LMFDB - Embedded newform 4005.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.679643$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.858442 q^{2} -1.26308 q^{4} +1.00000 q^{5} -2.67964 q^{7} -2.80116 q^{8} +O(q^{10})$	\(q+0.858442 q^{2} -1.26308 q^{4} +1.00000 q^{5} -2.67964 q^{7} -2.80116 q^{8} +0.858442 q^{10} -1.71688 q^{11} +4.06424 q^{13} -2.30032 q^{14} +0.121519 q^{16} +6.80116 q^{17} -0.640714 q^{19} -1.26308 q^{20} -1.47385 q^{22} -1.77585 q^{23} +1.00000 q^{25} +3.48891 q^{26} +3.38460 q^{28} -3.02531 q^{29} -5.88544 q^{31} +5.70664 q^{32} +5.83840 q^{34} -2.67964 q^{35} -2.08259 q^{37} -0.550015 q^{38} -2.80116 q^{40} -2.26308 q^{41} -3.10814 q^{43} +2.16856 q^{44} -1.52447 q^{46} -2.12963 q^{47} +0.180488 q^{49} +0.858442 q^{50} -5.13345 q^{52} +12.2581 q^{53} -1.71688 q^{55} +7.50612 q^{56} -2.59705 q^{58} +7.36456 q^{59} -13.2953 q^{61} -5.05231 q^{62} +4.65578 q^{64} +4.06424 q^{65} -2.77416 q^{67} -8.59039 q^{68} -2.30032 q^{70} -5.60233 q^{71} -2.50726 q^{73} -1.78778 q^{74} +0.809271 q^{76} +4.60064 q^{77} -15.5823 q^{79} +0.121519 q^{80} -1.94272 q^{82} +3.41825 q^{83} +6.80116 q^{85} -2.66816 q^{86} +4.80927 q^{88} -1.00000 q^{89} -10.8907 q^{91} +2.24304 q^{92} -1.82816 q^{94} -0.640714 q^{95} -15.9043 q^{97} +0.154938 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8	$ 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8} - 5 q^{13} + q^{14} - 10 q^{16} + 13 q^{17} - 10 q^{19} + 2 q^{20} - 20 q^{22} - 3 q^{23} + 4 q^{25} + 3 q^{26} - 4 q^{28} - 2 q^{29} - 2 q^{31} + q^{32} + 6 q^{34} - 7 q^{35} - 9 q^{37} - 2 q^{38} + 3 q^{40} - 2 q^{41} - 19 q^{43} - 6 q^{44} - 5 q^{47} - 7 q^{49} - 17 q^{52} + 3 q^{53} + 2 q^{56} - 6 q^{58} - 2 q^{59} - 4 q^{61} + 8 q^{62} + q^{64} - 5 q^{65} - 15 q^{67} + q^{68} + q^{70} + 6 q^{71} - 21 q^{73} - 10 q^{74} - 4 q^{76} - 2 q^{77} - 20 q^{79} - 10 q^{80} + 3 q^{82} + 9 q^{83} + 13 q^{85} - 16 q^{86} + 12 q^{88} - 4 q^{89} + 2 q^{91} - 12 q^{92} + 25 q^{94} - 10 q^{95} - 17 q^{97} - 13 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8 - 5 * q^13 + q^14 - 10 * q^16 + 13 * q^17 - 10 * q^19 + 2 * q^20 - 20 * q^22 - 3 * q^23 + 4 * q^25 + 3 * q^26 - 4 * q^28 - 2 * q^29 - 2 * q^31 + q^32 + 6 * q^34 - 7 * q^35 - 9 * q^37 - 2 * q^38 + 3 * q^40 - 2 * q^41 - 19 * q^43 - 6 * q^44 - 5 * q^47 - 7 * q^49 - 17 * q^52 + 3 * q^53 + 2 * q^56 - 6 * q^58 - 2 * q^59 - 4 * q^61 + 8 * q^62 + q^64 - 5 * q^65 - 15 * q^67 + q^68 + q^70 + 6 * q^71 - 21 * q^73 - 10 * q^74 - 4 * q^76 - 2 * q^77 - 20 * q^79 - 10 * q^80 + 3 * q^82 + 9 * q^83 + 13 * q^85 - 16 * q^86 + 12 * q^88 - 4 * q^89 + 2 * q^91 - 12 * q^92 + 25 * q^94 - 10 * q^95 - 17 * q^97 - 13 * q^98

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 $

$2$	0.858442	0.607010	0.303505	−	0.952830i	$-0.401843\pi$
$2$	0.858442	0.607010	0.303505	+	0.952830i	$0.401843\pi$
$3$	0	0
$3$	0	0
$4$	−1.26308	−0.631539
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.67964	−1.01281	−0.506405	−	0.862296i	$-0.669026\pi$
$7$	−2.67964	−1.01281	−0.506405	+	0.862296i	$0.669026\pi$
$8$	−2.80116	−0.990361
$9$	0	0
$10$	0.858442	0.271463
$11$	−1.71688	−0.517660	−0.258830	−	0.965923i	$-0.583337\pi$
$11$	−1.71688	−0.517660	−0.258830	+	0.965923i	$0.583337\pi$
$12$	0	0
$13$	4.06424	1.12722	0.563609	−	0.826042i	$-0.309413\pi$
$13$	4.06424	1.12722	0.563609	+	0.826042i	$0.309413\pi$
$14$	−2.30032	−0.614786
$15$	0	0
$16$	0.121519	0.0303798
$17$	6.80116	1.64952	0.824762	−	0.565480i	$-0.191309\pi$
$17$	6.80116	1.64952	0.824762	+	0.565480i	$0.191309\pi$
$18$	0	0
$19$	−0.640714	−0.146990	−0.0734949	−	0.997296i	$-0.523415\pi$
$19$	−0.640714	−0.146990	−0.0734949	+	0.997296i	$0.523415\pi$
$20$	−1.26308	−0.282433
$21$	0	0
$22$	−1.47385	−0.314225
$23$	−1.77585	−0.370291	−0.185145	−	0.982711i	$-0.559276\pi$
$23$	−1.77585	−0.370291	−0.185145	+	0.982711i	$0.559276\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.48891	0.684232
$27$	0	0
$28$	3.38460	0.639629
$29$	−3.02531	−0.561786	−0.280893	−	0.959739i	$-0.590631\pi$
$29$	−3.02531	−0.561786	−0.280893	+	0.959739i	$0.590631\pi$
$30$	0	0
$31$	−5.88544	−1.05706	−0.528528	−	0.848916i	$-0.677256\pi$
$31$	−5.88544	−1.05706	−0.528528	+	0.848916i	$0.677256\pi$
$32$	5.70664	1.00880
$33$	0	0
$34$	5.83840	1.00128
$35$	−2.67964	−0.452942
$36$	0	0
$37$	−2.08259	−0.342376	−0.171188	−	0.985238i	$-0.554760\pi$
$37$	−2.08259	−0.342376	−0.171188	+	0.985238i	$0.554760\pi$
$38$	−0.550015	−0.0892243
$39$	0	0
$40$	−2.80116	−0.442903
$41$	−2.26308	−0.353433	−0.176717	−	0.984262i	$-0.556548\pi$
$41$	−2.26308	−0.353433	−0.176717	+	0.984262i	$0.556548\pi$
$42$	0	0
$43$	−3.10814	−0.473987	−0.236993	−	0.971511i	$-0.576162\pi$
$43$	−3.10814	−0.473987	−0.236993	+	0.971511i	$0.576162\pi$
$44$	2.16856	0.326922
$45$	0	0
$46$	−1.52447	−0.224770
$47$	−2.12963	−0.310638	−0.155319	−	0.987864i	$-0.549641\pi$
$47$	−2.12963	−0.310638	−0.155319	+	0.987864i	$0.549641\pi$
$48$	0	0
$49$	0.180488	0.0257839
$50$	0.858442	0.121402
$51$	0	0
$52$	−5.13345	−0.711881
$53$	12.2581	1.68378	0.841890	−	0.539649i	$-0.181443\pi$
$53$	12.2581	1.68378	0.841890	+	0.539649i	$0.181443\pi$
$54$	0	0
$55$	−1.71688	−0.231505
$56$	7.50612	1.00305
$57$	0	0
$58$	−2.59705	−0.341010
$59$	7.36456	0.958784	0.479392	−	0.877601i	$-0.340857\pi$
$59$	7.36456	0.958784	0.479392	+	0.877601i	$0.340857\pi$
$60$	0	0
$61$	−13.2953	−1.70229	−0.851147	−	0.524927i	$-0.824093\pi$
$61$	−13.2953	−1.70229	−0.851147	+	0.524927i	$0.824093\pi$
$62$	−5.05231	−0.641644
$63$	0	0
$64$	4.65578	0.581973
$65$	4.06424	0.504107
$66$	0	0
$67$	−2.77416	−0.338918	−0.169459	−	0.985537i	$-0.554202\pi$
$67$	−2.77416	−0.338918	−0.169459	+	0.985537i	$0.554202\pi$
$68$	−8.59039	−1.04174
$69$	0	0
$70$	−2.30032	−0.274941
$71$	−5.60233	−0.664874	−0.332437	−	0.943126i	$-0.607871\pi$
$71$	−5.60233	−0.664874	−0.332437	+	0.943126i	$0.607871\pi$
$72$	0	0
$73$	−2.50726	−0.293453	−0.146727	−	0.989177i	$-0.546874\pi$
$73$	−2.50726	−0.293453	−0.146727	+	0.989177i	$0.546874\pi$
$74$	−1.78778	−0.207825
$75$	0	0
$76$	0.809271	0.0928297
$77$	4.60064	0.524291
$78$	0	0
$79$	−15.5823	−1.75314	−0.876572	−	0.481271i	$-0.840175\pi$
$79$	−15.5823	−1.75314	−0.876572	+	0.481271i	$0.840175\pi$
$80$	0.121519	0.0135863
$81$	0	0
$82$	−1.94272	−0.214538
$83$	3.41825	0.375202	0.187601	−	0.982245i	$-0.439929\pi$
$83$	3.41825	0.375202	0.187601	+	0.982245i	$0.439929\pi$
$84$	0	0
$85$	6.80116	0.737690
$86$	−2.66816	−0.287715
$87$	0	0
$88$	4.80927	0.512670
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−10.8907	−1.14166
$92$	2.24304	0.233853
$93$	0	0
$94$	−1.82816	−0.188560
$95$	−0.640714	−0.0657358
$96$	0	0
$97$	−15.9043	−1.61484	−0.807420	−	0.589977i	$-0.799137\pi$
$97$	−15.9043	−1.61484	−0.807420	+	0.589977i	$0.799137\pi$
$98$	0.154938	0.0156511
$99$	0	0
$100$	−1.26308	−0.126308
$101$	−16.5503	−1.64682	−0.823409	−	0.567448i	$-0.807931\pi$
$101$	−16.5503	−1.64682	−0.823409	+	0.567448i	$0.807931\pi$
$102$	0	0
$103$	−3.62878	−0.357555	−0.178777	−	0.983890i	$-0.557214\pi$
$103$	−3.62878	−0.357555	−0.178777	+	0.983890i	$0.557214\pi$
$104$	−11.3846	−1.11635
$105$	0	0
$106$	10.5229	1.02207
$107$	−8.61212	−0.832565	−0.416283	−	0.909235i	$-0.636667\pi$
$107$	−8.61212	−0.832565	−0.416283	+	0.909235i	$0.636667\pi$
$108$	0	0
$109$	3.21222	0.307675	0.153837	−	0.988096i	$-0.450837\pi$
$109$	3.21222	0.307675	0.153837	+	0.988096i	$0.450837\pi$
$110$	−1.47385	−0.140526
$111$	0	0
$112$	−0.325629	−0.0307690
$113$	2.17353	0.204468	0.102234	−	0.994760i	$-0.467401\pi$
$113$	2.17353	0.204468	0.102234	+	0.994760i	$0.467401\pi$
$114$	0	0
$115$	−1.77585	−0.165599
$116$	3.82120	0.354790
$117$	0	0
$118$	6.32205	0.581991
$119$	−18.2247	−1.67065
$120$	0	0
$121$	−8.05231	−0.732028
$122$	−11.4133	−1.03331
$123$	0	0
$124$	7.43377	0.667572
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.82671	−0.162094	−0.0810472	−	0.996710i	$-0.525826\pi$
$127$	−1.82671	−0.162094	−0.0810472	+	0.996710i	$0.525826\pi$
$128$	−7.41657	−0.655538
$129$	0	0
$130$	3.48891	0.305998
$131$	−5.17025	−0.451727	−0.225863	−	0.974159i	$-0.572520\pi$
$131$	−5.17025	−0.451727	−0.225863	+	0.974159i	$0.572520\pi$
$132$	0	0
$133$	1.71688	0.148873
$134$	−2.38146	−0.205727
$135$	0	0
$136$	−19.0512	−1.63362
$137$	1.54833	0.132282	0.0661412	−	0.997810i	$-0.478931\pi$
$137$	1.54833	0.132282	0.0661412	+	0.997810i	$0.478931\pi$
$138$	0	0
$139$	−2.14707	−0.182112	−0.0910560	−	0.995846i	$-0.529024\pi$
$139$	−2.14707	−0.182112	−0.0910560	+	0.995846i	$0.529024\pi$
$140$	3.38460	0.286051
$141$	0	0
$142$	−4.80927	−0.403585
$143$	−6.97783	−0.583515
$144$	0	0
$145$	−3.02531	−0.251238
$146$	−2.15234	−0.178129
$147$	0	0
$148$	2.63047	0.216223
$149$	15.1457	1.24078	0.620391	−	0.784292i	$-0.286974\pi$
$149$	15.1457	1.24078	0.620391	+	0.784292i	$0.286974\pi$
$150$	0	0
$151$	0.644091	0.0524154	0.0262077	−	0.999657i	$-0.491657\pi$
$151$	0.644091	0.0524154	0.0262077	+	0.999657i	$0.491657\pi$
$152$	1.79474	0.145573
$153$	0	0
$154$	3.94938	0.318250
$155$	−5.88544	−0.472730
$156$	0	0
$157$	−8.13345	−0.649120	−0.324560	−	0.945865i	$-0.605216\pi$
$157$	−8.13345	−0.649120	−0.324560	+	0.945865i	$0.605216\pi$
$158$	−13.3765	−1.06418
$159$	0	0
$160$	5.70664	0.451150
$161$	4.75865	0.375034
$162$	0	0
$163$	−15.3090	−1.19909	−0.599545	−	0.800341i	$-0.704652\pi$
$163$	−15.3090	−1.19909	−0.599545	+	0.800341i	$0.704652\pi$
$164$	2.85844	0.223207
$165$	0	0
$166$	2.93437	0.227752
$167$	−12.8735	−0.996182	−0.498091	−	0.867125i	$-0.665965\pi$
$167$	−12.8735	−0.996182	−0.498091	+	0.867125i	$0.665965\pi$
$168$	0	0
$169$	3.51805	0.270619
$170$	5.83840	0.447785
$171$	0	0
$172$	3.92582	0.299341
$173$	7.80836	0.593659	0.296829	−	0.954931i	$-0.404071\pi$
$173$	7.80836	0.593659	0.296829	+	0.954931i	$0.404071\pi$
$174$	0	0
$175$	−2.67964	−0.202562
$176$	−0.208635	−0.0157264
$177$	0	0
$178$	−0.858442	−0.0643429
$179$	−12.2430	−0.915088	−0.457544	−	0.889187i	$-0.651271\pi$
$179$	−12.2430	−0.915088	−0.457544	+	0.889187i	$0.651271\pi$
$180$	0	0
$181$	24.1285	1.79346	0.896728	−	0.442582i	$-0.145938\pi$
$181$	24.1285	1.79346	0.896728	+	0.442582i	$0.145938\pi$
$182$	−9.34904	−0.692997
$183$	0	0
$184$	4.97445	0.366721
$185$	−2.08259	−0.153115
$186$	0	0
$187$	−11.6768	−0.853893
$188$	2.68988	0.196180
$189$	0	0
$190$	−0.550015	−0.0399023
$191$	6.85800	0.496227	0.248114	−	0.968731i	$-0.420189\pi$
$191$	6.85800	0.496227	0.248114	+	0.968731i	$0.420189\pi$
$192$	0	0
$193$	−16.9261	−1.21836	−0.609182	−	0.793030i	$-0.708502\pi$
$193$	−16.9261	−1.21836	−0.609182	+	0.793030i	$0.708502\pi$
$194$	−13.6529	−0.980224
$195$	0	0
$196$	−0.227970	−0.0162836
$197$	−3.42612	−0.244101	−0.122051	−	0.992524i	$-0.538947\pi$
$197$	−3.42612	−0.244101	−0.122051	+	0.992524i	$0.538947\pi$
$198$	0	0
$199$	18.5207	1.31290	0.656450	−	0.754370i	$-0.272057\pi$
$199$	18.5207	1.31290	0.656450	+	0.754370i	$0.272057\pi$
$200$	−2.80116	−0.198072
$201$	0	0
$202$	−14.2075	−0.999635
$203$	8.10675	0.568982
$204$	0	0
$205$	−2.26308	−0.158060
$206$	−3.11510	−0.217039
$207$	0	0
$208$	0.493884	0.0342447
$209$	1.10003	0.0760907
$210$	0	0
$211$	−8.16518	−0.562114	−0.281057	−	0.959691i	$-0.590685\pi$
$211$	−8.16518	−0.562114	−0.281057	+	0.959691i	$0.590685\pi$
$212$	−15.4829	−1.06337
$213$	0	0
$214$	−7.39301	−0.505376
$215$	−3.10814	−0.211973
$216$	0	0
$217$	15.7709	1.07060
$218$	2.75750	0.186762
$219$	0	0
$220$	2.16856	0.146204
$221$	27.6416	1.85937
$222$	0	0
$223$	−2.69630	−0.180558	−0.0902789	−	0.995917i	$-0.528776\pi$
$223$	−2.69630	−0.180558	−0.0902789	+	0.995917i	$0.528776\pi$
$224$	−15.2918	−1.02172
$225$	0	0
$226$	1.86585	0.124114
$227$	29.5912	1.96404	0.982020	−	0.188778i	$-0.0604528\pi$
$227$	29.5912	1.96404	0.982020	+	0.188778i	$0.0604528\pi$
$228$	0	0
$229$	18.8471	1.24545	0.622724	−	0.782441i	$-0.286026\pi$
$229$	18.8471	1.24545	0.622724	+	0.782441i	$0.286026\pi$
$230$	−1.52447	−0.100520
$231$	0	0
$232$	8.47439	0.556371
$233$	−28.1666	−1.84525	−0.922626	−	0.385695i	$-0.873962\pi$
$233$	−28.1666	−1.84525	−0.922626	+	0.385695i	$0.873962\pi$
$234$	0	0
$235$	−2.12963	−0.138922
$236$	−9.30201	−0.605509
$237$	0	0
$238$	−15.6448	−1.01410
$239$	−14.8432	−0.960129	−0.480064	−	0.877233i	$-0.659387\pi$
$239$	−14.8432	−0.960129	−0.480064	+	0.877233i	$0.659387\pi$
$240$	0	0
$241$	1.52447	0.0981995	0.0490997	−	0.998794i	$-0.484365\pi$
$241$	1.52447	0.0981995	0.0490997	+	0.998794i	$0.484365\pi$
$242$	−6.91244	−0.444349
$243$	0	0
$244$	16.7931	1.07507
$245$	0.180488	0.0115309
$246$	0	0
$247$	−2.60401	−0.165689
$248$	16.4861	1.04687
$249$	0	0
$250$	0.858442	0.0542926
$251$	−24.6890	−1.55836	−0.779179	−	0.626801i	$-0.784364\pi$
$251$	−24.6890	−1.55836	−0.779179	+	0.626801i	$0.784364\pi$
$252$	0	0
$253$	3.04893	0.191685
$254$	−1.56813	−0.0983930
$255$	0	0
$256$	−15.6783	−0.979891
$257$	3.62166	0.225913	0.112956	−	0.993600i	$-0.463968\pi$
$257$	3.62166	0.225913	0.112956	+	0.993600i	$0.463968\pi$
$258$	0	0
$259$	5.58060	0.346761
$260$	−5.13345	−0.318363
$261$	0	0
$262$	−4.43836	−0.274203
$263$	21.2634	1.31116	0.655578	−	0.755127i	$-0.272425\pi$
$263$	21.2634	1.31116	0.655578	+	0.755127i	$0.272425\pi$
$264$	0	0
$265$	12.2581	0.753010
$266$	1.47385	0.0903672
$267$	0	0
$268$	3.50398	0.214040
$269$	8.70526	0.530769	0.265384	−	0.964143i	$-0.414501\pi$
$269$	8.70526	0.530769	0.265384	+	0.964143i	$0.414501\pi$
$270$	0	0
$271$	24.3832	1.48117	0.740587	−	0.671960i	$-0.234547\pi$
$271$	24.3832	1.48117	0.740587	+	0.671960i	$0.234547\pi$
$272$	0.826473	0.0501123
$273$	0	0
$274$	1.32915	0.0802968
$275$	−1.71688	−0.103532
$276$	0	0
$277$	−18.9695	−1.13977	−0.569883	−	0.821726i	$-0.693011\pi$
$277$	−18.9695	−1.13977	−0.569883	+	0.821726i	$0.693011\pi$
$278$	−1.84313	−0.110544
$279$	0	0
$280$	7.50612	0.448576
$281$	23.4558	1.39926	0.699628	−	0.714508i	$-0.253349\pi$
$281$	23.4558	1.39926	0.699628	+	0.714508i	$0.253349\pi$
$282$	0	0
$283$	0.563937	0.0335225	0.0167613	−	0.999860i	$-0.494664\pi$
$283$	0.563937	0.0335225	0.0167613	+	0.999860i	$0.494664\pi$
$284$	7.07617	0.419893
$285$	0	0
$286$	−5.99006	−0.354200
$287$	6.06424	0.357961
$288$	0	0
$289$	29.2558	1.72093
$290$	−2.59705	−0.152504
$291$	0	0
$292$	3.16687	0.185327
$293$	28.1397	1.64394	0.821970	−	0.569531i	$-0.192875\pi$
$293$	28.1397	1.64394	0.821970	+	0.569531i	$0.192875\pi$
$294$	0	0
$295$	7.36456	0.428781
$296$	5.83367	0.339075
$297$	0	0
$298$	13.0017	0.753168
$299$	−7.21749	−0.417398
$300$	0	0
$301$	8.32870	0.480059
$302$	0.552915	0.0318167
$303$	0	0
$304$	−0.0778591	−0.00446553
$305$	−13.2953	−0.761289
$306$	0	0
$307$	−14.3254	−0.817593	−0.408797	−	0.912626i	$-0.634051\pi$
$307$	−14.3254	−0.817593	−0.408797	+	0.912626i	$0.634051\pi$
$308$	−5.81096	−0.331110
$309$	0	0
$310$	−5.05231	−0.286952
$311$	−29.3493	−1.66425	−0.832124	−	0.554589i	$-0.812875\pi$
$311$	−29.3493	−1.66425	−0.832124	+	0.554589i	$0.812875\pi$
$312$	0	0
$313$	9.57036	0.540949	0.270474	−	0.962727i	$-0.412820\pi$
$313$	9.57036	0.540949	0.270474	+	0.962727i	$0.412820\pi$
$314$	−6.98209	−0.394022
$315$	0	0
$316$	19.6816	1.10718
$317$	−3.13200	−0.175911	−0.0879553	−	0.996124i	$-0.528033\pi$
$317$	−3.13200	−0.175911	−0.0879553	+	0.996124i	$0.528033\pi$
$318$	0	0
$319$	5.19411	0.290814
$320$	4.65578	0.260266
$321$	0	0
$322$	4.08502	0.227650
$323$	−4.35760	−0.242463
$324$	0	0
$325$	4.06424	0.225443
$326$	−13.1419	−0.727860
$327$	0	0
$328$	6.33925	0.350026
$329$	5.70664	0.314617
$330$	0	0
$331$	−18.1927	−0.999962	−0.499981	−	0.866036i	$-0.666660\pi$
$331$	−18.1927	−0.999962	−0.499981	+	0.866036i	$0.666660\pi$
$332$	−4.31752	−0.236955
$333$	0	0
$334$	−11.0512	−0.604693
$335$	−2.77416	−0.151569
$336$	0	0
$337$	−12.0551	−0.656686	−0.328343	−	0.944559i	$-0.606490\pi$
$337$	−12.0551	−0.656686	−0.328343	+	0.944559i	$0.606490\pi$
$338$	3.02004	0.164268
$339$	0	0
$340$	−8.59039	−0.465880
$341$	10.1046	0.547196
$342$	0	0
$343$	18.2739	0.986696
$344$	8.70640	0.469418
$345$	0	0
$346$	6.70303	0.360357
$347$	−17.5006	−0.939482	−0.469741	−	0.882804i	$-0.655653\pi$
$347$	−17.5006	−0.939482	−0.469741	+	0.882804i	$0.655653\pi$
$348$	0	0
$349$	7.98378	0.427362	0.213681	−	0.976903i	$-0.431455\pi$
$349$	7.98378	0.427362	0.213681	+	0.976903i	$0.431455\pi$
$350$	−2.30032	−0.122957
$351$	0	0
$352$	−9.79764	−0.522216
$353$	−12.5723	−0.669155	−0.334578	−	0.942368i	$-0.608594\pi$
$353$	−12.5723	−0.669155	−0.334578	+	0.942368i	$0.608594\pi$
$354$	0	0
$355$	−5.60233	−0.297341
$356$	1.26308	0.0669430
$357$	0	0
$358$	−10.5099	−0.555467
$359$	−8.08665	−0.426797	−0.213399	−	0.976965i	$-0.568453\pi$
$359$	−8.08665	−0.426797	−0.213399	+	0.976965i	$0.568453\pi$
$360$	0	0
$361$	−18.5895	−0.978394
$362$	20.7129	1.08865
$363$	0	0
$364$	13.7558	0.721001
$365$	−2.50726	−0.131236
$366$	0	0
$367$	12.7129	0.663608	0.331804	−	0.943348i	$-0.392343\pi$
$367$	12.7129	0.663608	0.331804	+	0.943348i	$0.392343\pi$
$368$	−0.215800	−0.0112494
$369$	0	0
$370$	−1.78778	−0.0929424
$371$	−32.8474	−1.70535
$372$	0	0
$373$	−13.6474	−0.706634	−0.353317	−	0.935504i	$-0.614946\pi$
$373$	−13.6474	−0.706634	−0.353317	+	0.935504i	$0.614946\pi$
$374$	−10.0239	−0.518321
$375$	0	0
$376$	5.96543	0.307644
$377$	−12.2956	−0.633255
$378$	0	0
$379$	7.08901	0.364138	0.182069	−	0.983286i	$-0.441721\pi$
$379$	7.08901	0.364138	0.182069	+	0.983286i	$0.441721\pi$
$380$	0.809271	0.0415147
$381$	0	0
$382$	5.88719	0.301215
$383$	−25.3937	−1.29756	−0.648778	−	0.760977i	$-0.724720\pi$
$383$	−25.3937	−1.29756	−0.648778	+	0.760977i	$0.724720\pi$
$384$	0	0
$385$	4.60064	0.234470
$386$	−14.5300	−0.739560
$387$	0	0
$388$	20.0884	1.01983
$389$	8.77799	0.445062	0.222531	−	0.974926i	$-0.428568\pi$
$389$	8.77799	0.445062	0.222531	+	0.974926i	$0.428568\pi$
$390$	0	0
$391$	−12.0779	−0.610804
$392$	−0.505575	−0.0255354
$393$	0	0
$394$	−2.94113	−0.148172
$395$	−15.5823	−0.784030
$396$	0	0
$397$	−11.2197	−0.563102	−0.281551	−	0.959546i	$-0.590849\pi$
$397$	−11.2197	−0.563102	−0.281551	+	0.959546i	$0.590849\pi$
$398$	15.8990	0.796943
$399$	0	0
$400$	0.121519	0.00607597
$401$	−22.3081	−1.11401	−0.557006	−	0.830509i	$-0.688050\pi$
$401$	−22.3081	−1.11401	−0.557006	+	0.830509i	$0.688050\pi$
$402$	0	0
$403$	−23.9198	−1.19153
$404$	20.9043	1.04003
$405$	0	0
$406$	6.95918	0.345378
$407$	3.57556	0.177234
$408$	0	0
$409$	30.6150	1.51381	0.756907	−	0.653522i	$-0.226709\pi$
$409$	30.6150	1.51381	0.756907	+	0.653522i	$0.226709\pi$
$410$	−1.94272	−0.0959441
$411$	0	0
$412$	4.58343	0.225810
$413$	−19.7344	−0.971066
$414$	0	0
$415$	3.41825	0.167796
$416$	23.1932	1.13714
$417$	0	0
$418$	0.944313	0.0461878
$419$	15.4207	0.753350	0.376675	−	0.926345i	$-0.377067\pi$
$419$	15.4207	0.753350	0.376675	+	0.926345i	$0.377067\pi$
$420$	0	0
$421$	−14.1629	−0.690257	−0.345128	−	0.938555i	$-0.612165\pi$
$421$	−14.1629	−0.690257	−0.345128	+	0.938555i	$0.612165\pi$
$422$	−7.00933	−0.341209
$423$	0	0
$424$	−34.3370	−1.66755
$425$	6.80116	0.329905
$426$	0	0
$427$	35.6268	1.72410
$428$	10.8778	0.525797
$429$	0	0
$430$	−2.66816	−0.128670
$431$	3.53028	0.170048	0.0850238	−	0.996379i	$-0.472903\pi$
$431$	3.53028	0.170048	0.0850238	+	0.996379i	$0.472903\pi$
$432$	0	0
$433$	2.50396	0.120333	0.0601664	−	0.998188i	$-0.480837\pi$
$433$	2.50396	0.120333	0.0601664	+	0.998188i	$0.480837\pi$
$434$	13.5384	0.649863
$435$	0	0
$436$	−4.05728	−0.194308
$437$	1.13781	0.0544290
$438$	0	0
$439$	11.4878	0.548281	0.274141	−	0.961690i	$-0.411607\pi$
$439$	11.4878	0.548281	0.274141	+	0.961690i	$0.411607\pi$
$440$	4.80927	0.229273
$441$	0	0
$442$	23.7287	1.12866
$443$	25.6165	1.21708	0.608539	−	0.793524i	$-0.291756\pi$
$443$	25.6165	1.21708	0.608539	+	0.793524i	$0.291756\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−2.31462	−0.109600
$447$	0	0
$448$	−12.4758	−0.589428
$449$	−33.0696	−1.56065	−0.780326	−	0.625373i	$-0.784947\pi$
$449$	−33.0696	−1.56065	−0.780326	+	0.625373i	$0.784947\pi$
$450$	0	0
$451$	3.88544	0.182958
$452$	−2.74533	−0.129130
$453$	0	0
$454$	25.4024	1.19219
$455$	−10.8907	−0.510564
$456$	0	0
$457$	13.6205	0.637141	0.318570	−	0.947899i	$-0.396797\pi$
$457$	13.6205	0.637141	0.318570	+	0.947899i	$0.396797\pi$
$458$	16.1791	0.756000
$459$	0	0
$460$	2.24304	0.104582
$461$	−36.6546	−1.70718	−0.853588	−	0.520948i	$-0.825578\pi$
$461$	−36.6546	−1.70718	−0.853588	+	0.520948i	$0.825578\pi$
$462$	0	0
$463$	31.5967	1.46842	0.734210	−	0.678922i	$-0.237553\pi$
$463$	31.5967	1.46842	0.734210	+	0.678922i	$0.237553\pi$
$464$	−0.367634	−0.0170670
$465$	0	0
$466$	−24.1794	−1.12009
$467$	2.48151	0.114831	0.0574153	−	0.998350i	$-0.481714\pi$
$467$	2.48151	0.114831	0.0574153	+	0.998350i	$0.481714\pi$
$468$	0	0
$469$	7.43377	0.343260
$470$	−1.82816	−0.0843268
$471$	0	0
$472$	−20.6293	−0.949542
$473$	5.33631	0.245364
$474$	0	0
$475$	−0.640714	−0.0293980
$476$	23.0192	1.05508
$477$	0	0
$478$	−12.7421	−0.582808
$479$	−10.7152	−0.489590	−0.244795	−	0.969575i	$-0.578721\pi$
$479$	−10.7152	−0.489590	−0.244795	+	0.969575i	$0.578721\pi$
$480$	0	0
$481$	−8.46414	−0.385932
$482$	1.30867	0.0596081
$483$	0	0
$484$	10.1707	0.462304
$485$	−15.9043	−0.722179
$486$	0	0
$487$	−42.8799	−1.94307	−0.971536	−	0.236892i	$-0.923871\pi$
$487$	−42.8799	−1.94307	−0.971536	+	0.236892i	$0.923871\pi$
$488$	37.2424	1.68589
$489$	0	0
$490$	0.154938	0.00699939
$491$	−0.574101	−0.0259088	−0.0129544	−	0.999916i	$-0.504124\pi$
$491$	−0.574101	−0.0259088	−0.0129544	+	0.999916i	$0.504124\pi$
$492$	0	0
$493$	−20.5756	−0.926680
$494$	−2.23539	−0.100575
$495$	0	0
$496$	−0.715195	−0.0321132
$497$	15.0122	0.673391
$498$	0	0
$499$	−0.624497	−0.0279563	−0.0139782	−	0.999902i	$-0.504450\pi$
$499$	−0.624497	−0.0279563	−0.0139782	+	0.999902i	$0.504450\pi$
$500$	−1.26308	−0.0564865
$501$	0	0
$502$	−21.1941	−0.945939
$503$	7.28770	0.324943	0.162471	−	0.986713i	$-0.448053\pi$
$503$	7.28770	0.324943	0.162471	+	0.986713i	$0.448053\pi$
$504$	0	0
$505$	−16.5503	−0.736479
$506$	2.61733	0.116355
$507$	0	0
$508$	2.30728	0.102369
$509$	31.5600	1.39887	0.699435	−	0.714696i	$-0.253435\pi$
$509$	31.5600	1.39887	0.699435	+	0.714696i	$0.253435\pi$
$510$	0	0
$511$	6.71857	0.297212
$512$	1.37426	0.0607342
$513$	0	0
$514$	3.10899	0.137131
$515$	−3.62878	−0.159903
$516$	0	0
$517$	3.65632	0.160805
$518$	4.79062	0.210488
$519$	0	0
$520$	−11.3846	−0.499248
$521$	24.0151	1.05212	0.526060	−	0.850448i	$-0.323669\pi$
$521$	24.0151	1.05212	0.526060	+	0.850448i	$0.323669\pi$
$522$	0	0
$523$	1.15065	0.0503145	0.0251572	−	0.999684i	$-0.491991\pi$
$523$	1.15065	0.0503145	0.0251572	+	0.999684i	$0.491991\pi$
$524$	6.53042	0.285283
$525$	0	0
$526$	18.2534	0.795885
$527$	−40.0278	−1.74364
$528$	0	0
$529$	−19.8463	−0.862885
$530$	10.5229	0.457084
$531$	0	0
$532$	−2.16856	−0.0940189
$533$	−9.19769	−0.398396
$534$	0	0
$535$	−8.61212	−0.372335
$536$	7.77088	0.335651
$537$	0	0
$538$	7.47296	0.322182
$539$	−0.309876	−0.0133473
$540$	0	0
$541$	41.7825	1.79637	0.898185	−	0.439617i	$-0.144886\pi$
$541$	41.7825	1.79637	0.898185	+	0.439617i	$0.144886\pi$
$542$	20.9316	0.899088
$543$	0	0
$544$	38.8118	1.66404
$545$	3.21222	0.137596
$546$	0	0
$547$	−37.7129	−1.61249	−0.806244	−	0.591583i	$-0.798503\pi$
$547$	−37.7129	−1.61249	−0.806244	+	0.591583i	$0.798503\pi$
$548$	−1.95566	−0.0835415
$549$	0	0
$550$	−1.47385	−0.0628450
$551$	1.93836	0.0825768
$552$	0	0
$553$	41.7550	1.77560
$554$	−16.2842	−0.691849
$555$	0	0
$556$	2.71191	0.115011
$557$	1.92284	0.0814735	0.0407368	−	0.999170i	$-0.487029\pi$
$557$	1.92284	0.0814735	0.0407368	+	0.999170i	$0.487029\pi$
$558$	0	0
$559$	−12.6322	−0.534286
$560$	−0.325629	−0.0137603
$561$	0	0
$562$	20.1354	0.849362
$563$	12.7179	0.535994	0.267997	−	0.963420i	$-0.413638\pi$
$563$	12.7179	0.535994	0.267997	+	0.963420i	$0.413638\pi$
$564$	0	0
$565$	2.17353	0.0914410
$566$	0.484107	0.0203485
$567$	0	0
$568$	15.6930	0.658465
$569$	−0.943023	−0.0395336	−0.0197668	−	0.999805i	$-0.506292\pi$
$569$	−0.943023	−0.0395336	−0.0197668	+	0.999805i	$0.506292\pi$
$570$	0	0
$571$	−33.4897	−1.40150	−0.700751	−	0.713406i	$-0.747152\pi$
$571$	−33.4897	−1.40150	−0.700751	+	0.713406i	$0.747152\pi$
$572$	8.81354	0.368513
$573$	0	0
$574$	5.20580	0.217286
$575$	−1.77585	−0.0740582
$576$	0	0
$577$	18.5824	0.773595	0.386797	−	0.922165i	$-0.373581\pi$
$577$	18.5824	0.773595	0.386797	+	0.922165i	$0.373581\pi$
$578$	25.1144	1.04462
$579$	0	0
$580$	3.82120	0.158667
$581$	−9.15970	−0.380008
$582$	0	0
$583$	−21.0457	−0.871626
$584$	7.02325	0.290624
$585$	0	0
$586$	24.1563	0.997888
$587$	−18.4192	−0.760243	−0.380122	−	0.924936i	$-0.624118\pi$
$587$	−18.4192	−0.760243	−0.380122	+	0.924936i	$0.624118\pi$
$588$	0	0
$589$	3.77088	0.155377
$590$	6.32205	0.260275
$591$	0	0
$592$	−0.253075	−0.0104013
$593$	6.74036	0.276794	0.138397	−	0.990377i	$-0.455805\pi$
$593$	6.74036	0.276794	0.138397	+	0.990377i	$0.455805\pi$
$594$	0	0
$595$	−18.2247	−0.747139
$596$	−19.1302	−0.783602
$597$	0	0
$598$	−6.19580	−0.253365
$599$	25.7740	1.05310	0.526549	−	0.850145i	$-0.323486\pi$
$599$	25.7740	1.05310	0.526549	+	0.850145i	$0.323486\pi$
$600$	0	0
$601$	3.09263	0.126151	0.0630754	−	0.998009i	$-0.479909\pi$
$601$	3.09263	0.126151	0.0630754	+	0.998009i	$0.479909\pi$
$602$	7.14971	0.291400
$603$	0	0
$604$	−0.813537	−0.0331024
$605$	−8.05231	−0.327373
$606$	0	0
$607$	−8.61615	−0.349719	−0.174859	−	0.984593i	$-0.555947\pi$
$607$	−8.61615	−0.349719	−0.174859	+	0.984593i	$0.555947\pi$
$608$	−3.65632	−0.148284
$609$	0	0
$610$	−11.4133	−0.462110
$611$	−8.65532	−0.350157
$612$	0	0
$613$	−3.58053	−0.144616	−0.0723082	−	0.997382i	$-0.523037\pi$
$613$	−3.58053	−0.144616	−0.0723082	+	0.997382i	$0.523037\pi$
$614$	−12.2975	−0.496287
$615$	0	0
$616$	−12.8871	−0.519237
$617$	20.3799	0.820463	0.410231	−	0.911982i	$-0.365448\pi$
$617$	20.3799	0.820463	0.410231	+	0.911982i	$0.365448\pi$
$618$	0	0
$619$	22.7725	0.915306	0.457653	−	0.889131i	$-0.348690\pi$
$619$	22.7725	0.915306	0.457653	+	0.889131i	$0.348690\pi$
$620$	7.43377	0.298547
$621$	0	0
$622$	−25.1947	−1.01022
$623$	2.67964	0.107358
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.21560	0.328361
$627$	0	0
$628$	10.2732	0.409944
$629$	−14.1640	−0.564757
$630$	0	0
$631$	−3.82388	−0.152226	−0.0761130	−	0.997099i	$-0.524251\pi$
$631$	−3.82388	−0.152226	−0.0761130	+	0.997099i	$0.524251\pi$
$632$	43.6485	1.73624
$633$	0	0
$634$	−2.68864	−0.106780
$635$	−1.82671	−0.0724909
$636$	0	0
$637$	0.733545	0.0290641
$638$	4.45884	0.176527
$639$	0	0
$640$	−7.41657	−0.293166
$641$	4.08015	0.161156	0.0805782	−	0.996748i	$-0.474323\pi$
$641$	4.08015	0.161156	0.0805782	+	0.996748i	$0.474323\pi$
$642$	0	0
$643$	25.7841	1.01683	0.508413	−	0.861113i	$-0.330232\pi$
$643$	25.7841	1.01683	0.508413	+	0.861113i	$0.330232\pi$
$644$	−6.01054	−0.236849
$645$	0	0
$646$	−3.74074	−0.147178
$647$	2.25996	0.0888482	0.0444241	−	0.999013i	$-0.485855\pi$
$647$	2.25996	0.0888482	0.0444241	+	0.999013i	$0.485855\pi$
$648$	0	0
$649$	−12.6441	−0.496324
$650$	3.48891	0.136846
$651$	0	0
$652$	19.3364	0.757272
$653$	12.8176	0.501592	0.250796	−	0.968040i	$-0.419308\pi$
$653$	12.8176	0.501592	0.250796	+	0.968040i	$0.419308\pi$
$654$	0	0
$655$	−5.17025	−0.202018
$656$	−0.275008	−0.0107372
$657$	0	0
$658$	4.89882	0.190976
$659$	29.8764	1.16382	0.581909	−	0.813254i	$-0.302306\pi$
$659$	29.8764	1.16382	0.581909	+	0.813254i	$0.302306\pi$
$660$	0	0
$661$	−23.1785	−0.901539	−0.450769	−	0.892640i	$-0.648850\pi$
$661$	−23.1785	−0.901539	−0.450769	+	0.892640i	$0.648850\pi$
$662$	−15.6174	−0.606987
$663$	0	0
$664$	−9.57509	−0.371585
$665$	1.71688	0.0665779
$666$	0	0
$667$	5.37250	0.208024
$668$	16.2602	0.629128
$669$	0	0
$670$	−2.38146	−0.0920038
$671$	22.8266	0.881210
$672$	0	0
$673$	−42.3904	−1.63403	−0.817015	−	0.576616i	$-0.804373\pi$
$673$	−42.3904	−1.63403	−0.817015	+	0.576616i	$0.804373\pi$
$674$	−10.3486	−0.398615
$675$	0	0
$676$	−4.44357	−0.170906
$677$	36.2036	1.39142	0.695708	−	0.718325i	$-0.255091\pi$
$677$	36.2036	1.39142	0.695708	+	0.718325i	$0.255091\pi$
$678$	0	0
$679$	42.6179	1.63553
$680$	−19.0512	−0.730579
$681$	0	0
$682$	8.67423	0.332153
$683$	18.6530	0.713739	0.356869	−	0.934154i	$-0.383844\pi$
$683$	18.6530	0.713739	0.356869	+	0.934154i	$0.383844\pi$
$684$	0	0
$685$	1.54833	0.0591585
$686$	15.6870	0.598934
$687$	0	0
$688$	−0.377699	−0.0143996
$689$	49.8199	1.89799
$690$	0	0
$691$	−42.1371	−1.60297	−0.801486	−	0.598014i	$-0.795957\pi$
$691$	−42.1371	−1.60297	−0.801486	+	0.598014i	$0.795957\pi$
$692$	−9.86257	−0.374918
$693$	0	0
$694$	−15.0233	−0.570275
$695$	−2.14707	−0.0814430
$696$	0	0
$697$	−15.3916	−0.582997
$698$	6.85361	0.259413
$699$	0	0
$700$	3.38460	0.127926
$701$	−7.89261	−0.298100	−0.149050	−	0.988830i	$-0.547621\pi$
$701$	−7.89261	−0.298100	−0.149050	+	0.988830i	$0.547621\pi$
$702$	0	0
$703$	1.33434	0.0503257
$704$	−7.99344	−0.301264
$705$	0	0
$706$	−10.7926	−0.406184
$707$	44.3489	1.66791
$708$	0	0
$709$	−4.12172	−0.154795	−0.0773973	−	0.997000i	$-0.524661\pi$
$709$	−4.12172	−0.154795	−0.0773973	+	0.997000i	$0.524661\pi$
$710$	−4.80927	−0.180489
$711$	0	0
$712$	2.80116	0.104978
$713$	10.4517	0.391418
$714$	0	0
$715$	−6.97783	−0.260956
$716$	15.4639	0.577913
$717$	0	0
$718$	−6.94192	−0.259070
$719$	39.2944	1.46543	0.732717	−	0.680533i	$-0.238252\pi$
$719$	39.2944	1.46543	0.732717	+	0.680533i	$0.238252\pi$
$720$	0	0
$721$	9.72384	0.362135
$722$	−15.9580	−0.593895
$723$	0	0
$724$	−30.4761	−1.13264
$725$	−3.02531	−0.112357
$726$	0	0
$727$	2.61212	0.0968783	0.0484391	−	0.998826i	$-0.484575\pi$
$727$	2.61212	0.0968783	0.0484391	+	0.998826i	$0.484575\pi$
$728$	30.5067	1.13065
$729$	0	0
$730$	−2.15234	−0.0796617
$731$	−21.1390	−0.781853
$732$	0	0
$733$	23.3520	0.862527	0.431263	−	0.902226i	$-0.358068\pi$
$733$	23.3520	0.862527	0.431263	+	0.902226i	$0.358068\pi$
$734$	10.9133	0.402817
$735$	0	0
$736$	−10.1342	−0.373550
$737$	4.76292	0.175444
$738$	0	0
$739$	−7.59468	−0.279375	−0.139687	−	0.990196i	$-0.544610\pi$
$739$	−7.59468	−0.279375	−0.139687	+	0.990196i	$0.544610\pi$
$740$	2.63047	0.0966981
$741$	0	0
$742$	−28.1975	−1.03516
$743$	−14.7861	−0.542450	−0.271225	−	0.962516i	$-0.587429\pi$
$743$	−14.7861	−0.542450	−0.271225	+	0.962516i	$0.587429\pi$
$744$	0	0
$745$	15.1457	0.554895
$746$	−11.7155	−0.428934
$747$	0	0
$748$	14.7487	0.539266
$749$	23.0774	0.843230
$750$	0	0
$751$	28.4680	1.03881	0.519407	−	0.854527i	$-0.326153\pi$
$751$	28.4680	1.03881	0.519407	+	0.854527i	$0.326153\pi$
$752$	−0.258791	−0.00943714
$753$	0	0
$754$	−10.5550	−0.384392
$755$	0.644091	0.0234409
$756$	0	0
$757$	4.58175	0.166526	0.0832632	−	0.996528i	$-0.473466\pi$
$757$	4.58175	0.166526	0.0832632	+	0.996528i	$0.473466\pi$
$758$	6.08550	0.221035
$759$	0	0
$760$	1.79474	0.0651022
$761$	30.1190	1.09181	0.545907	−	0.837846i	$-0.316185\pi$
$761$	30.1190	1.09181	0.545907	+	0.837846i	$0.316185\pi$
$762$	0	0
$763$	−8.60760	−0.311616
$764$	−8.66218	−0.313387
$765$	0	0
$766$	−21.7990	−0.787630
$767$	29.9313	1.08076
$768$	0	0
$769$	−44.0926	−1.59002	−0.795010	−	0.606596i	$-0.792535\pi$
$769$	−44.0926	−1.59002	−0.795010	+	0.606596i	$0.792535\pi$
$770$	3.94938	0.142326
$771$	0	0
$772$	21.3789	0.769444
$773$	−8.50787	−0.306007	−0.153003	−	0.988226i	$-0.548895\pi$
$773$	−8.50787	−0.306007	−0.153003	+	0.988226i	$0.548895\pi$
$774$	0	0
$775$	−5.88544	−0.211411
$776$	44.5506	1.59927
$777$	0	0
$778$	7.53539	0.270157
$779$	1.44998	0.0519511
$780$	0	0
$781$	9.61854	0.344178
$782$	−10.3681	−0.370764
$783$	0	0
$784$	0.0219327	0.000783312 0
$785$	−8.13345	−0.290295
$786$	0	0
$787$	53.8888	1.92093	0.960463	−	0.278407i	$-0.0898063\pi$
$787$	53.8888	1.92093	0.960463	+	0.278407i	$0.0898063\pi$
$788$	4.32746	0.154159
$789$	0	0
$790$	−13.3765	−0.475914
$791$	−5.82428	−0.207087
$792$	0	0
$793$	−54.0355	−1.91886
$794$	−9.63148	−0.341808
$795$	0	0
$796$	−23.3931	−0.829147
$797$	−11.9731	−0.424108	−0.212054	−	0.977258i	$-0.568015\pi$
$797$	−11.9731	−0.424108	−0.212054	+	0.977258i	$0.568015\pi$
$798$	0	0
$799$	−14.4839	−0.512405
$800$	5.70664	0.201760
$801$	0	0
$802$	−19.1502	−0.676216
$803$	4.30468	0.151909
$804$	0	0
$805$	4.75865	0.167720
$806$	−20.5338	−0.723272
$807$	0	0
$808$	46.3601	1.63094
$809$	14.2516	0.501060	0.250530	−	0.968109i	$-0.419395\pi$
$809$	14.2516	0.501060	0.250530	+	0.968109i	$0.419395\pi$
$810$	0	0
$811$	−2.03464	−0.0714460	−0.0357230	−	0.999362i	$-0.511373\pi$
$811$	−2.03464	−0.0714460	−0.0357230	+	0.999362i	$0.511373\pi$
$812$	−10.2395	−0.359334
$813$	0	0
$814$	3.06941	0.107583
$815$	−15.3090	−0.536250
$816$	0	0
$817$	1.99143	0.0696712
$818$	26.2812	0.918901
$819$	0	0
$820$	2.85844	0.0998211
$821$	−31.9682	−1.11570	−0.557848	−	0.829943i	$-0.688373\pi$
$821$	−31.9682	−1.11570	−0.557848	+	0.829943i	$0.688373\pi$
$822$	0	0
$823$	32.8282	1.14432	0.572159	−	0.820143i	$-0.306106\pi$
$823$	32.8282	1.14432	0.572159	+	0.820143i	$0.306106\pi$
$824$	10.1648	0.354108
$825$	0	0
$826$	−16.9408	−0.589447
$827$	−25.1229	−0.873609	−0.436804	−	0.899556i	$-0.643890\pi$
$827$	−25.1229	−0.873609	−0.436804	+	0.899556i	$0.643890\pi$
$828$	0	0
$829$	21.6453	0.751772	0.375886	−	0.926666i	$-0.377338\pi$
$829$	21.6453	0.751772	0.375886	+	0.926666i	$0.377338\pi$
$830$	2.93437	0.101854
$831$	0	0
$832$	18.9222	0.656010
$833$	1.22753	0.0425312
$834$	0	0
$835$	−12.8735	−0.445506
$836$	−1.38942	−0.0480542
$837$	0	0
$838$	13.2378	0.457291
$839$	0.653997	0.0225785	0.0112892	−	0.999936i	$-0.496406\pi$
$839$	0.653997	0.0225785	0.0112892	+	0.999936i	$0.496406\pi$
$840$	0	0
$841$	−19.8475	−0.684396
$842$	−12.1580	−0.418993
$843$	0	0
$844$	10.3133	0.354997
$845$	3.51805	0.121024
$846$	0	0
$847$	21.5773	0.741405
$848$	1.48960	0.0511530
$849$	0	0
$850$	5.83840	0.200256
$851$	3.69837	0.126779
$852$	0	0
$853$	−52.8191	−1.80849	−0.904245	−	0.427014i	$-0.859566\pi$
$853$	−52.8191	−1.80849	−0.904245	+	0.427014i	$0.859566\pi$
$854$	30.5835	1.04655
$855$	0	0
$856$	24.1240	0.824540
$857$	−36.5349	−1.24801	−0.624004	−	0.781421i	$-0.714495\pi$
$857$	−36.5349	−1.24801	−0.624004	+	0.781421i	$0.714495\pi$
$858$	0	0
$859$	41.3493	1.41082	0.705411	−	0.708799i	$-0.250763\pi$
$859$	41.3493	1.41082	0.705411	+	0.708799i	$0.250763\pi$
$860$	3.92582	0.133869
$861$	0	0
$862$	3.03054	0.103221
$863$	−3.49771	−0.119063	−0.0595316	−	0.998226i	$-0.518961\pi$
$863$	−3.49771	−0.119063	−0.0595316	+	0.998226i	$0.518961\pi$
$864$	0	0
$865$	7.80836	0.265492
$866$	2.14951	0.0730432
$867$	0	0
$868$	−19.9198	−0.676124
$869$	26.7530	0.907533
$870$	0	0
$871$	−11.2749	−0.382034
$872$	−8.99794	−0.304709
$873$	0	0
$874$	0.976746	0.0330389
$875$	−2.67964	−0.0905885
$876$	0	0
$877$	43.5384	1.47019	0.735093	−	0.677966i	$-0.237138\pi$
$877$	43.5384	1.47019	0.735093	+	0.677966i	$0.237138\pi$
$878$	9.86158	0.332812
$879$	0	0
$880$	−0.208635	−0.00703307
$881$	26.5745	0.895317	0.447658	−	0.894205i	$-0.352258\pi$
$881$	26.5745	0.895317	0.447658	+	0.894205i	$0.352258\pi$
$882$	0	0
$883$	52.7185	1.77412	0.887060	−	0.461655i	$-0.152744\pi$
$883$	52.7185	1.77412	0.887060	+	0.461655i	$0.152744\pi$
$884$	−34.9134	−1.17427
$885$	0	0
$886$	21.9903	0.738779
$887$	−15.3777	−0.516333	−0.258166	−	0.966100i	$-0.583118\pi$
$887$	−15.3777	−0.516333	−0.258166	+	0.966100i	$0.583118\pi$
$888$	0	0
$889$	4.89494	0.164171
$890$	−0.858442	−0.0287750
$891$	0	0
$892$	3.40564	0.114029
$893$	1.36448	0.0456606
$894$	0	0
$895$	−12.2430	−0.409240
$896$	19.8737	0.663935
$897$	0	0
$898$	−28.3883	−0.947331
$899$	17.8053	0.593840
$900$	0	0
$901$	83.3694	2.77744
$902$	3.33543	0.111058
$903$	0	0
$904$	−6.08840	−0.202497
$905$	24.1285	0.802058
$906$	0	0
$907$	−14.6695	−0.487094	−0.243547	−	0.969889i	$-0.578311\pi$
$907$	−14.6695	−0.487094	−0.243547	+	0.969889i	$0.578311\pi$
$908$	−37.3760	−1.24037
$909$	0	0
$910$	−9.34904	−0.309918
$911$	−50.0656	−1.65875	−0.829374	−	0.558694i	$-0.811303\pi$
$911$	−50.0656	−1.65875	−0.829374	+	0.558694i	$0.811303\pi$
$912$	0	0
$913$	−5.86875	−0.194227
$914$	11.6924	0.386751
$915$	0	0
$916$	−23.8053	−0.786549
$917$	13.8544	0.457513
$918$	0	0
$919$	46.4705	1.53292	0.766460	−	0.642293i	$-0.222017\pi$
$919$	46.4705	1.53292	0.766460	+	0.642293i	$0.222017\pi$
$920$	4.97445	0.164003
$921$	0	0
$922$	−31.4659	−1.03627
$923$	−22.7692	−0.749457
$924$	0	0
$925$	−2.08259	−0.0684751
$926$	27.1239	0.891346
$927$	0	0
$928$	−17.2644	−0.566731
$929$	−7.46620	−0.244958	−0.122479	−	0.992471i	$-0.539084\pi$
$929$	−7.46620	−0.244958	−0.122479	+	0.992471i	$0.539084\pi$
$930$	0	0
$931$	−0.115641	−0.00378998
$932$	35.5766	1.16535
$933$	0	0
$934$	2.13023	0.0697033
$935$	−11.6768	−0.381872
$936$	0	0
$937$	6.16311	0.201340	0.100670	−	0.994920i	$-0.467901\pi$
$937$	6.16311	0.201340	0.100670	+	0.994920i	$0.467901\pi$
$938$	6.38146	0.208362
$939$	0	0
$940$	2.68988	0.0877344
$941$	−12.6344	−0.411869	−0.205935	−	0.978566i	$-0.566023\pi$
$941$	−12.6344	−0.411869	−0.205935	+	0.978566i	$0.566023\pi$
$942$	0	0
$943$	4.01889	0.130873
$944$	0.894936	0.0291277
$945$	0	0
$946$	4.58092	0.148938
$947$	0.314034	0.0102047	0.00510237	−	0.999987i	$-0.498376\pi$
$947$	0.314034	0.0102047	0.00510237	+	0.999987i	$0.498376\pi$
$948$	0	0
$949$	−10.1901	−0.330785
$950$	−0.550015	−0.0178449
$951$	0	0
$952$	51.0503	1.65455
$953$	13.8623	0.449043	0.224522	−	0.974469i	$-0.427918\pi$
$953$	13.8623	0.449043	0.224522	+	0.974469i	$0.427918\pi$
$954$	0	0
$955$	6.85800	0.221920
$956$	18.7481	0.606358
$957$	0	0
$958$	−9.19837	−0.297186
$959$	−4.14896	−0.133977
$960$	0	0
$961$	3.63842	0.117368
$962$	−7.26598	−0.234264
$963$	0	0
$964$	−1.92552	−0.0620168
$965$	−16.9261	−0.544869
$966$	0	0
$967$	25.6170	0.823788	0.411894	−	0.911232i	$-0.364867\pi$
$967$	25.6170	0.823788	0.411894	+	0.911232i	$0.364867\pi$
$968$	22.5558	0.724972
$969$	0	0
$970$	−13.6529	−0.438370
$971$	−1.88484	−0.0604873	−0.0302437	−	0.999543i	$-0.509628\pi$
$971$	−1.88484	−0.0604873	−0.0302437	+	0.999543i	$0.509628\pi$
$972$	0	0
$973$	5.75338	0.184445
$974$	−36.8099	−1.17946
$975$	0	0
$976$	−1.61564	−0.0517154
$977$	18.4582	0.590529	0.295265	−	0.955416i	$-0.404592\pi$
$977$	18.4582	0.590529	0.295265	+	0.955416i	$0.404592\pi$
$978$	0	0
$979$	1.71688	0.0548718
$980$	−0.227970	−0.00728223
$981$	0	0
$982$	−0.492833	−0.0157269
$983$	−58.9507	−1.88024	−0.940118	−	0.340849i	$-0.889286\pi$
$983$	−58.9507	−1.88024	−0.940118	+	0.340849i	$0.889286\pi$
$984$	0	0
$985$	−3.42612	−0.109165
$986$	−17.6630	−0.562504
$987$	0	0
$988$	3.28907	0.104639
$989$	5.51960	0.175513
$990$	0	0
$991$	41.5111	1.31865	0.659323	−	0.751860i	$-0.270843\pi$
$991$	41.5111	1.31865	0.659323	+	0.751860i	$0.270843\pi$
$992$	−33.5861	−1.06636
$993$	0	0
$994$	12.8871	0.408755
$995$	18.5207	0.587146
$996$	0	0
$997$	−35.5897	−1.12714	−0.563569	−	0.826069i	$-0.690572\pi$
$997$	−35.5897	−1.12714	−0.563569	+	0.826069i	$0.690572\pi$
$998$	−0.536094	−0.0169698
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.m.1.3		4
3.2	odd	2		1335.2.a.f.1.2	✓	4
15.14	odd	2		6675.2.a.r.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.f.1.2	✓	4	3.2	odd	2
4005.2.a.m.1.3		4	1.1	even	1	trivial
6675.2.a.r.1.3		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+0.858442 q^{2} -1.26308 q^{4} +1.00000 q^{5} -2.67964 q^{7} -2.80116 q^{8} +O(q^{10})\)	\(q+0.858442 q^{2} -1.26308 q^{4} +1.00000 q^{5} -2.67964 q^{7} -2.80116 q^{8} +0.858442 q^{10} -1.71688 q^{11} +4.06424 q^{13} -2.30032 q^{14} +0.121519 q^{16} +6.80116 q^{17} -0.640714 q^{19} -1.26308 q^{20} -1.47385 q^{22} -1.77585 q^{23} +1.00000 q^{25} +3.48891 q^{26} +3.38460 q^{28} -3.02531 q^{29} -5.88544 q^{31} +5.70664 q^{32} +5.83840 q^{34} -2.67964 q^{35} -2.08259 q^{37} -0.550015 q^{38} -2.80116 q^{40} -2.26308 q^{41} -3.10814 q^{43} +2.16856 q^{44} -1.52447 q^{46} -2.12963 q^{47} +0.180488 q^{49} +0.858442 q^{50} -5.13345 q^{52} +12.2581 q^{53} -1.71688 q^{55} +7.50612 q^{56} -2.59705 q^{58} +7.36456 q^{59} -13.2953 q^{61} -5.05231 q^{62} +4.65578 q^{64} +4.06424 q^{65} -2.77416 q^{67} -8.59039 q^{68} -2.30032 q^{70} -5.60233 q^{71} -2.50726 q^{73} -1.78778 q^{74} +0.809271 q^{76} +4.60064 q^{77} -15.5823 q^{79} +0.121519 q^{80} -1.94272 q^{82} +3.41825 q^{83} +6.80116 q^{85} -2.66816 q^{86} +4.80927 q^{88} -1.00000 q^{89} -10.8907 q^{91} +2.24304 q^{92} -1.82816 q^{94} -0.640714 q^{95} -15.9043 q^{97} +0.154938 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8	\( 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8} - 5 q^{13} + q^{14} - 10 q^{16} + 13 q^{17} - 10 q^{19} + 2 q^{20} - 20 q^{22} - 3 q^{23} + 4 q^{25} + 3 q^{26} - 4 q^{28} - 2 q^{29} - 2 q^{31} + q^{32} + 6 q^{34} - 7 q^{35} - 9 q^{37} - 2 q^{38} + 3 q^{40} - 2 q^{41} - 19 q^{43} - 6 q^{44} - 5 q^{47} - 7 q^{49} - 17 q^{52} + 3 q^{53} + 2 q^{56} - 6 q^{58} - 2 q^{59} - 4 q^{61} + 8 q^{62} + q^{64} - 5 q^{65} - 15 q^{67} + q^{68} + q^{70} + 6 q^{71} - 21 q^{73} - 10 q^{74} - 4 q^{76} - 2 q^{77} - 20 q^{79} - 10 q^{80} + 3 q^{82} + 9 q^{83} + 13 q^{85} - 16 q^{86} + 12 q^{88} - 4 q^{89} + 2 q^{91} - 12 q^{92} + 25 q^{94} - 10 q^{95} - 17 q^{97} - 13 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8 - 5 * q^13 + q^14 - 10 * q^16 + 13 * q^17 - 10 * q^19 + 2 * q^20 - 20 * q^22 - 3 * q^23 + 4 * q^25 + 3 * q^26 - 4 * q^28 - 2 * q^29 - 2 * q^31 + q^32 + 6 * q^34 - 7 * q^35 - 9 * q^37 - 2 * q^38 + 3 * q^40 - 2 * q^41 - 19 * q^43 - 6 * q^44 - 5 * q^47 - 7 * q^49 - 17 * q^52 + 3 * q^53 + 2 * q^56 - 6 * q^58 - 2 * q^59 - 4 * q^61 + 8 * q^62 + q^64 - 5 * q^65 - 15 * q^67 + q^68 + q^70 + 6 * q^71 - 21 * q^73 - 10 * q^74 - 4 * q^76 - 2 * q^77 - 20 * q^79 - 10 * q^80 + 3 * q^82 + 9 * q^83 + 13 * q^85 - 16 * q^86 + 12 * q^88 - 4 * q^89 + 2 * q^91 - 12 * q^92 + 25 * q^94 - 10 * q^95 - 17 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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