LMFDB - Embedded newform 1352.4.a.k.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1352,4,Mod(1,1352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1352.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1352.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,3,0,-9] 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:	$79.7705823278$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 83x^{3} + 112x^{2} + 804x + 144 $ x^5 - 2x^4 - 83x^3 + 112x^2 + 804x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-8.27625$ of defining polynomial
Character	$\chi$	$=$	1352.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.27625 q^{3} -2.27999 q^{5} +28.5449 q^{7} +59.0489 q^{9} +0.604833 q^{11} -21.1498 q^{15} +41.0109 q^{17} +129.485 q^{19} +264.790 q^{21} -73.8974 q^{23} -119.802 q^{25} +297.293 q^{27} -214.532 q^{29} +126.175 q^{31} +5.61058 q^{33} -65.0822 q^{35} +309.470 q^{37} +88.1488 q^{41} +245.304 q^{43} -134.631 q^{45} -68.5441 q^{47} +471.814 q^{49} +380.428 q^{51} -613.275 q^{53} -1.37901 q^{55} +1201.14 q^{57} -840.276 q^{59} +587.808 q^{61} +1685.55 q^{63} -606.554 q^{67} -685.491 q^{69} -507.627 q^{71} +177.127 q^{73} -1111.31 q^{75} +17.2649 q^{77} +143.122 q^{79} +1163.45 q^{81} +773.287 q^{83} -93.5046 q^{85} -1990.06 q^{87} +1402.90 q^{89} +1170.43 q^{93} -295.226 q^{95} -468.225 q^{97} +35.7147 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 9 q^{5} + 17 q^{7} + 36 q^{9} + 4 q^{11} + 19 q^{15} - 29 q^{17} + 46 q^{19} + 113 q^{21} - 90 q^{23} - 14 q^{25} + 177 q^{27} - 196 q^{29} + 84 q^{31} + 352 q^{33} + 77 q^{35} + 327 q^{37}+ \cdots - 2844 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 9 * q^5 + 17 * q^7 + 36 * q^9 + 4 * q^11 + 19 * q^15 - 29 * q^17 + 46 * q^19 + 113 * q^21 - 90 * q^23 - 14 * q^25 + 177 * q^27 - 196 * q^29 + 84 * q^31 + 352 * q^33 + 77 * q^35 + 327 * q^37 + 480 * q^41 + 117 * q^43 - 198 * q^45 - 647 * q^47 + 64 * q^49 + 255 * q^51 - 622 * q^53 - 288 * q^55 + 538 * q^57 + 610 * q^59 - 28 * q^61 + 1906 * q^63 + 1216 * q^67 - 874 * q^69 - 1381 * q^71 - 906 * q^73 - 236 * q^75 + 152 * q^77 + 954 * q^79 + 1141 * q^81 + 586 * q^83 + 2995 * q^85 - 32 * q^87 + 2618 * q^89 + 3952 * q^93 + 1170 * q^95 - 1188 * q^97 - 2844 * 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
$3$	9.27625	1.78522	0.892608	−	0.450834i	$-0.148873\pi$
$3$	9.27625	1.78522	0.892608	+	0.450834i	$0.148873\pi$
$4$	0	0
$5$	−2.27999	−0.203929	−0.101964	−	0.994788i	$-0.532513\pi$
$5$	−2.27999	−0.203929	−0.101964	+	0.994788i	$0.532513\pi$
$6$	0	0
$7$	28.5449	1.54128	0.770641	−	0.637269i	$-0.219936\pi$
$7$	28.5449	1.54128	0.770641	+	0.637269i	$0.219936\pi$
$8$	0	0
$9$	59.0489	2.18699
$10$	0	0
$11$	0.604833	0.0165785	0.00828927	−	0.999966i	$-0.497361\pi$
$11$	0.604833	0.0165785	0.00828927	+	0.999966i	$0.497361\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−21.1498	−0.364057
$16$	0	0
$17$	41.0109	0.585095	0.292548	−	0.956251i	$-0.405497\pi$
$17$	41.0109	0.585095	0.292548	+	0.956251i	$0.405497\pi$
$18$	0	0
$19$	129.485	1.56347	0.781737	−	0.623609i	$-0.214334\pi$
$19$	129.485	1.56347	0.781737	+	0.623609i	$0.214334\pi$
$20$	0	0
$21$	264.790	2.75152
$22$	0	0
$23$	−73.8974	−0.669943	−0.334971	−	0.942228i	$-0.608727\pi$
$23$	−73.8974	−0.669943	−0.334971	+	0.942228i	$0.608727\pi$
$24$	0	0
$25$	−119.802	−0.958413
$26$	0	0
$27$	297.293	2.11904
$28$	0	0
$29$	−214.532	−1.37371	−0.686856	−	0.726793i	$-0.741010\pi$
$29$	−214.532	−1.37371	−0.686856	+	0.726793i	$0.741010\pi$
$30$	0	0
$31$	126.175	0.731023	0.365512	−	0.930807i	$-0.380894\pi$
$31$	126.175	0.731023	0.365512	+	0.930807i	$0.380894\pi$
$32$	0	0
$33$	5.61058	0.0295963
$34$	0	0
$35$	−65.0822	−0.314312
$36$	0	0
$37$	309.470	1.37504	0.687522	−	0.726164i	$-0.258699\pi$
$37$	309.470	1.37504	0.687522	+	0.726164i	$0.258699\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	88.1488	0.335769	0.167885	−	0.985807i	$-0.446306\pi$
$41$	88.1488	0.335769	0.167885	+	0.985807i	$0.446306\pi$
$42$	0	0
$43$	245.304	0.869967	0.434983	−	0.900438i	$-0.356754\pi$
$43$	245.304	0.869967	0.434983	+	0.900438i	$0.356754\pi$
$44$	0	0
$45$	−134.631	−0.445991
$46$	0	0
$47$	−68.5441	−0.212727	−0.106364	−	0.994327i	$-0.533921\pi$
$47$	−68.5441	−0.212727	−0.106364	+	0.994327i	$0.533921\pi$
$48$	0	0
$49$	471.814	1.37555
$50$	0	0
$51$	380.428	1.04452
$52$	0	0
$53$	−613.275	−1.58943	−0.794716	−	0.606982i	$-0.792380\pi$
$53$	−613.275	−1.58943	−0.794716	+	0.606982i	$0.792380\pi$
$54$	0	0
$55$	−1.37901	−0.00338084
$56$	0	0
$57$	1201.14	2.79114
$58$	0	0
$59$	−840.276	−1.85415	−0.927073	−	0.374882i	$-0.877683\pi$
$59$	−840.276	−1.85415	−0.927073	+	0.374882i	$0.877683\pi$
$60$	0	0
$61$	587.808	1.23379	0.616894	−	0.787046i	$-0.288391\pi$
$61$	587.808	1.23379	0.616894	+	0.787046i	$0.288391\pi$
$62$	0	0
$63$	1685.55	3.37078
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−606.554	−1.10601	−0.553003	−	0.833180i	$-0.686518\pi$
$67$	−606.554	−1.10601	−0.553003	+	0.833180i	$0.686518\pi$
$68$	0	0
$69$	−685.491	−1.19599
$70$	0	0
$71$	−507.627	−0.848510	−0.424255	−	0.905543i	$-0.639464\pi$
$71$	−507.627	−0.848510	−0.424255	+	0.905543i	$0.639464\pi$
$72$	0	0
$73$	177.127	0.283989	0.141994	−	0.989867i	$-0.454649\pi$
$73$	177.127	0.283989	0.141994	+	0.989867i	$0.454649\pi$
$74$	0	0
$75$	−1111.31	−1.71097
$76$	0	0
$77$	17.2649	0.0255522
$78$	0	0
$79$	143.122	0.203829	0.101914	−	0.994793i	$-0.467503\pi$
$79$	143.122	0.203829	0.101914	+	0.994793i	$0.467503\pi$
$80$	0	0
$81$	1163.45	1.59595
$82$	0	0
$83$	773.287	1.02264	0.511321	−	0.859390i	$-0.329156\pi$
$83$	773.287	1.02264	0.511321	+	0.859390i	$0.329156\pi$
$84$	0	0
$85$	−93.5046	−0.119318
$86$	0	0
$87$	−1990.06	−2.45237
$88$	0	0
$89$	1402.90	1.67086	0.835432	−	0.549593i	$-0.185217\pi$
$89$	1402.90	1.67086	0.835432	+	0.549593i	$0.185217\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1170.43	1.30503
$94$	0	0
$95$	−295.226	−0.318837
$96$	0	0
$97$	−468.225	−0.490114	−0.245057	−	0.969509i	$-0.578807\pi$
$97$	−468.225	−0.490114	−0.245057	+	0.969509i	$0.578807\pi$
$98$	0	0
$99$	35.7147	0.0362572
$100$	0	0
$101$	399.959	0.394033	0.197017	−	0.980400i	$-0.436875\pi$
$101$	399.959	0.394033	0.197017	+	0.980400i	$0.436875\pi$
$102$	0	0
$103$	−1404.11	−1.34321	−0.671607	−	0.740907i	$-0.734396\pi$
$103$	−1404.11	−1.34321	−0.671607	+	0.740907i	$0.734396\pi$
$104$	0	0
$105$	−603.719	−0.561114
$106$	0	0
$107$	−1596.54	−1.44246	−0.721232	−	0.692694i	$-0.756424\pi$
$107$	−1596.54	−1.44246	−0.721232	+	0.692694i	$0.756424\pi$
$108$	0	0
$109$	−247.807	−0.217758	−0.108879	−	0.994055i	$-0.534726\pi$
$109$	−247.807	−0.217758	−0.108879	+	0.994055i	$0.534726\pi$
$110$	0	0
$111$	2870.73	2.45475
$112$	0	0
$113$	423.360	0.352445	0.176223	−	0.984350i	$-0.443612\pi$
$113$	423.360	0.352445	0.176223	+	0.984350i	$0.443612\pi$
$114$	0	0
$115$	168.486	0.136621
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1170.66	0.901797
$120$	0	0
$121$	−1330.63	−0.999725
$122$	0	0
$123$	817.691	0.599420
$124$	0	0
$125$	558.146	0.399377
$126$	0	0
$127$	640.855	0.447769	0.223884	−	0.974616i	$-0.428126\pi$
$127$	640.855	0.447769	0.223884	+	0.974616i	$0.428126\pi$
$128$	0	0
$129$	2275.51	1.55308
$130$	0	0
$131$	809.348	0.539794	0.269897	−	0.962889i	$-0.413010\pi$
$131$	809.348	0.539794	0.269897	+	0.962889i	$0.413010\pi$
$132$	0	0
$133$	3696.15	2.40975
$134$	0	0
$135$	−677.826	−0.432133
$136$	0	0
$137$	1666.32	1.03915	0.519573	−	0.854426i	$-0.326091\pi$
$137$	1666.32	1.03915	0.519573	+	0.854426i	$0.326091\pi$
$138$	0	0
$139$	2473.29	1.50922	0.754609	−	0.656174i	$-0.227826\pi$
$139$	2473.29	1.50922	0.754609	+	0.656174i	$0.227826\pi$
$140$	0	0
$141$	−635.833	−0.379764
$142$	0	0
$143$	0	0
$144$	0	0
$145$	489.132	0.280139
$146$	0	0
$147$	4376.66	2.45565
$148$	0	0
$149$	2014.63	1.10769	0.553843	−	0.832621i	$-0.313161\pi$
$149$	2014.63	1.10769	0.553843	+	0.832621i	$0.313161\pi$
$150$	0	0
$151$	−389.182	−0.209743	−0.104872	−	0.994486i	$-0.533443\pi$
$151$	−389.182	−0.209743	−0.104872	+	0.994486i	$0.533443\pi$
$152$	0	0
$153$	2421.65	1.27960
$154$	0	0
$155$	−287.678	−0.149077
$156$	0	0
$157$	−1485.77	−0.755270	−0.377635	−	0.925954i	$-0.623263\pi$
$157$	−1485.77	−0.755270	−0.377635	+	0.925954i	$0.623263\pi$
$158$	0	0
$159$	−5688.90	−2.83748
$160$	0	0
$161$	−2109.40	−1.03257
$162$	0	0
$163$	2345.72	1.12718	0.563591	−	0.826054i	$-0.309419\pi$
$163$	2345.72	1.12718	0.563591	+	0.826054i	$0.309419\pi$
$164$	0	0
$165$	−12.7921	−0.00603553
$166$	0	0
$167$	−1978.23	−0.916648	−0.458324	−	0.888785i	$-0.651550\pi$
$167$	−1978.23	−0.916648	−0.458324	+	0.888785i	$0.651550\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7645.97	3.41931
$172$	0	0
$173$	1892.71	0.831791	0.415895	−	0.909412i	$-0.363468\pi$
$173$	1892.71	0.831791	0.415895	+	0.909412i	$0.363468\pi$
$174$	0	0
$175$	−3419.73	−1.47718
$176$	0	0
$177$	−7794.61	−3.31005
$178$	0	0
$179$	−4002.27	−1.67119	−0.835597	−	0.549343i	$-0.814878\pi$
$179$	−4002.27	−1.67119	−0.835597	+	0.549343i	$0.814878\pi$
$180$	0	0
$181$	−1128.60	−0.463469	−0.231734	−	0.972779i	$-0.574440\pi$
$181$	−1128.60	−0.463469	−0.231734	+	0.972779i	$0.574440\pi$
$182$	0	0
$183$	5452.65	2.20258
$184$	0	0
$185$	−705.590	−0.280411
$186$	0	0
$187$	24.8048	0.00970002
$188$	0	0
$189$	8486.22	3.26604
$190$	0	0
$191$	−3133.94	−1.18724	−0.593622	−	0.804744i	$-0.702303\pi$
$191$	−3133.94	−1.18724	−0.593622	+	0.804744i	$0.702303\pi$
$192$	0	0
$193$	2115.69	0.789070	0.394535	−	0.918881i	$-0.370906\pi$
$193$	2115.69	0.789070	0.394535	+	0.918881i	$0.370906\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.4068	0.00521038	0.00260519	−	0.999997i	$-0.499171\pi$
$197$	14.4068	0.00521038	0.00260519	+	0.999997i	$0.499171\pi$
$198$	0	0
$199$	−1412.24	−0.503072	−0.251536	−	0.967848i	$-0.580936\pi$
$199$	−1412.24	−0.503072	−0.251536	+	0.967848i	$0.580936\pi$
$200$	0	0
$201$	−5626.55	−1.97446
$202$	0	0
$203$	−6123.81	−2.11728
$204$	0	0
$205$	−200.979	−0.0684729
$206$	0	0
$207$	−4363.56	−1.46516
$208$	0	0
$209$	78.3170	0.0259201
$210$	0	0
$211$	2908.92	0.949093	0.474546	−	0.880231i	$-0.342612\pi$
$211$	2908.92	0.949093	0.474546	+	0.880231i	$0.342612\pi$
$212$	0	0
$213$	−4708.87	−1.51477
$214$	0	0
$215$	−559.292	−0.177411
$216$	0	0
$217$	3601.66	1.12671
$218$	0	0
$219$	1643.08	0.506981
$220$	0	0
$221$	0	0
$222$	0	0
$223$	354.193	0.106361	0.0531805	−	0.998585i	$-0.483064\pi$
$223$	354.193	0.106361	0.0531805	+	0.998585i	$0.483064\pi$
$224$	0	0
$225$	−7074.15	−2.09604
$226$	0	0
$227$	−4122.96	−1.20551	−0.602754	−	0.797927i	$-0.705930\pi$
$227$	−4122.96	−1.20551	−0.602754	+	0.797927i	$0.705930\pi$
$228$	0	0
$229$	−4250.04	−1.22642	−0.613210	−	0.789920i	$-0.710122\pi$
$229$	−4250.04	−1.22642	−0.613210	+	0.789920i	$0.710122\pi$
$230$	0	0
$231$	160.154	0.0456162
$232$	0	0
$233$	1590.88	0.447304	0.223652	−	0.974669i	$-0.428202\pi$
$233$	1590.88	0.447304	0.223652	+	0.974669i	$0.428202\pi$
$234$	0	0
$235$	156.280	0.0433812
$236$	0	0
$237$	1327.64	0.363879
$238$	0	0
$239$	−1030.76	−0.278972	−0.139486	−	0.990224i	$-0.544545\pi$
$239$	−1030.76	−0.278972	−0.139486	+	0.990224i	$0.544545\pi$
$240$	0	0
$241$	−4582.38	−1.22480	−0.612401	−	0.790547i	$-0.709796\pi$
$241$	−4582.38	−1.22480	−0.612401	+	0.790547i	$0.709796\pi$
$242$	0	0
$243$	2765.52	0.730076
$244$	0	0
$245$	−1075.73	−0.280514
$246$	0	0
$247$	0	0
$248$	0	0
$249$	7173.20	1.82564
$250$	0	0
$251$	3084.18	0.775584	0.387792	−	0.921747i	$-0.373238\pi$
$251$	3084.18	0.775584	0.387792	+	0.921747i	$0.373238\pi$
$252$	0	0
$253$	−44.6956	−0.0111067
$254$	0	0
$255$	−867.372	−0.213008
$256$	0	0
$257$	2328.69	0.565212	0.282606	−	0.959236i	$-0.408801\pi$
$257$	2328.69	0.565212	0.282606	+	0.959236i	$0.408801\pi$
$258$	0	0
$259$	8833.81	2.11933
$260$	0	0
$261$	−12667.9	−3.00430
$262$	0	0
$263$	3795.43	0.889873	0.444936	−	0.895562i	$-0.353226\pi$
$263$	3795.43	0.889873	0.444936	+	0.895562i	$0.353226\pi$
$264$	0	0
$265$	1398.26	0.324131
$266$	0	0
$267$	13013.6	2.98285
$268$	0	0
$269$	−6178.54	−1.40042	−0.700209	−	0.713938i	$-0.746910\pi$
$269$	−6178.54	−1.40042	−0.700209	+	0.713938i	$0.746910\pi$
$270$	0	0
$271$	−322.020	−0.0721819	−0.0360910	−	0.999349i	$-0.511491\pi$
$271$	−322.020	−0.0721819	−0.0360910	+	0.999349i	$0.511491\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−72.4600	−0.0158891
$276$	0	0
$277$	−4183.15	−0.907369	−0.453684	−	0.891162i	$-0.649891\pi$
$277$	−4183.15	−0.907369	−0.453684	+	0.891162i	$0.649891\pi$
$278$	0	0
$279$	7450.49	1.59874
$280$	0	0
$281$	−392.013	−0.0832226	−0.0416113	−	0.999134i	$-0.513249\pi$
$281$	−392.013	−0.0832226	−0.0416113	+	0.999134i	$0.513249\pi$
$282$	0	0
$283$	4119.24	0.865241	0.432621	−	0.901576i	$-0.357589\pi$
$283$	4119.24	0.865241	0.432621	+	0.901576i	$0.357589\pi$
$284$	0	0
$285$	−2738.59	−0.569193
$286$	0	0
$287$	2516.20	0.517515
$288$	0	0
$289$	−3231.10	−0.657664
$290$	0	0
$291$	−4343.37	−0.874959
$292$	0	0
$293$	4885.80	0.974169	0.487085	−	0.873355i	$-0.338060\pi$
$293$	4885.80	0.974169	0.487085	+	0.873355i	$0.338060\pi$
$294$	0	0
$295$	1915.82	0.378113
$296$	0	0
$297$	179.813	0.0351306
$298$	0	0
$299$	0	0
$300$	0	0
$301$	7002.20	1.34086
$302$	0	0
$303$	3710.12	0.703435
$304$	0	0
$305$	−1340.20	−0.251605
$306$	0	0
$307$	2024.02	0.376277	0.188139	−	0.982142i	$-0.439755\pi$
$307$	2024.02	0.376277	0.188139	+	0.982142i	$0.439755\pi$
$308$	0	0
$309$	−13024.9	−2.39793
$310$	0	0
$311$	1272.97	0.232102	0.116051	−	0.993243i	$-0.462976\pi$
$311$	1272.97	0.232102	0.116051	+	0.993243i	$0.462976\pi$
$312$	0	0
$313$	375.693	0.0678448	0.0339224	−	0.999424i	$-0.489200\pi$
$313$	375.693	0.0678448	0.0339224	+	0.999424i	$0.489200\pi$
$314$	0	0
$315$	−3843.03	−0.687398
$316$	0	0
$317$	10369.7	1.83728	0.918641	−	0.395094i	$-0.129288\pi$
$317$	10369.7	1.83728	0.918641	+	0.395094i	$0.129288\pi$
$318$	0	0
$319$	−129.756	−0.0227741
$320$	0	0
$321$	−14809.9	−2.57511
$322$	0	0
$323$	5310.32	0.914780
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2298.72	−0.388744
$328$	0	0
$329$	−1956.59	−0.327873
$330$	0	0
$331$	2272.03	0.377287	0.188644	−	0.982046i	$-0.439591\pi$
$331$	2272.03	0.377287	0.188644	+	0.982046i	$0.439591\pi$
$332$	0	0
$333$	18273.9	3.00721
$334$	0	0
$335$	1382.94	0.225546
$336$	0	0
$337$	2984.47	0.482417	0.241209	−	0.970473i	$-0.422456\pi$
$337$	2984.47	0.482417	0.241209	+	0.970473i	$0.422456\pi$
$338$	0	0
$339$	3927.19	0.629191
$340$	0	0
$341$	76.3148	0.0121193
$342$	0	0
$343$	3676.98	0.578829
$344$	0	0
$345$	1562.91	0.243897
$346$	0	0
$347$	4329.42	0.669786	0.334893	−	0.942256i	$-0.391300\pi$
$347$	4329.42	0.669786	0.334893	+	0.942256i	$0.391300\pi$
$348$	0	0
$349$	4025.97	0.617494	0.308747	−	0.951144i	$-0.400090\pi$
$349$	4025.97	0.617494	0.308747	+	0.951144i	$0.400090\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6907.56	−1.04151	−0.520754	−	0.853707i	$-0.674349\pi$
$353$	−6907.56	−1.04151	−0.520754	+	0.853707i	$0.674349\pi$
$354$	0	0
$355$	1157.38	0.173036
$356$	0	0
$357$	10859.3	1.60990
$358$	0	0
$359$	−1518.19	−0.223195	−0.111597	−	0.993754i	$-0.535597\pi$
$359$	−1518.19	−0.223195	−0.111597	+	0.993754i	$0.535597\pi$
$360$	0	0
$361$	9907.47	1.44445
$362$	0	0
$363$	−12343.3	−1.78472
$364$	0	0
$365$	−403.848	−0.0579134
$366$	0	0
$367$	456.984	0.0649983	0.0324992	−	0.999472i	$-0.489653\pi$
$367$	456.984	0.0649983	0.0324992	+	0.999472i	$0.489653\pi$
$368$	0	0
$369$	5205.09	0.734325
$370$	0	0
$371$	−17505.9	−2.44976
$372$	0	0
$373$	908.186	0.126070	0.0630350	−	0.998011i	$-0.479922\pi$
$373$	908.186	0.126070	0.0630350	+	0.998011i	$0.479922\pi$
$374$	0	0
$375$	5177.50	0.712973
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−9874.70	−1.33834	−0.669168	−	0.743111i	$-0.733349\pi$
$379$	−9874.70	−1.33834	−0.669168	+	0.743111i	$0.733349\pi$
$380$	0	0
$381$	5944.73	0.799364
$382$	0	0
$383$	−7726.79	−1.03086	−0.515432	−	0.856931i	$-0.672368\pi$
$383$	−7726.79	−1.03086	−0.515432	+	0.856931i	$0.672368\pi$
$384$	0	0
$385$	−39.3639	−0.00521083
$386$	0	0
$387$	14484.9	1.90261
$388$	0	0
$389$	−6461.27	−0.842157	−0.421079	−	0.907024i	$-0.638348\pi$
$389$	−6461.27	−0.842157	−0.421079	+	0.907024i	$0.638348\pi$
$390$	0	0
$391$	−3030.60	−0.391980
$392$	0	0
$393$	7507.72	0.963649
$394$	0	0
$395$	−326.317	−0.0415666
$396$	0	0
$397$	−9730.13	−1.23008	−0.615039	−	0.788497i	$-0.710860\pi$
$397$	−9730.13	−1.23008	−0.615039	+	0.788497i	$0.710860\pi$
$398$	0	0
$399$	34286.5	4.30193
$400$	0	0
$401$	10566.7	1.31590	0.657950	−	0.753061i	$-0.271424\pi$
$401$	10566.7	1.31590	0.657950	+	0.753061i	$0.271424\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2652.65	−0.325460
$406$	0	0
$407$	187.178	0.0227962
$408$	0	0
$409$	−10627.3	−1.28481	−0.642405	−	0.766366i	$-0.722063\pi$
$409$	−10627.3	−1.28481	−0.642405	+	0.766366i	$0.722063\pi$
$410$	0	0
$411$	15457.2	1.85510
$412$	0	0
$413$	−23985.6	−2.85776
$414$	0	0
$415$	−1763.09	−0.208546
$416$	0	0
$417$	22942.8	2.69428
$418$	0	0
$419$	1384.56	0.161433	0.0807164	−	0.996737i	$-0.474279\pi$
$419$	1384.56	0.161433	0.0807164	+	0.996737i	$0.474279\pi$
$420$	0	0
$421$	4777.19	0.553031	0.276515	−	0.961010i	$-0.410820\pi$
$421$	4777.19	0.553031	0.276515	+	0.961010i	$0.410820\pi$
$422$	0	0
$423$	−4047.45	−0.465234
$424$	0	0
$425$	−4913.18	−0.560763
$426$	0	0
$427$	16778.9	1.90162
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8535.98	0.953976	0.476988	−	0.878910i	$-0.341729\pi$
$431$	8535.98	0.953976	0.476988	+	0.878910i	$0.341729\pi$
$432$	0	0
$433$	−637.729	−0.0707789	−0.0353895	−	0.999374i	$-0.511267\pi$
$433$	−637.729	−0.0707789	−0.0353895	+	0.999374i	$0.511267\pi$
$434$	0	0
$435$	4537.31	0.500109
$436$	0	0
$437$	−9568.64	−1.04744
$438$	0	0
$439$	1780.56	0.193580	0.0967901	−	0.995305i	$-0.469142\pi$
$439$	1780.56	0.193580	0.0967901	+	0.995305i	$0.469142\pi$
$440$	0	0
$441$	27860.1	3.00832
$442$	0	0
$443$	5257.36	0.563848	0.281924	−	0.959437i	$-0.409027\pi$
$443$	5257.36	0.563848	0.281924	+	0.959437i	$0.409027\pi$
$444$	0	0
$445$	−3198.60	−0.340737
$446$	0	0
$447$	18688.2	1.97746
$448$	0	0
$449$	−4175.11	−0.438832	−0.219416	−	0.975631i	$-0.570415\pi$
$449$	−4175.11	−0.438832	−0.219416	+	0.975631i	$0.570415\pi$
$450$	0	0
$451$	53.3153	0.00556656
$452$	0	0
$453$	−3610.15	−0.374437
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9325.52	−0.954550	−0.477275	−	0.878754i	$-0.658375\pi$
$457$	−9325.52	−0.954550	−0.477275	+	0.878754i	$0.658375\pi$
$458$	0	0
$459$	12192.3	1.23984
$460$	0	0
$461$	−17815.8	−1.79993	−0.899963	−	0.435966i	$-0.856407\pi$
$461$	−17815.8	−1.79993	−0.899963	+	0.435966i	$0.856407\pi$
$462$	0	0
$463$	−13941.4	−1.39938	−0.699689	−	0.714448i	$-0.746678\pi$
$463$	−13941.4	−1.39938	−0.699689	+	0.714448i	$0.746678\pi$
$464$	0	0
$465$	−2668.57	−0.266134
$466$	0	0
$467$	−17855.6	−1.76929	−0.884644	−	0.466268i	$-0.845598\pi$
$467$	−17855.6	−1.76929	−0.884644	+	0.466268i	$0.845598\pi$
$468$	0	0
$469$	−17314.0	−1.70467
$470$	0	0
$471$	−13782.4	−1.34832
$472$	0	0
$473$	148.368	0.0144228
$474$	0	0
$475$	−15512.6	−1.49845
$476$	0	0
$477$	−36213.2	−3.47608
$478$	0	0
$479$	−5268.36	−0.502542	−0.251271	−	0.967917i	$-0.580849\pi$
$479$	−5268.36	−0.502542	−0.251271	+	0.967917i	$0.580849\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−19567.3	−1.84336
$484$	0	0
$485$	1067.55	0.0999482
$486$	0	0
$487$	5676.35	0.528172	0.264086	−	0.964499i	$-0.414930\pi$
$487$	5676.35	0.528172	0.264086	+	0.964499i	$0.414930\pi$
$488$	0	0
$489$	21759.5	2.01226
$490$	0	0
$491$	12342.0	1.13439	0.567197	−	0.823582i	$-0.308028\pi$
$491$	12342.0	1.13439	0.567197	+	0.823582i	$0.308028\pi$
$492$	0	0
$493$	−8798.18	−0.803752
$494$	0	0
$495$	−81.4292	−0.00739388
$496$	0	0
$497$	−14490.2	−1.30779
$498$	0	0
$499$	16955.3	1.52109	0.760545	−	0.649285i	$-0.224932\pi$
$499$	16955.3	1.52109	0.760545	+	0.649285i	$0.224932\pi$
$500$	0	0
$501$	−18350.6	−1.63641
$502$	0	0
$503$	10583.7	0.938178	0.469089	−	0.883151i	$-0.344582\pi$
$503$	10583.7	0.938178	0.469089	+	0.883151i	$0.344582\pi$
$504$	0	0
$505$	−911.902	−0.0803547
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13349.6	−1.16249	−0.581246	−	0.813728i	$-0.697435\pi$
$509$	−13349.6	−1.16249	−0.581246	+	0.813728i	$0.697435\pi$
$510$	0	0
$511$	5056.09	0.437707
$512$	0	0
$513$	38495.1	3.31306
$514$	0	0
$515$	3201.36	0.273920
$516$	0	0
$517$	−41.4577	−0.00352671
$518$	0	0
$519$	17557.2	1.48493
$520$	0	0
$521$	−7321.30	−0.615647	−0.307823	−	0.951443i	$-0.599601\pi$
$521$	−7321.30	−0.615647	−0.307823	+	0.951443i	$0.599601\pi$
$522$	0	0
$523$	6772.54	0.566238	0.283119	−	0.959085i	$-0.408631\pi$
$523$	6772.54	0.566238	0.283119	+	0.959085i	$0.408631\pi$
$524$	0	0
$525$	−31722.3	−2.63709
$526$	0	0
$527$	5174.56	0.427718
$528$	0	0
$529$	−6706.17	−0.551177
$530$	0	0
$531$	−49617.3	−4.05501
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3640.10	0.294160
$536$	0	0
$537$	−37126.1	−2.98344
$538$	0	0
$539$	285.368	0.0228046
$540$	0	0
$541$	−2816.12	−0.223798	−0.111899	−	0.993720i	$-0.535693\pi$
$541$	−2816.12	−0.223798	−0.111899	+	0.993720i	$0.535693\pi$
$542$	0	0
$543$	−10469.1	−0.827391
$544$	0	0
$545$	564.998	0.0444070
$546$	0	0
$547$	−946.909	−0.0740163	−0.0370082	−	0.999315i	$-0.511783\pi$
$547$	−946.909	−0.0740163	−0.0370082	+	0.999315i	$0.511783\pi$
$548$	0	0
$549$	34709.4	2.69829
$550$	0	0
$551$	−27778.8	−2.14776
$552$	0	0
$553$	4085.41	0.314158
$554$	0	0
$555$	−6545.23	−0.500594
$556$	0	0
$557$	−6635.00	−0.504729	−0.252364	−	0.967632i	$-0.581208\pi$
$557$	−6635.00	−0.504729	−0.252364	+	0.967632i	$0.581208\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	230.095	0.0173166
$562$	0	0
$563$	−14327.3	−1.07251	−0.536254	−	0.844056i	$-0.680161\pi$
$563$	−14327.3	−1.07251	−0.536254	+	0.844056i	$0.680161\pi$
$564$	0	0
$565$	−965.256	−0.0718737
$566$	0	0
$567$	33210.6	2.45981
$568$	0	0
$569$	20996.9	1.54699	0.773493	−	0.633805i	$-0.218508\pi$
$569$	20996.9	1.54699	0.773493	+	0.633805i	$0.218508\pi$
$570$	0	0
$571$	−3078.19	−0.225601	−0.112800	−	0.993618i	$-0.535982\pi$
$571$	−3078.19	−0.225601	−0.112800	+	0.993618i	$0.535982\pi$
$572$	0	0
$573$	−29071.2	−2.11949
$574$	0	0
$575$	8853.04	0.642082
$576$	0	0
$577$	2220.02	0.160174	0.0800871	−	0.996788i	$-0.474480\pi$
$577$	2220.02	0.160174	0.0800871	+	0.996788i	$0.474480\pi$
$578$	0	0
$579$	19625.6	1.40866
$580$	0	0
$581$	22073.4	1.57618
$582$	0	0
$583$	−370.929	−0.0263504
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21570.1	1.51669	0.758343	−	0.651855i	$-0.226009\pi$
$587$	21570.1	1.51669	0.758343	+	0.651855i	$0.226009\pi$
$588$	0	0
$589$	16337.8	1.14293
$590$	0	0
$591$	133.642	0.00930166
$592$	0	0
$593$	−17778.7	−1.23117	−0.615584	−	0.788071i	$-0.711080\pi$
$593$	−17778.7	−1.23117	−0.615584	+	0.788071i	$0.711080\pi$
$594$	0	0
$595$	−2669.08	−0.183902
$596$	0	0
$597$	−13100.3	−0.898092
$598$	0	0
$599$	−6024.84	−0.410965	−0.205483	−	0.978661i	$-0.565876\pi$
$599$	−6024.84	−0.410965	−0.205483	+	0.978661i	$0.565876\pi$
$600$	0	0
$601$	−14058.2	−0.954153	−0.477077	−	0.878862i	$-0.658304\pi$
$601$	−14058.2	−0.954153	−0.477077	+	0.878862i	$0.658304\pi$
$602$	0	0
$603$	−35816.3	−2.41883
$604$	0	0
$605$	3033.83	0.203873
$606$	0	0
$607$	−16830.7	−1.12543	−0.562716	−	0.826650i	$-0.690243\pi$
$607$	−16830.7	−1.12543	−0.562716	+	0.826650i	$0.690243\pi$
$608$	0	0
$609$	−56806.1	−3.77980
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−9954.11	−0.655861	−0.327931	−	0.944702i	$-0.606351\pi$
$613$	−9954.11	−0.655861	−0.327931	+	0.944702i	$0.606351\pi$
$614$	0	0
$615$	−1864.33	−0.122239
$616$	0	0
$617$	19391.0	1.26524	0.632621	−	0.774462i	$-0.281979\pi$
$617$	19391.0	1.26524	0.632621	+	0.774462i	$0.281979\pi$
$618$	0	0
$619$	60.1151	0.00390344	0.00195172	−	0.999998i	$-0.499379\pi$
$619$	60.1151	0.00390344	0.00195172	+	0.999998i	$0.499379\pi$
$620$	0	0
$621$	−21969.2	−1.41964
$622$	0	0
$623$	40045.7	2.57527
$624$	0	0
$625$	13702.6	0.876969
$626$	0	0
$627$	726.488	0.0462730
$628$	0	0
$629$	12691.7	0.804531
$630$	0	0
$631$	−15910.7	−1.00379	−0.501896	−	0.864928i	$-0.667364\pi$
$631$	−15910.7	−1.00379	−0.501896	+	0.864928i	$0.667364\pi$
$632$	0	0
$633$	26983.9	1.69434
$634$	0	0
$635$	−1461.14	−0.0913129
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−29974.8	−1.85569
$640$	0	0
$641$	−10812.8	−0.666271	−0.333135	−	0.942879i	$-0.608107\pi$
$641$	−10812.8	−0.666271	−0.333135	+	0.942879i	$0.608107\pi$
$642$	0	0
$643$	−15062.1	−0.923783	−0.461891	−	0.886937i	$-0.652829\pi$
$643$	−15062.1	−0.923783	−0.461891	+	0.886937i	$0.652829\pi$
$644$	0	0
$645$	−5188.13	−0.316717
$646$	0	0
$647$	−21953.9	−1.33400	−0.667001	−	0.745057i	$-0.732422\pi$
$647$	−21953.9	−1.33400	−0.667001	+	0.745057i	$0.732422\pi$
$648$	0	0
$649$	−508.226	−0.0307390
$650$	0	0
$651$	33409.9	2.01143
$652$	0	0
$653$	−9906.28	−0.593664	−0.296832	−	0.954930i	$-0.595930\pi$
$653$	−9906.28	−0.593664	−0.296832	+	0.954930i	$0.595930\pi$
$654$	0	0
$655$	−1845.31	−0.110080
$656$	0	0
$657$	10459.2	0.621082
$658$	0	0
$659$	−17156.6	−1.01415	−0.507077	−	0.861901i	$-0.669274\pi$
$659$	−17156.6	−1.01415	−0.507077	+	0.861901i	$0.669274\pi$
$660$	0	0
$661$	1654.70	0.0973681	0.0486840	−	0.998814i	$-0.484497\pi$
$661$	1654.70	0.0973681	0.0486840	+	0.998814i	$0.484497\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8427.20	−0.491418
$666$	0	0
$667$	15853.4	0.920309
$668$	0	0
$669$	3285.58	0.189877
$670$	0	0
$671$	355.525	0.0204544
$672$	0	0
$673$	1227.71	0.0703190	0.0351595	−	0.999382i	$-0.488806\pi$
$673$	1227.71	0.0703190	0.0351595	+	0.999382i	$0.488806\pi$
$674$	0	0
$675$	−35616.2	−2.03092
$676$	0	0
$677$	−16939.3	−0.961638	−0.480819	−	0.876820i	$-0.659660\pi$
$677$	−16939.3	−0.961638	−0.480819	+	0.876820i	$0.659660\pi$
$678$	0	0
$679$	−13365.5	−0.755404
$680$	0	0
$681$	−38245.6	−2.15209
$682$	0	0
$683$	−16572.3	−0.928434	−0.464217	−	0.885721i	$-0.653664\pi$
$683$	−16572.3	−0.928434	−0.464217	+	0.885721i	$0.653664\pi$
$684$	0	0
$685$	−3799.19	−0.211912
$686$	0	0
$687$	−39424.4	−2.18942
$688$	0	0
$689$	0	0
$690$	0	0
$691$	3836.25	0.211198	0.105599	−	0.994409i	$-0.466324\pi$
$691$	3836.25	0.211198	0.105599	+	0.994409i	$0.466324\pi$
$692$	0	0
$693$	1019.47	0.0558825
$694$	0	0
$695$	−5639.07	−0.307773
$696$	0	0
$697$	3615.07	0.196457
$698$	0	0
$699$	14757.4	0.798533
$700$	0	0
$701$	−16178.6	−0.871692	−0.435846	−	0.900021i	$-0.643551\pi$
$701$	−16178.6	−0.871692	−0.435846	+	0.900021i	$0.643551\pi$
$702$	0	0
$703$	40071.9	2.14984
$704$	0	0
$705$	1449.69	0.0774448
$706$	0	0
$707$	11416.8	0.607317
$708$	0	0
$709$	−26689.0	−1.41372	−0.706859	−	0.707354i	$-0.749888\pi$
$709$	−26689.0	−1.41372	−0.706859	+	0.707354i	$0.749888\pi$
$710$	0	0
$711$	8451.19	0.445773
$712$	0	0
$713$	−9324.02	−0.489744
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9561.60	−0.498026
$718$	0	0
$719$	25906.0	1.34371	0.671857	−	0.740681i	$-0.265497\pi$
$719$	25906.0	1.34371	0.671857	+	0.740681i	$0.265497\pi$
$720$	0	0
$721$	−40080.3	−2.07027
$722$	0	0
$723$	−42507.3	−2.18654
$724$	0	0
$725$	25701.3	1.31658
$726$	0	0
$727$	−8386.29	−0.427827	−0.213914	−	0.976853i	$-0.568621\pi$
$727$	−8386.29	−0.427827	−0.213914	+	0.976853i	$0.568621\pi$
$728$	0	0
$729$	−5759.42	−0.292609
$730$	0	0
$731$	10060.2	0.509013
$732$	0	0
$733$	−289.490	−0.0145874	−0.00729371	−	0.999973i	$-0.502322\pi$
$733$	−289.490	−0.0145874	−0.00729371	+	0.999973i	$0.502322\pi$
$734$	0	0
$735$	−9978.76	−0.500778
$736$	0	0
$737$	−366.864	−0.0183359
$738$	0	0
$739$	17933.9	0.892705	0.446352	−	0.894857i	$-0.352723\pi$
$739$	17933.9	0.892705	0.446352	+	0.894857i	$0.352723\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22954.9	−1.13342	−0.566711	−	0.823917i	$-0.691784\pi$
$743$	−22954.9	−1.13342	−0.566711	+	0.823917i	$0.691784\pi$
$744$	0	0
$745$	−4593.35	−0.225889
$746$	0	0
$747$	45661.7	2.23651
$748$	0	0
$749$	−45573.2	−2.22324
$750$	0	0
$751$	−29914.5	−1.45352	−0.726761	−	0.686890i	$-0.758975\pi$
$751$	−29914.5	−1.45352	−0.726761	+	0.686890i	$0.758975\pi$
$752$	0	0
$753$	28609.6	1.38459
$754$	0	0
$755$	887.333	0.0427726
$756$	0	0
$757$	9422.87	0.452417	0.226209	−	0.974079i	$-0.427367\pi$
$757$	9422.87	0.452417	0.226209	+	0.974079i	$0.427367\pi$
$758$	0	0
$759$	−414.608	−0.0198278
$760$	0	0
$761$	−16008.9	−0.762577	−0.381288	−	0.924456i	$-0.624520\pi$
$761$	−16008.9	−0.762577	−0.381288	+	0.924456i	$0.624520\pi$
$762$	0	0
$763$	−7073.63	−0.335626
$764$	0	0
$765$	−5521.34	−0.260947
$766$	0	0
$767$	0	0
$768$	0	0
$769$	23058.4	1.08128	0.540642	−	0.841253i	$-0.318182\pi$
$769$	23058.4	1.08128	0.540642	+	0.841253i	$0.318182\pi$
$770$	0	0
$771$	21601.5	1.00903
$772$	0	0
$773$	−17638.7	−0.820726	−0.410363	−	0.911922i	$-0.634598\pi$
$773$	−17638.7	−0.820726	−0.410363	+	0.911922i	$0.634598\pi$
$774$	0	0
$775$	−15116.0	−0.700622
$776$	0	0
$777$	81944.7	3.78346
$778$	0	0
$779$	11414.0	0.524966
$780$	0	0
$781$	−307.029	−0.0140671
$782$	0	0
$783$	−63779.0	−2.91095
$784$	0	0
$785$	3387.55	0.154021
$786$	0	0
$787$	2401.87	0.108789	0.0543947	−	0.998520i	$-0.482677\pi$
$787$	2401.87	0.108789	0.0543947	+	0.998520i	$0.482677\pi$
$788$	0	0
$789$	35207.4	1.58861
$790$	0	0
$791$	12084.8	0.543218
$792$	0	0
$793$	0	0
$794$	0	0
$795$	12970.6	0.578643
$796$	0	0
$797$	3281.78	0.145855	0.0729276	−	0.997337i	$-0.476766\pi$
$797$	3281.78	0.145855	0.0729276	+	0.997337i	$0.476766\pi$
$798$	0	0
$799$	−2811.06	−0.124466
$800$	0	0
$801$	82839.6	3.65417
$802$	0	0
$803$	107.132	0.00470812
$804$	0	0
$805$	4809.41	0.210571
$806$	0	0
$807$	−57313.7	−2.50005
$808$	0	0
$809$	36413.5	1.58249	0.791243	−	0.611501i	$-0.209434\pi$
$809$	36413.5	1.58249	0.791243	+	0.611501i	$0.209434\pi$
$810$	0	0
$811$	31898.2	1.38113	0.690566	−	0.723269i	$-0.257361\pi$
$811$	31898.2	1.38113	0.690566	+	0.723269i	$0.257361\pi$
$812$	0	0
$813$	−2987.14	−0.128860
$814$	0	0
$815$	−5348.22	−0.229865
$816$	0	0
$817$	31763.3	1.36017
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29162.8	1.23970	0.619848	−	0.784722i	$-0.287194\pi$
$821$	29162.8	1.23970	0.619848	+	0.784722i	$0.287194\pi$
$822$	0	0
$823$	−30040.6	−1.27236	−0.636179	−	0.771541i	$-0.719486\pi$
$823$	−30040.6	−1.27236	−0.636179	+	0.771541i	$0.719486\pi$
$824$	0	0
$825$	−672.157	−0.0283655
$826$	0	0
$827$	−6175.43	−0.259662	−0.129831	−	0.991536i	$-0.541444\pi$
$827$	−6175.43	−0.259662	−0.129831	+	0.991536i	$0.541444\pi$
$828$	0	0
$829$	−46596.2	−1.95217	−0.976087	−	0.217381i	$-0.930249\pi$
$829$	−46596.2	−1.95217	−0.976087	+	0.217381i	$0.930249\pi$
$830$	0	0
$831$	−38804.0	−1.61985
$832$	0	0
$833$	19349.5	0.804828
$834$	0	0
$835$	4510.35	0.186931
$836$	0	0
$837$	37511.0	1.54907
$838$	0	0
$839$	124.642	0.00512887	0.00256443	−	0.999997i	$-0.499184\pi$
$839$	124.642	0.00512887	0.00256443	+	0.999997i	$0.499184\pi$
$840$	0	0
$841$	21635.1	0.887086
$842$	0	0
$843$	−3636.41	−0.148570
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−37982.9	−1.54086
$848$	0	0
$849$	38211.1	1.54464
$850$	0	0
$851$	−22869.1	−0.921200
$852$	0	0
$853$	33839.8	1.35833	0.679163	−	0.733988i	$-0.262343\pi$
$853$	33839.8	1.35833	0.679163	+	0.733988i	$0.262343\pi$
$854$	0	0
$855$	−17432.7	−0.697295
$856$	0	0
$857$	585.021	0.0233185	0.0116592	−	0.999932i	$-0.496289\pi$
$857$	585.021	0.0233185	0.0116592	+	0.999932i	$0.496289\pi$
$858$	0	0
$859$	21226.5	0.843120	0.421560	−	0.906800i	$-0.361483\pi$
$859$	21226.5	0.843120	0.421560	+	0.906800i	$0.361483\pi$
$860$	0	0
$861$	23340.9	0.923876
$862$	0	0
$863$	−38443.2	−1.51636	−0.758181	−	0.652044i	$-0.773912\pi$
$863$	−38443.2	−1.51636	−0.758181	+	0.652044i	$0.773912\pi$
$864$	0	0
$865$	−4315.35	−0.169626
$866$	0	0
$867$	−29972.5	−1.17407
$868$	0	0
$869$	86.5649	0.00337919
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−27648.1	−1.07188
$874$	0	0
$875$	15932.2	0.615552
$876$	0	0
$877$	15392.1	0.592652	0.296326	−	0.955087i	$-0.404239\pi$
$877$	15392.1	0.592652	0.296326	+	0.955087i	$0.404239\pi$
$878$	0	0
$879$	45321.9	1.73910
$880$	0	0
$881$	−16605.9	−0.635037	−0.317518	−	0.948252i	$-0.602850\pi$
$881$	−16605.9	−0.635037	−0.317518	+	0.948252i	$0.602850\pi$
$882$	0	0
$883$	20909.9	0.796913	0.398456	−	0.917187i	$-0.369546\pi$
$883$	20909.9	0.796913	0.398456	+	0.917187i	$0.369546\pi$
$884$	0	0
$885$	17771.6	0.675014
$886$	0	0
$887$	−266.852	−0.0101015	−0.00505074	−	0.999987i	$-0.501608\pi$
$887$	−266.852	−0.0101015	−0.00505074	+	0.999987i	$0.501608\pi$
$888$	0	0
$889$	18293.2	0.690138
$890$	0	0
$891$	703.692	0.0264585
$892$	0	0
$893$	−8875.46	−0.332594
$894$	0	0
$895$	9125.14	0.340804
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−27068.6	−1.00422
$900$	0	0
$901$	−25151.0	−0.929968
$902$	0	0
$903$	64954.2	2.39373
$904$	0	0
$905$	2573.19	0.0945145
$906$	0	0
$907$	−43847.3	−1.60521	−0.802606	−	0.596510i	$-0.796554\pi$
$907$	−43847.3	−1.60521	−0.802606	+	0.596510i	$0.796554\pi$
$908$	0	0
$909$	23617.1	0.861749
$910$	0	0
$911$	9417.62	0.342502	0.171251	−	0.985227i	$-0.445219\pi$
$911$	9417.62	0.342502	0.171251	+	0.985227i	$0.445219\pi$
$912$	0	0
$913$	467.709	0.0169539
$914$	0	0
$915$	−12432.0	−0.449169
$916$	0	0
$917$	23102.8	0.831976
$918$	0	0
$919$	−43296.8	−1.55411	−0.777056	−	0.629431i	$-0.783288\pi$
$919$	−43296.8	−1.55411	−0.777056	+	0.629431i	$0.783288\pi$
$920$	0	0
$921$	18775.3	0.671736
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−37075.1	−1.31786
$926$	0	0
$927$	−82911.1	−2.93760
$928$	0	0
$929$	31999.6	1.13011	0.565055	−	0.825053i	$-0.308855\pi$
$929$	31999.6	1.13011	0.565055	+	0.825053i	$0.308855\pi$
$930$	0	0
$931$	61093.0	2.15064
$932$	0	0
$933$	11808.4	0.414352
$934$	0	0
$935$	−56.5547	−0.00197811
$936$	0	0
$937$	1997.33	0.0696371	0.0348186	−	0.999394i	$-0.488915\pi$
$937$	1997.33	0.0696371	0.0348186	+	0.999394i	$0.488915\pi$
$938$	0	0
$939$	3485.02	0.121118
$940$	0	0
$941$	31241.0	1.08228	0.541141	−	0.840932i	$-0.317993\pi$
$941$	31241.0	1.08228	0.541141	+	0.840932i	$0.317993\pi$
$942$	0	0
$943$	−6513.97	−0.224946
$944$	0	0
$945$	−19348.5	−0.666039
$946$	0	0
$947$	2759.74	0.0946986	0.0473493	−	0.998878i	$-0.484923\pi$
$947$	2759.74	0.0946986	0.0473493	+	0.998878i	$0.484923\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	96191.6	3.27994
$952$	0	0
$953$	−11233.8	−0.381846	−0.190923	−	0.981605i	$-0.561148\pi$
$953$	−11233.8	−0.381846	−0.190923	+	0.981605i	$0.561148\pi$
$954$	0	0
$955$	7145.35	0.242113
$956$	0	0
$957$	−1203.65	−0.0406568
$958$	0	0
$959$	47564.9	1.60162
$960$	0	0
$961$	−13870.8	−0.465605
$962$	0	0
$963$	−94274.1	−3.15466
$964$	0	0
$965$	−4823.75	−0.160914
$966$	0	0
$967$	32213.3	1.07126	0.535630	−	0.844453i	$-0.320074\pi$
$967$	32213.3	1.07126	0.535630	+	0.844453i	$0.320074\pi$
$968$	0	0
$969$	49259.9	1.63308
$970$	0	0
$971$	24606.5	0.813245	0.406622	−	0.913596i	$-0.366706\pi$
$971$	24606.5	0.813245	0.406622	+	0.913596i	$0.366706\pi$
$972$	0	0
$973$	70599.8	2.32613
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32852.8	1.07580	0.537898	−	0.843010i	$-0.319218\pi$
$977$	32852.8	1.07580	0.537898	+	0.843010i	$0.319218\pi$
$978$	0	0
$979$	848.519	0.0277005
$980$	0	0
$981$	−14632.7	−0.476235
$982$	0	0
$983$	−29422.7	−0.954666	−0.477333	−	0.878722i	$-0.658396\pi$
$983$	−29422.7	−0.954666	−0.477333	+	0.878722i	$0.658396\pi$
$984$	0	0
$985$	−32.8475	−0.00106255
$986$	0	0
$987$	−18149.8	−0.585324
$988$	0	0
$989$	−18127.4	−0.582828
$990$	0	0
$991$	−1107.84	−0.0355113	−0.0177557	−	0.999842i	$-0.505652\pi$
$991$	−1107.84	−0.0355113	−0.0177557	+	0.999842i	$0.505652\pi$
$992$	0	0
$993$	21075.9	0.673539
$994$	0	0
$995$	3219.90	0.102591
$996$	0	0
$997$	15369.7	0.488227	0.244114	−	0.969747i	$-0.421503\pi$
$997$	15369.7	0.488227	0.244114	+	0.969747i	$0.421503\pi$
$998$	0	0
$999$	92003.5	2.91377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.4.a.k.1.5		5
13.5	odd	4		104.4.f.a.25.10	yes	10
13.8	odd	4		104.4.f.a.25.9	✓	10
13.12	even	2		1352.4.a.l.1.5		5
39.5	even	4		936.4.c.a.649.5		10
39.8	even	4		936.4.c.a.649.6		10
52.31	even	4		208.4.f.e.129.2		10
52.47	even	4		208.4.f.e.129.1		10
104.5	odd	4		832.4.f.k.129.1		10
104.21	odd	4		832.4.f.k.129.2		10
104.83	even	4		832.4.f.l.129.9		10
104.99	even	4		832.4.f.l.129.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.f.a.25.9	✓	10	13.8	odd	4
104.4.f.a.25.10	yes	10	13.5	odd	4
208.4.f.e.129.1		10	52.47	even	4
208.4.f.e.129.2		10	52.31	even	4
832.4.f.k.129.1		10	104.5	odd	4
832.4.f.k.129.2		10	104.21	odd	4
832.4.f.l.129.9		10	104.83	even	4
832.4.f.l.129.10		10	104.99	even	4
936.4.c.a.649.5		10	39.5	even	4
936.4.c.a.649.6		10	39.8	even	4
1352.4.a.k.1.5		5	1.1	even	1	trivial
1352.4.a.l.1.5		5	13.12	even	2

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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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q+9.27625 q^{3} -2.27999 q^{5} +28.5449 q^{7} +59.0489 q^{9} +0.604833 q^{11} -21.1498 q^{15} +41.0109 q^{17} +129.485 q^{19} +264.790 q^{21} -73.8974 q^{23} -119.802 q^{25} +297.293 q^{27} -214.532 q^{29} +126.175 q^{31} +5.61058 q^{33} -65.0822 q^{35} +309.470 q^{37} +88.1488 q^{41} +245.304 q^{43} -134.631 q^{45} -68.5441 q^{47} +471.814 q^{49} +380.428 q^{51} -613.275 q^{53} -1.37901 q^{55} +1201.14 q^{57} -840.276 q^{59} +587.808 q^{61} +1685.55 q^{63} -606.554 q^{67} -685.491 q^{69} -507.627 q^{71} +177.127 q^{73} -1111.31 q^{75} +17.2649 q^{77} +143.122 q^{79} +1163.45 q^{81} +773.287 q^{83} -93.5046 q^{85} -1990.06 q^{87} +1402.90 q^{89} +1170.43 q^{93} -295.226 q^{95} -468.225 q^{97} +35.7147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 9 q^{5} + 17 q^{7} + 36 q^{9} + 4 q^{11} + 19 q^{15} - 29 q^{17} + 46 q^{19} + 113 q^{21} - 90 q^{23} - 14 q^{25} + 177 q^{27} - 196 q^{29} + 84 q^{31} + 352 q^{33} + 77 q^{35} + 327 q^{37}+ \cdots - 2844 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 9 * q^5 + 17 * q^7 + 36 * q^9 + 4 * q^11 + 19 * q^15 - 29 * q^17 + 46 * q^19 + 113 * q^21 - 90 * q^23 - 14 * q^25 + 177 * q^27 - 196 * q^29 + 84 * q^31 + 352 * q^33 + 77 * q^35 + 327 * q^37 + 480 * q^41 + 117 * q^43 - 198 * q^45 - 647 * q^47 + 64 * q^49 + 255 * q^51 - 622 * q^53 - 288 * q^55 + 538 * q^57 + 610 * q^59 - 28 * q^61 + 1906 * q^63 + 1216 * q^67 - 874 * q^69 - 1381 * q^71 - 906 * q^73 - 236 * q^75 + 152 * q^77 + 954 * q^79 + 1141 * q^81 + 586 * q^83 + 2995 * q^85 - 32 * q^87 + 2618 * q^89 + 3952 * q^93 + 1170 * q^95 - 1188 * q^97 - 2844 * q^99

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