LMFDB - Embedded newform 8004.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.07005$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.07005 q^{5} -0.993636 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.07005 q^{5} -0.993636 q^{7} +1.00000 q^{9} +0.404264 q^{11} +5.75915 q^{13} -3.07005 q^{15} -1.51036 q^{17} +4.89096 q^{19} -0.993636 q^{21} -1.00000 q^{23} +4.42519 q^{25} +1.00000 q^{27} +1.00000 q^{29} -0.870033 q^{31} +0.404264 q^{33} +3.05051 q^{35} -1.02547 q^{37} +5.75915 q^{39} -2.16975 q^{41} +7.95596 q^{43} -3.07005 q^{45} +0.0449544 q^{47} -6.01269 q^{49} -1.51036 q^{51} +3.90013 q^{53} -1.24111 q^{55} +4.89096 q^{57} -14.4947 q^{59} -10.3453 q^{61} -0.993636 q^{63} -17.6809 q^{65} -9.21650 q^{67} -1.00000 q^{69} -1.42710 q^{71} +4.05700 q^{73} +4.42519 q^{75} -0.401691 q^{77} +11.5151 q^{79} +1.00000 q^{81} +12.0939 q^{83} +4.63688 q^{85} +1.00000 q^{87} -3.79745 q^{89} -5.72250 q^{91} -0.870033 q^{93} -15.0155 q^{95} +15.1905 q^{97} +0.404264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.07005	−1.37297	−0.686484	−	0.727145i	$-0.740847\pi$
$5$	−3.07005	−1.37297	−0.686484	+	0.727145i	$0.740847\pi$
$6$	0	0
$7$	−0.993636	−0.375559	−0.187780	−	0.982211i	$-0.560129\pi$
$7$	−0.993636	−0.375559	−0.187780	+	0.982211i	$0.560129\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.404264	0.121890	0.0609451	−	0.998141i	$-0.480589\pi$
$11$	0.404264	0.121890	0.0609451	+	0.998141i	$0.480589\pi$
$12$	0	0
$13$	5.75915	1.59730	0.798650	−	0.601796i	$-0.205548\pi$
$13$	5.75915	1.59730	0.798650	+	0.601796i	$0.205548\pi$
$14$	0	0
$15$	−3.07005	−0.792683
$16$	0	0
$17$	−1.51036	−0.366316	−0.183158	−	0.983083i	$-0.558632\pi$
$17$	−1.51036	−0.366316	−0.183158	+	0.983083i	$0.558632\pi$
$18$	0	0
$19$	4.89096	1.12206	0.561032	−	0.827794i	$-0.310405\pi$
$19$	4.89096	1.12206	0.561032	+	0.827794i	$0.310405\pi$
$20$	0	0
$21$	−0.993636	−0.216829
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	4.42519	0.885039
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.870033	−0.156262	−0.0781312	−	0.996943i	$-0.524895\pi$
$31$	−0.870033	−0.156262	−0.0781312	+	0.996943i	$0.524895\pi$
$32$	0	0
$33$	0.404264	0.0703733
$34$	0	0
$35$	3.05051	0.515630
$36$	0	0
$37$	−1.02547	−0.168587	−0.0842935	−	0.996441i	$-0.526863\pi$
$37$	−1.02547	−0.168587	−0.0842935	+	0.996441i	$0.526863\pi$
$38$	0	0
$39$	5.75915	0.922202
$40$	0	0
$41$	−2.16975	−0.338858	−0.169429	−	0.985542i	$-0.554192\pi$
$41$	−2.16975	−0.338858	−0.169429	+	0.985542i	$0.554192\pi$
$42$	0	0
$43$	7.95596	1.21327	0.606636	−	0.794979i	$-0.292518\pi$
$43$	7.95596	1.21327	0.606636	+	0.794979i	$0.292518\pi$
$44$	0	0
$45$	−3.07005	−0.457656
$46$	0	0
$47$	0.0449544	0.00655728	0.00327864	−	0.999995i	$-0.498956\pi$
$47$	0.0449544	0.00655728	0.00327864	+	0.999995i	$0.498956\pi$
$48$	0	0
$49$	−6.01269	−0.858955
$50$	0	0
$51$	−1.51036	−0.211493
$52$	0	0
$53$	3.90013	0.535724	0.267862	−	0.963457i	$-0.413683\pi$
$53$	3.90013	0.535724	0.267862	+	0.963457i	$0.413683\pi$
$54$	0	0
$55$	−1.24111	−0.167351
$56$	0	0
$57$	4.89096	0.647823
$58$	0	0
$59$	−14.4947	−1.88705	−0.943527	−	0.331297i	$-0.892514\pi$
$59$	−14.4947	−1.88705	−0.943527	+	0.331297i	$0.892514\pi$
$60$	0	0
$61$	−10.3453	−1.32458	−0.662288	−	0.749249i	$-0.730414\pi$
$61$	−10.3453	−1.32458	−0.662288	+	0.749249i	$0.730414\pi$
$62$	0	0
$63$	−0.993636	−0.125186
$64$	0	0
$65$	−17.6809	−2.19304
$66$	0	0
$67$	−9.21650	−1.12598	−0.562988	−	0.826465i	$-0.690348\pi$
$67$	−9.21650	−1.12598	−0.562988	+	0.826465i	$0.690348\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−1.42710	−0.169366	−0.0846830	−	0.996408i	$-0.526988\pi$
$71$	−1.42710	−0.169366	−0.0846830	+	0.996408i	$0.526988\pi$
$72$	0	0
$73$	4.05700	0.474836	0.237418	−	0.971408i	$-0.423699\pi$
$73$	4.05700	0.474836	0.237418	+	0.971408i	$0.423699\pi$
$74$	0	0
$75$	4.42519	0.510977
$76$	0	0
$77$	−0.401691	−0.0457770
$78$	0	0
$79$	11.5151	1.29554	0.647772	−	0.761834i	$-0.275701\pi$
$79$	11.5151	1.29554	0.647772	+	0.761834i	$0.275701\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0939	1.32748	0.663742	−	0.747962i	$-0.268967\pi$
$83$	12.0939	1.32748	0.663742	+	0.747962i	$0.268967\pi$
$84$	0	0
$85$	4.63688	0.502940
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−3.79745	−0.402529	−0.201265	−	0.979537i	$-0.564505\pi$
$89$	−3.79745	−0.402529	−0.201265	+	0.979537i	$0.564505\pi$
$90$	0	0
$91$	−5.72250	−0.599881
$92$	0	0
$93$	−0.870033	−0.0902182
$94$	0	0
$95$	−15.0155	−1.54056
$96$	0	0
$97$	15.1905	1.54237	0.771183	−	0.636613i	$-0.219665\pi$
$97$	15.1905	1.54237	0.771183	+	0.636613i	$0.219665\pi$
$98$	0	0
$99$	0.404264	0.0406301
$100$	0	0
$101$	−3.42410	−0.340711	−0.170356	−	0.985383i	$-0.554492\pi$
$101$	−3.42410	−0.340711	−0.170356	+	0.985383i	$0.554492\pi$
$102$	0	0
$103$	1.23332	0.121522	0.0607611	−	0.998152i	$-0.480647\pi$
$103$	1.23332	0.121522	0.0607611	+	0.998152i	$0.480647\pi$
$104$	0	0
$105$	3.05051	0.297699
$106$	0	0
$107$	−12.3428	−1.19322	−0.596612	−	0.802530i	$-0.703487\pi$
$107$	−12.3428	−1.19322	−0.596612	+	0.802530i	$0.703487\pi$
$108$	0	0
$109$	12.0044	1.14982	0.574908	−	0.818218i	$-0.305038\pi$
$109$	12.0044	1.14982	0.574908	+	0.818218i	$0.305038\pi$
$110$	0	0
$111$	−1.02547	−0.0973337
$112$	0	0
$113$	15.9728	1.50259	0.751296	−	0.659965i	$-0.229429\pi$
$113$	15.9728	1.50259	0.751296	+	0.659965i	$0.229429\pi$
$114$	0	0
$115$	3.07005	0.286283
$116$	0	0
$117$	5.75915	0.532433
$118$	0	0
$119$	1.50075	0.137573
$120$	0	0
$121$	−10.8366	−0.985143
$122$	0	0
$123$	−2.16975	−0.195640
$124$	0	0
$125$	1.76469	0.157838
$126$	0	0
$127$	18.8192	1.66994	0.834968	−	0.550298i	$-0.185486\pi$
$127$	18.8192	1.66994	0.834968	+	0.550298i	$0.185486\pi$
$128$	0	0
$129$	7.95596	0.700483
$130$	0	0
$131$	−15.1934	−1.32746	−0.663728	−	0.747974i	$-0.731027\pi$
$131$	−15.1934	−1.32746	−0.663728	+	0.747974i	$0.731027\pi$
$132$	0	0
$133$	−4.85984	−0.421401
$134$	0	0
$135$	−3.07005	−0.264228
$136$	0	0
$137$	13.7808	1.17737	0.588685	−	0.808362i	$-0.299646\pi$
$137$	13.7808	1.17737	0.588685	+	0.808362i	$0.299646\pi$
$138$	0	0
$139$	17.0983	1.45026	0.725130	−	0.688612i	$-0.241780\pi$
$139$	17.0983	1.45026	0.725130	+	0.688612i	$0.241780\pi$
$140$	0	0
$141$	0.0449544	0.00378585
$142$	0	0
$143$	2.32822	0.194695
$144$	0	0
$145$	−3.07005	−0.254954
$146$	0	0
$147$	−6.01269	−0.495918
$148$	0	0
$149$	8.27974	0.678303	0.339151	−	0.940732i	$-0.389860\pi$
$149$	8.27974	0.678303	0.339151	+	0.940732i	$0.389860\pi$
$150$	0	0
$151$	−7.37035	−0.599791	−0.299895	−	0.953972i	$-0.596952\pi$
$151$	−7.37035	−0.599791	−0.299895	+	0.953972i	$0.596952\pi$
$152$	0	0
$153$	−1.51036	−0.122105
$154$	0	0
$155$	2.67104	0.214543
$156$	0	0
$157$	5.13924	0.410156	0.205078	−	0.978746i	$-0.434255\pi$
$157$	5.13924	0.410156	0.205078	+	0.978746i	$0.434255\pi$
$158$	0	0
$159$	3.90013	0.309300
$160$	0	0
$161$	0.993636	0.0783095
$162$	0	0
$163$	14.3608	1.12482	0.562412	−	0.826857i	$-0.309873\pi$
$163$	14.3608	1.12482	0.562412	+	0.826857i	$0.309873\pi$
$164$	0	0
$165$	−1.24111	−0.0966203
$166$	0	0
$167$	−7.65819	−0.592608	−0.296304	−	0.955094i	$-0.595754\pi$
$167$	−7.65819	−0.592608	−0.296304	+	0.955094i	$0.595754\pi$
$168$	0	0
$169$	20.1678	1.55137
$170$	0	0
$171$	4.89096	0.374021
$172$	0	0
$173$	−8.41236	−0.639580	−0.319790	−	0.947488i	$-0.603612\pi$
$173$	−8.41236	−0.639580	−0.319790	+	0.947488i	$0.603612\pi$
$174$	0	0
$175$	−4.39703	−0.332384
$176$	0	0
$177$	−14.4947	−1.08949
$178$	0	0
$179$	2.84505	0.212649	0.106325	−	0.994331i	$-0.466092\pi$
$179$	2.84505	0.212649	0.106325	+	0.994331i	$0.466092\pi$
$180$	0	0
$181$	−13.8690	−1.03088	−0.515439	−	0.856927i	$-0.672371\pi$
$181$	−13.8690	−1.03088	−0.515439	+	0.856927i	$0.672371\pi$
$182$	0	0
$183$	−10.3453	−0.764744
$184$	0	0
$185$	3.14825	0.231464
$186$	0	0
$187$	−0.610585	−0.0446504
$188$	0	0
$189$	−0.993636	−0.0722764
$190$	0	0
$191$	22.4422	1.62386	0.811930	−	0.583755i	$-0.198417\pi$
$191$	22.4422	1.62386	0.811930	+	0.583755i	$0.198417\pi$
$192$	0	0
$193$	14.8174	1.06658	0.533288	−	0.845934i	$-0.320956\pi$
$193$	14.8174	1.06658	0.533288	+	0.845934i	$0.320956\pi$
$194$	0	0
$195$	−17.6809	−1.26615
$196$	0	0
$197$	−1.99182	−0.141911	−0.0709557	−	0.997479i	$-0.522605\pi$
$197$	−1.99182	−0.141911	−0.0709557	+	0.997479i	$0.522605\pi$
$198$	0	0
$199$	8.18018	0.579878	0.289939	−	0.957045i	$-0.406365\pi$
$199$	8.18018	0.579878	0.289939	+	0.957045i	$0.406365\pi$
$200$	0	0
$201$	−9.21650	−0.650082
$202$	0	0
$203$	−0.993636	−0.0697396
$204$	0	0
$205$	6.66123	0.465241
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	1.97724	0.136769
$210$	0	0
$211$	−7.17184	−0.493730	−0.246865	−	0.969050i	$-0.579400\pi$
$211$	−7.17184	−0.493730	−0.246865	+	0.969050i	$0.579400\pi$
$212$	0	0
$213$	−1.42710	−0.0977835
$214$	0	0
$215$	−24.4252	−1.66578
$216$	0	0
$217$	0.864496	0.0586858
$218$	0	0
$219$	4.05700	0.274147
$220$	0	0
$221$	−8.69839	−0.585117
$222$	0	0
$223$	−8.28635	−0.554895	−0.277447	−	0.960741i	$-0.589488\pi$
$223$	−8.28635	−0.554895	−0.277447	+	0.960741i	$0.589488\pi$
$224$	0	0
$225$	4.42519	0.295013
$226$	0	0
$227$	5.96730	0.396063	0.198032	−	0.980196i	$-0.436545\pi$
$227$	5.96730	0.396063	0.198032	+	0.980196i	$0.436545\pi$
$228$	0	0
$229$	−10.7711	−0.711772	−0.355886	−	0.934529i	$-0.615821\pi$
$229$	−10.7711	−0.711772	−0.355886	+	0.934529i	$0.615821\pi$
$230$	0	0
$231$	−0.401691	−0.0264294
$232$	0	0
$233$	5.93086	0.388544	0.194272	−	0.980948i	$-0.437766\pi$
$233$	5.93086	0.388544	0.194272	+	0.980948i	$0.437766\pi$
$234$	0	0
$235$	−0.138012	−0.00900293
$236$	0	0
$237$	11.5151	0.747983
$238$	0	0
$239$	−10.7858	−0.697677	−0.348838	−	0.937183i	$-0.613424\pi$
$239$	−10.7858	−0.697677	−0.348838	+	0.937183i	$0.613424\pi$
$240$	0	0
$241$	27.2357	1.75441	0.877203	−	0.480120i	$-0.159407\pi$
$241$	27.2357	1.75441	0.877203	+	0.480120i	$0.159407\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	18.4592	1.17932
$246$	0	0
$247$	28.1678	1.79227
$248$	0	0
$249$	12.0939	0.766423
$250$	0	0
$251$	28.7919	1.81733	0.908664	−	0.417528i	$-0.137104\pi$
$251$	28.7919	1.81733	0.908664	+	0.417528i	$0.137104\pi$
$252$	0	0
$253$	−0.404264	−0.0254159
$254$	0	0
$255$	4.63688	0.290373
$256$	0	0
$257$	30.5727	1.90707	0.953536	−	0.301279i	$-0.0974134\pi$
$257$	30.5727	1.90707	0.953536	+	0.301279i	$0.0974134\pi$
$258$	0	0
$259$	1.01895	0.0633144
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	25.8711	1.59528	0.797639	−	0.603135i	$-0.206082\pi$
$263$	25.8711	1.59528	0.797639	+	0.603135i	$0.206082\pi$
$264$	0	0
$265$	−11.9736	−0.735531
$266$	0	0
$267$	−3.79745	−0.232400
$268$	0	0
$269$	2.54811	0.155361	0.0776804	−	0.996978i	$-0.475249\pi$
$269$	2.54811	0.155361	0.0776804	+	0.996978i	$0.475249\pi$
$270$	0	0
$271$	−7.72407	−0.469204	−0.234602	−	0.972092i	$-0.575379\pi$
$271$	−7.72407	−0.469204	−0.234602	+	0.972092i	$0.575379\pi$
$272$	0	0
$273$	−5.72250	−0.346341
$274$	0	0
$275$	1.78895	0.107878
$276$	0	0
$277$	−25.5527	−1.53531	−0.767656	−	0.640862i	$-0.778577\pi$
$277$	−25.5527	−1.53531	−0.767656	+	0.640862i	$0.778577\pi$
$278$	0	0
$279$	−0.870033	−0.0520875
$280$	0	0
$281$	−8.66057	−0.516646	−0.258323	−	0.966059i	$-0.583170\pi$
$281$	−8.66057	−0.516646	−0.258323	+	0.966059i	$0.583170\pi$
$282$	0	0
$283$	21.4008	1.27214	0.636072	−	0.771630i	$-0.280558\pi$
$283$	21.4008	1.27214	0.636072	+	0.771630i	$0.280558\pi$
$284$	0	0
$285$	−15.0155	−0.889440
$286$	0	0
$287$	2.15594	0.127261
$288$	0	0
$289$	−14.7188	−0.865812
$290$	0	0
$291$	15.1905	0.890486
$292$	0	0
$293$	15.5059	0.905862	0.452931	−	0.891546i	$-0.350378\pi$
$293$	15.5059	0.905862	0.452931	+	0.891546i	$0.350378\pi$
$294$	0	0
$295$	44.4995	2.59086
$296$	0	0
$297$	0.404264	0.0234578
$298$	0	0
$299$	−5.75915	−0.333060
$300$	0	0
$301$	−7.90533	−0.455656
$302$	0	0
$303$	−3.42410	−0.196710
$304$	0	0
$305$	31.7605	1.81860
$306$	0	0
$307$	20.9358	1.19487	0.597434	−	0.801918i	$-0.296187\pi$
$307$	20.9358	1.19487	0.597434	+	0.801918i	$0.296187\pi$
$308$	0	0
$309$	1.23332	0.0701609
$310$	0	0
$311$	5.15575	0.292356	0.146178	−	0.989258i	$-0.453303\pi$
$311$	5.15575	0.292356	0.146178	+	0.989258i	$0.453303\pi$
$312$	0	0
$313$	−5.57721	−0.315243	−0.157621	−	0.987500i	$-0.550383\pi$
$313$	−5.57721	−0.315243	−0.157621	+	0.987500i	$0.550383\pi$
$314$	0	0
$315$	3.05051	0.171877
$316$	0	0
$317$	−7.41356	−0.416387	−0.208194	−	0.978088i	$-0.566758\pi$
$317$	−7.41356	−0.416387	−0.208194	+	0.978088i	$0.566758\pi$
$318$	0	0
$319$	0.404264	0.0226344
$320$	0	0
$321$	−12.3428	−0.688908
$322$	0	0
$323$	−7.38711	−0.411030
$324$	0	0
$325$	25.4853	1.41367
$326$	0	0
$327$	12.0044	0.663847
$328$	0	0
$329$	−0.0446684	−0.00246265
$330$	0	0
$331$	18.9038	1.03905	0.519523	−	0.854457i	$-0.326110\pi$
$331$	18.9038	1.03905	0.519523	+	0.854457i	$0.326110\pi$
$332$	0	0
$333$	−1.02547	−0.0561956
$334$	0	0
$335$	28.2951	1.54593
$336$	0	0
$337$	25.2482	1.37536	0.687679	−	0.726015i	$-0.258630\pi$
$337$	25.2482	1.37536	0.687679	+	0.726015i	$0.258630\pi$
$338$	0	0
$339$	15.9728	0.867522
$340$	0	0
$341$	−0.351723	−0.0190469
$342$	0	0
$343$	12.9299	0.698148
$344$	0	0
$345$	3.07005	0.165286
$346$	0	0
$347$	5.06765	0.272046	0.136023	−	0.990706i	$-0.456568\pi$
$347$	5.06765	0.272046	0.136023	+	0.990706i	$0.456568\pi$
$348$	0	0
$349$	−5.41220	−0.289709	−0.144854	−	0.989453i	$-0.546271\pi$
$349$	−5.41220	−0.289709	−0.144854	+	0.989453i	$0.546271\pi$
$350$	0	0
$351$	5.75915	0.307401
$352$	0	0
$353$	−15.6365	−0.832248	−0.416124	−	0.909308i	$-0.636612\pi$
$353$	−15.6365	−0.832248	−0.416124	+	0.909308i	$0.636612\pi$
$354$	0	0
$355$	4.38127	0.232534
$356$	0	0
$357$	1.50075	0.0794281
$358$	0	0
$359$	−16.8534	−0.889488	−0.444744	−	0.895658i	$-0.646705\pi$
$359$	−16.8534	−0.889488	−0.444744	+	0.895658i	$0.646705\pi$
$360$	0	0
$361$	4.92149	0.259026
$362$	0	0
$363$	−10.8366	−0.568772
$364$	0	0
$365$	−12.4552	−0.651935
$366$	0	0
$367$	−19.5471	−1.02035	−0.510174	−	0.860071i	$-0.670419\pi$
$367$	−19.5471	−1.02035	−0.510174	+	0.860071i	$0.670419\pi$
$368$	0	0
$369$	−2.16975	−0.112953
$370$	0	0
$371$	−3.87531	−0.201196
$372$	0	0
$373$	−31.9742	−1.65556	−0.827781	−	0.561052i	$-0.810397\pi$
$373$	−31.9742	−1.65556	−0.827781	+	0.561052i	$0.810397\pi$
$374$	0	0
$375$	1.76469	0.0911280
$376$	0	0
$377$	5.75915	0.296611
$378$	0	0
$379$	0.873405	0.0448638	0.0224319	−	0.999748i	$-0.492859\pi$
$379$	0.873405	0.0448638	0.0224319	+	0.999748i	$0.492859\pi$
$380$	0	0
$381$	18.8192	0.964138
$382$	0	0
$383$	−38.6100	−1.97288	−0.986439	−	0.164129i	$-0.947519\pi$
$383$	−38.6100	−1.97288	−0.986439	+	0.164129i	$0.947519\pi$
$384$	0	0
$385$	1.23321	0.0628503
$386$	0	0
$387$	7.95596	0.404424
$388$	0	0
$389$	−0.857566	−0.0434803	−0.0217402	−	0.999764i	$-0.506921\pi$
$389$	−0.857566	−0.0434803	−0.0217402	+	0.999764i	$0.506921\pi$
$390$	0	0
$391$	1.51036	0.0763822
$392$	0	0
$393$	−15.1934	−0.766407
$394$	0	0
$395$	−35.3518	−1.77874
$396$	0	0
$397$	13.7510	0.690145	0.345073	−	0.938576i	$-0.387854\pi$
$397$	13.7510	0.690145	0.345073	+	0.938576i	$0.387854\pi$
$398$	0	0
$399$	−4.85984	−0.243296
$400$	0	0
$401$	15.2096	0.759531	0.379765	−	0.925083i	$-0.376005\pi$
$401$	15.2096	0.759531	0.379765	+	0.925083i	$0.376005\pi$
$402$	0	0
$403$	−5.01065	−0.249598
$404$	0	0
$405$	−3.07005	−0.152552
$406$	0	0
$407$	−0.414562	−0.0205491
$408$	0	0
$409$	20.1591	0.996802	0.498401	−	0.866947i	$-0.333921\pi$
$409$	20.1591	0.996802	0.498401	+	0.866947i	$0.333921\pi$
$410$	0	0
$411$	13.7808	0.679755
$412$	0	0
$413$	14.4025	0.708700
$414$	0	0
$415$	−37.1290	−1.82259
$416$	0	0
$417$	17.0983	0.837308
$418$	0	0
$419$	30.2935	1.47993	0.739966	−	0.672644i	$-0.234841\pi$
$419$	30.2935	1.47993	0.739966	+	0.672644i	$0.234841\pi$
$420$	0	0
$421$	−34.8451	−1.69825	−0.849124	−	0.528193i	$-0.822870\pi$
$421$	−34.8451	−1.69825	−0.849124	+	0.528193i	$0.822870\pi$
$422$	0	0
$423$	0.0449544	0.00218576
$424$	0	0
$425$	−6.68364	−0.324204
$426$	0	0
$427$	10.2794	0.497457
$428$	0	0
$429$	2.32822	0.112407
$430$	0	0
$431$	−8.28156	−0.398909	−0.199454	−	0.979907i	$-0.563917\pi$
$431$	−8.28156	−0.398909	−0.199454	+	0.979907i	$0.563917\pi$
$432$	0	0
$433$	15.3947	0.739821	0.369910	−	0.929067i	$-0.379388\pi$
$433$	15.3947	0.739821	0.369910	+	0.929067i	$0.379388\pi$
$434$	0	0
$435$	−3.07005	−0.147198
$436$	0	0
$437$	−4.89096	−0.233966
$438$	0	0
$439$	21.4865	1.02549	0.512747	−	0.858540i	$-0.328628\pi$
$439$	21.4865	1.02549	0.512747	+	0.858540i	$0.328628\pi$
$440$	0	0
$441$	−6.01269	−0.286318
$442$	0	0
$443$	19.0695	0.906018	0.453009	−	0.891506i	$-0.350351\pi$
$443$	19.0695	0.906018	0.453009	+	0.891506i	$0.350351\pi$
$444$	0	0
$445$	11.6584	0.552659
$446$	0	0
$447$	8.27974	0.391618
$448$	0	0
$449$	16.3653	0.772325	0.386163	−	0.922431i	$-0.373800\pi$
$449$	16.3653	0.772325	0.386163	+	0.922431i	$0.373800\pi$
$450$	0	0
$451$	−0.877151	−0.0413034
$452$	0	0
$453$	−7.37035	−0.346289
$454$	0	0
$455$	17.5683	0.823617
$456$	0	0
$457$	23.0922	1.08021	0.540103	−	0.841599i	$-0.318385\pi$
$457$	23.0922	1.08021	0.540103	+	0.841599i	$0.318385\pi$
$458$	0	0
$459$	−1.51036	−0.0704976
$460$	0	0
$461$	3.08471	0.143669	0.0718345	−	0.997417i	$-0.477115\pi$
$461$	3.08471	0.143669	0.0718345	+	0.997417i	$0.477115\pi$
$462$	0	0
$463$	−7.17067	−0.333249	−0.166625	−	0.986020i	$-0.553287\pi$
$463$	−7.17067	−0.333249	−0.166625	+	0.986020i	$0.553287\pi$
$464$	0	0
$465$	2.67104	0.123867
$466$	0	0
$467$	1.23935	0.0573501	0.0286751	−	0.999589i	$-0.490871\pi$
$467$	1.23935	0.0573501	0.0286751	+	0.999589i	$0.490871\pi$
$468$	0	0
$469$	9.15785	0.422870
$470$	0	0
$471$	5.13924	0.236804
$472$	0	0
$473$	3.21631	0.147886
$474$	0	0
$475$	21.6434	0.993069
$476$	0	0
$477$	3.90013	0.178575
$478$	0	0
$479$	23.7766	1.08638	0.543190	−	0.839610i	$-0.317216\pi$
$479$	23.7766	1.08638	0.543190	+	0.839610i	$0.317216\pi$
$480$	0	0
$481$	−5.90586	−0.269284
$482$	0	0
$483$	0.993636	0.0452120
$484$	0	0
$485$	−46.6357	−2.11762
$486$	0	0
$487$	−37.7932	−1.71257	−0.856286	−	0.516502i	$-0.827234\pi$
$487$	−37.7932	−1.71257	−0.856286	+	0.516502i	$0.827234\pi$
$488$	0	0
$489$	14.3608	0.649418
$490$	0	0
$491$	−14.7412	−0.665263	−0.332631	−	0.943057i	$-0.607936\pi$
$491$	−14.7412	−0.665263	−0.332631	+	0.943057i	$0.607936\pi$
$492$	0	0
$493$	−1.51036	−0.0680232
$494$	0	0
$495$	−1.24111	−0.0557837
$496$	0	0
$497$	1.41802	0.0636069
$498$	0	0
$499$	42.0123	1.88073	0.940365	−	0.340168i	$-0.110484\pi$
$499$	42.0123	1.88073	0.940365	+	0.340168i	$0.110484\pi$
$500$	0	0
$501$	−7.65819	−0.342143
$502$	0	0
$503$	8.22320	0.366654	0.183327	−	0.983052i	$-0.441313\pi$
$503$	8.22320	0.366654	0.183327	+	0.983052i	$0.441313\pi$
$504$	0	0
$505$	10.5122	0.467785
$506$	0	0
$507$	20.1678	0.895683
$508$	0	0
$509$	−13.5775	−0.601811	−0.300905	−	0.953654i	$-0.597289\pi$
$509$	−13.5775	−0.601811	−0.300905	+	0.953654i	$0.597289\pi$
$510$	0	0
$511$	−4.03119	−0.178329
$512$	0	0
$513$	4.89096	0.215941
$514$	0	0
$515$	−3.78634	−0.166846
$516$	0	0
$517$	0.0181735	0.000799268 0
$518$	0	0
$519$	−8.41236	−0.369262
$520$	0	0
$521$	29.5640	1.29522	0.647610	−	0.761972i	$-0.275768\pi$
$521$	29.5640	1.29522	0.647610	+	0.761972i	$0.275768\pi$
$522$	0	0
$523$	−13.8546	−0.605817	−0.302909	−	0.953020i	$-0.597958\pi$
$523$	−13.8546	−0.605817	−0.302909	+	0.953020i	$0.597958\pi$
$524$	0	0
$525$	−4.39703	−0.191902
$526$	0	0
$527$	1.31406	0.0572415
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−14.4947	−0.629018
$532$	0	0
$533$	−12.4959	−0.541258
$534$	0	0
$535$	37.8930	1.63826
$536$	0	0
$537$	2.84505	0.122773
$538$	0	0
$539$	−2.43071	−0.104698
$540$	0	0
$541$	27.9516	1.20173	0.600867	−	0.799349i	$-0.294822\pi$
$541$	27.9516	1.20173	0.600867	+	0.799349i	$0.294822\pi$
$542$	0	0
$543$	−13.8690	−0.595177
$544$	0	0
$545$	−36.8542	−1.57866
$546$	0	0
$547$	28.1557	1.20385	0.601926	−	0.798552i	$-0.294400\pi$
$547$	28.1557	1.20385	0.601926	+	0.798552i	$0.294400\pi$
$548$	0	0
$549$	−10.3453	−0.441525
$550$	0	0
$551$	4.89096	0.208362
$552$	0	0
$553$	−11.4418	−0.486554
$554$	0	0
$555$	3.14825	0.133636
$556$	0	0
$557$	9.34421	0.395927	0.197963	−	0.980209i	$-0.436567\pi$
$557$	9.34421	0.395927	0.197963	+	0.980209i	$0.436567\pi$
$558$	0	0
$559$	45.8196	1.93796
$560$	0	0
$561$	−0.610585	−0.0257789
$562$	0	0
$563$	−8.66289	−0.365097	−0.182549	−	0.983197i	$-0.558435\pi$
$563$	−8.66289	−0.365097	−0.182549	+	0.983197i	$0.558435\pi$
$564$	0	0
$565$	−49.0372	−2.06301
$566$	0	0
$567$	−0.993636	−0.0417288
$568$	0	0
$569$	37.6901	1.58005	0.790026	−	0.613073i	$-0.210067\pi$
$569$	37.6901	1.58005	0.790026	+	0.613073i	$0.210067\pi$
$570$	0	0
$571$	17.9710	0.752063	0.376032	−	0.926607i	$-0.377288\pi$
$571$	17.9710	0.752063	0.376032	+	0.926607i	$0.377288\pi$
$572$	0	0
$573$	22.4422	0.937536
$574$	0	0
$575$	−4.42519	−0.184543
$576$	0	0
$577$	−47.5776	−1.98068	−0.990340	−	0.138660i	$-0.955721\pi$
$577$	−47.5776	−1.98068	−0.990340	+	0.138660i	$0.955721\pi$
$578$	0	0
$579$	14.8174	0.615788
$580$	0	0
$581$	−12.0170	−0.498549
$582$	0	0
$583$	1.57668	0.0652995
$584$	0	0
$585$	−17.6809	−0.731014
$586$	0	0
$587$	−46.1318	−1.90406	−0.952031	−	0.306002i	$-0.901009\pi$
$587$	−46.1318	−1.90406	−0.952031	+	0.306002i	$0.901009\pi$
$588$	0	0
$589$	−4.25529	−0.175336
$590$	0	0
$591$	−1.99182	−0.0819326
$592$	0	0
$593$	12.5854	0.516822	0.258411	−	0.966035i	$-0.416801\pi$
$593$	12.5854	0.516822	0.258411	+	0.966035i	$0.416801\pi$
$594$	0	0
$595$	−4.60737	−0.188884
$596$	0	0
$597$	8.18018	0.334792
$598$	0	0
$599$	−25.6057	−1.04622	−0.523111	−	0.852265i	$-0.675229\pi$
$599$	−25.6057	−1.04622	−0.523111	+	0.852265i	$0.675229\pi$
$600$	0	0
$601$	−24.5766	−1.00250	−0.501250	−	0.865303i	$-0.667126\pi$
$601$	−24.5766	−1.00250	−0.501250	+	0.865303i	$0.667126\pi$
$602$	0	0
$603$	−9.21650	−0.375325
$604$	0	0
$605$	33.2688	1.35257
$606$	0	0
$607$	0.156141	0.00633756	0.00316878	−	0.999995i	$-0.498991\pi$
$607$	0.156141	0.00633756	0.00316878	+	0.999995i	$0.498991\pi$
$608$	0	0
$609$	−0.993636	−0.0402642
$610$	0	0
$611$	0.258899	0.0104739
$612$	0	0
$613$	11.4574	0.462759	0.231380	−	0.972864i	$-0.425676\pi$
$613$	11.4574	0.462759	0.231380	+	0.972864i	$0.425676\pi$
$614$	0	0
$615$	6.66123	0.268607
$616$	0	0
$617$	−43.5582	−1.75359	−0.876794	−	0.480867i	$-0.840322\pi$
$617$	−43.5582	−1.75359	−0.876794	+	0.480867i	$0.840322\pi$
$618$	0	0
$619$	−32.2541	−1.29640	−0.648201	−	0.761470i	$-0.724478\pi$
$619$	−32.2541	−1.29640	−0.648201	+	0.761470i	$0.724478\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	3.77329	0.151174
$624$	0	0
$625$	−27.5436	−1.10175
$626$	0	0
$627$	1.97724	0.0789633
$628$	0	0
$629$	1.54884	0.0617561
$630$	0	0
$631$	−21.7500	−0.865853	−0.432927	−	0.901429i	$-0.642519\pi$
$631$	−21.7500	−0.865853	−0.432927	+	0.901429i	$0.642519\pi$
$632$	0	0
$633$	−7.17184	−0.285055
$634$	0	0
$635$	−57.7759	−2.29277
$636$	0	0
$637$	−34.6280	−1.37201
$638$	0	0
$639$	−1.42710	−0.0564553
$640$	0	0
$641$	−47.8500	−1.88996	−0.944981	−	0.327124i	$-0.893921\pi$
$641$	−47.8500	−1.88996	−0.944981	+	0.327124i	$0.893921\pi$
$642$	0	0
$643$	18.2611	0.720148	0.360074	−	0.932924i	$-0.382752\pi$
$643$	18.2611	0.720148	0.360074	+	0.932924i	$0.382752\pi$
$644$	0	0
$645$	−24.4252	−0.961741
$646$	0	0
$647$	−46.9877	−1.84728	−0.923638	−	0.383266i	$-0.874799\pi$
$647$	−46.9877	−1.84728	−0.923638	+	0.383266i	$0.874799\pi$
$648$	0	0
$649$	−5.85970	−0.230013
$650$	0	0
$651$	0.864496	0.0338823
$652$	0	0
$653$	33.4138	1.30758	0.653792	−	0.756674i	$-0.273177\pi$
$653$	33.4138	1.30758	0.653792	+	0.756674i	$0.273177\pi$
$654$	0	0
$655$	46.6446	1.82255
$656$	0	0
$657$	4.05700	0.158279
$658$	0	0
$659$	−15.5359	−0.605191	−0.302595	−	0.953119i	$-0.597853\pi$
$659$	−15.5359	−0.605191	−0.302595	+	0.953119i	$0.597853\pi$
$660$	0	0
$661$	23.6064	0.918184	0.459092	−	0.888389i	$-0.348175\pi$
$661$	23.6064	0.918184	0.459092	+	0.888389i	$0.348175\pi$
$662$	0	0
$663$	−8.69839	−0.337818
$664$	0	0
$665$	14.9199	0.578570
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−8.28635	−0.320369
$670$	0	0
$671$	−4.18222	−0.161453
$672$	0	0
$673$	37.2739	1.43680	0.718401	−	0.695629i	$-0.244874\pi$
$673$	37.2739	1.43680	0.718401	+	0.695629i	$0.244874\pi$
$674$	0	0
$675$	4.42519	0.170326
$676$	0	0
$677$	−23.2853	−0.894926	−0.447463	−	0.894303i	$-0.647672\pi$
$677$	−23.2853	−0.894926	−0.447463	+	0.894303i	$0.647672\pi$
$678$	0	0
$679$	−15.0939	−0.579250
$680$	0	0
$681$	5.96730	0.228667
$682$	0	0
$683$	−16.1121	−0.616514	−0.308257	−	0.951303i	$-0.599746\pi$
$683$	−16.1121	−0.616514	−0.308257	+	0.951303i	$0.599746\pi$
$684$	0	0
$685$	−42.3076	−1.61649
$686$	0	0
$687$	−10.7711	−0.410942
$688$	0	0
$689$	22.4614	0.855712
$690$	0	0
$691$	17.6459	0.671280	0.335640	−	0.941990i	$-0.391047\pi$
$691$	17.6459	0.671280	0.335640	+	0.941990i	$0.391047\pi$
$692$	0	0
$693$	−0.401691	−0.0152590
$694$	0	0
$695$	−52.4927	−1.99116
$696$	0	0
$697$	3.27710	0.124129
$698$	0	0
$699$	5.93086	0.224326
$700$	0	0
$701$	42.8627	1.61890	0.809451	−	0.587188i	$-0.199765\pi$
$701$	42.8627	1.61890	0.809451	+	0.587188i	$0.199765\pi$
$702$	0	0
$703$	−5.01555	−0.189165
$704$	0	0
$705$	−0.138012	−0.00519784
$706$	0	0
$707$	3.40231	0.127957
$708$	0	0
$709$	−11.6228	−0.436503	−0.218252	−	0.975893i	$-0.570035\pi$
$709$	−11.6228	−0.436503	−0.218252	+	0.975893i	$0.570035\pi$
$710$	0	0
$711$	11.5151	0.431848
$712$	0	0
$713$	0.870033	0.0325830
$714$	0	0
$715$	−7.14774	−0.267310
$716$	0	0
$717$	−10.7858	−0.402804
$718$	0	0
$719$	35.3936	1.31996	0.659980	−	0.751283i	$-0.270565\pi$
$719$	35.3936	1.31996	0.659980	+	0.751283i	$0.270565\pi$
$720$	0	0
$721$	−1.22547	−0.0456388
$722$	0	0
$723$	27.2357	1.01291
$724$	0	0
$725$	4.42519	0.164348
$726$	0	0
$727$	−2.37684	−0.0881520	−0.0440760	−	0.999028i	$-0.514034\pi$
$727$	−2.37684	−0.0881520	−0.0440760	+	0.999028i	$0.514034\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0164	−0.444442
$732$	0	0
$733$	41.4694	1.53171	0.765854	−	0.643014i	$-0.222316\pi$
$733$	41.4694	1.53171	0.765854	+	0.643014i	$0.222316\pi$
$734$	0	0
$735$	18.4592	0.680879
$736$	0	0
$737$	−3.72590	−0.137245
$738$	0	0
$739$	10.5714	0.388875	0.194438	−	0.980915i	$-0.437712\pi$
$739$	10.5714	0.388875	0.194438	+	0.980915i	$0.437712\pi$
$740$	0	0
$741$	28.1678	1.03477
$742$	0	0
$743$	4.94357	0.181362	0.0906810	−	0.995880i	$-0.471096\pi$
$743$	4.94357	0.181362	0.0906810	+	0.995880i	$0.471096\pi$
$744$	0	0
$745$	−25.4192	−0.931287
$746$	0	0
$747$	12.0939	0.442494
$748$	0	0
$749$	12.2643	0.448126
$750$	0	0
$751$	21.0716	0.768915	0.384458	−	0.923143i	$-0.374388\pi$
$751$	21.0716	0.768915	0.384458	+	0.923143i	$0.374388\pi$
$752$	0	0
$753$	28.7919	1.04923
$754$	0	0
$755$	22.6273	0.823493
$756$	0	0
$757$	−13.3180	−0.484051	−0.242026	−	0.970270i	$-0.577812\pi$
$757$	−13.3180	−0.484051	−0.242026	+	0.970270i	$0.577812\pi$
$758$	0	0
$759$	−0.404264	−0.0146739
$760$	0	0
$761$	−35.0682	−1.27122	−0.635611	−	0.772009i	$-0.719252\pi$
$761$	−35.0682	−1.27122	−0.635611	+	0.772009i	$0.719252\pi$
$762$	0	0
$763$	−11.9280	−0.431824
$764$	0	0
$765$	4.63688	0.167647
$766$	0	0
$767$	−83.4773	−3.01419
$768$	0	0
$769$	46.8926	1.69099	0.845496	−	0.533982i	$-0.179305\pi$
$769$	46.8926	1.69099	0.845496	+	0.533982i	$0.179305\pi$
$770$	0	0
$771$	30.5727	1.10105
$772$	0	0
$773$	−34.4835	−1.24029	−0.620143	−	0.784489i	$-0.712925\pi$
$773$	−34.4835	−1.24029	−0.620143	+	0.784489i	$0.712925\pi$
$774$	0	0
$775$	−3.85006	−0.138298
$776$	0	0
$777$	1.01895	0.0365546
$778$	0	0
$779$	−10.6122	−0.380220
$780$	0	0
$781$	−0.576926	−0.0206441
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−15.7777	−0.563131
$786$	0	0
$787$	−29.7952	−1.06208	−0.531041	−	0.847346i	$-0.678199\pi$
$787$	−29.7952	−1.06208	−0.531041	+	0.847346i	$0.678199\pi$
$788$	0	0
$789$	25.8711	0.921034
$790$	0	0
$791$	−15.8711	−0.564312
$792$	0	0
$793$	−59.5799	−2.11575
$794$	0	0
$795$	−11.9736	−0.424659
$796$	0	0
$797$	−21.5612	−0.763737	−0.381869	−	0.924217i	$-0.624719\pi$
$797$	−21.5612	−0.763737	−0.381869	+	0.924217i	$0.624719\pi$
$798$	0	0
$799$	−0.0678974	−0.00240204
$800$	0	0
$801$	−3.79745	−0.134176
$802$	0	0
$803$	1.64010	0.0578779
$804$	0	0
$805$	−3.05051	−0.107516
$806$	0	0
$807$	2.54811	0.0896976
$808$	0	0
$809$	23.7298	0.834295	0.417147	−	0.908839i	$-0.363030\pi$
$809$	23.7298	0.834295	0.417147	+	0.908839i	$0.363030\pi$
$810$	0	0
$811$	46.6119	1.63676	0.818382	−	0.574675i	$-0.194871\pi$
$811$	46.6119	1.63676	0.818382	+	0.574675i	$0.194871\pi$
$812$	0	0
$813$	−7.72407	−0.270895
$814$	0	0
$815$	−44.0883	−1.54435
$816$	0	0
$817$	38.9123	1.36137
$818$	0	0
$819$	−5.72250	−0.199960
$820$	0	0
$821$	13.0593	0.455772	0.227886	−	0.973688i	$-0.426819\pi$
$821$	13.0593	0.455772	0.227886	+	0.973688i	$0.426819\pi$
$822$	0	0
$823$	−28.5169	−0.994036	−0.497018	−	0.867740i	$-0.665572\pi$
$823$	−28.5169	−0.994036	−0.497018	+	0.867740i	$0.665572\pi$
$824$	0	0
$825$	1.78895	0.0622831
$826$	0	0
$827$	26.5811	0.924317	0.462158	−	0.886797i	$-0.347075\pi$
$827$	26.5811	0.924317	0.462158	+	0.886797i	$0.347075\pi$
$828$	0	0
$829$	−39.5070	−1.37213	−0.686067	−	0.727538i	$-0.740664\pi$
$829$	−39.5070	−1.37213	−0.686067	+	0.727538i	$0.740664\pi$
$830$	0	0
$831$	−25.5527	−0.886413
$832$	0	0
$833$	9.08133	0.314649
$834$	0	0
$835$	23.5110	0.813632
$836$	0	0
$837$	−0.870033	−0.0300727
$838$	0	0
$839$	−37.5817	−1.29746	−0.648732	−	0.761017i	$-0.724700\pi$
$839$	−37.5817	−1.29746	−0.648732	+	0.761017i	$0.724700\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−8.66057	−0.298286
$844$	0	0
$845$	−61.9161	−2.12998
$846$	0	0
$847$	10.7676	0.369979
$848$	0	0
$849$	21.4008	0.734473
$850$	0	0
$851$	1.02547	0.0351528
$852$	0	0
$853$	−7.50961	−0.257124	−0.128562	−	0.991701i	$-0.541036\pi$
$853$	−7.50961	−0.257124	−0.128562	+	0.991701i	$0.541036\pi$
$854$	0	0
$855$	−15.0155	−0.513519
$856$	0	0
$857$	38.3139	1.30878	0.654389	−	0.756158i	$-0.272926\pi$
$857$	38.3139	1.30878	0.654389	+	0.756158i	$0.272926\pi$
$858$	0	0
$859$	−20.9561	−0.715013	−0.357507	−	0.933911i	$-0.616373\pi$
$859$	−20.9561	−0.715013	−0.357507	+	0.933911i	$0.616373\pi$
$860$	0	0
$861$	2.15594	0.0734743
$862$	0	0
$863$	−27.0371	−0.920352	−0.460176	−	0.887828i	$-0.652214\pi$
$863$	−27.0371	−0.920352	−0.460176	+	0.887828i	$0.652214\pi$
$864$	0	0
$865$	25.8263	0.878122
$866$	0	0
$867$	−14.7188	−0.499877
$868$	0	0
$869$	4.65512	0.157914
$870$	0	0
$871$	−53.0792	−1.79852
$872$	0	0
$873$	15.1905	0.514122
$874$	0	0
$875$	−1.75346	−0.0592776
$876$	0	0
$877$	21.2596	0.717885	0.358943	−	0.933360i	$-0.383137\pi$
$877$	21.2596	0.717885	0.358943	+	0.933360i	$0.383137\pi$
$878$	0	0
$879$	15.5059	0.523000
$880$	0	0
$881$	−46.9018	−1.58016	−0.790082	−	0.613002i	$-0.789962\pi$
$881$	−46.9018	−1.58016	−0.790082	+	0.613002i	$0.789962\pi$
$882$	0	0
$883$	−56.4756	−1.90056	−0.950278	−	0.311402i	$-0.899201\pi$
$883$	−56.4756	−1.90056	−0.950278	+	0.311402i	$0.899201\pi$
$884$	0	0
$885$	44.4995	1.49583
$886$	0	0
$887$	31.1027	1.04433	0.522164	−	0.852845i	$-0.325125\pi$
$887$	31.1027	1.04433	0.522164	+	0.852845i	$0.325125\pi$
$888$	0	0
$889$	−18.6995	−0.627160
$890$	0	0
$891$	0.404264	0.0135434
$892$	0	0
$893$	0.219870	0.00735768
$894$	0	0
$895$	−8.73444	−0.291960
$896$	0	0
$897$	−5.75915	−0.192292
$898$	0	0
$899$	−0.870033	−0.0290172
$900$	0	0
$901$	−5.89060	−0.196244
$902$	0	0
$903$	−7.90533	−0.263073
$904$	0	0
$905$	42.5786	1.41536
$906$	0	0
$907$	41.0484	1.36299	0.681494	−	0.731824i	$-0.261330\pi$
$907$	41.0484	1.36299	0.681494	+	0.731824i	$0.261330\pi$
$908$	0	0
$909$	−3.42410	−0.113570
$910$	0	0
$911$	30.2575	1.00247	0.501237	−	0.865310i	$-0.332878\pi$
$911$	30.2575	1.00247	0.501237	+	0.865310i	$0.332878\pi$
$912$	0	0
$913$	4.88915	0.161807
$914$	0	0
$915$	31.7605	1.04997
$916$	0	0
$917$	15.0967	0.498539
$918$	0	0
$919$	20.0551	0.661557	0.330778	−	0.943708i	$-0.392689\pi$
$919$	20.0551	0.661557	0.330778	+	0.943708i	$0.392689\pi$
$920$	0	0
$921$	20.9358	0.689857
$922$	0	0
$923$	−8.21890	−0.270528
$924$	0	0
$925$	−4.53792	−0.149206
$926$	0	0
$927$	1.23332	0.0405074
$928$	0	0
$929$	−6.27776	−0.205967	−0.102983	−	0.994683i	$-0.532839\pi$
$929$	−6.27776	−0.205967	−0.102983	+	0.994683i	$0.532839\pi$
$930$	0	0
$931$	−29.4078	−0.963802
$932$	0	0
$933$	5.15575	0.168792
$934$	0	0
$935$	1.87452	0.0613035
$936$	0	0
$937$	−33.7132	−1.10136	−0.550681	−	0.834716i	$-0.685632\pi$
$937$	−33.7132	−1.10136	−0.550681	+	0.834716i	$0.685632\pi$
$938$	0	0
$939$	−5.57721	−0.182005
$940$	0	0
$941$	−22.0762	−0.719663	−0.359831	−	0.933017i	$-0.617166\pi$
$941$	−22.0762	−0.719663	−0.359831	+	0.933017i	$0.617166\pi$
$942$	0	0
$943$	2.16975	0.0706567
$944$	0	0
$945$	3.05051	0.0992331
$946$	0	0
$947$	3.21276	0.104401	0.0522003	−	0.998637i	$-0.483377\pi$
$947$	3.21276	0.104401	0.0522003	+	0.998637i	$0.483377\pi$
$948$	0	0
$949$	23.3649	0.758456
$950$	0	0
$951$	−7.41356	−0.240401
$952$	0	0
$953$	13.9369	0.451461	0.225730	−	0.974190i	$-0.427523\pi$
$953$	13.9369	0.451461	0.225730	+	0.974190i	$0.427523\pi$
$954$	0	0
$955$	−68.8986	−2.22951
$956$	0	0
$957$	0.404264	0.0130680
$958$	0	0
$959$	−13.6931	−0.442172
$960$	0	0
$961$	−30.2430	−0.975582
$962$	0	0
$963$	−12.3428	−0.397741
$964$	0	0
$965$	−45.4900	−1.46437
$966$	0	0
$967$	−44.4618	−1.42980	−0.714898	−	0.699229i	$-0.753527\pi$
$967$	−44.4618	−1.42980	−0.714898	+	0.699229i	$0.753527\pi$
$968$	0	0
$969$	−7.38711	−0.237308
$970$	0	0
$971$	−56.9177	−1.82658	−0.913288	−	0.407314i	$-0.866465\pi$
$971$	−56.9177	−1.82658	−0.913288	+	0.407314i	$0.866465\pi$
$972$	0	0
$973$	−16.9895	−0.544659
$974$	0	0
$975$	25.4853	0.816184
$976$	0	0
$977$	−35.3775	−1.13183	−0.565913	−	0.824465i	$-0.691476\pi$
$977$	−35.3775	−1.13183	−0.565913	+	0.824465i	$0.691476\pi$
$978$	0	0
$979$	−1.53517	−0.0490644
$980$	0	0
$981$	12.0044	0.383272
$982$	0	0
$983$	42.4731	1.35468	0.677341	−	0.735669i	$-0.263132\pi$
$983$	42.4731	1.35468	0.677341	+	0.735669i	$0.263132\pi$
$984$	0	0
$985$	6.11499	0.194840
$986$	0	0
$987$	−0.0446684	−0.00142181
$988$	0	0
$989$	−7.95596	−0.252985
$990$	0	0
$991$	−30.7348	−0.976322	−0.488161	−	0.872754i	$-0.662332\pi$
$991$	−30.7348	−0.976322	−0.488161	+	0.872754i	$0.662332\pi$
$992$	0	0
$993$	18.9038	0.599893
$994$	0	0
$995$	−25.1135	−0.796153
$996$	0	0
$997$	42.1180	1.33389	0.666945	−	0.745107i	$-0.267602\pi$
$997$	42.1180	1.33389	0.666945	+	0.745107i	$0.267602\pi$
$998$	0	0
$999$	−1.02547	−0.0324446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.3	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.3	✓	18	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.07005 q^{5} -0.993636 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.07005 q^{5} -0.993636 q^{7} +1.00000 q^{9} +0.404264 q^{11} +5.75915 q^{13} -3.07005 q^{15} -1.51036 q^{17} +4.89096 q^{19} -0.993636 q^{21} -1.00000 q^{23} +4.42519 q^{25} +1.00000 q^{27} +1.00000 q^{29} -0.870033 q^{31} +0.404264 q^{33} +3.05051 q^{35} -1.02547 q^{37} +5.75915 q^{39} -2.16975 q^{41} +7.95596 q^{43} -3.07005 q^{45} +0.0449544 q^{47} -6.01269 q^{49} -1.51036 q^{51} +3.90013 q^{53} -1.24111 q^{55} +4.89096 q^{57} -14.4947 q^{59} -10.3453 q^{61} -0.993636 q^{63} -17.6809 q^{65} -9.21650 q^{67} -1.00000 q^{69} -1.42710 q^{71} +4.05700 q^{73} +4.42519 q^{75} -0.401691 q^{77} +11.5151 q^{79} +1.00000 q^{81} +12.0939 q^{83} +4.63688 q^{85} +1.00000 q^{87} -3.79745 q^{89} -5.72250 q^{91} -0.870033 q^{93} -15.0155 q^{95} +15.1905 q^{97} +0.404264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

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Database

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