LMFDB - Embedded newform 1331.2.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1331,2,Mod(1,1331)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1331, 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("1331.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1331 = 11^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1331.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:	$10.6280885090$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	1331.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00638 q^{2} -2.20960 q^{3} +2.02557 q^{4} -3.12320 q^{5} -4.43331 q^{6} +0.406578 q^{7} +0.0513041 q^{8} +1.88235 q^{9} -6.26633 q^{10} -4.47571 q^{12} +1.59878 q^{13} +0.815751 q^{14} +6.90104 q^{15} -3.94821 q^{16} +0.870785 q^{17} +3.77672 q^{18} +7.69273 q^{19} -6.32626 q^{20} -0.898376 q^{21} +5.45997 q^{23} -0.113362 q^{24} +4.75438 q^{25} +3.20776 q^{26} +2.46956 q^{27} +0.823552 q^{28} -9.13391 q^{29} +13.8461 q^{30} +6.93772 q^{31} -8.02422 q^{32} +1.74713 q^{34} -1.26982 q^{35} +3.81284 q^{36} +7.90215 q^{37} +15.4346 q^{38} -3.53266 q^{39} -0.160233 q^{40} -2.89614 q^{41} -1.80249 q^{42} +5.29356 q^{43} -5.87896 q^{45} +10.9548 q^{46} +4.44000 q^{47} +8.72397 q^{48} -6.83469 q^{49} +9.53910 q^{50} -1.92409 q^{51} +3.23843 q^{52} +4.44799 q^{53} +4.95488 q^{54} +0.0208591 q^{56} -16.9979 q^{57} -18.3261 q^{58} -3.39440 q^{59} +13.9785 q^{60} +14.1754 q^{61} +13.9197 q^{62} +0.765323 q^{63} -8.20324 q^{64} -4.99330 q^{65} +5.40601 q^{67} +1.76384 q^{68} -12.0644 q^{69} -2.54775 q^{70} -1.81572 q^{71} +0.0965723 q^{72} +5.78453 q^{73} +15.8547 q^{74} -10.5053 q^{75} +15.5822 q^{76} -7.08787 q^{78} -15.7267 q^{79} +12.3310 q^{80} -11.1038 q^{81} -5.81077 q^{82} -9.70962 q^{83} -1.81972 q^{84} -2.71964 q^{85} +10.6209 q^{86} +20.1823 q^{87} -1.07269 q^{89} -11.7954 q^{90} +0.650027 q^{91} +11.0596 q^{92} -15.3296 q^{93} +8.90833 q^{94} -24.0259 q^{95} +17.7303 q^{96} +9.44446 q^{97} -13.7130 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} - 3 q^{3} + 25 q^{4} - 3 q^{5} + 15 q^{6} + 19 q^{7} + 9 q^{8} + 22 q^{9} + 25 q^{10} - 12 q^{12} + 46 q^{13} - 12 q^{14} - 15 q^{15} + 13 q^{16} + 14 q^{17} + 6 q^{18} + 45 q^{19} - 24 q^{20}+ \cdots - 39 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 - 3 * q^3 + 25 * q^4 - 3 * q^5 + 15 * q^6 + 19 * q^7 + 9 * q^8 + 22 * q^9 + 25 * q^10 - 12 * q^12 + 46 * q^13 - 12 * q^14 - 15 * q^15 + 13 * q^16 + 14 * q^17 + 6 * q^18 + 45 * q^19 - 24 * q^20 + 49 * q^21 + 2 * q^23 + 36 * q^24 + 16 * q^25 - 7 * q^26 - 21 * q^27 + 40 * q^28 + 40 * q^29 - 15 * q^30 - 4 * q^31 + 3 * q^32 - 24 * q^34 - q^35 + 38 * q^36 - 21 * q^37 + 30 * q^38 + 18 * q^39 + 54 * q^40 + 49 * q^41 + 13 * q^42 + 14 * q^43 - 42 * q^45 + 40 * q^46 - 2 * q^47 - 5 * q^48 + 18 * q^49 - 12 * q^50 + 7 * q^51 + 57 * q^52 - 25 * q^53 + 33 * q^54 - 11 * q^56 + 4 * q^57 + 23 * q^58 - 22 * q^59 - 12 * q^60 + 128 * q^61 - 8 * q^62 + 12 * q^63 + 31 * q^64 + 16 * q^67 - 6 * q^68 - 57 * q^69 + 38 * q^70 - 34 * q^71 - 15 * q^72 + 71 * q^73 - 21 * q^74 - 10 * q^75 + 95 * q^76 + 75 * q^78 + 53 * q^79 - 29 * q^80 + 21 * q^81 + 33 * q^82 - 6 * q^83 + 18 * q^84 + 101 * q^85 + 38 * q^86 - 16 * q^87 + 5 * q^89 - 20 * q^90 + 16 * q^91 + 53 * q^92 + 7 * q^93 + 57 * q^94 - 16 * q^95 + 32 * q^96 + 14 * q^97 - 39 * q^98

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 $

$2$	2.00638	1.41873	0.709363	−	0.704843i	$-0.248983\pi$
$2$	2.00638	1.41873	0.709363	+	0.704843i	$0.248983\pi$
$3$	−2.20960	−1.27572	−0.637858	−	0.770154i	$-0.720179\pi$
$3$	−2.20960	−1.27572	−0.637858	+	0.770154i	$0.720179\pi$
$4$	2.02557	1.01279
$5$	−3.12320	−1.39674	−0.698369	−	0.715738i	$-0.746090\pi$
$5$	−3.12320	−1.39674	−0.698369	+	0.715738i	$0.746090\pi$
$6$	−4.43331	−1.80989
$7$	0.406578	0.153672	0.0768360	−	0.997044i	$-0.475518\pi$
$7$	0.406578	0.153672	0.0768360	+	0.997044i	$0.475518\pi$
$8$	0.0513041	0.0181387
$9$	1.88235	0.627451
$10$	−6.26633	−1.98159
$11$	0	0
$11$	0	0
$12$	−4.47571	−1.29203
$13$	1.59878	0.443421	0.221710	−	0.975113i	$-0.428836\pi$
$13$	1.59878	0.443421	0.221710	+	0.975113i	$0.428836\pi$
$14$	0.815751	0.218019
$15$	6.90104	1.78184
$16$	−3.94821	−0.987051
$17$	0.870785	0.211196	0.105598	−	0.994409i	$-0.466324\pi$
$17$	0.870785	0.211196	0.105598	+	0.994409i	$0.466324\pi$
$18$	3.77672	0.890181
$19$	7.69273	1.76483	0.882417	−	0.470469i	$-0.155915\pi$
$19$	7.69273	1.76483	0.882417	+	0.470469i	$0.155915\pi$
$20$	−6.32626	−1.41460
$21$	−0.898376	−0.196042
$22$	0	0
$23$	5.45997	1.13848	0.569242	−	0.822170i	$-0.307237\pi$
$23$	5.45997	1.13848	0.569242	+	0.822170i	$0.307237\pi$
$24$	−0.113362	−0.0231399
$25$	4.75438	0.950876
$26$	3.20776	0.629093
$27$	2.46956	0.475267
$28$	0.823552	0.155637
$29$	−9.13391	−1.69612	−0.848062	−	0.529897i	$-0.822231\pi$
$29$	−9.13391	−1.69612	−0.848062	+	0.529897i	$0.822231\pi$
$30$	13.8461	2.52794
$31$	6.93772	1.24605	0.623026	−	0.782201i	$-0.285903\pi$
$31$	6.93772	1.24605	0.623026	+	0.782201i	$0.285903\pi$
$32$	−8.02422	−1.41849
$33$	0	0
$34$	1.74713	0.299630
$35$	−1.26982	−0.214639
$36$	3.81284	0.635473
$37$	7.90215	1.29911	0.649553	−	0.760317i	$-0.274956\pi$
$37$	7.90215	1.29911	0.649553	+	0.760317i	$0.274956\pi$
$38$	15.4346	2.50382
$39$	−3.53266	−0.565679
$40$	−0.160233	−0.0253350
$41$	−2.89614	−0.452301	−0.226151	−	0.974092i	$-0.572614\pi$
$41$	−2.89614	−0.452301	−0.226151	+	0.974092i	$0.572614\pi$
$42$	−1.80249	−0.278130
$43$	5.29356	0.807261	0.403630	−	0.914922i	$-0.367748\pi$
$43$	5.29356	0.807261	0.403630	+	0.914922i	$0.367748\pi$
$44$	0	0
$45$	−5.87896	−0.876384
$46$	10.9548	1.61520
$47$	4.44000	0.647640	0.323820	−	0.946119i	$-0.395033\pi$
$47$	4.44000	0.647640	0.323820	+	0.946119i	$0.395033\pi$
$48$	8.72397	1.25920
$49$	−6.83469	−0.976385
$50$	9.53910	1.34903
$51$	−1.92409	−0.269427
$52$	3.23843	0.449090
$53$	4.44799	0.610979	0.305489	−	0.952196i	$-0.401180\pi$
$53$	4.44799	0.610979	0.305489	+	0.952196i	$0.401180\pi$
$54$	4.95488	0.674274
$55$	0	0
$56$	0.0208591	0.00278741
$57$	−16.9979	−2.25143
$58$	−18.3261	−2.40634
$59$	−3.39440	−0.441913	−0.220956	−	0.975284i	$-0.570918\pi$
$59$	−3.39440	−0.441913	−0.220956	+	0.975284i	$0.570918\pi$
$60$	13.9785	1.80462
$61$	14.1754	1.81497	0.907486	−	0.420081i	$-0.137998\pi$
$61$	14.1754	1.81497	0.907486	+	0.420081i	$0.137998\pi$
$62$	13.9197	1.76781
$63$	0.765323	0.0964216
$64$	−8.20324	−1.02540
$65$	−4.99330	−0.619342
$66$	0	0
$67$	5.40601	0.660449	0.330224	−	0.943902i	$-0.392876\pi$
$67$	5.40601	0.660449	0.330224	+	0.943902i	$0.392876\pi$
$68$	1.76384	0.213897
$69$	−12.0644	−1.45238
$70$	−2.54775	−0.304515
$71$	−1.81572	−0.215486	−0.107743	−	0.994179i	$-0.534362\pi$
$71$	−1.81572	−0.215486	−0.107743	+	0.994179i	$0.534362\pi$
$72$	0.0965723	0.0113812
$73$	5.78453	0.677028	0.338514	−	0.940961i	$-0.390076\pi$
$73$	5.78453	0.677028	0.338514	+	0.940961i	$0.390076\pi$
$74$	15.8547	1.84308
$75$	−10.5053	−1.21305
$76$	15.5822	1.78740
$77$	0	0
$78$	−7.08787	−0.802543
$79$	−15.7267	−1.76939	−0.884696	−	0.466168i	$-0.845634\pi$
$79$	−15.7267	−1.76939	−0.884696	+	0.466168i	$0.845634\pi$
$80$	12.3310	1.37865
$81$	−11.1038	−1.23376
$82$	−5.81077	−0.641692
$83$	−9.70962	−1.06577	−0.532885	−	0.846188i	$-0.678892\pi$
$83$	−9.70962	−1.06577	−0.532885	+	0.846188i	$0.678892\pi$
$84$	−1.81972	−0.198548
$85$	−2.71964	−0.294986
$86$	10.6209	1.14528
$87$	20.1823	2.16377
$88$	0	0
$89$	−1.07269	−0.113705	−0.0568525	−	0.998383i	$-0.518106\pi$
$89$	−1.07269	−0.113705	−0.0568525	+	0.998383i	$0.518106\pi$
$90$	−11.7954	−1.24335
$91$	0.650027	0.0681413
$92$	11.0596	1.15304
$93$	−15.3296	−1.58961
$94$	8.90833	0.918824
$95$	−24.0259	−2.46501
$96$	17.7303	1.80960
$97$	9.44446	0.958940	0.479470	−	0.877558i	$-0.340829\pi$
$97$	9.44446	0.958940	0.479470	+	0.877558i	$0.340829\pi$
$98$	−13.7130	−1.38522
$99$	0	0
$100$	9.63033	0.963033
$101$	12.2477	1.21869	0.609346	−	0.792904i	$-0.291432\pi$
$101$	12.2477	1.21869	0.609346	+	0.792904i	$0.291432\pi$
$102$	−3.86046	−0.382243
$103$	−16.4415	−1.62003	−0.810013	−	0.586412i	$-0.800540\pi$
$103$	−16.4415	−1.62003	−0.810013	+	0.586412i	$0.800540\pi$
$104$	0.0820237	0.00804309
$105$	2.80581	0.273819
$106$	8.92437	0.866812
$107$	12.8539	1.24263	0.621314	−	0.783561i	$-0.286599\pi$
$107$	12.8539	1.24263	0.621314	+	0.783561i	$0.286599\pi$
$108$	5.00227	0.481343
$109$	10.1598	0.973130	0.486565	−	0.873644i	$-0.338250\pi$
$109$	10.1598	0.973130	0.486565	+	0.873644i	$0.338250\pi$
$110$	0	0
$111$	−17.4606	−1.65729
$112$	−1.60525	−0.151682
$113$	12.0980	1.13808	0.569040	−	0.822310i	$-0.307315\pi$
$113$	12.0980	1.13808	0.569040	+	0.822310i	$0.307315\pi$
$114$	−34.1043	−3.19416
$115$	−17.0526	−1.59016
$116$	−18.5014	−1.71781
$117$	3.00946	0.278225
$118$	−6.81046	−0.626954
$119$	0.354042	0.0324550
$120$	0.354051	0.0323203
$121$	0	0
$122$	28.4413	2.57495
$123$	6.39933	0.577008
$124$	14.0528	1.26198
$125$	0.767128	0.0686140
$126$	1.53553	0.136796
$127$	−15.4030	−1.36680	−0.683400	−	0.730044i	$-0.739499\pi$
$127$	−15.4030	−1.36680	−0.683400	+	0.730044i	$0.739499\pi$
$128$	−0.410399	−0.0362745
$129$	−11.6967	−1.02984
$130$	−10.0185	−0.878677
$131$	10.1481	0.886647	0.443324	−	0.896362i	$-0.353799\pi$
$131$	10.1481	0.886647	0.443324	+	0.896362i	$0.353799\pi$
$132$	0	0
$133$	3.12769	0.271205
$134$	10.8465	0.936996
$135$	−7.71293	−0.663823
$136$	0.0446748	0.00383083
$137$	−8.01446	−0.684721	−0.342361	−	0.939569i	$-0.611226\pi$
$137$	−8.01446	−0.684721	−0.342361	+	0.939569i	$0.611226\pi$
$138$	−24.2058	−2.06053
$139$	15.7916	1.33943	0.669713	−	0.742620i	$-0.266417\pi$
$139$	15.7916	1.33943	0.669713	+	0.742620i	$0.266417\pi$
$140$	−2.57212	−0.217384
$141$	−9.81064	−0.826205
$142$	−3.64303	−0.305716
$143$	0	0
$144$	−7.43191	−0.619326
$145$	28.5270	2.36904
$146$	11.6060	0.960518
$147$	15.1020	1.24559
$148$	16.0064	1.31571
$149$	−6.41132	−0.525236	−0.262618	−	0.964900i	$-0.584586\pi$
$149$	−6.41132	−0.525236	−0.262618	+	0.964900i	$0.584586\pi$
$150$	−21.0776	−1.72098
$151$	1.99522	0.162369	0.0811845	−	0.996699i	$-0.474130\pi$
$151$	1.99522	0.162369	0.0811845	+	0.996699i	$0.474130\pi$
$152$	0.394668	0.0320118
$153$	1.63912	0.132515
$154$	0	0
$155$	−21.6679	−1.74041
$156$	−7.15566	−0.572911
$157$	−3.05648	−0.243934	−0.121967	−	0.992534i	$-0.538920\pi$
$157$	−3.05648	−0.243934	−0.121967	+	0.992534i	$0.538920\pi$
$158$	−31.5538	−2.51028
$159$	−9.82830	−0.779435
$160$	25.0612	1.98126
$161$	2.21990	0.174953
$162$	−22.2785	−1.75036
$163$	−10.4582	−0.819148	−0.409574	−	0.912277i	$-0.634323\pi$
$163$	−10.4582	−0.819148	−0.409574	+	0.912277i	$0.634323\pi$
$164$	−5.86634	−0.458084
$165$	0	0
$166$	−19.4812	−1.51204
$167$	−2.90239	−0.224594	−0.112297	−	0.993675i	$-0.535821\pi$
$167$	−2.90239	−0.224594	−0.112297	+	0.993675i	$0.535821\pi$
$168$	−0.0460904	−0.00355595
$169$	−10.4439	−0.803378
$170$	−5.45663	−0.418504
$171$	14.4804	1.10735
$172$	10.7225	0.817582
$173$	−5.93003	−0.450851	−0.225426	−	0.974260i	$-0.572377\pi$
$173$	−5.93003	−0.450851	−0.225426	+	0.974260i	$0.572377\pi$
$174$	40.4935	3.06980
$175$	1.93302	0.146123
$176$	0	0
$177$	7.50028	0.563755
$178$	−2.15223	−0.161316
$179$	7.49585	0.560266	0.280133	−	0.959961i	$-0.409621\pi$
$179$	7.49585	0.560266	0.280133	+	0.959961i	$0.409621\pi$
$180$	−11.9083	−0.887589
$181$	22.4980	1.67226	0.836132	−	0.548529i	$-0.184812\pi$
$181$	22.4980	1.67226	0.836132	+	0.548529i	$0.184812\pi$
$182$	1.30420	0.0966739
$183$	−31.3220	−2.31539
$184$	0.280119	0.0206506
$185$	−24.6800	−1.81451
$186$	−30.7571	−2.25522
$187$	0	0
$188$	8.99353	0.655920
$189$	1.00407	0.0730352
$190$	−48.2052	−3.49717
$191$	−10.4593	−0.756810	−0.378405	−	0.925640i	$-0.623527\pi$
$191$	−10.4593	−0.756810	−0.378405	+	0.925640i	$0.623527\pi$
$192$	18.1259	1.30813
$193$	−7.13825	−0.513823	−0.256911	−	0.966435i	$-0.582705\pi$
$193$	−7.13825	−0.513823	−0.256911	+	0.966435i	$0.582705\pi$
$194$	18.9492	1.36047
$195$	11.0332	0.790105
$196$	−13.8442	−0.988868
$197$	2.45154	0.174665	0.0873324	−	0.996179i	$-0.472166\pi$
$197$	2.45154	0.174665	0.0873324	+	0.996179i	$0.472166\pi$
$198$	0	0
$199$	24.7018	1.75107	0.875533	−	0.483158i	$-0.160510\pi$
$199$	24.7018	1.75107	0.875533	+	0.483158i	$0.160510\pi$
$200$	0.243919	0.0172477
$201$	−11.9451	−0.842545
$202$	24.5736	1.72899
$203$	−3.71365	−0.260647
$204$	−3.89738	−0.272871
$205$	9.04523	0.631746
$206$	−32.9879	−2.29837
$207$	10.2776	0.714342
$208$	−6.31229	−0.437679
$209$	0	0
$210$	5.62953	0.388474
$211$	11.1941	0.770633	0.385316	−	0.922785i	$-0.374092\pi$
$211$	11.1941	0.770633	0.385316	+	0.922785i	$0.374092\pi$
$212$	9.00972	0.618790
$213$	4.01202	0.274899
$214$	25.7897	1.76295
$215$	−16.5329	−1.12753
$216$	0.126698	0.00862074
$217$	2.82072	0.191483
$218$	20.3844	1.38061
$219$	−12.7815	−0.863695
$220$	0	0
$221$	1.39219	0.0936488
$222$	−35.0327	−2.35124
$223$	−12.6036	−0.844000	−0.422000	−	0.906596i	$-0.638672\pi$
$223$	−12.6036	−0.844000	−0.422000	+	0.906596i	$0.638672\pi$
$224$	−3.26247	−0.217983
$225$	8.94941	0.596627
$226$	24.2731	1.61462
$227$	−2.93048	−0.194503	−0.0972513	−	0.995260i	$-0.531005\pi$
$227$	−2.93048	−0.194503	−0.0972513	+	0.995260i	$0.531005\pi$
$228$	−34.4304	−2.28021
$229$	−0.306170	−0.0202323	−0.0101161	−	0.999949i	$-0.503220\pi$
$229$	−0.306170	−0.0202323	−0.0101161	+	0.999949i	$0.503220\pi$
$230$	−34.2140	−2.25601
$231$	0	0
$232$	−0.468607	−0.0307655
$233$	0.541467	0.0354727	0.0177363	−	0.999843i	$-0.494354\pi$
$233$	0.541467	0.0354727	0.0177363	+	0.999843i	$0.494354\pi$
$234$	6.03813	0.394725
$235$	−13.8670	−0.904583
$236$	−6.87559	−0.447563
$237$	34.7498	2.25724
$238$	0.710344	0.0460447
$239$	18.7371	1.21200	0.606002	−	0.795463i	$-0.292772\pi$
$239$	18.7371	1.21200	0.606002	+	0.795463i	$0.292772\pi$
$240$	−27.2467	−1.75877
$241$	11.4587	0.738120	0.369060	−	0.929405i	$-0.379680\pi$
$241$	11.4587	0.738120	0.369060	+	0.929405i	$0.379680\pi$
$242$	0	0
$243$	17.1263	1.09866
$244$	28.7133	1.83818
$245$	21.3461	1.36375
$246$	12.8395	0.818617
$247$	12.2989	0.782563
$248$	0.355933	0.0226018
$249$	21.4544	1.35962
$250$	1.53915	0.0973445
$251$	−2.01853	−0.127409	−0.0637043	−	0.997969i	$-0.520291\pi$
$251$	−2.01853	−0.127409	−0.0637043	+	0.997969i	$0.520291\pi$
$252$	1.55022	0.0976544
$253$	0	0
$254$	−30.9044	−1.93911
$255$	6.00932	0.376318
$256$	15.5831	0.973941
$257$	−1.96540	−0.122599	−0.0612993	−	0.998119i	$-0.519524\pi$
$257$	−1.96540	−0.122599	−0.0612993	+	0.998119i	$0.519524\pi$
$258$	−23.4680	−1.46105
$259$	3.21284	0.199636
$260$	−10.1143	−0.627261
$261$	−17.1932	−1.06423
$262$	20.3611	1.25791
$263$	−12.1494	−0.749163	−0.374581	−	0.927194i	$-0.622214\pi$
$263$	−12.1494	−0.749163	−0.374581	+	0.927194i	$0.622214\pi$
$264$	0	0
$265$	−13.8920	−0.853377
$266$	6.27535	0.384766
$267$	2.37022	0.145055
$268$	10.9502	0.668893
$269$	1.70893	0.104195	0.0520977	−	0.998642i	$-0.483409\pi$
$269$	1.70893	0.104195	0.0520977	+	0.998642i	$0.483409\pi$
$270$	−15.4751	−0.941784
$271$	9.57352	0.581550	0.290775	−	0.956791i	$-0.406087\pi$
$271$	9.57352	0.581550	0.290775	+	0.956791i	$0.406087\pi$
$272$	−3.43804	−0.208462
$273$	−1.43630	−0.0869290
$274$	−16.0801	−0.971432
$275$	0	0
$276$	−24.4373	−1.47095
$277$	1.77483	0.106639	0.0533197	−	0.998577i	$-0.483020\pi$
$277$	1.77483	0.106639	0.0533197	+	0.998577i	$0.483020\pi$
$278$	31.6840	1.90028
$279$	13.0592	0.781836
$280$	−0.0651471	−0.00389329
$281$	24.8616	1.48312	0.741559	−	0.670887i	$-0.234086\pi$
$281$	24.8616	1.48312	0.741559	+	0.670887i	$0.234086\pi$
$282$	−19.6839	−1.17216
$283$	−15.8723	−0.943512	−0.471756	−	0.881729i	$-0.656380\pi$
$283$	−15.8723	−0.943512	−0.471756	+	0.881729i	$0.656380\pi$
$284$	−3.67787	−0.218241
$285$	53.0878	3.14465
$286$	0	0
$287$	−1.17751	−0.0695061
$288$	−15.1044	−0.890035
$289$	−16.2417	−0.955396
$290$	57.2361	3.36102
$291$	−20.8685	−1.22333
$292$	11.7170	0.685684
$293$	23.7771	1.38907	0.694536	−	0.719458i	$-0.255610\pi$
$293$	23.7771	1.38907	0.694536	+	0.719458i	$0.255610\pi$
$294$	30.3003	1.76715
$295$	10.6014	0.617236
$296$	0.405412	0.0235641
$297$	0	0
$298$	−12.8636	−0.745166
$299$	8.72927	0.504827
$300$	−21.2792	−1.22856
$301$	2.15225	0.124053
$302$	4.00318	0.230357
$303$	−27.0626	−1.55471
$304$	−30.3725	−1.74198
$305$	−44.2726	−2.53504
$306$	3.28871	0.188003
$307$	−19.8456	−1.13265	−0.566324	−	0.824183i	$-0.691635\pi$
$307$	−19.8456	−1.13265	−0.566324	+	0.824183i	$0.691635\pi$
$308$	0	0
$309$	36.3291	2.06669
$310$	−43.4741	−2.46916
$311$	19.5861	1.11063	0.555313	−	0.831641i	$-0.312598\pi$
$311$	19.5861	1.11063	0.555313	+	0.831641i	$0.312598\pi$
$312$	−0.181240	−0.0102607
$313$	−14.6641	−0.828865	−0.414432	−	0.910080i	$-0.636020\pi$
$313$	−14.6641	−0.828865	−0.414432	+	0.910080i	$0.636020\pi$
$314$	−6.13247	−0.346075
$315$	−2.39026	−0.134676
$316$	−31.8555	−1.79201
$317$	−12.7428	−0.715706	−0.357853	−	0.933778i	$-0.616491\pi$
$317$	−12.7428	−0.715706	−0.357853	+	0.933778i	$0.616491\pi$
$318$	−19.7193	−1.10581
$319$	0	0
$320$	25.6204	1.43222
$321$	−28.4019	−1.58524
$322$	4.45398	0.248211
$323$	6.69871	0.372726
$324$	−22.4915	−1.24953
$325$	7.60118	0.421638
$326$	−20.9831	−1.16215
$327$	−22.4491	−1.24144
$328$	−0.148584	−0.00820417
$329$	1.80520	0.0995242
$330$	0	0
$331$	−0.335176	−0.0184230	−0.00921148	−	0.999958i	$-0.502932\pi$
$331$	−0.335176	−0.0184230	−0.00921148	+	0.999958i	$0.502932\pi$
$332$	−19.6675	−1.07940
$333$	14.8746	0.815125
$334$	−5.82331	−0.318637
$335$	−16.8840	−0.922474
$336$	3.54697	0.193503
$337$	−0.812254	−0.0442463	−0.0221231	−	0.999755i	$-0.507043\pi$
$337$	−0.812254	−0.0442463	−0.0221231	+	0.999755i	$0.507043\pi$
$338$	−20.9545	−1.13977
$339$	−26.7317	−1.45187
$340$	−5.50881	−0.298757
$341$	0	0
$342$	29.0533	1.57102
$343$	−5.62488	−0.303715
$344$	0.271581	0.0146427
$345$	37.6795	2.02860
$346$	−11.8979	−0.639635
$347$	−8.90224	−0.477897	−0.238949	−	0.971032i	$-0.576803\pi$
$347$	−8.90224	−0.477897	−0.238949	+	0.971032i	$0.576803\pi$
$348$	40.8807	2.19144
$349$	10.8292	0.579676	0.289838	−	0.957076i	$-0.406399\pi$
$349$	10.8292	0.579676	0.289838	+	0.957076i	$0.406399\pi$
$350$	3.87839	0.207309
$351$	3.94827	0.210743
$352$	0	0
$353$	−18.8915	−1.00549	−0.502747	−	0.864433i	$-0.667677\pi$
$353$	−18.8915	−1.00549	−0.502747	+	0.864433i	$0.667677\pi$
$354$	15.0484	0.799815
$355$	5.67086	0.300978
$356$	−2.17281	−0.115159
$357$	−0.782293	−0.0414033
$358$	15.0395	0.794864
$359$	−17.0952	−0.902251	−0.451126	−	0.892460i	$-0.648977\pi$
$359$	−17.0952	−0.902251	−0.451126	+	0.892460i	$0.648977\pi$
$360$	−0.301615	−0.0158965
$361$	40.1781	2.11464
$362$	45.1396	2.37248
$363$	0	0
$364$	1.31668	0.0690125
$365$	−18.0662	−0.945631
$366$	−62.8439	−3.28490
$367$	33.3076	1.73864	0.869322	−	0.494247i	$-0.164556\pi$
$367$	33.3076	1.73864	0.869322	+	0.494247i	$0.164556\pi$
$368$	−21.5571	−1.12374
$369$	−5.45156	−0.283797
$370$	−49.5175	−2.57429
$371$	1.80845	0.0938903
$372$	−31.0512	−1.60993
$373$	22.9460	1.18810	0.594049	−	0.804429i	$-0.297528\pi$
$373$	22.9460	1.18810	0.594049	+	0.804429i	$0.297528\pi$
$374$	0	0
$375$	−1.69505	−0.0875319
$376$	0.227790	0.0117474
$377$	−14.6031	−0.752096
$378$	2.01455	0.103617
$379$	24.3737	1.25199	0.625997	−	0.779826i	$-0.284692\pi$
$379$	24.3737	1.25199	0.625997	+	0.779826i	$0.284692\pi$
$380$	−48.6662	−2.49652
$381$	34.0346	1.74365
$382$	−20.9854	−1.07371
$383$	11.5191	0.588599	0.294299	−	0.955713i	$-0.404914\pi$
$383$	11.5191	0.588599	0.294299	+	0.955713i	$0.404914\pi$
$384$	0.906820	0.0462759
$385$	0	0
$386$	−14.3221	−0.728974
$387$	9.96435	0.506516
$388$	19.1304	0.971200
$389$	−5.39102	−0.273336	−0.136668	−	0.990617i	$-0.543639\pi$
$389$	−5.39102	−0.273336	−0.136668	+	0.990617i	$0.543639\pi$
$390$	22.1368	1.12094
$391$	4.75446	0.240444
$392$	−0.350648	−0.0177104
$393$	−22.4234	−1.13111
$394$	4.91872	0.247802
$395$	49.1176	2.47138
$396$	0	0
$397$	−17.8854	−0.897644	−0.448822	−	0.893621i	$-0.648156\pi$
$397$	−17.8854	−0.897644	−0.448822	+	0.893621i	$0.648156\pi$
$398$	49.5613	2.48428
$399$	−6.91097	−0.345981
$400$	−18.7713	−0.938563
$401$	25.7031	1.28355	0.641776	−	0.766892i	$-0.278198\pi$
$401$	25.7031	1.28355	0.641776	+	0.766892i	$0.278198\pi$
$402$	−23.9665	−1.19534
$403$	11.0919	0.552525
$404$	24.8086	1.23427
$405$	34.6794	1.72323
$406$	−7.45099	−0.369787
$407$	0	0
$408$	−0.0987137	−0.00488706
$409$	6.70143	0.331364	0.165682	−	0.986179i	$-0.447017\pi$
$409$	6.70143	0.331364	0.165682	+	0.986179i	$0.447017\pi$
$410$	18.1482	0.896275
$411$	17.7088	0.873510
$412$	−33.3033	−1.64074
$413$	−1.38009	−0.0679096
$414$	20.6208	1.01346
$415$	30.3251	1.48860
$416$	−12.8289	−0.628990
$417$	−34.8932	−1.70873
$418$	0	0
$419$	−0.602761	−0.0294468	−0.0147234	−	0.999892i	$-0.504687\pi$
$419$	−0.602761	−0.0294468	−0.0147234	+	0.999892i	$0.504687\pi$
$420$	5.68336	0.277320
$421$	−35.9824	−1.75368	−0.876838	−	0.480787i	$-0.840351\pi$
$421$	−35.9824	−1.75368	−0.876838	+	0.480787i	$0.840351\pi$
$422$	22.4596	1.09332
$423$	8.35764	0.406362
$424$	0.228200	0.0110824
$425$	4.14004	0.200822
$426$	8.04966	0.390007
$427$	5.76340	0.278910
$428$	26.0364	1.25852
$429$	0	0
$430$	−33.1712	−1.59966
$431$	−25.0443	−1.20634	−0.603172	−	0.797611i	$-0.706097\pi$
$431$	−25.0443	−1.20634	−0.603172	+	0.797611i	$0.706097\pi$
$432$	−9.75033	−0.469113
$433$	4.64149	0.223056	0.111528	−	0.993761i	$-0.464426\pi$
$433$	4.64149	0.223056	0.111528	+	0.993761i	$0.464426\pi$
$434$	5.65945	0.271662
$435$	−63.0334	−3.02222
$436$	20.5793	0.985571
$437$	42.0021	2.00923
$438$	−25.6446	−1.22535
$439$	−7.86376	−0.375317	−0.187658	−	0.982234i	$-0.560090\pi$
$439$	−7.86376	−0.375317	−0.187658	+	0.982234i	$0.560090\pi$
$440$	0	0
$441$	−12.8653	−0.612633
$442$	2.79327	0.132862
$443$	−18.1876	−0.864117	−0.432058	−	0.901846i	$-0.642213\pi$
$443$	−18.1876	−0.864117	−0.432058	+	0.901846i	$0.642213\pi$
$444$	−35.3677	−1.67848
$445$	3.35023	0.158816
$446$	−25.2877	−1.19741
$447$	14.1665	0.670051
$448$	−3.33526	−0.157576
$449$	−19.4471	−0.917766	−0.458883	−	0.888497i	$-0.651750\pi$
$449$	−19.4471	−0.917766	−0.458883	+	0.888497i	$0.651750\pi$
$450$	17.9559	0.846451
$451$	0	0
$452$	24.5053	1.15263
$453$	−4.40865	−0.207137
$454$	−5.87966	−0.275946
$455$	−2.03016	−0.0951755
$456$	−0.872061	−0.0408380
$457$	39.8414	1.86370	0.931851	−	0.362840i	$-0.118193\pi$
$457$	39.8414	1.86370	0.931851	+	0.362840i	$0.118193\pi$
$458$	−0.614293	−0.0287040
$459$	2.15046	0.100375
$460$	−34.5412	−1.61049
$461$	29.8434	1.38995	0.694973	−	0.719036i	$-0.255417\pi$
$461$	29.8434	1.38995	0.694973	+	0.719036i	$0.255417\pi$
$462$	0	0
$463$	13.4996	0.627378	0.313689	−	0.949526i	$-0.398435\pi$
$463$	13.4996	0.627378	0.313689	+	0.949526i	$0.398435\pi$
$464$	36.0625	1.67416
$465$	47.8775	2.22026
$466$	1.08639	0.0503260
$467$	38.5271	1.78282	0.891410	−	0.453198i	$-0.149717\pi$
$467$	38.5271	1.78282	0.891410	+	0.453198i	$0.149717\pi$
$468$	6.09587	0.281782
$469$	2.19796	0.101493
$470$	−27.8225	−1.28336
$471$	6.75361	0.311190
$472$	−0.174146	−0.00801574
$473$	0	0
$474$	69.7214	3.20241
$475$	36.5741	1.67814
$476$	0.717137	0.0328699
$477$	8.37268	0.383359
$478$	37.5939	1.71950
$479$	26.7572	1.22257	0.611285	−	0.791411i	$-0.290653\pi$
$479$	26.7572	1.22257	0.611285	+	0.791411i	$0.290653\pi$
$480$	−55.3754	−2.52753
$481$	12.6338	0.576050
$482$	22.9906	1.04719
$483$	−4.90511	−0.223190
$484$	0	0
$485$	−29.4969	−1.33939
$486$	34.3620	1.55869
$487$	0.881973	0.0399660	0.0199830	−	0.999800i	$-0.493639\pi$
$487$	0.881973	0.0399660	0.0199830	+	0.999800i	$0.493639\pi$
$488$	0.727255	0.0329213
$489$	23.1084	1.04500
$490$	42.8285	1.93479
$491$	−14.6021	−0.658985	−0.329492	−	0.944158i	$-0.606878\pi$
$491$	−14.6021	−0.658985	−0.329492	+	0.944158i	$0.606878\pi$
$492$	12.9623	0.584385
$493$	−7.95367	−0.358215
$494$	24.6764	1.11024
$495$	0	0
$496$	−27.3915	−1.22992
$497$	−0.738232	−0.0331142
$498$	43.0458	1.92893
$499$	−20.1804	−0.903399	−0.451699	−	0.892170i	$-0.649182\pi$
$499$	−20.1804	−0.903399	−0.451699	+	0.892170i	$0.649182\pi$
$500$	1.55387	0.0694912
$501$	6.41314	0.286518
$502$	−4.04995	−0.180758
$503$	−1.25691	−0.0560427	−0.0280214	−	0.999607i	$-0.508921\pi$
$503$	−1.25691	−0.0560427	−0.0280214	+	0.999607i	$0.508921\pi$
$504$	0.0392642	0.00174897
$505$	−38.2520	−1.70219
$506$	0	0
$507$	23.0769	1.02488
$508$	−31.2000	−1.38427
$509$	24.4025	1.08162	0.540812	−	0.841144i	$-0.318117\pi$
$509$	24.4025	1.08162	0.540812	+	0.841144i	$0.318117\pi$
$510$	12.0570	0.533893
$511$	2.35186	0.104040
$512$	32.0864	1.41803
$513$	18.9977	0.838767
$514$	−3.94335	−0.173934
$515$	51.3500	2.26275
$516$	−23.6924	−1.04300
$517$	0	0
$518$	6.44618	0.283229
$519$	13.1030	0.575158
$520$	−0.256176	−0.0112341
$521$	−19.7666	−0.865992	−0.432996	−	0.901396i	$-0.642544\pi$
$521$	−19.7666	−0.865992	−0.432996	+	0.901396i	$0.642544\pi$
$522$	−34.4962	−1.50986
$523$	−2.73269	−0.119492	−0.0597460	−	0.998214i	$-0.519029\pi$
$523$	−2.73269	−0.119492	−0.0597460	+	0.998214i	$0.519029\pi$
$524$	20.5558	0.897983
$525$	−4.27122	−0.186411
$526$	−24.3763	−1.06286
$527$	6.04126	0.263162
$528$	0	0
$529$	6.81132	0.296144
$530$	−27.8726	−1.21071
$531$	−6.38945	−0.277279
$532$	6.33536	0.274673
$533$	−4.63028	−0.200560
$534$	4.75557	0.205794
$535$	−40.1452	−1.73563
$536$	0.277350	0.0119797
$537$	−16.5629	−0.714740
$538$	3.42877	0.147825
$539$	0	0
$540$	−15.6231	−0.672310
$541$	−2.09370	−0.0900151	−0.0450076	−	0.998987i	$-0.514331\pi$
$541$	−2.09370	−0.0900151	−0.0450076	+	0.998987i	$0.514331\pi$
$542$	19.2082	0.825061
$543$	−49.7117	−2.13333
$544$	−6.98737	−0.299581
$545$	−31.7310	−1.35921
$546$	−2.88177	−0.123328
$547$	−42.0902	−1.79965	−0.899824	−	0.436252i	$-0.856305\pi$
$547$	−42.0902	−1.79965	−0.899824	+	0.436252i	$0.856305\pi$
$548$	−16.2338	−0.693475
$549$	26.6831	1.13881
$550$	0	0
$551$	−70.2647	−2.99338
$552$	−0.618952	−0.0263443
$553$	−6.39413	−0.271906
$554$	3.56100	0.151292
$555$	54.5330	2.31480
$556$	31.9870	1.35655
$557$	−2.95969	−0.125406	−0.0627030	−	0.998032i	$-0.519972\pi$
$557$	−2.95969	−0.125406	−0.0627030	+	0.998032i	$0.519972\pi$
$558$	26.2018	1.10921
$559$	8.46322	0.357956
$560$	5.01353	0.211860
$561$	0	0
$562$	49.8819	2.10414
$563$	−3.18105	−0.134065	−0.0670327	−	0.997751i	$-0.521353\pi$
$563$	−3.18105	−0.134065	−0.0670327	+	0.997751i	$0.521353\pi$
$564$	−19.8721	−0.836768
$565$	−37.7843	−1.58960
$566$	−31.8460	−1.33859
$567$	−4.51456	−0.189594
$568$	−0.0931539	−0.00390865
$569$	29.8147	1.24990	0.624950	−	0.780665i	$-0.285119\pi$
$569$	29.8147	1.24990	0.624950	+	0.780665i	$0.285119\pi$
$570$	106.514	4.46140
$571$	4.76387	0.199362	0.0996808	−	0.995019i	$-0.468218\pi$
$571$	4.76387	0.199362	0.0996808	+	0.995019i	$0.468218\pi$
$572$	0	0
$573$	23.1110	0.965474
$574$	−2.36253	−0.0986101
$575$	25.9588	1.08256
$576$	−15.4414	−0.643391
$577$	−21.4605	−0.893412	−0.446706	−	0.894681i	$-0.647403\pi$
$577$	−21.4605	−0.893412	−0.446706	+	0.894681i	$0.647403\pi$
$578$	−32.5871	−1.35545
$579$	15.7727	0.655492
$580$	57.7835	2.39933
$581$	−3.94772	−0.163779
$582$	−41.8702	−1.73558
$583$	0	0
$584$	0.296770	0.0122804
$585$	−9.39914	−0.388607
$586$	47.7059	1.97071
$587$	−27.4088	−1.13128	−0.565642	−	0.824651i	$-0.691371\pi$
$587$	−27.4088	−1.13128	−0.565642	+	0.824651i	$0.691371\pi$
$588$	30.5901	1.26151
$589$	53.3700	2.19907
$590$	21.2704	0.875690
$591$	−5.41693	−0.222823
$592$	−31.1993	−1.28228
$593$	−47.2222	−1.93918	−0.969592	−	0.244725i	$-0.921302\pi$
$593$	−47.2222	−1.93918	−0.969592	+	0.244725i	$0.921302\pi$
$594$	0	0
$595$	−1.10574	−0.0453311
$596$	−12.9866	−0.531951
$597$	−54.5813	−2.23386
$598$	17.5143	0.716212
$599$	12.8208	0.523843	0.261922	−	0.965089i	$-0.415644\pi$
$599$	12.8208	0.523843	0.261922	+	0.965089i	$0.415644\pi$
$600$	−0.538964	−0.0220031
$601$	9.51727	0.388218	0.194109	−	0.980980i	$-0.437819\pi$
$601$	9.51727	0.388218	0.194109	+	0.980980i	$0.437819\pi$
$602$	4.31823	0.175998
$603$	10.1760	0.414399
$604$	4.04146	0.164445
$605$	0	0
$606$	−54.2979	−2.20570
$607$	−41.3334	−1.67767	−0.838836	−	0.544384i	$-0.816763\pi$
$607$	−41.3334	−1.67767	−0.838836	+	0.544384i	$0.816763\pi$
$608$	−61.7281	−2.50341
$609$	8.20569	0.332511
$610$	−88.8277	−3.59653
$611$	7.09856	0.287177
$612$	3.32016	0.134210
$613$	14.1112	0.569947	0.284974	−	0.958535i	$-0.408015\pi$
$613$	14.1112	0.569947	0.284974	+	0.958535i	$0.408015\pi$
$614$	−39.8179	−1.60692
$615$	−19.9864	−0.805929
$616$	0	0
$617$	44.1279	1.77652	0.888262	−	0.459338i	$-0.151913\pi$
$617$	44.1279	1.77652	0.888262	+	0.459338i	$0.151913\pi$
$618$	72.8901	2.93207
$619$	−16.7648	−0.673832	−0.336916	−	0.941535i	$-0.609384\pi$
$619$	−16.7648	−0.673832	−0.336916	+	0.941535i	$0.609384\pi$
$620$	−43.8898	−1.76266
$621$	13.4837	0.541084
$622$	39.2972	1.57568
$623$	−0.436132	−0.0174733
$624$	13.9477	0.558354
$625$	−26.1678	−1.04671
$626$	−29.4218	−1.17593
$627$	0	0
$628$	−6.19111	−0.247052
$629$	6.88108	0.274366
$630$	−4.79577	−0.191068
$631$	−2.51127	−0.0999723	−0.0499861	−	0.998750i	$-0.515918\pi$
$631$	−2.51127	−0.0999723	−0.0499861	+	0.998750i	$0.515918\pi$
$632$	−0.806844	−0.0320945
$633$	−24.7345	−0.983108
$634$	−25.5669	−1.01539
$635$	48.1068	1.90906
$636$	−19.9079	−0.789400
$637$	−10.9271	−0.432949
$638$	0	0
$639$	−3.41783	−0.135207
$640$	1.28176	0.0506659
$641$	12.3479	0.487711	0.243855	−	0.969812i	$-0.421588\pi$
$641$	12.3479	0.487711	0.243855	+	0.969812i	$0.421588\pi$
$642$	−56.9851	−2.24902
$643$	−21.5404	−0.849470	−0.424735	−	0.905318i	$-0.639633\pi$
$643$	−21.5404	−0.849470	−0.424735	+	0.905318i	$0.639633\pi$
$644$	4.49657	0.177190
$645$	36.5311	1.43841
$646$	13.4402	0.528797
$647$	−23.5227	−0.924773	−0.462386	−	0.886679i	$-0.653007\pi$
$647$	−23.5227	−0.924773	−0.462386	+	0.886679i	$0.653007\pi$
$648$	−0.569671	−0.0223788
$649$	0	0
$650$	15.2509	0.598189
$651$	−6.23268	−0.244278
$652$	−21.1838	−0.829621
$653$	−22.5920	−0.884094	−0.442047	−	0.896992i	$-0.645748\pi$
$653$	−22.5920	−0.884094	−0.442047	+	0.896992i	$0.645748\pi$
$654$	−45.0414	−1.76126
$655$	−31.6947	−1.23841
$656$	11.4346	0.446445
$657$	10.8885	0.424802
$658$	3.62193	0.141198
$659$	−3.76231	−0.146559	−0.0732794	−	0.997311i	$-0.523346\pi$
$659$	−3.76231	−0.146559	−0.0732794	+	0.997311i	$0.523346\pi$
$660$	0	0
$661$	−28.4373	−1.10608	−0.553041	−	0.833154i	$-0.686533\pi$
$661$	−28.4373	−1.10608	−0.553041	+	0.833154i	$0.686533\pi$
$662$	−0.672492	−0.0261371
$663$	−3.07619	−0.119469
$664$	−0.498143	−0.0193317
$665$	−9.76841	−0.378803
$666$	29.8442	1.15644
$667$	−49.8709	−1.93101
$668$	−5.87900	−0.227465
$669$	27.8490	1.07670
$670$	−33.8758	−1.30874
$671$	0	0
$672$	7.20877	0.278084
$673$	15.5578	0.599710	0.299855	−	0.953985i	$-0.403062\pi$
$673$	15.5578	0.599710	0.299855	+	0.953985i	$0.403062\pi$
$674$	−1.62969	−0.0627734
$675$	11.7412	0.451920
$676$	−21.1549	−0.813650
$677$	−12.3763	−0.475661	−0.237830	−	0.971307i	$-0.576436\pi$
$677$	−12.3763	−0.475661	−0.237830	+	0.971307i	$0.576436\pi$
$678$	−53.6340	−2.05980
$679$	3.83991	0.147362
$680$	−0.139528	−0.00535067
$681$	6.47520	0.248130
$682$	0	0
$683$	−26.3400	−1.00787	−0.503936	−	0.863741i	$-0.668115\pi$
$683$	−26.3400	−1.00787	−0.503936	+	0.863741i	$0.668115\pi$
$684$	29.3311	1.12150
$685$	25.0307	0.956376
$686$	−11.2857	−0.430889
$687$	0.676514	0.0258106
$688$	−20.9001	−0.796808
$689$	7.11134	0.270920
$690$	75.5995	2.87802
$691$	−19.4579	−0.740213	−0.370107	−	0.928989i	$-0.620679\pi$
$691$	−19.4579	−0.740213	−0.370107	+	0.928989i	$0.620679\pi$
$692$	−12.0117	−0.456616
$693$	0	0
$694$	−17.8613	−0.678006
$695$	−49.3203	−1.87083
$696$	1.03544	0.0392481
$697$	−2.52192	−0.0955245
$698$	21.7276	0.822402
$699$	−1.19643	−0.0452530
$700$	3.91548	0.147991
$701$	−17.9455	−0.677793	−0.338896	−	0.940824i	$-0.610054\pi$
$701$	−17.9455	−0.677793	−0.338896	+	0.940824i	$0.610054\pi$
$702$	7.92174	0.298987
$703$	60.7891	2.29270
$704$	0	0
$705$	30.6406	1.15399
$706$	−37.9036	−1.42652
$707$	4.97965	0.187279
$708$	15.1923	0.570963
$709$	−23.8992	−0.897555	−0.448777	−	0.893644i	$-0.648140\pi$
$709$	−23.8992	−0.897555	−0.448777	+	0.893644i	$0.648140\pi$
$710$	11.3779	0.427005
$711$	−29.6032	−1.11021
$712$	−0.0550334	−0.00206246
$713$	37.8798	1.41861
$714$	−1.56958	−0.0587400
$715$	0	0
$716$	15.1834	0.567429
$717$	−41.4017	−1.54617
$718$	−34.2995	−1.28005
$719$	27.8427	1.03836	0.519178	−	0.854666i	$-0.326238\pi$
$719$	27.8427	1.03836	0.519178	+	0.854666i	$0.326238\pi$
$720$	23.2113	0.865036
$721$	−6.68474	−0.248953
$722$	80.6126	3.00009
$723$	−25.3192	−0.941632
$724$	45.5713	1.69364
$725$	−43.4261	−1.61280
$726$	0	0
$727$	21.3953	0.793509	0.396755	−	0.917925i	$-0.370136\pi$
$727$	21.3953	0.793509	0.396755	+	0.917925i	$0.370136\pi$
$728$	0.0333490	0.00123600
$729$	−4.53102	−0.167816
$730$	−36.2478	−1.34159
$731$	4.60956	0.170491
$732$	−63.4449	−2.34499
$733$	10.2227	0.377583	0.188792	−	0.982017i	$-0.439543\pi$
$733$	10.2227	0.377583	0.188792	+	0.982017i	$0.439543\pi$
$734$	66.8278	2.46666
$735$	−47.1665	−1.73976
$736$	−43.8120	−1.61493
$737$	0	0
$738$	−10.9379	−0.402630
$739$	22.9382	0.843794	0.421897	−	0.906644i	$-0.361364\pi$
$739$	22.9382	0.843794	0.421897	+	0.906644i	$0.361364\pi$
$740$	−49.9911	−1.83771
$741$	−27.1758	−0.998328
$742$	3.62845	0.133205
$743$	44.7185	1.64056	0.820281	−	0.571960i	$-0.193817\pi$
$743$	44.7185	1.64056	0.820281	+	0.571960i	$0.193817\pi$
$744$	−0.786472	−0.0288335
$745$	20.0238	0.733616
$746$	46.0384	1.68559
$747$	−18.2769	−0.668718
$748$	0	0
$749$	5.22609	0.190957
$750$	−3.40092	−0.124184
$751$	−0.689344	−0.0251545	−0.0125773	−	0.999921i	$-0.504004\pi$
$751$	−0.689344	−0.0251545	−0.0125773	+	0.999921i	$0.504004\pi$
$752$	−17.5300	−0.639254
$753$	4.46016	0.162537
$754$	−29.2993	−1.06702
$755$	−6.23148	−0.226787
$756$	2.03381	0.0739690
$757$	−40.1068	−1.45771	−0.728853	−	0.684670i	$-0.759946\pi$
$757$	−40.1068	−1.45771	−0.728853	+	0.684670i	$0.759946\pi$
$758$	48.9030	1.77624
$759$	0	0
$760$	−1.23263	−0.0447121
$761$	−23.5917	−0.855198	−0.427599	−	0.903968i	$-0.640641\pi$
$761$	−23.5917	−0.855198	−0.427599	+	0.903968i	$0.640641\pi$
$762$	68.2865	2.47376
$763$	4.13074	0.149543
$764$	−21.1861	−0.766486
$765$	−5.11931	−0.185089
$766$	23.1117	0.835061
$767$	−5.42688	−0.195953
$768$	−34.4324	−1.24247
$769$	−31.8809	−1.14965	−0.574827	−	0.818275i	$-0.694930\pi$
$769$	−31.8809	−1.14965	−0.574827	+	0.818275i	$0.694930\pi$
$770$	0	0
$771$	4.34277	0.156401
$772$	−14.4590	−0.520392
$773$	−8.95710	−0.322165	−0.161082	−	0.986941i	$-0.551498\pi$
$773$	−8.95710	−0.322165	−0.161082	+	0.986941i	$0.551498\pi$
$774$	19.9923	0.718608
$775$	32.9845	1.18484
$776$	0.484539	0.0173939
$777$	−7.09910	−0.254679
$778$	−10.8164	−0.387788
$779$	−22.2792	−0.798237
$780$	22.3485	0.800206
$781$	0	0
$782$	9.53927	0.341124
$783$	−22.5567	−0.806112
$784$	26.9848	0.963742
$785$	9.54599	0.340711
$786$	−44.9899	−1.60474
$787$	17.7766	0.633666	0.316833	−	0.948481i	$-0.397380\pi$
$787$	17.7766	0.633666	0.316833	+	0.948481i	$0.397380\pi$
$788$	4.96576	0.176898
$789$	26.8453	0.955718
$790$	98.5487	3.50621
$791$	4.91876	0.174891
$792$	0	0
$793$	22.6633	0.804796
$794$	−35.8850	−1.27351
$795$	30.6957	1.08867
$796$	50.0353	1.77345
$797$	36.7438	1.30153	0.650767	−	0.759278i	$-0.274448\pi$
$797$	36.7438	1.30153	0.650767	+	0.759278i	$0.274448\pi$
$798$	−13.8660	−0.490852
$799$	3.86628	0.136779
$800$	−38.1502	−1.34881
$801$	−2.01918	−0.0713443
$802$	51.5703	1.82101
$803$	0	0
$804$	−24.1957	−0.853317
$805$	−6.93321	−0.244363
$806$	22.2545	0.783882
$807$	−3.77606	−0.132924
$808$	0.628357	0.0221055
$809$	−42.8794	−1.50756	−0.753780	−	0.657127i	$-0.771771\pi$
$809$	−42.8794	−1.50756	−0.753780	+	0.657127i	$0.771771\pi$
$810$	69.5802	2.44480
$811$	−18.8186	−0.660812	−0.330406	−	0.943839i	$-0.607186\pi$
$811$	−18.8186	−0.660812	−0.330406	+	0.943839i	$0.607186\pi$
$812$	−7.52225	−0.263979
$813$	−21.1537	−0.741893
$814$	0	0
$815$	32.6630	1.14413
$816$	7.59671	0.265938
$817$	40.7219	1.42468
$818$	13.4456	0.470115
$819$	1.22358	0.0427553
$820$	18.3218	0.639823
$821$	−21.8437	−0.762351	−0.381175	−	0.924503i	$-0.624481\pi$
$821$	−21.8437	−0.762351	−0.381175	+	0.924503i	$0.624481\pi$
$822$	35.5306	1.23927
$823$	−38.2787	−1.33431	−0.667156	−	0.744918i	$-0.732489\pi$
$823$	−38.2787	−1.33431	−0.667156	+	0.744918i	$0.732489\pi$
$824$	−0.843514	−0.0293852
$825$	0	0
$826$	−2.76898	−0.0963452
$827$	22.6917	0.789069	0.394534	−	0.918881i	$-0.370906\pi$
$827$	22.6917	0.789069	0.394534	+	0.918881i	$0.370906\pi$
$828$	20.8180	0.723475
$829$	27.5376	0.956421	0.478210	−	0.878245i	$-0.341286\pi$
$829$	27.5376	0.956421	0.478210	+	0.878245i	$0.341286\pi$
$830$	60.8437	2.11192
$831$	−3.92168	−0.136042
$832$	−13.1151	−0.454686
$833$	−5.95155	−0.206209
$834$	−70.0091	−2.42422
$835$	9.06475	0.313699
$836$	0	0
$837$	17.1331	0.592207
$838$	−1.20937	−0.0417770
$839$	−14.7079	−0.507773	−0.253886	−	0.967234i	$-0.581709\pi$
$839$	−14.7079	−0.507773	−0.253886	+	0.967234i	$0.581709\pi$
$840$	0.143949	0.00496673
$841$	54.4283	1.87684
$842$	−72.1945	−2.48799
$843$	−54.9343	−1.89204
$844$	22.6744	0.780485
$845$	32.6184	1.12211
$846$	16.7686	0.576517
$847$	0	0
$848$	−17.5616	−0.603067
$849$	35.0716	1.20365
$850$	8.30651	0.284911
$851$	43.1455	1.47901
$852$	8.12664	0.278414
$853$	22.3637	0.765719	0.382859	−	0.923807i	$-0.374939\pi$
$853$	22.3637	0.765719	0.382859	+	0.923807i	$0.374939\pi$
$854$	11.5636	0.395698
$855$	−45.2253	−1.54667
$856$	0.659455	0.0225397
$857$	6.64311	0.226924	0.113462	−	0.993542i	$-0.463806\pi$
$857$	6.64311	0.226924	0.113462	+	0.993542i	$0.463806\pi$
$858$	0	0
$859$	2.01848	0.0688696	0.0344348	−	0.999407i	$-0.489037\pi$
$859$	2.01848	0.0688696	0.0344348	+	0.999407i	$0.489037\pi$
$860$	−33.4885	−1.14195
$861$	2.60183	0.0886700
$862$	−50.2485	−1.71147
$863$	−4.91548	−0.167325	−0.0836624	−	0.996494i	$-0.526662\pi$
$863$	−4.91548	−0.167325	−0.0836624	+	0.996494i	$0.526662\pi$
$864$	−19.8163	−0.674164
$865$	18.5207	0.629721
$866$	9.31261	0.316455
$867$	35.8878	1.21881
$868$	5.71357	0.193931
$869$	0	0
$870$	−126.469	−4.28771
$871$	8.64299	0.292857
$872$	0.521238	0.0176513
$873$	17.7778	0.601687
$874$	84.2723	2.85055
$875$	0.311897	0.0105440
$876$	−25.8899	−0.874738
$877$	30.3758	1.02572	0.512859	−	0.858473i	$-0.328586\pi$
$877$	30.3758	1.02572	0.512859	+	0.858473i	$0.328586\pi$
$878$	−15.7777	−0.532472
$879$	−52.5380	−1.77206
$880$	0	0
$881$	−39.5513	−1.33252	−0.666259	−	0.745720i	$-0.732106\pi$
$881$	−39.5513	−1.33252	−0.666259	+	0.745720i	$0.732106\pi$
$882$	−25.8127	−0.869159
$883$	−16.0981	−0.541743	−0.270871	−	0.962616i	$-0.587312\pi$
$883$	−16.0981	−0.541743	−0.270871	+	0.962616i	$0.587312\pi$
$884$	2.81998	0.0948462
$885$	−23.4249	−0.787418
$886$	−36.4912	−1.22595
$887$	−33.6348	−1.12934	−0.564672	−	0.825315i	$-0.690997\pi$
$887$	−33.6348	−1.12934	−0.564672	+	0.825315i	$0.690997\pi$
$888$	−0.895801	−0.0300611
$889$	−6.26254	−0.210039
$890$	6.72184	0.225316
$891$	0	0
$892$	−25.5295	−0.854791
$893$	34.1557	1.14298
$894$	28.4234	0.950620
$895$	−23.4110	−0.782544
$896$	−0.166859	−0.00557437
$897$	−19.2882	−0.644016
$898$	−39.0184	−1.30206
$899$	−63.3685	−2.11346
$900$	18.1277	0.604255
$901$	3.87324	0.129036
$902$	0	0
$903$	−4.75561	−0.158257
$904$	0.620674	0.0206433
$905$	−70.2658	−2.33571
$906$	−8.84545	−0.293870
$907$	−26.3625	−0.875353	−0.437676	−	0.899133i	$-0.644198\pi$
$907$	−26.3625	−0.875353	−0.437676	+	0.899133i	$0.644198\pi$
$908$	−5.93589	−0.196989
$909$	23.0545	0.764669
$910$	−4.07329	−0.135028
$911$	−45.1795	−1.49686	−0.748431	−	0.663213i	$-0.769193\pi$
$911$	−45.1795	−1.49686	−0.748431	+	0.663213i	$0.769193\pi$
$912$	67.1112	2.22227
$913$	0	0
$914$	79.9371	2.64408
$915$	97.8249	3.23399
$916$	−0.620168	−0.0204909
$917$	4.12601	0.136253
$918$	4.31464	0.142404
$919$	26.6358	0.878634	0.439317	−	0.898332i	$-0.355221\pi$
$919$	26.6358	0.878634	0.439317	+	0.898332i	$0.355221\pi$
$920$	−0.874867	−0.0288435
$921$	43.8509	1.44494
$922$	59.8772	1.97195
$923$	−2.90293	−0.0955511
$924$	0	0
$925$	37.5698	1.23529
$926$	27.0853	0.890078
$927$	−30.9486	−1.01649
$928$	73.2925	2.40594
$929$	−30.8037	−1.01064	−0.505318	−	0.862933i	$-0.668625\pi$
$929$	−30.8037	−1.01064	−0.505318	+	0.862933i	$0.668625\pi$
$930$	96.0605	3.14995
$931$	−52.5775	−1.72316
$932$	1.09678	0.0359262
$933$	−43.2776	−1.41684
$934$	77.3000	2.52933
$935$	0	0
$936$	0.154397	0.00504664
$937$	39.1478	1.27890	0.639451	−	0.768832i	$-0.279162\pi$
$937$	39.1478	1.27890	0.639451	+	0.768832i	$0.279162\pi$
$938$	4.40995	0.143990
$939$	32.4019	1.05740
$940$	−28.0886	−0.916149
$941$	12.5043	0.407628	0.203814	−	0.979010i	$-0.434666\pi$
$941$	12.5043	0.407628	0.203814	+	0.979010i	$0.434666\pi$
$942$	13.5503	0.441493
$943$	−15.8129	−0.514938
$944$	13.4018	0.436191
$945$	−3.13591	−0.102011
$946$	0	0
$947$	3.43405	0.111592	0.0557959	−	0.998442i	$-0.482230\pi$
$947$	3.43405	0.111592	0.0557959	+	0.998442i	$0.482230\pi$
$948$	70.3881	2.28610
$949$	9.24817	0.300208
$950$	73.3817	2.38082
$951$	28.1565	0.913037
$952$	0.0181638	0.000588692 0
$953$	36.5001	1.18236	0.591178	−	0.806541i	$-0.298663\pi$
$953$	36.5001	1.18236	0.591178	+	0.806541i	$0.298663\pi$
$954$	16.7988	0.543881
$955$	32.6665	1.05706
$956$	37.9534	1.22750
$957$	0	0
$958$	53.6852	1.73449
$959$	−3.25850	−0.105222
$960$	−56.6109	−1.82711
$961$	17.1319	0.552643
$962$	25.3482	0.817258
$963$	24.1955	0.779688
$964$	23.2104	0.747557
$965$	22.2942	0.717675
$966$	−9.84153	−0.316646
$967$	−14.4299	−0.464036	−0.232018	−	0.972712i	$-0.574533\pi$
$967$	−14.4299	−0.464036	−0.232018	+	0.972712i	$0.574533\pi$
$968$	0	0
$969$	−14.8015	−0.475493
$970$	−59.1821	−1.90022
$971$	−19.9607	−0.640570	−0.320285	−	0.947321i	$-0.603779\pi$
$971$	−19.9607	−0.640570	−0.320285	+	0.947321i	$0.603779\pi$
$972$	34.6906	1.11270
$973$	6.42051	0.205832
$974$	1.76958	0.0567009
$975$	−16.7956	−0.537890
$976$	−55.9674	−1.79147
$977$	30.1293	0.963921	0.481961	−	0.876193i	$-0.339925\pi$
$977$	30.1293	0.963921	0.481961	+	0.876193i	$0.339925\pi$
$978$	46.3644	1.48257
$979$	0	0
$980$	43.2381	1.38119
$981$	19.1243	0.610591
$982$	−29.2975	−0.934920
$983$	−34.4650	−1.09926	−0.549631	−	0.835408i	$-0.685232\pi$
$983$	−34.4650	−1.09926	−0.549631	+	0.835408i	$0.685232\pi$
$984$	0.328312	0.0104662
$985$	−7.65664	−0.243961
$986$	−15.9581	−0.508210
$987$	−3.98879	−0.126965
$988$	24.9124	0.792569
$989$	28.9027	0.919053
$990$	0	0
$991$	33.9663	1.07898	0.539488	−	0.841994i	$-0.318618\pi$
$991$	33.9663	1.07898	0.539488	+	0.841994i	$0.318618\pi$
$992$	−55.6698	−1.76752
$993$	0.740607	0.0235025
$994$	−1.48118	−0.0469800
$995$	−77.1488	−2.44578
$996$	43.4574	1.37700
$997$	34.6930	1.09874	0.549369	−	0.835580i	$-0.314868\pi$
$997$	34.6930	1.09874	0.549369	+	0.835580i	$0.314868\pi$
$998$	−40.4896	−1.28168
$999$	19.5148	0.617422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1331.2.a.e.1.20	yes	25
11.10	odd	2		1331.2.a.d.1.6	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1331.2.a.d.1.6	✓	25	11.10	odd	2
1331.2.a.e.1.20	yes	25	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:	\( N \)	\(=\)	\( 1331 = 11^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1331.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00638 q^{2} -2.20960 q^{3} +2.02557 q^{4} -3.12320 q^{5} -4.43331 q^{6} +0.406578 q^{7} +0.0513041 q^{8} +1.88235 q^{9} -6.26633 q^{10} -4.47571 q^{12} +1.59878 q^{13} +0.815751 q^{14} +6.90104 q^{15} -3.94821 q^{16} +0.870785 q^{17} +3.77672 q^{18} +7.69273 q^{19} -6.32626 q^{20} -0.898376 q^{21} +5.45997 q^{23} -0.113362 q^{24} +4.75438 q^{25} +3.20776 q^{26} +2.46956 q^{27} +0.823552 q^{28} -9.13391 q^{29} +13.8461 q^{30} +6.93772 q^{31} -8.02422 q^{32} +1.74713 q^{34} -1.26982 q^{35} +3.81284 q^{36} +7.90215 q^{37} +15.4346 q^{38} -3.53266 q^{39} -0.160233 q^{40} -2.89614 q^{41} -1.80249 q^{42} +5.29356 q^{43} -5.87896 q^{45} +10.9548 q^{46} +4.44000 q^{47} +8.72397 q^{48} -6.83469 q^{49} +9.53910 q^{50} -1.92409 q^{51} +3.23843 q^{52} +4.44799 q^{53} +4.95488 q^{54} +0.0208591 q^{56} -16.9979 q^{57} -18.3261 q^{58} -3.39440 q^{59} +13.9785 q^{60} +14.1754 q^{61} +13.9197 q^{62} +0.765323 q^{63} -8.20324 q^{64} -4.99330 q^{65} +5.40601 q^{67} +1.76384 q^{68} -12.0644 q^{69} -2.54775 q^{70} -1.81572 q^{71} +0.0965723 q^{72} +5.78453 q^{73} +15.8547 q^{74} -10.5053 q^{75} +15.5822 q^{76} -7.08787 q^{78} -15.7267 q^{79} +12.3310 q^{80} -11.1038 q^{81} -5.81077 q^{82} -9.70962 q^{83} -1.81972 q^{84} -2.71964 q^{85} +10.6209 q^{86} +20.1823 q^{87} -1.07269 q^{89} -11.7954 q^{90} +0.650027 q^{91} +11.0596 q^{92} -15.3296 q^{93} +8.90833 q^{94} -24.0259 q^{95} +17.7303 q^{96} +9.44446 q^{97} -13.7130 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} - 3 q^{3} + 25 q^{4} - 3 q^{5} + 15 q^{6} + 19 q^{7} + 9 q^{8} + 22 q^{9} + 25 q^{10} - 12 q^{12} + 46 q^{13} - 12 q^{14} - 15 q^{15} + 13 q^{16} + 14 q^{17} + 6 q^{18} + 45 q^{19} - 24 q^{20}+ \cdots - 39 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 - 3 * q^3 + 25 * q^4 - 3 * q^5 + 15 * q^6 + 19 * q^7 + 9 * q^8 + 22 * q^9 + 25 * q^10 - 12 * q^12 + 46 * q^13 - 12 * q^14 - 15 * q^15 + 13 * q^16 + 14 * q^17 + 6 * q^18 + 45 * q^19 - 24 * q^20 + 49 * q^21 + 2 * q^23 + 36 * q^24 + 16 * q^25 - 7 * q^26 - 21 * q^27 + 40 * q^28 + 40 * q^29 - 15 * q^30 - 4 * q^31 + 3 * q^32 - 24 * q^34 - q^35 + 38 * q^36 - 21 * q^37 + 30 * q^38 + 18 * q^39 + 54 * q^40 + 49 * q^41 + 13 * q^42 + 14 * q^43 - 42 * q^45 + 40 * q^46 - 2 * q^47 - 5 * q^48 + 18 * q^49 - 12 * q^50 + 7 * q^51 + 57 * q^52 - 25 * q^53 + 33 * q^54 - 11 * q^56 + 4 * q^57 + 23 * q^58 - 22 * q^59 - 12 * q^60 + 128 * q^61 - 8 * q^62 + 12 * q^63 + 31 * q^64 + 16 * q^67 - 6 * q^68 - 57 * q^69 + 38 * q^70 - 34 * q^71 - 15 * q^72 + 71 * q^73 - 21 * q^74 - 10 * q^75 + 95 * q^76 + 75 * q^78 + 53 * q^79 - 29 * q^80 + 21 * q^81 + 33 * q^82 - 6 * q^83 + 18 * q^84 + 101 * q^85 + 38 * q^86 - 16 * q^87 + 5 * q^89 - 20 * q^90 + 16 * q^91 + 53 * q^92 + 7 * q^93 + 57 * q^94 - 16 * q^95 + 32 * q^96 + 14 * q^97 - 39 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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