LMFDB - Embedded newform 400.4.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.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:	$23.6007640023$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.89898 q^{3} -16.6969 q^{7} -18.5959 q^{9} +19.1918 q^{11} +61.7980 q^{13} -30.3837 q^{17} -59.1918 q^{19} -48.4041 q^{21} -205.687 q^{23} -132.182 q^{27} +8.38367 q^{29} -331.151 q^{31} +55.6367 q^{33} +266.565 q^{37} +179.151 q^{39} -320.788 q^{41} -83.1214 q^{43} +276.434 q^{47} -64.2122 q^{49} -88.0816 q^{51} +390.888 q^{53} -171.596 q^{57} -779.110 q^{59} -483.171 q^{61} +310.495 q^{63} -123.707 q^{67} -596.282 q^{69} -187.233 q^{71} -778.706 q^{73} -320.445 q^{77} +446.384 q^{79} +118.898 q^{81} +1054.05 q^{83} +24.3041 q^{87} -94.8490 q^{89} -1031.84 q^{91} -960.000 q^{93} +252.041 q^{97} -356.890 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 4 q^{7} + 2 q^{9} - 40 q^{11} + 104 q^{13} + 96 q^{17} - 40 q^{19} - 136 q^{21} - 284 q^{23} - 88 q^{27} - 140 q^{29} - 192 q^{31} + 464 q^{33} + 200 q^{37} - 112 q^{39} - 524 q^{41} - 372 q^{43}+ \cdots - 1576 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 4 * q^7 + 2 * q^9 - 40 * q^11 + 104 * q^13 + 96 * q^17 - 40 * q^19 - 136 * q^21 - 284 * q^23 - 88 * q^27 - 140 * q^29 - 192 * q^31 + 464 * q^33 + 200 * q^37 - 112 * q^39 - 524 * q^41 - 372 * q^43 - 84 * q^47 - 246 * q^49 - 960 * q^51 - 296 * q^53 - 304 * q^57 - 696 * q^59 - 692 * q^61 + 572 * q^63 - 316 * q^67 - 56 * q^69 - 688 * q^71 - 656 * q^73 - 1072 * q^77 + 736 * q^79 - 742 * q^81 + 1628 * q^83 + 1048 * q^87 - 660 * q^89 - 496 * q^91 - 1920 * q^93 + 896 * q^97 - 1576 * 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
$3$	2.89898	0.557909	0.278954	−	0.960304i	$-0.410012\pi$
$3$	2.89898	0.557909	0.278954	+	0.960304i	$0.410012\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−16.6969	−0.901550	−0.450775	−	0.892638i	$-0.648852\pi$
$7$	−16.6969	−0.901550	−0.450775	+	0.892638i	$0.648852\pi$
$8$	0	0
$9$	−18.5959	−0.688738
$10$	0	0
$11$	19.1918	0.526051	0.263025	−	0.964789i	$-0.415280\pi$
$11$	19.1918	0.526051	0.263025	+	0.964789i	$0.415280\pi$
$12$	0	0
$13$	61.7980	1.31844	0.659218	−	0.751952i	$-0.270887\pi$
$13$	61.7980	1.31844	0.659218	+	0.751952i	$0.270887\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−30.3837	−0.433478	−0.216739	−	0.976230i	$-0.569542\pi$
$17$	−30.3837	−0.433478	−0.216739	+	0.976230i	$0.569542\pi$
$18$	0	0
$19$	−59.1918	−0.714713	−0.357356	−	0.933968i	$-0.616322\pi$
$19$	−59.1918	−0.714713	−0.357356	+	0.933968i	$0.616322\pi$
$20$	0	0
$21$	−48.4041	−0.502983
$22$	0	0
$23$	−205.687	−1.86472	−0.932362	−	0.361526i	$-0.882256\pi$
$23$	−205.687	−1.86472	−0.932362	+	0.361526i	$0.882256\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−132.182	−0.942162
$28$	0	0
$29$	8.38367	0.0536831	0.0268415	−	0.999640i	$-0.491455\pi$
$29$	8.38367	0.0536831	0.0268415	+	0.999640i	$0.491455\pi$
$30$	0	0
$31$	−331.151	−1.91860	−0.959298	−	0.282396i	$-0.908871\pi$
$31$	−331.151	−1.91860	−0.959298	+	0.282396i	$0.908871\pi$
$32$	0	0
$33$	55.6367	0.293488
$34$	0	0
$35$	0	0
$36$	0	0
$37$	266.565	1.18441	0.592204	−	0.805788i	$-0.298258\pi$
$37$	266.565	1.18441	0.592204	+	0.805788i	$0.298258\pi$
$38$	0	0
$39$	179.151	0.735567
$40$	0	0
$41$	−320.788	−1.22192	−0.610959	−	0.791662i	$-0.709216\pi$
$41$	−320.788	−1.22192	−0.610959	+	0.791662i	$0.709216\pi$
$42$	0	0
$43$	−83.1214	−0.294788	−0.147394	−	0.989078i	$-0.547089\pi$
$43$	−83.1214	−0.294788	−0.147394	+	0.989078i	$0.547089\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	276.434	0.857915	0.428957	−	0.903325i	$-0.358881\pi$
$47$	276.434	0.857915	0.428957	+	0.903325i	$0.358881\pi$
$48$	0	0
$49$	−64.2122	−0.187208
$50$	0	0
$51$	−88.0816	−0.241841
$52$	0	0
$53$	390.888	1.01307	0.506534	−	0.862220i	$-0.330927\pi$
$53$	390.888	1.01307	0.506534	+	0.862220i	$0.330927\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−171.596	−0.398744
$58$	0	0
$59$	−779.110	−1.71918	−0.859589	−	0.510986i	$-0.829280\pi$
$59$	−779.110	−1.71918	−0.859589	+	0.510986i	$0.829280\pi$
$60$	0	0
$61$	−483.171	−1.01416	−0.507080	−	0.861899i	$-0.669275\pi$
$61$	−483.171	−1.01416	−0.507080	+	0.861899i	$0.669275\pi$
$62$	0	0
$63$	310.495	0.620931
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−123.707	−0.225571	−0.112785	−	0.993619i	$-0.535977\pi$
$67$	−123.707	−0.225571	−0.112785	+	0.993619i	$0.535977\pi$
$68$	0	0
$69$	−596.282	−1.04035
$70$	0	0
$71$	−187.233	−0.312964	−0.156482	−	0.987681i	$-0.550015\pi$
$71$	−187.233	−0.312964	−0.156482	+	0.987681i	$0.550015\pi$
$72$	0	0
$73$	−778.706	−1.24850	−0.624251	−	0.781224i	$-0.714596\pi$
$73$	−778.706	−1.24850	−0.624251	+	0.781224i	$0.714596\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−320.445	−0.474261
$78$	0	0
$79$	446.384	0.635723	0.317861	−	0.948137i	$-0.397035\pi$
$79$	446.384	0.635723	0.317861	+	0.948137i	$0.397035\pi$
$80$	0	0
$81$	118.898	0.163097
$82$	0	0
$83$	1054.05	1.39394	0.696970	−	0.717100i	$-0.254531\pi$
$83$	1054.05	1.39394	0.696970	+	0.717100i	$0.254531\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	24.3041	0.0299503
$88$	0	0
$89$	−94.8490	−0.112966	−0.0564830	−	0.998404i	$-0.517989\pi$
$89$	−94.8490	−0.112966	−0.0564830	+	0.998404i	$0.517989\pi$
$90$	0	0
$91$	−1031.84	−1.18864
$92$	0	0
$93$	−960.000	−1.07040
$94$	0	0
$95$	0	0
$96$	0	0
$97$	252.041	0.263823	0.131912	−	0.991261i	$-0.457888\pi$
$97$	252.041	0.263823	0.131912	+	0.991261i	$0.457888\pi$
$98$	0	0
$99$	−356.890	−0.362311
$100$	0	0
$101$	37.9184	0.0373566	0.0186783	−	0.999826i	$-0.494054\pi$
$101$	37.9184	0.0373566	0.0186783	+	0.999826i	$0.494054\pi$
$102$	0	0
$103$	−94.4133	−0.0903186	−0.0451593	−	0.998980i	$-0.514380\pi$
$103$	−94.4133	−0.0903186	−0.0451593	+	0.998980i	$0.514380\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	901.464	0.814466	0.407233	−	0.913324i	$-0.366494\pi$
$107$	901.464	0.814466	0.407233	+	0.913324i	$0.366494\pi$
$108$	0	0
$109$	1415.69	1.24403	0.622013	−	0.783007i	$-0.286315\pi$
$109$	1415.69	1.24403	0.622013	+	0.783007i	$0.286315\pi$
$110$	0	0
$111$	772.767	0.660791
$112$	0	0
$113$	−293.576	−0.244401	−0.122200	−	0.992505i	$-0.538995\pi$
$113$	−293.576	−0.244401	−0.122200	+	0.992505i	$0.538995\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1149.19	−0.908057
$118$	0	0
$119$	507.314	0.390802
$120$	0	0
$121$	−962.673	−0.723271
$122$	0	0
$123$	−929.957	−0.681719
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−774.717	−0.541300	−0.270650	−	0.962678i	$-0.587239\pi$
$127$	−774.717	−0.541300	−0.270650	+	0.962678i	$0.587239\pi$
$128$	0	0
$129$	−240.967	−0.164465
$130$	0	0
$131$	−334.343	−0.222990	−0.111495	−	0.993765i	$-0.535564\pi$
$131$	−334.343	−0.222990	−0.111495	+	0.993765i	$0.535564\pi$
$132$	0	0
$133$	988.322	0.644349
$134$	0	0
$135$	0	0
$136$	0	0
$137$	323.514	0.201750	0.100875	−	0.994899i	$-0.467836\pi$
$137$	323.514	0.201750	0.100875	+	0.994899i	$0.467836\pi$
$138$	0	0
$139$	396.482	0.241936	0.120968	−	0.992656i	$-0.461400\pi$
$139$	396.482	0.241936	0.120968	+	0.992656i	$0.461400\pi$
$140$	0	0
$141$	801.376	0.478638
$142$	0	0
$143$	1186.02	0.693564
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−186.150	−0.104445
$148$	0	0
$149$	−1682.89	−0.925284	−0.462642	−	0.886545i	$-0.653099\pi$
$149$	−1682.89	−0.925284	−0.462642	+	0.886545i	$0.653099\pi$
$150$	0	0
$151$	2924.52	1.57612	0.788060	−	0.615598i	$-0.211085\pi$
$151$	2924.52	1.57612	0.788060	+	0.615598i	$0.211085\pi$
$152$	0	0
$153$	565.012	0.298553
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2768.42	1.40729	0.703644	−	0.710553i	$-0.251555\pi$
$157$	2768.42	1.40729	0.703644	+	0.710553i	$0.251555\pi$
$158$	0	0
$159$	1133.18	0.565199
$160$	0	0
$161$	3434.34	1.68114
$162$	0	0
$163$	−2816.33	−1.35333	−0.676663	−	0.736292i	$-0.736575\pi$
$163$	−2816.33	−1.35333	−0.676663	+	0.736292i	$0.736575\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1836.41	−0.850933	−0.425466	−	0.904974i	$-0.639890\pi$
$167$	−1836.41	−0.850933	−0.425466	+	0.904974i	$0.639890\pi$
$168$	0	0
$169$	1621.99	0.738274
$170$	0	0
$171$	1100.73	0.492249
$172$	0	0
$173$	1224.22	0.538011	0.269006	−	0.963139i	$-0.413305\pi$
$173$	1224.22	0.538011	0.269006	+	0.963139i	$0.413305\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2258.62	−0.959145
$178$	0	0
$179$	−2729.58	−1.13977	−0.569883	−	0.821726i	$-0.693011\pi$
$179$	−2729.58	−1.13977	−0.569883	+	0.821726i	$0.693011\pi$
$180$	0	0
$181$	2642.36	1.08511	0.542555	−	0.840020i	$-0.317457\pi$
$181$	2642.36	1.08511	0.542555	+	0.840020i	$0.317457\pi$
$182$	0	0
$183$	−1400.70	−0.565809
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−583.118	−0.228031
$188$	0	0
$189$	2207.03	0.849406
$190$	0	0
$191$	2339.23	0.886183	0.443091	−	0.896476i	$-0.353882\pi$
$191$	2339.23	0.886183	0.443091	+	0.896476i	$0.353882\pi$
$192$	0	0
$193$	4601.61	1.71622	0.858111	−	0.513464i	$-0.171638\pi$
$193$	4601.61	1.71622	0.858111	+	0.513464i	$0.171638\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−823.941	−0.297987	−0.148993	−	0.988838i	$-0.547603\pi$
$197$	−823.941	−0.297987	−0.148993	+	0.988838i	$0.547603\pi$
$198$	0	0
$199$	3329.70	1.18611	0.593055	−	0.805162i	$-0.297922\pi$
$199$	3329.70	1.18611	0.593055	+	0.805162i	$0.297922\pi$
$200$	0	0
$201$	−358.624	−0.125848
$202$	0	0
$203$	−139.982	−0.0483980
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3824.93	1.28431
$208$	0	0
$209$	−1136.00	−0.375975
$210$	0	0
$211$	−1018.78	−0.332398	−0.166199	−	0.986092i	$-0.553149\pi$
$211$	−1018.78	−0.332398	−0.166199	+	0.986092i	$0.553149\pi$
$212$	0	0
$213$	−542.784	−0.174605
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5529.21	1.72971
$218$	0	0
$219$	−2257.45	−0.696550
$220$	0	0
$221$	−1877.65	−0.571513
$222$	0	0
$223$	99.1581	0.0297763	0.0148882	−	0.999889i	$-0.495261\pi$
$223$	99.1581	0.0297763	0.0148882	+	0.999889i	$0.495261\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1197.59	0.350163	0.175081	−	0.984554i	$-0.443981\pi$
$227$	1197.59	0.350163	0.175081	+	0.984554i	$0.443981\pi$
$228$	0	0
$229$	−453.592	−0.130892	−0.0654458	−	0.997856i	$-0.520847\pi$
$229$	−453.592	−0.130892	−0.0654458	+	0.997856i	$0.520847\pi$
$230$	0	0
$231$	−928.963	−0.264594
$232$	0	0
$233$	−3788.49	−1.06520	−0.532601	−	0.846367i	$-0.678785\pi$
$233$	−3788.49	−1.06520	−0.532601	+	0.846367i	$0.678785\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1294.06	0.354675
$238$	0	0
$239$	6000.47	1.62401	0.812005	−	0.583651i	$-0.198376\pi$
$239$	6000.47	1.62401	0.812005	+	0.583651i	$0.198376\pi$
$240$	0	0
$241$	1842.53	0.492480	0.246240	−	0.969209i	$-0.420805\pi$
$241$	1842.53	0.492480	0.246240	+	0.969209i	$0.420805\pi$
$242$	0	0
$243$	3913.59	1.03316
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3657.93	−0.942303
$248$	0	0
$249$	3055.67	0.777691
$250$	0	0
$251$	−1149.46	−0.289057	−0.144529	−	0.989501i	$-0.546167\pi$
$251$	−1149.46	−0.289057	−0.144529	+	0.989501i	$0.546167\pi$
$252$	0	0
$253$	−3947.51	−0.980939
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5407.67	−1.31253	−0.656267	−	0.754528i	$-0.727866\pi$
$257$	−5407.67	−1.31253	−0.656267	+	0.754528i	$0.727866\pi$
$258$	0	0
$259$	−4450.82	−1.06780
$260$	0	0
$261$	−155.902	−0.0369735
$262$	0	0
$263$	−2067.34	−0.484705	−0.242352	−	0.970188i	$-0.577919\pi$
$263$	−2067.34	−0.484705	−0.242352	+	0.970188i	$0.577919\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−274.965	−0.0630247
$268$	0	0
$269$	−592.388	−0.134270	−0.0671348	−	0.997744i	$-0.521386\pi$
$269$	−592.388	−0.134270	−0.0671348	+	0.997744i	$0.521386\pi$
$270$	0	0
$271$	2583.27	0.579049	0.289524	−	0.957171i	$-0.406503\pi$
$271$	2583.27	0.579049	0.289524	+	0.957171i	$0.406503\pi$
$272$	0	0
$273$	−2991.27	−0.663151
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4488.09	−0.973513	−0.486756	−	0.873538i	$-0.661820\pi$
$277$	−4488.09	−0.973513	−0.486756	+	0.873538i	$0.661820\pi$
$278$	0	0
$279$	6158.06	1.32141
$280$	0	0
$281$	−6280.54	−1.33333	−0.666665	−	0.745357i	$-0.732279\pi$
$281$	−6280.54	−1.33333	−0.666665	+	0.745357i	$0.732279\pi$
$282$	0	0
$283$	5233.40	1.09927	0.549635	−	0.835405i	$-0.314767\pi$
$283$	5233.40	1.09927	0.549635	+	0.835405i	$0.314767\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5356.17	1.10162
$288$	0	0
$289$	−3989.83	−0.812097
$290$	0	0
$291$	730.661	0.147189
$292$	0	0
$293$	−2438.21	−0.486149	−0.243074	−	0.970008i	$-0.578156\pi$
$293$	−2438.21	−0.486149	−0.243074	+	0.970008i	$0.578156\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2536.81	−0.495625
$298$	0	0
$299$	−12711.0	−2.45852
$300$	0	0
$301$	1387.87	0.265766
$302$	0	0
$303$	109.925	0.0208416
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7910.44	−1.47060	−0.735298	−	0.677744i	$-0.762958\pi$
$307$	−7910.44	−1.47060	−0.735298	+	0.677744i	$0.762958\pi$
$308$	0	0
$309$	−273.702	−0.0503895
$310$	0	0
$311$	−5419.71	−0.988178	−0.494089	−	0.869411i	$-0.664498\pi$
$311$	−5419.71	−0.988178	−0.494089	+	0.869411i	$0.664498\pi$
$312$	0	0
$313$	5570.86	1.00602	0.503009	−	0.864281i	$-0.332226\pi$
$313$	5570.86	1.00602	0.503009	+	0.864281i	$0.332226\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3724.09	−0.659829	−0.329915	−	0.944011i	$-0.607020\pi$
$317$	−3724.09	−0.659829	−0.329915	+	0.944011i	$0.607020\pi$
$318$	0	0
$319$	160.898	0.0282400
$320$	0	0
$321$	2613.33	0.454398
$322$	0	0
$323$	1798.47	0.309812
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4104.07	0.694053
$328$	0	0
$329$	−4615.60	−0.773453
$330$	0	0
$331$	−3223.96	−0.535362	−0.267681	−	0.963508i	$-0.586257\pi$
$331$	−3223.96	−0.535362	−0.267681	+	0.963508i	$0.586257\pi$
$332$	0	0
$333$	−4957.03	−0.815746
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9524.43	1.53955	0.769776	−	0.638314i	$-0.220368\pi$
$337$	9524.43	1.53955	0.769776	+	0.638314i	$0.220368\pi$
$338$	0	0
$339$	−851.069	−0.136353
$340$	0	0
$341$	−6355.40	−1.00928
$342$	0	0
$343$	6799.20	1.07033
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6042.30	0.934777	0.467388	−	0.884052i	$-0.345195\pi$
$347$	6042.30	0.934777	0.467388	+	0.884052i	$0.345195\pi$
$348$	0	0
$349$	−1626.20	−0.249422	−0.124711	−	0.992193i	$-0.539800\pi$
$349$	−1626.20	−0.249422	−0.124711	+	0.992193i	$0.539800\pi$
$350$	0	0
$351$	−8168.55	−1.24218
$352$	0	0
$353$	886.955	0.133733	0.0668667	−	0.997762i	$-0.478700\pi$
$353$	886.955	0.133733	0.0668667	+	0.997762i	$0.478700\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1470.69	0.218032
$358$	0	0
$359$	−1722.27	−0.253198	−0.126599	−	0.991954i	$-0.540406\pi$
$359$	−1722.27	−0.253198	−0.126599	+	0.991954i	$0.540406\pi$
$360$	0	0
$361$	−3355.33	−0.489186
$362$	0	0
$363$	−2790.77	−0.403519
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−9271.77	−1.31875	−0.659377	−	0.751813i	$-0.729180\pi$
$367$	−9271.77	−1.31875	−0.659377	+	0.751813i	$0.729180\pi$
$368$	0	0
$369$	5965.34	0.841581
$370$	0	0
$371$	−6526.63	−0.913331
$372$	0	0
$373$	5697.15	0.790850	0.395425	−	0.918498i	$-0.370597\pi$
$373$	5697.15	0.790850	0.395425	+	0.918498i	$0.370597\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	518.094	0.0707777
$378$	0	0
$379$	8526.24	1.15558	0.577789	−	0.816186i	$-0.303916\pi$
$379$	8526.24	1.15558	0.577789	+	0.816186i	$0.303916\pi$
$380$	0	0
$381$	−2245.89	−0.301996
$382$	0	0
$383$	−4069.23	−0.542893	−0.271447	−	0.962453i	$-0.587502\pi$
$383$	−4069.23	−0.542893	−0.271447	+	0.962453i	$0.587502\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1545.72	0.203032
$388$	0	0
$389$	−2394.17	−0.312054	−0.156027	−	0.987753i	$-0.549869\pi$
$389$	−2394.17	−0.312054	−0.156027	+	0.987753i	$0.549869\pi$
$390$	0	0
$391$	6249.52	0.808316
$392$	0	0
$393$	−969.253	−0.124408
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3497.79	0.442190	0.221095	−	0.975252i	$-0.429037\pi$
$397$	3497.79	0.442190	0.221095	+	0.975252i	$0.429037\pi$
$398$	0	0
$399$	2865.13	0.359488
$400$	0	0
$401$	−8608.89	−1.07209	−0.536044	−	0.844190i	$-0.680082\pi$
$401$	−8608.89	−1.07209	−0.536044	+	0.844190i	$0.680082\pi$
$402$	0	0
$403$	−20464.5	−2.52955
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5115.88	0.623058
$408$	0	0
$409$	1385.67	0.167523	0.0837615	−	0.996486i	$-0.473307\pi$
$409$	1385.67	0.167523	0.0837615	+	0.996486i	$0.473307\pi$
$410$	0	0
$411$	937.861	0.112558
$412$	0	0
$413$	13008.8	1.54992
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1149.39	0.134978
$418$	0	0
$419$	−3738.23	−0.435858	−0.217929	−	0.975965i	$-0.569930\pi$
$419$	−3738.23	−0.435858	−0.217929	+	0.975965i	$0.569930\pi$
$420$	0	0
$421$	−8993.95	−1.04118	−0.520592	−	0.853806i	$-0.674289\pi$
$421$	−8993.95	−1.04118	−0.520592	+	0.853806i	$0.674289\pi$
$422$	0	0
$423$	−5140.54	−0.590878
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8067.48	0.914316
$428$	0	0
$429$	3438.24	0.386946
$430$	0	0
$431$	−462.547	−0.0516940	−0.0258470	−	0.999666i	$-0.508228\pi$
$431$	−462.547	−0.0516940	−0.0258470	+	0.999666i	$0.508228\pi$
$432$	0	0
$433$	−5231.82	−0.580659	−0.290329	−	0.956927i	$-0.593765\pi$
$433$	−5231.82	−0.580659	−0.290329	+	0.956927i	$0.593765\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12175.0	1.33274
$438$	0	0
$439$	−8995.77	−0.978006	−0.489003	−	0.872282i	$-0.662639\pi$
$439$	−8995.77	−0.978006	−0.489003	+	0.872282i	$0.662639\pi$
$440$	0	0
$441$	1194.09	0.128937
$442$	0	0
$443$	5549.52	0.595182	0.297591	−	0.954694i	$-0.403817\pi$
$443$	5549.52	0.595182	0.297591	+	0.954694i	$0.403817\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4878.65	−0.516224
$448$	0	0
$449$	−5951.29	−0.625521	−0.312760	−	0.949832i	$-0.601254\pi$
$449$	−5951.29	−0.625521	−0.312760	+	0.949832i	$0.601254\pi$
$450$	0	0
$451$	−6156.51	−0.642791
$452$	0	0
$453$	8478.13	0.879332
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13912.7	−1.42409	−0.712047	−	0.702132i	$-0.752232\pi$
$457$	−13912.7	−1.42409	−0.712047	+	0.702132i	$0.752232\pi$
$458$	0	0
$459$	4016.16	0.408406
$460$	0	0
$461$	17467.6	1.76475	0.882374	−	0.470549i	$-0.155944\pi$
$461$	17467.6	1.76475	0.882374	+	0.470549i	$0.155944\pi$
$462$	0	0
$463$	−1575.36	−0.158127	−0.0790637	−	0.996870i	$-0.525193\pi$
$463$	−1575.36	−0.158127	−0.0790637	+	0.996870i	$0.525193\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15618.3	−1.54760	−0.773800	−	0.633430i	$-0.781646\pi$
$467$	−15618.3	−1.54760	−0.773800	+	0.633430i	$0.781646\pi$
$468$	0	0
$469$	2065.53	0.203363
$470$	0	0
$471$	8025.60	0.785138
$472$	0	0
$473$	−1595.25	−0.155074
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7268.92	−0.697738
$478$	0	0
$479$	−9527.90	−0.908854	−0.454427	−	0.890784i	$-0.650156\pi$
$479$	−9527.90	−0.908854	−0.454427	+	0.890784i	$0.650156\pi$
$480$	0	0
$481$	16473.2	1.56157
$482$	0	0
$483$	9956.08	0.937924
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15729.6	−1.46361	−0.731805	−	0.681514i	$-0.761322\pi$
$487$	−15729.6	−1.46361	−0.731805	+	0.681514i	$0.761322\pi$
$488$	0	0
$489$	−8164.49	−0.755033
$490$	0	0
$491$	−2566.49	−0.235894	−0.117947	−	0.993020i	$-0.537631\pi$
$491$	−2566.49	−0.235894	−0.117947	+	0.993020i	$0.537631\pi$
$492$	0	0
$493$	−254.727	−0.0232704
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3126.21	0.282152
$498$	0	0
$499$	13560.4	1.21652	0.608261	−	0.793737i	$-0.291867\pi$
$499$	13560.4	1.21652	0.608261	+	0.793737i	$0.291867\pi$
$500$	0	0
$501$	−5323.72	−0.474743
$502$	0	0
$503$	5222.51	0.462943	0.231471	−	0.972842i	$-0.425646\pi$
$503$	5222.51	0.462943	0.231471	+	0.972842i	$0.425646\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4702.11	0.411890
$508$	0	0
$509$	−11875.9	−1.03416	−0.517082	−	0.855936i	$-0.672982\pi$
$509$	−11875.9	−1.03416	−0.517082	+	0.855936i	$0.672982\pi$
$510$	0	0
$511$	13002.0	1.12559
$512$	0	0
$513$	7824.07	0.673375
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5305.27	0.451307
$518$	0	0
$519$	3549.00	0.300161
$520$	0	0
$521$	2456.04	0.206528	0.103264	−	0.994654i	$-0.467071\pi$
$521$	2456.04	0.206528	0.103264	+	0.994654i	$0.467071\pi$
$522$	0	0
$523$	−634.460	−0.0530459	−0.0265229	−	0.999648i	$-0.508444\pi$
$523$	−634.460	−0.0530459	−0.0265229	+	0.999648i	$0.508444\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10061.6	0.831669
$528$	0	0
$529$	30140.0	2.47720
$530$	0	0
$531$	14488.3	1.18406
$532$	0	0
$533$	−19824.0	−1.61102
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7912.98	−0.635885
$538$	0	0
$539$	−1232.35	−0.0984807
$540$	0	0
$541$	18078.4	1.43669	0.718347	−	0.695685i	$-0.244899\pi$
$541$	18078.4	1.43669	0.718347	+	0.695685i	$0.244899\pi$
$542$	0	0
$543$	7660.14	0.605393
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24815.3	1.93972	0.969860	−	0.243661i	$-0.0783486\pi$
$547$	24815.3	1.93972	0.969860	+	0.243661i	$0.0783486\pi$
$548$	0	0
$549$	8985.02	0.698490
$550$	0	0
$551$	−496.245	−0.0383680
$552$	0	0
$553$	−7453.24	−0.573136
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10073.7	−0.766310	−0.383155	−	0.923684i	$-0.625162\pi$
$557$	−10073.7	−0.766310	−0.383155	+	0.923684i	$0.625162\pi$
$558$	0	0
$559$	−5136.73	−0.388660
$560$	0	0
$561$	−1690.45	−0.127221
$562$	0	0
$563$	−7505.06	−0.561813	−0.280906	−	0.959735i	$-0.590635\pi$
$563$	−7505.06	−0.561813	−0.280906	+	0.959735i	$0.590635\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1985.23	−0.147040
$568$	0	0
$569$	−15251.4	−1.12368	−0.561838	−	0.827247i	$-0.689906\pi$
$569$	−15251.4	−1.12368	−0.561838	+	0.827247i	$0.689906\pi$
$570$	0	0
$571$	2683.78	0.196695	0.0983474	−	0.995152i	$-0.468644\pi$
$571$	2683.78	0.196695	0.0983474	+	0.995152i	$0.468644\pi$
$572$	0	0
$573$	6781.39	0.494409
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7369.88	−0.531737	−0.265868	−	0.964009i	$-0.585659\pi$
$577$	−7369.88	−0.531737	−0.265868	+	0.964009i	$0.585659\pi$
$578$	0	0
$579$	13340.0	0.957496
$580$	0	0
$581$	−17599.4	−1.25671
$582$	0	0
$583$	7501.85	0.532925
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20865.4	−1.46713	−0.733567	−	0.679618i	$-0.762146\pi$
$587$	−20865.4	−1.46713	−0.733567	+	0.679618i	$0.762146\pi$
$588$	0	0
$589$	19601.4	1.37124
$590$	0	0
$591$	−2388.59	−0.166249
$592$	0	0
$593$	−25894.0	−1.79316	−0.896578	−	0.442886i	$-0.853954\pi$
$593$	−25894.0	−1.79316	−0.896578	+	0.442886i	$0.853954\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9652.73	0.661742
$598$	0	0
$599$	−5632.42	−0.384198	−0.192099	−	0.981376i	$-0.561529\pi$
$599$	−5632.42	−0.384198	−0.192099	+	0.981376i	$0.561529\pi$
$600$	0	0
$601$	−13079.7	−0.887742	−0.443871	−	0.896091i	$-0.646395\pi$
$601$	−13079.7	−0.887742	−0.443871	+	0.896091i	$0.646395\pi$
$602$	0	0
$603$	2300.45	0.155359
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18890.2	1.26314	0.631572	−	0.775317i	$-0.282410\pi$
$607$	18890.2	1.26314	0.631572	+	0.775317i	$0.282410\pi$
$608$	0	0
$609$	−405.804	−0.0270017
$610$	0	0
$611$	17083.0	1.13111
$612$	0	0
$613$	−13631.5	−0.898155	−0.449078	−	0.893493i	$-0.648247\pi$
$613$	−13631.5	−0.898155	−0.449078	+	0.893493i	$0.648247\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14982.2	0.977568	0.488784	−	0.872405i	$-0.337441\pi$
$617$	14982.2	0.977568	0.488784	+	0.872405i	$0.337441\pi$
$618$	0	0
$619$	25049.9	1.62656	0.813279	−	0.581873i	$-0.197680\pi$
$619$	25049.9	1.62656	0.813279	+	0.581873i	$0.197680\pi$
$620$	0	0
$621$	27188.0	1.75687
$622$	0	0
$623$	1583.69	0.101844
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3293.24	−0.209760
$628$	0	0
$629$	−8099.23	−0.513414
$630$	0	0
$631$	−18711.0	−1.18046	−0.590232	−	0.807233i	$-0.700964\pi$
$631$	−18711.0	−1.18046	−0.590232	+	0.807233i	$0.700964\pi$
$632$	0	0
$633$	−2953.43	−0.185448
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3968.19	−0.246821
$638$	0	0
$639$	3481.76	0.215550
$640$	0	0
$641$	25792.2	1.58929	0.794643	−	0.607077i	$-0.207658\pi$
$641$	25792.2	1.58929	0.794643	+	0.607077i	$0.207658\pi$
$642$	0	0
$643$	20256.7	1.24237	0.621186	−	0.783663i	$-0.286651\pi$
$643$	20256.7	1.24237	0.621186	+	0.783663i	$0.286651\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6655.43	0.404408	0.202204	−	0.979343i	$-0.435190\pi$
$647$	6655.43	0.404408	0.202204	+	0.979343i	$0.435190\pi$
$648$	0	0
$649$	−14952.6	−0.904375
$650$	0	0
$651$	16029.1	0.965021
$652$	0	0
$653$	8490.91	0.508844	0.254422	−	0.967093i	$-0.418115\pi$
$653$	8490.91	0.508844	0.254422	+	0.967093i	$0.418115\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14480.8	0.859890
$658$	0	0
$659$	−15543.8	−0.918816	−0.459408	−	0.888225i	$-0.651938\pi$
$659$	−15543.8	−0.918816	−0.459408	+	0.888225i	$0.651938\pi$
$660$	0	0
$661$	−13519.8	−0.795553	−0.397777	−	0.917482i	$-0.630218\pi$
$661$	−13519.8	−0.795553	−0.397777	+	0.917482i	$0.630218\pi$
$662$	0	0
$663$	−5443.27	−0.318852
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1724.41	−0.100104
$668$	0	0
$669$	287.457	0.0166125
$670$	0	0
$671$	−9272.95	−0.533499
$672$	0	0
$673$	11565.3	0.662421	0.331211	−	0.943557i	$-0.392543\pi$
$673$	11565.3	0.662421	0.331211	+	0.943557i	$0.392543\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28227.5	−1.60247	−0.801235	−	0.598350i	$-0.795823\pi$
$677$	−28227.5	−1.60247	−0.801235	+	0.598350i	$0.795823\pi$
$678$	0	0
$679$	−4208.31	−0.237850
$680$	0	0
$681$	3471.79	0.195359
$682$	0	0
$683$	6425.99	0.360005	0.180003	−	0.983666i	$-0.442389\pi$
$683$	6425.99	0.360005	0.180003	+	0.983666i	$0.442389\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1314.95	−0.0730256
$688$	0	0
$689$	24156.1	1.33566
$690$	0	0
$691$	19066.9	1.04969	0.524846	−	0.851197i	$-0.324123\pi$
$691$	19066.9	1.04969	0.524846	+	0.851197i	$0.324123\pi$
$692$	0	0
$693$	5958.97	0.326641
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9746.71	0.529674
$698$	0	0
$699$	−10982.7	−0.594285
$700$	0	0
$701$	4796.00	0.258406	0.129203	−	0.991618i	$-0.458758\pi$
$701$	4796.00	0.258406	0.129203	+	0.991618i	$0.458758\pi$
$702$	0	0
$703$	−15778.5	−0.846511
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−633.121	−0.0336789
$708$	0	0
$709$	−9805.33	−0.519389	−0.259695	−	0.965691i	$-0.583622\pi$
$709$	−9805.33	−0.519389	−0.259695	+	0.965691i	$0.583622\pi$
$710$	0	0
$711$	−8300.91	−0.437846
$712$	0	0
$713$	68113.4	3.57765
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17395.2	0.906049
$718$	0	0
$719$	−27539.7	−1.42845	−0.714227	−	0.699915i	$-0.753221\pi$
$719$	−27539.7	−1.42845	−0.714227	+	0.699915i	$0.753221\pi$
$720$	0	0
$721$	1576.41	0.0814267
$722$	0	0
$723$	5341.45	0.274759
$724$	0	0
$725$	0	0
$726$	0	0
$727$	16543.8	0.843985	0.421993	−	0.906599i	$-0.361331\pi$
$727$	16543.8	0.843985	0.421993	+	0.906599i	$0.361331\pi$
$728$	0	0
$729$	8135.16	0.413309
$730$	0	0
$731$	2525.53	0.127784
$732$	0	0
$733$	−16718.6	−0.842450	−0.421225	−	0.906956i	$-0.638400\pi$
$733$	−16718.6	−0.842450	−0.421225	+	0.906956i	$0.638400\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2374.17	−0.118662
$738$	0	0
$739$	24022.6	1.19579	0.597893	−	0.801576i	$-0.296005\pi$
$739$	24022.6	1.19579	0.597893	+	0.801576i	$0.296005\pi$
$740$	0	0
$741$	−10604.3	−0.525719
$742$	0	0
$743$	−26201.8	−1.29374	−0.646871	−	0.762600i	$-0.723923\pi$
$743$	−26201.8	−1.29374	−0.646871	+	0.762600i	$0.723923\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−19601.0	−0.960059
$748$	0	0
$749$	−15051.7	−0.734282
$750$	0	0
$751$	−24451.7	−1.18809	−0.594045	−	0.804432i	$-0.702470\pi$
$751$	−24451.7	−1.18809	−0.594045	+	0.804432i	$0.702470\pi$
$752$	0	0
$753$	−3332.26	−0.161268
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−21403.8	−1.02765	−0.513827	−	0.857894i	$-0.671773\pi$
$757$	−21403.8	−1.02765	−0.513827	+	0.857894i	$0.671773\pi$
$758$	0	0
$759$	−11443.7	−0.547275
$760$	0	0
$761$	−28935.9	−1.37835	−0.689176	−	0.724594i	$-0.742027\pi$
$761$	−28935.9	−1.37835	−0.689176	+	0.724594i	$0.742027\pi$
$762$	0	0
$763$	−23637.8	−1.12155
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−48147.4	−2.26663
$768$	0	0
$769$	−11479.3	−0.538301	−0.269151	−	0.963098i	$-0.586743\pi$
$769$	−11479.3	−0.538301	−0.269151	+	0.963098i	$0.586743\pi$
$770$	0	0
$771$	−15676.7	−0.732275
$772$	0	0
$773$	−2512.27	−0.116895	−0.0584477	−	0.998290i	$-0.518615\pi$
$773$	−2512.27	−0.116895	−0.0584477	+	0.998290i	$0.518615\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12902.8	−0.595736
$778$	0	0
$779$	18988.0	0.873320
$780$	0	0
$781$	−3593.34	−0.164635
$782$	0	0
$783$	−1108.17	−0.0505781
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2650.18	0.120036	0.0600182	−	0.998197i	$-0.480884\pi$
$787$	2650.18	0.120036	0.0600182	+	0.998197i	$0.480884\pi$
$788$	0	0
$789$	−5993.16	−0.270421
$790$	0	0
$791$	4901.81	0.220339
$792$	0	0
$793$	−29859.0	−1.33711
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24516.7	1.08962	0.544810	−	0.838560i	$-0.316602\pi$
$797$	24516.7	1.08962	0.544810	+	0.838560i	$0.316602\pi$
$798$	0	0
$799$	−8399.07	−0.371887
$800$	0	0
$801$	1763.80	0.0778039
$802$	0	0
$803$	−14944.8	−0.656775
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1717.32	−0.0749102
$808$	0	0
$809$	−29725.5	−1.29183	−0.645917	−	0.763408i	$-0.723525\pi$
$809$	−29725.5	−1.29183	−0.645917	+	0.763408i	$0.723525\pi$
$810$	0	0
$811$	2050.38	0.0887773	0.0443887	−	0.999014i	$-0.485866\pi$
$811$	2050.38	0.0887773	0.0443887	+	0.999014i	$0.485866\pi$
$812$	0	0
$813$	7488.83	0.323056
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4920.11	0.210689
$818$	0	0
$819$	19188.0	0.818658
$820$	0	0
$821$	−33863.4	−1.43952	−0.719758	−	0.694225i	$-0.755747\pi$
$821$	−33863.4	−1.43952	−0.719758	+	0.694225i	$0.755747\pi$
$822$	0	0
$823$	1216.52	0.0515251	0.0257625	−	0.999668i	$-0.491799\pi$
$823$	1216.52	0.0515251	0.0257625	+	0.999668i	$0.491799\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25654.4	1.07871	0.539355	−	0.842079i	$-0.318668\pi$
$827$	25654.4	1.07871	0.539355	+	0.842079i	$0.318668\pi$
$828$	0	0
$829$	24544.6	1.02831	0.514154	−	0.857698i	$-0.328106\pi$
$829$	24544.6	1.02831	0.514154	+	0.857698i	$0.328106\pi$
$830$	0	0
$831$	−13010.9	−0.543131
$832$	0	0
$833$	1951.00	0.0811504
$834$	0	0
$835$	0	0
$836$	0	0
$837$	43772.1	1.80763
$838$	0	0
$839$	−16548.6	−0.680954	−0.340477	−	0.940253i	$-0.610589\pi$
$839$	−16548.6	−0.680954	−0.340477	+	0.940253i	$0.610589\pi$
$840$	0	0
$841$	−24318.7	−0.997118
$842$	0	0
$843$	−18207.2	−0.743877
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16073.7	0.652065
$848$	0	0
$849$	15171.5	0.613292
$850$	0	0
$851$	−54828.9	−2.20859
$852$	0	0
$853$	24363.5	0.977950	0.488975	−	0.872298i	$-0.337371\pi$
$853$	24363.5	0.977950	0.488975	+	0.872298i	$0.337371\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	575.718	0.0229477	0.0114738	−	0.999934i	$-0.496348\pi$
$857$	575.718	0.0229477	0.0114738	+	0.999934i	$0.496348\pi$
$858$	0	0
$859$	−1531.31	−0.0608236	−0.0304118	−	0.999537i	$-0.509682\pi$
$859$	−1531.31	−0.0608236	−0.0304118	+	0.999537i	$0.509682\pi$
$860$	0	0
$861$	15527.4	0.614604
$862$	0	0
$863$	7706.51	0.303978	0.151989	−	0.988382i	$-0.451432\pi$
$863$	7706.51	0.303978	0.151989	+	0.988382i	$0.451432\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11566.4	−0.453076
$868$	0	0
$869$	8566.92	0.334422
$870$	0	0
$871$	−7644.85	−0.297400
$872$	0	0
$873$	−4686.93	−0.181705
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−43618.9	−1.67948	−0.839741	−	0.542988i	$-0.817293\pi$
$877$	−43618.9	−1.67948	−0.839741	+	0.542988i	$0.817293\pi$
$878$	0	0
$879$	−7068.31	−0.271227
$880$	0	0
$881$	−13416.4	−0.513066	−0.256533	−	0.966536i	$-0.582580\pi$
$881$	−13416.4	−0.513066	−0.256533	+	0.966536i	$0.582580\pi$
$882$	0	0
$883$	29538.2	1.12575	0.562875	−	0.826542i	$-0.309695\pi$
$883$	29538.2	1.12575	0.562875	+	0.826542i	$0.309695\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1950.88	−0.0738490	−0.0369245	−	0.999318i	$-0.511756\pi$
$887$	−1950.88	−0.0738490	−0.0369245	+	0.999318i	$0.511756\pi$
$888$	0	0
$889$	12935.4	0.488009
$890$	0	0
$891$	2281.87	0.0857974
$892$	0	0
$893$	−16362.6	−0.613162
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−36849.0	−1.37163
$898$	0	0
$899$	−2776.26	−0.102996
$900$	0	0
$901$	−11876.6	−0.439142
$902$	0	0
$903$	4023.42	0.148273
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3507.55	0.128408	0.0642042	−	0.997937i	$-0.479549\pi$
$907$	3507.55	0.128408	0.0642042	+	0.997937i	$0.479549\pi$
$908$	0	0
$909$	−705.127	−0.0257289
$910$	0	0
$911$	33841.4	1.23075	0.615377	−	0.788233i	$-0.289004\pi$
$911$	33841.4	1.23075	0.615377	+	0.788233i	$0.289004\pi$
$912$	0	0
$913$	20229.2	0.733283
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5582.50	0.201036
$918$	0	0
$919$	−25440.7	−0.913178	−0.456589	−	0.889678i	$-0.650929\pi$
$919$	−25440.7	−0.913178	−0.456589	+	0.889678i	$0.650929\pi$
$920$	0	0
$921$	−22932.2	−0.820458
$922$	0	0
$923$	−11570.6	−0.412623
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1755.70	0.0622058
$928$	0	0
$929$	−26416.7	−0.932941	−0.466471	−	0.884537i	$-0.654475\pi$
$929$	−26416.7	−0.932941	−0.466471	+	0.884537i	$0.654475\pi$
$930$	0	0
$931$	3800.84	0.133800
$932$	0	0
$933$	−15711.6	−0.551313
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1936.70	0.0675231	0.0337616	−	0.999430i	$-0.489251\pi$
$937$	1936.70	0.0675231	0.0337616	+	0.999430i	$0.489251\pi$
$938$	0	0
$939$	16149.8	0.561266
$940$	0	0
$941$	23459.1	0.812694	0.406347	−	0.913719i	$-0.366802\pi$
$941$	23459.1	0.812694	0.406347	+	0.913719i	$0.366802\pi$
$942$	0	0
$943$	65981.8	2.27854
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23606.8	0.810049	0.405025	−	0.914306i	$-0.367263\pi$
$947$	23606.8	0.810049	0.405025	+	0.914306i	$0.367263\pi$
$948$	0	0
$949$	−48122.4	−1.64607
$950$	0	0
$951$	−10796.1	−0.368125
$952$	0	0
$953$	−30164.2	−1.02530	−0.512652	−	0.858596i	$-0.671337\pi$
$953$	−30164.2	−1.02530	−0.512652	+	0.858596i	$0.671337\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	466.440	0.0157553
$958$	0	0
$959$	−5401.70	−0.181887
$960$	0	0
$961$	79870.0	2.68101
$962$	0	0
$963$	−16763.6	−0.560953
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9034.04	0.300429	0.150215	−	0.988653i	$-0.452004\pi$
$967$	9034.04	0.300429	0.150215	+	0.988653i	$0.452004\pi$
$968$	0	0
$969$	5213.71	0.172847
$970$	0	0
$971$	−36159.0	−1.19506	−0.597528	−	0.801848i	$-0.703850\pi$
$971$	−36159.0	−1.19506	−0.597528	+	0.801848i	$0.703850\pi$
$972$	0	0
$973$	−6620.03	−0.218118
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38584.0	1.26347	0.631736	−	0.775184i	$-0.282343\pi$
$977$	38584.0	1.26347	0.631736	+	0.775184i	$0.282343\pi$
$978$	0	0
$979$	−1820.33	−0.0594258
$980$	0	0
$981$	−26326.1	−0.856808
$982$	0	0
$983$	53046.3	1.72117	0.860587	−	0.509303i	$-0.170097\pi$
$983$	53046.3	1.72117	0.860587	+	0.509303i	$0.170097\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13380.5	−0.431516
$988$	0	0
$989$	17097.0	0.549699
$990$	0	0
$991$	16425.7	0.526517	0.263259	−	0.964725i	$-0.415203\pi$
$991$	16425.7	0.526517	0.263259	+	0.964725i	$0.415203\pi$
$992$	0	0
$993$	−9346.19	−0.298683
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−20024.6	−0.636092	−0.318046	−	0.948075i	$-0.603027\pi$
$997$	−20024.6	−0.636092	−0.318046	+	0.948075i	$0.603027\pi$
$998$	0	0
$999$	−35235.0	−1.11590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.a.v.1.2		2
4.3	odd	2		200.4.a.l.1.1		2
5.2	odd	4		80.4.c.c.49.2		4
5.3	odd	4		80.4.c.c.49.3		4
5.4	even	2		400.4.a.x.1.1		2
8.3	odd	2		1600.4.a.ce.1.2		2
8.5	even	2		1600.4.a.cm.1.1		2
12.11	even	2		1800.4.a.bp.1.2		2
15.2	even	4		720.4.f.m.289.1		4
15.8	even	4		720.4.f.m.289.2		4
20.3	even	4		40.4.c.a.9.2	✓	4
20.7	even	4		40.4.c.a.9.3	yes	4
20.19	odd	2		200.4.a.k.1.2		2
40.3	even	4		320.4.c.g.129.3		4
40.13	odd	4		320.4.c.h.129.2		4
40.19	odd	2		1600.4.a.cl.1.1		2
40.27	even	4		320.4.c.g.129.2		4
40.29	even	2		1600.4.a.cf.1.2		2
40.37	odd	4		320.4.c.h.129.3		4
60.23	odd	4		360.4.f.e.289.2		4
60.47	odd	4		360.4.f.e.289.1		4
60.59	even	2		1800.4.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.c.a.9.2	✓	4	20.3	even	4
40.4.c.a.9.3	yes	4	20.7	even	4
80.4.c.c.49.2		4	5.2	odd	4
80.4.c.c.49.3		4	5.3	odd	4
200.4.a.k.1.2		2	20.19	odd	2
200.4.a.l.1.1		2	4.3	odd	2
320.4.c.g.129.2		4	40.27	even	4
320.4.c.g.129.3		4	40.3	even	4
320.4.c.h.129.2		4	40.13	odd	4
320.4.c.h.129.3		4	40.37	odd	4
360.4.f.e.289.1		4	60.47	odd	4
360.4.f.e.289.2		4	60.23	odd	4
400.4.a.v.1.2		2	1.1	even	1	trivial
400.4.a.x.1.1		2	5.4	even	2
720.4.f.m.289.1		4	15.2	even	4
720.4.f.m.289.2		4	15.8	even	4
1600.4.a.ce.1.2		2	8.3	odd	2
1600.4.a.cf.1.2		2	40.29	even	2
1600.4.a.cl.1.1		2	40.19	odd	2
1600.4.a.cm.1.1		2	8.5	even	2
1800.4.a.bk.1.1		2	60.59	even	2
1800.4.a.bp.1.2		2	12.11	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+2.89898 q^{3} -16.6969 q^{7} -18.5959 q^{9} +19.1918 q^{11} +61.7980 q^{13} -30.3837 q^{17} -59.1918 q^{19} -48.4041 q^{21} -205.687 q^{23} -132.182 q^{27} +8.38367 q^{29} -331.151 q^{31} +55.6367 q^{33} +266.565 q^{37} +179.151 q^{39} -320.788 q^{41} -83.1214 q^{43} +276.434 q^{47} -64.2122 q^{49} -88.0816 q^{51} +390.888 q^{53} -171.596 q^{57} -779.110 q^{59} -483.171 q^{61} +310.495 q^{63} -123.707 q^{67} -596.282 q^{69} -187.233 q^{71} -778.706 q^{73} -320.445 q^{77} +446.384 q^{79} +118.898 q^{81} +1054.05 q^{83} +24.3041 q^{87} -94.8490 q^{89} -1031.84 q^{91} -960.000 q^{93} +252.041 q^{97} -356.890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 4 q^{7} + 2 q^{9} - 40 q^{11} + 104 q^{13} + 96 q^{17} - 40 q^{19} - 136 q^{21} - 284 q^{23} - 88 q^{27} - 140 q^{29} - 192 q^{31} + 464 q^{33} + 200 q^{37} - 112 q^{39} - 524 q^{41} - 372 q^{43}+ \cdots - 1576 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 4 * q^7 + 2 * q^9 - 40 * q^11 + 104 * q^13 + 96 * q^17 - 40 * q^19 - 136 * q^21 - 284 * q^23 - 88 * q^27 - 140 * q^29 - 192 * q^31 + 464 * q^33 + 200 * q^37 - 112 * q^39 - 524 * q^41 - 372 * q^43 - 84 * q^47 - 246 * q^49 - 960 * q^51 - 296 * q^53 - 304 * q^57 - 696 * q^59 - 692 * q^61 + 572 * q^63 - 316 * q^67 - 56 * q^69 - 688 * q^71 - 656 * q^73 - 1072 * q^77 + 736 * q^79 - 742 * q^81 + 1628 * q^83 + 1048 * q^87 - 660 * q^89 - 496 * q^91 - 1920 * q^93 + 896 * q^97 - 1576 * q^99

Introduction

L-functions

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