LMFDB - Embedded newform 1452.4.a.s.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,4,Mod(1,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1452.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,18,0,-12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$85.6707733283$
Analytic rank:	$0$
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} - 57x^{4} - 42x^{3} + 603x^{2} + 630x - 882 $ x^6 - 57x^4 - 42x^3 + 603x^2 + 630x - 882
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.65070$ of defining polynomial
Character	$\chi$	$=$	1452.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +14.3567 q^{5} -3.76593 q^{7} +9.00000 q^{9} -34.4466 q^{13} +43.0702 q^{15} -45.4425 q^{17} +21.9272 q^{19} -11.2978 q^{21} -0.925093 q^{23} +81.1158 q^{25} +27.0000 q^{27} +34.8415 q^{29} +186.302 q^{31} -54.0665 q^{35} +295.783 q^{37} -103.340 q^{39} +341.613 q^{41} -6.89371 q^{43} +129.211 q^{45} +307.418 q^{47} -328.818 q^{49} -136.328 q^{51} +480.495 q^{53} +65.7816 q^{57} +444.925 q^{59} +530.344 q^{61} -33.8934 q^{63} -494.540 q^{65} -227.084 q^{67} -2.77528 q^{69} +464.058 q^{71} +421.231 q^{73} +243.347 q^{75} -893.034 q^{79} +81.0000 q^{81} +409.671 q^{83} -652.406 q^{85} +104.525 q^{87} +908.963 q^{89} +129.723 q^{91} +558.905 q^{93} +314.803 q^{95} -437.148 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{3} - 12 q^{5} + 54 q^{9} - 36 q^{15} + 156 q^{23} + 150 q^{25} + 162 q^{27} + 150 q^{31} + 366 q^{37} - 108 q^{45} + 540 q^{47} + 1308 q^{49} + 360 q^{53} + 2508 q^{59} + 1014 q^{67} + 468 q^{69}+ \cdots + 5790 q^{97}+O(q^{100}) $ 6 * q + 18 * q^3 - 12 * q^5 + 54 * q^9 - 36 * q^15 + 156 * q^23 + 150 * q^25 + 162 * q^27 + 150 * q^31 + 366 * q^37 - 108 * q^45 + 540 * q^47 + 1308 * q^49 + 360 * q^53 + 2508 * q^59 + 1014 * q^67 + 468 * q^69 - 1176 * q^71 + 450 * q^75 + 486 * q^81 + 2724 * q^89 + 1728 * q^91 + 450 * q^93 + 5790 * q^97

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
$5$	14.3567	1.28411	0.642053	−	0.766661i	$-0.278083\pi$
$5$	14.3567	1.28411	0.642053	+	0.766661i	$0.278083\pi$
$6$	0	0
$7$	−3.76593	−0.203341	−0.101671	−	0.994818i	$-0.532419\pi$
$7$	−3.76593	−0.203341	−0.101671	+	0.994818i	$0.532419\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−34.4466	−0.734904	−0.367452	−	0.930042i	$-0.619770\pi$
$13$	−34.4466	−0.734904	−0.367452	+	0.930042i	$0.619770\pi$
$14$	0	0
$15$	43.0702	0.741378
$16$	0	0
$17$	−45.4425	−0.648319	−0.324160	−	0.946002i	$-0.605082\pi$
$17$	−45.4425	−0.648319	−0.324160	+	0.946002i	$0.605082\pi$
$18$	0	0
$19$	21.9272	0.264760	0.132380	−	0.991199i	$-0.457738\pi$
$19$	21.9272	0.264760	0.132380	+	0.991199i	$0.457738\pi$
$20$	0	0
$21$	−11.2978	−0.117399
$22$	0	0
$23$	−0.925093	−0.00838675	−0.00419338	−	0.999991i	$-0.501335\pi$
$23$	−0.925093	−0.00838675	−0.00419338	+	0.999991i	$0.501335\pi$
$24$	0	0
$25$	81.1158	0.648926
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	34.8415	0.223100	0.111550	−	0.993759i	$-0.464418\pi$
$29$	34.8415	0.223100	0.111550	+	0.993759i	$0.464418\pi$
$30$	0	0
$31$	186.302	1.07938	0.539690	−	0.841864i	$-0.318541\pi$
$31$	186.302	1.07938	0.539690	+	0.841864i	$0.318541\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−54.0665	−0.261112
$36$	0	0
$37$	295.783	1.31423	0.657113	−	0.753792i	$-0.271777\pi$
$37$	295.783	1.31423	0.657113	+	0.753792i	$0.271777\pi$
$38$	0	0
$39$	−103.340	−0.424297
$40$	0	0
$41$	341.613	1.30124	0.650622	−	0.759401i	$-0.274508\pi$
$41$	341.613	1.30124	0.650622	+	0.759401i	$0.274508\pi$
$42$	0	0
$43$	−6.89371	−0.0244484	−0.0122242	−	0.999925i	$-0.503891\pi$
$43$	−6.89371	−0.0244484	−0.0122242	+	0.999925i	$0.503891\pi$
$44$	0	0
$45$	129.211	0.428035
$46$	0	0
$47$	307.418	0.954073	0.477037	−	0.878883i	$-0.341711\pi$
$47$	307.418	0.954073	0.477037	+	0.878883i	$0.341711\pi$
$48$	0	0
$49$	−328.818	−0.958652
$50$	0	0
$51$	−136.328	−0.374307
$52$	0	0
$53$	480.495	1.24530	0.622652	−	0.782499i	$-0.286055\pi$
$53$	480.495	1.24530	0.622652	+	0.782499i	$0.286055\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	65.7816	0.152860
$58$	0	0
$59$	444.925	0.981768	0.490884	−	0.871225i	$-0.336674\pi$
$59$	444.925	0.981768	0.490884	+	0.871225i	$0.336674\pi$
$60$	0	0
$61$	530.344	1.11317	0.556587	−	0.830790i	$-0.312111\pi$
$61$	530.344	1.11317	0.556587	+	0.830790i	$0.312111\pi$
$62$	0	0
$63$	−33.8934	−0.0677804
$64$	0	0
$65$	−494.540	−0.943694
$66$	0	0
$67$	−227.084	−0.414071	−0.207036	−	0.978333i	$-0.566382\pi$
$67$	−227.084	−0.414071	−0.207036	+	0.978333i	$0.566382\pi$
$68$	0	0
$69$	−2.77528	−0.00484209
$70$	0	0
$71$	464.058	0.775683	0.387842	−	0.921726i	$-0.373221\pi$
$71$	464.058	0.775683	0.387842	+	0.921726i	$0.373221\pi$
$72$	0	0
$73$	421.231	0.675361	0.337681	−	0.941261i	$-0.390358\pi$
$73$	421.231	0.675361	0.337681	+	0.941261i	$0.390358\pi$
$74$	0	0
$75$	243.347	0.374658
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−893.034	−1.27182	−0.635912	−	0.771761i	$-0.719376\pi$
$79$	−893.034	−1.27182	−0.635912	+	0.771761i	$0.719376\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	409.671	0.541774	0.270887	−	0.962611i	$-0.412683\pi$
$83$	409.671	0.541774	0.270887	+	0.962611i	$0.412683\pi$
$84$	0	0
$85$	−652.406	−0.832510
$86$	0	0
$87$	104.525	0.128807
$88$	0	0
$89$	908.963	1.08258	0.541292	−	0.840835i	$-0.317935\pi$
$89$	908.963	1.08258	0.541292	+	0.840835i	$0.317935\pi$
$90$	0	0
$91$	129.723	0.149436
$92$	0	0
$93$	558.905	0.623180
$94$	0	0
$95$	314.803	0.339980
$96$	0	0
$97$	−437.148	−0.457584	−0.228792	−	0.973475i	$-0.573478\pi$
$97$	−437.148	−0.457584	−0.228792	+	0.973475i	$0.573478\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1839.31	1.81206	0.906032	−	0.423209i	$-0.139096\pi$
$101$	1839.31	1.81206	0.906032	+	0.423209i	$0.139096\pi$
$102$	0	0
$103$	758.747	0.725840	0.362920	−	0.931820i	$-0.381780\pi$
$103$	758.747	0.725840	0.362920	+	0.931820i	$0.381780\pi$
$104$	0	0
$105$	−162.199	−0.150753
$106$	0	0
$107$	−1086.12	−0.981299	−0.490650	−	0.871357i	$-0.663241\pi$
$107$	−1086.12	−0.981299	−0.490650	+	0.871357i	$0.663241\pi$
$108$	0	0
$109$	−2094.95	−1.84092	−0.920459	−	0.390840i	$-0.872184\pi$
$109$	−2094.95	−1.84092	−0.920459	+	0.390840i	$0.872184\pi$
$110$	0	0
$111$	887.348	0.758769
$112$	0	0
$113$	−1066.72	−0.888042	−0.444021	−	0.896016i	$-0.646448\pi$
$113$	−1066.72	−0.888042	−0.444021	+	0.896016i	$0.646448\pi$
$114$	0	0
$115$	−13.2813	−0.0107695
$116$	0	0
$117$	−310.019	−0.244968
$118$	0	0
$119$	171.133	0.131830
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1024.84	0.751274
$124$	0	0
$125$	−630.034	−0.450816
$126$	0	0
$127$	2758.48	1.92737	0.963683	−	0.267049i	$-0.0860486\pi$
$127$	2758.48	1.92737	0.963683	+	0.267049i	$0.0860486\pi$
$128$	0	0
$129$	−20.6811	−0.0141153
$130$	0	0
$131$	−1342.82	−0.895592	−0.447796	−	0.894136i	$-0.647791\pi$
$131$	−1342.82	−0.895592	−0.447796	+	0.894136i	$0.647791\pi$
$132$	0	0
$133$	−82.5764	−0.0538367
$134$	0	0
$135$	387.632	0.247126
$136$	0	0
$137$	−845.784	−0.527447	−0.263723	−	0.964598i	$-0.584951\pi$
$137$	−845.784	−0.527447	−0.263723	+	0.964598i	$0.584951\pi$
$138$	0	0
$139$	2955.44	1.80343	0.901715	−	0.432330i	$-0.142309\pi$
$139$	2955.44	1.80343	0.901715	+	0.432330i	$0.142309\pi$
$140$	0	0
$141$	922.253	0.550835
$142$	0	0
$143$	0	0
$144$	0	0
$145$	500.211	0.286484
$146$	0	0
$147$	−986.453	−0.553478
$148$	0	0
$149$	−911.559	−0.501193	−0.250597	−	0.968092i	$-0.580627\pi$
$149$	−911.559	−0.501193	−0.250597	+	0.968092i	$0.580627\pi$
$150$	0	0
$151$	−3476.33	−1.87351	−0.936755	−	0.349986i	$-0.886186\pi$
$151$	−3476.33	−1.87351	−0.936755	+	0.349986i	$0.886186\pi$
$152$	0	0
$153$	−408.983	−0.216106
$154$	0	0
$155$	2674.68	1.38604
$156$	0	0
$157$	−1954.12	−0.993349	−0.496675	−	0.867937i	$-0.665446\pi$
$157$	−1954.12	−0.993349	−0.496675	+	0.867937i	$0.665446\pi$
$158$	0	0
$159$	1441.49	0.718976
$160$	0	0
$161$	3.48384	0.00170537
$162$	0	0
$163$	1326.73	0.637532	0.318766	−	0.947833i	$-0.396732\pi$
$163$	1326.73	0.637532	0.318766	+	0.947833i	$0.396732\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1493.67	0.692119	0.346060	−	0.938212i	$-0.387519\pi$
$167$	1493.67	0.692119	0.346060	+	0.938212i	$0.387519\pi$
$168$	0	0
$169$	−1010.43	−0.459916
$170$	0	0
$171$	197.345	0.0882535
$172$	0	0
$173$	−431.962	−0.189835	−0.0949175	−	0.995485i	$-0.530259\pi$
$173$	−431.962	−0.189835	−0.0949175	+	0.995485i	$0.530259\pi$
$174$	0	0
$175$	−305.477	−0.131953
$176$	0	0
$177$	1334.78	0.566824
$178$	0	0
$179$	−1220.71	−0.509722	−0.254861	−	0.966978i	$-0.582030\pi$
$179$	−1220.71	−0.509722	−0.254861	+	0.966978i	$0.582030\pi$
$180$	0	0
$181$	−1437.21	−0.590206	−0.295103	−	0.955465i	$-0.595354\pi$
$181$	−1437.21	−0.590206	−0.295103	+	0.955465i	$0.595354\pi$
$182$	0	0
$183$	1591.03	0.642691
$184$	0	0
$185$	4246.47	1.68760
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−101.680	−0.0391330
$190$	0	0
$191$	1993.91	0.755364	0.377682	−	0.925935i	$-0.376721\pi$
$191$	1993.91	0.755364	0.377682	+	0.925935i	$0.376721\pi$
$192$	0	0
$193$	77.7041	0.0289806	0.0144903	−	0.999895i	$-0.495387\pi$
$193$	77.7041	0.0289806	0.0144903	+	0.999895i	$0.495387\pi$
$194$	0	0
$195$	−1483.62	−0.544842
$196$	0	0
$197$	−3276.96	−1.18515	−0.592574	−	0.805516i	$-0.701888\pi$
$197$	−3276.96	−1.18515	−0.592574	+	0.805516i	$0.701888\pi$
$198$	0	0
$199$	1832.21	0.652672	0.326336	−	0.945254i	$-0.394186\pi$
$199$	1832.21	0.652672	0.326336	+	0.945254i	$0.394186\pi$
$200$	0	0
$201$	−681.253	−0.239064
$202$	0	0
$203$	−131.211	−0.0453655
$204$	0	0
$205$	4904.45	1.67093
$206$	0	0
$207$	−8.32584	−0.00279558
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4348.66	−1.41883	−0.709417	−	0.704789i	$-0.751042\pi$
$211$	−4348.66	−1.41883	−0.709417	+	0.704789i	$0.751042\pi$
$212$	0	0
$213$	1392.17	0.447841
$214$	0	0
$215$	−98.9711	−0.0313943
$216$	0	0
$217$	−701.600	−0.219482
$218$	0	0
$219$	1263.69	0.389920
$220$	0	0
$221$	1565.34	0.476453
$222$	0	0
$223$	3044.23	0.914156	0.457078	−	0.889427i	$-0.348896\pi$
$223$	3044.23	0.914156	0.457078	+	0.889427i	$0.348896\pi$
$224$	0	0
$225$	730.042	0.216309
$226$	0	0
$227$	5139.88	1.50285	0.751423	−	0.659821i	$-0.229368\pi$
$227$	5139.88	1.50285	0.751423	+	0.659821i	$0.229368\pi$
$228$	0	0
$229$	3667.42	1.05830	0.529148	−	0.848529i	$-0.322512\pi$
$229$	3667.42	1.05830	0.529148	+	0.848529i	$0.322512\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1182.61	−0.332512	−0.166256	−	0.986083i	$-0.553168\pi$
$233$	−1182.61	−0.332512	−0.166256	+	0.986083i	$0.553168\pi$
$234$	0	0
$235$	4413.51	1.22513
$236$	0	0
$237$	−2679.10	−0.734288
$238$	0	0
$239$	204.573	0.0553670	0.0276835	−	0.999617i	$-0.491187\pi$
$239$	204.573	0.0553670	0.0276835	+	0.999617i	$0.491187\pi$
$240$	0	0
$241$	−321.769	−0.0860040	−0.0430020	−	0.999075i	$-0.513692\pi$
$241$	−321.769	−0.0860040	−0.0430020	+	0.999075i	$0.513692\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−4720.75	−1.23101
$246$	0	0
$247$	−755.317	−0.194574
$248$	0	0
$249$	1229.01	0.312794
$250$	0	0
$251$	2815.22	0.707949	0.353975	−	0.935255i	$-0.384830\pi$
$251$	2815.22	0.707949	0.353975	+	0.935255i	$0.384830\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1957.22	−0.480650
$256$	0	0
$257$	1397.55	0.339210	0.169605	−	0.985512i	$-0.445751\pi$
$257$	1397.55	0.339210	0.169605	+	0.985512i	$0.445751\pi$
$258$	0	0
$259$	−1113.90	−0.267236
$260$	0	0
$261$	313.574	0.0743668
$262$	0	0
$263$	−6659.39	−1.56135	−0.780676	−	0.624936i	$-0.785125\pi$
$263$	−6659.39	−1.56135	−0.780676	+	0.624936i	$0.785125\pi$
$264$	0	0
$265$	6898.34	1.59910
$266$	0	0
$267$	2726.89	0.625030
$268$	0	0
$269$	−4643.24	−1.05243	−0.526215	−	0.850352i	$-0.676389\pi$
$269$	−4643.24	−1.05243	−0.526215	+	0.850352i	$0.676389\pi$
$270$	0	0
$271$	2266.26	0.507991	0.253996	−	0.967205i	$-0.418255\pi$
$271$	2266.26	0.507991	0.253996	+	0.967205i	$0.418255\pi$
$272$	0	0
$273$	389.170	0.0862771
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3318.90	0.719903	0.359952	−	0.932971i	$-0.382793\pi$
$277$	3318.90	0.719903	0.359952	+	0.932971i	$0.382793\pi$
$278$	0	0
$279$	1676.72	0.359793
$280$	0	0
$281$	−2107.00	−0.447307	−0.223654	−	0.974669i	$-0.571798\pi$
$281$	−2107.00	−0.447307	−0.223654	+	0.974669i	$0.571798\pi$
$282$	0	0
$283$	860.604	0.180769	0.0903844	−	0.995907i	$-0.471190\pi$
$283$	860.604	0.180769	0.0903844	+	0.995907i	$0.471190\pi$
$284$	0	0
$285$	944.409	0.196288
$286$	0	0
$287$	−1286.49	−0.264597
$288$	0	0
$289$	−2847.98	−0.579682
$290$	0	0
$291$	−1311.44	−0.264186
$292$	0	0
$293$	4615.86	0.920346	0.460173	−	0.887829i	$-0.347787\pi$
$293$	4615.86	0.920346	0.460173	+	0.887829i	$0.347787\pi$
$294$	0	0
$295$	6387.67	1.26069
$296$	0	0
$297$	0	0
$298$	0	0
$299$	31.8663	0.00616346
$300$	0	0
$301$	25.9612	0.00497137
$302$	0	0
$303$	5517.94	1.04620
$304$	0	0
$305$	7614.01	1.42943
$306$	0	0
$307$	1376.56	0.255910	0.127955	−	0.991780i	$-0.459159\pi$
$307$	1376.56	0.255910	0.127955	+	0.991780i	$0.459159\pi$
$308$	0	0
$309$	2276.24	0.419064
$310$	0	0
$311$	−4324.67	−0.788520	−0.394260	−	0.918999i	$-0.628999\pi$
$311$	−4324.67	−0.788520	−0.394260	+	0.918999i	$0.628999\pi$
$312$	0	0
$313$	7184.28	1.29738	0.648689	−	0.761054i	$-0.275318\pi$
$313$	7184.28	1.29738	0.648689	+	0.761054i	$0.275318\pi$
$314$	0	0
$315$	−486.598	−0.0870372
$316$	0	0
$317$	8606.78	1.52494	0.762468	−	0.647026i	$-0.223987\pi$
$317$	8606.78	1.52494	0.762468	+	0.647026i	$0.223987\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3258.36	−0.566553
$322$	0	0
$323$	−996.428	−0.171649
$324$	0	0
$325$	−2794.16	−0.476899
$326$	0	0
$327$	−6284.85	−1.06285
$328$	0	0
$329$	−1157.71	−0.194002
$330$	0	0
$331$	3651.76	0.606402	0.303201	−	0.952927i	$-0.401945\pi$
$331$	3651.76	0.606402	0.303201	+	0.952927i	$0.401945\pi$
$332$	0	0
$333$	2662.04	0.438075
$334$	0	0
$335$	−3260.19	−0.531711
$336$	0	0
$337$	−2835.73	−0.458374	−0.229187	−	0.973382i	$-0.573607\pi$
$337$	−2835.73	−0.458374	−0.229187	+	0.973382i	$0.573607\pi$
$338$	0	0
$339$	−3200.16	−0.512711
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2530.02	0.398275
$344$	0	0
$345$	−39.8440	−0.00621776
$346$	0	0
$347$	9539.30	1.47578	0.737891	−	0.674920i	$-0.235822\pi$
$347$	9539.30	1.47578	0.737891	+	0.674920i	$0.235822\pi$
$348$	0	0
$349$	−9866.52	−1.51330	−0.756652	−	0.653818i	$-0.773166\pi$
$349$	−9866.52	−1.51330	−0.756652	+	0.653818i	$0.773166\pi$
$350$	0	0
$351$	−930.057	−0.141432
$352$	0	0
$353$	−2902.64	−0.437654	−0.218827	−	0.975764i	$-0.570223\pi$
$353$	−2902.64	−0.437654	−0.218827	+	0.975764i	$0.570223\pi$
$354$	0	0
$355$	6662.35	0.996059
$356$	0	0
$357$	513.400	0.0761121
$358$	0	0
$359$	−7147.49	−1.05078	−0.525390	−	0.850861i	$-0.676081\pi$
$359$	−7147.49	−1.05078	−0.525390	+	0.850861i	$0.676081\pi$
$360$	0	0
$361$	−6378.20	−0.929902
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6047.50	0.867235
$366$	0	0
$367$	2084.01	0.296415	0.148207	−	0.988956i	$-0.452650\pi$
$367$	2084.01	0.296415	0.148207	+	0.988956i	$0.452650\pi$
$368$	0	0
$369$	3074.52	0.433748
$370$	0	0
$371$	−1809.51	−0.253222
$372$	0	0
$373$	371.957	0.0516332	0.0258166	−	0.999667i	$-0.491781\pi$
$373$	371.957	0.0516332	0.0258166	+	0.999667i	$0.491781\pi$
$374$	0	0
$375$	−1890.10	−0.260279
$376$	0	0
$377$	−1200.17	−0.163957
$378$	0	0
$379$	9829.64	1.33223	0.666114	−	0.745850i	$-0.267956\pi$
$379$	9829.64	1.33223	0.666114	+	0.745850i	$0.267956\pi$
$380$	0	0
$381$	8275.44	1.11277
$382$	0	0
$383$	−6956.98	−0.928160	−0.464080	−	0.885793i	$-0.653615\pi$
$383$	−6956.98	−0.928160	−0.464080	+	0.885793i	$0.653615\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−62.0434	−0.00814946
$388$	0	0
$389$	1105.26	0.144059	0.0720297	−	0.997402i	$-0.477052\pi$
$389$	1105.26	0.144059	0.0720297	+	0.997402i	$0.477052\pi$
$390$	0	0
$391$	42.0386	0.00543729
$392$	0	0
$393$	−4028.45	−0.517070
$394$	0	0
$395$	−12821.0	−1.63316
$396$	0	0
$397$	11085.8	1.40146	0.700730	−	0.713426i	$-0.252858\pi$
$397$	11085.8	1.40146	0.700730	+	0.713426i	$0.252858\pi$
$398$	0	0
$399$	−247.729	−0.0310826
$400$	0	0
$401$	−6798.34	−0.846616	−0.423308	−	0.905986i	$-0.639131\pi$
$401$	−6798.34	−0.846616	−0.423308	+	0.905986i	$0.639131\pi$
$402$	0	0
$403$	−6417.45	−0.793241
$404$	0	0
$405$	1162.90	0.142678
$406$	0	0
$407$	0	0
$408$	0	0
$409$	8173.41	0.988140	0.494070	−	0.869422i	$-0.335509\pi$
$409$	8173.41	0.988140	0.494070	+	0.869422i	$0.335509\pi$
$410$	0	0
$411$	−2537.35	−0.304522
$412$	0	0
$413$	−1675.56	−0.199634
$414$	0	0
$415$	5881.54	0.695695
$416$	0	0
$417$	8866.31	1.04121
$418$	0	0
$419$	−13224.0	−1.54185	−0.770927	−	0.636924i	$-0.780207\pi$
$419$	−13224.0	−1.54185	−0.770927	+	0.636924i	$0.780207\pi$
$420$	0	0
$421$	1729.58	0.200225	0.100113	−	0.994976i	$-0.468080\pi$
$421$	1729.58	0.200225	0.100113	+	0.994976i	$0.468080\pi$
$422$	0	0
$423$	2766.76	0.318024
$424$	0	0
$425$	−3686.11	−0.420711
$426$	0	0
$427$	−1997.24	−0.226354
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8946.40	0.999844	0.499922	−	0.866070i	$-0.333362\pi$
$431$	8946.40	0.999844	0.499922	+	0.866070i	$0.333362\pi$
$432$	0	0
$433$	−14201.3	−1.57615	−0.788073	−	0.615581i	$-0.788921\pi$
$433$	−14201.3	−1.57615	−0.788073	+	0.615581i	$0.788921\pi$
$434$	0	0
$435$	1500.63	0.165402
$436$	0	0
$437$	−20.2847	−0.00222048
$438$	0	0
$439$	−3959.72	−0.430495	−0.215247	−	0.976560i	$-0.569056\pi$
$439$	−3959.72	−0.430495	−0.215247	+	0.976560i	$0.569056\pi$
$440$	0	0
$441$	−2959.36	−0.319551
$442$	0	0
$443$	−8521.51	−0.913926	−0.456963	−	0.889486i	$-0.651063\pi$
$443$	−8521.51	−0.913926	−0.456963	+	0.889486i	$0.651063\pi$
$444$	0	0
$445$	13049.7	1.39015
$446$	0	0
$447$	−2734.68	−0.289364
$448$	0	0
$449$	−14904.6	−1.56658	−0.783288	−	0.621658i	$-0.786459\pi$
$449$	−14904.6	−1.56658	−0.783288	+	0.621658i	$0.786459\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−10429.0	−1.08167
$454$	0	0
$455$	1862.40	0.191892
$456$	0	0
$457$	−2071.71	−0.212058	−0.106029	−	0.994363i	$-0.533814\pi$
$457$	−2071.71	−0.212058	−0.106029	+	0.994363i	$0.533814\pi$
$458$	0	0
$459$	−1226.95	−0.124769
$460$	0	0
$461$	−9335.44	−0.943156	−0.471578	−	0.881824i	$-0.656315\pi$
$461$	−9335.44	−0.943156	−0.471578	+	0.881824i	$0.656315\pi$
$462$	0	0
$463$	−6087.54	−0.611041	−0.305521	−	0.952185i	$-0.598830\pi$
$463$	−6087.54	−0.611041	−0.305521	+	0.952185i	$0.598830\pi$
$464$	0	0
$465$	8024.05	0.800229
$466$	0	0
$467$	−15648.2	−1.55056	−0.775282	−	0.631616i	$-0.782392\pi$
$467$	−15648.2	−1.55056	−0.775282	+	0.631616i	$0.782392\pi$
$468$	0	0
$469$	855.184	0.0841978
$470$	0	0
$471$	−5862.36	−0.573510
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1778.64	0.171810
$476$	0	0
$477$	4324.46	0.415101
$478$	0	0
$479$	11423.8	1.08970	0.544851	−	0.838533i	$-0.316586\pi$
$479$	11423.8	1.08970	0.544851	+	0.838533i	$0.316586\pi$
$480$	0	0
$481$	−10188.7	−0.965830
$482$	0	0
$483$	10.4515	0.000984597 0
$484$	0	0
$485$	−6276.01	−0.587586
$486$	0	0
$487$	9042.53	0.841389	0.420694	−	0.907202i	$-0.361786\pi$
$487$	9042.53	0.841389	0.420694	+	0.907202i	$0.361786\pi$
$488$	0	0
$489$	3980.20	0.368079
$490$	0	0
$491$	−14756.3	−1.35630	−0.678150	−	0.734923i	$-0.737218\pi$
$491$	−14756.3	−1.35630	−0.678150	+	0.734923i	$0.737218\pi$
$492$	0	0
$493$	−1583.29	−0.144640
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1747.61	−0.157728
$498$	0	0
$499$	11208.9	1.00557	0.502787	−	0.864411i	$-0.332308\pi$
$499$	11208.9	1.00557	0.502787	+	0.864411i	$0.332308\pi$
$500$	0	0
$501$	4481.02	0.399595
$502$	0	0
$503$	16075.5	1.42499	0.712495	−	0.701677i	$-0.247565\pi$
$503$	16075.5	1.42499	0.712495	+	0.701677i	$0.247565\pi$
$504$	0	0
$505$	26406.5	2.32688
$506$	0	0
$507$	−3031.30	−0.265532
$508$	0	0
$509$	−8451.55	−0.735969	−0.367985	−	0.929832i	$-0.619952\pi$
$509$	−8451.55	−0.735969	−0.367985	+	0.929832i	$0.619952\pi$
$510$	0	0
$511$	−1586.33	−0.137329
$512$	0	0
$513$	592.035	0.0509532
$514$	0	0
$515$	10893.1	0.932055
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−1295.89	−0.109601
$520$	0	0
$521$	−8139.22	−0.684426	−0.342213	−	0.939622i	$-0.611176\pi$
$521$	−8139.22	−0.684426	−0.342213	+	0.939622i	$0.611176\pi$
$522$	0	0
$523$	1771.23	0.148089	0.0740444	−	0.997255i	$-0.476409\pi$
$523$	1771.23	0.148089	0.0740444	+	0.997255i	$0.476409\pi$
$524$	0	0
$525$	−916.430	−0.0761834
$526$	0	0
$527$	−8466.02	−0.699783
$528$	0	0
$529$	−12166.1	−0.999930
$530$	0	0
$531$	4004.33	0.327256
$532$	0	0
$533$	−11767.4	−0.956290
$534$	0	0
$535$	−15593.1	−1.26009
$536$	0	0
$537$	−3662.13	−0.294288
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11268.6	−0.895520	−0.447760	−	0.894154i	$-0.647778\pi$
$541$	−11268.6	−0.895520	−0.447760	+	0.894154i	$0.647778\pi$
$542$	0	0
$543$	−4311.64	−0.340756
$544$	0	0
$545$	−30076.7	−2.36393
$546$	0	0
$547$	19112.2	1.49393	0.746963	−	0.664865i	$-0.231511\pi$
$547$	19112.2	1.49393	0.746963	+	0.664865i	$0.231511\pi$
$548$	0	0
$549$	4773.09	0.371058
$550$	0	0
$551$	763.978	0.0590682
$552$	0	0
$553$	3363.10	0.258614
$554$	0	0
$555$	12739.4	0.974339
$556$	0	0
$557$	−4836.24	−0.367896	−0.183948	−	0.982936i	$-0.558888\pi$
$557$	−4836.24	−0.367896	−0.183948	+	0.982936i	$0.558888\pi$
$558$	0	0
$559$	237.465	0.0179672
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7374.72	0.552056	0.276028	−	0.961150i	$-0.410982\pi$
$563$	7374.72	0.552056	0.276028	+	0.961150i	$0.410982\pi$
$564$	0	0
$565$	−15314.6	−1.14034
$566$	0	0
$567$	−305.041	−0.0225935
$568$	0	0
$569$	19402.6	1.42952	0.714761	−	0.699369i	$-0.246535\pi$
$569$	19402.6	1.42952	0.714761	+	0.699369i	$0.246535\pi$
$570$	0	0
$571$	17441.4	1.27828	0.639141	−	0.769090i	$-0.279290\pi$
$571$	17441.4	1.27828	0.639141	+	0.769090i	$0.279290\pi$
$572$	0	0
$573$	5981.74	0.436110
$574$	0	0
$575$	−75.0397	−0.00544238
$576$	0	0
$577$	4011.75	0.289447	0.144724	−	0.989472i	$-0.453771\pi$
$577$	4011.75	0.289447	0.144724	+	0.989472i	$0.453771\pi$
$578$	0	0
$579$	233.112	0.0167320
$580$	0	0
$581$	−1542.79	−0.110165
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4450.86	−0.314565
$586$	0	0
$587$	10874.4	0.764628	0.382314	−	0.924033i	$-0.375127\pi$
$587$	10874.4	0.764628	0.382314	+	0.924033i	$0.375127\pi$
$588$	0	0
$589$	4085.08	0.285777
$590$	0	0
$591$	−9830.89	−0.684245
$592$	0	0
$593$	28041.7	1.94188	0.970938	−	0.239330i	$-0.0769278\pi$
$593$	28041.7	1.94188	0.970938	+	0.239330i	$0.0769278\pi$
$594$	0	0
$595$	2456.92	0.169284
$596$	0	0
$597$	5496.62	0.376820
$598$	0	0
$599$	−11472.4	−0.782555	−0.391278	−	0.920273i	$-0.627967\pi$
$599$	−11472.4	−0.782555	−0.391278	+	0.920273i	$0.627967\pi$
$600$	0	0
$601$	2005.60	0.136124	0.0680618	−	0.997681i	$-0.478319\pi$
$601$	2005.60	0.136124	0.0680618	+	0.997681i	$0.478319\pi$
$602$	0	0
$603$	−2043.76	−0.138024
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18394.0	−1.22996	−0.614982	−	0.788541i	$-0.710837\pi$
$607$	−18394.0	−1.22996	−0.614982	+	0.788541i	$0.710837\pi$
$608$	0	0
$609$	−393.633	−0.0261918
$610$	0	0
$611$	−10589.5	−0.701153
$612$	0	0
$613$	−8351.93	−0.550296	−0.275148	−	0.961402i	$-0.588727\pi$
$613$	−8351.93	−0.550296	−0.275148	+	0.961402i	$0.588727\pi$
$614$	0	0
$615$	14713.4	0.964715
$616$	0	0
$617$	−18241.1	−1.19021	−0.595103	−	0.803649i	$-0.702889\pi$
$617$	−18241.1	−1.19021	−0.595103	+	0.803649i	$0.702889\pi$
$618$	0	0
$619$	20683.9	1.34307	0.671533	−	0.740975i	$-0.265636\pi$
$619$	20683.9	1.34307	0.671533	+	0.740975i	$0.265636\pi$
$620$	0	0
$621$	−24.9775	−0.00161403
$622$	0	0
$623$	−3423.09	−0.220134
$624$	0	0
$625$	−19184.7	−1.22782
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13441.1	−0.852038
$630$	0	0
$631$	13889.5	0.876277	0.438138	−	0.898908i	$-0.355638\pi$
$631$	13889.5	0.876277	0.438138	+	0.898908i	$0.355638\pi$
$632$	0	0
$633$	−13046.0	−0.819164
$634$	0	0
$635$	39602.8	2.47494
$636$	0	0
$637$	11326.6	0.704518
$638$	0	0
$639$	4176.52	0.258561
$640$	0	0
$641$	−14975.3	−0.922759	−0.461379	−	0.887203i	$-0.652645\pi$
$641$	−14975.3	−0.922759	−0.461379	+	0.887203i	$0.652645\pi$
$642$	0	0
$643$	−5037.90	−0.308982	−0.154491	−	0.987994i	$-0.549374\pi$
$643$	−5037.90	−0.308982	−0.154491	+	0.987994i	$0.549374\pi$
$644$	0	0
$645$	−296.913	−0.0181255
$646$	0	0
$647$	−11042.8	−0.670998	−0.335499	−	0.942040i	$-0.608905\pi$
$647$	−11042.8	−0.670998	−0.335499	+	0.942040i	$0.608905\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2104.80	−0.126718
$652$	0	0
$653$	−10095.2	−0.604983	−0.302492	−	0.953152i	$-0.597818\pi$
$653$	−10095.2	−0.604983	−0.302492	+	0.953152i	$0.597818\pi$
$654$	0	0
$655$	−19278.5	−1.15003
$656$	0	0
$657$	3791.08	0.225120
$658$	0	0
$659$	−24381.0	−1.44120	−0.720599	−	0.693352i	$-0.756133\pi$
$659$	−24381.0	−1.44120	−0.720599	+	0.693352i	$0.756133\pi$
$660$	0	0
$661$	−978.393	−0.0575720	−0.0287860	−	0.999586i	$-0.509164\pi$
$661$	−978.393	−0.0575720	−0.0287860	+	0.999586i	$0.509164\pi$
$662$	0	0
$663$	4696.01	0.275080
$664$	0	0
$665$	−1185.53	−0.0691320
$666$	0	0
$667$	−32.2317	−0.00187109
$668$	0	0
$669$	9132.70	0.527788
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−25311.2	−1.44974	−0.724871	−	0.688884i	$-0.758101\pi$
$673$	−25311.2	−1.44974	−0.724871	+	0.688884i	$0.758101\pi$
$674$	0	0
$675$	2190.13	0.124886
$676$	0	0
$677$	−11922.3	−0.676827	−0.338413	−	0.940998i	$-0.609890\pi$
$677$	−11922.3	−0.676827	−0.338413	+	0.940998i	$0.609890\pi$
$678$	0	0
$679$	1646.27	0.0930457
$680$	0	0
$681$	15419.6	0.867669
$682$	0	0
$683$	−33145.2	−1.85691	−0.928453	−	0.371451i	$-0.878860\pi$
$683$	−33145.2	−1.85691	−0.928453	+	0.371451i	$0.878860\pi$
$684$	0	0
$685$	−12142.7	−0.677297
$686$	0	0
$687$	11002.3	0.611008
$688$	0	0
$689$	−16551.4	−0.915179
$690$	0	0
$691$	−9402.26	−0.517625	−0.258813	−	0.965928i	$-0.583331\pi$
$691$	−9402.26	−0.517625	−0.258813	+	0.965928i	$0.583331\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	42430.4	2.31579
$696$	0	0
$697$	−15523.8	−0.843622
$698$	0	0
$699$	−3547.82	−0.191976
$700$	0	0
$701$	26254.9	1.41460	0.707299	−	0.706914i	$-0.249913\pi$
$701$	26254.9	1.41460	0.707299	+	0.706914i	$0.249913\pi$
$702$	0	0
$703$	6485.69	0.347955
$704$	0	0
$705$	13240.5	0.707330
$706$	0	0
$707$	−6926.73	−0.368467
$708$	0	0
$709$	−17533.8	−0.928766	−0.464383	−	0.885635i	$-0.653724\pi$
$709$	−17533.8	−0.928766	−0.464383	+	0.885635i	$0.653724\pi$
$710$	0	0
$711$	−8037.30	−0.423941
$712$	0	0
$713$	−172.346	−0.00905249
$714$	0	0
$715$	0	0
$716$	0	0
$717$	613.718	0.0319661
$718$	0	0
$719$	−20952.5	−1.08678	−0.543392	−	0.839479i	$-0.682860\pi$
$719$	−20952.5	−1.08678	−0.543392	+	0.839479i	$0.682860\pi$
$720$	0	0
$721$	−2857.39	−0.147593
$722$	0	0
$723$	−965.307	−0.0496544
$724$	0	0
$725$	2826.20	0.144776
$726$	0	0
$727$	−33896.1	−1.72921	−0.864606	−	0.502450i	$-0.832432\pi$
$727$	−33896.1	−1.72921	−0.864606	+	0.502450i	$0.832432\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	313.268	0.0158504
$732$	0	0
$733$	24580.7	1.23862	0.619311	−	0.785146i	$-0.287412\pi$
$733$	24580.7	1.23862	0.619311	+	0.785146i	$0.287412\pi$
$734$	0	0
$735$	−14162.2	−0.710724
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−19835.6	−0.987369	−0.493685	−	0.869641i	$-0.664350\pi$
$739$	−19835.6	−0.987369	−0.493685	+	0.869641i	$0.664350\pi$
$740$	0	0
$741$	−2265.95	−0.112337
$742$	0	0
$743$	7477.99	0.369234	0.184617	−	0.982811i	$-0.440896\pi$
$743$	7477.99	0.369234	0.184617	+	0.982811i	$0.440896\pi$
$744$	0	0
$745$	−13087.0	−0.643585
$746$	0	0
$747$	3687.04	0.180591
$748$	0	0
$749$	4090.25	0.199539
$750$	0	0
$751$	2407.76	0.116991	0.0584956	−	0.998288i	$-0.481370\pi$
$751$	2407.76	0.116991	0.0584956	+	0.998288i	$0.481370\pi$
$752$	0	0
$753$	8445.66	0.408735
$754$	0	0
$755$	−49908.8	−2.40578
$756$	0	0
$757$	−25107.8	−1.20549	−0.602747	−	0.797932i	$-0.705927\pi$
$757$	−25107.8	−1.20549	−0.602747	+	0.797932i	$0.705927\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−578.905	−0.0275759	−0.0137880	−	0.999905i	$-0.504389\pi$
$761$	−578.905	−0.0275759	−0.0137880	+	0.999905i	$0.504389\pi$
$762$	0	0
$763$	7889.45	0.374334
$764$	0	0
$765$	−5871.66	−0.277503
$766$	0	0
$767$	−15326.1	−0.721505
$768$	0	0
$769$	14516.6	0.680731	0.340366	−	0.940293i	$-0.389449\pi$
$769$	14516.6	0.680731	0.340366	+	0.940293i	$0.389449\pi$
$770$	0	0
$771$	4192.66	0.195843
$772$	0	0
$773$	−27386.0	−1.27426	−0.637132	−	0.770755i	$-0.719879\pi$
$773$	−27386.0	−1.27426	−0.637132	+	0.770755i	$0.719879\pi$
$774$	0	0
$775$	15112.0	0.700438
$776$	0	0
$777$	−3341.69	−0.154289
$778$	0	0
$779$	7490.63	0.344518
$780$	0	0
$781$	0	0
$782$	0	0
$783$	940.722	0.0429357
$784$	0	0
$785$	−28054.8	−1.27556
$786$	0	0
$787$	26195.6	1.18650	0.593248	−	0.805020i	$-0.297845\pi$
$787$	26195.6	1.18650	0.593248	+	0.805020i	$0.297845\pi$
$788$	0	0
$789$	−19978.2	−0.901447
$790$	0	0
$791$	4017.20	0.180576
$792$	0	0
$793$	−18268.5	−0.818076
$794$	0	0
$795$	20695.0	0.923241
$796$	0	0
$797$	−27947.2	−1.24208	−0.621042	−	0.783777i	$-0.713291\pi$
$797$	−27947.2	−1.24208	−0.621042	+	0.783777i	$0.713291\pi$
$798$	0	0
$799$	−13969.8	−0.618544
$800$	0	0
$801$	8180.67	0.360861
$802$	0	0
$803$	0	0
$804$	0	0
$805$	50.0165	0.00218988
$806$	0	0
$807$	−13929.7	−0.607620
$808$	0	0
$809$	−22600.6	−0.982195	−0.491097	−	0.871105i	$-0.663404\pi$
$809$	−22600.6	−0.982195	−0.491097	+	0.871105i	$0.663404\pi$
$810$	0	0
$811$	−14249.2	−0.616964	−0.308482	−	0.951230i	$-0.599821\pi$
$811$	−14249.2	−0.616964	−0.308482	+	0.951230i	$0.599821\pi$
$812$	0	0
$813$	6798.79	0.293289
$814$	0	0
$815$	19047.5	0.818658
$816$	0	0
$817$	−151.160	−0.00647297
$818$	0	0
$819$	1167.51	0.0498121
$820$	0	0
$821$	40191.2	1.70850	0.854252	−	0.519858i	$-0.174015\pi$
$821$	40191.2	1.70850	0.854252	+	0.519858i	$0.174015\pi$
$822$	0	0
$823$	−32296.0	−1.36788	−0.683941	−	0.729537i	$-0.739736\pi$
$823$	−32296.0	−1.36788	−0.683941	+	0.729537i	$0.739736\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19874.0	0.835655	0.417828	−	0.908526i	$-0.362792\pi$
$827$	19874.0	0.835655	0.417828	+	0.908526i	$0.362792\pi$
$828$	0	0
$829$	−10614.5	−0.444700	−0.222350	−	0.974967i	$-0.571373\pi$
$829$	−10614.5	−0.444700	−0.222350	+	0.974967i	$0.571373\pi$
$830$	0	0
$831$	9956.69	0.415636
$832$	0	0
$833$	14942.3	0.621513
$834$	0	0
$835$	21444.3	0.888754
$836$	0	0
$837$	5030.15	0.207727
$838$	0	0
$839$	−2032.06	−0.0836167	−0.0418084	−	0.999126i	$-0.513312\pi$
$839$	−2032.06	−0.0836167	−0.0418084	+	0.999126i	$0.513312\pi$
$840$	0	0
$841$	−23175.1	−0.950226
$842$	0	0
$843$	−6321.01	−0.258253
$844$	0	0
$845$	−14506.5	−0.590580
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2581.81	0.104367
$850$	0	0
$851$	−273.627	−0.0110221
$852$	0	0
$853$	23107.0	0.927511	0.463756	−	0.885963i	$-0.346502\pi$
$853$	23107.0	0.927511	0.463756	+	0.885963i	$0.346502\pi$
$854$	0	0
$855$	2833.23	0.113327
$856$	0	0
$857$	−30614.3	−1.22026	−0.610132	−	0.792300i	$-0.708884\pi$
$857$	−30614.3	−1.22026	−0.610132	+	0.792300i	$0.708884\pi$
$858$	0	0
$859$	−9473.55	−0.376290	−0.188145	−	0.982141i	$-0.560248\pi$
$859$	−9473.55	−0.376290	−0.188145	+	0.982141i	$0.560248\pi$
$860$	0	0
$861$	−3859.48	−0.152765
$862$	0	0
$863$	12421.9	0.489974	0.244987	−	0.969526i	$-0.421216\pi$
$863$	12421.9	0.489974	0.244987	+	0.969526i	$0.421216\pi$
$864$	0	0
$865$	−6201.56	−0.243768
$866$	0	0
$867$	−8543.93	−0.334680
$868$	0	0
$869$	0	0
$870$	0	0
$871$	7822.27	0.304303
$872$	0	0
$873$	−3934.33	−0.152528
$874$	0	0
$875$	2372.67	0.0916694
$876$	0	0
$877$	−28407.3	−1.09378	−0.546892	−	0.837203i	$-0.684189\pi$
$877$	−28407.3	−1.09378	−0.546892	+	0.837203i	$0.684189\pi$
$878$	0	0
$879$	13847.6	0.531362
$880$	0	0
$881$	−42248.3	−1.61564	−0.807822	−	0.589427i	$-0.799354\pi$
$881$	−42248.3	−1.61564	−0.807822	+	0.589427i	$0.799354\pi$
$882$	0	0
$883$	40931.6	1.55998	0.779988	−	0.625794i	$-0.215225\pi$
$883$	40931.6	1.55998	0.779988	+	0.625794i	$0.215225\pi$
$884$	0	0
$885$	19163.0	0.727862
$886$	0	0
$887$	−13691.8	−0.518293	−0.259147	−	0.965838i	$-0.583441\pi$
$887$	−13691.8	−0.518293	−0.259147	+	0.965838i	$0.583441\pi$
$888$	0	0
$889$	−10388.2	−0.391913
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6740.81	0.252601
$894$	0	0
$895$	−17525.4	−0.654536
$896$	0	0
$897$	95.5988	0.00355847
$898$	0	0
$899$	6491.04	0.240810
$900$	0	0
$901$	−21834.9	−0.807354
$902$	0	0
$903$	77.8837	0.00287022
$904$	0	0
$905$	−20633.7	−0.757887
$906$	0	0
$907$	38992.3	1.42747	0.713736	−	0.700414i	$-0.247001\pi$
$907$	38992.3	1.42747	0.713736	+	0.700414i	$0.247001\pi$
$908$	0	0
$909$	16553.8	0.604021
$910$	0	0
$911$	19075.5	0.693743	0.346872	−	0.937913i	$-0.387244\pi$
$911$	19075.5	0.693743	0.346872	+	0.937913i	$0.387244\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	22842.0	0.825283
$916$	0	0
$917$	5056.96	0.182111
$918$	0	0
$919$	31574.9	1.13336	0.566681	−	0.823937i	$-0.308227\pi$
$919$	31574.9	1.13336	0.566681	+	0.823937i	$0.308227\pi$
$920$	0	0
$921$	4129.67	0.147749
$922$	0	0
$923$	−15985.2	−0.570053
$924$	0	0
$925$	23992.6	0.852836
$926$	0	0
$927$	6828.72	0.241947
$928$	0	0
$929$	−30478.8	−1.07640	−0.538201	−	0.842817i	$-0.680896\pi$
$929$	−30478.8	−1.07640	−0.538201	+	0.842817i	$0.680896\pi$
$930$	0	0
$931$	−7210.06	−0.253813
$932$	0	0
$933$	−12974.0	−0.455252
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12422.6	0.433115	0.216557	−	0.976270i	$-0.430517\pi$
$937$	12422.6	0.433115	0.216557	+	0.976270i	$0.430517\pi$
$938$	0	0
$939$	21552.8	0.749042
$940$	0	0
$941$	5844.18	0.202460	0.101230	−	0.994863i	$-0.467722\pi$
$941$	5844.18	0.202460	0.101230	+	0.994863i	$0.467722\pi$
$942$	0	0
$943$	−316.024	−0.0109132
$944$	0	0
$945$	−1459.80	−0.0502509
$946$	0	0
$947$	44610.3	1.53077	0.765385	−	0.643572i	$-0.222548\pi$
$947$	44610.3	1.53077	0.765385	+	0.643572i	$0.222548\pi$
$948$	0	0
$949$	−14510.0	−0.496326
$950$	0	0
$951$	25820.3	0.880422
$952$	0	0
$953$	−56139.2	−1.90821	−0.954107	−	0.299466i	$-0.903191\pi$
$953$	−56139.2	−1.90821	−0.954107	+	0.299466i	$0.903191\pi$
$954$	0	0
$955$	28626.1	0.969967
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3185.17	0.107252
$960$	0	0
$961$	4917.34	0.165061
$962$	0	0
$963$	−9775.07	−0.327100
$964$	0	0
$965$	1115.58	0.0372142
$966$	0	0
$967$	−17998.7	−0.598550	−0.299275	−	0.954167i	$-0.596745\pi$
$967$	−17998.7	−0.598550	−0.299275	+	0.954167i	$0.596745\pi$
$968$	0	0
$969$	−2989.28	−0.0991018
$970$	0	0
$971$	4024.47	0.133009	0.0665043	−	0.997786i	$-0.478815\pi$
$971$	4024.47	0.133009	0.0665043	+	0.997786i	$0.478815\pi$
$972$	0	0
$973$	−11130.0	−0.366712
$974$	0	0
$975$	−8382.48	−0.275338
$976$	0	0
$977$	−15981.6	−0.523333	−0.261666	−	0.965158i	$-0.584272\pi$
$977$	−15981.6	−0.523333	−0.261666	+	0.965158i	$0.584272\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−18854.6	−0.613639
$982$	0	0
$983$	59406.7	1.92755	0.963775	−	0.266718i	$-0.0859393\pi$
$983$	59406.7	1.92755	0.963775	+	0.266718i	$0.0859393\pi$
$984$	0	0
$985$	−47046.5	−1.52185
$986$	0	0
$987$	−3473.14	−0.112007
$988$	0	0
$989$	6.37732	0.000205043 0
$990$	0	0
$991$	13485.5	0.432270	0.216135	−	0.976363i	$-0.430655\pi$
$991$	13485.5	0.432270	0.216135	+	0.976363i	$0.430655\pi$
$992$	0	0
$993$	10955.3	0.350106
$994$	0	0
$995$	26304.5	0.838100
$996$	0	0
$997$	57247.5	1.81850	0.909250	−	0.416250i	$-0.136656\pi$
$997$	57247.5	1.81850	0.909250	+	0.416250i	$0.136656\pi$
$998$	0	0
$999$	7986.13	0.252923

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.4.a.s.1.5	✓	6
11.10	odd	2	inner	1452.4.a.s.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1452.4.a.s.1.5	✓	6	1.1	even	1	trivial
1452.4.a.s.1.6	yes	6	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1452.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +14.3567 q^{5} -3.76593 q^{7} +9.00000 q^{9} -34.4466 q^{13} +43.0702 q^{15} -45.4425 q^{17} +21.9272 q^{19} -11.2978 q^{21} -0.925093 q^{23} +81.1158 q^{25} +27.0000 q^{27} +34.8415 q^{29} +186.302 q^{31} -54.0665 q^{35} +295.783 q^{37} -103.340 q^{39} +341.613 q^{41} -6.89371 q^{43} +129.211 q^{45} +307.418 q^{47} -328.818 q^{49} -136.328 q^{51} +480.495 q^{53} +65.7816 q^{57} +444.925 q^{59} +530.344 q^{61} -33.8934 q^{63} -494.540 q^{65} -227.084 q^{67} -2.77528 q^{69} +464.058 q^{71} +421.231 q^{73} +243.347 q^{75} -893.034 q^{79} +81.0000 q^{81} +409.671 q^{83} -652.406 q^{85} +104.525 q^{87} +908.963 q^{89} +129.723 q^{91} +558.905 q^{93} +314.803 q^{95} -437.148 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{3} - 12 q^{5} + 54 q^{9} - 36 q^{15} + 156 q^{23} + 150 q^{25} + 162 q^{27} + 150 q^{31} + 366 q^{37} - 108 q^{45} + 540 q^{47} + 1308 q^{49} + 360 q^{53} + 2508 q^{59} + 1014 q^{67} + 468 q^{69}+ \cdots + 5790 q^{97}+O(q^{100}) \) 6 * q + 18 * q^3 - 12 * q^5 + 54 * q^9 - 36 * q^15 + 156 * q^23 + 150 * q^25 + 162 * q^27 + 150 * q^31 + 366 * q^37 - 108 * q^45 + 540 * q^47 + 1308 * q^49 + 360 * q^53 + 2508 * q^59 + 1014 * q^67 + 468 * q^69 - 1176 * q^71 + 450 * q^75 + 486 * q^81 + 2724 * q^89 + 1728 * q^91 + 450 * q^93 + 5790 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more