LMFDB - Embedded newform 1936.4.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,4,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, 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("1936.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1936.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:	$114.227697771$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4166757.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 42x^{2} + 43x + 64 $ x^4 - 2x^3 - 42x^2 + 43*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 968)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.839327$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.40272 q^{3} -12.7069 q^{5} -3.35731 q^{7} -21.2269 q^{9} +O(q^{10})$	$q+2.40272 q^{3} -12.7069 q^{5} -3.35731 q^{7} -21.2269 q^{9} -62.5103 q^{13} -30.5311 q^{15} +19.9228 q^{17} +140.234 q^{19} -8.06668 q^{21} +20.4780 q^{23} +36.4642 q^{25} -115.876 q^{27} -88.6056 q^{29} -99.8021 q^{31} +42.6609 q^{35} -195.072 q^{37} -150.195 q^{39} -368.401 q^{41} -315.347 q^{43} +269.727 q^{45} +86.7418 q^{47} -331.728 q^{49} +47.8690 q^{51} -144.778 q^{53} +336.943 q^{57} -361.875 q^{59} -120.063 q^{61} +71.2654 q^{63} +794.309 q^{65} +948.587 q^{67} +49.2030 q^{69} -376.128 q^{71} +1115.12 q^{73} +87.6134 q^{75} -730.730 q^{79} +294.709 q^{81} +847.568 q^{83} -253.157 q^{85} -212.895 q^{87} +967.458 q^{89} +209.866 q^{91} -239.797 q^{93} -1781.93 q^{95} +382.208 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} + 9 q^{5} + 8 q^{7} + 31 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 + 9 * q^5 + 8 * q^7 + 31 * q^9	$ 4 q + 3 q^{3} + 9 q^{5} + 8 q^{7} + 31 q^{9} + 80 q^{13} + 47 q^{15} - 16 q^{17} + 8 q^{19} + 192 q^{21} + 71 q^{23} + 11 q^{25} + 81 q^{27} + 240 q^{29} + 115 q^{31} - 168 q^{35} + 315 q^{37} + 232 q^{39} - 592 q^{41} - 624 q^{43} + 108 q^{45} + 304 q^{47} + 36 q^{49} - 1256 q^{51} + 184 q^{53} + 592 q^{57} + 805 q^{59} + 240 q^{61} + 2288 q^{63} + 1424 q^{65} - 119 q^{67} + 1499 q^{69} - 723 q^{71} + 1936 q^{73} + 300 q^{75} - 1520 q^{79} + 700 q^{81} - 2016 q^{83} - 3040 q^{85} - 3152 q^{87} + 367 q^{89} + 2848 q^{91} - 1625 q^{93} - 3400 q^{95} + 881 q^{97}+O(q^{100}) $ 4 * q + 3 * q^3 + 9 * q^5 + 8 * q^7 + 31 * q^9 + 80 * q^13 + 47 * q^15 - 16 * q^17 + 8 * q^19 + 192 * q^21 + 71 * q^23 + 11 * q^25 + 81 * q^27 + 240 * q^29 + 115 * q^31 - 168 * q^35 + 315 * q^37 + 232 * q^39 - 592 * q^41 - 624 * q^43 + 108 * q^45 + 304 * q^47 + 36 * q^49 - 1256 * q^51 + 184 * q^53 + 592 * q^57 + 805 * q^59 + 240 * q^61 + 2288 * q^63 + 1424 * q^65 - 119 * q^67 + 1499 * q^69 - 723 * q^71 + 1936 * q^73 + 300 * q^75 - 1520 * q^79 + 700 * q^81 - 2016 * q^83 - 3040 * q^85 - 3152 * q^87 + 367 * q^89 + 2848 * q^91 - 1625 * q^93 - 3400 * q^95 + 881 * q^97

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
$3$	2.40272	0.462404	0.231202	−	0.972906i	$-0.425734\pi$
$3$	2.40272	0.462404	0.231202	+	0.972906i	$0.425734\pi$
$4$	0	0
$5$	−12.7069	−1.13654	−0.568268	−	0.822844i	$-0.692386\pi$
$5$	−12.7069	−1.13654	−0.568268	+	0.822844i	$0.692386\pi$
$6$	0	0
$7$	−3.35731	−0.181278	−0.0906388	−	0.995884i	$-0.528891\pi$
$7$	−3.35731	−0.181278	−0.0906388	+	0.995884i	$0.528891\pi$
$8$	0	0
$9$	−21.2269	−0.786182
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−62.5103	−1.33363	−0.666816	−	0.745222i	$-0.732343\pi$
$13$	−62.5103	−1.33363	−0.666816	+	0.745222i	$0.732343\pi$
$14$	0	0
$15$	−30.5311	−0.525539
$16$	0	0
$17$	19.9228	0.284235	0.142118	−	0.989850i	$-0.454609\pi$
$17$	19.9228	0.284235	0.142118	+	0.989850i	$0.454609\pi$
$18$	0	0
$19$	140.234	1.69326	0.846628	−	0.532185i	$-0.178629\pi$
$19$	140.234	1.69326	0.846628	+	0.532185i	$0.178629\pi$
$20$	0	0
$21$	−8.06668	−0.0838236
$22$	0	0
$23$	20.4780	0.185651	0.0928253	−	0.995682i	$-0.470410\pi$
$23$	20.4780	0.185651	0.0928253	+	0.995682i	$0.470410\pi$
$24$	0	0
$25$	36.4642	0.291714
$26$	0	0
$27$	−115.876	−0.825938
$28$	0	0
$29$	−88.6056	−0.567367	−0.283684	−	0.958918i	$-0.591557\pi$
$29$	−88.6056	−0.567367	−0.283684	+	0.958918i	$0.591557\pi$
$30$	0	0
$31$	−99.8021	−0.578225	−0.289113	−	0.957295i	$-0.593360\pi$
$31$	−99.8021	−0.578225	−0.289113	+	0.957295i	$0.593360\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	42.6609	0.206029
$36$	0	0
$37$	−195.072	−0.866748	−0.433374	−	0.901214i	$-0.642677\pi$
$37$	−195.072	−0.866748	−0.433374	+	0.901214i	$0.642677\pi$
$38$	0	0
$39$	−150.195	−0.616677
$40$	0	0
$41$	−368.401	−1.40328	−0.701642	−	0.712530i	$-0.747549\pi$
$41$	−368.401	−1.40328	−0.701642	+	0.712530i	$0.747549\pi$
$42$	0	0
$43$	−315.347	−1.11837	−0.559186	−	0.829042i	$-0.688886\pi$
$43$	−315.347	−1.11837	−0.559186	+	0.829042i	$0.688886\pi$
$44$	0	0
$45$	269.727	0.893524
$46$	0	0
$47$	86.7418	0.269204	0.134602	−	0.990900i	$-0.457024\pi$
$47$	86.7418	0.269204	0.134602	+	0.990900i	$0.457024\pi$
$48$	0	0
$49$	−331.728	−0.967138
$50$	0	0
$51$	47.8690	0.131431
$52$	0	0
$53$	−144.778	−0.375223	−0.187611	−	0.982243i	$-0.560075\pi$
$53$	−144.778	−0.375223	−0.187611	+	0.982243i	$0.560075\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	336.943	0.782969
$58$	0	0
$59$	−361.875	−0.798511	−0.399256	−	0.916840i	$-0.630731\pi$
$59$	−361.875	−0.798511	−0.399256	+	0.916840i	$0.630731\pi$
$60$	0	0
$61$	−120.063	−0.252009	−0.126005	−	0.992030i	$-0.540215\pi$
$61$	−120.063	−0.252009	−0.126005	+	0.992030i	$0.540215\pi$
$62$	0	0
$63$	71.2654	0.142517
$64$	0	0
$65$	794.309	1.51572
$66$	0	0
$67$	948.587	1.72968	0.864838	−	0.502051i	$-0.167421\pi$
$67$	948.587	1.72968	0.864838	+	0.502051i	$0.167421\pi$
$68$	0	0
$69$	49.2030	0.0858456
$70$	0	0
$71$	−376.128	−0.628706	−0.314353	−	0.949306i	$-0.601788\pi$
$71$	−376.128	−0.628706	−0.314353	+	0.949306i	$0.601788\pi$
$72$	0	0
$73$	1115.12	1.78787	0.893936	−	0.448195i	$-0.147933\pi$
$73$	1115.12	1.78787	0.893936	+	0.448195i	$0.147933\pi$
$74$	0	0
$75$	87.6134	0.134890
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−730.730	−1.04068	−0.520339	−	0.853960i	$-0.674194\pi$
$79$	−730.730	−1.04068	−0.520339	+	0.853960i	$0.674194\pi$
$80$	0	0
$81$	294.709	0.404265
$82$	0	0
$83$	847.568	1.12088	0.560438	−	0.828197i	$-0.310633\pi$
$83$	847.568	1.12088	0.560438	+	0.828197i	$0.310633\pi$
$84$	0	0
$85$	−253.157	−0.323043
$86$	0	0
$87$	−212.895	−0.262353
$88$	0	0
$89$	967.458	1.15225	0.576126	−	0.817361i	$-0.304564\pi$
$89$	967.458	1.15225	0.576126	+	0.817361i	$0.304564\pi$
$90$	0	0
$91$	209.866	0.241758
$92$	0	0
$93$	−239.797	−0.267374
$94$	0	0
$95$	−1781.93	−1.92445
$96$	0	0
$97$	382.208	0.400076	0.200038	−	0.979788i	$-0.435893\pi$
$97$	382.208	0.400076	0.200038	+	0.979788i	$0.435893\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	688.639	0.678437	0.339218	−	0.940708i	$-0.389837\pi$
$101$	688.639	0.678437	0.339218	+	0.940708i	$0.389837\pi$
$102$	0	0
$103$	1699.38	1.62568	0.812839	−	0.582488i	$-0.197921\pi$
$103$	1699.38	1.62568	0.812839	+	0.582488i	$0.197921\pi$
$104$	0	0
$105$	102.502	0.0952685
$106$	0	0
$107$	1239.18	1.11959	0.559794	−	0.828632i	$-0.310880\pi$
$107$	1239.18	1.11959	0.559794	+	0.828632i	$0.310880\pi$
$108$	0	0
$109$	606.029	0.532542	0.266271	−	0.963898i	$-0.414208\pi$
$109$	606.029	0.532542	0.266271	+	0.963898i	$0.414208\pi$
$110$	0	0
$111$	−468.705	−0.400788
$112$	0	0
$113$	815.236	0.678681	0.339340	−	0.940664i	$-0.389796\pi$
$113$	815.236	0.678681	0.339340	+	0.940664i	$0.389796\pi$
$114$	0	0
$115$	−260.211	−0.210999
$116$	0	0
$117$	1326.90	1.04848
$118$	0	0
$119$	−66.8871	−0.0515255
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−885.166	−0.648884
$124$	0	0
$125$	1125.01	0.804993
$126$	0	0
$127$	−2001.73	−1.39862	−0.699312	−	0.714817i	$-0.746510\pi$
$127$	−2001.73	−1.39862	−0.699312	+	0.714817i	$0.746510\pi$
$128$	0	0
$129$	−757.692	−0.517140
$130$	0	0
$131$	−2148.68	−1.43306	−0.716532	−	0.697554i	$-0.754272\pi$
$131$	−2148.68	−1.43306	−0.716532	+	0.697554i	$0.754272\pi$
$132$	0	0
$133$	−470.809	−0.306950
$134$	0	0
$135$	1472.42	0.938708
$136$	0	0
$137$	−64.2589	−0.0400731	−0.0200365	−	0.999799i	$-0.506378\pi$
$137$	−64.2589	−0.0400731	−0.0200365	+	0.999799i	$0.506378\pi$
$138$	0	0
$139$	−2237.52	−1.36535	−0.682676	−	0.730721i	$-0.739184\pi$
$139$	−2237.52	−1.36535	−0.682676	+	0.730721i	$0.739184\pi$
$140$	0	0
$141$	208.416	0.124481
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1125.90	0.644833
$146$	0	0
$147$	−797.052	−0.447209
$148$	0	0
$149$	2216.34	1.21859	0.609293	−	0.792945i	$-0.291453\pi$
$149$	2216.34	1.21859	0.609293	+	0.792945i	$0.291453\pi$
$150$	0	0
$151$	2100.82	1.13220	0.566099	−	0.824337i	$-0.308452\pi$
$151$	2100.82	1.13220	0.566099	+	0.824337i	$0.308452\pi$
$152$	0	0
$153$	−422.900	−0.223461
$154$	0	0
$155$	1268.17	0.657174
$156$	0	0
$157$	2969.83	1.50967	0.754834	−	0.655916i	$-0.227717\pi$
$157$	2969.83	1.50967	0.754834	+	0.655916i	$0.227717\pi$
$158$	0	0
$159$	−347.862	−0.173505
$160$	0	0
$161$	−68.7511	−0.0336543
$162$	0	0
$163$	2951.07	1.41807	0.709037	−	0.705171i	$-0.249130\pi$
$163$	2951.07	1.41807	0.709037	+	0.705171i	$0.249130\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−59.7218	−0.0276731	−0.0138366	−	0.999904i	$-0.504404\pi$
$167$	−59.7218	−0.0276731	−0.0138366	+	0.999904i	$0.504404\pi$
$168$	0	0
$169$	1710.53	0.778576
$170$	0	0
$171$	−2976.74	−1.33121
$172$	0	0
$173$	−4539.79	−1.99511	−0.997554	−	0.0699021i	$-0.977731\pi$
$173$	−4539.79	−1.99511	−0.997554	+	0.0699021i	$0.977731\pi$
$174$	0	0
$175$	−122.422	−0.0528812
$176$	0	0
$177$	−869.486	−0.369235
$178$	0	0
$179$	4406.02	1.83978	0.919892	−	0.392172i	$-0.128276\pi$
$179$	4406.02	1.83978	0.919892	+	0.392172i	$0.128276\pi$
$180$	0	0
$181$	−1246.38	−0.511838	−0.255919	−	0.966698i	$-0.582378\pi$
$181$	−1246.38	−0.511838	−0.255919	+	0.966698i	$0.582378\pi$
$182$	0	0
$183$	−288.479	−0.116530
$184$	0	0
$185$	2478.76	0.985090
$186$	0	0
$187$	0	0
$188$	0	0
$189$	389.031	0.149724
$190$	0	0
$191$	4041.94	1.53123	0.765613	−	0.643301i	$-0.222436\pi$
$191$	4041.94	1.53123	0.765613	+	0.643301i	$0.222436\pi$
$192$	0	0
$193$	1436.78	0.535862	0.267931	−	0.963438i	$-0.413660\pi$
$193$	1436.78	0.535862	0.267931	+	0.963438i	$0.413660\pi$
$194$	0	0
$195$	1908.50	0.700876
$196$	0	0
$197$	3420.28	1.23698	0.618489	−	0.785794i	$-0.287745\pi$
$197$	3420.28	1.23698	0.618489	+	0.785794i	$0.287745\pi$
$198$	0	0
$199$	−4567.18	−1.62693	−0.813464	−	0.581616i	$-0.802421\pi$
$199$	−4567.18	−1.62693	−0.813464	+	0.581616i	$0.802421\pi$
$200$	0	0
$201$	2279.19	0.799810
$202$	0	0
$203$	297.476	0.102851
$204$	0	0
$205$	4681.22	1.59488
$206$	0	0
$207$	−434.685	−0.145955
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−674.302	−0.220004	−0.110002	−	0.993931i	$-0.535086\pi$
$211$	−674.302	−0.220004	−0.110002	+	0.993931i	$0.535086\pi$
$212$	0	0
$213$	−903.731	−0.290717
$214$	0	0
$215$	4007.07	1.27107
$216$	0	0
$217$	335.066	0.104819
$218$	0	0
$219$	2679.32	0.826720
$220$	0	0
$221$	−1245.38	−0.379065
$222$	0	0
$223$	940.591	0.282451	0.141226	−	0.989977i	$-0.454896\pi$
$223$	940.591	0.282451	0.141226	+	0.989977i	$0.454896\pi$
$224$	0	0
$225$	−774.023	−0.229340
$226$	0	0
$227$	22.6822	0.00663204	0.00331602	−	0.999995i	$-0.498944\pi$
$227$	22.6822	0.00663204	0.00331602	+	0.999995i	$0.498944\pi$
$228$	0	0
$229$	4104.86	1.18453	0.592263	−	0.805745i	$-0.298235\pi$
$229$	4104.86	1.18453	0.592263	+	0.805745i	$0.298235\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1644.33	−0.462333	−0.231167	−	0.972914i	$-0.574254\pi$
$233$	−1644.33	−0.462333	−0.231167	+	0.972914i	$0.574254\pi$
$234$	0	0
$235$	−1102.22	−0.305960
$236$	0	0
$237$	−1755.74	−0.481214
$238$	0	0
$239$	47.8817	0.0129590	0.00647951	−	0.999979i	$-0.497937\pi$
$239$	47.8817	0.0129590	0.00647951	+	0.999979i	$0.497937\pi$
$240$	0	0
$241$	1816.44	0.485508	0.242754	−	0.970088i	$-0.421949\pi$
$241$	1816.44	0.485508	0.242754	+	0.970088i	$0.421949\pi$
$242$	0	0
$243$	3836.75	1.01287
$244$	0	0
$245$	4215.23	1.09919
$246$	0	0
$247$	−8766.06	−2.25818
$248$	0	0
$249$	2036.47	0.518297
$250$	0	0
$251$	5112.30	1.28560	0.642800	−	0.766034i	$-0.277773\pi$
$251$	5112.30	1.28560	0.642800	+	0.766034i	$0.277773\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−608.265	−0.149377
$256$	0	0
$257$	4416.62	1.07199	0.535995	−	0.844221i	$-0.319937\pi$
$257$	4416.62	1.07199	0.535995	+	0.844221i	$0.319937\pi$
$258$	0	0
$259$	654.918	0.157122
$260$	0	0
$261$	1880.82	0.446054
$262$	0	0
$263$	−7177.97	−1.68294	−0.841468	−	0.540306i	$-0.818308\pi$
$263$	−7177.97	−1.68294	−0.841468	+	0.540306i	$0.818308\pi$
$264$	0	0
$265$	1839.67	0.426454
$266$	0	0
$267$	2324.53	0.532806
$268$	0	0
$269$	−7184.21	−1.62836	−0.814180	−	0.580613i	$-0.802813\pi$
$269$	−7184.21	−1.62836	−0.814180	+	0.580613i	$0.802813\pi$
$270$	0	0
$271$	145.327	0.0325757	0.0162878	−	0.999867i	$-0.494815\pi$
$271$	145.327	0.0325757	0.0162878	+	0.999867i	$0.494815\pi$
$272$	0	0
$273$	504.250	0.111790
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3129.03	0.678718	0.339359	−	0.940657i	$-0.389790\pi$
$277$	3129.03	0.678718	0.339359	+	0.940657i	$0.389790\pi$
$278$	0	0
$279$	2118.49	0.454590
$280$	0	0
$281$	−6072.48	−1.28916	−0.644580	−	0.764537i	$-0.722968\pi$
$281$	−6072.48	−1.28916	−0.644580	+	0.764537i	$0.722968\pi$
$282$	0	0
$283$	−6126.24	−1.28681	−0.643405	−	0.765526i	$-0.722479\pi$
$283$	−6126.24	−1.28681	−0.643405	+	0.765526i	$0.722479\pi$
$284$	0	0
$285$	−4281.49	−0.889872
$286$	0	0
$287$	1236.84	0.254384
$288$	0	0
$289$	−4516.08	−0.919210
$290$	0	0
$291$	918.341	0.184997
$292$	0	0
$293$	−8937.24	−1.78198	−0.890989	−	0.454026i	$-0.849987\pi$
$293$	−8937.24	−1.78198	−0.890989	+	0.454026i	$0.849987\pi$
$294$	0	0
$295$	4598.30	0.907537
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1280.09	−0.247590
$300$	0	0
$301$	1058.72	0.202736
$302$	0	0
$303$	1654.61	0.313712
$304$	0	0
$305$	1525.63	0.286417
$306$	0	0
$307$	−565.553	−0.105139	−0.0525697	−	0.998617i	$-0.516741\pi$
$307$	−565.553	−0.105139	−0.0525697	+	0.998617i	$0.516741\pi$
$308$	0	0
$309$	4083.14	0.751721
$310$	0	0
$311$	−2138.36	−0.389889	−0.194945	−	0.980814i	$-0.562453\pi$
$311$	−2138.36	−0.389889	−0.194945	+	0.980814i	$0.562453\pi$
$312$	0	0
$313$	−8327.37	−1.50380	−0.751902	−	0.659274i	$-0.770864\pi$
$313$	−8327.37	−1.50380	−0.751902	+	0.659274i	$0.770864\pi$
$314$	0	0
$315$	−905.559	−0.161976
$316$	0	0
$317$	1611.96	0.285604	0.142802	−	0.989751i	$-0.454389\pi$
$317$	1611.96	0.285604	0.142802	+	0.989751i	$0.454389\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2977.40	0.517702
$322$	0	0
$323$	2793.86	0.481283
$324$	0	0
$325$	−2279.39	−0.389039
$326$	0	0
$327$	1456.12	0.246250
$328$	0	0
$329$	−291.219	−0.0488007
$330$	0	0
$331$	2849.31	0.473149	0.236574	−	0.971613i	$-0.423975\pi$
$331$	2849.31	0.473149	0.236574	+	0.971613i	$0.423975\pi$
$332$	0	0
$333$	4140.78	0.681422
$334$	0	0
$335$	−12053.6	−1.96584
$336$	0	0
$337$	−4535.60	−0.733145	−0.366573	−	0.930389i	$-0.619469\pi$
$337$	−4535.60	−0.733145	−0.366573	+	0.930389i	$0.619469\pi$
$338$	0	0
$339$	1958.79	0.313825
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2265.27	0.356598
$344$	0	0
$345$	−625.216	−0.0975666
$346$	0	0
$347$	−662.855	−0.102547	−0.0512736	−	0.998685i	$-0.516328\pi$
$347$	−662.855	−0.102547	−0.0512736	+	0.998685i	$0.516328\pi$
$348$	0	0
$349$	9865.61	1.51316	0.756582	−	0.653899i	$-0.226868\pi$
$349$	9865.61	1.51316	0.756582	+	0.653899i	$0.226868\pi$
$350$	0	0
$351$	7243.43	1.10150
$352$	0	0
$353$	5460.87	0.823379	0.411689	−	0.911324i	$-0.364939\pi$
$353$	5460.87	0.823379	0.411689	+	0.911324i	$0.364939\pi$
$354$	0	0
$355$	4779.40	0.714547
$356$	0	0
$357$	−160.711	−0.0238256
$358$	0	0
$359$	9976.70	1.46671	0.733356	−	0.679844i	$-0.237953\pi$
$359$	9976.70	1.46671	0.733356	+	0.679844i	$0.237953\pi$
$360$	0	0
$361$	12806.6	1.86712
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14169.6	−2.03198
$366$	0	0
$367$	−2792.01	−0.397116	−0.198558	−	0.980089i	$-0.563626\pi$
$367$	−2792.01	−0.397116	−0.198558	+	0.980089i	$0.563626\pi$
$368$	0	0
$369$	7820.02	1.10324
$370$	0	0
$371$	486.065	0.0680195
$372$	0	0
$373$	−3716.19	−0.515863	−0.257932	−	0.966163i	$-0.583041\pi$
$373$	−3716.19	−0.515863	−0.257932	+	0.966163i	$0.583041\pi$
$374$	0	0
$375$	2703.09	0.372232
$376$	0	0
$377$	5538.76	0.756659
$378$	0	0
$379$	−3627.28	−0.491612	−0.245806	−	0.969319i	$-0.579053\pi$
$379$	−3627.28	−0.491612	−0.245806	+	0.969319i	$0.579053\pi$
$380$	0	0
$381$	−4809.61	−0.646729
$382$	0	0
$383$	936.474	0.124939	0.0624694	−	0.998047i	$-0.480102\pi$
$383$	936.474	0.124939	0.0624694	+	0.998047i	$0.480102\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6693.85	0.879245
$388$	0	0
$389$	−4504.90	−0.587165	−0.293583	−	0.955934i	$-0.594848\pi$
$389$	−4504.90	−0.587165	−0.293583	+	0.955934i	$0.594848\pi$
$390$	0	0
$391$	407.980	0.0527684
$392$	0	0
$393$	−5162.69	−0.662655
$394$	0	0
$395$	9285.28	1.18277
$396$	0	0
$397$	−6656.90	−0.841562	−0.420781	−	0.907162i	$-0.638244\pi$
$397$	−6656.90	−0.841562	−0.420781	+	0.907162i	$0.638244\pi$
$398$	0	0
$399$	−1131.22	−0.141935
$400$	0	0
$401$	−2338.87	−0.291266	−0.145633	−	0.989339i	$-0.546522\pi$
$401$	−2338.87	−0.291266	−0.145633	+	0.989339i	$0.546522\pi$
$402$	0	0
$403$	6238.65	0.771140
$404$	0	0
$405$	−3744.83	−0.459462
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−6475.30	−0.782843	−0.391421	−	0.920212i	$-0.628017\pi$
$409$	−6475.30	−0.782843	−0.391421	+	0.920212i	$0.628017\pi$
$410$	0	0
$411$	−154.396	−0.0185300
$412$	0	0
$413$	1214.93	0.144752
$414$	0	0
$415$	−10769.9	−1.27391
$416$	0	0
$417$	−5376.14	−0.631345
$418$	0	0
$419$	2938.24	0.342583	0.171291	−	0.985220i	$-0.445206\pi$
$419$	2938.24	0.342583	0.171291	+	0.985220i	$0.445206\pi$
$420$	0	0
$421$	−770.062	−0.0891461	−0.0445731	−	0.999006i	$-0.514193\pi$
$421$	−770.062	−0.0891461	−0.0445731	+	0.999006i	$0.514193\pi$
$422$	0	0
$423$	−1841.26	−0.211643
$424$	0	0
$425$	726.471	0.0829153
$426$	0	0
$427$	403.090	0.0456836
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10137.9	1.13301	0.566505	−	0.824059i	$-0.308295\pi$
$431$	10137.9	1.13301	0.566505	+	0.824059i	$0.308295\pi$
$432$	0	0
$433$	3146.25	0.349189	0.174595	−	0.984640i	$-0.444139\pi$
$433$	3146.25	0.349189	0.174595	+	0.984640i	$0.444139\pi$
$434$	0	0
$435$	2705.22	0.298174
$436$	0	0
$437$	2871.71	0.314354
$438$	0	0
$439$	13151.6	1.42983	0.714913	−	0.699214i	$-0.246466\pi$
$439$	13151.6	1.42983	0.714913	+	0.699214i	$0.246466\pi$
$440$	0	0
$441$	7041.57	0.760347
$442$	0	0
$443$	15612.6	1.67444	0.837222	−	0.546863i	$-0.184178\pi$
$443$	15612.6	1.67444	0.837222	+	0.546863i	$0.184178\pi$
$444$	0	0
$445$	−12293.4	−1.30957
$446$	0	0
$447$	5325.24	0.563479
$448$	0	0
$449$	−7508.81	−0.789226	−0.394613	−	0.918847i	$-0.629121\pi$
$449$	−7508.81	−0.789226	−0.394613	+	0.918847i	$0.629121\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	5047.68	0.523533
$454$	0	0
$455$	−2666.74	−0.274766
$456$	0	0
$457$	−663.197	−0.0678841	−0.0339421	−	0.999424i	$-0.510806\pi$
$457$	−663.197	−0.0678841	−0.0339421	+	0.999424i	$0.510806\pi$
$458$	0	0
$459$	−2308.58	−0.234761
$460$	0	0
$461$	−8634.41	−0.872331	−0.436165	−	0.899866i	$-0.643664\pi$
$461$	−8634.41	−0.872331	−0.436165	+	0.899866i	$0.643664\pi$
$462$	0	0
$463$	−7614.58	−0.764319	−0.382159	−	0.924096i	$-0.624819\pi$
$463$	−7614.58	−0.764319	−0.382159	+	0.924096i	$0.624819\pi$
$464$	0	0
$465$	3047.06	0.303880
$466$	0	0
$467$	8733.09	0.865352	0.432676	−	0.901550i	$-0.357569\pi$
$467$	8733.09	0.865352	0.432676	+	0.901550i	$0.357569\pi$
$468$	0	0
$469$	−3184.70	−0.313552
$470$	0	0
$471$	7135.67	0.698077
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5113.52	0.493946
$476$	0	0
$477$	3073.19	0.294993
$478$	0	0
$479$	−11602.5	−1.10674	−0.553371	−	0.832935i	$-0.686659\pi$
$479$	−11602.5	−1.10674	−0.553371	+	0.832935i	$0.686659\pi$
$480$	0	0
$481$	12194.0	1.15592
$482$	0	0
$483$	−165.190	−0.0155619
$484$	0	0
$485$	−4856.67	−0.454701
$486$	0	0
$487$	−5302.20	−0.493358	−0.246679	−	0.969097i	$-0.579339\pi$
$487$	−5302.20	−0.493358	−0.246679	+	0.969097i	$0.579339\pi$
$488$	0	0
$489$	7090.61	0.655723
$490$	0	0
$491$	12529.8	1.15166	0.575828	−	0.817571i	$-0.304680\pi$
$491$	12529.8	1.15166	0.575828	+	0.817571i	$0.304680\pi$
$492$	0	0
$493$	−1765.27	−0.161266
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1262.78	0.113970
$498$	0	0
$499$	12460.5	1.11785	0.558925	−	0.829218i	$-0.311214\pi$
$499$	12460.5	1.11785	0.558925	+	0.829218i	$0.311214\pi$
$500$	0	0
$501$	−143.495	−0.0127962
$502$	0	0
$503$	−12022.6	−1.06573	−0.532865	−	0.846200i	$-0.678885\pi$
$503$	−12022.6	−1.06573	−0.532865	+	0.846200i	$0.678885\pi$
$504$	0	0
$505$	−8750.44	−0.771068
$506$	0	0
$507$	4109.93	0.360017
$508$	0	0
$509$	14501.8	1.26283	0.631417	−	0.775444i	$-0.282474\pi$
$509$	14501.8	1.26283	0.631417	+	0.775444i	$0.282474\pi$
$510$	0	0
$511$	−3743.80	−0.324101
$512$	0	0
$513$	−16249.7	−1.39853
$514$	0	0
$515$	−21593.8	−1.84764
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−10907.9	−0.922546
$520$	0	0
$521$	13480.3	1.13355	0.566777	−	0.823871i	$-0.308190\pi$
$521$	13480.3	1.13355	0.566777	+	0.823871i	$0.308190\pi$
$522$	0	0
$523$	14160.7	1.18394	0.591972	−	0.805959i	$-0.298350\pi$
$523$	14160.7	1.18394	0.591972	+	0.805959i	$0.298350\pi$
$524$	0	0
$525$	−294.145	−0.0244525
$526$	0	0
$527$	−1988.34	−0.164352
$528$	0	0
$529$	−11747.7	−0.965534
$530$	0	0
$531$	7681.50	0.627775
$532$	0	0
$533$	23028.9	1.87146
$534$	0	0
$535$	−15746.1	−1.27245
$536$	0	0
$537$	10586.4	0.850724
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8553.19	0.679723	0.339862	−	0.940475i	$-0.389620\pi$
$541$	8553.19	0.679723	0.339862	+	0.940475i	$0.389620\pi$
$542$	0	0
$543$	−2994.71	−0.236676
$544$	0	0
$545$	−7700.72	−0.605253
$546$	0	0
$547$	−15375.3	−1.20183	−0.600915	−	0.799313i	$-0.705197\pi$
$547$	−15375.3	−1.20183	−0.600915	+	0.799313i	$0.705197\pi$
$548$	0	0
$549$	2548.58	0.198125
$550$	0	0
$551$	−12425.5	−0.960698
$552$	0	0
$553$	2453.29	0.188652
$554$	0	0
$555$	5955.76	0.455510
$556$	0	0
$557$	5025.32	0.382279	0.191140	−	0.981563i	$-0.438782\pi$
$557$	5025.32	0.382279	0.191140	+	0.981563i	$0.438782\pi$
$558$	0	0
$559$	19712.4	1.49150
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2375.32	−0.177811	−0.0889057	−	0.996040i	$-0.528337\pi$
$563$	−2375.32	−0.177811	−0.0889057	+	0.996040i	$0.528337\pi$
$564$	0	0
$565$	−10359.1	−0.771345
$566$	0	0
$567$	−989.430	−0.0732842
$568$	0	0
$569$	22475.3	1.65591	0.827954	−	0.560796i	$-0.189505\pi$
$569$	22475.3	1.65591	0.827954	+	0.560796i	$0.189505\pi$
$570$	0	0
$571$	−16030.7	−1.17489	−0.587446	−	0.809264i	$-0.699866\pi$
$571$	−16030.7	−1.17489	−0.587446	+	0.809264i	$0.699866\pi$
$572$	0	0
$573$	9711.65	0.708046
$574$	0	0
$575$	746.715	0.0541568
$576$	0	0
$577$	5630.99	0.406276	0.203138	−	0.979150i	$-0.434886\pi$
$577$	5630.99	0.406276	0.203138	+	0.979150i	$0.434886\pi$
$578$	0	0
$579$	3452.17	0.247785
$580$	0	0
$581$	−2845.55	−0.203190
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−16860.7	−1.19163
$586$	0	0
$587$	17078.5	1.20086	0.600430	−	0.799677i	$-0.294996\pi$
$587$	17078.5	1.20086	0.600430	+	0.799677i	$0.294996\pi$
$588$	0	0
$589$	−13995.6	−0.979083
$590$	0	0
$591$	8217.97	0.571984
$592$	0	0
$593$	−1991.01	−0.137877	−0.0689385	−	0.997621i	$-0.521961\pi$
$593$	−1991.01	−0.137877	−0.0689385	+	0.997621i	$0.521961\pi$
$594$	0	0
$595$	849.925	0.0585605
$596$	0	0
$597$	−10973.7	−0.752298
$598$	0	0
$599$	23440.9	1.59895	0.799475	−	0.600700i	$-0.205111\pi$
$599$	23440.9	1.59895	0.799475	+	0.600700i	$0.205111\pi$
$600$	0	0
$601$	−1558.40	−0.105771	−0.0528856	−	0.998601i	$-0.516842\pi$
$601$	−1558.40	−0.105771	−0.0528856	+	0.998601i	$0.516842\pi$
$602$	0	0
$603$	−20135.6	−1.35984
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13443.1	−0.898911	−0.449455	−	0.893303i	$-0.648382\pi$
$607$	−13443.1	−0.898911	−0.449455	+	0.893303i	$0.648382\pi$
$608$	0	0
$609$	714.753	0.0475587
$610$	0	0
$611$	−5422.25	−0.359019
$612$	0	0
$613$	5289.59	0.348523	0.174262	−	0.984699i	$-0.444246\pi$
$613$	5289.59	0.348523	0.174262	+	0.984699i	$0.444246\pi$
$614$	0	0
$615$	11247.7	0.737480
$616$	0	0
$617$	7141.26	0.465959	0.232979	−	0.972482i	$-0.425153\pi$
$617$	7141.26	0.465959	0.232979	+	0.972482i	$0.425153\pi$
$618$	0	0
$619$	−14748.1	−0.957635	−0.478817	−	0.877914i	$-0.658934\pi$
$619$	−14748.1	−0.957635	−0.478817	+	0.877914i	$0.658934\pi$
$620$	0	0
$621$	−2372.91	−0.153336
$622$	0	0
$623$	−3248.06	−0.208877
$624$	0	0
$625$	−18853.4	−1.20662
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3886.39	−0.246360
$630$	0	0
$631$	−12780.1	−0.806289	−0.403145	−	0.915136i	$-0.632083\pi$
$631$	−12780.1	−0.806289	−0.403145	+	0.915136i	$0.632083\pi$
$632$	0	0
$633$	−1620.16	−0.101731
$634$	0	0
$635$	25435.8	1.58959
$636$	0	0
$637$	20736.4	1.28981
$638$	0	0
$639$	7984.04	0.494278
$640$	0	0
$641$	−19650.6	−1.21085	−0.605423	−	0.795904i	$-0.706996\pi$
$641$	−19650.6	−1.21085	−0.605423	+	0.795904i	$0.706996\pi$
$642$	0	0
$643$	4028.87	0.247096	0.123548	−	0.992339i	$-0.460573\pi$
$643$	4028.87	0.247096	0.123548	+	0.992339i	$0.460573\pi$
$644$	0	0
$645$	9627.89	0.587748
$646$	0	0
$647$	17005.8	1.03333	0.516667	−	0.856187i	$-0.327173\pi$
$647$	17005.8	1.03333	0.516667	+	0.856187i	$0.327173\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	805.072	0.0484689
$652$	0	0
$653$	14589.9	0.874344	0.437172	−	0.899378i	$-0.355980\pi$
$653$	14589.9	0.874344	0.437172	+	0.899378i	$0.355980\pi$
$654$	0	0
$655$	27303.0	1.62873
$656$	0	0
$657$	−23670.5	−1.40559
$658$	0	0
$659$	12250.5	0.724146	0.362073	−	0.932150i	$-0.382069\pi$
$659$	12250.5	0.724146	0.362073	+	0.932150i	$0.382069\pi$
$660$	0	0
$661$	−7328.40	−0.431228	−0.215614	−	0.976479i	$-0.569175\pi$
$661$	−7328.40	−0.431228	−0.215614	+	0.976479i	$0.569175\pi$
$662$	0	0
$663$	−2992.31	−0.175281
$664$	0	0
$665$	5982.50	0.348859
$666$	0	0
$667$	−1814.47	−0.105332
$668$	0	0
$669$	2259.98	0.130607
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6348.37	0.363613	0.181807	−	0.983334i	$-0.441805\pi$
$673$	6348.37	0.363613	0.181807	+	0.983334i	$0.441805\pi$
$674$	0	0
$675$	−4225.33	−0.240938
$676$	0	0
$677$	20424.9	1.15952	0.579758	−	0.814789i	$-0.303147\pi$
$677$	20424.9	1.15952	0.579758	+	0.814789i	$0.303147\pi$
$678$	0	0
$679$	−1283.19	−0.0725249
$680$	0	0
$681$	54.4991	0.00306668
$682$	0	0
$683$	23874.5	1.33753	0.668765	−	0.743474i	$-0.266823\pi$
$683$	23874.5	1.33753	0.668765	+	0.743474i	$0.266823\pi$
$684$	0	0
$685$	816.529	0.0455445
$686$	0	0
$687$	9862.83	0.547730
$688$	0	0
$689$	9050.12	0.500409
$690$	0	0
$691$	−25987.0	−1.43067	−0.715334	−	0.698783i	$-0.753725\pi$
$691$	−25987.0	−1.43067	−0.715334	+	0.698783i	$0.753725\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28431.9	1.55177
$696$	0	0
$697$	−7339.59	−0.398862
$698$	0	0
$699$	−3950.87	−0.213785
$700$	0	0
$701$	−26368.7	−1.42073	−0.710365	−	0.703833i	$-0.751470\pi$
$701$	−26368.7	−1.42073	−0.710365	+	0.703833i	$0.751470\pi$
$702$	0	0
$703$	−27355.8	−1.46763
$704$	0	0
$705$	−2648.32	−0.141477
$706$	0	0
$707$	−2311.97	−0.122985
$708$	0	0
$709$	12322.1	0.652703	0.326352	−	0.945248i	$-0.394181\pi$
$709$	12322.1	0.652703	0.326352	+	0.945248i	$0.394181\pi$
$710$	0	0
$711$	15511.1	0.818162
$712$	0	0
$713$	−2043.75	−0.107348
$714$	0	0
$715$	0	0
$716$	0	0
$717$	115.046	0.00599231
$718$	0	0
$719$	−15276.6	−0.792382	−0.396191	−	0.918168i	$-0.629668\pi$
$719$	−15276.6	−0.792382	−0.396191	+	0.918168i	$0.629668\pi$
$720$	0	0
$721$	−5705.34	−0.294699
$722$	0	0
$723$	4364.41	0.224501
$724$	0	0
$725$	−3230.93	−0.165509
$726$	0	0
$727$	−30869.8	−1.57482	−0.787411	−	0.616428i	$-0.788579\pi$
$727$	−30869.8	−1.57482	−0.787411	+	0.616428i	$0.788579\pi$
$728$	0	0
$729$	1261.51	0.0640914
$730$	0	0
$731$	−6282.61	−0.317881
$732$	0	0
$733$	6134.39	0.309112	0.154556	−	0.987984i	$-0.450605\pi$
$733$	6134.39	0.309112	0.154556	+	0.987984i	$0.450605\pi$
$734$	0	0
$735$	10128.0	0.508269
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−10235.4	−0.509494	−0.254747	−	0.967008i	$-0.581992\pi$
$739$	−10235.4	−0.509494	−0.254747	+	0.967008i	$0.581992\pi$
$740$	0	0
$741$	−21062.4	−1.04419
$742$	0	0
$743$	−34343.0	−1.69572	−0.847862	−	0.530217i	$-0.822110\pi$
$743$	−34343.0	−1.69572	−0.847862	+	0.530217i	$0.822110\pi$
$744$	0	0
$745$	−28162.7	−1.38497
$746$	0	0
$747$	−17991.3	−0.881212
$748$	0	0
$749$	−4160.31	−0.202956
$750$	0	0
$751$	−34348.7	−1.66898	−0.834488	−	0.551026i	$-0.814236\pi$
$751$	−34348.7	−1.66898	−0.834488	+	0.551026i	$0.814236\pi$
$752$	0	0
$753$	12283.4	0.594467
$754$	0	0
$755$	−26694.8	−1.28678
$756$	0	0
$757$	−9772.73	−0.469215	−0.234608	−	0.972090i	$-0.575380\pi$
$757$	−9772.73	−0.469215	−0.234608	+	0.972090i	$0.575380\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32013.6	1.52496	0.762478	−	0.647015i	$-0.223983\pi$
$761$	32013.6	1.52496	0.762478	+	0.647015i	$0.223983\pi$
$762$	0	0
$763$	−2034.63	−0.0965379
$764$	0	0
$765$	5373.73	0.253971
$766$	0	0
$767$	22620.9	1.06492
$768$	0	0
$769$	28162.1	1.32061	0.660306	−	0.750997i	$-0.270427\pi$
$769$	28162.1	1.32061	0.660306	+	0.750997i	$0.270427\pi$
$770$	0	0
$771$	10611.9	0.495693
$772$	0	0
$773$	−21730.0	−1.01109	−0.505546	−	0.862800i	$-0.668709\pi$
$773$	−21730.0	−1.01109	−0.505546	+	0.862800i	$0.668709\pi$
$774$	0	0
$775$	−3639.20	−0.168676
$776$	0	0
$777$	1573.59	0.0726539
$778$	0	0
$779$	−51662.4	−2.37612
$780$	0	0
$781$	0	0
$782$	0	0
$783$	10267.3	0.468610
$784$	0	0
$785$	−37737.1	−1.71579
$786$	0	0
$787$	−16813.6	−0.761548	−0.380774	−	0.924668i	$-0.624342\pi$
$787$	−16813.6	−0.761548	−0.380774	+	0.924668i	$0.624342\pi$
$788$	0	0
$789$	−17246.7	−0.778197
$790$	0	0
$791$	−2737.00	−0.123030
$792$	0	0
$793$	7505.20	0.336088
$794$	0	0
$795$	4420.23	0.197194
$796$	0	0
$797$	−16019.3	−0.711959	−0.355979	−	0.934494i	$-0.615853\pi$
$797$	−16019.3	−0.711959	−0.355979	+	0.934494i	$0.615853\pi$
$798$	0	0
$799$	1728.14	0.0765172
$800$	0	0
$801$	−20536.2	−0.905879
$802$	0	0
$803$	0	0
$804$	0	0
$805$	873.610	0.0382493
$806$	0	0
$807$	−17261.7	−0.752960
$808$	0	0
$809$	39625.9	1.72209	0.861047	−	0.508526i	$-0.169809\pi$
$809$	39625.9	1.72209	0.861047	+	0.508526i	$0.169809\pi$
$810$	0	0
$811$	−3210.13	−0.138992	−0.0694962	−	0.997582i	$-0.522139\pi$
$811$	−3210.13	−0.138992	−0.0694962	+	0.997582i	$0.522139\pi$
$812$	0	0
$813$	349.182	0.0150631
$814$	0	0
$815$	−37498.9	−1.61169
$816$	0	0
$817$	−44222.4	−1.89369
$818$	0	0
$819$	−4454.82	−0.190066
$820$	0	0
$821$	−1929.57	−0.0820250	−0.0410125	−	0.999159i	$-0.513058\pi$
$821$	−1929.57	−0.0820250	−0.0410125	+	0.999159i	$0.513058\pi$
$822$	0	0
$823$	10985.0	0.465264	0.232632	−	0.972565i	$-0.425266\pi$
$823$	10985.0	0.465264	0.232632	+	0.972565i	$0.425266\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17543.0	−0.737641	−0.368820	−	0.929501i	$-0.620238\pi$
$827$	−17543.0	−0.737641	−0.368820	+	0.929501i	$0.620238\pi$
$828$	0	0
$829$	−2459.32	−0.103034	−0.0515172	−	0.998672i	$-0.516406\pi$
$829$	−2459.32	−0.103034	−0.0515172	+	0.998672i	$0.516406\pi$
$830$	0	0
$831$	7518.18	0.313842
$832$	0	0
$833$	−6608.97	−0.274895
$834$	0	0
$835$	758.877	0.0314515
$836$	0	0
$837$	11564.7	0.477578
$838$	0	0
$839$	−32977.3	−1.35698	−0.678488	−	0.734611i	$-0.737365\pi$
$839$	−32977.3	−1.35698	−0.678488	+	0.734611i	$0.737365\pi$
$840$	0	0
$841$	−16538.0	−0.678095
$842$	0	0
$843$	−14590.5	−0.596113
$844$	0	0
$845$	−21735.5	−0.884880
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−14719.7	−0.595026
$850$	0	0
$851$	−3994.69	−0.160912
$852$	0	0
$853$	−26067.0	−1.04633	−0.523165	−	0.852232i	$-0.675249\pi$
$853$	−26067.0	−1.04633	−0.523165	+	0.852232i	$0.675249\pi$
$854$	0	0
$855$	37825.0	1.51297
$856$	0	0
$857$	22264.0	0.887425	0.443712	−	0.896169i	$-0.353661\pi$
$857$	22264.0	0.887425	0.443712	+	0.896169i	$0.353661\pi$
$858$	0	0
$859$	31573.0	1.25408	0.627042	−	0.778985i	$-0.284265\pi$
$859$	31573.0	1.25408	0.627042	+	0.778985i	$0.284265\pi$
$860$	0	0
$861$	2971.78	0.117628
$862$	0	0
$863$	6498.99	0.256348	0.128174	−	0.991752i	$-0.459088\pi$
$863$	6498.99	0.256348	0.128174	+	0.991752i	$0.459088\pi$
$864$	0	0
$865$	57686.4	2.26751
$866$	0	0
$867$	−10850.9	−0.425047
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−59296.4	−2.30675
$872$	0	0
$873$	−8113.11	−0.314533
$874$	0	0
$875$	−3777.01	−0.145927
$876$	0	0
$877$	37708.2	1.45190	0.725949	−	0.687749i	$-0.241401\pi$
$877$	37708.2	1.45190	0.725949	+	0.687749i	$0.241401\pi$
$878$	0	0
$879$	−21473.7	−0.823994
$880$	0	0
$881$	−33193.8	−1.26938	−0.634692	−	0.772766i	$-0.718873\pi$
$881$	−33193.8	−1.26938	−0.634692	+	0.772766i	$0.718873\pi$
$882$	0	0
$883$	−5662.28	−0.215800	−0.107900	−	0.994162i	$-0.534413\pi$
$883$	−5662.28	−0.215800	−0.107900	+	0.994162i	$0.534413\pi$
$884$	0	0
$885$	11048.4	0.419649
$886$	0	0
$887$	−47693.1	−1.80539	−0.902693	−	0.430285i	$-0.858413\pi$
$887$	−47693.1	−1.80539	−0.902693	+	0.430285i	$0.858413\pi$
$888$	0	0
$889$	6720.44	0.253539
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12164.1	0.455831
$894$	0	0
$895$	−55986.7	−2.09098
$896$	0	0
$897$	−3075.69	−0.114487
$898$	0	0
$899$	8843.02	0.328066
$900$	0	0
$901$	−2884.39	−0.106651
$902$	0	0
$903$	2543.81	0.0937460
$904$	0	0
$905$	15837.6	0.581723
$906$	0	0
$907$	14712.1	0.538596	0.269298	−	0.963057i	$-0.413208\pi$
$907$	14712.1	0.538596	0.269298	+	0.963057i	$0.413208\pi$
$908$	0	0
$909$	−14617.7	−0.533375
$910$	0	0
$911$	34152.0	1.24205	0.621025	−	0.783791i	$-0.286717\pi$
$911$	34152.0	1.24205	0.621025	+	0.783791i	$0.286717\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3665.67	0.132441
$916$	0	0
$917$	7213.80	0.259783
$918$	0	0
$919$	40081.2	1.43869	0.719346	−	0.694652i	$-0.244442\pi$
$919$	40081.2	1.43869	0.719346	+	0.694652i	$0.244442\pi$
$920$	0	0
$921$	−1358.87	−0.0486169
$922$	0	0
$923$	23511.8	0.838463
$924$	0	0
$925$	−7113.16	−0.252842
$926$	0	0
$927$	−36072.6	−1.27808
$928$	0	0
$929$	42666.5	1.50683	0.753414	−	0.657547i	$-0.228406\pi$
$929$	42666.5	1.50683	0.753414	+	0.657547i	$0.228406\pi$
$930$	0	0
$931$	−46519.6	−1.63761
$932$	0	0
$933$	−5137.90	−0.180286
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33994.9	1.18524	0.592618	−	0.805484i	$-0.298095\pi$
$937$	33994.9	1.18524	0.592618	+	0.805484i	$0.298095\pi$
$938$	0	0
$939$	−20008.4	−0.695366
$940$	0	0
$941$	27688.9	0.959227	0.479613	−	0.877480i	$-0.340777\pi$
$941$	27688.9	0.959227	0.479613	+	0.877480i	$0.340777\pi$
$942$	0	0
$943$	−7544.13	−0.260520
$944$	0	0
$945$	−4943.37	−0.170167
$946$	0	0
$947$	−6781.85	−0.232714	−0.116357	−	0.993207i	$-0.537122\pi$
$947$	−6781.85	−0.232714	−0.116357	+	0.993207i	$0.537122\pi$
$948$	0	0
$949$	−69706.3	−2.38436
$950$	0	0
$951$	3873.08	0.132064
$952$	0	0
$953$	16829.0	0.572031	0.286015	−	0.958225i	$-0.407669\pi$
$953$	16829.0	0.572031	0.286015	+	0.958225i	$0.407669\pi$
$954$	0	0
$955$	−51360.3	−1.74029
$956$	0	0
$957$	0	0
$958$	0	0
$959$	215.737	0.00726435
$960$	0	0
$961$	−19830.6	−0.665656
$962$	0	0
$963$	−26304.0	−0.880201
$964$	0	0
$965$	−18256.9	−0.609026
$966$	0	0
$967$	−30373.0	−1.01006	−0.505030	−	0.863102i	$-0.668519\pi$
$967$	−30373.0	−1.01006	−0.505030	+	0.863102i	$0.668519\pi$
$968$	0	0
$969$	6712.86	0.222547
$970$	0	0
$971$	719.651	0.0237845	0.0118922	−	0.999929i	$-0.496214\pi$
$971$	719.651	0.0237845	0.0118922	+	0.999929i	$0.496214\pi$
$972$	0	0
$973$	7512.05	0.247508
$974$	0	0
$975$	−5476.74	−0.179893
$976$	0	0
$977$	−14550.3	−0.476464	−0.238232	−	0.971208i	$-0.576568\pi$
$977$	−14550.3	−0.476464	−0.238232	+	0.971208i	$0.576568\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−12864.1	−0.418675
$982$	0	0
$983$	39178.9	1.27122	0.635611	−	0.772009i	$-0.280748\pi$
$983$	39178.9	1.27122	0.635611	+	0.772009i	$0.280748\pi$
$984$	0	0
$985$	−43461.0	−1.40587
$986$	0	0
$987$	−699.719	−0.0225656
$988$	0	0
$989$	−6457.69	−0.207627
$990$	0	0
$991$	−35777.3	−1.14682	−0.573412	−	0.819267i	$-0.694381\pi$
$991$	−35777.3	−1.14682	−0.573412	+	0.819267i	$0.694381\pi$
$992$	0	0
$993$	6846.10	0.218786
$994$	0	0
$995$	58034.5	1.84906
$996$	0	0
$997$	45607.6	1.44875	0.724376	−	0.689405i	$-0.242128\pi$
$997$	45607.6	1.44875	0.724376	+	0.689405i	$0.242128\pi$
$998$	0	0
$999$	22604.2	0.715880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bp.1.3		4
4.3	odd	2		968.4.a.l.1.2	✓	4
11.10	odd	2		1936.4.a.bo.1.3		4
44.43	even	2		968.4.a.m.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.4.a.l.1.2	✓	4	4.3	odd	2
968.4.a.m.1.2	yes	4	44.43	even	2
1936.4.a.bo.1.3		4	11.10	odd	2
1936.4.a.bp.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q+2.40272 q^{3} -12.7069 q^{5} -3.35731 q^{7} -21.2269 q^{9} +O(q^{10})\)	\(q+2.40272 q^{3} -12.7069 q^{5} -3.35731 q^{7} -21.2269 q^{9} -62.5103 q^{13} -30.5311 q^{15} +19.9228 q^{17} +140.234 q^{19} -8.06668 q^{21} +20.4780 q^{23} +36.4642 q^{25} -115.876 q^{27} -88.6056 q^{29} -99.8021 q^{31} +42.6609 q^{35} -195.072 q^{37} -150.195 q^{39} -368.401 q^{41} -315.347 q^{43} +269.727 q^{45} +86.7418 q^{47} -331.728 q^{49} +47.8690 q^{51} -144.778 q^{53} +336.943 q^{57} -361.875 q^{59} -120.063 q^{61} +71.2654 q^{63} +794.309 q^{65} +948.587 q^{67} +49.2030 q^{69} -376.128 q^{71} +1115.12 q^{73} +87.6134 q^{75} -730.730 q^{79} +294.709 q^{81} +847.568 q^{83} -253.157 q^{85} -212.895 q^{87} +967.458 q^{89} +209.866 q^{91} -239.797 q^{93} -1781.93 q^{95} +382.208 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} + 9 q^{5} + 8 q^{7} + 31 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 + 9 * q^5 + 8 * q^7 + 31 * q^9	\( 4 q + 3 q^{3} + 9 q^{5} + 8 q^{7} + 31 q^{9} + 80 q^{13} + 47 q^{15} - 16 q^{17} + 8 q^{19} + 192 q^{21} + 71 q^{23} + 11 q^{25} + 81 q^{27} + 240 q^{29} + 115 q^{31} - 168 q^{35} + 315 q^{37} + 232 q^{39} - 592 q^{41} - 624 q^{43} + 108 q^{45} + 304 q^{47} + 36 q^{49} - 1256 q^{51} + 184 q^{53} + 592 q^{57} + 805 q^{59} + 240 q^{61} + 2288 q^{63} + 1424 q^{65} - 119 q^{67} + 1499 q^{69} - 723 q^{71} + 1936 q^{73} + 300 q^{75} - 1520 q^{79} + 700 q^{81} - 2016 q^{83} - 3040 q^{85} - 3152 q^{87} + 367 q^{89} + 2848 q^{91} - 1625 q^{93} - 3400 q^{95} + 881 q^{97}+O(q^{100}) \) 4 * q + 3 * q^3 + 9 * q^5 + 8 * q^7 + 31 * q^9 + 80 * q^13 + 47 * q^15 - 16 * q^17 + 8 * q^19 + 192 * q^21 + 71 * q^23 + 11 * q^25 + 81 * q^27 + 240 * q^29 + 115 * q^31 - 168 * q^35 + 315 * q^37 + 232 * q^39 - 592 * q^41 - 624 * q^43 + 108 * q^45 + 304 * q^47 + 36 * q^49 - 1256 * q^51 + 184 * q^53 + 592 * q^57 + 805 * q^59 + 240 * q^61 + 2288 * q^63 + 1424 * q^65 - 119 * q^67 + 1499 * q^69 - 723 * q^71 + 1936 * q^73 + 300 * q^75 - 1520 * q^79 + 700 * q^81 - 2016 * q^83 - 3040 * q^85 - 3152 * q^87 + 367 * q^89 + 2848 * q^91 - 1625 * q^93 - 3400 * q^95 + 881 * q^97

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