LMFDB - Embedded newform 1352.4.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,4,Mod(1,1352)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	yes
Analytic conductor:	$79.7705823278$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 123x^{4} + 243x^{3} + 3138x^{2} - 3264x - 4096 $ x^6 - 3x^5 - 123x^4 + 243x^3 + 3138x^2 - 3264*x - 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.753179$ 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)

$f(q)$	$=$	$q+0.753179 q^{3} +7.26675 q^{5} +2.85313 q^{7} -26.4327 q^{9} +O(q^{10})$	$q+0.753179 q^{3} +7.26675 q^{5} +2.85313 q^{7} -26.4327 q^{9} -58.0893 q^{11} +5.47317 q^{15} +73.0634 q^{17} +142.815 q^{19} +2.14891 q^{21} +48.8276 q^{23} -72.1943 q^{25} -40.2444 q^{27} +38.8864 q^{29} -153.866 q^{31} -43.7516 q^{33} +20.7330 q^{35} +356.246 q^{37} -3.88294 q^{41} +12.1821 q^{43} -192.080 q^{45} -258.312 q^{47} -334.860 q^{49} +55.0298 q^{51} -302.642 q^{53} -422.120 q^{55} +107.565 q^{57} -110.686 q^{59} -415.136 q^{61} -75.4159 q^{63} -933.920 q^{67} +36.7759 q^{69} -661.899 q^{71} -436.800 q^{73} -54.3753 q^{75} -165.736 q^{77} -318.158 q^{79} +683.372 q^{81} -595.656 q^{83} +530.934 q^{85} +29.2884 q^{87} -1146.68 q^{89} -115.889 q^{93} +1037.80 q^{95} +669.840 q^{97} +1535.46 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 9 q^{5} - q^{7} + 93 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 - 9 * q^5 - q^7 + 93 * q^9	$ 6 q - 3 q^{3} - 9 q^{5} - q^{7} + 93 q^{9} - 11 q^{11} - 168 q^{15} + 50 q^{17} - 211 q^{19} + 83 q^{21} - 103 q^{23} + 393 q^{25} - 243 q^{27} - 48 q^{29} - 190 q^{31} + 133 q^{33} - 226 q^{35} - 476 q^{37} - 10 q^{41} + 13 q^{43} + 57 q^{45} - 122 q^{47} - 31 q^{49} - 745 q^{51} - 483 q^{53} + 1510 q^{55} - 1177 q^{57} + 731 q^{59} - 704 q^{61} + 518 q^{63} - 901 q^{67} + 3479 q^{69} + 673 q^{71} - 1551 q^{73} - 2739 q^{75} + 133 q^{77} - 840 q^{79} + 3018 q^{81} - 2016 q^{83} + 1373 q^{85} - 1485 q^{87} - 295 q^{89} - 3220 q^{93} + 3128 q^{95} - 2575 q^{97} - 458 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 - 9 * q^5 - q^7 + 93 * q^9 - 11 * q^11 - 168 * q^15 + 50 * q^17 - 211 * q^19 + 83 * q^21 - 103 * q^23 + 393 * q^25 - 243 * q^27 - 48 * q^29 - 190 * q^31 + 133 * q^33 - 226 * q^35 - 476 * q^37 - 10 * q^41 + 13 * q^43 + 57 * q^45 - 122 * q^47 - 31 * q^49 - 745 * q^51 - 483 * q^53 + 1510 * q^55 - 1177 * q^57 + 731 * q^59 - 704 * q^61 + 518 * q^63 - 901 * q^67 + 3479 * q^69 + 673 * q^71 - 1551 * q^73 - 2739 * q^75 + 133 * q^77 - 840 * q^79 + 3018 * q^81 - 2016 * q^83 + 1373 * q^85 - 1485 * q^87 - 295 * q^89 - 3220 * q^93 + 3128 * q^95 - 2575 * q^97 - 458 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.753179	0.144949	0.0724747	−	0.997370i	$-0.476910\pi$
$3$	0.753179	0.144949	0.0724747	+	0.997370i	$0.476910\pi$
$4$	0	0
$5$	7.26675	0.649958	0.324979	−	0.945721i	$-0.394643\pi$
$5$	7.26675	0.649958	0.324979	+	0.945721i	$0.394643\pi$
$6$	0	0
$7$	2.85313	0.154054	0.0770271	−	0.997029i	$-0.475457\pi$
$7$	2.85313	0.154054	0.0770271	+	0.997029i	$0.475457\pi$
$8$	0	0
$9$	−26.4327	−0.978990
$10$	0	0
$11$	−58.0893	−1.59223	−0.796117	−	0.605143i	$-0.793116\pi$
$11$	−58.0893	−1.59223	−0.796117	+	0.605143i	$0.793116\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	5.47317	0.0942110
$16$	0	0
$17$	73.0634	1.04238	0.521190	−	0.853440i	$-0.325488\pi$
$17$	73.0634	1.04238	0.521190	+	0.853440i	$0.325488\pi$
$18$	0	0
$19$	142.815	1.72442	0.862212	−	0.506548i	$-0.169079\pi$
$19$	142.815	1.72442	0.862212	+	0.506548i	$0.169079\pi$
$20$	0	0
$21$	2.14891	0.0223301
$22$	0	0
$23$	48.8276	0.442663	0.221332	−	0.975199i	$-0.428960\pi$
$23$	48.8276	0.442663	0.221332	+	0.975199i	$0.428960\pi$
$24$	0	0
$25$	−72.1943	−0.577555
$26$	0	0
$27$	−40.2444	−0.286853
$28$	0	0
$29$	38.8864	0.249001	0.124500	−	0.992220i	$-0.460267\pi$
$29$	38.8864	0.249001	0.124500	+	0.992220i	$0.460267\pi$
$30$	0	0
$31$	−153.866	−0.891459	−0.445730	−	0.895168i	$-0.647056\pi$
$31$	−153.866	−0.891459	−0.445730	+	0.895168i	$0.647056\pi$
$32$	0	0
$33$	−43.7516	−0.230793
$34$	0	0
$35$	20.7330	0.100129
$36$	0	0
$37$	356.246	1.58288	0.791439	−	0.611249i	$-0.209332\pi$
$37$	356.246	1.58288	0.791439	+	0.611249i	$0.209332\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.88294	−0.0147906	−0.00739528	−	0.999973i	$-0.502354\pi$
$41$	−3.88294	−0.0147906	−0.00739528	+	0.999973i	$0.502354\pi$
$42$	0	0
$43$	12.1821	0.0432034	0.0216017	−	0.999767i	$-0.493123\pi$
$43$	12.1821	0.0432034	0.0216017	+	0.999767i	$0.493123\pi$
$44$	0	0
$45$	−192.080	−0.636302
$46$	0	0
$47$	−258.312	−0.801675	−0.400837	−	0.916149i	$-0.631281\pi$
$47$	−258.312	−0.801675	−0.400837	+	0.916149i	$0.631281\pi$
$48$	0	0
$49$	−334.860	−0.976267
$50$	0	0
$51$	55.0298	0.151092
$52$	0	0
$53$	−302.642	−0.784361	−0.392180	−	0.919888i	$-0.628279\pi$
$53$	−302.642	−0.784361	−0.392180	+	0.919888i	$0.628279\pi$
$54$	0	0
$55$	−422.120	−1.03489
$56$	0	0
$57$	107.565	0.249954
$58$	0	0
$59$	−110.686	−0.244238	−0.122119	−	0.992515i	$-0.538969\pi$
$59$	−110.686	−0.244238	−0.122119	+	0.992515i	$0.538969\pi$
$60$	0	0
$61$	−415.136	−0.871355	−0.435678	−	0.900103i	$-0.643491\pi$
$61$	−415.136	−0.871355	−0.435678	+	0.900103i	$0.643491\pi$
$62$	0	0
$63$	−75.4159	−0.150818
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−933.920	−1.70293	−0.851466	−	0.524409i	$-0.824286\pi$
$67$	−933.920	−1.70293	−0.851466	+	0.524409i	$0.824286\pi$
$68$	0	0
$69$	36.7759	0.0641638
$70$	0	0
$71$	−661.899	−1.10638	−0.553190	−	0.833055i	$-0.686590\pi$
$71$	−661.899	−1.10638	−0.553190	+	0.833055i	$0.686590\pi$
$72$	0	0
$73$	−436.800	−0.700323	−0.350162	−	0.936689i	$-0.613873\pi$
$73$	−436.800	−0.700323	−0.350162	+	0.936689i	$0.613873\pi$
$74$	0	0
$75$	−54.3753	−0.0837162
$76$	0	0
$77$	−165.736	−0.245290
$78$	0	0
$79$	−318.158	−0.453109	−0.226554	−	0.973999i	$-0.572746\pi$
$79$	−318.158	−0.453109	−0.226554	+	0.973999i	$0.572746\pi$
$80$	0	0
$81$	683.372	0.937410
$82$	0	0
$83$	−595.656	−0.787732	−0.393866	−	0.919168i	$-0.628863\pi$
$83$	−595.656	−0.787732	−0.393866	+	0.919168i	$0.628863\pi$
$84$	0	0
$85$	530.934	0.677504
$86$	0	0
$87$	29.2884	0.0360925
$88$	0	0
$89$	−1146.68	−1.36571	−0.682853	−	0.730556i	$-0.739261\pi$
$89$	−1146.68	−1.36571	−0.682853	+	0.730556i	$0.739261\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−115.889	−0.129216
$94$	0	0
$95$	1037.80	1.12080
$96$	0	0
$97$	669.840	0.701154	0.350577	−	0.936534i	$-0.385985\pi$
$97$	669.840	0.701154	0.350577	+	0.936534i	$0.385985\pi$
$98$	0	0
$99$	1535.46	1.55878
$100$	0	0
$101$	1433.86	1.41261	0.706307	−	0.707906i	$-0.250360\pi$
$101$	1433.86	1.41261	0.706307	+	0.707906i	$0.250360\pi$
$102$	0	0
$103$	794.312	0.759863	0.379931	−	0.925015i	$-0.375948\pi$
$103$	794.312	0.759863	0.379931	+	0.925015i	$0.375948\pi$
$104$	0	0
$105$	15.6156	0.0145136
$106$	0	0
$107$	1564.54	1.41355	0.706776	−	0.707438i	$-0.250149\pi$
$107$	1564.54	1.41355	0.706776	+	0.707438i	$0.250149\pi$
$108$	0	0
$109$	−2083.84	−1.83115	−0.915575	−	0.402147i	$-0.868264\pi$
$109$	−2083.84	−1.83115	−0.915575	+	0.402147i	$0.868264\pi$
$110$	0	0
$111$	268.317	0.229437
$112$	0	0
$113$	198.368	0.165140	0.0825701	−	0.996585i	$-0.473687\pi$
$113$	198.368	0.165140	0.0825701	+	0.996585i	$0.473687\pi$
$114$	0	0
$115$	354.818	0.287712
$116$	0	0
$117$	0	0
$118$	0	0
$119$	208.459	0.160583
$120$	0	0
$121$	2043.36	1.53521
$122$	0	0
$123$	−2.92455	−0.00214388
$124$	0	0
$125$	−1432.96	−1.02534
$126$	0	0
$127$	−2167.88	−1.51471	−0.757356	−	0.653003i	$-0.773509\pi$
$127$	−2167.88	−1.51471	−0.757356	+	0.653003i	$0.773509\pi$
$128$	0	0
$129$	9.17527	0.00626231
$130$	0	0
$131$	−800.420	−0.533840	−0.266920	−	0.963719i	$-0.586006\pi$
$131$	−800.420	−0.533840	−0.266920	+	0.963719i	$0.586006\pi$
$132$	0	0
$133$	407.470	0.265655
$134$	0	0
$135$	−292.446	−0.186443
$136$	0	0
$137$	1135.55	0.708149	0.354074	−	0.935217i	$-0.384796\pi$
$137$	1135.55	0.708149	0.354074	+	0.935217i	$0.384796\pi$
$138$	0	0
$139$	−2006.92	−1.22464	−0.612318	−	0.790611i	$-0.709763\pi$
$139$	−2006.92	−1.22464	−0.612318	+	0.790611i	$0.709763\pi$
$140$	0	0
$141$	−194.555	−0.116202
$142$	0	0
$143$	0	0
$144$	0	0
$145$	282.578	0.161840
$146$	0	0
$147$	−252.209	−0.141509
$148$	0	0
$149$	−2587.13	−1.42246	−0.711229	−	0.702960i	$-0.751861\pi$
$149$	−2587.13	−1.42246	−0.711229	+	0.702960i	$0.751861\pi$
$150$	0	0
$151$	−1582.79	−0.853016	−0.426508	−	0.904484i	$-0.640256\pi$
$151$	−1582.79	−0.853016	−0.426508	+	0.904484i	$0.640256\pi$
$152$	0	0
$153$	−1931.26	−1.02048
$154$	0	0
$155$	−1118.11	−0.579411
$156$	0	0
$157$	−270.338	−0.137423	−0.0687113	−	0.997637i	$-0.521889\pi$
$157$	−270.338	−0.137423	−0.0687113	+	0.997637i	$0.521889\pi$
$158$	0	0
$159$	−227.944	−0.113693
$160$	0	0
$161$	139.311	0.0681942
$162$	0	0
$163$	142.203	0.0683325	0.0341662	−	0.999416i	$-0.489122\pi$
$163$	142.203	0.0683325	0.0341662	+	0.999416i	$0.489122\pi$
$164$	0	0
$165$	−317.932	−0.150006
$166$	0	0
$167$	−242.913	−0.112558	−0.0562789	−	0.998415i	$-0.517924\pi$
$167$	−242.913	−0.112558	−0.0562789	+	0.998415i	$0.517924\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3774.99	−1.68819
$172$	0	0
$173$	3472.16	1.52592	0.762958	−	0.646448i	$-0.223746\pi$
$173$	3472.16	1.52592	0.762958	+	0.646448i	$0.223746\pi$
$174$	0	0
$175$	−205.979	−0.0889748
$176$	0	0
$177$	−83.3661	−0.0354022
$178$	0	0
$179$	−834.865	−0.348607	−0.174304	−	0.984692i	$-0.555767\pi$
$179$	−834.865	−0.348607	−0.174304	+	0.984692i	$0.555767\pi$
$180$	0	0
$181$	269.143	0.110526	0.0552632	−	0.998472i	$-0.482400\pi$
$181$	269.143	0.110526	0.0552632	+	0.998472i	$0.482400\pi$
$182$	0	0
$183$	−312.671	−0.126302
$184$	0	0
$185$	2588.75	1.02880
$186$	0	0
$187$	−4244.20	−1.65971
$188$	0	0
$189$	−114.822	−0.0441910
$190$	0	0
$191$	−1320.03	−0.500072	−0.250036	−	0.968237i	$-0.580442\pi$
$191$	−1320.03	−0.500072	−0.250036	+	0.968237i	$0.580442\pi$
$192$	0	0
$193$	1299.38	0.484617	0.242309	−	0.970199i	$-0.422095\pi$
$193$	1299.38	0.484617	0.242309	+	0.970199i	$0.422095\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2470.61	−0.893521	−0.446761	−	0.894653i	$-0.647422\pi$
$197$	−2470.61	−0.893521	−0.446761	+	0.894653i	$0.647422\pi$
$198$	0	0
$199$	−1401.88	−0.499380	−0.249690	−	0.968326i	$-0.580329\pi$
$199$	−1401.88	−0.499380	−0.249690	+	0.968326i	$0.580329\pi$
$200$	0	0
$201$	−703.409	−0.246839
$202$	0	0
$203$	110.948	0.0383596
$204$	0	0
$205$	−28.2163	−0.00961325
$206$	0	0
$207$	−1290.65	−0.433363
$208$	0	0
$209$	−8296.03	−2.74569
$210$	0	0
$211$	2691.00	0.877991	0.438996	−	0.898489i	$-0.355334\pi$
$211$	2691.00	0.877991	0.438996	+	0.898489i	$0.355334\pi$
$212$	0	0
$213$	−498.528	−0.160369
$214$	0	0
$215$	88.5240	0.0280804
$216$	0	0
$217$	−439.000	−0.137333
$218$	0	0
$219$	−328.989	−0.101511
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5439.78	−1.63352	−0.816759	−	0.576979i	$-0.804231\pi$
$223$	−5439.78	−1.63352	−0.816759	+	0.576979i	$0.804231\pi$
$224$	0	0
$225$	1908.29	0.565420
$226$	0	0
$227$	4040.78	1.18148	0.590740	−	0.806862i	$-0.298836\pi$
$227$	4040.78	1.18148	0.590740	+	0.806862i	$0.298836\pi$
$228$	0	0
$229$	−1072.07	−0.309365	−0.154682	−	0.987964i	$-0.549435\pi$
$229$	−1072.07	−0.309365	−0.154682	+	0.987964i	$0.549435\pi$
$230$	0	0
$231$	−124.829	−0.0355547
$232$	0	0
$233$	4062.66	1.14229	0.571145	−	0.820849i	$-0.306499\pi$
$233$	4062.66	1.14229	0.571145	+	0.820849i	$0.306499\pi$
$234$	0	0
$235$	−1877.09	−0.521055
$236$	0	0
$237$	−239.630	−0.0656778
$238$	0	0
$239$	−4371.13	−1.18303	−0.591516	−	0.806293i	$-0.701470\pi$
$239$	−4371.13	−1.18303	−0.591516	+	0.806293i	$0.701470\pi$
$240$	0	0
$241$	3447.22	0.921390	0.460695	−	0.887558i	$-0.347600\pi$
$241$	3447.22	0.921390	0.460695	+	0.887558i	$0.347600\pi$
$242$	0	0
$243$	1601.30	0.422730
$244$	0	0
$245$	−2433.34	−0.634533
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−448.636	−0.114181
$250$	0	0
$251$	6733.95	1.69340	0.846700	−	0.532071i	$-0.178586\pi$
$251$	6733.95	1.69340	0.846700	+	0.532071i	$0.178586\pi$
$252$	0	0
$253$	−2836.36	−0.704823
$254$	0	0
$255$	399.888	0.0982038
$256$	0	0
$257$	3000.95	0.728381	0.364190	−	0.931325i	$-0.381346\pi$
$257$	3000.95	0.728381	0.364190	+	0.931325i	$0.381346\pi$
$258$	0	0
$259$	1016.41	0.243849
$260$	0	0
$261$	−1027.87	−0.243769
$262$	0	0
$263$	−3161.43	−0.741224	−0.370612	−	0.928788i	$-0.620852\pi$
$263$	−3161.43	−0.741224	−0.370612	+	0.928788i	$0.620852\pi$
$264$	0	0
$265$	−2199.23	−0.509802
$266$	0	0
$267$	−863.656	−0.197958
$268$	0	0
$269$	−4342.67	−0.984303	−0.492151	−	0.870510i	$-0.663789\pi$
$269$	−4342.67	−0.984303	−0.492151	+	0.870510i	$0.663789\pi$
$270$	0	0
$271$	−5871.90	−1.31621	−0.658104	−	0.752927i	$-0.728641\pi$
$271$	−5871.90	−1.31621	−0.658104	+	0.752927i	$0.728641\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4193.72	0.919602
$276$	0	0
$277$	5414.31	1.17442	0.587210	−	0.809435i	$-0.300226\pi$
$277$	5414.31	1.17442	0.587210	+	0.809435i	$0.300226\pi$
$278$	0	0
$279$	4067.11	0.872729
$280$	0	0
$281$	−6679.68	−1.41806	−0.709032	−	0.705176i	$-0.750868\pi$
$281$	−6679.68	−1.41806	−0.709032	+	0.705176i	$0.750868\pi$
$282$	0	0
$283$	−8244.76	−1.73180	−0.865901	−	0.500215i	$-0.833254\pi$
$283$	−8244.76	−1.73180	−0.865901	+	0.500215i	$0.833254\pi$
$284$	0	0
$285$	781.651	0.162460
$286$	0	0
$287$	−11.0785	−0.00227855
$288$	0	0
$289$	425.259	0.0865579
$290$	0	0
$291$	504.510	0.101632
$292$	0	0
$293$	539.918	0.107653	0.0538265	−	0.998550i	$-0.482858\pi$
$293$	539.918	0.107653	0.0538265	+	0.998550i	$0.482858\pi$
$294$	0	0
$295$	−804.325	−0.158744
$296$	0	0
$297$	2337.77	0.456738
$298$	0	0
$299$	0	0
$300$	0	0
$301$	34.7569	0.00665567
$302$	0	0
$303$	1079.95	0.204758
$304$	0	0
$305$	−3016.69	−0.566344
$306$	0	0
$307$	4032.04	0.749578	0.374789	−	0.927110i	$-0.377715\pi$
$307$	4032.04	0.749578	0.374789	+	0.927110i	$0.377715\pi$
$308$	0	0
$309$	598.259	0.110142
$310$	0	0
$311$	−4146.61	−0.756054	−0.378027	−	0.925794i	$-0.623397\pi$
$311$	−4146.61	−0.756054	−0.378027	+	0.925794i	$0.623397\pi$
$312$	0	0
$313$	4998.17	0.902597	0.451299	−	0.892373i	$-0.350961\pi$
$313$	4998.17	0.902597	0.451299	+	0.892373i	$0.350961\pi$
$314$	0	0
$315$	−548.028	−0.0980251
$316$	0	0
$317$	−65.9112	−0.0116780	−0.00583902	−	0.999983i	$-0.501859\pi$
$317$	−65.9112	−0.0116780	−0.00583902	+	0.999983i	$0.501859\pi$
$318$	0	0
$319$	−2258.88	−0.396467
$320$	0	0
$321$	1178.38	0.204893
$322$	0	0
$323$	10434.6	1.79751
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1569.50	−0.265424
$328$	0	0
$329$	−736.997	−0.123501
$330$	0	0
$331$	−4073.09	−0.676367	−0.338184	−	0.941080i	$-0.609812\pi$
$331$	−4073.09	−0.676367	−0.338184	+	0.941080i	$0.609812\pi$
$332$	0	0
$333$	−9416.55	−1.54962
$334$	0	0
$335$	−6786.57	−1.10683
$336$	0	0
$337$	−6440.94	−1.04113	−0.520564	−	0.853822i	$-0.674278\pi$
$337$	−6440.94	−1.04113	−0.520564	+	0.853822i	$0.674278\pi$
$338$	0	0
$339$	149.406	0.0239370
$340$	0	0
$341$	8937.99	1.41941
$342$	0	0
$343$	−1934.02	−0.304452
$344$	0	0
$345$	267.241	0.0417037
$346$	0	0
$347$	−8303.40	−1.28458	−0.642291	−	0.766461i	$-0.722016\pi$
$347$	−8303.40	−1.28458	−0.642291	+	0.766461i	$0.722016\pi$
$348$	0	0
$349$	650.474	0.0997681	0.0498841	−	0.998755i	$-0.484115\pi$
$349$	650.474	0.0997681	0.0498841	+	0.998755i	$0.484115\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3042.67	0.458768	0.229384	−	0.973336i	$-0.426329\pi$
$353$	3042.67	0.458768	0.229384	+	0.973336i	$0.426329\pi$
$354$	0	0
$355$	−4809.86	−0.719100
$356$	0	0
$357$	157.007	0.0232764
$358$	0	0
$359$	13014.1	1.91325	0.956627	−	0.291317i	$-0.0940934\pi$
$359$	13014.1	1.91325	0.956627	+	0.291317i	$0.0940934\pi$
$360$	0	0
$361$	13537.2	1.97364
$362$	0	0
$363$	1539.02	0.222528
$364$	0	0
$365$	−3174.12	−0.455181
$366$	0	0
$367$	8850.83	1.25888	0.629441	−	0.777048i	$-0.283284\pi$
$367$	8850.83	1.25888	0.629441	+	0.777048i	$0.283284\pi$
$368$	0	0
$369$	102.637	0.0144798
$370$	0	0
$371$	−863.476	−0.120834
$372$	0	0
$373$	10671.7	1.48139	0.740694	−	0.671843i	$-0.234497\pi$
$373$	10671.7	1.48139	0.740694	+	0.671843i	$0.234497\pi$
$374$	0	0
$375$	−1079.28	−0.148623
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7451.00	−1.00985	−0.504923	−	0.863164i	$-0.668479\pi$
$379$	−7451.00	−1.00985	−0.504923	+	0.863164i	$0.668479\pi$
$380$	0	0
$381$	−1632.80	−0.219556
$382$	0	0
$383$	−641.269	−0.0855543	−0.0427772	−	0.999085i	$-0.513621\pi$
$383$	−641.269	−0.0855543	−0.0427772	+	0.999085i	$0.513621\pi$
$384$	0	0
$385$	−1204.36	−0.159428
$386$	0	0
$387$	−322.005	−0.0422957
$388$	0	0
$389$	4522.54	0.589465	0.294732	−	0.955580i	$-0.404769\pi$
$389$	4522.54	0.589465	0.294732	+	0.955580i	$0.404769\pi$
$390$	0	0
$391$	3567.51	0.461424
$392$	0	0
$393$	−602.859	−0.0773797
$394$	0	0
$395$	−2311.98	−0.294502
$396$	0	0
$397$	7380.01	0.932978	0.466489	−	0.884527i	$-0.345519\pi$
$397$	7380.01	0.932978	0.466489	+	0.884527i	$0.345519\pi$
$398$	0	0
$399$	306.898	0.0385065
$400$	0	0
$401$	−9770.19	−1.21671	−0.608354	−	0.793666i	$-0.708170\pi$
$401$	−9770.19	−1.21671	−0.608354	+	0.793666i	$0.708170\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4965.90	0.609277
$406$	0	0
$407$	−20694.1	−2.52031
$408$	0	0
$409$	−7925.84	−0.958209	−0.479104	−	0.877758i	$-0.659038\pi$
$409$	−7925.84	−0.958209	−0.479104	+	0.877758i	$0.659038\pi$
$410$	0	0
$411$	855.271	0.102646
$412$	0	0
$413$	−315.800	−0.0376259
$414$	0	0
$415$	−4328.48	−0.511992
$416$	0	0
$417$	−1511.57	−0.177510
$418$	0	0
$419$	−15583.2	−1.81692	−0.908462	−	0.417967i	$-0.862743\pi$
$419$	−15583.2	−1.81692	−0.908462	+	0.417967i	$0.862743\pi$
$420$	0	0
$421$	15726.7	1.82060	0.910298	−	0.413954i	$-0.135852\pi$
$421$	15726.7	1.82060	0.910298	+	0.413954i	$0.135852\pi$
$422$	0	0
$423$	6827.89	0.784831
$424$	0	0
$425$	−5274.76	−0.602032
$426$	0	0
$427$	−1184.43	−0.134236
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9844.23	−1.10019	−0.550093	−	0.835104i	$-0.685408\pi$
$431$	−9844.23	−1.10019	−0.550093	+	0.835104i	$0.685408\pi$
$432$	0	0
$433$	12317.4	1.36706	0.683531	−	0.729921i	$-0.260443\pi$
$433$	12317.4	1.36706	0.683531	+	0.729921i	$0.260443\pi$
$434$	0	0
$435$	212.832	0.0234586
$436$	0	0
$437$	6973.32	0.763339
$438$	0	0
$439$	9182.20	0.998274	0.499137	−	0.866523i	$-0.333650\pi$
$439$	9182.20	0.998274	0.499137	+	0.866523i	$0.333650\pi$
$440$	0	0
$441$	8851.25	0.955756
$442$	0	0
$443$	8288.47	0.888933	0.444466	−	0.895795i	$-0.353393\pi$
$443$	8288.47	0.888933	0.444466	+	0.895795i	$0.353393\pi$
$444$	0	0
$445$	−8332.64	−0.887652
$446$	0	0
$447$	−1948.58	−0.206184
$448$	0	0
$449$	−10295.4	−1.08212	−0.541058	−	0.840986i	$-0.681976\pi$
$449$	−10295.4	−1.08212	−0.541058	+	0.840986i	$0.681976\pi$
$450$	0	0
$451$	225.557	0.0235500
$452$	0	0
$453$	−1192.12	−0.123644
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12210.5	1.24985	0.624924	−	0.780685i	$-0.285130\pi$
$457$	12210.5	1.24985	0.624924	+	0.780685i	$0.285130\pi$
$458$	0	0
$459$	−2940.39	−0.299010
$460$	0	0
$461$	9114.70	0.920854	0.460427	−	0.887697i	$-0.347696\pi$
$461$	9114.70	0.920854	0.460427	+	0.887697i	$0.347696\pi$
$462$	0	0
$463$	−0.376664	−3.78079e−5 0	−1.89040e−5	−	1.00000i	$-0.500006\pi$
$463$	−0.376664	−3.78079e−5 0	−1.89040e−5		1.00000i	$0.500006\pi$
$464$	0	0
$465$	−842.137	−0.0839853
$466$	0	0
$467$	5771.35	0.571877	0.285938	−	0.958248i	$-0.407695\pi$
$467$	5771.35	0.571877	0.285938	+	0.958248i	$0.407695\pi$
$468$	0	0
$469$	−2664.59	−0.262344
$470$	0	0
$471$	−203.613	−0.0199193
$472$	0	0
$473$	−707.647	−0.0687899
$474$	0	0
$475$	−10310.4	−0.995949
$476$	0	0
$477$	7999.66	0.767881
$478$	0	0
$479$	−8609.07	−0.821207	−0.410604	−	0.911814i	$-0.634682\pi$
$479$	−8609.07	−0.821207	−0.410604	+	0.911814i	$0.634682\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	104.926	0.00988470
$484$	0	0
$485$	4867.56	0.455721
$486$	0	0
$487$	3635.16	0.338244	0.169122	−	0.985595i	$-0.445907\pi$
$487$	3635.16	0.338244	0.169122	+	0.985595i	$0.445907\pi$
$488$	0	0
$489$	107.104	0.00990475
$490$	0	0
$491$	−17387.5	−1.59814	−0.799068	−	0.601241i	$-0.794673\pi$
$491$	−17387.5	−1.59814	−0.799068	+	0.601241i	$0.794673\pi$
$492$	0	0
$493$	2841.17	0.259554
$494$	0	0
$495$	11157.8	1.01314
$496$	0	0
$497$	−1888.48	−0.170443
$498$	0	0
$499$	1174.90	0.105403	0.0527014	−	0.998610i	$-0.483217\pi$
$499$	1174.90	0.105403	0.0527014	+	0.998610i	$0.483217\pi$
$500$	0	0
$501$	−182.957	−0.0163152
$502$	0	0
$503$	21573.0	1.91231	0.956155	−	0.292862i	$-0.0946077\pi$
$503$	21573.0	1.91231	0.956155	+	0.292862i	$0.0946077\pi$
$504$	0	0
$505$	10419.5	0.918140
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21860.2	1.90361	0.951804	−	0.306708i	$-0.0992274\pi$
$509$	21860.2	1.90361	0.951804	+	0.306708i	$0.0992274\pi$
$510$	0	0
$511$	−1246.25	−0.107888
$512$	0	0
$513$	−5747.51	−0.494657
$514$	0	0
$515$	5772.07	0.493879
$516$	0	0
$517$	15005.2	1.27645
$518$	0	0
$519$	2615.16	0.221180
$520$	0	0
$521$	18858.4	1.58580	0.792899	−	0.609353i	$-0.208571\pi$
$521$	18858.4	1.58580	0.792899	+	0.609353i	$0.208571\pi$
$522$	0	0
$523$	−14433.7	−1.20677	−0.603386	−	0.797450i	$-0.706182\pi$
$523$	−14433.7	−1.20677	−0.603386	+	0.797450i	$0.706182\pi$
$524$	0	0
$525$	−155.139	−0.0128968
$526$	0	0
$527$	−11242.0	−0.929240
$528$	0	0
$529$	−9782.87	−0.804049
$530$	0	0
$531$	2925.72	0.239107
$532$	0	0
$533$	0	0
$534$	0	0
$535$	11369.1	0.918749
$536$	0	0
$537$	−628.803	−0.0505304
$538$	0	0
$539$	19451.8	1.55445
$540$	0	0
$541$	−9419.76	−0.748590	−0.374295	−	0.927310i	$-0.622115\pi$
$541$	−9419.76	−0.748590	−0.374295	+	0.927310i	$0.622115\pi$
$542$	0	0
$543$	202.713	0.0160207
$544$	0	0
$545$	−15142.7	−1.19017
$546$	0	0
$547$	−12270.8	−0.959162	−0.479581	−	0.877498i	$-0.659211\pi$
$547$	−12270.8	−0.959162	−0.479581	+	0.877498i	$0.659211\pi$
$548$	0	0
$549$	10973.2	0.853048
$550$	0	0
$551$	5553.57	0.429383
$552$	0	0
$553$	−907.745	−0.0698034
$554$	0	0
$555$	1949.79	0.149124
$556$	0	0
$557$	18279.2	1.39051	0.695254	−	0.718765i	$-0.255292\pi$
$557$	18279.2	1.39051	0.695254	+	0.718765i	$0.255292\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3196.64	−0.240575
$562$	0	0
$563$	−23994.1	−1.79615	−0.898073	−	0.439847i	$-0.855033\pi$
$563$	−23994.1	−1.79615	−0.898073	+	0.439847i	$0.855033\pi$
$564$	0	0
$565$	1441.49	0.107334
$566$	0	0
$567$	1949.75	0.144412
$568$	0	0
$569$	10263.6	0.756191	0.378096	−	0.925767i	$-0.376579\pi$
$569$	10263.6	0.756191	0.378096	+	0.925767i	$0.376579\pi$
$570$	0	0
$571$	−18173.6	−1.33195	−0.665974	−	0.745975i	$-0.731984\pi$
$571$	−18173.6	−1.33195	−0.665974	+	0.745975i	$0.731984\pi$
$572$	0	0
$573$	−994.216	−0.0724851
$574$	0	0
$575$	−3525.07	−0.255662
$576$	0	0
$577$	21688.8	1.56484	0.782422	−	0.622748i	$-0.213984\pi$
$577$	21688.8	1.56484	0.782422	+	0.622748i	$0.213984\pi$
$578$	0	0
$579$	978.663	0.0702450
$580$	0	0
$581$	−1699.48	−0.121353
$582$	0	0
$583$	17580.3	1.24889
$584$	0	0
$585$	0	0
$586$	0	0
$587$	929.325	0.0653447	0.0326724	−	0.999466i	$-0.489598\pi$
$587$	929.325	0.0653447	0.0326724	+	0.999466i	$0.489598\pi$
$588$	0	0
$589$	−21974.5	−1.53725
$590$	0	0
$591$	−1860.81	−0.129515
$592$	0	0
$593$	−8355.69	−0.578629	−0.289315	−	0.957234i	$-0.593427\pi$
$593$	−8355.69	−0.578629	−0.289315	+	0.957234i	$0.593427\pi$
$594$	0	0
$595$	1514.82	0.104372
$596$	0	0
$597$	−1055.87	−0.0723848
$598$	0	0
$599$	241.352	0.0164631	0.00823154	−	0.999966i	$-0.497380\pi$
$599$	241.352	0.0164631	0.00823154	+	0.999966i	$0.497380\pi$
$600$	0	0
$601$	−8607.03	−0.584173	−0.292087	−	0.956392i	$-0.594350\pi$
$601$	−8607.03	−0.584173	−0.292087	+	0.956392i	$0.594350\pi$
$602$	0	0
$603$	24686.1	1.66715
$604$	0	0
$605$	14848.6	0.997821
$606$	0	0
$607$	−4892.10	−0.327124	−0.163562	−	0.986533i	$-0.552298\pi$
$607$	−4892.10	−0.327124	−0.163562	+	0.986533i	$0.552298\pi$
$608$	0	0
$609$	83.5635	0.00556021
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−3875.72	−0.255365	−0.127682	−	0.991815i	$-0.540754\pi$
$613$	−3875.72	−0.255365	−0.127682	+	0.991815i	$0.540754\pi$
$614$	0	0
$615$	−21.2520	−0.00139343
$616$	0	0
$617$	8663.76	0.565300	0.282650	−	0.959223i	$-0.408787\pi$
$617$	8663.76	0.565300	0.282650	+	0.959223i	$0.408787\pi$
$618$	0	0
$619$	3269.09	0.212271	0.106136	−	0.994352i	$-0.466152\pi$
$619$	3269.09	0.212271	0.106136	+	0.994352i	$0.466152\pi$
$620$	0	0
$621$	−1965.04	−0.126979
$622$	0	0
$623$	−3271.62	−0.210393
$624$	0	0
$625$	−1388.69	−0.0888761
$626$	0	0
$627$	−6248.40	−0.397985
$628$	0	0
$629$	26028.5	1.64996
$630$	0	0
$631$	−11170.0	−0.704710	−0.352355	−	0.935866i	$-0.614619\pi$
$631$	−11170.0	−0.704710	−0.352355	+	0.935866i	$0.614619\pi$
$632$	0	0
$633$	2026.81	0.127264
$634$	0	0
$635$	−15753.5	−0.984499
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17495.8	1.08313
$640$	0	0
$641$	−20320.3	−1.25211	−0.626057	−	0.779778i	$-0.715332\pi$
$641$	−20320.3	−1.25211	−0.626057	+	0.779778i	$0.715332\pi$
$642$	0	0
$643$	10873.6	0.666892	0.333446	−	0.942769i	$-0.391788\pi$
$643$	10873.6	0.666892	0.333446	+	0.942769i	$0.391788\pi$
$644$	0	0
$645$	66.6744	0.00407024
$646$	0	0
$647$	7533.72	0.457776	0.228888	−	0.973453i	$-0.426491\pi$
$647$	7533.72	0.457776	0.228888	+	0.973453i	$0.426491\pi$
$648$	0	0
$649$	6429.65	0.388884
$650$	0	0
$651$	−330.646	−0.0199063
$652$	0	0
$653$	−3839.50	−0.230094	−0.115047	−	0.993360i	$-0.536702\pi$
$653$	−3839.50	−0.230094	−0.115047	+	0.993360i	$0.536702\pi$
$654$	0	0
$655$	−5816.45	−0.346973
$656$	0	0
$657$	11545.8	0.685609
$658$	0	0
$659$	−5500.42	−0.325138	−0.162569	−	0.986697i	$-0.551978\pi$
$659$	−5500.42	−0.325138	−0.162569	+	0.986697i	$0.551978\pi$
$660$	0	0
$661$	−13859.1	−0.815517	−0.407759	−	0.913090i	$-0.633690\pi$
$661$	−13859.1	−0.815517	−0.407759	+	0.913090i	$0.633690\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2960.98	0.172665
$666$	0	0
$667$	1898.73	0.110223
$668$	0	0
$669$	−4097.13	−0.236777
$670$	0	0
$671$	24114.9	1.38740
$672$	0	0
$673$	2069.86	0.118554	0.0592772	−	0.998242i	$-0.481120\pi$
$673$	2069.86	0.118554	0.0592772	+	0.998242i	$0.481120\pi$
$674$	0	0
$675$	2905.42	0.165673
$676$	0	0
$677$	7030.18	0.399102	0.199551	−	0.979887i	$-0.436052\pi$
$677$	7030.18	0.399102	0.199551	+	0.979887i	$0.436052\pi$
$678$	0	0
$679$	1911.14	0.108016
$680$	0	0
$681$	3043.43	0.171255
$682$	0	0
$683$	−3831.34	−0.214644	−0.107322	−	0.994224i	$-0.534228\pi$
$683$	−3831.34	−0.214644	−0.107322	+	0.994224i	$0.534228\pi$
$684$	0	0
$685$	8251.74	0.460267
$686$	0	0
$687$	−807.463	−0.0448422
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−8186.94	−0.450718	−0.225359	−	0.974276i	$-0.572355\pi$
$691$	−8186.94	−0.450718	−0.225359	+	0.974276i	$0.572355\pi$
$692$	0	0
$693$	4380.85	0.240137
$694$	0	0
$695$	−14583.8	−0.795962
$696$	0	0
$697$	−283.701	−0.0154174
$698$	0	0
$699$	3059.91	0.165574
$700$	0	0
$701$	−2297.51	−0.123789	−0.0618943	−	0.998083i	$-0.519714\pi$
$701$	−2297.51	−0.123789	−0.0618943	+	0.998083i	$0.519714\pi$
$702$	0	0
$703$	50877.3	2.72955
$704$	0	0
$705$	−1413.79	−0.0755266
$706$	0	0
$707$	4090.97	0.217619
$708$	0	0
$709$	−28328.7	−1.50057	−0.750285	−	0.661114i	$-0.770084\pi$
$709$	−28328.7	−1.50057	−0.750285	+	0.661114i	$0.770084\pi$
$710$	0	0
$711$	8409.79	0.443589
$712$	0	0
$713$	−7512.93	−0.394616
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3292.24	−0.171480
$718$	0	0
$719$	1170.26	0.0607001	0.0303500	−	0.999539i	$-0.490338\pi$
$719$	1170.26	0.0607001	0.0303500	+	0.999539i	$0.490338\pi$
$720$	0	0
$721$	2266.27	0.117060
$722$	0	0
$723$	2596.38	0.133555
$724$	0	0
$725$	−2807.38	−0.143812
$726$	0	0
$727$	30312.6	1.54640	0.773201	−	0.634162i	$-0.218655\pi$
$727$	30312.6	1.54640	0.773201	+	0.634162i	$0.218655\pi$
$728$	0	0
$729$	−17245.0	−0.876136
$730$	0	0
$731$	890.062	0.0450344
$732$	0	0
$733$	−27438.1	−1.38260	−0.691302	−	0.722565i	$-0.742963\pi$
$733$	−27438.1	−1.38260	−0.691302	+	0.722565i	$0.742963\pi$
$734$	0	0
$735$	−1832.74	−0.0919751
$736$	0	0
$737$	54250.7	2.71147
$738$	0	0
$739$	17327.6	0.862526	0.431263	−	0.902226i	$-0.358068\pi$
$739$	17327.6	0.862526	0.431263	+	0.902226i	$0.358068\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17296.0	0.854007	0.427004	−	0.904250i	$-0.359569\pi$
$743$	17296.0	0.854007	0.427004	+	0.904250i	$0.359569\pi$
$744$	0	0
$745$	−18800.1	−0.924538
$746$	0	0
$747$	15744.8	0.771181
$748$	0	0
$749$	4463.83	0.217764
$750$	0	0
$751$	−11089.6	−0.538835	−0.269417	−	0.963024i	$-0.586831\pi$
$751$	−11089.6	−0.538835	−0.269417	+	0.963024i	$0.586831\pi$
$752$	0	0
$753$	5071.87	0.245457
$754$	0	0
$755$	−11501.7	−0.554425
$756$	0	0
$757$	−23761.4	−1.14085	−0.570425	−	0.821350i	$-0.693221\pi$
$757$	−23761.4	−1.14085	−0.570425	+	0.821350i	$0.693221\pi$
$758$	0	0
$759$	−2136.29	−0.102164
$760$	0	0
$761$	−17689.7	−0.842641	−0.421321	−	0.906912i	$-0.638433\pi$
$761$	−17689.7	−0.842641	−0.421321	+	0.906912i	$0.638433\pi$
$762$	0	0
$763$	−5945.45	−0.282097
$764$	0	0
$765$	−14034.0	−0.663269
$766$	0	0
$767$	0	0
$768$	0	0
$769$	28732.2	1.34735	0.673673	−	0.739029i	$-0.264716\pi$
$769$	28732.2	1.34735	0.673673	+	0.739029i	$0.264716\pi$
$770$	0	0
$771$	2260.25	0.105578
$772$	0	0
$773$	−5188.28	−0.241410	−0.120705	−	0.992688i	$-0.538515\pi$
$773$	−5188.28	−0.241410	−0.120705	+	0.992688i	$0.538515\pi$
$774$	0	0
$775$	11108.3	0.514866
$776$	0	0
$777$	765.542	0.0353458
$778$	0	0
$779$	−554.543	−0.0255052
$780$	0	0
$781$	38449.2	1.76162
$782$	0	0
$783$	−1564.96	−0.0714267
$784$	0	0
$785$	−1964.48	−0.0893189
$786$	0	0
$787$	15807.0	0.715956	0.357978	−	0.933730i	$-0.383466\pi$
$787$	15807.0	0.715956	0.357978	+	0.933730i	$0.383466\pi$
$788$	0	0
$789$	−2381.12	−0.107440
$790$	0	0
$791$	565.967	0.0254406
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1656.41	−0.0738954
$796$	0	0
$797$	22603.4	1.00458	0.502292	−	0.864698i	$-0.332490\pi$
$797$	22603.4	1.00458	0.502292	+	0.864698i	$0.332490\pi$
$798$	0	0
$799$	−18873.2	−0.835650
$800$	0	0
$801$	30309.9	1.33701
$802$	0	0
$803$	25373.4	1.11508
$804$	0	0
$805$	1012.34	0.0443233
$806$	0	0
$807$	−3270.81	−0.142674
$808$	0	0
$809$	−9902.20	−0.430337	−0.215169	−	0.976577i	$-0.569030\pi$
$809$	−9902.20	−0.430337	−0.215169	+	0.976577i	$0.569030\pi$
$810$	0	0
$811$	−3136.80	−0.135818	−0.0679088	−	0.997692i	$-0.521633\pi$
$811$	−3136.80	−0.135818	−0.0679088	+	0.997692i	$0.521633\pi$
$812$	0	0
$813$	−4422.59	−0.190784
$814$	0	0
$815$	1033.35	0.0444133
$816$	0	0
$817$	1739.78	0.0745010
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−773.946	−0.0329000	−0.0164500	−	0.999865i	$-0.505236\pi$
$821$	−773.946	−0.0329000	−0.0164500	+	0.999865i	$0.505236\pi$
$822$	0	0
$823$	−12381.7	−0.524423	−0.262212	−	0.965010i	$-0.584452\pi$
$823$	−12381.7	−0.524423	−0.262212	+	0.965010i	$0.584452\pi$
$824$	0	0
$825$	3158.62	0.133296
$826$	0	0
$827$	−39362.3	−1.65509	−0.827545	−	0.561399i	$-0.810263\pi$
$827$	−39362.3	−1.65509	−0.827545	+	0.561399i	$0.810263\pi$
$828$	0	0
$829$	11667.1	0.488800	0.244400	−	0.969674i	$-0.421409\pi$
$829$	11667.1	0.488800	0.244400	+	0.969674i	$0.421409\pi$
$830$	0	0
$831$	4077.94	0.170231
$832$	0	0
$833$	−24466.0	−1.01764
$834$	0	0
$835$	−1765.19	−0.0731578
$836$	0	0
$837$	6192.26	0.255718
$838$	0	0
$839$	46018.4	1.89360	0.946801	−	0.321821i	$-0.104295\pi$
$839$	46018.4	1.89360	0.946801	+	0.321821i	$0.104295\pi$
$840$	0	0
$841$	−22876.8	−0.937999
$842$	0	0
$843$	−5030.99	−0.205548
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5829.97	0.236505
$848$	0	0
$849$	−6209.78	−0.251024
$850$	0	0
$851$	17394.6	0.700682
$852$	0	0
$853$	6667.50	0.267633	0.133817	−	0.991006i	$-0.457277\pi$
$853$	6667.50	0.267633	0.133817	+	0.991006i	$0.457277\pi$
$854$	0	0
$855$	−27431.9	−1.09725
$856$	0	0
$857$	23981.7	0.955890	0.477945	−	0.878390i	$-0.341382\pi$
$857$	23981.7	0.955890	0.477945	+	0.878390i	$0.341382\pi$
$858$	0	0
$859$	4760.36	0.189082	0.0945409	−	0.995521i	$-0.469862\pi$
$859$	4760.36	0.189082	0.0945409	+	0.995521i	$0.469862\pi$
$860$	0	0
$861$	−8.34410	−0.000330274 0
$862$	0	0
$863$	49009.9	1.93316	0.966580	−	0.256363i	$-0.0825244\pi$
$863$	49009.9	1.93316	0.966580	+	0.256363i	$0.0825244\pi$
$864$	0	0
$865$	25231.3	0.991781
$866$	0	0
$867$	320.296	0.0125465
$868$	0	0
$869$	18481.6	0.721455
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−17705.7	−0.686423
$874$	0	0
$875$	−4088.42	−0.157959
$876$	0	0
$877$	−48161.3	−1.85438	−0.927189	−	0.374593i	$-0.877782\pi$
$877$	−48161.3	−1.85438	−0.927189	+	0.374593i	$0.877782\pi$
$878$	0	0
$879$	406.655	0.0156042
$880$	0	0
$881$	−25752.2	−0.984804	−0.492402	−	0.870368i	$-0.663881\pi$
$881$	−25752.2	−0.984804	−0.492402	+	0.870368i	$0.663881\pi$
$882$	0	0
$883$	−41104.6	−1.56657	−0.783285	−	0.621663i	$-0.786457\pi$
$883$	−41104.6	−1.56657	−0.783285	+	0.621663i	$0.786457\pi$
$884$	0	0
$885$	−605.801	−0.0230099
$886$	0	0
$887$	48960.9	1.85338	0.926689	−	0.375829i	$-0.122642\pi$
$887$	48960.9	1.85338	0.926689	+	0.375829i	$0.122642\pi$
$888$	0	0
$889$	−6185.24	−0.233348
$890$	0	0
$891$	−39696.6	−1.49258
$892$	0	0
$893$	−36890.9	−1.38243
$894$	0	0
$895$	−6066.75	−0.226580
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5983.31	−0.221974
$900$	0	0
$901$	−22112.1	−0.817603
$902$	0	0
$903$	26.1782	0.000964735 0
$904$	0	0
$905$	1955.80	0.0718375
$906$	0	0
$907$	42398.2	1.55216	0.776081	−	0.630634i	$-0.217205\pi$
$907$	42398.2	1.55216	0.776081	+	0.630634i	$0.217205\pi$
$908$	0	0
$909$	−37900.7	−1.38293
$910$	0	0
$911$	−8150.26	−0.296411	−0.148205	−	0.988957i	$-0.547350\pi$
$911$	−8150.26	−0.296411	−0.148205	+	0.988957i	$0.547350\pi$
$912$	0	0
$913$	34601.2	1.25425
$914$	0	0
$915$	−2272.11	−0.0820913
$916$	0	0
$917$	−2283.70	−0.0822403
$918$	0	0
$919$	−27015.2	−0.969693	−0.484847	−	0.874599i	$-0.661125\pi$
$919$	−27015.2	−0.969693	−0.484847	+	0.874599i	$0.661125\pi$
$920$	0	0
$921$	3036.85	0.108651
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−25718.9	−0.914198
$926$	0	0
$927$	−20995.8	−0.743898
$928$	0	0
$929$	18722.9	0.661227	0.330613	−	0.943766i	$-0.392744\pi$
$929$	18722.9	0.661227	0.330613	+	0.943766i	$0.392744\pi$
$930$	0	0
$931$	−47823.0	−1.68350
$932$	0	0
$933$	−3123.14	−0.109590
$934$	0	0
$935$	−30841.5	−1.07874
$936$	0	0
$937$	15955.0	0.556274	0.278137	−	0.960541i	$-0.410283\pi$
$937$	15955.0	0.556274	0.278137	+	0.960541i	$0.410283\pi$
$938$	0	0
$939$	3764.51	0.130831
$940$	0	0
$941$	−9021.18	−0.312521	−0.156260	−	0.987716i	$-0.549944\pi$
$941$	−9021.18	−0.312521	−0.156260	+	0.987716i	$0.549944\pi$
$942$	0	0
$943$	−189.594	−0.00654724
$944$	0	0
$945$	−834.385	−0.0287223
$946$	0	0
$947$	12497.9	0.428857	0.214428	−	0.976740i	$-0.431211\pi$
$947$	12497.9	0.428857	0.214428	+	0.976740i	$0.431211\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−49.6429	−0.00169272
$952$	0	0
$953$	36334.4	1.23503	0.617516	−	0.786558i	$-0.288139\pi$
$953$	36334.4	1.23503	0.617516	+	0.786558i	$0.288139\pi$
$954$	0	0
$955$	−9592.30	−0.325026
$956$	0	0
$957$	−1701.34	−0.0574677
$958$	0	0
$959$	3239.86	0.109093
$960$	0	0
$961$	−6116.11	−0.205301
$962$	0	0
$963$	−41355.1	−1.38385
$964$	0	0
$965$	9442.24	0.314981
$966$	0	0
$967$	−40930.2	−1.36114	−0.680572	−	0.732681i	$-0.738269\pi$
$967$	−40930.2	−1.36114	−0.680572	+	0.732681i	$0.738269\pi$
$968$	0	0
$969$	7859.09	0.260547
$970$	0	0
$971$	29443.0	0.973092	0.486546	−	0.873655i	$-0.338257\pi$
$971$	29443.0	0.973092	0.486546	+	0.873655i	$0.338257\pi$
$972$	0	0
$973$	−5725.99	−0.188661
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26831.2	−0.878614	−0.439307	−	0.898337i	$-0.644776\pi$
$977$	−26831.2	−0.878614	−0.439307	+	0.898337i	$0.644776\pi$
$978$	0	0
$979$	66609.8	2.17452
$980$	0	0
$981$	55081.5	1.79268
$982$	0	0
$983$	−3409.34	−0.110622	−0.0553108	−	0.998469i	$-0.517615\pi$
$983$	−3409.34	−0.110622	−0.0553108	+	0.998469i	$0.517615\pi$
$984$	0	0
$985$	−17953.3	−0.580751
$986$	0	0
$987$	−555.091	−0.0179015
$988$	0	0
$989$	594.820	0.0191246
$990$	0	0
$991$	−23214.8	−0.744138	−0.372069	−	0.928205i	$-0.621352\pi$
$991$	−23214.8	−0.744138	−0.372069	+	0.928205i	$0.621352\pi$
$992$	0	0
$993$	−3067.77	−0.0980390
$994$	0	0
$995$	−10187.1	−0.324576
$996$	0	0
$997$	−13047.0	−0.414446	−0.207223	−	0.978294i	$-0.566443\pi$
$997$	−13047.0	−0.414446	−0.207223	+	0.978294i	$0.566443\pi$
$998$	0	0
$999$	−14336.9	−0.454054

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.4.a.m.1.4		6
13.4	even	6		104.4.i.c.81.3	yes	12
13.10	even	6		104.4.i.c.9.3	✓	12
13.12	even	2		1352.4.a.n.1.4		6
52.23	odd	6		208.4.i.h.113.4		12
52.43	odd	6		208.4.i.h.81.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.i.c.9.3	✓	12	13.10	even	6
104.4.i.c.81.3	yes	12	13.4	even	6
208.4.i.h.81.4		12	52.43	odd	6
208.4.i.h.113.4		12	52.23	odd	6
1352.4.a.m.1.4		6	1.1	even	1	trivial
1352.4.a.n.1.4		6	13.12	even	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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q+0.753179 q^{3} +7.26675 q^{5} +2.85313 q^{7} -26.4327 q^{9} +O(q^{10})\)	\(q+0.753179 q^{3} +7.26675 q^{5} +2.85313 q^{7} -26.4327 q^{9} -58.0893 q^{11} +5.47317 q^{15} +73.0634 q^{17} +142.815 q^{19} +2.14891 q^{21} +48.8276 q^{23} -72.1943 q^{25} -40.2444 q^{27} +38.8864 q^{29} -153.866 q^{31} -43.7516 q^{33} +20.7330 q^{35} +356.246 q^{37} -3.88294 q^{41} +12.1821 q^{43} -192.080 q^{45} -258.312 q^{47} -334.860 q^{49} +55.0298 q^{51} -302.642 q^{53} -422.120 q^{55} +107.565 q^{57} -110.686 q^{59} -415.136 q^{61} -75.4159 q^{63} -933.920 q^{67} +36.7759 q^{69} -661.899 q^{71} -436.800 q^{73} -54.3753 q^{75} -165.736 q^{77} -318.158 q^{79} +683.372 q^{81} -595.656 q^{83} +530.934 q^{85} +29.2884 q^{87} -1146.68 q^{89} -115.889 q^{93} +1037.80 q^{95} +669.840 q^{97} +1535.46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 9 q^{5} - q^{7} + 93 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 - 9 * q^5 - q^7 + 93 * q^9	\( 6 q - 3 q^{3} - 9 q^{5} - q^{7} + 93 q^{9} - 11 q^{11} - 168 q^{15} + 50 q^{17} - 211 q^{19} + 83 q^{21} - 103 q^{23} + 393 q^{25} - 243 q^{27} - 48 q^{29} - 190 q^{31} + 133 q^{33} - 226 q^{35} - 476 q^{37} - 10 q^{41} + 13 q^{43} + 57 q^{45} - 122 q^{47} - 31 q^{49} - 745 q^{51} - 483 q^{53} + 1510 q^{55} - 1177 q^{57} + 731 q^{59} - 704 q^{61} + 518 q^{63} - 901 q^{67} + 3479 q^{69} + 673 q^{71} - 1551 q^{73} - 2739 q^{75} + 133 q^{77} - 840 q^{79} + 3018 q^{81} - 2016 q^{83} + 1373 q^{85} - 1485 q^{87} - 295 q^{89} - 3220 q^{93} + 3128 q^{95} - 2575 q^{97} - 458 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 - 9 * q^5 - q^7 + 93 * q^9 - 11 * q^11 - 168 * q^15 + 50 * q^17 - 211 * q^19 + 83 * q^21 - 103 * q^23 + 393 * q^25 - 243 * q^27 - 48 * q^29 - 190 * q^31 + 133 * q^33 - 226 * q^35 - 476 * q^37 - 10 * q^41 + 13 * q^43 + 57 * q^45 - 122 * q^47 - 31 * q^49 - 745 * q^51 - 483 * q^53 + 1510 * q^55 - 1177 * q^57 + 731 * q^59 - 704 * q^61 + 518 * q^63 - 901 * q^67 + 3479 * q^69 + 673 * q^71 - 1551 * q^73 - 2739 * q^75 + 133 * q^77 - 840 * q^79 + 3018 * q^81 - 2016 * q^83 + 1373 * q^85 - 1485 * q^87 - 295 * q^89 - 3220 * q^93 + 3128 * q^95 - 2575 * q^97 - 458 * q^99

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