LMFDB - Embedded newform 2205.4.a.ch.1.3

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 62x^{10} + 1410x^{8} - 14366x^{6} + 63209x^{4} - 95056x^{2} + 64 $ x^12 - 62x^10 + 1410x^8 - 14366x^6 + 63209x^4 - 95056*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.83386$ 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)

$f(q)$	$=$	$q-3.83386 q^{2} +6.69846 q^{4} -5.00000 q^{5} +4.98993 q^{8} +19.1693 q^{10} +6.91200 q^{11} +50.6795 q^{13} -72.7183 q^{16} +47.2568 q^{17} -39.0480 q^{19} -33.4923 q^{20} -26.4996 q^{22} -10.2225 q^{23} +25.0000 q^{25} -194.298 q^{26} +208.830 q^{29} -118.990 q^{31} +238.872 q^{32} -181.176 q^{34} +124.558 q^{37} +149.705 q^{38} -24.9496 q^{40} +224.312 q^{41} +211.980 q^{43} +46.2997 q^{44} +39.1916 q^{46} +497.596 q^{47} -95.8464 q^{50} +339.474 q^{52} -383.745 q^{53} -34.5600 q^{55} -800.625 q^{58} +82.5591 q^{59} -304.035 q^{61} +456.191 q^{62} -334.055 q^{64} -253.397 q^{65} +256.984 q^{67} +316.548 q^{68} +415.831 q^{71} +508.576 q^{73} -477.536 q^{74} -261.562 q^{76} -1045.91 q^{79} +363.592 q^{80} -859.979 q^{82} -49.1650 q^{83} -236.284 q^{85} -812.703 q^{86} +34.4904 q^{88} +227.405 q^{89} -68.4750 q^{92} -1907.71 q^{94} +195.240 q^{95} -1445.51 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 28 q^{4} - 60 q^{5} - 160 q^{16} + 136 q^{17} - 140 q^{20} - 400 q^{22} + 300 q^{25} + 256 q^{26} - 280 q^{37} + 436 q^{38} + 376 q^{41} - 1856 q^{43} + 196 q^{46} + 968 q^{47} - 352 q^{58} + 3288 q^{59}+ \cdots + 4040 q^{89}+O(q^{100}) $ 12 * q + 28 * q^4 - 60 * q^5 - 160 * q^16 + 136 * q^17 - 140 * q^20 - 400 * q^22 + 300 * q^25 + 256 * q^26 - 280 * q^37 + 436 * q^38 + 376 * q^41 - 1856 * q^43 + 196 * q^46 + 968 * q^47 - 352 * q^58 + 3288 * q^59 + 1036 * q^62 - 2116 * q^64 - 1216 * q^67 + 2112 * q^68 - 472 * q^79 + 800 * q^80 + 4312 * q^83 - 680 * q^85 - 1224 * q^88 + 4040 * q^89

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 $

$2$	−3.83386	−1.35547	−0.677737	−	0.735305i	$-0.737039\pi$
$2$	−3.83386	−1.35547	−0.677737	+	0.735305i	$0.737039\pi$
$3$	0	0
$3$	0	0
$4$	6.69846	0.837307
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	4.98993	0.220526
$9$	0	0
$10$	19.1693	0.606186
$11$	6.91200	0.189459	0.0947294	−	0.995503i	$-0.469801\pi$
$11$	6.91200	0.189459	0.0947294	+	0.995503i	$0.469801\pi$
$12$	0	0
$13$	50.6795	1.08123	0.540614	−	0.841271i	$-0.318192\pi$
$13$	50.6795	1.08123	0.540614	+	0.841271i	$0.318192\pi$
$14$	0	0
$15$	0	0
$16$	−72.7183	−1.13622
$17$	47.2568	0.674204	0.337102	−	0.941468i	$-0.390553\pi$
$17$	47.2568	0.674204	0.337102	+	0.941468i	$0.390553\pi$
$18$	0	0
$19$	−39.0480	−0.471486	−0.235743	−	0.971815i	$-0.575752\pi$
$19$	−39.0480	−0.471486	−0.235743	+	0.971815i	$0.575752\pi$
$20$	−33.4923	−0.374455
$21$	0	0
$22$	−26.4996	−0.256806
$23$	−10.2225	−0.0926757	−0.0463378	−	0.998926i	$-0.514755\pi$
$23$	−10.2225	−0.0926757	−0.0463378	+	0.998926i	$0.514755\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−194.298	−1.46557
$27$	0	0
$28$	0	0
$29$	208.830	1.33720	0.668600	−	0.743622i	$-0.266894\pi$
$29$	208.830	1.33720	0.668600	+	0.743622i	$0.266894\pi$
$30$	0	0
$31$	−118.990	−0.689395	−0.344697	−	0.938714i	$-0.612018\pi$
$31$	−118.990	−0.689395	−0.344697	+	0.938714i	$0.612018\pi$
$32$	238.872	1.31960
$33$	0	0
$34$	−181.176	−0.913865
$35$	0	0
$36$	0	0
$37$	124.558	0.553436	0.276718	−	0.960951i	$-0.410753\pi$
$37$	124.558	0.553436	0.276718	+	0.960951i	$0.410753\pi$
$38$	149.705	0.639086
$39$	0	0
$40$	−24.9496	−0.0986221
$41$	224.312	0.854429	0.427215	−	0.904150i	$-0.359495\pi$
$41$	224.312	0.854429	0.427215	+	0.904150i	$0.359495\pi$
$42$	0	0
$43$	211.980	0.751784	0.375892	−	0.926663i	$-0.377336\pi$
$43$	211.980	0.751784	0.375892	+	0.926663i	$0.377336\pi$
$44$	46.2997	0.158635
$45$	0	0
$46$	39.1916	0.125619
$47$	497.596	1.54429	0.772147	−	0.635444i	$-0.219183\pi$
$47$	497.596	1.54429	0.772147	+	0.635444i	$0.219183\pi$
$48$	0	0
$49$	0	0
$50$	−95.8464	−0.271095
$51$	0	0
$52$	339.474	0.905319
$53$	−383.745	−0.994556	−0.497278	−	0.867591i	$-0.665667\pi$
$53$	−383.745	−0.994556	−0.497278	+	0.867591i	$0.665667\pi$
$54$	0	0
$55$	−34.5600	−0.0847285
$56$	0	0
$57$	0	0
$58$	−800.625	−1.81254
$59$	82.5591	0.182174	0.0910872	−	0.995843i	$-0.470966\pi$
$59$	82.5591	0.182174	0.0910872	+	0.995843i	$0.470966\pi$
$60$	0	0
$61$	−304.035	−0.638159	−0.319080	−	0.947728i	$-0.603374\pi$
$61$	−304.035	−0.638159	−0.319080	+	0.947728i	$0.603374\pi$
$62$	456.191	0.934456
$63$	0	0
$64$	−334.055	−0.652452
$65$	−253.397	−0.483540
$66$	0	0
$67$	256.984	0.468590	0.234295	−	0.972166i	$-0.424722\pi$
$67$	256.984	0.468590	0.234295	+	0.972166i	$0.424722\pi$
$68$	316.548	0.564516
$69$	0	0
$70$	0	0
$71$	415.831	0.695072	0.347536	−	0.937667i	$-0.387018\pi$
$71$	415.831	0.695072	0.347536	+	0.937667i	$0.387018\pi$
$72$	0	0
$73$	508.576	0.815401	0.407701	−	0.913116i	$-0.366331\pi$
$73$	508.576	0.815401	0.407701	+	0.913116i	$0.366331\pi$
$74$	−477.536	−0.750168
$75$	0	0
$76$	−261.562	−0.394778
$77$	0	0
$78$	0	0
$79$	−1045.91	−1.48954	−0.744772	−	0.667319i	$-0.767442\pi$
$79$	−1045.91	−1.48954	−0.744772	+	0.667319i	$0.767442\pi$
$80$	363.592	0.508135
$81$	0	0
$82$	−859.979	−1.15816
$83$	−49.1650	−0.0650188	−0.0325094	−	0.999471i	$-0.510350\pi$
$83$	−49.1650	−0.0650188	−0.0325094	+	0.999471i	$0.510350\pi$
$84$	0	0
$85$	−236.284	−0.301513
$86$	−812.703	−1.01902
$87$	0	0
$88$	34.4904	0.0417805
$89$	227.405	0.270841	0.135420	−	0.990788i	$-0.456762\pi$
$89$	227.405	0.270841	0.135420	+	0.990788i	$0.456762\pi$
$90$	0	0
$91$	0	0
$92$	−68.4750	−0.0775980
$93$	0	0
$94$	−1907.71	−2.09325
$95$	195.240	0.210855
$96$	0	0
$97$	−1445.51	−1.51309	−0.756544	−	0.653942i	$-0.773114\pi$
$97$	−1445.51	−1.51309	−0.756544	+	0.653942i	$0.773114\pi$
$98$	0	0
$99$	0	0
$100$	167.461	0.167461
$101$	832.866	0.820527	0.410264	−	0.911967i	$-0.365437\pi$
$101$	832.866	0.820527	0.410264	+	0.911967i	$0.365437\pi$
$102$	0	0
$103$	−553.252	−0.529258	−0.264629	−	0.964350i	$-0.585249\pi$
$103$	−553.252	−0.529258	−0.264629	+	0.964350i	$0.585249\pi$
$104$	252.887	0.238439
$105$	0	0
$106$	1471.22	1.34809
$107$	−617.632	−0.558026	−0.279013	−	0.960287i	$-0.590007\pi$
$107$	−617.632	−0.558026	−0.279013	+	0.960287i	$0.590007\pi$
$108$	0	0
$109$	744.353	0.654092	0.327046	−	0.945008i	$-0.393947\pi$
$109$	744.353	0.654092	0.327046	+	0.945008i	$0.393947\pi$
$110$	132.498	0.114847
$111$	0	0
$112$	0	0
$113$	462.167	0.384752	0.192376	−	0.981321i	$-0.438381\pi$
$113$	462.167	0.384752	0.192376	+	0.981321i	$0.438381\pi$
$114$	0	0
$115$	51.1125	0.0414458
$116$	1398.84	1.11965
$117$	0	0
$118$	−316.520	−0.246932
$119$	0	0
$120$	0	0
$121$	−1283.22	−0.964105
$122$	1165.63	0.865008
$123$	0	0
$124$	−797.049	−0.577235
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1906.87	1.33234	0.666172	−	0.745798i	$-0.267932\pi$
$127$	1906.87	1.33234	0.666172	+	0.745798i	$0.267932\pi$
$128$	−630.258	−0.435215
$129$	0	0
$130$	971.489	0.655425
$131$	1248.53	0.832707	0.416354	−	0.909203i	$-0.363308\pi$
$131$	1248.53	0.832707	0.416354	+	0.909203i	$0.363308\pi$
$132$	0	0
$133$	0	0
$134$	−985.239	−0.635162
$135$	0	0
$136$	235.808	0.148679
$137$	−972.392	−0.606402	−0.303201	−	0.952927i	$-0.598055\pi$
$137$	−972.392	−0.606402	−0.303201	+	0.952927i	$0.598055\pi$
$138$	0	0
$139$	−1743.55	−1.06393	−0.531965	−	0.846766i	$-0.678546\pi$
$139$	−1743.55	−1.06393	−0.531965	+	0.846766i	$0.678546\pi$
$140$	0	0
$141$	0	0
$142$	−1594.24	−0.942151
$143$	350.296	0.204848
$144$	0	0
$145$	−1044.15	−0.598014
$146$	−1949.81	−1.10525
$147$	0	0
$148$	834.344	0.463396
$149$	−612.327	−0.336670	−0.168335	−	0.985730i	$-0.553839\pi$
$149$	−612.327	−0.336670	−0.168335	+	0.985730i	$0.553839\pi$
$150$	0	0
$151$	19.1695	0.0103311	0.00516553	−	0.999987i	$-0.498356\pi$
$151$	19.1695	0.0103311	0.00516553	+	0.999987i	$0.498356\pi$
$152$	−194.847	−0.103975
$153$	0	0
$154$	0	0
$155$	594.950	0.308307
$156$	0	0
$157$	597.230	0.303593	0.151797	−	0.988412i	$-0.451494\pi$
$157$	597.230	0.303593	0.151797	+	0.988412i	$0.451494\pi$
$158$	4009.86	2.01904
$159$	0	0
$160$	−1194.36	−0.590141
$161$	0	0
$162$	0	0
$163$	−3878.90	−1.86392	−0.931960	−	0.362560i	$-0.881903\pi$
$163$	−3878.90	−1.86392	−0.931960	+	0.362560i	$0.881903\pi$
$164$	1502.54	0.715420
$165$	0	0
$166$	188.492	0.0881312
$167$	1287.58	0.596623	0.298312	−	0.954469i	$-0.403577\pi$
$167$	1287.58	0.596623	0.298312	+	0.954469i	$0.403577\pi$
$168$	0	0
$169$	371.409	0.169053
$170$	905.880	0.408693
$171$	0	0
$172$	1419.94	0.629474
$173$	685.838	0.301406	0.150703	−	0.988579i	$-0.451846\pi$
$173$	685.838	0.301406	0.150703	+	0.988579i	$0.451846\pi$
$174$	0	0
$175$	0	0
$176$	−502.629	−0.215268
$177$	0	0
$178$	−871.836	−0.367117
$179$	3379.24	1.41104	0.705520	−	0.708690i	$-0.250714\pi$
$179$	3379.24	1.41104	0.705520	+	0.708690i	$0.250714\pi$
$180$	0	0
$181$	−3472.90	−1.42618	−0.713091	−	0.701072i	$-0.752705\pi$
$181$	−3472.90	−1.42618	−0.713091	+	0.701072i	$0.752705\pi$
$182$	0	0
$183$	0	0
$184$	−51.0096	−0.0204374
$185$	−622.788	−0.247504
$186$	0	0
$187$	326.639	0.127734
$188$	3333.13	1.29305
$189$	0	0
$190$	−748.523	−0.285808
$191$	2457.20	0.930874	0.465437	−	0.885081i	$-0.345897\pi$
$191$	2457.20	0.930874	0.465437	+	0.885081i	$0.345897\pi$
$192$	0	0
$193$	16.3387	0.00609371	0.00304685	−	0.999995i	$-0.499030\pi$
$193$	16.3387	0.00609371	0.00304685	+	0.999995i	$0.499030\pi$
$194$	5541.89	2.05095
$195$	0	0
$196$	0	0
$197$	−3225.44	−1.16651	−0.583257	−	0.812288i	$-0.698222\pi$
$197$	−3225.44	−1.16651	−0.583257	+	0.812288i	$0.698222\pi$
$198$	0	0
$199$	−269.647	−0.0960542	−0.0480271	−	0.998846i	$-0.515293\pi$
$199$	−269.647	−0.0960542	−0.0480271	+	0.998846i	$0.515293\pi$
$200$	124.748	0.0441052
$201$	0	0
$202$	−3193.09	−1.11220
$203$	0	0
$204$	0	0
$205$	−1121.56	−0.382112
$206$	2121.09	0.717394
$207$	0	0
$208$	−3685.33	−1.22852
$209$	−269.900	−0.0893271
$210$	0	0
$211$	4808.38	1.56883	0.784413	−	0.620239i	$-0.212964\pi$
$211$	4808.38	1.56883	0.784413	+	0.620239i	$0.212964\pi$
$212$	−2570.50	−0.832749
$213$	0	0
$214$	2367.91	0.756389
$215$	−1059.90	−0.336208
$216$	0	0
$217$	0	0
$218$	−2853.74	−0.886604
$219$	0	0
$220$	−231.499	−0.0709438
$221$	2394.95	0.728968
$222$	0	0
$223$	215.945	0.0648464	0.0324232	−	0.999474i	$-0.489678\pi$
$223$	215.945	0.0648464	0.0324232	+	0.999474i	$0.489678\pi$
$224$	0	0
$225$	0	0
$226$	−1771.88	−0.521521
$227$	−4785.58	−1.39925	−0.699626	−	0.714510i	$-0.746650\pi$
$227$	−4785.58	−1.39925	−0.699626	+	0.714510i	$0.746650\pi$
$228$	0	0
$229$	124.766	0.0360032	0.0180016	−	0.999838i	$-0.494270\pi$
$229$	124.766	0.0360032	0.0180016	+	0.999838i	$0.494270\pi$
$230$	−195.958	−0.0561787
$231$	0	0
$232$	1042.05	0.294887
$233$	−1630.01	−0.458308	−0.229154	−	0.973390i	$-0.573596\pi$
$233$	−1630.01	−0.458308	−0.229154	+	0.973390i	$0.573596\pi$
$234$	0	0
$235$	−2487.98	−0.690630
$236$	553.019	0.152536
$237$	0	0
$238$	0	0
$239$	−3382.98	−0.915594	−0.457797	−	0.889057i	$-0.651361\pi$
$239$	−3382.98	−0.915594	−0.457797	+	0.889057i	$0.651361\pi$
$240$	0	0
$241$	−2469.16	−0.659970	−0.329985	−	0.943986i	$-0.607044\pi$
$241$	−2469.16	−0.659970	−0.329985	+	0.943986i	$0.607044\pi$
$242$	4919.70	1.30682
$243$	0	0
$244$	−2036.57	−0.534335
$245$	0	0
$246$	0	0
$247$	−1978.93	−0.509783
$248$	−593.752	−0.152029
$249$	0	0
$250$	479.232	0.121237
$251$	2340.15	0.588482	0.294241	−	0.955731i	$-0.404933\pi$
$251$	2340.15	0.588482	0.294241	+	0.955731i	$0.404933\pi$
$252$	0	0
$253$	−70.6580	−0.0175582
$254$	−7310.68	−1.80596
$255$	0	0
$256$	5088.76	1.24237
$257$	4632.62	1.12442	0.562208	−	0.826996i	$-0.309952\pi$
$257$	4632.62	1.12442	0.562208	+	0.826996i	$0.309952\pi$
$258$	0	0
$259$	0	0
$260$	−1697.37	−0.404871
$261$	0	0
$262$	−4786.69	−1.12871
$263$	2478.45	0.581093	0.290547	−	0.956861i	$-0.406163\pi$
$263$	2478.45	0.581093	0.290547	+	0.956861i	$0.406163\pi$
$264$	0	0
$265$	1918.73	0.444779
$266$	0	0
$267$	0	0
$268$	1721.39	0.392354
$269$	−1648.14	−0.373564	−0.186782	−	0.982401i	$-0.559806\pi$
$269$	−1648.14	−0.373564	−0.186782	+	0.982401i	$0.559806\pi$
$270$	0	0
$271$	5053.83	1.13284	0.566418	−	0.824118i	$-0.308329\pi$
$271$	5053.83	1.13284	0.566418	+	0.824118i	$0.308329\pi$
$272$	−3436.44	−0.766046
$273$	0	0
$274$	3728.01	0.821961
$275$	172.800	0.0378917
$276$	0	0
$277$	−1596.93	−0.346391	−0.173196	−	0.984887i	$-0.555409\pi$
$277$	−1596.93	−0.346391	−0.173196	+	0.984887i	$0.555409\pi$
$278$	6684.54	1.44213
$279$	0	0
$280$	0	0
$281$	−5530.70	−1.17414	−0.587071	−	0.809536i	$-0.699719\pi$
$281$	−5530.70	−1.17414	−0.587071	+	0.809536i	$0.699719\pi$
$282$	0	0
$283$	6353.81	1.33461	0.667305	−	0.744785i	$-0.267448\pi$
$283$	6353.81	1.33461	0.667305	+	0.744785i	$0.267448\pi$
$284$	2785.43	0.581989
$285$	0	0
$286$	−1342.99	−0.277666
$287$	0	0
$288$	0	0
$289$	−2679.79	−0.545449
$290$	4003.13	0.810592
$291$	0	0
$292$	3406.67	0.682741
$293$	−5328.53	−1.06244	−0.531222	−	0.847233i	$-0.678267\pi$
$293$	−5328.53	−1.06244	−0.531222	+	0.847233i	$0.678267\pi$
$294$	0	0
$295$	−412.796	−0.0814708
$296$	621.534	0.122047
$297$	0	0
$298$	2347.57	0.456347
$299$	−518.071	−0.100203
$300$	0	0
$301$	0	0
$302$	−73.4930	−0.0140035
$303$	0	0
$304$	2839.51	0.535713
$305$	1520.18	0.285393
$306$	0	0
$307$	1687.89	0.313789	0.156894	−	0.987615i	$-0.449852\pi$
$307$	1687.89	0.313789	0.156894	+	0.987615i	$0.449852\pi$
$308$	0	0
$309$	0	0
$310$	−2280.95	−0.417901
$311$	9821.39	1.79074	0.895370	−	0.445323i	$-0.146911\pi$
$311$	9821.39	1.79074	0.895370	+	0.445323i	$0.146911\pi$
$312$	0	0
$313$	−9673.64	−1.74692	−0.873461	−	0.486894i	$-0.838130\pi$
$313$	−9673.64	−1.74692	−0.873461	+	0.486894i	$0.838130\pi$
$314$	−2289.70	−0.411513
$315$	0	0
$316$	−7005.97	−1.24721
$317$	−6702.55	−1.18755	−0.593774	−	0.804632i	$-0.702363\pi$
$317$	−6702.55	−1.18755	−0.593774	+	0.804632i	$0.702363\pi$
$318$	0	0
$319$	1443.43	0.253344
$320$	1670.28	0.291785
$321$	0	0
$322$	0	0
$323$	−1845.29	−0.317878
$324$	0	0
$325$	1266.99	0.216245
$326$	14871.2	2.52649
$327$	0	0
$328$	1119.30	0.188424
$329$	0	0
$330$	0	0
$331$	8980.64	1.49130	0.745650	−	0.666338i	$-0.232139\pi$
$331$	8980.64	1.49130	0.745650	+	0.666338i	$0.232139\pi$
$332$	−329.330	−0.0544407
$333$	0	0
$334$	−4936.41	−0.808707
$335$	−1284.92	−0.209560
$336$	0	0
$337$	−3418.88	−0.552636	−0.276318	−	0.961066i	$-0.589114\pi$
$337$	−3418.88	−0.552636	−0.276318	+	0.961066i	$0.589114\pi$
$338$	−1423.93	−0.229146
$339$	0	0
$340$	−1582.74	−0.252459
$341$	−822.459	−0.130612
$342$	0	0
$343$	0	0
$344$	1057.77	0.165788
$345$	0	0
$346$	−2629.41	−0.408548
$347$	2309.32	0.357265	0.178633	−	0.983916i	$-0.442833\pi$
$347$	2309.32	0.357265	0.178633	+	0.983916i	$0.442833\pi$
$348$	0	0
$349$	6230.80	0.955665	0.477833	−	0.878451i	$-0.341423\pi$
$349$	6230.80	0.955665	0.477833	+	0.878451i	$0.341423\pi$
$350$	0	0
$351$	0	0
$352$	1651.08	0.250009
$353$	5599.25	0.844243	0.422122	−	0.906539i	$-0.361285\pi$
$353$	5599.25	0.844243	0.422122	+	0.906539i	$0.361285\pi$
$354$	0	0
$355$	−2079.16	−0.310846
$356$	1523.26	0.226777
$357$	0	0
$358$	−12955.5	−1.91263
$359$	4973.12	0.731118	0.365559	−	0.930788i	$-0.380878\pi$
$359$	4973.12	0.731118	0.365559	+	0.930788i	$0.380878\pi$
$360$	0	0
$361$	−5334.25	−0.777701
$362$	13314.6	1.93315
$363$	0	0
$364$	0	0
$365$	−2542.88	−0.364658
$366$	0	0
$367$	1894.42	0.269449	0.134725	−	0.990883i	$-0.456985\pi$
$367$	1894.42	0.269449	0.134725	+	0.990883i	$0.456985\pi$
$368$	743.364	0.105300
$369$	0	0
$370$	2387.68	0.335485
$371$	0	0
$372$	0	0
$373$	−3437.39	−0.477162	−0.238581	−	0.971123i	$-0.576682\pi$
$373$	−3437.39	−0.477162	−0.238581	+	0.971123i	$0.576682\pi$
$374$	−1252.29	−0.173140
$375$	0	0
$376$	2482.97	0.340557
$377$	10583.4	1.44582
$378$	0	0
$379$	8160.86	1.10606	0.553028	−	0.833163i	$-0.313472\pi$
$379$	8160.86	1.10606	0.553028	+	0.833163i	$0.313472\pi$
$380$	1307.81	0.176550
$381$	0	0
$382$	−9420.57	−1.26178
$383$	1002.83	0.133792	0.0668958	−	0.997760i	$-0.478691\pi$
$383$	1002.83	0.133792	0.0668958	+	0.997760i	$0.478691\pi$
$384$	0	0
$385$	0	0
$386$	−62.6402	−0.00825985
$387$	0	0
$388$	−9682.71	−1.26692
$389$	−4240.23	−0.552669	−0.276334	−	0.961062i	$-0.589120\pi$
$389$	−4240.23	−0.552669	−0.276334	+	0.961062i	$0.589120\pi$
$390$	0	0
$391$	−483.083	−0.0624823
$392$	0	0
$393$	0	0
$394$	12365.9	1.58118
$395$	5229.54	0.666144
$396$	0	0
$397$	14094.6	1.78183	0.890914	−	0.454173i	$-0.150065\pi$
$397$	14094.6	1.78183	0.890914	+	0.454173i	$0.150065\pi$
$398$	1033.79	0.130199
$399$	0	0
$400$	−1817.96	−0.227245
$401$	11501.8	1.43235	0.716174	−	0.697922i	$-0.245892\pi$
$401$	11501.8	1.43235	0.716174	+	0.697922i	$0.245892\pi$
$402$	0	0
$403$	−6030.35	−0.745393
$404$	5578.92	0.687033
$405$	0	0
$406$	0	0
$407$	860.942	0.104853
$408$	0	0
$409$	11433.0	1.38221	0.691106	−	0.722754i	$-0.257124\pi$
$409$	11433.0	1.38221	0.691106	+	0.722754i	$0.257124\pi$
$410$	4299.89	0.517943
$411$	0	0
$412$	−3705.93	−0.443151
$413$	0	0
$414$	0	0
$415$	245.825	0.0290773
$416$	12105.9	1.42678
$417$	0	0
$418$	1034.76	0.121080
$419$	9735.78	1.13514	0.567571	−	0.823325i	$-0.307883\pi$
$419$	9735.78	1.13514	0.567571	+	0.823325i	$0.307883\pi$
$420$	0	0
$421$	14881.6	1.72277	0.861386	−	0.507952i	$-0.169597\pi$
$421$	14881.6	1.72277	0.861386	+	0.507952i	$0.169597\pi$
$422$	−18434.6	−2.12650
$423$	0	0
$424$	−1914.86	−0.219325
$425$	1181.42	0.134841
$426$	0	0
$427$	0	0
$428$	−4137.18	−0.467239
$429$	0	0
$430$	4063.51	0.455721
$431$	4569.39	0.510672	0.255336	−	0.966852i	$-0.417814\pi$
$431$	4569.39	0.510672	0.255336	+	0.966852i	$0.417814\pi$
$432$	0	0
$433$	15362.9	1.70507	0.852534	−	0.522672i	$-0.175065\pi$
$433$	15362.9	1.70507	0.852534	+	0.522672i	$0.175065\pi$
$434$	0	0
$435$	0	0
$436$	4986.01	0.547676
$437$	399.169	0.0436953
$438$	0	0
$439$	7027.32	0.763999	0.381999	−	0.924163i	$-0.375236\pi$
$439$	7027.32	0.763999	0.381999	+	0.924163i	$0.375236\pi$
$440$	−172.452	−0.0186848
$441$	0	0
$442$	−9181.90	−0.988096
$443$	−9699.65	−1.04028	−0.520140	−	0.854081i	$-0.674120\pi$
$443$	−9699.65	−1.04028	−0.520140	+	0.854081i	$0.674120\pi$
$444$	0	0
$445$	−1137.02	−0.121124
$446$	−827.903	−0.0878976
$447$	0	0
$448$	0	0
$449$	−6017.85	−0.632516	−0.316258	−	0.948673i	$-0.602427\pi$
$449$	−6017.85	−0.632516	−0.316258	+	0.948673i	$0.602427\pi$
$450$	0	0
$451$	1550.44	0.161879
$452$	3095.80	0.322156
$453$	0	0
$454$	18347.2	1.89665
$455$	0	0
$456$	0	0
$457$	5845.64	0.598354	0.299177	−	0.954198i	$-0.403288\pi$
$457$	5845.64	0.598354	0.299177	+	0.954198i	$0.403288\pi$
$458$	−478.333	−0.0488014
$459$	0	0
$460$	342.375	0.0347029
$461$	−10489.1	−1.05971	−0.529856	−	0.848088i	$-0.677754\pi$
$461$	−10489.1	−1.05971	−0.529856	+	0.848088i	$0.677754\pi$
$462$	0	0
$463$	−19322.0	−1.93946	−0.969732	−	0.244172i	$-0.921484\pi$
$463$	−19322.0	−1.93946	−0.969732	+	0.244172i	$0.921484\pi$
$464$	−15185.8	−1.51936
$465$	0	0
$466$	6249.24	0.621224
$467$	9694.28	0.960594	0.480297	−	0.877106i	$-0.340529\pi$
$467$	9694.28	0.960594	0.480297	+	0.877106i	$0.340529\pi$
$468$	0	0
$469$	0	0
$470$	9538.56	0.936130
$471$	0	0
$472$	411.964	0.0401741
$473$	1465.21	0.142432
$474$	0	0
$475$	−976.201	−0.0942972
$476$	0	0
$477$	0	0
$478$	12969.9	1.24106
$479$	−86.9716	−0.00829611	−0.00414805	−	0.999991i	$-0.501320\pi$
$479$	−86.9716	−0.00829611	−0.00414805	+	0.999991i	$0.501320\pi$
$480$	0	0
$481$	6312.51	0.598390
$482$	9466.42	0.894572
$483$	0	0
$484$	−8595.62	−0.807252
$485$	7227.56	0.676674
$486$	0	0
$487$	−5395.22	−0.502014	−0.251007	−	0.967985i	$-0.580762\pi$
$487$	−5395.22	−0.502014	−0.251007	+	0.967985i	$0.580762\pi$
$488$	−1517.11	−0.140731
$489$	0	0
$490$	0	0
$491$	19386.7	1.78190	0.890948	−	0.454105i	$-0.150041\pi$
$491$	19386.7	1.78190	0.890948	+	0.454105i	$0.150041\pi$
$492$	0	0
$493$	9868.65	0.901545
$494$	7586.95	0.690998
$495$	0	0
$496$	8652.76	0.783307
$497$	0	0
$498$	0	0
$499$	632.060	0.0567032	0.0283516	−	0.999598i	$-0.490974\pi$
$499$	632.060	0.0567032	0.0283516	+	0.999598i	$0.490974\pi$
$500$	−837.307	−0.0748910
$501$	0	0
$502$	−8971.80	−0.797671
$503$	17621.1	1.56200	0.781001	−	0.624530i	$-0.214709\pi$
$503$	17621.1	1.56200	0.781001	+	0.624530i	$0.214709\pi$
$504$	0	0
$505$	−4164.33	−0.366951
$506$	270.893	0.0237997
$507$	0	0
$508$	12773.1	1.11558
$509$	12494.0	1.08799	0.543997	−	0.839087i	$-0.316910\pi$
$509$	12494.0	1.08799	0.543997	+	0.839087i	$0.316910\pi$
$510$	0	0
$511$	0	0
$512$	−14467.5	−1.24879
$513$	0	0
$514$	−17760.8	−1.52412
$515$	2766.26	0.236691
$516$	0	0
$517$	3439.38	0.292580
$518$	0	0
$519$	0	0
$520$	−1264.43	−0.106633
$521$	18223.4	1.53240	0.766201	−	0.642601i	$-0.222145\pi$
$521$	18223.4	1.53240	0.766201	+	0.642601i	$0.222145\pi$
$522$	0	0
$523$	−4912.90	−0.410757	−0.205379	−	0.978683i	$-0.565843\pi$
$523$	−4912.90	−0.410757	−0.205379	+	0.978683i	$0.565843\pi$
$524$	8363.23	0.697232
$525$	0	0
$526$	−9502.00	−0.787656
$527$	−5623.09	−0.464793
$528$	0	0
$529$	−12062.5	−0.991411
$530$	−7356.12	−0.602886
$531$	0	0
$532$	0	0
$533$	11368.0	0.923832
$534$	0	0
$535$	3088.16	0.249557
$536$	1282.33	0.103336
$537$	0	0
$538$	6318.72	0.506356
$539$	0	0
$540$	0	0
$541$	17891.3	1.42182	0.710911	−	0.703282i	$-0.248283\pi$
$541$	17891.3	1.42182	0.710911	+	0.703282i	$0.248283\pi$
$542$	−19375.7	−1.53553
$543$	0	0
$544$	11288.3	0.889676
$545$	−3721.76	−0.292519
$546$	0	0
$547$	12538.8	0.980112	0.490056	−	0.871691i	$-0.336976\pi$
$547$	12538.8	0.980112	0.490056	+	0.871691i	$0.336976\pi$
$548$	−6513.53	−0.507745
$549$	0	0
$550$	−662.490	−0.0513612
$551$	−8154.41	−0.630471
$552$	0	0
$553$	0	0
$554$	6122.41	0.469524
$555$	0	0
$556$	−11679.1	−0.890837
$557$	−23865.4	−1.81546	−0.907729	−	0.419557i	$-0.862185\pi$
$557$	−23865.4	−1.81546	−0.907729	+	0.419557i	$0.862185\pi$
$558$	0	0
$559$	10743.1	0.812849
$560$	0	0
$561$	0	0
$562$	21203.9	1.59152
$563$	−2946.76	−0.220588	−0.110294	−	0.993899i	$-0.535179\pi$
$563$	−2946.76	−0.220588	−0.110294	+	0.993899i	$0.535179\pi$
$564$	0	0
$565$	−2310.83	−0.172066
$566$	−24359.6	−1.80903
$567$	0	0
$568$	2074.97	0.153281
$569$	−17425.2	−1.28384	−0.641919	−	0.766773i	$-0.721861\pi$
$569$	−17425.2	−1.28384	−0.641919	+	0.766773i	$0.721861\pi$
$570$	0	0
$571$	−12642.8	−0.926595	−0.463298	−	0.886203i	$-0.653334\pi$
$571$	−12642.8	−0.926595	−0.463298	+	0.886203i	$0.653334\pi$
$572$	2346.45	0.171521
$573$	0	0
$574$	0	0
$575$	−255.563	−0.0185351
$576$	0	0
$577$	25122.4	1.81258	0.906292	−	0.422652i	$-0.138901\pi$
$577$	25122.4	1.81258	0.906292	+	0.422652i	$0.138901\pi$
$578$	10273.9	0.739342
$579$	0	0
$580$	−6994.20	−0.500721
$581$	0	0
$582$	0	0
$583$	−2652.45	−0.188427
$584$	2537.76	0.179817
$585$	0	0
$586$	20428.8	1.44011
$587$	−186.944	−0.0131448	−0.00657239	−	0.999978i	$-0.502092\pi$
$587$	−186.944	−0.0131448	−0.00657239	+	0.999978i	$0.502092\pi$
$588$	0	0
$589$	4646.32	0.325040
$590$	1582.60	0.110432
$591$	0	0
$592$	−9057.62	−0.628827
$593$	22217.8	1.53857	0.769287	−	0.638903i	$-0.220612\pi$
$593$	22217.8	1.53857	0.769287	+	0.638903i	$0.220612\pi$
$594$	0	0
$595$	0	0
$596$	−4101.65	−0.281896
$597$	0	0
$598$	1986.21	0.135823
$599$	−6776.55	−0.462241	−0.231121	−	0.972925i	$-0.574239\pi$
$599$	−6776.55	−0.462241	−0.231121	+	0.972925i	$0.574239\pi$
$600$	0	0
$601$	10022.2	0.680225	0.340112	−	0.940385i	$-0.389535\pi$
$601$	10022.2	0.680225	0.340112	+	0.940385i	$0.389535\pi$
$602$	0	0
$603$	0	0
$604$	128.406	0.00865027
$605$	6416.12	0.431161
$606$	0	0
$607$	21953.3	1.46797	0.733984	−	0.679166i	$-0.237658\pi$
$607$	21953.3	1.46797	0.733984	+	0.679166i	$0.237658\pi$
$608$	−9327.49	−0.622170
$609$	0	0
$610$	−5828.14	−0.386843
$611$	25217.9	1.66973
$612$	0	0
$613$	−7593.83	−0.500346	−0.250173	−	0.968201i	$-0.580487\pi$
$613$	−7593.83	−0.500346	−0.250173	+	0.968201i	$0.580487\pi$
$614$	−6471.14	−0.425332
$615$	0	0
$616$	0	0
$617$	1728.07	0.112754	0.0563772	−	0.998410i	$-0.482045\pi$
$617$	1728.07	0.112754	0.0563772	+	0.998410i	$0.482045\pi$
$618$	0	0
$619$	−13883.7	−0.901506	−0.450753	−	0.892649i	$-0.648845\pi$
$619$	−13883.7	−0.901506	−0.450753	+	0.892649i	$0.648845\pi$
$620$	3985.25	0.258147
$621$	0	0
$622$	−37653.8	−2.42730
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	37087.4	2.36791
$627$	0	0
$628$	4000.52	0.254201
$629$	5886.20	0.373129
$630$	0	0
$631$	9664.38	0.609719	0.304860	−	0.952397i	$-0.401390\pi$
$631$	9664.38	0.609719	0.304860	+	0.952397i	$0.401390\pi$
$632$	−5219.01	−0.328483
$633$	0	0
$634$	25696.6	1.60969
$635$	−9534.37	−0.595842
$636$	0	0
$637$	0	0
$638$	−5533.92	−0.343401
$639$	0	0
$640$	3151.29	0.194634
$641$	26764.8	1.64921	0.824607	−	0.565706i	$-0.191396\pi$
$641$	26764.8	1.64921	0.824607	+	0.565706i	$0.191396\pi$
$642$	0	0
$643$	27294.1	1.67399	0.836993	−	0.547213i	$-0.184311\pi$
$643$	27294.1	1.67399	0.836993	+	0.547213i	$0.184311\pi$
$644$	0	0
$645$	0	0
$646$	7074.56	0.430874
$647$	−26652.5	−1.61950	−0.809750	−	0.586774i	$-0.800398\pi$
$647$	−26652.5	−1.61950	−0.809750	+	0.586774i	$0.800398\pi$
$648$	0	0
$649$	570.649	0.0345145
$650$	−4857.45	−0.293115
$651$	0	0
$652$	−25982.7	−1.56067
$653$	24985.9	1.49736	0.748679	−	0.662933i	$-0.230688\pi$
$653$	24985.9	1.49736	0.748679	+	0.662933i	$0.230688\pi$
$654$	0	0
$655$	−6242.65	−0.372398
$656$	−16311.6	−0.970823
$657$	0	0
$658$	0	0
$659$	−13634.4	−0.805952	−0.402976	−	0.915211i	$-0.632024\pi$
$659$	−13634.4	−0.805952	−0.402976	+	0.915211i	$0.632024\pi$
$660$	0	0
$661$	6338.24	0.372964	0.186482	−	0.982458i	$-0.440291\pi$
$661$	6338.24	0.372964	0.186482	+	0.982458i	$0.440291\pi$
$662$	−34430.5	−2.02142
$663$	0	0
$664$	−245.330	−0.0143383
$665$	0	0
$666$	0	0
$667$	−2134.77	−0.123926
$668$	8624.81	0.499557
$669$	0	0
$670$	4926.19	0.284053
$671$	−2101.49	−0.120905
$672$	0	0
$673$	20551.3	1.17711	0.588555	−	0.808457i	$-0.299697\pi$
$673$	20551.3	1.17711	0.588555	+	0.808457i	$0.299697\pi$
$674$	13107.5	0.749083
$675$	0	0
$676$	2487.87	0.141549
$677$	7368.92	0.418332	0.209166	−	0.977880i	$-0.432925\pi$
$677$	7368.92	0.418332	0.209166	+	0.977880i	$0.432925\pi$
$678$	0	0
$679$	0	0
$680$	−1179.04	−0.0664914
$681$	0	0
$682$	3153.19	0.177041
$683$	−17858.4	−1.00049	−0.500244	−	0.865885i	$-0.666756\pi$
$683$	−17858.4	−1.00049	−0.500244	+	0.865885i	$0.666756\pi$
$684$	0	0
$685$	4861.96	0.271191
$686$	0	0
$687$	0	0
$688$	−15414.9	−0.854195
$689$	−19448.0	−1.07534
$690$	0	0
$691$	−9915.06	−0.545856	−0.272928	−	0.962034i	$-0.587992\pi$
$691$	−9915.06	−0.545856	−0.272928	+	0.962034i	$0.587992\pi$
$692$	4594.06	0.252370
$693$	0	0
$694$	−8853.61	−0.484263
$695$	8717.77	0.475804
$696$	0	0
$697$	10600.3	0.576059
$698$	−23888.0	−1.29538
$699$	0	0
$700$	0	0
$701$	−19140.4	−1.03127	−0.515636	−	0.856808i	$-0.672444\pi$
$701$	−19140.4	−1.03127	−0.515636	+	0.856808i	$0.672444\pi$
$702$	0	0
$703$	−4863.73	−0.260937
$704$	−2308.99	−0.123613
$705$	0	0
$706$	−21466.7	−1.14435
$707$	0	0
$708$	0	0
$709$	14674.2	0.777292	0.388646	−	0.921387i	$-0.372943\pi$
$709$	14674.2	0.777292	0.388646	+	0.921387i	$0.372943\pi$
$710$	7971.19	0.421343
$711$	0	0
$712$	1134.73	0.0597274
$713$	1216.38	0.0638901
$714$	0	0
$715$	−1751.48	−0.0916108
$716$	22635.7	1.18147
$717$	0	0
$718$	−19066.2	−0.991010
$719$	12916.1	0.669944	0.334972	−	0.942228i	$-0.391273\pi$
$719$	12916.1	0.669944	0.334972	+	0.942228i	$0.391273\pi$
$720$	0	0
$721$	0	0
$722$	20450.8	1.05415
$723$	0	0
$724$	−23263.1	−1.19415
$725$	5220.76	0.267440
$726$	0	0
$727$	3413.11	0.174120	0.0870599	−	0.996203i	$-0.472253\pi$
$727$	3413.11	0.174120	0.0870599	+	0.996203i	$0.472253\pi$
$728$	0	0
$729$	0	0
$730$	9749.03	0.494285
$731$	10017.5	0.506856
$732$	0	0
$733$	5637.02	0.284049	0.142025	−	0.989863i	$-0.454639\pi$
$733$	5637.02	0.284049	0.142025	+	0.989863i	$0.454639\pi$
$734$	−7262.93	−0.365231
$735$	0	0
$736$	−2441.87	−0.122294
$737$	1776.27	0.0887785
$738$	0	0
$739$	20017.8	0.996435	0.498218	−	0.867052i	$-0.333988\pi$
$739$	20017.8	0.996435	0.498218	+	0.867052i	$0.333988\pi$
$740$	−4171.72	−0.207237
$741$	0	0
$742$	0	0
$743$	−24349.1	−1.20226	−0.601131	−	0.799151i	$-0.705283\pi$
$743$	−24349.1	−1.20226	−0.601131	+	0.799151i	$0.705283\pi$
$744$	0	0
$745$	3061.64	0.150563
$746$	13178.5	0.646780
$747$	0	0
$748$	2187.98	0.106952
$749$	0	0
$750$	0	0
$751$	−36084.4	−1.75331	−0.876656	−	0.481118i	$-0.840231\pi$
$751$	−36084.4	−1.75331	−0.876656	+	0.481118i	$0.840231\pi$
$752$	−36184.4	−1.75466
$753$	0	0
$754$	−40575.3	−1.95977
$755$	−95.8474	−0.00462019
$756$	0	0
$757$	7500.30	0.360110	0.180055	−	0.983657i	$-0.442372\pi$
$757$	7500.30	0.360110	0.180055	+	0.983657i	$0.442372\pi$
$758$	−31287.6	−1.49923
$759$	0	0
$760$	974.234	0.0464989
$761$	−34893.7	−1.66215	−0.831076	−	0.556159i	$-0.812274\pi$
$761$	−34893.7	−1.66215	−0.831076	+	0.556159i	$0.812274\pi$
$762$	0	0
$763$	0	0
$764$	16459.5	0.779428
$765$	0	0
$766$	−3844.70	−0.181351
$767$	4184.05	0.196972
$768$	0	0
$769$	12125.8	0.568617	0.284309	−	0.958733i	$-0.408236\pi$
$769$	12125.8	0.568617	0.284309	+	0.958733i	$0.408236\pi$
$770$	0	0
$771$	0	0
$772$	109.444	0.00510230
$773$	−26707.7	−1.24270	−0.621352	−	0.783531i	$-0.713416\pi$
$773$	−26707.7	−1.24270	−0.621352	+	0.783531i	$0.713416\pi$
$774$	0	0
$775$	−2974.75	−0.137879
$776$	−7213.01	−0.333675
$777$	0	0
$778$	16256.4	0.749127
$779$	−8758.93	−0.402851
$780$	0	0
$781$	2874.23	0.131687
$782$	1852.07	0.0846931
$783$	0	0
$784$	0	0
$785$	−2986.15	−0.135771
$786$	0	0
$787$	10852.4	0.491546	0.245773	−	0.969327i	$-0.420958\pi$
$787$	10852.4	0.491546	0.245773	+	0.969327i	$0.420958\pi$
$788$	−21605.5	−0.976731
$789$	0	0
$790$	−20049.3	−0.902940
$791$	0	0
$792$	0	0
$793$	−15408.3	−0.689995
$794$	−54036.5	−2.41522
$795$	0	0
$796$	−1806.22	−0.0804269
$797$	22975.8	1.02113	0.510567	−	0.859838i	$-0.329436\pi$
$797$	22975.8	1.02113	0.510567	+	0.859838i	$0.329436\pi$
$798$	0	0
$799$	23514.8	1.04117
$800$	5971.81	0.263919
$801$	0	0
$802$	−44096.2	−1.94151
$803$	3515.27	0.154485
$804$	0	0
$805$	0	0
$806$	23119.5	1.01036
$807$	0	0
$808$	4155.94	0.180947
$809$	−23370.8	−1.01567	−0.507834	−	0.861455i	$-0.669554\pi$
$809$	−23370.8	−1.01567	−0.507834	+	0.861455i	$0.669554\pi$
$810$	0	0
$811$	7794.74	0.337497	0.168749	−	0.985659i	$-0.446027\pi$
$811$	7794.74	0.337497	0.168749	+	0.985659i	$0.446027\pi$
$812$	0	0
$813$	0	0
$814$	−3300.73	−0.142126
$815$	19394.5	0.833571
$816$	0	0
$817$	−8277.42	−0.354456
$818$	−43832.4	−1.87355
$819$	0	0
$820$	−7512.71	−0.319945
$821$	33309.4	1.41597	0.707983	−	0.706230i	$-0.249605\pi$
$821$	33309.4	1.41597	0.707983	+	0.706230i	$0.249605\pi$
$822$	0	0
$823$	16372.1	0.693432	0.346716	−	0.937970i	$-0.387297\pi$
$823$	16372.1	0.693432	0.346716	+	0.937970i	$0.387297\pi$
$824$	−2760.69	−0.116715
$825$	0	0
$826$	0	0
$827$	−23585.7	−0.991724	−0.495862	−	0.868401i	$-0.665148\pi$
$827$	−23585.7	−0.991724	−0.495862	+	0.868401i	$0.665148\pi$
$828$	0	0
$829$	−613.688	−0.0257108	−0.0128554	−	0.999917i	$-0.504092\pi$
$829$	−613.688	−0.0257108	−0.0128554	+	0.999917i	$0.504092\pi$
$830$	−942.458	−0.0394135
$831$	0	0
$832$	−16929.7	−0.705449
$833$	0	0
$834$	0	0
$835$	−6437.91	−0.266818
$836$	−1807.91	−0.0747942
$837$	0	0
$838$	−37325.6	−1.53865
$839$	841.107	0.0346105	0.0173053	−	0.999850i	$-0.494491\pi$
$839$	841.107	0.0346105	0.0173053	+	0.999850i	$0.494491\pi$
$840$	0	0
$841$	19221.1	0.788104
$842$	−57054.1	−2.33517
$843$	0	0
$844$	32208.7	1.31359
$845$	−1857.04	−0.0756027
$846$	0	0
$847$	0	0
$848$	27905.3	1.13004
$849$	0	0
$850$	−4529.40	−0.182773
$851$	−1273.29	−0.0512901
$852$	0	0
$853$	−33849.9	−1.35873	−0.679367	−	0.733799i	$-0.737746\pi$
$853$	−33849.9	−1.35873	−0.679367	+	0.733799i	$0.737746\pi$
$854$	0	0
$855$	0	0
$856$	−3081.94	−0.123059
$857$	−29964.6	−1.19436	−0.597182	−	0.802106i	$-0.703713\pi$
$857$	−29964.6	−1.19436	−0.597182	+	0.802106i	$0.703713\pi$
$858$	0	0
$859$	−3904.40	−0.155083	−0.0775415	−	0.996989i	$-0.524707\pi$
$859$	−3904.40	−0.155083	−0.0775415	+	0.996989i	$0.524707\pi$
$860$	−7099.71	−0.281509
$861$	0	0
$862$	−17518.4	−0.692202
$863$	31103.5	1.22685	0.613426	−	0.789752i	$-0.289791\pi$
$863$	31103.5	1.22685	0.613426	+	0.789752i	$0.289791\pi$
$864$	0	0
$865$	−3429.19	−0.134793
$866$	−58899.2	−2.31117
$867$	0	0
$868$	0	0
$869$	−7229.32	−0.282207
$870$	0	0
$871$	13023.8	0.506653
$872$	3714.27	0.144244
$873$	0	0
$874$	−1530.36	−0.0592278
$875$	0	0
$876$	0	0
$877$	−47325.0	−1.82218	−0.911089	−	0.412209i	$-0.864757\pi$
$877$	−47325.0	−1.82218	−0.911089	+	0.412209i	$0.864757\pi$
$878$	−26941.7	−1.03558
$879$	0	0
$880$	2513.15	0.0962706
$881$	−1181.40	−0.0451786	−0.0225893	−	0.999745i	$-0.507191\pi$
$881$	−1181.40	−0.0451786	−0.0225893	+	0.999745i	$0.507191\pi$
$882$	0	0
$883$	35232.8	1.34278	0.671392	−	0.741103i	$-0.265697\pi$
$883$	35232.8	1.34278	0.671392	+	0.741103i	$0.265697\pi$
$884$	16042.5	0.610370
$885$	0	0
$886$	37187.1	1.41007
$887$	21861.5	0.827552	0.413776	−	0.910379i	$-0.364210\pi$
$887$	21861.5	0.827552	0.413776	+	0.910379i	$0.364210\pi$
$888$	0	0
$889$	0	0
$890$	4359.18	0.164180
$891$	0	0
$892$	1446.50	0.0542964
$893$	−19430.1	−0.728113
$894$	0	0
$895$	−16896.2	−0.631036
$896$	0	0
$897$	0	0
$898$	23071.6	0.857358
$899$	−24848.7	−0.921859
$900$	0	0
$901$	−18134.6	−0.670534
$902$	−5944.17	−0.219423
$903$	0	0
$904$	2306.18	0.0848478
$905$	17364.5	0.637808
$906$	0	0
$907$	−33763.9	−1.23607	−0.618034	−	0.786152i	$-0.712071\pi$
$907$	−33763.9	−1.23607	−0.618034	+	0.786152i	$0.712071\pi$
$908$	−32056.0	−1.17160
$909$	0	0
$910$	0	0
$911$	29395.3	1.06906	0.534529	−	0.845150i	$-0.320489\pi$
$911$	29395.3	1.06906	0.534529	+	0.845150i	$0.320489\pi$
$912$	0	0
$913$	−339.829	−0.0123184
$914$	−22411.4	−0.811052
$915$	0	0
$916$	835.737	0.0301458
$917$	0	0
$918$	0	0
$919$	−39532.9	−1.41901	−0.709504	−	0.704701i	$-0.751081\pi$
$919$	−39532.9	−1.41901	−0.709504	+	0.704701i	$0.751081\pi$
$920$	255.048	0.00913987
$921$	0	0
$922$	40213.8	1.43641
$923$	21074.1	0.751531
$924$	0	0
$925$	3113.94	0.110687
$926$	74078.0	2.62889
$927$	0	0
$928$	49883.7	1.76456
$929$	−41709.3	−1.47302	−0.736511	−	0.676426i	$-0.763528\pi$
$929$	−41709.3	−1.47302	−0.736511	+	0.676426i	$0.763528\pi$
$930$	0	0
$931$	0	0
$932$	−10918.6	−0.383745
$933$	0	0
$934$	−37166.5	−1.30206
$935$	−1633.20	−0.0571243
$936$	0	0
$937$	−50786.4	−1.77067	−0.885336	−	0.464951i	$-0.846072\pi$
$937$	−50786.4	−1.77067	−0.885336	+	0.464951i	$0.846072\pi$
$938$	0	0
$939$	0	0
$940$	−16665.6	−0.578269
$941$	39550.7	1.37015	0.685077	−	0.728471i	$-0.259769\pi$
$941$	39550.7	1.37015	0.685077	+	0.728471i	$0.259769\pi$
$942$	0	0
$943$	−2293.03	−0.0791848
$944$	−6003.56	−0.206991
$945$	0	0
$946$	−5617.40	−0.193063
$947$	−38334.1	−1.31541	−0.657704	−	0.753276i	$-0.728472\pi$
$947$	−38334.1	−1.31541	−0.657704	+	0.753276i	$0.728472\pi$
$948$	0	0
$949$	25774.3	0.881634
$950$	3742.61	0.127817
$951$	0	0
$952$	0	0
$953$	−8101.55	−0.275378	−0.137689	−	0.990476i	$-0.543967\pi$
$953$	−8101.55	−0.275378	−0.137689	+	0.990476i	$0.543967\pi$
$954$	0	0
$955$	−12286.0	−0.416300
$956$	−22660.8	−0.766633
$957$	0	0
$958$	333.437	0.0112451
$959$	0	0
$960$	0	0
$961$	−15632.4	−0.524735
$962$	−24201.3	−0.811102
$963$	0	0
$964$	−16539.6	−0.552598
$965$	−81.6935	−0.00272519
$966$	0	0
$967$	−20290.6	−0.674769	−0.337384	−	0.941367i	$-0.609542\pi$
$967$	−20290.6	−0.674769	−0.337384	+	0.941367i	$0.609542\pi$
$968$	−6403.20	−0.212610
$969$	0	0
$970$	−27709.4	−0.917213
$971$	15756.5	0.520754	0.260377	−	0.965507i	$-0.416153\pi$
$971$	15756.5	0.520754	0.260377	+	0.965507i	$0.416153\pi$
$972$	0	0
$973$	0	0
$974$	20684.5	0.680466
$975$	0	0
$976$	22108.9	0.725092
$977$	−30984.3	−1.01461	−0.507306	−	0.861766i	$-0.669358\pi$
$977$	−30984.3	−1.01461	−0.507306	+	0.861766i	$0.669358\pi$
$978$	0	0
$979$	1571.82	0.0513132
$980$	0	0
$981$	0	0
$982$	−74326.0	−2.41531
$983$	6365.00	0.206523	0.103261	−	0.994654i	$-0.467072\pi$
$983$	6365.00	0.206523	0.103261	+	0.994654i	$0.467072\pi$
$984$	0	0
$985$	16127.2	0.521681
$986$	−37835.0	−1.22202
$987$	0	0
$988$	−13255.8	−0.426845
$989$	−2166.97	−0.0696721
$990$	0	0
$991$	−19976.6	−0.640341	−0.320170	−	0.947360i	$-0.603740\pi$
$991$	−19976.6	−0.640341	−0.320170	+	0.947360i	$0.603740\pi$
$992$	−28423.4	−0.909722
$993$	0	0
$994$	0	0
$995$	1348.24	0.0429568
$996$	0	0
$997$	−2500.19	−0.0794200	−0.0397100	−	0.999211i	$-0.512643\pi$
$997$	−2500.19	−0.0794200	−0.0397100	+	0.999211i	$0.512643\pi$
$998$	−2423.23	−0.0768597
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.ch.1.3	✓	12
3.2	odd	2		2205.4.a.ci.1.10	yes	12
7.6	odd	2		2205.4.a.ci.1.3	yes	12
21.20	even	2	inner	2205.4.a.ch.1.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.4.a.ch.1.3	✓	12	1.1	even	1	trivial
2205.4.a.ch.1.10	yes	12	21.20	even	2	inner
2205.4.a.ci.1.3	yes	12	7.6	odd	2
2205.4.a.ci.1.10	yes	12	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.83386 q^{2} +6.69846 q^{4} -5.00000 q^{5} +4.98993 q^{8} +19.1693 q^{10} +6.91200 q^{11} +50.6795 q^{13} -72.7183 q^{16} +47.2568 q^{17} -39.0480 q^{19} -33.4923 q^{20} -26.4996 q^{22} -10.2225 q^{23} +25.0000 q^{25} -194.298 q^{26} +208.830 q^{29} -118.990 q^{31} +238.872 q^{32} -181.176 q^{34} +124.558 q^{37} +149.705 q^{38} -24.9496 q^{40} +224.312 q^{41} +211.980 q^{43} +46.2997 q^{44} +39.1916 q^{46} +497.596 q^{47} -95.8464 q^{50} +339.474 q^{52} -383.745 q^{53} -34.5600 q^{55} -800.625 q^{58} +82.5591 q^{59} -304.035 q^{61} +456.191 q^{62} -334.055 q^{64} -253.397 q^{65} +256.984 q^{67} +316.548 q^{68} +415.831 q^{71} +508.576 q^{73} -477.536 q^{74} -261.562 q^{76} -1045.91 q^{79} +363.592 q^{80} -859.979 q^{82} -49.1650 q^{83} -236.284 q^{85} -812.703 q^{86} +34.4904 q^{88} +227.405 q^{89} -68.4750 q^{92} -1907.71 q^{94} +195.240 q^{95} -1445.51 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 28 q^{4} - 60 q^{5} - 160 q^{16} + 136 q^{17} - 140 q^{20} - 400 q^{22} + 300 q^{25} + 256 q^{26} - 280 q^{37} + 436 q^{38} + 376 q^{41} - 1856 q^{43} + 196 q^{46} + 968 q^{47} - 352 q^{58} + 3288 q^{59}+ \cdots + 4040 q^{89}+O(q^{100}) \) 12 * q + 28 * q^4 - 60 * q^5 - 160 * q^16 + 136 * q^17 - 140 * q^20 - 400 * q^22 + 300 * q^25 + 256 * q^26 - 280 * q^37 + 436 * q^38 + 376 * q^41 - 1856 * q^43 + 196 * q^46 + 968 * q^47 - 352 * q^58 + 3288 * q^59 + 1036 * q^62 - 2116 * q^64 - 1216 * q^67 + 2112 * q^68 - 472 * q^79 + 800 * q^80 + 4312 * q^83 - 680 * q^85 - 1224 * q^88 + 4040 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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