LMFDB - Embedded newform 2205.4.a.v.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,4,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, 4); 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, 4, names="a")

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:traces = [2,-3,0,9,-10,0,0,-51,0,15,-62,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$130.099211563$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ 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)

$f(q)$	$=$	$q+1.70156 q^{2} -5.10469 q^{4} -5.00000 q^{5} -22.2984 q^{8} -8.50781 q^{10} -37.4031 q^{11} -29.0156 q^{13} +2.89531 q^{16} +58.4187 q^{17} +54.5969 q^{19} +25.5234 q^{20} -63.6437 q^{22} -161.675 q^{23} +25.0000 q^{25} -49.3719 q^{26} -137.581 q^{29} -154.659 q^{31} +183.314 q^{32} +99.4031 q^{34} -350.125 q^{37} +92.9000 q^{38} +111.492 q^{40} +353.769 q^{41} -518.156 q^{43} +190.931 q^{44} -275.100 q^{46} -542.219 q^{47} +42.5391 q^{50} +148.116 q^{52} -305.309 q^{53} +187.016 q^{55} -234.103 q^{58} +14.6813 q^{59} +171.069 q^{61} -263.163 q^{62} +288.758 q^{64} +145.078 q^{65} +551.956 q^{67} -298.209 q^{68} +120.334 q^{71} -284.659 q^{73} -595.759 q^{74} -278.700 q^{76} +941.612 q^{79} -14.4766 q^{80} +601.959 q^{82} +377.150 q^{83} -292.094 q^{85} -881.675 q^{86} +834.031 q^{88} -677.725 q^{89} +825.300 q^{92} -922.619 q^{94} -272.984 q^{95} +1225.03 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 9 q^{4} - 10 q^{5} - 51 q^{8} + 15 q^{10} - 62 q^{11} + 6 q^{13} + 25 q^{16} + 40 q^{17} + 122 q^{19} - 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} - 214 q^{26} - 352 q^{29} - 66 q^{31}+ \cdots + 670 q^{97}+O(q^{100}) $ 2 * q - 3 * q^2 + 9 * q^4 - 10 * q^5 - 51 * q^8 + 15 * q^10 - 62 * q^11 + 6 * q^13 + 25 * q^16 + 40 * q^17 + 122 * q^19 - 45 * q^20 + 52 * q^22 - 16 * q^23 + 50 * q^25 - 214 * q^26 - 352 * q^29 - 66 * q^31 + 309 * q^32 + 186 * q^34 - 188 * q^37 - 224 * q^38 + 255 * q^40 + 16 * q^41 - 396 * q^43 - 156 * q^44 - 960 * q^46 - 188 * q^47 - 75 * q^50 + 642 * q^52 - 982 * q^53 + 310 * q^55 + 774 * q^58 + 516 * q^59 + 880 * q^61 - 680 * q^62 - 479 * q^64 - 30 * q^65 - 356 * q^67 - 558 * q^68 - 310 * q^71 - 326 * q^73 - 1358 * q^74 + 672 * q^76 + 1832 * q^79 - 125 * q^80 + 2190 * q^82 - 680 * q^83 - 200 * q^85 - 1456 * q^86 + 1540 * q^88 + 796 * q^89 + 2880 * q^92 - 2588 * q^94 - 610 * q^95 + 670 * q^97

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 $

$2$	1.70156	0.601593	0.300797	−	0.953688i	$-0.402747\pi$
$2$	1.70156	0.601593	0.300797	+	0.953688i	$0.402747\pi$
$3$	0	0
$3$	0	0
$4$	−5.10469	−0.638086
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−22.2984	−0.985461
$9$	0	0
$10$	−8.50781	−0.269041
$11$	−37.4031	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$11$	−37.4031	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$12$	0	0
$13$	−29.0156	−0.619037	−0.309519	−	0.950893i	$-0.600168\pi$
$13$	−29.0156	−0.619037	−0.309519	+	0.950893i	$0.600168\pi$
$14$	0	0
$15$	0	0
$16$	2.89531	0.0452393
$17$	58.4187	0.833449	0.416724	−	0.909033i	$-0.363178\pi$
$17$	58.4187	0.833449	0.416724	+	0.909033i	$0.363178\pi$
$18$	0	0
$19$	54.5969	0.659231	0.329615	−	0.944115i	$-0.393081\pi$
$19$	54.5969	0.659231	0.329615	+	0.944115i	$0.393081\pi$
$20$	25.5234	0.285361
$21$	0	0
$22$	−63.6437	−0.616768
$23$	−161.675	−1.46572	−0.732860	−	0.680379i	$-0.761815\pi$
$23$	−161.675	−1.46572	−0.732860	+	0.680379i	$0.761815\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−49.3719	−0.372409
$27$	0	0
$28$	0	0
$29$	−137.581	−0.880972	−0.440486	−	0.897759i	$-0.645194\pi$
$29$	−137.581	−0.880972	−0.440486	+	0.897759i	$0.645194\pi$
$30$	0	0
$31$	−154.659	−0.896053	−0.448026	−	0.894020i	$-0.647873\pi$
$31$	−154.659	−0.896053	−0.448026	+	0.894020i	$0.647873\pi$
$32$	183.314	1.01268
$33$	0	0
$34$	99.4031	0.501397
$35$	0	0
$36$	0	0
$37$	−350.125	−1.55568	−0.777840	−	0.628462i	$-0.783685\pi$
$37$	−350.125	−1.55568	−0.777840	+	0.628462i	$0.783685\pi$
$38$	92.9000	0.396589
$39$	0	0
$40$	111.492	0.440712
$41$	353.769	1.34755	0.673773	−	0.738938i	$-0.264673\pi$
$41$	353.769	1.34755	0.673773	+	0.738938i	$0.264673\pi$
$42$	0	0
$43$	−518.156	−1.83763	−0.918815	−	0.394689i	$-0.870852\pi$
$43$	−518.156	−1.83763	−0.918815	+	0.394689i	$0.870852\pi$
$44$	190.931	0.654181
$45$	0	0
$46$	−275.100	−0.881767
$47$	−542.219	−1.68278	−0.841391	−	0.540427i	$-0.818263\pi$
$47$	−542.219	−1.68278	−0.841391	+	0.540427i	$0.818263\pi$
$48$	0	0
$49$	0	0
$50$	42.5391	0.120319
$51$	0	0
$52$	148.116	0.394999
$53$	−305.309	−0.791273	−0.395637	−	0.918407i	$-0.629476\pi$
$53$	−305.309	−0.791273	−0.395637	+	0.918407i	$0.629476\pi$
$54$	0	0
$55$	187.016	0.458494
$56$	0	0
$57$	0	0
$58$	−234.103	−0.529987
$59$	14.6813	0.0323956	0.0161978	−	0.999869i	$-0.494844\pi$
$59$	14.6813	0.0323956	0.0161978	+	0.999869i	$0.494844\pi$
$60$	0	0
$61$	171.069	0.359067	0.179534	−	0.983752i	$-0.442541\pi$
$61$	171.069	0.359067	0.179534	+	0.983752i	$0.442541\pi$
$62$	−263.163	−0.539059
$63$	0	0
$64$	288.758	0.563980
$65$	145.078	0.276842
$66$	0	0
$67$	551.956	1.00645	0.503225	−	0.864155i	$-0.332147\pi$
$67$	551.956	1.00645	0.503225	+	0.864155i	$0.332147\pi$
$68$	−298.209	−0.531812
$69$	0	0
$70$	0	0
$71$	120.334	0.201142	0.100571	−	0.994930i	$-0.467933\pi$
$71$	120.334	0.201142	0.100571	+	0.994930i	$0.467933\pi$
$72$	0	0
$73$	−284.659	−0.456395	−0.228198	−	0.973615i	$-0.573283\pi$
$73$	−284.659	−0.456395	−0.228198	+	0.973615i	$0.573283\pi$
$74$	−595.759	−0.935887
$75$	0	0
$76$	−278.700	−0.420646
$77$	0	0
$78$	0	0
$79$	941.612	1.34101	0.670504	−	0.741906i	$-0.266078\pi$
$79$	941.612	1.34101	0.670504	+	0.741906i	$0.266078\pi$
$80$	−14.4766	−0.0202316
$81$	0	0
$82$	601.959	0.810674
$83$	377.150	0.498766	0.249383	−	0.968405i	$-0.419772\pi$
$83$	377.150	0.498766	0.249383	+	0.968405i	$0.419772\pi$
$84$	0	0
$85$	−292.094	−0.372730
$86$	−881.675	−1.10551
$87$	0	0
$88$	834.031	1.01032
$89$	−677.725	−0.807176	−0.403588	−	0.914941i	$-0.632237\pi$
$89$	−677.725	−0.807176	−0.403588	+	0.914941i	$0.632237\pi$
$90$	0	0
$91$	0	0
$92$	825.300	0.935255
$93$	0	0
$94$	−922.619	−1.01235
$95$	−272.984	−0.294817
$96$	0	0
$97$	1225.03	1.28230	0.641151	−	0.767414i	$-0.278457\pi$
$97$	1225.03	1.28230	0.641151	+	0.767414i	$0.278457\pi$
$98$	0	0
$99$	0	0
$100$	−127.617	−0.127617
$101$	−338.144	−0.333134	−0.166567	−	0.986030i	$-0.553268\pi$
$101$	−338.144	−0.333134	−0.166567	+	0.986030i	$0.553268\pi$
$102$	0	0
$103$	566.700	0.542122	0.271061	−	0.962562i	$-0.412625\pi$
$103$	566.700	0.542122	0.271061	+	0.962562i	$0.412625\pi$
$104$	647.003	0.610037
$105$	0	0
$106$	−519.503	−0.476024
$107$	562.531	0.508242	0.254121	−	0.967172i	$-0.418214\pi$
$107$	562.531	0.508242	0.254121	+	0.967172i	$0.418214\pi$
$108$	0	0
$109$	1830.79	1.60879	0.804396	−	0.594094i	$-0.202489\pi$
$109$	1830.79	1.60879	0.804396	+	0.594094i	$0.202489\pi$
$110$	318.219	0.275827
$111$	0	0
$112$	0	0
$113$	31.8032	0.0264761	0.0132380	−	0.999912i	$-0.495786\pi$
$113$	31.8032	0.0264761	0.0132380	+	0.999912i	$0.495786\pi$
$114$	0	0
$115$	808.375	0.655490
$116$	702.309	0.562136
$117$	0	0
$118$	24.9811	0.0194890
$119$	0	0
$120$	0	0
$121$	67.9937	0.0510847
$122$	291.084	0.216012
$123$	0	0
$124$	789.488	0.571759
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2220.81	1.55169	0.775847	−	0.630921i	$-0.217323\pi$
$127$	2220.81	1.55169	0.775847	+	0.630921i	$0.217323\pi$
$128$	−975.173	−0.673390
$129$	0	0
$130$	246.859	0.166546
$131$	646.512	0.431191	0.215596	−	0.976483i	$-0.430831\pi$
$131$	646.512	0.431191	0.215596	+	0.976483i	$0.430831\pi$
$132$	0	0
$133$	0	0
$134$	939.188	0.605474
$135$	0	0
$136$	−1302.65	−0.821331
$137$	896.009	0.558768	0.279384	−	0.960179i	$-0.409870\pi$
$137$	896.009	0.558768	0.279384	+	0.960179i	$0.409870\pi$
$138$	0	0
$139$	2313.61	1.41178	0.705891	−	0.708320i	$-0.250547\pi$
$139$	2313.61	1.41178	0.705891	+	0.708320i	$0.250547\pi$
$140$	0	0
$141$	0	0
$142$	204.756	0.121005
$143$	1085.27	0.634652
$144$	0	0
$145$	687.906	0.393983
$146$	−484.366	−0.274564
$147$	0	0
$148$	1787.28	0.992658
$149$	−819.337	−0.450488	−0.225244	−	0.974302i	$-0.572318\pi$
$149$	−819.337	−0.450488	−0.225244	+	0.974302i	$0.572318\pi$
$150$	0	0
$151$	534.744	0.288191	0.144095	−	0.989564i	$-0.453973\pi$
$151$	534.744	0.288191	0.144095	+	0.989564i	$0.453973\pi$
$152$	−1217.43	−0.649646
$153$	0	0
$154$	0	0
$155$	773.297	0.400727
$156$	0	0
$157$	−1564.76	−0.795423	−0.397711	−	0.917511i	$-0.630195\pi$
$157$	−1564.76	−0.795423	−0.397711	+	0.917511i	$0.630195\pi$
$158$	1602.21	0.806741
$159$	0	0
$160$	−916.570	−0.452883
$161$	0	0
$162$	0	0
$163$	−1114.31	−0.535455	−0.267728	−	0.963495i	$-0.586273\pi$
$163$	−1114.31	−0.535455	−0.267728	+	0.963495i	$0.586273\pi$
$164$	−1805.88	−0.859850
$165$	0	0
$166$	641.744	0.300054
$167$	−1774.47	−0.822231	−0.411115	−	0.911583i	$-0.634861\pi$
$167$	−1774.47	−0.822231	−0.411115	+	0.911583i	$0.634861\pi$
$168$	0	0
$169$	−1355.09	−0.616793
$170$	−497.016	−0.224232
$171$	0	0
$172$	2645.02	1.17257
$173$	−4215.88	−1.85276	−0.926380	−	0.376590i	$-0.877097\pi$
$173$	−4215.88	−1.85276	−0.926380	+	0.376590i	$0.877097\pi$
$174$	0	0
$175$	0	0
$176$	−108.294	−0.0463804
$177$	0	0
$178$	−1153.19	−0.485592
$179$	2430.70	1.01497	0.507483	−	0.861662i	$-0.330576\pi$
$179$	2430.70	1.01497	0.507483	+	0.861662i	$0.330576\pi$
$180$	0	0
$181$	2700.91	1.10916	0.554578	−	0.832132i	$-0.312880\pi$
$181$	2700.91	1.10916	0.554578	+	0.832132i	$0.312880\pi$
$182$	0	0
$183$	0	0
$184$	3605.10	1.44441
$185$	1750.62	0.695722
$186$	0	0
$187$	−2185.04	−0.854472
$188$	2767.86	1.07376
$189$	0	0
$190$	−464.500	−0.177360
$191$	−3611.10	−1.36801	−0.684005	−	0.729478i	$-0.739763\pi$
$191$	−3611.10	−1.36801	−0.684005	+	0.729478i	$0.739763\pi$
$192$	0	0
$193$	−4468.33	−1.66651	−0.833257	−	0.552886i	$-0.813526\pi$
$193$	−4468.33	−1.66651	−0.833257	+	0.552886i	$0.813526\pi$
$194$	2084.47	0.771425
$195$	0	0
$196$	0	0
$197$	434.422	0.157113	0.0785566	−	0.996910i	$-0.474969\pi$
$197$	434.422	0.157113	0.0785566	+	0.996910i	$0.474969\pi$
$198$	0	0
$199$	468.915	0.167038	0.0835189	−	0.996506i	$-0.473384\pi$
$199$	468.915	0.167038	0.0835189	+	0.996506i	$0.473384\pi$
$200$	−557.461	−0.197092
$201$	0	0
$202$	−575.372	−0.200411
$203$	0	0
$204$	0	0
$205$	−1768.84	−0.602641
$206$	964.275	0.326137
$207$	0	0
$208$	−84.0093	−0.0280048
$209$	−2042.09	−0.675859
$210$	0	0
$211$	3735.51	1.21878	0.609392	−	0.792869i	$-0.291414\pi$
$211$	3735.51	1.21878	0.609392	+	0.792869i	$0.291414\pi$
$212$	1558.51	0.504900
$213$	0	0
$214$	957.182	0.305755
$215$	2590.78	0.821813
$216$	0	0
$217$	0	0
$218$	3115.21	0.967838
$219$	0	0
$220$	−954.656	−0.292559
$221$	−1695.06	−0.515936
$222$	0	0
$223$	−842.806	−0.253087	−0.126544	−	0.991961i	$-0.540388\pi$
$223$	−842.806	−0.253087	−0.126544	+	0.991961i	$0.540388\pi$
$224$	0	0
$225$	0	0
$226$	54.1152	0.0159278
$227$	992.150	0.290094	0.145047	−	0.989425i	$-0.453667\pi$
$227$	992.150	0.290094	0.145047	+	0.989425i	$0.453667\pi$
$228$	0	0
$229$	6411.39	1.85012	0.925059	−	0.379825i	$-0.124016\pi$
$229$	6411.39	1.85012	0.925059	+	0.379825i	$0.124016\pi$
$230$	1375.50	0.394338
$231$	0	0
$232$	3067.85	0.868164
$233$	2274.35	0.639476	0.319738	−	0.947506i	$-0.396405\pi$
$233$	2274.35	0.639476	0.319738	+	0.947506i	$0.396405\pi$
$234$	0	0
$235$	2711.09	0.752563
$236$	−74.9433	−0.0206712
$237$	0	0
$238$	0	0
$239$	−2863.12	−0.774893	−0.387447	−	0.921892i	$-0.626643\pi$
$239$	−2863.12	−0.774893	−0.387447	+	0.921892i	$0.626643\pi$
$240$	0	0
$241$	5364.23	1.43378	0.716889	−	0.697187i	$-0.245565\pi$
$241$	5364.23	1.43378	0.716889	+	0.697187i	$0.245565\pi$
$242$	115.696	0.0307322
$243$	0	0
$244$	−873.252	−0.229116
$245$	0	0
$246$	0	0
$247$	−1584.16	−0.408088
$248$	3448.66	0.883025
$249$	0	0
$250$	−212.695	−0.0538081
$251$	5569.81	1.40065	0.700325	−	0.713824i	$-0.253039\pi$
$251$	5569.81	1.40065	0.700325	+	0.713824i	$0.253039\pi$
$252$	0	0
$253$	6047.15	1.50269
$254$	3778.85	0.933489
$255$	0	0
$256$	−3969.38	−0.969087
$257$	2095.36	0.508580	0.254290	−	0.967128i	$-0.418158\pi$
$257$	2095.36	0.508580	0.254290	+	0.967128i	$0.418158\pi$
$258$	0	0
$259$	0	0
$260$	−740.578	−0.176649
$261$	0	0
$262$	1100.08	0.259402
$263$	7465.88	1.75044	0.875220	−	0.483724i	$-0.160716\pi$
$263$	7465.88	1.75044	0.875220	+	0.483724i	$0.160716\pi$
$264$	0	0
$265$	1526.55	0.353868
$266$	0	0
$267$	0	0
$268$	−2817.56	−0.642202
$269$	−6521.38	−1.47812	−0.739062	−	0.673637i	$-0.764731\pi$
$269$	−6521.38	−1.47812	−0.739062	+	0.673637i	$0.764731\pi$
$270$	0	0
$271$	−2409.70	−0.540144	−0.270072	−	0.962840i	$-0.587048\pi$
$271$	−2409.70	−0.540144	−0.270072	+	0.962840i	$0.587048\pi$
$272$	169.141	0.0377046
$273$	0	0
$274$	1524.62	0.336151
$275$	−935.078	−0.205045
$276$	0	0
$277$	−2219.83	−0.481503	−0.240752	−	0.970587i	$-0.577394\pi$
$277$	−2219.83	−0.481503	−0.240752	+	0.970587i	$0.577394\pi$
$278$	3936.75	0.849319
$279$	0	0
$280$	0	0
$281$	−5838.56	−1.23950	−0.619749	−	0.784800i	$-0.712766\pi$
$281$	−5838.56	−1.23950	−0.619749	+	0.784800i	$0.712766\pi$
$282$	0	0
$283$	3645.04	0.765636	0.382818	−	0.923824i	$-0.374954\pi$
$283$	3645.04	0.765636	0.382818	+	0.923824i	$0.374954\pi$
$284$	−614.269	−0.128346
$285$	0	0
$286$	1846.66	0.381802
$287$	0	0
$288$	0	0
$289$	−1500.25	−0.305363
$290$	1170.52	0.237017
$291$	0	0
$292$	1453.10	0.291219
$293$	3777.91	0.753268	0.376634	−	0.926362i	$-0.377081\pi$
$293$	3777.91	0.753268	0.376634	+	0.926362i	$0.377081\pi$
$294$	0	0
$295$	−73.4064	−0.0144877
$296$	7807.24	1.53306
$297$	0	0
$298$	−1394.15	−0.271011
$299$	4691.10	0.907336
$300$	0	0
$301$	0	0
$302$	909.900	0.173374
$303$	0	0
$304$	158.075	0.0298231
$305$	−855.344	−0.160580
$306$	0	0
$307$	4799.64	0.892281	0.446140	−	0.894963i	$-0.352798\pi$
$307$	4799.64	0.892281	0.446140	+	0.894963i	$0.352798\pi$
$308$	0	0
$309$	0	0
$310$	1315.81	0.241075
$311$	580.113	0.105772	0.0528861	−	0.998601i	$-0.483158\pi$
$311$	580.113	0.105772	0.0528861	+	0.998601i	$0.483158\pi$
$312$	0	0
$313$	6114.78	1.10424	0.552121	−	0.833764i	$-0.313818\pi$
$313$	6114.78	1.10424	0.552121	+	0.833764i	$0.313818\pi$
$314$	−2662.53	−0.478521
$315$	0	0
$316$	−4806.64	−0.855679
$317$	4300.63	0.761979	0.380989	−	0.924579i	$-0.375583\pi$
$317$	4300.63	0.761979	0.380989	+	0.924579i	$0.375583\pi$
$318$	0	0
$319$	5145.97	0.903194
$320$	−1443.79	−0.252220
$321$	0	0
$322$	0	0
$323$	3189.48	0.549435
$324$	0	0
$325$	−725.391	−0.123807
$326$	−1896.06	−0.322126
$327$	0	0
$328$	−7888.49	−1.32795
$329$	0	0
$330$	0	0
$331$	−6687.54	−1.11051	−0.555257	−	0.831679i	$-0.687380\pi$
$331$	−6687.54	−1.11051	−0.555257	+	0.831679i	$0.687380\pi$
$332$	−1925.23	−0.318256
$333$	0	0
$334$	−3019.37	−0.494648
$335$	−2759.78	−0.450098
$336$	0	0
$337$	5869.28	0.948723	0.474362	−	0.880330i	$-0.342679\pi$
$337$	5869.28	0.948723	0.474362	+	0.880330i	$0.342679\pi$
$338$	−2305.78	−0.371058
$339$	0	0
$340$	1491.05	0.237833
$341$	5784.74	0.918655
$342$	0	0
$343$	0	0
$344$	11554.1	1.81091
$345$	0	0
$346$	−7173.58	−1.11461
$347$	−1937.22	−0.299699	−0.149850	−	0.988709i	$-0.547879\pi$
$347$	−1937.22	−0.299699	−0.149850	+	0.988709i	$0.547879\pi$
$348$	0	0
$349$	9748.82	1.49525	0.747625	−	0.664121i	$-0.231194\pi$
$349$	9748.82	1.49525	0.747625	+	0.664121i	$0.231194\pi$
$350$	0	0
$351$	0	0
$352$	−6856.52	−1.03822
$353$	−4576.61	−0.690052	−0.345026	−	0.938593i	$-0.612130\pi$
$353$	−4576.61	−0.690052	−0.345026	+	0.938593i	$0.612130\pi$
$354$	0	0
$355$	−601.672	−0.0899533
$356$	3459.57	0.515048
$357$	0	0
$358$	4135.98	0.610596
$359$	−10849.9	−1.59509	−0.797546	−	0.603258i	$-0.793869\pi$
$359$	−10849.9	−1.59509	−0.797546	+	0.603258i	$0.793869\pi$
$360$	0	0
$361$	−3878.18	−0.565415
$362$	4595.77	0.667261
$363$	0	0
$364$	0	0
$365$	1423.30	0.204106
$366$	0	0
$367$	−11467.7	−1.63108	−0.815541	−	0.578699i	$-0.803561\pi$
$367$	−11467.7	−1.63108	−0.815541	+	0.578699i	$0.803561\pi$
$368$	−468.100	−0.0663081
$369$	0	0
$370$	2978.80	0.418541
$371$	0	0
$372$	0	0
$373$	539.982	0.0749576	0.0374788	−	0.999297i	$-0.488067\pi$
$373$	539.982	0.0749576	0.0374788	+	0.999297i	$0.488067\pi$
$374$	−3717.99	−0.514044
$375$	0	0
$376$	12090.6	1.65832
$377$	3992.01	0.545355
$378$	0	0
$379$	8577.57	1.16253	0.581267	−	0.813713i	$-0.302557\pi$
$379$	8577.57	1.16253	0.581267	+	0.813713i	$0.302557\pi$
$380$	1393.50	0.188118
$381$	0	0
$382$	−6144.51	−0.822985
$383$	8627.96	1.15109	0.575546	−	0.817770i	$-0.304790\pi$
$383$	8627.96	1.15109	0.575546	+	0.817770i	$0.304790\pi$
$384$	0	0
$385$	0	0
$386$	−7603.13	−1.00256
$387$	0	0
$388$	−6253.42	−0.818219
$389$	−9234.06	−1.20356	−0.601781	−	0.798661i	$-0.705542\pi$
$389$	−9234.06	−1.20356	−0.601781	+	0.798661i	$0.705542\pi$
$390$	0	0
$391$	−9444.85	−1.22160
$392$	0	0
$393$	0	0
$394$	739.196	0.0945182
$395$	−4708.06	−0.599717
$396$	0	0
$397$	−11618.0	−1.46874	−0.734372	−	0.678747i	$-0.762523\pi$
$397$	−11618.0	−1.46874	−0.734372	+	0.678747i	$0.762523\pi$
$398$	797.889	0.100489
$399$	0	0
$400$	72.3828	0.00904786
$401$	−11157.1	−1.38942	−0.694711	−	0.719289i	$-0.744468\pi$
$401$	−11157.1	−1.38942	−0.694711	+	0.719289i	$0.744468\pi$
$402$	0	0
$403$	4487.54	0.554690
$404$	1726.12	0.212568
$405$	0	0
$406$	0	0
$407$	13095.8	1.59492
$408$	0	0
$409$	7428.08	0.898031	0.449015	−	0.893524i	$-0.351775\pi$
$409$	7428.08	0.898031	0.449015	+	0.893524i	$0.351775\pi$
$410$	−3009.80	−0.362545
$411$	0	0
$412$	−2892.83	−0.345921
$413$	0	0
$414$	0	0
$415$	−1885.75	−0.223055
$416$	−5318.97	−0.626885
$417$	0	0
$418$	−3474.75	−0.406592
$419$	9644.74	1.12453	0.562263	−	0.826959i	$-0.309931\pi$
$419$	9644.74	1.12453	0.562263	+	0.826959i	$0.309931\pi$
$420$	0	0
$421$	9918.18	1.14818	0.574088	−	0.818793i	$-0.305357\pi$
$421$	9918.18	1.14818	0.574088	+	0.818793i	$0.305357\pi$
$422$	6356.21	0.733212
$423$	0	0
$424$	6807.92	0.779769
$425$	1460.47	0.166690
$426$	0	0
$427$	0	0
$428$	−2871.54	−0.324302
$429$	0	0
$430$	4408.37	0.494397
$431$	16324.1	1.82437	0.912185	−	0.409779i	$-0.134394\pi$
$431$	16324.1	1.82437	0.912185	+	0.409779i	$0.134394\pi$
$432$	0	0
$433$	5168.75	0.573659	0.286829	−	0.957982i	$-0.407399\pi$
$433$	5168.75	0.573659	0.286829	+	0.957982i	$0.407399\pi$
$434$	0	0
$435$	0	0
$436$	−9345.63	−1.02655
$437$	−8826.95	−0.966248
$438$	0	0
$439$	−18339.0	−1.99378	−0.996892	−	0.0787782i	$-0.974898\pi$
$439$	−18339.0	−1.99378	−0.996892	+	0.0787782i	$0.974898\pi$
$440$	−4170.16	−0.451828
$441$	0	0
$442$	−2884.24	−0.310383
$443$	1613.28	0.173023	0.0865113	−	0.996251i	$-0.472428\pi$
$443$	1613.28	0.173023	0.0865113	+	0.996251i	$0.472428\pi$
$444$	0	0
$445$	3388.62	0.360980
$446$	−1434.09	−0.152256
$447$	0	0
$448$	0	0
$449$	886.750	0.0932034	0.0466017	−	0.998914i	$-0.485161\pi$
$449$	886.750	0.0932034	0.0466017	+	0.998914i	$0.485161\pi$
$450$	0	0
$451$	−13232.1	−1.38154
$452$	−162.345	−0.0168940
$453$	0	0
$454$	1688.21	0.174518
$455$	0	0
$456$	0	0
$457$	7391.22	0.756557	0.378279	−	0.925692i	$-0.376516\pi$
$457$	7391.22	0.756557	0.378279	+	0.925692i	$0.376516\pi$
$458$	10909.4	1.11302
$459$	0	0
$460$	−4126.50	−0.418259
$461$	−7133.35	−0.720679	−0.360340	−	0.932821i	$-0.617339\pi$
$461$	−7133.35	−0.720679	−0.360340	+	0.932821i	$0.617339\pi$
$462$	0	0
$463$	14461.8	1.45162	0.725808	−	0.687897i	$-0.241466\pi$
$463$	14461.8	1.45162	0.725808	+	0.687897i	$0.241466\pi$
$464$	−398.341	−0.0398546
$465$	0	0
$466$	3869.95	0.384704
$467$	−16393.5	−1.62441	−0.812206	−	0.583370i	$-0.801734\pi$
$467$	−16393.5	−1.62441	−0.812206	+	0.583370i	$0.801734\pi$
$468$	0	0
$469$	0	0
$470$	4613.09	0.452737
$471$	0	0
$472$	−327.370	−0.0319246
$473$	19380.7	1.88398
$474$	0	0
$475$	1364.92	0.131846
$476$	0	0
$477$	0	0
$478$	−4871.77	−0.466171
$479$	−12991.4	−1.23923	−0.619617	−	0.784904i	$-0.712712\pi$
$479$	−12991.4	−1.23923	−0.619617	+	0.784904i	$0.712712\pi$
$480$	0	0
$481$	10159.1	0.963025
$482$	9127.57	0.862551
$483$	0	0
$484$	−347.087	−0.0325964
$485$	−6125.17	−0.573463
$486$	0	0
$487$	−12863.5	−1.19692	−0.598461	−	0.801152i	$-0.704221\pi$
$487$	−12863.5	−1.19692	−0.598461	+	0.801152i	$0.704221\pi$
$488$	−3814.57	−0.353847
$489$	0	0
$490$	0	0
$491$	4898.10	0.450200	0.225100	−	0.974336i	$-0.427729\pi$
$491$	4898.10	0.450200	0.225100	+	0.974336i	$0.427729\pi$
$492$	0	0
$493$	−8037.32	−0.734245
$494$	−2695.55	−0.245503
$495$	0	0
$496$	−447.787	−0.0405368
$497$	0	0
$498$	0	0
$499$	10308.0	0.924746	0.462373	−	0.886686i	$-0.346998\pi$
$499$	10308.0	0.924746	0.462373	+	0.886686i	$0.346998\pi$
$500$	638.086	0.0570721
$501$	0	0
$502$	9477.37	0.842621
$503$	−15119.6	−1.34026	−0.670130	−	0.742244i	$-0.733762\pi$
$503$	−15119.6	−1.34026	−0.670130	+	0.742244i	$0.733762\pi$
$504$	0	0
$505$	1690.72	0.148982
$506$	10289.6	0.904009
$507$	0	0
$508$	−11336.6	−0.990114
$509$	14183.8	1.23514	0.617571	−	0.786515i	$-0.288117\pi$
$509$	14183.8	1.23514	0.617571	+	0.786515i	$0.288117\pi$
$510$	0	0
$511$	0	0
$512$	1047.24	0.0903943
$513$	0	0
$514$	3565.39	0.305958
$515$	−2833.50	−0.242444
$516$	0	0
$517$	20280.7	1.72523
$518$	0	0
$519$	0	0
$520$	−3235.02	−0.272817
$521$	−7464.08	−0.627653	−0.313827	−	0.949480i	$-0.601611\pi$
$521$	−7464.08	−0.627653	−0.313827	+	0.949480i	$0.601611\pi$
$522$	0	0
$523$	16642.9	1.39148	0.695739	−	0.718295i	$-0.255077\pi$
$523$	16642.9	1.39148	0.695739	+	0.718295i	$0.255077\pi$
$524$	−3300.24	−0.275137
$525$	0	0
$526$	12703.7	1.05305
$527$	−9035.01	−0.746814
$528$	0	0
$529$	13971.8	1.14834
$530$	2597.51	0.212885
$531$	0	0
$532$	0	0
$533$	−10264.8	−0.834181
$534$	0	0
$535$	−2812.66	−0.227293
$536$	−12307.8	−0.991818
$537$	0	0
$538$	−11096.5	−0.889230
$539$	0	0
$540$	0	0
$541$	67.8755	0.00539408	0.00269704	−	0.999996i	$-0.499142\pi$
$541$	67.8755	0.00539408	0.00269704	+	0.999996i	$0.499142\pi$
$542$	−4100.26	−0.324947
$543$	0	0
$544$	10709.0	0.844014
$545$	−9153.97	−0.719473
$546$	0	0
$547$	−9212.91	−0.720138	−0.360069	−	0.932926i	$-0.617247\pi$
$547$	−9212.91	−0.720138	−0.360069	+	0.932926i	$0.617247\pi$
$548$	−4573.85	−0.356542
$549$	0	0
$550$	−1591.09	−0.123354
$551$	−7511.51	−0.580764
$552$	0	0
$553$	0	0
$554$	−3777.17	−0.289669
$555$	0	0
$556$	−11810.2	−0.900838
$557$	−16699.6	−1.27035	−0.635173	−	0.772370i	$-0.719071\pi$
$557$	−16699.6	−1.27035	−0.635173	+	0.772370i	$0.719071\pi$
$558$	0	0
$559$	15034.6	1.13756
$560$	0	0
$561$	0	0
$562$	−9934.67	−0.745674
$563$	14772.8	1.10586	0.552931	−	0.833227i	$-0.313509\pi$
$563$	14772.8	1.10586	0.552931	+	0.833227i	$0.313509\pi$
$564$	0	0
$565$	−159.016	−0.0118405
$566$	6202.26	0.460601
$567$	0	0
$568$	−2683.27	−0.198217
$569$	−5663.76	−0.417289	−0.208644	−	0.977992i	$-0.566905\pi$
$569$	−5663.76	−0.417289	−0.208644	+	0.977992i	$0.566905\pi$
$570$	0	0
$571$	5579.58	0.408929	0.204464	−	0.978874i	$-0.434455\pi$
$571$	5579.58	0.408929	0.204464	+	0.978874i	$0.434455\pi$
$572$	−5539.99	−0.404962
$573$	0	0
$574$	0	0
$575$	−4041.87	−0.293144
$576$	0	0
$577$	2301.23	0.166034	0.0830170	−	0.996548i	$-0.473544\pi$
$577$	2301.23	0.166034	0.0830170	+	0.996548i	$0.473544\pi$
$578$	−2552.77	−0.183704
$579$	0	0
$580$	−3511.55	−0.251395
$581$	0	0
$582$	0	0
$583$	11419.5	0.811232
$584$	6347.46	0.449760
$585$	0	0
$586$	6428.34	0.453161
$587$	−16470.4	−1.15810	−0.579052	−	0.815291i	$-0.696577\pi$
$587$	−16470.4	−1.15810	−0.579052	+	0.815291i	$0.696577\pi$
$588$	0	0
$589$	−8443.92	−0.590706
$590$	−124.906	−0.00871573
$591$	0	0
$592$	−1013.72	−0.0703779
$593$	−13570.0	−0.939715	−0.469858	−	0.882742i	$-0.655695\pi$
$593$	−13570.0	−0.939715	−0.469858	+	0.882742i	$0.655695\pi$
$594$	0	0
$595$	0	0
$596$	4182.46	0.287450
$597$	0	0
$598$	7982.20	0.545847
$599$	−27814.1	−1.89725	−0.948625	−	0.316403i	$-0.897525\pi$
$599$	−27814.1	−1.89725	−0.948625	+	0.316403i	$0.897525\pi$
$600$	0	0
$601$	−20646.1	−1.40128	−0.700641	−	0.713514i	$-0.747102\pi$
$601$	−20646.1	−1.40128	−0.700641	+	0.713514i	$0.747102\pi$
$602$	0	0
$603$	0	0
$604$	−2729.70	−0.183891
$605$	−339.969	−0.0228458
$606$	0	0
$607$	−3315.28	−0.221686	−0.110843	−	0.993838i	$-0.535355\pi$
$607$	−3315.28	−0.221686	−0.110843	+	0.993838i	$0.535355\pi$
$608$	10008.4	0.667588
$609$	0	0
$610$	−1455.42	−0.0966037
$611$	15732.8	1.04170
$612$	0	0
$613$	−11113.9	−0.732278	−0.366139	−	0.930560i	$-0.619320\pi$
$613$	−11113.9	−0.732278	−0.366139	+	0.930560i	$0.619320\pi$
$614$	8166.89	0.536790
$615$	0	0
$616$	0	0
$617$	−7871.34	−0.513595	−0.256797	−	0.966465i	$-0.582667\pi$
$617$	−7871.34	−0.513595	−0.256797	+	0.966465i	$0.582667\pi$
$618$	0	0
$619$	19107.1	1.24068	0.620339	−	0.784334i	$-0.286995\pi$
$619$	19107.1	1.24068	0.620339	+	0.784334i	$0.286995\pi$
$620$	−3947.44	−0.255698
$621$	0	0
$622$	987.098	0.0636319
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	10404.7	0.664305
$627$	0	0
$628$	7987.60	0.507548
$629$	−20453.9	−1.29658
$630$	0	0
$631$	−25769.9	−1.62580	−0.812902	−	0.582401i	$-0.802113\pi$
$631$	−25769.9	−1.62580	−0.812902	+	0.582401i	$0.802113\pi$
$632$	−20996.5	−1.32151
$633$	0	0
$634$	7317.79	0.458401
$635$	−11104.1	−0.693939
$636$	0	0
$637$	0	0
$638$	8756.19	0.543355
$639$	0	0
$640$	4875.87	0.301149
$641$	1954.61	0.120440	0.0602202	−	0.998185i	$-0.480820\pi$
$641$	1954.61	0.120440	0.0602202	+	0.998185i	$0.480820\pi$
$642$	0	0
$643$	19396.5	1.18961	0.594807	−	0.803868i	$-0.297228\pi$
$643$	19396.5	1.18961	0.594807	+	0.803868i	$0.297228\pi$
$644$	0	0
$645$	0	0
$646$	5427.10	0.330536
$647$	31264.3	1.89973	0.949865	−	0.312661i	$-0.101220\pi$
$647$	31264.3	1.89973	0.949865	+	0.312661i	$0.101220\pi$
$648$	0	0
$649$	−549.126	−0.0332127
$650$	−1234.30	−0.0744817
$651$	0	0
$652$	5688.18	0.341666
$653$	−6442.75	−0.386102	−0.193051	−	0.981189i	$-0.561838\pi$
$653$	−6442.75	−0.386102	−0.193051	+	0.981189i	$0.561838\pi$
$654$	0	0
$655$	−3232.56	−0.192835
$656$	1024.27	0.0609620
$657$	0	0
$658$	0	0
$659$	3584.70	0.211897	0.105949	−	0.994372i	$-0.466212\pi$
$659$	3584.70	0.211897	0.105949	+	0.994372i	$0.466212\pi$
$660$	0	0
$661$	−6294.43	−0.370386	−0.185193	−	0.982702i	$-0.559291\pi$
$661$	−6294.43	−0.370386	−0.185193	+	0.982702i	$0.559291\pi$
$662$	−11379.3	−0.668078
$663$	0	0
$664$	−8409.85	−0.491514
$665$	0	0
$666$	0	0
$667$	22243.4	1.29126
$668$	9058.11	0.524654
$669$	0	0
$670$	−4695.94	−0.270776
$671$	−6398.51	−0.368125
$672$	0	0
$673$	−10233.9	−0.586162	−0.293081	−	0.956088i	$-0.594681\pi$
$673$	−10233.9	−0.586162	−0.293081	+	0.956088i	$0.594681\pi$
$674$	9986.94	0.570745
$675$	0	0
$676$	6917.33	0.393567
$677$	−7100.75	−0.403108	−0.201554	−	0.979477i	$-0.564599\pi$
$677$	−7100.75	−0.403108	−0.201554	+	0.979477i	$0.564599\pi$
$678$	0	0
$679$	0	0
$680$	6513.23	0.367310
$681$	0	0
$682$	9843.10	0.552657
$683$	35274.6	1.97620	0.988100	−	0.153813i	$-0.0491554\pi$
$683$	35274.6	1.97620	0.988100	+	0.153813i	$0.0491554\pi$
$684$	0	0
$685$	−4480.05	−0.249889
$686$	0	0
$687$	0	0
$688$	−1500.22	−0.0831330
$689$	8858.74	0.489828
$690$	0	0
$691$	−4945.12	−0.272245	−0.136122	−	0.990692i	$-0.543464\pi$
$691$	−4945.12	−0.272245	−0.136122	+	0.990692i	$0.543464\pi$
$692$	21520.8	1.18222
$693$	0	0
$694$	−3296.31	−0.180297
$695$	−11568.0	−0.631368
$696$	0	0
$697$	20666.7	1.12311
$698$	16588.2	0.899532
$699$	0	0
$700$	0	0
$701$	15300.4	0.824379	0.412190	−	0.911098i	$-0.364764\pi$
$701$	15300.4	0.824379	0.412190	+	0.911098i	$0.364764\pi$
$702$	0	0
$703$	−19115.7	−1.02555
$704$	−10800.4	−0.578206
$705$	0	0
$706$	−7787.38	−0.415130
$707$	0	0
$708$	0	0
$709$	−28297.4	−1.49892	−0.749458	−	0.662052i	$-0.769686\pi$
$709$	−28297.4	−1.49892	−0.749458	+	0.662052i	$0.769686\pi$
$710$	−1023.78	−0.0541153
$711$	0	0
$712$	15112.2	0.795441
$713$	25004.5	1.31336
$714$	0	0
$715$	−5426.37	−0.283825
$716$	−12407.9	−0.647635
$717$	0	0
$718$	−18461.9	−0.959597
$719$	8548.96	0.443425	0.221712	−	0.975112i	$-0.428835\pi$
$719$	8548.96	0.443425	0.221712	+	0.975112i	$0.428835\pi$
$720$	0	0
$721$	0	0
$722$	−6598.97	−0.340150
$723$	0	0
$724$	−13787.3	−0.707737
$725$	−3439.53	−0.176194
$726$	0	0
$727$	−14345.3	−0.731827	−0.365913	−	0.930649i	$-0.619243\pi$
$727$	−14345.3	−0.731827	−0.365913	+	0.930649i	$0.619243\pi$
$728$	0	0
$729$	0	0
$730$	2421.83	0.122789
$731$	−30270.0	−1.53157
$732$	0	0
$733$	22624.0	1.14002	0.570012	−	0.821637i	$-0.306939\pi$
$733$	22624.0	1.14002	0.570012	+	0.821637i	$0.306939\pi$
$734$	−19512.9	−0.981248
$735$	0	0
$736$	−29637.3	−1.48430
$737$	−20644.9	−1.03184
$738$	0	0
$739$	14837.3	0.738566	0.369283	−	0.929317i	$-0.379603\pi$
$739$	14837.3	0.738566	0.369283	+	0.929317i	$0.379603\pi$
$740$	−8936.39	−0.443930
$741$	0	0
$742$	0	0
$743$	−13073.1	−0.645497	−0.322749	−	0.946485i	$-0.604607\pi$
$743$	−13073.1	−0.645497	−0.322749	+	0.946485i	$0.604607\pi$
$744$	0	0
$745$	4096.69	0.201464
$746$	918.813	0.0450940
$747$	0	0
$748$	11154.0	0.545226
$749$	0	0
$750$	0	0
$751$	16213.3	0.787791	0.393895	−	0.919155i	$-0.371127\pi$
$751$	16213.3	0.787791	0.393895	+	0.919155i	$0.371127\pi$
$752$	−1569.89	−0.0761278
$753$	0	0
$754$	6792.65	0.328082
$755$	−2673.72	−0.128883
$756$	0	0
$757$	19903.9	0.955642	0.477821	−	0.878457i	$-0.341427\pi$
$757$	19903.9	0.955642	0.477821	+	0.878457i	$0.341427\pi$
$758$	14595.3	0.699372
$759$	0	0
$760$	6087.13	0.290531
$761$	−30125.5	−1.43502	−0.717509	−	0.696549i	$-0.754718\pi$
$761$	−30125.5	−1.43502	−0.717509	+	0.696549i	$0.754718\pi$
$762$	0	0
$763$	0	0
$764$	18433.5	0.872907
$765$	0	0
$766$	14681.0	0.692489
$767$	−425.986	−0.0200541
$768$	0	0
$769$	36049.1	1.69046	0.845230	−	0.534402i	$-0.179463\pi$
$769$	36049.1	1.69046	0.845230	+	0.534402i	$0.179463\pi$
$770$	0	0
$771$	0	0
$772$	22809.4	1.06338
$773$	−9644.77	−0.448769	−0.224384	−	0.974501i	$-0.572037\pi$
$773$	−9644.77	−0.448769	−0.224384	+	0.974501i	$0.572037\pi$
$774$	0	0
$775$	−3866.48	−0.179211
$776$	−27316.4	−1.26366
$777$	0	0
$778$	−15712.3	−0.724054
$779$	19314.7	0.888344
$780$	0	0
$781$	−4500.88	−0.206215
$782$	−16071.0	−0.734908
$783$	0	0
$784$	0	0
$785$	7823.80	0.355724
$786$	0	0
$787$	25218.6	1.14224	0.571122	−	0.820865i	$-0.306508\pi$
$787$	25218.6	1.14224	0.571122	+	0.820865i	$0.306508\pi$
$788$	−2217.59	−0.100252
$789$	0	0
$790$	−8011.06	−0.360786
$791$	0	0
$792$	0	0
$793$	−4963.67	−0.222276
$794$	−19768.8	−0.883586
$795$	0	0
$796$	−2393.67	−0.106584
$797$	32042.3	1.42409	0.712044	−	0.702135i	$-0.247770\pi$
$797$	32042.3	1.42409	0.712044	+	0.702135i	$0.247770\pi$
$798$	0	0
$799$	−31675.7	−1.40251
$800$	4582.85	0.202535
$801$	0	0
$802$	−18984.5	−0.835867
$803$	10647.1	0.467908
$804$	0	0
$805$	0	0
$806$	7635.82	0.333698
$807$	0	0
$808$	7540.07	0.328291
$809$	−3427.90	−0.148972	−0.0744860	−	0.997222i	$-0.523732\pi$
$809$	−3427.90	−0.148972	−0.0744860	+	0.997222i	$0.523732\pi$
$810$	0	0
$811$	−23094.4	−0.999943	−0.499972	−	0.866042i	$-0.666656\pi$
$811$	−23094.4	−0.999943	−0.499972	+	0.866042i	$0.666656\pi$
$812$	0	0
$813$	0	0
$814$	22283.3	0.959494
$815$	5571.53	0.239463
$816$	0	0
$817$	−28289.7	−1.21142
$818$	12639.3	0.540249
$819$	0	0
$820$	9029.39	0.384537
$821$	−474.741	−0.0201810	−0.0100905	−	0.999949i	$-0.503212\pi$
$821$	−474.741	−0.0201810	−0.0100905	+	0.999949i	$0.503212\pi$
$822$	0	0
$823$	24159.8	1.02328	0.511638	−	0.859201i	$-0.329039\pi$
$823$	24159.8	1.02328	0.511638	+	0.859201i	$0.329039\pi$
$824$	−12636.5	−0.534240
$825$	0	0
$826$	0	0
$827$	7566.35	0.318147	0.159074	−	0.987267i	$-0.449149\pi$
$827$	7566.35	0.318147	0.159074	+	0.987267i	$0.449149\pi$
$828$	0	0
$829$	10580.9	0.443295	0.221648	−	0.975127i	$-0.428857\pi$
$829$	10580.9	0.443295	0.221648	+	0.975127i	$0.428857\pi$
$830$	−3208.72	−0.134188
$831$	0	0
$832$	−8378.49	−0.349125
$833$	0	0
$834$	0	0
$835$	8872.34	0.367713
$836$	10424.2	0.431256
$837$	0	0
$838$	16411.1	0.676507
$839$	15315.6	0.630218	0.315109	−	0.949055i	$-0.397959\pi$
$839$	15315.6	0.630218	0.315109	+	0.949055i	$0.397959\pi$
$840$	0	0
$841$	−5460.40	−0.223888
$842$	16876.4	0.690735
$843$	0	0
$844$	−19068.6	−0.777688
$845$	6775.47	0.275838
$846$	0	0
$847$	0	0
$848$	−883.966	−0.0357966
$849$	0	0
$850$	2485.08	0.100279
$851$	56606.4	2.28019
$852$	0	0
$853$	−18598.2	−0.746528	−0.373264	−	0.927725i	$-0.621761\pi$
$853$	−18598.2	−0.746528	−0.373264	+	0.927725i	$0.621761\pi$
$854$	0	0
$855$	0	0
$856$	−12543.6	−0.500853
$857$	41775.3	1.66513	0.832566	−	0.553926i	$-0.186871\pi$
$857$	41775.3	1.66513	0.832566	+	0.553926i	$0.186871\pi$
$858$	0	0
$859$	−32414.7	−1.28752	−0.643758	−	0.765229i	$-0.722626\pi$
$859$	−32414.7	−1.28752	−0.643758	+	0.765229i	$0.722626\pi$
$860$	−13225.1	−0.524387
$861$	0	0
$862$	27776.4	1.09753
$863$	31299.7	1.23459	0.617297	−	0.786730i	$-0.288228\pi$
$863$	31299.7	1.23459	0.617297	+	0.786730i	$0.288228\pi$
$864$	0	0
$865$	21079.4	0.828580
$866$	8794.94	0.345109
$867$	0	0
$868$	0	0
$869$	−35219.2	−1.37483
$870$	0	0
$871$	−16015.4	−0.623030
$872$	−40823.8	−1.58540
$873$	0	0
$874$	−15019.6	−0.581288
$875$	0	0
$876$	0	0
$877$	−19973.0	−0.769031	−0.384515	−	0.923119i	$-0.625631\pi$
$877$	−19973.0	−0.769031	−0.384515	+	0.923119i	$0.625631\pi$
$878$	−31204.9	−1.19945
$879$	0	0
$880$	541.469	0.0207419
$881$	17367.9	0.664176	0.332088	−	0.943248i	$-0.392247\pi$
$881$	17367.9	0.664176	0.332088	+	0.943248i	$0.392247\pi$
$882$	0	0
$883$	−14364.2	−0.547446	−0.273723	−	0.961809i	$-0.588255\pi$
$883$	−14364.2	−0.547446	−0.273723	+	0.961809i	$0.588255\pi$
$884$	8652.73	0.329211
$885$	0	0
$886$	2745.09	0.104089
$887$	−33738.1	−1.27713	−0.638564	−	0.769568i	$-0.720471\pi$
$887$	−33738.1	−1.27713	−0.638564	+	0.769568i	$0.720471\pi$
$888$	0	0
$889$	0	0
$890$	5765.95	0.217163
$891$	0	0
$892$	4302.26	0.161491
$893$	−29603.4	−1.10934
$894$	0	0
$895$	−12153.5	−0.453906
$896$	0	0
$897$	0	0
$898$	1508.86	0.0560705
$899$	21278.2	0.789398
$900$	0	0
$901$	−17835.8	−0.659485
$902$	−22515.2	−0.831123
$903$	0	0
$904$	−709.162	−0.0260911
$905$	−13504.6	−0.496030
$906$	0	0
$907$	−32998.6	−1.20805	−0.604024	−	0.796966i	$-0.706437\pi$
$907$	−32998.6	−1.20805	−0.604024	+	0.796966i	$0.706437\pi$
$908$	−5064.62	−0.185105
$909$	0	0
$910$	0	0
$911$	−33446.3	−1.21638	−0.608192	−	0.793790i	$-0.708105\pi$
$911$	−33446.3	−1.21638	−0.608192	+	0.793790i	$0.708105\pi$
$912$	0	0
$913$	−14106.6	−0.511347
$914$	12576.6	0.455139
$915$	0	0
$916$	−32728.2	−1.18053
$917$	0	0
$918$	0	0
$919$	41708.7	1.49711	0.748554	−	0.663074i	$-0.230748\pi$
$919$	41708.7	1.49711	0.748554	+	0.663074i	$0.230748\pi$
$920$	−18025.5	−0.645960
$921$	0	0
$922$	−12137.8	−0.433556
$923$	−3491.58	−0.124514
$924$	0	0
$925$	−8753.12	−0.311136
$926$	24607.7	0.873283
$927$	0	0
$928$	−25220.6	−0.892140
$929$	49024.6	1.73137	0.865686	−	0.500587i	$-0.166883\pi$
$929$	49024.6	1.73137	0.865686	+	0.500587i	$0.166883\pi$
$930$	0	0
$931$	0	0
$932$	−11609.9	−0.408040
$933$	0	0
$934$	−27894.6	−0.977235
$935$	10925.2	0.382131
$936$	0	0
$937$	5447.58	0.189930	0.0949651	−	0.995481i	$-0.469726\pi$
$937$	5447.58	0.189930	0.0949651	+	0.995481i	$0.469726\pi$
$938$	0	0
$939$	0	0
$940$	−13839.3	−0.480200
$941$	4125.75	0.142928	0.0714642	−	0.997443i	$-0.477233\pi$
$941$	4125.75	0.142928	0.0714642	+	0.997443i	$0.477233\pi$
$942$	0	0
$943$	−57195.5	−1.97513
$944$	42.5069	0.00146555
$945$	0	0
$946$	32977.4	1.13339
$947$	−17332.3	−0.594747	−0.297374	−	0.954761i	$-0.596111\pi$
$947$	−17332.3	−0.594747	−0.297374	+	0.954761i	$0.596111\pi$
$948$	0	0
$949$	8259.57	0.282526
$950$	2322.50	0.0793177
$951$	0	0
$952$	0	0
$953$	−56839.4	−1.93201	−0.966007	−	0.258517i	$-0.916766\pi$
$953$	−56839.4	−1.93201	−0.966007	+	0.258517i	$0.916766\pi$
$954$	0	0
$955$	18055.5	0.611792
$956$	14615.3	0.494449
$957$	0	0
$958$	−22105.7	−0.745515
$959$	0	0
$960$	0	0
$961$	−5871.48	−0.197089
$962$	17286.3	0.579349
$963$	0	0
$964$	−27382.7	−0.914873
$965$	22341.6	0.745287
$966$	0	0
$967$	−13284.2	−0.441769	−0.220884	−	0.975300i	$-0.570894\pi$
$967$	−13284.2	−0.441769	−0.220884	+	0.975300i	$0.570894\pi$
$968$	−1516.15	−0.0503420
$969$	0	0
$970$	−10422.4	−0.344992
$971$	12153.0	0.401658	0.200829	−	0.979626i	$-0.435636\pi$
$971$	12153.0	0.401658	0.200829	+	0.979626i	$0.435636\pi$
$972$	0	0
$973$	0	0
$974$	−21888.1	−0.720060
$975$	0	0
$976$	495.298	0.0162440
$977$	37999.3	1.24433	0.622163	−	0.782888i	$-0.286254\pi$
$977$	37999.3	1.24433	0.622163	+	0.782888i	$0.286254\pi$
$978$	0	0
$979$	25349.0	0.827537
$980$	0	0
$981$	0	0
$982$	8334.42	0.270837
$983$	22375.0	0.725993	0.362996	−	0.931791i	$-0.381754\pi$
$983$	22375.0	0.725993	0.362996	+	0.931791i	$0.381754\pi$
$984$	0	0
$985$	−2172.11	−0.0702631
$986$	−13676.0	−0.441717
$987$	0	0
$988$	8086.65	0.260395
$989$	83772.9	2.69345
$990$	0	0
$991$	18985.3	0.608564	0.304282	−	0.952582i	$-0.401583\pi$
$991$	18985.3	0.608564	0.304282	+	0.952582i	$0.401583\pi$
$992$	−28351.2	−0.907412
$993$	0	0
$994$	0	0
$995$	−2344.58	−0.0747016
$996$	0	0
$997$	56476.5	1.79401	0.897006	−	0.442019i	$-0.145738\pi$
$997$	56476.5	1.79401	0.897006	+	0.442019i	$0.145738\pi$
$998$	17539.6	0.556321
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.v.1.2		2
3.2	odd	2		735.4.a.q.1.1		2
7.6	odd	2		315.4.a.g.1.2		2
21.20	even	2		105.4.a.g.1.1	✓	2
35.34	odd	2		1575.4.a.y.1.1		2
84.83	odd	2		1680.4.a.y.1.1		2
105.62	odd	4		525.4.d.j.274.2		4
105.83	odd	4		525.4.d.j.274.3		4
105.104	even	2		525.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.g.1.1	✓	2	21.20	even	2
315.4.a.g.1.2		2	7.6	odd	2
525.4.a.i.1.2		2	105.104	even	2
525.4.d.j.274.2		4	105.62	odd	4
525.4.d.j.274.3		4	105.83	odd	4
735.4.a.q.1.1		2	3.2	odd	2
1575.4.a.y.1.1		2	35.34	odd	2
1680.4.a.y.1.1		2	84.83	odd	2
2205.4.a.v.1.2		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70156 q^{2} -5.10469 q^{4} -5.00000 q^{5} -22.2984 q^{8} -8.50781 q^{10} -37.4031 q^{11} -29.0156 q^{13} +2.89531 q^{16} +58.4187 q^{17} +54.5969 q^{19} +25.5234 q^{20} -63.6437 q^{22} -161.675 q^{23} +25.0000 q^{25} -49.3719 q^{26} -137.581 q^{29} -154.659 q^{31} +183.314 q^{32} +99.4031 q^{34} -350.125 q^{37} +92.9000 q^{38} +111.492 q^{40} +353.769 q^{41} -518.156 q^{43} +190.931 q^{44} -275.100 q^{46} -542.219 q^{47} +42.5391 q^{50} +148.116 q^{52} -305.309 q^{53} +187.016 q^{55} -234.103 q^{58} +14.6813 q^{59} +171.069 q^{61} -263.163 q^{62} +288.758 q^{64} +145.078 q^{65} +551.956 q^{67} -298.209 q^{68} +120.334 q^{71} -284.659 q^{73} -595.759 q^{74} -278.700 q^{76} +941.612 q^{79} -14.4766 q^{80} +601.959 q^{82} +377.150 q^{83} -292.094 q^{85} -881.675 q^{86} +834.031 q^{88} -677.725 q^{89} +825.300 q^{92} -922.619 q^{94} -272.984 q^{95} +1225.03 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 9 q^{4} - 10 q^{5} - 51 q^{8} + 15 q^{10} - 62 q^{11} + 6 q^{13} + 25 q^{16} + 40 q^{17} + 122 q^{19} - 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} - 214 q^{26} - 352 q^{29} - 66 q^{31}+ \cdots + 670 q^{97}+O(q^{100}) \) 2 * q - 3 * q^2 + 9 * q^4 - 10 * q^5 - 51 * q^8 + 15 * q^10 - 62 * q^11 + 6 * q^13 + 25 * q^16 + 40 * q^17 + 122 * q^19 - 45 * q^20 + 52 * q^22 - 16 * q^23 + 50 * q^25 - 214 * q^26 - 352 * q^29 - 66 * q^31 + 309 * q^32 + 186 * q^34 - 188 * q^37 - 224 * q^38 + 255 * q^40 + 16 * q^41 - 396 * q^43 - 156 * q^44 - 960 * q^46 - 188 * q^47 - 75 * q^50 + 642 * q^52 - 982 * q^53 + 310 * q^55 + 774 * q^58 + 516 * q^59 + 880 * q^61 - 680 * q^62 - 479 * q^64 - 30 * q^65 - 356 * q^67 - 558 * q^68 - 310 * q^71 - 326 * q^73 - 1358 * q^74 + 672 * q^76 + 1832 * q^79 - 125 * q^80 + 2190 * q^82 - 680 * q^83 - 200 * q^85 - 1456 * q^86 + 1540 * q^88 + 796 * q^89 + 2880 * q^92 - 2588 * q^94 - 610 * q^95 + 670 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more