LMFDB - Embedded newform 8049.2.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, 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("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$1$
Dimension:	$95$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55942 q^{2} +1.00000 q^{3} +4.55064 q^{4} -0.0700614 q^{5} -2.55942 q^{6} -1.78635 q^{7} -6.52816 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.55942 q^{2} +1.00000 q^{3} +4.55064 q^{4} -0.0700614 q^{5} -2.55942 q^{6} -1.78635 q^{7} -6.52816 q^{8} +1.00000 q^{9} +0.179317 q^{10} +2.28047 q^{11} +4.55064 q^{12} +1.17228 q^{13} +4.57201 q^{14} -0.0700614 q^{15} +7.60703 q^{16} -4.31139 q^{17} -2.55942 q^{18} +2.82394 q^{19} -0.318824 q^{20} -1.78635 q^{21} -5.83668 q^{22} +5.45574 q^{23} -6.52816 q^{24} -4.99509 q^{25} -3.00035 q^{26} +1.00000 q^{27} -8.12902 q^{28} +4.97140 q^{29} +0.179317 q^{30} -4.76697 q^{31} -6.41328 q^{32} +2.28047 q^{33} +11.0347 q^{34} +0.125154 q^{35} +4.55064 q^{36} -6.87040 q^{37} -7.22765 q^{38} +1.17228 q^{39} +0.457372 q^{40} -8.45559 q^{41} +4.57201 q^{42} +7.83134 q^{43} +10.3776 q^{44} -0.0700614 q^{45} -13.9635 q^{46} +4.96531 q^{47} +7.60703 q^{48} -3.80897 q^{49} +12.7845 q^{50} -4.31139 q^{51} +5.33460 q^{52} +6.11547 q^{53} -2.55942 q^{54} -0.159773 q^{55} +11.6616 q^{56} +2.82394 q^{57} -12.7239 q^{58} +3.85854 q^{59} -0.318824 q^{60} -9.59660 q^{61} +12.2007 q^{62} -1.78635 q^{63} +1.20023 q^{64} -0.0821313 q^{65} -5.83668 q^{66} -10.8418 q^{67} -19.6196 q^{68} +5.45574 q^{69} -0.320322 q^{70} -11.0571 q^{71} -6.52816 q^{72} +4.81766 q^{73} +17.5843 q^{74} -4.99509 q^{75} +12.8507 q^{76} -4.07371 q^{77} -3.00035 q^{78} -12.0961 q^{79} -0.532959 q^{80} +1.00000 q^{81} +21.6414 q^{82} -16.4860 q^{83} -8.12902 q^{84} +0.302062 q^{85} -20.0437 q^{86} +4.97140 q^{87} -14.8873 q^{88} +1.42185 q^{89} +0.179317 q^{90} -2.09409 q^{91} +24.8271 q^{92} -4.76697 q^{93} -12.7083 q^{94} -0.197849 q^{95} -6.41328 q^{96} +3.20050 q^{97} +9.74875 q^{98} +2.28047 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 95 q - 9 q^{2} + 95 q^{3} + 65 q^{4} - 15 q^{5} - 9 q^{6} - 36 q^{7} - 27 q^{8} + 95 q^{9}+O(q^{10}) $ 95 * q - 9 * q^2 + 95 * q^3 + 65 * q^4 - 15 * q^5 - 9 * q^6 - 36 * q^7 - 27 * q^8 + 95 * q^9	\( 95 q - 9 q^{2} + 95 q^{3} + 65 q^{4} - 15 q^{5} - 9 q^{6} - 36 q^{7} - 27 q^{8} + 95 q^{9} - 36 q^{10} - 48 q^{11} + 65 q^{12} - 73 q^{13} - 17 q^{14} - 15 q^{15} + 13 q^{16} - 9 q^{17} - 9 q^{18} - 66 q^{19} - 35 q^{20} - 36 q^{21} - 37 q^{22} - 58 q^{23} - 27 q^{24} + 24 q^{25} - 25 q^{26} + 95 q^{27} - 75 q^{28} - 31 q^{29} - 36 q^{30} - 129 q^{31} - 53 q^{32} - 48 q^{33} - 61 q^{34} - 38 q^{35} + 65 q^{36} - 127 q^{37} + q^{38} - 73 q^{39} - 74 q^{40} - 31 q^{41} - 17 q^{42} - 62 q^{43} - 76 q^{44} - 15 q^{45} - 60 q^{46} - 75 q^{47} + 13 q^{48} + 5 q^{49} - 30 q^{50} - 9 q^{51} - 137 q^{52} - 28 q^{53} - 9 q^{54} - 117 q^{55} - 23 q^{56} - 66 q^{57} - 90 q^{58} - 60 q^{59} - 35 q^{60} - 96 q^{61} + 10 q^{62} - 36 q^{63} - 75 q^{64} - 28 q^{65} - 37 q^{66} - 116 q^{67} + 3 q^{68} - 58 q^{69} - 73 q^{70} - 144 q^{71} - 27 q^{72} - 121 q^{73} - 16 q^{74} + 24 q^{75} - 118 q^{76} - 3 q^{77} - 25 q^{78} - 135 q^{79} - 36 q^{80} + 95 q^{81} - 102 q^{82} - 21 q^{83} - 75 q^{84} - 129 q^{85} - 46 q^{86} - 31 q^{87} - 77 q^{88} - 63 q^{89} - 36 q^{90} - 123 q^{91} - 42 q^{92} - 129 q^{93} - 44 q^{94} - 80 q^{95} - 53 q^{96} - 144 q^{97} + 10 q^{98} - 48 q^{99}+O(q^{100}) \) 95 * q - 9 * q^2 + 95 * q^3 + 65 * q^4 - 15 * q^5 - 9 * q^6 - 36 * q^7 - 27 * q^8 + 95 * q^9 - 36 * q^10 - 48 * q^11 + 65 * q^12 - 73 * q^13 - 17 * q^14 - 15 * q^15 + 13 * q^16 - 9 * q^17 - 9 * q^18 - 66 * q^19 - 35 * q^20 - 36 * q^21 - 37 * q^22 - 58 * q^23 - 27 * q^24 + 24 * q^25 - 25 * q^26 + 95 * q^27 - 75 * q^28 - 31 * q^29 - 36 * q^30 - 129 * q^31 - 53 * q^32 - 48 * q^33 - 61 * q^34 - 38 * q^35 + 65 * q^36 - 127 * q^37 + q^38 - 73 * q^39 - 74 * q^40 - 31 * q^41 - 17 * q^42 - 62 * q^43 - 76 * q^44 - 15 * q^45 - 60 * q^46 - 75 * q^47 + 13 * q^48 + 5 * q^49 - 30 * q^50 - 9 * q^51 - 137 * q^52 - 28 * q^53 - 9 * q^54 - 117 * q^55 - 23 * q^56 - 66 * q^57 - 90 * q^58 - 60 * q^59 - 35 * q^60 - 96 * q^61 + 10 * q^62 - 36 * q^63 - 75 * q^64 - 28 * q^65 - 37 * q^66 - 116 * q^67 + 3 * q^68 - 58 * q^69 - 73 * q^70 - 144 * q^71 - 27 * q^72 - 121 * q^73 - 16 * q^74 + 24 * q^75 - 118 * q^76 - 3 * q^77 - 25 * q^78 - 135 * q^79 - 36 * q^80 + 95 * q^81 - 102 * q^82 - 21 * q^83 - 75 * q^84 - 129 * q^85 - 46 * q^86 - 31 * q^87 - 77 * q^88 - 63 * q^89 - 36 * q^90 - 123 * q^91 - 42 * q^92 - 129 * q^93 - 44 * q^94 - 80 * q^95 - 53 * q^96 - 144 * q^97 + 10 * q^98 - 48 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.55942	−1.80978	−0.904892	−	0.425641i	$-0.860049\pi$
$2$	−2.55942	−1.80978	−0.904892	+	0.425641i	$0.860049\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.55064	2.27532
$5$	−0.0700614	−0.0313324	−0.0156662	−	0.999877i	$-0.504987\pi$
$5$	−0.0700614	−0.0313324	−0.0156662	+	0.999877i	$0.504987\pi$
$6$	−2.55942	−1.04488
$7$	−1.78635	−0.675176	−0.337588	−	0.941294i	$-0.609611\pi$
$7$	−1.78635	−0.675176	−0.337588	+	0.941294i	$0.609611\pi$
$8$	−6.52816	−2.30805
$9$	1.00000	0.333333
$10$	0.179317	0.0567049
$11$	2.28047	0.687587	0.343794	−	0.939045i	$-0.388288\pi$
$11$	2.28047	0.687587	0.343794	+	0.939045i	$0.388288\pi$
$12$	4.55064	1.31366
$13$	1.17228	0.325131	0.162565	−	0.986698i	$-0.448023\pi$
$13$	1.17228	0.325131	0.162565	+	0.986698i	$0.448023\pi$
$14$	4.57201	1.22192
$15$	−0.0700614	−0.0180898
$16$	7.60703	1.90176
$17$	−4.31139	−1.04567	−0.522833	−	0.852435i	$-0.675125\pi$
$17$	−4.31139	−1.04567	−0.522833	+	0.852435i	$0.675125\pi$
$18$	−2.55942	−0.603261
$19$	2.82394	0.647856	0.323928	−	0.946082i	$-0.394996\pi$
$19$	2.82394	0.647856	0.323928	+	0.946082i	$0.394996\pi$
$20$	−0.318824	−0.0712912
$21$	−1.78635	−0.389813
$22$	−5.83668	−1.24438
$23$	5.45574	1.13760	0.568800	−	0.822476i	$-0.307408\pi$
$23$	5.45574	1.13760	0.568800	+	0.822476i	$0.307408\pi$
$24$	−6.52816	−1.33255
$25$	−4.99509	−0.999018
$26$	−3.00035	−0.588417
$27$	1.00000	0.192450
$28$	−8.12902	−1.53624
$29$	4.97140	0.923166	0.461583	−	0.887097i	$-0.347282\pi$
$29$	4.97140	0.923166	0.461583	+	0.887097i	$0.347282\pi$
$30$	0.179317	0.0327386
$31$	−4.76697	−0.856172	−0.428086	−	0.903738i	$-0.640812\pi$
$31$	−4.76697	−0.856172	−0.428086	+	0.903738i	$0.640812\pi$
$32$	−6.41328	−1.13372
$33$	2.28047	0.396979
$34$	11.0347	1.89243
$35$	0.125154	0.0211549
$36$	4.55064	0.758440
$37$	−6.87040	−1.12949	−0.564744	−	0.825266i	$-0.691025\pi$
$37$	−6.87040	−1.12949	−0.564744	+	0.825266i	$0.691025\pi$
$38$	−7.22765	−1.17248
$39$	1.17228	0.187714
$40$	0.457372	0.0723168
$41$	−8.45559	−1.32054	−0.660271	−	0.751028i	$-0.729558\pi$
$41$	−8.45559	−1.32054	−0.660271	+	0.751028i	$0.729558\pi$
$42$	4.57201	0.705477
$43$	7.83134	1.19427	0.597134	−	0.802142i	$-0.296306\pi$
$43$	7.83134	1.19427	0.597134	+	0.802142i	$0.296306\pi$
$44$	10.3776	1.56448
$45$	−0.0700614	−0.0104441
$46$	−13.9635	−2.05881
$47$	4.96531	0.724265	0.362132	−	0.932127i	$-0.382049\pi$
$47$	4.96531	0.724265	0.362132	+	0.932127i	$0.382049\pi$
$48$	7.60703	1.09798
$49$	−3.80897	−0.544138
$50$	12.7845	1.80801
$51$	−4.31139	−0.603715
$52$	5.33460	0.739776
$53$	6.11547	0.840024	0.420012	−	0.907518i	$-0.362026\pi$
$53$	6.11547	0.840024	0.420012	+	0.907518i	$0.362026\pi$
$54$	−2.55942	−0.348293
$55$	−0.159773	−0.0215438
$56$	11.6616	1.55834
$57$	2.82394	0.374040
$58$	−12.7239	−1.67073
$59$	3.85854	0.502339	0.251170	−	0.967943i	$-0.419185\pi$
$59$	3.85854	0.502339	0.251170	+	0.967943i	$0.419185\pi$
$60$	−0.318824	−0.0411600
$61$	−9.59660	−1.22872	−0.614359	−	0.789026i	$-0.710585\pi$
$61$	−9.59660	−1.22872	−0.614359	+	0.789026i	$0.710585\pi$
$62$	12.2007	1.54949
$63$	−1.78635	−0.225059
$64$	1.20023	0.150029
$65$	−0.0821313	−0.0101871
$66$	−5.83668	−0.718446
$67$	−10.8418	−1.32453	−0.662266	−	0.749269i	$-0.730405\pi$
$67$	−10.8418	−1.32453	−0.662266	+	0.749269i	$0.730405\pi$
$68$	−19.6196	−2.37922
$69$	5.45574	0.656794
$70$	−0.320322	−0.0382858
$71$	−11.0571	−1.31224	−0.656118	−	0.754659i	$-0.727802\pi$
$71$	−11.0571	−1.31224	−0.656118	+	0.754659i	$0.727802\pi$
$72$	−6.52816	−0.769351
$73$	4.81766	0.563864	0.281932	−	0.959434i	$-0.409025\pi$
$73$	4.81766	0.563864	0.281932	+	0.959434i	$0.409025\pi$
$74$	17.5843	2.04413
$75$	−4.99509	−0.576783
$76$	12.8507	1.47408
$77$	−4.07371	−0.464242
$78$	−3.00035	−0.339722
$79$	−12.0961	−1.36092	−0.680460	−	0.732785i	$-0.738220\pi$
$79$	−12.0961	−1.36092	−0.680460	+	0.732785i	$0.738220\pi$
$80$	−0.532959	−0.0595867
$81$	1.00000	0.111111
$82$	21.6414	2.38989
$83$	−16.4860	−1.80957	−0.904785	−	0.425868i	$-0.859969\pi$
$83$	−16.4860	−1.80957	−0.904785	+	0.425868i	$0.859969\pi$
$84$	−8.12902	−0.886948
$85$	0.302062	0.0327632
$86$	−20.0437	−2.16137
$87$	4.97140	0.532990
$88$	−14.8873	−1.58699
$89$	1.42185	0.150716	0.0753580	−	0.997157i	$-0.475990\pi$
$89$	1.42185	0.150716	0.0753580	+	0.997157i	$0.475990\pi$
$90$	0.179317	0.0189016
$91$	−2.09409	−0.219520
$92$	24.8271	2.58840
$93$	−4.76697	−0.494311
$94$	−12.7083	−1.31076
$95$	−0.197849	−0.0202989
$96$	−6.41328	−0.654553
$97$	3.20050	0.324961	0.162481	−	0.986712i	$-0.448051\pi$
$97$	3.20050	0.324961	0.162481	+	0.986712i	$0.448051\pi$
$98$	9.74875	0.984772
$99$	2.28047	0.229196
$100$	−22.7309	−2.27309
$101$	7.38681	0.735015	0.367507	−	0.930021i	$-0.380211\pi$
$101$	7.38681	0.735015	0.367507	+	0.930021i	$0.380211\pi$
$102$	11.0347	1.09259
$103$	3.41466	0.336456	0.168228	−	0.985748i	$-0.446195\pi$
$103$	3.41466	0.336456	0.168228	+	0.985748i	$0.446195\pi$
$104$	−7.65280	−0.750419
$105$	0.125154	0.0122138
$106$	−15.6521	−1.52026
$107$	13.7416	1.32845	0.664226	−	0.747532i	$-0.268761\pi$
$107$	13.7416	1.32845	0.664226	+	0.747532i	$0.268761\pi$
$108$	4.55064	0.437885
$109$	4.38863	0.420354	0.210177	−	0.977663i	$-0.432596\pi$
$109$	4.38863	0.420354	0.210177	+	0.977663i	$0.432596\pi$
$110$	0.408926	0.0389896
$111$	−6.87040	−0.652110
$112$	−13.5888	−1.28402
$113$	17.2985	1.62730	0.813652	−	0.581352i	$-0.197476\pi$
$113$	17.2985	1.62730	0.813652	+	0.581352i	$0.197476\pi$
$114$	−7.22765	−0.676932
$115$	−0.382237	−0.0356438
$116$	22.6230	2.10050
$117$	1.17228	0.108377
$118$	−9.87563	−0.909126
$119$	7.70164	0.706008
$120$	0.457372	0.0417521
$121$	−5.79946	−0.527224
$122$	24.5617	2.22372
$123$	−8.45559	−0.762415
$124$	−21.6927	−1.94807
$125$	0.700270	0.0626341
$126$	4.57201	0.407307
$127$	9.27127	0.822692	0.411346	−	0.911479i	$-0.365059\pi$
$127$	9.27127	0.822692	0.411346	+	0.911479i	$0.365059\pi$
$128$	9.75466	0.862199
$129$	7.83134	0.689511
$130$	0.210209	0.0184365
$131$	−5.89899	−0.515397	−0.257698	−	0.966225i	$-0.582964\pi$
$131$	−5.89899	−0.515397	−0.257698	+	0.966225i	$0.582964\pi$
$132$	10.3776	0.903253
$133$	−5.04453	−0.437417
$134$	27.7486	2.39712
$135$	−0.0700614	−0.00602992
$136$	28.1454	2.41345
$137$	−21.1192	−1.80434	−0.902168	−	0.431384i	$-0.858025\pi$
$137$	−21.1192	−1.80434	−0.902168	+	0.431384i	$0.858025\pi$
$138$	−13.9635	−1.18866
$139$	9.71783	0.824256	0.412128	−	0.911126i	$-0.364786\pi$
$139$	9.71783	0.824256	0.412128	+	0.911126i	$0.364786\pi$
$140$	0.569530	0.0481341
$141$	4.96531	0.418154
$142$	28.2998	2.37486
$143$	2.67334	0.223556
$144$	7.60703	0.633919
$145$	−0.348303	−0.0289250
$146$	−12.3304	−1.02047
$147$	−3.80897	−0.314158
$148$	−31.2647	−2.56994
$149$	3.84783	0.315227	0.157613	−	0.987501i	$-0.449620\pi$
$149$	3.84783	0.315227	0.157613	+	0.987501i	$0.449620\pi$
$150$	12.7845	1.04385
$151$	1.25490	0.102123	0.0510613	−	0.998696i	$-0.483740\pi$
$151$	1.25490	0.102123	0.0510613	+	0.998696i	$0.483740\pi$
$152$	−18.4351	−1.49529
$153$	−4.31139	−0.348555
$154$	10.4263	0.840178
$155$	0.333980	0.0268259
$156$	5.33460	0.427110
$157$	8.86344	0.707380	0.353690	−	0.935363i	$-0.384927\pi$
$157$	8.86344	0.707380	0.353690	+	0.935363i	$0.384927\pi$
$158$	30.9591	2.46297
$159$	6.11547	0.484988
$160$	0.449324	0.0355222
$161$	−9.74584	−0.768080
$162$	−2.55942	−0.201087
$163$	−15.0633	−1.17985	−0.589923	−	0.807460i	$-0.700842\pi$
$163$	−15.0633	−1.17985	−0.589923	+	0.807460i	$0.700842\pi$
$164$	−38.4783	−3.00465
$165$	−0.159773	−0.0124383
$166$	42.1946	3.27493
$167$	7.14312	0.552752	0.276376	−	0.961050i	$-0.410867\pi$
$167$	7.14312	0.552752	0.276376	+	0.961050i	$0.410867\pi$
$168$	11.6616	0.899708
$169$	−11.6258	−0.894290
$170$	−0.773104	−0.0592944
$171$	2.82394	0.215952
$172$	35.6376	2.71734
$173$	−16.8895	−1.28408	−0.642042	−	0.766669i	$-0.721913\pi$
$173$	−16.8895	−1.28408	−0.642042	+	0.766669i	$0.721913\pi$
$174$	−12.7239	−0.964597
$175$	8.92296	0.674513
$176$	17.3476	1.30762
$177$	3.85854	0.290026
$178$	−3.63912	−0.272763
$179$	3.72883	0.278706	0.139353	−	0.990243i	$-0.455498\pi$
$179$	3.72883	0.278706	0.139353	+	0.990243i	$0.455498\pi$
$180$	−0.318824	−0.0237637
$181$	−11.6088	−0.862873	−0.431437	−	0.902143i	$-0.641993\pi$
$181$	−11.6088	−0.862873	−0.431437	+	0.902143i	$0.641993\pi$
$182$	5.35966	0.397284
$183$	−9.59660	−0.709401
$184$	−35.6159	−2.62564
$185$	0.481350	0.0353896
$186$	12.2007	0.894597
$187$	−9.83199	−0.718986
$188$	22.5953	1.64793
$189$	−1.78635	−0.129938
$190$	0.506379	0.0367366
$191$	1.43353	0.103727	0.0518634	−	0.998654i	$-0.483484\pi$
$191$	1.43353	0.103727	0.0518634	+	0.998654i	$0.483484\pi$
$192$	1.20023	0.0866194
$193$	−12.3095	−0.886056	−0.443028	−	0.896508i	$-0.646096\pi$
$193$	−12.3095	−0.886056	−0.443028	+	0.896508i	$0.646096\pi$
$194$	−8.19142	−0.588109
$195$	−0.0821313	−0.00588154
$196$	−17.3332	−1.23809
$197$	7.32089	0.521592	0.260796	−	0.965394i	$-0.416015\pi$
$197$	7.32089	0.521592	0.260796	+	0.965394i	$0.416015\pi$
$198$	−5.83668	−0.414795
$199$	2.47396	0.175374	0.0876872	−	0.996148i	$-0.472052\pi$
$199$	2.47396	0.175374	0.0876872	+	0.996148i	$0.472052\pi$
$200$	32.6087	2.30579
$201$	−10.8418	−0.764719
$202$	−18.9060	−1.33022
$203$	−8.88064	−0.623299
$204$	−19.6196	−1.37365
$205$	0.592410	0.0413757
$206$	−8.73955	−0.608913
$207$	5.45574	0.379200
$208$	8.91754	0.618320
$209$	6.43990	0.445458
$210$	−0.320322	−0.0221043
$211$	−20.1290	−1.38574	−0.692870	−	0.721062i	$-0.743654\pi$
$211$	−20.1290	−1.38574	−0.692870	+	0.721062i	$0.743654\pi$
$212$	27.8293	1.91132
$213$	−11.0571	−0.757619
$214$	−35.1706	−2.40421
$215$	−0.548674	−0.0374193
$216$	−6.52816	−0.444185
$217$	8.51545	0.578067
$218$	−11.2324	−0.760751
$219$	4.81766	0.325547
$220$	−0.727068	−0.0490189
$221$	−5.05414	−0.339978
$222$	17.5843	1.18018
$223$	5.07858	0.340087	0.170044	−	0.985437i	$-0.445609\pi$
$223$	5.07858	0.340087	0.170044	+	0.985437i	$0.445609\pi$
$224$	11.4563	0.765459
$225$	−4.99509	−0.333006
$226$	−44.2741	−2.94507
$227$	11.2959	0.749733	0.374866	−	0.927079i	$-0.377689\pi$
$227$	11.2959	0.749733	0.374866	+	0.927079i	$0.377689\pi$
$228$	12.8507	0.851060
$229$	−2.11425	−0.139713	−0.0698567	−	0.997557i	$-0.522254\pi$
$229$	−2.11425	−0.139713	−0.0698567	+	0.997557i	$0.522254\pi$
$230$	0.978305	0.0645075
$231$	−4.07371	−0.268030
$232$	−32.4541	−2.13072
$233$	18.4918	1.21144	0.605718	−	0.795679i	$-0.292886\pi$
$233$	18.4918	1.21144	0.605718	+	0.795679i	$0.292886\pi$
$234$	−3.00035	−0.196139
$235$	−0.347876	−0.0226930
$236$	17.5588	1.14298
$237$	−12.0961	−0.785727
$238$	−19.7117	−1.27772
$239$	8.50268	0.549993	0.274996	−	0.961445i	$-0.411323\pi$
$239$	8.50268	0.549993	0.274996	+	0.961445i	$0.411323\pi$
$240$	−0.532959	−0.0344024
$241$	−11.0537	−0.712034	−0.356017	−	0.934479i	$-0.615866\pi$
$241$	−11.0537	−0.712034	−0.356017	+	0.934479i	$0.615866\pi$
$242$	14.8433	0.954162
$243$	1.00000	0.0641500
$244$	−43.6706	−2.79573
$245$	0.266861	0.0170492
$246$	21.6414	1.37981
$247$	3.31044	0.210638
$248$	31.1195	1.97609
$249$	−16.4860	−1.04476
$250$	−1.79229	−0.113354
$251$	13.5078	0.852607	0.426304	−	0.904580i	$-0.359815\pi$
$251$	13.5078	0.852607	0.426304	+	0.904580i	$0.359815\pi$
$252$	−8.12902	−0.512080
$253$	12.4416	0.782199
$254$	−23.7291	−1.48889
$255$	0.302062	0.0189159
$256$	−27.3668	−1.71042
$257$	−29.9189	−1.86629	−0.933143	−	0.359504i	$-0.882946\pi$
$257$	−29.9189	−1.86629	−0.933143	+	0.359504i	$0.882946\pi$
$258$	−20.0437	−1.24787
$259$	12.2729	0.762602
$260$	−0.373750	−0.0231790
$261$	4.97140	0.307722
$262$	15.0980	0.932757
$263$	−11.9618	−0.737594	−0.368797	−	0.929510i	$-0.620230\pi$
$263$	−11.9618	−0.737594	−0.368797	+	0.929510i	$0.620230\pi$
$264$	−14.8873	−0.916247
$265$	−0.428458	−0.0263200
$266$	12.9111	0.791630
$267$	1.42185	0.0870159
$268$	−49.3369	−3.01373
$269$	−25.5698	−1.55902	−0.779509	−	0.626391i	$-0.784531\pi$
$269$	−25.5698	−1.55902	−0.779509	+	0.626391i	$0.784531\pi$
$270$	0.179317	0.0109129
$271$	−3.36085	−0.204157	−0.102078	−	0.994776i	$-0.532549\pi$
$271$	−3.36085	−0.204157	−0.102078	+	0.994776i	$0.532549\pi$
$272$	−32.7969	−1.98860
$273$	−2.09409	−0.126740
$274$	54.0530	3.26546
$275$	−11.3911	−0.686912
$276$	24.8271	1.49442
$277$	−8.61440	−0.517589	−0.258795	−	0.965932i	$-0.583325\pi$
$277$	−8.61440	−0.517589	−0.258795	+	0.965932i	$0.583325\pi$
$278$	−24.8720	−1.49173
$279$	−4.76697	−0.285391
$280$	−0.817025	−0.0488266
$281$	−20.4222	−1.21829	−0.609143	−	0.793061i	$-0.708486\pi$
$281$	−20.4222	−1.21829	−0.609143	+	0.793061i	$0.708486\pi$
$282$	−12.7083	−0.756769
$283$	19.6078	1.16556	0.582780	−	0.812630i	$-0.301965\pi$
$283$	19.6078	1.16556	0.582780	+	0.812630i	$0.301965\pi$
$284$	−50.3168	−2.98575
$285$	−0.197849	−0.0117196
$286$	−6.84220	−0.404588
$287$	15.1046	0.891597
$288$	−6.41328	−0.377906
$289$	1.58809	0.0934168
$290$	0.891455	0.0523480
$291$	3.20050	0.187616
$292$	21.9234	1.28297
$293$	−1.01467	−0.0592775	−0.0296388	−	0.999561i	$-0.509436\pi$
$293$	−1.01467	−0.0592775	−0.0296388	+	0.999561i	$0.509436\pi$
$294$	9.74875	0.568559
$295$	−0.270335	−0.0157395
$296$	44.8511	2.60692
$297$	2.28047	0.132326
$298$	−9.84822	−0.570492
$299$	6.39563	0.369869
$300$	−22.7309	−1.31237
$301$	−13.9895	−0.806340
$302$	−3.21183	−0.184820
$303$	7.38681	0.424361
$304$	21.4818	1.23207
$305$	0.672351	0.0384987
$306$	11.0347	0.630810
$307$	25.0394	1.42908	0.714538	−	0.699597i	$-0.246637\pi$
$307$	25.0394	1.42908	0.714538	+	0.699597i	$0.246637\pi$
$308$	−18.5380	−1.05630
$309$	3.41466	0.194253
$310$	−0.854796	−0.0485492
$311$	3.01371	0.170892	0.0854458	−	0.996343i	$-0.472769\pi$
$311$	3.01371	0.170892	0.0854458	+	0.996343i	$0.472769\pi$
$312$	−7.65280	−0.433255
$313$	−20.7651	−1.17371	−0.586857	−	0.809691i	$-0.699635\pi$
$313$	−20.7651	−1.17371	−0.586857	+	0.809691i	$0.699635\pi$
$314$	−22.6853	−1.28020
$315$	0.125154	0.00705162
$316$	−55.0451	−3.09653
$317$	1.90471	0.106979	0.0534896	−	0.998568i	$-0.482966\pi$
$317$	1.90471	0.106979	0.0534896	+	0.998568i	$0.482966\pi$
$318$	−15.6521	−0.877724
$319$	11.3371	0.634757
$320$	−0.0840900	−0.00470078
$321$	13.7416	0.766982
$322$	24.9437	1.39006
$323$	−12.1751	−0.677441
$324$	4.55064	0.252813
$325$	−5.85562	−0.324812
$326$	38.5532	2.13527
$327$	4.38863	0.242692
$328$	55.1994	3.04788
$329$	−8.86976	−0.489006
$330$	0.408926	0.0225106
$331$	12.1898	0.670009	0.335005	−	0.942216i	$-0.391262\pi$
$331$	12.1898	0.670009	0.335005	+	0.942216i	$0.391262\pi$
$332$	−75.0217	−4.11735
$333$	−6.87040	−0.376496
$334$	−18.2823	−1.00036
$335$	0.759589	0.0415008
$336$	−13.5888	−0.741330
$337$	−5.62067	−0.306177	−0.153089	−	0.988212i	$-0.548922\pi$
$337$	−5.62067	−0.306177	−0.153089	+	0.988212i	$0.548922\pi$
$338$	29.7552	1.61847
$339$	17.2985	0.939525
$340$	1.37457	0.0745468
$341$	−10.8709	−0.588693
$342$	−7.22765	−0.390827
$343$	19.3086	1.04256
$344$	−51.1242	−2.75643
$345$	−0.382237	−0.0205789
$346$	43.2273	2.32392
$347$	3.24544	0.174225	0.0871123	−	0.996198i	$-0.472236\pi$
$347$	3.24544	0.174225	0.0871123	+	0.996198i	$0.472236\pi$
$348$	22.6230	1.21272
$349$	−17.1918	−0.920254	−0.460127	−	0.887853i	$-0.652196\pi$
$349$	−17.1918	−0.920254	−0.460127	+	0.887853i	$0.652196\pi$
$350$	−22.8376	−1.22072
$351$	1.17228	0.0625714
$352$	−14.6253	−0.779531
$353$	8.52270	0.453618	0.226809	−	0.973939i	$-0.427171\pi$
$353$	8.52270	0.453618	0.226809	+	0.973939i	$0.427171\pi$
$354$	−9.87563	−0.524884
$355$	0.774675	0.0411155
$356$	6.47033	0.342927
$357$	7.70164	0.407614
$358$	−9.54364	−0.504397
$359$	−29.2749	−1.54507	−0.772535	−	0.634973i	$-0.781011\pi$
$359$	−29.2749	−1.54507	−0.772535	+	0.634973i	$0.781011\pi$
$360$	0.457372	0.0241056
$361$	−11.0254	−0.580282
$362$	29.7117	1.56161
$363$	−5.79946	−0.304393
$364$	−9.52945	−0.499479
$365$	−0.337532	−0.0176672
$366$	24.5617	1.28386
$367$	−18.1747	−0.948711	−0.474355	−	0.880333i	$-0.657319\pi$
$367$	−18.1747	−0.948711	−0.474355	+	0.880333i	$0.657319\pi$
$368$	41.5020	2.16344
$369$	−8.45559	−0.440180
$370$	−1.23198	−0.0640475
$371$	−10.9243	−0.567164
$372$	−21.6927	−1.12472
$373$	−7.11056	−0.368171	−0.184085	−	0.982910i	$-0.558932\pi$
$373$	−7.11056	−0.368171	−0.184085	+	0.982910i	$0.558932\pi$
$374$	25.1642	1.30121
$375$	0.700270	0.0361618
$376$	−32.4143	−1.67164
$377$	5.82785	0.300150
$378$	4.57201	0.235159
$379$	−17.7839	−0.913496	−0.456748	−	0.889596i	$-0.650986\pi$
$379$	−17.7839	−0.913496	−0.456748	+	0.889596i	$0.650986\pi$
$380$	−0.900340	−0.0461865
$381$	9.27127	0.474981
$382$	−3.66902	−0.187723
$383$	35.0643	1.79170	0.895851	−	0.444354i	$-0.146567\pi$
$383$	35.0643	1.79170	0.895851	+	0.444354i	$0.146567\pi$
$384$	9.75466	0.497791
$385$	0.285410	0.0145458
$386$	31.5051	1.60357
$387$	7.83134	0.398089
$388$	14.5643	0.739390
$389$	22.6487	1.14834	0.574168	−	0.818737i	$-0.305326\pi$
$389$	22.6487	1.14834	0.574168	+	0.818737i	$0.305326\pi$
$390$	0.210209	0.0106443
$391$	−23.5218	−1.18955
$392$	24.8655	1.25590
$393$	−5.89899	−0.297564
$394$	−18.7372	−0.943969
$395$	0.847471	0.0426409
$396$	10.3776	0.521493
$397$	3.62680	0.182024	0.0910119	−	0.995850i	$-0.470990\pi$
$397$	3.62680	0.182024	0.0910119	+	0.995850i	$0.470990\pi$
$398$	−6.33191	−0.317390
$399$	−5.04453	−0.252543
$400$	−37.9978	−1.89989
$401$	2.94594	0.147113	0.0735567	−	0.997291i	$-0.476565\pi$
$401$	2.94594	0.147113	0.0735567	+	0.997291i	$0.476565\pi$
$402$	27.7486	1.38398
$403$	−5.58820	−0.278368
$404$	33.6147	1.67239
$405$	−0.0700614	−0.00348138
$406$	22.7293	1.12804
$407$	−15.6677	−0.776621
$408$	28.1454	1.39341
$409$	21.7119	1.07359	0.536793	−	0.843714i	$-0.319636\pi$
$409$	21.7119	1.07359	0.536793	+	0.843714i	$0.319636\pi$
$410$	−1.51623	−0.0748811
$411$	−21.1192	−1.04173
$412$	15.5389	0.765545
$413$	−6.89269	−0.339167
$414$	−13.9635	−0.686270
$415$	1.15503	0.0566982
$416$	−7.51814	−0.368607
$417$	9.71783	0.475884
$418$	−16.4824	−0.806182
$419$	−22.1351	−1.08137	−0.540686	−	0.841224i	$-0.681835\pi$
$419$	−22.1351	−1.08137	−0.540686	+	0.841224i	$0.681835\pi$
$420$	0.569530	0.0277902
$421$	−30.1692	−1.47036	−0.735178	−	0.677874i	$-0.762901\pi$
$421$	−30.1692	−1.47036	−0.735178	+	0.677874i	$0.762901\pi$
$422$	51.5187	2.50789
$423$	4.96531	0.241422
$424$	−39.9228	−1.93882
$425$	21.5358	1.04464
$426$	28.2998	1.37113
$427$	17.1428	0.829601
$428$	62.5331	3.02265
$429$	2.67334	0.129070
$430$	1.40429	0.0677208
$431$	8.12719	0.391473	0.195737	−	0.980657i	$-0.437290\pi$
$431$	8.12719	0.391473	0.195737	+	0.980657i	$0.437290\pi$
$432$	7.60703	0.365993
$433$	−20.4105	−0.980864	−0.490432	−	0.871480i	$-0.663161\pi$
$433$	−20.4105	−0.980864	−0.490432	+	0.871480i	$0.663161\pi$
$434$	−21.7946	−1.04618
$435$	−0.348303	−0.0166999
$436$	19.9711	0.956440
$437$	15.4067	0.737001
$438$	−12.3304	−0.589170
$439$	−16.6715	−0.795685	−0.397842	−	0.917454i	$-0.630241\pi$
$439$	−16.6715	−0.795685	−0.397842	+	0.917454i	$0.630241\pi$
$440$	1.04302	0.0497241
$441$	−3.80897	−0.181379
$442$	12.9357	0.615287
$443$	2.60898	0.123956	0.0619782	−	0.998078i	$-0.480259\pi$
$443$	2.60898	0.123956	0.0619782	+	0.998078i	$0.480259\pi$
$444$	−31.2647	−1.48376
$445$	−0.0996169	−0.00472229
$446$	−12.9982	−0.615484
$447$	3.84783	0.181996
$448$	−2.14403	−0.101296
$449$	−4.87710	−0.230165	−0.115082	−	0.993356i	$-0.536713\pi$
$449$	−4.87710	−0.230165	−0.115082	+	0.993356i	$0.536713\pi$
$450$	12.7845	0.602669
$451$	−19.2827	−0.907987
$452$	78.7192	3.70264
$453$	1.25490	0.0589606
$454$	−28.9109	−1.35685
$455$	0.146715	0.00687810
$456$	−18.4351	−0.863304
$457$	23.0303	1.07731	0.538657	−	0.842525i	$-0.318932\pi$
$457$	23.0303	1.07731	0.538657	+	0.842525i	$0.318932\pi$
$458$	5.41125	0.252851
$459$	−4.31139	−0.201238
$460$	−1.73942	−0.0811009
$461$	−29.2895	−1.36415	−0.682075	−	0.731282i	$-0.738922\pi$
$461$	−29.2895	−1.36415	−0.682075	+	0.731282i	$0.738922\pi$
$462$	10.4263	0.485077
$463$	8.29054	0.385294	0.192647	−	0.981268i	$-0.438293\pi$
$463$	8.29054	0.385294	0.192647	+	0.981268i	$0.438293\pi$
$464$	37.8176	1.75564
$465$	0.333980	0.0154880
$466$	−47.3282	−2.19244
$467$	−6.96550	−0.322325	−0.161162	−	0.986928i	$-0.551524\pi$
$467$	−6.96550	−0.322325	−0.161162	+	0.986928i	$0.551524\pi$
$468$	5.33460	0.246592
$469$	19.3671	0.894292
$470$	0.890362	0.0410694
$471$	8.86344	0.408406
$472$	−25.1892	−1.15943
$473$	17.8591	0.821163
$474$	30.9591	1.42200
$475$	−14.1058	−0.647220
$476$	35.0474	1.60639
$477$	6.11547	0.280008
$478$	−21.7619	−0.995368
$479$	−26.2087	−1.19751	−0.598753	−	0.800934i	$-0.704337\pi$
$479$	−26.2087	−1.19751	−0.598753	+	0.800934i	$0.704337\pi$
$480$	0.449324	0.0205087
$481$	−8.05401	−0.367231
$482$	28.2912	1.28863
$483$	−9.74584	−0.443451
$484$	−26.3913	−1.19960
$485$	−0.224231	−0.0101818
$486$	−2.55942	−0.116098
$487$	1.20848	0.0547615	0.0273808	−	0.999625i	$-0.491283\pi$
$487$	1.20848	0.0547615	0.0273808	+	0.999625i	$0.491283\pi$
$488$	62.6481	2.83595
$489$	−15.0633	−0.681184
$490$	−0.683011	−0.0308553
$491$	17.9848	0.811641	0.405821	−	0.913953i	$-0.366986\pi$
$491$	17.9848	0.811641	0.405821	+	0.913953i	$0.366986\pi$
$492$	−38.4783	−1.73474
$493$	−21.4336	−0.965323
$494$	−8.47280	−0.381209
$495$	−0.159773	−0.00718125
$496$	−36.2625	−1.62823
$497$	19.7518	0.885989
$498$	42.1946	1.89078
$499$	−21.8471	−0.978012	−0.489006	−	0.872280i	$-0.662640\pi$
$499$	−21.8471	−0.978012	−0.489006	+	0.872280i	$0.662640\pi$
$500$	3.18668	0.142512
$501$	7.14312	0.319131
$502$	−34.5723	−1.54304
$503$	20.4675	0.912601	0.456300	−	0.889826i	$-0.349174\pi$
$503$	20.4675	0.912601	0.456300	+	0.889826i	$0.349174\pi$
$504$	11.6616	0.519447
$505$	−0.517530	−0.0230298
$506$	−31.8434	−1.41561
$507$	−11.6258	−0.516319
$508$	42.1902	1.87189
$509$	−31.9407	−1.41575	−0.707873	−	0.706340i	$-0.750345\pi$
$509$	−31.9407	−1.41575	−0.707873	+	0.706340i	$0.750345\pi$
$510$	−0.773104	−0.0342336
$511$	−8.60601	−0.380707
$512$	50.5338	2.23330
$513$	2.82394	0.124680
$514$	76.5750	3.37758
$515$	−0.239236	−0.0105420
$516$	35.6376	1.56886
$517$	11.3232	0.497995
$518$	−31.4116	−1.38015
$519$	−16.8895	−0.741366
$520$	0.536166	0.0235124
$521$	−12.8660	−0.563670	−0.281835	−	0.959463i	$-0.590943\pi$
$521$	−12.8660	−0.563670	−0.281835	+	0.959463i	$0.590943\pi$
$522$	−12.7239	−0.556910
$523$	36.0832	1.57781	0.788905	−	0.614515i	$-0.210648\pi$
$523$	36.0832	1.57781	0.788905	+	0.614515i	$0.210648\pi$
$524$	−26.8442	−1.17269
$525$	8.92296	0.389430
$526$	30.6152	1.33489
$527$	20.5523	0.895270
$528$	17.3476	0.754957
$529$	6.76509	0.294135
$530$	1.09661	0.0476335
$531$	3.85854	0.167446
$532$	−22.9559	−0.995262
$533$	−9.91228	−0.429349
$534$	−3.63912	−0.157480
$535$	−0.962756	−0.0416236
$536$	70.7767	3.05709
$537$	3.72883	0.160911
$538$	65.4439	2.82149
$539$	−8.68623	−0.374142
$540$	−0.318824	−0.0137200
$541$	10.6792	0.459133	0.229567	−	0.973293i	$-0.426269\pi$
$541$	10.6792	0.459133	0.229567	+	0.973293i	$0.426269\pi$
$542$	8.60182	0.369480
$543$	−11.6088	−0.498180
$544$	27.6502	1.18549
$545$	−0.307473	−0.0131707
$546$	5.35966	0.229372
$547$	10.1169	0.432567	0.216284	−	0.976331i	$-0.430606\pi$
$547$	10.1169	0.432567	0.216284	+	0.976331i	$0.430606\pi$
$548$	−96.1059	−4.10544
$549$	−9.59660	−0.409573
$550$	29.1547	1.24316
$551$	14.0389	0.598079
$552$	−35.6159	−1.51591
$553$	21.6079	0.918860
$554$	22.0479	0.936725
$555$	0.481350	0.0204322
$556$	44.2223	1.87544
$557$	−25.0966	−1.06338	−0.531689	−	0.846940i	$-0.678443\pi$
$557$	−25.0966	−1.06338	−0.531689	+	0.846940i	$0.678443\pi$
$558$	12.2007	0.516496
$559$	9.18048	0.388293
$560$	0.952050	0.0402315
$561$	−9.83199	−0.415107
$562$	52.2690	2.20483
$563$	40.5515	1.70904	0.854521	−	0.519417i	$-0.173851\pi$
$563$	40.5515	1.70904	0.854521	+	0.519417i	$0.173851\pi$
$564$	22.5953	0.951435
$565$	−1.21196	−0.0509874
$566$	−50.1845	−2.10941
$567$	−1.78635	−0.0750195
$568$	72.1824	3.02871
$569$	34.1484	1.43158	0.715789	−	0.698317i	$-0.246067\pi$
$569$	34.1484	1.43158	0.715789	+	0.698317i	$0.246067\pi$
$570$	0.506379	0.0212099
$571$	−30.7120	−1.28526	−0.642628	−	0.766178i	$-0.722156\pi$
$571$	−30.7120	−1.28526	−0.642628	+	0.766178i	$0.722156\pi$
$572$	12.1654	0.508661
$573$	1.43353	0.0598867
$574$	−38.6591	−1.61360
$575$	−27.2519	−1.13648
$576$	1.20023	0.0500097
$577$	5.27316	0.219525	0.109762	−	0.993958i	$-0.464991\pi$
$577$	5.27316	0.219525	0.109762	+	0.993958i	$0.464991\pi$
$578$	−4.06458	−0.169064
$579$	−12.3095	−0.511564
$580$	−1.58500	−0.0658136
$581$	29.4497	1.22178
$582$	−8.19142	−0.339545
$583$	13.9461	0.577590
$584$	−31.4504	−1.30143
$585$	−0.0821313	−0.00339571
$586$	2.59696	0.107280
$587$	22.9968	0.949178	0.474589	−	0.880208i	$-0.342597\pi$
$587$	22.9968	0.949178	0.474589	+	0.880208i	$0.342597\pi$
$588$	−17.3332	−0.714810
$589$	−13.4616	−0.554677
$590$	0.691901	0.0284851
$591$	7.32089	0.301141
$592$	−52.2634	−2.14801
$593$	−24.8102	−1.01883	−0.509416	−	0.860520i	$-0.670138\pi$
$593$	−24.8102	−1.01883	−0.509416	+	0.860520i	$0.670138\pi$
$594$	−5.83668	−0.239482
$595$	−0.539587	−0.0221209
$596$	17.5101	0.717241
$597$	2.47396	0.101253
$598$	−16.3691	−0.669383
$599$	36.4740	1.49029	0.745144	−	0.666903i	$-0.232381\pi$
$599$	36.4740	1.49029	0.745144	+	0.666903i	$0.232381\pi$
$600$	32.6087	1.33125
$601$	25.2570	1.03026	0.515128	−	0.857113i	$-0.327744\pi$
$601$	25.2570	1.03026	0.515128	+	0.857113i	$0.327744\pi$
$602$	35.8050	1.45930
$603$	−10.8418	−0.441511
$604$	5.71062	0.232362
$605$	0.406318	0.0165192
$606$	−18.9060	−0.768002
$607$	−22.7760	−0.924448	−0.462224	−	0.886763i	$-0.652948\pi$
$607$	−22.7760	−0.924448	−0.462224	+	0.886763i	$0.652948\pi$
$608$	−18.1107	−0.734487
$609$	−8.88064	−0.359862
$610$	−1.72083	−0.0696743
$611$	5.82071	0.235481
$612$	−19.6196	−0.793074
$613$	14.3314	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$613$	14.3314	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$614$	−64.0864	−2.58632
$615$	0.592410	0.0238883
$616$	26.5938	1.07149
$617$	0.457923	0.0184353	0.00921764	−	0.999958i	$-0.497066\pi$
$617$	0.457923	0.0184353	0.00921764	+	0.999958i	$0.497066\pi$
$618$	−8.73955	−0.351556
$619$	46.8675	1.88376	0.941882	−	0.335944i	$-0.109055\pi$
$619$	46.8675	1.88376	0.941882	+	0.335944i	$0.109055\pi$
$620$	1.51982	0.0610376
$621$	5.45574	0.218931
$622$	−7.71334	−0.309277
$623$	−2.53992	−0.101760
$624$	8.91754	0.356987
$625$	24.9264	0.997056
$626$	53.1467	2.12417
$627$	6.43990	0.257185
$628$	40.3343	1.60951
$629$	29.6210	1.18107
$630$	−0.320322	−0.0127619
$631$	6.98082	0.277902	0.138951	−	0.990299i	$-0.455627\pi$
$631$	6.98082	0.277902	0.138951	+	0.990299i	$0.455627\pi$
$632$	78.9654	3.14107
$633$	−20.1290	−0.800058
$634$	−4.87495	−0.193609
$635$	−0.649558	−0.0257769
$636$	27.8293	1.10350
$637$	−4.46516	−0.176916
$638$	−29.0165	−1.14877
$639$	−11.0571	−0.437412
$640$	−0.683425	−0.0270148
$641$	−15.4679	−0.610945	−0.305472	−	0.952201i	$-0.598814\pi$
$641$	−15.4679	−0.610945	−0.305472	+	0.952201i	$0.598814\pi$
$642$	−35.1706	−1.38807
$643$	−25.3342	−0.999085	−0.499542	−	0.866289i	$-0.666499\pi$
$643$	−25.3342	−0.999085	−0.499542	+	0.866289i	$0.666499\pi$
$644$	−44.3498	−1.74763
$645$	−0.548674	−0.0216040
$646$	31.1612	1.22602
$647$	−34.3483	−1.35037	−0.675185	−	0.737648i	$-0.735936\pi$
$647$	−34.3483	−1.35037	−0.675185	+	0.737648i	$0.735936\pi$
$648$	−6.52816	−0.256450
$649$	8.79928	0.345402
$650$	14.9870	0.587839
$651$	8.51545	0.333747
$652$	−68.5475	−2.68453
$653$	31.3243	1.22582	0.612908	−	0.790155i	$-0.290000\pi$
$653$	31.3243	1.22582	0.612908	+	0.790155i	$0.290000\pi$
$654$	−11.2324	−0.439220
$655$	0.413291	0.0161486
$656$	−64.3219	−2.51135
$657$	4.81766	0.187955
$658$	22.7015	0.884995
$659$	−4.17117	−0.162486	−0.0812429	−	0.996694i	$-0.525889\pi$
$659$	−4.17117	−0.162486	−0.0812429	+	0.996694i	$0.525889\pi$
$660$	−0.727068	−0.0283011
$661$	−42.6927	−1.66055	−0.830276	−	0.557353i	$-0.811817\pi$
$661$	−42.6927	−1.66055	−0.830276	+	0.557353i	$0.811817\pi$
$662$	−31.1987	−1.21257
$663$	−5.05414	−0.196286
$664$	107.623	4.17658
$665$	0.353427	0.0137053
$666$	17.5843	0.681376
$667$	27.1227	1.05019
$668$	32.5058	1.25769
$669$	5.07858	0.196349
$670$	−1.94411	−0.0751075
$671$	−21.8847	−0.844851
$672$	11.4563	0.441938
$673$	−24.2899	−0.936307	−0.468153	−	0.883647i	$-0.655081\pi$
$673$	−24.2899	−0.936307	−0.468153	+	0.883647i	$0.655081\pi$
$674$	14.3857	0.554115
$675$	−4.99509	−0.192261
$676$	−52.9047	−2.03480
$677$	15.7842	0.606634	0.303317	−	0.952890i	$-0.401906\pi$
$677$	15.7842	0.606634	0.303317	+	0.952890i	$0.401906\pi$
$678$	−44.2741	−1.70034
$679$	−5.71719	−0.219406
$680$	−1.97191	−0.0756192
$681$	11.2959	0.432858
$682$	27.8233	1.06541
$683$	−27.1634	−1.03938	−0.519690	−	0.854355i	$-0.673953\pi$
$683$	−27.1634	−1.03938	−0.519690	+	0.854355i	$0.673953\pi$
$684$	12.8507	0.491360
$685$	1.47964	0.0565342
$686$	−49.4187	−1.88682
$687$	−2.11425	−0.0806636
$688$	59.5732	2.27121
$689$	7.16902	0.273118
$690$	0.978305	0.0372434
$691$	5.90227	0.224533	0.112267	−	0.993678i	$-0.464189\pi$
$691$	5.90227	0.224533	0.112267	+	0.993678i	$0.464189\pi$
$692$	−76.8580	−2.92170
$693$	−4.07371	−0.154747
$694$	−8.30646	−0.315309
$695$	−0.680845	−0.0258259
$696$	−32.4541	−1.23017
$697$	36.4553	1.38084
$698$	44.0010	1.66546
$699$	18.4918	0.699423
$700$	40.6052	1.53473
$701$	−49.3586	−1.86425	−0.932123	−	0.362141i	$-0.882046\pi$
$701$	−49.3586	−1.86425	−0.932123	+	0.362141i	$0.882046\pi$
$702$	−3.00035	−0.113241
$703$	−19.4016	−0.731745
$704$	2.73710	0.103158
$705$	−0.347876	−0.0131018
$706$	−21.8132	−0.820950
$707$	−13.1954	−0.496264
$708$	17.5588	0.659901
$709$	21.6217	0.812021	0.406010	−	0.913869i	$-0.366920\pi$
$709$	21.6217	0.812021	0.406010	+	0.913869i	$0.366920\pi$
$710$	−1.98272	−0.0744102
$711$	−12.0961	−0.453640
$712$	−9.28207	−0.347860
$713$	−26.0073	−0.973982
$714$	−19.7117	−0.737693
$715$	−0.187298	−0.00700454
$716$	16.9685	0.634144
$717$	8.50268	0.317538
$718$	74.9268	2.79624
$719$	34.3374	1.28057	0.640285	−	0.768138i	$-0.278816\pi$
$719$	34.3374	1.28057	0.640285	+	0.768138i	$0.278816\pi$
$720$	−0.532959	−0.0198622
$721$	−6.09976	−0.227167
$722$	28.2186	1.05019
$723$	−11.0537	−0.411093
$724$	−52.8273	−1.96331
$725$	−24.8326	−0.922260
$726$	14.8433	0.550885
$727$	−19.0236	−0.705547	−0.352774	−	0.935709i	$-0.614761\pi$
$727$	−19.0236	−0.705547	−0.352774	+	0.935709i	$0.614761\pi$
$728$	13.6706	0.506664
$729$	1.00000	0.0370370
$730$	0.863886	0.0319739
$731$	−33.7639	−1.24880
$732$	−43.6706	−1.61411
$733$	−45.2697	−1.67207	−0.836037	−	0.548673i	$-0.815133\pi$
$733$	−45.2697	−1.67207	−0.836037	+	0.548673i	$0.815133\pi$
$734$	46.5167	1.71696
$735$	0.266861	0.00984333
$736$	−34.9892	−1.28972
$737$	−24.7243	−0.910731
$738$	21.6414	0.796632
$739$	−48.8351	−1.79643	−0.898214	−	0.439559i	$-0.855135\pi$
$739$	−48.8351	−1.79643	−0.898214	+	0.439559i	$0.855135\pi$
$740$	2.19045	0.0805225
$741$	3.31044	0.121612
$742$	27.9600	1.02644
$743$	−44.9429	−1.64879	−0.824397	−	0.566012i	$-0.808486\pi$
$743$	−44.9429	−1.64879	−0.824397	+	0.566012i	$0.808486\pi$
$744$	31.1195	1.14090
$745$	−0.269584	−0.00987681
$746$	18.1989	0.666309
$747$	−16.4860	−0.603190
$748$	−44.7418	−1.63592
$749$	−24.5473	−0.896938
$750$	−1.79229	−0.0654450
$751$	44.1717	1.61185	0.805925	−	0.592018i	$-0.201669\pi$
$751$	44.1717	1.61185	0.805925	+	0.592018i	$0.201669\pi$
$752$	37.7713	1.37738
$753$	13.5078	0.492253
$754$	−14.9159	−0.543206
$755$	−0.0879204	−0.00319975
$756$	−8.12902	−0.295649
$757$	−24.0293	−0.873359	−0.436679	−	0.899617i	$-0.643846\pi$
$757$	−24.0293	−0.873359	−0.436679	+	0.899617i	$0.643846\pi$
$758$	45.5164	1.65323
$759$	12.4416	0.451603
$760$	1.29159	0.0468509
$761$	10.4809	0.379933	0.189966	−	0.981791i	$-0.439162\pi$
$761$	10.4809	0.379933	0.189966	+	0.981791i	$0.439162\pi$
$762$	−23.7291	−0.859614
$763$	−7.83961	−0.283813
$764$	6.52350	0.236012
$765$	0.302062	0.0109211
$766$	−89.7444	−3.24260
$767$	4.52327	0.163326
$768$	−27.3668	−0.987513
$769$	20.0296	0.722284	0.361142	−	0.932511i	$-0.382387\pi$
$769$	20.0296	0.722284	0.361142	+	0.932511i	$0.382387\pi$
$770$	−0.730483	−0.0263248
$771$	−29.9189	−1.07750
$772$	−56.0160	−2.01606
$773$	−30.6187	−1.10128	−0.550638	−	0.834744i	$-0.685616\pi$
$773$	−30.6187	−1.10128	−0.550638	+	0.834744i	$0.685616\pi$
$774$	−20.0437	−0.720456
$775$	23.8114	0.855332
$776$	−20.8933	−0.750027
$777$	12.2729	0.440289
$778$	−57.9676	−2.07824
$779$	−23.8781	−0.855521
$780$	−0.373750	−0.0133824
$781$	−25.2153	−0.902276
$782$	60.2023	2.15283
$783$	4.97140	0.177663
$784$	−28.9749	−1.03482
$785$	−0.620985	−0.0221639
$786$	15.0980	0.538527
$787$	−19.8653	−0.708123	−0.354061	−	0.935222i	$-0.615200\pi$
$787$	−19.8653	−0.708123	−0.354061	+	0.935222i	$0.615200\pi$
$788$	33.3147	1.18679
$789$	−11.9618	−0.425850
$790$	−2.16904	−0.0771708
$791$	−30.9011	−1.09872
$792$	−14.8873	−0.528996
$793$	−11.2499	−0.399494
$794$	−9.28250	−0.329424
$795$	−0.428458	−0.0151958
$796$	11.2581	0.399033
$797$	−1.03439	−0.0366401	−0.0183200	−	0.999832i	$-0.505832\pi$
$797$	−1.03439	−0.0366401	−0.0183200	+	0.999832i	$0.505832\pi$
$798$	12.9111	0.457048
$799$	−21.4074	−0.757339
$800$	32.0349	1.13261
$801$	1.42185	0.0502386
$802$	−7.53991	−0.266244
$803$	10.9865	0.387706
$804$	−49.3369	−1.73998
$805$	0.682807	0.0240658
$806$	14.3026	0.503786
$807$	−25.5698	−0.900100
$808$	−48.2222	−1.69645
$809$	55.1011	1.93725	0.968625	−	0.248525i	$-0.0799459\pi$
$809$	55.1011	1.93725	0.968625	+	0.248525i	$0.0799459\pi$
$810$	0.179317	0.00630054
$811$	0.735101	0.0258129	0.0129064	−	0.999917i	$-0.495892\pi$
$811$	0.735101	0.0258129	0.0129064	+	0.999917i	$0.495892\pi$
$812$	−40.4126	−1.41820
$813$	−3.36085	−0.117870
$814$	40.1003	1.40552
$815$	1.05535	0.0369674
$816$	−32.7969	−1.14812
$817$	22.1152	0.773714
$818$	−55.5700	−1.94296
$819$	−2.09409	−0.0731734
$820$	2.69585	0.0941430
$821$	39.6378	1.38337	0.691685	−	0.722199i	$-0.256869\pi$
$821$	39.6378	1.38337	0.691685	+	0.722199i	$0.256869\pi$
$822$	54.0530	1.88531
$823$	−54.6014	−1.90329	−0.951643	−	0.307207i	$-0.900606\pi$
$823$	−54.6014	−1.90329	−0.951643	+	0.307207i	$0.900606\pi$
$824$	−22.2914	−0.776558
$825$	−11.3911	−0.396589
$826$	17.6413	0.613819
$827$	−45.3727	−1.57776	−0.788882	−	0.614545i	$-0.789340\pi$
$827$	−45.3727	−1.57776	−0.788882	+	0.614545i	$0.789340\pi$
$828$	24.8271	0.862801
$829$	−32.6447	−1.13380	−0.566899	−	0.823788i	$-0.691857\pi$
$829$	−32.6447	−1.13380	−0.566899	+	0.823788i	$0.691857\pi$
$830$	−2.95621	−0.102612
$831$	−8.61440	−0.298830
$832$	1.40700	0.0487791
$833$	16.4219	0.568986
$834$	−24.8720	−0.861248
$835$	−0.500457	−0.0173190
$836$	29.3057	1.01356
$837$	−4.76697	−0.164770
$838$	56.6532	1.95705
$839$	−15.3862	−0.531189	−0.265595	−	0.964085i	$-0.585568\pi$
$839$	−15.3862	−0.531189	−0.265595	+	0.964085i	$0.585568\pi$
$840$	−0.817025	−0.0281900
$841$	−4.28518	−0.147765
$842$	77.2156	2.66103
$843$	−20.4222	−0.703377
$844$	−91.6000	−3.15300
$845$	0.814518	0.0280203
$846$	−12.7083	−0.436921
$847$	10.3599	0.355969
$848$	46.5206	1.59752
$849$	19.6078	0.672936
$850$	−55.1192	−1.89057
$851$	−37.4831	−1.28491
$852$	−50.3168	−1.72383
$853$	25.7044	0.880102	0.440051	−	0.897973i	$-0.354960\pi$
$853$	25.7044	0.880102	0.440051	+	0.897973i	$0.354960\pi$
$854$	−43.8758	−1.50140
$855$	−0.197849	−0.00676630
$856$	−89.7074	−3.06614
$857$	1.55740	0.0531998	0.0265999	−	0.999646i	$-0.491532\pi$
$857$	1.55740	0.0531998	0.0265999	+	0.999646i	$0.491532\pi$
$858$	−6.84220	−0.233589
$859$	−3.62390	−0.123646	−0.0618230	−	0.998087i	$-0.519691\pi$
$859$	−3.62390	−0.123646	−0.0618230	+	0.998087i	$0.519691\pi$
$860$	−2.49682	−0.0851408
$861$	15.1046	0.514764
$862$	−20.8009	−0.708482
$863$	−21.6917	−0.738393	−0.369197	−	0.929351i	$-0.620367\pi$
$863$	−21.6917	−0.738393	−0.369197	+	0.929351i	$0.620367\pi$
$864$	−6.41328	−0.218184
$865$	1.18330	0.0402335
$866$	52.2390	1.77515
$867$	1.58809	0.0539342
$868$	38.7507	1.31529
$869$	−27.5848	−0.935751
$870$	0.891455	0.0302231
$871$	−12.7095	−0.430646
$872$	−28.6497	−0.970200
$873$	3.20050	0.108320
$874$	−39.4322	−1.33381
$875$	−1.25092	−0.0422890
$876$	21.9234	0.740724
$877$	15.8725	0.535976	0.267988	−	0.963422i	$-0.413641\pi$
$877$	15.8725	0.535976	0.267988	+	0.963422i	$0.413641\pi$
$878$	42.6693	1.44002
$879$	−1.01467	−0.0342239
$880$	−1.21540	−0.0409710
$881$	−42.1628	−1.42050	−0.710251	−	0.703948i	$-0.751419\pi$
$881$	−42.1628	−1.42050	−0.710251	+	0.703948i	$0.751419\pi$
$882$	9.74875	0.328257
$883$	−4.97366	−0.167377	−0.0836885	−	0.996492i	$-0.526670\pi$
$883$	−4.97366	−0.167377	−0.0836885	+	0.996492i	$0.526670\pi$
$884$	−22.9996	−0.773559
$885$	−0.270335	−0.00908720
$886$	−6.67748	−0.224334
$887$	−38.6854	−1.29893	−0.649464	−	0.760392i	$-0.725007\pi$
$887$	−38.6854	−1.29893	−0.649464	+	0.760392i	$0.725007\pi$
$888$	44.8511	1.50510
$889$	−16.5617	−0.555461
$890$	0.254962	0.00854633
$891$	2.28047	0.0763986
$892$	23.1108	0.773807
$893$	14.0217	0.469219
$894$	−9.84822	−0.329374
$895$	−0.261247	−0.00873252
$896$	−17.4252	−0.582135
$897$	6.39563	0.213544
$898$	12.4826	0.416548
$899$	−23.6985	−0.790389
$900$	−22.7309	−0.757695
$901$	−26.3662	−0.878385
$902$	49.3526	1.64326
$903$	−13.9895	−0.465541
$904$	−112.927	−3.75591
$905$	0.813327	0.0270359
$906$	−3.21183	−0.106706
$907$	−22.8374	−0.758303	−0.379151	−	0.925335i	$-0.623784\pi$
$907$	−22.8374	−0.758303	−0.379151	+	0.925335i	$0.623784\pi$
$908$	51.4034	1.70588
$909$	7.38681	0.245005
$910$	−0.375505	−0.0124479
$911$	22.3211	0.739532	0.369766	−	0.929125i	$-0.379438\pi$
$911$	22.3211	0.739532	0.369766	+	0.929125i	$0.379438\pi$
$912$	21.4818	0.711333
$913$	−37.5957	−1.24424
$914$	−58.9443	−1.94971
$915$	0.672351	0.0222272
$916$	−9.62118	−0.317893
$917$	10.5376	0.347983
$918$	11.0347	0.364198
$919$	5.90212	0.194693	0.0973465	−	0.995251i	$-0.468964\pi$
$919$	5.90212	0.194693	0.0973465	+	0.995251i	$0.468964\pi$
$920$	2.49530	0.0822677
$921$	25.0394	0.825077
$922$	74.9643	2.46882
$923$	−12.9620	−0.426648
$924$	−18.5380	−0.609854
$925$	34.3183	1.12838
$926$	−21.2190	−0.697299
$927$	3.41466	0.112152
$928$	−31.8830	−1.04661
$929$	−4.92134	−0.161464	−0.0807319	−	0.996736i	$-0.525726\pi$
$929$	−4.92134	−0.161464	−0.0807319	+	0.996736i	$0.525726\pi$
$930$	−0.854796	−0.0280299
$931$	−10.7563	−0.352523
$932$	84.1494	2.75640
$933$	3.01371	0.0986643
$934$	17.8276	0.583339
$935$	0.688843	0.0225276
$936$	−7.65280	−0.250140
$937$	49.6571	1.62223	0.811113	−	0.584889i	$-0.198862\pi$
$937$	49.6571	1.62223	0.811113	+	0.584889i	$0.198862\pi$
$938$	−49.5687	−1.61848
$939$	−20.7651	−0.677644
$940$	−1.58306	−0.0516337
$941$	29.3465	0.956669	0.478335	−	0.878178i	$-0.341241\pi$
$941$	29.3465	0.956669	0.478335	+	0.878178i	$0.341241\pi$
$942$	−22.6853	−0.739126
$943$	−46.1315	−1.50225
$944$	29.3520	0.955328
$945$	0.125154	0.00407126
$946$	−45.7090	−1.48613
$947$	−50.5318	−1.64206	−0.821031	−	0.570884i	$-0.806601\pi$
$947$	−50.5318	−1.64206	−0.821031	+	0.570884i	$0.806601\pi$
$948$	−55.0451	−1.78778
$949$	5.64762	0.183330
$950$	36.1028	1.17133
$951$	1.90471	0.0617644
$952$	−50.2775	−1.62950
$953$	−35.7963	−1.15955	−0.579777	−	0.814775i	$-0.696860\pi$
$953$	−35.7963	−1.15955	−0.579777	+	0.814775i	$0.696860\pi$
$954$	−15.6521	−0.506754
$955$	−0.100435	−0.00325001
$956$	38.6926	1.25141
$957$	11.3371	0.366477
$958$	67.0791	2.16723
$959$	37.7262	1.21824
$960$	−0.0840900	−0.00271399
$961$	−8.27603	−0.266969
$962$	20.6136	0.664609
$963$	13.7416	0.442817
$964$	−50.3016	−1.62011
$965$	0.862419	0.0277623
$966$	24.9437	0.802551
$967$	41.2688	1.32712	0.663558	−	0.748125i	$-0.269046\pi$
$967$	41.2688	1.32712	0.663558	+	0.748125i	$0.269046\pi$
$968$	37.8598	1.21686
$969$	−12.1751	−0.391121
$970$	0.573902	0.0184269
$971$	−37.1772	−1.19307	−0.596537	−	0.802585i	$-0.703457\pi$
$971$	−37.1772	−1.19307	−0.596537	+	0.802585i	$0.703457\pi$
$972$	4.55064	0.145962
$973$	−17.3594	−0.556517
$974$	−3.09301	−0.0991065
$975$	−5.85562	−0.187530
$976$	−73.0016	−2.33672
$977$	−24.2860	−0.776977	−0.388489	−	0.921454i	$-0.627003\pi$
$977$	−24.2860	−0.776977	−0.388489	+	0.921454i	$0.627003\pi$
$978$	38.5532	1.23280
$979$	3.24249	0.103630
$980$	1.21439	0.0387923
$981$	4.38863	0.140118
$982$	−46.0306	−1.46890
$983$	−11.3720	−0.362709	−0.181355	−	0.983418i	$-0.558048\pi$
$983$	−11.3720	−0.362709	−0.181355	+	0.983418i	$0.558048\pi$
$984$	55.1994	1.75969
$985$	−0.512912	−0.0163427
$986$	54.8577	1.74703
$987$	−8.86976	−0.282328
$988$	15.0646	0.479269
$989$	42.7257	1.35860
$990$	0.408926	0.0129965
$991$	−18.9959	−0.603424	−0.301712	−	0.953399i	$-0.597558\pi$
$991$	−18.9959	−0.603424	−0.301712	+	0.953399i	$0.597558\pi$
$992$	30.5719	0.970659
$993$	12.1898	0.386830
$994$	−50.5532	−1.60345
$995$	−0.173329	−0.00549490
$996$	−75.0217	−2.37715
$997$	−53.7230	−1.70143	−0.850713	−	0.525631i	$-0.823829\pi$
$997$	−53.7230	−1.70143	−0.850713	+	0.525631i	$0.823829\pi$
$998$	55.9160	1.76999
$999$	−6.87040	−0.217370

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.a.1.4	✓	95

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.a.1.4	✓	95	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55942 q^{2} +1.00000 q^{3} +4.55064 q^{4} -0.0700614 q^{5} -2.55942 q^{6} -1.78635 q^{7} -6.52816 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.55942 q^{2} +1.00000 q^{3} +4.55064 q^{4} -0.0700614 q^{5} -2.55942 q^{6} -1.78635 q^{7} -6.52816 q^{8} +1.00000 q^{9} +0.179317 q^{10} +2.28047 q^{11} +4.55064 q^{12} +1.17228 q^{13} +4.57201 q^{14} -0.0700614 q^{15} +7.60703 q^{16} -4.31139 q^{17} -2.55942 q^{18} +2.82394 q^{19} -0.318824 q^{20} -1.78635 q^{21} -5.83668 q^{22} +5.45574 q^{23} -6.52816 q^{24} -4.99509 q^{25} -3.00035 q^{26} +1.00000 q^{27} -8.12902 q^{28} +4.97140 q^{29} +0.179317 q^{30} -4.76697 q^{31} -6.41328 q^{32} +2.28047 q^{33} +11.0347 q^{34} +0.125154 q^{35} +4.55064 q^{36} -6.87040 q^{37} -7.22765 q^{38} +1.17228 q^{39} +0.457372 q^{40} -8.45559 q^{41} +4.57201 q^{42} +7.83134 q^{43} +10.3776 q^{44} -0.0700614 q^{45} -13.9635 q^{46} +4.96531 q^{47} +7.60703 q^{48} -3.80897 q^{49} +12.7845 q^{50} -4.31139 q^{51} +5.33460 q^{52} +6.11547 q^{53} -2.55942 q^{54} -0.159773 q^{55} +11.6616 q^{56} +2.82394 q^{57} -12.7239 q^{58} +3.85854 q^{59} -0.318824 q^{60} -9.59660 q^{61} +12.2007 q^{62} -1.78635 q^{63} +1.20023 q^{64} -0.0821313 q^{65} -5.83668 q^{66} -10.8418 q^{67} -19.6196 q^{68} +5.45574 q^{69} -0.320322 q^{70} -11.0571 q^{71} -6.52816 q^{72} +4.81766 q^{73} +17.5843 q^{74} -4.99509 q^{75} +12.8507 q^{76} -4.07371 q^{77} -3.00035 q^{78} -12.0961 q^{79} -0.532959 q^{80} +1.00000 q^{81} +21.6414 q^{82} -16.4860 q^{83} -8.12902 q^{84} +0.302062 q^{85} -20.0437 q^{86} +4.97140 q^{87} -14.8873 q^{88} +1.42185 q^{89} +0.179317 q^{90} -2.09409 q^{91} +24.8271 q^{92} -4.76697 q^{93} -12.7083 q^{94} -0.197849 q^{95} -6.41328 q^{96} +3.20050 q^{97} +9.74875 q^{98} +2.28047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 95 q - 9 q^{2} + 95 q^{3} + 65 q^{4} - 15 q^{5} - 9 q^{6} - 36 q^{7} - 27 q^{8} + 95 q^{9}+O(q^{10}) \) 95 * q - 9 * q^2 + 95 * q^3 + 65 * q^4 - 15 * q^5 - 9 * q^6 - 36 * q^7 - 27 * q^8 + 95 * q^9	\( 95 q - 9 q^{2} + 95 q^{3} + 65 q^{4} - 15 q^{5} - 9 q^{6} - 36 q^{7} - 27 q^{8} + 95 q^{9} - 36 q^{10} - 48 q^{11} + 65 q^{12} - 73 q^{13} - 17 q^{14} - 15 q^{15} + 13 q^{16} - 9 q^{17} - 9 q^{18} - 66 q^{19} - 35 q^{20} - 36 q^{21} - 37 q^{22} - 58 q^{23} - 27 q^{24} + 24 q^{25} - 25 q^{26} + 95 q^{27} - 75 q^{28} - 31 q^{29} - 36 q^{30} - 129 q^{31} - 53 q^{32} - 48 q^{33} - 61 q^{34} - 38 q^{35} + 65 q^{36} - 127 q^{37} + q^{38} - 73 q^{39} - 74 q^{40} - 31 q^{41} - 17 q^{42} - 62 q^{43} - 76 q^{44} - 15 q^{45} - 60 q^{46} - 75 q^{47} + 13 q^{48} + 5 q^{49} - 30 q^{50} - 9 q^{51} - 137 q^{52} - 28 q^{53} - 9 q^{54} - 117 q^{55} - 23 q^{56} - 66 q^{57} - 90 q^{58} - 60 q^{59} - 35 q^{60} - 96 q^{61} + 10 q^{62} - 36 q^{63} - 75 q^{64} - 28 q^{65} - 37 q^{66} - 116 q^{67} + 3 q^{68} - 58 q^{69} - 73 q^{70} - 144 q^{71} - 27 q^{72} - 121 q^{73} - 16 q^{74} + 24 q^{75} - 118 q^{76} - 3 q^{77} - 25 q^{78} - 135 q^{79} - 36 q^{80} + 95 q^{81} - 102 q^{82} - 21 q^{83} - 75 q^{84} - 129 q^{85} - 46 q^{86} - 31 q^{87} - 77 q^{88} - 63 q^{89} - 36 q^{90} - 123 q^{91} - 42 q^{92} - 129 q^{93} - 44 q^{94} - 80 q^{95} - 53 q^{96} - 144 q^{97} + 10 q^{98} - 48 q^{99}+O(q^{100}) \) 95 * q - 9 * q^2 + 95 * q^3 + 65 * q^4 - 15 * q^5 - 9 * q^6 - 36 * q^7 - 27 * q^8 + 95 * q^9 - 36 * q^10 - 48 * q^11 + 65 * q^12 - 73 * q^13 - 17 * q^14 - 15 * q^15 + 13 * q^16 - 9 * q^17 - 9 * q^18 - 66 * q^19 - 35 * q^20 - 36 * q^21 - 37 * q^22 - 58 * q^23 - 27 * q^24 + 24 * q^25 - 25 * q^26 + 95 * q^27 - 75 * q^28 - 31 * q^29 - 36 * q^30 - 129 * q^31 - 53 * q^32 - 48 * q^33 - 61 * q^34 - 38 * q^35 + 65 * q^36 - 127 * q^37 + q^38 - 73 * q^39 - 74 * q^40 - 31 * q^41 - 17 * q^42 - 62 * q^43 - 76 * q^44 - 15 * q^45 - 60 * q^46 - 75 * q^47 + 13 * q^48 + 5 * q^49 - 30 * q^50 - 9 * q^51 - 137 * q^52 - 28 * q^53 - 9 * q^54 - 117 * q^55 - 23 * q^56 - 66 * q^57 - 90 * q^58 - 60 * q^59 - 35 * q^60 - 96 * q^61 + 10 * q^62 - 36 * q^63 - 75 * q^64 - 28 * q^65 - 37 * q^66 - 116 * q^67 + 3 * q^68 - 58 * q^69 - 73 * q^70 - 144 * q^71 - 27 * q^72 - 121 * q^73 - 16 * q^74 + 24 * q^75 - 118 * q^76 - 3 * q^77 - 25 * q^78 - 135 * q^79 - 36 * q^80 + 95 * q^81 - 102 * q^82 - 21 * q^83 - 75 * q^84 - 129 * q^85 - 46 * q^86 - 31 * q^87 - 77 * q^88 - 63 * q^89 - 36 * q^90 - 123 * q^91 - 42 * q^92 - 129 * q^93 - 44 * q^94 - 80 * q^95 - 53 * q^96 - 144 * q^97 + 10 * q^98 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more