LMFDB - Embedded newform 624.4.a.s.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,4,Mod(1,624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("624.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-9,0,-4,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.13916.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 6 $ x^3 - x^2 - 16*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.68690$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +3.18652 q^{5} -23.9341 q^{7} +9.00000 q^{9} +6.74761 q^{11} +13.0000 q^{13} -9.55955 q^{15} +104.244 q^{17} +137.915 q^{19} +71.8024 q^{21} -110.244 q^{23} -114.846 q^{25} -27.0000 q^{27} -57.6048 q^{29} -319.893 q^{31} -20.2428 q^{33} -76.2666 q^{35} -2.88089 q^{37} -39.0000 q^{39} +319.145 q^{41} -344.705 q^{43} +28.6787 q^{45} +439.549 q^{47} +229.843 q^{49} -312.733 q^{51} -97.2632 q^{53} +21.5014 q^{55} -413.745 q^{57} -448.114 q^{59} -264.592 q^{61} -215.407 q^{63} +41.4247 q^{65} +712.701 q^{67} +330.733 q^{69} -1134.68 q^{71} -666.573 q^{73} +344.538 q^{75} -161.498 q^{77} -828.682 q^{79} +81.0000 q^{81} -734.136 q^{83} +332.177 q^{85} +172.814 q^{87} +153.255 q^{89} -311.144 q^{91} +959.679 q^{93} +439.469 q^{95} -569.044 q^{97} +60.7285 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 4 q^{5} - 6 q^{7} + 27 q^{9} - 32 q^{11} + 39 q^{13} + 12 q^{15} + 158 q^{17} - 70 q^{19} + 18 q^{21} - 176 q^{23} + 209 q^{25} - 81 q^{27} + 222 q^{29} - 54 q^{31} + 96 q^{33} - 496 q^{35}+ \cdots - 288 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 4 * q^5 - 6 * q^7 + 27 * q^9 - 32 * q^11 + 39 * q^13 + 12 * q^15 + 158 * q^17 - 70 * q^19 + 18 * q^21 - 176 * q^23 + 209 * q^25 - 81 * q^27 + 222 * q^29 - 54 * q^31 + 96 * q^33 - 496 * q^35 - 90 * q^37 - 117 * q^39 + 104 * q^41 - 140 * q^43 - 36 * q^45 - 328 * q^47 - 65 * q^49 - 474 * q^51 - 358 * q^53 - 32 * q^55 + 210 * q^57 - 1060 * q^59 - 186 * q^61 - 54 * q^63 - 52 * q^65 - 354 * q^67 + 528 * q^69 - 1692 * q^71 - 974 * q^73 - 627 * q^75 - 384 * q^77 - 776 * q^79 + 243 * q^81 - 2340 * q^83 - 2824 * q^85 - 666 * q^87 - 684 * q^89 - 78 * q^91 + 162 * q^93 - 742 * q^97 - 288 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	3.18652	0.285011	0.142505	−	0.989794i	$-0.454484\pi$
$5$	3.18652	0.285011	0.142505	+	0.989794i	$0.454484\pi$
$6$	0	0
$7$	−23.9341	−1.29232	−0.646161	−	0.763201i	$-0.723626\pi$
$7$	−23.9341	−1.29232	−0.646161	+	0.763201i	$0.723626\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	6.74761	0.184953	0.0924765	−	0.995715i	$-0.470522\pi$
$11$	6.74761	0.184953	0.0924765	+	0.995715i	$0.470522\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−9.55955	−0.164551
$16$	0	0
$17$	104.244	1.48723	0.743617	−	0.668606i	$-0.233109\pi$
$17$	104.244	1.48723	0.743617	+	0.668606i	$0.233109\pi$
$18$	0	0
$19$	137.915	1.66526	0.832628	−	0.553832i	$-0.186835\pi$
$19$	137.915	1.66526	0.832628	+	0.553832i	$0.186835\pi$
$20$	0	0
$21$	71.8024	0.746122
$22$	0	0
$23$	−110.244	−0.999458	−0.499729	−	0.866182i	$-0.666567\pi$
$23$	−110.244	−0.999458	−0.499729	+	0.866182i	$0.666567\pi$
$24$	0	0
$25$	−114.846	−0.918769
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−57.6048	−0.368860	−0.184430	−	0.982846i	$-0.559044\pi$
$29$	−57.6048	−0.368860	−0.184430	+	0.982846i	$0.559044\pi$
$30$	0	0
$31$	−319.893	−1.85337	−0.926685	−	0.375839i	$-0.877354\pi$
$31$	−319.893	−1.85337	−0.926685	+	0.375839i	$0.877354\pi$
$32$	0	0
$33$	−20.2428	−0.106783
$34$	0	0
$35$	−76.2666	−0.368326
$36$	0	0
$37$	−2.88089	−0.0128004	−0.00640021	−	0.999980i	$-0.502037\pi$
$37$	−2.88089	−0.0128004	−0.00640021	+	0.999980i	$0.502037\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	319.145	1.21566	0.607831	−	0.794067i	$-0.292040\pi$
$41$	319.145	1.21566	0.607831	+	0.794067i	$0.292040\pi$
$42$	0	0
$43$	−344.705	−1.22249	−0.611244	−	0.791442i	$-0.709331\pi$
$43$	−344.705	−1.22249	−0.611244	+	0.791442i	$0.709331\pi$
$44$	0	0
$45$	28.6787	0.0950036
$46$	0	0
$47$	439.549	1.36415	0.682073	−	0.731284i	$-0.261079\pi$
$47$	439.549	1.36415	0.682073	+	0.731284i	$0.261079\pi$
$48$	0	0
$49$	229.843	0.670095
$50$	0	0
$51$	−312.733	−0.858655
$52$	0	0
$53$	−97.2632	−0.252078	−0.126039	−	0.992025i	$-0.540226\pi$
$53$	−97.2632	−0.252078	−0.126039	+	0.992025i	$0.540226\pi$
$54$	0	0
$55$	21.5014	0.0527136
$56$	0	0
$57$	−413.745	−0.961437
$58$	0	0
$59$	−448.114	−0.988805	−0.494403	−	0.869233i	$-0.664613\pi$
$59$	−448.114	−0.988805	−0.494403	+	0.869233i	$0.664613\pi$
$60$	0	0
$61$	−264.592	−0.555370	−0.277685	−	0.960672i	$-0.589567\pi$
$61$	−264.592	−0.555370	−0.277685	+	0.960672i	$0.589567\pi$
$62$	0	0
$63$	−215.407	−0.430774
$64$	0	0
$65$	41.4247	0.0790478
$66$	0	0
$67$	712.701	1.29956	0.649778	−	0.760124i	$-0.274862\pi$
$67$	712.701	1.29956	0.649778	+	0.760124i	$0.274862\pi$
$68$	0	0
$69$	330.733	0.577038
$70$	0	0
$71$	−1134.68	−1.89665	−0.948323	−	0.317308i	$-0.897221\pi$
$71$	−1134.68	−1.89665	−0.948323	+	0.317308i	$0.897221\pi$
$72$	0	0
$73$	−666.573	−1.06872	−0.534359	−	0.845257i	$-0.679447\pi$
$73$	−666.573	−1.06872	−0.534359	+	0.845257i	$0.679447\pi$
$74$	0	0
$75$	344.538	0.530451
$76$	0	0
$77$	−161.498	−0.239019
$78$	0	0
$79$	−828.682	−1.18018	−0.590089	−	0.807338i	$-0.700907\pi$
$79$	−828.682	−1.18018	−0.590089	+	0.807338i	$0.700907\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−734.136	−0.970867	−0.485433	−	0.874274i	$-0.661338\pi$
$83$	−734.136	−0.970867	−0.485433	+	0.874274i	$0.661338\pi$
$84$	0	0
$85$	332.177	0.423878
$86$	0	0
$87$	172.814	0.212961
$88$	0	0
$89$	153.255	0.182528	0.0912639	−	0.995827i	$-0.470909\pi$
$89$	153.255	0.182528	0.0912639	+	0.995827i	$0.470909\pi$
$90$	0	0
$91$	−311.144	−0.358426
$92$	0	0
$93$	959.679	1.07004
$94$	0	0
$95$	439.469	0.474616
$96$	0	0
$97$	−569.044	−0.595646	−0.297823	−	0.954621i	$-0.596261\pi$
$97$	−569.044	−0.595646	−0.297823	+	0.954621i	$0.596261\pi$
$98$	0	0
$99$	60.7285	0.0616510
$100$	0	0
$101$	−1504.47	−1.48218	−0.741090	−	0.671406i	$-0.765691\pi$
$101$	−1504.47	−1.48218	−0.741090	+	0.671406i	$0.765691\pi$
$102$	0	0
$103$	1564.26	1.49642	0.748211	−	0.663461i	$-0.230913\pi$
$103$	1564.26	1.49642	0.748211	+	0.663461i	$0.230913\pi$
$104$	0	0
$105$	228.800	0.212653
$106$	0	0
$107$	−1486.11	−1.34268	−0.671342	−	0.741148i	$-0.734282\pi$
$107$	−1486.11	−1.34268	−0.671342	+	0.741148i	$0.734282\pi$
$108$	0	0
$109$	−1482.89	−1.30307	−0.651537	−	0.758617i	$-0.725875\pi$
$109$	−1482.89	−1.30307	−0.651537	+	0.758617i	$0.725875\pi$
$110$	0	0
$111$	8.64267	0.00739033
$112$	0	0
$113$	−534.504	−0.444973	−0.222486	−	0.974936i	$-0.571417\pi$
$113$	−534.504	−0.444973	−0.222486	+	0.974936i	$0.571417\pi$
$114$	0	0
$115$	−351.296	−0.284856
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−2495.00	−1.92198
$120$	0	0
$121$	−1285.47	−0.965792
$122$	0	0
$123$	−957.436	−0.701862
$124$	0	0
$125$	−764.274	−0.546870
$126$	0	0
$127$	1810.76	1.26519	0.632594	−	0.774484i	$-0.281990\pi$
$127$	1810.76	1.26519	0.632594	+	0.774484i	$0.281990\pi$
$128$	0	0
$129$	1034.11	0.705804
$130$	0	0
$131$	1210.44	0.807303	0.403652	−	0.914913i	$-0.367741\pi$
$131$	1210.44	0.807303	0.403652	+	0.914913i	$0.367741\pi$
$132$	0	0
$133$	−3300.88	−2.15205
$134$	0	0
$135$	−86.0360	−0.0548504
$136$	0	0
$137$	−1894.68	−1.18156	−0.590780	−	0.806833i	$-0.701180\pi$
$137$	−1894.68	−1.18156	−0.590780	+	0.806833i	$0.701180\pi$
$138$	0	0
$139$	799.458	0.487835	0.243918	−	0.969796i	$-0.421567\pi$
$139$	799.458	0.487835	0.243918	+	0.969796i	$0.421567\pi$
$140$	0	0
$141$	−1318.65	−0.787590
$142$	0	0
$143$	87.7190	0.0512967
$144$	0	0
$145$	−183.559	−0.105129
$146$	0	0
$147$	−689.528	−0.386880
$148$	0	0
$149$	−1936.45	−1.06470	−0.532350	−	0.846524i	$-0.678691\pi$
$149$	−1936.45	−1.06470	−0.532350	+	0.846524i	$0.678691\pi$
$150$	0	0
$151$	2200.40	1.18587	0.592933	−	0.805252i	$-0.297970\pi$
$151$	2200.40	1.18587	0.592933	+	0.805252i	$0.297970\pi$
$152$	0	0
$153$	938.199	0.495745
$154$	0	0
$155$	−1019.34	−0.528230
$156$	0	0
$157$	2438.48	1.23957	0.619783	−	0.784773i	$-0.287220\pi$
$157$	2438.48	1.23957	0.619783	+	0.784773i	$0.287220\pi$
$158$	0	0
$159$	291.790	0.145537
$160$	0	0
$161$	2638.60	1.29162
$162$	0	0
$163$	946.712	0.454921	0.227461	−	0.973787i	$-0.426958\pi$
$163$	946.712	0.454921	0.227461	+	0.973787i	$0.426958\pi$
$164$	0	0
$165$	−64.5042	−0.0304342
$166$	0	0
$167$	2027.84	0.939633	0.469816	−	0.882764i	$-0.344320\pi$
$167$	2027.84	0.939633	0.469816	+	0.882764i	$0.344320\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	1241.24	0.555086
$172$	0	0
$173$	1050.48	0.461655	0.230828	−	0.972995i	$-0.425857\pi$
$173$	1050.48	0.461655	0.230828	+	0.972995i	$0.425857\pi$
$174$	0	0
$175$	2748.74	1.18734
$176$	0	0
$177$	1344.34	0.570887
$178$	0	0
$179$	−2332.61	−0.974009	−0.487005	−	0.873399i	$-0.661911\pi$
$179$	−2332.61	−0.974009	−0.487005	+	0.873399i	$0.661911\pi$
$180$	0	0
$181$	1058.06	0.434502	0.217251	−	0.976116i	$-0.430291\pi$
$181$	1058.06	0.434502	0.217251	+	0.976116i	$0.430291\pi$
$182$	0	0
$183$	793.777	0.320643
$184$	0	0
$185$	−9.18001	−0.00364826
$186$	0	0
$187$	703.401	0.275068
$188$	0	0
$189$	646.222	0.248707
$190$	0	0
$191$	102.458	0.0388146	0.0194073	−	0.999812i	$-0.493822\pi$
$191$	102.458	0.0388146	0.0194073	+	0.999812i	$0.493822\pi$
$192$	0	0
$193$	102.396	0.0381899	0.0190949	−	0.999818i	$-0.493922\pi$
$193$	102.396	0.0381899	0.0190949	+	0.999818i	$0.493922\pi$
$194$	0	0
$195$	−124.274	−0.0456383
$196$	0	0
$197$	−4990.28	−1.80478	−0.902392	−	0.430916i	$-0.858191\pi$
$197$	−4990.28	−1.80478	−0.902392	+	0.430916i	$0.858191\pi$
$198$	0	0
$199$	2934.60	1.04537	0.522684	−	0.852526i	$-0.324931\pi$
$199$	2934.60	1.04537	0.522684	+	0.852526i	$0.324931\pi$
$200$	0	0
$201$	−2138.10	−0.750299
$202$	0	0
$203$	1378.72	0.476686
$204$	0	0
$205$	1016.96	0.346477
$206$	0	0
$207$	−992.199	−0.333153
$208$	0	0
$209$	930.598	0.307994
$210$	0	0
$211$	−4955.08	−1.61669	−0.808345	−	0.588709i	$-0.799636\pi$
$211$	−4955.08	−1.61669	−0.808345	+	0.588709i	$0.799636\pi$
$212$	0	0
$213$	3404.04	1.09503
$214$	0	0
$215$	−1098.41	−0.348422
$216$	0	0
$217$	7656.36	2.39515
$218$	0	0
$219$	1999.72	0.617025
$220$	0	0
$221$	1355.18	0.412484
$222$	0	0
$223$	−1258.69	−0.377975	−0.188987	−	0.981980i	$-0.560521\pi$
$223$	−1258.69	−0.377975	−0.188987	+	0.981980i	$0.560521\pi$
$224$	0	0
$225$	−1033.61	−0.306256
$226$	0	0
$227$	102.354	0.0299273	0.0149636	−	0.999888i	$-0.495237\pi$
$227$	102.354	0.0299273	0.0149636	+	0.999888i	$0.495237\pi$
$228$	0	0
$229$	−2489.73	−0.718455	−0.359228	−	0.933250i	$-0.616960\pi$
$229$	−2489.73	−0.718455	−0.359228	+	0.933250i	$0.616960\pi$
$230$	0	0
$231$	484.495	0.137998
$232$	0	0
$233$	4274.64	1.20189	0.600946	−	0.799289i	$-0.294791\pi$
$233$	4274.64	1.20189	0.600946	+	0.799289i	$0.294791\pi$
$234$	0	0
$235$	1400.63	0.388796
$236$	0	0
$237$	2486.05	0.681376
$238$	0	0
$239$	−2348.81	−0.635697	−0.317848	−	0.948142i	$-0.602960\pi$
$239$	−2348.81	−0.635697	−0.317848	+	0.948142i	$0.602960\pi$
$240$	0	0
$241$	−3130.05	−0.836615	−0.418308	−	0.908305i	$-0.637377\pi$
$241$	−3130.05	−0.836615	−0.418308	+	0.908305i	$0.637377\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	732.398	0.190984
$246$	0	0
$247$	1792.90	0.461859
$248$	0	0
$249$	2202.41	0.560530
$250$	0	0
$251$	1571.61	0.395216	0.197608	−	0.980281i	$-0.436683\pi$
$251$	1571.61	0.395216	0.197608	+	0.980281i	$0.436683\pi$
$252$	0	0
$253$	−743.887	−0.184853
$254$	0	0
$255$	−996.530	−0.244726
$256$	0	0
$257$	2968.71	0.720555	0.360278	−	0.932845i	$-0.382682\pi$
$257$	2968.71	0.720555	0.360278	+	0.932845i	$0.382682\pi$
$258$	0	0
$259$	68.9516	0.0165423
$260$	0	0
$261$	−518.443	−0.122953
$262$	0	0
$263$	−2982.14	−0.699188	−0.349594	−	0.936901i	$-0.613680\pi$
$263$	−2982.14	−0.699188	−0.349594	+	0.936901i	$0.613680\pi$
$264$	0	0
$265$	−309.931	−0.0718449
$266$	0	0
$267$	−459.765	−0.105383
$268$	0	0
$269$	−1706.67	−0.386832	−0.193416	−	0.981117i	$-0.561957\pi$
$269$	−1706.67	−0.386832	−0.193416	+	0.981117i	$0.561957\pi$
$270$	0	0
$271$	−3988.26	−0.893984	−0.446992	−	0.894538i	$-0.647505\pi$
$271$	−3988.26	−0.893984	−0.446992	+	0.894538i	$0.647505\pi$
$272$	0	0
$273$	933.431	0.206937
$274$	0	0
$275$	−774.937	−0.169929
$276$	0	0
$277$	−5098.99	−1.10602	−0.553012	−	0.833173i	$-0.686522\pi$
$277$	−5098.99	−1.10602	−0.553012	+	0.833173i	$0.686522\pi$
$278$	0	0
$279$	−2879.04	−0.617790
$280$	0	0
$281$	8080.89	1.71554	0.857768	−	0.514038i	$-0.171851\pi$
$281$	8080.89	1.71554	0.857768	+	0.514038i	$0.171851\pi$
$282$	0	0
$283$	228.187	0.0479305	0.0239652	−	0.999713i	$-0.492371\pi$
$283$	228.187	0.0479305	0.0239652	+	0.999713i	$0.492371\pi$
$284$	0	0
$285$	−1318.41	−0.274020
$286$	0	0
$287$	−7638.47	−1.57103
$288$	0	0
$289$	5953.89	1.21186
$290$	0	0
$291$	1707.13	0.343896
$292$	0	0
$293$	−1857.24	−0.370310	−0.185155	−	0.982709i	$-0.559279\pi$
$293$	−1857.24	−0.370310	−0.185155	+	0.982709i	$0.559279\pi$
$294$	0	0
$295$	−1427.92	−0.281820
$296$	0	0
$297$	−182.186	−0.0355942
$298$	0	0
$299$	−1433.18	−0.277200
$300$	0	0
$301$	8250.21	1.57985
$302$	0	0
$303$	4513.41	0.855737
$304$	0	0
$305$	−843.128	−0.158286
$306$	0	0
$307$	8204.32	1.52523	0.762614	−	0.646854i	$-0.223916\pi$
$307$	8204.32	1.52523	0.762614	+	0.646854i	$0.223916\pi$
$308$	0	0
$309$	−4692.79	−0.863960
$310$	0	0
$311$	−3194.10	−0.582382	−0.291191	−	0.956665i	$-0.594051\pi$
$311$	−3194.10	−0.582382	−0.291191	+	0.956665i	$0.594051\pi$
$312$	0	0
$313$	7309.24	1.31995	0.659973	−	0.751290i	$-0.270568\pi$
$313$	7309.24	1.31995	0.659973	+	0.751290i	$0.270568\pi$
$314$	0	0
$315$	−686.399	−0.122775
$316$	0	0
$317$	−8262.59	−1.46395	−0.731976	−	0.681330i	$-0.761402\pi$
$317$	−8262.59	−1.46395	−0.731976	+	0.681330i	$0.761402\pi$
$318$	0	0
$319$	−388.695	−0.0682218
$320$	0	0
$321$	4458.32	0.775199
$322$	0	0
$323$	14376.9	2.47663
$324$	0	0
$325$	−1493.00	−0.254821
$326$	0	0
$327$	4448.67	0.752330
$328$	0	0
$329$	−10520.2	−1.76291
$330$	0	0
$331$	2736.73	0.454454	0.227227	−	0.973842i	$-0.427034\pi$
$331$	2736.73	0.454454	0.227227	+	0.973842i	$0.427034\pi$
$332$	0	0
$333$	−25.9280	−0.00426681
$334$	0	0
$335$	2271.03	0.370387
$336$	0	0
$337$	5166.62	0.835144	0.417572	−	0.908644i	$-0.362881\pi$
$337$	5166.62	0.835144	0.417572	+	0.908644i	$0.362881\pi$
$338$	0	0
$339$	1603.51	0.256905
$340$	0	0
$341$	−2158.51	−0.342786
$342$	0	0
$343$	2708.32	0.426343
$344$	0	0
$345$	1053.89	0.164462
$346$	0	0
$347$	−10218.8	−1.58091	−0.790456	−	0.612519i	$-0.790156\pi$
$347$	−10218.8	−1.58091	−0.790456	+	0.612519i	$0.790156\pi$
$348$	0	0
$349$	1341.71	0.205788	0.102894	−	0.994692i	$-0.467190\pi$
$349$	1341.71	0.205788	0.102894	+	0.994692i	$0.467190\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	5268.23	0.794333	0.397166	−	0.917747i	$-0.369994\pi$
$353$	5268.23	0.794333	0.397166	+	0.917747i	$0.369994\pi$
$354$	0	0
$355$	−3615.68	−0.540564
$356$	0	0
$357$	7485.00	1.10966
$358$	0	0
$359$	2034.67	0.299125	0.149562	−	0.988752i	$-0.452214\pi$
$359$	2034.67	0.299125	0.149562	+	0.988752i	$0.452214\pi$
$360$	0	0
$361$	12161.6	1.77308
$362$	0	0
$363$	3856.41	0.557601
$364$	0	0
$365$	−2124.05	−0.304596
$366$	0	0
$367$	2765.13	0.393293	0.196647	−	0.980474i	$-0.436995\pi$
$367$	2765.13	0.393293	0.196647	+	0.980474i	$0.436995\pi$
$368$	0	0
$369$	2872.31	0.405220
$370$	0	0
$371$	2327.91	0.325766
$372$	0	0
$373$	−1160.30	−0.161067	−0.0805337	−	0.996752i	$-0.525662\pi$
$373$	−1160.30	−0.161067	−0.0805337	+	0.996752i	$0.525662\pi$
$374$	0	0
$375$	2292.82	0.315736
$376$	0	0
$377$	−748.862	−0.102303
$378$	0	0
$379$	−2577.88	−0.349385	−0.174692	−	0.984623i	$-0.555893\pi$
$379$	−2577.88	−0.349385	−0.174692	+	0.984623i	$0.555893\pi$
$380$	0	0
$381$	−5432.27	−0.730456
$382$	0	0
$383$	−814.563	−0.108674	−0.0543371	−	0.998523i	$-0.517305\pi$
$383$	−814.563	−0.108674	−0.0543371	+	0.998523i	$0.517305\pi$
$384$	0	0
$385$	−514.617	−0.0681229
$386$	0	0
$387$	−3102.34	−0.407496
$388$	0	0
$389$	−1736.18	−0.226293	−0.113147	−	0.993578i	$-0.536093\pi$
$389$	−1736.18	−0.226293	−0.113147	+	0.993578i	$0.536093\pi$
$390$	0	0
$391$	−11492.4	−1.48643
$392$	0	0
$393$	−3631.32	−0.466097
$394$	0	0
$395$	−2640.61	−0.336363
$396$	0	0
$397$	1.18770	0.000150149 0	7.50746e−5	−	1.00000i	$-0.499976\pi$
$397$	1.18770	0.000150149 0	7.50746e−5		1.00000i	$0.499976\pi$
$398$	0	0
$399$	9902.63	1.24249
$400$	0	0
$401$	−6552.48	−0.815998	−0.407999	−	0.912982i	$-0.633773\pi$
$401$	−6552.48	−0.815998	−0.407999	+	0.912982i	$0.633773\pi$
$402$	0	0
$403$	−4158.61	−0.514032
$404$	0	0
$405$	258.108	0.0316679
$406$	0	0
$407$	−19.4391	−0.00236748
$408$	0	0
$409$	−2735.87	−0.330759	−0.165379	−	0.986230i	$-0.552885\pi$
$409$	−2735.87	−0.330759	−0.165379	+	0.986230i	$0.552885\pi$
$410$	0	0
$411$	5684.05	0.682174
$412$	0	0
$413$	10725.2	1.27785
$414$	0	0
$415$	−2339.34	−0.276708
$416$	0	0
$417$	−2398.37	−0.281652
$418$	0	0
$419$	9666.00	1.12700	0.563502	−	0.826115i	$-0.309454\pi$
$419$	9666.00	1.12700	0.563502	+	0.826115i	$0.309454\pi$
$420$	0	0
$421$	9774.14	1.13150	0.565751	−	0.824576i	$-0.308586\pi$
$421$	9774.14	1.13150	0.565751	+	0.824576i	$0.308586\pi$
$422$	0	0
$423$	3955.94	0.454715
$424$	0	0
$425$	−11972.1	−1.36642
$426$	0	0
$427$	6332.78	0.717716
$428$	0	0
$429$	−263.157	−0.0296162
$430$	0	0
$431$	−5984.17	−0.668787	−0.334394	−	0.942434i	$-0.608531\pi$
$431$	−5984.17	−0.668787	−0.334394	+	0.942434i	$0.608531\pi$
$432$	0	0
$433$	−1179.51	−0.130909	−0.0654544	−	0.997856i	$-0.520850\pi$
$433$	−1179.51	−0.130909	−0.0654544	+	0.997856i	$0.520850\pi$
$434$	0	0
$435$	550.676	0.0606963
$436$	0	0
$437$	−15204.4	−1.66436
$438$	0	0
$439$	−11989.5	−1.30348	−0.651741	−	0.758442i	$-0.725961\pi$
$439$	−11989.5	−1.30348	−0.651741	+	0.758442i	$0.725961\pi$
$440$	0	0
$441$	2068.58	0.223365
$442$	0	0
$443$	−7682.86	−0.823981	−0.411991	−	0.911188i	$-0.635166\pi$
$443$	−7682.86	−0.823981	−0.411991	+	0.911188i	$0.635166\pi$
$444$	0	0
$445$	488.349	0.0520224
$446$	0	0
$447$	5809.36	0.614705
$448$	0	0
$449$	−12713.8	−1.33631	−0.668155	−	0.744022i	$-0.732916\pi$
$449$	−12713.8	−1.33631	−0.668155	+	0.744022i	$0.732916\pi$
$450$	0	0
$451$	2153.47	0.224840
$452$	0	0
$453$	−6601.19	−0.684659
$454$	0	0
$455$	−991.465	−0.102155
$456$	0	0
$457$	−1707.47	−0.174775	−0.0873875	−	0.996174i	$-0.527852\pi$
$457$	−1707.47	−0.174775	−0.0873875	+	0.996174i	$0.527852\pi$
$458$	0	0
$459$	−2814.60	−0.286218
$460$	0	0
$461$	−12514.3	−1.26432	−0.632159	−	0.774838i	$-0.717831\pi$
$461$	−12514.3	−1.26432	−0.632159	+	0.774838i	$0.717831\pi$
$462$	0	0
$463$	10447.6	1.04868	0.524340	−	0.851509i	$-0.324312\pi$
$463$	10447.6	1.04868	0.524340	+	0.851509i	$0.324312\pi$
$464$	0	0
$465$	3058.03	0.304974
$466$	0	0
$467$	15170.6	1.50324	0.751618	−	0.659599i	$-0.229274\pi$
$467$	15170.6	1.50324	0.751618	+	0.659599i	$0.229274\pi$
$468$	0	0
$469$	−17057.9	−1.67944
$470$	0	0
$471$	−7315.44	−0.715664
$472$	0	0
$473$	−2325.94	−0.226103
$474$	0	0
$475$	−15839.0	−1.52999
$476$	0	0
$477$	−875.369	−0.0840259
$478$	0	0
$479$	5177.65	0.493889	0.246945	−	0.969030i	$-0.420573\pi$
$479$	5177.65	0.493889	0.246945	+	0.969030i	$0.420573\pi$
$480$	0	0
$481$	−37.4516	−0.00355020
$482$	0	0
$483$	−7915.81	−0.745718
$484$	0	0
$485$	−1813.27	−0.169766
$486$	0	0
$487$	4056.52	0.377451	0.188725	−	0.982030i	$-0.439564\pi$
$487$	4056.52	0.377451	0.188725	+	0.982030i	$0.439564\pi$
$488$	0	0
$489$	−2840.13	−0.262649
$490$	0	0
$491$	4328.53	0.397849	0.198924	−	0.980015i	$-0.436255\pi$
$491$	4328.53	0.397849	0.198924	+	0.980015i	$0.436255\pi$
$492$	0	0
$493$	−6004.98	−0.548581
$494$	0	0
$495$	193.513	0.0175712
$496$	0	0
$497$	27157.6	2.45108
$498$	0	0
$499$	12764.9	1.14516	0.572581	−	0.819848i	$-0.305942\pi$
$499$	12764.9	1.14516	0.572581	+	0.819848i	$0.305942\pi$
$500$	0	0
$501$	−6083.51	−0.542497
$502$	0	0
$503$	16718.4	1.48198	0.740991	−	0.671515i	$-0.234356\pi$
$503$	16718.4	1.48198	0.740991	+	0.671515i	$0.234356\pi$
$504$	0	0
$505$	−4794.02	−0.422437
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	6994.30	0.609071	0.304535	−	0.952501i	$-0.401499\pi$
$509$	6994.30	0.609071	0.304535	+	0.952501i	$0.401499\pi$
$510$	0	0
$511$	15953.8	1.38113
$512$	0	0
$513$	−3723.71	−0.320479
$514$	0	0
$515$	4984.56	0.426497
$516$	0	0
$517$	2965.91	0.252303
$518$	0	0
$519$	−3151.43	−0.266537
$520$	0	0
$521$	−5281.01	−0.444079	−0.222040	−	0.975038i	$-0.571271\pi$
$521$	−5281.01	−0.444079	−0.222040	+	0.975038i	$0.571271\pi$
$522$	0	0
$523$	−3410.20	−0.285120	−0.142560	−	0.989786i	$-0.545533\pi$
$523$	−3410.20	−0.285120	−0.142560	+	0.989786i	$0.545533\pi$
$524$	0	0
$525$	−8246.23	−0.685514
$526$	0	0
$527$	−33347.0	−2.75639
$528$	0	0
$529$	−13.1752	−0.00108286
$530$	0	0
$531$	−4033.03	−0.329602
$532$	0	0
$533$	4148.89	0.337164
$534$	0	0
$535$	−4735.50	−0.382680
$536$	0	0
$537$	6997.84	0.562344
$538$	0	0
$539$	1550.89	0.123936
$540$	0	0
$541$	1280.98	0.101800	0.0508998	−	0.998704i	$-0.483791\pi$
$541$	1280.98	0.101800	0.0508998	+	0.998704i	$0.483791\pi$
$542$	0	0
$543$	−3174.18	−0.250860
$544$	0	0
$545$	−4725.25	−0.371390
$546$	0	0
$547$	−14379.3	−1.12397	−0.561986	−	0.827147i	$-0.689963\pi$
$547$	−14379.3	−1.12397	−0.561986	+	0.827147i	$0.689963\pi$
$548$	0	0
$549$	−2381.33	−0.185123
$550$	0	0
$551$	−7944.57	−0.614247
$552$	0	0
$553$	19833.8	1.52517
$554$	0	0
$555$	27.5400	0.00210632
$556$	0	0
$557$	−17781.2	−1.35263	−0.676313	−	0.736614i	$-0.736423\pi$
$557$	−17781.2	−1.35263	−0.676313	+	0.736614i	$0.736423\pi$
$558$	0	0
$559$	−4481.16	−0.339057
$560$	0	0
$561$	−2110.20	−0.158811
$562$	0	0
$563$	−11953.0	−0.894775	−0.447387	−	0.894340i	$-0.647645\pi$
$563$	−11953.0	−0.894775	−0.447387	+	0.894340i	$0.647645\pi$
$564$	0	0
$565$	−1703.21	−0.126822
$566$	0	0
$567$	−1938.66	−0.143591
$568$	0	0
$569$	18111.3	1.33439	0.667193	−	0.744885i	$-0.267496\pi$
$569$	18111.3	1.33439	0.667193	+	0.744885i	$0.267496\pi$
$570$	0	0
$571$	−10493.0	−0.769032	−0.384516	−	0.923118i	$-0.625632\pi$
$571$	−10493.0	−0.769032	−0.384516	+	0.923118i	$0.625632\pi$
$572$	0	0
$573$	−307.374	−0.0224096
$574$	0	0
$575$	12661.1	0.918271
$576$	0	0
$577$	8839.27	0.637754	0.318877	−	0.947796i	$-0.396694\pi$
$577$	8839.27	0.637754	0.318877	+	0.947796i	$0.396694\pi$
$578$	0	0
$579$	−307.189	−0.0220489
$580$	0	0
$581$	17570.9	1.25467
$582$	0	0
$583$	−656.294	−0.0466225
$584$	0	0
$585$	372.823	0.0263493
$586$	0	0
$587$	−22816.3	−1.60431	−0.802155	−	0.597116i	$-0.796313\pi$
$587$	−22816.3	−1.60431	−0.802155	+	0.597116i	$0.796313\pi$
$588$	0	0
$589$	−44118.0	−3.08634
$590$	0	0
$591$	14970.8	1.04199
$592$	0	0
$593$	−28015.1	−1.94004	−0.970018	−	0.243033i	$-0.921858\pi$
$593$	−28015.1	−1.94004	−0.970018	+	0.243033i	$0.921858\pi$
$594$	0	0
$595$	−7950.36	−0.547787
$596$	0	0
$597$	−8803.80	−0.603544
$598$	0	0
$599$	10404.4	0.709702	0.354851	−	0.934923i	$-0.384532\pi$
$599$	10404.4	0.709702	0.354851	+	0.934923i	$0.384532\pi$
$600$	0	0
$601$	−6801.55	−0.461632	−0.230816	−	0.972997i	$-0.574140\pi$
$601$	−6801.55	−0.461632	−0.230816	+	0.972997i	$0.574140\pi$
$602$	0	0
$603$	6414.30	0.433185
$604$	0	0
$605$	−4096.17	−0.275261
$606$	0	0
$607$	13046.5	0.872394	0.436197	−	0.899851i	$-0.356325\pi$
$607$	13046.5	0.872394	0.436197	+	0.899851i	$0.356325\pi$
$608$	0	0
$609$	−4136.16	−0.275215
$610$	0	0
$611$	5714.14	0.378346
$612$	0	0
$613$	15725.0	1.03610	0.518049	−	0.855351i	$-0.326658\pi$
$613$	15725.0	1.03610	0.518049	+	0.855351i	$0.326658\pi$
$614$	0	0
$615$	−3050.89	−0.200038
$616$	0	0
$617$	27568.7	1.79882	0.899411	−	0.437104i	$-0.143996\pi$
$617$	27568.7	1.79882	0.899411	+	0.437104i	$0.143996\pi$
$618$	0	0
$619$	−9433.05	−0.612514	−0.306257	−	0.951949i	$-0.599077\pi$
$619$	−9433.05	−0.612514	−0.306257	+	0.951949i	$0.599077\pi$
$620$	0	0
$621$	2976.60	0.192346
$622$	0	0
$623$	−3668.02	−0.235885
$624$	0	0
$625$	11920.4	0.762905
$626$	0	0
$627$	−2791.79	−0.177821
$628$	0	0
$629$	−300.317	−0.0190372
$630$	0	0
$631$	−29379.6	−1.85354	−0.926771	−	0.375627i	$-0.877428\pi$
$631$	−29379.6	−1.85354	−0.926771	+	0.375627i	$0.877428\pi$
$632$	0	0
$633$	14865.2	0.933396
$634$	0	0
$635$	5770.01	0.360592
$636$	0	0
$637$	2987.96	0.185851
$638$	0	0
$639$	−10212.1	−0.632215
$640$	0	0
$641$	−28264.3	−1.74161	−0.870805	−	0.491629i	$-0.836401\pi$
$641$	−28264.3	−1.74161	−0.870805	+	0.491629i	$0.836401\pi$
$642$	0	0
$643$	−27550.9	−1.68974	−0.844869	−	0.534973i	$-0.820322\pi$
$643$	−27550.9	−1.68974	−0.844869	+	0.534973i	$0.820322\pi$
$644$	0	0
$645$	3295.22	0.201162
$646$	0	0
$647$	−23186.0	−1.40887	−0.704433	−	0.709770i	$-0.748799\pi$
$647$	−23186.0	−1.40887	−0.704433	+	0.709770i	$0.748799\pi$
$648$	0	0
$649$	−3023.70	−0.182882
$650$	0	0
$651$	−22969.1	−1.38284
$652$	0	0
$653$	14294.6	0.856647	0.428323	−	0.903625i	$-0.359104\pi$
$653$	14294.6	0.856647	0.428323	+	0.903625i	$0.359104\pi$
$654$	0	0
$655$	3857.09	0.230090
$656$	0	0
$657$	−5999.16	−0.356240
$658$	0	0
$659$	−2487.32	−0.147029	−0.0735146	−	0.997294i	$-0.523422\pi$
$659$	−2487.32	−0.147029	−0.0735146	+	0.997294i	$0.523422\pi$
$660$	0	0
$661$	−2542.95	−0.149636	−0.0748180	−	0.997197i	$-0.523838\pi$
$661$	−2542.95	−0.149636	−0.0748180	+	0.997197i	$0.523838\pi$
$662$	0	0
$663$	−4065.53	−0.238148
$664$	0	0
$665$	−10518.3	−0.613357
$666$	0	0
$667$	6350.61	0.368660
$668$	0	0
$669$	3776.08	0.218224
$670$	0	0
$671$	−1785.37	−0.102717
$672$	0	0
$673$	−15068.1	−0.863049	−0.431524	−	0.902101i	$-0.642024\pi$
$673$	−15068.1	−0.863049	−0.431524	+	0.902101i	$0.642024\pi$
$674$	0	0
$675$	3100.84	0.176817
$676$	0	0
$677$	28291.2	1.60608	0.803042	−	0.595923i	$-0.203214\pi$
$677$	28291.2	1.60608	0.803042	+	0.595923i	$0.203214\pi$
$678$	0	0
$679$	13619.6	0.769766
$680$	0	0
$681$	−307.063	−0.0172785
$682$	0	0
$683$	22052.3	1.23545	0.617723	−	0.786396i	$-0.288055\pi$
$683$	22052.3	1.23545	0.617723	+	0.786396i	$0.288055\pi$
$684$	0	0
$685$	−6037.44	−0.336757
$686$	0	0
$687$	7469.20	0.414800
$688$	0	0
$689$	−1264.42	−0.0699138
$690$	0	0
$691$	−22010.1	−1.21173	−0.605864	−	0.795568i	$-0.707173\pi$
$691$	−22010.1	−1.21173	−0.605864	+	0.795568i	$0.707173\pi$
$692$	0	0
$693$	−1453.48	−0.0796729
$694$	0	0
$695$	2547.49	0.139038
$696$	0	0
$697$	33269.1	1.80797
$698$	0	0
$699$	−12823.9	−0.693913
$700$	0	0
$701$	18959.7	1.02154	0.510769	−	0.859718i	$-0.329361\pi$
$701$	18959.7	1.02154	0.510769	+	0.859718i	$0.329361\pi$
$702$	0	0
$703$	−397.318	−0.0213160
$704$	0	0
$705$	−4201.89	−0.224472
$706$	0	0
$707$	36008.1	1.91545
$708$	0	0
$709$	−21466.2	−1.13707	−0.568534	−	0.822660i	$-0.692489\pi$
$709$	−21466.2	−1.13707	−0.568534	+	0.822660i	$0.692489\pi$
$710$	0	0
$711$	−7458.14	−0.393392
$712$	0	0
$713$	35266.4	1.85237
$714$	0	0
$715$	279.518	0.0146201
$716$	0	0
$717$	7046.42	0.367020
$718$	0	0
$719$	5589.46	0.289919	0.144959	−	0.989438i	$-0.453695\pi$
$719$	5589.46	0.289919	0.144959	+	0.989438i	$0.453695\pi$
$720$	0	0
$721$	−37439.3	−1.93386
$722$	0	0
$723$	9390.15	0.483020
$724$	0	0
$725$	6615.69	0.338897
$726$	0	0
$727$	−33648.8	−1.71660	−0.858298	−	0.513152i	$-0.828478\pi$
$727$	−33648.8	−1.71660	−0.858298	+	0.513152i	$0.828478\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−35933.5	−1.81813
$732$	0	0
$733$	1542.91	0.0777470	0.0388735	−	0.999244i	$-0.487623\pi$
$733$	1542.91	0.0777470	0.0388735	+	0.999244i	$0.487623\pi$
$734$	0	0
$735$	−2197.19	−0.110265
$736$	0	0
$737$	4809.03	0.240357
$738$	0	0
$739$	30154.3	1.50101	0.750503	−	0.660867i	$-0.229811\pi$
$739$	30154.3	1.50101	0.750503	+	0.660867i	$0.229811\pi$
$740$	0	0
$741$	−5378.69	−0.266655
$742$	0	0
$743$	−25634.5	−1.26573	−0.632866	−	0.774261i	$-0.718122\pi$
$743$	−25634.5	−1.26573	−0.632866	+	0.774261i	$0.718122\pi$
$744$	0	0
$745$	−6170.54	−0.303451
$746$	0	0
$747$	−6607.23	−0.323622
$748$	0	0
$749$	35568.6	1.73518
$750$	0	0
$751$	−4193.68	−0.203768	−0.101884	−	0.994796i	$-0.532487\pi$
$751$	−4193.68	−0.203768	−0.101884	+	0.994796i	$0.532487\pi$
$752$	0	0
$753$	−4714.83	−0.228178
$754$	0	0
$755$	7011.60	0.337984
$756$	0	0
$757$	21998.9	1.05623	0.528113	−	0.849174i	$-0.322900\pi$
$757$	21998.9	1.05623	0.528113	+	0.849174i	$0.322900\pi$
$758$	0	0
$759$	2231.66	0.106725
$760$	0	0
$761$	−6212.65	−0.295938	−0.147969	−	0.988992i	$-0.547274\pi$
$761$	−6212.65	−0.295938	−0.147969	+	0.988992i	$0.547274\pi$
$762$	0	0
$763$	35491.7	1.68399
$764$	0	0
$765$	2989.59	0.141293
$766$	0	0
$767$	−5825.48	−0.274245
$768$	0	0
$769$	12573.3	0.589603	0.294802	−	0.955558i	$-0.404746\pi$
$769$	12573.3	0.589603	0.294802	+	0.955558i	$0.404746\pi$
$770$	0	0
$771$	−8906.12	−0.416013
$772$	0	0
$773$	−35282.4	−1.64168	−0.820840	−	0.571159i	$-0.806494\pi$
$773$	−35282.4	−1.64168	−0.820840	+	0.571159i	$0.806494\pi$
$774$	0	0
$775$	36738.5	1.70282
$776$	0	0
$777$	−206.855	−0.00955068
$778$	0	0
$779$	44014.9	2.02439
$780$	0	0
$781$	−7656.39	−0.350790
$782$	0	0
$783$	1555.33	0.0709872
$784$	0	0
$785$	7770.26	0.353290
$786$	0	0
$787$	35759.1	1.61966	0.809831	−	0.586663i	$-0.199559\pi$
$787$	35759.1	1.61966	0.809831	+	0.586663i	$0.199559\pi$
$788$	0	0
$789$	8946.41	0.403676
$790$	0	0
$791$	12792.9	0.575048
$792$	0	0
$793$	−3439.70	−0.154032
$794$	0	0
$795$	929.793	0.0414797
$796$	0	0
$797$	−24603.7	−1.09349	−0.546743	−	0.837301i	$-0.684133\pi$
$797$	−24603.7	−1.09349	−0.546743	+	0.837301i	$0.684133\pi$
$798$	0	0
$799$	45820.5	2.02880
$800$	0	0
$801$	1379.29	0.0608426
$802$	0	0
$803$	−4497.78	−0.197663
$804$	0	0
$805$	8407.96	0.368126
$806$	0	0
$807$	5120.02	0.223338
$808$	0	0
$809$	33283.3	1.44645	0.723226	−	0.690612i	$-0.242659\pi$
$809$	33283.3	1.44645	0.723226	+	0.690612i	$0.242659\pi$
$810$	0	0
$811$	−40351.9	−1.74716	−0.873581	−	0.486679i	$-0.838208\pi$
$811$	−40351.9	−1.74716	−0.873581	+	0.486679i	$0.838208\pi$
$812$	0	0
$813$	11964.8	0.516142
$814$	0	0
$815$	3016.71	0.129658
$816$	0	0
$817$	−47540.0	−2.03576
$818$	0	0
$819$	−2800.29	−0.119475
$820$	0	0
$821$	−18080.9	−0.768607	−0.384303	−	0.923207i	$-0.625558\pi$
$821$	−18080.9	−0.768607	−0.384303	+	0.923207i	$0.625558\pi$
$822$	0	0
$823$	22784.9	0.965043	0.482522	−	0.875884i	$-0.339721\pi$
$823$	22784.9	0.965043	0.482522	+	0.875884i	$0.339721\pi$
$824$	0	0
$825$	2324.81	0.0981086
$826$	0	0
$827$	26389.3	1.10961	0.554804	−	0.831981i	$-0.312793\pi$
$827$	26389.3	1.10961	0.554804	+	0.831981i	$0.312793\pi$
$828$	0	0
$829$	23934.1	1.00274	0.501368	−	0.865234i	$-0.332830\pi$
$829$	23934.1	1.00274	0.501368	+	0.865234i	$0.332830\pi$
$830$	0	0
$831$	15297.0	0.638564
$832$	0	0
$833$	23959.8	0.996589
$834$	0	0
$835$	6461.74	0.267806
$836$	0	0
$837$	8637.11	0.356681
$838$	0	0
$839$	−17716.6	−0.729016	−0.364508	−	0.931200i	$-0.618763\pi$
$839$	−17716.6	−0.729016	−0.364508	+	0.931200i	$0.618763\pi$
$840$	0	0
$841$	−21070.7	−0.863942
$842$	0	0
$843$	−24242.7	−0.990465
$844$	0	0
$845$	538.522	0.0219239
$846$	0	0
$847$	30766.6	1.24811
$848$	0	0
$849$	−684.562	−0.0276727
$850$	0	0
$851$	317.602	0.0127935
$852$	0	0
$853$	27077.6	1.08689	0.543446	−	0.839444i	$-0.317119\pi$
$853$	27077.6	1.08689	0.543446	+	0.839444i	$0.317119\pi$
$854$	0	0
$855$	3955.22	0.158205
$856$	0	0
$857$	−22069.8	−0.879684	−0.439842	−	0.898075i	$-0.644966\pi$
$857$	−22069.8	−0.879684	−0.439842	+	0.898075i	$0.644966\pi$
$858$	0	0
$859$	24577.5	0.976221	0.488110	−	0.872782i	$-0.337686\pi$
$859$	24577.5	0.976221	0.488110	+	0.872782i	$0.337686\pi$
$860$	0	0
$861$	22915.4	0.907032
$862$	0	0
$863$	−24322.4	−0.959381	−0.479690	−	0.877438i	$-0.659251\pi$
$863$	−24322.4	−0.959381	−0.479690	+	0.877438i	$0.659251\pi$
$864$	0	0
$865$	3347.37	0.131577
$866$	0	0
$867$	−17861.7	−0.699671
$868$	0	0
$869$	−5591.63	−0.218277
$870$	0	0
$871$	9265.11	0.360432
$872$	0	0
$873$	−5121.40	−0.198549
$874$	0	0
$875$	18292.2	0.706732
$876$	0	0
$877$	−18624.2	−0.717096	−0.358548	−	0.933511i	$-0.616728\pi$
$877$	−18624.2	−0.717096	−0.358548	+	0.933511i	$0.616728\pi$
$878$	0	0
$879$	5571.71	0.213799
$880$	0	0
$881$	46666.0	1.78458	0.892291	−	0.451460i	$-0.149097\pi$
$881$	46666.0	1.78458	0.892291	+	0.451460i	$0.149097\pi$
$882$	0	0
$883$	−29108.5	−1.10937	−0.554687	−	0.832059i	$-0.687162\pi$
$883$	−29108.5	−1.10937	−0.554687	+	0.832059i	$0.687162\pi$
$884$	0	0
$885$	4283.77	0.162709
$886$	0	0
$887$	18025.7	0.682351	0.341176	−	0.940000i	$-0.389175\pi$
$887$	18025.7	0.682351	0.341176	+	0.940000i	$0.389175\pi$
$888$	0	0
$889$	−43338.9	−1.63503
$890$	0	0
$891$	546.557	0.0205503
$892$	0	0
$893$	60620.4	2.27165
$894$	0	0
$895$	−7432.91	−0.277603
$896$	0	0
$897$	4299.53	0.160041
$898$	0	0
$899$	18427.4	0.683634
$900$	0	0
$901$	−10139.1	−0.374899
$902$	0	0
$903$	−24750.6	−0.912126
$904$	0	0
$905$	3371.52	0.123838
$906$	0	0
$907$	6690.51	0.244933	0.122467	−	0.992473i	$-0.460920\pi$
$907$	6690.51	0.244933	0.122467	+	0.992473i	$0.460920\pi$
$908$	0	0
$909$	−13540.2	−0.494060
$910$	0	0
$911$	2333.45	0.0848633	0.0424317	−	0.999099i	$-0.486490\pi$
$911$	2333.45	0.0848633	0.0424317	+	0.999099i	$0.486490\pi$
$912$	0	0
$913$	−4953.67	−0.179565
$914$	0	0
$915$	2529.38	0.0913867
$916$	0	0
$917$	−28970.9	−1.04330
$918$	0	0
$919$	−5069.08	−0.181952	−0.0909758	−	0.995853i	$-0.528999\pi$
$919$	−5069.08	−0.181952	−0.0909758	+	0.995853i	$0.528999\pi$
$920$	0	0
$921$	−24613.0	−0.880591
$922$	0	0
$923$	−14750.8	−0.526035
$924$	0	0
$925$	330.859	0.0117606
$926$	0	0
$927$	14078.4	0.498807
$928$	0	0
$929$	1321.66	0.0466763	0.0233382	−	0.999728i	$-0.492571\pi$
$929$	1321.66	0.0466763	0.0233382	+	0.999728i	$0.492571\pi$
$930$	0	0
$931$	31698.8	1.11588
$932$	0	0
$933$	9582.30	0.336238
$934$	0	0
$935$	2241.40	0.0783975
$936$	0	0
$937$	−52868.3	−1.84326	−0.921629	−	0.388072i	$-0.873141\pi$
$937$	−52868.3	−1.84326	−0.921629	+	0.388072i	$0.873141\pi$
$938$	0	0
$939$	−21927.7	−0.762071
$940$	0	0
$941$	−29982.2	−1.03867	−0.519337	−	0.854570i	$-0.673821\pi$
$941$	−29982.2	−1.03867	−0.519337	+	0.854570i	$0.673821\pi$
$942$	0	0
$943$	−35184.0	−1.21500
$944$	0	0
$945$	2059.20	0.0708843
$946$	0	0
$947$	20676.2	0.709488	0.354744	−	0.934963i	$-0.384568\pi$
$947$	20676.2	0.709488	0.354744	+	0.934963i	$0.384568\pi$
$948$	0	0
$949$	−8665.45	−0.296409
$950$	0	0
$951$	24787.8	0.845214
$952$	0	0
$953$	34705.5	1.17967	0.589833	−	0.807526i	$-0.299194\pi$
$953$	34705.5	1.17967	0.589833	+	0.807526i	$0.299194\pi$
$954$	0	0
$955$	326.484	0.0110626
$956$	0	0
$957$	1166.09	0.0393879
$958$	0	0
$959$	45347.6	1.52696
$960$	0	0
$961$	72540.5	2.43498
$962$	0	0
$963$	−13374.9	−0.447561
$964$	0	0
$965$	326.288	0.0108845
$966$	0	0
$967$	−22925.4	−0.762390	−0.381195	−	0.924495i	$-0.624487\pi$
$967$	−22925.4	−0.762390	−0.381195	+	0.924495i	$0.624487\pi$
$968$	0	0
$969$	−43130.6	−1.42988
$970$	0	0
$971$	18359.7	0.606787	0.303393	−	0.952865i	$-0.401880\pi$
$971$	18359.7	0.606787	0.303393	+	0.952865i	$0.401880\pi$
$972$	0	0
$973$	−19134.3	−0.630440
$974$	0	0
$975$	4479.00	0.147121
$976$	0	0
$977$	14567.7	0.477032	0.238516	−	0.971139i	$-0.423339\pi$
$977$	14567.7	0.477032	0.238516	+	0.971139i	$0.423339\pi$
$978$	0	0
$979$	1034.10	0.0337591
$980$	0	0
$981$	−13346.0	−0.434358
$982$	0	0
$983$	15297.6	0.496355	0.248178	−	0.968715i	$-0.420168\pi$
$983$	15297.6	0.496355	0.248178	+	0.968715i	$0.420168\pi$
$984$	0	0
$985$	−15901.6	−0.514383
$986$	0	0
$987$	31560.7	1.01782
$988$	0	0
$989$	38001.8	1.22183
$990$	0	0
$991$	11376.1	0.364655	0.182328	−	0.983238i	$-0.441637\pi$
$991$	11376.1	0.364655	0.182328	+	0.983238i	$0.441637\pi$
$992$	0	0
$993$	−8210.20	−0.262379
$994$	0	0
$995$	9351.16	0.297941
$996$	0	0
$997$	46713.5	1.48388	0.741941	−	0.670465i	$-0.233906\pi$
$997$	46713.5	1.48388	0.741941	+	0.670465i	$0.233906\pi$
$998$	0	0
$999$	77.7840	0.00246344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.s.1.2		3
3.2	odd	2		1872.4.a.bl.1.2		3
4.3	odd	2		312.4.a.h.1.2	✓	3
8.3	odd	2		2496.4.a.bm.1.2		3
8.5	even	2		2496.4.a.bq.1.2		3
12.11	even	2		936.4.a.l.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.h.1.2	✓	3	4.3	odd	2
624.4.a.s.1.2		3	1.1	even	1	trivial
936.4.a.l.1.2		3	12.11	even	2
1872.4.a.bl.1.2		3	3.2	odd	2
2496.4.a.bm.1.2		3	8.3	odd	2
2496.4.a.bq.1.2		3	8.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +3.18652 q^{5} -23.9341 q^{7} +9.00000 q^{9} +6.74761 q^{11} +13.0000 q^{13} -9.55955 q^{15} +104.244 q^{17} +137.915 q^{19} +71.8024 q^{21} -110.244 q^{23} -114.846 q^{25} -27.0000 q^{27} -57.6048 q^{29} -319.893 q^{31} -20.2428 q^{33} -76.2666 q^{35} -2.88089 q^{37} -39.0000 q^{39} +319.145 q^{41} -344.705 q^{43} +28.6787 q^{45} +439.549 q^{47} +229.843 q^{49} -312.733 q^{51} -97.2632 q^{53} +21.5014 q^{55} -413.745 q^{57} -448.114 q^{59} -264.592 q^{61} -215.407 q^{63} +41.4247 q^{65} +712.701 q^{67} +330.733 q^{69} -1134.68 q^{71} -666.573 q^{73} +344.538 q^{75} -161.498 q^{77} -828.682 q^{79} +81.0000 q^{81} -734.136 q^{83} +332.177 q^{85} +172.814 q^{87} +153.255 q^{89} -311.144 q^{91} +959.679 q^{93} +439.469 q^{95} -569.044 q^{97} +60.7285 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 4 q^{5} - 6 q^{7} + 27 q^{9} - 32 q^{11} + 39 q^{13} + 12 q^{15} + 158 q^{17} - 70 q^{19} + 18 q^{21} - 176 q^{23} + 209 q^{25} - 81 q^{27} + 222 q^{29} - 54 q^{31} + 96 q^{33} - 496 q^{35}+ \cdots - 288 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 4 * q^5 - 6 * q^7 + 27 * q^9 - 32 * q^11 + 39 * q^13 + 12 * q^15 + 158 * q^17 - 70 * q^19 + 18 * q^21 - 176 * q^23 + 209 * q^25 - 81 * q^27 + 222 * q^29 - 54 * q^31 + 96 * q^33 - 496 * q^35 - 90 * q^37 - 117 * q^39 + 104 * q^41 - 140 * q^43 - 36 * q^45 - 328 * q^47 - 65 * q^49 - 474 * q^51 - 358 * q^53 - 32 * q^55 + 210 * q^57 - 1060 * q^59 - 186 * q^61 - 54 * q^63 - 52 * q^65 - 354 * q^67 + 528 * q^69 - 1692 * q^71 - 974 * q^73 - 627 * q^75 - 384 * q^77 - 776 * q^79 + 243 * q^81 - 2340 * q^83 - 2824 * q^85 - 666 * q^87 - 684 * q^89 - 78 * q^91 + 162 * q^93 - 742 * q^97 - 288 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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