LMFDB - Embedded newform 2205.4.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.56155 q^{2} -5.56155 q^{4} -5.00000 q^{5} +21.1771 q^{8} +O(q^{10})$	$q-1.56155 q^{2} -5.56155 q^{4} -5.00000 q^{5} +21.1771 q^{8} +7.80776 q^{10} -10.2462 q^{11} -34.3542 q^{13} +11.4233 q^{16} +82.6004 q^{17} -90.7083 q^{19} +27.8078 q^{20} +16.0000 q^{22} -12.1383 q^{23} +25.0000 q^{25} +53.6458 q^{26} -105.153 q^{29} +142.108 q^{31} -187.255 q^{32} -128.985 q^{34} +64.8466 q^{37} +141.646 q^{38} -105.885 q^{40} +195.201 q^{41} -319.218 q^{43} +56.9848 q^{44} +18.9545 q^{46} +318.847 q^{47} -39.0388 q^{50} +191.062 q^{52} -296.799 q^{53} +51.2311 q^{55} +164.203 q^{58} +284.000 q^{59} +494.540 q^{61} -221.909 q^{62} +201.022 q^{64} +171.771 q^{65} +549.032 q^{67} -459.386 q^{68} -740.972 q^{71} +556.850 q^{73} -101.261 q^{74} +504.479 q^{76} -376.189 q^{79} -57.1165 q^{80} -304.816 q^{82} -752.466 q^{83} -413.002 q^{85} +498.475 q^{86} -216.985 q^{88} +945.299 q^{89} +67.5076 q^{92} -497.896 q^{94} +453.542 q^{95} +180.668 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 7 q^{4} - 10 q^{5} - 3 q^{8}+O(q^{10}) $ 2 * q + q^2 - 7 * q^4 - 10 * q^5 - 3 * q^8	$ 2 q + q^{2} - 7 q^{4} - 10 q^{5} - 3 q^{8} - 5 q^{10} - 4 q^{11} + 22 q^{13} - 39 q^{16} + 58 q^{17} + 35 q^{20} + 32 q^{22} - 82 q^{23} + 50 q^{25} + 198 q^{26} - 334 q^{29} + 210 q^{31} - 123 q^{32} - 192 q^{34} + 6 q^{37} + 374 q^{38} + 15 q^{40} + 176 q^{41} + 46 q^{43} + 48 q^{44} - 160 q^{46} + 514 q^{47} + 25 q^{50} + 110 q^{52} - 808 q^{53} + 20 q^{55} - 422 q^{58} + 568 q^{59} + 618 q^{61} - 48 q^{62} + 769 q^{64} - 110 q^{65} + 694 q^{67} - 424 q^{68} - 814 q^{71} - 82 q^{73} - 252 q^{74} + 374 q^{76} + 600 q^{79} + 195 q^{80} - 354 q^{82} - 268 q^{83} - 290 q^{85} + 1434 q^{86} - 368 q^{88} - 72 q^{89} + 168 q^{92} + 2 q^{94} - 1626 q^{97}+O(q^{100}) $ 2 * q + q^2 - 7 * q^4 - 10 * q^5 - 3 * q^8 - 5 * q^10 - 4 * q^11 + 22 * q^13 - 39 * q^16 + 58 * q^17 + 35 * q^20 + 32 * q^22 - 82 * q^23 + 50 * q^25 + 198 * q^26 - 334 * q^29 + 210 * q^31 - 123 * q^32 - 192 * q^34 + 6 * q^37 + 374 * q^38 + 15 * q^40 + 176 * q^41 + 46 * q^43 + 48 * q^44 - 160 * q^46 + 514 * q^47 + 25 * q^50 + 110 * q^52 - 808 * q^53 + 20 * q^55 - 422 * q^58 + 568 * q^59 + 618 * q^61 - 48 * q^62 + 769 * q^64 - 110 * q^65 + 694 * q^67 - 424 * q^68 - 814 * q^71 - 82 * q^73 - 252 * q^74 + 374 * q^76 + 600 * q^79 + 195 * q^80 - 354 * q^82 - 268 * q^83 - 290 * q^85 + 1434 * q^86 - 368 * q^88 - 72 * q^89 + 168 * q^92 + 2 * q^94 - 1626 * q^97

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 $

$2$	−1.56155	−0.552092	−0.276046	−	0.961144i	$-0.589024\pi$
$2$	−1.56155	−0.552092	−0.276046	+	0.961144i	$0.589024\pi$
$3$	0	0
$3$	0	0
$4$	−5.56155	−0.695194
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.1771	0.935904
$9$	0	0
$10$	7.80776	0.246903
$11$	−10.2462	−0.280850	−0.140425	−	0.990091i	$-0.544847\pi$
$11$	−10.2462	−0.280850	−0.140425	+	0.990091i	$0.544847\pi$
$12$	0	0
$13$	−34.3542	−0.732933	−0.366467	−	0.930431i	$-0.619433\pi$
$13$	−34.3542	−0.732933	−0.366467	+	0.930431i	$0.619433\pi$
$14$	0	0
$15$	0	0
$16$	11.4233	0.178489
$17$	82.6004	1.17844	0.589222	−	0.807972i	$-0.299435\pi$
$17$	82.6004	1.17844	0.589222	+	0.807972i	$0.299435\pi$
$18$	0	0
$19$	−90.7083	−1.09526	−0.547629	−	0.836721i	$-0.684470\pi$
$19$	−90.7083	−1.09526	−0.547629	+	0.836721i	$0.684470\pi$
$20$	27.8078	0.310900
$21$	0	0
$22$	16.0000	0.155055
$23$	−12.1383	−0.110044	−0.0550218	−	0.998485i	$-0.517523\pi$
$23$	−12.1383	−0.110044	−0.0550218	+	0.998485i	$0.517523\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	53.6458	0.404647
$27$	0	0
$28$	0	0
$29$	−105.153	−0.673328	−0.336664	−	0.941625i	$-0.609299\pi$
$29$	−105.153	−0.673328	−0.336664	+	0.941625i	$0.609299\pi$
$30$	0	0
$31$	142.108	0.823334	0.411667	−	0.911334i	$-0.364947\pi$
$31$	142.108	0.823334	0.411667	+	0.911334i	$0.364947\pi$
$32$	−187.255	−1.03445
$33$	0	0
$34$	−128.985	−0.650609
$35$	0	0
$36$	0	0
$37$	64.8466	0.288127	0.144064	−	0.989568i	$-0.453983\pi$
$37$	64.8466	0.288127	0.144064	+	0.989568i	$0.453983\pi$
$38$	141.646	0.604684
$39$	0	0
$40$	−105.885	−0.418549
$41$	195.201	0.743542	0.371771	−	0.928324i	$-0.378751\pi$
$41$	195.201	0.743542	0.371771	+	0.928324i	$0.378751\pi$
$42$	0	0
$43$	−319.218	−1.13210	−0.566049	−	0.824371i	$-0.691529\pi$
$43$	−319.218	−1.13210	−0.566049	+	0.824371i	$0.691529\pi$
$44$	56.9848	0.195245
$45$	0	0
$46$	18.9545	0.0607542
$47$	318.847	0.989544	0.494772	−	0.869023i	$-0.335252\pi$
$47$	318.847	0.989544	0.494772	+	0.869023i	$0.335252\pi$
$48$	0	0
$49$	0	0
$50$	−39.0388	−0.110418
$51$	0	0
$52$	191.062	0.509531
$53$	−296.799	−0.769217	−0.384609	−	0.923080i	$-0.625664\pi$
$53$	−296.799	−0.769217	−0.384609	+	0.923080i	$0.625664\pi$
$54$	0	0
$55$	51.2311	0.125600
$56$	0	0
$57$	0	0
$58$	164.203	0.371739
$59$	284.000	0.626672	0.313336	−	0.949642i	$-0.398553\pi$
$59$	284.000	0.626672	0.313336	+	0.949642i	$0.398553\pi$
$60$	0	0
$61$	494.540	1.03802	0.519011	−	0.854768i	$-0.326300\pi$
$61$	494.540	1.03802	0.519011	+	0.854768i	$0.326300\pi$
$62$	−221.909	−0.454556
$63$	0	0
$64$	201.022	0.392621
$65$	171.771	0.327778
$66$	0	0
$67$	549.032	1.00112	0.500559	−	0.865702i	$-0.333128\pi$
$67$	549.032	1.00112	0.500559	+	0.865702i	$0.333128\pi$
$68$	−459.386	−0.819247
$69$	0	0
$70$	0	0
$71$	−740.972	−1.23855	−0.619276	−	0.785174i	$-0.712574\pi$
$71$	−740.972	−1.23855	−0.619276	+	0.785174i	$0.712574\pi$
$72$	0	0
$73$	556.850	0.892800	0.446400	−	0.894834i	$-0.352706\pi$
$73$	556.850	0.892800	0.446400	+	0.894834i	$0.352706\pi$
$74$	−101.261	−0.159073
$75$	0	0
$76$	504.479	0.761417
$77$	0	0
$78$	0	0
$79$	−376.189	−0.535755	−0.267877	−	0.963453i	$-0.586322\pi$
$79$	−376.189	−0.535755	−0.267877	+	0.963453i	$0.586322\pi$
$80$	−57.1165	−0.0798227
$81$	0	0
$82$	−304.816	−0.410504
$83$	−752.466	−0.995107	−0.497553	−	0.867433i	$-0.665768\pi$
$83$	−752.466	−0.995107	−0.497553	+	0.867433i	$0.665768\pi$
$84$	0	0
$85$	−413.002	−0.527016
$86$	498.475	0.625023
$87$	0	0
$88$	−216.985	−0.262848
$89$	945.299	1.12586	0.562930	−	0.826505i	$-0.309674\pi$
$89$	945.299	1.12586	0.562930	+	0.826505i	$0.309674\pi$
$90$	0	0
$91$	0	0
$92$	67.5076	0.0765016
$93$	0	0
$94$	−497.896	−0.546319
$95$	453.542	0.489815
$96$	0	0
$97$	180.668	0.189114	0.0945572	−	0.995519i	$-0.469856\pi$
$97$	180.668	0.189114	0.0945572	+	0.995519i	$0.469856\pi$
$98$	0	0
$99$	0	0
$100$	−139.039	−0.139039
$101$	1463.05	1.44138	0.720689	−	0.693258i	$-0.243826\pi$
$101$	1463.05	1.44138	0.720689	+	0.693258i	$0.243826\pi$
$102$	0	0
$103$	−535.049	−0.511844	−0.255922	−	0.966697i	$-0.582379\pi$
$103$	−535.049	−0.511844	−0.255922	+	0.966697i	$0.582379\pi$
$104$	−727.521	−0.685955
$105$	0	0
$106$	463.468	0.424679
$107$	327.153	0.295581	0.147790	−	0.989019i	$-0.452784\pi$
$107$	327.153	0.295581	0.147790	+	0.989019i	$0.452784\pi$
$108$	0	0
$109$	1621.70	1.42506	0.712528	−	0.701644i	$-0.247550\pi$
$109$	1621.70	1.42506	0.712528	+	0.701644i	$0.247550\pi$
$110$	−80.0000	−0.0693427
$111$	0	0
$112$	0	0
$113$	450.716	0.375219	0.187610	−	0.982244i	$-0.439926\pi$
$113$	450.716	0.375219	0.187610	+	0.982244i	$0.439926\pi$
$114$	0	0
$115$	60.6913	0.0492130
$116$	584.816	0.468093
$117$	0	0
$118$	−443.481	−0.345981
$119$	0	0
$120$	0	0
$121$	−1226.02	−0.921123
$122$	−772.250	−0.573084
$123$	0	0
$124$	−790.341	−0.572377
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1710.10	1.19486	0.597428	−	0.801923i	$-0.296189\pi$
$127$	1710.10	1.19486	0.597428	+	0.801923i	$0.296189\pi$
$128$	1184.13	0.817683
$129$	0	0
$130$	−268.229	−0.180964
$131$	−1584.49	−1.05678	−0.528388	−	0.849003i	$-0.677203\pi$
$131$	−1584.49	−1.05678	−0.528388	+	0.849003i	$0.677203\pi$
$132$	0	0
$133$	0	0
$134$	−857.343	−0.552710
$135$	0	0
$136$	1749.23	1.10291
$137$	−2095.24	−1.30663	−0.653316	−	0.757085i	$-0.726623\pi$
$137$	−2095.24	−1.30663	−0.653316	+	0.757085i	$0.726623\pi$
$138$	0	0
$139$	1385.00	0.845135	0.422568	−	0.906331i	$-0.361129\pi$
$139$	1385.00	0.845135	0.422568	+	0.906331i	$0.361129\pi$
$140$	0	0
$141$	0	0
$142$	1157.07	0.683795
$143$	352.000	0.205844
$144$	0	0
$145$	525.767	0.301121
$146$	−869.551	−0.492908
$147$	0	0
$148$	−360.648	−0.200304
$149$	2502.89	1.37614	0.688069	−	0.725645i	$-0.258459\pi$
$149$	2502.89	1.37614	0.688069	+	0.725645i	$0.258459\pi$
$150$	0	0
$151$	395.212	0.212993	0.106496	−	0.994313i	$-0.466037\pi$
$151$	395.212	0.212993	0.106496	+	0.994313i	$0.466037\pi$
$152$	−1920.94	−1.02506
$153$	0	0
$154$	0	0
$155$	−710.540	−0.368206
$156$	0	0
$157$	−1732.35	−0.880615	−0.440308	−	0.897847i	$-0.645131\pi$
$157$	−1732.35	−0.880615	−0.440308	+	0.897847i	$0.645131\pi$
$158$	587.439	0.295786
$159$	0	0
$160$	936.274	0.462618
$161$	0	0
$162$	0	0
$163$	3303.26	1.58731	0.793654	−	0.608369i	$-0.208176\pi$
$163$	3303.26	1.58731	0.793654	+	0.608369i	$0.208176\pi$
$164$	−1085.62	−0.516906
$165$	0	0
$166$	1175.02	0.549391
$167$	−601.941	−0.278920	−0.139460	−	0.990228i	$-0.544537\pi$
$167$	−601.941	−0.278920	−0.139460	+	0.990228i	$0.544537\pi$
$168$	0	0
$169$	−1016.79	−0.462809
$170$	644.924	0.290961
$171$	0	0
$172$	1775.35	0.787028
$173$	−3741.17	−1.64414	−0.822068	−	0.569389i	$-0.807180\pi$
$173$	−3741.17	−1.64414	−0.822068	+	0.569389i	$0.807180\pi$
$174$	0	0
$175$	0	0
$176$	−117.045	−0.0501286
$177$	0	0
$178$	−1476.13	−0.621578
$179$	968.352	0.404347	0.202173	−	0.979350i	$-0.435200\pi$
$179$	968.352	0.404347	0.202173	+	0.979350i	$0.435200\pi$
$180$	0	0
$181$	1895.65	0.778465	0.389233	−	0.921139i	$-0.372740\pi$
$181$	1895.65	0.778465	0.389233	+	0.921139i	$0.372740\pi$
$182$	0	0
$183$	0	0
$184$	−257.053	−0.102990
$185$	−324.233	−0.128854
$186$	0	0
$187$	−846.341	−0.330966
$188$	−1773.28	−0.687925
$189$	0	0
$190$	−708.229	−0.270423
$191$	−2729.88	−1.03417	−0.517086	−	0.855933i	$-0.672983\pi$
$191$	−2729.88	−1.03417	−0.517086	+	0.855933i	$0.672983\pi$
$192$	0	0
$193$	1460.44	0.544689	0.272345	−	0.962200i	$-0.412201\pi$
$193$	1460.44	0.544689	0.272345	+	0.962200i	$0.412201\pi$
$194$	−282.123	−0.104409
$195$	0	0
$196$	0	0
$197$	−4831.42	−1.74733	−0.873665	−	0.486527i	$-0.838263\pi$
$197$	−4831.42	−1.74733	−0.873665	+	0.486527i	$0.838263\pi$
$198$	0	0
$199$	873.070	0.311006	0.155503	−	0.987835i	$-0.450300\pi$
$199$	873.070	0.311006	0.155503	+	0.987835i	$0.450300\pi$
$200$	529.427	0.187181
$201$	0	0
$202$	−2284.63	−0.795774
$203$	0	0
$204$	0	0
$205$	−976.004	−0.332522
$206$	835.508	0.282585
$207$	0	0
$208$	−392.438	−0.130820
$209$	929.417	0.307603
$210$	0	0
$211$	−5148.02	−1.67964	−0.839821	−	0.542863i	$-0.817340\pi$
$211$	−5148.02	−1.67964	−0.839821	+	0.542863i	$0.817340\pi$
$212$	1650.66	0.534755
$213$	0	0
$214$	−510.867	−0.163188
$215$	1596.09	0.506290
$216$	0	0
$217$	0	0
$218$	−2532.38	−0.786762
$219$	0	0
$220$	−284.924	−0.0873163
$221$	−2837.67	−0.863720
$222$	0	0
$223$	2435.31	0.731304	0.365652	−	0.930752i	$-0.380846\pi$
$223$	2435.31	0.731304	0.365652	+	0.930752i	$0.380846\pi$
$224$	0	0
$225$	0	0
$226$	−703.817	−0.207156
$227$	3508.47	1.02584	0.512919	−	0.858437i	$-0.328564\pi$
$227$	3508.47	1.02584	0.512919	+	0.858437i	$0.328564\pi$
$228$	0	0
$229$	−3429.54	−0.989653	−0.494827	−	0.868992i	$-0.664768\pi$
$229$	−3429.54	−0.989653	−0.494827	+	0.868992i	$0.664768\pi$
$230$	−94.7727	−0.0271701
$231$	0	0
$232$	−2226.84	−0.630170
$233$	−3989.71	−1.12178	−0.560890	−	0.827890i	$-0.689541\pi$
$233$	−3989.71	−1.12178	−0.560890	+	0.827890i	$0.689541\pi$
$234$	0	0
$235$	−1594.23	−0.442537
$236$	−1579.48	−0.435659
$237$	0	0
$238$	0	0
$239$	−5934.71	−1.60621	−0.803105	−	0.595837i	$-0.796820\pi$
$239$	−5934.71	−1.60621	−0.803105	+	0.595837i	$0.796820\pi$
$240$	0	0
$241$	6481.90	1.73251	0.866257	−	0.499598i	$-0.166519\pi$
$241$	6481.90	1.73251	0.866257	+	0.499598i	$0.166519\pi$
$242$	1914.49	0.508545
$243$	0	0
$244$	−2750.41	−0.721627
$245$	0	0
$246$	0	0
$247$	3116.21	0.802751
$248$	3009.43	0.770561
$249$	0	0
$250$	195.194	0.0493806
$251$	−5257.70	−1.32216	−0.661081	−	0.750314i	$-0.729902\pi$
$251$	−5257.70	−1.32216	−0.661081	+	0.750314i	$0.729902\pi$
$252$	0	0
$253$	124.371	0.0309057
$254$	−2670.41	−0.659671
$255$	0	0
$256$	−3457.26	−0.844057
$257$	−4108.52	−0.997209	−0.498604	−	0.866830i	$-0.666154\pi$
$257$	−4108.52	−0.997209	−0.498604	+	0.866830i	$0.666154\pi$
$258$	0	0
$259$	0	0
$260$	−955.312	−0.227869
$261$	0	0
$262$	2474.27	0.583438
$263$	−6236.89	−1.46229	−0.731146	−	0.682221i	$-0.761014\pi$
$263$	−6236.89	−1.46229	−0.731146	+	0.682221i	$0.761014\pi$
$264$	0	0
$265$	1484.00	0.344004
$266$	0	0
$267$	0	0
$268$	−3053.47	−0.695972
$269$	4356.67	0.987475	0.493738	−	0.869611i	$-0.335630\pi$
$269$	4356.67	0.987475	0.493738	+	0.869611i	$0.335630\pi$
$270$	0	0
$271$	63.0970	0.0141434	0.00707171	−	0.999975i	$-0.497749\pi$
$271$	63.0970	0.0141434	0.00707171	+	0.999975i	$0.497749\pi$
$272$	943.568	0.210339
$273$	0	0
$274$	3271.83	0.721382
$275$	−256.155	−0.0561700
$276$	0	0
$277$	5804.64	1.25909	0.629543	−	0.776966i	$-0.283242\pi$
$277$	5804.64	1.25909	0.629543	+	0.776966i	$0.283242\pi$
$278$	−2162.74	−0.466593
$279$	0	0
$280$	0	0
$281$	−8430.89	−1.78984	−0.894920	−	0.446227i	$-0.852767\pi$
$281$	−8430.89	−1.78984	−0.894920	+	0.446227i	$0.852767\pi$
$282$	0	0
$283$	−7415.73	−1.55767	−0.778833	−	0.627232i	$-0.784188\pi$
$283$	−7415.73	−1.55767	−0.778833	+	0.627232i	$0.784188\pi$
$284$	4120.95	0.861034
$285$	0	0
$286$	−549.667	−0.113645
$287$	0	0
$288$	0	0
$289$	1909.82	0.388728
$290$	−821.013	−0.166247
$291$	0	0
$292$	−3096.95	−0.620669
$293$	482.022	0.0961094	0.0480547	−	0.998845i	$-0.484698\pi$
$293$	482.022	0.0961094	0.0480547	+	0.998845i	$0.484698\pi$
$294$	0	0
$295$	−1420.00	−0.280256
$296$	1373.26	0.269659
$297$	0	0
$298$	−3908.39	−0.759755
$299$	417.000	0.0806546
$300$	0	0
$301$	0	0
$302$	−617.144	−0.117592
$303$	0	0
$304$	−1036.19	−0.195492
$305$	−2472.70	−0.464217
$306$	0	0
$307$	−2757.90	−0.512709	−0.256354	−	0.966583i	$-0.582521\pi$
$307$	−2757.90	−0.512709	−0.256354	+	0.966583i	$0.582521\pi$
$308$	0	0
$309$	0	0
$310$	1109.55	0.203284
$311$	−5817.25	−1.06066	−0.530331	−	0.847791i	$-0.677932\pi$
$311$	−5817.25	−1.06066	−0.530331	+	0.847791i	$0.677932\pi$
$312$	0	0
$313$	−2232.83	−0.403218	−0.201609	−	0.979466i	$-0.564617\pi$
$313$	−2232.83	−0.403218	−0.201609	+	0.979466i	$0.564617\pi$
$314$	2705.16	0.486181
$315$	0	0
$316$	2092.20	0.372453
$317$	81.4013	0.0144226	0.00721128	−	0.999974i	$-0.497705\pi$
$317$	81.4013	0.0144226	0.00721128	+	0.999974i	$0.497705\pi$
$318$	0	0
$319$	1077.42	0.189104
$320$	−1005.11	−0.175585
$321$	0	0
$322$	0	0
$323$	−7492.54	−1.29070
$324$	0	0
$325$	−858.854	−0.146587
$326$	−5158.21	−0.876341
$327$	0	0
$328$	4133.78	0.695884
$329$	0	0
$330$	0	0
$331$	−6550.18	−1.08771	−0.543853	−	0.839181i	$-0.683035\pi$
$331$	−6550.18	−1.08771	−0.543853	+	0.839181i	$0.683035\pi$
$332$	4184.88	0.691792
$333$	0	0
$334$	939.963	0.153990
$335$	−2745.16	−0.447714
$336$	0	0
$337$	4124.63	0.666715	0.333358	−	0.942800i	$-0.391818\pi$
$337$	4124.63	0.666715	0.333358	+	0.942800i	$0.391818\pi$
$338$	1587.77	0.255513
$339$	0	0
$340$	2296.93	0.366378
$341$	−1456.07	−0.231233
$342$	0	0
$343$	0	0
$344$	−6760.10	−1.05954
$345$	0	0
$346$	5842.03	0.907715
$347$	1228.44	0.190047	0.0950233	−	0.995475i	$-0.469707\pi$
$347$	1228.44	0.190047	0.0950233	+	0.995475i	$0.469707\pi$
$348$	0	0
$349$	−3320.93	−0.509357	−0.254678	−	0.967026i	$-0.581970\pi$
$349$	−3320.93	−0.509357	−0.254678	+	0.967026i	$0.581970\pi$
$350$	0	0
$351$	0	0
$352$	1918.65	0.290524
$353$	4589.25	0.691958	0.345979	−	0.938242i	$-0.387547\pi$
$353$	4589.25	0.691958	0.345979	+	0.938242i	$0.387547\pi$
$354$	0	0
$355$	3704.86	0.553897
$356$	−5257.33	−0.782691
$357$	0	0
$358$	−1512.13	−0.223237
$359$	12381.6	1.82026	0.910130	−	0.414323i	$-0.135982\pi$
$359$	12381.6	1.82026	0.910130	+	0.414323i	$0.135982\pi$
$360$	0	0
$361$	1369.00	0.199592
$362$	−2960.15	−0.429785
$363$	0	0
$364$	0	0
$365$	−2784.25	−0.399272
$366$	0	0
$367$	−5612.33	−0.798259	−0.399130	−	0.916894i	$-0.630688\pi$
$367$	−5612.33	−0.798259	−0.399130	+	0.916894i	$0.630688\pi$
$368$	−138.659	−0.0196416
$369$	0	0
$370$	506.307	0.0711396
$371$	0	0
$372$	0	0
$373$	−3691.05	−0.512374	−0.256187	−	0.966627i	$-0.582466\pi$
$373$	−3691.05	−0.512374	−0.256187	+	0.966627i	$0.582466\pi$
$374$	1321.61	0.182724
$375$	0	0
$376$	6752.24	0.926118
$377$	3612.46	0.493504
$378$	0	0
$379$	5350.16	0.725117	0.362559	−	0.931961i	$-0.381903\pi$
$379$	5350.16	0.725117	0.362559	+	0.931961i	$0.381903\pi$
$380$	−2522.40	−0.340516
$381$	0	0
$382$	4262.85	0.570959
$383$	−7300.09	−0.973936	−0.486968	−	0.873420i	$-0.661897\pi$
$383$	−7300.09	−0.973936	−0.486968	+	0.873420i	$0.661897\pi$
$384$	0	0
$385$	0	0
$386$	−2280.56	−0.300719
$387$	0	0
$388$	−1004.80	−0.131471
$389$	−9102.27	−1.18638	−0.593192	−	0.805061i	$-0.702133\pi$
$389$	−9102.27	−1.18638	−0.593192	+	0.805061i	$0.702133\pi$
$390$	0	0
$391$	−1002.62	−0.129680
$392$	0	0
$393$	0	0
$394$	7544.51	0.964688
$395$	1880.95	0.239597
$396$	0	0
$397$	11520.1	1.45637	0.728185	−	0.685381i	$-0.240364\pi$
$397$	11520.1	1.45637	0.728185	+	0.685381i	$0.240364\pi$
$398$	−1363.34	−0.171704
$399$	0	0
$400$	285.582	0.0356978
$401$	3812.93	0.474834	0.237417	−	0.971408i	$-0.423699\pi$
$401$	3812.93	0.474834	0.237417	+	0.971408i	$0.423699\pi$
$402$	0	0
$403$	−4882.00	−0.603448
$404$	−8136.85	−1.00204
$405$	0	0
$406$	0	0
$407$	−664.432	−0.0809205
$408$	0	0
$409$	4098.82	0.495535	0.247767	−	0.968820i	$-0.420303\pi$
$409$	4098.82	0.495535	0.247767	+	0.968820i	$0.420303\pi$
$410$	1524.08	0.183583
$411$	0	0
$412$	2975.70	0.355831
$413$	0	0
$414$	0	0
$415$	3762.33	0.445025
$416$	6432.98	0.758180
$417$	0	0
$418$	−1451.33	−0.169825
$419$	−14312.4	−1.66876	−0.834378	−	0.551193i	$-0.814173\pi$
$419$	−14312.4	−1.66876	−0.834378	+	0.551193i	$0.814173\pi$
$420$	0	0
$421$	6679.57	0.773259	0.386630	−	0.922235i	$-0.373639\pi$
$421$	6679.57	0.773259	0.386630	+	0.922235i	$0.373639\pi$
$422$	8038.91	0.927317
$423$	0	0
$424$	−6285.34	−0.719913
$425$	2065.01	0.235689
$426$	0	0
$427$	0	0
$428$	−1819.48	−0.205486
$429$	0	0
$430$	−2492.38	−0.279519
$431$	11503.8	1.28566	0.642828	−	0.766010i	$-0.277761\pi$
$431$	11503.8	1.28566	0.642828	+	0.766010i	$0.277761\pi$
$432$	0	0
$433$	−3243.99	−0.360037	−0.180019	−	0.983663i	$-0.557616\pi$
$433$	−3243.99	−0.360037	−0.180019	+	0.983663i	$0.557616\pi$
$434$	0	0
$435$	0	0
$436$	−9019.19	−0.990691
$437$	1101.04	0.120526
$438$	0	0
$439$	2744.21	0.298346	0.149173	−	0.988811i	$-0.452339\pi$
$439$	2744.21	0.298346	0.149173	+	0.988811i	$0.452339\pi$
$440$	1084.92	0.117549
$441$	0	0
$442$	4431.17	0.476853
$443$	8180.27	0.877328	0.438664	−	0.898651i	$-0.355452\pi$
$443$	8180.27	0.877328	0.438664	+	0.898651i	$0.355452\pi$
$444$	0	0
$445$	−4726.50	−0.503500
$446$	−3802.87	−0.403747
$447$	0	0
$448$	0	0
$449$	6725.70	0.706916	0.353458	−	0.935450i	$-0.385006\pi$
$449$	6725.70	0.706916	0.353458	+	0.935450i	$0.385006\pi$
$450$	0	0
$451$	−2000.07	−0.208824
$452$	−2506.68	−0.260850
$453$	0	0
$454$	−5478.65	−0.566357
$455$	0	0
$456$	0	0
$457$	−10807.0	−1.10619	−0.553095	−	0.833118i	$-0.686553\pi$
$457$	−10807.0	−1.10619	−0.553095	+	0.833118i	$0.686553\pi$
$458$	5355.41	0.546380
$459$	0	0
$460$	−337.538	−0.0342126
$461$	−17775.8	−1.79588	−0.897942	−	0.440113i	$-0.854938\pi$
$461$	−17775.8	−1.79588	−0.897942	+	0.440113i	$0.854938\pi$
$462$	0	0
$463$	−4418.93	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$463$	−4418.93	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$464$	−1201.20	−0.120182
$465$	0	0
$466$	6230.15	0.619326
$467$	−3373.08	−0.334235	−0.167117	−	0.985937i	$-0.553446\pi$
$467$	−3373.08	−0.334235	−0.167117	+	0.985937i	$0.553446\pi$
$468$	0	0
$469$	0	0
$470$	2489.48	0.244321
$471$	0	0
$472$	6014.29	0.586505
$473$	3270.77	0.317950
$474$	0	0
$475$	−2267.71	−0.219052
$476$	0	0
$477$	0	0
$478$	9267.36	0.886776
$479$	−4620.02	−0.440697	−0.220349	−	0.975421i	$-0.570720\pi$
$479$	−4620.02	−0.440697	−0.220349	+	0.975421i	$0.570720\pi$
$480$	0	0
$481$	−2227.75	−0.211178
$482$	−10121.8	−0.956508
$483$	0	0
$484$	6818.55	0.640360
$485$	−903.342	−0.0845746
$486$	0	0
$487$	−9108.62	−0.847538	−0.423769	−	0.905770i	$-0.639293\pi$
$487$	−9108.62	−0.847538	−0.423769	+	0.905770i	$0.639293\pi$
$488$	10472.9	0.971488
$489$	0	0
$490$	0	0
$491$	−2696.84	−0.247875	−0.123938	−	0.992290i	$-0.539552\pi$
$491$	−2696.84	−0.247875	−0.123938	+	0.992290i	$0.539552\pi$
$492$	0	0
$493$	−8685.71	−0.793478
$494$	−4866.12	−0.443193
$495$	0	0
$496$	1623.34	0.146956
$497$	0	0
$498$	0	0
$499$	−20054.2	−1.79910	−0.899549	−	0.436820i	$-0.856104\pi$
$499$	−20054.2	−1.79910	−0.899549	+	0.436820i	$0.856104\pi$
$500$	695.194	0.0621801
$501$	0	0
$502$	8210.17	0.729956
$503$	−8106.49	−0.718590	−0.359295	−	0.933224i	$-0.616983\pi$
$503$	−8106.49	−0.718590	−0.359295	+	0.933224i	$0.616983\pi$
$504$	0	0
$505$	−7315.26	−0.644604
$506$	−194.212	−0.0170628
$507$	0	0
$508$	−9510.80	−0.830657
$509$	−12455.8	−1.08466	−0.542332	−	0.840164i	$-0.682458\pi$
$509$	−12455.8	−1.08466	−0.542332	+	0.840164i	$0.682458\pi$
$510$	0	0
$511$	0	0
$512$	−4074.36	−0.351686
$513$	0	0
$514$	6415.68	0.550551
$515$	2675.25	0.228904
$516$	0	0
$517$	−3266.97	−0.277913
$518$	0	0
$519$	0	0
$520$	3637.60	0.306768
$521$	5539.61	0.465825	0.232913	−	0.972498i	$-0.425174\pi$
$521$	5539.61	0.465825	0.232913	+	0.972498i	$0.425174\pi$
$522$	0	0
$523$	−15504.2	−1.29628	−0.648138	−	0.761523i	$-0.724452\pi$
$523$	−15504.2	−1.29628	−0.648138	+	0.761523i	$0.724452\pi$
$524$	8812.24	0.734665
$525$	0	0
$526$	9739.23	0.807321
$527$	11738.2	0.970252
$528$	0	0
$529$	−12019.7	−0.987890
$530$	−2317.34	−0.189922
$531$	0	0
$532$	0	0
$533$	−6705.96	−0.544967
$534$	0	0
$535$	−1635.77	−0.132188
$536$	11626.9	0.936951
$537$	0	0
$538$	−6803.17	−0.545178
$539$	0	0
$540$	0	0
$541$	7853.16	0.624092	0.312046	−	0.950067i	$-0.398986\pi$
$541$	7853.16	0.624092	0.312046	+	0.950067i	$0.398986\pi$
$542$	−98.5293	−0.00780848
$543$	0	0
$544$	−15467.3	−1.21904
$545$	−8108.52	−0.637304
$546$	0	0
$547$	−9177.75	−0.717390	−0.358695	−	0.933455i	$-0.616778\pi$
$547$	−9177.75	−0.717390	−0.358695	+	0.933455i	$0.616778\pi$
$548$	11652.8	0.908363
$549$	0	0
$550$	400.000	0.0310110
$551$	9538.29	0.737468
$552$	0	0
$553$	0	0
$554$	−9064.25	−0.695132
$555$	0	0
$556$	−7702.73	−0.587533
$557$	−25030.2	−1.90407	−0.952033	−	0.305996i	$-0.901010\pi$
$557$	−25030.2	−1.90407	−0.952033	+	0.305996i	$0.901010\pi$
$558$	0	0
$559$	10966.5	0.829753
$560$	0	0
$561$	0	0
$562$	13165.3	0.988157
$563$	−19486.1	−1.45869	−0.729345	−	0.684146i	$-0.760175\pi$
$563$	−19486.1	−1.45869	−0.729345	+	0.684146i	$0.760175\pi$
$564$	0	0
$565$	−2253.58	−0.167803
$566$	11580.0	0.859975
$567$	0	0
$568$	−15691.6	−1.15916
$569$	−5813.63	−0.428331	−0.214165	−	0.976797i	$-0.568703\pi$
$569$	−5813.63	−0.428331	−0.214165	+	0.976797i	$0.568703\pi$
$570$	0	0
$571$	268.770	0.0196982	0.00984910	−	0.999951i	$-0.496865\pi$
$571$	268.770	0.0196982	0.00984910	+	0.999951i	$0.496865\pi$
$572$	−1957.67	−0.143102
$573$	0	0
$574$	0	0
$575$	−303.457	−0.0220087
$576$	0	0
$577$	−8283.57	−0.597660	−0.298830	−	0.954306i	$-0.596596\pi$
$577$	−8283.57	−0.597660	−0.298830	+	0.954306i	$0.596596\pi$
$578$	−2982.29	−0.214614
$579$	0	0
$580$	−2924.08	−0.209338
$581$	0	0
$582$	0	0
$583$	3041.07	0.216035
$584$	11792.5	0.835575
$585$	0	0
$586$	−752.703	−0.0530612
$587$	6671.87	0.469127	0.234563	−	0.972101i	$-0.424634\pi$
$587$	6671.87	0.469127	0.234563	+	0.972101i	$0.424634\pi$
$588$	0	0
$589$	−12890.4	−0.901763
$590$	2217.40	0.154727
$591$	0	0
$592$	740.761	0.0514275
$593$	11409.0	0.790071	0.395035	−	0.918666i	$-0.370732\pi$
$593$	11409.0	0.790071	0.395035	+	0.918666i	$0.370732\pi$
$594$	0	0
$595$	0	0
$596$	−13919.9	−0.956683
$597$	0	0
$598$	−651.167	−0.0445288
$599$	−13118.3	−0.894826	−0.447413	−	0.894327i	$-0.647655\pi$
$599$	−13118.3	−0.894826	−0.447413	+	0.894327i	$0.647655\pi$
$600$	0	0
$601$	−21970.2	−1.49115	−0.745577	−	0.666420i	$-0.767826\pi$
$601$	−21970.2	−1.49115	−0.745577	+	0.666420i	$0.767826\pi$
$602$	0	0
$603$	0	0
$604$	−2197.99	−0.148071
$605$	6130.08	0.411939
$606$	0	0
$607$	21115.1	1.41192	0.705959	−	0.708253i	$-0.250516\pi$
$607$	21115.1	1.41192	0.705959	+	0.708253i	$0.250516\pi$
$608$	16985.6	1.13299
$609$	0	0
$610$	3861.25	0.256291
$611$	−10953.7	−0.725269
$612$	0	0
$613$	−1963.61	−0.129379	−0.0646895	−	0.997905i	$-0.520606\pi$
$613$	−1963.61	−0.129379	−0.0646895	+	0.997905i	$0.520606\pi$
$614$	4306.60	0.283062
$615$	0	0
$616$	0	0
$617$	17352.1	1.13221	0.566103	−	0.824335i	$-0.308450\pi$
$617$	17352.1	1.13221	0.566103	+	0.824335i	$0.308450\pi$
$618$	0	0
$619$	3614.68	0.234711	0.117356	−	0.993090i	$-0.462558\pi$
$619$	3614.68	0.234711	0.117356	+	0.993090i	$0.462558\pi$
$620$	3951.70	0.255975
$621$	0	0
$622$	9083.94	0.585583
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	3486.68	0.222613
$627$	0	0
$628$	9634.56	0.612199
$629$	5356.35	0.339542
$630$	0	0
$631$	16368.8	1.03269	0.516347	−	0.856379i	$-0.327291\pi$
$631$	16368.8	1.03269	0.516347	+	0.856379i	$0.327291\pi$
$632$	−7966.59	−0.501415
$633$	0	0
$634$	−127.112	−0.00796258
$635$	−8550.49	−0.534356
$636$	0	0
$637$	0	0
$638$	−1682.45	−0.104403
$639$	0	0
$640$	−5920.66	−0.365679
$641$	−849.649	−0.0523543	−0.0261772	−	0.999657i	$-0.508333\pi$
$641$	−849.649	−0.0523543	−0.0261772	+	0.999657i	$0.508333\pi$
$642$	0	0
$643$	1335.12	0.0818850	0.0409425	−	0.999162i	$-0.486964\pi$
$643$	1335.12	0.0818850	0.0409425	+	0.999162i	$0.486964\pi$
$644$	0	0
$645$	0	0
$646$	11700.0	0.712586
$647$	5944.39	0.361202	0.180601	−	0.983556i	$-0.442196\pi$
$647$	5944.39	0.361202	0.180601	+	0.983556i	$0.442196\pi$
$648$	0	0
$649$	−2909.92	−0.176001
$650$	1341.15	0.0809293
$651$	0	0
$652$	−18371.3	−1.10349
$653$	−5561.29	−0.333277	−0.166639	−	0.986018i	$-0.553291\pi$
$653$	−5561.29	−0.333277	−0.166639	+	0.986018i	$0.553291\pi$
$654$	0	0
$655$	7922.46	0.472605
$656$	2229.84	0.132714
$657$	0	0
$658$	0	0
$659$	27399.5	1.61963	0.809813	−	0.586688i	$-0.199569\pi$
$659$	27399.5	1.61963	0.809813	+	0.586688i	$0.199569\pi$
$660$	0	0
$661$	−3931.52	−0.231344	−0.115672	−	0.993287i	$-0.536902\pi$
$661$	−3931.52	−0.231344	−0.115672	+	0.993287i	$0.536902\pi$
$662$	10228.5	0.600514
$663$	0	0
$664$	−15935.0	−0.931324
$665$	0	0
$666$	0	0
$667$	1276.38	0.0740954
$668$	3347.73	0.193903
$669$	0	0
$670$	4286.71	0.247179
$671$	−5067.16	−0.291528
$672$	0	0
$673$	11427.3	0.654515	0.327257	−	0.944935i	$-0.393876\pi$
$673$	11427.3	0.654515	0.327257	+	0.944935i	$0.393876\pi$
$674$	−6440.83	−0.368088
$675$	0	0
$676$	5654.94	0.321742
$677$	16259.3	0.923035	0.461518	−	0.887131i	$-0.347305\pi$
$677$	16259.3	0.923035	0.461518	+	0.887131i	$0.347305\pi$
$678$	0	0
$679$	0	0
$680$	−8746.17	−0.493236
$681$	0	0
$682$	2273.73	0.127662
$683$	10873.9	0.609192	0.304596	−	0.952482i	$-0.401479\pi$
$683$	10873.9	0.609192	0.304596	+	0.952482i	$0.401479\pi$
$684$	0	0
$685$	10476.2	0.584344
$686$	0	0
$687$	0	0
$688$	−3646.52	−0.202067
$689$	10196.3	0.563785
$690$	0	0
$691$	−21613.9	−1.18991	−0.594957	−	0.803758i	$-0.702831\pi$
$691$	−21613.9	−1.18991	−0.594957	+	0.803758i	$0.702831\pi$
$692$	20806.7	1.14299
$693$	0	0
$694$	−1918.28	−0.104923
$695$	−6924.98	−0.377956
$696$	0	0
$697$	16123.7	0.876222
$698$	5185.81	0.281212
$699$	0	0
$700$	0	0
$701$	−20667.7	−1.11357	−0.556783	−	0.830658i	$-0.687964\pi$
$701$	−20667.7	−1.11357	−0.556783	+	0.830658i	$0.687964\pi$
$702$	0	0
$703$	−5882.12	−0.315574
$704$	−2059.71	−0.110267
$705$	0	0
$706$	−7166.36	−0.382025
$707$	0	0
$708$	0	0
$709$	−22813.3	−1.20842	−0.604212	−	0.796824i	$-0.706512\pi$
$709$	−22813.3	−1.20842	−0.604212	+	0.796824i	$0.706512\pi$
$710$	−5785.33	−0.305802
$711$	0	0
$712$	20018.7	1.05370
$713$	−1724.94	−0.0906026
$714$	0	0
$715$	−1760.00	−0.0920563
$716$	−5385.54	−0.281099
$717$	0	0
$718$	−19334.4	−1.00495
$719$	−9551.14	−0.495407	−0.247703	−	0.968836i	$-0.579676\pi$
$719$	−9551.14	−0.495407	−0.247703	+	0.968836i	$0.579676\pi$
$720$	0	0
$721$	0	0
$722$	−2137.77	−0.110193
$723$	0	0
$724$	−10542.7	−0.541185
$725$	−2628.84	−0.134666
$726$	0	0
$727$	30003.0	1.53060	0.765302	−	0.643672i	$-0.222590\pi$
$727$	30003.0	1.53060	0.765302	+	0.643672i	$0.222590\pi$
$728$	0	0
$729$	0	0
$730$	4347.76	0.220435
$731$	−26367.5	−1.33411
$732$	0	0
$733$	−16863.7	−0.849762	−0.424881	−	0.905249i	$-0.639684\pi$
$733$	−16863.7	−0.849762	−0.424881	+	0.905249i	$0.639684\pi$
$734$	8763.95	0.440713
$735$	0	0
$736$	2272.95	0.113834
$737$	−5625.50	−0.281164
$738$	0	0
$739$	−30778.9	−1.53210	−0.766048	−	0.642783i	$-0.777780\pi$
$739$	−30778.9	−1.53210	−0.766048	+	0.642783i	$0.777780\pi$
$740$	1803.24	0.0895789
$741$	0	0
$742$	0	0
$743$	28510.9	1.40776	0.703879	−	0.710320i	$-0.251450\pi$
$743$	28510.9	1.40776	0.703879	+	0.710320i	$0.251450\pi$
$744$	0	0
$745$	−12514.4	−0.615428
$746$	5763.77	0.282878
$747$	0	0
$748$	4706.97	0.230085
$749$	0	0
$750$	0	0
$751$	12480.0	0.606395	0.303197	−	0.952928i	$-0.401946\pi$
$751$	12480.0	0.606395	0.303197	+	0.952928i	$0.401946\pi$
$752$	3642.28	0.176623
$753$	0	0
$754$	−5641.04	−0.272460
$755$	−1976.06	−0.0952532
$756$	0	0
$757$	−11272.1	−0.541203	−0.270601	−	0.962691i	$-0.587223\pi$
$757$	−11272.1	−0.541203	−0.270601	+	0.962691i	$0.587223\pi$
$758$	−8354.56	−0.400332
$759$	0	0
$760$	9604.69	0.458419
$761$	28775.4	1.37071	0.685354	−	0.728210i	$-0.259647\pi$
$761$	28775.4	1.37071	0.685354	+	0.728210i	$0.259647\pi$
$762$	0	0
$763$	0	0
$764$	15182.4	0.718951
$765$	0	0
$766$	11399.5	0.537702
$767$	−9756.58	−0.459309
$768$	0	0
$769$	17298.3	0.811175	0.405587	−	0.914056i	$-0.367067\pi$
$769$	17298.3	0.811175	0.405587	+	0.914056i	$0.367067\pi$
$770$	0	0
$771$	0	0
$772$	−8122.33	−0.378665
$773$	−5348.18	−0.248849	−0.124425	−	0.992229i	$-0.539709\pi$
$773$	−5348.18	−0.248849	−0.124425	+	0.992229i	$0.539709\pi$
$774$	0	0
$775$	3552.70	0.164667
$776$	3826.03	0.176993
$777$	0	0
$778$	14213.7	0.654994
$779$	−17706.3	−0.814371
$780$	0	0
$781$	7592.15	0.347847
$782$	1565.65	0.0715954
$783$	0	0
$784$	0	0
$785$	8661.75	0.393823
$786$	0	0
$787$	21615.1	0.979030	0.489515	−	0.871995i	$-0.337174\pi$
$787$	21615.1	0.979030	0.489515	+	0.871995i	$0.337174\pi$
$788$	26870.2	1.21473
$789$	0	0
$790$	−2937.20	−0.132279
$791$	0	0
$792$	0	0
$793$	−16989.5	−0.760800
$794$	−17989.3	−0.804050
$795$	0	0
$796$	−4855.62	−0.216210
$797$	38266.0	1.70069	0.850345	−	0.526225i	$-0.176393\pi$
$797$	38266.0	1.70069	0.850345	+	0.526225i	$0.176393\pi$
$798$	0	0
$799$	26336.8	1.16612
$800$	−4681.37	−0.206889
$801$	0	0
$802$	−5954.09	−0.262152
$803$	−5705.61	−0.250743
$804$	0	0
$805$	0	0
$806$	7623.50	0.333159
$807$	0	0
$808$	30983.2	1.34899
$809$	4724.05	0.205301	0.102651	−	0.994717i	$-0.467268\pi$
$809$	4724.05	0.205301	0.102651	+	0.994717i	$0.467268\pi$
$810$	0	0
$811$	−9687.39	−0.419446	−0.209723	−	0.977761i	$-0.567256\pi$
$811$	−9687.39	−0.419446	−0.209723	+	0.977761i	$0.567256\pi$
$812$	0	0
$813$	0	0
$814$	1037.55	0.0446756
$815$	−16516.3	−0.709866
$816$	0	0
$817$	28955.7	1.23994
$818$	−6400.53	−0.273581
$819$	0	0
$820$	5428.10	0.231167
$821$	−32250.8	−1.37096	−0.685482	−	0.728090i	$-0.740408\pi$
$821$	−32250.8	−1.37096	−0.685482	+	0.728090i	$0.740408\pi$
$822$	0	0
$823$	5858.93	0.248153	0.124076	−	0.992273i	$-0.460403\pi$
$823$	5858.93	0.248153	0.124076	+	0.992273i	$0.460403\pi$
$824$	−11330.8	−0.479037
$825$	0	0
$826$	0	0
$827$	16375.7	0.688560	0.344280	−	0.938867i	$-0.388123\pi$
$827$	16375.7	0.688560	0.344280	+	0.938867i	$0.388123\pi$
$828$	0	0
$829$	15152.9	0.634841	0.317421	−	0.948285i	$-0.397183\pi$
$829$	15152.9	0.634841	0.317421	+	0.948285i	$0.397183\pi$
$830$	−5875.08	−0.245695
$831$	0	0
$832$	−6905.94	−0.287765
$833$	0	0
$834$	0	0
$835$	3009.71	0.124737
$836$	−5169.00	−0.213844
$837$	0	0
$838$	22349.6	0.921307
$839$	8829.87	0.363338	0.181669	−	0.983360i	$-0.441850\pi$
$839$	8829.87	0.363338	0.181669	+	0.983360i	$0.441850\pi$
$840$	0	0
$841$	−13331.8	−0.546630
$842$	−10430.5	−0.426911
$843$	0	0
$844$	28631.0	1.16768
$845$	5083.96	0.206975
$846$	0	0
$847$	0	0
$848$	−3390.42	−0.137297
$849$	0	0
$850$	−3224.62	−0.130122
$851$	−787.125	−0.0317066
$852$	0	0
$853$	4227.64	0.169697	0.0848486	−	0.996394i	$-0.472959\pi$
$853$	4227.64	0.169697	0.0848486	+	0.996394i	$0.472959\pi$
$854$	0	0
$855$	0	0
$856$	6928.15	0.276635
$857$	−32233.1	−1.28479	−0.642394	−	0.766375i	$-0.722059\pi$
$857$	−32233.1	−1.28479	−0.642394	+	0.766375i	$0.722059\pi$
$858$	0	0
$859$	−37392.3	−1.48523	−0.742613	−	0.669720i	$-0.766414\pi$
$859$	−37392.3	−1.48523	−0.742613	+	0.669720i	$0.766414\pi$
$860$	−8876.73	−0.351970
$861$	0	0
$862$	−17963.8	−0.709801
$863$	8372.73	0.330256	0.165128	−	0.986272i	$-0.447196\pi$
$863$	8372.73	0.330256	0.165128	+	0.986272i	$0.447196\pi$
$864$	0	0
$865$	18705.8	0.735280
$866$	5065.66	0.198774
$867$	0	0
$868$	0	0
$869$	3854.52	0.150467
$870$	0	0
$871$	−18861.5	−0.733753
$872$	34343.0	1.33372
$873$	0	0
$874$	−1719.33	−0.0665416
$875$	0	0
$876$	0	0
$877$	38583.0	1.48558	0.742790	−	0.669524i	$-0.233502\pi$
$877$	38583.0	1.48558	0.742790	+	0.669524i	$0.233502\pi$
$878$	−4285.23	−0.164715
$879$	0	0
$880$	585.227	0.0224182
$881$	−41611.7	−1.59130	−0.795650	−	0.605757i	$-0.792870\pi$
$881$	−41611.7	−1.59130	−0.795650	+	0.605757i	$0.792870\pi$
$882$	0	0
$883$	43408.1	1.65436	0.827180	−	0.561937i	$-0.189944\pi$
$883$	43408.1	1.65436	0.827180	+	0.561937i	$0.189944\pi$
$884$	15781.8	0.600453
$885$	0	0
$886$	−12773.9	−0.484366
$887$	−19523.6	−0.739050	−0.369525	−	0.929221i	$-0.620480\pi$
$887$	−19523.6	−0.739050	−0.369525	+	0.929221i	$0.620480\pi$
$888$	0	0
$889$	0	0
$890$	7380.67	0.277978
$891$	0	0
$892$	−13544.1	−0.508398
$893$	−28922.0	−1.08381
$894$	0	0
$895$	−4841.76	−0.180829
$896$	0	0
$897$	0	0
$898$	−10502.5	−0.390283
$899$	−14943.1	−0.554373
$900$	0	0
$901$	−24515.7	−0.906479
$902$	3123.21	0.115290
$903$	0	0
$904$	9544.84	0.351169
$905$	−9478.23	−0.348140
$906$	0	0
$907$	44114.7	1.61500	0.807499	−	0.589869i	$-0.200820\pi$
$907$	44114.7	1.61500	0.807499	+	0.589869i	$0.200820\pi$
$908$	−19512.5	−0.713156
$909$	0	0
$910$	0	0
$911$	49231.9	1.79048	0.895240	−	0.445585i	$-0.147004\pi$
$911$	49231.9	1.79048	0.895240	+	0.445585i	$0.147004\pi$
$912$	0	0
$913$	7709.92	0.279476
$914$	16875.7	0.610719
$915$	0	0
$916$	19073.6	0.688001
$917$	0	0
$918$	0	0
$919$	−46993.6	−1.68681	−0.843403	−	0.537282i	$-0.819451\pi$
$919$	−46993.6	−1.68681	−0.843403	+	0.537282i	$0.819451\pi$
$920$	1285.26	0.0460586
$921$	0	0
$922$	27757.9	0.991494
$923$	25455.5	0.907775
$924$	0	0
$925$	1621.16	0.0576255
$926$	6900.40	0.244883
$927$	0	0
$928$	19690.5	0.696521
$929$	33023.5	1.16627	0.583135	−	0.812375i	$-0.301826\pi$
$929$	33023.5	1.16627	0.583135	+	0.812375i	$0.301826\pi$
$930$	0	0
$931$	0	0
$932$	22189.0	0.779855
$933$	0	0
$934$	5267.25	0.184529
$935$	4231.70	0.148012
$936$	0	0
$937$	23431.0	0.816924	0.408462	−	0.912775i	$-0.366065\pi$
$937$	23431.0	0.816924	0.408462	+	0.912775i	$0.366065\pi$
$938$	0	0
$939$	0	0
$940$	8866.41	0.307649
$941$	−43688.0	−1.51348	−0.756742	−	0.653714i	$-0.773210\pi$
$941$	−43688.0	−1.51348	−0.756742	+	0.653714i	$0.773210\pi$
$942$	0	0
$943$	−2369.40	−0.0818221
$944$	3244.21	0.111854
$945$	0	0
$946$	−5107.48	−0.175538
$947$	34884.0	1.19702	0.598511	−	0.801115i	$-0.295759\pi$
$947$	34884.0	1.19702	0.598511	+	0.801115i	$0.295759\pi$
$948$	0	0
$949$	−19130.1	−0.654363
$950$	3541.15	0.120937
$951$	0	0
$952$	0	0
$953$	34486.4	1.17222	0.586108	−	0.810233i	$-0.300659\pi$
$953$	34486.4	1.17222	0.586108	+	0.810233i	$0.300659\pi$
$954$	0	0
$955$	13649.4	0.462496
$956$	33006.2	1.11663
$957$	0	0
$958$	7214.40	0.243306
$959$	0	0
$960$	0	0
$961$	−9596.33	−0.322122
$962$	3478.75	0.116590
$963$	0	0
$964$	−36049.4	−1.20443
$965$	−7302.21	−0.243592
$966$	0	0
$967$	−46341.6	−1.54110	−0.770550	−	0.637379i	$-0.780019\pi$
$967$	−46341.6	−1.54110	−0.770550	+	0.637379i	$0.780019\pi$
$968$	−25963.4	−0.862083
$969$	0	0
$970$	1410.62	0.0466930
$971$	34551.4	1.14192	0.570962	−	0.820977i	$-0.306571\pi$
$971$	34551.4	1.14192	0.570962	+	0.820977i	$0.306571\pi$
$972$	0	0
$973$	0	0
$974$	14223.6	0.467919
$975$	0	0
$976$	5649.27	0.185275
$977$	970.127	0.0317678	0.0158839	−	0.999874i	$-0.494944\pi$
$977$	970.127	0.0317678	0.0158839	+	0.999874i	$0.494944\pi$
$978$	0	0
$979$	−9685.73	−0.316198
$980$	0	0
$981$	0	0
$982$	4211.26	0.136850
$983$	−18342.2	−0.595142	−0.297571	−	0.954700i	$-0.596176\pi$
$983$	−18342.2	−0.595142	−0.297571	+	0.954700i	$0.596176\pi$
$984$	0	0
$985$	24157.1	0.781430
$986$	13563.2	0.438073
$987$	0	0
$988$	−17331.0	−0.558068
$989$	3874.75	0.124580
$990$	0	0
$991$	48979.6	1.57002	0.785010	−	0.619484i	$-0.212658\pi$
$991$	48979.6	1.57002	0.785010	+	0.619484i	$0.212658\pi$
$992$	−26610.4	−0.851694
$993$	0	0
$994$	0	0
$995$	−4365.35	−0.139086
$996$	0	0
$997$	−31492.9	−1.00039	−0.500196	−	0.865912i	$-0.666739\pi$
$997$	−31492.9	−1.00039	−0.500196	+	0.865912i	$0.666739\pi$
$998$	31315.7	0.993268
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.ba.1.1		2
3.2	odd	2		2205.4.a.y.1.2		2
7.6	odd	2		315.4.a.j.1.1	yes	2
21.20	even	2		315.4.a.h.1.2	✓	2
35.34	odd	2		1575.4.a.r.1.2		2
105.104	even	2		1575.4.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.h.1.2	✓	2	21.20	even	2
315.4.a.j.1.1	yes	2	7.6	odd	2
1575.4.a.r.1.2		2	35.34	odd	2
1575.4.a.u.1.1		2	105.104	even	2
2205.4.a.y.1.2		2	3.2	odd	2
2205.4.a.ba.1.1		2	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} -5.56155 q^{4} -5.00000 q^{5} +21.1771 q^{8} +O(q^{10})\)	\(q-1.56155 q^{2} -5.56155 q^{4} -5.00000 q^{5} +21.1771 q^{8} +7.80776 q^{10} -10.2462 q^{11} -34.3542 q^{13} +11.4233 q^{16} +82.6004 q^{17} -90.7083 q^{19} +27.8078 q^{20} +16.0000 q^{22} -12.1383 q^{23} +25.0000 q^{25} +53.6458 q^{26} -105.153 q^{29} +142.108 q^{31} -187.255 q^{32} -128.985 q^{34} +64.8466 q^{37} +141.646 q^{38} -105.885 q^{40} +195.201 q^{41} -319.218 q^{43} +56.9848 q^{44} +18.9545 q^{46} +318.847 q^{47} -39.0388 q^{50} +191.062 q^{52} -296.799 q^{53} +51.2311 q^{55} +164.203 q^{58} +284.000 q^{59} +494.540 q^{61} -221.909 q^{62} +201.022 q^{64} +171.771 q^{65} +549.032 q^{67} -459.386 q^{68} -740.972 q^{71} +556.850 q^{73} -101.261 q^{74} +504.479 q^{76} -376.189 q^{79} -57.1165 q^{80} -304.816 q^{82} -752.466 q^{83} -413.002 q^{85} +498.475 q^{86} -216.985 q^{88} +945.299 q^{89} +67.5076 q^{92} -497.896 q^{94} +453.542 q^{95} +180.668 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 7 q^{4} - 10 q^{5} - 3 q^{8}+O(q^{10}) \) 2 * q + q^2 - 7 * q^4 - 10 * q^5 - 3 * q^8	\( 2 q + q^{2} - 7 q^{4} - 10 q^{5} - 3 q^{8} - 5 q^{10} - 4 q^{11} + 22 q^{13} - 39 q^{16} + 58 q^{17} + 35 q^{20} + 32 q^{22} - 82 q^{23} + 50 q^{25} + 198 q^{26} - 334 q^{29} + 210 q^{31} - 123 q^{32} - 192 q^{34} + 6 q^{37} + 374 q^{38} + 15 q^{40} + 176 q^{41} + 46 q^{43} + 48 q^{44} - 160 q^{46} + 514 q^{47} + 25 q^{50} + 110 q^{52} - 808 q^{53} + 20 q^{55} - 422 q^{58} + 568 q^{59} + 618 q^{61} - 48 q^{62} + 769 q^{64} - 110 q^{65} + 694 q^{67} - 424 q^{68} - 814 q^{71} - 82 q^{73} - 252 q^{74} + 374 q^{76} + 600 q^{79} + 195 q^{80} - 354 q^{82} - 268 q^{83} - 290 q^{85} + 1434 q^{86} - 368 q^{88} - 72 q^{89} + 168 q^{92} + 2 q^{94} - 1626 q^{97}+O(q^{100}) \) 2 * q + q^2 - 7 * q^4 - 10 * q^5 - 3 * q^8 - 5 * q^10 - 4 * q^11 + 22 * q^13 - 39 * q^16 + 58 * q^17 + 35 * q^20 + 32 * q^22 - 82 * q^23 + 50 * q^25 + 198 * q^26 - 334 * q^29 + 210 * q^31 - 123 * q^32 - 192 * q^34 + 6 * q^37 + 374 * q^38 + 15 * q^40 + 176 * q^41 + 46 * q^43 + 48 * q^44 - 160 * q^46 + 514 * q^47 + 25 * q^50 + 110 * q^52 - 808 * q^53 + 20 * q^55 - 422 * q^58 + 568 * q^59 + 618 * q^61 - 48 * q^62 + 769 * q^64 - 110 * q^65 + 694 * q^67 - 424 * q^68 - 814 * q^71 - 82 * q^73 - 252 * q^74 + 374 * q^76 + 600 * q^79 + 195 * q^80 - 354 * q^82 - 268 * q^83 - 290 * q^85 + 1434 * q^86 - 368 * q^88 - 72 * q^89 + 168 * q^92 + 2 * q^94 - 1626 * q^97

Introduction

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