LMFDB - Embedded newform 1336.2.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$10.6680137100$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 5x^{4} + 25x^{3} - 4x^{2} - 17x + 1 $ x^7 - x^6 - 9x^5 + 5x^4 + 25x^3 - 4x^2 - 17*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.41051$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41051 q^{3} +0.329231 q^{5} -2.82300 q^{7} -1.01045 q^{9} +O(q^{10})$	$q+1.41051 q^{3} +0.329231 q^{5} -2.82300 q^{7} -1.01045 q^{9} +0.647542 q^{11} -5.15026 q^{13} +0.464384 q^{15} +0.793615 q^{17} -4.24097 q^{19} -3.98187 q^{21} -1.35328 q^{23} -4.89161 q^{25} -5.65680 q^{27} +7.17555 q^{29} -0.216464 q^{31} +0.913365 q^{33} -0.929418 q^{35} +1.38463 q^{37} -7.26450 q^{39} -10.2237 q^{41} -9.36903 q^{43} -0.332673 q^{45} +8.80147 q^{47} +0.969317 q^{49} +1.11940 q^{51} +8.56574 q^{53} +0.213191 q^{55} -5.98194 q^{57} -11.9520 q^{59} +9.46622 q^{61} +2.85251 q^{63} -1.69562 q^{65} -5.91351 q^{67} -1.90881 q^{69} -2.07131 q^{71} -9.19625 q^{73} -6.89967 q^{75} -1.82801 q^{77} +2.65223 q^{79} -4.94762 q^{81} +9.39541 q^{83} +0.261283 q^{85} +10.1212 q^{87} +16.5954 q^{89} +14.5392 q^{91} -0.305325 q^{93} -1.39626 q^{95} +16.5225 q^{97} -0.654311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 6 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 7 * q - 6 * q^3 - q^7 + 3 * q^9	$ 7 q - 6 q^{3} - q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 9 q^{15} - 9 q^{17} - 3 q^{19} + 7 q^{21} - 10 q^{23} + 3 q^{25} - 12 q^{27} - 5 q^{29} - 21 q^{31} + 8 q^{33} - 12 q^{35} + 19 q^{37} - 27 q^{39} - 22 q^{41} - 19 q^{43} + 13 q^{45} - 13 q^{47} - 14 q^{49} + 4 q^{51} + 5 q^{53} - 17 q^{55} + 5 q^{57} - 18 q^{59} + 26 q^{61} - 20 q^{63} - 20 q^{65} - 27 q^{67} - 3 q^{69} - 46 q^{71} - 25 q^{73} - 19 q^{75} - 19 q^{77} - 22 q^{79} - 9 q^{81} + q^{83} - 11 q^{85} + 9 q^{87} - 3 q^{89} + 33 q^{93} - 40 q^{95} + 11 q^{97} - 21 q^{99}+O(q^{100}) $ 7 * q - 6 * q^3 - q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 9 * q^15 - 9 * q^17 - 3 * q^19 + 7 * q^21 - 10 * q^23 + 3 * q^25 - 12 * q^27 - 5 * q^29 - 21 * q^31 + 8 * q^33 - 12 * q^35 + 19 * q^37 - 27 * q^39 - 22 * q^41 - 19 * q^43 + 13 * q^45 - 13 * q^47 - 14 * q^49 + 4 * q^51 + 5 * q^53 - 17 * q^55 + 5 * q^57 - 18 * q^59 + 26 * q^61 - 20 * q^63 - 20 * q^65 - 27 * q^67 - 3 * q^69 - 46 * q^71 - 25 * q^73 - 19 * q^75 - 19 * q^77 - 22 * q^79 - 9 * q^81 + q^83 - 11 * q^85 + 9 * q^87 - 3 * q^89 + 33 * q^93 - 40 * q^95 + 11 * q^97 - 21 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.41051	0.814360	0.407180	−	0.913348i	$-0.366512\pi$
$3$	1.41051	0.814360	0.407180	+	0.913348i	$0.366512\pi$
$4$	0	0
$5$	0.329231	0.147237	0.0736183	−	0.997286i	$-0.476545\pi$
$5$	0.329231	0.147237	0.0736183	+	0.997286i	$0.476545\pi$
$6$	0	0
$7$	−2.82300	−1.06699	−0.533496	−	0.845802i	$-0.679122\pi$
$7$	−2.82300	−1.06699	−0.533496	+	0.845802i	$0.679122\pi$
$8$	0	0
$9$	−1.01045	−0.336818
$10$	0	0
$11$	0.647542	0.195241	0.0976206	−	0.995224i	$-0.468877\pi$
$11$	0.647542	0.195241	0.0976206	+	0.995224i	$0.468877\pi$
$12$	0	0
$13$	−5.15026	−1.42842	−0.714212	−	0.699929i	$-0.753215\pi$
$13$	−5.15026	−1.42842	−0.714212	+	0.699929i	$0.753215\pi$
$14$	0	0
$15$	0.464384	0.119904
$16$	0	0
$17$	0.793615	0.192480	0.0962400	−	0.995358i	$-0.469318\pi$
$17$	0.793615	0.192480	0.0962400	+	0.995358i	$0.469318\pi$
$18$	0	0
$19$	−4.24097	−0.972944	−0.486472	−	0.873696i	$-0.661716\pi$
$19$	−4.24097	−0.972944	−0.486472	+	0.873696i	$0.661716\pi$
$20$	0	0
$21$	−3.98187	−0.868916
$22$	0	0
$23$	−1.35328	−0.282178	−0.141089	−	0.989997i	$-0.545060\pi$
$23$	−1.35328	−0.282178	−0.141089	+	0.989997i	$0.545060\pi$
$24$	0	0
$25$	−4.89161	−0.978321
$26$	0	0
$27$	−5.65680	−1.08865
$28$	0	0
$29$	7.17555	1.33247	0.666233	−	0.745744i	$-0.267906\pi$
$29$	7.17555	1.33247	0.666233	+	0.745744i	$0.267906\pi$
$30$	0	0
$31$	−0.216464	−0.0388781	−0.0194390	−	0.999811i	$-0.506188\pi$
$31$	−0.216464	−0.0388781	−0.0194390	+	0.999811i	$0.506188\pi$
$32$	0	0
$33$	0.913365	0.158997
$34$	0	0
$35$	−0.929418	−0.157100
$36$	0	0
$37$	1.38463	0.227632	0.113816	−	0.993502i	$-0.463693\pi$
$37$	1.38463	0.227632	0.113816	+	0.993502i	$0.463693\pi$
$38$	0	0
$39$	−7.26450	−1.16325
$40$	0	0
$41$	−10.2237	−1.59668	−0.798340	−	0.602207i	$-0.794288\pi$
$41$	−10.2237	−1.59668	−0.798340	+	0.602207i	$0.794288\pi$
$42$	0	0
$43$	−9.36903	−1.42876	−0.714382	−	0.699756i	$-0.753292\pi$
$43$	−9.36903	−1.42876	−0.714382	+	0.699756i	$0.753292\pi$
$44$	0	0
$45$	−0.332673	−0.0495919
$46$	0	0
$47$	8.80147	1.28383	0.641913	−	0.766777i	$-0.278141\pi$
$47$	8.80147	1.28383	0.641913	+	0.766777i	$0.278141\pi$
$48$	0	0
$49$	0.969317	0.138474
$50$	0	0
$51$	1.11940	0.156748
$52$	0	0
$53$	8.56574	1.17659	0.588297	−	0.808645i	$-0.299798\pi$
$53$	8.56574	1.17659	0.588297	+	0.808645i	$0.299798\pi$
$54$	0	0
$55$	0.213191	0.0287466
$56$	0	0
$57$	−5.98194	−0.792327
$58$	0	0
$59$	−11.9520	−1.55602	−0.778010	−	0.628252i	$-0.783770\pi$
$59$	−11.9520	−1.55602	−0.778010	+	0.628252i	$0.783770\pi$
$60$	0	0
$61$	9.46622	1.21203	0.606013	−	0.795455i	$-0.292768\pi$
$61$	9.46622	1.21203	0.606013	+	0.795455i	$0.292768\pi$
$62$	0	0
$63$	2.85251	0.359383
$64$	0	0
$65$	−1.69562	−0.210316
$66$	0	0
$67$	−5.91351	−0.722450	−0.361225	−	0.932479i	$-0.617641\pi$
$67$	−5.91351	−0.722450	−0.361225	+	0.932479i	$0.617641\pi$
$68$	0	0
$69$	−1.90881	−0.229794
$70$	0	0
$71$	−2.07131	−0.245819	−0.122910	−	0.992418i	$-0.539223\pi$
$71$	−2.07131	−0.245819	−0.122910	+	0.992418i	$0.539223\pi$
$72$	0	0
$73$	−9.19625	−1.07634	−0.538170	−	0.842836i	$-0.680884\pi$
$73$	−9.19625	−1.07634	−0.538170	+	0.842836i	$0.680884\pi$
$74$	0	0
$75$	−6.89967	−0.796706
$76$	0	0
$77$	−1.82801	−0.208321
$78$	0	0
$79$	2.65223	0.298399	0.149200	−	0.988807i	$-0.452330\pi$
$79$	2.65223	0.298399	0.149200	+	0.988807i	$0.452330\pi$
$80$	0	0
$81$	−4.94762	−0.549735
$82$	0	0
$83$	9.39541	1.03128	0.515640	−	0.856805i	$-0.327554\pi$
$83$	9.39541	1.03128	0.515640	+	0.856805i	$0.327554\pi$
$84$	0	0
$85$	0.261283	0.0283401
$86$	0	0
$87$	10.1212	1.08511
$88$	0	0
$89$	16.5954	1.75911	0.879553	−	0.475802i	$-0.157842\pi$
$89$	16.5954	1.75911	0.879553	+	0.475802i	$0.157842\pi$
$90$	0	0
$91$	14.5392	1.52412
$92$	0	0
$93$	−0.305325	−0.0316607
$94$	0	0
$95$	−1.39626	−0.143253
$96$	0	0
$97$	16.5225	1.67761	0.838804	−	0.544434i	$-0.183256\pi$
$97$	16.5225	1.67761	0.838804	+	0.544434i	$0.183256\pi$
$98$	0	0
$99$	−0.654311	−0.0657607
$100$	0	0
$101$	−17.2324	−1.71469	−0.857345	−	0.514742i	$-0.827888\pi$
$101$	−17.2324	−1.71469	−0.857345	+	0.514742i	$0.827888\pi$
$102$	0	0
$103$	2.84305	0.280134	0.140067	−	0.990142i	$-0.455268\pi$
$103$	2.84305	0.280134	0.140067	+	0.990142i	$0.455268\pi$
$104$	0	0
$105$	−1.31096	−0.127936
$106$	0	0
$107$	−8.64017	−0.835277	−0.417638	−	0.908613i	$-0.637142\pi$
$107$	−8.64017	−0.835277	−0.417638	+	0.908613i	$0.637142\pi$
$108$	0	0
$109$	−0.656825	−0.0629124	−0.0314562	−	0.999505i	$-0.510014\pi$
$109$	−0.656825	−0.0629124	−0.0314562	+	0.999505i	$0.510014\pi$
$110$	0	0
$111$	1.95304	0.185374
$112$	0	0
$113$	8.92888	0.839959	0.419979	−	0.907534i	$-0.362037\pi$
$113$	8.92888	0.839959	0.419979	+	0.907534i	$0.362037\pi$
$114$	0	0
$115$	−0.445541	−0.0415469
$116$	0	0
$117$	5.20410	0.481119
$118$	0	0
$119$	−2.24037	−0.205375
$120$	0	0
$121$	−10.5807	−0.961881
$122$	0	0
$123$	−14.4207	−1.30027
$124$	0	0
$125$	−3.25662	−0.291281
$126$	0	0
$127$	−8.03244	−0.712764	−0.356382	−	0.934340i	$-0.615990\pi$
$127$	−8.03244	−0.712764	−0.356382	+	0.934340i	$0.615990\pi$
$128$	0	0
$129$	−13.2151	−1.16353
$130$	0	0
$131$	−8.35545	−0.730019	−0.365010	−	0.931004i	$-0.618934\pi$
$131$	−8.35545	−0.730019	−0.365010	+	0.931004i	$0.618934\pi$
$132$	0	0
$133$	11.9722	1.03812
$134$	0	0
$135$	−1.86239	−0.160289
$136$	0	0
$137$	17.0743	1.45876	0.729378	−	0.684111i	$-0.239809\pi$
$137$	17.0743	1.45876	0.729378	+	0.684111i	$0.239809\pi$
$138$	0	0
$139$	−0.384365	−0.0326014	−0.0163007	−	0.999867i	$-0.505189\pi$
$139$	−0.384365	−0.0326014	−0.0163007	+	0.999867i	$0.505189\pi$
$140$	0	0
$141$	12.4146	1.04550
$142$	0	0
$143$	−3.33500	−0.278887
$144$	0	0
$145$	2.36241	0.196188
$146$	0	0
$147$	1.36723	0.112768
$148$	0	0
$149$	−16.2155	−1.32843	−0.664213	−	0.747543i	$-0.731233\pi$
$149$	−16.2155	−1.32843	−0.664213	+	0.747543i	$0.731233\pi$
$150$	0	0
$151$	2.46481	0.200584	0.100292	−	0.994958i	$-0.468022\pi$
$151$	2.46481	0.200584	0.100292	+	0.994958i	$0.468022\pi$
$152$	0	0
$153$	−0.801912	−0.0648307
$154$	0	0
$155$	−0.0712666	−0.00572427
$156$	0	0
$157$	11.7288	0.936059	0.468029	−	0.883713i	$-0.344964\pi$
$157$	11.7288	0.936059	0.468029	+	0.883713i	$0.344964\pi$
$158$	0	0
$159$	12.0821	0.958172
$160$	0	0
$161$	3.82030	0.301082
$162$	0	0
$163$	−19.4671	−1.52478	−0.762389	−	0.647119i	$-0.775974\pi$
$163$	−19.4671	−1.52478	−0.762389	+	0.647119i	$0.775974\pi$
$164$	0	0
$165$	0.300708	0.0234101
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	13.5251	1.04040
$170$	0	0
$171$	4.28530	0.327705
$172$	0	0
$173$	−0.511083	−0.0388570	−0.0194285	−	0.999811i	$-0.506185\pi$
$173$	−0.511083	−0.0388570	−0.0194285	+	0.999811i	$0.506185\pi$
$174$	0	0
$175$	13.8090	1.04386
$176$	0	0
$177$	−16.8585	−1.26716
$178$	0	0
$179$	−1.10589	−0.0826581	−0.0413291	−	0.999146i	$-0.513159\pi$
$179$	−1.10589	−0.0826581	−0.0413291	+	0.999146i	$0.513159\pi$
$180$	0	0
$181$	16.6149	1.23498	0.617488	−	0.786580i	$-0.288150\pi$
$181$	16.6149	1.23498	0.617488	+	0.786580i	$0.288150\pi$
$182$	0	0
$183$	13.3522	0.987025
$184$	0	0
$185$	0.455864	0.0335158
$186$	0	0
$187$	0.513899	0.0375800
$188$	0	0
$189$	15.9691	1.16158
$190$	0	0
$191$	−25.7912	−1.86619	−0.933094	−	0.359634i	$-0.882902\pi$
$191$	−25.7912	−1.86619	−0.933094	+	0.359634i	$0.882902\pi$
$192$	0	0
$193$	7.38704	0.531731	0.265865	−	0.964010i	$-0.414342\pi$
$193$	7.38704	0.531731	0.265865	+	0.964010i	$0.414342\pi$
$194$	0	0
$195$	−2.39170	−0.171273
$196$	0	0
$197$	−4.97110	−0.354176	−0.177088	−	0.984195i	$-0.556668\pi$
$197$	−4.97110	−0.354176	−0.177088	+	0.984195i	$0.556668\pi$
$198$	0	0
$199$	−23.9845	−1.70022	−0.850109	−	0.526606i	$-0.823464\pi$
$199$	−23.9845	−1.70022	−0.850109	+	0.526606i	$0.823464\pi$
$200$	0	0
$201$	−8.34108	−0.588334
$202$	0	0
$203$	−20.2566	−1.42173
$204$	0	0
$205$	−3.36597	−0.235090
$206$	0	0
$207$	1.36742	0.0950426
$208$	0	0
$209$	−2.74620	−0.189959
$210$	0	0
$211$	−7.10921	−0.489418	−0.244709	−	0.969597i	$-0.578692\pi$
$211$	−7.10921	−0.489418	−0.244709	+	0.969597i	$0.578692\pi$
$212$	0	0
$213$	−2.92161	−0.200185
$214$	0	0
$215$	−3.08457	−0.210366
$216$	0	0
$217$	0.611077	0.0414826
$218$	0	0
$219$	−12.9714	−0.876528
$220$	0	0
$221$	−4.08732	−0.274943
$222$	0	0
$223$	−21.7635	−1.45739	−0.728695	−	0.684838i	$-0.759873\pi$
$223$	−21.7635	−1.45739	−0.728695	+	0.684838i	$0.759873\pi$
$224$	0	0
$225$	4.94275	0.329516
$226$	0	0
$227$	25.4687	1.69042	0.845209	−	0.534437i	$-0.179476\pi$
$227$	25.4687	1.69042	0.845209	+	0.534437i	$0.179476\pi$
$228$	0	0
$229$	29.0113	1.91712	0.958559	−	0.284895i	$-0.0919587\pi$
$229$	29.0113	1.91712	0.958559	+	0.284895i	$0.0919587\pi$
$230$	0	0
$231$	−2.57843	−0.169648
$232$	0	0
$233$	13.2842	0.870280	0.435140	−	0.900363i	$-0.356699\pi$
$233$	13.2842	0.870280	0.435140	+	0.900363i	$0.356699\pi$
$234$	0	0
$235$	2.89772	0.189026
$236$	0	0
$237$	3.74100	0.243004
$238$	0	0
$239$	7.56619	0.489416	0.244708	−	0.969597i	$-0.421308\pi$
$239$	7.56619	0.489416	0.244708	+	0.969597i	$0.421308\pi$
$240$	0	0
$241$	17.8601	1.15047	0.575234	−	0.817989i	$-0.304911\pi$
$241$	17.8601	1.15047	0.575234	+	0.817989i	$0.304911\pi$
$242$	0	0
$243$	9.99171	0.640968
$244$	0	0
$245$	0.319129	0.0203884
$246$	0	0
$247$	21.8421	1.38978
$248$	0	0
$249$	13.2523	0.839833
$250$	0	0
$251$	−6.72508	−0.424483	−0.212242	−	0.977217i	$-0.568076\pi$
$251$	−6.72508	−0.424483	−0.212242	+	0.977217i	$0.568076\pi$
$252$	0	0
$253$	−0.876303	−0.0550927
$254$	0	0
$255$	0.368542	0.0230790
$256$	0	0
$257$	−15.0042	−0.935936	−0.467968	−	0.883745i	$-0.655014\pi$
$257$	−15.0042	−0.935936	−0.467968	+	0.883745i	$0.655014\pi$
$258$	0	0
$259$	−3.90881	−0.242882
$260$	0	0
$261$	−7.25056	−0.448799
$262$	0	0
$263$	−6.50797	−0.401299	−0.200649	−	0.979663i	$-0.564305\pi$
$263$	−6.50797	−0.401299	−0.200649	+	0.979663i	$0.564305\pi$
$264$	0	0
$265$	2.82011	0.173238
$266$	0	0
$267$	23.4080	1.43254
$268$	0	0
$269$	−7.32212	−0.446438	−0.223219	−	0.974768i	$-0.571656\pi$
$269$	−7.32212	−0.446438	−0.223219	+	0.974768i	$0.571656\pi$
$270$	0	0
$271$	−23.1868	−1.40850	−0.704250	−	0.709952i	$-0.748717\pi$
$271$	−23.1868	−1.40850	−0.704250	+	0.709952i	$0.748717\pi$
$272$	0	0
$273$	20.5077	1.24118
$274$	0	0
$275$	−3.16752	−0.191009
$276$	0	0
$277$	12.2596	0.736610	0.368305	−	0.929705i	$-0.379938\pi$
$277$	12.2596	0.736610	0.368305	+	0.929705i	$0.379938\pi$
$278$	0	0
$279$	0.218727	0.0130948
$280$	0	0
$281$	−20.9434	−1.24938	−0.624691	−	0.780872i	$-0.714775\pi$
$281$	−20.9434	−1.24938	−0.624691	+	0.780872i	$0.714775\pi$
$282$	0	0
$283$	−1.65633	−0.0984587	−0.0492293	−	0.998788i	$-0.515677\pi$
$283$	−1.65633	−0.0984587	−0.0492293	+	0.998788i	$0.515677\pi$
$284$	0	0
$285$	−1.96944	−0.116659
$286$	0	0
$287$	28.8616	1.70365
$288$	0	0
$289$	−16.3702	−0.962951
$290$	0	0
$291$	23.3052	1.36618
$292$	0	0
$293$	14.3663	0.839288	0.419644	−	0.907689i	$-0.362155\pi$
$293$	14.3663	0.839288	0.419644	+	0.907689i	$0.362155\pi$
$294$	0	0
$295$	−3.93497	−0.229103
$296$	0	0
$297$	−3.66301	−0.212549
$298$	0	0
$299$	6.96972	0.403069
$300$	0	0
$301$	26.4487	1.52448
$302$	0	0
$303$	−24.3065	−1.39637
$304$	0	0
$305$	3.11657	0.178454
$306$	0	0
$307$	34.4919	1.96856	0.984278	−	0.176625i	$-0.0565180\pi$
$307$	34.4919	1.96856	0.984278	+	0.176625i	$0.0565180\pi$
$308$	0	0
$309$	4.01016	0.228130
$310$	0	0
$311$	16.9100	0.958875	0.479438	−	0.877576i	$-0.340841\pi$
$311$	16.9100	0.958875	0.479438	+	0.877576i	$0.340841\pi$
$312$	0	0
$313$	−23.2680	−1.31518	−0.657592	−	0.753375i	$-0.728425\pi$
$313$	−23.2680	−1.31518	−0.657592	+	0.753375i	$0.728425\pi$
$314$	0	0
$315$	0.939135	0.0529142
$316$	0	0
$317$	6.71210	0.376989	0.188494	−	0.982074i	$-0.439639\pi$
$317$	6.71210	0.376989	0.188494	+	0.982074i	$0.439639\pi$
$318$	0	0
$319$	4.64646	0.260152
$320$	0	0
$321$	−12.1871	−0.680216
$322$	0	0
$323$	−3.36569	−0.187272
$324$	0	0
$325$	25.1930	1.39746
$326$	0	0
$327$	−0.926460	−0.0512333
$328$	0	0
$329$	−24.8465	−1.36983
$330$	0	0
$331$	22.0175	1.21019	0.605095	−	0.796153i	$-0.293135\pi$
$331$	22.0175	1.21019	0.605095	+	0.796153i	$0.293135\pi$
$332$	0	0
$333$	−1.39911	−0.0766706
$334$	0	0
$335$	−1.94691	−0.106371
$336$	0	0
$337$	−4.21446	−0.229576	−0.114788	−	0.993390i	$-0.536619\pi$
$337$	−4.21446	−0.229576	−0.114788	+	0.993390i	$0.536619\pi$
$338$	0	0
$339$	12.5943	0.684028
$340$	0	0
$341$	−0.140169	−0.00759060
$342$	0	0
$343$	17.0246	0.919242
$344$	0	0
$345$	−0.628441	−0.0338341
$346$	0	0
$347$	15.4626	0.830076	0.415038	−	0.909804i	$-0.363768\pi$
$347$	15.4626	0.830076	0.415038	+	0.909804i	$0.363768\pi$
$348$	0	0
$349$	32.1550	1.72122	0.860608	−	0.509268i	$-0.170084\pi$
$349$	32.1550	1.72122	0.860608	+	0.509268i	$0.170084\pi$
$350$	0	0
$351$	29.1339	1.55506
$352$	0	0
$353$	−27.8632	−1.48301	−0.741504	−	0.670949i	$-0.765887\pi$
$353$	−27.8632	−1.48301	−0.741504	+	0.670949i	$0.765887\pi$
$354$	0	0
$355$	−0.681939	−0.0361936
$356$	0	0
$357$	−3.16008	−0.167249
$358$	0	0
$359$	3.08719	0.162935	0.0814677	−	0.996676i	$-0.474039\pi$
$359$	3.08719	0.162935	0.0814677	+	0.996676i	$0.474039\pi$
$360$	0	0
$361$	−1.01421	−0.0533794
$362$	0	0
$363$	−14.9242	−0.783317
$364$	0	0
$365$	−3.02769	−0.158477
$366$	0	0
$367$	14.1281	0.737481	0.368740	−	0.929532i	$-0.379789\pi$
$367$	14.1281	0.737481	0.368740	+	0.929532i	$0.379789\pi$
$368$	0	0
$369$	10.3306	0.537791
$370$	0	0
$371$	−24.1811	−1.25542
$372$	0	0
$373$	−12.5143	−0.647967	−0.323983	−	0.946063i	$-0.605022\pi$
$373$	−12.5143	−0.647967	−0.323983	+	0.946063i	$0.605022\pi$
$374$	0	0
$375$	−4.59351	−0.237208
$376$	0	0
$377$	−36.9559	−1.90333
$378$	0	0
$379$	−13.7772	−0.707690	−0.353845	−	0.935304i	$-0.615126\pi$
$379$	−13.7772	−0.707690	−0.353845	+	0.935304i	$0.615126\pi$
$380$	0	0
$381$	−11.3299	−0.580446
$382$	0	0
$383$	−7.33238	−0.374667	−0.187334	−	0.982296i	$-0.559985\pi$
$383$	−7.33238	−0.374667	−0.187334	+	0.982296i	$0.559985\pi$
$384$	0	0
$385$	−0.601837	−0.0306724
$386$	0	0
$387$	9.46698	0.481233
$388$	0	0
$389$	3.00680	0.152451	0.0762253	−	0.997091i	$-0.475713\pi$
$389$	3.00680	0.152451	0.0762253	+	0.997091i	$0.475713\pi$
$390$	0	0
$391$	−1.07398	−0.0543136
$392$	0	0
$393$	−11.7855	−0.594498
$394$	0	0
$395$	0.873196	0.0439353
$396$	0	0
$397$	−12.0098	−0.602753	−0.301377	−	0.953505i	$-0.597446\pi$
$397$	−12.0098	−0.602753	−0.301377	+	0.953505i	$0.597446\pi$
$398$	0	0
$399$	16.8870	0.845407
$400$	0	0
$401$	9.76872	0.487827	0.243913	−	0.969797i	$-0.421569\pi$
$401$	9.76872	0.487827	0.243913	+	0.969797i	$0.421569\pi$
$402$	0	0
$403$	1.11484	0.0555344
$404$	0	0
$405$	−1.62891	−0.0809411
$406$	0	0
$407$	0.896607	0.0444432
$408$	0	0
$409$	−23.1213	−1.14328	−0.571638	−	0.820506i	$-0.693692\pi$
$409$	−23.1213	−1.14328	−0.571638	+	0.820506i	$0.693692\pi$
$410$	0	0
$411$	24.0835	1.18795
$412$	0	0
$413$	33.7405	1.66026
$414$	0	0
$415$	3.09326	0.151842
$416$	0	0
$417$	−0.542152	−0.0265493
$418$	0	0
$419$	38.2877	1.87048	0.935239	−	0.354018i	$-0.115185\pi$
$419$	38.2877	1.87048	0.935239	+	0.354018i	$0.115185\pi$
$420$	0	0
$421$	−18.7105	−0.911894	−0.455947	−	0.890007i	$-0.650699\pi$
$421$	−18.7105	−0.911894	−0.455947	+	0.890007i	$0.650699\pi$
$422$	0	0
$423$	−8.89348	−0.432416
$424$	0	0
$425$	−3.88205	−0.188307
$426$	0	0
$427$	−26.7231	−1.29322
$428$	0	0
$429$	−4.70407	−0.227114
$430$	0	0
$431$	−18.4796	−0.890133	−0.445066	−	0.895498i	$-0.646820\pi$
$431$	−18.4796	−0.890133	−0.445066	+	0.895498i	$0.646820\pi$
$432$	0	0
$433$	18.2643	0.877725	0.438862	−	0.898554i	$-0.355382\pi$
$433$	18.2643	0.877725	0.438862	+	0.898554i	$0.355382\pi$
$434$	0	0
$435$	3.33221	0.159767
$436$	0	0
$437$	5.73920	0.274543
$438$	0	0
$439$	17.0033	0.811524	0.405762	−	0.913979i	$-0.367006\pi$
$439$	17.0033	0.811524	0.405762	+	0.913979i	$0.367006\pi$
$440$	0	0
$441$	−0.979450	−0.0466405
$442$	0	0
$443$	−10.9180	−0.518730	−0.259365	−	0.965779i	$-0.583513\pi$
$443$	−10.9180	−0.518730	−0.259365	+	0.965779i	$0.583513\pi$
$444$	0	0
$445$	5.46371	0.259005
$446$	0	0
$447$	−22.8722	−1.08182
$448$	0	0
$449$	−0.792350	−0.0373933	−0.0186967	−	0.999825i	$-0.505952\pi$
$449$	−0.792350	−0.0373933	−0.0186967	+	0.999825i	$0.505952\pi$
$450$	0	0
$451$	−6.62030	−0.311738
$452$	0	0
$453$	3.47665	0.163347
$454$	0	0
$455$	4.78674	0.224406
$456$	0	0
$457$	22.0068	1.02943	0.514717	−	0.857360i	$-0.327897\pi$
$457$	22.0068	1.02943	0.514717	+	0.857360i	$0.327897\pi$
$458$	0	0
$459$	−4.48932	−0.209543
$460$	0	0
$461$	−26.9234	−1.25395	−0.626974	−	0.779040i	$-0.715707\pi$
$461$	−26.9234	−1.25395	−0.626974	+	0.779040i	$0.715707\pi$
$462$	0	0
$463$	3.24420	0.150771	0.0753853	−	0.997154i	$-0.475981\pi$
$463$	3.24420	0.150771	0.0753853	+	0.997154i	$0.475981\pi$
$464$	0	0
$465$	−0.100522	−0.00466162
$466$	0	0
$467$	−9.92142	−0.459108	−0.229554	−	0.973296i	$-0.573727\pi$
$467$	−9.92142	−0.459108	−0.229554	+	0.973296i	$0.573727\pi$
$468$	0	0
$469$	16.6938	0.770849
$470$	0	0
$471$	16.5436	0.762289
$472$	0	0
$473$	−6.06684	−0.278953
$474$	0	0
$475$	20.7451	0.951852
$476$	0	0
$477$	−8.65529	−0.396298
$478$	0	0
$479$	5.28167	0.241326	0.120663	−	0.992694i	$-0.461498\pi$
$479$	5.28167	0.241326	0.120663	+	0.992694i	$0.461498\pi$
$480$	0	0
$481$	−7.13121	−0.325155
$482$	0	0
$483$	5.38858	0.245189
$484$	0	0
$485$	5.43972	0.247005
$486$	0	0
$487$	−7.66934	−0.347531	−0.173766	−	0.984787i	$-0.555593\pi$
$487$	−7.66934	−0.347531	−0.173766	+	0.984787i	$0.555593\pi$
$488$	0	0
$489$	−27.4585	−1.24172
$490$	0	0
$491$	7.30724	0.329771	0.164886	−	0.986313i	$-0.447275\pi$
$491$	7.30724	0.329771	0.164886	+	0.986313i	$0.447275\pi$
$492$	0	0
$493$	5.69462	0.256473
$494$	0	0
$495$	−0.215419	−0.00968238
$496$	0	0
$497$	5.84730	0.262287
$498$	0	0
$499$	15.0671	0.674495	0.337247	−	0.941416i	$-0.390504\pi$
$499$	15.0671	0.674495	0.337247	+	0.941416i	$0.390504\pi$
$500$	0	0
$501$	−1.41051	−0.0630171
$502$	0	0
$503$	−29.1807	−1.30110	−0.650552	−	0.759462i	$-0.725462\pi$
$503$	−29.1807	−1.30110	−0.650552	+	0.759462i	$0.725462\pi$
$504$	0	0
$505$	−5.67345	−0.252465
$506$	0	0
$507$	19.0774	0.847256
$508$	0	0
$509$	2.01779	0.0894371	0.0447185	−	0.999000i	$-0.485761\pi$
$509$	2.01779	0.0894371	0.0447185	+	0.999000i	$0.485761\pi$
$510$	0	0
$511$	25.9610	1.14845
$512$	0	0
$513$	23.9903	1.05920
$514$	0	0
$515$	0.936020	0.0412460
$516$	0	0
$517$	5.69932	0.250656
$518$	0	0
$519$	−0.720890	−0.0316435
$520$	0	0
$521$	−21.0308	−0.921375	−0.460688	−	0.887562i	$-0.652397\pi$
$521$	−21.0308	−0.921375	−0.460688	+	0.887562i	$0.652397\pi$
$522$	0	0
$523$	−39.6730	−1.73478	−0.867389	−	0.497631i	$-0.834203\pi$
$523$	−39.6730	−1.73478	−0.867389	+	0.497631i	$0.834203\pi$
$524$	0	0
$525$	19.4778	0.850079
$526$	0	0
$527$	−0.171789	−0.00748325
$528$	0	0
$529$	−21.1686	−0.920376
$530$	0	0
$531$	12.0770	0.524095
$532$	0	0
$533$	52.6549	2.28074
$534$	0	0
$535$	−2.84461	−0.122983
$536$	0	0
$537$	−1.55987	−0.0673135
$538$	0	0
$539$	0.627673	0.0270358
$540$	0	0
$541$	−5.89168	−0.253303	−0.126651	−	0.991947i	$-0.540423\pi$
$541$	−5.89168	−0.253303	−0.126651	+	0.991947i	$0.540423\pi$
$542$	0	0
$543$	23.4355	1.00572
$544$	0	0
$545$	−0.216247	−0.00926301
$546$	0	0
$547$	26.8378	1.14750	0.573751	−	0.819030i	$-0.305488\pi$
$547$	26.8378	1.14750	0.573751	+	0.819030i	$0.305488\pi$
$548$	0	0
$549$	−9.56518	−0.408232
$550$	0	0
$551$	−30.4313	−1.29641
$552$	0	0
$553$	−7.48724	−0.318390
$554$	0	0
$555$	0.643001	0.0272939
$556$	0	0
$557$	−36.2807	−1.53726	−0.768631	−	0.639693i	$-0.779062\pi$
$557$	−36.2807	−1.53726	−0.768631	+	0.639693i	$0.779062\pi$
$558$	0	0
$559$	48.2529	2.04088
$560$	0	0
$561$	0.724861	0.0306036
$562$	0	0
$563$	18.5002	0.779692	0.389846	−	0.920880i	$-0.372528\pi$
$563$	18.5002	0.779692	0.389846	+	0.920880i	$0.372528\pi$
$564$	0	0
$565$	2.93966	0.123673
$566$	0	0
$567$	13.9671	0.586564
$568$	0	0
$569$	−44.0253	−1.84564	−0.922819	−	0.385234i	$-0.874121\pi$
$569$	−44.0253	−1.84564	−0.922819	+	0.385234i	$0.874121\pi$
$570$	0	0
$571$	−22.7302	−0.951230	−0.475615	−	0.879654i	$-0.657774\pi$
$571$	−22.7302	−0.951230	−0.475615	+	0.879654i	$0.657774\pi$
$572$	0	0
$573$	−36.3788	−1.51975
$574$	0	0
$575$	6.61970	0.276061
$576$	0	0
$577$	10.5931	0.440998	0.220499	−	0.975387i	$-0.429231\pi$
$577$	10.5931	0.440998	0.220499	+	0.975387i	$0.429231\pi$
$578$	0	0
$579$	10.4195	0.433020
$580$	0	0
$581$	−26.5232	−1.10037
$582$	0	0
$583$	5.54667	0.229720
$584$	0	0
$585$	1.71335	0.0708383
$586$	0	0
$587$	−20.3725	−0.840861	−0.420431	−	0.907325i	$-0.638121\pi$
$587$	−20.3725	−0.840861	−0.420431	+	0.907325i	$0.638121\pi$
$588$	0	0
$589$	0.918016	0.0378262
$590$	0	0
$591$	−7.01180	−0.288427
$592$	0	0
$593$	2.41852	0.0993168	0.0496584	−	0.998766i	$-0.484187\pi$
$593$	2.41852	0.0993168	0.0496584	+	0.998766i	$0.484187\pi$
$594$	0	0
$595$	−0.737600	−0.0302387
$596$	0	0
$597$	−33.8305	−1.38459
$598$	0	0
$599$	27.0111	1.10364	0.551821	−	0.833962i	$-0.313933\pi$
$599$	27.0111	1.10364	0.551821	+	0.833962i	$0.313933\pi$
$600$	0	0
$601$	−30.9292	−1.26163	−0.630814	−	0.775934i	$-0.717279\pi$
$601$	−30.9292	−1.26163	−0.630814	+	0.775934i	$0.717279\pi$
$602$	0	0
$603$	5.97533	0.243334
$604$	0	0
$605$	−3.48349	−0.141624
$606$	0	0
$607$	−11.0153	−0.447095	−0.223548	−	0.974693i	$-0.571764\pi$
$607$	−11.0153	−0.447095	−0.223548	+	0.974693i	$0.571764\pi$
$608$	0	0
$609$	−28.5721	−1.15780
$610$	0	0
$611$	−45.3298	−1.83385
$612$	0	0
$613$	−3.88888	−0.157070	−0.0785351	−	0.996911i	$-0.525024\pi$
$613$	−3.88888	−0.157070	−0.0785351	+	0.996911i	$0.525024\pi$
$614$	0	0
$615$	−4.74774	−0.191447
$616$	0	0
$617$	3.83701	0.154472	0.0772360	−	0.997013i	$-0.475391\pi$
$617$	3.83701	0.154472	0.0772360	+	0.997013i	$0.475391\pi$
$618$	0	0
$619$	18.5819	0.746869	0.373434	−	0.927657i	$-0.378180\pi$
$619$	18.5819	0.746869	0.373434	+	0.927657i	$0.378180\pi$
$620$	0	0
$621$	7.65521	0.307193
$622$	0	0
$623$	−46.8487	−1.87695
$624$	0	0
$625$	23.3859	0.935434
$626$	0	0
$627$	−3.87355	−0.154695
$628$	0	0
$629$	1.09887	0.0438146
$630$	0	0
$631$	−36.3695	−1.44785	−0.723923	−	0.689880i	$-0.757663\pi$
$631$	−36.3695	−1.44785	−0.723923	+	0.689880i	$0.757663\pi$
$632$	0	0
$633$	−10.0276	−0.398563
$634$	0	0
$635$	−2.64453	−0.104945
$636$	0	0
$637$	−4.99223	−0.197799
$638$	0	0
$639$	2.09296	0.0827964
$640$	0	0
$641$	16.9793	0.670644	0.335322	−	0.942104i	$-0.391155\pi$
$641$	16.9793	0.670644	0.335322	+	0.942104i	$0.391155\pi$
$642$	0	0
$643$	9.73145	0.383771	0.191886	−	0.981417i	$-0.438540\pi$
$643$	9.73145	0.383771	0.191886	+	0.981417i	$0.438540\pi$
$644$	0	0
$645$	−4.35083	−0.171314
$646$	0	0
$647$	−27.0734	−1.06436	−0.532182	−	0.846630i	$-0.678628\pi$
$647$	−27.0734	−1.06436	−0.532182	+	0.846630i	$0.678628\pi$
$648$	0	0
$649$	−7.73942	−0.303799
$650$	0	0
$651$	0.861932	0.0337818
$652$	0	0
$653$	−38.3977	−1.50262	−0.751309	−	0.659951i	$-0.770577\pi$
$653$	−38.3977	−1.50262	−0.751309	+	0.659951i	$0.770577\pi$
$654$	0	0
$655$	−2.75087	−0.107485
$656$	0	0
$657$	9.29239	0.362531
$658$	0	0
$659$	−34.3823	−1.33934	−0.669671	−	0.742657i	$-0.733565\pi$
$659$	−34.3823	−1.33934	−0.669671	+	0.742657i	$0.733565\pi$
$660$	0	0
$661$	−13.8262	−0.537776	−0.268888	−	0.963171i	$-0.586656\pi$
$661$	−13.8262	−0.537776	−0.268888	+	0.963171i	$0.586656\pi$
$662$	0	0
$663$	−5.76522	−0.223903
$664$	0	0
$665$	3.94163	0.152850
$666$	0	0
$667$	−9.71050	−0.375992
$668$	0	0
$669$	−30.6976	−1.18684
$670$	0	0
$671$	6.12977	0.236637
$672$	0	0
$673$	−24.1053	−0.929190	−0.464595	−	0.885523i	$-0.653800\pi$
$673$	−24.1053	−0.929190	−0.464595	+	0.885523i	$0.653800\pi$
$674$	0	0
$675$	27.6708	1.06505
$676$	0	0
$677$	12.0213	0.462015	0.231008	−	0.972952i	$-0.425798\pi$
$677$	12.0213	0.462015	0.231008	+	0.972952i	$0.425798\pi$
$678$	0	0
$679$	−46.6430	−1.78999
$680$	0	0
$681$	35.9239	1.37661
$682$	0	0
$683$	−1.74922	−0.0669320	−0.0334660	−	0.999440i	$-0.510655\pi$
$683$	−1.74922	−0.0669320	−0.0334660	+	0.999440i	$0.510655\pi$
$684$	0	0
$685$	5.62139	0.214782
$686$	0	0
$687$	40.9207	1.56122
$688$	0	0
$689$	−44.1158	−1.68068
$690$	0	0
$691$	−47.0997	−1.79176	−0.895878	−	0.444299i	$-0.853453\pi$
$691$	−47.0997	−1.79176	−0.895878	+	0.444299i	$0.853453\pi$
$692$	0	0
$693$	1.84712	0.0701662
$694$	0	0
$695$	−0.126545	−0.00480012
$696$	0	0
$697$	−8.11371	−0.307329
$698$	0	0
$699$	18.7376	0.708721
$700$	0	0
$701$	5.93907	0.224316	0.112158	−	0.993690i	$-0.464224\pi$
$701$	5.93907	0.224316	0.112158	+	0.993690i	$0.464224\pi$
$702$	0	0
$703$	−5.87218	−0.221473
$704$	0	0
$705$	4.08726	0.153935
$706$	0	0
$707$	48.6471	1.82956
$708$	0	0
$709$	31.9710	1.20070	0.600348	−	0.799739i	$-0.295029\pi$
$709$	31.9710	1.20070	0.600348	+	0.799739i	$0.295029\pi$
$710$	0	0
$711$	−2.67996	−0.100506
$712$	0	0
$713$	0.292936	0.0109705
$714$	0	0
$715$	−1.09799	−0.0410624
$716$	0	0
$717$	10.6722	0.398561
$718$	0	0
$719$	4.14992	0.154766	0.0773829	−	0.997001i	$-0.475344\pi$
$719$	4.14992	0.154766	0.0773829	+	0.997001i	$0.475344\pi$
$720$	0	0
$721$	−8.02593	−0.298901
$722$	0	0
$723$	25.1918	0.936895
$724$	0	0
$725$	−35.1000	−1.30358
$726$	0	0
$727$	−24.0918	−0.893517	−0.446758	−	0.894655i	$-0.647422\pi$
$727$	−24.0918	−0.893517	−0.446758	+	0.894655i	$0.647422\pi$
$728$	0	0
$729$	28.9363	1.07171
$730$	0	0
$731$	−7.43540	−0.275008
$732$	0	0
$733$	6.22385	0.229883	0.114942	−	0.993372i	$-0.463332\pi$
$733$	6.22385	0.229883	0.114942	+	0.993372i	$0.463332\pi$
$734$	0	0
$735$	0.450136	0.0166035
$736$	0	0
$737$	−3.82924	−0.141052
$738$	0	0
$739$	−48.4232	−1.78128	−0.890638	−	0.454713i	$-0.849742\pi$
$739$	−48.4232	−1.78128	−0.890638	+	0.454713i	$0.849742\pi$
$740$	0	0
$741$	30.8085	1.13178
$742$	0	0
$743$	−11.2780	−0.413749	−0.206874	−	0.978368i	$-0.566329\pi$
$743$	−11.2780	−0.413749	−0.206874	+	0.978368i	$0.566329\pi$
$744$	0	0
$745$	−5.33865	−0.195593
$746$	0	0
$747$	−9.49363	−0.347354
$748$	0	0
$749$	24.3912	0.891234
$750$	0	0
$751$	−15.6289	−0.570308	−0.285154	−	0.958482i	$-0.592045\pi$
$751$	−15.6289	−0.570308	−0.285154	+	0.958482i	$0.592045\pi$
$752$	0	0
$753$	−9.48581	−0.345682
$754$	0	0
$755$	0.811492	0.0295332
$756$	0	0
$757$	18.3833	0.668153	0.334076	−	0.942546i	$-0.391576\pi$
$757$	18.3833	0.668153	0.334076	+	0.942546i	$0.391576\pi$
$758$	0	0
$759$	−1.23604	−0.0448653
$760$	0	0
$761$	29.1937	1.05827	0.529135	−	0.848537i	$-0.322516\pi$
$761$	29.1937	1.05827	0.529135	+	0.848537i	$0.322516\pi$
$762$	0	0
$763$	1.85422	0.0671271
$764$	0	0
$765$	−0.264014	−0.00954545
$766$	0	0
$767$	61.5559	2.22265
$768$	0	0
$769$	−29.0552	−1.04776	−0.523878	−	0.851793i	$-0.675515\pi$
$769$	−29.0552	−1.04776	−0.523878	+	0.851793i	$0.675515\pi$
$770$	0	0
$771$	−21.1636	−0.762189
$772$	0	0
$773$	8.08621	0.290841	0.145420	−	0.989370i	$-0.453547\pi$
$773$	8.08621	0.290841	0.145420	+	0.989370i	$0.453547\pi$
$774$	0	0
$775$	1.05886	0.0380352
$776$	0	0
$777$	−5.51343	−0.197793
$778$	0	0
$779$	43.3585	1.55348
$780$	0	0
$781$	−1.34126	−0.0479940
$782$	0	0
$783$	−40.5906	−1.45059
$784$	0	0
$785$	3.86148	0.137822
$786$	0	0
$787$	−26.6098	−0.948539	−0.474269	−	0.880380i	$-0.657288\pi$
$787$	−26.6098	−0.948539	−0.474269	+	0.880380i	$0.657288\pi$
$788$	0	0
$789$	−9.17957	−0.326802
$790$	0	0
$791$	−25.2062	−0.896230
$792$	0	0
$793$	−48.7534	−1.73129
$794$	0	0
$795$	3.97780	0.141078
$796$	0	0
$797$	−27.5352	−0.975347	−0.487673	−	0.873026i	$-0.662154\pi$
$797$	−27.5352	−0.975347	−0.487673	+	0.873026i	$0.662154\pi$
$798$	0	0
$799$	6.98498	0.247111
$800$	0	0
$801$	−16.7689	−0.592498
$802$	0	0
$803$	−5.95495	−0.210146
$804$	0	0
$805$	1.25776	0.0443302
$806$	0	0
$807$	−10.3279	−0.363561
$808$	0	0
$809$	−36.4041	−1.27990	−0.639949	−	0.768417i	$-0.721045\pi$
$809$	−36.4041	−1.27990	−0.639949	+	0.768417i	$0.721045\pi$
$810$	0	0
$811$	0.861339	0.0302457	0.0151229	−	0.999886i	$-0.495186\pi$
$811$	0.861339	0.0302457	0.0151229	+	0.999886i	$0.495186\pi$
$812$	0	0
$813$	−32.7053	−1.14703
$814$	0	0
$815$	−6.40916	−0.224503
$816$	0	0
$817$	39.7337	1.39011
$818$	0	0
$819$	−14.6912	−0.513351
$820$	0	0
$821$	−29.7097	−1.03688	−0.518438	−	0.855115i	$-0.673486\pi$
$821$	−29.7097	−1.03688	−0.518438	+	0.855115i	$0.673486\pi$
$822$	0	0
$823$	−5.28036	−0.184062	−0.0920310	−	0.995756i	$-0.529336\pi$
$823$	−5.28036	−0.184062	−0.0920310	+	0.995756i	$0.529336\pi$
$824$	0	0
$825$	−4.46782	−0.155550
$826$	0	0
$827$	−14.8674	−0.516990	−0.258495	−	0.966013i	$-0.583226\pi$
$827$	−14.8674	−0.516990	−0.258495	+	0.966013i	$0.583226\pi$
$828$	0	0
$829$	9.28332	0.322423	0.161212	−	0.986920i	$-0.448460\pi$
$829$	9.28332	0.322423	0.161212	+	0.986920i	$0.448460\pi$
$830$	0	0
$831$	17.2924	0.599866
$832$	0	0
$833$	0.769265	0.0266534
$834$	0	0
$835$	−0.329231	−0.0113935
$836$	0	0
$837$	1.22449	0.0423246
$838$	0	0
$839$	10.8525	0.374669	0.187335	−	0.982296i	$-0.440015\pi$
$839$	10.8525	0.374669	0.187335	+	0.982296i	$0.440015\pi$
$840$	0	0
$841$	22.4885	0.775465
$842$	0	0
$843$	−29.5410	−1.01745
$844$	0	0
$845$	4.45289	0.153184
$846$	0	0
$847$	29.8693	1.02632
$848$	0	0
$849$	−2.33628	−0.0801808
$850$	0	0
$851$	−1.87379	−0.0642327
$852$	0	0
$853$	−13.5222	−0.462992	−0.231496	−	0.972836i	$-0.574362\pi$
$853$	−13.5222	−0.462992	−0.231496	+	0.972836i	$0.574362\pi$
$854$	0	0
$855$	1.41085	0.0482502
$856$	0	0
$857$	−13.8181	−0.472017	−0.236009	−	0.971751i	$-0.575839\pi$
$857$	−13.8181	−0.472017	−0.236009	+	0.971751i	$0.575839\pi$
$858$	0	0
$859$	45.5521	1.55422	0.777109	−	0.629366i	$-0.216685\pi$
$859$	45.5521	1.55422	0.777109	+	0.629366i	$0.216685\pi$
$860$	0	0
$861$	40.7096	1.38738
$862$	0	0
$863$	−26.8287	−0.913259	−0.456629	−	0.889657i	$-0.650943\pi$
$863$	−26.8287	−0.913259	−0.456629	+	0.889657i	$0.650943\pi$
$864$	0	0
$865$	−0.168264	−0.00572116
$866$	0	0
$867$	−23.0903	−0.784189
$868$	0	0
$869$	1.71743	0.0582598
$870$	0	0
$871$	30.4561	1.03197
$872$	0	0
$873$	−16.6952	−0.565048
$874$	0	0
$875$	9.19344	0.310795
$876$	0	0
$877$	22.8097	0.770229	0.385115	−	0.922869i	$-0.374162\pi$
$877$	22.8097	0.770229	0.385115	+	0.922869i	$0.374162\pi$
$878$	0	0
$879$	20.2638	0.683482
$880$	0	0
$881$	−31.2081	−1.05143	−0.525714	−	0.850662i	$-0.676202\pi$
$881$	−31.2081	−1.05143	−0.525714	+	0.850662i	$0.676202\pi$
$882$	0	0
$883$	18.1892	0.612115	0.306057	−	0.952013i	$-0.400990\pi$
$883$	18.1892	0.612115	0.306057	+	0.952013i	$0.400990\pi$
$884$	0	0
$885$	−5.55033	−0.186572
$886$	0	0
$887$	−12.7151	−0.426931	−0.213465	−	0.976951i	$-0.568475\pi$
$887$	−12.7151	−0.426931	−0.213465	+	0.976951i	$0.568475\pi$
$888$	0	0
$889$	22.6756	0.760514
$890$	0	0
$891$	−3.20379	−0.107331
$892$	0	0
$893$	−37.3267	−1.24909
$894$	0	0
$895$	−0.364093	−0.0121703
$896$	0	0
$897$	9.83088	0.328244
$898$	0	0
$899$	−1.55325	−0.0518037
$900$	0	0
$901$	6.79790	0.226471
$902$	0	0
$903$	37.3063	1.24148
$904$	0	0
$905$	5.47014	0.181834
$906$	0	0
$907$	30.1321	1.00052	0.500260	−	0.865875i	$-0.333238\pi$
$907$	30.1321	1.00052	0.500260	+	0.865875i	$0.333238\pi$
$908$	0	0
$909$	17.4126	0.577539
$910$	0	0
$911$	14.8609	0.492363	0.246181	−	0.969224i	$-0.420824\pi$
$911$	14.8609	0.492363	0.246181	+	0.969224i	$0.420824\pi$
$912$	0	0
$913$	6.08392	0.201348
$914$	0	0
$915$	4.39596	0.145326
$916$	0	0
$917$	23.5874	0.778925
$918$	0	0
$919$	−48.1307	−1.58768	−0.793842	−	0.608124i	$-0.791922\pi$
$919$	−48.1307	−1.58768	−0.793842	+	0.608124i	$0.791922\pi$
$920$	0	0
$921$	48.6512	1.60311
$922$	0	0
$923$	10.6678	0.351134
$924$	0	0
$925$	−6.77308	−0.222697
$926$	0	0
$927$	−2.87277	−0.0943543
$928$	0	0
$929$	−0.726072	−0.0238217	−0.0119108	−	0.999929i	$-0.503791\pi$
$929$	−0.726072	−0.0238217	−0.0119108	+	0.999929i	$0.503791\pi$
$930$	0	0
$931$	−4.11084	−0.134727
$932$	0	0
$933$	23.8517	0.780870
$934$	0	0
$935$	0.169191	0.00553315
$936$	0	0
$937$	22.2524	0.726954	0.363477	−	0.931603i	$-0.381590\pi$
$937$	22.2524	0.726954	0.363477	+	0.931603i	$0.381590\pi$
$938$	0	0
$939$	−32.8198	−1.07103
$940$	0	0
$941$	−38.5010	−1.25510	−0.627548	−	0.778578i	$-0.715941\pi$
$941$	−38.5010	−1.25510	−0.627548	+	0.778578i	$0.715941\pi$
$942$	0	0
$943$	13.8355	0.450547
$944$	0	0
$945$	5.25753	0.171027
$946$	0	0
$947$	24.5539	0.797894	0.398947	−	0.916974i	$-0.369376\pi$
$947$	24.5539	0.797894	0.398947	+	0.916974i	$0.369376\pi$
$948$	0	0
$949$	47.3630	1.53747
$950$	0	0
$951$	9.46750	0.307005
$952$	0	0
$953$	−12.8421	−0.415995	−0.207998	−	0.978129i	$-0.566695\pi$
$953$	−12.8421	−0.415995	−0.207998	+	0.978129i	$0.566695\pi$
$954$	0	0
$955$	−8.49127	−0.274771
$956$	0	0
$957$	6.55390	0.211857
$958$	0	0
$959$	−48.2007	−1.55648
$960$	0	0
$961$	−30.9531	−0.998488
$962$	0	0
$963$	8.73050	0.281336
$964$	0	0
$965$	2.43204	0.0782902
$966$	0	0
$967$	−29.1811	−0.938400	−0.469200	−	0.883092i	$-0.655458\pi$
$967$	−29.1811	−0.938400	−0.469200	+	0.883092i	$0.655458\pi$
$968$	0	0
$969$	−4.74735	−0.152507
$970$	0	0
$971$	38.7444	1.24337	0.621684	−	0.783268i	$-0.286449\pi$
$971$	38.7444	1.24337	0.621684	+	0.783268i	$0.286449\pi$
$972$	0	0
$973$	1.08506	0.0347855
$974$	0	0
$975$	35.5351	1.13803
$976$	0	0
$977$	−8.44847	−0.270291	−0.135145	−	0.990826i	$-0.543150\pi$
$977$	−8.44847	−0.270291	−0.135145	+	0.990826i	$0.543150\pi$
$978$	0	0
$979$	10.7462	0.343450
$980$	0	0
$981$	0.663692	0.0211900
$982$	0	0
$983$	−13.5256	−0.431401	−0.215701	−	0.976460i	$-0.569203\pi$
$983$	−13.5256	−0.431401	−0.215701	+	0.976460i	$0.569203\pi$
$984$	0	0
$985$	−1.63664	−0.0521477
$986$	0	0
$987$	−35.0463	−1.11554
$988$	0	0
$989$	12.6789	0.403165
$990$	0	0
$991$	55.1837	1.75297	0.876483	−	0.481432i	$-0.159883\pi$
$991$	55.1837	1.75297	0.876483	+	0.481432i	$0.159883\pi$
$992$	0	0
$993$	31.0559	0.985530
$994$	0	0
$995$	−7.89645	−0.250334
$996$	0	0
$997$	27.5827	0.873554	0.436777	−	0.899570i	$-0.356120\pi$
$997$	27.5827	0.873554	0.436777	+	0.899570i	$0.356120\pi$
$998$	0	0
$999$	−7.83258	−0.247812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.b.1.7	✓	7
4.3	odd	2		2672.2.a.l.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.b.1.7	✓	7	1.1	even	1	trivial
2672.2.a.l.1.1		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41051 q^{3} +0.329231 q^{5} -2.82300 q^{7} -1.01045 q^{9} +O(q^{10})\)	\(q+1.41051 q^{3} +0.329231 q^{5} -2.82300 q^{7} -1.01045 q^{9} +0.647542 q^{11} -5.15026 q^{13} +0.464384 q^{15} +0.793615 q^{17} -4.24097 q^{19} -3.98187 q^{21} -1.35328 q^{23} -4.89161 q^{25} -5.65680 q^{27} +7.17555 q^{29} -0.216464 q^{31} +0.913365 q^{33} -0.929418 q^{35} +1.38463 q^{37} -7.26450 q^{39} -10.2237 q^{41} -9.36903 q^{43} -0.332673 q^{45} +8.80147 q^{47} +0.969317 q^{49} +1.11940 q^{51} +8.56574 q^{53} +0.213191 q^{55} -5.98194 q^{57} -11.9520 q^{59} +9.46622 q^{61} +2.85251 q^{63} -1.69562 q^{65} -5.91351 q^{67} -1.90881 q^{69} -2.07131 q^{71} -9.19625 q^{73} -6.89967 q^{75} -1.82801 q^{77} +2.65223 q^{79} -4.94762 q^{81} +9.39541 q^{83} +0.261283 q^{85} +10.1212 q^{87} +16.5954 q^{89} +14.5392 q^{91} -0.305325 q^{93} -1.39626 q^{95} +16.5225 q^{97} -0.654311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 6 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 7 * q - 6 * q^3 - q^7 + 3 * q^9	\( 7 q - 6 q^{3} - q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 9 q^{15} - 9 q^{17} - 3 q^{19} + 7 q^{21} - 10 q^{23} + 3 q^{25} - 12 q^{27} - 5 q^{29} - 21 q^{31} + 8 q^{33} - 12 q^{35} + 19 q^{37} - 27 q^{39} - 22 q^{41} - 19 q^{43} + 13 q^{45} - 13 q^{47} - 14 q^{49} + 4 q^{51} + 5 q^{53} - 17 q^{55} + 5 q^{57} - 18 q^{59} + 26 q^{61} - 20 q^{63} - 20 q^{65} - 27 q^{67} - 3 q^{69} - 46 q^{71} - 25 q^{73} - 19 q^{75} - 19 q^{77} - 22 q^{79} - 9 q^{81} + q^{83} - 11 q^{85} + 9 q^{87} - 3 q^{89} + 33 q^{93} - 40 q^{95} + 11 q^{97} - 21 q^{99}+O(q^{100}) \) 7 * q - 6 * q^3 - q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 9 * q^15 - 9 * q^17 - 3 * q^19 + 7 * q^21 - 10 * q^23 + 3 * q^25 - 12 * q^27 - 5 * q^29 - 21 * q^31 + 8 * q^33 - 12 * q^35 + 19 * q^37 - 27 * q^39 - 22 * q^41 - 19 * q^43 + 13 * q^45 - 13 * q^47 - 14 * q^49 + 4 * q^51 + 5 * q^53 - 17 * q^55 + 5 * q^57 - 18 * q^59 + 26 * q^61 - 20 * q^63 - 20 * q^65 - 27 * q^67 - 3 * q^69 - 46 * q^71 - 25 * q^73 - 19 * q^75 - 19 * q^77 - 22 * q^79 - 9 * q^81 + q^83 - 11 * q^85 + 9 * q^87 - 3 * q^89 + 33 * q^93 - 40 * q^95 + 11 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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