LMFDB - Embedded newform 4005.2.a.w.1.9

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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.184653$ 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)

$f(q)$	$=$	$q-0.184653 q^{2} -1.96590 q^{4} -1.00000 q^{5} +2.68806 q^{7} +0.732314 q^{8} +O(q^{10})$	\(q-0.184653 q^{2} -1.96590 q^{4} -1.00000 q^{5} +2.68806 q^{7} +0.732314 q^{8} +0.184653 q^{10} +4.77393 q^{11} +6.17995 q^{13} -0.496358 q^{14} +3.79658 q^{16} -5.76151 q^{17} +7.05215 q^{19} +1.96590 q^{20} -0.881519 q^{22} +0.693354 q^{23} +1.00000 q^{25} -1.14114 q^{26} -5.28447 q^{28} +8.42661 q^{29} +3.38619 q^{31} -2.16568 q^{32} +1.06388 q^{34} -2.68806 q^{35} -11.0755 q^{37} -1.30220 q^{38} -0.732314 q^{40} +2.52468 q^{41} -4.09538 q^{43} -9.38509 q^{44} -0.128030 q^{46} -0.000463914 q^{47} +0.225689 q^{49} -0.184653 q^{50} -12.1492 q^{52} -4.86876 q^{53} -4.77393 q^{55} +1.96851 q^{56} -1.55599 q^{58} -0.626017 q^{59} +9.47749 q^{61} -0.625268 q^{62} -7.19327 q^{64} -6.17995 q^{65} -0.700531 q^{67} +11.3266 q^{68} +0.496358 q^{70} -9.55727 q^{71} +9.93261 q^{73} +2.04512 q^{74} -13.8639 q^{76} +12.8326 q^{77} +3.17751 q^{79} -3.79658 q^{80} -0.466189 q^{82} +5.19031 q^{83} +5.76151 q^{85} +0.756222 q^{86} +3.49602 q^{88} +1.00000 q^{89} +16.6121 q^{91} -1.36307 q^{92} +8.56628e-5 q^{94} -7.05215 q^{95} +4.50587 q^{97} -0.0416741 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * 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 $

$2$	−0.184653	−0.130569	−0.0652845	−	0.997867i	$-0.520796\pi$
$2$	−0.184653	−0.130569	−0.0652845	+	0.997867i	$0.520796\pi$
$3$	0	0
$3$	0	0
$4$	−1.96590	−0.982952
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.68806	1.01599	0.507996	−	0.861359i	$-0.330386\pi$
$7$	2.68806	1.01599	0.507996	+	0.861359i	$0.330386\pi$
$8$	0.732314	0.258912
$9$	0	0
$10$	0.184653	0.0583923
$11$	4.77393	1.43940	0.719698	−	0.694288i	$-0.244280\pi$
$11$	4.77393	1.43940	0.719698	+	0.694288i	$0.244280\pi$
$12$	0	0
$13$	6.17995	1.71401	0.857005	−	0.515308i	$-0.172322\pi$
$13$	6.17995	1.71401	0.857005	+	0.515308i	$0.172322\pi$
$14$	−0.496358	−0.132657
$15$	0	0
$16$	3.79658	0.949146
$17$	−5.76151	−1.39737	−0.698686	−	0.715429i	$-0.746231\pi$
$17$	−5.76151	−1.39737	−0.698686	+	0.715429i	$0.746231\pi$
$18$	0	0
$19$	7.05215	1.61787	0.808937	−	0.587895i	$-0.200043\pi$
$19$	7.05215	1.61787	0.808937	+	0.587895i	$0.200043\pi$
$20$	1.96590	0.439589
$21$	0	0
$22$	−0.881519	−0.187940
$23$	0.693354	0.144574	0.0722872	−	0.997384i	$-0.476970\pi$
$23$	0.693354	0.144574	0.0722872	+	0.997384i	$0.476970\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.14114	−0.223797
$27$	0	0
$28$	−5.28447	−0.998672
$29$	8.42661	1.56478	0.782391	−	0.622788i	$-0.214000\pi$
$29$	8.42661	1.56478	0.782391	+	0.622788i	$0.214000\pi$
$30$	0	0
$31$	3.38619	0.608177	0.304089	−	0.952644i	$-0.401648\pi$
$31$	3.38619	0.608177	0.304089	+	0.952644i	$0.401648\pi$
$32$	−2.16568	−0.382841
$33$	0	0
$34$	1.06388	0.182453
$35$	−2.68806	−0.454366
$36$	0	0
$37$	−11.0755	−1.82081	−0.910403	−	0.413724i	$-0.864228\pi$
$37$	−11.0755	−1.82081	−0.910403	+	0.413724i	$0.864228\pi$
$38$	−1.30220	−0.211244
$39$	0	0
$40$	−0.732314	−0.115789
$41$	2.52468	0.394289	0.197144	−	0.980374i	$-0.436833\pi$
$41$	2.52468	0.394289	0.197144	+	0.980374i	$0.436833\pi$
$42$	0	0
$43$	−4.09538	−0.624540	−0.312270	−	0.949993i	$-0.601089\pi$
$43$	−4.09538	−0.624540	−0.312270	+	0.949993i	$0.601089\pi$
$44$	−9.38509	−1.41486
$45$	0	0
$46$	−0.128030	−0.0188769
$47$	−0.000463914 0	−6.76687e−5 0	−3.38344e−5	−	1.00000i	$-0.500011\pi$
$47$	−0.000463914 0	−6.76687e−5 0	−3.38344e−5		1.00000i	$0.500011\pi$
$48$	0	0
$49$	0.225689	0.0322413
$50$	−0.184653	−0.0261138
$51$	0	0
$52$	−12.1492	−1.68479
$53$	−4.86876	−0.668775	−0.334388	−	0.942436i	$-0.608529\pi$
$53$	−4.86876	−0.668775	−0.334388	+	0.942436i	$0.608529\pi$
$54$	0	0
$55$	−4.77393	−0.643717
$56$	1.96851	0.263053
$57$	0	0
$58$	−1.55599	−0.204312
$59$	−0.626017	−0.0815004	−0.0407502	−	0.999169i	$-0.512975\pi$
$59$	−0.626017	−0.0815004	−0.0407502	+	0.999169i	$0.512975\pi$
$60$	0	0
$61$	9.47749	1.21347	0.606734	−	0.794905i	$-0.292479\pi$
$61$	9.47749	1.21347	0.606734	+	0.794905i	$0.292479\pi$
$62$	−0.625268	−0.0794091
$63$	0	0
$64$	−7.19327	−0.899159
$65$	−6.17995	−0.766529
$66$	0	0
$67$	−0.700531	−0.0855835	−0.0427917	−	0.999084i	$-0.513625\pi$
$67$	−0.700531	−0.0855835	−0.0427917	+	0.999084i	$0.513625\pi$
$68$	11.3266	1.37355
$69$	0	0
$70$	0.496358	0.0593261
$71$	−9.55727	−1.13424	−0.567119	−	0.823636i	$-0.691942\pi$
$71$	−9.55727	−1.13424	−0.567119	+	0.823636i	$0.691942\pi$
$72$	0	0
$73$	9.93261	1.16252	0.581262	−	0.813716i	$-0.302559\pi$
$73$	9.93261	1.16252	0.581262	+	0.813716i	$0.302559\pi$
$74$	2.04512	0.237741
$75$	0	0
$76$	−13.8639	−1.59029
$77$	12.8326	1.46242
$78$	0	0
$79$	3.17751	0.357498	0.178749	−	0.983895i	$-0.442795\pi$
$79$	3.17751	0.357498	0.178749	+	0.983895i	$0.442795\pi$
$80$	−3.79658	−0.424471
$81$	0	0
$82$	−0.466189	−0.0514819
$83$	5.19031	0.569711	0.284855	−	0.958571i	$-0.408054\pi$
$83$	5.19031	0.569711	0.284855	+	0.958571i	$0.408054\pi$
$84$	0	0
$85$	5.76151	0.624923
$86$	0.756222	0.0815456
$87$	0	0
$88$	3.49602	0.372677
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	16.6121	1.74142
$92$	−1.36307	−0.142110
$93$	0	0
$94$	8.56628e−5 0	8.83544e−6 0
$95$	−7.05215	−0.723536
$96$	0	0
$97$	4.50587	0.457502	0.228751	−	0.973485i	$-0.426536\pi$
$97$	4.50587	0.457502	0.228751	+	0.973485i	$0.426536\pi$
$98$	−0.0416741	−0.00420972
$99$	0	0
$100$	−1.96590	−0.196590
$101$	−15.1712	−1.50959	−0.754797	−	0.655959i	$-0.772265\pi$
$101$	−15.1712	−1.50959	−0.754797	+	0.655959i	$0.772265\pi$
$102$	0	0
$103$	−8.10341	−0.798453	−0.399227	−	0.916852i	$-0.630721\pi$
$103$	−8.10341	−0.798453	−0.399227	+	0.916852i	$0.630721\pi$
$104$	4.52567	0.443778
$105$	0	0
$106$	0.899028	0.0873213
$107$	−4.89092	−0.472823	−0.236412	−	0.971653i	$-0.575971\pi$
$107$	−4.89092	−0.472823	−0.236412	+	0.971653i	$0.575971\pi$
$108$	0	0
$109$	0.433771	0.0415478	0.0207739	−	0.999784i	$-0.493387\pi$
$109$	0.433771	0.0415478	0.0207739	+	0.999784i	$0.493387\pi$
$110$	0.881519	0.0840495
$111$	0	0
$112$	10.2055	0.964325
$113$	9.04799	0.851163	0.425582	−	0.904920i	$-0.360070\pi$
$113$	9.04799	0.851163	0.425582	+	0.904920i	$0.360070\pi$
$114$	0	0
$115$	−0.693354	−0.0646556
$116$	−16.5659	−1.53810
$117$	0	0
$118$	0.115596	0.0106414
$119$	−15.4873	−1.41972
$120$	0	0
$121$	11.7904	1.07186
$122$	−1.75004	−0.158441
$123$	0	0
$124$	−6.65692	−0.597809
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.24081	0.819989	0.409995	−	0.912088i	$-0.365531\pi$
$127$	9.24081	0.819989	0.409995	+	0.912088i	$0.365531\pi$
$128$	5.65961	0.500243
$129$	0	0
$130$	1.14114	0.100085
$131$	−17.2012	−1.50288	−0.751439	−	0.659803i	$-0.770640\pi$
$131$	−17.2012	−1.50288	−0.751439	+	0.659803i	$0.770640\pi$
$132$	0	0
$133$	18.9566	1.64375
$134$	0.129355	0.0111746
$135$	0	0
$136$	−4.21923	−0.361796
$137$	1.59260	0.136065	0.0680323	−	0.997683i	$-0.478328\pi$
$137$	1.59260	0.136065	0.0680323	+	0.997683i	$0.478328\pi$
$138$	0	0
$139$	20.4231	1.73227	0.866133	−	0.499813i	$-0.166598\pi$
$139$	20.4231	1.73227	0.866133	+	0.499813i	$0.166598\pi$
$140$	5.28447	0.446620
$141$	0	0
$142$	1.76477	0.148096
$143$	29.5027	2.46714
$144$	0	0
$145$	−8.42661	−0.699792
$146$	−1.83408	−0.151790
$147$	0	0
$148$	21.7734	1.78976
$149$	15.7449	1.28987	0.644937	−	0.764235i	$-0.276883\pi$
$149$	15.7449	1.28987	0.644937	+	0.764235i	$0.276883\pi$
$150$	0	0
$151$	−8.23393	−0.670068	−0.335034	−	0.942206i	$-0.608748\pi$
$151$	−8.23393	−0.670068	−0.335034	+	0.942206i	$0.608748\pi$
$152$	5.16439	0.418887
$153$	0	0
$154$	−2.36958	−0.190946
$155$	−3.38619	−0.271985
$156$	0	0
$157$	−1.47711	−0.117886	−0.0589430	−	0.998261i	$-0.518773\pi$
$157$	−1.47711	−0.117886	−0.0589430	+	0.998261i	$0.518773\pi$
$158$	−0.586735	−0.0466781
$159$	0	0
$160$	2.16568	0.171212
$161$	1.86378	0.146886
$162$	0	0
$163$	3.65680	0.286422	0.143211	−	0.989692i	$-0.454257\pi$
$163$	3.65680	0.286422	0.143211	+	0.989692i	$0.454257\pi$
$164$	−4.96328	−0.387567
$165$	0	0
$166$	−0.958404	−0.0743866
$167$	−18.7535	−1.45119	−0.725595	−	0.688122i	$-0.758436\pi$
$167$	−18.7535	−1.45119	−0.725595	+	0.688122i	$0.758436\pi$
$168$	0	0
$169$	25.1918	1.93783
$170$	−1.06388	−0.0815957
$171$	0	0
$172$	8.05112	0.613892
$173$	−8.85431	−0.673180	−0.336590	−	0.941651i	$-0.609274\pi$
$173$	−8.85431	−0.673180	−0.336590	+	0.941651i	$0.609274\pi$
$174$	0	0
$175$	2.68806	0.203199
$176$	18.1246	1.36620
$177$	0	0
$178$	−0.184653	−0.0138403
$179$	−19.7770	−1.47820	−0.739102	−	0.673593i	$-0.764750\pi$
$179$	−19.7770	−1.47820	−0.739102	+	0.673593i	$0.764750\pi$
$180$	0	0
$181$	−4.97505	−0.369793	−0.184896	−	0.982758i	$-0.559195\pi$
$181$	−4.97505	−0.369793	−0.184896	+	0.982758i	$0.559195\pi$
$182$	−3.06747	−0.227376
$183$	0	0
$184$	0.507753	0.0374320
$185$	11.0755	0.814289
$186$	0	0
$187$	−27.5051	−2.01137
$188$	0.000912009 0	6.65151e−5 0
$189$	0	0
$190$	1.30220	0.0944714
$191$	20.9581	1.51648	0.758239	−	0.651977i	$-0.226060\pi$
$191$	20.9581	1.51648	0.758239	+	0.651977i	$0.226060\pi$
$192$	0	0
$193$	3.33384	0.239975	0.119988	−	0.992775i	$-0.461715\pi$
$193$	3.33384	0.239975	0.119988	+	0.992775i	$0.461715\pi$
$194$	−0.832020	−0.0597356
$195$	0	0
$196$	−0.443683	−0.0316917
$197$	19.4744	1.38749	0.693745	−	0.720221i	$-0.255959\pi$
$197$	19.4744	1.38749	0.693745	+	0.720221i	$0.255959\pi$
$198$	0	0
$199$	20.3793	1.44465	0.722326	−	0.691553i	$-0.243073\pi$
$199$	20.3793	1.44465	0.722326	+	0.691553i	$0.243073\pi$
$200$	0.732314	0.0517824
$201$	0	0
$202$	2.80141	0.197106
$203$	22.6513	1.58981
$204$	0	0
$205$	−2.52468	−0.176331
$206$	1.49632	0.104253
$207$	0	0
$208$	23.4627	1.62685
$209$	33.6665	2.32876
$210$	0	0
$211$	10.2749	0.707355	0.353677	−	0.935368i	$-0.384931\pi$
$211$	10.2749	0.707355	0.353677	+	0.935368i	$0.384931\pi$
$212$	9.57151	0.657374
$213$	0	0
$214$	0.903121	0.0617361
$215$	4.09538	0.279303
$216$	0	0
$217$	9.10229	0.617904
$218$	−0.0800970	−0.00542485
$219$	0	0
$220$	9.38509	0.632743
$221$	−35.6059	−2.39511
$222$	0	0
$223$	−24.2527	−1.62408	−0.812041	−	0.583600i	$-0.801643\pi$
$223$	−24.2527	−1.62408	−0.812041	+	0.583600i	$0.801643\pi$
$224$	−5.82148	−0.388964
$225$	0	0
$226$	−1.67073	−0.111136
$227$	−15.8353	−1.05103	−0.525513	−	0.850785i	$-0.676127\pi$
$227$	−15.8353	−1.05103	−0.525513	+	0.850785i	$0.676127\pi$
$228$	0	0
$229$	−7.75066	−0.512178	−0.256089	−	0.966653i	$-0.582434\pi$
$229$	−7.75066	−0.512178	−0.256089	+	0.966653i	$0.582434\pi$
$230$	0.128030	0.00844202
$231$	0	0
$232$	6.17092	0.405141
$233$	29.9910	1.96477	0.982387	−	0.186859i	$-0.0598308\pi$
$233$	29.9910	1.96477	0.982387	+	0.186859i	$0.0598308\pi$
$234$	0	0
$235$	0.000463914 0	3.02624e−5 0
$236$	1.23069	0.0801110
$237$	0	0
$238$	2.85977	0.185371
$239$	17.5556	1.13558	0.567788	−	0.823175i	$-0.307799\pi$
$239$	17.5556	1.13558	0.567788	+	0.823175i	$0.307799\pi$
$240$	0	0
$241$	−8.10166	−0.521874	−0.260937	−	0.965356i	$-0.584031\pi$
$241$	−8.10166	−0.521874	−0.260937	+	0.965356i	$0.584031\pi$
$242$	−2.17714	−0.139952
$243$	0	0
$244$	−18.6318	−1.19278
$245$	−0.225689	−0.0144188
$246$	0	0
$247$	43.5820	2.77305
$248$	2.47975	0.157464
$249$	0	0
$250$	0.184653	0.0116785
$251$	−29.9728	−1.89187	−0.945934	−	0.324358i	$-0.894852\pi$
$251$	−29.9728	−1.89187	−0.945934	+	0.324358i	$0.894852\pi$
$252$	0	0
$253$	3.31003	0.208100
$254$	−1.70634	−0.107065
$255$	0	0
$256$	13.3415	0.833842
$257$	−23.3364	−1.45569	−0.727844	−	0.685743i	$-0.759477\pi$
$257$	−23.3364	−1.45569	−0.727844	+	0.685743i	$0.759477\pi$
$258$	0	0
$259$	−29.7717	−1.84992
$260$	12.1492	0.753461
$261$	0	0
$262$	3.17625	0.196229
$263$	13.3215	0.821439	0.410719	−	0.911762i	$-0.365278\pi$
$263$	13.3215	0.821439	0.410719	+	0.911762i	$0.365278\pi$
$264$	0	0
$265$	4.86876	0.299085
$266$	−3.50039	−0.214623
$267$	0	0
$268$	1.37718	0.0841244
$269$	31.1426	1.89880	0.949399	−	0.314073i	$-0.101694\pi$
$269$	31.1426	1.89880	0.949399	+	0.314073i	$0.101694\pi$
$270$	0	0
$271$	2.56847	0.156024	0.0780118	−	0.996952i	$-0.475143\pi$
$271$	2.56847	0.156024	0.0780118	+	0.996952i	$0.475143\pi$
$272$	−21.8741	−1.32631
$273$	0	0
$274$	−0.294077	−0.0177658
$275$	4.77393	0.287879
$276$	0	0
$277$	1.98643	0.119353	0.0596764	−	0.998218i	$-0.480993\pi$
$277$	1.98643	0.119353	0.0596764	+	0.998218i	$0.480993\pi$
$278$	−3.77118	−0.226180
$279$	0	0
$280$	−1.96851	−0.117641
$281$	12.8856	0.768693	0.384346	−	0.923189i	$-0.374427\pi$
$281$	12.8856	0.768693	0.384346	+	0.923189i	$0.374427\pi$
$282$	0	0
$283$	12.0575	0.716744	0.358372	−	0.933579i	$-0.383332\pi$
$283$	12.0575	0.716744	0.358372	+	0.933579i	$0.383332\pi$
$284$	18.7887	1.11490
$285$	0	0
$286$	−5.44775	−0.322132
$287$	6.78650	0.400595
$288$	0	0
$289$	16.1950	0.952647
$290$	1.55599	0.0913711
$291$	0	0
$292$	−19.5266	−1.14271
$293$	−28.3591	−1.65676	−0.828380	−	0.560167i	$-0.810737\pi$
$293$	−28.3591	−1.65676	−0.828380	+	0.560167i	$0.810737\pi$
$294$	0	0
$295$	0.626017	0.0364481
$296$	−8.11076	−0.471429
$297$	0	0
$298$	−2.90734	−0.168418
$299$	4.28489	0.247802
$300$	0	0
$301$	−11.0086	−0.634528
$302$	1.52042	0.0874901
$303$	0	0
$304$	26.7741	1.53560
$305$	−9.47749	−0.542680
$306$	0	0
$307$	11.0518	0.630759	0.315380	−	0.948966i	$-0.397868\pi$
$307$	11.0518	0.630759	0.315380	+	0.948966i	$0.397868\pi$
$308$	−25.2277	−1.43748
$309$	0	0
$310$	0.625268	0.0355128
$311$	−1.97471	−0.111975	−0.0559877	−	0.998431i	$-0.517831\pi$
$311$	−1.97471	−0.111975	−0.0559877	+	0.998431i	$0.517831\pi$
$312$	0	0
$313$	−18.9196	−1.06940	−0.534699	−	0.845043i	$-0.679575\pi$
$313$	−18.9196	−1.06940	−0.534699	+	0.845043i	$0.679575\pi$
$314$	0.272752	0.0153923
$315$	0	0
$316$	−6.24667	−0.351403
$317$	−23.0612	−1.29525	−0.647624	−	0.761960i	$-0.724237\pi$
$317$	−23.0612	−1.29525	−0.647624	+	0.761960i	$0.724237\pi$
$318$	0	0
$319$	40.2281	2.25234
$320$	7.19327	0.402116
$321$	0	0
$322$	−0.344152	−0.0191788
$323$	−40.6310	−2.26077
$324$	0	0
$325$	6.17995	0.342802
$326$	−0.675237	−0.0373979
$327$	0	0
$328$	1.84886	0.102086
$329$	−0.00124703	−6.87510e−5 0
$330$	0	0
$331$	23.4611	1.28954	0.644768	−	0.764378i	$-0.276954\pi$
$331$	23.4611	1.28954	0.644768	+	0.764378i	$0.276954\pi$
$332$	−10.2036	−0.559998
$333$	0	0
$334$	3.46288	0.189481
$335$	0.700531	0.0382741
$336$	0	0
$337$	29.7838	1.62243	0.811214	−	0.584750i	$-0.198807\pi$
$337$	29.7838	1.62243	0.811214	+	0.584750i	$0.198807\pi$
$338$	−4.65173	−0.253021
$339$	0	0
$340$	−11.3266	−0.614270
$341$	16.1654	0.875408
$342$	0	0
$343$	−18.2098	−0.983236
$344$	−2.99911	−0.161701
$345$	0	0
$346$	1.63497	0.0878965
$347$	−32.5350	−1.74657	−0.873286	−	0.487209i	$-0.838015\pi$
$347$	−32.5350	−1.74657	−0.873286	+	0.487209i	$0.838015\pi$
$348$	0	0
$349$	14.6360	0.783450	0.391725	−	0.920082i	$-0.371879\pi$
$349$	14.6360	0.783450	0.391725	+	0.920082i	$0.371879\pi$
$350$	−0.496358	−0.0265314
$351$	0	0
$352$	−10.3388	−0.551060
$353$	28.4723	1.51543	0.757713	−	0.652588i	$-0.226317\pi$
$353$	28.4723	1.51543	0.757713	+	0.652588i	$0.226317\pi$
$354$	0	0
$355$	9.55727	0.507247
$356$	−1.96590	−0.104193
$357$	0	0
$358$	3.65188	0.193008
$359$	−16.4143	−0.866314	−0.433157	−	0.901318i	$-0.642601\pi$
$359$	−16.4143	−0.866314	−0.433157	+	0.901318i	$0.642601\pi$
$360$	0	0
$361$	30.7329	1.61752
$362$	0.918656	0.0482835
$363$	0	0
$364$	−32.6578	−1.71173
$365$	−9.93261	−0.519897
$366$	0	0
$367$	6.83968	0.357028	0.178514	−	0.983937i	$-0.442871\pi$
$367$	6.83968	0.357028	0.178514	+	0.983937i	$0.442871\pi$
$368$	2.63238	0.137222
$369$	0	0
$370$	−2.04512	−0.106321
$371$	−13.0875	−0.679471
$372$	0	0
$373$	−16.1458	−0.835995	−0.417998	−	0.908448i	$-0.637268\pi$
$373$	−16.1458	−0.835995	−0.417998	+	0.908448i	$0.637268\pi$
$374$	5.07888	0.262623
$375$	0	0
$376$	−0.000339730 0	−1.75203e−5 0
$377$	52.0760	2.68205
$378$	0	0
$379$	−31.5449	−1.62035	−0.810177	−	0.586186i	$-0.800629\pi$
$379$	−31.5449	−1.62035	−0.810177	+	0.586186i	$0.800629\pi$
$380$	13.8639	0.711201
$381$	0	0
$382$	−3.86997	−0.198005
$383$	17.4210	0.890173	0.445086	−	0.895488i	$-0.353173\pi$
$383$	17.4210	0.890173	0.445086	+	0.895488i	$0.353173\pi$
$384$	0	0
$385$	−12.8326	−0.654012
$386$	−0.615602	−0.0313333
$387$	0	0
$388$	−8.85811	−0.449702
$389$	−16.7007	−0.846757	−0.423379	−	0.905953i	$-0.639156\pi$
$389$	−16.7007	−0.846757	−0.423379	+	0.905953i	$0.639156\pi$
$390$	0	0
$391$	−3.99477	−0.202024
$392$	0.165275	0.00834767
$393$	0	0
$394$	−3.59599	−0.181163
$395$	−3.17751	−0.159878
$396$	0	0
$397$	23.8217	1.19558	0.597789	−	0.801653i	$-0.296046\pi$
$397$	23.8217	1.19558	0.597789	+	0.801653i	$0.296046\pi$
$398$	−3.76309	−0.188627
$399$	0	0
$400$	3.79658	0.189829
$401$	−26.5898	−1.32783	−0.663915	−	0.747808i	$-0.731106\pi$
$401$	−26.5898	−1.32783	−0.663915	+	0.747808i	$0.731106\pi$
$402$	0	0
$403$	20.9265	1.04242
$404$	29.8252	1.48386
$405$	0	0
$406$	−4.18261	−0.207580
$407$	−52.8738	−2.62086
$408$	0	0
$409$	−37.1757	−1.83822	−0.919110	−	0.394001i	$-0.871091\pi$
$409$	−37.1757	−1.83822	−0.919110	+	0.394001i	$0.871091\pi$
$410$	0.466189	0.0230234
$411$	0	0
$412$	15.9305	0.784841
$413$	−1.68277	−0.0828038
$414$	0	0
$415$	−5.19031	−0.254782
$416$	−13.3838	−0.656194
$417$	0	0
$418$	−6.21661	−0.304064
$419$	8.31983	0.406450	0.203225	−	0.979132i	$-0.434858\pi$
$419$	8.31983	0.406450	0.203225	+	0.979132i	$0.434858\pi$
$420$	0	0
$421$	23.9691	1.16818	0.584091	−	0.811688i	$-0.301451\pi$
$421$	23.9691	1.16818	0.584091	+	0.811688i	$0.301451\pi$
$422$	−1.89729	−0.0923586
$423$	0	0
$424$	−3.56546	−0.173154
$425$	−5.76151	−0.279474
$426$	0	0
$427$	25.4761	1.23288
$428$	9.61508	0.464762
$429$	0	0
$430$	−0.756222	−0.0364683
$431$	−5.90823	−0.284590	−0.142295	−	0.989824i	$-0.545448\pi$
$431$	−5.90823	−0.284590	−0.142295	+	0.989824i	$0.545448\pi$
$432$	0	0
$433$	−10.6461	−0.511620	−0.255810	−	0.966727i	$-0.582342\pi$
$433$	−10.6461	−0.511620	−0.255810	+	0.966727i	$0.582342\pi$
$434$	−1.68076	−0.0806791
$435$	0	0
$436$	−0.852752	−0.0408394
$437$	4.88964	0.233903
$438$	0	0
$439$	−13.5222	−0.645381	−0.322691	−	0.946505i	$-0.604587\pi$
$439$	−13.5222	−0.645381	−0.322691	+	0.946505i	$0.604587\pi$
$440$	−3.49602	−0.166666
$441$	0	0
$442$	6.57471	0.312727
$443$	−6.96098	−0.330726	−0.165363	−	0.986233i	$-0.552880\pi$
$443$	−6.96098	−0.330726	−0.165363	+	0.986233i	$0.552880\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	4.47833	0.212055
$447$	0	0
$448$	−19.3360	−0.913539
$449$	−13.7574	−0.649252	−0.324626	−	0.945842i	$-0.605238\pi$
$449$	−13.7574	−0.649252	−0.324626	+	0.945842i	$0.605238\pi$
$450$	0	0
$451$	12.0527	0.567538
$452$	−17.7875	−0.836652
$453$	0	0
$454$	2.92403	0.137232
$455$	−16.6121	−0.778788
$456$	0	0
$457$	−8.26612	−0.386673	−0.193336	−	0.981133i	$-0.561931\pi$
$457$	−8.26612	−0.386673	−0.193336	+	0.981133i	$0.561931\pi$
$458$	1.43118	0.0668746
$459$	0	0
$460$	1.36307	0.0635533
$461$	36.1821	1.68517	0.842583	−	0.538566i	$-0.181034\pi$
$461$	36.1821	1.68517	0.842583	+	0.538566i	$0.181034\pi$
$462$	0	0
$463$	19.9172	0.925630	0.462815	−	0.886455i	$-0.346839\pi$
$463$	19.9172	0.925630	0.462815	+	0.886455i	$0.346839\pi$
$464$	31.9923	1.48521
$465$	0	0
$466$	−5.53791	−0.256539
$467$	29.6612	1.37256	0.686279	−	0.727339i	$-0.259243\pi$
$467$	29.6612	1.37256	0.686279	+	0.727339i	$0.259243\pi$
$468$	0	0
$469$	−1.88307	−0.0869522
$470$	−8.56628e−5 0	−3.95133e−6 0
$471$	0	0
$472$	−0.458441	−0.0211014
$473$	−19.5511	−0.898960
$474$	0	0
$475$	7.05215	0.323575
$476$	30.4466	1.39552
$477$	0	0
$478$	−3.24168	−0.148271
$479$	−5.26624	−0.240621	−0.120310	−	0.992736i	$-0.538389\pi$
$479$	−5.26624	−0.240621	−0.120310	+	0.992736i	$0.538389\pi$
$480$	0	0
$481$	−68.4462	−3.12088
$482$	1.49599	0.0681405
$483$	0	0
$484$	−23.1789	−1.05359
$485$	−4.50587	−0.204601
$486$	0	0
$487$	−26.6319	−1.20681	−0.603404	−	0.797436i	$-0.706189\pi$
$487$	−26.6319	−1.20681	−0.603404	+	0.797436i	$0.706189\pi$
$488$	6.94050	0.314182
$489$	0	0
$490$	0.0416741	0.00188264
$491$	22.3859	1.01026	0.505130	−	0.863043i	$-0.331445\pi$
$491$	22.3859	1.01026	0.505130	+	0.863043i	$0.331445\pi$
$492$	0	0
$493$	−48.5500	−2.18658
$494$	−8.04752	−0.362075
$495$	0	0
$496$	12.8559	0.577249
$497$	−25.6905	−1.15238
$498$	0	0
$499$	−1.11761	−0.0500311	−0.0250155	−	0.999687i	$-0.507964\pi$
$499$	−1.11761	−0.0500311	−0.0250155	+	0.999687i	$0.507964\pi$
$500$	1.96590	0.0879179
$501$	0	0
$502$	5.53456	0.247019
$503$	1.12607	0.0502089	0.0251044	−	0.999685i	$-0.492008\pi$
$503$	1.12607	0.0502089	0.0251044	+	0.999685i	$0.492008\pi$
$504$	0	0
$505$	15.1712	0.675111
$506$	−0.611205	−0.0271714
$507$	0	0
$508$	−18.1665	−0.806010
$509$	36.8802	1.63469	0.817344	−	0.576150i	$-0.195446\pi$
$509$	36.8802	1.63469	0.817344	+	0.576150i	$0.195446\pi$
$510$	0	0
$511$	26.6995	1.18112
$512$	−13.7828	−0.609117
$513$	0	0
$514$	4.30913	0.190068
$515$	8.10341	0.357079
$516$	0	0
$517$	−0.00221469	−9.74021e−5 0
$518$	5.49742	0.241543
$519$	0	0
$520$	−4.52567	−0.198464
$521$	−22.0602	−0.966475	−0.483238	−	0.875489i	$-0.660539\pi$
$521$	−22.0602	−0.966475	−0.483238	+	0.875489i	$0.660539\pi$
$522$	0	0
$523$	18.6487	0.815452	0.407726	−	0.913104i	$-0.366322\pi$
$523$	18.6487	0.815452	0.407726	+	0.913104i	$0.366322\pi$
$524$	33.8159	1.47726
$525$	0	0
$526$	−2.45985	−0.107255
$527$	−19.5096	−0.849850
$528$	0	0
$529$	−22.5193	−0.979098
$530$	−0.899028	−0.0390513
$531$	0	0
$532$	−37.2669	−1.61573
$533$	15.6024	0.675815
$534$	0	0
$535$	4.89092	0.211453
$536$	−0.513009	−0.0221586
$537$	0	0
$538$	−5.75056	−0.247924
$539$	1.07743	0.0464080
$540$	0	0
$541$	−17.5335	−0.753822	−0.376911	−	0.926249i	$-0.623014\pi$
$541$	−17.5335	−0.753822	−0.376911	+	0.926249i	$0.623014\pi$
$542$	−0.474275	−0.0203718
$543$	0	0
$544$	12.4776	0.534971
$545$	−0.433771	−0.0185807
$546$	0	0
$547$	−28.8701	−1.23440	−0.617199	−	0.786807i	$-0.711733\pi$
$547$	−28.8701	−1.23440	−0.617199	+	0.786807i	$0.711733\pi$
$548$	−3.13089	−0.133745
$549$	0	0
$550$	−0.881519	−0.0375881
$551$	59.4257	2.53162
$552$	0	0
$553$	8.54135	0.363215
$554$	−0.366799	−0.0155838
$555$	0	0
$556$	−40.1499	−1.70273
$557$	3.82995	0.162280	0.0811402	−	0.996703i	$-0.474144\pi$
$557$	3.82995	0.162280	0.0811402	+	0.996703i	$0.474144\pi$
$558$	0	0
$559$	−25.3093	−1.07047
$560$	−10.2055	−0.431259
$561$	0	0
$562$	−2.37937	−0.100367
$563$	22.3939	0.943789	0.471894	−	0.881655i	$-0.343570\pi$
$563$	22.3939	0.943789	0.471894	+	0.881655i	$0.343570\pi$
$564$	0	0
$565$	−9.04799	−0.380652
$566$	−2.22645	−0.0935845
$567$	0	0
$568$	−6.99892	−0.293668
$569$	10.0251	0.420274	0.210137	−	0.977672i	$-0.432609\pi$
$569$	10.0251	0.420274	0.210137	+	0.977672i	$0.432609\pi$
$570$	0	0
$571$	−3.64573	−0.152569	−0.0762845	−	0.997086i	$-0.524306\pi$
$571$	−3.64573	−0.152569	−0.0762845	+	0.997086i	$0.524306\pi$
$572$	−57.9994	−2.42508
$573$	0	0
$574$	−1.25315	−0.0523053
$575$	0.693354	0.0289149
$576$	0	0
$577$	−44.9128	−1.86974	−0.934871	−	0.354987i	$-0.884485\pi$
$577$	−44.9128	−1.86974	−0.934871	+	0.354987i	$0.884485\pi$
$578$	−2.99045	−0.124386
$579$	0	0
$580$	16.5659	0.687861
$581$	13.9519	0.578822
$582$	0	0
$583$	−23.2431	−0.962632
$584$	7.27379	0.300992
$585$	0	0
$586$	5.23659	0.216321
$587$	−22.4040	−0.924714	−0.462357	−	0.886694i	$-0.652996\pi$
$587$	−22.4040	−0.924714	−0.462357	+	0.886694i	$0.652996\pi$
$588$	0	0
$589$	23.8799	0.983955
$590$	−0.115596	−0.00475899
$591$	0	0
$592$	−42.0492	−1.72821
$593$	6.88700	0.282815	0.141408	−	0.989951i	$-0.454837\pi$
$593$	6.88700	0.282815	0.141408	+	0.989951i	$0.454837\pi$
$594$	0	0
$595$	15.4873	0.634918
$596$	−30.9530	−1.26788
$597$	0	0
$598$	−0.791217	−0.0323553
$599$	−0.775048	−0.0316676	−0.0158338	−	0.999875i	$-0.505040\pi$
$599$	−0.775048	−0.0316676	−0.0158338	+	0.999875i	$0.505040\pi$
$600$	0	0
$601$	−14.8595	−0.606132	−0.303066	−	0.952970i	$-0.598010\pi$
$601$	−14.8595	−0.606132	−0.303066	+	0.952970i	$0.598010\pi$
$602$	2.03277	0.0828497
$603$	0	0
$604$	16.1871	0.658644
$605$	−11.7904	−0.479350
$606$	0	0
$607$	−7.49110	−0.304054	−0.152027	−	0.988376i	$-0.548580\pi$
$607$	−7.49110	−0.304054	−0.152027	+	0.988376i	$0.548580\pi$
$608$	−15.2727	−0.619389
$609$	0	0
$610$	1.75004	0.0708572
$611$	−0.00286696	−0.000115985 0
$612$	0	0
$613$	26.2005	1.05823	0.529115	−	0.848550i	$-0.322524\pi$
$613$	26.2005	1.05823	0.529115	+	0.848550i	$0.322524\pi$
$614$	−2.04074	−0.0823577
$615$	0	0
$616$	9.39752	0.378637
$617$	22.7521	0.915966	0.457983	−	0.888961i	$-0.348572\pi$
$617$	22.7521	0.915966	0.457983	+	0.888961i	$0.348572\pi$
$618$	0	0
$619$	−26.1884	−1.05260	−0.526300	−	0.850299i	$-0.676421\pi$
$619$	−26.1884	−1.05260	−0.526300	+	0.850299i	$0.676421\pi$
$620$	6.65692	0.267348
$621$	0	0
$622$	0.364635	0.0146205
$623$	2.68806	0.107695
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.49355	0.139630
$627$	0	0
$628$	2.90385	0.115876
$629$	63.8117	2.54434
$630$	0	0
$631$	43.0886	1.71533	0.857665	−	0.514209i	$-0.171915\pi$
$631$	43.0886	1.71533	0.857665	+	0.514209i	$0.171915\pi$
$632$	2.32693	0.0925605
$633$	0	0
$634$	4.25832	0.169119
$635$	−9.24081	−0.366710
$636$	0	0
$637$	1.39475	0.0552620
$638$	−7.42821	−0.294086
$639$	0	0
$640$	−5.65961	−0.223716
$641$	46.3238	1.82968	0.914840	−	0.403816i	$-0.132316\pi$
$641$	46.3238	1.82968	0.914840	+	0.403816i	$0.132316\pi$
$642$	0	0
$643$	23.7463	0.936462	0.468231	−	0.883606i	$-0.344892\pi$
$643$	23.7463	0.936462	0.468231	+	0.883606i	$0.344892\pi$
$644$	−3.66401	−0.144382
$645$	0	0
$646$	7.50263	0.295187
$647$	10.6381	0.418229	0.209114	−	0.977891i	$-0.432942\pi$
$647$	10.6381	0.418229	0.209114	+	0.977891i	$0.432942\pi$
$648$	0	0
$649$	−2.98856	−0.117311
$650$	−1.14114	−0.0447593
$651$	0	0
$652$	−7.18891	−0.281539
$653$	−34.6932	−1.35765	−0.678825	−	0.734300i	$-0.737511\pi$
$653$	−34.6932	−1.35765	−0.678825	+	0.734300i	$0.737511\pi$
$654$	0	0
$655$	17.2012	0.672107
$656$	9.58516	0.374238
$657$	0	0
$658$	0.000230267 0	8.97675e−6 0
$659$	−32.4911	−1.26567	−0.632836	−	0.774286i	$-0.718109\pi$
$659$	−32.4911	−1.26567	−0.632836	+	0.774286i	$0.718109\pi$
$660$	0	0
$661$	18.3885	0.715228	0.357614	−	0.933869i	$-0.383590\pi$
$661$	18.3885	0.715228	0.357614	+	0.933869i	$0.383590\pi$
$662$	−4.33214	−0.168374
$663$	0	0
$664$	3.80094	0.147505
$665$	−18.9566	−0.735107
$666$	0	0
$667$	5.84262	0.226227
$668$	36.8676	1.42645
$669$	0	0
$670$	−0.129355	−0.00499741
$671$	45.2449	1.74666
$672$	0	0
$673$	−3.33876	−0.128700	−0.0643498	−	0.997927i	$-0.520497\pi$
$673$	−3.33876	−0.128700	−0.0643498	+	0.997927i	$0.520497\pi$
$674$	−5.49966	−0.211839
$675$	0	0
$676$	−49.5247	−1.90479
$677$	−28.2976	−1.08757	−0.543784	−	0.839226i	$-0.683009\pi$
$677$	−28.2976	−1.08757	−0.543784	+	0.839226i	$0.683009\pi$
$678$	0	0
$679$	12.1121	0.464819
$680$	4.21923	0.161800
$681$	0	0
$682$	−2.98499	−0.114301
$683$	−10.4031	−0.398062	−0.199031	−	0.979993i	$-0.563779\pi$
$683$	−10.4031	−0.398062	−0.199031	+	0.979993i	$0.563779\pi$
$684$	0	0
$685$	−1.59260	−0.0608500
$686$	3.36248	0.128380
$687$	0	0
$688$	−15.5485	−0.592779
$689$	−30.0887	−1.14629
$690$	0	0
$691$	−27.7506	−1.05568	−0.527841	−	0.849343i	$-0.676998\pi$
$691$	−27.7506	−1.05568	−0.527841	+	0.849343i	$0.676998\pi$
$692$	17.4067	0.661704
$693$	0	0
$694$	6.00767	0.228048
$695$	−20.4231	−0.774693
$696$	0	0
$697$	−14.5460	−0.550968
$698$	−2.70258	−0.102294
$699$	0	0
$700$	−5.28447	−0.199734
$701$	−0.723879	−0.0273405	−0.0136703	−	0.999907i	$-0.504352\pi$
$701$	−0.723879	−0.0273405	−0.0136703	+	0.999907i	$0.504352\pi$
$702$	0	0
$703$	−78.1063	−2.94583
$704$	−34.3402	−1.29424
$705$	0	0
$706$	−5.25748	−0.197868
$707$	−40.7812	−1.53374
$708$	0	0
$709$	−16.0462	−0.602628	−0.301314	−	0.953525i	$-0.597425\pi$
$709$	−16.0462	−0.602628	−0.301314	+	0.953525i	$0.597425\pi$
$710$	−1.76477	−0.0662308
$711$	0	0
$712$	0.732314	0.0274446
$713$	2.34783	0.0879268
$714$	0	0
$715$	−29.5027	−1.10334
$716$	38.8797	1.45300
$717$	0	0
$718$	3.03094	0.113114
$719$	−14.5697	−0.543358	−0.271679	−	0.962388i	$-0.587579\pi$
$719$	−14.5697	−0.543358	−0.271679	+	0.962388i	$0.587579\pi$
$720$	0	0
$721$	−21.7825	−0.811223
$722$	−5.67490	−0.211198
$723$	0	0
$724$	9.78048	0.363489
$725$	8.42661	0.312956
$726$	0	0
$727$	28.4856	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$727$	28.4856	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$728$	12.1653	0.450875
$729$	0	0
$730$	1.83408	0.0678824
$731$	23.5956	0.872714
$732$	0	0
$733$	36.0947	1.33319	0.666594	−	0.745421i	$-0.267752\pi$
$733$	36.0947	1.33319	0.666594	+	0.745421i	$0.267752\pi$
$734$	−1.26296	−0.0466169
$735$	0	0
$736$	−1.50158	−0.0553490
$737$	−3.34429	−0.123188
$738$	0	0
$739$	−21.1001	−0.776180	−0.388090	−	0.921622i	$-0.626865\pi$
$739$	−21.1001	−0.776180	−0.388090	+	0.921622i	$0.626865\pi$
$740$	−21.7734	−0.800407
$741$	0	0
$742$	2.41665	0.0887179
$743$	30.0034	1.10072	0.550360	−	0.834928i	$-0.314491\pi$
$743$	30.0034	1.10072	0.550360	+	0.834928i	$0.314491\pi$
$744$	0	0
$745$	−15.7449	−0.576850
$746$	2.98135	0.109155
$747$	0	0
$748$	54.0723	1.97708
$749$	−13.1471	−0.480385
$750$	0	0
$751$	15.5615	0.567848	0.283924	−	0.958847i	$-0.408364\pi$
$751$	15.5615	0.567848	0.283924	+	0.958847i	$0.408364\pi$
$752$	−0.00176129	−6.42275e−5 0
$753$	0	0
$754$	−9.61597	−0.350193
$755$	8.23393	0.299664
$756$	0	0
$757$	−14.7246	−0.535173	−0.267587	−	0.963534i	$-0.586226\pi$
$757$	−14.7246	−0.535173	−0.267587	+	0.963534i	$0.586226\pi$
$758$	5.82485	0.211568
$759$	0	0
$760$	−5.16439	−0.187332
$761$	2.46904	0.0895025	0.0447513	−	0.998998i	$-0.485750\pi$
$761$	2.46904	0.0895025	0.0447513	+	0.998998i	$0.485750\pi$
$762$	0	0
$763$	1.16601	0.0422122
$764$	−41.2017	−1.49062
$765$	0	0
$766$	−3.21684	−0.116229
$767$	−3.86875	−0.139693
$768$	0	0
$769$	28.9418	1.04367	0.521834	−	0.853047i	$-0.325248\pi$
$769$	28.9418	1.04367	0.521834	+	0.853047i	$0.325248\pi$
$770$	2.36958	0.0853937
$771$	0	0
$772$	−6.55401	−0.235884
$773$	−1.45433	−0.0523087	−0.0261544	−	0.999658i	$-0.508326\pi$
$773$	−1.45433	−0.0523087	−0.0261544	+	0.999658i	$0.508326\pi$
$774$	0	0
$775$	3.38619	0.121635
$776$	3.29971	0.118453
$777$	0	0
$778$	3.08382	0.110560
$779$	17.8044	0.637910
$780$	0	0
$781$	−45.6258	−1.63262
$782$	0.737644	0.0263781
$783$	0	0
$784$	0.856848	0.0306017
$785$	1.47711	0.0527202
$786$	0	0
$787$	−53.2350	−1.89762	−0.948812	−	0.315842i	$-0.897713\pi$
$787$	−53.2350	−1.89762	−0.948812	+	0.315842i	$0.897713\pi$
$788$	−38.2847	−1.36384
$789$	0	0
$790$	0.586735	0.0208751
$791$	24.3216	0.864776
$792$	0	0
$793$	58.5705	2.07990
$794$	−4.39874	−0.156106
$795$	0	0
$796$	−40.0638	−1.42002
$797$	−35.0813	−1.24264	−0.621322	−	0.783555i	$-0.713404\pi$
$797$	−35.0813	−1.24264	−0.621322	+	0.783555i	$0.713404\pi$
$798$	0	0
$799$	0.00267284	9.45584e−5 0
$800$	−2.16568	−0.0765682
$801$	0	0
$802$	4.90987	0.173373
$803$	47.4176	1.67333
$804$	0	0
$805$	−1.86378	−0.0656896
$806$	−3.86413	−0.136108
$807$	0	0
$808$	−11.1101	−0.390852
$809$	11.0271	0.387694	0.193847	−	0.981032i	$-0.437903\pi$
$809$	11.0271	0.387694	0.193847	+	0.981032i	$0.437903\pi$
$810$	0	0
$811$	49.7416	1.74666	0.873331	−	0.487127i	$-0.161955\pi$
$811$	49.7416	1.74666	0.873331	+	0.487127i	$0.161955\pi$
$812$	−44.5302	−1.56270
$813$	0	0
$814$	9.76329	0.342203
$815$	−3.65680	−0.128092
$816$	0	0
$817$	−28.8813	−1.01043
$818$	6.86459	0.240015
$819$	0	0
$820$	4.96328	0.173325
$821$	0.510540	0.0178180	0.00890898	−	0.999960i	$-0.497164\pi$
$821$	0.510540	0.0178180	0.00890898	+	0.999960i	$0.497164\pi$
$822$	0	0
$823$	9.62127	0.335376	0.167688	−	0.985840i	$-0.446370\pi$
$823$	9.62127	0.335376	0.167688	+	0.985840i	$0.446370\pi$
$824$	−5.93424	−0.206729
$825$	0	0
$826$	0.310728	0.0108116
$827$	2.60332	0.0905263	0.0452632	−	0.998975i	$-0.485587\pi$
$827$	2.60332	0.0905263	0.0452632	+	0.998975i	$0.485587\pi$
$828$	0	0
$829$	−22.3197	−0.775197	−0.387598	−	0.921828i	$-0.626695\pi$
$829$	−22.3197	−0.775197	−0.387598	+	0.921828i	$0.626695\pi$
$830$	0.958404	0.0332667
$831$	0	0
$832$	−44.4541	−1.54117
$833$	−1.30031	−0.0450531
$834$	0	0
$835$	18.7535	0.648992
$836$	−66.1851	−2.28906
$837$	0	0
$838$	−1.53628	−0.0530698
$839$	−18.6074	−0.642398	−0.321199	−	0.947012i	$-0.604086\pi$
$839$	−18.6074	−0.642398	−0.321199	+	0.947012i	$0.604086\pi$
$840$	0	0
$841$	42.0077	1.44854
$842$	−4.42595	−0.152528
$843$	0	0
$844$	−20.1995	−0.695295
$845$	−25.1918	−0.866624
$846$	0	0
$847$	31.6935	1.08900
$848$	−18.4846	−0.634765
$849$	0	0
$850$	1.06388	0.0364907
$851$	−7.67926	−0.263242
$852$	0	0
$853$	−24.6245	−0.843125	−0.421563	−	0.906799i	$-0.638518\pi$
$853$	−24.6245	−0.843125	−0.421563	+	0.906799i	$0.638518\pi$
$854$	−4.70423	−0.160975
$855$	0	0
$856$	−3.58169	−0.122420
$857$	46.5289	1.58940	0.794698	−	0.607005i	$-0.207629\pi$
$857$	46.5289	1.58940	0.794698	+	0.607005i	$0.207629\pi$
$858$	0	0
$859$	22.0849	0.753528	0.376764	−	0.926309i	$-0.377037\pi$
$859$	22.0849	0.753528	0.376764	+	0.926309i	$0.377037\pi$
$860$	−8.05112	−0.274541
$861$	0	0
$862$	1.09097	0.0371586
$863$	43.1821	1.46994	0.734968	−	0.678102i	$-0.237197\pi$
$863$	43.1821	1.46994	0.734968	+	0.678102i	$0.237197\pi$
$864$	0	0
$865$	8.85431	0.301055
$866$	1.96583	0.0668017
$867$	0	0
$868$	−17.8942	−0.607370
$869$	15.1692	0.514580
$870$	0	0
$871$	−4.32925	−0.146691
$872$	0.317657	0.0107572
$873$	0	0
$874$	−0.902884	−0.0305405
$875$	−2.68806	−0.0908732
$876$	0	0
$877$	−11.6322	−0.392791	−0.196395	−	0.980525i	$-0.562924\pi$
$877$	−11.6322	−0.392791	−0.196395	+	0.980525i	$0.562924\pi$
$878$	2.49691	0.0842668
$879$	0	0
$880$	−18.1246	−0.610981
$881$	−25.6870	−0.865418	−0.432709	−	0.901534i	$-0.642442\pi$
$881$	−25.6870	−0.865418	−0.432709	+	0.901534i	$0.642442\pi$
$882$	0	0
$883$	−17.7170	−0.596224	−0.298112	−	0.954531i	$-0.596357\pi$
$883$	−17.7170	−0.596224	−0.298112	+	0.954531i	$0.596357\pi$
$884$	69.9977	2.35428
$885$	0	0
$886$	1.28536	0.0431826
$887$	8.83850	0.296768	0.148384	−	0.988930i	$-0.452593\pi$
$887$	8.83850	0.296768	0.148384	+	0.988930i	$0.452593\pi$
$888$	0	0
$889$	24.8399	0.833103
$890$	0.184653	0.00618957
$891$	0	0
$892$	47.6785	1.59639
$893$	−0.00327159	−0.000109480 0
$894$	0	0
$895$	19.7770	0.661073
$896$	15.2134	0.508244
$897$	0	0
$898$	2.54034	0.0847722
$899$	28.5341	0.951665
$900$	0	0
$901$	28.0514	0.934527
$902$	−2.22555	−0.0741028
$903$	0	0
$904$	6.62597	0.220376
$905$	4.97505	0.165376
$906$	0	0
$907$	14.1203	0.468858	0.234429	−	0.972133i	$-0.424678\pi$
$907$	14.1203	0.468858	0.234429	+	0.972133i	$0.424678\pi$
$908$	31.1307	1.03311
$909$	0	0
$910$	3.06747	0.101686
$911$	20.4661	0.678072	0.339036	−	0.940773i	$-0.389899\pi$
$911$	20.4661	0.678072	0.339036	+	0.940773i	$0.389899\pi$
$912$	0	0
$913$	24.7782	0.820039
$914$	1.52636	0.0504875
$915$	0	0
$916$	15.2371	0.503447
$917$	−46.2380	−1.52691
$918$	0	0
$919$	−20.3308	−0.670651	−0.335325	−	0.942102i	$-0.608846\pi$
$919$	−20.3308	−0.670651	−0.335325	+	0.942102i	$0.608846\pi$
$920$	−0.507753	−0.0167401
$921$	0	0
$922$	−6.68111	−0.220031
$923$	−59.0634	−1.94410
$924$	0	0
$925$	−11.0755	−0.364161
$926$	−3.67776	−0.120859
$927$	0	0
$928$	−18.2493	−0.599063
$929$	−28.3276	−0.929397	−0.464699	−	0.885469i	$-0.653837\pi$
$929$	−28.3276	−0.929397	−0.464699	+	0.885469i	$0.653837\pi$
$930$	0	0
$931$	1.59160	0.0521624
$932$	−58.9593	−1.93128
$933$	0	0
$934$	−5.47702	−0.179214
$935$	27.5051	0.899512
$936$	0	0
$937$	52.6912	1.72135	0.860673	−	0.509158i	$-0.170043\pi$
$937$	52.6912	1.72135	0.860673	+	0.509158i	$0.170043\pi$
$938$	0.347714	0.0113533
$939$	0	0
$940$	−0.000912009 0	−2.97465e−5 0
$941$	−8.46142	−0.275834	−0.137917	−	0.990444i	$-0.544041\pi$
$941$	−8.46142	−0.275834	−0.137917	+	0.990444i	$0.544041\pi$
$942$	0	0
$943$	1.75050	0.0570040
$944$	−2.37672	−0.0773558
$945$	0	0
$946$	3.61016	0.117376
$947$	39.8894	1.29623	0.648116	−	0.761542i	$-0.275557\pi$
$947$	39.8894	1.29623	0.648116	+	0.761542i	$0.275557\pi$
$948$	0	0
$949$	61.3831	1.99258
$950$	−1.30220	−0.0422489
$951$	0	0
$952$	−11.3416	−0.367583
$953$	−26.4028	−0.855271	−0.427635	−	0.903951i	$-0.640653\pi$
$953$	−26.4028	−0.855271	−0.427635	+	0.903951i	$0.640653\pi$
$954$	0	0
$955$	−20.9581	−0.678190
$956$	−34.5126	−1.11622
$957$	0	0
$958$	0.972425	0.0314176
$959$	4.28100	0.138241
$960$	0	0
$961$	−19.5337	−0.630120
$962$	12.6388	0.407490
$963$	0	0
$964$	15.9271	0.512976
$965$	−3.33384	−0.107320
$966$	0	0
$967$	16.6234	0.534574	0.267287	−	0.963617i	$-0.413873\pi$
$967$	16.6234	0.534574	0.267287	+	0.963617i	$0.413873\pi$
$968$	8.63431	0.277517
$969$	0	0
$970$	0.832020	0.0267146
$971$	35.2837	1.13231	0.566154	−	0.824299i	$-0.308431\pi$
$971$	35.2837	1.13231	0.566154	+	0.824299i	$0.308431\pi$
$972$	0	0
$973$	54.8987	1.75997
$974$	4.91765	0.157572
$975$	0	0
$976$	35.9821	1.15176
$977$	25.3175	0.809979	0.404989	−	0.914321i	$-0.367275\pi$
$977$	25.3175	0.809979	0.404989	+	0.914321i	$0.367275\pi$
$978$	0	0
$979$	4.77393	0.152576
$980$	0.443683	0.0141729
$981$	0	0
$982$	−4.13361	−0.131909
$983$	42.5424	1.35689	0.678445	−	0.734651i	$-0.262654\pi$
$983$	42.5424	1.35689	0.678445	+	0.734651i	$0.262654\pi$
$984$	0	0
$985$	−19.4744	−0.620504
$986$	8.96488	0.285500
$987$	0	0
$988$	−85.6779	−2.72578
$989$	−2.83955	−0.0902924
$990$	0	0
$991$	47.5917	1.51180	0.755900	−	0.654687i	$-0.227200\pi$
$991$	47.5917	1.51180	0.755900	+	0.654687i	$0.227200\pi$
$992$	−7.33339	−0.232835
$993$	0	0
$994$	4.74382	0.150465
$995$	−20.3793	−0.646068
$996$	0	0
$997$	−7.77852	−0.246348	−0.123174	−	0.992385i	$-0.539307\pi$
$997$	−7.77852	−0.246348	−0.123174	+	0.992385i	$0.539307\pi$
$998$	0.206369	0.00653251
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.w.1.9	✓	17
3.2	odd	2		4005.2.a.x.1.9	yes	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.9	✓	17	1.1	even	1	trivial
4005.2.a.x.1.9	yes	17	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.184653 q^{2} -1.96590 q^{4} -1.00000 q^{5} +2.68806 q^{7} +0.732314 q^{8} +O(q^{10})\)	\(q-0.184653 q^{2} -1.96590 q^{4} -1.00000 q^{5} +2.68806 q^{7} +0.732314 q^{8} +0.184653 q^{10} +4.77393 q^{11} +6.17995 q^{13} -0.496358 q^{14} +3.79658 q^{16} -5.76151 q^{17} +7.05215 q^{19} +1.96590 q^{20} -0.881519 q^{22} +0.693354 q^{23} +1.00000 q^{25} -1.14114 q^{26} -5.28447 q^{28} +8.42661 q^{29} +3.38619 q^{31} -2.16568 q^{32} +1.06388 q^{34} -2.68806 q^{35} -11.0755 q^{37} -1.30220 q^{38} -0.732314 q^{40} +2.52468 q^{41} -4.09538 q^{43} -9.38509 q^{44} -0.128030 q^{46} -0.000463914 q^{47} +0.225689 q^{49} -0.184653 q^{50} -12.1492 q^{52} -4.86876 q^{53} -4.77393 q^{55} +1.96851 q^{56} -1.55599 q^{58} -0.626017 q^{59} +9.47749 q^{61} -0.625268 q^{62} -7.19327 q^{64} -6.17995 q^{65} -0.700531 q^{67} +11.3266 q^{68} +0.496358 q^{70} -9.55727 q^{71} +9.93261 q^{73} +2.04512 q^{74} -13.8639 q^{76} +12.8326 q^{77} +3.17751 q^{79} -3.79658 q^{80} -0.466189 q^{82} +5.19031 q^{83} +5.76151 q^{85} +0.756222 q^{86} +3.49602 q^{88} +1.00000 q^{89} +16.6121 q^{91} -1.36307 q^{92} +8.56628e-5 q^{94} -7.05215 q^{95} +4.50587 q^{97} -0.0416741 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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