LMFDB - Embedded newform 2205.2.a.bf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,2,Mod(1,2205)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2205.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4,0,8,-4,0,0,-12,0,4,-8,0,0,0,0,12,8,0,8,-8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.27133$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.68554 q^{2} +0.841058 q^{4} -1.00000 q^{5} -1.95345 q^{8} -1.68554 q^{10} -1.77522 q^{11} +3.05320 q^{13} -4.97474 q^{16} +7.59587 q^{17} +3.12845 q^{19} -0.841058 q^{20} -2.99222 q^{22} +4.18165 q^{23} +1.00000 q^{25} +5.14631 q^{26} -7.08532 q^{29} +0.871553 q^{31} -4.47824 q^{32} +12.8032 q^{34} +9.91375 q^{37} +5.27314 q^{38} +1.95345 q^{40} +2.76744 q^{41} +0.317883 q^{43} -1.49307 q^{44} +7.04836 q^{46} +10.4853 q^{47} +1.68554 q^{50} +2.56792 q^{52} -3.63899 q^{53} +1.77522 q^{55} -11.9426 q^{58} -9.57060 q^{59} +6.58579 q^{61} +1.46904 q^{62} +2.40120 q^{64} -3.05320 q^{65} +14.0202 q^{67} +6.38857 q^{68} -6.13853 q^{71} +7.68897 q^{73} +16.7101 q^{74} +2.63121 q^{76} +14.0811 q^{79} +4.97474 q^{80} +4.66464 q^{82} +12.8284 q^{83} -7.59587 q^{85} +0.535806 q^{86} +3.46780 q^{88} -4.82843 q^{89} +3.51701 q^{92} +17.6734 q^{94} -3.12845 q^{95} -16.6238 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 8 q^{4} - 4 q^{5} - 12 q^{8} + 4 q^{10} - 8 q^{11} + 12 q^{16} + 8 q^{17} + 8 q^{19} - 8 q^{20} + 4 q^{25} - 8 q^{29} + 8 q^{31} - 28 q^{32} + 8 q^{34} + 8 q^{37} + 4 q^{38} + 12 q^{40}+ \cdots - 8 q^{95}+O(q^{100}) $ 4 * q - 4 * q^2 + 8 * q^4 - 4 * q^5 - 12 * q^8 + 4 * q^10 - 8 * q^11 + 12 * q^16 + 8 * q^17 + 8 * q^19 - 8 * q^20 + 4 * q^25 - 8 * q^29 + 8 * q^31 - 28 * q^32 + 8 * q^34 + 8 * q^37 + 4 * q^38 + 12 * q^40 - 8 * q^43 + 16 * q^44 + 12 * q^46 + 8 * q^47 - 4 * q^50 + 32 * q^52 - 8 * q^53 + 8 * q^55 - 24 * q^58 + 16 * q^59 + 32 * q^61 - 20 * q^62 + 24 * q^64 + 24 * q^68 + 8 * q^71 + 32 * q^74 - 8 * q^76 - 12 * q^80 + 8 * q^82 + 40 * q^83 - 8 * q^85 + 32 * q^86 - 40 * q^88 - 8 * q^89 + 8 * q^92 + 16 * q^94 - 8 * q^95

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 $

$2$	1.68554	1.19186	0.595930	−	0.803037i	$-0.296784\pi$
$2$	1.68554	1.19186	0.595930	+	0.803037i	$0.296784\pi$
$3$	0	0
$3$	0	0
$4$	0.841058	0.420529
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.95345	−0.690648
$9$	0	0
$10$	−1.68554	−0.533016
$11$	−1.77522	−0.535250	−0.267625	−	0.963523i	$-0.586239\pi$
$11$	−1.77522	−0.535250	−0.267625	+	0.963523i	$0.586239\pi$
$12$	0	0
$13$	3.05320	0.846807	0.423403	−	0.905941i	$-0.360835\pi$
$13$	3.05320	0.846807	0.423403	+	0.905941i	$0.360835\pi$
$14$	0	0
$15$	0	0
$16$	−4.97474	−1.24368
$17$	7.59587	1.84227	0.921134	−	0.389246i	$-0.127264\pi$
$17$	7.59587	1.84227	0.921134	+	0.389246i	$0.127264\pi$
$18$	0	0
$19$	3.12845	0.717715	0.358858	−	0.933392i	$-0.383166\pi$
$19$	3.12845	0.717715	0.358858	+	0.933392i	$0.383166\pi$
$20$	−0.841058	−0.188066
$21$	0	0
$22$	−2.99222	−0.637943
$23$	4.18165	0.871935	0.435967	−	0.899962i	$-0.356406\pi$
$23$	4.18165	0.871935	0.435967	+	0.899962i	$0.356406\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.14631	1.00927
$27$	0	0
$28$	0	0
$29$	−7.08532	−1.31571	−0.657856	−	0.753144i	$-0.728536\pi$
$29$	−7.08532	−1.31571	−0.657856	+	0.753144i	$0.728536\pi$
$30$	0	0
$31$	0.871553	0.156536	0.0782678	−	0.996932i	$-0.475061\pi$
$31$	0.871553	0.156536	0.0782678	+	0.996932i	$0.475061\pi$
$32$	−4.47824	−0.791649
$33$	0	0
$34$	12.8032	2.19572
$35$	0	0
$36$	0	0
$37$	9.91375	1.62981	0.814905	−	0.579594i	$-0.196789\pi$
$37$	9.91375	1.62981	0.814905	+	0.579594i	$0.196789\pi$
$38$	5.27314	0.855415
$39$	0	0
$40$	1.95345	0.308867
$41$	2.76744	0.432201	0.216101	−	0.976371i	$-0.430666\pi$
$41$	2.76744	0.432201	0.216101	+	0.976371i	$0.430666\pi$
$42$	0	0
$43$	0.317883	0.0484767	0.0242384	−	0.999706i	$-0.492284\pi$
$43$	0.317883	0.0484767	0.0242384	+	0.999706i	$0.492284\pi$
$44$	−1.49307	−0.225088
$45$	0	0
$46$	7.04836	1.03922
$47$	10.4853	1.52944	0.764718	−	0.644365i	$-0.222878\pi$
$47$	10.4853	1.52944	0.764718	+	0.644365i	$0.222878\pi$
$48$	0	0
$49$	0	0
$50$	1.68554	0.238372
$51$	0	0
$52$	2.56792	0.356107
$53$	−3.63899	−0.499854	−0.249927	−	0.968265i	$-0.580407\pi$
$53$	−3.63899	−0.499854	−0.249927	+	0.968265i	$0.580407\pi$
$54$	0	0
$55$	1.77522	0.239371
$56$	0	0
$57$	0	0
$58$	−11.9426	−1.56814
$59$	−9.57060	−1.24599	−0.622993	−	0.782227i	$-0.714084\pi$
$59$	−9.57060	−1.24599	−0.622993	+	0.782227i	$0.714084\pi$
$60$	0	0
$61$	6.58579	0.843224	0.421612	−	0.906776i	$-0.361465\pi$
$61$	6.58579	0.843224	0.421612	+	0.906776i	$0.361465\pi$
$62$	1.46904	0.186568
$63$	0	0
$64$	2.40120	0.300150
$65$	−3.05320	−0.378703
$66$	0	0
$67$	14.0202	1.71283	0.856417	−	0.516284i	$-0.172685\pi$
$67$	14.0202	1.71283	0.856417	+	0.516284i	$0.172685\pi$
$68$	6.38857	0.774727
$69$	0	0
$70$	0	0
$71$	−6.13853	−0.728509	−0.364255	−	0.931299i	$-0.618676\pi$
$71$	−6.13853	−0.728509	−0.364255	+	0.931299i	$0.618676\pi$
$72$	0	0
$73$	7.68897	0.899926	0.449963	−	0.893047i	$-0.351437\pi$
$73$	7.68897	0.899926	0.449963	+	0.893047i	$0.351437\pi$
$74$	16.7101	1.94250
$75$	0	0
$76$	2.63121	0.301820
$77$	0	0
$78$	0	0
$79$	14.0811	1.58425	0.792126	−	0.610357i	$-0.208974\pi$
$79$	14.0811	1.58425	0.792126	+	0.610357i	$0.208974\pi$
$80$	4.97474	0.556193
$81$	0	0
$82$	4.66464	0.515123
$83$	12.8284	1.40810	0.704051	−	0.710149i	$-0.251372\pi$
$83$	12.8284	1.40810	0.704051	+	0.710149i	$0.251372\pi$
$84$	0	0
$85$	−7.59587	−0.823887
$86$	0.535806	0.0577775
$87$	0	0
$88$	3.46780	0.369669
$89$	−4.82843	−0.511812	−0.255906	−	0.966702i	$-0.582374\pi$
$89$	−4.82843	−0.511812	−0.255906	+	0.966702i	$0.582374\pi$
$90$	0	0
$91$	0	0
$92$	3.51701	0.366674
$93$	0	0
$94$	17.6734	1.82287
$95$	−3.12845	−0.320972
$96$	0	0
$97$	−16.6238	−1.68789	−0.843946	−	0.536428i	$-0.819773\pi$
$97$	−16.6238	−1.68789	−0.843946	+	0.536428i	$0.819773\pi$
$98$	0	0
$99$	0	0
$100$	0.841058	0.0841058
$101$	−12.4243	−1.23626	−0.618132	−	0.786075i	$-0.712110\pi$
$101$	−12.4243	−1.23626	−0.618132	+	0.786075i	$0.712110\pi$
$102$	0	0
$103$	3.91375	0.385633	0.192817	−	0.981235i	$-0.438238\pi$
$103$	3.91375	0.385633	0.192817	+	0.981235i	$0.438238\pi$
$104$	−5.96427	−0.584845
$105$	0	0
$106$	−6.13368	−0.595756
$107$	−3.73210	−0.360795	−0.180398	−	0.983594i	$-0.557738\pi$
$107$	−3.73210	−0.360795	−0.180398	+	0.983594i	$0.557738\pi$
$108$	0	0
$109$	7.65685	0.733394	0.366697	−	0.930341i	$-0.380489\pi$
$109$	7.65685	0.733394	0.366697	+	0.930341i	$0.380489\pi$
$110$	2.99222	0.285297
$111$	0	0
$112$	0	0
$113$	−19.4023	−1.82521	−0.912605	−	0.408842i	$-0.865933\pi$
$113$	−19.4023	−1.82521	−0.912605	+	0.408842i	$0.865933\pi$
$114$	0	0
$115$	−4.18165	−0.389941
$116$	−5.95917	−0.553295
$117$	0	0
$118$	−16.1317	−1.48504
$119$	0	0
$120$	0	0
$121$	−7.84858	−0.713508
$122$	11.1006	1.00500
$123$	0	0
$124$	0.733027	0.0658277
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.1463	−1.34402	−0.672009	−	0.740543i	$-0.734568\pi$
$127$	−15.1463	−1.34402	−0.672009	+	0.740543i	$0.734568\pi$
$128$	13.0038	1.14939
$129$	0	0
$130$	−5.14631	−0.451361
$131$	13.5252	1.18170	0.590850	−	0.806781i	$-0.298792\pi$
$131$	13.5252	1.18170	0.590850	+	0.806781i	$0.298792\pi$
$132$	0	0
$133$	0	0
$134$	23.6316	2.04146
$135$	0	0
$136$	−14.8381	−1.27236
$137$	−2.73988	−0.234084	−0.117042	−	0.993127i	$-0.537341\pi$
$137$	−2.73988	−0.234084	−0.117042	+	0.993127i	$0.537341\pi$
$138$	0	0
$139$	6.67889	0.566496	0.283248	−	0.959047i	$-0.408588\pi$
$139$	6.67889	0.566496	0.283248	+	0.959047i	$0.408588\pi$
$140$	0	0
$141$	0	0
$142$	−10.3468	−0.868280
$143$	−5.42012	−0.453253
$144$	0	0
$145$	7.08532	0.588404
$146$	12.9601	1.07259
$147$	0	0
$148$	8.33804	0.685383
$149$	−1.36423	−0.111762	−0.0558812	−	0.998437i	$-0.517797\pi$
$149$	−1.36423	−0.111762	−0.0558812	+	0.998437i	$0.517797\pi$
$150$	0	0
$151$	−23.8022	−1.93700	−0.968499	−	0.249017i	$-0.919893\pi$
$151$	−23.8022	−1.93700	−0.968499	+	0.249017i	$0.919893\pi$
$152$	−6.11126	−0.495688
$153$	0	0
$154$	0	0
$155$	−0.871553	−0.0700048
$156$	0	0
$157$	12.7101	1.01437	0.507187	−	0.861836i	$-0.330686\pi$
$157$	12.7101	1.01437	0.507187	+	0.861836i	$0.330686\pi$
$158$	23.7344	1.88821
$159$	0	0
$160$	4.47824	0.354036
$161$	0	0
$162$	0	0
$163$	−9.70227	−0.759941	−0.379970	−	0.924999i	$-0.624066\pi$
$163$	−9.70227	−0.759941	−0.379970	+	0.924999i	$0.624066\pi$
$164$	2.32758	0.181753
$165$	0	0
$166$	21.6229	1.67826
$167$	−12.8486	−0.994253	−0.497127	−	0.867678i	$-0.665612\pi$
$167$	−12.8486	−0.994253	−0.497127	+	0.867678i	$0.665612\pi$
$168$	0	0
$169$	−3.67794	−0.282919
$170$	−12.8032	−0.981958
$171$	0	0
$172$	0.267358	0.0203859
$173$	−15.9949	−1.21607	−0.608035	−	0.793910i	$-0.708042\pi$
$173$	−15.9949	−1.21607	−0.608035	+	0.793910i	$0.708042\pi$
$174$	0	0
$175$	0	0
$176$	8.83127	0.665682
$177$	0	0
$178$	−8.13853	−0.610008
$179$	24.8807	1.85967	0.929835	−	0.367976i	$-0.119949\pi$
$179$	24.8807	1.85967	0.929835	+	0.367976i	$0.119949\pi$
$180$	0	0
$181$	3.18583	0.236801	0.118400	−	0.992966i	$-0.462223\pi$
$181$	3.18583	0.236801	0.118400	+	0.992966i	$0.462223\pi$
$182$	0	0
$183$	0	0
$184$	−8.16864	−0.602200
$185$	−9.91375	−0.728873
$186$	0	0
$187$	−13.4844	−0.986073
$188$	8.81873	0.643172
$189$	0	0
$190$	−5.27314	−0.382553
$191$	2.34676	0.169805	0.0849026	−	0.996389i	$-0.472942\pi$
$191$	2.34676	0.169805	0.0849026	+	0.996389i	$0.472942\pi$
$192$	0	0
$193$	23.1307	1.66499	0.832494	−	0.554035i	$-0.186912\pi$
$193$	23.1307	1.66499	0.832494	+	0.554035i	$0.186912\pi$
$194$	−28.0202	−2.01173
$195$	0	0
$196$	0	0
$197$	3.03895	0.216516	0.108258	−	0.994123i	$-0.465473\pi$
$197$	3.03895	0.216516	0.108258	+	0.994123i	$0.465473\pi$
$198$	0	0
$199$	4.42200	0.313467	0.156734	−	0.987641i	$-0.449904\pi$
$199$	4.42200	0.313467	0.156734	+	0.987641i	$0.449904\pi$
$200$	−1.95345	−0.138130
$201$	0	0
$202$	−20.9417	−1.47345
$203$	0	0
$204$	0	0
$205$	−2.76744	−0.193286
$206$	6.59680	0.459621
$207$	0	0
$208$	−15.1889	−1.05316
$209$	−5.55369	−0.384157
$210$	0	0
$211$	4.57153	0.314717	0.157359	−	0.987542i	$-0.449702\pi$
$211$	4.57153	0.314717	0.157359	+	0.987542i	$0.449702\pi$
$212$	−3.06060	−0.210203
$213$	0	0
$214$	−6.29061	−0.430017
$215$	−0.317883	−0.0216795
$216$	0	0
$217$	0	0
$218$	12.9060	0.874102
$219$	0	0
$220$	1.49307	0.100662
$221$	23.1917	1.56004
$222$	0	0
$223$	4.53581	0.303740	0.151870	−	0.988400i	$-0.451470\pi$
$223$	4.53581	0.303740	0.151870	+	0.988400i	$0.451470\pi$
$224$	0	0
$225$	0	0
$226$	−32.7034	−2.17539
$227$	8.44955	0.560817	0.280408	−	0.959881i	$-0.409530\pi$
$227$	8.44955	0.560817	0.280408	+	0.959881i	$0.409530\pi$
$228$	0	0
$229$	19.0711	1.26025	0.630126	−	0.776493i	$-0.283003\pi$
$229$	19.0711	1.26025	0.630126	+	0.776493i	$0.283003\pi$
$230$	−7.04836	−0.464755
$231$	0	0
$232$	13.8408	0.908693
$233$	−9.08303	−0.595049	−0.297524	−	0.954714i	$-0.596161\pi$
$233$	−9.08303	−0.595049	−0.297524	+	0.954714i	$0.596161\pi$
$234$	0	0
$235$	−10.4853	−0.683984
$236$	−8.04944	−0.523974
$237$	0	0
$238$	0	0
$239$	17.1532	1.10955	0.554773	−	0.832002i	$-0.312805\pi$
$239$	17.1532	1.10955	0.554773	+	0.832002i	$0.312805\pi$
$240$	0	0
$241$	−19.9839	−1.28728	−0.643638	−	0.765330i	$-0.722576\pi$
$241$	−19.9839	−1.28728	−0.643638	+	0.765330i	$0.722576\pi$
$242$	−13.2291	−0.850401
$243$	0	0
$244$	5.53903	0.354600
$245$	0	0
$246$	0	0
$247$	9.55179	0.607766
$248$	−1.70253	−0.108111
$249$	0	0
$250$	−1.68554	−0.106603
$251$	−4.93484	−0.311484	−0.155742	−	0.987798i	$-0.549777\pi$
$251$	−4.93484	−0.311484	−0.155742	+	0.987798i	$0.549777\pi$
$252$	0	0
$253$	−7.42336	−0.466703
$254$	−25.5298	−1.60188
$255$	0	0
$256$	17.1161	1.06976
$257$	−1.55045	−0.0967141	−0.0483571	−	0.998830i	$-0.515399\pi$
$257$	−1.55045	−0.0967141	−0.0483571	+	0.998830i	$0.515399\pi$
$258$	0	0
$259$	0	0
$260$	−2.56792	−0.159256
$261$	0	0
$262$	22.7973	1.40842
$263$	−13.1165	−0.808797	−0.404399	−	0.914583i	$-0.632519\pi$
$263$	−13.1165	−0.808797	−0.404399	+	0.914583i	$0.632519\pi$
$264$	0	0
$265$	3.63899	0.223541
$266$	0	0
$267$	0	0
$268$	11.7918	0.720297
$269$	4.69676	0.286366	0.143183	−	0.989696i	$-0.454266\pi$
$269$	4.69676	0.286366	0.143183	+	0.989696i	$0.454266\pi$
$270$	0	0
$271$	−17.6339	−1.07118	−0.535591	−	0.844477i	$-0.679911\pi$
$271$	−17.6339	−1.07118	−0.535591	+	0.844477i	$0.679911\pi$
$272$	−37.7874	−2.29120
$273$	0	0
$274$	−4.61819	−0.278995
$275$	−1.77522	−0.107050
$276$	0	0
$277$	−15.8528	−0.952500	−0.476250	−	0.879310i	$-0.658004\pi$
$277$	−15.8528	−0.952500	−0.476250	+	0.879310i	$0.658004\pi$
$278$	11.2576	0.675184
$279$	0	0
$280$	0	0
$281$	−16.6779	−0.994923	−0.497461	−	0.867486i	$-0.665734\pi$
$281$	−16.6779	−0.994923	−0.497461	+	0.867486i	$0.665734\pi$
$282$	0	0
$283$	12.8129	0.761645	0.380823	−	0.924648i	$-0.375641\pi$
$283$	12.8129	0.761645	0.380823	+	0.924648i	$0.375641\pi$
$284$	−5.16286	−0.306359
$285$	0	0
$286$	−9.13585	−0.540214
$287$	0	0
$288$	0	0
$289$	40.6972	2.39395
$290$	11.9426	0.701295
$291$	0	0
$292$	6.46687	0.378445
$293$	6.57153	0.383913	0.191957	−	0.981403i	$-0.438517\pi$
$293$	6.57153	0.383913	0.191957	+	0.981403i	$0.438517\pi$
$294$	0	0
$295$	9.57060	0.557222
$296$	−19.3660	−1.12562
$297$	0	0
$298$	−2.29948	−0.133205
$299$	12.7674	0.738360
$300$	0	0
$301$	0	0
$302$	−40.1197	−2.30863
$303$	0	0
$304$	−15.5632	−0.892611
$305$	−6.58579	−0.377101
$306$	0	0
$307$	15.3137	0.874000	0.437000	−	0.899462i	$-0.356041\pi$
$307$	15.3137	0.874000	0.437000	+	0.899462i	$0.356041\pi$
$308$	0	0
$309$	0	0
$310$	−1.46904	−0.0834359
$311$	−15.7464	−0.892894	−0.446447	−	0.894810i	$-0.647311\pi$
$311$	−15.7464	−0.892894	−0.446447	+	0.894810i	$0.647311\pi$
$312$	0	0
$313$	−20.0165	−1.13140	−0.565701	−	0.824610i	$-0.691394\pi$
$313$	−20.0165	−1.13140	−0.565701	+	0.824610i	$0.691394\pi$
$314$	21.4234	1.20899
$315$	0	0
$316$	11.8431	0.666225
$317$	22.8664	1.28431	0.642154	−	0.766576i	$-0.278041\pi$
$317$	22.8664	1.28431	0.642154	+	0.766576i	$0.278041\pi$
$318$	0	0
$319$	12.5780	0.704234
$320$	−2.40120	−0.134231
$321$	0	0
$322$	0	0
$323$	23.7633	1.32222
$324$	0	0
$325$	3.05320	0.169361
$326$	−16.3536	−0.905743
$327$	0	0
$328$	−5.40604	−0.298499
$329$	0	0
$330$	0	0
$331$	−12.3990	−0.681512	−0.340756	−	0.940152i	$-0.610683\pi$
$331$	−12.3990	−0.681512	−0.340756	+	0.940152i	$0.610683\pi$
$332$	10.7895	0.592148
$333$	0	0
$334$	−21.6569	−1.18501
$335$	−14.0202	−0.766003
$336$	0	0
$337$	0.625303	0.0340624	0.0170312	−	0.999855i	$-0.494579\pi$
$337$	0.625303	0.0340624	0.0170312	+	0.999855i	$0.494579\pi$
$338$	−6.19933	−0.337199
$339$	0	0
$340$	−6.38857	−0.346469
$341$	−1.54720	−0.0837856
$342$	0	0
$343$	0	0
$344$	−0.620968	−0.0334804
$345$	0	0
$346$	−26.9601	−1.44938
$347$	1.71653	0.0921481	0.0460740	−	0.998938i	$-0.485329\pi$
$347$	1.71653	0.0921481	0.0460740	+	0.998938i	$0.485329\pi$
$348$	0	0
$349$	−15.0491	−0.805557	−0.402779	−	0.915297i	$-0.631956\pi$
$349$	−15.0491	−0.805557	−0.402779	+	0.915297i	$0.631956\pi$
$350$	0	0
$351$	0	0
$352$	7.94988	0.423730
$353$	−22.4633	−1.19560	−0.597799	−	0.801646i	$-0.703958\pi$
$353$	−22.4633	−1.19560	−0.597799	+	0.801646i	$0.703958\pi$
$354$	0	0
$355$	6.13853	0.325799
$356$	−4.06099	−0.215232
$357$	0	0
$358$	41.9375	2.21647
$359$	−3.43208	−0.181138	−0.0905690	−	0.995890i	$-0.528869\pi$
$359$	−3.43208	−0.181138	−0.0905690	+	0.995890i	$0.528869\pi$
$360$	0	0
$361$	−9.21282	−0.484885
$362$	5.36985	0.282233
$363$	0	0
$364$	0	0
$365$	−7.68897	−0.402459
$366$	0	0
$367$	24.1908	1.26275	0.631375	−	0.775478i	$-0.282491\pi$
$367$	24.1908	1.26275	0.631375	+	0.775478i	$0.282491\pi$
$368$	−20.8026	−1.08441
$369$	0	0
$370$	−16.7101	−0.868715
$371$	0	0
$372$	0	0
$373$	−0.856934	−0.0443704	−0.0221852	−	0.999754i	$-0.507062\pi$
$373$	−0.856934	−0.0443704	−0.0221852	+	0.999754i	$0.507062\pi$
$374$	−22.7285	−1.17526
$375$	0	0
$376$	−20.4824	−1.05630
$377$	−21.6329	−1.11415
$378$	0	0
$379$	5.63159	0.289275	0.144638	−	0.989485i	$-0.453798\pi$
$379$	5.63159	0.289275	0.144638	+	0.989485i	$0.453798\pi$
$380$	−2.63121	−0.134978
$381$	0	0
$382$	3.95556	0.202384
$383$	5.39996	0.275925	0.137963	−	0.990437i	$-0.455945\pi$
$383$	5.39996	0.275925	0.137963	+	0.990437i	$0.455945\pi$
$384$	0	0
$385$	0	0
$386$	38.9879	1.98443
$387$	0	0
$388$	−13.9816	−0.709808
$389$	−14.2485	−0.722430	−0.361215	−	0.932483i	$-0.617638\pi$
$389$	−14.2485	−0.722430	−0.361215	+	0.932483i	$0.617638\pi$
$390$	0	0
$391$	31.7633	1.60634
$392$	0	0
$393$	0	0
$394$	5.12229	0.258057
$395$	−14.0811	−0.708499
$396$	0	0
$397$	−11.6532	−0.584860	−0.292430	−	0.956287i	$-0.594464\pi$
$397$	−11.6532	−0.584860	−0.292430	+	0.956287i	$0.594464\pi$
$398$	7.45347	0.373609
$399$	0	0
$400$	−4.97474	−0.248737
$401$	2.65778	0.132723	0.0663617	−	0.997796i	$-0.478861\pi$
$401$	2.65778	0.132723	0.0663617	+	0.997796i	$0.478861\pi$
$402$	0	0
$403$	2.66103	0.132555
$404$	−10.4496	−0.519885
$405$	0	0
$406$	0	0
$407$	−17.5991	−0.872356
$408$	0	0
$409$	−26.9206	−1.33114	−0.665569	−	0.746337i	$-0.731811\pi$
$409$	−26.9206	−1.33114	−0.665569	+	0.746337i	$0.731811\pi$
$410$	−4.66464	−0.230370
$411$	0	0
$412$	3.29169	0.162170
$413$	0	0
$414$	0	0
$415$	−12.8284	−0.629723
$416$	−13.6730	−0.670374
$417$	0	0
$418$	−9.36099	−0.457861
$419$	15.1917	0.742165	0.371082	−	0.928600i	$-0.378987\pi$
$419$	15.1917	0.742165	0.371082	+	0.928600i	$0.378987\pi$
$420$	0	0
$421$	−0.757744	−0.0369302	−0.0184651	−	0.999830i	$-0.505878\pi$
$421$	−0.757744	−0.0369302	−0.0184651	+	0.999830i	$0.505878\pi$
$422$	7.70552	0.375099
$423$	0	0
$424$	7.10858	0.345223
$425$	7.59587	0.368454
$426$	0	0
$427$	0	0
$428$	−3.13891	−0.151725
$429$	0	0
$430$	−0.535806	−0.0258389
$431$	−23.1091	−1.11313	−0.556563	−	0.830806i	$-0.687880\pi$
$431$	−23.1091	−1.11313	−0.556563	+	0.830806i	$0.687880\pi$
$432$	0	0
$433$	15.4522	0.742587	0.371294	−	0.928516i	$-0.378914\pi$
$433$	15.4522	0.742587	0.371294	+	0.928516i	$0.378914\pi$
$434$	0	0
$435$	0	0
$436$	6.43986	0.308413
$437$	13.0821	0.625801
$438$	0	0
$439$	−28.5843	−1.36425	−0.682127	−	0.731234i	$-0.738945\pi$
$439$	−28.5843	−1.36425	−0.682127	+	0.731234i	$0.738945\pi$
$440$	−3.46780	−0.165321
$441$	0	0
$442$	39.0907	1.85935
$443$	22.1110	1.05052	0.525262	−	0.850941i	$-0.323967\pi$
$443$	22.1110	1.05052	0.525262	+	0.850941i	$0.323967\pi$
$444$	0	0
$445$	4.82843	0.228889
$446$	7.64530	0.362015
$447$	0	0
$448$	0	0
$449$	26.4393	1.24775	0.623875	−	0.781524i	$-0.285557\pi$
$449$	26.4393	1.24775	0.623875	+	0.781524i	$0.285557\pi$
$450$	0	0
$451$	−4.91282	−0.231336
$452$	−16.3184	−0.767554
$453$	0	0
$454$	14.2421	0.668415
$455$	0	0
$456$	0	0
$457$	35.0063	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$457$	35.0063	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$458$	32.1451	1.50204
$459$	0	0
$460$	−3.51701	−0.163982
$461$	8.77790	0.408828	0.204414	−	0.978885i	$-0.434471\pi$
$461$	8.77790	0.408828	0.204414	+	0.978885i	$0.434471\pi$
$462$	0	0
$463$	−42.1655	−1.95960	−0.979799	−	0.199983i	$-0.935911\pi$
$463$	−42.1655	−1.95960	−0.979799	+	0.199983i	$0.935911\pi$
$464$	35.2476	1.63633
$465$	0	0
$466$	−15.3098	−0.709215
$467$	11.1275	0.514919	0.257460	−	0.966289i	$-0.417115\pi$
$467$	11.1275	0.514919	0.257460	+	0.966289i	$0.417115\pi$
$468$	0	0
$469$	0	0
$470$	−17.6734	−0.815213
$471$	0	0
$472$	18.6957	0.860538
$473$	−0.564314	−0.0259472
$474$	0	0
$475$	3.12845	0.143543
$476$	0	0
$477$	0	0
$478$	28.9124	1.32242
$479$	24.5949	1.12377	0.561886	−	0.827215i	$-0.310076\pi$
$479$	24.5949	1.12377	0.561886	+	0.827215i	$0.310076\pi$
$480$	0	0
$481$	30.2687	1.38013
$482$	−33.6837	−1.53425
$483$	0	0
$484$	−6.60112	−0.300051
$485$	16.6238	0.754848
$486$	0	0
$487$	−33.1665	−1.50292	−0.751458	−	0.659781i	$-0.770649\pi$
$487$	−33.1665	−1.50292	−0.751458	+	0.659781i	$0.770649\pi$
$488$	−12.8650	−0.582371
$489$	0	0
$490$	0	0
$491$	−25.8513	−1.16665	−0.583326	−	0.812238i	$-0.698249\pi$
$491$	−25.8513	−1.16665	−0.583326	+	0.812238i	$0.698249\pi$
$492$	0	0
$493$	−53.8191	−2.42389
$494$	16.1000	0.724371
$495$	0	0
$496$	−4.33575	−0.194681
$497$	0	0
$498$	0	0
$499$	0.610504	0.0273299	0.0136650	−	0.999907i	$-0.495650\pi$
$499$	0.610504	0.0273299	0.0136650	+	0.999907i	$0.495650\pi$
$500$	−0.841058	−0.0376133
$501$	0	0
$502$	−8.31788	−0.371245
$503$	9.49080	0.423174	0.211587	−	0.977359i	$-0.432137\pi$
$503$	9.49080	0.423174	0.211587	+	0.977359i	$0.432137\pi$
$504$	0	0
$505$	12.4243	0.552874
$506$	−12.5124	−0.556244
$507$	0	0
$508$	−12.7389	−0.565199
$509$	−21.6673	−0.960387	−0.480193	−	0.877163i	$-0.659434\pi$
$509$	−21.6673	−0.960387	−0.480193	+	0.877163i	$0.659434\pi$
$510$	0	0
$511$	0	0
$512$	2.84232	0.125614
$513$	0	0
$514$	−2.61334	−0.115270
$515$	−3.91375	−0.172460
$516$	0	0
$517$	−18.6137	−0.818630
$518$	0	0
$519$	0	0
$520$	5.96427	0.261551
$521$	34.3939	1.50683	0.753413	−	0.657548i	$-0.228406\pi$
$521$	34.3939	1.50683	0.753413	+	0.657548i	$0.228406\pi$
$522$	0	0
$523$	22.7779	0.996008	0.498004	−	0.867175i	$-0.334066\pi$
$523$	22.7779	0.996008	0.498004	+	0.867175i	$0.334066\pi$
$524$	11.3755	0.496940
$525$	0	0
$526$	−22.1084	−0.963973
$527$	6.62020	0.288380
$528$	0	0
$529$	−5.51379	−0.239730
$530$	6.13368	0.266430
$531$	0	0
$532$	0	0
$533$	8.44955	0.365991
$534$	0	0
$535$	3.73210	0.161353
$536$	−27.3876	−1.18297
$537$	0	0
$538$	7.91659	0.341308
$539$	0	0
$540$	0	0
$541$	27.4045	1.17821	0.589107	−	0.808055i	$-0.299480\pi$
$541$	27.4045	1.17821	0.589107	+	0.808055i	$0.299480\pi$
$542$	−29.7227	−1.27670
$543$	0	0
$544$	−34.0161	−1.45843
$545$	−7.65685	−0.327984
$546$	0	0
$547$	−18.1421	−0.775702	−0.387851	−	0.921722i	$-0.626782\pi$
$547$	−18.1421	−0.775702	−0.387851	+	0.921722i	$0.626782\pi$
$548$	−2.30440	−0.0984391
$549$	0	0
$550$	−2.99222	−0.127589
$551$	−22.1661	−0.944306
$552$	0	0
$553$	0	0
$554$	−26.7205	−1.13525
$555$	0	0
$556$	5.61734	0.238228
$557$	−22.0582	−0.934635	−0.467318	−	0.884090i	$-0.654780\pi$
$557$	−22.0582	−0.934635	−0.467318	+	0.884090i	$0.654780\pi$
$558$	0	0
$559$	0.970563	0.0410504
$560$	0	0
$561$	0	0
$562$	−28.1114	−1.18581
$563$	−10.1577	−0.428096	−0.214048	−	0.976823i	$-0.568665\pi$
$563$	−10.1577	−0.428096	−0.214048	+	0.976823i	$0.568665\pi$
$564$	0	0
$565$	19.4023	0.816259
$566$	21.5966	0.907774
$567$	0	0
$568$	11.9913	0.503143
$569$	42.5256	1.78277	0.891383	−	0.453251i	$-0.149736\pi$
$569$	42.5256	1.78277	0.891383	+	0.453251i	$0.149736\pi$
$570$	0	0
$571$	−22.7688	−0.952844	−0.476422	−	0.879217i	$-0.658067\pi$
$571$	−22.7688	−0.952844	−0.476422	+	0.879217i	$0.658067\pi$
$572$	−4.55864	−0.190606
$573$	0	0
$574$	0	0
$575$	4.18165	0.174387
$576$	0	0
$577$	−40.4467	−1.68382	−0.841909	−	0.539619i	$-0.818568\pi$
$577$	−40.4467	−1.68382	−0.841909	+	0.539619i	$0.818568\pi$
$578$	68.5969	2.85325
$579$	0	0
$580$	5.95917	0.247441
$581$	0	0
$582$	0	0
$583$	6.46002	0.267547
$584$	−15.0200	−0.621532
$585$	0	0
$586$	11.0766	0.457570
$587$	21.7918	0.899443	0.449721	−	0.893169i	$-0.351523\pi$
$587$	21.7918	0.899443	0.449721	+	0.893169i	$0.351523\pi$
$588$	0	0
$589$	2.72661	0.112348
$590$	16.1317	0.664130
$591$	0	0
$592$	−49.3183	−2.02697
$593$	−2.09084	−0.0858605	−0.0429303	−	0.999078i	$-0.513669\pi$
$593$	−2.09084	−0.0858605	−0.0429303	+	0.999078i	$0.513669\pi$
$594$	0	0
$595$	0	0
$596$	−1.14740	−0.0469993
$597$	0	0
$598$	21.5201	0.880021
$599$	21.7247	0.887647	0.443824	−	0.896114i	$-0.353622\pi$
$599$	21.7247	0.887647	0.443824	+	0.896114i	$0.353622\pi$
$600$	0	0
$601$	15.6711	0.639238	0.319619	−	0.947546i	$-0.396445\pi$
$601$	15.6711	0.639238	0.319619	+	0.947546i	$0.396445\pi$
$602$	0	0
$603$	0	0
$604$	−20.0191	−0.814564
$605$	7.84858	0.319090
$606$	0	0
$607$	15.3688	0.623801	0.311901	−	0.950115i	$-0.399034\pi$
$607$	15.3688	0.623801	0.311901	+	0.950115i	$0.399034\pi$
$608$	−14.0100	−0.568179
$609$	0	0
$610$	−11.1006	−0.449451
$611$	32.0137	1.29514
$612$	0	0
$613$	−24.9958	−1.00957	−0.504786	−	0.863245i	$-0.668429\pi$
$613$	−24.9958	−1.00957	−0.504786	+	0.863245i	$0.668429\pi$
$614$	25.8119	1.04168
$615$	0	0
$616$	0	0
$617$	−23.9738	−0.965148	−0.482574	−	0.875855i	$-0.660298\pi$
$617$	−23.9738	−0.965148	−0.482574	+	0.875855i	$0.660298\pi$
$618$	0	0
$619$	−7.31466	−0.294001	−0.147000	−	0.989136i	$-0.546962\pi$
$619$	−7.31466	−0.294001	−0.147000	+	0.989136i	$0.546962\pi$
$620$	−0.733027	−0.0294391
$621$	0	0
$622$	−26.5412	−1.06420
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−33.7388	−1.34847
$627$	0	0
$628$	10.6899	0.426573
$629$	75.3035	3.00255
$630$	0	0
$631$	1.18204	0.0470561	0.0235281	−	0.999723i	$-0.492510\pi$
$631$	1.18204	0.0470561	0.0235281	+	0.999723i	$0.492510\pi$
$632$	−27.5068	−1.09416
$633$	0	0
$634$	38.5424	1.53071
$635$	15.1463	0.601063
$636$	0	0
$637$	0	0
$638$	21.2008	0.839348
$639$	0	0
$640$	−13.0038	−0.514021
$641$	40.6917	1.60722	0.803612	−	0.595154i	$-0.202909\pi$
$641$	40.6917	1.60722	0.803612	+	0.595154i	$0.202909\pi$
$642$	0	0
$643$	20.2771	0.799649	0.399824	−	0.916592i	$-0.369071\pi$
$643$	20.2771	0.799649	0.399824	+	0.916592i	$0.369071\pi$
$644$	0	0
$645$	0	0
$646$	40.0540	1.57590
$647$	39.8990	1.56859	0.784295	−	0.620388i	$-0.213025\pi$
$647$	39.8990	1.56859	0.784295	+	0.620388i	$0.213025\pi$
$648$	0	0
$649$	16.9900	0.666914
$650$	5.14631	0.201855
$651$	0	0
$652$	−8.16018	−0.319577
$653$	−7.10318	−0.277969	−0.138985	−	0.990295i	$-0.544384\pi$
$653$	−7.10318	−0.277969	−0.138985	+	0.990295i	$0.544384\pi$
$654$	0	0
$655$	−13.5252	−0.528473
$656$	−13.7673	−0.537522
$657$	0	0
$658$	0	0
$659$	20.8404	0.811826	0.405913	−	0.913912i	$-0.366954\pi$
$659$	20.8404	0.811826	0.405913	+	0.913912i	$0.366954\pi$
$660$	0	0
$661$	41.5545	1.61628	0.808141	−	0.588989i	$-0.200474\pi$
$661$	41.5545	1.61628	0.808141	+	0.588989i	$0.200474\pi$
$662$	−20.8991	−0.812267
$663$	0	0
$664$	−25.0597	−0.972503
$665$	0	0
$666$	0	0
$667$	−29.6283	−1.14721
$668$	−10.8064	−0.418113
$669$	0	0
$670$	−23.6316	−0.912968
$671$	−11.6912	−0.451335
$672$	0	0
$673$	−5.34943	−0.206206	−0.103103	−	0.994671i	$-0.532877\pi$
$673$	−5.34943	−0.206206	−0.103103	+	0.994671i	$0.532877\pi$
$674$	1.05397	0.0405976
$675$	0	0
$676$	−3.09336	−0.118976
$677$	−2.83488	−0.108953	−0.0544766	−	0.998515i	$-0.517349\pi$
$677$	−2.83488	−0.108953	−0.0544766	+	0.998515i	$0.517349\pi$
$678$	0	0
$679$	0	0
$680$	14.8381	0.569016
$681$	0	0
$682$	−2.60787	−0.0998607
$683$	−23.1724	−0.886666	−0.443333	−	0.896357i	$-0.646204\pi$
$683$	−23.1724	−0.886666	−0.443333	+	0.896357i	$0.646204\pi$
$684$	0	0
$685$	2.73988	0.104685
$686$	0	0
$687$	0	0
$688$	−1.58139	−0.0602898
$689$	−11.1106	−0.423280
$690$	0	0
$691$	−38.9678	−1.48240	−0.741202	−	0.671283i	$-0.765744\pi$
$691$	−38.9678	−1.48240	−0.741202	+	0.671283i	$0.765744\pi$
$692$	−13.4526	−0.511393
$693$	0	0
$694$	2.89328	0.109828
$695$	−6.67889	−0.253345
$696$	0	0
$697$	21.0211	0.796230
$698$	−25.3658	−0.960111
$699$	0	0
$700$	0	0
$701$	−20.2421	−0.764533	−0.382267	−	0.924052i	$-0.624856\pi$
$701$	−20.2421	−0.764533	−0.382267	+	0.924052i	$0.624856\pi$
$702$	0	0
$703$	31.0146	1.16974
$704$	−4.26266	−0.160655
$705$	0	0
$706$	−37.8628	−1.42499
$707$	0	0
$708$	0	0
$709$	−32.2128	−1.20978	−0.604889	−	0.796310i	$-0.706782\pi$
$709$	−32.2128	−1.20978	−0.604889	+	0.796310i	$0.706782\pi$
$710$	10.3468	0.388307
$711$	0	0
$712$	9.43208	0.353482
$713$	3.64453	0.136489
$714$	0	0
$715$	5.42012	0.202701
$716$	20.9261	0.782046
$717$	0	0
$718$	−5.78492	−0.215891
$719$	8.45867	0.315455	0.157728	−	0.987483i	$-0.449583\pi$
$719$	8.45867	0.315455	0.157728	+	0.987483i	$0.449583\pi$
$720$	0	0
$721$	0	0
$722$	−15.5286	−0.577915
$723$	0	0
$724$	2.67947	0.0995816
$725$	−7.08532	−0.263142
$726$	0	0
$727$	−11.6550	−0.432260	−0.216130	−	0.976365i	$-0.569343\pi$
$727$	−11.6550	−0.432260	−0.216130	+	0.976365i	$0.569343\pi$
$728$	0	0
$729$	0	0
$730$	−12.9601	−0.479675
$731$	2.41460	0.0893072
$732$	0	0
$733$	13.0182	0.480840	0.240420	−	0.970669i	$-0.422715\pi$
$733$	13.0182	0.480840	0.240420	+	0.970669i	$0.422715\pi$
$734$	40.7747	1.50502
$735$	0	0
$736$	−18.7265	−0.690266
$737$	−24.8889	−0.916794
$738$	0	0
$739$	−24.3413	−0.895409	−0.447704	−	0.894182i	$-0.647758\pi$
$739$	−24.3413	−0.895409	−0.447704	+	0.894182i	$0.647758\pi$
$740$	−8.33804	−0.306512
$741$	0	0
$742$	0	0
$743$	12.1174	0.444545	0.222272	−	0.974985i	$-0.428653\pi$
$743$	12.1174	0.444545	0.222272	+	0.974985i	$0.428653\pi$
$744$	0	0
$745$	1.36423	0.0499816
$746$	−1.44440	−0.0528833
$747$	0	0
$748$	−11.3411	−0.414673
$749$	0	0
$750$	0	0
$751$	11.3779	0.415187	0.207594	−	0.978215i	$-0.433437\pi$
$751$	11.3779	0.415187	0.207594	+	0.978215i	$0.433437\pi$
$752$	−52.1615	−1.90214
$753$	0	0
$754$	−36.4633	−1.32791
$755$	23.8022	0.866252
$756$	0	0
$757$	−30.5664	−1.11096	−0.555478	−	0.831531i	$-0.687465\pi$
$757$	−30.5664	−1.11096	−0.555478	+	0.831531i	$0.687465\pi$
$758$	9.49230	0.344776
$759$	0	0
$760$	6.11126	0.221679
$761$	−10.3128	−0.373838	−0.186919	−	0.982375i	$-0.559850\pi$
$761$	−10.3128	−0.373838	−0.186919	+	0.982375i	$0.559850\pi$
$762$	0	0
$763$	0	0
$764$	1.97376	0.0714081
$765$	0	0
$766$	9.10187	0.328864
$767$	−29.2210	−1.05511
$768$	0	0
$769$	4.12788	0.148855	0.0744276	−	0.997226i	$-0.476287\pi$
$769$	4.12788	0.148855	0.0744276	+	0.997226i	$0.476287\pi$
$770$	0	0
$771$	0	0
$772$	19.4543	0.700176
$773$	13.1041	0.471323	0.235661	−	0.971835i	$-0.424274\pi$
$773$	13.1041	0.471323	0.235661	+	0.971835i	$0.424274\pi$
$774$	0	0
$775$	0.871553	0.0313071
$776$	32.4737	1.16574
$777$	0	0
$778$	−24.0165	−0.861035
$779$	8.65778	0.310197
$780$	0	0
$781$	10.8972	0.389934
$782$	53.5384	1.91453
$783$	0	0
$784$	0	0
$785$	−12.7101	−0.453641
$786$	0	0
$787$	8.39444	0.299230	0.149615	−	0.988744i	$-0.452197\pi$
$787$	8.39444	0.299230	0.149615	+	0.988744i	$0.452197\pi$
$788$	2.55594	0.0910514
$789$	0	0
$790$	−23.7344	−0.844432
$791$	0	0
$792$	0	0
$793$	20.1078	0.714047
$794$	−19.6421	−0.697070
$795$	0	0
$796$	3.71916	0.131822
$797$	16.8032	0.595199	0.297599	−	0.954691i	$-0.403814\pi$
$797$	16.8032	0.595199	0.297599	+	0.954691i	$0.403814\pi$
$798$	0	0
$799$	79.6448	2.81763
$800$	−4.47824	−0.158330
$801$	0	0
$802$	4.47981	0.158188
$803$	−13.6496	−0.481685
$804$	0	0
$805$	0	0
$806$	4.48528	0.157987
$807$	0	0
$808$	24.2702	0.853823
$809$	−19.6926	−0.692354	−0.346177	−	0.938169i	$-0.612520\pi$
$809$	−19.6926	−0.692354	−0.346177	+	0.938169i	$0.612520\pi$
$810$	0	0
$811$	17.3568	0.609481	0.304740	−	0.952435i	$-0.401430\pi$
$811$	17.3568	0.609481	0.304740	+	0.952435i	$0.401430\pi$
$812$	0	0
$813$	0	0
$814$	−29.6641	−1.03973
$815$	9.70227	0.339856
$816$	0	0
$817$	0.994481	0.0347925
$818$	−45.3758	−1.58653
$819$	0	0
$820$	−2.32758	−0.0812825
$821$	−35.9696	−1.25535	−0.627674	−	0.778476i	$-0.715993\pi$
$821$	−35.9696	−1.25535	−0.627674	+	0.778476i	$0.715993\pi$
$822$	0	0
$823$	−31.1463	−1.08569	−0.542846	−	0.839832i	$-0.682653\pi$
$823$	−31.1463	−1.08569	−0.542846	+	0.839832i	$0.682653\pi$
$824$	−7.64530	−0.266337
$825$	0	0
$826$	0	0
$827$	52.2577	1.81718	0.908589	−	0.417691i	$-0.137161\pi$
$827$	52.2577	1.81718	0.908589	+	0.417691i	$0.137161\pi$
$828$	0	0
$829$	4.55083	0.158057	0.0790284	−	0.996872i	$-0.474818\pi$
$829$	4.55083	0.158057	0.0790284	+	0.996872i	$0.474818\pi$
$830$	−21.6229	−0.750541
$831$	0	0
$832$	7.33135	0.254169
$833$	0	0
$834$	0	0
$835$	12.8486	0.444644
$836$	−4.67098	−0.161549
$837$	0	0
$838$	25.6063	0.884556
$839$	−43.3994	−1.49832	−0.749158	−	0.662392i	$-0.769541\pi$
$839$	−43.3994	−1.49832	−0.749158	+	0.662392i	$0.769541\pi$
$840$	0	0
$841$	21.2018	0.731096
$842$	−1.27721	−0.0440156
$843$	0	0
$844$	3.84493	0.132348
$845$	3.67794	0.126525
$846$	0	0
$847$	0	0
$848$	18.1030	0.621660
$849$	0	0
$850$	12.8032	0.439145
$851$	41.4558	1.42109
$852$	0	0
$853$	−11.9238	−0.408263	−0.204132	−	0.978943i	$-0.565437\pi$
$853$	−11.9238	−0.408263	−0.204132	+	0.978943i	$0.565437\pi$
$854$	0	0
$855$	0	0
$856$	7.29045	0.249183
$857$	34.5003	1.17851	0.589254	−	0.807947i	$-0.299422\pi$
$857$	34.5003	1.17851	0.589254	+	0.807947i	$0.299422\pi$
$858$	0	0
$859$	48.5311	1.65586	0.827930	−	0.560831i	$-0.189518\pi$
$859$	48.5311	1.65586	0.827930	+	0.560831i	$0.189518\pi$
$860$	−0.267358	−0.00911685
$861$	0	0
$862$	−38.9514	−1.32669
$863$	−37.2431	−1.26777	−0.633884	−	0.773428i	$-0.718540\pi$
$863$	−37.2431	−1.26777	−0.633884	+	0.773428i	$0.718540\pi$
$864$	0	0
$865$	15.9949	0.543843
$866$	26.0454	0.885059
$867$	0	0
$868$	0	0
$869$	−24.9972	−0.847971
$870$	0	0
$871$	42.8064	1.45044
$872$	−14.9573	−0.506517
$873$	0	0
$874$	22.0504	0.745866
$875$	0	0
$876$	0	0
$877$	22.4399	0.757740	0.378870	−	0.925450i	$-0.376313\pi$
$877$	22.4399	0.757740	0.378870	+	0.925450i	$0.376313\pi$
$878$	−48.1801	−1.62600
$879$	0	0
$880$	−8.83127	−0.297702
$881$	−15.4844	−0.521681	−0.260841	−	0.965382i	$-0.584000\pi$
$881$	−15.4844	−0.521681	−0.260841	+	0.965382i	$0.584000\pi$
$882$	0	0
$883$	7.19173	0.242021	0.121011	−	0.992651i	$-0.461387\pi$
$883$	7.19173	0.242021	0.121011	+	0.992651i	$0.461387\pi$
$884$	19.5056	0.656044
$885$	0	0
$886$	37.2690	1.25208
$887$	−26.8201	−0.900530	−0.450265	−	0.892895i	$-0.648670\pi$
$887$	−26.8201	−0.900530	−0.450265	+	0.892895i	$0.648670\pi$
$888$	0	0
$889$	0	0
$890$	8.13853	0.272804
$891$	0	0
$892$	3.81488	0.127732
$893$	32.8026	1.09770
$894$	0	0
$895$	−24.8807	−0.831670
$896$	0	0
$897$	0	0
$898$	44.5647	1.48714
$899$	−6.17523	−0.205956
$900$	0	0
$901$	−27.6413	−0.920865
$902$	−8.28077	−0.275720
$903$	0	0
$904$	37.9013	1.26058
$905$	−3.18583	−0.105900
$906$	0	0
$907$	22.5054	0.747281	0.373640	−	0.927574i	$-0.378109\pi$
$907$	22.5054	0.747281	0.373640	+	0.927574i	$0.378109\pi$
$908$	7.10657	0.235840
$909$	0	0
$910$	0	0
$911$	41.8798	1.38754	0.693769	−	0.720197i	$-0.255949\pi$
$911$	41.8798	1.38754	0.693769	+	0.720197i	$0.255949\pi$
$912$	0	0
$913$	−22.7733	−0.753687
$914$	59.0046	1.95170
$915$	0	0
$916$	16.0399	0.529973
$917$	0	0
$918$	0	0
$919$	−13.8119	−0.455613	−0.227807	−	0.973706i	$-0.573155\pi$
$919$	−13.8119	−0.455613	−0.227807	+	0.973706i	$0.573155\pi$
$920$	8.16864	0.269312
$921$	0	0
$922$	14.7955	0.487265
$923$	−18.7422	−0.616906
$924$	0	0
$925$	9.91375	0.325962
$926$	−71.0719	−2.33557
$927$	0	0
$928$	31.7298	1.04158
$929$	−1.12291	−0.0368414	−0.0184207	−	0.999830i	$-0.505864\pi$
$929$	−1.12291	−0.0368414	−0.0184207	+	0.999830i	$0.505864\pi$
$930$	0	0
$931$	0	0
$932$	−7.63936	−0.250235
$933$	0	0
$934$	18.7559	0.613711
$935$	13.4844	0.440985
$936$	0	0
$937$	1.43208	0.0467839	0.0233920	−	0.999726i	$-0.492553\pi$
$937$	1.43208	0.0467839	0.0233920	+	0.999726i	$0.492553\pi$
$938$	0	0
$939$	0	0
$940$	−8.81873	−0.287635
$941$	19.6972	0.642109	0.321055	−	0.947061i	$-0.395963\pi$
$941$	19.6972	0.642109	0.321055	+	0.947061i	$0.395963\pi$
$942$	0	0
$943$	11.5725	0.376851
$944$	47.6112	1.54961
$945$	0	0
$946$	−0.951175	−0.0309254
$947$	−25.1807	−0.818264	−0.409132	−	0.912475i	$-0.634168\pi$
$947$	−25.1807	−0.818264	−0.409132	+	0.912475i	$0.634168\pi$
$948$	0	0
$949$	23.4760	0.762063
$950$	5.27314	0.171083
$951$	0	0
$952$	0	0
$953$	−47.0884	−1.52534	−0.762671	−	0.646786i	$-0.776113\pi$
$953$	−47.0884	−1.52534	−0.762671	+	0.646786i	$0.776113\pi$
$954$	0	0
$955$	−2.34676	−0.0759392
$956$	14.4268	0.466596
$957$	0	0
$958$	41.4558	1.33938
$959$	0	0
$960$	0	0
$961$	−30.2404	−0.975497
$962$	51.0192	1.64493
$963$	0	0
$964$	−16.8076	−0.541337
$965$	−23.1307	−0.744605
$966$	0	0
$967$	−2.32113	−0.0746425	−0.0373212	−	0.999303i	$-0.511882\pi$
$967$	−2.32113	−0.0746425	−0.0373212	+	0.999303i	$0.511882\pi$
$968$	15.3318	0.492783
$969$	0	0
$970$	28.0202	0.899673
$971$	−12.5592	−0.403044	−0.201522	−	0.979484i	$-0.564589\pi$
$971$	−12.5592	−0.403044	−0.201522	+	0.979484i	$0.564589\pi$
$972$	0	0
$973$	0	0
$974$	−55.9035	−1.79126
$975$	0	0
$976$	−32.7626	−1.04870
$977$	9.66637	0.309255	0.154627	−	0.987973i	$-0.450582\pi$
$977$	9.66637	0.309255	0.154627	+	0.987973i	$0.450582\pi$
$978$	0	0
$979$	8.57153	0.273947
$980$	0	0
$981$	0	0
$982$	−43.5734	−1.39048
$983$	−46.0559	−1.46895	−0.734477	−	0.678633i	$-0.762573\pi$
$983$	−46.0559	−1.46895	−0.734477	+	0.678633i	$0.762573\pi$
$984$	0	0
$985$	−3.03895	−0.0968290
$986$	−90.7145	−2.88894
$987$	0	0
$988$	8.03361	0.255583
$989$	1.32928	0.0422686
$990$	0	0
$991$	27.8138	0.883534	0.441767	−	0.897130i	$-0.354352\pi$
$991$	27.8138	0.883534	0.441767	+	0.897130i	$0.354352\pi$
$992$	−3.90303	−0.123921
$993$	0	0
$994$	0	0
$995$	−4.42200	−0.140187
$996$	0	0
$997$	−46.5505	−1.47427	−0.737134	−	0.675746i	$-0.763822\pi$
$997$	−46.5505	−1.47427	−0.737134	+	0.675746i	$0.763822\pi$
$998$	1.02903	0.0325734
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.bf.1.4		4
3.2	odd	2		735.2.a.n.1.1	✓	4
7.6	odd	2		2205.2.a.bg.1.4		4
15.14	odd	2		3675.2.a.bl.1.4		4
21.2	odd	6		735.2.i.n.361.4		8
21.5	even	6		735.2.i.m.361.4		8
21.11	odd	6		735.2.i.n.226.4		8
21.17	even	6		735.2.i.m.226.4		8
21.20	even	2		735.2.a.o.1.1	yes	4
105.104	even	2		3675.2.a.bk.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.a.n.1.1	✓	4	3.2	odd	2
735.2.a.o.1.1	yes	4	21.20	even	2
735.2.i.m.226.4		8	21.17	even	6
735.2.i.m.361.4		8	21.5	even	6
735.2.i.n.226.4		8	21.11	odd	6
735.2.i.n.361.4		8	21.2	odd	6
2205.2.a.bf.1.4		4	1.1	even	1	trivial
2205.2.a.bg.1.4		4	7.6	odd	2
3675.2.a.bk.1.4		4	105.104	even	2
3675.2.a.bl.1.4		4	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q+1.68554 q^{2} +0.841058 q^{4} -1.00000 q^{5} -1.95345 q^{8} -1.68554 q^{10} -1.77522 q^{11} +3.05320 q^{13} -4.97474 q^{16} +7.59587 q^{17} +3.12845 q^{19} -0.841058 q^{20} -2.99222 q^{22} +4.18165 q^{23} +1.00000 q^{25} +5.14631 q^{26} -7.08532 q^{29} +0.871553 q^{31} -4.47824 q^{32} +12.8032 q^{34} +9.91375 q^{37} +5.27314 q^{38} +1.95345 q^{40} +2.76744 q^{41} +0.317883 q^{43} -1.49307 q^{44} +7.04836 q^{46} +10.4853 q^{47} +1.68554 q^{50} +2.56792 q^{52} -3.63899 q^{53} +1.77522 q^{55} -11.9426 q^{58} -9.57060 q^{59} +6.58579 q^{61} +1.46904 q^{62} +2.40120 q^{64} -3.05320 q^{65} +14.0202 q^{67} +6.38857 q^{68} -6.13853 q^{71} +7.68897 q^{73} +16.7101 q^{74} +2.63121 q^{76} +14.0811 q^{79} +4.97474 q^{80} +4.66464 q^{82} +12.8284 q^{83} -7.59587 q^{85} +0.535806 q^{86} +3.46780 q^{88} -4.82843 q^{89} +3.51701 q^{92} +17.6734 q^{94} -3.12845 q^{95} -16.6238 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 8 q^{4} - 4 q^{5} - 12 q^{8} + 4 q^{10} - 8 q^{11} + 12 q^{16} + 8 q^{17} + 8 q^{19} - 8 q^{20} + 4 q^{25} - 8 q^{29} + 8 q^{31} - 28 q^{32} + 8 q^{34} + 8 q^{37} + 4 q^{38} + 12 q^{40}+ \cdots - 8 q^{95}+O(q^{100}) \) 4 * q - 4 * q^2 + 8 * q^4 - 4 * q^5 - 12 * q^8 + 4 * q^10 - 8 * q^11 + 12 * q^16 + 8 * q^17 + 8 * q^19 - 8 * q^20 + 4 * q^25 - 8 * q^29 + 8 * q^31 - 28 * q^32 + 8 * q^34 + 8 * q^37 + 4 * q^38 + 12 * q^40 - 8 * q^43 + 16 * q^44 + 12 * q^46 + 8 * q^47 - 4 * q^50 + 32 * q^52 - 8 * q^53 + 8 * q^55 - 24 * q^58 + 16 * q^59 + 32 * q^61 - 20 * q^62 + 24 * q^64 + 24 * q^68 + 8 * q^71 + 32 * q^74 - 8 * q^76 - 12 * q^80 + 8 * q^82 + 40 * q^83 - 8 * q^85 + 32 * q^86 - 40 * q^88 - 8 * q^89 + 8 * q^92 + 16 * q^94 - 8 * q^95

Introduction

L-functions

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