LMFDB - Embedded newform 920.4.a.b.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,4,Mod(1,920)] 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("920.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.2817572053$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 $ x^6 - 2x^5 - 81x^4 + 161x^3 + 1520x^2 - 3915*x + 588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$7.55347$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+7.55347 q^{3} -5.00000 q^{5} +4.93892 q^{7} +30.0549 q^{9} -25.3248 q^{11} -23.7663 q^{13} -37.7674 q^{15} -86.0484 q^{17} +5.67472 q^{19} +37.3060 q^{21} -23.0000 q^{23} +25.0000 q^{25} +23.0753 q^{27} -198.859 q^{29} -214.255 q^{31} -191.290 q^{33} -24.6946 q^{35} +0.432409 q^{37} -179.518 q^{39} +215.124 q^{41} -127.168 q^{43} -150.275 q^{45} +413.634 q^{47} -318.607 q^{49} -649.964 q^{51} +302.342 q^{53} +126.624 q^{55} +42.8638 q^{57} -426.766 q^{59} +693.800 q^{61} +148.439 q^{63} +118.831 q^{65} -569.947 q^{67} -173.730 q^{69} +250.438 q^{71} -782.465 q^{73} +188.837 q^{75} -125.077 q^{77} -1130.12 q^{79} -637.184 q^{81} -1052.18 q^{83} +430.242 q^{85} -1502.08 q^{87} +901.023 q^{89} -117.380 q^{91} -1618.37 q^{93} -28.3736 q^{95} +75.8213 q^{97} -761.135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 30 q^{5} + 28 q^{7} + 4 q^{9} - 3 q^{11} - 28 q^{13} - 10 q^{15} + 24 q^{17} - 3 q^{19} + 60 q^{21} - 138 q^{23} + 150 q^{25} - 97 q^{27} + 76 q^{29} - 381 q^{31} - 3 q^{33} - 140 q^{35}+ \cdots - 525 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 30 * q^5 + 28 * q^7 + 4 * q^9 - 3 * q^11 - 28 * q^13 - 10 * q^15 + 24 * q^17 - 3 * q^19 + 60 * q^21 - 138 * q^23 + 150 * q^25 - 97 * q^27 + 76 * q^29 - 381 * q^31 - 3 * q^33 - 140 * q^35 + 131 * q^37 + 41 * q^39 - 95 * q^41 - 202 * q^43 - 20 * q^45 - 119 * q^47 - 578 * q^49 - 997 * q^51 + 137 * q^53 + 15 * q^55 - 894 * q^57 - 39 * q^59 - 573 * q^61 - 355 * q^63 + 140 * q^65 - 563 * q^67 - 46 * q^69 - 83 * q^71 - 1799 * q^73 + 50 * q^75 - 1410 * q^77 - 1636 * q^79 - 2006 * q^81 - 1191 * q^83 - 120 * q^85 - 1821 * q^87 - 1370 * q^89 - 1251 * q^91 - 2215 * q^93 + 15 * q^95 - 3021 * q^97 - 525 * 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
$3$	7.55347	1.45367	0.726833	−	0.686814i	$-0.240991\pi$
$3$	7.55347	1.45367	0.726833	+	0.686814i	$0.240991\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	4.93892	0.266677	0.133338	−	0.991071i	$-0.457430\pi$
$7$	4.93892	0.266677	0.133338	+	0.991071i	$0.457430\pi$
$8$	0	0
$9$	30.0549	1.11315
$10$	0	0
$11$	−25.3248	−0.694156	−0.347078	−	0.937836i	$-0.612826\pi$
$11$	−25.3248	−0.694156	−0.347078	+	0.937836i	$0.612826\pi$
$12$	0	0
$13$	−23.7663	−0.507044	−0.253522	−	0.967330i	$-0.581589\pi$
$13$	−23.7663	−0.507044	−0.253522	+	0.967330i	$0.581589\pi$
$14$	0	0
$15$	−37.7674	−0.650099
$16$	0	0
$17$	−86.0484	−1.22763	−0.613817	−	0.789448i	$-0.710367\pi$
$17$	−86.0484	−1.22763	−0.613817	+	0.789448i	$0.710367\pi$
$18$	0	0
$19$	5.67472	0.0685194	0.0342597	−	0.999413i	$-0.489093\pi$
$19$	5.67472	0.0685194	0.0342597	+	0.999413i	$0.489093\pi$
$20$	0	0
$21$	37.3060	0.387659
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	23.0753	0.164476
$28$	0	0
$29$	−198.859	−1.27335	−0.636677	−	0.771130i	$-0.719692\pi$
$29$	−198.859	−1.27335	−0.636677	+	0.771130i	$0.719692\pi$
$30$	0	0
$31$	−214.255	−1.24133	−0.620666	−	0.784075i	$-0.713138\pi$
$31$	−214.255	−1.24133	−0.620666	+	0.784075i	$0.713138\pi$
$32$	0	0
$33$	−191.290	−1.00907
$34$	0	0
$35$	−24.6946	−0.119262
$36$	0	0
$37$	0.432409	0.00192129	0.000960644	−	1.00000i	$-0.499694\pi$
$37$	0.432409	0.00192129	0.000960644		1.00000i	$0.499694\pi$
$38$	0	0
$39$	−179.518	−0.737073
$40$	0	0
$41$	215.124	0.819431	0.409715	−	0.912213i	$-0.365628\pi$
$41$	215.124	0.819431	0.409715	+	0.912213i	$0.365628\pi$
$42$	0	0
$43$	−127.168	−0.450998	−0.225499	−	0.974243i	$-0.572401\pi$
$43$	−127.168	−0.450998	−0.225499	+	0.974243i	$0.572401\pi$
$44$	0	0
$45$	−150.275	−0.497814
$46$	0	0
$47$	413.634	1.28372	0.641859	−	0.766823i	$-0.278163\pi$
$47$	413.634	1.28372	0.641859	+	0.766823i	$0.278163\pi$
$48$	0	0
$49$	−318.607	−0.928883
$50$	0	0
$51$	−649.964	−1.78457
$52$	0	0
$53$	302.342	0.783581	0.391791	−	0.920054i	$-0.371856\pi$
$53$	302.342	0.783581	0.391791	+	0.920054i	$0.371856\pi$
$54$	0	0
$55$	126.624	0.310436
$56$	0	0
$57$	42.8638	0.0996044
$58$	0	0
$59$	−426.766	−0.941699	−0.470850	−	0.882214i	$-0.656053\pi$
$59$	−426.766	−0.941699	−0.470850	+	0.882214i	$0.656053\pi$
$60$	0	0
$61$	693.800	1.45626	0.728131	−	0.685438i	$-0.240389\pi$
$61$	693.800	1.45626	0.728131	+	0.685438i	$0.240389\pi$
$62$	0	0
$63$	148.439	0.296850
$64$	0	0
$65$	118.831	0.226757
$66$	0	0
$67$	−569.947	−1.03926	−0.519628	−	0.854393i	$-0.673929\pi$
$67$	−569.947	−1.03926	−0.519628	+	0.854393i	$0.673929\pi$
$68$	0	0
$69$	−173.730	−0.303110
$70$	0	0
$71$	250.438	0.418613	0.209306	−	0.977850i	$-0.432879\pi$
$71$	250.438	0.418613	0.209306	+	0.977850i	$0.432879\pi$
$72$	0	0
$73$	−782.465	−1.25453	−0.627264	−	0.778806i	$-0.715825\pi$
$73$	−782.465	−1.25453	−0.627264	+	0.778806i	$0.715825\pi$
$74$	0	0
$75$	188.837	0.290733
$76$	0	0
$77$	−125.077	−0.185115
$78$	0	0
$79$	−1130.12	−1.60947	−0.804735	−	0.593635i	$-0.797692\pi$
$79$	−1130.12	−1.60947	−0.804735	+	0.593635i	$0.797692\pi$
$80$	0	0
$81$	−637.184	−0.874053
$82$	0	0
$83$	−1052.18	−1.39147	−0.695733	−	0.718300i	$-0.744920\pi$
$83$	−1052.18	−1.39147	−0.695733	+	0.718300i	$0.744920\pi$
$84$	0	0
$85$	430.242	0.549015
$86$	0	0
$87$	−1502.08	−1.85103
$88$	0	0
$89$	901.023	1.07313	0.536563	−	0.843860i	$-0.319722\pi$
$89$	901.023	1.07313	0.536563	+	0.843860i	$0.319722\pi$
$90$	0	0
$91$	−117.380	−0.135217
$92$	0	0
$93$	−1618.37	−1.80448
$94$	0	0
$95$	−28.3736	−0.0306428
$96$	0	0
$97$	75.8213	0.0793658	0.0396829	−	0.999212i	$-0.487365\pi$
$97$	75.8213	0.0793658	0.0396829	+	0.999212i	$0.487365\pi$
$98$	0	0
$99$	−761.135	−0.772697
$100$	0	0
$101$	−39.1964	−0.0386157	−0.0193079	−	0.999814i	$-0.506146\pi$
$101$	−39.1964	−0.0386157	−0.0193079	+	0.999814i	$0.506146\pi$
$102$	0	0
$103$	−1160.67	−1.11033	−0.555165	−	0.831740i	$-0.687345\pi$
$103$	−1160.67	−1.11033	−0.555165	+	0.831740i	$0.687345\pi$
$104$	0	0
$105$	−186.530	−0.173366
$106$	0	0
$107$	2097.40	1.89498	0.947491	−	0.319781i	$-0.103609\pi$
$107$	2097.40	1.89498	0.947491	+	0.319781i	$0.103609\pi$
$108$	0	0
$109$	1317.62	1.15784	0.578922	−	0.815383i	$-0.303474\pi$
$109$	1317.62	1.15784	0.578922	+	0.815383i	$0.303474\pi$
$110$	0	0
$111$	3.26619	0.00279291
$112$	0	0
$113$	1483.69	1.23517	0.617584	−	0.786505i	$-0.288111\pi$
$113$	1483.69	1.23517	0.617584	+	0.786505i	$0.288111\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−714.293	−0.564414
$118$	0	0
$119$	−424.986	−0.327382
$120$	0	0
$121$	−689.654	−0.518147
$122$	0	0
$123$	1624.93	1.19118
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	164.209	0.114733	0.0573667	−	0.998353i	$-0.481730\pi$
$127$	164.209	0.114733	0.0573667	+	0.998353i	$0.481730\pi$
$128$	0	0
$129$	−960.559	−0.655601
$130$	0	0
$131$	222.370	0.148310	0.0741549	−	0.997247i	$-0.476374\pi$
$131$	222.370	0.148310	0.0741549	+	0.997247i	$0.476374\pi$
$132$	0	0
$133$	28.0270	0.0182726
$134$	0	0
$135$	−115.376	−0.0735557
$136$	0	0
$137$	−2400.82	−1.49720	−0.748599	−	0.663023i	$-0.769273\pi$
$137$	−2400.82	−1.49720	−0.748599	+	0.663023i	$0.769273\pi$
$138$	0	0
$139$	248.735	0.151780	0.0758901	−	0.997116i	$-0.475820\pi$
$139$	248.735	0.151780	0.0758901	+	0.997116i	$0.475820\pi$
$140$	0	0
$141$	3124.37	1.86610
$142$	0	0
$143$	601.876	0.351968
$144$	0	0
$145$	994.297	0.569461
$146$	0	0
$147$	−2406.59	−1.35029
$148$	0	0
$149$	556.747	0.306111	0.153055	−	0.988218i	$-0.451089\pi$
$149$	556.747	0.306111	0.153055	+	0.988218i	$0.451089\pi$
$150$	0	0
$151$	1113.72	0.600221	0.300110	−	0.953904i	$-0.402976\pi$
$151$	1113.72	0.600221	0.300110	+	0.953904i	$0.402976\pi$
$152$	0	0
$153$	−2586.18	−1.36654
$154$	0	0
$155$	1071.27	0.555140
$156$	0	0
$157$	−860.641	−0.437495	−0.218747	−	0.975782i	$-0.570197\pi$
$157$	−860.641	−0.437495	−0.218747	+	0.975782i	$0.570197\pi$
$158$	0	0
$159$	2283.73	1.13907
$160$	0	0
$161$	−113.595	−0.0556060
$162$	0	0
$163$	1545.13	0.742476	0.371238	−	0.928538i	$-0.378933\pi$
$163$	1545.13	0.742476	0.371238	+	0.928538i	$0.378933\pi$
$164$	0	0
$165$	956.451	0.451271
$166$	0	0
$167$	109.841	0.0508965	0.0254483	−	0.999676i	$-0.491899\pi$
$167$	109.841	0.0508965	0.0254483	+	0.999676i	$0.491899\pi$
$168$	0	0
$169$	−1632.17	−0.742906
$170$	0	0
$171$	170.553	0.0762721
$172$	0	0
$173$	−853.947	−0.375286	−0.187643	−	0.982237i	$-0.560085\pi$
$173$	−853.947	−0.375286	−0.187643	+	0.982237i	$0.560085\pi$
$174$	0	0
$175$	123.473	0.0533354
$176$	0	0
$177$	−3223.57	−1.36892
$178$	0	0
$179$	−2035.16	−0.849803	−0.424902	−	0.905239i	$-0.639691\pi$
$179$	−2035.16	−0.849803	−0.424902	+	0.905239i	$0.639691\pi$
$180$	0	0
$181$	1006.29	0.413243	0.206621	−	0.978421i	$-0.433753\pi$
$181$	1006.29	0.413243	0.206621	+	0.978421i	$0.433753\pi$
$182$	0	0
$183$	5240.60	2.11692
$184$	0	0
$185$	−2.16205	−0.000859226 0
$186$	0	0
$187$	2179.16	0.852171
$188$	0	0
$189$	113.967	0.0438618
$190$	0	0
$191$	2400.21	0.909281	0.454641	−	0.890675i	$-0.349768\pi$
$191$	2400.21	0.909281	0.454641	+	0.890675i	$0.349768\pi$
$192$	0	0
$193$	−2883.63	−1.07548	−0.537742	−	0.843110i	$-0.680722\pi$
$193$	−2883.63	−1.07548	−0.537742	+	0.843110i	$0.680722\pi$
$194$	0	0
$195$	897.588	0.329629
$196$	0	0
$197$	−4894.79	−1.77025	−0.885125	−	0.465353i	$-0.845928\pi$
$197$	−4894.79	−1.77025	−0.885125	+	0.465353i	$0.845928\pi$
$198$	0	0
$199$	−751.449	−0.267682	−0.133841	−	0.991003i	$-0.542731\pi$
$199$	−751.449	−0.267682	−0.133841	+	0.991003i	$0.542731\pi$
$200$	0	0
$201$	−4305.08	−1.51073
$202$	0	0
$203$	−982.152	−0.339574
$204$	0	0
$205$	−1075.62	−0.366461
$206$	0	0
$207$	−691.263	−0.232107
$208$	0	0
$209$	−143.711	−0.0475632
$210$	0	0
$211$	1258.11	0.410484	0.205242	−	0.978711i	$-0.434202\pi$
$211$	1258.11	0.410484	0.205242	+	0.978711i	$0.434202\pi$
$212$	0	0
$213$	1891.67	0.608523
$214$	0	0
$215$	635.840	0.201693
$216$	0	0
$217$	−1058.19	−0.331034
$218$	0	0
$219$	−5910.33	−1.82367
$220$	0	0
$221$	2045.05	0.622465
$222$	0	0
$223$	402.761	0.120946	0.0604728	−	0.998170i	$-0.480739\pi$
$223$	402.761	0.120946	0.0604728	+	0.998170i	$0.480739\pi$
$224$	0	0
$225$	751.373	0.222629
$226$	0	0
$227$	−4056.30	−1.18602	−0.593008	−	0.805196i	$-0.702060\pi$
$227$	−4056.30	−1.18602	−0.593008	+	0.805196i	$0.702060\pi$
$228$	0	0
$229$	398.823	0.115087	0.0575437	−	0.998343i	$-0.481673\pi$
$229$	398.823	0.115087	0.0575437	+	0.998343i	$0.481673\pi$
$230$	0	0
$231$	−944.768	−0.269096
$232$	0	0
$233$	6395.65	1.79825	0.899127	−	0.437689i	$-0.144203\pi$
$233$	6395.65	1.79825	0.899127	+	0.437689i	$0.144203\pi$
$234$	0	0
$235$	−2068.17	−0.574096
$236$	0	0
$237$	−8536.30	−2.33963
$238$	0	0
$239$	1325.15	0.358647	0.179324	−	0.983790i	$-0.442609\pi$
$239$	1325.15	0.358647	0.179324	+	0.983790i	$0.442609\pi$
$240$	0	0
$241$	−1610.06	−0.430344	−0.215172	−	0.976576i	$-0.569031\pi$
$241$	−1610.06	−0.430344	−0.215172	+	0.976576i	$0.569031\pi$
$242$	0	0
$243$	−5435.99	−1.43506
$244$	0	0
$245$	1593.04	0.415409
$246$	0	0
$247$	−134.867	−0.0347424
$248$	0	0
$249$	−7947.61	−2.02273
$250$	0	0
$251$	1891.58	0.475680	0.237840	−	0.971304i	$-0.423561\pi$
$251$	1891.58	0.475680	0.237840	+	0.971304i	$0.423561\pi$
$252$	0	0
$253$	582.471	0.144742
$254$	0	0
$255$	3249.82	0.798085
$256$	0	0
$257$	−2772.37	−0.672901	−0.336451	−	0.941701i	$-0.609227\pi$
$257$	−2772.37	−0.672901	−0.336451	+	0.941701i	$0.609227\pi$
$258$	0	0
$259$	2.13564	0.000512363 0
$260$	0	0
$261$	−5976.71	−1.41743
$262$	0	0
$263$	3133.14	0.734592	0.367296	−	0.930104i	$-0.380284\pi$
$263$	3133.14	0.734592	0.367296	+	0.930104i	$0.380284\pi$
$264$	0	0
$265$	−1511.71	−0.350428
$266$	0	0
$267$	6805.85	1.55997
$268$	0	0
$269$	3459.75	0.784182	0.392091	−	0.919926i	$-0.371752\pi$
$269$	3459.75	0.784182	0.392091	+	0.919926i	$0.371752\pi$
$270$	0	0
$271$	169.887	0.0380809	0.0190404	−	0.999819i	$-0.493939\pi$
$271$	169.887	0.0380809	0.0190404	+	0.999819i	$0.493939\pi$
$272$	0	0
$273$	−886.624	−0.196560
$274$	0	0
$275$	−633.120	−0.138831
$276$	0	0
$277$	−4742.37	−1.02867	−0.514334	−	0.857590i	$-0.671961\pi$
$277$	−4742.37	−1.02867	−0.514334	+	0.857590i	$0.671961\pi$
$278$	0	0
$279$	−6439.41	−1.38178
$280$	0	0
$281$	−2489.30	−0.528467	−0.264234	−	0.964459i	$-0.585119\pi$
$281$	−2489.30	−0.528467	−0.264234	+	0.964459i	$0.585119\pi$
$282$	0	0
$283$	5867.18	1.23239	0.616197	−	0.787592i	$-0.288672\pi$
$283$	5867.18	1.23239	0.616197	+	0.787592i	$0.288672\pi$
$284$	0	0
$285$	−214.319	−0.0445444
$286$	0	0
$287$	1062.48	0.218523
$288$	0	0
$289$	2491.32	0.507088
$290$	0	0
$291$	572.714	0.115371
$292$	0	0
$293$	7462.95	1.48802	0.744010	−	0.668168i	$-0.232921\pi$
$293$	7462.95	1.48802	0.744010	+	0.668168i	$0.232921\pi$
$294$	0	0
$295$	2133.83	0.421141
$296$	0	0
$297$	−584.377	−0.114172
$298$	0	0
$299$	546.624	0.105726
$300$	0	0
$301$	−628.073	−0.120271
$302$	0	0
$303$	−296.069	−0.0561344
$304$	0	0
$305$	−3469.00	−0.651261
$306$	0	0
$307$	−7285.04	−1.35433	−0.677164	−	0.735832i	$-0.736791\pi$
$307$	−7285.04	−1.35433	−0.677164	+	0.735832i	$0.736791\pi$
$308$	0	0
$309$	−8767.07	−1.61405
$310$	0	0
$311$	4198.11	0.765444	0.382722	−	0.923864i	$-0.374987\pi$
$311$	4198.11	0.765444	0.382722	+	0.923864i	$0.374987\pi$
$312$	0	0
$313$	−7610.41	−1.37433	−0.687166	−	0.726501i	$-0.741145\pi$
$313$	−7610.41	−1.37433	−0.687166	+	0.726501i	$0.741145\pi$
$314$	0	0
$315$	−742.195	−0.132755
$316$	0	0
$317$	−6039.30	−1.07003	−0.535017	−	0.844841i	$-0.679695\pi$
$317$	−6039.30	−1.07003	−0.535017	+	0.844841i	$0.679695\pi$
$318$	0	0
$319$	5036.08	0.883907
$320$	0	0
$321$	15842.6	2.75467
$322$	0	0
$323$	−488.300	−0.0841169
$324$	0	0
$325$	−594.156	−0.101409
$326$	0	0
$327$	9952.59	1.68312
$328$	0	0
$329$	2042.91	0.342338
$330$	0	0
$331$	2824.86	0.469089	0.234544	−	0.972105i	$-0.424640\pi$
$331$	2824.86	0.469089	0.234544	+	0.972105i	$0.424640\pi$
$332$	0	0
$333$	12.9960	0.00213867
$334$	0	0
$335$	2849.73	0.464769
$336$	0	0
$337$	3175.13	0.513235	0.256618	−	0.966513i	$-0.417392\pi$
$337$	3175.13	0.513235	0.256618	+	0.966513i	$0.417392\pi$
$338$	0	0
$339$	11207.0	1.79552
$340$	0	0
$341$	5425.96	0.861678
$342$	0	0
$343$	−3267.63	−0.514389
$344$	0	0
$345$	868.649	0.135555
$346$	0	0
$347$	−1395.28	−0.215857	−0.107928	−	0.994159i	$-0.534422\pi$
$347$	−1395.28	−0.215857	−0.107928	+	0.994159i	$0.534422\pi$
$348$	0	0
$349$	248.886	0.0381735	0.0190867	−	0.999818i	$-0.493924\pi$
$349$	248.886	0.0381735	0.0190867	+	0.999818i	$0.493924\pi$
$350$	0	0
$351$	−548.413	−0.0833963
$352$	0	0
$353$	−3289.01	−0.495910	−0.247955	−	0.968772i	$-0.579759\pi$
$353$	−3289.01	−0.495910	−0.247955	+	0.968772i	$0.579759\pi$
$354$	0	0
$355$	−1252.19	−0.187209
$356$	0	0
$357$	−3210.12	−0.475904
$358$	0	0
$359$	6061.01	0.891052	0.445526	−	0.895269i	$-0.353017\pi$
$359$	6061.01	0.891052	0.445526	+	0.895269i	$0.353017\pi$
$360$	0	0
$361$	−6826.80	−0.995305
$362$	0	0
$363$	−5209.28	−0.753213
$364$	0	0
$365$	3912.32	0.561042
$366$	0	0
$367$	12067.3	1.71636	0.858182	−	0.513346i	$-0.171594\pi$
$367$	12067.3	1.71636	0.858182	+	0.513346i	$0.171594\pi$
$368$	0	0
$369$	6465.52	0.912146
$370$	0	0
$371$	1493.24	0.208963
$372$	0	0
$373$	5665.83	0.786503	0.393252	−	0.919431i	$-0.371350\pi$
$373$	5665.83	0.786503	0.393252	+	0.919431i	$0.371350\pi$
$374$	0	0
$375$	−944.184	−0.130020
$376$	0	0
$377$	4726.14	0.645647
$378$	0	0
$379$	−820.068	−0.111145	−0.0555727	−	0.998455i	$-0.517698\pi$
$379$	−820.068	−0.111145	−0.0555727	+	0.998455i	$0.517698\pi$
$380$	0	0
$381$	1240.34	0.166784
$382$	0	0
$383$	8037.19	1.07227	0.536137	−	0.844131i	$-0.319883\pi$
$383$	8037.19	1.07227	0.536137	+	0.844131i	$0.319883\pi$
$384$	0	0
$385$	625.387	0.0827861
$386$	0	0
$387$	−3822.02	−0.502027
$388$	0	0
$389$	−3224.48	−0.420277	−0.210138	−	0.977672i	$-0.567391\pi$
$389$	−3224.48	−0.420277	−0.210138	+	0.977672i	$0.567391\pi$
$390$	0	0
$391$	1979.11	0.255980
$392$	0	0
$393$	1679.67	0.215593
$394$	0	0
$395$	5650.58	0.719776
$396$	0	0
$397$	−6180.53	−0.781340	−0.390670	−	0.920531i	$-0.627757\pi$
$397$	−6180.53	−0.781340	−0.390670	+	0.920531i	$0.627757\pi$
$398$	0	0
$399$	211.701	0.0265622
$400$	0	0
$401$	−5156.49	−0.642152	−0.321076	−	0.947053i	$-0.604044\pi$
$401$	−5156.49	−0.642152	−0.321076	+	0.947053i	$0.604044\pi$
$402$	0	0
$403$	5092.03	0.629410
$404$	0	0
$405$	3185.92	0.390888
$406$	0	0
$407$	−10.9507	−0.00133367
$408$	0	0
$409$	4306.36	0.520626	0.260313	−	0.965524i	$-0.416174\pi$
$409$	4306.36	0.520626	0.260313	+	0.965524i	$0.416174\pi$
$410$	0	0
$411$	−18134.6	−2.17643
$412$	0	0
$413$	−2107.77	−0.251129
$414$	0	0
$415$	5260.90	0.622283
$416$	0	0
$417$	1878.81	0.220638
$418$	0	0
$419$	−6086.11	−0.709608	−0.354804	−	0.934941i	$-0.615452\pi$
$419$	−6086.11	−0.709608	−0.354804	+	0.934941i	$0.615452\pi$
$420$	0	0
$421$	−1120.54	−0.129719	−0.0648594	−	0.997894i	$-0.520660\pi$
$421$	−1120.54	−0.129719	−0.0648594	+	0.997894i	$0.520660\pi$
$422$	0	0
$423$	12431.7	1.42896
$424$	0	0
$425$	−2151.21	−0.245527
$426$	0	0
$427$	3426.63	0.388352
$428$	0	0
$429$	4546.25	0.511644
$430$	0	0
$431$	8278.90	0.925245	0.462623	−	0.886555i	$-0.346908\pi$
$431$	8278.90	0.925245	0.462623	+	0.886555i	$0.346908\pi$
$432$	0	0
$433$	14843.9	1.64747	0.823734	−	0.566976i	$-0.191887\pi$
$433$	14843.9	1.64747	0.823734	+	0.566976i	$0.191887\pi$
$434$	0	0
$435$	7510.40	0.827807
$436$	0	0
$437$	−130.518	−0.0142873
$438$	0	0
$439$	4226.81	0.459532	0.229766	−	0.973246i	$-0.426204\pi$
$439$	4226.81	0.459532	0.229766	+	0.973246i	$0.426204\pi$
$440$	0	0
$441$	−9575.71	−1.03398
$442$	0	0
$443$	1327.36	0.142359	0.0711793	−	0.997464i	$-0.477324\pi$
$443$	1327.36	0.142359	0.0711793	+	0.997464i	$0.477324\pi$
$444$	0	0
$445$	−4505.12	−0.479917
$446$	0	0
$447$	4205.38	0.444983
$448$	0	0
$449$	−8822.31	−0.927284	−0.463642	−	0.886022i	$-0.653458\pi$
$449$	−8822.31	−0.927284	−0.463642	+	0.886022i	$0.653458\pi$
$450$	0	0
$451$	−5447.97	−0.568813
$452$	0	0
$453$	8412.46	0.872521
$454$	0	0
$455$	586.899	0.0604708
$456$	0	0
$457$	10364.9	1.06094	0.530470	−	0.847703i	$-0.322015\pi$
$457$	10364.9	1.06094	0.530470	+	0.847703i	$0.322015\pi$
$458$	0	0
$459$	−1985.59	−0.201916
$460$	0	0
$461$	16974.6	1.71494	0.857469	−	0.514535i	$-0.172035\pi$
$461$	16974.6	1.71494	0.857469	+	0.514535i	$0.172035\pi$
$462$	0	0
$463$	12837.0	1.28852	0.644260	−	0.764807i	$-0.277166\pi$
$463$	12837.0	1.28852	0.644260	+	0.764807i	$0.277166\pi$
$464$	0	0
$465$	8091.83	0.806989
$466$	0	0
$467$	18616.1	1.84465	0.922326	−	0.386413i	$-0.126286\pi$
$467$	18616.1	1.84465	0.922326	+	0.386413i	$0.126286\pi$
$468$	0	0
$469$	−2814.93	−0.277145
$470$	0	0
$471$	−6500.83	−0.635971
$472$	0	0
$473$	3220.51	0.313063
$474$	0	0
$475$	141.868	0.0137039
$476$	0	0
$477$	9086.85	0.872240
$478$	0	0
$479$	−10882.8	−1.03810	−0.519048	−	0.854745i	$-0.673713\pi$
$479$	−10882.8	−1.03810	−0.519048	+	0.854745i	$0.673713\pi$
$480$	0	0
$481$	−10.2767	−0.000974178 0
$482$	0	0
$483$	−858.039	−0.0808325
$484$	0	0
$485$	−379.106	−0.0354935
$486$	0	0
$487$	−2974.99	−0.276817	−0.138408	−	0.990375i	$-0.544199\pi$
$487$	−2974.99	−0.276817	−0.138408	+	0.990375i	$0.544199\pi$
$488$	0	0
$489$	11671.1	1.07931
$490$	0	0
$491$	6366.96	0.585207	0.292604	−	0.956234i	$-0.405478\pi$
$491$	6366.96	0.585207	0.292604	+	0.956234i	$0.405478\pi$
$492$	0	0
$493$	17111.5	1.56321
$494$	0	0
$495$	3805.68	0.345561
$496$	0	0
$497$	1236.89	0.111634
$498$	0	0
$499$	−4739.07	−0.425150	−0.212575	−	0.977145i	$-0.568185\pi$
$499$	−4739.07	−0.425150	−0.212575	+	0.977145i	$0.568185\pi$
$500$	0	0
$501$	829.678	0.0739866
$502$	0	0
$503$	10254.9	0.909035	0.454517	−	0.890738i	$-0.349812\pi$
$503$	10254.9	0.909035	0.454517	+	0.890738i	$0.349812\pi$
$504$	0	0
$505$	195.982	0.0172695
$506$	0	0
$507$	−12328.5	−1.07994
$508$	0	0
$509$	9219.13	0.802811	0.401405	−	0.915900i	$-0.368522\pi$
$509$	9219.13	0.802811	0.401405	+	0.915900i	$0.368522\pi$
$510$	0	0
$511$	−3864.54	−0.334554
$512$	0	0
$513$	130.946	0.0112698
$514$	0	0
$515$	5803.34	0.496555
$516$	0	0
$517$	−10475.2	−0.891101
$518$	0	0
$519$	−6450.27	−0.545540
$520$	0	0
$521$	−10904.6	−0.916963	−0.458482	−	0.888704i	$-0.651607\pi$
$521$	−10904.6	−0.916963	−0.458482	+	0.888704i	$0.651607\pi$
$522$	0	0
$523$	−15804.3	−1.32137	−0.660683	−	0.750665i	$-0.729733\pi$
$523$	−15804.3	−1.32137	−0.660683	+	0.750665i	$0.729733\pi$
$524$	0	0
$525$	932.651	0.0775318
$526$	0	0
$527$	18436.3	1.52390
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−12826.4	−1.04825
$532$	0	0
$533$	−5112.68	−0.415487
$534$	0	0
$535$	−10487.0	−0.847462
$536$	0	0
$537$	−15372.5	−1.23533
$538$	0	0
$539$	8068.67	0.644790
$540$	0	0
$541$	−4716.49	−0.374820	−0.187410	−	0.982282i	$-0.560009\pi$
$541$	−4716.49	−0.374820	−0.187410	+	0.982282i	$0.560009\pi$
$542$	0	0
$543$	7600.98	0.600717
$544$	0	0
$545$	−6588.09	−0.517803
$546$	0	0
$547$	13363.9	1.04461	0.522304	−	0.852759i	$-0.325072\pi$
$547$	13363.9	1.04461	0.522304	+	0.852759i	$0.325072\pi$
$548$	0	0
$549$	20852.1	1.62103
$550$	0	0
$551$	−1128.47	−0.0872495
$552$	0	0
$553$	−5581.56	−0.429208
$554$	0	0
$555$	−16.3310	−0.00124903
$556$	0	0
$557$	6080.36	0.462537	0.231268	−	0.972890i	$-0.425712\pi$
$557$	6080.36	0.462537	0.231268	+	0.972890i	$0.425712\pi$
$558$	0	0
$559$	3022.31	0.228676
$560$	0	0
$561$	16460.2	1.23877
$562$	0	0
$563$	−17362.3	−1.29971	−0.649854	−	0.760059i	$-0.725170\pi$
$563$	−17362.3	−1.29971	−0.649854	+	0.760059i	$0.725170\pi$
$564$	0	0
$565$	−7418.46	−0.552384
$566$	0	0
$567$	−3147.01	−0.233090
$568$	0	0
$569$	−5264.30	−0.387857	−0.193929	−	0.981016i	$-0.562123\pi$
$569$	−5264.30	−0.387857	−0.193929	+	0.981016i	$0.562123\pi$
$570$	0	0
$571$	−19270.6	−1.41235	−0.706174	−	0.708038i	$-0.749581\pi$
$571$	−19270.6	−1.41235	−0.706174	+	0.708038i	$0.749581\pi$
$572$	0	0
$573$	18129.9	1.32179
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−18225.4	−1.31496	−0.657481	−	0.753471i	$-0.728378\pi$
$577$	−18225.4	−1.31496	−0.657481	+	0.753471i	$0.728378\pi$
$578$	0	0
$579$	−21781.4	−1.56339
$580$	0	0
$581$	−5196.63	−0.371072
$582$	0	0
$583$	−7656.75	−0.543928
$584$	0	0
$585$	3571.46	0.252413
$586$	0	0
$587$	26714.8	1.87843	0.939214	−	0.343333i	$-0.111556\pi$
$587$	26714.8	1.87843	0.939214	+	0.343333i	$0.111556\pi$
$588$	0	0
$589$	−1215.83	−0.0850553
$590$	0	0
$591$	−36972.7	−2.57335
$592$	0	0
$593$	7707.57	0.533747	0.266874	−	0.963732i	$-0.414009\pi$
$593$	7707.57	0.533747	0.266874	+	0.963732i	$0.414009\pi$
$594$	0	0
$595$	2124.93	0.146410
$596$	0	0
$597$	−5676.05	−0.389121
$598$	0	0
$599$	−16530.7	−1.12759	−0.563795	−	0.825915i	$-0.690659\pi$
$599$	−16530.7	−1.12759	−0.563795	+	0.825915i	$0.690659\pi$
$600$	0	0
$601$	2351.15	0.159576	0.0797880	−	0.996812i	$-0.474576\pi$
$601$	2351.15	0.159576	0.0797880	+	0.996812i	$0.474576\pi$
$602$	0	0
$603$	−17129.7	−1.15684
$604$	0	0
$605$	3448.27	0.231722
$606$	0	0
$607$	−19245.5	−1.28691	−0.643453	−	0.765486i	$-0.722499\pi$
$607$	−19245.5	−1.28691	−0.643453	+	0.765486i	$0.722499\pi$
$608$	0	0
$609$	−7418.66	−0.493627
$610$	0	0
$611$	−9830.53	−0.650901
$612$	0	0
$613$	29069.6	1.91535	0.957675	−	0.287853i	$-0.0929413\pi$
$613$	29069.6	1.91535	0.957675	+	0.287853i	$0.0929413\pi$
$614$	0	0
$615$	−8124.65	−0.532711
$616$	0	0
$617$	−9134.94	−0.596043	−0.298022	−	0.954559i	$-0.596327\pi$
$617$	−9134.94	−0.596043	−0.298022	+	0.954559i	$0.596327\pi$
$618$	0	0
$619$	5251.25	0.340978	0.170489	−	0.985360i	$-0.445465\pi$
$619$	5251.25	0.340978	0.170489	+	0.985360i	$0.445465\pi$
$620$	0	0
$621$	−530.731	−0.0342955
$622$	0	0
$623$	4450.09	0.286178
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−1085.52	−0.0691410
$628$	0	0
$629$	−37.2081	−0.00235864
$630$	0	0
$631$	−20458.1	−1.29069	−0.645344	−	0.763892i	$-0.723286\pi$
$631$	−20458.1	−1.29069	−0.645344	+	0.763892i	$0.723286\pi$
$632$	0	0
$633$	9503.12	0.596706
$634$	0	0
$635$	−821.043	−0.0513104
$636$	0	0
$637$	7572.09	0.470985
$638$	0	0
$639$	7526.89	0.465977
$640$	0	0
$641$	−12101.2	−0.745664	−0.372832	−	0.927899i	$-0.621613\pi$
$641$	−12101.2	−0.745664	−0.372832	+	0.927899i	$0.621613\pi$
$642$	0	0
$643$	10876.0	0.667042	0.333521	−	0.942743i	$-0.391763\pi$
$643$	10876.0	0.667042	0.333521	+	0.942743i	$0.391763\pi$
$644$	0	0
$645$	4802.80	0.293194
$646$	0	0
$647$	−8656.66	−0.526010	−0.263005	−	0.964794i	$-0.584714\pi$
$647$	−8656.66	−0.526010	−0.263005	+	0.964794i	$0.584714\pi$
$648$	0	0
$649$	10807.8	0.653686
$650$	0	0
$651$	−7992.99	−0.481214
$652$	0	0
$653$	20070.1	1.20276	0.601381	−	0.798962i	$-0.294617\pi$
$653$	20070.1	1.20276	0.601381	+	0.798962i	$0.294617\pi$
$654$	0	0
$655$	−1111.85	−0.0663262
$656$	0	0
$657$	−23516.9	−1.39647
$658$	0	0
$659$	−24532.2	−1.45014	−0.725068	−	0.688677i	$-0.758192\pi$
$659$	−24532.2	−1.45014	−0.725068	+	0.688677i	$0.758192\pi$
$660$	0	0
$661$	305.872	0.0179985	0.00899926	−	0.999960i	$-0.497135\pi$
$661$	305.872	0.0179985	0.00899926	+	0.999960i	$0.497135\pi$
$662$	0	0
$663$	15447.2	0.904856
$664$	0	0
$665$	−140.135	−0.00817173
$666$	0	0
$667$	4573.77	0.265513
$668$	0	0
$669$	3042.24	0.175814
$670$	0	0
$671$	−17570.4	−1.01087
$672$	0	0
$673$	−1428.57	−0.0818234	−0.0409117	−	0.999163i	$-0.513026\pi$
$673$	−1428.57	−0.0818234	−0.0409117	+	0.999163i	$0.513026\pi$
$674$	0	0
$675$	576.882	0.0328951
$676$	0	0
$677$	−14353.7	−0.814858	−0.407429	−	0.913237i	$-0.633575\pi$
$677$	−14353.7	−0.814858	−0.407429	+	0.913237i	$0.633575\pi$
$678$	0	0
$679$	374.476	0.0211650
$680$	0	0
$681$	−30639.1	−1.72407
$682$	0	0
$683$	−3227.68	−0.180825	−0.0904126	−	0.995904i	$-0.528819\pi$
$683$	−3227.68	−0.180825	−0.0904126	+	0.995904i	$0.528819\pi$
$684$	0	0
$685$	12004.1	0.669568
$686$	0	0
$687$	3012.50	0.167299
$688$	0	0
$689$	−7185.53	−0.397310
$690$	0	0
$691$	−15581.0	−0.857784	−0.428892	−	0.903356i	$-0.641096\pi$
$691$	−15581.0	−0.857784	−0.428892	+	0.903356i	$0.641096\pi$
$692$	0	0
$693$	−3759.19	−0.206060
$694$	0	0
$695$	−1243.68	−0.0678782
$696$	0	0
$697$	−18511.0	−1.00596
$698$	0	0
$699$	48309.4	2.61406
$700$	0	0
$701$	−14792.3	−0.797000	−0.398500	−	0.917168i	$-0.630469\pi$
$701$	−14792.3	−0.797000	−0.398500	+	0.917168i	$0.630469\pi$
$702$	0	0
$703$	2.45380	0.000131646 0
$704$	0	0
$705$	−15621.9	−0.834544
$706$	0	0
$707$	−193.588	−0.0102979
$708$	0	0
$709$	11123.5	0.589215	0.294607	−	0.955618i	$-0.404811\pi$
$709$	11123.5	0.589215	0.294607	+	0.955618i	$0.404811\pi$
$710$	0	0
$711$	−33965.6	−1.79157
$712$	0	0
$713$	4927.86	0.258836
$714$	0	0
$715$	−3009.38	−0.157405
$716$	0	0
$717$	10009.5	0.521354
$718$	0	0
$719$	−4074.76	−0.211353	−0.105677	−	0.994401i	$-0.533701\pi$
$719$	−4074.76	−0.211353	−0.105677	+	0.994401i	$0.533701\pi$
$720$	0	0
$721$	−5732.45	−0.296099
$722$	0	0
$723$	−12161.5	−0.625576
$724$	0	0
$725$	−4971.49	−0.254671
$726$	0	0
$727$	341.016	0.0173970	0.00869848	−	0.999962i	$-0.497231\pi$
$727$	341.016	0.0173970	0.00869848	+	0.999962i	$0.497231\pi$
$728$	0	0
$729$	−23856.6	−1.21204
$730$	0	0
$731$	10942.6	0.553661
$732$	0	0
$733$	−6235.62	−0.314213	−0.157106	−	0.987582i	$-0.550217\pi$
$733$	−6235.62	−0.314213	−0.157106	+	0.987582i	$0.550217\pi$
$734$	0	0
$735$	12032.9	0.603866
$736$	0	0
$737$	14433.8	0.721406
$738$	0	0
$739$	27056.2	1.34679	0.673397	−	0.739281i	$-0.264835\pi$
$739$	27056.2	1.34679	0.673397	+	0.739281i	$0.264835\pi$
$740$	0	0
$741$	−1018.71	−0.0505038
$742$	0	0
$743$	34029.6	1.68025	0.840124	−	0.542394i	$-0.182482\pi$
$743$	34029.6	1.68025	0.840124	+	0.542394i	$0.182482\pi$
$744$	0	0
$745$	−2783.74	−0.136897
$746$	0	0
$747$	−31623.2	−1.54890
$748$	0	0
$749$	10358.9	0.505348
$750$	0	0
$751$	−8377.99	−0.407080	−0.203540	−	0.979067i	$-0.565245\pi$
$751$	−8377.99	−0.407080	−0.203540	+	0.979067i	$0.565245\pi$
$752$	0	0
$753$	14288.0	0.691479
$754$	0	0
$755$	−5568.60	−0.268427
$756$	0	0
$757$	−5812.74	−0.279085	−0.139543	−	0.990216i	$-0.544563\pi$
$757$	−5812.74	−0.279085	−0.139543	+	0.990216i	$0.544563\pi$
$758$	0	0
$759$	4399.68	0.210406
$760$	0	0
$761$	−825.051	−0.0393010	−0.0196505	−	0.999807i	$-0.506255\pi$
$761$	−825.051	−0.0393010	−0.0196505	+	0.999807i	$0.506255\pi$
$762$	0	0
$763$	6507.62	0.308770
$764$	0	0
$765$	12930.9	0.611134
$766$	0	0
$767$	10142.6	0.477483
$768$	0	0
$769$	1916.89	0.0898893	0.0449446	−	0.998989i	$-0.485689\pi$
$769$	1916.89	0.0898893	0.0449446	+	0.998989i	$0.485689\pi$
$770$	0	0
$771$	−20941.0	−0.978174
$772$	0	0
$773$	−12874.3	−0.599039	−0.299520	−	0.954090i	$-0.596826\pi$
$773$	−12874.3	−0.599039	−0.299520	+	0.954090i	$0.596826\pi$
$774$	0	0
$775$	−5356.37	−0.248266
$776$	0	0
$777$	16.1315	0.000744805 0
$778$	0	0
$779$	1220.77	0.0561469
$780$	0	0
$781$	−6342.29	−0.290583
$782$	0	0
$783$	−4588.74	−0.209436
$784$	0	0
$785$	4303.21	0.195654
$786$	0	0
$787$	23099.8	1.04628	0.523139	−	0.852248i	$-0.324761\pi$
$787$	23099.8	1.04628	0.523139	+	0.852248i	$0.324761\pi$
$788$	0	0
$789$	23666.1	1.06785
$790$	0	0
$791$	7327.85	0.329391
$792$	0	0
$793$	−16489.0	−0.738389
$794$	0	0
$795$	−11418.6	−0.509406
$796$	0	0
$797$	1192.59	0.0530034	0.0265017	−	0.999649i	$-0.491563\pi$
$797$	1192.59	0.0530034	0.0265017	+	0.999649i	$0.491563\pi$
$798$	0	0
$799$	−35592.5	−1.57594
$800$	0	0
$801$	27080.2	1.19455
$802$	0	0
$803$	19815.8	0.870839
$804$	0	0
$805$	567.976	0.0248677
$806$	0	0
$807$	26133.2	1.13994
$808$	0	0
$809$	−20412.6	−0.887104	−0.443552	−	0.896249i	$-0.646282\pi$
$809$	−20412.6	−0.887104	−0.443552	+	0.896249i	$0.646282\pi$
$810$	0	0
$811$	−25753.0	−1.11505	−0.557527	−	0.830159i	$-0.688250\pi$
$811$	−25753.0	−1.11505	−0.557527	+	0.830159i	$0.688250\pi$
$812$	0	0
$813$	1283.24	0.0553569
$814$	0	0
$815$	−7725.63	−0.332046
$816$	0	0
$817$	−721.642	−0.0309022
$818$	0	0
$819$	−3527.84	−0.150516
$820$	0	0
$821$	−22757.9	−0.967425	−0.483712	−	0.875227i	$-0.660712\pi$
$821$	−22757.9	−0.967425	−0.483712	+	0.875227i	$0.660712\pi$
$822$	0	0
$823$	−21532.3	−0.911991	−0.455995	−	0.889982i	$-0.650717\pi$
$823$	−21532.3	−0.911991	−0.455995	+	0.889982i	$0.650717\pi$
$824$	0	0
$825$	−4782.26	−0.201814
$826$	0	0
$827$	21257.1	0.893811	0.446906	−	0.894581i	$-0.352526\pi$
$827$	21257.1	0.893811	0.446906	+	0.894581i	$0.352526\pi$
$828$	0	0
$829$	−39959.2	−1.67411	−0.837057	−	0.547115i	$-0.815726\pi$
$829$	−39959.2	−1.67411	−0.837057	+	0.547115i	$0.815726\pi$
$830$	0	0
$831$	−35821.3	−1.49534
$832$	0	0
$833$	27415.6	1.14033
$834$	0	0
$835$	−549.203	−0.0227616
$836$	0	0
$837$	−4943.99	−0.204169
$838$	0	0
$839$	−18083.9	−0.744132	−0.372066	−	0.928206i	$-0.621350\pi$
$839$	−18083.9	−0.744132	−0.372066	+	0.928206i	$0.621350\pi$
$840$	0	0
$841$	15156.1	0.621431
$842$	0	0
$843$	−18802.9	−0.768215
$844$	0	0
$845$	8160.83	0.332238
$846$	0	0
$847$	−3406.15	−0.138178
$848$	0	0
$849$	44317.6	1.79149
$850$	0	0
$851$	−9.94541	−0.000400616 0
$852$	0	0
$853$	−37161.5	−1.49166	−0.745829	−	0.666137i	$-0.767946\pi$
$853$	−37161.5	−1.49166	−0.745829	+	0.666137i	$0.767946\pi$
$854$	0	0
$855$	−852.766	−0.0341099
$856$	0	0
$857$	11094.9	0.442235	0.221118	−	0.975247i	$-0.429030\pi$
$857$	11094.9	0.442235	0.221118	+	0.975247i	$0.429030\pi$
$858$	0	0
$859$	−27181.7	−1.07966	−0.539830	−	0.841774i	$-0.681511\pi$
$859$	−27181.7	−1.07966	−0.539830	+	0.841774i	$0.681511\pi$
$860$	0	0
$861$	8025.41	0.317660
$862$	0	0
$863$	−5902.79	−0.232831	−0.116416	−	0.993201i	$-0.537140\pi$
$863$	−5902.79	−0.232831	−0.116416	+	0.993201i	$0.537140\pi$
$864$	0	0
$865$	4269.74	0.167833
$866$	0	0
$867$	18818.1	0.737136
$868$	0	0
$869$	28620.0	1.11722
$870$	0	0
$871$	13545.5	0.526948
$872$	0	0
$873$	2278.80	0.0883457
$874$	0	0
$875$	−617.366	−0.0238523
$876$	0	0
$877$	−35394.3	−1.36281	−0.681403	−	0.731908i	$-0.738630\pi$
$877$	−35394.3	−1.36281	−0.681403	+	0.731908i	$0.738630\pi$
$878$	0	0
$879$	56371.2	2.16309
$880$	0	0
$881$	−46842.6	−1.79134	−0.895668	−	0.444724i	$-0.853302\pi$
$881$	−46842.6	−1.79134	−0.895668	+	0.444724i	$0.853302\pi$
$882$	0	0
$883$	24311.0	0.926537	0.463268	−	0.886218i	$-0.346677\pi$
$883$	24311.0	0.926537	0.463268	+	0.886218i	$0.346677\pi$
$884$	0	0
$885$	16117.8	0.612198
$886$	0	0
$887$	−2411.91	−0.0913009	−0.0456504	−	0.998957i	$-0.514536\pi$
$887$	−2411.91	−0.0913009	−0.0456504	+	0.998957i	$0.514536\pi$
$888$	0	0
$889$	811.014	0.0305968
$890$	0	0
$891$	16136.6	0.606729
$892$	0	0
$893$	2347.26	0.0879596
$894$	0	0
$895$	10175.8	0.380044
$896$	0	0
$897$	4128.91	0.153690
$898$	0	0
$899$	42606.6	1.58065
$900$	0	0
$901$	−26016.0	−0.961952
$902$	0	0
$903$	−4744.13	−0.174834
$904$	0	0
$905$	−5031.45	−0.184808
$906$	0	0
$907$	−26207.5	−0.959435	−0.479717	−	0.877423i	$-0.659261\pi$
$907$	−26207.5	−0.959435	−0.479717	+	0.877423i	$0.659261\pi$
$908$	0	0
$909$	−1178.05	−0.0429849
$910$	0	0
$911$	12187.3	0.443231	0.221615	−	0.975134i	$-0.428867\pi$
$911$	12187.3	0.443231	0.221615	+	0.975134i	$0.428867\pi$
$912$	0	0
$913$	26646.2	0.965895
$914$	0	0
$915$	−26203.0	−0.946715
$916$	0	0
$917$	1098.27	0.0395508
$918$	0	0
$919$	−14643.3	−0.525614	−0.262807	−	0.964848i	$-0.584648\pi$
$919$	−14643.3	−0.525614	−0.262807	+	0.964848i	$0.584648\pi$
$920$	0	0
$921$	−55027.3	−1.96874
$922$	0	0
$923$	−5951.97	−0.212255
$924$	0	0
$925$	10.8102	0.000384258 0
$926$	0	0
$927$	−34883.8	−1.23596
$928$	0	0
$929$	−19281.5	−0.680954	−0.340477	−	0.940253i	$-0.610589\pi$
$929$	−19281.5	−0.680954	−0.340477	+	0.940253i	$0.610589\pi$
$930$	0	0
$931$	−1808.00	−0.0636466
$932$	0	0
$933$	31710.3	1.11270
$934$	0	0
$935$	−10895.8	−0.381102
$936$	0	0
$937$	−26610.7	−0.927784	−0.463892	−	0.885892i	$-0.653547\pi$
$937$	−26610.7	−0.927784	−0.463892	+	0.885892i	$0.653547\pi$
$938$	0	0
$939$	−57485.0	−1.99782
$940$	0	0
$941$	50369.2	1.74494	0.872470	−	0.488668i	$-0.162517\pi$
$941$	50369.2	1.74494	0.872470	+	0.488668i	$0.162517\pi$
$942$	0	0
$943$	−4947.84	−0.170863
$944$	0	0
$945$	−569.835	−0.0196156
$946$	0	0
$947$	27235.5	0.934568	0.467284	−	0.884107i	$-0.345233\pi$
$947$	27235.5	0.934568	0.467284	+	0.884107i	$0.345233\pi$
$948$	0	0
$949$	18596.3	0.636101
$950$	0	0
$951$	−45617.7	−1.55547
$952$	0	0
$953$	−37676.1	−1.28064	−0.640319	−	0.768109i	$-0.721198\pi$
$953$	−37676.1	−1.28064	−0.640319	+	0.768109i	$0.721198\pi$
$954$	0	0
$955$	−12001.0	−0.406643
$956$	0	0
$957$	38039.9	1.28491
$958$	0	0
$959$	−11857.5	−0.399268
$960$	0	0
$961$	16114.1	0.540904
$962$	0	0
$963$	63037.1	2.10939
$964$	0	0
$965$	14418.2	0.480971
$966$	0	0
$967$	−43160.6	−1.43532	−0.717658	−	0.696396i	$-0.754786\pi$
$967$	−43160.6	−1.43532	−0.717658	+	0.696396i	$0.754786\pi$
$968$	0	0
$969$	−3688.36	−0.122278
$970$	0	0
$971$	10023.6	0.331281	0.165641	−	0.986186i	$-0.447031\pi$
$971$	10023.6	0.331281	0.165641	+	0.986186i	$0.447031\pi$
$972$	0	0
$973$	1228.48	0.0404763
$974$	0	0
$975$	−4487.94	−0.147415
$976$	0	0
$977$	−41061.0	−1.34458	−0.672291	−	0.740287i	$-0.734690\pi$
$977$	−41061.0	−1.34458	−0.672291	+	0.740287i	$0.734690\pi$
$978$	0	0
$979$	−22818.2	−0.744918
$980$	0	0
$981$	39600.9	1.28885
$982$	0	0
$983$	16922.9	0.549091	0.274545	−	0.961574i	$-0.411473\pi$
$983$	16922.9	0.549091	0.274545	+	0.961574i	$0.411473\pi$
$984$	0	0
$985$	24474.0	0.791680
$986$	0	0
$987$	15431.0	0.497645
$988$	0	0
$989$	2924.86	0.0940397
$990$	0	0
$991$	−12342.5	−0.395632	−0.197816	−	0.980239i	$-0.563385\pi$
$991$	−12342.5	−0.395632	−0.197816	+	0.980239i	$0.563385\pi$
$992$	0	0
$993$	21337.5	0.681898
$994$	0	0
$995$	3757.25	0.119711
$996$	0	0
$997$	39279.5	1.24774	0.623869	−	0.781529i	$-0.285560\pi$
$997$	39279.5	1.24774	0.623869	+	0.781529i	$0.285560\pi$
$998$	0	0
$999$	9.97796	0.000316005 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.4.a.b.1.6	✓	6
4.3	odd	2		1840.4.a.s.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.b.1.6	✓	6	1.1	even	1	trivial
1840.4.a.s.1.1		6	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q+7.55347 q^{3} -5.00000 q^{5} +4.93892 q^{7} +30.0549 q^{9} -25.3248 q^{11} -23.7663 q^{13} -37.7674 q^{15} -86.0484 q^{17} +5.67472 q^{19} +37.3060 q^{21} -23.0000 q^{23} +25.0000 q^{25} +23.0753 q^{27} -198.859 q^{29} -214.255 q^{31} -191.290 q^{33} -24.6946 q^{35} +0.432409 q^{37} -179.518 q^{39} +215.124 q^{41} -127.168 q^{43} -150.275 q^{45} +413.634 q^{47} -318.607 q^{49} -649.964 q^{51} +302.342 q^{53} +126.624 q^{55} +42.8638 q^{57} -426.766 q^{59} +693.800 q^{61} +148.439 q^{63} +118.831 q^{65} -569.947 q^{67} -173.730 q^{69} +250.438 q^{71} -782.465 q^{73} +188.837 q^{75} -125.077 q^{77} -1130.12 q^{79} -637.184 q^{81} -1052.18 q^{83} +430.242 q^{85} -1502.08 q^{87} +901.023 q^{89} -117.380 q^{91} -1618.37 q^{93} -28.3736 q^{95} +75.8213 q^{97} -761.135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 30 q^{5} + 28 q^{7} + 4 q^{9} - 3 q^{11} - 28 q^{13} - 10 q^{15} + 24 q^{17} - 3 q^{19} + 60 q^{21} - 138 q^{23} + 150 q^{25} - 97 q^{27} + 76 q^{29} - 381 q^{31} - 3 q^{33} - 140 q^{35}+ \cdots - 525 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 30 * q^5 + 28 * q^7 + 4 * q^9 - 3 * q^11 - 28 * q^13 - 10 * q^15 + 24 * q^17 - 3 * q^19 + 60 * q^21 - 138 * q^23 + 150 * q^25 - 97 * q^27 + 76 * q^29 - 381 * q^31 - 3 * q^33 - 140 * q^35 + 131 * q^37 + 41 * q^39 - 95 * q^41 - 202 * q^43 - 20 * q^45 - 119 * q^47 - 578 * q^49 - 997 * q^51 + 137 * q^53 + 15 * q^55 - 894 * q^57 - 39 * q^59 - 573 * q^61 - 355 * q^63 + 140 * q^65 - 563 * q^67 - 46 * q^69 - 83 * q^71 - 1799 * q^73 + 50 * q^75 - 1410 * q^77 - 1636 * q^79 - 2006 * q^81 - 1191 * q^83 - 120 * q^85 - 1821 * q^87 - 1370 * q^89 - 1251 * q^91 - 2215 * q^93 + 15 * q^95 - 3021 * q^97 - 525 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more