LMFDB - Embedded newform 1344.4.b.j.895.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,4,Mod(895,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.895"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,72,0,0,0,-20] 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:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.5
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.j.895.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -13.1053i q^{5} +(-9.87442 + 15.6683i) q^{7} +9.00000 q^{9} -66.5093i q^{11} +33.0750i q^{13} -39.3160i q^{15} +86.2229i q^{17} -21.0063 q^{19} +(-29.6233 + 47.0049i) q^{21} +218.669i q^{23} -46.7501 q^{25} +27.0000 q^{27} +133.471 q^{29} +324.428 q^{31} -199.528i q^{33} +(205.339 + 129.408i) q^{35} +73.1277 q^{37} +99.2251i q^{39} -309.181i q^{41} +28.7269i q^{43} -117.948i q^{45} +162.730 q^{47} +(-147.992 - 309.431i) q^{49} +258.669i q^{51} -575.828 q^{53} -871.627 q^{55} -63.0188 q^{57} +861.347 q^{59} -367.037i q^{61} +(-88.8698 + 141.015i) q^{63} +433.460 q^{65} -818.701i q^{67} +656.006i q^{69} +407.234i q^{71} -515.657i q^{73} -140.250 q^{75} +(1042.09 + 656.740i) q^{77} -1135.55i q^{79} +81.0000 q^{81} +941.158 q^{83} +1129.98 q^{85} +400.412 q^{87} +628.192i q^{89} +(-518.230 - 326.597i) q^{91} +973.285 q^{93} +275.294i q^{95} -522.331i q^{97} -598.583i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59}+ \cdots + 1392 q^{93}+O(q^{100}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$		−	13.1053i		−	1.17218i	−0.810247	−	0.586089i	$-0.800667\pi$
$5$		−	13.1053i		−	1.17218i	0.810247	−	0.586089i	$-0.199333\pi$
$6$		0			0
$7$	−9.87442	+	15.6683i	−0.533169	+	0.846009i
$7$	−9.87442	+	15.6683i	−0.533169	+	0.846009i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	66.5093i		−	1.82303i	−0.411270	−	0.911513i	$-0.634915\pi$
$11$		−	66.5093i		−	1.82303i	0.411270	−	0.911513i	$-0.365085\pi$
$12$		0			0
$13$			33.0750i			0.705643i	0.935691	+	0.352822i	$0.114778\pi$
$13$			33.0750i			0.705643i	−0.935691	+	0.352822i	$0.885222\pi$
$14$		0			0
$15$		−	39.3160i		−	0.676757i
$16$		0			0
$17$			86.2229i			1.23013i	0.788478	+	0.615063i	$0.210869\pi$
$17$			86.2229i			1.23013i	−0.788478	+	0.615063i	$0.789131\pi$
$18$		0			0
$19$	−21.0063			−0.253640			−0.126820	−	0.991926i	$-0.540477\pi$
$19$	−21.0063			−0.253640			−0.126820	+	0.991926i	$0.540477\pi$
$20$		0			0
$21$	−29.6233	+	47.0049i	−0.307825	+	0.488444i
$22$		0			0
$23$			218.669i			1.98242i	0.132307	+	0.991209i	$0.457761\pi$
$23$			218.669i			1.98242i	−0.132307	+	0.991209i	$0.542239\pi$
$24$		0			0
$25$	−46.7501			−0.374001
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	133.471			0.854652			0.427326	−	0.904098i	$-0.359456\pi$
$29$	133.471			0.854652			0.427326	+	0.904098i	$0.359456\pi$
$30$		0			0
$31$	324.428			1.87965			0.939823	−	0.341662i	$-0.110990\pi$
$31$	324.428			1.87965			0.939823	+	0.341662i	$0.110990\pi$
$32$		0			0
$33$		−	199.528i		−	1.05253i
$34$		0			0
$35$	205.339	+	129.408i	0.991673	+	0.624968i
$36$		0			0
$37$	73.1277			0.324922			0.162461	−	0.986715i	$-0.448057\pi$
$37$	73.1277			0.324922			0.162461	+	0.986715i	$0.448057\pi$
$38$		0			0
$39$			99.2251i			0.407403i
$40$		0			0
$41$		−	309.181i		−	1.17771i	−0.808240	−	0.588854i	$-0.799579\pi$
$41$		−	309.181i		−	1.17771i	0.808240	−	0.588854i	$-0.200421\pi$
$42$		0			0
$43$			28.7269i			0.101879i	0.998702	+	0.0509396i	$0.0162216\pi$
$43$			28.7269i			0.101879i	−0.998702	+	0.0509396i	$0.983778\pi$
$44$		0			0
$45$		−	117.948i		−	0.390726i
$46$		0			0
$47$	162.730			0.505036			0.252518	−	0.967592i	$-0.418741\pi$
$47$	162.730			0.505036			0.252518	+	0.967592i	$0.418741\pi$
$48$		0			0
$49$	−147.992	−	309.431i	−0.431463	−	0.902131i
$50$		0			0
$51$			258.669i			0.710213i
$52$		0			0
$53$	−575.828			−1.49238			−0.746189	−	0.665734i	$-0.768118\pi$
$53$	−575.828			−1.49238			−0.746189	+	0.665734i	$0.768118\pi$
$54$		0			0
$55$	−871.627			−2.13691
$56$		0			0
$57$	−63.0188			−0.146439
$58$		0			0
$59$	861.347			1.90064			0.950320	−	0.311273i	$-0.100755\pi$
$59$	861.347			1.90064			0.950320	+	0.311273i	$0.100755\pi$
$60$		0			0
$61$		−	367.037i		−	0.770399i	−0.922833	−	0.385199i	$-0.874133\pi$
$61$		−	367.037i		−	0.770399i	0.922833	−	0.385199i	$-0.125867\pi$
$62$		0			0
$63$	−88.8698	+	141.015i	−0.177723	+	0.282003i
$64$		0			0
$65$	433.460			0.827140
$66$		0			0
$67$		−	818.701i		−	1.49284i	−0.665476	−	0.746419i	$-0.731771\pi$
$67$		−	818.701i		−	1.49284i	0.665476	−	0.746419i	$-0.268229\pi$
$68$		0			0
$69$			656.006i			1.14455i
$70$		0			0
$71$			407.234i			0.680701i	0.940299	+	0.340351i	$0.110546\pi$
$71$			407.234i			0.680701i	−0.940299	+	0.340351i	$0.889454\pi$
$72$		0			0
$73$		−	515.657i		−	0.826754i	−0.910560	−	0.413377i	$-0.864349\pi$
$73$		−	515.657i		−	0.826754i	0.910560	−	0.413377i	$-0.135651\pi$
$74$		0			0
$75$	−140.250			−0.215929
$76$		0			0
$77$	1042.09	+	656.740i	1.54230	+	0.971981i
$78$		0			0
$79$		−	1135.55i		−	1.61721i	−0.588355	−	0.808603i	$-0.700224\pi$
$79$		−	1135.55i		−	1.61721i	0.588355	−	0.808603i	$-0.299776\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	941.158			1.24464			0.622322	−	0.782761i	$-0.286189\pi$
$83$	941.158			1.24464			0.622322	+	0.782761i	$0.286189\pi$
$84$		0			0
$85$	1129.98			1.44193
$86$		0			0
$87$	400.412			0.493434
$88$		0			0
$89$			628.192i			0.748182i	0.927392	+	0.374091i	$0.122045\pi$
$89$			628.192i			0.748182i	−0.927392	+	0.374091i	$0.877955\pi$
$90$		0			0
$91$	−518.230	−	326.597i	−0.596981	−	0.376227i
$92$		0			0
$93$	973.285			1.08521
$94$		0			0
$95$			275.294i			0.297312i
$96$		0			0
$97$		−	522.331i		−	0.546750i	−0.961908	−	0.273375i	$-0.911860\pi$
$97$		−	522.331i		−	0.546750i	0.961908	−	0.273375i	$-0.0881399\pi$
$98$		0			0
$99$		−	598.583i		−	0.607676i
$100$		0			0
$101$			1086.40i			1.07031i	0.844755	+	0.535153i	$0.179746\pi$
$101$			1086.40i			1.07031i	−0.844755	+	0.535153i	$0.820254\pi$
$102$		0			0
$103$	−196.731			−0.188199			−0.0940995	−	0.995563i	$-0.529997\pi$
$103$	−196.731			−0.188199			−0.0940995	+	0.995563i	$0.529997\pi$
$104$		0			0
$105$	616.016	+	388.223i	0.572543	+	0.360826i
$106$		0			0
$107$		−	870.161i		−	0.786183i	−0.919499	−	0.393092i	$-0.871406\pi$
$107$		−	870.161i		−	0.786183i	0.919499	−	0.393092i	$-0.128594\pi$
$108$		0			0
$109$	−277.220			−0.243604			−0.121802	−	0.992554i	$-0.538867\pi$
$109$	−277.220			−0.243604			−0.121802	+	0.992554i	$0.538867\pi$
$110$		0			0
$111$	219.383			0.187594
$112$		0			0
$113$	958.381			0.797849			0.398924	−	0.916984i	$-0.369384\pi$
$113$	958.381			0.797849			0.398924	+	0.916984i	$0.369384\pi$
$114$		0			0
$115$	2865.73			2.32375
$116$		0			0
$117$			297.675i			0.235214i
$118$		0			0
$119$	−1350.97	−	851.402i	−1.04070	−	0.655864i
$120$		0			0
$121$	−3092.48			−2.32343
$122$		0			0
$123$		−	927.544i		−	0.679950i
$124$		0			0
$125$		−	1025.49i		−	0.733782i
$126$		0			0
$127$			63.9927i			0.0447121i	0.999750	+	0.0223560i	$0.00711674\pi$
$127$			63.9927i			0.0447121i	−0.999750	+	0.0223560i	$0.992883\pi$
$128$		0			0
$129$			86.1807i			0.0588200i
$130$		0			0
$131$	−219.879			−0.146648			−0.0733240	−	0.997308i	$-0.523361\pi$
$131$	−219.879			−0.146648			−0.0733240	+	0.997308i	$0.523361\pi$
$132$		0			0
$133$	207.425	−	329.133i	0.135233	−	0.214582i
$134$		0			0
$135$		−	353.844i		−	0.225586i
$136$		0			0
$137$	−1493.47			−0.931358			−0.465679	−	0.884954i	$-0.654190\pi$
$137$	−1493.47			−0.931358			−0.465679	+	0.884954i	$0.654190\pi$
$138$		0			0
$139$	1149.51			0.701442			0.350721	−	0.936480i	$-0.385936\pi$
$139$	1149.51			0.701442			0.350721	+	0.936480i	$0.385936\pi$
$140$		0			0
$141$	488.191			0.291582
$142$		0			0
$143$	2199.80			1.28641
$144$		0			0
$145$		−	1749.18i		−	1.00180i
$146$		0			0
$147$	−443.975	−	928.293i	−0.249105	−	0.520845i
$148$		0			0
$149$	229.427			0.126144			0.0630719	−	0.998009i	$-0.479910\pi$
$149$	229.427			0.126144			0.0630719	+	0.998009i	$0.479910\pi$
$150$		0			0
$151$		−	407.635i		−	0.219688i	−0.993949	−	0.109844i	$-0.964965\pi$
$151$		−	407.635i		−	0.219688i	0.993949	−	0.109844i	$-0.0350351\pi$
$152$		0			0
$153$			776.007i			0.410042i
$154$		0			0
$155$		−	4251.74i		−	2.20328i
$156$		0			0
$157$			167.822i			0.0853101i	0.999090	+	0.0426551i	$0.0135817\pi$
$157$			167.822i			0.0853101i	−0.999090	+	0.0426551i	$0.986418\pi$
$158$		0			0
$159$	−1727.48			−0.861625
$160$		0			0
$161$	−3426.17	−	2159.23i	−1.67714	−	1.05696i
$162$		0			0
$163$			3096.94i			1.48817i	0.668086	+	0.744084i	$0.267114\pi$
$163$			3096.94i			1.48817i	−0.668086	+	0.744084i	$0.732886\pi$
$164$		0			0
$165$	−2614.88			−1.23375
$166$		0			0
$167$	3764.16			1.74419			0.872095	−	0.489337i	$-0.162761\pi$
$167$	3764.16			1.74419			0.872095	+	0.489337i	$0.162761\pi$
$168$		0			0
$169$	1103.04			0.502067
$170$		0			0
$171$	−189.056			−0.0845468
$172$		0			0
$173$			3438.40i			1.51108i	0.655103	+	0.755539i	$0.272625\pi$
$173$			3438.40i			1.51108i	−0.655103	+	0.755539i	$0.727375\pi$
$174$		0			0
$175$	461.630	−	732.495i	0.199405	−	0.316408i
$176$		0			0
$177$	2584.04			1.09734
$178$		0			0
$179$		−	1128.66i		−	0.471286i	−0.971840	−	0.235643i	$-0.924280\pi$
$179$		−	1128.66i		−	0.471286i	0.971840	−	0.235643i	$-0.0757196\pi$
$180$		0			0
$181$		−	1613.94i		−	0.662781i	−0.943494	−	0.331391i	$-0.892482\pi$
$181$		−	1613.94i		−	0.662781i	0.943494	−	0.331391i	$-0.107518\pi$
$182$		0			0
$183$		−	1101.11i		−	0.444790i
$184$		0			0
$185$		−	958.364i		−	0.380866i
$186$		0			0
$187$	5734.62			2.24255
$188$		0			0
$189$	−266.609	+	423.044i	−0.102608	+	0.162815i
$190$		0			0
$191$		−	2819.97i		−	1.06830i	−0.845388	−	0.534152i	$-0.820631\pi$
$191$		−	2819.97i		−	1.06830i	0.845388	−	0.534152i	$-0.179369\pi$
$192$		0			0
$193$	1749.41			0.652463			0.326232	−	0.945290i	$-0.394221\pi$
$193$	1749.41			0.652463			0.326232	+	0.945290i	$0.394221\pi$
$194$		0			0
$195$	1300.38			0.477549
$196$		0			0
$197$	−3168.95			−1.14608			−0.573041	−	0.819527i	$-0.694236\pi$
$197$	−3168.95			−1.14608			−0.573041	+	0.819527i	$0.694236\pi$
$198$		0			0
$199$	978.805			0.348672			0.174336	−	0.984686i	$-0.444222\pi$
$199$	978.805			0.348672			0.174336	+	0.984686i	$0.444222\pi$
$200$		0			0
$201$		−	2456.10i		−	0.861891i
$202$		0			0
$203$	−1317.95	+	2091.26i	−0.455674	+	0.723043i
$204$		0			0
$205$	−4051.93			−1.38048
$206$		0			0
$207$			1968.02i			0.660806i
$208$		0			0
$209$			1397.11i			0.462393i
$210$		0			0
$211$			1183.70i			0.386204i	0.981179	+	0.193102i	$0.0618548\pi$
$211$			1183.70i			0.386204i	−0.981179	+	0.193102i	$0.938145\pi$
$212$		0			0
$213$			1221.70i			0.393003i
$214$		0			0
$215$	376.476			0.119421
$216$		0			0
$217$	−3203.54	+	5083.24i	−1.00217	+	1.59020i
$218$		0			0
$219$		−	1546.97i		−	0.477327i
$220$		0			0
$221$	−2851.83			−0.868030
$222$		0			0
$223$	1724.50			0.517852			0.258926	−	0.965897i	$-0.416631\pi$
$223$	1724.50			0.517852			0.258926	+	0.965897i	$0.416631\pi$
$224$		0			0
$225$	−420.751			−0.124667
$226$		0			0
$227$	−2296.41			−0.671445			−0.335723	−	0.941961i	$-0.608980\pi$
$227$	−2296.41			−0.671445			−0.335723	+	0.941961i	$0.608980\pi$
$228$		0			0
$229$		−	4404.32i		−	1.27094i	−0.772125	−	0.635471i	$-0.780806\pi$
$229$		−	4404.32i		−	1.27094i	0.772125	−	0.635471i	$-0.219194\pi$
$230$		0			0
$231$	3126.26	+	1970.22i	0.890446	+	0.561173i
$232$		0			0
$233$	4222.07			1.18711			0.593556	−	0.804793i	$-0.297723\pi$
$233$	4222.07			1.18711			0.593556	+	0.804793i	$0.297723\pi$
$234$		0			0
$235$		−	2132.64i		−	0.591992i
$236$		0			0
$237$		−	3406.65i		−	0.933694i
$238$		0			0
$239$			4649.91i			1.25848i	0.777210	+	0.629242i	$0.216634\pi$
$239$			4649.91i			1.25848i	−0.777210	+	0.629242i	$0.783366\pi$
$240$		0			0
$241$		−	14.3928i		−	0.00384697i	−0.999998	−	0.00192348i	$-0.999388\pi$
$241$		−	14.3928i		−	0.00384697i	0.999998	−	0.00192348i	$-0.000612264\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	−4055.20	+	1939.48i	−1.05746	+	0.505751i
$246$		0			0
$247$		−	694.783i		−	0.178980i
$248$		0			0
$249$	2823.47			0.718596
$250$		0			0
$251$	6957.71			1.74967			0.874834	−	0.484422i	$-0.160970\pi$
$251$	6957.71			1.74967			0.874834	+	0.484422i	$0.160970\pi$
$252$		0			0
$253$	14543.5			3.61400
$254$		0			0
$255$	3389.94			0.832496
$256$		0			0
$257$			4915.09i			1.19298i	0.802622	+	0.596488i	$0.203438\pi$
$257$			4915.09i			1.19298i	−0.802622	+	0.596488i	$0.796562\pi$
$258$		0			0
$259$	−722.093	+	1145.79i	−0.173238	+	0.274887i
$260$		0			0
$261$	1201.24			0.284884
$262$		0			0
$263$		−	6930.51i		−	1.62492i	−0.583018	−	0.812459i	$-0.698128\pi$
$263$		−	6930.51i		−	1.62492i	0.583018	−	0.812459i	$-0.301872\pi$
$264$		0			0
$265$			7546.42i			1.74933i
$266$		0			0
$267$			1884.57i			0.431963i
$268$		0			0
$269$			481.497i			0.109135i	0.998510	+	0.0545676i	$0.0173780\pi$
$269$			481.497i			0.109135i	−0.998510	+	0.0545676i	$0.982622\pi$
$270$		0			0
$271$	5156.02			1.15574			0.577871	−	0.816128i	$-0.303884\pi$
$271$	5156.02			1.15574			0.577871	+	0.816128i	$0.303884\pi$
$272$		0			0
$273$	−1554.69	−	979.790i	−0.344667	−	0.217215i
$274$		0			0
$275$			3109.31i			0.681814i
$276$		0			0
$277$	2065.98			0.448134			0.224067	−	0.974574i	$-0.428067\pi$
$277$	2065.98			0.448134			0.224067	+	0.974574i	$0.428067\pi$
$278$		0			0
$279$	2919.85			0.626549
$280$		0			0
$281$	4260.27			0.904436			0.452218	−	0.891907i	$-0.350633\pi$
$281$	4260.27			0.904436			0.452218	+	0.891907i	$0.350633\pi$
$282$		0			0
$283$	−3195.17			−0.671141			−0.335571	−	0.942015i	$-0.608929\pi$
$283$	−3195.17			−0.671141			−0.335571	+	0.942015i	$0.608929\pi$
$284$		0			0
$285$			825.883i			0.171653i
$286$		0			0
$287$	4844.35	+	3052.98i	0.996351	+	0.627916i
$288$		0			0
$289$	−2521.40			−0.513209
$290$		0			0
$291$		−	1566.99i		−	0.315666i
$292$		0			0
$293$			6349.41i			1.26599i	0.774154	+	0.632997i	$0.218176\pi$
$293$			6349.41i			1.26599i	−0.774154	+	0.632997i	$0.781824\pi$
$294$		0			0
$295$		−	11288.2i		−	2.22789i
$296$		0			0
$297$		−	1795.75i		−	0.350842i
$298$		0			0
$299$	−7232.48			−1.39888
$300$		0			0
$301$	−450.102	−	283.661i	−0.0861908	−	0.0543188i
$302$		0			0
$303$			3259.20i			0.617942i
$304$		0			0
$305$	−4810.15			−0.903044
$306$		0			0
$307$	−684.156			−0.127188			−0.0635942	−	0.997976i	$-0.520256\pi$
$307$	−684.156			−0.127188			−0.0635942	+	0.997976i	$0.520256\pi$
$308$		0			0
$309$	−590.193			−0.108657
$310$		0			0
$311$	4151.64			0.756970			0.378485	−	0.925607i	$-0.376445\pi$
$311$	4151.64			0.756970			0.378485	+	0.925607i	$0.376445\pi$
$312$		0			0
$313$			2788.88i			0.503632i	0.967775	+	0.251816i	$0.0810277\pi$
$313$			2788.88i			0.503632i	−0.967775	+	0.251816i	$0.918972\pi$
$314$		0			0
$315$	1848.05	+	1164.67i	0.330558	+	0.208323i
$316$		0			0
$317$	−1123.95			−0.199139			−0.0995697	−	0.995031i	$-0.531747\pi$
$317$	−1123.95			−0.199139			−0.0995697	+	0.995031i	$0.531747\pi$
$318$		0			0
$319$		−	8877.04i		−	1.55805i
$320$		0			0
$321$		−	2610.48i		−	0.453903i
$322$		0			0
$323$		−	1811.22i		−	0.312010i
$324$		0			0
$325$		−	1546.26i		−	0.263911i
$326$		0			0
$327$	−831.659			−0.140645
$328$		0			0
$329$	−1606.87	+	2549.71i	−0.269269	+	0.427265i
$330$		0			0
$331$			4265.14i			0.708258i	0.935197	+	0.354129i	$0.115223\pi$
$331$			4265.14i			0.708258i	−0.935197	+	0.354129i	$0.884777\pi$
$332$		0			0
$333$	658.149			0.108307
$334$		0			0
$335$	−10729.4			−1.74987
$336$		0			0
$337$	5217.89			0.843433			0.421716	−	0.906728i	$-0.361428\pi$
$337$	5217.89			0.843433			0.421716	+	0.906728i	$0.361428\pi$
$338$		0			0
$339$	2875.14			0.460638
$340$		0			0
$341$		−	21577.5i		−	3.42664i
$342$		0			0
$343$	6309.59	+	736.671i	0.993253	+	0.115966i
$344$		0			0
$345$	8597.19			1.34162
$346$		0			0
$347$		−	2596.17i		−	0.401642i	−0.979628	−	0.200821i	$-0.935639\pi$
$347$		−	2596.17i		−	0.401642i	0.979628	−	0.200821i	$-0.0643609\pi$
$348$		0			0
$349$		−	9467.68i		−	1.45213i	−0.687626	−	0.726065i	$-0.741347\pi$
$349$		−	9467.68i		−	1.45213i	0.687626	−	0.726065i	$-0.258653\pi$
$350$		0			0
$351$			893.026i			0.135801i
$352$		0			0
$353$			2276.11i			0.343188i	0.985168	+	0.171594i	$0.0548917\pi$
$353$			2276.11i			0.343188i	−0.985168	+	0.171594i	$0.945108\pi$
$354$		0			0
$355$	5336.94			0.797903
$356$		0			0
$357$	−4052.90	−	2554.20i	−0.600847	−	0.378663i
$358$		0			0
$359$			4663.75i			0.685636i	0.939402	+	0.342818i	$0.111381\pi$
$359$			4663.75i			0.685636i	−0.939402	+	0.342818i	$0.888619\pi$
$360$		0			0
$361$	−6417.74			−0.935667
$362$		0			0
$363$	−9277.44			−1.34143
$364$		0			0
$365$	−6757.86			−0.969103
$366$		0			0
$367$	−4677.32			−0.665270			−0.332635	−	0.943056i	$-0.607938\pi$
$367$	−4677.32			−0.665270			−0.332635	+	0.943056i	$0.607938\pi$
$368$		0			0
$369$		−	2782.63i		−	0.392569i
$370$		0			0
$371$	5685.97	−	9022.25i	0.795689	−	1.26257i
$372$		0			0
$373$	−551.009			−0.0764884			−0.0382442	−	0.999268i	$-0.512176\pi$
$373$	−551.009			−0.0764884			−0.0382442	+	0.999268i	$0.512176\pi$
$374$		0			0
$375$		−	3076.48i		−	0.423649i
$376$		0			0
$377$			4414.55i			0.603080i
$378$		0			0
$379$		−	573.945i		−	0.0777878i	−0.999243	−	0.0388939i	$-0.987617\pi$
$379$		−	573.945i		−	0.0777878i	0.999243	−	0.0388939i	$-0.0123834\pi$
$380$		0			0
$381$			191.978i			0.0258145i
$382$		0			0
$383$	10376.9			1.38443			0.692215	−	0.721692i	$-0.256635\pi$
$383$	10376.9			1.38443			0.692215	+	0.721692i	$0.256635\pi$
$384$		0			0
$385$	8606.81	−	13656.9i	1.13933	−	1.80785i
$386$		0			0
$387$			258.542i			0.0339598i
$388$		0			0
$389$	−8040.94			−1.04805			−0.524025	−	0.851703i	$-0.675570\pi$
$389$	−8040.94			−1.04805			−0.524025	+	0.851703i	$0.675570\pi$
$390$		0			0
$391$	−18854.3			−2.43862
$392$		0			0
$393$	−659.636			−0.0846673
$394$		0			0
$395$	−14881.8			−1.89565
$396$		0			0
$397$			6608.26i			0.835413i	0.908582	+	0.417706i	$0.137166\pi$
$397$			6608.26i			0.835413i	−0.908582	+	0.417706i	$0.862834\pi$
$398$		0			0
$399$	622.274	−	987.398i	0.0780768	−	0.123889i
$400$		0			0
$401$	−292.364			−0.0364089			−0.0182045	−	0.999834i	$-0.505795\pi$
$401$	−292.364			−0.0364089			−0.0182045	+	0.999834i	$0.505795\pi$
$402$		0			0
$403$			10730.5i			1.32636i
$404$		0			0
$405$		−	1061.53i		−	0.130242i
$406$		0			0
$407$		−	4863.67i		−	0.592342i
$408$		0			0
$409$		−	1334.80i		−	0.161374i	−0.996740	−	0.0806868i	$-0.974289\pi$
$409$		−	1334.80i		−	0.161374i	0.996740	−	0.0806868i	$-0.0257114\pi$
$410$		0			0
$411$	−4480.42			−0.537720
$412$		0			0
$413$	−8505.30	+	13495.8i	−1.01336	+	1.60796i
$414$		0			0
$415$		−	12334.2i		−	1.45894i
$416$		0			0
$417$	3448.54			0.404978
$418$		0			0
$419$	8852.36			1.03214			0.516069	−	0.856547i	$-0.327395\pi$
$419$	8852.36			1.03214			0.516069	+	0.856547i	$0.327395\pi$
$420$		0			0
$421$	6270.01			0.725847			0.362924	−	0.931819i	$-0.381779\pi$
$421$	6270.01			0.725847			0.362924	+	0.931819i	$0.381779\pi$
$422$		0			0
$423$	1464.57			0.168345
$424$		0			0
$425$		−	4030.93i		−	0.460068i
$426$		0			0
$427$	5750.86	+	3624.28i	0.651764	+	0.410752i
$428$		0			0
$429$	6599.39			0.742707
$430$		0			0
$431$		−	205.022i		−	0.0229131i	−0.999934	−	0.0114566i	$-0.996353\pi$
$431$		−	205.022i		−	0.0229131i	0.999934	−	0.0114566i	$-0.00364682\pi$
$432$		0			0
$433$			12003.1i			1.33217i	0.745875	+	0.666086i	$0.232032\pi$
$433$			12003.1i			1.33217i	−0.745875	+	0.666086i	$0.767968\pi$
$434$		0			0
$435$		−	5247.54i		−	0.578392i
$436$		0			0
$437$		−	4593.41i		−	0.502821i
$438$		0			0
$439$	−2757.16			−0.299754			−0.149877	−	0.988705i	$-0.547888\pi$
$439$	−2757.16			−0.299754			−0.149877	+	0.988705i	$0.547888\pi$
$440$		0			0
$441$	−1331.93	−	2784.88i	−0.143821	−	0.300710i
$442$		0			0
$443$		−	3626.94i		−	0.388987i	−0.980904	−	0.194493i	$-0.937694\pi$
$443$		−	3626.94i		−	0.388987i	0.980904	−	0.194493i	$-0.0623063\pi$
$444$		0			0
$445$	8232.67			0.877002
$446$		0			0
$447$	688.282			0.0728291
$448$		0			0
$449$	−16303.5			−1.71360			−0.856801	−	0.515647i	$-0.827552\pi$
$449$	−16303.5			−1.71360			−0.856801	+	0.515647i	$0.827552\pi$
$450$		0			0
$451$	−20563.4			−2.14699
$452$		0			0
$453$		−	1222.90i		−	0.126837i
$454$		0			0
$455$	−4280.16	+	6791.58i	−0.441005	+	0.699768i
$456$		0			0
$457$	11048.3			1.13089			0.565447	−	0.824784i	$-0.308704\pi$
$457$	11048.3			1.13089			0.565447	+	0.824784i	$0.308704\pi$
$458$		0			0
$459$			2328.02i			0.236738i
$460$		0			0
$461$			5823.95i			0.588391i	0.955745	+	0.294196i	$0.0950518\pi$
$461$			5823.95i			0.588391i	−0.955745	+	0.294196i	$0.904948\pi$
$462$		0			0
$463$			15638.6i			1.56974i	0.619661	+	0.784869i	$0.287270\pi$
$463$			15638.6i			1.56974i	−0.619661	+	0.784869i	$0.712730\pi$
$464$		0			0
$465$		−	12755.2i		−	1.27206i
$466$		0			0
$467$	−730.574			−0.0723917			−0.0361959	−	0.999345i	$-0.511524\pi$
$467$	−730.574			−0.0723917			−0.0361959	+	0.999345i	$0.511524\pi$
$468$		0			0
$469$	12827.7	+	8084.19i	1.26296	+	0.795935i
$470$		0			0
$471$			503.467i			0.0492538i
$472$		0			0
$473$	1910.60			0.185729
$474$		0			0
$475$	982.045			0.0948617
$476$		0			0
$477$	−5182.45			−0.497459
$478$		0			0
$479$	−1211.82			−0.115594			−0.0577971	−	0.998328i	$-0.518408\pi$
$479$	−1211.82			−0.115594			−0.0577971	+	0.998328i	$0.518408\pi$
$480$		0			0
$481$			2418.70i			0.229279i
$482$		0			0
$483$	−10278.5	−	6477.68i	−0.968299	−	0.610238i
$484$		0			0
$485$	−6845.33			−0.640888
$486$		0			0
$487$			5068.72i			0.471634i	0.971798	+	0.235817i	$0.0757766\pi$
$487$			5068.72i			0.471634i	−0.971798	+	0.235817i	$0.924223\pi$
$488$		0			0
$489$			9290.83i			0.859194i
$490$		0			0
$491$			11619.0i			1.06794i	0.845503	+	0.533971i	$0.179301\pi$
$491$			11619.0i			1.06794i	−0.845503	+	0.533971i	$0.820699\pi$
$492$		0			0
$493$			11508.2i			1.05133i
$494$		0			0
$495$	−7844.64			−0.712304
$496$		0			0
$497$	−6380.67	−	4021.20i	−0.575879	−	0.362928i
$498$		0			0
$499$		−	6132.84i		−	0.550187i	−0.961417	−	0.275094i	$-0.911291\pi$
$499$		−	6132.84i		−	0.550187i	0.961417	−	0.275094i	$-0.0887089\pi$
$500$		0			0
$501$	11292.5			1.00701
$502$		0			0
$503$	−5868.53			−0.520208			−0.260104	−	0.965581i	$-0.583757\pi$
$503$	−5868.53			−0.520208			−0.260104	+	0.965581i	$0.583757\pi$
$504$		0			0
$505$	14237.7			1.25459
$506$		0			0
$507$	3309.13			0.289869
$508$		0			0
$509$			8716.70i			0.759059i	0.925180	+	0.379529i	$0.123914\pi$
$509$			8716.70i			0.759059i	−0.925180	+	0.379529i	$0.876086\pi$
$510$		0			0
$511$	8079.47	+	5091.81i	0.699442	+	0.440799i
$512$		0			0
$513$	−567.169			−0.0488131
$514$		0			0
$515$			2578.23i			0.220603i
$516$		0			0
$517$		−	10823.1i		−	0.920693i
$518$		0			0
$519$			10315.2i			0.872422i
$520$		0			0
$521$		−	6289.58i		−	0.528890i	−0.964401	−	0.264445i	$-0.914811\pi$
$521$		−	6289.58i		−	0.528890i	0.964401	−	0.264445i	$-0.0851887\pi$
$522$		0			0
$523$	19253.1			1.60971			0.804854	−	0.593473i	$-0.202243\pi$
$523$	19253.1			1.60971			0.804854	+	0.593473i	$0.202243\pi$
$524$		0			0
$525$	1384.89	−	2197.48i	0.115127	−	0.182678i
$526$		0			0
$527$			27973.2i			2.31220i
$528$		0			0
$529$	−35649.1			−2.92998
$530$		0			0
$531$	7752.12			0.633547
$532$		0			0
$533$	10226.2			0.831041
$534$		0			0
$535$	−11403.8			−0.921546
$536$		0			0
$537$		−	3385.99i		−	0.272097i
$538$		0			0
$539$	−20580.0	+	9842.82i	−1.64461	+	0.786568i
$540$		0			0
$541$	6806.51			0.540915			0.270457	−	0.962732i	$-0.412825\pi$
$541$	6806.51			0.540915			0.270457	+	0.962732i	$0.412825\pi$
$542$		0			0
$543$		−	4841.83i		−	0.382657i
$544$		0			0
$545$			3633.06i			0.285547i
$546$		0			0
$547$			2333.60i			0.182409i	0.995832	+	0.0912044i	$0.0290717\pi$
$547$			2333.60i			0.182409i	−0.995832	+	0.0912044i	$0.970928\pi$
$548$		0			0
$549$		−	3303.34i		−	0.256800i
$550$		0			0
$551$	−2803.72			−0.216774
$552$		0			0
$553$	17792.1	+	11212.9i	1.36817	+	0.862243i
$554$		0			0
$555$		−	2875.09i		−	0.219893i
$556$		0			0
$557$	−3035.04			−0.230878			−0.115439	−	0.993315i	$-0.536827\pi$
$557$	−3035.04			−0.230878			−0.115439	+	0.993315i	$0.536827\pi$
$558$		0			0
$559$	−950.143			−0.0718904
$560$		0			0
$561$	17203.9			1.29474
$562$		0			0
$563$	9471.73			0.709034			0.354517	−	0.935050i	$-0.384645\pi$
$563$	9471.73			0.709034			0.354517	+	0.935050i	$0.384645\pi$
$564$		0			0
$565$		−	12559.9i		−	0.935220i
$566$		0			0
$567$	−799.828	+	1269.13i	−0.0592409	+	0.0940010i
$568$		0			0
$569$	−4580.84			−0.337502			−0.168751	−	0.985659i	$-0.553973\pi$
$569$	−4580.84			−0.337502			−0.168751	+	0.985659i	$0.553973\pi$
$570$		0			0
$571$		−	15298.8i		−	1.12125i	−0.828068	−	0.560627i	$-0.810560\pi$
$571$		−	15298.8i		−	1.12125i	0.828068	−	0.560627i	$-0.189440\pi$
$572$		0			0
$573$		−	8459.92i		−	0.616786i
$574$		0			0
$575$		−	10222.8i		−	0.741426i
$576$		0			0
$577$		−	22630.9i		−	1.63282i	−0.577473	−	0.816410i	$-0.695961\pi$
$577$		−	22630.9i		−	1.63282i	0.577473	−	0.816410i	$-0.304039\pi$
$578$		0			0
$579$	5248.23			0.376700
$580$		0			0
$581$	−9293.38	+	14746.3i	−0.663605	+	1.05298i
$582$		0			0
$583$			38297.9i			2.72065i
$584$		0			0
$585$	3901.14			0.275713
$586$		0			0
$587$	17070.7			1.20031			0.600157	−	0.799883i	$-0.295105\pi$
$587$	17070.7			1.20031			0.600157	+	0.799883i	$0.295105\pi$
$588$		0			0
$589$	−6815.02			−0.476754
$590$		0			0
$591$	−9506.84			−0.661691
$592$		0			0
$593$			2588.94i			0.179283i	0.995974	+	0.0896416i	$0.0285722\pi$
$593$			2588.94i			0.179283i	−0.995974	+	0.0896416i	$0.971428\pi$
$594$		0			0
$595$	−11157.9	+	17704.9i	−0.768790	+	1.21988i
$596$		0			0
$597$	2936.42			0.201306
$598$		0			0
$599$		−	432.117i		−	0.0294755i	−0.999891	−	0.0147378i	$-0.995309\pi$
$599$		−	432.117i		−	0.0294755i	0.999891	−	0.0147378i	$-0.00469134\pi$
$600$		0			0
$601$			11299.7i			0.766927i	0.923556	+	0.383463i	$0.125269\pi$
$601$			11299.7i			0.766927i	−0.923556	+	0.383463i	$0.874731\pi$
$602$		0			0
$603$		−	7368.31i		−	0.497613i
$604$		0			0
$605$			40528.0i			2.72347i
$606$		0			0
$607$	−20570.8			−1.37552			−0.687761	−	0.725938i	$-0.741406\pi$
$607$	−20570.8			−1.37552			−0.687761	+	0.725938i	$0.741406\pi$
$608$		0			0
$609$	−3953.84	+	6273.79i	−0.263083	+	0.417449i
$610$		0			0
$611$			5382.32i			0.356375i
$612$		0			0
$613$	−11492.6			−0.757231			−0.378615	−	0.925554i	$-0.623600\pi$
$613$	−11492.6			−0.757231			−0.378615	+	0.925554i	$0.623600\pi$
$614$		0			0
$615$	−12155.8			−0.797022
$616$		0			0
$617$	−26688.7			−1.74140			−0.870701	−	0.491812i	$-0.836335\pi$
$617$	−26688.7			−1.74140			−0.870701	+	0.491812i	$0.836335\pi$
$618$		0			0
$619$	−4597.10			−0.298502			−0.149251	−	0.988799i	$-0.547686\pi$
$619$	−4597.10			−0.298502			−0.149251	+	0.988799i	$0.547686\pi$
$620$		0			0
$621$			5904.06i			0.381516i
$622$		0			0
$623$	−9842.70	−	6203.03i	−0.632969	−	0.398907i
$624$		0			0
$625$	−19283.2			−1.23412
$626$		0			0
$627$			4191.33i			0.266963i
$628$		0			0
$629$			6305.28i			0.399695i
$630$		0			0
$631$		−	5107.87i		−	0.322252i	−0.986934	−	0.161126i	$-0.948487\pi$
$631$		−	5107.87i		−	0.322252i	0.986934	−	0.161126i	$-0.0515126\pi$
$632$		0			0
$633$			3551.09i			0.222975i
$634$		0			0
$635$	838.647			0.0524105
$636$		0			0
$637$	10234.4	−	4894.83i	0.636583	−	0.304459i
$638$		0			0
$639$			3665.11i			0.226900i
$640$		0			0
$641$	1581.52			0.0974512			0.0487256	−	0.998812i	$-0.484484\pi$
$641$	1581.52			0.0974512			0.0487256	+	0.998812i	$0.484484\pi$
$642$		0			0
$643$	2210.56			0.135577			0.0677884	−	0.997700i	$-0.478406\pi$
$643$	2210.56			0.135577			0.0677884	+	0.997700i	$0.478406\pi$
$644$		0			0
$645$	1129.43			0.0689475
$646$		0			0
$647$	12507.1			0.759976			0.379988	−	0.924991i	$-0.375928\pi$
$647$	12507.1			0.759976			0.379988	+	0.924991i	$0.375928\pi$
$648$		0			0
$649$		−	57287.5i		−	3.46492i
$650$		0			0
$651$	−9610.62	+	15249.7i	−0.578602	+	0.918101i
$652$		0			0
$653$	−990.222			−0.0593421			−0.0296710	−	0.999560i	$-0.509446\pi$
$653$	−990.222			−0.0593421			−0.0296710	+	0.999560i	$0.509446\pi$
$654$		0			0
$655$			2881.59i			0.171898i
$656$		0			0
$657$		−	4640.91i		−	0.275585i
$658$		0			0
$659$			25900.2i			1.53100i	0.643435	+	0.765501i	$0.277509\pi$
$659$			25900.2i			1.53100i	−0.643435	+	0.765501i	$0.722491\pi$
$660$		0			0
$661$		−	22104.7i		−	1.30071i	−0.759628	−	0.650357i	$-0.774619\pi$
$661$		−	22104.7i		−	1.30071i	0.759628	−	0.650357i	$-0.225381\pi$
$662$		0			0
$663$	−8555.48			−0.501157
$664$		0			0
$665$	−4313.40	−	2718.37i	−0.251528	−	0.158517i
$666$		0			0
$667$			29185.9i			1.69428i
$668$		0			0
$669$	5173.49			0.298982
$670$		0			0
$671$	−24411.4			−1.40446
$672$		0			0
$673$	−6847.62			−0.392209			−0.196104	−	0.980583i	$-0.562829\pi$
$673$	−6847.62			−0.392209			−0.196104	+	0.980583i	$0.562829\pi$
$674$		0			0
$675$	−1262.25			−0.0719765
$676$		0			0
$677$		−	8884.62i		−	0.504378i	−0.967678	−	0.252189i	$-0.918850\pi$
$677$		−	8884.62i		−	0.504378i	0.967678	−	0.252189i	$-0.0811504\pi$
$678$		0			0
$679$	8184.05	+	5157.72i	0.462555	+	0.291510i
$680$		0			0
$681$	−6889.23			−0.387659
$682$		0			0
$683$		−	24173.3i		−	1.35427i	−0.735859	−	0.677135i	$-0.763221\pi$
$683$		−	24173.3i		−	1.35427i	0.735859	−	0.677135i	$-0.236779\pi$
$684$		0			0
$685$			19572.5i			1.09172i
$686$		0			0
$687$		−	13213.0i		−	0.733778i
$688$		0			0
$689$		−	19045.5i		−	1.05309i
$690$		0			0
$691$	−11158.2			−0.614296			−0.307148	−	0.951662i	$-0.599375\pi$
$691$	−11158.2			−0.614296			−0.307148	+	0.951662i	$0.599375\pi$
$692$		0			0
$693$	9378.79	+	5910.66i	0.514099	+	0.323994i
$694$		0			0
$695$		−	15064.8i		−	0.822215i
$696$		0			0
$697$	26658.5			1.44873
$698$		0			0
$699$	12666.2			0.685380
$700$		0			0
$701$	−4440.80			−0.239268			−0.119634	−	0.992818i	$-0.538172\pi$
$701$	−4440.80			−0.239268			−0.119634	+	0.992818i	$0.538172\pi$
$702$		0			0
$703$	−1536.14			−0.0824134
$704$		0			0
$705$		−	6397.92i		−	0.341786i
$706$		0			0
$707$	−17022.1	−	10727.6i	−0.905489	−	0.570654i
$708$		0			0
$709$	−19194.5			−1.01673			−0.508367	−	0.861140i	$-0.669751\pi$
$709$	−19194.5			−1.01673			−0.508367	+	0.861140i	$0.669751\pi$
$710$		0			0
$711$		−	10219.9i		−	0.539068i
$712$		0			0
$713$			70942.3i			3.72624i
$714$		0			0
$715$		−	28829.1i		−	1.50790i
$716$		0			0
$717$			13949.7i			0.726586i
$718$		0			0
$719$	−35198.1			−1.82569			−0.912843	−	0.408310i	$-0.866118\pi$
$719$	−35198.1			−1.82569			−0.912843	+	0.408310i	$0.866118\pi$
$720$		0			0
$721$	1942.61	−	3082.44i	0.100342	−	0.159218i
$722$		0			0
$723$		−	43.1783i		−	0.00222105i
$724$		0			0
$725$	−6239.77			−0.319641
$726$		0			0
$727$	−13232.3			−0.675047			−0.337524	−	0.941317i	$-0.609589\pi$
$727$	−13232.3			−0.675047			−0.337524	+	0.941317i	$0.609589\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−2476.92			−0.125324
$732$		0			0
$733$		−	20075.6i		−	1.01161i	−0.862649	−	0.505804i	$-0.831196\pi$
$733$		−	20075.6i		−	1.01161i	0.862649	−	0.505804i	$-0.168804\pi$
$734$		0			0
$735$	−12165.6	+	5818.45i	−0.610523	+	0.291995i
$736$		0			0
$737$	−54451.2			−2.72149
$738$		0			0
$739$		−	15808.7i		−	0.786917i	−0.919342	−	0.393459i	$-0.871278\pi$
$739$		−	15808.7i		−	0.786917i	0.919342	−	0.393459i	$-0.128722\pi$
$740$		0			0
$741$		−	2084.35i		−	0.103334i
$742$		0			0
$743$			4785.70i			0.236299i	0.992996	+	0.118150i	$0.0376962\pi$
$743$			4785.70i			0.236299i	−0.992996	+	0.118150i	$0.962304\pi$
$744$		0			0
$745$		−	3006.73i		−	0.147863i
$746$		0			0
$747$	8470.42			0.414881
$748$		0			0
$749$	13633.9	+	8592.33i	0.665118	+	0.419168i
$750$		0			0
$751$		−	26285.5i		−	1.27719i	−0.769543	−	0.638595i	$-0.779516\pi$
$751$		−	26285.5i		−	1.27719i	0.769543	−	0.638595i	$-0.220484\pi$
$752$		0			0
$753$	20873.1			1.01017
$754$		0			0
$755$	−5342.20			−0.257513
$756$		0			0
$757$	−26989.1			−1.29582			−0.647910	−	0.761717i	$-0.724357\pi$
$757$	−26989.1			−1.29582			−0.647910	+	0.761717i	$0.724357\pi$
$758$		0			0
$759$	43630.5			2.08654
$760$		0			0
$761$			26835.7i			1.27831i	0.769079	+	0.639154i	$0.220715\pi$
$761$			26835.7i			1.27831i	−0.769079	+	0.639154i	$0.779285\pi$
$762$		0			0
$763$	2737.38	−	4343.56i	0.129882	−	0.206091i
$764$		0			0
$765$	10169.8			0.480642
$766$		0			0
$767$			28489.1i			1.34117i
$768$		0			0
$769$		−	7482.83i		−	0.350894i	−0.984489	−	0.175447i	$-0.943863\pi$
$769$		−	7482.83i		−	0.350894i	0.984489	−	0.175447i	$-0.0561371\pi$
$770$		0			0
$771$			14745.3i			0.688765i
$772$		0			0
$773$		−	755.076i		−	0.0351335i	−0.999846	−	0.0175668i	$-0.994408\pi$
$773$		−	755.076i		−	0.0351335i	0.999846	−	0.0175668i	$-0.00559196\pi$
$774$		0			0
$775$	−15167.1			−0.702989
$776$		0			0
$777$	−2166.28	+	3437.36i	−0.100019	+	0.158706i
$778$		0			0
$779$			6494.74i			0.298714i
$780$		0			0
$781$	27084.8			1.24094
$782$		0			0
$783$	3603.71			0.164478
$784$		0			0
$785$	2199.37			0.0999987
$786$		0			0
$787$	−38108.4			−1.72607			−0.863036	−	0.505143i	$-0.831440\pi$
$787$	−38108.4			−1.72607			−0.863036	+	0.505143i	$0.831440\pi$
$788$		0			0
$789$		−	20791.5i		−	0.938147i
$790$		0			0
$791$	−9463.45	+	15016.2i	−0.425388	+	0.674987i
$792$		0			0
$793$	12139.8			0.543627
$794$		0			0
$795$			22639.3i			1.00998i
$796$		0			0
$797$		−	20013.0i		−	0.889455i	−0.895666	−	0.444728i	$-0.853301\pi$
$797$		−	20013.0i		−	0.889455i	0.895666	−	0.444728i	$-0.146699\pi$
$798$		0			0
$799$			14031.1i			0.621257i
$800$		0			0
$801$			5653.72i			0.249394i
$802$		0			0
$803$	−34295.9			−1.50720
$804$		0			0
$805$	−28297.4	+	44901.1i	−1.23895	+	1.96591i
$806$		0			0
$807$			1444.49i			0.0630093i
$808$		0			0
$809$	15288.5			0.664420			0.332210	−	0.943206i	$-0.392206\pi$
$809$	15288.5			0.664420			0.332210	+	0.943206i	$0.392206\pi$
$810$		0			0
$811$	15348.1			0.664542			0.332271	−	0.943184i	$-0.392185\pi$
$811$	15348.1			0.664542			0.332271	+	0.943184i	$0.392185\pi$
$812$		0			0
$813$	15468.1			0.667268
$814$		0			0
$815$	40586.5			1.74440
$816$		0			0
$817$		−	603.444i		−	0.0258407i
$818$		0			0
$819$	−4664.07	−	2939.37i	−0.198994	−	0.125409i
$820$		0			0
$821$	15311.7			0.650891			0.325445	−	0.945561i	$-0.394486\pi$
$821$	15311.7			0.650891			0.325445	+	0.945561i	$0.394486\pi$
$822$		0			0
$823$			30450.5i			1.28972i	0.764301	+	0.644859i	$0.223084\pi$
$823$			30450.5i			1.28972i	−0.764301	+	0.644859i	$0.776916\pi$
$824$		0			0
$825$			9327.94i			0.393645i
$826$		0			0
$827$		−	8352.35i		−	0.351197i	−0.984462	−	0.175598i	$-0.943814\pi$
$827$		−	8352.35i		−	0.351197i	0.984462	−	0.175598i	$-0.0561860\pi$
$828$		0			0
$829$			18715.9i			0.784114i	0.919941	+	0.392057i	$0.128236\pi$
$829$			18715.9i			0.784114i	−0.919941	+	0.392057i	$0.871764\pi$
$830$		0			0
$831$	6197.95			0.258730
$832$		0			0
$833$	26680.0	−	12760.3i	1.10973	−	0.530753i
$834$		0			0
$835$		−	49330.6i		−	2.04450i
$836$		0			0
$837$	8759.56			0.361738
$838$		0			0
$839$	−19309.6			−0.794566			−0.397283	−	0.917696i	$-0.630047\pi$
$839$	−19309.6			−0.794566			−0.397283	+	0.917696i	$0.630047\pi$
$840$		0			0
$841$	−6574.54			−0.269570
$842$		0			0
$843$	12780.8			0.522177
$844$		0			0
$845$		−	14455.7i		−	0.588512i
$846$		0			0
$847$	30536.5	−	48453.9i	1.23878	−	1.96564i
$848$		0			0
$849$	−9585.50			−0.387484
$850$		0			0
$851$			15990.7i			0.644131i
$852$		0			0
$853$		−	27138.9i		−	1.08935i	−0.838646	−	0.544677i	$-0.816652\pi$
$853$		−	27138.9i		−	1.08935i	0.838646	−	0.544677i	$-0.183348\pi$
$854$		0			0
$855$			2477.65i			0.0991039i
$856$		0			0
$857$			25760.4i			1.02679i	0.858152	+	0.513395i	$0.171613\pi$
$857$			25760.4i			1.02679i	−0.858152	+	0.513395i	$0.828387\pi$
$858$		0			0
$859$	−42892.5			−1.70369			−0.851846	−	0.523792i	$-0.824517\pi$
$859$	−42892.5			−1.70369			−0.851846	+	0.523792i	$0.824517\pi$
$860$		0			0
$861$	14533.0	+	9158.95i	0.575243	+	0.362528i
$862$		0			0
$863$		−	26017.9i		−	1.02626i	−0.858312	−	0.513128i	$-0.828487\pi$
$863$		−	26017.9i		−	1.02626i	0.858312	−	0.513128i	$-0.171513\pi$
$864$		0			0
$865$	45061.4			1.77125
$866$		0			0
$867$	−7564.19			−0.296301
$868$		0			0
$869$	−75524.5			−2.94821
$870$		0			0
$871$	27078.6			1.05341
$872$		0			0
$873$		−	4700.98i		−	0.182250i
$874$		0			0
$875$	16067.7	+	10126.1i	0.620786	+	0.391230i
$876$		0			0
$877$	4450.82			0.171372			0.0856861	−	0.996322i	$-0.472692\pi$
$877$	4450.82			0.171372			0.0856861	+	0.996322i	$0.472692\pi$
$878$		0			0
$879$			19048.2i			0.730922i
$880$		0			0
$881$		−	15737.7i		−	0.601837i	−0.953650	−	0.300918i	$-0.902707\pi$
$881$		−	15737.7i		−	0.601837i	0.953650	−	0.300918i	$-0.0972931\pi$
$882$		0			0
$883$			5677.78i			0.216390i	0.994130	+	0.108195i	$0.0345071\pi$
$883$			5677.78i			0.216390i	−0.994130	+	0.108195i	$0.965493\pi$
$884$		0			0
$885$		−	33864.7i		−	1.28627i
$886$		0			0
$887$	−7460.00			−0.282393			−0.141196	−	0.989982i	$-0.545095\pi$
$887$	−7460.00			−0.282393			−0.141196	+	0.989982i	$0.545095\pi$
$888$		0			0
$889$	−1002.66	−	631.891i	−0.0378268	−	0.0238391i
$890$		0			0
$891$		−	5387.25i		−	0.202559i
$892$		0			0
$893$	−3418.36			−0.128097
$894$		0			0
$895$	−14791.5			−0.552431
$896$		0			0
$897$	−21697.4			−0.807644
$898$		0			0
$899$	43301.7			1.60644
$900$		0			0
$901$		−	49649.6i		−	1.83581i
$902$		0			0
$903$	−1350.30	−	850.984i	−0.0497623	−	0.0313610i
$904$		0			0
$905$	−21151.3			−0.776897
$906$		0			0
$907$		−	34731.8i		−	1.27150i	−0.771894	−	0.635751i	$-0.780691\pi$
$907$		−	34731.8i		−	1.27150i	0.771894	−	0.635751i	$-0.219309\pi$
$908$		0			0
$909$			9777.61i			0.356769i
$910$		0			0
$911$			1692.78i			0.0615636i	0.999526	+	0.0307818i	$0.00979970\pi$
$911$			1692.78i			0.0615636i	−0.999526	+	0.0307818i	$0.990200\pi$
$912$		0			0
$913$		−	62595.7i		−	2.26902i
$914$		0			0
$915$	−14430.5			−0.521373
$916$		0			0
$917$	2171.17	−	3445.13i	0.0781881	−	0.124066i
$918$		0			0
$919$			7532.50i			0.270375i	0.990820	+	0.135187i	$0.0431636\pi$
$919$			7532.50i			0.270375i	−0.990820	+	0.135187i	$0.956836\pi$
$920$		0			0
$921$	−2052.47			−0.0734323
$922$		0			0
$923$	−13469.3			−0.480332
$924$		0			0
$925$	−3418.73			−0.121521
$926$		0			0
$927$	−1770.58			−0.0627330
$928$		0			0
$929$		−	21633.4i		−	0.764012i	−0.924160	−	0.382006i	$-0.875233\pi$
$929$		−	21633.4i		−	0.764012i	0.924160	−	0.382006i	$-0.124767\pi$
$930$		0			0
$931$	3108.75	+	6499.99i	0.109436	+	0.228817i
$932$		0			0
$933$	12454.9			0.437037
$934$		0			0
$935$		−	75154.2i		−	2.62867i
$936$		0			0
$937$			33386.1i			1.16401i	0.813185	+	0.582005i	$0.197732\pi$
$937$			33386.1i			1.16401i	−0.813185	+	0.582005i	$0.802268\pi$
$938$		0			0
$939$			8366.63i			0.290772i
$940$		0			0
$941$		−	22939.9i		−	0.794709i	−0.917665	−	0.397354i	$-0.869928\pi$
$941$		−	22939.9i		−	0.794709i	0.917665	−	0.397354i	$-0.130072\pi$
$942$		0			0
$943$	67608.3			2.33471
$944$		0			0
$945$	5544.14	+	3494.01i	0.190848	+	0.120275i
$946$		0			0
$947$			3074.78i			0.105509i	0.998608	+	0.0527545i	$0.0168001\pi$
$947$			3074.78i			0.105509i	−0.998608	+	0.0527545i	$0.983200\pi$
$948$		0			0
$949$	17055.4			0.583394
$950$		0			0
$951$	−3371.84			−0.114973
$952$		0			0
$953$	26530.3			0.901786			0.450893	−	0.892578i	$-0.351106\pi$
$953$	26530.3			0.901786			0.450893	+	0.892578i	$0.351106\pi$
$954$		0			0
$955$	−36956.7			−1.25224
$956$		0			0
$957$		−	26631.1i		−	0.899543i
$958$		0			0
$959$	14747.2	−	23400.2i	0.496571	−	0.787937i
$960$		0			0
$961$	75462.6			2.53307
$962$		0			0
$963$		−	7831.45i		−	0.262061i
$964$		0			0
$965$		−	22926.6i		−	0.764803i
$966$		0			0
$967$		−	5995.46i		−	0.199381i	−0.995019	−	0.0996903i	$-0.968215\pi$
$967$		−	5995.46i		−	0.199381i	0.995019	−	0.0996903i	$-0.0317852\pi$
$968$		0			0
$969$		−	5433.67i		−	0.180139i
$970$		0			0
$971$	16996.3			0.561726			0.280863	−	0.959748i	$-0.409379\pi$
$971$	16996.3			0.561726			0.280863	+	0.959748i	$0.409379\pi$
$972$		0			0
$973$	−11350.8	+	18010.9i	−0.373987	+	0.593427i
$974$		0			0
$975$		−	4638.78i		−	0.152369i
$976$		0			0
$977$	8989.54			0.294371			0.147186	−	0.989109i	$-0.452979\pi$
$977$	8989.54			0.294371			0.147186	+	0.989109i	$0.452979\pi$
$978$		0			0
$979$	41780.6			1.36396
$980$		0			0
$981$	−2494.98			−0.0812013
$982$		0			0
$983$	−1169.17			−0.0379356			−0.0189678	−	0.999820i	$-0.506038\pi$
$983$	−1169.17			−0.0379356			−0.0189678	+	0.999820i	$0.506038\pi$
$984$		0			0
$985$			41530.1i			1.34341i
$986$		0			0
$987$	−4820.61	+	7649.13i	−0.155463	+	0.246681i
$988$		0			0
$989$	−6281.67			−0.201967
$990$		0			0
$991$			55448.1i			1.77736i	0.458525	+	0.888681i	$0.348378\pi$
$991$			55448.1i			1.77736i	−0.458525	+	0.888681i	$0.651622\pi$
$992$		0			0
$993$			12795.4i			0.408913i
$994$		0			0
$995$		−	12827.6i		−	0.408705i
$996$		0			0
$997$			48292.0i			1.53403i	0.641631	+	0.767013i	$0.278258\pi$
$997$			48292.0i			1.53403i	−0.641631	+	0.767013i	$0.721742\pi$
$998$		0			0
$999$	1974.45			0.0625313

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.j.895.5		24
4.3	odd	2		1344.4.b.i.895.5		24
7.6	odd	2		1344.4.b.i.895.20		24
8.3	odd	2		672.4.b.b.223.20	yes	24
8.5	even	2		672.4.b.a.223.20	yes	24
28.27	even	2	inner	1344.4.b.j.895.20		24
56.13	odd	2		672.4.b.b.223.5	yes	24
56.27	even	2		672.4.b.a.223.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.5	✓	24	56.27	even	2
672.4.b.a.223.20	yes	24	8.5	even	2
672.4.b.b.223.5	yes	24	56.13	odd	2
672.4.b.b.223.20	yes	24	8.3	odd	2
1344.4.b.i.895.5		24	4.3	odd	2
1344.4.b.i.895.20		24	7.6	odd	2
1344.4.b.j.895.5		24	1.1	even	1	trivial
1344.4.b.j.895.20		24	28.27	even	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -13.1053i q^{5} +(-9.87442 + 15.6683i) q^{7} +9.00000 q^{9} -66.5093i q^{11} +33.0750i q^{13} -39.3160i q^{15} +86.2229i q^{17} -21.0063 q^{19} +(-29.6233 + 47.0049i) q^{21} +218.669i q^{23} -46.7501 q^{25} +27.0000 q^{27} +133.471 q^{29} +324.428 q^{31} -199.528i q^{33} +(205.339 + 129.408i) q^{35} +73.1277 q^{37} +99.2251i q^{39} -309.181i q^{41} +28.7269i q^{43} -117.948i q^{45} +162.730 q^{47} +(-147.992 - 309.431i) q^{49} +258.669i q^{51} -575.828 q^{53} -871.627 q^{55} -63.0188 q^{57} +861.347 q^{59} -367.037i q^{61} +(-88.8698 + 141.015i) q^{63} +433.460 q^{65} -818.701i q^{67} +656.006i q^{69} +407.234i q^{71} -515.657i q^{73} -140.250 q^{75} +(1042.09 + 656.740i) q^{77} -1135.55i q^{79} +81.0000 q^{81} +941.158 q^{83} +1129.98 q^{85} +400.412 q^{87} +628.192i q^{89} +(-518.230 - 326.597i) q^{91} +973.285 q^{93} +275.294i q^{95} -522.331i q^{97} -598.583i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59}+ \cdots + 1392 q^{93}+O(q^{100}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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