LMFDB - Embedded newform 2151.4.a.a.1.9

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:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.83829 q^{2} -4.62067 q^{4} -9.40555 q^{5} +19.7665 q^{7} +23.2005 q^{8} +O(q^{10})$	\(q-1.83829 q^{2} -4.62067 q^{4} -9.40555 q^{5} +19.7665 q^{7} +23.2005 q^{8} +17.2902 q^{10} -23.4581 q^{11} -4.74130 q^{13} -36.3367 q^{14} -5.68401 q^{16} +90.5088 q^{17} -133.353 q^{19} +43.4600 q^{20} +43.1228 q^{22} +97.8372 q^{23} -36.5357 q^{25} +8.71592 q^{26} -91.3346 q^{28} +6.66731 q^{29} -70.0295 q^{31} -175.155 q^{32} -166.382 q^{34} -185.915 q^{35} -34.2348 q^{37} +245.142 q^{38} -218.214 q^{40} +135.636 q^{41} -43.9979 q^{43} +108.392 q^{44} -179.854 q^{46} -87.6641 q^{47} +47.7149 q^{49} +67.1633 q^{50} +21.9080 q^{52} +486.395 q^{53} +220.636 q^{55} +458.593 q^{56} -12.2565 q^{58} +610.897 q^{59} -101.818 q^{61} +128.735 q^{62} +367.459 q^{64} +44.5946 q^{65} +162.483 q^{67} -418.211 q^{68} +341.766 q^{70} -1086.07 q^{71} -534.959 q^{73} +62.9336 q^{74} +616.181 q^{76} -463.684 q^{77} -337.326 q^{79} +53.4612 q^{80} -249.340 q^{82} +120.417 q^{83} -851.285 q^{85} +80.8810 q^{86} -544.239 q^{88} -290.120 q^{89} -93.7190 q^{91} -452.073 q^{92} +161.152 q^{94} +1254.26 q^{95} +801.960 q^{97} -87.7140 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * 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 $

$2$	−1.83829	−0.649935	−0.324968	−	0.945725i	$-0.605353\pi$
$2$	−1.83829	−0.649935	−0.324968	+	0.945725i	$0.605353\pi$
$3$	0	0
$3$	0	0
$4$	−4.62067	−0.577584
$5$	−9.40555	−0.841258	−0.420629	−	0.907233i	$-0.638191\pi$
$5$	−9.40555	−0.841258	−0.420629	+	0.907233i	$0.638191\pi$
$6$	0	0
$7$	19.7665	1.06729	0.533646	−	0.845708i	$-0.320822\pi$
$7$	19.7665	1.06729	0.533646	+	0.845708i	$0.320822\pi$
$8$	23.2005	1.02533
$9$	0	0
$10$	17.2902	0.546763
$11$	−23.4581	−0.642988	−0.321494	−	0.946912i	$-0.604185\pi$
$11$	−23.4581	−0.642988	−0.321494	+	0.946912i	$0.604185\pi$
$12$	0	0
$13$	−4.74130	−0.101154	−0.0505770	−	0.998720i	$-0.516106\pi$
$13$	−4.74130	−0.101154	−0.0505770	+	0.998720i	$0.516106\pi$
$14$	−36.3367	−0.693670
$15$	0	0
$16$	−5.68401	−0.0888126
$17$	90.5088	1.29127	0.645635	−	0.763646i	$-0.276593\pi$
$17$	90.5088	1.29127	0.645635	+	0.763646i	$0.276593\pi$
$18$	0	0
$19$	−133.353	−1.61017	−0.805087	−	0.593157i	$-0.797881\pi$
$19$	−133.353	−1.61017	−0.805087	+	0.593157i	$0.797881\pi$
$20$	43.4600	0.485897
$21$	0	0
$22$	43.1228	0.417901
$23$	97.8372	0.886976	0.443488	−	0.896280i	$-0.353741\pi$
$23$	97.8372	0.886976	0.443488	+	0.896280i	$0.353741\pi$
$24$	0	0
$25$	−36.5357	−0.292285
$26$	8.71592	0.0657435
$27$	0	0
$28$	−91.3346	−0.616450
$29$	6.66731	0.0426927	0.0213463	−	0.999772i	$-0.493205\pi$
$29$	6.66731	0.0426927	0.0213463	+	0.999772i	$0.493205\pi$
$30$	0	0
$31$	−70.0295	−0.405731	−0.202866	−	0.979207i	$-0.565026\pi$
$31$	−70.0295	−0.405731	−0.202866	+	0.979207i	$0.565026\pi$
$32$	−175.155	−0.967605
$33$	0	0
$34$	−166.382	−0.839242
$35$	−185.915	−0.897867
$36$	0	0
$37$	−34.2348	−0.152113	−0.0760563	−	0.997104i	$-0.524233\pi$
$37$	−34.2348	−0.152113	−0.0760563	+	0.997104i	$0.524233\pi$
$38$	245.142	1.04651
$39$	0	0
$40$	−218.214	−0.862565
$41$	135.636	0.516655	0.258328	−	0.966057i	$-0.416829\pi$
$41$	135.636	0.516655	0.258328	+	0.966057i	$0.416829\pi$
$42$	0	0
$43$	−43.9979	−0.156037	−0.0780187	−	0.996952i	$-0.524859\pi$
$43$	−43.9979	−0.156037	−0.0780187	+	0.996952i	$0.524859\pi$
$44$	108.392	0.371380
$45$	0	0
$46$	−179.854	−0.576477
$47$	−87.6641	−0.272066	−0.136033	−	0.990704i	$-0.543435\pi$
$47$	−87.6641	−0.272066	−0.136033	+	0.990704i	$0.543435\pi$
$48$	0	0
$49$	47.7149	0.139110
$50$	67.1633	0.189967
$51$	0	0
$52$	21.9080	0.0584249
$53$	486.395	1.26059	0.630297	−	0.776354i	$-0.282933\pi$
$53$	486.395	1.26059	0.630297	+	0.776354i	$0.282933\pi$
$54$	0	0
$55$	220.636	0.540919
$56$	458.593	1.09432
$57$	0	0
$58$	−12.2565	−0.0277475
$59$	610.897	1.34800	0.674000	−	0.738731i	$-0.264575\pi$
$59$	610.897	1.34800	0.674000	+	0.738731i	$0.264575\pi$
$60$	0	0
$61$	−101.818	−0.213713	−0.106857	−	0.994274i	$-0.534079\pi$
$61$	−101.818	−0.213713	−0.106857	+	0.994274i	$0.534079\pi$
$62$	128.735	0.263699
$63$	0	0
$64$	367.459	0.717693
$65$	44.5946	0.0850965
$66$	0	0
$67$	162.483	0.296276	0.148138	−	0.988967i	$-0.452672\pi$
$67$	162.483	0.296276	0.148138	+	0.988967i	$0.452672\pi$
$68$	−418.211	−0.745817
$69$	0	0
$70$	341.766	0.583556
$71$	−1086.07	−1.81539	−0.907696	−	0.419628i	$-0.862161\pi$
$71$	−1086.07	−1.81539	−0.907696	+	0.419628i	$0.862161\pi$
$72$	0	0
$73$	−534.959	−0.857701	−0.428851	−	0.903375i	$-0.641081\pi$
$73$	−534.959	−0.857701	−0.428851	+	0.903375i	$0.641081\pi$
$74$	62.9336	0.0988633
$75$	0	0
$76$	616.181	0.930010
$77$	−463.684	−0.686255
$78$	0	0
$79$	−337.326	−0.480406	−0.240203	−	0.970723i	$-0.577214\pi$
$79$	−337.326	−0.480406	−0.240203	+	0.970723i	$0.577214\pi$
$80$	53.4612	0.0747143
$81$	0	0
$82$	−249.340	−0.335792
$83$	120.417	0.159247	0.0796237	−	0.996825i	$-0.474628\pi$
$83$	120.417	0.159247	0.0796237	+	0.996825i	$0.474628\pi$
$84$	0	0
$85$	−851.285	−1.08629
$86$	80.8810	0.101414
$87$	0	0
$88$	−544.239	−0.659273
$89$	−290.120	−0.345536	−0.172768	−	0.984963i	$-0.555271\pi$
$89$	−290.120	−0.345536	−0.172768	+	0.984963i	$0.555271\pi$
$90$	0	0
$91$	−93.7190	−0.107961
$92$	−452.073	−0.512303
$93$	0	0
$94$	161.152	0.176826
$95$	1254.26	1.35457
$96$	0	0
$97$	801.960	0.839450	0.419725	−	0.907651i	$-0.362127\pi$
$97$	801.960	0.839450	0.419725	+	0.907651i	$0.362127\pi$
$98$	−87.7140	−0.0904128
$99$	0	0
$100$	168.819	0.168819
$101$	1142.05	1.12513	0.562566	−	0.826752i	$-0.309814\pi$
$101$	1142.05	1.12513	0.562566	+	0.826752i	$0.309814\pi$
$102$	0	0
$103$	1693.06	1.61963	0.809817	−	0.586683i	$-0.199566\pi$
$103$	1693.06	1.61963	0.809817	+	0.586683i	$0.199566\pi$
$104$	−110.001	−0.103716
$105$	0	0
$106$	−894.137	−0.819305
$107$	1575.49	1.42344	0.711720	−	0.702463i	$-0.247916\pi$
$107$	1575.49	1.42344	0.711720	+	0.702463i	$0.247916\pi$
$108$	0	0
$109$	769.260	0.675979	0.337990	−	0.941150i	$-0.390253\pi$
$109$	769.260	0.675979	0.337990	+	0.941150i	$0.390253\pi$
$110$	−405.594	−0.351562
$111$	0	0
$112$	−112.353	−0.0947889
$113$	1156.20	0.962535	0.481267	−	0.876574i	$-0.340177\pi$
$113$	1156.20	0.962535	0.481267	+	0.876574i	$0.340177\pi$
$114$	0	0
$115$	−920.212	−0.746176
$116$	−30.8075	−0.0246586
$117$	0	0
$118$	−1123.01	−0.876113
$119$	1789.04	1.37816
$120$	0	0
$121$	−780.720	−0.586566
$122$	187.172	0.138900
$123$	0	0
$124$	323.584	0.234344
$125$	1519.33	1.08715
$126$	0	0
$127$	1147.31	0.801631	0.400816	−	0.916159i	$-0.368727\pi$
$127$	1147.31	0.801631	0.400816	+	0.916159i	$0.368727\pi$
$128$	725.744	0.501151
$129$	0	0
$130$	−81.9780	−0.0553073
$131$	−652.853	−0.435420	−0.217710	−	0.976014i	$-0.569859\pi$
$131$	−652.853	−0.435420	−0.217710	+	0.976014i	$0.569859\pi$
$132$	0	0
$133$	−2635.92	−1.71852
$134$	−298.692	−0.192560
$135$	0	0
$136$	2099.85	1.32398
$137$	−1678.68	−1.04686	−0.523428	−	0.852070i	$-0.675347\pi$
$137$	−1678.68	−1.04686	−0.523428	+	0.852070i	$0.675347\pi$
$138$	0	0
$139$	1599.19	0.975836	0.487918	−	0.872890i	$-0.337757\pi$
$139$	1599.19	0.975836	0.487918	+	0.872890i	$0.337757\pi$
$140$	859.052	0.518594
$141$	0	0
$142$	1996.52	1.17989
$143$	111.222	0.0650408
$144$	0	0
$145$	−62.7097	−0.0359156
$146$	983.412	0.557450
$147$	0	0
$148$	158.188	0.0878578
$149$	−952.731	−0.523831	−0.261915	−	0.965091i	$-0.584354\pi$
$149$	−952.731	−0.523831	−0.261915	+	0.965091i	$0.584354\pi$
$150$	0	0
$151$	−1912.38	−1.03065	−0.515323	−	0.856996i	$-0.672328\pi$
$151$	−1912.38	−1.03065	−0.515323	+	0.856996i	$0.672328\pi$
$152$	−3093.86	−1.65096
$153$	0	0
$154$	852.387	0.446022
$155$	658.666	0.341325
$156$	0	0
$157$	−604.720	−0.307401	−0.153700	−	0.988118i	$-0.549119\pi$
$157$	−604.720	−0.307401	−0.153700	+	0.988118i	$0.549119\pi$
$158$	620.104	0.312233
$159$	0	0
$160$	1647.43	0.814005
$161$	1933.90	0.946662
$162$	0	0
$163$	3040.57	1.46108	0.730538	−	0.682872i	$-0.239269\pi$
$163$	3040.57	1.46108	0.730538	+	0.682872i	$0.239269\pi$
$164$	−626.732	−0.298412
$165$	0	0
$166$	−221.363	−0.103500
$167$	−1307.90	−0.606038	−0.303019	−	0.952985i	$-0.597995\pi$
$167$	−1307.90	−0.606038	−0.303019	+	0.952985i	$0.597995\pi$
$168$	0	0
$169$	−2174.52	−0.989768
$170$	1564.91	0.706019
$171$	0	0
$172$	203.300	0.0901248
$173$	−433.349	−0.190444	−0.0952222	−	0.995456i	$-0.530356\pi$
$173$	−433.349	−0.190444	−0.0952222	+	0.995456i	$0.530356\pi$
$174$	0	0
$175$	−722.183	−0.311954
$176$	133.336	0.0571055
$177$	0	0
$178$	533.327	0.224576
$179$	−4340.00	−1.81222	−0.906108	−	0.423047i	$-0.860961\pi$
$179$	−4340.00	−1.81222	−0.906108	+	0.423047i	$0.860961\pi$
$180$	0	0
$181$	−934.218	−0.383646	−0.191823	−	0.981430i	$-0.561440\pi$
$181$	−934.218	−0.383646	−0.191823	+	0.981430i	$0.561440\pi$
$182$	172.283	0.0701675
$183$	0	0
$184$	2269.87	0.909441
$185$	321.997	0.127966
$186$	0	0
$187$	−2123.16	−0.830272
$188$	405.067	0.157141
$189$	0	0
$190$	−2305.70	−0.880383
$191$	2665.21	1.00967	0.504837	−	0.863215i	$-0.331553\pi$
$191$	2665.21	1.00967	0.504837	+	0.863215i	$0.331553\pi$
$192$	0	0
$193$	−4884.75	−1.82183	−0.910913	−	0.412600i	$-0.864621\pi$
$193$	−4884.75	−1.82183	−0.910913	+	0.412600i	$0.864621\pi$
$194$	−1474.24	−0.545588
$195$	0	0
$196$	−220.475	−0.0803480
$197$	−2037.99	−0.737061	−0.368531	−	0.929616i	$-0.620139\pi$
$197$	−2037.99	−0.737061	−0.368531	+	0.929616i	$0.620139\pi$
$198$	0	0
$199$	−1480.90	−0.527528	−0.263764	−	0.964587i	$-0.584964\pi$
$199$	−1480.90	−0.527528	−0.263764	+	0.964587i	$0.584964\pi$
$200$	−847.646	−0.299688
$201$	0	0
$202$	−2099.43	−0.731263
$203$	131.789	0.0455655
$204$	0	0
$205$	−1275.74	−0.434640
$206$	−3112.35	−1.05266
$207$	0	0
$208$	26.9496	0.00898375
$209$	3128.20	1.03532
$210$	0	0
$211$	3106.91	1.01369	0.506844	−	0.862038i	$-0.330812\pi$
$211$	3106.91	1.01369	0.506844	+	0.862038i	$0.330812\pi$
$212$	−2247.47	−0.728099
$213$	0	0
$214$	−2896.21	−0.925144
$215$	413.824	0.131268
$216$	0	0
$217$	−1384.24	−0.433034
$218$	−1414.13	−0.439343
$219$	0	0
$220$	−1019.49	−0.312426
$221$	−429.130	−0.130617
$222$	0	0
$223$	−4647.28	−1.39554	−0.697768	−	0.716324i	$-0.745823\pi$
$223$	−4647.28	−1.39554	−0.697768	+	0.716324i	$0.745823\pi$
$224$	−3462.21	−1.03272
$225$	0	0
$226$	−2125.44	−0.625586
$227$	410.193	0.119936	0.0599680	−	0.998200i	$-0.480900\pi$
$227$	410.193	0.119936	0.0599680	+	0.998200i	$0.480900\pi$
$228$	0	0
$229$	3766.00	1.08674	0.543371	−	0.839492i	$-0.317148\pi$
$229$	3766.00	1.08674	0.543371	+	0.839492i	$0.317148\pi$
$230$	1691.62	0.484966
$231$	0	0
$232$	154.685	0.0437740
$233$	−1284.83	−0.361253	−0.180627	−	0.983552i	$-0.557813\pi$
$233$	−1284.83	−0.361253	−0.180627	+	0.983552i	$0.557813\pi$
$234$	0	0
$235$	824.529	0.228878
$236$	−2822.75	−0.778583
$237$	0	0
$238$	−3288.79	−0.895716
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−3797.31	−1.01496	−0.507482	−	0.861662i	$-0.669424\pi$
$241$	−3797.31	−1.01496	−0.507482	+	0.861662i	$0.669424\pi$
$242$	1435.19	0.381230
$243$	0	0
$244$	470.469	0.123437
$245$	−448.785	−0.117028
$246$	0	0
$247$	632.267	0.162875
$248$	−1624.72	−0.416008
$249$	0	0
$250$	−2792.98	−0.706574
$251$	−1423.65	−0.358007	−0.179004	−	0.983848i	$-0.557287\pi$
$251$	−1423.65	−0.358007	−0.179004	+	0.983848i	$0.557287\pi$
$252$	0	0
$253$	−2295.07	−0.570315
$254$	−2109.09	−0.521009
$255$	0	0
$256$	−4273.80	−1.04341
$257$	4349.34	1.05566	0.527830	−	0.849350i	$-0.323006\pi$
$257$	4349.34	1.05566	0.527830	+	0.849350i	$0.323006\pi$
$258$	0	0
$259$	−676.702	−0.162348
$260$	−206.057	−0.0491504
$261$	0	0
$262$	1200.14	0.282995
$263$	−6373.06	−1.49422	−0.747110	−	0.664701i	$-0.768559\pi$
$263$	−6373.06	−1.49422	−0.747110	+	0.664701i	$0.768559\pi$
$264$	0	0
$265$	−4574.81	−1.06048
$266$	4845.61	1.11693
$267$	0	0
$268$	−750.781	−0.171124
$269$	3650.71	0.827463	0.413732	−	0.910399i	$-0.364225\pi$
$269$	3650.71	0.827463	0.413732	+	0.910399i	$0.364225\pi$
$270$	0	0
$271$	6705.13	1.50298	0.751491	−	0.659744i	$-0.229335\pi$
$271$	6705.13	1.50298	0.751491	+	0.659744i	$0.229335\pi$
$272$	−514.453	−0.114681
$273$	0	0
$274$	3085.91	0.680389
$275$	857.055	0.187936
$276$	0	0
$277$	−676.282	−0.146693	−0.0733463	−	0.997307i	$-0.523368\pi$
$277$	−676.282	−0.146693	−0.0733463	+	0.997307i	$0.523368\pi$
$278$	−2939.77	−0.634230
$279$	0	0
$280$	−4313.32	−0.920608
$281$	−2473.70	−0.525155	−0.262578	−	0.964911i	$-0.584573\pi$
$281$	−2473.70	−0.525155	−0.262578	+	0.964911i	$0.584573\pi$
$282$	0	0
$283$	−3175.82	−0.667078	−0.333539	−	0.942736i	$-0.608243\pi$
$283$	−3175.82	−0.667078	−0.333539	+	0.942736i	$0.608243\pi$
$284$	5018.38	1.04854
$285$	0	0
$286$	−204.458	−0.0422723
$287$	2681.06	0.551421
$288$	0	0
$289$	3278.84	0.667380
$290$	115.279	0.0233428
$291$	0	0
$292$	2471.87	0.495395
$293$	−7084.14	−1.41249	−0.706245	−	0.707967i	$-0.749612\pi$
$293$	−7084.14	−1.41249	−0.706245	+	0.707967i	$0.749612\pi$
$294$	0	0
$295$	−5745.82	−1.13402
$296$	−794.265	−0.155965
$297$	0	0
$298$	1751.40	0.340456
$299$	−463.876	−0.0897212
$300$	0	0
$301$	−869.684	−0.166537
$302$	3515.52	0.669853
$303$	0	0
$304$	757.980	0.143004
$305$	957.657	0.179788
$306$	0	0
$307$	4463.90	0.829864	0.414932	−	0.909852i	$-0.363805\pi$
$307$	4463.90	0.829864	0.414932	+	0.909852i	$0.363805\pi$
$308$	2142.53	0.396370
$309$	0	0
$310$	−1210.82	−0.221839
$311$	−2880.21	−0.525150	−0.262575	−	0.964912i	$-0.584572\pi$
$311$	−2880.21	−0.525150	−0.262575	+	0.964912i	$0.584572\pi$
$312$	0	0
$313$	−8551.50	−1.54428	−0.772140	−	0.635453i	$-0.780813\pi$
$313$	−8551.50	−1.54428	−0.772140	+	0.635453i	$0.780813\pi$
$314$	1111.65	0.199791
$315$	0	0
$316$	1558.67	0.277475
$317$	−3275.50	−0.580348	−0.290174	−	0.956974i	$-0.593713\pi$
$317$	−3275.50	−0.580348	−0.290174	+	0.956974i	$0.593713\pi$
$318$	0	0
$319$	−156.402	−0.0274509
$320$	−3456.15	−0.603765
$321$	0	0
$322$	−3555.08	−0.615269
$323$	−12069.6	−2.07917
$324$	0	0
$325$	173.227	0.0295658
$326$	−5589.46	−0.949605
$327$	0	0
$328$	3146.84	0.529741
$329$	−1732.81	−0.290374
$330$	0	0
$331$	−480.848	−0.0798483	−0.0399241	−	0.999203i	$-0.512712\pi$
$331$	−480.848	−0.0798483	−0.0399241	+	0.999203i	$0.512712\pi$
$332$	−556.410	−0.0919787
$333$	0	0
$334$	2404.31	0.393885
$335$	−1528.24	−0.249244
$336$	0	0
$337$	−2328.95	−0.376456	−0.188228	−	0.982125i	$-0.560274\pi$
$337$	−2328.95	−0.376456	−0.188228	+	0.982125i	$0.560274\pi$
$338$	3997.41	0.643285
$339$	0	0
$340$	3933.51	0.627425
$341$	1642.76	0.260880
$342$	0	0
$343$	−5836.76	−0.918820
$344$	−1020.77	−0.159990
$345$	0	0
$346$	796.622	0.123777
$347$	−10665.9	−1.65007	−0.825033	−	0.565085i	$-0.808843\pi$
$347$	−10665.9	−1.65007	−0.825033	+	0.565085i	$0.808843\pi$
$348$	0	0
$349$	−3550.79	−0.544612	−0.272306	−	0.962211i	$-0.587786\pi$
$349$	−3550.79	−0.544612	−0.272306	+	0.962211i	$0.587786\pi$
$350$	1327.58	0.202750
$351$	0	0
$352$	4108.80	0.622159
$353$	−11360.0	−1.71284	−0.856421	−	0.516278i	$-0.827317\pi$
$353$	−11360.0	−1.71284	−0.856421	+	0.516278i	$0.827317\pi$
$354$	0	0
$355$	10215.1	1.52721
$356$	1340.55	0.199576
$357$	0	0
$358$	7978.19	1.17782
$359$	−4601.20	−0.676441	−0.338220	−	0.941067i	$-0.609825\pi$
$359$	−4601.20	−0.676441	−0.338220	+	0.941067i	$0.609825\pi$
$360$	0	0
$361$	10924.0	1.59266
$362$	1717.37	0.249345
$363$	0	0
$364$	433.045	0.0623564
$365$	5031.58	0.721548
$366$	0	0
$367$	−6665.05	−0.947991	−0.473996	−	0.880527i	$-0.657189\pi$
$367$	−6665.05	−0.947991	−0.473996	+	0.880527i	$0.657189\pi$
$368$	−556.107	−0.0787747
$369$	0	0
$370$	−591.925	−0.0831696
$371$	9614.33	1.34542
$372$	0	0
$373$	5742.24	0.797110	0.398555	−	0.917144i	$-0.369512\pi$
$373$	5742.24	0.797110	0.398555	+	0.917144i	$0.369512\pi$
$374$	3902.99	0.539623
$375$	0	0
$376$	−2033.85	−0.278957
$377$	−31.6117	−0.00431853
$378$	0	0
$379$	−5259.49	−0.712828	−0.356414	−	0.934328i	$-0.616001\pi$
$379$	−5259.49	−0.712828	−0.356414	+	0.934328i	$0.616001\pi$
$380$	−5795.52	−0.782378
$381$	0	0
$382$	−4899.44	−0.656223
$383$	10722.1	1.43048	0.715239	−	0.698879i	$-0.246318\pi$
$383$	10722.1	1.43048	0.715239	+	0.698879i	$0.246318\pi$
$384$	0	0
$385$	4361.20	0.577318
$386$	8979.62	1.18407
$387$	0	0
$388$	−3705.59	−0.484853
$389$	−2864.69	−0.373382	−0.186691	−	0.982419i	$-0.559776\pi$
$389$	−2864.69	−0.373382	−0.186691	+	0.982419i	$0.559776\pi$
$390$	0	0
$391$	8855.12	1.14533
$392$	1107.01	0.142634
$393$	0	0
$394$	3746.43	0.479042
$395$	3172.73	0.404145
$396$	0	0
$397$	−4474.49	−0.565664	−0.282832	−	0.959170i	$-0.591274\pi$
$397$	−4474.49	−0.565664	−0.282832	+	0.959170i	$0.591274\pi$
$398$	2722.33	0.342859
$399$	0	0
$400$	207.669	0.0259586
$401$	10640.0	1.32502	0.662511	−	0.749052i	$-0.269491\pi$
$401$	10640.0	1.32502	0.662511	+	0.749052i	$0.269491\pi$
$402$	0	0
$403$	332.031	0.0410413
$404$	−5277.04	−0.649858
$405$	0	0
$406$	−242.268	−0.0296147
$407$	803.082	0.0978066
$408$	0	0
$409$	−6188.63	−0.748185	−0.374093	−	0.927391i	$-0.622046\pi$
$409$	−6188.63	−0.748185	−0.374093	+	0.927391i	$0.622046\pi$
$410$	2345.18	0.282488
$411$	0	0
$412$	−7823.08	−0.935475
$413$	12075.3	1.43871
$414$	0	0
$415$	−1132.59	−0.133968
$416$	830.464	0.0978771
$417$	0	0
$418$	−5750.56	−0.672892
$419$	16605.5	1.93612	0.968058	−	0.250727i	$-0.0806696\pi$
$419$	16605.5	1.93612	0.968058	+	0.250727i	$0.0806696\pi$
$420$	0	0
$421$	−7363.22	−0.852402	−0.426201	−	0.904628i	$-0.640148\pi$
$421$	−7363.22	−0.852402	−0.426201	+	0.904628i	$0.640148\pi$
$422$	−5711.41	−0.658832
$423$	0	0
$424$	11284.6	1.29252
$425$	−3306.80	−0.377419
$426$	0	0
$427$	−2012.59	−0.228094
$428$	−7279.81	−0.822156
$429$	0	0
$430$	−760.731	−0.0853155
$431$	15124.0	1.69025	0.845124	−	0.534570i	$-0.179527\pi$
$431$	15124.0	1.69025	0.845124	+	0.534570i	$0.179527\pi$
$432$	0	0
$433$	9093.33	1.00923	0.504616	−	0.863344i	$-0.331634\pi$
$433$	9093.33	1.00923	0.504616	+	0.863344i	$0.331634\pi$
$434$	2544.64	0.281444
$435$	0	0
$436$	−3554.50	−0.390435
$437$	−13046.9	−1.42819
$438$	0	0
$439$	−18168.5	−1.97525	−0.987625	−	0.156833i	$-0.949872\pi$
$439$	−18168.5	−1.97525	−0.987625	+	0.156833i	$0.949872\pi$
$440$	5118.87	0.554619
$441$	0	0
$442$	788.867	0.0848927
$443$	−15977.4	−1.71357	−0.856785	−	0.515674i	$-0.827542\pi$
$443$	−15977.4	−1.71357	−0.856785	+	0.515674i	$0.827542\pi$
$444$	0	0
$445$	2728.74	0.290685
$446$	8543.06	0.907009
$447$	0	0
$448$	7263.38	0.765988
$449$	−14102.0	−1.48222	−0.741110	−	0.671384i	$-0.765700\pi$
$449$	−14102.0	−1.48222	−0.741110	+	0.671384i	$0.765700\pi$
$450$	0	0
$451$	−3181.77	−0.332203
$452$	−5342.44	−0.555945
$453$	0	0
$454$	−754.056	−0.0779506
$455$	881.479	0.0908228
$456$	0	0
$457$	14092.4	1.44248	0.721240	−	0.692685i	$-0.243573\pi$
$457$	14092.4	1.44248	0.721240	+	0.692685i	$0.243573\pi$
$458$	−6923.01	−0.706313
$459$	0	0
$460$	4252.00	0.430979
$461$	4162.15	0.420501	0.210250	−	0.977648i	$-0.432572\pi$
$461$	4162.15	0.420501	0.210250	+	0.977648i	$0.432572\pi$
$462$	0	0
$463$	−12556.5	−1.26036	−0.630181	−	0.776448i	$-0.717019\pi$
$463$	−12556.5	−1.26036	−0.630181	+	0.776448i	$0.717019\pi$
$464$	−37.8970	−0.00379165
$465$	0	0
$466$	2361.90	0.234791
$467$	−19265.8	−1.90902	−0.954510	−	0.298178i	$-0.903621\pi$
$467$	−19265.8	−1.90902	−0.954510	+	0.298178i	$0.903621\pi$
$468$	0	0
$469$	3211.72	0.316212
$470$	−1515.73	−0.148756
$471$	0	0
$472$	14173.1	1.38214
$473$	1032.10	0.100330
$474$	0	0
$475$	4872.14	0.470630
$476$	−8266.58	−0.796004
$477$	0	0
$478$	439.352	0.0420408
$479$	7706.01	0.735066	0.367533	−	0.930010i	$-0.380202\pi$
$479$	7706.01	0.735066	0.367533	+	0.930010i	$0.380202\pi$
$480$	0	0
$481$	162.318	0.0153868
$482$	6980.57	0.659661
$483$	0	0
$484$	3607.45	0.338791
$485$	−7542.87	−0.706194
$486$	0	0
$487$	−9662.57	−0.899081	−0.449541	−	0.893260i	$-0.648412\pi$
$487$	−9662.57	−0.899081	−0.449541	+	0.893260i	$0.648412\pi$
$488$	−2362.24	−0.219126
$489$	0	0
$490$	824.999	0.0760605
$491$	515.342	0.0473667	0.0236834	−	0.999720i	$-0.492461\pi$
$491$	515.342	0.0473667	0.0236834	+	0.999720i	$0.492461\pi$
$492$	0	0
$493$	603.450	0.0551278
$494$	−1162.29	−0.105858
$495$	0	0
$496$	398.048	0.0360341
$497$	−21467.8	−1.93755
$498$	0	0
$499$	11382.6	1.02115	0.510574	−	0.859834i	$-0.329433\pi$
$499$	11382.6	1.02115	0.510574	+	0.859834i	$0.329433\pi$
$500$	−7020.33	−0.627918
$501$	0	0
$502$	2617.08	0.232681
$503$	3285.45	0.291234	0.145617	−	0.989341i	$-0.453483\pi$
$503$	3285.45	0.291234	0.145617	+	0.989341i	$0.453483\pi$
$504$	0	0
$505$	−10741.6	−0.946526
$506$	4219.01	0.370668
$507$	0	0
$508$	−5301.34	−0.463010
$509$	−1671.84	−0.145586	−0.0727930	−	0.997347i	$-0.523191\pi$
$509$	−1671.84	−0.145586	−0.0727930	+	0.997347i	$0.523191\pi$
$510$	0	0
$511$	−10574.3	−0.915417
$512$	2050.56	0.176998
$513$	0	0
$514$	−7995.38	−0.686111
$515$	−15924.2	−1.36253
$516$	0	0
$517$	2056.43	0.174935
$518$	1243.98	0.105516
$519$	0	0
$520$	1034.62	0.0872518
$521$	16592.2	1.39524	0.697618	−	0.716470i	$-0.254243\pi$
$521$	16592.2	1.39524	0.697618	+	0.716470i	$0.254243\pi$
$522$	0	0
$523$	−20773.2	−1.73680	−0.868401	−	0.495862i	$-0.834852\pi$
$523$	−20773.2	−1.73680	−0.868401	+	0.495862i	$0.834852\pi$
$524$	3016.62	0.251492
$525$	0	0
$526$	11715.6	0.971146
$527$	−6338.29	−0.523909
$528$	0	0
$529$	−2594.89	−0.213273
$530$	8409.85	0.689247
$531$	0	0
$532$	12179.7	0.992592
$533$	−643.094	−0.0522617
$534$	0	0
$535$	−14818.3	−1.19748
$536$	3769.69	0.303780
$537$	0	0
$538$	−6711.08	−0.537798
$539$	−1119.30	−0.0894464
$540$	0	0
$541$	−12797.9	−1.01705	−0.508525	−	0.861047i	$-0.669809\pi$
$541$	−12797.9	−1.01705	−0.508525	+	0.861047i	$0.669809\pi$
$542$	−12326.0	−0.976840
$543$	0	0
$544$	−15853.1	−1.24944
$545$	−7235.31	−0.568673
$546$	0	0
$547$	10730.1	0.838732	0.419366	−	0.907817i	$-0.362252\pi$
$547$	10730.1	0.838732	0.419366	+	0.907817i	$0.362252\pi$
$548$	7756.63	0.604648
$549$	0	0
$550$	−1575.52	−0.122146
$551$	−889.106	−0.0687426
$552$	0	0
$553$	−6667.75	−0.512733
$554$	1243.21	0.0953408
$555$	0	0
$556$	−7389.31	−0.563627
$557$	1827.28	0.139003	0.0695014	−	0.997582i	$-0.477859\pi$
$557$	1827.28	0.139003	0.0695014	+	0.997582i	$0.477859\pi$
$558$	0	0
$559$	208.607	0.0157838
$560$	1056.74	0.0797419
$561$	0	0
$562$	4547.39	0.341317
$563$	−6995.91	−0.523699	−0.261849	−	0.965109i	$-0.584332\pi$
$563$	−6995.91	−0.523699	−0.261849	+	0.965109i	$0.584332\pi$
$564$	0	0
$565$	−10874.7	−0.809740
$566$	5838.10	0.433557
$567$	0	0
$568$	−25197.4	−1.86137
$569$	14156.3	1.04299	0.521497	−	0.853253i	$-0.325374\pi$
$569$	14156.3	1.04299	0.521497	+	0.853253i	$0.325374\pi$
$570$	0	0
$571$	20040.4	1.46876	0.734382	−	0.678737i	$-0.237472\pi$
$571$	20040.4	1.46876	0.734382	+	0.678737i	$0.237472\pi$
$572$	−513.919	−0.0375665
$573$	0	0
$574$	−4928.58	−0.358388
$575$	−3574.55	−0.259250
$576$	0	0
$577$	−3825.34	−0.275998	−0.137999	−	0.990432i	$-0.544067\pi$
$577$	−3825.34	−0.275998	−0.137999	+	0.990432i	$0.544067\pi$
$578$	−6027.47	−0.433754
$579$	0	0
$580$	289.761	0.0207443
$581$	2380.23	0.169963
$582$	0	0
$583$	−11409.9	−0.810547
$584$	−12411.3	−0.879425
$585$	0	0
$586$	13022.7	0.918028
$587$	14023.0	0.986017	0.493009	−	0.870024i	$-0.335897\pi$
$587$	14023.0	0.986017	0.493009	+	0.870024i	$0.335897\pi$
$588$	0	0
$589$	9338.65	0.653298
$590$	10562.5	0.737037
$591$	0	0
$592$	194.591	0.0135095
$593$	−23026.2	−1.59455	−0.797277	−	0.603613i	$-0.793727\pi$
$593$	−23026.2	−1.59455	−0.797277	+	0.603613i	$0.793727\pi$
$594$	0	0
$595$	−16826.9	−1.15939
$596$	4402.26	0.302556
$597$	0	0
$598$	852.740	0.0583130
$599$	8711.86	0.594252	0.297126	−	0.954838i	$-0.403972\pi$
$599$	8711.86	0.594252	0.297126	+	0.954838i	$0.403972\pi$
$600$	0	0
$601$	7756.87	0.526472	0.263236	−	0.964732i	$-0.415210\pi$
$601$	7756.87	0.526472	0.263236	+	0.964732i	$0.415210\pi$
$602$	1598.74	0.108239
$603$	0	0
$604$	8836.49	0.595284
$605$	7343.10	0.493454
$606$	0	0
$607$	310.614	0.0207701	0.0103850	−	0.999946i	$-0.496694\pi$
$607$	310.614	0.0207701	0.0103850	+	0.999946i	$0.496694\pi$
$608$	23357.5	1.55801
$609$	0	0
$610$	−1760.46	−0.116850
$611$	415.642	0.0275206
$612$	0	0
$613$	18863.2	1.24286	0.621432	−	0.783468i	$-0.286551\pi$
$613$	18863.2	1.24286	0.621432	+	0.783468i	$0.286551\pi$
$614$	−8205.97	−0.539358
$615$	0	0
$616$	−10757.7	−0.703637
$617$	−7238.78	−0.472322	−0.236161	−	0.971714i	$-0.575889\pi$
$617$	−7238.78	−0.472322	−0.236161	+	0.971714i	$0.575889\pi$
$618$	0	0
$619$	−15392.1	−0.999455	−0.499728	−	0.866183i	$-0.666567\pi$
$619$	−15392.1	−0.999455	−0.499728	+	0.866183i	$0.666567\pi$
$620$	−3043.48	−0.197144
$621$	0	0
$622$	5294.67	0.341313
$623$	−5734.67	−0.368787
$624$	0	0
$625$	−9723.19	−0.622284
$626$	15720.2	1.00368
$627$	0	0
$628$	2794.21	0.177550
$629$	−3098.55	−0.196419
$630$	0	0
$631$	−24256.4	−1.53032	−0.765160	−	0.643841i	$-0.777340\pi$
$631$	−24256.4	−1.53032	−0.765160	+	0.643841i	$0.777340\pi$
$632$	−7826.13	−0.492574
$633$	0	0
$634$	6021.33	0.377189
$635$	−10791.1	−0.674379
$636$	0	0
$637$	−226.231	−0.0140716
$638$	287.513	0.0178413
$639$	0	0
$640$	−6826.02	−0.421597
$641$	−22682.4	−1.39766	−0.698829	−	0.715288i	$-0.746295\pi$
$641$	−22682.4	−1.39766	−0.698829	+	0.715288i	$0.746295\pi$
$642$	0	0
$643$	−10117.0	−0.620488	−0.310244	−	0.950657i	$-0.600411\pi$
$643$	−10117.0	−0.620488	−0.310244	+	0.950657i	$0.600411\pi$
$644$	−8935.91	−0.546777
$645$	0	0
$646$	22187.5	1.35133
$647$	−8849.81	−0.537747	−0.268873	−	0.963176i	$-0.586651\pi$
$647$	−8849.81	−0.537747	−0.268873	+	0.963176i	$0.586651\pi$
$648$	0	0
$649$	−14330.4	−0.866748
$650$	−318.442	−0.0192159
$651$	0	0
$652$	−14049.5	−0.843894
$653$	−19223.5	−1.15203	−0.576013	−	0.817440i	$-0.695392\pi$
$653$	−19223.5	−1.15203	−0.576013	+	0.817440i	$0.695392\pi$
$654$	0	0
$655$	6140.44	0.366300
$656$	−770.959	−0.0458855
$657$	0	0
$658$	3185.42	0.188724
$659$	13958.9	0.825133	0.412566	−	0.910928i	$-0.364632\pi$
$659$	13958.9	0.825133	0.412566	+	0.910928i	$0.364632\pi$
$660$	0	0
$661$	−23187.0	−1.36440	−0.682201	−	0.731165i	$-0.738977\pi$
$661$	−23187.0	−1.36440	−0.682201	+	0.731165i	$0.738977\pi$
$662$	883.940	0.0518962
$663$	0	0
$664$	2793.75	0.163281
$665$	24792.3	1.44572
$666$	0	0
$667$	652.311	0.0378674
$668$	6043.38	0.350038
$669$	0	0
$670$	2809.36	0.161993
$671$	2388.46	0.137415
$672$	0	0
$673$	19919.6	1.14093	0.570464	−	0.821322i	$-0.306763\pi$
$673$	19919.6	1.14093	0.570464	+	0.821322i	$0.306763\pi$
$674$	4281.29	0.244672
$675$	0	0
$676$	10047.7	0.571674
$677$	9614.32	0.545802	0.272901	−	0.962042i	$-0.412017\pi$
$677$	9614.32	0.545802	0.272901	+	0.962042i	$0.412017\pi$
$678$	0	0
$679$	15851.9	0.895938
$680$	−19750.2	−1.11380
$681$	0	0
$682$	−3019.87	−0.169555
$683$	−615.384	−0.0344759	−0.0172379	−	0.999851i	$-0.505487\pi$
$683$	−615.384	−0.0344759	−0.0172379	+	0.999851i	$0.505487\pi$
$684$	0	0
$685$	15788.9	0.880676
$686$	10729.7	0.597173
$687$	0	0
$688$	250.084	0.0138581
$689$	−2306.15	−0.127514
$690$	0	0
$691$	−3925.33	−0.216102	−0.108051	−	0.994145i	$-0.534461\pi$
$691$	−3925.33	−0.216102	−0.108051	+	0.994145i	$0.534461\pi$
$692$	2002.36	0.109998
$693$	0	0
$694$	19607.0	1.07244
$695$	−15041.2	−0.820929
$696$	0	0
$697$	12276.3	0.667141
$698$	6527.40	0.353962
$699$	0	0
$700$	3336.97	0.180179
$701$	6707.26	0.361383	0.180692	−	0.983540i	$-0.442166\pi$
$701$	6707.26	0.361383	0.180692	+	0.983540i	$0.442166\pi$
$702$	0	0
$703$	4565.31	0.244928
$704$	−8619.87	−0.461468
$705$	0	0
$706$	20883.1	1.11324
$707$	22574.4	1.20084
$708$	0	0
$709$	−27557.1	−1.45970	−0.729850	−	0.683608i	$-0.760410\pi$
$709$	−27557.1	−1.45970	−0.729850	+	0.683608i	$0.760410\pi$
$710$	−18778.3	−0.992590
$711$	0	0
$712$	−6730.94	−0.354287
$713$	−6851.49	−0.359874
$714$	0	0
$715$	−1046.10	−0.0547161
$716$	20053.7	1.04671
$717$	0	0
$718$	8458.37	0.439643
$719$	4677.86	0.242635	0.121318	−	0.992614i	$-0.461288\pi$
$719$	4677.86	0.242635	0.121318	+	0.992614i	$0.461288\pi$
$720$	0	0
$721$	33465.9	1.72862
$722$	−20081.6	−1.03512
$723$	0	0
$724$	4316.71	0.221588
$725$	−243.595	−0.0124784
$726$	0	0
$727$	−28082.4	−1.43262	−0.716312	−	0.697780i	$-0.754171\pi$
$727$	−28082.4	−1.43262	−0.716312	+	0.697780i	$0.754171\pi$
$728$	−2174.33	−0.110695
$729$	0	0
$730$	−9249.53	−0.468960
$731$	−3982.19	−0.201487
$732$	0	0
$733$	−279.921	−0.0141052	−0.00705260	−	0.999975i	$-0.502245\pi$
$733$	−279.921	−0.0141052	−0.00705260	+	0.999975i	$0.502245\pi$
$734$	12252.3	0.616133
$735$	0	0
$736$	−17136.7	−0.858243
$737$	−3811.54	−0.190502
$738$	0	0
$739$	23887.2	1.18904	0.594522	−	0.804079i	$-0.297341\pi$
$739$	23887.2	1.18904	0.594522	+	0.804079i	$0.297341\pi$
$740$	−1487.84	−0.0739111
$741$	0	0
$742$	−17674.0	−0.874437
$743$	−17811.6	−0.879469	−0.439734	−	0.898128i	$-0.644927\pi$
$743$	−17811.6	−0.879469	−0.439734	+	0.898128i	$0.644927\pi$
$744$	0	0
$745$	8960.96	0.440677
$746$	−10555.9	−0.518070
$747$	0	0
$748$	9810.42	0.479552
$749$	31141.9	1.51923
$750$	0	0
$751$	4143.90	0.201349	0.100675	−	0.994919i	$-0.467900\pi$
$751$	4143.90	0.201349	0.100675	+	0.994919i	$0.467900\pi$
$752$	498.283	0.0241629
$753$	0	0
$754$	58.1117	0.00280677
$755$	17987.0	0.867038
$756$	0	0
$757$	23473.1	1.12701	0.563503	−	0.826114i	$-0.309453\pi$
$757$	23473.1	1.12701	0.563503	+	0.826114i	$0.309453\pi$
$758$	9668.48	0.463292
$759$	0	0
$760$	29099.4	1.38888
$761$	−20048.4	−0.954996	−0.477498	−	0.878633i	$-0.658456\pi$
$761$	−20048.4	−0.954996	−0.477498	+	0.878633i	$0.658456\pi$
$762$	0	0
$763$	15205.6	0.721467
$764$	−12315.1	−0.583172
$765$	0	0
$766$	−19710.4	−0.929719
$767$	−2896.45	−0.136356
$768$	0	0
$769$	−31240.6	−1.46497	−0.732487	−	0.680781i	$-0.761641\pi$
$769$	−31240.6	−1.46497	−0.732487	+	0.680781i	$0.761641\pi$
$770$	−8017.17	−0.375219
$771$	0	0
$772$	22570.8	1.05226
$773$	21150.0	0.984103	0.492052	−	0.870566i	$-0.336247\pi$
$773$	21150.0	0.984103	0.492052	+	0.870566i	$0.336247\pi$
$774$	0	0
$775$	2558.58	0.118589
$776$	18605.9	0.860711
$777$	0	0
$778$	5266.14	0.242674
$779$	−18087.5	−0.831904
$780$	0	0
$781$	25477.1	1.16728
$782$	−16278.3	−0.744388
$783$	0	0
$784$	−271.212	−0.0123548
$785$	5687.72	0.258603
$786$	0	0
$787$	−12155.1	−0.550552	−0.275276	−	0.961365i	$-0.588769\pi$
$787$	−12155.1	−0.550552	−0.275276	+	0.961365i	$0.588769\pi$
$788$	9416.90	0.425715
$789$	0	0
$790$	−5832.42	−0.262668
$791$	22854.1	1.02731
$792$	0	0
$793$	482.751	0.0216179
$794$	8225.44	0.367645
$795$	0	0
$796$	6842.74	0.304692
$797$	28448.1	1.26434	0.632172	−	0.774828i	$-0.282164\pi$
$797$	28448.1	1.26434	0.632172	+	0.774828i	$0.282164\pi$
$798$	0	0
$799$	−7934.37	−0.351311
$800$	6399.41	0.282817
$801$	0	0
$802$	−19559.4	−0.861179
$803$	12549.1	0.551492
$804$	0	0
$805$	−18189.4	−0.796387
$806$	−610.372	−0.0266742
$807$	0	0
$808$	26496.2	1.15363
$809$	11107.8	0.482730	0.241365	−	0.970434i	$-0.422405\pi$
$809$	11107.8	0.482730	0.241365	+	0.970434i	$0.422405\pi$
$810$	0	0
$811$	−131.869	−0.00570968	−0.00285484	−	0.999996i	$-0.500909\pi$
$811$	−131.869	−0.00570968	−0.00285484	+	0.999996i	$0.500909\pi$
$812$	−608.956	−0.0263179
$813$	0	0
$814$	−1476.30	−0.0635679
$815$	−28598.2	−1.22914
$816$	0	0
$817$	5867.25	0.251247
$818$	11376.5	0.486272
$819$	0	0
$820$	5894.75	0.251041
$821$	21629.0	0.919436	0.459718	−	0.888065i	$-0.347950\pi$
$821$	21629.0	0.919436	0.459718	+	0.888065i	$0.347950\pi$
$822$	0	0
$823$	−41312.9	−1.74979	−0.874895	−	0.484313i	$-0.839069\pi$
$823$	−41312.9	−1.74979	−0.874895	+	0.484313i	$0.839069\pi$
$824$	39279.9	1.66066
$825$	0	0
$826$	−22198.0	−0.935067
$827$	−10466.4	−0.440086	−0.220043	−	0.975490i	$-0.570620\pi$
$827$	−10466.4	−0.440086	−0.220043	+	0.975490i	$0.570620\pi$
$828$	0	0
$829$	−36170.8	−1.51540	−0.757699	−	0.652604i	$-0.773677\pi$
$829$	−36170.8	−1.51540	−0.757699	+	0.652604i	$0.773677\pi$
$830$	2082.04	0.0870706
$831$	0	0
$832$	−1742.24	−0.0725975
$833$	4318.62	0.179629
$834$	0	0
$835$	12301.5	0.509834
$836$	−14454.4	−0.597986
$837$	0	0
$838$	−30525.8	−1.25835
$839$	−378.230	−0.0155637	−0.00778185	−	0.999970i	$-0.502477\pi$
$839$	−378.230	−0.0155637	−0.00778185	+	0.999970i	$0.502477\pi$
$840$	0	0
$841$	−24344.5	−0.998177
$842$	13535.8	0.554006
$843$	0	0
$844$	−14356.0	−0.585490
$845$	20452.6	0.832650
$846$	0	0
$847$	−15432.1	−0.626037
$848$	−2764.67	−0.111957
$849$	0	0
$850$	6078.87	0.245298
$851$	−3349.43	−0.134920
$852$	0	0
$853$	−17315.2	−0.695029	−0.347515	−	0.937675i	$-0.612974\pi$
$853$	−17315.2	−0.695029	−0.347515	+	0.937675i	$0.612974\pi$
$854$	3699.74	0.148246
$855$	0	0
$856$	36552.1	1.45949
$857$	−2859.56	−0.113980	−0.0569899	−	0.998375i	$-0.518150\pi$
$857$	−2859.56	−0.113980	−0.0569899	+	0.998375i	$0.518150\pi$
$858$	0	0
$859$	27598.3	1.09621	0.548104	−	0.836410i	$-0.315350\pi$
$859$	27598.3	1.09621	0.548104	+	0.836410i	$0.315350\pi$
$860$	−1912.15	−0.0758182
$861$	0	0
$862$	−27802.3	−1.09855
$863$	−25077.1	−0.989147	−0.494573	−	0.869136i	$-0.664676\pi$
$863$	−25077.1	−0.989147	−0.494573	+	0.869136i	$0.664676\pi$
$864$	0	0
$865$	4075.88	0.160213
$866$	−16716.2	−0.655936
$867$	0	0
$868$	6396.12	0.250113
$869$	7913.00	0.308895
$870$	0	0
$871$	−770.382	−0.0299694
$872$	17847.2	0.693100
$873$	0	0
$874$	23984.0	0.928228
$875$	30031.9	1.16030
$876$	0	0
$877$	−7281.42	−0.280360	−0.140180	−	0.990126i	$-0.544768\pi$
$877$	−7281.42	−0.280360	−0.140180	+	0.990126i	$0.544768\pi$
$878$	33399.0	1.28379
$879$	0	0
$880$	−1254.10	−0.0480404
$881$	−25566.0	−0.977686	−0.488843	−	0.872372i	$-0.662581\pi$
$881$	−25566.0	−0.977686	−0.488843	+	0.872372i	$0.662581\pi$
$882$	0	0
$883$	51507.2	1.96303	0.981515	−	0.191383i	$-0.0612974\pi$
$883$	51507.2	1.96303	0.981515	+	0.191383i	$0.0612974\pi$
$884$	1982.87	0.0754424
$885$	0	0
$886$	29371.3	1.11371
$887$	37615.8	1.42392	0.711959	−	0.702221i	$-0.247808\pi$
$887$	37615.8	1.42392	0.711959	+	0.702221i	$0.247808\pi$
$888$	0	0
$889$	22678.3	0.855574
$890$	−5016.23	−0.188926
$891$	0	0
$892$	21473.5	0.806040
$893$	11690.3	0.438074
$894$	0	0
$895$	40820.0	1.52454
$896$	14345.4	0.534874
$897$	0	0
$898$	25923.7	0.963347
$899$	−466.909	−0.0173218
$900$	0	0
$901$	44023.0	1.62777
$902$	5849.03	0.215910
$903$	0	0
$904$	26824.5	0.986914
$905$	8786.83	0.322745
$906$	0	0
$907$	−19812.1	−0.725305	−0.362652	−	0.931924i	$-0.618129\pi$
$907$	−19812.1	−0.725305	−0.362652	+	0.931924i	$0.618129\pi$
$908$	−1895.37	−0.0692731
$909$	0	0
$910$	−1620.42	−0.0590289
$911$	−1762.20	−0.0640882	−0.0320441	−	0.999486i	$-0.510202\pi$
$911$	−1762.20	−0.0640882	−0.0320441	+	0.999486i	$0.510202\pi$
$912$	0	0
$913$	−2824.76	−0.102394
$914$	−25905.9	−0.937519
$915$	0	0
$916$	−17401.4	−0.627685
$917$	−12904.6	−0.464720
$918$	0	0
$919$	−37762.2	−1.35545	−0.677726	−	0.735315i	$-0.737034\pi$
$919$	−37762.2	−1.35545	−0.677726	+	0.735315i	$0.737034\pi$
$920$	−21349.4	−0.765075
$921$	0	0
$922$	−7651.27	−0.273298
$923$	5149.39	0.183634
$924$	0	0
$925$	1250.79	0.0444603
$926$	23082.5	0.819154
$927$	0	0
$928$	−1167.81	−0.0413097
$929$	18625.0	0.657766	0.328883	−	0.944371i	$-0.393328\pi$
$929$	18625.0	0.657766	0.328883	+	0.944371i	$0.393328\pi$
$930$	0	0
$931$	−6362.93	−0.223992
$932$	5936.78	0.208654
$933$	0	0
$934$	35416.1	1.24074
$935$	19969.5	0.698472
$936$	0	0
$937$	28383.2	0.989582	0.494791	−	0.869012i	$-0.335245\pi$
$937$	28383.2	0.989582	0.494791	+	0.869012i	$0.335245\pi$
$938$	−5904.09	−0.205518
$939$	0	0
$940$	−3809.88	−0.132196
$941$	9483.18	0.328526	0.164263	−	0.986417i	$-0.447475\pi$
$941$	9483.18	0.328526	0.164263	+	0.986417i	$0.447475\pi$
$942$	0	0
$943$	13270.3	0.458261
$944$	−3472.34	−0.119719
$945$	0	0
$946$	−1897.31	−0.0652082
$947$	−22338.0	−0.766513	−0.383256	−	0.923642i	$-0.625197\pi$
$947$	−22338.0	−0.766513	−0.383256	+	0.923642i	$0.625197\pi$
$948$	0	0
$949$	2536.40	0.0867599
$950$	−8956.43	−0.305879
$951$	0	0
$952$	41506.7	1.41307
$953$	52132.5	1.77202	0.886011	−	0.463664i	$-0.153465\pi$
$953$	52132.5	1.77202	0.886011	+	0.463664i	$0.153465\pi$
$954$	0	0
$955$	−25067.8	−0.849397
$956$	1104.34	0.0373608
$957$	0	0
$958$	−14165.9	−0.477746
$959$	−33181.7	−1.11730
$960$	0	0
$961$	−24886.9	−0.835382
$962$	−298.388	−0.0100004
$963$	0	0
$964$	17546.1	0.586227
$965$	45943.8	1.53262
$966$	0	0
$967$	−34241.1	−1.13870	−0.569348	−	0.822097i	$-0.692804\pi$
$967$	−34241.1	−1.13870	−0.569348	+	0.822097i	$0.692804\pi$
$968$	−18113.1	−0.601423
$969$	0	0
$970$	13866.0	0.458980
$971$	32908.5	1.08763	0.543813	−	0.839207i	$-0.316980\pi$
$971$	32908.5	1.08763	0.543813	+	0.839207i	$0.316980\pi$
$972$	0	0
$973$	31610.3	1.04150
$974$	17762.6	0.584345
$975$	0	0
$976$	578.736	0.0189804
$977$	9026.06	0.295567	0.147784	−	0.989020i	$-0.452786\pi$
$977$	9026.06	0.295567	0.147784	+	0.989020i	$0.452786\pi$
$978$	0	0
$979$	6805.66	0.222175
$980$	2073.69	0.0675934
$981$	0	0
$982$	−947.351	−0.0307853
$983$	−32325.0	−1.04884	−0.524419	−	0.851460i	$-0.675718\pi$
$983$	−32325.0	−1.04884	−0.524419	+	0.851460i	$0.675718\pi$
$984$	0	0
$985$	19168.4	0.620058
$986$	−1109.32	−0.0358295
$987$	0	0
$988$	−2921.50	−0.0940742
$989$	−4304.63	−0.138402
$990$	0	0
$991$	−34414.8	−1.10315	−0.551575	−	0.834125i	$-0.685973\pi$
$991$	−34414.8	−1.10315	−0.551575	+	0.834125i	$0.685973\pi$
$992$	12266.0	0.392588
$993$	0	0
$994$	39464.2	1.25928
$995$	13928.6	0.443787
$996$	0	0
$997$	11741.3	0.372971	0.186486	−	0.982458i	$-0.440290\pi$
$997$	11741.3	0.372971	0.186486	+	0.982458i	$0.440290\pi$
$998$	−20924.5	−0.663680
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.9		22
3.2	odd	2		239.4.a.a.1.14	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.14	✓	22	3.2	odd	2
2151.4.a.a.1.9		22	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83829 q^{2} -4.62067 q^{4} -9.40555 q^{5} +19.7665 q^{7} +23.2005 q^{8} +O(q^{10})\)	\(q-1.83829 q^{2} -4.62067 q^{4} -9.40555 q^{5} +19.7665 q^{7} +23.2005 q^{8} +17.2902 q^{10} -23.4581 q^{11} -4.74130 q^{13} -36.3367 q^{14} -5.68401 q^{16} +90.5088 q^{17} -133.353 q^{19} +43.4600 q^{20} +43.1228 q^{22} +97.8372 q^{23} -36.5357 q^{25} +8.71592 q^{26} -91.3346 q^{28} +6.66731 q^{29} -70.0295 q^{31} -175.155 q^{32} -166.382 q^{34} -185.915 q^{35} -34.2348 q^{37} +245.142 q^{38} -218.214 q^{40} +135.636 q^{41} -43.9979 q^{43} +108.392 q^{44} -179.854 q^{46} -87.6641 q^{47} +47.7149 q^{49} +67.1633 q^{50} +21.9080 q^{52} +486.395 q^{53} +220.636 q^{55} +458.593 q^{56} -12.2565 q^{58} +610.897 q^{59} -101.818 q^{61} +128.735 q^{62} +367.459 q^{64} +44.5946 q^{65} +162.483 q^{67} -418.211 q^{68} +341.766 q^{70} -1086.07 q^{71} -534.959 q^{73} +62.9336 q^{74} +616.181 q^{76} -463.684 q^{77} -337.326 q^{79} +53.4612 q^{80} -249.340 q^{82} +120.417 q^{83} -851.285 q^{85} +80.8810 q^{86} -544.239 q^{88} -290.120 q^{89} -93.7190 q^{91} -452.073 q^{92} +161.152 q^{94} +1254.26 q^{95} +801.960 q^{97} -87.7140 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

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