LMFDB - Embedded newform 2151.4.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.21293 q^{2} +2.32292 q^{4} +13.8674 q^{5} +29.9628 q^{7} +18.2401 q^{8} +O(q^{10})$	\(q-3.21293 q^{2} +2.32292 q^{4} +13.8674 q^{5} +29.9628 q^{7} +18.2401 q^{8} -44.5549 q^{10} +38.5636 q^{11} -23.0445 q^{13} -96.2684 q^{14} -77.1874 q^{16} -22.8946 q^{17} +2.58195 q^{19} +32.2128 q^{20} -123.902 q^{22} -49.7956 q^{23} +67.3044 q^{25} +74.0404 q^{26} +69.6011 q^{28} +235.884 q^{29} -309.636 q^{31} +102.077 q^{32} +73.5586 q^{34} +415.506 q^{35} -239.205 q^{37} -8.29563 q^{38} +252.942 q^{40} -228.968 q^{41} -115.502 q^{43} +89.5800 q^{44} +159.990 q^{46} +327.249 q^{47} +554.769 q^{49} -216.244 q^{50} -53.5305 q^{52} +267.203 q^{53} +534.776 q^{55} +546.523 q^{56} -757.880 q^{58} +837.268 q^{59} +894.801 q^{61} +994.838 q^{62} +289.532 q^{64} -319.567 q^{65} -548.064 q^{67} -53.1822 q^{68} -1334.99 q^{70} -881.420 q^{71} -271.324 q^{73} +768.548 q^{74} +5.99766 q^{76} +1155.47 q^{77} +1217.39 q^{79} -1070.39 q^{80} +735.658 q^{82} +706.087 q^{83} -317.488 q^{85} +371.100 q^{86} +703.402 q^{88} +1490.27 q^{89} -690.478 q^{91} -115.671 q^{92} -1051.43 q^{94} +35.8049 q^{95} +1761.03 q^{97} -1782.43 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

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 $

$2$	−3.21293	−1.13594	−0.567971	−	0.823048i	$-0.692271\pi$
$2$	−3.21293	−1.13594	−0.567971	+	0.823048i	$0.692271\pi$
$3$	0	0
$3$	0	0
$4$	2.32292	0.290365
$5$	13.8674	1.24034	0.620168	−	0.784469i	$-0.287064\pi$
$5$	13.8674	1.24034	0.620168	+	0.784469i	$0.287064\pi$
$6$	0	0
$7$	29.9628	1.61784	0.808920	−	0.587919i	$-0.200053\pi$
$7$	29.9628	1.61784	0.808920	+	0.587919i	$0.200053\pi$
$8$	18.2401	0.806105
$9$	0	0
$10$	−44.5549	−1.40895
$11$	38.5636	1.05703	0.528516	−	0.848923i	$-0.322749\pi$
$11$	38.5636	1.05703	0.528516	+	0.848923i	$0.322749\pi$
$12$	0	0
$13$	−23.0445	−0.491646	−0.245823	−	0.969315i	$-0.579058\pi$
$13$	−23.0445	−0.491646	−0.245823	+	0.969315i	$0.579058\pi$
$14$	−96.2684	−1.83777
$15$	0	0
$16$	−77.1874	−1.20605
$17$	−22.8946	−0.326632	−0.163316	−	0.986574i	$-0.552219\pi$
$17$	−22.8946	−0.326632	−0.163316	+	0.986574i	$0.552219\pi$
$18$	0	0
$19$	2.58195	0.0311758	0.0155879	−	0.999879i	$-0.495038\pi$
$19$	2.58195	0.0311758	0.0155879	+	0.999879i	$0.495038\pi$
$20$	32.2128	0.360150
$21$	0	0
$22$	−123.902	−1.20073
$23$	−49.7956	−0.451439	−0.225720	−	0.974192i	$-0.572473\pi$
$23$	−49.7956	−0.451439	−0.225720	+	0.974192i	$0.572473\pi$
$24$	0	0
$25$	67.3044	0.538435
$26$	74.0404	0.558482
$27$	0	0
$28$	69.6011	0.469763
$29$	235.884	1.51044	0.755218	−	0.655474i	$-0.227531\pi$
$29$	235.884	1.51044	0.755218	+	0.655474i	$0.227531\pi$
$30$	0	0
$31$	−309.636	−1.79394	−0.896972	−	0.442088i	$-0.854238\pi$
$31$	−309.636	−1.79394	−0.896972	+	0.442088i	$0.854238\pi$
$32$	102.077	0.563902
$33$	0	0
$34$	73.5586	0.371035
$35$	415.506	2.00667
$36$	0	0
$37$	−239.205	−1.06284	−0.531419	−	0.847109i	$-0.678341\pi$
$37$	−239.205	−1.06284	−0.531419	+	0.847109i	$0.678341\pi$
$38$	−8.29563	−0.0354139
$39$	0	0
$40$	252.942	0.999841
$41$	−228.968	−0.872166	−0.436083	−	0.899906i	$-0.643635\pi$
$41$	−228.968	−0.872166	−0.436083	+	0.899906i	$0.643635\pi$
$42$	0	0
$43$	−115.502	−0.409625	−0.204813	−	0.978801i	$-0.565659\pi$
$43$	−115.502	−0.409625	−0.204813	+	0.978801i	$0.565659\pi$
$44$	89.5800	0.306925
$45$	0	0
$46$	159.990	0.512809
$47$	327.249	1.01562	0.507810	−	0.861469i	$-0.330455\pi$
$47$	327.249	1.01562	0.507810	+	0.861469i	$0.330455\pi$
$48$	0	0
$49$	554.769	1.61740
$50$	−216.244	−0.611631
$51$	0	0
$52$	−53.5305	−0.142757
$53$	267.203	0.692512	0.346256	−	0.938140i	$-0.387453\pi$
$53$	267.203	0.692512	0.346256	+	0.938140i	$0.387453\pi$
$54$	0	0
$55$	534.776	1.31108
$56$	546.523	1.30415
$57$	0	0
$58$	−757.880	−1.71577
$59$	837.268	1.84751	0.923755	−	0.382985i	$-0.125104\pi$
$59$	837.268	1.84751	0.923755	+	0.382985i	$0.125104\pi$
$60$	0	0
$61$	894.801	1.87816	0.939078	−	0.343704i	$-0.111682\pi$
$61$	894.801	1.87816	0.939078	+	0.343704i	$0.111682\pi$
$62$	994.838	2.03782
$63$	0	0
$64$	289.532	0.565493
$65$	−319.567	−0.609807
$66$	0	0
$67$	−548.064	−0.999354	−0.499677	−	0.866212i	$-0.666548\pi$
$67$	−548.064	−0.999354	−0.499677	+	0.866212i	$0.666548\pi$
$68$	−53.1822	−0.0948425
$69$	0	0
$70$	−1334.99	−2.27946
$71$	−881.420	−1.47331	−0.736657	−	0.676266i	$-0.763597\pi$
$71$	−881.420	−1.47331	−0.736657	+	0.676266i	$0.763597\pi$
$72$	0	0
$73$	−271.324	−0.435014	−0.217507	−	0.976059i	$-0.569793\pi$
$73$	−271.324	−0.435014	−0.217507	+	0.976059i	$0.569793\pi$
$74$	768.548	1.20732
$75$	0	0
$76$	5.99766	0.00905236
$77$	1155.47	1.71011
$78$	0	0
$79$	1217.39	1.73376	0.866882	−	0.498513i	$-0.166120\pi$
$79$	1217.39	1.73376	0.866882	+	0.498513i	$0.166120\pi$
$80$	−1070.39	−1.49591
$81$	0	0
$82$	735.658	0.990730
$83$	706.087	0.933773	0.466886	−	0.884317i	$-0.345376\pi$
$83$	706.087	0.933773	0.466886	+	0.884317i	$0.345376\pi$
$84$	0	0
$85$	−317.488	−0.405134
$86$	371.100	0.465311
$87$	0	0
$88$	703.402	0.852078
$89$	1490.27	1.77492	0.887460	−	0.460885i	$-0.152468\pi$
$89$	1490.27	1.77492	0.887460	+	0.460885i	$0.152468\pi$
$90$	0	0
$91$	−690.478	−0.795404
$92$	−115.671	−0.131082
$93$	0	0
$94$	−1051.43	−1.15369
$95$	35.8049	0.0386685
$96$	0	0
$97$	1761.03	1.84336	0.921678	−	0.387955i	$-0.126818\pi$
$97$	1761.03	1.84336	0.921678	+	0.387955i	$0.126818\pi$
$98$	−1782.43	−1.83728
$99$	0	0
$100$	156.343	0.156343
$101$	837.163	0.824761	0.412380	−	0.911012i	$-0.364697\pi$
$101$	837.163	0.824761	0.412380	+	0.911012i	$0.364697\pi$
$102$	0	0
$103$	−608.226	−0.581847	−0.290924	−	0.956746i	$-0.593963\pi$
$103$	−608.226	−0.581847	−0.290924	+	0.956746i	$0.593963\pi$
$104$	−420.334	−0.396318
$105$	0	0
$106$	−858.504	−0.786653
$107$	927.372	0.837873	0.418937	−	0.908015i	$-0.362403\pi$
$107$	927.372	0.837873	0.418937	+	0.908015i	$0.362403\pi$
$108$	0	0
$109$	−1608.49	−1.41345	−0.706724	−	0.707490i	$-0.749828\pi$
$109$	−1608.49	−1.41345	−0.706724	+	0.707490i	$0.749828\pi$
$110$	−1718.20	−1.48931
$111$	0	0
$112$	−2312.75	−1.95120
$113$	−1814.44	−1.51052	−0.755259	−	0.655427i	$-0.772489\pi$
$113$	−1814.44	−1.51052	−0.755259	+	0.655427i	$0.772489\pi$
$114$	0	0
$115$	−690.535	−0.559936
$116$	547.940	0.438577
$117$	0	0
$118$	−2690.08	−2.09866
$119$	−685.985	−0.528438
$120$	0	0
$121$	156.148	0.117316
$122$	−2874.93	−2.13348
$123$	0	0
$124$	−719.259	−0.520898
$125$	−800.087	−0.572496
$126$	0	0
$127$	2564.07	1.79153	0.895767	−	0.444524i	$-0.146627\pi$
$127$	2564.07	1.79153	0.895767	+	0.444524i	$0.146627\pi$
$128$	−1746.86	−1.20627
$129$	0	0
$130$	1026.75	0.692705
$131$	−1202.21	−0.801816	−0.400908	−	0.916118i	$-0.631305\pi$
$131$	−1202.21	−0.801816	−0.400908	+	0.916118i	$0.631305\pi$
$132$	0	0
$133$	77.3625	0.0504374
$134$	1760.89	1.13521
$135$	0	0
$136$	−417.598	−0.263300
$137$	−321.091	−0.200238	−0.100119	−	0.994975i	$-0.531922\pi$
$137$	−321.091	−0.200238	−0.100119	+	0.994975i	$0.531922\pi$
$138$	0	0
$139$	−307.658	−0.187735	−0.0938676	−	0.995585i	$-0.529923\pi$
$139$	−307.658	−0.187735	−0.0938676	+	0.995585i	$0.529923\pi$
$140$	965.186	0.582665
$141$	0	0
$142$	2831.94	1.67360
$143$	−888.679	−0.519686
$144$	0	0
$145$	3271.10	1.87345
$146$	871.744	0.494151
$147$	0	0
$148$	−555.653	−0.308611
$149$	1563.88	0.859851	0.429925	−	0.902864i	$-0.358540\pi$
$149$	1563.88	0.859851	0.429925	+	0.902864i	$0.358540\pi$
$150$	0	0
$151$	3135.74	1.68995	0.844975	−	0.534805i	$-0.179615\pi$
$151$	3135.74	1.68995	0.844975	+	0.534805i	$0.179615\pi$
$152$	47.0950	0.0251310
$153$	0	0
$154$	−3712.45	−1.94258
$155$	−4293.84	−2.22509
$156$	0	0
$157$	−1828.16	−0.929319	−0.464660	−	0.885489i	$-0.653823\pi$
$157$	−1828.16	−0.929319	−0.464660	+	0.885489i	$0.653823\pi$
$158$	−3911.40	−1.96946
$159$	0	0
$160$	1415.54	0.699428
$161$	−1492.02	−0.730356
$162$	0	0
$163$	1241.34	0.596497	0.298249	−	0.954488i	$-0.403598\pi$
$163$	1241.34	0.596497	0.298249	+	0.954488i	$0.403598\pi$
$164$	−531.874	−0.253246
$165$	0	0
$166$	−2268.61	−1.06071
$167$	3028.63	1.40337	0.701684	−	0.712488i	$-0.252432\pi$
$167$	3028.63	1.40337	0.701684	+	0.712488i	$0.252432\pi$
$168$	0	0
$169$	−1665.95	−0.758284
$170$	1020.07	0.460209
$171$	0	0
$172$	−268.302	−0.118941
$173$	−628.663	−0.276280	−0.138140	−	0.990413i	$-0.544112\pi$
$173$	−628.663	−0.276280	−0.138140	+	0.990413i	$0.544112\pi$
$174$	0	0
$175$	2016.63	0.871102
$176$	−2976.62	−1.27484
$177$	0	0
$178$	−4788.12	−2.01621
$179$	2440.20	1.01894	0.509468	−	0.860490i	$-0.329842\pi$
$179$	2440.20	1.01894	0.509468	+	0.860490i	$0.329842\pi$
$180$	0	0
$181$	3851.13	1.58150	0.790752	−	0.612137i	$-0.209690\pi$
$181$	3851.13	1.58150	0.790752	+	0.612137i	$0.209690\pi$
$182$	2218.46	0.903533
$183$	0	0
$184$	−908.275	−0.363907
$185$	−3317.15	−1.31828
$186$	0	0
$187$	−882.896	−0.345261
$188$	760.172	0.294900
$189$	0	0
$190$	−115.039	−0.0439252
$191$	−391.100	−0.148162	−0.0740811	−	0.997252i	$-0.523602\pi$
$191$	−391.100	−0.148162	−0.0740811	+	0.997252i	$0.523602\pi$
$192$	0	0
$193$	1038.84	0.387448	0.193724	−	0.981056i	$-0.437943\pi$
$193$	1038.84	0.387448	0.193724	+	0.981056i	$0.437943\pi$
$194$	−5658.07	−2.09395
$195$	0	0
$196$	1288.68	0.469637
$197$	−2246.33	−0.812408	−0.406204	−	0.913782i	$-0.633148\pi$
$197$	−2246.33	−0.812408	−0.406204	+	0.913782i	$0.633148\pi$
$198$	0	0
$199$	2520.75	0.897948	0.448974	−	0.893545i	$-0.351790\pi$
$199$	2520.75	0.897948	0.448974	+	0.893545i	$0.351790\pi$
$200$	1227.64	0.434035
$201$	0	0
$202$	−2689.75	−0.936881
$203$	7067.76	2.44364
$204$	0	0
$205$	−3175.19	−1.08178
$206$	1954.19	0.660945
$207$	0	0
$208$	1778.75	0.592951
$209$	99.5692	0.0329538
$210$	0	0
$211$	646.167	0.210824	0.105412	−	0.994429i	$-0.466384\pi$
$211$	646.167	0.210824	0.105412	+	0.994429i	$0.466384\pi$
$212$	620.690	0.201081
$213$	0	0
$214$	−2979.58	−0.951776
$215$	−1601.71	−0.508073
$216$	0	0
$217$	−9277.56	−2.90231
$218$	5167.98	1.60559
$219$	0	0
$220$	1242.24	0.380690
$221$	527.594	0.160588
$222$	0	0
$223$	−5084.30	−1.52677	−0.763385	−	0.645943i	$-0.776464\pi$
$223$	−5084.30	−1.52677	−0.763385	+	0.645943i	$0.776464\pi$
$224$	3058.52	0.912303
$225$	0	0
$226$	5829.68	1.71586
$227$	−2692.89	−0.787371	−0.393686	−	0.919245i	$-0.628800\pi$
$227$	−2692.89	−0.787371	−0.393686	+	0.919245i	$0.628800\pi$
$228$	0	0
$229$	804.429	0.232132	0.116066	−	0.993242i	$-0.462972\pi$
$229$	804.429	0.232132	0.116066	+	0.993242i	$0.462972\pi$
$230$	2218.64	0.636055
$231$	0	0
$232$	4302.55	1.21757
$233$	5065.65	1.42430	0.712149	−	0.702028i	$-0.247722\pi$
$233$	5065.65	1.42430	0.712149	+	0.702028i	$0.247722\pi$
$234$	0	0
$235$	4538.09	1.25971
$236$	1944.91	0.536452
$237$	0	0
$238$	2204.02	0.600276
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	3530.53	0.943656	0.471828	−	0.881691i	$-0.343594\pi$
$241$	3530.53	0.943656	0.471828	+	0.881691i	$0.343594\pi$
$242$	−501.692	−0.133264
$243$	0	0
$244$	2078.55	0.545350
$245$	7693.20	2.00612
$246$	0	0
$247$	−59.4999	−0.0153275
$248$	−5647.78	−1.44611
$249$	0	0
$250$	2570.62	0.650322
$251$	−1582.12	−0.397859	−0.198930	−	0.980014i	$-0.563747\pi$
$251$	−1582.12	−0.397859	−0.198930	+	0.980014i	$0.563747\pi$
$252$	0	0
$253$	−1920.30	−0.477185
$254$	−8238.19	−2.03508
$255$	0	0
$256$	3296.29	0.804759
$257$	−926.736	−0.224935	−0.112467	−	0.993655i	$-0.535875\pi$
$257$	−926.736	−0.224935	−0.112467	+	0.993655i	$0.535875\pi$
$258$	0	0
$259$	−7167.24	−1.71950
$260$	−742.329	−0.177066
$261$	0	0
$262$	3862.63	0.910817
$263$	−4673.04	−1.09564	−0.547818	−	0.836598i	$-0.684541\pi$
$263$	−4673.04	−1.09564	−0.547818	+	0.836598i	$0.684541\pi$
$264$	0	0
$265$	3705.40	0.858948
$266$	−248.560	−0.0572940
$267$	0	0
$268$	−1273.11	−0.290177
$269$	1330.80	0.301637	0.150818	−	0.988562i	$-0.451809\pi$
$269$	1330.80	0.301637	0.150818	+	0.988562i	$0.451809\pi$
$270$	0	0
$271$	−2451.82	−0.549585	−0.274793	−	0.961504i	$-0.588609\pi$
$271$	−2451.82	−0.549585	−0.274793	+	0.961504i	$0.588609\pi$
$272$	1767.17	0.393936
$273$	0	0
$274$	1031.64	0.227459
$275$	2595.50	0.569143
$276$	0	0
$277$	5874.15	1.27416	0.637082	−	0.770796i	$-0.280141\pi$
$277$	5874.15	1.27416	0.637082	+	0.770796i	$0.280141\pi$
$278$	988.483	0.213256
$279$	0	0
$280$	7578.85	1.61758
$281$	−1014.98	−0.215476	−0.107738	−	0.994179i	$-0.534361\pi$
$281$	−1014.98	−0.215476	−0.107738	+	0.994179i	$0.534361\pi$
$282$	0	0
$283$	2274.39	0.477734	0.238867	−	0.971052i	$-0.423224\pi$
$283$	2274.39	0.477734	0.238867	+	0.971052i	$0.423224\pi$
$284$	−2047.47	−0.427799
$285$	0	0
$286$	2855.26	0.590333
$287$	−6860.52	−1.41102
$288$	0	0
$289$	−4388.84	−0.893311
$290$	−10509.8	−2.12813
$291$	0	0
$292$	−630.263	−0.126313
$293$	5955.27	1.18741	0.593704	−	0.804684i	$-0.297665\pi$
$293$	5955.27	1.18741	0.593704	+	0.804684i	$0.297665\pi$
$294$	0	0
$295$	11610.7	2.29153
$296$	−4363.11	−0.856759
$297$	0	0
$298$	−5024.63	−0.976741
$299$	1147.52	0.221948
$300$	0	0
$301$	−3460.76	−0.662708
$302$	−10074.9	−1.91969
$303$	0	0
$304$	−199.294	−0.0375997
$305$	12408.6	2.32955
$306$	0	0
$307$	4153.39	0.772139	0.386069	−	0.922470i	$-0.373833\pi$
$307$	4153.39	0.772139	0.386069	+	0.922470i	$0.373833\pi$
$308$	2684.07	0.496555
$309$	0	0
$310$	13795.8	2.52758
$311$	−3405.16	−0.620865	−0.310433	−	0.950595i	$-0.600474\pi$
$311$	−3405.16	−0.620865	−0.310433	+	0.950595i	$0.600474\pi$
$312$	0	0
$313$	−696.349	−0.125751	−0.0628754	−	0.998021i	$-0.520027\pi$
$313$	−696.349	−0.125751	−0.0628754	+	0.998021i	$0.520027\pi$
$314$	5873.75	1.05565
$315$	0	0
$316$	2827.90	0.503424
$317$	8758.96	1.55190	0.775950	−	0.630794i	$-0.217271\pi$
$317$	8758.96	1.55190	0.775950	+	0.630794i	$0.217271\pi$
$318$	0	0
$319$	9096.54	1.59658
$320$	4015.06	0.701402
$321$	0	0
$322$	4793.74	0.829642
$323$	−59.1127	−0.0101830
$324$	0	0
$325$	−1551.00	−0.264720
$326$	−3988.33	−0.677586
$327$	0	0
$328$	−4176.39	−0.703057
$329$	9805.29	1.64311
$330$	0	0
$331$	8912.00	1.47990	0.739951	−	0.672660i	$-0.234849\pi$
$331$	8912.00	1.47990	0.739951	+	0.672660i	$0.234849\pi$
$332$	1640.18	0.271135
$333$	0	0
$334$	−9730.78	−1.59415
$335$	−7600.22	−1.23954
$336$	0	0
$337$	−524.604	−0.0847982	−0.0423991	−	0.999101i	$-0.513500\pi$
$337$	−524.604	−0.0847982	−0.0423991	+	0.999101i	$0.513500\pi$
$338$	5352.58	0.861367
$339$	0	0
$340$	−737.498	−0.117637
$341$	−11940.7	−1.89625
$342$	0	0
$343$	6345.20	0.998859
$344$	−2106.76	−0.330201
$345$	0	0
$346$	2019.85	0.313838
$347$	5749.51	0.889480	0.444740	−	0.895660i	$-0.353296\pi$
$347$	5749.51	0.889480	0.444740	+	0.895660i	$0.353296\pi$
$348$	0	0
$349$	−5064.05	−0.776712	−0.388356	−	0.921509i	$-0.626957\pi$
$349$	−5064.05	−0.776712	−0.388356	+	0.921509i	$0.626957\pi$
$350$	−6479.29	−0.989521
$351$	0	0
$352$	3936.46	0.596062
$353$	2838.93	0.428048	0.214024	−	0.976828i	$-0.431343\pi$
$353$	2838.93	0.428048	0.214024	+	0.976828i	$0.431343\pi$
$354$	0	0
$355$	−12223.0	−1.82741
$356$	3461.76	0.515374
$357$	0	0
$358$	−7840.20	−1.15745
$359$	9504.83	1.39734	0.698671	−	0.715443i	$-0.253775\pi$
$359$	9504.83	1.39734	0.698671	+	0.715443i	$0.253775\pi$
$360$	0	0
$361$	−6852.33	−0.999028
$362$	−12373.4	−1.79650
$363$	0	0
$364$	−1603.92	−0.230957
$365$	−3762.55	−0.539564
$366$	0	0
$367$	3488.50	0.496180	0.248090	−	0.968737i	$-0.420197\pi$
$367$	3488.50	0.496180	0.248090	+	0.968737i	$0.420197\pi$
$368$	3843.59	0.544459
$369$	0	0
$370$	10657.8	1.49749
$371$	8006.14	1.12037
$372$	0	0
$373$	1725.93	0.239585	0.119793	−	0.992799i	$-0.461777\pi$
$373$	1725.93	0.239585	0.119793	+	0.992799i	$0.461777\pi$
$374$	2836.68	0.392196
$375$	0	0
$376$	5969.04	0.818696
$377$	−5435.84	−0.742600
$378$	0	0
$379$	−1618.97	−0.219422	−0.109711	−	0.993964i	$-0.534993\pi$
$379$	−1618.97	−0.219422	−0.109711	+	0.993964i	$0.534993\pi$
$380$	83.1719	0.0112280
$381$	0	0
$382$	1256.58	0.168304
$383$	−2885.61	−0.384981	−0.192491	−	0.981299i	$-0.561657\pi$
$383$	−2885.61	−0.384981	−0.192491	+	0.981299i	$0.561657\pi$
$384$	0	0
$385$	16023.4	2.12111
$386$	−3337.73	−0.440119
$387$	0	0
$388$	4090.73	0.535246
$389$	−10620.7	−1.38430	−0.692151	−	0.721753i	$-0.743337\pi$
$389$	−10620.7	−1.38430	−0.692151	+	0.721753i	$0.743337\pi$
$390$	0	0
$391$	1140.05	0.147455
$392$	10119.0	1.30380
$393$	0	0
$394$	7217.30	0.922849
$395$	16882.1	2.15045
$396$	0	0
$397$	−2740.42	−0.346443	−0.173221	−	0.984883i	$-0.555418\pi$
$397$	−2740.42	−0.346443	−0.173221	+	0.984883i	$0.555418\pi$
$398$	−8099.01	−1.02002
$399$	0	0
$400$	−5195.05	−0.649382
$401$	−73.4973	−0.00915281	−0.00457641	−	0.999990i	$-0.501457\pi$
$401$	−73.4973	−0.00915281	−0.00457641	+	0.999990i	$0.501457\pi$
$402$	0	0
$403$	7135.41	0.881985
$404$	1944.66	0.239481
$405$	0	0
$406$	−22708.2	−2.77584
$407$	−9224.59	−1.12345
$408$	0	0
$409$	−11105.7	−1.34264	−0.671320	−	0.741167i	$-0.734272\pi$
$409$	−11105.7	−1.34264	−0.671320	+	0.741167i	$0.734272\pi$
$410$	10201.7	1.22884
$411$	0	0
$412$	−1412.86	−0.168948
$413$	25086.9	2.98897
$414$	0	0
$415$	9791.59	1.15819
$416$	−2352.32	−0.277240
$417$	0	0
$418$	−319.909	−0.0374336
$419$	−3968.34	−0.462687	−0.231344	−	0.972872i	$-0.574312\pi$
$419$	−3968.34	−0.462687	−0.231344	+	0.972872i	$0.574312\pi$
$420$	0	0
$421$	−8533.07	−0.987830	−0.493915	−	0.869510i	$-0.664435\pi$
$421$	−8533.07	−0.987830	−0.493915	+	0.869510i	$0.664435\pi$
$422$	−2076.09	−0.239484
$423$	0	0
$424$	4873.80	0.558237
$425$	−1540.91	−0.175870
$426$	0	0
$427$	26810.7	3.03855
$428$	2154.21	0.243289
$429$	0	0
$430$	5146.19	0.577142
$431$	9715.38	1.08578	0.542892	−	0.839802i	$-0.317329\pi$
$431$	9715.38	1.08578	0.542892	+	0.839802i	$0.317329\pi$
$432$	0	0
$433$	−761.637	−0.0845311	−0.0422655	−	0.999106i	$-0.513458\pi$
$433$	−761.637	−0.0845311	−0.0422655	+	0.999106i	$0.513458\pi$
$434$	29808.1	3.29686
$435$	0	0
$436$	−3736.40	−0.410415
$437$	−128.570	−0.0140740
$438$	0	0
$439$	−9439.48	−1.02625	−0.513123	−	0.858315i	$-0.671511\pi$
$439$	−9439.48	−1.02625	−0.513123	+	0.858315i	$0.671511\pi$
$440$	9754.34	1.05686
$441$	0	0
$442$	−1695.12	−0.182418
$443$	4396.66	0.471538	0.235769	−	0.971809i	$-0.424239\pi$
$443$	4396.66	0.471538	0.235769	+	0.971809i	$0.424239\pi$
$444$	0	0
$445$	20666.1	2.20150
$446$	16335.5	1.73432
$447$	0	0
$448$	8675.20	0.914877
$449$	14050.4	1.47680	0.738398	−	0.674365i	$-0.235582\pi$
$449$	14050.4	1.47680	0.738398	+	0.674365i	$0.235582\pi$
$450$	0	0
$451$	−8829.82	−0.921907
$452$	−4214.80	−0.438601
$453$	0	0
$454$	8652.06	0.894408
$455$	−9575.13	−0.986569
$456$	0	0
$457$	−12456.3	−1.27501	−0.637506	−	0.770446i	$-0.720034\pi$
$457$	−12456.3	−1.27501	−0.637506	+	0.770446i	$0.720034\pi$
$458$	−2584.58	−0.263688
$459$	0	0
$460$	−1604.06	−0.162586
$461$	10573.3	1.06821	0.534107	−	0.845417i	$-0.320648\pi$
$461$	10573.3	1.06821	0.534107	+	0.845417i	$0.320648\pi$
$462$	0	0
$463$	−13126.5	−1.31758	−0.658790	−	0.752327i	$-0.728932\pi$
$463$	−13126.5	−1.31758	−0.658790	+	0.752327i	$0.728932\pi$
$464$	−18207.3	−1.82167
$465$	0	0
$466$	−16275.6	−1.61792
$467$	−13874.5	−1.37481	−0.687405	−	0.726274i	$-0.741250\pi$
$467$	−13874.5	−1.37481	−0.687405	+	0.726274i	$0.741250\pi$
$468$	0	0
$469$	−16421.5	−1.61679
$470$	−14580.6	−1.43096
$471$	0	0
$472$	15271.8	1.48929
$473$	−4454.17	−0.432987
$474$	0	0
$475$	173.777	0.0167862
$476$	−1593.49	−0.153440
$477$	0	0
$478$	−767.890	−0.0734780
$479$	−4815.44	−0.459338	−0.229669	−	0.973269i	$-0.573764\pi$
$479$	−4815.44	−0.459338	−0.229669	+	0.973269i	$0.573764\pi$
$480$	0	0
$481$	5512.36	0.522540
$482$	−11343.3	−1.07194
$483$	0	0
$484$	362.718	0.0340645
$485$	24420.9	2.28638
$486$	0	0
$487$	−7976.98	−0.742241	−0.371121	−	0.928585i	$-0.621026\pi$
$487$	−7976.98	−0.742241	−0.371121	+	0.928585i	$0.621026\pi$
$488$	16321.2	1.51399
$489$	0	0
$490$	−24717.7	−2.27884
$491$	1960.14	0.180163	0.0900813	−	0.995934i	$-0.471287\pi$
$491$	1960.14	0.180163	0.0900813	+	0.995934i	$0.471287\pi$
$492$	0	0
$493$	−5400.47	−0.493357
$494$	191.169	0.0174111
$495$	0	0
$496$	23900.0	2.16359
$497$	−26409.8	−2.38359
$498$	0	0
$499$	15144.3	1.35862	0.679310	−	0.733851i	$-0.262279\pi$
$499$	15144.3	1.35862	0.679310	+	0.733851i	$0.262279\pi$
$500$	−1858.54	−0.166233
$501$	0	0
$502$	5083.25	0.451945
$503$	290.617	0.0257614	0.0128807	−	0.999917i	$-0.495900\pi$
$503$	290.617	0.0257614	0.0128807	+	0.999917i	$0.495900\pi$
$504$	0	0
$505$	11609.3	1.02298
$506$	6169.77	0.542055
$507$	0	0
$508$	5956.14	0.520198
$509$	−10356.5	−0.901857	−0.450928	−	0.892560i	$-0.648907\pi$
$509$	−10356.5	−0.901857	−0.450928	+	0.892560i	$0.648907\pi$
$510$	0	0
$511$	−8129.62	−0.703783
$512$	3384.16	0.292109
$513$	0	0
$514$	2977.54	0.255513
$515$	−8434.51	−0.721687
$516$	0	0
$517$	12619.9	1.07354
$518$	23027.9	1.95325
$519$	0	0
$520$	−5828.93	−0.491568
$521$	−2168.01	−0.182307	−0.0911536	−	0.995837i	$-0.529055\pi$
$521$	−2168.01	−0.182307	−0.0911536	+	0.995837i	$0.529055\pi$
$522$	0	0
$523$	10192.4	0.852169	0.426085	−	0.904683i	$-0.359893\pi$
$523$	10192.4	0.852169	0.426085	+	0.904683i	$0.359893\pi$
$524$	−2792.64	−0.232819
$525$	0	0
$526$	15014.2	1.24458
$527$	7088.98	0.585960
$528$	0	0
$529$	−9687.40	−0.796203
$530$	−11905.2	−0.975715
$531$	0	0
$532$	179.707	0.0146453
$533$	5276.46	0.428797
$534$	0	0
$535$	12860.2	1.03925
$536$	−9996.73	−0.805584
$537$	0	0
$538$	−4275.76	−0.342642
$539$	21393.9	1.70965
$540$	0	0
$541$	−3054.78	−0.242764	−0.121382	−	0.992606i	$-0.538733\pi$
$541$	−3054.78	−0.242764	−0.121382	+	0.992606i	$0.538733\pi$
$542$	7877.53	0.624297
$543$	0	0
$544$	−2337.01	−0.184189
$545$	−22305.6	−1.75315
$546$	0	0
$547$	21995.6	1.71931	0.859656	−	0.510873i	$-0.170678\pi$
$547$	21995.6	1.71931	0.859656	+	0.510873i	$0.170678\pi$
$548$	−745.868	−0.0581422
$549$	0	0
$550$	−8339.15	−0.646514
$551$	609.042	0.0470891
$552$	0	0
$553$	36476.5	2.80495
$554$	−18873.2	−1.44738
$555$	0	0
$556$	−714.664	−0.0545117
$557$	18380.1	1.39819	0.699095	−	0.715029i	$-0.253587\pi$
$557$	18380.1	1.39819	0.699095	+	0.715029i	$0.253587\pi$
$558$	0	0
$559$	2661.69	0.201391
$560$	−32071.8	−2.42014
$561$	0	0
$562$	3261.07	0.244769
$563$	13798.3	1.03291	0.516457	−	0.856313i	$-0.327251\pi$
$563$	13798.3	1.03291	0.516457	+	0.856313i	$0.327251\pi$
$564$	0	0
$565$	−25161.6	−1.87355
$566$	−7307.46	−0.542678
$567$	0	0
$568$	−16077.2	−1.18765
$569$	−5526.46	−0.407173	−0.203586	−	0.979057i	$-0.565260\pi$
$569$	−5526.46	−0.407173	−0.203586	+	0.979057i	$0.565260\pi$
$570$	0	0
$571$	3815.17	0.279615	0.139807	−	0.990179i	$-0.455352\pi$
$571$	3815.17	0.279615	0.139807	+	0.990179i	$0.455352\pi$
$572$	−2064.33	−0.150898
$573$	0	0
$574$	22042.4	1.60284
$575$	−3351.46	−0.243071
$576$	0	0
$577$	−26061.1	−1.88031	−0.940155	−	0.340747i	$-0.889320\pi$
$577$	−26061.1	−1.88031	−0.940155	+	0.340747i	$0.889320\pi$
$578$	14101.0	1.01475
$579$	0	0
$580$	7598.50	0.543984
$581$	21156.4	1.51069
$582$	0	0
$583$	10304.3	0.732007
$584$	−4948.96	−0.350667
$585$	0	0
$586$	−19133.9	−1.34883
$587$	11328.2	0.796536	0.398268	−	0.917269i	$-0.369611\pi$
$587$	11328.2	0.796536	0.398268	+	0.917269i	$0.369611\pi$
$588$	0	0
$589$	−799.465	−0.0559276
$590$	−37304.4	−2.60305
$591$	0	0
$592$	18463.6	1.28184
$593$	−17745.3	−1.22886	−0.614429	−	0.788972i	$-0.710614\pi$
$593$	−17745.3	−1.22886	−0.614429	+	0.788972i	$0.710614\pi$
$594$	0	0
$595$	−9512.82	−0.655442
$596$	3632.76	0.249670
$597$	0	0
$598$	−3686.89	−0.252120
$599$	−9409.78	−0.641859	−0.320929	−	0.947103i	$-0.603995\pi$
$599$	−9409.78	−0.641859	−0.320929	+	0.947103i	$0.603995\pi$
$600$	0	0
$601$	−13763.2	−0.934129	−0.467064	−	0.884223i	$-0.654688\pi$
$601$	−13763.2	−0.934129	−0.467064	+	0.884223i	$0.654688\pi$
$602$	11119.2	0.752798
$603$	0	0
$604$	7284.06	0.490702
$605$	2165.36	0.145511
$606$	0	0
$607$	−13643.2	−0.912288	−0.456144	−	0.889906i	$-0.650770\pi$
$607$	−13643.2	−0.912288	−0.456144	+	0.889906i	$0.650770\pi$
$608$	263.558	0.0175801
$609$	0	0
$610$	−39867.8	−2.64623
$611$	−7541.29	−0.499326
$612$	0	0
$613$	22573.0	1.48730	0.743651	−	0.668567i	$-0.233092\pi$
$613$	22573.0	1.48730	0.743651	+	0.668567i	$0.233092\pi$
$614$	−13344.6	−0.877105
$615$	0	0
$616$	21075.9	1.37853
$617$	−5364.22	−0.350009	−0.175004	−	0.984568i	$-0.555994\pi$
$617$	−5364.22	−0.350009	−0.175004	+	0.984568i	$0.555994\pi$
$618$	0	0
$619$	764.179	0.0496202	0.0248101	−	0.999692i	$-0.492102\pi$
$619$	764.179	0.0496202	0.0248101	+	0.999692i	$0.492102\pi$
$620$	−9974.24	−0.646089
$621$	0	0
$622$	10940.6	0.705267
$623$	44652.5	2.87153
$624$	0	0
$625$	−19508.2	−1.24852
$626$	2237.32	0.142846
$627$	0	0
$628$	−4246.67	−0.269842
$629$	5476.49	0.347157
$630$	0	0
$631$	−29378.7	−1.85348	−0.926740	−	0.375703i	$-0.877401\pi$
$631$	−29378.7	−1.85348	−0.926740	+	0.375703i	$0.877401\pi$
$632$	22205.3	1.39760
$633$	0	0
$634$	−28141.9	−1.76287
$635$	35557.0	2.22211
$636$	0	0
$637$	−12784.4	−0.795190
$638$	−29226.6	−1.81362
$639$	0	0
$640$	−24224.4	−1.49618
$641$	21357.5	1.31603	0.658013	−	0.753007i	$-0.271397\pi$
$641$	21357.5	1.31603	0.658013	+	0.753007i	$0.271397\pi$
$642$	0	0
$643$	18858.7	1.15663	0.578317	−	0.815812i	$-0.303710\pi$
$643$	18858.7	1.15663	0.578317	+	0.815812i	$0.303710\pi$
$644$	−3465.83	−0.212070
$645$	0	0
$646$	189.925	0.0115673
$647$	1608.74	0.0977531	0.0488766	−	0.998805i	$-0.484436\pi$
$647$	1608.74	0.0977531	0.0488766	+	0.998805i	$0.484436\pi$
$648$	0	0
$649$	32288.0	1.95288
$650$	4983.25	0.300706
$651$	0	0
$652$	2883.52	0.173202
$653$	−27539.0	−1.65036	−0.825181	−	0.564869i	$-0.808927\pi$
$653$	−27539.0	−1.65036	−0.825181	+	0.564869i	$0.808927\pi$
$654$	0	0
$655$	−16671.6	−0.994522
$656$	17673.4	1.05188
$657$	0	0
$658$	−31503.7	−1.86648
$659$	14364.0	0.849078	0.424539	−	0.905410i	$-0.360436\pi$
$659$	14364.0	0.849078	0.424539	+	0.905410i	$0.360436\pi$
$660$	0	0
$661$	−22031.9	−1.29643	−0.648217	−	0.761456i	$-0.724485\pi$
$661$	−22031.9	−1.29643	−0.648217	+	0.761456i	$0.724485\pi$
$662$	−28633.6	−1.68108
$663$	0	0
$664$	12879.1	0.752719
$665$	1072.82	0.0625594
$666$	0	0
$667$	−11746.0	−0.681870
$668$	7035.26	0.407489
$669$	0	0
$670$	24419.0	1.40804
$671$	34506.7	1.98527
$672$	0	0
$673$	1606.02	0.0919872	0.0459936	−	0.998942i	$-0.485355\pi$
$673$	1606.02	0.0919872	0.0459936	+	0.998942i	$0.485355\pi$
$674$	1685.51	0.0963258
$675$	0	0
$676$	−3869.86	−0.220179
$677$	4082.11	0.231741	0.115870	−	0.993264i	$-0.463034\pi$
$677$	4082.11	0.231741	0.115870	+	0.993264i	$0.463034\pi$
$678$	0	0
$679$	52765.4	2.98225
$680$	−5791.00	−0.326580
$681$	0	0
$682$	38364.5	2.15404
$683$	9092.52	0.509393	0.254697	−	0.967021i	$-0.418024\pi$
$683$	9092.52	0.509393	0.254697	+	0.967021i	$0.418024\pi$
$684$	0	0
$685$	−4452.69	−0.248363
$686$	−20386.7	−1.13465
$687$	0	0
$688$	8915.30	0.494030
$689$	−6157.56	−0.340471
$690$	0	0
$691$	−1482.40	−0.0816108	−0.0408054	−	0.999167i	$-0.512992\pi$
$691$	−1482.40	−0.0816108	−0.0408054	+	0.999167i	$0.512992\pi$
$692$	−1460.33	−0.0802218
$693$	0	0
$694$	−18472.8	−1.01040
$695$	−4266.41	−0.232855
$696$	0	0
$697$	5242.13	0.284878
$698$	16270.4	0.882300
$699$	0	0
$700$	4684.46	0.252937
$701$	3845.95	0.207217	0.103609	−	0.994618i	$-0.466961\pi$
$701$	3845.95	0.207217	0.103609	+	0.994618i	$0.466961\pi$
$702$	0	0
$703$	−617.615	−0.0331349
$704$	11165.4	0.597744
$705$	0	0
$706$	−9121.28	−0.486238
$707$	25083.8	1.33433
$708$	0	0
$709$	−4375.01	−0.231744	−0.115872	−	0.993264i	$-0.536966\pi$
$709$	−4375.01	−0.231744	−0.115872	+	0.993264i	$0.536966\pi$
$710$	39271.6	2.07583
$711$	0	0
$712$	27182.5	1.43077
$713$	15418.5	0.809856
$714$	0	0
$715$	−12323.7	−0.644585
$716$	5668.39	0.295863
$717$	0	0
$718$	−30538.4	−1.58730
$719$	17145.3	0.889306	0.444653	−	0.895703i	$-0.353327\pi$
$719$	17145.3	0.889306	0.444653	+	0.895703i	$0.353327\pi$
$720$	0	0
$721$	−18224.2	−0.941336
$722$	22016.1	1.13484
$723$	0	0
$724$	8945.86	0.459213
$725$	15876.1	0.813272
$726$	0	0
$727$	−124.425	−0.00634754	−0.00317377	−	0.999995i	$-0.501010\pi$
$727$	−124.425	−0.00634754	−0.00317377	+	0.999995i	$0.501010\pi$
$728$	−12594.4	−0.641179
$729$	0	0
$730$	12088.8	0.612914
$731$	2644.37	0.133797
$732$	0	0
$733$	21562.9	1.08655	0.543276	−	0.839554i	$-0.317184\pi$
$733$	21562.9	1.08655	0.543276	+	0.839554i	$0.317184\pi$
$734$	−11208.3	−0.563632
$735$	0	0
$736$	−5082.99	−0.254567
$737$	−21135.3	−1.05635
$738$	0	0
$739$	−17886.8	−0.890362	−0.445181	−	0.895441i	$-0.646861\pi$
$739$	−17886.8	−0.890362	−0.445181	+	0.895441i	$0.646861\pi$
$740$	−7705.46	−0.382781
$741$	0	0
$742$	−25723.2	−1.27268
$743$	14639.9	0.722863	0.361432	−	0.932399i	$-0.382288\pi$
$743$	14639.9	0.722863	0.361432	+	0.932399i	$0.382288\pi$
$744$	0	0
$745$	21686.9	1.06650
$746$	−5545.30	−0.272155
$747$	0	0
$748$	−2050.89	−0.100252
$749$	27786.7	1.35554
$750$	0	0
$751$	−14352.0	−0.697354	−0.348677	−	0.937243i	$-0.613369\pi$
$751$	−14352.0	−0.697354	−0.348677	+	0.937243i	$0.613369\pi$
$752$	−25259.5	−1.22489
$753$	0	0
$754$	17465.0	0.843551
$755$	43484.5	2.09611
$756$	0	0
$757$	−8088.58	−0.388355	−0.194177	−	0.980966i	$-0.562204\pi$
$757$	−8088.58	−0.388355	−0.194177	+	0.980966i	$0.562204\pi$
$758$	5201.65	0.249251
$759$	0	0
$760$	653.084	0.0311709
$761$	−37989.8	−1.80963	−0.904816	−	0.425804i	$-0.859991\pi$
$761$	−37989.8	−1.80963	−0.904816	+	0.425804i	$0.859991\pi$
$762$	0	0
$763$	−48195.0	−2.28673
$764$	−908.493	−0.0430211
$765$	0	0
$766$	9271.27	0.437317
$767$	−19294.4	−0.908321
$768$	0	0
$769$	4301.94	0.201732	0.100866	−	0.994900i	$-0.467839\pi$
$769$	4301.94	0.201732	0.100866	+	0.994900i	$0.467839\pi$
$770$	−51482.0	−2.40946
$771$	0	0
$772$	2413.15	0.112501
$773$	−22422.8	−1.04333	−0.521664	−	0.853151i	$-0.674688\pi$
$773$	−22422.8	−1.04333	−0.521664	+	0.853151i	$0.674688\pi$
$774$	0	0
$775$	−20839.9	−0.965922
$776$	32121.3	1.48594
$777$	0	0
$778$	34123.7	1.57249
$779$	−591.185	−0.0271905
$780$	0	0
$781$	−33990.7	−1.55734
$782$	−3662.90	−0.167500
$783$	0	0
$784$	−42821.2	−1.95067
$785$	−25351.8	−1.15267
$786$	0	0
$787$	−412.633	−0.0186897	−0.00934484	−	0.999956i	$-0.502975\pi$
$787$	−412.633	−0.0186897	−0.00934484	+	0.999956i	$0.502975\pi$
$788$	−5218.04	−0.235895
$789$	0	0
$790$	−54240.9	−2.44279
$791$	−54365.8	−2.44377
$792$	0	0
$793$	−20620.3	−0.923388
$794$	8804.78	0.393539
$795$	0	0
$796$	5855.51	0.260732
$797$	−29309.9	−1.30265	−0.651325	−	0.758799i	$-0.725786\pi$
$797$	−29309.9	−1.30265	−0.651325	+	0.758799i	$0.725786\pi$
$798$	0	0
$799$	−7492.22	−0.331734
$800$	6870.24	0.303625
$801$	0	0
$802$	236.142	0.0103971
$803$	−10463.2	−0.459824
$804$	0	0
$805$	−20690.4	−0.905887
$806$	−22925.6	−1.00188
$807$	0	0
$808$	15269.9	0.664844
$809$	34918.2	1.51750	0.758750	−	0.651382i	$-0.225811\pi$
$809$	34918.2	1.51750	0.758750	+	0.651382i	$0.225811\pi$
$810$	0	0
$811$	−15898.2	−0.688364	−0.344182	−	0.938903i	$-0.611844\pi$
$811$	−15898.2	−0.688364	−0.344182	+	0.938903i	$0.611844\pi$
$812$	16417.8	0.709547
$813$	0	0
$814$	29637.9	1.27618
$815$	17214.1	0.739857
$816$	0	0
$817$	−298.221	−0.0127704
$818$	35681.7	1.52516
$819$	0	0
$820$	−7375.70	−0.314111
$821$	−6503.93	−0.276479	−0.138239	−	0.990399i	$-0.544144\pi$
$821$	−6503.93	−0.276479	−0.138239	+	0.990399i	$0.544144\pi$
$822$	0	0
$823$	−15776.0	−0.668187	−0.334094	−	0.942540i	$-0.608430\pi$
$823$	−15776.0	−0.668187	−0.334094	+	0.942540i	$0.608430\pi$
$824$	−11094.1	−0.469030
$825$	0	0
$826$	−80602.5	−3.39530
$827$	−42182.2	−1.77366	−0.886830	−	0.462095i	$-0.847098\pi$
$827$	−42182.2	−1.77366	−0.886830	+	0.462095i	$0.847098\pi$
$828$	0	0
$829$	24658.5	1.03308	0.516542	−	0.856262i	$-0.327219\pi$
$829$	24658.5	1.03308	0.516542	+	0.856262i	$0.327219\pi$
$830$	−31459.7	−1.31564
$831$	0	0
$832$	−6672.14	−0.278023
$833$	−12701.2	−0.528296
$834$	0	0
$835$	41999.2	1.74065
$836$	231.291	0.00956863
$837$	0	0
$838$	12750.0	0.525586
$839$	−18151.7	−0.746921	−0.373461	−	0.927646i	$-0.621829\pi$
$839$	−18151.7	−0.746921	−0.373461	+	0.927646i	$0.621829\pi$
$840$	0	0
$841$	31252.5	1.28142
$842$	27416.2	1.12212
$843$	0	0
$844$	1500.99	0.0612160
$845$	−23102.4	−0.940528
$846$	0	0
$847$	4678.62	0.189799
$848$	−20624.7	−0.835206
$849$	0	0
$850$	4950.82	0.199779
$851$	11911.3	0.479807
$852$	0	0
$853$	−15137.9	−0.607632	−0.303816	−	0.952731i	$-0.598261\pi$
$853$	−15137.9	−0.607632	−0.303816	+	0.952731i	$0.598261\pi$
$854$	−86141.0	−3.45162
$855$	0	0
$856$	16915.3	0.675414
$857$	999.329	0.0398325	0.0199162	−	0.999802i	$-0.493660\pi$
$857$	999.329	0.0398325	0.0199162	+	0.999802i	$0.493660\pi$
$858$	0	0
$859$	−26847.5	−1.06639	−0.533193	−	0.845994i	$-0.679008\pi$
$859$	−26847.5	−1.06639	−0.533193	+	0.845994i	$0.679008\pi$
$860$	−3720.64	−0.147527
$861$	0	0
$862$	−31214.8	−1.23339
$863$	17092.2	0.674187	0.337094	−	0.941471i	$-0.390556\pi$
$863$	17092.2	0.674187	0.337094	+	0.941471i	$0.390556\pi$
$864$	0	0
$865$	−8717.91	−0.342680
$866$	2447.09	0.0960224
$867$	0	0
$868$	−21551.0	−0.842729
$869$	46947.0	1.83264
$870$	0	0
$871$	12629.9	0.491329
$872$	−29339.0	−1.13939
$873$	0	0
$874$	413.086	0.0159872
$875$	−23972.8	−0.926206
$876$	0	0
$877$	−16808.6	−0.647190	−0.323595	−	0.946196i	$-0.604892\pi$
$877$	−16808.6	−0.647190	−0.323595	+	0.946196i	$0.604892\pi$
$878$	30328.4	1.16576
$879$	0	0
$880$	−41277.9	−1.58123
$881$	13942.3	0.533178	0.266589	−	0.963810i	$-0.414103\pi$
$881$	13942.3	0.533178	0.266589	+	0.963810i	$0.414103\pi$
$882$	0	0
$883$	−27976.7	−1.06624	−0.533120	−	0.846040i	$-0.678981\pi$
$883$	−27976.7	−1.06624	−0.533120	+	0.846040i	$0.678981\pi$
$884$	1225.56	0.0466290
$885$	0	0
$886$	−14126.1	−0.535640
$887$	4743.70	0.179569	0.0897845	−	0.995961i	$-0.471382\pi$
$887$	4743.70	0.179569	0.0897845	+	0.995961i	$0.471382\pi$
$888$	0	0
$889$	76826.9	2.89841
$890$	−66398.7	−2.50077
$891$	0	0
$892$	−11810.4	−0.443320
$893$	844.941	0.0316628
$894$	0	0
$895$	33839.3	1.26382
$896$	−52341.0	−1.95155
$897$	0	0
$898$	−45143.1	−1.67756
$899$	−73038.3	−2.70964
$900$	0	0
$901$	−6117.49	−0.226197
$902$	28369.6	1.04723
$903$	0	0
$904$	−33095.6	−1.21764
$905$	53405.1	1.96160
$906$	0	0
$907$	−41083.2	−1.50402	−0.752010	−	0.659152i	$-0.770916\pi$
$907$	−41083.2	−1.50402	−0.752010	+	0.659152i	$0.770916\pi$
$908$	−6255.36	−0.228625
$909$	0	0
$910$	30764.2	1.12069
$911$	−46373.8	−1.68653	−0.843267	−	0.537494i	$-0.819371\pi$
$911$	−46373.8	−1.68653	−0.843267	+	0.537494i	$0.819371\pi$
$912$	0	0
$913$	27229.2	0.987028
$914$	40021.2	1.44834
$915$	0	0
$916$	1868.62	0.0674029
$917$	−36021.7	−1.29721
$918$	0	0
$919$	−14502.3	−0.520551	−0.260276	−	0.965534i	$-0.583813\pi$
$919$	−14502.3	−0.520551	−0.260276	+	0.965534i	$0.583813\pi$
$920$	−12595.4	−0.451367
$921$	0	0
$922$	−33971.2	−1.21343
$923$	20311.9	0.724350
$924$	0	0
$925$	−16099.5	−0.572270
$926$	42174.5	1.49670
$927$	0	0
$928$	24078.4	0.851738
$929$	20759.6	0.733156	0.366578	−	0.930387i	$-0.380529\pi$
$929$	20759.6	0.733156	0.366578	+	0.930387i	$0.380529\pi$
$930$	0	0
$931$	1432.39	0.0504239
$932$	11767.1	0.413566
$933$	0	0
$934$	44577.9	1.56171
$935$	−12243.5	−0.428240
$936$	0	0
$937$	−48039.8	−1.67491	−0.837456	−	0.546504i	$-0.815958\pi$
$937$	−48039.8	−1.67491	−0.837456	+	0.546504i	$0.815958\pi$
$938$	52761.3	1.83658
$939$	0	0
$940$	10541.6	0.365776
$941$	−1246.76	−0.0431915	−0.0215958	−	0.999767i	$-0.506875\pi$
$941$	−1246.76	−0.0431915	−0.0215958	+	0.999767i	$0.506875\pi$
$942$	0	0
$943$	11401.6	0.393730
$944$	−64626.6	−2.22819
$945$	0	0
$946$	14310.9	0.491848
$947$	−35735.3	−1.22623	−0.613116	−	0.789993i	$-0.710084\pi$
$947$	−35735.3	−1.22623	−0.613116	+	0.789993i	$0.710084\pi$
$948$	0	0
$949$	6252.53	0.213873
$950$	−558.333	−0.0190681
$951$	0	0
$952$	−12512.4	−0.425977
$953$	−53690.0	−1.82496	−0.912482	−	0.409117i	$-0.865837\pi$
$953$	−53690.0	−1.82496	−0.912482	+	0.409117i	$0.865837\pi$
$954$	0	0
$955$	−5423.53	−0.183771
$956$	555.177	0.0187821
$957$	0	0
$958$	15471.7	0.521781
$959$	−9620.79	−0.323954
$960$	0	0
$961$	66083.3	2.21823
$962$	−17710.8	−0.593576
$963$	0	0
$964$	8201.12	0.274004
$965$	14406.0	0.480566
$966$	0	0
$967$	10192.2	0.338946	0.169473	−	0.985535i	$-0.445794\pi$
$967$	10192.2	0.338946	0.169473	+	0.985535i	$0.445794\pi$
$968$	2848.14	0.0945690
$969$	0	0
$970$	−78462.6	−2.59720
$971$	18620.5	0.615409	0.307704	−	0.951482i	$-0.400439\pi$
$971$	18620.5	0.615409	0.307704	+	0.951482i	$0.400439\pi$
$972$	0	0
$973$	−9218.29	−0.303725
$974$	25629.5	0.843143
$975$	0	0
$976$	−69067.4	−2.26516
$977$	9933.43	0.325280	0.162640	−	0.986685i	$-0.447999\pi$
$977$	9933.43	0.325280	0.162640	+	0.986685i	$0.447999\pi$
$978$	0	0
$979$	57469.9	1.87615
$980$	17870.7	0.582508
$981$	0	0
$982$	−6297.79	−0.204654
$983$	−26489.0	−0.859480	−0.429740	−	0.902953i	$-0.641395\pi$
$983$	−26489.0	−0.859480	−0.429740	+	0.902953i	$0.641395\pi$
$984$	0	0
$985$	−31150.7	−1.00766
$986$	17351.3	0.560425
$987$	0	0
$988$	−138.213	−0.00445056
$989$	5751.49	0.184921
$990$	0	0
$991$	−10937.7	−0.350604	−0.175302	−	0.984515i	$-0.556090\pi$
$991$	−10937.7	−0.350604	−0.175302	+	0.984515i	$0.556090\pi$
$992$	−31606.7	−1.01161
$993$	0	0
$994$	84852.9	2.70762
$995$	34956.3	1.11376
$996$	0	0
$997$	−4379.40	−0.139114	−0.0695571	−	0.997578i	$-0.522159\pi$
$997$	−4379.40	−0.139114	−0.0695571	+	0.997578i	$0.522159\pi$
$998$	−48657.5	−1.54331
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.6		28
3.2	odd	2		717.4.a.a.1.23	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.23	✓	28	3.2	odd	2
2151.4.a.c.1.6		28	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-3.21293 q^{2} +2.32292 q^{4} +13.8674 q^{5} +29.9628 q^{7} +18.2401 q^{8} +O(q^{10})\)	\(q-3.21293 q^{2} +2.32292 q^{4} +13.8674 q^{5} +29.9628 q^{7} +18.2401 q^{8} -44.5549 q^{10} +38.5636 q^{11} -23.0445 q^{13} -96.2684 q^{14} -77.1874 q^{16} -22.8946 q^{17} +2.58195 q^{19} +32.2128 q^{20} -123.902 q^{22} -49.7956 q^{23} +67.3044 q^{25} +74.0404 q^{26} +69.6011 q^{28} +235.884 q^{29} -309.636 q^{31} +102.077 q^{32} +73.5586 q^{34} +415.506 q^{35} -239.205 q^{37} -8.29563 q^{38} +252.942 q^{40} -228.968 q^{41} -115.502 q^{43} +89.5800 q^{44} +159.990 q^{46} +327.249 q^{47} +554.769 q^{49} -216.244 q^{50} -53.5305 q^{52} +267.203 q^{53} +534.776 q^{55} +546.523 q^{56} -757.880 q^{58} +837.268 q^{59} +894.801 q^{61} +994.838 q^{62} +289.532 q^{64} -319.567 q^{65} -548.064 q^{67} -53.1822 q^{68} -1334.99 q^{70} -881.420 q^{71} -271.324 q^{73} +768.548 q^{74} +5.99766 q^{76} +1155.47 q^{77} +1217.39 q^{79} -1070.39 q^{80} +735.658 q^{82} +706.087 q^{83} -317.488 q^{85} +371.100 q^{86} +703.402 q^{88} +1490.27 q^{89} -690.478 q^{91} -115.671 q^{92} -1051.43 q^{94} +35.8049 q^{95} +1761.03 q^{97} -1782.43 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

Introduction

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