LMFDB - Embedded newform 4140.2.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.14345904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12 $ x^5 - 2x^4 - 13x^3 + 34x^2 - 11x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.23445$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.81459 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.81459 q^{7} +4.70892 q^{11} +4.24002 q^{13} -3.85252 q^{17} +6.70892 q^{19} -1.00000 q^{23} +1.00000 q^{25} +1.42543 q^{29} -9.03035 q^{31} -2.81459 q^{35} +5.28349 q^{37} +10.3777 q^{41} +12.1739 q^{43} -10.0287 q^{47} +0.921920 q^{49} +9.17229 q^{53} -4.70892 q^{55} -6.36323 q^{59} -10.4140 q^{61} -4.24002 q^{65} +6.50517 q^{67} -8.74852 q^{71} -0.240022 q^{73} +13.2537 q^{77} -4.48337 q^{79} -13.1109 q^{83} +3.85252 q^{85} +7.62918 q^{89} +11.9339 q^{91} -6.70892 q^{95} -0.385284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 4 q^{7}+O(q^{10}) $ 5 * q - 5 * q^5 + 4 * q^7	$ 5 q - 5 q^{5} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 5 q^{23} + 5 q^{25} - 2 q^{29} + 8 q^{31} - 4 q^{35} + 8 q^{37} - 2 q^{41} + 18 q^{43} + 4 q^{47} + 13 q^{49} + 2 q^{53} + 4 q^{55} - 6 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 10 q^{71} + 18 q^{73} + 16 q^{77} + 14 q^{79} - 8 q^{83} + 2 q^{85} + 18 q^{89} + 36 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 5 * q^23 + 5 * q^25 - 2 * q^29 + 8 * q^31 - 4 * q^35 + 8 * q^37 - 2 * q^41 + 18 * q^43 + 4 * q^47 + 13 * q^49 + 2 * q^53 + 4 * q^55 - 6 * q^59 + 10 * q^61 - 2 * q^65 + 16 * q^67 - 10 * q^71 + 18 * q^73 + 16 * q^77 + 14 * q^79 - 8 * q^83 + 2 * q^85 + 18 * q^89 + 36 * q^91 - 6 * q^95 + 6 * q^97

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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.81459	1.06382	0.531908	−	0.846802i	$-0.321475\pi$
$7$	2.81459	1.06382	0.531908	+	0.846802i	$0.321475\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.70892	1.41979	0.709897	−	0.704306i	$-0.248742\pi$
$11$	4.70892	1.41979	0.709897	+	0.704306i	$0.248742\pi$
$12$	0	0
$13$	4.24002	1.17597	0.587985	−	0.808872i	$-0.299921\pi$
$13$	4.24002	1.17597	0.587985	+	0.808872i	$0.299921\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.85252	−0.934374	−0.467187	−	0.884159i	$-0.654733\pi$
$17$	−3.85252	−0.934374	−0.467187	+	0.884159i	$0.654733\pi$
$18$	0	0
$19$	6.70892	1.53913	0.769566	−	0.638567i	$-0.220473\pi$
$19$	6.70892	1.53913	0.769566	+	0.638567i	$0.220473\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.42543	0.264696	0.132348	−	0.991203i	$-0.457748\pi$
$29$	1.42543	0.264696	0.132348	+	0.991203i	$0.457748\pi$
$30$	0	0
$31$	−9.03035	−1.62190	−0.810949	−	0.585117i	$-0.801049\pi$
$31$	−9.03035	−1.62190	−0.810949	+	0.585117i	$0.801049\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.81459	−0.475753
$36$	0	0
$37$	5.28349	0.868601	0.434300	−	0.900768i	$-0.356996\pi$
$37$	5.28349	0.868601	0.434300	+	0.900768i	$0.356996\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.3777	1.62072	0.810362	−	0.585930i	$-0.199270\pi$
$41$	10.3777	1.62072	0.810362	+	0.585930i	$0.199270\pi$
$42$	0	0
$43$	12.1739	1.85651	0.928255	−	0.371945i	$-0.121309\pi$
$43$	12.1739	1.85651	0.928255	+	0.371945i	$0.121309\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.0287	−1.46283	−0.731417	−	0.681930i	$-0.761141\pi$
$47$	−10.0287	−1.46283	−0.731417	+	0.681930i	$0.761141\pi$
$48$	0	0
$49$	0.921920	0.131703
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.17229	1.25991	0.629955	−	0.776632i	$-0.283073\pi$
$53$	9.17229	1.25991	0.629955	+	0.776632i	$0.283073\pi$
$54$	0	0
$55$	−4.70892	−0.634951
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.36323	−0.828422	−0.414211	−	0.910181i	$-0.635942\pi$
$59$	−6.36323	−0.828422	−0.414211	+	0.910181i	$0.635942\pi$
$60$	0	0
$61$	−10.4140	−1.33337	−0.666686	−	0.745339i	$-0.732288\pi$
$61$	−10.4140	−1.33337	−0.666686	+	0.745339i	$0.732288\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.24002	−0.525910
$66$	0	0
$67$	6.50517	0.794733	0.397367	−	0.917660i	$-0.369924\pi$
$67$	6.50517	0.794733	0.397367	+	0.917660i	$0.369924\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.74852	−1.03826	−0.519129	−	0.854696i	$-0.673744\pi$
$71$	−8.74852	−1.03826	−0.519129	+	0.854696i	$0.673744\pi$
$72$	0	0
$73$	−0.240022	−0.0280924	−0.0140462	−	0.999901i	$-0.504471\pi$
$73$	−0.240022	−0.0280924	−0.0140462	+	0.999901i	$0.504471\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.2537	1.51040
$78$	0	0
$79$	−4.48337	−0.504418	−0.252209	−	0.967673i	$-0.581157\pi$
$79$	−4.48337	−0.504418	−0.252209	+	0.967673i	$0.581157\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.1109	−1.43911	−0.719553	−	0.694437i	$-0.755653\pi$
$83$	−13.1109	−1.43911	−0.719553	+	0.694437i	$0.755653\pi$
$84$	0	0
$85$	3.85252	0.417865
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.62918	0.808692	0.404346	−	0.914606i	$-0.367499\pi$
$89$	7.62918	0.808692	0.404346	+	0.914606i	$0.367499\pi$
$90$	0	0
$91$	11.9339	1.25102
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.70892	−0.688321
$96$	0	0
$97$	−0.385284	−0.0391196	−0.0195598	−	0.999809i	$-0.506226\pi$
$97$	−0.385284	−0.0391196	−0.0195598	+	0.999809i	$0.506226\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.5372	−1.64551	−0.822755	−	0.568396i	$-0.807564\pi$
$101$	−16.5372	−1.64551	−0.822755	+	0.568396i	$0.807564\pi$
$102$	0	0
$103$	−17.5918	−1.73337	−0.866685	−	0.498855i	$-0.833754\pi$
$103$	−17.5918	−1.73337	−0.866685	+	0.498855i	$0.833754\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.9506	1.34866	0.674328	−	0.738432i	$-0.264433\pi$
$107$	13.9506	1.34866	0.674328	+	0.738432i	$0.264433\pi$
$108$	0	0
$109$	6.70892	0.642598	0.321299	−	0.946978i	$-0.395881\pi$
$109$	6.70892	0.642598	0.321299	+	0.946978i	$0.395881\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.1756	−1.05131	−0.525656	−	0.850697i	$-0.676180\pi$
$113$	−11.1756	−1.05131	−0.525656	+	0.850697i	$0.676180\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.8433	−0.994001
$120$	0	0
$121$	11.1739	1.01581
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.0942	1.33939	0.669697	−	0.742634i	$-0.266424\pi$
$127$	15.0942	1.33939	0.669697	+	0.742634i	$0.266424\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.62918	0.142342	0.0711711	−	0.997464i	$-0.477326\pi$
$131$	1.62918	0.142342	0.0711711	+	0.997464i	$0.477326\pi$
$132$	0	0
$133$	18.8829	1.63735
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.61804	0.479981	0.239991	−	0.970775i	$-0.422856\pi$
$137$	5.61804	0.479981	0.239991	+	0.970775i	$0.422856\pi$
$138$	0	0
$139$	4.07808	0.345898	0.172949	−	0.984931i	$-0.444670\pi$
$139$	4.07808	0.345898	0.172949	+	0.984931i	$0.444670\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.9659	1.66964
$144$	0	0
$145$	−1.42543	−0.118376
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.1268	1.64885	0.824424	−	0.565972i	$-0.191499\pi$
$149$	20.1268	1.64885	0.824424	+	0.565972i	$0.191499\pi$
$150$	0	0
$151$	18.1739	1.47897	0.739487	−	0.673170i	$-0.235068\pi$
$151$	18.1739	1.47897	0.739487	+	0.673170i	$0.235068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.03035	0.725335
$156$	0	0
$157$	15.0833	1.20378	0.601889	−	0.798580i	$-0.294415\pi$
$157$	15.0833	1.20378	0.601889	+	0.798580i	$0.294415\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.81459	−0.221821
$162$	0	0
$163$	0.294954	0.0231026	0.0115513	−	0.999933i	$-0.496323\pi$
$163$	0.294954	0.0231026	0.0115513	+	0.999933i	$0.496323\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.24334	−0.328360	−0.164180	−	0.986430i	$-0.552498\pi$
$167$	−4.24334	−0.328360	−0.164180	+	0.986430i	$0.552498\pi$
$168$	0	0
$169$	4.97778	0.382906
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−24.9116	−1.89399	−0.946995	−	0.321248i	$-0.895898\pi$
$173$	−24.9116	−1.89399	−0.946995	+	0.321248i	$0.895898\pi$
$174$	0	0
$175$	2.81459	0.212763
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.03256	0.525638	0.262819	−	0.964845i	$-0.415348\pi$
$179$	7.03256	0.525638	0.262819	+	0.964845i	$0.415348\pi$
$180$	0	0
$181$	−15.5806	−1.15810	−0.579050	−	0.815292i	$-0.696576\pi$
$181$	−15.5806	−1.15810	−0.579050	+	0.815292i	$0.696576\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.28349	−0.388450
$186$	0	0
$187$	−18.1412	−1.32662
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.8940	1.22241	0.611204	−	0.791473i	$-0.290686\pi$
$191$	16.8940	1.22241	0.611204	+	0.791473i	$0.290686\pi$
$192$	0	0
$193$	−1.22500	−0.0881776	−0.0440888	−	0.999028i	$-0.514038\pi$
$193$	−1.22500	−0.0881776	−0.0440888	+	0.999028i	$0.514038\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.1092	1.29023	0.645114	−	0.764086i	$-0.276810\pi$
$197$	18.1092	1.29023	0.645114	+	0.764086i	$0.276810\pi$
$198$	0	0
$199$	22.1962	1.57344	0.786722	−	0.617307i	$-0.211776\pi$
$199$	22.1962	1.57344	0.786722	+	0.617307i	$0.211776\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.01200	0.281588
$204$	0	0
$205$	−10.3777	−0.724810
$206$	0	0
$207$	0	0
$208$	0	0
$209$	31.5918	2.18525
$210$	0	0
$211$	−23.6843	−1.63050	−0.815248	−	0.579111i	$-0.803400\pi$
$211$	−23.6843	−1.63050	−0.815248	+	0.579111i	$0.803400\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.1739	−0.830256
$216$	0	0
$217$	−25.4167	−1.72540
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−16.3348	−1.09880
$222$	0	0
$223$	15.7050	1.05169	0.525844	−	0.850581i	$-0.323750\pi$
$223$	15.7050	1.05169	0.525844	+	0.850581i	$0.323750\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.3206	0.685000	0.342500	−	0.939518i	$-0.388726\pi$
$227$	10.3206	0.685000	0.342500	+	0.939518i	$0.388726\pi$
$228$	0	0
$229$	3.54556	0.234297	0.117149	−	0.993114i	$-0.462625\pi$
$229$	3.54556	0.234297	0.117149	+	0.993114i	$0.462625\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.22168	0.0800350	0.0400175	−	0.999199i	$-0.487259\pi$
$233$	1.22168	0.0800350	0.0400175	+	0.999199i	$0.487259\pi$
$234$	0	0
$235$	10.0287	0.654199
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.8211	0.829325	0.414663	−	0.909975i	$-0.363900\pi$
$239$	12.8211	0.829325	0.414663	+	0.909975i	$0.363900\pi$
$240$	0	0
$241$	17.6390	1.13623	0.568113	−	0.822951i	$-0.307674\pi$
$241$	17.6390	1.13623	0.568113	+	0.822951i	$0.307674\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.921920	−0.0588993
$246$	0	0
$247$	28.4460	1.80997
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.1596	1.20934	0.604671	−	0.796476i	$-0.293305\pi$
$251$	19.1596	1.20934	0.604671	+	0.796476i	$0.293305\pi$
$252$	0	0
$253$	−4.70892	−0.296047
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.2034	−0.823607	−0.411804	−	0.911273i	$-0.635101\pi$
$257$	−13.2034	−0.823607	−0.411804	+	0.911273i	$0.635101\pi$
$258$	0	0
$259$	14.8709	0.924031
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−26.3029	−1.62191	−0.810954	−	0.585110i	$-0.801051\pi$
$263$	−26.3029	−1.62191	−0.810954	+	0.585110i	$0.801051\pi$
$264$	0	0
$265$	−9.17229	−0.563449
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.53466	−0.459396	−0.229698	−	0.973262i	$-0.573774\pi$
$269$	−7.53466	−0.459396	−0.229698	+	0.973262i	$0.573774\pi$
$270$	0	0
$271$	2.21355	0.134464	0.0672319	−	0.997737i	$-0.478583\pi$
$271$	2.21355	0.134464	0.0672319	+	0.997737i	$0.478583\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.70892	0.283959
$276$	0	0
$277$	10.4867	0.630084	0.315042	−	0.949078i	$-0.397981\pi$
$277$	10.4867	0.630084	0.315042	+	0.949078i	$0.397981\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−23.2682	−1.38806	−0.694031	−	0.719945i	$-0.744167\pi$
$281$	−23.2682	−1.38806	−0.694031	+	0.719945i	$0.744167\pi$
$282$	0	0
$283$	23.3297	1.38681	0.693404	−	0.720549i	$-0.256110\pi$
$283$	23.3297	1.38681	0.693404	+	0.720549i	$0.256110\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	29.2090	1.72415
$288$	0	0
$289$	−2.15807	−0.126945
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.2371	1.53279	0.766394	−	0.642371i	$-0.222049\pi$
$293$	26.2371	1.53279	0.766394	+	0.642371i	$0.222049\pi$
$294$	0	0
$295$	6.36323	0.370482
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.24002	−0.245207
$300$	0	0
$301$	34.2647	1.97498
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.4140	0.596302
$306$	0	0
$307$	13.3814	0.763717	0.381859	−	0.924221i	$-0.375284\pi$
$307$	13.3814	0.763717	0.381859	+	0.924221i	$0.375284\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.3309	−1.09616	−0.548078	−	0.836428i	$-0.684640\pi$
$311$	−19.3309	−1.09616	−0.548078	+	0.836428i	$0.684640\pi$
$312$	0	0
$313$	−8.90492	−0.503336	−0.251668	−	0.967814i	$-0.580979\pi$
$313$	−8.90492	−0.503336	−0.251668	+	0.967814i	$0.580979\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.32696	−0.243026	−0.121513	−	0.992590i	$-0.538775\pi$
$317$	−4.32696	−0.243026	−0.121513	+	0.992590i	$0.538775\pi$
$318$	0	0
$319$	6.71224	0.375814
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−25.8463	−1.43813
$324$	0	0
$325$	4.24002	0.235194
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−28.2266	−1.55619
$330$	0	0
$331$	17.9681	0.987619	0.493809	−	0.869570i	$-0.335604\pi$
$331$	17.9681	0.987619	0.493809	+	0.869570i	$0.335604\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.50517	−0.355416
$336$	0	0
$337$	4.93448	0.268798	0.134399	−	0.990927i	$-0.457090\pi$
$337$	4.93448	0.268798	0.134399	+	0.990927i	$0.457090\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−42.5232	−2.30276
$342$	0	0
$343$	−17.1073	−0.923708
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.06220	0.379119	0.189559	−	0.981869i	$-0.439294\pi$
$347$	7.06220	0.379119	0.189559	+	0.981869i	$0.439294\pi$
$348$	0	0
$349$	−36.3349	−1.94496	−0.972482	−	0.232977i	$-0.925153\pi$
$349$	−36.3349	−1.94496	−0.972482	+	0.232977i	$0.925153\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.83252	0.0975353	0.0487676	−	0.998810i	$-0.484471\pi$
$353$	1.83252	0.0975353	0.0487676	+	0.998810i	$0.484471\pi$
$354$	0	0
$355$	8.74852	0.464323
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.6587	−0.879211	−0.439605	−	0.898191i	$-0.644882\pi$
$359$	−16.6587	−0.879211	−0.439605	+	0.898191i	$0.644882\pi$
$360$	0	0
$361$	26.0096	1.36893
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.240022	0.0125633
$366$	0	0
$367$	14.9060	0.778088	0.389044	−	0.921219i	$-0.372805\pi$
$367$	14.9060	0.778088	0.389044	+	0.921219i	$0.372805\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	25.8162	1.34031
$372$	0	0
$373$	9.99929	0.517744	0.258872	−	0.965912i	$-0.416649\pi$
$373$	9.99929	0.517744	0.258872	+	0.965912i	$0.416649\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.04386	0.311275
$378$	0	0
$379$	−25.7624	−1.32333	−0.661663	−	0.749801i	$-0.730149\pi$
$379$	−25.7624	−1.32333	−0.661663	+	0.749801i	$0.730149\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.56089	−0.386343	−0.193172	−	0.981165i	$-0.561877\pi$
$383$	−7.56089	−0.386343	−0.193172	+	0.981165i	$0.561877\pi$
$384$	0	0
$385$	−13.2537	−0.675470
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.1884	1.02359	0.511797	−	0.859107i	$-0.328980\pi$
$389$	20.1884	1.02359	0.511797	+	0.859107i	$0.328980\pi$
$390$	0	0
$391$	3.85252	0.194830
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.48337	0.225583
$396$	0	0
$397$	−15.3453	−0.770159	−0.385079	−	0.922883i	$-0.625826\pi$
$397$	−15.3453	−0.770159	−0.385079	+	0.922883i	$0.625826\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.8542	−1.14128	−0.570642	−	0.821199i	$-0.693305\pi$
$401$	−22.8542	−1.14128	−0.570642	+	0.821199i	$0.693305\pi$
$402$	0	0
$403$	−38.2889	−1.90730
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.8795	1.23323
$408$	0	0
$409$	−21.9793	−1.08681	−0.543403	−	0.839472i	$-0.682864\pi$
$409$	−21.9793	−1.08681	−0.543403	+	0.839472i	$0.682864\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−17.9099	−0.881288
$414$	0	0
$415$	13.1109	0.643588
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−25.8207	−1.26143	−0.630713	−	0.776016i	$-0.717237\pi$
$419$	−25.8207	−1.26143	−0.630713	+	0.776016i	$0.717237\pi$
$420$	0	0
$421$	−24.0431	−1.17179	−0.585896	−	0.810386i	$-0.699257\pi$
$421$	−24.0431	−1.17179	−0.585896	+	0.810386i	$0.699257\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.85252	−0.186875
$426$	0	0
$427$	−29.3111	−1.41846
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−39.4060	−1.89812	−0.949060	−	0.315096i	$-0.897963\pi$
$431$	−39.4060	−1.89812	−0.949060	+	0.315096i	$0.897963\pi$
$432$	0	0
$433$	29.2717	1.40671	0.703354	−	0.710839i	$-0.251685\pi$
$433$	29.2717	1.40671	0.703354	+	0.710839i	$0.251685\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.70892	−0.320931
$438$	0	0
$439$	32.5296	1.55255	0.776276	−	0.630393i	$-0.217106\pi$
$439$	32.5296	1.55255	0.776276	+	0.630393i	$0.217106\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.0909	−0.812012	−0.406006	−	0.913870i	$-0.633079\pi$
$443$	−17.0909	−0.812012	−0.406006	+	0.913870i	$0.633079\pi$
$444$	0	0
$445$	−7.62918	−0.361658
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.7630	1.16864	0.584319	−	0.811524i	$-0.301362\pi$
$449$	24.7630	1.16864	0.584319	+	0.811524i	$0.301362\pi$
$450$	0	0
$451$	48.8678	2.30109
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−11.9339	−0.559471
$456$	0	0
$457$	−30.4397	−1.42391	−0.711956	−	0.702225i	$-0.752190\pi$
$457$	−30.4397	−1.42391	−0.711956	+	0.702225i	$0.752190\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.0241	0.979190	0.489595	−	0.871950i	$-0.337145\pi$
$461$	21.0241	0.979190	0.489595	+	0.871950i	$0.337145\pi$
$462$	0	0
$463$	−6.73114	−0.312823	−0.156411	−	0.987692i	$-0.549993\pi$
$463$	−6.73114	−0.312823	−0.156411	+	0.987692i	$0.549993\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.7256	1.05162	0.525808	−	0.850603i	$-0.323763\pi$
$467$	22.7256	1.05162	0.525808	+	0.850603i	$0.323763\pi$
$468$	0	0
$469$	18.3094	0.845449
$470$	0	0
$471$	0	0
$472$	0	0
$473$	57.3262	2.63586
$474$	0	0
$475$	6.70892	0.307826
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−23.2759	−1.06350	−0.531752	−	0.846900i	$-0.678466\pi$
$479$	−23.2759	−1.06350	−0.531752	+	0.846900i	$0.678466\pi$
$480$	0	0
$481$	22.4021	1.02145
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.385284	0.0174948
$486$	0	0
$487$	19.6579	0.890783	0.445391	−	0.895336i	$-0.353065\pi$
$487$	19.6579	0.890783	0.445391	+	0.895336i	$0.353065\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.1494	−0.773940	−0.386970	−	0.922092i	$-0.626478\pi$
$491$	−17.1494	−0.773940	−0.386970	+	0.922092i	$0.626478\pi$
$492$	0	0
$493$	−5.49151	−0.247325
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.6235	−1.10451
$498$	0	0
$499$	−29.2265	−1.30836	−0.654179	−	0.756340i	$-0.726986\pi$
$499$	−29.2265	−1.30836	−0.654179	+	0.756340i	$0.726986\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	37.9543	1.69230	0.846150	−	0.532945i	$-0.178915\pi$
$503$	37.9543	1.69230	0.846150	+	0.532945i	$0.178915\pi$
$504$	0	0
$505$	16.5372	0.735895
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22.8502	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$509$	−22.8502	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$510$	0	0
$511$	−0.675563	−0.0298851
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.5918	0.775187
$516$	0	0
$517$	−47.2243	−2.07692
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.08566	0.0913742	0.0456871	−	0.998956i	$-0.485452\pi$
$521$	2.08566	0.0913742	0.0456871	+	0.998956i	$0.485452\pi$
$522$	0	0
$523$	−32.4682	−1.41973	−0.709867	−	0.704335i	$-0.751245\pi$
$523$	−32.4682	−1.41973	−0.709867	+	0.704335i	$0.751245\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34.7896	1.51546
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	44.0017	1.90592
$534$	0	0
$535$	−13.9506	−0.603137
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.34125	0.186991
$540$	0	0
$541$	−34.9746	−1.50367	−0.751837	−	0.659349i	$-0.770832\pi$
$541$	−34.9746	−1.50367	−0.751837	+	0.659349i	$0.770832\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.70892	−0.287379
$546$	0	0
$547$	−15.2321	−0.651278	−0.325639	−	0.945494i	$-0.605579\pi$
$547$	−15.2321	−0.651278	−0.325639	+	0.945494i	$0.605579\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.56311	0.407402
$552$	0	0
$553$	−12.6188	−0.536608
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.75112	0.413168	0.206584	−	0.978429i	$-0.433765\pi$
$557$	9.75112	0.413168	0.206584	+	0.978429i	$0.433765\pi$
$558$	0	0
$559$	51.6178	2.18320
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.1405	0.553807	0.276904	−	0.960898i	$-0.410692\pi$
$563$	13.1405	0.553807	0.276904	+	0.960898i	$0.410692\pi$
$564$	0	0
$565$	11.1756	0.470161
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0.855292	0.0358557	0.0179279	−	0.999839i	$-0.494293\pi$
$569$	0.855292	0.0358557	0.0179279	+	0.999839i	$0.494293\pi$
$570$	0	0
$571$	−32.9699	−1.37975	−0.689873	−	0.723930i	$-0.742334\pi$
$571$	−32.9699	−1.37975	−0.689873	+	0.723930i	$0.742334\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	18.5943	0.774091	0.387046	−	0.922061i	$-0.373496\pi$
$577$	18.5943	0.774091	0.387046	+	0.922061i	$0.373496\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−36.9018	−1.53094
$582$	0	0
$583$	43.1916	1.78881
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.7002	1.84498	0.922488	−	0.386027i	$-0.126153\pi$
$587$	44.7002	1.84498	0.922488	+	0.386027i	$0.126153\pi$
$588$	0	0
$589$	−60.5839	−2.49632
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15.0501	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$593$	−15.0501	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$594$	0	0
$595$	10.8433	0.444531
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17.4970	0.714909	0.357455	−	0.933930i	$-0.383645\pi$
$599$	17.4970	0.714909	0.357455	+	0.933930i	$0.383645\pi$
$600$	0	0
$601$	−26.4515	−1.07898	−0.539490	−	0.841992i	$-0.681383\pi$
$601$	−26.4515	−1.07898	−0.539490	+	0.841992i	$0.681383\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.1739	−0.454286
$606$	0	0
$607$	−35.0390	−1.42219	−0.711095	−	0.703096i	$-0.751800\pi$
$607$	−35.0390	−1.42219	−0.711095	+	0.703096i	$0.751800\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−42.5218	−1.72025
$612$	0	0
$613$	−10.4911	−0.423732	−0.211866	−	0.977299i	$-0.567954\pi$
$613$	−10.4911	−0.423732	−0.211866	+	0.977299i	$0.567954\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.1126	0.447377	0.223688	−	0.974661i	$-0.428190\pi$
$617$	11.1126	0.447377	0.223688	+	0.974661i	$0.428190\pi$
$618$	0	0
$619$	6.36307	0.255753	0.127877	−	0.991790i	$-0.459184\pi$
$619$	6.36307	0.255753	0.127877	+	0.991790i	$0.459184\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	21.4730	0.860298
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.3548	−0.811598
$630$	0	0
$631$	−24.3955	−0.971168	−0.485584	−	0.874190i	$-0.661393\pi$
$631$	−24.3955	−0.971168	−0.485584	+	0.874190i	$0.661393\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−15.0942	−0.598995
$636$	0	0
$637$	3.90896	0.154879
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−35.2130	−1.39083	−0.695414	−	0.718609i	$-0.744779\pi$
$641$	−35.2130	−1.39083	−0.695414	+	0.718609i	$0.744779\pi$
$642$	0	0
$643$	21.4200	0.844724	0.422362	−	0.906427i	$-0.361201\pi$
$643$	21.4200	0.844724	0.422362	+	0.906427i	$0.361201\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.795930	−0.0312912	−0.0156456	−	0.999878i	$-0.504980\pi$
$647$	−0.795930	−0.0312912	−0.0156456	+	0.999878i	$0.504980\pi$
$648$	0	0
$649$	−29.9640	−1.17619
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.4936	1.66290	0.831451	−	0.555599i	$-0.187511\pi$
$653$	42.4936	1.66290	0.831451	+	0.555599i	$0.187511\pi$
$654$	0	0
$655$	−1.62918	−0.0636574
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.5631	0.839979	0.419990	−	0.907529i	$-0.362034\pi$
$659$	21.5631	0.839979	0.419990	+	0.907529i	$0.362034\pi$
$660$	0	0
$661$	−21.8328	−0.849196	−0.424598	−	0.905382i	$-0.639585\pi$
$661$	−21.8328	−0.849196	−0.424598	+	0.905382i	$0.639585\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−18.8829	−0.732246
$666$	0	0
$667$	−1.42543	−0.0551929
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−49.0386	−1.89311
$672$	0	0
$673$	6.32696	0.243886	0.121943	−	0.992537i	$-0.461087\pi$
$673$	6.32696	0.243886	0.121943	+	0.992537i	$0.461087\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.40504	0.246166	0.123083	−	0.992396i	$-0.460722\pi$
$677$	6.40504	0.246166	0.123083	+	0.992396i	$0.460722\pi$
$678$	0	0
$679$	−1.08442	−0.0416161
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.19300	0.122177	0.0610883	−	0.998132i	$-0.480543\pi$
$683$	3.19300	0.122177	0.0610883	+	0.998132i	$0.480543\pi$
$684$	0	0
$685$	−5.61804	−0.214654
$686$	0	0
$687$	0	0
$688$	0	0
$689$	38.8907	1.48162
$690$	0	0
$691$	17.4257	0.662904	0.331452	−	0.943472i	$-0.392462\pi$
$691$	17.4257	0.662904	0.331452	+	0.943472i	$0.392462\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.07808	−0.154690
$696$	0	0
$697$	−39.9803	−1.51436
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.53418	−0.284562	−0.142281	−	0.989826i	$-0.545444\pi$
$701$	−7.53418	−0.284562	−0.142281	+	0.989826i	$0.545444\pi$
$702$	0	0
$703$	35.4465	1.33689
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−46.5454	−1.75052
$708$	0	0
$709$	−27.8803	−1.04707	−0.523534	−	0.852005i	$-0.675387\pi$
$709$	−27.8803	−1.04707	−0.523534	+	0.852005i	$0.675387\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.03035	0.338189
$714$	0	0
$715$	−19.9659	−0.746683
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.22333	0.157504	0.0787518	−	0.996894i	$-0.474907\pi$
$719$	4.22333	0.157504	0.0787518	+	0.996894i	$0.474907\pi$
$720$	0	0
$721$	−49.5137	−1.84399
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.42543	0.0529392
$726$	0	0
$727$	−7.65770	−0.284008	−0.142004	−	0.989866i	$-0.545355\pi$
$727$	−7.65770	−0.284008	−0.142004	+	0.989866i	$0.545355\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−46.9004	−1.73467
$732$	0	0
$733$	−32.3457	−1.19472	−0.597358	−	0.801975i	$-0.703783\pi$
$733$	−32.3457	−1.19472	−0.597358	+	0.801975i	$0.703783\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.6323	1.12836
$738$	0	0
$739$	8.58366	0.315755	0.157878	−	0.987459i	$-0.449535\pi$
$739$	8.58366	0.315755	0.157878	+	0.987459i	$0.449535\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.9601	1.20919	0.604594	−	0.796534i	$-0.293335\pi$
$743$	32.9601	1.20919	0.604594	+	0.796534i	$0.293335\pi$
$744$	0	0
$745$	−20.1268	−0.737388
$746$	0	0
$747$	0	0
$748$	0	0
$749$	39.2652	1.43472
$750$	0	0
$751$	7.31841	0.267053	0.133526	−	0.991045i	$-0.457370\pi$
$751$	7.31841	0.267053	0.133526	+	0.991045i	$0.457370\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.1739	−0.661418
$756$	0	0
$757$	9.65028	0.350745	0.175373	−	0.984502i	$-0.443887\pi$
$757$	9.65028	0.350745	0.175373	+	0.984502i	$0.443887\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26.8784	0.974340	0.487170	−	0.873307i	$-0.338029\pi$
$761$	26.8784	0.974340	0.487170	+	0.873307i	$0.338029\pi$
$762$	0	0
$763$	18.8829	0.683606
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.9802	−0.974200
$768$	0	0
$769$	2.03983	0.0735581	0.0367790	−	0.999323i	$-0.488290\pi$
$769$	2.03983	0.0735581	0.0367790	+	0.999323i	$0.488290\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.54889	−0.343450	−0.171725	−	0.985145i	$-0.554934\pi$
$773$	−9.54889	−0.343450	−0.171725	+	0.985145i	$0.554934\pi$
$774$	0	0
$775$	−9.03035	−0.324380
$776$	0	0
$777$	0	0
$778$	0	0
$779$	69.6232	2.49451
$780$	0	0
$781$	−41.1961	−1.47411
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−15.0833	−0.538346
$786$	0	0
$787$	40.2557	1.43496	0.717481	−	0.696578i	$-0.245295\pi$
$787$	40.2557	1.43496	0.717481	+	0.696578i	$0.245295\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−31.4548	−1.11840
$792$	0	0
$793$	−44.1555	−1.56801
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.6875	−1.51207	−0.756034	−	0.654532i	$-0.772866\pi$
$797$	−42.6875	−1.51207	−0.756034	+	0.654532i	$0.772866\pi$
$798$	0	0
$799$	38.6357	1.36683
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.13024	−0.0398854
$804$	0	0
$805$	2.81459	0.0992013
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28.3699	0.997434	0.498717	−	0.866765i	$-0.333805\pi$
$809$	28.3699	0.997434	0.498717	+	0.866765i	$0.333805\pi$
$810$	0	0
$811$	−23.0415	−0.809096	−0.404548	−	0.914517i	$-0.632571\pi$
$811$	−23.0415	−0.809096	−0.404548	+	0.914517i	$0.632571\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.294954	−0.0103318
$816$	0	0
$817$	81.6741	2.85741
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.5296	0.437286	0.218643	−	0.975805i	$-0.429837\pi$
$821$	12.5296	0.437286	0.218643	+	0.975805i	$0.429837\pi$
$822$	0	0
$823$	7.53814	0.262763	0.131382	−	0.991332i	$-0.458059\pi$
$823$	7.53814	0.262763	0.131382	+	0.991332i	$0.458059\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.5405	0.783810	0.391905	−	0.920006i	$-0.371816\pi$
$827$	22.5405	0.783810	0.391905	+	0.920006i	$0.371816\pi$
$828$	0	0
$829$	−5.28862	−0.183681	−0.0918406	−	0.995774i	$-0.529275\pi$
$829$	−5.28862	−0.183681	−0.0918406	+	0.995774i	$0.529275\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.55172	−0.123060
$834$	0	0
$835$	4.24334	0.146847
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.8187	−1.58184	−0.790918	−	0.611922i	$-0.790397\pi$
$839$	−45.8187	−1.58184	−0.790918	+	0.611922i	$0.790397\pi$
$840$	0	0
$841$	−26.9681	−0.929936
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.97778	−0.171241
$846$	0	0
$847$	31.4501	1.08064
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.28349	−0.181116
$852$	0	0
$853$	−24.5910	−0.841980	−0.420990	−	0.907065i	$-0.638317\pi$
$853$	−24.5910	−0.841980	−0.420990	+	0.907065i	$0.638317\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.53965	−0.291709	−0.145854	−	0.989306i	$-0.546593\pi$
$857$	−8.53965	−0.291709	−0.145854	+	0.989306i	$0.546593\pi$
$858$	0	0
$859$	−43.0704	−1.46954	−0.734772	−	0.678314i	$-0.762711\pi$
$859$	−43.0704	−1.46954	−0.734772	+	0.678314i	$0.762711\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.952975	0.0324396	0.0162198	−	0.999868i	$-0.494837\pi$
$863$	0.952975	0.0324396	0.0162198	+	0.999868i	$0.494837\pi$
$864$	0	0
$865$	24.9116	0.847018
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21.1118	−0.716169
$870$	0	0
$871$	27.5821	0.934583
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.81459	−0.0951505
$876$	0	0
$877$	−38.1895	−1.28957	−0.644784	−	0.764365i	$-0.723053\pi$
$877$	−38.1895	−1.28957	−0.644784	+	0.764365i	$0.723053\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.4747	0.959336	0.479668	−	0.877450i	$-0.340757\pi$
$881$	28.4747	0.959336	0.479668	+	0.877450i	$0.340757\pi$
$882$	0	0
$883$	−15.9768	−0.537663	−0.268832	−	0.963187i	$-0.586637\pi$
$883$	−15.9768	−0.537663	−0.268832	+	0.963187i	$0.586637\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.16280	0.274080	0.137040	−	0.990566i	$-0.456241\pi$
$887$	8.16280	0.274080	0.137040	+	0.990566i	$0.456241\pi$
$888$	0	0
$889$	42.4840	1.42487
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−67.2817	−2.25149
$894$	0	0
$895$	−7.03256	−0.235073
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.8721	−0.429310
$900$	0	0
$901$	−35.3364	−1.17723
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.5806	0.517918
$906$	0	0
$907$	28.0245	0.930538	0.465269	−	0.885169i	$-0.345958\pi$
$907$	28.0245	0.930538	0.465269	+	0.885169i	$0.345958\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.0367	−0.862634	−0.431317	−	0.902201i	$-0.641951\pi$
$911$	−26.0367	−0.862634	−0.431317	+	0.902201i	$0.641951\pi$
$912$	0	0
$913$	−61.7381	−2.04323
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.58548	0.151426
$918$	0	0
$919$	−30.0044	−0.989754	−0.494877	−	0.868963i	$-0.664787\pi$
$919$	−30.0044	−0.989754	−0.494877	+	0.868963i	$0.664787\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−37.0939	−1.22096
$924$	0	0
$925$	5.28349	0.173720
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.2038	1.18781	0.593903	−	0.804536i	$-0.297586\pi$
$929$	36.2038	1.18781	0.593903	+	0.804536i	$0.297586\pi$
$930$	0	0
$931$	6.18509	0.202708
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18.1412	0.593282
$936$	0	0
$937$	−6.68356	−0.218342	−0.109171	−	0.994023i	$-0.534820\pi$
$937$	−6.68356	−0.218342	−0.109171	+	0.994023i	$0.534820\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−60.6068	−1.97573	−0.987863	−	0.155329i	$-0.950356\pi$
$941$	−60.6068	−1.97573	−0.987863	+	0.155329i	$0.950356\pi$
$942$	0	0
$943$	−10.3777	−0.337944
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.2369	0.722604	0.361302	−	0.932449i	$-0.382332\pi$
$947$	22.2369	0.722604	0.361302	+	0.932449i	$0.382332\pi$
$948$	0	0
$949$	−1.01770	−0.0330358
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22.3412	−0.723704	−0.361852	−	0.932236i	$-0.617855\pi$
$953$	−22.3412	−0.723704	−0.361852	+	0.932236i	$0.617855\pi$
$954$	0	0
$955$	−16.8940	−0.546677
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.8125	0.510612
$960$	0	0
$961$	50.5471	1.63055
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.22500	0.0394342
$966$	0	0
$967$	−30.6654	−0.986132	−0.493066	−	0.869992i	$-0.664124\pi$
$967$	−30.6654	−0.986132	−0.493066	+	0.869992i	$0.664124\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.2779	0.618659	0.309329	−	0.950955i	$-0.399895\pi$
$971$	19.2779	0.618659	0.309329	+	0.950955i	$0.399895\pi$
$972$	0	0
$973$	11.4781	0.367972
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.9499	−0.894196	−0.447098	−	0.894485i	$-0.647543\pi$
$977$	−27.9499	−0.894196	−0.447098	+	0.894485i	$0.647543\pi$
$978$	0	0
$979$	35.9252	1.14817
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55.4140	1.76743	0.883717	−	0.468022i	$-0.155033\pi$
$983$	55.4140	1.76743	0.883717	+	0.468022i	$0.155033\pi$
$984$	0	0
$985$	−18.1092	−0.577008
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.1739	−0.387109
$990$	0	0
$991$	−2.89148	−0.0918510	−0.0459255	−	0.998945i	$-0.514624\pi$
$991$	−2.89148	−0.0918510	−0.0459255	+	0.998945i	$0.514624\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22.1962	−0.703666
$996$	0	0
$997$	−49.7428	−1.57537	−0.787685	−	0.616078i	$-0.788721\pi$
$997$	−49.7428	−1.57537	−0.787685	+	0.616078i	$0.788721\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.t.1.4	✓	5
3.2	odd	2		4140.2.a.u.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.a.t.1.4	✓	5	1.1	even	1	trivial
4140.2.a.u.1.4	yes	5	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.81459 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.81459 q^{7} +4.70892 q^{11} +4.24002 q^{13} -3.85252 q^{17} +6.70892 q^{19} -1.00000 q^{23} +1.00000 q^{25} +1.42543 q^{29} -9.03035 q^{31} -2.81459 q^{35} +5.28349 q^{37} +10.3777 q^{41} +12.1739 q^{43} -10.0287 q^{47} +0.921920 q^{49} +9.17229 q^{53} -4.70892 q^{55} -6.36323 q^{59} -10.4140 q^{61} -4.24002 q^{65} +6.50517 q^{67} -8.74852 q^{71} -0.240022 q^{73} +13.2537 q^{77} -4.48337 q^{79} -13.1109 q^{83} +3.85252 q^{85} +7.62918 q^{89} +11.9339 q^{91} -6.70892 q^{95} -0.385284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 4 q^{7}+O(q^{10}) \) 5 * q - 5 * q^5 + 4 * q^7	\( 5 q - 5 q^{5} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 5 q^{23} + 5 q^{25} - 2 q^{29} + 8 q^{31} - 4 q^{35} + 8 q^{37} - 2 q^{41} + 18 q^{43} + 4 q^{47} + 13 q^{49} + 2 q^{53} + 4 q^{55} - 6 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 10 q^{71} + 18 q^{73} + 16 q^{77} + 14 q^{79} - 8 q^{83} + 2 q^{85} + 18 q^{89} + 36 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 5 * q^23 + 5 * q^25 - 2 * q^29 + 8 * q^31 - 4 * q^35 + 8 * q^37 - 2 * q^41 + 18 * q^43 + 4 * q^47 + 13 * q^49 + 2 * q^53 + 4 * q^55 - 6 * q^59 + 10 * q^61 - 2 * q^65 + 16 * q^67 - 10 * q^71 + 18 * q^73 + 16 * q^77 + 14 * q^79 - 8 * q^83 + 2 * q^85 + 18 * q^89 + 36 * q^91 - 6 * q^95 + 6 * q^97

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