LMFDB - Embedded newform 840.4.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [840,4,Mod(1,840)] 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("840.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,10,0,-14,0,18,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.5616044048$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2881}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 720 $ x^2 - x - 720
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-26.3375$ of defining polynomial
Character	$\chi$	$=$	840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +42.6749 q^{11} -64.6749 q^{13} +15.0000 q^{15} -14.0000 q^{17} +53.3251 q^{19} -21.0000 q^{21} +37.3499 q^{23} +25.0000 q^{25} +27.0000 q^{27} +294.000 q^{29} +228.025 q^{31} +128.025 q^{33} -35.0000 q^{35} -234.050 q^{37} -194.025 q^{39} +12.6501 q^{41} -433.400 q^{43} +45.0000 q^{45} +96.0000 q^{47} +49.0000 q^{49} -42.0000 q^{51} +607.375 q^{53} +213.375 q^{55} +159.975 q^{57} +588.050 q^{59} +758.099 q^{61} -63.0000 q^{63} -323.375 q^{65} -329.400 q^{67} +112.050 q^{69} +134.625 q^{71} +384.774 q^{73} +75.0000 q^{75} -298.725 q^{77} -21.3002 q^{79} +81.0000 q^{81} -468.099 q^{83} -70.0000 q^{85} +882.000 q^{87} -1104.70 q^{89} +452.725 q^{91} +684.075 q^{93} +266.625 q^{95} +240.625 q^{97} +384.075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9} - 22 q^{11} - 22 q^{13} + 30 q^{15} - 28 q^{17} + 214 q^{19} - 42 q^{21} - 140 q^{23} + 50 q^{25} + 54 q^{27} + 588 q^{29} + 134 q^{31} - 66 q^{33} - 70 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 - 22 * q^11 - 22 * q^13 + 30 * q^15 - 28 * q^17 + 214 * q^19 - 42 * q^21 - 140 * q^23 + 50 * q^25 + 54 * q^27 + 588 * q^29 + 134 * q^31 - 66 * q^33 - 70 * q^35 + 176 * q^37 - 66 * q^39 + 240 * q^41 - 8 * q^43 + 90 * q^45 + 192 * q^47 + 98 * q^49 - 84 * q^51 + 678 * q^53 - 110 * q^55 + 642 * q^57 + 532 * q^59 + 228 * q^61 - 126 * q^63 - 110 * q^65 + 200 * q^67 - 420 * q^69 + 806 * q^71 - 626 * q^73 + 150 * q^75 + 154 * q^77 - 472 * q^79 + 162 * q^81 + 352 * q^83 - 140 * q^85 + 1764 * q^87 - 1780 * q^89 + 154 * q^91 + 402 * q^93 + 1070 * q^95 + 1018 * q^97 - 198 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	42.6749	1.16973	0.584863	−	0.811132i	$-0.301148\pi$
$11$	42.6749	1.16973	0.584863	+	0.811132i	$0.301148\pi$
$12$	0	0
$13$	−64.6749	−1.37982	−0.689908	−	0.723897i	$-0.742349\pi$
$13$	−64.6749	−1.37982	−0.689908	+	0.723897i	$0.742349\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	53.3251	0.643874	0.321937	−	0.946761i	$-0.395666\pi$
$19$	53.3251	0.643874	0.321937	+	0.946761i	$0.395666\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	37.3499	0.338608	0.169304	−	0.985564i	$-0.445848\pi$
$23$	37.3499	0.338608	0.169304	+	0.985564i	$0.445848\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	294.000	1.88257	0.941283	−	0.337618i	$-0.109621\pi$
$29$	294.000	1.88257	0.941283	+	0.337618i	$0.109621\pi$
$30$	0	0
$31$	228.025	1.32111	0.660556	−	0.750777i	$-0.270321\pi$
$31$	228.025	1.32111	0.660556	+	0.750777i	$0.270321\pi$
$32$	0	0
$33$	128.025	0.675341
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−234.050	−1.03993	−0.519967	−	0.854187i	$-0.674056\pi$
$37$	−234.050	−1.03993	−0.519967	+	0.854187i	$0.674056\pi$
$38$	0	0
$39$	−194.025	−0.796637
$40$	0	0
$41$	12.6501	0.0481857	0.0240929	−	0.999710i	$-0.492330\pi$
$41$	12.6501	0.0481857	0.0240929	+	0.999710i	$0.492330\pi$
$42$	0	0
$43$	−433.400	−1.53704	−0.768521	−	0.639824i	$-0.779007\pi$
$43$	−433.400	−1.53704	−0.768521	+	0.639824i	$0.779007\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	96.0000	0.297937	0.148969	−	0.988842i	$-0.452405\pi$
$47$	96.0000	0.297937	0.148969	+	0.988842i	$0.452405\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−42.0000	−0.115317
$52$	0	0
$53$	607.375	1.57414	0.787069	−	0.616865i	$-0.211597\pi$
$53$	607.375	1.57414	0.787069	+	0.616865i	$0.211597\pi$
$54$	0	0
$55$	213.375	0.523117
$56$	0	0
$57$	159.975	0.371741
$58$	0	0
$59$	588.050	1.29759	0.648793	−	0.760965i	$-0.275274\pi$
$59$	588.050	1.29759	0.648793	+	0.760965i	$0.275274\pi$
$60$	0	0
$61$	758.099	1.59122	0.795612	−	0.605806i	$-0.207149\pi$
$61$	758.099	1.59122	0.795612	+	0.605806i	$0.207149\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−323.375	−0.617072
$66$	0	0
$67$	−329.400	−0.600635	−0.300318	−	0.953839i	$-0.597093\pi$
$67$	−329.400	−0.600635	−0.300318	+	0.953839i	$0.597093\pi$
$68$	0	0
$69$	112.050	0.195496
$70$	0	0
$71$	134.625	0.225029	0.112515	−	0.993650i	$-0.464109\pi$
$71$	134.625	0.225029	0.112515	+	0.993650i	$0.464109\pi$
$72$	0	0
$73$	384.774	0.616910	0.308455	−	0.951239i	$-0.400188\pi$
$73$	384.774	0.616910	0.308455	+	0.951239i	$0.400188\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−298.725	−0.442115
$78$	0	0
$79$	−21.3002	−0.0303349	−0.0151675	−	0.999885i	$-0.504828\pi$
$79$	−21.3002	−0.0303349	−0.0151675	+	0.999885i	$0.504828\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−468.099	−0.619043	−0.309522	−	0.950892i	$-0.600169\pi$
$83$	−468.099	−0.619043	−0.309522	+	0.950892i	$0.600169\pi$
$84$	0	0
$85$	−70.0000	−0.0893243
$86$	0	0
$87$	882.000	1.08690
$88$	0	0
$89$	−1104.70	−1.31571	−0.657854	−	0.753146i	$-0.728535\pi$
$89$	−1104.70	−1.31571	−0.657854	+	0.753146i	$0.728535\pi$
$90$	0	0
$91$	452.725	0.521521
$92$	0	0
$93$	684.075	0.762744
$94$	0	0
$95$	266.625	0.287949
$96$	0	0
$97$	240.625	0.251874	0.125937	−	0.992038i	$-0.459806\pi$
$97$	240.625	0.251874	0.125937	+	0.992038i	$0.459806\pi$
$98$	0	0
$99$	384.075	0.389909
$100$	0	0
$101$	1494.15	1.47201	0.736007	−	0.676974i	$-0.236709\pi$
$101$	1494.15	1.47201	0.736007	+	0.676974i	$0.236709\pi$
$102$	0	0
$103$	968.000	0.926018	0.463009	−	0.886354i	$-0.346770\pi$
$103$	968.000	0.926018	0.463009	+	0.886354i	$0.346770\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	116.099	0.104895	0.0524474	−	0.998624i	$-0.483298\pi$
$107$	116.099	0.104895	0.0524474	+	0.998624i	$0.483298\pi$
$108$	0	0
$109$	248.551	0.218411	0.109206	−	0.994019i	$-0.465169\pi$
$109$	248.551	0.218411	0.109206	+	0.994019i	$0.465169\pi$
$110$	0	0
$111$	−702.149	−0.600406
$112$	0	0
$113$	−1991.62	−1.65802	−0.829010	−	0.559235i	$-0.811095\pi$
$113$	−1991.62	−1.65802	−0.829010	+	0.559235i	$0.811095\pi$
$114$	0	0
$115$	186.749	0.151430
$116$	0	0
$117$	−582.075	−0.459939
$118$	0	0
$119$	98.0000	0.0754928
$120$	0	0
$121$	490.151	0.368258
$122$	0	0
$123$	37.9503	0.0278200
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	901.598	0.629952	0.314976	−	0.949100i	$-0.398004\pi$
$127$	901.598	0.629952	0.314976	+	0.949100i	$0.398004\pi$
$128$	0	0
$129$	−1300.20	−0.887412
$130$	0	0
$131$	−745.300	−0.497078	−0.248539	−	0.968622i	$-0.579950\pi$
$131$	−745.300	−0.497078	−0.248539	+	0.968622i	$0.579950\pi$
$132$	0	0
$133$	−373.275	−0.243361
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	2430.22	1.51553	0.757766	−	0.652526i	$-0.226291\pi$
$137$	2430.22	1.51553	0.757766	+	0.652526i	$0.226291\pi$
$138$	0	0
$139$	−424.174	−0.258834	−0.129417	−	0.991590i	$-0.541311\pi$
$139$	−424.174	−0.258834	−0.129417	+	0.991590i	$0.541311\pi$
$140$	0	0
$141$	288.000	0.172014
$142$	0	0
$143$	−2760.00	−1.61401
$144$	0	0
$145$	1470.00	0.841909
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−217.851	−0.119779	−0.0598894	−	0.998205i	$-0.519075\pi$
$149$	−217.851	−0.119779	−0.0598894	+	0.998205i	$0.519075\pi$
$150$	0	0
$151$	−229.350	−0.123604	−0.0618021	−	0.998088i	$-0.519685\pi$
$151$	−229.350	−0.123604	−0.0618021	+	0.998088i	$0.519685\pi$
$152$	0	0
$153$	−126.000	−0.0665784
$154$	0	0
$155$	1140.12	0.590819
$156$	0	0
$157$	1490.07	0.757458	0.378729	−	0.925508i	$-0.376361\pi$
$157$	1490.07	0.757458	0.378729	+	0.925508i	$0.376361\pi$
$158$	0	0
$159$	1822.12	0.908829
$160$	0	0
$161$	−261.449	−0.127982
$162$	0	0
$163$	838.700	0.403019	0.201509	−	0.979487i	$-0.435415\pi$
$163$	838.700	0.403019	0.201509	+	0.979487i	$0.435415\pi$
$164$	0	0
$165$	640.124	0.302022
$166$	0	0
$167$	3037.60	1.40752	0.703762	−	0.710436i	$-0.251502\pi$
$167$	3037.60	1.40752	0.703762	+	0.710436i	$0.251502\pi$
$168$	0	0
$169$	1985.85	0.903891
$170$	0	0
$171$	479.925	0.214625
$172$	0	0
$173$	3725.95	1.63745	0.818725	−	0.574186i	$-0.194681\pi$
$173$	3725.95	1.63745	0.818725	+	0.574186i	$0.194681\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	1764.15	0.749161
$178$	0	0
$179$	−1928.32	−0.805193	−0.402597	−	0.915377i	$-0.631892\pi$
$179$	−1928.32	−0.805193	−0.402597	+	0.915377i	$0.631892\pi$
$180$	0	0
$181$	0.451362	0.000185356 0	9.26782e−5	−	1.00000i	$-0.499970\pi$
$181$	0.451362	0.000185356 0	9.26782e−5		1.00000i	$0.499970\pi$
$182$	0	0
$183$	2274.30	0.918694
$184$	0	0
$185$	−1170.25	−0.465072
$186$	0	0
$187$	−597.449	−0.233635
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−1204.12	−0.456164	−0.228082	−	0.973642i	$-0.573245\pi$
$191$	−1204.12	−0.456164	−0.228082	+	0.973642i	$0.573245\pi$
$192$	0	0
$193$	3522.40	1.31372	0.656859	−	0.754013i	$-0.271885\pi$
$193$	3522.40	1.31372	0.656859	+	0.754013i	$0.271885\pi$
$194$	0	0
$195$	−970.124	−0.356267
$196$	0	0
$197$	1815.42	0.656567	0.328283	−	0.944579i	$-0.393530\pi$
$197$	1815.42	0.656567	0.328283	+	0.944579i	$0.393530\pi$
$198$	0	0
$199$	−5145.97	−1.83311	−0.916553	−	0.399913i	$-0.869040\pi$
$199$	−5145.97	−1.83311	−0.916553	+	0.399913i	$0.869040\pi$
$200$	0	0
$201$	−988.199	−0.346777
$202$	0	0
$203$	−2058.00	−0.711543
$204$	0	0
$205$	63.2505	0.0215493
$206$	0	0
$207$	336.149	0.112869
$208$	0	0
$209$	2275.64	0.753156
$210$	0	0
$211$	1894.50	0.618118	0.309059	−	0.951043i	$-0.399986\pi$
$211$	1894.50	0.618118	0.309059	+	0.951043i	$0.399986\pi$
$212$	0	0
$213$	403.876	0.129921
$214$	0	0
$215$	−2167.00	−0.687386
$216$	0	0
$217$	−1596.17	−0.499333
$218$	0	0
$219$	1154.32	0.356173
$220$	0	0
$221$	905.449	0.275598
$222$	0	0
$223$	−4285.35	−1.28685	−0.643427	−	0.765508i	$-0.722488\pi$
$223$	−4285.35	−1.28685	−0.643427	+	0.765508i	$0.722488\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−1582.50	−0.462706	−0.231353	−	0.972870i	$-0.574315\pi$
$227$	−1582.50	−0.462706	−0.231353	+	0.972870i	$0.574315\pi$
$228$	0	0
$229$	3107.50	0.896722	0.448361	−	0.893853i	$-0.352008\pi$
$229$	3107.50	0.896722	0.448361	+	0.893853i	$0.352008\pi$
$230$	0	0
$231$	−896.174	−0.255255
$232$	0	0
$233$	336.675	0.0946623	0.0473311	−	0.998879i	$-0.484928\pi$
$233$	336.675	0.0946623	0.0473311	+	0.998879i	$0.484928\pi$
$234$	0	0
$235$	480.000	0.133241
$236$	0	0
$237$	−63.9006	−0.0175139
$238$	0	0
$239$	−4545.57	−1.23025	−0.615123	−	0.788431i	$-0.710894\pi$
$239$	−4545.57	−1.23025	−0.615123	+	0.788431i	$0.710894\pi$
$240$	0	0
$241$	4314.05	1.15308	0.576540	−	0.817069i	$-0.304402\pi$
$241$	4314.05	1.15308	0.576540	+	0.817069i	$0.304402\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−3448.79	−0.888427
$248$	0	0
$249$	−1404.30	−0.357405
$250$	0	0
$251$	2204.75	0.554431	0.277216	−	0.960808i	$-0.410588\pi$
$251$	2204.75	0.554431	0.277216	+	0.960808i	$0.410588\pi$
$252$	0	0
$253$	1593.90	0.396079
$254$	0	0
$255$	−210.000	−0.0515714
$256$	0	0
$257$	−5835.95	−1.41648	−0.708242	−	0.705970i	$-0.750511\pi$
$257$	−5835.95	−1.41648	−0.708242	+	0.705970i	$0.750511\pi$
$258$	0	0
$259$	1638.35	0.393058
$260$	0	0
$261$	2646.00	0.627522
$262$	0	0
$263$	−7768.60	−1.82142	−0.910708	−	0.413052i	$-0.864463\pi$
$263$	−7768.60	−1.82142	−0.910708	+	0.413052i	$0.864463\pi$
$264$	0	0
$265$	3036.87	0.703976
$266$	0	0
$267$	−3314.10	−0.759624
$268$	0	0
$269$	−7987.04	−1.81033	−0.905165	−	0.425061i	$-0.860253\pi$
$269$	−7987.04	−1.81033	−0.905165	+	0.425061i	$0.860253\pi$
$270$	0	0
$271$	636.472	0.142668	0.0713338	−	0.997452i	$-0.477274\pi$
$271$	636.472	0.142668	0.0713338	+	0.997452i	$0.477274\pi$
$272$	0	0
$273$	1358.17	0.301100
$274$	0	0
$275$	1066.87	0.233945
$276$	0	0
$277$	3621.95	0.785639	0.392819	−	0.919616i	$-0.371500\pi$
$277$	3621.95	0.785639	0.392819	+	0.919616i	$0.371500\pi$
$278$	0	0
$279$	2052.22	0.440371
$280$	0	0
$281$	−173.553	−0.0368445	−0.0184222	−	0.999830i	$-0.505864\pi$
$281$	−173.553	−0.0368445	−0.0184222	+	0.999830i	$0.505864\pi$
$282$	0	0
$283$	3620.89	0.760565	0.380282	−	0.924870i	$-0.375827\pi$
$283$	3620.89	0.760565	0.380282	+	0.924870i	$0.375827\pi$
$284$	0	0
$285$	799.876	0.166248
$286$	0	0
$287$	−88.5507	−0.0182125
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	721.876	0.145420
$292$	0	0
$293$	−1444.75	−0.288065	−0.144033	−	0.989573i	$-0.546007\pi$
$293$	−1444.75	−0.288065	−0.144033	+	0.989573i	$0.546007\pi$
$294$	0	0
$295$	2940.25	0.580298
$296$	0	0
$297$	1152.22	0.225114
$298$	0	0
$299$	−2415.60	−0.467217
$300$	0	0
$301$	3033.80	0.580947
$302$	0	0
$303$	4482.45	0.849868
$304$	0	0
$305$	3790.50	0.711617
$306$	0	0
$307$	−10316.8	−1.91796	−0.958980	−	0.283475i	$-0.908513\pi$
$307$	−10316.8	−1.91796	−0.958980	+	0.283475i	$0.908513\pi$
$308$	0	0
$309$	2904.00	0.534637
$310$	0	0
$311$	−4878.00	−0.889408	−0.444704	−	0.895678i	$-0.646691\pi$
$311$	−4878.00	−0.889408	−0.444704	+	0.895678i	$0.646691\pi$
$312$	0	0
$313$	−3596.48	−0.649472	−0.324736	−	0.945805i	$-0.605276\pi$
$313$	−3596.48	−0.649472	−0.324736	+	0.945805i	$0.605276\pi$
$314$	0	0
$315$	−315.000	−0.0563436
$316$	0	0
$317$	−8995.72	−1.59385	−0.796924	−	0.604079i	$-0.793541\pi$
$317$	−8995.72	−1.59385	−0.796924	+	0.604079i	$0.793541\pi$
$318$	0	0
$319$	12546.4	2.20209
$320$	0	0
$321$	348.298	0.0605611
$322$	0	0
$323$	−746.551	−0.128604
$324$	0	0
$325$	−1616.87	−0.275963
$326$	0	0
$327$	745.652	0.126100
$328$	0	0
$329$	−672.000	−0.112610
$330$	0	0
$331$	2746.10	0.456009	0.228005	−	0.973660i	$-0.426780\pi$
$331$	2746.10	0.456009	0.228005	+	0.973660i	$0.426780\pi$
$332$	0	0
$333$	−2106.45	−0.346644
$334$	0	0
$335$	−1647.00	−0.268612
$336$	0	0
$337$	12083.2	1.95316	0.976578	−	0.215165i	$-0.0690288\pi$
$337$	12083.2	1.95316	0.976578	+	0.215165i	$0.0690288\pi$
$338$	0	0
$339$	−5974.87	−0.957258
$340$	0	0
$341$	9730.95	1.54534
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	560.248	0.0874283
$346$	0	0
$347$	−10143.0	−1.56919	−0.784593	−	0.620012i	$-0.787128\pi$
$347$	−10143.0	−1.56919	−0.784593	+	0.620012i	$0.787128\pi$
$348$	0	0
$349$	−1903.80	−0.292000	−0.146000	−	0.989285i	$-0.546640\pi$
$349$	−1903.80	−0.292000	−0.146000	+	0.989285i	$0.546640\pi$
$350$	0	0
$351$	−1746.22	−0.265546
$352$	0	0
$353$	3042.79	0.458787	0.229393	−	0.973334i	$-0.426326\pi$
$353$	3042.79	0.458787	0.229393	+	0.973334i	$0.426326\pi$
$354$	0	0
$355$	673.126	0.100636
$356$	0	0
$357$	294.000	0.0435858
$358$	0	0
$359$	−4407.62	−0.647981	−0.323991	−	0.946060i	$-0.605025\pi$
$359$	−4407.62	−0.647981	−0.323991	+	0.946060i	$0.605025\pi$
$360$	0	0
$361$	−4015.44	−0.585426
$362$	0	0
$363$	1470.45	0.212614
$364$	0	0
$365$	1923.87	0.275891
$366$	0	0
$367$	−7853.35	−1.11701	−0.558503	−	0.829502i	$-0.688624\pi$
$367$	−7853.35	−1.11701	−0.558503	+	0.829502i	$0.688624\pi$
$368$	0	0
$369$	113.851	0.0160619
$370$	0	0
$371$	−4251.62	−0.594968
$372$	0	0
$373$	−12600.3	−1.74911	−0.874556	−	0.484925i	$-0.838847\pi$
$373$	−12600.3	−1.74911	−0.874556	+	0.484925i	$0.838847\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−19014.4	−2.59759
$378$	0	0
$379$	−12191.9	−1.65240	−0.826198	−	0.563380i	$-0.809501\pi$
$379$	−12191.9	−1.65240	−0.826198	+	0.563380i	$0.809501\pi$
$380$	0	0
$381$	2704.79	0.363703
$382$	0	0
$383$	10778.8	1.43804	0.719022	−	0.694987i	$-0.244590\pi$
$383$	10778.8	1.43804	0.719022	+	0.694987i	$0.244590\pi$
$384$	0	0
$385$	−1493.62	−0.197720
$386$	0	0
$387$	−3900.60	−0.512347
$388$	0	0
$389$	−164.153	−0.0213956	−0.0106978	−	0.999943i	$-0.503405\pi$
$389$	−164.153	−0.0213956	−0.0106978	+	0.999943i	$0.503405\pi$
$390$	0	0
$391$	−522.899	−0.0676320
$392$	0	0
$393$	−2235.90	−0.286988
$394$	0	0
$395$	−106.501	−0.0135662
$396$	0	0
$397$	14960.8	1.89134	0.945670	−	0.325129i	$-0.105408\pi$
$397$	14960.8	1.89134	0.945670	+	0.325129i	$0.105408\pi$
$398$	0	0
$399$	−1119.83	−0.140505
$400$	0	0
$401$	2023.80	0.252029	0.126014	−	0.992028i	$-0.459781\pi$
$401$	2023.80	0.252029	0.126014	+	0.992028i	$0.459781\pi$
$402$	0	0
$403$	−14747.5	−1.82289
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−9988.06	−1.21644
$408$	0	0
$409$	−8059.10	−0.974320	−0.487160	−	0.873313i	$-0.661967\pi$
$409$	−8059.10	−0.974320	−0.487160	+	0.873313i	$0.661967\pi$
$410$	0	0
$411$	7290.67	0.874993
$412$	0	0
$413$	−4116.35	−0.490441
$414$	0	0
$415$	−2340.50	−0.276845
$416$	0	0
$417$	−1272.52	−0.149438
$418$	0	0
$419$	977.996	0.114029	0.0570146	−	0.998373i	$-0.481842\pi$
$419$	977.996	0.114029	0.0570146	+	0.998373i	$0.481842\pi$
$420$	0	0
$421$	−663.797	−0.0768444	−0.0384222	−	0.999262i	$-0.512233\pi$
$421$	−663.797	−0.0768444	−0.0384222	+	0.999262i	$0.512233\pi$
$422$	0	0
$423$	864.000	0.0993123
$424$	0	0
$425$	−350.000	−0.0399470
$426$	0	0
$427$	−5306.70	−0.601426
$428$	0	0
$429$	−8280.00	−0.931847
$430$	0	0
$431$	5130.02	0.573328	0.286664	−	0.958031i	$-0.407454\pi$
$431$	5130.02	0.573328	0.286664	+	0.958031i	$0.407454\pi$
$432$	0	0
$433$	12236.2	1.35805	0.679024	−	0.734116i	$-0.262403\pi$
$433$	12236.2	1.35805	0.679024	+	0.734116i	$0.262403\pi$
$434$	0	0
$435$	4410.00	0.486077
$436$	0	0
$437$	1991.69	0.218021
$438$	0	0
$439$	7296.63	0.793278	0.396639	−	0.917975i	$-0.370176\pi$
$439$	7296.63	0.793278	0.396639	+	0.917975i	$0.370176\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−1810.86	−0.194213	−0.0971066	−	0.995274i	$-0.530959\pi$
$443$	−1810.86	−0.194213	−0.0971066	+	0.995274i	$0.530959\pi$
$444$	0	0
$445$	−5523.50	−0.588402
$446$	0	0
$447$	−653.553	−0.0691543
$448$	0	0
$449$	−15610.5	−1.64077	−0.820384	−	0.571814i	$-0.806240\pi$
$449$	−15610.5	−1.64077	−0.820384	+	0.571814i	$0.806240\pi$
$450$	0	0
$451$	539.843	0.0563641
$452$	0	0
$453$	−688.050	−0.0713629
$454$	0	0
$455$	2263.62	0.233231
$456$	0	0
$457$	2093.84	0.214323	0.107162	−	0.994242i	$-0.465824\pi$
$457$	2093.84	0.214323	0.107162	+	0.994242i	$0.465824\pi$
$458$	0	0
$459$	−378.000	−0.0384391
$460$	0	0
$461$	−10575.9	−1.06848	−0.534239	−	0.845333i	$-0.679402\pi$
$461$	−10575.9	−1.06848	−0.534239	+	0.845333i	$0.679402\pi$
$462$	0	0
$463$	−15367.1	−1.54248	−0.771241	−	0.636543i	$-0.780364\pi$
$463$	−15367.1	−1.54248	−0.771241	+	0.636543i	$0.780364\pi$
$464$	0	0
$465$	3420.37	0.341110
$466$	0	0
$467$	4263.00	0.422415	0.211208	−	0.977441i	$-0.432260\pi$
$467$	4263.00	0.422415	0.211208	+	0.977441i	$0.432260\pi$
$468$	0	0
$469$	2305.80	0.227019
$470$	0	0
$471$	4470.22	0.437319
$472$	0	0
$473$	−18495.3	−1.79792
$474$	0	0
$475$	1333.13	0.128775
$476$	0	0
$477$	5466.37	0.524713
$478$	0	0
$479$	11270.5	1.07508	0.537539	−	0.843239i	$-0.319354\pi$
$479$	11270.5	1.07508	0.537539	+	0.843239i	$0.319354\pi$
$480$	0	0
$481$	15137.2	1.43492
$482$	0	0
$483$	−784.348	−0.0738904
$484$	0	0
$485$	1203.13	0.112642
$486$	0	0
$487$	−667.909	−0.0621475	−0.0310738	−	0.999517i	$-0.509893\pi$
$487$	−667.909	−0.0621475	−0.0310738	+	0.999517i	$0.509893\pi$
$488$	0	0
$489$	2516.10	0.232683
$490$	0	0
$491$	−4715.92	−0.433455	−0.216727	−	0.976232i	$-0.569538\pi$
$491$	−4715.92	−0.433455	−0.216727	+	0.976232i	$0.569538\pi$
$492$	0	0
$493$	−4116.00	−0.376015
$494$	0	0
$495$	1920.37	0.174372
$496$	0	0
$497$	−942.377	−0.0850531
$498$	0	0
$499$	−11138.9	−0.999293	−0.499647	−	0.866229i	$-0.666537\pi$
$499$	−11138.9	−0.999293	−0.499647	+	0.866229i	$0.666537\pi$
$500$	0	0
$501$	9112.79	0.812634
$502$	0	0
$503$	6345.09	0.562453	0.281226	−	0.959641i	$-0.409259\pi$
$503$	6345.09	0.562453	0.281226	+	0.959641i	$0.409259\pi$
$504$	0	0
$505$	7470.75	0.658305
$506$	0	0
$507$	5957.55	0.521862
$508$	0	0
$509$	−18247.5	−1.58901	−0.794507	−	0.607255i	$-0.792271\pi$
$509$	−18247.5	−1.58901	−0.794507	+	0.607255i	$0.792271\pi$
$510$	0	0
$511$	−2693.42	−0.233170
$512$	0	0
$513$	1439.78	0.123914
$514$	0	0
$515$	4840.00	0.414128
$516$	0	0
$517$	4096.79	0.348505
$518$	0	0
$519$	11177.9	0.945382
$520$	0	0
$521$	−13328.2	−1.12077	−0.560384	−	0.828233i	$-0.689347\pi$
$521$	−13328.2	−1.12077	−0.560384	+	0.828233i	$0.689347\pi$
$522$	0	0
$523$	−1688.65	−0.141185	−0.0705924	−	0.997505i	$-0.522489\pi$
$523$	−1688.65	−0.141185	−0.0705924	+	0.997505i	$0.522489\pi$
$524$	0	0
$525$	−525.000	−0.0436436
$526$	0	0
$527$	−3192.35	−0.263873
$528$	0	0
$529$	−10772.0	−0.885344
$530$	0	0
$531$	5292.45	0.432529
$532$	0	0
$533$	−818.145	−0.0664874
$534$	0	0
$535$	580.497	0.0469104
$536$	0	0
$537$	−5784.97	−0.464879
$538$	0	0
$539$	2091.07	0.167104
$540$	0	0
$541$	5579.50	0.443404	0.221702	−	0.975115i	$-0.428839\pi$
$541$	5579.50	0.443404	0.221702	+	0.975115i	$0.428839\pi$
$542$	0	0
$543$	1.35409	0.000107016 0
$544$	0	0
$545$	1242.75	0.0976765
$546$	0	0
$547$	−7416.70	−0.579736	−0.289868	−	0.957067i	$-0.593611\pi$
$547$	−7416.70	−0.579736	−0.289868	+	0.957067i	$0.593611\pi$
$548$	0	0
$549$	6822.89	0.530408
$550$	0	0
$551$	15677.6	1.21214
$552$	0	0
$553$	149.101	0.0114655
$554$	0	0
$555$	−3510.75	−0.268510
$556$	0	0
$557$	−11055.2	−0.840978	−0.420489	−	0.907298i	$-0.638141\pi$
$557$	−11055.2	−0.840978	−0.420489	+	0.907298i	$0.638141\pi$
$558$	0	0
$559$	28030.1	2.12083
$560$	0	0
$561$	−1792.35	−0.134889
$562$	0	0
$563$	−9941.09	−0.744169	−0.372084	−	0.928199i	$-0.621357\pi$
$563$	−9941.09	−0.744169	−0.372084	+	0.928199i	$0.621357\pi$
$564$	0	0
$565$	−9958.12	−0.741489
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	8321.98	0.613138	0.306569	−	0.951848i	$-0.400819\pi$
$569$	8321.98	0.613138	0.306569	+	0.951848i	$0.400819\pi$
$570$	0	0
$571$	9419.06	0.690324	0.345162	−	0.938543i	$-0.387824\pi$
$571$	9419.06	0.690324	0.345162	+	0.938543i	$0.387824\pi$
$572$	0	0
$573$	−3612.37	−0.263367
$574$	0	0
$575$	933.747	0.0677217
$576$	0	0
$577$	−23276.3	−1.67938	−0.839691	−	0.543064i	$-0.817264\pi$
$577$	−23276.3	−1.67938	−0.839691	+	0.543064i	$0.817264\pi$
$578$	0	0
$579$	10567.2	0.758476
$580$	0	0
$581$	3276.70	0.233976
$582$	0	0
$583$	25919.7	1.84131
$584$	0	0
$585$	−2910.37	−0.205691
$586$	0	0
$587$	−13074.8	−0.919343	−0.459672	−	0.888089i	$-0.652033\pi$
$587$	−13074.8	−0.919343	−0.459672	+	0.888089i	$0.652033\pi$
$588$	0	0
$589$	12159.4	0.850630
$590$	0	0
$591$	5446.27	0.379069
$592$	0	0
$593$	12548.1	0.868951	0.434476	−	0.900684i	$-0.356934\pi$
$593$	12548.1	0.868951	0.434476	+	0.900684i	$0.356934\pi$
$594$	0	0
$595$	490.000	0.0337614
$596$	0	0
$597$	−15437.9	−1.05834
$598$	0	0
$599$	−8168.02	−0.557155	−0.278578	−	0.960414i	$-0.589863\pi$
$599$	−8168.02	−0.557155	−0.278578	+	0.960414i	$0.589863\pi$
$600$	0	0
$601$	8870.85	0.602079	0.301040	−	0.953612i	$-0.402666\pi$
$601$	8870.85	0.602079	0.301040	+	0.953612i	$0.402666\pi$
$602$	0	0
$603$	−2964.60	−0.200212
$604$	0	0
$605$	2450.76	0.164690
$606$	0	0
$607$	−21269.9	−1.42227	−0.711137	−	0.703054i	$-0.751819\pi$
$607$	−21269.9	−1.42227	−0.711137	+	0.703054i	$0.751819\pi$
$608$	0	0
$609$	−6174.00	−0.410810
$610$	0	0
$611$	−6208.79	−0.411098
$612$	0	0
$613$	19271.0	1.26974	0.634870	−	0.772619i	$-0.281054\pi$
$613$	19271.0	1.26974	0.634870	+	0.772619i	$0.281054\pi$
$614$	0	0
$615$	189.752	0.0124415
$616$	0	0
$617$	5430.37	0.354325	0.177163	−	0.984182i	$-0.443308\pi$
$617$	5430.37	0.354325	0.177163	+	0.984182i	$0.443308\pi$
$618$	0	0
$619$	5118.91	0.332385	0.166193	−	0.986093i	$-0.446853\pi$
$619$	5118.91	0.332385	0.166193	+	0.986093i	$0.446853\pi$
$620$	0	0
$621$	1008.45	0.0651652
$622$	0	0
$623$	7732.90	0.497291
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	6826.93	0.434835
$628$	0	0
$629$	3276.70	0.207711
$630$	0	0
$631$	30035.6	1.89493	0.947463	−	0.319866i	$-0.103638\pi$
$631$	30035.6	1.89493	0.947463	+	0.319866i	$0.103638\pi$
$632$	0	0
$633$	5683.50	0.356870
$634$	0	0
$635$	4507.99	0.281723
$636$	0	0
$637$	−3169.07	−0.197117
$638$	0	0
$639$	1211.63	0.0750098
$640$	0	0
$641$	−14304.4	−0.881416	−0.440708	−	0.897650i	$-0.645273\pi$
$641$	−14304.4	−0.881416	−0.440708	+	0.897650i	$0.645273\pi$
$642$	0	0
$643$	−9179.06	−0.562965	−0.281483	−	0.959566i	$-0.590826\pi$
$643$	−9179.06	−0.562965	−0.281483	+	0.959566i	$0.590826\pi$
$644$	0	0
$645$	−6500.99	−0.396863
$646$	0	0
$647$	8125.60	0.493741	0.246870	−	0.969049i	$-0.420598\pi$
$647$	8125.60	0.493741	0.246870	+	0.969049i	$0.420598\pi$
$648$	0	0
$649$	25095.0	1.51782
$650$	0	0
$651$	−4788.52	−0.288290
$652$	0	0
$653$	11530.1	0.690978	0.345489	−	0.938423i	$-0.387713\pi$
$653$	11530.1	0.690978	0.345489	+	0.938423i	$0.387713\pi$
$654$	0	0
$655$	−3726.50	−0.222300
$656$	0	0
$657$	3462.97	0.205637
$658$	0	0
$659$	8613.62	0.509164	0.254582	−	0.967051i	$-0.418062\pi$
$659$	8613.62	0.509164	0.254582	+	0.967051i	$0.418062\pi$
$660$	0	0
$661$	30559.5	1.79822	0.899112	−	0.437718i	$-0.144213\pi$
$661$	30559.5	1.79822	0.899112	+	0.437718i	$0.144213\pi$
$662$	0	0
$663$	2716.35	0.159116
$664$	0	0
$665$	−1866.38	−0.108835
$666$	0	0
$667$	10980.9	0.637453
$668$	0	0
$669$	−12856.0	−0.742965
$670$	0	0
$671$	32351.9	1.86130
$672$	0	0
$673$	20623.0	1.18121	0.590607	−	0.806960i	$-0.298889\pi$
$673$	20623.0	1.18121	0.590607	+	0.806960i	$0.298889\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	17307.6	0.982551	0.491275	−	0.871004i	$-0.336531\pi$
$677$	17307.6	0.982551	0.491275	+	0.871004i	$0.336531\pi$
$678$	0	0
$679$	−1684.38	−0.0951995
$680$	0	0
$681$	−4747.50	−0.267143
$682$	0	0
$683$	−4549.64	−0.254886	−0.127443	−	0.991846i	$-0.540677\pi$
$683$	−4549.64	−0.254886	−0.127443	+	0.991846i	$0.540677\pi$
$684$	0	0
$685$	12151.1	0.677767
$686$	0	0
$687$	9322.50	0.517723
$688$	0	0
$689$	−39281.9	−2.17202
$690$	0	0
$691$	13761.7	0.757625	0.378813	−	0.925473i	$-0.376332\pi$
$691$	13761.7	0.757625	0.378813	+	0.925473i	$0.376332\pi$
$692$	0	0
$693$	−2688.52	−0.147372
$694$	0	0
$695$	−2120.87	−0.115754
$696$	0	0
$697$	−177.101	−0.00962439
$698$	0	0
$699$	1010.02	0.0546533
$700$	0	0
$701$	32136.0	1.73147	0.865734	−	0.500504i	$-0.166852\pi$
$701$	32136.0	1.73147	0.865734	+	0.500504i	$0.166852\pi$
$702$	0	0
$703$	−12480.7	−0.669586
$704$	0	0
$705$	1440.00	0.0769270
$706$	0	0
$707$	−10459.0	−0.556369
$708$	0	0
$709$	−14508.4	−0.768511	−0.384256	−	0.923227i	$-0.625542\pi$
$709$	−14508.4	−0.768511	−0.384256	+	0.923227i	$0.625542\pi$
$710$	0	0
$711$	−191.702	−0.0101116
$712$	0	0
$713$	8516.70	0.447340
$714$	0	0
$715$	−13800.0	−0.721805
$716$	0	0
$717$	−13636.7	−0.710283
$718$	0	0
$719$	27588.2	1.43097	0.715485	−	0.698628i	$-0.246206\pi$
$719$	27588.2	1.43097	0.715485	+	0.698628i	$0.246206\pi$
$720$	0	0
$721$	−6776.00	−0.350002
$722$	0	0
$723$	12942.1	0.665731
$724$	0	0
$725$	7350.00	0.376513
$726$	0	0
$727$	15720.5	0.801984	0.400992	−	0.916082i	$-0.368666\pi$
$727$	15720.5	0.801984	0.400992	+	0.916082i	$0.368666\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	6067.59	0.307001
$732$	0	0
$733$	1204.03	0.0606710	0.0303355	−	0.999540i	$-0.490342\pi$
$733$	1204.03	0.0606710	0.0303355	+	0.999540i	$0.490342\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	−14057.1	−0.702578
$738$	0	0
$739$	−34173.9	−1.70109	−0.850547	−	0.525899i	$-0.823729\pi$
$739$	−34173.9	−1.70109	−0.850547	+	0.525899i	$0.823729\pi$
$740$	0	0
$741$	−10346.4	−0.512934
$742$	0	0
$743$	27756.0	1.37048	0.685241	−	0.728317i	$-0.259697\pi$
$743$	27756.0	1.37048	0.685241	+	0.728317i	$0.259697\pi$
$744$	0	0
$745$	−1089.25	−0.0535667
$746$	0	0
$747$	−4212.89	−0.206348
$748$	0	0
$749$	−812.696	−0.0396465
$750$	0	0
$751$	−6453.50	−0.313571	−0.156785	−	0.987633i	$-0.550113\pi$
$751$	−6453.50	−0.313571	−0.156785	+	0.987633i	$0.550113\pi$
$752$	0	0
$753$	6614.24	0.320101
$754$	0	0
$755$	−1146.75	−0.0552775
$756$	0	0
$757$	−483.888	−0.0232328	−0.0116164	−	0.999933i	$-0.503698\pi$
$757$	−483.888	−0.0232328	−0.0116164	+	0.999933i	$0.503698\pi$
$758$	0	0
$759$	4781.71	0.228676
$760$	0	0
$761$	24641.6	1.17379	0.586897	−	0.809662i	$-0.300349\pi$
$761$	24641.6	1.17379	0.586897	+	0.809662i	$0.300349\pi$
$762$	0	0
$763$	−1739.86	−0.0825517
$764$	0	0
$765$	−630.000	−0.0297748
$766$	0	0
$767$	−38032.1	−1.79043
$768$	0	0
$769$	7129.75	0.334337	0.167169	−	0.985928i	$-0.446538\pi$
$769$	7129.75	0.334337	0.167169	+	0.985928i	$0.446538\pi$
$770$	0	0
$771$	−17507.8	−0.817807
$772$	0	0
$773$	−14834.7	−0.690253	−0.345127	−	0.938556i	$-0.612164\pi$
$773$	−14834.7	−0.690253	−0.345127	+	0.938556i	$0.612164\pi$
$774$	0	0
$775$	5700.62	0.264222
$776$	0	0
$777$	4915.04	0.226932
$778$	0	0
$779$	674.568	0.0310255
$780$	0	0
$781$	5745.13	0.263222
$782$	0	0
$783$	7938.00	0.362300
$784$	0	0
$785$	7450.37	0.338745
$786$	0	0
$787$	−37440.1	−1.69580	−0.847901	−	0.530154i	$-0.822134\pi$
$787$	−37440.1	−1.69580	−0.847901	+	0.530154i	$0.822134\pi$
$788$	0	0
$789$	−23305.8	−1.05159
$790$	0	0
$791$	13941.4	0.626672
$792$	0	0
$793$	−49030.0	−2.19560
$794$	0	0
$795$	9110.62	0.406441
$796$	0	0
$797$	−10964.3	−0.487298	−0.243649	−	0.969863i	$-0.578345\pi$
$797$	−10964.3	−0.487298	−0.243649	+	0.969863i	$0.578345\pi$
$798$	0	0
$799$	−1344.00	−0.0595085
$800$	0	0
$801$	−9942.30	−0.438569
$802$	0	0
$803$	16420.2	0.721615
$804$	0	0
$805$	−1307.25	−0.0572353
$806$	0	0
$807$	−23961.1	−1.04519
$808$	0	0
$809$	15621.2	0.678879	0.339440	−	0.940628i	$-0.389763\pi$
$809$	15621.2	0.678879	0.339440	+	0.940628i	$0.389763\pi$
$810$	0	0
$811$	30591.5	1.32455	0.662277	−	0.749259i	$-0.269590\pi$
$811$	30591.5	1.32455	0.662277	+	0.749259i	$0.269590\pi$
$812$	0	0
$813$	1909.42	0.0823692
$814$	0	0
$815$	4193.50	0.180235
$816$	0	0
$817$	−23111.1	−0.989662
$818$	0	0
$819$	4074.52	0.173840
$820$	0	0
$821$	−39126.0	−1.66322	−0.831611	−	0.555358i	$-0.812581\pi$
$821$	−39126.0	−1.66322	−0.831611	+	0.555358i	$0.812581\pi$
$822$	0	0
$823$	−24681.8	−1.04539	−0.522693	−	0.852521i	$-0.675073\pi$
$823$	−24681.8	−1.04539	−0.522693	+	0.852521i	$0.675073\pi$
$824$	0	0
$825$	3200.62	0.135068
$826$	0	0
$827$	−21801.3	−0.916693	−0.458346	−	0.888774i	$-0.651558\pi$
$827$	−21801.3	−0.916693	−0.458346	+	0.888774i	$0.651558\pi$
$828$	0	0
$829$	−7571.94	−0.317231	−0.158615	−	0.987340i	$-0.550703\pi$
$829$	−7571.94	−0.317231	−0.158615	+	0.987340i	$0.550703\pi$
$830$	0	0
$831$	10865.9	0.453589
$832$	0	0
$833$	−686.000	−0.0285336
$834$	0	0
$835$	15188.0	0.629464
$836$	0	0
$837$	6156.67	0.254248
$838$	0	0
$839$	15347.3	0.631521	0.315761	−	0.948839i	$-0.397740\pi$
$839$	15347.3	0.631521	0.315761	+	0.948839i	$0.397740\pi$
$840$	0	0
$841$	62047.0	2.54406
$842$	0	0
$843$	−520.658	−0.0212722
$844$	0	0
$845$	9929.24	0.404232
$846$	0	0
$847$	−3431.06	−0.139188
$848$	0	0
$849$	10862.7	0.439112
$850$	0	0
$851$	−8741.73	−0.352130
$852$	0	0
$853$	−37886.3	−1.52075	−0.760376	−	0.649483i	$-0.774985\pi$
$853$	−37886.3	−1.52075	−0.760376	+	0.649483i	$0.774985\pi$
$854$	0	0
$855$	2399.63	0.0959831
$856$	0	0
$857$	5381.53	0.214503	0.107252	−	0.994232i	$-0.465795\pi$
$857$	5381.53	0.214503	0.107252	+	0.994232i	$0.465795\pi$
$858$	0	0
$859$	29448.1	1.16968	0.584840	−	0.811149i	$-0.301157\pi$
$859$	29448.1	1.16968	0.584840	+	0.811149i	$0.301157\pi$
$860$	0	0
$861$	−265.652	−0.0105150
$862$	0	0
$863$	27156.1	1.07115	0.535577	−	0.844486i	$-0.320094\pi$
$863$	27156.1	1.07115	0.535577	+	0.844486i	$0.320094\pi$
$864$	0	0
$865$	18629.8	0.732290
$866$	0	0
$867$	−14151.0	−0.554317
$868$	0	0
$869$	−908.985	−0.0354836
$870$	0	0
$871$	21303.9	0.828766
$872$	0	0
$873$	2165.63	0.0839581
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	8688.67	0.334544	0.167272	−	0.985911i	$-0.446504\pi$
$877$	8688.67	0.334544	0.167272	+	0.985911i	$0.446504\pi$
$878$	0	0
$879$	−4334.25	−0.166315
$880$	0	0
$881$	30771.8	1.17676	0.588381	−	0.808584i	$-0.299765\pi$
$881$	30771.8	1.17676	0.588381	+	0.808584i	$0.299765\pi$
$882$	0	0
$883$	43804.5	1.66946	0.834732	−	0.550656i	$-0.185622\pi$
$883$	43804.5	1.66946	0.834732	+	0.550656i	$0.185622\pi$
$884$	0	0
$885$	8820.75	0.335035
$886$	0	0
$887$	40119.7	1.51870	0.759350	−	0.650683i	$-0.225517\pi$
$887$	40119.7	1.51870	0.759350	+	0.650683i	$0.225517\pi$
$888$	0	0
$889$	−6311.19	−0.238100
$890$	0	0
$891$	3456.67	0.129970
$892$	0	0
$893$	5119.21	0.191834
$894$	0	0
$895$	−9641.61	−0.360093
$896$	0	0
$897$	−7246.81	−0.269748
$898$	0	0
$899$	67039.3	2.48708
$900$	0	0
$901$	−8503.25	−0.314411
$902$	0	0
$903$	9101.39	0.335410
$904$	0	0
$905$	2.25681	8.28939e−5 0
$906$	0	0
$907$	−24417.7	−0.893910	−0.446955	−	0.894556i	$-0.647492\pi$
$907$	−24417.7	−0.893910	−0.446955	+	0.894556i	$0.647492\pi$
$908$	0	0
$909$	13447.3	0.490671
$910$	0	0
$911$	9147.15	0.332666	0.166333	−	0.986070i	$-0.446807\pi$
$911$	9147.15	0.332666	0.166333	+	0.986070i	$0.446807\pi$
$912$	0	0
$913$	−19976.1	−0.724111
$914$	0	0
$915$	11371.5	0.410852
$916$	0	0
$917$	5217.10	0.187878
$918$	0	0
$919$	−49339.8	−1.77102	−0.885511	−	0.464619i	$-0.846191\pi$
$919$	−49339.8	−1.77102	−0.885511	+	0.464619i	$0.846191\pi$
$920$	0	0
$921$	−30950.5	−1.10733
$922$	0	0
$923$	−8706.88	−0.310499
$924$	0	0
$925$	−5851.24	−0.207987
$926$	0	0
$927$	8712.00	0.308673
$928$	0	0
$929$	2526.82	0.0892382	0.0446191	−	0.999004i	$-0.485793\pi$
$929$	2526.82	0.0892382	0.0446191	+	0.999004i	$0.485793\pi$
$930$	0	0
$931$	2612.93	0.0919820
$932$	0	0
$933$	−14634.0	−0.513500
$934$	0	0
$935$	−2987.25	−0.104485
$936$	0	0
$937$	47067.7	1.64102	0.820510	−	0.571632i	$-0.193689\pi$
$937$	47067.7	1.64102	0.820510	+	0.571632i	$0.193689\pi$
$938$	0	0
$939$	−10789.4	−0.374973
$940$	0	0
$941$	−41999.4	−1.45498	−0.727492	−	0.686116i	$-0.759314\pi$
$941$	−41999.4	−1.45498	−0.727492	+	0.686116i	$0.759314\pi$
$942$	0	0
$943$	472.480	0.0163161
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	−26732.6	−0.917312	−0.458656	−	0.888614i	$-0.651669\pi$
$947$	−26732.6	−0.917312	−0.458656	+	0.888614i	$0.651669\pi$
$948$	0	0
$949$	−24885.3	−0.851222
$950$	0	0
$951$	−26987.2	−0.920209
$952$	0	0
$953$	−19555.5	−0.664706	−0.332353	−	0.943155i	$-0.607843\pi$
$953$	−19555.5	−0.664706	−0.332353	+	0.943155i	$0.607843\pi$
$954$	0	0
$955$	−6020.62	−0.204003
$956$	0	0
$957$	37639.3	1.27138
$958$	0	0
$959$	−17011.6	−0.572818
$960$	0	0
$961$	22204.3	0.745337
$962$	0	0
$963$	1044.89	0.0349650
$964$	0	0
$965$	17612.0	0.587513
$966$	0	0
$967$	38903.5	1.29375	0.646873	−	0.762598i	$-0.276076\pi$
$967$	38903.5	1.29375	0.646873	+	0.762598i	$0.276076\pi$
$968$	0	0
$969$	−2239.65	−0.0742497
$970$	0	0
$971$	−796.132	−0.0263122	−0.0131561	−	0.999913i	$-0.504188\pi$
$971$	−796.132	−0.0263122	−0.0131561	+	0.999913i	$0.504188\pi$
$972$	0	0
$973$	2969.22	0.0978302
$974$	0	0
$975$	−4850.62	−0.159327
$976$	0	0
$977$	6186.46	0.202582	0.101291	−	0.994857i	$-0.467703\pi$
$977$	6186.46	0.202582	0.101291	+	0.994857i	$0.467703\pi$
$978$	0	0
$979$	−47143.0	−1.53902
$980$	0	0
$981$	2236.96	0.0728038
$982$	0	0
$983$	−3542.18	−0.114932	−0.0574659	−	0.998347i	$-0.518302\pi$
$983$	−3542.18	−0.114932	−0.0574659	+	0.998347i	$0.518302\pi$
$984$	0	0
$985$	9077.12	0.293626
$986$	0	0
$987$	−2016.00	−0.0650152
$988$	0	0
$989$	−16187.4	−0.520455
$990$	0	0
$991$	−21923.2	−0.702737	−0.351369	−	0.936237i	$-0.614284\pi$
$991$	−21923.2	−0.702737	−0.351369	+	0.936237i	$0.614284\pi$
$992$	0	0
$993$	8238.29	0.263277
$994$	0	0
$995$	−25729.9	−0.819790
$996$	0	0
$997$	−126.571	−0.00402062	−0.00201031	−	0.999998i	$-0.500640\pi$
$997$	−126.571	−0.00402062	−0.00201031	+	0.999998i	$0.500640\pi$
$998$	0	0
$999$	−6319.34	−0.200135

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.4.a.o.1.2	✓	2
4.3	odd	2		1680.4.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.4.a.o.1.2	✓	2	1.1	even	1	trivial
1680.4.a.bf.1.1		2	4.3	odd	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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	840.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +42.6749 q^{11} -64.6749 q^{13} +15.0000 q^{15} -14.0000 q^{17} +53.3251 q^{19} -21.0000 q^{21} +37.3499 q^{23} +25.0000 q^{25} +27.0000 q^{27} +294.000 q^{29} +228.025 q^{31} +128.025 q^{33} -35.0000 q^{35} -234.050 q^{37} -194.025 q^{39} +12.6501 q^{41} -433.400 q^{43} +45.0000 q^{45} +96.0000 q^{47} +49.0000 q^{49} -42.0000 q^{51} +607.375 q^{53} +213.375 q^{55} +159.975 q^{57} +588.050 q^{59} +758.099 q^{61} -63.0000 q^{63} -323.375 q^{65} -329.400 q^{67} +112.050 q^{69} +134.625 q^{71} +384.774 q^{73} +75.0000 q^{75} -298.725 q^{77} -21.3002 q^{79} +81.0000 q^{81} -468.099 q^{83} -70.0000 q^{85} +882.000 q^{87} -1104.70 q^{89} +452.725 q^{91} +684.075 q^{93} +266.625 q^{95} +240.625 q^{97} +384.075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9} - 22 q^{11} - 22 q^{13} + 30 q^{15} - 28 q^{17} + 214 q^{19} - 42 q^{21} - 140 q^{23} + 50 q^{25} + 54 q^{27} + 588 q^{29} + 134 q^{31} - 66 q^{33} - 70 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 - 22 * q^11 - 22 * q^13 + 30 * q^15 - 28 * q^17 + 214 * q^19 - 42 * q^21 - 140 * q^23 + 50 * q^25 + 54 * q^27 + 588 * q^29 + 134 * q^31 - 66 * q^33 - 70 * q^35 + 176 * q^37 - 66 * q^39 + 240 * q^41 - 8 * q^43 + 90 * q^45 + 192 * q^47 + 98 * q^49 - 84 * q^51 + 678 * q^53 - 110 * q^55 + 642 * q^57 + 532 * q^59 + 228 * q^61 - 126 * q^63 - 110 * q^65 + 200 * q^67 - 420 * q^69 + 806 * q^71 - 626 * q^73 + 150 * q^75 + 154 * q^77 - 472 * q^79 + 162 * q^81 + 352 * q^83 - 140 * q^85 + 1764 * q^87 - 1780 * q^89 + 154 * q^91 + 402 * q^93 + 1070 * q^95 + 1018 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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