LMFDB - Embedded newform 1160.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,2,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1160.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:	$9.26264663447$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.580484.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - 3x^{2} + 8x + 3 $ x^5 - 7x^3 - 3x^2 + 8*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.366905$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.02751 q^{3} +1.00000 q^{5} +0.377000 q^{7} -1.94422 q^{9} +2.73381 q^{11} -6.02446 q^{13} -1.02751 q^{15} +7.94117 q^{17} +6.56722 q^{19} -0.387372 q^{21} -6.75827 q^{23} +1.00000 q^{25} +5.08025 q^{27} +1.00000 q^{29} +5.67803 q^{31} -2.80903 q^{33} +0.377000 q^{35} -4.40955 q^{37} +6.19022 q^{39} +1.24600 q^{41} +9.94117 q^{43} -1.94422 q^{45} +2.30408 q^{47} -6.85787 q^{49} -8.15966 q^{51} +7.10471 q^{53} +2.73381 q^{55} -6.74790 q^{57} +13.4921 q^{59} -9.85787 q^{61} -0.732969 q^{63} -6.02446 q^{65} +2.02019 q^{67} +6.94422 q^{69} +4.11005 q^{71} +1.29874 q^{73} -1.02751 q^{75} +1.03065 q^{77} +6.04465 q^{79} +0.612628 q^{81} +5.09960 q^{83} +7.94117 q^{85} -1.02751 q^{87} +11.7218 q^{89} -2.27122 q^{91} -5.83425 q^{93} +6.56722 q^{95} +5.77542 q^{97} -5.31512 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 8 q^{9} + 10 q^{11} + q^{13} + 3 q^{15} - q^{17} + 10 q^{19} - 8 q^{21} + q^{23} + 5 q^{25} + 12 q^{27} + 5 q^{29} + 7 q^{31} - 8 q^{33} + 7 q^{35} - 8 q^{37} + 3 q^{39}+ \cdots + 32 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 + 5 * q^5 + 7 * q^7 + 8 * q^9 + 10 * q^11 + q^13 + 3 * q^15 - q^17 + 10 * q^19 - 8 * q^21 + q^23 + 5 * q^25 + 12 * q^27 + 5 * q^29 + 7 * q^31 - 8 * q^33 + 7 * q^35 - 8 * q^37 + 3 * q^39 - 4 * q^41 + 9 * q^43 + 8 * q^45 + 8 * q^47 + 16 * q^49 + 2 * q^51 - 9 * q^53 + 10 * q^55 + 2 * q^57 + 29 * q^59 + q^61 + 16 * q^63 + q^65 + 24 * q^67 + 17 * q^69 - 12 * q^71 - 9 * q^73 + 3 * q^75 + 2 * q^77 + 13 * q^79 - 3 * q^81 + 10 * q^83 - q^85 + 3 * q^87 + 4 * q^89 - 4 * q^91 - 26 * q^93 + 10 * q^95 - 15 * q^97 + 32 * 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
$3$	−1.02751	−0.593235	−0.296618	−	0.954996i	$-0.595859\pi$
$3$	−1.02751	−0.593235	−0.296618	+	0.954996i	$0.595859\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.377000	0.142493	0.0712463	−	0.997459i	$-0.477302\pi$
$7$	0.377000	0.142493	0.0712463	+	0.997459i	$0.477302\pi$
$8$	0	0
$9$	−1.94422	−0.648072
$10$	0	0
$11$	2.73381	0.824275	0.412137	−	0.911122i	$-0.364782\pi$
$11$	2.73381	0.824275	0.412137	+	0.911122i	$0.364782\pi$
$12$	0	0
$13$	−6.02446	−1.67089	−0.835443	−	0.549577i	$-0.814789\pi$
$13$	−6.02446	−1.67089	−0.835443	+	0.549577i	$0.814789\pi$
$14$	0	0
$15$	−1.02751	−0.265303
$16$	0	0
$17$	7.94117	1.92602	0.963008	−	0.269473i	$-0.0868494\pi$
$17$	7.94117	1.92602	0.963008	+	0.269473i	$0.0868494\pi$
$18$	0	0
$19$	6.56722	1.50662	0.753311	−	0.657664i	$-0.228455\pi$
$19$	6.56722	1.50662	0.753311	+	0.657664i	$0.228455\pi$
$20$	0	0
$21$	−0.387372	−0.0845316
$22$	0	0
$23$	−6.75827	−1.40920	−0.704599	−	0.709606i	$-0.748873\pi$
$23$	−6.75827	−1.40920	−0.704599	+	0.709606i	$0.748873\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.08025	0.977694
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.67803	1.01980	0.509902	−	0.860233i	$-0.329682\pi$
$31$	5.67803	1.01980	0.509902	+	0.860233i	$0.329682\pi$
$32$	0	0
$33$	−2.80903	−0.488989
$34$	0	0
$35$	0.377000	0.0637246
$36$	0	0
$37$	−4.40955	−0.724925	−0.362462	−	0.931998i	$-0.618064\pi$
$37$	−4.40955	−0.724925	−0.362462	+	0.931998i	$0.618064\pi$
$38$	0	0
$39$	6.19022	0.991228
$40$	0	0
$41$	1.24600	0.194593	0.0972963	−	0.995255i	$-0.468981\pi$
$41$	1.24600	0.194593	0.0972963	+	0.995255i	$0.468981\pi$
$42$	0	0
$43$	9.94117	1.51601	0.758007	−	0.652246i	$-0.226173\pi$
$43$	9.94117	1.51601	0.758007	+	0.652246i	$0.226173\pi$
$44$	0	0
$45$	−1.94422	−0.289827
$46$	0	0
$47$	2.30408	0.336084	0.168042	−	0.985780i	$-0.446256\pi$
$47$	2.30408	0.336084	0.168042	+	0.985780i	$0.446256\pi$
$48$	0	0
$49$	−6.85787	−0.979696
$50$	0	0
$51$	−8.15966	−1.14258
$52$	0	0
$53$	7.10471	0.975907	0.487954	−	0.872870i	$-0.337744\pi$
$53$	7.10471	0.975907	0.487954	+	0.872870i	$0.337744\pi$
$54$	0	0
$55$	2.73381	0.368627
$56$	0	0
$57$	−6.74790	−0.893781
$58$	0	0
$59$	13.4921	1.75652	0.878260	−	0.478184i	$-0.158705\pi$
$59$	13.4921	1.75652	0.878260	+	0.478184i	$0.158705\pi$
$60$	0	0
$61$	−9.85787	−1.26217	−0.631086	−	0.775713i	$-0.717390\pi$
$61$	−9.85787	−1.26217	−0.631086	+	0.775713i	$0.717390\pi$
$62$	0	0
$63$	−0.732969	−0.0923454
$64$	0	0
$65$	−6.02446	−0.747243
$66$	0	0
$67$	2.02019	0.246805	0.123403	−	0.992357i	$-0.460619\pi$
$67$	2.02019	0.246805	0.123403	+	0.992357i	$0.460619\pi$
$68$	0	0
$69$	6.94422	0.835985
$70$	0	0
$71$	4.11005	0.487774	0.243887	−	0.969804i	$-0.421577\pi$
$71$	4.11005	0.487774	0.243887	+	0.969804i	$0.421577\pi$
$72$	0	0
$73$	1.29874	0.152005	0.0760027	−	0.997108i	$-0.475784\pi$
$73$	1.29874	0.152005	0.0760027	+	0.997108i	$0.475784\pi$
$74$	0	0
$75$	−1.02751	−0.118647
$76$	0	0
$77$	1.03065	0.117453
$78$	0	0
$79$	6.04465	0.680077	0.340038	−	0.940412i	$-0.389560\pi$
$79$	6.04465	0.680077	0.340038	+	0.940412i	$0.389560\pi$
$80$	0	0
$81$	0.612628	0.0680697
$82$	0	0
$83$	5.09960	0.559753	0.279877	−	0.960036i	$-0.409706\pi$
$83$	5.09960	0.559753	0.279877	+	0.960036i	$0.409706\pi$
$84$	0	0
$85$	7.94117	0.861341
$86$	0	0
$87$	−1.02751	−0.110161
$88$	0	0
$89$	11.7218	1.24251	0.621256	−	0.783607i	$-0.286623\pi$
$89$	11.7218	1.24251	0.621256	+	0.783607i	$0.286623\pi$
$90$	0	0
$91$	−2.27122	−0.238089
$92$	0	0
$93$	−5.83425	−0.604983
$94$	0	0
$95$	6.56722	0.673782
$96$	0	0
$97$	5.77542	0.586405	0.293202	−	0.956050i	$-0.405279\pi$
$97$	5.77542	0.586405	0.293202	+	0.956050i	$0.405279\pi$
$98$	0	0
$99$	−5.31512	−0.534190
$100$	0	0
$101$	15.4355	1.53589	0.767947	−	0.640513i	$-0.221278\pi$
$101$	15.4355	1.53589	0.767947	+	0.640513i	$0.221278\pi$
$102$	0	0
$103$	4.51219	0.444599	0.222300	−	0.974978i	$-0.428644\pi$
$103$	4.51219	0.444599	0.222300	+	0.974978i	$0.428644\pi$
$104$	0	0
$105$	−0.387372	−0.0378037
$106$	0	0
$107$	−9.20965	−0.890330	−0.445165	−	0.895448i	$-0.646855\pi$
$107$	−9.20965	−0.890330	−0.445165	+	0.895448i	$0.646855\pi$
$108$	0	0
$109$	16.8580	1.61470	0.807350	−	0.590073i	$-0.200901\pi$
$109$	16.8580	1.61470	0.807350	+	0.590073i	$0.200901\pi$
$110$	0	0
$111$	4.53087	0.430051
$112$	0	0
$113$	−19.9954	−1.88100	−0.940502	−	0.339787i	$-0.889645\pi$
$113$	−19.9954	−1.88100	−0.940502	+	0.339787i	$0.889645\pi$
$114$	0	0
$115$	−6.75827	−0.630212
$116$	0	0
$117$	11.7129	1.08285
$118$	0	0
$119$	2.99382	0.274443
$120$	0	0
$121$	−3.52628	−0.320571
$122$	0	0
$123$	−1.28028	−0.115439
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.83036	−0.251154	−0.125577	−	0.992084i	$-0.540078\pi$
$127$	−2.83036	−0.251154	−0.125577	+	0.992084i	$0.540078\pi$
$128$	0	0
$129$	−10.2147	−0.899353
$130$	0	0
$131$	6.67269	0.582995	0.291498	−	0.956572i	$-0.405846\pi$
$131$	6.67269	0.582995	0.291498	+	0.956572i	$0.405846\pi$
$132$	0	0
$133$	2.47584	0.214683
$134$	0	0
$135$	5.08025	0.437238
$136$	0	0
$137$	−7.19411	−0.614634	−0.307317	−	0.951607i	$-0.599431\pi$
$137$	−7.19411	−0.614634	−0.307317	+	0.951607i	$0.599431\pi$
$138$	0	0
$139$	−17.7618	−1.50654	−0.753268	−	0.657714i	$-0.771524\pi$
$139$	−17.7618	−1.50654	−0.753268	+	0.657714i	$0.771524\pi$
$140$	0	0
$141$	−2.36747	−0.199377
$142$	0	0
$143$	−16.4697	−1.37727
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	7.04655	0.581190
$148$	0	0
$149$	−16.5409	−1.35509	−0.677543	−	0.735483i	$-0.736955\pi$
$149$	−16.5409	−1.35509	−0.677543	+	0.735483i	$0.736955\pi$
$150$	0	0
$151$	8.04893	0.655013	0.327506	−	0.944849i	$-0.393792\pi$
$151$	8.04893	0.655013	0.327506	+	0.944849i	$0.393792\pi$
$152$	0	0
$153$	−15.4393	−1.24820
$154$	0	0
$155$	5.67803	0.456070
$156$	0	0
$157$	−2.22830	−0.177838	−0.0889190	−	0.996039i	$-0.528341\pi$
$157$	−2.22830	−0.177838	−0.0889190	+	0.996039i	$0.528341\pi$
$158$	0	0
$159$	−7.30019	−0.578942
$160$	0	0
$161$	−2.54787	−0.200800
$162$	0	0
$163$	13.1574	1.03057	0.515285	−	0.857019i	$-0.327686\pi$
$163$	13.1574	1.03057	0.515285	+	0.857019i	$0.327686\pi$
$164$	0	0
$165$	−2.80903	−0.218682
$166$	0	0
$167$	−19.2350	−1.48845	−0.744223	−	0.667932i	$-0.767180\pi$
$167$	−19.2350	−1.48845	−0.744223	+	0.667932i	$0.767180\pi$
$168$	0	0
$169$	23.2942	1.79186
$170$	0	0
$171$	−12.7681	−0.976400
$172$	0	0
$173$	−24.4553	−1.85931	−0.929653	−	0.368437i	$-0.879893\pi$
$173$	−24.4553	−1.85931	−0.929653	+	0.368437i	$0.879893\pi$
$174$	0	0
$175$	0.377000	0.0284985
$176$	0	0
$177$	−13.8633	−1.04203
$178$	0	0
$179$	2.50792	0.187450	0.0937252	−	0.995598i	$-0.470122\pi$
$179$	2.50792	0.187450	0.0937252	+	0.995598i	$0.470122\pi$
$180$	0	0
$181$	15.6831	1.16572	0.582859	−	0.812573i	$-0.301934\pi$
$181$	15.6831	1.16572	0.582859	+	0.812573i	$0.301934\pi$
$182$	0	0
$183$	10.1291	0.748764
$184$	0	0
$185$	−4.40955	−0.324196
$186$	0	0
$187$	21.7096	1.58757
$188$	0	0
$189$	1.91525	0.139314
$190$	0	0
$191$	−13.6780	−0.989707	−0.494854	−	0.868976i	$-0.664778\pi$
$191$	−13.6780	−0.989707	−0.494854	+	0.868976i	$0.664778\pi$
$192$	0	0
$193$	1.68611	0.121369	0.0606843	−	0.998157i	$-0.480672\pi$
$193$	1.68611	0.121369	0.0606843	+	0.998157i	$0.480672\pi$
$194$	0	0
$195$	6.19022	0.443291
$196$	0	0
$197$	6.08033	0.433206	0.216603	−	0.976260i	$-0.430502\pi$
$197$	6.08033	0.433206	0.216603	+	0.976260i	$0.430502\pi$
$198$	0	0
$199$	−9.99849	−0.708774	−0.354387	−	0.935099i	$-0.615310\pi$
$199$	−9.99849	−0.708774	−0.354387	+	0.935099i	$0.615310\pi$
$200$	0	0
$201$	−2.07577	−0.146414
$202$	0	0
$203$	0.377000	0.0264602
$204$	0	0
$205$	1.24600	0.0870244
$206$	0	0
$207$	13.1395	0.913262
$208$	0	0
$209$	17.9535	1.24187
$210$	0	0
$211$	8.20143	0.564610	0.282305	−	0.959325i	$-0.408901\pi$
$211$	8.20143	0.564610	0.282305	+	0.959325i	$0.408901\pi$
$212$	0	0
$213$	−4.22313	−0.289364
$214$	0	0
$215$	9.94117	0.677982
$216$	0	0
$217$	2.14061	0.145314
$218$	0	0
$219$	−1.33447	−0.0901750
$220$	0	0
$221$	−47.8413	−3.21815
$222$	0	0
$223$	−18.8683	−1.26352	−0.631758	−	0.775165i	$-0.717666\pi$
$223$	−18.8683	−1.26352	−0.631758	+	0.775165i	$0.717666\pi$
$224$	0	0
$225$	−1.94422	−0.129614
$226$	0	0
$227$	−27.6018	−1.83200	−0.915999	−	0.401181i	$-0.868600\pi$
$227$	−27.6018	−1.83200	−0.915999	+	0.401181i	$0.868600\pi$
$228$	0	0
$229$	−0.0253054	−0.00167222	−0.000836112	−	1.00000i	$-0.500266\pi$
$229$	−0.0253054	−0.00167222	−0.000836112		1.00000i	$0.500266\pi$
$230$	0	0
$231$	−1.05900	−0.0696772
$232$	0	0
$233$	−24.6021	−1.61173	−0.805867	−	0.592097i	$-0.798300\pi$
$233$	−24.6021	−1.61173	−0.805867	+	0.592097i	$0.798300\pi$
$234$	0	0
$235$	2.30408	0.150301
$236$	0	0
$237$	−6.21096	−0.403445
$238$	0	0
$239$	18.3806	1.18894	0.594471	−	0.804117i	$-0.297361\pi$
$239$	18.3806	1.18894	0.594471	+	0.804117i	$0.297361\pi$
$240$	0	0
$241$	−11.8725	−0.764776	−0.382388	−	0.924002i	$-0.624898\pi$
$241$	−11.8725	−0.764776	−0.382388	+	0.924002i	$0.624898\pi$
$242$	0	0
$243$	−15.8702	−1.01808
$244$	0	0
$245$	−6.85787	−0.438133
$246$	0	0
$247$	−39.5640	−2.51739
$248$	0	0
$249$	−5.23990	−0.332065
$250$	0	0
$251$	13.7359	0.867004	0.433502	−	0.901153i	$-0.357278\pi$
$251$	13.7359	0.867004	0.433502	+	0.901153i	$0.357278\pi$
$252$	0	0
$253$	−18.4758	−1.16157
$254$	0	0
$255$	−8.15966	−0.510977
$256$	0	0
$257$	−6.75400	−0.421303	−0.210651	−	0.977561i	$-0.567559\pi$
$257$	−6.75400	−0.421303	−0.210651	+	0.977561i	$0.567559\pi$
$258$	0	0
$259$	−1.66240	−0.103296
$260$	0	0
$261$	−1.94422	−0.120344
$262$	0	0
$263$	−32.1849	−1.98461	−0.992303	−	0.123834i	$-0.960481\pi$
$263$	−32.1849	−1.98461	−0.992303	+	0.123834i	$0.960481\pi$
$264$	0	0
$265$	7.10471	0.436439
$266$	0	0
$267$	−12.0443	−0.737102
$268$	0	0
$269$	2.93812	0.179140	0.0895701	−	0.995981i	$-0.471451\pi$
$269$	2.93812	0.179140	0.0895701	+	0.995981i	$0.471451\pi$
$270$	0	0
$271$	−5.07885	−0.308518	−0.154259	−	0.988030i	$-0.549299\pi$
$271$	−5.07885	−0.308518	−0.154259	+	0.988030i	$0.549299\pi$
$272$	0	0
$273$	2.33371	0.141243
$274$	0	0
$275$	2.73381	0.164855
$276$	0	0
$277$	21.0183	1.26287	0.631433	−	0.775430i	$-0.282467\pi$
$277$	21.0183	1.26287	0.631433	+	0.775430i	$0.282467\pi$
$278$	0	0
$279$	−11.0393	−0.660906
$280$	0	0
$281$	−4.34065	−0.258941	−0.129471	−	0.991583i	$-0.541328\pi$
$281$	−4.34065	−0.258941	−0.129471	+	0.991583i	$0.541328\pi$
$282$	0	0
$283$	19.3136	1.14807	0.574037	−	0.818829i	$-0.305376\pi$
$283$	19.3136	1.14807	0.574037	+	0.818829i	$0.305376\pi$
$284$	0	0
$285$	−6.74790	−0.399711
$286$	0	0
$287$	0.469742	0.0277280
$288$	0	0
$289$	46.0621	2.70954
$290$	0	0
$291$	−5.93432	−0.347876
$292$	0	0
$293$	11.5379	0.674050	0.337025	−	0.941496i	$-0.390579\pi$
$293$	11.5379	0.674050	0.337025	+	0.941496i	$0.390579\pi$
$294$	0	0
$295$	13.4921	0.785540
$296$	0	0
$297$	13.8884	0.805889
$298$	0	0
$299$	40.7150	2.35461
$300$	0	0
$301$	3.74782	0.216021
$302$	0	0
$303$	−15.8602	−0.911146
$304$	0	0
$305$	−9.85787	−0.564460
$306$	0	0
$307$	23.7595	1.35603	0.678013	−	0.735050i	$-0.262841\pi$
$307$	23.7595	1.35603	0.678013	+	0.735050i	$0.262841\pi$
$308$	0	0
$309$	−4.63634	−0.263752
$310$	0	0
$311$	−31.9982	−1.81445	−0.907225	−	0.420645i	$-0.861804\pi$
$311$	−31.9982	−1.81445	−0.907225	+	0.420645i	$0.861804\pi$
$312$	0	0
$313$	−15.4554	−0.873592	−0.436796	−	0.899561i	$-0.643887\pi$
$313$	−15.4554	−0.873592	−0.436796	+	0.899561i	$0.643887\pi$
$314$	0	0
$315$	−0.732969	−0.0412981
$316$	0	0
$317$	29.8903	1.67881	0.839403	−	0.543509i	$-0.182905\pi$
$317$	29.8903	1.67881	0.839403	+	0.543509i	$0.182905\pi$
$318$	0	0
$319$	2.73381	0.153064
$320$	0	0
$321$	9.46304	0.528175
$322$	0	0
$323$	52.1514	2.90178
$324$	0	0
$325$	−6.02446	−0.334177
$326$	0	0
$327$	−17.3218	−0.957896
$328$	0	0
$329$	0.868636	0.0478895
$330$	0	0
$331$	−17.0431	−0.936771	−0.468386	−	0.883524i	$-0.655164\pi$
$331$	−17.0431	−0.936771	−0.468386	+	0.883524i	$0.655164\pi$
$332$	0	0
$333$	8.57311	0.469804
$334$	0	0
$335$	2.02019	0.110375
$336$	0	0
$337$	−9.40879	−0.512529	−0.256265	−	0.966607i	$-0.582492\pi$
$337$	−9.40879	−0.512529	−0.256265	+	0.966607i	$0.582492\pi$
$338$	0	0
$339$	20.5455	1.11588
$340$	0	0
$341$	15.5226	0.840598
$342$	0	0
$343$	−5.22441	−0.282092
$344$	0	0
$345$	6.94422	0.373864
$346$	0	0
$347$	−19.2439	−1.03307	−0.516534	−	0.856267i	$-0.672778\pi$
$347$	−19.2439	−1.03307	−0.516534	+	0.856267i	$0.672778\pi$
$348$	0	0
$349$	−24.4907	−1.31095	−0.655477	−	0.755215i	$-0.727533\pi$
$349$	−24.4907	−1.31095	−0.655477	+	0.755215i	$0.727533\pi$
$350$	0	0
$351$	−30.6058	−1.63362
$352$	0	0
$353$	−3.52416	−0.187572	−0.0937861	−	0.995592i	$-0.529897\pi$
$353$	−3.52416	−0.187572	−0.0937861	+	0.995592i	$0.529897\pi$
$354$	0	0
$355$	4.11005	0.218139
$356$	0	0
$357$	−3.07619	−0.162809
$358$	0	0
$359$	3.34110	0.176336	0.0881682	−	0.996106i	$-0.471899\pi$
$359$	3.34110	0.176336	0.0881682	+	0.996106i	$0.471899\pi$
$360$	0	0
$361$	24.1283	1.26991
$362$	0	0
$363$	3.62330	0.190174
$364$	0	0
$365$	1.29874	0.0679789
$366$	0	0
$367$	29.6036	1.54529	0.772647	−	0.634835i	$-0.218932\pi$
$367$	29.6036	1.54529	0.772647	+	0.634835i	$0.218932\pi$
$368$	0	0
$369$	−2.42249	−0.126110
$370$	0	0
$371$	2.67848	0.139059
$372$	0	0
$373$	13.7471	0.711800	0.355900	−	0.934524i	$-0.384174\pi$
$373$	13.7471	0.711800	0.355900	+	0.934524i	$0.384174\pi$
$374$	0	0
$375$	−1.02751	−0.0530606
$376$	0	0
$377$	−6.02446	−0.310276
$378$	0	0
$379$	8.32485	0.427619	0.213809	−	0.976875i	$-0.431413\pi$
$379$	8.32485	0.427619	0.213809	+	0.976875i	$0.431413\pi$
$380$	0	0
$381$	2.90823	0.148993
$382$	0	0
$383$	14.5910	0.745566	0.372783	−	0.927919i	$-0.378404\pi$
$383$	14.5910	0.745566	0.372783	+	0.927919i	$0.378404\pi$
$384$	0	0
$385$	1.03065	0.0525266
$386$	0	0
$387$	−19.3278	−0.982486
$388$	0	0
$389$	−3.91086	−0.198288	−0.0991442	−	0.995073i	$-0.531611\pi$
$389$	−3.91086	−0.198288	−0.0991442	+	0.995073i	$0.531611\pi$
$390$	0	0
$391$	−53.6686	−2.71414
$392$	0	0
$393$	−6.85627	−0.345853
$394$	0	0
$395$	6.04465	0.304140
$396$	0	0
$397$	4.59028	0.230380	0.115190	−	0.993343i	$-0.463252\pi$
$397$	4.59028	0.230380	0.115190	+	0.993343i	$0.463252\pi$
$398$	0	0
$399$	−2.54396	−0.127357
$400$	0	0
$401$	−21.9068	−1.09397	−0.546987	−	0.837141i	$-0.684225\pi$
$401$	−21.9068	−1.09397	−0.546987	+	0.837141i	$0.684225\pi$
$402$	0	0
$403$	−34.2071	−1.70398
$404$	0	0
$405$	0.612628	0.0304417
$406$	0	0
$407$	−12.0549	−0.597537
$408$	0	0
$409$	−4.58589	−0.226758	−0.113379	−	0.993552i	$-0.536167\pi$
$409$	−4.58589	−0.226758	−0.113379	+	0.993552i	$0.536167\pi$
$410$	0	0
$411$	7.39204	0.364622
$412$	0	0
$413$	5.08651	0.250291
$414$	0	0
$415$	5.09960	0.250329
$416$	0	0
$417$	18.2505	0.893730
$418$	0	0
$419$	−16.7272	−0.817176	−0.408588	−	0.912719i	$-0.633979\pi$
$419$	−16.7272	−0.817176	−0.408588	+	0.912719i	$0.633979\pi$
$420$	0	0
$421$	2.20529	0.107479	0.0537396	−	0.998555i	$-0.482886\pi$
$421$	2.20529	0.107479	0.0537396	+	0.998555i	$0.482886\pi$
$422$	0	0
$423$	−4.47962	−0.217807
$424$	0	0
$425$	7.94117	0.385203
$426$	0	0
$427$	−3.71642	−0.179850
$428$	0	0
$429$	16.9229	0.817044
$430$	0	0
$431$	−1.55693	−0.0749946	−0.0374973	−	0.999297i	$-0.511939\pi$
$431$	−1.55693	−0.0749946	−0.0374973	+	0.999297i	$0.511939\pi$
$432$	0	0
$433$	−13.7999	−0.663180	−0.331590	−	0.943424i	$-0.607585\pi$
$433$	−13.7999	−0.663180	−0.331590	+	0.943424i	$0.607585\pi$
$434$	0	0
$435$	−1.02751	−0.0492655
$436$	0	0
$437$	−44.3831	−2.12313
$438$	0	0
$439$	−4.69288	−0.223979	−0.111989	−	0.993709i	$-0.535722\pi$
$439$	−4.69288	−0.223979	−0.111989	+	0.993709i	$0.535722\pi$
$440$	0	0
$441$	13.3332	0.634914
$442$	0	0
$443$	38.3821	1.82359	0.911793	−	0.410650i	$-0.134698\pi$
$443$	38.3821	1.82359	0.911793	+	0.410650i	$0.134698\pi$
$444$	0	0
$445$	11.7218	0.555668
$446$	0	0
$447$	16.9960	0.803884
$448$	0	0
$449$	−22.7967	−1.07584	−0.537921	−	0.842996i	$-0.680790\pi$
$449$	−22.7967	−1.07584	−0.537921	+	0.842996i	$0.680790\pi$
$450$	0	0
$451$	3.40633	0.160398
$452$	0	0
$453$	−8.27038	−0.388576
$454$	0	0
$455$	−2.27122	−0.106477
$456$	0	0
$457$	−20.0636	−0.938535	−0.469267	−	0.883056i	$-0.655482\pi$
$457$	−20.0636	−0.938535	−0.469267	+	0.883056i	$0.655482\pi$
$458$	0	0
$459$	40.3431	1.88305
$460$	0	0
$461$	30.3996	1.41585	0.707926	−	0.706286i	$-0.249631\pi$
$461$	30.3996	1.41585	0.707926	+	0.706286i	$0.249631\pi$
$462$	0	0
$463$	37.6896	1.75159	0.875793	−	0.482687i	$-0.160339\pi$
$463$	37.6896	1.75159	0.875793	+	0.482687i	$0.160339\pi$
$464$	0	0
$465$	−5.83425	−0.270557
$466$	0	0
$467$	2.92288	0.135255	0.0676275	−	0.997711i	$-0.478457\pi$
$467$	2.92288	0.135255	0.0676275	+	0.997711i	$0.478457\pi$
$468$	0	0
$469$	0.761611	0.0351679
$470$	0	0
$471$	2.28961	0.105500
$472$	0	0
$473$	27.1773	1.24961
$474$	0	0
$475$	6.56722	0.301325
$476$	0	0
$477$	−13.8131	−0.632458
$478$	0	0
$479$	−26.4648	−1.20921	−0.604604	−	0.796526i	$-0.706668\pi$
$479$	−26.4648	−1.20921	−0.604604	+	0.796526i	$0.706668\pi$
$480$	0	0
$481$	26.5651	1.21127
$482$	0	0
$483$	2.61797	0.119122
$484$	0	0
$485$	5.77542	0.262248
$486$	0	0
$487$	19.8927	0.901425	0.450712	−	0.892669i	$-0.351170\pi$
$487$	19.8927	0.901425	0.450712	+	0.892669i	$0.351170\pi$
$488$	0	0
$489$	−13.5195	−0.611371
$490$	0	0
$491$	−6.42305	−0.289868	−0.144934	−	0.989441i	$-0.546297\pi$
$491$	−6.42305	−0.289868	−0.144934	+	0.989441i	$0.546297\pi$
$492$	0	0
$493$	7.94117	0.357652
$494$	0	0
$495$	−5.31512	−0.238897
$496$	0	0
$497$	1.54949	0.0695041
$498$	0	0
$499$	−3.65090	−0.163437	−0.0817183	−	0.996655i	$-0.526041\pi$
$499$	−3.65090	−0.163437	−0.0817183	+	0.996655i	$0.526041\pi$
$500$	0	0
$501$	19.7642	0.882998
$502$	0	0
$503$	−16.2801	−0.725895	−0.362948	−	0.931810i	$-0.618230\pi$
$503$	−16.2801	−0.725895	−0.362948	+	0.931810i	$0.618230\pi$
$504$	0	0
$505$	15.4355	0.686873
$506$	0	0
$507$	−23.9351	−1.06299
$508$	0	0
$509$	−22.5053	−0.997530	−0.498765	−	0.866737i	$-0.666213\pi$
$509$	−22.5053	−0.997530	−0.498765	+	0.866737i	$0.666213\pi$
$510$	0	0
$511$	0.489623	0.0216596
$512$	0	0
$513$	33.3631	1.47302
$514$	0	0
$515$	4.51219	0.198831
$516$	0	0
$517$	6.29891	0.277026
$518$	0	0
$519$	25.1282	1.10300
$520$	0	0
$521$	20.9320	0.917049	0.458524	−	0.888682i	$-0.348378\pi$
$521$	20.9320	0.917049	0.458524	+	0.888682i	$0.348378\pi$
$522$	0	0
$523$	−20.1034	−0.879060	−0.439530	−	0.898228i	$-0.644855\pi$
$523$	−20.1034	−0.879060	−0.439530	+	0.898228i	$0.644855\pi$
$524$	0	0
$525$	−0.387372	−0.0169063
$526$	0	0
$527$	45.0902	1.96416
$528$	0	0
$529$	22.6743	0.985838
$530$	0	0
$531$	−26.2315	−1.13835
$532$	0	0
$533$	−7.50649	−0.325142
$534$	0	0
$535$	−9.20965	−0.398168
$536$	0	0
$537$	−2.57692	−0.111202
$538$	0	0
$539$	−18.7481	−0.807539
$540$	0	0
$541$	10.4926	0.451112	0.225556	−	0.974230i	$-0.427580\pi$
$541$	10.4926	0.451112	0.225556	+	0.974230i	$0.427580\pi$
$542$	0	0
$543$	−16.1146	−0.691545
$544$	0	0
$545$	16.8580	0.722115
$546$	0	0
$547$	2.90040	0.124012	0.0620061	−	0.998076i	$-0.480250\pi$
$547$	2.90040	0.124012	0.0620061	+	0.998076i	$0.480250\pi$
$548$	0	0
$549$	19.1658	0.817978
$550$	0	0
$551$	6.56722	0.279773
$552$	0	0
$553$	2.27883	0.0969058
$554$	0	0
$555$	4.53087	0.192325
$556$	0	0
$557$	24.0005	1.01693	0.508467	−	0.861081i	$-0.330212\pi$
$557$	24.0005	1.01693	0.508467	+	0.861081i	$0.330212\pi$
$558$	0	0
$559$	−59.8902	−2.53309
$560$	0	0
$561$	−22.3069	−0.941800
$562$	0	0
$563$	−1.09719	−0.0462409	−0.0231205	−	0.999733i	$-0.507360\pi$
$563$	−1.09719	−0.0462409	−0.0231205	+	0.999733i	$0.507360\pi$
$564$	0	0
$565$	−19.9954	−0.841211
$566$	0	0
$567$	0.230961	0.00969943
$568$	0	0
$569$	40.9362	1.71613	0.858067	−	0.513538i	$-0.171665\pi$
$569$	40.9362	1.71613	0.858067	+	0.513538i	$0.171665\pi$
$570$	0	0
$571$	−12.5374	−0.524673	−0.262336	−	0.964976i	$-0.584493\pi$
$571$	−12.5374	−0.524673	−0.262336	+	0.964976i	$0.584493\pi$
$572$	0	0
$573$	14.0544	0.587129
$574$	0	0
$575$	−6.75827	−0.281840
$576$	0	0
$577$	0.983392	0.0409391	0.0204696	−	0.999790i	$-0.493484\pi$
$577$	0.983392	0.0409391	0.0204696	+	0.999790i	$0.493484\pi$
$578$	0	0
$579$	−1.73250	−0.0720001
$580$	0	0
$581$	1.92255	0.0797607
$582$	0	0
$583$	19.4229	0.804416
$584$	0	0
$585$	11.7129	0.484267
$586$	0	0
$587$	−7.40309	−0.305558	−0.152779	−	0.988260i	$-0.548822\pi$
$587$	−7.40309	−0.305558	−0.152779	+	0.988260i	$0.548822\pi$
$588$	0	0
$589$	37.2888	1.53646
$590$	0	0
$591$	−6.24762	−0.256993
$592$	0	0
$593$	−7.03065	−0.288714	−0.144357	−	0.989526i	$-0.546111\pi$
$593$	−7.03065	−0.288714	−0.144357	+	0.989526i	$0.546111\pi$
$594$	0	0
$595$	2.99382	0.122735
$596$	0	0
$597$	10.2736	0.420470
$598$	0	0
$599$	−26.8048	−1.09522	−0.547608	−	0.836735i	$-0.684462\pi$
$599$	−26.8048	−1.09522	−0.547608	+	0.836735i	$0.684462\pi$
$600$	0	0
$601$	−24.1350	−0.984489	−0.492244	−	0.870457i	$-0.663823\pi$
$601$	−24.1350	−0.984489	−0.492244	+	0.870457i	$0.663823\pi$
$602$	0	0
$603$	−3.92769	−0.159948
$604$	0	0
$605$	−3.52628	−0.143364
$606$	0	0
$607$	28.2431	1.14635	0.573176	−	0.819432i	$-0.305711\pi$
$607$	28.2431	1.14635	0.573176	+	0.819432i	$0.305711\pi$
$608$	0	0
$609$	−0.387372	−0.0156971
$610$	0	0
$611$	−13.8808	−0.561558
$612$	0	0
$613$	−36.3416	−1.46782	−0.733911	−	0.679245i	$-0.762307\pi$
$613$	−36.3416	−1.46782	−0.733911	+	0.679245i	$0.762307\pi$
$614$	0	0
$615$	−1.28028	−0.0516259
$616$	0	0
$617$	19.8932	0.800869	0.400435	−	0.916325i	$-0.368859\pi$
$617$	19.8932	0.800869	0.400435	+	0.916325i	$0.368859\pi$
$618$	0	0
$619$	36.5535	1.46921	0.734605	−	0.678495i	$-0.237368\pi$
$619$	36.5535	1.46921	0.734605	+	0.678495i	$0.237368\pi$
$620$	0	0
$621$	−34.3337	−1.37776
$622$	0	0
$623$	4.41913	0.177049
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−18.4475	−0.736722
$628$	0	0
$629$	−35.0169	−1.39622
$630$	0	0
$631$	45.8848	1.82664	0.913322	−	0.407238i	$-0.133508\pi$
$631$	45.8848	1.82664	0.913322	+	0.407238i	$0.133508\pi$
$632$	0	0
$633$	−8.42708	−0.334946
$634$	0	0
$635$	−2.83036	−0.112319
$636$	0	0
$637$	41.3150	1.63696
$638$	0	0
$639$	−7.99083	−0.316112
$640$	0	0
$641$	11.2948	0.446116	0.223058	−	0.974805i	$-0.428396\pi$
$641$	11.2948	0.446116	0.223058	+	0.974805i	$0.428396\pi$
$642$	0	0
$643$	3.77039	0.148689	0.0743447	−	0.997233i	$-0.476313\pi$
$643$	3.77039	0.148689	0.0743447	+	0.997233i	$0.476313\pi$
$644$	0	0
$645$	−10.2147	−0.402203
$646$	0	0
$647$	−23.1694	−0.910885	−0.455442	−	0.890265i	$-0.650519\pi$
$647$	−23.1694	−0.910885	−0.455442	+	0.890265i	$0.650519\pi$
$648$	0	0
$649$	36.8848	1.44785
$650$	0	0
$651$	−2.19951	−0.0862056
$652$	0	0
$653$	−42.8732	−1.67776	−0.838879	−	0.544318i	$-0.816788\pi$
$653$	−42.8732	−1.67776	−0.838879	+	0.544318i	$0.816788\pi$
$654$	0	0
$655$	6.67269	0.260723
$656$	0	0
$657$	−2.52502	−0.0985105
$658$	0	0
$659$	−48.0213	−1.87064	−0.935322	−	0.353797i	$-0.884890\pi$
$659$	−48.0213	−1.87064	−0.935322	+	0.353797i	$0.884890\pi$
$660$	0	0
$661$	20.8563	0.811215	0.405608	−	0.914047i	$-0.367060\pi$
$661$	20.8563	0.811215	0.405608	+	0.914047i	$0.367060\pi$
$662$	0	0
$663$	49.1576	1.90912
$664$	0	0
$665$	2.47584	0.0960089
$666$	0	0
$667$	−6.75827	−0.261681
$668$	0	0
$669$	19.3875	0.749562
$670$	0	0
$671$	−26.9495	−1.04038
$672$	0	0
$673$	16.3865	0.631654	0.315827	−	0.948817i	$-0.397718\pi$
$673$	16.3865	0.631654	0.315827	+	0.948817i	$0.397718\pi$
$674$	0	0
$675$	5.08025	0.195539
$676$	0	0
$677$	−3.22315	−0.123876	−0.0619379	−	0.998080i	$-0.519728\pi$
$677$	−3.22315	−0.123876	−0.0619379	+	0.998080i	$0.519728\pi$
$678$	0	0
$679$	2.17733	0.0835583
$680$	0	0
$681$	28.3612	1.08681
$682$	0	0
$683$	3.97736	0.152189	0.0760947	−	0.997101i	$-0.475755\pi$
$683$	3.97736	0.152189	0.0760947	+	0.997101i	$0.475755\pi$
$684$	0	0
$685$	−7.19411	−0.274873
$686$	0	0
$687$	0.0260016	0.000992022 0
$688$	0	0
$689$	−42.8021	−1.63063
$690$	0	0
$691$	31.5280	1.19938	0.599690	−	0.800232i	$-0.295290\pi$
$691$	31.5280	1.19938	0.599690	+	0.800232i	$0.295290\pi$
$692$	0	0
$693$	−2.00380	−0.0761180
$694$	0	0
$695$	−17.7618	−0.673743
$696$	0	0
$697$	9.89470	0.374788
$698$	0	0
$699$	25.2789	0.956137
$700$	0	0
$701$	6.94989	0.262494	0.131247	−	0.991350i	$-0.458102\pi$
$701$	6.94989	0.262494	0.131247	+	0.991350i	$0.458102\pi$
$702$	0	0
$703$	−28.9584	−1.09219
$704$	0	0
$705$	−2.36747	−0.0891640
$706$	0	0
$707$	5.81920	0.218853
$708$	0	0
$709$	−1.18946	−0.0446711	−0.0223356	−	0.999751i	$-0.507110\pi$
$709$	−1.18946	−0.0446711	−0.0223356	+	0.999751i	$0.507110\pi$
$710$	0	0
$711$	−11.7521	−0.440739
$712$	0	0
$713$	−38.3737	−1.43710
$714$	0	0
$715$	−16.4697	−0.615933
$716$	0	0
$717$	−18.8863	−0.705322
$718$	0	0
$719$	−6.44307	−0.240286	−0.120143	−	0.992757i	$-0.538335\pi$
$719$	−6.44307	−0.240286	−0.120143	+	0.992757i	$0.538335\pi$
$720$	0	0
$721$	1.70109	0.0633521
$722$	0	0
$723$	12.1992	0.453692
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	16.8319	0.624260	0.312130	−	0.950039i	$-0.398958\pi$
$727$	16.8319	0.624260	0.312130	+	0.950039i	$0.398958\pi$
$728$	0	0
$729$	14.4690	0.535888
$730$	0	0
$731$	78.9445	2.91987
$732$	0	0
$733$	4.68161	0.172919	0.0864596	−	0.996255i	$-0.472445\pi$
$733$	4.68161	0.172919	0.0864596	+	0.996255i	$0.472445\pi$
$734$	0	0
$735$	7.04655	0.259916
$736$	0	0
$737$	5.52281	0.203435
$738$	0	0
$739$	46.1607	1.69805	0.849025	−	0.528353i	$-0.177190\pi$
$739$	46.1607	1.69805	0.849025	+	0.528353i	$0.177190\pi$
$740$	0	0
$741$	40.6525	1.49341
$742$	0	0
$743$	26.7173	0.980162	0.490081	−	0.871677i	$-0.336967\pi$
$743$	26.7173	0.980162	0.490081	+	0.871677i	$0.336967\pi$
$744$	0	0
$745$	−16.5409	−0.606013
$746$	0	0
$747$	−9.91472	−0.362761
$748$	0	0
$749$	−3.47204	−0.126865
$750$	0	0
$751$	19.0472	0.695042	0.347521	−	0.937672i	$-0.387024\pi$
$751$	19.0472	0.695042	0.347521	+	0.937672i	$0.387024\pi$
$752$	0	0
$753$	−14.1139	−0.514337
$754$	0	0
$755$	8.04893	0.292931
$756$	0	0
$757$	−50.6932	−1.84248	−0.921238	−	0.388999i	$-0.872821\pi$
$757$	−50.6932	−1.84248	−0.921238	+	0.388999i	$0.872821\pi$
$758$	0	0
$759$	18.9842	0.689082
$760$	0	0
$761$	7.48118	0.271193	0.135596	−	0.990764i	$-0.456705\pi$
$761$	7.48118	0.271193	0.135596	+	0.990764i	$0.456705\pi$
$762$	0	0
$763$	6.35545	0.230083
$764$	0	0
$765$	−15.4393	−0.558211
$766$	0	0
$767$	−81.2826	−2.93494
$768$	0	0
$769$	21.1168	0.761490	0.380745	−	0.924680i	$-0.375668\pi$
$769$	21.1168	0.761490	0.380745	+	0.924680i	$0.375668\pi$
$770$	0	0
$771$	6.93982	0.249932
$772$	0	0
$773$	19.7478	0.710277	0.355139	−	0.934814i	$-0.384434\pi$
$773$	19.7478	0.710277	0.355139	+	0.934814i	$0.384434\pi$
$774$	0	0
$775$	5.67803	0.203961
$776$	0	0
$777$	1.70814	0.0612790
$778$	0	0
$779$	8.18275	0.293178
$780$	0	0
$781$	11.2361	0.402059
$782$	0	0
$783$	5.08025	0.181553
$784$	0	0
$785$	−2.22830	−0.0795316
$786$	0	0
$787$	37.0417	1.32039	0.660197	−	0.751092i	$-0.270473\pi$
$787$	37.0417	1.32039	0.660197	+	0.751092i	$0.270473\pi$
$788$	0	0
$789$	33.0704	1.17734
$790$	0	0
$791$	−7.53824	−0.268029
$792$	0	0
$793$	59.3884	2.10894
$794$	0	0
$795$	−7.30019	−0.258911
$796$	0	0
$797$	−23.8221	−0.843823	−0.421912	−	0.906637i	$-0.638641\pi$
$797$	−23.8221	−0.843823	−0.421912	+	0.906637i	$0.638641\pi$
$798$	0	0
$799$	18.2971	0.647303
$800$	0	0
$801$	−22.7898	−0.805238
$802$	0	0
$803$	3.55050	0.125294
$804$	0	0
$805$	−2.54787	−0.0898006
$806$	0	0
$807$	−3.01896	−0.106272
$808$	0	0
$809$	−2.16659	−0.0761734	−0.0380867	−	0.999274i	$-0.512126\pi$
$809$	−2.16659	−0.0761734	−0.0380867	+	0.999274i	$0.512126\pi$
$810$	0	0
$811$	−55.8612	−1.96155	−0.980775	−	0.195140i	$-0.937484\pi$
$811$	−55.8612	−1.96155	−0.980775	+	0.195140i	$0.937484\pi$
$812$	0	0
$813$	5.21859	0.183024
$814$	0	0
$815$	13.1574	0.460885
$816$	0	0
$817$	65.2858	2.28406
$818$	0	0
$819$	4.41575	0.154299
$820$	0	0
$821$	26.4922	0.924583	0.462292	−	0.886728i	$-0.347027\pi$
$821$	26.4922	0.924583	0.462292	+	0.886728i	$0.347027\pi$
$822$	0	0
$823$	24.3545	0.848945	0.424473	−	0.905441i	$-0.360460\pi$
$823$	24.3545	0.848945	0.424473	+	0.905441i	$0.360460\pi$
$824$	0	0
$825$	−2.80903	−0.0977977
$826$	0	0
$827$	12.3963	0.431060	0.215530	−	0.976497i	$-0.430852\pi$
$827$	12.3963	0.431060	0.215530	+	0.976497i	$0.430852\pi$
$828$	0	0
$829$	−9.68037	−0.336213	−0.168107	−	0.985769i	$-0.553765\pi$
$829$	−9.68037	−0.336213	−0.168107	+	0.985769i	$0.553765\pi$
$830$	0	0
$831$	−21.5966	−0.749177
$832$	0	0
$833$	−54.4595	−1.88691
$834$	0	0
$835$	−19.2350	−0.665653
$836$	0	0
$837$	28.8458	0.997056
$838$	0	0
$839$	−14.3130	−0.494140	−0.247070	−	0.968998i	$-0.579468\pi$
$839$	−14.3130	−0.494140	−0.247070	+	0.968998i	$0.579468\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	4.46007	0.153613
$844$	0	0
$845$	23.2942	0.801344
$846$	0	0
$847$	−1.32941	−0.0456790
$848$	0	0
$849$	−19.8450	−0.681078
$850$	0	0
$851$	29.8009	1.02156
$852$	0	0
$853$	−34.0794	−1.16686	−0.583429	−	0.812164i	$-0.698289\pi$
$853$	−34.0794	−1.16686	−0.583429	+	0.812164i	$0.698289\pi$
$854$	0	0
$855$	−12.7681	−0.436660
$856$	0	0
$857$	44.0576	1.50498	0.752490	−	0.658603i	$-0.228852\pi$
$857$	44.0576	1.50498	0.752490	+	0.658603i	$0.228852\pi$
$858$	0	0
$859$	−37.0066	−1.26265	−0.631325	−	0.775518i	$-0.717489\pi$
$859$	−37.0066	−1.26265	−0.631325	+	0.775518i	$0.717489\pi$
$860$	0	0
$861$	−0.482666	−0.0164492
$862$	0	0
$863$	16.5009	0.561696	0.280848	−	0.959752i	$-0.409384\pi$
$863$	16.5009	0.561696	0.280848	+	0.959752i	$0.409384\pi$
$864$	0	0
$865$	−24.4553	−0.831507
$866$	0	0
$867$	−47.3295	−1.60739
$868$	0	0
$869$	16.5249	0.560570
$870$	0	0
$871$	−12.1706	−0.412384
$872$	0	0
$873$	−11.2287	−0.380033
$874$	0	0
$875$	0.377000	0.0127449
$876$	0	0
$877$	40.5683	1.36989	0.684947	−	0.728593i	$-0.259825\pi$
$877$	40.5683	1.36989	0.684947	+	0.728593i	$0.259825\pi$
$878$	0	0
$879$	−11.8553	−0.399870
$880$	0	0
$881$	−23.4052	−0.788540	−0.394270	−	0.918995i	$-0.629002\pi$
$881$	−23.4052	−0.788540	−0.394270	+	0.918995i	$0.629002\pi$
$882$	0	0
$883$	−49.4633	−1.66457	−0.832285	−	0.554347i	$-0.812968\pi$
$883$	−49.4633	−1.66457	−0.832285	+	0.554347i	$0.812968\pi$
$884$	0	0
$885$	−13.8633	−0.466010
$886$	0	0
$887$	19.5071	0.654983	0.327492	−	0.944854i	$-0.393797\pi$
$887$	19.5071	0.654983	0.327492	+	0.944854i	$0.393797\pi$
$888$	0	0
$889$	−1.06704	−0.0357875
$890$	0	0
$891$	1.67481	0.0561082
$892$	0	0
$893$	15.1314	0.506352
$894$	0	0
$895$	2.50792	0.0838304
$896$	0	0
$897$	−41.8352	−1.39684
$898$	0	0
$899$	5.67803	0.189373
$900$	0	0
$901$	56.4197	1.87961
$902$	0	0
$903$	−3.85093	−0.128151
$904$	0	0
$905$	15.6831	0.521325
$906$	0	0
$907$	−42.1613	−1.39994	−0.699971	−	0.714171i	$-0.746804\pi$
$907$	−42.1613	−1.39994	−0.699971	+	0.714171i	$0.746804\pi$
$908$	0	0
$909$	−30.0100	−0.995370
$910$	0	0
$911$	−1.35574	−0.0449178	−0.0224589	−	0.999748i	$-0.507149\pi$
$911$	−1.35574	−0.0449178	−0.0224589	+	0.999748i	$0.507149\pi$
$912$	0	0
$913$	13.9413	0.461391
$914$	0	0
$915$	10.1291	0.334857
$916$	0	0
$917$	2.51560	0.0830725
$918$	0	0
$919$	−57.4928	−1.89651	−0.948256	−	0.317507i	$-0.897154\pi$
$919$	−57.4928	−1.89651	−0.948256	+	0.317507i	$0.897154\pi$
$920$	0	0
$921$	−24.4132	−0.804442
$922$	0	0
$923$	−24.7609	−0.815014
$924$	0	0
$925$	−4.40955	−0.144985
$926$	0	0
$927$	−8.77268	−0.288132
$928$	0	0
$929$	−45.7321	−1.50042	−0.750211	−	0.661199i	$-0.770048\pi$
$929$	−45.7321	−1.50042	−0.750211	+	0.661199i	$0.770048\pi$
$930$	0	0
$931$	−45.0371	−1.47603
$932$	0	0
$933$	32.8786	1.07640
$934$	0	0
$935$	21.7096	0.709981
$936$	0	0
$937$	−34.4638	−1.12588	−0.562942	−	0.826497i	$-0.690331\pi$
$937$	−34.4638	−1.12588	−0.562942	+	0.826497i	$0.690331\pi$
$938$	0	0
$939$	15.8807	0.518246
$940$	0	0
$941$	−19.6164	−0.639475	−0.319738	−	0.947506i	$-0.603595\pi$
$941$	−19.6164	−0.639475	−0.319738	+	0.947506i	$0.603595\pi$
$942$	0	0
$943$	−8.42081	−0.274219
$944$	0	0
$945$	1.91525	0.0623032
$946$	0	0
$947$	7.25310	0.235694	0.117847	−	0.993032i	$-0.462401\pi$
$947$	7.25310	0.235694	0.117847	+	0.993032i	$0.462401\pi$
$948$	0	0
$949$	−7.82418	−0.253984
$950$	0	0
$951$	−30.7127	−0.995927
$952$	0	0
$953$	−1.48853	−0.0482183	−0.0241092	−	0.999709i	$-0.507675\pi$
$953$	−1.48853	−0.0482183	−0.0241092	+	0.999709i	$0.507675\pi$
$954$	0	0
$955$	−13.6780	−0.442611
$956$	0	0
$957$	−2.80903	−0.0908029
$958$	0	0
$959$	−2.71218	−0.0875808
$960$	0	0
$961$	1.23999	0.0399996
$962$	0	0
$963$	17.9056	0.576998
$964$	0	0
$965$	1.68611	0.0542777
$966$	0	0
$967$	14.0886	0.453057	0.226529	−	0.974004i	$-0.427262\pi$
$967$	14.0886	0.453057	0.226529	+	0.974004i	$0.427262\pi$
$968$	0	0
$969$	−53.5862	−1.72144
$970$	0	0
$971$	32.4110	1.04012	0.520060	−	0.854130i	$-0.325910\pi$
$971$	32.4110	1.04012	0.520060	+	0.854130i	$0.325910\pi$
$972$	0	0
$973$	−6.69619	−0.214670
$974$	0	0
$975$	6.19022	0.198246
$976$	0	0
$977$	−6.42216	−0.205463	−0.102732	−	0.994709i	$-0.532758\pi$
$977$	−6.42216	−0.205463	−0.102732	+	0.994709i	$0.532758\pi$
$978$	0	0
$979$	32.0453	1.02417
$980$	0	0
$981$	−32.7755	−1.04644
$982$	0	0
$983$	0.340039	0.0108456	0.00542278	−	0.999985i	$-0.498274\pi$
$983$	0.340039	0.0108456	0.00542278	+	0.999985i	$0.498274\pi$
$984$	0	0
$985$	6.08033	0.193735
$986$	0	0
$987$	−0.892535	−0.0284097
$988$	0	0
$989$	−67.1851	−2.13636
$990$	0	0
$991$	−7.94972	−0.252531	−0.126266	−	0.991996i	$-0.540299\pi$
$991$	−7.94972	−0.252531	−0.126266	+	0.991996i	$0.540299\pi$
$992$	0	0
$993$	17.5120	0.555725
$994$	0	0
$995$	−9.99849	−0.316973
$996$	0	0
$997$	−1.56820	−0.0496653	−0.0248326	−	0.999692i	$-0.507905\pi$
$997$	−1.56820	−0.0496653	−0.0248326	+	0.999692i	$0.507905\pi$
$998$	0	0
$999$	−22.4016	−0.708755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.2.a.j.1.2	✓	5
4.3	odd	2		2320.2.a.u.1.4		5
5.4	even	2		5800.2.a.t.1.4		5
8.3	odd	2		9280.2.a.cl.1.2		5
8.5	even	2		9280.2.a.cg.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.j.1.2	✓	5	1.1	even	1	trivial
2320.2.a.u.1.4		5	4.3	odd	2
5800.2.a.t.1.4		5	5.4	even	2
9280.2.a.cg.1.4		5	8.5	even	2
9280.2.a.cl.1.2		5	8.3	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:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q-1.02751 q^{3} +1.00000 q^{5} +0.377000 q^{7} -1.94422 q^{9} +2.73381 q^{11} -6.02446 q^{13} -1.02751 q^{15} +7.94117 q^{17} +6.56722 q^{19} -0.387372 q^{21} -6.75827 q^{23} +1.00000 q^{25} +5.08025 q^{27} +1.00000 q^{29} +5.67803 q^{31} -2.80903 q^{33} +0.377000 q^{35} -4.40955 q^{37} +6.19022 q^{39} +1.24600 q^{41} +9.94117 q^{43} -1.94422 q^{45} +2.30408 q^{47} -6.85787 q^{49} -8.15966 q^{51} +7.10471 q^{53} +2.73381 q^{55} -6.74790 q^{57} +13.4921 q^{59} -9.85787 q^{61} -0.732969 q^{63} -6.02446 q^{65} +2.02019 q^{67} +6.94422 q^{69} +4.11005 q^{71} +1.29874 q^{73} -1.02751 q^{75} +1.03065 q^{77} +6.04465 q^{79} +0.612628 q^{81} +5.09960 q^{83} +7.94117 q^{85} -1.02751 q^{87} +11.7218 q^{89} -2.27122 q^{91} -5.83425 q^{93} +6.56722 q^{95} +5.77542 q^{97} -5.31512 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 8 q^{9} + 10 q^{11} + q^{13} + 3 q^{15} - q^{17} + 10 q^{19} - 8 q^{21} + q^{23} + 5 q^{25} + 12 q^{27} + 5 q^{29} + 7 q^{31} - 8 q^{33} + 7 q^{35} - 8 q^{37} + 3 q^{39}+ \cdots + 32 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 + 5 * q^5 + 7 * q^7 + 8 * q^9 + 10 * q^11 + q^13 + 3 * q^15 - q^17 + 10 * q^19 - 8 * q^21 + q^23 + 5 * q^25 + 12 * q^27 + 5 * q^29 + 7 * q^31 - 8 * q^33 + 7 * q^35 - 8 * q^37 + 3 * q^39 - 4 * q^41 + 9 * q^43 + 8 * q^45 + 8 * q^47 + 16 * q^49 + 2 * q^51 - 9 * q^53 + 10 * q^55 + 2 * q^57 + 29 * q^59 + q^61 + 16 * q^63 + q^65 + 24 * q^67 + 17 * q^69 - 12 * q^71 - 9 * q^73 + 3 * q^75 + 2 * q^77 + 13 * q^79 - 3 * q^81 + 10 * q^83 - q^85 + 3 * q^87 + 4 * q^89 - 4 * q^91 - 26 * q^93 + 10 * q^95 - 15 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more