LMFDB - Embedded newform 363.4.a.v.1.2

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			1.2
Root			$3.80972$ 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)

$f(q)$	$=$	\(q-2.80972 q^{2} -3.00000 q^{3} -0.105474 q^{4} +12.1997 q^{5} +8.42916 q^{6} +6.70654 q^{7} +22.7741 q^{8} +9.00000 q^{9} -34.2777 q^{10} +0.316423 q^{12} +78.8371 q^{13} -18.8435 q^{14} -36.5991 q^{15} -63.1451 q^{16} -67.3763 q^{17} -25.2875 q^{18} +47.1974 q^{19} -1.28676 q^{20} -20.1196 q^{21} -21.9125 q^{23} -68.3223 q^{24} +23.8328 q^{25} -221.510 q^{26} -27.0000 q^{27} -0.707368 q^{28} +81.8261 q^{29} +102.833 q^{30} -67.1775 q^{31} -4.77291 q^{32} +189.309 q^{34} +81.8178 q^{35} -0.949269 q^{36} -382.927 q^{37} -132.611 q^{38} -236.511 q^{39} +277.837 q^{40} +368.938 q^{41} +56.5305 q^{42} +153.423 q^{43} +109.797 q^{45} +61.5680 q^{46} +381.947 q^{47} +189.435 q^{48} -298.022 q^{49} -66.9634 q^{50} +202.129 q^{51} -8.31528 q^{52} +586.639 q^{53} +75.8624 q^{54} +152.736 q^{56} -141.592 q^{57} -229.909 q^{58} +119.466 q^{59} +3.86027 q^{60} +865.731 q^{61} +188.750 q^{62} +60.3589 q^{63} +518.571 q^{64} +961.789 q^{65} -884.849 q^{67} +7.10647 q^{68} +65.7375 q^{69} -229.885 q^{70} +673.884 q^{71} +204.967 q^{72} +290.003 q^{73} +1075.92 q^{74} -71.4983 q^{75} -4.97811 q^{76} +664.530 q^{78} -738.693 q^{79} -770.351 q^{80} +81.0000 q^{81} -1036.61 q^{82} -445.439 q^{83} +2.12210 q^{84} -821.971 q^{85} -431.077 q^{86} -245.478 q^{87} +304.691 q^{89} -308.500 q^{90} +528.724 q^{91} +2.31121 q^{92} +201.533 q^{93} -1073.16 q^{94} +575.794 q^{95} +14.3187 q^{96} +1680.23 q^{97} +837.359 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.80972	−0.993386	−0.496693	−	0.867926i	$-0.665453\pi$
$2$	−2.80972	−0.993386	−0.496693	+	0.867926i	$0.665453\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−0.105474	−0.0131843
$5$	12.1997	1.09117	0.545587	−	0.838054i	$-0.316307\pi$
$5$	12.1997	1.09117	0.545587	+	0.838054i	$0.316307\pi$
$6$	8.42916	0.573532
$7$	6.70654	0.362119	0.181060	−	0.983472i	$-0.442047\pi$
$7$	6.70654	0.362119	0.181060	+	0.983472i	$0.442047\pi$
$8$	22.7741	1.00648
$9$	9.00000	0.333333
$10$	−34.2777	−1.08396
$11$	0	0
$11$	0	0
$12$	0.316423	0.00761195
$13$	78.8371	1.68196	0.840980	−	0.541067i	$-0.181979\pi$
$13$	78.8371	1.68196	0.840980	+	0.541067i	$0.181979\pi$
$14$	−18.8435	−0.359724
$15$	−36.5991	−0.629990
$16$	−63.1451	−0.986642
$17$	−67.3763	−0.961244	−0.480622	−	0.876928i	$-0.659589\pi$
$17$	−67.3763	−0.961244	−0.480622	+	0.876928i	$0.659589\pi$
$18$	−25.2875	−0.331129
$19$	47.1974	0.569885	0.284943	−	0.958545i	$-0.408025\pi$
$19$	47.1974	0.569885	0.284943	+	0.958545i	$0.408025\pi$
$20$	−1.28676	−0.0143864
$21$	−20.1196	−0.209070
$22$	0	0
$23$	−21.9125	−0.198655	−0.0993277	−	0.995055i	$-0.531669\pi$
$23$	−21.9125	−0.198655	−0.0993277	+	0.995055i	$0.531669\pi$
$24$	−68.3223	−0.581093
$25$	23.8328	0.190662
$26$	−221.510	−1.67083
$27$	−27.0000	−0.192450
$28$	−0.707368	−0.00477428
$29$	81.8261	0.523956	0.261978	−	0.965074i	$-0.415625\pi$
$29$	81.8261	0.523956	0.261978	+	0.965074i	$0.415625\pi$
$30$	102.833	0.625823
$31$	−67.1775	−0.389208	−0.194604	−	0.980882i	$-0.562342\pi$
$31$	−67.1775	−0.389208	−0.194604	+	0.980882i	$0.562342\pi$
$32$	−4.77291	−0.0263668
$33$	0	0
$34$	189.309	0.954887
$35$	81.8178	0.395135
$36$	−0.949269	−0.00439476
$37$	−382.927	−1.70143	−0.850714	−	0.525629i	$-0.823830\pi$
$37$	−382.927	−1.70143	−0.850714	+	0.525629i	$0.823830\pi$
$38$	−132.611	−0.566116
$39$	−236.511	−0.971079
$40$	277.837	1.09825
$41$	368.938	1.40533	0.702664	−	0.711522i	$-0.251994\pi$
$41$	368.938	1.40533	0.702664	+	0.711522i	$0.251994\pi$
$42$	56.5305	0.207687
$43$	153.423	0.544113	0.272056	−	0.962281i	$-0.412296\pi$
$43$	153.423	0.544113	0.272056	+	0.962281i	$0.412296\pi$
$44$	0	0
$45$	109.797	0.363725
$46$	61.5680	0.197341
$47$	381.947	1.18538	0.592689	−	0.805432i	$-0.298066\pi$
$47$	381.947	1.18538	0.592689	+	0.805432i	$0.298066\pi$
$48$	189.435	0.569638
$49$	−298.022	−0.868870
$50$	−66.9634	−0.189401
$51$	202.129	0.554975
$52$	−8.31528	−0.0221754
$53$	586.639	1.52040	0.760198	−	0.649691i	$-0.225102\pi$
$53$	586.639	1.52040	0.760198	+	0.649691i	$0.225102\pi$
$54$	75.8624	0.191177
$55$	0	0
$56$	152.736	0.364467
$57$	−141.592	−0.329023
$58$	−229.909	−0.520491
$59$	119.466	0.263613	0.131807	−	0.991275i	$-0.457922\pi$
$59$	119.466	0.263613	0.131807	+	0.991275i	$0.457922\pi$
$60$	3.86027	0.00830597
$61$	865.731	1.81714	0.908570	−	0.417733i	$-0.137175\pi$
$61$	865.731	1.81714	0.908570	+	0.417733i	$0.137175\pi$
$62$	188.750	0.386633
$63$	60.3589	0.120706
$64$	518.571	1.01283
$65$	961.789	1.83531
$66$	0	0
$67$	−884.849	−1.61346	−0.806728	−	0.590923i	$-0.798764\pi$
$67$	−884.849	−1.61346	−0.806728	+	0.590923i	$0.798764\pi$
$68$	7.10647	0.0126733
$69$	65.7375	0.114694
$70$	−229.885	−0.392522
$71$	673.884	1.12641	0.563206	−	0.826316i	$-0.309568\pi$
$71$	673.884	1.12641	0.563206	+	0.826316i	$0.309568\pi$
$72$	204.967	0.335494
$73$	290.003	0.464962	0.232481	−	0.972601i	$-0.425316\pi$
$73$	290.003	0.464962	0.232481	+	0.972601i	$0.425316\pi$
$74$	1075.92	1.69017
$75$	−71.4983	−0.110079
$76$	−4.97811	−0.00751353
$77$	0	0
$78$	664.530	0.964657
$79$	−738.693	−1.05202	−0.526010	−	0.850479i	$-0.676312\pi$
$79$	−738.693	−1.05202	−0.526010	+	0.850479i	$0.676312\pi$
$80$	−770.351	−1.07660
$81$	81.0000	0.111111
$82$	−1036.61	−1.39603
$83$	−445.439	−0.589075	−0.294538	−	0.955640i	$-0.595166\pi$
$83$	−445.439	−0.589075	−0.294538	+	0.955640i	$0.595166\pi$
$84$	2.12210	0.00275643
$85$	−821.971	−1.04889
$86$	−431.077	−0.540514
$87$	−245.478	−0.302506
$88$	0	0
$89$	304.691	0.362890	0.181445	−	0.983401i	$-0.441923\pi$
$89$	304.691	0.362890	0.181445	+	0.983401i	$0.441923\pi$
$90$	−308.500	−0.361319
$91$	528.724	0.609070
$92$	2.31121	0.00261913
$93$	201.533	0.224709
$94$	−1073.16	−1.17754
$95$	575.794	0.621844
$96$	14.3187	0.0152229
$97$	1680.23	1.75877	0.879387	−	0.476107i	$-0.157952\pi$
$97$	1680.23	1.75877	0.879387	+	0.476107i	$0.157952\pi$
$98$	837.359	0.863123
$99$	0	0
$100$	−2.51374	−0.00251374
$101$	570.857	0.562400	0.281200	−	0.959649i	$-0.409268\pi$
$101$	570.857	0.562400	0.281200	+	0.959649i	$0.409268\pi$
$102$	−567.926	−0.551304
$103$	981.463	0.938897	0.469448	−	0.882960i	$-0.344453\pi$
$103$	981.463	0.938897	0.469448	+	0.882960i	$0.344453\pi$
$104$	1795.44	1.69286
$105$	−245.453	−0.228131
$106$	−1648.29	−1.51034
$107$	−663.315	−0.599300	−0.299650	−	0.954049i	$-0.596870\pi$
$107$	−663.315	−0.599300	−0.299650	+	0.954049i	$0.596870\pi$
$108$	2.84781	0.00253732
$109$	−507.146	−0.445650	−0.222825	−	0.974859i	$-0.571528\pi$
$109$	−507.146	−0.445650	−0.222825	+	0.974859i	$0.571528\pi$
$110$	0	0
$111$	1148.78	0.982320
$112$	−423.485	−0.357282
$113$	1136.88	0.946449	0.473224	−	0.880942i	$-0.343090\pi$
$113$	1136.88	0.946449	0.473224	+	0.880942i	$0.343090\pi$
$114$	397.834	0.326847
$115$	−267.326	−0.216768
$116$	−8.63055	−0.00690799
$117$	709.534	0.560653
$118$	−335.667	−0.261870
$119$	−451.862	−0.348085
$120$	−833.512	−0.634074
$121$	0	0
$122$	−2432.46	−1.80512
$123$	−1106.81	−0.811366
$124$	7.08550	0.00513143
$125$	−1234.21	−0.883129
$126$	−169.592	−0.119908
$127$	1428.00	0.997754	0.498877	−	0.866673i	$-0.333746\pi$
$127$	1428.00	0.997754	0.498877	+	0.866673i	$0.333746\pi$
$128$	−1418.86	−0.979769
$129$	−460.270	−0.314144
$130$	−2702.36	−1.82317
$131$	−746.926	−0.498162	−0.249081	−	0.968483i	$-0.580129\pi$
$131$	−746.926	−0.498162	−0.249081	+	0.968483i	$0.580129\pi$
$132$	0	0
$133$	316.531	0.206366
$134$	2486.18	1.60278
$135$	−329.392	−0.209997
$136$	−1534.44	−0.967476
$137$	−145.916	−0.0909961	−0.0454980	−	0.998964i	$-0.514487\pi$
$137$	−145.916	−0.0909961	−0.0454980	+	0.998964i	$0.514487\pi$
$138$	−184.704	−0.113935
$139$	−545.621	−0.332942	−0.166471	−	0.986046i	$-0.553237\pi$
$139$	−545.621	−0.332942	−0.166471	+	0.986046i	$0.553237\pi$
$140$	−8.62968	−0.00520958
$141$	−1145.84	−0.684378
$142$	−1893.43	−1.11896
$143$	0	0
$144$	−568.306	−0.328881
$145$	998.255	0.571728
$146$	−814.826	−0.461887
$147$	894.067	0.501642
$148$	40.3890	0.0224321
$149$	709.609	0.390157	0.195079	−	0.980788i	$-0.437504\pi$
$149$	709.609	0.390157	0.195079	+	0.980788i	$0.437504\pi$
$150$	200.890	0.109351
$151$	−1849.74	−0.996886	−0.498443	−	0.866923i	$-0.666095\pi$
$151$	−1849.74	−0.996886	−0.498443	+	0.866923i	$0.666095\pi$
$152$	1074.88	0.573580
$153$	−606.387	−0.320415
$154$	0	0
$155$	−819.546	−0.424693
$156$	24.9459	0.0128030
$157$	−1031.79	−0.524498	−0.262249	−	0.965000i	$-0.584464\pi$
$157$	−1031.79	−0.524498	−0.262249	+	0.965000i	$0.584464\pi$
$158$	2075.52	1.04506
$159$	−1759.92	−0.877801
$160$	−58.2281	−0.0287708
$161$	−146.957	−0.0719369
$162$	−227.587	−0.110376
$163$	−3134.37	−1.50615	−0.753076	−	0.657933i	$-0.771431\pi$
$163$	−3134.37	−1.50615	−0.753076	+	0.657933i	$0.771431\pi$
$164$	−38.9135	−0.0185282
$165$	0	0
$166$	1251.56	0.585179
$167$	1862.40	0.862973	0.431487	−	0.902119i	$-0.357989\pi$
$167$	1862.40	0.862973	0.431487	+	0.902119i	$0.357989\pi$
$168$	−458.207	−0.210425
$169$	4018.28	1.82899
$170$	2309.51	1.04195
$171$	424.776	0.189962
$172$	−16.1822	−0.00717374
$173$	2769.23	1.21700	0.608500	−	0.793554i	$-0.291772\pi$
$173$	2769.23	1.21700	0.608500	+	0.793554i	$0.291772\pi$
$174$	689.726	0.300506
$175$	159.835	0.0690424
$176$	0	0
$177$	−358.399	−0.152197
$178$	−856.097	−0.360490
$179$	3474.79	1.45094	0.725469	−	0.688255i	$-0.241623\pi$
$179$	3474.79	1.45094	0.725469	+	0.688255i	$0.241623\pi$
$180$	−11.5808	−0.00479545
$181$	−1027.42	−0.421919	−0.210959	−	0.977495i	$-0.567659\pi$
$181$	−1027.42	−0.421919	−0.210959	+	0.977495i	$0.567659\pi$
$182$	−1485.57	−0.605041
$183$	−2597.19	−1.04913
$184$	−499.038	−0.199943
$185$	−4671.60	−1.85656
$186$	−566.250	−0.223223
$187$	0	0
$188$	−40.2856	−0.0156284
$189$	−181.077	−0.0696899
$190$	−1617.82	−0.617731
$191$	−3070.47	−1.16320	−0.581601	−	0.813474i	$-0.697574\pi$
$191$	−3070.47	−1.16320	−0.581601	+	0.813474i	$0.697574\pi$
$192$	−1555.71	−0.584760
$193$	757.933	0.282680	0.141340	−	0.989961i	$-0.454859\pi$
$193$	757.933	0.282680	0.141340	+	0.989961i	$0.454859\pi$
$194$	−4720.97	−1.74714
$195$	−2885.37	−1.05962
$196$	31.4337	0.0114554
$197$	2783.97	1.00685	0.503425	−	0.864039i	$-0.332073\pi$
$197$	2783.97	1.00685	0.503425	+	0.864039i	$0.332073\pi$
$198$	0	0
$199$	1770.84	0.630813	0.315406	−	0.948957i	$-0.397859\pi$
$199$	1770.84	0.630813	0.315406	+	0.948957i	$0.397859\pi$
$200$	542.770	0.191898
$201$	2654.55	0.931529
$202$	−1603.95	−0.558680
$203$	548.770	0.189735
$204$	−21.3194	−0.00731694
$205$	4500.93	1.53346
$206$	−2757.63	−0.932687
$207$	−197.213	−0.0662184
$208$	−4978.17	−1.65949
$209$	0	0
$210$	689.655	0.226623
$211$	−2282.14	−0.744591	−0.372295	−	0.928114i	$-0.621429\pi$
$211$	−2282.14	−0.744591	−0.372295	+	0.928114i	$0.621429\pi$
$212$	−61.8753	−0.0200453
$213$	−2021.65	−0.650335
$214$	1863.73	0.595336
$215$	1871.72	0.593722
$216$	−614.901	−0.193698
$217$	−450.529	−0.140940
$218$	1424.94	0.442702
$219$	−870.008	−0.268446
$220$	0	0
$221$	−5311.75	−1.61677
$222$	−3227.75	−0.975823
$223$	667.357	0.200402	0.100201	−	0.994967i	$-0.468051\pi$
$223$	667.357	0.200402	0.100201	+	0.994967i	$0.468051\pi$
$224$	−32.0097	−0.00954794
$225$	214.495	0.0635540
$226$	−3194.32	−0.940189
$227$	1579.65	0.461873	0.230937	−	0.972969i	$-0.425821\pi$
$227$	1579.65	0.461873	0.230937	+	0.972969i	$0.425821\pi$
$228$	14.9343	0.00433794
$229$	1556.70	0.449212	0.224606	−	0.974450i	$-0.427890\pi$
$229$	1556.70	0.449212	0.224606	+	0.974450i	$0.427890\pi$
$230$	751.111	0.215334
$231$	0	0
$232$	1863.52	0.527353
$233$	2080.25	0.584899	0.292450	−	0.956281i	$-0.405530\pi$
$233$	2080.25	0.584899	0.292450	+	0.956281i	$0.405530\pi$
$234$	−1993.59	−0.556945
$235$	4659.64	1.29345
$236$	−12.6006	−0.00347555
$237$	2216.08	0.607384
$238$	1269.61	0.345783
$239$	−3686.69	−0.997791	−0.498896	−	0.866662i	$-0.666261\pi$
$239$	−3686.69	−0.997791	−0.498896	+	0.866662i	$0.666261\pi$
$240$	2311.05	0.621574
$241$	4069.62	1.08775	0.543874	−	0.839167i	$-0.316957\pi$
$241$	4069.62	1.08775	0.543874	+	0.839167i	$0.316957\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−91.3124	−0.0239577
$245$	−3635.78	−0.948089
$246$	3109.84	0.806000
$247$	3720.90	0.958523
$248$	−1529.91	−0.391731
$249$	1336.32	0.340103
$250$	3467.79	0.877288
$251$	6203.06	1.55989	0.779947	−	0.625845i	$-0.215246\pi$
$251$	6203.06	1.55989	0.779947	+	0.625845i	$0.215246\pi$
$252$	−6.36631	−0.00159143
$253$	0	0
$254$	−4012.29	−0.991155
$255$	2465.91	0.605574
$256$	−161.980	−0.0395460
$257$	1109.34	0.269256	0.134628	−	0.990896i	$-0.457016\pi$
$257$	1109.34	0.269256	0.134628	+	0.990896i	$0.457016\pi$
$258$	1293.23	0.312066
$259$	−2568.12	−0.616120
$260$	−101.444	−0.0241973
$261$	736.435	0.174652
$262$	2098.65	0.494867
$263$	6866.55	1.60992	0.804961	−	0.593327i	$-0.202186\pi$
$263$	6866.55	1.60992	0.804961	+	0.593327i	$0.202186\pi$
$264$	0	0
$265$	7156.82	1.65902
$266$	−889.364	−0.205001
$267$	−914.074	−0.209515
$268$	93.3289	0.0212723
$269$	−2101.39	−0.476298	−0.238149	−	0.971229i	$-0.576541\pi$
$269$	−2101.39	−0.476298	−0.238149	+	0.971229i	$0.576541\pi$
$270$	925.499	0.208608
$271$	−3640.81	−0.816101	−0.408050	−	0.912959i	$-0.633791\pi$
$271$	−3640.81	−0.816101	−0.408050	+	0.912959i	$0.633791\pi$
$272$	4254.48	0.948404
$273$	−1586.17	−0.351647
$274$	409.984	0.0903942
$275$	0	0
$276$	−6.93362	−0.00151215
$277$	−5725.88	−1.24200	−0.621002	−	0.783809i	$-0.713274\pi$
$277$	−5725.88	−1.24200	−0.621002	+	0.783809i	$0.713274\pi$
$278$	1533.04	0.330740
$279$	−604.598	−0.129736
$280$	1863.33	0.397697
$281$	3402.31	0.722294	0.361147	−	0.932509i	$-0.382385\pi$
$281$	3402.31	0.722294	0.361147	+	0.932509i	$0.382385\pi$
$282$	3219.49	0.679851
$283$	6487.92	1.36278	0.681390	−	0.731921i	$-0.261376\pi$
$283$	6487.92	1.36278	0.681390	+	0.731921i	$0.261376\pi$
$284$	−71.0774	−0.0148510
$285$	−1727.38	−0.359022
$286$	0	0
$287$	2474.30	0.508896
$288$	−42.9562	−0.00878895
$289$	−373.435	−0.0760095
$290$	−2804.82	−0.567946
$291$	−5040.68	−1.01543
$292$	−30.5878	−0.00613019
$293$	4009.60	0.799465	0.399733	−	0.916632i	$-0.369103\pi$
$293$	4009.60	0.799465	0.399733	+	0.916632i	$0.369103\pi$
$294$	−2512.08	−0.498324
$295$	1457.45	0.287648
$296$	−8720.83	−1.71246
$297$	0	0
$298$	−1993.80	−0.387577
$299$	−1727.52	−0.334130
$300$	7.54123	0.00145131
$301$	1028.94	0.197034
$302$	5197.25	0.990292
$303$	−1712.57	−0.324702
$304$	−2980.28	−0.562272
$305$	10561.7	1.98282
$306$	1703.78	0.318296
$307$	8966.59	1.66694	0.833470	−	0.552565i	$-0.186351\pi$
$307$	8966.59	1.66694	0.833470	+	0.552565i	$0.186351\pi$
$308$	0	0
$309$	−2944.39	−0.542072
$310$	2302.69	0.421885
$311$	−1397.40	−0.254788	−0.127394	−	0.991852i	$-0.540661\pi$
$311$	−1397.40	−0.254788	−0.127394	+	0.991852i	$0.540661\pi$
$312$	−5386.33	−0.977375
$313$	−3037.00	−0.548438	−0.274219	−	0.961667i	$-0.588419\pi$
$313$	−3037.00	−0.548438	−0.274219	+	0.961667i	$0.588419\pi$
$314$	2899.05	0.521029
$315$	736.360	0.131712
$316$	77.9132	0.0138701
$317$	2997.34	0.531064	0.265532	−	0.964102i	$-0.414452\pi$
$317$	2997.34	0.531064	0.265532	+	0.964102i	$0.414452\pi$
$318$	4944.87	0.871996
$319$	0	0
$320$	6326.41	1.10518
$321$	1989.94	0.346006
$322$	412.908	0.0714611
$323$	−3179.98	−0.547799
$324$	−8.54342	−0.00146492
$325$	1878.90	0.320686
$326$	8806.71	1.49619
$327$	1521.44	0.257296
$328$	8402.24	1.41444
$329$	2561.55	0.429248
$330$	0	0
$331$	−7286.34	−1.20995	−0.604975	−	0.796245i	$-0.706817\pi$
$331$	−7286.34	−1.20995	−0.604975	+	0.796245i	$0.706817\pi$
$332$	46.9823	0.00776654
$333$	−3446.34	−0.567143
$334$	−5232.81	−0.857266
$335$	−10794.9	−1.76056
$336$	1270.46	0.206277
$337$	1724.12	0.278691	0.139346	−	0.990244i	$-0.455500\pi$
$337$	1724.12	0.278691	0.139346	+	0.990244i	$0.455500\pi$
$338$	−11290.2	−1.81689
$339$	−3410.64	−0.546433
$340$	86.6968	0.0138288
$341$	0	0
$342$	−1193.50	−0.188705
$343$	−4299.04	−0.676754
$344$	3494.08	0.547640
$345$	801.978	0.125151
$346$	−7780.77	−1.20895
$347$	−5450.67	−0.843249	−0.421624	−	0.906771i	$-0.638540\pi$
$347$	−5450.67	−0.843249	−0.421624	+	0.906771i	$0.638540\pi$
$348$	25.8917	0.00398833
$349$	−900.949	−0.138185	−0.0690927	−	0.997610i	$-0.522010\pi$
$349$	−900.949	−0.138185	−0.0690927	+	0.997610i	$0.522010\pi$
$350$	−449.093	−0.0685857
$351$	−2128.60	−0.323693
$352$	0	0
$353$	10056.2	1.51625	0.758127	−	0.652107i	$-0.226115\pi$
$353$	10056.2	1.51625	0.758127	+	0.652107i	$0.226115\pi$
$354$	1007.00	0.151191
$355$	8221.18	1.22911
$356$	−32.1371	−0.00478445
$357$	1355.59	0.200967
$358$	−9763.18	−1.44134
$359$	358.995	0.0527772	0.0263886	−	0.999652i	$-0.491599\pi$
$359$	358.995	0.0527772	0.0263886	+	0.999652i	$0.491599\pi$
$360$	2500.54	0.366083
$361$	−4631.41	−0.675231
$362$	2886.75	0.419128
$363$	0	0
$364$	−55.7668	−0.00803015
$365$	3537.95	0.507355
$366$	7297.39	1.04219
$367$	−11961.7	−1.70135	−0.850675	−	0.525693i	$-0.823806\pi$
$367$	−11961.7	−1.70135	−0.850675	+	0.525693i	$0.823806\pi$
$368$	1383.67	0.196002
$369$	3320.44	0.468443
$370$	13125.9	1.84428
$371$	3934.32	0.550565
$372$	−21.2565	−0.00296263
$373$	−6531.90	−0.906727	−0.453363	−	0.891326i	$-0.649776\pi$
$373$	−6531.90	−0.906727	−0.453363	+	0.891326i	$0.649776\pi$
$374$	0	0
$375$	3702.63	0.509875
$376$	8698.51	1.19306
$377$	6450.93	0.881273
$378$	508.775	0.0692290
$379$	660.161	0.0894728	0.0447364	−	0.998999i	$-0.485755\pi$
$379$	660.161	0.0894728	0.0447364	+	0.998999i	$0.485755\pi$
$380$	−60.7314	−0.00819857
$381$	−4284.01	−0.576053
$382$	8627.17	1.15551
$383$	−7538.45	−1.00574	−0.502868	−	0.864363i	$-0.667722\pi$
$383$	−7538.45	−1.00574	−0.502868	+	0.864363i	$0.667722\pi$
$384$	4256.57	0.565670
$385$	0	0
$386$	−2129.58	−0.280810
$387$	1380.81	0.181371
$388$	−177.221	−0.0231882
$389$	−4293.42	−0.559602	−0.279801	−	0.960058i	$-0.590268\pi$
$389$	−4293.42	−0.559602	−0.279801	+	0.960058i	$0.590268\pi$
$390$	8107.07	1.05261
$391$	1476.38	0.190956
$392$	−6787.19	−0.874503
$393$	2240.78	0.287614
$394$	−7822.17	−1.00019
$395$	−9011.84	−1.14794
$396$	0	0
$397$	1240.68	0.156846	0.0784231	−	0.996920i	$-0.475011\pi$
$397$	1240.68	0.156846	0.0784231	+	0.996920i	$0.475011\pi$
$398$	−4975.57	−0.626641
$399$	−949.593	−0.119146
$400$	−1504.92	−0.188115
$401$	−7802.31	−0.971643	−0.485821	−	0.874058i	$-0.661479\pi$
$401$	−7802.31	−0.971643	−0.485821	+	0.874058i	$0.661479\pi$
$402$	−7458.54	−0.925368
$403$	−5296.08	−0.654631
$404$	−60.2108	−0.00741485
$405$	988.176	0.121242
$406$	−1541.89	−0.188480
$407$	0	0
$408$	4603.31	0.558573
$409$	3631.11	0.438990	0.219495	−	0.975614i	$-0.429559\pi$
$409$	3631.11	0.438990	0.219495	+	0.975614i	$0.429559\pi$
$410$	−12646.4	−1.52332
$411$	437.749	0.0525366
$412$	−103.519	−0.0123787
$413$	801.205	0.0954594
$414$	554.112	0.0657805
$415$	−5434.22	−0.642784
$416$	−376.282	−0.0443480
$417$	1636.86	0.192224
$418$	0	0
$419$	−5274.50	−0.614979	−0.307489	−	0.951552i	$-0.599489\pi$
$419$	−5274.50	−0.614979	−0.307489	+	0.951552i	$0.599489\pi$
$420$	25.8890	0.00300775
$421$	−9729.64	−1.12635	−0.563176	−	0.826337i	$-0.690421\pi$
$421$	−9729.64	−1.12635	−0.563176	+	0.826337i	$0.690421\pi$
$422$	6412.16	0.739666
$423$	3437.53	0.395126
$424$	13360.2	1.53025
$425$	−1605.76	−0.183273
$426$	5680.28	0.646033
$427$	5806.06	0.658021
$428$	69.9627	0.00790134
$429$	0	0
$430$	−5259.01	−0.589795
$431$	−15030.2	−1.67977	−0.839883	−	0.542768i	$-0.817376\pi$
$431$	−15030.2	−1.67977	−0.839883	+	0.542768i	$0.817376\pi$
$432$	1704.92	0.189879
$433$	7264.80	0.806291	0.403146	−	0.915136i	$-0.367917\pi$
$433$	7264.80	0.806291	0.403146	+	0.915136i	$0.367917\pi$
$434$	1265.86	0.140007
$435$	−2994.76	−0.330087
$436$	53.4909	0.00587557
$437$	−1034.21	−0.113211
$438$	2444.48	0.266670
$439$	−1279.92	−0.139151	−0.0695756	−	0.997577i	$-0.522164\pi$
$439$	−1279.92	−0.139151	−0.0695756	+	0.997577i	$0.522164\pi$
$440$	0	0
$441$	−2682.20	−0.289623
$442$	14924.5	1.60608
$443$	−2552.44	−0.273748	−0.136874	−	0.990588i	$-0.543705\pi$
$443$	−2552.44	−0.273748	−0.136874	+	0.990588i	$0.543705\pi$
$444$	−121.167	−0.0129512
$445$	3717.14	0.395976
$446$	−1875.09	−0.199076
$447$	−2128.83	−0.225257
$448$	3477.82	0.366767
$449$	1816.63	0.190941	0.0954703	−	0.995432i	$-0.469565\pi$
$449$	1816.63	0.190941	0.0954703	+	0.995432i	$0.469565\pi$
$450$	−602.670	−0.0631337
$451$	0	0
$452$	−119.912	−0.0124783
$453$	5549.22	0.575552
$454$	−4438.38	−0.458818
$455$	6450.28	0.664601
$456$	−3224.63	−0.331156
$457$	8219.57	0.841346	0.420673	−	0.907212i	$-0.361794\pi$
$457$	8219.57	0.841346	0.420673	+	0.907212i	$0.361794\pi$
$458$	−4373.89	−0.446241
$459$	1819.16	0.184992
$460$	28.1960	0.00285793
$461$	−12922.7	−1.30558	−0.652789	−	0.757539i	$-0.726401\pi$
$461$	−12922.7	−1.30558	−0.652789	+	0.757539i	$0.726401\pi$
$462$	0	0
$463$	−12352.7	−1.23991	−0.619954	−	0.784638i	$-0.712849\pi$
$463$	−12352.7	−1.23991	−0.619954	+	0.784638i	$0.712849\pi$
$464$	−5166.92	−0.516957
$465$	2458.64	0.245197
$466$	−5844.91	−0.581031
$467$	−8484.64	−0.840733	−0.420366	−	0.907354i	$-0.638098\pi$
$467$	−8484.64	−0.840733	−0.420366	+	0.907354i	$0.638098\pi$
$468$	−74.8376	−0.00739181
$469$	−5934.28	−0.584263
$470$	−13092.3	−1.28490
$471$	3095.38	0.302819
$472$	2720.74	0.265322
$473$	0	0
$474$	−6226.57	−0.603366
$475$	1124.84	0.108655
$476$	47.6598	0.00458925
$477$	5279.75	0.506799
$478$	10358.6	0.991192
$479$	−11531.4	−1.09996	−0.549980	−	0.835178i	$-0.685365\pi$
$479$	−11531.4	−1.09996	−0.549980	+	0.835178i	$0.685365\pi$
$480$	174.684	0.0166108
$481$	−30188.9	−2.86173
$482$	−11434.5	−1.08055
$483$	440.871	0.0415328
$484$	0	0
$485$	20498.3	1.91913
$486$	682.762	0.0637257
$487$	1749.21	0.162760	0.0813802	−	0.996683i	$-0.474067\pi$
$487$	1749.21	0.162760	0.0813802	+	0.996683i	$0.474067\pi$
$488$	19716.3	1.82892
$489$	9403.11	0.869578
$490$	10215.5	0.941818
$491$	1691.32	0.155454	0.0777271	−	0.996975i	$-0.475234\pi$
$491$	1691.32	0.155454	0.0777271	+	0.996975i	$0.475234\pi$
$492$	116.740	0.0106973
$493$	−5513.14	−0.503650
$494$	−10454.7	−0.952184
$495$	0	0
$496$	4241.93	0.384009
$497$	4519.43	0.407896
$498$	−3754.67	−0.337853
$499$	4017.14	0.360385	0.180192	−	0.983631i	$-0.442328\pi$
$499$	4017.14	0.360385	0.180192	+	0.983631i	$0.442328\pi$
$500$	130.177	0.0116434
$501$	−5587.19	−0.498238
$502$	−17428.9	−1.54958
$503$	10632.0	0.942465	0.471232	−	0.882009i	$-0.343809\pi$
$503$	10632.0	0.942465	0.471232	+	0.882009i	$0.343809\pi$
$504$	1374.62	0.121489
$505$	6964.29	0.613677
$506$	0	0
$507$	−12054.8	−1.05597
$508$	−150.618	−0.0131547
$509$	−11426.0	−0.994990	−0.497495	−	0.867467i	$-0.665747\pi$
$509$	−11426.0	−0.994990	−0.497495	+	0.867467i	$0.665747\pi$
$510$	−6928.52	−0.601569
$511$	1944.91	0.168372
$512$	11806.0	1.01905
$513$	−1274.33	−0.109674
$514$	−3116.94	−0.267475
$515$	11973.6	1.02450
$516$	48.5467	0.00414176
$517$	0	0
$518$	7215.69	0.612045
$519$	−8307.70	−0.702635
$520$	21903.9	1.84721
$521$	3567.00	0.299948	0.149974	−	0.988690i	$-0.452081\pi$
$521$	3567.00	0.299948	0.149974	+	0.988690i	$0.452081\pi$
$522$	−2069.18	−0.173497
$523$	−9974.64	−0.833959	−0.416979	−	0.908916i	$-0.636911\pi$
$523$	−9974.64	−0.833959	−0.416979	+	0.908916i	$0.636911\pi$
$524$	78.7815	0.00656791
$525$	−479.506	−0.0398616
$526$	−19293.1	−1.59927
$527$	4526.17	0.374124
$528$	0	0
$529$	−11686.8	−0.960536
$530$	−20108.7	−1.64805
$531$	1075.20	0.0878711
$532$	−33.3859	−0.00272079
$533$	29086.0	2.36370
$534$	2568.29	0.208129
$535$	−8092.24	−0.653941
$536$	−20151.7	−1.62392
$537$	−10424.4	−0.837699
$538$	5904.32	0.473148
$539$	0	0
$540$	34.7424	0.00276866
$541$	4082.16	0.324410	0.162205	−	0.986757i	$-0.448139\pi$
$541$	4082.16	0.324410	0.162205	+	0.986757i	$0.448139\pi$
$542$	10229.6	0.810703
$543$	3082.25	0.243595
$544$	321.581	0.0253450
$545$	−6187.03	−0.486282
$546$	4456.70	0.349321
$547$	−4088.15	−0.319555	−0.159778	−	0.987153i	$-0.551078\pi$
$547$	−4088.15	−0.319555	−0.159778	+	0.987153i	$0.551078\pi$
$548$	15.3904	0.00119972
$549$	7791.58	0.605713
$550$	0	0
$551$	3861.98	0.298595
$552$	1497.11	0.115437
$553$	−4954.08	−0.380956
$554$	16088.1	1.23379
$555$	14014.8	1.07188
$556$	57.5490	0.00438960
$557$	4043.80	0.307615	0.153807	−	0.988101i	$-0.450846\pi$
$557$	4043.80	0.307615	0.153807	+	0.988101i	$0.450846\pi$
$558$	1698.75	0.128878
$559$	12095.4	0.915175
$560$	−5166.39	−0.389857
$561$	0	0
$562$	−9559.53	−0.717517
$563$	−21672.9	−1.62238	−0.811192	−	0.584780i	$-0.801181\pi$
$563$	−21672.9	−1.62238	−0.811192	+	0.584780i	$0.801181\pi$
$564$	120.857	0.00902303
$565$	13869.6	1.03274
$566$	−18229.2	−1.35377
$567$	543.230	0.0402355
$568$	15347.1	1.13372
$569$	−17864.3	−1.31618	−0.658092	−	0.752937i	$-0.728636\pi$
$569$	−17864.3	−1.31618	−0.658092	+	0.752937i	$0.728636\pi$
$570$	4853.46	0.356647
$571$	1307.77	0.0958468	0.0479234	−	0.998851i	$-0.484740\pi$
$571$	1307.77	0.0958468	0.0479234	+	0.998851i	$0.484740\pi$
$572$	0	0
$573$	9211.42	0.671575
$574$	−6952.09	−0.505530
$575$	−522.235	−0.0378760
$576$	4667.14	0.337611
$577$	2240.24	0.161633	0.0808167	−	0.996729i	$-0.474247\pi$
$577$	2240.24	0.161633	0.0808167	+	0.996729i	$0.474247\pi$
$578$	1049.25	0.0755067
$579$	−2273.80	−0.163205
$580$	−105.290	−0.00753782
$581$	−2987.35	−0.213316
$582$	14162.9	1.00871
$583$	0	0
$584$	6604.55	0.467976
$585$	8656.10	0.611770
$586$	−11265.8	−0.794177
$587$	−12380.9	−0.870552	−0.435276	−	0.900297i	$-0.643349\pi$
$587$	−12380.9	−0.870552	−0.435276	+	0.900297i	$0.643349\pi$
$588$	−94.3011	−0.00661379
$589$	−3170.60	−0.221804
$590$	−4095.03	−0.285746
$591$	−8351.90	−0.581305
$592$	24180.0	1.67870
$593$	−9317.16	−0.645210	−0.322605	−	0.946534i	$-0.604559\pi$
$593$	−9317.16	−0.645210	−0.322605	+	0.946534i	$0.604559\pi$
$594$	0	0
$595$	−5512.58	−0.379822
$596$	−74.8455	−0.00514394
$597$	−5312.53	−0.364200
$598$	4853.84	0.331920
$599$	−6260.19	−0.427019	−0.213510	−	0.976941i	$-0.568489\pi$
$599$	−6260.19	−0.427019	−0.213510	+	0.976941i	$0.568489\pi$
$600$	−1628.31	−0.110792
$601$	−1888.27	−0.128160	−0.0640799	−	0.997945i	$-0.520411\pi$
$601$	−1888.27	−0.128160	−0.0640799	+	0.997945i	$0.520411\pi$
$602$	−2891.03	−0.195730
$603$	−7963.64	−0.537819
$604$	195.100	0.0131432
$605$	0	0
$606$	4811.85	0.322554
$607$	20651.6	1.38093	0.690463	−	0.723367i	$-0.257407\pi$
$607$	20651.6	1.38093	0.690463	+	0.723367i	$0.257407\pi$
$608$	−225.269	−0.0150261
$609$	−1646.31	−0.109543
$610$	−29675.3	−1.96970
$611$	30111.6	1.99376
$612$	63.9582	0.00422444
$613$	16808.1	1.10746	0.553729	−	0.832697i	$-0.313204\pi$
$613$	16808.1	1.10746	0.553729	+	0.832697i	$0.313204\pi$
$614$	−25193.6	−1.65591
$615$	−13502.8	−0.885342
$616$	0	0
$617$	−20733.1	−1.35281	−0.676403	−	0.736531i	$-0.736462\pi$
$617$	−20733.1	−1.35281	−0.676403	+	0.736531i	$0.736462\pi$
$618$	8272.90	0.538487
$619$	8833.91	0.573610	0.286805	−	0.957989i	$-0.407407\pi$
$619$	8833.91	0.573610	0.286805	+	0.957989i	$0.407407\pi$
$620$	86.4410	0.00559928
$621$	591.638	0.0382312
$622$	3926.29	0.253103
$623$	2043.43	0.131409
$624$	14934.5	0.958108
$625$	−18036.1	−1.15431
$626$	8533.11	0.544811
$627$	0	0
$628$	108.828	0.00691513
$629$	25800.2	1.63549
$630$	−2068.97	−0.130841
$631$	−2967.94	−0.187246	−0.0936228	−	0.995608i	$-0.529845\pi$
$631$	−2967.94	−0.187246	−0.0936228	+	0.995608i	$0.529845\pi$
$632$	−16823.1	−1.05884
$633$	6846.41	0.429890
$634$	−8421.68	−0.527551
$635$	17421.2	1.08872
$636$	185.626	0.0115732
$637$	−23495.2	−1.46140
$638$	0	0
$639$	6064.96	0.375471
$640$	−17309.6	−1.06910
$641$	−4084.03	−0.251653	−0.125826	−	0.992052i	$-0.540158\pi$
$641$	−4084.03	−0.251653	−0.125826	+	0.992052i	$0.540158\pi$
$642$	−5591.19	−0.343717
$643$	20973.2	1.28632	0.643159	−	0.765733i	$-0.277623\pi$
$643$	20973.2	1.28632	0.643159	+	0.765733i	$0.277623\pi$
$644$	15.5002	0.000948437 0
$645$	−5615.16	−0.342785
$646$	8934.86	0.544176
$647$	21289.7	1.29364	0.646819	−	0.762644i	$-0.276099\pi$
$647$	21289.7	1.29364	0.646819	+	0.762644i	$0.276099\pi$
$648$	1844.70	0.111831
$649$	0	0
$650$	−5279.19	−0.318565
$651$	1351.59	0.0813715
$652$	330.596	0.0198576
$653$	−12892.8	−0.772643	−0.386321	−	0.922364i	$-0.626254\pi$
$653$	−12892.8	−0.772643	−0.386321	+	0.922364i	$0.626254\pi$
$654$	−4274.82	−0.255594
$655$	−9112.28	−0.543582
$656$	−23296.6	−1.38656
$657$	2610.02	0.154987
$658$	−7197.22	−0.426409
$659$	−1601.22	−0.0946508	−0.0473254	−	0.998880i	$-0.515070\pi$
$659$	−1601.22	−0.0946508	−0.0473254	+	0.998880i	$0.515070\pi$
$660$	0	0
$661$	2829.75	0.166512	0.0832561	−	0.996528i	$-0.473468\pi$
$661$	2829.75	0.166512	0.0832561	+	0.996528i	$0.473468\pi$
$662$	20472.6	1.20195
$663$	15935.2	0.933445
$664$	−10144.5	−0.592894
$665$	3861.58	0.225182
$666$	9683.26	0.563392
$667$	−1793.02	−0.104087
$668$	−196.435	−0.0113777
$669$	−2002.07	−0.115702
$670$	30330.6	1.74892
$671$	0	0
$672$	96.0292	0.00551251
$673$	−11657.2	−0.667683	−0.333842	−	0.942629i	$-0.608345\pi$
$673$	−11657.2	−0.667683	−0.333842	+	0.942629i	$0.608345\pi$
$674$	−4844.30	−0.276848
$675$	−643.484	−0.0366929
$676$	−423.826	−0.0241139
$677$	−11435.1	−0.649166	−0.324583	−	0.945857i	$-0.605224\pi$
$677$	−11435.1	−0.649166	−0.324583	+	0.945857i	$0.605224\pi$
$678$	9582.95	0.542818
$679$	11268.5	0.636886
$680$	−18719.7	−1.05569
$681$	−4738.96	−0.266663
$682$	0	0
$683$	24568.3	1.37640	0.688199	−	0.725522i	$-0.258402\pi$
$683$	24568.3	1.37640	0.688199	+	0.725522i	$0.258402\pi$
$684$	−44.8030	−0.00250451
$685$	−1780.13	−0.0992926
$686$	12079.1	0.672278
$687$	−4670.10	−0.259353
$688$	−9687.93	−0.536844
$689$	46248.9	2.55724
$690$	−2253.33	−0.124323
$691$	32559.0	1.79248	0.896240	−	0.443570i	$-0.146288\pi$
$691$	32559.0	1.79248	0.896240	+	0.443570i	$0.146288\pi$
$692$	−292.083	−0.0160453
$693$	0	0
$694$	15314.9	0.837672
$695$	−6656.41	−0.363298
$696$	−5590.55	−0.304467
$697$	−24857.7	−1.35086
$698$	2531.42	0.137271
$699$	−6240.74	−0.337692
$700$	−16.8585	−0.000910275 0
$701$	−6004.68	−0.323529	−0.161764	−	0.986829i	$-0.551718\pi$
$701$	−6004.68	−0.323529	−0.161764	+	0.986829i	$0.551718\pi$
$702$	5980.77	0.321552
$703$	−18073.1	−0.969618
$704$	0	0
$705$	−13978.9	−0.746776
$706$	−28255.1	−1.50623
$707$	3828.48	0.203656
$708$	37.8018	0.00200661
$709$	19015.2	1.00724	0.503618	−	0.863926i	$-0.332002\pi$
$709$	19015.2	1.00724	0.503618	+	0.863926i	$0.332002\pi$
$710$	−23099.2	−1.22098
$711$	−6648.24	−0.350673
$712$	6939.07	0.365243
$713$	1472.03	0.0773182
$714$	−3808.82	−0.199638
$715$	0	0
$716$	−366.501	−0.0191296
$717$	11060.1	0.576075
$718$	−1008.67	−0.0524281
$719$	−21816.5	−1.13160	−0.565798	−	0.824544i	$-0.691432\pi$
$719$	−21816.5	−1.13160	−0.565798	+	0.824544i	$0.691432\pi$
$720$	−6933.16	−0.358866
$721$	6582.22	0.339993
$722$	13013.0	0.670765
$723$	−12208.9	−0.628012
$724$	108.366	0.00556270
$725$	1950.14	0.0998986
$726$	0	0
$727$	−2988.61	−0.152464	−0.0762321	−	0.997090i	$-0.524289\pi$
$727$	−2988.61	−0.152464	−0.0762321	+	0.997090i	$0.524289\pi$
$728$	12041.2	0.613018
$729$	729.000	0.0370370
$730$	−9940.64	−0.503999
$731$	−10337.1	−0.523025
$732$	273.937	0.0138320
$733$	22653.7	1.14152	0.570761	−	0.821117i	$-0.306648\pi$
$733$	22653.7	1.14152	0.570761	+	0.821117i	$0.306648\pi$
$734$	33609.0	1.69010
$735$	10907.4	0.547379
$736$	104.586	0.00523791
$737$	0	0
$738$	−9329.51	−0.465344
$739$	33676.1	1.67631	0.838157	−	0.545429i	$-0.183633\pi$
$739$	33676.1	1.67631	0.838157	+	0.545429i	$0.183633\pi$
$740$	492.733	0.0244774
$741$	−11162.7	−0.553404
$742$	−11054.3	−0.546923
$743$	−37527.9	−1.85298	−0.926491	−	0.376317i	$-0.877191\pi$
$743$	−37527.9	−1.85298	−0.926491	+	0.376317i	$0.877191\pi$
$744$	4589.72	0.226166
$745$	8657.01	0.425729
$746$	18352.8	0.900730
$747$	−4008.95	−0.196358
$748$	0	0
$749$	−4448.55	−0.217018
$750$	−10403.4	−0.506502
$751$	−18617.9	−0.904630	−0.452315	−	0.891858i	$-0.649402\pi$
$751$	−18617.9	−0.904630	−0.452315	+	0.891858i	$0.649402\pi$
$752$	−24118.1	−1.16954
$753$	−18609.2	−0.900606
$754$	−18125.3	−0.875444
$755$	−22566.3	−1.08778
$756$	19.0989	0.000918811 0
$757$	−40533.2	−1.94611	−0.973053	−	0.230581i	$-0.925937\pi$
$757$	−40533.2	−1.94611	−0.973053	+	0.230581i	$0.925937\pi$
$758$	−1854.87	−0.0888810
$759$	0	0
$760$	13113.2	0.625876
$761$	−10978.4	−0.522954	−0.261477	−	0.965210i	$-0.584210\pi$
$761$	−10978.4	−0.522954	−0.261477	+	0.965210i	$0.584210\pi$
$762$	12036.9	0.572243
$763$	−3401.20	−0.161378
$764$	323.856	0.0153360
$765$	−7397.74	−0.349628
$766$	21180.9	0.999084
$767$	9418.37	0.443387
$768$	485.941	0.0228319
$769$	−11205.6	−0.525466	−0.262733	−	0.964869i	$-0.584624\pi$
$769$	−11205.6	−0.525466	−0.262733	+	0.964869i	$0.584624\pi$
$770$	0	0
$771$	−3328.02	−0.155455
$772$	−79.9425	−0.00372693
$773$	11187.2	0.520536	0.260268	−	0.965536i	$-0.416189\pi$
$773$	11187.2	0.520536	0.260268	+	0.965536i	$0.416189\pi$
$774$	−3879.69	−0.180171
$775$	−1601.03	−0.0742071
$776$	38265.7	1.77018
$777$	7704.35	0.355717
$778$	12063.3	0.555901
$779$	17412.9	0.800875
$780$	304.332	0.0139703
$781$	0	0
$782$	−4148.22	−0.189693
$783$	−2209.31	−0.100835
$784$	18818.6	0.857263
$785$	−12587.6	−0.572318
$786$	−6295.96	−0.285712
$787$	−18540.0	−0.839746	−0.419873	−	0.907583i	$-0.637925\pi$
$787$	−18540.0	−0.839746	−0.419873	+	0.907583i	$0.637925\pi$
$788$	−293.637	−0.0132746
$789$	−20599.7	−0.929489
$790$	25320.7	1.14034
$791$	7624.54	0.342727
$792$	0	0
$793$	68251.7	3.05636
$794$	−3485.96	−0.155809
$795$	−21470.5	−0.957835
$796$	−186.778	−0.00831682
$797$	2336.61	0.103848	0.0519241	−	0.998651i	$-0.483465\pi$
$797$	2336.61	0.103848	0.0519241	+	0.998651i	$0.483465\pi$
$798$	2668.09	0.118358
$799$	−25734.2	−1.13944
$800$	−113.752	−0.00502716
$801$	2742.22	0.120963
$802$	21922.3	0.965216
$803$	0	0
$804$	−279.987	−0.0122815
$805$	−1792.83	−0.0784957
$806$	14880.5	0.650302
$807$	6304.18	0.274991
$808$	13000.8	0.566046
$809$	20445.1	0.888518	0.444259	−	0.895898i	$-0.353467\pi$
$809$	20445.1	0.888518	0.444259	+	0.895898i	$0.353467\pi$
$810$	−2776.50	−0.120440
$811$	2552.90	0.110536	0.0552678	−	0.998472i	$-0.482399\pi$
$811$	2552.90	0.110536	0.0552678	+	0.998472i	$0.482399\pi$
$812$	−57.8812	−0.00250152
$813$	10922.4	0.471176
$814$	0	0
$815$	−38238.4	−1.64348
$816$	−12763.4	−0.547561
$817$	7241.18	0.310082
$818$	−10202.4	−0.436086
$819$	4758.52	0.203023
$820$	−474.733	−0.0202176
$821$	−11464.4	−0.487346	−0.243673	−	0.969857i	$-0.578352\pi$
$821$	−11464.4	−0.487346	−0.243673	+	0.969857i	$0.578352\pi$
$822$	−1229.95	−0.0521891
$823$	1340.56	0.0567790	0.0283895	−	0.999597i	$-0.490962\pi$
$823$	1340.56	0.0567790	0.0283895	+	0.999597i	$0.490962\pi$
$824$	22351.9	0.944984
$825$	0	0
$826$	−2251.16	−0.0948280
$827$	10155.1	0.426999	0.213500	−	0.976943i	$-0.431514\pi$
$827$	10155.1	0.426999	0.213500	+	0.976943i	$0.431514\pi$
$828$	20.8008	0.000873043 0
$829$	28863.4	1.20925	0.604625	−	0.796510i	$-0.293323\pi$
$829$	28863.4	1.20925	0.604625	+	0.796510i	$0.293323\pi$
$830$	15268.6	0.638533
$831$	17177.6	0.717071
$832$	40882.6	1.70355
$833$	20079.6	0.835196
$834$	−4599.12	−0.190953
$835$	22720.7	0.941654
$836$	0	0
$837$	1813.79	0.0749030
$838$	14819.9	0.610911
$839$	−2469.12	−0.101601	−0.0508007	−	0.998709i	$-0.516177\pi$
$839$	−2469.12	−0.101601	−0.0508007	+	0.998709i	$0.516177\pi$
$840$	−5589.98	−0.229610
$841$	−17693.5	−0.725470
$842$	27337.6	1.11890
$843$	−10206.9	−0.417017
$844$	240.707	0.00981690
$845$	49021.9	1.99574
$846$	−9658.48	−0.392512
$847$	0	0
$848$	−37043.3	−1.50009
$849$	−19463.8	−0.786801
$850$	4511.74	0.182061
$851$	8390.89	0.337998
$852$	213.232	0.00857420
$853$	9927.90	0.398505	0.199253	−	0.979948i	$-0.436149\pi$
$853$	9927.90	0.398505	0.199253	+	0.979948i	$0.436149\pi$
$854$	−16313.4	−0.653669
$855$	5182.14	0.207281
$856$	−15106.4	−0.603185
$857$	32734.5	1.30477	0.652385	−	0.757888i	$-0.273769\pi$
$857$	32734.5	1.30477	0.652385	+	0.757888i	$0.273769\pi$
$858$	0	0
$859$	1731.02	0.0687563	0.0343782	−	0.999409i	$-0.489055\pi$
$859$	1731.02	0.0687563	0.0343782	+	0.999409i	$0.489055\pi$
$860$	−197.418	−0.00782780
$861$	−7422.90	−0.293811
$862$	42230.6	1.66866
$863$	−12057.4	−0.475597	−0.237798	−	0.971315i	$-0.576426\pi$
$863$	−12057.4	−0.475597	−0.237798	+	0.971315i	$0.576426\pi$
$864$	128.869	0.00507430
$865$	33783.8	1.32796
$866$	−20412.1	−0.800958
$867$	1120.30	0.0438841
$868$	47.5192	0.00185819
$869$	0	0
$870$	8414.45	0.327904
$871$	−69758.9	−2.71377
$872$	−11549.8	−0.448539
$873$	15122.0	0.586258
$874$	2905.85	0.112462
$875$	−8277.28	−0.319798
$876$	91.7635	0.00353927
$877$	24565.9	0.945874	0.472937	−	0.881096i	$-0.343194\pi$
$877$	24565.9	0.945874	0.472937	+	0.881096i	$0.343194\pi$
$878$	3596.22	0.138231
$879$	−12028.8	−0.461571
$880$	0	0
$881$	29413.2	1.12481	0.562404	−	0.826863i	$-0.309877\pi$
$881$	29413.2	1.12481	0.562404	+	0.826863i	$0.309877\pi$
$882$	7536.23	0.287708
$883$	−45396.6	−1.73014	−0.865072	−	0.501648i	$-0.832727\pi$
$883$	−45396.6	−1.73014	−0.865072	+	0.501648i	$0.832727\pi$
$884$	560.253	0.0213160
$885$	−4372.36	−0.166074
$886$	7171.65	0.271937
$887$	13952.6	0.528163	0.264082	−	0.964500i	$-0.414931\pi$
$887$	13952.6	0.528163	0.264082	+	0.964500i	$0.414931\pi$
$888$	26162.5	0.988688
$889$	9576.96	0.361306
$890$	−10444.1	−0.393357
$891$	0	0
$892$	−70.3890	−0.00264215
$893$	18026.9	0.675529
$894$	5981.40	0.223767
$895$	42391.4	1.58323
$896$	−9515.62	−0.354793
$897$	5182.55	0.192910
$898$	−5104.24	−0.189678
$899$	−5496.88	−0.203928
$900$	−22.6237	−0.000837914 0
$901$	−39525.5	−1.46147
$902$	0	0
$903$	−3086.82	−0.113757
$904$	25891.4	0.952585
$905$	−12534.2	−0.460387
$906$	−15591.8	−0.571745
$907$	11631.2	0.425808	0.212904	−	0.977073i	$-0.431708\pi$
$907$	11631.2	0.425808	0.212904	+	0.977073i	$0.431708\pi$
$908$	−166.613	−0.00608947
$909$	5137.72	0.187467
$910$	−18123.5	−0.660206
$911$	−23556.1	−0.856695	−0.428348	−	0.903614i	$-0.640904\pi$
$911$	−23556.1	−0.856695	−0.428348	+	0.903614i	$0.640904\pi$
$912$	8940.84	0.324628
$913$	0	0
$914$	−23094.7	−0.835781
$915$	−31685.0	−1.14478
$916$	−164.192	−0.00592255
$917$	−5009.29	−0.180394
$918$	−5111.33	−0.183768
$919$	4266.91	0.153158	0.0765791	−	0.997064i	$-0.475600\pi$
$919$	4266.91	0.153158	0.0765791	+	0.997064i	$0.475600\pi$
$920$	−6088.11	−0.218173
$921$	−26899.8	−0.962408
$922$	36309.3	1.29694
$923$	53127.0	1.89458
$924$	0	0
$925$	−9126.21	−0.324398
$926$	34707.6	1.23171
$927$	8833.16	0.312966
$928$	−390.549	−0.0138151
$929$	24778.5	0.875087	0.437543	−	0.899197i	$-0.355849\pi$
$929$	24778.5	0.875087	0.437543	+	0.899197i	$0.355849\pi$
$930$	−6908.08	−0.243575
$931$	−14065.9	−0.495156
$932$	−219.413	−0.00771148
$933$	4192.19	0.147102
$934$	23839.5	0.835172
$935$	0	0
$936$	16159.0	0.564288
$937$	14524.9	0.506413	0.253206	−	0.967412i	$-0.418515\pi$
$937$	14524.9	0.506413	0.253206	+	0.967412i	$0.418515\pi$
$938$	16673.7	0.580399
$939$	9110.99	0.316641
$940$	−491.473	−0.0170533
$941$	−27484.4	−0.952141	−0.476071	−	0.879407i	$-0.657939\pi$
$941$	−27484.4	−0.952141	−0.476071	+	0.879407i	$0.657939\pi$
$942$	−8697.16	−0.300816
$943$	−8084.35	−0.279176
$944$	−7543.70	−0.260092
$945$	−2209.08	−0.0760438
$946$	0	0
$947$	−36072.2	−1.23779	−0.618896	−	0.785473i	$-0.712420\pi$
$947$	−36072.2	−1.23779	−0.618896	+	0.785473i	$0.712420\pi$
$948$	−233.740	−0.00800792
$949$	22863.0	0.782047
$950$	−3160.49	−0.107937
$951$	−8992.01	−0.306610
$952$	−10290.8	−0.350342
$953$	17861.7	0.607133	0.303567	−	0.952810i	$-0.401822\pi$
$953$	17861.7	0.607133	0.303567	+	0.952810i	$0.401822\pi$
$954$	−14834.6	−0.503447
$955$	−37458.9	−1.26926
$956$	388.851	0.0131552
$957$	0	0
$958$	32399.9	1.09269
$959$	−978.593	−0.0329514
$960$	−18979.2	−0.638075
$961$	−25278.2	−0.848517
$962$	84822.2	2.84280
$963$	−5969.83	−0.199767
$964$	−429.240	−0.0143412
$965$	9246.56	0.308453
$966$	−1238.72	−0.0412581
$967$	−26773.3	−0.890352	−0.445176	−	0.895443i	$-0.646859\pi$
$967$	−26773.3	−0.890352	−0.445176	+	0.895443i	$0.646859\pi$
$968$	0	0
$969$	9539.95	0.316272
$970$	−57594.4	−1.90644
$971$	53427.8	1.76579	0.882893	−	0.469574i	$-0.155592\pi$
$971$	53427.8	1.76579	0.882893	+	0.469574i	$0.155592\pi$
$972$	25.6303	0.000845772 0
$973$	−3659.23	−0.120565
$974$	−4914.79	−0.161684
$975$	−5636.71	−0.185148
$976$	−54666.7	−1.79287
$977$	−37120.9	−1.21556	−0.607780	−	0.794105i	$-0.707940\pi$
$977$	−37120.9	−1.21556	−0.607780	+	0.794105i	$0.707940\pi$
$978$	−26420.1	−0.863826
$979$	0	0
$980$	383.482	0.0124999
$981$	−4564.32	−0.148550
$982$	−4752.12	−0.154426
$983$	39035.8	1.26658	0.633290	−	0.773914i	$-0.281704\pi$
$983$	39035.8	1.26658	0.633290	+	0.773914i	$0.281704\pi$
$984$	−25206.7	−0.816627
$985$	33963.6	1.09865
$986$	15490.4	0.500319
$987$	−7684.64	−0.247826
$988$	−392.459	−0.0126374
$989$	−3361.89	−0.108091
$990$	0	0
$991$	−34917.1	−1.11925	−0.559626	−	0.828745i	$-0.689055\pi$
$991$	−34917.1	−1.11925	−0.559626	+	0.828745i	$0.689055\pi$
$992$	320.632	0.0102622
$993$	21859.0	0.698565
$994$	−12698.3	−0.405198
$995$	21603.8	0.688327
$996$	−140.947	−0.00448401
$997$	−54943.4	−1.74531	−0.872656	−	0.488335i	$-0.837604\pi$
$997$	−54943.4	−1.74531	−0.872656	+	0.488335i	$0.837604\pi$
$998$	−11287.0	−0.358001
$999$	10339.0	0.327440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.v.1.2		6
3.2	odd	2		1089.4.a.bi.1.5		6
11.3	even	5		33.4.e.c.31.1	yes	12
11.4	even	5		33.4.e.c.16.1	✓	12
11.10	odd	2		363.4.a.u.1.5		6
33.14	odd	10		99.4.f.d.64.3		12
33.26	odd	10		99.4.f.d.82.3		12
33.32	even	2		1089.4.a.bk.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.c.16.1	✓	12	11.4	even	5
33.4.e.c.31.1	yes	12	11.3	even	5
99.4.f.d.64.3		12	33.14	odd	10
99.4.f.d.82.3		12	33.26	odd	10
363.4.a.u.1.5		6	11.10	odd	2
363.4.a.v.1.2		6	1.1	even	1	trivial
1089.4.a.bi.1.5		6	3.2	odd	2
1089.4.a.bk.1.2		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)

\(f(q)\)	\(=\)	\(q-2.80972 q^{2} -3.00000 q^{3} -0.105474 q^{4} +12.1997 q^{5} +8.42916 q^{6} +6.70654 q^{7} +22.7741 q^{8} +9.00000 q^{9} -34.2777 q^{10} +0.316423 q^{12} +78.8371 q^{13} -18.8435 q^{14} -36.5991 q^{15} -63.1451 q^{16} -67.3763 q^{17} -25.2875 q^{18} +47.1974 q^{19} -1.28676 q^{20} -20.1196 q^{21} -21.9125 q^{23} -68.3223 q^{24} +23.8328 q^{25} -221.510 q^{26} -27.0000 q^{27} -0.707368 q^{28} +81.8261 q^{29} +102.833 q^{30} -67.1775 q^{31} -4.77291 q^{32} +189.309 q^{34} +81.8178 q^{35} -0.949269 q^{36} -382.927 q^{37} -132.611 q^{38} -236.511 q^{39} +277.837 q^{40} +368.938 q^{41} +56.5305 q^{42} +153.423 q^{43} +109.797 q^{45} +61.5680 q^{46} +381.947 q^{47} +189.435 q^{48} -298.022 q^{49} -66.9634 q^{50} +202.129 q^{51} -8.31528 q^{52} +586.639 q^{53} +75.8624 q^{54} +152.736 q^{56} -141.592 q^{57} -229.909 q^{58} +119.466 q^{59} +3.86027 q^{60} +865.731 q^{61} +188.750 q^{62} +60.3589 q^{63} +518.571 q^{64} +961.789 q^{65} -884.849 q^{67} +7.10647 q^{68} +65.7375 q^{69} -229.885 q^{70} +673.884 q^{71} +204.967 q^{72} +290.003 q^{73} +1075.92 q^{74} -71.4983 q^{75} -4.97811 q^{76} +664.530 q^{78} -738.693 q^{79} -770.351 q^{80} +81.0000 q^{81} -1036.61 q^{82} -445.439 q^{83} +2.12210 q^{84} -821.971 q^{85} -431.077 q^{86} -245.478 q^{87} +304.691 q^{89} -308.500 q^{90} +528.724 q^{91} +2.31121 q^{92} +201.533 q^{93} -1073.16 q^{94} +575.794 q^{95} +14.3187 q^{96} +1680.23 q^{97} +837.359 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) \) 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

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Database

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