LMFDB - Embedded newform 363.4.a.v.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	363.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:	$21.4176933321$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 30x^{4} + 3x^{3} + 211x^{2} + 208x + 44 $ x^6 - x^5 - 30x^4 + 3x^3 + 211x^2 + 208x + 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.808830$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.80883 q^{2} -3.00000 q^{3} -4.72813 q^{4} -21.4614 q^{5} -5.42649 q^{6} -18.7206 q^{7} -23.0230 q^{8} +9.00000 q^{9} -38.8200 q^{10} +14.1844 q^{12} +13.8787 q^{13} -33.8624 q^{14} +64.3841 q^{15} -3.81969 q^{16} -21.7681 q^{17} +16.2795 q^{18} -126.266 q^{19} +101.472 q^{20} +56.1618 q^{21} +1.06620 q^{23} +69.0691 q^{24} +335.590 q^{25} +25.1042 q^{26} -27.0000 q^{27} +88.5134 q^{28} +130.455 q^{29} +116.460 q^{30} -192.869 q^{31} +177.275 q^{32} -39.3748 q^{34} +401.769 q^{35} -42.5532 q^{36} +41.7438 q^{37} -228.394 q^{38} -41.6360 q^{39} +494.106 q^{40} -59.2714 q^{41} +101.587 q^{42} -172.713 q^{43} -193.152 q^{45} +1.92857 q^{46} -133.951 q^{47} +11.4591 q^{48} +7.46030 q^{49} +607.026 q^{50} +65.3044 q^{51} -65.6203 q^{52} -40.2812 q^{53} -48.8384 q^{54} +431.005 q^{56} +378.798 q^{57} +235.970 q^{58} +200.872 q^{59} -304.417 q^{60} +205.256 q^{61} -348.868 q^{62} -168.485 q^{63} +351.218 q^{64} -297.856 q^{65} -474.187 q^{67} +102.923 q^{68} -3.19860 q^{69} +726.733 q^{70} -422.982 q^{71} -207.207 q^{72} -1087.23 q^{73} +75.5075 q^{74} -1006.77 q^{75} +597.003 q^{76} -75.3125 q^{78} +575.020 q^{79} +81.9757 q^{80} +81.0000 q^{81} -107.212 q^{82} +671.681 q^{83} -265.540 q^{84} +467.174 q^{85} -312.409 q^{86} -391.364 q^{87} -311.151 q^{89} -349.380 q^{90} -259.817 q^{91} -5.04113 q^{92} +578.608 q^{93} -242.295 q^{94} +2709.84 q^{95} -531.825 q^{96} +684.390 q^{97} +13.4944 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) $ 6 * q + 5 * q^2 - 18 * q^3 + 17 * q^4 + 9 * q^5 - 15 * q^6 + q^7 + 24 * q^8 + 54 * q^9 + 50 * q^10 - 51 * q^12 + 66 * q^13 - 42 * q^14 - 27 * q^15 - 71 * q^16 + 80 * q^17 + 45 * q^18 - 90 * q^19 + 455 * q^20 - 3 * q^21 - 196 * q^23 - 72 * q^24 + 351 * q^25 + 93 * q^26 - 162 * q^27 - 52 * q^28 + 579 * q^29 - 150 * q^30 + 129 * q^31 + 657 * q^32 + 530 * q^34 + 618 * q^35 + 153 * q^36 + 192 * q^37 - 769 * q^38 - 198 * q^39 + 1476 * q^40 + 990 * q^41 + 126 * q^42 + 208 * q^43 + 81 * q^45 + 493 * q^46 + 680 * q^47 + 213 * q^48 - 963 * q^49 + 1656 * q^50 - 240 * q^51 + 887 * q^52 + 197 * q^53 - 135 * q^54 - 429 * q^56 + 270 * q^57 + 395 * q^58 - 1229 * q^59 - 1365 * q^60 + 928 * q^61 + 2117 * q^62 + 9 * q^63 + 844 * q^64 + 2278 * q^65 - 264 * q^67 + 3279 * q^68 + 588 * q^69 + 665 * q^70 - 786 * q^71 + 216 * q^72 + 593 * q^73 + 1543 * q^74 - 1053 * q^75 - 434 * q^76 - 279 * q^78 - 421 * q^79 + 18 * q^80 + 486 * q^81 - 739 * q^82 + 2043 * q^83 + 156 * q^84 + 1900 * q^85 + 2683 * q^86 - 1737 * q^87 + 158 * q^89 + 450 * q^90 + 698 * q^91 - 3915 * q^92 - 387 * q^93 - 4934 * q^94 + 1038 * q^95 - 1971 * q^96 - 1711 * q^97 + 1405 * 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 $

$2$	1.80883	0.639518	0.319759	−	0.947499i	$-0.396398\pi$
$2$	1.80883	0.639518	0.319759	+	0.947499i	$0.396398\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.72813	−0.591017
$5$	−21.4614	−1.91956	−0.959782	−	0.280748i	$-0.909418\pi$
$5$	−21.4614	−1.91956	−0.959782	+	0.280748i	$0.909418\pi$
$6$	−5.42649	−0.369226
$7$	−18.7206	−1.01082	−0.505408	−	0.862880i	$-0.668658\pi$
$7$	−18.7206	−1.01082	−0.505408	+	0.862880i	$0.668658\pi$
$8$	−23.0230	−1.01748
$9$	9.00000	0.333333
$10$	−38.8200	−1.22760
$11$	0	0
$11$	0	0
$12$	14.1844	0.341224
$13$	13.8787	0.296096	0.148048	−	0.988980i	$-0.452701\pi$
$13$	13.8787	0.296096	0.148048	+	0.988980i	$0.452701\pi$
$14$	−33.8624	−0.646435
$15$	64.3841	1.10826
$16$	−3.81969	−0.0596826
$17$	−21.7681	−0.310561	−0.155281	−	0.987870i	$-0.549628\pi$
$17$	−21.7681	−0.310561	−0.155281	+	0.987870i	$0.549628\pi$
$18$	16.2795	0.213173
$19$	−126.266	−1.52460	−0.762301	−	0.647223i	$-0.775930\pi$
$19$	−126.266	−1.52460	−0.762301	+	0.647223i	$0.775930\pi$
$20$	101.472	1.13449
$21$	56.1618	0.583595
$22$	0	0
$23$	1.06620	0.00966600	0.00483300	−	0.999988i	$-0.498462\pi$
$23$	1.06620	0.00966600	0.00483300	+	0.999988i	$0.498462\pi$
$24$	69.0691	0.587445
$25$	335.590	2.68472
$26$	25.1042	0.189359
$27$	−27.0000	−0.192450
$28$	88.5134	0.597409
$29$	130.455	0.835338	0.417669	−	0.908599i	$-0.362847\pi$
$29$	130.455	0.835338	0.417669	+	0.908599i	$0.362847\pi$
$30$	116.460	0.708752
$31$	−192.869	−1.11743	−0.558716	−	0.829359i	$-0.688706\pi$
$31$	−192.869	−1.11743	−0.558716	+	0.829359i	$0.688706\pi$
$32$	177.275	0.979316
$33$	0	0
$34$	−39.3748	−0.198610
$35$	401.769	1.94033
$36$	−42.5532	−0.197006
$37$	41.7438	0.185477	0.0927384	−	0.995691i	$-0.470438\pi$
$37$	41.7438	0.185477	0.0927384	+	0.995691i	$0.470438\pi$
$38$	−228.394	−0.975010
$39$	−41.6360	−0.170951
$40$	494.106	1.95312
$41$	−59.2714	−0.225772	−0.112886	−	0.993608i	$-0.536009\pi$
$41$	−59.2714	−0.225772	−0.112886	+	0.993608i	$0.536009\pi$
$42$	101.587	0.373220
$43$	−172.713	−0.612524	−0.306262	−	0.951947i	$-0.599078\pi$
$43$	−172.713	−0.612524	−0.306262	+	0.951947i	$0.599078\pi$
$44$	0	0
$45$	−193.152	−0.639854
$46$	1.92857	0.00618158
$47$	−133.951	−0.415719	−0.207860	−	0.978159i	$-0.566650\pi$
$47$	−133.951	−0.415719	−0.207860	+	0.978159i	$0.566650\pi$
$48$	11.4591	0.0344578
$49$	7.46030	0.0217501
$50$	607.026	1.71693
$51$	65.3044	0.179303
$52$	−65.6203	−0.174998
$53$	−40.2812	−0.104397	−0.0521985	−	0.998637i	$-0.516623\pi$
$53$	−40.2812	−0.104397	−0.0521985	+	0.998637i	$0.516623\pi$
$54$	−48.8384	−0.123075
$55$	0	0
$56$	431.005	1.02849
$57$	378.798	0.880229
$58$	235.970	0.534214
$59$	200.872	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$59$	200.872	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$60$	−304.417	−0.655000
$61$	205.256	0.430824	0.215412	−	0.976523i	$-0.430891\pi$
$61$	205.256	0.430824	0.215412	+	0.976523i	$0.430891\pi$
$62$	−348.868	−0.714617
$63$	−168.485	−0.336939
$64$	351.218	0.685973
$65$	−297.856	−0.568376
$66$	0	0
$67$	−474.187	−0.864644	−0.432322	−	0.901719i	$-0.642306\pi$
$67$	−474.187	−0.864644	−0.432322	+	0.901719i	$0.642306\pi$
$68$	102.923	0.183547
$69$	−3.19860	−0.00558067
$70$	726.733	1.24087
$71$	−422.982	−0.707024	−0.353512	−	0.935430i	$-0.615013\pi$
$71$	−422.982	−0.707024	−0.353512	+	0.935430i	$0.615013\pi$
$72$	−207.207	−0.339161
$73$	−1087.23	−1.74316	−0.871582	−	0.490250i	$-0.836905\pi$
$73$	−1087.23	−1.74316	−0.871582	+	0.490250i	$0.836905\pi$
$74$	75.5075	0.118616
$75$	−1006.77	−1.55003
$76$	597.003	0.901065
$77$	0	0
$78$	−75.3125	−0.109326
$79$	575.020	0.818922	0.409461	−	0.912328i	$-0.365717\pi$
$79$	575.020	0.818922	0.409461	+	0.912328i	$0.365717\pi$
$80$	81.9757	0.114565
$81$	81.0000	0.111111
$82$	−107.212	−0.144385
$83$	671.681	0.888272	0.444136	−	0.895959i	$-0.353511\pi$
$83$	671.681	0.888272	0.444136	+	0.895959i	$0.353511\pi$
$84$	−265.540	−0.344915
$85$	467.174	0.596142
$86$	−312.409	−0.391720
$87$	−391.364	−0.482283
$88$	0	0
$89$	−311.151	−0.370584	−0.185292	−	0.982684i	$-0.559323\pi$
$89$	−311.151	−0.370584	−0.185292	+	0.982684i	$0.559323\pi$
$90$	−349.380	−0.409198
$91$	−259.817	−0.299299
$92$	−5.04113	−0.00571277
$93$	578.608	0.645149
$94$	−242.295	−0.265860
$95$	2709.84	2.92657
$96$	−531.825	−0.565408
$97$	684.390	0.716384	0.358192	−	0.933648i	$-0.383393\pi$
$97$	684.390	0.716384	0.358192	+	0.933648i	$0.383393\pi$
$98$	13.4944	0.0139096
$99$	0	0
$100$	−1586.72	−1.58672
$101$	−1161.20	−1.14399	−0.571997	−	0.820256i	$-0.693831\pi$
$101$	−1161.20	−1.14399	−0.571997	+	0.820256i	$0.693831\pi$
$102$	118.124	0.114667
$103$	449.261	0.429777	0.214888	−	0.976639i	$-0.431061\pi$
$103$	449.261	0.429777	0.214888	+	0.976639i	$0.431061\pi$
$104$	−319.529	−0.301273
$105$	−1205.31	−1.12025
$106$	−72.8618	−0.0667638
$107$	610.486	0.551570	0.275785	−	0.961219i	$-0.411062\pi$
$107$	610.486	0.551570	0.275785	+	0.961219i	$0.411062\pi$
$108$	127.660	0.113741
$109$	1128.39	0.991560	0.495780	−	0.868448i	$-0.334882\pi$
$109$	1128.39	0.991560	0.495780	+	0.868448i	$0.334882\pi$
$110$	0	0
$111$	−125.231	−0.107085
$112$	71.5068	0.0603282
$113$	−246.156	−0.204924	−0.102462	−	0.994737i	$-0.532672\pi$
$113$	−246.156	−0.204924	−0.102462	+	0.994737i	$0.532672\pi$
$114$	685.182	0.562922
$115$	−22.8821	−0.0185545
$116$	−616.806	−0.493699
$117$	124.908	0.0986988
$118$	363.343	0.283462
$119$	407.512	0.313921
$120$	−1482.32	−1.12764
$121$	0	0
$122$	371.272	0.275520
$123$	177.814	0.130349
$124$	911.912	0.660421
$125$	−4519.56	−3.23393
$126$	−304.761	−0.215478
$127$	555.585	0.388190	0.194095	−	0.980983i	$-0.437823\pi$
$127$	555.585	0.388190	0.194095	+	0.980983i	$0.437823\pi$
$128$	−782.907	−0.540624
$129$	518.140	0.353641
$130$	−538.770	−0.363487
$131$	1958.51	1.30623	0.653113	−	0.757260i	$-0.273463\pi$
$131$	1958.51	1.30623	0.653113	+	0.757260i	$0.273463\pi$
$132$	0	0
$133$	2363.77	1.54109
$134$	−857.723	−0.552955
$135$	579.457	0.369420
$136$	501.168	0.315991
$137$	2498.83	1.55831	0.779157	−	0.626829i	$-0.215647\pi$
$137$	2498.83	1.55831	0.779157	+	0.626829i	$0.215647\pi$
$138$	−5.78572	−0.00356894
$139$	−2222.29	−1.35606	−0.678031	−	0.735033i	$-0.737166\pi$
$139$	−2222.29	−1.35606	−0.678031	+	0.735033i	$0.737166\pi$
$140$	−1899.62	−1.14677
$141$	401.854	0.240016
$142$	−765.102	−0.452155
$143$	0	0
$144$	−34.3772	−0.0198942
$145$	−2799.73	−1.60348
$146$	−1966.62	−1.11478
$147$	−22.3809	−0.0125574
$148$	−197.370	−0.109620
$149$	2440.81	1.34200	0.671002	−	0.741455i	$-0.265864\pi$
$149$	2440.81	1.34200	0.671002	+	0.741455i	$0.265864\pi$
$150$	−1821.08	−0.991269
$151$	−1656.65	−0.892825	−0.446413	−	0.894827i	$-0.647299\pi$
$151$	−1656.65	−0.892825	−0.446413	+	0.894827i	$0.647299\pi$
$152$	2907.03	1.55126
$153$	−195.913	−0.103520
$154$	0	0
$155$	4139.24	2.14498
$156$	196.861	0.101035
$157$	−2600.65	−1.32200	−0.661001	−	0.750385i	$-0.729868\pi$
$157$	−2600.65	−1.32200	−0.661001	+	0.750385i	$0.729868\pi$
$158$	1040.11	0.523715
$159$	120.843	0.0602737
$160$	−3804.57	−1.87986
$161$	−19.9599	−0.00977055
$162$	146.515	0.0710576
$163$	−1566.09	−0.752549	−0.376275	−	0.926508i	$-0.622795\pi$
$163$	−1566.09	−0.752549	−0.376275	+	0.926508i	$0.622795\pi$
$164$	280.243	0.133435
$165$	0	0
$166$	1214.96	0.568066
$167$	2782.40	1.28927	0.644636	−	0.764490i	$-0.277009\pi$
$167$	2782.40	1.28927	0.644636	+	0.764490i	$0.277009\pi$
$168$	−1293.01	−0.593799
$169$	−2004.38	−0.912327
$170$	845.038	0.381244
$171$	−1136.39	−0.508200
$172$	816.612	0.362012
$173$	3207.19	1.40947	0.704734	−	0.709472i	$-0.251066\pi$
$173$	3207.19	1.40947	0.704734	+	0.709472i	$0.251066\pi$
$174$	−707.910	−0.308428
$175$	−6282.45	−2.71376
$176$	0	0
$177$	−602.616	−0.255906
$178$	−562.820	−0.236995
$179$	−831.063	−0.347020	−0.173510	−	0.984832i	$-0.555511\pi$
$179$	−831.063	−0.347020	−0.173510	+	0.984832i	$0.555511\pi$
$180$	913.250	0.378165
$181$	4101.12	1.68416	0.842082	−	0.539350i	$-0.181330\pi$
$181$	4101.12	1.68416	0.842082	+	0.539350i	$0.181330\pi$
$182$	−469.965	−0.191407
$183$	−615.767	−0.248736
$184$	−24.5471	−0.00983500
$185$	−895.879	−0.356034
$186$	1046.60	0.412585
$187$	0	0
$188$	633.339	0.245697
$189$	505.456	0.194532
$190$	4901.64	1.87159
$191$	−1839.44	−0.696843	−0.348421	−	0.937338i	$-0.613282\pi$
$191$	−1839.44	−0.696843	−0.348421	+	0.937338i	$0.613282\pi$
$192$	−1053.65	−0.396047
$193$	−3237.11	−1.20732	−0.603659	−	0.797243i	$-0.706291\pi$
$193$	−3237.11	−1.20732	−0.603659	+	0.797243i	$0.706291\pi$
$194$	1237.94	0.458140
$195$	893.567	0.328152
$196$	−35.2733	−0.0128547
$197$	3195.60	1.15572	0.577860	−	0.816136i	$-0.303888\pi$
$197$	3195.60	1.15572	0.577860	+	0.816136i	$0.303888\pi$
$198$	0	0
$199$	−2966.11	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$199$	−2966.11	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$200$	−7726.31	−2.73166
$201$	1422.56	0.499202
$202$	−2100.41	−0.731604
$203$	−2442.19	−0.844373
$204$	−308.768	−0.105971
$205$	1272.05	0.433383
$206$	812.637	0.274850
$207$	9.59580	0.00322200
$208$	−53.0122	−0.0176718
$209$	0	0
$210$	−2180.20	−0.716419
$211$	3573.66	1.16598	0.582988	−	0.812481i	$-0.301883\pi$
$211$	3573.66	1.16598	0.582988	+	0.812481i	$0.301883\pi$
$212$	190.455	0.0617004
$213$	1268.95	0.408201
$214$	1104.27	0.352739
$215$	3706.67	1.17578
$216$	621.622	0.195815
$217$	3610.63	1.12952
$218$	2041.06	0.634121
$219$	3261.70	1.00642
$220$	0	0
$221$	−302.113	−0.0919561
$222$	−226.522	−0.0684828
$223$	−1283.84	−0.385525	−0.192762	−	0.981245i	$-0.561745\pi$
$223$	−1283.84	−0.385525	−0.192762	+	0.981245i	$0.561745\pi$
$224$	−3318.69	−0.989909
$225$	3020.31	0.894908
$226$	−445.255	−0.131053
$227$	−1122.08	−0.328085	−0.164043	−	0.986453i	$-0.552453\pi$
$227$	−1122.08	−0.328085	−0.164043	+	0.986453i	$0.552453\pi$
$228$	−1791.01	−0.520230
$229$	−4803.11	−1.38602	−0.693010	−	0.720928i	$-0.743716\pi$
$229$	−4803.11	−1.38602	−0.693010	+	0.720928i	$0.743716\pi$
$230$	−41.3898	−0.0118659
$231$	0	0
$232$	−3003.46	−0.849943
$233$	−105.139	−0.0295619	−0.0147809	−	0.999891i	$-0.504705\pi$
$233$	−105.139	−0.0295619	−0.0147809	+	0.999891i	$0.504705\pi$
$234$	225.938	0.0631197
$235$	2874.78	0.797999
$236$	−949.749	−0.261964
$237$	−1725.06	−0.472805
$238$	737.120	0.200758
$239$	−1090.69	−0.295193	−0.147596	−	0.989048i	$-0.547154\pi$
$239$	−1090.69	−0.295193	−0.147596	+	0.989048i	$0.547154\pi$
$240$	−245.927	−0.0661438
$241$	1068.58	0.285615	0.142808	−	0.989750i	$-0.454387\pi$
$241$	1068.58	0.285615	0.142808	+	0.989750i	$0.454387\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−970.475	−0.254624
$245$	−160.108	−0.0417508
$246$	321.636	0.0833608
$247$	−1752.41	−0.451429
$248$	4440.44	1.13697
$249$	−2015.04	−0.512844
$250$	−8175.11	−2.06816
$251$	1992.38	0.501026	0.250513	−	0.968113i	$-0.419401\pi$
$251$	1992.38	0.501026	0.250513	+	0.968113i	$0.419401\pi$
$252$	796.621	0.199136
$253$	0	0
$254$	1004.96	0.248255
$255$	−1401.52	−0.344183
$256$	−4225.89	−1.03171
$257$	7985.30	1.93817	0.969084	−	0.246730i	$-0.0793560\pi$
$257$	7985.30	1.93817	0.969084	+	0.246730i	$0.0793560\pi$
$258$	937.227	0.226160
$259$	−781.469	−0.187483
$260$	1408.30	0.335920
$261$	1174.09	0.278446
$262$	3542.61	0.835356
$263$	−3975.12	−0.932002	−0.466001	−	0.884784i	$-0.654306\pi$
$263$	−3975.12	−0.932002	−0.466001	+	0.884784i	$0.654306\pi$
$264$	0	0
$265$	864.489	0.200397
$266$	4275.67	0.985556
$267$	933.454	0.213957
$268$	2242.02	0.511019
$269$	−1127.52	−0.255561	−0.127781	−	0.991802i	$-0.540785\pi$
$269$	−1127.52	−0.255561	−0.127781	+	0.991802i	$0.540785\pi$
$270$	1048.14	0.236251
$271$	−3949.98	−0.885404	−0.442702	−	0.896669i	$-0.645980\pi$
$271$	−3949.98	−0.885404	−0.442702	+	0.896669i	$0.645980\pi$
$272$	83.1474	0.0185351
$273$	779.451	0.172800
$274$	4519.95	0.996570
$275$	0	0
$276$	15.1234	0.00329827
$277$	656.095	0.142314	0.0711569	−	0.997465i	$-0.477331\pi$
$277$	656.095	0.142314	0.0711569	+	0.997465i	$0.477331\pi$
$278$	−4019.75	−0.867226
$279$	−1735.82	−0.372477
$280$	−9249.95	−1.97425
$281$	859.863	0.182545	0.0912725	−	0.995826i	$-0.470907\pi$
$281$	859.863	0.182545	0.0912725	+	0.995826i	$0.470907\pi$
$282$	726.885	0.153494
$283$	7716.50	1.62084	0.810421	−	0.585848i	$-0.199238\pi$
$283$	7716.50	1.62084	0.810421	+	0.585848i	$0.199238\pi$
$284$	1999.91	0.417863
$285$	−8129.53	−1.68965
$286$	0	0
$287$	1109.60	0.228214
$288$	1595.48	0.326439
$289$	−4439.15	−0.903552
$290$	−5064.24	−1.02546
$291$	−2053.17	−0.413604
$292$	5140.58	1.03024
$293$	−581.812	−0.116006	−0.0580031	−	0.998316i	$-0.518473\pi$
$293$	−581.812	−0.116006	−0.0580031	+	0.998316i	$0.518473\pi$
$294$	−40.4832	−0.00803071
$295$	−4310.99	−0.850832
$296$	−961.069	−0.188720
$297$	0	0
$298$	4415.01	0.858236
$299$	14.7974	0.00286207
$300$	4760.15	0.916091
$301$	3233.30	0.619150
$302$	−2996.61	−0.570978
$303$	3483.59	0.660485
$304$	482.297	0.0909921
$305$	−4405.06	−0.826994
$306$	−354.373	−0.0662032
$307$	2369.06	0.440421	0.220211	−	0.975452i	$-0.429326\pi$
$307$	2369.06	0.440421	0.220211	+	0.975452i	$0.429326\pi$
$308$	0	0
$309$	−1347.78	−0.248132
$310$	7487.19	1.37175
$311$	10507.8	1.91589	0.957946	−	0.286949i	$-0.0926412\pi$
$311$	10507.8	1.91589	0.957946	+	0.286949i	$0.0926412\pi$
$312$	958.588	0.173940
$313$	−7261.22	−1.31127	−0.655637	−	0.755077i	$-0.727600\pi$
$313$	−7261.22	−1.31127	−0.655637	+	0.755077i	$0.727600\pi$
$314$	−4704.13	−0.845444
$315$	3615.92	0.646775
$316$	−2718.77	−0.483996
$317$	−2943.52	−0.521528	−0.260764	−	0.965403i	$-0.583975\pi$
$317$	−2943.52	−0.521528	−0.260764	+	0.965403i	$0.583975\pi$
$318$	218.585	0.0385461
$319$	0	0
$320$	−7537.62	−1.31677
$321$	−1831.46	−0.318449
$322$	−36.1040	−0.00624844
$323$	2748.57	0.473482
$324$	−382.979	−0.0656685
$325$	4657.55	0.794937
$326$	−2832.79	−0.481269
$327$	−3385.17	−0.572477
$328$	1364.61	0.229719
$329$	2507.65	0.420216
$330$	0	0
$331$	−6904.52	−1.14655	−0.573273	−	0.819364i	$-0.694327\pi$
$331$	−6904.52	−1.14655	−0.573273	+	0.819364i	$0.694327\pi$
$332$	−3175.80	−0.524983
$333$	375.694	0.0618256
$334$	5032.88	0.824512
$335$	10176.7	1.65974
$336$	−214.520	−0.0348305
$337$	10251.2	1.65702	0.828510	−	0.559974i	$-0.189189\pi$
$337$	10251.2	1.65702	0.828510	+	0.559974i	$0.189189\pi$
$338$	−3625.59	−0.583449
$339$	738.468	0.118313
$340$	−2208.86	−0.352330
$341$	0	0
$342$	−2055.54	−0.325003
$343$	6281.50	0.988831
$344$	3976.39	0.623234
$345$	68.6463	0.0107124
$346$	5801.26	0.901380
$347$	5259.88	0.813733	0.406866	−	0.913488i	$-0.366621\pi$
$347$	5259.88	0.813733	0.406866	+	0.913488i	$0.366621\pi$
$348$	1850.42	0.285037
$349$	10566.1	1.62061	0.810304	−	0.586010i	$-0.199302\pi$
$349$	10566.1	1.62061	0.810304	+	0.586010i	$0.199302\pi$
$350$	−11363.9	−1.73550
$351$	−374.724	−0.0569838
$352$	0	0
$353$	−1822.89	−0.274852	−0.137426	−	0.990512i	$-0.543883\pi$
$353$	−1822.89	−0.274852	−0.137426	+	0.990512i	$0.543883\pi$
$354$	−1090.03	−0.163657
$355$	9077.77	1.35718
$356$	1471.16	0.219021
$357$	−1222.54	−0.181242
$358$	−1503.25	−0.221925
$359$	11524.0	1.69419	0.847093	−	0.531445i	$-0.178351\pi$
$359$	11524.0	1.69419	0.847093	+	0.531445i	$0.178351\pi$
$360$	4446.95	0.651042
$361$	9084.12	1.32441
$362$	7418.22	1.07705
$363$	0	0
$364$	1228.45	0.176891
$365$	23333.5	3.34611
$366$	−1113.82	−0.159071
$367$	−2579.26	−0.366857	−0.183428	−	0.983033i	$-0.558720\pi$
$367$	−2579.26	−0.366857	−0.183428	+	0.983033i	$0.558720\pi$
$368$	−4.07255	−0.000576892 0
$369$	−533.443	−0.0752572
$370$	−1620.49	−0.227690
$371$	754.087	0.105526
$372$	−2735.74	−0.381294
$373$	6084.16	0.844573	0.422287	−	0.906462i	$-0.361228\pi$
$373$	6084.16	0.844573	0.422287	+	0.906462i	$0.361228\pi$
$374$	0	0
$375$	13558.7	1.86711
$376$	3083.96	0.422988
$377$	1810.54	0.247341
$378$	914.284	0.124407
$379$	5898.93	0.799493	0.399746	−	0.916626i	$-0.369098\pi$
$379$	5898.93	0.799493	0.399746	+	0.916626i	$0.369098\pi$
$380$	−12812.5	−1.72965
$381$	−1666.75	−0.224122
$382$	−3327.23	−0.445643
$383$	−1599.92	−0.213452	−0.106726	−	0.994288i	$-0.534037\pi$
$383$	−1599.92	−0.213452	−0.106726	+	0.994288i	$0.534037\pi$
$384$	2348.72	0.312129
$385$	0	0
$386$	−5855.38	−0.772102
$387$	−1554.42	−0.204175
$388$	−3235.89	−0.423395
$389$	2514.60	0.327751	0.163875	−	0.986481i	$-0.447600\pi$
$389$	2514.60	0.327751	0.163875	+	0.986481i	$0.447600\pi$
$390$	1616.31	0.209859
$391$	−23.2092	−0.00300189
$392$	−171.759	−0.0221304
$393$	−5875.53	−0.754150
$394$	5780.29	0.739104
$395$	−12340.7	−1.57197
$396$	0	0
$397$	5486.84	0.693644	0.346822	−	0.937931i	$-0.387261\pi$
$397$	5486.84	0.693644	0.346822	+	0.937931i	$0.387261\pi$
$398$	−5365.19	−0.675710
$399$	−7091.32	−0.889750
$400$	−1281.85	−0.160231
$401$	−14713.0	−1.83225	−0.916127	−	0.400889i	$-0.868701\pi$
$401$	−14713.0	−1.83225	−0.916127	+	0.400889i	$0.868701\pi$
$402$	2573.17	0.319249
$403$	−2676.77	−0.330867
$404$	5490.29	0.676119
$405$	−1738.37	−0.213285
$406$	−4417.50	−0.539992
$407$	0	0
$408$	−1503.50	−0.182438
$409$	−7953.60	−0.961565	−0.480783	−	0.876840i	$-0.659647\pi$
$409$	−7953.60	−0.961565	−0.480783	+	0.876840i	$0.659647\pi$
$410$	2300.91	0.277156
$411$	−7496.48	−0.899693
$412$	−2124.17	−0.254005
$413$	−3760.44	−0.448037
$414$	17.3572	0.00206053
$415$	−14415.2	−1.70509
$416$	2460.34	0.289972
$417$	6666.88	0.782923
$418$	0	0
$419$	7289.12	0.849873	0.424936	−	0.905223i	$-0.360296\pi$
$419$	7289.12	0.849873	0.424936	+	0.905223i	$0.360296\pi$
$420$	5698.86	0.662085
$421$	5556.26	0.643220	0.321610	−	0.946872i	$-0.395776\pi$
$421$	5556.26	0.643220	0.321610	+	0.946872i	$0.395776\pi$
$422$	6464.15	0.745663
$423$	−1205.56	−0.138573
$424$	927.394	0.106222
$425$	−7305.17	−0.833771
$426$	2295.31	0.261052
$427$	−3842.50	−0.435484
$428$	−2886.46	−0.325987
$429$	0	0
$430$	6704.73	0.751932
$431$	−13047.6	−1.45819	−0.729094	−	0.684414i	$-0.760058\pi$
$431$	−13047.6	−1.45819	−0.729094	+	0.684414i	$0.760058\pi$
$432$	103.132	0.0114859
$433$	−14567.2	−1.61675	−0.808375	−	0.588668i	$-0.799653\pi$
$433$	−14567.2	−1.61675	−0.808375	+	0.588668i	$0.799653\pi$
$434$	6531.01	0.722347
$435$	8399.20	0.925772
$436$	−5335.17	−0.586029
$437$	−134.625	−0.0147368
$438$	5899.86	0.643621
$439$	−9451.64	−1.02757	−0.513784	−	0.857920i	$-0.671757\pi$
$439$	−9451.64	−1.02757	−0.513784	+	0.857920i	$0.671757\pi$
$440$	0	0
$441$	67.1427	0.00725005
$442$	−546.471	−0.0588076
$443$	−12814.6	−1.37435	−0.687176	−	0.726491i	$-0.741150\pi$
$443$	−12814.6	−1.37435	−0.687176	+	0.726491i	$0.741150\pi$
$444$	592.111	0.0632891
$445$	6677.73	0.711359
$446$	−2322.24	−0.246550
$447$	−7322.42	−0.774807
$448$	−6575.01	−0.693393
$449$	6073.88	0.638406	0.319203	−	0.947686i	$-0.396585\pi$
$449$	6073.88	0.638406	0.319203	+	0.947686i	$0.396585\pi$
$450$	5463.23	0.572310
$451$	0	0
$452$	1163.86	0.121114
$453$	4969.96	0.515473
$454$	−2029.66	−0.209816
$455$	5576.03	0.574524
$456$	−8721.08	−0.895619
$457$	15380.8	1.57436	0.787182	−	0.616721i	$-0.211539\pi$
$457$	15380.8	1.57436	0.787182	+	0.616721i	$0.211539\pi$
$458$	−8688.02	−0.886385
$459$	587.739	0.0597676
$460$	108.190	0.0109660
$461$	−3156.66	−0.318916	−0.159458	−	0.987205i	$-0.550975\pi$
$461$	−3156.66	−0.318916	−0.159458	+	0.987205i	$0.550975\pi$
$462$	0	0
$463$	4925.34	0.494384	0.247192	−	0.968966i	$-0.420492\pi$
$463$	4925.34	0.494384	0.247192	+	0.968966i	$0.420492\pi$
$464$	−498.295	−0.0498551
$465$	−12417.7	−1.23840
$466$	−190.179	−0.0189053
$467$	−11651.6	−1.15455	−0.577274	−	0.816551i	$-0.695884\pi$
$467$	−11651.6	−1.15455	−0.577274	+	0.816551i	$0.695884\pi$
$468$	−590.582	−0.0583327
$469$	8877.05	0.873996
$470$	5199.98	0.510335
$471$	7801.94	0.763258
$472$	−4624.68	−0.450992
$473$	0	0
$474$	−3120.34	−0.302367
$475$	−42373.7	−4.09313
$476$	−1926.77	−0.185532
$477$	−362.530	−0.0347990
$478$	−1972.88	−0.188781
$479$	−13119.8	−1.25148	−0.625741	−	0.780031i	$-0.715203\pi$
$479$	−13119.8	−1.25148	−0.625741	+	0.780031i	$0.715203\pi$
$480$	11413.7	1.08534
$481$	579.349	0.0549190
$482$	1932.88	0.182656
$483$	59.8796	0.00564103
$484$	0	0
$485$	−14687.9	−1.37514
$486$	−439.546	−0.0410251
$487$	2777.27	0.258419	0.129210	−	0.991617i	$-0.458756\pi$
$487$	2777.27	0.258419	0.129210	+	0.991617i	$0.458756\pi$
$488$	−4725.60	−0.438357
$489$	4698.27	0.434485
$490$	−289.609	−0.0267004
$491$	−12658.1	−1.16344	−0.581722	−	0.813387i	$-0.697621\pi$
$491$	−12658.1	−1.16344	−0.581722	+	0.813387i	$0.697621\pi$
$492$	−840.729	−0.0770386
$493$	−2839.75	−0.259424
$494$	−3169.81	−0.288697
$495$	0	0
$496$	736.701	0.0666912
$497$	7918.47	0.714672
$498$	−3644.87	−0.327973
$499$	−10139.0	−0.909591	−0.454796	−	0.890596i	$-0.650288\pi$
$499$	−10139.0	−0.909591	−0.454796	+	0.890596i	$0.650288\pi$
$500$	21369.1	1.91131
$501$	−8347.19	−0.744361
$502$	3603.87	0.320415
$503$	−150.229	−0.0133169	−0.00665843	−	0.999978i	$-0.502119\pi$
$503$	−150.229	−0.0133169	−0.00665843	+	0.999978i	$0.502119\pi$
$504$	3879.04	0.342830
$505$	24920.9	2.19597
$506$	0	0
$507$	6013.15	0.526732
$508$	−2626.88	−0.229427
$509$	7313.72	0.636886	0.318443	−	0.947942i	$-0.396840\pi$
$509$	7313.72	0.636886	0.318443	+	0.947942i	$0.396840\pi$
$510$	−2535.11	−0.220111
$511$	20353.6	1.76202
$512$	−1380.66	−0.119174
$513$	3409.18	0.293410
$514$	14444.1	1.23949
$515$	−9641.76	−0.824984
$516$	−2449.84	−0.209008
$517$	0	0
$518$	−1413.54	−0.119899
$519$	−9621.56	−0.813757
$520$	6857.54	0.578313
$521$	17688.4	1.48742	0.743708	−	0.668504i	$-0.233065\pi$
$521$	17688.4	1.48742	0.743708	+	0.668504i	$0.233065\pi$
$522$	2123.73	0.178071
$523$	−21439.3	−1.79249	−0.896246	−	0.443557i	$-0.853716\pi$
$523$	−21439.3	−1.79249	−0.896246	+	0.443557i	$0.853716\pi$
$524$	−9260.09	−0.772002
$525$	18847.3	1.56679
$526$	−7190.32	−0.596032
$527$	4198.40	0.347031
$528$	0	0
$529$	−12165.9	−0.999907
$530$	1563.71	0.128157
$531$	1807.85	0.147747
$532$	−11176.2	−0.910811
$533$	−822.609	−0.0668502
$534$	1688.46	0.136829
$535$	−13101.9	−1.05877
$536$	10917.2	0.879761
$537$	2493.19	0.200352
$538$	−2039.49	−0.163436
$539$	0	0
$540$	−2739.75	−0.218333
$541$	7188.86	0.571300	0.285650	−	0.958334i	$-0.407791\pi$
$541$	7188.86	0.571300	0.285650	+	0.958334i	$0.407791\pi$
$542$	−7144.85	−0.566232
$543$	−12303.3	−0.972352
$544$	−3858.95	−0.304138
$545$	−24216.8	−1.90336
$546$	1409.89	0.110509
$547$	−5564.78	−0.434978	−0.217489	−	0.976063i	$-0.569787\pi$
$547$	−5564.78	−0.434978	−0.217489	+	0.976063i	$0.569787\pi$
$548$	−11814.8	−0.920990
$549$	1847.30	0.143608
$550$	0	0
$551$	−16472.0	−1.27356
$552$	73.6414	0.00567824
$553$	−10764.7	−0.827780
$554$	1186.76	0.0910123
$555$	2687.64	0.205557
$556$	10507.3	0.801455
$557$	−1479.13	−0.112519	−0.0562593	−	0.998416i	$-0.517917\pi$
$557$	−1479.13	−0.112519	−0.0562593	+	0.998416i	$0.517917\pi$
$558$	−3139.81	−0.238206
$559$	−2397.03	−0.181366
$560$	−1534.63	−0.115804
$561$	0	0
$562$	1555.35	0.116741
$563$	−5810.38	−0.434952	−0.217476	−	0.976066i	$-0.569782\pi$
$563$	−5810.38	−0.434952	−0.217476	+	0.976066i	$0.569782\pi$
$564$	−1900.02	−0.141853
$565$	5282.85	0.393365
$566$	13957.8	1.03656
$567$	−1516.37	−0.112313
$568$	9738.32	0.719386
$569$	−17138.1	−1.26268	−0.631341	−	0.775505i	$-0.717495\pi$
$569$	−17138.1	−1.26268	−0.631341	+	0.775505i	$0.717495\pi$
$570$	−14704.9	−1.08056
$571$	−4065.88	−0.297989	−0.148994	−	0.988838i	$-0.547604\pi$
$571$	−4065.88	−0.297989	−0.148994	+	0.988838i	$0.547604\pi$
$572$	0	0
$573$	5518.31	0.402322
$574$	2007.07	0.145947
$575$	357.806	0.0259505
$576$	3160.96	0.228658
$577$	−5452.43	−0.393392	−0.196696	−	0.980464i	$-0.563021\pi$
$577$	−5452.43	−0.393392	−0.196696	+	0.980464i	$0.563021\pi$
$578$	−8029.67	−0.577838
$579$	9711.33	0.697045
$580$	13237.5	0.947686
$581$	−12574.3	−0.897880
$582$	−3713.83	−0.264507
$583$	0	0
$584$	25031.4	1.77364
$585$	−2680.70	−0.189459
$586$	−1052.40	−0.0741880
$587$	−13440.4	−0.945051	−0.472526	−	0.881317i	$-0.656658\pi$
$587$	−13440.4	−0.945051	−0.472526	+	0.881317i	$0.656658\pi$
$588$	105.820	0.00742166
$589$	24352.9	1.70364
$590$	−7797.84	−0.544122
$591$	−9586.79	−0.667255
$592$	−159.448	−0.0110697
$593$	−12395.2	−0.858367	−0.429183	−	0.903217i	$-0.641199\pi$
$593$	−12395.2	−0.858367	−0.429183	+	0.903217i	$0.641199\pi$
$594$	0	0
$595$	−8745.76	−0.602591
$596$	−11540.5	−0.793147
$597$	8898.32	0.610024
$598$	26.7661	0.00183034
$599$	13298.5	0.907118	0.453559	−	0.891226i	$-0.350154\pi$
$599$	13298.5	0.907118	0.453559	+	0.891226i	$0.350154\pi$
$600$	23178.9	1.57713
$601$	−13300.2	−0.902709	−0.451355	−	0.892345i	$-0.649059\pi$
$601$	−13300.2	−0.902709	−0.451355	+	0.892345i	$0.649059\pi$
$602$	5848.48	0.395957
$603$	−4267.68	−0.288215
$604$	7832.88	0.527675
$605$	0	0
$606$	6301.22	0.422392
$607$	1402.79	0.0938018	0.0469009	−	0.998900i	$-0.485066\pi$
$607$	1402.79	0.0938018	0.0469009	+	0.998900i	$0.485066\pi$
$608$	−22383.8	−1.49307
$609$	7326.56	0.487499
$610$	−7968.01	−0.528878
$611$	−1859.07	−0.123093
$612$	926.303	0.0611823
$613$	17095.6	1.12640	0.563200	−	0.826321i	$-0.309570\pi$
$613$	17095.6	1.12640	0.563200	+	0.826321i	$0.309570\pi$
$614$	4285.22	0.281657
$615$	−3816.14	−0.250214
$616$	0	0
$617$	−5567.24	−0.363255	−0.181628	−	0.983367i	$-0.558137\pi$
$617$	−5567.24	−0.363255	−0.181628	+	0.983367i	$0.558137\pi$
$618$	−2437.91	−0.158685
$619$	−6908.19	−0.448568	−0.224284	−	0.974524i	$-0.572004\pi$
$619$	−6908.19	−0.448568	−0.224284	+	0.974524i	$0.572004\pi$
$620$	−19570.9	−1.26772
$621$	−28.7874	−0.00186022
$622$	19006.8	1.22525
$623$	5824.93	0.374592
$624$	159.037	0.0102028
$625$	55047.1	3.52302
$626$	−13134.3	−0.838583
$627$	0	0
$628$	12296.2	0.781325
$629$	−908.684	−0.0576019
$630$	6540.59	0.413625
$631$	24021.9	1.51552	0.757761	−	0.652532i	$-0.226293\pi$
$631$	24021.9	1.51552	0.757761	+	0.652532i	$0.226293\pi$
$632$	−13238.7	−0.833240
$633$	−10721.0	−0.673177
$634$	−5324.33	−0.333527
$635$	−11923.6	−0.745156
$636$	−571.364	−0.0356227
$637$	103.539	0.00644014
$638$	0	0
$639$	−3806.84	−0.235675
$640$	16802.3	1.03776
$641$	16928.4	1.04311	0.521554	−	0.853218i	$-0.325352\pi$
$641$	16928.4	1.04311	0.521554	+	0.853218i	$0.325352\pi$
$642$	−3312.80	−0.203654
$643$	331.007	0.0203012	0.0101506	−	0.999948i	$-0.496769\pi$
$643$	331.007	0.0203012	0.0101506	+	0.999948i	$0.496769\pi$
$644$	94.3730	0.00577456
$645$	−11120.0	−0.678836
$646$	4971.70	0.302800
$647$	14400.6	0.875035	0.437517	−	0.899210i	$-0.355858\pi$
$647$	14400.6	0.875035	0.437517	+	0.899210i	$0.355858\pi$
$648$	−1864.87	−0.113054
$649$	0	0
$650$	8424.72	0.508377
$651$	−10831.9	−0.652128
$652$	7404.68	0.444769
$653$	13145.4	0.787779	0.393889	−	0.919158i	$-0.371129\pi$
$653$	13145.4	0.787779	0.393889	+	0.919158i	$0.371129\pi$
$654$	−6123.19	−0.366110
$655$	−42032.3	−2.50738
$656$	226.398	0.0134746
$657$	−9785.09	−0.581055
$658$	4535.91	0.268736
$659$	−3909.26	−0.231082	−0.115541	−	0.993303i	$-0.536860\pi$
$659$	−3909.26	−0.231082	−0.115541	+	0.993303i	$0.536860\pi$
$660$	0	0
$661$	20672.6	1.21645	0.608224	−	0.793765i	$-0.291882\pi$
$661$	20672.6	1.21645	0.608224	+	0.793765i	$0.291882\pi$
$662$	−12489.1	−0.733237
$663$	906.338	0.0530909
$664$	−15464.1	−0.903802
$665$	−50729.8	−2.95822
$666$	679.567	0.0395386
$667$	139.091	0.00807437
$668$	−13155.5	−0.761981
$669$	3851.51	0.222583
$670$	18407.9	1.06143
$671$	0	0
$672$	9956.08	0.571524
$673$	19328.2	1.10705	0.553526	−	0.832832i	$-0.313282\pi$
$673$	19328.2	1.10705	0.553526	+	0.832832i	$0.313282\pi$
$674$	18542.6	1.05969
$675$	−9060.94	−0.516675
$676$	9476.99	0.539200
$677$	31564.4	1.79190	0.895952	−	0.444150i	$-0.146494\pi$
$677$	31564.4	1.79190	0.895952	+	0.444150i	$0.146494\pi$
$678$	1335.76	0.0756633
$679$	−12812.2	−0.724133
$680$	−10755.8	−0.606565
$681$	3366.25	0.189420
$682$	0	0
$683$	−17726.5	−0.993096	−0.496548	−	0.868009i	$-0.665399\pi$
$683$	−17726.5	−0.993096	−0.496548	+	0.868009i	$0.665399\pi$
$684$	5373.02	0.300355
$685$	−53628.2	−2.99128
$686$	11362.2	0.632375
$687$	14409.3	0.800219
$688$	659.711	0.0365570
$689$	−559.049	−0.0309116
$690$	124.169	0.00685080
$691$	−23126.9	−1.27321	−0.636606	−	0.771189i	$-0.719662\pi$
$691$	−23126.9	−1.27321	−0.636606	+	0.771189i	$0.719662\pi$
$692$	−15164.0	−0.833019
$693$	0	0
$694$	9514.23	0.520397
$695$	47693.5	2.60305
$696$	9010.38	0.490715
$697$	1290.23	0.0701160
$698$	19112.3	1.03641
$699$	315.418	0.0170675
$700$	29704.3	1.60388
$701$	−2215.31	−0.119359	−0.0596797	−	0.998218i	$-0.519008\pi$
$701$	−2215.31	−0.119359	−0.0596797	+	0.998218i	$0.519008\pi$
$702$	−677.813	−0.0364422
$703$	−5270.83	−0.282778
$704$	0	0
$705$	−8624.33	−0.460725
$706$	−3297.30	−0.175773
$707$	21738.3	1.15637
$708$	2849.25	0.151245
$709$	−10763.9	−0.570165	−0.285082	−	0.958503i	$-0.592021\pi$
$709$	−10763.9	−0.570165	−0.285082	+	0.958503i	$0.592021\pi$
$710$	16420.1	0.867939
$711$	5175.18	0.272974
$712$	7163.65	0.377063
$713$	−205.637	−0.0108011
$714$	−2211.36	−0.115908
$715$	0	0
$716$	3929.38	0.205094
$717$	3272.08	0.170430
$718$	20844.9	1.08346
$719$	−15669.7	−0.812768	−0.406384	−	0.913702i	$-0.633211\pi$
$719$	−15669.7	−0.812768	−0.406384	+	0.913702i	$0.633211\pi$
$720$	737.781	0.0381882
$721$	−8410.43	−0.434426
$722$	16431.6	0.846983
$723$	−3205.74	−0.164900
$724$	−19390.6	−0.995369
$725$	43779.3	2.24265
$726$	0	0
$727$	7671.66	0.391370	0.195685	−	0.980667i	$-0.437307\pi$
$727$	7671.66	0.391370	0.195685	+	0.980667i	$0.437307\pi$
$728$	5981.78	0.304532
$729$	729.000	0.0370370
$730$	42206.3	2.13990
$731$	3759.65	0.190226
$732$	2911.43	0.147007
$733$	19563.6	0.985807	0.492904	−	0.870084i	$-0.335935\pi$
$733$	19563.6	0.985807	0.492904	+	0.870084i	$0.335935\pi$
$734$	−4665.45	−0.234611
$735$	480.325	0.0241048
$736$	189.011	0.00946606
$737$	0	0
$738$	−964.907	−0.0481284
$739$	5972.99	0.297321	0.148660	−	0.988888i	$-0.452504\pi$
$739$	5972.99	0.297321	0.148660	+	0.988888i	$0.452504\pi$
$740$	4235.84	0.210422
$741$	5257.22	0.260633
$742$	1364.02	0.0674859
$743$	1319.52	0.0651526	0.0325763	−	0.999469i	$-0.489629\pi$
$743$	1319.52	0.0651526	0.0325763	+	0.999469i	$0.489629\pi$
$744$	−13321.3	−0.656429
$745$	−52383.1	−2.57606
$746$	11005.2	0.540120
$747$	6045.13	0.296091
$748$	0	0
$749$	−11428.7	−0.557536
$750$	24525.3	1.19405
$751$	32452.2	1.57683	0.788414	−	0.615145i	$-0.210902\pi$
$751$	32452.2	1.57683	0.788414	+	0.615145i	$0.210902\pi$
$752$	511.652	0.0248112
$753$	−5977.13	−0.289268
$754$	3274.95	0.158179
$755$	35554.1	1.71383
$756$	−2389.86	−0.114972
$757$	2891.12	0.138810	0.0694051	−	0.997589i	$-0.477890\pi$
$757$	2891.12	0.138810	0.0694051	+	0.997589i	$0.477890\pi$
$758$	10670.2	0.511290
$759$	0	0
$760$	−62388.8	−2.97774
$761$	−22428.7	−1.06838	−0.534191	−	0.845364i	$-0.679384\pi$
$761$	−22428.7	−1.06838	−0.534191	+	0.845364i	$0.679384\pi$
$762$	−3014.88	−0.143330
$763$	−21124.1	−1.00229
$764$	8697.10	0.411846
$765$	4204.56	0.198714
$766$	−2893.98	−0.136506
$767$	2787.84	0.131243
$768$	12677.7	0.595659
$769$	5386.04	0.252569	0.126285	−	0.991994i	$-0.459695\pi$
$769$	5386.04	0.252569	0.126285	+	0.991994i	$0.459695\pi$
$770$	0	0
$771$	−23955.9	−1.11900
$772$	15305.5	0.713545
$773$	7509.87	0.349432	0.174716	−	0.984619i	$-0.444099\pi$
$773$	7509.87	0.349432	0.174716	+	0.984619i	$0.444099\pi$
$774$	−2811.68	−0.130573
$775$	−64725.1	−2.99999
$776$	−15756.7	−0.728909
$777$	2344.41	0.108243
$778$	4548.48	0.209603
$779$	7483.97	0.344212
$780$	−4224.90	−0.193943
$781$	0	0
$782$	−41.9814	−0.00191976
$783$	−3522.27	−0.160761
$784$	−28.4960	−0.00129810
$785$	55813.5	2.53767
$786$	−10627.8	−0.482293
$787$	759.591	0.0344047	0.0172023	−	0.999852i	$-0.494524\pi$
$787$	759.591	0.0344047	0.0172023	+	0.999852i	$0.494524\pi$
$788$	−15109.2	−0.683050
$789$	11925.4	0.538092
$790$	−22322.3	−1.00530
$791$	4608.19	0.207141
$792$	0	0
$793$	2848.68	0.127566
$794$	9924.76	0.443598
$795$	−2593.47	−0.115699
$796$	14024.2	0.624464
$797$	26047.6	1.15766	0.578830	−	0.815448i	$-0.303510\pi$
$797$	26047.6	1.15766	0.578830	+	0.815448i	$0.303510\pi$
$798$	−12827.0	−0.569011
$799$	2915.87	0.129106
$800$	59491.8	2.62919
$801$	−2800.36	−0.123528
$802$	−26613.4	−1.17176
$803$	0	0
$804$	−6726.05	−0.295037
$805$	428.366	0.0187552
$806$	−4841.83	−0.211596
$807$	3382.55	0.147548
$808$	26734.3	1.16399
$809$	11238.3	0.488403	0.244202	−	0.969724i	$-0.421474\pi$
$809$	11238.3	0.488403	0.244202	+	0.969724i	$0.421474\pi$
$810$	−3144.42	−0.136399
$811$	3933.59	0.170317	0.0851585	−	0.996367i	$-0.472860\pi$
$811$	3933.59	0.170317	0.0851585	+	0.996367i	$0.472860\pi$
$812$	11547.0	0.499039
$813$	11850.0	0.511188
$814$	0	0
$815$	33610.4	1.44457
$816$	−249.442	−0.0107013
$817$	21807.8	0.933855
$818$	−14386.7	−0.614938
$819$	−2338.35	−0.0997664
$820$	−6014.40	−0.256137
$821$	12639.9	0.537314	0.268657	−	0.963236i	$-0.413420\pi$
$821$	12639.9	0.537314	0.268657	+	0.963236i	$0.413420\pi$
$822$	−13559.9	−0.575370
$823$	−27042.8	−1.14538	−0.572692	−	0.819770i	$-0.694101\pi$
$823$	−27042.8	−1.14538	−0.572692	+	0.819770i	$0.694101\pi$
$824$	−10343.4	−0.437291
$825$	0	0
$826$	−6802.00	−0.286528
$827$	7525.49	0.316429	0.158215	−	0.987405i	$-0.449426\pi$
$827$	7525.49	0.316429	0.158215	+	0.987405i	$0.449426\pi$
$828$	−45.3702	−0.00190426
$829$	−25018.8	−1.04818	−0.524089	−	0.851663i	$-0.675594\pi$
$829$	−25018.8	−1.04818	−0.524089	+	0.851663i	$0.675594\pi$
$830$	−26074.6	−1.09044
$831$	−1968.29	−0.0821650
$832$	4874.44	0.203114
$833$	−162.397	−0.00675475
$834$	12059.3	0.500693
$835$	−59714.0	−2.47484
$836$	0	0
$837$	5207.47	0.215050
$838$	13184.8	0.543509
$839$	12384.0	0.509585	0.254793	−	0.966996i	$-0.417993\pi$
$839$	12384.0	0.509585	0.254793	+	0.966996i	$0.417993\pi$
$840$	27749.8	1.13983
$841$	−7370.61	−0.302210
$842$	10050.3	0.411351
$843$	−2579.59	−0.105392
$844$	−16896.7	−0.689111
$845$	43016.8	1.75127
$846$	−2180.66	−0.0886200
$847$	0	0
$848$	153.861	0.00623069
$849$	−23149.5	−0.935794
$850$	−13213.8	−0.533212
$851$	44.5072	0.00179282
$852$	−5999.74	−0.241253
$853$	21243.7	0.852719	0.426360	−	0.904554i	$-0.359796\pi$
$853$	21243.7	0.852719	0.426360	+	0.904554i	$0.359796\pi$
$854$	−6950.44	−0.278500
$855$	24388.6	0.975523
$856$	−14055.2	−0.561213
$857$	44030.0	1.75500	0.877501	−	0.479575i	$-0.159209\pi$
$857$	44030.0	1.75500	0.877501	+	0.479575i	$0.159209\pi$
$858$	0	0
$859$	16585.2	0.658764	0.329382	−	0.944197i	$-0.393160\pi$
$859$	16585.2	0.658764	0.329382	+	0.944197i	$0.393160\pi$
$860$	−17525.6	−0.694905
$861$	−3328.79	−0.131759
$862$	−23600.8	−0.932537
$863$	−36409.9	−1.43616	−0.718081	−	0.695960i	$-0.754979\pi$
$863$	−36409.9	−1.43616	−0.718081	+	0.695960i	$0.754979\pi$
$864$	−4786.43	−0.188469
$865$	−68830.6	−2.70556
$866$	−26349.5	−1.03394
$867$	13317.4	0.521666
$868$	−17071.5	−0.667564
$869$	0	0
$870$	15192.7	0.592048
$871$	−6581.09	−0.256018
$872$	−25978.9	−1.00890
$873$	6159.51	0.238795
$874$	−243.513	−0.00942444
$875$	84608.8	3.26891
$876$	−15421.7	−0.594809
$877$	12665.7	0.487675	0.243837	−	0.969816i	$-0.421594\pi$
$877$	12665.7	0.487675	0.243837	+	0.969816i	$0.421594\pi$
$878$	−17096.4	−0.657148
$879$	1745.44	0.0669762
$880$	0	0
$881$	−33584.8	−1.28434	−0.642169	−	0.766563i	$-0.721965\pi$
$881$	−33584.8	−1.28434	−0.642169	+	0.766563i	$0.721965\pi$
$882$	121.450	0.00463653
$883$	−16487.8	−0.628381	−0.314190	−	0.949360i	$-0.601733\pi$
$883$	−16487.8	−0.628381	−0.314190	+	0.949360i	$0.601733\pi$
$884$	1428.43	0.0543476
$885$	12933.0	0.491228
$886$	−23179.4	−0.878923
$887$	9359.13	0.354283	0.177141	−	0.984185i	$-0.443315\pi$
$887$	9359.13	0.354283	0.177141	+	0.984185i	$0.443315\pi$
$888$	2883.21	0.108957
$889$	−10400.9	−0.392389
$890$	12078.9	0.454927
$891$	0	0
$892$	6070.15	0.227852
$893$	16913.5	0.633806
$894$	−13245.0	−0.495503
$895$	17835.7	0.666126
$896$	14656.5	0.546472
$897$	−44.3923	−0.00165242
$898$	10986.6	0.408272
$899$	−25160.7	−0.933433
$900$	−14280.4	−0.528905
$901$	876.845	0.0324217
$902$	0	0
$903$	−9699.89	−0.357466
$904$	5667.26	0.208507
$905$	−88015.6	−3.23286
$906$	8989.82	0.329654
$907$	36421.5	1.33336	0.666678	−	0.745345i	$-0.267715\pi$
$907$	36421.5	1.33336	0.666678	+	0.745345i	$0.267715\pi$
$908$	5305.36	0.193904
$909$	−10450.8	−0.381331
$910$	10086.1	0.367418
$911$	−49025.8	−1.78298	−0.891492	−	0.453037i	$-0.850341\pi$
$911$	−49025.8	−1.78298	−0.891492	+	0.453037i	$0.850341\pi$
$912$	−1446.89	−0.0525343
$913$	0	0
$914$	27821.3	1.00683
$915$	13215.2	0.477465
$916$	22709.8	0.819161
$917$	−36664.4	−1.32036
$918$	1063.12	0.0382224
$919$	12936.3	0.464340	0.232170	−	0.972675i	$-0.425417\pi$
$919$	12936.3	0.464340	0.232170	+	0.972675i	$0.425417\pi$
$920$	526.815	0.0188789
$921$	−7107.18	−0.254277
$922$	−5709.86	−0.203952
$923$	−5870.43	−0.209347
$924$	0	0
$925$	14008.8	0.497954
$926$	8909.10	0.316168
$927$	4043.35	0.143259
$928$	23126.3	0.818060
$929$	−2273.34	−0.0802864	−0.0401432	−	0.999194i	$-0.512781\pi$
$929$	−2273.34	−0.0802864	−0.0401432	+	0.999194i	$0.512781\pi$
$930$	−22461.6	−0.791982
$931$	−941.982	−0.0331603
$932$	497.113	0.0174716
$933$	−31523.4	−1.10614
$934$	−21075.8	−0.738354
$935$	0	0
$936$	−2875.76	−0.100424
$937$	−17198.9	−0.599642	−0.299821	−	0.953996i	$-0.596927\pi$
$937$	−17198.9	−0.599642	−0.299821	+	0.953996i	$0.596927\pi$
$938$	16057.1	0.558936
$939$	21783.7	0.757064
$940$	−13592.3	−0.471631
$941$	−33205.7	−1.15035	−0.575173	−	0.818032i	$-0.695065\pi$
$941$	−33205.7	−1.15035	−0.575173	+	0.818032i	$0.695065\pi$
$942$	14112.4	0.488117
$943$	−63.1951	−0.00218231
$944$	−767.268	−0.0264539
$945$	−10847.8	−0.373416
$946$	0	0
$947$	23096.7	0.792546	0.396273	−	0.918133i	$-0.370303\pi$
$947$	23096.7	0.792546	0.396273	+	0.918133i	$0.370303\pi$
$948$	8156.32	0.279435
$949$	−15089.4	−0.516145
$950$	−76646.8	−2.61763
$951$	8830.56	0.301105
$952$	−9382.16	−0.319409
$953$	8790.13	0.298783	0.149391	−	0.988778i	$-0.452269\pi$
$953$	8790.13	0.298783	0.149391	+	0.988778i	$0.452269\pi$
$954$	−655.756	−0.0222546
$955$	39476.8	1.33763
$956$	5156.94	0.174464
$957$	0	0
$958$	−23731.5	−0.800346
$959$	−46779.5	−1.57517
$960$	22612.9	0.760236
$961$	7407.61	0.248653
$962$	1047.94	0.0351217
$963$	5494.38	0.183857
$964$	−5052.39	−0.168803
$965$	69472.8	2.31752
$966$	108.312	0.00360754
$967$	29582.8	0.983785	0.491892	−	0.870656i	$-0.336305\pi$
$967$	29582.8	0.983785	0.491892	+	0.870656i	$0.336305\pi$
$968$	0	0
$969$	−8245.72	−0.273365
$970$	−26568.0	−0.879429
$971$	34522.7	1.14097	0.570487	−	0.821307i	$-0.306755\pi$
$971$	34522.7	1.14097	0.570487	+	0.821307i	$0.306755\pi$
$972$	1148.94	0.0379137
$973$	41602.7	1.37073
$974$	5023.61	0.165264
$975$	−13972.7	−0.458957
$976$	−784.012	−0.0257127
$977$	58011.5	1.89964	0.949822	−	0.312790i	$-0.101264\pi$
$977$	58011.5	1.89964	0.949822	+	0.312790i	$0.101264\pi$
$978$	8498.37	0.277861
$979$	0	0
$980$	757.013	0.0246754
$981$	10155.5	0.330520
$982$	−22896.3	−0.744044
$983$	−21992.2	−0.713574	−0.356787	−	0.934186i	$-0.616128\pi$
$983$	−21992.2	−0.713574	−0.356787	+	0.934186i	$0.616128\pi$
$984$	−4093.82	−0.132628
$985$	−68581.9	−2.21848
$986$	−5136.63	−0.165906
$987$	−7522.94	−0.242612
$988$	8285.61	0.266802
$989$	−184.147	−0.00592066
$990$	0	0
$991$	−48417.3	−1.55199	−0.775997	−	0.630737i	$-0.782753\pi$
$991$	−48417.3	−1.55199	−0.775997	+	0.630737i	$0.782753\pi$
$992$	−34190.9	−1.09432
$993$	20713.6	0.661959
$994$	14323.2	0.457045
$995$	63656.7	2.02820
$996$	9527.39	0.303099
$997$	25882.6	0.822178	0.411089	−	0.911595i	$-0.365149\pi$
$997$	25882.6	0.822178	0.411089	+	0.911595i	$0.365149\pi$
$998$	−18339.8	−0.581700
$999$	−1127.08	−0.0356950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.v.1.4		6
3.2	odd	2		1089.4.a.bi.1.3		6
11.5	even	5		33.4.e.c.25.2	yes	12
11.9	even	5		33.4.e.c.4.2	✓	12
11.10	odd	2		363.4.a.u.1.3		6
33.5	odd	10		99.4.f.d.91.2		12
33.20	odd	10		99.4.f.d.37.2		12
33.32	even	2		1089.4.a.bk.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.c.4.2	✓	12	11.9	even	5
33.4.e.c.25.2	yes	12	11.5	even	5
99.4.f.d.37.2		12	33.20	odd	10
99.4.f.d.91.2		12	33.5	odd	10
363.4.a.u.1.3		6	11.10	odd	2
363.4.a.v.1.4		6	1.1	even	1	trivial
1089.4.a.bi.1.3		6	3.2	odd	2
1089.4.a.bk.1.4		6	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+1.80883 q^{2} -3.00000 q^{3} -4.72813 q^{4} -21.4614 q^{5} -5.42649 q^{6} -18.7206 q^{7} -23.0230 q^{8} +9.00000 q^{9} -38.8200 q^{10} +14.1844 q^{12} +13.8787 q^{13} -33.8624 q^{14} +64.3841 q^{15} -3.81969 q^{16} -21.7681 q^{17} +16.2795 q^{18} -126.266 q^{19} +101.472 q^{20} +56.1618 q^{21} +1.06620 q^{23} +69.0691 q^{24} +335.590 q^{25} +25.1042 q^{26} -27.0000 q^{27} +88.5134 q^{28} +130.455 q^{29} +116.460 q^{30} -192.869 q^{31} +177.275 q^{32} -39.3748 q^{34} +401.769 q^{35} -42.5532 q^{36} +41.7438 q^{37} -228.394 q^{38} -41.6360 q^{39} +494.106 q^{40} -59.2714 q^{41} +101.587 q^{42} -172.713 q^{43} -193.152 q^{45} +1.92857 q^{46} -133.951 q^{47} +11.4591 q^{48} +7.46030 q^{49} +607.026 q^{50} +65.3044 q^{51} -65.6203 q^{52} -40.2812 q^{53} -48.8384 q^{54} +431.005 q^{56} +378.798 q^{57} +235.970 q^{58} +200.872 q^{59} -304.417 q^{60} +205.256 q^{61} -348.868 q^{62} -168.485 q^{63} +351.218 q^{64} -297.856 q^{65} -474.187 q^{67} +102.923 q^{68} -3.19860 q^{69} +726.733 q^{70} -422.982 q^{71} -207.207 q^{72} -1087.23 q^{73} +75.5075 q^{74} -1006.77 q^{75} +597.003 q^{76} -75.3125 q^{78} +575.020 q^{79} +81.9757 q^{80} +81.0000 q^{81} -107.212 q^{82} +671.681 q^{83} -265.540 q^{84} +467.174 q^{85} -312.409 q^{86} -391.364 q^{87} -311.151 q^{89} -349.380 q^{90} -259.817 q^{91} -5.04113 q^{92} +578.608 q^{93} -242.295 q^{94} +2709.84 q^{95} -531.825 q^{96} +684.390 q^{97} +13.4944 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) \) 6 * q + 5 * q^2 - 18 * q^3 + 17 * q^4 + 9 * q^5 - 15 * q^6 + q^7 + 24 * q^8 + 54 * q^9 + 50 * q^10 - 51 * q^12 + 66 * q^13 - 42 * q^14 - 27 * q^15 - 71 * q^16 + 80 * q^17 + 45 * q^18 - 90 * q^19 + 455 * q^20 - 3 * q^21 - 196 * q^23 - 72 * q^24 + 351 * q^25 + 93 * q^26 - 162 * q^27 - 52 * q^28 + 579 * q^29 - 150 * q^30 + 129 * q^31 + 657 * q^32 + 530 * q^34 + 618 * q^35 + 153 * q^36 + 192 * q^37 - 769 * q^38 - 198 * q^39 + 1476 * q^40 + 990 * q^41 + 126 * q^42 + 208 * q^43 + 81 * q^45 + 493 * q^46 + 680 * q^47 + 213 * q^48 - 963 * q^49 + 1656 * q^50 - 240 * q^51 + 887 * q^52 + 197 * q^53 - 135 * q^54 - 429 * q^56 + 270 * q^57 + 395 * q^58 - 1229 * q^59 - 1365 * q^60 + 928 * q^61 + 2117 * q^62 + 9 * q^63 + 844 * q^64 + 2278 * q^65 - 264 * q^67 + 3279 * q^68 + 588 * q^69 + 665 * q^70 - 786 * q^71 + 216 * q^72 + 593 * q^73 + 1543 * q^74 - 1053 * q^75 - 434 * q^76 - 279 * q^78 - 421 * q^79 + 18 * q^80 + 486 * q^81 - 739 * q^82 + 2043 * q^83 + 156 * q^84 + 1900 * q^85 + 2683 * q^86 - 1737 * q^87 + 158 * q^89 + 450 * q^90 + 698 * q^91 - 3915 * q^92 - 387 * q^93 - 4934 * q^94 + 1038 * q^95 - 1971 * q^96 - 1711 * q^97 + 1405 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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