LMFDB - Embedded newform 363.4.a.v.1.3

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			1.3
Root			$-0.303175$ 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)

$f(q)$	$=$	\(q+1.30317 q^{2} -3.00000 q^{3} -6.30174 q^{4} -8.10194 q^{5} -3.90952 q^{6} +0.543555 q^{7} -18.6377 q^{8} +9.00000 q^{9} -10.5582 q^{10} +18.9052 q^{12} -69.6511 q^{13} +0.708348 q^{14} +24.3058 q^{15} +26.1258 q^{16} -42.5773 q^{17} +11.7286 q^{18} +75.4539 q^{19} +51.0563 q^{20} -1.63067 q^{21} +37.9924 q^{23} +55.9130 q^{24} -59.3586 q^{25} -90.7675 q^{26} -27.0000 q^{27} -3.42534 q^{28} +120.108 q^{29} +31.6747 q^{30} +277.188 q^{31} +183.148 q^{32} -55.4856 q^{34} -4.40385 q^{35} -56.7156 q^{36} -14.8922 q^{37} +98.3296 q^{38} +208.953 q^{39} +151.001 q^{40} +284.880 q^{41} -2.12504 q^{42} +312.130 q^{43} -72.9175 q^{45} +49.5107 q^{46} +337.320 q^{47} -78.3773 q^{48} -342.705 q^{49} -77.3546 q^{50} +127.732 q^{51} +438.923 q^{52} +169.723 q^{53} -35.1857 q^{54} -10.1306 q^{56} -226.362 q^{57} +156.522 q^{58} -801.918 q^{59} -153.169 q^{60} -386.778 q^{61} +361.224 q^{62} +4.89200 q^{63} +29.6673 q^{64} +564.309 q^{65} +981.580 q^{67} +268.311 q^{68} -113.977 q^{69} -5.73899 q^{70} -535.379 q^{71} -167.739 q^{72} +456.833 q^{73} -19.4072 q^{74} +178.076 q^{75} -475.490 q^{76} +272.302 q^{78} -619.503 q^{79} -211.669 q^{80} +81.0000 q^{81} +371.249 q^{82} +198.439 q^{83} +10.2760 q^{84} +344.959 q^{85} +406.759 q^{86} -360.325 q^{87} -1317.67 q^{89} -95.0242 q^{90} -37.8592 q^{91} -239.418 q^{92} -831.563 q^{93} +439.587 q^{94} -611.323 q^{95} -549.443 q^{96} -929.812 q^{97} -446.604 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.30317	0.460742	0.230371	−	0.973103i	$-0.426006\pi$
$2$	1.30317	0.460742	0.230371	+	0.973103i	$0.426006\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−6.30174	−0.787717
$5$	−8.10194	−0.724660	−0.362330	−	0.932050i	$-0.618019\pi$
$5$	−8.10194	−0.724660	−0.362330	+	0.932050i	$0.618019\pi$
$6$	−3.90952	−0.266009
$7$	0.543555	0.0293492	0.0146746	−	0.999892i	$-0.495329\pi$
$7$	0.543555	0.0293492	0.0146746	+	0.999892i	$0.495329\pi$
$8$	−18.6377	−0.823676
$9$	9.00000	0.333333
$10$	−10.5582	−0.333881
$11$	0	0
$11$	0	0
$12$	18.9052	0.454789
$13$	−69.6511	−1.48598	−0.742989	−	0.669303i	$-0.766593\pi$
$13$	−69.6511	−1.48598	−0.742989	+	0.669303i	$0.766593\pi$
$14$	0.708348	0.0135224
$15$	24.3058	0.418382
$16$	26.1258	0.408215
$17$	−42.5773	−0.607441	−0.303721	−	0.952761i	$-0.598229\pi$
$17$	−42.5773	−0.607441	−0.303721	+	0.952761i	$0.598229\pi$
$18$	11.7286	0.153581
$19$	75.4539	0.911069	0.455534	−	0.890218i	$-0.349448\pi$
$19$	75.4539	0.911069	0.455534	+	0.890218i	$0.349448\pi$
$20$	51.0563	0.570827
$21$	−1.63067	−0.0169448
$22$	0	0
$23$	37.9924	0.344433	0.172217	−	0.985059i	$-0.444907\pi$
$23$	37.9924	0.344433	0.172217	+	0.985059i	$0.444907\pi$
$24$	55.9130	0.475550
$25$	−59.3586	−0.474869
$26$	−90.7675	−0.684653
$27$	−27.0000	−0.192450
$28$	−3.42534	−0.0231189
$29$	120.108	0.769088	0.384544	−	0.923107i	$-0.374359\pi$
$29$	120.108	0.769088	0.384544	+	0.923107i	$0.374359\pi$
$30$	31.6747	0.192766
$31$	277.188	1.60595	0.802974	−	0.596014i	$-0.203250\pi$
$31$	277.188	1.60595	0.802974	+	0.596014i	$0.203250\pi$
$32$	183.148	1.01176
$33$	0	0
$34$	−55.4856	−0.279874
$35$	−4.40385	−0.0212682
$36$	−56.7156	−0.262572
$37$	−14.8922	−0.0661694	−0.0330847	−	0.999453i	$-0.510533\pi$
$37$	−14.8922	−0.0661694	−0.0330847	+	0.999453i	$0.510533\pi$
$38$	98.3296	0.419768
$39$	208.953	0.857930
$40$	151.001	0.596885
$41$	284.880	1.08514	0.542571	−	0.840010i	$-0.317451\pi$
$41$	284.880	1.08514	0.542571	+	0.840010i	$0.317451\pi$
$42$	−2.12504	−0.00780717
$43$	312.130	1.10696	0.553481	−	0.832862i	$-0.313299\pi$
$43$	312.130	1.10696	0.553481	+	0.832862i	$0.313299\pi$
$44$	0	0
$45$	−72.9175	−0.241553
$46$	49.5107	0.158695
$47$	337.320	1.04688	0.523438	−	0.852064i	$-0.324649\pi$
$47$	337.320	1.04688	0.523438	+	0.852064i	$0.324649\pi$
$48$	−78.3773	−0.235683
$49$	−342.705	−0.999139
$50$	−77.3546	−0.218792
$51$	127.732	0.350707
$52$	438.923	1.17053
$53$	169.723	0.439874	0.219937	−	0.975514i	$-0.429415\pi$
$53$	169.723	0.439874	0.219937	+	0.975514i	$0.429415\pi$
$54$	−35.1857	−0.0886698
$55$	0	0
$56$	−10.1306	−0.0241743
$57$	−226.362	−0.526006
$58$	156.522	0.354351
$59$	−801.918	−1.76951	−0.884753	−	0.466061i	$-0.845673\pi$
$59$	−801.918	−1.76951	−0.884753	+	0.466061i	$0.845673\pi$
$60$	−153.169	−0.329567
$61$	−386.778	−0.811833	−0.405916	−	0.913910i	$-0.633048\pi$
$61$	−386.778	−0.811833	−0.405916	+	0.913910i	$0.633048\pi$
$62$	361.224	0.739927
$63$	4.89200	0.00978308
$64$	29.6673	0.0579440
$65$	564.309	1.07683
$66$	0	0
$67$	981.580	1.78984	0.894919	−	0.446229i	$-0.147233\pi$
$67$	981.580	1.78984	0.894919	+	0.446229i	$0.147233\pi$
$68$	268.311	0.478492
$69$	−113.977	−0.198859
$70$	−5.73899	−0.00979915
$71$	−535.379	−0.894899	−0.447450	−	0.894309i	$-0.647668\pi$
$71$	−535.379	−0.894899	−0.447450	+	0.894309i	$0.647668\pi$
$72$	−167.739	−0.274559
$73$	456.833	0.732442	0.366221	−	0.930528i	$-0.380652\pi$
$73$	456.833	0.732442	0.366221	+	0.930528i	$0.380652\pi$
$74$	−19.4072	−0.0304870
$75$	178.076	0.274165
$76$	−475.490	−0.717664
$77$	0	0
$78$	272.302	0.395284
$79$	−619.503	−0.882273	−0.441137	−	0.897440i	$-0.645425\pi$
$79$	−619.503	−0.882273	−0.441137	+	0.897440i	$0.645425\pi$
$80$	−211.669	−0.295817
$81$	81.0000	0.111111
$82$	371.249	0.499970
$83$	198.439	0.262428	0.131214	−	0.991354i	$-0.458112\pi$
$83$	198.439	0.262428	0.131214	+	0.991354i	$0.458112\pi$
$84$	10.2760	0.0133477
$85$	344.959	0.440188
$86$	406.759	0.510023
$87$	−360.325	−0.444033
$88$	0	0
$89$	−1317.67	−1.56935	−0.784676	−	0.619906i	$-0.787171\pi$
$89$	−1317.67	−1.56935	−0.784676	+	0.619906i	$0.787171\pi$
$90$	−95.0242	−0.111294
$91$	−37.8592	−0.0436123
$92$	−239.418	−0.271316
$93$	−831.563	−0.927194
$94$	439.587	0.482339
$95$	−611.323	−0.660215
$96$	−549.443	−0.584139
$97$	−929.812	−0.973279	−0.486640	−	0.873603i	$-0.661778\pi$
$97$	−929.812	−0.973279	−0.486640	+	0.873603i	$0.661778\pi$
$98$	−446.604	−0.460345
$99$	0	0
$100$	374.062	0.374062
$101$	1524.29	1.50171	0.750856	−	0.660466i	$-0.229641\pi$
$101$	1524.29	1.50171	0.750856	+	0.660466i	$0.229641\pi$
$102$	166.457	0.161585
$103$	−142.828	−0.136633	−0.0683166	−	0.997664i	$-0.521763\pi$
$103$	−142.828	−0.136633	−0.0683166	+	0.997664i	$0.521763\pi$
$104$	1298.13	1.22397
$105$	13.2116	0.0122792
$106$	221.179	0.202668
$107$	1548.47	1.39903	0.699513	−	0.714619i	$-0.253400\pi$
$107$	1548.47	1.39903	0.699513	+	0.714619i	$0.253400\pi$
$108$	170.147	0.151596
$109$	−650.244	−0.571395	−0.285697	−	0.958320i	$-0.592225\pi$
$109$	−650.244	−0.571395	−0.285697	+	0.958320i	$0.592225\pi$
$110$	0	0
$111$	44.6767	0.0382029
$112$	14.2008	0.0119808
$113$	79.4957	0.0661799	0.0330899	−	0.999452i	$-0.489465\pi$
$113$	79.4957	0.0661799	0.0330899	+	0.999452i	$0.489465\pi$
$114$	−294.989	−0.242353
$115$	−307.812	−0.249597
$116$	−756.891	−0.605824
$117$	−626.859	−0.495326
$118$	−1045.04	−0.815285
$119$	−23.1431	−0.0178279
$120$	−453.004	−0.344612
$121$	0	0
$122$	−504.039	−0.374045
$123$	−854.641	−0.626507
$124$	−1746.76	−1.26503
$125$	1493.66	1.06878
$126$	6.37513	0.00450747
$127$	−410.724	−0.286975	−0.143488	−	0.989652i	$-0.545832\pi$
$127$	−410.724	−0.286975	−0.143488	+	0.989652i	$0.545832\pi$
$128$	−1426.52	−0.985060
$129$	−936.389	−0.639104
$130$	735.393	0.496140
$131$	2714.85	1.81067	0.905334	−	0.424699i	$-0.139620\pi$
$131$	2714.85	1.81067	0.905334	+	0.424699i	$0.139620\pi$
$132$	0	0
$133$	41.0134	0.0267392
$134$	1279.17	0.824653
$135$	218.752	0.139461
$136$	793.541	0.500335
$137$	2658.27	1.65775	0.828875	−	0.559434i	$-0.188981\pi$
$137$	2658.27	1.65775	0.828875	+	0.559434i	$0.188981\pi$
$138$	−148.532	−0.0916224
$139$	−199.838	−0.121943	−0.0609714	−	0.998140i	$-0.519420\pi$
$139$	−199.838	−0.121943	−0.0609714	+	0.998140i	$0.519420\pi$
$140$	27.7519	0.0167533
$141$	−1011.96	−0.604414
$142$	−697.693	−0.412318
$143$	0	0
$144$	235.132	0.136072
$145$	−973.110	−0.557327
$146$	595.333	0.337466
$147$	1028.11	0.576853
$148$	93.8469	0.0521228
$149$	2513.68	1.38207	0.691035	−	0.722822i	$-0.257155\pi$
$149$	2513.68	1.38207	0.691035	+	0.722822i	$0.257155\pi$
$150$	232.064	0.126319
$151$	−2970.67	−1.60099	−0.800495	−	0.599339i	$-0.795430\pi$
$151$	−2970.67	−1.60099	−0.800495	+	0.599339i	$0.795430\pi$
$152$	−1406.28	−0.750426
$153$	−383.195	−0.202480
$154$	0	0
$155$	−2245.76	−1.16377
$156$	−1316.77	−0.675806
$157$	2052.60	1.04341	0.521705	−	0.853126i	$-0.325296\pi$
$157$	2052.60	1.04341	0.521705	+	0.853126i	$0.325296\pi$
$158$	−807.321	−0.406500
$159$	−509.170	−0.253961
$160$	−1483.85	−0.733180
$161$	20.6510	0.0101088
$162$	105.557	0.0511935
$163$	−2317.68	−1.11371	−0.556856	−	0.830609i	$-0.687992\pi$
$163$	−2317.68	−1.11371	−0.556856	+	0.830609i	$0.687992\pi$
$164$	−1795.24	−0.854785
$165$	0	0
$166$	258.601	0.120912
$167$	1237.88	0.573591	0.286795	−	0.957992i	$-0.407410\pi$
$167$	1237.88	0.573591	0.286795	+	0.957992i	$0.407410\pi$
$168$	30.3918	0.0139570
$169$	2654.27	1.20813
$170$	449.541	0.202813
$171$	679.085	0.303690
$172$	−1966.96	−0.871972
$173$	−2290.75	−1.00672	−0.503360	−	0.864077i	$-0.667903\pi$
$173$	−2290.75	−1.00672	−0.503360	+	0.864077i	$0.667903\pi$
$174$	−469.566	−0.204585
$175$	−32.2647	−0.0139370
$176$	0	0
$177$	2405.75	1.02162
$178$	−1717.15	−0.723066
$179$	540.198	0.225566	0.112783	−	0.993620i	$-0.464024\pi$
$179$	540.198	0.225566	0.112783	+	0.993620i	$0.464024\pi$
$180$	459.507	0.190276
$181$	−2240.37	−0.920029	−0.460015	−	0.887911i	$-0.652156\pi$
$181$	−2240.37	−0.920029	−0.460015	+	0.887911i	$0.652156\pi$
$182$	−49.3372	−0.0200940
$183$	1160.33	0.468712
$184$	−708.089	−0.283701
$185$	120.656	0.0479503
$186$	−1083.67	−0.427197
$187$	0	0
$188$	−2125.70	−0.824642
$189$	−14.6760	−0.00564826
$190$	−796.660	−0.304189
$191$	632.258	0.239521	0.119761	−	0.992803i	$-0.461787\pi$
$191$	632.258	0.239521	0.119761	+	0.992803i	$0.461787\pi$
$192$	−89.0020	−0.0334540
$193$	4124.77	1.53838	0.769190	−	0.639020i	$-0.220660\pi$
$193$	4124.77	1.53838	0.769190	+	0.639020i	$0.220660\pi$
$194$	−1211.71	−0.448431
$195$	−1692.93	−0.621707
$196$	2159.63	0.787038
$197$	3297.43	1.19255	0.596274	−	0.802781i	$-0.296647\pi$
$197$	3297.43	1.19255	0.596274	+	0.802781i	$0.296647\pi$
$198$	0	0
$199$	47.3923	0.0168822	0.00844109	−	0.999964i	$-0.497313\pi$
$199$	47.3923	0.0168822	0.00844109	+	0.999964i	$0.497313\pi$
$200$	1106.30	0.391138
$201$	−2944.74	−1.03336
$202$	1986.42	0.691901
$203$	65.2855	0.0225721
$204$	−804.932	−0.276257
$205$	−2308.08	−0.786359
$206$	−186.129	−0.0629527
$207$	341.931	0.114811
$208$	−1819.69	−0.606599
$209$	0	0
$210$	17.2170	0.00565754
$211$	−1518.83	−0.495547	−0.247774	−	0.968818i	$-0.579699\pi$
$211$	−1518.83	−0.495547	−0.247774	+	0.968818i	$0.579699\pi$
$212$	−1069.55	−0.346496
$213$	1606.14	0.516670
$214$	2017.92	0.644590
$215$	−2528.86	−0.802170
$216$	503.217	0.158517
$217$	150.667	0.0471333
$218$	−847.381	−0.263266
$219$	−1370.50	−0.422875
$220$	0	0
$221$	2965.55	0.902645
$222$	58.2215	0.0176017
$223$	1023.29	0.307284	0.153642	−	0.988127i	$-0.450900\pi$
$223$	1023.29	0.307284	0.153642	+	0.988127i	$0.450900\pi$
$224$	99.5509	0.0296943
$225$	−534.227	−0.158290
$226$	103.597	0.0304918
$227$	2544.42	0.743959	0.371980	−	0.928241i	$-0.378679\pi$
$227$	2544.42	0.743959	0.371980	+	0.928241i	$0.378679\pi$
$228$	1426.47	0.414344
$229$	−3578.08	−1.03251	−0.516257	−	0.856433i	$-0.672675\pi$
$229$	−3578.08	−1.03251	−0.516257	+	0.856433i	$0.672675\pi$
$230$	−401.133	−0.115000
$231$	0	0
$232$	−2238.54	−0.633479
$233$	1729.65	0.486323	0.243162	−	0.969986i	$-0.421815\pi$
$233$	1729.65	0.486323	0.243162	+	0.969986i	$0.421815\pi$
$234$	−816.907	−0.228218
$235$	−2732.94	−0.758628
$236$	5053.48	1.39387
$237$	1858.51	0.509381
$238$	−30.1595	−0.00821408
$239$	5385.30	1.45752	0.728758	−	0.684771i	$-0.240098\pi$
$239$	5385.30	1.45752	0.728758	+	0.684771i	$0.240098\pi$
$240$	635.008	0.170790
$241$	−3515.37	−0.939605	−0.469803	−	0.882771i	$-0.655675\pi$
$241$	−3515.37	−0.939605	−0.469803	+	0.882771i	$0.655675\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	2437.37	0.639495
$245$	2776.57	0.724035
$246$	−1113.75	−0.288658
$247$	−5255.44	−1.35383
$248$	−5166.13	−1.32278
$249$	−595.318	−0.151513
$250$	1946.50	0.492431
$251$	−1443.12	−0.362903	−0.181452	−	0.983400i	$-0.558080\pi$
$251$	−1443.12	−0.362903	−0.181452	+	0.983400i	$0.558080\pi$
$252$	−30.8281	−0.00770630
$253$	0	0
$254$	−535.245	−0.132221
$255$	−1034.88	−0.254143
$256$	−2096.34	−0.511803
$257$	−6125.91	−1.48686	−0.743431	−	0.668812i	$-0.766803\pi$
$257$	−6125.91	−1.48686	−0.743431	+	0.668812i	$0.766803\pi$
$258$	−1220.28	−0.294462
$259$	−8.09475	−0.00194202
$260$	−3556.12	−0.848236
$261$	1080.97	0.256363
$262$	3537.93	0.834251
$263$	2135.26	0.500631	0.250315	−	0.968164i	$-0.419466\pi$
$263$	2135.26	0.500631	0.250315	+	0.968164i	$0.419466\pi$
$264$	0	0
$265$	−1375.09	−0.318759
$266$	53.4476	0.0123199
$267$	3953.00	0.906066
$268$	−6185.66	−1.40989
$269$	−2.07944	−0.000471323 0	−0.000235662	−	1.00000i	$-0.500075\pi$
$269$	−2.07944	−0.000471323 0	−0.000235662		1.00000i	$0.500075\pi$
$270$	285.073	0.0642554
$271$	4514.27	1.01189	0.505946	−	0.862565i	$-0.331144\pi$
$271$	4514.27	1.01189	0.505946	+	0.862565i	$0.331144\pi$
$272$	−1112.36	−0.247967
$273$	113.578	0.0251796
$274$	3464.20	0.763795
$275$	0	0
$276$	718.254	0.156644
$277$	4832.57	1.04823	0.524117	−	0.851646i	$-0.324395\pi$
$277$	4832.57	1.04823	0.524117	+	0.851646i	$0.324395\pi$
$278$	−260.424	−0.0561842
$279$	2494.69	0.535316
$280$	82.0775	0.0175181
$281$	196.899	0.0418008	0.0209004	−	0.999782i	$-0.493347\pi$
$281$	196.899	0.0418008	0.0209004	+	0.999782i	$0.493347\pi$
$282$	−1318.76	−0.278479
$283$	−2738.90	−0.575303	−0.287652	−	0.957735i	$-0.592875\pi$
$283$	−2738.90	−0.575303	−0.287652	+	0.957735i	$0.592875\pi$
$284$	3373.82	0.704927
$285$	1833.97	0.381175
$286$	0	0
$287$	154.848	0.0318481
$288$	1648.33	0.337253
$289$	−3100.18	−0.631015
$290$	−1268.13	−0.256784
$291$	2789.44	0.561923
$292$	−2878.84	−0.576957
$293$	6640.68	1.32407	0.662035	−	0.749473i	$-0.269693\pi$
$293$	6640.68	1.32407	0.662035	+	0.749473i	$0.269693\pi$
$294$	1339.81	0.265780
$295$	6497.09	1.28229
$296$	277.556	0.0545022
$297$	0	0
$298$	3275.76	0.636777
$299$	−2646.21	−0.511820
$300$	−1122.19	−0.215965
$301$	169.660	0.0324885
$302$	−3871.30	−0.737643
$303$	−4572.88	−0.867014
$304$	1971.29	0.371912
$305$	3133.65	0.588302
$306$	−499.371	−0.0932912
$307$	3278.70	0.609530	0.304765	−	0.952428i	$-0.401422\pi$
$307$	3278.70	0.609530	0.304765	+	0.952428i	$0.401422\pi$
$308$	0	0
$309$	428.483	0.0788853
$310$	−2926.62	−0.536195
$311$	6562.80	1.19660	0.598299	−	0.801273i	$-0.295843\pi$
$311$	6562.80	1.19660	0.598299	+	0.801273i	$0.295843\pi$
$312$	−3894.40	−0.706657
$313$	677.351	0.122320	0.0611599	−	0.998128i	$-0.480520\pi$
$313$	677.351	0.122320	0.0611599	+	0.998128i	$0.480520\pi$
$314$	2674.90	0.480743
$315$	−39.6347	−0.00708940
$316$	3903.95	0.694982
$317$	3805.72	0.674293	0.337146	−	0.941452i	$-0.390538\pi$
$317$	3805.72	0.674293	0.337146	+	0.941452i	$0.390538\pi$
$318$	−663.538	−0.117011
$319$	0	0
$320$	−240.363	−0.0419897
$321$	−4645.40	−0.807729
$322$	26.9118	0.00465757
$323$	−3212.62	−0.553421
$324$	−510.441	−0.0875241
$325$	4134.39	0.705645
$326$	−3020.35	−0.513134
$327$	1950.73	0.329895
$328$	−5309.50	−0.893806
$329$	183.352	0.0307250
$330$	0	0
$331$	3339.46	0.554542	0.277271	−	0.960792i	$-0.410570\pi$
$331$	3339.46	0.554542	0.277271	+	0.960792i	$0.410570\pi$
$332$	−1250.51	−0.206719
$333$	−134.030	−0.0220565
$334$	1613.17	0.264277
$335$	−7952.70	−1.29702
$336$	−42.6024	−0.00691712
$337$	9635.36	1.55748	0.778741	−	0.627345i	$-0.215859\pi$
$337$	9635.36	1.55748	0.778741	+	0.627345i	$0.215859\pi$
$338$	3458.98	0.556638
$339$	−238.487	−0.0382090
$340$	−2173.84	−0.346744
$341$	0	0
$342$	884.966	0.139923
$343$	−372.718	−0.0586732
$344$	−5817.37	−0.911777
$345$	923.436	0.144105
$346$	−2985.25	−0.463838
$347$	2663.13	0.412001	0.206001	−	0.978552i	$-0.433955\pi$
$347$	2663.13	0.412001	0.206001	+	0.978552i	$0.433955\pi$
$348$	2270.67	0.349772
$349$	5393.98	0.827316	0.413658	−	0.910432i	$-0.364251\pi$
$349$	5393.98	0.827316	0.413658	+	0.910432i	$0.364251\pi$
$350$	−42.0465	−0.00642137
$351$	1880.58	0.285977
$352$	0	0
$353$	3003.14	0.452807	0.226403	−	0.974034i	$-0.427303\pi$
$353$	3003.14	0.452807	0.226403	+	0.974034i	$0.427303\pi$
$354$	3135.12	0.470705
$355$	4337.61	0.648497
$356$	8303.58	1.23620
$357$	69.4293	0.0102930
$358$	703.972	0.103928
$359$	2027.48	0.298067	0.149034	−	0.988832i	$-0.452384\pi$
$359$	2027.48	0.298067	0.149034	+	0.988832i	$0.452384\pi$
$360$	1359.01	0.198962
$361$	−1165.71	−0.169954
$362$	−2919.59	−0.423896
$363$	0	0
$364$	238.579	0.0343542
$365$	−3701.23	−0.530771
$366$	1512.12	0.215955
$367$	−2441.40	−0.347248	−0.173624	−	0.984812i	$-0.555548\pi$
$367$	−2441.40	−0.347248	−0.173624	+	0.984812i	$0.555548\pi$
$368$	992.580	0.140603
$369$	2563.92	0.361714
$370$	157.236	0.0220927
$371$	92.2541	0.0129100
$372$	5240.29	0.730367
$373$	10410.5	1.44513	0.722565	−	0.691302i	$-0.242963\pi$
$373$	10410.5	1.44513	0.722565	+	0.691302i	$0.242963\pi$
$374$	0	0
$375$	−4480.99	−0.617059
$376$	−6286.85	−0.862286
$377$	−8365.67	−1.14285
$378$	−19.1254	−0.00260239
$379$	4880.16	0.661417	0.330708	−	0.943733i	$-0.392712\pi$
$379$	4880.16	0.661417	0.330708	+	0.943733i	$0.392712\pi$
$380$	3852.40	0.520062
$381$	1232.17	0.165685
$382$	823.942	0.110357
$383$	11959.2	1.59553	0.797763	−	0.602971i	$-0.206017\pi$
$383$	11959.2	1.59553	0.797763	+	0.602971i	$0.206017\pi$
$384$	4279.56	0.568725
$385$	0	0
$386$	5375.30	0.708796
$387$	2809.17	0.368987
$388$	5859.43	0.766669
$389$	7177.52	0.935513	0.467757	−	0.883857i	$-0.345062\pi$
$389$	7177.52	0.935513	0.467757	+	0.883857i	$0.345062\pi$
$390$	−2206.18	−0.286447
$391$	−1617.61	−0.209223
$392$	6387.21	0.822966
$393$	−8144.55	−1.04539
$394$	4297.12	0.549457
$395$	5019.18	0.639348
$396$	0	0
$397$	−2881.37	−0.364262	−0.182131	−	0.983274i	$-0.558299\pi$
$397$	−2881.37	−0.364262	−0.182131	+	0.983274i	$0.558299\pi$
$398$	61.7605	0.00777833
$399$	−123.040	−0.0154379
$400$	−1550.79	−0.193848
$401$	−3963.18	−0.493545	−0.246773	−	0.969073i	$-0.579370\pi$
$401$	−3963.18	−0.493545	−0.246773	+	0.969073i	$0.579370\pi$
$402$	−3837.51	−0.476114
$403$	−19306.4	−2.38640
$404$	−9605.69	−1.18292
$405$	−656.257	−0.0805177
$406$	85.0784	0.0103999
$407$	0	0
$408$	−2380.62	−0.288869
$409$	9041.81	1.09313	0.546563	−	0.837418i	$-0.315936\pi$
$409$	9041.81	1.09313	0.546563	+	0.837418i	$0.315936\pi$
$410$	−3007.84	−0.362308
$411$	−7974.82	−0.957102
$412$	900.062	0.107628
$413$	−435.887	−0.0519336
$414$	445.596	0.0528982
$415$	−1607.74	−0.190171
$416$	−12756.4	−1.50345
$417$	599.514	0.0704037
$418$	0	0
$419$	−840.109	−0.0979523	−0.0489761	−	0.998800i	$-0.515596\pi$
$419$	−840.109	−0.0979523	−0.0489761	+	0.998800i	$0.515596\pi$
$420$	−83.2558	−0.00967254
$421$	2004.95	0.232103	0.116052	−	0.993243i	$-0.462976\pi$
$421$	2004.95	0.232103	0.116052	+	0.993243i	$0.462976\pi$
$422$	−1979.30	−0.228319
$423$	3035.88	0.348958
$424$	−3163.25	−0.362313
$425$	2527.33	0.288455
$426$	2093.08	0.238052
$427$	−210.235	−0.0238267
$428$	−9758.02	−1.10204
$429$	0	0
$430$	−3295.54	−0.369593
$431$	−11919.4	−1.33211	−0.666054	−	0.745904i	$-0.732018\pi$
$431$	−11919.4	−1.33211	−0.666054	+	0.745904i	$0.732018\pi$
$432$	−705.396	−0.0785610
$433$	960.857	0.106642	0.0533208	−	0.998577i	$-0.483019\pi$
$433$	960.857	0.106642	0.0533208	+	0.998577i	$0.483019\pi$
$434$	196.345	0.0217163
$435$	2919.33	0.321773
$436$	4097.66	0.450097
$437$	2866.67	0.313802
$438$	−1786.00	−0.194836
$439$	−7590.46	−0.825223	−0.412611	−	0.910907i	$-0.635383\pi$
$439$	−7590.46	−0.825223	−0.412611	+	0.910907i	$0.635383\pi$
$440$	0	0
$441$	−3084.34	−0.333046
$442$	3864.63	0.415886
$443$	424.716	0.0455505	0.0227753	−	0.999741i	$-0.492750\pi$
$443$	424.716	0.0455505	0.0227753	+	0.999741i	$0.492750\pi$
$444$	−281.541	−0.0300931
$445$	10675.6	1.13725
$446$	1333.52	0.141578
$447$	−7541.03	−0.797938
$448$	16.1258	0.00170061
$449$	−4733.53	−0.497525	−0.248763	−	0.968564i	$-0.580024\pi$
$449$	−4733.53	−0.497525	−0.248763	+	0.968564i	$0.580024\pi$
$450$	−696.191	−0.0729306
$451$	0	0
$452$	−500.961	−0.0521310
$453$	8912.00	0.924332
$454$	3315.82	0.342773
$455$	306.733	0.0316041
$456$	4218.85	0.433258
$457$	−15170.8	−1.55286	−0.776431	−	0.630202i	$-0.782972\pi$
$457$	−15170.8	−1.55286	−0.776431	+	0.630202i	$0.782972\pi$
$458$	−4662.86	−0.475723
$459$	1149.59	0.116902
$460$	1939.75	0.196612
$461$	−12952.3	−1.30856	−0.654282	−	0.756251i	$-0.727029\pi$
$461$	−12952.3	−1.30856	−0.654282	+	0.756251i	$0.727029\pi$
$462$	0	0
$463$	−6774.30	−0.679975	−0.339988	−	0.940430i	$-0.610423\pi$
$463$	−6774.30	−0.679975	−0.339988	+	0.940430i	$0.610423\pi$
$464$	3137.92	0.313953
$465$	6737.27	0.671900
$466$	2254.04	0.224069
$467$	5187.23	0.513996	0.256998	−	0.966412i	$-0.417267\pi$
$467$	5187.23	0.513996	0.256998	+	0.966412i	$0.417267\pi$
$468$	3950.30	0.390177
$469$	533.543	0.0525304
$470$	−3561.50	−0.349532
$471$	−6157.80	−0.602413
$472$	14945.9	1.45750
$473$	0	0
$474$	2421.96	0.234693
$475$	−4478.83	−0.432638
$476$	145.842	0.0140434
$477$	1527.51	0.146625
$478$	7017.99	0.671539
$479$	−2753.01	−0.262606	−0.131303	−	0.991342i	$-0.541916\pi$
$479$	−2753.01	−0.262606	−0.131303	+	0.991342i	$0.541916\pi$
$480$	4451.56	0.423302
$481$	1037.26	0.0983264
$482$	−4581.14	−0.432915
$483$	−61.9529	−0.00583634
$484$	0	0
$485$	7533.28	0.705296
$486$	−316.671	−0.0295566
$487$	−9320.38	−0.867242	−0.433621	−	0.901095i	$-0.642764\pi$
$487$	−9320.38	−0.867242	−0.433621	+	0.901095i	$0.642764\pi$
$488$	7208.63	0.668687
$489$	6953.05	0.643002
$490$	3618.36	0.333593
$491$	−9787.37	−0.899588	−0.449794	−	0.893132i	$-0.648503\pi$
$491$	−9787.37	−0.899588	−0.449794	+	0.893132i	$0.648503\pi$
$492$	5385.72	0.493510
$493$	−5113.88	−0.467176
$494$	−6848.76	−0.623766
$495$	0	0
$496$	7241.74	0.655572
$497$	−291.008	−0.0262646
$498$	−775.804	−0.0698084
$499$	8749.24	0.784909	0.392454	−	0.919771i	$-0.371626\pi$
$499$	8749.24	0.784909	0.392454	+	0.919771i	$0.371626\pi$
$500$	−9412.66	−0.841894
$501$	−3713.63	−0.331163
$502$	−1880.63	−0.167205
$503$	−20829.8	−1.84643	−0.923214	−	0.384287i	$-0.874447\pi$
$503$	−20829.8	−1.84643	−0.923214	+	0.384287i	$0.874447\pi$
$504$	−91.1754	−0.00805808
$505$	−12349.7	−1.08823
$506$	0	0
$507$	−7962.81	−0.697516
$508$	2588.27	0.226055
$509$	−2497.25	−0.217463	−0.108732	−	0.994071i	$-0.534679\pi$
$509$	−2497.25	−0.217463	−0.108732	+	0.994071i	$0.534679\pi$
$510$	−1348.62	−0.117094
$511$	248.314	0.0214966
$512$	8680.26	0.749252
$513$	−2037.25	−0.175335
$514$	−7983.13	−0.685060
$515$	1157.18	0.0990126
$516$	5900.88	0.503433
$517$	0	0
$518$	−10.5489	−0.000894770 0
$519$	6872.26	0.581231
$520$	−10517.4	−0.886958
$521$	1557.78	0.130994	0.0654968	−	0.997853i	$-0.479137\pi$
$521$	1557.78	0.130994	0.0654968	+	0.997853i	$0.479137\pi$
$522$	1408.70	0.118117
$523$	678.171	0.0567005	0.0283502	−	0.999598i	$-0.490975\pi$
$523$	678.171	0.0567005	0.0283502	+	0.999598i	$0.490975\pi$
$524$	−17108.3	−1.42629
$525$	96.7940	0.00804655
$526$	2782.62	0.230661
$527$	−11801.9	−0.975519
$528$	0	0
$529$	−10723.6	−0.881366
$530$	−1791.98	−0.146865
$531$	−7217.26	−0.589835
$532$	−258.455	−0.0210629
$533$	−19842.2	−1.61250
$534$	5151.45	0.417462
$535$	−12545.6	−1.01382
$536$	−18294.4	−1.47425
$537$	−1620.59	−0.130231
$538$	−2.70988	−0.000217158 0
$539$	0	0
$540$	−1378.52	−0.109856
$541$	10074.0	0.800580	0.400290	−	0.916388i	$-0.368909\pi$
$541$	10074.0	0.800580	0.400290	+	0.916388i	$0.368909\pi$
$542$	5882.89	0.466221
$543$	6721.10	0.531179
$544$	−7797.93	−0.614584
$545$	5268.24	0.414067
$546$	148.011	0.0116013
$547$	8890.78	0.694958	0.347479	−	0.937688i	$-0.387038\pi$
$547$	8890.78	0.694958	0.347479	+	0.937688i	$0.387038\pi$
$548$	−16751.7	−1.30584
$549$	−3481.00	−0.270611
$550$	0	0
$551$	9062.64	0.700692
$552$	2124.27	0.163795
$553$	−336.734	−0.0258940
$554$	6297.68	0.482965
$555$	−361.968	−0.0276841
$556$	1259.33	0.0960564
$557$	13329.0	1.01395	0.506974	−	0.861961i	$-0.330764\pi$
$557$	13329.0	1.01395	0.506974	+	0.861961i	$0.330764\pi$
$558$	3251.02	0.246642
$559$	−21740.2	−1.64492
$560$	−115.054	−0.00868200
$561$	0	0
$562$	256.594	0.0192594
$563$	−13841.1	−1.03611	−0.518057	−	0.855346i	$-0.673345\pi$
$563$	−13841.1	−1.03611	−0.518057	+	0.855346i	$0.673345\pi$
$564$	6377.10	0.476107
$565$	−644.069	−0.0479579
$566$	−3569.27	−0.265066
$567$	44.0280	0.00326103
$568$	9978.22	0.737107
$569$	860.552	0.0634028	0.0317014	−	0.999497i	$-0.489907\pi$
$569$	860.552	0.0634028	0.0317014	+	0.999497i	$0.489907\pi$
$570$	2389.98	0.175623
$571$	−20409.1	−1.49579	−0.747895	−	0.663817i	$-0.768935\pi$
$571$	−20409.1	−1.49579	−0.747895	+	0.663817i	$0.768935\pi$
$572$	0	0
$573$	−1896.77	−0.138288
$574$	201.794	0.0146737
$575$	−2255.17	−0.163560
$576$	267.006	0.0193147
$577$	17478.0	1.26104	0.630518	−	0.776175i	$-0.282843\pi$
$577$	17478.0	1.26104	0.630518	+	0.776175i	$0.282843\pi$
$578$	−4040.07	−0.290735
$579$	−12374.3	−0.888185
$580$	6132.28	0.439016
$581$	107.863	0.00770207
$582$	3635.12	0.258901
$583$	0	0
$584$	−8514.29	−0.603294
$585$	5078.78	0.358943
$586$	8653.97	0.610055
$587$	−5473.09	−0.384836	−0.192418	−	0.981313i	$-0.561633\pi$
$587$	−5473.09	−0.384836	−0.192418	+	0.981313i	$0.561633\pi$
$588$	−6478.90	−0.454397
$589$	20914.9	1.46313
$590$	8466.84	0.590804
$591$	−9892.28	−0.688518
$592$	−389.071	−0.0270114
$593$	−5410.65	−0.374686	−0.187343	−	0.982295i	$-0.559988\pi$
$593$	−5410.65	−0.374686	−0.187343	+	0.982295i	$0.559988\pi$
$594$	0	0
$595$	187.504	0.0129192
$596$	−15840.5	−1.08868
$597$	−142.177	−0.00974693
$598$	−3448.47	−0.235817
$599$	1011.78	0.0690155	0.0345078	−	0.999404i	$-0.489014\pi$
$599$	1011.78	0.0690155	0.0345078	+	0.999404i	$0.489014\pi$
$600$	−3318.91	−0.225824
$601$	−6054.83	−0.410951	−0.205476	−	0.978662i	$-0.565874\pi$
$601$	−6054.83	−0.410951	−0.205476	+	0.978662i	$0.565874\pi$
$602$	221.096	0.0149688
$603$	8834.22	0.596612
$604$	18720.4	1.26113
$605$	0	0
$606$	−5959.26	−0.399469
$607$	−6920.07	−0.462730	−0.231365	−	0.972867i	$-0.574319\pi$
$607$	−6920.07	−0.462730	−0.231365	+	0.972867i	$0.574319\pi$
$608$	13819.2	0.921781
$609$	−195.857	−0.0130320
$610$	4083.69	0.271056
$611$	−23494.7	−1.55563
$612$	2414.80	0.159497
$613$	−7242.95	−0.477227	−0.238613	−	0.971115i	$-0.576693\pi$
$613$	−7242.95	−0.477227	−0.238613	+	0.971115i	$0.576693\pi$
$614$	4272.73	0.280836
$615$	6924.25	0.454004
$616$	0	0
$617$	26187.8	1.70872	0.854361	−	0.519680i	$-0.173949\pi$
$617$	26187.8	1.70872	0.854361	+	0.519680i	$0.173949\pi$
$618$	558.388	0.0363457
$619$	8431.55	0.547484	0.273742	−	0.961803i	$-0.411739\pi$
$619$	8431.55	0.547484	0.273742	+	0.961803i	$0.411739\pi$
$620$	14152.2	0.916718
$621$	−1025.79	−0.0662862
$622$	8552.48	0.551323
$623$	−716.224	−0.0460593
$624$	5459.06	0.350220
$625$	−4681.74	−0.299631
$626$	882.706	0.0563579
$627$	0	0
$628$	−12935.0	−0.821912
$629$	634.071	0.0401940
$630$	−51.6509	−0.00326638
$631$	11427.4	0.720947	0.360473	−	0.932770i	$-0.382615\pi$
$631$	11427.4	0.720947	0.360473	+	0.932770i	$0.382615\pi$
$632$	11546.1	0.726707
$633$	4556.48	0.286104
$634$	4959.52	0.310675
$635$	3327.66	0.207959
$636$	3208.66	0.200050
$637$	23869.7	1.48470
$638$	0	0
$639$	−4818.41	−0.298300
$640$	11557.6	0.713833
$641$	−22759.2	−1.40239	−0.701195	−	0.712969i	$-0.747350\pi$
$641$	−22759.2	−1.40239	−0.701195	+	0.712969i	$0.747350\pi$
$642$	−6053.77	−0.372154
$643$	−722.522	−0.0443134	−0.0221567	−	0.999755i	$-0.507053\pi$
$643$	−722.522	−0.0443134	−0.0221567	+	0.999755i	$0.507053\pi$
$644$	−130.137	−0.00796291
$645$	7586.57	0.463133
$646$	−4186.61	−0.254984
$647$	12986.6	0.789112	0.394556	−	0.918872i	$-0.370898\pi$
$647$	12986.6	0.789112	0.394556	+	0.918872i	$0.370898\pi$
$648$	−1509.65	−0.0915196
$649$	0	0
$650$	5387.83	0.325120
$651$	−452.001	−0.0272124
$652$	14605.4	0.877290
$653$	−17876.8	−1.07132	−0.535662	−	0.844432i	$-0.679938\pi$
$653$	−17876.8	−1.07132	−0.535662	+	0.844432i	$0.679938\pi$
$654$	2542.14	0.151996
$655$	−21995.6	−1.31212
$656$	7442.72	0.442971
$657$	4111.49	0.244147
$658$	238.940	0.0141563
$659$	14584.5	0.862114	0.431057	−	0.902325i	$-0.358141\pi$
$659$	14584.5	0.862114	0.431057	+	0.902325i	$0.358141\pi$
$660$	0	0
$661$	28799.8	1.69468	0.847340	−	0.531051i	$-0.178203\pi$
$661$	28799.8	1.69468	0.847340	+	0.531051i	$0.178203\pi$
$662$	4351.90	0.255501
$663$	−8896.66	−0.521142
$664$	−3698.45	−0.216156
$665$	−332.288	−0.0193768
$666$	−174.665	−0.0101623
$667$	4563.20	0.264899
$668$	−7800.76	−0.451827
$669$	−3069.86	−0.177410
$670$	−10363.8	−0.597593
$671$	0	0
$672$	−298.653	−0.0171440
$673$	9585.30	0.549014	0.274507	−	0.961585i	$-0.411485\pi$
$673$	9585.30	0.549014	0.274507	+	0.961585i	$0.411485\pi$
$674$	12556.6	0.717597
$675$	1602.68	0.0913885
$676$	−16726.5	−0.951667
$677$	−978.240	−0.0555345	−0.0277672	−	0.999614i	$-0.508840\pi$
$677$	−978.240	−0.0555345	−0.0277672	+	0.999614i	$0.508840\pi$
$678$	−310.790	−0.0176045
$679$	−505.404	−0.0285650
$680$	−6429.22	−0.362573
$681$	−7633.25	−0.429525
$682$	0	0
$683$	−32472.0	−1.81919	−0.909594	−	0.415499i	$-0.863607\pi$
$683$	−32472.0	−1.81919	−0.909594	+	0.415499i	$0.863607\pi$
$684$	−4279.41	−0.239221
$685$	−21537.2	−1.20130
$686$	−485.717	−0.0270332
$687$	10734.2	0.596123
$688$	8154.63	0.451878
$689$	−11821.4	−0.653643
$690$	1203.40	0.0663951
$691$	29595.8	1.62934	0.814672	−	0.579922i	$-0.196917\pi$
$691$	29595.8	1.62934	0.814672	+	0.579922i	$0.196917\pi$
$692$	14435.7	0.793011
$693$	0	0
$694$	3470.53	0.189826
$695$	1619.08	0.0883670
$696$	6715.61	0.365739
$697$	−12129.4	−0.659160
$698$	7029.30	0.381179
$699$	−5188.96	−0.280779
$700$	203.323	0.0109784
$701$	26738.3	1.44064	0.720321	−	0.693641i	$-0.243994\pi$
$701$	26738.3	1.44064	0.720321	+	0.693641i	$0.243994\pi$
$702$	2450.72	0.131761
$703$	−1123.68	−0.0602849
$704$	0	0
$705$	8198.83	0.437994
$706$	3913.61	0.208627
$707$	828.538	0.0440741
$708$	−15160.4	−0.804751
$709$	3143.76	0.166525	0.0832625	−	0.996528i	$-0.473466\pi$
$709$	3143.76	0.166525	0.0832625	+	0.996528i	$0.473466\pi$
$710$	5652.67	0.298790
$711$	−5575.53	−0.294091
$712$	24558.2	1.29264
$713$	10531.0	0.553141
$714$	90.4785	0.00474240
$715$	0	0
$716$	−3404.18	−0.177682
$717$	−16155.9	−0.841497
$718$	2642.15	0.137332
$719$	−8528.65	−0.442371	−0.221186	−	0.975232i	$-0.570993\pi$
$719$	−8528.65	−0.442371	−0.221186	+	0.975232i	$0.570993\pi$
$720$	−1905.02	−0.0986056
$721$	−77.6347	−0.00401008
$722$	−1519.13	−0.0783047
$723$	10546.1	0.542481
$724$	14118.2	0.724722
$725$	−7129.46	−0.365216
$726$	0	0
$727$	−34617.2	−1.76600	−0.882999	−	0.469374i	$-0.844480\pi$
$727$	−34617.2	−1.76600	−0.882999	+	0.469374i	$0.844480\pi$
$728$	705.607	0.0359224
$729$	729.000	0.0370370
$730$	−4823.35	−0.244548
$731$	−13289.6	−0.672414
$732$	−7312.11	−0.369212
$733$	9800.28	0.493836	0.246918	−	0.969036i	$-0.420582\pi$
$733$	9800.28	0.493836	0.246918	+	0.969036i	$0.420582\pi$
$734$	−3181.57	−0.159992
$735$	−8329.72	−0.418022
$736$	6958.22	0.348483
$737$	0	0
$738$	3341.24	0.166657
$739$	−1846.17	−0.0918978	−0.0459489	−	0.998944i	$-0.514631\pi$
$739$	−1846.17	−0.0918978	−0.0459489	+	0.998944i	$0.514631\pi$
$740$	−760.342	−0.0377713
$741$	15766.3	0.781634
$742$	120.223	0.00594816
$743$	21690.8	1.07101	0.535504	−	0.844533i	$-0.320122\pi$
$743$	21690.8	1.07101	0.535504	+	0.844533i	$0.320122\pi$
$744$	15498.4	0.763708
$745$	−20365.6	−1.00153
$746$	13566.7	0.665832
$747$	1785.95	0.0874761
$748$	0	0
$749$	841.677	0.0410604
$750$	−5839.51	−0.284305
$751$	−12211.7	−0.593359	−0.296680	−	0.954977i	$-0.595879\pi$
$751$	−12211.7	−0.593359	−0.296680	+	0.954977i	$0.595879\pi$
$752$	8812.74	0.427350
$753$	4329.35	0.209522
$754$	−10901.9	−0.526558
$755$	24068.2	1.16017
$756$	92.4842	0.00444923
$757$	−746.885	−0.0358600	−0.0179300	−	0.999839i	$-0.505708\pi$
$757$	−746.885	−0.0358600	−0.0179300	+	0.999839i	$0.505708\pi$
$758$	6359.70	0.304742
$759$	0	0
$760$	11393.6	0.543803
$761$	4937.60	0.235201	0.117600	−	0.993061i	$-0.462480\pi$
$761$	4937.60	0.235201	0.117600	+	0.993061i	$0.462480\pi$
$762$	1605.73	0.0763381
$763$	−353.443	−0.0167700
$764$	−3984.32	−0.188675
$765$	3104.63	0.146729
$766$	15584.9	0.735126
$767$	55854.4	2.62945
$768$	6289.03	0.295489
$769$	−2022.76	−0.0948538	−0.0474269	−	0.998875i	$-0.515102\pi$
$769$	−2022.76	−0.0948538	−0.0474269	+	0.998875i	$0.515102\pi$
$770$	0	0
$771$	18377.7	0.858440
$772$	−25993.2	−1.21181
$773$	−10864.5	−0.505521	−0.252761	−	0.967529i	$-0.581338\pi$
$773$	−10864.5	−0.505521	−0.252761	+	0.967529i	$0.581338\pi$
$774$	3660.84	0.170008
$775$	−16453.5	−0.762614
$776$	17329.5	0.801667
$777$	24.2843	0.00112123
$778$	9353.56	0.431030
$779$	21495.3	0.988639
$780$	10668.4	0.489729
$781$	0	0
$782$	−2108.03	−0.0963977
$783$	−3242.92	−0.148011
$784$	−8953.42	−0.407863
$785$	−16630.1	−0.756117
$786$	−10613.8	−0.481655
$787$	−18860.6	−0.854266	−0.427133	−	0.904189i	$-0.640476\pi$
$787$	−18860.6	−0.854266	−0.427133	+	0.904189i	$0.640476\pi$
$788$	−20779.5	−0.939390
$789$	−6405.78	−0.289039
$790$	6540.87	0.294574
$791$	43.2103	0.00194233
$792$	0	0
$793$	26939.5	1.20637
$794$	−3754.93	−0.167831
$795$	4125.27	0.184035
$796$	−298.654	−0.0132984
$797$	35879.7	1.59463	0.797317	−	0.603560i	$-0.206252\pi$
$797$	35879.7	1.59463	0.797317	+	0.603560i	$0.206252\pi$
$798$	−160.343	−0.00711287
$799$	−14362.2	−0.635916
$800$	−10871.4	−0.480452
$801$	−11859.0	−0.523117
$802$	−5164.71	−0.227397
$803$	0	0
$804$	18557.0	0.813998
$805$	−167.313	−0.00732547
$806$	−25159.6	−1.09952
$807$	6.23833	0.000272118 0
$808$	−28409.3	−1.23692
$809$	−35351.3	−1.53632	−0.768161	−	0.640256i	$-0.778828\pi$
$809$	−35351.3	−1.53632	−0.768161	+	0.640256i	$0.778828\pi$
$810$	−855.218	−0.0370979
$811$	−9326.21	−0.403807	−0.201904	−	0.979405i	$-0.564713\pi$
$811$	−9326.21	−0.403807	−0.201904	+	0.979405i	$0.564713\pi$
$812$	−411.412	−0.0177805
$813$	−13542.8	−0.584216
$814$	0	0
$815$	18777.7	0.807062
$816$	3337.09	0.143164
$817$	23551.4	1.00852
$818$	11783.1	0.503649
$819$	−340.733	−0.0145374
$820$	14544.9	0.619428
$821$	46531.1	1.97801	0.989005	−	0.147881i	$-0.0472453\pi$
$821$	46531.1	1.97801	0.989005	+	0.147881i	$0.0472453\pi$
$822$	−10392.6	−0.440977
$823$	17984.1	0.761710	0.380855	−	0.924635i	$-0.375630\pi$
$823$	17984.1	0.761710	0.380855	+	0.924635i	$0.375630\pi$
$824$	2661.97	0.112542
$825$	0	0
$826$	−568.037	−0.0239280
$827$	43270.0	1.81940	0.909701	−	0.415264i	$-0.136311\pi$
$827$	43270.0	1.81940	0.909701	+	0.415264i	$0.136311\pi$
$828$	−2154.76	−0.0904386
$829$	−35904.2	−1.50423	−0.752114	−	0.659033i	$-0.770966\pi$
$829$	−35904.2	−1.50423	−0.752114	+	0.659033i	$0.770966\pi$
$830$	−2095.17	−0.0876199
$831$	−14497.7	−0.605198
$832$	−2066.36	−0.0861036
$833$	14591.4	0.606918
$834$	781.272	0.0324379
$835$	−10029.2	−0.415658
$836$	0	0
$837$	−7484.07	−0.309065
$838$	−1094.81	−0.0451307
$839$	29387.6	1.20926	0.604632	−	0.796505i	$-0.293320\pi$
$839$	29387.6	1.20926	0.604632	+	0.796505i	$0.293320\pi$
$840$	−246.233	−0.0101141
$841$	−9963.00	−0.408504
$842$	2612.80	0.106940
$843$	−590.697	−0.0241337
$844$	9571.25	0.390351
$845$	−21504.7	−0.875485
$846$	3956.28	0.160780
$847$	0	0
$848$	4434.15	0.179563
$849$	8216.70	0.332151
$850$	3293.55	0.132903
$851$	−565.792	−0.0227909
$852$	−10121.5	−0.406990
$853$	17281.7	0.693686	0.346843	−	0.937923i	$-0.387254\pi$
$853$	17281.7	0.693686	0.346843	+	0.937923i	$0.387254\pi$
$854$	−273.973	−0.0109779
$855$	−5501.91	−0.220072
$856$	−28859.8	−1.15234
$857$	25834.0	1.02972	0.514862	−	0.857273i	$-0.327843\pi$
$857$	25834.0	1.02972	0.514862	+	0.857273i	$0.327843\pi$
$858$	0	0
$859$	−29861.7	−1.18611	−0.593055	−	0.805162i	$-0.702078\pi$
$859$	−29861.7	−1.18611	−0.593055	+	0.805162i	$0.702078\pi$
$860$	15936.2	0.631883
$861$	−464.545	−0.0183875
$862$	−15533.1	−0.613758
$863$	−1581.90	−0.0623970	−0.0311985	−	0.999513i	$-0.509932\pi$
$863$	−1581.90	−0.0623970	−0.0311985	+	0.999513i	$0.509932\pi$
$864$	−4944.99	−0.194713
$865$	18559.5	0.729530
$866$	1252.16	0.0491342
$867$	9300.53	0.364317
$868$	−949.463	−0.0371277
$869$	0	0
$870$	3804.40	0.148254
$871$	−68368.1	−2.65966
$872$	12119.0	0.470644
$873$	−8368.31	−0.324426
$874$	3735.78	0.144582
$875$	811.888	0.0313678
$876$	8636.52	0.333106
$877$	−21886.5	−0.842708	−0.421354	−	0.906896i	$-0.638445\pi$
$877$	−21886.5	−0.842708	−0.421354	+	0.906896i	$0.638445\pi$
$878$	−9891.69	−0.380215
$879$	−19922.0	−0.764452
$880$	0	0
$881$	−2989.02	−0.114305	−0.0571524	−	0.998365i	$-0.518202\pi$
$881$	−2989.02	−0.114305	−0.0571524	+	0.998365i	$0.518202\pi$
$882$	−4019.43	−0.153448
$883$	9041.43	0.344585	0.172292	−	0.985046i	$-0.444883\pi$
$883$	9041.43	0.344585	0.172292	+	0.985046i	$0.444883\pi$
$884$	−18688.1	−0.711029
$885$	−19491.3	−0.740330
$886$	553.480	0.0209870
$887$	−32723.5	−1.23872	−0.619361	−	0.785106i	$-0.712608\pi$
$887$	−32723.5	−1.23872	−0.619361	+	0.785106i	$0.712608\pi$
$888$	−832.669	−0.0314668
$889$	−223.251	−0.00842250
$890$	13912.2	0.523977
$891$	0	0
$892$	−6448.47	−0.242052
$893$	25452.1	0.953776
$894$	−9827.27	−0.367643
$895$	−4376.65	−0.163458
$896$	−775.393	−0.0289108
$897$	7938.63	0.295500
$898$	−6168.61	−0.229231
$899$	33292.5	1.23512
$900$	3366.56	0.124687
$901$	−7226.36	−0.267198
$902$	0	0
$903$	−508.979	−0.0187572
$904$	−1481.61	−0.0545108
$905$	18151.3	0.666708
$906$	11613.9	0.425878
$907$	−10904.5	−0.399204	−0.199602	−	0.979877i	$-0.563965\pi$
$907$	−10904.5	−0.399204	−0.199602	+	0.979877i	$0.563965\pi$
$908$	−16034.2	−0.586029
$909$	13718.6	0.500571
$910$	399.727	0.0145613
$911$	23757.0	0.864000	0.432000	−	0.901873i	$-0.357808\pi$
$911$	23757.0	0.864000	0.432000	+	0.901873i	$0.357808\pi$
$912$	−5913.87	−0.214723
$913$	0	0
$914$	−19770.1	−0.715468
$915$	−9400.95	−0.339657
$916$	22548.1	0.813329
$917$	1475.67	0.0531417
$918$	1498.11	0.0538617
$919$	16951.8	0.608473	0.304236	−	0.952597i	$-0.401599\pi$
$919$	16951.8	0.608473	0.304236	+	0.952597i	$0.401599\pi$
$920$	5736.90	0.205587
$921$	−9836.11	−0.351912
$922$	−16879.1	−0.602910
$923$	37289.7	1.32980
$924$	0	0
$925$	883.982	0.0314218
$926$	−8828.10	−0.313293
$927$	−1285.45	−0.0455444
$928$	21997.6	0.778131
$929$	38495.6	1.35953	0.679763	−	0.733432i	$-0.262082\pi$
$929$	38495.6	1.35953	0.679763	+	0.733432i	$0.262082\pi$
$930$	8779.85	0.309573
$931$	−25858.4	−0.910284
$932$	−10899.8	−0.383085
$933$	−19688.4	−0.690857
$934$	6759.86	0.236819
$935$	0	0
$936$	11683.2	0.407988
$937$	48776.4	1.70059	0.850297	−	0.526304i	$-0.176423\pi$
$937$	48776.4	1.70059	0.850297	+	0.526304i	$0.176423\pi$
$938$	695.300	0.0242029
$939$	−2032.05	−0.0706214
$940$	17222.3	0.597584
$941$	7854.68	0.272110	0.136055	−	0.990701i	$-0.456558\pi$
$941$	7854.68	0.272110	0.136055	+	0.990701i	$0.456558\pi$
$942$	−8024.70	−0.277557
$943$	10823.3	0.373759
$944$	−20950.7	−0.722339
$945$	118.904	0.00409307
$946$	0	0
$947$	−38877.6	−1.33406	−0.667029	−	0.745032i	$-0.732434\pi$
$947$	−38877.6	−1.33406	−0.667029	+	0.745032i	$0.732434\pi$
$948$	−11711.8	−0.401248
$949$	−31818.9	−1.08839
$950$	−5836.70	−0.199334
$951$	−11417.2	−0.389303
$952$	431.333	0.0146844
$953$	−12903.2	−0.438590	−0.219295	−	0.975659i	$-0.570376\pi$
$953$	−12903.2	−0.438590	−0.219295	+	0.975659i	$0.570376\pi$
$954$	1990.61	0.0675561
$955$	−5122.51	−0.173571
$956$	−33936.8	−1.14811
$957$	0	0
$958$	−3587.66	−0.120994
$959$	1444.92	0.0486537
$960$	721.089	0.0242428
$961$	47042.0	1.57907
$962$	1351.73	0.0453031
$963$	13936.2	0.466342
$964$	22152.9	0.740143
$965$	−33418.6	−1.11480
$966$	−80.7354	−0.00268905
$967$	−5006.71	−0.166499	−0.0832497	−	0.996529i	$-0.526530\pi$
$967$	−5006.71	−0.166499	−0.0832497	+	0.996529i	$0.526530\pi$
$968$	0	0
$969$	9637.86	0.319518
$970$	9817.18	0.324959
$971$	−15655.8	−0.517425	−0.258712	−	0.965954i	$-0.583298\pi$
$971$	−15655.8	−0.517425	−0.258712	+	0.965954i	$0.583298\pi$
$972$	1531.32	0.0505321
$973$	−108.623	−0.00357893
$974$	−12146.1	−0.399575
$975$	−12403.2	−0.407404
$976$	−10104.9	−0.331402
$977$	−11179.4	−0.366080	−0.183040	−	0.983105i	$-0.558594\pi$
$977$	−11179.4	−0.366080	−0.183040	+	0.983105i	$0.558594\pi$
$978$	9061.04	0.296258
$979$	0	0
$980$	−17497.2	−0.570335
$981$	−5852.19	−0.190465
$982$	−12754.7	−0.414478
$983$	−27168.3	−0.881521	−0.440760	−	0.897625i	$-0.645291\pi$
$983$	−27168.3	−0.881521	−0.440760	+	0.897625i	$0.645291\pi$
$984$	15928.5	0.516039
$985$	−26715.5	−0.864191
$986$	−6664.28	−0.215247
$987$	−550.056	−0.0177391
$988$	33118.4	1.06643
$989$	11858.6	0.381274
$990$	0	0
$991$	59150.4	1.89604	0.948020	−	0.318212i	$-0.103082\pi$
$991$	59150.4	1.89604	0.948020	+	0.318212i	$0.103082\pi$
$992$	50766.3	1.62483
$993$	−10018.4	−0.320165
$994$	−379.235	−0.0121012
$995$	−383.970	−0.0122338
$996$	3751.54	0.119349
$997$	−43855.7	−1.39310	−0.696552	−	0.717506i	$-0.745284\pi$
$997$	−43855.7	−1.39310	−0.696552	+	0.717506i	$0.745284\pi$
$998$	11401.8	0.361640
$999$	402.090	0.0127343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.v.1.3		6
3.2	odd	2		1089.4.a.bi.1.4		6
11.3	even	5		33.4.e.c.31.2	yes	12
11.4	even	5		33.4.e.c.16.2	✓	12
11.10	odd	2		363.4.a.u.1.4		6
33.14	odd	10		99.4.f.d.64.2		12
33.26	odd	10		99.4.f.d.82.2		12
33.32	even	2		1089.4.a.bk.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.c.16.2	✓	12	11.4	even	5
33.4.e.c.31.2	yes	12	11.3	even	5
99.4.f.d.64.2		12	33.14	odd	10
99.4.f.d.82.2		12	33.26	odd	10
363.4.a.u.1.4		6	11.10	odd	2
363.4.a.v.1.3		6	1.1	even	1	trivial
1089.4.a.bi.1.4		6	3.2	odd	2
1089.4.a.bk.1.3		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)

\(f(q)\)	\(=\)	\(q+1.30317 q^{2} -3.00000 q^{3} -6.30174 q^{4} -8.10194 q^{5} -3.90952 q^{6} +0.543555 q^{7} -18.6377 q^{8} +9.00000 q^{9} -10.5582 q^{10} +18.9052 q^{12} -69.6511 q^{13} +0.708348 q^{14} +24.3058 q^{15} +26.1258 q^{16} -42.5773 q^{17} +11.7286 q^{18} +75.4539 q^{19} +51.0563 q^{20} -1.63067 q^{21} +37.9924 q^{23} +55.9130 q^{24} -59.3586 q^{25} -90.7675 q^{26} -27.0000 q^{27} -3.42534 q^{28} +120.108 q^{29} +31.6747 q^{30} +277.188 q^{31} +183.148 q^{32} -55.4856 q^{34} -4.40385 q^{35} -56.7156 q^{36} -14.8922 q^{37} +98.3296 q^{38} +208.953 q^{39} +151.001 q^{40} +284.880 q^{41} -2.12504 q^{42} +312.130 q^{43} -72.9175 q^{45} +49.5107 q^{46} +337.320 q^{47} -78.3773 q^{48} -342.705 q^{49} -77.3546 q^{50} +127.732 q^{51} +438.923 q^{52} +169.723 q^{53} -35.1857 q^{54} -10.1306 q^{56} -226.362 q^{57} +156.522 q^{58} -801.918 q^{59} -153.169 q^{60} -386.778 q^{61} +361.224 q^{62} +4.89200 q^{63} +29.6673 q^{64} +564.309 q^{65} +981.580 q^{67} +268.311 q^{68} -113.977 q^{69} -5.73899 q^{70} -535.379 q^{71} -167.739 q^{72} +456.833 q^{73} -19.4072 q^{74} +178.076 q^{75} -475.490 q^{76} +272.302 q^{78} -619.503 q^{79} -211.669 q^{80} +81.0000 q^{81} +371.249 q^{82} +198.439 q^{83} +10.2760 q^{84} +344.959 q^{85} +406.759 q^{86} -360.325 q^{87} -1317.67 q^{89} -95.0242 q^{90} -37.8592 q^{91} -239.418 q^{92} -831.563 q^{93} +439.587 q^{94} -611.323 q^{95} -549.443 q^{96} -929.812 q^{97} -446.604 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) \) 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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