1.01254	0.506268	−	0.862376i	$-0.331025\pi$
$941$	31.0603	1.01254	0.506268	+	0.862376i	$0.331025\pi$
$942$	3.01271	0.0981595
$943$	−27.7269	−0.902913
$944$	4.37336	0.142341
$945$	−7.81377	−0.254182
$946$	−1.00000	−0.0325128
$947$	5.88068	0.191096	0.0955482	−	0.995425i	$-0.469540\pi$
$947$	5.88068	0.191096	0.0955482	+	0.995425i	$0.469540\pi$
$948$	1.69610	0.0550868
$949$	−23.8064	−0.772789
$950$	−2.78382	−0.0903190
$951$	−5.06458	−0.164230
$952$	4.57091	0.148144
$953$	−28.1041	−0.910381	−0.455190	−	0.890394i	$-0.650429\pi$
$953$	−28.1041	−0.910381	−0.455190	+	0.890394i	$0.650429\pi$
$954$	20.6500	0.668570
$955$	−4.74647	−0.153592
$956$	−7.32611	−0.236943
$957$	0.654754	0.0211652
$958$	17.2296	0.556663
$959$	−2.33230	−0.0753139
$960$	−0.342987	−0.0110699
$961$	−23.8306	−0.768729
$962$	30.3820	0.979555
$963$	−27.0349	−0.871188
$964$	9.79369	0.315433
$965$	−19.1907	−0.617772
$966$	−4.09711	−0.131822
$967$	−16.3611	−0.526136	−0.263068	−	0.964777i	$-0.584734\pi$
$967$	−16.3611	−0.526136	−0.263068	+	0.964777i	$0.584734\pi$
$968$	1.00000	0.0321412
$969$	−1.12691	−0.0362017
$970$	16.4470	0.528081
$971$	−44.8721	−1.44002	−0.720008	−	0.693966i	$-0.755862\pi$
$971$	−44.8721	−1.44002	−0.720008	+	0.693966i	$0.755862\pi$
$972$	−8.78121	−0.281658
$973$	−47.9392	−1.53686
$974$	−2.57189	−0.0824085
$975$	−1.74388	−0.0558489
$976$	9.48586	0.303635
$977$	−54.2467	−1.73551	−0.867753	−	0.496995i	$-0.834437\pi$
$977$	−54.2467	−1.73551	−0.867753	+	0.496995i	$0.834437\pi$
$978$	−2.96026	−0.0946587
$979$	−4.58827	−0.146642
$980$	7.99899	0.255518
$981$	−2.35441	−0.0751704
$982$	39.5104	1.26083
$983$	−44.6995	−1.42569	−0.712847	−	0.701320i	$-0.752595\pi$
$983$	−44.6995	−1.42569	−0.712847	+	0.701320i	$0.752595\pi$
$984$	−3.08326	−0.0982909
$985$	21.1138	0.672741
$986$	−2.25305	−0.0717519
$987$	12.7878	0.407041
$988$	−14.1540	−0.450299
$989$	−3.08439	−0.0980778
$990$	2.88236	0.0916074
$991$	7.79017	0.247463	0.123731	−	0.992316i	$-0.460514\pi$
$991$	7.79017	0.247463	0.123731	+	0.992316i	$0.460514\pi$
$992$	2.67757	0.0850130
$993$	−11.5198	−0.365569
$994$	41.9172	1.32953
$995$	0.404858	0.0128349
$996$	3.62494	0.114861
$997$	2.00926	0.0636340	0.0318170	−	0.999494i	$-0.489871\pi$
$997$	2.00926	0.0636340	0.0318170	+	0.999494i	$0.489871\pi$
$998$	−15.7140	−0.497419
$999$	12.0561	0.381439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4730.2.a.be.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4730.2.a.be.1.6	✓	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4730.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.342987 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.342987 q^{6} -3.87285 q^{7} +1.00000 q^{8} -2.88236 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.342987 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.342987 q^{6} -3.87285 q^{7} +1.00000 q^{8} -2.88236 q^{9} +1.00000 q^{10} -1.00000 q^{11} -0.342987 q^{12} +5.08439 q^{13} -3.87285 q^{14} -0.342987 q^{15} +1.00000 q^{16} -1.18024 q^{17} -2.88236 q^{18} -2.78382 q^{19} +1.00000 q^{20} +1.32834 q^{21} -1.00000 q^{22} -3.08439 q^{23} -0.342987 q^{24} +1.00000 q^{25} +5.08439 q^{26} +2.01757 q^{27} -3.87285 q^{28} +1.90897 q^{29} -0.342987 q^{30} +2.67757 q^{31} +1.00000 q^{32} +0.342987 q^{33} -1.18024 q^{34} -3.87285 q^{35} -2.88236 q^{36} +5.97555 q^{37} -2.78382 q^{38} -1.74388 q^{39} +1.00000 q^{40} +8.98944 q^{41} +1.32834 q^{42} +1.00000 q^{43} -1.00000 q^{44} -2.88236 q^{45} -3.08439 q^{46} +9.62693 q^{47} -0.342987 q^{48} +7.99899 q^{49} +1.00000 q^{50} +0.404809 q^{51} +5.08439 q^{52} -7.16428 q^{53} +2.01757 q^{54} -1.00000 q^{55} -3.87285 q^{56} +0.954814 q^{57} +1.90897 q^{58} +4.37336 q^{59} -0.342987 q^{60} +9.48586 q^{61} +2.67757 q^{62} +11.1630 q^{63} +1.00000 q^{64} +5.08439 q^{65} +0.342987 q^{66} +8.49930 q^{67} -1.18024 q^{68} +1.05791 q^{69} -3.87285 q^{70} -10.8233 q^{71} -2.88236 q^{72} -4.68226 q^{73} +5.97555 q^{74} -0.342987 q^{75} -2.78382 q^{76} +3.87285 q^{77} -1.74388 q^{78} -4.94508 q^{79} +1.00000 q^{80} +7.95508 q^{81} +8.98944 q^{82} -10.5687 q^{83} +1.32834 q^{84} -1.18024 q^{85} +1.00000 q^{86} -0.654754 q^{87} -1.00000 q^{88} +4.58827 q^{89} -2.88236 q^{90} -19.6911 q^{91} -3.08439 q^{92} -0.918373 q^{93} +9.62693 q^{94} -2.78382 q^{95} -0.342987 q^{96} +16.4470 q^{97} +7.99899 q^{98} +2.88236 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 12 q^{5} + 3 q^{6} + 8 q^{7} + 12 q^{8} + 25 q^{9}+O(q^{10}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 12 * q^5 + 3 * q^6 + 8 * q^7 + 12 * q^8 + 25 * q^9	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 12 q^{5} + 3 q^{6} + 8 q^{7} + 12 q^{8} + 25 q^{9} + 12 q^{10} - 12 q^{11} + 3 q^{12} + 16 q^{13} + 8 q^{14} + 3 q^{15} + 12 q^{16} + 18 q^{17} + 25 q^{18} - 4 q^{19} + 12 q^{20} + 4 q^{21} - 12 q^{22} + 8 q^{23} + 3 q^{24} + 12 q^{25} + 16 q^{26} + 6 q^{27} + 8 q^{28} + 20 q^{29} + 3 q^{30} + 5 q^{31} + 12 q^{32} - 3 q^{33} + 18 q^{34} + 8 q^{35} + 25 q^{36} + 19 q^{37} - 4 q^{38} + 6 q^{39} + 12 q^{40} + 16 q^{41} + 4 q^{42} + 12 q^{43} - 12 q^{44} + 25 q^{45} + 8 q^{46} - q^{47} + 3 q^{48} + 52 q^{49} + 12 q^{50} + q^{51} + 16 q^{52} + 11 q^{53} + 6 q^{54} - 12 q^{55} + 8 q^{56} + 9 q^{57} + 20 q^{58} - 11 q^{59} + 3 q^{60} + 18 q^{61} + 5 q^{62} + 15 q^{63} + 12 q^{64} + 16 q^{65} - 3 q^{66} - 10 q^{67} + 18 q^{68} + 8 q^{70} - 2 q^{71} + 25 q^{72} + 29 q^{73} + 19 q^{74} + 3 q^{75} - 4 q^{76} - 8 q^{77} + 6 q^{78} + 2 q^{79} + 12 q^{80} - 8 q^{81} + 16 q^{82} + 26 q^{83} + 4 q^{84} + 18 q^{85} + 12 q^{86} - 4 q^{87} - 12 q^{88} + 41 q^{89} + 25 q^{90} - 4 q^{91} + 8 q^{92} + 5 q^{93} - q^{94} - 4 q^{95} + 3 q^{96} - 7 q^{97} + 52 q^{98} - 25 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 12 * q^5 + 3 * q^6 + 8 * q^7 + 12 * q^8 + 25 * q^9 + 12 * q^10 - 12 * q^11 + 3 * q^12 + 16 * q^13 + 8 * q^14 + 3 * q^15 + 12 * q^16 + 18 * q^17 + 25 * q^18 - 4 * q^19 + 12 * q^20 + 4 * q^21 - 12 * q^22 + 8 * q^23 + 3 * q^24 + 12 * q^25 + 16 * q^26 + 6 * q^27 + 8 * q^28 + 20 * q^29 + 3 * q^30 + 5 * q^31 + 12 * q^32 - 3 * q^33 + 18 * q^34 + 8 * q^35 + 25 * q^36 + 19 * q^37 - 4 * q^38 + 6 * q^39 + 12 * q^40 + 16 * q^41 + 4 * q^42 + 12 * q^43 - 12 * q^44 + 25 * q^45 + 8 * q^46 - q^47 + 3 * q^48 + 52 * q^49 + 12 * q^50 + q^51 + 16 * q^52 + 11 * q^53 + 6 * q^54 - 12 * q^55 + 8 * q^56 + 9 * q^57 + 20 * q^58 - 11 * q^59 + 3 * q^60 + 18 * q^61 + 5 * q^62 + 15 * q^63 + 12 * q^64 + 16 * q^65 - 3 * q^66 - 10 * q^67 + 18 * q^68 + 8 * q^70 - 2 * q^71 + 25 * q^72 + 29 * q^73 + 19 * q^74 + 3 * q^75 - 4 * q^76 - 8 * q^77 + 6 * q^78 + 2 * q^79 + 12 * q^80 - 8 * q^81 + 16 * q^82 + 26 * q^83 + 4 * q^84 + 18 * q^85 + 12 * q^86 - 4 * q^87 - 12 * q^88 + 41 * q^89 + 25 * q^90 - 4 * q^91 + 8 * q^92 + 5 * q^93 - q^94 - 4 * q^95 + 3 * q^96 - 7 * q^97 + 52 * q^98 - 25 * q^99

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