LMFDB - Embedded newform 2205.4.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$130.099211563$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ 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)

$f(q)$	$=$	$q-4.31662 q^{2} +10.6332 q^{4} +5.00000 q^{5} -11.3668 q^{8} +O(q^{10})$	$q-4.31662 q^{2} +10.6332 q^{4} +5.00000 q^{5} -11.3668 q^{8} -21.5831 q^{10} -19.7335 q^{11} -71.3325 q^{13} -36.0000 q^{16} -31.3325 q^{17} -136.332 q^{19} +53.1662 q^{20} +85.1821 q^{22} +100.865 q^{23} +25.0000 q^{25} +307.916 q^{26} +288.198 q^{29} +208.997 q^{31} +246.332 q^{32} +135.251 q^{34} +309.931 q^{37} +588.496 q^{38} -56.8338 q^{40} -181.662 q^{41} -18.2005 q^{43} -209.831 q^{44} -435.398 q^{46} -147.665 q^{47} -107.916 q^{50} -758.496 q^{52} +127.995 q^{53} -98.6675 q^{55} -1244.04 q^{58} +322.665 q^{59} -341.003 q^{61} -902.164 q^{62} -775.325 q^{64} -356.662 q^{65} -84.3960 q^{67} -333.166 q^{68} +315.736 q^{71} +1093.32 q^{73} -1337.86 q^{74} -1449.66 q^{76} +1233.19 q^{79} -180.000 q^{80} +784.169 q^{82} -643.325 q^{83} -156.662 q^{85} +78.5647 q^{86} +224.306 q^{88} +1140.32 q^{89} +1072.53 q^{92} +637.414 q^{94} -681.662 q^{95} -1411.99 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8	$ 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8} - 10 q^{10} - 66 q^{11} - 10 q^{13} - 72 q^{16} + 70 q^{17} - 140 q^{19} + 40 q^{20} - 22 q^{22} + 16 q^{23} + 50 q^{25} + 450 q^{26} + 258 q^{29} + 20 q^{31} + 360 q^{32} + 370 q^{34} + 328 q^{37} + 580 q^{38} - 180 q^{40} + 300 q^{41} - 116 q^{43} - 88 q^{44} - 632 q^{46} - 30 q^{47} - 50 q^{50} - 920 q^{52} - 540 q^{53} - 330 q^{55} - 1314 q^{58} + 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} - 50 q^{65} + 468 q^{67} - 600 q^{68} + 1056 q^{71} + 860 q^{73} - 1296 q^{74} - 1440 q^{76} + 158 q^{79} - 360 q^{80} + 1900 q^{82} + 40 q^{83} + 350 q^{85} - 148 q^{86} + 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 910 q^{94} - 700 q^{95} - 1630 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8 - 10 * q^10 - 66 * q^11 - 10 * q^13 - 72 * q^16 + 70 * q^17 - 140 * q^19 + 40 * q^20 - 22 * q^22 + 16 * q^23 + 50 * q^25 + 450 * q^26 + 258 * q^29 + 20 * q^31 + 360 * q^32 + 370 * q^34 + 328 * q^37 + 580 * q^38 - 180 * q^40 + 300 * q^41 - 116 * q^43 - 88 * q^44 - 632 * q^46 - 30 * q^47 - 50 * q^50 - 920 * q^52 - 540 * q^53 - 330 * q^55 - 1314 * q^58 + 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 - 50 * q^65 + 468 * q^67 - 600 * q^68 + 1056 * q^71 + 860 * q^73 - 1296 * q^74 - 1440 * q^76 + 158 * q^79 - 360 * q^80 + 1900 * q^82 + 40 * q^83 + 350 * q^85 - 148 * q^86 + 1364 * q^88 - 240 * q^89 + 1296 * q^92 + 910 * q^94 - 700 * q^95 - 1630 * 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 $

$2$	−4.31662	−1.52616	−0.763079	−	0.646306i	$-0.776313\pi$
$2$	−4.31662	−1.52616	−0.763079	+	0.646306i	$0.776313\pi$
$3$	0	0
$3$	0	0
$4$	10.6332	1.32916
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−11.3668	−0.502344
$9$	0	0
$10$	−21.5831	−0.682518
$11$	−19.7335	−0.540898	−0.270449	−	0.962734i	$-0.587172\pi$
$11$	−19.7335	−0.540898	−0.270449	+	0.962734i	$0.587172\pi$
$12$	0	0
$13$	−71.3325	−1.52185	−0.760926	−	0.648839i	$-0.775255\pi$
$13$	−71.3325	−1.52185	−0.760926	+	0.648839i	$0.775255\pi$
$14$	0	0
$15$	0	0
$16$	−36.0000	−0.562500
$17$	−31.3325	−0.447014	−0.223507	−	0.974702i	$-0.571751\pi$
$17$	−31.3325	−0.447014	−0.223507	+	0.974702i	$0.571751\pi$
$18$	0	0
$19$	−136.332	−1.64615	−0.823074	−	0.567934i	$-0.807743\pi$
$19$	−136.332	−1.64615	−0.823074	+	0.567934i	$0.807743\pi$
$20$	53.1662	0.594417
$21$	0	0
$22$	85.1821	0.825495
$23$	100.865	0.914431	0.457215	−	0.889356i	$-0.348847\pi$
$23$	100.865	0.914431	0.457215	+	0.889356i	$0.348847\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	307.916	2.32259
$27$	0	0
$28$	0	0
$29$	288.198	1.84541	0.922707	−	0.385501i	$-0.125971\pi$
$29$	288.198	1.84541	0.922707	+	0.385501i	$0.125971\pi$
$30$	0	0
$31$	208.997	1.21087	0.605436	−	0.795894i	$-0.292999\pi$
$31$	208.997	1.21087	0.605436	+	0.795894i	$0.292999\pi$
$32$	246.332	1.36081
$33$	0	0
$34$	135.251	0.682214
$35$	0	0
$36$	0	0
$37$	309.931	1.37709	0.688546	−	0.725192i	$-0.258249\pi$
$37$	309.931	1.37709	0.688546	+	0.725192i	$0.258249\pi$
$38$	588.496	2.51228
$39$	0	0
$40$	−56.8338	−0.224655
$41$	−181.662	−0.691973	−0.345987	−	0.938239i	$-0.612456\pi$
$41$	−181.662	−0.691973	−0.345987	+	0.938239i	$0.612456\pi$
$42$	0	0
$43$	−18.2005	−0.0645477	−0.0322738	−	0.999479i	$-0.510275\pi$
$43$	−18.2005	−0.0645477	−0.0322738	+	0.999479i	$0.510275\pi$
$44$	−209.831	−0.718937
$45$	0	0
$46$	−435.398	−1.39557
$47$	−147.665	−0.458280	−0.229140	−	0.973393i	$-0.573591\pi$
$47$	−147.665	−0.458280	−0.229140	+	0.973393i	$0.573591\pi$
$48$	0	0
$49$	0	0
$50$	−107.916	−0.305231
$51$	0	0
$52$	−758.496	−2.02278
$53$	127.995	0.331726	0.165863	−	0.986149i	$-0.446959\pi$
$53$	127.995	0.331726	0.165863	+	0.986149i	$0.446959\pi$
$54$	0	0
$55$	−98.6675	−0.241897
$56$	0	0
$57$	0	0
$58$	−1244.04	−2.81639
$59$	322.665	0.711990	0.355995	−	0.934488i	$-0.384142\pi$
$59$	322.665	0.711990	0.355995	+	0.934488i	$0.384142\pi$
$60$	0	0
$61$	−341.003	−0.715752	−0.357876	−	0.933769i	$-0.616499\pi$
$61$	−341.003	−0.715752	−0.357876	+	0.933769i	$0.616499\pi$
$62$	−902.164	−1.84798
$63$	0	0
$64$	−775.325	−1.51431
$65$	−356.662	−0.680593
$66$	0	0
$67$	−84.3960	−0.153890	−0.0769449	−	0.997035i	$-0.524517\pi$
$67$	−84.3960	−0.153890	−0.0769449	+	0.997035i	$0.524517\pi$
$68$	−333.166	−0.594152
$69$	0	0
$70$	0	0
$71$	315.736	0.527760	0.263880	−	0.964555i	$-0.414998\pi$
$71$	315.736	0.527760	0.263880	+	0.964555i	$0.414998\pi$
$72$	0	0
$73$	1093.32	1.75293	0.876466	−	0.481464i	$-0.159895\pi$
$73$	1093.32	1.75293	0.876466	+	0.481464i	$0.159895\pi$
$74$	−1337.86	−2.10166
$75$	0	0
$76$	−1449.66	−2.18799
$77$	0	0
$78$	0	0
$79$	1233.19	1.75626	0.878128	−	0.478426i	$-0.158793\pi$
$79$	1233.19	1.75626	0.878128	+	0.478426i	$0.158793\pi$
$80$	−180.000	−0.251558
$81$	0	0
$82$	784.169	1.05606
$83$	−643.325	−0.850772	−0.425386	−	0.905012i	$-0.639862\pi$
$83$	−643.325	−0.850772	−0.425386	+	0.905012i	$0.639862\pi$
$84$	0	0
$85$	−156.662	−0.199911
$86$	78.5647	0.0985099
$87$	0	0
$88$	224.306	0.271717
$89$	1140.32	1.35813	0.679064	−	0.734079i	$-0.262386\pi$
$89$	1140.32	1.35813	0.679064	+	0.734079i	$0.262386\pi$
$90$	0	0
$91$	0	0
$92$	1072.53	1.21542
$93$	0	0
$94$	637.414	0.699407
$95$	−681.662	−0.736180
$96$	0	0
$97$	−1411.99	−1.47800	−0.739001	−	0.673705i	$-0.764702\pi$
$97$	−1411.99	−1.47800	−0.739001	+	0.673705i	$0.764702\pi$
$98$	0	0
$99$	0	0
$100$	265.831	0.265831
$101$	−545.673	−0.537589	−0.268794	−	0.963198i	$-0.586625\pi$
$101$	−545.673	−0.537589	−0.268794	+	0.963198i	$0.586625\pi$
$102$	0	0
$103$	−780.990	−0.747119	−0.373559	−	0.927606i	$-0.621863\pi$
$103$	−780.990	−0.747119	−0.373559	+	0.927606i	$0.621863\pi$
$104$	810.819	0.764493
$105$	0	0
$106$	−552.506	−0.506266
$107$	620.660	0.560761	0.280381	−	0.959889i	$-0.409539\pi$
$107$	620.660	0.560761	0.280381	+	0.959889i	$0.409539\pi$
$108$	0	0
$109$	4.18794	0.00368011	0.00184005	−	0.999998i	$-0.499414\pi$
$109$	4.18794	0.00368011	0.00184005	+	0.999998i	$0.499414\pi$
$110$	425.911	0.369173
$111$	0	0
$112$	0	0
$113$	−1413.53	−1.17676	−0.588379	−	0.808585i	$-0.700234\pi$
$113$	−1413.53	−1.17676	−0.588379	+	0.808585i	$0.700234\pi$
$114$	0	0
$115$	504.327	0.408946
$116$	3064.48	2.45284
$117$	0	0
$118$	−1392.82	−1.08661
$119$	0	0
$120$	0	0
$121$	−941.589	−0.707430
$122$	1471.98	1.09235
$123$	0	0
$124$	2222.32	1.60944
$125$	125.000	0.0894427
$126$	0	0
$127$	−2046.26	−1.42974	−0.714868	−	0.699259i	$-0.753513\pi$
$127$	−2046.26	−1.42974	−0.714868	+	0.699259i	$0.753513\pi$
$128$	1376.13	0.950262
$129$	0	0
$130$	1539.58	1.03869
$131$	1140.32	0.760534	0.380267	−	0.924877i	$-0.375832\pi$
$131$	1140.32	0.760534	0.380267	+	0.924877i	$0.375832\pi$
$132$	0	0
$133$	0	0
$134$	364.306	0.234860
$135$	0	0
$136$	356.149	0.224555
$137$	490.343	0.305787	0.152893	−	0.988243i	$-0.451141\pi$
$137$	490.343	0.305787	0.152893	+	0.988243i	$0.451141\pi$
$138$	0	0
$139$	−2800.00	−1.70858	−0.854291	−	0.519795i	$-0.826008\pi$
$139$	−2800.00	−1.70858	−0.854291	+	0.519795i	$0.826008\pi$
$140$	0	0
$141$	0	0
$142$	−1362.91	−0.805445
$143$	1407.64	0.823166
$144$	0	0
$145$	1440.99	0.825294
$146$	−4719.47	−2.67525
$147$	0	0
$148$	3295.58	1.83037
$149$	−1166.12	−0.641154	−0.320577	−	0.947222i	$-0.603877\pi$
$149$	−1166.12	−0.641154	−0.320577	+	0.947222i	$0.603877\pi$
$150$	0	0
$151$	959.581	0.517150	0.258575	−	0.965991i	$-0.416747\pi$
$151$	959.581	0.517150	0.258575	+	0.965991i	$0.416747\pi$
$152$	1549.66	0.826933
$153$	0	0
$154$	0	0
$155$	1044.99	0.541519
$156$	0	0
$157$	1020.66	0.518838	0.259419	−	0.965765i	$-0.416469\pi$
$157$	1020.66	0.518838	0.259419	+	0.965765i	$0.416469\pi$
$158$	−5323.20	−2.68032
$159$	0	0
$160$	1231.66	0.608572
$161$	0	0
$162$	0	0
$163$	−1566.02	−0.752514	−0.376257	−	0.926515i	$-0.622789\pi$
$163$	−1566.02	−0.752514	−0.376257	+	0.926515i	$0.622789\pi$
$164$	−1931.66	−0.919741
$165$	0	0
$166$	2776.99	1.29841
$167$	−1130.30	−0.523746	−0.261873	−	0.965102i	$-0.584340\pi$
$167$	−1130.30	−0.523746	−0.261873	+	0.965102i	$0.584340\pi$
$168$	0	0
$169$	2891.32	1.31603
$170$	676.253	0.305096
$171$	0	0
$172$	−193.530	−0.0857940
$173$	2543.34	1.11772	0.558862	−	0.829260i	$-0.311238\pi$
$173$	2543.34	1.11772	0.558862	+	0.829260i	$0.311238\pi$
$174$	0	0
$175$	0	0
$176$	710.406	0.304255
$177$	0	0
$178$	−4922.32	−2.07272
$179$	1210.65	0.505521	0.252760	−	0.967529i	$-0.418662\pi$
$179$	1210.65	0.505521	0.252760	+	0.967529i	$0.418662\pi$
$180$	0	0
$181$	3031.32	1.24484	0.622421	−	0.782683i	$-0.286149\pi$
$181$	3031.32	1.24484	0.622421	+	0.782683i	$0.286149\pi$
$182$	0	0
$183$	0	0
$184$	−1146.51	−0.459359
$185$	1549.66	0.615854
$186$	0	0
$187$	618.300	0.241789
$188$	−1570.16	−0.609125
$189$	0	0
$190$	2942.48	1.12353
$191$	2168.64	0.821555	0.410778	−	0.911736i	$-0.365257\pi$
$191$	2168.64	0.821555	0.410778	+	0.911736i	$0.365257\pi$
$192$	0	0
$193$	1490.48	0.555892	0.277946	−	0.960597i	$-0.410346\pi$
$193$	1490.48	0.555892	0.277946	+	0.960597i	$0.410346\pi$
$194$	6095.04	2.25566
$195$	0	0
$196$	0	0
$197$	−3380.57	−1.22262	−0.611308	−	0.791393i	$-0.709356\pi$
$197$	−3380.57	−1.22262	−0.611308	+	0.791393i	$0.709356\pi$
$198$	0	0
$199$	−4595.33	−1.63696	−0.818478	−	0.574538i	$-0.805182\pi$
$199$	−4595.33	−1.63696	−0.818478	+	0.574538i	$0.805182\pi$
$200$	−284.169	−0.100469
$201$	0	0
$202$	2355.46	0.820445
$203$	0	0
$204$	0	0
$205$	−908.312	−0.309460
$206$	3371.24	1.14022
$207$	0	0
$208$	2567.97	0.856042
$209$	2690.32	0.890398
$210$	0	0
$211$	−4988.66	−1.62765	−0.813824	−	0.581112i	$-0.802618\pi$
$211$	−4988.66	−1.62765	−0.813824	+	0.581112i	$0.802618\pi$
$212$	1361.00	0.440915
$213$	0	0
$214$	−2679.16	−0.855810
$215$	−91.0025	−0.0288666
$216$	0	0
$217$	0	0
$218$	−18.0778	−0.00561643
$219$	0	0
$220$	−1049.16	−0.321519
$221$	2235.03	0.680290
$222$	0	0
$223$	−3792.97	−1.13900	−0.569498	−	0.821993i	$-0.692862\pi$
$223$	−3792.97	−1.13900	−0.569498	+	0.821993i	$0.692862\pi$
$224$	0	0
$225$	0	0
$226$	6101.68	1.79592
$227$	3910.96	1.14352	0.571762	−	0.820420i	$-0.306260\pi$
$227$	3910.96	1.14352	0.571762	+	0.820420i	$0.306260\pi$
$228$	0	0
$229$	−354.327	−0.102247	−0.0511236	−	0.998692i	$-0.516280\pi$
$229$	−354.327	−0.102247	−0.0511236	+	0.998692i	$0.516280\pi$
$230$	−2176.99	−0.624116
$231$	0	0
$232$	−3275.87	−0.927033
$233$	−6492.48	−1.82548	−0.912739	−	0.408543i	$-0.866037\pi$
$233$	−6492.48	−1.82548	−0.912739	+	0.408543i	$0.866037\pi$
$234$	0	0
$235$	−738.325	−0.204949
$236$	3430.98	0.946346
$237$	0	0
$238$	0	0
$239$	342.688	0.0927474	0.0463737	−	0.998924i	$-0.485234\pi$
$239$	342.688	0.0927474	0.0463737	+	0.998924i	$0.485234\pi$
$240$	0	0
$241$	−2313.67	−0.618408	−0.309204	−	0.950996i	$-0.600063\pi$
$241$	−2313.67	−0.618408	−0.309204	+	0.950996i	$0.600063\pi$
$242$	4064.49	1.07965
$243$	0	0
$244$	−3625.96	−0.951347
$245$	0	0
$246$	0	0
$247$	9724.94	2.50519
$248$	−2375.62	−0.608275
$249$	0	0
$250$	−539.578	−0.136504
$251$	3989.29	1.00319	0.501597	−	0.865101i	$-0.332746\pi$
$251$	3989.29	1.00319	0.501597	+	0.865101i	$0.332746\pi$
$252$	0	0
$253$	−1990.43	−0.494614
$254$	8832.95	2.18200
$255$	0	0
$256$	262.376	0.0640566
$257$	2291.32	0.556142	0.278071	−	0.960560i	$-0.410305\pi$
$257$	2291.32	0.556142	0.278071	+	0.960560i	$0.410305\pi$
$258$	0	0
$259$	0	0
$260$	−3792.48	−0.904614
$261$	0	0
$262$	−4922.32	−1.16070
$263$	−6360.47	−1.49127	−0.745634	−	0.666356i	$-0.767853\pi$
$263$	−6360.47	−1.49127	−0.745634	+	0.666356i	$0.767853\pi$
$264$	0	0
$265$	639.975	0.148352
$266$	0	0
$267$	0	0
$268$	−897.404	−0.204543
$269$	−991.345	−0.224697	−0.112348	−	0.993669i	$-0.535837\pi$
$269$	−991.345	−0.224697	−0.112348	+	0.993669i	$0.535837\pi$
$270$	0	0
$271$	−730.977	−0.163851	−0.0819257	−	0.996638i	$-0.526107\pi$
$271$	−730.977	−0.163851	−0.0819257	+	0.996638i	$0.526107\pi$
$272$	1127.97	0.251446
$273$	0	0
$274$	−2116.62	−0.466679
$275$	−493.338	−0.108180
$276$	0	0
$277$	−3538.63	−0.767566	−0.383783	−	0.923423i	$-0.625379\pi$
$277$	−3538.63	−0.767566	−0.383783	+	0.923423i	$0.625379\pi$
$278$	12086.5	2.60756
$279$	0	0
$280$	0	0
$281$	4663.20	0.989975	0.494988	−	0.868900i	$-0.335173\pi$
$281$	4663.20	0.989975	0.494988	+	0.868900i	$0.335173\pi$
$282$	0	0
$283$	2104.95	0.442142	0.221071	−	0.975258i	$-0.429045\pi$
$283$	2104.95	0.442142	0.221071	+	0.975258i	$0.429045\pi$
$284$	3357.30	0.701476
$285$	0	0
$286$	−6076.25	−1.25628
$287$	0	0
$288$	0	0
$289$	−3931.27	−0.800178
$290$	−6220.21	−1.25953
$291$	0	0
$292$	11625.6	2.32992
$293$	−6594.66	−1.31489	−0.657447	−	0.753501i	$-0.728364\pi$
$293$	−6594.66	−1.31489	−0.657447	+	0.753501i	$0.728364\pi$
$294$	0	0
$295$	1613.32	0.318412
$296$	−3522.91	−0.691774
$297$	0	0
$298$	5033.69	0.978503
$299$	−7194.99	−1.39163
$300$	0	0
$301$	0	0
$302$	−4142.15	−0.789252
$303$	0	0
$304$	4907.97	0.925958
$305$	−1705.01	−0.320094
$306$	0	0
$307$	2672.97	0.496920	0.248460	−	0.968642i	$-0.420076\pi$
$307$	2672.97	0.496920	0.248460	+	0.968642i	$0.420076\pi$
$308$	0	0
$309$	0	0
$310$	−4510.82	−0.826443
$311$	−855.698	−0.156020	−0.0780099	−	0.996953i	$-0.524857\pi$
$311$	−855.698	−0.156020	−0.0780099	+	0.996953i	$0.524857\pi$
$312$	0	0
$313$	3349.99	0.604960	0.302480	−	0.953156i	$-0.402185\pi$
$313$	3349.99	0.604960	0.302480	+	0.953156i	$0.402185\pi$
$314$	−4405.81	−0.791828
$315$	0	0
$316$	13112.8	2.33434
$317$	−7633.05	−1.35241	−0.676206	−	0.736712i	$-0.736377\pi$
$317$	−7633.05	−1.35241	−0.676206	+	0.736712i	$0.736377\pi$
$318$	0	0
$319$	−5687.16	−0.998180
$320$	−3876.62	−0.677218
$321$	0	0
$322$	0	0
$323$	4271.64	0.735852
$324$	0	0
$325$	−1783.31	−0.304370
$326$	6759.90	1.14845
$327$	0	0
$328$	2064.91	0.347609
$329$	0	0
$330$	0	0
$331$	−3321.70	−0.551593	−0.275796	−	0.961216i	$-0.588942\pi$
$331$	−3321.70	−0.551593	−0.275796	+	0.961216i	$0.588942\pi$
$332$	−6840.63	−1.13081
$333$	0	0
$334$	4879.10	0.799319
$335$	−421.980	−0.0688216
$336$	0	0
$337$	−2233.98	−0.361107	−0.180553	−	0.983565i	$-0.557789\pi$
$337$	−2233.98	−0.361107	−0.180553	+	0.983565i	$0.557789\pi$
$338$	−12480.8	−2.00847
$339$	0	0
$340$	−1665.83	−0.265713
$341$	−4124.25	−0.654958
$342$	0	0
$343$	0	0
$344$	206.881	0.0324252
$345$	0	0
$346$	−10978.6	−1.70582
$347$	−2528.61	−0.391190	−0.195595	−	0.980685i	$-0.562664\pi$
$347$	−2528.61	−0.391190	−0.195595	+	0.980685i	$0.562664\pi$
$348$	0	0
$349$	−1291.00	−0.198011	−0.0990054	−	0.995087i	$-0.531566\pi$
$349$	−1291.00	−0.198011	−0.0990054	+	0.995087i	$0.531566\pi$
$350$	0	0
$351$	0	0
$352$	−4861.00	−0.736058
$353$	−7768.64	−1.17134	−0.585670	−	0.810550i	$-0.699169\pi$
$353$	−7768.64	−1.17134	−0.585670	+	0.810550i	$0.699169\pi$
$354$	0	0
$355$	1578.68	0.236022
$356$	12125.3	1.80516
$357$	0	0
$358$	−5225.92	−0.771504
$359$	2284.14	0.335800	0.167900	−	0.985804i	$-0.446301\pi$
$359$	2284.14	0.335800	0.167900	+	0.985804i	$0.446301\pi$
$360$	0	0
$361$	11727.5	1.70980
$362$	−13085.1	−1.89982
$363$	0	0
$364$	0	0
$365$	5466.62	0.783935
$366$	0	0
$367$	−10707.0	−1.52289	−0.761446	−	0.648229i	$-0.775510\pi$
$367$	−10707.0	−1.52289	−0.761446	+	0.648229i	$0.775510\pi$
$368$	−3631.16	−0.514367
$369$	0	0
$370$	−6689.29	−0.939891
$371$	0	0
$372$	0	0
$373$	−830.429	−0.115276	−0.0576381	−	0.998338i	$-0.518357\pi$
$373$	−830.429	−0.115276	−0.0576381	+	0.998338i	$0.518357\pi$
$374$	−2668.97	−0.369008
$375$	0	0
$376$	1678.47	0.230214
$377$	−20557.9	−2.80845
$378$	0	0
$379$	5253.17	0.711972	0.355986	−	0.934491i	$-0.384145\pi$
$379$	5253.17	0.711972	0.355986	+	0.934491i	$0.384145\pi$
$380$	−7248.29	−0.978498
$381$	0	0
$382$	−9361.19	−1.25382
$383$	−11243.9	−1.50010	−0.750048	−	0.661383i	$-0.769970\pi$
$383$	−11243.9	−1.50010	−0.750048	+	0.661383i	$0.769970\pi$
$384$	0	0
$385$	0	0
$386$	−6433.84	−0.848378
$387$	0	0
$388$	−15014.1	−1.96449
$389$	8506.85	1.10878	0.554389	−	0.832258i	$-0.312952\pi$
$389$	8506.85	1.10878	0.554389	+	0.832258i	$0.312952\pi$
$390$	0	0
$391$	−3160.37	−0.408764
$392$	0	0
$393$	0	0
$394$	14592.6	1.86590
$395$	6165.93	0.785421
$396$	0	0
$397$	−3123.24	−0.394838	−0.197419	−	0.980319i	$-0.563256\pi$
$397$	−3123.24	−0.394838	−0.197419	+	0.980319i	$0.563256\pi$
$398$	19836.3	2.49825
$399$	0	0
$400$	−900.000	−0.112500
$401$	11255.3	1.40166	0.700828	−	0.713330i	$-0.252814\pi$
$401$	11255.3	1.40166	0.700828	+	0.713330i	$0.252814\pi$
$402$	0	0
$403$	−14908.3	−1.84277
$404$	−5802.27	−0.714539
$405$	0	0
$406$	0	0
$407$	−6116.03	−0.744866
$408$	0	0
$409$	7919.92	0.957494	0.478747	−	0.877953i	$-0.341091\pi$
$409$	7919.92	0.957494	0.478747	+	0.877953i	$0.341091\pi$
$410$	3920.84	0.472285
$411$	0	0
$412$	−8304.46	−0.993037
$413$	0	0
$414$	0	0
$415$	−3216.62	−0.380477
$416$	−17571.5	−2.07095
$417$	0	0
$418$	−11613.1	−1.35889
$419$	−5257.28	−0.612972	−0.306486	−	0.951875i	$-0.599153\pi$
$419$	−5257.28	−0.612972	−0.306486	+	0.951875i	$0.599153\pi$
$420$	0	0
$421$	1457.36	0.168711	0.0843556	−	0.996436i	$-0.473117\pi$
$421$	1457.36	0.168711	0.0843556	+	0.996436i	$0.473117\pi$
$422$	21534.2	2.48405
$423$	0	0
$424$	−1454.89	−0.166640
$425$	−783.312	−0.0894029
$426$	0	0
$427$	0	0
$428$	6599.63	0.745339
$429$	0	0
$430$	392.824	0.0440550
$431$	15291.2	1.70893	0.854467	−	0.519506i	$-0.173884\pi$
$431$	15291.2	1.70893	0.854467	+	0.519506i	$0.173884\pi$
$432$	0	0
$433$	−187.260	−0.0207832	−0.0103916	−	0.999946i	$-0.503308\pi$
$433$	−187.260	−0.0207832	−0.0103916	+	0.999946i	$0.503308\pi$
$434$	0	0
$435$	0	0
$436$	44.5314	0.00489144
$437$	−13751.2	−1.50529
$438$	0	0
$439$	−3587.92	−0.390073	−0.195037	−	0.980796i	$-0.562483\pi$
$439$	−3587.92	−0.390073	−0.195037	+	0.980796i	$0.562483\pi$
$440$	1121.53	0.121515
$441$	0	0
$442$	−9647.76	−1.03823
$443$	−4915.65	−0.527200	−0.263600	−	0.964632i	$-0.584910\pi$
$443$	−4915.65	−0.527200	−0.263600	+	0.964632i	$0.584910\pi$
$444$	0	0
$445$	5701.59	0.607373
$446$	16372.8	1.73829
$447$	0	0
$448$	0	0
$449$	−7091.12	−0.745324	−0.372662	−	0.927967i	$-0.621555\pi$
$449$	−7091.12	−0.745324	−0.372662	+	0.927967i	$0.621555\pi$
$450$	0	0
$451$	3584.84	0.374287
$452$	−15030.4	−1.56410
$453$	0	0
$454$	−16882.2	−1.74520
$455$	0	0
$456$	0	0
$457$	5051.81	0.517098	0.258549	−	0.965998i	$-0.416756\pi$
$457$	5051.81	0.517098	0.258549	+	0.965998i	$0.416756\pi$
$458$	1529.50	0.156045
$459$	0	0
$460$	5362.64	0.543553
$461$	−16681.3	−1.68531	−0.842653	−	0.538456i	$-0.819008\pi$
$461$	−16681.3	−1.68531	−0.842653	+	0.538456i	$0.819008\pi$
$462$	0	0
$463$	15569.6	1.56280	0.781402	−	0.624027i	$-0.214505\pi$
$463$	15569.6	1.56280	0.781402	+	0.624027i	$0.214505\pi$
$464$	−10375.1	−1.03805
$465$	0	0
$466$	28025.6	2.78597
$467$	3328.35	0.329802	0.164901	−	0.986310i	$-0.447269\pi$
$467$	3328.35	0.329802	0.164901	+	0.986310i	$0.447269\pi$
$468$	0	0
$469$	0	0
$470$	3187.07	0.312784
$471$	0	0
$472$	−3667.65	−0.357664
$473$	359.160	0.0349137
$474$	0	0
$475$	−3408.31	−0.329230
$476$	0	0
$477$	0	0
$478$	−1479.25	−0.141547
$479$	−14607.5	−1.39339	−0.696695	−	0.717367i	$-0.745347\pi$
$479$	−14607.5	−1.39339	−0.696695	+	0.717367i	$0.745347\pi$
$480$	0	0
$481$	−22108.2	−2.09573
$482$	9987.23	0.943789
$483$	0	0
$484$	−10012.2	−0.940285
$485$	−7059.96	−0.660982
$486$	0	0
$487$	−1879.49	−0.174882	−0.0874412	−	0.996170i	$-0.527869\pi$
$487$	−1879.49	−0.174882	−0.0874412	+	0.996170i	$0.527869\pi$
$488$	3876.09	0.359554
$489$	0	0
$490$	0	0
$491$	−3221.13	−0.296064	−0.148032	−	0.988983i	$-0.547294\pi$
$491$	−3221.13	−0.296064	−0.148032	+	0.988983i	$0.547294\pi$
$492$	0	0
$493$	−9029.96	−0.824927
$494$	−41978.9	−3.82332
$495$	0	0
$496$	−7523.91	−0.681116
$497$	0	0
$498$	0	0
$499$	9713.81	0.871443	0.435721	−	0.900082i	$-0.356493\pi$
$499$	9713.81	0.871443	0.435721	+	0.900082i	$0.356493\pi$
$500$	1329.16	0.118883
$501$	0	0
$502$	−17220.3	−1.53103
$503$	2078.32	0.184230	0.0921152	−	0.995748i	$-0.470637\pi$
$503$	2078.32	0.184230	0.0921152	+	0.995748i	$0.470637\pi$
$504$	0	0
$505$	−2728.36	−0.240417
$506$	8591.94	0.754858
$507$	0	0
$508$	−21758.4	−1.90034
$509$	−18974.5	−1.65232	−0.826158	−	0.563439i	$-0.809478\pi$
$509$	−18974.5	−1.65232	−0.826158	+	0.563439i	$0.809478\pi$
$510$	0	0
$511$	0	0
$512$	−12141.6	−1.04802
$513$	0	0
$514$	−9890.77	−0.848761
$515$	−3904.95	−0.334122
$516$	0	0
$517$	2913.95	0.247883
$518$	0	0
$519$	0	0
$520$	4054.09	0.341892
$521$	17523.6	1.47355	0.736777	−	0.676136i	$-0.236347\pi$
$521$	17523.6	1.47355	0.736777	+	0.676136i	$0.236347\pi$
$522$	0	0
$523$	−15218.6	−1.27239	−0.636197	−	0.771527i	$-0.719493\pi$
$523$	−15218.6	−1.27239	−0.636197	+	0.771527i	$0.719493\pi$
$524$	12125.3	1.01087
$525$	0	0
$526$	27455.8	2.27591
$527$	−6548.41	−0.541278
$528$	0	0
$529$	−1993.15	−0.163816
$530$	−2762.53	−0.226409
$531$	0	0
$532$	0	0
$533$	12958.4	1.05308
$534$	0	0
$535$	3103.30	0.250780
$536$	959.308	0.0773056
$537$	0	0
$538$	4279.26	0.342922
$539$	0	0
$540$	0	0
$541$	−15559.3	−1.23650	−0.618249	−	0.785983i	$-0.712158\pi$
$541$	−15559.3	−1.23650	−0.618249	+	0.785983i	$0.712158\pi$
$542$	3155.36	0.250063
$543$	0	0
$544$	−7718.21	−0.608301
$545$	20.9397	0.00164579
$546$	0	0
$547$	−8690.70	−0.679319	−0.339660	−	0.940548i	$-0.610312\pi$
$547$	−8690.70	−0.679319	−0.339660	+	0.940548i	$0.610312\pi$
$548$	5213.93	0.406438
$549$	0	0
$550$	2129.55	0.165099
$551$	−39290.8	−3.03783
$552$	0	0
$553$	0	0
$554$	15274.9	1.17143
$555$	0	0
$556$	−29773.1	−2.27097
$557$	7376.26	0.561117	0.280559	−	0.959837i	$-0.409480\pi$
$557$	7376.26	0.561117	0.280559	+	0.959837i	$0.409480\pi$
$558$	0	0
$559$	1298.29	0.0982320
$560$	0	0
$561$	0	0
$562$	−20129.3	−1.51086
$563$	12875.4	0.963825	0.481913	−	0.876219i	$-0.339942\pi$
$563$	12875.4	0.963825	0.481913	+	0.876219i	$0.339942\pi$
$564$	0	0
$565$	−7067.65	−0.526263
$566$	−9086.28	−0.674779
$567$	0	0
$568$	−3588.89	−0.265117
$569$	12064.6	0.888882	0.444441	−	0.895808i	$-0.353402\pi$
$569$	12064.6	0.888882	0.444441	+	0.895808i	$0.353402\pi$
$570$	0	0
$571$	23745.6	1.74032	0.870158	−	0.492772i	$-0.164016\pi$
$571$	23745.6	1.74032	0.870158	+	0.492772i	$0.164016\pi$
$572$	14967.8	1.09412
$573$	0	0
$574$	0	0
$575$	2521.64	0.182886
$576$	0	0
$577$	9846.08	0.710394	0.355197	−	0.934791i	$-0.384414\pi$
$577$	9846.08	0.710394	0.355197	+	0.934791i	$0.384414\pi$
$578$	16969.8	1.22120
$579$	0	0
$580$	15322.4	1.09695
$581$	0	0
$582$	0	0
$583$	−2525.79	−0.179430
$584$	−12427.6	−0.880575
$585$	0	0
$586$	28466.7	2.00674
$587$	10074.7	0.708392	0.354196	−	0.935171i	$-0.384755\pi$
$587$	10074.7	0.708392	0.354196	+	0.935171i	$0.384755\pi$
$588$	0	0
$589$	−28493.1	−1.99328
$590$	−6964.12	−0.485946
$591$	0	0
$592$	−11157.5	−0.774615
$593$	−7387.25	−0.511565	−0.255782	−	0.966734i	$-0.582333\pi$
$593$	−7387.25	−0.511565	−0.255782	+	0.966734i	$0.582333\pi$
$594$	0	0
$595$	0	0
$596$	−12399.6	−0.852194
$597$	0	0
$598$	31058.1	2.12384
$599$	−1252.73	−0.0854510	−0.0427255	−	0.999087i	$-0.513604\pi$
$599$	−1252.73	−0.0854510	−0.0427255	+	0.999087i	$0.513604\pi$
$600$	0	0
$601$	1800.81	0.122224	0.0611120	−	0.998131i	$-0.480535\pi$
$601$	1800.81	0.122224	0.0611120	+	0.998131i	$0.480535\pi$
$602$	0	0
$603$	0	0
$604$	10203.5	0.687373
$605$	−4707.94	−0.316372
$606$	0	0
$607$	−2497.06	−0.166973	−0.0834863	−	0.996509i	$-0.526605\pi$
$607$	−2497.06	−0.166973	−0.0834863	+	0.996509i	$0.526605\pi$
$608$	−33583.1	−2.24009
$609$	0	0
$610$	7359.90	0.488514
$611$	10533.3	0.697434
$612$	0	0
$613$	−19750.8	−1.30135	−0.650674	−	0.759357i	$-0.725513\pi$
$613$	−19750.8	−1.30135	−0.650674	+	0.759357i	$0.725513\pi$
$614$	−11538.2	−0.758378
$615$	0	0
$616$	0	0
$617$	−16797.4	−1.09601	−0.548004	−	0.836476i	$-0.684612\pi$
$617$	−16797.4	−1.09601	−0.548004	+	0.836476i	$0.684612\pi$
$618$	0	0
$619$	−26547.4	−1.72379	−0.861897	−	0.507084i	$-0.830723\pi$
$619$	−26547.4	−1.72379	−0.861897	+	0.507084i	$0.830723\pi$
$620$	11111.6	0.719763
$621$	0	0
$622$	3693.73	0.238111
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−14460.6	−0.923264
$627$	0	0
$628$	10852.9	0.689616
$629$	−9710.93	−0.615580
$630$	0	0
$631$	5394.86	0.340358	0.170179	−	0.985413i	$-0.445565\pi$
$631$	5394.86	0.340358	0.170179	+	0.985413i	$0.445565\pi$
$632$	−14017.3	−0.882245
$633$	0	0
$634$	32949.0	2.06399
$635$	−10231.3	−0.639398
$636$	0	0
$637$	0	0
$638$	24549.3	1.52338
$639$	0	0
$640$	6880.63	0.424970
$641$	−2452.41	−0.151114	−0.0755572	−	0.997141i	$-0.524074\pi$
$641$	−2452.41	−0.151114	−0.0755572	+	0.997141i	$0.524074\pi$
$642$	0	0
$643$	−7074.97	−0.433919	−0.216959	−	0.976181i	$-0.569614\pi$
$643$	−7074.97	−0.433919	−0.216959	+	0.976181i	$0.569614\pi$
$644$	0	0
$645$	0	0
$646$	−18439.1	−1.12303
$647$	3341.37	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$647$	3341.37	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$648$	0	0
$649$	−6367.31	−0.385114
$650$	7697.89	0.464517
$651$	0	0
$652$	−16651.8	−1.00021
$653$	23061.6	1.38204	0.691019	−	0.722837i	$-0.257162\pi$
$653$	23061.6	1.38204	0.691019	+	0.722837i	$0.257162\pi$
$654$	0	0
$655$	5701.59	0.340121
$656$	6539.85	0.389235
$657$	0	0
$658$	0	0
$659$	−1742.64	−0.103010	−0.0515049	−	0.998673i	$-0.516402\pi$
$659$	−1742.64	−0.103010	−0.0515049	+	0.998673i	$0.516402\pi$
$660$	0	0
$661$	12576.5	0.740046	0.370023	−	0.929023i	$-0.379350\pi$
$661$	12576.5	0.740046	0.370023	+	0.929023i	$0.379350\pi$
$662$	14338.5	0.841817
$663$	0	0
$664$	7312.51	0.427380
$665$	0	0
$666$	0	0
$667$	29069.2	1.68750
$668$	−12018.8	−0.696141
$669$	0	0
$670$	1821.53	0.105033
$671$	6729.17	0.387149
$672$	0	0
$673$	−10680.8	−0.611760	−0.305880	−	0.952070i	$-0.598951\pi$
$673$	−10680.8	−0.611760	−0.305880	+	0.952070i	$0.598951\pi$
$674$	9643.27	0.551105
$675$	0	0
$676$	30744.2	1.74921
$677$	−29559.1	−1.67806	−0.839032	−	0.544082i	$-0.816878\pi$
$677$	−29559.1	−1.67806	−0.839032	+	0.544082i	$0.816878\pi$
$678$	0	0
$679$	0	0
$680$	1780.74	0.100424
$681$	0	0
$682$	17802.8	0.999569
$683$	−10250.4	−0.574263	−0.287132	−	0.957891i	$-0.592702\pi$
$683$	−10250.4	−0.574263	−0.287132	+	0.957891i	$0.592702\pi$
$684$	0	0
$685$	2451.71	0.136752
$686$	0	0
$687$	0	0
$688$	655.218	0.0363081
$689$	−9130.20	−0.504837
$690$	0	0
$691$	8874.04	0.488544	0.244272	−	0.969707i	$-0.421451\pi$
$691$	8874.04	0.488544	0.244272	+	0.969707i	$0.421451\pi$
$692$	27043.9	1.48563
$693$	0	0
$694$	10915.1	0.597018
$695$	−14000.0	−0.764101
$696$	0	0
$697$	5691.94	0.309322
$698$	5572.77	0.302196
$699$	0	0
$700$	0	0
$701$	22086.2	1.18999	0.594996	−	0.803729i	$-0.297154\pi$
$701$	22086.2	1.18999	0.594996	+	0.803729i	$0.297154\pi$
$702$	0	0
$703$	−42253.7	−2.26690
$704$	15299.9	0.819085
$705$	0	0
$706$	33534.3	1.78765
$707$	0	0
$708$	0	0
$709$	−27878.9	−1.47675	−0.738373	−	0.674392i	$-0.764406\pi$
$709$	−27878.9	−1.47675	−0.738373	+	0.674392i	$0.764406\pi$
$710$	−6814.57	−0.360206
$711$	0	0
$712$	−12961.7	−0.682248
$713$	21080.6	1.10726
$714$	0	0
$715$	7038.20	0.368131
$716$	12873.1	0.671916
$717$	0	0
$718$	−9859.76	−0.512483
$719$	25863.3	1.34150	0.670750	−	0.741684i	$-0.265973\pi$
$719$	25863.3	1.34150	0.670750	+	0.741684i	$0.265973\pi$
$720$	0	0
$721$	0	0
$722$	−50623.4	−2.60943
$723$	0	0
$724$	32232.8	1.65459
$725$	7204.95	0.369083
$726$	0	0
$727$	−29157.0	−1.48744	−0.743722	−	0.668489i	$-0.766941\pi$
$727$	−29157.0	−1.48744	−0.743722	+	0.668489i	$0.766941\pi$
$728$	0	0
$729$	0	0
$730$	−23597.4	−1.19641
$731$	570.267	0.0288538
$732$	0	0
$733$	−11006.1	−0.554595	−0.277297	−	0.960784i	$-0.589439\pi$
$733$	−11006.1	−0.554595	−0.277297	+	0.960784i	$0.589439\pi$
$734$	46218.1	2.32417
$735$	0	0
$736$	24846.4	1.24436
$737$	1665.43	0.0832386
$738$	0	0
$739$	−37214.4	−1.85244	−0.926221	−	0.376982i	$-0.876962\pi$
$739$	−37214.4	−1.85244	−0.926221	+	0.376982i	$0.876962\pi$
$740$	16477.9	0.818567
$741$	0	0
$742$	0	0
$743$	−11214.5	−0.553730	−0.276865	−	0.960909i	$-0.589296\pi$
$743$	−11214.5	−0.553730	−0.276865	+	0.960909i	$0.589296\pi$
$744$	0	0
$745$	−5830.58	−0.286733
$746$	3584.65	0.175930
$747$	0	0
$748$	6574.54	0.321375
$749$	0	0
$750$	0	0
$751$	6965.26	0.338437	0.169218	−	0.985579i	$-0.445876\pi$
$751$	6965.26	0.338437	0.169218	+	0.985579i	$0.445876\pi$
$752$	5315.94	0.257782
$753$	0	0
$754$	88740.7	4.28613
$755$	4797.91	0.231276
$756$	0	0
$757$	−19352.8	−0.929180	−0.464590	−	0.885526i	$-0.653798\pi$
$757$	−19352.8	−0.929180	−0.464590	+	0.885526i	$0.653798\pi$
$758$	−22676.0	−1.08658
$759$	0	0
$760$	7748.29	0.369816
$761$	32383.6	1.54258	0.771291	−	0.636483i	$-0.219612\pi$
$761$	32383.6	1.54258	0.771291	+	0.636483i	$0.219612\pi$
$762$	0	0
$763$	0	0
$764$	23059.7	1.09198
$765$	0	0
$766$	48535.7	2.28938
$767$	−23016.5	−1.08354
$768$	0	0
$769$	25353.9	1.18893	0.594463	−	0.804123i	$-0.297365\pi$
$769$	25353.9	1.18893	0.594463	+	0.804123i	$0.297365\pi$
$770$	0	0
$771$	0	0
$772$	15848.6	0.738867
$773$	26117.0	1.21522	0.607610	−	0.794236i	$-0.292128\pi$
$773$	26117.0	1.21522	0.607610	+	0.794236i	$0.292128\pi$
$774$	0	0
$775$	5224.94	0.242175
$776$	16049.8	0.742465
$777$	0	0
$778$	−36720.9	−1.69217
$779$	24766.5	1.13909
$780$	0	0
$781$	−6230.58	−0.285464
$782$	13642.1	0.623838
$783$	0	0
$784$	0	0
$785$	5103.30	0.232031
$786$	0	0
$787$	2273.38	0.102970	0.0514849	−	0.998674i	$-0.483605\pi$
$787$	2273.38	0.102970	0.0514849	+	0.998674i	$0.483605\pi$
$788$	−35946.4	−1.62505
$789$	0	0
$790$	−26616.0	−1.19868
$791$	0	0
$792$	0	0
$793$	24324.6	1.08927
$794$	13481.8	0.602585
$795$	0	0
$796$	−48863.3	−2.17577
$797$	2937.42	0.130551	0.0652753	−	0.997867i	$-0.479207\pi$
$797$	2937.42	0.130551	0.0652753	+	0.997867i	$0.479207\pi$
$798$	0	0
$799$	4626.71	0.204858
$800$	6158.31	0.272162
$801$	0	0
$802$	−48585.0	−2.13915
$803$	−21575.1	−0.948157
$804$	0	0
$805$	0	0
$806$	64353.6	2.81236
$807$	0	0
$808$	6202.52	0.270054
$809$	4317.51	0.187634	0.0938169	−	0.995589i	$-0.470093\pi$
$809$	4317.51	0.187634	0.0938169	+	0.995589i	$0.470093\pi$
$810$	0	0
$811$	−1286.12	−0.0556863	−0.0278432	−	0.999612i	$-0.508864\pi$
$811$	−1286.12	−0.0556863	−0.0278432	+	0.999612i	$0.508864\pi$
$812$	0	0
$813$	0	0
$814$	26400.6	1.13678
$815$	−7830.08	−0.336534
$816$	0	0
$817$	2481.32	0.106255
$818$	−34187.3	−1.46129
$819$	0	0
$820$	−9658.31	−0.411321
$821$	−26350.2	−1.12013	−0.560066	−	0.828448i	$-0.689224\pi$
$821$	−26350.2	−1.12013	−0.560066	+	0.828448i	$0.689224\pi$
$822$	0	0
$823$	9820.05	0.415924	0.207962	−	0.978137i	$-0.433317\pi$
$823$	9820.05	0.415924	0.207962	+	0.978137i	$0.433317\pi$
$824$	8877.32	0.375311
$825$	0	0
$826$	0	0
$827$	30370.7	1.27702	0.638509	−	0.769615i	$-0.279552\pi$
$827$	30370.7	1.27702	0.638509	+	0.769615i	$0.279552\pi$
$828$	0	0
$829$	30817.7	1.29112	0.645562	−	0.763708i	$-0.276623\pi$
$829$	30817.7	1.29112	0.645562	+	0.763708i	$0.276623\pi$
$830$	13885.0	0.580668
$831$	0	0
$832$	55305.9	2.30455
$833$	0	0
$834$	0	0
$835$	−5651.52	−0.234226
$836$	28606.8	1.18348
$837$	0	0
$838$	22693.7	0.935491
$839$	−24746.0	−1.01827	−0.509134	−	0.860688i	$-0.670034\pi$
$839$	−24746.0	−1.01827	−0.509134	+	0.860688i	$0.670034\pi$
$840$	0	0
$841$	58669.1	2.40556
$842$	−6290.88	−0.257480
$843$	0	0
$844$	−53045.7	−2.16340
$845$	14456.6	0.588548
$846$	0	0
$847$	0	0
$848$	−4607.82	−0.186596
$849$	0	0
$850$	3381.27	0.136443
$851$	31261.4	1.25926
$852$	0	0
$853$	11812.1	0.474135	0.237067	−	0.971493i	$-0.423814\pi$
$853$	11812.1	0.474135	0.237067	+	0.971493i	$0.423814\pi$
$854$	0	0
$855$	0	0
$856$	−7054.89	−0.281695
$857$	−23440.6	−0.934322	−0.467161	−	0.884172i	$-0.654723\pi$
$857$	−23440.6	−0.934322	−0.467161	+	0.884172i	$0.654723\pi$
$858$	0	0
$859$	−8945.58	−0.355319	−0.177660	−	0.984092i	$-0.556853\pi$
$859$	−8945.58	−0.355319	−0.177660	+	0.984092i	$0.556853\pi$
$860$	−967.652	−0.0383682
$861$	0	0
$862$	−66006.3	−2.60810
$863$	−19313.5	−0.761806	−0.380903	−	0.924615i	$-0.624387\pi$
$863$	−19313.5	−0.761806	−0.380903	+	0.924615i	$0.624387\pi$
$864$	0	0
$865$	12716.7	0.499862
$866$	808.330	0.0317184
$867$	0	0
$868$	0	0
$869$	−24335.1	−0.949955
$870$	0	0
$871$	6020.18	0.234197
$872$	−47.6033	−0.00184868
$873$	0	0
$874$	59359.0	2.29731
$875$	0	0
$876$	0	0
$877$	−12154.8	−0.468001	−0.234001	−	0.972236i	$-0.575182\pi$
$877$	−12154.8	−0.468001	−0.234001	+	0.972236i	$0.575182\pi$
$878$	15487.7	0.595313
$879$	0	0
$880$	3552.03	0.136067
$881$	−29390.4	−1.12394	−0.561968	−	0.827159i	$-0.689956\pi$
$881$	−29390.4	−1.12394	−0.561968	+	0.827159i	$0.689956\pi$
$882$	0	0
$883$	4180.02	0.159308	0.0796540	−	0.996823i	$-0.474618\pi$
$883$	4180.02	0.159308	0.0796540	+	0.996823i	$0.474618\pi$
$884$	23765.6	0.904211
$885$	0	0
$886$	21219.0	0.804590
$887$	21825.8	0.826198	0.413099	−	0.910686i	$-0.364446\pi$
$887$	21825.8	0.826198	0.413099	+	0.910686i	$0.364446\pi$
$888$	0	0
$889$	0	0
$890$	−24611.6	−0.926947
$891$	0	0
$892$	−40331.6	−1.51390
$893$	20131.5	0.754397
$894$	0	0
$895$	6053.25	0.226076
$896$	0	0
$897$	0	0
$898$	30609.7	1.13748
$899$	60232.7	2.23456
$900$	0	0
$901$	−4010.40	−0.148286
$902$	−15474.4	−0.571221
$903$	0	0
$904$	16067.2	0.591138
$905$	15156.6	0.556710
$906$	0	0
$907$	−8356.11	−0.305910	−0.152955	−	0.988233i	$-0.548879\pi$
$907$	−8356.11	−0.305910	−0.152955	+	0.988233i	$0.548879\pi$
$908$	41586.3	1.51992
$909$	0	0
$910$	0	0
$911$	4419.80	0.160740	0.0803701	−	0.996765i	$-0.474390\pi$
$911$	4419.80	0.160740	0.0803701	+	0.996765i	$0.474390\pi$
$912$	0	0
$913$	12695.1	0.460181
$914$	−21806.8	−0.789173
$915$	0	0
$916$	−3767.65	−0.135903
$917$	0	0
$918$	0	0
$919$	−39257.6	−1.40913	−0.704563	−	0.709641i	$-0.748857\pi$
$919$	−39257.6	−1.40913	−0.704563	+	0.709641i	$0.748857\pi$
$920$	−5732.56	−0.205432
$921$	0	0
$922$	72007.0	2.57204
$923$	−22522.2	−0.803173
$924$	0	0
$925$	7748.29	0.275419
$926$	−67207.9	−2.38509
$927$	0	0
$928$	70992.5	2.51125
$929$	−13399.9	−0.473235	−0.236618	−	0.971603i	$-0.576039\pi$
$929$	−13399.9	−0.473235	−0.236618	+	0.971603i	$0.576039\pi$
$930$	0	0
$931$	0	0
$932$	−69036.1	−2.42635
$933$	0	0
$934$	−14367.2	−0.503330
$935$	3091.50	0.108131
$936$	0	0
$937$	−27539.8	−0.960176	−0.480088	−	0.877220i	$-0.659395\pi$
$937$	−27539.8	−0.960176	−0.480088	+	0.877220i	$0.659395\pi$
$938$	0	0
$939$	0	0
$940$	−7850.79	−0.272409
$941$	14363.8	0.497605	0.248802	−	0.968554i	$-0.419963\pi$
$941$	14363.8	0.497605	0.248802	+	0.968554i	$0.419963\pi$
$942$	0	0
$943$	−18323.5	−0.632762
$944$	−11615.9	−0.400494
$945$	0	0
$946$	−1550.36	−0.0532838
$947$	6372.12	0.218655	0.109327	−	0.994006i	$-0.465130\pi$
$947$	6372.12	0.218655	0.109327	+	0.994006i	$0.465130\pi$
$948$	0	0
$949$	−77989.6	−2.66770
$950$	14712.4	0.502456
$951$	0	0
$952$	0	0
$953$	−958.776	−0.0325895	−0.0162948	−	0.999867i	$-0.505187\pi$
$953$	−958.776	−0.0325895	−0.0162948	+	0.999867i	$0.505187\pi$
$954$	0	0
$955$	10843.2	0.367411
$956$	3643.88	0.123276
$957$	0	0
$958$	63055.1	2.12653
$959$	0	0
$960$	0	0
$961$	13888.9	0.466213
$962$	95432.7	3.19842
$963$	0	0
$964$	−24601.8	−0.821961
$965$	7452.40	0.248602
$966$	0	0
$967$	−40104.5	−1.33369	−0.666843	−	0.745198i	$-0.732355\pi$
$967$	−40104.5	−1.33369	−0.666843	+	0.745198i	$0.732355\pi$
$968$	10702.8	0.355373
$969$	0	0
$970$	30475.2	1.00876
$971$	12397.4	0.409732	0.204866	−	0.978790i	$-0.434324\pi$
$971$	12397.4	0.409732	0.204866	+	0.978790i	$0.434324\pi$
$972$	0	0
$973$	0	0
$974$	8113.05	0.266898
$975$	0	0
$976$	12276.1	0.402611
$977$	−44982.9	−1.47301	−0.736506	−	0.676432i	$-0.763525\pi$
$977$	−44982.9	−1.47301	−0.736506	+	0.676432i	$0.763525\pi$
$978$	0	0
$979$	−22502.5	−0.734608
$980$	0	0
$981$	0	0
$982$	13904.4	0.451841
$983$	−7895.76	−0.256191	−0.128095	−	0.991762i	$-0.540886\pi$
$983$	−7895.76	−0.256191	−0.128095	+	0.991762i	$0.540886\pi$
$984$	0	0
$985$	−16902.8	−0.546771
$986$	38979.0	1.25897
$987$	0	0
$988$	103408.	3.32979
$989$	−1835.80	−0.0590244
$990$	0	0
$991$	54534.9	1.74809	0.874046	−	0.485844i	$-0.161488\pi$
$991$	54534.9	1.74809	0.874046	+	0.485844i	$0.161488\pi$
$992$	51482.9	1.64776
$993$	0	0
$994$	0	0
$995$	−22976.6	−0.732069
$996$	0	0
$997$	6028.06	0.191485	0.0957425	−	0.995406i	$-0.469477\pi$
$997$	6028.06	0.191485	0.0957425	+	0.995406i	$0.469477\pi$
$998$	−41930.9	−1.32996
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.x.1.1		2
3.2	odd	2		245.4.a.i.1.2	✓	2
7.6	odd	2		2205.4.a.w.1.1		2
15.14	odd	2		1225.4.a.q.1.1		2
21.2	odd	6		245.4.e.k.116.1		4
21.5	even	6		245.4.e.j.116.1		4
21.11	odd	6		245.4.e.k.226.1		4
21.17	even	6		245.4.e.j.226.1		4
21.20	even	2		245.4.a.j.1.2	yes	2
105.104	even	2		1225.4.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.i.1.2	✓	2	3.2	odd	2
245.4.a.j.1.2	yes	2	21.20	even	2
245.4.e.j.116.1		4	21.5	even	6
245.4.e.j.226.1		4	21.17	even	6
245.4.e.k.116.1		4	21.2	odd	6
245.4.e.k.226.1		4	21.11	odd	6
1225.4.a.p.1.1		2	105.104	even	2
1225.4.a.q.1.1		2	15.14	odd	2
2205.4.a.w.1.1		2	7.6	odd	2
2205.4.a.x.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-4.31662 q^{2} +10.6332 q^{4} +5.00000 q^{5} -11.3668 q^{8} +O(q^{10})\)	\(q-4.31662 q^{2} +10.6332 q^{4} +5.00000 q^{5} -11.3668 q^{8} -21.5831 q^{10} -19.7335 q^{11} -71.3325 q^{13} -36.0000 q^{16} -31.3325 q^{17} -136.332 q^{19} +53.1662 q^{20} +85.1821 q^{22} +100.865 q^{23} +25.0000 q^{25} +307.916 q^{26} +288.198 q^{29} +208.997 q^{31} +246.332 q^{32} +135.251 q^{34} +309.931 q^{37} +588.496 q^{38} -56.8338 q^{40} -181.662 q^{41} -18.2005 q^{43} -209.831 q^{44} -435.398 q^{46} -147.665 q^{47} -107.916 q^{50} -758.496 q^{52} +127.995 q^{53} -98.6675 q^{55} -1244.04 q^{58} +322.665 q^{59} -341.003 q^{61} -902.164 q^{62} -775.325 q^{64} -356.662 q^{65} -84.3960 q^{67} -333.166 q^{68} +315.736 q^{71} +1093.32 q^{73} -1337.86 q^{74} -1449.66 q^{76} +1233.19 q^{79} -180.000 q^{80} +784.169 q^{82} -643.325 q^{83} -156.662 q^{85} +78.5647 q^{86} +224.306 q^{88} +1140.32 q^{89} +1072.53 q^{92} +637.414 q^{94} -681.662 q^{95} -1411.99 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8	\( 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8} - 10 q^{10} - 66 q^{11} - 10 q^{13} - 72 q^{16} + 70 q^{17} - 140 q^{19} + 40 q^{20} - 22 q^{22} + 16 q^{23} + 50 q^{25} + 450 q^{26} + 258 q^{29} + 20 q^{31} + 360 q^{32} + 370 q^{34} + 328 q^{37} + 580 q^{38} - 180 q^{40} + 300 q^{41} - 116 q^{43} - 88 q^{44} - 632 q^{46} - 30 q^{47} - 50 q^{50} - 920 q^{52} - 540 q^{53} - 330 q^{55} - 1314 q^{58} + 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} - 50 q^{65} + 468 q^{67} - 600 q^{68} + 1056 q^{71} + 860 q^{73} - 1296 q^{74} - 1440 q^{76} + 158 q^{79} - 360 q^{80} + 1900 q^{82} + 40 q^{83} + 350 q^{85} - 148 q^{86} + 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 910 q^{94} - 700 q^{95} - 1630 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8 - 10 * q^10 - 66 * q^11 - 10 * q^13 - 72 * q^16 + 70 * q^17 - 140 * q^19 + 40 * q^20 - 22 * q^22 + 16 * q^23 + 50 * q^25 + 450 * q^26 + 258 * q^29 + 20 * q^31 + 360 * q^32 + 370 * q^34 + 328 * q^37 + 580 * q^38 - 180 * q^40 + 300 * q^41 - 116 * q^43 - 88 * q^44 - 632 * q^46 - 30 * q^47 - 50 * q^50 - 920 * q^52 - 540 * q^53 - 330 * q^55 - 1314 * q^58 + 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 - 50 * q^65 + 468 * q^67 - 600 * q^68 + 1056 * q^71 + 860 * q^73 - 1296 * q^74 - 1440 * q^76 + 158 * q^79 - 360 * q^80 + 1900 * q^82 + 40 * q^83 + 350 * q^85 - 148 * q^86 + 1364 * q^88 - 240 * q^89 + 1296 * q^92 + 910 * q^94 - 700 * q^95 - 1630 * q^97

Introduction

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