LMFDB - Embedded newform 2352.4.a.ch.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.58461.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 65x - 126 $ x^3 - x^2 - 65*x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.20369$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +0.239289 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +0.239289 q^{5} +9.00000 q^{9} -8.09691 q^{11} +43.7693 q^{13} -0.717867 q^{15} -60.8148 q^{17} +98.8861 q^{19} +213.875 q^{23} -124.943 q^{25} -27.0000 q^{27} +110.313 q^{29} +80.0138 q^{31} +24.2907 q^{33} -2.88064 q^{37} -131.308 q^{39} -242.307 q^{41} -367.006 q^{43} +2.15360 q^{45} -89.3910 q^{47} +182.444 q^{51} +11.4127 q^{53} -1.93750 q^{55} -296.658 q^{57} -400.097 q^{59} +480.381 q^{61} +10.4735 q^{65} +240.999 q^{67} -641.624 q^{69} +978.496 q^{71} -622.215 q^{73} +374.828 q^{75} +545.957 q^{79} +81.0000 q^{81} +845.995 q^{83} -14.5523 q^{85} -330.940 q^{87} -203.319 q^{89} -240.042 q^{93} +23.6623 q^{95} -1199.17 q^{97} -72.8722 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9	$ 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9} + 19 q^{11} - 22 q^{13} + 33 q^{15} - 104 q^{17} - 202 q^{19} + 280 q^{23} + 186 q^{25} - 81 q^{27} - 73 q^{29} + 131 q^{31} - 57 q^{33} - 326 q^{37} + 66 q^{39} - 516 q^{41} - 36 q^{43} - 99 q^{45} + 126 q^{47} + 312 q^{51} - 385 q^{53} + 611 q^{55} + 606 q^{57} - 285 q^{59} - 34 q^{61} - 920 q^{65} + 100 q^{67} - 840 q^{69} - 34 q^{71} - 108 q^{73} - 558 q^{75} + 2463 q^{79} + 243 q^{81} + 115 q^{83} - 500 q^{85} + 219 q^{87} - 110 q^{89} - 393 q^{93} + 2064 q^{95} - 2941 q^{97} + 171 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9 + 19 * q^11 - 22 * q^13 + 33 * q^15 - 104 * q^17 - 202 * q^19 + 280 * q^23 + 186 * q^25 - 81 * q^27 - 73 * q^29 + 131 * q^31 - 57 * q^33 - 326 * q^37 + 66 * q^39 - 516 * q^41 - 36 * q^43 - 99 * q^45 + 126 * q^47 + 312 * q^51 - 385 * q^53 + 611 * q^55 + 606 * q^57 - 285 * q^59 - 34 * q^61 - 920 * q^65 + 100 * q^67 - 840 * q^69 - 34 * q^71 - 108 * q^73 - 558 * q^75 + 2463 * q^79 + 243 * q^81 + 115 * q^83 - 500 * q^85 + 219 * q^87 - 110 * q^89 - 393 * q^93 + 2064 * q^95 - 2941 * q^97 + 171 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0.239289	0.0214026	0.0107013	−	0.999943i	$-0.496594\pi$
$5$	0.239289	0.0214026	0.0107013	+	0.999943i	$0.496594\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−8.09691	−0.221937	−0.110969	−	0.993824i	$-0.535395\pi$
$11$	−8.09691	−0.221937	−0.110969	+	0.993824i	$0.535395\pi$
$12$	0	0
$13$	43.7693	0.933802	0.466901	−	0.884310i	$-0.345370\pi$
$13$	43.7693	0.933802	0.466901	+	0.884310i	$0.345370\pi$
$14$	0	0
$15$	−0.717867	−0.0123568
$16$	0	0
$17$	−60.8148	−0.867632	−0.433816	−	0.901001i	$-0.642833\pi$
$17$	−60.8148	−0.867632	−0.433816	+	0.901001i	$0.642833\pi$
$18$	0	0
$19$	98.8861	1.19400	0.597000	−	0.802241i	$-0.296359\pi$
$19$	98.8861	1.19400	0.597000	+	0.802241i	$0.296359\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	213.875	1.93896	0.969478	−	0.245179i	$-0.0788466\pi$
$23$	213.875	1.93896	0.969478	+	0.245179i	$0.0788466\pi$
$24$	0	0
$25$	−124.943	−0.999542
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	110.313	0.706367	0.353184	−	0.935554i	$-0.385099\pi$
$29$	110.313	0.706367	0.353184	+	0.935554i	$0.385099\pi$
$30$	0	0
$31$	80.0138	0.463578	0.231789	−	0.972766i	$-0.425542\pi$
$31$	80.0138	0.463578	0.231789	+	0.972766i	$0.425542\pi$
$32$	0	0
$33$	24.2907	0.128136
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.88064	−0.0127993	−0.00639966	−	0.999980i	$-0.502037\pi$
$37$	−2.88064	−0.0127993	−0.00639966	+	0.999980i	$0.502037\pi$
$38$	0	0
$39$	−131.308	−0.539131
$40$	0	0
$41$	−242.307	−0.922976	−0.461488	−	0.887146i	$-0.652684\pi$
$41$	−242.307	−0.922976	−0.461488	+	0.887146i	$0.652684\pi$
$42$	0	0
$43$	−367.006	−1.30158	−0.650790	−	0.759258i	$-0.725562\pi$
$43$	−367.006	−1.30158	−0.650790	+	0.759258i	$0.725562\pi$
$44$	0	0
$45$	2.15360	0.00713422
$46$	0	0
$47$	−89.3910	−0.277426	−0.138713	−	0.990333i	$-0.544297\pi$
$47$	−89.3910	−0.277426	−0.138713	+	0.990333i	$0.544297\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	182.444	0.500928
$52$	0	0
$53$	11.4127	0.0295784	0.0147892	−	0.999891i	$-0.495292\pi$
$53$	11.4127	0.0295784	0.0147892	+	0.999891i	$0.495292\pi$
$54$	0	0
$55$	−1.93750	−0.00475005
$56$	0	0
$57$	−296.658	−0.689357
$58$	0	0
$59$	−400.097	−0.882851	−0.441425	−	0.897298i	$-0.645527\pi$
$59$	−400.097	−0.882851	−0.441425	+	0.897298i	$0.645527\pi$
$60$	0	0
$61$	480.381	1.00830	0.504152	−	0.863615i	$-0.331805\pi$
$61$	480.381	1.00830	0.504152	+	0.863615i	$0.331805\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.4735	0.0199858
$66$	0	0
$67$	240.999	0.439444	0.219722	−	0.975562i	$-0.429485\pi$
$67$	240.999	0.439444	0.219722	+	0.975562i	$0.429485\pi$
$68$	0	0
$69$	−641.624	−1.11946
$70$	0	0
$71$	978.496	1.63558	0.817790	−	0.575517i	$-0.195199\pi$
$71$	978.496	1.63558	0.817790	+	0.575517i	$0.195199\pi$
$72$	0	0
$73$	−622.215	−0.997600	−0.498800	−	0.866717i	$-0.666226\pi$
$73$	−622.215	−0.997600	−0.498800	+	0.866717i	$0.666226\pi$
$74$	0	0
$75$	374.828	0.577086
$76$	0	0
$77$	0	0
$78$	0	0
$79$	545.957	0.777531	0.388766	−	0.921337i	$-0.372902\pi$
$79$	545.957	0.777531	0.388766	+	0.921337i	$0.372902\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	845.995	1.11879	0.559397	−	0.828900i	$-0.311033\pi$
$83$	845.995	1.11879	0.559397	+	0.828900i	$0.311033\pi$
$84$	0	0
$85$	−14.5523	−0.0185696
$86$	0	0
$87$	−330.940	−0.407821
$88$	0	0
$89$	−203.319	−0.242154	−0.121077	−	0.992643i	$-0.538635\pi$
$89$	−203.319	−0.242154	−0.121077	+	0.992643i	$0.538635\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−240.042	−0.267647
$94$	0	0
$95$	23.6623	0.0255548
$96$	0	0
$97$	−1199.17	−1.25523	−0.627617	−	0.778522i	$-0.715970\pi$
$97$	−1199.17	−1.25523	−0.627617	+	0.778522i	$0.715970\pi$
$98$	0	0
$99$	−72.8722	−0.0739791
$100$	0	0
$101$	367.585	0.362139	0.181070	−	0.983470i	$-0.442044\pi$
$101$	367.585	0.362139	0.181070	+	0.983470i	$0.442044\pi$
$102$	0	0
$103$	−945.228	−0.904233	−0.452117	−	0.891959i	$-0.649331\pi$
$103$	−945.228	−0.904233	−0.452117	+	0.891959i	$0.649331\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	713.157	0.644331	0.322166	−	0.946683i	$-0.395589\pi$
$107$	713.157	0.644331	0.322166	+	0.946683i	$0.395589\pi$
$108$	0	0
$109$	−933.622	−0.820411	−0.410205	−	0.911993i	$-0.634543\pi$
$109$	−933.622	−0.820411	−0.410205	+	0.911993i	$0.634543\pi$
$110$	0	0
$111$	8.64192	0.00738969
$112$	0	0
$113$	1191.38	0.991822	0.495911	−	0.868373i	$-0.334834\pi$
$113$	1191.38	0.991822	0.495911	+	0.868373i	$0.334834\pi$
$114$	0	0
$115$	51.1779	0.0414988
$116$	0	0
$117$	393.924	0.311267
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1265.44	−0.950744
$122$	0	0
$123$	726.922	0.532881
$124$	0	0
$125$	−59.8085	−0.0427955
$126$	0	0
$127$	1196.63	0.836090	0.418045	−	0.908426i	$-0.362715\pi$
$127$	1196.63	0.836090	0.418045	+	0.908426i	$0.362715\pi$
$128$	0	0
$129$	1101.02	0.751467
$130$	0	0
$131$	−505.508	−0.337148	−0.168574	−	0.985689i	$-0.553916\pi$
$131$	−505.508	−0.337148	−0.168574	+	0.985689i	$0.553916\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.46080	−0.00411894
$136$	0	0
$137$	−739.542	−0.461192	−0.230596	−	0.973050i	$-0.574068\pi$
$137$	−739.542	−0.461192	−0.230596	+	0.973050i	$0.574068\pi$
$138$	0	0
$139$	−3095.88	−1.88913	−0.944566	−	0.328322i	$-0.893517\pi$
$139$	−3095.88	−1.88913	−0.944566	+	0.328322i	$0.893517\pi$
$140$	0	0
$141$	268.173	0.160172
$142$	0	0
$143$	−354.396	−0.207246
$144$	0	0
$145$	26.3967	0.0151181
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2989.16	−1.64350	−0.821751	−	0.569847i	$-0.807002\pi$
$149$	−2989.16	−1.64350	−0.821751	+	0.569847i	$0.807002\pi$
$150$	0	0
$151$	1769.68	0.953741	0.476871	−	0.878974i	$-0.341771\pi$
$151$	1769.68	0.953741	0.476871	+	0.878974i	$0.341771\pi$
$152$	0	0
$153$	−547.333	−0.289211
$154$	0	0
$155$	19.1464	0.00992179
$156$	0	0
$157$	2328.02	1.18341	0.591707	−	0.806153i	$-0.298454\pi$
$157$	2328.02	1.18341	0.591707	+	0.806153i	$0.298454\pi$
$158$	0	0
$159$	−34.2381	−0.0170771
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2597.36	1.24810	0.624051	−	0.781384i	$-0.285486\pi$
$163$	2597.36	1.24810	0.624051	+	0.781384i	$0.285486\pi$
$164$	0	0
$165$	5.81250	0.00274244
$166$	0	0
$167$	3276.61	1.51827	0.759137	−	0.650931i	$-0.225621\pi$
$167$	3276.61	1.51827	0.759137	+	0.650931i	$0.225621\pi$
$168$	0	0
$169$	−281.247	−0.128014
$170$	0	0
$171$	889.974	0.398000
$172$	0	0
$173$	−230.036	−0.101094	−0.0505471	−	0.998722i	$-0.516096\pi$
$173$	−230.036	−0.101094	−0.0505471	+	0.998722i	$0.516096\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1200.29	0.509714
$178$	0	0
$179$	3730.56	1.55774	0.778869	−	0.627186i	$-0.215794\pi$
$179$	3730.56	1.55774	0.778869	+	0.627186i	$0.215794\pi$
$180$	0	0
$181$	−1649.91	−0.677550	−0.338775	−	0.940867i	$-0.610013\pi$
$181$	−1649.91	−0.677550	−0.338775	+	0.940867i	$0.610013\pi$
$182$	0	0
$183$	−1441.14	−0.582144
$184$	0	0
$185$	−0.689305	−0.000273939 0
$186$	0	0
$187$	492.412	0.192560
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1809.65	−0.685560	−0.342780	−	0.939416i	$-0.611368\pi$
$191$	−1809.65	−0.685560	−0.342780	+	0.939416i	$0.611368\pi$
$192$	0	0
$193$	4367.23	1.62881	0.814404	−	0.580298i	$-0.197064\pi$
$193$	4367.23	1.62881	0.814404	+	0.580298i	$0.197064\pi$
$194$	0	0
$195$	−31.4205	−0.0115388
$196$	0	0
$197$	398.287	0.144044	0.0720222	−	0.997403i	$-0.477055\pi$
$197$	398.287	0.144044	0.0720222	+	0.997403i	$0.477055\pi$
$198$	0	0
$199$	−4343.35	−1.54720	−0.773598	−	0.633677i	$-0.781545\pi$
$199$	−4343.35	−1.54720	−0.773598	+	0.633677i	$0.781545\pi$
$200$	0	0
$201$	−722.998	−0.253713
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−57.9814	−0.0197541
$206$	0	0
$207$	1924.87	0.646319
$208$	0	0
$209$	−800.672	−0.264993
$210$	0	0
$211$	2723.91	0.888728	0.444364	−	0.895846i	$-0.353430\pi$
$211$	2723.91	0.888728	0.444364	+	0.895846i	$0.353430\pi$
$212$	0	0
$213$	−2935.49	−0.944302
$214$	0	0
$215$	−87.8205	−0.0278572
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1866.65	0.575965
$220$	0	0
$221$	−2661.82	−0.810197
$222$	0	0
$223$	4896.87	1.47049	0.735243	−	0.677803i	$-0.237068\pi$
$223$	4896.87	1.47049	0.735243	+	0.677803i	$0.237068\pi$
$224$	0	0
$225$	−1124.48	−0.333181
$226$	0	0
$227$	−3231.55	−0.944869	−0.472434	−	0.881366i	$-0.656625\pi$
$227$	−3231.55	−0.944869	−0.472434	+	0.881366i	$0.656625\pi$
$228$	0	0
$229$	5918.53	1.70789	0.853947	−	0.520360i	$-0.174202\pi$
$229$	5918.53	1.70789	0.853947	+	0.520360i	$0.174202\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−521.944	−0.146754	−0.0733771	−	0.997304i	$-0.523378\pi$
$233$	−521.944	−0.146754	−0.0733771	+	0.997304i	$0.523378\pi$
$234$	0	0
$235$	−21.3903	−0.00593765
$236$	0	0
$237$	−1637.87	−0.448908
$238$	0	0
$239$	6626.70	1.79350	0.896748	−	0.442542i	$-0.145923\pi$
$239$	6626.70	1.79350	0.896748	+	0.442542i	$0.145923\pi$
$240$	0	0
$241$	2025.39	0.541357	0.270678	−	0.962670i	$-0.412752\pi$
$241$	2025.39	0.541357	0.270678	+	0.962670i	$0.412752\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4328.17	1.11496
$248$	0	0
$249$	−2537.98	−0.645936
$250$	0	0
$251$	−3352.64	−0.843096	−0.421548	−	0.906806i	$-0.638513\pi$
$251$	−3352.64	−0.843096	−0.421548	+	0.906806i	$0.638513\pi$
$252$	0	0
$253$	−1731.73	−0.430327
$254$	0	0
$255$	43.6569	0.0107212
$256$	0	0
$257$	6826.73	1.65696	0.828482	−	0.560015i	$-0.189205\pi$
$257$	6826.73	1.65696	0.828482	+	0.560015i	$0.189205\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	992.819	0.235456
$262$	0	0
$263$	1794.16	0.420657	0.210329	−	0.977631i	$-0.432547\pi$
$263$	1794.16	0.420657	0.210329	+	0.977631i	$0.432547\pi$
$264$	0	0
$265$	2.73094	0.000633057 0
$266$	0	0
$267$	609.956	0.139808
$268$	0	0
$269$	−1466.66	−0.332431	−0.166216	−	0.986089i	$-0.553155\pi$
$269$	−1466.66	−0.332431	−0.166216	+	0.986089i	$0.553155\pi$
$270$	0	0
$271$	1756.51	0.393729	0.196865	−	0.980431i	$-0.436924\pi$
$271$	1756.51	0.393729	0.196865	+	0.980431i	$0.436924\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1011.65	0.221836
$276$	0	0
$277$	7708.00	1.67194	0.835972	−	0.548771i	$-0.184904\pi$
$277$	7708.00	1.67194	0.835972	+	0.548771i	$0.184904\pi$
$278$	0	0
$279$	720.125	0.154526
$280$	0	0
$281$	6837.65	1.45160	0.725800	−	0.687905i	$-0.241470\pi$
$281$	6837.65	1.45160	0.725800	+	0.687905i	$0.241470\pi$
$282$	0	0
$283$	−3248.60	−0.682365	−0.341182	−	0.939997i	$-0.610827\pi$
$283$	−3248.60	−0.682365	−0.341182	+	0.939997i	$0.610827\pi$
$284$	0	0
$285$	−70.9870	−0.0147541
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1214.56	−0.247214
$290$	0	0
$291$	3597.52	0.724710
$292$	0	0
$293$	−9674.19	−1.92892	−0.964458	−	0.264237i	$-0.914880\pi$
$293$	−9674.19	−1.92892	−0.964458	+	0.264237i	$0.914880\pi$
$294$	0	0
$295$	−95.7388	−0.0188953
$296$	0	0
$297$	218.617	0.0427119
$298$	0	0
$299$	9361.15	1.81060
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1102.76	−0.209081
$304$	0	0
$305$	114.950	0.0215804
$306$	0	0
$307$	−4959.70	−0.922035	−0.461018	−	0.887391i	$-0.652516\pi$
$307$	−4959.70	−0.922035	−0.461018	+	0.887391i	$0.652516\pi$
$308$	0	0
$309$	2835.68	0.522059
$310$	0	0
$311$	6757.74	1.23214	0.616071	−	0.787690i	$-0.288723\pi$
$311$	6757.74	1.23214	0.616071	+	0.787690i	$0.288723\pi$
$312$	0	0
$313$	3382.02	0.610744	0.305372	−	0.952233i	$-0.401219\pi$
$313$	3382.02	0.610744	0.305372	+	0.952233i	$0.401219\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	867.009	0.153615	0.0768077	−	0.997046i	$-0.475527\pi$
$317$	867.009	0.153615	0.0768077	+	0.997046i	$0.475527\pi$
$318$	0	0
$319$	−893.196	−0.156769
$320$	0	0
$321$	−2139.47	−0.372005
$322$	0	0
$323$	−6013.73	−1.03595
$324$	0	0
$325$	−5468.66	−0.933374
$326$	0	0
$327$	2800.87	0.473664
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1379.21	0.229028	0.114514	−	0.993422i	$-0.463469\pi$
$331$	1379.21	0.229028	0.114514	+	0.993422i	$0.463469\pi$
$332$	0	0
$333$	−25.9258	−0.00426644
$334$	0	0
$335$	57.6685	0.00940527
$336$	0	0
$337$	−2109.75	−0.341025	−0.170513	−	0.985355i	$-0.554542\pi$
$337$	−2109.75	−0.341025	−0.170513	+	0.985355i	$0.554542\pi$
$338$	0	0
$339$	−3574.15	−0.572629
$340$	0	0
$341$	−647.865	−0.102885
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−153.534	−0.0239593
$346$	0	0
$347$	4188.30	0.647953	0.323976	−	0.946065i	$-0.394980\pi$
$347$	4188.30	0.647953	0.323976	+	0.946065i	$0.394980\pi$
$348$	0	0
$349$	10456.8	1.60385	0.801923	−	0.597428i	$-0.203810\pi$
$349$	10456.8	1.60385	0.801923	+	0.597428i	$0.203810\pi$
$350$	0	0
$351$	−1181.77	−0.179710
$352$	0	0
$353$	−4581.24	−0.690750	−0.345375	−	0.938465i	$-0.612248\pi$
$353$	−4581.24	−0.690750	−0.345375	+	0.938465i	$0.612248\pi$
$354$	0	0
$355$	234.143	0.0350057
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7780.68	1.14387	0.571934	−	0.820300i	$-0.306193\pi$
$359$	7780.68	1.14387	0.571934	+	0.820300i	$0.306193\pi$
$360$	0	0
$361$	2919.45	0.425638
$362$	0	0
$363$	3796.32	0.548912
$364$	0	0
$365$	−148.889	−0.0213513
$366$	0	0
$367$	4126.59	0.586938	0.293469	−	0.955969i	$-0.405190\pi$
$367$	4126.59	0.586938	0.293469	+	0.955969i	$0.405190\pi$
$368$	0	0
$369$	−2180.76	−0.307659
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5129.82	−0.712097	−0.356048	−	0.934468i	$-0.615876\pi$
$373$	−5129.82	−0.712097	−0.356048	+	0.934468i	$0.615876\pi$
$374$	0	0
$375$	179.426	0.0247080
$376$	0	0
$377$	4828.33	0.659607
$378$	0	0
$379$	1017.94	0.137963	0.0689815	−	0.997618i	$-0.478025\pi$
$379$	1017.94	0.137963	0.0689815	+	0.997618i	$0.478025\pi$
$380$	0	0
$381$	−3589.88	−0.482717
$382$	0	0
$383$	10366.6	1.38305	0.691525	−	0.722352i	$-0.256939\pi$
$383$	10366.6	1.38305	0.691525	+	0.722352i	$0.256939\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3303.06	−0.433860
$388$	0	0
$389$	13006.6	1.69527	0.847637	−	0.530576i	$-0.178024\pi$
$389$	13006.6	1.69527	0.847637	+	0.530576i	$0.178024\pi$
$390$	0	0
$391$	−13006.8	−1.68230
$392$	0	0
$393$	1516.52	0.194653
$394$	0	0
$395$	130.641	0.0166412
$396$	0	0
$397$	1954.49	0.247086	0.123543	−	0.992339i	$-0.460574\pi$
$397$	1954.49	0.247086	0.123543	+	0.992339i	$0.460574\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10846.1	1.35069	0.675345	−	0.737502i	$-0.263995\pi$
$401$	10846.1	1.35069	0.675345	+	0.737502i	$0.263995\pi$
$402$	0	0
$403$	3502.15	0.432890
$404$	0	0
$405$	19.3824	0.00237807
$406$	0	0
$407$	23.3243	0.00284065
$408$	0	0
$409$	−5900.36	−0.713335	−0.356667	−	0.934231i	$-0.616087\pi$
$409$	−5900.36	−0.713335	−0.356667	+	0.934231i	$0.616087\pi$
$410$	0	0
$411$	2218.63	0.266269
$412$	0	0
$413$	0	0
$414$	0	0
$415$	202.437	0.0239452
$416$	0	0
$417$	9287.65	1.09069
$418$	0	0
$419$	8994.03	1.04866	0.524328	−	0.851516i	$-0.324316\pi$
$419$	8994.03	1.04866	0.524328	+	0.851516i	$0.324316\pi$
$420$	0	0
$421$	8032.54	0.929886	0.464943	−	0.885341i	$-0.346075\pi$
$421$	8032.54	0.929886	0.464943	+	0.885341i	$0.346075\pi$
$422$	0	0
$423$	−804.519	−0.0924753
$424$	0	0
$425$	7598.37	0.867235
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1063.19	0.119653
$430$	0	0
$431$	7285.93	0.814271	0.407136	−	0.913368i	$-0.366528\pi$
$431$	7285.93	0.814271	0.407136	+	0.913368i	$0.366528\pi$
$432$	0	0
$433$	11152.5	1.23777	0.618887	−	0.785480i	$-0.287584\pi$
$433$	11152.5	1.23777	0.618887	+	0.785480i	$0.287584\pi$
$434$	0	0
$435$	−79.1902	−0.00872845
$436$	0	0
$437$	21149.2	2.31511
$438$	0	0
$439$	−10898.9	−1.18491	−0.592457	−	0.805602i	$-0.701842\pi$
$439$	−10898.9	−1.18491	−0.592457	+	0.805602i	$0.701842\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17184.4	−1.84302	−0.921509	−	0.388356i	$-0.873043\pi$
$443$	−17184.4	−1.84302	−0.921509	+	0.388356i	$0.873043\pi$
$444$	0	0
$445$	−48.6519	−0.00518274
$446$	0	0
$447$	8967.49	0.948876
$448$	0	0
$449$	5273.80	0.554312	0.277156	−	0.960825i	$-0.410608\pi$
$449$	5273.80	0.554312	0.277156	+	0.960825i	$0.410608\pi$
$450$	0	0
$451$	1961.94	0.204843
$452$	0	0
$453$	−5309.05	−0.550643
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4945.05	0.506170	0.253085	−	0.967444i	$-0.418555\pi$
$457$	4945.05	0.506170	0.253085	+	0.967444i	$0.418555\pi$
$458$	0	0
$459$	1642.00	0.166976
$460$	0	0
$461$	−6611.17	−0.667924	−0.333962	−	0.942587i	$-0.608386\pi$
$461$	−6611.17	−0.667924	−0.333962	+	0.942587i	$0.608386\pi$
$462$	0	0
$463$	4339.15	0.435545	0.217772	−	0.976000i	$-0.430121\pi$
$463$	4339.15	0.435545	0.217772	+	0.976000i	$0.430121\pi$
$464$	0	0
$465$	−57.4393	−0.00572835
$466$	0	0
$467$	12296.5	1.21844	0.609221	−	0.793001i	$-0.291482\pi$
$467$	12296.5	1.21844	0.609221	+	0.793001i	$0.291482\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6984.05	−0.683244
$472$	0	0
$473$	2971.62	0.288869
$474$	0	0
$475$	−12355.1	−1.19345
$476$	0	0
$477$	102.714	0.00985948
$478$	0	0
$479$	3899.71	0.371988	0.185994	−	0.982551i	$-0.440449\pi$
$479$	3899.71	0.371988	0.185994	+	0.982551i	$0.440449\pi$
$480$	0	0
$481$	−126.084	−0.0119520
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−286.949	−0.0268653
$486$	0	0
$487$	−9992.44	−0.929776	−0.464888	−	0.885370i	$-0.653905\pi$
$487$	−9992.44	−0.929776	−0.464888	+	0.885370i	$0.653905\pi$
$488$	0	0
$489$	−7792.07	−0.720592
$490$	0	0
$491$	18654.9	1.71463	0.857317	−	0.514788i	$-0.172130\pi$
$491$	18654.9	1.71463	0.857317	+	0.514788i	$0.172130\pi$
$492$	0	0
$493$	−6708.67	−0.612867
$494$	0	0
$495$	−17.4375	−0.00158335
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17669.0	−1.58512	−0.792558	−	0.609796i	$-0.791251\pi$
$499$	−17669.0	−1.58512	−0.792558	+	0.609796i	$0.791251\pi$
$500$	0	0
$501$	−9829.84	−0.876576
$502$	0	0
$503$	−11137.5	−0.987270	−0.493635	−	0.869669i	$-0.664332\pi$
$503$	−11137.5	−0.987270	−0.493635	+	0.869669i	$0.664332\pi$
$504$	0	0
$505$	87.9590	0.00775074
$506$	0	0
$507$	843.741	0.0739090
$508$	0	0
$509$	10872.5	0.946786	0.473393	−	0.880851i	$-0.343029\pi$
$509$	10872.5	0.946786	0.473393	+	0.880851i	$0.343029\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2669.92	−0.229786
$514$	0	0
$515$	−226.182	−0.0193530
$516$	0	0
$517$	723.791	0.0615712
$518$	0	0
$519$	690.107	0.0583667
$520$	0	0
$521$	−20457.2	−1.72025	−0.860123	−	0.510086i	$-0.829613\pi$
$521$	−20457.2	−1.72025	−0.860123	+	0.510086i	$0.829613\pi$
$522$	0	0
$523$	−14914.1	−1.24694	−0.623470	−	0.781847i	$-0.714278\pi$
$523$	−14914.1	−1.24694	−0.623470	+	0.781847i	$0.714278\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4866.02	−0.402215
$528$	0	0
$529$	33575.4	2.75955
$530$	0	0
$531$	−3600.87	−0.294284
$532$	0	0
$533$	−10605.6	−0.861877
$534$	0	0
$535$	170.650	0.0137904
$536$	0	0
$537$	−11191.7	−0.899361
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3615.88	0.287355	0.143677	−	0.989625i	$-0.454107\pi$
$541$	3615.88	0.287355	0.143677	+	0.989625i	$0.454107\pi$
$542$	0	0
$543$	4949.72	0.391184
$544$	0	0
$545$	−223.405	−0.0175590
$546$	0	0
$547$	13365.1	1.04470	0.522350	−	0.852731i	$-0.325055\pi$
$547$	13365.1	1.04470	0.522350	+	0.852731i	$0.325055\pi$
$548$	0	0
$549$	4323.43	0.336101
$550$	0	0
$551$	10908.4	0.843403
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.06792	0.000158159 0
$556$	0	0
$557$	2933.95	0.223188	0.111594	−	0.993754i	$-0.464404\pi$
$557$	2933.95	0.223188	0.111594	+	0.993754i	$0.464404\pi$
$558$	0	0
$559$	−16063.6	−1.21542
$560$	0	0
$561$	−1477.24	−0.111175
$562$	0	0
$563$	−954.247	−0.0714329	−0.0357164	−	0.999362i	$-0.511371\pi$
$563$	−954.247	−0.0714329	−0.0357164	+	0.999362i	$0.511371\pi$
$564$	0	0
$565$	285.085	0.0212276
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13258.6	0.976853	0.488426	−	0.872605i	$-0.337571\pi$
$569$	13258.6	0.976853	0.488426	+	0.872605i	$0.337571\pi$
$570$	0	0
$571$	−2773.75	−0.203288	−0.101644	−	0.994821i	$-0.532410\pi$
$571$	−2773.75	−0.203288	−0.101644	+	0.994821i	$0.532410\pi$
$572$	0	0
$573$	5428.96	0.395808
$574$	0	0
$575$	−26722.1	−1.93807
$576$	0	0
$577$	10361.0	0.747546	0.373773	−	0.927520i	$-0.378064\pi$
$577$	10361.0	0.747546	0.373773	+	0.927520i	$0.378064\pi$
$578$	0	0
$579$	−13101.7	−0.940393
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−92.4078	−0.00656456
$584$	0	0
$585$	94.2616	0.00666194
$586$	0	0
$587$	−4373.97	−0.307552	−0.153776	−	0.988106i	$-0.549143\pi$
$587$	−4373.97	−0.307552	−0.153776	+	0.988106i	$0.549143\pi$
$588$	0	0
$589$	7912.25	0.553512
$590$	0	0
$591$	−1194.86	−0.0831641
$592$	0	0
$593$	−21102.2	−1.46132	−0.730662	−	0.682740i	$-0.760788\pi$
$593$	−21102.2	−1.46132	−0.730662	+	0.682740i	$0.760788\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13030.0	0.893274
$598$	0	0
$599$	−5381.42	−0.367076	−0.183538	−	0.983013i	$-0.558755\pi$
$599$	−5381.42	−0.367076	−0.183538	+	0.983013i	$0.558755\pi$
$600$	0	0
$601$	26223.5	1.77983	0.889915	−	0.456126i	$-0.150763\pi$
$601$	26223.5	1.77983	0.889915	+	0.456126i	$0.150763\pi$
$602$	0	0
$603$	2168.99	0.146481
$604$	0	0
$605$	−302.806	−0.0203484
$606$	0	0
$607$	19022.2	1.27197	0.635986	−	0.771700i	$-0.280593\pi$
$607$	19022.2	1.27197	0.635986	+	0.771700i	$0.280593\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3912.58	−0.259061
$612$	0	0
$613$	267.855	0.0176485	0.00882426	−	0.999961i	$-0.497191\pi$
$613$	267.855	0.0176485	0.00882426	+	0.999961i	$0.497191\pi$
$614$	0	0
$615$	173.944	0.0114051
$616$	0	0
$617$	18568.5	1.21157	0.605785	−	0.795628i	$-0.292859\pi$
$617$	18568.5	1.21157	0.605785	+	0.795628i	$0.292859\pi$
$618$	0	0
$619$	−22873.8	−1.48526	−0.742630	−	0.669702i	$-0.766422\pi$
$619$	−22873.8	−1.48526	−0.742630	+	0.669702i	$0.766422\pi$
$620$	0	0
$621$	−5774.62	−0.373152
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15603.5	0.998626
$626$	0	0
$627$	2402.02	0.152994
$628$	0	0
$629$	175.186	0.0111051
$630$	0	0
$631$	−16406.4	−1.03507	−0.517535	−	0.855662i	$-0.673150\pi$
$631$	−16406.4	−1.03507	−0.517535	+	0.855662i	$0.673150\pi$
$632$	0	0
$633$	−8171.73	−0.513107
$634$	0	0
$635$	286.340	0.0178945
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8806.47	0.545193
$640$	0	0
$641$	−26155.9	−1.61169	−0.805847	−	0.592124i	$-0.798290\pi$
$641$	−26155.9	−1.61169	−0.805847	+	0.592124i	$0.798290\pi$
$642$	0	0
$643$	−4898.20	−0.300414	−0.150207	−	0.988655i	$-0.547994\pi$
$643$	−4898.20	−0.300414	−0.150207	+	0.988655i	$0.547994\pi$
$644$	0	0
$645$	263.461	0.0160834
$646$	0	0
$647$	11929.2	0.724863	0.362432	−	0.932010i	$-0.381947\pi$
$647$	11929.2	0.724863	0.362432	+	0.932010i	$0.381947\pi$
$648$	0	0
$649$	3239.55	0.195938
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7249.87	0.434470	0.217235	−	0.976119i	$-0.430296\pi$
$653$	7249.87	0.434470	0.217235	+	0.976119i	$0.430296\pi$
$654$	0	0
$655$	−120.962	−0.00721587
$656$	0	0
$657$	−5599.94	−0.332533
$658$	0	0
$659$	12078.7	0.713992	0.356996	−	0.934106i	$-0.383801\pi$
$659$	12078.7	0.713992	0.356996	+	0.934106i	$0.383801\pi$
$660$	0	0
$661$	−1816.64	−0.106897	−0.0534486	−	0.998571i	$-0.517021\pi$
$661$	−1816.64	−0.106897	−0.0534486	+	0.998571i	$0.517021\pi$
$662$	0	0
$663$	7985.46	0.467767
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23593.2	1.36961
$668$	0	0
$669$	−14690.6	−0.848986
$670$	0	0
$671$	−3889.60	−0.223780
$672$	0	0
$673$	−21386.6	−1.22495	−0.612477	−	0.790488i	$-0.709827\pi$
$673$	−21386.6	−1.22495	−0.612477	+	0.790488i	$0.709827\pi$
$674$	0	0
$675$	3373.45	0.192362
$676$	0	0
$677$	14029.5	0.796449	0.398225	−	0.917288i	$-0.369626\pi$
$677$	14029.5	0.796449	0.398225	+	0.917288i	$0.369626\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9694.64	0.545520
$682$	0	0
$683$	−12032.6	−0.674107	−0.337053	−	0.941486i	$-0.609430\pi$
$683$	−12032.6	−0.674107	−0.337053	+	0.941486i	$0.609430\pi$
$684$	0	0
$685$	−176.964	−0.00987074
$686$	0	0
$687$	−17755.6	−0.986053
$688$	0	0
$689$	499.527	0.0276204
$690$	0	0
$691$	−10875.1	−0.598710	−0.299355	−	0.954142i	$-0.596771\pi$
$691$	−10875.1	−0.598710	−0.299355	+	0.954142i	$0.596771\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−740.810	−0.0404324
$696$	0	0
$697$	14735.9	0.800804
$698$	0	0
$699$	1565.83	0.0847285
$700$	0	0
$701$	31811.2	1.71397	0.856984	−	0.515343i	$-0.172335\pi$
$701$	31811.2	1.71397	0.856984	+	0.515343i	$0.172335\pi$
$702$	0	0
$703$	−284.855	−0.0152824
$704$	0	0
$705$	64.1708	0.00342810
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−26927.7	−1.42636	−0.713180	−	0.700981i	$-0.752746\pi$
$709$	−26927.7	−1.42636	−0.713180	+	0.700981i	$0.752746\pi$
$710$	0	0
$711$	4913.61	0.259177
$712$	0	0
$713$	17112.9	0.898857
$714$	0	0
$715$	−84.8031	−0.00443560
$716$	0	0
$717$	−19880.1	−1.03548
$718$	0	0
$719$	14549.4	0.754661	0.377331	−	0.926079i	$-0.376842\pi$
$719$	14549.4	0.754661	0.377331	+	0.926079i	$0.376842\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6076.18	−0.312552
$724$	0	0
$725$	−13782.8	−0.706043
$726$	0	0
$727$	−8370.11	−0.427001	−0.213501	−	0.976943i	$-0.568487\pi$
$727$	−8370.11	−0.427001	−0.213501	+	0.976943i	$0.568487\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	22319.4	1.12929
$732$	0	0
$733$	7667.66	0.386373	0.193187	−	0.981162i	$-0.438118\pi$
$733$	7667.66	0.386373	0.193187	+	0.981162i	$0.438118\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1951.35	−0.0975291
$738$	0	0
$739$	−37924.7	−1.88780	−0.943899	−	0.330236i	$-0.892872\pi$
$739$	−37924.7	−1.88780	−0.943899	+	0.330236i	$0.892872\pi$
$740$	0	0
$741$	−12984.5	−0.643723
$742$	0	0
$743$	27008.1	1.33356	0.666778	−	0.745256i	$-0.267673\pi$
$743$	27008.1	1.33356	0.666778	+	0.745256i	$0.267673\pi$
$744$	0	0
$745$	−715.273	−0.0351753
$746$	0	0
$747$	7613.95	0.372932
$748$	0	0
$749$	0	0
$750$	0	0
$751$	17693.1	0.859693	0.429847	−	0.902902i	$-0.358568\pi$
$751$	17693.1	0.859693	0.429847	+	0.902902i	$0.358568\pi$
$752$	0	0
$753$	10057.9	0.486761
$754$	0	0
$755$	423.466	0.0204126
$756$	0	0
$757$	−39291.5	−1.88649	−0.943247	−	0.332093i	$-0.892245\pi$
$757$	−39291.5	−1.88649	−0.943247	+	0.332093i	$0.892245\pi$
$758$	0	0
$759$	5195.18	0.248449
$760$	0	0
$761$	21335.0	1.01629	0.508143	−	0.861273i	$-0.330332\pi$
$761$	21335.0	1.01629	0.508143	+	0.861273i	$0.330332\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−130.971	−0.00618988
$766$	0	0
$767$	−17512.0	−0.824408
$768$	0	0
$769$	−17085.6	−0.801201	−0.400600	−	0.916253i	$-0.631198\pi$
$769$	−17085.6	−0.801201	−0.400600	+	0.916253i	$0.631198\pi$
$770$	0	0
$771$	−20480.2	−0.956649
$772$	0	0
$773$	32736.1	1.52320	0.761601	−	0.648047i	$-0.224414\pi$
$773$	32736.1	1.52320	0.761601	+	0.648047i	$0.224414\pi$
$774$	0	0
$775$	−9997.15	−0.463365
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−23960.8	−1.10203
$780$	0	0
$781$	−7922.80	−0.362996
$782$	0	0
$783$	−2978.46	−0.135940
$784$	0	0
$785$	557.069	0.0253282
$786$	0	0
$787$	−10474.6	−0.474435	−0.237217	−	0.971457i	$-0.576235\pi$
$787$	−10474.6	−0.474435	−0.237217	+	0.971457i	$0.576235\pi$
$788$	0	0
$789$	−5382.49	−0.242867
$790$	0	0
$791$	0	0
$792$	0	0
$793$	21026.0	0.941556
$794$	0	0
$795$	−8.19281	−0.000365496 0
$796$	0	0
$797$	−41837.3	−1.85941	−0.929706	−	0.368302i	$-0.879939\pi$
$797$	−41837.3	−1.85941	−0.929706	+	0.368302i	$0.879939\pi$
$798$	0	0
$799$	5436.29	0.240704
$800$	0	0
$801$	−1829.87	−0.0807181
$802$	0	0
$803$	5038.02	0.221405
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4399.99	0.191929
$808$	0	0
$809$	−37729.8	−1.63969	−0.819846	−	0.572584i	$-0.805941\pi$
$809$	−37729.8	−1.63969	−0.819846	+	0.572584i	$0.805941\pi$
$810$	0	0
$811$	9550.14	0.413503	0.206751	−	0.978394i	$-0.433711\pi$
$811$	9550.14	0.413503	0.206751	+	0.978394i	$0.433711\pi$
$812$	0	0
$813$	−5269.54	−0.227320
$814$	0	0
$815$	621.518	0.0267127
$816$	0	0
$817$	−36291.8	−1.55409
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−41279.7	−1.75478	−0.877389	−	0.479779i	$-0.840717\pi$
$821$	−41279.7	−1.75478	−0.877389	+	0.479779i	$0.840717\pi$
$822$	0	0
$823$	−10479.7	−0.443864	−0.221932	−	0.975062i	$-0.571236\pi$
$823$	−10479.7	−0.443864	−0.221932	+	0.975062i	$0.571236\pi$
$824$	0	0
$825$	−3034.95	−0.128077
$826$	0	0
$827$	−2353.89	−0.0989757	−0.0494879	−	0.998775i	$-0.515759\pi$
$827$	−2353.89	−0.0989757	−0.0494879	+	0.998775i	$0.515759\pi$
$828$	0	0
$829$	−12428.8	−0.520712	−0.260356	−	0.965513i	$-0.583840\pi$
$829$	−12428.8	−0.520712	−0.260356	+	0.965513i	$0.583840\pi$
$830$	0	0
$831$	−23124.0	−0.965298
$832$	0	0
$833$	0	0
$834$	0	0
$835$	784.057	0.0324951
$836$	0	0
$837$	−2160.37	−0.0892156
$838$	0	0
$839$	44513.6	1.83168	0.915840	−	0.401544i	$-0.131526\pi$
$839$	44513.6	1.83168	0.915840	+	0.401544i	$0.131526\pi$
$840$	0	0
$841$	−12220.0	−0.501046
$842$	0	0
$843$	−20512.9	−0.838082
$844$	0	0
$845$	−67.2993	−0.00273984
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9745.80	0.393963
$850$	0	0
$851$	−616.097	−0.0248173
$852$	0	0
$853$	−17576.4	−0.705516	−0.352758	−	0.935715i	$-0.614756\pi$
$853$	−17576.4	−0.705516	−0.352758	+	0.935715i	$0.614756\pi$
$854$	0	0
$855$	212.961	0.00851826
$856$	0	0
$857$	781.198	0.0311379	0.0155690	−	0.999879i	$-0.495044\pi$
$857$	781.198	0.0311379	0.0155690	+	0.999879i	$0.495044\pi$
$858$	0	0
$859$	−35266.3	−1.40078	−0.700391	−	0.713759i	$-0.746991\pi$
$859$	−35266.3	−1.40078	−0.700391	+	0.713759i	$0.746991\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3977.79	0.156901	0.0784506	−	0.996918i	$-0.475003\pi$
$863$	3977.79	0.156901	0.0784506	+	0.996918i	$0.475003\pi$
$864$	0	0
$865$	−55.0450	−0.00216368
$866$	0	0
$867$	3643.69	0.142729
$868$	0	0
$869$	−4420.57	−0.172563
$870$	0	0
$871$	10548.4	0.410354
$872$	0	0
$873$	−10792.6	−0.418411
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−15620.3	−0.601437	−0.300718	−	0.953713i	$-0.597226\pi$
$877$	−15620.3	−0.601437	−0.300718	+	0.953713i	$0.597226\pi$
$878$	0	0
$879$	29022.6	1.11366
$880$	0	0
$881$	30711.6	1.17446	0.587231	−	0.809419i	$-0.300218\pi$
$881$	30711.6	1.17446	0.587231	+	0.809419i	$0.300218\pi$
$882$	0	0
$883$	−15042.1	−0.573279	−0.286640	−	0.958038i	$-0.592538\pi$
$883$	−15042.1	−0.573279	−0.286640	+	0.958038i	$0.592538\pi$
$884$	0	0
$885$	287.216	0.0109092
$886$	0	0
$887$	−16272.7	−0.615990	−0.307995	−	0.951388i	$-0.599658\pi$
$887$	−16272.7	−0.615990	−0.307995	+	0.951388i	$0.599658\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−655.850	−0.0246597
$892$	0	0
$893$	−8839.52	−0.331247
$894$	0	0
$895$	892.682	0.0333397
$896$	0	0
$897$	−28083.5	−1.04535
$898$	0	0
$899$	8826.58	0.327456
$900$	0	0
$901$	−694.062	−0.0256632
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−394.804	−0.0145014
$906$	0	0
$907$	12727.0	0.465923	0.232961	−	0.972486i	$-0.425158\pi$
$907$	12727.0	0.465923	0.232961	+	0.972486i	$0.425158\pi$
$908$	0	0
$909$	3308.27	0.120713
$910$	0	0
$911$	23984.8	0.872283	0.436142	−	0.899878i	$-0.356345\pi$
$911$	23984.8	0.872283	0.436142	+	0.899878i	$0.356345\pi$
$912$	0	0
$913$	−6849.94	−0.248302
$914$	0	0
$915$	−344.850	−0.0124594
$916$	0	0
$917$	0	0
$918$	0	0
$919$	9613.65	0.345076	0.172538	−	0.985003i	$-0.444803\pi$
$919$	9613.65	0.345076	0.172538	+	0.985003i	$0.444803\pi$
$920$	0	0
$921$	14879.1	0.532337
$922$	0	0
$923$	42828.1	1.52731
$924$	0	0
$925$	359.915	0.0127934
$926$	0	0
$927$	−8507.05	−0.301411
$928$	0	0
$929$	−18698.0	−0.660344	−0.330172	−	0.943921i	$-0.607107\pi$
$929$	−18698.0	−0.660344	−0.330172	+	0.943921i	$0.607107\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−20273.2	−0.711378
$934$	0	0
$935$	117.829	0.00412129
$936$	0	0
$937$	6833.17	0.238239	0.119120	−	0.992880i	$-0.461993\pi$
$937$	6833.17	0.238239	0.119120	+	0.992880i	$0.461993\pi$
$938$	0	0
$939$	−10146.1	−0.352613
$940$	0	0
$941$	−6888.64	−0.238643	−0.119322	−	0.992856i	$-0.538072\pi$
$941$	−6888.64	−0.238643	−0.119322	+	0.992856i	$0.538072\pi$
$942$	0	0
$943$	−51823.4	−1.78961
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21171.7	0.726492	0.363246	−	0.931693i	$-0.381668\pi$
$947$	21171.7	0.726492	0.363246	+	0.931693i	$0.381668\pi$
$948$	0	0
$949$	−27233.9	−0.931561
$950$	0	0
$951$	−2601.03	−0.0886899
$952$	0	0
$953$	50173.4	1.70543	0.852715	−	0.522376i	$-0.174954\pi$
$953$	50173.4	1.70543	0.852715	+	0.522376i	$0.174954\pi$
$954$	0	0
$955$	−433.030	−0.0146728
$956$	0	0
$957$	2679.59	0.0905108
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23388.8	−0.785096
$962$	0	0
$963$	6418.41	0.214777
$964$	0	0
$965$	1045.03	0.0348608
$966$	0	0
$967$	−30613.9	−1.01807	−0.509036	−	0.860745i	$-0.669998\pi$
$967$	−30613.9	−1.01807	−0.509036	+	0.860745i	$0.669998\pi$
$968$	0	0
$969$	18041.2	0.598108
$970$	0	0
$971$	6056.78	0.200177	0.100088	−	0.994979i	$-0.468087\pi$
$971$	6056.78	0.200177	0.100088	+	0.994979i	$0.468087\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	16406.0	0.538884
$976$	0	0
$977$	25920.1	0.848779	0.424389	−	0.905480i	$-0.360489\pi$
$977$	25920.1	0.848779	0.424389	+	0.905480i	$0.360489\pi$
$978$	0	0
$979$	1646.25	0.0537431
$980$	0	0
$981$	−8402.60	−0.273470
$982$	0	0
$983$	3354.44	0.108840	0.0544201	−	0.998518i	$-0.482669\pi$
$983$	3354.44	0.108840	0.0544201	+	0.998518i	$0.482669\pi$
$984$	0	0
$985$	95.3055	0.00308293
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−78493.4	−2.52371
$990$	0	0
$991$	8625.53	0.276487	0.138244	−	0.990398i	$-0.455854\pi$
$991$	8625.53	0.276487	0.138244	+	0.990398i	$0.455854\pi$
$992$	0	0
$993$	−4137.63	−0.132229
$994$	0	0
$995$	−1039.32	−0.0331141
$996$	0	0
$997$	62201.1	1.97586	0.987928	−	0.154913i	$-0.0495098\pi$
$997$	62201.1	1.97586	0.987928	+	0.154913i	$0.0495098\pi$
$998$	0	0
$999$	77.7773	0.00246323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.ch.1.2		3
4.3	odd	2		1176.4.a.y.1.2		3
7.2	even	3		336.4.q.l.193.2		6
7.4	even	3		336.4.q.l.289.2		6
7.6	odd	2		2352.4.a.cj.1.2		3
28.11	odd	6		168.4.q.e.121.2	yes	6
28.23	odd	6		168.4.q.e.25.2	✓	6
28.27	even	2		1176.4.a.x.1.2		3
84.11	even	6		504.4.s.g.289.2		6
84.23	even	6		504.4.s.g.361.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.e.25.2	✓	6	28.23	odd	6
168.4.q.e.121.2	yes	6	28.11	odd	6
336.4.q.l.193.2		6	7.2	even	3
336.4.q.l.289.2		6	7.4	even	3
504.4.s.g.289.2		6	84.11	even	6
504.4.s.g.361.2		6	84.23	even	6
1176.4.a.x.1.2		3	28.27	even	2
1176.4.a.y.1.2		3	4.3	odd	2
2352.4.a.ch.1.2		3	1.1	even	1	trivial
2352.4.a.cj.1.2		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +0.239289 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +0.239289 q^{5} +9.00000 q^{9} -8.09691 q^{11} +43.7693 q^{13} -0.717867 q^{15} -60.8148 q^{17} +98.8861 q^{19} +213.875 q^{23} -124.943 q^{25} -27.0000 q^{27} +110.313 q^{29} +80.0138 q^{31} +24.2907 q^{33} -2.88064 q^{37} -131.308 q^{39} -242.307 q^{41} -367.006 q^{43} +2.15360 q^{45} -89.3910 q^{47} +182.444 q^{51} +11.4127 q^{53} -1.93750 q^{55} -296.658 q^{57} -400.097 q^{59} +480.381 q^{61} +10.4735 q^{65} +240.999 q^{67} -641.624 q^{69} +978.496 q^{71} -622.215 q^{73} +374.828 q^{75} +545.957 q^{79} +81.0000 q^{81} +845.995 q^{83} -14.5523 q^{85} -330.940 q^{87} -203.319 q^{89} -240.042 q^{93} +23.6623 q^{95} -1199.17 q^{97} -72.8722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9	\( 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9} + 19 q^{11} - 22 q^{13} + 33 q^{15} - 104 q^{17} - 202 q^{19} + 280 q^{23} + 186 q^{25} - 81 q^{27} - 73 q^{29} + 131 q^{31} - 57 q^{33} - 326 q^{37} + 66 q^{39} - 516 q^{41} - 36 q^{43} - 99 q^{45} + 126 q^{47} + 312 q^{51} - 385 q^{53} + 611 q^{55} + 606 q^{57} - 285 q^{59} - 34 q^{61} - 920 q^{65} + 100 q^{67} - 840 q^{69} - 34 q^{71} - 108 q^{73} - 558 q^{75} + 2463 q^{79} + 243 q^{81} + 115 q^{83} - 500 q^{85} + 219 q^{87} - 110 q^{89} - 393 q^{93} + 2064 q^{95} - 2941 q^{97} + 171 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9 + 19 * q^11 - 22 * q^13 + 33 * q^15 - 104 * q^17 - 202 * q^19 + 280 * q^23 + 186 * q^25 - 81 * q^27 - 73 * q^29 + 131 * q^31 - 57 * q^33 - 326 * q^37 + 66 * q^39 - 516 * q^41 - 36 * q^43 - 99 * q^45 + 126 * q^47 + 312 * q^51 - 385 * q^53 + 611 * q^55 + 606 * q^57 - 285 * q^59 - 34 * q^61 - 920 * q^65 + 100 * q^67 - 840 * q^69 - 34 * q^71 - 108 * q^73 - 558 * q^75 + 2463 * q^79 + 243 * q^81 + 115 * q^83 - 500 * q^85 + 219 * q^87 - 110 * q^89 - 393 * q^93 + 2064 * q^95 - 2941 * q^97 + 171 * q^99

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