LMFDB - Embedded newform 8009.2.a.b.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.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:	$63.9521869788$
Analytic rank:	$0$
Dimension:	$361$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69652 q^{2} -0.708536 q^{3} +5.27124 q^{4} +1.66526 q^{5} +1.91059 q^{6} +4.37208 q^{7} -8.82098 q^{8} -2.49798 q^{9} +O(q^{10})$	\(q-2.69652 q^{2} -0.708536 q^{3} +5.27124 q^{4} +1.66526 q^{5} +1.91059 q^{6} +4.37208 q^{7} -8.82098 q^{8} -2.49798 q^{9} -4.49040 q^{10} +5.54765 q^{11} -3.73487 q^{12} +5.07684 q^{13} -11.7894 q^{14} -1.17990 q^{15} +13.2435 q^{16} +4.98886 q^{17} +6.73585 q^{18} +4.12562 q^{19} +8.77797 q^{20} -3.09778 q^{21} -14.9594 q^{22} -4.15175 q^{23} +6.24998 q^{24} -2.22692 q^{25} -13.6898 q^{26} +3.89552 q^{27} +23.0463 q^{28} +3.65409 q^{29} +3.18162 q^{30} +2.67360 q^{31} -18.0694 q^{32} -3.93071 q^{33} -13.4526 q^{34} +7.28064 q^{35} -13.1674 q^{36} -2.47381 q^{37} -11.1248 q^{38} -3.59713 q^{39} -14.6892 q^{40} +2.99414 q^{41} +8.35323 q^{42} +12.2890 q^{43} +29.2430 q^{44} -4.15977 q^{45} +11.1953 q^{46} +10.7397 q^{47} -9.38349 q^{48} +12.1151 q^{49} +6.00494 q^{50} -3.53479 q^{51} +26.7613 q^{52} -10.3287 q^{53} -10.5044 q^{54} +9.23827 q^{55} -38.5660 q^{56} -2.92315 q^{57} -9.85335 q^{58} -7.34368 q^{59} -6.21951 q^{60} -0.906326 q^{61} -7.20943 q^{62} -10.9213 q^{63} +22.2377 q^{64} +8.45425 q^{65} +10.5993 q^{66} +13.8216 q^{67} +26.2975 q^{68} +2.94166 q^{69} -19.6324 q^{70} -7.59229 q^{71} +22.0346 q^{72} -3.08204 q^{73} +6.67069 q^{74} +1.57785 q^{75} +21.7471 q^{76} +24.2548 q^{77} +9.69974 q^{78} -16.3398 q^{79} +22.0538 q^{80} +4.73381 q^{81} -8.07377 q^{82} -16.3701 q^{83} -16.3291 q^{84} +8.30774 q^{85} -33.1377 q^{86} -2.58906 q^{87} -48.9357 q^{88} +12.5881 q^{89} +11.2169 q^{90} +22.1964 q^{91} -21.8849 q^{92} -1.89434 q^{93} -28.9597 q^{94} +6.87021 q^{95} +12.8029 q^{96} +4.94266 q^{97} -32.6686 q^{98} -13.8579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) $ 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * 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 $

$2$	−2.69652	−1.90673	−0.953365	−	0.301820i	$-0.902406\pi$
$2$	−2.69652	−1.90673	−0.953365	+	0.301820i	$0.902406\pi$
$3$	−0.708536	−0.409074	−0.204537	−	0.978859i	$-0.565569\pi$
$3$	−0.708536	−0.409074	−0.204537	+	0.978859i	$0.565569\pi$
$4$	5.27124	2.63562
$5$	1.66526	0.744726	0.372363	−	0.928087i	$-0.378548\pi$
$5$	1.66526	0.744726	0.372363	+	0.928087i	$0.378548\pi$
$6$	1.91059	0.779993
$7$	4.37208	1.65249	0.826245	−	0.563310i	$-0.190473\pi$
$7$	4.37208	1.65249	0.826245	+	0.563310i	$0.190473\pi$
$8$	−8.82098	−3.11869
$9$	−2.49798	−0.832659
$10$	−4.49040	−1.41999
$11$	5.54765	1.67268	0.836340	−	0.548211i	$-0.184691\pi$
$11$	5.54765	1.67268	0.836340	+	0.548211i	$0.184691\pi$
$12$	−3.73487	−1.07816
$13$	5.07684	1.40806	0.704031	−	0.710169i	$-0.251381\pi$
$13$	5.07684	1.40806	0.704031	+	0.710169i	$0.251381\pi$
$14$	−11.7894	−3.15085
$15$	−1.17990	−0.304648
$16$	13.2435	3.31087
$17$	4.98886	1.20998	0.604989	−	0.796234i	$-0.293178\pi$
$17$	4.98886	1.20998	0.604989	+	0.796234i	$0.293178\pi$
$18$	6.73585	1.58766
$19$	4.12562	0.946482	0.473241	−	0.880933i	$-0.343084\pi$
$19$	4.12562	0.946482	0.473241	+	0.880933i	$0.343084\pi$
$20$	8.77797	1.96281
$21$	−3.09778	−0.675990
$22$	−14.9594	−3.18935
$23$	−4.15175	−0.865699	−0.432849	−	0.901466i	$-0.642492\pi$
$23$	−4.15175	−0.865699	−0.432849	+	0.901466i	$0.642492\pi$
$24$	6.24998	1.27577
$25$	−2.22692	−0.445384
$26$	−13.6898	−2.68480
$27$	3.89552	0.749692
$28$	23.0463	4.35534
$29$	3.65409	0.678548	0.339274	−	0.940688i	$-0.389819\pi$
$29$	3.65409	0.678548	0.339274	+	0.940688i	$0.389819\pi$
$30$	3.18162	0.580881
$31$	2.67360	0.480193	0.240097	−	0.970749i	$-0.422821\pi$
$31$	2.67360	0.480193	0.240097	+	0.970749i	$0.422821\pi$
$32$	−18.0694	−3.19425
$33$	−3.93071	−0.684249
$34$	−13.4526	−2.30710
$35$	7.28064	1.23065
$36$	−13.1674	−2.19457
$37$	−2.47381	−0.406692	−0.203346	−	0.979107i	$-0.565182\pi$
$37$	−2.47381	−0.406692	−0.203346	+	0.979107i	$0.565182\pi$
$38$	−11.1248	−1.80469
$39$	−3.59713	−0.576001
$40$	−14.6892	−2.32256
$41$	2.99414	0.467606	0.233803	−	0.972284i	$-0.424883\pi$
$41$	2.99414	0.467606	0.233803	+	0.972284i	$0.424883\pi$
$42$	8.35323	1.28893
$43$	12.2890	1.87406	0.937031	−	0.349247i	$-0.113562\pi$
$43$	12.2890	1.87406	0.937031	+	0.349247i	$0.113562\pi$
$44$	29.2430	4.40855
$45$	−4.15977	−0.620102
$46$	11.1953	1.65065
$47$	10.7397	1.56654	0.783270	−	0.621682i	$-0.213550\pi$
$47$	10.7397	1.56654	0.783270	+	0.621682i	$0.213550\pi$
$48$	−9.38349	−1.35439
$49$	12.1151	1.73073
$50$	6.00494	0.849227
$51$	−3.53479	−0.494970
$52$	26.7613	3.71112
$53$	−10.3287	−1.41875	−0.709376	−	0.704830i	$-0.751023\pi$
$53$	−10.3287	−1.41875	−0.709376	+	0.704830i	$0.751023\pi$
$54$	−10.5044	−1.42946
$55$	9.23827	1.24569
$56$	−38.5660	−5.15360
$57$	−2.92315	−0.387181
$58$	−9.85335	−1.29381
$59$	−7.34368	−0.956065	−0.478033	−	0.878342i	$-0.658650\pi$
$59$	−7.34368	−0.956065	−0.478033	+	0.878342i	$0.658650\pi$
$60$	−6.21951	−0.802935
$61$	−0.906326	−0.116043	−0.0580216	−	0.998315i	$-0.518479\pi$
$61$	−0.906326	−0.116043	−0.0580216	+	0.998315i	$0.518479\pi$
$62$	−7.20943	−0.915599
$63$	−10.9213	−1.37596
$64$	22.2377	2.77971
$65$	8.45425	1.04862
$66$	10.5993	1.30468
$67$	13.8216	1.68858	0.844288	−	0.535890i	$-0.180024\pi$
$67$	13.8216	1.68858	0.844288	+	0.535890i	$0.180024\pi$
$68$	26.2975	3.18904
$69$	2.94166	0.354135
$70$	−19.6324	−2.34652
$71$	−7.59229	−0.901039	−0.450520	−	0.892767i	$-0.648761\pi$
$71$	−7.59229	−0.901039	−0.450520	+	0.892767i	$0.648761\pi$
$72$	22.0346	2.59680
$73$	−3.08204	−0.360725	−0.180363	−	0.983600i	$-0.557727\pi$
$73$	−3.08204	−0.360725	−0.180363	+	0.983600i	$0.557727\pi$
$74$	6.67069	0.775452
$75$	1.57785	0.182195
$76$	21.7471	2.49457
$77$	24.2548	2.76409
$78$	9.69974	1.09828
$79$	−16.3398	−1.83837	−0.919187	−	0.393821i	$-0.871153\pi$
$79$	−16.3398	−1.83837	−0.919187	+	0.393821i	$0.871153\pi$
$80$	22.0538	2.46569
$81$	4.73381	0.525979
$82$	−8.07377	−0.891599
$83$	−16.3701	−1.79685	−0.898427	−	0.439123i	$-0.855289\pi$
$83$	−16.3701	−1.79685	−0.898427	+	0.439123i	$0.855289\pi$
$84$	−16.3291	−1.78165
$85$	8.30774	0.901101
$86$	−33.1377	−3.57333
$87$	−2.58906	−0.277576
$88$	−48.9357	−5.21656
$89$	12.5881	1.33433	0.667165	−	0.744910i	$-0.267507\pi$
$89$	12.5881	1.33433	0.667165	+	0.744910i	$0.267507\pi$
$90$	11.2169	1.18237
$91$	22.1964	2.32681
$92$	−21.8849	−2.28165
$93$	−1.89434	−0.196434
$94$	−28.9597	−2.98697
$95$	6.87021	0.704869
$96$	12.8029	1.30669
$97$	4.94266	0.501851	0.250926	−	0.968006i	$-0.419265\pi$
$97$	4.94266	0.501851	0.250926	+	0.968006i	$0.419265\pi$
$98$	−32.6686	−3.30003
$99$	−13.8579	−1.39277
$100$	−11.7386	−1.17386
$101$	11.0758	1.10208	0.551042	−	0.834478i	$-0.314230\pi$
$101$	11.0758	1.10208	0.551042	+	0.834478i	$0.314230\pi$
$102$	9.53165	0.943774
$103$	1.42249	0.140162	0.0700808	−	0.997541i	$-0.477674\pi$
$103$	1.42249	0.140162	0.0700808	+	0.997541i	$0.477674\pi$
$104$	−44.7827	−4.39131
$105$	−5.15859	−0.503427
$106$	27.8515	2.70518
$107$	−12.8606	−1.24328	−0.621639	−	0.783304i	$-0.713533\pi$
$107$	−12.8606	−1.24328	−0.621639	+	0.783304i	$0.713533\pi$
$108$	20.5342	1.97590
$109$	3.81790	0.365689	0.182844	−	0.983142i	$-0.441470\pi$
$109$	3.81790	0.365689	0.182844	+	0.983142i	$0.441470\pi$
$110$	−24.9112	−2.37519
$111$	1.75279	0.166367
$112$	57.9016	5.47119
$113$	7.27407	0.684287	0.342144	−	0.939648i	$-0.388847\pi$
$113$	7.27407	0.684287	0.342144	+	0.939648i	$0.388847\pi$
$114$	7.88234	0.738249
$115$	−6.91372	−0.644708
$116$	19.2616	1.78840
$117$	−12.6818	−1.17244
$118$	19.8024	1.82296
$119$	21.8117	1.99948
$120$	10.4078	0.950100
$121$	19.7764	1.79786
$122$	2.44393	0.221263
$123$	−2.12146	−0.191285
$124$	14.0932	1.26561
$125$	−12.0347	−1.07641
$126$	29.4497	2.62359
$127$	−11.7091	−1.03901	−0.519507	−	0.854466i	$-0.673884\pi$
$127$	−11.7091	−1.03901	−0.519507	+	0.854466i	$0.673884\pi$
$128$	−23.8255	−2.10590
$129$	−8.70724	−0.766629
$130$	−22.7971	−1.99944
$131$	−1.00057	−0.0874200	−0.0437100	−	0.999044i	$-0.513918\pi$
$131$	−1.00057	−0.0874200	−0.0437100	+	0.999044i	$0.513918\pi$
$132$	−20.7197	−1.80342
$133$	18.0375	1.56405
$134$	−37.2702	−3.21966
$135$	6.48704	0.558315
$136$	−44.0067	−3.77354
$137$	−1.21352	−0.103678	−0.0518392	−	0.998655i	$-0.516508\pi$
$137$	−1.21352	−0.103678	−0.0518392	+	0.998655i	$0.516508\pi$
$138$	−7.93227	−0.675239
$139$	20.5186	1.74037	0.870184	−	0.492727i	$-0.164000\pi$
$139$	20.5186	1.74037	0.870184	+	0.492727i	$0.164000\pi$
$140$	38.3780	3.24353
$141$	−7.60943	−0.640830
$142$	20.4728	1.71804
$143$	28.1646	2.35524
$144$	−33.0819	−2.75683
$145$	6.08501	0.505332
$146$	8.31079	0.687805
$147$	−8.58397	−0.707994
$148$	−13.0401	−1.07189
$149$	−12.7647	−1.04573	−0.522863	−	0.852417i	$-0.675136\pi$
$149$	−12.7647	−1.04573	−0.522863	+	0.852417i	$0.675136\pi$
$150$	−4.25472	−0.347396
$151$	−10.4148	−0.847547	−0.423774	−	0.905768i	$-0.639295\pi$
$151$	−10.4148	−0.847547	−0.423774	+	0.905768i	$0.639295\pi$
$152$	−36.3920	−2.95178
$153$	−12.4621	−1.00750
$154$	−65.4036	−5.27037
$155$	4.45224	0.357612
$156$	−18.9613	−1.51812
$157$	1.61882	0.129196	0.0645981	−	0.997911i	$-0.479423\pi$
$157$	1.61882	0.129196	0.0645981	+	0.997911i	$0.479423\pi$
$158$	44.0607	3.50528
$159$	7.31824	0.580374
$160$	−30.0902	−2.37884
$161$	−18.1518	−1.43056
$162$	−12.7648	−1.00290
$163$	1.14485	0.0896718	0.0448359	−	0.998994i	$-0.485724\pi$
$163$	1.14485	0.0896718	0.0448359	+	0.998994i	$0.485724\pi$
$164$	15.7828	1.23243
$165$	−6.54565	−0.509578
$166$	44.1424	3.42612
$167$	3.05060	0.236063	0.118031	−	0.993010i	$-0.462342\pi$
$167$	3.05060	0.236063	0.118031	+	0.993010i	$0.462342\pi$
$168$	27.3254	2.10820
$169$	12.7743	0.982641
$170$	−22.4020	−1.71816
$171$	−10.3057	−0.788096
$172$	64.7785	4.93931
$173$	−10.1547	−0.772048	−0.386024	−	0.922489i	$-0.626152\pi$
$173$	−10.1547	−0.772048	−0.386024	+	0.922489i	$0.626152\pi$
$174$	6.98146	0.529263
$175$	−9.73627	−0.735993
$176$	73.4703	5.53803
$177$	5.20326	0.391101
$178$	−33.9440	−2.54421
$179$	1.04352	0.0779962	0.0389981	−	0.999239i	$-0.487583\pi$
$179$	1.04352	0.0779962	0.0389981	+	0.999239i	$0.487583\pi$
$180$	−21.9272	−1.63435
$181$	−2.23235	−0.165929	−0.0829646	−	0.996552i	$-0.526439\pi$
$181$	−2.23235	−0.165929	−0.0829646	+	0.996552i	$0.526439\pi$
$182$	−59.8530	−4.43660
$183$	0.642165	0.0474702
$184$	36.6225	2.69984
$185$	−4.11953	−0.302874
$186$	5.10815	0.374547
$187$	27.6765	2.02391
$188$	56.6113	4.12880
$189$	17.0315	1.23886
$190$	−18.5257	−1.34399
$191$	13.9997	1.01298	0.506490	−	0.862246i	$-0.330943\pi$
$191$	13.9997	1.01298	0.506490	+	0.862246i	$0.330943\pi$
$192$	−15.7562	−1.13711
$193$	−14.0547	−1.01168	−0.505839	−	0.862628i	$-0.668817\pi$
$193$	−14.0547	−1.01168	−0.505839	+	0.862628i	$0.668817\pi$
$194$	−13.3280	−0.956895
$195$	−5.99014	−0.428963
$196$	63.8615	4.56153
$197$	21.2533	1.51424	0.757118	−	0.653278i	$-0.226607\pi$
$197$	21.2533	1.51424	0.757118	+	0.653278i	$0.226607\pi$
$198$	37.3682	2.65564
$199$	−6.59245	−0.467327	−0.233663	−	0.972318i	$-0.575071\pi$
$199$	−6.59245	−0.467327	−0.233663	+	0.972318i	$0.575071\pi$
$200$	19.6436	1.38901
$201$	−9.79309	−0.690752
$202$	−29.8662	−2.10138
$203$	15.9760	1.12129
$204$	−18.6327	−1.30455
$205$	4.98601	0.348238
$206$	−3.83577	−0.267251
$207$	10.3710	0.720832
$208$	67.2351	4.66192
$209$	22.8875	1.58316
$210$	13.9103	0.959900
$211$	9.10184	0.626596	0.313298	−	0.949655i	$-0.398566\pi$
$211$	9.10184	0.626596	0.313298	+	0.949655i	$0.398566\pi$
$212$	−54.4449	−3.73929
$213$	5.37941	0.368591
$214$	34.6788	2.37060
$215$	20.4644	1.39566
$216$	−34.3623	−2.33806
$217$	11.6892	0.793515
$218$	−10.2951	−0.697270
$219$	2.18374	0.147563
$220$	48.6971	3.28316
$221$	25.3277	1.70372
$222$	−4.72643	−0.317217
$223$	5.03201	0.336968	0.168484	−	0.985704i	$-0.446113\pi$
$223$	5.03201	0.336968	0.168484	+	0.985704i	$0.446113\pi$
$224$	−79.0010	−5.27848
$225$	5.56279	0.370853
$226$	−19.6147	−1.30475
$227$	4.27205	0.283546	0.141773	−	0.989899i	$-0.454720\pi$
$227$	4.27205	0.283546	0.141773	+	0.989899i	$0.454720\pi$
$228$	−15.4086	−1.02046
$229$	28.7260	1.89827	0.949133	−	0.314874i	$-0.101962\pi$
$229$	28.7260	1.89827	0.949133	+	0.314874i	$0.101962\pi$
$230$	18.6430	1.22928
$231$	−17.1854	−1.13072
$232$	−32.2327	−2.11618
$233$	11.8295	0.774975	0.387488	−	0.921875i	$-0.373343\pi$
$233$	11.8295	0.774975	0.387488	+	0.921875i	$0.373343\pi$
$234$	34.1969	2.23552
$235$	17.8843	1.16664
$236$	−38.7103	−2.51983
$237$	11.5774	0.752031
$238$	−58.8158	−3.81246
$239$	−20.7689	−1.34343	−0.671715	−	0.740810i	$-0.734442\pi$
$239$	−20.7689	−1.34343	−0.671715	+	0.740810i	$0.734442\pi$
$240$	−15.6259	−1.00865
$241$	10.5553	0.679927	0.339964	−	0.940439i	$-0.389585\pi$
$241$	10.5553	0.679927	0.339964	+	0.940439i	$0.389585\pi$
$242$	−53.3277	−3.42803
$243$	−15.0406	−0.964857
$244$	−4.77746	−0.305846
$245$	20.1747	1.28892
$246$	5.72056	0.364729
$247$	20.9451	1.33271
$248$	−23.5838	−1.49757
$249$	11.5988	0.735046
$250$	32.4518	2.05243
$251$	−20.3447	−1.28415	−0.642073	−	0.766644i	$-0.721925\pi$
$251$	−20.3447	−1.28415	−0.642073	+	0.766644i	$0.721925\pi$
$252$	−57.5691	−3.62651
$253$	−23.0324	−1.44804
$254$	31.5738	1.98112
$255$	−5.88634	−0.368617
$256$	19.7708	1.23568
$257$	−18.1371	−1.13136	−0.565682	−	0.824624i	$-0.691387\pi$
$257$	−18.1371	−1.13136	−0.565682	+	0.824624i	$0.691387\pi$
$258$	23.4793	1.46176
$259$	−10.8157	−0.672055
$260$	44.5644	2.76376
$261$	−9.12784	−0.564999
$262$	2.69805	0.166686
$263$	−22.7771	−1.40449	−0.702247	−	0.711933i	$-0.747820\pi$
$263$	−22.7771	−1.40449	−0.702247	+	0.711933i	$0.747820\pi$
$264$	34.6727	2.13396
$265$	−17.1999	−1.05658
$266$	−48.6386	−2.98223
$267$	−8.91909	−0.545840
$268$	72.8569	4.45044
$269$	−5.33215	−0.325107	−0.162554	−	0.986700i	$-0.551973\pi$
$269$	−5.33215	−0.325107	−0.162554	+	0.986700i	$0.551973\pi$
$270$	−17.4924	−1.06456
$271$	1.53302	0.0931244	0.0465622	−	0.998915i	$-0.485173\pi$
$271$	1.53302	0.0931244	0.0465622	+	0.998915i	$0.485173\pi$
$272$	66.0700	4.00608
$273$	−15.7269	−0.951837
$274$	3.27230	0.197687
$275$	−12.3542	−0.744985
$276$	15.5062	0.933364
$277$	−15.4320	−0.927220	−0.463610	−	0.886039i	$-0.653446\pi$
$277$	−15.4320	−0.927220	−0.463610	+	0.886039i	$0.653446\pi$
$278$	−55.3290	−3.31841
$279$	−6.67860	−0.399837
$280$	−64.2223	−3.83802
$281$	−33.2090	−1.98108	−0.990540	−	0.137223i	$-0.956182\pi$
$281$	−33.2090	−1.98108	−0.990540	+	0.137223i	$0.956182\pi$
$282$	20.5190	1.22189
$283$	−24.7806	−1.47305	−0.736527	−	0.676408i	$-0.763536\pi$
$283$	−24.7806	−1.47305	−0.736527	+	0.676408i	$0.763536\pi$
$284$	−40.0208	−2.37480
$285$	−4.86780	−0.288343
$286$	−75.9464	−4.49080
$287$	13.0906	0.772715
$288$	45.1370	2.65972
$289$	7.88877	0.464045
$290$	−16.4084	−0.963532
$291$	−3.50206	−0.205294
$292$	−16.2462	−0.950734
$293$	−8.29484	−0.484590	−0.242295	−	0.970203i	$-0.577900\pi$
$293$	−8.29484	−0.484590	−0.242295	+	0.970203i	$0.577900\pi$
$294$	23.1469	1.34995
$295$	−12.2291	−0.712006
$296$	21.8214	1.26835
$297$	21.6110	1.25400
$298$	34.4203	1.99392
$299$	−21.0778	−1.21896
$300$	8.31724	0.480196
$301$	53.7287	3.09687
$302$	28.0839	1.61604
$303$	−7.84761	−0.450834
$304$	54.6376	3.13368
$305$	−1.50927	−0.0864203
$306$	33.6043	1.92103
$307$	−16.9118	−0.965209	−0.482604	−	0.875838i	$-0.660309\pi$
$307$	−16.9118	−0.965209	−0.482604	+	0.875838i	$0.660309\pi$
$308$	127.853	7.28509
$309$	−1.00788	−0.0573365
$310$	−12.0056	−0.681870
$311$	−16.4125	−0.930667	−0.465334	−	0.885135i	$-0.654066\pi$
$311$	−16.4125	−0.930667	−0.465334	+	0.885135i	$0.654066\pi$
$312$	31.7302	1.79637
$313$	−31.0431	−1.75466	−0.877331	−	0.479886i	$-0.840678\pi$
$313$	−31.0431	−1.75466	−0.877331	+	0.479886i	$0.840678\pi$
$314$	−4.36519	−0.246342
$315$	−18.1869	−1.02471
$316$	−86.1312	−4.84526
$317$	−11.0421	−0.620183	−0.310092	−	0.950707i	$-0.600360\pi$
$317$	−11.0421	−0.620183	−0.310092	+	0.950707i	$0.600360\pi$
$318$	−19.7338	−1.10662
$319$	20.2716	1.13499
$320$	37.0314	2.07012
$321$	9.11218	0.508593
$322$	48.9467	2.72769
$323$	20.5821	1.14522
$324$	24.9531	1.38628
$325$	−11.3057	−0.627128
$326$	−3.08712	−0.170980
$327$	−2.70512	−0.149594
$328$	−26.4112	−1.45832
$329$	46.9546	2.58869
$330$	17.6505	0.971628
$331$	1.05485	0.0579799	0.0289900	−	0.999580i	$-0.490771\pi$
$331$	1.05485	0.0579799	0.0289900	+	0.999580i	$0.490771\pi$
$332$	−86.2908	−4.73583
$333$	6.17953	0.338636
$334$	−8.22602	−0.450108
$335$	23.0165	1.25753
$336$	−41.0254	−2.23812
$337$	−29.3368	−1.59808	−0.799040	−	0.601278i	$-0.794658\pi$
$337$	−29.3368	−1.59808	−0.799040	+	0.601278i	$0.794658\pi$
$338$	−34.4463	−1.87363
$339$	−5.15395	−0.279924
$340$	43.7921	2.37496
$341$	14.8322	0.803210
$342$	27.7895	1.50269
$343$	22.3635	1.20752
$344$	−108.401	−5.84461
$345$	4.89863	0.263733
$346$	27.3824	1.47209
$347$	−8.28082	−0.444538	−0.222269	−	0.974985i	$-0.571346\pi$
$347$	−8.28082	−0.444538	−0.222269	+	0.974985i	$0.571346\pi$
$348$	−13.6476	−0.731586
$349$	20.5196	1.09839	0.549194	−	0.835695i	$-0.314935\pi$
$349$	20.5196	1.09839	0.549194	+	0.835695i	$0.314935\pi$
$350$	26.2541	1.40334
$351$	19.7769	1.05561
$352$	−100.243	−5.34297
$353$	−16.0657	−0.855092	−0.427546	−	0.903994i	$-0.640622\pi$
$353$	−16.0657	−0.855092	−0.427546	+	0.903994i	$0.640622\pi$
$354$	−14.0307	−0.745725
$355$	−12.6431	−0.671027
$356$	66.3546	3.51679
$357$	−15.4544	−0.817933
$358$	−2.81387	−0.148718
$359$	−10.6511	−0.562146	−0.281073	−	0.959686i	$-0.590690\pi$
$359$	−10.6511	−0.562146	−0.281073	+	0.959686i	$0.590690\pi$
$360$	36.6932	1.93390
$361$	−1.97928	−0.104173
$362$	6.01958	0.316382
$363$	−14.0123	−0.735457
$364$	117.002	6.13259
$365$	−5.13238	−0.268641
$366$	−1.73161	−0.0905129
$367$	−15.6566	−0.817269	−0.408635	−	0.912698i	$-0.633995\pi$
$367$	−15.6566	−0.817269	−0.408635	+	0.912698i	$0.633995\pi$
$368$	−54.9836	−2.86622
$369$	−7.47929	−0.389356
$370$	11.1084	0.577499
$371$	−45.1578	−2.34448
$372$	−9.98555	−0.517726
$373$	−3.53302	−0.182933	−0.0914664	−	0.995808i	$-0.529155\pi$
$373$	−3.53302	−0.182933	−0.0914664	+	0.995808i	$0.529155\pi$
$374$	−74.6303	−3.85904
$375$	8.52701	0.440333
$376$	−94.7342	−4.88554
$377$	18.5513	0.955439
$378$	−45.9259	−2.36217
$379$	23.0797	1.18553	0.592764	−	0.805377i	$-0.298037\pi$
$379$	23.0797	1.18553	0.592764	+	0.805377i	$0.298037\pi$
$380$	36.2145	1.85777
$381$	8.29632	0.425033
$382$	−37.7504	−1.93148
$383$	−29.4117	−1.50287	−0.751435	−	0.659807i	$-0.770638\pi$
$383$	−29.4117	−1.50287	−0.751435	+	0.659807i	$0.770638\pi$
$384$	16.8813	0.861469
$385$	40.3904	2.05849
$386$	37.8988	1.92900
$387$	−30.6977	−1.56045
$388$	26.0540	1.32269
$389$	16.8240	0.853012	0.426506	−	0.904485i	$-0.359744\pi$
$389$	16.8240	0.853012	0.426506	+	0.904485i	$0.359744\pi$
$390$	16.1526	0.817917
$391$	−20.7125	−1.04748
$392$	−106.867	−5.39759
$393$	0.708939	0.0357612
$394$	−57.3101	−2.88724
$395$	−27.2100	−1.36908
$396$	−73.0483	−3.67082
$397$	15.7668	0.791311	0.395656	−	0.918399i	$-0.370517\pi$
$397$	15.7668	0.791311	0.395656	+	0.918399i	$0.370517\pi$
$398$	17.7767	0.891066
$399$	−12.7802	−0.639812
$400$	−29.4922	−1.47461
$401$	−13.8852	−0.693396	−0.346698	−	0.937977i	$-0.612697\pi$
$401$	−13.8852	−0.693396	−0.346698	+	0.937977i	$0.612697\pi$
$402$	26.4073	1.31708
$403$	13.5735	0.676142
$404$	58.3832	2.90467
$405$	7.88302	0.391710
$406$	−43.0796	−2.13801
$407$	−13.7239	−0.680266
$408$	31.1803	1.54366
$409$	−13.8742	−0.686036	−0.343018	−	0.939329i	$-0.611449\pi$
$409$	−13.8742	−0.686036	−0.343018	+	0.939329i	$0.611449\pi$
$410$	−13.4449	−0.663996
$411$	0.859826	0.0424121
$412$	7.49826	0.369413
$413$	−32.1071	−1.57989
$414$	−27.9655	−1.37443
$415$	−27.2605	−1.33816
$416$	−91.7357	−4.49771
$417$	−14.5382	−0.711939
$418$	−61.7167	−3.01866
$419$	−25.6107	−1.25116	−0.625582	−	0.780158i	$-0.715139\pi$
$419$	−25.6107	−1.25116	−0.625582	+	0.780158i	$0.715139\pi$
$420$	−27.1922	−1.32684
$421$	−7.47777	−0.364444	−0.182222	−	0.983257i	$-0.558329\pi$
$421$	−7.47777	−0.364444	−0.182222	+	0.983257i	$0.558329\pi$
$422$	−24.5433	−1.19475
$423$	−26.8274	−1.30439
$424$	91.1090	4.42464
$425$	−11.1098	−0.538904
$426$	−14.5057	−0.702804
$427$	−3.96253	−0.191760
$428$	−67.7912	−3.27681
$429$	−19.9556	−0.963466
$430$	−55.1828	−2.66115
$431$	20.1485	0.970521	0.485260	−	0.874370i	$-0.338725\pi$
$431$	20.1485	0.970521	0.485260	+	0.874370i	$0.338725\pi$
$432$	51.5902	2.48214
$433$	−2.24252	−0.107769	−0.0538844	−	0.998547i	$-0.517160\pi$
$433$	−2.24252	−0.107769	−0.0538844	+	0.998547i	$0.517160\pi$
$434$	−31.5202	−1.51302
$435$	−4.31145	−0.206718
$436$	20.1251	0.963817
$437$	−17.1285	−0.819368
$438$	−5.88849	−0.281363
$439$	−0.0516499	−0.00246512	−0.00123256	−	0.999999i	$-0.500392\pi$
$439$	−0.0516499	−0.00246512	−0.00123256	+	0.999999i	$0.500392\pi$
$440$	−81.4905	−3.88491
$441$	−30.2632	−1.44110
$442$	−68.2967	−3.24854
$443$	24.5128	1.16464	0.582318	−	0.812961i	$-0.302146\pi$
$443$	24.5128	1.16464	0.582318	+	0.812961i	$0.302146\pi$
$444$	9.23936	0.438480
$445$	20.9623	0.993710
$446$	−13.5689	−0.642507
$447$	9.04426	0.427779
$448$	97.2249	4.59344
$449$	−19.0962	−0.901205	−0.450603	−	0.892725i	$-0.648791\pi$
$449$	−19.0962	−0.901205	−0.450603	+	0.892725i	$0.648791\pi$
$450$	−15.0002	−0.707116
$451$	16.6104	0.782155
$452$	38.3434	1.80352
$453$	7.37929	0.346709
$454$	−11.5197	−0.540646
$455$	36.9626	1.73284
$456$	25.7850	1.20750
$457$	11.8763	0.555550	0.277775	−	0.960646i	$-0.410403\pi$
$457$	11.8763	0.555550	0.277775	+	0.960646i	$0.410403\pi$
$458$	−77.4603	−3.61948
$459$	19.4342	0.907111
$460$	−36.4439	−1.69921
$461$	−31.5616	−1.46997	−0.734984	−	0.678084i	$-0.762811\pi$
$461$	−31.5616	−1.46997	−0.734984	+	0.678084i	$0.762811\pi$
$462$	46.3408	2.15597
$463$	−7.89741	−0.367024	−0.183512	−	0.983018i	$-0.558747\pi$
$463$	−7.89741	−0.367024	−0.183512	+	0.983018i	$0.558747\pi$
$464$	48.3930	2.24659
$465$	−3.15457	−0.146290
$466$	−31.8985	−1.47767
$467$	−15.1992	−0.703333	−0.351667	−	0.936125i	$-0.614385\pi$
$467$	−15.1992	−0.703333	−0.351667	+	0.936125i	$0.614385\pi$
$468$	−66.8490	−3.09010
$469$	60.4291	2.79035
$470$	−48.2254	−2.22447
$471$	−1.14699	−0.0528507
$472$	64.7784	2.98167
$473$	68.1753	3.13471
$474$	−31.2186	−1.43392
$475$	−9.18742	−0.421548
$476$	114.975	5.26986
$477$	25.8008	1.18134
$478$	56.0039	2.56156
$479$	30.4724	1.39232	0.696161	−	0.717886i	$-0.254890\pi$
$479$	30.4724	1.39232	0.696161	+	0.717886i	$0.254890\pi$
$480$	21.3200	0.973122
$481$	−12.5592	−0.572648
$482$	−28.4626	−1.29644
$483$	12.8612	0.585204
$484$	104.246	4.73847
$485$	8.23080	0.373741
$486$	40.5574	1.83972
$487$	−9.09093	−0.411949	−0.205975	−	0.978557i	$-0.566036\pi$
$487$	−9.09093	−0.411949	−0.205975	+	0.978557i	$0.566036\pi$
$488$	7.99468	0.361902
$489$	−0.811170	−0.0366824
$490$	−54.4016	−2.45761
$491$	−23.0008	−1.03801	−0.519006	−	0.854771i	$-0.673698\pi$
$491$	−23.0008	−1.03801	−0.519006	+	0.854771i	$0.673698\pi$
$492$	−11.1827	−0.504155
$493$	18.2298	0.821028
$494$	−56.4790	−2.54111
$495$	−23.0770	−1.03723
$496$	35.4078	1.58986
$497$	−33.1941	−1.48896
$498$	−31.2765	−1.40153
$499$	12.9001	0.577489	0.288745	−	0.957406i	$-0.406762\pi$
$499$	12.9001	0.577489	0.288745	+	0.957406i	$0.406762\pi$
$500$	−63.4377	−2.83702
$501$	−2.16146	−0.0965670
$502$	54.8599	2.44852
$503$	5.75556	0.256628	0.128314	−	0.991734i	$-0.459044\pi$
$503$	5.75556	0.256628	0.128314	+	0.991734i	$0.459044\pi$
$504$	96.3370	4.29119
$505$	18.4441	0.820750
$506$	62.1075	2.76102
$507$	−9.05108	−0.401973
$508$	−61.7214	−2.73845
$509$	−27.6805	−1.22692	−0.613458	−	0.789727i	$-0.710222\pi$
$509$	−27.6805	−1.22692	−0.613458	+	0.789727i	$0.710222\pi$
$510$	15.8726	0.702853
$511$	−13.4749	−0.596095
$512$	−5.66135	−0.250199
$513$	16.0714	0.709570
$514$	48.9072	2.15721
$515$	2.36880	0.104382
$516$	−45.8979	−2.02054
$517$	59.5799	2.62032
$518$	29.1648	1.28143
$519$	7.19498	0.315825
$520$	−74.5747	−3.27032
$521$	−2.49775	−0.109428	−0.0547142	−	0.998502i	$-0.517425\pi$
$521$	−2.49775	−0.109428	−0.0547142	+	0.998502i	$0.517425\pi$
$522$	24.6134	1.07730
$523$	6.36858	0.278479	0.139239	−	0.990259i	$-0.455534\pi$
$523$	6.36858	0.278479	0.139239	+	0.990259i	$0.455534\pi$
$524$	−5.27423	−0.230406
$525$	6.89850	0.301075
$526$	61.4189	2.67799
$527$	13.3382	0.581023
$528$	−52.0564	−2.26546
$529$	−5.76300	−0.250565
$530$	46.3799	2.01462
$531$	18.3443	0.796076
$532$	95.0801	4.12225
$533$	15.2008	0.658419
$534$	24.0505	1.04077
$535$	−21.4162	−0.925901
$536$	−121.920	−5.26614
$537$	−0.739371	−0.0319062
$538$	14.3783	0.619892
$539$	67.2102	2.89495
$540$	34.1947	1.47151
$541$	2.79638	0.120226	0.0601129	−	0.998192i	$-0.480854\pi$
$541$	2.79638	0.120226	0.0601129	+	0.998192i	$0.480854\pi$
$542$	−4.13383	−0.177563
$543$	1.58170	0.0678773
$544$	−90.1460	−3.86498
$545$	6.35779	0.272338
$546$	42.4080	1.81490
$547$	−16.1419	−0.690179	−0.345089	−	0.938570i	$-0.612151\pi$
$547$	−16.1419	−0.690179	−0.345089	+	0.938570i	$0.612151\pi$
$548$	−6.39678	−0.273257
$549$	2.26398	0.0966244
$550$	33.3133	1.42049
$551$	15.0754	0.642234
$552$	−25.9483	−1.10443
$553$	−71.4390	−3.03790
$554$	41.6128	1.76796
$555$	2.91884	0.123898
$556$	108.159	4.58695
$557$	33.3977	1.41510	0.707552	−	0.706661i	$-0.249799\pi$
$557$	33.3977	1.41510	0.707552	+	0.706661i	$0.249799\pi$
$558$	18.0090	0.762381
$559$	62.3895	2.63880
$560$	96.4210	4.07453
$561$	−19.6098	−0.827926
$562$	89.5487	3.77739
$563$	0.570138	0.0240284	0.0120142	−	0.999928i	$-0.496176\pi$
$563$	0.570138	0.0240284	0.0120142	+	0.999928i	$0.496176\pi$
$564$	−40.1111	−1.68898
$565$	12.1132	0.509606
$566$	66.8215	2.80872
$567$	20.6966	0.869176
$568$	66.9714	2.81006
$569$	−3.23053	−0.135431	−0.0677155	−	0.997705i	$-0.521571\pi$
$569$	−3.23053	−0.135431	−0.0677155	+	0.997705i	$0.521571\pi$
$570$	13.1261	0.549793
$571$	−12.2300	−0.511810	−0.255905	−	0.966702i	$-0.582373\pi$
$571$	−12.2300	−0.511810	−0.255905	+	0.966702i	$0.582373\pi$
$572$	148.462	6.20751
$573$	−9.91927	−0.414383
$574$	−35.2992	−1.47336
$575$	9.24561	0.385568
$576$	−55.5492	−2.31455
$577$	1.33477	0.0555673	0.0277836	−	0.999614i	$-0.491155\pi$
$577$	1.33477	0.0555673	0.0277836	+	0.999614i	$0.491155\pi$
$578$	−21.2723	−0.884809
$579$	9.95826	0.413851
$580$	32.0755	1.33186
$581$	−71.5715	−2.96928
$582$	9.44338	0.391441
$583$	−57.2999	−2.37312
$584$	27.1866	1.12499
$585$	−21.1185	−0.873143
$586$	22.3672	0.923982
$587$	29.6178	1.22246	0.611230	−	0.791453i	$-0.290675\pi$
$587$	29.6178	1.22246	0.611230	+	0.791453i	$0.290675\pi$
$588$	−45.2482	−1.86600
$589$	11.0303	0.454494
$590$	32.9761	1.35760
$591$	−15.0587	−0.619434
$592$	−32.7619	−1.34651
$593$	−34.7325	−1.42629	−0.713147	−	0.701014i	$-0.752731\pi$
$593$	−34.7325	−1.42629	−0.713147	+	0.701014i	$0.752731\pi$
$594$	−58.2745	−2.39103
$595$	36.3221	1.48906
$596$	−67.2858	−2.75613
$597$	4.67099	0.191171
$598$	56.8367	2.32423
$599$	−41.3534	−1.68965	−0.844827	−	0.535039i	$-0.820297\pi$
$599$	−41.3534	−1.68965	−0.844827	+	0.535039i	$0.820297\pi$
$600$	−13.9182	−0.568208
$601$	−24.0624	−0.981527	−0.490763	−	0.871293i	$-0.663282\pi$
$601$	−24.0624	−0.981527	−0.490763	+	0.871293i	$0.663282\pi$
$602$	−144.881	−5.90489
$603$	−34.5260	−1.40601
$604$	−54.8991	−2.23381
$605$	32.9329	1.33891
$606$	21.1613	0.859618
$607$	−42.7504	−1.73518	−0.867592	−	0.497276i	$-0.834334\pi$
$607$	−42.7504	−1.73518	−0.867592	+	0.497276i	$0.834334\pi$
$608$	−74.5476	−3.02330
$609$	−11.3196	−0.458692
$610$	4.06977	0.164780
$611$	54.5235	2.20579
$612$	−65.6905	−2.65538
$613$	34.6890	1.40108	0.700538	−	0.713615i	$-0.252943\pi$
$613$	34.6890	1.40108	0.700538	+	0.713615i	$0.252943\pi$
$614$	45.6031	1.84039
$615$	−3.53277	−0.142455
$616$	−213.951	−8.62032
$617$	−13.1949	−0.531205	−0.265602	−	0.964083i	$-0.585571\pi$
$617$	−13.1949	−0.531205	−0.265602	+	0.964083i	$0.585571\pi$
$618$	2.71778	0.109325
$619$	−16.9232	−0.680200	−0.340100	−	0.940389i	$-0.610461\pi$
$619$	−16.9232	−0.680200	−0.340100	+	0.940389i	$0.610461\pi$
$620$	23.4688	0.942530
$621$	−16.1732	−0.649008
$622$	44.2567	1.77453
$623$	55.0360	2.20497
$624$	−47.6385	−1.90707
$625$	−8.90623	−0.356249
$626$	83.7086	3.34567
$627$	−16.2166	−0.647629
$628$	8.53320	0.340512
$629$	−12.3415	−0.492089
$630$	49.0413	1.95385
$631$	−27.9100	−1.11108	−0.555540	−	0.831490i	$-0.687488\pi$
$631$	−27.9100	−1.11108	−0.555540	+	0.831490i	$0.687488\pi$
$632$	144.133	5.73331
$633$	−6.44898	−0.256324
$634$	29.7752	1.18252
$635$	−19.4986	−0.773780
$636$	38.5762	1.52965
$637$	61.5063	2.43697
$638$	−54.6630	−2.16413
$639$	18.9654	0.750258
$640$	−39.6757	−1.56832
$641$	30.6063	1.20887	0.604437	−	0.796653i	$-0.293398\pi$
$641$	30.6063	1.20887	0.604437	+	0.796653i	$0.293398\pi$
$642$	−24.5712	−0.969749
$643$	−21.6917	−0.855437	−0.427719	−	0.903912i	$-0.640683\pi$
$643$	−21.6917	−0.855437	−0.427719	+	0.903912i	$0.640683\pi$
$644$	−95.6823	−3.77041
$645$	−14.4998	−0.570928
$646$	−55.5003	−2.18363
$647$	28.7460	1.13012	0.565060	−	0.825050i	$-0.308853\pi$
$647$	28.7460	1.13012	0.565060	+	0.825050i	$0.308853\pi$
$648$	−41.7569	−1.64036
$649$	−40.7402	−1.59919
$650$	30.4861	1.19576
$651$	−8.28223	−0.324606
$652$	6.03479	0.236341
$653$	−44.1231	−1.72667	−0.863335	−	0.504631i	$-0.831628\pi$
$653$	−44.1231	−1.72667	−0.863335	+	0.504631i	$0.831628\pi$
$654$	7.29443	0.285235
$655$	−1.66620	−0.0651039
$656$	39.6529	1.54818
$657$	7.69885	0.300361
$658$	−126.614	−4.93594
$659$	−9.71619	−0.378489	−0.189245	−	0.981930i	$-0.560604\pi$
$659$	−9.71619	−0.378489	−0.189245	+	0.981930i	$0.560604\pi$
$660$	−34.5037	−1.34305
$661$	28.6291	1.11354	0.556772	−	0.830665i	$-0.312040\pi$
$661$	28.6291	1.11354	0.556772	+	0.830665i	$0.312040\pi$
$662$	−2.84443	−0.110552
$663$	−17.9456	−0.696949
$664$	144.400	5.60382
$665$	30.0371	1.16479
$666$	−16.6632	−0.645687
$667$	−15.1709	−0.587419
$668$	16.0805	0.622172
$669$	−3.56536	−0.137845
$670$	−62.0645	−2.39776
$671$	−5.02798	−0.194103
$672$	55.9751	2.15929
$673$	−10.0868	−0.388819	−0.194410	−	0.980920i	$-0.562279\pi$
$673$	−10.0868	−0.388819	−0.194410	+	0.980920i	$0.562279\pi$
$674$	79.1075	3.04711
$675$	−8.67500	−0.333901
$676$	67.3366	2.58987
$677$	4.50963	0.173319	0.0866595	−	0.996238i	$-0.472381\pi$
$677$	4.50963	0.173319	0.0866595	+	0.996238i	$0.472381\pi$
$678$	13.8977	0.533739
$679$	21.6097	0.829305
$680$	−73.2824	−2.81025
$681$	−3.02690	−0.115991
$682$	−39.9954	−1.53150
$683$	49.9175	1.91004	0.955020	−	0.296541i	$-0.0958330\pi$
$683$	49.9175	1.91004	0.955020	+	0.296541i	$0.0958330\pi$
$684$	−54.3238	−2.07712
$685$	−2.02083	−0.0772119
$686$	−60.3038	−2.30241
$687$	−20.3534	−0.776531
$688$	162.750	6.20478
$689$	−52.4371	−1.99769
$690$	−13.2093	−0.502868
$691$	20.6687	0.786276	0.393138	−	0.919479i	$-0.371389\pi$
$691$	20.6687	0.786276	0.393138	+	0.919479i	$0.371389\pi$
$692$	−53.5279	−2.03483
$693$	−60.5879	−2.30154
$694$	22.3294	0.847613
$695$	34.1688	1.29610
$696$	22.8380	0.865673
$697$	14.9374	0.565793
$698$	−55.3315	−2.09433
$699$	−8.38162	−0.317022
$700$	−51.3222	−1.93980
$701$	0.0143892	0.000543474 0	0.000271737	−	1.00000i	$-0.499914\pi$
$701$	0.0143892	0.000543474 0	0.000271737		1.00000i	$0.499914\pi$
$702$	−53.3289	−2.01277
$703$	−10.2060	−0.384927
$704$	123.367	4.64956
$705$	−12.6717	−0.477242
$706$	43.3216	1.63043
$707$	48.4243	1.82118
$708$	27.4276	1.03079
$709$	37.5954	1.41193	0.705963	−	0.708249i	$-0.250514\pi$
$709$	37.5954	1.41193	0.705963	+	0.708249i	$0.250514\pi$
$710$	34.0925	1.27947
$711$	40.8165	1.53074
$712$	−111.039	−4.16136
$713$	−11.1001	−0.415703
$714$	41.6731	1.55958
$715$	46.9012	1.75401
$716$	5.50063	0.205568
$717$	14.7155	0.549562
$718$	28.7211	1.07186
$719$	36.3132	1.35425	0.677127	−	0.735866i	$-0.263225\pi$
$719$	36.3132	1.35425	0.677127	+	0.735866i	$0.263225\pi$
$720$	−55.0899	−2.05308
$721$	6.21922	0.231616
$722$	5.33717	0.198629
$723$	−7.47882	−0.278140
$724$	−11.7672	−0.437326
$725$	−8.13737	−0.302215
$726$	37.7846	1.40232
$727$	−37.2500	−1.38153	−0.690763	−	0.723082i	$-0.742725\pi$
$727$	−37.2500	−1.38153	−0.690763	+	0.723082i	$0.742725\pi$
$728$	−195.794	−7.25659
$729$	−3.54461	−0.131282
$730$	13.8396	0.512226
$731$	61.3084	2.26757
$732$	3.38501	0.125113
$733$	−45.4303	−1.67800	−0.839002	−	0.544128i	$-0.816861\pi$
$733$	−45.4303	−1.67800	−0.839002	+	0.544128i	$0.816861\pi$
$734$	42.2185	1.55831
$735$	−14.2945	−0.527261
$736$	75.0197	2.76526
$737$	76.6773	2.82445
$738$	20.1681	0.742397
$739$	43.6052	1.60404	0.802022	−	0.597294i	$-0.203757\pi$
$739$	43.6052	1.60404	0.802022	+	0.597294i	$0.203757\pi$
$740$	−21.7151	−0.798261
$741$	−14.8404	−0.545175
$742$	121.769	4.47028
$743$	−33.9439	−1.24528	−0.622641	−	0.782508i	$-0.713940\pi$
$743$	−33.9439	−1.24528	−0.622641	+	0.782508i	$0.713940\pi$
$744$	16.7100	0.612617
$745$	−21.2565	−0.778778
$746$	9.52687	0.348803
$747$	40.8922	1.49617
$748$	145.889	5.33425
$749$	−56.2274	−2.05451
$750$	−22.9933	−0.839596
$751$	29.7380	1.08516	0.542578	−	0.840005i	$-0.317448\pi$
$751$	29.7380	1.08516	0.542578	+	0.840005i	$0.317448\pi$
$752$	142.230	5.18661
$753$	14.4150	0.525310
$754$	−50.0239	−1.82176
$755$	−17.3434	−0.631190
$756$	89.7772	3.26516
$757$	−41.5958	−1.51183	−0.755913	−	0.654672i	$-0.772807\pi$
$757$	−41.5958	−1.51183	−0.755913	+	0.654672i	$0.772807\pi$
$758$	−62.2351	−2.26048
$759$	16.3193	0.592354
$760$	−60.6020	−2.19827
$761$	4.66081	0.168954	0.0844772	−	0.996425i	$-0.473078\pi$
$761$	4.66081	0.168954	0.0844772	+	0.996425i	$0.473078\pi$
$762$	−22.3712	−0.810424
$763$	16.6922	0.604297
$764$	73.7956	2.66983
$765$	−20.7525	−0.750310
$766$	79.3094	2.86557
$767$	−37.2827	−1.34620
$768$	−14.0083	−0.505482
$769$	5.81000	0.209514	0.104757	−	0.994498i	$-0.466594\pi$
$769$	5.81000	0.209514	0.104757	+	0.994498i	$0.466594\pi$
$770$	−108.914	−3.92498
$771$	12.8508	0.462811
$772$	−74.0856	−2.66640
$773$	−8.04286	−0.289282	−0.144641	−	0.989484i	$-0.546203\pi$
$773$	−8.04286	−0.289282	−0.144641	+	0.989484i	$0.546203\pi$
$774$	82.7772	2.97536
$775$	−5.95390	−0.213870
$776$	−43.5991	−1.56512
$777$	7.66332	0.274920
$778$	−45.3664	−1.62646
$779$	12.3527	0.442581
$780$	−31.5755	−1.13058
$781$	−42.1194	−1.50715
$782$	55.8518	1.99725
$783$	14.2346	0.508703
$784$	160.446	5.73021
$785$	2.69576	0.0962156
$786$	−1.91167	−0.0681870
$787$	−24.2629	−0.864877	−0.432439	−	0.901663i	$-0.642347\pi$
$787$	−24.2629	−0.864877	−0.432439	+	0.901663i	$0.642347\pi$
$788$	112.031	3.99095
$789$	16.1384	0.574542
$790$	73.3724	2.61047
$791$	31.8028	1.13078
$792$	122.240	4.34362
$793$	−4.60128	−0.163396
$794$	−42.5155	−1.50882
$795$	12.1868	0.432220
$796$	−34.7504	−1.23170
$797$	−22.5623	−0.799199	−0.399599	−	0.916690i	$-0.630851\pi$
$797$	−22.5623	−0.799199	−0.399599	+	0.916690i	$0.630851\pi$
$798$	34.4622	1.21995
$799$	53.5787	1.89548
$800$	40.2392	1.42267
$801$	−31.4446	−1.11104
$802$	37.4419	1.32212
$803$	−17.0981	−0.603378
$804$	−51.6218	−1.82056
$805$	−30.2274	−1.06537
$806$	−36.6012	−1.28922
$807$	3.77803	0.132993
$808$	−97.6994	−3.43705
$809$	−7.61049	−0.267570	−0.133785	−	0.991010i	$-0.542713\pi$
$809$	−7.61049	−0.267570	−0.133785	+	0.991010i	$0.542713\pi$
$810$	−21.2567	−0.746886
$811$	−14.5113	−0.509560	−0.254780	−	0.966999i	$-0.582003\pi$
$811$	−14.5113	−0.509560	−0.254780	+	0.966999i	$0.582003\pi$
$812$	84.2133	2.95531
$813$	−1.08620	−0.0380947
$814$	37.0067	1.29708
$815$	1.90647	0.0667809
$816$	−46.8130	−1.63878
$817$	50.6999	1.77376
$818$	37.4122	1.30809
$819$	−55.4460	−1.93744
$820$	26.2825	0.917823
$821$	43.6131	1.52211	0.761053	−	0.648690i	$-0.224683\pi$
$821$	43.6131	1.52211	0.761053	+	0.648690i	$0.224683\pi$
$822$	−2.31854	−0.0808684
$823$	−20.2488	−0.705830	−0.352915	−	0.935655i	$-0.614809\pi$
$823$	−20.2488	−0.705830	−0.352915	+	0.935655i	$0.614809\pi$
$824$	−12.5477	−0.437120
$825$	8.75338	0.304754
$826$	86.5777	3.01242
$827$	−35.7814	−1.24424	−0.622120	−	0.782922i	$-0.713728\pi$
$827$	−35.7814	−1.24424	−0.622120	+	0.782922i	$0.713728\pi$
$828$	54.6678	1.89984
$829$	20.3152	0.705576	0.352788	−	0.935703i	$-0.385234\pi$
$829$	20.3152	0.705576	0.352788	+	0.935703i	$0.385234\pi$
$830$	73.5085	2.55152
$831$	10.9341	0.379301
$832$	112.897	3.91401
$833$	60.4405	2.09414
$834$	39.2026	1.35748
$835$	5.08004	0.175802
$836$	120.645	4.17261
$837$	10.4151	0.359997
$838$	69.0599	2.38563
$839$	6.17026	0.213021	0.106511	−	0.994312i	$-0.466032\pi$
$839$	6.17026	0.213021	0.106511	+	0.994312i	$0.466032\pi$
$840$	45.5038	1.57003
$841$	−15.6476	−0.539572
$842$	20.1640	0.694897
$843$	23.5298	0.810408
$844$	47.9780	1.65147
$845$	21.2725	0.731798
$846$	72.3407	2.48712
$847$	86.4642	2.97095
$848$	−136.788	−4.69731
$849$	17.5580	0.602588
$850$	29.9578	1.02755
$851$	10.2706	0.352073
$852$	28.3562	0.971467
$853$	13.7129	0.469521	0.234760	−	0.972053i	$-0.424569\pi$
$853$	13.7129	0.469521	0.234760	+	0.972053i	$0.424569\pi$
$854$	10.6851	0.365635
$855$	−17.1616	−0.586915
$856$	113.443	3.87740
$857$	32.9562	1.12576	0.562881	−	0.826538i	$-0.309693\pi$
$857$	32.9562	1.12576	0.562881	+	0.826538i	$0.309693\pi$
$858$	53.8108	1.83707
$859$	−38.8596	−1.32587	−0.662936	−	0.748676i	$-0.730690\pi$
$859$	−38.8596	−1.32587	−0.662936	+	0.748676i	$0.730690\pi$
$860$	107.873	3.67843
$861$	−9.27518	−0.316097
$862$	−54.3310	−1.85052
$863$	−13.6999	−0.466350	−0.233175	−	0.972435i	$-0.574912\pi$
$863$	−13.6999	−0.466350	−0.233175	+	0.972435i	$0.574912\pi$
$864$	−70.3898	−2.39471
$865$	−16.9102	−0.574964
$866$	6.04702	0.205486
$867$	−5.58948	−0.189829
$868$	61.6166	2.09140
$869$	−90.6477	−3.07501
$870$	11.6259	0.394156
$871$	70.1700	2.37762
$872$	−33.6776	−1.14047
$873$	−12.3467	−0.417871
$874$	46.1875	1.56231
$875$	−52.6166	−1.77876
$876$	11.5110	0.388920
$877$	47.2450	1.59535	0.797674	−	0.603088i	$-0.206063\pi$
$877$	47.2450	1.59535	0.797674	+	0.603088i	$0.206063\pi$
$878$	0.139275	0.00470031
$879$	5.87720	0.198233
$880$	122.347	4.12431
$881$	−49.4332	−1.66545	−0.832723	−	0.553689i	$-0.813219\pi$
$881$	−49.4332	−1.66545	−0.832723	+	0.553689i	$0.813219\pi$
$882$	81.6054	2.74780
$883$	−39.8292	−1.34036	−0.670179	−	0.742199i	$-0.733783\pi$
$883$	−39.8292	−1.34036	−0.670179	+	0.742199i	$0.733783\pi$
$884$	133.508	4.49037
$885$	8.66477	0.291263
$886$	−66.0993	−2.22065
$887$	3.87335	0.130054	0.0650271	−	0.997883i	$-0.479287\pi$
$887$	3.87335	0.130054	0.0650271	+	0.997883i	$0.479287\pi$
$888$	−15.4613	−0.518847
$889$	−51.1931	−1.71696
$890$	−56.5254	−1.89474
$891$	26.2616	0.879795
$892$	26.5249	0.888120
$893$	44.3077	1.48270
$894$	−24.3881	−0.815659
$895$	1.73773	0.0580858
$896$	−104.167	−3.47998
$897$	14.9344	0.498644
$898$	51.4933	1.71836
$899$	9.76960	0.325834
$900$	29.3228	0.977427
$901$	−51.5284	−1.71666
$902$	−44.7905	−1.49136
$903$	−38.0687	−1.26685
$904$	−64.1644	−2.13408
$905$	−3.71743	−0.123572
$906$	−19.8984	−0.661081
$907$	37.1900	1.23487	0.617437	−	0.786621i	$-0.288171\pi$
$907$	37.1900	1.23487	0.617437	+	0.786621i	$0.288171\pi$
$908$	22.5190	0.747320
$909$	−27.6671	−0.917660
$910$	−99.6706	−3.30405
$911$	52.3814	1.73547	0.867736	−	0.497025i	$-0.165574\pi$
$911$	52.3814	1.73547	0.867736	+	0.497025i	$0.165574\pi$
$912$	−38.7127	−1.28191
$913$	−90.8157	−3.00556
$914$	−32.0247	−1.05928
$915$	1.06937	0.0353523
$916$	151.422	5.00311
$917$	−4.37456	−0.144461
$918$	−52.4048	−1.72962
$919$	48.5227	1.60062	0.800308	−	0.599589i	$-0.204669\pi$
$919$	48.5227	1.60062	0.800308	+	0.599589i	$0.204669\pi$
$920$	60.9858	2.01064
$921$	11.9826	0.394841
$922$	85.1065	2.80283
$923$	−38.5449	−1.26872
$924$	−90.5883	−2.98014
$925$	5.50898	0.181134
$926$	21.2955	0.699815
$927$	−3.55334	−0.116707
$928$	−66.0274	−2.16746
$929$	−41.5552	−1.36338	−0.681691	−	0.731640i	$-0.738755\pi$
$929$	−41.5552	−1.36338	−0.681691	+	0.731640i	$0.738755\pi$
$930$	8.50637	0.278935
$931$	49.9822	1.63810
$932$	62.3560	2.04254
$933$	11.6289	0.380712
$934$	40.9849	1.34107
$935$	46.0885	1.50725
$936$	111.866	3.65646
$937$	40.2229	1.31403	0.657013	−	0.753880i	$-0.271820\pi$
$937$	40.2229	1.31403	0.657013	+	0.753880i	$0.271820\pi$
$938$	−162.948	−5.32045
$939$	21.9952	0.717786
$940$	94.2723	3.07482
$941$	−5.16760	−0.168459	−0.0842295	−	0.996446i	$-0.526843\pi$
$941$	−5.16760	−0.168459	−0.0842295	+	0.996446i	$0.526843\pi$
$942$	3.09290	0.100772
$943$	−12.4309	−0.404806
$944$	−97.2559	−3.16541
$945$	28.3618	0.922610
$946$	−183.836	−5.97704
$947$	−19.2997	−0.627156	−0.313578	−	0.949562i	$-0.601528\pi$
$947$	−19.2997	−0.627156	−0.313578	+	0.949562i	$0.601528\pi$
$948$	61.0271	1.98207
$949$	−15.6470	−0.507924
$950$	24.7741	0.803778
$951$	7.82370	0.253701
$952$	−192.401	−6.23574
$953$	0.740568	0.0239894	0.0119947	−	0.999928i	$-0.496182\pi$
$953$	0.740568	0.0239894	0.0119947	+	0.999928i	$0.496182\pi$
$954$	−69.5724	−2.25249
$955$	23.3130	0.754392
$956$	−109.478	−3.54077
$957$	−14.3632	−0.464296
$958$	−82.1697	−2.65478
$959$	−5.30562	−0.171328
$960$	−26.2381	−0.846832
$961$	−23.8518	−0.769414
$962$	33.8661	1.09189
$963$	32.1254	1.03523
$964$	55.6396	1.79203
$965$	−23.4047	−0.753423
$966$	−34.6805	−1.11583
$967$	−20.6175	−0.663013	−0.331507	−	0.943453i	$-0.607557\pi$
$967$	−20.6175	−0.663013	−0.331507	+	0.943453i	$0.607557\pi$
$968$	−174.448	−5.60696
$969$	−14.5832	−0.468480
$970$	−22.1946	−0.712624
$971$	59.7813	1.91847	0.959237	−	0.282603i	$-0.0911979\pi$
$971$	59.7813	1.91847	0.959237	+	0.282603i	$0.0911979\pi$
$972$	−79.2828	−2.54300
$973$	89.7091	2.87594
$974$	24.5139	0.785476
$975$	8.01051	0.256542
$976$	−12.0029	−0.384204
$977$	−0.145522	−0.00465565	−0.00232782	−	0.999997i	$-0.500741\pi$
$977$	−0.145522	−0.00465565	−0.00232782	+	0.999997i	$0.500741\pi$
$978$	2.18734	0.0699434
$979$	69.8341	2.23191
$980$	106.346	3.39709
$981$	−9.53703	−0.304494
$982$	62.0222	1.97921
$983$	13.7667	0.439091	0.219546	−	0.975602i	$-0.429543\pi$
$983$	13.7667	0.439091	0.219546	+	0.975602i	$0.429543\pi$
$984$	18.7133	0.596559
$985$	35.3922	1.12769
$986$	−49.1570	−1.56548
$987$	−33.2690	−1.05897
$988$	110.407	3.51251
$989$	−51.0210	−1.62237
$990$	62.2276	1.97772
$991$	24.1414	0.766877	0.383438	−	0.923566i	$-0.374740\pi$
$991$	24.1414	0.766877	0.383438	+	0.923566i	$0.374740\pi$
$992$	−48.3105	−1.53386
$993$	−0.747401	−0.0237181
$994$	89.5087	2.83904
$995$	−10.9781	−0.348030
$996$	61.1402	1.93730
$997$	6.96956	0.220728	0.110364	−	0.993891i	$-0.464798\pi$
$997$	6.96956	0.220728	0.110364	+	0.993891i	$0.464798\pi$
$998$	−34.7855	−1.10112
$999$	−9.63678	−0.304894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.b.1.13	✓	361

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.b.1.13	✓	361	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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69652 q^{2} -0.708536 q^{3} +5.27124 q^{4} +1.66526 q^{5} +1.91059 q^{6} +4.37208 q^{7} -8.82098 q^{8} -2.49798 q^{9} +O(q^{10})\)	\(q-2.69652 q^{2} -0.708536 q^{3} +5.27124 q^{4} +1.66526 q^{5} +1.91059 q^{6} +4.37208 q^{7} -8.82098 q^{8} -2.49798 q^{9} -4.49040 q^{10} +5.54765 q^{11} -3.73487 q^{12} +5.07684 q^{13} -11.7894 q^{14} -1.17990 q^{15} +13.2435 q^{16} +4.98886 q^{17} +6.73585 q^{18} +4.12562 q^{19} +8.77797 q^{20} -3.09778 q^{21} -14.9594 q^{22} -4.15175 q^{23} +6.24998 q^{24} -2.22692 q^{25} -13.6898 q^{26} +3.89552 q^{27} +23.0463 q^{28} +3.65409 q^{29} +3.18162 q^{30} +2.67360 q^{31} -18.0694 q^{32} -3.93071 q^{33} -13.4526 q^{34} +7.28064 q^{35} -13.1674 q^{36} -2.47381 q^{37} -11.1248 q^{38} -3.59713 q^{39} -14.6892 q^{40} +2.99414 q^{41} +8.35323 q^{42} +12.2890 q^{43} +29.2430 q^{44} -4.15977 q^{45} +11.1953 q^{46} +10.7397 q^{47} -9.38349 q^{48} +12.1151 q^{49} +6.00494 q^{50} -3.53479 q^{51} +26.7613 q^{52} -10.3287 q^{53} -10.5044 q^{54} +9.23827 q^{55} -38.5660 q^{56} -2.92315 q^{57} -9.85335 q^{58} -7.34368 q^{59} -6.21951 q^{60} -0.906326 q^{61} -7.20943 q^{62} -10.9213 q^{63} +22.2377 q^{64} +8.45425 q^{65} +10.5993 q^{66} +13.8216 q^{67} +26.2975 q^{68} +2.94166 q^{69} -19.6324 q^{70} -7.59229 q^{71} +22.0346 q^{72} -3.08204 q^{73} +6.67069 q^{74} +1.57785 q^{75} +21.7471 q^{76} +24.2548 q^{77} +9.69974 q^{78} -16.3398 q^{79} +22.0538 q^{80} +4.73381 q^{81} -8.07377 q^{82} -16.3701 q^{83} -16.3291 q^{84} +8.30774 q^{85} -33.1377 q^{86} -2.58906 q^{87} -48.9357 q^{88} +12.5881 q^{89} +11.2169 q^{90} +22.1964 q^{91} -21.8849 q^{92} -1.89434 q^{93} -28.9597 q^{94} +6.87021 q^{95} +12.8029 q^{96} +4.94266 q^{97} -32.6686 q^{98} -13.8579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * q^99

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