LMFDB - Embedded newform 2205.4.a.bk.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 = [3,-2,0,14,15,0,0,-66,0,-10,20,0,0] 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:	$3$
Coefficient field:	3.3.22952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 14 $ x^3 - x^2 - 18*x + 14
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			1.2
Root			$0.770205$ 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)

$f(q)$	$=$	$q-0.229795 q^{2} -7.94719 q^{4} +5.00000 q^{5} +3.66459 q^{8} -1.14898 q^{10} +51.0059 q^{11} -46.0867 q^{13} +62.7354 q^{16} +72.7575 q^{17} +123.876 q^{19} -39.7360 q^{20} -11.7209 q^{22} -156.124 q^{23} +25.0000 q^{25} +10.5905 q^{26} -191.155 q^{29} +116.298 q^{31} -43.7331 q^{32} -16.7194 q^{34} +83.1918 q^{37} -28.4660 q^{38} +18.3230 q^{40} +466.186 q^{41} +422.695 q^{43} -405.354 q^{44} +35.8766 q^{46} +268.534 q^{47} -5.74489 q^{50} +366.260 q^{52} -310.545 q^{53} +255.030 q^{55} +43.9266 q^{58} -709.612 q^{59} -402.425 q^{61} -26.7247 q^{62} -491.834 q^{64} -230.434 q^{65} -114.306 q^{67} -578.218 q^{68} -214.461 q^{71} -402.954 q^{73} -19.1171 q^{74} -984.463 q^{76} -1376.43 q^{79} +313.677 q^{80} -107.127 q^{82} +1152.33 q^{83} +363.788 q^{85} -97.1334 q^{86} +186.916 q^{88} +366.795 q^{89} +1240.75 q^{92} -61.7080 q^{94} +619.378 q^{95} +1068.22 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 14 q^{4} + 15 q^{5} - 66 q^{8} - 10 q^{10} + 20 q^{11} + 114 q^{16} + 234 q^{17} + 82 q^{19} + 70 q^{20} - 236 q^{22} - 30 q^{23} + 75 q^{25} + 76 q^{26} + 32 q^{29} + 362 q^{31} - 430 q^{32}+ \cdots + 3052 q^{97}+O(q^{100}) $ 3 * q - 2 * q^2 + 14 * q^4 + 15 * q^5 - 66 * q^8 - 10 * q^10 + 20 * q^11 + 114 * q^16 + 234 * q^17 + 82 * q^19 + 70 * q^20 - 236 * q^22 - 30 * q^23 + 75 * q^25 + 76 * q^26 + 32 * q^29 + 362 * q^31 - 430 * q^32 - 596 * q^34 + 302 * q^37 + 160 * q^38 - 330 * q^40 + 626 * q^41 - 68 * q^43 - 300 * q^44 - 412 * q^46 + 936 * q^47 - 50 * q^50 + 684 * q^52 - 298 * q^53 + 100 * q^55 - 1184 * q^58 + 428 * q^59 - 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 - 416 * q^71 + 312 * q^73 - 1668 * q^74 - 1712 * q^76 - 488 * q^79 + 570 * q^80 + 1476 * q^82 + 948 * q^83 + 1170 * q^85 + 1928 * q^86 + 124 * q^88 - 142 * q^89 + 3220 * q^92 - 328 * q^94 + 410 * q^95 + 3052 * 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 $

$2$	−0.229795	−0.0812450	−0.0406225	−	0.999175i	$-0.512934\pi$
$2$	−0.229795	−0.0812450	−0.0406225	+	0.999175i	$0.512934\pi$
$3$	0	0
$3$	0	0
$4$	−7.94719	−0.993399
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.66459	0.161954
$9$	0	0
$10$	−1.14898	−0.0363339
$11$	51.0059	1.39808	0.699039	−	0.715083i	$-0.253611\pi$
$11$	51.0059	1.39808	0.699039	+	0.715083i	$0.253611\pi$
$12$	0	0
$13$	−46.0867	−0.983243	−0.491622	−	0.870809i	$-0.663596\pi$
$13$	−46.0867	−0.983243	−0.491622	+	0.870809i	$0.663596\pi$
$14$	0	0
$15$	0	0
$16$	62.7354	0.980241
$17$	72.7575	1.03802	0.519009	−	0.854769i	$-0.326301\pi$
$17$	72.7575	1.03802	0.519009	+	0.854769i	$0.326301\pi$
$18$	0	0
$19$	123.876	1.49574	0.747868	−	0.663847i	$-0.231078\pi$
$19$	123.876	1.49574	0.747868	+	0.663847i	$0.231078\pi$
$20$	−39.7360	−0.444262
$21$	0	0
$22$	−11.7209	−0.113587
$23$	−156.124	−1.41539	−0.707697	−	0.706516i	$-0.750266\pi$
$23$	−156.124	−1.41539	−0.707697	+	0.706516i	$0.750266\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	10.5905	0.0798835
$27$	0	0
$28$	0	0
$29$	−191.155	−1.22402	−0.612011	−	0.790850i	$-0.709639\pi$
$29$	−191.155	−1.22402	−0.612011	+	0.790850i	$0.709639\pi$
$30$	0	0
$31$	116.298	0.673798	0.336899	−	0.941541i	$-0.390622\pi$
$31$	116.298	0.673798	0.336899	+	0.941541i	$0.390622\pi$
$32$	−43.7331	−0.241593
$33$	0	0
$34$	−16.7194	−0.0843337
$35$	0	0
$36$	0	0
$37$	83.1918	0.369639	0.184820	−	0.982772i	$-0.440830\pi$
$37$	83.1918	0.369639	0.184820	+	0.982772i	$0.440830\pi$
$38$	−28.4660	−0.121521
$39$	0	0
$40$	18.3230	0.0724279
$41$	466.186	1.77576	0.887879	−	0.460077i	$-0.152178\pi$
$41$	466.186	1.77576	0.887879	+	0.460077i	$0.152178\pi$
$42$	0	0
$43$	422.695	1.49908	0.749539	−	0.661960i	$-0.230275\pi$
$43$	422.695	1.49908	0.749539	+	0.661960i	$0.230275\pi$
$44$	−405.354	−1.38885
$45$	0	0
$46$	35.8766	0.114994
$47$	268.534	0.833400	0.416700	−	0.909044i	$-0.363187\pi$
$47$	268.534	0.833400	0.416700	+	0.909044i	$0.363187\pi$
$48$	0	0
$49$	0	0
$50$	−5.74489	−0.0162490
$51$	0	0
$52$	366.260	0.976753
$53$	−310.545	−0.804841	−0.402421	−	0.915455i	$-0.631831\pi$
$53$	−310.545	−0.804841	−0.402421	+	0.915455i	$0.631831\pi$
$54$	0	0
$55$	255.030	0.625240
$56$	0	0
$57$	0	0
$58$	43.9266	0.0994455
$59$	−709.612	−1.56582	−0.782912	−	0.622133i	$-0.786266\pi$
$59$	−709.612	−1.56582	−0.782912	+	0.622133i	$0.786266\pi$
$60$	0	0
$61$	−402.425	−0.844675	−0.422338	−	0.906439i	$-0.638790\pi$
$61$	−402.425	−0.844675	−0.422338	+	0.906439i	$0.638790\pi$
$62$	−26.7247	−0.0547427
$63$	0	0
$64$	−491.834	−0.960613
$65$	−230.434	−0.439720
$66$	0	0
$67$	−114.306	−0.208428	−0.104214	−	0.994555i	$-0.533233\pi$
$67$	−114.306	−0.208428	−0.104214	+	0.994555i	$0.533233\pi$
$68$	−578.218	−1.03117
$69$	0	0
$70$	0	0
$71$	−214.461	−0.358476	−0.179238	−	0.983806i	$-0.557363\pi$
$71$	−214.461	−0.358476	−0.179238	+	0.983806i	$0.557363\pi$
$72$	0	0
$73$	−402.954	−0.646058	−0.323029	−	0.946389i	$-0.604701\pi$
$73$	−402.954	−0.646058	−0.323029	+	0.946389i	$0.604701\pi$
$74$	−19.1171	−0.0300313
$75$	0	0
$76$	−984.463	−1.48586
$77$	0	0
$78$	0	0
$79$	−1376.43	−1.96026	−0.980128	−	0.198367i	$-0.936436\pi$
$79$	−1376.43	−1.96026	−0.980128	+	0.198367i	$0.936436\pi$
$80$	313.677	0.438377
$81$	0	0
$82$	−107.127	−0.144271
$83$	1152.33	1.52391	0.761955	−	0.647630i	$-0.224240\pi$
$83$	1152.33	1.52391	0.761955	+	0.647630i	$0.224240\pi$
$84$	0	0
$85$	363.788	0.464216
$86$	−97.1334	−0.121793
$87$	0	0
$88$	186.916	0.226424
$89$	366.795	0.436856	0.218428	−	0.975853i	$-0.429907\pi$
$89$	366.795	0.436856	0.218428	+	0.975853i	$0.429907\pi$
$90$	0	0
$91$	0	0
$92$	1240.75	1.40605
$93$	0	0
$94$	−61.7080	−0.0677095
$95$	619.378	0.668914
$96$	0	0
$97$	1068.22	1.11816	0.559079	−	0.829115i	$-0.311155\pi$
$97$	1068.22	1.11816	0.559079	+	0.829115i	$0.311155\pi$
$98$	0	0
$99$	0	0
$100$	−198.680	−0.198680
$101$	−393.710	−0.387877	−0.193939	−	0.981014i	$-0.562126\pi$
$101$	−393.710	−0.387877	−0.193939	+	0.981014i	$0.562126\pi$
$102$	0	0
$103$	1419.35	1.35779	0.678895	−	0.734235i	$-0.262459\pi$
$103$	1419.35	1.35779	0.678895	+	0.734235i	$0.262459\pi$
$104$	−168.889	−0.159240
$105$	0	0
$106$	71.3617	0.0653893
$107$	−2178.53	−1.96829	−0.984143	−	0.177377i	$-0.943239\pi$
$107$	−2178.53	−1.96829	−0.984143	+	0.177377i	$0.943239\pi$
$108$	0	0
$109$	249.368	0.219130	0.109565	−	0.993980i	$-0.465054\pi$
$109$	249.368	0.219130	0.109565	+	0.993980i	$0.465054\pi$
$110$	−58.6046	−0.0507976
$111$	0	0
$112$	0	0
$113$	805.709	0.670750	0.335375	−	0.942085i	$-0.391137\pi$
$113$	805.709	0.670750	0.335375	+	0.942085i	$0.391137\pi$
$114$	0	0
$115$	−780.619	−0.632984
$116$	1519.15	1.21594
$117$	0	0
$118$	163.066	0.127215
$119$	0	0
$120$	0	0
$121$	1270.60	0.954623
$122$	92.4753	0.0686256
$123$	0	0
$124$	−924.242	−0.669350
$125$	125.000	0.0894427
$126$	0	0
$127$	−458.507	−0.320362	−0.160181	−	0.987088i	$-0.551208\pi$
$127$	−458.507	−0.320362	−0.160181	+	0.987088i	$0.551208\pi$
$128$	462.886	0.319638
$129$	0	0
$130$	52.9526	0.0357250
$131$	−1048.12	−0.699041	−0.349521	−	0.936929i	$-0.613656\pi$
$131$	−1048.12	−0.699041	−0.349521	+	0.936929i	$0.613656\pi$
$132$	0	0
$133$	0	0
$134$	26.2670	0.0169338
$135$	0	0
$136$	266.627	0.168111
$137$	1559.36	0.972445	0.486223	−	0.873835i	$-0.338374\pi$
$137$	1559.36	0.972445	0.486223	+	0.873835i	$0.338374\pi$
$138$	0	0
$139$	−230.969	−0.140939	−0.0704696	−	0.997514i	$-0.522450\pi$
$139$	−230.969	−0.140939	−0.0704696	+	0.997514i	$0.522450\pi$
$140$	0	0
$141$	0	0
$142$	49.2821	0.0291244
$143$	−2350.70	−1.37465
$144$	0	0
$145$	−955.776	−0.547399
$146$	92.5970	0.0524889
$147$	0	0
$148$	−661.142	−0.367199
$149$	1754.58	0.964701	0.482351	−	0.875978i	$-0.339783\pi$
$149$	1754.58	0.964701	0.482351	+	0.875978i	$0.339783\pi$
$150$	0	0
$151$	1707.28	0.920107	0.460054	−	0.887891i	$-0.347830\pi$
$151$	1707.28	0.920107	0.460054	+	0.887891i	$0.347830\pi$
$152$	453.953	0.242240
$153$	0	0
$154$	0	0
$155$	581.490	0.301331
$156$	0	0
$157$	1815.33	0.922798	0.461399	−	0.887193i	$-0.347348\pi$
$157$	1815.33	0.922798	0.461399	+	0.887193i	$0.347348\pi$
$158$	316.297	0.159261
$159$	0	0
$160$	−218.665	−0.108044
$161$	0	0
$162$	0	0
$163$	1763.58	0.847449	0.423724	−	0.905791i	$-0.360723\pi$
$163$	1763.58	0.847449	0.423724	+	0.905791i	$0.360723\pi$
$164$	−3704.87	−1.76404
$165$	0	0
$166$	−264.800	−0.123810
$167$	−1184.52	−0.548866	−0.274433	−	0.961606i	$-0.588490\pi$
$167$	−1184.52	−0.548866	−0.274433	+	0.961606i	$0.588490\pi$
$168$	0	0
$169$	−73.0133	−0.0332332
$170$	−83.5968	−0.0377152
$171$	0	0
$172$	−3359.24	−1.48918
$173$	3232.75	1.42070	0.710351	−	0.703847i	$-0.248536\pi$
$173$	3232.75	1.42070	0.710351	+	0.703847i	$0.248536\pi$
$174$	0	0
$175$	0	0
$176$	3199.88	1.37045
$177$	0	0
$178$	−84.2878	−0.0354924
$179$	−1520.16	−0.634760	−0.317380	−	0.948298i	$-0.602803\pi$
$179$	−1520.16	−0.634760	−0.317380	+	0.948298i	$0.602803\pi$
$180$	0	0
$181$	−2173.37	−0.892514	−0.446257	−	0.894905i	$-0.647243\pi$
$181$	−2173.37	−0.892514	−0.446257	+	0.894905i	$0.647243\pi$
$182$	0	0
$183$	0	0
$184$	−572.130	−0.229228
$185$	415.959	0.165308
$186$	0	0
$187$	3711.06	1.45123
$188$	−2134.10	−0.827899
$189$	0	0
$190$	−142.330	−0.0543459
$191$	−1195.92	−0.453057	−0.226528	−	0.974005i	$-0.572738\pi$
$191$	−1195.92	−0.453057	−0.226528	+	0.974005i	$0.572738\pi$
$192$	0	0
$193$	−4678.84	−1.74503	−0.872514	−	0.488588i	$-0.837512\pi$
$193$	−4678.84	−1.74503	−0.872514	+	0.488588i	$0.837512\pi$
$194$	−245.472	−0.0908446
$195$	0	0
$196$	0	0
$197$	4802.92	1.73702	0.868512	−	0.495668i	$-0.165077\pi$
$197$	4802.92	1.73702	0.868512	+	0.495668i	$0.165077\pi$
$198$	0	0
$199$	−1348.64	−0.480414	−0.240207	−	0.970722i	$-0.577215\pi$
$199$	−1348.64	−0.480414	−0.240207	+	0.970722i	$0.577215\pi$
$200$	91.6148	0.0323907
$201$	0	0
$202$	90.4727	0.0315131
$203$	0	0
$204$	0	0
$205$	2330.93	0.794143
$206$	−326.159	−0.110314
$207$	0	0
$208$	−2891.27	−0.963815
$209$	6318.38	2.09116
$210$	0	0
$211$	5550.35	1.81091	0.905455	−	0.424442i	$-0.139530\pi$
$211$	5550.35	1.81091	0.905455	+	0.424442i	$0.139530\pi$
$212$	2467.96	0.799529
$213$	0	0
$214$	500.617	0.159913
$215$	2113.47	0.670408
$216$	0	0
$217$	0	0
$218$	−57.3037	−0.0178032
$219$	0	0
$220$	−2026.77	−0.621113
$221$	−3353.16	−1.02062
$222$	0	0
$223$	771.304	0.231616	0.115808	−	0.993272i	$-0.463054\pi$
$223$	771.304	0.231616	0.115808	+	0.993272i	$0.463054\pi$
$224$	0	0
$225$	0	0
$226$	−185.148	−0.0544951
$227$	5509.84	1.61102	0.805508	−	0.592585i	$-0.201892\pi$
$227$	5509.84	1.61102	0.805508	+	0.592585i	$0.201892\pi$
$228$	0	0
$229$	1378.43	0.397771	0.198885	−	0.980023i	$-0.436268\pi$
$229$	1378.43	0.397771	0.198885	+	0.980023i	$0.436268\pi$
$230$	179.383	0.0514267
$231$	0	0
$232$	−700.506	−0.198235
$233$	−2540.67	−0.714354	−0.357177	−	0.934037i	$-0.616261\pi$
$233$	−2540.67	−0.714354	−0.357177	+	0.934037i	$0.616261\pi$
$234$	0	0
$235$	1342.67	0.372708
$236$	5639.42	1.55549
$237$	0	0
$238$	0	0
$239$	5082.14	1.37547	0.687733	−	0.725964i	$-0.258606\pi$
$239$	5082.14	1.37547	0.687733	+	0.725964i	$0.258606\pi$
$240$	0	0
$241$	−681.272	−0.182094	−0.0910468	−	0.995847i	$-0.529021\pi$
$241$	−681.272	−0.182094	−0.0910468	+	0.995847i	$0.529021\pi$
$242$	−291.979	−0.0775583
$243$	0	0
$244$	3198.15	0.839100
$245$	0	0
$246$	0	0
$247$	−5709.02	−1.47067
$248$	426.185	0.109124
$249$	0	0
$250$	−28.7244	−0.00726677
$251$	1044.10	0.262561	0.131280	−	0.991345i	$-0.458091\pi$
$251$	1044.10	0.262561	0.131280	+	0.991345i	$0.458091\pi$
$252$	0	0
$253$	−7963.24	−1.97883
$254$	105.363	0.0260278
$255$	0	0
$256$	3828.30	0.934644
$257$	5205.91	1.26356	0.631781	−	0.775147i	$-0.282324\pi$
$257$	5205.91	1.26356	0.631781	+	0.775147i	$0.282324\pi$
$258$	0	0
$259$	0	0
$260$	1831.30	0.436817
$261$	0	0
$262$	240.853	0.0567936
$263$	2984.55	0.699754	0.349877	−	0.936796i	$-0.386223\pi$
$263$	2984.55	0.699754	0.349877	+	0.936796i	$0.386223\pi$
$264$	0	0
$265$	−1552.72	−0.359936
$266$	0	0
$267$	0	0
$268$	908.412	0.207053
$269$	3507.88	0.795089	0.397545	−	0.917583i	$-0.369862\pi$
$269$	3507.88	0.795089	0.397545	+	0.917583i	$0.369862\pi$
$270$	0	0
$271$	−1612.27	−0.361396	−0.180698	−	0.983539i	$-0.557836\pi$
$271$	−1612.27	−0.361396	−0.180698	+	0.983539i	$0.557836\pi$
$272$	4564.48	1.01751
$273$	0	0
$274$	−358.334	−0.0790063
$275$	1275.15	0.279616
$276$	0	0
$277$	7326.89	1.58928	0.794639	−	0.607082i	$-0.207660\pi$
$277$	7326.89	1.58928	0.794639	+	0.607082i	$0.207660\pi$
$278$	53.0756	0.0114506
$279$	0	0
$280$	0	0
$281$	3499.91	0.743016	0.371508	−	0.928430i	$-0.378841\pi$
$281$	3499.91	0.743016	0.371508	+	0.928430i	$0.378841\pi$
$282$	0	0
$283$	−6261.51	−1.31522	−0.657612	−	0.753357i	$-0.728433\pi$
$283$	−6261.51	−1.31522	−0.657612	+	0.753357i	$0.728433\pi$
$284$	1704.36	0.356110
$285$	0	0
$286$	540.179	0.111683
$287$	0	0
$288$	0	0
$289$	380.660	0.0774802
$290$	219.633	0.0444734
$291$	0	0
$292$	3202.35	0.641793
$293$	5860.41	1.16849	0.584247	−	0.811576i	$-0.301390\pi$
$293$	5860.41	1.16849	0.584247	+	0.811576i	$0.301390\pi$
$294$	0	0
$295$	−3548.06	−0.700258
$296$	304.864	0.0598644
$297$	0	0
$298$	−403.194	−0.0783771
$299$	7195.24	1.39168
$300$	0	0
$301$	0	0
$302$	−392.324	−0.0747541
$303$	0	0
$304$	7771.39	1.46618
$305$	−2012.12	−0.377750
$306$	0	0
$307$	3135.86	0.582974	0.291487	−	0.956575i	$-0.405850\pi$
$307$	3135.86	0.582974	0.291487	+	0.956575i	$0.405850\pi$
$308$	0	0
$309$	0	0
$310$	−133.624	−0.0244817
$311$	−5903.72	−1.07643	−0.538214	−	0.842808i	$-0.680901\pi$
$311$	−5903.72	−1.07643	−0.538214	+	0.842808i	$0.680901\pi$
$312$	0	0
$313$	−6127.89	−1.10661	−0.553304	−	0.832979i	$-0.686633\pi$
$313$	−6127.89	−1.10661	−0.553304	+	0.832979i	$0.686633\pi$
$314$	−417.155	−0.0749727
$315$	0	0
$316$	10938.7	1.94732
$317$	−2846.55	−0.504348	−0.252174	−	0.967682i	$-0.581146\pi$
$317$	−2846.55	−0.504348	−0.252174	+	0.967682i	$0.581146\pi$
$318$	0	0
$319$	−9750.04	−1.71128
$320$	−2459.17	−0.429599
$321$	0	0
$322$	0	0
$323$	9012.88	1.55260
$324$	0	0
$325$	−1152.17	−0.196649
$326$	−405.262	−0.0688509
$327$	0	0
$328$	1708.38	0.287590
$329$	0	0
$330$	0	0
$331$	−2166.44	−0.359753	−0.179877	−	0.983689i	$-0.557570\pi$
$331$	−2166.44	−0.359753	−0.179877	+	0.983689i	$0.557570\pi$
$332$	−9157.79	−1.51385
$333$	0	0
$334$	272.197	0.0445926
$335$	−571.530	−0.0932120
$336$	0	0
$337$	−4382.45	−0.708390	−0.354195	−	0.935172i	$-0.615245\pi$
$337$	−4382.45	−0.708390	−0.354195	+	0.935172i	$0.615245\pi$
$338$	16.7781	0.00270003
$339$	0	0
$340$	−2891.09	−0.461151
$341$	5931.88	0.942022
$342$	0	0
$343$	0	0
$344$	1549.00	0.242781
$345$	0	0
$346$	−742.871	−0.115425
$347$	5931.63	0.917656	0.458828	−	0.888525i	$-0.348269\pi$
$347$	5931.63	0.917656	0.458828	+	0.888525i	$0.348269\pi$
$348$	0	0
$349$	−1340.59	−0.205616	−0.102808	−	0.994701i	$-0.532783\pi$
$349$	−1340.59	−0.205616	−0.102808	+	0.994701i	$0.532783\pi$
$350$	0	0
$351$	0	0
$352$	−2230.64	−0.337766
$353$	716.976	0.108104	0.0540521	−	0.998538i	$-0.482786\pi$
$353$	716.976	0.108104	0.0540521	+	0.998538i	$0.482786\pi$
$354$	0	0
$355$	−1072.30	−0.160315
$356$	−2914.99	−0.433973
$357$	0	0
$358$	349.326	0.0515711
$359$	−3962.70	−0.582572	−0.291286	−	0.956636i	$-0.594083\pi$
$359$	−3962.70	−0.582572	−0.291286	+	0.956636i	$0.594083\pi$
$360$	0	0
$361$	8486.14	1.23723
$362$	499.430	0.0725123
$363$	0	0
$364$	0	0
$365$	−2014.77	−0.288926
$366$	0	0
$367$	7257.92	1.03232	0.516158	−	0.856493i	$-0.327362\pi$
$367$	7257.92	1.03232	0.516158	+	0.856493i	$0.327362\pi$
$368$	−9794.50	−1.38743
$369$	0	0
$370$	−95.5855	−0.0134304
$371$	0	0
$372$	0	0
$373$	1949.21	0.270580	0.135290	−	0.990806i	$-0.456803\pi$
$373$	1949.21	0.270580	0.135290	+	0.990806i	$0.456803\pi$
$374$	−852.786	−0.117905
$375$	0	0
$376$	984.070	0.134972
$377$	8809.71	1.20351
$378$	0	0
$379$	10147.5	1.37531	0.687655	−	0.726038i	$-0.258640\pi$
$379$	10147.5	1.37531	0.687655	+	0.726038i	$0.258640\pi$
$380$	−4922.31	−0.664498
$381$	0	0
$382$	274.817	0.0368086
$383$	1593.28	0.212566	0.106283	−	0.994336i	$-0.466105\pi$
$383$	1593.28	0.212566	0.106283	+	0.994336i	$0.466105\pi$
$384$	0	0
$385$	0	0
$386$	1075.18	0.141775
$387$	0	0
$388$	−8489.35	−1.11078
$389$	10208.7	1.33060	0.665300	−	0.746576i	$-0.268304\pi$
$389$	10208.7	1.33060	0.665300	+	0.746576i	$0.268304\pi$
$390$	0	0
$391$	−11359.2	−1.46920
$392$	0	0
$393$	0	0
$394$	−1103.69	−0.141124
$395$	−6882.14	−0.876653
$396$	0	0
$397$	8550.26	1.08092	0.540460	−	0.841370i	$-0.318250\pi$
$397$	8550.26	1.08092	0.540460	+	0.841370i	$0.318250\pi$
$398$	309.911	0.0390312
$399$	0	0
$400$	1568.39	0.196048
$401$	−6422.21	−0.799775	−0.399888	−	0.916564i	$-0.630951\pi$
$401$	−6422.21	−0.799775	−0.399888	+	0.916564i	$0.630951\pi$
$402$	0	0
$403$	−5359.79	−0.662507
$404$	3128.89	0.385317
$405$	0	0
$406$	0	0
$407$	4243.27	0.516785
$408$	0	0
$409$	8536.12	1.03199	0.515995	−	0.856591i	$-0.327422\pi$
$409$	8536.12	1.03199	0.515995	+	0.856591i	$0.327422\pi$
$410$	−535.637	−0.0645201
$411$	0	0
$412$	−11279.8	−1.34883
$413$	0	0
$414$	0	0
$415$	5761.65	0.681514
$416$	2015.51	0.237545
$417$	0	0
$418$	−1451.94	−0.169896
$419$	−9948.69	−1.15997	−0.579983	−	0.814629i	$-0.696941\pi$
$419$	−9948.69	−1.15997	−0.579983	+	0.814629i	$0.696941\pi$
$420$	0	0
$421$	−7989.63	−0.924918	−0.462459	−	0.886641i	$-0.653033\pi$
$421$	−7989.63	−0.924918	−0.462459	+	0.886641i	$0.653033\pi$
$422$	−1275.45	−0.147127
$423$	0	0
$424$	−1138.02	−0.130347
$425$	1818.94	0.207603
$426$	0	0
$427$	0	0
$428$	17313.2	1.95529
$429$	0	0
$430$	−485.667	−0.0544673
$431$	9893.26	1.10566	0.552832	−	0.833292i	$-0.313547\pi$
$431$	9893.26	1.10566	0.552832	+	0.833292i	$0.313547\pi$
$432$	0	0
$433$	10225.7	1.13491	0.567456	−	0.823404i	$-0.307928\pi$
$433$	10225.7	1.13491	0.567456	+	0.823404i	$0.307928\pi$
$434$	0	0
$435$	0	0
$436$	−1981.78	−0.217684
$437$	−19339.9	−2.11706
$438$	0	0
$439$	−3871.07	−0.420857	−0.210428	−	0.977609i	$-0.567486\pi$
$439$	−3871.07	−0.420857	−0.210428	+	0.977609i	$0.567486\pi$
$440$	934.579	0.101260
$441$	0	0
$442$	770.540	0.0829205
$443$	−394.626	−0.0423233	−0.0211617	−	0.999776i	$-0.506736\pi$
$443$	−394.626	−0.0423233	−0.0211617	+	0.999776i	$0.506736\pi$
$444$	0	0
$445$	1833.98	0.195368
$446$	−177.242	−0.0188176
$447$	0	0
$448$	0	0
$449$	14437.4	1.51747	0.758735	−	0.651400i	$-0.225818\pi$
$449$	14437.4	1.51747	0.758735	+	0.651400i	$0.225818\pi$
$450$	0	0
$451$	23778.3	2.48265
$452$	−6403.13	−0.666323
$453$	0	0
$454$	−1266.14	−0.130887
$455$	0	0
$456$	0	0
$457$	10334.8	1.05786	0.528928	−	0.848667i	$-0.322594\pi$
$457$	10334.8	1.05786	0.528928	+	0.848667i	$0.322594\pi$
$458$	−316.758	−0.0323169
$459$	0	0
$460$	6203.73	0.628806
$461$	14401.9	1.45502	0.727511	−	0.686096i	$-0.240677\pi$
$461$	14401.9	1.45502	0.727511	+	0.686096i	$0.240677\pi$
$462$	0	0
$463$	1265.08	0.126984	0.0634919	−	0.997982i	$-0.479776\pi$
$463$	1265.08	0.126984	0.0634919	+	0.997982i	$0.479776\pi$
$464$	−11992.2	−1.19984
$465$	0	0
$466$	583.833	0.0580377
$467$	−13178.9	−1.30588	−0.652941	−	0.757409i	$-0.726465\pi$
$467$	−13178.9	−1.30588	−0.652941	+	0.757409i	$0.726465\pi$
$468$	0	0
$469$	0	0
$470$	−308.540	−0.0302806
$471$	0	0
$472$	−2600.44	−0.253591
$473$	21559.9	2.09583
$474$	0	0
$475$	3096.89	0.299147
$476$	0	0
$477$	0	0
$478$	−1167.85	−0.111750
$479$	883.505	0.0842763	0.0421382	−	0.999112i	$-0.486583\pi$
$479$	883.505	0.0842763	0.0421382	+	0.999112i	$0.486583\pi$
$480$	0	0
$481$	−3834.04	−0.363445
$482$	156.553	0.0147942
$483$	0	0
$484$	−10097.7	−0.948322
$485$	5341.10	0.500055
$486$	0	0
$487$	4959.35	0.461458	0.230729	−	0.973018i	$-0.425889\pi$
$487$	4959.35	0.461458	0.230729	+	0.973018i	$0.425889\pi$
$488$	−1474.72	−0.136798
$489$	0	0
$490$	0	0
$491$	18088.8	1.66260	0.831301	−	0.555822i	$-0.187596\pi$
$491$	18088.8	1.66260	0.831301	+	0.555822i	$0.187596\pi$
$492$	0	0
$493$	−13908.0	−1.27056
$494$	1311.91	0.119485
$495$	0	0
$496$	7296.00	0.660484
$497$	0	0
$498$	0	0
$499$	9428.10	0.845811	0.422906	−	0.906174i	$-0.361010\pi$
$499$	9428.10	0.845811	0.422906	+	0.906174i	$0.361010\pi$
$500$	−993.399	−0.0888523
$501$	0	0
$502$	−239.928	−0.0213317
$503$	8543.95	0.757367	0.378684	−	0.925526i	$-0.376377\pi$
$503$	8543.95	0.757367	0.378684	+	0.925526i	$0.376377\pi$
$504$	0	0
$505$	−1968.55	−0.173464
$506$	1829.92	0.160770
$507$	0	0
$508$	3643.85	0.318247
$509$	−4081.74	−0.355442	−0.177721	−	0.984081i	$-0.556872\pi$
$509$	−4081.74	−0.355442	−0.177721	+	0.984081i	$0.556872\pi$
$510$	0	0
$511$	0	0
$512$	−4582.81	−0.395573
$513$	0	0
$514$	−1196.29	−0.102658
$515$	7096.73	0.607222
$516$	0	0
$517$	13696.8	1.16516
$518$	0	0
$519$	0	0
$520$	−844.445	−0.0712142
$521$	−2593.89	−0.218120	−0.109060	−	0.994035i	$-0.534784\pi$
$521$	−2593.89	−0.218120	−0.109060	+	0.994035i	$0.534784\pi$
$522$	0	0
$523$	3216.08	0.268890	0.134445	−	0.990921i	$-0.457075\pi$
$523$	3216.08	0.268890	0.134445	+	0.990921i	$0.457075\pi$
$524$	8329.59	0.694427
$525$	0	0
$526$	−685.836	−0.0568515
$527$	8461.55	0.699414
$528$	0	0
$529$	12207.7	1.00334
$530$	356.809	0.0292430
$531$	0	0
$532$	0	0
$533$	−21485.0	−1.74600
$534$	0	0
$535$	−10892.7	−0.880244
$536$	−418.885	−0.0337557
$537$	0	0
$538$	−806.094	−0.0645970
$539$	0	0
$540$	0	0
$541$	−14686.4	−1.16713	−0.583567	−	0.812065i	$-0.698343\pi$
$541$	−14686.4	−1.16713	−0.583567	+	0.812065i	$0.698343\pi$
$542$	370.492	0.0293616
$543$	0	0
$544$	−3181.91	−0.250778
$545$	1246.84	0.0979979
$546$	0	0
$547$	8296.56	0.648510	0.324255	−	0.945970i	$-0.394886\pi$
$547$	8296.56	0.648510	0.324255	+	0.945970i	$0.394886\pi$
$548$	−12392.5	−0.966027
$549$	0	0
$550$	−293.023	−0.0227174
$551$	−23679.4	−1.83081
$552$	0	0
$553$	0	0
$554$	−1683.69	−0.129121
$555$	0	0
$556$	1835.56	0.140009
$557$	−19711.8	−1.49949	−0.749746	−	0.661726i	$-0.769824\pi$
$557$	−19711.8	−1.49949	−0.749746	+	0.661726i	$0.769824\pi$
$558$	0	0
$559$	−19480.6	−1.47396
$560$	0	0
$561$	0	0
$562$	−804.264	−0.0603663
$563$	−11500.3	−0.860891	−0.430446	−	0.902617i	$-0.641644\pi$
$563$	−11500.3	−0.860891	−0.430446	+	0.902617i	$0.641644\pi$
$564$	0	0
$565$	4028.55	0.299969
$566$	1438.87	0.106855
$567$	0	0
$568$	−785.911	−0.0580565
$569$	−6191.04	−0.456137	−0.228068	−	0.973645i	$-0.573241\pi$
$569$	−6191.04	−0.456137	−0.228068	+	0.973645i	$0.573241\pi$
$570$	0	0
$571$	−15992.6	−1.17210	−0.586050	−	0.810275i	$-0.699318\pi$
$571$	−15992.6	−1.17210	−0.586050	+	0.810275i	$0.699318\pi$
$572$	18681.4	1.36558
$573$	0	0
$574$	0	0
$575$	−3903.10	−0.283079
$576$	0	0
$577$	−8285.26	−0.597781	−0.298891	−	0.954287i	$-0.596617\pi$
$577$	−8285.26	−0.597781	−0.298891	+	0.954287i	$0.596617\pi$
$578$	−87.4740	−0.00629488
$579$	0	0
$580$	7595.73	0.543786
$581$	0	0
$582$	0	0
$583$	−15839.6	−1.12523
$584$	−1476.66	−0.104631
$585$	0	0
$586$	−1346.70	−0.0949343
$587$	5419.34	0.381056	0.190528	−	0.981682i	$-0.438980\pi$
$587$	5419.34	0.381056	0.190528	+	0.981682i	$0.438980\pi$
$588$	0	0
$589$	14406.5	1.00782
$590$	815.328	0.0568924
$591$	0	0
$592$	5219.08	0.362336
$593$	16791.7	1.16282	0.581412	−	0.813609i	$-0.302500\pi$
$593$	16791.7	1.16282	0.581412	+	0.813609i	$0.302500\pi$
$594$	0	0
$595$	0	0
$596$	−13944.0	−0.958333
$597$	0	0
$598$	−1653.43	−0.113067
$599$	−3005.34	−0.205000	−0.102500	−	0.994733i	$-0.532684\pi$
$599$	−3005.34	−0.205000	−0.102500	+	0.994733i	$0.532684\pi$
$600$	0	0
$601$	1088.83	0.0739007	0.0369503	−	0.999317i	$-0.488236\pi$
$601$	1088.83	0.0739007	0.0369503	+	0.999317i	$0.488236\pi$
$602$	0	0
$603$	0	0
$604$	−13568.1	−0.914034
$605$	6353.02	0.426920
$606$	0	0
$607$	2227.51	0.148949	0.0744743	−	0.997223i	$-0.476272\pi$
$607$	2227.51	0.148949	0.0744743	+	0.997223i	$0.476272\pi$
$608$	−5417.46	−0.361360
$609$	0	0
$610$	462.377	0.0306903
$611$	−12375.9	−0.819434
$612$	0	0
$613$	−20003.8	−1.31802	−0.659010	−	0.752134i	$-0.729024\pi$
$613$	−20003.8	−1.31802	−0.659010	+	0.752134i	$0.729024\pi$
$614$	−720.606	−0.0473637
$615$	0	0
$616$	0	0
$617$	5297.13	0.345631	0.172816	−	0.984954i	$-0.444714\pi$
$617$	5297.13	0.345631	0.172816	+	0.984954i	$0.444714\pi$
$618$	0	0
$619$	−24122.7	−1.56636	−0.783178	−	0.621797i	$-0.786403\pi$
$619$	−24122.7	−1.56636	−0.783178	+	0.621797i	$0.786403\pi$
$620$	−4621.21	−0.299342
$621$	0	0
$622$	1356.65	0.0874544
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	1408.16	0.0899064
$627$	0	0
$628$	−14426.8	−0.916707
$629$	6052.83	0.383692
$630$	0	0
$631$	−13099.2	−0.826423	−0.413211	−	0.910635i	$-0.635593\pi$
$631$	−13099.2	−0.826423	−0.413211	+	0.910635i	$0.635593\pi$
$632$	−5044.05	−0.317471
$633$	0	0
$634$	654.125	0.0409757
$635$	−2292.54	−0.143270
$636$	0	0
$637$	0	0
$638$	2240.51	0.139033
$639$	0	0
$640$	2314.43	0.142947
$641$	7992.28	0.492475	0.246237	−	0.969210i	$-0.420806\pi$
$641$	7992.28	0.492475	0.246237	+	0.969210i	$0.420806\pi$
$642$	0	0
$643$	7618.97	0.467283	0.233642	−	0.972323i	$-0.424936\pi$
$643$	7618.97	0.467283	0.233642	+	0.972323i	$0.424936\pi$
$644$	0	0
$645$	0	0
$646$	−2071.12	−0.126141
$647$	5037.17	0.306076	0.153038	−	0.988220i	$-0.451094\pi$
$647$	5037.17	0.306076	0.153038	+	0.988220i	$0.451094\pi$
$648$	0	0
$649$	−36194.4	−2.18914
$650$	264.763	0.0159767
$651$	0	0
$652$	−14015.5	−0.841855
$653$	8818.74	0.528490	0.264245	−	0.964456i	$-0.414877\pi$
$653$	8818.74	0.528490	0.264245	+	0.964456i	$0.414877\pi$
$654$	0	0
$655$	−5240.59	−0.312621
$656$	29246.4	1.74067
$657$	0	0
$658$	0	0
$659$	−2787.13	−0.164752	−0.0823758	−	0.996601i	$-0.526251\pi$
$659$	−2787.13	−0.164752	−0.0823758	+	0.996601i	$0.526251\pi$
$660$	0	0
$661$	14848.3	0.873726	0.436863	−	0.899528i	$-0.356090\pi$
$661$	14848.3	0.873726	0.436863	+	0.899528i	$0.356090\pi$
$662$	497.838	0.0292281
$663$	0	0
$664$	4222.82	0.246803
$665$	0	0
$666$	0	0
$667$	29843.9	1.73247
$668$	9413.58	0.545243
$669$	0	0
$670$	131.335	0.00757301
$671$	−20526.0	−1.18092
$672$	0	0
$673$	−2238.59	−0.128219	−0.0641093	−	0.997943i	$-0.520421\pi$
$673$	−2238.59	−0.128219	−0.0641093	+	0.997943i	$0.520421\pi$
$674$	1007.07	0.0575531
$675$	0	0
$676$	580.250	0.0330138
$677$	−5505.03	−0.312519	−0.156259	−	0.987716i	$-0.549944\pi$
$677$	−5505.03	−0.312519	−0.156259	+	0.987716i	$0.549944\pi$
$678$	0	0
$679$	0	0
$680$	1333.13	0.0751814
$681$	0	0
$682$	−1363.12	−0.0765345
$683$	−8638.56	−0.483961	−0.241980	−	0.970281i	$-0.577797\pi$
$683$	−8638.56	−0.483961	−0.241980	+	0.970281i	$0.577797\pi$
$684$	0	0
$685$	7796.79	0.434891
$686$	0	0
$687$	0	0
$688$	26518.0	1.46946
$689$	14312.0	0.791355
$690$	0	0
$691$	32838.2	1.80785	0.903925	−	0.427691i	$-0.140673\pi$
$691$	32838.2	1.80785	0.903925	+	0.427691i	$0.140673\pi$
$692$	−25691.3	−1.41132
$693$	0	0
$694$	−1363.06	−0.0745549
$695$	−1154.85	−0.0630299
$696$	0	0
$697$	33918.6	1.84327
$698$	308.061	0.0167052
$699$	0	0
$700$	0	0
$701$	−32942.3	−1.77491	−0.887455	−	0.460894i	$-0.847529\pi$
$701$	−32942.3	−1.77491	−0.887455	+	0.460894i	$0.847529\pi$
$702$	0	0
$703$	10305.4	0.552883
$704$	−25086.4	−1.34301
$705$	0	0
$706$	−164.758	−0.00878292
$707$	0	0
$708$	0	0
$709$	−34495.5	−1.82723	−0.913614	−	0.406583i	$-0.866720\pi$
$709$	−34495.5	−1.82723	−0.913614	+	0.406583i	$0.866720\pi$
$710$	246.410	0.0130248
$711$	0	0
$712$	1344.15	0.0707505
$713$	−18156.9	−0.953690
$714$	0	0
$715$	−11753.5	−0.614762
$716$	12081.0	0.630570
$717$	0	0
$718$	910.611	0.0473310
$719$	−6884.11	−0.357071	−0.178535	−	0.983933i	$-0.557136\pi$
$719$	−6884.11	−0.357071	−0.178535	+	0.983933i	$0.557136\pi$
$720$	0	0
$721$	0	0
$722$	−1950.08	−0.100518
$723$	0	0
$724$	17272.2	0.886623
$725$	−4778.88	−0.244804
$726$	0	0
$727$	−13950.4	−0.711682	−0.355841	−	0.934546i	$-0.615806\pi$
$727$	−13950.4	−0.711682	−0.355841	+	0.934546i	$0.615806\pi$
$728$	0	0
$729$	0	0
$730$	462.985	0.0234738
$731$	30754.2	1.55607
$732$	0	0
$733$	4055.50	0.204357	0.102178	−	0.994766i	$-0.467419\pi$
$733$	4055.50	0.204357	0.102178	+	0.994766i	$0.467419\pi$
$734$	−1667.84	−0.0838705
$735$	0	0
$736$	6827.77	0.341950
$737$	−5830.28	−0.291399
$738$	0	0
$739$	−11122.2	−0.553638	−0.276819	−	0.960922i	$-0.589280\pi$
$739$	−11122.2	−0.553638	−0.276819	+	0.960922i	$0.589280\pi$
$740$	−3305.71	−0.164217
$741$	0	0
$742$	0	0
$743$	−4496.34	−0.222012	−0.111006	−	0.993820i	$-0.535407\pi$
$743$	−4496.34	−0.222012	−0.111006	+	0.993820i	$0.535407\pi$
$744$	0	0
$745$	8772.88	0.431427
$746$	−447.920	−0.0219833
$747$	0	0
$748$	−29492.6	−1.44165
$749$	0	0
$750$	0	0
$751$	1524.74	0.0740862	0.0370431	−	0.999314i	$-0.488206\pi$
$751$	1524.74	0.0740862	0.0370431	+	0.999314i	$0.488206\pi$
$752$	16846.6	0.816933
$753$	0	0
$754$	−2024.43	−0.0977791
$755$	8536.38	0.411485
$756$	0	0
$757$	19727.4	0.947166	0.473583	−	0.880749i	$-0.342960\pi$
$757$	19727.4	0.947166	0.473583	+	0.880749i	$0.342960\pi$
$758$	−2331.85	−0.111737
$759$	0	0
$760$	2269.77	0.108333
$761$	−6327.41	−0.301404	−0.150702	−	0.988579i	$-0.548153\pi$
$761$	−6327.41	−0.301404	−0.150702	+	0.988579i	$0.548153\pi$
$762$	0	0
$763$	0	0
$764$	9504.22	0.450066
$765$	0	0
$766$	−366.128	−0.0172699
$767$	32703.7	1.53959
$768$	0	0
$769$	−6909.63	−0.324015	−0.162008	−	0.986790i	$-0.551797\pi$
$769$	−6909.63	−0.324015	−0.162008	+	0.986790i	$0.551797\pi$
$770$	0	0
$771$	0	0
$772$	37183.7	1.73351
$773$	15447.7	0.718777	0.359388	−	0.933188i	$-0.382985\pi$
$773$	15447.7	0.718777	0.359388	+	0.933188i	$0.382985\pi$
$774$	0	0
$775$	2907.45	0.134760
$776$	3914.59	0.181090
$777$	0	0
$778$	−2345.92	−0.108105
$779$	57749.1	2.65607
$780$	0	0
$781$	−10938.8	−0.501178
$782$	2610.29	0.119365
$783$	0	0
$784$	0	0
$785$	9076.66	0.412688
$786$	0	0
$787$	−28152.9	−1.27515	−0.637575	−	0.770388i	$-0.720062\pi$
$787$	−28152.9	−1.27515	−0.637575	+	0.770388i	$0.720062\pi$
$788$	−38169.7	−1.72556
$789$	0	0
$790$	1581.48	0.0712236
$791$	0	0
$792$	0	0
$793$	18546.4	0.830521
$794$	−1964.81	−0.0878193
$795$	0	0
$796$	10717.9	0.477243
$797$	10603.4	0.471259	0.235629	−	0.971843i	$-0.424285\pi$
$797$	10603.4	0.471259	0.235629	+	0.971843i	$0.424285\pi$
$798$	0	0
$799$	19537.9	0.865083
$800$	−1093.33	−0.0483187
$801$	0	0
$802$	1475.79	0.0649777
$803$	−20553.0	−0.903239
$804$	0	0
$805$	0	0
$806$	1231.66	0.0538253
$807$	0	0
$808$	−1442.79	−0.0628181
$809$	−8022.99	−0.348669	−0.174335	−	0.984686i	$-0.555777\pi$
$809$	−8022.99	−0.348669	−0.174335	+	0.984686i	$0.555777\pi$
$810$	0	0
$811$	33370.2	1.44487	0.722433	−	0.691441i	$-0.243024\pi$
$811$	33370.2	1.44487	0.722433	+	0.691441i	$0.243024\pi$
$812$	0	0
$813$	0	0
$814$	−975.085	−0.0419861
$815$	8817.89	0.378991
$816$	0	0
$817$	52361.5	2.24223
$818$	−1961.56	−0.0838440
$819$	0	0
$820$	−18524.4	−0.788901
$821$	29452.2	1.25200	0.625999	−	0.779824i	$-0.284692\pi$
$821$	29452.2	1.25200	0.625999	+	0.779824i	$0.284692\pi$
$822$	0	0
$823$	−3399.31	−0.143976	−0.0719882	−	0.997405i	$-0.522934\pi$
$823$	−3399.31	−0.143976	−0.0719882	+	0.997405i	$0.522934\pi$
$824$	5201.33	0.219899
$825$	0	0
$826$	0	0
$827$	−18959.8	−0.797217	−0.398608	−	0.917121i	$-0.630507\pi$
$827$	−18959.8	−0.797217	−0.398608	+	0.917121i	$0.630507\pi$
$828$	0	0
$829$	3162.26	0.132485	0.0662424	−	0.997804i	$-0.478899\pi$
$829$	3162.26	0.132485	0.0662424	+	0.997804i	$0.478899\pi$
$830$	−1324.00	−0.0553695
$831$	0	0
$832$	22667.0	0.944516
$833$	0	0
$834$	0	0
$835$	−5922.58	−0.245460
$836$	−50213.4	−2.07735
$837$	0	0
$838$	2286.16	0.0942413
$839$	10141.2	0.417296	0.208648	−	0.977991i	$-0.433094\pi$
$839$	10141.2	0.417296	0.208648	+	0.977991i	$0.433094\pi$
$840$	0	0
$841$	12151.3	0.498228
$842$	1835.98	0.0751450
$843$	0	0
$844$	−44109.7	−1.79896
$845$	−365.066	−0.0148623
$846$	0	0
$847$	0	0
$848$	−19482.2	−0.788939
$849$	0	0
$850$	−417.984	−0.0168667
$851$	−12988.2	−0.523185
$852$	0	0
$853$	−32027.8	−1.28559	−0.642796	−	0.766038i	$-0.722226\pi$
$853$	−32027.8	−1.28559	−0.642796	+	0.766038i	$0.722226\pi$
$854$	0	0
$855$	0	0
$856$	−7983.43	−0.318771
$857$	49716.8	1.98167	0.990837	−	0.135067i	$-0.0431249\pi$
$857$	49716.8	1.98167	0.990837	+	0.135067i	$0.0431249\pi$
$858$	0	0
$859$	−40559.0	−1.61101	−0.805504	−	0.592590i	$-0.798105\pi$
$859$	−40559.0	−1.61101	−0.805504	+	0.592590i	$0.798105\pi$
$860$	−16796.2	−0.665983
$861$	0	0
$862$	−2273.43	−0.0898297
$863$	41262.0	1.62755	0.813774	−	0.581182i	$-0.197410\pi$
$863$	41262.0	1.62755	0.813774	+	0.581182i	$0.197410\pi$
$864$	0	0
$865$	16163.8	0.635357
$866$	−2349.83	−0.0922059
$867$	0	0
$868$	0	0
$869$	−70205.9	−2.74059
$870$	0	0
$871$	5267.99	0.204936
$872$	913.834	0.0354889
$873$	0	0
$874$	4444.23	0.172000
$875$	0	0
$876$	0	0
$877$	−25132.2	−0.967677	−0.483838	−	0.875157i	$-0.660758\pi$
$877$	−25132.2	−0.967677	−0.483838	+	0.875157i	$0.660758\pi$
$878$	889.554	0.0341925
$879$	0	0
$880$	15999.4	0.612886
$881$	−42008.7	−1.60648	−0.803240	−	0.595655i	$-0.796893\pi$
$881$	−42008.7	−1.60648	−0.803240	+	0.595655i	$0.796893\pi$
$882$	0	0
$883$	−2280.54	−0.0869153	−0.0434576	−	0.999055i	$-0.513837\pi$
$883$	−2280.54	−0.0869153	−0.0434576	+	0.999055i	$0.513837\pi$
$884$	26648.2	1.01389
$885$	0	0
$886$	90.6832	0.00343856
$887$	−22386.4	−0.847419	−0.423709	−	0.905798i	$-0.639272\pi$
$887$	−22386.4	−0.847419	−0.423709	+	0.905798i	$0.639272\pi$
$888$	0	0
$889$	0	0
$890$	−421.439	−0.0158727
$891$	0	0
$892$	−6129.70	−0.230087
$893$	33264.8	1.24655
$894$	0	0
$895$	−7600.80	−0.283873
$896$	0	0
$897$	0	0
$898$	−3317.65	−0.123287
$899$	−22230.9	−0.824743
$900$	0	0
$901$	−22594.5	−0.835439
$902$	−5464.14	−0.201703
$903$	0	0
$904$	2952.60	0.108630
$905$	−10866.8	−0.399145
$906$	0	0
$907$	19570.3	0.716451	0.358226	−	0.933635i	$-0.383382\pi$
$907$	19570.3	0.716451	0.358226	+	0.933635i	$0.383382\pi$
$908$	−43787.7	−1.60038
$909$	0	0
$910$	0	0
$911$	15605.6	0.567550	0.283775	−	0.958891i	$-0.408413\pi$
$911$	15605.6	0.567550	0.283775	+	0.958891i	$0.408413\pi$
$912$	0	0
$913$	58775.6	2.13055
$914$	−2374.88	−0.0859454
$915$	0	0
$916$	−10954.7	−0.395145
$917$	0	0
$918$	0	0
$919$	−905.249	−0.0324934	−0.0162467	−	0.999868i	$-0.505172\pi$
$919$	−905.249	−0.0324934	−0.0162467	+	0.999868i	$0.505172\pi$
$920$	−2860.65	−0.102514
$921$	0	0
$922$	−3309.50	−0.118213
$923$	9883.79	0.352469
$924$	0	0
$925$	2079.80	0.0739278
$926$	−290.711	−0.0103168
$927$	0	0
$928$	8359.80	0.295715
$929$	−44804.2	−1.58232	−0.791161	−	0.611608i	$-0.790523\pi$
$929$	−44804.2	−1.58232	−0.791161	+	0.611608i	$0.790523\pi$
$930$	0	0
$931$	0	0
$932$	20191.2	0.709639
$933$	0	0
$934$	3028.45	0.106096
$935$	18555.3	0.649010
$936$	0	0
$937$	−22958.8	−0.800461	−0.400230	−	0.916415i	$-0.631070\pi$
$937$	−22958.8	−0.800461	−0.400230	+	0.916415i	$0.631070\pi$
$938$	0	0
$939$	0	0
$940$	−10670.5	−0.370247
$941$	27191.5	0.941994	0.470997	−	0.882135i	$-0.343894\pi$
$941$	27191.5	0.941994	0.470997	+	0.882135i	$0.343894\pi$
$942$	0	0
$943$	−72782.8	−2.51340
$944$	−44517.8	−1.53488
$945$	0	0
$946$	−4954.38	−0.170276
$947$	45681.4	1.56752	0.783762	−	0.621061i	$-0.213298\pi$
$947$	45681.4	1.56752	0.783762	+	0.621061i	$0.213298\pi$
$948$	0	0
$949$	18570.8	0.635232
$950$	−711.651	−0.0243042
$951$	0	0
$952$	0	0
$953$	−21670.3	−0.736591	−0.368295	−	0.929709i	$-0.620058\pi$
$953$	−21670.3	−0.736591	−0.368295	+	0.929709i	$0.620058\pi$
$954$	0	0
$955$	−5979.60	−0.202613
$956$	−40388.7	−1.36639
$957$	0	0
$958$	−203.025	−0.00684703
$959$	0	0
$960$	0	0
$961$	−16265.8	−0.545997
$962$	881.045	0.0295281
$963$	0	0
$964$	5414.20	0.180892
$965$	−23394.2	−0.780401
$966$	0	0
$967$	−46935.4	−1.56085	−0.780424	−	0.625250i	$-0.784997\pi$
$967$	−46935.4	−1.56085	−0.780424	+	0.625250i	$0.784997\pi$
$968$	4656.24	0.154605
$969$	0	0
$970$	−1227.36	−0.0406270
$971$	−6475.64	−0.214020	−0.107010	−	0.994258i	$-0.534128\pi$
$971$	−6475.64	−0.214020	−0.107010	+	0.994258i	$0.534128\pi$
$972$	0	0
$973$	0	0
$974$	−1139.64	−0.0374911
$975$	0	0
$976$	−25246.3	−0.827985
$977$	−1025.22	−0.0335717	−0.0167859	−	0.999859i	$-0.505343\pi$
$977$	−1025.22	−0.0335717	−0.0167859	+	0.999859i	$0.505343\pi$
$978$	0	0
$979$	18708.7	0.610759
$980$	0	0
$981$	0	0
$982$	−4156.73	−0.135078
$983$	−7379.49	−0.239440	−0.119720	−	0.992808i	$-0.538200\pi$
$983$	−7379.49	−0.239440	−0.119720	+	0.992808i	$0.538200\pi$
$984$	0	0
$985$	24014.6	0.776821
$986$	3195.99	0.103226
$987$	0	0
$988$	45370.7	1.46096
$989$	−65992.8	−2.12179
$990$	0	0
$991$	−27747.3	−0.889428	−0.444714	−	0.895673i	$-0.646695\pi$
$991$	−27747.3	−0.889428	−0.444714	+	0.895673i	$0.646695\pi$
$992$	−5086.07	−0.162785
$993$	0	0
$994$	0	0
$995$	−6743.19	−0.214848
$996$	0	0
$997$	−48509.6	−1.54094	−0.770469	−	0.637478i	$-0.779978\pi$
$997$	−48509.6	−1.54094	−0.770469	+	0.637478i	$0.779978\pi$
$998$	−2166.54	−0.0687179
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bk.1.2		3
3.2	odd	2		2205.4.a.bl.1.2		3
7.6	odd	2		315.4.a.n.1.2	✓	3
21.20	even	2		315.4.a.o.1.2	yes	3
35.34	odd	2		1575.4.a.be.1.2		3
105.104	even	2		1575.4.a.bb.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.n.1.2	✓	3	7.6	odd	2
315.4.a.o.1.2	yes	3	21.20	even	2
1575.4.a.bb.1.2		3	105.104	even	2
1575.4.a.be.1.2		3	35.34	odd	2
2205.4.a.bk.1.2		3	1.1	even	1	trivial
2205.4.a.bl.1.2		3	3.2	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-0.229795 q^{2} -7.94719 q^{4} +5.00000 q^{5} +3.66459 q^{8} -1.14898 q^{10} +51.0059 q^{11} -46.0867 q^{13} +62.7354 q^{16} +72.7575 q^{17} +123.876 q^{19} -39.7360 q^{20} -11.7209 q^{22} -156.124 q^{23} +25.0000 q^{25} +10.5905 q^{26} -191.155 q^{29} +116.298 q^{31} -43.7331 q^{32} -16.7194 q^{34} +83.1918 q^{37} -28.4660 q^{38} +18.3230 q^{40} +466.186 q^{41} +422.695 q^{43} -405.354 q^{44} +35.8766 q^{46} +268.534 q^{47} -5.74489 q^{50} +366.260 q^{52} -310.545 q^{53} +255.030 q^{55} +43.9266 q^{58} -709.612 q^{59} -402.425 q^{61} -26.7247 q^{62} -491.834 q^{64} -230.434 q^{65} -114.306 q^{67} -578.218 q^{68} -214.461 q^{71} -402.954 q^{73} -19.1171 q^{74} -984.463 q^{76} -1376.43 q^{79} +313.677 q^{80} -107.127 q^{82} +1152.33 q^{83} +363.788 q^{85} -97.1334 q^{86} +186.916 q^{88} +366.795 q^{89} +1240.75 q^{92} -61.7080 q^{94} +619.378 q^{95} +1068.22 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 14 q^{4} + 15 q^{5} - 66 q^{8} - 10 q^{10} + 20 q^{11} + 114 q^{16} + 234 q^{17} + 82 q^{19} + 70 q^{20} - 236 q^{22} - 30 q^{23} + 75 q^{25} + 76 q^{26} + 32 q^{29} + 362 q^{31} - 430 q^{32}+ \cdots + 3052 q^{97}+O(q^{100}) \) 3 * q - 2 * q^2 + 14 * q^4 + 15 * q^5 - 66 * q^8 - 10 * q^10 + 20 * q^11 + 114 * q^16 + 234 * q^17 + 82 * q^19 + 70 * q^20 - 236 * q^22 - 30 * q^23 + 75 * q^25 + 76 * q^26 + 32 * q^29 + 362 * q^31 - 430 * q^32 - 596 * q^34 + 302 * q^37 + 160 * q^38 - 330 * q^40 + 626 * q^41 - 68 * q^43 - 300 * q^44 - 412 * q^46 + 936 * q^47 - 50 * q^50 + 684 * q^52 - 298 * q^53 + 100 * q^55 - 1184 * q^58 + 428 * q^59 - 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 - 416 * q^71 + 312 * q^73 - 1668 * q^74 - 1712 * q^76 - 488 * q^79 + 570 * q^80 + 1476 * q^82 + 948 * q^83 + 1170 * q^85 + 1928 * q^86 + 124 * q^88 - 142 * q^89 + 3220 * q^92 - 328 * q^94 + 410 * q^95 + 3052 * q^97

Introduction

L-functions

Modular forms

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