LMFDB - Embedded newform 585.4.b.e.181.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [585,4,Mod(181,585)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("585.181"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	585.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,-56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$34.5161173534$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 84x^{12} + 2674x^{10} + 40048x^{8} + 278769x^{6} + 727552x^{4} + 339456x^{2} + 9216 $ x^14 + 84x^12 + 2674x^10 + 40048x^8 + 278769x^6 + 727552x^4 + 339456x^2 + 9216
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.1
Root			$-5.37688i$ of defining polynomial
Character	$\chi$	$=$	585.181
Dual form			585.4.b.e.181.14

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.37688i q^{2} -20.9108 q^{4} -5.00000i q^{5} -0.201771i q^{7} +69.4197i q^{8} -26.8844 q^{10} -10.4218i q^{11} +(39.9303 + 24.5473i) q^{13} -1.08490 q^{14} +205.975 q^{16} +54.7775 q^{17} +129.099i q^{19} +104.554i q^{20} -56.0369 q^{22} +175.846 q^{23} -25.0000 q^{25} +(131.988 - 214.700i) q^{26} +4.21919i q^{28} -107.045 q^{29} +22.2028i q^{31} -552.144i q^{32} -294.532i q^{34} -1.00886 q^{35} +205.591i q^{37} +694.150 q^{38} +347.099 q^{40} -285.238i q^{41} +321.835 q^{43} +217.929i q^{44} -945.503i q^{46} +379.907i q^{47} +342.959 q^{49} +134.422i q^{50} +(-834.975 - 513.303i) q^{52} -506.192 q^{53} -52.1092 q^{55} +14.0069 q^{56} +575.566i q^{58} -678.186i q^{59} -186.634 q^{61} +119.382 q^{62} -1321.01 q^{64} +(122.736 - 199.652i) q^{65} -925.768i q^{67} -1145.44 q^{68} +5.42449i q^{70} +376.232i q^{71} +671.223i q^{73} +1105.44 q^{74} -2699.57i q^{76} -2.10282 q^{77} -593.491 q^{79} -1029.87i q^{80} -1533.69 q^{82} +425.750i q^{83} -273.888i q^{85} -1730.47i q^{86} +723.481 q^{88} -141.676i q^{89} +(4.95293 - 8.05678i) q^{91} -3677.08 q^{92} +2042.71 q^{94} +645.496 q^{95} -551.467i q^{97} -1844.05i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 56 q^{4} - 20 q^{10} - 4 q^{13} + 152 q^{14} + 280 q^{16} + 100 q^{17} + 648 q^{22} + 532 q^{23} - 350 q^{25} + 344 q^{26} - 588 q^{29} - 540 q^{35} - 148 q^{38} + 240 q^{40} + 728 q^{43} - 1302 q^{49}+ \cdots + 3120 q^{95}+O(q^{100}) $ 14 * q - 56 * q^4 - 20 * q^10 - 4 * q^13 + 152 * q^14 + 280 * q^16 + 100 * q^17 + 648 * q^22 + 532 * q^23 - 350 * q^25 + 344 * q^26 - 588 * q^29 - 540 * q^35 - 148 * q^38 + 240 * q^40 + 728 * q^43 - 1302 * q^49 - 528 * q^52 - 1040 * q^53 - 40 * q^55 - 512 * q^56 - 1460 * q^61 - 2964 * q^62 + 1296 * q^64 + 30 * q^65 - 1604 * q^68 + 5928 * q^74 + 3568 * q^77 - 1024 * q^79 - 6660 * q^82 + 1132 * q^88 + 916 * q^91 - 6044 * q^92 + 2656 * q^94 + 3120 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$496$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	5.37688i		−	1.90101i	−0.310705	−	0.950506i	$-0.600565\pi$
$2$		−	5.37688i		−	1.90101i	0.310705	−	0.950506i	$-0.399435\pi$
$3$		0			0
$3$		0			0
$4$	−20.9108			−2.61385
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$		−	0.201771i		−	0.0108946i	−0.999985	−	0.00544731i	$-0.998266\pi$
$7$		−	0.201771i		−	0.0108946i	0.999985	−	0.00544731i	$-0.00173394\pi$
$8$			69.4197i			3.06795i
$9$		0			0
$10$	−26.8844			−0.850159
$11$		−	10.4218i		−	0.285664i	−0.989747	−	0.142832i	$-0.954379\pi$
$11$		−	10.4218i		−	0.285664i	0.989747	−	0.142832i	$-0.0456208\pi$
$12$		0			0
$13$	39.9303	+	24.5473i	0.851898	+	0.523707i
$13$	39.9303	+	24.5473i	0.851898	+	0.523707i
$14$	−1.08490			−0.0207108
$15$		0			0
$16$	205.975			3.21836
$17$	54.7775			0.781500			0.390750	−	0.920497i	$-0.372216\pi$
$17$	54.7775			0.781500			0.390750	+	0.920497i	$0.372216\pi$
$18$		0			0
$19$			129.099i			1.55881i	0.626521	+	0.779405i	$0.284478\pi$
$19$			129.099i			1.55881i	−0.626521	+	0.779405i	$0.715522\pi$
$20$			104.554i			1.16895i
$21$		0			0
$22$	−56.0369			−0.543050
$23$	175.846			1.59419			0.797097	−	0.603852i	$-0.206368\pi$
$23$	175.846			1.59419			0.797097	+	0.603852i	$0.206368\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$	131.988	−	214.700i	0.995574	−	1.61947i
$27$		0			0
$28$			4.21919i			0.0284769i
$29$	−107.045			−0.685438			−0.342719	−	0.939438i	$-0.611348\pi$
$29$	−107.045			−0.685438			−0.342719	+	0.939438i	$0.611348\pi$
$30$		0			0
$31$			22.2028i			0.128637i	0.997929	+	0.0643184i	$0.0204873\pi$
$31$			22.2028i			0.128637i	−0.997929	+	0.0643184i	$0.979513\pi$
$32$		−	552.144i		−	3.05019i
$33$		0			0
$34$		−	294.532i		−	1.48564i
$35$	−1.00886			−0.00487222
$36$		0			0
$37$			205.591i			0.913486i	0.889599	+	0.456743i	$0.150984\pi$
$37$			205.591i			0.913486i	−0.889599	+	0.456743i	$0.849016\pi$
$38$	694.150			2.96332
$39$		0			0
$40$	347.099			1.37203
$41$		−	285.238i		−	1.08651i	−0.839569	−	0.543253i	$-0.817193\pi$
$41$		−	285.238i		−	1.08651i	0.839569	−	0.543253i	$-0.182807\pi$
$42$		0			0
$43$	321.835			1.14138			0.570690	−	0.821166i	$-0.306676\pi$
$43$	321.835			1.14138			0.570690	+	0.821166i	$0.306676\pi$
$44$			217.929i			0.746682i
$45$		0			0
$46$		−	945.503i		−	3.03058i
$47$			379.907i			1.17905i	0.807752	+	0.589523i	$0.200684\pi$
$47$			379.907i			1.17905i	−0.807752	+	0.589523i	$0.799316\pi$
$48$		0			0
$49$	342.959			0.999881
$50$			134.422i			0.380203i
$51$		0			0
$52$	−834.975	−	513.303i	−2.22673	−	1.36889i
$53$	−506.192			−1.31190			−0.655951	−	0.754804i	$-0.727732\pi$
$53$	−506.192			−1.31190			−0.655951	+	0.754804i	$0.727732\pi$
$54$		0			0
$55$	−52.1092			−0.127753
$56$	14.0069			0.0334241
$57$		0			0
$58$			575.566i			1.30303i
$59$		−	678.186i		−	1.49648i	−0.663428	−	0.748240i	$-0.730899\pi$
$59$		−	678.186i		−	1.49648i	0.663428	−	0.748240i	$-0.269101\pi$
$60$		0			0
$61$	−186.634			−0.391738			−0.195869	−	0.980630i	$-0.562753\pi$
$61$	−186.634			−0.391738			−0.195869	+	0.980630i	$0.562753\pi$
$62$	119.382			0.244540
$63$		0			0
$64$	−1321.01			−2.58009
$65$	122.736	−	199.652i	0.234209	−	0.380981i
$66$		0			0
$67$		−	925.768i		−	1.68807i	−0.536289	−	0.844034i	$-0.680174\pi$
$67$		−	925.768i		−	1.68807i	0.536289	−	0.844034i	$-0.319826\pi$
$68$	−1145.44			−2.04272
$69$		0			0
$70$			5.42449i			0.00926215i
$71$			376.232i			0.628881i	0.949277	+	0.314441i	$0.101817\pi$
$71$			376.232i			0.628881i	−0.949277	+	0.314441i	$0.898183\pi$
$72$		0			0
$73$			671.223i			1.07617i	0.842889	+	0.538087i	$0.180853\pi$
$73$			671.223i			1.07617i	−0.842889	+	0.538087i	$0.819147\pi$
$74$	1105.44			1.73655
$75$		0			0
$76$		−	2699.57i		−	4.07449i
$77$	−2.10282			−0.00311220
$78$		0			0
$79$	−593.491			−0.845227			−0.422613	−	0.906310i	$-0.638887\pi$
$79$	−593.491			−0.845227			−0.422613	+	0.906310i	$0.638887\pi$
$80$		−	1029.87i		−	1.43929i
$81$		0			0
$82$	−1533.69			−2.06546
$83$			425.750i			0.563037i	0.959556	+	0.281519i	$0.0908381\pi$
$83$			425.750i			0.563037i	−0.959556	+	0.281519i	$0.909162\pi$
$84$		0			0
$85$		−	273.888i		−	0.349498i
$86$		−	1730.47i		−	2.16978i
$87$		0			0
$88$	723.481			0.876401
$89$		−	141.676i		−	0.168737i	−0.996435	−	0.0843685i	$-0.973113\pi$
$89$		−	141.676i		−	0.168737i	0.996435	−	0.0843685i	$-0.0268873\pi$
$90$		0			0
$91$	4.95293	−	8.05678i	0.00570559	−	0.00928110i
$92$	−3677.08			−4.16698
$93$		0			0
$94$	2042.71			2.24138
$95$	645.496			0.697121
$96$		0			0
$97$		−	551.467i		−	0.577247i	−0.957443	−	0.288623i	$-0.906802\pi$
$97$		−	551.467i		−	0.577247i	0.957443	−	0.288623i	$-0.0931976\pi$
$98$		−	1844.05i		−	1.90079i
$99$		0			0
$100$	522.770			0.522770
$101$	1386.39			1.36585			0.682924	−	0.730489i	$-0.260708\pi$
$101$	1386.39			1.36585			0.682924	+	0.730489i	$0.260708\pi$
$102$		0			0
$103$	1156.61			1.10645			0.553223	−	0.833033i	$-0.313398\pi$
$103$	1156.61			1.10645			0.553223	+	0.833033i	$0.313398\pi$
$104$	−1704.07	+	2771.95i	−1.60671	+	2.61358i
$105$		0			0
$106$			2721.73i			2.49394i
$107$	1145.31			1.03478			0.517390	−	0.855750i	$-0.326904\pi$
$107$	1145.31			1.03478			0.517390	+	0.855750i	$0.326904\pi$
$108$		0			0
$109$			2033.95i			1.78731i	0.448750	+	0.893657i	$0.351869\pi$
$109$			2033.95i			1.78731i	−0.448750	+	0.893657i	$0.648131\pi$
$110$			280.185i			0.242860i
$111$		0			0
$112$		−	41.5598i		−	0.0350628i
$113$	1218.91			1.01474			0.507370	−	0.861728i	$-0.330618\pi$
$113$	1218.91			1.01474			0.507370	+	0.861728i	$0.330618\pi$
$114$		0			0
$115$		−	879.231i		−	0.712945i
$116$	2238.39			1.79163
$117$		0			0
$118$	−3646.52			−2.84483
$119$		−	11.0525i		−	0.00851414i
$120$		0			0
$121$	1222.39			0.918396
$122$			1003.51i			0.744699i
$123$		0			0
$124$		−	464.278i		−	0.336237i
$125$			125.000i			0.0894427i
$126$		0			0
$127$	−1219.03			−0.851746			−0.425873	−	0.904783i	$-0.640033\pi$
$127$	−1219.03			−0.851746			−0.425873	+	0.904783i	$0.640033\pi$
$128$			2685.75i			1.85460i
$129$		0			0
$130$	−1073.50	−	659.938i	−0.724249	−	0.445234i
$131$	−807.392			−0.538490			−0.269245	−	0.963072i	$-0.586774\pi$
$131$	−807.392			−0.538490			−0.269245	+	0.963072i	$0.586774\pi$
$132$		0			0
$133$	26.0485			0.0169826
$134$	−4977.74			−3.20904
$135$		0			0
$136$			3802.64i			2.39760i
$137$		−	24.8642i		−	0.0155057i	−0.999970	−	0.00775287i	$-0.997532\pi$
$137$		−	24.8642i		−	0.0155057i	0.999970	−	0.00775287i	$-0.00246784\pi$
$138$		0			0
$139$	−1351.45			−0.824665			−0.412332	−	0.911033i	$-0.635286\pi$
$139$	−1351.45			−0.824665			−0.412332	+	0.911033i	$0.635286\pi$
$140$	21.0960			0.0127352
$141$		0			0
$142$	2022.96			1.19551
$143$	255.828	−	416.147i	0.149604	−	0.243357i
$144$		0			0
$145$			535.223i			0.306537i
$146$	3609.08			2.04582
$147$		0			0
$148$		−	4299.08i		−	2.38772i
$149$		−	318.875i		−	0.175324i	−0.996150	−	0.0876621i	$-0.972060\pi$
$149$		−	318.875i		−	0.175324i	0.996150	−	0.0876621i	$-0.0279396\pi$
$150$		0			0
$151$			472.671i			0.254738i	0.991855	+	0.127369i	$0.0406533\pi$
$151$			472.671i			0.254738i	−0.991855	+	0.127369i	$0.959347\pi$
$152$	−8962.03			−4.78234
$153$		0			0
$154$			11.3066i			0.00591632i
$155$	111.014			0.0575281
$156$		0			0
$157$	2157.92			1.09695			0.548474	−	0.836168i	$-0.315209\pi$
$157$	2157.92			1.09695			0.548474	+	0.836168i	$0.315209\pi$
$158$			3191.13i			1.60679i
$159$		0			0
$160$	−2760.72			−1.36409
$161$		−	35.4807i		−	0.0173681i
$162$		0			0
$163$		−	2482.96i		−	1.19313i	−0.802564	−	0.596567i	$-0.796531\pi$
$163$		−	2482.96i		−	1.19313i	0.802564	−	0.596567i	$-0.203469\pi$
$164$			5964.56i			2.83996i
$165$		0			0
$166$	2289.20			1.07034
$167$			1563.31i			0.724389i	0.932103	+	0.362194i	$0.117972\pi$
$167$			1563.31i			0.724389i	−0.932103	+	0.362194i	$0.882028\pi$
$168$		0			0
$169$	991.862	+	1960.36i	0.451462	+	0.892290i
$170$	−1472.66			−0.664399
$171$		0			0
$172$	−6729.82			−2.98339
$173$	−2785.12			−1.22398			−0.611991	−	0.790865i	$-0.709631\pi$
$173$	−2785.12			−1.22398			−0.611991	+	0.790865i	$0.709631\pi$
$174$		0			0
$175$			5.04428i			0.00217892i
$176$		−	2146.64i		−	0.919368i
$177$		0			0
$178$	−761.772			−0.320771
$179$	3352.21			1.39975			0.699877	−	0.714264i	$-0.253238\pi$
$179$	3352.21			1.39975			0.699877	+	0.714264i	$0.253238\pi$
$180$		0			0
$181$	3474.25			1.42674			0.713368	−	0.700789i	$-0.247169\pi$
$181$	3474.25			1.42674			0.713368	+	0.700789i	$0.247169\pi$
$182$	−43.3203	−	26.6313i	−0.0176435	−	0.0108464i
$183$		0			0
$184$			12207.2i			4.89090i
$185$	1027.96			0.408524
$186$		0			0
$187$		−	570.882i		−	0.223246i
$188$		−	7944.16i		−	3.08185i
$189$		0			0
$190$		−	3470.75i		−	1.32524i
$191$	823.112			0.311824			0.155912	−	0.987771i	$-0.450168\pi$
$191$	823.112			0.311824			0.155912	+	0.987771i	$0.450168\pi$
$192$		0			0
$193$		−	186.060i		−	0.0693933i	−0.999398	−	0.0346967i	$-0.988953\pi$
$193$		−	186.060i		−	0.0693933i	0.999398	−	0.0346967i	$-0.0110465\pi$
$194$	−2965.17			−1.09735
$195$		0			0
$196$	−7171.55			−2.61354
$197$		−	3035.00i		−	1.09764i	−0.835942	−	0.548819i	$-0.815078\pi$
$197$		−	3035.00i		−	1.09764i	0.835942	−	0.548819i	$-0.184922\pi$
$198$		0			0
$199$	2690.11			0.958276			0.479138	−	0.877740i	$-0.340949\pi$
$199$	2690.11			0.958276			0.479138	+	0.877740i	$0.340949\pi$
$200$		−	1735.49i		−	0.613589i
$201$		0			0
$202$		−	7454.43i		−	2.59650i
$203$			21.5985i			0.00746758i
$204$		0			0
$205$	−1426.19			−0.485900
$206$		−	6218.93i		−	2.10337i
$207$		0			0
$208$	8224.64	+	5056.12i	2.74171	+	1.68548i
$209$	1345.45			0.445295
$210$		0			0
$211$	1411.57			0.460554			0.230277	−	0.973125i	$-0.426037\pi$
$211$	1411.57			0.460554			0.230277	+	0.973125i	$0.426037\pi$
$212$	10584.9			3.42911
$213$		0			0
$214$		−	6158.20i		−	1.96713i
$215$		−	1609.17i		−	0.510441i
$216$		0			0
$217$	4.47988			0.00140145
$218$	10936.3			3.39771
$219$		0			0
$220$	1089.64			0.333926
$221$	2187.28	+	1344.64i	0.665759	+	0.409277i
$222$		0			0
$223$			474.236i			0.142409i	0.997462	+	0.0712044i	$0.0226843\pi$
$223$			474.236i			0.142409i	−0.997462	+	0.0712044i	$0.977316\pi$
$224$	−111.407			−0.0332307
$225$		0			0
$226$		−	6553.94i		−	1.92903i
$227$		−	1561.74i		−	0.456635i	−0.973587	−	0.228318i	$-0.926678\pi$
$227$		−	1561.74i		−	0.456635i	0.973587	−	0.228318i	$-0.0733225\pi$
$228$		0			0
$229$		−	2928.82i		−	0.845162i	−0.906325	−	0.422581i	$-0.861124\pi$
$229$		−	2928.82i		−	0.845162i	0.906325	−	0.422581i	$-0.138876\pi$
$230$	−4727.51			−1.35532
$231$		0			0
$232$		−	7431.01i		−	2.10289i
$233$	−2785.91			−0.783310			−0.391655	−	0.920112i	$-0.628097\pi$
$233$	−2785.91			−0.783310			−0.391655	+	0.920112i	$0.628097\pi$
$234$		0			0
$235$	1899.54			0.527285
$236$			14181.4i			3.91157i
$237$		0			0
$238$	−59.4280			−0.0161855
$239$		−	2488.61i		−	0.673534i	−0.941588	−	0.336767i	$-0.890667\pi$
$239$		−	2488.61i		−	0.673534i	0.941588	−	0.336767i	$-0.109333\pi$
$240$		0			0
$241$			4840.64i			1.29383i	0.762562	+	0.646915i	$0.223941\pi$
$241$			4840.64i			1.29383i	−0.762562	+	0.646915i	$0.776059\pi$
$242$		−	6572.61i		−	1.74588i
$243$		0			0
$244$	3902.66			1.02394
$245$		−	1714.80i		−	0.447161i
$246$		0			0
$247$	−3169.03	+	5154.97i	−0.816359	+	1.32795i
$248$	−1541.31			−0.394651
$249$		0			0
$250$	672.109			0.170032
$251$	−8.77945			−0.00220779			−0.00110389	−	0.999999i	$-0.500351\pi$
$251$	−8.77945			−0.00220779			−0.00110389	+	0.999999i	$0.500351\pi$
$252$		0			0
$253$		−	1832.64i		−	0.455403i
$254$			6554.59i			1.61918i
$255$		0			0
$256$	3872.87			0.945525
$257$	−1600.04			−0.388357			−0.194179	−	0.980966i	$-0.562204\pi$
$257$	−1600.04			−0.388357			−0.194179	+	0.980966i	$0.562204\pi$
$258$		0			0
$259$	41.4824			0.00995208
$260$	−2566.52	+	4174.87i	−0.612187	+	0.995826i
$261$		0			0
$262$			4341.25i			1.02368i
$263$	3050.24			0.715155			0.357578	−	0.933883i	$-0.383603\pi$
$263$	3050.24			0.715155			0.357578	+	0.933883i	$0.383603\pi$
$264$		0			0
$265$			2530.96i			0.586700i
$266$		−	140.059i		−	0.0322842i
$267$		0			0
$268$			19358.6i			4.41236i
$269$	6168.58			1.39816			0.699080	−	0.715043i	$-0.253593\pi$
$269$	6168.58			1.39816			0.699080	+	0.715043i	$0.253593\pi$
$270$		0			0
$271$		−	2424.93i		−	0.543556i	−0.962360	−	0.271778i	$-0.912388\pi$
$271$		−	2424.93i		−	0.543556i	0.962360	−	0.271778i	$-0.0876117\pi$
$272$	11282.8			2.51515
$273$		0			0
$274$	−133.691			−0.0294766
$275$			260.546i			0.0571328i
$276$		0			0
$277$	4442.72			0.963673			0.481836	−	0.876261i	$-0.339970\pi$
$277$	4442.72			0.963673			0.481836	+	0.876261i	$0.339970\pi$
$278$			7266.57i			1.56770i
$279$		0			0
$280$		−	70.0345i		−	0.0149477i
$281$			357.622i			0.0759215i	0.999279	+	0.0379608i	$0.0120862\pi$
$281$			357.622i			0.0759215i	−0.999279	+	0.0379608i	$0.987914\pi$
$282$		0			0
$283$	1573.71			0.330556			0.165278	−	0.986247i	$-0.447148\pi$
$283$	1573.71			0.330556			0.165278	+	0.986247i	$0.447148\pi$
$284$		−	7867.32i		−	1.64380i
$285$		0			0
$286$	−2237.57	−	1375.55i	−0.462624	−	0.284399i
$287$	−57.5528			−0.0118371
$288$		0			0
$289$	−1912.42			−0.389257
$290$	2877.83			0.582731
$291$		0			0
$292$		−	14035.8i		−	2.81296i
$293$		−	4370.32i		−	0.871389i	−0.900095	−	0.435694i	$-0.856503\pi$
$293$		−	4370.32i		−	0.871389i	0.900095	−	0.435694i	$-0.143497\pi$
$294$		0			0
$295$	−3390.93			−0.669246
$296$	−14272.1			−2.80253
$297$		0			0
$298$	−1714.55			−0.333293
$299$	7021.59	+	4316.54i	1.35809	+	0.834890i
$300$		0			0
$301$		−	64.9369i		−	0.0124349i
$302$	2541.50			0.484260
$303$		0			0
$304$			26591.2i			5.01680i
$305$			933.170i			0.175191i
$306$		0			0
$307$			3998.97i			0.743430i	0.928347	+	0.371715i	$0.121230\pi$
$307$			3998.97i			0.743430i	−0.928347	+	0.371715i	$0.878770\pi$
$308$	43.9717			0.00813481
$309$		0			0
$310$		−	596.908i		−	0.109362i
$311$	−2116.08			−0.385825			−0.192913	−	0.981216i	$-0.561793\pi$
$311$	−2116.08			−0.385825			−0.192913	+	0.981216i	$0.561793\pi$
$312$		0			0
$313$	1729.75			0.312368			0.156184	−	0.987728i	$-0.450081\pi$
$313$	1729.75			0.312368			0.156184	+	0.987728i	$0.450081\pi$
$314$		−	11602.9i		−	2.08531i
$315$		0			0
$316$	12410.4			2.20930
$317$		−	4636.84i		−	0.821548i	−0.911737	−	0.410774i	$-0.865259\pi$
$317$		−	4636.84i		−	0.821548i	0.911737	−	0.410774i	$-0.134741\pi$
$318$		0			0
$319$			1115.60i			0.195805i
$320$			6605.04i			1.15385i
$321$		0			0
$322$	−190.775			−0.0330170
$323$			7071.73i			1.21821i
$324$		0			0
$325$	−998.258	−	613.682i	−0.170380	−	0.104741i
$326$	−13350.6			−2.26816
$327$		0			0
$328$	19801.2			3.33334
$329$	76.6543			0.0128453
$330$		0			0
$331$		−	2278.33i		−	0.378333i	−0.981945	−	0.189167i	$-0.939421\pi$
$331$		−	2278.33i		−	0.378333i	0.981945	−	0.189167i	$-0.0605787\pi$
$332$		−	8902.76i		−	1.47169i
$333$		0			0
$334$	8405.75			1.37707
$335$	−4628.84			−0.754927
$336$		0			0
$337$	−419.703			−0.0678418			−0.0339209	−	0.999425i	$-0.510799\pi$
$337$	−419.703			−0.0678418			−0.0339209	+	0.999425i	$0.510799\pi$
$338$	10540.6	−	5333.12i	1.69626	−	0.858235i
$339$		0			0
$340$			5727.21i			0.913534i
$341$	231.394			0.0367469
$342$		0			0
$343$		−	138.407i		−	0.0217879i
$344$			22341.7i			3.50169i
$345$		0			0
$346$			14975.3i			2.32681i
$347$	495.311			0.0766273			0.0383137	−	0.999266i	$-0.487801\pi$
$347$	495.311			0.0766273			0.0383137	+	0.999266i	$0.487801\pi$
$348$		0			0
$349$		−	4453.77i		−	0.683109i	−0.939862	−	0.341554i	$-0.889047\pi$
$349$		−	4453.77i		−	0.683109i	0.939862	−	0.341554i	$-0.110953\pi$
$350$	27.1224			0.00414216
$351$		0			0
$352$	−5754.35			−0.871329
$353$			11893.7i			1.79330i	0.442736	+	0.896652i	$0.354008\pi$
$353$			11893.7i			1.79330i	−0.442736	+	0.896652i	$0.645992\pi$
$354$		0			0
$355$	1881.16			0.281244
$356$			2962.55i			0.441053i
$357$		0			0
$358$		−	18024.4i		−	2.66095i
$359$			5141.43i			0.755861i	0.925834	+	0.377931i	$0.123364\pi$
$359$			5141.43i			0.755861i	−0.925834	+	0.377931i	$0.876636\pi$
$360$		0			0
$361$	−9807.58			−1.42989
$362$		−	18680.6i		−	2.71224i
$363$		0			0
$364$	−103.570	+	168.474i	−0.0149135	+	0.0242594i
$365$	3356.12			0.481280
$366$		0			0
$367$	9410.74			1.33852			0.669260	−	0.743028i	$-0.266611\pi$
$367$	9410.74			1.33852			0.669260	+	0.743028i	$0.266611\pi$
$368$	36219.9			5.13068
$369$		0			0
$370$		−	5527.19i		−	0.776608i
$371$			102.135i			0.0142927i
$372$		0			0
$373$	−4992.80			−0.693076			−0.346538	−	0.938036i	$-0.612643\pi$
$373$	−4992.80			−0.693076			−0.346538	+	0.938036i	$0.612643\pi$
$374$	−3069.56			−0.424394
$375$		0			0
$376$	−26373.1			−3.61725
$377$	−4274.33	−	2627.66i	−0.583923	−	0.358969i
$378$		0			0
$379$			2436.78i			0.330261i	0.986272	+	0.165131i	$0.0528046\pi$
$379$			2436.78i			0.330261i	−0.986272	+	0.165131i	$0.947195\pi$
$380$	−13497.8			−1.82217
$381$		0			0
$382$		−	4425.77i		−	0.592781i
$383$		−	10395.4i		−	1.38690i	−0.720506	−	0.693449i	$-0.756090\pi$
$383$		−	10395.4i		−	1.38690i	0.720506	−	0.693449i	$-0.243910\pi$
$384$		0			0
$385$			10.5141i			0.00139182i
$386$	−1000.42			−0.131918
$387$		0			0
$388$			11531.6i			1.50884i
$389$	−13872.3			−1.80811			−0.904056	−	0.427413i	$-0.859425\pi$
$389$	−13872.3			−1.80811			−0.904056	+	0.427413i	$0.859425\pi$
$390$		0			0
$391$	9632.42			1.24586
$392$			23808.1i			3.06758i
$393$		0			0
$394$	−16318.8			−2.08662
$395$			2967.45i			0.377997i
$396$		0			0
$397$		−	2216.16i		−	0.280166i	−0.990140	−	0.140083i	$-0.955263\pi$
$397$		−	2216.16i		−	0.280166i	0.990140	−	0.140083i	$-0.0447369\pi$
$398$		−	14464.4i		−	1.82169i
$399$		0			0
$400$	−5149.37			−0.643672
$401$			12127.7i			1.51030i	0.655554	+	0.755148i	$0.272435\pi$
$401$			12127.7i			1.51030i	−0.655554	+	0.755148i	$0.727565\pi$
$402$		0			0
$403$	−545.018	+	886.565i	−0.0673680	+	0.109585i
$404$	−28990.5			−3.57012
$405$		0			0
$406$	116.133			0.0141960
$407$	2142.64			0.260950
$408$		0			0
$409$			2755.96i			0.333187i	0.986026	+	0.166594i	$0.0532768\pi$
$409$			2755.96i			0.333187i	−0.986026	+	0.166594i	$0.946723\pi$
$410$			7668.45i			0.923702i
$411$		0			0
$412$	−24185.6			−2.89208
$413$	−136.838			−0.0163036
$414$		0			0
$415$	2128.75			0.251798
$416$	13553.6	−	22047.3i	1.59741	−	2.59845i
$417$		0			0
$418$		−	7234.32i		−	0.846512i
$419$	−417.932			−0.0487287			−0.0243643	−	0.999703i	$-0.507756\pi$
$419$	−417.932			−0.0487287			−0.0243643	+	0.999703i	$0.507756\pi$
$420$		0			0
$421$			7623.50i			0.882534i	0.897376	+	0.441267i	$0.145471\pi$
$421$			7623.50i			0.882534i	−0.897376	+	0.441267i	$0.854529\pi$
$422$		−	7589.86i		−	0.875518i
$423$		0			0
$424$		−	35139.7i		−	4.02484i
$425$	−1369.44			−0.156300
$426$		0			0
$427$			37.6573i			0.00426783i
$428$	−23949.4			−2.70476
$429$		0			0
$430$	−8652.33			−0.970354
$431$		−	6348.22i		−	0.709473i	−0.934966	−	0.354736i	$-0.884571\pi$
$431$		−	6348.22i		−	0.709473i	0.934966	−	0.354736i	$-0.115429\pi$
$432$		0			0
$433$	−4286.27			−0.475716			−0.237858	−	0.971300i	$-0.576445\pi$
$433$	−4286.27			−0.475716			−0.237858	+	0.971300i	$0.576445\pi$
$434$		−	24.0878i		−	0.00266417i
$435$		0			0
$436$		−	42531.6i		−	4.67177i
$437$			22701.6i			2.48504i
$438$		0			0
$439$	−898.674			−0.0977024			−0.0488512	−	0.998806i	$-0.515556\pi$
$439$	−898.674			−0.0977024			−0.0488512	+	0.998806i	$0.515556\pi$
$440$		−	3617.40i		−	0.391939i
$441$		0			0
$442$	7229.96	−	11760.8i	0.778041	−	1.26562i
$443$	−4981.83			−0.534297			−0.267149	−	0.963655i	$-0.586081\pi$
$443$	−4981.83			−0.534297			−0.267149	+	0.963655i	$0.586081\pi$
$444$		0			0
$445$	−708.378			−0.0754615
$446$	2549.91			0.270721
$447$		0			0
$448$			266.541i			0.0281091i
$449$			4494.76i			0.472429i	0.971701	+	0.236215i	$0.0759068\pi$
$449$			4494.76i			0.472429i	−0.971701	+	0.236215i	$0.924093\pi$
$450$		0			0
$451$	−2972.71			−0.310375
$452$	−25488.4			−2.65238
$453$		0			0
$454$	−8397.27			−0.868069
$455$	−40.2839	−	24.7647i	−0.00415064	−	0.00255162i
$456$		0			0
$457$			2924.71i			0.299370i	0.988734	+	0.149685i	$0.0478259\pi$
$457$			2924.71i			0.299370i	−0.988734	+	0.149685i	$0.952174\pi$
$458$	−15747.9			−1.60666
$459$		0			0
$460$			18385.4i			1.86353i
$461$			10122.2i			1.02264i	0.859390	+	0.511320i	$0.170843\pi$
$461$			10122.2i			1.02264i	−0.859390	+	0.511320i	$0.829157\pi$
$462$		0			0
$463$		−	5225.27i		−	0.524490i	−0.965001	−	0.262245i	$-0.915537\pi$
$463$		−	5225.27i		−	0.524490i	0.965001	−	0.262245i	$-0.0844628\pi$
$464$	−22048.5			−2.20598
$465$		0			0
$466$			14979.5i			1.48908i
$467$	5596.68			0.554569			0.277284	−	0.960788i	$-0.410566\pi$
$467$	5596.68			0.554569			0.277284	+	0.960788i	$0.410566\pi$
$468$		0			0
$469$	−186.793			−0.0183909
$470$		−	10213.6i		−	1.00238i
$471$		0			0
$472$	47079.5			4.59112
$473$		−	3354.11i		−	0.326051i
$474$		0			0
$475$		−	3227.48i		−	0.311762i
$476$			231.117i			0.0222547i
$477$		0			0
$478$	−13380.9			−1.28040
$479$		−	14342.8i		−	1.36814i	−0.729418	−	0.684069i	$-0.760209\pi$
$479$		−	14342.8i		−	1.36814i	0.729418	−	0.684069i	$-0.239791\pi$
$480$		0			0
$481$	−5046.71	+	8209.33i	−0.478399	+	0.778198i
$482$	26027.5			2.45959
$483$		0			0
$484$	−25561.0			−2.40055
$485$	−2757.33			−0.258153
$486$		0			0
$487$			19369.6i			1.80230i	0.433512	+	0.901148i	$0.357274\pi$
$487$			19369.6i			1.80230i	−0.433512	+	0.901148i	$0.642726\pi$
$488$		−	12956.1i		−	1.20183i
$489$		0			0
$490$	−9220.25			−0.850058
$491$	−12486.5			−1.14768			−0.573838	−	0.818969i	$-0.694546\pi$
$491$	−12486.5			−1.14768			−0.573838	+	0.818969i	$0.694546\pi$
$492$		0			0
$493$	−5863.64			−0.535670
$494$	27717.6	+	17039.5i	2.52444	+	1.55191i
$495$		0			0
$496$			4573.22i			0.413999i
$497$	75.9128			0.00685142
$498$		0			0
$499$			15364.4i			1.37837i	0.724585	+	0.689186i	$0.242032\pi$
$499$			15364.4i			1.37837i	−0.724585	+	0.689186i	$0.757968\pi$
$500$		−	2613.85i		−	0.233790i
$501$		0			0
$502$			47.2060i			0.00419703i
$503$	−15778.4			−1.39866			−0.699328	−	0.714800i	$-0.746517\pi$
$503$	−15778.4			−1.39866			−0.699328	+	0.714800i	$0.746517\pi$
$504$		0			0
$505$		−	6931.94i		−	0.610826i
$506$	−9853.87			−0.865727
$507$		0			0
$508$	25491.0			2.22634
$509$			6252.54i			0.544478i	0.962230	+	0.272239i	$0.0877641\pi$
$509$			6252.54i			0.544478i	−0.962230	+	0.272239i	$0.912236\pi$
$510$		0			0
$511$	135.433			0.0117245
$512$			662.049i			0.0571459i
$513$		0			0
$514$			8603.22i			0.738272i
$515$		−	5783.04i		−	0.494817i
$516$		0			0
$517$	3959.33			0.336811
$518$		−	223.046i		−	0.0189190i
$519$		0			0
$520$	13859.8	+	8520.33i	1.16883	+	0.718541i
$521$	13016.7			1.09457			0.547287	−	0.836945i	$-0.315660\pi$
$521$	13016.7			1.09457			0.547287	+	0.836945i	$0.315660\pi$
$522$		0			0
$523$	−16344.2			−1.36650			−0.683251	−	0.730183i	$-0.739435\pi$
$523$	−16344.2			−1.36650			−0.683251	+	0.730183i	$0.739435\pi$
$524$	16883.2			1.40753
$525$		0			0
$526$		−	16400.8i		−	1.35952i
$527$			1216.21i			0.100530i
$528$		0			0
$529$	18754.9			1.54145
$530$	13608.6			1.11532
$531$		0			0
$532$	−544.694			−0.0443900
$533$	7001.82	−	11389.7i	0.569010	−	0.925592i
$534$		0			0
$535$		−	5726.56i		−	0.462768i
$536$	64266.6			5.17891
$537$		0			0
$538$		−	33167.7i		−	2.65792i
$539$		−	3574.26i		−	0.285630i
$540$		0			0
$541$			21385.8i			1.69953i	0.527158	+	0.849767i	$0.323258\pi$
$541$			21385.8i			1.69953i	−0.527158	+	0.849767i	$0.676742\pi$
$542$	−13038.5			−1.03331
$543$		0			0
$544$		−	30245.1i		−	2.38373i
$545$	10169.8			0.799312
$546$		0			0
$547$	−22836.7			−1.78505			−0.892527	−	0.450993i	$-0.851070\pi$
$547$	−22836.7			−1.78505			−0.892527	+	0.450993i	$0.851070\pi$
$548$			519.929i			0.0405297i
$549$		0			0
$550$	1400.92			0.108610
$551$		−	13819.4i		−	1.06847i
$552$		0			0
$553$			119.749i			0.00920842i
$554$		−	23888.0i		−	1.83195i
$555$		0			0
$556$	28259.9			2.15555
$557$			14375.7i			1.09357i	0.837274	+	0.546784i	$0.184148\pi$
$557$			14375.7i			1.09357i	−0.837274	+	0.546784i	$0.815852\pi$
$558$		0			0
$559$	12851.0	+	7900.17i	0.972340	+	0.597749i
$560$	−207.799			−0.0156805
$561$		0			0
$562$	1922.89			0.144328
$563$	−2088.93			−0.156373			−0.0781866	−	0.996939i	$-0.524913\pi$
$563$	−2088.93			−0.156373			−0.0781866	+	0.996939i	$0.524913\pi$
$564$		0			0
$565$		−	6094.57i		−	0.453806i
$566$		−	8461.65i		−	0.628392i
$567$		0			0
$568$	−26118.0			−1.92937
$569$	−7775.20			−0.572853			−0.286426	−	0.958102i	$-0.592467\pi$
$569$	−7775.20			−0.572853			−0.286426	+	0.958102i	$0.592467\pi$
$570$		0			0
$571$	21491.9			1.57515			0.787573	−	0.616221i	$-0.211337\pi$
$571$	21491.9			1.57515			0.787573	+	0.616221i	$0.211337\pi$
$572$	−5349.56	+	8701.97i	−0.391043	+	0.636097i
$573$		0			0
$574$			309.454i			0.0225024i
$575$	−4396.15			−0.318839
$576$		0			0
$577$		−	4348.75i		−	0.313762i	−0.987617	−	0.156881i	$-0.949856\pi$
$577$		−	4348.75i		−	0.313762i	0.987617	−	0.156881i	$-0.0501440\pi$
$578$			10282.9i			0.739983i
$579$		0			0
$580$		−	11191.9i		−	0.801242i
$581$	85.9040			0.00613407
$582$		0			0
$583$			5275.45i			0.374763i
$584$	−46596.1			−3.30165
$585$		0			0
$586$	−23498.7			−1.65652
$587$			5800.67i			0.407870i	0.978984	+	0.203935i	$0.0653731\pi$
$587$			5800.67i			0.407870i	−0.978984	+	0.203935i	$0.934627\pi$
$588$		0			0
$589$	−2866.36			−0.200520
$590$			18232.6i			1.27225i
$591$		0			0
$592$			42346.6i			2.93993i
$593$		−	4577.14i		−	0.316966i	−0.987362	−	0.158483i	$-0.949340\pi$
$593$		−	4577.14i		−	0.316966i	0.987362	−	0.158483i	$-0.0506603\pi$
$594$		0			0
$595$	−55.2626			−0.00380764
$596$			6667.94i			0.458271i
$597$		0			0
$598$	23209.5	−	37754.2i	1.58714	−	2.58175i
$599$	−17537.7			−1.19628			−0.598140	−	0.801392i	$-0.704093\pi$
$599$	−17537.7			−1.19628			−0.598140	+	0.801392i	$0.704093\pi$
$600$		0			0
$601$	13316.0			0.903780			0.451890	−	0.892074i	$-0.350750\pi$
$601$	13316.0			0.903780			0.451890	+	0.892074i	$0.350750\pi$
$602$	−349.158			−0.0236389
$603$		0			0
$604$		−	9883.94i		−	0.665847i
$605$		−	6111.93i		−	0.410719i
$606$		0			0
$607$	2103.20			0.140636			0.0703182	−	0.997525i	$-0.477599\pi$
$607$	2103.20			0.140636			0.0703182	+	0.997525i	$0.477599\pi$
$608$	71281.3			4.75467
$609$		0			0
$610$	5017.54			0.333040
$611$	−9325.69	+	15169.8i	−0.617475	+	1.00443i
$612$		0			0
$613$			6984.21i			0.460178i	0.973170	+	0.230089i	$0.0739018\pi$
$613$			6984.21i			0.460178i	−0.973170	+	0.230089i	$0.926098\pi$
$614$	21502.0			1.41327
$615$		0			0
$616$		−	145.977i		−	0.00954805i
$617$			13605.0i			0.887713i	0.896098	+	0.443856i	$0.146390\pi$
$617$			13605.0i			0.887713i	−0.896098	+	0.443856i	$0.853610\pi$
$618$		0			0
$619$		−	24505.8i		−	1.59123i	−0.605801	−	0.795616i	$-0.707147\pi$
$619$		−	24505.8i		−	1.59123i	0.605801	−	0.795616i	$-0.292853\pi$
$620$	−2321.39			−0.150370
$621$		0			0
$622$			11377.9i			0.733458i
$623$	−28.5860			−0.00183832
$624$		0			0
$625$	625.000			0.0400000
$626$		−	9300.65i		−	0.593816i
$627$		0			0
$628$	−45123.8			−2.86726
$629$			11261.8i			0.713890i
$630$		0			0
$631$			18694.2i			1.17940i	0.807621	+	0.589702i	$0.200755\pi$
$631$			18694.2i			1.17940i	−0.807621	+	0.589702i	$0.799245\pi$
$632$		−	41200.0i		−	2.59311i
$633$		0			0
$634$	−24931.7			−1.56177
$635$			6095.17i			0.380912i
$636$		0			0
$637$	13694.5	+	8418.72i	0.851797	+	0.523645i
$638$	5998.45			0.372227
$639$		0			0
$640$	13428.7			0.829403
$641$	22658.3			1.39617			0.698087	−	0.716013i	$-0.254035\pi$
$641$	22658.3			1.39617			0.698087	+	0.716013i	$0.254035\pi$
$642$		0			0
$643$		−	18187.4i		−	1.11546i	−0.830022	−	0.557730i	$-0.811672\pi$
$643$		−	18187.4i		−	1.11546i	0.830022	−	0.557730i	$-0.188328\pi$
$644$			741.929i			0.0453976i
$645$		0			0
$646$	38023.8			2.31583
$647$	−4755.08			−0.288936			−0.144468	−	0.989509i	$-0.546147\pi$
$647$	−4755.08			−0.288936			−0.144468	+	0.989509i	$0.546147\pi$
$648$		0			0
$649$	−7067.94			−0.427490
$650$	−3299.69	+	5367.51i	−0.199115	+	0.323894i
$651$		0			0
$652$			51920.7i			3.11867i
$653$	−731.494			−0.0438370			−0.0219185	−	0.999760i	$-0.506977\pi$
$653$	−731.494			−0.0438370			−0.0219185	+	0.999760i	$0.506977\pi$
$654$		0			0
$655$			4036.96i			0.240820i
$656$		−	58751.9i		−	3.49676i
$657$		0			0
$658$		−	412.161i		−	0.0244190i
$659$	−16005.3			−0.946099			−0.473050	−	0.881036i	$-0.656847\pi$
$659$	−16005.3			−0.946099			−0.473050	+	0.881036i	$0.656847\pi$
$660$		0			0
$661$			12332.1i			0.725664i	0.931855	+	0.362832i	$0.118190\pi$
$661$			12332.1i			0.725664i	−0.931855	+	0.362832i	$0.881810\pi$
$662$	−12250.3			−0.719217
$663$		0			0
$664$	−29555.4			−1.72737
$665$		−	130.242i		−	0.00759486i
$666$		0			0
$667$	−18823.4			−1.09272
$668$		−	32690.2i		−	1.89344i
$669$		0			0
$670$			24888.7i			1.43513i
$671$			1945.07i			0.111905i
$672$		0			0
$673$	12183.0			0.697799			0.348899	−	0.937160i	$-0.386556\pi$
$673$	12183.0			0.697799			0.348899	+	0.937160i	$0.386556\pi$
$674$			2256.69i			0.128968i
$675$		0			0
$676$	−20740.6	−	40992.7i	−1.18005	−	2.33231i
$677$	−25830.0			−1.46637			−0.733183	−	0.680032i	$-0.761966\pi$
$677$	−25830.0			−1.46637			−0.733183	+	0.680032i	$0.761966\pi$
$678$		0			0
$679$	−111.270			−0.00628888
$680$	19013.2			1.07224
$681$		0			0
$682$		−	1244.18i		−	0.0698562i
$683$			6832.74i			0.382793i	0.981513	+	0.191396i	$0.0613016\pi$
$683$			6832.74i			0.382793i	−0.981513	+	0.191396i	$0.938698\pi$
$684$		0			0
$685$	−124.321			−0.00693438
$686$	−744.196			−0.0414191
$687$		0			0
$688$	66289.9			3.67337
$689$	−20212.4	−	12425.6i	−1.11761	−	0.687052i
$690$		0			0
$691$		−	26068.7i		−	1.43517i	−0.696472	−	0.717584i	$-0.745248\pi$
$691$		−	26068.7i		−	1.43517i	0.696472	−	0.717584i	$-0.254752\pi$
$692$	58239.1			3.19930
$693$		0			0
$694$		−	2663.23i		−	0.145670i
$695$			6757.24i			0.368801i
$696$		0			0
$697$		−	15624.6i		−	0.849104i
$698$	−23947.4			−1.29860
$699$		0			0
$700$		−	105.480i		−	0.00569537i
$701$	31003.6			1.67045			0.835227	−	0.549905i	$-0.185336\pi$
$701$	31003.6			1.67045			0.835227	+	0.549905i	$0.185336\pi$
$702$		0			0
$703$	−26541.7			−1.42395
$704$			13767.3i			0.737039i
$705$		0			0
$706$	63950.8			3.40909
$707$		−	279.733i		−	0.0148804i
$708$		0			0
$709$			19868.2i			1.05242i	0.850355	+	0.526209i	$0.176387\pi$
$709$			19868.2i			1.05242i	−0.850355	+	0.526209i	$0.823613\pi$
$710$		−	10114.8i		−	0.534649i
$711$		0			0
$712$	9835.08			0.517676
$713$			3904.27i			0.205072i
$714$		0			0
$715$	−2080.74	−	1279.14i	−0.108832	−	0.0669050i
$716$	−70097.4			−3.65874
$717$		0			0
$718$	27644.8			1.43690
$719$	−2878.38			−0.149298			−0.0746491	−	0.997210i	$-0.523784\pi$
$719$	−2878.38			−0.149298			−0.0746491	+	0.997210i	$0.523784\pi$
$720$		0			0
$721$		−	233.370i		−	0.0120543i
$722$			52734.2i			2.71823i
$723$		0			0
$724$	−72649.4			−3.72927
$725$	2676.12			0.137088
$726$		0			0
$727$	−22798.9			−1.16309			−0.581543	−	0.813516i	$-0.697551\pi$
$727$	−22798.9			−1.16309			−0.581543	+	0.813516i	$0.697551\pi$
$728$	559.300	+	343.831i	0.0284739	+	0.0175044i
$729$		0			0
$730$		−	18045.4i		−	0.914919i
$731$	17629.3			0.891989
$732$		0			0
$733$		−	27940.5i		−	1.40792i	−0.710240	−	0.703960i	$-0.751414\pi$
$733$		−	27940.5i		−	1.40792i	0.710240	−	0.703960i	$-0.248586\pi$
$734$		−	50600.4i		−	2.54454i
$735$		0			0
$736$		−	97092.3i		−	4.86259i
$737$	−9648.21			−0.482220
$738$		0			0
$739$		−	30468.8i		−	1.51666i	−0.651871	−	0.758330i	$-0.726016\pi$
$739$		−	30468.8i		−	1.51666i	0.651871	−	0.758330i	$-0.273984\pi$
$740$	−21495.4			−1.06782
$741$		0			0
$742$	549.166			0.0271705
$743$		−	17528.7i		−	0.865501i	−0.901514	−	0.432750i	$-0.857543\pi$
$743$		−	17528.7i		−	0.865501i	0.901514	−	0.432750i	$-0.142457\pi$
$744$		0			0
$745$	−1594.38			−0.0784073
$746$			26845.7i			1.31755i
$747$		0			0
$748$			11937.6i			0.583532i
$749$		−	231.091i		−	0.0112735i
$750$		0			0
$751$	−4371.31			−0.212399			−0.106199	−	0.994345i	$-0.533868\pi$
$751$	−4371.31			−0.212399			−0.106199	+	0.994345i	$0.533868\pi$
$752$			78251.4i			3.79459i
$753$		0			0
$754$	−14128.6	+	22982.5i	−0.682404	+	1.11005i
$755$	2363.36			0.113922
$756$		0			0
$757$	2920.51			0.140221			0.0701107	−	0.997539i	$-0.477665\pi$
$757$	2920.51			0.140221			0.0701107	+	0.997539i	$0.477665\pi$
$758$	13102.3			0.627831
$759$		0			0
$760$			44810.1i			2.13873i
$761$		−	40352.4i		−	1.92217i	−0.276246	−	0.961087i	$-0.589091\pi$
$761$		−	40352.4i		−	1.92217i	0.276246	−	0.961087i	$-0.410909\pi$
$762$		0			0
$763$	410.393			0.0194721
$764$	−17211.9			−0.815060
$765$		0			0
$766$	−55895.0			−2.63651
$767$	16647.6	−	27080.2i	0.783717	−	1.27485i
$768$		0			0
$769$		−	27282.1i		−	1.27935i	−0.768646	−	0.639674i	$-0.779069\pi$
$769$		−	27282.1i		−	1.27935i	0.768646	−	0.639674i	$-0.220931\pi$
$770$	56.5331			0.00264586
$771$		0			0
$772$			3890.67i			0.181384i
$773$		−	10121.3i		−	0.470942i	−0.971881	−	0.235471i	$-0.924337\pi$
$773$		−	10121.3i		−	0.470942i	0.971881	−	0.235471i	$-0.0756633\pi$
$774$		0			0
$775$		−	555.070i		−	0.0257273i
$776$	38282.7			1.77096
$777$		0			0
$778$			74589.9i			3.43725i
$779$	36824.0			1.69365
$780$		0			0
$781$	3921.03			0.179649
$782$		−	51792.3i		−	2.36840i
$783$		0			0
$784$	70641.0			3.21798
$785$		−	10789.6i		−	0.490570i
$786$		0			0
$787$		−	17973.2i		−	0.814072i	−0.913412	−	0.407036i	$-0.866562\pi$
$787$		−	17973.2i		−	0.814072i	0.913412	−	0.407036i	$-0.133438\pi$
$788$			63464.2i			2.86906i
$789$		0			0
$790$	15955.6			0.718577
$791$		−	245.941i		−	0.0110552i
$792$		0			0
$793$	−7452.35	−	4581.36i	−0.333721	−	0.205156i
$794$	−11916.0			−0.532598
$795$		0			0
$796$	−56252.3			−2.50479
$797$	1050.61			0.0466931			0.0233465	−	0.999727i	$-0.492568\pi$
$797$	1050.61			0.0466931			0.0233465	+	0.999727i	$0.492568\pi$
$798$		0			0
$799$			20810.4i			0.921425i
$800$			13803.6i			0.610038i
$801$		0			0
$802$	65209.2			2.87109
$803$	6995.38			0.307424
$804$		0			0
$805$	−177.403			−0.00776726
$806$	4766.95	+	2930.49i	0.208323	+	0.128067i
$807$		0			0
$808$			96242.6i			4.19035i
$809$	10131.7			0.440310			0.220155	−	0.975465i	$-0.429344\pi$
$809$	10131.7			0.440310			0.220155	+	0.975465i	$0.429344\pi$
$810$		0			0
$811$			3463.42i			0.149959i	0.997185	+	0.0749797i	$0.0238892\pi$
$811$			3463.42i			0.149959i	−0.997185	+	0.0749797i	$0.976111\pi$
$812$		−	451.642i		−	0.0195191i
$813$		0			0
$814$		−	11520.7i		−	0.496069i
$815$	−12414.8			−0.533585
$816$		0			0
$817$			41548.6i			1.77919i
$818$	14818.5			0.633393
$819$		0			0
$820$	29822.8			1.27007
$821$			9876.00i			0.419823i	0.977720	+	0.209912i	$0.0673176\pi$
$821$			9876.00i			0.419823i	−0.977720	+	0.209912i	$0.932682\pi$
$822$		0			0
$823$	−7199.95			−0.304951			−0.152475	−	0.988307i	$-0.548724\pi$
$823$	−7199.95			−0.304951			−0.152475	+	0.988307i	$0.548724\pi$
$824$			80291.3i			3.39452i
$825$		0			0
$826$			735.763i			0.0309933i
$827$			16864.4i			0.709110i	0.935035	+	0.354555i	$0.115368\pi$
$827$			16864.4i			0.709110i	−0.935035	+	0.354555i	$0.884632\pi$
$828$		0			0
$829$	26061.1			1.09184			0.545922	−	0.837836i	$-0.316180\pi$
$829$	26061.1			1.09184			0.545922	+	0.837836i	$0.316180\pi$
$830$		−	11446.0i		−	0.478671i
$831$		0			0
$832$	−52748.3	−	32427.2i	−2.19798	−	1.35121i
$833$	18786.5			0.781407
$834$		0			0
$835$	7816.57			0.323957
$836$	−28134.4			−1.16393
$837$		0			0
$838$			2247.17i			0.0926338i
$839$			35491.9i			1.46045i	0.683208	+	0.730224i	$0.260584\pi$
$839$			35491.9i			1.46045i	−0.683208	+	0.730224i	$0.739416\pi$
$840$		0			0
$841$	−12930.4			−0.530175
$842$	40990.6			1.67771
$843$		0			0
$844$	−29517.1			−1.20382
$845$	9801.81	−	4959.31i	0.399044	−	0.201900i
$846$		0			0
$847$		−	246.642i		−	0.0100056i
$848$	−104263.			−4.22217
$849$		0			0
$850$			7363.30i			0.297128i
$851$			36152.4i			1.45627i
$852$		0			0
$853$		−	10763.1i		−	0.432029i	−0.976390	−	0.216014i	$-0.930694\pi$
$853$		−	10763.1i		−	0.432029i	0.976390	−	0.216014i	$-0.0693058\pi$
$854$	202.479			0.00811321
$855$		0			0
$856$			79507.2i			3.17465i
$857$	3344.32			0.133302			0.0666511	−	0.997776i	$-0.478769\pi$
$857$	3344.32			0.133302			0.0666511	+	0.997776i	$0.478769\pi$
$858$		0			0
$859$	−31361.2			−1.24567			−0.622836	−	0.782353i	$-0.714020\pi$
$859$	−31361.2			−1.24567			−0.622836	+	0.782353i	$0.714020\pi$
$860$			33649.1i			1.33421i
$861$		0			0
$862$	−34133.6			−1.34872
$863$		−	9576.65i		−	0.377744i	−0.982002	−	0.188872i	$-0.939517\pi$
$863$		−	9576.65i		−	0.377744i	0.982002	−	0.188872i	$-0.0604831\pi$
$864$		0			0
$865$			13925.6i			0.547382i
$866$			23046.8i			0.904342i
$867$		0			0
$868$	−93.6778			−0.00366317
$869$			6185.26i			0.241451i
$870$		0			0
$871$	22725.1	−	36966.2i	0.884054	−	1.43806i
$872$	−141196.			−5.48339
$873$		0			0
$874$	122064.			4.72410
$875$	25.2214			0.000974444
$876$		0			0
$877$			28993.5i			1.11635i	0.829722	+	0.558177i	$0.188499\pi$
$877$			28993.5i			1.11635i	−0.829722	+	0.558177i	$0.811501\pi$
$878$			4832.06i			0.185734i
$879$		0			0
$880$	−10733.2			−0.411154
$881$	−23068.0			−0.882156			−0.441078	−	0.897469i	$-0.645404\pi$
$881$	−23068.0			−0.882156			−0.441078	+	0.897469i	$0.645404\pi$
$882$		0			0
$883$	−29675.0			−1.13097			−0.565483	−	0.824760i	$-0.691310\pi$
$883$	−29675.0			−1.13097			−0.565483	+	0.824760i	$0.691310\pi$
$884$	−45737.9	−	28117.5i	−1.74019	−	1.06979i
$885$		0			0
$886$			26786.7i			1.01571i
$887$	5118.12			0.193742			0.0968712	−	0.995297i	$-0.469117\pi$
$887$	5118.12			0.193742			0.0968712	+	0.995297i	$0.469117\pi$
$888$		0			0
$889$			245.966i			0.00927944i
$890$			3808.86i			0.143453i
$891$		0			0
$892$		−	9916.64i		−	0.372235i
$893$	−49045.7			−1.83791
$894$		0			0
$895$		−	16761.0i		−	0.625989i
$896$	541.906			0.0202052
$897$		0			0
$898$	24167.8			0.898094
$899$		−	2376.69i		−	0.0881725i
$900$		0			0
$901$	−27727.9			−1.02525
$902$			15983.9i			0.590027i
$903$		0			0
$904$			84616.6i			3.11317i
$905$		−	17371.3i		−	0.638056i
$906$		0			0
$907$	37154.2			1.36018			0.680091	−	0.733128i	$-0.261940\pi$
$907$	37154.2			1.36018			0.680091	+	0.733128i	$0.261940\pi$
$908$			32657.2i			1.19358i
$909$		0			0
$910$	−133.156	+	216.602i	−0.00485065	+	0.00789041i
$911$	−7888.56			−0.286893			−0.143447	−	0.989658i	$-0.545819\pi$
$911$	−7888.56			−0.286893			−0.143447	+	0.989658i	$0.545819\pi$
$912$		0			0
$913$	4437.09			0.160839
$914$	15725.8			0.569106
$915$		0			0
$916$			61244.1i			2.20913i
$917$			162.908i			0.00586664i
$918$		0			0
$919$	−32292.7			−1.15913			−0.579563	−	0.814927i	$-0.696777\pi$
$919$	−32292.7			−1.15913			−0.579563	+	0.814927i	$0.696777\pi$
$920$	61035.9			2.18728
$921$		0			0
$922$	54425.7			1.94405
$923$	−9235.48	+	15023.1i	−0.329350	+	0.535743i
$924$		0			0
$925$		−	5139.78i		−	0.182697i
$926$	−28095.6			−0.997062
$927$		0			0
$928$			59104.0i			2.09072i
$929$			7931.61i			0.280116i	0.990143	+	0.140058i	$0.0447289\pi$
$929$			7931.61i			0.280116i	−0.990143	+	0.140058i	$0.955271\pi$
$930$		0			0
$931$			44275.7i			1.55862i
$932$	58255.6			2.04745
$933$		0			0
$934$		−	30092.7i		−	1.05424i
$935$	−2854.41			−0.0998388
$936$		0			0
$937$	35368.6			1.23313			0.616565	−	0.787304i	$-0.288524\pi$
$937$	35368.6			1.23313			0.616565	+	0.787304i	$0.288524\pi$
$938$			1004.36i			0.0349613i
$939$		0			0
$940$	−39720.8			−1.37824
$941$		−	21319.4i		−	0.738569i	−0.929316	−	0.369285i	$-0.879603\pi$
$941$		−	21319.4i		−	0.738569i	0.929316	−	0.369285i	$-0.120397\pi$
$942$		0			0
$943$		−	50158.0i		−	1.73210i
$944$		−	139689.i		−	4.81621i
$945$		0			0
$946$	−18034.6			−0.619827
$947$		−	16623.5i		−	0.570422i	−0.958465	−	0.285211i	$-0.907936\pi$
$947$		−	16623.5i		−	0.570422i	0.958465	−	0.285211i	$-0.0920637\pi$
$948$		0			0
$949$	−16476.7	+	26802.2i	−0.563600	+	0.916792i
$950$	−17353.7			−0.592663
$951$		0			0
$952$	767.263			0.0261209
$953$	−25047.2			−0.851374			−0.425687	−	0.904870i	$-0.639968\pi$
$953$	−25047.2			−0.851374			−0.425687	+	0.904870i	$0.639968\pi$
$954$		0			0
$955$		−	4115.56i		−	0.139452i
$956$			52038.8i			1.76052i
$957$		0			0
$958$	−77119.3			−2.60085
$959$	−5.01687			−0.000168929
$960$		0			0
$961$	29298.0			0.983453
$962$	44140.5	+	27135.5i	1.47936	+	0.909443i
$963$		0			0
$964$		−	101222.i		−	3.38188i
$965$	−930.301			−0.0310336
$966$		0			0
$967$			13656.8i			0.454161i	0.973876	+	0.227081i	$0.0729181\pi$
$967$			13656.8i			0.454161i	−0.973876	+	0.227081i	$0.927082\pi$
$968$			84857.7i			2.81759i
$969$		0			0
$970$			14825.8i			0.490752i
$971$	−2265.67			−0.0748803			−0.0374401	−	0.999299i	$-0.511920\pi$
$971$	−2265.67			−0.0748803			−0.0374401	+	0.999299i	$0.511920\pi$
$972$		0			0
$973$			272.683i			0.00898440i
$974$	104148.			3.42619
$975$		0			0
$976$	−38441.9			−1.26075
$977$		−	20144.9i		−	0.659666i	−0.944039	−	0.329833i	$-0.893008\pi$
$977$		−	20144.9i		−	0.659666i	0.944039	−	0.329833i	$-0.106992\pi$
$978$		0			0
$979$	−1476.52			−0.0482020
$980$			35857.8i			1.16881i
$981$		0			0
$982$			67138.5i			2.18175i
$983$		−	10526.6i		−	0.341552i	−0.985310	−	0.170776i	$-0.945373\pi$
$983$		−	10526.6i		−	0.341552i	0.985310	−	0.170776i	$-0.0546274\pi$
$984$		0			0
$985$	−15175.0			−0.490878
$986$			31528.1i			1.01831i
$987$		0			0
$988$	66267.0	−	107795.i	2.13384	−	3.47105i
$989$	56593.4			1.81958
$990$		0			0
$991$	55992.7			1.79482			0.897410	−	0.441198i	$-0.145446\pi$
$991$	55992.7			1.79482			0.897410	+	0.441198i	$0.145446\pi$
$992$	12259.1			0.392367
$993$		0			0
$994$		−	408.174i		−	0.0130246i
$995$		−	13450.6i		−	0.428554i
$996$		0			0
$997$	48578.5			1.54312			0.771562	−	0.636154i	$-0.219476\pi$
$997$	48578.5			1.54312			0.771562	+	0.636154i	$0.219476\pi$
$998$	82612.7			2.62030
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.4.b.e.181.1		14
3.2	odd	2		65.4.c.a.51.14	yes	14
12.11	even	2		1040.4.k.d.961.8		14
13.12	even	2	inner	585.4.b.e.181.14		14
15.2	even	4		325.4.d.c.324.2		14
15.8	even	4		325.4.d.d.324.13		14
15.14	odd	2		325.4.c.e.51.1		14
39.5	even	4		845.4.a.l.1.7		7
39.8	even	4		845.4.a.i.1.1		7
39.38	odd	2		65.4.c.a.51.1	✓	14
156.155	even	2		1040.4.k.d.961.7		14
195.38	even	4		325.4.d.c.324.1		14
195.77	even	4		325.4.d.d.324.14		14
195.194	odd	2		325.4.c.e.51.14		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.4.c.a.51.1	✓	14	39.38	odd	2
65.4.c.a.51.14	yes	14	3.2	odd	2
325.4.c.e.51.1		14	15.14	odd	2
325.4.c.e.51.14		14	195.194	odd	2
325.4.d.c.324.1		14	195.38	even	4
325.4.d.c.324.2		14	15.2	even	4
325.4.d.d.324.13		14	15.8	even	4
325.4.d.d.324.14		14	195.77	even	4
585.4.b.e.181.1		14	1.1	even	1	trivial
585.4.b.e.181.14		14	13.12	even	2	inner
845.4.a.i.1.1		7	39.8	even	4
845.4.a.l.1.7		7	39.5	even	4
1040.4.k.d.961.7		14	156.155	even	2
1040.4.k.d.961.8		14	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	585.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.37688i q^{2} -20.9108 q^{4} -5.00000i q^{5} -0.201771i q^{7} +69.4197i q^{8} -26.8844 q^{10} -10.4218i q^{11} +(39.9303 + 24.5473i) q^{13} -1.08490 q^{14} +205.975 q^{16} +54.7775 q^{17} +129.099i q^{19} +104.554i q^{20} -56.0369 q^{22} +175.846 q^{23} -25.0000 q^{25} +(131.988 - 214.700i) q^{26} +4.21919i q^{28} -107.045 q^{29} +22.2028i q^{31} -552.144i q^{32} -294.532i q^{34} -1.00886 q^{35} +205.591i q^{37} +694.150 q^{38} +347.099 q^{40} -285.238i q^{41} +321.835 q^{43} +217.929i q^{44} -945.503i q^{46} +379.907i q^{47} +342.959 q^{49} +134.422i q^{50} +(-834.975 - 513.303i) q^{52} -506.192 q^{53} -52.1092 q^{55} +14.0069 q^{56} +575.566i q^{58} -678.186i q^{59} -186.634 q^{61} +119.382 q^{62} -1321.01 q^{64} +(122.736 - 199.652i) q^{65} -925.768i q^{67} -1145.44 q^{68} +5.42449i q^{70} +376.232i q^{71} +671.223i q^{73} +1105.44 q^{74} -2699.57i q^{76} -2.10282 q^{77} -593.491 q^{79} -1029.87i q^{80} -1533.69 q^{82} +425.750i q^{83} -273.888i q^{85} -1730.47i q^{86} +723.481 q^{88} -141.676i q^{89} +(4.95293 - 8.05678i) q^{91} -3677.08 q^{92} +2042.71 q^{94} +645.496 q^{95} -551.467i q^{97} -1844.05i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 56 q^{4} - 20 q^{10} - 4 q^{13} + 152 q^{14} + 280 q^{16} + 100 q^{17} + 648 q^{22} + 532 q^{23} - 350 q^{25} + 344 q^{26} - 588 q^{29} - 540 q^{35} - 148 q^{38} + 240 q^{40} + 728 q^{43} - 1302 q^{49}+ \cdots + 3120 q^{95}+O(q^{100}) \) 14 * q - 56 * q^4 - 20 * q^10 - 4 * q^13 + 152 * q^14 + 280 * q^16 + 100 * q^17 + 648 * q^22 + 532 * q^23 - 350 * q^25 + 344 * q^26 - 588 * q^29 - 540 * q^35 - 148 * q^38 + 240 * q^40 + 728 * q^43 - 1302 * q^49 - 528 * q^52 - 1040 * q^53 - 40 * q^55 - 512 * q^56 - 1460 * q^61 - 2964 * q^62 + 1296 * q^64 + 30 * q^65 - 1604 * q^68 + 5928 * q^74 + 3568 * q^77 - 1024 * q^79 - 6660 * q^82 + 1132 * q^88 + 916 * q^91 - 6044 * q^92 + 2656 * q^94 + 3120 * q^95

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