LMFDB - Embedded newform 543.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [543,2,Mod(1,543)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(543, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("543.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 543 = 3 \cdot 181 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	543.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$4.33587682976$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 21x^{5} + 5x^{4} - 35x^{3} + 10x^{2} + 4x - 1 $ x^8 - 3x^7 - 6x^6 + 21x^5 + 5x^4 - 35x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.57409$ of defining polynomial
Character	$\chi$	$=$	543.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.57409 q^{2} -1.00000 q^{3} +0.477759 q^{4} +2.61079 q^{5} +1.57409 q^{6} +3.43627 q^{7} +2.39614 q^{8} +1.00000 q^{9} -4.10962 q^{10} +0.961440 q^{11} -0.477759 q^{12} -2.19899 q^{13} -5.40900 q^{14} -2.61079 q^{15} -4.72726 q^{16} +1.38921 q^{17} -1.57409 q^{18} -1.29785 q^{19} +1.24733 q^{20} -3.43627 q^{21} -1.51339 q^{22} -1.03393 q^{23} -2.39614 q^{24} +1.81623 q^{25} +3.46140 q^{26} -1.00000 q^{27} +1.64171 q^{28} +10.3621 q^{29} +4.10962 q^{30} +1.46354 q^{31} +2.64885 q^{32} -0.961440 q^{33} -2.18674 q^{34} +8.97139 q^{35} +0.477759 q^{36} +0.102931 q^{37} +2.04293 q^{38} +2.19899 q^{39} +6.25583 q^{40} +2.19969 q^{41} +5.40900 q^{42} -5.89954 q^{43} +0.459336 q^{44} +2.61079 q^{45} +1.62749 q^{46} +4.32862 q^{47} +4.72726 q^{48} +4.80796 q^{49} -2.85891 q^{50} -1.38921 q^{51} -1.05059 q^{52} +9.97667 q^{53} +1.57409 q^{54} +2.51012 q^{55} +8.23380 q^{56} +1.29785 q^{57} -16.3109 q^{58} +3.19012 q^{59} -1.24733 q^{60} -7.75991 q^{61} -2.30375 q^{62} +3.43627 q^{63} +5.28500 q^{64} -5.74110 q^{65} +1.51339 q^{66} +4.55325 q^{67} +0.663707 q^{68} +1.03393 q^{69} -14.1218 q^{70} -13.5850 q^{71} +2.39614 q^{72} +1.86716 q^{73} -0.162023 q^{74} -1.81623 q^{75} -0.620058 q^{76} +3.30377 q^{77} -3.46140 q^{78} +1.27450 q^{79} -12.3419 q^{80} +1.00000 q^{81} -3.46252 q^{82} +6.39088 q^{83} -1.64171 q^{84} +3.62693 q^{85} +9.28640 q^{86} -10.3621 q^{87} +2.30375 q^{88} +0.429627 q^{89} -4.10962 q^{90} -7.55632 q^{91} -0.493968 q^{92} -1.46354 q^{93} -6.81364 q^{94} -3.38841 q^{95} -2.64885 q^{96} +7.74609 q^{97} -7.56817 q^{98} +0.961440 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} - 8 q^{3} + 5 q^{4} + 5 q^{5} - 3 q^{6} - 3 q^{7} + 6 q^{8} + 8 q^{9} + 2 q^{10} + 4 q^{11} - 5 q^{12} + 13 q^{13} + 5 q^{14} - 5 q^{15} + 3 q^{16} + 27 q^{17} + 3 q^{18} - 10 q^{19} + 21 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) $ 8 * q + 3 * q^2 - 8 * q^3 + 5 * q^4 + 5 * q^5 - 3 * q^6 - 3 * q^7 + 6 * q^8 + 8 * q^9 + 2 * q^10 + 4 * q^11 - 5 * q^12 + 13 * q^13 + 5 * q^14 - 5 * q^15 + 3 * q^16 + 27 * q^17 + 3 * q^18 - 10 * q^19 + 21 * q^20 + 3 * q^21 + 11 * q^22 + 3 * q^23 - 6 * q^24 + 13 * q^25 + 11 * q^26 - 8 * q^27 - 3 * q^28 + 16 * q^29 - 2 * q^30 - q^31 + 10 * q^32 - 4 * q^33 + 10 * q^34 + 14 * q^35 + 5 * q^36 + 3 * q^37 + 14 * q^38 - 13 * q^39 + 11 * q^40 + 20 * q^41 - 5 * q^42 - 4 * q^43 + 9 * q^44 + 5 * q^45 - 17 * q^46 + 12 * q^47 - 3 * q^48 + 5 * q^49 + 2 * q^50 - 27 * q^51 + 21 * q^52 + 23 * q^53 - 3 * q^54 - 13 * q^55 - 8 * q^56 + 10 * q^57 - 21 * q^58 + 7 * q^59 - 21 * q^60 + q^61 + q^62 - 3 * q^63 - 32 * q^64 + 10 * q^65 - 11 * q^66 - 25 * q^67 - q^68 - 3 * q^69 - 10 * q^70 + 11 * q^71 + 6 * q^72 + 23 * q^73 + 2 * q^74 - 13 * q^75 - 19 * q^76 + 20 * q^77 - 11 * q^78 - 26 * q^79 - 15 * q^80 + 8 * q^81 - 22 * q^82 + 16 * q^83 + 3 * q^84 - 33 * q^85 - 21 * q^86 - 16 * q^87 - q^88 + 14 * q^89 + 2 * q^90 - 35 * q^91 - 18 * q^92 + q^93 - 52 * q^94 + 10 * q^95 - 10 * q^96 + 28 * q^97 - 51 * q^98 + 4 * 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 $

$2$	−1.57409	−1.11305	−0.556525	−	0.830831i	$-0.687866\pi$
$2$	−1.57409	−1.11305	−0.556525	+	0.830831i	$0.687866\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.477759	0.238880
$5$	2.61079	1.16758	0.583791	−	0.811904i	$-0.301569\pi$
$5$	2.61079	1.16758	0.583791	+	0.811904i	$0.301569\pi$
$6$	1.57409	0.642620
$7$	3.43627	1.29879	0.649394	−	0.760452i	$-0.275022\pi$
$7$	3.43627	1.29879	0.649394	+	0.760452i	$0.275022\pi$
$8$	2.39614	0.847165
$9$	1.00000	0.333333
$10$	−4.10962	−1.29958
$11$	0.961440	0.289885	0.144942	−	0.989440i	$-0.453700\pi$
$11$	0.961440	0.289885	0.144942	+	0.989440i	$0.453700\pi$
$12$	−0.477759	−0.137917
$13$	−2.19899	−0.609889	−0.304945	−	0.952370i	$-0.598638\pi$
$13$	−2.19899	−0.609889	−0.304945	+	0.952370i	$0.598638\pi$
$14$	−5.40900	−1.44562
$15$	−2.61079	−0.674103
$16$	−4.72726	−1.18182
$17$	1.38921	0.336933	0.168466	−	0.985707i	$-0.446119\pi$
$17$	1.38921	0.336933	0.168466	+	0.985707i	$0.446119\pi$
$18$	−1.57409	−0.371017
$19$	−1.29785	−0.297746	−0.148873	−	0.988856i	$-0.547565\pi$
$19$	−1.29785	−0.297746	−0.148873	+	0.988856i	$0.547565\pi$
$20$	1.24733	0.278911
$21$	−3.43627	−0.749856
$22$	−1.51339	−0.322656
$23$	−1.03393	−0.215589	−0.107794	−	0.994173i	$-0.534379\pi$
$23$	−1.03393	−0.215589	−0.107794	+	0.994173i	$0.534379\pi$
$24$	−2.39614	−0.489111
$25$	1.81623	0.363246
$26$	3.46140	0.678837
$27$	−1.00000	−0.192450
$28$	1.64171	0.310254
$29$	10.3621	1.92420	0.962099	−	0.272701i	$-0.0879171\pi$
$29$	10.3621	1.92420	0.962099	+	0.272701i	$0.0879171\pi$
$30$	4.10962	0.750310
$31$	1.46354	0.262860	0.131430	−	0.991325i	$-0.458043\pi$
$31$	1.46354	0.262860	0.131430	+	0.991325i	$0.458043\pi$
$32$	2.64885	0.468255
$33$	−0.961440	−0.167365
$34$	−2.18674	−0.375023
$35$	8.97139	1.51644
$36$	0.477759	0.0796265
$37$	0.102931	0.0169218	0.00846089	−	0.999964i	$-0.497307\pi$
$37$	0.102931	0.0169218	0.00846089	+	0.999964i	$0.497307\pi$
$38$	2.04293	0.331407
$39$	2.19899	0.352120
$40$	6.25583	0.989134
$41$	2.19969	0.343534	0.171767	−	0.985138i	$-0.445052\pi$
$41$	2.19969	0.343534	0.171767	+	0.985138i	$0.445052\pi$
$42$	5.40900	0.834627
$43$	−5.89954	−0.899671	−0.449835	−	0.893111i	$-0.648517\pi$
$43$	−5.89954	−0.899671	−0.449835	+	0.893111i	$0.648517\pi$
$44$	0.459336	0.0692476
$45$	2.61079	0.389194
$46$	1.62749	0.239961
$47$	4.32862	0.631394	0.315697	−	0.948860i	$-0.397762\pi$
$47$	4.32862	0.631394	0.315697	+	0.948860i	$0.397762\pi$
$48$	4.72726	0.682322
$49$	4.80796	0.686852
$50$	−2.85891	−0.404311
$51$	−1.38921	−0.194528
$52$	−1.05059	−0.145690
$53$	9.97667	1.37040	0.685200	−	0.728355i	$-0.259715\pi$
$53$	9.97667	1.37040	0.685200	+	0.728355i	$0.259715\pi$
$54$	1.57409	0.214207
$55$	2.51012	0.338464
$56$	8.23380	1.10029
$57$	1.29785	0.171904
$58$	−16.3109	−2.14173
$59$	3.19012	0.415319	0.207659	−	0.978201i	$-0.433415\pi$
$59$	3.19012	0.415319	0.207659	+	0.978201i	$0.433415\pi$
$60$	−1.24733	−0.161029
$61$	−7.75991	−0.993555	−0.496778	−	0.867878i	$-0.665484\pi$
$61$	−7.75991	−0.993555	−0.496778	+	0.867878i	$0.665484\pi$
$62$	−2.30375	−0.292576
$63$	3.43627	0.432930
$64$	5.28500	0.660625
$65$	−5.74110	−0.712095
$66$	1.51339	0.186286
$67$	4.55325	0.556268	0.278134	−	0.960542i	$-0.410284\pi$
$67$	4.55325	0.556268	0.278134	+	0.960542i	$0.410284\pi$
$68$	0.663707	0.0804863
$69$	1.03393	0.124470
$70$	−14.1218	−1.68787
$71$	−13.5850	−1.61224	−0.806121	−	0.591751i	$-0.798437\pi$
$71$	−13.5850	−1.61224	−0.806121	+	0.591751i	$0.798437\pi$
$72$	2.39614	0.282388
$73$	1.86716	0.218535	0.109267	−	0.994012i	$-0.465149\pi$
$73$	1.86716	0.218535	0.109267	+	0.994012i	$0.465149\pi$
$74$	−0.162023	−0.0188348
$75$	−1.81623	−0.209720
$76$	−0.620058	−0.0711255
$77$	3.30377	0.376499
$78$	−3.46140	−0.391927
$79$	1.27450	0.143393	0.0716964	−	0.997427i	$-0.477159\pi$
$79$	1.27450	0.143393	0.0716964	+	0.997427i	$0.477159\pi$
$80$	−12.3419	−1.37987
$81$	1.00000	0.111111
$82$	−3.46252	−0.382371
$83$	6.39088	0.701490	0.350745	−	0.936471i	$-0.385928\pi$
$83$	6.39088	0.701490	0.350745	+	0.936471i	$0.385928\pi$
$84$	−1.64171	−0.179125
$85$	3.62693	0.393396
$86$	9.28640	1.00138
$87$	−10.3621	−1.11094
$88$	2.30375	0.245580
$89$	0.429627	0.0455404	0.0227702	−	0.999741i	$-0.492751\pi$
$89$	0.429627	0.0455404	0.0227702	+	0.999741i	$0.492751\pi$
$90$	−4.10962	−0.433192
$91$	−7.55632	−0.792118
$92$	−0.493968	−0.0514997
$93$	−1.46354	−0.151762
$94$	−6.81364	−0.702773
$95$	−3.38841	−0.347643
$96$	−2.64885	−0.270347
$97$	7.74609	0.786496	0.393248	−	0.919432i	$-0.371351\pi$
$97$	7.74609	0.786496	0.393248	+	0.919432i	$0.371351\pi$
$98$	−7.56817	−0.764500
$99$	0.961440	0.0966283
$100$	0.867720	0.0867720
$101$	8.32864	0.828731	0.414365	−	0.910111i	$-0.364004\pi$
$101$	8.32864	0.828731	0.414365	+	0.910111i	$0.364004\pi$
$102$	2.18674	0.216520
$103$	13.5440	1.33453	0.667265	−	0.744821i	$-0.267465\pi$
$103$	13.5440	1.33453	0.667265	+	0.744821i	$0.267465\pi$
$104$	−5.26909	−0.516677
$105$	−8.97139	−0.875518
$106$	−15.7042	−1.52532
$107$	−11.6097	−1.12235	−0.561175	−	0.827697i	$-0.689650\pi$
$107$	−11.6097	−1.12235	−0.561175	+	0.827697i	$0.689650\pi$
$108$	−0.477759	−0.0459724
$109$	−4.66299	−0.446633	−0.223317	−	0.974746i	$-0.571688\pi$
$109$	−4.66299	−0.446633	−0.223317	+	0.974746i	$0.571688\pi$
$110$	−3.95115	−0.376727
$111$	−0.102931	−0.00976980
$112$	−16.2442	−1.53493
$113$	10.9955	1.03437	0.517187	−	0.855872i	$-0.326979\pi$
$113$	10.9955	1.03437	0.517187	+	0.855872i	$0.326979\pi$
$114$	−2.04293	−0.191338
$115$	−2.69937	−0.251717
$116$	4.95060	0.459651
$117$	−2.19899	−0.203296
$118$	−5.02154	−0.462270
$119$	4.77370	0.437604
$120$	−6.25583	−0.571077
$121$	−10.0756	−0.915967
$122$	12.2148	1.10588
$123$	−2.19969	−0.198340
$124$	0.699221	0.0627919
$125$	−8.31216	−0.743462
$126$	−5.40900	−0.481872
$127$	8.07470	0.716513	0.358257	−	0.933623i	$-0.383371\pi$
$127$	8.07470	0.716513	0.358257	+	0.933623i	$0.383371\pi$
$128$	−13.6168	−1.20356
$129$	5.89954	0.519425
$130$	9.03700	0.792598
$131$	−18.8695	−1.64864	−0.824320	−	0.566125i	$-0.808442\pi$
$131$	−18.8695	−1.64864	−0.824320	+	0.566125i	$0.808442\pi$
$132$	−0.459336	−0.0399801
$133$	−4.45975	−0.386710
$134$	−7.16723	−0.619154
$135$	−2.61079	−0.224701
$136$	3.32875	0.285438
$137$	17.6058	1.50417	0.752085	−	0.659067i	$-0.229048\pi$
$137$	17.6058	1.50417	0.752085	+	0.659067i	$0.229048\pi$
$138$	−1.62749	−0.138541
$139$	−7.03888	−0.597030	−0.298515	−	0.954405i	$-0.596491\pi$
$139$	−7.03888	−0.597030	−0.298515	+	0.954405i	$0.596491\pi$
$140$	4.28616	0.362247
$141$	−4.32862	−0.364536
$142$	21.3840	1.79451
$143$	−2.11419	−0.176798
$144$	−4.72726	−0.393939
$145$	27.0533	2.24666
$146$	−2.93908	−0.243240
$147$	−4.80796	−0.396554
$148$	0.0491763	0.00404227
$149$	−6.74021	−0.552180	−0.276090	−	0.961132i	$-0.589039\pi$
$149$	−6.74021	−0.552180	−0.276090	+	0.961132i	$0.589039\pi$
$150$	2.85891	0.233429
$151$	3.82904	0.311603	0.155802	−	0.987788i	$-0.450204\pi$
$151$	3.82904	0.311603	0.155802	+	0.987788i	$0.450204\pi$
$152$	−3.10983	−0.252240
$153$	1.38921	0.112311
$154$	−5.20043	−0.419062
$155$	3.82100	0.306910
$156$	1.05059	0.0841142
$157$	−2.71018	−0.216296	−0.108148	−	0.994135i	$-0.534492\pi$
$157$	−2.71018	−0.216296	−0.108148	+	0.994135i	$0.534492\pi$
$158$	−2.00618	−0.159603
$159$	−9.97667	−0.791201
$160$	6.91560	0.546726
$161$	−3.55285	−0.280004
$162$	−1.57409	−0.123672
$163$	−24.5296	−1.92131	−0.960654	−	0.277749i	$-0.910412\pi$
$163$	−24.5296	−1.92131	−0.960654	+	0.277749i	$0.910412\pi$
$164$	1.05092	0.0820633
$165$	−2.51012	−0.195412
$166$	−10.0598	−0.780793
$167$	−7.97488	−0.617115	−0.308557	−	0.951206i	$-0.599846\pi$
$167$	−7.97488	−0.617115	−0.308557	+	0.951206i	$0.599846\pi$
$168$	−8.23380	−0.635252
$169$	−8.16445	−0.628035
$170$	−5.70912	−0.437870
$171$	−1.29785	−0.0992488
$172$	−2.81856	−0.214913
$173$	23.6725	1.79979	0.899895	−	0.436107i	$-0.143643\pi$
$173$	23.6725	1.79979	0.899895	+	0.436107i	$0.143643\pi$
$174$	16.3109	1.23653
$175$	6.24106	0.471779
$176$	−4.54498	−0.342591
$177$	−3.19012	−0.239784
$178$	−0.676272	−0.0506887
$179$	−14.7917	−1.10558	−0.552790	−	0.833320i	$-0.686437\pi$
$179$	−14.7917	−1.10558	−0.552790	+	0.833320i	$0.686437\pi$
$180$	1.24733	0.0929704
$181$	1.00000	0.0743294
$181$	1.00000	0.0743294
$182$	11.8943	0.881666
$183$	7.75991	0.573629
$184$	−2.47744	−0.182639
$185$	0.268732	0.0197576
$186$	2.30375	0.168919
$187$	1.33564	0.0976717
$188$	2.06804	0.150827
$189$	−3.43627	−0.249952
$190$	5.33366	0.386944
$191$	−16.3286	−1.18150	−0.590749	−	0.806855i	$-0.701168\pi$
$191$	−16.3286	−1.18150	−0.590749	+	0.806855i	$0.701168\pi$
$192$	−5.28500	−0.381412
$193$	21.4797	1.54614	0.773070	−	0.634321i	$-0.218720\pi$
$193$	21.4797	1.54614	0.773070	+	0.634321i	$0.218720\pi$
$194$	−12.1930	−0.875409
$195$	5.74110	0.411128
$196$	2.29705	0.164075
$197$	−0.649364	−0.0462653	−0.0231326	−	0.999732i	$-0.507364\pi$
$197$	−0.649364	−0.0462653	−0.0231326	+	0.999732i	$0.507364\pi$
$198$	−1.51339	−0.107552
$199$	13.6175	0.965321	0.482661	−	0.875807i	$-0.339670\pi$
$199$	13.6175	0.965321	0.482661	+	0.875807i	$0.339670\pi$
$200$	4.35195	0.307729
$201$	−4.55325	−0.321162
$202$	−13.1100	−0.922418
$203$	35.6071	2.49913
$204$	−0.663707	−0.0464688
$205$	5.74294	0.401104
$206$	−21.3195	−1.48540
$207$	−1.03393	−0.0718629
$208$	10.3952	0.720777
$209$	−1.24780	−0.0863122
$210$	14.1218	0.974495
$211$	6.80702	0.468615	0.234307	−	0.972163i	$-0.424718\pi$
$211$	6.80702	0.468615	0.234307	+	0.972163i	$0.424718\pi$
$212$	4.76644	0.327361
$213$	13.5850	0.930828
$214$	18.2747	1.24923
$215$	−15.4025	−1.05044
$216$	−2.39614	−0.163037
$217$	5.02913	0.341400
$218$	7.33996	0.497125
$219$	−1.86716	−0.126171
$220$	1.19923	0.0808522
$221$	−3.05485	−0.205492
$222$	0.162023	0.0108743
$223$	−0.115112	−0.00770848	−0.00385424	−	0.999993i	$-0.501227\pi$
$223$	−0.115112	−0.00770848	−0.00385424	+	0.999993i	$0.501227\pi$
$224$	9.10217	0.608164
$225$	1.81623	0.121082
$226$	−17.3080	−1.15131
$227$	−10.4534	−0.693815	−0.346908	−	0.937899i	$-0.612768\pi$
$227$	−10.4534	−0.693815	−0.346908	+	0.937899i	$0.612768\pi$
$228$	0.620058	0.0410643
$229$	22.5933	1.49301	0.746504	−	0.665381i	$-0.231731\pi$
$229$	22.5933	1.49301	0.746504	+	0.665381i	$0.231731\pi$
$230$	4.24905	0.280174
$231$	−3.30377	−0.217372
$232$	24.8291	1.63011
$233$	−8.32806	−0.545590	−0.272795	−	0.962072i	$-0.587948\pi$
$233$	−8.32806	−0.545590	−0.272795	+	0.962072i	$0.587948\pi$
$234$	3.46140	0.226279
$235$	11.3011	0.737204
$236$	1.52411	0.0992112
$237$	−1.27450	−0.0827878
$238$	−7.51423	−0.487075
$239$	4.65178	0.300899	0.150449	−	0.988618i	$-0.451928\pi$
$239$	4.65178	0.300899	0.150449	+	0.988618i	$0.451928\pi$
$240$	12.3419	0.796666
$241$	−15.4970	−0.998251	−0.499125	−	0.866530i	$-0.666345\pi$
$241$	−15.4970	−0.998251	−0.499125	+	0.866530i	$0.666345\pi$
$242$	15.8600	1.01952
$243$	−1.00000	−0.0641500
$244$	−3.70737	−0.237340
$245$	12.5526	0.801956
$246$	3.46252	0.220762
$247$	2.85395	0.181592
$248$	3.50686	0.222686
$249$	−6.39088	−0.405005
$250$	13.0841	0.827510
$251$	−22.2885	−1.40684	−0.703420	−	0.710775i	$-0.748344\pi$
$251$	−22.2885	−1.40684	−0.703420	+	0.710775i	$0.748344\pi$
$252$	1.64171	0.103418
$253$	−0.994058	−0.0624959
$254$	−12.7103	−0.797515
$255$	−3.62693	−0.227127
$256$	10.8640	0.679001
$257$	−21.9037	−1.36631	−0.683157	−	0.730271i	$-0.739394\pi$
$257$	−21.9037	−1.36631	−0.683157	+	0.730271i	$0.739394\pi$
$258$	−9.28640	−0.578146
$259$	0.353700	0.0219778
$260$	−2.74286	−0.170105
$261$	10.3621	0.641399
$262$	29.7023	1.83502
$263$	−4.62072	−0.284926	−0.142463	−	0.989800i	$-0.545502\pi$
$263$	−4.62072	−0.284926	−0.142463	+	0.989800i	$0.545502\pi$
$264$	−2.30375	−0.141786
$265$	26.0470	1.60005
$266$	7.02005	0.430427
$267$	−0.429627	−0.0262928
$268$	2.17536	0.132881
$269$	−15.4154	−0.939895	−0.469948	−	0.882694i	$-0.655727\pi$
$269$	−15.4154	−0.939895	−0.469948	+	0.882694i	$0.655727\pi$
$270$	4.10962	0.250103
$271$	−29.3414	−1.78237	−0.891183	−	0.453645i	$-0.850124\pi$
$271$	−29.3414	−1.78237	−0.891183	+	0.453645i	$0.850124\pi$
$272$	−6.56716	−0.398192
$273$	7.55632	0.457329
$274$	−27.7132	−1.67421
$275$	1.74619	0.105299
$276$	0.493968	0.0297334
$277$	2.86762	0.172299	0.0861493	−	0.996282i	$-0.472544\pi$
$277$	2.86762	0.172299	0.0861493	+	0.996282i	$0.472544\pi$
$278$	11.0798	0.664524
$279$	1.46354	0.0876200
$280$	21.4967	1.28468
$281$	32.5003	1.93881	0.969404	−	0.245471i	$-0.0789426\pi$
$281$	32.5003	1.93881	0.969404	+	0.245471i	$0.0789426\pi$
$282$	6.81364	0.405746
$283$	−20.1377	−1.19706	−0.598532	−	0.801099i	$-0.704249\pi$
$283$	−20.1377	−1.19706	−0.598532	+	0.801099i	$0.704249\pi$
$284$	−6.49035	−0.385131
$285$	3.38841	0.200712
$286$	3.32793	0.196785
$287$	7.55875	0.446179
$288$	2.64885	0.156085
$289$	−15.0701	−0.886476
$290$	−42.5844	−2.50064
$291$	−7.74609	−0.454084
$292$	0.892054	0.0522035
$293$	18.9168	1.10513	0.552567	−	0.833468i	$-0.313648\pi$
$293$	18.9168	1.10513	0.552567	+	0.833468i	$0.313648\pi$
$294$	7.56817	0.441385
$295$	8.32875	0.484918
$296$	0.246638	0.0143355
$297$	−0.961440	−0.0557884
$298$	10.6097	0.614603
$299$	2.27359	0.131485
$300$	−0.867720	−0.0500978
$301$	−20.2724	−1.16848
$302$	−6.02726	−0.346830
$303$	−8.32864	−0.478468
$304$	6.13526	0.351881
$305$	−20.2595	−1.16006
$306$	−2.18674	−0.125008
$307$	6.24336	0.356327	0.178164	−	0.984001i	$-0.442984\pi$
$307$	6.24336	0.356327	0.178164	+	0.984001i	$0.442984\pi$
$308$	1.57840	0.0899380
$309$	−13.5440	−0.770491
$310$	−6.01460	−0.341607
$311$	7.45699	0.422847	0.211424	−	0.977395i	$-0.432190\pi$
$311$	7.45699	0.422847	0.211424	+	0.977395i	$0.432190\pi$
$312$	5.26909	0.298304
$313$	28.5353	1.61291	0.806457	−	0.591293i	$-0.201382\pi$
$313$	28.5353	1.61291	0.806457	+	0.591293i	$0.201382\pi$
$314$	4.26606	0.240748
$315$	8.97139	0.505480
$316$	0.608905	0.0342536
$317$	9.42774	0.529515	0.264757	−	0.964315i	$-0.414708\pi$
$317$	9.42774	0.529515	0.264757	+	0.964315i	$0.414708\pi$
$318$	15.7042	0.880646
$319$	9.96255	0.557796
$320$	13.7980	0.771333
$321$	11.6097	0.647989
$322$	5.59251	0.311658
$323$	−1.80298	−0.100321
$324$	0.477759	0.0265422
$325$	−3.99386	−0.221540
$326$	38.6118	2.13851
$327$	4.66299	0.257864
$328$	5.27078	0.291030
$329$	14.8743	0.820048
$330$	3.95115	0.217504
$331$	−16.6458	−0.914934	−0.457467	−	0.889227i	$-0.651243\pi$
$331$	−16.6458	−0.914934	−0.457467	+	0.889227i	$0.651243\pi$
$332$	3.05330	0.167572
$333$	0.102931	0.00564059
$334$	12.5532	0.686879
$335$	11.8876	0.649489
$336$	16.2442	0.886192
$337$	−34.6785	−1.88906	−0.944530	−	0.328426i	$-0.893482\pi$
$337$	−34.6785	−1.88906	−0.944530	+	0.328426i	$0.893482\pi$
$338$	12.8516	0.699034
$339$	−10.9955	−0.597196
$340$	1.73280	0.0939743
$341$	1.40711	0.0761992
$342$	2.04293	0.110469
$343$	−7.53243	−0.406713
$344$	−14.1361	−0.762169
$345$	2.69937	0.145329
$346$	−37.2627	−2.00325
$347$	−12.1433	−0.651887	−0.325944	−	0.945389i	$-0.605682\pi$
$347$	−12.1433	−0.651887	−0.325944	+	0.945389i	$0.605682\pi$
$348$	−4.95060	−0.265380
$349$	9.00204	0.481868	0.240934	−	0.970541i	$-0.422546\pi$
$349$	9.00204	0.481868	0.240934	+	0.970541i	$0.422546\pi$
$350$	−9.82398	−0.525114
$351$	2.19899	0.117373
$352$	2.54671	0.135740
$353$	9.71810	0.517242	0.258621	−	0.965979i	$-0.416732\pi$
$353$	9.71810	0.517242	0.258621	+	0.965979i	$0.416732\pi$
$354$	5.02154	0.266892
$355$	−35.4676	−1.88242
$356$	0.205258	0.0108787
$357$	−4.77370	−0.252651
$358$	23.2834	1.23057
$359$	27.7228	1.46315	0.731577	−	0.681759i	$-0.238785\pi$
$359$	27.7228	1.46315	0.731577	+	0.681759i	$0.238785\pi$
$360$	6.25583	0.329711
$361$	−17.3156	−0.911347
$362$	−1.57409	−0.0827323
$363$	10.0756	0.528834
$364$	−3.61010	−0.189221
$365$	4.87477	0.255157
$366$	−12.2148	−0.638478
$367$	7.47223	0.390047	0.195024	−	0.980799i	$-0.437522\pi$
$367$	7.47223	0.390047	0.195024	+	0.980799i	$0.437522\pi$
$368$	4.88765	0.254786
$369$	2.19969	0.114511
$370$	−0.423008	−0.0219911
$371$	34.2825	1.77986
$372$	−0.699221	−0.0362529
$373$	−33.2179	−1.71996	−0.859978	−	0.510331i	$-0.829523\pi$
$373$	−33.2179	−1.71996	−0.859978	+	0.510331i	$0.829523\pi$
$374$	−2.10242	−0.108713
$375$	8.31216	0.429238
$376$	10.3720	0.534895
$377$	−22.7862	−1.17355
$378$	5.40900	0.278209
$379$	−4.98384	−0.256003	−0.128001	−	0.991774i	$-0.540856\pi$
$379$	−4.98384	−0.256003	−0.128001	+	0.991774i	$0.540856\pi$
$380$	−1.61884	−0.0830448
$381$	−8.07470	−0.413679
$382$	25.7027	1.31507
$383$	−18.1335	−0.926577	−0.463289	−	0.886207i	$-0.653331\pi$
$383$	−18.1335	−0.926577	−0.463289	+	0.886207i	$0.653331\pi$
$384$	13.6168	0.694878
$385$	8.62545	0.439594
$386$	−33.8109	−1.72093
$387$	−5.89954	−0.299890
$388$	3.70076	0.187878
$389$	−26.2513	−1.33099	−0.665496	−	0.746402i	$-0.731780\pi$
$389$	−26.2513	−1.33099	−0.665496	+	0.746402i	$0.731780\pi$
$390$	−9.03700	−0.457606
$391$	−1.43634	−0.0726389
$392$	11.5206	0.581877
$393$	18.8695	0.951842
$394$	1.02216	0.0514955
$395$	3.32746	0.167423
$396$	0.459336	0.0230825
$397$	−0.305813	−0.0153483	−0.00767416	−	0.999971i	$-0.502443\pi$
$397$	−0.305813	−0.0153483	−0.00767416	+	0.999971i	$0.502443\pi$
$398$	−21.4352	−1.07445
$399$	4.45975	0.223267
$400$	−8.58579	−0.429290
$401$	−30.9766	−1.54690	−0.773448	−	0.633860i	$-0.781469\pi$
$401$	−30.9766	−1.54690	−0.773448	+	0.633860i	$0.781469\pi$
$402$	7.16723	0.357469
$403$	−3.21831	−0.160316
$404$	3.97908	0.197967
$405$	2.61079	0.129731
$406$	−56.0487	−2.78165
$407$	0.0989621	0.00490537
$408$	−3.32875	−0.164797
$409$	−4.93720	−0.244129	−0.122065	−	0.992522i	$-0.538951\pi$
$409$	−4.93720	−0.244129	−0.122065	+	0.992522i	$0.538951\pi$
$410$	−9.03991	−0.446449
$411$	−17.6058	−0.868432
$412$	6.47077	0.318792
$413$	10.9621	0.539411
$414$	1.62749	0.0799870
$415$	16.6852	0.819046
$416$	−5.82479	−0.285584
$417$	7.03888	0.344695
$418$	1.96415	0.0960698
$419$	−26.3828	−1.28888	−0.644442	−	0.764653i	$-0.722910\pi$
$419$	−26.3828	−1.28888	−0.644442	+	0.764653i	$0.722910\pi$
$420$	−4.28616	−0.209143
$421$	−18.4529	−0.899337	−0.449668	−	0.893196i	$-0.648458\pi$
$421$	−18.4529	−0.899337	−0.449668	+	0.893196i	$0.648458\pi$
$422$	−10.7149	−0.521591
$423$	4.32862	0.210465
$424$	23.9055	1.16096
$425$	2.52312	0.122389
$426$	−21.3840	−1.03606
$427$	−26.6652	−1.29042
$428$	−5.54662	−0.268106
$429$	2.11419	0.102074
$430$	24.2448	1.16919
$431$	41.1292	1.98112	0.990562	−	0.137066i	$-0.0437673\pi$
$431$	41.1292	1.98112	0.990562	+	0.137066i	$0.0437673\pi$
$432$	4.72726	0.227441
$433$	−25.8354	−1.24157	−0.620784	−	0.783982i	$-0.713186\pi$
$433$	−25.8354	−1.24157	−0.620784	+	0.783982i	$0.713186\pi$
$434$	−7.91630	−0.379995
$435$	−27.0533	−1.29711
$436$	−2.22778	−0.106692
$437$	1.34188	0.0641907
$438$	2.93908	0.140435
$439$	−12.4780	−0.595542	−0.297771	−	0.954637i	$-0.596243\pi$
$439$	−12.4780	−0.595542	−0.297771	+	0.954637i	$0.596243\pi$
$440$	6.01460	0.286735
$441$	4.80796	0.228951
$442$	4.80861	0.228722
$443$	1.99221	0.0946526	0.0473263	−	0.998879i	$-0.484930\pi$
$443$	1.99221	0.0946526	0.0473263	+	0.998879i	$0.484930\pi$
$444$	−0.0491763	−0.00233380
$445$	1.12167	0.0531721
$446$	0.181197	0.00857992
$447$	6.74021	0.318801
$448$	18.1607	0.858012
$449$	8.59158	0.405462	0.202731	−	0.979234i	$-0.435018\pi$
$449$	8.59158	0.405462	0.202731	+	0.979234i	$0.435018\pi$
$450$	−2.85891	−0.134770
$451$	2.11487	0.0995855
$452$	5.25322	0.247091
$453$	−3.82904	−0.179904
$454$	16.4546	0.772251
$455$	−19.7280	−0.924861
$456$	3.10983	0.145631
$457$	−18.2645	−0.854378	−0.427189	−	0.904162i	$-0.640496\pi$
$457$	−18.2645	−0.854378	−0.427189	+	0.904162i	$0.640496\pi$
$458$	−35.5639	−1.66179
$459$	−1.38921	−0.0648427
$460$	−1.28965	−0.0601301
$461$	−9.20255	−0.428605	−0.214303	−	0.976767i	$-0.568748\pi$
$461$	−9.20255	−0.428605	−0.214303	+	0.976767i	$0.568748\pi$
$462$	5.20043	0.241946
$463$	−7.76606	−0.360919	−0.180460	−	0.983582i	$-0.557759\pi$
$463$	−7.76606	−0.360919	−0.180460	+	0.983582i	$0.557759\pi$
$464$	−48.9845	−2.27405
$465$	−3.82100	−0.177195
$466$	13.1091	0.607268
$467$	−8.59615	−0.397782	−0.198891	−	0.980022i	$-0.563734\pi$
$467$	−8.59615	−0.397782	−0.198891	+	0.980022i	$0.563734\pi$
$468$	−1.05059	−0.0485634
$469$	15.6462	0.722475
$470$	−17.7890	−0.820545
$471$	2.71018	0.124878
$472$	7.64400	0.351844
$473$	−5.67205	−0.260801
$474$	2.00618	0.0921470
$475$	−2.35719	−0.108155
$476$	2.28068	0.104535
$477$	9.97667	0.456800
$478$	−7.32232	−0.334915
$479$	21.8628	0.998935	0.499468	−	0.866333i	$-0.333529\pi$
$479$	21.8628	0.998935	0.499468	+	0.866333i	$0.333529\pi$
$480$	−6.91560	−0.315652
$481$	−0.226344	−0.0103204
$482$	24.3937	1.11110
$483$	3.55285	0.161660
$484$	−4.81372	−0.218806
$485$	20.2234	0.918298
$486$	1.57409	0.0714022
$487$	−35.5574	−1.61126	−0.805629	−	0.592420i	$-0.798173\pi$
$487$	−35.5574	−1.61126	−0.805629	+	0.592420i	$0.798173\pi$
$488$	−18.5939	−0.841705
$489$	24.5296	1.10927
$490$	−19.7589	−0.892616
$491$	−23.8887	−1.07808	−0.539041	−	0.842279i	$-0.681213\pi$
$491$	−23.8887	−1.07808	−0.539041	+	0.842279i	$0.681213\pi$
$492$	−1.05092	−0.0473793
$493$	14.3952	0.648325
$494$	−4.49237	−0.202121
$495$	2.51012	0.112821
$496$	−6.91855	−0.310652
$497$	−46.6817	−2.09396
$498$	10.0598	0.450791
$499$	25.1828	1.12734	0.563668	−	0.826002i	$-0.309390\pi$
$499$	25.1828	1.12734	0.563668	+	0.826002i	$0.309390\pi$
$500$	−3.97121	−0.177598
$501$	7.97488	0.356291
$502$	35.0841	1.56588
$503$	−8.57635	−0.382400	−0.191200	−	0.981551i	$-0.561238\pi$
$503$	−8.57635	−0.382400	−0.191200	+	0.981551i	$0.561238\pi$
$504$	8.23380	0.366763
$505$	21.7443	0.967610
$506$	1.56474	0.0695610
$507$	8.16445	0.362596
$508$	3.85776	0.171160
$509$	2.70721	0.119995	0.0599975	−	0.998199i	$-0.480891\pi$
$509$	2.70721	0.119995	0.0599975	+	0.998199i	$0.480891\pi$
$510$	5.70912	0.252804
$511$	6.41608	0.283831
$512$	10.1326	0.447802
$513$	1.29785	0.0573013
$514$	34.4784	1.52078
$515$	35.3605	1.55817
$516$	2.81856	0.124080
$517$	4.16171	0.183032
$518$	−0.556755	−0.0244624
$519$	−23.6725	−1.03911
$520$	−13.7565	−0.603262
$521$	−26.8480	−1.17623	−0.588117	−	0.808776i	$-0.700131\pi$
$521$	−26.8480	−1.17623	−0.588117	+	0.808776i	$0.700131\pi$
$522$	−16.3109	−0.713909
$523$	−26.7930	−1.17157	−0.585787	−	0.810465i	$-0.699215\pi$
$523$	−26.7930	−1.17157	−0.585787	+	0.810465i	$0.699215\pi$
$524$	−9.01509	−0.393826
$525$	−6.24106	−0.272382
$526$	7.27343	0.317137
$527$	2.03317	0.0885662
$528$	4.54498	0.197795
$529$	−21.9310	−0.953522
$530$	−41.0003	−1.78094
$531$	3.19012	0.138440
$532$	−2.13069	−0.0923770
$533$	−4.83710	−0.209518
$534$	0.676272	0.0292651
$535$	−30.3104	−1.31043
$536$	10.9103	0.471251
$537$	14.7917	0.638307
$538$	24.2653	1.04615
$539$	4.62257	0.199108
$540$	−1.24733	−0.0536765
$541$	−3.30744	−0.142198	−0.0710991	−	0.997469i	$-0.522651\pi$
$541$	−3.30744	−0.142198	−0.0710991	+	0.997469i	$0.522651\pi$
$542$	46.1861	1.98386
$543$	−1.00000	−0.0429141
$544$	3.67981	0.157770
$545$	−12.1741	−0.521481
$546$	−11.8943	−0.509030
$547$	5.26057	0.224926	0.112463	−	0.993656i	$-0.464126\pi$
$547$	5.26057	0.224926	0.112463	+	0.993656i	$0.464126\pi$
$548$	8.41135	0.359315
$549$	−7.75991	−0.331185
$550$	−2.74867	−0.117204
$551$	−13.4484	−0.572923
$552$	2.47744	0.105447
$553$	4.37954	0.186237
$554$	−4.51389	−0.191777
$555$	−0.268732	−0.0114070
$556$	−3.36289	−0.142618
$557$	−9.12524	−0.386649	−0.193324	−	0.981135i	$-0.561927\pi$
$557$	−9.12524	−0.386649	−0.193324	+	0.981135i	$0.561927\pi$
$558$	−2.30375	−0.0975254
$559$	12.9730	0.548700
$560$	−42.4101	−1.79215
$561$	−1.33564	−0.0563908
$562$	−51.1585	−2.15799
$563$	−8.22149	−0.346495	−0.173247	−	0.984878i	$-0.555426\pi$
$563$	−8.22149	−0.346495	−0.173247	+	0.984878i	$0.555426\pi$
$564$	−2.06804	−0.0870801
$565$	28.7071	1.20772
$566$	31.6986	1.33239
$567$	3.43627	0.144310
$568$	−32.5516	−1.36583
$569$	34.5974	1.45040	0.725198	−	0.688540i	$-0.241748\pi$
$569$	34.5974	1.45040	0.725198	+	0.688540i	$0.241748\pi$
$570$	−5.33366	−0.223402
$571$	−26.2483	−1.09846	−0.549229	−	0.835672i	$-0.685078\pi$
$571$	−26.2483	−1.09846	−0.549229	+	0.835672i	$0.685078\pi$
$572$	−1.01008	−0.0422334
$573$	16.3286	0.682139
$574$	−11.8981	−0.496619
$575$	−1.87785	−0.0783117
$576$	5.28500	0.220208
$577$	−28.7241	−1.19580	−0.597900	−	0.801570i	$-0.703998\pi$
$577$	−28.7241	−1.19580	−0.597900	+	0.801570i	$0.703998\pi$
$578$	23.7217	0.986692
$579$	−21.4797	−0.892664
$580$	12.9250	0.536680
$581$	21.9608	0.911087
$582$	12.1930	0.505418
$583$	9.59197	0.397259
$584$	4.47399	0.185135
$585$	−5.74110	−0.237365
$586$	−29.7768	−1.23007
$587$	17.2536	0.712134	0.356067	−	0.934460i	$-0.384117\pi$
$587$	17.2536	0.712134	0.356067	+	0.934460i	$0.384117\pi$
$588$	−2.29705	−0.0947287
$589$	−1.89945	−0.0782656
$590$	−13.1102	−0.539738
$591$	0.649364	0.0267113
$592$	−0.486583	−0.0199984
$593$	37.5049	1.54014	0.770072	−	0.637957i	$-0.220220\pi$
$593$	37.5049	1.54014	0.770072	+	0.637957i	$0.220220\pi$
$594$	1.51339	0.0620952
$595$	12.4631	0.510939
$596$	−3.22020	−0.131904
$597$	−13.6175	−0.557328
$598$	−3.57884	−0.146350
$599$	28.1640	1.15075	0.575375	−	0.817890i	$-0.304856\pi$
$599$	28.1640	1.15075	0.575375	+	0.817890i	$0.304856\pi$
$600$	−4.35195	−0.177667
$601$	28.6925	1.17039	0.585196	−	0.810892i	$-0.301018\pi$
$601$	28.6925	1.17039	0.585196	+	0.810892i	$0.301018\pi$
$602$	31.9106	1.30058
$603$	4.55325	0.185423
$604$	1.82936	0.0744356
$605$	−26.3054	−1.06947
$606$	13.1100	0.532559
$607$	44.1968	1.79389	0.896946	−	0.442141i	$-0.145781\pi$
$607$	44.1968	1.79389	0.896946	+	0.442141i	$0.145781\pi$
$608$	−3.43780	−0.139421
$609$	−35.6071	−1.44287
$610$	31.8903	1.29120
$611$	−9.51858	−0.385081
$612$	0.663707	0.0268288
$613$	16.4282	0.663528	0.331764	−	0.943362i	$-0.392356\pi$
$613$	16.4282	0.663528	0.331764	+	0.943362i	$0.392356\pi$
$614$	−9.82760	−0.396610
$615$	−5.74294	−0.231578
$616$	7.91630	0.318957
$617$	29.6980	1.19560	0.597798	−	0.801647i	$-0.296042\pi$
$617$	29.6980	1.19560	0.597798	+	0.801647i	$0.296042\pi$
$618$	21.3195	0.857595
$619$	22.7157	0.913021	0.456511	−	0.889718i	$-0.349099\pi$
$619$	22.7157	0.913021	0.456511	+	0.889718i	$0.349099\pi$
$620$	1.82552	0.0733146
$621$	1.03393	0.0414901
$622$	−11.7380	−0.470650
$623$	1.47632	0.0591473
$624$	−10.3952	−0.416141
$625$	−30.7825	−1.23130
$626$	−44.9172	−1.79525
$627$	1.24780	0.0498324
$628$	−1.29481	−0.0516686
$629$	0.142993	0.00570150
$630$	−14.1218	−0.562625
$631$	36.5102	1.45345	0.726724	−	0.686930i	$-0.241042\pi$
$631$	36.5102	1.45345	0.726724	+	0.686930i	$0.241042\pi$
$632$	3.05389	0.121477
$633$	−6.80702	−0.270555
$634$	−14.8401	−0.589376
$635$	21.0813	0.836588
$636$	−4.76644	−0.189002
$637$	−10.5727	−0.418904
$638$	−15.6820	−0.620855
$639$	−13.5850	−0.537414
$640$	−35.5505	−1.40526
$641$	3.50020	0.138250	0.0691249	−	0.997608i	$-0.477979\pi$
$641$	3.50020	0.138250	0.0691249	+	0.997608i	$0.477979\pi$
$642$	−18.2747	−0.721243
$643$	46.8651	1.84818	0.924089	−	0.382177i	$-0.124825\pi$
$643$	46.8651	1.84818	0.924089	+	0.382177i	$0.124825\pi$
$644$	−1.69741	−0.0668872
$645$	15.4025	0.606471
$646$	2.83805	0.111662
$647$	10.0298	0.394310	0.197155	−	0.980372i	$-0.436830\pi$
$647$	10.0298	0.394310	0.197155	+	0.980372i	$0.436830\pi$
$648$	2.39614	0.0941294
$649$	3.06711	0.120395
$650$	6.28670	0.246585
$651$	−5.02913	−0.197107
$652$	−11.7192	−0.458961
$653$	−6.80902	−0.266458	−0.133229	−	0.991085i	$-0.542535\pi$
$653$	−6.80902	−0.266458	−0.133229	+	0.991085i	$0.542535\pi$
$654$	−7.33996	−0.287015
$655$	−49.2644	−1.92492
$656$	−10.3985	−0.405995
$657$	1.86716	0.0728450
$658$	−23.4135	−0.912754
$659$	39.5734	1.54156	0.770780	−	0.637101i	$-0.219867\pi$
$659$	39.5734	1.54156	0.770780	+	0.637101i	$0.219867\pi$
$660$	−1.19923	−0.0466800
$661$	−29.3280	−1.14073	−0.570363	−	0.821393i	$-0.693198\pi$
$661$	−29.3280	−1.14073	−0.570363	+	0.821393i	$0.693198\pi$
$662$	26.2019	1.01837
$663$	3.05485	0.118641
$664$	15.3135	0.594277
$665$	−11.6435	−0.451515
$666$	−0.162023	−0.00627826
$667$	−10.7137	−0.414835
$668$	−3.81007	−0.147416
$669$	0.115112	0.00445049
$670$	−18.7121	−0.722913
$671$	−7.46069	−0.288017
$672$	−9.10217	−0.351124
$673$	44.8722	1.72970	0.864848	−	0.502034i	$-0.167415\pi$
$673$	44.8722	1.72970	0.864848	+	0.502034i	$0.167415\pi$
$674$	54.5871	2.10262
$675$	−1.81623	−0.0699067
$676$	−3.90064	−0.150025
$677$	19.9829	0.768006	0.384003	−	0.923332i	$-0.374545\pi$
$677$	19.9829	0.768006	0.384003	+	0.923332i	$0.374545\pi$
$678$	17.3080	0.664709
$679$	26.6177	1.02149
$680$	8.69066	0.333272
$681$	10.4534	0.400574
$682$	−2.21491	−0.0848135
$683$	20.2488	0.774800	0.387400	−	0.921912i	$-0.373373\pi$
$683$	20.2488	0.774800	0.387400	+	0.921912i	$0.373373\pi$
$684$	−0.620058	−0.0237085
$685$	45.9652	1.75624
$686$	11.8567	0.452692
$687$	−22.5933	−0.861989
$688$	27.8887	1.06325
$689$	−21.9386	−0.835793
$690$	−4.24905	−0.161758
$691$	−36.5673	−1.39108	−0.695542	−	0.718485i	$-0.744836\pi$
$691$	−36.5673	−1.39108	−0.695542	+	0.718485i	$0.744836\pi$
$692$	11.3098	0.429933
$693$	3.30377	0.125500
$694$	19.1147	0.725583
$695$	−18.3770	−0.697081
$696$	−24.8291	−0.941146
$697$	3.05584	0.115748
$698$	−14.1700	−0.536343
$699$	8.32806	0.314996
$700$	2.98172	0.112698
$701$	−18.3854	−0.694407	−0.347204	−	0.937790i	$-0.612869\pi$
$701$	−18.3854	−0.694407	−0.347204	+	0.937790i	$0.612869\pi$
$702$	−3.46140	−0.130642
$703$	−0.133589	−0.00503840
$704$	5.08121	0.191505
$705$	−11.3011	−0.425625
$706$	−15.2972	−0.575716
$707$	28.6195	1.07635
$708$	−1.52411	−0.0572796
$709$	−14.3088	−0.537380	−0.268690	−	0.963227i	$-0.586591\pi$
$709$	−14.3088	−0.537380	−0.268690	+	0.963227i	$0.586591\pi$
$710$	55.8291	2.09523
$711$	1.27450	0.0477976
$712$	1.02945	0.0385802
$713$	−1.51320	−0.0566696
$714$	7.51423	0.281213
$715$	−5.51972	−0.206426
$716$	−7.06685	−0.264101
$717$	−4.65178	−0.173724
$718$	−43.6382	−1.62856
$719$	−14.5481	−0.542552	−0.271276	−	0.962502i	$-0.587446\pi$
$719$	−14.5481	−0.542552	−0.271276	+	0.962502i	$0.587446\pi$
$720$	−12.3419	−0.459955
$721$	46.5409	1.73327
$722$	27.2563	1.01437
$723$	15.4970	0.576340
$724$	0.477759	0.0177558
$725$	18.8200	0.698957
$726$	−15.8600	−0.588618
$727$	21.9157	0.812810	0.406405	−	0.913693i	$-0.366782\pi$
$727$	21.9157	0.812810	0.406405	+	0.913693i	$0.366782\pi$
$728$	−18.1060	−0.671054
$729$	1.00000	0.0370370
$730$	−7.67333	−0.284003
$731$	−8.19569	−0.303129
$732$	3.70737	0.137028
$733$	−37.9899	−1.40319	−0.701595	−	0.712576i	$-0.747528\pi$
$733$	−37.9899	−1.40319	−0.701595	+	0.712576i	$0.747528\pi$
$734$	−11.7620	−0.434142
$735$	−12.5526	−0.463009
$736$	−2.73872	−0.100950
$737$	4.37768	0.161254
$738$	−3.46252	−0.127457
$739$	49.5268	1.82187	0.910937	−	0.412545i	$-0.135360\pi$
$739$	49.5268	1.82187	0.910937	+	0.412545i	$0.135360\pi$
$740$	0.128389	0.00471967
$741$	−2.85395	−0.104842
$742$	−53.9638	−1.98107
$743$	24.6189	0.903179	0.451590	−	0.892226i	$-0.350857\pi$
$743$	24.6189	0.903179	0.451590	+	0.892226i	$0.350857\pi$
$744$	−3.50686	−0.128568
$745$	−17.5973	−0.644714
$746$	52.2879	1.91440
$747$	6.39088	0.233830
$748$	0.638114	0.0233318
$749$	−39.8940	−1.45769
$750$	−13.0841	−0.477763
$751$	−19.0575	−0.695418	−0.347709	−	0.937603i	$-0.613040\pi$
$751$	−19.0575	−0.695418	−0.347709	+	0.937603i	$0.613040\pi$
$752$	−20.4625	−0.746192
$753$	22.2885	0.812239
$754$	35.8675	1.30622
$755$	9.99683	0.363822
$756$	−1.64171	−0.0597084
$757$	−5.82521	−0.211721	−0.105860	−	0.994381i	$-0.533760\pi$
$757$	−5.82521	−0.211721	−0.105860	+	0.994381i	$0.533760\pi$
$758$	7.84502	0.284944
$759$	0.994058	0.0360820
$760$	−8.11911	−0.294511
$761$	−40.9900	−1.48589	−0.742943	−	0.669355i	$-0.766571\pi$
$761$	−40.9900	−1.48589	−0.742943	+	0.669355i	$0.766571\pi$
$762$	12.7103	0.460446
$763$	−16.0233	−0.580082
$764$	−7.80115	−0.282236
$765$	3.62693	0.131132
$766$	28.5437	1.03133
$767$	−7.01504	−0.253299
$768$	−10.8640	−0.392021
$769$	1.22318	0.0441088	0.0220544	−	0.999757i	$-0.492979\pi$
$769$	1.22318	0.0441088	0.0220544	+	0.999757i	$0.492979\pi$
$770$	−13.5772	−0.489289
$771$	21.9037	0.788842
$772$	10.2621	0.369341
$773$	−21.7320	−0.781645	−0.390823	−	0.920466i	$-0.627809\pi$
$773$	−21.7320	−0.781645	−0.390823	+	0.920466i	$0.627809\pi$
$774$	9.28640	0.333793
$775$	2.65813	0.0954828
$776$	18.5607	0.666292
$777$	−0.353700	−0.0126889
$778$	41.3218	1.48146
$779$	−2.85487	−0.102286
$780$	2.74286	0.0982102
$781$	−13.0611	−0.467365
$782$	2.26093	0.0808507
$783$	−10.3621	−0.370312
$784$	−22.7285	−0.811733
$785$	−7.07571	−0.252543
$786$	−29.7023	−1.05945
$787$	−17.8059	−0.634712	−0.317356	−	0.948307i	$-0.602795\pi$
$787$	−17.8059	−0.634712	−0.317356	+	0.948307i	$0.602795\pi$
$788$	−0.310239	−0.0110518
$789$	4.62072	0.164502
$790$	−5.23772	−0.186350
$791$	37.7837	1.34343
$792$	2.30375	0.0818601
$793$	17.0640	0.605959
$794$	0.481378	0.0170835
$795$	−26.0470	−0.923792
$796$	6.50590	0.230595
$797$	9.71471	0.344113	0.172056	−	0.985087i	$-0.444959\pi$
$797$	9.71471	0.344113	0.172056	+	0.985087i	$0.444959\pi$
$798$	−7.02005	−0.248507
$799$	6.01336	0.212737
$800$	4.81092	0.170092
$801$	0.429627	0.0151801
$802$	48.7599	1.72177
$803$	1.79517	0.0633500
$804$	−2.17536	−0.0767190
$805$	−9.27576	−0.326927
$806$	5.06591	0.178439
$807$	15.4154	0.542649
$808$	19.9566	0.702072
$809$	−29.1858	−1.02612	−0.513060	−	0.858353i	$-0.671488\pi$
$809$	−29.1858	−1.02612	−0.513060	+	0.858353i	$0.671488\pi$
$810$	−4.10962	−0.144397
$811$	−42.0110	−1.47521	−0.737603	−	0.675235i	$-0.764042\pi$
$811$	−42.0110	−1.47521	−0.737603	+	0.675235i	$0.764042\pi$
$812$	17.0116	0.596990
$813$	29.3414	1.02905
$814$	−0.155775	−0.00545992
$815$	−64.0417	−2.24328
$816$	6.56716	0.229897
$817$	7.65669	0.267874
$818$	7.77160	0.271728
$819$	−7.55632	−0.264039
$820$	2.74374	0.0958156
$821$	−2.14389	−0.0748222	−0.0374111	−	0.999300i	$-0.511911\pi$
$821$	−2.14389	−0.0748222	−0.0374111	+	0.999300i	$0.511911\pi$
$822$	27.7132	0.966608
$823$	30.0484	1.04742	0.523711	−	0.851896i	$-0.324547\pi$
$823$	30.0484	1.04742	0.523711	+	0.851896i	$0.324547\pi$
$824$	32.4534	1.13057
$825$	−1.74619	−0.0607947
$826$	−17.2554	−0.600392
$827$	32.3020	1.12325	0.561625	−	0.827392i	$-0.310176\pi$
$827$	32.3020	1.12325	0.561625	+	0.827392i	$0.310176\pi$
$828$	−0.493968	−0.0171666
$829$	30.2603	1.05098	0.525491	−	0.850799i	$-0.323882\pi$
$829$	30.2603	1.05098	0.525491	+	0.850799i	$0.323882\pi$
$830$	−26.2641	−0.911639
$831$	−2.86762	−0.0994767
$832$	−11.6216	−0.402908
$833$	6.67927	0.231423
$834$	−11.0798	−0.383663
$835$	−20.8207	−0.720531
$836$	−0.596148	−0.0206182
$837$	−1.46354	−0.0505874
$838$	41.5289	1.43459
$839$	−0.881464	−0.0304315	−0.0152158	−	0.999884i	$-0.504844\pi$
$839$	−0.881464	−0.0304315	−0.0152158	+	0.999884i	$0.504844\pi$
$840$	−21.4967	−0.741708
$841$	78.3735	2.70254
$842$	29.0464	1.00101
$843$	−32.5003	−1.11937
$844$	3.25212	0.111942
$845$	−21.3157	−0.733282
$846$	−6.81364	−0.234258
$847$	−34.6226	−1.18965
$848$	−47.1623	−1.61956
$849$	20.1377	0.691126
$850$	−3.97162	−0.136225
$851$	−0.106423	−0.00364814
$852$	6.49035	0.222356
$853$	−3.10967	−0.106473	−0.0532365	−	0.998582i	$-0.516954\pi$
$853$	−3.10967	−0.106473	−0.0532365	+	0.998582i	$0.516954\pi$
$854$	41.9734	1.43630
$855$	−3.38841	−0.115881
$856$	−27.8184	−0.950815
$857$	31.5844	1.07890	0.539451	−	0.842017i	$-0.318632\pi$
$857$	31.5844	1.07890	0.539451	+	0.842017i	$0.318632\pi$
$858$	−3.32793	−0.113614
$859$	−10.4583	−0.356831	−0.178416	−	0.983955i	$-0.557097\pi$
$859$	−10.4583	−0.356831	−0.178416	+	0.983955i	$0.557097\pi$
$860$	−7.35866	−0.250928
$861$	−7.55875	−0.257601
$862$	−64.7411	−2.20509
$863$	29.2721	0.996433	0.498217	−	0.867053i	$-0.333988\pi$
$863$	29.2721	0.996433	0.498217	+	0.867053i	$0.333988\pi$
$864$	−2.64885	−0.0901157
$865$	61.8040	2.10140
$866$	40.6672	1.38193
$867$	15.0701	0.511807
$868$	2.40271	0.0815534
$869$	1.22536	0.0415674
$870$	42.5844	1.44375
$871$	−10.0125	−0.339262
$872$	−11.1732	−0.378372
$873$	7.74609	0.262165
$874$	−2.11224	−0.0714475
$875$	−28.5628	−0.965600
$876$	−0.892054	−0.0301397
$877$	−17.6778	−0.596937	−0.298468	−	0.954419i	$-0.596476\pi$
$877$	−17.6778	−0.596937	−0.298468	+	0.954419i	$0.596476\pi$
$878$	19.6415	0.662868
$879$	−18.9168	−0.638049
$880$	−11.8660	−0.400002
$881$	−3.04541	−0.102602	−0.0513012	−	0.998683i	$-0.516337\pi$
$881$	−3.04541	−0.102602	−0.0513012	+	0.998683i	$0.516337\pi$
$882$	−7.56817	−0.254833
$883$	−15.7628	−0.530459	−0.265230	−	0.964185i	$-0.585448\pi$
$883$	−15.7628	−0.530459	−0.265230	+	0.964185i	$0.585448\pi$
$884$	−1.45948	−0.0490878
$885$	−8.32875	−0.279968
$886$	−3.13591	−0.105353
$887$	−21.8406	−0.733335	−0.366667	−	0.930352i	$-0.619501\pi$
$887$	−21.8406	−0.733335	−0.366667	+	0.930352i	$0.619501\pi$
$888$	−0.246638	−0.00827663
$889$	27.7469	0.930600
$890$	−1.76560	−0.0591832
$891$	0.961440	0.0322094
$892$	−0.0549958	−0.00184140
$893$	−5.61788	−0.187995
$894$	−10.6097	−0.354841
$895$	−38.6179	−1.29086
$896$	−46.7909	−1.56317
$897$	−2.27359	−0.0759130
$898$	−13.5239	−0.451299
$899$	15.1654	0.505795
$900$	0.867720	0.0289240
$901$	13.8597	0.461733
$902$	−3.32900	−0.110844
$903$	20.2724	0.674624
$904$	26.3469	0.876286
$905$	2.61079	0.0867856
$906$	6.02726	0.200242
$907$	−44.5734	−1.48003	−0.740017	−	0.672588i	$-0.765183\pi$
$907$	−44.5734	−1.48003	−0.740017	+	0.672588i	$0.765183\pi$
$908$	−4.99420	−0.165738
$909$	8.32864	0.276244
$910$	31.0536	1.02942
$911$	−21.3786	−0.708305	−0.354153	−	0.935188i	$-0.615231\pi$
$911$	−21.3786	−0.708305	−0.354153	+	0.935188i	$0.615231\pi$
$912$	−6.13526	−0.203159
$913$	6.14444	0.203351
$914$	28.7500	0.950965
$915$	20.2595	0.669759
$916$	10.7942	0.356649
$917$	−64.8409	−2.14123
$918$	2.18674	0.0721732
$919$	−45.8883	−1.51371	−0.756857	−	0.653581i	$-0.773266\pi$
$919$	−45.8883	−1.51371	−0.756857	+	0.653581i	$0.773266\pi$
$920$	−6.46807	−0.213246
$921$	−6.24336	−0.205726
$922$	14.4856	0.477059
$923$	29.8732	0.983289
$924$	−1.57840	−0.0519257
$925$	0.186947	0.00614677
$926$	12.2245	0.401721
$927$	13.5440	0.444843
$928$	27.4477	0.901015
$929$	9.13738	0.299788	0.149894	−	0.988702i	$-0.452107\pi$
$929$	9.13738	0.299788	0.149894	+	0.988702i	$0.452107\pi$
$930$	6.01460	0.197227
$931$	−6.24000	−0.204508
$932$	−3.97881	−0.130330
$933$	−7.45699	−0.244131
$934$	13.5311	0.442752
$935$	3.48708	0.114040
$936$	−5.26909	−0.172226
$937$	34.6066	1.13055	0.565275	−	0.824903i	$-0.308770\pi$
$937$	34.6066	1.13055	0.565275	+	0.824903i	$0.308770\pi$
$938$	−24.6286	−0.804151
$939$	−28.5353	−0.931216
$940$	5.39921	0.176103
$941$	15.0445	0.490437	0.245219	−	0.969468i	$-0.421140\pi$
$941$	15.0445	0.490437	0.245219	+	0.969468i	$0.421140\pi$
$942$	−4.26606	−0.138996
$943$	−2.27432	−0.0740621
$944$	−15.0806	−0.490830
$945$	−8.97139	−0.291839
$946$	8.92831	0.290284
$947$	20.2433	0.657820	0.328910	−	0.944361i	$-0.393319\pi$
$947$	20.2433	0.657820	0.328910	+	0.944361i	$0.393319\pi$
$948$	−0.608905	−0.0197763
$949$	−4.10587	−0.133282
$950$	3.71042	0.120382
$951$	−9.42774	−0.305715
$952$	11.4385	0.370723
$953$	−40.3903	−1.30837	−0.654185	−	0.756334i	$-0.726988\pi$
$953$	−40.3903	−1.30837	−0.654185	+	0.756334i	$0.726988\pi$
$954$	−15.7042	−0.508441
$955$	−42.6307	−1.37950
$956$	2.22243	0.0718785
$957$	−9.96255	−0.322044
$958$	−34.4140	−1.11186
$959$	60.4985	1.95360
$960$	−13.7980	−0.445329
$961$	−28.8580	−0.930905
$962$	0.356286	0.0114871
$963$	−11.6097	−0.374116
$964$	−7.40384	−0.238462
$965$	56.0789	1.80524
$966$	−5.59251	−0.179936
$967$	−21.2417	−0.683087	−0.341543	−	0.939866i	$-0.610950\pi$
$967$	−21.2417	−0.683087	−0.341543	+	0.939866i	$0.610950\pi$
$968$	−24.1427	−0.775975
$969$	1.80298	0.0579201
$970$	−31.8335	−1.02211
$971$	24.9595	0.800990	0.400495	−	0.916299i	$-0.368838\pi$
$971$	24.9595	0.800990	0.400495	+	0.916299i	$0.368838\pi$
$972$	−0.477759	−0.0153241
$973$	−24.1875	−0.775416
$974$	55.9705	1.79341
$975$	3.99386	0.127906
$976$	36.6832	1.17420
$977$	−25.7236	−0.822972	−0.411486	−	0.911416i	$-0.634990\pi$
$977$	−25.7236	−0.822972	−0.411486	+	0.911416i	$0.634990\pi$
$978$	−38.6118	−1.23467
$979$	0.413061	0.0132015
$980$	5.99711	0.191571
$981$	−4.66299	−0.148878
$982$	37.6030	1.19996
$983$	−48.1254	−1.53496	−0.767480	−	0.641073i	$-0.778490\pi$
$983$	−48.1254	−1.53496	−0.767480	+	0.641073i	$0.778490\pi$
$984$	−5.27078	−0.168026
$985$	−1.69535	−0.0540185
$986$	−22.6593	−0.721618
$987$	−14.8743	−0.473455
$988$	1.36350	0.0433787
$989$	6.09969	0.193959
$990$	−3.95115	−0.125576
$991$	12.6416	0.401572	0.200786	−	0.979635i	$-0.435650\pi$
$991$	12.6416	0.401572	0.200786	+	0.979635i	$0.435650\pi$
$992$	3.87671	0.123086
$993$	16.6458	0.528238
$994$	73.4812	2.33068
$995$	35.5525	1.12709
$996$	−3.05330	−0.0967475
$997$	34.3583	1.08814	0.544069	−	0.839041i	$-0.316883\pi$
$997$	34.3583	1.08814	0.544069	+	0.839041i	$0.316883\pi$
$998$	−39.6399	−1.25478
$999$	−0.102931	−0.00325660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	543.2.a.d.1.2	✓	8
3.2	odd	2		1629.2.a.e.1.7		8
4.3	odd	2		8688.2.a.bf.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
543.2.a.d.1.2	✓	8	1.1	even	1	trivial
1629.2.a.e.1.7		8	3.2	odd	2
8688.2.a.bf.1.6		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 543 = 3 \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	543.a (trivial)

\(f(q)\)	\(=\)	\(q-1.57409 q^{2} -1.00000 q^{3} +0.477759 q^{4} +2.61079 q^{5} +1.57409 q^{6} +3.43627 q^{7} +2.39614 q^{8} +1.00000 q^{9} -4.10962 q^{10} +0.961440 q^{11} -0.477759 q^{12} -2.19899 q^{13} -5.40900 q^{14} -2.61079 q^{15} -4.72726 q^{16} +1.38921 q^{17} -1.57409 q^{18} -1.29785 q^{19} +1.24733 q^{20} -3.43627 q^{21} -1.51339 q^{22} -1.03393 q^{23} -2.39614 q^{24} +1.81623 q^{25} +3.46140 q^{26} -1.00000 q^{27} +1.64171 q^{28} +10.3621 q^{29} +4.10962 q^{30} +1.46354 q^{31} +2.64885 q^{32} -0.961440 q^{33} -2.18674 q^{34} +8.97139 q^{35} +0.477759 q^{36} +0.102931 q^{37} +2.04293 q^{38} +2.19899 q^{39} +6.25583 q^{40} +2.19969 q^{41} +5.40900 q^{42} -5.89954 q^{43} +0.459336 q^{44} +2.61079 q^{45} +1.62749 q^{46} +4.32862 q^{47} +4.72726 q^{48} +4.80796 q^{49} -2.85891 q^{50} -1.38921 q^{51} -1.05059 q^{52} +9.97667 q^{53} +1.57409 q^{54} +2.51012 q^{55} +8.23380 q^{56} +1.29785 q^{57} -16.3109 q^{58} +3.19012 q^{59} -1.24733 q^{60} -7.75991 q^{61} -2.30375 q^{62} +3.43627 q^{63} +5.28500 q^{64} -5.74110 q^{65} +1.51339 q^{66} +4.55325 q^{67} +0.663707 q^{68} +1.03393 q^{69} -14.1218 q^{70} -13.5850 q^{71} +2.39614 q^{72} +1.86716 q^{73} -0.162023 q^{74} -1.81623 q^{75} -0.620058 q^{76} +3.30377 q^{77} -3.46140 q^{78} +1.27450 q^{79} -12.3419 q^{80} +1.00000 q^{81} -3.46252 q^{82} +6.39088 q^{83} -1.64171 q^{84} +3.62693 q^{85} +9.28640 q^{86} -10.3621 q^{87} +2.30375 q^{88} +0.429627 q^{89} -4.10962 q^{90} -7.55632 q^{91} -0.493968 q^{92} -1.46354 q^{93} -6.81364 q^{94} -3.38841 q^{95} -2.64885 q^{96} +7.74609 q^{97} -7.56817 q^{98} +0.961440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} - 8 q^{3} + 5 q^{4} + 5 q^{5} - 3 q^{6} - 3 q^{7} + 6 q^{8} + 8 q^{9} + 2 q^{10} + 4 q^{11} - 5 q^{12} + 13 q^{13} + 5 q^{14} - 5 q^{15} + 3 q^{16} + 27 q^{17} + 3 q^{18} - 10 q^{19} + 21 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) \) 8 * q + 3 * q^2 - 8 * q^3 + 5 * q^4 + 5 * q^5 - 3 * q^6 - 3 * q^7 + 6 * q^8 + 8 * q^9 + 2 * q^10 + 4 * q^11 - 5 * q^12 + 13 * q^13 + 5 * q^14 - 5 * q^15 + 3 * q^16 + 27 * q^17 + 3 * q^18 - 10 * q^19 + 21 * q^20 + 3 * q^21 + 11 * q^22 + 3 * q^23 - 6 * q^24 + 13 * q^25 + 11 * q^26 - 8 * q^27 - 3 * q^28 + 16 * q^29 - 2 * q^30 - q^31 + 10 * q^32 - 4 * q^33 + 10 * q^34 + 14 * q^35 + 5 * q^36 + 3 * q^37 + 14 * q^38 - 13 * q^39 + 11 * q^40 + 20 * q^41 - 5 * q^42 - 4 * q^43 + 9 * q^44 + 5 * q^45 - 17 * q^46 + 12 * q^47 - 3 * q^48 + 5 * q^49 + 2 * q^50 - 27 * q^51 + 21 * q^52 + 23 * q^53 - 3 * q^54 - 13 * q^55 - 8 * q^56 + 10 * q^57 - 21 * q^58 + 7 * q^59 - 21 * q^60 + q^61 + q^62 - 3 * q^63 - 32 * q^64 + 10 * q^65 - 11 * q^66 - 25 * q^67 - q^68 - 3 * q^69 - 10 * q^70 + 11 * q^71 + 6 * q^72 + 23 * q^73 + 2 * q^74 - 13 * q^75 - 19 * q^76 + 20 * q^77 - 11 * q^78 - 26 * q^79 - 15 * q^80 + 8 * q^81 - 22 * q^82 + 16 * q^83 + 3 * q^84 - 33 * q^85 - 21 * q^86 - 16 * q^87 - q^88 + 14 * q^89 + 2 * q^90 - 35 * q^91 - 18 * q^92 + q^93 - 52 * q^94 + 10 * q^95 - 10 * q^96 + 28 * q^97 - 51 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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