LMFDB - Embedded newform 8027.2.a.f.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8027,2,Mod(1,8027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8027, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8027 = 23 \cdot 349 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8027.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:	$64.0959177025$
Analytic rank:	$0$
Dimension:	$176$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43642 q^{2} -3.01973 q^{3} +3.93613 q^{4} +2.61235 q^{5} +7.35732 q^{6} -1.67075 q^{7} -4.71721 q^{8} +6.11876 q^{9} +O(q^{10})$	\(q-2.43642 q^{2} -3.01973 q^{3} +3.93613 q^{4} +2.61235 q^{5} +7.35732 q^{6} -1.67075 q^{7} -4.71721 q^{8} +6.11876 q^{9} -6.36478 q^{10} -3.15935 q^{11} -11.8860 q^{12} +5.74300 q^{13} +4.07065 q^{14} -7.88859 q^{15} +3.62083 q^{16} -4.74863 q^{17} -14.9078 q^{18} +0.791197 q^{19} +10.2825 q^{20} +5.04522 q^{21} +7.69748 q^{22} +1.00000 q^{23} +14.2447 q^{24} +1.82438 q^{25} -13.9923 q^{26} -9.41780 q^{27} -6.57630 q^{28} -0.131990 q^{29} +19.2199 q^{30} -9.74098 q^{31} +0.612559 q^{32} +9.54037 q^{33} +11.5696 q^{34} -4.36460 q^{35} +24.0842 q^{36} -4.75435 q^{37} -1.92769 q^{38} -17.3423 q^{39} -12.3230 q^{40} -8.38920 q^{41} -12.2923 q^{42} +5.86775 q^{43} -12.4356 q^{44} +15.9843 q^{45} -2.43642 q^{46} -1.61269 q^{47} -10.9339 q^{48} -4.20858 q^{49} -4.44495 q^{50} +14.3396 q^{51} +22.6052 q^{52} +5.04031 q^{53} +22.9457 q^{54} -8.25332 q^{55} +7.88130 q^{56} -2.38920 q^{57} +0.321584 q^{58} -9.46781 q^{59} -31.0505 q^{60} +1.98037 q^{61} +23.7331 q^{62} -10.2229 q^{63} -8.73411 q^{64} +15.0027 q^{65} -23.2443 q^{66} -13.4173 q^{67} -18.6912 q^{68} -3.01973 q^{69} +10.6340 q^{70} -6.91127 q^{71} -28.8635 q^{72} +10.8566 q^{73} +11.5836 q^{74} -5.50914 q^{75} +3.11425 q^{76} +5.27849 q^{77} +42.2530 q^{78} +3.41290 q^{79} +9.45889 q^{80} +10.0829 q^{81} +20.4396 q^{82} -11.1177 q^{83} +19.8586 q^{84} -12.4051 q^{85} -14.2963 q^{86} +0.398575 q^{87} +14.9033 q^{88} +5.66608 q^{89} -38.9445 q^{90} -9.59514 q^{91} +3.93613 q^{92} +29.4151 q^{93} +3.92918 q^{94} +2.06689 q^{95} -1.84976 q^{96} +5.87627 q^{97} +10.2539 q^{98} -19.3313 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) $ 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.43642	−1.72281	−0.861403	−	0.507922i	$-0.830414\pi$
$2$	−2.43642	−1.72281	−0.861403	+	0.507922i	$0.830414\pi$
$3$	−3.01973	−1.74344	−0.871720	−	0.490004i	$-0.836995\pi$
$3$	−3.01973	−1.74344	−0.871720	+	0.490004i	$0.836995\pi$
$4$	3.93613	1.96806
$5$	2.61235	1.16828	0.584140	−	0.811653i	$-0.301432\pi$
$5$	2.61235	1.16828	0.584140	+	0.811653i	$0.301432\pi$
$6$	7.35732	3.00361
$7$	−1.67075	−0.631486	−0.315743	−	0.948845i	$-0.602254\pi$
$7$	−1.67075	−0.631486	−0.315743	+	0.948845i	$0.602254\pi$
$8$	−4.71721	−1.66778
$9$	6.11876	2.03959
$10$	−6.36478	−2.01272
$11$	−3.15935	−0.952579	−0.476289	−	0.879289i	$-0.658019\pi$
$11$	−3.15935	−0.952579	−0.476289	+	0.879289i	$0.658019\pi$
$12$	−11.8860	−3.43120
$13$	5.74300	1.59282	0.796410	−	0.604757i	$-0.206730\pi$
$13$	5.74300	1.59282	0.796410	+	0.604757i	$0.206730\pi$
$14$	4.07065	1.08793
$15$	−7.88859	−2.03683
$16$	3.62083	0.905208
$17$	−4.74863	−1.15171	−0.575856	−	0.817551i	$-0.695331\pi$
$17$	−4.74863	−1.15171	−0.575856	+	0.817551i	$0.695331\pi$
$18$	−14.9078	−3.51381
$19$	0.791197	0.181513	0.0907565	−	0.995873i	$-0.471071\pi$
$19$	0.791197	0.181513	0.0907565	+	0.995873i	$0.471071\pi$
$20$	10.2825	2.29925
$21$	5.04522	1.10096
$22$	7.69748	1.64111
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	14.2447	2.90768
$25$	1.82438	0.364876
$26$	−13.9923	−2.74412
$27$	−9.41780	−1.81246
$28$	−6.57630	−1.24280
$29$	−0.131990	−0.0245100	−0.0122550	−	0.999925i	$-0.503901\pi$
$29$	−0.131990	−0.0245100	−0.0122550	+	0.999925i	$0.503901\pi$
$30$	19.2199	3.50906
$31$	−9.74098	−1.74953	−0.874766	−	0.484546i	$-0.838985\pi$
$31$	−9.74098	−1.74953	−0.874766	+	0.484546i	$0.838985\pi$
$32$	0.612559	0.108286
$33$	9.54037	1.66076
$34$	11.5696	1.98418
$35$	−4.36460	−0.737752
$36$	24.0842	4.01403
$37$	−4.75435	−0.781610	−0.390805	−	0.920473i	$-0.627803\pi$
$37$	−4.75435	−0.781610	−0.390805	+	0.920473i	$0.627803\pi$
$38$	−1.92769	−0.312712
$39$	−17.3423	−2.77699
$40$	−12.3230	−1.94844
$41$	−8.38920	−1.31017	−0.655087	−	0.755554i	$-0.727368\pi$
$41$	−8.38920	−1.31017	−0.655087	+	0.755554i	$0.727368\pi$
$42$	−12.2923	−1.89674
$43$	5.86775	0.894824	0.447412	−	0.894328i	$-0.352346\pi$
$43$	5.86775	0.894824	0.447412	+	0.894328i	$0.352346\pi$
$44$	−12.4356	−1.87473
$45$	15.9843	2.38281
$46$	−2.43642	−0.359230
$47$	−1.61269	−0.235235	−0.117617	−	0.993059i	$-0.537526\pi$
$47$	−1.61269	−0.235235	−0.117617	+	0.993059i	$0.537526\pi$
$48$	−10.9339	−1.57818
$49$	−4.20858	−0.601226
$50$	−4.44495	−0.628611
$51$	14.3396	2.00794
$52$	22.6052	3.13477
$53$	5.04031	0.692340	0.346170	−	0.938172i	$-0.387482\pi$
$53$	5.04031	0.692340	0.346170	+	0.938172i	$0.387482\pi$
$54$	22.9457	3.12251
$55$	−8.25332	−1.11288
$56$	7.88130	1.05318
$57$	−2.38920	−0.316457
$58$	0.321584	0.0422260
$59$	−9.46781	−1.23260	−0.616302	−	0.787510i	$-0.711370\pi$
$59$	−9.46781	−1.23260	−0.616302	+	0.787510i	$0.711370\pi$
$60$	−31.0505	−4.00860
$61$	1.98037	0.253560	0.126780	−	0.991931i	$-0.459536\pi$
$61$	1.98037	0.253560	0.126780	+	0.991931i	$0.459536\pi$
$62$	23.7331	3.01410
$63$	−10.2229	−1.28797
$64$	−8.73411	−1.09176
$65$	15.0027	1.86086
$66$	−23.2443	−2.86118
$67$	−13.4173	−1.63919	−0.819593	−	0.572947i	$-0.805800\pi$
$67$	−13.4173	−1.63919	−0.819593	+	0.572947i	$0.805800\pi$
$68$	−18.6912	−2.26664
$69$	−3.01973	−0.363533
$70$	10.6340	1.27100
$71$	−6.91127	−0.820217	−0.410109	−	0.912037i	$-0.634509\pi$
$71$	−6.91127	−0.820217	−0.410109	+	0.912037i	$0.634509\pi$
$72$	−28.8635	−3.40159
$73$	10.8566	1.27067	0.635334	−	0.772238i	$-0.280862\pi$
$73$	10.8566	1.27067	0.635334	+	0.772238i	$0.280862\pi$
$74$	11.5836	1.34656
$75$	−5.50914	−0.636140
$76$	3.11425	0.357229
$77$	5.27849	0.601540
$78$	42.2530	4.78421
$79$	3.41290	0.383981	0.191990	−	0.981397i	$-0.438506\pi$
$79$	3.41290	0.383981	0.191990	+	0.981397i	$0.438506\pi$
$80$	9.45889	1.05754
$81$	10.0829	1.12033
$82$	20.4396	2.25718
$83$	−11.1177	−1.22033	−0.610164	−	0.792275i	$-0.708896\pi$
$83$	−11.1177	−1.22033	−0.610164	+	0.792275i	$0.708896\pi$
$84$	19.8586	2.16676
$85$	−12.4051	−1.34552
$86$	−14.2963	−1.54161
$87$	0.398575	0.0427317
$88$	14.9033	1.58870
$89$	5.66608	0.600603	0.300302	−	0.953844i	$-0.402913\pi$
$89$	5.66608	0.600603	0.300302	+	0.953844i	$0.402913\pi$
$90$	−38.9445	−4.10511
$91$	−9.59514	−1.00584
$92$	3.93613	0.410369
$93$	29.4151	3.05020
$94$	3.92918	0.405264
$95$	2.06689	0.212058
$96$	−1.84976	−0.188791
$97$	5.87627	0.596645	0.298323	−	0.954465i	$-0.403573\pi$
$97$	5.87627	0.596645	0.298323	+	0.954465i	$0.403573\pi$
$98$	10.2539	1.03580
$99$	−19.3313	−1.94287
$100$	7.18099	0.718099
$101$	−15.3945	−1.53181	−0.765904	−	0.642955i	$-0.777708\pi$
$101$	−15.3945	−1.53181	−0.765904	+	0.642955i	$0.777708\pi$
$102$	−34.9372	−3.45930
$103$	3.10900	0.306339	0.153169	−	0.988200i	$-0.451052\pi$
$103$	3.10900	0.306339	0.153169	+	0.988200i	$0.451052\pi$
$104$	−27.0909	−2.65648
$105$	13.1799	1.28623
$106$	−12.2803	−1.19277
$107$	−2.29765	−0.222122	−0.111061	−	0.993814i	$-0.535425\pi$
$107$	−2.29765	−0.222122	−0.111061	+	0.993814i	$0.535425\pi$
$108$	−37.0697	−3.56703
$109$	20.7668	1.98910	0.994548	−	0.104277i	$-0.0332527\pi$
$109$	20.7668	1.98910	0.994548	+	0.104277i	$0.0332527\pi$
$110$	20.1085	1.91727
$111$	14.3568	1.36269
$112$	−6.04952	−0.571626
$113$	−8.23562	−0.774742	−0.387371	−	0.921924i	$-0.626617\pi$
$113$	−8.23562	−0.774742	−0.387371	+	0.921924i	$0.626617\pi$
$114$	5.82109	0.545195
$115$	2.61235	0.243603
$116$	−0.519531	−0.0482372
$117$	35.1400	3.24870
$118$	23.0675	2.12354
$119$	7.93380	0.727290
$120$	37.2121	3.39699
$121$	−1.01853	−0.0925937
$122$	−4.82499	−0.436835
$123$	25.3331	2.28421
$124$	−38.3417	−3.44319
$125$	−8.29583	−0.742002
$126$	24.9073	2.21892
$127$	10.7850	0.957017	0.478508	−	0.878083i	$-0.341178\pi$
$127$	10.7850	0.957017	0.478508	+	0.878083i	$0.341178\pi$
$128$	20.0548	1.77261
$129$	−17.7190	−1.56007
$130$	−36.5529	−3.20590
$131$	16.6325	1.45319	0.726593	−	0.687069i	$-0.241103\pi$
$131$	16.6325	1.45319	0.726593	+	0.687069i	$0.241103\pi$
$132$	37.5521	3.26849
$133$	−1.32190	−0.114623
$134$	32.6902	2.82400
$135$	−24.6026	−2.11746
$136$	22.4003	1.92081
$137$	13.3033	1.13658	0.568289	−	0.822829i	$-0.307606\pi$
$137$	13.3033	1.13658	0.568289	+	0.822829i	$0.307606\pi$
$138$	7.35732	0.626296
$139$	−0.408853	−0.0346785	−0.0173392	−	0.999850i	$-0.505520\pi$
$139$	−0.408853	−0.0346785	−0.0173392	+	0.999850i	$0.505520\pi$
$140$	−17.1796	−1.45194
$141$	4.86988	0.410118
$142$	16.8387	1.41308
$143$	−18.1441	−1.51729
$144$	22.1550	1.84625
$145$	−0.344805	−0.0286345
$146$	−26.4512	−2.18911
$147$	12.7088	1.04820
$148$	−18.7137	−1.53826
$149$	11.9972	0.982847	0.491423	−	0.870921i	$-0.336477\pi$
$149$	11.9972	0.982847	0.491423	+	0.870921i	$0.336477\pi$
$150$	13.4225	1.09595
$151$	14.6009	1.18820	0.594102	−	0.804389i	$-0.297507\pi$
$151$	14.6009	1.18820	0.594102	+	0.804389i	$0.297507\pi$
$152$	−3.73224	−0.302725
$153$	−29.0557	−2.34902
$154$	−12.8606	−1.03634
$155$	−25.4469	−2.04394
$156$	−68.2614	−5.46529
$157$	0.968319	0.0772803	0.0386401	−	0.999253i	$-0.487697\pi$
$157$	0.968319	0.0772803	0.0386401	+	0.999253i	$0.487697\pi$
$158$	−8.31524	−0.661525
$159$	−15.2204	−1.20705
$160$	1.60022	0.126508
$161$	−1.67075	−0.131674
$162$	−24.5662	−1.93010
$163$	−15.6204	−1.22348	−0.611741	−	0.791058i	$-0.709530\pi$
$163$	−15.6204	−1.22348	−0.611741	+	0.791058i	$0.709530\pi$
$164$	−33.0210	−2.57850
$165$	24.9228	1.94024
$166$	27.0874	2.10239
$167$	20.8315	1.61199	0.805995	−	0.591922i	$-0.201631\pi$
$167$	20.8315	1.61199	0.805995	+	0.591922i	$0.201631\pi$
$168$	−23.7994	−1.83616
$169$	19.9820	1.53708
$170$	30.2240	2.31807
$171$	4.84114	0.370212
$172$	23.0962	1.76107
$173$	1.88450	0.143276	0.0716378	−	0.997431i	$-0.477177\pi$
$173$	1.88450	0.143276	0.0716378	+	0.997431i	$0.477177\pi$
$174$	−0.971095	−0.0736185
$175$	−3.04809	−0.230414
$176$	−11.4395	−0.862282
$177$	28.5902	2.14897
$178$	−13.8049	−1.03472
$179$	19.5433	1.46074	0.730369	−	0.683053i	$-0.239348\pi$
$179$	19.5433	1.46074	0.730369	+	0.683053i	$0.239348\pi$
$180$	62.9164	4.68951
$181$	11.3746	0.845466	0.422733	−	0.906254i	$-0.361071\pi$
$181$	11.3746	0.845466	0.422733	+	0.906254i	$0.361071\pi$
$182$	23.3778	1.73287
$183$	−5.98017	−0.442067
$184$	−4.71721	−0.347757
$185$	−12.4200	−0.913139
$186$	−71.6674	−5.25491
$187$	15.0026	1.09710
$188$	−6.34774	−0.462957
$189$	15.7348	1.14454
$190$	−5.03579	−0.365335
$191$	−13.2328	−0.957495	−0.478748	−	0.877953i	$-0.658909\pi$
$191$	−13.2328	−0.957495	−0.478748	+	0.877953i	$0.658909\pi$
$192$	26.3747	1.90343
$193$	16.7239	1.20381	0.601905	−	0.798568i	$-0.294409\pi$
$193$	16.7239	1.20381	0.601905	+	0.798568i	$0.294409\pi$
$194$	−14.3170	−1.02790
$195$	−45.3042	−3.24430
$196$	−16.5655	−1.18325
$197$	−4.94853	−0.352568	−0.176284	−	0.984339i	$-0.556408\pi$
$197$	−4.94853	−0.352568	−0.176284	+	0.984339i	$0.556408\pi$
$198$	47.0990	3.34718
$199$	16.6286	1.17877	0.589385	−	0.807852i	$-0.299370\pi$
$199$	16.6286	1.17877	0.589385	+	0.807852i	$0.299370\pi$
$200$	−8.60599	−0.608535
$201$	40.5166	2.85782
$202$	37.5074	2.63901
$203$	0.220524	0.0154777
$204$	56.4424	3.95176
$205$	−21.9155	−1.53065
$206$	−7.57482	−0.527763
$207$	6.11876	0.425283
$208$	20.7944	1.44183
$209$	−2.49967	−0.172905
$210$	−32.1117	−2.21592
$211$	−15.3954	−1.05986	−0.529930	−	0.848041i	$-0.677782\pi$
$211$	−15.3954	−1.05986	−0.529930	+	0.848041i	$0.677782\pi$
$212$	19.8393	1.36257
$213$	20.8702	1.43000
$214$	5.59802	0.382673
$215$	15.3286	1.04540
$216$	44.4257	3.02279
$217$	16.2748	1.10480
$218$	−50.5965	−3.42683
$219$	−32.7840	−2.21533
$220$	−32.4861	−2.19021
$221$	−27.2714	−1.83447
$222$	−34.9792	−2.34765
$223$	−9.25134	−0.619516	−0.309758	−	0.950815i	$-0.600248\pi$
$223$	−9.25134	−0.619516	−0.309758	+	0.950815i	$0.600248\pi$
$224$	−1.02344	−0.0683812
$225$	11.1629	0.744197
$226$	20.0654	1.33473
$227$	9.21489	0.611613	0.305807	−	0.952094i	$-0.401074\pi$
$227$	9.21489	0.611613	0.305807	+	0.952094i	$0.401074\pi$
$228$	−9.40419	−0.622808
$229$	−21.5800	−1.42604	−0.713022	−	0.701142i	$-0.752674\pi$
$229$	−21.5800	−1.42604	−0.713022	+	0.701142i	$0.752674\pi$
$230$	−6.36478	−0.419681
$231$	−15.9396	−1.04875
$232$	0.622626	0.0408774
$233$	24.0900	1.57819	0.789093	−	0.614274i	$-0.210551\pi$
$233$	24.0900	1.57819	0.789093	+	0.614274i	$0.210551\pi$
$234$	−85.6157	−5.59687
$235$	−4.21291	−0.274820
$236$	−37.2665	−2.42584
$237$	−10.3060	−0.669448
$238$	−19.3300	−1.25298
$239$	−6.93888	−0.448839	−0.224419	−	0.974493i	$-0.572049\pi$
$239$	−6.93888	−0.448839	−0.224419	+	0.974493i	$0.572049\pi$
$240$	−28.5633	−1.84375
$241$	−22.7903	−1.46805	−0.734027	−	0.679121i	$-0.762361\pi$
$241$	−22.7903	−1.46805	−0.734027	+	0.679121i	$0.762361\pi$
$242$	2.48157	0.159521
$243$	−2.19430	−0.140765
$244$	7.79497	0.499022
$245$	−10.9943	−0.702399
$246$	−61.7220	−3.93525
$247$	4.54384	0.289118
$248$	45.9502	2.91784
$249$	33.5725	2.12757
$250$	20.2121	1.27833
$251$	−19.4117	−1.22526	−0.612628	−	0.790372i	$-0.709887\pi$
$251$	−19.4117	−1.22526	−0.612628	+	0.790372i	$0.709887\pi$
$252$	−40.2388	−2.53481
$253$	−3.15935	−0.198626
$254$	−26.2768	−1.64875
$255$	37.4600	2.34584
$256$	−31.3937	−1.96210
$257$	−2.91235	−0.181668	−0.0908338	−	0.995866i	$-0.528953\pi$
$257$	−2.91235	−0.181668	−0.0908338	+	0.995866i	$0.528953\pi$
$258$	43.1709	2.68770
$259$	7.94335	0.493576
$260$	59.0526	3.66229
$261$	−0.807618	−0.0499903
$262$	−40.5236	−2.50356
$263$	−9.63250	−0.593965	−0.296983	−	0.954883i	$-0.595980\pi$
$263$	−9.63250	−0.593965	−0.296983	+	0.954883i	$0.595980\pi$
$264$	−45.0039	−2.76980
$265$	13.1671	0.808847
$266$	3.22069	0.197473
$267$	−17.1100	−1.04712
$268$	−52.8122	−3.22602
$269$	−14.9550	−0.911820	−0.455910	−	0.890026i	$-0.650686\pi$
$269$	−14.9550	−0.911820	−0.455910	+	0.890026i	$0.650686\pi$
$270$	59.9422	3.64797
$271$	−20.1036	−1.22121	−0.610604	−	0.791936i	$-0.709073\pi$
$271$	−20.1036	−1.22121	−0.610604	+	0.791936i	$0.709073\pi$
$272$	−17.1940	−1.04254
$273$	28.9747	1.75363
$274$	−32.4124	−1.95810
$275$	−5.76385	−0.347573
$276$	−11.8860	−0.715455
$277$	26.8898	1.61565	0.807825	−	0.589422i	$-0.200644\pi$
$277$	26.8898	1.61565	0.807825	+	0.589422i	$0.200644\pi$
$278$	0.996136	0.0597443
$279$	−59.6027	−3.56832
$280$	20.5887	1.23041
$281$	−28.4525	−1.69733	−0.848666	−	0.528929i	$-0.822594\pi$
$281$	−28.4525	−1.69733	−0.848666	+	0.528929i	$0.822594\pi$
$282$	−11.8651	−0.706553
$283$	−0.0438726	−0.00260796	−0.00130398	−	0.999999i	$-0.500415\pi$
$283$	−0.0438726	−0.00260796	−0.00130398	+	0.999999i	$0.500415\pi$
$284$	−27.2036	−1.61424
$285$	−6.24143	−0.369710
$286$	44.2066	2.61399
$287$	14.0163	0.827356
$288$	3.74810	0.220859
$289$	5.54952	0.326442
$290$	0.840090	0.0493318
$291$	−17.7447	−1.04022
$292$	42.7329	2.50075
$293$	−4.43306	−0.258982	−0.129491	−	0.991581i	$-0.541334\pi$
$293$	−4.43306	−0.258982	−0.129491	+	0.991581i	$0.541334\pi$
$294$	−30.9638	−1.80585
$295$	−24.7332	−1.44002
$296$	22.4272	1.30356
$297$	29.7541	1.72651
$298$	−29.2301	−1.69326
$299$	5.74300	0.332126
$300$	−21.6846	−1.25196
$301$	−9.80358	−0.565069
$302$	−35.5739	−2.04705
$303$	46.4871	2.67062
$304$	2.86479	0.164307
$305$	5.17341	0.296229
$306$	70.7919	4.04690
$307$	−23.7391	−1.35486	−0.677430	−	0.735587i	$-0.736906\pi$
$307$	−23.7391	−1.35486	−0.677430	+	0.735587i	$0.736906\pi$
$308$	20.7768	1.18387
$309$	−9.38834	−0.534084
$310$	61.9991	3.52131
$311$	−9.43499	−0.535009	−0.267505	−	0.963557i	$-0.586199\pi$
$311$	−9.43499	−0.535009	−0.267505	+	0.963557i	$0.586199\pi$
$312$	81.8072	4.63142
$313$	−0.414487	−0.0234282	−0.0117141	−	0.999931i	$-0.503729\pi$
$313$	−0.414487	−0.0234282	−0.0117141	+	0.999931i	$0.503729\pi$
$314$	−2.35923	−0.133139
$315$	−26.7059	−1.50471
$316$	13.4336	0.755699
$317$	−24.1531	−1.35657	−0.678287	−	0.734797i	$-0.737277\pi$
$317$	−24.1531	−1.35657	−0.678287	+	0.734797i	$0.737277\pi$
$318$	37.0832	2.07952
$319$	0.417003	0.0233477
$320$	−22.8166	−1.27549
$321$	6.93827	0.387256
$322$	4.07065	0.226849
$323$	−3.75710	−0.209051
$324$	39.6877	2.20487
$325$	10.4774	0.581182
$326$	38.0577	2.10782
$327$	−62.7100	−3.46787
$328$	39.5736	2.18509
$329$	2.69440	0.148547
$330$	−60.7223	−3.34265
$331$	7.63321	0.419559	0.209780	−	0.977749i	$-0.432725\pi$
$331$	7.63321	0.419559	0.209780	+	0.977749i	$0.432725\pi$
$332$	−43.7607	−2.40168
$333$	−29.0907	−1.59416
$334$	−50.7542	−2.77715
$335$	−35.0507	−1.91503
$336$	18.2679	0.996597
$337$	−29.6998	−1.61785	−0.808927	−	0.587909i	$-0.799951\pi$
$337$	−29.6998	−1.61785	−0.808927	+	0.587909i	$0.799951\pi$
$338$	−48.6845	−2.64809
$339$	24.8693	1.35072
$340$	−48.8280	−2.64807
$341$	30.7751	1.66657
$342$	−11.7950	−0.637803
$343$	18.7268	1.01115
$344$	−27.6794	−1.49237
$345$	−7.88859	−0.424708
$346$	−4.59142	−0.246836
$347$	14.3919	0.772600	0.386300	−	0.922373i	$-0.373753\pi$
$347$	14.3919	0.772600	0.386300	+	0.922373i	$0.373753\pi$
$348$	1.56884	0.0840988
$349$	−1.00000	−0.0535288
$349$	−1.00000	−0.0535288
$350$	7.42642	0.396959
$351$	−54.0864	−2.88692
$352$	−1.93529	−0.103151
$353$	11.5712	0.615871	0.307936	−	0.951407i	$-0.400362\pi$
$353$	11.5712	0.615871	0.307936	+	0.951407i	$0.400362\pi$
$354$	−69.6576	−3.70226
$355$	−18.0547	−0.958242
$356$	22.3024	1.18203
$357$	−23.9579	−1.26799
$358$	−47.6157	−2.51657
$359$	9.25082	0.488239	0.244120	−	0.969745i	$-0.421501\pi$
$359$	9.25082	0.488239	0.244120	+	0.969745i	$0.421501\pi$
$360$	−75.4015	−3.97401
$361$	−18.3740	−0.967053
$362$	−27.7132	−1.45657
$363$	3.07569	0.161432
$364$	−37.7677	−1.97956
$365$	28.3612	1.48449
$366$	14.5702	0.761595
$367$	9.40578	0.490978	0.245489	−	0.969399i	$-0.421052\pi$
$367$	9.40578	0.490978	0.245489	+	0.969399i	$0.421052\pi$
$368$	3.62083	0.188749
$369$	−51.3315	−2.67221
$370$	30.2604	1.57316
$371$	−8.42113	−0.437203
$372$	115.782	6.00299
$373$	−19.0790	−0.987874	−0.493937	−	0.869498i	$-0.664443\pi$
$373$	−19.0790	−0.987874	−0.493937	+	0.869498i	$0.664443\pi$
$374$	−36.5525	−1.89009
$375$	25.0512	1.29364
$376$	7.60738	0.392321
$377$	−0.758021	−0.0390400
$378$	−38.3366	−1.97182
$379$	24.1949	1.24281	0.621405	−	0.783490i	$-0.286562\pi$
$379$	24.1949	1.24281	0.621405	+	0.783490i	$0.286562\pi$
$380$	8.13552	0.417343
$381$	−32.5679	−1.66850
$382$	32.2407	1.64958
$383$	21.6310	1.10529	0.552647	−	0.833416i	$-0.313618\pi$
$383$	21.6310	1.10529	0.552647	+	0.833416i	$0.313618\pi$
$384$	−60.5601	−3.09045
$385$	13.7893	0.702767
$386$	−40.7463	−2.07393
$387$	35.9034	1.82507
$388$	23.1297	1.17424
$389$	3.23130	0.163833	0.0819167	−	0.996639i	$-0.473896\pi$
$389$	3.23130	0.163833	0.0819167	+	0.996639i	$0.473896\pi$
$390$	110.380	5.58930
$391$	−4.74863	−0.240149
$392$	19.8527	1.00272
$393$	−50.2255	−2.53354
$394$	12.0567	0.607407
$395$	8.91569	0.448597
$396$	−76.0903	−3.82368
$397$	20.5294	1.03034	0.515170	−	0.857088i	$-0.327729\pi$
$397$	20.5294	1.03034	0.515170	+	0.857088i	$0.327729\pi$
$398$	−40.5142	−2.03079
$399$	3.99177	0.199838
$400$	6.60578	0.330289
$401$	24.9096	1.24392	0.621962	−	0.783047i	$-0.286336\pi$
$401$	24.9096	1.24392	0.621962	+	0.783047i	$0.286336\pi$
$402$	−98.7154	−4.92348
$403$	−55.9424	−2.78669
$404$	−60.5946	−3.01469
$405$	26.3402	1.30885
$406$	−0.537287	−0.0266651
$407$	15.0206	0.744545
$408$	−67.6428	−3.34882
$409$	16.9284	0.837057	0.418529	−	0.908204i	$-0.362546\pi$
$409$	16.9284	0.837057	0.418529	+	0.908204i	$0.362546\pi$
$410$	53.3954	2.63701
$411$	−40.1723	−1.98156
$412$	12.2374	0.602894
$413$	15.8184	0.778372
$414$	−14.9078	−0.732681
$415$	−29.0434	−1.42568
$416$	3.51792	0.172480
$417$	1.23463	0.0604599
$418$	6.09023	0.297883
$419$	−31.2105	−1.52473	−0.762367	−	0.647145i	$-0.775963\pi$
$419$	−31.2105	−1.52473	−0.762367	+	0.647145i	$0.775963\pi$
$420$	51.8777	2.53137
$421$	−25.1222	−1.22438	−0.612191	−	0.790710i	$-0.709712\pi$
$421$	−25.1222	−1.22438	−0.612191	+	0.790710i	$0.709712\pi$
$422$	37.5095	1.82593
$423$	−9.86764	−0.479781
$424$	−23.7762	−1.15467
$425$	−8.66332	−0.420233
$426$	−50.8484	−2.46361
$427$	−3.30870	−0.160119
$428$	−9.04382	−0.437150
$429$	54.7903	2.64530
$430$	−37.3469	−1.80103
$431$	24.5573	1.18289	0.591443	−	0.806347i	$-0.298558\pi$
$431$	24.5573	1.18289	0.591443	+	0.806347i	$0.298558\pi$
$432$	−34.1003	−1.64065
$433$	24.5274	1.17871	0.589356	−	0.807873i	$-0.299381\pi$
$433$	24.5274	1.17871	0.589356	+	0.807873i	$0.299381\pi$
$434$	−39.6521	−1.90336
$435$	1.04122	0.0499226
$436$	81.7406	3.91467
$437$	0.791197	0.0378481
$438$	79.8754	3.81659
$439$	8.54063	0.407622	0.203811	−	0.979010i	$-0.434667\pi$
$439$	8.54063	0.407622	0.203811	+	0.979010i	$0.434667\pi$
$440$	38.9326	1.85604
$441$	−25.7513	−1.22625
$442$	66.4445	3.16044
$443$	26.9284	1.27941	0.639703	−	0.768622i	$-0.279057\pi$
$443$	26.9284	1.27941	0.639703	+	0.768622i	$0.279057\pi$
$444$	56.5103	2.68186
$445$	14.8018	0.701672
$446$	22.5401	1.06731
$447$	−36.2282	−1.71354
$448$	14.5926	0.689434
$449$	35.6900	1.68432	0.842158	−	0.539231i	$-0.181285\pi$
$449$	35.6900	1.68432	0.842158	+	0.539231i	$0.181285\pi$
$450$	−27.1976	−1.28211
$451$	26.5044	1.24804
$452$	−32.4164	−1.52474
$453$	−44.0908	−2.07156
$454$	−22.4513	−1.05369
$455$	−25.0659	−1.17511
$456$	11.2704	0.527783
$457$	38.2768	1.79051	0.895257	−	0.445549i	$-0.146992\pi$
$457$	38.2768	1.79051	0.895257	+	0.445549i	$0.146992\pi$
$458$	52.5778	2.45680
$459$	44.7217	2.08743
$460$	10.2825	0.479426
$461$	2.61753	0.121911	0.0609553	−	0.998140i	$-0.480585\pi$
$461$	2.61753	0.121911	0.0609553	+	0.998140i	$0.480585\pi$
$462$	38.8355	1.80679
$463$	19.9385	0.926621	0.463310	−	0.886196i	$-0.346661\pi$
$463$	19.9385	0.926621	0.463310	+	0.886196i	$0.346661\pi$
$464$	−0.477915	−0.0221867
$465$	76.8426	3.56349
$466$	−58.6932	−2.71891
$467$	−10.2731	−0.475382	−0.237691	−	0.971341i	$-0.576391\pi$
$467$	−10.2731	−0.475382	−0.237691	+	0.971341i	$0.576391\pi$
$468$	138.316	6.39364
$469$	22.4170	1.03512
$470$	10.2644	0.473461
$471$	−2.92406	−0.134734
$472$	44.6616	2.05572
$473$	−18.5383	−0.852390
$474$	25.1098	1.15333
$475$	1.44345	0.0662298
$476$	31.2284	1.43135
$477$	30.8405	1.41209
$478$	16.9060	0.773262
$479$	−9.58116	−0.437775	−0.218887	−	0.975750i	$-0.570243\pi$
$479$	−9.58116	−0.437775	−0.218887	+	0.975750i	$0.570243\pi$
$480$	−4.83223	−0.220560
$481$	−27.3042	−1.24496
$482$	55.5267	2.52917
$483$	5.04522	0.229566
$484$	−4.00907	−0.182230
$485$	15.3509	0.697048
$486$	5.34624	0.242510
$487$	36.1071	1.63617	0.818084	−	0.575099i	$-0.195036\pi$
$487$	36.1071	1.63617	0.818084	+	0.575099i	$0.195036\pi$
$488$	−9.34180	−0.422883
$489$	47.1693	2.13307
$490$	26.7867	1.21010
$491$	−11.2738	−0.508779	−0.254389	−	0.967102i	$-0.581875\pi$
$491$	−11.2738	−0.508779	−0.254389	+	0.967102i	$0.581875\pi$
$492$	99.7143	4.49547
$493$	0.626774	0.0282285
$494$	−11.0707	−0.498094
$495$	−50.5001	−2.26981
$496$	−35.2705	−1.58369
$497$	11.5470	0.517955
$498$	−81.7965	−3.66539
$499$	21.8883	0.979856	0.489928	−	0.871763i	$-0.337023\pi$
$499$	21.8883	0.979856	0.489928	+	0.871763i	$0.337023\pi$
$500$	−32.6534	−1.46031
$501$	−62.9055	−2.81041
$502$	47.2950	2.11088
$503$	15.1308	0.674650	0.337325	−	0.941388i	$-0.390478\pi$
$503$	15.1308	0.674650	0.337325	+	0.941388i	$0.390478\pi$
$504$	48.2238	2.14806
$505$	−40.2158	−1.78958
$506$	7.69748	0.342195
$507$	−60.3403	−2.67980
$508$	42.4512	1.88347
$509$	−8.78028	−0.389179	−0.194590	−	0.980885i	$-0.562337\pi$
$509$	−8.78028	−0.389179	−0.194590	+	0.980885i	$0.562337\pi$
$510$	−91.2682	−4.04143
$511$	−18.1387	−0.802409
$512$	36.3784	1.60771
$513$	−7.45134	−0.328985
$514$	7.09571	0.312978
$515$	8.12180	0.357889
$516$	−69.7443	−3.07032
$517$	5.09504	0.224080
$518$	−19.3533	−0.850335
$519$	−5.69066	−0.249792
$520$	−70.7710	−3.10351
$521$	−30.4706	−1.33494	−0.667470	−	0.744636i	$-0.732623\pi$
$521$	−30.4706	−1.33494	−0.667470	+	0.744636i	$0.732623\pi$
$522$	1.96769	0.0861236
$523$	17.8875	0.782164	0.391082	−	0.920356i	$-0.372101\pi$
$523$	17.8875	0.782164	0.391082	+	0.920356i	$0.372101\pi$
$524$	65.4675	2.85996
$525$	9.20441	0.401714
$526$	23.4688	1.02329
$527$	46.2563	2.01496
$528$	34.5441	1.50334
$529$	1.00000	0.0434783
$530$	−32.0805	−1.39349
$531$	−57.9312	−2.51400
$532$	−5.20315	−0.225585
$533$	−48.1792	−2.08687
$534$	41.6871	1.80398
$535$	−6.00226	−0.259500
$536$	63.2922	2.73381
$537$	−59.0156	−2.54671
$538$	36.4365	1.57089
$539$	13.2964	0.572715
$540$	−96.8390	−4.16729
$541$	−19.1321	−0.822552	−0.411276	−	0.911511i	$-0.634917\pi$
$541$	−19.1321	−0.822552	−0.411276	+	0.911511i	$0.634917\pi$
$542$	48.9808	2.10391
$543$	−34.3481	−1.47402
$544$	−2.90882	−0.124715
$545$	54.2501	2.32382
$546$	−70.5945	−3.02116
$547$	−17.5666	−0.751094	−0.375547	−	0.926803i	$-0.622545\pi$
$547$	−17.5666	−0.751094	−0.375547	+	0.926803i	$0.622545\pi$
$548$	52.3634	2.23686
$549$	12.1174	0.517157
$550$	14.0431	0.598802
$551$	−0.104430	−0.00444889
$552$	14.2447	0.606294
$553$	−5.70211	−0.242479
$554$	−65.5147	−2.78345
$555$	37.5051	1.59200
$556$	−1.60930	−0.0682494
$557$	−18.6109	−0.788570	−0.394285	−	0.918988i	$-0.629008\pi$
$557$	−18.6109	−0.788570	−0.394285	+	0.918988i	$0.629008\pi$
$558$	145.217	6.14752
$559$	33.6985	1.42529
$560$	−15.8035	−0.667819
$561$	−45.3037	−1.91272
$562$	69.3221	2.92417
$563$	−30.2806	−1.27617	−0.638087	−	0.769965i	$-0.720274\pi$
$563$	−30.2806	−1.27617	−0.638087	+	0.769965i	$0.720274\pi$
$564$	19.1684	0.807137
$565$	−21.5143	−0.905115
$566$	0.106892	0.00449301
$567$	−16.8461	−0.707470
$568$	32.6019	1.36795
$569$	−29.4342	−1.23395	−0.616973	−	0.786985i	$-0.711641\pi$
$569$	−29.4342	−1.23395	−0.616973	+	0.786985i	$0.711641\pi$
$570$	15.2067	0.636940
$571$	40.2500	1.68441	0.842204	−	0.539159i	$-0.181258\pi$
$571$	40.2500	1.68441	0.842204	+	0.539159i	$0.181258\pi$
$572$	−71.4175	−2.98612
$573$	39.9596	1.66934
$574$	−34.1495	−1.42537
$575$	1.82438	0.0760820
$576$	−53.4419	−2.22675
$577$	24.1173	1.00402	0.502009	−	0.864863i	$-0.332595\pi$
$577$	24.1173	1.00402	0.502009	+	0.864863i	$0.332595\pi$
$578$	−13.5209	−0.562397
$579$	−50.5015	−2.09877
$580$	−1.35720	−0.0563546
$581$	18.5750	0.770619
$582$	43.2336	1.79209
$583$	−15.9241	−0.659509
$584$	−51.2128	−2.11920
$585$	91.7981	3.79538
$586$	10.8008	0.446176
$587$	−36.9619	−1.52558	−0.762791	−	0.646645i	$-0.776172\pi$
$587$	−36.9619	−1.52558	−0.762791	+	0.646645i	$0.776172\pi$
$588$	50.0233	2.06293
$589$	−7.70703	−0.317563
$590$	60.2605	2.48088
$591$	14.9432	0.614682
$592$	−17.2147	−0.707520
$593$	−26.0090	−1.06806	−0.534031	−	0.845465i	$-0.679323\pi$
$593$	−26.0090	−1.06806	−0.534031	+	0.845465i	$0.679323\pi$
$594$	−72.4934	−2.97444
$595$	20.7259	0.849678
$596$	47.2224	1.93430
$597$	−50.2139	−2.05512
$598$	−13.9923	−0.572189
$599$	31.8299	1.30053	0.650267	−	0.759705i	$-0.274657\pi$
$599$	31.8299	1.30053	0.650267	+	0.759705i	$0.274657\pi$
$600$	25.9877	1.06095
$601$	8.37677	0.341695	0.170848	−	0.985297i	$-0.445349\pi$
$601$	8.37677	0.341695	0.170848	+	0.985297i	$0.445349\pi$
$602$	23.8856	0.973504
$603$	−82.0973	−3.34326
$604$	57.4710	2.33846
$605$	−2.66076	−0.108175
$606$	−113.262	−4.60096
$607$	−1.94966	−0.0791343	−0.0395672	−	0.999217i	$-0.512598\pi$
$607$	−1.94966	−0.0791343	−0.0395672	+	0.999217i	$0.512598\pi$
$608$	0.484655	0.0196554
$609$	−0.665921	−0.0269845
$610$	−12.6046	−0.510345
$611$	−9.26166	−0.374687
$612$	−114.367	−4.62301
$613$	28.3980	1.14698	0.573492	−	0.819211i	$-0.305588\pi$
$613$	28.3980	1.14698	0.573492	+	0.819211i	$0.305588\pi$
$614$	57.8383	2.33416
$615$	66.1790	2.66859
$616$	−24.8997	−1.00324
$617$	−5.77207	−0.232375	−0.116187	−	0.993227i	$-0.537067\pi$
$617$	−5.77207	−0.232375	−0.116187	+	0.993227i	$0.537067\pi$
$618$	22.8739	0.920123
$619$	17.8778	0.718569	0.359285	−	0.933228i	$-0.383021\pi$
$619$	17.8778	0.718569	0.359285	+	0.933228i	$0.383021\pi$
$620$	−100.162	−4.02260
$621$	−9.41780	−0.377923
$622$	22.9876	0.921717
$623$	−9.46663	−0.379273
$624$	−62.7935	−2.51375
$625$	−30.7935	−1.23174
$626$	1.00986	0.0403622
$627$	7.54831	0.301451
$628$	3.81143	0.152092
$629$	22.5767	0.900190
$630$	65.0668	2.59232
$631$	−10.3415	−0.411690	−0.205845	−	0.978585i	$-0.565994\pi$
$631$	−10.3415	−0.411690	−0.205845	+	0.978585i	$0.565994\pi$
$632$	−16.0993	−0.640398
$633$	46.4898	1.84780
$634$	58.8471	2.33712
$635$	28.1743	1.11806
$636$	−59.9093	−2.37556
$637$	−24.1699	−0.957645
$638$	−1.01599	−0.0402236
$639$	−42.2884	−1.67290
$640$	52.3903	2.07091
$641$	−6.50092	−0.256771	−0.128385	−	0.991724i	$-0.540979\pi$
$641$	−6.50092	−0.256771	−0.128385	+	0.991724i	$0.540979\pi$
$642$	−16.9045	−0.667168
$643$	−13.3659	−0.527099	−0.263549	−	0.964646i	$-0.584893\pi$
$643$	−13.3659	−0.527099	−0.263549	+	0.964646i	$0.584893\pi$
$644$	−6.57630	−0.259143
$645$	−46.2883	−1.82260
$646$	9.15387	0.360154
$647$	−46.5640	−1.83062	−0.915310	−	0.402749i	$-0.868055\pi$
$647$	−46.5640	−1.83062	−0.915310	+	0.402749i	$0.868055\pi$
$648$	−47.5633	−1.86846
$649$	29.9121	1.17415
$650$	−25.5273	−1.00127
$651$	−49.1454	−1.92616
$652$	−61.4837	−2.40789
$653$	−15.9703	−0.624966	−0.312483	−	0.949923i	$-0.601161\pi$
$653$	−15.9703	−0.624966	−0.312483	+	0.949923i	$0.601161\pi$
$654$	152.788	5.97447
$655$	43.4498	1.69773
$656$	−30.3759	−1.18598
$657$	66.4289	2.59164
$658$	−6.56469	−0.255918
$659$	8.67092	0.337771	0.168886	−	0.985636i	$-0.445983\pi$
$659$	8.67092	0.337771	0.168886	+	0.985636i	$0.445983\pi$
$660$	98.0992	3.81851
$661$	39.7126	1.54464	0.772321	−	0.635232i	$-0.219096\pi$
$661$	39.7126	1.54464	0.772321	+	0.635232i	$0.219096\pi$
$662$	−18.5977	−0.722819
$663$	82.3522	3.19829
$664$	52.4445	2.03524
$665$	−3.45326	−0.133912
$666$	70.8771	2.74643
$667$	−0.131990	−0.00511069
$668$	81.9954	3.17250
$669$	27.9365	1.08009
$670$	85.3982	3.29922
$671$	−6.25666	−0.241536
$672$	3.09050	0.119219
$673$	−26.5903	−1.02498	−0.512491	−	0.858693i	$-0.671277\pi$
$673$	−26.5903	−1.02498	−0.512491	+	0.858693i	$0.671277\pi$
$674$	72.3612	2.78725
$675$	−17.1817	−0.661323
$676$	78.6517	3.02507
$677$	22.3917	0.860584	0.430292	−	0.902690i	$-0.358411\pi$
$677$	22.3917	0.860584	0.430292	+	0.902690i	$0.358411\pi$
$678$	−60.5921	−2.32702
$679$	−9.81781	−0.376773
$680$	58.5174	2.24404
$681$	−27.8265	−1.06631
$682$	−74.9810	−2.87117
$683$	32.1430	1.22992	0.614959	−	0.788559i	$-0.289172\pi$
$683$	32.1430	1.22992	0.614959	+	0.788559i	$0.289172\pi$
$684$	19.0554	0.728600
$685$	34.7529	1.32784
$686$	−45.6262	−1.74202
$687$	65.1656	2.48622
$688$	21.2462	0.810002
$689$	28.9465	1.10277
$690$	19.2199	0.731689
$691$	−6.05487	−0.230338	−0.115169	−	0.993346i	$-0.536741\pi$
$691$	−6.05487	−0.230338	−0.115169	+	0.993346i	$0.536741\pi$
$692$	7.41761	0.281975
$693$	32.2978	1.22689
$694$	−35.0648	−1.33104
$695$	−1.06807	−0.0405141
$696$	−1.88016	−0.0712674
$697$	39.8373	1.50894
$698$	2.43642	0.0922197
$699$	−72.7452	−2.75147
$700$	−11.9977	−0.453470
$701$	−28.9957	−1.09515	−0.547576	−	0.836756i	$-0.684449\pi$
$701$	−28.9957	−1.09515	−0.547576	+	0.836756i	$0.684449\pi$
$702$	131.777	4.97360
$703$	−3.76163	−0.141872
$704$	27.5941	1.03999
$705$	12.7218	0.479132
$706$	−28.1922	−1.06103
$707$	25.7204	0.967315
$708$	112.535	4.22931
$709$	26.2953	0.987539	0.493770	−	0.869593i	$-0.335619\pi$
$709$	26.2953	0.987539	0.493770	+	0.869593i	$0.335619\pi$
$710$	43.9887	1.65087
$711$	20.8827	0.783162
$712$	−26.7281	−1.00168
$713$	−9.74098	−0.364802
$714$	58.3715	2.18450
$715$	−47.3988	−1.77262
$716$	76.9250	2.87482
$717$	20.9535	0.782524
$718$	−22.5388	−0.841142
$719$	3.09876	0.115564	0.0577822	−	0.998329i	$-0.481597\pi$
$719$	3.09876	0.115564	0.0577822	+	0.998329i	$0.481597\pi$
$720$	57.8767	2.15694
$721$	−5.19438	−0.193449
$722$	44.7667	1.66605
$723$	68.8206	2.55946
$724$	44.7718	1.66393
$725$	−0.240801	−0.00894312
$726$	−7.49366	−0.278116
$727$	25.7756	0.955965	0.477983	−	0.878369i	$-0.341368\pi$
$727$	25.7756	0.955965	0.477983	+	0.878369i	$0.341368\pi$
$728$	45.2623	1.67753
$729$	−23.6226	−0.874911
$730$	−69.0998	−2.55750
$731$	−27.8638	−1.03058
$732$	−23.5387	−0.870015
$733$	−34.9384	−1.29048	−0.645240	−	0.763980i	$-0.723243\pi$
$733$	−34.9384	−1.29048	−0.645240	+	0.763980i	$0.723243\pi$
$734$	−22.9164	−0.845860
$735$	33.1998	1.22459
$736$	0.612559	0.0225792
$737$	42.3899	1.56145
$738$	125.065	4.60370
$739$	−5.29026	−0.194605	−0.0973027	−	0.995255i	$-0.531021\pi$
$739$	−5.29026	−0.194605	−0.0973027	+	0.995255i	$0.531021\pi$
$740$	−48.8868	−1.79711
$741$	−13.7212	−0.504060
$742$	20.5174	0.753217
$743$	27.7567	1.01830	0.509148	−	0.860679i	$-0.329961\pi$
$743$	27.7567	1.01830	0.509148	+	0.860679i	$0.329961\pi$
$744$	−138.757	−5.08708
$745$	31.3408	1.14824
$746$	46.4844	1.70192
$747$	−68.0266	−2.48896
$748$	59.0520	2.15916
$749$	3.83880	0.140267
$750$	−61.0351	−2.22869
$751$	−42.0831	−1.53563	−0.767817	−	0.640669i	$-0.778657\pi$
$751$	−42.0831	−1.53563	−0.767817	+	0.640669i	$0.778657\pi$
$752$	−5.83927	−0.212936
$753$	58.6180	2.13616
$754$	1.84685	0.0672585
$755$	38.1427	1.38815
$756$	61.9343	2.25253
$757$	49.6104	1.80312	0.901560	−	0.432655i	$-0.142423\pi$
$757$	49.6104	1.80312	0.901560	+	0.432655i	$0.142423\pi$
$758$	−58.9489	−2.14112
$759$	9.54037	0.346293
$760$	−9.74993	−0.353667
$761$	31.9028	1.15648	0.578238	−	0.815868i	$-0.303740\pi$
$761$	31.9028	1.15648	0.578238	+	0.815868i	$0.303740\pi$
$762$	79.3489	2.87451
$763$	−34.6962	−1.25609
$764$	−52.0861	−1.88441
$765$	−75.9038	−2.74431
$766$	−52.7022	−1.90421
$767$	−54.3736	−1.96332
$768$	94.8004	3.42081
$769$	35.7865	1.29049	0.645247	−	0.763974i	$-0.276754\pi$
$769$	35.7865	1.29049	0.645247	+	0.763974i	$0.276754\pi$
$770$	−33.5964	−1.21073
$771$	8.79452	0.316727
$772$	65.8272	2.36917
$773$	32.8032	1.17985	0.589925	−	0.807458i	$-0.299157\pi$
$773$	32.8032	1.17985	0.589925	+	0.807458i	$0.299157\pi$
$774$	−87.4756	−3.14424
$775$	−17.7713	−0.638362
$776$	−27.7196	−0.995076
$777$	−23.9868	−0.860520
$778$	−7.87279	−0.282253
$779$	−6.63751	−0.237814
$780$	−178.323	−6.38498
$781$	21.8351	0.781321
$782$	11.5696	0.413730
$783$	1.24306	0.0444233
$784$	−15.2386	−0.544234
$785$	2.52959	0.0902849
$786$	122.370	4.36480
$787$	15.6011	0.556119	0.278060	−	0.960564i	$-0.410309\pi$
$787$	15.6011	0.556119	0.278060	+	0.960564i	$0.410309\pi$
$788$	−19.4781	−0.693877
$789$	29.0875	1.03554
$790$	−21.7223	−0.772846
$791$	13.7597	0.489239
$792$	91.1897	3.24028
$793$	11.3732	0.403875
$794$	−50.0181	−1.77508
$795$	−39.7610	−1.41018
$796$	65.4523	2.31989
$797$	−0.905480	−0.0320738	−0.0160369	−	0.999871i	$-0.505105\pi$
$797$	−0.905480	−0.0320738	−0.0160369	+	0.999871i	$0.505105\pi$
$798$	−9.72561	−0.344283
$799$	7.65806	0.270923
$800$	1.11754	0.0395110
$801$	34.6694	1.22498
$802$	−60.6901	−2.14304
$803$	−34.2997	−1.21041
$804$	159.479	5.62437
$805$	−4.36460	−0.153832
$806$	136.299	4.80093
$807$	45.1599	1.58970
$808$	72.6190	2.55473
$809$	−51.1243	−1.79744	−0.898718	−	0.438528i	$-0.855500\pi$
$809$	−51.1243	−1.79744	−0.898718	+	0.438528i	$0.855500\pi$
$810$	−64.1756	−2.25490
$811$	−29.7633	−1.04513	−0.522566	−	0.852599i	$-0.675025\pi$
$811$	−29.7633	−1.04513	−0.522566	+	0.852599i	$0.675025\pi$
$812$	0.868009	0.0304611
$813$	60.7075	2.12910
$814$	−36.5965	−1.28271
$815$	−40.8059	−1.42937
$816$	51.9212	1.81761
$817$	4.64255	0.162422
$818$	−41.2447	−1.44209
$819$	−58.7103	−2.05151
$820$	−86.2624	−3.01241
$821$	−28.7463	−1.00325	−0.501627	−	0.865084i	$-0.667265\pi$
$821$	−28.7463	−1.00325	−0.501627	+	0.865084i	$0.667265\pi$
$822$	97.8765	3.41384
$823$	46.2762	1.61309	0.806544	−	0.591174i	$-0.201335\pi$
$823$	46.2762	1.61309	0.806544	+	0.591174i	$0.201335\pi$
$824$	−14.6658	−0.510907
$825$	17.4053	0.605974
$826$	−38.5402	−1.34098
$827$	−35.7122	−1.24183	−0.620917	−	0.783876i	$-0.713240\pi$
$827$	−35.7122	−1.24183	−0.620917	+	0.783876i	$0.713240\pi$
$828$	24.0842	0.836984
$829$	−0.802378	−0.0278678	−0.0139339	−	0.999903i	$-0.504435\pi$
$829$	−0.802378	−0.0278678	−0.0139339	+	0.999903i	$0.504435\pi$
$830$	70.7617	2.45618
$831$	−81.1999	−2.81679
$832$	−50.1600	−1.73898
$833$	19.9850	0.692439
$834$	−3.00806	−0.104161
$835$	54.4192	1.88325
$836$	−9.83900	−0.340289
$837$	91.7386	3.17095
$838$	76.0418	2.62682
$839$	18.3045	0.631941	0.315970	−	0.948769i	$-0.397670\pi$
$839$	18.3045	0.631941	0.315970	+	0.948769i	$0.397670\pi$
$840$	−62.1723	−2.14515
$841$	−28.9826	−0.999399
$842$	61.2082	2.10937
$843$	85.9187	2.95920
$844$	−60.5981	−2.08587
$845$	52.2000	1.79574
$846$	24.0417	0.826570
$847$	1.70172	0.0584716
$848$	18.2501	0.626712
$849$	0.132483	0.00454682
$850$	21.1075	0.723980
$851$	−4.75435	−0.162977
$852$	82.1476	2.81433
$853$	17.7658	0.608291	0.304145	−	0.952626i	$-0.401629\pi$
$853$	17.7658	0.608291	0.304145	+	0.952626i	$0.401629\pi$
$854$	8.06138	0.275855
$855$	12.6468	0.432510
$856$	10.8385	0.370451
$857$	−16.2061	−0.553590	−0.276795	−	0.960929i	$-0.589272\pi$
$857$	−16.2061	−0.553590	−0.276795	+	0.960929i	$0.589272\pi$
$858$	−133.492	−4.55734
$859$	38.4894	1.31324	0.656621	−	0.754221i	$-0.271985\pi$
$859$	38.4894	1.31324	0.656621	+	0.754221i	$0.271985\pi$
$860$	60.3354	2.05742
$861$	−42.3254	−1.44245
$862$	−59.8319	−2.03788
$863$	26.5984	0.905421	0.452711	−	0.891657i	$-0.350457\pi$
$863$	26.5984	0.905421	0.452711	+	0.891657i	$0.350457\pi$
$864$	−5.76896	−0.196264
$865$	4.92296	0.167386
$866$	−59.7590	−2.03069
$867$	−16.7580	−0.569133
$868$	64.0596	2.17432
$869$	−10.7825	−0.365772
$870$	−2.53684	−0.0860070
$871$	−77.0556	−2.61093
$872$	−97.9612	−3.31739
$873$	35.9555	1.21691
$874$	−1.92769	−0.0652049
$875$	13.8603	0.468564
$876$	−129.042	−4.35992
$877$	22.7766	0.769111	0.384555	−	0.923102i	$-0.374355\pi$
$877$	22.7766	0.769111	0.384555	+	0.923102i	$0.374355\pi$
$878$	−20.8085	−0.702254
$879$	13.3866	0.451520
$880$	−29.8839	−1.00739
$881$	−38.2881	−1.28996	−0.644979	−	0.764201i	$-0.723134\pi$
$881$	−38.2881	−1.28996	−0.644979	+	0.764201i	$0.723134\pi$
$882$	62.7408	2.11259
$883$	−29.3563	−0.987920	−0.493960	−	0.869485i	$-0.664451\pi$
$883$	−29.3563	−0.987920	−0.493960	+	0.869485i	$0.664451\pi$
$884$	−107.344	−3.61036
$885$	74.6877	2.51060
$886$	−65.6088	−2.20417
$887$	−2.03266	−0.0682499	−0.0341250	−	0.999418i	$-0.510864\pi$
$887$	−2.03266	−0.0682499	−0.0341250	+	0.999418i	$0.510864\pi$
$888$	−67.7242	−2.27268
$889$	−18.0191	−0.604343
$890$	−36.0633	−1.20885
$891$	−31.8555	−1.06720
$892$	−36.4144	−1.21925
$893$	−1.27595	−0.0426982
$894$	88.2670	2.95209
$895$	51.0541	1.70655
$896$	−33.5067	−1.11938
$897$	−17.3423	−0.579042
$898$	−86.9557	−2.90175
$899$	1.28572	0.0428810
$900$	43.9388	1.46463
$901$	−23.9346	−0.797377
$902$	−64.5758	−2.15014
$903$	29.6041	0.985164
$904$	38.8491	1.29210
$905$	29.7144	0.987740
$906$	107.423	3.56891
$907$	19.1723	0.636606	0.318303	−	0.947989i	$-0.396887\pi$
$907$	19.1723	0.636606	0.318303	+	0.947989i	$0.396887\pi$
$908$	36.2709	1.20369
$909$	−94.1951	−3.12425
$910$	61.0709	2.02448
$911$	−28.8864	−0.957051	−0.478525	−	0.878074i	$-0.658829\pi$
$911$	−28.8864	−0.957051	−0.478525	+	0.878074i	$0.658829\pi$
$912$	−8.65090	−0.286460
$913$	35.1247	1.16246
$914$	−93.2583	−3.08471
$915$	−15.6223	−0.516457
$916$	−84.9414	−2.80654
$917$	−27.7888	−0.917666
$918$	−108.961	−3.59624
$919$	−50.8723	−1.67812	−0.839061	−	0.544038i	$-0.816895\pi$
$919$	−50.8723	−1.67812	−0.839061	+	0.544038i	$0.816895\pi$
$920$	−12.3230	−0.406278
$921$	71.6855	2.36212
$922$	−6.37740	−0.210028
$923$	−39.6914	−1.30646
$924$	−62.7403	−2.06400
$925$	−8.67374	−0.285191
$926$	−48.5785	−1.59639
$927$	19.0232	0.624805
$928$	−0.0808519	−0.00265409
$929$	46.2201	1.51643	0.758216	−	0.652003i	$-0.226071\pi$
$929$	46.2201	1.51643	0.758216	+	0.652003i	$0.226071\pi$
$930$	−187.221	−6.13920
$931$	−3.32982	−0.109130
$932$	94.8211	3.10597
$933$	28.4911	0.932757
$934$	25.0295	0.818991
$935$	39.1920	1.28172
$936$	−165.763	−5.41813
$937$	−28.3378	−0.925754	−0.462877	−	0.886422i	$-0.653183\pi$
$937$	−28.3378	−0.925754	−0.462877	+	0.886422i	$0.653183\pi$
$938$	−54.6172	−1.78332
$939$	1.25164	0.0408457
$940$	−16.5825	−0.540862
$941$	30.9464	1.00882	0.504412	−	0.863463i	$-0.331709\pi$
$941$	30.9464	1.00882	0.504412	+	0.863463i	$0.331709\pi$
$942$	7.12423	0.232120
$943$	−8.38920	−0.273190
$944$	−34.2813	−1.11576
$945$	41.1049	1.33714
$946$	45.1669	1.46850
$947$	33.3038	1.08223	0.541114	−	0.840949i	$-0.318003\pi$
$947$	33.3038	1.08223	0.541114	+	0.840949i	$0.318003\pi$
$948$	−40.5658	−1.31752
$949$	62.3494	2.02395
$950$	−3.51683	−0.114101
$951$	72.9359	2.36511
$952$	−37.4254	−1.21296
$953$	−1.16448	−0.0377212	−0.0188606	−	0.999822i	$-0.506004\pi$
$953$	−1.16448	−0.0377212	−0.0188606	+	0.999822i	$0.506004\pi$
$954$	−75.1402	−2.43275
$955$	−34.5688	−1.11862
$956$	−27.3123	−0.883343
$957$	−1.25924	−0.0407054
$958$	23.3437	0.754201
$959$	−22.2265	−0.717732
$960$	68.8999	2.22373
$961$	63.8866	2.06086
$962$	66.5244	2.14483
$963$	−14.0587	−0.453037
$964$	−89.7055	−2.88922
$965$	43.6886	1.40639
$966$	−12.2923	−0.395497
$967$	25.4670	0.818964	0.409482	−	0.912318i	$-0.365709\pi$
$967$	25.4670	0.818964	0.409482	+	0.912318i	$0.365709\pi$
$968$	4.80462	0.154426
$969$	11.3454	0.364468
$970$	−37.4012	−1.20088
$971$	39.3817	1.26382	0.631909	−	0.775043i	$-0.282272\pi$
$971$	39.3817	1.26382	0.631909	+	0.775043i	$0.282272\pi$
$972$	−8.63706	−0.277034
$973$	0.683093	0.0218990
$974$	−87.9719	−2.81880
$975$	−31.6389	−1.01326
$976$	7.17057	0.229524
$977$	−30.8832	−0.988041	−0.494021	−	0.869450i	$-0.664473\pi$
$977$	−30.8832	−0.988041	−0.494021	+	0.869450i	$0.664473\pi$
$978$	−114.924	−3.67486
$979$	−17.9011	−0.572122
$980$	−43.2749	−1.38237
$981$	127.067	4.05693
$982$	27.4676	0.876528
$983$	−6.00724	−0.191601	−0.0958005	−	0.995401i	$-0.530541\pi$
$983$	−6.00724	−0.191601	−0.0958005	+	0.995401i	$0.530541\pi$
$984$	−119.502	−3.80957
$985$	−12.9273	−0.411898
$986$	−1.52708	−0.0486322
$987$	−8.13637	−0.258984
$988$	17.8851	0.569002
$989$	5.86775	0.186584
$990$	123.039	3.91044
$991$	−19.7731	−0.628113	−0.314056	−	0.949404i	$-0.601688\pi$
$991$	−19.7731	−0.628113	−0.314056	+	0.949404i	$0.601688\pi$
$992$	−5.96692	−0.189450
$993$	−23.0502	−0.731477
$994$	−28.1334	−0.892337
$995$	43.4398	1.37713
$996$	132.145	4.18719
$997$	−17.7526	−0.562229	−0.281115	−	0.959674i	$-0.590704\pi$
$997$	−17.7526	−0.562229	−0.281115	+	0.959674i	$0.590704\pi$
$998$	−53.3291	−1.68810
$999$	44.7755	1.41663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8027.2.a.f.1.15	✓	176

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8027.2.a.f.1.15	✓	176	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:	\( N \)	\(=\)	\( 8027 = 23 \cdot 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43642 q^{2} -3.01973 q^{3} +3.93613 q^{4} +2.61235 q^{5} +7.35732 q^{6} -1.67075 q^{7} -4.71721 q^{8} +6.11876 q^{9} +O(q^{10})\)	\(q-2.43642 q^{2} -3.01973 q^{3} +3.93613 q^{4} +2.61235 q^{5} +7.35732 q^{6} -1.67075 q^{7} -4.71721 q^{8} +6.11876 q^{9} -6.36478 q^{10} -3.15935 q^{11} -11.8860 q^{12} +5.74300 q^{13} +4.07065 q^{14} -7.88859 q^{15} +3.62083 q^{16} -4.74863 q^{17} -14.9078 q^{18} +0.791197 q^{19} +10.2825 q^{20} +5.04522 q^{21} +7.69748 q^{22} +1.00000 q^{23} +14.2447 q^{24} +1.82438 q^{25} -13.9923 q^{26} -9.41780 q^{27} -6.57630 q^{28} -0.131990 q^{29} +19.2199 q^{30} -9.74098 q^{31} +0.612559 q^{32} +9.54037 q^{33} +11.5696 q^{34} -4.36460 q^{35} +24.0842 q^{36} -4.75435 q^{37} -1.92769 q^{38} -17.3423 q^{39} -12.3230 q^{40} -8.38920 q^{41} -12.2923 q^{42} +5.86775 q^{43} -12.4356 q^{44} +15.9843 q^{45} -2.43642 q^{46} -1.61269 q^{47} -10.9339 q^{48} -4.20858 q^{49} -4.44495 q^{50} +14.3396 q^{51} +22.6052 q^{52} +5.04031 q^{53} +22.9457 q^{54} -8.25332 q^{55} +7.88130 q^{56} -2.38920 q^{57} +0.321584 q^{58} -9.46781 q^{59} -31.0505 q^{60} +1.98037 q^{61} +23.7331 q^{62} -10.2229 q^{63} -8.73411 q^{64} +15.0027 q^{65} -23.2443 q^{66} -13.4173 q^{67} -18.6912 q^{68} -3.01973 q^{69} +10.6340 q^{70} -6.91127 q^{71} -28.8635 q^{72} +10.8566 q^{73} +11.5836 q^{74} -5.50914 q^{75} +3.11425 q^{76} +5.27849 q^{77} +42.2530 q^{78} +3.41290 q^{79} +9.45889 q^{80} +10.0829 q^{81} +20.4396 q^{82} -11.1177 q^{83} +19.8586 q^{84} -12.4051 q^{85} -14.2963 q^{86} +0.398575 q^{87} +14.9033 q^{88} +5.66608 q^{89} -38.9445 q^{90} -9.59514 q^{91} +3.93613 q^{92} +29.4151 q^{93} +3.92918 q^{94} +2.06689 q^{95} -1.84976 q^{96} +5.87627 q^{97} +10.2539 q^{98} -19.3313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * q^99

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