LMFDB - Embedded newform 546.4.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,4,Mod(1,546)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("546.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	546.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,6,-9,12,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.2150428631$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.118088.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 50x - 32 $ x^3 - 50*x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.37163$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.57278 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +13.1456 q^{10} +18.0593 q^{11} -12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -19.7183 q^{15} +16.0000 q^{16} -42.3389 q^{17} +18.0000 q^{18} +77.7068 q^{19} +26.2911 q^{20} +21.0000 q^{21} +36.1186 q^{22} +43.8983 q^{23} -24.0000 q^{24} -81.7985 q^{25} +26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +121.075 q^{29} -39.4367 q^{30} +248.214 q^{31} +32.0000 q^{32} -54.1779 q^{33} -84.6779 q^{34} -46.0095 q^{35} +36.0000 q^{36} -279.817 q^{37} +155.414 q^{38} -39.0000 q^{39} +52.5823 q^{40} +425.980 q^{41} +42.0000 q^{42} +179.185 q^{43} +72.2372 q^{44} +59.1550 q^{45} +87.7966 q^{46} -216.321 q^{47} -48.0000 q^{48} +49.0000 q^{49} -163.597 q^{50} +127.017 q^{51} +52.0000 q^{52} +713.003 q^{53} -54.0000 q^{54} +118.700 q^{55} -56.0000 q^{56} -233.121 q^{57} +242.150 q^{58} +403.004 q^{59} -78.8734 q^{60} +246.765 q^{61} +496.429 q^{62} -63.0000 q^{63} +64.0000 q^{64} +85.4462 q^{65} -108.356 q^{66} -742.301 q^{67} -169.356 q^{68} -131.695 q^{69} -92.0190 q^{70} -65.6086 q^{71} +72.0000 q^{72} +965.099 q^{73} -559.635 q^{74} +245.396 q^{75} +310.827 q^{76} -126.415 q^{77} -78.0000 q^{78} +382.593 q^{79} +105.165 q^{80} +81.0000 q^{81} +851.961 q^{82} -1347.88 q^{83} +84.0000 q^{84} -278.285 q^{85} +358.371 q^{86} -363.224 q^{87} +144.474 q^{88} +830.091 q^{89} +118.310 q^{90} -91.0000 q^{91} +175.593 q^{92} -744.643 q^{93} -432.643 q^{94} +510.750 q^{95} -96.0000 q^{96} +376.407 q^{97} +98.0000 q^{98} +162.534 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 14 q^{10} - 47 q^{11} - 36 q^{12} + 39 q^{13} - 42 q^{14} - 21 q^{15} + 48 q^{16} + 119 q^{17} + 54 q^{18} + 101 q^{19}+ \cdots - 423 q^{99}+O(q^{100}) $ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 + 24 * q^8 + 27 * q^9 + 14 * q^10 - 47 * q^11 - 36 * q^12 + 39 * q^13 - 42 * q^14 - 21 * q^15 + 48 * q^16 + 119 * q^17 + 54 * q^18 + 101 * q^19 + 28 * q^20 + 63 * q^21 - 94 * q^22 - 27 * q^23 - 72 * q^24 + 266 * q^25 + 78 * q^26 - 81 * q^27 - 84 * q^28 + 345 * q^29 - 42 * q^30 + 152 * q^31 + 96 * q^32 + 141 * q^33 + 238 * q^34 - 49 * q^35 + 108 * q^36 + 227 * q^37 + 202 * q^38 - 117 * q^39 + 56 * q^40 + 452 * q^41 + 126 * q^42 - 415 * q^43 - 188 * q^44 + 63 * q^45 - 54 * q^46 + 240 * q^47 - 144 * q^48 + 147 * q^49 + 532 * q^50 - 357 * q^51 + 156 * q^52 + 874 * q^53 - 162 * q^54 + 1123 * q^55 - 168 * q^56 - 303 * q^57 + 690 * q^58 + 948 * q^59 - 84 * q^60 + 951 * q^61 + 304 * q^62 - 189 * q^63 + 192 * q^64 + 91 * q^65 + 282 * q^66 + 742 * q^67 + 476 * q^68 + 81 * q^69 - 98 * q^70 + 732 * q^71 + 216 * q^72 - 585 * q^73 + 454 * q^74 - 798 * q^75 + 404 * q^76 + 329 * q^77 - 234 * q^78 - 618 * q^79 + 112 * q^80 + 243 * q^81 + 904 * q^82 + 774 * q^83 + 252 * q^84 + 1273 * q^85 - 830 * q^86 - 1035 * q^87 - 376 * q^88 + 730 * q^89 + 126 * q^90 - 273 * q^91 - 108 * q^92 - 456 * q^93 + 480 * q^94 - 2617 * q^95 - 288 * q^96 + 2518 * q^97 + 294 * q^98 - 423 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	6.57278	0.587888	0.293944	−	0.955823i	$-0.405032\pi$
$5$	6.57278	0.587888	0.293944	+	0.955823i	$0.405032\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	13.1456	0.415699
$11$	18.0593	0.495008	0.247504	−	0.968887i	$-0.420390\pi$
$11$	18.0593	0.495008	0.247504	+	0.968887i	$0.420390\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−14.0000	−0.267261
$15$	−19.7183	−0.339417
$16$	16.0000	0.250000
$17$	−42.3389	−0.604041	−0.302021	−	0.953301i	$-0.597661\pi$
$17$	−42.3389	−0.604041	−0.302021	+	0.953301i	$0.597661\pi$
$18$	18.0000	0.235702
$19$	77.7068	0.938272	0.469136	−	0.883126i	$-0.344565\pi$
$19$	77.7068	0.938272	0.469136	+	0.883126i	$0.344565\pi$
$20$	26.2911	0.293944
$21$	21.0000	0.218218
$22$	36.1186	0.350023
$23$	43.8983	0.397975	0.198988	−	0.980002i	$-0.436235\pi$
$23$	43.8983	0.397975	0.198988	+	0.980002i	$0.436235\pi$
$24$	−24.0000	−0.204124
$25$	−81.7985	−0.654388
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	121.075	0.775276	0.387638	−	0.921812i	$-0.373291\pi$
$29$	121.075	0.775276	0.387638	+	0.921812i	$0.373291\pi$
$30$	−39.4367	−0.240004
$31$	248.214	1.43808	0.719042	−	0.694967i	$-0.244581\pi$
$31$	248.214	1.43808	0.719042	+	0.694967i	$0.244581\pi$
$32$	32.0000	0.176777
$33$	−54.1779	−0.285793
$34$	−84.6779	−0.427122
$35$	−46.0095	−0.222201
$36$	36.0000	0.166667
$37$	−279.817	−1.24329	−0.621645	−	0.783299i	$-0.713535\pi$
$37$	−279.817	−1.24329	−0.621645	+	0.783299i	$0.713535\pi$
$38$	155.414	0.663459
$39$	−39.0000	−0.160128
$40$	52.5823	0.207850
$41$	425.980	1.62261	0.811304	−	0.584624i	$-0.198758\pi$
$41$	425.980	1.62261	0.811304	+	0.584624i	$0.198758\pi$
$42$	42.0000	0.154303
$43$	179.185	0.635477	0.317738	−	0.948178i	$-0.397077\pi$
$43$	179.185	0.635477	0.317738	+	0.948178i	$0.397077\pi$
$44$	72.2372	0.247504
$45$	59.1550	0.195963
$46$	87.7966	0.281411
$47$	−216.321	−0.671356	−0.335678	−	0.941977i	$-0.608965\pi$
$47$	−216.321	−0.671356	−0.335678	+	0.941977i	$0.608965\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−163.597	−0.462722
$51$	127.017	0.348743
$52$	52.0000	0.138675
$53$	713.003	1.84790	0.923948	−	0.382518i	$-0.124943\pi$
$53$	713.003	1.84790	0.923948	+	0.382518i	$0.124943\pi$
$54$	−54.0000	−0.136083
$55$	118.700	0.291009
$56$	−56.0000	−0.133631
$57$	−233.121	−0.541712
$58$	242.150	0.548203
$59$	403.004	0.889264	0.444632	−	0.895713i	$-0.353334\pi$
$59$	403.004	0.889264	0.444632	+	0.895713i	$0.353334\pi$
$60$	−78.8734	−0.169709
$61$	246.765	0.517950	0.258975	−	0.965884i	$-0.416615\pi$
$61$	246.765	0.517950	0.258975	+	0.965884i	$0.416615\pi$
$62$	496.429	1.01688
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	85.4462	0.163051
$66$	−108.356	−0.202086
$67$	−742.301	−1.35353	−0.676765	−	0.736199i	$-0.736619\pi$
$67$	−742.301	−1.35353	−0.676765	+	0.736199i	$0.736619\pi$
$68$	−169.356	−0.302021
$69$	−131.695	−0.229771
$70$	−92.0190	−0.157120
$71$	−65.6086	−0.109666	−0.0548331	−	0.998496i	$-0.517463\pi$
$71$	−65.6086	−0.109666	−0.0548331	+	0.998496i	$0.517463\pi$
$72$	72.0000	0.117851
$73$	965.099	1.54735	0.773673	−	0.633585i	$-0.218417\pi$
$73$	965.099	1.54735	0.773673	+	0.633585i	$0.218417\pi$
$74$	−559.635	−0.879138
$75$	245.396	0.377811
$76$	310.827	0.469136
$77$	−126.415	−0.187095
$78$	−78.0000	−0.113228
$79$	382.593	0.544874	0.272437	−	0.962174i	$-0.412170\pi$
$79$	382.593	0.544874	0.272437	+	0.962174i	$0.412170\pi$
$80$	105.165	0.146972
$81$	81.0000	0.111111
$82$	851.961	1.14736
$83$	−1347.88	−1.78252	−0.891258	−	0.453496i	$-0.850177\pi$
$83$	−1347.88	−1.78252	−0.891258	+	0.453496i	$0.850177\pi$
$84$	84.0000	0.109109
$85$	−278.285	−0.355108
$86$	358.371	0.449350
$87$	−363.224	−0.447606
$88$	144.474	0.175012
$89$	830.091	0.988646	0.494323	−	0.869278i	$-0.335416\pi$
$89$	830.091	0.988646	0.494323	+	0.869278i	$0.335416\pi$
$90$	118.310	0.138566
$91$	−91.0000	−0.104828
$92$	175.593	0.198988
$93$	−744.643	−0.830278
$94$	−432.643	−0.474720
$95$	510.750	0.551599
$96$	−96.0000	−0.102062
$97$	376.407	0.394003	0.197002	−	0.980403i	$-0.436880\pi$
$97$	376.407	0.394003	0.197002	+	0.980403i	$0.436880\pi$
$98$	98.0000	0.101015
$99$	162.534	0.165003
$100$	−327.194	−0.327194
$101$	646.395	0.636819	0.318409	−	0.947953i	$-0.396851\pi$
$101$	646.395	0.636819	0.318409	+	0.947953i	$0.396851\pi$
$102$	254.034	0.246599
$103$	41.7528	0.0399420	0.0199710	−	0.999801i	$-0.493643\pi$
$103$	41.7528	0.0399420	0.0199710	+	0.999801i	$0.493643\pi$
$104$	104.000	0.0980581
$105$	138.028	0.128288
$106$	1426.01	1.30666
$107$	−536.924	−0.485107	−0.242554	−	0.970138i	$-0.577985\pi$
$107$	−536.924	−0.485107	−0.242554	+	0.970138i	$0.577985\pi$
$108$	−108.000	−0.0962250
$109$	1959.48	1.72187	0.860935	−	0.508715i	$-0.169879\pi$
$109$	1959.48	1.72187	0.860935	+	0.508715i	$0.169879\pi$
$110$	237.400	0.205774
$111$	839.452	0.717813
$112$	−112.000	−0.0944911
$113$	1503.52	1.25168	0.625838	−	0.779953i	$-0.284757\pi$
$113$	1503.52	1.25168	0.625838	+	0.779953i	$0.284757\pi$
$114$	−466.241	−0.383048
$115$	288.534	0.233965
$116$	484.299	0.387638
$117$	117.000	0.0924500
$118$	806.007	0.628805
$119$	296.373	0.228306
$120$	−157.747	−0.120002
$121$	−1004.86	−0.754967
$122$	493.529	0.366246
$123$	−1277.94	−0.936814
$124$	992.857	0.719042
$125$	−1359.24	−0.972594
$126$	−126.000	−0.0890871
$127$	−506.342	−0.353784	−0.176892	−	0.984230i	$-0.556604\pi$
$127$	−506.342	−0.353784	−0.176892	+	0.984230i	$0.556604\pi$
$128$	128.000	0.0883883
$129$	−537.556	−0.366893
$130$	170.892	0.115294
$131$	−1140.76	−0.760833	−0.380416	−	0.924815i	$-0.624219\pi$
$131$	−1140.76	−0.760833	−0.380416	+	0.924815i	$0.624219\pi$
$132$	−216.712	−0.142896
$133$	−543.948	−0.354634
$134$	−1484.60	−0.957090
$135$	−177.465	−0.113139
$136$	−338.712	−0.213561
$137$	−2042.68	−1.27386	−0.636928	−	0.770923i	$-0.719795\pi$
$137$	−2042.68	−1.27386	−0.636928	+	0.770923i	$0.719795\pi$
$138$	−263.390	−0.162473
$139$	85.0793	0.0519161	0.0259580	−	0.999663i	$-0.491736\pi$
$139$	85.0793	0.0519161	0.0259580	+	0.999663i	$0.491736\pi$
$140$	−184.038	−0.111100
$141$	648.964	0.387607
$142$	−131.217	−0.0775458
$143$	234.771	0.137291
$144$	144.000	0.0833333
$145$	795.798	0.455775
$146$	1930.20	1.09414
$147$	−147.000	−0.0824786
$148$	−1119.27	−0.621645
$149$	−397.408	−0.218503	−0.109251	−	0.994014i	$-0.534845\pi$
$149$	−397.408	−0.218503	−0.109251	+	0.994014i	$0.534845\pi$
$150$	490.791	0.267153
$151$	−3099.79	−1.67058	−0.835290	−	0.549810i	$-0.814700\pi$
$151$	−3099.79	−1.67058	−0.835290	+	0.549810i	$0.814700\pi$
$152$	621.655	0.331729
$153$	−381.050	−0.201347
$154$	−252.830	−0.132296
$155$	1631.46	0.845431
$156$	−156.000	−0.0800641
$157$	2827.44	1.43729	0.718645	−	0.695377i	$-0.244763\pi$
$157$	2827.44	1.43729	0.718645	+	0.695377i	$0.244763\pi$
$158$	765.186	0.385284
$159$	−2139.01	−1.06688
$160$	210.329	0.103925
$161$	−307.288	−0.150420
$162$	162.000	0.0785674
$163$	−3011.00	−1.44687	−0.723436	−	0.690392i	$-0.757438\pi$
$163$	−3011.00	−1.44687	−0.723436	+	0.690392i	$0.757438\pi$
$164$	1703.92	0.811304
$165$	−356.100	−0.168014
$166$	−2695.76	−1.26043
$167$	1853.19	0.858708	0.429354	−	0.903136i	$-0.358741\pi$
$167$	1853.19	0.858708	0.429354	+	0.903136i	$0.358741\pi$
$168$	168.000	0.0771517
$169$	169.000	0.0769231
$170$	−556.569	−0.251100
$171$	699.362	0.312757
$172$	716.741	0.317738
$173$	−4400.17	−1.93375	−0.966874	−	0.255255i	$-0.917841\pi$
$173$	−4400.17	−1.93375	−0.966874	+	0.255255i	$0.917841\pi$
$174$	−726.449	−0.316505
$175$	572.590	0.247336
$176$	288.949	0.123752
$177$	−1209.01	−0.513417
$178$	1660.18	0.699078
$179$	−3617.60	−1.51057	−0.755285	−	0.655397i	$-0.772502\pi$
$179$	−3617.60	−1.51057	−0.755285	+	0.655397i	$0.772502\pi$
$180$	236.620	0.0979813
$181$	396.014	0.162627	0.0813135	−	0.996689i	$-0.474089\pi$
$181$	396.014	0.162627	0.0813135	+	0.996689i	$0.474089\pi$
$182$	−182.000	−0.0741249
$183$	−740.294	−0.299039
$184$	351.186	0.140705
$185$	−1839.18	−0.730914
$186$	−1489.29	−0.587095
$187$	−764.612	−0.299005
$188$	−865.286	−0.335678
$189$	189.000	0.0727393
$190$	1021.50	0.390039
$191$	−2672.76	−1.01253	−0.506267	−	0.862377i	$-0.668975\pi$
$191$	−2672.76	−1.01253	−0.506267	+	0.862377i	$0.668975\pi$
$192$	−192.000	−0.0721688
$193$	1025.21	0.382363	0.191182	−	0.981555i	$-0.438768\pi$
$193$	1025.21	0.382363	0.191182	+	0.981555i	$0.438768\pi$
$194$	752.814	0.278603
$195$	−256.339	−0.0941373
$196$	196.000	0.0714286
$197$	−3848.98	−1.39202	−0.696011	−	0.718032i	$-0.745043\pi$
$197$	−3848.98	−1.39202	−0.696011	+	0.718032i	$0.745043\pi$
$198$	325.068	0.116674
$199$	−4509.60	−1.60642	−0.803209	−	0.595697i	$-0.796876\pi$
$199$	−4509.60	−1.60642	−0.803209	+	0.595697i	$0.796876\pi$
$200$	−654.388	−0.231361
$201$	2226.90	0.781461
$202$	1292.79	0.450299
$203$	−847.523	−0.293027
$204$	508.067	0.174372
$205$	2799.88	0.953912
$206$	83.5057	0.0282433
$207$	395.085	0.132658
$208$	208.000	0.0693375
$209$	1403.33	0.464452
$210$	276.057	0.0907130
$211$	3323.88	1.08448	0.542240	−	0.840223i	$-0.317576\pi$
$211$	3323.88	1.08448	0.542240	+	0.840223i	$0.317576\pi$
$212$	2852.01	0.923948
$213$	196.826	0.0633158
$214$	−1073.85	−0.343022
$215$	1177.75	0.373589
$216$	−216.000	−0.0680414
$217$	−1737.50	−0.543545
$218$	3918.95	1.21755
$219$	−2895.30	−0.893361
$220$	474.800	0.145505
$221$	−550.406	−0.167531
$222$	1678.90	0.507571
$223$	1938.33	0.582062	0.291031	−	0.956714i	$-0.406002\pi$
$223$	1938.33	0.582062	0.291031	+	0.956714i	$0.406002\pi$
$224$	−224.000	−0.0668153
$225$	−736.187	−0.218129
$226$	3007.04	0.885069
$227$	−2557.44	−0.747768	−0.373884	−	0.927475i	$-0.621974\pi$
$227$	−2557.44	−0.747768	−0.373884	+	0.927475i	$0.621974\pi$
$228$	−932.482	−0.270856
$229$	−3873.48	−1.11776	−0.558879	−	0.829249i	$-0.688768\pi$
$229$	−3873.48	−1.11776	−0.558879	+	0.829249i	$0.688768\pi$
$230$	577.068	0.165438
$231$	379.246	0.108020
$232$	968.598	0.274102
$233$	−226.544	−0.0636970	−0.0318485	−	0.999493i	$-0.510139\pi$
$233$	−226.544	−0.0636970	−0.0318485	+	0.999493i	$0.510139\pi$
$234$	234.000	0.0653720
$235$	−1421.83	−0.394682
$236$	1612.01	0.444632
$237$	−1147.78	−0.314583
$238$	592.745	0.161437
$239$	−4141.26	−1.12082	−0.560410	−	0.828216i	$-0.689356\pi$
$239$	−4141.26	−1.12082	−0.560410	+	0.828216i	$0.689356\pi$
$240$	−315.494	−0.0848543
$241$	2450.96	0.655104	0.327552	−	0.944833i	$-0.393776\pi$
$241$	2450.96	0.655104	0.327552	+	0.944833i	$0.393776\pi$
$242$	−2009.72	−0.533842
$243$	−243.000	−0.0641500
$244$	987.059	0.258975
$245$	322.066	0.0839839
$246$	−2555.88	−0.662427
$247$	1010.19	0.260230
$248$	1985.71	0.508439
$249$	4043.63	1.02914
$250$	−2718.48	−0.687728
$251$	737.540	0.185471	0.0927353	−	0.995691i	$-0.470439\pi$
$251$	737.540	0.185471	0.0927353	+	0.995691i	$0.470439\pi$
$252$	−252.000	−0.0629941
$253$	792.773	0.197001
$254$	−1012.68	−0.250163
$255$	834.854	0.205022
$256$	256.000	0.0625000
$257$	3381.82	0.820825	0.410412	−	0.911900i	$-0.365385\pi$
$257$	3381.82	0.820825	0.410412	+	0.911900i	$0.365385\pi$
$258$	−1075.11	−0.259432
$259$	1958.72	0.469919
$260$	341.785	0.0815253
$261$	1089.67	0.258425
$262$	−2281.53	−0.537990
$263$	3031.47	0.710755	0.355377	−	0.934723i	$-0.384352\pi$
$263$	3031.47	0.710755	0.355377	+	0.934723i	$0.384352\pi$
$264$	−433.423	−0.101043
$265$	4686.41	1.08635
$266$	−1087.90	−0.250764
$267$	−2490.27	−0.570795
$268$	−2969.20	−0.676765
$269$	6048.97	1.37105	0.685524	−	0.728050i	$-0.259573\pi$
$269$	6048.97	1.37105	0.685524	+	0.728050i	$0.259573\pi$
$270$	−354.930	−0.0800014
$271$	7457.62	1.67165	0.835827	−	0.548993i	$-0.184989\pi$
$271$	7457.62	1.67165	0.835827	+	0.548993i	$0.184989\pi$
$272$	−677.423	−0.151010
$273$	273.000	0.0605228
$274$	−4085.37	−0.900752
$275$	−1477.23	−0.323927
$276$	−526.780	−0.114886
$277$	−3427.82	−0.743531	−0.371765	−	0.928327i	$-0.621247\pi$
$277$	−3427.82	−0.743531	−0.371765	+	0.928327i	$0.621247\pi$
$278$	170.159	0.0367102
$279$	2233.93	0.479361
$280$	−368.076	−0.0785598
$281$	−3646.53	−0.774142	−0.387071	−	0.922050i	$-0.626513\pi$
$281$	−3646.53	−0.774142	−0.387071	+	0.922050i	$0.626513\pi$
$282$	1297.93	0.274080
$283$	−3135.21	−0.658547	−0.329273	−	0.944235i	$-0.606804\pi$
$283$	−3135.21	−0.658547	−0.329273	+	0.944235i	$0.606804\pi$
$284$	−262.434	−0.0548331
$285$	−1532.25	−0.318466
$286$	469.542	0.0970790
$287$	−2981.86	−0.613289
$288$	288.000	0.0589256
$289$	−3120.41	−0.635134
$290$	1591.60	0.322282
$291$	−1129.22	−0.227478
$292$	3860.40	0.773673
$293$	−6551.35	−1.30626	−0.653129	−	0.757246i	$-0.726544\pi$
$293$	−6551.35	−1.30626	−0.653129	+	0.757246i	$0.726544\pi$
$294$	−294.000	−0.0583212
$295$	2648.85	0.522787
$296$	−2238.54	−0.439569
$297$	−487.601	−0.0952643
$298$	−794.816	−0.154505
$299$	570.678	0.110378
$300$	981.582	0.188906
$301$	−1254.30	−0.240188
$302$	−6199.58	−1.18128
$303$	−1939.19	−0.367668
$304$	1243.31	0.234568
$305$	1621.93	0.304497
$306$	−762.101	−0.142374
$307$	8630.62	1.60448	0.802240	−	0.597002i	$-0.203641\pi$
$307$	8630.62	1.60448	0.802240	+	0.597002i	$0.203641\pi$
$308$	−505.661	−0.0935477
$309$	−125.259	−0.0230605
$310$	3262.92	0.597810
$311$	−5438.07	−0.991526	−0.495763	−	0.868458i	$-0.665112\pi$
$311$	−5438.07	−0.991526	−0.495763	+	0.868458i	$0.665112\pi$
$312$	−312.000	−0.0566139
$313$	4103.38	0.741012	0.370506	−	0.928830i	$-0.379184\pi$
$313$	4103.38	0.741012	0.370506	+	0.928830i	$0.379184\pi$
$314$	5654.89	1.01632
$315$	−414.085	−0.0740669
$316$	1530.37	0.272437
$317$	−4303.28	−0.762449	−0.381225	−	0.924482i	$-0.624498\pi$
$317$	−4303.28	−0.762449	−0.381225	+	0.924482i	$0.624498\pi$
$318$	−4278.02	−0.754400
$319$	2186.53	0.383768
$320$	420.658	0.0734859
$321$	1610.77	0.280077
$322$	−614.576	−0.106363
$323$	−3290.03	−0.566755
$324$	324.000	0.0555556
$325$	−1063.38	−0.181495
$326$	−6022.01	−1.02309
$327$	−5878.43	−0.994122
$328$	3407.84	0.573679
$329$	1514.25	0.253749
$330$	−712.200	−0.118804
$331$	−4518.12	−0.750266	−0.375133	−	0.926971i	$-0.622403\pi$
$331$	−4518.12	−0.750266	−0.375133	+	0.926971i	$0.622403\pi$
$332$	−5391.51	−0.891258
$333$	−2518.36	−0.414430
$334$	3706.38	0.607198
$335$	−4878.98	−0.795723
$336$	336.000	0.0545545
$337$	7466.07	1.20683	0.603416	−	0.797426i	$-0.293806\pi$
$337$	7466.07	1.20683	0.603416	+	0.797426i	$0.293806\pi$
$338$	338.000	0.0543928
$339$	−4510.57	−0.722656
$340$	−1113.14	−0.177554
$341$	4482.58	0.711863
$342$	1398.72	0.221153
$343$	−343.000	−0.0539949
$344$	1433.48	0.224675
$345$	−865.602	−0.135080
$346$	−8800.33	−1.36737
$347$	504.632	0.0780694	0.0390347	−	0.999238i	$-0.487572\pi$
$347$	504.632	0.0780694	0.0390347	+	0.999238i	$0.487572\pi$
$348$	−1452.90	−0.223803
$349$	8535.35	1.30913	0.654566	−	0.756005i	$-0.272851\pi$
$349$	8535.35	1.30913	0.654566	+	0.756005i	$0.272851\pi$
$350$	1145.18	0.174893
$351$	−351.000	−0.0533761
$352$	577.898	0.0875059
$353$	−4330.89	−0.653002	−0.326501	−	0.945197i	$-0.605870\pi$
$353$	−4330.89	−0.653002	−0.326501	+	0.945197i	$0.605870\pi$
$354$	−2418.02	−0.363041
$355$	−431.231	−0.0644714
$356$	3320.36	0.494323
$357$	−889.118	−0.131813
$358$	−7235.20	−1.06813
$359$	−6906.32	−1.01532	−0.507662	−	0.861556i	$-0.669490\pi$
$359$	−6906.32	−1.01532	−0.507662	+	0.861556i	$0.669490\pi$
$360$	473.240	0.0692832
$361$	−820.646	−0.119645
$362$	792.028	0.114995
$363$	3014.58	0.435880
$364$	−364.000	−0.0524142
$365$	6343.39	0.909666
$366$	−1480.59	−0.211452
$367$	766.383	0.109005	0.0545025	−	0.998514i	$-0.482643\pi$
$367$	766.383	0.109005	0.0545025	+	0.998514i	$0.482643\pi$
$368$	702.373	0.0994938
$369$	3833.82	0.540870
$370$	−3678.36	−0.516834
$371$	−4991.02	−0.698439
$372$	−2978.57	−0.415139
$373$	5357.17	0.743657	0.371828	−	0.928302i	$-0.378731\pi$
$373$	5357.17	0.743657	0.371828	+	0.928302i	$0.378731\pi$
$374$	−1529.22	−0.211429
$375$	4077.73	0.561528
$376$	−1730.57	−0.237360
$377$	1573.97	0.215023
$378$	378.000	0.0514344
$379$	10300.5	1.39605	0.698024	−	0.716074i	$-0.254063\pi$
$379$	10300.5	1.39605	0.698024	+	0.716074i	$0.254063\pi$
$380$	2043.00	0.275799
$381$	1519.02	0.204257
$382$	−5345.51	−0.715969
$383$	−3326.55	−0.443809	−0.221904	−	0.975068i	$-0.571227\pi$
$383$	−3326.55	−0.443809	−0.221904	+	0.975068i	$0.571227\pi$
$384$	−384.000	−0.0510310
$385$	−830.899	−0.109991
$386$	2050.42	0.270372
$387$	1612.67	0.211826
$388$	1505.63	0.197002
$389$	−7172.37	−0.934843	−0.467421	−	0.884035i	$-0.654817\pi$
$389$	−7172.37	−0.934843	−0.467421	+	0.884035i	$0.654817\pi$
$390$	−512.677	−0.0665652
$391$	−1858.61	−0.240393
$392$	392.000	0.0505076
$393$	3422.29	0.439267
$394$	−7697.95	−0.984308
$395$	2514.70	0.320325
$396$	650.135	0.0825013
$397$	−2251.67	−0.284655	−0.142328	−	0.989820i	$-0.545459\pi$
$397$	−2251.67	−0.284655	−0.142328	+	0.989820i	$0.545459\pi$
$398$	−9019.21	−1.13591
$399$	1631.84	0.204748
$400$	−1308.78	−0.163597
$401$	−10811.6	−1.34639	−0.673197	−	0.739463i	$-0.735080\pi$
$401$	−10811.6	−1.34639	−0.673197	+	0.739463i	$0.735080\pi$
$402$	4453.80	0.552576
$403$	3226.79	0.398853
$404$	2585.58	0.318409
$405$	532.395	0.0653208
$406$	−1695.05	−0.207201
$407$	−5053.31	−0.615438
$408$	1016.13	0.123299
$409$	−4439.26	−0.536693	−0.268347	−	0.963322i	$-0.586477\pi$
$409$	−4439.26	−0.536693	−0.268347	+	0.963322i	$0.586477\pi$
$410$	5599.75	0.674517
$411$	6128.05	0.735461
$412$	167.011	0.0199710
$413$	−2821.02	−0.336110
$414$	790.170	0.0938037
$415$	−8859.31	−1.04792
$416$	416.000	0.0490290
$417$	−255.238	−0.0299738
$418$	2806.66	0.328417
$419$	4859.95	0.566644	0.283322	−	0.959025i	$-0.408563\pi$
$419$	4859.95	0.566644	0.283322	+	0.959025i	$0.408563\pi$
$420$	552.114	0.0641438
$421$	12316.8	1.42585	0.712927	−	0.701238i	$-0.247369\pi$
$421$	12316.8	1.42585	0.712927	+	0.701238i	$0.247369\pi$
$422$	6647.76	0.766844
$423$	−1946.89	−0.223785
$424$	5704.02	0.653330
$425$	3463.26	0.395278
$426$	393.651	0.0447711
$427$	−1727.35	−0.195767
$428$	−2147.70	−0.242554
$429$	−704.313	−0.0792647
$430$	2355.49	0.264167
$431$	−6239.42	−0.697314	−0.348657	−	0.937250i	$-0.613362\pi$
$431$	−6239.42	−0.697314	−0.348657	+	0.937250i	$0.613362\pi$
$432$	−432.000	−0.0481125
$433$	5522.10	0.612876	0.306438	−	0.951891i	$-0.400863\pi$
$433$	5522.10	0.612876	0.306438	+	0.951891i	$0.400863\pi$
$434$	−3475.00	−0.384344
$435$	−2387.39	−0.263142
$436$	7837.91	0.860935
$437$	3411.20	0.373409
$438$	−5790.60	−0.631702
$439$	11311.7	1.22979	0.614896	−	0.788608i	$-0.289198\pi$
$439$	11311.7	1.22979	0.614896	+	0.788608i	$0.289198\pi$
$440$	949.599	0.102887
$441$	441.000	0.0476190
$442$	−1100.81	−0.118462
$443$	−2739.63	−0.293823	−0.146912	−	0.989150i	$-0.546933\pi$
$443$	−2739.63	−0.293823	−0.146912	+	0.989150i	$0.546933\pi$
$444$	3357.81	0.358907
$445$	5456.01	0.581212
$446$	3876.65	0.411580
$447$	1192.22	0.126153
$448$	−448.000	−0.0472456
$449$	−8871.18	−0.932421	−0.466210	−	0.884674i	$-0.654381\pi$
$449$	−8871.18	−0.932421	−0.466210	+	0.884674i	$0.654381\pi$
$450$	−1472.37	−0.154241
$451$	7692.91	0.803204
$452$	6014.09	0.625838
$453$	9299.38	0.964510
$454$	−5114.89	−0.528752
$455$	−598.123	−0.0616274
$456$	−1864.96	−0.191524
$457$	1435.24	0.146909	0.0734547	−	0.997299i	$-0.476598\pi$
$457$	1435.24	0.146909	0.0734547	+	0.997299i	$0.476598\pi$
$458$	−7746.95	−0.790374
$459$	1143.15	0.116248
$460$	1154.14	0.116982
$461$	151.976	0.0153541	0.00767706	−	0.999971i	$-0.497556\pi$
$461$	151.976	0.0153541	0.00767706	+	0.999971i	$0.497556\pi$
$462$	758.491	0.0763814
$463$	−1057.75	−0.106172	−0.0530860	−	0.998590i	$-0.516906\pi$
$463$	−1057.75	−0.106172	−0.0530860	+	0.998590i	$0.516906\pi$
$464$	1937.20	0.193819
$465$	−4894.37	−0.488110
$466$	−453.088	−0.0450406
$467$	15537.7	1.53961	0.769806	−	0.638278i	$-0.220353\pi$
$467$	15537.7	1.53961	0.769806	+	0.638278i	$0.220353\pi$
$468$	468.000	0.0462250
$469$	5196.11	0.511586
$470$	−2843.67	−0.279082
$471$	−8482.33	−0.829820
$472$	3224.03	0.314402
$473$	3235.96	0.314566
$474$	−2295.56	−0.222444
$475$	−6356.31	−0.613994
$476$	1185.49	0.114153
$477$	6417.03	0.615965
$478$	−8282.52	−0.792539
$479$	2046.03	0.195168	0.0975842	−	0.995227i	$-0.468888\pi$
$479$	2046.03	0.195168	0.0975842	+	0.995227i	$0.468888\pi$
$480$	−630.987	−0.0600010
$481$	−3637.63	−0.344826
$482$	4901.92	0.463228
$483$	921.864	0.0868453
$484$	−4019.44	−0.377484
$485$	2474.04	0.231630
$486$	−486.000	−0.0453609
$487$	10798.4	1.00477	0.502383	−	0.864645i	$-0.332457\pi$
$487$	10798.4	1.00477	0.502383	+	0.864645i	$0.332457\pi$
$488$	1974.12	0.183123
$489$	9033.01	0.835351
$490$	644.133	0.0593856
$491$	15026.5	1.38114	0.690568	−	0.723267i	$-0.257360\pi$
$491$	15026.5	1.38114	0.690568	+	0.723267i	$0.257360\pi$
$492$	−5111.76	−0.468407
$493$	−5126.18	−0.468299
$494$	2020.38	0.184010
$495$	1068.30	0.0970030
$496$	3971.43	0.359521
$497$	459.260	0.0414499
$498$	8087.27	0.727709
$499$	2603.35	0.233551	0.116776	−	0.993158i	$-0.462744\pi$
$499$	2603.35	0.233551	0.116776	+	0.993158i	$0.462744\pi$
$500$	−5436.97	−0.486297
$501$	−5559.58	−0.495775
$502$	1475.08	0.131148
$503$	4785.14	0.424173	0.212086	−	0.977251i	$-0.431974\pi$
$503$	4785.14	0.424173	0.212086	+	0.977251i	$0.431974\pi$
$504$	−504.000	−0.0445435
$505$	4248.61	0.374378
$506$	1585.55	0.139301
$507$	−507.000	−0.0444116
$508$	−2025.37	−0.176892
$509$	17105.9	1.48960	0.744799	−	0.667289i	$-0.232545\pi$
$509$	17105.9	1.48960	0.744799	+	0.667289i	$0.232545\pi$
$510$	1669.71	0.144972
$511$	−6755.69	−0.584842
$512$	512.000	0.0441942
$513$	−2098.08	−0.180571
$514$	6763.64	0.580411
$515$	274.432	0.0234814
$516$	−2150.22	−0.183446
$517$	−3906.62	−0.332326
$518$	3917.44	0.332283
$519$	13200.5	1.11645
$520$	683.569	0.0576471
$521$	−16650.7	−1.40016	−0.700079	−	0.714065i	$-0.746852\pi$
$521$	−16650.7	−1.40016	−0.700079	+	0.714065i	$0.746852\pi$
$522$	2179.35	0.182734
$523$	−18616.3	−1.55647	−0.778234	−	0.627975i	$-0.783884\pi$
$523$	−18616.3	−1.55647	−0.778234	+	0.627975i	$0.783884\pi$
$524$	−4563.06	−0.380416
$525$	−1717.77	−0.142799
$526$	6062.94	0.502579
$527$	−10509.1	−0.868662
$528$	−866.847	−0.0714482
$529$	−10239.9	−0.841616
$530$	9372.83	0.768169
$531$	3627.03	0.296421
$532$	−2175.79	−0.177317
$533$	5537.75	0.450031
$534$	−4980.55	−0.403613
$535$	−3529.09	−0.285188
$536$	−5938.41	−0.478545
$537$	10852.8	0.872128
$538$	12097.9	0.969478
$539$	884.906	0.0707154
$540$	−709.860	−0.0565695
$541$	6924.25	0.550271	0.275136	−	0.961405i	$-0.411277\pi$
$541$	6924.25	0.550271	0.275136	+	0.961405i	$0.411277\pi$
$542$	14915.2	1.18204
$543$	−1188.04	−0.0938927
$544$	−1354.85	−0.106780
$545$	12879.2	1.01227
$546$	546.000	0.0427960
$547$	11206.9	0.876000	0.438000	−	0.898975i	$-0.355687\pi$
$547$	11206.9	0.876000	0.438000	+	0.898975i	$0.355687\pi$
$548$	−8170.74	−0.636928
$549$	2220.88	0.172650
$550$	−2954.45	−0.229051
$551$	9408.34	0.727420
$552$	−1053.56	−0.0812363
$553$	−2678.15	−0.205943
$554$	−6855.65	−0.525756
$555$	5517.54	0.421993
$556$	340.317	0.0259580
$557$	24497.1	1.86351	0.931754	−	0.363091i	$-0.118279\pi$
$557$	24497.1	1.86351	0.931754	+	0.363091i	$0.118279\pi$
$558$	4467.86	0.338960
$559$	2329.41	0.176250
$560$	−736.152	−0.0555501
$561$	2293.84	0.172631
$562$	−7293.06	−0.547401
$563$	−12058.6	−0.902680	−0.451340	−	0.892352i	$-0.649054\pi$
$563$	−12058.6	−0.902680	−0.451340	+	0.892352i	$0.649054\pi$
$564$	2595.86	0.193804
$565$	9882.32	0.735845
$566$	−6270.42	−0.465663
$567$	−567.000	−0.0419961
$568$	−524.869	−0.0387729
$569$	4121.76	0.303678	0.151839	−	0.988405i	$-0.451480\pi$
$569$	4121.76	0.303678	0.151839	+	0.988405i	$0.451480\pi$
$570$	−3064.50	−0.225189
$571$	−11260.1	−0.825257	−0.412629	−	0.910899i	$-0.635389\pi$
$571$	−11260.1	−0.825257	−0.412629	+	0.910899i	$0.635389\pi$
$572$	939.084	0.0686453
$573$	8018.27	0.584586
$574$	−5963.73	−0.433660
$575$	−3590.82	−0.260430
$576$	576.000	0.0416667
$577$	−5111.73	−0.368811	−0.184406	−	0.982850i	$-0.559036\pi$
$577$	−5111.73	−0.368811	−0.184406	+	0.982850i	$0.559036\pi$
$578$	−6240.83	−0.449108
$579$	−3075.63	−0.220758
$580$	3183.19	0.227888
$581$	9435.15	0.673728
$582$	−2258.44	−0.160851
$583$	12876.3	0.914723
$584$	7720.79	0.547070
$585$	769.016	0.0543502
$586$	−13102.7	−0.923664
$587$	−12902.9	−0.907253	−0.453627	−	0.891192i	$-0.649870\pi$
$587$	−12902.9	−0.907253	−0.453627	+	0.891192i	$0.649870\pi$
$588$	−588.000	−0.0412393
$589$	19287.9	1.34931
$590$	5297.71	0.369666
$591$	11546.9	0.803684
$592$	−4477.08	−0.310822
$593$	−559.726	−0.0387609	−0.0193804	−	0.999812i	$-0.506169\pi$
$593$	−559.726	−0.0387609	−0.0193804	+	0.999812i	$0.506169\pi$
$594$	−975.203	−0.0673621
$595$	1947.99	0.134218
$596$	−1589.63	−0.109251
$597$	13528.8	0.927466
$598$	1141.36	0.0780494
$599$	23453.1	1.59978	0.799889	−	0.600148i	$-0.204892\pi$
$599$	23453.1	1.59978	0.799889	+	0.600148i	$0.204892\pi$
$600$	1963.16	0.133576
$601$	−11687.3	−0.793239	−0.396619	−	0.917983i	$-0.629817\pi$
$601$	−11687.3	−0.793239	−0.396619	+	0.917983i	$0.629817\pi$
$602$	−2508.59	−0.169838
$603$	−6680.71	−0.451176
$604$	−12399.2	−0.835290
$605$	−6604.73	−0.443836
$606$	−3878.37	−0.259980
$607$	4531.12	0.302986	0.151493	−	0.988458i	$-0.451592\pi$
$607$	4531.12	0.302986	0.151493	+	0.988458i	$0.451592\pi$
$608$	2486.62	0.165865
$609$	2542.57	0.169179
$610$	3243.86	0.215312
$611$	−2812.18	−0.186201
$612$	−1524.20	−0.100674
$613$	−6577.62	−0.433389	−0.216695	−	0.976239i	$-0.569528\pi$
$613$	−6577.62	−0.433389	−0.216695	+	0.976239i	$0.569528\pi$
$614$	17261.2	1.13454
$615$	−8399.63	−0.550741
$616$	−1011.32	−0.0661482
$617$	10214.3	0.666473	0.333236	−	0.942843i	$-0.391859\pi$
$617$	10214.3	0.666473	0.333236	+	0.942843i	$0.391859\pi$
$618$	−250.517	−0.0163063
$619$	−17248.0	−1.11996	−0.559980	−	0.828506i	$-0.689191\pi$
$619$	−17248.0	−1.11996	−0.559980	+	0.828506i	$0.689191\pi$
$620$	6525.83	0.422716
$621$	−1185.25	−0.0765904
$622$	−10876.1	−0.701115
$623$	−5810.64	−0.373673
$624$	−624.000	−0.0400320
$625$	1290.82	0.0826123
$626$	8206.76	0.523975
$627$	−4210.00	−0.268152
$628$	11309.8	0.718645
$629$	11847.2	0.750998
$630$	−828.171	−0.0523732
$631$	9107.60	0.574592	0.287296	−	0.957842i	$-0.407244\pi$
$631$	9107.60	0.574592	0.287296	+	0.957842i	$0.407244\pi$
$632$	3060.74	0.192642
$633$	−9971.64	−0.626125
$634$	−8606.57	−0.539133
$635$	−3328.07	−0.207985
$636$	−8556.03	−0.533442
$637$	637.000	0.0396214
$638$	4373.05	0.271365
$639$	−590.477	−0.0365554
$640$	841.316	0.0519624
$641$	10871.5	0.669887	0.334944	−	0.942238i	$-0.391283\pi$
$641$	10871.5	0.669887	0.334944	+	0.942238i	$0.391283\pi$
$642$	3221.55	0.198044
$643$	−8150.23	−0.499866	−0.249933	−	0.968263i	$-0.580409\pi$
$643$	−8150.23	−0.499866	−0.249933	+	0.968263i	$0.580409\pi$
$644$	−1229.15	−0.0752102
$645$	−3533.24	−0.215692
$646$	−6580.05	−0.400756
$647$	−12505.3	−0.759865	−0.379933	−	0.925014i	$-0.624053\pi$
$647$	−12505.3	−0.759865	−0.379933	+	0.925014i	$0.624053\pi$
$648$	648.000	0.0392837
$649$	7277.97	0.440193
$650$	−2126.76	−0.128336
$651$	5212.50	0.313816
$652$	−12044.0	−0.723436
$653$	−11681.2	−0.700033	−0.350016	−	0.936744i	$-0.613824\pi$
$653$	−11681.2	−0.700033	−0.350016	+	0.936744i	$0.613824\pi$
$654$	−11756.9	−0.702950
$655$	−7498.00	−0.447284
$656$	6815.69	0.405652
$657$	8685.89	0.515782
$658$	3028.50	0.179427
$659$	−7375.64	−0.435985	−0.217992	−	0.975950i	$-0.569951\pi$
$659$	−7375.64	−0.435985	−0.217992	+	0.975950i	$0.569951\pi$
$660$	−1424.40	−0.0840071
$661$	13919.0	0.819044	0.409522	−	0.912300i	$-0.365696\pi$
$661$	13919.0	0.819044	0.409522	+	0.912300i	$0.365696\pi$
$662$	−9036.23	−0.530518
$663$	1651.22	0.0967240
$664$	−10783.0	−0.630215
$665$	−3575.25	−0.208485
$666$	−5036.71	−0.293046
$667$	5314.98	0.308541
$668$	7412.77	0.429354
$669$	−5814.98	−0.336054
$670$	−9757.96	−0.562661
$671$	4456.40	0.256390
$672$	672.000	0.0385758
$673$	1972.29	0.112966	0.0564829	−	0.998404i	$-0.482011\pi$
$673$	1972.29	0.112966	0.0564829	+	0.998404i	$0.482011\pi$
$674$	14932.1	0.853360
$675$	2208.56	0.125937
$676$	676.000	0.0384615
$677$	6306.27	0.358005	0.179003	−	0.983849i	$-0.442713\pi$
$677$	6306.27	0.358005	0.179003	+	0.983849i	$0.442713\pi$
$678$	−9021.13	−0.510995
$679$	−2634.85	−0.148919
$680$	−2226.28	−0.125550
$681$	7672.33	0.431724
$682$	8965.16	0.503363
$683$	−16430.3	−0.920478	−0.460239	−	0.887795i	$-0.652236\pi$
$683$	−16430.3	−0.920478	−0.460239	+	0.887795i	$0.652236\pi$
$684$	2797.45	0.156379
$685$	−13426.1	−0.748884
$686$	−686.000	−0.0381802
$687$	11620.4	0.645338
$688$	2866.96	0.158869
$689$	9269.04	0.512514
$690$	−1731.20	−0.0955157
$691$	−30739.4	−1.69230	−0.846151	−	0.532943i	$-0.821086\pi$
$691$	−30739.4	−1.69230	−0.846151	+	0.532943i	$0.821086\pi$
$692$	−17600.7	−0.966874
$693$	−1137.74	−0.0623651
$694$	1009.26	0.0552034
$695$	559.208	0.0305208
$696$	−2905.79	−0.158253
$697$	−18035.6	−0.980123
$698$	17070.7	0.925696
$699$	679.632	0.0367755
$700$	2290.36	0.123668
$701$	−28051.1	−1.51138	−0.755689	−	0.654930i	$-0.772698\pi$
$701$	−28051.1	−1.51138	−0.755689	+	0.654930i	$0.772698\pi$
$702$	−702.000	−0.0377426
$703$	−21743.7	−1.16654
$704$	1155.80	0.0618760
$705$	4265.50	0.227870
$706$	−8661.77	−0.461742
$707$	−4524.77	−0.240695
$708$	−4836.04	−0.256708
$709$	−35964.1	−1.90502	−0.952510	−	0.304506i	$-0.901509\pi$
$709$	−35964.1	−1.90502	−0.952510	+	0.304506i	$0.901509\pi$
$710$	−862.462	−0.0455882
$711$	3443.34	0.181625
$712$	6640.73	0.349539
$713$	10896.2	0.572322
$714$	−1778.24	−0.0932056
$715$	1543.10	0.0807114
$716$	−14470.4	−0.755285
$717$	12423.8	0.647105
$718$	−13812.6	−0.717943
$719$	30052.3	1.55878	0.779389	−	0.626541i	$-0.215530\pi$
$719$	30052.3	1.55878	0.779389	+	0.626541i	$0.215530\pi$
$720$	946.481	0.0489906
$721$	−292.270	−0.0150967
$722$	−1641.29	−0.0846019
$723$	−7352.87	−0.378224
$724$	1584.06	0.0813135
$725$	−9903.74	−0.507332
$726$	6029.17	0.308214
$727$	−20439.2	−1.04270	−0.521352	−	0.853342i	$-0.674572\pi$
$727$	−20439.2	−1.04270	−0.521352	+	0.853342i	$0.674572\pi$
$728$	−728.000	−0.0370625
$729$	729.000	0.0370370
$730$	12686.8	0.643231
$731$	−7586.52	−0.383854
$732$	−2961.18	−0.149519
$733$	25363.0	1.27804	0.639021	−	0.769189i	$-0.279340\pi$
$733$	25363.0	1.27804	0.639021	+	0.769189i	$0.279340\pi$
$734$	1532.77	0.0770782
$735$	−966.199	−0.0484881
$736$	1404.75	0.0703527
$737$	−13405.4	−0.670008
$738$	7667.65	0.382453
$739$	11265.6	0.560776	0.280388	−	0.959887i	$-0.409537\pi$
$739$	11265.6	0.560776	0.280388	+	0.959887i	$0.409537\pi$
$740$	−7356.72	−0.365457
$741$	−3030.57	−0.150244
$742$	−9982.04	−0.493871
$743$	21110.7	1.04236	0.521182	−	0.853445i	$-0.325491\pi$
$743$	21110.7	1.04236	0.521182	+	0.853445i	$0.325491\pi$
$744$	−5957.14	−0.293548
$745$	−2612.08	−0.128455
$746$	10714.3	0.525845
$747$	−12130.9	−0.594172
$748$	−3058.45	−0.149503
$749$	3758.47	0.183353
$750$	8155.45	0.397060
$751$	−33643.2	−1.63470	−0.817348	−	0.576145i	$-0.804557\pi$
$751$	−33643.2	−1.63470	−0.817348	+	0.576145i	$0.804557\pi$
$752$	−3461.14	−0.167839
$753$	−2212.62	−0.107082
$754$	3147.94	0.152044
$755$	−20374.3	−0.982113
$756$	756.000	0.0363696
$757$	−557.749	−0.0267791	−0.0133895	−	0.999910i	$-0.504262\pi$
$757$	−557.749	−0.0267791	−0.0133895	+	0.999910i	$0.504262\pi$
$758$	20601.0	0.987155
$759$	−2378.32	−0.113739
$760$	4086.00	0.195020
$761$	−15083.4	−0.718492	−0.359246	−	0.933243i	$-0.616966\pi$
$761$	−15083.4	−0.718492	−0.359246	+	0.933243i	$0.616966\pi$
$762$	3038.05	0.144432
$763$	−13716.3	−0.650806
$764$	−10691.0	−0.506267
$765$	−2504.56	−0.118369
$766$	−6653.10	−0.313820
$767$	5239.05	0.246638
$768$	−768.000	−0.0360844
$769$	−18531.2	−0.868987	−0.434494	−	0.900675i	$-0.643073\pi$
$769$	−18531.2	−0.868987	−0.434494	+	0.900675i	$0.643073\pi$
$770$	−1661.80	−0.0777754
$771$	−10145.5	−0.473904
$772$	4100.83	0.191182
$773$	−18436.6	−0.857852	−0.428926	−	0.903340i	$-0.641108\pi$
$773$	−18436.6	−0.857852	−0.428926	+	0.903340i	$0.641108\pi$
$774$	3225.34	0.149783
$775$	−20303.6	−0.941065
$776$	3011.26	0.139301
$777$	−5876.17	−0.271308
$778$	−14344.7	−0.661034
$779$	33101.6	1.52245
$780$	−1025.35	−0.0470687
$781$	−1184.85	−0.0542857
$782$	−3717.22	−0.169984
$783$	−3269.02	−0.149202
$784$	784.000	0.0357143
$785$	18584.2	0.844965
$786$	6844.59	0.310609
$787$	−43153.4	−1.95458	−0.977288	−	0.211915i	$-0.932030\pi$
$787$	−43153.4	−1.95458	−0.977288	+	0.211915i	$0.932030\pi$
$788$	−15395.9	−0.696011
$789$	−9094.41	−0.410354
$790$	5029.40	0.226504
$791$	−10524.7	−0.473089
$792$	1300.27	0.0583372
$793$	3207.94	0.143654
$794$	−4503.34	−0.201282
$795$	−14059.2	−0.627207
$796$	−18038.4	−0.803209
$797$	−28095.4	−1.24867	−0.624336	−	0.781156i	$-0.714630\pi$
$797$	−28095.4	−1.24867	−0.624336	+	0.781156i	$0.714630\pi$
$798$	3263.69	0.144779
$799$	9158.82	0.405527
$800$	−2617.55	−0.115681
$801$	7470.82	0.329549
$802$	−21623.2	−0.952045
$803$	17429.0	0.765949
$804$	8907.61	0.390730
$805$	−2019.74	−0.0884303
$806$	6453.57	0.282031
$807$	−18146.9	−0.791575
$808$	5171.16	0.225149
$809$	73.9427	0.00321346	0.00160673	−	0.999999i	$-0.499489\pi$
$809$	73.9427	0.00321346	0.00160673	+	0.999999i	$0.499489\pi$
$810$	1064.79	0.0461888
$811$	18635.3	0.806872	0.403436	−	0.915008i	$-0.367816\pi$
$811$	18635.3	0.806872	0.403436	+	0.915008i	$0.367816\pi$
$812$	−3390.09	−0.146513
$813$	−22372.9	−0.965130
$814$	−10106.6	−0.435180
$815$	−19790.7	−0.850597
$816$	2032.27	0.0871859
$817$	13923.9	0.596250
$818$	−8878.53	−0.379499
$819$	−819.000	−0.0349428
$820$	11199.5	0.476956
$821$	34847.8	1.48136	0.740680	−	0.671858i	$-0.234504\pi$
$821$	34847.8	1.48136	0.740680	+	0.671858i	$0.234504\pi$
$822$	12256.1	0.520050
$823$	−23482.4	−0.994587	−0.497293	−	0.867582i	$-0.665673\pi$
$823$	−23482.4	−0.994587	−0.497293	+	0.867582i	$0.665673\pi$
$824$	334.023	0.0141216
$825$	4431.68	0.187020
$826$	−5642.05	−0.237666
$827$	26051.3	1.09540	0.547698	−	0.836676i	$-0.315504\pi$
$827$	26051.3	1.09540	0.547698	+	0.836676i	$0.315504\pi$
$828$	1580.34	0.0663292
$829$	42125.9	1.76489	0.882445	−	0.470416i	$-0.155896\pi$
$829$	42125.9	1.76489	0.882445	+	0.470416i	$0.155896\pi$
$830$	−17718.6	−0.740991
$831$	10283.5	0.429278
$832$	832.000	0.0346688
$833$	−2074.61	−0.0862916
$834$	−510.476	−0.0211947
$835$	12180.6	0.504824
$836$	5613.33	0.232226
$837$	−6701.78	−0.276759
$838$	9719.90	0.400678
$839$	−44837.3	−1.84500	−0.922501	−	0.385995i	$-0.873858\pi$
$839$	−44837.3	−1.84500	−0.922501	+	0.385995i	$0.873858\pi$
$840$	1104.23	0.0453565
$841$	−9729.90	−0.398946
$842$	24633.6	1.00823
$843$	10939.6	0.446951
$844$	13295.5	0.542240
$845$	1110.80	0.0452221
$846$	−3893.78	−0.158240
$847$	7034.03	0.285351
$848$	11408.0	0.461974
$849$	9405.62	0.380212
$850$	6926.53	0.279503
$851$	−12283.5	−0.494798
$852$	787.303	0.0316579
$853$	6832.88	0.274271	0.137136	−	0.990552i	$-0.456210\pi$
$853$	6832.88	0.274271	0.137136	+	0.990552i	$0.456210\pi$
$854$	−3454.71	−0.138428
$855$	4596.75	0.183866
$856$	−4295.40	−0.171511
$857$	−34793.5	−1.38684	−0.693421	−	0.720533i	$-0.743897\pi$
$857$	−34793.5	−1.38684	−0.693421	+	0.720533i	$0.743897\pi$
$858$	−1408.63	−0.0560486
$859$	12163.1	0.483118	0.241559	−	0.970386i	$-0.422341\pi$
$859$	12163.1	0.483118	0.241559	+	0.970386i	$0.422341\pi$
$860$	4710.98	0.186794
$861$	8945.59	0.354082
$862$	−12478.8	−0.493076
$863$	−1008.87	−0.0397941	−0.0198971	−	0.999802i	$-0.506334\pi$
$863$	−1008.87	−0.0397941	−0.0198971	+	0.999802i	$0.506334\pi$
$864$	−864.000	−0.0340207
$865$	−28921.3	−1.13683
$866$	11044.2	0.433369
$867$	9361.24	0.366695
$868$	−6950.00	−0.271772
$869$	6909.36	0.269717
$870$	−4774.79	−0.186070
$871$	−9649.91	−0.375401
$872$	15675.8	0.608773
$873$	3387.66	0.131334
$874$	6822.40	0.264040
$875$	9514.69	0.367606
$876$	−11581.2	−0.446681
$877$	−35771.9	−1.37734	−0.688672	−	0.725073i	$-0.741806\pi$
$877$	−35771.9	−1.37734	−0.688672	+	0.725073i	$0.741806\pi$
$878$	22623.4	0.869595
$879$	19654.0	0.754169
$880$	1899.20	0.0727523
$881$	14557.1	0.556689	0.278344	−	0.960481i	$-0.410214\pi$
$881$	14557.1	0.556689	0.278344	+	0.960481i	$0.410214\pi$
$882$	882.000	0.0336718
$883$	24368.1	0.928712	0.464356	−	0.885649i	$-0.346286\pi$
$883$	24368.1	0.928712	0.464356	+	0.885649i	$0.346286\pi$
$884$	−2201.63	−0.0837655
$885$	−7946.56	−0.301831
$886$	−5479.26	−0.207764
$887$	−14649.3	−0.554537	−0.277268	−	0.960792i	$-0.589429\pi$
$887$	−14649.3	−0.554537	−0.277268	+	0.960792i	$0.589429\pi$
$888$	6715.62	0.253785
$889$	3544.39	0.133718
$890$	10912.0	0.410979
$891$	1462.80	0.0550009
$892$	7753.31	0.291031
$893$	−16809.7	−0.629914
$894$	2384.45	0.0892034
$895$	−23777.7	−0.888045
$896$	−896.000	−0.0334077
$897$	−1712.03	−0.0637270
$898$	−17742.4	−0.659321
$899$	30052.5	1.11491
$900$	−2944.75	−0.109065
$901$	−30187.8	−1.11621
$902$	15385.8	0.567951
$903$	3762.89	0.138672
$904$	12028.2	0.442534
$905$	2602.91	0.0956064
$906$	18598.8	0.682011
$907$	16643.1	0.609287	0.304644	−	0.952466i	$-0.401463\pi$
$907$	16643.1	0.609287	0.304644	+	0.952466i	$0.401463\pi$
$908$	−10229.8	−0.373884
$909$	5817.56	0.212273
$910$	−1196.25	−0.0435771
$911$	−29484.9	−1.07231	−0.536157	−	0.844119i	$-0.680124\pi$
$911$	−29484.9	−1.07231	−0.536157	+	0.844119i	$0.680124\pi$
$912$	−3729.93	−0.135428
$913$	−24341.8	−0.882360
$914$	2870.48	0.103881
$915$	−4865.79	−0.175801
$916$	−15493.9	−0.558879
$917$	7985.35	0.287568
$918$	2286.30	0.0821996
$919$	−43199.5	−1.55062	−0.775310	−	0.631580i	$-0.782407\pi$
$919$	−43199.5	−1.55062	−0.775310	+	0.631580i	$0.782407\pi$
$920$	2308.27	0.0827190
$921$	−25891.8	−0.926347
$922$	303.953	0.0108570
$923$	−852.911	−0.0304159
$924$	1516.98	0.0540098
$925$	22888.7	0.813594
$926$	−2115.49	−0.0750749
$927$	375.776	0.0133140
$928$	3874.39	0.137051
$929$	14700.1	0.519154	0.259577	−	0.965722i	$-0.416417\pi$
$929$	14700.1	0.519154	0.259577	+	0.965722i	$0.416417\pi$
$930$	−9788.75	−0.345146
$931$	3807.64	0.134039
$932$	−906.176	−0.0318485
$933$	16314.2	0.572458
$934$	31075.4	1.08867
$935$	−5025.63	−0.175781
$936$	936.000	0.0326860
$937$	319.330	0.0111335	0.00556673	−	0.999985i	$-0.498228\pi$
$937$	319.330	0.0111335	0.00556673	+	0.999985i	$0.498228\pi$
$938$	10392.2	0.361746
$939$	−12310.1	−0.427824
$940$	−5687.33	−0.197341
$941$	−17188.4	−0.595459	−0.297729	−	0.954650i	$-0.596229\pi$
$941$	−17188.4	−0.595459	−0.297729	+	0.954650i	$0.596229\pi$
$942$	−16964.7	−0.586771
$943$	18699.8	0.645758
$944$	6448.06	0.222316
$945$	1242.26	0.0427625
$946$	6471.93	0.222432
$947$	−9298.04	−0.319056	−0.159528	−	0.987193i	$-0.550997\pi$
$947$	−9298.04	−0.319056	−0.159528	+	0.987193i	$0.550997\pi$
$948$	−4591.12	−0.157292
$949$	12546.3	0.429157
$950$	−12712.6	−0.434160
$951$	12909.8	0.440200
$952$	2370.98	0.0807184
$953$	−38827.1	−1.31976	−0.659882	−	0.751369i	$-0.729394\pi$
$953$	−38827.1	−1.31976	−0.659882	+	0.751369i	$0.729394\pi$
$954$	12834.1	0.435553
$955$	−17567.4	−0.595256
$956$	−16565.0	−0.560410
$957$	−6559.58	−0.221569
$958$	4092.07	0.138005
$959$	14298.8	0.481472
$960$	−1261.97	−0.0424271
$961$	31819.3	1.06808
$962$	−7275.25	−0.243829
$963$	−4832.32	−0.161702
$964$	9803.83	0.327552
$965$	6738.47	0.224787
$966$	1843.73	0.0614089
$967$	16033.4	0.533196	0.266598	−	0.963808i	$-0.414100\pi$
$967$	16033.4	0.533196	0.266598	+	0.963808i	$0.414100\pi$
$968$	−8038.89	−0.266921
$969$	9870.08	0.327216
$970$	4948.08	0.163787
$971$	5182.02	0.171266	0.0856328	−	0.996327i	$-0.472709\pi$
$971$	5182.02	0.171266	0.0856328	+	0.996327i	$0.472709\pi$
$972$	−972.000	−0.0320750
$973$	−595.555	−0.0196224
$974$	21596.8	0.710477
$975$	3190.14	0.104786
$976$	3948.23	0.129488
$977$	30393.7	0.995271	0.497636	−	0.867386i	$-0.334202\pi$
$977$	30393.7	0.995271	0.497636	+	0.867386i	$0.334202\pi$
$978$	18066.0	0.590683
$979$	14990.9	0.489387
$980$	1288.27	0.0419920
$981$	17635.3	0.573957
$982$	30053.1	0.976611
$983$	−29485.7	−0.956713	−0.478356	−	0.878166i	$-0.658767\pi$
$983$	−29485.7	−0.956713	−0.478356	+	0.878166i	$0.658767\pi$
$984$	−10223.5	−0.331214
$985$	−25298.5	−0.818352
$986$	−10252.4	−0.331137
$987$	−4542.75	−0.146502
$988$	4040.76	0.130115
$989$	7865.93	0.252904
$990$	2136.60	0.0685915
$991$	−48854.3	−1.56600	−0.783000	−	0.622021i	$-0.786312\pi$
$991$	−48854.3	−1.56600	−0.783000	+	0.622021i	$0.786312\pi$
$992$	7942.86	0.254220
$993$	13554.3	0.433166
$994$	918.520	0.0293095
$995$	−29640.6	−0.944393
$996$	16174.5	0.514568
$997$	3620.03	0.114993	0.0574963	−	0.998346i	$-0.481688\pi$
$997$	3620.03	0.114993	0.0574963	+	0.998346i	$0.481688\pi$
$998$	5206.71	0.165146
$999$	7555.07	0.239271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.4.a.p.1.2	✓	3
3.2	odd	2		1638.4.a.v.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.p.1.2	✓	3	1.1	even	1	trivial
1638.4.a.v.1.2		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.57278 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +13.1456 q^{10} +18.0593 q^{11} -12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -19.7183 q^{15} +16.0000 q^{16} -42.3389 q^{17} +18.0000 q^{18} +77.7068 q^{19} +26.2911 q^{20} +21.0000 q^{21} +36.1186 q^{22} +43.8983 q^{23} -24.0000 q^{24} -81.7985 q^{25} +26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +121.075 q^{29} -39.4367 q^{30} +248.214 q^{31} +32.0000 q^{32} -54.1779 q^{33} -84.6779 q^{34} -46.0095 q^{35} +36.0000 q^{36} -279.817 q^{37} +155.414 q^{38} -39.0000 q^{39} +52.5823 q^{40} +425.980 q^{41} +42.0000 q^{42} +179.185 q^{43} +72.2372 q^{44} +59.1550 q^{45} +87.7966 q^{46} -216.321 q^{47} -48.0000 q^{48} +49.0000 q^{49} -163.597 q^{50} +127.017 q^{51} +52.0000 q^{52} +713.003 q^{53} -54.0000 q^{54} +118.700 q^{55} -56.0000 q^{56} -233.121 q^{57} +242.150 q^{58} +403.004 q^{59} -78.8734 q^{60} +246.765 q^{61} +496.429 q^{62} -63.0000 q^{63} +64.0000 q^{64} +85.4462 q^{65} -108.356 q^{66} -742.301 q^{67} -169.356 q^{68} -131.695 q^{69} -92.0190 q^{70} -65.6086 q^{71} +72.0000 q^{72} +965.099 q^{73} -559.635 q^{74} +245.396 q^{75} +310.827 q^{76} -126.415 q^{77} -78.0000 q^{78} +382.593 q^{79} +105.165 q^{80} +81.0000 q^{81} +851.961 q^{82} -1347.88 q^{83} +84.0000 q^{84} -278.285 q^{85} +358.371 q^{86} -363.224 q^{87} +144.474 q^{88} +830.091 q^{89} +118.310 q^{90} -91.0000 q^{91} +175.593 q^{92} -744.643 q^{93} -432.643 q^{94} +510.750 q^{95} -96.0000 q^{96} +376.407 q^{97} +98.0000 q^{98} +162.534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 14 q^{10} - 47 q^{11} - 36 q^{12} + 39 q^{13} - 42 q^{14} - 21 q^{15} + 48 q^{16} + 119 q^{17} + 54 q^{18} + 101 q^{19}+ \cdots - 423 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 + 24 * q^8 + 27 * q^9 + 14 * q^10 - 47 * q^11 - 36 * q^12 + 39 * q^13 - 42 * q^14 - 21 * q^15 + 48 * q^16 + 119 * q^17 + 54 * q^18 + 101 * q^19 + 28 * q^20 + 63 * q^21 - 94 * q^22 - 27 * q^23 - 72 * q^24 + 266 * q^25 + 78 * q^26 - 81 * q^27 - 84 * q^28 + 345 * q^29 - 42 * q^30 + 152 * q^31 + 96 * q^32 + 141 * q^33 + 238 * q^34 - 49 * q^35 + 108 * q^36 + 227 * q^37 + 202 * q^38 - 117 * q^39 + 56 * q^40 + 452 * q^41 + 126 * q^42 - 415 * q^43 - 188 * q^44 + 63 * q^45 - 54 * q^46 + 240 * q^47 - 144 * q^48 + 147 * q^49 + 532 * q^50 - 357 * q^51 + 156 * q^52 + 874 * q^53 - 162 * q^54 + 1123 * q^55 - 168 * q^56 - 303 * q^57 + 690 * q^58 + 948 * q^59 - 84 * q^60 + 951 * q^61 + 304 * q^62 - 189 * q^63 + 192 * q^64 + 91 * q^65 + 282 * q^66 + 742 * q^67 + 476 * q^68 + 81 * q^69 - 98 * q^70 + 732 * q^71 + 216 * q^72 - 585 * q^73 + 454 * q^74 - 798 * q^75 + 404 * q^76 + 329 * q^77 - 234 * q^78 - 618 * q^79 + 112 * q^80 + 243 * q^81 + 904 * q^82 + 774 * q^83 + 252 * q^84 + 1273 * q^85 - 830 * q^86 - 1035 * q^87 - 376 * q^88 + 730 * q^89 + 126 * q^90 - 273 * q^91 - 108 * q^92 - 456 * q^93 + 480 * q^94 - 2617 * q^95 - 288 * q^96 + 2518 * q^97 + 294 * q^98 - 423 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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