LMFDB - Embedded newform 4005.2.a.v.1.5

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 58x^8 - 202x^7 - 95x^6 + 432x^5 + 4x^4 - 368x^3 + 94x^2 + 77*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.551952$ 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)

$f(q)$	$=$	$q-0.551952 q^{2} -1.69535 q^{4} -1.00000 q^{5} +1.42988 q^{7} +2.03966 q^{8} +O(q^{10})$	\(q-0.551952 q^{2} -1.69535 q^{4} -1.00000 q^{5} +1.42988 q^{7} +2.03966 q^{8} +0.551952 q^{10} +2.78245 q^{11} -4.03796 q^{13} -0.789227 q^{14} +2.26491 q^{16} -2.89530 q^{17} +3.03728 q^{19} +1.69535 q^{20} -1.53578 q^{22} +0.295578 q^{23} +1.00000 q^{25} +2.22876 q^{26} -2.42415 q^{28} -5.45690 q^{29} -9.06927 q^{31} -5.32943 q^{32} +1.59807 q^{34} -1.42988 q^{35} +7.41912 q^{37} -1.67643 q^{38} -2.03966 q^{40} -0.989375 q^{41} -2.15300 q^{43} -4.71723 q^{44} -0.163145 q^{46} +11.7541 q^{47} -4.95544 q^{49} -0.551952 q^{50} +6.84575 q^{52} +5.13864 q^{53} -2.78245 q^{55} +2.91647 q^{56} +3.01195 q^{58} +8.42517 q^{59} -0.169549 q^{61} +5.00580 q^{62} -1.58822 q^{64} +4.03796 q^{65} -1.78557 q^{67} +4.90854 q^{68} +0.789227 q^{70} +7.10711 q^{71} -8.61435 q^{73} -4.09500 q^{74} -5.14925 q^{76} +3.97858 q^{77} +10.3374 q^{79} -2.26491 q^{80} +0.546088 q^{82} -3.55317 q^{83} +2.89530 q^{85} +1.18835 q^{86} +5.67525 q^{88} -1.00000 q^{89} -5.77381 q^{91} -0.501109 q^{92} -6.48770 q^{94} -3.03728 q^{95} -5.97854 q^{97} +2.73516 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8}+O(q^{10}) $ 12 * q + 3 * q^2 + 11 * q^4 - 12 * q^5 - 8 * q^7 + 9 * q^8	$ 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8} - 3 q^{10} - 12 q^{13} - 4 q^{14} + q^{16} - 24 q^{19} - 11 q^{20} - 16 q^{22} + 24 q^{23} + 12 q^{25} + q^{26} - 44 q^{28} - 8 q^{29} - 12 q^{31} + 31 q^{32} - 18 q^{34} + 8 q^{35} - 10 q^{37} - 2 q^{38} - 9 q^{40} + 10 q^{41} - 42 q^{43} - 42 q^{44} - 24 q^{46} + 22 q^{47} - 4 q^{49} + 3 q^{50} - 30 q^{52} - 8 q^{53} - 27 q^{56} - 12 q^{58} - 4 q^{59} - 52 q^{61} + 14 q^{62} + 7 q^{64} + 12 q^{65} - 40 q^{67} - 23 q^{68} + 4 q^{70} - 2 q^{71} - 8 q^{73} - 26 q^{74} - 46 q^{76} + 12 q^{77} - 26 q^{79} - q^{80} - 26 q^{82} + 14 q^{83} - 32 q^{86} - 60 q^{88} - 12 q^{89} - 24 q^{91} + 38 q^{92} - 26 q^{94} + 24 q^{95} - 6 q^{97} - 30 q^{98}+O(q^{100}) $ 12 * q + 3 * q^2 + 11 * q^4 - 12 * q^5 - 8 * q^7 + 9 * q^8 - 3 * q^10 - 12 * q^13 - 4 * q^14 + q^16 - 24 * q^19 - 11 * q^20 - 16 * q^22 + 24 * q^23 + 12 * q^25 + q^26 - 44 * q^28 - 8 * q^29 - 12 * q^31 + 31 * q^32 - 18 * q^34 + 8 * q^35 - 10 * q^37 - 2 * q^38 - 9 * q^40 + 10 * q^41 - 42 * q^43 - 42 * q^44 - 24 * q^46 + 22 * q^47 - 4 * q^49 + 3 * q^50 - 30 * q^52 - 8 * q^53 - 27 * q^56 - 12 * q^58 - 4 * q^59 - 52 * q^61 + 14 * q^62 + 7 * q^64 + 12 * q^65 - 40 * q^67 - 23 * q^68 + 4 * q^70 - 2 * q^71 - 8 * q^73 - 26 * q^74 - 46 * q^76 + 12 * q^77 - 26 * q^79 - q^80 - 26 * q^82 + 14 * q^83 - 32 * q^86 - 60 * q^88 - 12 * q^89 - 24 * q^91 + 38 * q^92 - 26 * q^94 + 24 * q^95 - 6 * q^97 - 30 * 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 $

$2$	−0.551952	−0.390289	−0.195145	−	0.980774i	$-0.562518\pi$
$2$	−0.551952	−0.390289	−0.195145	+	0.980774i	$0.562518\pi$
$3$	0	0
$3$	0	0
$4$	−1.69535	−0.847674
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.42988	0.540445	0.270222	−	0.962798i	$-0.412903\pi$
$7$	1.42988	0.540445	0.270222	+	0.962798i	$0.412903\pi$
$8$	2.03966	0.721127
$9$	0	0
$10$	0.551952	0.174543
$11$	2.78245	0.838941	0.419471	−	0.907769i	$-0.362216\pi$
$11$	2.78245	0.838941	0.419471	+	0.907769i	$0.362216\pi$
$12$	0	0
$13$	−4.03796	−1.11993	−0.559964	−	0.828517i	$-0.689185\pi$
$13$	−4.03796	−1.11993	−0.559964	+	0.828517i	$0.689185\pi$
$14$	−0.789227	−0.210930
$15$	0	0
$16$	2.26491	0.566226
$17$	−2.89530	−0.702213	−0.351107	−	0.936335i	$-0.614195\pi$
$17$	−2.89530	−0.702213	−0.351107	+	0.936335i	$0.614195\pi$
$18$	0	0
$19$	3.03728	0.696801	0.348400	−	0.937346i	$-0.386725\pi$
$19$	3.03728	0.696801	0.348400	+	0.937346i	$0.386725\pi$
$20$	1.69535	0.379092
$21$	0	0
$22$	−1.53578	−0.327430
$23$	0.295578	0.0616324	0.0308162	−	0.999525i	$-0.490189\pi$
$23$	0.295578	0.0616324	0.0308162	+	0.999525i	$0.490189\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.22876	0.437096
$27$	0	0
$28$	−2.42415	−0.458121
$29$	−5.45690	−1.01332	−0.506660	−	0.862146i	$-0.669120\pi$
$29$	−5.45690	−1.01332	−0.506660	+	0.862146i	$0.669120\pi$
$30$	0	0
$31$	−9.06927	−1.62889	−0.814445	−	0.580241i	$-0.802958\pi$
$31$	−9.06927	−1.62889	−0.814445	+	0.580241i	$0.802958\pi$
$32$	−5.32943	−0.942119
$33$	0	0
$34$	1.59807	0.274066
$35$	−1.42988	−0.241694
$36$	0	0
$37$	7.41912	1.21970	0.609848	−	0.792518i	$-0.291230\pi$
$37$	7.41912	1.21970	0.609848	+	0.792518i	$0.291230\pi$
$38$	−1.67643	−0.271954
$39$	0	0
$40$	−2.03966	−0.322498
$41$	−0.989375	−0.154514	−0.0772572	−	0.997011i	$-0.524616\pi$
$41$	−0.989375	−0.154514	−0.0772572	+	0.997011i	$0.524616\pi$
$42$	0	0
$43$	−2.15300	−0.328329	−0.164164	−	0.986433i	$-0.552493\pi$
$43$	−2.15300	−0.328329	−0.164164	+	0.986433i	$0.552493\pi$
$44$	−4.71723	−0.711149
$45$	0	0
$46$	−0.163145	−0.0240544
$47$	11.7541	1.71451	0.857256	−	0.514891i	$-0.172168\pi$
$47$	11.7541	1.71451	0.857256	+	0.514891i	$0.172168\pi$
$48$	0	0
$49$	−4.95544	−0.707919
$50$	−0.551952	−0.0780578
$51$	0	0
$52$	6.84575	0.949335
$53$	5.13864	0.705847	0.352924	−	0.935652i	$-0.385188\pi$
$53$	5.13864	0.705847	0.352924	+	0.935652i	$0.385188\pi$
$54$	0	0
$55$	−2.78245	−0.375186
$56$	2.91647	0.389729
$57$	0	0
$58$	3.01195	0.395488
$59$	8.42517	1.09686	0.548432	−	0.836195i	$-0.315225\pi$
$59$	8.42517	1.09686	0.548432	+	0.836195i	$0.315225\pi$
$60$	0	0
$61$	−0.169549	−0.0217085	−0.0108542	−	0.999941i	$-0.503455\pi$
$61$	−0.169549	−0.0217085	−0.0108542	+	0.999941i	$0.503455\pi$
$62$	5.00580	0.635738
$63$	0	0
$64$	−1.58822	−0.198528
$65$	4.03796	0.500847
$66$	0	0
$67$	−1.78557	−0.218142	−0.109071	−	0.994034i	$-0.534788\pi$
$67$	−1.78557	−0.218142	−0.109071	+	0.994034i	$0.534788\pi$
$68$	4.90854	0.595248
$69$	0	0
$70$	0.789227	0.0943306
$71$	7.10711	0.843459	0.421730	−	0.906722i	$-0.361423\pi$
$71$	7.10711	0.843459	0.421730	+	0.906722i	$0.361423\pi$
$72$	0	0
$73$	−8.61435	−1.00823	−0.504117	−	0.863635i	$-0.668182\pi$
$73$	−8.61435	−1.00823	−0.504117	+	0.863635i	$0.668182\pi$
$74$	−4.09500	−0.476034
$75$	0	0
$76$	−5.14925	−0.590660
$77$	3.97858	0.453402
$78$	0	0
$79$	10.3374	1.16304	0.581522	−	0.813531i	$-0.302458\pi$
$79$	10.3374	1.16304	0.581522	+	0.813531i	$0.302458\pi$
$80$	−2.26491	−0.253224
$81$	0	0
$82$	0.546088	0.0603053
$83$	−3.55317	−0.390011	−0.195005	−	0.980802i	$-0.562472\pi$
$83$	−3.55317	−0.390011	−0.195005	+	0.980802i	$0.562472\pi$
$84$	0	0
$85$	2.89530	0.314039
$86$	1.18835	0.128143
$87$	0	0
$88$	5.67525	0.604983
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−5.77381	−0.605260
$92$	−0.501109	−0.0522442
$93$	0	0
$94$	−6.48770	−0.669155
$95$	−3.03728	−0.311619
$96$	0	0
$97$	−5.97854	−0.607029	−0.303514	−	0.952827i	$-0.598160\pi$
$97$	−5.97854	−0.607029	−0.303514	+	0.952827i	$0.598160\pi$
$98$	2.73516	0.276293
$99$	0	0
$100$	−1.69535	−0.169535
$101$	−4.94918	−0.492461	−0.246231	−	0.969211i	$-0.579192\pi$
$101$	−4.94918	−0.492461	−0.246231	+	0.969211i	$0.579192\pi$
$102$	0	0
$103$	−11.5238	−1.13548	−0.567739	−	0.823209i	$-0.692182\pi$
$103$	−11.5238	−1.13548	−0.567739	+	0.823209i	$0.692182\pi$
$104$	−8.23605	−0.807611
$105$	0	0
$106$	−2.83629	−0.275484
$107$	11.3431	1.09658	0.548291	−	0.836288i	$-0.315279\pi$
$107$	11.3431	1.09658	0.548291	+	0.836288i	$0.315279\pi$
$108$	0	0
$109$	−8.64772	−0.828301	−0.414151	−	0.910208i	$-0.635921\pi$
$109$	−8.64772	−0.828301	−0.414151	+	0.910208i	$0.635921\pi$
$110$	1.53578	0.146431
$111$	0	0
$112$	3.23855	0.306014
$113$	−13.8383	−1.30179	−0.650897	−	0.759166i	$-0.725607\pi$
$113$	−13.8383	−1.30179	−0.650897	+	0.759166i	$0.725607\pi$
$114$	0	0
$115$	−0.295578	−0.0275628
$116$	9.25135	0.858966
$117$	0	0
$118$	−4.65029	−0.428094
$119$	−4.13994	−0.379508
$120$	0	0
$121$	−3.25795	−0.296177
$122$	0.0935827	0.00847258
$123$	0	0
$124$	15.3756	1.38077
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.89347	0.434226	0.217113	−	0.976147i	$-0.430336\pi$
$127$	4.89347	0.434226	0.217113	+	0.976147i	$0.430336\pi$
$128$	11.5355	1.01960
$129$	0	0
$130$	−2.22876	−0.195475
$131$	−3.76602	−0.329038	−0.164519	−	0.986374i	$-0.552607\pi$
$131$	−3.76602	−0.329038	−0.164519	+	0.986374i	$0.552607\pi$
$132$	0	0
$133$	4.34296	0.376582
$134$	0.985550	0.0851385
$135$	0	0
$136$	−5.90541	−0.506385
$137$	7.47314	0.638474	0.319237	−	0.947675i	$-0.396573\pi$
$137$	7.47314	0.638474	0.319237	+	0.947675i	$0.396573\pi$
$138$	0	0
$139$	−18.5104	−1.57003	−0.785014	−	0.619478i	$-0.787344\pi$
$139$	−18.5104	−1.57003	−0.785014	+	0.619478i	$0.787344\pi$
$140$	2.42415	0.204878
$141$	0	0
$142$	−3.92279	−0.329193
$143$	−11.2354	−0.939555
$144$	0	0
$145$	5.45690	0.453171
$146$	4.75471	0.393503
$147$	0	0
$148$	−12.5780	−1.03391
$149$	−4.10832	−0.336566	−0.168283	−	0.985739i	$-0.553822\pi$
$149$	−4.10832	−0.336566	−0.168283	+	0.985739i	$0.553822\pi$
$150$	0	0
$151$	−14.4308	−1.17436	−0.587181	−	0.809456i	$-0.699762\pi$
$151$	−14.4308	−1.17436	−0.587181	+	0.809456i	$0.699762\pi$
$152$	6.19501	0.502482
$153$	0	0
$154$	−2.19599	−0.176958
$155$	9.06927	0.728461
$156$	0	0
$157$	−21.1088	−1.68467	−0.842335	−	0.538955i	$-0.818819\pi$
$157$	−21.1088	−1.68467	−0.842335	+	0.538955i	$0.818819\pi$
$158$	−5.70573	−0.453923
$159$	0	0
$160$	5.32943	0.421328
$161$	0.422643	0.0333089
$162$	0	0
$163$	10.2291	0.801207	0.400604	−	0.916251i	$-0.368800\pi$
$163$	10.2291	0.801207	0.400604	+	0.916251i	$0.368800\pi$
$164$	1.67734	0.130978
$165$	0	0
$166$	1.96118	0.152217
$167$	18.4477	1.42753	0.713764	−	0.700387i	$-0.246989\pi$
$167$	18.4477	1.42753	0.713764	+	0.700387i	$0.246989\pi$
$168$	0	0
$169$	3.30513	0.254241
$170$	−1.59807	−0.122566
$171$	0	0
$172$	3.65008	0.278316
$173$	−9.83819	−0.747984	−0.373992	−	0.927432i	$-0.622011\pi$
$173$	−9.83819	−0.747984	−0.373992	+	0.927432i	$0.622011\pi$
$174$	0	0
$175$	1.42988	0.108089
$176$	6.30200	0.475031
$177$	0	0
$178$	0.551952	0.0413706
$179$	−7.95789	−0.594801	−0.297400	−	0.954753i	$-0.596120\pi$
$179$	−7.95789	−0.594801	−0.297400	+	0.954753i	$0.596120\pi$
$180$	0	0
$181$	−8.32451	−0.618756	−0.309378	−	0.950939i	$-0.600121\pi$
$181$	−8.32451	−0.618756	−0.309378	+	0.950939i	$0.600121\pi$
$182$	3.18687	0.236226
$183$	0	0
$184$	0.602878	0.0444448
$185$	−7.41912	−0.545465
$186$	0	0
$187$	−8.05604	−0.589116
$188$	−19.9273	−1.45335
$189$	0	0
$190$	1.67643	0.121621
$191$	7.44520	0.538716	0.269358	−	0.963040i	$-0.413189\pi$
$191$	7.44520	0.538716	0.269358	+	0.963040i	$0.413189\pi$
$192$	0	0
$193$	2.99419	0.215527	0.107763	−	0.994177i	$-0.465631\pi$
$193$	2.99419	0.215527	0.107763	+	0.994177i	$0.465631\pi$
$194$	3.29987	0.236917
$195$	0	0
$196$	8.40119	0.600085
$197$	−24.6825	−1.75855	−0.879277	−	0.476311i	$-0.841974\pi$
$197$	−24.6825	−1.75855	−0.879277	+	0.476311i	$0.841974\pi$
$198$	0	0
$199$	5.21104	0.369401	0.184700	−	0.982795i	$-0.440869\pi$
$199$	5.21104	0.369401	0.184700	+	0.982795i	$0.440869\pi$
$200$	2.03966	0.144225
$201$	0	0
$202$	2.73171	0.192202
$203$	−7.80272	−0.547644
$204$	0	0
$205$	0.989375	0.0691010
$206$	6.36061	0.443165
$207$	0	0
$208$	−9.14560	−0.634133
$209$	8.45110	0.584575
$210$	0	0
$211$	15.6780	1.07932	0.539659	−	0.841884i	$-0.318553\pi$
$211$	15.6780	1.07932	0.539659	+	0.841884i	$0.318553\pi$
$212$	−8.71180	−0.598329
$213$	0	0
$214$	−6.26087	−0.427984
$215$	2.15300	0.146833
$216$	0	0
$217$	−12.9680	−0.880325
$218$	4.77313	0.323277
$219$	0	0
$220$	4.71723	0.318036
$221$	11.6911	0.786429
$222$	0	0
$223$	−17.3314	−1.16060	−0.580298	−	0.814404i	$-0.697064\pi$
$223$	−17.3314	−1.16060	−0.580298	+	0.814404i	$0.697064\pi$
$224$	−7.62046	−0.509163
$225$	0	0
$226$	7.63806	0.508076
$227$	−0.908105	−0.0602730	−0.0301365	−	0.999546i	$-0.509594\pi$
$227$	−0.908105	−0.0602730	−0.0301365	+	0.999546i	$0.509594\pi$
$228$	0	0
$229$	−13.6515	−0.902117	−0.451059	−	0.892494i	$-0.648953\pi$
$229$	−13.6515	−0.902117	−0.451059	+	0.892494i	$0.648953\pi$
$230$	0.163145	0.0107575
$231$	0	0
$232$	−11.1302	−0.730733
$233$	−20.0588	−1.31410	−0.657049	−	0.753848i	$-0.728196\pi$
$233$	−20.0588	−1.31410	−0.657049	+	0.753848i	$0.728196\pi$
$234$	0	0
$235$	−11.7541	−0.766753
$236$	−14.2836	−0.929784
$237$	0	0
$238$	2.28505	0.148118
$239$	−16.8013	−1.08679	−0.543394	−	0.839478i	$-0.682861\pi$
$239$	−16.8013	−1.08679	−0.543394	+	0.839478i	$0.682861\pi$
$240$	0	0
$241$	−28.3201	−1.82426	−0.912128	−	0.409906i	$-0.865561\pi$
$241$	−28.3201	−1.82426	−0.912128	+	0.409906i	$0.865561\pi$
$242$	1.79823	0.115595
$243$	0	0
$244$	0.287444	0.0184017
$245$	4.95544	0.316591
$246$	0	0
$247$	−12.2644	−0.780367
$248$	−18.4982	−1.17464
$249$	0	0
$250$	0.551952	0.0349085
$251$	18.1058	1.14283	0.571415	−	0.820662i	$-0.306395\pi$
$251$	18.1058	1.14283	0.571415	+	0.820662i	$0.306395\pi$
$252$	0	0
$253$	0.822434	0.0517060
$254$	−2.70096	−0.169473
$255$	0	0
$256$	−3.19059	−0.199412
$257$	8.31669	0.518781	0.259390	−	0.965773i	$-0.416478\pi$
$257$	8.31669	0.518781	0.259390	+	0.965773i	$0.416478\pi$
$258$	0	0
$259$	10.6085	0.659178
$260$	−6.84575	−0.424556
$261$	0	0
$262$	2.07866	0.128420
$263$	−16.7359	−1.03198	−0.515989	−	0.856595i	$-0.672575\pi$
$263$	−16.7359	−1.03198	−0.515989	+	0.856595i	$0.672575\pi$
$264$	0	0
$265$	−5.13864	−0.315664
$266$	−2.39710	−0.146976
$267$	0	0
$268$	3.02717	0.184914
$269$	4.15546	0.253363	0.126682	−	0.991943i	$-0.459567\pi$
$269$	4.15546	0.253363	0.126682	+	0.991943i	$0.459567\pi$
$270$	0	0
$271$	−4.68684	−0.284705	−0.142353	−	0.989816i	$-0.545467\pi$
$271$	−4.68684	−0.284705	−0.142353	+	0.989816i	$0.545467\pi$
$272$	−6.55758	−0.397612
$273$	0	0
$274$	−4.12482	−0.249189
$275$	2.78245	0.167788
$276$	0	0
$277$	−9.36419	−0.562640	−0.281320	−	0.959614i	$-0.590772\pi$
$277$	−9.36419	−0.562640	−0.281320	+	0.959614i	$0.590772\pi$
$278$	10.2168	0.612765
$279$	0	0
$280$	−2.91647	−0.174292
$281$	−9.77872	−0.583350	−0.291675	−	0.956518i	$-0.594213\pi$
$281$	−9.77872	−0.583350	−0.291675	+	0.956518i	$0.594213\pi$
$282$	0	0
$283$	−17.8177	−1.05915	−0.529575	−	0.848263i	$-0.677649\pi$
$283$	−17.8177	−1.05915	−0.529575	+	0.848263i	$0.677649\pi$
$284$	−12.0490	−0.714979
$285$	0	0
$286$	6.20143	0.366698
$287$	−1.41469	−0.0835066
$288$	0	0
$289$	−8.61724	−0.506897
$290$	−3.01195	−0.176868
$291$	0	0
$292$	14.6043	0.854654
$293$	4.48428	0.261975	0.130987	−	0.991384i	$-0.458185\pi$
$293$	4.48428	0.261975	0.130987	+	0.991384i	$0.458185\pi$
$294$	0	0
$295$	−8.42517	−0.490533
$296$	15.1325	0.879556
$297$	0	0
$298$	2.26759	0.131358
$299$	−1.19353	−0.0690239
$300$	0	0
$301$	−3.07853	−0.177444
$302$	7.96511	0.458340
$303$	0	0
$304$	6.87916	0.394547
$305$	0.169549	0.00970833
$306$	0	0
$307$	5.29269	0.302070	0.151035	−	0.988528i	$-0.451739\pi$
$307$	5.29269	0.302070	0.151035	+	0.988528i	$0.451739\pi$
$308$	−6.74509	−0.384337
$309$	0	0
$310$	−5.00580	−0.284311
$311$	14.2373	0.807324	0.403662	−	0.914908i	$-0.367737\pi$
$311$	14.2373	0.807324	0.403662	+	0.914908i	$0.367737\pi$
$312$	0	0
$313$	−1.06301	−0.0600849	−0.0300425	−	0.999549i	$-0.509564\pi$
$313$	−1.06301	−0.0600849	−0.0300425	+	0.999549i	$0.509564\pi$
$314$	11.6511	0.657508
$315$	0	0
$316$	−17.5254	−0.985882
$317$	−13.9762	−0.784983	−0.392491	−	0.919756i	$-0.628387\pi$
$317$	−13.9762	−0.784983	−0.392491	+	0.919756i	$0.628387\pi$
$318$	0	0
$319$	−15.1836	−0.850117
$320$	1.58822	0.0887843
$321$	0	0
$322$	−0.233278	−0.0130001
$323$	−8.79384	−0.489303
$324$	0	0
$325$	−4.03796	−0.223986
$326$	−5.64599	−0.312702
$327$	0	0
$328$	−2.01799	−0.111425
$329$	16.8070	0.926599
$330$	0	0
$331$	−16.4282	−0.902978	−0.451489	−	0.892277i	$-0.649107\pi$
$331$	−16.4282	−0.902978	−0.451489	+	0.892277i	$0.649107\pi$
$332$	6.02386	0.330602
$333$	0	0
$334$	−10.1823	−0.557148
$335$	1.78557	0.0975562
$336$	0	0
$337$	−17.2981	−0.942290	−0.471145	−	0.882056i	$-0.656159\pi$
$337$	−17.2981	−0.942290	−0.471145	+	0.882056i	$0.656159\pi$
$338$	−1.82427	−0.0992274
$339$	0	0
$340$	−4.90854	−0.266203
$341$	−25.2348	−1.36654
$342$	0	0
$343$	−17.0949	−0.923036
$344$	−4.39137	−0.236767
$345$	0	0
$346$	5.43021	0.291930
$347$	12.5182	0.672013	0.336007	−	0.941860i	$-0.390924\pi$
$347$	12.5182	0.672013	0.336007	+	0.941860i	$0.390924\pi$
$348$	0	0
$349$	−17.3441	−0.928406	−0.464203	−	0.885729i	$-0.653659\pi$
$349$	−17.3441	−0.928406	−0.464203	+	0.885729i	$0.653659\pi$
$350$	−0.789227	−0.0421859
$351$	0	0
$352$	−14.8289	−0.790383
$353$	18.8004	1.00064	0.500322	−	0.865839i	$-0.333215\pi$
$353$	18.8004	1.00064	0.500322	+	0.865839i	$0.333215\pi$
$354$	0	0
$355$	−7.10711	−0.377206
$356$	1.69535	0.0898533
$357$	0	0
$358$	4.39237	0.232144
$359$	4.36780	0.230523	0.115262	−	0.993335i	$-0.463229\pi$
$359$	4.36780	0.230523	0.115262	+	0.993335i	$0.463229\pi$
$360$	0	0
$361$	−9.77491	−0.514469
$362$	4.59473	0.241494
$363$	0	0
$364$	9.78862	0.513063
$365$	8.61435	0.450896
$366$	0	0
$367$	28.2686	1.47561	0.737803	−	0.675016i	$-0.235863\pi$
$367$	28.2686	1.47561	0.737803	+	0.675016i	$0.235863\pi$
$368$	0.669457	0.0348979
$369$	0	0
$370$	4.09500	0.212889
$371$	7.34766	0.381471
$372$	0	0
$373$	5.18493	0.268465	0.134233	−	0.990950i	$-0.457143\pi$
$373$	5.18493	0.268465	0.134233	+	0.990950i	$0.457143\pi$
$374$	4.44655	0.229925
$375$	0	0
$376$	23.9743	1.23638
$377$	22.0347	1.13485
$378$	0	0
$379$	21.3421	1.09627	0.548135	−	0.836390i	$-0.315338\pi$
$379$	21.3421	1.09627	0.548135	+	0.836390i	$0.315338\pi$
$380$	5.14925	0.264151
$381$	0	0
$382$	−4.10939	−0.210255
$383$	15.0995	0.771550	0.385775	−	0.922593i	$-0.373934\pi$
$383$	15.0995	0.771550	0.385775	+	0.922593i	$0.373934\pi$
$384$	0	0
$385$	−3.97858	−0.202767
$386$	−1.65265	−0.0841177
$387$	0	0
$388$	10.1357	0.514563
$389$	34.4448	1.74642	0.873210	−	0.487344i	$-0.162034\pi$
$389$	34.4448	1.74642	0.873210	+	0.487344i	$0.162034\pi$
$390$	0	0
$391$	−0.855788	−0.0432791
$392$	−10.1074	−0.510500
$393$	0	0
$394$	13.6235	0.686344
$395$	−10.3374	−0.520129
$396$	0	0
$397$	18.4108	0.924013	0.462007	−	0.886876i	$-0.347130\pi$
$397$	18.4108	0.924013	0.462007	+	0.886876i	$0.347130\pi$
$398$	−2.87624	−0.144173
$399$	0	0
$400$	2.26491	0.113245
$401$	−37.1555	−1.85546	−0.927729	−	0.373254i	$-0.878242\pi$
$401$	−37.1555	−1.85546	−0.927729	+	0.373254i	$0.878242\pi$
$402$	0	0
$403$	36.6214	1.82424
$404$	8.39058	0.417447
$405$	0	0
$406$	4.30673	0.213739
$407$	20.6434	1.02325
$408$	0	0
$409$	−8.88118	−0.439146	−0.219573	−	0.975596i	$-0.570466\pi$
$409$	−8.88118	−0.439146	−0.219573	+	0.975596i	$0.570466\pi$
$410$	−0.546088	−0.0269694
$411$	0	0
$412$	19.5369	0.962516
$413$	12.0470	0.592795
$414$	0	0
$415$	3.55317	0.174418
$416$	21.5200	1.05511
$417$	0	0
$418$	−4.66460	−0.228153
$419$	−27.9097	−1.36348	−0.681740	−	0.731595i	$-0.738776\pi$
$419$	−27.9097	−1.36348	−0.681740	+	0.731595i	$0.738776\pi$
$420$	0	0
$421$	−5.94913	−0.289943	−0.144971	−	0.989436i	$-0.546309\pi$
$421$	−5.94913	−0.289943	−0.144971	+	0.989436i	$0.546309\pi$
$422$	−8.65351	−0.421246
$423$	0	0
$424$	10.4811	0.509005
$425$	−2.89530	−0.140443
$426$	0	0
$427$	−0.242435	−0.0117322
$428$	−19.2306	−0.929545
$429$	0	0
$430$	−1.18835	−0.0573074
$431$	−20.8153	−1.00264	−0.501318	−	0.865263i	$-0.667151\pi$
$431$	−20.8153	−1.00264	−0.501318	+	0.865263i	$0.667151\pi$
$432$	0	0
$433$	−10.8377	−0.520828	−0.260414	−	0.965497i	$-0.583859\pi$
$433$	−10.8377	−0.520828	−0.260414	+	0.965497i	$0.583859\pi$
$434$	7.15771	0.343581
$435$	0	0
$436$	14.6609	0.702130
$437$	0.897756	0.0429455
$438$	0	0
$439$	24.8760	1.18727	0.593633	−	0.804736i	$-0.297693\pi$
$439$	24.8760	1.18727	0.593633	+	0.804736i	$0.297693\pi$
$440$	−5.67525	−0.270557
$441$	0	0
$442$	−6.45293	−0.306935
$443$	−3.34356	−0.158857	−0.0794286	−	0.996841i	$-0.525310\pi$
$443$	−3.34356	−0.158857	−0.0794286	+	0.996841i	$0.525310\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	9.56609	0.452968
$447$	0	0
$448$	−2.27097	−0.107293
$449$	−3.19475	−0.150770	−0.0753848	−	0.997155i	$-0.524019\pi$
$449$	−3.19475	−0.150770	−0.0753848	+	0.997155i	$0.524019\pi$
$450$	0	0
$451$	−2.75289	−0.129629
$452$	23.4607	1.10350
$453$	0	0
$454$	0.501230	0.0235239
$455$	5.77381	0.270680
$456$	0	0
$457$	20.8897	0.977179	0.488589	−	0.872514i	$-0.337512\pi$
$457$	20.8897	0.977179	0.488589	+	0.872514i	$0.337512\pi$
$458$	7.53498	0.352086
$459$	0	0
$460$	0.501109	0.0233643
$461$	−21.6712	−1.00933	−0.504664	−	0.863316i	$-0.668384\pi$
$461$	−21.6712	−1.00933	−0.504664	+	0.863316i	$0.668384\pi$
$462$	0	0
$463$	40.0723	1.86232	0.931160	−	0.364612i	$-0.118798\pi$
$463$	40.0723	1.86232	0.931160	+	0.364612i	$0.118798\pi$
$464$	−12.3594	−0.573769
$465$	0	0
$466$	11.0715	0.512878
$467$	26.6727	1.23426	0.617132	−	0.786860i	$-0.288295\pi$
$467$	26.6727	1.23426	0.617132	+	0.786860i	$0.288295\pi$
$468$	0	0
$469$	−2.55316	−0.117894
$470$	6.48770	0.299255
$471$	0	0
$472$	17.1845	0.790979
$473$	−5.99061	−0.275449
$474$	0	0
$475$	3.03728	0.139360
$476$	7.01864	0.321699
$477$	0	0
$478$	9.27354	0.424162
$479$	−37.1401	−1.69698	−0.848488	−	0.529214i	$-0.822487\pi$
$479$	−37.1401	−1.69698	−0.848488	+	0.529214i	$0.822487\pi$
$480$	0	0
$481$	−29.9581	−1.36597
$482$	15.6313	0.711987
$483$	0	0
$484$	5.52336	0.251062
$485$	5.97854	0.271471
$486$	0	0
$487$	10.6586	0.482987	0.241493	−	0.970402i	$-0.422363\pi$
$487$	10.6586	0.482987	0.241493	+	0.970402i	$0.422363\pi$
$488$	−0.345821	−0.0156546
$489$	0	0
$490$	−2.73516	−0.123562
$491$	0.469262	0.0211775	0.0105888	−	0.999944i	$-0.496629\pi$
$491$	0.469262	0.0211775	0.0105888	+	0.999944i	$0.496629\pi$
$492$	0	0
$493$	15.7994	0.711567
$494$	6.76938	0.304569
$495$	0	0
$496$	−20.5410	−0.922320
$497$	10.1623	0.455843
$498$	0	0
$499$	17.2866	0.773854	0.386927	−	0.922110i	$-0.373537\pi$
$499$	17.2866	0.773854	0.386927	+	0.922110i	$0.373537\pi$
$500$	1.69535	0.0758183
$501$	0	0
$502$	−9.99354	−0.446034
$503$	−24.8649	−1.10867	−0.554336	−	0.832293i	$-0.687028\pi$
$503$	−24.8649	−1.10867	−0.554336	+	0.832293i	$0.687028\pi$
$504$	0	0
$505$	4.94918	0.220235
$506$	−0.453944	−0.0201803
$507$	0	0
$508$	−8.29615	−0.368082
$509$	−41.0043	−1.81748	−0.908741	−	0.417359i	$-0.862956\pi$
$509$	−41.0043	−1.81748	−0.908741	+	0.417359i	$0.862956\pi$
$510$	0	0
$511$	−12.3175	−0.544895
$512$	−21.3099	−0.941774
$513$	0	0
$514$	−4.59041	−0.202474
$515$	11.5238	0.507801
$516$	0	0
$517$	32.7052	1.43837
$518$	−5.85537	−0.257270
$519$	0	0
$520$	8.23605	0.361175
$521$	−22.7890	−0.998404	−0.499202	−	0.866486i	$-0.666373\pi$
$521$	−22.7890	−0.998404	−0.499202	+	0.866486i	$0.666373\pi$
$522$	0	0
$523$	−9.98082	−0.436431	−0.218215	−	0.975901i	$-0.570024\pi$
$523$	−9.98082	−0.436431	−0.218215	+	0.975901i	$0.570024\pi$
$524$	6.38471	0.278918
$525$	0	0
$526$	9.23739	0.402769
$527$	26.2583	1.14383
$528$	0	0
$529$	−22.9126	−0.996201
$530$	2.83629	0.123200
$531$	0	0
$532$	−7.36283	−0.319219
$533$	3.99506	0.173045
$534$	0	0
$535$	−11.3431	−0.490406
$536$	−3.64195	−0.157308
$537$	0	0
$538$	−2.29362	−0.0988848
$539$	−13.7883	−0.593903
$540$	0	0
$541$	12.4639	0.535866	0.267933	−	0.963438i	$-0.413659\pi$
$541$	12.4639	0.535866	0.267933	+	0.963438i	$0.413659\pi$
$542$	2.58691	0.111117
$543$	0	0
$544$	15.4303	0.661569
$545$	8.64772	0.370428
$546$	0	0
$547$	−32.8642	−1.40517	−0.702585	−	0.711600i	$-0.747971\pi$
$547$	−32.8642	−1.40517	−0.702585	+	0.711600i	$0.747971\pi$
$548$	−12.6696	−0.541218
$549$	0	0
$550$	−1.53578	−0.0654859
$551$	−16.5741	−0.706082
$552$	0	0
$553$	14.7812	0.628561
$554$	5.16858	0.219592
$555$	0	0
$556$	31.3815	1.33087
$557$	8.74956	0.370731	0.185365	−	0.982670i	$-0.440653\pi$
$557$	8.74956	0.370731	0.185365	+	0.982670i	$0.440653\pi$
$558$	0	0
$559$	8.69372	0.367705
$560$	−3.23855	−0.136854
$561$	0	0
$562$	5.39739	0.227675
$563$	0.386452	0.0162870	0.00814351	−	0.999967i	$-0.497408\pi$
$563$	0.386452	0.0162870	0.00814351	+	0.999967i	$0.497408\pi$
$564$	0	0
$565$	13.8383	0.582180
$566$	9.83450	0.413375
$567$	0	0
$568$	14.4961	0.608241
$569$	−28.0280	−1.17499	−0.587497	−	0.809226i	$-0.699887\pi$
$569$	−28.0280	−1.17499	−0.587497	+	0.809226i	$0.699887\pi$
$570$	0	0
$571$	11.1039	0.464682	0.232341	−	0.972634i	$-0.425361\pi$
$571$	11.1039	0.464682	0.232341	+	0.972634i	$0.425361\pi$
$572$	19.0480	0.796437
$573$	0	0
$574$	0.780842	0.0325917
$575$	0.295578	0.0123265
$576$	0	0
$577$	−24.4339	−1.01720	−0.508598	−	0.861004i	$-0.669836\pi$
$577$	−24.4339	−1.01720	−0.508598	+	0.861004i	$0.669836\pi$
$578$	4.75630	0.197836
$579$	0	0
$580$	−9.25135	−0.384141
$581$	−5.08061	−0.210779
$582$	0	0
$583$	14.2980	0.592164
$584$	−17.5703	−0.727065
$585$	0	0
$586$	−2.47511	−0.102246
$587$	0.517464	0.0213580	0.0106790	−	0.999943i	$-0.496601\pi$
$587$	0.517464	0.0213580	0.0106790	+	0.999943i	$0.496601\pi$
$588$	0	0
$589$	−27.5459	−1.13501
$590$	4.65029	0.191450
$591$	0	0
$592$	16.8036	0.690624
$593$	−28.4818	−1.16961	−0.584804	−	0.811175i	$-0.698828\pi$
$593$	−28.4818	−1.16961	−0.584804	+	0.811175i	$0.698828\pi$
$594$	0	0
$595$	4.13994	0.169721
$596$	6.96503	0.285299
$597$	0	0
$598$	0.658774	0.0269393
$599$	36.0488	1.47291	0.736457	−	0.676485i	$-0.236497\pi$
$599$	36.0488	1.47291	0.736457	+	0.676485i	$0.236497\pi$
$600$	0	0
$601$	36.7194	1.49781	0.748907	−	0.662675i	$-0.230579\pi$
$601$	36.7194	1.49781	0.748907	+	0.662675i	$0.230579\pi$
$602$	1.69920	0.0692543
$603$	0	0
$604$	24.4652	0.995476
$605$	3.25795	0.132454
$606$	0	0
$607$	−2.15714	−0.0875557	−0.0437779	−	0.999041i	$-0.513939\pi$
$607$	−2.15714	−0.0875557	−0.0437779	+	0.999041i	$0.513939\pi$
$608$	−16.1870	−0.656469
$609$	0	0
$610$	−0.0935827	−0.00378905
$611$	−47.4626	−1.92013
$612$	0	0
$613$	8.93902	0.361044	0.180522	−	0.983571i	$-0.442221\pi$
$613$	8.93902	0.361044	0.180522	+	0.983571i	$0.442221\pi$
$614$	−2.92131	−0.117895
$615$	0	0
$616$	8.11494	0.326960
$617$	26.4905	1.06647	0.533234	−	0.845968i	$-0.320977\pi$
$617$	26.4905	1.06647	0.533234	+	0.845968i	$0.320977\pi$
$618$	0	0
$619$	−18.4180	−0.740280	−0.370140	−	0.928976i	$-0.620690\pi$
$619$	−18.4180	−0.740280	−0.370140	+	0.928976i	$0.620690\pi$
$620$	−15.3756	−0.617498
$621$	0	0
$622$	−7.85831	−0.315090
$623$	−1.42988	−0.0572870
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.586731	0.0234505
$627$	0	0
$628$	35.7869	1.42805
$629$	−21.4806	−0.856487
$630$	0	0
$631$	1.32424	0.0527171	0.0263586	−	0.999653i	$-0.491609\pi$
$631$	1.32424	0.0527171	0.0263586	+	0.999653i	$0.491609\pi$
$632$	21.0846	0.838702
$633$	0	0
$634$	7.71421	0.306370
$635$	−4.89347	−0.194192
$636$	0	0
$637$	20.0099	0.792819
$638$	8.38060	0.331791
$639$	0	0
$640$	−11.5355	−0.455980
$641$	17.7818	0.702340	0.351170	−	0.936312i	$-0.385784\pi$
$641$	17.7818	0.702340	0.351170	+	0.936312i	$0.385784\pi$
$642$	0	0
$643$	−36.1805	−1.42682	−0.713410	−	0.700747i	$-0.752850\pi$
$643$	−36.1805	−1.42682	−0.713410	+	0.700747i	$0.752850\pi$
$644$	−0.716527	−0.0282351
$645$	0	0
$646$	4.85378	0.190969
$647$	17.3083	0.680460	0.340230	−	0.940342i	$-0.389495\pi$
$647$	17.3083	0.680460	0.340230	+	0.940342i	$0.389495\pi$
$648$	0	0
$649$	23.4427	0.920205
$650$	2.22876	0.0874192
$651$	0	0
$652$	−17.3419	−0.679163
$653$	15.3746	0.601654	0.300827	−	0.953679i	$-0.402737\pi$
$653$	15.3746	0.601654	0.300827	+	0.953679i	$0.402737\pi$
$654$	0	0
$655$	3.76602	0.147150
$656$	−2.24084	−0.0874902
$657$	0	0
$658$	−9.27665	−0.361641
$659$	−32.1049	−1.25063	−0.625315	−	0.780372i	$-0.715029\pi$
$659$	−32.1049	−1.25063	−0.625315	+	0.780372i	$0.715029\pi$
$660$	0	0
$661$	20.3705	0.792322	0.396161	−	0.918181i	$-0.370342\pi$
$661$	20.3705	0.792322	0.396161	+	0.918181i	$0.370342\pi$
$662$	9.06760	0.352422
$663$	0	0
$664$	−7.24724	−0.281247
$665$	−4.34296	−0.168413
$666$	0	0
$667$	−1.61294	−0.0624534
$668$	−31.2753	−1.21008
$669$	0	0
$670$	−0.985550	−0.0380751
$671$	−0.471761	−0.0182121
$672$	0	0
$673$	22.9359	0.884115	0.442058	−	0.896987i	$-0.354249\pi$
$673$	22.9359	0.884115	0.442058	+	0.896987i	$0.354249\pi$
$674$	9.54774	0.367765
$675$	0	0
$676$	−5.60335	−0.215514
$677$	6.77239	0.260284	0.130142	−	0.991495i	$-0.458457\pi$
$677$	6.77239	0.260284	0.130142	+	0.991495i	$0.458457\pi$
$678$	0	0
$679$	−8.54861	−0.328065
$680$	5.90541	0.226462
$681$	0	0
$682$	13.9284	0.533347
$683$	21.3100	0.815404	0.407702	−	0.913115i	$-0.366330\pi$
$683$	21.3100	0.815404	0.407702	+	0.913115i	$0.366330\pi$
$684$	0	0
$685$	−7.47314	−0.285534
$686$	9.43555	0.360251
$687$	0	0
$688$	−4.87633	−0.185909
$689$	−20.7496	−0.790499
$690$	0	0
$691$	−35.2195	−1.33981	−0.669907	−	0.742445i	$-0.733666\pi$
$691$	−35.2195	−1.33981	−0.669907	+	0.742445i	$0.733666\pi$
$692$	16.6792	0.634047
$693$	0	0
$694$	−6.90946	−0.262279
$695$	18.5104	0.702138
$696$	0	0
$697$	2.86454	0.108502
$698$	9.57309	0.362347
$699$	0	0
$700$	−2.42415	−0.0916243
$701$	−11.1696	−0.421869	−0.210935	−	0.977500i	$-0.567651\pi$
$701$	−11.1696	−0.421869	−0.210935	+	0.977500i	$0.567651\pi$
$702$	0	0
$703$	22.5340	0.849885
$704$	−4.41915	−0.166553
$705$	0	0
$706$	−10.3769	−0.390541
$707$	−7.07674	−0.266148
$708$	0	0
$709$	38.6537	1.45167	0.725835	−	0.687869i	$-0.241454\pi$
$709$	38.6537	1.45167	0.725835	+	0.687869i	$0.241454\pi$
$710$	3.92279	0.147220
$711$	0	0
$712$	−2.03966	−0.0764393
$713$	−2.68068	−0.100392
$714$	0	0
$715$	11.2354	0.420182
$716$	13.4914	0.504197
$717$	0	0
$718$	−2.41081	−0.0899708
$719$	−46.1121	−1.71969	−0.859846	−	0.510554i	$-0.829440\pi$
$719$	−46.1121	−1.71969	−0.859846	+	0.510554i	$0.829440\pi$
$720$	0	0
$721$	−16.4777	−0.613663
$722$	5.39528	0.200792
$723$	0	0
$724$	14.1130	0.524504
$725$	−5.45690	−0.202664
$726$	0	0
$727$	40.9106	1.51729	0.758646	−	0.651503i	$-0.225861\pi$
$727$	40.9106	1.51729	0.758646	+	0.651503i	$0.225861\pi$
$728$	−11.7766	−0.436469
$729$	0	0
$730$	−4.75471	−0.175980
$731$	6.23357	0.230557
$732$	0	0
$733$	−18.4438	−0.681238	−0.340619	−	0.940201i	$-0.610637\pi$
$733$	−18.4438	−0.681238	−0.340619	+	0.940201i	$0.610637\pi$
$734$	−15.6029	−0.575913
$735$	0	0
$736$	−1.57526	−0.0580650
$737$	−4.96827	−0.183009
$738$	0	0
$739$	−37.3912	−1.37546	−0.687730	−	0.725967i	$-0.741393\pi$
$739$	−37.3912	−1.37546	−0.687730	+	0.725967i	$0.741393\pi$
$740$	12.5780	0.462376
$741$	0	0
$742$	−4.05556	−0.148884
$743$	49.3566	1.81072	0.905359	−	0.424647i	$-0.139602\pi$
$743$	49.3566	1.81072	0.905359	+	0.424647i	$0.139602\pi$
$744$	0	0
$745$	4.10832	0.150517
$746$	−2.86183	−0.104779
$747$	0	0
$748$	13.6578	0.499378
$749$	16.2193	0.592642
$750$	0	0
$751$	6.20496	0.226422	0.113211	−	0.993571i	$-0.463886\pi$
$751$	6.20496	0.226422	0.113211	+	0.993571i	$0.463886\pi$
$752$	26.6219	0.970802
$753$	0	0
$754$	−12.1621	−0.442918
$755$	14.4308	0.525190
$756$	0	0
$757$	6.60784	0.240166	0.120083	−	0.992764i	$-0.461684\pi$
$757$	6.60784	0.240166	0.120083	+	0.992764i	$0.461684\pi$
$758$	−11.7798	−0.427862
$759$	0	0
$760$	−6.19501	−0.224717
$761$	2.19817	0.0796837	0.0398418	−	0.999206i	$-0.487315\pi$
$761$	2.19817	0.0796837	0.0398418	+	0.999206i	$0.487315\pi$
$762$	0	0
$763$	−12.3652	−0.447651
$764$	−12.6222	−0.456656
$765$	0	0
$766$	−8.33421	−0.301127
$767$	−34.0205	−1.22841
$768$	0	0
$769$	46.8772	1.69043	0.845217	−	0.534423i	$-0.179471\pi$
$769$	46.8772	1.69043	0.845217	+	0.534423i	$0.179471\pi$
$770$	2.19599	0.0791379
$771$	0	0
$772$	−5.07620	−0.182697
$773$	10.3063	0.370693	0.185347	−	0.982673i	$-0.440659\pi$
$773$	10.3063	0.370693	0.185347	+	0.982673i	$0.440659\pi$
$774$	0	0
$775$	−9.06927	−0.325778
$776$	−12.1942	−0.437745
$777$	0	0
$778$	−19.0119	−0.681609
$779$	−3.00501	−0.107666
$780$	0	0
$781$	19.7752	0.707613
$782$	0.472354	0.0168913
$783$	0	0
$784$	−11.2236	−0.400843
$785$	21.1088	0.753407
$786$	0	0
$787$	6.60036	0.235278	0.117639	−	0.993056i	$-0.462468\pi$
$787$	6.60036	0.235278	0.117639	+	0.993056i	$0.462468\pi$
$788$	41.8454	1.49068
$789$	0	0
$790$	5.70573	0.203001
$791$	−19.7871	−0.703548
$792$	0	0
$793$	0.684631	0.0243120
$794$	−10.1619	−0.360632
$795$	0	0
$796$	−8.83453	−0.313131
$797$	−12.2499	−0.433914	−0.216957	−	0.976181i	$-0.569613\pi$
$797$	−12.2499	−0.433914	−0.216957	+	0.976181i	$0.569613\pi$
$798$	0	0
$799$	−34.0316	−1.20395
$800$	−5.32943	−0.188424
$801$	0	0
$802$	20.5081	0.724165
$803$	−23.9690	−0.845849
$804$	0	0
$805$	−0.422643	−0.0148962
$806$	−20.2132	−0.711981
$807$	0	0
$808$	−10.0946	−0.355127
$809$	5.22817	0.183813	0.0919063	−	0.995768i	$-0.470704\pi$
$809$	5.22817	0.183813	0.0919063	+	0.995768i	$0.470704\pi$
$810$	0	0
$811$	−3.29397	−0.115667	−0.0578335	−	0.998326i	$-0.518419\pi$
$811$	−3.29397	−0.115667	−0.0578335	+	0.998326i	$0.518419\pi$
$812$	13.2283	0.464224
$813$	0	0
$814$	−11.3941	−0.399365
$815$	−10.2291	−0.358311
$816$	0	0
$817$	−6.53926	−0.228780
$818$	4.90198	0.171394
$819$	0	0
$820$	−1.67734	−0.0585751
$821$	7.38126	0.257608	0.128804	−	0.991670i	$-0.458886\pi$
$821$	7.38126	0.257608	0.128804	+	0.991670i	$0.458886\pi$
$822$	0	0
$823$	−8.09872	−0.282303	−0.141152	−	0.989988i	$-0.545081\pi$
$823$	−8.09872	−0.282303	−0.141152	+	0.989988i	$0.545081\pi$
$824$	−23.5047	−0.818824
$825$	0	0
$826$	−6.64937	−0.231361
$827$	13.9887	0.486435	0.243217	−	0.969972i	$-0.421797\pi$
$827$	13.9887	0.486435	0.243217	+	0.969972i	$0.421797\pi$
$828$	0	0
$829$	−5.22166	−0.181356	−0.0906779	−	0.995880i	$-0.528903\pi$
$829$	−5.22166	−0.181356	−0.0906779	+	0.995880i	$0.528903\pi$
$830$	−1.96118	−0.0680735
$831$	0	0
$832$	6.41318	0.222337
$833$	14.3475	0.497110
$834$	0	0
$835$	−18.4477	−0.638409
$836$	−14.3276	−0.495529
$837$	0	0
$838$	15.4048	0.532151
$839$	−49.2959	−1.70188	−0.850941	−	0.525261i	$-0.823968\pi$
$839$	−49.2959	−1.70188	−0.850941	+	0.525261i	$0.823968\pi$
$840$	0	0
$841$	0.777747	0.0268188
$842$	3.28364	0.113162
$843$	0	0
$844$	−26.5797	−0.914911
$845$	−3.30513	−0.113700
$846$	0	0
$847$	−4.65848	−0.160067
$848$	11.6385	0.399669
$849$	0	0
$850$	1.59807	0.0548132
$851$	2.19293	0.0751728
$852$	0	0
$853$	40.6425	1.39157	0.695787	−	0.718248i	$-0.255056\pi$
$853$	40.6425	1.39157	0.695787	+	0.718248i	$0.255056\pi$
$854$	0.133812	0.00457896
$855$	0	0
$856$	23.1361	0.790775
$857$	−23.7625	−0.811712	−0.405856	−	0.913937i	$-0.633027\pi$
$857$	−23.7625	−0.811712	−0.405856	+	0.913937i	$0.633027\pi$
$858$	0	0
$859$	26.5889	0.907202	0.453601	−	0.891205i	$-0.350139\pi$
$859$	26.5889	0.907202	0.453601	+	0.891205i	$0.350139\pi$
$860$	−3.65008	−0.124467
$861$	0	0
$862$	11.4890	0.391318
$863$	27.6487	0.941172	0.470586	−	0.882354i	$-0.344043\pi$
$863$	27.6487	0.941172	0.470586	+	0.882354i	$0.344043\pi$
$864$	0	0
$865$	9.83819	0.334509
$866$	5.98191	0.203273
$867$	0	0
$868$	21.9853	0.746229
$869$	28.7632	0.975725
$870$	0	0
$871$	7.21007	0.244304
$872$	−17.6384	−0.597310
$873$	0	0
$874$	−0.495518	−0.0167611
$875$	−1.42988	−0.0483389
$876$	0	0
$877$	26.2469	0.886296	0.443148	−	0.896448i	$-0.353862\pi$
$877$	26.2469	0.886296	0.443148	+	0.896448i	$0.353862\pi$
$878$	−13.7304	−0.463377
$879$	0	0
$880$	−6.30200	−0.212440
$881$	−7.39271	−0.249067	−0.124533	−	0.992215i	$-0.539743\pi$
$881$	−7.39271	−0.249067	−0.124533	+	0.992215i	$0.539743\pi$
$882$	0	0
$883$	−57.3507	−1.93001	−0.965003	−	0.262239i	$-0.915539\pi$
$883$	−57.3507	−1.93001	−0.965003	+	0.262239i	$0.915539\pi$
$884$	−19.8205	−0.666636
$885$	0	0
$886$	1.84548	0.0620002
$887$	−6.26180	−0.210251	−0.105125	−	0.994459i	$-0.533524\pi$
$887$	−6.26180	−0.210251	−0.105125	+	0.994459i	$0.533524\pi$
$888$	0	0
$889$	6.99709	0.234675
$890$	−0.551952	−0.0185015
$891$	0	0
$892$	29.3827	0.983807
$893$	35.7005	1.19467
$894$	0	0
$895$	7.95789	0.266003
$896$	16.4944	0.551039
$897$	0	0
$898$	1.76335	0.0588437
$899$	49.4901	1.65059
$900$	0	0
$901$	−14.8779	−0.495655
$902$	1.51946	0.0505926
$903$	0	0
$904$	−28.2253	−0.938759
$905$	8.32451	0.276716
$906$	0	0
$907$	−15.4960	−0.514536	−0.257268	−	0.966340i	$-0.582822\pi$
$907$	−15.4960	−0.514536	−0.257268	+	0.966340i	$0.582822\pi$
$908$	1.53955	0.0510919
$909$	0	0
$910$	−3.18687	−0.105644
$911$	37.2213	1.23320	0.616599	−	0.787278i	$-0.288510\pi$
$911$	37.2213	1.23320	0.616599	+	0.787278i	$0.288510\pi$
$912$	0	0
$913$	−9.88653	−0.327196
$914$	−11.5301	−0.381382
$915$	0	0
$916$	23.1441	0.764702
$917$	−5.38496	−0.177827
$918$	0	0
$919$	18.6875	0.616444	0.308222	−	0.951314i	$-0.400266\pi$
$919$	18.6875	0.616444	0.308222	+	0.951314i	$0.400266\pi$
$920$	−0.602878	−0.0198763
$921$	0	0
$922$	11.9615	0.393929
$923$	−28.6983	−0.944614
$924$	0	0
$925$	7.41912	0.243939
$926$	−22.1180	−0.726843
$927$	0	0
$928$	29.0822	0.954669
$929$	−49.1555	−1.61274	−0.806370	−	0.591411i	$-0.798571\pi$
$929$	−49.1555	−1.61274	−0.806370	+	0.591411i	$0.798571\pi$
$930$	0	0
$931$	−15.0511	−0.493279
$932$	34.0067	1.11393
$933$	0	0
$934$	−14.7220	−0.481720
$935$	8.05604	0.263461
$936$	0	0
$937$	15.4987	0.506320	0.253160	−	0.967424i	$-0.418530\pi$
$937$	15.4987	0.506320	0.253160	+	0.967424i	$0.418530\pi$
$938$	1.40922	0.0460127
$939$	0	0
$940$	19.9273	0.649957
$941$	−27.2614	−0.888695	−0.444347	−	0.895855i	$-0.646564\pi$
$941$	−27.2614	−0.888695	−0.444347	+	0.895855i	$0.646564\pi$
$942$	0	0
$943$	−0.292438	−0.00952309
$944$	19.0822	0.621074
$945$	0	0
$946$	3.30653	0.107505
$947$	9.72009	0.315861	0.157930	−	0.987450i	$-0.449518\pi$
$947$	9.72009	0.315861	0.157930	+	0.987450i	$0.449518\pi$
$948$	0	0
$949$	34.7844	1.12915
$950$	−1.67643	−0.0543907
$951$	0	0
$952$	−8.44405	−0.273673
$953$	−29.8061	−0.965513	−0.482757	−	0.875755i	$-0.660364\pi$
$953$	−29.8061	−0.965513	−0.482757	+	0.875755i	$0.660364\pi$
$954$	0	0
$955$	−7.44520	−0.240921
$956$	28.4841	0.921243
$957$	0	0
$958$	20.4996	0.662311
$959$	10.6857	0.345060
$960$	0	0
$961$	51.2517	1.65328
$962$	16.5355	0.533124
$963$	0	0
$964$	48.0124	1.54637
$965$	−2.99419	−0.0963865
$966$	0	0
$967$	36.1992	1.16409	0.582045	−	0.813157i	$-0.302253\pi$
$967$	36.1992	1.16409	0.582045	+	0.813157i	$0.302253\pi$
$968$	−6.64509	−0.213581
$969$	0	0
$970$	−3.29987	−0.105952
$971$	21.9223	0.703521	0.351761	−	0.936090i	$-0.385583\pi$
$971$	21.9223	0.703521	0.351761	+	0.936090i	$0.385583\pi$
$972$	0	0
$973$	−26.4676	−0.848514
$974$	−5.88303	−0.188504
$975$	0	0
$976$	−0.384012	−0.0122919
$977$	1.83190	0.0586076	0.0293038	−	0.999571i	$-0.490671\pi$
$977$	1.83190	0.0586076	0.0293038	+	0.999571i	$0.490671\pi$
$978$	0	0
$979$	−2.78245	−0.0889276
$980$	−8.40119	−0.268366
$981$	0	0
$982$	−0.259010	−0.00826535
$983$	44.7941	1.42871	0.714355	−	0.699784i	$-0.246720\pi$
$983$	44.7941	1.42871	0.714355	+	0.699784i	$0.246720\pi$
$984$	0	0
$985$	24.6825	0.786449
$986$	−8.72049	−0.277717
$987$	0	0
$988$	20.7925	0.661497
$989$	−0.636380	−0.0202357
$990$	0	0
$991$	−38.0579	−1.20895	−0.604474	−	0.796625i	$-0.706617\pi$
$991$	−38.0579	−1.20895	−0.604474	+	0.796625i	$0.706617\pi$
$992$	48.3341	1.53461
$993$	0	0
$994$	−5.60912	−0.177911
$995$	−5.21104	−0.165201
$996$	0	0
$997$	38.0064	1.20368	0.601838	−	0.798619i	$-0.294435\pi$
$997$	38.0064	1.20368	0.601838	+	0.798619i	$0.294435\pi$
$998$	−9.54137	−0.302027
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.v.1.5	yes	12
3.2	odd	2		4005.2.a.u.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.8	✓	12	3.2	odd	2
4005.2.a.v.1.5	yes	12	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)

\(f(q)\)	\(=\)	\(q-0.551952 q^{2} -1.69535 q^{4} -1.00000 q^{5} +1.42988 q^{7} +2.03966 q^{8} +O(q^{10})\)	\(q-0.551952 q^{2} -1.69535 q^{4} -1.00000 q^{5} +1.42988 q^{7} +2.03966 q^{8} +0.551952 q^{10} +2.78245 q^{11} -4.03796 q^{13} -0.789227 q^{14} +2.26491 q^{16} -2.89530 q^{17} +3.03728 q^{19} +1.69535 q^{20} -1.53578 q^{22} +0.295578 q^{23} +1.00000 q^{25} +2.22876 q^{26} -2.42415 q^{28} -5.45690 q^{29} -9.06927 q^{31} -5.32943 q^{32} +1.59807 q^{34} -1.42988 q^{35} +7.41912 q^{37} -1.67643 q^{38} -2.03966 q^{40} -0.989375 q^{41} -2.15300 q^{43} -4.71723 q^{44} -0.163145 q^{46} +11.7541 q^{47} -4.95544 q^{49} -0.551952 q^{50} +6.84575 q^{52} +5.13864 q^{53} -2.78245 q^{55} +2.91647 q^{56} +3.01195 q^{58} +8.42517 q^{59} -0.169549 q^{61} +5.00580 q^{62} -1.58822 q^{64} +4.03796 q^{65} -1.78557 q^{67} +4.90854 q^{68} +0.789227 q^{70} +7.10711 q^{71} -8.61435 q^{73} -4.09500 q^{74} -5.14925 q^{76} +3.97858 q^{77} +10.3374 q^{79} -2.26491 q^{80} +0.546088 q^{82} -3.55317 q^{83} +2.89530 q^{85} +1.18835 q^{86} +5.67525 q^{88} -1.00000 q^{89} -5.77381 q^{91} -0.501109 q^{92} -6.48770 q^{94} -3.03728 q^{95} -5.97854 q^{97} +2.73516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8}+O(q^{10}) \) 12 * q + 3 * q^2 + 11 * q^4 - 12 * q^5 - 8 * q^7 + 9 * q^8	\( 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8} - 3 q^{10} - 12 q^{13} - 4 q^{14} + q^{16} - 24 q^{19} - 11 q^{20} - 16 q^{22} + 24 q^{23} + 12 q^{25} + q^{26} - 44 q^{28} - 8 q^{29} - 12 q^{31} + 31 q^{32} - 18 q^{34} + 8 q^{35} - 10 q^{37} - 2 q^{38} - 9 q^{40} + 10 q^{41} - 42 q^{43} - 42 q^{44} - 24 q^{46} + 22 q^{47} - 4 q^{49} + 3 q^{50} - 30 q^{52} - 8 q^{53} - 27 q^{56} - 12 q^{58} - 4 q^{59} - 52 q^{61} + 14 q^{62} + 7 q^{64} + 12 q^{65} - 40 q^{67} - 23 q^{68} + 4 q^{70} - 2 q^{71} - 8 q^{73} - 26 q^{74} - 46 q^{76} + 12 q^{77} - 26 q^{79} - q^{80} - 26 q^{82} + 14 q^{83} - 32 q^{86} - 60 q^{88} - 12 q^{89} - 24 q^{91} + 38 q^{92} - 26 q^{94} + 24 q^{95} - 6 q^{97} - 30 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 + 11 * q^4 - 12 * q^5 - 8 * q^7 + 9 * q^8 - 3 * q^10 - 12 * q^13 - 4 * q^14 + q^16 - 24 * q^19 - 11 * q^20 - 16 * q^22 + 24 * q^23 + 12 * q^25 + q^26 - 44 * q^28 - 8 * q^29 - 12 * q^31 + 31 * q^32 - 18 * q^34 + 8 * q^35 - 10 * q^37 - 2 * q^38 - 9 * q^40 + 10 * q^41 - 42 * q^43 - 42 * q^44 - 24 * q^46 + 22 * q^47 - 4 * q^49 + 3 * q^50 - 30 * q^52 - 8 * q^53 - 27 * q^56 - 12 * q^58 - 4 * q^59 - 52 * q^61 + 14 * q^62 + 7 * q^64 + 12 * q^65 - 40 * q^67 - 23 * q^68 + 4 * q^70 - 2 * q^71 - 8 * q^73 - 26 * q^74 - 46 * q^76 + 12 * q^77 - 26 * q^79 - q^80 - 26 * q^82 + 14 * q^83 - 32 * q^86 - 60 * q^88 - 12 * q^89 - 24 * q^91 + 38 * q^92 - 26 * q^94 + 24 * q^95 - 6 * q^97 - 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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