LMFDB - Embedded newform 552.4.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [552,4,Mod(1,552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("552.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	552.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-12,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.5690543232$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1623576.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 22x^{2} + 17x + 91 $ x^4 - x^3 - 22x^2 + 17x + 91
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.94329$ of defining polynomial
Character	$\chi$	$=$	552.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -8.67405 q^{5} -19.8035 q^{7} +9.00000 q^{9} +46.0239 q^{11} +61.0443 q^{13} +26.0222 q^{15} -40.1964 q^{17} +96.6739 q^{19} +59.4104 q^{21} +23.0000 q^{23} -49.7608 q^{25} -27.0000 q^{27} +2.45709 q^{29} -98.5456 q^{31} -138.072 q^{33} +171.776 q^{35} -137.979 q^{37} -183.133 q^{39} -366.840 q^{41} -405.130 q^{43} -78.0665 q^{45} +112.205 q^{47} +49.1777 q^{49} +120.589 q^{51} -250.039 q^{53} -399.213 q^{55} -290.022 q^{57} -123.308 q^{59} +643.722 q^{61} -178.231 q^{63} -529.501 q^{65} -12.0270 q^{67} -69.0000 q^{69} +332.539 q^{71} +71.5191 q^{73} +149.282 q^{75} -911.433 q^{77} +1075.74 q^{79} +81.0000 q^{81} -188.768 q^{83} +348.666 q^{85} -7.37128 q^{87} -1228.86 q^{89} -1208.89 q^{91} +295.637 q^{93} -838.555 q^{95} -1048.33 q^{97} +414.215 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 14 q^{5} + 14 q^{7} + 36 q^{9} - 16 q^{11} + 4 q^{13} + 42 q^{15} - 22 q^{17} + 134 q^{19} - 42 q^{21} + 92 q^{23} + 24 q^{25} - 108 q^{27} - 180 q^{29} + 276 q^{31} + 48 q^{33} - 96 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 - 14 * q^5 + 14 * q^7 + 36 * q^9 - 16 * q^11 + 4 * q^13 + 42 * q^15 - 22 * q^17 + 134 * q^19 - 42 * q^21 + 92 * q^23 + 24 * q^25 - 108 * q^27 - 180 * q^29 + 276 * q^31 + 48 * q^33 - 96 * q^35 - 244 * q^37 - 12 * q^39 - 416 * q^41 - 242 * q^43 - 126 * q^45 - 384 * q^47 - 296 * q^49 + 66 * q^51 - 1426 * q^53 - 296 * q^55 - 402 * q^57 - 1000 * q^59 - 1140 * q^61 + 126 * q^63 - 1916 * q^65 - 646 * q^67 - 276 * q^69 - 432 * q^71 - 1568 * q^73 - 72 * q^75 - 2472 * q^77 + 778 * q^79 + 324 * q^81 + 112 * q^83 - 1772 * q^85 + 540 * q^87 - 2458 * q^89 - 316 * q^91 - 828 * q^93 + 952 * q^95 - 980 * q^97 - 144 * 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
$5$	−8.67405	−0.775831	−0.387915	−	0.921695i	$-0.626805\pi$
$5$	−8.67405	−0.775831	−0.387915	+	0.921695i	$0.626805\pi$
$6$	0	0
$7$	−19.8035	−1.06929	−0.534644	−	0.845078i	$-0.679554\pi$
$7$	−19.8035	−1.06929	−0.534644	+	0.845078i	$0.679554\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	46.0239	1.26152	0.630760	−	0.775978i	$-0.282743\pi$
$11$	46.0239	1.26152	0.630760	+	0.775978i	$0.282743\pi$
$12$	0	0
$13$	61.0443	1.30236	0.651178	−	0.758925i	$-0.274275\pi$
$13$	61.0443	1.30236	0.651178	+	0.758925i	$0.274275\pi$
$14$	0	0
$15$	26.0222	0.447926
$16$	0	0
$17$	−40.1964	−0.573474	−0.286737	−	0.958009i	$-0.592571\pi$
$17$	−40.1964	−0.573474	−0.286737	+	0.958009i	$0.592571\pi$
$18$	0	0
$19$	96.6739	1.16729	0.583645	−	0.812009i	$-0.301626\pi$
$19$	96.6739	1.16729	0.583645	+	0.812009i	$0.301626\pi$
$20$	0	0
$21$	59.4104	0.617353
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−49.7608	−0.398087
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	2.45709	0.0157335	0.00786673	−	0.999969i	$-0.497496\pi$
$29$	2.45709	0.0157335	0.00786673	+	0.999969i	$0.497496\pi$
$30$	0	0
$31$	−98.5456	−0.570946	−0.285473	−	0.958387i	$-0.592151\pi$
$31$	−98.5456	−0.570946	−0.285473	+	0.958387i	$0.592151\pi$
$32$	0	0
$33$	−138.072	−0.728339
$34$	0	0
$35$	171.776	0.829586
$36$	0	0
$37$	−137.979	−0.613072	−0.306536	−	0.951859i	$-0.599170\pi$
$37$	−137.979	−0.613072	−0.306536	+	0.951859i	$0.599170\pi$
$38$	0	0
$39$	−183.133	−0.751916
$40$	0	0
$41$	−366.840	−1.39733	−0.698667	−	0.715447i	$-0.746223\pi$
$41$	−366.840	−1.39733	−0.698667	+	0.715447i	$0.746223\pi$
$42$	0	0
$43$	−405.130	−1.43678	−0.718392	−	0.695639i	$-0.755121\pi$
$43$	−405.130	−1.43678	−0.718392	+	0.695639i	$0.755121\pi$
$44$	0	0
$45$	−78.0665	−0.258610
$46$	0	0
$47$	112.205	0.348228	0.174114	−	0.984726i	$-0.444294\pi$
$47$	112.205	0.348228	0.174114	+	0.984726i	$0.444294\pi$
$48$	0	0
$49$	49.1777	0.143375
$50$	0	0
$51$	120.589	0.331095
$52$	0	0
$53$	−250.039	−0.648027	−0.324014	−	0.946052i	$-0.605032\pi$
$53$	−250.039	−0.648027	−0.324014	+	0.946052i	$0.605032\pi$
$54$	0	0
$55$	−399.213	−0.978726
$56$	0	0
$57$	−290.022	−0.673936
$58$	0	0
$59$	−123.308	−0.272090	−0.136045	−	0.990703i	$-0.543439\pi$
$59$	−123.308	−0.272090	−0.136045	+	0.990703i	$0.543439\pi$
$60$	0	0
$61$	643.722	1.35115	0.675575	−	0.737291i	$-0.263895\pi$
$61$	643.722	1.35115	0.675575	+	0.737291i	$0.263895\pi$
$62$	0	0
$63$	−178.231	−0.356429
$64$	0	0
$65$	−529.501	−1.01041
$66$	0	0
$67$	−12.0270	−0.0219303	−0.0109651	−	0.999940i	$-0.503490\pi$
$67$	−12.0270	−0.0219303	−0.0109651	+	0.999940i	$0.503490\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	332.539	0.555846	0.277923	−	0.960603i	$-0.410354\pi$
$71$	332.539	0.555846	0.277923	+	0.960603i	$0.410354\pi$
$72$	0	0
$73$	71.5191	0.114667	0.0573334	−	0.998355i	$-0.481740\pi$
$73$	71.5191	0.114667	0.0573334	+	0.998355i	$0.481740\pi$
$74$	0	0
$75$	149.282	0.229835
$76$	0	0
$77$	−911.433	−1.34893
$78$	0	0
$79$	1075.74	1.53203	0.766017	−	0.642821i	$-0.222236\pi$
$79$	1075.74	1.53203	0.766017	+	0.642821i	$0.222236\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−188.768	−0.249638	−0.124819	−	0.992180i	$-0.539835\pi$
$83$	−188.768	−0.249638	−0.124819	+	0.992180i	$0.539835\pi$
$84$	0	0
$85$	348.666	0.444919
$86$	0	0
$87$	−7.37128	−0.00908372
$88$	0	0
$89$	−1228.86	−1.46359	−0.731793	−	0.681527i	$-0.761316\pi$
$89$	−1228.86	−1.46359	−0.731793	+	0.681527i	$0.761316\pi$
$90$	0	0
$91$	−1208.89	−1.39259
$92$	0	0
$93$	295.637	0.329636
$94$	0	0
$95$	−838.555	−0.905620
$96$	0	0
$97$	−1048.33	−1.09734	−0.548670	−	0.836039i	$-0.684866\pi$
$97$	−1048.33	−1.09734	−0.548670	+	0.836039i	$0.684866\pi$
$98$	0	0
$99$	414.215	0.420507
$100$	0	0
$101$	−1629.34	−1.60520	−0.802602	−	0.596515i	$-0.796551\pi$
$101$	−1629.34	−1.60520	−0.802602	+	0.596515i	$0.796551\pi$
$102$	0	0
$103$	1126.13	1.07729	0.538646	−	0.842532i	$-0.318936\pi$
$103$	1126.13	1.07729	0.538646	+	0.842532i	$0.318936\pi$
$104$	0	0
$105$	−515.329	−0.478962
$106$	0	0
$107$	−1163.68	−1.05137	−0.525687	−	0.850678i	$-0.676192\pi$
$107$	−1163.68	−1.05137	−0.525687	+	0.850678i	$0.676192\pi$
$108$	0	0
$109$	−1598.16	−1.40437	−0.702185	−	0.711995i	$-0.747792\pi$
$109$	−1598.16	−1.40437	−0.702185	+	0.711995i	$0.747792\pi$
$110$	0	0
$111$	413.938	0.353958
$112$	0	0
$113$	−177.305	−0.147605	−0.0738027	−	0.997273i	$-0.523514\pi$
$113$	−177.305	−0.147605	−0.0738027	+	0.997273i	$0.523514\pi$
$114$	0	0
$115$	−199.503	−0.161772
$116$	0	0
$117$	549.398	0.434119
$118$	0	0
$119$	796.029	0.613209
$120$	0	0
$121$	787.197	0.591433
$122$	0	0
$123$	1100.52	0.806751
$124$	0	0
$125$	1515.88	1.08468
$126$	0	0
$127$	−497.140	−0.347355	−0.173677	−	0.984803i	$-0.555565\pi$
$127$	−497.140	−0.347355	−0.173677	+	0.984803i	$0.555565\pi$
$128$	0	0
$129$	1215.39	0.829527
$130$	0	0
$131$	−1931.02	−1.28789	−0.643945	−	0.765071i	$-0.722704\pi$
$131$	−1931.02	−1.28789	−0.643945	+	0.765071i	$0.722704\pi$
$132$	0	0
$133$	−1914.48	−1.24817
$134$	0	0
$135$	234.199	0.149309
$136$	0	0
$137$	−2615.29	−1.63095	−0.815473	−	0.578795i	$-0.803523\pi$
$137$	−2615.29	−1.63095	−0.815473	+	0.578795i	$0.803523\pi$
$138$	0	0
$139$	633.621	0.386641	0.193320	−	0.981136i	$-0.438074\pi$
$139$	633.621	0.386641	0.193320	+	0.981136i	$0.438074\pi$
$140$	0	0
$141$	−336.614	−0.201050
$142$	0	0
$143$	2809.49	1.64295
$144$	0	0
$145$	−21.3129	−0.0122065
$146$	0	0
$147$	−147.533	−0.0827777
$148$	0	0
$149$	1115.41	0.613274	0.306637	−	0.951826i	$-0.400796\pi$
$149$	1115.41	0.613274	0.306637	+	0.951826i	$0.400796\pi$
$150$	0	0
$151$	2054.35	1.10716	0.553578	−	0.832798i	$-0.313262\pi$
$151$	2054.35	1.10716	0.553578	+	0.832798i	$0.313262\pi$
$152$	0	0
$153$	−361.768	−0.191158
$154$	0	0
$155$	854.790	0.442957
$156$	0	0
$157$	−1656.55	−0.842083	−0.421041	−	0.907041i	$-0.638335\pi$
$157$	−1656.55	−0.842083	−0.421041	+	0.907041i	$0.638335\pi$
$158$	0	0
$159$	750.116	0.374139
$160$	0	0
$161$	−455.480	−0.222962
$162$	0	0
$163$	−441.687	−0.212243	−0.106121	−	0.994353i	$-0.533843\pi$
$163$	−441.687	−0.212243	−0.106121	+	0.994353i	$0.533843\pi$
$164$	0	0
$165$	1197.64	0.565068
$166$	0	0
$167$	−850.795	−0.394230	−0.197115	−	0.980380i	$-0.563157\pi$
$167$	−850.795	−0.394230	−0.197115	+	0.980380i	$0.563157\pi$
$168$	0	0
$169$	1529.40	0.696131
$170$	0	0
$171$	870.065	0.389097
$172$	0	0
$173$	−2187.80	−0.961475	−0.480738	−	0.876865i	$-0.659631\pi$
$173$	−2187.80	−0.961475	−0.480738	+	0.876865i	$0.659631\pi$
$174$	0	0
$175$	985.437	0.425669
$176$	0	0
$177$	369.924	0.157092
$178$	0	0
$179$	853.200	0.356264	0.178132	−	0.984007i	$-0.442995\pi$
$179$	853.200	0.356264	0.178132	+	0.984007i	$0.442995\pi$
$180$	0	0
$181$	−1476.82	−0.606471	−0.303236	−	0.952916i	$-0.598067\pi$
$181$	−1476.82	−0.606471	−0.303236	+	0.952916i	$0.598067\pi$
$182$	0	0
$183$	−1931.17	−0.780087
$184$	0	0
$185$	1196.84	0.475640
$186$	0	0
$187$	−1849.99	−0.723449
$188$	0	0
$189$	534.694	0.205784
$190$	0	0
$191$	−545.456	−0.206638	−0.103319	−	0.994648i	$-0.532946\pi$
$191$	−545.456	−0.206638	−0.103319	+	0.994648i	$0.532946\pi$
$192$	0	0
$193$	3683.88	1.37395	0.686973	−	0.726683i	$-0.258939\pi$
$193$	3683.88	1.37395	0.686973	+	0.726683i	$0.258939\pi$
$194$	0	0
$195$	1588.50	0.583359
$196$	0	0
$197$	−999.094	−0.361332	−0.180666	−	0.983544i	$-0.557825\pi$
$197$	−999.094	−0.361332	−0.180666	+	0.983544i	$0.557825\pi$
$198$	0	0
$199$	4943.50	1.76098	0.880491	−	0.474063i	$-0.157213\pi$
$199$	4943.50	1.76098	0.880491	+	0.474063i	$0.157213\pi$
$200$	0	0
$201$	36.0809	0.0126615
$202$	0	0
$203$	−48.6590	−0.0168236
$204$	0	0
$205$	3181.99	1.08410
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	4449.31	1.47256
$210$	0	0
$211$	−885.122	−0.288788	−0.144394	−	0.989520i	$-0.546123\pi$
$211$	−885.122	−0.288788	−0.144394	+	0.989520i	$0.546123\pi$
$212$	0	0
$213$	−997.616	−0.320918
$214$	0	0
$215$	3514.12	1.11470
$216$	0	0
$217$	1951.55	0.610505
$218$	0	0
$219$	−214.557	−0.0662029
$220$	0	0
$221$	−2453.76	−0.746868
$222$	0	0
$223$	−3715.27	−1.11566	−0.557831	−	0.829955i	$-0.688366\pi$
$223$	−3715.27	−1.11566	−0.557831	+	0.829955i	$0.688366\pi$
$224$	0	0
$225$	−447.847	−0.132696
$226$	0	0
$227$	−1088.85	−0.318367	−0.159184	−	0.987249i	$-0.550886\pi$
$227$	−1088.85	−0.318367	−0.159184	+	0.987249i	$0.550886\pi$
$228$	0	0
$229$	2163.74	0.624383	0.312192	−	0.950019i	$-0.398937\pi$
$229$	2163.74	0.624383	0.312192	+	0.950019i	$0.398937\pi$
$230$	0	0
$231$	2734.30	0.778804
$232$	0	0
$233$	−3172.39	−0.891976	−0.445988	−	0.895039i	$-0.647148\pi$
$233$	−3172.39	−0.891976	−0.445988	+	0.895039i	$0.647148\pi$
$234$	0	0
$235$	−973.268	−0.270166
$236$	0	0
$237$	−3227.23	−0.884520
$238$	0	0
$239$	−6385.22	−1.72814	−0.864071	−	0.503371i	$-0.832093\pi$
$239$	−6385.22	−1.72814	−0.864071	+	0.503371i	$0.832093\pi$
$240$	0	0
$241$	5630.66	1.50499	0.752495	−	0.658598i	$-0.228850\pi$
$241$	5630.66	1.50499	0.752495	+	0.658598i	$0.228850\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−426.570	−0.111235
$246$	0	0
$247$	5901.39	1.52023
$248$	0	0
$249$	566.303	0.144128
$250$	0	0
$251$	180.007	0.0452666	0.0226333	−	0.999744i	$-0.492795\pi$
$251$	180.007	0.0452666	0.0226333	+	0.999744i	$0.492795\pi$
$252$	0	0
$253$	1058.55	0.263045
$254$	0	0
$255$	−1046.00	−0.256874
$256$	0	0
$257$	−1261.82	−0.306264	−0.153132	−	0.988206i	$-0.548936\pi$
$257$	−1261.82	−0.306264	−0.153132	+	0.988206i	$0.548936\pi$
$258$	0	0
$259$	2732.47	0.655551
$260$	0	0
$261$	22.1138	0.00524449
$262$	0	0
$263$	2168.60	0.508447	0.254223	−	0.967146i	$-0.418180\pi$
$263$	2168.60	0.508447	0.254223	+	0.967146i	$0.418180\pi$
$264$	0	0
$265$	2168.85	0.502760
$266$	0	0
$267$	3686.59	0.845001
$268$	0	0
$269$	3314.38	0.751232	0.375616	−	0.926775i	$-0.377431\pi$
$269$	3314.38	0.751232	0.375616	+	0.926775i	$0.377431\pi$
$270$	0	0
$271$	−7146.46	−1.60191	−0.800953	−	0.598728i	$-0.795673\pi$
$271$	−7146.46	−1.60191	−0.800953	+	0.598728i	$0.795673\pi$
$272$	0	0
$273$	3626.67	0.804014
$274$	0	0
$275$	−2290.19	−0.502194
$276$	0	0
$277$	−4381.36	−0.950363	−0.475182	−	0.879888i	$-0.657618\pi$
$277$	−4381.36	−0.950363	−0.475182	+	0.879888i	$0.657618\pi$
$278$	0	0
$279$	−886.910	−0.190315
$280$	0	0
$281$	−3484.10	−0.739658	−0.369829	−	0.929100i	$-0.620584\pi$
$281$	−3484.10	−0.739658	−0.369829	+	0.929100i	$0.620584\pi$
$282$	0	0
$283$	7754.76	1.62888	0.814439	−	0.580249i	$-0.197045\pi$
$283$	7754.76	1.62888	0.814439	+	0.580249i	$0.197045\pi$
$284$	0	0
$285$	2515.66	0.522860
$286$	0	0
$287$	7264.70	1.49415
$288$	0	0
$289$	−3297.25	−0.671127
$290$	0	0
$291$	3144.99	0.633549
$292$	0	0
$293$	−5165.64	−1.02997	−0.514983	−	0.857200i	$-0.672202\pi$
$293$	−5165.64	−1.02997	−0.514983	+	0.857200i	$0.672202\pi$
$294$	0	0
$295$	1069.58	0.211096
$296$	0	0
$297$	−1242.64	−0.242780
$298$	0	0
$299$	1404.02	0.271560
$300$	0	0
$301$	8022.98	1.53633
$302$	0	0
$303$	4888.02	0.926764
$304$	0	0
$305$	−5583.68	−1.04826
$306$	0	0
$307$	5113.44	0.950617	0.475309	−	0.879819i	$-0.342336\pi$
$307$	5113.44	0.950617	0.475309	+	0.879819i	$0.342336\pi$
$308$	0	0
$309$	−3378.40	−0.621975
$310$	0	0
$311$	873.505	0.159267	0.0796333	−	0.996824i	$-0.474625\pi$
$311$	873.505	0.159267	0.0796333	+	0.996824i	$0.474625\pi$
$312$	0	0
$313$	−4695.50	−0.847941	−0.423970	−	0.905676i	$-0.639364\pi$
$313$	−4695.50	−0.847941	−0.423970	+	0.905676i	$0.639364\pi$
$314$	0	0
$315$	1545.99	0.276529
$316$	0	0
$317$	−7580.48	−1.34310	−0.671549	−	0.740960i	$-0.734371\pi$
$317$	−7580.48	−1.34310	−0.671549	+	0.740960i	$0.734371\pi$
$318$	0	0
$319$	113.085	0.0198481
$320$	0	0
$321$	3491.04	0.607011
$322$	0	0
$323$	−3885.95	−0.669411
$324$	0	0
$325$	−3037.61	−0.518451
$326$	0	0
$327$	4794.49	0.810813
$328$	0	0
$329$	−2222.04	−0.372356
$330$	0	0
$331$	−1816.51	−0.301644	−0.150822	−	0.988561i	$-0.548192\pi$
$331$	−1816.51	−0.301644	−0.150822	+	0.988561i	$0.548192\pi$
$332$	0	0
$333$	−1241.82	−0.204357
$334$	0	0
$335$	104.323	0.0170142
$336$	0	0
$337$	4946.46	0.799558	0.399779	−	0.916612i	$-0.369087\pi$
$337$	4946.46	0.799558	0.399779	+	0.916612i	$0.369087\pi$
$338$	0	0
$339$	531.914	0.0852200
$340$	0	0
$341$	−4535.45	−0.720259
$342$	0	0
$343$	5818.70	0.915978
$344$	0	0
$345$	598.510	0.0933990
$346$	0	0
$347$	−8830.59	−1.36614	−0.683071	−	0.730352i	$-0.739356\pi$
$347$	−8830.59	−1.36614	−0.683071	+	0.730352i	$0.739356\pi$
$348$	0	0
$349$	3170.24	0.486244	0.243122	−	0.969996i	$-0.421828\pi$
$349$	3170.24	0.486244	0.243122	+	0.969996i	$0.421828\pi$
$350$	0	0
$351$	−1648.19	−0.250639
$352$	0	0
$353$	7498.78	1.13065	0.565325	−	0.824868i	$-0.308751\pi$
$353$	7498.78	1.13065	0.565325	+	0.824868i	$0.308751\pi$
$354$	0	0
$355$	−2884.46	−0.431242
$356$	0	0
$357$	−2388.09	−0.354036
$358$	0	0
$359$	5761.34	0.846997	0.423498	−	0.905897i	$-0.360802\pi$
$359$	5761.34	0.846997	0.423498	+	0.905897i	$0.360802\pi$
$360$	0	0
$361$	2486.85	0.362567
$362$	0	0
$363$	−2361.59	−0.341464
$364$	0	0
$365$	−620.361	−0.0889621
$366$	0	0
$367$	3504.14	0.498405	0.249203	−	0.968451i	$-0.419831\pi$
$367$	3504.14	0.498405	0.249203	+	0.968451i	$0.419831\pi$
$368$	0	0
$369$	−3301.56	−0.465778
$370$	0	0
$371$	4951.64	0.692927
$372$	0	0
$373$	−8701.82	−1.20794	−0.603972	−	0.797006i	$-0.706416\pi$
$373$	−8701.82	−1.20794	−0.603972	+	0.797006i	$0.706416\pi$
$374$	0	0
$375$	−4547.65	−0.626239
$376$	0	0
$377$	149.991	0.0204906
$378$	0	0
$379$	−10063.2	−1.36388	−0.681940	−	0.731408i	$-0.738864\pi$
$379$	−10063.2	−1.36388	−0.681940	+	0.731408i	$0.738864\pi$
$380$	0	0
$381$	1491.42	0.200545
$382$	0	0
$383$	−13910.3	−1.85582	−0.927912	−	0.372799i	$-0.878398\pi$
$383$	−13910.3	−1.85582	−0.927912	+	0.372799i	$0.878398\pi$
$384$	0	0
$385$	7905.81	1.04654
$386$	0	0
$387$	−3646.17	−0.478928
$388$	0	0
$389$	11518.6	1.50133	0.750667	−	0.660681i	$-0.229732\pi$
$389$	11518.6	1.50133	0.750667	+	0.660681i	$0.229732\pi$
$390$	0	0
$391$	−924.517	−0.119578
$392$	0	0
$393$	5793.05	0.743564
$394$	0	0
$395$	−9331.06	−1.18860
$396$	0	0
$397$	7851.67	0.992605	0.496302	−	0.868150i	$-0.334691\pi$
$397$	7851.67	0.992605	0.496302	+	0.868150i	$0.334691\pi$
$398$	0	0
$399$	5743.44	0.720631
$400$	0	0
$401$	−1145.46	−0.142647	−0.0713236	−	0.997453i	$-0.522722\pi$
$401$	−1145.46	−0.142647	−0.0713236	+	0.997453i	$0.522722\pi$
$402$	0	0
$403$	−6015.64	−0.743574
$404$	0	0
$405$	−702.598	−0.0862034
$406$	0	0
$407$	−6350.35	−0.773403
$408$	0	0
$409$	−1521.03	−0.183888	−0.0919441	−	0.995764i	$-0.529308\pi$
$409$	−1521.03	−0.183888	−0.0919441	+	0.995764i	$0.529308\pi$
$410$	0	0
$411$	7845.88	0.941627
$412$	0	0
$413$	2441.93	0.290943
$414$	0	0
$415$	1637.38	0.193677
$416$	0	0
$417$	−1900.86	−0.223227
$418$	0	0
$419$	9961.20	1.16142	0.580712	−	0.814109i	$-0.302774\pi$
$419$	9961.20	1.16142	0.580712	+	0.814109i	$0.302774\pi$
$420$	0	0
$421$	16828.4	1.94813	0.974067	−	0.226260i	$-0.0726499\pi$
$421$	16828.4	1.94813	0.974067	+	0.226260i	$0.0726499\pi$
$422$	0	0
$423$	1009.84	0.116076
$424$	0	0
$425$	2000.21	0.228292
$426$	0	0
$427$	−12747.9	−1.44477
$428$	0	0
$429$	−8428.48	−0.948557
$430$	0	0
$431$	−12717.8	−1.42133	−0.710666	−	0.703530i	$-0.751606\pi$
$431$	−12717.8	−1.42133	−0.710666	+	0.703530i	$0.751606\pi$
$432$	0	0
$433$	−11248.1	−1.24838	−0.624190	−	0.781272i	$-0.714571\pi$
$433$	−11248.1	−1.24838	−0.624190	+	0.781272i	$0.714571\pi$
$434$	0	0
$435$	63.9388	0.00704743
$436$	0	0
$437$	2223.50	0.243397
$438$	0	0
$439$	−12938.2	−1.40662	−0.703312	−	0.710881i	$-0.748296\pi$
$439$	−12938.2	−1.40662	−0.703312	+	0.710881i	$0.748296\pi$
$440$	0	0
$441$	442.599	0.0477917
$442$	0	0
$443$	13580.7	1.45652	0.728258	−	0.685303i	$-0.240330\pi$
$443$	13580.7	1.45652	0.728258	+	0.685303i	$0.240330\pi$
$444$	0	0
$445$	10659.2	1.13549
$446$	0	0
$447$	−3346.23	−0.354074
$448$	0	0
$449$	−1491.04	−0.156718	−0.0783592	−	0.996925i	$-0.524968\pi$
$449$	−1491.04	−0.156718	−0.0783592	+	0.996925i	$0.524968\pi$
$450$	0	0
$451$	−16883.4	−1.76277
$452$	0	0
$453$	−6163.04	−0.639216
$454$	0	0
$455$	10486.0	1.08042
$456$	0	0
$457$	12837.2	1.31400	0.657000	−	0.753890i	$-0.271825\pi$
$457$	12837.2	1.31400	0.657000	+	0.753890i	$0.271825\pi$
$458$	0	0
$459$	1085.30	0.110365
$460$	0	0
$461$	4401.50	0.444682	0.222341	−	0.974969i	$-0.428630\pi$
$461$	4401.50	0.444682	0.222341	+	0.974969i	$0.428630\pi$
$462$	0	0
$463$	−7008.56	−0.703489	−0.351745	−	0.936096i	$-0.614411\pi$
$463$	−7008.56	−0.703489	−0.351745	+	0.936096i	$0.614411\pi$
$464$	0	0
$465$	−2564.37	−0.255741
$466$	0	0
$467$	−11424.2	−1.13201	−0.566004	−	0.824402i	$-0.691511\pi$
$467$	−11424.2	−1.13201	−0.566004	+	0.824402i	$0.691511\pi$
$468$	0	0
$469$	238.176	0.0234498
$470$	0	0
$471$	4969.65	0.486177
$472$	0	0
$473$	−18645.6	−1.81253
$474$	0	0
$475$	−4810.58	−0.464683
$476$	0	0
$477$	−2250.35	−0.216009
$478$	0	0
$479$	9251.61	0.882499	0.441249	−	0.897384i	$-0.354535\pi$
$479$	9251.61	0.882499	0.441249	+	0.897384i	$0.354535\pi$
$480$	0	0
$481$	−8422.85	−0.798439
$482$	0	0
$483$	1366.44	0.128727
$484$	0	0
$485$	9093.28	0.851350
$486$	0	0
$487$	10761.6	1.00135	0.500673	−	0.865636i	$-0.333086\pi$
$487$	10761.6	1.00135	0.500673	+	0.865636i	$0.333086\pi$
$488$	0	0
$489$	1325.06	0.122538
$490$	0	0
$491$	−11506.4	−1.05759	−0.528794	−	0.848750i	$-0.677356\pi$
$491$	−11506.4	−1.05759	−0.528794	+	0.848750i	$0.677356\pi$
$492$	0	0
$493$	−98.7663	−0.00902274
$494$	0	0
$495$	−3592.92	−0.326242
$496$	0	0
$497$	−6585.42	−0.594359
$498$	0	0
$499$	−21982.2	−1.97206	−0.986029	−	0.166574i	$-0.946729\pi$
$499$	−21982.2	−1.97206	−0.986029	+	0.166574i	$0.946729\pi$
$500$	0	0
$501$	2552.38	0.227609
$502$	0	0
$503$	−4490.69	−0.398071	−0.199036	−	0.979992i	$-0.563781\pi$
$503$	−4490.69	−0.398071	−0.199036	+	0.979992i	$0.563781\pi$
$504$	0	0
$505$	14133.0	1.24537
$506$	0	0
$507$	−4588.20	−0.401912
$508$	0	0
$509$	18802.4	1.63733	0.818666	−	0.574270i	$-0.194714\pi$
$509$	18802.4	1.63733	0.818666	+	0.574270i	$0.194714\pi$
$510$	0	0
$511$	−1416.33	−0.122612
$512$	0	0
$513$	−2610.20	−0.224645
$514$	0	0
$515$	−9768.12	−0.835796
$516$	0	0
$517$	5164.09	0.439297
$518$	0	0
$519$	6563.39	0.555108
$520$	0	0
$521$	−10094.8	−0.848866	−0.424433	−	0.905459i	$-0.639527\pi$
$521$	−10094.8	−0.848866	−0.424433	+	0.905459i	$0.639527\pi$
$522$	0	0
$523$	−14732.3	−1.23174	−0.615870	−	0.787848i	$-0.711195\pi$
$523$	−14732.3	−1.23174	−0.615870	+	0.787848i	$0.711195\pi$
$524$	0	0
$525$	−2956.31	−0.245760
$526$	0	0
$527$	3961.18	0.327423
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−1109.77	−0.0906968
$532$	0	0
$533$	−22393.4	−1.81983
$534$	0	0
$535$	10093.8	0.815689
$536$	0	0
$537$	−2559.60	−0.205689
$538$	0	0
$539$	2263.35	0.180871
$540$	0	0
$541$	2074.91	0.164894	0.0824468	−	0.996595i	$-0.473727\pi$
$541$	2074.91	0.164894	0.0824468	+	0.996595i	$0.473727\pi$
$542$	0	0
$543$	4430.47	0.350146
$544$	0	0
$545$	13862.6	1.08955
$546$	0	0
$547$	−18548.2	−1.44984	−0.724919	−	0.688834i	$-0.758123\pi$
$547$	−18548.2	−1.44984	−0.724919	+	0.688834i	$0.758123\pi$
$548$	0	0
$549$	5793.50	0.450383
$550$	0	0
$551$	237.537	0.0183655
$552$	0	0
$553$	−21303.5	−1.63818
$554$	0	0
$555$	−3590.52	−0.274611
$556$	0	0
$557$	−2369.39	−0.180241	−0.0901207	−	0.995931i	$-0.528725\pi$
$557$	−2369.39	−0.180241	−0.0901207	+	0.995931i	$0.528725\pi$
$558$	0	0
$559$	−24730.8	−1.87120
$560$	0	0
$561$	5549.98	0.417684
$562$	0	0
$563$	8946.87	0.669743	0.334872	−	0.942264i	$-0.391307\pi$
$563$	8946.87	0.669743	0.334872	+	0.942264i	$0.391307\pi$
$564$	0	0
$565$	1537.95	0.114517
$566$	0	0
$567$	−1604.08	−0.118810
$568$	0	0
$569$	−18755.8	−1.38187	−0.690934	−	0.722918i	$-0.742800\pi$
$569$	−18755.8	−1.38187	−0.690934	+	0.722918i	$0.742800\pi$
$570$	0	0
$571$	−16337.1	−1.19735	−0.598674	−	0.800992i	$-0.704306\pi$
$571$	−16337.1	−1.19735	−0.598674	+	0.800992i	$0.704306\pi$
$572$	0	0
$573$	1636.37	0.119302
$574$	0	0
$575$	−1144.50	−0.0830068
$576$	0	0
$577$	−12440.9	−0.897613	−0.448807	−	0.893629i	$-0.648151\pi$
$577$	−12440.9	−0.897613	−0.448807	+	0.893629i	$0.648151\pi$
$578$	0	0
$579$	−11051.6	−0.793248
$580$	0	0
$581$	3738.25	0.266934
$582$	0	0
$583$	−11507.7	−0.817500
$584$	0	0
$585$	−4765.51	−0.336803
$586$	0	0
$587$	15761.8	1.10828	0.554141	−	0.832423i	$-0.313047\pi$
$587$	15761.8	1.10828	0.554141	+	0.832423i	$0.313047\pi$
$588$	0	0
$589$	−9526.79	−0.666459
$590$	0	0
$591$	2997.28	0.208615
$592$	0	0
$593$	14371.7	0.995233	0.497617	−	0.867397i	$-0.334209\pi$
$593$	14371.7	0.995233	0.497617	+	0.867397i	$0.334209\pi$
$594$	0	0
$595$	−6904.79	−0.475746
$596$	0	0
$597$	−14830.5	−1.01670
$598$	0	0
$599$	−9596.41	−0.654589	−0.327294	−	0.944922i	$-0.606137\pi$
$599$	−9596.41	−0.654589	−0.327294	+	0.944922i	$0.606137\pi$
$600$	0	0
$601$	26034.3	1.76699	0.883494	−	0.468443i	$-0.155185\pi$
$601$	26034.3	1.76699	0.883494	+	0.468443i	$0.155185\pi$
$602$	0	0
$603$	−108.243	−0.00731009
$604$	0	0
$605$	−6828.19	−0.458852
$606$	0	0
$607$	−1428.99	−0.0955535	−0.0477767	−	0.998858i	$-0.515214\pi$
$607$	−1428.99	−0.0955535	−0.0477767	+	0.998858i	$0.515214\pi$
$608$	0	0
$609$	145.977	0.00971311
$610$	0	0
$611$	6849.44	0.453517
$612$	0	0
$613$	6750.66	0.444790	0.222395	−	0.974957i	$-0.428612\pi$
$613$	6750.66	0.444790	0.222395	+	0.974957i	$0.428612\pi$
$614$	0	0
$615$	−9545.96	−0.625903
$616$	0	0
$617$	21435.4	1.39863	0.699315	−	0.714813i	$-0.253488\pi$
$617$	21435.4	1.39863	0.699315	+	0.714813i	$0.253488\pi$
$618$	0	0
$619$	14378.4	0.933627	0.466813	−	0.884356i	$-0.345402\pi$
$619$	14378.4	0.933627	0.466813	+	0.884356i	$0.345402\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	24335.7	1.56499
$624$	0	0
$625$	−6928.76	−0.443440
$626$	0	0
$627$	−13347.9	−0.850183
$628$	0	0
$629$	5546.28	0.351581
$630$	0	0
$631$	−5285.58	−0.333464	−0.166732	−	0.986002i	$-0.553321\pi$
$631$	−5285.58	−0.333464	−0.166732	+	0.986002i	$0.553321\pi$
$632$	0	0
$633$	2655.37	0.166732
$634$	0	0
$635$	4312.22	0.269488
$636$	0	0
$637$	3002.02	0.186726
$638$	0	0
$639$	2992.85	0.185282
$640$	0	0
$641$	19010.8	1.17142	0.585710	−	0.810521i	$-0.300816\pi$
$641$	19010.8	1.17142	0.585710	+	0.810521i	$0.300816\pi$
$642$	0	0
$643$	22018.1	1.35040	0.675202	−	0.737633i	$-0.264057\pi$
$643$	22018.1	1.35040	0.675202	+	0.737633i	$0.264057\pi$
$644$	0	0
$645$	−10542.3	−0.643573
$646$	0	0
$647$	4748.99	0.288566	0.144283	−	0.989536i	$-0.453912\pi$
$647$	4748.99	0.288566	0.144283	+	0.989536i	$0.453912\pi$
$648$	0	0
$649$	−5675.11	−0.343248
$650$	0	0
$651$	−5854.64	−0.352475
$652$	0	0
$653$	−31371.1	−1.88001	−0.940005	−	0.341161i	$-0.889180\pi$
$653$	−31371.1	−1.88001	−0.940005	+	0.341161i	$0.889180\pi$
$654$	0	0
$655$	16749.7	0.999185
$656$	0	0
$657$	643.672	0.0382223
$658$	0	0
$659$	−10422.1	−0.616065	−0.308032	−	0.951376i	$-0.599671\pi$
$659$	−10422.1	−0.616065	−0.308032	+	0.951376i	$0.599671\pi$
$660$	0	0
$661$	19988.9	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$661$	19988.9	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$662$	0	0
$663$	7361.28	0.431204
$664$	0	0
$665$	16606.3	0.968368
$666$	0	0
$667$	56.5131	0.00328065
$668$	0	0
$669$	11145.8	0.644128
$670$	0	0
$671$	29626.6	1.70450
$672$	0	0
$673$	−338.677	−0.0193983	−0.00969915	−	0.999953i	$-0.503087\pi$
$673$	−338.677	−0.0193983	−0.00969915	+	0.999953i	$0.503087\pi$
$674$	0	0
$675$	1343.54	0.0766118
$676$	0	0
$677$	14911.2	0.846507	0.423254	−	0.906011i	$-0.360888\pi$
$677$	14911.2	0.846507	0.423254	+	0.906011i	$0.360888\pi$
$678$	0	0
$679$	20760.6	1.17337
$680$	0	0
$681$	3266.54	0.183809
$682$	0	0
$683$	−18813.5	−1.05399	−0.526997	−	0.849867i	$-0.676682\pi$
$683$	−18813.5	−1.05399	−0.526997	+	0.849867i	$0.676682\pi$
$684$	0	0
$685$	22685.2	1.26534
$686$	0	0
$687$	−6491.21	−0.360488
$688$	0	0
$689$	−15263.4	−0.843962
$690$	0	0
$691$	5438.39	0.299401	0.149701	−	0.988731i	$-0.452169\pi$
$691$	5438.39	0.299401	0.149701	+	0.988731i	$0.452169\pi$
$692$	0	0
$693$	−8202.89	−0.449642
$694$	0	0
$695$	−5496.06	−0.299968
$696$	0	0
$697$	14745.6	0.801335
$698$	0	0
$699$	9517.18	0.514983
$700$	0	0
$701$	2569.04	0.138418	0.0692092	−	0.997602i	$-0.477952\pi$
$701$	2569.04	0.138418	0.0692092	+	0.997602i	$0.477952\pi$
$702$	0	0
$703$	−13339.0	−0.715634
$704$	0	0
$705$	2919.80	0.155980
$706$	0	0
$707$	32266.6	1.71642
$708$	0	0
$709$	11457.4	0.606901	0.303450	−	0.952847i	$-0.401861\pi$
$709$	11457.4	0.606901	0.303450	+	0.952847i	$0.401861\pi$
$710$	0	0
$711$	9681.69	0.510678
$712$	0	0
$713$	−2266.55	−0.119050
$714$	0	0
$715$	−24369.7	−1.27465
$716$	0	0
$717$	19155.7	0.997743
$718$	0	0
$719$	35811.2	1.85749	0.928743	−	0.370723i	$-0.120890\pi$
$719$	35811.2	1.85749	0.928743	+	0.370723i	$0.120890\pi$
$720$	0	0
$721$	−22301.3	−1.15193
$722$	0	0
$723$	−16892.0	−0.868906
$724$	0	0
$725$	−122.267	−0.00626328
$726$	0	0
$727$	27988.9	1.42785	0.713927	−	0.700220i	$-0.246915\pi$
$727$	27988.9	1.42785	0.713927	+	0.700220i	$0.246915\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	16284.8	0.823958
$732$	0	0
$733$	−4337.73	−0.218578	−0.109289	−	0.994010i	$-0.534857\pi$
$733$	−4337.73	−0.218578	−0.109289	+	0.994010i	$0.534857\pi$
$734$	0	0
$735$	1279.71	0.0642215
$736$	0	0
$737$	−553.528	−0.0276655
$738$	0	0
$739$	21813.4	1.08582	0.542910	−	0.839791i	$-0.317323\pi$
$739$	21813.4	1.08582	0.542910	+	0.839791i	$0.317323\pi$
$740$	0	0
$741$	−17704.2	−0.877704
$742$	0	0
$743$	1591.64	0.0785888	0.0392944	−	0.999228i	$-0.487489\pi$
$743$	1591.64	0.0785888	0.0392944	+	0.999228i	$0.487489\pi$
$744$	0	0
$745$	−9675.12	−0.475797
$746$	0	0
$747$	−1698.91	−0.0832126
$748$	0	0
$749$	23044.9	1.12422
$750$	0	0
$751$	27206.5	1.32194	0.660971	−	0.750411i	$-0.270145\pi$
$751$	27206.5	1.32194	0.660971	+	0.750411i	$0.270145\pi$
$752$	0	0
$753$	−540.020	−0.0261347
$754$	0	0
$755$	−17819.5	−0.858965
$756$	0	0
$757$	−14705.7	−0.706062	−0.353031	−	0.935612i	$-0.614849\pi$
$757$	−14705.7	−0.706062	−0.353031	+	0.935612i	$0.614849\pi$
$758$	0	0
$759$	−3175.65	−0.151869
$760$	0	0
$761$	−29220.7	−1.39192	−0.695959	−	0.718081i	$-0.745021\pi$
$761$	−29220.7	−1.39192	−0.695959	+	0.718081i	$0.745021\pi$
$762$	0	0
$763$	31649.2	1.50167
$764$	0	0
$765$	3137.99	0.148306
$766$	0	0
$767$	−7527.25	−0.354359
$768$	0	0
$769$	−18693.9	−0.876617	−0.438309	−	0.898825i	$-0.644422\pi$
$769$	−18693.9	−0.876617	−0.438309	+	0.898825i	$0.644422\pi$
$770$	0	0
$771$	3785.45	0.176822
$772$	0	0
$773$	−9047.60	−0.420983	−0.210491	−	0.977596i	$-0.567506\pi$
$773$	−9047.60	−0.420983	−0.210491	+	0.977596i	$0.567506\pi$
$774$	0	0
$775$	4903.71	0.227286
$776$	0	0
$777$	−8197.42	−0.378482
$778$	0	0
$779$	−35463.8	−1.63110
$780$	0	0
$781$	15304.7	0.701211
$782$	0	0
$783$	−66.3415	−0.00302791
$784$	0	0
$785$	14369.0	0.653314
$786$	0	0
$787$	6860.04	0.310717	0.155358	−	0.987858i	$-0.450347\pi$
$787$	6860.04	0.310717	0.155358	+	0.987858i	$0.450347\pi$
$788$	0	0
$789$	−6505.79	−0.293552
$790$	0	0
$791$	3511.25	0.157833
$792$	0	0
$793$	39295.5	1.75968
$794$	0	0
$795$	−6506.55	−0.290268
$796$	0	0
$797$	18943.8	0.841936	0.420968	−	0.907076i	$-0.361690\pi$
$797$	18943.8	0.841936	0.420968	+	0.907076i	$0.361690\pi$
$798$	0	0
$799$	−4510.22	−0.199700
$800$	0	0
$801$	−11059.8	−0.487862
$802$	0	0
$803$	3291.59	0.144655
$804$	0	0
$805$	3950.86	0.172981
$806$	0	0
$807$	−9943.14	−0.433724
$808$	0	0
$809$	21889.6	0.951295	0.475648	−	0.879636i	$-0.342214\pi$
$809$	21889.6	0.951295	0.475648	+	0.879636i	$0.342214\pi$
$810$	0	0
$811$	−11468.8	−0.496578	−0.248289	−	0.968686i	$-0.579868\pi$
$811$	−11468.8	−0.496578	−0.248289	+	0.968686i	$0.579868\pi$
$812$	0	0
$813$	21439.4	0.924860
$814$	0	0
$815$	3831.21	0.164664
$816$	0	0
$817$	−39165.5	−1.67714
$818$	0	0
$819$	−10880.0	−0.464198
$820$	0	0
$821$	−26782.7	−1.13852	−0.569258	−	0.822159i	$-0.692769\pi$
$821$	−26782.7	−1.13852	−0.569258	+	0.822159i	$0.692769\pi$
$822$	0	0
$823$	−10312.0	−0.436761	−0.218381	−	0.975864i	$-0.570077\pi$
$823$	−10312.0	−0.436761	−0.218381	+	0.975864i	$0.570077\pi$
$824$	0	0
$825$	6870.56	0.289942
$826$	0	0
$827$	750.162	0.0315425	0.0157713	−	0.999876i	$-0.494980\pi$
$827$	750.162	0.0315425	0.0157713	+	0.999876i	$0.494980\pi$
$828$	0	0
$829$	−6918.42	−0.289851	−0.144926	−	0.989443i	$-0.546294\pi$
$829$	−6918.42	−0.289851	−0.144926	+	0.989443i	$0.546294\pi$
$830$	0	0
$831$	13144.1	0.548693
$832$	0	0
$833$	−1976.77	−0.0822220
$834$	0	0
$835$	7379.84	0.305856
$836$	0	0
$837$	2660.73	0.109879
$838$	0	0
$839$	38385.2	1.57950	0.789752	−	0.613427i	$-0.210209\pi$
$839$	38385.2	1.57950	0.789752	+	0.613427i	$0.210209\pi$
$840$	0	0
$841$	−24383.0	−0.999752
$842$	0	0
$843$	10452.3	0.427042
$844$	0	0
$845$	−13266.1	−0.540080
$846$	0	0
$847$	−15589.2	−0.632411
$848$	0	0
$849$	−23264.3	−0.940433
$850$	0	0
$851$	−3173.53	−0.127834
$852$	0	0
$853$	−9691.88	−0.389031	−0.194516	−	0.980899i	$-0.562314\pi$
$853$	−9691.88	−0.389031	−0.194516	+	0.980899i	$0.562314\pi$
$854$	0	0
$855$	−7546.99	−0.301873
$856$	0	0
$857$	35563.4	1.41753	0.708764	−	0.705445i	$-0.249253\pi$
$857$	35563.4	1.41753	0.708764	+	0.705445i	$0.249253\pi$
$858$	0	0
$859$	−22900.8	−0.909623	−0.454812	−	0.890588i	$-0.650293\pi$
$859$	−22900.8	−0.909623	−0.454812	+	0.890588i	$0.650293\pi$
$860$	0	0
$861$	−21794.1	−0.862649
$862$	0	0
$863$	6214.81	0.245138	0.122569	−	0.992460i	$-0.460887\pi$
$863$	6214.81	0.245138	0.122569	+	0.992460i	$0.460887\pi$
$864$	0	0
$865$	18977.1	0.745942
$866$	0	0
$867$	9891.75	0.387476
$868$	0	0
$869$	49509.9	1.93269
$870$	0	0
$871$	−734.178	−0.0285610
$872$	0	0
$873$	−9434.98	−0.365780
$874$	0	0
$875$	−30019.8	−1.15983
$876$	0	0
$877$	−21745.3	−0.837271	−0.418636	−	0.908154i	$-0.637492\pi$
$877$	−21745.3	−0.837271	−0.418636	+	0.908154i	$0.637492\pi$
$878$	0	0
$879$	15496.9	0.594651
$880$	0	0
$881$	−11420.6	−0.436741	−0.218371	−	0.975866i	$-0.570074\pi$
$881$	−11420.6	−0.436741	−0.218371	+	0.975866i	$0.570074\pi$
$882$	0	0
$883$	−32830.6	−1.25123	−0.625616	−	0.780131i	$-0.715152\pi$
$883$	−32830.6	−1.25123	−0.625616	+	0.780131i	$0.715152\pi$
$884$	0	0
$885$	−3208.74	−0.121876
$886$	0	0
$887$	43694.4	1.65402	0.827009	−	0.562189i	$-0.190041\pi$
$887$	43694.4	1.65402	0.827009	+	0.562189i	$0.190041\pi$
$888$	0	0
$889$	9845.10	0.371422
$890$	0	0
$891$	3727.93	0.140169
$892$	0	0
$893$	10847.3	0.406483
$894$	0	0
$895$	−7400.70	−0.276400
$896$	0	0
$897$	−4212.05	−0.156785
$898$	0	0
$899$	−242.136	−0.00898295
$900$	0	0
$901$	10050.7	0.371627
$902$	0	0
$903$	−24068.9	−0.887003
$904$	0	0
$905$	12810.0	0.470519
$906$	0	0
$907$	−13315.4	−0.487465	−0.243732	−	0.969843i	$-0.578372\pi$
$907$	−13315.4	−0.487465	−0.243732	+	0.969843i	$0.578372\pi$
$908$	0	0
$909$	−14664.1	−0.535068
$910$	0	0
$911$	−329.128	−0.0119698	−0.00598491	−	0.999982i	$-0.501905\pi$
$911$	−329.128	−0.0119698	−0.00598491	+	0.999982i	$0.501905\pi$
$912$	0	0
$913$	−8687.82	−0.314923
$914$	0	0
$915$	16751.0	0.605216
$916$	0	0
$917$	38240.8	1.37713
$918$	0	0
$919$	−14903.4	−0.534949	−0.267474	−	0.963565i	$-0.586189\pi$
$919$	−14903.4	−0.534949	−0.267474	+	0.963565i	$0.586189\pi$
$920$	0	0
$921$	−15340.3	−0.548839
$922$	0	0
$923$	20299.6	0.723909
$924$	0	0
$925$	6865.97	0.244056
$926$	0	0
$927$	10135.2	0.359097
$928$	0	0
$929$	41848.7	1.47795	0.738973	−	0.673735i	$-0.235311\pi$
$929$	41848.7	1.47795	0.738973	+	0.673735i	$0.235311\pi$
$930$	0	0
$931$	4754.20	0.167361
$932$	0	0
$933$	−2620.52	−0.0919526
$934$	0	0
$935$	16046.9	0.561274
$936$	0	0
$937$	1604.21	0.0559309	0.0279654	−	0.999609i	$-0.491097\pi$
$937$	1604.21	0.0559309	0.0279654	+	0.999609i	$0.491097\pi$
$938$	0	0
$939$	14086.5	0.489559
$940$	0	0
$941$	14998.1	0.519578	0.259789	−	0.965665i	$-0.416347\pi$
$941$	14998.1	0.519578	0.259789	+	0.965665i	$0.416347\pi$
$942$	0	0
$943$	−8437.31	−0.291364
$944$	0	0
$945$	−4637.96	−0.159654
$946$	0	0
$947$	44654.6	1.53229	0.766145	−	0.642668i	$-0.222172\pi$
$947$	44654.6	1.53229	0.766145	+	0.642668i	$0.222172\pi$
$948$	0	0
$949$	4365.83	0.149337
$950$	0	0
$951$	22741.4	0.775438
$952$	0	0
$953$	7443.66	0.253015	0.126508	−	0.991966i	$-0.459623\pi$
$953$	7443.66	0.253015	0.126508	+	0.991966i	$0.459623\pi$
$954$	0	0
$955$	4731.31	0.160316
$956$	0	0
$957$	−339.255	−0.0114593
$958$	0	0
$959$	51791.9	1.74395
$960$	0	0
$961$	−20079.8	−0.674021
$962$	0	0
$963$	−10473.1	−0.350458
$964$	0	0
$965$	−31954.2	−1.06595
$966$	0	0
$967$	10913.3	0.362926	0.181463	−	0.983398i	$-0.441917\pi$
$967$	10913.3	0.362926	0.181463	+	0.983398i	$0.441917\pi$
$968$	0	0
$969$	11657.8	0.386485
$970$	0	0
$971$	13606.9	0.449706	0.224853	−	0.974393i	$-0.427810\pi$
$971$	13606.9	0.449706	0.224853	+	0.974393i	$0.427810\pi$
$972$	0	0
$973$	−12547.9	−0.413430
$974$	0	0
$975$	9112.84	0.299328
$976$	0	0
$977$	−17616.4	−0.576865	−0.288433	−	0.957500i	$-0.593134\pi$
$977$	−17616.4	−0.576865	−0.288433	+	0.957500i	$0.593134\pi$
$978$	0	0
$979$	−56557.0	−1.84634
$980$	0	0
$981$	−14383.5	−0.468123
$982$	0	0
$983$	1411.26	0.0457905	0.0228953	−	0.999738i	$-0.492712\pi$
$983$	1411.26	0.0457905	0.0228953	+	0.999738i	$0.492712\pi$
$984$	0	0
$985$	8666.19	0.280333
$986$	0	0
$987$	6666.12	0.214980
$988$	0	0
$989$	−9317.98	−0.299590
$990$	0	0
$991$	23423.4	0.750827	0.375414	−	0.926857i	$-0.377501\pi$
$991$	23423.4	0.750827	0.375414	+	0.926857i	$0.377501\pi$
$992$	0	0
$993$	5449.52	0.174154
$994$	0	0
$995$	−42880.2	−1.36622
$996$	0	0
$997$	−3214.85	−0.102122	−0.0510608	−	0.998696i	$-0.516260\pi$
$997$	−3214.85	−0.102122	−0.0510608	+	0.998696i	$0.516260\pi$
$998$	0	0
$999$	3725.45	0.117986

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.4.a.g.1.2	✓	4
3.2	odd	2		1656.4.a.m.1.3		4
4.3	odd	2		1104.4.a.u.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.4.a.g.1.2	✓	4	1.1	even	1	trivial
1104.4.a.u.1.2		4	4.3	odd	2
1656.4.a.m.1.3		4	3.2	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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	552.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -8.67405 q^{5} -19.8035 q^{7} +9.00000 q^{9} +46.0239 q^{11} +61.0443 q^{13} +26.0222 q^{15} -40.1964 q^{17} +96.6739 q^{19} +59.4104 q^{21} +23.0000 q^{23} -49.7608 q^{25} -27.0000 q^{27} +2.45709 q^{29} -98.5456 q^{31} -138.072 q^{33} +171.776 q^{35} -137.979 q^{37} -183.133 q^{39} -366.840 q^{41} -405.130 q^{43} -78.0665 q^{45} +112.205 q^{47} +49.1777 q^{49} +120.589 q^{51} -250.039 q^{53} -399.213 q^{55} -290.022 q^{57} -123.308 q^{59} +643.722 q^{61} -178.231 q^{63} -529.501 q^{65} -12.0270 q^{67} -69.0000 q^{69} +332.539 q^{71} +71.5191 q^{73} +149.282 q^{75} -911.433 q^{77} +1075.74 q^{79} +81.0000 q^{81} -188.768 q^{83} +348.666 q^{85} -7.37128 q^{87} -1228.86 q^{89} -1208.89 q^{91} +295.637 q^{93} -838.555 q^{95} -1048.33 q^{97} +414.215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 14 q^{5} + 14 q^{7} + 36 q^{9} - 16 q^{11} + 4 q^{13} + 42 q^{15} - 22 q^{17} + 134 q^{19} - 42 q^{21} + 92 q^{23} + 24 q^{25} - 108 q^{27} - 180 q^{29} + 276 q^{31} + 48 q^{33} - 96 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 - 14 * q^5 + 14 * q^7 + 36 * q^9 - 16 * q^11 + 4 * q^13 + 42 * q^15 - 22 * q^17 + 134 * q^19 - 42 * q^21 + 92 * q^23 + 24 * q^25 - 108 * q^27 - 180 * q^29 + 276 * q^31 + 48 * q^33 - 96 * q^35 - 244 * q^37 - 12 * q^39 - 416 * q^41 - 242 * q^43 - 126 * q^45 - 384 * q^47 - 296 * q^49 + 66 * q^51 - 1426 * q^53 - 296 * q^55 - 402 * q^57 - 1000 * q^59 - 1140 * q^61 + 126 * q^63 - 1916 * q^65 - 646 * q^67 - 276 * q^69 - 432 * q^71 - 1568 * q^73 - 72 * q^75 - 2472 * q^77 + 778 * q^79 + 324 * q^81 + 112 * q^83 - 1772 * q^85 + 540 * q^87 - 2458 * q^89 - 316 * q^91 - 828 * q^93 + 952 * q^95 - 980 * q^97 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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