LMFDB - Embedded newform 2312.4.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,4,Mod(1,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$136.412415933$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	2312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.52797 q^{3} +9.40407 q^{5} -4.13788 q^{7} +45.7263 q^{9} -2.00416 q^{11} -58.7886 q^{13} -80.1977 q^{15} +83.4503 q^{19} +35.2877 q^{21} +39.2877 q^{23} -36.5634 q^{25} -159.698 q^{27} +157.900 q^{29} +120.205 q^{31} +17.0915 q^{33} -38.9129 q^{35} -337.772 q^{37} +501.347 q^{39} -271.819 q^{41} +239.225 q^{43} +430.014 q^{45} -150.254 q^{47} -325.878 q^{49} -726.333 q^{53} -18.8473 q^{55} -711.662 q^{57} +634.284 q^{59} +617.478 q^{61} -189.210 q^{63} -552.852 q^{65} -712.549 q^{67} -335.045 q^{69} +830.269 q^{71} +510.246 q^{73} +311.812 q^{75} +8.29299 q^{77} +299.690 q^{79} +127.286 q^{81} -614.259 q^{83} -1346.57 q^{87} +1268.61 q^{89} +243.260 q^{91} -1025.10 q^{93} +784.772 q^{95} -1027.92 q^{97} -91.6430 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 88 q^{9} - 168 q^{13} - 120 q^{15} + 88 q^{19} - 64 q^{21} + 144 q^{25} - 520 q^{33} + 512 q^{35} - 616 q^{43} - 984 q^{47} + 272 q^{49} - 1640 q^{53} - 2296 q^{55} + 1304 q^{59} - 1960 q^{67} - 2408 q^{69}+ \cdots + 616 q^{93}+O(q^{100}) $ 24 * q + 88 * q^9 - 168 * q^13 - 120 * q^15 + 88 * q^19 - 64 * q^21 + 144 * q^25 - 520 * q^33 + 512 * q^35 - 616 * q^43 - 984 * q^47 + 272 * q^49 - 1640 * q^53 - 2296 * q^55 + 1304 * q^59 - 1960 * q^67 - 2408 * q^69 - 5248 * q^77 - 3560 * q^81 + 696 * q^83 + 1176 * q^87 - 5504 * q^89 + 616 * q^93

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
$3$	−8.52797	−1.64121	−0.820605	−	0.571496i	$-0.806363\pi$
$3$	−8.52797	−1.64121	−0.820605	+	0.571496i	$0.806363\pi$
$4$	0	0
$5$	9.40407	0.841126	0.420563	−	0.907263i	$-0.361833\pi$
$5$	9.40407	0.841126	0.420563	+	0.907263i	$0.361833\pi$
$6$	0	0
$7$	−4.13788	−0.223425	−0.111712	−	0.993741i	$-0.535633\pi$
$7$	−4.13788	−0.223425	−0.111712	+	0.993741i	$0.535633\pi$
$8$	0	0
$9$	45.7263	1.69357
$10$	0	0
$11$	−2.00416	−0.0549344	−0.0274672	−	0.999623i	$-0.508744\pi$
$11$	−2.00416	−0.0549344	−0.0274672	+	0.999623i	$0.508744\pi$
$12$	0	0
$13$	−58.7886	−1.25423	−0.627116	−	0.778926i	$-0.715765\pi$
$13$	−58.7886	−1.25423	−0.627116	+	0.778926i	$0.715765\pi$
$14$	0	0
$15$	−80.1977	−1.38046
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	83.4503	1.00762	0.503811	−	0.863814i	$-0.331931\pi$
$19$	83.4503	1.00762	0.503811	+	0.863814i	$0.331931\pi$
$20$	0	0
$21$	35.2877	0.366686
$22$	0	0
$23$	39.2877	0.356176	0.178088	−	0.984015i	$-0.443009\pi$
$23$	39.2877	0.356176	0.178088	+	0.984015i	$0.443009\pi$
$24$	0	0
$25$	−36.5634	−0.292507
$26$	0	0
$27$	−159.698	−1.13829
$28$	0	0
$29$	157.900	1.01108	0.505540	−	0.862803i	$-0.331293\pi$
$29$	157.900	1.01108	0.505540	+	0.862803i	$0.331293\pi$
$30$	0	0
$31$	120.205	0.696433	0.348216	−	0.937414i	$-0.386787\pi$
$31$	120.205	0.696433	0.348216	+	0.937414i	$0.386787\pi$
$32$	0	0
$33$	17.0915	0.0901588
$34$	0	0
$35$	−38.9129	−0.187928
$36$	0	0
$37$	−337.772	−1.50080	−0.750398	−	0.660986i	$-0.770138\pi$
$37$	−337.772	−1.50080	−0.750398	+	0.660986i	$0.770138\pi$
$38$	0	0
$39$	501.347	2.05846
$40$	0	0
$41$	−271.819	−1.03539	−0.517695	−	0.855566i	$-0.673210\pi$
$41$	−271.819	−1.03539	−0.517695	+	0.855566i	$0.673210\pi$
$42$	0	0
$43$	239.225	0.848405	0.424202	−	0.905567i	$-0.360554\pi$
$43$	239.225	0.848405	0.424202	+	0.905567i	$0.360554\pi$
$44$	0	0
$45$	430.014	1.42450
$46$	0	0
$47$	−150.254	−0.466314	−0.233157	−	0.972439i	$-0.574906\pi$
$47$	−150.254	−0.466314	−0.233157	+	0.972439i	$0.574906\pi$
$48$	0	0
$49$	−325.878	−0.950081
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−726.333	−1.88244	−0.941222	−	0.337788i	$-0.890321\pi$
$53$	−726.333	−1.88244	−0.941222	+	0.337788i	$0.890321\pi$
$54$	0	0
$55$	−18.8473	−0.0462067
$56$	0	0
$57$	−711.662	−1.65372
$58$	0	0
$59$	634.284	1.39961	0.699803	−	0.714336i	$-0.253271\pi$
$59$	634.284	1.39961	0.699803	+	0.714336i	$0.253271\pi$
$60$	0	0
$61$	617.478	1.29607	0.648033	−	0.761612i	$-0.275592\pi$
$61$	617.478	1.29607	0.648033	+	0.761612i	$0.275592\pi$
$62$	0	0
$63$	−189.210	−0.378385
$64$	0	0
$65$	−552.852	−1.05497
$66$	0	0
$67$	−712.549	−1.29928	−0.649640	−	0.760242i	$-0.725080\pi$
$67$	−712.549	−1.29928	−0.649640	+	0.760242i	$0.725080\pi$
$68$	0	0
$69$	−335.045	−0.584560
$70$	0	0
$71$	830.269	1.38781	0.693907	−	0.720064i	$-0.255888\pi$
$71$	830.269	1.38781	0.693907	+	0.720064i	$0.255888\pi$
$72$	0	0
$73$	510.246	0.818080	0.409040	−	0.912516i	$-0.365864\pi$
$73$	510.246	0.818080	0.409040	+	0.912516i	$0.365864\pi$
$74$	0	0
$75$	311.812	0.480066
$76$	0	0
$77$	8.29299	0.0122737
$78$	0	0
$79$	299.690	0.426807	0.213404	−	0.976964i	$-0.431545\pi$
$79$	299.690	0.426807	0.213404	+	0.976964i	$0.431545\pi$
$80$	0	0
$81$	127.286	0.174604
$82$	0	0
$83$	−614.259	−0.812333	−0.406167	−	0.913799i	$-0.633135\pi$
$83$	−614.259	−0.812333	−0.406167	+	0.913799i	$0.633135\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1346.57	−1.65939
$88$	0	0
$89$	1268.61	1.51093	0.755466	−	0.655188i	$-0.227411\pi$
$89$	1268.61	1.51093	0.755466	+	0.655188i	$0.227411\pi$
$90$	0	0
$91$	243.260	0.280226
$92$	0	0
$93$	−1025.10	−1.14299
$94$	0	0
$95$	784.772	0.847536
$96$	0	0
$97$	−1027.92	−1.07598	−0.537989	−	0.842952i	$-0.680816\pi$
$97$	−1027.92	−1.07598	−0.537989	+	0.842952i	$0.680816\pi$
$98$	0	0
$99$	−91.6430	−0.0930351
$100$	0	0
$101$	−53.5154	−0.0527226	−0.0263613	−	0.999652i	$-0.508392\pi$
$101$	−53.5154	−0.0527226	−0.0263613	+	0.999652i	$0.508392\pi$
$102$	0	0
$103$	1364.04	1.30488	0.652442	−	0.757839i	$-0.273745\pi$
$103$	1364.04	1.30488	0.652442	+	0.757839i	$0.273745\pi$
$104$	0	0
$105$	331.848	0.308429
$106$	0	0
$107$	536.496	0.484720	0.242360	−	0.970186i	$-0.422078\pi$
$107$	536.496	0.484720	0.242360	+	0.970186i	$0.422078\pi$
$108$	0	0
$109$	1359.64	1.19477	0.597383	−	0.801956i	$-0.296207\pi$
$109$	1359.64	1.19477	0.597383	+	0.801956i	$0.296207\pi$
$110$	0	0
$111$	2880.51	2.46312
$112$	0	0
$113$	1289.88	1.07382	0.536910	−	0.843640i	$-0.319591\pi$
$113$	1289.88	1.07382	0.536910	+	0.843640i	$0.319591\pi$
$114$	0	0
$115$	369.465	0.299589
$116$	0	0
$117$	−2688.19	−2.12413
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1326.98	−0.996982
$122$	0	0
$123$	2318.06	1.69929
$124$	0	0
$125$	−1519.35	−1.08716
$126$	0	0
$127$	−1061.22	−0.741483	−0.370741	−	0.928736i	$-0.620896\pi$
$127$	−1061.22	−0.741483	−0.370741	+	0.928736i	$0.620896\pi$
$128$	0	0
$129$	−2040.10	−1.39241
$130$	0	0
$131$	1.15632	0.000771206 0	0.000385603	−	1.00000i	$-0.499877\pi$
$131$	1.15632	0.000771206 0	0.000385603		1.00000i	$0.499877\pi$
$132$	0	0
$133$	−345.307	−0.225127
$134$	0	0
$135$	−1501.81	−0.957445
$136$	0	0
$137$	581.471	0.362616	0.181308	−	0.983426i	$-0.441967\pi$
$137$	581.471	0.362616	0.181308	+	0.983426i	$0.441967\pi$
$138$	0	0
$139$	2521.16	1.53843	0.769217	−	0.638987i	$-0.220646\pi$
$139$	2521.16	1.53843	0.769217	+	0.638987i	$0.220646\pi$
$140$	0	0
$141$	1281.36	0.765319
$142$	0	0
$143$	117.822	0.0689005
$144$	0	0
$145$	1484.90	0.850445
$146$	0	0
$147$	2779.08	1.55928
$148$	0	0
$149$	1889.71	1.03900	0.519501	−	0.854470i	$-0.326118\pi$
$149$	1889.71	1.03900	0.519501	+	0.854470i	$0.326118\pi$
$150$	0	0
$151$	−1128.94	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$151$	−1128.94	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1130.41	0.585787
$156$	0	0
$157$	2012.97	1.02327	0.511633	−	0.859204i	$-0.329041\pi$
$157$	2012.97	1.02327	0.511633	+	0.859204i	$0.329041\pi$
$158$	0	0
$159$	6194.15	3.08949
$160$	0	0
$161$	−162.568	−0.0795785
$162$	0	0
$163$	2800.61	1.34577	0.672886	−	0.739746i	$-0.265054\pi$
$163$	2800.61	1.34577	0.672886	+	0.739746i	$0.265054\pi$
$164$	0	0
$165$	160.729	0.0758349
$166$	0	0
$167$	−2088.33	−0.967665	−0.483832	−	0.875161i	$-0.660756\pi$
$167$	−2088.33	−0.967665	−0.483832	+	0.875161i	$0.660756\pi$
$168$	0	0
$169$	1259.10	0.573098
$170$	0	0
$171$	3815.87	1.70647
$172$	0	0
$173$	−956.725	−0.420453	−0.210227	−	0.977653i	$-0.567420\pi$
$173$	−956.725	−0.420453	−0.210227	+	0.977653i	$0.567420\pi$
$174$	0	0
$175$	151.295	0.0653533
$176$	0	0
$177$	−5409.16	−2.29705
$178$	0	0
$179$	−4186.09	−1.74795	−0.873975	−	0.485972i	$-0.838466\pi$
$179$	−4186.09	−1.74795	−0.873975	+	0.485972i	$0.838466\pi$
$180$	0	0
$181$	−465.328	−0.191092	−0.0955458	−	0.995425i	$-0.530460\pi$
$181$	−465.328	−0.191092	−0.0955458	+	0.995425i	$0.530460\pi$
$182$	0	0
$183$	−5265.84	−2.12711
$184$	0	0
$185$	−3176.44	−1.26236
$186$	0	0
$187$	0	0
$188$	0	0
$189$	660.809	0.254322
$190$	0	0
$191$	−2273.42	−0.861249	−0.430625	−	0.902531i	$-0.641707\pi$
$191$	−2273.42	−0.861249	−0.430625	+	0.902531i	$0.641707\pi$
$192$	0	0
$193$	−1048.40	−0.391015	−0.195507	−	0.980702i	$-0.562635\pi$
$193$	−1048.40	−0.391015	−0.195507	+	0.980702i	$0.562635\pi$
$194$	0	0
$195$	4714.71	1.73142
$196$	0	0
$197$	−3159.17	−1.14254	−0.571272	−	0.820761i	$-0.693550\pi$
$197$	−3159.17	−1.14254	−0.571272	+	0.820761i	$0.693550\pi$
$198$	0	0
$199$	−952.107	−0.339161	−0.169581	−	0.985516i	$-0.554241\pi$
$199$	−952.107	−0.339161	−0.169581	+	0.985516i	$0.554241\pi$
$200$	0	0
$201$	6076.60	2.13239
$202$	0	0
$203$	−653.372	−0.225900
$204$	0	0
$205$	−2556.20	−0.870892
$206$	0	0
$207$	1796.48	0.603209
$208$	0	0
$209$	−167.248	−0.0553530
$210$	0	0
$211$	4718.27	1.53943	0.769713	−	0.638390i	$-0.220399\pi$
$211$	4718.27	1.53943	0.769713	+	0.638390i	$0.220399\pi$
$212$	0	0
$213$	−7080.51	−2.27769
$214$	0	0
$215$	2249.69	0.713615
$216$	0	0
$217$	−497.393	−0.155600
$218$	0	0
$219$	−4351.37	−1.34264
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3891.86	−1.16869	−0.584346	−	0.811505i	$-0.698649\pi$
$223$	−3891.86	−1.16869	−0.584346	+	0.811505i	$0.698649\pi$
$224$	0	0
$225$	−1671.91	−0.495381
$226$	0	0
$227$	−2996.14	−0.876038	−0.438019	−	0.898966i	$-0.644320\pi$
$227$	−2996.14	−0.876038	−0.438019	+	0.898966i	$0.644320\pi$
$228$	0	0
$229$	−3927.50	−1.13335	−0.566674	−	0.823942i	$-0.691770\pi$
$229$	−3927.50	−1.13335	−0.566674	+	0.823942i	$0.691770\pi$
$230$	0	0
$231$	−70.7224	−0.0201437
$232$	0	0
$233$	2817.81	0.792278	0.396139	−	0.918191i	$-0.370350\pi$
$233$	2817.81	0.792278	0.396139	+	0.918191i	$0.370350\pi$
$234$	0	0
$235$	−1413.00	−0.392229
$236$	0	0
$237$	−2555.75	−0.700480
$238$	0	0
$239$	−4737.45	−1.28218	−0.641089	−	0.767467i	$-0.721517\pi$
$239$	−4737.45	−1.28218	−0.641089	+	0.767467i	$0.721517\pi$
$240$	0	0
$241$	−3996.77	−1.06828	−0.534138	−	0.845397i	$-0.679364\pi$
$241$	−3996.77	−1.06828	−0.534138	+	0.845397i	$0.679364\pi$
$242$	0	0
$243$	3226.34	0.851728
$244$	0	0
$245$	−3064.58	−0.799138
$246$	0	0
$247$	−4905.92	−1.26379
$248$	0	0
$249$	5238.38	1.33321
$250$	0	0
$251$	−6549.14	−1.64693	−0.823463	−	0.567370i	$-0.807961\pi$
$251$	−6549.14	−1.64693	−0.823463	+	0.567370i	$0.807961\pi$
$252$	0	0
$253$	−78.7390	−0.0195663
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1024.97	0.248779	0.124389	−	0.992233i	$-0.460303\pi$
$257$	1024.97	0.248779	0.124389	+	0.992233i	$0.460303\pi$
$258$	0	0
$259$	1397.66	0.335315
$260$	0	0
$261$	7220.19	1.71233
$262$	0	0
$263$	−1434.01	−0.336217	−0.168108	−	0.985769i	$-0.553766\pi$
$263$	−1434.01	−0.336217	−0.168108	+	0.985769i	$0.553766\pi$
$264$	0	0
$265$	−6830.49	−1.58337
$266$	0	0
$267$	−10818.7	−2.47975
$268$	0	0
$269$	1259.89	0.285565	0.142782	−	0.989754i	$-0.454395\pi$
$269$	1259.89	0.285565	0.142782	+	0.989754i	$0.454395\pi$
$270$	0	0
$271$	546.986	0.122609	0.0613044	−	0.998119i	$-0.480474\pi$
$271$	546.986	0.122609	0.0613044	+	0.998119i	$0.480474\pi$
$272$	0	0
$273$	−2074.52	−0.459910
$274$	0	0
$275$	73.2791	0.0160687
$276$	0	0
$277$	−7462.88	−1.61878	−0.809388	−	0.587274i	$-0.800201\pi$
$277$	−7462.88	−1.61878	−0.809388	+	0.587274i	$0.800201\pi$
$278$	0	0
$279$	5496.52	1.17946
$280$	0	0
$281$	−4464.00	−0.947687	−0.473844	−	0.880609i	$-0.657134\pi$
$281$	−4464.00	−0.947687	−0.473844	+	0.880609i	$0.657134\pi$
$282$	0	0
$283$	−5935.97	−1.24684	−0.623422	−	0.781885i	$-0.714258\pi$
$283$	−5935.97	−1.24684	−0.623422	+	0.781885i	$0.714258\pi$
$284$	0	0
$285$	−6692.52	−1.39098
$286$	0	0
$287$	1124.75	0.231331
$288$	0	0
$289$	0	0
$290$	0	0
$291$	8766.11	1.76591
$292$	0	0
$293$	670.673	0.133724	0.0668620	−	0.997762i	$-0.478701\pi$
$293$	670.673	0.133724	0.0668620	+	0.997762i	$0.478701\pi$
$294$	0	0
$295$	5964.85	1.17724
$296$	0	0
$297$	320.060	0.0625312
$298$	0	0
$299$	−2309.67	−0.446728
$300$	0	0
$301$	−989.883	−0.189554
$302$	0	0
$303$	456.378	0.0865288
$304$	0	0
$305$	5806.81	1.09015
$306$	0	0
$307$	9301.80	1.72926	0.864628	−	0.502412i	$-0.167554\pi$
$307$	9301.80	1.72926	0.864628	+	0.502412i	$0.167554\pi$
$308$	0	0
$309$	−11632.5	−2.14159
$310$	0	0
$311$	−9072.42	−1.65418	−0.827089	−	0.562071i	$-0.810005\pi$
$311$	−9072.42	−1.65418	−0.827089	+	0.562071i	$0.810005\pi$
$312$	0	0
$313$	3447.23	0.622520	0.311260	−	0.950325i	$-0.399249\pi$
$313$	3447.23	0.622520	0.311260	+	0.950325i	$0.399249\pi$
$314$	0	0
$315$	−1779.34	−0.318269
$316$	0	0
$317$	6060.27	1.07375	0.536875	−	0.843662i	$-0.319605\pi$
$317$	6060.27	1.07375	0.536875	+	0.843662i	$0.319605\pi$
$318$	0	0
$319$	−316.458	−0.0555430
$320$	0	0
$321$	−4575.23	−0.795527
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2149.51	0.366872
$326$	0	0
$327$	−11594.9	−1.96086
$328$	0	0
$329$	621.732	0.104186
$330$	0	0
$331$	10905.6	1.81096	0.905479	−	0.424390i	$-0.139512\pi$
$331$	10905.6	1.81096	0.905479	+	0.424390i	$0.139512\pi$
$332$	0	0
$333$	−15445.1	−2.54170
$334$	0	0
$335$	−6700.87	−1.09286
$336$	0	0
$337$	−3583.49	−0.579243	−0.289622	−	0.957141i	$-0.593529\pi$
$337$	−3583.49	−0.579243	−0.289622	+	0.957141i	$0.593529\pi$
$338$	0	0
$339$	−11000.1	−1.76236
$340$	0	0
$341$	−240.910	−0.0382581
$342$	0	0
$343$	2767.74	0.435696
$344$	0	0
$345$	−3150.78	−0.491688
$346$	0	0
$347$	5075.02	0.785133	0.392566	−	0.919724i	$-0.371587\pi$
$347$	5075.02	0.785133	0.392566	+	0.919724i	$0.371587\pi$
$348$	0	0
$349$	191.428	0.0293608	0.0146804	−	0.999892i	$-0.495327\pi$
$349$	191.428	0.0293608	0.0146804	+	0.999892i	$0.495327\pi$
$350$	0	0
$351$	9388.40	1.42768
$352$	0	0
$353$	2272.17	0.342594	0.171297	−	0.985219i	$-0.445204\pi$
$353$	2272.17	0.342594	0.171297	+	0.985219i	$0.445204\pi$
$354$	0	0
$355$	7807.91	1.16733
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8848.23	−1.30081	−0.650407	−	0.759586i	$-0.725401\pi$
$359$	−8848.23	−1.30081	−0.650407	+	0.759586i	$0.725401\pi$
$360$	0	0
$361$	104.949	0.0153009
$362$	0	0
$363$	11316.5	1.63626
$364$	0	0
$365$	4798.39	0.688108
$366$	0	0
$367$	−2380.72	−0.338617	−0.169309	−	0.985563i	$-0.554153\pi$
$367$	−2380.72	−0.338617	−0.169309	+	0.985563i	$0.554153\pi$
$368$	0	0
$369$	−12429.3	−1.75350
$370$	0	0
$371$	3005.48	0.420584
$372$	0	0
$373$	−6927.12	−0.961589	−0.480794	−	0.876833i	$-0.659652\pi$
$373$	−6927.12	−0.961589	−0.480794	+	0.876833i	$0.659652\pi$
$374$	0	0
$375$	12957.0	1.78426
$376$	0	0
$377$	−9282.72	−1.26813
$378$	0	0
$379$	−12553.8	−1.70145	−0.850723	−	0.525615i	$-0.823835\pi$
$379$	−12553.8	−1.70145	−0.850723	+	0.525615i	$0.823835\pi$
$380$	0	0
$381$	9050.08	1.21693
$382$	0	0
$383$	5106.77	0.681315	0.340658	−	0.940187i	$-0.389350\pi$
$383$	5106.77	0.681315	0.340658	+	0.940187i	$0.389350\pi$
$384$	0	0
$385$	77.9879	0.0103237
$386$	0	0
$387$	10938.9	1.43683
$388$	0	0
$389$	9980.35	1.30083	0.650416	−	0.759578i	$-0.274595\pi$
$389$	9980.35	1.30083	0.650416	+	0.759578i	$0.274595\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.86105	−0.00126571
$394$	0	0
$395$	2818.31	0.358999
$396$	0	0
$397$	3314.57	0.419027	0.209513	−	0.977806i	$-0.432812\pi$
$397$	3314.57	0.419027	0.209513	+	0.977806i	$0.432812\pi$
$398$	0	0
$399$	2944.77	0.369481
$400$	0	0
$401$	−8451.38	−1.05247	−0.526237	−	0.850338i	$-0.676397\pi$
$401$	−8451.38	−1.05247	−0.526237	+	0.850338i	$0.676397\pi$
$402$	0	0
$403$	−7066.67	−0.873488
$404$	0	0
$405$	1197.01	0.146864
$406$	0	0
$407$	676.951	0.0824453
$408$	0	0
$409$	−44.1623	−0.00533908	−0.00266954	−	0.999996i	$-0.500850\pi$
$409$	−44.1623	−0.00533908	−0.00266954	+	0.999996i	$0.500850\pi$
$410$	0	0
$411$	−4958.77	−0.595129
$412$	0	0
$413$	−2624.59	−0.312706
$414$	0	0
$415$	−5776.54	−0.683275
$416$	0	0
$417$	−21500.4	−2.52489
$418$	0	0
$419$	−11515.4	−1.34263	−0.671316	−	0.741171i	$-0.734271\pi$
$419$	−11515.4	−1.34263	−0.671316	+	0.741171i	$0.734271\pi$
$420$	0	0
$421$	−10857.1	−1.25688	−0.628438	−	0.777860i	$-0.716305\pi$
$421$	−10857.1	−1.25688	−0.628438	+	0.777860i	$0.716305\pi$
$422$	0	0
$423$	−6870.55	−0.789735
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2555.05	−0.289573
$428$	0	0
$429$	−1004.78	−0.113080
$430$	0	0
$431$	−1723.74	−0.192644	−0.0963220	−	0.995350i	$-0.530708\pi$
$431$	−1723.74	−0.192644	−0.0963220	+	0.995350i	$0.530708\pi$
$432$	0	0
$433$	2261.87	0.251036	0.125518	−	0.992091i	$-0.459941\pi$
$433$	2261.87	0.251036	0.125518	+	0.992091i	$0.459941\pi$
$434$	0	0
$435$	−12663.2	−1.39576
$436$	0	0
$437$	3278.57	0.358891
$438$	0	0
$439$	7812.41	0.849353	0.424676	−	0.905345i	$-0.360388\pi$
$439$	7812.41	0.849353	0.424676	+	0.905345i	$0.360388\pi$
$440$	0	0
$441$	−14901.2	−1.60903
$442$	0	0
$443$	4323.15	0.463654	0.231827	−	0.972757i	$-0.425530\pi$
$443$	4323.15	0.463654	0.231827	+	0.972757i	$0.425530\pi$
$444$	0	0
$445$	11930.1	1.27088
$446$	0	0
$447$	−16115.4	−1.70522
$448$	0	0
$449$	−1534.50	−0.161286	−0.0806430	−	0.996743i	$-0.525697\pi$
$449$	−1534.50	−0.161286	−0.0806430	+	0.996743i	$0.525697\pi$
$450$	0	0
$451$	544.769	0.0568784
$452$	0	0
$453$	9627.59	0.998551
$454$	0	0
$455$	2287.64	0.235705
$456$	0	0
$457$	−12303.6	−1.25939	−0.629693	−	0.776844i	$-0.716819\pi$
$457$	−12303.6	−1.25939	−0.629693	+	0.776844i	$0.716819\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2668.02	0.269549	0.134774	−	0.990876i	$-0.456969\pi$
$461$	2668.02	0.269549	0.134774	+	0.990876i	$0.456969\pi$
$462$	0	0
$463$	−15499.8	−1.55580	−0.777900	−	0.628388i	$-0.783715\pi$
$463$	−15499.8	−1.55580	−0.777900	+	0.628388i	$0.783715\pi$
$464$	0	0
$465$	−9640.14	−0.961400
$466$	0	0
$467$	−12417.5	−1.23044	−0.615220	−	0.788355i	$-0.710933\pi$
$467$	−12417.5	−1.23044	−0.615220	+	0.788355i	$0.710933\pi$
$468$	0	0
$469$	2948.44	0.290291
$470$	0	0
$471$	−17166.6	−1.67939
$472$	0	0
$473$	−479.445	−0.0466066
$474$	0	0
$475$	−3051.23	−0.294737
$476$	0	0
$477$	−33212.6	−3.18805
$478$	0	0
$479$	−7455.49	−0.711169	−0.355585	−	0.934644i	$-0.615718\pi$
$479$	−7455.49	−0.711169	−0.355585	+	0.934644i	$0.615718\pi$
$480$	0	0
$481$	19857.2	1.88235
$482$	0	0
$483$	1386.37	0.130605
$484$	0	0
$485$	−9666.67	−0.905033
$486$	0	0
$487$	10935.4	1.01751	0.508757	−	0.860910i	$-0.330105\pi$
$487$	10935.4	1.01751	0.508757	+	0.860910i	$0.330105\pi$
$488$	0	0
$489$	−23883.5	−2.20869
$490$	0	0
$491$	−5498.02	−0.505340	−0.252670	−	0.967552i	$-0.581309\pi$
$491$	−5498.02	−0.505340	−0.252670	+	0.967552i	$0.581309\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−861.818	−0.0782542
$496$	0	0
$497$	−3435.55	−0.310072
$498$	0	0
$499$	839.358	0.0753002	0.0376501	−	0.999291i	$-0.488013\pi$
$499$	839.358	0.0753002	0.0376501	+	0.999291i	$0.488013\pi$
$500$	0	0
$501$	17809.2	1.58814
$502$	0	0
$503$	−10982.9	−0.973566	−0.486783	−	0.873523i	$-0.661830\pi$
$503$	−10982.9	−0.973566	−0.486783	+	0.873523i	$0.661830\pi$
$504$	0	0
$505$	−503.263	−0.0443463
$506$	0	0
$507$	−10737.5	−0.940574
$508$	0	0
$509$	2774.81	0.241633	0.120817	−	0.992675i	$-0.461449\pi$
$509$	2774.81	0.241633	0.120817	+	0.992675i	$0.461449\pi$
$510$	0	0
$511$	−2111.34	−0.182779
$512$	0	0
$513$	−13326.8	−1.14696
$514$	0	0
$515$	12827.5	1.09757
$516$	0	0
$517$	301.133	0.0256167
$518$	0	0
$519$	8158.92	0.690052
$520$	0	0
$521$	−15691.0	−1.31945	−0.659727	−	0.751506i	$-0.729328\pi$
$521$	−15691.0	−1.31945	−0.659727	+	0.751506i	$0.729328\pi$
$522$	0	0
$523$	−5670.76	−0.474121	−0.237060	−	0.971495i	$-0.576184\pi$
$523$	−5670.76	−0.474121	−0.237060	+	0.971495i	$0.576184\pi$
$524$	0	0
$525$	−1290.24	−0.107258
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−10623.5	−0.873138
$530$	0	0
$531$	29003.5	2.37033
$532$	0	0
$533$	15979.8	1.29862
$534$	0	0
$535$	5045.25	0.407711
$536$	0	0
$537$	35698.8	2.86875
$538$	0	0
$539$	653.113	0.0521921
$540$	0	0
$541$	−757.935	−0.0602332	−0.0301166	−	0.999546i	$-0.509588\pi$
$541$	−757.935	−0.0602332	−0.0301166	+	0.999546i	$0.509588\pi$
$542$	0	0
$543$	3968.31	0.313621
$544$	0	0
$545$	12786.1	1.00495
$546$	0	0
$547$	2541.09	0.198627	0.0993136	−	0.995056i	$-0.468335\pi$
$547$	2541.09	0.198627	0.0993136	+	0.995056i	$0.468335\pi$
$548$	0	0
$549$	28235.0	2.19497
$550$	0	0
$551$	13176.8	1.01879
$552$	0	0
$553$	−1240.08	−0.0953592
$554$	0	0
$555$	27088.6	2.07179
$556$	0	0
$557$	−17606.6	−1.33934	−0.669672	−	0.742657i	$-0.733565\pi$
$557$	−17606.6	−1.33934	−0.669672	+	0.742657i	$0.733565\pi$
$558$	0	0
$559$	−14063.7	−1.06410
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7476.24	−0.559656	−0.279828	−	0.960050i	$-0.590277\pi$
$563$	−7476.24	−0.559656	−0.279828	+	0.960050i	$0.590277\pi$
$564$	0	0
$565$	12130.1	0.903218
$566$	0	0
$567$	−526.694	−0.0390107
$568$	0	0
$569$	−17176.1	−1.26548	−0.632742	−	0.774363i	$-0.718071\pi$
$569$	−17176.1	−1.26548	−0.632742	+	0.774363i	$0.718071\pi$
$570$	0	0
$571$	−14747.8	−1.08087	−0.540433	−	0.841387i	$-0.681740\pi$
$571$	−14747.8	−1.08087	−0.540433	+	0.841387i	$0.681740\pi$
$572$	0	0
$573$	19387.6	1.41349
$574$	0	0
$575$	−1436.49	−0.104184
$576$	0	0
$577$	21154.7	1.52631	0.763156	−	0.646214i	$-0.223649\pi$
$577$	21154.7	1.52631	0.763156	+	0.646214i	$0.223649\pi$
$578$	0	0
$579$	8940.77	0.641737
$580$	0	0
$581$	2541.73	0.181495
$582$	0	0
$583$	1455.69	0.103411
$584$	0	0
$585$	−25279.9	−1.78666
$586$	0	0
$587$	−282.370	−0.0198546	−0.00992731	−	0.999951i	$-0.503160\pi$
$587$	−282.370	−0.0198546	−0.00992731	+	0.999951i	$0.503160\pi$
$588$	0	0
$589$	10031.1	0.701740
$590$	0	0
$591$	26941.3	1.87515
$592$	0	0
$593$	−26808.9	−1.85651	−0.928253	−	0.371950i	$-0.878689\pi$
$593$	−26808.9	−1.85651	−0.928253	+	0.371950i	$0.878689\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8119.54	0.556634
$598$	0	0
$599$	24971.8	1.70337	0.851687	−	0.524050i	$-0.175580\pi$
$599$	24971.8	1.70337	0.851687	+	0.524050i	$0.175580\pi$
$600$	0	0
$601$	−13110.3	−0.889815	−0.444908	−	0.895576i	$-0.646764\pi$
$601$	−13110.3	−0.889815	−0.444908	+	0.895576i	$0.646764\pi$
$602$	0	0
$603$	−32582.3	−2.20042
$604$	0	0
$605$	−12479.0	−0.838587
$606$	0	0
$607$	21659.7	1.44834	0.724170	−	0.689622i	$-0.242223\pi$
$607$	21659.7	1.44834	0.724170	+	0.689622i	$0.242223\pi$
$608$	0	0
$609$	5571.94	0.370749
$610$	0	0
$611$	8833.21	0.584866
$612$	0	0
$613$	13819.3	0.910535	0.455268	−	0.890355i	$-0.349544\pi$
$613$	13819.3	0.910535	0.455268	+	0.890355i	$0.349544\pi$
$614$	0	0
$615$	21799.2	1.42932
$616$	0	0
$617$	−6805.24	−0.444033	−0.222017	−	0.975043i	$-0.571264\pi$
$617$	−6805.24	−0.444033	−0.222017	+	0.975043i	$0.571264\pi$
$618$	0	0
$619$	−9778.77	−0.634963	−0.317481	−	0.948265i	$-0.602837\pi$
$619$	−9778.77	−0.634963	−0.317481	+	0.948265i	$0.602837\pi$
$620$	0	0
$621$	−6274.15	−0.405432
$622$	0	0
$623$	−5249.38	−0.337579
$624$	0	0
$625$	−9717.69	−0.621932
$626$	0	0
$627$	1426.29	0.0908459
$628$	0	0
$629$	0	0
$630$	0	0
$631$	15422.1	0.972969	0.486485	−	0.873689i	$-0.338279\pi$
$631$	15422.1	0.972969	0.486485	+	0.873689i	$0.338279\pi$
$632$	0	0
$633$	−40237.3	−2.52652
$634$	0	0
$635$	−9979.82	−0.623680
$636$	0	0
$637$	19157.9	1.19162
$638$	0	0
$639$	37965.2	2.35036
$640$	0	0
$641$	−74.2747	−0.00457672	−0.00228836	−	0.999997i	$-0.500728\pi$
$641$	−74.2747	−0.00457672	−0.00228836	+	0.999997i	$0.500728\pi$
$642$	0	0
$643$	−7086.66	−0.434636	−0.217318	−	0.976101i	$-0.569731\pi$
$643$	−7086.66	−0.434636	−0.217318	+	0.976101i	$0.569731\pi$
$644$	0	0
$645$	−19185.3	−1.17119
$646$	0	0
$647$	−11222.7	−0.681932	−0.340966	−	0.940076i	$-0.610754\pi$
$647$	−11222.7	−0.681932	−0.340966	+	0.940076i	$0.610754\pi$
$648$	0	0
$649$	−1271.21	−0.0768865
$650$	0	0
$651$	4241.75	0.255372
$652$	0	0
$653$	−18981.6	−1.13753	−0.568765	−	0.822500i	$-0.692579\pi$
$653$	−18981.6	−1.13753	−0.568765	+	0.822500i	$0.692579\pi$
$654$	0	0
$655$	10.8741	0.000648681 0
$656$	0	0
$657$	23331.7	1.38547
$658$	0	0
$659$	26196.7	1.54853	0.774263	−	0.632864i	$-0.218121\pi$
$659$	26196.7	1.54853	0.774263	+	0.632864i	$0.218121\pi$
$660$	0	0
$661$	−6740.44	−0.396631	−0.198315	−	0.980138i	$-0.563547\pi$
$661$	−6740.44	−0.396631	−0.198315	+	0.980138i	$0.563547\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3247.29	−0.189360
$666$	0	0
$667$	6203.54	0.360123
$668$	0	0
$669$	33189.7	1.91807
$670$	0	0
$671$	−1237.53	−0.0711985
$672$	0	0
$673$	7131.24	0.408453	0.204227	−	0.978924i	$-0.434532\pi$
$673$	7131.24	0.408453	0.204227	+	0.978924i	$0.434532\pi$
$674$	0	0
$675$	5839.09	0.332958
$676$	0	0
$677$	−4296.22	−0.243895	−0.121948	−	0.992537i	$-0.538914\pi$
$677$	−4296.22	−0.243895	−0.121948	+	0.992537i	$0.538914\pi$
$678$	0	0
$679$	4253.43	0.240400
$680$	0	0
$681$	25551.0	1.43776
$682$	0	0
$683$	12331.2	0.690834	0.345417	−	0.938449i	$-0.387737\pi$
$683$	12331.2	0.690834	0.345417	+	0.938449i	$0.387737\pi$
$684$	0	0
$685$	5468.19	0.305006
$686$	0	0
$687$	33493.6	1.86006
$688$	0	0
$689$	42700.1	2.36102
$690$	0	0
$691$	−23842.4	−1.31260	−0.656300	−	0.754500i	$-0.727879\pi$
$691$	−23842.4	−1.31260	−0.656300	+	0.754500i	$0.727879\pi$
$692$	0	0
$693$	379.208	0.0207863
$694$	0	0
$695$	23709.2	1.29402
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−24030.2	−1.30029
$700$	0	0
$701$	33388.6	1.79896	0.899479	−	0.436964i	$-0.143946\pi$
$701$	33388.6	1.79896	0.899479	+	0.436964i	$0.143946\pi$
$702$	0	0
$703$	−28187.2	−1.51223
$704$	0	0
$705$	12050.0	0.643730
$706$	0	0
$707$	221.440	0.0117795
$708$	0	0
$709$	18906.2	1.00146	0.500731	−	0.865603i	$-0.333065\pi$
$709$	18906.2	1.00146	0.500731	+	0.865603i	$0.333065\pi$
$710$	0	0
$711$	13703.7	0.722827
$712$	0	0
$713$	4722.57	0.248053
$714$	0	0
$715$	1108.01	0.0579539
$716$	0	0
$717$	40400.9	2.10432
$718$	0	0
$719$	−25967.4	−1.34690	−0.673451	−	0.739232i	$-0.735189\pi$
$719$	−25967.4	−1.34690	−0.673451	+	0.739232i	$0.735189\pi$
$720$	0	0
$721$	−5644.24	−0.291543
$722$	0	0
$723$	34084.3	1.75326
$724$	0	0
$725$	−5773.37	−0.295748
$726$	0	0
$727$	3980.23	0.203052	0.101526	−	0.994833i	$-0.467628\pi$
$727$	3980.23	0.203052	0.101526	+	0.994833i	$0.467628\pi$
$728$	0	0
$729$	−30950.9	−1.57247
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−8241.98	−0.415313	−0.207656	−	0.978202i	$-0.566584\pi$
$733$	−8241.98	−0.415313	−0.207656	+	0.978202i	$0.566584\pi$
$734$	0	0
$735$	26134.7	1.31155
$736$	0	0
$737$	1428.07	0.0713751
$738$	0	0
$739$	−22817.5	−1.13580	−0.567901	−	0.823097i	$-0.692244\pi$
$739$	−22817.5	−1.13580	−0.567901	+	0.823097i	$0.692244\pi$
$740$	0	0
$741$	41837.6	2.07415
$742$	0	0
$743$	−15912.5	−0.785698	−0.392849	−	0.919603i	$-0.628510\pi$
$743$	−15912.5	−0.785698	−0.392849	+	0.919603i	$0.628510\pi$
$744$	0	0
$745$	17771.0	0.873931
$746$	0	0
$747$	−28087.8	−1.37574
$748$	0	0
$749$	−2219.96	−0.108298
$750$	0	0
$751$	33746.0	1.63969	0.819846	−	0.572584i	$-0.194059\pi$
$751$	33746.0	1.63969	0.819846	+	0.572584i	$0.194059\pi$
$752$	0	0
$753$	55850.9	2.70295
$754$	0	0
$755$	−10616.7	−0.511761
$756$	0	0
$757$	17855.6	0.857297	0.428648	−	0.903471i	$-0.358990\pi$
$757$	17855.6	0.857297	0.428648	+	0.903471i	$0.358990\pi$
$758$	0	0
$759$	671.484	0.0321124
$760$	0	0
$761$	15100.8	0.719322	0.359661	−	0.933083i	$-0.382892\pi$
$761$	15100.8	0.719322	0.359661	+	0.933083i	$0.382892\pi$
$762$	0	0
$763$	−5626.01	−0.266940
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−37288.7	−1.75543
$768$	0	0
$769$	−22563.7	−1.05809	−0.529044	−	0.848595i	$-0.677449\pi$
$769$	−22563.7	−1.05809	−0.529044	+	0.848595i	$0.677449\pi$
$770$	0	0
$771$	−8740.96	−0.408298
$772$	0	0
$773$	−12598.6	−0.586209	−0.293104	−	0.956080i	$-0.594688\pi$
$773$	−12598.6	−0.586209	−0.293104	+	0.956080i	$0.594688\pi$
$774$	0	0
$775$	−4395.10	−0.203712
$776$	0	0
$777$	−11919.2	−0.550321
$778$	0	0
$779$	−22683.3	−1.04328
$780$	0	0
$781$	−1664.00	−0.0762387
$782$	0	0
$783$	−25216.3	−1.15090
$784$	0	0
$785$	18930.1	0.860695
$786$	0	0
$787$	−33969.6	−1.53861	−0.769304	−	0.638882i	$-0.779397\pi$
$787$	−33969.6	−1.53861	−0.769304	+	0.638882i	$0.779397\pi$
$788$	0	0
$789$	12229.2	0.551802
$790$	0	0
$791$	−5337.37	−0.239918
$792$	0	0
$793$	−36300.7	−1.62557
$794$	0	0
$795$	58250.3	2.59865
$796$	0	0
$797$	2078.89	0.0923941	0.0461971	−	0.998932i	$-0.485290\pi$
$797$	2078.89	0.0923941	0.0461971	+	0.998932i	$0.485290\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	58009.1	2.55886
$802$	0	0
$803$	−1022.62	−0.0449407
$804$	0	0
$805$	−1528.80	−0.0669355
$806$	0	0
$807$	−10744.3	−0.468671
$808$	0	0
$809$	−2256.87	−0.0980809	−0.0490405	−	0.998797i	$-0.515616\pi$
$809$	−2256.87	−0.0980809	−0.0490405	+	0.998797i	$0.515616\pi$
$810$	0	0
$811$	8964.94	0.388165	0.194082	−	0.980985i	$-0.437827\pi$
$811$	8964.94	0.388165	0.194082	+	0.980985i	$0.437827\pi$
$812$	0	0
$813$	−4664.68	−0.201227
$814$	0	0
$815$	26337.2	1.13196
$816$	0	0
$817$	19963.4	0.854871
$818$	0	0
$819$	11123.4	0.474582
$820$	0	0
$821$	13908.9	0.591260	0.295630	−	0.955303i	$-0.404471\pi$
$821$	13908.9	0.591260	0.295630	+	0.955303i	$0.404471\pi$
$822$	0	0
$823$	18123.0	0.767592	0.383796	−	0.923418i	$-0.374617\pi$
$823$	18123.0	0.767592	0.383796	+	0.923418i	$0.374617\pi$
$824$	0	0
$825$	−624.922	−0.0263721
$826$	0	0
$827$	23470.8	0.986891	0.493445	−	0.869777i	$-0.335737\pi$
$827$	23470.8	0.986891	0.493445	+	0.869777i	$0.335737\pi$
$828$	0	0
$829$	−43790.8	−1.83464	−0.917321	−	0.398149i	$-0.869653\pi$
$829$	−43790.8	−1.83464	−0.917321	+	0.398149i	$0.869653\pi$
$830$	0	0
$831$	63643.2	2.65675
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−19638.8	−0.813928
$836$	0	0
$837$	−19196.4	−0.792742
$838$	0	0
$839$	−12876.3	−0.529844	−0.264922	−	0.964270i	$-0.585346\pi$
$839$	−12876.3	−0.529844	−0.264922	+	0.964270i	$0.585346\pi$
$840$	0	0
$841$	543.446	0.0222824
$842$	0	0
$843$	38068.9	1.55535
$844$	0	0
$845$	11840.6	0.482048
$846$	0	0
$847$	5490.90	0.222750
$848$	0	0
$849$	50621.8	2.04633
$850$	0	0
$851$	−13270.3	−0.534548
$852$	0	0
$853$	−25822.5	−1.03651	−0.518257	−	0.855225i	$-0.673419\pi$
$853$	−25822.5	−1.03651	−0.518257	+	0.855225i	$0.673419\pi$
$854$	0	0
$855$	35884.8	1.43536
$856$	0	0
$857$	−16951.4	−0.675669	−0.337835	−	0.941205i	$-0.609694\pi$
$857$	−16951.4	−0.675669	−0.337835	+	0.941205i	$0.609694\pi$
$858$	0	0
$859$	−12762.7	−0.506934	−0.253467	−	0.967344i	$-0.581571\pi$
$859$	−12762.7	−0.506934	−0.253467	+	0.967344i	$0.581571\pi$
$860$	0	0
$861$	−9591.86	−0.379663
$862$	0	0
$863$	894.306	0.0352752	0.0176376	−	0.999844i	$-0.494385\pi$
$863$	894.306	0.0352752	0.0176376	+	0.999844i	$0.494385\pi$
$864$	0	0
$865$	−8997.11	−0.353654
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−600.628	−0.0234464
$870$	0	0
$871$	41889.8	1.62960
$872$	0	0
$873$	−47003.2	−1.82224
$874$	0	0
$875$	6286.90	0.242898
$876$	0	0
$877$	−2985.37	−0.114947	−0.0574737	−	0.998347i	$-0.518305\pi$
$877$	−2985.37	−0.114947	−0.0574737	+	0.998347i	$0.518305\pi$
$878$	0	0
$879$	−5719.48	−0.219469
$880$	0	0
$881$	−2015.87	−0.0770901	−0.0385450	−	0.999257i	$-0.512272\pi$
$881$	−2015.87	−0.0770901	−0.0385450	+	0.999257i	$0.512272\pi$
$882$	0	0
$883$	14631.2	0.557620	0.278810	−	0.960346i	$-0.410060\pi$
$883$	14631.2	0.557620	0.278810	+	0.960346i	$0.410060\pi$
$884$	0	0
$885$	−50868.1	−1.93211
$886$	0	0
$887$	−41218.0	−1.56028	−0.780139	−	0.625607i	$-0.784851\pi$
$887$	−41218.0	−1.56028	−0.780139	+	0.625607i	$0.784851\pi$
$888$	0	0
$889$	4391.21	0.165665
$890$	0	0
$891$	−255.102	−0.00959174
$892$	0	0
$893$	−12538.7	−0.469868
$894$	0	0
$895$	−39366.3	−1.47025
$896$	0	0
$897$	19696.8	0.733174
$898$	0	0
$899$	18980.3	0.704149
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8441.69	0.311099
$904$	0	0
$905$	−4375.98	−0.160732
$906$	0	0
$907$	46611.2	1.70639	0.853196	−	0.521590i	$-0.174661\pi$
$907$	46611.2	1.70639	0.853196	+	0.521590i	$0.174661\pi$
$908$	0	0
$909$	−2447.06	−0.0892893
$910$	0	0
$911$	−30366.8	−1.10439	−0.552194	−	0.833716i	$-0.686209\pi$
$911$	−30366.8	−1.10439	−0.552194	+	0.833716i	$0.686209\pi$
$912$	0	0
$913$	1231.08	0.0446250
$914$	0	0
$915$	−49520.3	−1.78917
$916$	0	0
$917$	−4.78470	−0.000172306 0
$918$	0	0
$919$	−4035.91	−0.144867	−0.0724333	−	0.997373i	$-0.523076\pi$
$919$	−4035.91	−0.144867	−0.0724333	+	0.997373i	$0.523076\pi$
$920$	0	0
$921$	−79325.5	−2.83807
$922$	0	0
$923$	−48810.4	−1.74064
$924$	0	0
$925$	12350.1	0.438994
$926$	0	0
$927$	62372.6	2.20991
$928$	0	0
$929$	−6116.13	−0.216000	−0.108000	−	0.994151i	$-0.534445\pi$
$929$	−6116.13	−0.216000	−0.108000	+	0.994151i	$0.534445\pi$
$930$	0	0
$931$	−27194.6	−0.957322
$932$	0	0
$933$	77369.3	2.71485
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−31795.8	−1.10856	−0.554282	−	0.832329i	$-0.687007\pi$
$937$	−31795.8	−1.10856	−0.554282	+	0.832329i	$0.687007\pi$
$938$	0	0
$939$	−29397.8	−1.02169
$940$	0	0
$941$	−41083.2	−1.42324	−0.711622	−	0.702562i	$-0.752039\pi$
$941$	−41083.2	−1.42324	−0.711622	+	0.702562i	$0.752039\pi$
$942$	0	0
$943$	−10679.1	−0.368781
$944$	0	0
$945$	6214.30	0.213917
$946$	0	0
$947$	40161.2	1.37810	0.689052	−	0.724712i	$-0.258027\pi$
$947$	40161.2	1.37810	0.689052	+	0.724712i	$0.258027\pi$
$948$	0	0
$949$	−29996.7	−1.02606
$950$	0	0
$951$	−51681.8	−1.76225
$952$	0	0
$953$	34421.2	1.17000	0.585002	−	0.811032i	$-0.301094\pi$
$953$	34421.2	1.17000	0.585002	+	0.811032i	$0.301094\pi$
$954$	0	0
$955$	−21379.4	−0.724419
$956$	0	0
$957$	2698.74	0.0911577
$958$	0	0
$959$	−2406.06	−0.0810173
$960$	0	0
$961$	−15341.8	−0.514982
$962$	0	0
$963$	24532.0	0.820906
$964$	0	0
$965$	−9859.27	−0.328892
$966$	0	0
$967$	−21369.3	−0.710642	−0.355321	−	0.934744i	$-0.615628\pi$
$967$	−21369.3	−0.710642	−0.355321	+	0.934744i	$0.615628\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40323.8	1.33270	0.666351	−	0.745638i	$-0.267855\pi$
$971$	40323.8	1.33270	0.666351	+	0.745638i	$0.267855\pi$
$972$	0	0
$973$	−10432.3	−0.343724
$974$	0	0
$975$	−18331.0	−0.602114
$976$	0	0
$977$	43661.9	1.42975	0.714877	−	0.699250i	$-0.246483\pi$
$977$	43661.9	1.42975	0.714877	+	0.699250i	$0.246483\pi$
$978$	0	0
$979$	−2542.51	−0.0830021
$980$	0	0
$981$	62171.2	2.02342
$982$	0	0
$983$	4631.96	0.150291	0.0751457	−	0.997173i	$-0.476058\pi$
$983$	4631.96	0.150291	0.0751457	+	0.997173i	$0.476058\pi$
$984$	0	0
$985$	−29709.0	−0.961023
$986$	0	0
$987$	−5302.12	−0.170991
$988$	0	0
$989$	9398.59	0.302182
$990$	0	0
$991$	8846.42	0.283568	0.141784	−	0.989898i	$-0.454716\pi$
$991$	8846.42	0.283568	0.141784	+	0.989898i	$0.454716\pi$
$992$	0	0
$993$	−93002.9	−2.97216
$994$	0	0
$995$	−8953.68	−0.285277
$996$	0	0
$997$	−16194.4	−0.514424	−0.257212	−	0.966355i	$-0.582804\pi$
$997$	−16194.4	−0.514424	−0.257212	+	0.966355i	$0.582804\pi$
$998$	0	0
$999$	53941.4	1.70834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.r.1.2		24
17.5	odd	16		136.4.n.a.25.6	✓	24
17.7	odd	16		136.4.n.a.49.6	yes	24
17.16	even	2	inner	2312.4.a.r.1.23		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.n.a.25.6	✓	24	17.5	odd	16
136.4.n.a.49.6	yes	24	17.7	odd	16
2312.4.a.r.1.2		24	1.1	even	1	trivial
2312.4.a.r.1.23		24	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-8.52797 q^{3} +9.40407 q^{5} -4.13788 q^{7} +45.7263 q^{9} -2.00416 q^{11} -58.7886 q^{13} -80.1977 q^{15} +83.4503 q^{19} +35.2877 q^{21} +39.2877 q^{23} -36.5634 q^{25} -159.698 q^{27} +157.900 q^{29} +120.205 q^{31} +17.0915 q^{33} -38.9129 q^{35} -337.772 q^{37} +501.347 q^{39} -271.819 q^{41} +239.225 q^{43} +430.014 q^{45} -150.254 q^{47} -325.878 q^{49} -726.333 q^{53} -18.8473 q^{55} -711.662 q^{57} +634.284 q^{59} +617.478 q^{61} -189.210 q^{63} -552.852 q^{65} -712.549 q^{67} -335.045 q^{69} +830.269 q^{71} +510.246 q^{73} +311.812 q^{75} +8.29299 q^{77} +299.690 q^{79} +127.286 q^{81} -614.259 q^{83} -1346.57 q^{87} +1268.61 q^{89} +243.260 q^{91} -1025.10 q^{93} +784.772 q^{95} -1027.92 q^{97} -91.6430 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9} - 168 q^{13} - 120 q^{15} + 88 q^{19} - 64 q^{21} + 144 q^{25} - 520 q^{33} + 512 q^{35} - 616 q^{43} - 984 q^{47} + 272 q^{49} - 1640 q^{53} - 2296 q^{55} + 1304 q^{59} - 1960 q^{67} - 2408 q^{69}+ \cdots + 616 q^{93}+O(q^{100}) \) 24 * q + 88 * q^9 - 168 * q^13 - 120 * q^15 + 88 * q^19 - 64 * q^21 + 144 * q^25 - 520 * q^33 + 512 * q^35 - 616 * q^43 - 984 * q^47 + 272 * q^49 - 1640 * q^53 - 2296 * q^55 + 1304 * q^59 - 1960 * q^67 - 2408 * q^69 - 5248 * q^77 - 3560 * q^81 + 696 * q^83 + 1176 * q^87 - 5504 * q^89 + 616 * q^93

Introduction

L-functions

Modular forms

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