LMFDB - Embedded newform 4005.2.a.x.1.7

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			1.7
Root			$-0.320785$ 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)

$f(q)$	$=$	$q-0.320785 q^{2} -1.89710 q^{4} +1.00000 q^{5} +4.94917 q^{7} +1.25013 q^{8} +O(q^{10})$	\(q-0.320785 q^{2} -1.89710 q^{4} +1.00000 q^{5} +4.94917 q^{7} +1.25013 q^{8} -0.320785 q^{10} +5.87943 q^{11} -1.94474 q^{13} -1.58762 q^{14} +3.39317 q^{16} +2.22790 q^{17} +2.30334 q^{19} -1.89710 q^{20} -1.88603 q^{22} +6.59732 q^{23} +1.00000 q^{25} +0.623844 q^{26} -9.38905 q^{28} -1.58752 q^{29} +4.58756 q^{31} -3.58874 q^{32} -0.714676 q^{34} +4.94917 q^{35} +7.15048 q^{37} -0.738877 q^{38} +1.25013 q^{40} -0.124430 q^{41} +1.60004 q^{43} -11.1538 q^{44} -2.11632 q^{46} -0.375166 q^{47} +17.4943 q^{49} -0.320785 q^{50} +3.68936 q^{52} -1.94430 q^{53} +5.87943 q^{55} +6.18711 q^{56} +0.509253 q^{58} -13.0590 q^{59} -10.7175 q^{61} -1.47162 q^{62} -5.63513 q^{64} -1.94474 q^{65} +1.45401 q^{67} -4.22654 q^{68} -1.58762 q^{70} +3.17712 q^{71} -7.38258 q^{73} -2.29377 q^{74} -4.36966 q^{76} +29.0983 q^{77} -16.1705 q^{79} +3.39317 q^{80} +0.0399154 q^{82} -15.6129 q^{83} +2.22790 q^{85} -0.513270 q^{86} +7.35005 q^{88} -1.00000 q^{89} -9.62484 q^{91} -12.5158 q^{92} +0.120348 q^{94} +2.30334 q^{95} +0.637546 q^{97} -5.61190 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.320785	−0.226829	−0.113415	−	0.993548i	$-0.536179\pi$
$2$	−0.320785	−0.226829	−0.113415	+	0.993548i	$0.536179\pi$
$3$	0	0
$3$	0	0
$4$	−1.89710	−0.948548
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.94917	1.87061	0.935305	−	0.353843i	$-0.115125\pi$
$7$	4.94917	1.87061	0.935305	+	0.353843i	$0.115125\pi$
$8$	1.25013	0.441988
$9$	0	0
$10$	−0.320785	−0.101441
$11$	5.87943	1.77271	0.886357	−	0.463002i	$-0.153228\pi$
$11$	5.87943	1.77271	0.886357	+	0.463002i	$0.153228\pi$
$12$	0	0
$13$	−1.94474	−0.539374	−0.269687	−	0.962948i	$-0.586920\pi$
$13$	−1.94474	−0.539374	−0.269687	+	0.962948i	$0.586920\pi$
$14$	−1.58762	−0.424309
$15$	0	0
$16$	3.39317	0.848293
$17$	2.22790	0.540345	0.270172	−	0.962812i	$-0.412919\pi$
$17$	2.22790	0.540345	0.270172	+	0.962812i	$0.412919\pi$
$18$	0	0
$19$	2.30334	0.528422	0.264211	−	0.964465i	$-0.414888\pi$
$19$	2.30334	0.528422	0.264211	+	0.964465i	$0.414888\pi$
$20$	−1.89710	−0.424204
$21$	0	0
$22$	−1.88603	−0.402104
$23$	6.59732	1.37564	0.687818	−	0.725883i	$-0.258569\pi$
$23$	6.59732	1.37564	0.687818	+	0.725883i	$0.258569\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.623844	0.122346
$27$	0	0
$28$	−9.38905	−1.77436
$29$	−1.58752	−0.294795	−0.147398	−	0.989077i	$-0.547090\pi$
$29$	−1.58752	−0.294795	−0.147398	+	0.989077i	$0.547090\pi$
$30$	0	0
$31$	4.58756	0.823951	0.411975	−	0.911195i	$-0.364839\pi$
$31$	4.58756	0.823951	0.411975	+	0.911195i	$0.364839\pi$
$32$	−3.58874	−0.634405
$33$	0	0
$34$	−0.714676	−0.122566
$35$	4.94917	0.836562
$36$	0	0
$37$	7.15048	1.17553	0.587766	−	0.809031i	$-0.300008\pi$
$37$	7.15048	1.17553	0.587766	+	0.809031i	$0.300008\pi$
$38$	−0.738877	−0.119862
$39$	0	0
$40$	1.25013	0.197663
$41$	−0.124430	−0.0194328	−0.00971638	−	0.999953i	$-0.503093\pi$
$41$	−0.124430	−0.0194328	−0.00971638	+	0.999953i	$0.503093\pi$
$42$	0	0
$43$	1.60004	0.244004	0.122002	−	0.992530i	$-0.461069\pi$
$43$	1.60004	0.244004	0.122002	+	0.992530i	$0.461069\pi$
$44$	−11.1538	−1.68151
$45$	0	0
$46$	−2.11632	−0.312034
$47$	−0.375166	−0.0547236	−0.0273618	−	0.999626i	$-0.508711\pi$
$47$	−0.375166	−0.0547236	−0.0273618	+	0.999626i	$0.508711\pi$
$48$	0	0
$49$	17.4943	2.49918
$50$	−0.320785	−0.0453659
$51$	0	0
$52$	3.68936	0.511622
$53$	−1.94430	−0.267070	−0.133535	−	0.991044i	$-0.542633\pi$
$53$	−1.94430	−0.267070	−0.133535	+	0.991044i	$0.542633\pi$
$54$	0	0
$55$	5.87943	0.792782
$56$	6.18711	0.826787
$57$	0	0
$58$	0.509253	0.0668682
$59$	−13.0590	−1.70014	−0.850070	−	0.526670i	$-0.823440\pi$
$59$	−13.0590	−1.70014	−0.850070	+	0.526670i	$0.823440\pi$
$60$	0	0
$61$	−10.7175	−1.37224	−0.686120	−	0.727488i	$-0.740687\pi$
$61$	−10.7175	−1.37224	−0.686120	+	0.727488i	$0.740687\pi$
$62$	−1.47162	−0.186896
$63$	0	0
$64$	−5.63513	−0.704391
$65$	−1.94474	−0.241215
$66$	0	0
$67$	1.45401	0.177635	0.0888175	−	0.996048i	$-0.471691\pi$
$67$	1.45401	0.177635	0.0888175	+	0.996048i	$0.471691\pi$
$68$	−4.22654	−0.512543
$69$	0	0
$70$	−1.58762	−0.189757
$71$	3.17712	0.377055	0.188527	−	0.982068i	$-0.439629\pi$
$71$	3.17712	0.377055	0.188527	+	0.982068i	$0.439629\pi$
$72$	0	0
$73$	−7.38258	−0.864065	−0.432033	−	0.901858i	$-0.642203\pi$
$73$	−7.38258	−0.864065	−0.432033	+	0.901858i	$0.642203\pi$
$74$	−2.29377	−0.266645
$75$	0	0
$76$	−4.36966	−0.501234
$77$	29.0983	3.31606
$78$	0	0
$79$	−16.1705	−1.81932	−0.909659	−	0.415355i	$-0.863657\pi$
$79$	−16.1705	−1.81932	−0.909659	+	0.415355i	$0.863657\pi$
$80$	3.39317	0.379368
$81$	0	0
$82$	0.0399154	0.00440792
$83$	−15.6129	−1.71374	−0.856869	−	0.515533i	$-0.827594\pi$
$83$	−15.6129	−1.71374	−0.856869	+	0.515533i	$0.827594\pi$
$84$	0	0
$85$	2.22790	0.241649
$86$	−0.513270	−0.0553473
$87$	0	0
$88$	7.35005	0.783518
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−9.62484	−1.00896
$92$	−12.5158	−1.30486
$93$	0	0
$94$	0.120348	0.0124129
$95$	2.30334	0.236318
$96$	0	0
$97$	0.637546	0.0647329	0.0323665	−	0.999476i	$-0.489696\pi$
$97$	0.637546	0.0647329	0.0323665	+	0.999476i	$0.489696\pi$
$98$	−5.61190	−0.566887
$99$	0	0
$100$	−1.89710	−0.189710
$101$	10.0214	0.997169	0.498584	−	0.866841i	$-0.333853\pi$
$101$	10.0214	0.997169	0.498584	+	0.866841i	$0.333853\pi$
$102$	0	0
$103$	6.52538	0.642965	0.321482	−	0.946916i	$-0.395819\pi$
$103$	6.52538	0.642965	0.321482	+	0.946916i	$0.395819\pi$
$104$	−2.43118	−0.238397
$105$	0	0
$106$	0.623702	0.0605793
$107$	−14.2067	−1.37342	−0.686708	−	0.726933i	$-0.740945\pi$
$107$	−14.2067	−1.37342	−0.686708	+	0.726933i	$0.740945\pi$
$108$	0	0
$109$	7.28770	0.698035	0.349018	−	0.937116i	$-0.386515\pi$
$109$	7.28770	0.698035	0.349018	+	0.937116i	$0.386515\pi$
$110$	−1.88603	−0.179826
$111$	0	0
$112$	16.7934	1.58682
$113$	−17.5274	−1.64884	−0.824419	−	0.565979i	$-0.808498\pi$
$113$	−17.5274	−1.64884	−0.824419	+	0.565979i	$0.808498\pi$
$114$	0	0
$115$	6.59732	0.615203
$116$	3.01168	0.279628
$117$	0	0
$118$	4.18914	0.385642
$119$	11.0262	1.01077
$120$	0	0
$121$	23.5677	2.14252
$122$	3.43803	0.311264
$123$	0	0
$124$	−8.70305	−0.781557
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.8021	−1.66842	−0.834208	−	0.551450i	$-0.814075\pi$
$127$	−18.8021	−1.66842	−0.834208	+	0.551450i	$0.814075\pi$
$128$	8.98514	0.794182
$129$	0	0
$130$	0.623844	0.0547147
$131$	−3.23011	−0.282216	−0.141108	−	0.989994i	$-0.545066\pi$
$131$	−3.23011	−0.282216	−0.141108	+	0.989994i	$0.545066\pi$
$132$	0	0
$133$	11.3996	0.988472
$134$	−0.466423	−0.0402928
$135$	0	0
$136$	2.78516	0.238826
$137$	−17.8191	−1.52239	−0.761193	−	0.648525i	$-0.775386\pi$
$137$	−17.8191	−1.52239	−0.761193	+	0.648525i	$0.775386\pi$
$138$	0	0
$139$	8.62523	0.731583	0.365791	−	0.930697i	$-0.380798\pi$
$139$	8.62523	0.731583	0.365791	+	0.930697i	$0.380798\pi$
$140$	−9.38905	−0.793520
$141$	0	0
$142$	−1.01917	−0.0855271
$143$	−11.4340	−0.956156
$144$	0	0
$145$	−1.58752	−0.131837
$146$	2.36822	0.195995
$147$	0	0
$148$	−13.5652	−1.11505
$149$	5.37886	0.440653	0.220327	−	0.975426i	$-0.429288\pi$
$149$	5.37886	0.440653	0.220327	+	0.975426i	$0.429288\pi$
$150$	0	0
$151$	14.2223	1.15739	0.578696	−	0.815543i	$-0.303562\pi$
$151$	14.2223	1.15739	0.578696	+	0.815543i	$0.303562\pi$
$152$	2.87947	0.233556
$153$	0	0
$154$	−9.33430	−0.752179
$155$	4.58756	0.368482
$156$	0	0
$157$	20.5489	1.63998	0.819991	−	0.572376i	$-0.193978\pi$
$157$	20.5489	1.63998	0.819991	+	0.572376i	$0.193978\pi$
$158$	5.18724	0.412675
$159$	0	0
$160$	−3.58874	−0.283715
$161$	32.6512	2.57328
$162$	0	0
$163$	−7.31133	−0.572668	−0.286334	−	0.958130i	$-0.592437\pi$
$163$	−7.31133	−0.572668	−0.286334	+	0.958130i	$0.592437\pi$
$164$	0.236056	0.0184329
$165$	0	0
$166$	5.00839	0.388726
$167$	−6.56546	−0.508050	−0.254025	−	0.967198i	$-0.581755\pi$
$167$	−6.56546	−0.508050	−0.254025	+	0.967198i	$0.581755\pi$
$168$	0	0
$169$	−9.21799	−0.709076
$170$	−0.714676	−0.0548132
$171$	0	0
$172$	−3.03544	−0.231450
$173$	−14.0582	−1.06882	−0.534411	−	0.845225i	$-0.679466\pi$
$173$	−14.0582	−1.06882	−0.534411	+	0.845225i	$0.679466\pi$
$174$	0	0
$175$	4.94917	0.374122
$176$	19.9499	1.50378
$177$	0	0
$178$	0.320785	0.0240439
$179$	−8.78192	−0.656391	−0.328196	−	0.944610i	$-0.606441\pi$
$179$	−8.78192	−0.656391	−0.328196	+	0.944610i	$0.606441\pi$
$180$	0	0
$181$	7.81407	0.580815	0.290408	−	0.956903i	$-0.406209\pi$
$181$	7.81407	0.580815	0.290408	+	0.956903i	$0.406209\pi$
$182$	3.08751	0.228861
$183$	0	0
$184$	8.24751	0.608014
$185$	7.15048	0.525714
$186$	0	0
$187$	13.0988	0.957877
$188$	0.711727	0.0519080
$189$	0	0
$190$	−0.738877	−0.0536038
$191$	−19.3352	−1.39904	−0.699522	−	0.714612i	$-0.746603\pi$
$191$	−19.3352	−1.39904	−0.699522	+	0.714612i	$0.746603\pi$
$192$	0	0
$193$	17.4747	1.25785	0.628927	−	0.777465i	$-0.283495\pi$
$193$	17.4747	1.25785	0.628927	+	0.777465i	$0.283495\pi$
$194$	−0.204515	−0.0146833
$195$	0	0
$196$	−33.1883	−2.37059
$197$	−0.419335	−0.0298764	−0.0149382	−	0.999888i	$-0.504755\pi$
$197$	−0.419335	−0.0298764	−0.0149382	+	0.999888i	$0.504755\pi$
$198$	0	0
$199$	−8.00934	−0.567767	−0.283884	−	0.958859i	$-0.591623\pi$
$199$	−8.00934	−0.567767	−0.283884	+	0.958859i	$0.591623\pi$
$200$	1.25013	0.0883976
$201$	0	0
$202$	−3.21472	−0.226187
$203$	−7.85691	−0.551447
$204$	0	0
$205$	−0.124430	−0.00869059
$206$	−2.09324	−0.145843
$207$	0	0
$208$	−6.59883	−0.457547
$209$	13.5423	0.936742
$210$	0	0
$211$	28.8188	1.98397	0.991983	−	0.126372i	$-0.0403334\pi$
$211$	28.8188	1.98397	0.991983	+	0.126372i	$0.0403334\pi$
$212$	3.68852	0.253329
$213$	0	0
$214$	4.55731	0.311531
$215$	1.60004	0.109122
$216$	0	0
$217$	22.7046	1.54129
$218$	−2.33779	−0.158335
$219$	0	0
$220$	−11.1538	−0.751992
$221$	−4.33268	−0.291448
$222$	0	0
$223$	−1.27398	−0.0853123	−0.0426562	−	0.999090i	$-0.513582\pi$
$223$	−1.27398	−0.0853123	−0.0426562	+	0.999090i	$0.513582\pi$
$224$	−17.7613	−1.18672
$225$	0	0
$226$	5.62253	0.374005
$227$	9.85955	0.654401	0.327201	−	0.944955i	$-0.393895\pi$
$227$	9.85955	0.654401	0.327201	+	0.944955i	$0.393895\pi$
$228$	0	0
$229$	21.4183	1.41536	0.707679	−	0.706534i	$-0.249742\pi$
$229$	21.4183	1.41536	0.707679	+	0.706534i	$0.249742\pi$
$230$	−2.11632	−0.139546
$231$	0	0
$232$	−1.98461	−0.130296
$233$	7.17229	0.469872	0.234936	−	0.972011i	$-0.424512\pi$
$233$	7.17229	0.469872	0.234936	+	0.972011i	$0.424512\pi$
$234$	0	0
$235$	−0.375166	−0.0244731
$236$	24.7742	1.61267
$237$	0	0
$238$	−3.53705	−0.229273
$239$	−6.72584	−0.435058	−0.217529	−	0.976054i	$-0.569800\pi$
$239$	−6.72584	−0.435058	−0.217529	+	0.976054i	$0.569800\pi$
$240$	0	0
$241$	−27.2748	−1.75692	−0.878461	−	0.477814i	$-0.841429\pi$
$241$	−27.2748	−1.75692	−0.878461	+	0.477814i	$0.841429\pi$
$242$	−7.56017	−0.485986
$243$	0	0
$244$	20.3322	1.30164
$245$	17.4943	1.11767
$246$	0	0
$247$	−4.47940	−0.285017
$248$	5.73505	0.364176
$249$	0	0
$250$	−0.320785	−0.0202882
$251$	−7.27729	−0.459338	−0.229669	−	0.973269i	$-0.573764\pi$
$251$	−7.27729	−0.459338	−0.229669	+	0.973269i	$0.573764\pi$
$252$	0	0
$253$	38.7885	2.43861
$254$	6.03143	0.378446
$255$	0	0
$256$	8.38796	0.524247
$257$	8.60297	0.536638	0.268319	−	0.963330i	$-0.413532\pi$
$257$	8.60297	0.536638	0.268319	+	0.963330i	$0.413532\pi$
$258$	0	0
$259$	35.3889	2.19896
$260$	3.68936	0.228804
$261$	0	0
$262$	1.03617	0.0640148
$263$	17.3370	1.06905	0.534523	−	0.845154i	$-0.320491\pi$
$263$	17.3370	1.06905	0.534523	+	0.845154i	$0.320491\pi$
$264$	0	0
$265$	−1.94430	−0.119437
$266$	−3.65683	−0.224214
$267$	0	0
$268$	−2.75839	−0.168495
$269$	13.7941	0.841040	0.420520	−	0.907283i	$-0.361848\pi$
$269$	13.7941	0.841040	0.420520	+	0.907283i	$0.361848\pi$
$270$	0	0
$271$	−19.4639	−1.18235	−0.591175	−	0.806543i	$-0.701336\pi$
$271$	−19.4639	−1.18235	−0.591175	+	0.806543i	$0.701336\pi$
$272$	7.55964	0.458370
$273$	0	0
$274$	5.71610	0.345322
$275$	5.87943	0.354543
$276$	0	0
$277$	14.0990	0.847125	0.423562	−	0.905867i	$-0.360779\pi$
$277$	14.0990	0.847125	0.423562	+	0.905867i	$0.360779\pi$
$278$	−2.76685	−0.165944
$279$	0	0
$280$	6.18711	0.369750
$281$	−13.4962	−0.805118	−0.402559	−	0.915394i	$-0.631879\pi$
$281$	−13.4962	−0.805118	−0.402559	+	0.915394i	$0.631879\pi$
$282$	0	0
$283$	20.4543	1.21588	0.607942	−	0.793982i	$-0.291995\pi$
$283$	20.4543	1.21588	0.607942	+	0.793982i	$0.291995\pi$
$284$	−6.02731	−0.357655
$285$	0	0
$286$	3.66784	0.216884
$287$	−0.615827	−0.0363511
$288$	0	0
$289$	−12.0365	−0.708028
$290$	0.509253	0.0299044
$291$	0	0
$292$	14.0055	0.819608
$293$	−18.8803	−1.10300	−0.551500	−	0.834175i	$-0.685944\pi$
$293$	−18.8803	−1.10300	−0.551500	+	0.834175i	$0.685944\pi$
$294$	0	0
$295$	−13.0590	−0.760326
$296$	8.93904	0.519571
$297$	0	0
$298$	−1.72546	−0.0999530
$299$	−12.8301	−0.741982
$300$	0	0
$301$	7.91888	0.456437
$302$	−4.56230	−0.262531
$303$	0	0
$304$	7.81562	0.448257
$305$	−10.7175	−0.613684
$306$	0	0
$307$	1.82406	0.104105	0.0520523	−	0.998644i	$-0.483424\pi$
$307$	1.82406	0.104105	0.0520523	+	0.998644i	$0.483424\pi$
$308$	−55.2023	−3.14544
$309$	0	0
$310$	−1.47162	−0.0835825
$311$	17.8324	1.01119	0.505593	−	0.862772i	$-0.331274\pi$
$311$	17.8324	1.01119	0.505593	+	0.862772i	$0.331274\pi$
$312$	0	0
$313$	9.88377	0.558664	0.279332	−	0.960195i	$-0.409887\pi$
$313$	9.88377	0.558664	0.279332	+	0.960195i	$0.409887\pi$
$314$	−6.59179	−0.371996
$315$	0	0
$316$	30.6769	1.72571
$317$	9.00638	0.505849	0.252924	−	0.967486i	$-0.418608\pi$
$317$	9.00638	0.505849	0.252924	+	0.967486i	$0.418608\pi$
$318$	0	0
$319$	−9.33372	−0.522588
$320$	−5.63513	−0.315013
$321$	0	0
$322$	−10.4740	−0.583695
$323$	5.13160	0.285530
$324$	0	0
$325$	−1.94474	−0.107875
$326$	2.34537	0.129898
$327$	0	0
$328$	−0.155554	−0.00858904
$329$	−1.85676	−0.102367
$330$	0	0
$331$	−5.81108	−0.319406	−0.159703	−	0.987165i	$-0.551054\pi$
$331$	−5.81108	−0.319406	−0.159703	+	0.987165i	$0.551054\pi$
$332$	29.6192	1.62556
$333$	0	0
$334$	2.10610	0.115241
$335$	1.45401	0.0794408
$336$	0	0
$337$	15.2395	0.830151	0.415075	−	0.909787i	$-0.363755\pi$
$337$	15.2395	0.830151	0.415075	+	0.909787i	$0.363755\pi$
$338$	2.95699	0.160839
$339$	0	0
$340$	−4.22654	−0.229216
$341$	26.9723	1.46063
$342$	0	0
$343$	51.9379	2.80438
$344$	2.00026	0.107847
$345$	0	0
$346$	4.50965	0.242440
$347$	−4.48589	−0.240815	−0.120408	−	0.992725i	$-0.538420\pi$
$347$	−4.48589	−0.240815	−0.120408	+	0.992725i	$0.538420\pi$
$348$	0	0
$349$	26.3460	1.41027	0.705134	−	0.709074i	$-0.250887\pi$
$349$	26.3460	1.41027	0.705134	+	0.709074i	$0.250887\pi$
$350$	−1.58762	−0.0848618
$351$	0	0
$352$	−21.0997	−1.12462
$353$	−19.9944	−1.06420	−0.532098	−	0.846683i	$-0.678596\pi$
$353$	−19.9944	−1.06420	−0.532098	+	0.846683i	$0.678596\pi$
$354$	0	0
$355$	3.17712	0.168624
$356$	1.89710	0.100546
$357$	0	0
$358$	2.81711	0.148889
$359$	−27.9253	−1.47384	−0.736920	−	0.675979i	$-0.763721\pi$
$359$	−27.9253	−1.47384	−0.736920	+	0.675979i	$0.763721\pi$
$360$	0	0
$361$	−13.6946	−0.720770
$362$	−2.50664	−0.131746
$363$	0	0
$364$	18.2593	0.957045
$365$	−7.38258	−0.386422
$366$	0	0
$367$	−19.2231	−1.00344	−0.501720	−	0.865030i	$-0.667299\pi$
$367$	−19.2231	−1.00344	−0.501720	+	0.865030i	$0.667299\pi$
$368$	22.3858	1.16694
$369$	0	0
$370$	−2.29377	−0.119247
$371$	−9.62266	−0.499583
$372$	0	0
$373$	−13.5107	−0.699556	−0.349778	−	0.936833i	$-0.613743\pi$
$373$	−13.5107	−0.699556	−0.349778	+	0.936833i	$0.613743\pi$
$374$	−4.20189	−0.217275
$375$	0	0
$376$	−0.469007	−0.0241872
$377$	3.08732	0.159005
$378$	0	0
$379$	−8.00125	−0.410997	−0.205498	−	0.978657i	$-0.565881\pi$
$379$	−8.00125	−0.410997	−0.205498	+	0.978657i	$0.565881\pi$
$380$	−4.36966	−0.224159
$381$	0	0
$382$	6.20243	0.317344
$383$	13.2776	0.678454	0.339227	−	0.940705i	$-0.389834\pi$
$383$	13.2776	0.678454	0.339227	+	0.940705i	$0.389834\pi$
$384$	0	0
$385$	29.0983	1.48299
$386$	−5.60561	−0.285318
$387$	0	0
$388$	−1.20949	−0.0614023
$389$	3.69240	0.187212	0.0936060	−	0.995609i	$-0.470161\pi$
$389$	3.69240	0.187212	0.0936060	+	0.995609i	$0.470161\pi$
$390$	0	0
$391$	14.6981	0.743317
$392$	21.8701	1.10461
$393$	0	0
$394$	0.134517	0.00677684
$395$	−16.1705	−0.813624
$396$	0	0
$397$	9.86950	0.495336	0.247668	−	0.968845i	$-0.420336\pi$
$397$	9.86950	0.495336	0.247668	+	0.968845i	$0.420336\pi$
$398$	2.56928	0.128786
$399$	0	0
$400$	3.39317	0.169659
$401$	−10.7734	−0.537997	−0.268999	−	0.963141i	$-0.586693\pi$
$401$	−10.7734	−0.537997	−0.268999	+	0.963141i	$0.586693\pi$
$402$	0	0
$403$	−8.92162	−0.444418
$404$	−19.0116	−0.945863
$405$	0	0
$406$	2.52038	0.125084
$407$	42.0408	2.08388
$408$	0	0
$409$	−1.67665	−0.0829052	−0.0414526	−	0.999140i	$-0.513199\pi$
$409$	−1.67665	−0.0829052	−0.0414526	+	0.999140i	$0.513199\pi$
$410$	0.0399154	0.00197128
$411$	0	0
$412$	−12.3793	−0.609883
$413$	−64.6313	−3.18030
$414$	0	0
$415$	−15.6129	−0.766407
$416$	6.97916	0.342182
$417$	0	0
$418$	−4.34417	−0.212481
$419$	25.7607	1.25849	0.629245	−	0.777207i	$-0.283364\pi$
$419$	25.7607	1.25849	0.629245	+	0.777207i	$0.283364\pi$
$420$	0	0
$421$	−23.1878	−1.13010	−0.565051	−	0.825056i	$-0.691144\pi$
$421$	−23.1878	−1.13010	−0.565051	+	0.825056i	$0.691144\pi$
$422$	−9.24463	−0.450022
$423$	0	0
$424$	−2.43063	−0.118042
$425$	2.22790	0.108069
$426$	0	0
$427$	−53.0429	−2.56692
$428$	26.9516	1.30275
$429$	0	0
$430$	−0.513270	−0.0247521
$431$	−26.9715	−1.29917	−0.649586	−	0.760288i	$-0.725058\pi$
$431$	−26.9715	−1.29917	−0.649586	+	0.760288i	$0.725058\pi$
$432$	0	0
$433$	−21.3959	−1.02822	−0.514112	−	0.857723i	$-0.671878\pi$
$433$	−21.3959	−1.02822	−0.514112	+	0.857723i	$0.671878\pi$
$434$	−7.28330	−0.349610
$435$	0	0
$436$	−13.8255	−0.662120
$437$	15.1959	0.726917
$438$	0	0
$439$	9.48123	0.452514	0.226257	−	0.974068i	$-0.427351\pi$
$439$	9.48123	0.452514	0.226257	+	0.974068i	$0.427351\pi$
$440$	7.35005	0.350400
$441$	0	0
$442$	1.38986	0.0661089
$443$	6.33495	0.300983	0.150491	−	0.988611i	$-0.451914\pi$
$443$	6.33495	0.300983	0.150491	+	0.988611i	$0.451914\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	0.408675	0.0193513
$447$	0	0
$448$	−27.8892	−1.31764
$449$	−19.5223	−0.921313	−0.460657	−	0.887578i	$-0.652386\pi$
$449$	−19.5223	−0.921313	−0.460657	+	0.887578i	$0.652386\pi$
$450$	0	0
$451$	−0.731580	−0.0344487
$452$	33.2512	1.56400
$453$	0	0
$454$	−3.16280	−0.148437
$455$	−9.62484	−0.451220
$456$	0	0
$457$	−22.2963	−1.04298	−0.521489	−	0.853258i	$-0.674623\pi$
$457$	−22.2963	−1.04298	−0.521489	+	0.853258i	$0.674623\pi$
$458$	−6.87066	−0.321045
$459$	0	0
$460$	−12.5158	−0.583550
$461$	−25.7522	−1.19940	−0.599700	−	0.800225i	$-0.704713\pi$
$461$	−25.7522	−1.19940	−0.599700	+	0.800225i	$0.704713\pi$
$462$	0	0
$463$	−6.05779	−0.281529	−0.140765	−	0.990043i	$-0.544956\pi$
$463$	−6.05779	−0.281529	−0.140765	+	0.990043i	$0.544956\pi$
$464$	−5.38673	−0.250073
$465$	0	0
$466$	−2.30076	−0.106581
$467$	34.3665	1.59029	0.795145	−	0.606419i	$-0.207394\pi$
$467$	34.3665	1.59029	0.795145	+	0.606419i	$0.207394\pi$
$468$	0	0
$469$	7.19612	0.332286
$470$	0.120348	0.00555123
$471$	0	0
$472$	−16.3255	−0.751441
$473$	9.40734	0.432550
$474$	0	0
$475$	2.30334	0.105684
$476$	−20.9178	−0.958768
$477$	0	0
$478$	2.15755	0.0986840
$479$	−5.21190	−0.238138	−0.119069	−	0.992886i	$-0.537991\pi$
$479$	−5.21190	−0.238138	−0.119069	+	0.992886i	$0.537991\pi$
$480$	0	0
$481$	−13.9058	−0.634051
$482$	8.74934	0.398521
$483$	0	0
$484$	−44.7102	−2.03228
$485$	0.637546	0.0289495
$486$	0	0
$487$	−27.8736	−1.26307	−0.631537	−	0.775346i	$-0.717576\pi$
$487$	−27.8736	−1.26307	−0.631537	+	0.775346i	$0.717576\pi$
$488$	−13.3983	−0.606513
$489$	0	0
$490$	−5.61190	−0.253520
$491$	6.52348	0.294400	0.147200	−	0.989107i	$-0.452974\pi$
$491$	6.52348	0.294400	0.147200	+	0.989107i	$0.452974\pi$
$492$	0	0
$493$	−3.53684	−0.159291
$494$	1.43692	0.0646502
$495$	0	0
$496$	15.5664	0.698951
$497$	15.7241	0.705323
$498$	0	0
$499$	−12.9587	−0.580110	−0.290055	−	0.957010i	$-0.593674\pi$
$499$	−12.9587	−0.580110	−0.290055	+	0.957010i	$0.593674\pi$
$500$	−1.89710	−0.0848408
$501$	0	0
$502$	2.33445	0.104191
$503$	26.1919	1.16784	0.583920	−	0.811811i	$-0.301518\pi$
$503$	26.1919	1.16784	0.583920	+	0.811811i	$0.301518\pi$
$504$	0	0
$505$	10.0214	0.445947
$506$	−12.4428	−0.553148
$507$	0	0
$508$	35.6694	1.58257
$509$	−40.6691	−1.80263	−0.901313	−	0.433168i	$-0.857396\pi$
$509$	−40.6691	−1.80263	−0.901313	+	0.433168i	$0.857396\pi$
$510$	0	0
$511$	−36.5376	−1.61633
$512$	−20.6610	−0.913097
$513$	0	0
$514$	−2.75970	−0.121725
$515$	6.52538	0.287543
$516$	0	0
$517$	−2.20576	−0.0970094
$518$	−11.3522	−0.498789
$519$	0	0
$520$	−2.43118	−0.106614
$521$	12.2693	0.537529	0.268765	−	0.963206i	$-0.413385\pi$
$521$	12.2693	0.537529	0.268765	+	0.963206i	$0.413385\pi$
$522$	0	0
$523$	−7.98212	−0.349034	−0.174517	−	0.984654i	$-0.555836\pi$
$523$	−7.98212	−0.349034	−0.174517	+	0.984654i	$0.555836\pi$
$524$	6.12782	0.267695
$525$	0	0
$526$	−5.56146	−0.242491
$527$	10.2206	0.445217
$528$	0	0
$529$	20.5246	0.892374
$530$	0.623702	0.0270919
$531$	0	0
$532$	−21.6262	−0.937613
$533$	0.241985	0.0104815
$534$	0	0
$535$	−14.2067	−0.614211
$536$	1.81770	0.0785125
$537$	0	0
$538$	−4.42494	−0.190773
$539$	102.856	4.43033
$540$	0	0
$541$	−7.09887	−0.305204	−0.152602	−	0.988288i	$-0.548765\pi$
$541$	−7.09887	−0.305204	−0.152602	+	0.988288i	$0.548765\pi$
$542$	6.24374	0.268192
$543$	0	0
$544$	−7.99534	−0.342798
$545$	7.28770	0.312171
$546$	0	0
$547$	−0.253706	−0.0108477	−0.00542385	−	0.999985i	$-0.501726\pi$
$547$	−0.253706	−0.0108477	−0.00542385	+	0.999985i	$0.501726\pi$
$548$	33.8045	1.44406
$549$	0	0
$550$	−1.88603	−0.0804207
$551$	−3.65660	−0.155776
$552$	0	0
$553$	−80.0303	−3.40324
$554$	−4.52274	−0.192153
$555$	0	0
$556$	−16.3629	−0.693942
$557$	34.8395	1.47620	0.738099	−	0.674692i	$-0.235724\pi$
$557$	34.8395	1.47620	0.738099	+	0.674692i	$0.235724\pi$
$558$	0	0
$559$	−3.11167	−0.131610
$560$	16.7934	0.709649
$561$	0	0
$562$	4.32939	0.182624
$563$	19.2769	0.812426	0.406213	−	0.913778i	$-0.366849\pi$
$563$	19.2769	0.812426	0.406213	+	0.913778i	$0.366849\pi$
$564$	0	0
$565$	−17.5274	−0.737383
$566$	−6.56144	−0.275798
$567$	0	0
$568$	3.97182	0.166654
$569$	42.0671	1.76354	0.881772	−	0.471675i	$-0.156351\pi$
$569$	42.0671	1.76354	0.881772	+	0.471675i	$0.156351\pi$
$570$	0	0
$571$	−30.7641	−1.28744	−0.643719	−	0.765262i	$-0.722609\pi$
$571$	−30.7641	−1.28744	−0.643719	+	0.765262i	$0.722609\pi$
$572$	21.6913	0.906960
$573$	0	0
$574$	0.197548	0.00824549
$575$	6.59732	0.275127
$576$	0	0
$577$	41.6850	1.73537	0.867685	−	0.497115i	$-0.165607\pi$
$577$	41.6850	1.73537	0.867685	+	0.497115i	$0.165607\pi$
$578$	3.86112	0.160601
$579$	0	0
$580$	3.01168	0.125053
$581$	−77.2709	−3.20574
$582$	0	0
$583$	−11.4314	−0.473439
$584$	−9.22918	−0.381906
$585$	0	0
$586$	6.05652	0.250193
$587$	7.62687	0.314795	0.157397	−	0.987535i	$-0.449690\pi$
$587$	7.62687	0.314795	0.157397	+	0.987535i	$0.449690\pi$
$588$	0	0
$589$	10.5667	0.435394
$590$	4.18914	0.172464
$591$	0	0
$592$	24.2628	0.997195
$593$	40.7909	1.67508	0.837541	−	0.546374i	$-0.183992\pi$
$593$	40.7909	1.67508	0.837541	+	0.546374i	$0.183992\pi$
$594$	0	0
$595$	11.0262	0.452032
$596$	−10.2042	−0.417981
$597$	0	0
$598$	4.11569	0.168303
$599$	−39.5219	−1.61482	−0.807410	−	0.589991i	$-0.799131\pi$
$599$	−39.5219	−1.61482	−0.807410	+	0.589991i	$0.799131\pi$
$600$	0	0
$601$	16.0836	0.656064	0.328032	−	0.944667i	$-0.393615\pi$
$601$	16.0836	0.656064	0.328032	+	0.944667i	$0.393615\pi$
$602$	−2.54026	−0.103533
$603$	0	0
$604$	−26.9810	−1.09784
$605$	23.5677	0.958163
$606$	0	0
$607$	31.4738	1.27748	0.638740	−	0.769423i	$-0.279456\pi$
$607$	31.4738	1.27748	0.638740	+	0.769423i	$0.279456\pi$
$608$	−8.26608	−0.335234
$609$	0	0
$610$	3.43803	0.139202
$611$	0.729601	0.0295165
$612$	0	0
$613$	17.1611	0.693130	0.346565	−	0.938026i	$-0.387348\pi$
$613$	17.1611	0.693130	0.346565	+	0.938026i	$0.387348\pi$
$614$	−0.585131	−0.0236140
$615$	0	0
$616$	36.3767	1.46566
$617$	−24.8934	−1.00217	−0.501086	−	0.865397i	$-0.667066\pi$
$617$	−24.8934	−1.00217	−0.501086	+	0.865397i	$0.667066\pi$
$618$	0	0
$619$	19.9565	0.802119	0.401060	−	0.916052i	$-0.368642\pi$
$619$	19.9565	0.802119	0.401060	+	0.916052i	$0.368642\pi$
$620$	−8.70305	−0.349523
$621$	0	0
$622$	−5.72038	−0.229366
$623$	−4.94917	−0.198284
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.17056	−0.126721
$627$	0	0
$628$	−38.9833	−1.55560
$629$	15.9305	0.635192
$630$	0	0
$631$	−14.2055	−0.565512	−0.282756	−	0.959192i	$-0.591249\pi$
$631$	−14.2055	−0.565512	−0.282756	+	0.959192i	$0.591249\pi$
$632$	−20.2152	−0.804117
$633$	0	0
$634$	−2.88911	−0.114741
$635$	−18.8021	−0.746138
$636$	0	0
$637$	−34.0218	−1.34799
$638$	2.99412	0.118538
$639$	0	0
$640$	8.98514	0.355169
$641$	−30.0887	−1.18843	−0.594217	−	0.804305i	$-0.702538\pi$
$641$	−30.0887	−1.18843	−0.594217	+	0.804305i	$0.702538\pi$
$642$	0	0
$643$	3.91214	0.154280	0.0771398	−	0.997020i	$-0.475421\pi$
$643$	3.91214	0.154280	0.0771398	+	0.997020i	$0.475421\pi$
$644$	−61.9426	−2.44088
$645$	0	0
$646$	−1.64614	−0.0647666
$647$	−5.69020	−0.223705	−0.111853	−	0.993725i	$-0.535678\pi$
$647$	−5.69020	−0.223705	−0.111853	+	0.993725i	$0.535678\pi$
$648$	0	0
$649$	−76.7796	−3.01386
$650$	0.623844	0.0244692
$651$	0	0
$652$	13.8703	0.543203
$653$	−28.9307	−1.13215	−0.566074	−	0.824355i	$-0.691538\pi$
$653$	−28.9307	−1.13215	−0.566074	+	0.824355i	$0.691538\pi$
$654$	0	0
$655$	−3.23011	−0.126211
$656$	−0.422213	−0.0164847
$657$	0	0
$658$	0.595621	0.0232197
$659$	−18.0673	−0.703804	−0.351902	−	0.936037i	$-0.614465\pi$
$659$	−18.0673	−0.703804	−0.351902	+	0.936037i	$0.614465\pi$
$660$	0	0
$661$	−22.3854	−0.870689	−0.435345	−	0.900264i	$-0.643374\pi$
$661$	−22.3854	−0.870689	−0.435345	+	0.900264i	$0.643374\pi$
$662$	1.86411	0.0724506
$663$	0	0
$664$	−19.5182	−0.757452
$665$	11.3996	0.442058
$666$	0	0
$667$	−10.4734	−0.405531
$668$	12.4553	0.481910
$669$	0	0
$670$	−0.466423	−0.0180195
$671$	−63.0130	−2.43259
$672$	0	0
$673$	8.36104	0.322294	0.161147	−	0.986930i	$-0.448481\pi$
$673$	8.36104	0.322294	0.161147	+	0.986930i	$0.448481\pi$
$674$	−4.88862	−0.188303
$675$	0	0
$676$	17.4874	0.672593
$677$	26.5026	1.01858	0.509289	−	0.860595i	$-0.329908\pi$
$677$	26.5026	1.01858	0.509289	+	0.860595i	$0.329908\pi$
$678$	0	0
$679$	3.15532	0.121090
$680$	2.78516	0.106806
$681$	0	0
$682$	−8.65230	−0.331314
$683$	−38.2847	−1.46492	−0.732462	−	0.680808i	$-0.761629\pi$
$683$	−38.2847	−1.46492	−0.732462	+	0.680808i	$0.761629\pi$
$684$	0	0
$685$	−17.8191	−0.680832
$686$	−16.6609	−0.636116
$687$	0	0
$688$	5.42922	0.206987
$689$	3.78115	0.144050
$690$	0	0
$691$	−11.6822	−0.444411	−0.222206	−	0.975000i	$-0.571326\pi$
$691$	−11.6822	−0.444411	−0.222206	+	0.975000i	$0.571326\pi$
$692$	26.6697	1.01383
$693$	0	0
$694$	1.43901	0.0546239
$695$	8.62523	0.327174
$696$	0	0
$697$	−0.277218	−0.0105004
$698$	−8.45140	−0.319890
$699$	0	0
$700$	−9.38905	−0.354873
$701$	28.5249	1.07737	0.538685	−	0.842507i	$-0.318921\pi$
$701$	28.5249	1.07737	0.538685	+	0.842507i	$0.318921\pi$
$702$	0	0
$703$	16.4700	0.621177
$704$	−33.1313	−1.24868
$705$	0	0
$706$	6.41391	0.241391
$707$	49.5977	1.86531
$708$	0	0
$709$	−15.8589	−0.595594	−0.297797	−	0.954629i	$-0.596252\pi$
$709$	−15.8589	−0.595594	−0.297797	+	0.954629i	$0.596252\pi$
$710$	−1.01917	−0.0382489
$711$	0	0
$712$	−1.25013	−0.0468506
$713$	30.2656	1.13346
$714$	0	0
$715$	−11.4340	−0.427606
$716$	16.6602	0.622619
$717$	0	0
$718$	8.95802	0.334310
$719$	52.4499	1.95605	0.978026	−	0.208483i	$-0.0668526\pi$
$719$	52.4499	1.95605	0.978026	+	0.208483i	$0.0668526\pi$
$720$	0	0
$721$	32.2952	1.20274
$722$	4.39303	0.163492
$723$	0	0
$724$	−14.8240	−0.550931
$725$	−1.58752	−0.0589591
$726$	0	0
$727$	−34.9538	−1.29637	−0.648183	−	0.761485i	$-0.724471\pi$
$727$	−34.9538	−1.29637	−0.648183	+	0.761485i	$0.724471\pi$
$728$	−12.0323	−0.445947
$729$	0	0
$730$	2.36822	0.0876518
$731$	3.56473	0.131846
$732$	0	0
$733$	−23.1449	−0.854877	−0.427439	−	0.904044i	$-0.640584\pi$
$733$	−23.1449	−0.854877	−0.427439	+	0.904044i	$0.640584\pi$
$734$	6.16650	0.227610
$735$	0	0
$736$	−23.6761	−0.872711
$737$	8.54872	0.314896
$738$	0	0
$739$	18.6591	0.686386	0.343193	−	0.939265i	$-0.388491\pi$
$739$	18.6591	0.686386	0.343193	+	0.939265i	$0.388491\pi$
$740$	−13.5652	−0.498665
$741$	0	0
$742$	3.08680	0.113320
$743$	−47.8351	−1.75490	−0.877450	−	0.479667i	$-0.840757\pi$
$743$	−47.8351	−1.75490	−0.877450	+	0.479667i	$0.840757\pi$
$744$	0	0
$745$	5.37886	0.197066
$746$	4.33402	0.158680
$747$	0	0
$748$	−24.8496	−0.908593
$749$	−70.3115	−2.56913
$750$	0	0
$751$	30.6104	1.11699	0.558495	−	0.829508i	$-0.311379\pi$
$751$	30.6104	1.11699	0.558495	+	0.829508i	$0.311379\pi$
$752$	−1.27300	−0.0464216
$753$	0	0
$754$	−0.990365	−0.0360670
$755$	14.2223	0.517602
$756$	0	0
$757$	−7.48603	−0.272085	−0.136042	−	0.990703i	$-0.543438\pi$
$757$	−7.48603	−0.272085	−0.136042	+	0.990703i	$0.543438\pi$
$758$	2.56668	0.0932261
$759$	0	0
$760$	2.87947	0.104450
$761$	17.6983	0.641561	0.320781	−	0.947154i	$-0.396055\pi$
$761$	17.6983	0.641561	0.320781	+	0.947154i	$0.396055\pi$
$762$	0	0
$763$	36.0681	1.30575
$764$	36.6807	1.32706
$765$	0	0
$766$	−4.25926	−0.153893
$767$	25.3964	0.917011
$768$	0	0
$769$	0.472238	0.0170293	0.00851466	−	0.999964i	$-0.497290\pi$
$769$	0.472238	0.0170293	0.00851466	+	0.999964i	$0.497290\pi$
$770$	−9.33430	−0.336385
$771$	0	0
$772$	−33.1511	−1.19314
$773$	12.1244	0.436083	0.218041	−	0.975940i	$-0.430033\pi$
$773$	12.1244	0.436083	0.218041	+	0.975940i	$0.430033\pi$
$774$	0	0
$775$	4.58756	0.164790
$776$	0.797015	0.0286112
$777$	0	0
$778$	−1.18447	−0.0424652
$779$	−0.286605	−0.0102687
$780$	0	0
$781$	18.6797	0.668411
$782$	−4.71495	−0.168606
$783$	0	0
$784$	59.3610	2.12004
$785$	20.5489	0.733422
$786$	0	0
$787$	−22.6875	−0.808723	−0.404361	−	0.914599i	$-0.632506\pi$
$787$	−22.6875	−0.808723	−0.404361	+	0.914599i	$0.632506\pi$
$788$	0.795520	0.0283392
$789$	0	0
$790$	5.18724	0.184554
$791$	−86.7460	−3.08433
$792$	0	0
$793$	20.8428	0.740150
$794$	−3.16599	−0.112357
$795$	0	0
$796$	15.1945	0.538555
$797$	−36.6403	−1.29787	−0.648933	−	0.760846i	$-0.724784\pi$
$797$	−36.6403	−1.29787	−0.648933	+	0.760846i	$0.724784\pi$
$798$	0	0
$799$	−0.835832	−0.0295696
$800$	−3.58874	−0.126881
$801$	0	0
$802$	3.45594	0.122034
$803$	−43.4053	−1.53174
$804$	0	0
$805$	32.6512	1.15080
$806$	2.86192	0.100807
$807$	0	0
$808$	12.5281	0.440736
$809$	17.3921	0.611475	0.305738	−	0.952116i	$-0.401097\pi$
$809$	17.3921	0.611475	0.305738	+	0.952116i	$0.401097\pi$
$810$	0	0
$811$	39.2107	1.37687	0.688437	−	0.725296i	$-0.258297\pi$
$811$	39.2107	1.37687	0.688437	+	0.725296i	$0.258297\pi$
$812$	14.9053	0.523074
$813$	0	0
$814$	−13.4860	−0.472686
$815$	−7.31133	−0.256105
$816$	0	0
$817$	3.68544	0.128937
$818$	0.537845	0.0188053
$819$	0	0
$820$	0.236056	0.00824345
$821$	−12.0262	−0.419716	−0.209858	−	0.977732i	$-0.567300\pi$
$821$	−12.0262	−0.419716	−0.209858	+	0.977732i	$0.567300\pi$
$822$	0	0
$823$	39.6068	1.38061	0.690303	−	0.723520i	$-0.257477\pi$
$823$	39.6068	1.38061	0.690303	+	0.723520i	$0.257477\pi$
$824$	8.15758	0.284183
$825$	0	0
$826$	20.7328	0.721385
$827$	17.3926	0.604798	0.302399	−	0.953181i	$-0.402212\pi$
$827$	17.3926	0.604798	0.302399	+	0.953181i	$0.402212\pi$
$828$	0	0
$829$	12.3529	0.429034	0.214517	−	0.976720i	$-0.431182\pi$
$829$	12.3529	0.429034	0.214517	+	0.976720i	$0.431182\pi$
$830$	5.00839	0.173844
$831$	0	0
$832$	10.9589	0.379930
$833$	38.9754	1.35042
$834$	0	0
$835$	−6.56546	−0.227207
$836$	−25.6911	−0.888545
$837$	0	0
$838$	−8.26363	−0.285463
$839$	−47.4297	−1.63746	−0.818728	−	0.574181i	$-0.805320\pi$
$839$	−47.4297	−1.63746	−0.818728	+	0.574181i	$0.805320\pi$
$840$	0	0
$841$	−26.4798	−0.913096
$842$	7.43828	0.256340
$843$	0	0
$844$	−54.6720	−1.88189
$845$	−9.21799	−0.317108
$846$	0	0
$847$	116.641	4.00782
$848$	−6.59734	−0.226553
$849$	0	0
$850$	−0.714676	−0.0245132
$851$	47.1740	1.61710
$852$	0	0
$853$	31.0083	1.06171	0.530853	−	0.847464i	$-0.321872\pi$
$853$	31.0083	1.06171	0.530853	+	0.847464i	$0.321872\pi$
$854$	17.0154	0.582254
$855$	0	0
$856$	−17.7603	−0.607034
$857$	49.8375	1.70242	0.851209	−	0.524828i	$-0.175870\pi$
$857$	49.8375	1.70242	0.851209	+	0.524828i	$0.175870\pi$
$858$	0	0
$859$	1.46489	0.0499813	0.0249906	−	0.999688i	$-0.492044\pi$
$859$	1.46489	0.0499813	0.0249906	+	0.999688i	$0.492044\pi$
$860$	−3.03544	−0.103508
$861$	0	0
$862$	8.65205	0.294690
$863$	−21.5481	−0.733507	−0.366754	−	0.930318i	$-0.619531\pi$
$863$	−21.5481	−0.733507	−0.366754	+	0.930318i	$0.619531\pi$
$864$	0	0
$865$	−14.0582	−0.477992
$866$	6.86350	0.233231
$867$	0	0
$868$	−43.0729	−1.46199
$869$	−95.0731	−3.22513
$870$	0	0
$871$	−2.82766	−0.0958117
$872$	9.11058	0.308523
$873$	0	0
$874$	−4.87460	−0.164886
$875$	4.94917	0.167312
$876$	0	0
$877$	17.4280	0.588502	0.294251	−	0.955728i	$-0.404930\pi$
$877$	17.4280	0.588502	0.294251	+	0.955728i	$0.404930\pi$
$878$	−3.04144	−0.102644
$879$	0	0
$880$	19.9499	0.672511
$881$	7.99539	0.269372	0.134686	−	0.990888i	$-0.456997\pi$
$881$	7.99539	0.269372	0.134686	+	0.990888i	$0.456997\pi$
$882$	0	0
$883$	−4.70264	−0.158257	−0.0791283	−	0.996864i	$-0.525214\pi$
$883$	−4.70264	−0.158257	−0.0791283	+	0.996864i	$0.525214\pi$
$884$	8.21952	0.276452
$885$	0	0
$886$	−2.03216	−0.0682717
$887$	18.7994	0.631221	0.315611	−	0.948889i	$-0.397791\pi$
$887$	18.7994	0.631221	0.315611	+	0.948889i	$0.397791\pi$
$888$	0	0
$889$	−93.0547	−3.12096
$890$	0.320785	0.0107527
$891$	0	0
$892$	2.41687	0.0809229
$893$	−0.864135	−0.0289172
$894$	0	0
$895$	−8.78192	−0.293547
$896$	44.4690	1.48560
$897$	0	0
$898$	6.26246	0.208981
$899$	−7.28286	−0.242897
$900$	0	0
$901$	−4.33170	−0.144310
$902$	0.234680	0.00781398
$903$	0	0
$904$	−21.9115	−0.728767
$905$	7.81407	0.259748
$906$	0	0
$907$	43.8430	1.45578	0.727892	−	0.685692i	$-0.240500\pi$
$907$	43.8430	1.45578	0.727892	+	0.685692i	$0.240500\pi$
$908$	−18.7045	−0.620731
$909$	0	0
$910$	3.08751	0.102350
$911$	24.8252	0.822495	0.411247	−	0.911524i	$-0.365093\pi$
$911$	24.8252	0.822495	0.411247	+	0.911524i	$0.365093\pi$
$912$	0	0
$913$	−91.7950	−3.03797
$914$	7.15232	0.236578
$915$	0	0
$916$	−40.6325	−1.34254
$917$	−15.9863	−0.527915
$918$	0	0
$919$	−11.4806	−0.378710	−0.189355	−	0.981909i	$-0.560640\pi$
$919$	−11.4806	−0.378710	−0.189355	+	0.981909i	$0.560640\pi$
$920$	8.24751	0.271912
$921$	0	0
$922$	8.26092	0.272059
$923$	−6.17868	−0.203374
$924$	0	0
$925$	7.15048	0.235106
$926$	1.94325	0.0638591
$927$	0	0
$928$	5.69720	0.187020
$929$	−55.6737	−1.82659	−0.913297	−	0.407294i	$-0.866472\pi$
$929$	−55.6737	−1.82659	−0.913297	+	0.407294i	$0.866472\pi$
$930$	0	0
$931$	40.2952	1.32062
$932$	−13.6065	−0.445697
$933$	0	0
$934$	−11.0243	−0.360725
$935$	13.0988	0.428376
$936$	0	0
$937$	−39.7985	−1.30016	−0.650081	−	0.759865i	$-0.725265\pi$
$937$	−39.7985	−1.30016	−0.650081	+	0.759865i	$0.725265\pi$
$938$	−2.30841	−0.0753721
$939$	0	0
$940$	0.711727	0.0232140
$941$	14.5471	0.474223	0.237111	−	0.971482i	$-0.423799\pi$
$941$	14.5471	0.474223	0.237111	+	0.971482i	$0.423799\pi$
$942$	0	0
$943$	−0.820907	−0.0267324
$944$	−44.3115	−1.44222
$945$	0	0
$946$	−3.01773	−0.0981150
$947$	−21.3012	−0.692196	−0.346098	−	0.938198i	$-0.612494\pi$
$947$	−21.3012	−0.692196	−0.346098	+	0.938198i	$0.612494\pi$
$948$	0	0
$949$	14.3572	0.466054
$950$	−0.738877	−0.0239723
$951$	0	0
$952$	13.7842	0.446750
$953$	−20.2998	−0.657574	−0.328787	−	0.944404i	$-0.606640\pi$
$953$	−20.2998	−0.657574	−0.328787	+	0.944404i	$0.606640\pi$
$954$	0	0
$955$	−19.3352	−0.625671
$956$	12.7596	0.412674
$957$	0	0
$958$	1.67190	0.0540166
$959$	−88.1896	−2.84779
$960$	0	0
$961$	−9.95426	−0.321105
$962$	4.46078	0.143821
$963$	0	0
$964$	51.7429	1.66653
$965$	17.4747	0.562529
$966$	0	0
$967$	42.3607	1.36223	0.681114	−	0.732178i	$-0.261496\pi$
$967$	42.3607	1.36223	0.681114	+	0.732178i	$0.261496\pi$
$968$	29.4627	0.946967
$969$	0	0
$970$	−0.204515	−0.00656658
$971$	17.1018	0.548823	0.274411	−	0.961612i	$-0.411517\pi$
$971$	17.1018	0.548823	0.274411	+	0.961612i	$0.411517\pi$
$972$	0	0
$973$	42.6877	1.36851
$974$	8.94144	0.286502
$975$	0	0
$976$	−36.3664	−1.16406
$977$	10.2004	0.326341	0.163171	−	0.986598i	$-0.447828\pi$
$977$	10.2004	0.326341	0.163171	+	0.986598i	$0.447828\pi$
$978$	0	0
$979$	−5.87943	−0.187907
$980$	−33.1883	−1.06016
$981$	0	0
$982$	−2.09263	−0.0667787
$983$	37.3172	1.19024	0.595118	−	0.803639i	$-0.297105\pi$
$983$	37.3172	1.19024	0.595118	+	0.803639i	$0.297105\pi$
$984$	0	0
$985$	−0.419335	−0.0133611
$986$	1.13456	0.0361319
$987$	0	0
$988$	8.49785	0.270353
$989$	10.5560	0.335661
$990$	0	0
$991$	−51.9876	−1.65144	−0.825721	−	0.564079i	$-0.809231\pi$
$991$	−51.9876	−1.65144	−0.825721	+	0.564079i	$0.809231\pi$
$992$	−16.4636	−0.522719
$993$	0	0
$994$	−5.04406	−0.159988
$995$	−8.00934	−0.253913
$996$	0	0
$997$	59.6104	1.88788	0.943941	−	0.330116i	$-0.107088\pi$
$997$	59.6104	1.88788	0.943941	+	0.330116i	$0.107088\pi$
$998$	4.15695	0.131586
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.x.1.7	yes	17
3.2	odd	2		4005.2.a.w.1.11	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.11	✓	17	3.2	odd	2
4005.2.a.x.1.7	yes	17	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-0.320785 q^{2} -1.89710 q^{4} +1.00000 q^{5} +4.94917 q^{7} +1.25013 q^{8} +O(q^{10})\)	\(q-0.320785 q^{2} -1.89710 q^{4} +1.00000 q^{5} +4.94917 q^{7} +1.25013 q^{8} -0.320785 q^{10} +5.87943 q^{11} -1.94474 q^{13} -1.58762 q^{14} +3.39317 q^{16} +2.22790 q^{17} +2.30334 q^{19} -1.89710 q^{20} -1.88603 q^{22} +6.59732 q^{23} +1.00000 q^{25} +0.623844 q^{26} -9.38905 q^{28} -1.58752 q^{29} +4.58756 q^{31} -3.58874 q^{32} -0.714676 q^{34} +4.94917 q^{35} +7.15048 q^{37} -0.738877 q^{38} +1.25013 q^{40} -0.124430 q^{41} +1.60004 q^{43} -11.1538 q^{44} -2.11632 q^{46} -0.375166 q^{47} +17.4943 q^{49} -0.320785 q^{50} +3.68936 q^{52} -1.94430 q^{53} +5.87943 q^{55} +6.18711 q^{56} +0.509253 q^{58} -13.0590 q^{59} -10.7175 q^{61} -1.47162 q^{62} -5.63513 q^{64} -1.94474 q^{65} +1.45401 q^{67} -4.22654 q^{68} -1.58762 q^{70} +3.17712 q^{71} -7.38258 q^{73} -2.29377 q^{74} -4.36966 q^{76} +29.0983 q^{77} -16.1705 q^{79} +3.39317 q^{80} +0.0399154 q^{82} -15.6129 q^{83} +2.22790 q^{85} -0.513270 q^{86} +7.35005 q^{88} -1.00000 q^{89} -9.62484 q^{91} -12.5158 q^{92} +0.120348 q^{94} +2.30334 q^{95} +0.637546 q^{97} -5.61190 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 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

Varieties

Fields

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Groups

Database

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