LMFDB - Embedded newform 261.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,4,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, 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("261.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	261.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:	$15.3994985115$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{2} -7.82843 q^{4} -0.656854 q^{5} +6.14214 q^{7} +6.55635 q^{8} +O(q^{10})$	\(q-0.414214 q^{2} -7.82843 q^{4} -0.656854 q^{5} +6.14214 q^{7} +6.55635 q^{8} +0.272078 q^{10} +65.3259 q^{11} -49.7696 q^{13} -2.54416 q^{14} +59.9117 q^{16} -55.4558 q^{17} -64.7452 q^{19} +5.14214 q^{20} -27.0589 q^{22} -93.8823 q^{23} -124.569 q^{25} +20.6152 q^{26} -48.0833 q^{28} -29.0000 q^{29} -236.095 q^{31} -77.2670 q^{32} +22.9706 q^{34} -4.03449 q^{35} +76.8040 q^{37} +26.8183 q^{38} -4.30657 q^{40} -215.161 q^{41} +80.8305 q^{43} -511.399 q^{44} +38.8873 q^{46} +357.742 q^{47} -305.274 q^{49} +51.5980 q^{50} +389.617 q^{52} -328.466 q^{53} -42.9096 q^{55} +40.2700 q^{56} +12.0122 q^{58} +99.2750 q^{59} -725.730 q^{61} +97.7939 q^{62} -447.288 q^{64} +32.6913 q^{65} +844.479 q^{67} +434.132 q^{68} +1.67114 q^{70} +378.083 q^{71} -581.097 q^{73} -31.8133 q^{74} +506.853 q^{76} +401.241 q^{77} -353.247 q^{79} -39.3532 q^{80} +89.1228 q^{82} -696.510 q^{83} +36.4264 q^{85} -33.4811 q^{86} +428.299 q^{88} -1118.22 q^{89} -305.691 q^{91} +734.950 q^{92} -148.182 q^{94} +42.5281 q^{95} -805.415 q^{97} +126.449 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 10 q^{4} + 10 q^{5} - 16 q^{7} - 18 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 10 * q^4 + 10 * q^5 - 16 * q^7 - 18 * q^8	$ 2 q + 2 q^{2} - 10 q^{4} + 10 q^{5} - 16 q^{7} - 18 q^{8} + 26 q^{10} + 26 q^{11} - 26 q^{13} - 56 q^{14} + 18 q^{16} - 60 q^{17} - 220 q^{19} - 18 q^{20} - 122 q^{22} - 52 q^{23} - 136 q^{25} + 78 q^{26} - 58 q^{29} - 294 q^{31} + 18 q^{32} + 12 q^{34} - 240 q^{35} + 312 q^{37} - 348 q^{38} - 266 q^{40} - 40 q^{41} - 322 q^{43} - 426 q^{44} + 140 q^{46} + 130 q^{47} - 158 q^{49} + 24 q^{50} + 338 q^{52} - 1002 q^{53} - 462 q^{55} + 584 q^{56} - 58 q^{58} + 900 q^{59} - 948 q^{61} - 42 q^{62} + 118 q^{64} + 286 q^{65} + 320 q^{67} + 444 q^{68} - 568 q^{70} + 660 q^{71} + 648 q^{73} + 536 q^{74} + 844 q^{76} + 1272 q^{77} + 258 q^{79} - 486 q^{80} + 512 q^{82} - 1212 q^{83} - 12 q^{85} - 1006 q^{86} + 1394 q^{88} - 760 q^{89} - 832 q^{91} + 644 q^{92} - 698 q^{94} - 1612 q^{95} + 24 q^{97} + 482 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 10 * q^4 + 10 * q^5 - 16 * q^7 - 18 * q^8 + 26 * q^10 + 26 * q^11 - 26 * q^13 - 56 * q^14 + 18 * q^16 - 60 * q^17 - 220 * q^19 - 18 * q^20 - 122 * q^22 - 52 * q^23 - 136 * q^25 + 78 * q^26 - 58 * q^29 - 294 * q^31 + 18 * q^32 + 12 * q^34 - 240 * q^35 + 312 * q^37 - 348 * q^38 - 266 * q^40 - 40 * q^41 - 322 * q^43 - 426 * q^44 + 140 * q^46 + 130 * q^47 - 158 * q^49 + 24 * q^50 + 338 * q^52 - 1002 * q^53 - 462 * q^55 + 584 * q^56 - 58 * q^58 + 900 * q^59 - 948 * q^61 - 42 * q^62 + 118 * q^64 + 286 * q^65 + 320 * q^67 + 444 * q^68 - 568 * q^70 + 660 * q^71 + 648 * q^73 + 536 * q^74 + 844 * q^76 + 1272 * q^77 + 258 * q^79 - 486 * q^80 + 512 * q^82 - 1212 * q^83 - 12 * q^85 - 1006 * q^86 + 1394 * q^88 - 760 * q^89 - 832 * q^91 + 644 * q^92 - 698 * q^94 - 1612 * q^95 + 24 * q^97 + 482 * 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 $

$2$	−0.414214	−0.146447	−0.0732233	−	0.997316i	$-0.523329\pi$
$2$	−0.414214	−0.146447	−0.0732233	+	0.997316i	$0.523329\pi$
$3$	0	0
$3$	0	0
$4$	−7.82843	−0.978553
$5$	−0.656854	−0.0587508	−0.0293754	−	0.999568i	$-0.509352\pi$
$5$	−0.656854	−0.0587508	−0.0293754	+	0.999568i	$0.509352\pi$
$6$	0	0
$7$	6.14214	0.331644	0.165822	−	0.986156i	$-0.446972\pi$
$7$	6.14214	0.331644	0.165822	+	0.986156i	$0.446972\pi$
$8$	6.55635	0.289752
$9$	0	0
$10$	0.272078	0.00860386
$11$	65.3259	1.79059	0.895295	−	0.445473i	$-0.146964\pi$
$11$	65.3259	1.79059	0.895295	+	0.445473i	$0.146964\pi$
$12$	0	0
$13$	−49.7696	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−49.7696	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	−2.54416	−0.0485682
$15$	0	0
$16$	59.9117	0.936120
$17$	−55.4558	−0.791178	−0.395589	−	0.918428i	$-0.629459\pi$
$17$	−55.4558	−0.791178	−0.395589	+	0.918428i	$0.629459\pi$
$18$	0	0
$19$	−64.7452	−0.781766	−0.390883	−	0.920440i	$-0.627830\pi$
$19$	−64.7452	−0.781766	−0.390883	+	0.920440i	$0.627830\pi$
$20$	5.14214	0.0574908
$21$	0	0
$22$	−27.0589	−0.262226
$23$	−93.8823	−0.851122	−0.425561	−	0.904930i	$-0.639923\pi$
$23$	−93.8823	−0.851122	−0.425561	+	0.904930i	$0.639923\pi$
$24$	0	0
$25$	−124.569	−0.996548
$26$	20.6152	0.155499
$27$	0	0
$28$	−48.0833	−0.324532
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−236.095	−1.36787	−0.683935	−	0.729543i	$-0.739733\pi$
$31$	−236.095	−1.36787	−0.683935	+	0.729543i	$0.739733\pi$
$32$	−77.2670	−0.426844
$33$	0	0
$34$	22.9706	0.115865
$35$	−4.03449	−0.0194844
$36$	0	0
$37$	76.8040	0.341257	0.170628	−	0.985335i	$-0.445420\pi$
$37$	76.8040	0.341257	0.170628	+	0.985335i	$0.445420\pi$
$38$	26.8183	0.114487
$39$	0	0
$40$	−4.30657	−0.0170232
$41$	−215.161	−0.819575	−0.409788	−	0.912181i	$-0.634397\pi$
$41$	−215.161	−0.819575	−0.409788	+	0.912181i	$0.634397\pi$
$42$	0	0
$43$	80.8305	0.286664	0.143332	−	0.989675i	$-0.454218\pi$
$43$	80.8305	0.286664	0.143332	+	0.989675i	$0.454218\pi$
$44$	−511.399	−1.75219
$45$	0	0
$46$	38.8873	0.124644
$47$	357.742	1.11026	0.555128	−	0.831765i	$-0.312669\pi$
$47$	357.742	1.11026	0.555128	+	0.831765i	$0.312669\pi$
$48$	0	0
$49$	−305.274	−0.890012
$50$	51.5980	0.145941
$51$	0	0
$52$	389.617	1.03904
$53$	−328.466	−0.851288	−0.425644	−	0.904891i	$-0.639952\pi$
$53$	−328.466	−0.851288	−0.425644	+	0.904891i	$0.639952\pi$
$54$	0	0
$55$	−42.9096	−0.105199
$56$	40.2700	0.0960947
$57$	0	0
$58$	12.0122	0.0271945
$59$	99.2750	0.219059	0.109530	−	0.993984i	$-0.465066\pi$
$59$	99.2750	0.219059	0.109530	+	0.993984i	$0.465066\pi$
$60$	0	0
$61$	−725.730	−1.52328	−0.761641	−	0.647999i	$-0.775606\pi$
$61$	−725.730	−1.52328	−0.761641	+	0.647999i	$0.775606\pi$
$62$	97.7939	0.200320
$63$	0	0
$64$	−447.288	−0.873610
$65$	32.6913	0.0623825
$66$	0	0
$67$	844.479	1.53984	0.769922	−	0.638138i	$-0.220295\pi$
$67$	844.479	1.53984	0.769922	+	0.638138i	$0.220295\pi$
$68$	434.132	0.774209
$69$	0	0
$70$	1.67114	0.00285342
$71$	378.083	0.631975	0.315988	−	0.948763i	$-0.397664\pi$
$71$	378.083	0.631975	0.315988	+	0.948763i	$0.397664\pi$
$72$	0	0
$73$	−581.097	−0.931674	−0.465837	−	0.884870i	$-0.654247\pi$
$73$	−581.097	−0.931674	−0.465837	+	0.884870i	$0.654247\pi$
$74$	−31.8133	−0.0499759
$75$	0	0
$76$	506.853	0.765000
$77$	401.241	0.593839
$78$	0	0
$79$	−353.247	−0.503081	−0.251540	−	0.967847i	$-0.580937\pi$
$79$	−353.247	−0.503081	−0.251540	+	0.967847i	$0.580937\pi$
$80$	−39.3532	−0.0549978
$81$	0	0
$82$	89.1228	0.120024
$83$	−696.510	−0.921107	−0.460553	−	0.887632i	$-0.652349\pi$
$83$	−696.510	−0.921107	−0.460553	+	0.887632i	$0.652349\pi$
$84$	0	0
$85$	36.4264	0.0464823
$86$	−33.4811	−0.0419809
$87$	0	0
$88$	428.299	0.518828
$89$	−1118.22	−1.33181	−0.665905	−	0.746037i	$-0.731954\pi$
$89$	−1118.22	−1.33181	−0.665905	+	0.746037i	$0.731954\pi$
$90$	0	0
$91$	−305.691	−0.352145
$92$	734.950	0.832868
$93$	0	0
$94$	−148.182	−0.162593
$95$	42.5281	0.0459294
$96$	0	0
$97$	−805.415	−0.843068	−0.421534	−	0.906813i	$-0.638508\pi$
$97$	−805.415	−0.843068	−0.421534	+	0.906813i	$0.638508\pi$
$98$	126.449	0.130339
$99$	0	0
$100$	975.176	0.975176
$101$	1373.99	1.35363	0.676817	−	0.736151i	$-0.263359\pi$
$101$	1373.99	1.35363	0.676817	+	0.736151i	$0.263359\pi$
$102$	0	0
$103$	−634.672	−0.607147	−0.303573	−	0.952808i	$-0.598180\pi$
$103$	−634.672	−0.607147	−0.303573	+	0.952808i	$0.598180\pi$
$104$	−326.307	−0.307663
$105$	0	0
$106$	136.055	0.124668
$107$	180.956	0.163493	0.0817463	−	0.996653i	$-0.473950\pi$
$107$	180.956	0.163493	0.0817463	+	0.996653i	$0.473950\pi$
$108$	0	0
$109$	1038.14	0.912251	0.456125	−	0.889916i	$-0.349237\pi$
$109$	1038.14	0.912251	0.456125	+	0.889916i	$0.349237\pi$
$110$	17.7737	0.0154060
$111$	0	0
$112$	367.986	0.310459
$113$	184.765	0.153816	0.0769082	−	0.997038i	$-0.475495\pi$
$113$	184.765	0.153816	0.0769082	+	0.997038i	$0.475495\pi$
$114$	0	0
$115$	61.6670	0.0500041
$116$	227.024	0.181713
$117$	0	0
$118$	−41.1211	−0.0320805
$119$	−340.617	−0.262389
$120$	0	0
$121$	2936.47	2.20622
$122$	300.607	0.223079
$123$	0	0
$124$	1848.26	1.33853
$125$	163.930	0.117299
$126$	0	0
$127$	1999.58	1.39712	0.698558	−	0.715554i	$-0.253825\pi$
$127$	1999.58	1.39712	0.698558	+	0.715554i	$0.253825\pi$
$128$	803.409	0.554781
$129$	0	0
$130$	−13.5412	−0.00913570
$131$	−561.468	−0.374471	−0.187236	−	0.982315i	$-0.559953\pi$
$131$	−561.468	−0.374471	−0.187236	+	0.982315i	$0.559953\pi$
$132$	0	0
$133$	−397.674	−0.259268
$134$	−349.795	−0.225505
$135$	0	0
$136$	−363.588	−0.229246
$137$	250.489	0.156210	0.0781050	−	0.996945i	$-0.475113\pi$
$137$	250.489	0.156210	0.0781050	+	0.996945i	$0.475113\pi$
$138$	0	0
$139$	−242.244	−0.147819	−0.0739096	−	0.997265i	$-0.523548\pi$
$139$	−242.244	−0.147819	−0.0739096	+	0.997265i	$0.523548\pi$
$140$	31.5837	0.0190665
$141$	0	0
$142$	−156.607	−0.0925506
$143$	−3251.24	−1.90128
$144$	0	0
$145$	19.0488	0.0109098
$146$	240.698	0.136441
$147$	0	0
$148$	−601.255	−0.333938
$149$	1632.63	0.897652	0.448826	−	0.893619i	$-0.351842\pi$
$149$	1632.63	0.897652	0.448826	+	0.893619i	$0.351842\pi$
$150$	0	0
$151$	−121.582	−0.0655245	−0.0327623	−	0.999463i	$-0.510430\pi$
$151$	−121.582	−0.0655245	−0.0327623	+	0.999463i	$0.510430\pi$
$152$	−424.492	−0.226519
$153$	0	0
$154$	−166.199	−0.0869657
$155$	155.080	0.0803635
$156$	0	0
$157$	−753.163	−0.382860	−0.191430	−	0.981506i	$-0.561312\pi$
$157$	−753.163	−0.382860	−0.191430	+	0.981506i	$0.561312\pi$
$158$	146.320	0.0736745
$159$	0	0
$160$	50.7532	0.0250774
$161$	−576.638	−0.282270
$162$	0	0
$163$	537.917	0.258484	0.129242	−	0.991613i	$-0.458746\pi$
$163$	537.917	0.258484	0.129242	+	0.991613i	$0.458746\pi$
$164$	1684.38	0.801998
$165$	0	0
$166$	288.504	0.134893
$167$	−484.613	−0.224554	−0.112277	−	0.993677i	$-0.535814\pi$
$167$	−484.613	−0.224554	−0.112277	+	0.993677i	$0.535814\pi$
$168$	0	0
$169$	280.008	0.127450
$170$	−15.0883	−0.00680718
$171$	0	0
$172$	−632.776	−0.280516
$173$	3269.70	1.43694	0.718469	−	0.695559i	$-0.244843\pi$
$173$	3269.70	1.43694	0.718469	+	0.695559i	$0.244843\pi$
$174$	0	0
$175$	−765.117	−0.330499
$176$	3913.79	1.67621
$177$	0	0
$178$	463.182	0.195039
$179$	562.267	0.234781	0.117390	−	0.993086i	$-0.462547\pi$
$179$	562.267	0.234781	0.117390	+	0.993086i	$0.462547\pi$
$180$	0	0
$181$	−1507.32	−0.618998	−0.309499	−	0.950900i	$-0.600161\pi$
$181$	−1507.32	−0.618998	−0.309499	+	0.950900i	$0.600161\pi$
$182$	126.621	0.0515704
$183$	0	0
$184$	−615.525	−0.246615
$185$	−50.4491	−0.0200491
$186$	0	0
$187$	−3622.70	−1.41668
$188$	−2800.56	−1.08645
$189$	0	0
$190$	−17.6157	−0.00672621
$191$	−4532.00	−1.71688	−0.858439	−	0.512915i	$-0.828566\pi$
$191$	−4532.00	−1.71688	−0.858439	+	0.512915i	$0.828566\pi$
$192$	0	0
$193$	−2935.17	−1.09471	−0.547353	−	0.836902i	$-0.684364\pi$
$193$	−2935.17	−1.09471	−0.547353	+	0.836902i	$0.684364\pi$
$194$	333.614	0.123464
$195$	0	0
$196$	2389.82	0.870924
$197$	−2682.20	−0.970043	−0.485022	−	0.874502i	$-0.661188\pi$
$197$	−2682.20	−0.970043	−0.485022	+	0.874502i	$0.661188\pi$
$198$	0	0
$199$	−648.376	−0.230966	−0.115483	−	0.993309i	$-0.536842\pi$
$199$	−648.376	−0.230966	−0.115483	+	0.993309i	$0.536842\pi$
$200$	−816.715	−0.288752
$201$	0	0
$202$	−569.125	−0.198235
$203$	−178.122	−0.0615848
$204$	0	0
$205$	141.330	0.0481507
$206$	262.890	0.0889145
$207$	0	0
$208$	−2981.78	−0.993986
$209$	−4229.54	−1.39982
$210$	0	0
$211$	−4949.57	−1.61489	−0.807446	−	0.589941i	$-0.799151\pi$
$211$	−4949.57	−1.61489	−0.807446	+	0.589941i	$0.799151\pi$
$212$	2571.37	0.833031
$213$	0	0
$214$	−74.9545	−0.0239429
$215$	−53.0939	−0.0168417
$216$	0	0
$217$	−1450.13	−0.453646
$218$	−430.010	−0.133596
$219$	0	0
$220$	335.915	0.102943
$221$	2760.01	0.840084
$222$	0	0
$223$	2216.94	0.665729	0.332864	−	0.942975i	$-0.391985\pi$
$223$	2216.94	0.665729	0.332864	+	0.942975i	$0.391985\pi$
$224$	−474.585	−0.141560
$225$	0	0
$226$	−76.5323	−0.0225259
$227$	−4546.09	−1.32923	−0.664613	−	0.747187i	$-0.731404\pi$
$227$	−4546.09	−1.32923	−0.664613	+	0.747187i	$0.731404\pi$
$228$	0	0
$229$	3339.05	0.963539	0.481770	−	0.876298i	$-0.339994\pi$
$229$	3339.05	0.963539	0.481770	+	0.876298i	$0.339994\pi$
$230$	−25.5433	−0.00732293
$231$	0	0
$232$	−190.134	−0.0538057
$233$	3995.35	1.12336	0.561682	−	0.827353i	$-0.310154\pi$
$233$	3995.35	1.12336	0.561682	+	0.827353i	$0.310154\pi$
$234$	0	0
$235$	−234.984	−0.0652285
$236$	−777.167	−0.214361
$237$	0	0
$238$	141.088	0.0384260
$239$	1400.04	0.378915	0.189458	−	0.981889i	$-0.439327\pi$
$239$	1400.04	0.378915	0.189458	+	0.981889i	$0.439327\pi$
$240$	0	0
$241$	−2040.94	−0.545513	−0.272756	−	0.962083i	$-0.587935\pi$
$241$	−2040.94	−0.545513	−0.272756	+	0.962083i	$0.587935\pi$
$242$	−1216.33	−0.323093
$243$	0	0
$244$	5681.32	1.49061
$245$	200.521	0.0522890
$246$	0	0
$247$	3222.34	0.830091
$248$	−1547.92	−0.396344
$249$	0	0
$250$	−67.9021	−0.0171780
$251$	−802.648	−0.201843	−0.100922	−	0.994894i	$-0.532179\pi$
$251$	−802.648	−0.201843	−0.100922	+	0.994894i	$0.532179\pi$
$252$	0	0
$253$	−6132.94	−1.52401
$254$	−828.252	−0.204603
$255$	0	0
$256$	3245.52	0.792364
$257$	4464.10	1.08351	0.541756	−	0.840536i	$-0.317760\pi$
$257$	4464.10	1.08351	0.541756	+	0.840536i	$0.317760\pi$
$258$	0	0
$259$	471.741	0.113176
$260$	−255.922	−0.0610446
$261$	0	0
$262$	232.568	0.0548400
$263$	3815.21	0.894509	0.447255	−	0.894407i	$-0.352402\pi$
$263$	3815.21	0.894509	0.447255	+	0.894407i	$0.352402\pi$
$264$	0	0
$265$	215.754	0.0500139
$266$	164.722	0.0379690
$267$	0	0
$268$	−6610.95	−1.50682
$269$	−4523.98	−1.02540	−0.512699	−	0.858569i	$-0.671354\pi$
$269$	−4523.98	−1.02540	−0.512699	+	0.858569i	$0.671354\pi$
$270$	0	0
$271$	3962.65	0.888242	0.444121	−	0.895967i	$-0.353516\pi$
$271$	3962.65	0.888242	0.444121	+	0.895967i	$0.353516\pi$
$272$	−3322.45	−0.740637
$273$	0	0
$274$	−103.756	−0.0228764
$275$	−8137.55	−1.78441
$276$	0	0
$277$	2217.59	0.481019	0.240509	−	0.970647i	$-0.422685\pi$
$277$	2217.59	0.481019	0.240509	+	0.970647i	$0.422685\pi$
$278$	100.341	0.0216476
$279$	0	0
$280$	−26.4515	−0.00564564
$281$	2562.96	0.544105	0.272053	−	0.962282i	$-0.412298\pi$
$281$	2562.96	0.544105	0.272053	+	0.962282i	$0.412298\pi$
$282$	0	0
$283$	−3869.29	−0.812741	−0.406370	−	0.913708i	$-0.633206\pi$
$283$	−3869.29	−0.812741	−0.406370	+	0.913708i	$0.633206\pi$
$284$	−2959.80	−0.618421
$285$	0	0
$286$	1346.71	0.278435
$287$	−1321.55	−0.271807
$288$	0	0
$289$	−1837.65	−0.374038
$290$	−7.89026	−0.00159770
$291$	0	0
$292$	4549.07	0.911693
$293$	3883.83	0.774388	0.387194	−	0.921998i	$-0.373444\pi$
$293$	3883.83	0.774388	0.387194	+	0.921998i	$0.373444\pi$
$294$	0	0
$295$	−65.2092	−0.0128699
$296$	503.554	0.0988800
$297$	0	0
$298$	−676.257	−0.131458
$299$	4672.48	0.903734
$300$	0	0
$301$	496.472	0.0950703
$302$	50.3609	0.00959584
$303$	0	0
$304$	−3878.99	−0.731827
$305$	476.699	0.0894941
$306$	0	0
$307$	−403.210	−0.0749590	−0.0374795	−	0.999297i	$-0.511933\pi$
$307$	−403.210	−0.0749590	−0.0374795	+	0.999297i	$0.511933\pi$
$308$	−3141.08	−0.581103
$309$	0	0
$310$	−64.2364	−0.0117690
$311$	4838.71	0.882244	0.441122	−	0.897447i	$-0.354581\pi$
$311$	4838.71	0.882244	0.441122	+	0.897447i	$0.354581\pi$
$312$	0	0
$313$	−8544.28	−1.54298	−0.771488	−	0.636244i	$-0.780487\pi$
$313$	−8544.28	−1.54298	−0.771488	+	0.636244i	$0.780487\pi$
$314$	311.970	0.0560685
$315$	0	0
$316$	2765.37	0.492291
$317$	1773.06	0.314148	0.157074	−	0.987587i	$-0.449794\pi$
$317$	1773.06	0.314148	0.157074	+	0.987587i	$0.449794\pi$
$318$	0	0
$319$	−1894.45	−0.332504
$320$	293.803	0.0513253
$321$	0	0
$322$	238.851	0.0413374
$323$	3590.50	0.618516
$324$	0	0
$325$	6199.72	1.05815
$326$	−222.813	−0.0378541
$327$	0	0
$328$	−1410.67	−0.237474
$329$	2197.30	0.368210
$330$	0	0
$331$	801.875	0.133157	0.0665786	−	0.997781i	$-0.478792\pi$
$331$	801.875	0.133157	0.0665786	+	0.997781i	$0.478792\pi$
$332$	5452.58	0.901352
$333$	0	0
$334$	200.733	0.0328851
$335$	−554.700	−0.0904671
$336$	0	0
$337$	8193.23	1.32437	0.662186	−	0.749339i	$-0.269629\pi$
$337$	8193.23	1.32437	0.662186	+	0.749339i	$0.269629\pi$
$338$	−115.983	−0.0186647
$339$	0	0
$340$	−285.161	−0.0454854
$341$	−15423.1	−2.44930
$342$	0	0
$343$	−3981.79	−0.626811
$344$	529.953	0.0830615
$345$	0	0
$346$	−1354.35	−0.210435
$347$	10914.9	1.68860	0.844301	−	0.535869i	$-0.180016\pi$
$347$	10914.9	1.68860	0.844301	+	0.535869i	$0.180016\pi$
$348$	0	0
$349$	6697.83	1.02730	0.513649	−	0.858001i	$-0.328294\pi$
$349$	6697.83	1.02730	0.513649	+	0.858001i	$0.328294\pi$
$350$	316.922	0.0484005
$351$	0	0
$352$	−5047.54	−0.764303
$353$	3764.83	0.567654	0.283827	−	0.958875i	$-0.408396\pi$
$353$	3764.83	0.567654	0.283827	+	0.958875i	$0.408396\pi$
$354$	0	0
$355$	−248.346	−0.0371291
$356$	8753.90	1.30325
$357$	0	0
$358$	−232.898	−0.0343829
$359$	−6577.13	−0.966930	−0.483465	−	0.875364i	$-0.660622\pi$
$359$	−6577.13	−0.966930	−0.483465	+	0.875364i	$0.660622\pi$
$360$	0	0
$361$	−2667.06	−0.388841
$362$	624.354	0.0906501
$363$	0	0
$364$	2393.08	0.344592
$365$	381.696	0.0547366
$366$	0	0
$367$	2274.27	0.323477	0.161738	−	0.986834i	$-0.448290\pi$
$367$	2274.27	0.323477	0.161738	+	0.986834i	$0.448290\pi$
$368$	−5624.64	−0.796752
$369$	0	0
$370$	20.8967	0.00293613
$371$	−2017.48	−0.282325
$372$	0	0
$373$	1284.94	0.178369	0.0891844	−	0.996015i	$-0.471574\pi$
$373$	1284.94	0.178369	0.0891844	+	0.996015i	$0.471574\pi$
$374$	1500.57	0.207467
$375$	0	0
$376$	2345.48	0.321700
$377$	1443.32	0.197174
$378$	0	0
$379$	174.785	0.0236890	0.0118445	−	0.999930i	$-0.496230\pi$
$379$	174.785	0.0236890	0.0118445	+	0.999930i	$0.496230\pi$
$380$	−332.928	−0.0449444
$381$	0	0
$382$	1877.22	0.251431
$383$	5558.62	0.741599	0.370799	−	0.928713i	$-0.379084\pi$
$383$	5558.62	0.741599	0.370799	+	0.928713i	$0.379084\pi$
$384$	0	0
$385$	−263.557	−0.0348885
$386$	1215.79	0.160316
$387$	0	0
$388$	6305.14	0.824987
$389$	2556.05	0.333154	0.166577	−	0.986028i	$-0.446729\pi$
$389$	2556.05	0.333154	0.166577	+	0.986028i	$0.446729\pi$
$390$	0	0
$391$	5206.32	0.673388
$392$	−2001.48	−0.257883
$393$	0	0
$394$	1111.00	0.142060
$395$	232.032	0.0295564
$396$	0	0
$397$	5927.27	0.749323	0.374662	−	0.927162i	$-0.377759\pi$
$397$	5927.27	0.749323	0.374662	+	0.927162i	$0.377759\pi$
$398$	268.566	0.0338241
$399$	0	0
$400$	−7463.11	−0.932889
$401$	−4747.99	−0.591280	−0.295640	−	0.955299i	$-0.595533\pi$
$401$	−4747.99	−0.591280	−0.295640	+	0.955299i	$0.595533\pi$
$402$	0	0
$403$	11750.4	1.45243
$404$	−10756.2	−1.32460
$405$	0	0
$406$	73.7805	0.00901888
$407$	5017.29	0.611052
$408$	0	0
$409$	5200.19	0.628686	0.314343	−	0.949309i	$-0.398216\pi$
$409$	5200.19	0.628686	0.314343	+	0.949309i	$0.398216\pi$
$410$	−58.5407	−0.00705151
$411$	0	0
$412$	4968.48	0.594125
$413$	609.761	0.0726498
$414$	0	0
$415$	457.505	0.0541158
$416$	3845.55	0.453229
$417$	0	0
$418$	1751.93	0.204999
$419$	6425.59	0.749189	0.374595	−	0.927189i	$-0.377782\pi$
$419$	6425.59	0.749189	0.374595	+	0.927189i	$0.377782\pi$
$420$	0	0
$421$	10037.6	1.16201	0.581003	−	0.813902i	$-0.302661\pi$
$421$	10037.6	1.16201	0.581003	+	0.813902i	$0.302661\pi$
$422$	2050.18	0.236496
$423$	0	0
$424$	−2153.54	−0.246663
$425$	6908.05	0.788447
$426$	0	0
$427$	−4457.53	−0.505188
$428$	−1416.60	−0.159986
$429$	0	0
$430$	21.9922	0.00246641
$431$	−16646.8	−1.86044	−0.930218	−	0.367006i	$-0.880383\pi$
$431$	−16646.8	−1.86044	−0.930218	+	0.367006i	$0.880383\pi$
$432$	0	0
$433$	15089.1	1.67468	0.837340	−	0.546682i	$-0.184109\pi$
$433$	15089.1	1.67468	0.837340	+	0.546682i	$0.184109\pi$
$434$	600.664	0.0664350
$435$	0	0
$436$	−8126.97	−0.892686
$437$	6078.42	0.665378
$438$	0	0
$439$	−3777.24	−0.410656	−0.205328	−	0.978693i	$-0.565826\pi$
$439$	−3777.24	−0.410656	−0.205328	+	0.978693i	$0.565826\pi$
$440$	−281.330	−0.0304816
$441$	0	0
$442$	−1143.23	−0.123027
$443$	−7992.65	−0.857206	−0.428603	−	0.903493i	$-0.640994\pi$
$443$	−7992.65	−0.857206	−0.428603	+	0.903493i	$0.640994\pi$
$444$	0	0
$445$	734.507	0.0782449
$446$	−918.288	−0.0974937
$447$	0	0
$448$	−2747.31	−0.289728
$449$	−6433.54	−0.676209	−0.338104	−	0.941109i	$-0.609786\pi$
$449$	−6433.54	−0.676209	−0.338104	+	0.941109i	$0.609786\pi$
$450$	0	0
$451$	−14055.6	−1.46752
$452$	−1446.42	−0.150518
$453$	0	0
$454$	1883.05	0.194661
$455$	200.795	0.0206888
$456$	0	0
$457$	6975.18	0.713972	0.356986	−	0.934110i	$-0.383804\pi$
$457$	6975.18	0.713972	0.356986	+	0.934110i	$0.383804\pi$
$458$	−1383.08	−0.141107
$459$	0	0
$460$	−482.755	−0.0489317
$461$	14758.9	1.49109	0.745543	−	0.666458i	$-0.232190\pi$
$461$	14758.9	1.49109	0.745543	+	0.666458i	$0.232190\pi$
$462$	0	0
$463$	−18951.2	−1.90224	−0.951121	−	0.308818i	$-0.900067\pi$
$463$	−18951.2	−1.90224	−0.951121	+	0.308818i	$0.900067\pi$
$464$	−1737.44	−0.173833
$465$	0	0
$466$	−1654.93	−0.164513
$467$	−12442.4	−1.23290	−0.616449	−	0.787395i	$-0.711429\pi$
$467$	−12442.4	−1.23290	−0.616449	+	0.787395i	$0.711429\pi$
$468$	0	0
$469$	5186.91	0.510680
$470$	97.3338	0.00955249
$471$	0	0
$472$	650.882	0.0634730
$473$	5280.33	0.513297
$474$	0	0
$475$	8065.21	0.779068
$476$	2666.50	0.256762
$477$	0	0
$478$	−579.914	−0.0554909
$479$	−12947.5	−1.23504	−0.617522	−	0.786554i	$-0.711863\pi$
$479$	−12947.5	−1.23504	−0.617522	+	0.786554i	$0.711863\pi$
$480$	0	0
$481$	−3822.50	−0.362352
$482$	845.385	0.0798885
$483$	0	0
$484$	−22988.0	−2.15890
$485$	529.041	0.0495309
$486$	0	0
$487$	9844.72	0.916030	0.458015	−	0.888944i	$-0.348561\pi$
$487$	9844.72	0.916030	0.458015	+	0.888944i	$0.348561\pi$
$488$	−4758.14	−0.441375
$489$	0	0
$490$	−83.0584	−0.00765754
$491$	6809.50	0.625883	0.312941	−	0.949772i	$-0.398686\pi$
$491$	6809.50	0.625883	0.312941	+	0.949772i	$0.398686\pi$
$492$	0	0
$493$	1608.22	0.146918
$494$	−1334.74	−0.121564
$495$	0	0
$496$	−14144.9	−1.28049
$497$	2322.24	0.209591
$498$	0	0
$499$	−15953.8	−1.43124	−0.715622	−	0.698488i	$-0.753857\pi$
$499$	−15953.8	−1.43124	−0.715622	+	0.698488i	$0.753857\pi$
$500$	−1283.32	−0.114783
$501$	0	0
$502$	332.468	0.0295593
$503$	14582.7	1.29267	0.646334	−	0.763054i	$-0.276301\pi$
$503$	14582.7	1.29267	0.646334	+	0.763054i	$0.276301\pi$
$504$	0	0
$505$	−902.511	−0.0795272
$506$	2540.35	0.223186
$507$	0	0
$508$	−15653.5	−1.36715
$509$	−20906.4	−1.82055	−0.910273	−	0.414008i	$-0.864129\pi$
$509$	−20906.4	−1.82055	−0.910273	+	0.414008i	$0.864129\pi$
$510$	0	0
$511$	−3569.17	−0.308984
$512$	−7771.61	−0.670820
$513$	0	0
$514$	−1849.09	−0.158677
$515$	416.887	0.0356704
$516$	0	0
$517$	23369.8	1.98802
$518$	−195.401	−0.0165742
$519$	0	0
$520$	214.336	0.0180755
$521$	15131.7	1.27242	0.636212	−	0.771515i	$-0.280500\pi$
$521$	15131.7	1.27242	0.636212	+	0.771515i	$0.280500\pi$
$522$	0	0
$523$	12146.9	1.01558	0.507790	−	0.861481i	$-0.330463\pi$
$523$	12146.9	1.01558	0.507790	+	0.861481i	$0.330463\pi$
$524$	4395.41	0.366440
$525$	0	0
$526$	−1580.31	−0.130998
$527$	13092.9	1.08223
$528$	0	0
$529$	−3353.12	−0.275592
$530$	−89.3683	−0.00732436
$531$	0	0
$532$	3113.16	0.253708
$533$	10708.5	0.870237
$534$	0	0
$535$	−118.862	−0.00960532
$536$	5536.70	0.446174
$537$	0	0
$538$	1873.89	0.150166
$539$	−19942.3	−1.59365
$540$	0	0
$541$	−22291.8	−1.77153	−0.885767	−	0.464130i	$-0.846367\pi$
$541$	−22291.8	−1.77153	−0.885767	+	0.464130i	$0.846367\pi$
$542$	−1641.38	−0.130080
$543$	0	0
$544$	4284.91	0.337709
$545$	−681.904	−0.0535955
$546$	0	0
$547$	−15439.4	−1.20684	−0.603421	−	0.797423i	$-0.706196\pi$
$547$	−15439.4	−1.20684	−0.603421	+	0.797423i	$0.706196\pi$
$548$	−1960.94	−0.152860
$549$	0	0
$550$	3370.68	0.261321
$551$	1877.61	0.145170
$552$	0	0
$553$	−2169.69	−0.166844
$554$	−918.557	−0.0704436
$555$	0	0
$556$	1896.39	0.144649
$557$	−2336.99	−0.177776	−0.0888881	−	0.996042i	$-0.528331\pi$
$557$	−2336.99	−0.177776	−0.0888881	+	0.996042i	$0.528331\pi$
$558$	0	0
$559$	−4022.90	−0.304384
$560$	−241.713	−0.0182397
$561$	0	0
$562$	−1061.61	−0.0796824
$563$	−19833.3	−1.48468	−0.742340	−	0.670023i	$-0.766284\pi$
$563$	−19833.3	−1.48468	−0.742340	+	0.670023i	$0.766284\pi$
$564$	0	0
$565$	−121.364	−0.00903685
$566$	1602.71	0.119023
$567$	0	0
$568$	2478.85	0.183116
$569$	−11063.7	−0.815141	−0.407571	−	0.913174i	$-0.633624\pi$
$569$	−11063.7	−0.815141	−0.407571	+	0.913174i	$0.633624\pi$
$570$	0	0
$571$	−665.827	−0.0487986	−0.0243993	−	0.999702i	$-0.507767\pi$
$571$	−665.827	−0.0487986	−0.0243993	+	0.999702i	$0.507767\pi$
$572$	25452.1	1.86050
$573$	0	0
$574$	547.404	0.0398053
$575$	11694.8	0.848184
$576$	0	0
$577$	−7165.21	−0.516970	−0.258485	−	0.966015i	$-0.583223\pi$
$577$	−7165.21	−0.516970	−0.258485	+	0.966015i	$0.583223\pi$
$578$	761.179	0.0547766
$579$	0	0
$580$	−149.122	−0.0106758
$581$	−4278.06	−0.305480
$582$	0	0
$583$	−21457.3	−1.52431
$584$	−3809.87	−0.269955
$585$	0	0
$586$	−1608.73	−0.113406
$587$	10375.2	0.729525	0.364763	−	0.931101i	$-0.381150\pi$
$587$	10375.2	0.729525	0.364763	+	0.931101i	$0.381150\pi$
$588$	0	0
$589$	15286.0	1.06936
$590$	27.0105	0.00188476
$591$	0	0
$592$	4601.46	0.319457
$593$	−18931.5	−1.31100	−0.655501	−	0.755194i	$-0.727542\pi$
$593$	−18931.5	−1.31100	−0.655501	+	0.755194i	$0.727542\pi$
$594$	0	0
$595$	223.736	0.0154156
$596$	−12780.9	−0.878401
$597$	0	0
$598$	−1935.40	−0.132349
$599$	12244.2	0.835199	0.417600	−	0.908631i	$-0.362871\pi$
$599$	12244.2	0.835199	0.417600	+	0.908631i	$0.362871\pi$
$600$	0	0
$601$	15596.9	1.05859	0.529293	−	0.848439i	$-0.322457\pi$
$601$	15596.9	1.05859	0.529293	+	0.848439i	$0.322457\pi$
$602$	−205.645	−0.0139227
$603$	0	0
$604$	951.796	0.0641192
$605$	−1928.84	−0.129617
$606$	0	0
$607$	−10155.5	−0.679076	−0.339538	−	0.940592i	$-0.610271\pi$
$607$	−10155.5	−0.679076	−0.339538	+	0.940592i	$0.610271\pi$
$608$	5002.67	0.333692
$609$	0	0
$610$	−197.455	−0.0131061
$611$	−17804.7	−1.17889
$612$	0	0
$613$	−6227.67	−0.410331	−0.205166	−	0.978727i	$-0.565773\pi$
$613$	−6227.67	−0.410331	−0.205166	+	0.978727i	$0.565773\pi$
$614$	167.015	0.0109775
$615$	0	0
$616$	2630.67	0.172066
$617$	−14357.5	−0.936808	−0.468404	−	0.883514i	$-0.655171\pi$
$617$	−14357.5	−0.936808	−0.468404	+	0.883514i	$0.655171\pi$
$618$	0	0
$619$	−13220.6	−0.858453	−0.429227	−	0.903197i	$-0.641214\pi$
$619$	−13220.6	−0.858453	−0.429227	+	0.903197i	$0.641214\pi$
$620$	−1214.03	−0.0786400
$621$	0	0
$622$	−2004.26	−0.129202
$623$	−6868.26	−0.441687
$624$	0	0
$625$	15463.4	0.989657
$626$	3539.16	0.225963
$627$	0	0
$628$	5896.08	0.374649
$629$	−4259.23	−0.269995
$630$	0	0
$631$	1828.97	0.115389	0.0576943	−	0.998334i	$-0.481625\pi$
$631$	1828.97	0.115389	0.0576943	+	0.998334i	$0.481625\pi$
$632$	−2316.01	−0.145769
$633$	0	0
$634$	−734.426	−0.0460059
$635$	−1313.43	−0.0820817
$636$	0	0
$637$	15193.4	0.945028
$638$	784.707	0.0486941
$639$	0	0
$640$	−527.723	−0.0325939
$641$	22644.1	1.39530	0.697651	−	0.716437i	$-0.254228\pi$
$641$	22644.1	1.39530	0.697651	+	0.716437i	$0.254228\pi$
$642$	0	0
$643$	22728.4	1.39397	0.696983	−	0.717088i	$-0.254525\pi$
$643$	22728.4	1.39397	0.696983	+	0.717088i	$0.254525\pi$
$644$	4514.16	0.276216
$645$	0	0
$646$	−1487.23	−0.0905796
$647$	5844.85	0.355154	0.177577	−	0.984107i	$-0.443174\pi$
$647$	5844.85	0.355154	0.177577	+	0.984107i	$0.443174\pi$
$648$	0	0
$649$	6485.23	0.392246
$650$	−2568.01	−0.154962
$651$	0	0
$652$	−4211.04	−0.252941
$653$	−15174.1	−0.909355	−0.454677	−	0.890656i	$-0.650245\pi$
$653$	−15174.1	−0.909355	−0.454677	+	0.890656i	$0.650245\pi$
$654$	0	0
$655$	368.803	0.0220005
$656$	−12890.7	−0.767221
$657$	0	0
$658$	−910.152	−0.0539231
$659$	27857.0	1.64667	0.823333	−	0.567558i	$-0.192112\pi$
$659$	27857.0	1.64667	0.823333	+	0.567558i	$0.192112\pi$
$660$	0	0
$661$	−4966.64	−0.292254	−0.146127	−	0.989266i	$-0.546681\pi$
$661$	−4966.64	−0.292254	−0.146127	+	0.989266i	$0.546681\pi$
$662$	−332.148	−0.0195004
$663$	0	0
$664$	−4566.56	−0.266893
$665$	261.214	0.0152322
$666$	0	0
$667$	2722.59	0.158049
$668$	3793.76	0.219738
$669$	0	0
$670$	229.764	0.0132486
$671$	−47409.0	−2.72758
$672$	0	0
$673$	−2338.02	−0.133914	−0.0669569	−	0.997756i	$-0.521329\pi$
$673$	−2338.02	−0.133914	−0.0669569	+	0.997756i	$0.521329\pi$
$674$	−3393.75	−0.193950
$675$	0	0
$676$	−2192.03	−0.124717
$677$	−6342.30	−0.360051	−0.180025	−	0.983662i	$-0.557618\pi$
$677$	−6342.30	−0.360051	−0.180025	+	0.983662i	$0.557618\pi$
$678$	0	0
$679$	−4946.97	−0.279598
$680$	238.824	0.0134684
$681$	0	0
$682$	6388.48	0.358691
$683$	30366.6	1.70124	0.850620	−	0.525780i	$-0.176227\pi$
$683$	30366.6	1.70124	0.850620	+	0.525780i	$0.176227\pi$
$684$	0	0
$685$	−164.535	−0.00917746
$686$	1649.31	0.0917944
$687$	0	0
$688$	4842.69	0.268352
$689$	16347.6	0.903910
$690$	0	0
$691$	−11826.5	−0.651089	−0.325545	−	0.945527i	$-0.605548\pi$
$691$	−11826.5	−0.651089	−0.325545	+	0.945527i	$0.605548\pi$
$692$	−25596.6	−1.40612
$693$	0	0
$694$	−4521.12	−0.247290
$695$	159.119	0.00868450
$696$	0	0
$697$	11932.0	0.648429
$698$	−2774.33	−0.150444
$699$	0	0
$700$	5989.66	0.323411
$701$	2776.33	0.149587	0.0747936	−	0.997199i	$-0.476170\pi$
$701$	2776.33	0.149587	0.0747936	+	0.997199i	$0.476170\pi$
$702$	0	0
$703$	−4972.69	−0.266783
$704$	−29219.5	−1.56428
$705$	0	0
$706$	−1559.45	−0.0831310
$707$	8439.23	0.448925
$708$	0	0
$709$	−15962.7	−0.845543	−0.422772	−	0.906236i	$-0.638943\pi$
$709$	−15962.7	−0.845543	−0.422772	+	0.906236i	$0.638943\pi$
$710$	102.868	0.00543742
$711$	0	0
$712$	−7331.44	−0.385895
$713$	22165.2	1.16422
$714$	0	0
$715$	2135.59	0.111702
$716$	−4401.66	−0.229746
$717$	0	0
$718$	2724.34	0.141604
$719$	20832.9	1.08058	0.540289	−	0.841480i	$-0.318315\pi$
$719$	20832.9	1.08058	0.540289	+	0.841480i	$0.318315\pi$
$720$	0	0
$721$	−3898.24	−0.201357
$722$	1104.73	0.0569445
$723$	0	0
$724$	11800.0	0.605722
$725$	3612.49	0.185054
$726$	0	0
$727$	4452.04	0.227121	0.113561	−	0.993531i	$-0.463774\pi$
$727$	4452.04	0.227121	0.113561	+	0.993531i	$0.463774\pi$
$728$	−2004.22	−0.102035
$729$	0	0
$730$	−158.104	−0.00801599
$731$	−4482.52	−0.226802
$732$	0	0
$733$	−12107.2	−0.610082	−0.305041	−	0.952339i	$-0.598670\pi$
$733$	−12107.2	−0.610082	−0.305041	+	0.952339i	$0.598670\pi$
$734$	−942.034	−0.0473721
$735$	0	0
$736$	7254.00	0.363296
$737$	55166.4	2.75723
$738$	0	0
$739$	−4506.27	−0.224311	−0.112156	−	0.993691i	$-0.535775\pi$
$739$	−4506.27	−0.224311	−0.112156	+	0.993691i	$0.535775\pi$
$740$	394.937	0.0196191
$741$	0	0
$742$	835.669	0.0413455
$743$	−1177.19	−0.0581253	−0.0290626	−	0.999578i	$-0.509252\pi$
$743$	−1177.19	−0.0581253	−0.0290626	+	0.999578i	$0.509252\pi$
$744$	0	0
$745$	−1072.40	−0.0527378
$746$	−532.239	−0.0261215
$747$	0	0
$748$	28360.1	1.38629
$749$	1111.46	0.0542214
$750$	0	0
$751$	27631.6	1.34260	0.671300	−	0.741186i	$-0.265736\pi$
$751$	27631.6	1.34260	0.671300	+	0.741186i	$0.265736\pi$
$752$	21432.9	1.03933
$753$	0	0
$754$	−597.841	−0.0288755
$755$	79.8616	0.00384962
$756$	0	0
$757$	11336.6	0.544300	0.272150	−	0.962255i	$-0.412265\pi$
$757$	11336.6	0.544300	0.272150	+	0.962255i	$0.412265\pi$
$758$	−72.3984	−0.00346917
$759$	0	0
$760$	278.829	0.0133082
$761$	−4356.58	−0.207524	−0.103762	−	0.994602i	$-0.533088\pi$
$761$	−4356.58	−0.207524	−0.103762	+	0.994602i	$0.533088\pi$
$762$	0	0
$763$	6376.37	0.302543
$764$	35478.4	1.68006
$765$	0	0
$766$	−2302.46	−0.108605
$767$	−4940.87	−0.232601
$768$	0	0
$769$	−21718.1	−1.01843	−0.509217	−	0.860638i	$-0.670065\pi$
$769$	−21718.1	−1.01843	−0.509217	+	0.860638i	$0.670065\pi$
$770$	109.169	0.00510931
$771$	0	0
$772$	22977.8	1.07123
$773$	−22688.4	−1.05568	−0.527842	−	0.849343i	$-0.676999\pi$
$773$	−22688.4	−1.05568	−0.527842	+	0.849343i	$0.676999\pi$
$774$	0	0
$775$	29410.1	1.36315
$776$	−5280.58	−0.244281
$777$	0	0
$778$	−1058.75	−0.0487893
$779$	13930.7	0.640716
$780$	0	0
$781$	24698.6	1.13161
$782$	−2156.53	−0.0986155
$783$	0	0
$784$	−18289.5	−0.833158
$785$	494.718	0.0224933
$786$	0	0
$787$	−32890.9	−1.48975	−0.744875	−	0.667204i	$-0.767491\pi$
$787$	−32890.9	−1.48975	−0.744875	+	0.667204i	$0.767491\pi$
$788$	20997.4	0.949239
$789$	0	0
$790$	−96.1107	−0.00432844
$791$	1134.85	0.0510123
$792$	0	0
$793$	36119.3	1.61744
$794$	−2455.16	−0.109736
$795$	0	0
$796$	5075.76	0.226012
$797$	−30404.1	−1.35128	−0.675638	−	0.737233i	$-0.736132\pi$
$797$	−30404.1	−1.35128	−0.675638	+	0.737233i	$0.736132\pi$
$798$	0	0
$799$	−19838.9	−0.878410
$800$	9625.04	0.425371
$801$	0	0
$802$	1966.68	0.0865909
$803$	−37960.7	−1.66825
$804$	0	0
$805$	378.767	0.0165836
$806$	−4867.16	−0.212703
$807$	0	0
$808$	9008.36	0.392219
$809$	37889.3	1.64662	0.823311	−	0.567591i	$-0.192124\pi$
$809$	37889.3	1.64662	0.823311	+	0.567591i	$0.192124\pi$
$810$	0	0
$811$	−8123.23	−0.351720	−0.175860	−	0.984415i	$-0.556271\pi$
$811$	−8123.23	−0.351720	−0.175860	+	0.984415i	$0.556271\pi$
$812$	1394.41	0.0602640
$813$	0	0
$814$	−2078.23	−0.0894864
$815$	−353.333	−0.0151862
$816$	0	0
$817$	−5233.39	−0.224104
$818$	−2153.99	−0.0920690
$819$	0	0
$820$	−1106.39	−0.0471180
$821$	−13226.8	−0.562264	−0.281132	−	0.959669i	$-0.590710\pi$
$821$	−13226.8	−0.562264	−0.281132	+	0.959669i	$0.590710\pi$
$822$	0	0
$823$	−29575.2	−1.25265	−0.626323	−	0.779563i	$-0.715441\pi$
$823$	−29575.2	−1.25265	−0.626323	+	0.779563i	$0.715441\pi$
$824$	−4161.13	−0.175922
$825$	0	0
$826$	−252.571	−0.0106393
$827$	36661.2	1.54152	0.770758	−	0.637128i	$-0.219878\pi$
$827$	36661.2	1.54152	0.770758	+	0.637128i	$0.219878\pi$
$828$	0	0
$829$	11277.0	0.472455	0.236228	−	0.971698i	$-0.424089\pi$
$829$	11277.0	0.472455	0.236228	+	0.971698i	$0.424089\pi$
$830$	−189.505	−0.00792507
$831$	0	0
$832$	22261.3	0.927612
$833$	16929.2	0.704158
$834$	0	0
$835$	318.320	0.0131927
$836$	33110.6	1.36980
$837$	0	0
$838$	−2661.56	−0.109716
$839$	18965.7	0.780417	0.390208	−	0.920727i	$-0.372403\pi$
$839$	18965.7	0.780417	0.390208	+	0.920727i	$0.372403\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−4157.72	−0.170172
$843$	0	0
$844$	38747.3	1.58026
$845$	−183.925	−0.00748781
$846$	0	0
$847$	18036.2	0.731679
$848$	−19678.9	−0.796908
$849$	0	0
$850$	−2861.41	−0.115465
$851$	−7210.54	−0.290451
$852$	0	0
$853$	8067.23	0.323818	0.161909	−	0.986806i	$-0.448235\pi$
$853$	8067.23	0.323818	0.161909	+	0.986806i	$0.448235\pi$
$854$	1846.37	0.0739830
$855$	0	0
$856$	1186.41	0.0473724
$857$	−15281.7	−0.609118	−0.304559	−	0.952493i	$-0.598509\pi$
$857$	−15281.7	−0.609118	−0.304559	+	0.952493i	$0.598509\pi$
$858$	0	0
$859$	−36789.1	−1.46127	−0.730634	−	0.682770i	$-0.760775\pi$
$859$	−36789.1	−1.46127	−0.730634	+	0.682770i	$0.760775\pi$
$860$	415.641	0.0164805
$861$	0	0
$862$	6895.33	0.272455
$863$	−40907.8	−1.61358	−0.806788	−	0.590840i	$-0.798796\pi$
$863$	−40907.8	−1.61358	−0.806788	+	0.590840i	$0.798796\pi$
$864$	0	0
$865$	−2147.71	−0.0844213
$866$	−6250.12	−0.245251
$867$	0	0
$868$	11352.2	0.443917
$869$	−23076.2	−0.900812
$870$	0	0
$871$	−42029.4	−1.63503
$872$	6806.38	0.264327
$873$	0	0
$874$	−2517.76	−0.0974424
$875$	1006.88	0.0389015
$876$	0	0
$877$	2391.60	0.0920852	0.0460426	−	0.998939i	$-0.485339\pi$
$877$	2391.60	0.0920852	0.0460426	+	0.998939i	$0.485339\pi$
$878$	1564.59	0.0601392
$879$	0	0
$880$	−2570.79	−0.0984786
$881$	−5487.72	−0.209859	−0.104930	−	0.994480i	$-0.533462\pi$
$881$	−5487.72	−0.209859	−0.104930	+	0.994480i	$0.533462\pi$
$882$	0	0
$883$	170.008	0.00647931	0.00323966	−	0.999995i	$-0.498969\pi$
$883$	170.008	0.00647931	0.00323966	+	0.999995i	$0.498969\pi$
$884$	−21606.6	−0.822067
$885$	0	0
$886$	3310.66	0.125535
$887$	−25867.3	−0.979188	−0.489594	−	0.871950i	$-0.662855\pi$
$887$	−25867.3	−0.979188	−0.489594	+	0.871950i	$0.662855\pi$
$888$	0	0
$889$	12281.7	0.463345
$890$	−304.243	−0.0114587
$891$	0	0
$892$	−17355.2	−0.651451
$893$	−23162.1	−0.867961
$894$	0	0
$895$	−369.327	−0.0137936
$896$	4934.65	0.183990
$897$	0	0
$898$	2664.86	0.0990285
$899$	6846.77	0.254007
$900$	0	0
$901$	18215.4	0.673520
$902$	5822.03	0.214914
$903$	0	0
$904$	1211.39	0.0445687
$905$	990.093	0.0363666
$906$	0	0
$907$	11411.5	0.417765	0.208882	−	0.977941i	$-0.433017\pi$
$907$	11411.5	0.417765	0.208882	+	0.977941i	$0.433017\pi$
$908$	35588.7	1.30072
$909$	0	0
$910$	−83.1719	−0.00302980
$911$	38718.5	1.40813	0.704063	−	0.710138i	$-0.251367\pi$
$911$	38718.5	1.40813	0.704063	+	0.710138i	$0.251367\pi$
$912$	0	0
$913$	−45500.1	−1.64933
$914$	−2889.21	−0.104559
$915$	0	0
$916$	−26139.5	−0.942875
$917$	−3448.62	−0.124191
$918$	0	0
$919$	−48465.3	−1.73963	−0.869817	−	0.493374i	$-0.835763\pi$
$919$	−48465.3	−1.73963	−0.869817	+	0.493374i	$0.835763\pi$
$920$	404.310	0.0144888
$921$	0	0
$922$	−6113.34	−0.218364
$923$	−18817.0	−0.671040
$924$	0	0
$925$	−9567.37	−0.340079
$926$	7849.85	0.278577
$927$	0	0
$928$	2240.74	0.0792630
$929$	−10560.1	−0.372946	−0.186473	−	0.982460i	$-0.559706\pi$
$929$	−10560.1	−0.372946	−0.186473	+	0.982460i	$0.559706\pi$
$930$	0	0
$931$	19765.0	0.695782
$932$	−31277.3	−1.09927
$933$	0	0
$934$	5153.79	0.180554
$935$	2379.59	0.0832309
$936$	0	0
$937$	−23025.0	−0.802769	−0.401384	−	0.915910i	$-0.631471\pi$
$937$	−23025.0	−0.802769	−0.401384	+	0.915910i	$0.631471\pi$
$938$	−2148.49	−0.0747874
$939$	0	0
$940$	1839.56	0.0638296
$941$	−40778.1	−1.41268	−0.706338	−	0.707874i	$-0.749654\pi$
$941$	−40778.1	−1.41268	−0.706338	+	0.707874i	$0.749654\pi$
$942$	0	0
$943$	20199.8	0.697558
$944$	5947.74	0.205066
$945$	0	0
$946$	−2187.18	−0.0751707
$947$	−36129.8	−1.23977	−0.619884	−	0.784693i	$-0.712821\pi$
$947$	−36129.8	−1.23977	−0.619884	+	0.784693i	$0.712821\pi$
$948$	0	0
$949$	28920.9	0.989265
$950$	−3340.72	−0.114092
$951$	0	0
$952$	−2233.21	−0.0760280
$953$	−20831.4	−0.708075	−0.354037	−	0.935231i	$-0.615191\pi$
$953$	−20831.4	−0.708075	−0.354037	+	0.935231i	$0.615191\pi$
$954$	0	0
$955$	2976.86	0.100868
$956$	−10960.1	−0.370789
$957$	0	0
$958$	5363.02	0.180868
$959$	1538.54	0.0518061
$960$	0	0
$961$	25950.1	0.871071
$962$	1583.33	0.0530652
$963$	0	0
$964$	15977.4	0.533813
$965$	1927.98	0.0643149
$966$	0	0
$967$	49242.5	1.63757	0.818785	−	0.574100i	$-0.194648\pi$
$967$	49242.5	1.63757	0.818785	+	0.574100i	$0.194648\pi$
$968$	19252.5	0.639256
$969$	0	0
$970$	−219.136	−0.00725363
$971$	2352.05	0.0777351	0.0388675	−	0.999244i	$-0.487625\pi$
$971$	2352.05	0.0777351	0.0388675	+	0.999244i	$0.487625\pi$
$972$	0	0
$973$	−1487.89	−0.0490233
$974$	−4077.82	−0.134150
$975$	0	0
$976$	−43479.7	−1.42598
$977$	−18768.3	−0.614588	−0.307294	−	0.951615i	$-0.599423\pi$
$977$	−18768.3	−0.614588	−0.307294	+	0.951615i	$0.599423\pi$
$978$	0	0
$979$	−73048.7	−2.38473
$980$	−1569.76	−0.0511675
$981$	0	0
$982$	−2820.59	−0.0916584
$983$	49014.5	1.59036	0.795179	−	0.606375i	$-0.207377\pi$
$983$	49014.5	1.59036	0.795179	+	0.606375i	$0.207377\pi$
$984$	0	0
$985$	1761.81	0.0569908
$986$	−666.146	−0.0215156
$987$	0	0
$988$	−25225.8	−0.812288
$989$	−7588.55	−0.243986
$990$	0	0
$991$	−48860.6	−1.56620	−0.783102	−	0.621893i	$-0.786364\pi$
$991$	−48860.6	−1.56620	−0.783102	+	0.621893i	$0.786364\pi$
$992$	18242.4	0.583868
$993$	0	0
$994$	−961.903	−0.0306939
$995$	425.888	0.0135694
$996$	0	0
$997$	2934.57	0.0932184	0.0466092	−	0.998913i	$-0.485158\pi$
$997$	2934.57	0.0932184	0.0466092	+	0.998913i	$0.485158\pi$
$998$	6608.29	0.209601
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.4.a.b.1.1		2
3.2	odd	2		29.4.a.a.1.2	✓	2
12.11	even	2		464.4.a.f.1.2		2
15.14	odd	2		725.4.a.b.1.1		2
21.20	even	2		1421.4.a.c.1.2		2
24.5	odd	2		1856.4.a.n.1.2		2
24.11	even	2		1856.4.a.h.1.1		2
87.86	odd	2		841.4.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.a.1.2	✓	2	3.2	odd	2
261.4.a.b.1.1		2	1.1	even	1	trivial
464.4.a.f.1.2		2	12.11	even	2
725.4.a.b.1.1		2	15.14	odd	2
841.4.a.a.1.1		2	87.86	odd	2
1421.4.a.c.1.2		2	21.20	even	2
1856.4.a.h.1.1		2	24.11	even	2
1856.4.a.n.1.2		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -7.82843 q^{4} -0.656854 q^{5} +6.14214 q^{7} +6.55635 q^{8} +O(q^{10})\)	\(q-0.414214 q^{2} -7.82843 q^{4} -0.656854 q^{5} +6.14214 q^{7} +6.55635 q^{8} +0.272078 q^{10} +65.3259 q^{11} -49.7696 q^{13} -2.54416 q^{14} +59.9117 q^{16} -55.4558 q^{17} -64.7452 q^{19} +5.14214 q^{20} -27.0589 q^{22} -93.8823 q^{23} -124.569 q^{25} +20.6152 q^{26} -48.0833 q^{28} -29.0000 q^{29} -236.095 q^{31} -77.2670 q^{32} +22.9706 q^{34} -4.03449 q^{35} +76.8040 q^{37} +26.8183 q^{38} -4.30657 q^{40} -215.161 q^{41} +80.8305 q^{43} -511.399 q^{44} +38.8873 q^{46} +357.742 q^{47} -305.274 q^{49} +51.5980 q^{50} +389.617 q^{52} -328.466 q^{53} -42.9096 q^{55} +40.2700 q^{56} +12.0122 q^{58} +99.2750 q^{59} -725.730 q^{61} +97.7939 q^{62} -447.288 q^{64} +32.6913 q^{65} +844.479 q^{67} +434.132 q^{68} +1.67114 q^{70} +378.083 q^{71} -581.097 q^{73} -31.8133 q^{74} +506.853 q^{76} +401.241 q^{77} -353.247 q^{79} -39.3532 q^{80} +89.1228 q^{82} -696.510 q^{83} +36.4264 q^{85} -33.4811 q^{86} +428.299 q^{88} -1118.22 q^{89} -305.691 q^{91} +734.950 q^{92} -148.182 q^{94} +42.5281 q^{95} -805.415 q^{97} +126.449 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 10 q^{4} + 10 q^{5} - 16 q^{7} - 18 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 10 * q^4 + 10 * q^5 - 16 * q^7 - 18 * q^8	\( 2 q + 2 q^{2} - 10 q^{4} + 10 q^{5} - 16 q^{7} - 18 q^{8} + 26 q^{10} + 26 q^{11} - 26 q^{13} - 56 q^{14} + 18 q^{16} - 60 q^{17} - 220 q^{19} - 18 q^{20} - 122 q^{22} - 52 q^{23} - 136 q^{25} + 78 q^{26} - 58 q^{29} - 294 q^{31} + 18 q^{32} + 12 q^{34} - 240 q^{35} + 312 q^{37} - 348 q^{38} - 266 q^{40} - 40 q^{41} - 322 q^{43} - 426 q^{44} + 140 q^{46} + 130 q^{47} - 158 q^{49} + 24 q^{50} + 338 q^{52} - 1002 q^{53} - 462 q^{55} + 584 q^{56} - 58 q^{58} + 900 q^{59} - 948 q^{61} - 42 q^{62} + 118 q^{64} + 286 q^{65} + 320 q^{67} + 444 q^{68} - 568 q^{70} + 660 q^{71} + 648 q^{73} + 536 q^{74} + 844 q^{76} + 1272 q^{77} + 258 q^{79} - 486 q^{80} + 512 q^{82} - 1212 q^{83} - 12 q^{85} - 1006 q^{86} + 1394 q^{88} - 760 q^{89} - 832 q^{91} + 644 q^{92} - 698 q^{94} - 1612 q^{95} + 24 q^{97} + 482 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 10 * q^4 + 10 * q^5 - 16 * q^7 - 18 * q^8 + 26 * q^10 + 26 * q^11 - 26 * q^13 - 56 * q^14 + 18 * q^16 - 60 * q^17 - 220 * q^19 - 18 * q^20 - 122 * q^22 - 52 * q^23 - 136 * q^25 + 78 * q^26 - 58 * q^29 - 294 * q^31 + 18 * q^32 + 12 * q^34 - 240 * q^35 + 312 * q^37 - 348 * q^38 - 266 * q^40 - 40 * q^41 - 322 * q^43 - 426 * q^44 + 140 * q^46 + 130 * q^47 - 158 * q^49 + 24 * q^50 + 338 * q^52 - 1002 * q^53 - 462 * q^55 + 584 * q^56 - 58 * q^58 + 900 * q^59 - 948 * q^61 - 42 * q^62 + 118 * q^64 + 286 * q^65 + 320 * q^67 + 444 * q^68 - 568 * q^70 + 660 * q^71 + 648 * q^73 + 536 * q^74 + 844 * q^76 + 1272 * q^77 + 258 * q^79 - 486 * q^80 + 512 * q^82 - 1212 * q^83 - 12 * q^85 - 1006 * q^86 + 1394 * q^88 - 760 * q^89 - 832 * q^91 + 644 * q^92 - 698 * q^94 - 1612 * q^95 + 24 * q^97 + 482 * q^98

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