LMFDB - Embedded newform 368.4.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [368,4,Mod(1,368)] 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("368.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	368.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$21.7127028821$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.334189.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} - 5x + 4 $ x^4 - 2x^3 - 16x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.743529$ of defining polynomial
Character	$\chi$	$=$	368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.55870 q^{3} +10.0635 q^{5} -24.3381 q^{7} -24.5704 q^{9} -1.55839 q^{11} +85.6294 q^{13} -15.6860 q^{15} -35.1015 q^{17} +124.400 q^{19} +37.9359 q^{21} +23.0000 q^{23} -23.7259 q^{25} +80.3831 q^{27} +130.943 q^{29} +82.0830 q^{31} +2.42907 q^{33} -244.926 q^{35} -107.402 q^{37} -133.471 q^{39} +35.6655 q^{41} +227.986 q^{43} -247.265 q^{45} +268.071 q^{47} +249.342 q^{49} +54.7128 q^{51} +567.034 q^{53} -15.6829 q^{55} -193.902 q^{57} +422.803 q^{59} -57.0580 q^{61} +597.997 q^{63} +861.732 q^{65} +517.544 q^{67} -35.8502 q^{69} +418.494 q^{71} +586.385 q^{73} +36.9817 q^{75} +37.9282 q^{77} -595.986 q^{79} +538.108 q^{81} -346.074 q^{83} -353.244 q^{85} -204.102 q^{87} -322.588 q^{89} -2084.05 q^{91} -127.943 q^{93} +1251.90 q^{95} -1102.27 q^{97} +38.2903 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 7 q^{3} + 14 q^{5} - 16 q^{7} - 33 q^{9} - 8 q^{11} + 111 q^{13} - 10 q^{15} + 98 q^{17} - 96 q^{19} + 180 q^{21} + 92 q^{23} + 184 q^{25} + 155 q^{27} + 21 q^{29} + 193 q^{31} - 418 q^{33} + 752 q^{35}+ \cdots + 1498 q^{99}+O(q^{100}) $ 4 * q - 7 * q^3 + 14 * q^5 - 16 * q^7 - 33 * q^9 - 8 * q^11 + 111 * q^13 - 10 * q^15 + 98 * q^17 - 96 * q^19 + 180 * q^21 + 92 * q^23 + 184 * q^25 + 155 * q^27 + 21 * q^29 + 193 * q^31 - 418 * q^33 + 752 * q^35 + 170 * q^37 + 291 * q^39 - 125 * q^41 - 2 * q^43 + 168 * q^45 + 677 * q^47 + 1220 * q^49 + 340 * q^51 - 230 * q^53 + 972 * q^55 + 1322 * q^57 + 1140 * q^59 + 754 * q^61 + 1092 * q^63 + 1318 * q^65 - 488 * q^67 - 161 * q^69 + 401 * q^71 + 1509 * q^73 - 1401 * q^75 + 736 * q^77 + 838 * q^79 - 932 * q^81 - 142 * q^83 + 112 * q^85 - 2223 * q^87 + 2342 * q^89 - 292 * q^91 - 509 * q^93 + 956 * q^95 + 1062 * q^97 + 1498 * 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
$3$	−1.55870	−0.299973	−0.149986	−	0.988688i	$-0.547923\pi$
$3$	−1.55870	−0.299973	−0.149986	+	0.988688i	$0.547923\pi$
$4$	0	0
$5$	10.0635	0.900107	0.450054	−	0.893002i	$-0.351405\pi$
$5$	10.0635	0.900107	0.450054	+	0.893002i	$0.351405\pi$
$6$	0	0
$7$	−24.3381	−1.31413	−0.657066	−	0.753833i	$-0.728203\pi$
$7$	−24.3381	−1.31413	−0.657066	+	0.753833i	$0.728203\pi$
$8$	0	0
$9$	−24.5704	−0.910016
$10$	0	0
$11$	−1.55839	−0.0427156	−0.0213578	−	0.999772i	$-0.506799\pi$
$11$	−1.55839	−0.0427156	−0.0213578	+	0.999772i	$0.506799\pi$
$12$	0	0
$13$	85.6294	1.82687	0.913435	−	0.406984i	$-0.133419\pi$
$13$	85.6294	1.82687	0.913435	+	0.406984i	$0.133419\pi$
$14$	0	0
$15$	−15.6860	−0.270008
$16$	0	0
$17$	−35.1015	−0.500786	−0.250393	−	0.968144i	$-0.580560\pi$
$17$	−35.1015	−0.500786	−0.250393	+	0.968144i	$0.580560\pi$
$18$	0	0
$19$	124.400	1.50207	0.751033	−	0.660265i	$-0.229556\pi$
$19$	124.400	1.50207	0.751033	+	0.660265i	$0.229556\pi$
$20$	0	0
$21$	37.9359	0.394204
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−23.7259	−0.189807
$26$	0	0
$27$	80.3831	0.572953
$28$	0	0
$29$	130.943	0.838467	0.419234	−	0.907878i	$-0.362299\pi$
$29$	130.943	0.838467	0.419234	+	0.907878i	$0.362299\pi$
$30$	0	0
$31$	82.0830	0.475566	0.237783	−	0.971318i	$-0.423579\pi$
$31$	82.0830	0.475566	0.237783	+	0.971318i	$0.423579\pi$
$32$	0	0
$33$	2.42907	0.0128135
$34$	0	0
$35$	−244.926	−1.18286
$36$	0	0
$37$	−107.402	−0.477209	−0.238604	−	0.971117i	$-0.576690\pi$
$37$	−107.402	−0.477209	−0.238604	+	0.971117i	$0.576690\pi$
$38$	0	0
$39$	−133.471	−0.548012
$40$	0	0
$41$	35.6655	0.135854	0.0679270	−	0.997690i	$-0.478361\pi$
$41$	35.6655	0.135854	0.0679270	+	0.997690i	$0.478361\pi$
$42$	0	0
$43$	227.986	0.808548	0.404274	−	0.914638i	$-0.367524\pi$
$43$	227.986	0.808548	0.404274	+	0.914638i	$0.367524\pi$
$44$	0	0
$45$	−247.265	−0.819112
$46$	0	0
$47$	268.071	0.831961	0.415980	−	0.909374i	$-0.363438\pi$
$47$	268.071	0.831961	0.415980	+	0.909374i	$0.363438\pi$
$48$	0	0
$49$	249.342	0.726945
$50$	0	0
$51$	54.7128	0.150222
$52$	0	0
$53$	567.034	1.46959	0.734794	−	0.678291i	$-0.237279\pi$
$53$	567.034	1.46959	0.734794	+	0.678291i	$0.237279\pi$
$54$	0	0
$55$	−15.6829	−0.0384487
$56$	0	0
$57$	−193.902	−0.450579
$58$	0	0
$59$	422.803	0.932955	0.466477	−	0.884533i	$-0.345523\pi$
$59$	422.803	0.932955	0.466477	+	0.884533i	$0.345523\pi$
$60$	0	0
$61$	−57.0580	−0.119763	−0.0598814	−	0.998206i	$-0.519072\pi$
$61$	−57.0580	−0.119763	−0.0598814	+	0.998206i	$0.519072\pi$
$62$	0	0
$63$	597.997	1.19588
$64$	0	0
$65$	861.732	1.64438
$66$	0	0
$67$	517.544	0.943702	0.471851	−	0.881678i	$-0.343586\pi$
$67$	517.544	0.943702	0.471851	+	0.881678i	$0.343586\pi$
$68$	0	0
$69$	−35.8502	−0.0625486
$70$	0	0
$71$	418.494	0.699523	0.349761	−	0.936839i	$-0.386263\pi$
$71$	418.494	0.699523	0.349761	+	0.936839i	$0.386263\pi$
$72$	0	0
$73$	586.385	0.940154	0.470077	−	0.882625i	$-0.344226\pi$
$73$	586.385	0.940154	0.470077	+	0.882625i	$0.344226\pi$
$74$	0	0
$75$	36.9817	0.0569370
$76$	0	0
$77$	37.9282	0.0561340
$78$	0	0
$79$	−595.986	−0.848781	−0.424390	−	0.905479i	$-0.639512\pi$
$79$	−595.986	−0.848781	−0.424390	+	0.905479i	$0.639512\pi$
$80$	0	0
$81$	538.108	0.738146
$82$	0	0
$83$	−346.074	−0.457669	−0.228835	−	0.973465i	$-0.573492\pi$
$83$	−346.074	−0.457669	−0.228835	+	0.973465i	$0.573492\pi$
$84$	0	0
$85$	−353.244	−0.450761
$86$	0	0
$87$	−204.102	−0.251517
$88$	0	0
$89$	−322.588	−0.384206	−0.192103	−	0.981375i	$-0.561531\pi$
$89$	−322.588	−0.384206	−0.192103	+	0.981375i	$0.561531\pi$
$90$	0	0
$91$	−2084.05	−2.40075
$92$	0	0
$93$	−127.943	−0.142657
$94$	0	0
$95$	1251.90	1.35202
$96$	0	0
$97$	−1102.27	−1.15380	−0.576900	−	0.816815i	$-0.695738\pi$
$97$	−1102.27	−1.15380	−0.576900	+	0.816815i	$0.695738\pi$
$98$	0	0
$99$	38.2903	0.0388719
$100$	0	0
$101$	−281.413	−0.277244	−0.138622	−	0.990345i	$-0.544267\pi$
$101$	−281.413	−0.277244	−0.138622	+	0.990345i	$0.544267\pi$
$102$	0	0
$103$	−606.665	−0.580354	−0.290177	−	0.956973i	$-0.593714\pi$
$103$	−606.665	−0.580354	−0.290177	+	0.956973i	$0.593714\pi$
$104$	0	0
$105$	381.768	0.354826
$106$	0	0
$107$	−758.318	−0.685135	−0.342567	−	0.939493i	$-0.611296\pi$
$107$	−758.318	−0.685135	−0.342567	+	0.939493i	$0.611296\pi$
$108$	0	0
$109$	1730.34	1.52052	0.760261	−	0.649618i	$-0.225071\pi$
$109$	1730.34	1.52052	0.760261	+	0.649618i	$0.225071\pi$
$110$	0	0
$111$	167.407	0.143150
$112$	0	0
$113$	1712.46	1.42561	0.712807	−	0.701361i	$-0.247424\pi$
$113$	1712.46	1.42561	0.712807	+	0.701361i	$0.247424\pi$
$114$	0	0
$115$	231.461	0.187685
$116$	0	0
$117$	−2103.95	−1.66248
$118$	0	0
$119$	854.303	0.658099
$120$	0	0
$121$	−1328.57	−0.998175
$122$	0	0
$123$	−55.5920	−0.0407525
$124$	0	0
$125$	−1496.70	−1.07095
$126$	0	0
$127$	−1008.43	−0.704594	−0.352297	−	0.935888i	$-0.614599\pi$
$127$	−1008.43	−0.704594	−0.352297	+	0.935888i	$0.614599\pi$
$128$	0	0
$129$	−355.363	−0.242542
$130$	0	0
$131$	338.615	0.225839	0.112920	−	0.993604i	$-0.463980\pi$
$131$	338.615	0.225839	0.112920	+	0.993604i	$0.463980\pi$
$132$	0	0
$133$	−3027.65	−1.97391
$134$	0	0
$135$	808.935	0.515719
$136$	0	0
$137$	341.454	0.212937	0.106469	−	0.994316i	$-0.466046\pi$
$137$	341.454	0.212937	0.106469	+	0.994316i	$0.466046\pi$
$138$	0	0
$139$	510.426	0.311466	0.155733	−	0.987799i	$-0.450226\pi$
$139$	510.426	0.311466	0.155733	+	0.987799i	$0.450226\pi$
$140$	0	0
$141$	−417.843	−0.249566
$142$	0	0
$143$	−133.444	−0.0780360
$144$	0	0
$145$	1317.75	0.754710
$146$	0	0
$147$	−388.651	−0.218064
$148$	0	0
$149$	−1989.54	−1.09389	−0.546944	−	0.837169i	$-0.684209\pi$
$149$	−1989.54	−1.09389	−0.546944	+	0.837169i	$0.684209\pi$
$150$	0	0
$151$	−607.109	−0.327191	−0.163596	−	0.986527i	$-0.552309\pi$
$151$	−607.109	−0.327191	−0.163596	+	0.986527i	$0.552309\pi$
$152$	0	0
$153$	862.459	0.455723
$154$	0	0
$155$	826.043	0.428060
$156$	0	0
$157$	1119.43	0.569045	0.284523	−	0.958669i	$-0.408165\pi$
$157$	1119.43	0.569045	0.284523	+	0.958669i	$0.408165\pi$
$158$	0	0
$159$	−883.839	−0.440836
$160$	0	0
$161$	−559.776	−0.274016
$162$	0	0
$163$	2794.13	1.34266	0.671328	−	0.741160i	$-0.265724\pi$
$163$	2794.13	1.34266	0.671328	+	0.741160i	$0.265724\pi$
$164$	0	0
$165$	24.4449	0.0115335
$166$	0	0
$167$	−2676.11	−1.24002	−0.620010	−	0.784594i	$-0.712871\pi$
$167$	−2676.11	−1.24002	−0.620010	+	0.784594i	$0.712871\pi$
$168$	0	0
$169$	5135.39	2.33746
$170$	0	0
$171$	−3056.55	−1.36690
$172$	0	0
$173$	−60.7346	−0.0266911	−0.0133456	−	0.999911i	$-0.504248\pi$
$173$	−60.7346	−0.0266911	−0.0133456	+	0.999911i	$0.504248\pi$
$174$	0	0
$175$	577.443	0.249432
$176$	0	0
$177$	−659.026	−0.279861
$178$	0	0
$179$	862.735	0.360245	0.180123	−	0.983644i	$-0.442351\pi$
$179$	862.735	0.360245	0.180123	+	0.983644i	$0.442351\pi$
$180$	0	0
$181$	766.840	0.314910	0.157455	−	0.987526i	$-0.449671\pi$
$181$	766.840	0.314910	0.157455	+	0.987526i	$0.449671\pi$
$182$	0	0
$183$	88.9365	0.0359256
$184$	0	0
$185$	−1080.84	−0.429539
$186$	0	0
$187$	54.7018	0.0213914
$188$	0	0
$189$	−1956.37	−0.752936
$190$	0	0
$191$	−3786.27	−1.43437	−0.717186	−	0.696882i	$-0.754570\pi$
$191$	−3786.27	−1.43437	−0.717186	+	0.696882i	$0.754570\pi$
$192$	0	0
$193$	2713.36	1.01198	0.505990	−	0.862539i	$-0.331127\pi$
$193$	2713.36	1.01198	0.505990	+	0.862539i	$0.331127\pi$
$194$	0	0
$195$	−1343.18	−0.493269
$196$	0	0
$197$	−5275.02	−1.90777	−0.953883	−	0.300180i	$-0.902953\pi$
$197$	−5275.02	−1.90777	−0.953883	+	0.300180i	$0.902953\pi$
$198$	0	0
$199$	−2689.93	−0.958212	−0.479106	−	0.877757i	$-0.659039\pi$
$199$	−2689.93	−0.958212	−0.479106	+	0.877757i	$0.659039\pi$
$200$	0	0
$201$	−806.697	−0.283085
$202$	0	0
$203$	−3186.91	−1.10186
$204$	0	0
$205$	358.920	0.122283
$206$	0	0
$207$	−565.120	−0.189752
$208$	0	0
$209$	−193.863	−0.0641617
$210$	0	0
$211$	2900.32	0.946285	0.473143	−	0.880986i	$-0.343120\pi$
$211$	2900.32	0.946285	0.473143	+	0.880986i	$0.343120\pi$
$212$	0	0
$213$	−652.309	−0.209838
$214$	0	0
$215$	2294.34	0.727780
$216$	0	0
$217$	−1997.74	−0.624957
$218$	0	0
$219$	−914.002	−0.282021
$220$	0	0
$221$	−3005.72	−0.914871
$222$	0	0
$223$	−6307.92	−1.89421	−0.947106	−	0.320920i	$-0.896008\pi$
$223$	−6307.92	−1.89421	−0.947106	+	0.320920i	$0.896008\pi$
$224$	0	0
$225$	582.956	0.172728
$226$	0	0
$227$	5454.59	1.59486	0.797431	−	0.603410i	$-0.206192\pi$
$227$	5454.59	1.59486	0.797431	+	0.603410i	$0.206192\pi$
$228$	0	0
$229$	1535.31	0.443039	0.221520	−	0.975156i	$-0.428898\pi$
$229$	1535.31	0.443039	0.221520	+	0.975156i	$0.428898\pi$
$230$	0	0
$231$	−59.1189	−0.0168387
$232$	0	0
$233$	−2808.98	−0.789795	−0.394897	−	0.918725i	$-0.629220\pi$
$233$	−2808.98	−0.789795	−0.394897	+	0.918725i	$0.629220\pi$
$234$	0	0
$235$	2697.73	0.748854
$236$	0	0
$237$	928.966	0.254611
$238$	0	0
$239$	3051.86	0.825978	0.412989	−	0.910736i	$-0.364485\pi$
$239$	3051.86	0.825978	0.412989	+	0.910736i	$0.364485\pi$
$240$	0	0
$241$	994.590	0.265839	0.132919	−	0.991127i	$-0.457565\pi$
$241$	994.590	0.265839	0.132919	+	0.991127i	$0.457565\pi$
$242$	0	0
$243$	−3009.09	−0.794377
$244$	0	0
$245$	2509.25	0.654328
$246$	0	0
$247$	10652.3	2.74408
$248$	0	0
$249$	539.427	0.137288
$250$	0	0
$251$	−6960.36	−1.75034	−0.875168	−	0.483820i	$-0.839249\pi$
$251$	−6960.36	−1.75034	−0.875168	+	0.483820i	$0.839249\pi$
$252$	0	0
$253$	−35.8430	−0.00890683
$254$	0	0
$255$	550.603	0.135216
$256$	0	0
$257$	528.738	0.128334	0.0641669	−	0.997939i	$-0.479561\pi$
$257$	528.738	0.128334	0.0641669	+	0.997939i	$0.479561\pi$
$258$	0	0
$259$	2613.95	0.627116
$260$	0	0
$261$	−3217.33	−0.763019
$262$	0	0
$263$	620.103	0.145389	0.0726943	−	0.997354i	$-0.476840\pi$
$263$	620.103	0.145389	0.0726943	+	0.997354i	$0.476840\pi$
$264$	0	0
$265$	5706.35	1.32279
$266$	0	0
$267$	502.820	0.115251
$268$	0	0
$269$	−2448.16	−0.554895	−0.277448	−	0.960741i	$-0.589488\pi$
$269$	−2448.16	−0.554895	−0.277448	+	0.960741i	$0.589488\pi$
$270$	0	0
$271$	904.980	0.202855	0.101427	−	0.994843i	$-0.467659\pi$
$271$	904.980	0.202855	0.101427	+	0.994843i	$0.467659\pi$
$272$	0	0
$273$	3248.43	0.720160
$274$	0	0
$275$	36.9742	0.00810774
$276$	0	0
$277$	5045.12	1.09434	0.547169	−	0.837022i	$-0.315705\pi$
$277$	5045.12	1.09434	0.547169	+	0.837022i	$0.315705\pi$
$278$	0	0
$279$	−2016.82	−0.432773
$280$	0	0
$281$	3192.77	0.677811	0.338905	−	0.940820i	$-0.389943\pi$
$281$	3192.77	0.677811	0.338905	+	0.940820i	$0.389943\pi$
$282$	0	0
$283$	−239.398	−0.0502853	−0.0251426	−	0.999684i	$-0.508004\pi$
$283$	−239.398	−0.0502853	−0.0251426	+	0.999684i	$0.508004\pi$
$284$	0	0
$285$	−1951.34	−0.405569
$286$	0	0
$287$	−868.030	−0.178530
$288$	0	0
$289$	−3680.89	−0.749213
$290$	0	0
$291$	1718.11	0.346108
$292$	0	0
$293$	4584.70	0.914134	0.457067	−	0.889432i	$-0.348900\pi$
$293$	4584.70	0.914134	0.457067	+	0.889432i	$0.348900\pi$
$294$	0	0
$295$	4254.88	0.839759
$296$	0	0
$297$	−125.268	−0.0244741
$298$	0	0
$299$	1969.48	0.380929
$300$	0	0
$301$	−5548.75	−1.06254
$302$	0	0
$303$	438.640	0.0831657
$304$	0	0
$305$	−574.203	−0.107799
$306$	0	0
$307$	−483.762	−0.0899340	−0.0449670	−	0.998988i	$-0.514318\pi$
$307$	−483.762	−0.0899340	−0.0449670	+	0.998988i	$0.514318\pi$
$308$	0	0
$309$	945.611	0.174090
$310$	0	0
$311$	−1131.39	−0.206286	−0.103143	−	0.994667i	$-0.532890\pi$
$311$	−1131.39	−0.206286	−0.103143	+	0.994667i	$0.532890\pi$
$312$	0	0
$313$	8678.30	1.56718	0.783589	−	0.621280i	$-0.213387\pi$
$313$	8678.30	1.56718	0.783589	+	0.621280i	$0.213387\pi$
$314$	0	0
$315$	6017.95	1.07642
$316$	0	0
$317$	5835.70	1.03396	0.516981	−	0.855997i	$-0.327056\pi$
$317$	5835.70	1.03396	0.516981	+	0.855997i	$0.327056\pi$
$318$	0	0
$319$	−204.061	−0.0358157
$320$	0	0
$321$	1181.99	0.205522
$322$	0	0
$323$	−4366.61	−0.752213
$324$	0	0
$325$	−2031.64	−0.346754
$326$	0	0
$327$	−2697.09	−0.456115
$328$	0	0
$329$	−6524.33	−1.09331
$330$	0	0
$331$	4804.86	0.797882	0.398941	−	0.916977i	$-0.369378\pi$
$331$	4804.86	0.797882	0.398941	+	0.916977i	$0.369378\pi$
$332$	0	0
$333$	2638.91	0.434268
$334$	0	0
$335$	5208.30	0.849433
$336$	0	0
$337$	308.081	0.0497989	0.0248995	−	0.999690i	$-0.492073\pi$
$337$	308.081	0.0497989	0.0248995	+	0.999690i	$0.492073\pi$
$338$	0	0
$339$	−2669.21	−0.427645
$340$	0	0
$341$	−127.917	−0.0203141
$342$	0	0
$343$	2279.45	0.358831
$344$	0	0
$345$	−360.779	−0.0563005
$346$	0	0
$347$	−2133.10	−0.330003	−0.165001	−	0.986293i	$-0.552763\pi$
$347$	−2133.10	−0.330003	−0.165001	+	0.986293i	$0.552763\pi$
$348$	0	0
$349$	−10534.3	−1.61573	−0.807864	−	0.589369i	$-0.799376\pi$
$349$	−10534.3	−1.61573	−0.807864	+	0.589369i	$0.799376\pi$
$350$	0	0
$351$	6883.15	1.04671
$352$	0	0
$353$	3551.48	0.535485	0.267742	−	0.963491i	$-0.413722\pi$
$353$	3551.48	0.535485	0.267742	+	0.963491i	$0.413722\pi$
$354$	0	0
$355$	4211.52	0.629645
$356$	0	0
$357$	−1331.61	−0.197412
$358$	0	0
$359$	−11547.3	−1.69761	−0.848804	−	0.528707i	$-0.822677\pi$
$359$	−11547.3	−1.69761	−0.848804	+	0.528707i	$0.822677\pi$
$360$	0	0
$361$	8616.28	1.25620
$362$	0	0
$363$	2070.85	0.299425
$364$	0	0
$365$	5901.09	0.846239
$366$	0	0
$367$	7553.39	1.07434	0.537171	−	0.843473i	$-0.319493\pi$
$367$	7553.39	1.07434	0.537171	+	0.843473i	$0.319493\pi$
$368$	0	0
$369$	−876.317	−0.123629
$370$	0	0
$371$	−13800.5	−1.93123
$372$	0	0
$373$	−7766.30	−1.07808	−0.539040	−	0.842280i	$-0.681213\pi$
$373$	−7766.30	−1.07808	−0.539040	+	0.842280i	$0.681213\pi$
$374$	0	0
$375$	2332.92	0.321257
$376$	0	0
$377$	11212.6	1.53177
$378$	0	0
$379$	1196.96	0.162226	0.0811132	−	0.996705i	$-0.474152\pi$
$379$	1196.96	0.162226	0.0811132	+	0.996705i	$0.474152\pi$
$380$	0	0
$381$	1571.84	0.211359
$382$	0	0
$383$	−489.495	−0.0653055	−0.0326528	−	0.999467i	$-0.510396\pi$
$383$	−489.495	−0.0653055	−0.0326528	+	0.999467i	$0.510396\pi$
$384$	0	0
$385$	381.691	0.0505266
$386$	0	0
$387$	−5601.72	−0.735792
$388$	0	0
$389$	4892.37	0.637668	0.318834	−	0.947810i	$-0.396709\pi$
$389$	4892.37	0.637668	0.318834	+	0.947810i	$0.396709\pi$
$390$	0	0
$391$	−807.334	−0.104421
$392$	0	0
$393$	−527.801	−0.0677456
$394$	0	0
$395$	−5997.71	−0.763993
$396$	0	0
$397$	9231.40	1.16703	0.583515	−	0.812102i	$-0.301677\pi$
$397$	9231.40	1.16703	0.583515	+	0.812102i	$0.301677\pi$
$398$	0	0
$399$	4719.21	0.592120
$400$	0	0
$401$	−4350.38	−0.541765	−0.270883	−	0.962612i	$-0.587315\pi$
$401$	−4350.38	−0.541765	−0.270883	+	0.962612i	$0.587315\pi$
$402$	0	0
$403$	7028.72	0.868798
$404$	0	0
$405$	5415.26	0.664410
$406$	0	0
$407$	167.374	0.0203843
$408$	0	0
$409$	−2058.13	−0.248822	−0.124411	−	0.992231i	$-0.539704\pi$
$409$	−2058.13	−0.248822	−0.124411	+	0.992231i	$0.539704\pi$
$410$	0	0
$411$	−532.226	−0.0638753
$412$	0	0
$413$	−10290.2	−1.22603
$414$	0	0
$415$	−3482.72	−0.411951
$416$	0	0
$417$	−795.603	−0.0934313
$418$	0	0
$419$	15060.4	1.75596	0.877982	−	0.478694i	$-0.158890\pi$
$419$	15060.4	1.75596	0.877982	+	0.478694i	$0.158890\pi$
$420$	0	0
$421$	1109.01	0.128385	0.0641923	−	0.997938i	$-0.479553\pi$
$421$	1109.01	0.128385	0.0641923	+	0.997938i	$0.479553\pi$
$422$	0	0
$423$	−6586.62	−0.757098
$424$	0	0
$425$	832.815	0.0950528
$426$	0	0
$427$	1388.68	0.157384
$428$	0	0
$429$	208.000	0.0234087
$430$	0	0
$431$	7453.68	0.833019	0.416509	−	0.909131i	$-0.363253\pi$
$431$	7453.68	0.833019	0.416509	+	0.909131i	$0.363253\pi$
$432$	0	0
$433$	−3360.65	−0.372985	−0.186493	−	0.982456i	$-0.559712\pi$
$433$	−3360.65	−0.372985	−0.186493	+	0.982456i	$0.559712\pi$
$434$	0	0
$435$	−2053.98	−0.226392
$436$	0	0
$437$	2861.19	0.313202
$438$	0	0
$439$	−11073.4	−1.20389	−0.601943	−	0.798539i	$-0.705607\pi$
$439$	−11073.4	−1.20389	−0.601943	+	0.798539i	$0.705607\pi$
$440$	0	0
$441$	−6126.45	−0.661532
$442$	0	0
$443$	15911.1	1.70646	0.853229	−	0.521536i	$-0.174641\pi$
$443$	15911.1	1.70646	0.853229	+	0.521536i	$0.174641\pi$
$444$	0	0
$445$	−3246.37	−0.345826
$446$	0	0
$447$	3101.10	0.328136
$448$	0	0
$449$	−14842.1	−1.56001	−0.780003	−	0.625776i	$-0.784783\pi$
$449$	−14842.1	−1.56001	−0.780003	+	0.625776i	$0.784783\pi$
$450$	0	0
$451$	−55.5807	−0.00580309
$452$	0	0
$453$	946.304	0.0981484
$454$	0	0
$455$	−20972.9	−2.16093
$456$	0	0
$457$	14903.8	1.52554	0.762771	−	0.646669i	$-0.223839\pi$
$457$	14903.8	1.52554	0.762771	+	0.646669i	$0.223839\pi$
$458$	0	0
$459$	−2821.56	−0.286927
$460$	0	0
$461$	9271.90	0.936736	0.468368	−	0.883533i	$-0.344842\pi$
$461$	9271.90	0.936736	0.468368	+	0.883533i	$0.344842\pi$
$462$	0	0
$463$	6157.42	0.618055	0.309028	−	0.951053i	$-0.399996\pi$
$463$	6157.42	0.618055	0.309028	+	0.951053i	$0.399996\pi$
$464$	0	0
$465$	−1287.56	−0.128406
$466$	0	0
$467$	−3030.18	−0.300257	−0.150129	−	0.988666i	$-0.547969\pi$
$467$	−3030.18	−0.300257	−0.150129	+	0.988666i	$0.547969\pi$
$468$	0	0
$469$	−12596.0	−1.24015
$470$	0	0
$471$	−1744.86	−0.170698
$472$	0	0
$473$	−355.291	−0.0345376
$474$	0	0
$475$	−2951.50	−0.285103
$476$	0	0
$477$	−13932.3	−1.33735
$478$	0	0
$479$	11390.0	1.08648	0.543240	−	0.839577i	$-0.317197\pi$
$479$	11390.0	1.08648	0.543240	+	0.839577i	$0.317197\pi$
$480$	0	0
$481$	−9196.74	−0.871799
$482$	0	0
$483$	872.525	0.0821972
$484$	0	0
$485$	−11092.7	−1.03854
$486$	0	0
$487$	6027.22	0.560820	0.280410	−	0.959880i	$-0.409530\pi$
$487$	6027.22	0.560820	0.280410	+	0.959880i	$0.409530\pi$
$488$	0	0
$489$	−4355.22	−0.402760
$490$	0	0
$491$	−6253.16	−0.574748	−0.287374	−	0.957818i	$-0.592782\pi$
$491$	−6253.16	−0.574748	−0.287374	+	0.957818i	$0.592782\pi$
$492$	0	0
$493$	−4596.30	−0.419892
$494$	0	0
$495$	385.335	0.0349889
$496$	0	0
$497$	−10185.3	−0.919266
$498$	0	0
$499$	12372.4	1.10995	0.554973	−	0.831868i	$-0.312729\pi$
$499$	12372.4	1.10995	0.554973	+	0.831868i	$0.312729\pi$
$500$	0	0
$501$	4171.26	0.371972
$502$	0	0
$503$	−15826.3	−1.40290	−0.701450	−	0.712719i	$-0.747463\pi$
$503$	−15826.3	−1.40290	−0.701450	+	0.712719i	$0.747463\pi$
$504$	0	0
$505$	−2832.00	−0.249549
$506$	0	0
$507$	−8004.56	−0.701173
$508$	0	0
$509$	1293.91	0.112675	0.0563373	−	0.998412i	$-0.482058\pi$
$509$	1293.91	0.112675	0.0563373	+	0.998412i	$0.482058\pi$
$510$	0	0
$511$	−14271.5	−1.23549
$512$	0	0
$513$	9999.63	0.860613
$514$	0	0
$515$	−6105.18	−0.522381
$516$	0	0
$517$	−417.759	−0.0355377
$518$	0	0
$519$	94.6672	0.00800661
$520$	0	0
$521$	−16165.7	−1.35937	−0.679684	−	0.733505i	$-0.737883\pi$
$521$	−16165.7	−1.35937	−0.679684	+	0.733505i	$0.737883\pi$
$522$	0	0
$523$	−19513.2	−1.63146	−0.815729	−	0.578435i	$-0.803664\pi$
$523$	−19513.2	−1.63146	−0.815729	+	0.578435i	$0.803664\pi$
$524$	0	0
$525$	−900.063	−0.0748228
$526$	0	0
$527$	−2881.24	−0.238157
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−10388.5	−0.849004
$532$	0	0
$533$	3054.02	0.248188
$534$	0	0
$535$	−7631.34	−0.616694
$536$	0	0
$537$	−1344.75	−0.108064
$538$	0	0
$539$	−388.572	−0.0310519
$540$	0	0
$541$	−3124.23	−0.248283	−0.124141	−	0.992265i	$-0.539618\pi$
$541$	−3124.23	−0.248283	−0.124141	+	0.992265i	$0.539618\pi$
$542$	0	0
$543$	−1195.28	−0.0944645
$544$	0	0
$545$	17413.3	1.36863
$546$	0	0
$547$	1818.46	0.142142	0.0710709	−	0.997471i	$-0.477358\pi$
$547$	1818.46	0.142142	0.0710709	+	0.997471i	$0.477358\pi$
$548$	0	0
$549$	1401.94	0.108986
$550$	0	0
$551$	16289.3	1.25943
$552$	0	0
$553$	14505.2	1.11541
$554$	0	0
$555$	1684.70	0.128850
$556$	0	0
$557$	20165.9	1.53404	0.767018	−	0.641626i	$-0.221740\pi$
$557$	20165.9	1.53404	0.767018	+	0.641626i	$0.221740\pi$
$558$	0	0
$559$	19522.3	1.47711
$560$	0	0
$561$	−85.2639	−0.00641683
$562$	0	0
$563$	3770.72	0.282268	0.141134	−	0.989991i	$-0.454925\pi$
$563$	3770.72	0.282268	0.141134	+	0.989991i	$0.454925\pi$
$564$	0	0
$565$	17233.3	1.28320
$566$	0	0
$567$	−13096.5	−0.970022
$568$	0	0
$569$	−23371.9	−1.72197	−0.860986	−	0.508629i	$-0.830152\pi$
$569$	−23371.9	−1.72197	−0.860986	+	0.508629i	$0.830152\pi$
$570$	0	0
$571$	−5535.00	−0.405661	−0.202830	−	0.979214i	$-0.565014\pi$
$571$	−5535.00	−0.405661	−0.202830	+	0.979214i	$0.565014\pi$
$572$	0	0
$573$	5901.67	0.430272
$574$	0	0
$575$	−545.696	−0.0395776
$576$	0	0
$577$	8409.03	0.606711	0.303356	−	0.952877i	$-0.401893\pi$
$577$	8409.03	0.606711	0.303356	+	0.952877i	$0.401893\pi$
$578$	0	0
$579$	−4229.33	−0.303567
$580$	0	0
$581$	8422.78	0.601438
$582$	0	0
$583$	−883.660	−0.0627744
$584$	0	0
$585$	−21173.1	−1.49641
$586$	0	0
$587$	−22796.7	−1.60293	−0.801467	−	0.598039i	$-0.795947\pi$
$587$	−22796.7	−1.60293	−0.801467	+	0.598039i	$0.795947\pi$
$588$	0	0
$589$	10211.1	0.714331
$590$	0	0
$591$	8222.20	0.572278
$592$	0	0
$593$	−22303.3	−1.54450	−0.772249	−	0.635320i	$-0.780868\pi$
$593$	−22303.3	−1.54450	−0.772249	+	0.635320i	$0.780868\pi$
$594$	0	0
$595$	8597.28	0.592360
$596$	0	0
$597$	4192.81	0.287437
$598$	0	0
$599$	−2650.30	−0.180782	−0.0903910	−	0.995906i	$-0.528812\pi$
$599$	−2650.30	−0.180782	−0.0903910	+	0.995906i	$0.528812\pi$
$600$	0	0
$601$	16837.0	1.14275	0.571376	−	0.820688i	$-0.306410\pi$
$601$	16837.0	1.14275	0.571376	+	0.820688i	$0.306410\pi$
$602$	0	0
$603$	−12716.3	−0.858784
$604$	0	0
$605$	−13370.1	−0.898465
$606$	0	0
$607$	18425.8	1.23209	0.616046	−	0.787711i	$-0.288734\pi$
$607$	18425.8	1.23209	0.616046	+	0.787711i	$0.288734\pi$
$608$	0	0
$609$	4967.44	0.330527
$610$	0	0
$611$	22954.7	1.51988
$612$	0	0
$613$	−15686.0	−1.03352	−0.516762	−	0.856129i	$-0.672863\pi$
$613$	−15686.0	−1.03352	−0.516762	+	0.856129i	$0.672863\pi$
$614$	0	0
$615$	−559.450	−0.0366816
$616$	0	0
$617$	−19489.5	−1.27167	−0.635833	−	0.771827i	$-0.719343\pi$
$617$	−19489.5	−1.27167	−0.635833	+	0.771827i	$0.719343\pi$
$618$	0	0
$619$	−7827.43	−0.508257	−0.254128	−	0.967171i	$-0.581789\pi$
$619$	−7827.43	−0.508257	−0.254128	+	0.967171i	$0.581789\pi$
$620$	0	0
$621$	1848.81	0.119469
$622$	0	0
$623$	7851.18	0.504897
$624$	0	0
$625$	−12096.3	−0.774166
$626$	0	0
$627$	302.175	0.0192468
$628$	0	0
$629$	3769.96	0.238979
$630$	0	0
$631$	−30686.2	−1.93597	−0.967986	−	0.251003i	$-0.919240\pi$
$631$	−30686.2	−1.93597	−0.967986	+	0.251003i	$0.919240\pi$
$632$	0	0
$633$	−4520.74	−0.283860
$634$	0	0
$635$	−10148.3	−0.634210
$636$	0	0
$637$	21351.0	1.32803
$638$	0	0
$639$	−10282.6	−0.636577
$640$	0	0
$641$	−26453.5	−1.63003	−0.815016	−	0.579439i	$-0.803272\pi$
$641$	−26453.5	−1.63003	−0.815016	+	0.579439i	$0.803272\pi$
$642$	0	0
$643$	19559.5	1.19961	0.599807	−	0.800145i	$-0.295244\pi$
$643$	19559.5	1.19961	0.599807	+	0.800145i	$0.295244\pi$
$644$	0	0
$645$	−3576.20	−0.218314
$646$	0	0
$647$	7463.15	0.453488	0.226744	−	0.973954i	$-0.427192\pi$
$647$	7463.15	0.453488	0.226744	+	0.973954i	$0.427192\pi$
$648$	0	0
$649$	−658.892	−0.0398518
$650$	0	0
$651$	3113.89	0.187470
$652$	0	0
$653$	3667.38	0.219779	0.109890	−	0.993944i	$-0.464950\pi$
$653$	3667.38	0.219779	0.109890	+	0.993944i	$0.464950\pi$
$654$	0	0
$655$	3407.65	0.203279
$656$	0	0
$657$	−14407.7	−0.855555
$658$	0	0
$659$	−11117.2	−0.657155	−0.328578	−	0.944477i	$-0.606569\pi$
$659$	−11117.2	−0.657155	−0.328578	+	0.944477i	$0.606569\pi$
$660$	0	0
$661$	10886.9	0.640622	0.320311	−	0.947312i	$-0.396213\pi$
$661$	10886.9	0.640622	0.320311	+	0.947312i	$0.396213\pi$
$662$	0	0
$663$	4685.03	0.274436
$664$	0	0
$665$	−30468.8	−1.77673
$666$	0	0
$667$	3011.69	0.174832
$668$	0	0
$669$	9832.18	0.568212
$670$	0	0
$671$	88.9186	0.00511574
$672$	0	0
$673$	−5441.92	−0.311695	−0.155848	−	0.987781i	$-0.549811\pi$
$673$	−5441.92	−0.311695	−0.155848	+	0.987781i	$0.549811\pi$
$674$	0	0
$675$	−1907.16	−0.108751
$676$	0	0
$677$	28579.1	1.62243	0.811215	−	0.584749i	$-0.198807\pi$
$677$	28579.1	1.62243	0.811215	+	0.584749i	$0.198807\pi$
$678$	0	0
$679$	26827.1	1.51625
$680$	0	0
$681$	−8502.09	−0.478415
$682$	0	0
$683$	26971.1	1.51101	0.755504	−	0.655144i	$-0.227392\pi$
$683$	26971.1	1.51101	0.755504	+	0.655144i	$0.227392\pi$
$684$	0	0
$685$	3436.22	0.191666
$686$	0	0
$687$	−2393.09	−0.132900
$688$	0	0
$689$	48554.8	2.68475
$690$	0	0
$691$	27966.0	1.53962	0.769810	−	0.638273i	$-0.220351\pi$
$691$	27966.0	1.53962	0.769810	+	0.638273i	$0.220351\pi$
$692$	0	0
$693$	−931.913	−0.0510829
$694$	0	0
$695$	5136.67	0.280353
$696$	0	0
$697$	−1251.91	−0.0680338
$698$	0	0
$699$	4378.36	0.236917
$700$	0	0
$701$	2681.30	0.144467	0.0722336	−	0.997388i	$-0.476987\pi$
$701$	2681.30	0.144467	0.0722336	+	0.997388i	$0.476987\pi$
$702$	0	0
$703$	−13360.7	−0.716799
$704$	0	0
$705$	−4204.97	−0.224636
$706$	0	0
$707$	6849.06	0.364336
$708$	0	0
$709$	31519.3	1.66958	0.834790	−	0.550569i	$-0.185589\pi$
$709$	31519.3	1.66958	0.834790	+	0.550569i	$0.185589\pi$
$710$	0	0
$711$	14643.6	0.772404
$712$	0	0
$713$	1887.91	0.0991624
$714$	0	0
$715$	−1342.91	−0.0702407
$716$	0	0
$717$	−4756.95	−0.247771
$718$	0	0
$719$	5550.14	0.287880	0.143940	−	0.989586i	$-0.454023\pi$
$719$	5550.14	0.287880	0.143940	+	0.989586i	$0.454023\pi$
$720$	0	0
$721$	14765.1	0.762662
$722$	0	0
$723$	−1550.27	−0.0797444
$724$	0	0
$725$	−3106.75	−0.159147
$726$	0	0
$727$	−9562.34	−0.487823	−0.243912	−	0.969797i	$-0.578431\pi$
$727$	−9562.34	−0.487823	−0.243912	+	0.969797i	$0.578431\pi$
$728$	0	0
$729$	−9838.64	−0.499855
$730$	0	0
$731$	−8002.65	−0.404909
$732$	0	0
$733$	15084.1	0.760087	0.380043	−	0.924969i	$-0.375909\pi$
$733$	15084.1	0.760087	0.380043	+	0.924969i	$0.375909\pi$
$734$	0	0
$735$	−3911.19	−0.196281
$736$	0	0
$737$	−806.534	−0.0403108
$738$	0	0
$739$	14657.3	0.729607	0.364803	−	0.931085i	$-0.381136\pi$
$739$	14657.3	0.729607	0.364803	+	0.931085i	$0.381136\pi$
$740$	0	0
$741$	−16603.7	−0.823149
$742$	0	0
$743$	−25211.2	−1.24483	−0.622416	−	0.782687i	$-0.713849\pi$
$743$	−25211.2	−1.24483	−0.622416	+	0.782687i	$0.713849\pi$
$744$	0	0
$745$	−20021.7	−0.984616
$746$	0	0
$747$	8503.19	0.416487
$748$	0	0
$749$	18456.0	0.900358
$750$	0	0
$751$	−16991.7	−0.825614	−0.412807	−	0.910818i	$-0.635452\pi$
$751$	−16991.7	−0.825614	−0.412807	+	0.910818i	$0.635452\pi$
$752$	0	0
$753$	10849.1	0.525053
$754$	0	0
$755$	−6109.65	−0.294507
$756$	0	0
$757$	27649.0	1.32750	0.663751	−	0.747954i	$-0.268964\pi$
$757$	27649.0	1.32750	0.663751	+	0.747954i	$0.268964\pi$
$758$	0	0
$759$	55.8686	0.00267181
$760$	0	0
$761$	18471.8	0.879896	0.439948	−	0.898023i	$-0.354997\pi$
$761$	18471.8	0.879896	0.439948	+	0.898023i	$0.354997\pi$
$762$	0	0
$763$	−42113.2	−1.99817
$764$	0	0
$765$	8679.36	0.410200
$766$	0	0
$767$	36204.4	1.70439
$768$	0	0
$769$	−2326.91	−0.109117	−0.0545583	−	0.998511i	$-0.517375\pi$
$769$	−2326.91	−0.109117	−0.0545583	+	0.998511i	$0.517375\pi$
$770$	0	0
$771$	−824.146	−0.0384966
$772$	0	0
$773$	−40624.7	−1.89026	−0.945129	−	0.326698i	$-0.894064\pi$
$773$	−40624.7	−1.89026	−0.945129	+	0.326698i	$0.894064\pi$
$774$	0	0
$775$	−1947.49	−0.0902659
$776$	0	0
$777$	−4074.37	−0.188118
$778$	0	0
$779$	4436.78	0.204062
$780$	0	0
$781$	−652.177	−0.0298806
$782$	0	0
$783$	10525.6	0.480402
$784$	0	0
$785$	11265.4	0.512202
$786$	0	0
$787$	10522.2	0.476588	0.238294	−	0.971193i	$-0.423412\pi$
$787$	10522.2	0.476588	0.238294	+	0.971193i	$0.423412\pi$
$788$	0	0
$789$	−966.558	−0.0436126
$790$	0	0
$791$	−41677.9	−1.87345
$792$	0	0
$793$	−4885.84	−0.218791
$794$	0	0
$795$	−8894.51	−0.396800
$796$	0	0
$797$	−34463.9	−1.53171	−0.765856	−	0.643013i	$-0.777684\pi$
$797$	−34463.9	−1.53171	−0.765856	+	0.643013i	$0.777684\pi$
$798$	0	0
$799$	−9409.69	−0.416634
$800$	0	0
$801$	7926.14	0.349634
$802$	0	0
$803$	−913.817	−0.0401593
$804$	0	0
$805$	−5633.31	−0.246643
$806$	0	0
$807$	3815.95	0.166453
$808$	0	0
$809$	4092.42	0.177851	0.0889257	−	0.996038i	$-0.471657\pi$
$809$	4092.42	0.177851	0.0889257	+	0.996038i	$0.471657\pi$
$810$	0	0
$811$	5684.59	0.246132	0.123066	−	0.992398i	$-0.460727\pi$
$811$	5684.59	0.246132	0.123066	+	0.992398i	$0.460727\pi$
$812$	0	0
$813$	−1410.60	−0.0608509
$814$	0	0
$815$	28118.7	1.20853
$816$	0	0
$817$	28361.4	1.21449
$818$	0	0
$819$	51206.2	2.18472
$820$	0	0
$821$	−4718.22	−0.200569	−0.100284	−	0.994959i	$-0.531975\pi$
$821$	−4718.22	−0.200569	−0.100284	+	0.994959i	$0.531975\pi$
$822$	0	0
$823$	26909.4	1.13974	0.569868	−	0.821736i	$-0.306994\pi$
$823$	26909.4	1.13974	0.569868	+	0.821736i	$0.306994\pi$
$824$	0	0
$825$	−57.6319	−0.00243210
$826$	0	0
$827$	14401.1	0.605532	0.302766	−	0.953065i	$-0.402090\pi$
$827$	14401.1	0.605532	0.302766	+	0.953065i	$0.402090\pi$
$828$	0	0
$829$	−16061.0	−0.672883	−0.336442	−	0.941704i	$-0.609223\pi$
$829$	−16061.0	−0.672883	−0.336442	+	0.941704i	$0.609223\pi$
$830$	0	0
$831$	−7863.85	−0.328272
$832$	0	0
$833$	−8752.28	−0.364044
$834$	0	0
$835$	−26931.0	−1.11615
$836$	0	0
$837$	6598.08	0.272477
$838$	0	0
$839$	−34894.5	−1.43587	−0.717934	−	0.696112i	$-0.754912\pi$
$839$	−34894.5	−1.43587	−0.717934	+	0.696112i	$0.754912\pi$
$840$	0	0
$841$	−7242.88	−0.296973
$842$	0	0
$843$	−4976.59	−0.203325
$844$	0	0
$845$	51680.0	2.10396
$846$	0	0
$847$	32334.9	1.31173
$848$	0	0
$849$	373.151	0.0150842
$850$	0	0
$851$	−2470.24	−0.0995049
$852$	0	0
$853$	−26275.4	−1.05469	−0.527346	−	0.849651i	$-0.676813\pi$
$853$	−26275.4	−1.05469	−0.527346	+	0.849651i	$0.676813\pi$
$854$	0	0
$855$	−30759.6	−1.23036
$856$	0	0
$857$	25459.0	1.01478	0.507389	−	0.861717i	$-0.330611\pi$
$857$	25459.0	1.01478	0.507389	+	0.861717i	$0.330611\pi$
$858$	0	0
$859$	1582.86	0.0628715	0.0314357	−	0.999506i	$-0.489992\pi$
$859$	1582.86	0.0628715	0.0314357	+	0.999506i	$0.489992\pi$
$860$	0	0
$861$	1353.00	0.0535542
$862$	0	0
$863$	4277.66	0.168729	0.0843645	−	0.996435i	$-0.473114\pi$
$863$	4277.66	0.168729	0.0843645	+	0.996435i	$0.473114\pi$
$864$	0	0
$865$	−611.202	−0.0240249
$866$	0	0
$867$	5737.41	0.224744
$868$	0	0
$869$	928.778	0.0362562
$870$	0	0
$871$	44316.9	1.72402
$872$	0	0
$873$	27083.3	1.04998
$874$	0	0
$875$	36426.9	1.40738
$876$	0	0
$877$	−16453.2	−0.633505	−0.316753	−	0.948508i	$-0.602593\pi$
$877$	−16453.2	−0.633505	−0.316753	+	0.948508i	$0.602593\pi$
$878$	0	0
$879$	−7146.20	−0.274215
$880$	0	0
$881$	−3452.74	−0.132038	−0.0660191	−	0.997818i	$-0.521030\pi$
$881$	−3452.74	−0.132038	−0.0660191	+	0.997818i	$0.521030\pi$
$882$	0	0
$883$	−21075.1	−0.803210	−0.401605	−	0.915813i	$-0.631548\pi$
$883$	−21075.1	−0.803210	−0.401605	+	0.915813i	$0.631548\pi$
$884$	0	0
$885$	−6632.10	−0.251905
$886$	0	0
$887$	−41611.9	−1.57519	−0.787594	−	0.616195i	$-0.788673\pi$
$887$	−41611.9	−1.57519	−0.787594	+	0.616195i	$0.788673\pi$
$888$	0	0
$889$	24543.2	0.925930
$890$	0	0
$891$	−838.583	−0.0315304
$892$	0	0
$893$	33347.9	1.24966
$894$	0	0
$895$	8682.14	0.324259
$896$	0	0
$897$	−3069.83	−0.114268
$898$	0	0
$899$	10748.2	0.398746
$900$	0	0
$901$	−19903.7	−0.735949
$902$	0	0
$903$	8648.85	0.318733
$904$	0	0
$905$	7717.09	0.283453
$906$	0	0
$907$	−3328.73	−0.121862	−0.0609310	−	0.998142i	$-0.519407\pi$
$907$	−3328.73	−0.121862	−0.0609310	+	0.998142i	$0.519407\pi$
$908$	0	0
$909$	6914.45	0.252297
$910$	0	0
$911$	−39768.1	−1.44630	−0.723149	−	0.690692i	$-0.757306\pi$
$911$	−39768.1	−1.44630	−0.723149	+	0.690692i	$0.757306\pi$
$912$	0	0
$913$	539.318	0.0195496
$914$	0	0
$915$	895.013	0.0323368
$916$	0	0
$917$	−8241.24	−0.296783
$918$	0	0
$919$	2917.02	0.104705	0.0523524	−	0.998629i	$-0.483328\pi$
$919$	2917.02	0.104705	0.0523524	+	0.998629i	$0.483328\pi$
$920$	0	0
$921$	754.041	0.0269777
$922$	0	0
$923$	35835.4	1.27794
$924$	0	0
$925$	2548.20	0.0905777
$926$	0	0
$927$	14906.0	0.528132
$928$	0	0
$929$	−1564.80	−0.0552632	−0.0276316	−	0.999618i	$-0.508797\pi$
$929$	−1564.80	−0.0552632	−0.0276316	+	0.999618i	$0.508797\pi$
$930$	0	0
$931$	31018.1	1.09192
$932$	0	0
$933$	1763.50	0.0618802
$934$	0	0
$935$	550.491	0.0192545
$936$	0	0
$937$	21786.7	0.759596	0.379798	−	0.925069i	$-0.375993\pi$
$937$	21786.7	0.759596	0.379798	+	0.925069i	$0.375993\pi$
$938$	0	0
$939$	−13526.9	−0.470110
$940$	0	0
$941$	−34207.7	−1.18506	−0.592530	−	0.805549i	$-0.701871\pi$
$941$	−34207.7	−1.18506	−0.592530	+	0.805549i	$0.701871\pi$
$942$	0	0
$943$	820.307	0.0283275
$944$	0	0
$945$	−19687.9	−0.677723
$946$	0	0
$947$	−17863.6	−0.612976	−0.306488	−	0.951875i	$-0.599154\pi$
$947$	−17863.6	−0.612976	−0.306488	+	0.951875i	$0.599154\pi$
$948$	0	0
$949$	50211.8	1.71754
$950$	0	0
$951$	−9096.14	−0.310160
$952$	0	0
$953$	−2205.70	−0.0749733	−0.0374867	−	0.999297i	$-0.511935\pi$
$953$	−2205.70	−0.0749733	−0.0374867	+	0.999297i	$0.511935\pi$
$954$	0	0
$955$	−38103.1	−1.29109
$956$	0	0
$957$	318.070	0.0107437
$958$	0	0
$959$	−8310.33	−0.279828
$960$	0	0
$961$	−23053.4	−0.773837
$962$	0	0
$963$	18632.2	0.623484
$964$	0	0
$965$	27305.9	0.910891
$966$	0	0
$967$	55599.4	1.84897	0.924487	−	0.381215i	$-0.124494\pi$
$967$	55599.4	1.84897	0.924487	+	0.381215i	$0.124494\pi$
$968$	0	0
$969$	6806.26	0.225643
$970$	0	0
$971$	−46743.5	−1.54487	−0.772435	−	0.635094i	$-0.780961\pi$
$971$	−46743.5	−1.54487	−0.772435	+	0.635094i	$0.780961\pi$
$972$	0	0
$973$	−12422.8	−0.409308
$974$	0	0
$975$	3166.72	0.104017
$976$	0	0
$977$	−35978.0	−1.17814	−0.589068	−	0.808084i	$-0.700505\pi$
$977$	−35978.0	−1.17814	−0.589068	+	0.808084i	$0.700505\pi$
$978$	0	0
$979$	502.718	0.0164116
$980$	0	0
$981$	−42515.3	−1.38370
$982$	0	0
$983$	25329.0	0.821840	0.410920	−	0.911671i	$-0.365208\pi$
$983$	25329.0	0.821840	0.410920	+	0.911671i	$0.365208\pi$
$984$	0	0
$985$	−53085.2	−1.71719
$986$	0	0
$987$	10169.5	0.327962
$988$	0	0
$989$	5243.68	0.168594
$990$	0	0
$991$	7490.70	0.240111	0.120055	−	0.992767i	$-0.461693\pi$
$991$	7490.70	0.240111	0.120055	+	0.992767i	$0.461693\pi$
$992$	0	0
$993$	−7489.35	−0.239343
$994$	0	0
$995$	−27070.1	−0.862493
$996$	0	0
$997$	34489.1	1.09557	0.547784	−	0.836620i	$-0.315472\pi$
$997$	34489.1	1.09557	0.547784	+	0.836620i	$0.315472\pi$
$998$	0	0
$999$	−8633.27	−0.273418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.4.a.l.1.3		4
4.3	odd	2		23.4.a.b.1.1	✓	4
8.3	odd	2		1472.4.a.y.1.3		4
8.5	even	2		1472.4.a.bf.1.2		4
12.11	even	2		207.4.a.e.1.4		4
20.3	even	4		575.4.b.g.24.8		8
20.7	even	4		575.4.b.g.24.1		8
20.19	odd	2		575.4.a.i.1.4		4
28.27	even	2		1127.4.a.c.1.1		4
92.91	even	2		529.4.a.g.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.1	✓	4	4.3	odd	2
207.4.a.e.1.4		4	12.11	even	2
368.4.a.l.1.3		4	1.1	even	1	trivial
529.4.a.g.1.1		4	92.91	even	2
575.4.a.i.1.4		4	20.19	odd	2
575.4.b.g.24.1		8	20.7	even	4
575.4.b.g.24.8		8	20.3	even	4
1127.4.a.c.1.1		4	28.27	even	2
1472.4.a.y.1.3		4	8.3	odd	2
1472.4.a.bf.1.2		4	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:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.55870 q^{3} +10.0635 q^{5} -24.3381 q^{7} -24.5704 q^{9} -1.55839 q^{11} +85.6294 q^{13} -15.6860 q^{15} -35.1015 q^{17} +124.400 q^{19} +37.9359 q^{21} +23.0000 q^{23} -23.7259 q^{25} +80.3831 q^{27} +130.943 q^{29} +82.0830 q^{31} +2.42907 q^{33} -244.926 q^{35} -107.402 q^{37} -133.471 q^{39} +35.6655 q^{41} +227.986 q^{43} -247.265 q^{45} +268.071 q^{47} +249.342 q^{49} +54.7128 q^{51} +567.034 q^{53} -15.6829 q^{55} -193.902 q^{57} +422.803 q^{59} -57.0580 q^{61} +597.997 q^{63} +861.732 q^{65} +517.544 q^{67} -35.8502 q^{69} +418.494 q^{71} +586.385 q^{73} +36.9817 q^{75} +37.9282 q^{77} -595.986 q^{79} +538.108 q^{81} -346.074 q^{83} -353.244 q^{85} -204.102 q^{87} -322.588 q^{89} -2084.05 q^{91} -127.943 q^{93} +1251.90 q^{95} -1102.27 q^{97} +38.2903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 7 q^{3} + 14 q^{5} - 16 q^{7} - 33 q^{9} - 8 q^{11} + 111 q^{13} - 10 q^{15} + 98 q^{17} - 96 q^{19} + 180 q^{21} + 92 q^{23} + 184 q^{25} + 155 q^{27} + 21 q^{29} + 193 q^{31} - 418 q^{33} + 752 q^{35}+ \cdots + 1498 q^{99}+O(q^{100}) \) 4 * q - 7 * q^3 + 14 * q^5 - 16 * q^7 - 33 * q^9 - 8 * q^11 + 111 * q^13 - 10 * q^15 + 98 * q^17 - 96 * q^19 + 180 * q^21 + 92 * q^23 + 184 * q^25 + 155 * q^27 + 21 * q^29 + 193 * q^31 - 418 * q^33 + 752 * q^35 + 170 * q^37 + 291 * q^39 - 125 * q^41 - 2 * q^43 + 168 * q^45 + 677 * q^47 + 1220 * q^49 + 340 * q^51 - 230 * q^53 + 972 * q^55 + 1322 * q^57 + 1140 * q^59 + 754 * q^61 + 1092 * q^63 + 1318 * q^65 - 488 * q^67 - 161 * q^69 + 401 * q^71 + 1509 * q^73 - 1401 * q^75 + 736 * q^77 + 838 * q^79 - 932 * q^81 - 142 * q^83 + 112 * q^85 - 2223 * q^87 + 2342 * q^89 - 292 * q^91 - 509 * q^93 + 956 * q^95 + 1062 * q^97 + 1498 * q^99

Introduction

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