LMFDB - Embedded newform 1600.4.a.cu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.37485.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 12x - 3 $ x^4 - x^3 - 8x^2 + 12x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.03888$ of defining polynomial
Character	$\chi$	$=$	1600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.57295 q^{3} -22.1403 q^{7} +46.4955 q^{9} +O(q^{10})$	$q-8.57295 q^{3} -22.1403 q^{7} +46.4955 q^{9} +27.1347 q^{11} -70.3303 q^{13} +73.3212 q^{17} -110.033 q^{19} +189.808 q^{21} -107.870 q^{23} -167.134 q^{27} -68.6424 q^{29} +137.167 q^{31} -232.624 q^{33} +60.3121 q^{37} +602.938 q^{39} +95.1470 q^{41} -501.479 q^{43} -439.305 q^{47} +147.192 q^{49} -628.579 q^{51} -286.955 q^{53} +943.303 q^{57} -547.175 q^{59} -511.459 q^{61} -1029.42 q^{63} -301.547 q^{67} +924.762 q^{69} -82.8978 q^{71} +763.267 q^{73} -600.770 q^{77} -1011.45 q^{79} +177.450 q^{81} -704.554 q^{83} +588.468 q^{87} -743.212 q^{89} +1557.13 q^{91} -1175.93 q^{93} +1136.00 q^{97} +1261.64 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 76 q^{9}+O(q^{10}) $ 4 * q + 76 * q^9	$ 4 q + 76 q^{9} - 208 q^{13} + 136 q^{21} + 312 q^{29} + 96 q^{33} - 272 q^{37} - 96 q^{41} + 1212 q^{49} - 48 q^{53} + 3040 q^{57} - 1056 q^{61} + 1976 q^{69} + 1440 q^{73} - 4896 q^{77} - 500 q^{81} - 40 q^{89} - 2944 q^{93} + 4544 q^{97}+O(q^{100}) $ 4 * q + 76 * q^9 - 208 * q^13 + 136 * q^21 + 312 * q^29 + 96 * q^33 - 272 * q^37 - 96 * q^41 + 1212 * q^49 - 48 * q^53 + 3040 * q^57 - 1056 * q^61 + 1976 * q^69 + 1440 * q^73 - 4896 * q^77 - 500 * q^81 - 40 * q^89 - 2944 * q^93 + 4544 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−8.57295	−1.64986	−0.824932	−	0.565231i	$-0.808787\pi$
$3$	−8.57295	−1.64986	−0.824932	+	0.565231i	$0.808787\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−22.1403	−1.19546	−0.597732	−	0.801696i	$-0.703931\pi$
$7$	−22.1403	−1.19546	−0.597732	+	0.801696i	$0.703931\pi$
$8$	0	0
$9$	46.4955	1.72205
$10$	0	0
$11$	27.1347	0.743765	0.371882	−	0.928280i	$-0.378712\pi$
$11$	27.1347	0.743765	0.371882	+	0.928280i	$0.378712\pi$
$12$	0	0
$13$	−70.3303	−1.50047	−0.750235	−	0.661171i	$-0.770060\pi$
$13$	−70.3303	−1.50047	−0.750235	+	0.661171i	$0.770060\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	73.3212	1.04606	0.523030	−	0.852315i	$-0.324802\pi$
$17$	73.3212	1.04606	0.523030	+	0.852315i	$0.324802\pi$
$18$	0	0
$19$	−110.033	−1.32859	−0.664294	−	0.747471i	$-0.731268\pi$
$19$	−110.033	−1.32859	−0.664294	+	0.747471i	$0.731268\pi$
$20$	0	0
$21$	189.808	1.97235
$22$	0	0
$23$	−107.870	−0.977931	−0.488965	−	0.872303i	$-0.662626\pi$
$23$	−107.870	−0.977931	−0.488965	+	0.872303i	$0.662626\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−167.134	−1.19129
$28$	0	0
$29$	−68.6424	−0.439537	−0.219769	−	0.975552i	$-0.570530\pi$
$29$	−68.6424	−0.439537	−0.219769	+	0.975552i	$0.570530\pi$
$30$	0	0
$31$	137.167	0.794708	0.397354	−	0.917665i	$-0.369928\pi$
$31$	137.167	0.794708	0.397354	+	0.917665i	$0.369928\pi$
$32$	0	0
$33$	−232.624	−1.22711
$34$	0	0
$35$	0	0
$36$	0	0
$37$	60.3121	0.267980	0.133990	−	0.990983i	$-0.457221\pi$
$37$	60.3121	0.267980	0.133990	+	0.990983i	$0.457221\pi$
$38$	0	0
$39$	602.938	2.47557
$40$	0	0
$41$	95.1470	0.362426	0.181213	−	0.983444i	$-0.441998\pi$
$41$	95.1470	0.362426	0.181213	+	0.983444i	$0.441998\pi$
$42$	0	0
$43$	−501.479	−1.77848	−0.889241	−	0.457438i	$-0.848767\pi$
$43$	−501.479	−1.77848	−0.889241	+	0.457438i	$0.848767\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−439.305	−1.36339	−0.681694	−	0.731637i	$-0.738756\pi$
$47$	−439.305	−1.36339	−0.681694	+	0.731637i	$0.738756\pi$
$48$	0	0
$49$	147.192	0.429132
$50$	0	0
$51$	−628.579	−1.72586
$52$	0	0
$53$	−286.955	−0.743703	−0.371851	−	0.928292i	$-0.621277\pi$
$53$	−286.955	−0.743703	−0.371851	+	0.928292i	$0.621277\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	943.303	2.19199
$58$	0	0
$59$	−547.175	−1.20739	−0.603696	−	0.797215i	$-0.706306\pi$
$59$	−547.175	−1.20739	−0.603696	+	0.797215i	$0.706306\pi$
$60$	0	0
$61$	−511.459	−1.07353	−0.536767	−	0.843730i	$-0.680355\pi$
$61$	−511.459	−1.07353	−0.536767	+	0.843730i	$0.680355\pi$
$62$	0	0
$63$	−1029.42	−2.05865
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−301.547	−0.549848	−0.274924	−	0.961466i	$-0.588653\pi$
$67$	−301.547	−0.549848	−0.274924	+	0.961466i	$0.588653\pi$
$68$	0	0
$69$	924.762	1.61345
$70$	0	0
$71$	−82.8978	−0.138566	−0.0692828	−	0.997597i	$-0.522071\pi$
$71$	−82.8978	−0.138566	−0.0692828	+	0.997597i	$0.522071\pi$
$72$	0	0
$73$	763.267	1.22375	0.611874	−	0.790955i	$-0.290416\pi$
$73$	763.267	1.22375	0.611874	+	0.790955i	$0.290416\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−600.770	−0.889144
$78$	0	0
$79$	−1011.45	−1.44047	−0.720236	−	0.693729i	$-0.755966\pi$
$79$	−1011.45	−1.44047	−0.720236	+	0.693729i	$0.755966\pi$
$80$	0	0
$81$	177.450	0.243416
$82$	0	0
$83$	−704.554	−0.931745	−0.465872	−	0.884852i	$-0.654259\pi$
$83$	−704.554	−0.931745	−0.465872	+	0.884852i	$0.654259\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	588.468	0.725177
$88$	0	0
$89$	−743.212	−0.885172	−0.442586	−	0.896726i	$-0.645939\pi$
$89$	−743.212	−0.885172	−0.442586	+	0.896726i	$0.645939\pi$
$90$	0	0
$91$	1557.13	1.79376
$92$	0	0
$93$	−1175.93	−1.31116
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1136.00	1.18911	0.594553	−	0.804056i	$-0.297329\pi$
$97$	1136.00	1.18911	0.594553	+	0.804056i	$0.297329\pi$
$98$	0	0
$99$	1261.64	1.28080
$100$	0	0
$101$	−291.212	−0.286898	−0.143449	−	0.989658i	$-0.545819\pi$
$101$	−291.212	−0.286898	−0.143449	+	0.989658i	$0.545819\pi$
$102$	0	0
$103$	200.445	0.191751	0.0958757	−	0.995393i	$-0.469435\pi$
$103$	200.445	0.191751	0.0958757	+	0.995393i	$0.469435\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	754.342	0.681542	0.340771	−	0.940146i	$-0.389312\pi$
$107$	754.342	0.681542	0.340771	+	0.940146i	$0.389312\pi$
$108$	0	0
$109$	−501.680	−0.440846	−0.220423	−	0.975404i	$-0.570744\pi$
$109$	−501.680	−0.440846	−0.220423	+	0.975404i	$0.570744\pi$
$110$	0	0
$111$	−517.053	−0.442130
$112$	0	0
$113$	−497.248	−0.413958	−0.206979	−	0.978345i	$-0.566363\pi$
$113$	−497.248	−0.413958	−0.206979	+	0.978345i	$0.566363\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3270.04	−2.58389
$118$	0	0
$119$	−1623.35	−1.25053
$120$	0	0
$121$	−594.709	−0.446814
$122$	0	0
$123$	−815.690	−0.597954
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−915.221	−0.639470	−0.319735	−	0.947507i	$-0.603594\pi$
$127$	−915.221	−0.639470	−0.319735	+	0.947507i	$0.603594\pi$
$128$	0	0
$129$	4299.15	2.93426
$130$	0	0
$131$	607.419	0.405118	0.202559	−	0.979270i	$-0.435074\pi$
$131$	607.419	0.405118	0.202559	+	0.979270i	$0.435074\pi$
$132$	0	0
$133$	2436.15	1.58828
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1418.98	−0.884900	−0.442450	−	0.896793i	$-0.645891\pi$
$137$	−1418.98	−0.884900	−0.442450	+	0.896793i	$0.645891\pi$
$138$	0	0
$139$	3035.85	1.85250	0.926250	−	0.376910i	$-0.123013\pi$
$139$	3035.85	1.85250	0.926250	+	0.376910i	$0.123013\pi$
$140$	0	0
$141$	3766.14	2.24941
$142$	0	0
$143$	−1908.39	−1.11600
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1261.87	−0.708011
$148$	0	0
$149$	2649.26	1.45662	0.728308	−	0.685250i	$-0.240307\pi$
$149$	2649.26	1.45662	0.728308	+	0.685250i	$0.240307\pi$
$150$	0	0
$151$	−283.297	−0.152678	−0.0763390	−	0.997082i	$-0.524323\pi$
$151$	−283.297	−0.152678	−0.0763390	+	0.997082i	$0.524323\pi$
$152$	0	0
$153$	3409.10	1.80137
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1414.88	−0.719235	−0.359617	−	0.933100i	$-0.617093\pi$
$157$	−1414.88	−0.719235	−0.359617	+	0.933100i	$0.617093\pi$
$158$	0	0
$159$	2460.05	1.22701
$160$	0	0
$161$	2388.27	1.16908
$162$	0	0
$163$	1725.68	0.829239	0.414619	−	0.909995i	$-0.363915\pi$
$163$	1725.68	0.829239	0.414619	+	0.909995i	$0.363915\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1979.92	0.917428	0.458714	−	0.888584i	$-0.348310\pi$
$167$	1979.92	0.917428	0.458714	+	0.888584i	$0.348310\pi$
$168$	0	0
$169$	2749.35	1.25141
$170$	0	0
$171$	−5116.01	−2.28790
$172$	0	0
$173$	−1468.72	−0.645460	−0.322730	−	0.946491i	$-0.604601\pi$
$173$	−1468.72	−0.645460	−0.322730	+	0.946491i	$0.604601\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4690.90	1.99203
$178$	0	0
$179$	−2978.59	−1.24375	−0.621873	−	0.783118i	$-0.713628\pi$
$179$	−2978.59	−1.24375	−0.621873	+	0.783118i	$0.713628\pi$
$180$	0	0
$181$	1682.07	0.690760	0.345380	−	0.938463i	$-0.387750\pi$
$181$	1682.07	0.690760	0.345380	+	0.938463i	$0.387750\pi$
$182$	0	0
$183$	4384.71	1.77119
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1989.55	0.778022
$188$	0	0
$189$	3700.38	1.42414
$190$	0	0
$191$	−274.334	−0.103927	−0.0519637	−	0.998649i	$-0.516548\pi$
$191$	−274.334	−0.103927	−0.0519637	+	0.998649i	$0.516548\pi$
$192$	0	0
$193$	2898.50	1.08103	0.540514	−	0.841335i	$-0.318230\pi$
$193$	2898.50	1.08103	0.540514	+	0.841335i	$0.318230\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2330.37	−0.842801	−0.421400	−	0.906875i	$-0.638461\pi$
$197$	−2330.37	−0.842801	−0.421400	+	0.906875i	$0.638461\pi$
$198$	0	0
$199$	−1105.05	−0.393644	−0.196822	−	0.980439i	$-0.563062\pi$
$199$	−1105.05	−0.393644	−0.196822	+	0.980439i	$0.563062\pi$
$200$	0	0
$201$	2585.15	0.907175
$202$	0	0
$203$	1519.76	0.525451
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5015.45	−1.68405
$208$	0	0
$209$	−2985.70	−0.988158
$210$	0	0
$211$	4749.82	1.54972	0.774860	−	0.632133i	$-0.217820\pi$
$211$	4749.82	1.54972	0.774860	+	0.632133i	$0.217820\pi$
$212$	0	0
$213$	710.679	0.228615
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3036.92	−0.950044
$218$	0	0
$219$	−6543.45	−2.01902
$220$	0	0
$221$	−5156.70	−1.56958
$222$	0	0
$223$	1889.71	0.567462	0.283731	−	0.958904i	$-0.408428\pi$
$223$	1889.71	0.567462	0.283731	+	0.958904i	$0.408428\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4154.11	−1.21462	−0.607309	−	0.794466i	$-0.707751\pi$
$227$	−4154.11	−1.21462	−0.607309	+	0.794466i	$0.707751\pi$
$228$	0	0
$229$	888.642	0.256433	0.128216	−	0.991746i	$-0.459075\pi$
$229$	888.642	0.256433	0.128216	+	0.991746i	$0.459075\pi$
$230$	0	0
$231$	5150.37	1.46697
$232$	0	0
$233$	−4919.38	−1.38317	−0.691586	−	0.722294i	$-0.743088\pi$
$233$	−4919.38	−1.38317	−0.691586	+	0.722294i	$0.743088\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8671.13	2.37658
$238$	0	0
$239$	2178.00	0.589468	0.294734	−	0.955579i	$-0.404769\pi$
$239$	2178.00	0.589468	0.294734	+	0.955579i	$0.404769\pi$
$240$	0	0
$241$	−3156.12	−0.843583	−0.421792	−	0.906693i	$-0.638599\pi$
$241$	−3156.12	−0.843583	−0.421792	+	0.906693i	$0.638599\pi$
$242$	0	0
$243$	2991.34	0.789688
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7738.62	1.99351
$248$	0	0
$249$	6040.10	1.53725
$250$	0	0
$251$	2719.20	0.683801	0.341901	−	0.939736i	$-0.388929\pi$
$251$	2719.20	0.683801	0.341901	+	0.939736i	$0.388929\pi$
$252$	0	0
$253$	−2927.01	−0.727350
$254$	0	0
$255$	0	0
$256$	0	0
$257$	749.067	0.181811	0.0909056	−	0.995860i	$-0.471024\pi$
$257$	749.067	0.181811	0.0909056	+	0.995860i	$0.471024\pi$
$258$	0	0
$259$	−1335.33	−0.320360
$260$	0	0
$261$	−3191.56	−0.756907
$262$	0	0
$263$	2546.04	0.596942	0.298471	−	0.954419i	$-0.403523\pi$
$263$	2546.04	0.596942	0.298471	+	0.954419i	$0.403523\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6371.52	1.46041
$268$	0	0
$269$	7982.07	1.80920	0.904601	−	0.426260i	$-0.140169\pi$
$269$	7982.07	1.80920	0.904601	+	0.426260i	$0.140169\pi$
$270$	0	0
$271$	−1686.58	−0.378054	−0.189027	−	0.981972i	$-0.560533\pi$
$271$	−1686.58	−0.378054	−0.189027	+	0.981972i	$0.560533\pi$
$272$	0	0
$273$	−13349.2	−2.95946
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1423.07	−0.308679	−0.154339	−	0.988018i	$-0.549325\pi$
$277$	−1423.07	−0.308679	−0.154339	+	0.988018i	$0.549325\pi$
$278$	0	0
$279$	6377.65	1.36853
$280$	0	0
$281$	5418.67	1.15036	0.575180	−	0.818027i	$-0.304932\pi$
$281$	5418.67	1.15036	0.575180	+	0.818027i	$0.304932\pi$
$282$	0	0
$283$	8343.38	1.75252	0.876258	−	0.481842i	$-0.160032\pi$
$283$	8343.38	1.75252	0.876258	+	0.481842i	$0.160032\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2106.58	−0.433267
$288$	0	0
$289$	463.000	0.0942398
$290$	0	0
$291$	−9738.87	−1.96186
$292$	0	0
$293$	−2331.73	−0.464919	−0.232459	−	0.972606i	$-0.574677\pi$
$293$	−2331.73	−0.464919	−0.232459	+	0.972606i	$0.574677\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4535.12	−0.886041
$298$	0	0
$299$	7586.51	1.46736
$300$	0	0
$301$	11102.9	2.12611
$302$	0	0
$303$	2496.55	0.473343
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2100.00	0.390401	0.195200	−	0.980763i	$-0.437464\pi$
$307$	2100.00	0.390401	0.195200	+	0.980763i	$0.437464\pi$
$308$	0	0
$309$	−1718.40	−0.316364
$310$	0	0
$311$	−8501.38	−1.55006	−0.775030	−	0.631924i	$-0.782265\pi$
$311$	−8501.38	−1.55006	−0.775030	+	0.631924i	$0.782265\pi$
$312$	0	0
$313$	7257.73	1.31064	0.655321	−	0.755350i	$-0.272533\pi$
$313$	7257.73	1.31064	0.655321	+	0.755350i	$0.272533\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9639.14	1.70785	0.853925	−	0.520397i	$-0.174216\pi$
$317$	9639.14	1.70785	0.853925	+	0.520397i	$0.174216\pi$
$318$	0	0
$319$	−1862.59	−0.326912
$320$	0	0
$321$	−6466.93	−1.12445
$322$	0	0
$323$	−8067.72	−1.38978
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4300.88	0.727337
$328$	0	0
$329$	9726.34	1.62988
$330$	0	0
$331$	−360.466	−0.0598581	−0.0299290	−	0.999552i	$-0.509528\pi$
$331$	−360.466	−0.0598581	−0.0299290	+	0.999552i	$0.509528\pi$
$332$	0	0
$333$	2804.24	0.461476
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2820.47	−0.455908	−0.227954	−	0.973672i	$-0.573204\pi$
$337$	−2820.47	−0.455908	−0.227954	+	0.973672i	$0.573204\pi$
$338$	0	0
$339$	4262.89	0.682974
$340$	0	0
$341$	3721.99	0.591076
$342$	0	0
$343$	4335.24	0.682451
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8617.62	1.33319	0.666597	−	0.745419i	$-0.267750\pi$
$347$	8617.62	1.33319	0.666597	+	0.745419i	$0.267750\pi$
$348$	0	0
$349$	−735.067	−0.112743	−0.0563714	−	0.998410i	$-0.517953\pi$
$349$	−735.067	−0.112743	−0.0563714	+	0.998410i	$0.517953\pi$
$350$	0	0
$351$	11754.6	1.78750
$352$	0	0
$353$	4535.35	0.683830	0.341915	−	0.939731i	$-0.388924\pi$
$353$	4535.35	0.683830	0.341915	+	0.939731i	$0.388924\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13916.9	2.06320
$358$	0	0
$359$	−5426.44	−0.797762	−0.398881	−	0.917003i	$-0.630601\pi$
$359$	−5426.44	−0.797762	−0.398881	+	0.917003i	$0.630601\pi$
$360$	0	0
$361$	5248.15	0.765148
$362$	0	0
$363$	5098.41	0.737182
$364$	0	0
$365$	0	0
$366$	0	0
$367$	609.673	0.0867157	0.0433579	−	0.999060i	$-0.486194\pi$
$367$	609.673	0.0867157	0.0433579	+	0.999060i	$0.486194\pi$
$368$	0	0
$369$	4423.90	0.624117
$370$	0	0
$371$	6353.26	0.889069
$372$	0	0
$373$	5089.42	0.706489	0.353244	−	0.935531i	$-0.385078\pi$
$373$	5089.42	0.706489	0.353244	+	0.935531i	$0.385078\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4827.64	0.659513
$378$	0	0
$379$	−910.876	−0.123453	−0.0617263	−	0.998093i	$-0.519661\pi$
$379$	−910.876	−0.123453	−0.0617263	+	0.998093i	$0.519661\pi$
$380$	0	0
$381$	7846.14	1.05504
$382$	0	0
$383$	−7686.98	−1.02555	−0.512776	−	0.858522i	$-0.671383\pi$
$383$	−7686.98	−1.02555	−0.512776	+	0.858522i	$0.671383\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−23316.5	−3.06264
$388$	0	0
$389$	−4372.77	−0.569943	−0.284972	−	0.958536i	$-0.591984\pi$
$389$	−4372.77	−0.569943	−0.284972	+	0.958536i	$0.591984\pi$
$390$	0	0
$391$	−7909.14	−1.02297
$392$	0	0
$393$	−5207.38	−0.668390
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12591.9	−1.59186	−0.795928	−	0.605391i	$-0.793017\pi$
$397$	−12591.9	−1.59186	−0.795928	+	0.605391i	$0.793017\pi$
$398$	0	0
$399$	−20885.0	−2.62045
$400$	0	0
$401$	3614.48	0.450122	0.225061	−	0.974345i	$-0.427742\pi$
$401$	3614.48	0.450122	0.225061	+	0.974345i	$0.427742\pi$
$402$	0	0
$403$	−9647.01	−1.19244
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1636.55	0.199314
$408$	0	0
$409$	639.314	0.0772910	0.0386455	−	0.999253i	$-0.487696\pi$
$409$	639.314	0.0772910	0.0386455	+	0.999253i	$0.487696\pi$
$410$	0	0
$411$	12164.8	1.45997
$412$	0	0
$413$	12114.6	1.44339
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−26026.2	−3.05637
$418$	0	0
$419$	11018.4	1.28469	0.642346	−	0.766415i	$-0.277961\pi$
$419$	11018.4	1.28469	0.642346	+	0.766415i	$0.277961\pi$
$420$	0	0
$421$	−16513.1	−1.91164	−0.955820	−	0.293952i	$-0.905029\pi$
$421$	−16513.1	−1.91164	−0.955820	+	0.293952i	$0.905029\pi$
$422$	0	0
$423$	−20425.7	−2.34783
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11323.9	1.28337
$428$	0	0
$429$	16360.5	1.84124
$430$	0	0
$431$	−6106.80	−0.682492	−0.341246	−	0.939974i	$-0.610849\pi$
$431$	−6106.80	−0.682492	−0.341246	+	0.939974i	$0.610849\pi$
$432$	0	0
$433$	1757.88	0.195100	0.0975500	−	0.995231i	$-0.468899\pi$
$433$	1757.88	0.195100	0.0975500	+	0.995231i	$0.468899\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11869.2	1.29927
$438$	0	0
$439$	−1055.02	−0.114700	−0.0573500	−	0.998354i	$-0.518265\pi$
$439$	−1055.02	−0.114700	−0.0573500	+	0.998354i	$0.518265\pi$
$440$	0	0
$441$	6843.78	0.738989
$442$	0	0
$443$	−7775.51	−0.833918	−0.416959	−	0.908925i	$-0.636904\pi$
$443$	−7775.51	−0.833918	−0.416959	+	0.908925i	$0.636904\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−22712.0	−2.40322
$448$	0	0
$449$	−11245.1	−1.18194	−0.590969	−	0.806694i	$-0.701255\pi$
$449$	−11245.1	−1.18194	−0.590969	+	0.806694i	$0.701255\pi$
$450$	0	0
$451$	2581.78	0.269560
$452$	0	0
$453$	2428.69	0.251898
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13576.9	−1.38972	−0.694860	−	0.719145i	$-0.744534\pi$
$457$	−13576.9	−1.38972	−0.694860	+	0.719145i	$0.744534\pi$
$458$	0	0
$459$	−12254.4	−1.24616
$460$	0	0
$461$	9605.08	0.970397	0.485199	−	0.874404i	$-0.338747\pi$
$461$	9605.08	0.970397	0.485199	+	0.874404i	$0.338747\pi$
$462$	0	0
$463$	3226.71	0.323883	0.161942	−	0.986800i	$-0.448224\pi$
$463$	3226.71	0.323883	0.161942	+	0.986800i	$0.448224\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.6790	−0.00204906	−0.00102453	−	0.999999i	$-0.500326\pi$
$467$	−20.6790	−0.00204906	−0.00102453	+	0.999999i	$0.500326\pi$
$468$	0	0
$469$	6676.34	0.657323
$470$	0	0
$471$	12129.7	1.18664
$472$	0	0
$473$	−13607.5	−1.32277
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13342.1	−1.28070
$478$	0	0
$479$	12492.9	1.19168	0.595841	−	0.803102i	$-0.296819\pi$
$479$	12492.9	1.19168	0.595841	+	0.803102i	$0.296819\pi$
$480$	0	0
$481$	−4241.77	−0.402096
$482$	0	0
$483$	−20474.5	−1.92882
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4944.60	−0.460084	−0.230042	−	0.973181i	$-0.573886\pi$
$487$	−4944.60	−0.460084	−0.230042	+	0.973181i	$0.573886\pi$
$488$	0	0
$489$	−14794.2	−1.36813
$490$	0	0
$491$	−7712.73	−0.708902	−0.354451	−	0.935075i	$-0.615332\pi$
$491$	−7712.73	−0.708902	−0.354451	+	0.935075i	$0.615332\pi$
$492$	0	0
$493$	−5032.95	−0.459782
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1835.38	0.165650
$498$	0	0
$499$	−1492.41	−0.133886	−0.0669432	−	0.997757i	$-0.521325\pi$
$499$	−1492.41	−0.133886	−0.0669432	+	0.997757i	$0.521325\pi$
$500$	0	0
$501$	−16973.7	−1.51363
$502$	0	0
$503$	−3105.13	−0.275250	−0.137625	−	0.990484i	$-0.543947\pi$
$503$	−3105.13	−0.275250	−0.137625	+	0.990484i	$0.543947\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23570.1	−2.06466
$508$	0	0
$509$	−15363.5	−1.33787	−0.668933	−	0.743322i	$-0.733249\pi$
$509$	−15363.5	−1.33787	−0.668933	+	0.743322i	$0.733249\pi$
$510$	0	0
$511$	−16898.9	−1.46295
$512$	0	0
$513$	18390.1	1.58274
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11920.4	−1.01404
$518$	0	0
$519$	12591.2	1.06492
$520$	0	0
$521$	−9924.25	−0.834529	−0.417264	−	0.908785i	$-0.637011\pi$
$521$	−9924.25	−0.834529	−0.417264	+	0.908785i	$0.637011\pi$
$522$	0	0
$523$	455.146	0.0380538	0.0190269	−	0.999819i	$-0.493943\pi$
$523$	455.146	0.0380538	0.0190269	+	0.999819i	$0.493943\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10057.3	0.831312
$528$	0	0
$529$	−531.111	−0.0436517
$530$	0	0
$531$	−25441.1	−2.07919
$532$	0	0
$533$	−6691.72	−0.543809
$534$	0	0
$535$	0	0
$536$	0	0
$537$	25535.3	2.05201
$538$	0	0
$539$	3994.02	0.319174
$540$	0	0
$541$	−23383.9	−1.85832	−0.929160	−	0.369678i	$-0.879468\pi$
$541$	−23383.9	−1.85832	−0.929160	+	0.369678i	$0.879468\pi$
$542$	0	0
$543$	−14420.3	−1.13966
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6908.03	−0.539974	−0.269987	−	0.962864i	$-0.587019\pi$
$547$	−6908.03	−0.539974	−0.269987	+	0.962864i	$0.587019\pi$
$548$	0	0
$549$	−23780.5	−1.84868
$550$	0	0
$551$	7552.90	0.583964
$552$	0	0
$553$	22393.8	1.72203
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3221.81	0.245085	0.122543	−	0.992463i	$-0.460895\pi$
$557$	3221.81	0.245085	0.122543	+	0.992463i	$0.460895\pi$
$558$	0	0
$559$	35269.1	2.66856
$560$	0	0
$561$	−17056.3	−1.28363
$562$	0	0
$563$	14822.7	1.10959	0.554796	−	0.831986i	$-0.312796\pi$
$563$	14822.7	1.10959	0.554796	+	0.831986i	$0.312796\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3928.79	−0.290994
$568$	0	0
$569$	6434.10	0.474045	0.237022	−	0.971504i	$-0.423828\pi$
$569$	6434.10	0.474045	0.237022	+	0.971504i	$0.423828\pi$
$570$	0	0
$571$	−17760.3	−1.30165	−0.650827	−	0.759226i	$-0.725578\pi$
$571$	−17760.3	−1.30165	−0.650827	+	0.759226i	$0.725578\pi$
$572$	0	0
$573$	2351.85	0.171466
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1341.96	−0.0968227	−0.0484113	−	0.998827i	$-0.515416\pi$
$577$	−1341.96	−0.0968227	−0.0484113	+	0.998827i	$0.515416\pi$
$578$	0	0
$579$	−24848.7	−1.78355
$580$	0	0
$581$	15599.0	1.11387
$582$	0	0
$583$	−7786.42	−0.553140
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12957.3	0.911082	0.455541	−	0.890215i	$-0.349446\pi$
$587$	12957.3	0.911082	0.455541	+	0.890215i	$0.349446\pi$
$588$	0	0
$589$	−15092.8	−1.05584
$590$	0	0
$591$	19978.1	1.39051
$592$	0	0
$593$	−5966.63	−0.413187	−0.206594	−	0.978427i	$-0.566238\pi$
$593$	−5966.63	−0.413187	−0.206594	+	0.978427i	$0.566238\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9473.56	0.649459
$598$	0	0
$599$	−10311.9	−0.703396	−0.351698	−	0.936114i	$-0.614396\pi$
$599$	−10311.9	−0.703396	−0.351698	+	0.936114i	$0.614396\pi$
$600$	0	0
$601$	2594.60	0.176100	0.0880499	−	0.996116i	$-0.471936\pi$
$601$	2594.60	0.176100	0.0880499	+	0.996116i	$0.471936\pi$
$602$	0	0
$603$	−14020.6	−0.946868
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7796.08	0.521306	0.260653	−	0.965432i	$-0.416062\pi$
$607$	7796.08	0.521306	0.260653	+	0.965432i	$0.416062\pi$
$608$	0	0
$609$	−13028.9	−0.866922
$610$	0	0
$611$	30896.5	2.04572
$612$	0	0
$613$	−10443.5	−0.688106	−0.344053	−	0.938950i	$-0.611800\pi$
$613$	−10443.5	−0.688106	−0.344053	+	0.938950i	$0.611800\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18306.1	1.19445	0.597226	−	0.802073i	$-0.296270\pi$
$617$	18306.1	1.19445	0.597226	+	0.802073i	$0.296270\pi$
$618$	0	0
$619$	−149.857	−0.00973066	−0.00486533	−	0.999988i	$-0.501549\pi$
$619$	−149.857	−0.00973066	−0.00486533	+	0.999988i	$0.501549\pi$
$620$	0	0
$621$	18028.7	1.16500
$622$	0	0
$623$	16454.9	1.05819
$624$	0	0
$625$	0	0
$626$	0	0
$627$	25596.2	1.63033
$628$	0	0
$629$	4422.16	0.280323
$630$	0	0
$631$	24466.5	1.54358	0.771789	−	0.635879i	$-0.219362\pi$
$631$	24466.5	1.54358	0.771789	+	0.635879i	$0.219362\pi$
$632$	0	0
$633$	−40719.9	−2.55683
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−10352.1	−0.643901
$638$	0	0
$639$	−3854.37	−0.238618
$640$	0	0
$641$	−20274.7	−1.24930	−0.624652	−	0.780903i	$-0.714759\pi$
$641$	−20274.7	−1.24930	−0.624652	+	0.780903i	$0.714759\pi$
$642$	0	0
$643$	9167.25	0.562241	0.281121	−	0.959672i	$-0.409294\pi$
$643$	9167.25	0.562241	0.281121	+	0.959672i	$0.409294\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5459.82	0.331758	0.165879	−	0.986146i	$-0.446954\pi$
$647$	5459.82	0.331758	0.165879	+	0.986146i	$0.446954\pi$
$648$	0	0
$649$	−14847.4	−0.898016
$650$	0	0
$651$	26035.4	1.56744
$652$	0	0
$653$	16280.5	0.975659	0.487830	−	0.872939i	$-0.337789\pi$
$653$	16280.5	0.975659	0.487830	+	0.872939i	$0.337789\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	35488.4	2.10736
$658$	0	0
$659$	−23975.9	−1.41725	−0.708625	−	0.705585i	$-0.750684\pi$
$659$	−23975.9	−1.41725	−0.708625	+	0.705585i	$0.750684\pi$
$660$	0	0
$661$	13876.4	0.816532	0.408266	−	0.912863i	$-0.366134\pi$
$661$	13876.4	0.816532	0.408266	+	0.912863i	$0.366134\pi$
$662$	0	0
$663$	44208.2	2.58960
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7404.44	0.429837
$668$	0	0
$669$	−16200.3	−0.936236
$670$	0	0
$671$	−13878.3	−0.798458
$672$	0	0
$673$	30526.1	1.74843	0.874216	−	0.485537i	$-0.161376\pi$
$673$	30526.1	1.74843	0.874216	+	0.485537i	$0.161376\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6992.09	−0.396939	−0.198470	−	0.980107i	$-0.563597\pi$
$677$	−6992.09	−0.396939	−0.198470	+	0.980107i	$0.563597\pi$
$678$	0	0
$679$	−25151.4	−1.42153
$680$	0	0
$681$	35613.0	2.00395
$682$	0	0
$683$	22479.2	1.25936	0.629681	−	0.776854i	$-0.283186\pi$
$683$	22479.2	1.25936	0.629681	+	0.776854i	$0.283186\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7618.29	−0.423080
$688$	0	0
$689$	20181.6	1.11590
$690$	0	0
$691$	5536.97	0.304828	0.152414	−	0.988317i	$-0.451295\pi$
$691$	5536.97	0.304828	0.152414	+	0.988317i	$0.451295\pi$
$692$	0	0
$693$	−27933.1	−1.53115
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6976.29	0.379119
$698$	0	0
$699$	42173.6	2.28205
$700$	0	0
$701$	16934.8	0.912436	0.456218	−	0.889868i	$-0.349204\pi$
$701$	16934.8	0.912436	0.456218	+	0.889868i	$0.349204\pi$
$702$	0	0
$703$	−6636.29	−0.356035
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6447.52	0.342976
$708$	0	0
$709$	4112.16	0.217821	0.108911	−	0.994052i	$-0.465264\pi$
$709$	4112.16	0.217821	0.108911	+	0.994052i	$0.465264\pi$
$710$	0	0
$711$	−47027.9	−2.48057
$712$	0	0
$713$	−14796.2	−0.777169
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−18671.9	−0.972543
$718$	0	0
$719$	−34938.3	−1.81221	−0.906105	−	0.423053i	$-0.860958\pi$
$719$	−34938.3	−1.81221	−0.906105	+	0.423053i	$0.860958\pi$
$720$	0	0
$721$	−4437.90	−0.229232
$722$	0	0
$723$	27057.3	1.39180
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34679.2	1.76916	0.884580	−	0.466389i	$-0.154445\pi$
$727$	34679.2	1.76916	0.884580	+	0.466389i	$0.154445\pi$
$728$	0	0
$729$	−30435.7	−1.54629
$730$	0	0
$731$	−36769.0	−1.86040
$732$	0	0
$733$	−30802.4	−1.55213	−0.776065	−	0.630652i	$-0.782787\pi$
$733$	−30802.4	−1.55213	−0.776065	+	0.630652i	$0.782787\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8182.38	−0.408958
$738$	0	0
$739$	16364.2	0.814570	0.407285	−	0.913301i	$-0.366476\pi$
$739$	16364.2	0.814570	0.407285	+	0.913301i	$0.366476\pi$
$740$	0	0
$741$	−66342.8	−3.28902
$742$	0	0
$743$	−4464.06	−0.220418	−0.110209	−	0.993908i	$-0.535152\pi$
$743$	−4464.06	−0.220418	−0.110209	+	0.993908i	$0.535152\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−32758.5	−1.60451
$748$	0	0
$749$	−16701.3	−0.814758
$750$	0	0
$751$	−17873.6	−0.868463	−0.434231	−	0.900801i	$-0.642980\pi$
$751$	−17873.6	−0.868463	−0.434231	+	0.900801i	$0.642980\pi$
$752$	0	0
$753$	−23311.5	−1.12818
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−14305.3	−0.686834	−0.343417	−	0.939183i	$-0.611584\pi$
$757$	−14305.3	−0.686834	−0.343417	+	0.939183i	$0.611584\pi$
$758$	0	0
$759$	25093.1	1.20003
$760$	0	0
$761$	21819.9	1.03938	0.519692	−	0.854354i	$-0.326047\pi$
$761$	21819.9	1.03938	0.519692	+	0.854354i	$0.326047\pi$
$762$	0	0
$763$	11107.3	0.527016
$764$	0	0
$765$	0	0
$766$	0	0
$767$	38483.0	1.81166
$768$	0	0
$769$	−29446.6	−1.38085	−0.690423	−	0.723406i	$-0.742575\pi$
$769$	−29446.6	−1.38085	−0.690423	+	0.723406i	$0.742575\pi$
$770$	0	0
$771$	−6421.71	−0.299964
$772$	0	0
$773$	−30232.4	−1.40671	−0.703354	−	0.710840i	$-0.748315\pi$
$773$	−30232.4	−1.40671	−0.703354	+	0.710840i	$0.748315\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	11447.7	0.528551
$778$	0	0
$779$	−10469.3	−0.481515
$780$	0	0
$781$	−2249.41	−0.103060
$782$	0	0
$783$	11472.5	0.523617
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30871.6	1.39829	0.699145	−	0.714980i	$-0.253564\pi$
$787$	30871.6	1.39829	0.699145	+	0.714980i	$0.253564\pi$
$788$	0	0
$789$	−21827.1	−0.984873
$790$	0	0
$791$	11009.2	0.494871
$792$	0	0
$793$	35971.1	1.61081
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4612.35	0.204991	0.102496	−	0.994733i	$-0.467317\pi$
$797$	4612.35	0.204991	0.102496	+	0.994733i	$0.467317\pi$
$798$	0	0
$799$	−32210.4	−1.42618
$800$	0	0
$801$	−34556.0	−1.52431
$802$	0	0
$803$	20711.0	0.910181
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−68429.8	−2.98494
$808$	0	0
$809$	−5893.44	−0.256122	−0.128061	−	0.991766i	$-0.540875\pi$
$809$	−5893.44	−0.256122	−0.128061	+	0.991766i	$0.540875\pi$
$810$	0	0
$811$	28114.3	1.21730	0.608648	−	0.793441i	$-0.291712\pi$
$811$	28114.3	1.21730	0.608648	+	0.793441i	$0.291712\pi$
$812$	0	0
$813$	14459.0	0.623739
$814$	0	0
$815$	0	0
$816$	0	0
$817$	55178.9	2.36287
$818$	0	0
$819$	72399.6	3.08895
$820$	0	0
$821$	−28087.6	−1.19399	−0.596994	−	0.802246i	$-0.703638\pi$
$821$	−28087.6	−1.19399	−0.596994	+	0.802246i	$0.703638\pi$
$822$	0	0
$823$	43426.8	1.83932	0.919661	−	0.392712i	$-0.128463\pi$
$823$	43426.8	1.83932	0.919661	+	0.392712i	$0.128463\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9481.07	0.398657	0.199328	−	0.979933i	$-0.436124\pi$
$827$	9481.07	0.398657	0.199328	+	0.979933i	$0.436124\pi$
$828$	0	0
$829$	31634.3	1.32534	0.662668	−	0.748913i	$-0.269424\pi$
$829$	31634.3	1.32534	0.662668	+	0.748913i	$0.269424\pi$
$830$	0	0
$831$	12199.9	0.509278
$832$	0	0
$833$	10792.3	0.448898
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−22925.2	−0.946729
$838$	0	0
$839$	24661.7	1.01480	0.507399	−	0.861711i	$-0.330607\pi$
$839$	24661.7	1.01480	0.507399	+	0.861711i	$0.330607\pi$
$840$	0	0
$841$	−19677.2	−0.806807
$842$	0	0
$843$	−46454.0	−1.89794
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13167.0	0.534149
$848$	0	0
$849$	−71527.3	−2.89142
$850$	0	0
$851$	−6505.86	−0.262066
$852$	0	0
$853$	24125.3	0.968386	0.484193	−	0.874961i	$-0.339113\pi$
$853$	24125.3	0.968386	0.484193	+	0.874961i	$0.339113\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23911.4	−0.953089	−0.476544	−	0.879150i	$-0.658111\pi$
$857$	−23911.4	−0.953089	−0.476544	+	0.879150i	$0.658111\pi$
$858$	0	0
$859$	−22987.8	−0.913079	−0.456539	−	0.889703i	$-0.650911\pi$
$859$	−22987.8	−0.913079	−0.456539	+	0.889703i	$0.650911\pi$
$860$	0	0
$861$	18059.6	0.714832
$862$	0	0
$863$	29565.5	1.16619	0.583095	−	0.812404i	$-0.301841\pi$
$863$	29565.5	1.16619	0.583095	+	0.812404i	$0.301841\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3969.28	−0.155483
$868$	0	0
$869$	−27445.4	−1.07137
$870$	0	0
$871$	21207.9	0.825031
$872$	0	0
$873$	52818.8	2.04771
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10218.0	0.393427	0.196714	−	0.980461i	$-0.436973\pi$
$877$	10218.0	0.393427	0.196714	+	0.980461i	$0.436973\pi$
$878$	0	0
$879$	19989.8	0.767053
$880$	0	0
$881$	−3269.43	−0.125028	−0.0625141	−	0.998044i	$-0.519912\pi$
$881$	−3269.43	−0.125028	−0.0625141	+	0.998044i	$0.519912\pi$
$882$	0	0
$883$	13275.6	0.505956	0.252978	−	0.967472i	$-0.418590\pi$
$883$	13275.6	0.505956	0.252978	+	0.967472i	$0.418590\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23809.7	−0.901298	−0.450649	−	0.892701i	$-0.648807\pi$
$887$	−23809.7	−0.901298	−0.450649	+	0.892701i	$0.648807\pi$
$888$	0	0
$889$	20263.3	0.764463
$890$	0	0
$891$	4815.05	0.181044
$892$	0	0
$893$	48337.8	1.81138
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−65038.8	−2.42094
$898$	0	0
$899$	−9415.49	−0.349304
$900$	0	0
$901$	−21039.9	−0.777957
$902$	0	0
$903$	−95184.4	−3.50780
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−39404.3	−1.44255	−0.721277	−	0.692646i	$-0.756445\pi$
$907$	−39404.3	−1.44255	−0.721277	+	0.692646i	$0.756445\pi$
$908$	0	0
$909$	−13540.0	−0.494054
$910$	0	0
$911$	48325.4	1.75751	0.878755	−	0.477273i	$-0.158375\pi$
$911$	48325.4	1.75751	0.878755	+	0.477273i	$0.158375\pi$
$912$	0	0
$913$	−19117.8	−0.692999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13448.4	−0.484304
$918$	0	0
$919$	−35431.2	−1.27178	−0.635891	−	0.771779i	$-0.719367\pi$
$919$	−35431.2	−1.27178	−0.635891	+	0.771779i	$0.719367\pi$
$920$	0	0
$921$	−18003.2	−0.644109
$922$	0	0
$923$	5830.23	0.207914
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9319.76	0.330206
$928$	0	0
$929$	40430.1	1.42785	0.713923	−	0.700225i	$-0.246917\pi$
$929$	40430.1	1.42785	0.713923	+	0.700225i	$0.246917\pi$
$930$	0	0
$931$	−16196.0	−0.570141
$932$	0	0
$933$	72881.9	2.55739
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3421.64	−0.119296	−0.0596479	−	0.998219i	$-0.518998\pi$
$937$	−3421.64	−0.119296	−0.0596479	+	0.998219i	$0.518998\pi$
$938$	0	0
$939$	−62220.1	−2.16238
$940$	0	0
$941$	33075.2	1.14582	0.572912	−	0.819617i	$-0.305814\pi$
$941$	33075.2	1.14582	0.572912	+	0.819617i	$0.305814\pi$
$942$	0	0
$943$	−10263.5	−0.354427
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21896.5	0.751364	0.375682	−	0.926749i	$-0.377409\pi$
$947$	21896.5	0.751364	0.375682	+	0.926749i	$0.377409\pi$
$948$	0	0
$949$	−53680.8	−1.83620
$950$	0	0
$951$	−82635.9	−2.81772
$952$	0	0
$953$	−48287.2	−1.64132	−0.820660	−	0.571417i	$-0.806394\pi$
$953$	−48287.2	−1.64132	−0.820660	+	0.571417i	$0.806394\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	15967.9	0.539361
$958$	0	0
$959$	31416.5	1.05787
$960$	0	0
$961$	−10976.2	−0.368439
$962$	0	0
$963$	35073.5	1.17365
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−13708.7	−0.455886	−0.227943	−	0.973674i	$-0.573200\pi$
$967$	−13708.7	−0.455886	−0.227943	+	0.973674i	$0.573200\pi$
$968$	0	0
$969$	69164.1	2.29295
$970$	0	0
$971$	24503.4	0.809836	0.404918	−	0.914353i	$-0.367300\pi$
$971$	24503.4	0.809836	0.404918	+	0.914353i	$0.367300\pi$
$972$	0	0
$973$	−67214.6	−2.21460
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27660.1	−0.905758	−0.452879	−	0.891572i	$-0.649603\pi$
$977$	−27660.1	−0.905758	−0.452879	+	0.891572i	$0.649603\pi$
$978$	0	0
$979$	−20166.8	−0.658360
$980$	0	0
$981$	−23325.9	−0.759161
$982$	0	0
$983$	56556.7	1.83507	0.917537	−	0.397651i	$-0.130175\pi$
$983$	56556.7	1.83507	0.917537	+	0.397651i	$0.130175\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−83383.4	−2.68908
$988$	0	0
$989$	54094.4	1.73923
$990$	0	0
$991$	47154.6	1.51152	0.755760	−	0.654848i	$-0.227268\pi$
$991$	47154.6	1.51152	0.755760	+	0.654848i	$0.227268\pi$
$992$	0	0
$993$	3090.26	0.0987578
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13606.6	0.432223	0.216111	−	0.976369i	$-0.430663\pi$
$997$	13606.6	0.432223	0.216111	+	0.976369i	$0.430663\pi$
$998$	0	0
$999$	−10080.2	−0.319242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.cu.1.1		4
4.3	odd	2	inner	1600.4.a.cu.1.4		4
5.2	odd	4		320.4.c.j.129.7		8
5.3	odd	4		320.4.c.j.129.2		8
5.4	even	2		1600.4.a.cv.1.4		4
8.3	odd	2		800.4.a.z.1.1		4
8.5	even	2		800.4.a.z.1.4		4
20.3	even	4		320.4.c.j.129.8		8
20.7	even	4		320.4.c.j.129.1		8
20.19	odd	2		1600.4.a.cv.1.1		4
40.3	even	4		160.4.c.d.129.1	✓	8
40.13	odd	4		160.4.c.d.129.7	yes	8
40.19	odd	2		800.4.a.y.1.4		4
40.27	even	4		160.4.c.d.129.8	yes	8
40.29	even	2		800.4.a.y.1.1		4
40.37	odd	4		160.4.c.d.129.2	yes	8
120.53	even	4		1440.4.f.k.289.8		8
120.77	even	4		1440.4.f.k.289.5		8
120.83	odd	4		1440.4.f.k.289.7		8
120.107	odd	4		1440.4.f.k.289.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.d.129.1	✓	8	40.3	even	4
160.4.c.d.129.2	yes	8	40.37	odd	4
160.4.c.d.129.7	yes	8	40.13	odd	4
160.4.c.d.129.8	yes	8	40.27	even	4
320.4.c.j.129.1		8	20.7	even	4
320.4.c.j.129.2		8	5.3	odd	4
320.4.c.j.129.7		8	5.2	odd	4
320.4.c.j.129.8		8	20.3	even	4
800.4.a.y.1.1		4	40.29	even	2
800.4.a.y.1.4		4	40.19	odd	2
800.4.a.z.1.1		4	8.3	odd	2
800.4.a.z.1.4		4	8.5	even	2
1440.4.f.k.289.5		8	120.77	even	4
1440.4.f.k.289.6		8	120.107	odd	4
1440.4.f.k.289.7		8	120.83	odd	4
1440.4.f.k.289.8		8	120.53	even	4
1600.4.a.cu.1.1		4	1.1	even	1	trivial
1600.4.a.cu.1.4		4	4.3	odd	2	inner
1600.4.a.cv.1.1		4	20.19	odd	2
1600.4.a.cv.1.4		4	5.4	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

\(f(q)\)	\(=\)	\(q-8.57295 q^{3} -22.1403 q^{7} +46.4955 q^{9} +O(q^{10})\)	\(q-8.57295 q^{3} -22.1403 q^{7} +46.4955 q^{9} +27.1347 q^{11} -70.3303 q^{13} +73.3212 q^{17} -110.033 q^{19} +189.808 q^{21} -107.870 q^{23} -167.134 q^{27} -68.6424 q^{29} +137.167 q^{31} -232.624 q^{33} +60.3121 q^{37} +602.938 q^{39} +95.1470 q^{41} -501.479 q^{43} -439.305 q^{47} +147.192 q^{49} -628.579 q^{51} -286.955 q^{53} +943.303 q^{57} -547.175 q^{59} -511.459 q^{61} -1029.42 q^{63} -301.547 q^{67} +924.762 q^{69} -82.8978 q^{71} +763.267 q^{73} -600.770 q^{77} -1011.45 q^{79} +177.450 q^{81} -704.554 q^{83} +588.468 q^{87} -743.212 q^{89} +1557.13 q^{91} -1175.93 q^{93} +1136.00 q^{97} +1261.64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 76 q^{9}+O(q^{10}) \) 4 * q + 76 * q^9	\( 4 q + 76 q^{9} - 208 q^{13} + 136 q^{21} + 312 q^{29} + 96 q^{33} - 272 q^{37} - 96 q^{41} + 1212 q^{49} - 48 q^{53} + 3040 q^{57} - 1056 q^{61} + 1976 q^{69} + 1440 q^{73} - 4896 q^{77} - 500 q^{81} - 40 q^{89} - 2944 q^{93} + 4544 q^{97}+O(q^{100}) \) 4 * q + 76 * q^9 - 208 * q^13 + 136 * q^21 + 312 * q^29 + 96 * q^33 - 272 * q^37 - 96 * q^41 + 1212 * q^49 - 48 * q^53 + 3040 * q^57 - 1056 * q^61 + 1976 * q^69 + 1440 * q^73 - 4896 * q^77 - 500 * q^81 - 40 * q^89 - 2944 * q^93 + 4544 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more