LMFDB - Embedded newform 2205.4.a.cg.1.6

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 45x^{6} + 134x^{5} + 641x^{4} - 1130x^{3} - 2877x^{2} + 2584x + 3696 $ x^8 - 4x^7 - 45x^6 + 134x^5 + 641x^4 - 1130x^3 - 2877x^2 + 2584*x + 3696
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.02408$ 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)

$f(q)$	$=$	$q+3.02408 q^{2} +1.14504 q^{4} +5.00000 q^{5} -20.7299 q^{8} +O(q^{10})$	$q+3.02408 q^{2} +1.14504 q^{4} +5.00000 q^{5} -20.7299 q^{8} +15.1204 q^{10} -22.1595 q^{11} +35.9983 q^{13} -71.8492 q^{16} +85.8421 q^{17} +97.0029 q^{19} +5.72522 q^{20} -67.0121 q^{22} -98.2844 q^{23} +25.0000 q^{25} +108.862 q^{26} +200.052 q^{29} -296.976 q^{31} -51.4382 q^{32} +259.593 q^{34} -185.337 q^{37} +293.344 q^{38} -103.650 q^{40} -291.601 q^{41} +173.597 q^{43} -25.3736 q^{44} -297.220 q^{46} +52.9175 q^{47} +75.6019 q^{50} +41.2196 q^{52} +294.704 q^{53} -110.798 q^{55} +604.973 q^{58} -197.462 q^{59} +430.880 q^{61} -898.078 q^{62} +419.241 q^{64} +179.991 q^{65} -519.033 q^{67} +98.2929 q^{68} +980.318 q^{71} -55.7062 q^{73} -560.473 q^{74} +111.072 q^{76} +830.967 q^{79} -359.246 q^{80} -881.824 q^{82} +1280.48 q^{83} +429.210 q^{85} +524.969 q^{86} +459.365 q^{88} -506.166 q^{89} -112.540 q^{92} +160.026 q^{94} +485.014 q^{95} +1041.60 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 42 q^{4} + 40 q^{5} + 48 q^{8}+O(q^{10}) $ 8 * q + 4 * q^2 + 42 * q^4 + 40 * q^5 + 48 * q^8	$ 8 q + 4 q^{2} + 42 q^{4} + 40 q^{5} + 48 q^{8} + 20 q^{10} + 100 q^{11} + 102 q^{13} + 266 q^{16} - 56 q^{17} + 210 q^{20} - 70 q^{22} + 190 q^{23} + 200 q^{25} - 60 q^{26} + 296 q^{29} - 42 q^{31} + 718 q^{32} + 488 q^{34} + 314 q^{37} + 514 q^{38} + 240 q^{40} + 28 q^{41} + 714 q^{43} + 410 q^{44} - 650 q^{46} - 326 q^{47} + 100 q^{50} + 794 q^{52} + 1282 q^{53} + 500 q^{55} + 942 q^{58} + 924 q^{59} + 536 q^{61} - 50 q^{62} + 1902 q^{64} + 510 q^{65} + 2 q^{67} - 2690 q^{68} + 1516 q^{71} - 86 q^{73} + 4754 q^{74} - 30 q^{76} - 42 q^{79} + 1330 q^{80} + 3602 q^{82} - 572 q^{83} - 280 q^{85} + 1758 q^{86} - 4312 q^{88} - 940 q^{89} + 3844 q^{92} - 2866 q^{94} + 1720 q^{97}+O(q^{100}) $ 8 * q + 4 * q^2 + 42 * q^4 + 40 * q^5 + 48 * q^8 + 20 * q^10 + 100 * q^11 + 102 * q^13 + 266 * q^16 - 56 * q^17 + 210 * q^20 - 70 * q^22 + 190 * q^23 + 200 * q^25 - 60 * q^26 + 296 * q^29 - 42 * q^31 + 718 * q^32 + 488 * q^34 + 314 * q^37 + 514 * q^38 + 240 * q^40 + 28 * q^41 + 714 * q^43 + 410 * q^44 - 650 * q^46 - 326 * q^47 + 100 * q^50 + 794 * q^52 + 1282 * q^53 + 500 * q^55 + 942 * q^58 + 924 * q^59 + 536 * q^61 - 50 * q^62 + 1902 * q^64 + 510 * q^65 + 2 * q^67 - 2690 * q^68 + 1516 * q^71 - 86 * q^73 + 4754 * q^74 - 30 * q^76 - 42 * q^79 + 1330 * q^80 + 3602 * q^82 - 572 * q^83 - 280 * q^85 + 1758 * q^86 - 4312 * q^88 - 940 * q^89 + 3844 * q^92 - 2866 * q^94 + 1720 * 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 $

$2$	3.02408	1.06917	0.534586	−	0.845114i	$-0.320467\pi$
$2$	3.02408	1.06917	0.534586	+	0.845114i	$0.320467\pi$
$3$	0	0
$3$	0	0
$4$	1.14504	0.143130
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−20.7299	−0.916142
$9$	0	0
$10$	15.1204	0.478149
$11$	−22.1595	−0.607395	−0.303698	−	0.952769i	$-0.598221\pi$
$11$	−22.1595	−0.607395	−0.303698	+	0.952769i	$0.598221\pi$
$12$	0	0
$13$	35.9983	0.768010	0.384005	−	0.923331i	$-0.374545\pi$
$13$	35.9983	0.768010	0.384005	+	0.923331i	$0.374545\pi$
$14$	0	0
$15$	0	0
$16$	−71.8492	−1.12264
$17$	85.8421	1.22469	0.612346	−	0.790590i	$-0.290226\pi$
$17$	85.8421	1.22469	0.612346	+	0.790590i	$0.290226\pi$
$18$	0	0
$19$	97.0029	1.17126	0.585631	−	0.810578i	$-0.300847\pi$
$19$	97.0029	1.17126	0.585631	+	0.810578i	$0.300847\pi$
$20$	5.72522	0.0640099
$21$	0	0
$22$	−67.0121	−0.649410
$23$	−98.2844	−0.891031	−0.445516	−	0.895274i	$-0.646980\pi$
$23$	−98.2844	−0.891031	−0.445516	+	0.895274i	$0.646980\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	108.862	0.821135
$27$	0	0
$28$	0	0
$29$	200.052	1.28099	0.640496	−	0.767962i	$-0.278729\pi$
$29$	200.052	1.28099	0.640496	+	0.767962i	$0.278729\pi$
$30$	0	0
$31$	−296.976	−1.72059	−0.860297	−	0.509793i	$-0.829722\pi$
$31$	−296.976	−1.72059	−0.860297	+	0.509793i	$0.829722\pi$
$32$	−51.4382	−0.284159
$33$	0	0
$34$	259.593	1.30941
$35$	0	0
$36$	0	0
$37$	−185.337	−0.823492	−0.411746	−	0.911299i	$-0.635081\pi$
$37$	−185.337	−0.823492	−0.411746	+	0.911299i	$0.635081\pi$
$38$	293.344	1.25228
$39$	0	0
$40$	−103.650	−0.409711
$41$	−291.601	−1.11074	−0.555371	−	0.831603i	$-0.687424\pi$
$41$	−291.601	−1.11074	−0.555371	+	0.831603i	$0.687424\pi$
$42$	0	0
$43$	173.597	0.615656	0.307828	−	0.951442i	$-0.400398\pi$
$43$	173.597	0.615656	0.307828	+	0.951442i	$0.400398\pi$
$44$	−25.3736	−0.0869367
$45$	0	0
$46$	−297.220	−0.952666
$47$	52.9175	0.164230	0.0821150	−	0.996623i	$-0.473833\pi$
$47$	52.9175	0.164230	0.0821150	+	0.996623i	$0.473833\pi$
$48$	0	0
$49$	0	0
$50$	75.6019	0.213835
$51$	0	0
$52$	41.2196	0.109926
$53$	294.704	0.763788	0.381894	−	0.924206i	$-0.375272\pi$
$53$	294.704	0.763788	0.381894	+	0.924206i	$0.375272\pi$
$54$	0	0
$55$	−110.798	−0.271635
$56$	0	0
$57$	0	0
$58$	604.973	1.36960
$59$	−197.462	−0.435719	−0.217859	−	0.975980i	$-0.569907\pi$
$59$	−197.462	−0.435719	−0.217859	+	0.975980i	$0.569907\pi$
$60$	0	0
$61$	430.880	0.904401	0.452201	−	0.891916i	$-0.350639\pi$
$61$	430.880	0.904401	0.452201	+	0.891916i	$0.350639\pi$
$62$	−898.078	−1.83961
$63$	0	0
$64$	419.241	0.818829
$65$	179.991	0.343464
$66$	0	0
$67$	−519.033	−0.946417	−0.473209	−	0.880950i	$-0.656904\pi$
$67$	−519.033	−0.946417	−0.473209	+	0.880950i	$0.656904\pi$
$68$	98.2929	0.175291
$69$	0	0
$70$	0	0
$71$	980.318	1.63862	0.819312	−	0.573347i	$-0.194355\pi$
$71$	980.318	1.63862	0.819312	+	0.573347i	$0.194355\pi$
$72$	0	0
$73$	−55.7062	−0.0893140	−0.0446570	−	0.999002i	$-0.514219\pi$
$73$	−55.7062	−0.0893140	−0.0446570	+	0.999002i	$0.514219\pi$
$74$	−560.473	−0.880455
$75$	0	0
$76$	111.072	0.167643
$77$	0	0
$78$	0	0
$79$	830.967	1.18343	0.591716	−	0.806147i	$-0.298451\pi$
$79$	830.967	1.18343	0.591716	+	0.806147i	$0.298451\pi$
$80$	−359.246	−0.502062
$81$	0	0
$82$	−881.824	−1.18757
$83$	1280.48	1.69338	0.846690	−	0.532087i	$-0.178592\pi$
$83$	1280.48	1.69338	0.846690	+	0.532087i	$0.178592\pi$
$84$	0	0
$85$	429.210	0.547699
$86$	524.969	0.658243
$87$	0	0
$88$	459.365	0.556460
$89$	−506.166	−0.602848	−0.301424	−	0.953490i	$-0.597462\pi$
$89$	−506.166	−0.602848	−0.301424	+	0.953490i	$0.597462\pi$
$90$	0	0
$91$	0	0
$92$	−112.540	−0.127534
$93$	0	0
$94$	160.026	0.175590
$95$	485.014	0.523804
$96$	0	0
$97$	1041.60	1.09030	0.545149	−	0.838339i	$-0.316473\pi$
$97$	1041.60	1.09030	0.545149	+	0.838339i	$0.316473\pi$
$98$	0	0
$99$	0	0
$100$	28.6261	0.0286261
$101$	249.341	0.245647	0.122824	−	0.992429i	$-0.460805\pi$
$101$	249.341	0.245647	0.122824	+	0.992429i	$0.460805\pi$
$102$	0	0
$103$	1788.97	1.71138	0.855692	−	0.517485i	$-0.173132\pi$
$103$	1788.97	1.71138	0.855692	+	0.517485i	$0.173132\pi$
$104$	−746.241	−0.703606
$105$	0	0
$106$	891.208	0.816621
$107$	−327.084	−0.295517	−0.147759	−	0.989023i	$-0.547206\pi$
$107$	−327.084	−0.295517	−0.147759	+	0.989023i	$0.547206\pi$
$108$	0	0
$109$	1148.95	1.00963	0.504816	−	0.863227i	$-0.331560\pi$
$109$	1148.95	1.00963	0.504816	+	0.863227i	$0.331560\pi$
$110$	−335.060	−0.290425
$111$	0	0
$112$	0	0
$113$	−1417.50	−1.18006	−0.590032	−	0.807380i	$-0.700885\pi$
$113$	−1417.50	−1.18006	−0.590032	+	0.807380i	$0.700885\pi$
$114$	0	0
$115$	−491.422	−0.398481
$116$	229.068	0.183349
$117$	0	0
$118$	−597.142	−0.465859
$119$	0	0
$120$	0	0
$121$	−839.956	−0.631071
$122$	1303.01	0.966961
$123$	0	0
$124$	−340.050	−0.246269
$125$	125.000	0.0894427
$126$	0	0
$127$	2413.19	1.68611	0.843055	−	0.537828i	$-0.180755\pi$
$127$	2413.19	1.68611	0.843055	+	0.537828i	$0.180755\pi$
$128$	1679.32	1.15963
$129$	0	0
$130$	544.308	0.367223
$131$	2248.54	1.49966	0.749832	−	0.661628i	$-0.230134\pi$
$131$	2248.54	1.49966	0.749832	+	0.661628i	$0.230134\pi$
$132$	0	0
$133$	0	0
$134$	−1569.60	−1.01188
$135$	0	0
$136$	−1779.50	−1.12199
$137$	1788.25	1.11519	0.557595	−	0.830113i	$-0.311724\pi$
$137$	1788.25	1.11519	0.557595	+	0.830113i	$0.311724\pi$
$138$	0	0
$139$	874.354	0.533538	0.266769	−	0.963761i	$-0.414044\pi$
$139$	874.354	0.533538	0.266769	+	0.963761i	$0.414044\pi$
$140$	0	0
$141$	0	0
$142$	2964.56	1.75197
$143$	−797.704	−0.466485
$144$	0	0
$145$	1000.26	0.572877
$146$	−168.460	−0.0954921
$147$	0	0
$148$	−212.219	−0.117867
$149$	2678.32	1.47260	0.736298	−	0.676657i	$-0.236572\pi$
$149$	2678.32	1.47260	0.736298	+	0.676657i	$0.236572\pi$
$150$	0	0
$151$	2304.52	1.24198	0.620990	−	0.783819i	$-0.286731\pi$
$151$	2304.52	1.24198	0.620990	+	0.783819i	$0.286731\pi$
$152$	−2010.86	−1.07304
$153$	0	0
$154$	0	0
$155$	−1484.88	−0.769473
$156$	0	0
$157$	−2417.93	−1.22912	−0.614561	−	0.788870i	$-0.710667\pi$
$157$	−2417.93	−1.22912	−0.614561	+	0.788870i	$0.710667\pi$
$158$	2512.91	1.26529
$159$	0	0
$160$	−257.191	−0.127080
$161$	0	0
$162$	0	0
$163$	1025.39	0.492727	0.246363	−	0.969178i	$-0.420764\pi$
$163$	1025.39	0.492727	0.246363	+	0.969178i	$0.420764\pi$
$164$	−333.896	−0.158981
$165$	0	0
$166$	3872.26	1.81051
$167$	−756.612	−0.350589	−0.175295	−	0.984516i	$-0.556088\pi$
$167$	−756.612	−0.350589	−0.175295	+	0.984516i	$0.556088\pi$
$168$	0	0
$169$	−901.124	−0.410161
$170$	1297.97	0.585585
$171$	0	0
$172$	198.776	0.0881191
$173$	−493.967	−0.217084	−0.108542	−	0.994092i	$-0.534618\pi$
$173$	−493.967	−0.217084	−0.108542	+	0.994092i	$0.534618\pi$
$174$	0	0
$175$	0	0
$176$	1592.14	0.681888
$177$	0	0
$178$	−1530.69	−0.644549
$179$	875.260	0.365475	0.182737	−	0.983162i	$-0.441504\pi$
$179$	875.260	0.365475	0.182737	+	0.983162i	$0.441504\pi$
$180$	0	0
$181$	−2385.85	−0.979771	−0.489886	−	0.871787i	$-0.662961\pi$
$181$	−2385.85	−0.979771	−0.489886	+	0.871787i	$0.662961\pi$
$182$	0	0
$183$	0	0
$184$	2037.43	0.816311
$185$	−926.685	−0.368277
$186$	0	0
$187$	−1902.22	−0.743872
$188$	60.5928	0.0235063
$189$	0	0
$190$	1466.72	0.560037
$191$	282.807	0.107137	0.0535685	−	0.998564i	$-0.482940\pi$
$191$	282.807	0.107137	0.0535685	+	0.998564i	$0.482940\pi$
$192$	0	0
$193$	−4348.60	−1.62186	−0.810930	−	0.585143i	$-0.801038\pi$
$193$	−4348.60	−1.62186	−0.810930	+	0.585143i	$0.801038\pi$
$194$	3149.89	1.16572
$195$	0	0
$196$	0	0
$197$	1145.52	0.414289	0.207145	−	0.978310i	$-0.433583\pi$
$197$	1145.52	0.414289	0.207145	+	0.978310i	$0.433583\pi$
$198$	0	0
$199$	−345.518	−0.123081	−0.0615405	−	0.998105i	$-0.519601\pi$
$199$	−345.518	−0.123081	−0.0615405	+	0.998105i	$0.519601\pi$
$200$	−518.248	−0.183228
$201$	0	0
$202$	754.027	0.262640
$203$	0	0
$204$	0	0
$205$	−1458.00	−0.496739
$206$	5409.99	1.82977
$207$	0	0
$208$	−2586.45	−0.862202
$209$	−2149.54	−0.711419
$210$	0	0
$211$	4921.63	1.60578	0.802888	−	0.596129i	$-0.203295\pi$
$211$	4921.63	1.60578	0.802888	+	0.596129i	$0.203295\pi$
$212$	337.449	0.109321
$213$	0	0
$214$	−989.126	−0.315959
$215$	867.983	0.275330
$216$	0	0
$217$	0	0
$218$	3474.53	1.07947
$219$	0	0
$220$	−126.868	−0.0388793
$221$	3090.17	0.940575
$222$	0	0
$223$	2413.56	0.724771	0.362386	−	0.932028i	$-0.381962\pi$
$223$	2413.56	0.724771	0.362386	+	0.932028i	$0.381962\pi$
$224$	0	0
$225$	0	0
$226$	−4286.63	−1.26169
$227$	−2918.57	−0.853358	−0.426679	−	0.904403i	$-0.640317\pi$
$227$	−2918.57	−0.853358	−0.426679	+	0.904403i	$0.640317\pi$
$228$	0	0
$229$	−2376.86	−0.685883	−0.342942	−	0.939357i	$-0.611423\pi$
$229$	−2376.86	−0.685883	−0.342942	+	0.939357i	$0.611423\pi$
$230$	−1486.10	−0.426045
$231$	0	0
$232$	−4147.07	−1.17357
$233$	−90.9088	−0.0255607	−0.0127803	−	0.999918i	$-0.504068\pi$
$233$	−90.9088	−0.0255607	−0.0127803	+	0.999918i	$0.504068\pi$
$234$	0	0
$235$	264.587	0.0734458
$236$	−226.103	−0.0623646
$237$	0	0
$238$	0	0
$239$	4650.97	1.25877	0.629385	−	0.777093i	$-0.283307\pi$
$239$	4650.97	1.25877	0.629385	+	0.777093i	$0.283307\pi$
$240$	0	0
$241$	−1764.47	−0.471615	−0.235808	−	0.971800i	$-0.575774\pi$
$241$	−1764.47	−0.471615	−0.235808	+	0.971800i	$0.575774\pi$
$242$	−2540.09	−0.674724
$243$	0	0
$244$	493.376	0.129447
$245$	0	0
$246$	0	0
$247$	3491.94	0.899541
$248$	6156.28	1.57631
$249$	0	0
$250$	378.010	0.0956297
$251$	−3136.94	−0.788853	−0.394427	−	0.918927i	$-0.629057\pi$
$251$	−3136.94	−0.788853	−0.394427	+	0.918927i	$0.629057\pi$
$252$	0	0
$253$	2177.93	0.541208
$254$	7297.67	1.80274
$255$	0	0
$256$	1724.47	0.421014
$257$	4213.67	1.02273	0.511364	−	0.859364i	$-0.329140\pi$
$257$	4213.67	1.02273	0.511364	+	0.859364i	$0.329140\pi$
$258$	0	0
$259$	0	0
$260$	206.098	0.0491602
$261$	0	0
$262$	6799.76	1.60340
$263$	−2704.90	−0.634188	−0.317094	−	0.948394i	$-0.602707\pi$
$263$	−2704.90	−0.634188	−0.317094	+	0.948394i	$0.602707\pi$
$264$	0	0
$265$	1473.52	0.341576
$266$	0	0
$267$	0	0
$268$	−594.315	−0.135461
$269$	−1717.93	−0.389384	−0.194692	−	0.980864i	$-0.562371\pi$
$269$	−1717.93	−0.389384	−0.194692	+	0.980864i	$0.562371\pi$
$270$	0	0
$271$	3950.61	0.885543	0.442772	−	0.896634i	$-0.353995\pi$
$271$	3950.61	0.885543	0.442772	+	0.896634i	$0.353995\pi$
$272$	−6167.69	−1.37489
$273$	0	0
$274$	5407.82	1.19233
$275$	−553.988	−0.121479
$276$	0	0
$277$	−488.734	−0.106011	−0.0530057	−	0.998594i	$-0.516880\pi$
$277$	−488.734	−0.106011	−0.0530057	+	0.998594i	$0.516880\pi$
$278$	2644.11	0.570444
$279$	0	0
$280$	0	0
$281$	−8404.58	−1.78425	−0.892127	−	0.451786i	$-0.850787\pi$
$281$	−8404.58	−1.78425	−0.892127	+	0.451786i	$0.850787\pi$
$282$	0	0
$283$	−2370.21	−0.497859	−0.248929	−	0.968522i	$-0.580079\pi$
$283$	−2370.21	−0.497859	−0.248929	+	0.968522i	$0.580079\pi$
$284$	1122.51	0.234537
$285$	0	0
$286$	−2412.32	−0.498753
$287$	0	0
$288$	0	0
$289$	2455.86	0.499870
$290$	3024.87	0.612504
$291$	0	0
$292$	−63.7861	−0.0127835
$293$	−9033.59	−1.80119	−0.900593	−	0.434663i	$-0.856868\pi$
$293$	−9033.59	−1.80119	−0.900593	+	0.434663i	$0.856868\pi$
$294$	0	0
$295$	−987.312	−0.194859
$296$	3842.02	0.754435
$297$	0	0
$298$	8099.46	1.57446
$299$	−3538.07	−0.684321
$300$	0	0
$301$	0	0
$302$	6969.04	1.32789
$303$	0	0
$304$	−6969.58	−1.31491
$305$	2154.40	0.404460
$306$	0	0
$307$	3800.19	0.706477	0.353239	−	0.935533i	$-0.385080\pi$
$307$	3800.19	0.706477	0.353239	+	0.935533i	$0.385080\pi$
$308$	0	0
$309$	0	0
$310$	−4490.39	−0.822700
$311$	2934.67	0.535080	0.267540	−	0.963547i	$-0.413789\pi$
$311$	2934.67	0.535080	0.267540	+	0.963547i	$0.413789\pi$
$312$	0	0
$313$	−237.718	−0.0429285	−0.0214643	−	0.999770i	$-0.506833\pi$
$313$	−237.718	−0.0429285	−0.0214643	+	0.999770i	$0.506833\pi$
$314$	−7312.02	−1.31414
$315$	0	0
$316$	951.493	0.169385
$317$	7181.26	1.27236	0.636182	−	0.771539i	$-0.280513\pi$
$317$	7181.26	1.27236	0.636182	+	0.771539i	$0.280513\pi$
$318$	0	0
$319$	−4433.06	−0.778068
$320$	2096.20	0.366192
$321$	0	0
$322$	0	0
$323$	8326.93	1.43444
$324$	0	0
$325$	899.957	0.153602
$326$	3100.85	0.526810
$327$	0	0
$328$	6044.86	1.01760
$329$	0	0
$330$	0	0
$331$	−807.118	−0.134028	−0.0670139	−	0.997752i	$-0.521347\pi$
$331$	−807.118	−0.134028	−0.0670139	+	0.997752i	$0.521347\pi$
$332$	1466.20	0.242374
$333$	0	0
$334$	−2288.05	−0.374840
$335$	−2595.16	−0.423251
$336$	0	0
$337$	−7760.21	−1.25438	−0.627189	−	0.778867i	$-0.715795\pi$
$337$	−7760.21	−1.25438	−0.627189	+	0.778867i	$0.715795\pi$
$338$	−2725.07	−0.438533
$339$	0	0
$340$	491.464	0.0783924
$341$	6580.84	1.04508
$342$	0	0
$343$	0	0
$344$	−3598.64	−0.564028
$345$	0	0
$346$	−1493.79	−0.232101
$347$	8225.96	1.27260	0.636301	−	0.771441i	$-0.280464\pi$
$347$	8225.96	1.27260	0.636301	+	0.771441i	$0.280464\pi$
$348$	0	0
$349$	−3056.70	−0.468829	−0.234415	−	0.972137i	$-0.575317\pi$
$349$	−3056.70	−0.468829	−0.234415	+	0.972137i	$0.575317\pi$
$350$	0	0
$351$	0	0
$352$	1139.85	0.172597
$353$	7141.73	1.07682	0.538408	−	0.842684i	$-0.319026\pi$
$353$	7141.73	1.07682	0.538408	+	0.842684i	$0.319026\pi$
$354$	0	0
$355$	4901.59	0.732815
$356$	−579.582	−0.0862859
$357$	0	0
$358$	2646.85	0.390756
$359$	991.981	0.145835	0.0729175	−	0.997338i	$-0.476769\pi$
$359$	991.981	0.145835	0.0729175	+	0.997338i	$0.476769\pi$
$360$	0	0
$361$	2550.55	0.371855
$362$	−7214.98	−1.04754
$363$	0	0
$364$	0	0
$365$	−278.531	−0.0399424
$366$	0	0
$367$	9246.88	1.31521	0.657606	−	0.753362i	$-0.271569\pi$
$367$	9246.88	1.31521	0.657606	+	0.753362i	$0.271569\pi$
$368$	7061.66	1.00031
$369$	0	0
$370$	−2802.37	−0.393752
$371$	0	0
$372$	0	0
$373$	−2729.78	−0.378935	−0.189468	−	0.981887i	$-0.560676\pi$
$373$	−2729.78	−0.378935	−0.189468	+	0.981887i	$0.560676\pi$
$374$	−5752.46	−0.795327
$375$	0	0
$376$	−1096.97	−0.150458
$377$	7201.53	0.983814
$378$	0	0
$379$	−12382.1	−1.67817	−0.839084	−	0.544001i	$-0.816909\pi$
$379$	−12382.1	−1.67817	−0.839084	+	0.544001i	$0.816909\pi$
$380$	555.362	0.0749723
$381$	0	0
$382$	855.230	0.114548
$383$	8448.60	1.12716	0.563581	−	0.826061i	$-0.309423\pi$
$383$	8448.60	1.12716	0.563581	+	0.826061i	$0.309423\pi$
$384$	0	0
$385$	0	0
$386$	−13150.5	−1.73405
$387$	0	0
$388$	1192.68	0.156055
$389$	−556.013	−0.0724704	−0.0362352	−	0.999343i	$-0.511537\pi$
$389$	−556.013	−0.0724704	−0.0362352	+	0.999343i	$0.511537\pi$
$390$	0	0
$391$	−8436.94	−1.09124
$392$	0	0
$393$	0	0
$394$	3464.14	0.442947
$395$	4154.84	0.529247
$396$	0	0
$397$	1490.84	0.188471	0.0942357	−	0.995550i	$-0.469959\pi$
$397$	1490.84	0.188471	0.0942357	+	0.995550i	$0.469959\pi$
$398$	−1044.87	−0.131595
$399$	0	0
$400$	−1796.23	−0.224529
$401$	13949.1	1.73711	0.868557	−	0.495589i	$-0.165048\pi$
$401$	13949.1	1.73711	0.868557	+	0.495589i	$0.165048\pi$
$402$	0	0
$403$	−10690.6	−1.32143
$404$	285.507	0.0351596
$405$	0	0
$406$	0	0
$407$	4106.98	0.500185
$408$	0	0
$409$	2437.58	0.294696	0.147348	−	0.989085i	$-0.452926\pi$
$409$	2437.58	0.294696	0.147348	+	0.989085i	$0.452926\pi$
$410$	−4409.12	−0.531100
$411$	0	0
$412$	2048.45	0.244951
$413$	0	0
$414$	0	0
$415$	6402.38	0.757302
$416$	−1851.69	−0.218237
$417$	0	0
$418$	−6500.36	−0.760630
$419$	409.228	0.0477139	0.0238569	−	0.999715i	$-0.492405\pi$
$419$	409.228	0.0477139	0.0238569	+	0.999715i	$0.492405\pi$
$420$	0	0
$421$	−6585.66	−0.762388	−0.381194	−	0.924495i	$-0.624487\pi$
$421$	−6585.66	−0.762388	−0.381194	+	0.924495i	$0.624487\pi$
$422$	14883.4	1.71685
$423$	0	0
$424$	−6109.20	−0.699738
$425$	2146.05	0.244938
$426$	0	0
$427$	0	0
$428$	−374.525	−0.0422975
$429$	0	0
$430$	2624.85	0.294375
$431$	13490.9	1.50773	0.753866	−	0.657028i	$-0.228187\pi$
$431$	13490.9	1.50773	0.753866	+	0.657028i	$0.228187\pi$
$432$	0	0
$433$	−12067.5	−1.33933	−0.669663	−	0.742666i	$-0.733561\pi$
$433$	−12067.5	−1.33933	−0.669663	+	0.742666i	$0.733561\pi$
$434$	0	0
$435$	0	0
$436$	1315.60	0.144509
$437$	−9533.87	−1.04363
$438$	0	0
$439$	3668.25	0.398807	0.199403	−	0.979918i	$-0.436100\pi$
$439$	3668.25	0.398807	0.199403	+	0.979918i	$0.436100\pi$
$440$	2296.82	0.248856
$441$	0	0
$442$	9344.90	1.00564
$443$	12294.1	1.31853	0.659264	−	0.751911i	$-0.270868\pi$
$443$	12294.1	1.31853	0.659264	+	0.751911i	$0.270868\pi$
$444$	0	0
$445$	−2530.83	−0.269602
$446$	7298.80	0.774906
$447$	0	0
$448$	0	0
$449$	7971.90	0.837901	0.418950	−	0.908009i	$-0.362398\pi$
$449$	7971.90	0.837901	0.418950	+	0.908009i	$0.362398\pi$
$450$	0	0
$451$	6461.73	0.674659
$452$	−1623.10	−0.168903
$453$	0	0
$454$	−8825.98	−0.912387
$455$	0	0
$456$	0	0
$457$	−13781.4	−1.41065	−0.705324	−	0.708885i	$-0.749198\pi$
$457$	−13781.4	−1.41065	−0.705324	+	0.708885i	$0.749198\pi$
$458$	−7187.80	−0.733327
$459$	0	0
$460$	−562.699	−0.0570348
$461$	10974.6	1.10876	0.554380	−	0.832263i	$-0.312955\pi$
$461$	10974.6	1.10876	0.554380	+	0.832263i	$0.312955\pi$
$462$	0	0
$463$	−8803.53	−0.883661	−0.441830	−	0.897099i	$-0.645671\pi$
$463$	−8803.53	−0.883661	−0.441830	+	0.897099i	$0.645671\pi$
$464$	−14373.6	−1.43810
$465$	0	0
$466$	−274.915	−0.0273288
$467$	−7378.30	−0.731107	−0.365554	−	0.930790i	$-0.619120\pi$
$467$	−7378.30	−0.731107	−0.365554	+	0.930790i	$0.619120\pi$
$468$	0	0
$469$	0	0
$470$	800.132	0.0785263
$471$	0	0
$472$	4093.38	0.399180
$473$	−3846.81	−0.373947
$474$	0	0
$475$	2425.07	0.234252
$476$	0	0
$477$	0	0
$478$	14064.9	1.34584
$479$	12014.4	1.14603	0.573017	−	0.819543i	$-0.305773\pi$
$479$	12014.4	1.14603	0.573017	+	0.819543i	$0.305773\pi$
$480$	0	0
$481$	−6671.81	−0.632450
$482$	−5335.88	−0.504238
$483$	0	0
$484$	−961.786	−0.0903255
$485$	5208.02	0.487596
$486$	0	0
$487$	2736.76	0.254650	0.127325	−	0.991861i	$-0.459361\pi$
$487$	2736.76	0.254650	0.127325	+	0.991861i	$0.459361\pi$
$488$	−8932.10	−0.828560
$489$	0	0
$490$	0	0
$491$	2154.33	0.198012	0.0990058	−	0.995087i	$-0.468434\pi$
$491$	2154.33	0.198012	0.0990058	+	0.995087i	$0.468434\pi$
$492$	0	0
$493$	17172.9	1.56882
$494$	10559.9	0.961764
$495$	0	0
$496$	21337.5	1.93162
$497$	0	0
$498$	0	0
$499$	−14437.6	−1.29522	−0.647609	−	0.761973i	$-0.724231\pi$
$499$	−14437.6	−1.29522	−0.647609	+	0.761973i	$0.724231\pi$
$500$	143.130	0.0128020
$501$	0	0
$502$	−9486.36	−0.843420
$503$	−10779.5	−0.955532	−0.477766	−	0.878487i	$-0.658553\pi$
$503$	−10779.5	−0.955532	−0.477766	+	0.878487i	$0.658553\pi$
$504$	0	0
$505$	1246.71	0.109857
$506$	6586.24	0.578645
$507$	0	0
$508$	2763.21	0.241334
$509$	−20715.5	−1.80393	−0.901964	−	0.431811i	$-0.857875\pi$
$509$	−20715.5	−1.80393	−0.901964	+	0.431811i	$0.857875\pi$
$510$	0	0
$511$	0	0
$512$	−8219.63	−0.709492
$513$	0	0
$514$	12742.5	1.09347
$515$	8944.86	0.765354
$516$	0	0
$517$	−1172.63	−0.0997524
$518$	0	0
$519$	0	0
$520$	−3731.21	−0.314662
$521$	16612.0	1.39690	0.698448	−	0.715661i	$-0.253874\pi$
$521$	16612.0	1.39690	0.698448	+	0.715661i	$0.253874\pi$
$522$	0	0
$523$	10772.5	0.900667	0.450333	−	0.892860i	$-0.351305\pi$
$523$	10772.5	0.900667	0.450333	+	0.892860i	$0.351305\pi$
$524$	2574.68	0.214648
$525$	0	0
$526$	−8179.83	−0.678056
$527$	−25493.0	−2.10720
$528$	0	0
$529$	−2507.18	−0.206064
$530$	4456.04	0.365204
$531$	0	0
$532$	0	0
$533$	−10497.1	−0.853060
$534$	0	0
$535$	−1635.42	−0.132159
$536$	10759.5	0.867052
$537$	0	0
$538$	−5195.16	−0.416318
$539$	0	0
$540$	0	0
$541$	−12431.9	−0.987969	−0.493984	−	0.869471i	$-0.664460\pi$
$541$	−12431.9	−0.987969	−0.493984	+	0.869471i	$0.664460\pi$
$542$	11946.9	0.946799
$543$	0	0
$544$	−4415.57	−0.348007
$545$	5744.77	0.451521
$546$	0	0
$547$	10016.4	0.782947	0.391473	−	0.920189i	$-0.371966\pi$
$547$	10016.4	0.782947	0.391473	+	0.920189i	$0.371966\pi$
$548$	2047.63	0.159617
$549$	0	0
$550$	−1675.30	−0.129882
$551$	19405.6	1.50038
$552$	0	0
$553$	0	0
$554$	−1477.97	−0.113345
$555$	0	0
$556$	1001.17	0.0763655
$557$	−23857.0	−1.81481	−0.907407	−	0.420252i	$-0.861942\pi$
$557$	−23857.0	−1.81481	−0.907407	+	0.420252i	$0.861942\pi$
$558$	0	0
$559$	6249.18	0.472830
$560$	0	0
$561$	0	0
$562$	−25416.1	−1.90767
$563$	−15827.6	−1.18482	−0.592411	−	0.805636i	$-0.701824\pi$
$563$	−15827.6	−1.18482	−0.592411	+	0.805636i	$0.701824\pi$
$564$	0	0
$565$	−7087.51	−0.527741
$566$	−7167.68	−0.532297
$567$	0	0
$568$	−20321.9	−1.50121
$569$	15189.3	1.11910	0.559552	−	0.828795i	$-0.310973\pi$
$569$	15189.3	1.11910	0.559552	+	0.828795i	$0.310973\pi$
$570$	0	0
$571$	−15482.1	−1.13469	−0.567345	−	0.823481i	$-0.692029\pi$
$571$	−15482.1	−1.13469	−0.567345	+	0.823481i	$0.692029\pi$
$572$	−913.406	−0.0667682
$573$	0	0
$574$	0	0
$575$	−2457.11	−0.178206
$576$	0	0
$577$	−10789.8	−0.778481	−0.389241	−	0.921136i	$-0.627263\pi$
$577$	−10789.8	−0.778481	−0.389241	+	0.921136i	$0.627263\pi$
$578$	7426.72	0.534447
$579$	0	0
$580$	1145.34	0.0819961
$581$	0	0
$582$	0	0
$583$	−6530.50	−0.463921
$584$	1154.79	0.0818243
$585$	0	0
$586$	−27318.3	−1.92578
$587$	−8755.42	−0.615630	−0.307815	−	0.951446i	$-0.599598\pi$
$587$	−8755.42	−0.615630	−0.307815	+	0.951446i	$0.599598\pi$
$588$	0	0
$589$	−28807.5	−2.01527
$590$	−2985.71	−0.208338
$591$	0	0
$592$	13316.3	0.924488
$593$	−11281.5	−0.781244	−0.390622	−	0.920551i	$-0.627740\pi$
$593$	−11281.5	−0.781244	−0.390622	+	0.920551i	$0.627740\pi$
$594$	0	0
$595$	0	0
$596$	3066.80	0.210773
$597$	0	0
$598$	−10699.4	−0.731657
$599$	1099.29	0.0749845	0.0374922	−	0.999297i	$-0.488063\pi$
$599$	1099.29	0.0749845	0.0374922	+	0.999297i	$0.488063\pi$
$600$	0	0
$601$	−19709.3	−1.33770	−0.668852	−	0.743396i	$-0.733214\pi$
$601$	−19709.3	−1.33770	−0.668852	+	0.743396i	$0.733214\pi$
$602$	0	0
$603$	0	0
$604$	2638.77	0.177765
$605$	−4199.78	−0.282224
$606$	0	0
$607$	13968.7	0.934057	0.467028	−	0.884242i	$-0.345325\pi$
$607$	13968.7	0.934057	0.467028	+	0.884242i	$0.345325\pi$
$608$	−4989.66	−0.332825
$609$	0	0
$610$	6515.06	0.432438
$611$	1904.94	0.126130
$612$	0	0
$613$	5751.23	0.378940	0.189470	−	0.981887i	$-0.439323\pi$
$613$	5751.23	0.378940	0.189470	+	0.981887i	$0.439323\pi$
$614$	11492.1	0.755346
$615$	0	0
$616$	0	0
$617$	29541.3	1.92753	0.963767	−	0.266745i	$-0.0859482\pi$
$617$	29541.3	1.92753	0.963767	+	0.266745i	$0.0859482\pi$
$618$	0	0
$619$	−25408.0	−1.64981	−0.824907	−	0.565269i	$-0.808772\pi$
$619$	−25408.0	−1.64981	−0.824907	+	0.565269i	$0.808772\pi$
$620$	−1700.25	−0.110135
$621$	0	0
$622$	8874.66	0.572093
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−718.879	−0.0458980
$627$	0	0
$628$	−2768.64	−0.175925
$629$	−15909.7	−1.00852
$630$	0	0
$631$	−23162.9	−1.46133	−0.730667	−	0.682734i	$-0.760791\pi$
$631$	−23162.9	−1.46133	−0.730667	+	0.682734i	$0.760791\pi$
$632$	−17225.9	−1.08419
$633$	0	0
$634$	21716.7	1.36038
$635$	12065.9	0.754051
$636$	0	0
$637$	0	0
$638$	−13405.9	−0.831889
$639$	0	0
$640$	8396.61	0.518602
$641$	−15953.6	−0.983042	−0.491521	−	0.870866i	$-0.663559\pi$
$641$	−15953.6	−0.983042	−0.491521	+	0.870866i	$0.663559\pi$
$642$	0	0
$643$	1142.04	0.0700428	0.0350214	−	0.999387i	$-0.488850\pi$
$643$	1142.04	0.0700428	0.0350214	+	0.999387i	$0.488850\pi$
$644$	0	0
$645$	0	0
$646$	25181.3	1.53366
$647$	−19269.2	−1.17086	−0.585432	−	0.810721i	$-0.699075\pi$
$647$	−19269.2	−1.17086	−0.585432	+	0.810721i	$0.699075\pi$
$648$	0	0
$649$	4375.67	0.264653
$650$	2721.54	0.164227
$651$	0	0
$652$	1174.11	0.0705242
$653$	21703.4	1.30064	0.650320	−	0.759660i	$-0.274635\pi$
$653$	21703.4	1.30064	0.650320	+	0.759660i	$0.274635\pi$
$654$	0	0
$655$	11242.7	0.670670
$656$	20951.3	1.24697
$657$	0	0
$658$	0	0
$659$	25654.1	1.51645	0.758225	−	0.651993i	$-0.226067\pi$
$659$	25654.1	1.51645	0.758225	+	0.651993i	$0.226067\pi$
$660$	0	0
$661$	2403.21	0.141413	0.0707065	−	0.997497i	$-0.477475\pi$
$661$	2403.21	0.141413	0.0707065	+	0.997497i	$0.477475\pi$
$662$	−2440.79	−0.143299
$663$	0	0
$664$	−26544.2	−1.55138
$665$	0	0
$666$	0	0
$667$	−19662.0	−1.14140
$668$	−866.353	−0.0501800
$669$	0	0
$670$	−7847.98	−0.452528
$671$	−9548.08	−0.549329
$672$	0	0
$673$	20175.3	1.15557	0.577787	−	0.816187i	$-0.303916\pi$
$673$	20175.3	1.15557	0.577787	+	0.816187i	$0.303916\pi$
$674$	−23467.5	−1.34115
$675$	0	0
$676$	−1031.83	−0.0587065
$677$	−27147.8	−1.54117	−0.770587	−	0.637335i	$-0.780037\pi$
$677$	−27147.8	−1.54117	−0.770587	+	0.637335i	$0.780037\pi$
$678$	0	0
$679$	0	0
$680$	−8897.50	−0.501770
$681$	0	0
$682$	19901.0	1.11737
$683$	−16658.9	−0.933287	−0.466643	−	0.884446i	$-0.654537\pi$
$683$	−16658.9	−0.933287	−0.466643	+	0.884446i	$0.654537\pi$
$684$	0	0
$685$	8941.27	0.498728
$686$	0	0
$687$	0	0
$688$	−12472.8	−0.691163
$689$	10608.8	0.586596
$690$	0	0
$691$	−1496.62	−0.0823938	−0.0411969	−	0.999151i	$-0.513117\pi$
$691$	−1496.62	−0.0823938	−0.0411969	+	0.999151i	$0.513117\pi$
$692$	−565.613	−0.0310714
$693$	0	0
$694$	24875.9	1.36063
$695$	4371.77	0.238605
$696$	0	0
$697$	−25031.6	−1.36032
$698$	−9243.70	−0.501260
$699$	0	0
$700$	0	0
$701$	−1953.58	−0.105258	−0.0526290	−	0.998614i	$-0.516760\pi$
$701$	−1953.58	−0.105258	−0.0526290	+	0.998614i	$0.516760\pi$
$702$	0	0
$703$	−17978.2	−0.964525
$704$	−9290.17	−0.497353
$705$	0	0
$706$	21597.1	1.15130
$707$	0	0
$708$	0	0
$709$	23486.6	1.24409	0.622043	−	0.782983i	$-0.286303\pi$
$709$	23486.6	1.24409	0.622043	+	0.782983i	$0.286303\pi$
$710$	14822.8	0.783506
$711$	0	0
$712$	10492.8	0.552294
$713$	29188.1	1.53310
$714$	0	0
$715$	−3988.52	−0.208619
$716$	1002.21	0.0523106
$717$	0	0
$718$	2999.83	0.155923
$719$	20053.2	1.04014	0.520069	−	0.854124i	$-0.325906\pi$
$719$	20053.2	1.04014	0.520069	+	0.854124i	$0.325906\pi$
$720$	0	0
$721$	0	0
$722$	7713.07	0.397577
$723$	0	0
$724$	−2731.90	−0.140235
$725$	5001.30	0.256198
$726$	0	0
$727$	−28882.5	−1.47344	−0.736721	−	0.676197i	$-0.763627\pi$
$727$	−28882.5	−1.47344	−0.736721	+	0.676197i	$0.763627\pi$
$728$	0	0
$729$	0	0
$730$	−842.300	−0.0427054
$731$	14901.9	0.753989
$732$	0	0
$733$	1365.77	0.0688210	0.0344105	−	0.999408i	$-0.489045\pi$
$733$	1365.77	0.0688210	0.0344105	+	0.999408i	$0.489045\pi$
$734$	27963.3	1.40619
$735$	0	0
$736$	5055.58	0.253194
$737$	11501.5	0.574849
$738$	0	0
$739$	−8684.84	−0.432310	−0.216155	−	0.976359i	$-0.569352\pi$
$739$	−8684.84	−0.432310	−0.216155	+	0.976359i	$0.569352\pi$
$740$	−1061.09	−0.0527116
$741$	0	0
$742$	0	0
$743$	13165.8	0.650078	0.325039	−	0.945701i	$-0.394623\pi$
$743$	13165.8	0.650078	0.325039	+	0.945701i	$0.394623\pi$
$744$	0	0
$745$	13391.6	0.658565
$746$	−8255.08	−0.405147
$747$	0	0
$748$	−2178.12	−0.106471
$749$	0	0
$750$	0	0
$751$	26495.8	1.28741	0.643705	−	0.765274i	$-0.277396\pi$
$751$	26495.8	1.28741	0.643705	+	0.765274i	$0.277396\pi$
$752$	−3802.08	−0.184372
$753$	0	0
$754$	21778.0	1.05187
$755$	11522.6	0.555430
$756$	0	0
$757$	4927.95	0.236604	0.118302	−	0.992978i	$-0.462255\pi$
$757$	4927.95	0.236604	0.118302	+	0.992978i	$0.462255\pi$
$758$	−37444.4	−1.79425
$759$	0	0
$760$	−10054.3	−0.479879
$761$	31910.6	1.52005	0.760025	−	0.649894i	$-0.225187\pi$
$761$	31910.6	1.52005	0.760025	+	0.649894i	$0.225187\pi$
$762$	0	0
$763$	0	0
$764$	323.826	0.0153346
$765$	0	0
$766$	25549.2	1.20513
$767$	−7108.31	−0.334636
$768$	0	0
$769$	8265.63	0.387602	0.193801	−	0.981041i	$-0.437918\pi$
$769$	8265.63	0.387602	0.193801	+	0.981041i	$0.437918\pi$
$770$	0	0
$771$	0	0
$772$	−4979.33	−0.232137
$773$	−27988.1	−1.30228	−0.651141	−	0.758957i	$-0.725709\pi$
$773$	−27988.1	−1.30228	−0.651141	+	0.758957i	$0.725709\pi$
$774$	0	0
$775$	−7424.40	−0.344119
$776$	−21592.4	−0.998868
$777$	0	0
$778$	−1681.43	−0.0774834
$779$	−28286.1	−1.30097
$780$	0	0
$781$	−21723.4	−0.995293
$782$	−25514.0	−1.16672
$783$	0	0
$784$	0	0
$785$	−12089.7	−0.549680
$786$	0	0
$787$	20804.8	0.942328	0.471164	−	0.882046i	$-0.343834\pi$
$787$	20804.8	0.942328	0.471164	+	0.882046i	$0.343834\pi$
$788$	1311.67	0.0592974
$789$	0	0
$790$	12564.5	0.565856
$791$	0	0
$792$	0	0
$793$	15510.9	0.694589
$794$	4508.42	0.201508
$795$	0	0
$796$	−395.633	−0.0176166
$797$	−18327.5	−0.814545	−0.407273	−	0.913307i	$-0.633520\pi$
$797$	−18327.5	−0.814545	−0.407273	+	0.913307i	$0.633520\pi$
$798$	0	0
$799$	4542.54	0.201131
$800$	−1285.96	−0.0568318
$801$	0	0
$802$	42183.0	1.85728
$803$	1234.42	0.0542489
$804$	0	0
$805$	0	0
$806$	−32329.3	−1.41284
$807$	0	0
$808$	−5168.83	−0.225048
$809$	−20786.5	−0.903357	−0.451679	−	0.892181i	$-0.649175\pi$
$809$	−20786.5	−0.903357	−0.451679	+	0.892181i	$0.649175\pi$
$810$	0	0
$811$	−2566.26	−0.111114	−0.0555571	−	0.998456i	$-0.517693\pi$
$811$	−2566.26	−0.111114	−0.0555571	+	0.998456i	$0.517693\pi$
$812$	0	0
$813$	0	0
$814$	12419.8	0.534784
$815$	5126.93	0.220354
$816$	0	0
$817$	16839.4	0.721095
$818$	7371.44	0.315081
$819$	0	0
$820$	−1669.48	−0.0710984
$821$	−5471.54	−0.232592	−0.116296	−	0.993215i	$-0.537102\pi$
$821$	−5471.54	−0.232592	−0.116296	+	0.993215i	$0.537102\pi$
$822$	0	0
$823$	−36025.6	−1.52585	−0.762924	−	0.646489i	$-0.776237\pi$
$823$	−36025.6	−1.52585	−0.762924	+	0.646489i	$0.776237\pi$
$824$	−37085.2	−1.56787
$825$	0	0
$826$	0	0
$827$	17877.2	0.751696	0.375848	−	0.926681i	$-0.377351\pi$
$827$	17877.2	0.751696	0.375848	+	0.926681i	$0.377351\pi$
$828$	0	0
$829$	−40520.0	−1.69761	−0.848804	−	0.528707i	$-0.822677\pi$
$829$	−40520.0	−1.69761	−0.848804	+	0.528707i	$0.822677\pi$
$830$	19361.3	0.809687
$831$	0	0
$832$	15091.9	0.628869
$833$	0	0
$834$	0	0
$835$	−3783.06	−0.156788
$836$	−2461.31	−0.101826
$837$	0	0
$838$	1237.54	0.0510144
$839$	16699.4	0.687162	0.343581	−	0.939123i	$-0.388360\pi$
$839$	16699.4	0.687162	0.343581	+	0.939123i	$0.388360\pi$
$840$	0	0
$841$	15631.9	0.640939
$842$	−19915.5	−0.815124
$843$	0	0
$844$	5635.48	0.229835
$845$	−4505.62	−0.183430
$846$	0	0
$847$	0	0
$848$	−21174.3	−0.857462
$849$	0	0
$850$	6489.83	0.261881
$851$	18215.7	0.733757
$852$	0	0
$853$	−9970.55	−0.400217	−0.200109	−	0.979774i	$-0.564130\pi$
$853$	−9970.55	−0.400217	−0.200109	+	0.979774i	$0.564130\pi$
$854$	0	0
$855$	0	0
$856$	6780.42	0.270736
$857$	−44760.9	−1.78414	−0.892068	−	0.451901i	$-0.850746\pi$
$857$	−44760.9	−1.78414	−0.892068	+	0.451901i	$0.850746\pi$
$858$	0	0
$859$	−33091.0	−1.31438	−0.657189	−	0.753726i	$-0.728255\pi$
$859$	−33091.0	−1.31438	−0.657189	+	0.753726i	$0.728255\pi$
$860$	993.878	0.0394081
$861$	0	0
$862$	40797.5	1.61203
$863$	−12939.6	−0.510391	−0.255196	−	0.966889i	$-0.582140\pi$
$863$	−12939.6	−0.510391	−0.255196	+	0.966889i	$0.582140\pi$
$864$	0	0
$865$	−2469.83	−0.0970831
$866$	−36493.1	−1.43197
$867$	0	0
$868$	0	0
$869$	−18413.8	−0.718811
$870$	0	0
$871$	−18684.3	−0.726858
$872$	−23817.7	−0.924966
$873$	0	0
$874$	−28831.2	−1.11582
$875$	0	0
$876$	0	0
$877$	−40107.6	−1.54429	−0.772143	−	0.635449i	$-0.780815\pi$
$877$	−40107.6	−1.54429	−0.772143	+	0.635449i	$0.780815\pi$
$878$	11093.1	0.426393
$879$	0	0
$880$	7960.72	0.304950
$881$	52262.8	1.99861	0.999307	−	0.0372129i	$-0.0118480\pi$
$881$	52262.8	1.99861	0.999307	+	0.0372129i	$0.0118480\pi$
$882$	0	0
$883$	−10356.1	−0.394688	−0.197344	−	0.980334i	$-0.563232\pi$
$883$	−10356.1	−0.394688	−0.197344	+	0.980334i	$0.563232\pi$
$884$	3538.38	0.134625
$885$	0	0
$886$	37178.2	1.40973
$887$	−26596.6	−1.00679	−0.503397	−	0.864056i	$-0.667916\pi$
$887$	−26596.6	−1.00679	−0.503397	+	0.864056i	$0.667916\pi$
$888$	0	0
$889$	0	0
$890$	−7653.43	−0.288251
$891$	0	0
$892$	2763.63	0.103737
$893$	5133.14	0.192356
$894$	0	0
$895$	4376.30	0.163445
$896$	0	0
$897$	0	0
$898$	24107.7	0.895861
$899$	−59410.7	−2.20407
$900$	0	0
$901$	25298.0	0.935404
$902$	19540.8	0.721327
$903$	0	0
$904$	29384.7	1.08111
$905$	−11929.2	−0.438167
$906$	0	0
$907$	30836.3	1.12889	0.564445	−	0.825471i	$-0.309090\pi$
$907$	30836.3	1.12889	0.564445	+	0.825471i	$0.309090\pi$
$908$	−3341.89	−0.122141
$909$	0	0
$910$	0	0
$911$	−50063.0	−1.82070	−0.910352	−	0.413835i	$-0.864189\pi$
$911$	−50063.0	−1.82070	−0.910352	+	0.413835i	$0.864189\pi$
$912$	0	0
$913$	−28374.7	−1.02855
$914$	−41676.0	−1.50823
$915$	0	0
$916$	−2721.61	−0.0981707
$917$	0	0
$918$	0	0
$919$	41903.1	1.50409	0.752043	−	0.659115i	$-0.229069\pi$
$919$	41903.1	1.50409	0.752043	+	0.659115i	$0.229069\pi$
$920$	10187.1	0.365065
$921$	0	0
$922$	33188.1	1.18546
$923$	35289.8	1.25848
$924$	0	0
$925$	−4633.42	−0.164698
$926$	−26622.6	−0.944786
$927$	0	0
$928$	−10290.3	−0.364005
$929$	−26257.8	−0.927332	−0.463666	−	0.886010i	$-0.653466\pi$
$929$	−26257.8	−0.927332	−0.463666	+	0.886010i	$0.653466\pi$
$930$	0	0
$931$	0	0
$932$	−104.095	−0.00365851
$933$	0	0
$934$	−22312.6	−0.781680
$935$	−9511.09	−0.332670
$936$	0	0
$937$	42191.5	1.47101	0.735504	−	0.677520i	$-0.236945\pi$
$937$	42191.5	1.47101	0.735504	+	0.677520i	$0.236945\pi$
$938$	0	0
$939$	0	0
$940$	302.964	0.0105123
$941$	28028.2	0.970981	0.485491	−	0.874242i	$-0.338641\pi$
$941$	28028.2	0.970981	0.485491	+	0.874242i	$0.338641\pi$
$942$	0	0
$943$	28659.8	0.989705
$944$	14187.5	0.489157
$945$	0	0
$946$	−11633.1	−0.399813
$947$	−27776.0	−0.953115	−0.476558	−	0.879143i	$-0.658116\pi$
$947$	−27776.0	−0.953115	−0.476558	+	0.879143i	$0.658116\pi$
$948$	0	0
$949$	−2005.33	−0.0685940
$950$	7333.60	0.250456
$951$	0	0
$952$	0	0
$953$	26183.7	0.890004	0.445002	−	0.895530i	$-0.353203\pi$
$953$	26183.7	0.890004	0.445002	+	0.895530i	$0.353203\pi$
$954$	0	0
$955$	1414.03	0.0479132
$956$	5325.56	0.180168
$957$	0	0
$958$	36332.4	1.22531
$959$	0	0
$960$	0	0
$961$	58403.6	1.96045
$962$	−20176.1	−0.676198
$963$	0	0
$964$	−2020.39	−0.0675025
$965$	−21743.0	−0.725318
$966$	0	0
$967$	12347.8	0.410630	0.205315	−	0.978696i	$-0.434178\pi$
$967$	12347.8	0.410630	0.205315	+	0.978696i	$0.434178\pi$
$968$	17412.2	0.578151
$969$	0	0
$970$	15749.5	0.521325
$971$	59130.7	1.95427	0.977135	−	0.212620i	$-0.0681997\pi$
$971$	59130.7	1.95427	0.977135	+	0.212620i	$0.0681997\pi$
$972$	0	0
$973$	0	0
$974$	8276.18	0.272265
$975$	0	0
$976$	−30958.4	−1.01532
$977$	39916.1	1.30709	0.653546	−	0.756887i	$-0.273281\pi$
$977$	39916.1	1.30709	0.653546	+	0.756887i	$0.273281\pi$
$978$	0	0
$979$	11216.4	0.366167
$980$	0	0
$981$	0	0
$982$	6514.87	0.211709
$983$	−13689.2	−0.444167	−0.222084	−	0.975028i	$-0.571286\pi$
$983$	−13689.2	−0.444167	−0.222084	+	0.975028i	$0.571286\pi$
$984$	0	0
$985$	5727.60	0.185276
$986$	51932.2	1.67734
$987$	0	0
$988$	3998.42	0.128752
$989$	−17061.8	−0.548569
$990$	0	0
$991$	9050.75	0.290117	0.145059	−	0.989423i	$-0.453663\pi$
$991$	9050.75	0.290117	0.145059	+	0.989423i	$0.453663\pi$
$992$	15275.9	0.488922
$993$	0	0
$994$	0	0
$995$	−1727.59	−0.0550435
$996$	0	0
$997$	12171.9	0.386648	0.193324	−	0.981135i	$-0.438073\pi$
$997$	12171.9	0.386648	0.193324	+	0.981135i	$0.438073\pi$
$998$	−43660.3	−1.38481
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.cg.1.6		8
3.2	odd	2		2205.4.a.cb.1.3		8
7.3	odd	6		315.4.j.i.226.3	yes	16
7.5	odd	6		315.4.j.i.46.3	✓	16
7.6	odd	2		2205.4.a.cf.1.6		8
21.5	even	6		315.4.j.j.46.6	yes	16
21.17	even	6		315.4.j.j.226.6	yes	16
21.20	even	2		2205.4.a.cc.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.j.i.46.3	✓	16	7.5	odd	6
315.4.j.i.226.3	yes	16	7.3	odd	6
315.4.j.j.46.6	yes	16	21.5	even	6
315.4.j.j.226.6	yes	16	21.17	even	6
2205.4.a.cb.1.3		8	3.2	odd	2
2205.4.a.cc.1.3		8	21.20	even	2
2205.4.a.cf.1.6		8	7.6	odd	2
2205.4.a.cg.1.6		8	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)

\(f(q)\)	\(=\)	\(q+3.02408 q^{2} +1.14504 q^{4} +5.00000 q^{5} -20.7299 q^{8} +O(q^{10})\)	\(q+3.02408 q^{2} +1.14504 q^{4} +5.00000 q^{5} -20.7299 q^{8} +15.1204 q^{10} -22.1595 q^{11} +35.9983 q^{13} -71.8492 q^{16} +85.8421 q^{17} +97.0029 q^{19} +5.72522 q^{20} -67.0121 q^{22} -98.2844 q^{23} +25.0000 q^{25} +108.862 q^{26} +200.052 q^{29} -296.976 q^{31} -51.4382 q^{32} +259.593 q^{34} -185.337 q^{37} +293.344 q^{38} -103.650 q^{40} -291.601 q^{41} +173.597 q^{43} -25.3736 q^{44} -297.220 q^{46} +52.9175 q^{47} +75.6019 q^{50} +41.2196 q^{52} +294.704 q^{53} -110.798 q^{55} +604.973 q^{58} -197.462 q^{59} +430.880 q^{61} -898.078 q^{62} +419.241 q^{64} +179.991 q^{65} -519.033 q^{67} +98.2929 q^{68} +980.318 q^{71} -55.7062 q^{73} -560.473 q^{74} +111.072 q^{76} +830.967 q^{79} -359.246 q^{80} -881.824 q^{82} +1280.48 q^{83} +429.210 q^{85} +524.969 q^{86} +459.365 q^{88} -506.166 q^{89} -112.540 q^{92} +160.026 q^{94} +485.014 q^{95} +1041.60 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 42 q^{4} + 40 q^{5} + 48 q^{8}+O(q^{10}) \) 8 * q + 4 * q^2 + 42 * q^4 + 40 * q^5 + 48 * q^8	\( 8 q + 4 q^{2} + 42 q^{4} + 40 q^{5} + 48 q^{8} + 20 q^{10} + 100 q^{11} + 102 q^{13} + 266 q^{16} - 56 q^{17} + 210 q^{20} - 70 q^{22} + 190 q^{23} + 200 q^{25} - 60 q^{26} + 296 q^{29} - 42 q^{31} + 718 q^{32} + 488 q^{34} + 314 q^{37} + 514 q^{38} + 240 q^{40} + 28 q^{41} + 714 q^{43} + 410 q^{44} - 650 q^{46} - 326 q^{47} + 100 q^{50} + 794 q^{52} + 1282 q^{53} + 500 q^{55} + 942 q^{58} + 924 q^{59} + 536 q^{61} - 50 q^{62} + 1902 q^{64} + 510 q^{65} + 2 q^{67} - 2690 q^{68} + 1516 q^{71} - 86 q^{73} + 4754 q^{74} - 30 q^{76} - 42 q^{79} + 1330 q^{80} + 3602 q^{82} - 572 q^{83} - 280 q^{85} + 1758 q^{86} - 4312 q^{88} - 940 q^{89} + 3844 q^{92} - 2866 q^{94} + 1720 q^{97}+O(q^{100}) \) 8 * q + 4 * q^2 + 42 * q^4 + 40 * q^5 + 48 * q^8 + 20 * q^10 + 100 * q^11 + 102 * q^13 + 266 * q^16 - 56 * q^17 + 210 * q^20 - 70 * q^22 + 190 * q^23 + 200 * q^25 - 60 * q^26 + 296 * q^29 - 42 * q^31 + 718 * q^32 + 488 * q^34 + 314 * q^37 + 514 * q^38 + 240 * q^40 + 28 * q^41 + 714 * q^43 + 410 * q^44 - 650 * q^46 - 326 * q^47 + 100 * q^50 + 794 * q^52 + 1282 * q^53 + 500 * q^55 + 942 * q^58 + 924 * q^59 + 536 * q^61 - 50 * q^62 + 1902 * q^64 + 510 * q^65 + 2 * q^67 - 2690 * q^68 + 1516 * q^71 - 86 * q^73 + 4754 * q^74 - 30 * q^76 - 42 * q^79 + 1330 * q^80 + 3602 * q^82 - 572 * q^83 - 280 * q^85 + 1758 * q^86 - 4312 * q^88 - 940 * q^89 + 3844 * q^92 - 2866 * q^94 + 1720 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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