LMFDB - Embedded newform 4205.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4205,2,Mod(1,4205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, 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("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4205 = 5 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4205.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:	$33.5770940499$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	4205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.517638 q^{2} -1.41421 q^{3} -1.73205 q^{4} +1.00000 q^{5} -0.732051 q^{6} +0.732051 q^{7} -1.93185 q^{8} -1.00000 q^{9} +0.517638 q^{10} -5.27792 q^{11} +2.44949 q^{12} +1.46410 q^{13} +0.378937 q^{14} -1.41421 q^{15} +2.46410 q^{16} +6.31319 q^{17} -0.517638 q^{18} +4.24264 q^{19} -1.73205 q^{20} -1.03528 q^{21} -2.73205 q^{22} -8.19615 q^{23} +2.73205 q^{24} +1.00000 q^{25} +0.757875 q^{26} +5.65685 q^{27} -1.26795 q^{28} -0.732051 q^{30} +4.24264 q^{31} +5.13922 q^{32} +7.46410 q^{33} +3.26795 q^{34} +0.732051 q^{35} +1.73205 q^{36} +4.24264 q^{37} +2.19615 q^{38} -2.07055 q^{39} -1.93185 q^{40} -8.76268 q^{41} -0.535898 q^{42} +4.24264 q^{43} +9.14162 q^{44} -1.00000 q^{45} -4.24264 q^{46} +8.38375 q^{47} -3.48477 q^{48} -6.46410 q^{49} +0.517638 q^{50} -8.92820 q^{51} -2.53590 q^{52} +2.92820 q^{54} -5.27792 q^{55} -1.41421 q^{56} -6.00000 q^{57} +6.00000 q^{59} +2.44949 q^{60} -3.10583 q^{61} +2.19615 q^{62} -0.732051 q^{63} -2.26795 q^{64} +1.46410 q^{65} +3.86370 q^{66} -11.1244 q^{67} -10.9348 q^{68} +11.5911 q^{69} +0.378937 q^{70} -6.00000 q^{71} +1.93185 q^{72} -1.13681 q^{73} +2.19615 q^{74} -1.41421 q^{75} -7.34847 q^{76} -3.86370 q^{77} -1.07180 q^{78} -15.8338 q^{79} +2.46410 q^{80} -5.00000 q^{81} -4.53590 q^{82} +2.19615 q^{83} +1.79315 q^{84} +6.31319 q^{85} +2.19615 q^{86} +10.1962 q^{88} +2.07055 q^{89} -0.517638 q^{90} +1.07180 q^{91} +14.1962 q^{92} -6.00000 q^{93} +4.33975 q^{94} +4.24264 q^{95} -7.26795 q^{96} -7.34847 q^{97} -3.34607 q^{98} +5.27792 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{9} - 8 q^{13} - 4 q^{16} - 4 q^{22} - 12 q^{23} + 4 q^{24} + 4 q^{25} - 12 q^{28} + 4 q^{30} + 16 q^{33} + 20 q^{34} - 4 q^{35} - 12 q^{38} - 16 q^{42} - 4 q^{45}+ \cdots - 36 q^{96}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^9 - 8 * q^13 - 4 * q^16 - 4 * q^22 - 12 * q^23 + 4 * q^24 + 4 * q^25 - 12 * q^28 + 4 * q^30 + 16 * q^33 + 20 * q^34 - 4 * q^35 - 12 * q^38 - 16 * q^42 - 4 * q^45 - 12 * q^49 - 8 * q^51 - 24 * q^52 - 16 * q^54 - 24 * q^57 + 24 * q^59 - 12 * q^62 + 4 * q^63 - 16 * q^64 - 8 * q^65 + 4 * q^67 - 24 * q^71 - 12 * q^74 - 32 * q^78 - 4 * q^80 - 20 * q^81 - 32 * q^82 - 12 * q^83 - 12 * q^86 + 20 * q^88 + 32 * q^91 + 36 * q^92 - 24 * q^93 + 52 * q^94 - 36 * q^96

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 $

$2$	0.517638	0.366025	0.183013	−	0.983111i	$-0.441415\pi$
$2$	0.517638	0.366025	0.183013	+	0.983111i	$0.441415\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	−1.73205	−0.866025
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.732051	−0.298858
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	−1.93185	−0.683013
$9$	−1.00000	−0.333333
$10$	0.517638	0.163692
$11$	−5.27792	−1.59135	−0.795676	−	0.605723i	$-0.792884\pi$
$11$	−5.27792	−1.59135	−0.795676	+	0.605723i	$0.792884\pi$
$12$	2.44949	0.707107
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	0.378937	0.101275
$15$	−1.41421	−0.365148
$16$	2.46410	0.616025
$17$	6.31319	1.53117	0.765587	−	0.643332i	$-0.222449\pi$
$17$	6.31319	1.53117	0.765587	+	0.643332i	$0.222449\pi$
$18$	−0.517638	−0.122008
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	−1.73205	−0.387298
$21$	−1.03528	−0.225916
$22$	−2.73205	−0.582475
$23$	−8.19615	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$23$	−8.19615	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$24$	2.73205	0.557678
$25$	1.00000	0.200000
$26$	0.757875	0.148631
$27$	5.65685	1.08866
$28$	−1.26795	−0.239620
$29$	0	0
$29$	0	0
$30$	−0.732051	−0.133654
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	5.13922	0.908494
$33$	7.46410	1.29933
$34$	3.26795	0.560449
$35$	0.732051	0.123739
$36$	1.73205	0.288675
$37$	4.24264	0.697486	0.348743	−	0.937218i	$-0.386609\pi$
$37$	4.24264	0.697486	0.348743	+	0.937218i	$0.386609\pi$
$38$	2.19615	0.356263
$39$	−2.07055	−0.331554
$40$	−1.93185	−0.305453
$41$	−8.76268	−1.36850	−0.684251	−	0.729247i	$-0.739871\pi$
$41$	−8.76268	−1.36850	−0.684251	+	0.729247i	$0.739871\pi$
$42$	−0.535898	−0.0826909
$43$	4.24264	0.646997	0.323498	−	0.946229i	$-0.395141\pi$
$43$	4.24264	0.646997	0.323498	+	0.946229i	$0.395141\pi$
$44$	9.14162	1.37815
$45$	−1.00000	−0.149071
$46$	−4.24264	−0.625543
$47$	8.38375	1.22289	0.611447	−	0.791285i	$-0.290588\pi$
$47$	8.38375	1.22289	0.611447	+	0.791285i	$0.290588\pi$
$48$	−3.48477	−0.502983
$49$	−6.46410	−0.923443
$50$	0.517638	0.0732051
$51$	−8.92820	−1.25020
$52$	−2.53590	−0.351666
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	2.92820	0.398478
$55$	−5.27792	−0.711674
$56$	−1.41421	−0.188982
$57$	−6.00000	−0.794719
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	2.44949	0.316228
$61$	−3.10583	−0.397661	−0.198830	−	0.980034i	$-0.563714\pi$
$61$	−3.10583	−0.397661	−0.198830	+	0.980034i	$0.563714\pi$
$62$	2.19615	0.278912
$63$	−0.732051	−0.0922297
$64$	−2.26795	−0.283494
$65$	1.46410	0.181599
$66$	3.86370	0.475589
$67$	−11.1244	−1.35906	−0.679528	−	0.733649i	$-0.737815\pi$
$67$	−11.1244	−1.35906	−0.679528	+	0.733649i	$0.737815\pi$
$68$	−10.9348	−1.32604
$69$	11.5911	1.39541
$70$	0.378937	0.0452917
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.93185	0.227671
$73$	−1.13681	−0.133054	−0.0665269	−	0.997785i	$-0.521192\pi$
$73$	−1.13681	−0.133054	−0.0665269	+	0.997785i	$0.521192\pi$
$74$	2.19615	0.255298
$75$	−1.41421	−0.163299
$76$	−7.34847	−0.842927
$77$	−3.86370	−0.440310
$78$	−1.07180	−0.121357
$79$	−15.8338	−1.78144	−0.890718	−	0.454556i	$-0.849798\pi$
$79$	−15.8338	−1.78144	−0.890718	+	0.454556i	$0.849798\pi$
$80$	2.46410	0.275495
$81$	−5.00000	−0.555556
$82$	−4.53590	−0.500906
$83$	2.19615	0.241059	0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615	0.241059	0.120530	+	0.992710i	$0.461541\pi$
$84$	1.79315	0.195649
$85$	6.31319	0.684762
$86$	2.19615	0.236817
$87$	0	0
$88$	10.1962	1.08691
$89$	2.07055	0.219478	0.109739	−	0.993960i	$-0.464998\pi$
$89$	2.07055	0.219478	0.109739	+	0.993960i	$0.464998\pi$
$90$	−0.517638	−0.0545638
$91$	1.07180	0.112355
$92$	14.1962	1.48005
$93$	−6.00000	−0.622171
$94$	4.33975	0.447611
$95$	4.24264	0.435286
$96$	−7.26795	−0.741782
$97$	−7.34847	−0.746124	−0.373062	−	0.927806i	$-0.621692\pi$
$97$	−7.34847	−0.746124	−0.373062	+	0.927806i	$0.621692\pi$
$98$	−3.34607	−0.338004
$99$	5.27792	0.530451
$100$	−1.73205	−0.173205
$101$	−4.14110	−0.412055	−0.206028	−	0.978546i	$-0.566054\pi$
$101$	−4.14110	−0.412055	−0.206028	+	0.978546i	$0.566054\pi$
$102$	−4.62158	−0.457604
$103$	−0.196152	−0.0193275	−0.00966374	−	0.999953i	$-0.503076\pi$
$103$	−0.196152	−0.0193275	−0.00966374	+	0.999953i	$0.503076\pi$
$104$	−2.82843	−0.277350
$105$	−1.03528	−0.101033
$106$	0	0
$107$	2.19615	0.212310	0.106155	−	0.994350i	$-0.466146\pi$
$107$	2.19615	0.212310	0.106155	+	0.994350i	$0.466146\pi$
$108$	−9.79796	−0.942809
$109$	1.46410	0.140236	0.0701178	−	0.997539i	$-0.477662\pi$
$109$	1.46410	0.140236	0.0701178	+	0.997539i	$0.477662\pi$
$110$	−2.73205	−0.260491
$111$	−6.00000	−0.569495
$112$	1.80385	0.170448
$113$	13.7632	1.29473	0.647366	−	0.762179i	$-0.275870\pi$
$113$	13.7632	1.29473	0.647366	+	0.762179i	$0.275870\pi$
$114$	−3.10583	−0.290887
$115$	−8.19615	−0.764295
$116$	0	0
$117$	−1.46410	−0.135356
$118$	3.10583	0.285915
$119$	4.62158	0.423659
$120$	2.73205	0.249401
$121$	16.8564	1.53240
$122$	−1.60770	−0.145554
$123$	12.3923	1.11738
$124$	−7.34847	−0.659912
$125$	1.00000	0.0894427
$126$	−0.378937	−0.0337584
$127$	−12.7279	−1.12942	−0.564710	−	0.825289i	$-0.691012\pi$
$127$	−12.7279	−1.12942	−0.564710	+	0.825289i	$0.691012\pi$
$128$	−11.4524	−1.01226
$129$	−6.00000	−0.528271
$130$	0.757875	0.0664700
$131$	2.17209	0.189776	0.0948881	−	0.995488i	$-0.469751\pi$
$131$	2.17209	0.189776	0.0948881	+	0.995488i	$0.469751\pi$
$132$	−12.9282	−1.12526
$133$	3.10583	0.269309
$134$	−5.75839	−0.497449
$135$	5.65685	0.486864
$136$	−12.1962	−1.04581
$137$	9.89949	0.845771	0.422885	−	0.906183i	$-0.361017\pi$
$137$	9.89949	0.845771	0.422885	+	0.906183i	$0.361017\pi$
$138$	6.00000	0.510754
$139$	−13.4641	−1.14201	−0.571005	−	0.820947i	$-0.693446\pi$
$139$	−13.4641	−1.14201	−0.571005	+	0.820947i	$0.693446\pi$
$140$	−1.26795	−0.107161
$141$	−11.8564	−0.998490
$142$	−3.10583	−0.260635
$143$	−7.72741	−0.646198
$144$	−2.46410	−0.205342
$145$	0	0
$146$	−0.588457	−0.0487011
$147$	9.14162	0.753988
$148$	−7.34847	−0.604040
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−0.732051	−0.0597717
$151$	−18.3923	−1.49674	−0.748372	−	0.663279i	$-0.769164\pi$
$151$	−18.3923	−1.49674	−0.748372	+	0.663279i	$0.769164\pi$
$152$	−8.19615	−0.664796
$153$	−6.31319	−0.510391
$154$	−2.00000	−0.161165
$155$	4.24264	0.340777
$156$	3.58630	0.287134
$157$	7.34847	0.586472	0.293236	−	0.956040i	$-0.405268\pi$
$157$	7.34847	0.586472	0.293236	+	0.956040i	$0.405268\pi$
$158$	−8.19615	−0.652051
$159$	0	0
$160$	5.13922	0.406291
$161$	−6.00000	−0.472866
$162$	−2.58819	−0.203347
$163$	4.24264	0.332309	0.166155	−	0.986100i	$-0.446865\pi$
$163$	4.24264	0.332309	0.166155	+	0.986100i	$0.446865\pi$
$164$	15.1774	1.18516
$165$	7.46410	0.581080
$166$	1.13681	0.0882337
$167$	−3.80385	−0.294351	−0.147175	−	0.989110i	$-0.547018\pi$
$167$	−3.80385	−0.294351	−0.147175	+	0.989110i	$0.547018\pi$
$168$	2.00000	0.154303
$169$	−10.8564	−0.835108
$170$	3.26795	0.250640
$171$	−4.24264	−0.324443
$172$	−7.34847	−0.560316
$173$	−20.7846	−1.58022	−0.790112	−	0.612962i	$-0.789978\pi$
$173$	−20.7846	−1.58022	−0.790112	+	0.612962i	$0.789978\pi$
$174$	0	0
$175$	0.732051	0.0553378
$176$	−13.0053	−0.980313
$177$	−8.48528	−0.637793
$178$	1.07180	0.0803346
$179$	−16.3923	−1.22522	−0.612609	−	0.790386i	$-0.709880\pi$
$179$	−16.3923	−1.22522	−0.612609	+	0.790386i	$0.709880\pi$
$180$	1.73205	0.129099
$181$	−5.85641	−0.435303	−0.217652	−	0.976027i	$-0.569840\pi$
$181$	−5.85641	−0.435303	−0.217652	+	0.976027i	$0.569840\pi$
$182$	0.554803	0.0411247
$183$	4.39230	0.324689
$184$	15.8338	1.16728
$185$	4.24264	0.311925
$186$	−3.10583	−0.227730
$187$	−33.3205	−2.43664
$188$	−14.5211	−1.05906
$189$	4.14110	0.301221
$190$	2.19615	0.159326
$191$	19.2170	1.39049	0.695246	−	0.718772i	$-0.255295\pi$
$191$	19.2170	1.39049	0.695246	+	0.718772i	$0.255295\pi$
$192$	3.20736	0.231472
$193$	−12.7279	−0.916176	−0.458088	−	0.888907i	$-0.651466\pi$
$193$	−12.7279	−0.916176	−0.458088	+	0.888907i	$0.651466\pi$
$194$	−3.80385	−0.273100
$195$	−2.07055	−0.148275
$196$	11.1962	0.799725
$197$	−1.60770	−0.114544	−0.0572718	−	0.998359i	$-0.518240\pi$
$197$	−1.60770	−0.114544	−0.0572718	+	0.998359i	$0.518240\pi$
$198$	2.73205	0.194158
$199$	−19.4641	−1.37977	−0.689887	−	0.723917i	$-0.742340\pi$
$199$	−19.4641	−1.37977	−0.689887	+	0.723917i	$0.742340\pi$
$200$	−1.93185	−0.136603
$201$	15.7322	1.10966
$202$	−2.14359	−0.150823
$203$	0	0
$204$	15.4641	1.08270
$205$	−8.76268	−0.612012
$206$	−0.101536	−0.00707435
$207$	8.19615	0.569672
$208$	3.60770	0.250149
$209$	−22.3923	−1.54891
$210$	−0.535898	−0.0369805
$211$	−22.0454	−1.51767	−0.758834	−	0.651284i	$-0.774231\pi$
$211$	−22.0454	−1.51767	−0.758834	+	0.651284i	$0.774231\pi$
$212$	0	0
$213$	8.48528	0.581402
$214$	1.13681	0.0777109
$215$	4.24264	0.289346
$216$	−10.9282	−0.743570
$217$	3.10583	0.210837
$218$	0.757875	0.0513298
$219$	1.60770	0.108638
$220$	9.14162	0.616328
$221$	9.24316	0.621762
$222$	−3.10583	−0.208450
$223$	−9.66025	−0.646898	−0.323449	−	0.946246i	$-0.604842\pi$
$223$	−9.66025	−0.646898	−0.323449	+	0.946246i	$0.604842\pi$
$224$	3.76217	0.251370
$225$	−1.00000	−0.0666667
$226$	7.12436	0.473905
$227$	−15.8038	−1.04894	−0.524469	−	0.851429i	$-0.675736\pi$
$227$	−15.8038	−1.04894	−0.524469	+	0.851429i	$0.675736\pi$
$228$	10.3923	0.688247
$229$	−25.4558	−1.68217	−0.841085	−	0.540903i	$-0.818082\pi$
$229$	−25.4558	−1.68217	−0.841085	+	0.540903i	$0.818082\pi$
$230$	−4.24264	−0.279751
$231$	5.46410	0.359511
$232$	0	0
$233$	22.3923	1.46697	0.733484	−	0.679706i	$-0.237893\pi$
$233$	22.3923	1.46697	0.733484	+	0.679706i	$0.237893\pi$
$234$	−0.757875	−0.0495438
$235$	8.38375	0.546895
$236$	−10.3923	−0.676481
$237$	22.3923	1.45454
$238$	2.39230	0.155070
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−3.48477	−0.224941
$241$	−8.92820	−0.575116	−0.287558	−	0.957763i	$-0.592843\pi$
$241$	−8.92820	−0.575116	−0.287558	+	0.957763i	$0.592843\pi$
$242$	8.72552	0.560898
$243$	−9.89949	−0.635053
$244$	5.37945	0.344384
$245$	−6.46410	−0.412976
$246$	6.41473	0.408988
$247$	6.21166	0.395238
$248$	−8.19615	−0.520456
$249$	−3.10583	−0.196824
$250$	0.517638	0.0327383
$251$	−13.7632	−0.868725	−0.434363	−	0.900738i	$-0.643026\pi$
$251$	−13.7632	−0.868725	−0.434363	+	0.900738i	$0.643026\pi$
$252$	1.26795	0.0798733
$253$	43.2586	2.71965
$254$	−6.58846	−0.413397
$255$	−8.92820	−0.559106
$256$	−1.39230	−0.0870191
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	−3.10583	−0.193360
$259$	3.10583	0.192987
$260$	−2.53590	−0.157270
$261$	0	0
$262$	1.12436	0.0694629
$263$	−4.79744	−0.295823	−0.147912	−	0.989001i	$-0.547255\pi$
$263$	−4.79744	−0.295823	−0.147912	+	0.989001i	$0.547255\pi$
$264$	−14.4195	−0.887461
$265$	0	0
$266$	1.60770	0.0985741
$267$	−2.92820	−0.179203
$268$	19.2679	1.17698
$269$	32.7028	1.99392	0.996962	−	0.0778925i	$-0.0248191\pi$
$269$	32.7028	1.99392	0.996962	+	0.0778925i	$0.0248191\pi$
$270$	2.92820	0.178205
$271$	−9.62209	−0.584501	−0.292250	−	0.956342i	$-0.594404\pi$
$271$	−9.62209	−0.584501	−0.292250	+	0.956342i	$0.594404\pi$
$272$	15.5563	0.943242
$273$	−1.51575	−0.0917373
$274$	5.12436	0.309574
$275$	−5.27792	−0.318270
$276$	−20.0764	−1.20846
$277$	2.92820	0.175939	0.0879693	−	0.996123i	$-0.471962\pi$
$277$	2.92820	0.175939	0.0879693	+	0.996123i	$0.471962\pi$
$278$	−6.96953	−0.418005
$279$	−4.24264	−0.254000
$280$	−1.41421	−0.0845154
$281$	−4.39230	−0.262023	−0.131011	−	0.991381i	$-0.541822\pi$
$281$	−4.39230	−0.262023	−0.131011	+	0.991381i	$0.541822\pi$
$282$	−6.13733	−0.365473
$283$	20.9808	1.24718	0.623588	−	0.781753i	$-0.285674\pi$
$283$	20.9808	1.24718	0.623588	+	0.781753i	$0.285674\pi$
$284$	10.3923	0.616670
$285$	−6.00000	−0.355409
$286$	−4.00000	−0.236525
$287$	−6.41473	−0.378649
$288$	−5.13922	−0.302831
$289$	22.8564	1.34449
$290$	0	0
$291$	10.3923	0.609208
$292$	1.96902	0.115228
$293$	11.6926	0.683092	0.341546	−	0.939865i	$-0.389050\pi$
$293$	11.6926	0.683092	0.341546	+	0.939865i	$0.389050\pi$
$294$	4.73205	0.275979
$295$	6.00000	0.349334
$296$	−8.19615	−0.476392
$297$	−29.8564	−1.73244
$298$	9.31749	0.539747
$299$	−12.0000	−0.693978
$300$	2.44949	0.141421
$301$	3.10583	0.179017
$302$	−9.52056	−0.547847
$303$	5.85641	0.336442
$304$	10.4543	0.599595
$305$	−3.10583	−0.177839
$306$	−3.26795	−0.186816
$307$	−25.1512	−1.43546	−0.717728	−	0.696323i	$-0.754818\pi$
$307$	−25.1512	−1.43546	−0.717728	+	0.696323i	$0.754818\pi$
$308$	6.69213	0.381320
$309$	0.277401	0.0157808
$310$	2.19615	0.124733
$311$	0.101536	0.00575758	0.00287879	−	0.999996i	$-0.499084\pi$
$311$	0.101536	0.00575758	0.00287879	+	0.999996i	$0.499084\pi$
$312$	4.00000	0.226455
$313$	−25.4641	−1.43932	−0.719658	−	0.694329i	$-0.755701\pi$
$313$	−25.4641	−1.43932	−0.719658	+	0.694329i	$0.755701\pi$
$314$	3.80385	0.214664
$315$	−0.732051	−0.0412464
$316$	27.4249	1.54277
$317$	−21.0101	−1.18005	−0.590023	−	0.807386i	$-0.700881\pi$
$317$	−21.0101	−1.18005	−0.590023	+	0.807386i	$0.700881\pi$
$318$	0	0
$319$	0	0
$320$	−2.26795	−0.126782
$321$	−3.10583	−0.173350
$322$	−3.10583	−0.173081
$323$	26.7846	1.49034
$324$	8.66025	0.481125
$325$	1.46410	0.0812137
$326$	2.19615	0.121634
$327$	−2.07055	−0.114502
$328$	16.9282	0.934704
$329$	6.13733	0.338362
$330$	3.86370	0.212690
$331$	18.9396	1.04101	0.520507	−	0.853858i	$-0.325743\pi$
$331$	18.9396	1.04101	0.520507	+	0.853858i	$0.325743\pi$
$332$	−3.80385	−0.208763
$333$	−4.24264	−0.232495
$334$	−1.96902	−0.107740
$335$	−11.1244	−0.607788
$336$	−2.55103	−0.139170
$337$	−18.1074	−0.986372	−0.493186	−	0.869924i	$-0.664168\pi$
$337$	−18.1074	−0.986372	−0.493186	+	0.869924i	$0.664168\pi$
$338$	−5.61969	−0.305671
$339$	−19.4641	−1.05714
$340$	−10.9348	−0.593021
$341$	−22.3923	−1.21261
$342$	−2.19615	−0.118754
$343$	−9.85641	−0.532196
$344$	−8.19615	−0.441907
$345$	11.5911	0.624044
$346$	−10.7589	−0.578402
$347$	−26.1962	−1.40628	−0.703142	−	0.711050i	$-0.748220\pi$
$347$	−26.1962	−1.40628	−0.703142	+	0.711050i	$0.748220\pi$
$348$	0	0
$349$	18.3923	0.984518	0.492259	−	0.870449i	$-0.336171\pi$
$349$	18.3923	0.984518	0.492259	+	0.870449i	$0.336171\pi$
$350$	0.378937	0.0202551
$351$	8.28221	0.442072
$352$	−27.1244	−1.44573
$353$	−32.7846	−1.74495	−0.872474	−	0.488660i	$-0.837486\pi$
$353$	−32.7846	−1.74495	−0.872474	+	0.488660i	$0.837486\pi$
$354$	−4.39230	−0.233448
$355$	−6.00000	−0.318447
$356$	−3.58630	−0.190074
$357$	−6.53590	−0.345916
$358$	−8.48528	−0.448461
$359$	−7.07107	−0.373197	−0.186598	−	0.982436i	$-0.559746\pi$
$359$	−7.07107	−0.373197	−0.186598	+	0.982436i	$0.559746\pi$
$360$	1.93185	0.101818
$361$	−1.00000	−0.0526316
$362$	−3.03150	−0.159332
$363$	−23.8386	−1.25120
$364$	−1.85641	−0.0973021
$365$	−1.13681	−0.0595035
$366$	2.27362	0.118844
$367$	1.96902	0.102782	0.0513909	−	0.998679i	$-0.483635\pi$
$367$	1.96902	0.102782	0.0513909	+	0.998679i	$0.483635\pi$
$368$	−20.1962	−1.05280
$369$	8.76268	0.456167
$370$	2.19615	0.114173
$371$	0	0
$372$	10.3923	0.538816
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−17.2480	−0.891871
$375$	−1.41421	−0.0730297
$376$	−16.1962	−0.835253
$377$	0	0
$378$	2.14359	0.110255
$379$	−15.8338	−0.813325	−0.406663	−	0.913578i	$-0.633308\pi$
$379$	−15.8338	−0.813325	−0.406663	+	0.913578i	$0.633308\pi$
$380$	−7.34847	−0.376969
$381$	18.0000	0.922168
$382$	9.94744	0.508955
$383$	2.19615	0.112218	0.0561091	−	0.998425i	$-0.482131\pi$
$383$	2.19615	0.112218	0.0561091	+	0.998425i	$0.482131\pi$
$384$	16.1962	0.826506
$385$	−3.86370	−0.196913
$386$	−6.58846	−0.335344
$387$	−4.24264	−0.215666
$388$	12.7279	0.646162
$389$	32.7028	1.65810	0.829048	−	0.559177i	$-0.188883\pi$
$389$	32.7028	1.65810	0.829048	+	0.559177i	$0.188883\pi$
$390$	−1.07180	−0.0542725
$391$	−51.7439	−2.61680
$392$	12.4877	0.630723
$393$	−3.07180	−0.154952
$394$	−0.832204	−0.0419258
$395$	−15.8338	−0.796682
$396$	−9.14162	−0.459384
$397$	30.3923	1.52535	0.762673	−	0.646784i	$-0.223887\pi$
$397$	30.3923	1.52535	0.762673	+	0.646784i	$0.223887\pi$
$398$	−10.0754	−0.505032
$399$	−4.39230	−0.219890
$400$	2.46410	0.123205
$401$	8.78461	0.438682	0.219341	−	0.975648i	$-0.429609\pi$
$401$	8.78461	0.438682	0.219341	+	0.975648i	$0.429609\pi$
$402$	8.14359	0.406166
$403$	6.21166	0.309425
$404$	7.17260	0.356850
$405$	−5.00000	−0.248452
$406$	0	0
$407$	−22.3923	−1.10995
$408$	17.2480	0.853901
$409$	17.8028	0.880290	0.440145	−	0.897927i	$-0.354927\pi$
$409$	17.8028	0.880290	0.440145	+	0.897927i	$0.354927\pi$
$410$	−4.53590	−0.224012
$411$	−14.0000	−0.690569
$412$	0.339746	0.0167381
$413$	4.39230	0.216131
$414$	4.24264	0.208514
$415$	2.19615	0.107805
$416$	7.52433	0.368911
$417$	19.0411	0.932447
$418$	−11.5911	−0.566940
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	1.79315	0.0874968
$421$	−17.8028	−0.867654	−0.433827	−	0.900996i	$-0.642837\pi$
$421$	−17.8028	−0.867654	−0.433827	+	0.900996i	$0.642837\pi$
$422$	−11.4115	−0.555505
$423$	−8.38375	−0.407632
$424$	0	0
$425$	6.31319	0.306235
$426$	4.39230	0.212808
$427$	−2.27362	−0.110028
$428$	−3.80385	−0.183866
$429$	10.9282	0.527619
$430$	2.19615	0.105908
$431$	4.39230	0.211570	0.105785	−	0.994389i	$-0.466264\pi$
$431$	4.39230	0.211570	0.105785	+	0.994389i	$0.466264\pi$
$432$	13.9391	0.670644
$433$	12.7279	0.611665	0.305832	−	0.952085i	$-0.401065\pi$
$433$	12.7279	0.611665	0.305832	+	0.952085i	$0.401065\pi$
$434$	1.60770	0.0771718
$435$	0	0
$436$	−2.53590	−0.121448
$437$	−34.7733	−1.66343
$438$	0.832204	0.0397643
$439$	−8.92820	−0.426120	−0.213060	−	0.977039i	$-0.568343\pi$
$439$	−8.92820	−0.426120	−0.213060	+	0.977039i	$0.568343\pi$
$440$	10.1962	0.486082
$441$	6.46410	0.307814
$442$	4.78461	0.227581
$443$	−14.7985	−0.703097	−0.351548	−	0.936170i	$-0.614345\pi$
$443$	−14.7985	−0.703097	−0.351548	+	0.936170i	$0.614345\pi$
$444$	10.3923	0.493197
$445$	2.07055	0.0981536
$446$	−5.00052	−0.236781
$447$	−25.4558	−1.20402
$448$	−1.66025	−0.0784396
$449$	30.6322	1.44562	0.722812	−	0.691045i	$-0.242849\pi$
$449$	30.6322	1.44562	0.722812	+	0.691045i	$0.242849\pi$
$450$	−0.517638	−0.0244017
$451$	46.2487	2.17777
$452$	−23.8386	−1.12127
$453$	26.0106	1.22209
$454$	−8.18067	−0.383938
$455$	1.07180	0.0502466
$456$	11.5911	0.542803
$457$	−26.9282	−1.25965	−0.629824	−	0.776738i	$-0.716873\pi$
$457$	−26.9282	−1.25965	−0.629824	+	0.776738i	$0.716873\pi$
$458$	−13.1769	−0.615717
$459$	35.7128	1.66693
$460$	14.1962	0.661899
$461$	−22.6274	−1.05386	−0.526932	−	0.849907i	$-0.676658\pi$
$461$	−22.6274	−1.05386	−0.526932	+	0.849907i	$0.676658\pi$
$462$	2.82843	0.131590
$463$	21.6603	1.00664	0.503319	−	0.864101i	$-0.332112\pi$
$463$	21.6603	1.00664	0.503319	+	0.864101i	$0.332112\pi$
$464$	0	0
$465$	−6.00000	−0.278243
$466$	11.5911	0.536948
$467$	−21.7680	−1.00730	−0.503652	−	0.863907i	$-0.668010\pi$
$467$	−21.7680	−1.00730	−0.503652	+	0.863907i	$0.668010\pi$
$468$	2.53590	0.117222
$469$	−8.14359	−0.376036
$470$	4.33975	0.200178
$471$	−10.3923	−0.478852
$472$	−11.5911	−0.533524
$473$	−22.3923	−1.02960
$474$	11.5911	0.532397
$475$	4.24264	0.194666
$476$	−8.00481	−0.366900
$477$	0	0
$478$	3.10583	0.142057
$479$	−4.52004	−0.206526	−0.103263	−	0.994654i	$-0.532928\pi$
$479$	−4.52004	−0.206526	−0.103263	+	0.994654i	$0.532928\pi$
$480$	−7.26795	−0.331735
$481$	6.21166	0.283227
$482$	−4.62158	−0.210507
$483$	8.48528	0.386094
$484$	−29.1962	−1.32710
$485$	−7.34847	−0.333677
$486$	−5.12436	−0.232445
$487$	−22.5885	−1.02358	−0.511790	−	0.859110i	$-0.671018\pi$
$487$	−22.5885	−1.02358	−0.511790	+	0.859110i	$0.671018\pi$
$488$	6.00000	0.271607
$489$	−6.00000	−0.271329
$490$	−3.34607	−0.151160
$491$	−1.41421	−0.0638226	−0.0319113	−	0.999491i	$-0.510159\pi$
$491$	−1.41421	−0.0638226	−0.0319113	+	0.999491i	$0.510159\pi$
$492$	−21.4641	−0.967676
$493$	0	0
$494$	3.21539	0.144667
$495$	5.27792	0.237225
$496$	10.4543	0.469412
$497$	−4.39230	−0.197022
$498$	−1.60770	−0.0720425
$499$	5.85641	0.262169	0.131084	−	0.991371i	$-0.458154\pi$
$499$	5.85641	0.262169	0.131084	+	0.991371i	$0.458154\pi$
$500$	−1.73205	−0.0774597
$501$	5.37945	0.240336
$502$	−7.12436	−0.317976
$503$	−14.7985	−0.659831	−0.329916	−	0.944010i	$-0.607020\pi$
$503$	−14.7985	−0.659831	−0.329916	+	0.944010i	$0.607020\pi$
$504$	1.41421	0.0629941
$505$	−4.14110	−0.184277
$506$	22.3923	0.995459
$507$	15.3533	0.681863
$508$	22.0454	0.978107
$509$	20.7846	0.921262	0.460631	−	0.887592i	$-0.347623\pi$
$509$	20.7846	0.921262	0.460631	+	0.887592i	$0.347623\pi$
$510$	−4.62158	−0.204647
$511$	−0.832204	−0.0368145
$512$	22.1841	0.980408
$513$	24.0000	1.05963
$514$	6.21166	0.273984
$515$	−0.196152	−0.00864351
$516$	10.3923	0.457496
$517$	−44.2487	−1.94606
$518$	1.60770	0.0706381
$519$	29.3939	1.29025
$520$	−2.82843	−0.124035
$521$	32.7846	1.43632	0.718160	−	0.695878i	$-0.244985\pi$
$521$	32.7846	1.43632	0.718160	+	0.695878i	$0.244985\pi$
$522$	0	0
$523$	−21.6603	−0.947137	−0.473568	−	0.880757i	$-0.657034\pi$
$523$	−21.6603	−0.947137	−0.473568	+	0.880757i	$0.657034\pi$
$524$	−3.76217	−0.164351
$525$	−1.03528	−0.0451832
$526$	−2.48334	−0.108279
$527$	26.7846	1.16676
$528$	18.3923	0.800422
$529$	44.1769	1.92074
$530$	0	0
$531$	−6.00000	−0.260378
$532$	−5.37945	−0.233229
$533$	−12.8295	−0.555706
$534$	−1.51575	−0.0655929
$535$	2.19615	0.0949479
$536$	21.4906	0.928253
$537$	23.1822	1.00039
$538$	16.9282	0.729827
$539$	34.1170	1.46952
$540$	−9.79796	−0.421637
$541$	25.4558	1.09443	0.547216	−	0.836991i	$-0.315688\pi$
$541$	25.4558	1.09443	0.547216	+	0.836991i	$0.315688\pi$
$542$	−4.98076	−0.213942
$543$	8.28221	0.355424
$544$	32.4449	1.39106
$545$	1.46410	0.0627152
$546$	−0.784610	−0.0335782
$547$	21.5167	0.919986	0.459993	−	0.887923i	$-0.347852\pi$
$547$	21.5167	0.919986	0.459993	+	0.887923i	$0.347852\pi$
$548$	−17.1464	−0.732459
$549$	3.10583	0.132554
$550$	−2.73205	−0.116495
$551$	0	0
$552$	−22.3923	−0.953080
$553$	−11.5911	−0.492904
$554$	1.51575	0.0643980
$555$	−6.00000	−0.254686
$556$	23.3205	0.989010
$557$	22.3923	0.948792	0.474396	−	0.880311i	$-0.342666\pi$
$557$	22.3923	0.948792	0.474396	+	0.880311i	$0.342666\pi$
$558$	−2.19615	−0.0929705
$559$	6.21166	0.262725
$560$	1.80385	0.0762265
$561$	47.1223	1.98951
$562$	−2.27362	−0.0959071
$563$	16.8690	0.710945	0.355472	−	0.934687i	$-0.384320\pi$
$563$	16.8690	0.710945	0.355472	+	0.934687i	$0.384320\pi$
$564$	20.5359	0.864717
$565$	13.7632	0.579022
$566$	10.8604	0.456498
$567$	−3.66025	−0.153716
$568$	11.5911	0.486352
$569$	−15.9353	−0.668042	−0.334021	−	0.942566i	$-0.608406\pi$
$569$	−15.9353	−0.668042	−0.334021	+	0.942566i	$0.608406\pi$
$570$	−3.10583	−0.130089
$571$	−11.6077	−0.485767	−0.242883	−	0.970055i	$-0.578093\pi$
$571$	−11.6077	−0.485767	−0.242883	+	0.970055i	$0.578093\pi$
$572$	13.3843	0.559624
$573$	−27.1769	−1.13533
$574$	−3.32051	−0.138595
$575$	−8.19615	−0.341803
$576$	2.26795	0.0944979
$577$	44.3954	1.84821	0.924103	−	0.382144i	$-0.124814\pi$
$577$	44.3954	1.84821	0.924103	+	0.382144i	$0.124814\pi$
$578$	11.8313	0.492119
$579$	18.0000	0.748054
$580$	0	0
$581$	1.60770	0.0666984
$582$	5.37945	0.222985
$583$	0	0
$584$	2.19615	0.0908774
$585$	−1.46410	−0.0605332
$586$	6.05256	0.250029
$587$	−44.1962	−1.82417	−0.912085	−	0.410001i	$-0.865528\pi$
$587$	−44.1962	−1.82417	−0.912085	+	0.410001i	$0.865528\pi$
$588$	−15.8338	−0.652973
$589$	18.0000	0.741677
$590$	3.10583	0.127865
$591$	2.27362	0.0935244
$592$	10.4543	0.429669
$593$	19.6077	0.805192	0.402596	−	0.915378i	$-0.368108\pi$
$593$	19.6077	0.805192	0.402596	+	0.915378i	$0.368108\pi$
$594$	−15.4548	−0.634119
$595$	4.62158	0.189466
$596$	−31.1769	−1.27706
$597$	27.5264	1.12658
$598$	−6.21166	−0.254014
$599$	7.14540	0.291953	0.145977	−	0.989288i	$-0.453368\pi$
$599$	7.14540	0.291953	0.145977	+	0.989288i	$0.453368\pi$
$600$	2.73205	0.111536
$601$	−20.0764	−0.818933	−0.409467	−	0.912325i	$-0.634285\pi$
$601$	−20.0764	−0.818933	−0.409467	+	0.912325i	$0.634285\pi$
$602$	1.60770	0.0655248
$603$	11.1244	0.453019
$604$	31.8564	1.29622
$605$	16.8564	0.685310
$606$	3.03150	0.123146
$607$	−12.7279	−0.516610	−0.258305	−	0.966063i	$-0.583164\pi$
$607$	−12.7279	−0.516610	−0.258305	+	0.966063i	$0.583164\pi$
$608$	21.8038	0.884263
$609$	0	0
$610$	−1.60770	−0.0650937
$611$	12.2747	0.496579
$612$	10.9348	0.442012
$613$	−32.9282	−1.32996	−0.664979	−	0.746862i	$-0.731559\pi$
$613$	−32.9282	−1.32996	−0.664979	+	0.746862i	$0.731559\pi$
$614$	−13.0192	−0.525414
$615$	12.3923	0.499706
$616$	7.46410	0.300737
$617$	−6.79367	−0.273503	−0.136751	−	0.990605i	$-0.543666\pi$
$617$	−6.79367	−0.273503	−0.136751	+	0.990605i	$0.543666\pi$
$618$	0.143594	0.00577618
$619$	−13.5601	−0.545027	−0.272514	−	0.962152i	$-0.587855\pi$
$619$	−13.5601	−0.545027	−0.272514	+	0.962152i	$0.587855\pi$
$620$	−7.34847	−0.295122
$621$	−46.3644	−1.86054
$622$	0.0525589	0.00210742
$623$	1.51575	0.0607272
$624$	−5.10205	−0.204246
$625$	1.00000	0.0400000
$626$	−13.1812	−0.526826
$627$	31.6675	1.26468
$628$	−12.7279	−0.507899
$629$	26.7846	1.06797
$630$	−0.378937	−0.0150972
$631$	17.8564	0.710852	0.355426	−	0.934704i	$-0.384336\pi$
$631$	17.8564	0.710852	0.355426	+	0.934704i	$0.384336\pi$
$632$	30.5885	1.21674
$633$	31.1769	1.23917
$634$	−10.8756	−0.431927
$635$	−12.7279	−0.505092
$636$	0	0
$637$	−9.46410	−0.374981
$638$	0	0
$639$	6.00000	0.237356
$640$	−11.4524	−0.452696
$641$	15.9353	0.629406	0.314703	−	0.949190i	$-0.398095\pi$
$641$	15.9353	0.629406	0.314703	+	0.949190i	$0.398095\pi$
$642$	−1.60770	−0.0634507
$643$	−9.94744	−0.392289	−0.196144	−	0.980575i	$-0.562842\pi$
$643$	−9.94744	−0.392289	−0.196144	+	0.980575i	$0.562842\pi$
$644$	10.3923	0.409514
$645$	−6.00000	−0.236250
$646$	13.8647	0.545501
$647$	27.3731	1.07615	0.538073	−	0.842898i	$-0.319152\pi$
$647$	27.3731	1.07615	0.538073	+	0.842898i	$0.319152\pi$
$648$	9.65926	0.379452
$649$	−31.6675	−1.24306
$650$	0.757875	0.0297263
$651$	−4.39230	−0.172148
$652$	−7.34847	−0.287788
$653$	−26.3896	−1.03270	−0.516352	−	0.856376i	$-0.672710\pi$
$653$	−26.3896	−1.03270	−0.516352	+	0.856376i	$0.672710\pi$
$654$	−1.07180	−0.0419106
$655$	2.17209	0.0848705
$656$	−21.5921	−0.843031
$657$	1.13681	0.0443513
$658$	3.17691	0.123849
$659$	−10.6574	−0.415152	−0.207576	−	0.978219i	$-0.566557\pi$
$659$	−10.6574	−0.415152	−0.207576	+	0.978219i	$0.566557\pi$
$660$	−12.9282	−0.503230
$661$	−5.85641	−0.227788	−0.113894	−	0.993493i	$-0.536332\pi$
$661$	−5.85641	−0.227788	−0.113894	+	0.993493i	$0.536332\pi$
$662$	9.80385	0.381037
$663$	−13.0718	−0.507667
$664$	−4.24264	−0.164646
$665$	3.10583	0.120439
$666$	−2.19615	−0.0850992
$667$	0	0
$668$	6.58846	0.254915
$669$	13.6617	0.528190
$670$	−5.75839	−0.222466
$671$	16.3923	0.632818
$672$	−5.32051	−0.205243
$673$	−25.3205	−0.976034	−0.488017	−	0.872834i	$-0.662280\pi$
$673$	−25.3205	−0.976034	−0.488017	+	0.872834i	$0.662280\pi$
$674$	−9.37307	−0.361037
$675$	5.65685	0.217732
$676$	18.8038	0.723225
$677$	−38.8129	−1.49170	−0.745850	−	0.666113i	$-0.767957\pi$
$677$	−38.8129	−1.49170	−0.745850	+	0.666113i	$0.767957\pi$
$678$	−10.0754	−0.386942
$679$	−5.37945	−0.206444
$680$	−12.1962	−0.467701
$681$	22.3500	0.856454
$682$	−11.5911	−0.443847
$683$	32.1962	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$683$	32.1962	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$684$	7.34847	0.280976
$685$	9.89949	0.378240
$686$	−5.10205	−0.194797
$687$	36.0000	1.37349
$688$	10.4543	0.398566
$689$	0	0
$690$	6.00000	0.228416
$691$	−3.07180	−0.116857	−0.0584283	−	0.998292i	$-0.518609\pi$
$691$	−3.07180	−0.116857	−0.0584283	+	0.998292i	$0.518609\pi$
$692$	36.0000	1.36851
$693$	3.86370	0.146770
$694$	−13.5601	−0.514735
$695$	−13.4641	−0.510722
$696$	0	0
$697$	−55.3205	−2.09541
$698$	9.52056	0.360358
$699$	−31.6675	−1.19777
$700$	−1.26795	−0.0479240
$701$	−32.7846	−1.23826	−0.619129	−	0.785289i	$-0.712514\pi$
$701$	−32.7846	−1.23826	−0.619129	+	0.785289i	$0.712514\pi$
$702$	4.28719	0.161809
$703$	18.0000	0.678883
$704$	11.9700	0.451138
$705$	−11.8564	−0.446538
$706$	−16.9706	−0.638696
$707$	−3.03150	−0.114011
$708$	14.6969	0.552345
$709$	7.21539	0.270980	0.135490	−	0.990779i	$-0.456739\pi$
$709$	7.21539	0.270980	0.135490	+	0.990779i	$0.456739\pi$
$710$	−3.10583	−0.116560
$711$	15.8338	0.593812
$712$	−4.00000	−0.149906
$713$	−34.7733	−1.30227
$714$	−3.38323	−0.126614
$715$	−7.72741	−0.288989
$716$	28.3923	1.06107
$717$	−8.48528	−0.316889
$718$	−3.66025	−0.136599
$719$	−10.3923	−0.387568	−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923	−0.387568	−0.193784	+	0.981044i	$0.562076\pi$
$720$	−2.46410	−0.0918316
$721$	−0.143594	−0.00534770
$722$	−0.517638	−0.0192645
$723$	12.6264	0.469580
$724$	10.1436	0.376984
$725$	0	0
$726$	−12.3397	−0.457971
$727$	50.6071	1.87691	0.938456	−	0.345398i	$-0.112256\pi$
$727$	50.6071	1.87691	0.938456	+	0.345398i	$0.112256\pi$
$728$	−2.07055	−0.0767398
$729$	29.0000	1.07407
$730$	−0.588457	−0.0217798
$731$	26.7846	0.990665
$732$	−7.60770	−0.281189
$733$	15.0015	0.554095	0.277047	−	0.960856i	$-0.410644\pi$
$733$	15.0015	0.554095	0.277047	+	0.960856i	$0.410644\pi$
$734$	1.01924	0.0376208
$735$	9.14162	0.337194
$736$	−42.1218	−1.55263
$737$	58.7134	2.16274
$738$	4.53590	0.166969
$739$	16.6660	0.613067	0.306534	−	0.951860i	$-0.400831\pi$
$739$	16.6660	0.613067	0.306534	+	0.951860i	$0.400831\pi$
$740$	−7.34847	−0.270135
$741$	−8.78461	−0.322711
$742$	0	0
$743$	−0.101536	−0.00372499	−0.00186250	−	0.999998i	$-0.500593\pi$
$743$	−0.101536	−0.00372499	−0.00186250	+	0.999998i	$0.500593\pi$
$744$	11.5911	0.424951
$745$	18.0000	0.659469
$746$	−11.3880	−0.416946
$747$	−2.19615	−0.0803530
$748$	57.7128	2.11019
$749$	1.60770	0.0587439
$750$	−0.732051	−0.0267307
$751$	−18.9396	−0.691115	−0.345558	−	0.938398i	$-0.612310\pi$
$751$	−18.9396	−0.691115	−0.345558	+	0.938398i	$0.612310\pi$
$752$	20.6584	0.753334
$753$	19.4641	0.709311
$754$	0	0
$755$	−18.3923	−0.669365
$756$	−7.17260	−0.260865
$757$	30.5307	1.10966	0.554828	−	0.831965i	$-0.312784\pi$
$757$	30.5307	1.10966	0.554828	+	0.831965i	$0.312784\pi$
$758$	−8.19615	−0.297698
$759$	−61.1769	−2.22058
$760$	−8.19615	−0.297306
$761$	−10.3923	−0.376721	−0.188360	−	0.982100i	$-0.560317\pi$
$761$	−10.3923	−0.376721	−0.188360	+	0.982100i	$0.560317\pi$
$762$	9.31749	0.337537
$763$	1.07180	0.0388016
$764$	−33.2848	−1.20420
$765$	−6.31319	−0.228254
$766$	1.13681	0.0410747
$767$	8.78461	0.317194
$768$	1.96902	0.0710508
$769$	14.6969	0.529985	0.264993	−	0.964250i	$-0.414630\pi$
$769$	14.6969	0.529985	0.264993	+	0.964250i	$0.414630\pi$
$770$	−2.00000	−0.0720750
$771$	−16.9706	−0.611180
$772$	22.0454	0.793432
$773$	18.3848	0.661254	0.330627	−	0.943761i	$-0.392740\pi$
$773$	18.3848	0.661254	0.330627	+	0.943761i	$0.392740\pi$
$774$	−2.19615	−0.0789391
$775$	4.24264	0.152400
$776$	14.1962	0.509612
$777$	−4.39230	−0.157573
$778$	16.9282	0.606905
$779$	−37.1769	−1.33200
$780$	3.58630	0.128410
$781$	31.6675	1.13315
$782$	−26.7846	−0.957816
$783$	0	0
$784$	−15.9282	−0.568864
$785$	7.34847	0.262278
$786$	−1.59008	−0.0567162
$787$	−38.5885	−1.37553	−0.687765	−	0.725933i	$-0.741408\pi$
$787$	−38.5885	−1.37553	−0.687765	+	0.725933i	$0.741408\pi$
$788$	2.78461	0.0991976
$789$	6.78461	0.241539
$790$	−8.19615	−0.291606
$791$	10.0754	0.358239
$792$	−10.1962	−0.362305
$793$	−4.54725	−0.161478
$794$	15.7322	0.558315
$795$	0	0
$796$	33.7128	1.19492
$797$	6.79367	0.240644	0.120322	−	0.992735i	$-0.461607\pi$
$797$	6.79367	0.240644	0.120322	+	0.992735i	$0.461607\pi$
$798$	−2.27362	−0.0804854
$799$	52.9282	1.87247
$800$	5.13922	0.181699
$801$	−2.07055	−0.0731594
$802$	4.54725	0.160569
$803$	6.00000	0.211735
$804$	−27.2490	−0.960998
$805$	−6.00000	−0.211472
$806$	3.21539	0.113257
$807$	−46.2487	−1.62803
$808$	8.00000	0.281439
$809$	31.8706	1.12051	0.560255	−	0.828320i	$-0.310703\pi$
$809$	31.8706	1.12051	0.560255	+	0.828320i	$0.310703\pi$
$810$	−2.58819	−0.0909397
$811$	4.67949	0.164319	0.0821596	−	0.996619i	$-0.473818\pi$
$811$	4.67949	0.164319	0.0821596	+	0.996619i	$0.473818\pi$
$812$	0	0
$813$	13.6077	0.477243
$814$	−11.5911	−0.406268
$815$	4.24264	0.148613
$816$	−22.0000	−0.770154
$817$	18.0000	0.629740
$818$	9.21539	0.322209
$819$	−1.07180	−0.0374516
$820$	15.1774	0.530018
$821$	26.7846	0.934789	0.467395	−	0.884049i	$-0.345193\pi$
$821$	26.7846	0.934789	0.467395	+	0.884049i	$0.345193\pi$
$822$	−7.24693	−0.252766
$823$	50.6071	1.76405	0.882026	−	0.471201i	$-0.156179\pi$
$823$	50.6071	1.76405	0.882026	+	0.471201i	$0.156179\pi$
$824$	0.378937	0.0132009
$825$	7.46410	0.259867
$826$	2.27362	0.0791095
$827$	−33.0817	−1.15036	−0.575182	−	0.818025i	$-0.695069\pi$
$827$	−33.0817	−1.15036	−0.575182	+	0.818025i	$0.695069\pi$
$828$	−14.1962	−0.493350
$829$	−45.5322	−1.58140	−0.790700	−	0.612204i	$-0.790283\pi$
$829$	−45.5322	−1.58140	−0.790700	+	0.612204i	$0.790283\pi$
$830$	1.13681	0.0394593
$831$	−4.14110	−0.143653
$832$	−3.32051	−0.115118
$833$	−40.8091	−1.41395
$834$	9.85641	0.341299
$835$	−3.80385	−0.131638
$836$	38.7846	1.34139
$837$	24.0000	0.829561
$838$	9.31749	0.321867
$839$	13.7632	0.475158	0.237579	−	0.971368i	$-0.423646\pi$
$839$	13.7632	0.475158	0.237579	+	0.971368i	$0.423646\pi$
$840$	2.00000	0.0690066
$841$	0	0
$842$	−9.21539	−0.317583
$843$	6.21166	0.213941
$844$	38.1838	1.31434
$845$	−10.8564	−0.373472
$846$	−4.33975	−0.149204
$847$	12.3397	0.423999
$848$	0	0
$849$	−29.6713	−1.01832
$850$	3.26795	0.112090
$851$	−34.7733	−1.19201
$852$	−14.6969	−0.503509
$853$	−35.9101	−1.22954	−0.614770	−	0.788707i	$-0.710751\pi$
$853$	−35.9101	−1.22954	−0.614770	+	0.788707i	$0.710751\pi$
$854$	−1.17691	−0.0402732
$855$	−4.24264	−0.145095
$856$	−4.24264	−0.145010
$857$	−44.7846	−1.52981	−0.764907	−	0.644141i	$-0.777215\pi$
$857$	−44.7846	−1.52981	−0.764907	+	0.644141i	$0.777215\pi$
$858$	5.65685	0.193122
$859$	5.07484	0.173151	0.0865757	−	0.996245i	$-0.472408\pi$
$859$	5.07484	0.173151	0.0865757	+	0.996245i	$0.472408\pi$
$860$	−7.34847	−0.250581
$861$	9.07180	0.309166
$862$	2.27362	0.0774400
$863$	38.1962	1.30021	0.650106	−	0.759843i	$-0.274724\pi$
$863$	38.1962	1.30021	0.650106	+	0.759843i	$0.274724\pi$
$864$	29.0718	0.989043
$865$	−20.7846	−0.706698
$866$	6.58846	0.223885
$867$	−32.3238	−1.09778
$868$	−5.37945	−0.182591
$869$	83.5692	2.83489
$870$	0	0
$871$	−16.2872	−0.551870
$872$	−2.82843	−0.0957826
$873$	7.34847	0.248708
$874$	−18.0000	−0.608859
$875$	0.732051	0.0247478
$876$	−2.78461	−0.0940832
$877$	−24.7846	−0.836917	−0.418458	−	0.908236i	$-0.637429\pi$
$877$	−24.7846	−0.836917	−0.418458	+	0.908236i	$0.637429\pi$
$878$	−4.62158	−0.155971
$879$	−16.5359	−0.557742
$880$	−13.0053	−0.438409
$881$	30.6322	1.03203	0.516013	−	0.856581i	$-0.327416\pi$
$881$	30.6322	1.03203	0.516013	+	0.856581i	$0.327416\pi$
$882$	3.34607	0.112668
$883$	22.5885	0.760162	0.380081	−	0.924953i	$-0.375896\pi$
$883$	22.5885	0.760162	0.380081	+	0.924953i	$0.375896\pi$
$884$	−16.0096	−0.538462
$885$	−8.48528	−0.285230
$886$	−7.66025	−0.257351
$887$	21.0101	0.705451	0.352726	−	0.935727i	$-0.385255\pi$
$887$	21.0101	0.705451	0.352726	+	0.935727i	$0.385255\pi$
$888$	11.5911	0.388972
$889$	−9.31749	−0.312498
$890$	1.07180	0.0359267
$891$	26.3896	0.884084
$892$	16.7321	0.560230
$893$	35.5692	1.19028
$894$	−13.1769	−0.440702
$895$	−16.3923	−0.547934
$896$	−8.38375	−0.280081
$897$	16.9706	0.566631
$898$	15.8564	0.529135
$899$	0	0
$900$	1.73205	0.0577350
$901$	0	0
$902$	23.9401	0.797118
$903$	−4.39230	−0.146167
$904$	−26.5885	−0.884319
$905$	−5.85641	−0.194674
$906$	13.4641	0.447315
$907$	35.9101	1.19238	0.596188	−	0.802845i	$-0.296681\pi$
$907$	35.9101	1.19238	0.596188	+	0.802845i	$0.296681\pi$
$908$	27.3731	0.908407
$909$	4.14110	0.137352
$910$	0.554803	0.0183915
$911$	−21.0101	−0.696097	−0.348048	−	0.937477i	$-0.613156\pi$
$911$	−21.0101	−0.696097	−0.348048	+	0.937477i	$0.613156\pi$
$912$	−14.7846	−0.489567
$913$	−11.5911	−0.383610
$914$	−13.9391	−0.461063
$915$	4.39230	0.145205
$916$	44.0908	1.45680
$917$	1.59008	0.0525090
$918$	18.4863	0.610139
$919$	−23.8564	−0.786950	−0.393475	−	0.919335i	$-0.628727\pi$
$919$	−23.8564	−0.786950	−0.393475	+	0.919335i	$0.628727\pi$
$920$	15.8338	0.522023
$921$	35.5692	1.17205
$922$	−11.7128	−0.385741
$923$	−8.78461	−0.289149
$924$	−9.46410	−0.311346
$925$	4.24264	0.139497
$926$	11.2122	0.368455
$927$	0.196152	0.00644249
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−3.10583	−0.101844
$931$	−27.4249	−0.898814
$932$	−38.7846	−1.27043
$933$	−0.143594	−0.00470104
$934$	−11.2679	−0.368699
$935$	−33.3205	−1.08970
$936$	2.82843	0.0924500
$937$	46.2487	1.51088	0.755440	−	0.655218i	$-0.227423\pi$
$937$	46.2487	1.51088	0.755440	+	0.655218i	$0.227423\pi$
$938$	−4.21543	−0.137639
$939$	36.0117	1.17520
$940$	−14.5211	−0.473625
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−5.37945	−0.175272
$943$	71.8203	2.33879
$944$	14.7846	0.481198
$945$	4.14110	0.134710
$946$	−11.5911	−0.376859
$947$	−48.5365	−1.57723	−0.788613	−	0.614890i	$-0.789200\pi$
$947$	−48.5365	−1.57723	−0.788613	+	0.614890i	$0.789200\pi$
$948$	−38.7846	−1.25967
$949$	−1.66441	−0.0540290
$950$	2.19615	0.0712526
$951$	29.7128	0.963504
$952$	−8.92820	−0.289365
$953$	−47.5692	−1.54092	−0.770459	−	0.637489i	$-0.779973\pi$
$953$	−47.5692	−1.54092	−0.770459	+	0.637489i	$0.779973\pi$
$954$	0	0
$955$	19.2170	0.621847
$956$	−10.3923	−0.336111
$957$	0	0
$958$	−2.33975	−0.0755938
$959$	7.24693	0.234016
$960$	3.20736	0.103517
$961$	−13.0000	−0.419355
$962$	3.21539	0.103668
$963$	−2.19615	−0.0707700
$964$	15.4641	0.498065
$965$	−12.7279	−0.409726
$966$	4.39230	0.141320
$967$	59.0924	1.90028	0.950141	−	0.311820i	$-0.100939\pi$
$967$	59.0924	1.90028	0.950141	+	0.311820i	$0.100939\pi$
$968$	−32.5641	−1.04665
$969$	−37.8792	−1.21685
$970$	−3.80385	−0.122134
$971$	−36.4649	−1.17022	−0.585108	−	0.810955i	$-0.698948\pi$
$971$	−36.4649	−1.17022	−0.585108	+	0.810955i	$0.698948\pi$
$972$	17.1464	0.549972
$973$	−9.85641	−0.315982
$974$	−11.6926	−0.374657
$975$	−2.07055	−0.0663107
$976$	−7.65308	−0.244969
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	−3.10583	−0.0993134
$979$	−10.9282	−0.349267
$980$	11.1962	0.357648
$981$	−1.46410	−0.0467452
$982$	−0.732051	−0.0233607
$983$	0.859411	0.0274109	0.0137055	−	0.999906i	$-0.495637\pi$
$983$	0.859411	0.0274109	0.0137055	+	0.999906i	$0.495637\pi$
$984$	−23.9401	−0.763182
$985$	−1.60770	−0.0512254
$986$	0	0
$987$	−8.67949	−0.276271
$988$	−10.7589	−0.342286
$989$	−34.7733	−1.10573
$990$	2.73205	0.0868303
$991$	−61.3205	−1.94791	−0.973955	−	0.226741i	$-0.927193\pi$
$991$	−61.3205	−1.94791	−0.973955	+	0.226741i	$0.927193\pi$
$992$	21.8038	0.692273
$993$	−26.7846	−0.849984
$994$	−2.27362	−0.0721150
$995$	−19.4641	−0.617054
$996$	5.37945	0.170454
$997$	−30.5307	−0.966917	−0.483458	−	0.875367i	$-0.660620\pi$
$997$	−30.5307	−0.966917	−0.483458	+	0.875367i	$0.660620\pi$
$998$	3.03150	0.0959604
$999$	24.0000	0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4205.2.a.g.1.3		4
29.12	odd	4		145.2.c.a.86.3	yes	4
29.17	odd	4		145.2.c.a.86.2	✓	4
29.28	even	2	inner	4205.2.a.g.1.2		4
87.17	even	4		1305.2.d.a.811.3		4
87.41	even	4		1305.2.d.a.811.2		4
116.75	even	4		2320.2.g.e.1681.1		4
116.99	even	4		2320.2.g.e.1681.3		4
145.12	even	4		725.2.d.b.724.4		8
145.17	even	4		725.2.d.b.724.6		8
145.99	odd	4		725.2.c.d.376.2		4
145.104	odd	4		725.2.c.d.376.3		4
145.128	even	4		725.2.d.b.724.5		8
145.133	even	4		725.2.d.b.724.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.c.a.86.2	✓	4	29.17	odd	4
145.2.c.a.86.3	yes	4	29.12	odd	4
725.2.c.d.376.2		4	145.99	odd	4
725.2.c.d.376.3		4	145.104	odd	4
725.2.d.b.724.3		8	145.133	even	4
725.2.d.b.724.4		8	145.12	even	4
725.2.d.b.724.5		8	145.128	even	4
725.2.d.b.724.6		8	145.17	even	4
1305.2.d.a.811.2		4	87.41	even	4
1305.2.d.a.811.3		4	87.17	even	4
2320.2.g.e.1681.1		4	116.75	even	4
2320.2.g.e.1681.3		4	116.99	even	4
4205.2.a.g.1.2		4	29.28	even	2	inner
4205.2.a.g.1.3		4	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:	\( N \)	\(=\)	\( 4205 = 5 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4205.a (trivial)

\(f(q)\)	\(=\)	\(q+0.517638 q^{2} -1.41421 q^{3} -1.73205 q^{4} +1.00000 q^{5} -0.732051 q^{6} +0.732051 q^{7} -1.93185 q^{8} -1.00000 q^{9} +0.517638 q^{10} -5.27792 q^{11} +2.44949 q^{12} +1.46410 q^{13} +0.378937 q^{14} -1.41421 q^{15} +2.46410 q^{16} +6.31319 q^{17} -0.517638 q^{18} +4.24264 q^{19} -1.73205 q^{20} -1.03528 q^{21} -2.73205 q^{22} -8.19615 q^{23} +2.73205 q^{24} +1.00000 q^{25} +0.757875 q^{26} +5.65685 q^{27} -1.26795 q^{28} -0.732051 q^{30} +4.24264 q^{31} +5.13922 q^{32} +7.46410 q^{33} +3.26795 q^{34} +0.732051 q^{35} +1.73205 q^{36} +4.24264 q^{37} +2.19615 q^{38} -2.07055 q^{39} -1.93185 q^{40} -8.76268 q^{41} -0.535898 q^{42} +4.24264 q^{43} +9.14162 q^{44} -1.00000 q^{45} -4.24264 q^{46} +8.38375 q^{47} -3.48477 q^{48} -6.46410 q^{49} +0.517638 q^{50} -8.92820 q^{51} -2.53590 q^{52} +2.92820 q^{54} -5.27792 q^{55} -1.41421 q^{56} -6.00000 q^{57} +6.00000 q^{59} +2.44949 q^{60} -3.10583 q^{61} +2.19615 q^{62} -0.732051 q^{63} -2.26795 q^{64} +1.46410 q^{65} +3.86370 q^{66} -11.1244 q^{67} -10.9348 q^{68} +11.5911 q^{69} +0.378937 q^{70} -6.00000 q^{71} +1.93185 q^{72} -1.13681 q^{73} +2.19615 q^{74} -1.41421 q^{75} -7.34847 q^{76} -3.86370 q^{77} -1.07180 q^{78} -15.8338 q^{79} +2.46410 q^{80} -5.00000 q^{81} -4.53590 q^{82} +2.19615 q^{83} +1.79315 q^{84} +6.31319 q^{85} +2.19615 q^{86} +10.1962 q^{88} +2.07055 q^{89} -0.517638 q^{90} +1.07180 q^{91} +14.1962 q^{92} -6.00000 q^{93} +4.33975 q^{94} +4.24264 q^{95} -7.26795 q^{96} -7.34847 q^{97} -3.34607 q^{98} +5.27792 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{9} - 8 q^{13} - 4 q^{16} - 4 q^{22} - 12 q^{23} + 4 q^{24} + 4 q^{25} - 12 q^{28} + 4 q^{30} + 16 q^{33} + 20 q^{34} - 4 q^{35} - 12 q^{38} - 16 q^{42} - 4 q^{45}+ \cdots - 36 q^{96}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^9 - 8 * q^13 - 4 * q^16 - 4 * q^22 - 12 * q^23 + 4 * q^24 + 4 * q^25 - 12 * q^28 + 4 * q^30 + 16 * q^33 + 20 * q^34 - 4 * q^35 - 12 * q^38 - 16 * q^42 - 4 * q^45 - 12 * q^49 - 8 * q^51 - 24 * q^52 - 16 * q^54 - 24 * q^57 + 24 * q^59 - 12 * q^62 + 4 * q^63 - 16 * q^64 - 8 * q^65 + 4 * q^67 - 24 * q^71 - 12 * q^74 - 32 * q^78 - 4 * q^80 - 20 * q^81 - 32 * q^82 - 12 * q^83 - 12 * q^86 + 20 * q^88 + 32 * q^91 + 36 * q^92 - 24 * q^93 + 52 * q^94 - 36 * q^96

Introduction

L-functions

Modular forms

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Database

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