LMFDB - Embedded newform 2352.4.a.ci.1.1

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:	3.3.57516.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 24x + 6 $ x^3 - x^2 - 24*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.248072$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -12.4346 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -12.4346 q^{5} +9.00000 q^{9} -60.3115 q^{11} +36.4269 q^{13} -37.3038 q^{15} -48.7461 q^{17} +50.5500 q^{19} -138.792 q^{23} +29.6194 q^{25} +27.0000 q^{27} -61.1345 q^{29} +1.16935 q^{31} -180.935 q^{33} +69.5268 q^{37} +109.281 q^{39} +308.115 q^{41} -174.443 q^{43} -111.911 q^{45} -389.362 q^{47} -146.238 q^{51} +314.935 q^{53} +749.950 q^{55} +151.650 q^{57} -844.526 q^{59} -338.538 q^{61} -452.954 q^{65} +971.550 q^{67} -416.377 q^{69} +98.4698 q^{71} +710.235 q^{73} +88.8581 q^{75} +486.884 q^{79} +81.0000 q^{81} -605.688 q^{83} +606.139 q^{85} -183.403 q^{87} +218.069 q^{89} +3.50806 q^{93} -628.569 q^{95} -782.288 q^{97} -542.804 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 35 q^{11} + 62 q^{13} + 33 q^{15} + 48 q^{17} + 202 q^{19} - 216 q^{23} + 130 q^{25} + 81 q^{27} + 53 q^{29} + 95 q^{31} - 105 q^{33} + 262 q^{37} + 186 q^{39} + 244 q^{41} - 360 q^{43} + 99 q^{45} + 210 q^{47} + 144 q^{51} + 393 q^{53} + 1031 q^{55} + 606 q^{57} - 1143 q^{59} - 70 q^{61} - 472 q^{65} + 628 q^{67} - 648 q^{69} - 318 q^{71} + 988 q^{73} + 390 q^{75} - 861 q^{79} + 243 q^{81} - 519 q^{83} + 1800 q^{85} + 159 q^{87} + 1766 q^{89} + 285 q^{93} + 736 q^{95} + 19 q^{97} - 315 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 + 11 * q^5 + 27 * q^9 - 35 * q^11 + 62 * q^13 + 33 * q^15 + 48 * q^17 + 202 * q^19 - 216 * q^23 + 130 * q^25 + 81 * q^27 + 53 * q^29 + 95 * q^31 - 105 * q^33 + 262 * q^37 + 186 * q^39 + 244 * q^41 - 360 * q^43 + 99 * q^45 + 210 * q^47 + 144 * q^51 + 393 * q^53 + 1031 * q^55 + 606 * q^57 - 1143 * q^59 - 70 * q^61 - 472 * q^65 + 628 * q^67 - 648 * q^69 - 318 * q^71 + 988 * q^73 + 390 * q^75 - 861 * q^79 + 243 * q^81 - 519 * q^83 + 1800 * q^85 + 159 * q^87 + 1766 * q^89 + 285 * q^93 + 736 * q^95 + 19 * q^97 - 315 * 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
$5$	−12.4346	−1.11218	−0.556092	−	0.831120i	$-0.687700\pi$
$5$	−12.4346	−1.11218	−0.556092	+	0.831120i	$0.687700\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−60.3115	−1.65315	−0.826573	−	0.562829i	$-0.809713\pi$
$11$	−60.3115	−1.65315	−0.826573	+	0.562829i	$0.809713\pi$
$12$	0	0
$13$	36.4269	0.777154	0.388577	−	0.921416i	$-0.372967\pi$
$13$	36.4269	0.777154	0.388577	+	0.921416i	$0.372967\pi$
$14$	0	0
$15$	−37.3038	−0.642120
$16$	0	0
$17$	−48.7461	−0.695451	−0.347726	−	0.937596i	$-0.613046\pi$
$17$	−48.7461	−0.695451	−0.347726	+	0.937596i	$0.613046\pi$
$18$	0	0
$19$	50.5500	0.610366	0.305183	−	0.952294i	$-0.401282\pi$
$19$	50.5500	0.610366	0.305183	+	0.952294i	$0.401282\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−138.792	−1.25827	−0.629135	−	0.777296i	$-0.716591\pi$
$23$	−138.792	−1.25827	−0.629135	+	0.777296i	$0.716591\pi$
$24$	0	0
$25$	29.6194	0.236955
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−61.1345	−0.391462	−0.195731	−	0.980658i	$-0.562708\pi$
$29$	−61.1345	−0.391462	−0.195731	+	0.980658i	$0.562708\pi$
$30$	0	0
$31$	1.16935	0.00677490	0.00338745	−	0.999994i	$-0.498922\pi$
$31$	1.16935	0.00677490	0.00338745	+	0.999994i	$0.498922\pi$
$32$	0	0
$33$	−180.935	−0.954444
$34$	0	0
$35$	0	0
$36$	0	0
$37$	69.5268	0.308923	0.154461	−	0.987999i	$-0.450636\pi$
$37$	69.5268	0.308923	0.154461	+	0.987999i	$0.450636\pi$
$38$	0	0
$39$	109.281	0.448690
$40$	0	0
$41$	308.115	1.17365	0.586823	−	0.809715i	$-0.300378\pi$
$41$	308.115	1.17365	0.586823	+	0.809715i	$0.300378\pi$
$42$	0	0
$43$	−174.443	−0.618657	−0.309329	−	0.950955i	$-0.600104\pi$
$43$	−174.443	−0.618657	−0.309329	+	0.950955i	$0.600104\pi$
$44$	0	0
$45$	−111.911	−0.370728
$46$	0	0
$47$	−389.362	−1.20839	−0.604194	−	0.796837i	$-0.706505\pi$
$47$	−389.362	−1.20839	−0.604194	+	0.796837i	$0.706505\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−146.238	−0.401519
$52$	0	0
$53$	314.935	0.816220	0.408110	−	0.912933i	$-0.366188\pi$
$53$	314.935	0.816220	0.408110	+	0.912933i	$0.366188\pi$
$54$	0	0
$55$	749.950	1.83860
$56$	0	0
$57$	151.650	0.352395
$58$	0	0
$59$	−844.526	−1.86352	−0.931762	−	0.363068i	$-0.881729\pi$
$59$	−844.526	−1.86352	−0.931762	+	0.363068i	$0.881729\pi$
$60$	0	0
$61$	−338.538	−0.710579	−0.355290	−	0.934756i	$-0.615618\pi$
$61$	−338.538	−0.710579	−0.355290	+	0.934756i	$0.615618\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−452.954	−0.864339
$66$	0	0
$67$	971.550	1.77155	0.885774	−	0.464117i	$-0.153628\pi$
$67$	971.550	1.77155	0.885774	+	0.464117i	$0.153628\pi$
$68$	0	0
$69$	−416.377	−0.726463
$70$	0	0
$71$	98.4698	0.164595	0.0822973	−	0.996608i	$-0.473774\pi$
$71$	98.4698	0.164595	0.0822973	+	0.996608i	$0.473774\pi$
$72$	0	0
$73$	710.235	1.13872	0.569361	−	0.822088i	$-0.307191\pi$
$73$	710.235	1.13872	0.569361	+	0.822088i	$0.307191\pi$
$74$	0	0
$75$	88.8581	0.136806
$76$	0	0
$77$	0	0
$78$	0	0
$79$	486.884	0.693402	0.346701	−	0.937976i	$-0.387302\pi$
$79$	486.884	0.693402	0.346701	+	0.937976i	$0.387302\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−605.688	−0.800999	−0.400499	−	0.916297i	$-0.631163\pi$
$83$	−605.688	−0.800999	−0.400499	+	0.916297i	$0.631163\pi$
$84$	0	0
$85$	606.139	0.773470
$86$	0	0
$87$	−183.403	−0.226010
$88$	0	0
$89$	218.069	0.259722	0.129861	−	0.991532i	$-0.458547\pi$
$89$	218.069	0.259722	0.129861	+	0.991532i	$0.458547\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.50806	0.00391149
$94$	0	0
$95$	−628.569	−0.678840
$96$	0	0
$97$	−782.288	−0.818859	−0.409429	−	0.912342i	$-0.634272\pi$
$97$	−782.288	−0.818859	−0.409429	+	0.912342i	$0.634272\pi$
$98$	0	0
$99$	−542.804	−0.551049
$100$	0	0
$101$	−311.646	−0.307029	−0.153514	−	0.988146i	$-0.549059\pi$
$101$	−311.646	−0.307029	−0.153514	+	0.988146i	$0.549059\pi$
$102$	0	0
$103$	149.258	0.142784	0.0713922	−	0.997448i	$-0.477256\pi$
$103$	149.258	0.142784	0.0713922	+	0.997448i	$0.477256\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−851.519	−0.769341	−0.384670	−	0.923054i	$-0.625685\pi$
$107$	−851.519	−0.769341	−0.384670	+	0.923054i	$0.625685\pi$
$108$	0	0
$109$	1361.88	1.19674	0.598369	−	0.801221i	$-0.295816\pi$
$109$	1361.88	1.19674	0.598369	+	0.801221i	$0.295816\pi$
$110$	0	0
$111$	208.581	0.178357
$112$	0	0
$113$	1048.55	0.872917	0.436459	−	0.899724i	$-0.356233\pi$
$113$	1048.55	0.872917	0.436459	+	0.899724i	$0.356233\pi$
$114$	0	0
$115$	1725.83	1.39943
$116$	0	0
$117$	327.842	0.259051
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2306.48	1.73289
$122$	0	0
$123$	924.346	0.677605
$124$	0	0
$125$	1186.02	0.848647
$126$	0	0
$127$	−488.408	−0.341254	−0.170627	−	0.985336i	$-0.554579\pi$
$127$	−488.408	−0.341254	−0.170627	+	0.985336i	$0.554579\pi$
$128$	0	0
$129$	−523.328	−0.357182
$130$	0	0
$131$	1854.23	1.23668	0.618338	−	0.785912i	$-0.287806\pi$
$131$	1854.23	1.23668	0.618338	+	0.785912i	$0.287806\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−335.734	−0.214040
$136$	0	0
$137$	−511.115	−0.318741	−0.159370	−	0.987219i	$-0.550946\pi$
$137$	−511.115	−0.318741	−0.159370	+	0.987219i	$0.550946\pi$
$138$	0	0
$139$	−2266.10	−1.38279	−0.691397	−	0.722475i	$-0.743005\pi$
$139$	−2266.10	−1.38279	−0.691397	+	0.722475i	$0.743005\pi$
$140$	0	0
$141$	−1168.09	−0.697663
$142$	0	0
$143$	−2196.96	−1.28475
$144$	0	0
$145$	760.183	0.435378
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1507.90	0.829074	0.414537	−	0.910033i	$-0.363944\pi$
$149$	1507.90	0.829074	0.414537	+	0.910033i	$0.363944\pi$
$150$	0	0
$151$	−1591.83	−0.857887	−0.428943	−	0.903331i	$-0.641114\pi$
$151$	−1591.83	−0.857887	−0.428943	+	0.903331i	$0.641114\pi$
$152$	0	0
$153$	−438.715	−0.231817
$154$	0	0
$155$	−14.5404	−0.00753494
$156$	0	0
$157$	1164.16	0.591784	0.295892	−	0.955221i	$-0.404383\pi$
$157$	1164.16	0.591784	0.295892	+	0.955221i	$0.404383\pi$
$158$	0	0
$159$	944.805	0.471245
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1155.88	0.555432	0.277716	−	0.960663i	$-0.410423\pi$
$163$	1155.88	0.555432	0.277716	+	0.960663i	$0.410423\pi$
$164$	0	0
$165$	2249.85	1.06152
$166$	0	0
$167$	2890.61	1.33941	0.669707	−	0.742626i	$-0.266420\pi$
$167$	2890.61	1.33941	0.669707	+	0.742626i	$0.266420\pi$
$168$	0	0
$169$	−870.082	−0.396032
$170$	0	0
$171$	454.950	0.203455
$172$	0	0
$173$	1894.94	0.832770	0.416385	−	0.909188i	$-0.363297\pi$
$173$	1894.94	0.832770	0.416385	+	0.909188i	$0.363297\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2533.58	−1.07591
$178$	0	0
$179$	4288.49	1.79071	0.895355	−	0.445354i	$-0.146922\pi$
$179$	4288.49	1.79071	0.895355	+	0.445354i	$0.146922\pi$
$180$	0	0
$181$	383.732	0.157583	0.0787917	−	0.996891i	$-0.474894\pi$
$181$	383.732	0.157583	0.0787917	+	0.996891i	$0.474894\pi$
$182$	0	0
$183$	−1015.61	−0.410253
$184$	0	0
$185$	−864.539	−0.343579
$186$	0	0
$187$	2939.95	1.14968
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−385.311	−0.145969	−0.0729845	−	0.997333i	$-0.523252\pi$
$191$	−385.311	−0.145969	−0.0729845	+	0.997333i	$0.523252\pi$
$192$	0	0
$193$	630.224	0.235049	0.117525	−	0.993070i	$-0.462504\pi$
$193$	630.224	0.235049	0.117525	+	0.993070i	$0.462504\pi$
$194$	0	0
$195$	−1358.86	−0.499026
$196$	0	0
$197$	−1250.23	−0.452158	−0.226079	−	0.974109i	$-0.572591\pi$
$197$	−1250.23	−0.452158	−0.226079	+	0.974109i	$0.572591\pi$
$198$	0	0
$199$	1092.24	0.389081	0.194541	−	0.980894i	$-0.437678\pi$
$199$	1092.24	0.389081	0.194541	+	0.980894i	$0.437678\pi$
$200$	0	0
$201$	2914.65	1.02280
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3831.29	−1.30531
$206$	0	0
$207$	−1249.13	−0.419423
$208$	0	0
$209$	−3048.75	−1.00902
$210$	0	0
$211$	3620.05	1.18111	0.590556	−	0.806997i	$-0.298909\pi$
$211$	3620.05	1.18111	0.590556	+	0.806997i	$0.298909\pi$
$212$	0	0
$213$	295.409	0.0950287
$214$	0	0
$215$	2169.13	0.688061
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2130.70	0.657442
$220$	0	0
$221$	−1775.67	−0.540473
$222$	0	0
$223$	183.844	0.0552069	0.0276034	−	0.999619i	$-0.491212\pi$
$223$	183.844	0.0552069	0.0276034	+	0.999619i	$0.491212\pi$
$224$	0	0
$225$	266.574	0.0789850
$226$	0	0
$227$	−2279.52	−0.666506	−0.333253	−	0.942837i	$-0.608146\pi$
$227$	−2279.52	−0.666506	−0.333253	+	0.942837i	$0.608146\pi$
$228$	0	0
$229$	5412.67	1.56192	0.780960	−	0.624582i	$-0.214730\pi$
$229$	5412.67	1.56192	0.780960	+	0.624582i	$0.214730\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1138.37	0.320073	0.160036	−	0.987111i	$-0.448839\pi$
$233$	1138.37	0.320073	0.160036	+	0.987111i	$0.448839\pi$
$234$	0	0
$235$	4841.56	1.34395
$236$	0	0
$237$	1460.65	0.400336
$238$	0	0
$239$	6226.36	1.68515	0.842573	−	0.538583i	$-0.181040\pi$
$239$	6226.36	1.68515	0.842573	+	0.538583i	$0.181040\pi$
$240$	0	0
$241$	−3196.20	−0.854295	−0.427147	−	0.904182i	$-0.640481\pi$
$241$	−3196.20	−0.854295	−0.427147	+	0.904182i	$0.640481\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1841.38	0.474349
$248$	0	0
$249$	−1817.06	−0.462457
$250$	0	0
$251$	−239.608	−0.0602546	−0.0301273	−	0.999546i	$-0.509591\pi$
$251$	−239.608	−0.0602546	−0.0301273	+	0.999546i	$0.509591\pi$
$252$	0	0
$253$	8370.78	2.08010
$254$	0	0
$255$	1818.42	0.446563
$256$	0	0
$257$	699.117	0.169688	0.0848439	−	0.996394i	$-0.472961\pi$
$257$	699.117	0.169688	0.0848439	+	0.996394i	$0.472961\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−550.210	−0.130487
$262$	0	0
$263$	919.040	0.215477	0.107738	−	0.994179i	$-0.465639\pi$
$263$	919.040	0.215477	0.107738	+	0.994179i	$0.465639\pi$
$264$	0	0
$265$	−3916.09	−0.907787
$266$	0	0
$267$	654.206	0.149950
$268$	0	0
$269$	−2779.17	−0.629923	−0.314961	−	0.949104i	$-0.601992\pi$
$269$	−2779.17	−0.629923	−0.314961	+	0.949104i	$0.601992\pi$
$270$	0	0
$271$	−2226.98	−0.499186	−0.249593	−	0.968351i	$-0.580297\pi$
$271$	−2226.98	−0.499186	−0.249593	+	0.968351i	$0.580297\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1786.39	−0.391721
$276$	0	0
$277$	7307.69	1.58511	0.792557	−	0.609797i	$-0.208749\pi$
$277$	7307.69	1.58511	0.792557	+	0.609797i	$0.208749\pi$
$278$	0	0
$279$	10.5242	0.00225830
$280$	0	0
$281$	2730.61	0.579696	0.289848	−	0.957073i	$-0.406395\pi$
$281$	2730.61	0.579696	0.289848	+	0.957073i	$0.406395\pi$
$282$	0	0
$283$	1769.85	0.371755	0.185878	−	0.982573i	$-0.440487\pi$
$283$	1769.85	0.371755	0.185878	+	0.982573i	$0.440487\pi$
$284$	0	0
$285$	−1885.71	−0.391928
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2536.81	−0.516347
$290$	0	0
$291$	−2346.86	−0.472768
$292$	0	0
$293$	8228.81	1.64072	0.820362	−	0.571844i	$-0.193772\pi$
$293$	8228.81	1.64072	0.820362	+	0.571844i	$0.193772\pi$
$294$	0	0
$295$	10501.4	2.07258
$296$	0	0
$297$	−1628.41	−0.318148
$298$	0	0
$299$	−5055.78	−0.977870
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−934.937	−0.177263
$304$	0	0
$305$	4209.58	0.790295
$306$	0	0
$307$	−6019.62	−1.11908	−0.559541	−	0.828803i	$-0.689023\pi$
$307$	−6019.62	−1.11908	−0.559541	+	0.828803i	$0.689023\pi$
$308$	0	0
$309$	447.773	0.0824366
$310$	0	0
$311$	−1193.71	−0.217650	−0.108825	−	0.994061i	$-0.534709\pi$
$311$	−1193.71	−0.217650	−0.108825	+	0.994061i	$0.534709\pi$
$312$	0	0
$313$	−8846.04	−1.59747	−0.798734	−	0.601684i	$-0.794497\pi$
$313$	−8846.04	−1.59747	−0.798734	+	0.601684i	$0.794497\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6081.43	1.07750	0.538750	−	0.842466i	$-0.318897\pi$
$317$	6081.43	1.07750	0.538750	+	0.842466i	$0.318897\pi$
$318$	0	0
$319$	3687.11	0.647143
$320$	0	0
$321$	−2554.56	−0.444179
$322$	0	0
$323$	−2464.12	−0.424480
$324$	0	0
$325$	1078.94	0.184151
$326$	0	0
$327$	4085.64	0.690936
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3053.30	−0.507022	−0.253511	−	0.967333i	$-0.581585\pi$
$331$	−3053.30	−0.507022	−0.253511	+	0.967333i	$0.581585\pi$
$332$	0	0
$333$	625.742	0.102974
$334$	0	0
$335$	−12080.8	−1.97029
$336$	0	0
$337$	3865.80	0.624877	0.312438	−	0.949938i	$-0.398854\pi$
$337$	3865.80	0.624877	0.312438	+	0.949938i	$0.398854\pi$
$338$	0	0
$339$	3145.66	0.503979
$340$	0	0
$341$	−70.5255	−0.0111999
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5177.49	0.807961
$346$	0	0
$347$	99.5931	0.0154076	0.00770380	−	0.999970i	$-0.497548\pi$
$347$	99.5931	0.0154076	0.00770380	+	0.999970i	$0.497548\pi$
$348$	0	0
$349$	−3607.34	−0.553285	−0.276643	−	0.960973i	$-0.589222\pi$
$349$	−3607.34	−0.553285	−0.276643	+	0.960973i	$0.589222\pi$
$350$	0	0
$351$	983.526	0.149563
$352$	0	0
$353$	7130.73	1.07516	0.537579	−	0.843214i	$-0.319339\pi$
$353$	7130.73	1.07516	0.537579	+	0.843214i	$0.319339\pi$
$354$	0	0
$355$	−1224.43	−0.183060
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6500.29	0.955632	0.477816	−	0.878460i	$-0.341428\pi$
$359$	6500.29	0.955632	0.477816	+	0.878460i	$0.341428\pi$
$360$	0	0
$361$	−4303.70	−0.627453
$362$	0	0
$363$	6919.44	1.00049
$364$	0	0
$365$	−8831.49	−1.26647
$366$	0	0
$367$	−824.886	−0.117326	−0.0586631	−	0.998278i	$-0.518684\pi$
$367$	−824.886	−0.117326	−0.0586631	+	0.998278i	$0.518684\pi$
$368$	0	0
$369$	2773.04	0.391216
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1333.85	0.185159	0.0925793	−	0.995705i	$-0.470489\pi$
$373$	1333.85	0.185159	0.0925793	+	0.995705i	$0.470489\pi$
$374$	0	0
$375$	3558.06	0.489967
$376$	0	0
$377$	−2226.94	−0.304226
$378$	0	0
$379$	1338.29	0.181380	0.0906902	−	0.995879i	$-0.471093\pi$
$379$	1338.29	0.181380	0.0906902	+	0.995879i	$0.471093\pi$
$380$	0	0
$381$	−1465.22	−0.197023
$382$	0	0
$383$	−353.376	−0.0471453	−0.0235727	−	0.999722i	$-0.507504\pi$
$383$	−353.376	−0.0471453	−0.0235727	+	0.999722i	$0.507504\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1569.98	−0.206219
$388$	0	0
$389$	11737.2	1.52982	0.764908	−	0.644139i	$-0.222784\pi$
$389$	11737.2	1.52982	0.764908	+	0.644139i	$0.222784\pi$
$390$	0	0
$391$	6765.59	0.875066
$392$	0	0
$393$	5562.68	0.713996
$394$	0	0
$395$	−6054.21	−0.771191
$396$	0	0
$397$	13281.4	1.67903	0.839516	−	0.543335i	$-0.182839\pi$
$397$	13281.4	1.67903	0.839516	+	0.543335i	$0.182839\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7482.36	−0.931798	−0.465899	−	0.884838i	$-0.654269\pi$
$401$	−7482.36	−0.931798	−0.465899	+	0.884838i	$0.654269\pi$
$402$	0	0
$403$	42.5959	0.00526514
$404$	0	0
$405$	−1007.20	−0.123576
$406$	0	0
$407$	−4193.27	−0.510694
$408$	0	0
$409$	−13796.6	−1.66797	−0.833983	−	0.551791i	$-0.813945\pi$
$409$	−13796.6	−1.66797	−0.833983	+	0.551791i	$0.813945\pi$
$410$	0	0
$411$	−1533.35	−0.184025
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7531.49	0.890859
$416$	0	0
$417$	−6798.31	−0.798357
$418$	0	0
$419$	−9497.56	−1.10737	−0.553683	−	0.832728i	$-0.686778\pi$
$419$	−9497.56	−1.10737	−0.553683	+	0.832728i	$0.686778\pi$
$420$	0	0
$421$	624.367	0.0722797	0.0361399	−	0.999347i	$-0.488494\pi$
$421$	624.367	0.0722797	0.0361399	+	0.999347i	$0.488494\pi$
$422$	0	0
$423$	−3504.26	−0.402796
$424$	0	0
$425$	−1443.83	−0.164791
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6590.88	−0.741750
$430$	0	0
$431$	13397.3	1.49727	0.748636	−	0.662981i	$-0.230709\pi$
$431$	13397.3	1.49727	0.748636	+	0.662981i	$0.230709\pi$
$432$	0	0
$433$	−14057.3	−1.56016	−0.780079	−	0.625681i	$-0.784821\pi$
$433$	−14057.3	−1.56016	−0.780079	+	0.625681i	$0.784821\pi$
$434$	0	0
$435$	2280.55	0.251365
$436$	0	0
$437$	−7015.95	−0.768006
$438$	0	0
$439$	16368.8	1.77960	0.889798	−	0.456356i	$-0.150845\pi$
$439$	16368.8	1.77960	0.889798	+	0.456356i	$0.150845\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1178.71	0.126416	0.0632078	−	0.998000i	$-0.479867\pi$
$443$	1178.71	0.126416	0.0632078	+	0.998000i	$0.479867\pi$
$444$	0	0
$445$	−2711.60	−0.288858
$446$	0	0
$447$	4523.70	0.478666
$448$	0	0
$449$	−12400.9	−1.30342	−0.651709	−	0.758469i	$-0.725948\pi$
$449$	−12400.9	−1.30342	−0.651709	+	0.758469i	$0.725948\pi$
$450$	0	0
$451$	−18582.9	−1.94021
$452$	0	0
$453$	−4775.48	−0.495301
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9925.58	1.01597	0.507986	−	0.861365i	$-0.330390\pi$
$457$	9925.58	1.01597	0.507986	+	0.861365i	$0.330390\pi$
$458$	0	0
$459$	−1316.15	−0.133840
$460$	0	0
$461$	−16010.3	−1.61751	−0.808755	−	0.588146i	$-0.799858\pi$
$461$	−16010.3	−1.61751	−0.808755	+	0.588146i	$0.799858\pi$
$462$	0	0
$463$	−17372.4	−1.74377	−0.871883	−	0.489714i	$-0.837101\pi$
$463$	−17372.4	−1.74377	−0.871883	+	0.489714i	$0.837101\pi$
$464$	0	0
$465$	−43.6213	−0.00435030
$466$	0	0
$467$	−2108.06	−0.208886	−0.104443	−	0.994531i	$-0.533306\pi$
$467$	−2108.06	−0.208886	−0.104443	+	0.994531i	$0.533306\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3492.48	0.341667
$472$	0	0
$473$	10520.9	1.02273
$474$	0	0
$475$	1497.26	0.144629
$476$	0	0
$477$	2834.41	0.272073
$478$	0	0
$479$	2450.04	0.233706	0.116853	−	0.993149i	$-0.462719\pi$
$479$	2450.04	0.233706	0.116853	+	0.993149i	$0.462719\pi$
$480$	0	0
$481$	2532.65	0.240081
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9727.44	0.910722
$486$	0	0
$487$	−645.236	−0.0600379	−0.0300189	−	0.999549i	$-0.509557\pi$
$487$	−645.236	−0.0600379	−0.0300189	+	0.999549i	$0.509557\pi$
$488$	0	0
$489$	3467.64	0.320679
$490$	0	0
$491$	−11766.1	−1.08146	−0.540731	−	0.841196i	$-0.681852\pi$
$491$	−11766.1	−1.08146	−0.540731	+	0.841196i	$0.681852\pi$
$492$	0	0
$493$	2980.07	0.272242
$494$	0	0
$495$	6749.55	0.612868
$496$	0	0
$497$	0	0
$498$	0	0
$499$	44.0209	0.00394919	0.00197459	−	0.999998i	$-0.499371\pi$
$499$	44.0209	0.00394919	0.00197459	+	0.999998i	$0.499371\pi$
$500$	0	0
$501$	8671.83	0.773311
$502$	0	0
$503$	−8290.27	−0.734880	−0.367440	−	0.930047i	$-0.619766\pi$
$503$	−8290.27	−0.734880	−0.367440	+	0.930047i	$0.619766\pi$
$504$	0	0
$505$	3875.19	0.341473
$506$	0	0
$507$	−2610.24	−0.228649
$508$	0	0
$509$	6915.04	0.602168	0.301084	−	0.953598i	$-0.402651\pi$
$509$	6915.04	0.602168	0.301084	+	0.953598i	$0.402651\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1364.85	0.117465
$514$	0	0
$515$	−1855.96	−0.158803
$516$	0	0
$517$	23483.0	1.99764
$518$	0	0
$519$	5684.81	0.480800
$520$	0	0
$521$	13399.3	1.12674	0.563371	−	0.826204i	$-0.309504\pi$
$521$	13399.3	1.12674	0.563371	+	0.826204i	$0.309504\pi$
$522$	0	0
$523$	9936.99	0.830811	0.415406	−	0.909636i	$-0.363640\pi$
$523$	9936.99	0.830811	0.415406	+	0.909636i	$0.363640\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−57.0014	−0.00471161
$528$	0	0
$529$	7096.33	0.583244
$530$	0	0
$531$	−7600.74	−0.621175
$532$	0	0
$533$	11223.7	0.912104
$534$	0	0
$535$	10588.3	0.855649
$536$	0	0
$537$	12865.5	1.03387
$538$	0	0
$539$	0	0
$540$	0	0
$541$	9286.17	0.737973	0.368987	−	0.929435i	$-0.379705\pi$
$541$	9286.17	0.737973	0.368987	+	0.929435i	$0.379705\pi$
$542$	0	0
$543$	1151.20	0.0909809
$544$	0	0
$545$	−16934.4	−1.33099
$546$	0	0
$547$	16821.6	1.31488	0.657438	−	0.753508i	$-0.271640\pi$
$547$	16821.6	1.31488	0.657438	+	0.753508i	$0.271640\pi$
$548$	0	0
$549$	−3046.84	−0.236860
$550$	0	0
$551$	−3090.35	−0.238935
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2593.62	−0.198366
$556$	0	0
$557$	1805.94	0.137379	0.0686897	−	0.997638i	$-0.478118\pi$
$557$	1805.94	0.137379	0.0686897	+	0.997638i	$0.478118\pi$
$558$	0	0
$559$	−6354.40	−0.480792
$560$	0	0
$561$	8819.86	0.663770
$562$	0	0
$563$	−12214.9	−0.914381	−0.457190	−	0.889369i	$-0.651144\pi$
$563$	−12214.9	−0.914381	−0.457190	+	0.889369i	$0.651144\pi$
$564$	0	0
$565$	−13038.3	−0.970845
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4283.77	0.315615	0.157808	−	0.987470i	$-0.449557\pi$
$569$	4283.77	0.315615	0.157808	+	0.987470i	$0.449557\pi$
$570$	0	0
$571$	−6359.94	−0.466121	−0.233060	−	0.972462i	$-0.574874\pi$
$571$	−6359.94	−0.466121	−0.233060	+	0.972462i	$0.574874\pi$
$572$	0	0
$573$	−1155.93	−0.0842753
$574$	0	0
$575$	−4110.95	−0.298153
$576$	0	0
$577$	14468.7	1.04392	0.521959	−	0.852971i	$-0.325201\pi$
$577$	14468.7	1.04392	0.521959	+	0.852971i	$0.325201\pi$
$578$	0	0
$579$	1890.67	0.135706
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−18994.2	−1.34933
$584$	0	0
$585$	−4076.59	−0.288113
$586$	0	0
$587$	11132.6	0.782777	0.391388	−	0.920226i	$-0.371995\pi$
$587$	11132.6	0.782777	0.391388	+	0.920226i	$0.371995\pi$
$588$	0	0
$589$	59.1108	0.00413517
$590$	0	0
$591$	−3750.69	−0.261054
$592$	0	0
$593$	−19775.6	−1.36946	−0.684728	−	0.728799i	$-0.740079\pi$
$593$	−19775.6	−1.36946	−0.684728	+	0.728799i	$0.740079\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3276.73	0.224636
$598$	0	0
$599$	23891.0	1.62965	0.814825	−	0.579707i	$-0.196833\pi$
$599$	23891.0	1.62965	0.814825	+	0.579707i	$0.196833\pi$
$600$	0	0
$601$	19395.5	1.31641	0.658204	−	0.752840i	$-0.271317\pi$
$601$	19395.5	1.31641	0.658204	+	0.752840i	$0.271317\pi$
$602$	0	0
$603$	8743.95	0.590516
$604$	0	0
$605$	−28680.2	−1.92730
$606$	0	0
$607$	−14596.7	−0.976051	−0.488025	−	0.872829i	$-0.662283\pi$
$607$	−14596.7	−0.976051	−0.488025	+	0.872829i	$0.662283\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−14183.2	−0.939104
$612$	0	0
$613$	1979.80	0.130446	0.0652229	−	0.997871i	$-0.479224\pi$
$613$	1979.80	0.130446	0.0652229	+	0.997871i	$0.479224\pi$
$614$	0	0
$615$	−11493.9	−0.753622
$616$	0	0
$617$	16262.4	1.06110	0.530551	−	0.847653i	$-0.321985\pi$
$617$	16262.4	1.06110	0.530551	+	0.847653i	$0.321985\pi$
$618$	0	0
$619$	12021.0	0.780555	0.390278	−	0.920697i	$-0.372379\pi$
$619$	12021.0	0.780555	0.390278	+	0.920697i	$0.372379\pi$
$620$	0	0
$621$	−3747.39	−0.242154
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−18450.1	−1.18081
$626$	0	0
$627$	−9146.24	−0.582561
$628$	0	0
$629$	−3389.16	−0.214841
$630$	0	0
$631$	−25347.6	−1.59916	−0.799582	−	0.600557i	$-0.794945\pi$
$631$	−25347.6	−1.59916	−0.799582	+	0.600557i	$0.794945\pi$
$632$	0	0
$633$	10860.1	0.681915
$634$	0	0
$635$	6073.16	0.379537
$636$	0	0
$637$	0	0
$638$	0	0
$639$	886.228	0.0548648
$640$	0	0
$641$	−5111.60	−0.314971	−0.157485	−	0.987521i	$-0.550339\pi$
$641$	−5111.60	−0.314971	−0.157485	+	0.987521i	$0.550339\pi$
$642$	0	0
$643$	10931.3	0.670435	0.335217	−	0.942141i	$-0.391190\pi$
$643$	10931.3	0.670435	0.335217	+	0.942141i	$0.391190\pi$
$644$	0	0
$645$	6507.38	0.397252
$646$	0	0
$647$	18406.1	1.11842	0.559211	−	0.829025i	$-0.311104\pi$
$647$	18406.1	1.11842	0.559211	+	0.829025i	$0.311104\pi$
$648$	0	0
$649$	50934.7	3.08068
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19921.4	1.19385	0.596926	−	0.802296i	$-0.296389\pi$
$653$	19921.4	1.19385	0.596926	+	0.802296i	$0.296389\pi$
$654$	0	0
$655$	−23056.6	−1.37541
$656$	0	0
$657$	6392.11	0.379574
$658$	0	0
$659$	18858.8	1.11477	0.557385	−	0.830254i	$-0.311805\pi$
$659$	18858.8	1.11477	0.557385	+	0.830254i	$0.311805\pi$
$660$	0	0
$661$	25832.1	1.52005	0.760023	−	0.649896i	$-0.225188\pi$
$661$	25832.1	1.52005	0.760023	+	0.649896i	$0.225188\pi$
$662$	0	0
$663$	−5327.01	−0.312042
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8485.00	0.492564
$668$	0	0
$669$	551.533	0.0318737
$670$	0	0
$671$	20417.7	1.17469
$672$	0	0
$673$	−16275.0	−0.932178	−0.466089	−	0.884738i	$-0.654337\pi$
$673$	−16275.0	−0.932178	−0.466089	+	0.884738i	$0.654337\pi$
$674$	0	0
$675$	799.723	0.0456020
$676$	0	0
$677$	−26271.8	−1.49144	−0.745720	−	0.666259i	$-0.767895\pi$
$677$	−26271.8	−1.49144	−0.745720	+	0.666259i	$0.767895\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6838.55	−0.384808
$682$	0	0
$683$	8072.29	0.452237	0.226118	−	0.974100i	$-0.427396\pi$
$683$	8072.29	0.452237	0.226118	+	0.974100i	$0.427396\pi$
$684$	0	0
$685$	6355.51	0.354499
$686$	0	0
$687$	16238.0	0.901774
$688$	0	0
$689$	11472.1	0.634328
$690$	0	0
$691$	24485.3	1.34799	0.673997	−	0.738734i	$-0.264576\pi$
$691$	24485.3	1.34799	0.673997	+	0.738734i	$0.264576\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28178.1	1.53792
$696$	0	0
$697$	−15019.4	−0.816214
$698$	0	0
$699$	3415.10	0.184794
$700$	0	0
$701$	778.448	0.0419423	0.0209712	−	0.999780i	$-0.493324\pi$
$701$	778.448	0.0419423	0.0209712	+	0.999780i	$0.493324\pi$
$702$	0	0
$703$	3514.58	0.188556
$704$	0	0
$705$	14524.7	0.775930
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24172.0	−1.28039	−0.640197	−	0.768211i	$-0.721147\pi$
$709$	−24172.0	−1.28039	−0.640197	+	0.768211i	$0.721147\pi$
$710$	0	0
$711$	4381.96	0.231134
$712$	0	0
$713$	−162.297	−0.00852466
$714$	0	0
$715$	27318.3	1.42888
$716$	0	0
$717$	18679.1	0.972919
$718$	0	0
$719$	81.8835	0.00424720	0.00212360	−	0.999998i	$-0.499324\pi$
$719$	81.8835	0.00424720	0.00212360	+	0.999998i	$0.499324\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9588.59	−0.493227
$724$	0	0
$725$	−1810.76	−0.0927588
$726$	0	0
$727$	32542.9	1.66018	0.830088	−	0.557632i	$-0.188290\pi$
$727$	32542.9	1.66018	0.830088	+	0.557632i	$0.188290\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	8503.40	0.430246
$732$	0	0
$733$	−5068.94	−0.255424	−0.127712	−	0.991811i	$-0.540763\pi$
$733$	−5068.94	−0.255424	−0.127712	+	0.991811i	$0.540763\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−58595.6	−2.92863
$738$	0	0
$739$	38428.5	1.91287	0.956437	−	0.291939i	$-0.0943004\pi$
$739$	38428.5	1.91287	0.956437	+	0.291939i	$0.0943004\pi$
$740$	0	0
$741$	5524.14	0.273865
$742$	0	0
$743$	−21592.9	−1.06617	−0.533086	−	0.846061i	$-0.678968\pi$
$743$	−21592.9	−1.06617	−0.533086	+	0.846061i	$0.678968\pi$
$744$	0	0
$745$	−18750.1	−0.922083
$746$	0	0
$747$	−5451.19	−0.267000
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8112.60	−0.394185	−0.197093	−	0.980385i	$-0.563150\pi$
$751$	−8112.60	−0.394185	−0.197093	+	0.980385i	$0.563150\pi$
$752$	0	0
$753$	−718.823	−0.0347880
$754$	0	0
$755$	19793.7	0.954129
$756$	0	0
$757$	3108.01	0.149224	0.0746120	−	0.997213i	$-0.476228\pi$
$757$	3108.01	0.149224	0.0746120	+	0.997213i	$0.476228\pi$
$758$	0	0
$759$	25112.3	1.20095
$760$	0	0
$761$	−7211.93	−0.343538	−0.171769	−	0.985137i	$-0.554948\pi$
$761$	−7211.93	−0.343538	−0.171769	+	0.985137i	$0.554948\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5455.25	0.257823
$766$	0	0
$767$	−30763.5	−1.44825
$768$	0	0
$769$	−7533.07	−0.353250	−0.176625	−	0.984278i	$-0.556518\pi$
$769$	−7533.07	−0.353250	−0.176625	+	0.984278i	$0.556518\pi$
$770$	0	0
$771$	2097.35	0.0979693
$772$	0	0
$773$	−24832.6	−1.15546	−0.577728	−	0.816229i	$-0.696060\pi$
$773$	−24832.6	−1.15546	−0.577728	+	0.816229i	$0.696060\pi$
$774$	0	0
$775$	34.6355	0.00160535
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15575.2	0.716355
$780$	0	0
$781$	−5938.86	−0.272099
$782$	0	0
$783$	−1650.63	−0.0753368
$784$	0	0
$785$	−14475.9	−0.658173
$786$	0	0
$787$	−36313.1	−1.64476	−0.822378	−	0.568941i	$-0.807353\pi$
$787$	−36313.1	−1.64476	−0.822378	+	0.568941i	$0.807353\pi$
$788$	0	0
$789$	2757.12	0.124406
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12331.9	−0.552229
$794$	0	0
$795$	−11748.3	−0.524111
$796$	0	0
$797$	31665.7	1.40735	0.703675	−	0.710522i	$-0.251541\pi$
$797$	31665.7	1.40735	0.703675	+	0.710522i	$0.251541\pi$
$798$	0	0
$799$	18979.9	0.840375
$800$	0	0
$801$	1962.62	0.0865738
$802$	0	0
$803$	−42835.4	−1.88247
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8337.52	−0.363686
$808$	0	0
$809$	12384.6	0.538219	0.269110	−	0.963110i	$-0.413271\pi$
$809$	12384.6	0.538219	0.269110	+	0.963110i	$0.413271\pi$
$810$	0	0
$811$	−16742.4	−0.724914	−0.362457	−	0.932000i	$-0.618062\pi$
$811$	−16742.4	−0.724914	−0.362457	+	0.932000i	$0.618062\pi$
$812$	0	0
$813$	−6680.94	−0.288205
$814$	0	0
$815$	−14372.9	−0.617744
$816$	0	0
$817$	−8818.07	−0.377607
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26456.3	1.12464	0.562322	−	0.826918i	$-0.309908\pi$
$821$	26456.3	1.12464	0.562322	+	0.826918i	$0.309908\pi$
$822$	0	0
$823$	−23098.5	−0.978328	−0.489164	−	0.872192i	$-0.662698\pi$
$823$	−23098.5	−0.978328	−0.489164	+	0.872192i	$0.662698\pi$
$824$	0	0
$825$	−5359.17	−0.226160
$826$	0	0
$827$	−20647.6	−0.868183	−0.434092	−	0.900869i	$-0.642931\pi$
$827$	−20647.6	−0.868183	−0.434092	+	0.900869i	$0.642931\pi$
$828$	0	0
$829$	−23368.5	−0.979037	−0.489519	−	0.871993i	$-0.662827\pi$
$829$	−23368.5	−0.979037	−0.489519	+	0.871993i	$0.662827\pi$
$830$	0	0
$831$	21923.1	0.915166
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−35943.6	−1.48968
$836$	0	0
$837$	31.5725	0.00130383
$838$	0	0
$839$	−16735.5	−0.688645	−0.344322	−	0.938851i	$-0.611891\pi$
$839$	−16735.5	−0.688645	−0.344322	+	0.938851i	$0.611891\pi$
$840$	0	0
$841$	−20651.6	−0.846758
$842$	0	0
$843$	8191.84	0.334688
$844$	0	0
$845$	10819.1	0.440460
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5309.55	0.214633
$850$	0	0
$851$	−9649.80	−0.388708
$852$	0	0
$853$	−10294.5	−0.413219	−0.206609	−	0.978424i	$-0.566243\pi$
$853$	−10294.5	−0.413219	−0.206609	+	0.978424i	$0.566243\pi$
$854$	0	0
$855$	−5657.12	−0.226280
$856$	0	0
$857$	−32788.6	−1.30693	−0.653463	−	0.756958i	$-0.726685\pi$
$857$	−32788.6	−1.30693	−0.653463	+	0.756958i	$0.726685\pi$
$858$	0	0
$859$	4909.76	0.195016	0.0975081	−	0.995235i	$-0.468913\pi$
$859$	4909.76	0.195016	0.0975081	+	0.995235i	$0.468913\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17795.0	−0.701909	−0.350954	−	0.936393i	$-0.614143\pi$
$863$	−17795.0	−0.701909	−0.350954	+	0.936393i	$0.614143\pi$
$864$	0	0
$865$	−23562.8	−0.926195
$866$	0	0
$867$	−7610.44	−0.298113
$868$	0	0
$869$	−29364.7	−1.14629
$870$	0	0
$871$	35390.5	1.37677
$872$	0	0
$873$	−7040.59	−0.272953
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34672.2	1.33500	0.667501	−	0.744609i	$-0.267364\pi$
$877$	34672.2	1.33500	0.667501	+	0.744609i	$0.267364\pi$
$878$	0	0
$879$	24686.4	0.947273
$880$	0	0
$881$	40848.2	1.56210	0.781051	−	0.624467i	$-0.214684\pi$
$881$	40848.2	1.56210	0.781051	+	0.624467i	$0.214684\pi$
$882$	0	0
$883$	−30035.1	−1.14469	−0.572345	−	0.820013i	$-0.693966\pi$
$883$	−30035.1	−1.14469	−0.572345	+	0.820013i	$0.693966\pi$
$884$	0	0
$885$	31504.1	1.19661
$886$	0	0
$887$	−33210.7	−1.25717	−0.628583	−	0.777742i	$-0.716365\pi$
$887$	−33210.7	−1.25717	−0.628583	+	0.777742i	$0.716365\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4885.23	−0.183683
$892$	0	0
$893$	−19682.2	−0.737559
$894$	0	0
$895$	−53325.7	−1.99160
$896$	0	0
$897$	−15167.3	−0.564573
$898$	0	0
$899$	−71.4878	−0.00265211
$900$	0	0
$901$	−15351.9	−0.567641
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4771.56	−0.175262
$906$	0	0
$907$	−2497.83	−0.0914433	−0.0457217	−	0.998954i	$-0.514559\pi$
$907$	−2497.83	−0.0914433	−0.0457217	+	0.998954i	$0.514559\pi$
$908$	0	0
$909$	−2804.81	−0.102343
$910$	0	0
$911$	1895.00	0.0689180	0.0344590	−	0.999406i	$-0.489029\pi$
$911$	1895.00	0.0689180	0.0344590	+	0.999406i	$0.489029\pi$
$912$	0	0
$913$	36530.0	1.32417
$914$	0	0
$915$	12628.8	0.456277
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−6270.71	−0.225083	−0.112542	−	0.993647i	$-0.535899\pi$
$919$	−6270.71	−0.225083	−0.112542	+	0.993647i	$0.535899\pi$
$920$	0	0
$921$	−18058.9	−0.646102
$922$	0	0
$923$	3586.95	0.127915
$924$	0	0
$925$	2059.34	0.0732008
$926$	0	0
$927$	1343.32	0.0475948
$928$	0	0
$929$	−31552.6	−1.11432	−0.557161	−	0.830404i	$-0.688110\pi$
$929$	−31552.6	−1.11432	−0.557161	+	0.830404i	$0.688110\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3581.14	−0.125661
$934$	0	0
$935$	−36557.2	−1.27866
$936$	0	0
$937$	−22030.2	−0.768084	−0.384042	−	0.923316i	$-0.625468\pi$
$937$	−22030.2	−0.768084	−0.384042	+	0.923316i	$0.625468\pi$
$938$	0	0
$939$	−26538.1	−0.922299
$940$	0	0
$941$	−32538.6	−1.12724	−0.563618	−	0.826036i	$-0.690591\pi$
$941$	−32538.6	−1.12724	−0.563618	+	0.826036i	$0.690591\pi$
$942$	0	0
$943$	−42764.1	−1.47677
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40711.0	1.39697	0.698485	−	0.715625i	$-0.253858\pi$
$947$	40711.0	1.39697	0.698485	+	0.715625i	$0.253858\pi$
$948$	0	0
$949$	25871.7	0.884962
$950$	0	0
$951$	18244.3	0.622095
$952$	0	0
$953$	−52516.4	−1.78507	−0.892536	−	0.450976i	$-0.851076\pi$
$953$	−52516.4	−1.78507	−0.892536	+	0.450976i	$0.851076\pi$
$954$	0	0
$955$	4791.18	0.162345
$956$	0	0
$957$	11061.3	0.373628
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29789.6	−0.999954
$962$	0	0
$963$	−7663.67	−0.256447
$964$	0	0
$965$	−7836.58	−0.261418
$966$	0	0
$967$	−14721.6	−0.489570	−0.244785	−	0.969577i	$-0.578717\pi$
$967$	−14721.6	−0.489570	−0.244785	+	0.969577i	$0.578717\pi$
$968$	0	0
$969$	−7392.35	−0.245074
$970$	0	0
$971$	−13772.5	−0.455181	−0.227590	−	0.973757i	$-0.573085\pi$
$971$	−13772.5	−0.455181	−0.227590	+	0.973757i	$0.573085\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3236.83	0.106319
$976$	0	0
$977$	24782.1	0.811513	0.405757	−	0.913981i	$-0.367008\pi$
$977$	24782.1	0.811513	0.405757	+	0.913981i	$0.367008\pi$
$978$	0	0
$979$	−13152.0	−0.429358
$980$	0	0
$981$	12256.9	0.398912
$982$	0	0
$983$	−42804.7	−1.38887	−0.694435	−	0.719556i	$-0.744345\pi$
$983$	−42804.7	−1.38887	−0.694435	+	0.719556i	$0.744345\pi$
$984$	0	0
$985$	15546.1	0.502883
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24211.3	0.778438
$990$	0	0
$991$	−449.862	−0.0144201	−0.00721006	−	0.999974i	$-0.502295\pi$
$991$	−449.862	−0.0144201	−0.00721006	+	0.999974i	$0.502295\pi$
$992$	0	0
$993$	−9159.89	−0.292729
$994$	0	0
$995$	−13581.6	−0.432730
$996$	0	0
$997$	−21473.7	−0.682127	−0.341063	−	0.940040i	$-0.610787\pi$
$997$	−21473.7	−0.682127	−0.341063	+	0.940040i	$0.610787\pi$
$998$	0	0
$999$	1877.22	0.0594522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.ci.1.1		3
4.3	odd	2		147.4.a.l.1.2		3
7.2	even	3		336.4.q.k.193.3		6
7.4	even	3		336.4.q.k.289.3		6
7.6	odd	2		2352.4.a.cg.1.3		3
12.11	even	2		441.4.a.s.1.2		3
28.3	even	6		147.4.e.n.79.2		6
28.11	odd	6		21.4.e.b.16.2	yes	6
28.19	even	6		147.4.e.n.67.2		6
28.23	odd	6		21.4.e.b.4.2	✓	6
28.27	even	2		147.4.a.m.1.2		3
84.11	even	6		63.4.e.c.37.2		6
84.23	even	6		63.4.e.c.46.2		6
84.47	odd	6		441.4.e.w.361.2		6
84.59	odd	6		441.4.e.w.226.2		6
84.83	odd	2		441.4.a.t.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.b.4.2	✓	6	28.23	odd	6
21.4.e.b.16.2	yes	6	28.11	odd	6
63.4.e.c.37.2		6	84.11	even	6
63.4.e.c.46.2		6	84.23	even	6
147.4.a.l.1.2		3	4.3	odd	2
147.4.a.m.1.2		3	28.27	even	2
147.4.e.n.67.2		6	28.19	even	6
147.4.e.n.79.2		6	28.3	even	6
336.4.q.k.193.3		6	7.2	even	3
336.4.q.k.289.3		6	7.4	even	3
441.4.a.s.1.2		3	12.11	even	2
441.4.a.t.1.2		3	84.83	odd	2
441.4.e.w.226.2		6	84.59	odd	6
441.4.e.w.361.2		6	84.47	odd	6
2352.4.a.cg.1.3		3	7.6	odd	2
2352.4.a.ci.1.1		3	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -12.4346 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -12.4346 q^{5} +9.00000 q^{9} -60.3115 q^{11} +36.4269 q^{13} -37.3038 q^{15} -48.7461 q^{17} +50.5500 q^{19} -138.792 q^{23} +29.6194 q^{25} +27.0000 q^{27} -61.1345 q^{29} +1.16935 q^{31} -180.935 q^{33} +69.5268 q^{37} +109.281 q^{39} +308.115 q^{41} -174.443 q^{43} -111.911 q^{45} -389.362 q^{47} -146.238 q^{51} +314.935 q^{53} +749.950 q^{55} +151.650 q^{57} -844.526 q^{59} -338.538 q^{61} -452.954 q^{65} +971.550 q^{67} -416.377 q^{69} +98.4698 q^{71} +710.235 q^{73} +88.8581 q^{75} +486.884 q^{79} +81.0000 q^{81} -605.688 q^{83} +606.139 q^{85} -183.403 q^{87} +218.069 q^{89} +3.50806 q^{93} -628.569 q^{95} -782.288 q^{97} -542.804 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} + 11 q^{5} + 27 q^{9}+O(q^{10}) \) 3 * q + 9 * q^3 + 11 * q^5 + 27 * q^9	\( 3 q + 9 q^{3} + 11 q^{5} + 27 q^{9} - 35 q^{11} + 62 q^{13} + 33 q^{15} + 48 q^{17} + 202 q^{19} - 216 q^{23} + 130 q^{25} + 81 q^{27} + 53 q^{29} + 95 q^{31} - 105 q^{33} + 262 q^{37} + 186 q^{39} + 244 q^{41} - 360 q^{43} + 99 q^{45} + 210 q^{47} + 144 q^{51} + 393 q^{53} + 1031 q^{55} + 606 q^{57} - 1143 q^{59} - 70 q^{61} - 472 q^{65} + 628 q^{67} - 648 q^{69} - 318 q^{71} + 988 q^{73} + 390 q^{75} - 861 q^{79} + 243 q^{81} - 519 q^{83} + 1800 q^{85} + 159 q^{87} + 1766 q^{89} + 285 q^{93} + 736 q^{95} + 19 q^{97} - 315 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 + 11 * q^5 + 27 * q^9 - 35 * q^11 + 62 * q^13 + 33 * q^15 + 48 * q^17 + 202 * q^19 - 216 * q^23 + 130 * q^25 + 81 * q^27 + 53 * q^29 + 95 * q^31 - 105 * q^33 + 262 * q^37 + 186 * q^39 + 244 * q^41 - 360 * q^43 + 99 * q^45 + 210 * q^47 + 144 * q^51 + 393 * q^53 + 1031 * q^55 + 606 * q^57 - 1143 * q^59 - 70 * q^61 - 472 * q^65 + 628 * q^67 - 648 * q^69 - 318 * q^71 + 988 * q^73 + 390 * q^75 - 861 * q^79 + 243 * q^81 - 519 * q^83 + 1800 * q^85 + 159 * q^87 + 1766 * q^89 + 285 * q^93 + 736 * q^95 + 19 * q^97 - 315 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more