LMFDB - Embedded newform 704.4.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [704,4,Mod(1,704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("704.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	704.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.5373446440$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	704.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+7.92820 q^{3} -14.8564 q^{5} +3.07180 q^{7} +35.8564 q^{9} +11.0000 q^{11} -5.35898 q^{13} -117.785 q^{15} -41.2154 q^{17} -139.923 q^{19} +24.3538 q^{21} -111.354 q^{23} +95.7128 q^{25} +70.2154 q^{27} +24.9948 q^{29} +31.4974 q^{31} +87.2102 q^{33} -45.6359 q^{35} -13.1436 q^{37} -42.4871 q^{39} +261.072 q^{41} +57.7128 q^{43} -532.697 q^{45} -343.846 q^{47} -333.564 q^{49} -326.764 q^{51} +342.995 q^{53} -163.420 q^{55} -1109.34 q^{57} -88.3693 q^{59} -738.697 q^{61} +110.144 q^{63} +79.6152 q^{65} -342.359 q^{67} -882.836 q^{69} -207.364 q^{71} -1010.60 q^{73} +758.831 q^{75} +33.7898 q^{77} +1294.23 q^{79} -411.441 q^{81} -441.846 q^{83} +612.313 q^{85} +198.164 q^{87} -1489.11 q^{89} -16.4617 q^{91} +249.718 q^{93} +2078.75 q^{95} +1346.42 q^{97} +394.420 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 20 q^{7} + 44 q^{9} + 22 q^{11} - 80 q^{13} - 194 q^{15} - 124 q^{17} - 72 q^{19} - 76 q^{21} - 98 q^{23} + 136 q^{25} + 182 q^{27} - 144 q^{29} - 34 q^{31} + 22 q^{33} + 172 q^{35}+ \cdots + 484 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 20 * q^7 + 44 * q^9 + 22 * q^11 - 80 * q^13 - 194 * q^15 - 124 * q^17 - 72 * q^19 - 76 * q^21 - 98 * q^23 + 136 * q^25 + 182 * q^27 - 144 * q^29 - 34 * q^31 + 22 * q^33 + 172 * q^35 - 54 * q^37 + 400 * q^39 + 536 * q^41 + 60 * q^43 - 428 * q^45 - 272 * q^47 - 390 * q^49 + 164 * q^51 + 492 * q^53 - 22 * q^55 - 1512 * q^57 - 634 * q^59 - 840 * q^61 + 248 * q^63 - 880 * q^65 - 754 * q^67 - 962 * q^69 - 678 * q^71 - 400 * q^73 + 520 * q^75 + 220 * q^77 + 316 * q^79 - 1294 * q^81 - 468 * q^83 - 452 * q^85 + 1200 * q^87 - 1842 * q^89 - 1280 * q^91 + 638 * q^93 + 2952 * q^95 + 2194 * q^97 + 484 * 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
$3$	7.92820	1.52578	0.762892	−	0.646526i	$-0.223779\pi$
$3$	7.92820	1.52578	0.762892	+	0.646526i	$0.223779\pi$
$4$	0	0
$5$	−14.8564	−1.32880	−0.664399	−	0.747378i	$-0.731312\pi$
$5$	−14.8564	−1.32880	−0.664399	+	0.747378i	$0.731312\pi$
$6$	0	0
$7$	3.07180	0.165861	0.0829307	−	0.996555i	$-0.473572\pi$
$7$	3.07180	0.165861	0.0829307	+	0.996555i	$0.473572\pi$
$8$	0	0
$9$	35.8564	1.32802
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−5.35898	−0.114332	−0.0571659	−	0.998365i	$-0.518206\pi$
$13$	−5.35898	−0.114332	−0.0571659	+	0.998365i	$0.518206\pi$
$14$	0	0
$15$	−117.785	−2.02746
$16$	0	0
$17$	−41.2154	−0.588012	−0.294006	−	0.955804i	$-0.594989\pi$
$17$	−41.2154	−0.588012	−0.294006	+	0.955804i	$0.594989\pi$
$18$	0	0
$19$	−139.923	−1.68950	−0.844751	−	0.535159i	$-0.820252\pi$
$19$	−139.923	−1.68950	−0.844751	+	0.535159i	$0.820252\pi$
$20$	0	0
$21$	24.3538	0.253069
$22$	0	0
$23$	−111.354	−1.00952	−0.504758	−	0.863261i	$-0.668418\pi$
$23$	−111.354	−1.00952	−0.504758	+	0.863261i	$0.668418\pi$
$24$	0	0
$25$	95.7128	0.765703
$26$	0	0
$27$	70.2154	0.500480
$28$	0	0
$29$	24.9948	0.160049	0.0800246	−	0.996793i	$-0.474500\pi$
$29$	24.9948	0.160049	0.0800246	+	0.996793i	$0.474500\pi$
$30$	0	0
$31$	31.4974	0.182487	0.0912436	−	0.995829i	$-0.470916\pi$
$31$	31.4974	0.182487	0.0912436	+	0.995829i	$0.470916\pi$
$32$	0	0
$33$	87.2102	0.460041
$34$	0	0
$35$	−45.6359	−0.220396
$36$	0	0
$37$	−13.1436	−0.0583998	−0.0291999	−	0.999574i	$-0.509296\pi$
$37$	−13.1436	−0.0583998	−0.0291999	+	0.999574i	$0.509296\pi$
$38$	0	0
$39$	−42.4871	−0.174446
$40$	0	0
$41$	261.072	0.994453	0.497226	−	0.867621i	$-0.334352\pi$
$41$	261.072	0.994453	0.497226	+	0.867621i	$0.334352\pi$
$42$	0	0
$43$	57.7128	0.204677	0.102339	−	0.994750i	$-0.467367\pi$
$43$	57.7128	0.204677	0.102339	+	0.994750i	$0.467367\pi$
$44$	0	0
$45$	−532.697	−1.76466
$46$	0	0
$47$	−343.846	−1.06713	−0.533565	−	0.845759i	$-0.679148\pi$
$47$	−343.846	−1.06713	−0.533565	+	0.845759i	$0.679148\pi$
$48$	0	0
$49$	−333.564	−0.972490
$50$	0	0
$51$	−326.764	−0.897179
$52$	0	0
$53$	342.995	0.888943	0.444471	−	0.895793i	$-0.353392\pi$
$53$	342.995	0.888943	0.444471	+	0.895793i	$0.353392\pi$
$54$	0	0
$55$	−163.420	−0.400647
$56$	0	0
$57$	−1109.34	−2.57782
$58$	0	0
$59$	−88.3693	−0.194995	−0.0974975	−	0.995236i	$-0.531084\pi$
$59$	−88.3693	−0.194995	−0.0974975	+	0.995236i	$0.531084\pi$
$60$	0	0
$61$	−738.697	−1.55050	−0.775250	−	0.631654i	$-0.782376\pi$
$61$	−738.697	−1.55050	−0.775250	+	0.631654i	$0.782376\pi$
$62$	0	0
$63$	110.144	0.220266
$64$	0	0
$65$	79.6152	0.151924
$66$	0	0
$67$	−342.359	−0.624266	−0.312133	−	0.950038i	$-0.601043\pi$
$67$	−342.359	−0.624266	−0.312133	+	0.950038i	$0.601043\pi$
$68$	0	0
$69$	−882.836	−1.54030
$70$	0	0
$71$	−207.364	−0.346614	−0.173307	−	0.984868i	$-0.555445\pi$
$71$	−207.364	−0.346614	−0.173307	+	0.984868i	$0.555445\pi$
$72$	0	0
$73$	−1010.60	−1.62030	−0.810149	−	0.586224i	$-0.800614\pi$
$73$	−1010.60	−1.62030	−0.810149	+	0.586224i	$0.800614\pi$
$74$	0	0
$75$	758.831	1.16830
$76$	0	0
$77$	33.7898	0.0500091
$78$	0	0
$79$	1294.23	1.84319	0.921593	−	0.388157i	$-0.126888\pi$
$79$	1294.23	1.84319	0.921593	+	0.388157i	$0.126888\pi$
$80$	0	0
$81$	−411.441	−0.564391
$82$	0	0
$83$	−441.846	−0.584324	−0.292162	−	0.956369i	$-0.594375\pi$
$83$	−441.846	−0.584324	−0.292162	+	0.956369i	$0.594375\pi$
$84$	0	0
$85$	612.313	0.781349
$86$	0	0
$87$	198.164	0.244200
$88$	0	0
$89$	−1489.11	−1.77355	−0.886773	−	0.462205i	$-0.847058\pi$
$89$	−1489.11	−1.77355	−0.886773	+	0.462205i	$0.847058\pi$
$90$	0	0
$91$	−16.4617	−0.0189633
$92$	0	0
$93$	249.718	0.278436
$94$	0	0
$95$	2078.75	2.24501
$96$	0	0
$97$	1346.42	1.40936	0.704679	−	0.709526i	$-0.251091\pi$
$97$	1346.42	1.40936	0.704679	+	0.709526i	$0.251091\pi$
$98$	0	0
$99$	394.420	0.400412
$100$	0	0
$101$	161.461	0.159069	0.0795347	−	0.996832i	$-0.474657\pi$
$101$	161.461	0.159069	0.0795347	+	0.996832i	$0.474657\pi$
$102$	0	0
$103$	−34.7592	−0.0332517	−0.0166259	−	0.999862i	$-0.505292\pi$
$103$	−34.7592	−0.0332517	−0.0166259	+	0.999862i	$0.505292\pi$
$104$	0	0
$105$	−361.810	−0.336277
$106$	0	0
$107$	−832.179	−0.751867	−0.375934	−	0.926647i	$-0.622678\pi$
$107$	−832.179	−0.751867	−0.375934	+	0.926647i	$0.622678\pi$
$108$	0	0
$109$	−1044.26	−0.917629	−0.458815	−	0.888532i	$-0.651726\pi$
$109$	−1044.26	−0.917629	−0.458815	+	0.888532i	$0.651726\pi$
$110$	0	0
$111$	−104.205	−0.0891055
$112$	0	0
$113$	295.082	0.245654	0.122827	−	0.992428i	$-0.460804\pi$
$113$	295.082	0.245654	0.122827	+	0.992428i	$0.460804\pi$
$114$	0	0
$115$	1654.32	1.34144
$116$	0	0
$117$	−192.154	−0.151834
$118$	0	0
$119$	−126.605	−0.0975285
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	2069.83	1.51732
$124$	0	0
$125$	435.102	0.311334
$126$	0	0
$127$	−1317.60	−0.920618	−0.460309	−	0.887759i	$-0.652261\pi$
$127$	−1317.60	−0.920618	−0.460309	+	0.887759i	$0.652261\pi$
$128$	0	0
$129$	457.559	0.312293
$130$	0	0
$131$	1600.71	1.06759	0.533797	−	0.845612i	$-0.320765\pi$
$131$	1600.71	1.06759	0.533797	+	0.845612i	$0.320765\pi$
$132$	0	0
$133$	−429.815	−0.280223
$134$	0	0
$135$	−1043.15	−0.665036
$136$	0	0
$137$	1611.68	1.00507	0.502536	−	0.864556i	$-0.332400\pi$
$137$	1611.68	1.00507	0.502536	+	0.864556i	$0.332400\pi$
$138$	0	0
$139$	31.8619	0.0194424	0.00972120	−	0.999953i	$-0.496906\pi$
$139$	31.8619	0.0194424	0.00972120	+	0.999953i	$0.496906\pi$
$140$	0	0
$141$	−2726.08	−1.62821
$142$	0	0
$143$	−58.9488	−0.0344724
$144$	0	0
$145$	−371.334	−0.212673
$146$	0	0
$147$	−2644.56	−1.48381
$148$	0	0
$149$	2428.34	1.33515	0.667576	−	0.744542i	$-0.267332\pi$
$149$	2428.34	1.33515	0.667576	+	0.744542i	$0.267332\pi$
$150$	0	0
$151$	−2576.68	−1.38866	−0.694328	−	0.719659i	$-0.744298\pi$
$151$	−2576.68	−1.38866	−0.694328	+	0.719659i	$0.744298\pi$
$152$	0	0
$153$	−1477.84	−0.780889
$154$	0	0
$155$	−467.939	−0.242489
$156$	0	0
$157$	−2475.94	−1.25861	−0.629305	−	0.777158i	$-0.716660\pi$
$157$	−2475.94	−1.25861	−0.629305	+	0.777158i	$0.716660\pi$
$158$	0	0
$159$	2719.33	1.35633
$160$	0	0
$161$	−342.056	−0.167440
$162$	0	0
$163$	2725.11	1.30949	0.654745	−	0.755850i	$-0.272776\pi$
$163$	2725.11	1.30949	0.654745	+	0.755850i	$0.272776\pi$
$164$	0	0
$165$	−1295.63	−0.611301
$166$	0	0
$167$	2737.30	1.26837	0.634187	−	0.773180i	$-0.281335\pi$
$167$	2737.30	1.26837	0.634187	+	0.773180i	$0.281335\pi$
$168$	0	0
$169$	−2168.28	−0.986928
$170$	0	0
$171$	−5017.14	−2.24368
$172$	0	0
$173$	−2307.42	−1.01404	−0.507022	−	0.861933i	$-0.669254\pi$
$173$	−2307.42	−1.01404	−0.507022	+	0.861933i	$0.669254\pi$
$174$	0	0
$175$	294.010	0.127001
$176$	0	0
$177$	−700.610	−0.297520
$178$	0	0
$179$	1312.15	0.547905	0.273953	−	0.961743i	$-0.411669\pi$
$179$	1312.15	0.547905	0.273953	+	0.961743i	$0.411669\pi$
$180$	0	0
$181$	803.174	0.329831	0.164916	−	0.986308i	$-0.447265\pi$
$181$	803.174	0.329831	0.164916	+	0.986308i	$0.447265\pi$
$182$	0	0
$183$	−5856.54	−2.36573
$184$	0	0
$185$	195.267	0.0776015
$186$	0	0
$187$	−453.369	−0.177292
$188$	0	0
$189$	215.687	0.0830103
$190$	0	0
$191$	1718.25	0.650932	0.325466	−	0.945554i	$-0.394479\pi$
$191$	1718.25	0.650932	0.325466	+	0.945554i	$0.394479\pi$
$192$	0	0
$193$	1340.18	0.499837	0.249919	−	0.968267i	$-0.419596\pi$
$193$	1340.18	0.499837	0.249919	+	0.968267i	$0.419596\pi$
$194$	0	0
$195$	631.206	0.231803
$196$	0	0
$197$	3518.33	1.27244	0.636220	−	0.771508i	$-0.280497\pi$
$197$	3518.33	1.27244	0.636220	+	0.771508i	$0.280497\pi$
$198$	0	0
$199$	823.692	0.293417	0.146709	−	0.989180i	$-0.453132\pi$
$199$	823.692	0.293417	0.146709	+	0.989180i	$0.453132\pi$
$200$	0	0
$201$	−2714.29	−0.952494
$202$	0	0
$203$	76.7791	0.0265460
$204$	0	0
$205$	−3878.59	−1.32143
$206$	0	0
$207$	−3992.75	−1.34065
$208$	0	0
$209$	−1539.15	−0.509404
$210$	0	0
$211$	107.343	0.0350228	0.0175114	−	0.999847i	$-0.494426\pi$
$211$	107.343	0.0350228	0.0175114	+	0.999847i	$0.494426\pi$
$212$	0	0
$213$	−1644.03	−0.528858
$214$	0	0
$215$	−857.405	−0.271975
$216$	0	0
$217$	96.7537	0.0302676
$218$	0	0
$219$	−8012.24	−2.47222
$220$	0	0
$221$	220.873	0.0672285
$222$	0	0
$223$	−3933.68	−1.18125	−0.590625	−	0.806946i	$-0.701119\pi$
$223$	−3933.68	−1.18125	−0.590625	+	0.806946i	$0.701119\pi$
$224$	0	0
$225$	3431.92	1.01686
$226$	0	0
$227$	1771.90	0.518085	0.259042	−	0.965866i	$-0.416593\pi$
$227$	1771.90	0.518085	0.259042	+	0.965866i	$0.416593\pi$
$228$	0	0
$229$	−1915.37	−0.552713	−0.276356	−	0.961055i	$-0.589127\pi$
$229$	−1915.37	−0.552713	−0.276356	+	0.961055i	$0.589127\pi$
$230$	0	0
$231$	267.892	0.0763031
$232$	0	0
$233$	4396.32	1.23610	0.618052	−	0.786137i	$-0.287922\pi$
$233$	4396.32	1.23610	0.618052	+	0.786137i	$0.287922\pi$
$234$	0	0
$235$	5108.32	1.41800
$236$	0	0
$237$	10260.9	2.81230
$238$	0	0
$239$	−4084.49	−1.10546	−0.552728	−	0.833362i	$-0.686413\pi$
$239$	−4084.49	−1.10546	−0.552728	+	0.833362i	$0.686413\pi$
$240$	0	0
$241$	3908.58	1.04471	0.522353	−	0.852730i	$-0.325054\pi$
$241$	3908.58	1.04471	0.522353	+	0.852730i	$0.325054\pi$
$242$	0	0
$243$	−5157.80	−1.36162
$244$	0	0
$245$	4955.56	1.29224
$246$	0	0
$247$	749.845	0.193164
$248$	0	0
$249$	−3503.05	−0.891552
$250$	0	0
$251$	−1094.89	−0.275335	−0.137667	−	0.990479i	$-0.543960\pi$
$251$	−1094.89	−0.275335	−0.137667	+	0.990479i	$0.543960\pi$
$252$	0	0
$253$	−1224.89	−0.304381
$254$	0	0
$255$	4854.54	1.19217
$256$	0	0
$257$	783.179	0.190091	0.0950454	−	0.995473i	$-0.469700\pi$
$257$	783.179	0.190091	0.0950454	+	0.995473i	$0.469700\pi$
$258$	0	0
$259$	−40.3744	−0.00968628
$260$	0	0
$261$	896.225	0.212548
$262$	0	0
$263$	6180.06	1.44897	0.724484	−	0.689292i	$-0.242078\pi$
$263$	6180.06	1.44897	0.724484	+	0.689292i	$0.242078\pi$
$264$	0	0
$265$	−5095.67	−1.18122
$266$	0	0
$267$	−11806.0	−2.70605
$268$	0	0
$269$	−986.965	−0.223704	−0.111852	−	0.993725i	$-0.535678\pi$
$269$	−986.965	−0.223704	−0.111852	+	0.993725i	$0.535678\pi$
$270$	0	0
$271$	4576.99	1.02595	0.512975	−	0.858404i	$-0.328543\pi$
$271$	4576.99	1.02595	0.512975	+	0.858404i	$0.328543\pi$
$272$	0	0
$273$	−130.512	−0.0289338
$274$	0	0
$275$	1052.84	0.230868
$276$	0	0
$277$	−567.836	−0.123169	−0.0615847	−	0.998102i	$-0.519615\pi$
$277$	−567.836	−0.123169	−0.0615847	+	0.998102i	$0.519615\pi$
$278$	0	0
$279$	1129.38	0.242346
$280$	0	0
$281$	5311.01	1.12750	0.563752	−	0.825944i	$-0.309357\pi$
$281$	5311.01	1.12750	0.563752	+	0.825944i	$0.309357\pi$
$282$	0	0
$283$	4728.44	0.993204	0.496602	−	0.867978i	$-0.334581\pi$
$283$	4728.44	0.993204	0.496602	+	0.867978i	$0.334581\pi$
$284$	0	0
$285$	16480.8	3.42539
$286$	0	0
$287$	801.960	0.164941
$288$	0	0
$289$	−3214.29	−0.654242
$290$	0	0
$291$	10674.7	2.15038
$292$	0	0
$293$	−2328.92	−0.464358	−0.232179	−	0.972673i	$-0.574585\pi$
$293$	−2328.92	−0.464358	−0.232179	+	0.972673i	$0.574585\pi$
$294$	0	0
$295$	1312.85	0.259109
$296$	0	0
$297$	772.369	0.150900
$298$	0	0
$299$	596.743	0.115420
$300$	0	0
$301$	177.282	0.0339481
$302$	0	0
$303$	1280.10	0.242705
$304$	0	0
$305$	10974.4	2.06030
$306$	0	0
$307$	1678.07	0.311962	0.155981	−	0.987760i	$-0.450146\pi$
$307$	1678.07	0.311962	0.155981	+	0.987760i	$0.450146\pi$
$308$	0	0
$309$	−275.578	−0.0507349
$310$	0	0
$311$	3572.71	0.651413	0.325707	−	0.945471i	$-0.394398\pi$
$311$	3572.71	0.651413	0.325707	+	0.945471i	$0.394398\pi$
$312$	0	0
$313$	7184.36	1.29739	0.648697	−	0.761047i	$-0.275314\pi$
$313$	7184.36	1.29739	0.648697	+	0.761047i	$0.275314\pi$
$314$	0	0
$315$	−1636.34	−0.292690
$316$	0	0
$317$	15.7077	0.00278306	0.00139153	−	0.999999i	$-0.499557\pi$
$317$	15.7077	0.00278306	0.00139153	+	0.999999i	$0.499557\pi$
$318$	0	0
$319$	274.943	0.0482566
$320$	0	0
$321$	−6597.69	−1.14719
$322$	0	0
$323$	5766.98	0.993447
$324$	0	0
$325$	−512.923	−0.0875442
$326$	0	0
$327$	−8279.08	−1.40010
$328$	0	0
$329$	−1056.23	−0.176996
$330$	0	0
$331$	1318.95	0.219022	0.109511	−	0.993986i	$-0.465072\pi$
$331$	1318.95	0.219022	0.109511	+	0.993986i	$0.465072\pi$
$332$	0	0
$333$	−471.282	−0.0775558
$334$	0	0
$335$	5086.22	0.829523
$336$	0	0
$337$	−239.183	−0.0386621	−0.0193310	−	0.999813i	$-0.506154\pi$
$337$	−239.183	−0.0386621	−0.0193310	+	0.999813i	$0.506154\pi$
$338$	0	0
$339$	2339.47	0.374816
$340$	0	0
$341$	346.472	0.0550220
$342$	0	0
$343$	−2078.27	−0.327160
$344$	0	0
$345$	13115.8	2.04675
$346$	0	0
$347$	5862.79	0.907006	0.453503	−	0.891255i	$-0.350174\pi$
$347$	5862.79	0.907006	0.453503	+	0.891255i	$0.350174\pi$
$348$	0	0
$349$	−3491.73	−0.535553	−0.267776	−	0.963481i	$-0.586289\pi$
$349$	−3491.73	−0.535553	−0.267776	+	0.963481i	$0.586289\pi$
$350$	0	0
$351$	−376.283	−0.0572208
$352$	0	0
$353$	−10916.7	−1.64600	−0.822999	−	0.568043i	$-0.807701\pi$
$353$	−10916.7	−1.64600	−0.822999	+	0.568043i	$0.807701\pi$
$354$	0	0
$355$	3080.69	0.460580
$356$	0	0
$357$	−1003.75	−0.148807
$358$	0	0
$359$	−11500.7	−1.69077	−0.845384	−	0.534160i	$-0.820628\pi$
$359$	−11500.7	−1.69077	−0.845384	+	0.534160i	$0.820628\pi$
$360$	0	0
$361$	12719.5	1.85442
$362$	0	0
$363$	959.313	0.138708
$364$	0	0
$365$	15013.9	2.15305
$366$	0	0
$367$	6767.01	0.962493	0.481246	−	0.876585i	$-0.340184\pi$
$367$	6767.01	0.962493	0.481246	+	0.876585i	$0.340184\pi$
$368$	0	0
$369$	9361.10	1.32065
$370$	0	0
$371$	1053.61	0.147441
$372$	0	0
$373$	5310.22	0.737139	0.368569	−	0.929600i	$-0.379848\pi$
$373$	5310.22	0.737139	0.368569	+	0.929600i	$0.379848\pi$
$374$	0	0
$375$	3449.58	0.475028
$376$	0	0
$377$	−133.947	−0.0182987
$378$	0	0
$379$	838.267	0.113612	0.0568059	−	0.998385i	$-0.481908\pi$
$379$	838.267	0.113612	0.0568059	+	0.998385i	$0.481908\pi$
$380$	0	0
$381$	−10446.2	−1.40466
$382$	0	0
$383$	−2832.16	−0.377851	−0.188925	−	0.981991i	$-0.560500\pi$
$383$	−2832.16	−0.377851	−0.188925	+	0.981991i	$0.560500\pi$
$384$	0	0
$385$	−501.994	−0.0664520
$386$	0	0
$387$	2069.37	0.271814
$388$	0	0
$389$	−3111.25	−0.405519	−0.202759	−	0.979229i	$-0.564991\pi$
$389$	−3111.25	−0.405519	−0.202759	+	0.979229i	$0.564991\pi$
$390$	0	0
$391$	4589.49	0.593608
$392$	0	0
$393$	12690.8	1.62892
$394$	0	0
$395$	−19227.5	−2.44922
$396$	0	0
$397$	−14208.7	−1.79626	−0.898131	−	0.439728i	$-0.855075\pi$
$397$	−14208.7	−1.79626	−0.898131	+	0.439728i	$0.855075\pi$
$398$	0	0
$399$	−3407.66	−0.427560
$400$	0	0
$401$	−6261.68	−0.779784	−0.389892	−	0.920861i	$-0.627488\pi$
$401$	−6261.68	−0.779784	−0.389892	+	0.920861i	$0.627488\pi$
$402$	0	0
$403$	−168.794	−0.0208641
$404$	0	0
$405$	6112.54	0.749961
$406$	0	0
$407$	−144.580	−0.0176082
$408$	0	0
$409$	−4192.50	−0.506860	−0.253430	−	0.967354i	$-0.581559\pi$
$409$	−4192.50	−0.506860	−0.253430	+	0.967354i	$0.581559\pi$
$410$	0	0
$411$	12777.7	1.53352
$412$	0	0
$413$	−271.453	−0.0323421
$414$	0	0
$415$	6564.25	0.776448
$416$	0	0
$417$	252.608	0.0296649
$418$	0	0
$419$	9287.15	1.08283	0.541416	−	0.840755i	$-0.317888\pi$
$419$	9287.15	1.08283	0.541416	+	0.840755i	$0.317888\pi$
$420$	0	0
$421$	−13146.0	−1.52185	−0.760923	−	0.648842i	$-0.775254\pi$
$421$	−13146.0	−1.52185	−0.760923	+	0.648842i	$0.775254\pi$
$422$	0	0
$423$	−12329.1	−1.41716
$424$	0	0
$425$	−3944.84	−0.450242
$426$	0	0
$427$	−2269.13	−0.257168
$428$	0	0
$429$	−467.358	−0.0525974
$430$	0	0
$431$	4909.67	0.548701	0.274351	−	0.961630i	$-0.411537\pi$
$431$	4909.67	0.548701	0.274351	+	0.961630i	$0.411537\pi$
$432$	0	0
$433$	−11743.3	−1.30334	−0.651671	−	0.758502i	$-0.725932\pi$
$433$	−11743.3	−1.30334	−0.651671	+	0.758502i	$0.725932\pi$
$434$	0	0
$435$	−2944.01	−0.324493
$436$	0	0
$437$	15581.0	1.70558
$438$	0	0
$439$	−11824.2	−1.28551	−0.642754	−	0.766073i	$-0.722208\pi$
$439$	−11824.2	−1.28551	−0.642754	+	0.766073i	$0.722208\pi$
$440$	0	0
$441$	−11960.4	−1.29148
$442$	0	0
$443$	−10102.1	−1.08344	−0.541722	−	0.840558i	$-0.682228\pi$
$443$	−10102.1	−1.08344	−0.541722	+	0.840558i	$0.682228\pi$
$444$	0	0
$445$	22122.9	2.35668
$446$	0	0
$447$	19252.4	2.03715
$448$	0	0
$449$	−345.254	−0.0362885	−0.0181443	−	0.999835i	$-0.505776\pi$
$449$	−345.254	−0.0362885	−0.0181443	+	0.999835i	$0.505776\pi$
$450$	0	0
$451$	2871.79	0.299839
$452$	0	0
$453$	−20428.4	−2.11879
$454$	0	0
$455$	244.562	0.0251983
$456$	0	0
$457$	−10567.1	−1.08164	−0.540821	−	0.841138i	$-0.681886\pi$
$457$	−10567.1	−1.08164	−0.540821	+	0.841138i	$0.681886\pi$
$458$	0	0
$459$	−2893.95	−0.294288
$460$	0	0
$461$	−4733.96	−0.478270	−0.239135	−	0.970986i	$-0.576864\pi$
$461$	−4733.96	−0.478270	−0.239135	+	0.970986i	$0.576864\pi$
$462$	0	0
$463$	3431.20	0.344409	0.172204	−	0.985061i	$-0.444911\pi$
$463$	3431.20	0.344409	0.172204	+	0.985061i	$0.444911\pi$
$464$	0	0
$465$	−3709.91	−0.369985
$466$	0	0
$467$	−5116.96	−0.507034	−0.253517	−	0.967331i	$-0.581587\pi$
$467$	−5116.96	−0.507034	−0.253517	+	0.967331i	$0.581587\pi$
$468$	0	0
$469$	−1051.66	−0.103542
$470$	0	0
$471$	−19629.8	−1.92037
$472$	0	0
$473$	634.841	0.0617125
$474$	0	0
$475$	−13392.4	−1.29366
$476$	0	0
$477$	12298.6	1.18053
$478$	0	0
$479$	11566.9	1.10335	0.551675	−	0.834059i	$-0.313989\pi$
$479$	11566.9	1.10335	0.551675	+	0.834059i	$0.313989\pi$
$480$	0	0
$481$	70.4363	0.00667696
$482$	0	0
$483$	−2711.89	−0.255477
$484$	0	0
$485$	−20002.9	−1.87275
$486$	0	0
$487$	−18326.5	−1.70525	−0.852623	−	0.522527i	$-0.824990\pi$
$487$	−18326.5	−1.70525	−0.852623	+	0.522527i	$0.824990\pi$
$488$	0	0
$489$	21605.2	1.99800
$490$	0	0
$491$	7617.58	0.700156	0.350078	−	0.936721i	$-0.386155\pi$
$491$	7617.58	0.700156	0.350078	+	0.936721i	$0.386155\pi$
$492$	0	0
$493$	−1030.17	−0.0941108
$494$	0	0
$495$	−5859.67	−0.532066
$496$	0	0
$497$	−636.980	−0.0574899
$498$	0	0
$499$	−12909.1	−1.15810	−0.579050	−	0.815292i	$-0.696576\pi$
$499$	−12909.1	−1.15810	−0.579050	+	0.815292i	$0.696576\pi$
$500$	0	0
$501$	21701.8	1.93526
$502$	0	0
$503$	10165.7	0.901121	0.450561	−	0.892746i	$-0.351224\pi$
$503$	10165.7	0.901121	0.450561	+	0.892746i	$0.351224\pi$
$504$	0	0
$505$	−2398.74	−0.211371
$506$	0	0
$507$	−17190.6	−1.50584
$508$	0	0
$509$	−6449.93	−0.561666	−0.280833	−	0.959757i	$-0.590611\pi$
$509$	−6449.93	−0.561666	−0.280833	+	0.959757i	$0.590611\pi$
$510$	0	0
$511$	−3104.36	−0.268745
$512$	0	0
$513$	−9824.75	−0.845562
$514$	0	0
$515$	516.397	0.0441848
$516$	0	0
$517$	−3782.31	−0.321752
$518$	0	0
$519$	−18293.7	−1.54721
$520$	0	0
$521$	−19327.4	−1.62524	−0.812620	−	0.582794i	$-0.801959\pi$
$521$	−19327.4	−1.62524	−0.812620	+	0.582794i	$0.801959\pi$
$522$	0	0
$523$	−6259.09	−0.523310	−0.261655	−	0.965161i	$-0.584268\pi$
$523$	−6259.09	−0.523310	−0.261655	+	0.965161i	$0.584268\pi$
$524$	0	0
$525$	2330.97	0.193775
$526$	0	0
$527$	−1298.18	−0.107305
$528$	0	0
$529$	232.675	0.0191235
$530$	0	0
$531$	−3168.61	−0.258956
$532$	0	0
$533$	−1399.08	−0.113698
$534$	0	0
$535$	12363.2	0.999079
$536$	0	0
$537$	10403.0	0.835985
$538$	0	0
$539$	−3669.20	−0.293217
$540$	0	0
$541$	14008.2	1.11323	0.556616	−	0.830770i	$-0.312100\pi$
$541$	14008.2	1.11323	0.556616	+	0.830770i	$0.312100\pi$
$542$	0	0
$543$	6367.72	0.503251
$544$	0	0
$545$	15513.9	1.21934
$546$	0	0
$547$	4949.45	0.386879	0.193440	−	0.981112i	$-0.438036\pi$
$547$	4949.45	0.386879	0.193440	+	0.981112i	$0.438036\pi$
$548$	0	0
$549$	−26487.0	−2.05909
$550$	0	0
$551$	−3497.35	−0.270404
$552$	0	0
$553$	3975.60	0.305714
$554$	0	0
$555$	1548.11	0.118403
$556$	0	0
$557$	3801.58	0.289188	0.144594	−	0.989491i	$-0.453812\pi$
$557$	3801.58	0.289188	0.144594	+	0.989491i	$0.453812\pi$
$558$	0	0
$559$	−309.282	−0.0234011
$560$	0	0
$561$	−3594.40	−0.270510
$562$	0	0
$563$	9900.11	0.741101	0.370551	−	0.928812i	$-0.379169\pi$
$563$	9900.11	0.741101	0.370551	+	0.928812i	$0.379169\pi$
$564$	0	0
$565$	−4383.85	−0.326425
$566$	0	0
$567$	−1263.86	−0.0936107
$568$	0	0
$569$	5329.16	0.392636	0.196318	−	0.980540i	$-0.437102\pi$
$569$	5329.16	0.392636	0.196318	+	0.980540i	$0.437102\pi$
$570$	0	0
$571$	16962.6	1.24319	0.621597	−	0.783337i	$-0.286484\pi$
$571$	16962.6	1.24319	0.621597	+	0.783337i	$0.286484\pi$
$572$	0	0
$573$	13622.6	0.993181
$574$	0	0
$575$	−10658.0	−0.772989
$576$	0	0
$577$	−15487.0	−1.11738	−0.558692	−	0.829375i	$-0.688697\pi$
$577$	−15487.0	−1.11738	−0.558692	+	0.829375i	$0.688697\pi$
$578$	0	0
$579$	10625.3	0.762643
$580$	0	0
$581$	−1357.26	−0.0969169
$582$	0	0
$583$	3772.94	0.268026
$584$	0	0
$585$	2854.72	0.201757
$586$	0	0
$587$	−11084.2	−0.779373	−0.389686	−	0.920948i	$-0.627417\pi$
$587$	−11084.2	−0.779373	−0.389686	+	0.920948i	$0.627417\pi$
$588$	0	0
$589$	−4407.22	−0.308313
$590$	0	0
$591$	27894.0	1.94147
$592$	0	0
$593$	4349.68	0.301214	0.150607	−	0.988594i	$-0.451877\pi$
$593$	4349.68	0.301214	0.150607	+	0.988594i	$0.451877\pi$
$594$	0	0
$595$	1880.90	0.129596
$596$	0	0
$597$	6530.40	0.447691
$598$	0	0
$599$	13183.9	0.899299	0.449650	−	0.893205i	$-0.351549\pi$
$599$	13183.9	0.899299	0.449650	+	0.893205i	$0.351549\pi$
$600$	0	0
$601$	−18765.0	−1.27361	−0.636806	−	0.771024i	$-0.719745\pi$
$601$	−18765.0	−1.27361	−0.636806	+	0.771024i	$0.719745\pi$
$602$	0	0
$603$	−12275.8	−0.829034
$604$	0	0
$605$	−1797.63	−0.120800
$606$	0	0
$607$	21871.4	1.46249	0.731244	−	0.682116i	$-0.238940\pi$
$607$	21871.4	1.46249	0.731244	+	0.682116i	$0.238940\pi$
$608$	0	0
$609$	608.720	0.0405034
$610$	0	0
$611$	1842.67	0.122007
$612$	0	0
$613$	3527.85	0.232445	0.116222	−	0.993223i	$-0.462921\pi$
$613$	3527.85	0.232445	0.116222	+	0.993223i	$0.462921\pi$
$614$	0	0
$615$	−30750.2	−2.01621
$616$	0	0
$617$	−22728.1	−1.48298	−0.741490	−	0.670963i	$-0.765881\pi$
$617$	−22728.1	−1.48298	−0.741490	+	0.670963i	$0.765881\pi$
$618$	0	0
$619$	21443.3	1.39237	0.696187	−	0.717861i	$-0.254879\pi$
$619$	21443.3	1.39237	0.696187	+	0.717861i	$0.254879\pi$
$620$	0	0
$621$	−7818.75	−0.505243
$622$	0	0
$623$	−4574.25	−0.294163
$624$	0	0
$625$	−18428.2	−1.17940
$626$	0	0
$627$	−12202.7	−0.777240
$628$	0	0
$629$	541.718	0.0343398
$630$	0	0
$631$	21532.0	1.35844	0.679219	−	0.733936i	$-0.262319\pi$
$631$	21532.0	1.35844	0.679219	+	0.733936i	$0.262319\pi$
$632$	0	0
$633$	851.038	0.0534372
$634$	0	0
$635$	19574.9	1.22332
$636$	0	0
$637$	1787.56	0.111187
$638$	0	0
$639$	−7435.33	−0.460309
$640$	0	0
$641$	20148.3	1.24151	0.620756	−	0.784004i	$-0.286826\pi$
$641$	20148.3	1.24151	0.620756	+	0.784004i	$0.286826\pi$
$642$	0	0
$643$	−28869.7	−1.77062	−0.885310	−	0.465000i	$-0.846054\pi$
$643$	−28869.7	−1.77062	−0.885310	+	0.465000i	$0.846054\pi$
$644$	0	0
$645$	−6797.68	−0.414974
$646$	0	0
$647$	−1590.02	−0.0966155	−0.0483077	−	0.998833i	$-0.515383\pi$
$647$	−1590.02	−0.0966155	−0.0483077	+	0.998833i	$0.515383\pi$
$648$	0	0
$649$	−972.062	−0.0587932
$650$	0	0
$651$	767.083	0.0461818
$652$	0	0
$653$	−20028.1	−1.20024	−0.600122	−	0.799909i	$-0.704881\pi$
$653$	−20028.1	−1.20024	−0.600122	+	0.799909i	$0.704881\pi$
$654$	0	0
$655$	−23780.8	−1.41862
$656$	0	0
$657$	−36236.5	−2.15178
$658$	0	0
$659$	10520.7	0.621897	0.310948	−	0.950427i	$-0.399353\pi$
$659$	10520.7	0.621897	0.310948	+	0.950427i	$0.399353\pi$
$660$	0	0
$661$	−3295.83	−0.193938	−0.0969690	−	0.995287i	$-0.530915\pi$
$661$	−3295.83	−0.193938	−0.0969690	+	0.995287i	$0.530915\pi$
$662$	0	0
$663$	1751.12	0.102576
$664$	0	0
$665$	6385.51	0.372360
$666$	0	0
$667$	−2783.27	−0.161572
$668$	0	0
$669$	−31187.0	−1.80233
$670$	0	0
$671$	−8125.67	−0.467493
$672$	0	0
$673$	−1187.64	−0.0680239	−0.0340119	−	0.999421i	$-0.510828\pi$
$673$	−1187.64	−0.0680239	−0.0340119	+	0.999421i	$0.510828\pi$
$674$	0	0
$675$	6720.51	0.383219
$676$	0	0
$677$	−13221.4	−0.750574	−0.375287	−	0.926909i	$-0.622456\pi$
$677$	−13221.4	−0.750574	−0.375287	+	0.926909i	$0.622456\pi$
$678$	0	0
$679$	4135.91	0.233758
$680$	0	0
$681$	14048.0	0.790485
$682$	0	0
$683$	13831.4	0.774882	0.387441	−	0.921894i	$-0.373359\pi$
$683$	13831.4	0.774882	0.387441	+	0.921894i	$0.373359\pi$
$684$	0	0
$685$	−23943.7	−1.33554
$686$	0	0
$687$	−15185.4	−0.843320
$688$	0	0
$689$	−1838.10	−0.101635
$690$	0	0
$691$	9817.07	0.540462	0.270231	−	0.962796i	$-0.412900\pi$
$691$	9817.07	0.540462	0.270231	+	0.962796i	$0.412900\pi$
$692$	0	0
$693$	1211.58	0.0664128
$694$	0	0
$695$	−473.354	−0.0258350
$696$	0	0
$697$	−10760.2	−0.584750
$698$	0	0
$699$	34854.9	1.88603
$700$	0	0
$701$	−29949.8	−1.61368	−0.806838	−	0.590773i	$-0.798823\pi$
$701$	−29949.8	−1.61368	−0.806838	+	0.590773i	$0.798823\pi$
$702$	0	0
$703$	1839.09	0.0986667
$704$	0	0
$705$	40499.8	2.16356
$706$	0	0
$707$	495.976	0.0263835
$708$	0	0
$709$	−11307.5	−0.598959	−0.299479	−	0.954103i	$-0.596813\pi$
$709$	−11307.5	−0.598959	−0.299479	+	0.954103i	$0.596813\pi$
$710$	0	0
$711$	46406.3	2.44778
$712$	0	0
$713$	−3507.36	−0.184224
$714$	0	0
$715$	875.768	0.0458068
$716$	0	0
$717$	−32382.7	−1.68669
$718$	0	0
$719$	−32623.4	−1.69214	−0.846070	−	0.533071i	$-0.821038\pi$
$719$	−32623.4	−1.69214	−0.846070	+	0.533071i	$0.821038\pi$
$720$	0	0
$721$	−106.773	−0.00551518
$722$	0	0
$723$	30988.0	1.59399
$724$	0	0
$725$	2392.33	0.122550
$726$	0	0
$727$	−502.545	−0.0256373	−0.0128187	−	0.999918i	$-0.504080\pi$
$727$	−502.545	−0.0256373	−0.0128187	+	0.999918i	$0.504080\pi$
$728$	0	0
$729$	−29783.2	−1.51314
$730$	0	0
$731$	−2378.66	−0.120353
$732$	0	0
$733$	−8631.37	−0.434935	−0.217467	−	0.976068i	$-0.569780\pi$
$733$	−8631.37	−0.434935	−0.217467	+	0.976068i	$0.569780\pi$
$734$	0	0
$735$	39288.7	1.97168
$736$	0	0
$737$	−3765.95	−0.188223
$738$	0	0
$739$	18357.5	0.913792	0.456896	−	0.889520i	$-0.348961\pi$
$739$	18357.5	0.913792	0.456896	+	0.889520i	$0.348961\pi$
$740$	0	0
$741$	5944.93	0.294726
$742$	0	0
$743$	11182.6	0.552155	0.276078	−	0.961135i	$-0.410965\pi$
$743$	11182.6	0.552155	0.276078	+	0.961135i	$0.410965\pi$
$744$	0	0
$745$	−36076.4	−1.77415
$746$	0	0
$747$	−15843.0	−0.775991
$748$	0	0
$749$	−2556.29	−0.124706
$750$	0	0
$751$	16733.4	0.813063	0.406531	−	0.913637i	$-0.366738\pi$
$751$	16733.4	0.813063	0.406531	+	0.913637i	$0.366738\pi$
$752$	0	0
$753$	−8680.53	−0.420101
$754$	0	0
$755$	38280.2	1.84524
$756$	0	0
$757$	24402.4	1.17163	0.585813	−	0.810446i	$-0.300775\pi$
$757$	24402.4	1.17163	0.585813	+	0.810446i	$0.300775\pi$
$758$	0	0
$759$	−9711.19	−0.464419
$760$	0	0
$761$	8469.33	0.403434	0.201717	−	0.979444i	$-0.435348\pi$
$761$	8469.33	0.403434	0.201717	+	0.979444i	$0.435348\pi$
$762$	0	0
$763$	−3207.74	−0.152199
$764$	0	0
$765$	21955.3	1.03764
$766$	0	0
$767$	473.570	0.0222941
$768$	0	0
$769$	32834.7	1.53973	0.769864	−	0.638208i	$-0.220324\pi$
$769$	32834.7	1.53973	0.769864	+	0.638208i	$0.220324\pi$
$770$	0	0
$771$	6209.20	0.290038
$772$	0	0
$773$	35571.4	1.65513	0.827564	−	0.561371i	$-0.189726\pi$
$773$	35571.4	1.65513	0.827564	+	0.561371i	$0.189726\pi$
$774$	0	0
$775$	3014.71	0.139731
$776$	0	0
$777$	−320.097	−0.0147792
$778$	0	0
$779$	−36530.0	−1.68013
$780$	0	0
$781$	−2281.01	−0.104508
$782$	0	0
$783$	1755.02	0.0801014
$784$	0	0
$785$	36783.6	1.67244
$786$	0	0
$787$	−15729.6	−0.712452	−0.356226	−	0.934400i	$-0.615937\pi$
$787$	−15729.6	−0.712452	−0.356226	+	0.934400i	$0.615937\pi$
$788$	0	0
$789$	48996.7	2.21081
$790$	0	0
$791$	906.431	0.0407446
$792$	0	0
$793$	3958.67	0.177272
$794$	0	0
$795$	−40399.5	−1.80229
$796$	0	0
$797$	−7888.07	−0.350577	−0.175288	−	0.984517i	$-0.556086\pi$
$797$	−7888.07	−0.350577	−0.175288	+	0.984517i	$0.556086\pi$
$798$	0	0
$799$	14171.8	0.627485
$800$	0	0
$801$	−53394.2	−2.35530
$802$	0	0
$803$	−11116.6	−0.488538
$804$	0	0
$805$	5081.73	0.222494
$806$	0	0
$807$	−7824.86	−0.341323
$808$	0	0
$809$	5896.97	0.256275	0.128138	−	0.991756i	$-0.459100\pi$
$809$	5896.97	0.256275	0.128138	+	0.991756i	$0.459100\pi$
$810$	0	0
$811$	−14197.9	−0.614744	−0.307372	−	0.951589i	$-0.599450\pi$
$811$	−14197.9	−0.614744	−0.307372	+	0.951589i	$0.599450\pi$
$812$	0	0
$813$	36287.3	1.56538
$814$	0	0
$815$	−40485.3	−1.74005
$816$	0	0
$817$	−8075.35	−0.345803
$818$	0	0
$819$	−590.258	−0.0251835
$820$	0	0
$821$	19841.7	0.843459	0.421729	−	0.906722i	$-0.361423\pi$
$821$	19841.7	0.843459	0.421729	+	0.906722i	$0.361423\pi$
$822$	0	0
$823$	−28202.2	−1.19449	−0.597246	−	0.802058i	$-0.703738\pi$
$823$	−28202.2	−1.19449	−0.597246	+	0.802058i	$0.703738\pi$
$824$	0	0
$825$	8347.14	0.352255
$826$	0	0
$827$	−34031.0	−1.43092	−0.715462	−	0.698651i	$-0.753784\pi$
$827$	−34031.0	−1.43092	−0.715462	+	0.698651i	$0.753784\pi$
$828$	0	0
$829$	−4931.55	−0.206610	−0.103305	−	0.994650i	$-0.532942\pi$
$829$	−4931.55	−0.206610	−0.103305	+	0.994650i	$0.532942\pi$
$830$	0	0
$831$	−4501.92	−0.187930
$832$	0	0
$833$	13748.0	0.571836
$834$	0	0
$835$	−40666.4	−1.68541
$836$	0	0
$837$	2211.60	0.0913312
$838$	0	0
$839$	−38189.8	−1.57146	−0.785731	−	0.618568i	$-0.787713\pi$
$839$	−38189.8	−1.57146	−0.785731	+	0.618568i	$0.787713\pi$
$840$	0	0
$841$	−23764.3	−0.974384
$842$	0	0
$843$	42106.8	1.72033
$844$	0	0
$845$	32212.9	1.31143
$846$	0	0
$847$	371.687	0.0150783
$848$	0	0
$849$	37488.0	1.51541
$850$	0	0
$851$	1463.59	0.0589556
$852$	0	0
$853$	−42966.8	−1.72469	−0.862343	−	0.506325i	$-0.831003\pi$
$853$	−42966.8	−1.72469	−0.862343	+	0.506325i	$0.831003\pi$
$854$	0	0
$855$	74536.6	2.98140
$856$	0	0
$857$	−17281.5	−0.688828	−0.344414	−	0.938818i	$-0.611922\pi$
$857$	−17281.5	−0.688828	−0.344414	+	0.938818i	$0.611922\pi$
$858$	0	0
$859$	−9316.75	−0.370062	−0.185031	−	0.982733i	$-0.559239\pi$
$859$	−9316.75	−0.370062	−0.185031	+	0.982733i	$0.559239\pi$
$860$	0	0
$861$	6358.10	0.251665
$862$	0	0
$863$	−9647.65	−0.380544	−0.190272	−	0.981731i	$-0.560937\pi$
$863$	−9647.65	−0.380544	−0.190272	+	0.981731i	$0.560937\pi$
$864$	0	0
$865$	34279.9	1.34746
$866$	0	0
$867$	−25483.6	−0.998232
$868$	0	0
$869$	14236.5	0.555742
$870$	0	0
$871$	1834.70	0.0713735
$872$	0	0
$873$	48277.6	1.87165
$874$	0	0
$875$	1336.55	0.0516383
$876$	0	0
$877$	−19728.7	−0.759624	−0.379812	−	0.925064i	$-0.624011\pi$
$877$	−19728.7	−0.759624	−0.379812	+	0.925064i	$0.624011\pi$
$878$	0	0
$879$	−18464.1	−0.708509
$880$	0	0
$881$	19473.9	0.744712	0.372356	−	0.928090i	$-0.378550\pi$
$881$	19473.9	0.744712	0.372356	+	0.928090i	$0.378550\pi$
$882$	0	0
$883$	−49092.4	−1.87100	−0.935499	−	0.353329i	$-0.885050\pi$
$883$	−49092.4	−1.87100	−0.935499	+	0.353329i	$0.885050\pi$
$884$	0	0
$885$	10408.5	0.395344
$886$	0	0
$887$	9292.86	0.351774	0.175887	−	0.984410i	$-0.443721\pi$
$887$	9292.86	0.351774	0.175887	+	0.984410i	$0.443721\pi$
$888$	0	0
$889$	−4047.41	−0.152695
$890$	0	0
$891$	−4525.85	−0.170170
$892$	0	0
$893$	48112.0	1.80292
$894$	0	0
$895$	−19493.9	−0.728055
$896$	0	0
$897$	4731.10	0.176106
$898$	0	0
$899$	787.273	0.0292069
$900$	0	0
$901$	−14136.7	−0.522709
$902$	0	0
$903$	1405.53	0.0517974
$904$	0	0
$905$	−11932.3	−0.438279
$906$	0	0
$907$	−37688.7	−1.37975	−0.689875	−	0.723928i	$-0.742335\pi$
$907$	−37688.7	−1.37975	−0.689875	+	0.723928i	$0.742335\pi$
$908$	0	0
$909$	5789.42	0.211246
$910$	0	0
$911$	33049.6	1.20196	0.600979	−	0.799265i	$-0.294778\pi$
$911$	33049.6	1.20196	0.600979	+	0.799265i	$0.294778\pi$
$912$	0	0
$913$	−4860.31	−0.176180
$914$	0	0
$915$	87007.2	3.14357
$916$	0	0
$917$	4917.06	0.177073
$918$	0	0
$919$	−23148.0	−0.830883	−0.415442	−	0.909620i	$-0.636373\pi$
$919$	−23148.0	−0.830883	−0.415442	+	0.909620i	$0.636373\pi$
$920$	0	0
$921$	13304.1	0.475986
$922$	0	0
$923$	1111.26	0.0396290
$924$	0	0
$925$	−1258.01	−0.0447169
$926$	0	0
$927$	−1246.34	−0.0441588
$928$	0	0
$929$	−23177.9	−0.818561	−0.409280	−	0.912409i	$-0.634220\pi$
$929$	−23177.9	−0.818561	−0.409280	+	0.912409i	$0.634220\pi$
$930$	0	0
$931$	46673.3	1.64302
$932$	0	0
$933$	28325.1	0.993916
$934$	0	0
$935$	6735.44	0.235585
$936$	0	0
$937$	−34574.7	−1.20545	−0.602724	−	0.797950i	$-0.705918\pi$
$937$	−34574.7	−1.20545	−0.602724	+	0.797950i	$0.705918\pi$
$938$	0	0
$939$	56959.1	1.97954
$940$	0	0
$941$	−41831.2	−1.44916	−0.724578	−	0.689192i	$-0.757966\pi$
$941$	−41831.2	−1.44916	−0.724578	+	0.689192i	$0.757966\pi$
$942$	0	0
$943$	−29071.3	−1.00392
$944$	0	0
$945$	−3204.34	−0.110304
$946$	0	0
$947$	−27231.2	−0.934419	−0.467209	−	0.884147i	$-0.654741\pi$
$947$	−27231.2	−0.934419	−0.467209	+	0.884147i	$0.654741\pi$
$948$	0	0
$949$	5415.79	0.185252
$950$	0	0
$951$	124.534	0.00424635
$952$	0	0
$953$	40939.4	1.39156	0.695781	−	0.718254i	$-0.255058\pi$
$953$	40939.4	1.39156	0.695781	+	0.718254i	$0.255058\pi$
$954$	0	0
$955$	−25527.0	−0.864956
$956$	0	0
$957$	2179.81	0.0736292
$958$	0	0
$959$	4950.74	0.166703
$960$	0	0
$961$	−28798.9	−0.966698
$962$	0	0
$963$	−29839.0	−0.998491
$964$	0	0
$965$	−19910.3	−0.664182
$966$	0	0
$967$	−46173.1	−1.53550	−0.767750	−	0.640750i	$-0.778624\pi$
$967$	−46173.1	−1.53550	−0.767750	+	0.640750i	$0.778624\pi$
$968$	0	0
$969$	45721.8	1.51579
$970$	0	0
$971$	5153.91	0.170337	0.0851683	−	0.996367i	$-0.472857\pi$
$971$	5153.91	0.170337	0.0851683	+	0.996367i	$0.472857\pi$
$972$	0	0
$973$	97.8734	0.00322474
$974$	0	0
$975$	−4066.56	−0.133574
$976$	0	0
$977$	9692.13	0.317378	0.158689	−	0.987329i	$-0.449273\pi$
$977$	9692.13	0.317378	0.158689	+	0.987329i	$0.449273\pi$
$978$	0	0
$979$	−16380.2	−0.534744
$980$	0	0
$981$	−37443.3	−1.21863
$982$	0	0
$983$	32915.7	1.06800	0.534002	−	0.845483i	$-0.320687\pi$
$983$	32915.7	1.06800	0.534002	+	0.845483i	$0.320687\pi$
$984$	0	0
$985$	−52269.7	−1.69081
$986$	0	0
$987$	−8373.97	−0.270057
$988$	0	0
$989$	−6426.54	−0.206625
$990$	0	0
$991$	29477.9	0.944901	0.472451	−	0.881357i	$-0.343370\pi$
$991$	29477.9	0.944901	0.472451	+	0.881357i	$0.343370\pi$
$992$	0	0
$993$	10456.9	0.334180
$994$	0	0
$995$	−12237.1	−0.389892
$996$	0	0
$997$	31944.4	1.01473	0.507366	−	0.861731i	$-0.330619\pi$
$997$	31944.4	1.01473	0.507366	+	0.861731i	$0.330619\pi$
$998$	0	0
$999$	−922.883	−0.0292279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.4.a.p.1.2		2
4.3	odd	2		704.4.a.n.1.1		2
8.3	odd	2		176.4.a.i.1.2		2
8.5	even	2		11.4.a.a.1.2	✓	2
24.5	odd	2		99.4.a.c.1.1		2
24.11	even	2		1584.4.a.bc.1.1		2
40.13	odd	4		275.4.b.c.199.1		4
40.29	even	2		275.4.a.b.1.1		2
40.37	odd	4		275.4.b.c.199.4		4
56.13	odd	2		539.4.a.e.1.2		2
88.5	even	10		121.4.c.c.3.1		8
88.13	odd	10		121.4.c.f.81.2		8
88.21	odd	2		121.4.a.c.1.1		2
88.29	odd	10		121.4.c.f.27.1		8
88.37	even	10		121.4.c.c.27.2		8
88.43	even	2		1936.4.a.w.1.2		2
88.53	even	10		121.4.c.c.81.1		8
88.61	odd	10		121.4.c.f.3.2		8
88.69	even	10		121.4.c.c.9.2		8
88.85	odd	10		121.4.c.f.9.1		8
104.77	even	2		1859.4.a.a.1.1		2
120.29	odd	2		2475.4.a.q.1.2		2
264.197	even	2		1089.4.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.2	✓	2	8.5	even	2
99.4.a.c.1.1		2	24.5	odd	2
121.4.a.c.1.1		2	88.21	odd	2
121.4.c.c.3.1		8	88.5	even	10
121.4.c.c.9.2		8	88.69	even	10
121.4.c.c.27.2		8	88.37	even	10
121.4.c.c.81.1		8	88.53	even	10
121.4.c.f.3.2		8	88.61	odd	10
121.4.c.f.9.1		8	88.85	odd	10
121.4.c.f.27.1		8	88.29	odd	10
121.4.c.f.81.2		8	88.13	odd	10
176.4.a.i.1.2		2	8.3	odd	2
275.4.a.b.1.1		2	40.29	even	2
275.4.b.c.199.1		4	40.13	odd	4
275.4.b.c.199.4		4	40.37	odd	4
539.4.a.e.1.2		2	56.13	odd	2
704.4.a.n.1.1		2	4.3	odd	2
704.4.a.p.1.2		2	1.1	even	1	trivial
1089.4.a.v.1.2		2	264.197	even	2
1584.4.a.bc.1.1		2	24.11	even	2
1859.4.a.a.1.1		2	104.77	even	2
1936.4.a.w.1.2		2	88.43	even	2
2475.4.a.q.1.2		2	120.29	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

\(f(q)\)	\(=\)	\(q+7.92820 q^{3} -14.8564 q^{5} +3.07180 q^{7} +35.8564 q^{9} +11.0000 q^{11} -5.35898 q^{13} -117.785 q^{15} -41.2154 q^{17} -139.923 q^{19} +24.3538 q^{21} -111.354 q^{23} +95.7128 q^{25} +70.2154 q^{27} +24.9948 q^{29} +31.4974 q^{31} +87.2102 q^{33} -45.6359 q^{35} -13.1436 q^{37} -42.4871 q^{39} +261.072 q^{41} +57.7128 q^{43} -532.697 q^{45} -343.846 q^{47} -333.564 q^{49} -326.764 q^{51} +342.995 q^{53} -163.420 q^{55} -1109.34 q^{57} -88.3693 q^{59} -738.697 q^{61} +110.144 q^{63} +79.6152 q^{65} -342.359 q^{67} -882.836 q^{69} -207.364 q^{71} -1010.60 q^{73} +758.831 q^{75} +33.7898 q^{77} +1294.23 q^{79} -411.441 q^{81} -441.846 q^{83} +612.313 q^{85} +198.164 q^{87} -1489.11 q^{89} -16.4617 q^{91} +249.718 q^{93} +2078.75 q^{95} +1346.42 q^{97} +394.420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 20 q^{7} + 44 q^{9} + 22 q^{11} - 80 q^{13} - 194 q^{15} - 124 q^{17} - 72 q^{19} - 76 q^{21} - 98 q^{23} + 136 q^{25} + 182 q^{27} - 144 q^{29} - 34 q^{31} + 22 q^{33} + 172 q^{35}+ \cdots + 484 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 20 * q^7 + 44 * q^9 + 22 * q^11 - 80 * q^13 - 194 * q^15 - 124 * q^17 - 72 * q^19 - 76 * q^21 - 98 * q^23 + 136 * q^25 + 182 * q^27 - 144 * q^29 - 34 * q^31 + 22 * q^33 + 172 * q^35 - 54 * q^37 + 400 * q^39 + 536 * q^41 + 60 * q^43 - 428 * q^45 - 272 * q^47 - 390 * q^49 + 164 * q^51 + 492 * q^53 - 22 * q^55 - 1512 * q^57 - 634 * q^59 - 840 * q^61 + 248 * q^63 - 880 * q^65 - 754 * q^67 - 962 * q^69 - 678 * q^71 - 400 * q^73 + 520 * q^75 + 220 * q^77 + 316 * q^79 - 1294 * q^81 - 468 * q^83 - 452 * q^85 + 1200 * q^87 - 1842 * q^89 - 1280 * q^91 + 638 * q^93 + 2952 * q^95 + 2194 * q^97 + 484 * q^99

Introduction

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