LMFDB - Embedded newform 2160.4.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.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:	$127.444125612$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 141x^{2} + 200x + 3500 $ x^4 - x^3 - 141x^2 + 200x + 3500
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.73555$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} -11.2995 q^{7} +O(q^{10})$	$q+5.00000 q^{5} -11.2995 q^{7} -61.3933 q^{11} -75.7226 q^{13} +11.1657 q^{17} -71.5271 q^{19} -126.156 q^{23} +25.0000 q^{25} -235.502 q^{29} -110.923 q^{31} -56.4973 q^{35} +434.358 q^{37} -1.15792 q^{41} +77.6254 q^{43} +231.442 q^{47} -215.322 q^{49} +500.296 q^{53} -306.967 q^{55} +334.718 q^{59} +147.171 q^{61} -378.613 q^{65} -84.9722 q^{67} +101.308 q^{71} -50.4773 q^{73} +693.712 q^{77} +818.225 q^{79} -206.044 q^{83} +55.8283 q^{85} -648.508 q^{89} +855.625 q^{91} -357.636 q^{95} +1885.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{5} - 14 q^{7}+O(q^{10}) $ 4 * q + 20 * q^5 - 14 * q^7	$ 4 q + 20 q^{5} - 14 q^{7} + 4 q^{11} + 30 q^{13} - 28 q^{17} - 78 q^{19} - 182 q^{23} + 100 q^{25} + 202 q^{29} + 76 q^{31} - 70 q^{35} + 302 q^{37} + 380 q^{41} - 178 q^{43} - 114 q^{47} + 958 q^{49} - 256 q^{53} + 20 q^{55} + 204 q^{59} + 766 q^{61} + 150 q^{65} - 330 q^{67} + 1060 q^{71} + 1442 q^{73} + 216 q^{77} - 742 q^{79} + 768 q^{83} - 140 q^{85} - 400 q^{89} - 3066 q^{91} - 390 q^{95} + 3338 q^{97}+O(q^{100}) $ 4 * q + 20 * q^5 - 14 * q^7 + 4 * q^11 + 30 * q^13 - 28 * q^17 - 78 * q^19 - 182 * q^23 + 100 * q^25 + 202 * q^29 + 76 * q^31 - 70 * q^35 + 302 * q^37 + 380 * q^41 - 178 * q^43 - 114 * q^47 + 958 * q^49 - 256 * q^53 + 20 * q^55 + 204 * q^59 + 766 * q^61 + 150 * q^65 - 330 * q^67 + 1060 * q^71 + 1442 * q^73 + 216 * q^77 - 742 * q^79 + 768 * q^83 - 140 * q^85 - 400 * q^89 - 3066 * q^91 - 390 * q^95 + 3338 * 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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−11.2995	−0.610114	−0.305057	−	0.952334i	$-0.598675\pi$
$7$	−11.2995	−0.610114	−0.305057	+	0.952334i	$0.598675\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−61.3933	−1.68280	−0.841399	−	0.540414i	$-0.818268\pi$
$11$	−61.3933	−1.68280	−0.841399	+	0.540414i	$0.818268\pi$
$12$	0	0
$13$	−75.7226	−1.61551	−0.807757	−	0.589516i	$-0.799319\pi$
$13$	−75.7226	−1.61551	−0.807757	+	0.589516i	$0.799319\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	11.1657	0.159298	0.0796491	−	0.996823i	$-0.474620\pi$
$17$	11.1657	0.159298	0.0796491	+	0.996823i	$0.474620\pi$
$18$	0	0
$19$	−71.5271	−0.863655	−0.431828	−	0.901956i	$-0.642131\pi$
$19$	−71.5271	−0.863655	−0.431828	+	0.901956i	$0.642131\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−126.156	−1.14371	−0.571855	−	0.820355i	$-0.693776\pi$
$23$	−126.156	−1.14371	−0.571855	+	0.820355i	$0.693776\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−235.502	−1.50798	−0.753992	−	0.656883i	$-0.771874\pi$
$29$	−235.502	−1.50798	−0.753992	+	0.656883i	$0.771874\pi$
$30$	0	0
$31$	−110.923	−0.642654	−0.321327	−	0.946968i	$-0.604129\pi$
$31$	−110.923	−0.642654	−0.321327	+	0.946968i	$0.604129\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−56.4973	−0.272851
$36$	0	0
$37$	434.358	1.92995	0.964973	−	0.262349i	$-0.0844971\pi$
$37$	434.358	1.92995	0.964973	+	0.262349i	$0.0844971\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.15792	−0.00441066	−0.00220533	−	0.999998i	$-0.500702\pi$
$41$	−1.15792	−0.00441066	−0.00220533	+	0.999998i	$0.500702\pi$
$42$	0	0
$43$	77.6254	0.275297	0.137648	−	0.990481i	$-0.456046\pi$
$43$	77.6254	0.275297	0.137648	+	0.990481i	$0.456046\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	231.442	0.718282	0.359141	−	0.933283i	$-0.383070\pi$
$47$	231.442	0.718282	0.359141	+	0.933283i	$0.383070\pi$
$48$	0	0
$49$	−215.322	−0.627761
$50$	0	0
$51$	0	0
$52$	0	0
$53$	500.296	1.29662	0.648310	−	0.761376i	$-0.275476\pi$
$53$	500.296	1.29662	0.648310	+	0.761376i	$0.275476\pi$
$54$	0	0
$55$	−306.967	−0.752571
$56$	0	0
$57$	0	0
$58$	0	0
$59$	334.718	0.738587	0.369293	−	0.929313i	$-0.379600\pi$
$59$	334.718	0.738587	0.369293	+	0.929313i	$0.379600\pi$
$60$	0	0
$61$	147.171	0.308906	0.154453	−	0.988000i	$-0.450638\pi$
$61$	147.171	0.308906	0.154453	+	0.988000i	$0.450638\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−378.613	−0.722480
$66$	0	0
$67$	−84.9722	−0.154940	−0.0774702	−	0.996995i	$-0.524684\pi$
$67$	−84.9722	−0.154940	−0.0774702	+	0.996995i	$0.524684\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	101.308	0.169339	0.0846695	−	0.996409i	$-0.473017\pi$
$71$	101.308	0.169339	0.0846695	+	0.996409i	$0.473017\pi$
$72$	0	0
$73$	−50.4773	−0.0809304	−0.0404652	−	0.999181i	$-0.512884\pi$
$73$	−50.4773	−0.0809304	−0.0404652	+	0.999181i	$0.512884\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	693.712	1.02670
$78$	0	0
$79$	818.225	1.16528	0.582642	−	0.812729i	$-0.302019\pi$
$79$	818.225	1.16528	0.582642	+	0.812729i	$0.302019\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−206.044	−0.272486	−0.136243	−	0.990675i	$-0.543503\pi$
$83$	−206.044	−0.272486	−0.136243	+	0.990675i	$0.543503\pi$
$84$	0	0
$85$	55.8283	0.0712403
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−648.508	−0.772379	−0.386189	−	0.922420i	$-0.626209\pi$
$89$	−648.508	−0.772379	−0.386189	+	0.922420i	$0.626209\pi$
$90$	0	0
$91$	855.625	0.985647
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−357.636	−0.386238
$96$	0	0
$97$	1885.00	1.97313	0.986563	−	0.163381i	$-0.0522399\pi$
$97$	1885.00	1.97313	0.986563	+	0.163381i	$0.0522399\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1923.79	1.89529	0.947647	−	0.319321i	$-0.103455\pi$
$101$	1923.79	1.89529	0.947647	+	0.319321i	$0.103455\pi$
$102$	0	0
$103$	−1808.72	−1.73028	−0.865139	−	0.501531i	$-0.832770\pi$
$103$	−1808.72	−1.73028	−0.865139	+	0.501531i	$0.832770\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1342.31	1.21277	0.606385	−	0.795171i	$-0.292619\pi$
$107$	1342.31	1.21277	0.606385	+	0.795171i	$0.292619\pi$
$108$	0	0
$109$	1173.14	1.03088	0.515442	−	0.856924i	$-0.327628\pi$
$109$	1173.14	1.03088	0.515442	+	0.856924i	$0.327628\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−284.667	−0.236984	−0.118492	−	0.992955i	$-0.537806\pi$
$113$	−284.667	−0.236984	−0.118492	+	0.992955i	$0.537806\pi$
$114$	0	0
$115$	−630.779	−0.511483
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−126.166	−0.0971900
$120$	0	0
$121$	2438.14	1.83181
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2413.23	−1.68614	−0.843068	−	0.537807i	$-0.819253\pi$
$127$	−2413.23	−1.68614	−0.843068	+	0.537807i	$0.819253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	118.912	0.0793085	0.0396543	−	0.999213i	$-0.487374\pi$
$131$	118.912	0.0793085	0.0396543	+	0.999213i	$0.487374\pi$
$132$	0	0
$133$	808.218	0.526928
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1097.16	−0.684212	−0.342106	−	0.939661i	$-0.611140\pi$
$137$	−1097.16	−0.684212	−0.342106	+	0.939661i	$0.611140\pi$
$138$	0	0
$139$	355.233	0.216766	0.108383	−	0.994109i	$-0.465433\pi$
$139$	355.233	0.216766	0.108383	+	0.994109i	$0.465433\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4648.86	2.71858
$144$	0	0
$145$	−1177.51	−0.674391
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2001.55	−1.10049	−0.550246	−	0.835002i	$-0.685466\pi$
$149$	−2001.55	−1.10049	−0.550246	+	0.835002i	$0.685466\pi$
$150$	0	0
$151$	−2652.70	−1.42963	−0.714813	−	0.699316i	$-0.753488\pi$
$151$	−2652.70	−1.42963	−0.714813	+	0.699316i	$0.753488\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−554.613	−0.287404
$156$	0	0
$157$	−3087.09	−1.56928	−0.784640	−	0.619952i	$-0.787152\pi$
$157$	−3087.09	−1.56928	−0.784640	+	0.619952i	$0.787152\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1425.49	0.697793
$162$	0	0
$163$	−1139.42	−0.547523	−0.273761	−	0.961798i	$-0.588268\pi$
$163$	−1139.42	−0.547523	−0.273761	+	0.961798i	$0.588268\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2531.03	−1.17280	−0.586398	−	0.810023i	$-0.699455\pi$
$167$	−2531.03	−1.17280	−0.586398	+	0.810023i	$0.699455\pi$
$168$	0	0
$169$	3536.92	1.60988
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2872.89	1.26255	0.631277	−	0.775557i	$-0.282531\pi$
$173$	2872.89	1.26255	0.631277	+	0.775557i	$0.282531\pi$
$174$	0	0
$175$	−282.487	−0.122023
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3827.16	−1.59807	−0.799036	−	0.601283i	$-0.794657\pi$
$179$	−3827.16	−1.59807	−0.799036	+	0.601283i	$0.794657\pi$
$180$	0	0
$181$	−659.781	−0.270946	−0.135473	−	0.990781i	$-0.543255\pi$
$181$	−659.781	−0.270946	−0.135473	+	0.990781i	$0.543255\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2171.79	0.863098
$186$	0	0
$187$	−685.497	−0.268067
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4376.11	1.65782	0.828911	−	0.559380i	$-0.188961\pi$
$191$	4376.11	1.65782	0.828911	+	0.559380i	$0.188961\pi$
$192$	0	0
$193$	−3802.02	−1.41801	−0.709004	−	0.705204i	$-0.750855\pi$
$193$	−3802.02	−1.41801	−0.709004	+	0.705204i	$0.750855\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3340.27	1.20804	0.604021	−	0.796968i	$-0.293564\pi$
$197$	3340.27	1.20804	0.604021	+	0.796968i	$0.293564\pi$
$198$	0	0
$199$	−4274.83	−1.52279	−0.761393	−	0.648290i	$-0.775484\pi$
$199$	−4274.83	−1.52279	−0.761393	+	0.648290i	$0.775484\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2661.04	0.920042
$204$	0	0
$205$	−5.78960	−0.00197251
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4391.29	1.45336
$210$	0	0
$211$	126.348	0.0412233	0.0206117	−	0.999788i	$-0.493439\pi$
$211$	126.348	0.0412233	0.0206117	+	0.999788i	$0.493439\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	388.127	0.123117
$216$	0	0
$217$	1253.36	0.392092
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−845.493	−0.257348
$222$	0	0
$223$	−3423.31	−1.02799	−0.513996	−	0.857793i	$-0.671835\pi$
$223$	−3423.31	−1.02799	−0.513996	+	0.857793i	$0.671835\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2145.09	0.627200	0.313600	−	0.949555i	$-0.398465\pi$
$227$	2145.09	0.627200	0.313600	+	0.949555i	$0.398465\pi$
$228$	0	0
$229$	3524.09	1.01694	0.508468	−	0.861081i	$-0.330212\pi$
$229$	3524.09	1.01694	0.508468	+	0.861081i	$0.330212\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1479.58	0.416011	0.208005	−	0.978128i	$-0.433303\pi$
$233$	1479.58	0.416011	0.208005	+	0.978128i	$0.433303\pi$
$234$	0	0
$235$	1157.21	0.321226
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3075.33	0.832328	0.416164	−	0.909290i	$-0.363374\pi$
$239$	3075.33	0.832328	0.416164	+	0.909290i	$0.363374\pi$
$240$	0	0
$241$	2600.22	0.694998	0.347499	−	0.937680i	$-0.387031\pi$
$241$	2600.22	0.694998	0.347499	+	0.937680i	$0.387031\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1076.61	−0.280743
$246$	0	0
$247$	5416.22	1.39525
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1186.65	−0.298410	−0.149205	−	0.988806i	$-0.547671\pi$
$251$	−1186.65	−0.298410	−0.149205	+	0.988806i	$0.547671\pi$
$252$	0	0
$253$	7745.13	1.92463
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3364.26	0.816562	0.408281	−	0.912856i	$-0.366128\pi$
$257$	3364.26	0.816562	0.408281	+	0.912856i	$0.366128\pi$
$258$	0	0
$259$	−4908.01	−1.17749
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1655.19	0.388074	0.194037	−	0.980994i	$-0.437842\pi$
$263$	1655.19	0.388074	0.194037	+	0.980994i	$0.437842\pi$
$264$	0	0
$265$	2501.48	0.579867
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2646.65	−0.599886	−0.299943	−	0.953957i	$-0.596968\pi$
$269$	−2646.65	−0.599886	−0.299943	+	0.953957i	$0.596968\pi$
$270$	0	0
$271$	3798.31	0.851405	0.425703	−	0.904863i	$-0.360027\pi$
$271$	3798.31	0.851405	0.425703	+	0.904863i	$0.360027\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1534.83	−0.336560
$276$	0	0
$277$	−5471.39	−1.18680	−0.593400	−	0.804908i	$-0.702215\pi$
$277$	−5471.39	−1.18680	−0.593400	+	0.804908i	$0.702215\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3611.68	−0.766743	−0.383371	−	0.923594i	$-0.625237\pi$
$281$	−3611.68	−0.766743	−0.383371	+	0.923594i	$0.625237\pi$
$282$	0	0
$283$	1914.83	0.402208	0.201104	−	0.979570i	$-0.435547\pi$
$283$	1914.83	0.402208	0.201104	+	0.979570i	$0.435547\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.0839	0.00269100
$288$	0	0
$289$	−4788.33	−0.974624
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1706.82	0.340319	0.170159	−	0.985417i	$-0.445572\pi$
$293$	1706.82	0.340319	0.170159	+	0.985417i	$0.445572\pi$
$294$	0	0
$295$	1673.59	0.330306
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9552.85	1.84768
$300$	0	0
$301$	−877.126	−0.167962
$302$	0	0
$303$	0	0
$304$	0	0
$305$	735.853	0.138147
$306$	0	0
$307$	−5888.56	−1.09472	−0.547358	−	0.836898i	$-0.684366\pi$
$307$	−5888.56	−1.09472	−0.547358	+	0.836898i	$0.684366\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4902.98	0.893962	0.446981	−	0.894543i	$-0.352499\pi$
$311$	4902.98	0.893962	0.446981	+	0.894543i	$0.352499\pi$
$312$	0	0
$313$	6209.11	1.12128	0.560638	−	0.828061i	$-0.310556\pi$
$313$	6209.11	1.12128	0.560638	+	0.828061i	$0.310556\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2427.25	0.430056	0.215028	−	0.976608i	$-0.431016\pi$
$317$	2427.25	0.430056	0.215028	+	0.976608i	$0.431016\pi$
$318$	0	0
$319$	14458.2	2.53763
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−798.648	−0.137579
$324$	0	0
$325$	−1893.07	−0.323103
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2615.17	−0.438234
$330$	0	0
$331$	−6058.23	−1.00601	−0.503006	−	0.864283i	$-0.667773\pi$
$331$	−6058.23	−1.00601	−0.503006	+	0.864283i	$0.667773\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−424.861	−0.0692915
$336$	0	0
$337$	117.584	0.0190065	0.00950326	−	0.999955i	$-0.496975\pi$
$337$	117.584	0.0190065	0.00950326	+	0.999955i	$0.496975\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6809.90	1.08146
$342$	0	0
$343$	6308.74	0.993119
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1596.08	0.246922	0.123461	−	0.992349i	$-0.460601\pi$
$347$	1596.08	0.246922	0.123461	+	0.992349i	$0.460601\pi$
$348$	0	0
$349$	3555.89	0.545394	0.272697	−	0.962100i	$-0.412084\pi$
$349$	3555.89	0.545394	0.272697	+	0.962100i	$0.412084\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2973.17	−0.448289	−0.224145	−	0.974556i	$-0.571959\pi$
$353$	−2973.17	−0.448289	−0.224145	+	0.974556i	$0.571959\pi$
$354$	0	0
$355$	506.541	0.0757307
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12834.7	1.88688	0.943442	−	0.331537i	$-0.107567\pi$
$359$	12834.7	1.88688	0.943442	+	0.331537i	$0.107567\pi$
$360$	0	0
$361$	−1742.87	−0.254099
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−252.386	−0.0361932
$366$	0	0
$367$	−7503.89	−1.06730	−0.533651	−	0.845705i	$-0.679181\pi$
$367$	−7503.89	−1.06730	−0.533651	+	0.845705i	$0.679181\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5653.07	−0.791086
$372$	0	0
$373$	−10834.1	−1.50394	−0.751970	−	0.659197i	$-0.770896\pi$
$373$	−10834.1	−1.50394	−0.751970	+	0.659197i	$0.770896\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17832.8	2.43617
$378$	0	0
$379$	−1526.64	−0.206909	−0.103454	−	0.994634i	$-0.532990\pi$
$379$	−1526.64	−0.206909	−0.103454	+	0.994634i	$0.532990\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8794.30	−1.17328	−0.586642	−	0.809846i	$-0.699550\pi$
$383$	−8794.30	−1.17328	−0.586642	+	0.809846i	$0.699550\pi$
$384$	0	0
$385$	3468.56	0.459154
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7955.47	1.03691	0.518456	−	0.855104i	$-0.326507\pi$
$389$	7955.47	1.03691	0.518456	+	0.855104i	$0.326507\pi$
$390$	0	0
$391$	−1408.61	−0.182191
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4091.12	0.521131
$396$	0	0
$397$	−3359.28	−0.424679	−0.212339	−	0.977196i	$-0.568108\pi$
$397$	−3359.28	−0.424679	−0.212339	+	0.977196i	$0.568108\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3387.77	−0.421888	−0.210944	−	0.977498i	$-0.567654\pi$
$401$	−3387.77	−0.421888	−0.210944	+	0.977498i	$0.567654\pi$
$402$	0	0
$403$	8399.34	1.03822
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−26666.7	−3.24771
$408$	0	0
$409$	−7919.60	−0.957454	−0.478727	−	0.877964i	$-0.658902\pi$
$409$	−7919.60	−0.957454	−0.478727	+	0.877964i	$0.658902\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3782.14	−0.450622
$414$	0	0
$415$	−1030.22	−0.121859
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10671.6	1.24425	0.622127	−	0.782917i	$-0.286269\pi$
$419$	10671.6	1.24425	0.622127	+	0.782917i	$0.286269\pi$
$420$	0	0
$421$	−8445.59	−0.977703	−0.488852	−	0.872367i	$-0.662584\pi$
$421$	−8445.59	−0.977703	−0.488852	+	0.872367i	$0.662584\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	279.141	0.0318596
$426$	0	0
$427$	−1662.95	−0.188468
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11206.9	1.25248	0.626241	−	0.779630i	$-0.284593\pi$
$431$	11206.9	1.25248	0.626241	+	0.779630i	$0.284593\pi$
$432$	0	0
$433$	−5119.10	−0.568148	−0.284074	−	0.958802i	$-0.591686\pi$
$433$	−5119.10	−0.568148	−0.284074	+	0.958802i	$0.591686\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9023.57	0.987771
$438$	0	0
$439$	14018.6	1.52408	0.762039	−	0.647531i	$-0.224198\pi$
$439$	14018.6	1.52408	0.762039	+	0.647531i	$0.224198\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10393.3	1.11468	0.557338	−	0.830286i	$-0.311823\pi$
$443$	10393.3	1.11468	0.557338	+	0.830286i	$0.311823\pi$
$444$	0	0
$445$	−3242.54	−0.345418
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2522.28	−0.265108	−0.132554	−	0.991176i	$-0.542318\pi$
$449$	−2522.28	−0.265108	−0.132554	+	0.991176i	$0.542318\pi$
$450$	0	0
$451$	71.0886	0.00742225
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4278.13	0.440795
$456$	0	0
$457$	10879.1	1.11357	0.556785	−	0.830657i	$-0.312035\pi$
$457$	10879.1	1.11357	0.556785	+	0.830657i	$0.312035\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−288.174	−0.0291141	−0.0145570	−	0.999894i	$-0.504634\pi$
$461$	−288.174	−0.0291141	−0.0145570	+	0.999894i	$0.504634\pi$
$462$	0	0
$463$	3397.84	0.341060	0.170530	−	0.985352i	$-0.445452\pi$
$463$	3397.84	0.341060	0.170530	+	0.985352i	$0.445452\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16484.3	1.63341	0.816703	−	0.577058i	$-0.195799\pi$
$467$	16484.3	1.63341	0.816703	+	0.577058i	$0.195799\pi$
$468$	0	0
$469$	960.140	0.0945313
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4765.68	−0.463269
$474$	0	0
$475$	−1788.18	−0.172731
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8583.18	0.818738	0.409369	−	0.912369i	$-0.365749\pi$
$479$	8583.18	0.818738	0.409369	+	0.912369i	$0.365749\pi$
$480$	0	0
$481$	−32890.7	−3.11785
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9425.02	0.882409
$486$	0	0
$487$	6076.49	0.565404	0.282702	−	0.959208i	$-0.408769\pi$
$487$	6076.49	0.565404	0.282702	+	0.959208i	$0.408769\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1652.84	−0.151918	−0.0759589	−	0.997111i	$-0.524202\pi$
$491$	−1652.84	−0.151918	−0.0759589	+	0.997111i	$0.524202\pi$
$492$	0	0
$493$	−2629.53	−0.240219
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1144.73	−0.103316
$498$	0	0
$499$	18468.0	1.65680	0.828400	−	0.560137i	$-0.189252\pi$
$499$	18468.0	1.65680	0.828400	+	0.560137i	$0.189252\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3044.06	0.269837	0.134918	−	0.990857i	$-0.456923\pi$
$503$	3044.06	0.269837	0.134918	+	0.990857i	$0.456923\pi$
$504$	0	0
$505$	9618.97	0.847601
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8238.76	0.717440	0.358720	−	0.933445i	$-0.383213\pi$
$509$	8238.76	0.717440	0.358720	+	0.933445i	$0.383213\pi$
$510$	0	0
$511$	570.366	0.0493767
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9043.61	−0.773804
$516$	0	0
$517$	−14209.0	−1.20872
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11609.7	0.976261	0.488131	−	0.872771i	$-0.337679\pi$
$521$	11609.7	0.976261	0.488131	+	0.872771i	$0.337679\pi$
$522$	0	0
$523$	10413.6	0.870662	0.435331	−	0.900270i	$-0.356631\pi$
$523$	10413.6	0.870662	0.435331	+	0.900270i	$0.356631\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1238.52	−0.102374
$528$	0	0
$529$	3748.31	0.308072
$530$	0	0
$531$	0	0
$532$	0	0
$533$	87.6808	0.00712547
$534$	0	0
$535$	6711.57	0.542367
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13219.3	1.05640
$540$	0	0
$541$	15045.2	1.19565	0.597823	−	0.801628i	$-0.296032\pi$
$541$	15045.2	1.19565	0.597823	+	0.801628i	$0.296032\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5865.70	0.461026
$546$	0	0
$547$	8132.94	0.635721	0.317860	−	0.948138i	$-0.397036\pi$
$547$	8132.94	0.635721	0.317860	+	0.948138i	$0.397036\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	16844.8	1.30238
$552$	0	0
$553$	−9245.50	−0.710956
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19419.4	1.47725	0.738623	−	0.674118i	$-0.235476\pi$
$557$	19419.4	1.47725	0.738623	+	0.674118i	$0.235476\pi$
$558$	0	0
$559$	−5878.00	−0.444746
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5696.48	−0.426426	−0.213213	−	0.977006i	$-0.568393\pi$
$563$	−5696.48	−0.426426	−0.213213	+	0.977006i	$0.568393\pi$
$564$	0	0
$565$	−1423.33	−0.105982
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18615.2	−1.37151	−0.685754	−	0.727833i	$-0.740527\pi$
$569$	−18615.2	−1.37151	−0.685754	+	0.727833i	$0.740527\pi$
$570$	0	0
$571$	11250.1	0.824521	0.412260	−	0.911066i	$-0.364739\pi$
$571$	11250.1	0.824521	0.412260	+	0.911066i	$0.364739\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3153.90	−0.228742
$576$	0	0
$577$	2336.40	0.168571	0.0842856	−	0.996442i	$-0.473139\pi$
$577$	2336.40	0.168571	0.0842856	+	0.996442i	$0.473139\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2328.19	0.166247
$582$	0	0
$583$	−30714.8	−2.18195
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20277.2	−1.42578	−0.712889	−	0.701277i	$-0.752614\pi$
$587$	−20277.2	−1.42578	−0.712889	+	0.701277i	$0.752614\pi$
$588$	0	0
$589$	7933.97	0.555031
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25051.8	−1.73483	−0.867414	−	0.497587i	$-0.834220\pi$
$593$	−25051.8	−1.73483	−0.867414	+	0.497587i	$0.834220\pi$
$594$	0	0
$595$	−630.830	−0.0434647
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26572.3	−1.81254	−0.906271	−	0.422697i	$-0.861084\pi$
$599$	−26572.3	−1.81254	−0.906271	+	0.422697i	$0.861084\pi$
$600$	0	0
$601$	4224.74	0.286740	0.143370	−	0.989669i	$-0.454206\pi$
$601$	4224.74	0.286740	0.143370	+	0.989669i	$0.454206\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12190.7	0.819211
$606$	0	0
$607$	17428.2	1.16539	0.582693	−	0.812692i	$-0.301999\pi$
$607$	17428.2	1.16539	0.582693	+	0.812692i	$0.301999\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17525.4	−1.16039
$612$	0	0
$613$	−7100.27	−0.467826	−0.233913	−	0.972258i	$-0.575153\pi$
$613$	−7100.27	−0.467826	−0.233913	+	0.972258i	$0.575153\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26045.2	−1.69941	−0.849707	−	0.527255i	$-0.823221\pi$
$617$	−26045.2	−1.69941	−0.849707	+	0.527255i	$0.823221\pi$
$618$	0	0
$619$	−8865.32	−0.575650	−0.287825	−	0.957683i	$-0.592932\pi$
$619$	−8865.32	−0.575650	−0.287825	+	0.957683i	$0.592932\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7327.79	0.471239
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4849.89	0.307437
$630$	0	0
$631$	−177.883	−0.0112225	−0.00561126	−	0.999984i	$-0.501786\pi$
$631$	−177.883	−0.0112225	−0.00561126	+	0.999984i	$0.501786\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12066.1	−0.754063
$636$	0	0
$637$	16304.8	1.01416
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27742.7	−1.70947	−0.854736	−	0.519064i	$-0.826281\pi$
$641$	−27742.7	−1.70947	−0.854736	+	0.519064i	$0.826281\pi$
$642$	0	0
$643$	−9182.34	−0.563166	−0.281583	−	0.959537i	$-0.590860\pi$
$643$	−9182.34	−0.563166	−0.281583	+	0.959537i	$0.590860\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16429.9	−0.998341	−0.499171	−	0.866504i	$-0.666362\pi$
$647$	−16429.9	−0.998341	−0.499171	+	0.866504i	$0.666362\pi$
$648$	0	0
$649$	−20549.5	−1.24289
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10965.8	−0.657158	−0.328579	−	0.944477i	$-0.606570\pi$
$653$	−10965.8	−0.657158	−0.328579	+	0.944477i	$0.606570\pi$
$654$	0	0
$655$	594.561	0.0354678
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23252.7	1.37450	0.687250	−	0.726421i	$-0.258818\pi$
$659$	23252.7	1.37450	0.687250	+	0.726421i	$0.258818\pi$
$660$	0	0
$661$	−2215.22	−0.130351	−0.0651755	−	0.997874i	$-0.520761\pi$
$661$	−2215.22	−0.130351	−0.0651755	+	0.997874i	$0.520761\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4041.09	0.235649
$666$	0	0
$667$	29709.9	1.72470
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9035.30	−0.519827
$672$	0	0
$673$	−20009.1	−1.14605	−0.573026	−	0.819537i	$-0.694230\pi$
$673$	−20009.1	−1.14605	−0.573026	+	0.819537i	$0.694230\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18842.2	1.06967	0.534834	−	0.844957i	$-0.320374\pi$
$677$	18842.2	1.06967	0.534834	+	0.844957i	$0.320374\pi$
$678$	0	0
$679$	−21299.5	−1.20383
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20258.1	1.13493	0.567463	−	0.823399i	$-0.307925\pi$
$683$	20258.1	1.13493	0.567463	+	0.823399i	$0.307925\pi$
$684$	0	0
$685$	−5485.82	−0.305989
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−37883.7	−2.09471
$690$	0	0
$691$	−22738.2	−1.25181	−0.625905	−	0.779899i	$-0.715270\pi$
$691$	−22738.2	−1.25181	−0.625905	+	0.779899i	$0.715270\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1776.17	0.0969407
$696$	0	0
$697$	−12.9289	−0.000702610 0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8306.55	−0.447552	−0.223776	−	0.974641i	$-0.571838\pi$
$701$	−8306.55	−0.447552	−0.223776	+	0.974641i	$0.571838\pi$
$702$	0	0
$703$	−31068.4	−1.66681
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21737.8	−1.15634
$708$	0	0
$709$	24716.6	1.30924	0.654620	−	0.755958i	$-0.272829\pi$
$709$	24716.6	1.30924	0.654620	+	0.755958i	$0.272829\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13993.5	0.735009
$714$	0	0
$715$	23244.3	1.21579
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9888.23	0.512891	0.256446	−	0.966559i	$-0.417449\pi$
$719$	9888.23	0.512891	0.256446	+	0.966559i	$0.417449\pi$
$720$	0	0
$721$	20437.6	1.05567
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5887.54	−0.301597
$726$	0	0
$727$	−6335.29	−0.323195	−0.161598	−	0.986857i	$-0.551665\pi$
$727$	−6335.29	−0.323195	−0.161598	+	0.986857i	$0.551665\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	866.739	0.0438543
$732$	0	0
$733$	−13170.9	−0.663680	−0.331840	−	0.943336i	$-0.607669\pi$
$733$	−13170.9	−0.663680	−0.331840	+	0.943336i	$0.607669\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5216.73	0.260734
$738$	0	0
$739$	31361.7	1.56111	0.780554	−	0.625088i	$-0.214937\pi$
$739$	31361.7	1.56111	0.780554	+	0.625088i	$0.214937\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20002.1	0.987624	0.493812	−	0.869569i	$-0.335603\pi$
$743$	20002.1	0.987624	0.493812	+	0.869569i	$0.335603\pi$
$744$	0	0
$745$	−10007.8	−0.492155
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15167.4	−0.739927
$750$	0	0
$751$	25698.3	1.24866	0.624331	−	0.781160i	$-0.285372\pi$
$751$	25698.3	1.24866	0.624331	+	0.781160i	$0.285372\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13263.5	−0.639348
$756$	0	0
$757$	−33958.5	−1.63044	−0.815219	−	0.579153i	$-0.803384\pi$
$757$	−33958.5	−1.63044	−0.815219	+	0.579153i	$0.803384\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13216.4	−0.629559	−0.314780	−	0.949165i	$-0.601931\pi$
$761$	−13216.4	−0.629559	−0.314780	+	0.949165i	$0.601931\pi$
$762$	0	0
$763$	−13255.9	−0.628957
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25345.8	−1.19320
$768$	0	0
$769$	3107.27	0.145710	0.0728551	−	0.997343i	$-0.476789\pi$
$769$	3107.27	0.145710	0.0728551	+	0.997343i	$0.476789\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20568.2	0.957034	0.478517	−	0.878078i	$-0.341175\pi$
$773$	20568.2	0.957034	0.478517	+	0.878078i	$0.341175\pi$
$774$	0	0
$775$	−2773.06	−0.128531
$776$	0	0
$777$	0	0
$778$	0	0
$779$	82.8228	0.00380929
$780$	0	0
$781$	−6219.65	−0.284963
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−15435.5	−0.701803
$786$	0	0
$787$	−19353.9	−0.876609	−0.438304	−	0.898827i	$-0.644421\pi$
$787$	−19353.9	−0.876609	−0.438304	+	0.898827i	$0.644421\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3216.58	0.144587
$792$	0	0
$793$	−11144.1	−0.499042
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29243.9	1.29971	0.649856	−	0.760057i	$-0.274829\pi$
$797$	29243.9	1.29971	0.649856	+	0.760057i	$0.274829\pi$
$798$	0	0
$799$	2584.20	0.114421
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3098.97	0.136190
$804$	0	0
$805$	7127.47	0.312063
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31460.5	−1.36723	−0.683617	−	0.729841i	$-0.739594\pi$
$809$	−31460.5	−1.36723	−0.683617	+	0.729841i	$0.739594\pi$
$810$	0	0
$811$	−22991.8	−0.995503	−0.497751	−	0.867320i	$-0.665841\pi$
$811$	−22991.8	−0.995503	−0.497751	+	0.867320i	$0.665841\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5697.10	−0.244860
$816$	0	0
$817$	−5552.32	−0.237762
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32144.1	1.36643	0.683214	−	0.730218i	$-0.260581\pi$
$821$	32144.1	1.36643	0.683214	+	0.730218i	$0.260581\pi$
$822$	0	0
$823$	−908.840	−0.0384935	−0.0192468	−	0.999815i	$-0.506127\pi$
$823$	−908.840	−0.0384935	−0.0192468	+	0.999815i	$0.506127\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23098.9	0.971255	0.485628	−	0.874166i	$-0.338591\pi$
$827$	23098.9	0.971255	0.485628	+	0.874166i	$0.338591\pi$
$828$	0	0
$829$	42491.6	1.78021	0.890104	−	0.455757i	$-0.150631\pi$
$829$	42491.6	1.78021	0.890104	+	0.455757i	$0.150631\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2404.21	−0.100001
$834$	0	0
$835$	−12655.2	−0.524491
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12239.5	−0.503642	−0.251821	−	0.967774i	$-0.581029\pi$
$839$	−12239.5	−0.503642	−0.251821	+	0.967774i	$0.581029\pi$
$840$	0	0
$841$	31072.0	1.27402
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17684.6	0.719962
$846$	0	0
$847$	−27549.7	−1.11761
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−54796.8	−2.20730
$852$	0	0
$853$	9020.23	0.362071	0.181036	−	0.983477i	$-0.442055\pi$
$853$	9020.23	0.362071	0.181036	+	0.983477i	$0.442055\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39913.4	1.59092	0.795459	−	0.606008i	$-0.207230\pi$
$857$	39913.4	1.59092	0.795459	+	0.606008i	$0.207230\pi$
$858$	0	0
$859$	−26038.5	−1.03425	−0.517126	−	0.855910i	$-0.672998\pi$
$859$	−26038.5	−1.03425	−0.517126	+	0.855910i	$0.672998\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6044.84	−0.238434	−0.119217	−	0.992868i	$-0.538038\pi$
$863$	−6044.84	−0.238434	−0.119217	+	0.992868i	$0.538038\pi$
$864$	0	0
$865$	14364.5	0.564631
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−50233.5	−1.96094
$870$	0	0
$871$	6434.32	0.250308
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1412.43	−0.0545702
$876$	0	0
$877$	28918.4	1.11346	0.556731	−	0.830693i	$-0.312055\pi$
$877$	28918.4	1.11346	0.556731	+	0.830693i	$0.312055\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	25949.7	0.992359	0.496179	−	0.868220i	$-0.334736\pi$
$881$	25949.7	0.992359	0.496179	+	0.868220i	$0.334736\pi$
$882$	0	0
$883$	−39784.1	−1.51624	−0.758121	−	0.652114i	$-0.773882\pi$
$883$	−39784.1	−1.51624	−0.758121	+	0.652114i	$0.773882\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16227.6	−0.614284	−0.307142	−	0.951664i	$-0.599373\pi$
$887$	−16227.6	−0.614284	−0.307142	+	0.951664i	$0.599373\pi$
$888$	0	0
$889$	27268.2	1.02873
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−16554.4	−0.620348
$894$	0	0
$895$	−19135.8	−0.714680
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26122.4	0.969112
$900$	0	0
$901$	5586.13	0.206549
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3298.91	−0.121171
$906$	0	0
$907$	2107.66	0.0771596	0.0385798	−	0.999256i	$-0.487717\pi$
$907$	2107.66	0.0771596	0.0385798	+	0.999256i	$0.487717\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3471.16	−0.126240	−0.0631199	−	0.998006i	$-0.520105\pi$
$911$	−3471.16	−0.126240	−0.0631199	+	0.998006i	$0.520105\pi$
$912$	0	0
$913$	12649.8	0.458538
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1343.65	−0.0483872
$918$	0	0
$919$	7620.88	0.273547	0.136773	−	0.990602i	$-0.456327\pi$
$919$	7620.88	0.273547	0.136773	+	0.990602i	$0.456327\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7671.32	−0.273569
$924$	0	0
$925$	10858.9	0.385989
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23396.4	−0.826276	−0.413138	−	0.910669i	$-0.635567\pi$
$929$	−23396.4	−0.826276	−0.413138	+	0.910669i	$0.635567\pi$
$930$	0	0
$931$	15401.4	0.542169
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3427.48	−0.119883
$936$	0	0
$937$	23669.1	0.825226	0.412613	−	0.910906i	$-0.364616\pi$
$937$	23669.1	0.825226	0.412613	+	0.910906i	$0.364616\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−15682.0	−0.543272	−0.271636	−	0.962400i	$-0.587565\pi$
$941$	−15682.0	−0.543272	−0.271636	+	0.962400i	$0.587565\pi$
$942$	0	0
$943$	146.079	0.00504451
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28683.3	−0.984247	−0.492123	−	0.870526i	$-0.663779\pi$
$947$	−28683.3	−0.984247	−0.492123	+	0.870526i	$0.663779\pi$
$948$	0	0
$949$	3822.27	0.130744
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20850.6	0.708728	0.354364	−	0.935108i	$-0.384697\pi$
$953$	20850.6	0.708728	0.354364	+	0.935108i	$0.384697\pi$
$954$	0	0
$955$	21880.5	0.741401
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12397.4	0.417447
$960$	0	0
$961$	−17487.2	−0.586996
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19010.1	−0.634152
$966$	0	0
$967$	5002.87	0.166372	0.0831858	−	0.996534i	$-0.473491\pi$
$967$	5002.87	0.166372	0.0831858	+	0.996534i	$0.473491\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39413.3	1.30261	0.651304	−	0.758817i	$-0.274222\pi$
$971$	39413.3	1.30261	0.651304	+	0.758817i	$0.274222\pi$
$972$	0	0
$973$	−4013.94	−0.132252
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11631.3	−0.380879	−0.190439	−	0.981699i	$-0.560991\pi$
$977$	−11631.3	−0.380879	−0.190439	+	0.981699i	$0.560991\pi$
$978$	0	0
$979$	39814.1	1.29976
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38605.2	−1.25261	−0.626304	−	0.779579i	$-0.715433\pi$
$983$	−38605.2	−1.25261	−0.626304	+	0.779579i	$0.715433\pi$
$984$	0	0
$985$	16701.3	0.540253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9792.90	−0.314860
$990$	0	0
$991$	−382.345	−0.0122559	−0.00612795	−	0.999981i	$-0.501951\pi$
$991$	−382.345	−0.0122559	−0.00612795	+	0.999981i	$0.501951\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−21374.1	−0.681011
$996$	0	0
$997$	36072.1	1.14585	0.572926	−	0.819607i	$-0.305808\pi$
$997$	36072.1	1.14585	0.572926	+	0.819607i	$0.305808\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bv.1.2		4
3.2	odd	2		2160.4.a.bu.1.2		4
4.3	odd	2		1080.4.a.p.1.3	yes	4
12.11	even	2		1080.4.a.o.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.o.1.3	✓	4	12.11	even	2
1080.4.a.p.1.3	yes	4	4.3	odd	2
2160.4.a.bu.1.2		4	3.2	odd	2
2160.4.a.bv.1.2		4	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -11.2995 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} -11.2995 q^{7} -61.3933 q^{11} -75.7226 q^{13} +11.1657 q^{17} -71.5271 q^{19} -126.156 q^{23} +25.0000 q^{25} -235.502 q^{29} -110.923 q^{31} -56.4973 q^{35} +434.358 q^{37} -1.15792 q^{41} +77.6254 q^{43} +231.442 q^{47} -215.322 q^{49} +500.296 q^{53} -306.967 q^{55} +334.718 q^{59} +147.171 q^{61} -378.613 q^{65} -84.9722 q^{67} +101.308 q^{71} -50.4773 q^{73} +693.712 q^{77} +818.225 q^{79} -206.044 q^{83} +55.8283 q^{85} -648.508 q^{89} +855.625 q^{91} -357.636 q^{95} +1885.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{5} - 14 q^{7}+O(q^{10}) \) 4 * q + 20 * q^5 - 14 * q^7	\( 4 q + 20 q^{5} - 14 q^{7} + 4 q^{11} + 30 q^{13} - 28 q^{17} - 78 q^{19} - 182 q^{23} + 100 q^{25} + 202 q^{29} + 76 q^{31} - 70 q^{35} + 302 q^{37} + 380 q^{41} - 178 q^{43} - 114 q^{47} + 958 q^{49} - 256 q^{53} + 20 q^{55} + 204 q^{59} + 766 q^{61} + 150 q^{65} - 330 q^{67} + 1060 q^{71} + 1442 q^{73} + 216 q^{77} - 742 q^{79} + 768 q^{83} - 140 q^{85} - 400 q^{89} - 3066 q^{91} - 390 q^{95} + 3338 q^{97}+O(q^{100}) \) 4 * q + 20 * q^5 - 14 * q^7 + 4 * q^11 + 30 * q^13 - 28 * q^17 - 78 * q^19 - 182 * q^23 + 100 * q^25 + 202 * q^29 + 76 * q^31 - 70 * q^35 + 302 * q^37 + 380 * q^41 - 178 * q^43 - 114 * q^47 + 958 * q^49 - 256 * q^53 + 20 * q^55 + 204 * q^59 + 766 * q^61 + 150 * q^65 - 330 * q^67 + 1060 * q^71 + 1442 * q^73 + 216 * q^77 - 742 * q^79 + 768 * q^83 - 140 * q^85 - 400 * q^89 - 3066 * q^91 - 390 * q^95 + 3338 * q^97

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