LMFDB - Embedded newform 8016.2.a.z.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 $ x^8 - 2x^7 - 8x^6 + 15x^5 + 19x^4 - 31x^3 - 13x^2 + 14*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.09883$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.14293 q^{5} +1.03675 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.14293 q^{5} +1.03675 q^{7} +1.00000 q^{9} -3.15291 q^{11} -4.46759 q^{13} +4.14293 q^{15} -7.46321 q^{17} -4.84381 q^{19} +1.03675 q^{21} -5.72825 q^{23} +12.1639 q^{25} +1.00000 q^{27} +4.37985 q^{29} -11.0902 q^{31} -3.15291 q^{33} +4.29517 q^{35} +1.22509 q^{37} -4.46759 q^{39} -2.29440 q^{41} -0.00419434 q^{43} +4.14293 q^{45} -5.10859 q^{47} -5.92516 q^{49} -7.46321 q^{51} +5.05303 q^{53} -13.0623 q^{55} -4.84381 q^{57} -8.22992 q^{59} +2.02411 q^{61} +1.03675 q^{63} -18.5089 q^{65} +2.70164 q^{67} -5.72825 q^{69} +3.08019 q^{71} +10.6919 q^{73} +12.1639 q^{75} -3.26876 q^{77} +14.2639 q^{79} +1.00000 q^{81} +3.69318 q^{83} -30.9196 q^{85} +4.37985 q^{87} +13.1042 q^{89} -4.63175 q^{91} -11.0902 q^{93} -20.0676 q^{95} +2.58759 q^{97} -3.15291 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * 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
$5$	4.14293	1.85278	0.926388	−	0.376571i	$-0.122897\pi$
$5$	4.14293	1.85278	0.926388	+	0.376571i	$0.122897\pi$
$6$	0	0
$7$	1.03675	0.391853	0.195926	−	0.980619i	$-0.437229\pi$
$7$	1.03675	0.391853	0.195926	+	0.980619i	$0.437229\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.15291	−0.950637	−0.475319	−	0.879814i	$-0.657667\pi$
$11$	−3.15291	−0.950637	−0.475319	+	0.879814i	$0.657667\pi$
$12$	0	0
$13$	−4.46759	−1.23909	−0.619543	−	0.784963i	$-0.712682\pi$
$13$	−4.46759	−1.23909	−0.619543	+	0.784963i	$0.712682\pi$
$14$	0	0
$15$	4.14293	1.06970
$16$	0	0
$17$	−7.46321	−1.81009	−0.905047	−	0.425311i	$-0.860165\pi$
$17$	−7.46321	−1.81009	−0.905047	+	0.425311i	$0.860165\pi$
$18$	0	0
$19$	−4.84381	−1.11125	−0.555623	−	0.831435i	$-0.687520\pi$
$19$	−4.84381	−1.11125	−0.555623	+	0.831435i	$0.687520\pi$
$20$	0	0
$21$	1.03675	0.226236
$22$	0	0
$23$	−5.72825	−1.19442	−0.597211	−	0.802084i	$-0.703724\pi$
$23$	−5.72825	−1.19442	−0.597211	+	0.802084i	$0.703724\pi$
$24$	0	0
$25$	12.1639	2.43278
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.37985	0.813317	0.406659	−	0.913580i	$-0.366694\pi$
$29$	4.37985	0.813317	0.406659	+	0.913580i	$0.366694\pi$
$30$	0	0
$31$	−11.0902	−1.99186	−0.995929	−	0.0901418i	$-0.971268\pi$
$31$	−11.0902	−1.99186	−0.995929	+	0.0901418i	$0.971268\pi$
$32$	0	0
$33$	−3.15291	−0.548851
$34$	0	0
$35$	4.29517	0.726015
$36$	0	0
$37$	1.22509	0.201403	0.100701	−	0.994917i	$-0.467891\pi$
$37$	1.22509	0.201403	0.100701	+	0.994917i	$0.467891\pi$
$38$	0	0
$39$	−4.46759	−0.715386
$40$	0	0
$41$	−2.29440	−0.358325	−0.179162	−	0.983820i	$-0.557339\pi$
$41$	−2.29440	−0.358325	−0.179162	+	0.983820i	$0.557339\pi$
$42$	0	0
$43$	−0.00419434	−0.000639630 0	−0.000319815	−	1.00000i	$-0.500102\pi$
$43$	−0.00419434	−0.000639630 0	−0.000319815		1.00000i	$0.500102\pi$
$44$	0	0
$45$	4.14293	0.617592
$46$	0	0
$47$	−5.10859	−0.745164	−0.372582	−	0.927999i	$-0.621527\pi$
$47$	−5.10859	−0.745164	−0.372582	+	0.927999i	$0.621527\pi$
$48$	0	0
$49$	−5.92516	−0.846451
$50$	0	0
$51$	−7.46321	−1.04506
$52$	0	0
$53$	5.05303	0.694087	0.347043	−	0.937849i	$-0.387186\pi$
$53$	5.05303	0.694087	0.347043	+	0.937849i	$0.387186\pi$
$54$	0	0
$55$	−13.0623	−1.76132
$56$	0	0
$57$	−4.84381	−0.641578
$58$	0	0
$59$	−8.22992	−1.07144	−0.535722	−	0.844394i	$-0.679961\pi$
$59$	−8.22992	−1.07144	−0.535722	+	0.844394i	$0.679961\pi$
$60$	0	0
$61$	2.02411	0.259160	0.129580	−	0.991569i	$-0.458637\pi$
$61$	2.02411	0.259160	0.129580	+	0.991569i	$0.458637\pi$
$62$	0	0
$63$	1.03675	0.130618
$64$	0	0
$65$	−18.5089	−2.29575
$66$	0	0
$67$	2.70164	0.330058	0.165029	−	0.986289i	$-0.447228\pi$
$67$	2.70164	0.330058	0.165029	+	0.986289i	$0.447228\pi$
$68$	0	0
$69$	−5.72825	−0.689600
$70$	0	0
$71$	3.08019	0.365551	0.182776	−	0.983155i	$-0.441492\pi$
$71$	3.08019	0.365551	0.182776	+	0.983155i	$0.441492\pi$
$72$	0	0
$73$	10.6919	1.25139	0.625697	−	0.780066i	$-0.284815\pi$
$73$	10.6919	1.25139	0.625697	+	0.780066i	$0.284815\pi$
$74$	0	0
$75$	12.1639	1.40456
$76$	0	0
$77$	−3.26876	−0.372510
$78$	0	0
$79$	14.2639	1.60481	0.802405	−	0.596780i	$-0.203553\pi$
$79$	14.2639	1.60481	0.802405	+	0.596780i	$0.203553\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.69318	0.405379	0.202689	−	0.979243i	$-0.435032\pi$
$83$	3.69318	0.405379	0.202689	+	0.979243i	$0.435032\pi$
$84$	0	0
$85$	−30.9196	−3.35370
$86$	0	0
$87$	4.37985	0.469569
$88$	0	0
$89$	13.1042	1.38904	0.694521	−	0.719472i	$-0.255616\pi$
$89$	13.1042	1.38904	0.694521	+	0.719472i	$0.255616\pi$
$90$	0	0
$91$	−4.63175	−0.485539
$92$	0	0
$93$	−11.0902	−1.15000
$94$	0	0
$95$	−20.0676	−2.05889
$96$	0	0
$97$	2.58759	0.262730	0.131365	−	0.991334i	$-0.458064\pi$
$97$	2.58759	0.262730	0.131365	+	0.991334i	$0.458064\pi$
$98$	0	0
$99$	−3.15291	−0.316879
$100$	0	0
$101$	−19.1202	−1.90253	−0.951267	−	0.308367i	$-0.900217\pi$
$101$	−19.1202	−1.90253	−0.951267	+	0.308367i	$0.900217\pi$
$102$	0	0
$103$	−6.77219	−0.667284	−0.333642	−	0.942700i	$-0.608278\pi$
$103$	−6.77219	−0.667284	−0.333642	+	0.942700i	$0.608278\pi$
$104$	0	0
$105$	4.29517	0.419165
$106$	0	0
$107$	11.3637	1.09857	0.549287	−	0.835634i	$-0.314900\pi$
$107$	11.3637	1.09857	0.549287	+	0.835634i	$0.314900\pi$
$108$	0	0
$109$	2.13664	0.204653	0.102326	−	0.994751i	$-0.467371\pi$
$109$	2.13664	0.204653	0.102326	+	0.994751i	$0.467371\pi$
$110$	0	0
$111$	1.22509	0.116280
$112$	0	0
$113$	−14.1409	−1.33027	−0.665134	−	0.746724i	$-0.731626\pi$
$113$	−14.1409	−1.33027	−0.665134	+	0.746724i	$0.731626\pi$
$114$	0	0
$115$	−23.7317	−2.21300
$116$	0	0
$117$	−4.46759	−0.413028
$118$	0	0
$119$	−7.73745	−0.709291
$120$	0	0
$121$	−1.05918	−0.0962888
$122$	0	0
$123$	−2.29440	−0.206879
$124$	0	0
$125$	29.6795	2.65461
$126$	0	0
$127$	−19.8539	−1.76175	−0.880876	−	0.473348i	$-0.843045\pi$
$127$	−19.8539	−1.76175	−0.880876	+	0.473348i	$0.843045\pi$
$128$	0	0
$129$	−0.00419434	−0.000369291 0
$130$	0	0
$131$	5.11338	0.446758	0.223379	−	0.974732i	$-0.428291\pi$
$131$	5.11338	0.446758	0.223379	+	0.974732i	$0.428291\pi$
$132$	0	0
$133$	−5.02179	−0.435445
$134$	0	0
$135$	4.14293	0.356567
$136$	0	0
$137$	−15.8251	−1.35203	−0.676016	−	0.736887i	$-0.736295\pi$
$137$	−15.8251	−1.35203	−0.676016	+	0.736887i	$0.736295\pi$
$138$	0	0
$139$	−5.44253	−0.461630	−0.230815	−	0.972998i	$-0.574139\pi$
$139$	−5.44253	−0.461630	−0.230815	+	0.972998i	$0.574139\pi$
$140$	0	0
$141$	−5.10859	−0.430221
$142$	0	0
$143$	14.0859	1.17792
$144$	0	0
$145$	18.1454	1.50689
$146$	0	0
$147$	−5.92516	−0.488699
$148$	0	0
$149$	−15.6452	−1.28170	−0.640851	−	0.767665i	$-0.721418\pi$
$149$	−15.6452	−1.28170	−0.640851	+	0.767665i	$0.721418\pi$
$150$	0	0
$151$	−1.41219	−0.114923	−0.0574613	−	0.998348i	$-0.518301\pi$
$151$	−1.41219	−0.114923	−0.0574613	+	0.998348i	$0.518301\pi$
$152$	0	0
$153$	−7.46321	−0.603365
$154$	0	0
$155$	−45.9459	−3.69047
$156$	0	0
$157$	−22.9926	−1.83501	−0.917503	−	0.397728i	$-0.869799\pi$
$157$	−22.9926	−1.83501	−0.917503	+	0.397728i	$0.869799\pi$
$158$	0	0
$159$	5.05303	0.400731
$160$	0	0
$161$	−5.93873	−0.468038
$162$	0	0
$163$	0.534913	0.0418976	0.0209488	−	0.999781i	$-0.493331\pi$
$163$	0.534913	0.0418976	0.0209488	+	0.999781i	$0.493331\pi$
$164$	0	0
$165$	−13.0623	−1.01690
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	6.95933	0.535333
$170$	0	0
$171$	−4.84381	−0.370415
$172$	0	0
$173$	14.8264	1.12723	0.563616	−	0.826037i	$-0.309410\pi$
$173$	14.8264	1.12723	0.563616	+	0.826037i	$0.309410\pi$
$174$	0	0
$175$	12.6109	0.953291
$176$	0	0
$177$	−8.22992	−0.618599
$178$	0	0
$179$	−5.50306	−0.411318	−0.205659	−	0.978624i	$-0.565934\pi$
$179$	−5.50306	−0.411318	−0.205659	+	0.978624i	$0.565934\pi$
$180$	0	0
$181$	2.71026	0.201452	0.100726	−	0.994914i	$-0.467883\pi$
$181$	2.71026	0.201452	0.100726	+	0.994914i	$0.467883\pi$
$182$	0	0
$183$	2.02411	0.149626
$184$	0	0
$185$	5.07544	0.373154
$186$	0	0
$187$	23.5308	1.72074
$188$	0	0
$189$	1.03675	0.0754121
$190$	0	0
$191$	1.73921	0.125845	0.0629223	−	0.998018i	$-0.479958\pi$
$191$	1.73921	0.125845	0.0629223	+	0.998018i	$0.479958\pi$
$192$	0	0
$193$	−2.46438	−0.177390	−0.0886949	−	0.996059i	$-0.528270\pi$
$193$	−2.46438	−0.177390	−0.0886949	+	0.996059i	$0.528270\pi$
$194$	0	0
$195$	−18.5089	−1.32545
$196$	0	0
$197$	−14.6400	−1.04306	−0.521530	−	0.853233i	$-0.674639\pi$
$197$	−14.6400	−1.04306	−0.521530	+	0.853233i	$0.674639\pi$
$198$	0	0
$199$	−2.76829	−0.196239	−0.0981195	−	0.995175i	$-0.531283\pi$
$199$	−2.76829	−0.196239	−0.0981195	+	0.995175i	$0.531283\pi$
$200$	0	0
$201$	2.70164	0.190559
$202$	0	0
$203$	4.54079	0.318701
$204$	0	0
$205$	−9.50553	−0.663895
$206$	0	0
$207$	−5.72825	−0.398141
$208$	0	0
$209$	15.2721	1.05639
$210$	0	0
$211$	11.5671	0.796311	0.398155	−	0.917318i	$-0.369651\pi$
$211$	11.5671	0.796311	0.398155	+	0.917318i	$0.369651\pi$
$212$	0	0
$213$	3.08019	0.211051
$214$	0	0
$215$	−0.0173769	−0.00118509
$216$	0	0
$217$	−11.4977	−0.780515
$218$	0	0
$219$	10.6919	0.722492
$220$	0	0
$221$	33.3425	2.24286
$222$	0	0
$223$	0.906902	0.0607307	0.0303653	−	0.999539i	$-0.490333\pi$
$223$	0.906902	0.0607307	0.0303653	+	0.999539i	$0.490333\pi$
$224$	0	0
$225$	12.1639	0.810926
$226$	0	0
$227$	25.9521	1.72250	0.861251	−	0.508180i	$-0.169681\pi$
$227$	25.9521	1.72250	0.861251	+	0.508180i	$0.169681\pi$
$228$	0	0
$229$	−7.53366	−0.497838	−0.248919	−	0.968524i	$-0.580075\pi$
$229$	−7.53366	−0.497838	−0.248919	+	0.968524i	$0.580075\pi$
$230$	0	0
$231$	−3.26876	−0.215069
$232$	0	0
$233$	12.0894	0.792002	0.396001	−	0.918250i	$-0.370398\pi$
$233$	12.0894	0.792002	0.396001	+	0.918250i	$0.370398\pi$
$234$	0	0
$235$	−21.1645	−1.38062
$236$	0	0
$237$	14.2639	0.926538
$238$	0	0
$239$	24.7634	1.60181	0.800906	−	0.598791i	$-0.204352\pi$
$239$	24.7634	1.60181	0.800906	+	0.598791i	$0.204352\pi$
$240$	0	0
$241$	19.3715	1.24783	0.623915	−	0.781492i	$-0.285541\pi$
$241$	19.3715	1.24783	0.623915	+	0.781492i	$0.285541\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−24.5475	−1.56828
$246$	0	0
$247$	21.6401	1.37693
$248$	0	0
$249$	3.69318	0.234046
$250$	0	0
$251$	10.2306	0.645751	0.322876	−	0.946441i	$-0.395350\pi$
$251$	10.2306	0.645751	0.322876	+	0.946441i	$0.395350\pi$
$252$	0	0
$253$	18.0606	1.13546
$254$	0	0
$255$	−30.9196	−1.93626
$256$	0	0
$257$	−10.9078	−0.680410	−0.340205	−	0.940351i	$-0.610497\pi$
$257$	−10.9078	−0.680410	−0.340205	+	0.940351i	$0.610497\pi$
$258$	0	0
$259$	1.27010	0.0789202
$260$	0	0
$261$	4.37985	0.271106
$262$	0	0
$263$	6.25855	0.385919	0.192959	−	0.981207i	$-0.438191\pi$
$263$	6.25855	0.385919	0.192959	+	0.981207i	$0.438191\pi$
$264$	0	0
$265$	20.9344	1.28599
$266$	0	0
$267$	13.1042	0.801964
$268$	0	0
$269$	19.5130	1.18973	0.594864	−	0.803827i	$-0.297206\pi$
$269$	19.5130	1.18973	0.594864	+	0.803827i	$0.297206\pi$
$270$	0	0
$271$	−2.09461	−0.127239	−0.0636194	−	0.997974i	$-0.520264\pi$
$271$	−2.09461	−0.127239	−0.0636194	+	0.997974i	$0.520264\pi$
$272$	0	0
$273$	−4.63175	−0.280326
$274$	0	0
$275$	−38.3516	−2.31269
$276$	0	0
$277$	15.7238	0.944749	0.472374	−	0.881398i	$-0.343397\pi$
$277$	15.7238	0.944749	0.472374	+	0.881398i	$0.343397\pi$
$278$	0	0
$279$	−11.0902	−0.663953
$280$	0	0
$281$	4.80654	0.286734	0.143367	−	0.989670i	$-0.454207\pi$
$281$	4.80654	0.286734	0.143367	+	0.989670i	$0.454207\pi$
$282$	0	0
$283$	−8.76619	−0.521096	−0.260548	−	0.965461i	$-0.583903\pi$
$283$	−8.76619	−0.521096	−0.260548	+	0.965461i	$0.583903\pi$
$284$	0	0
$285$	−20.0676	−1.18870
$286$	0	0
$287$	−2.37870	−0.140411
$288$	0	0
$289$	38.6995	2.27644
$290$	0	0
$291$	2.58759	0.151687
$292$	0	0
$293$	11.2792	0.658941	0.329470	−	0.944166i	$-0.393130\pi$
$293$	11.2792	0.658941	0.329470	+	0.944166i	$0.393130\pi$
$294$	0	0
$295$	−34.0960	−1.98515
$296$	0	0
$297$	−3.15291	−0.182950
$298$	0	0
$299$	25.5914	1.47999
$300$	0	0
$301$	−0.00434846	−0.000250641 0
$302$	0	0
$303$	−19.1202	−1.09843
$304$	0	0
$305$	8.38574	0.480166
$306$	0	0
$307$	−0.138289	−0.00789255	−0.00394627	−	0.999992i	$-0.501256\pi$
$307$	−0.138289	−0.00789255	−0.00394627	+	0.999992i	$0.501256\pi$
$308$	0	0
$309$	−6.77219	−0.385256
$310$	0	0
$311$	−9.20714	−0.522089	−0.261045	−	0.965327i	$-0.584067\pi$
$311$	−9.20714	−0.522089	−0.261045	+	0.965327i	$0.584067\pi$
$312$	0	0
$313$	−27.8661	−1.57508	−0.787541	−	0.616262i	$-0.788646\pi$
$313$	−27.8661	−1.57508	−0.787541	+	0.616262i	$0.788646\pi$
$314$	0	0
$315$	4.29517	0.242005
$316$	0	0
$317$	3.46368	0.194540	0.0972698	−	0.995258i	$-0.468989\pi$
$317$	3.46368	0.194540	0.0972698	+	0.995258i	$0.468989\pi$
$318$	0	0
$319$	−13.8093	−0.773170
$320$	0	0
$321$	11.3637	0.634262
$322$	0	0
$323$	36.1504	2.01146
$324$	0	0
$325$	−54.3432	−3.01442
$326$	0	0
$327$	2.13664	0.118156
$328$	0	0
$329$	−5.29630	−0.291995
$330$	0	0
$331$	10.3902	0.571099	0.285550	−	0.958364i	$-0.407824\pi$
$331$	10.3902	0.571099	0.285550	+	0.958364i	$0.407824\pi$
$332$	0	0
$333$	1.22509	0.0671342
$334$	0	0
$335$	11.1927	0.611523
$336$	0	0
$337$	−7.38038	−0.402035	−0.201017	−	0.979588i	$-0.564425\pi$
$337$	−7.38038	−0.402035	−0.201017	+	0.979588i	$0.564425\pi$
$338$	0	0
$339$	−14.1409	−0.768031
$340$	0	0
$341$	34.9664	1.89353
$342$	0	0
$343$	−13.4001	−0.723537
$344$	0	0
$345$	−23.7317	−1.27767
$346$	0	0
$347$	28.4104	1.52515	0.762576	−	0.646899i	$-0.223934\pi$
$347$	28.4104	1.52515	0.762576	+	0.646899i	$0.223934\pi$
$348$	0	0
$349$	−26.9323	−1.44165	−0.720826	−	0.693116i	$-0.756237\pi$
$349$	−26.9323	−1.44165	−0.720826	+	0.693116i	$0.756237\pi$
$350$	0	0
$351$	−4.46759	−0.238462
$352$	0	0
$353$	−14.8008	−0.787769	−0.393884	−	0.919160i	$-0.628869\pi$
$353$	−14.8008	−0.787769	−0.393884	+	0.919160i	$0.628869\pi$
$354$	0	0
$355$	12.7610	0.677285
$356$	0	0
$357$	−7.73745	−0.409509
$358$	0	0
$359$	4.48702	0.236816	0.118408	−	0.992965i	$-0.462221\pi$
$359$	4.48702	0.236816	0.118408	+	0.992965i	$0.462221\pi$
$360$	0	0
$361$	4.46246	0.234866
$362$	0	0
$363$	−1.05918	−0.0555924
$364$	0	0
$365$	44.2959	2.31855
$366$	0	0
$367$	−10.0866	−0.526517	−0.263258	−	0.964725i	$-0.584797\pi$
$367$	−10.0866	−0.526517	−0.263258	+	0.964725i	$0.584797\pi$
$368$	0	0
$369$	−2.29440	−0.119442
$370$	0	0
$371$	5.23870	0.271980
$372$	0	0
$373$	18.5873	0.962414	0.481207	−	0.876607i	$-0.340199\pi$
$373$	18.5873	0.962414	0.481207	+	0.876607i	$0.340199\pi$
$374$	0	0
$375$	29.6795	1.53264
$376$	0	0
$377$	−19.5673	−1.00777
$378$	0	0
$379$	23.0209	1.18250	0.591252	−	0.806487i	$-0.298634\pi$
$379$	23.0209	1.18250	0.591252	+	0.806487i	$0.298634\pi$
$380$	0	0
$381$	−19.8539	−1.01715
$382$	0	0
$383$	9.84246	0.502926	0.251463	−	0.967867i	$-0.419088\pi$
$383$	9.84246	0.502926	0.251463	+	0.967867i	$0.419088\pi$
$384$	0	0
$385$	−13.5423	−0.690177
$386$	0	0
$387$	−0.00419434	−0.000213210 0
$388$	0	0
$389$	−26.3740	−1.33722	−0.668608	−	0.743615i	$-0.733110\pi$
$389$	−26.3740	−1.33722	−0.668608	+	0.743615i	$0.733110\pi$
$390$	0	0
$391$	42.7511	2.16202
$392$	0	0
$393$	5.11338	0.257936
$394$	0	0
$395$	59.0942	2.97335
$396$	0	0
$397$	9.19034	0.461250	0.230625	−	0.973043i	$-0.425923\pi$
$397$	9.19034	0.461250	0.230625	+	0.973043i	$0.425923\pi$
$398$	0	0
$399$	−5.02179	−0.251404
$400$	0	0
$401$	−30.4954	−1.52287	−0.761433	−	0.648244i	$-0.775504\pi$
$401$	−30.4954	−1.52287	−0.761433	+	0.648244i	$0.775504\pi$
$402$	0	0
$403$	49.5464	2.46808
$404$	0	0
$405$	4.14293	0.205864
$406$	0	0
$407$	−3.86258	−0.191461
$408$	0	0
$409$	10.0808	0.498464	0.249232	−	0.968444i	$-0.419822\pi$
$409$	10.0808	0.498464	0.249232	+	0.968444i	$0.419822\pi$
$410$	0	0
$411$	−15.8251	−0.780596
$412$	0	0
$413$	−8.53233	−0.419848
$414$	0	0
$415$	15.3006	0.751076
$416$	0	0
$417$	−5.44253	−0.266522
$418$	0	0
$419$	−16.7471	−0.818149	−0.409075	−	0.912501i	$-0.634148\pi$
$419$	−16.7471	−0.818149	−0.409075	+	0.912501i	$0.634148\pi$
$420$	0	0
$421$	−13.3802	−0.652113	−0.326057	−	0.945350i	$-0.605720\pi$
$421$	−13.3802	−0.652113	−0.326057	+	0.945350i	$0.605720\pi$
$422$	0	0
$423$	−5.10859	−0.248388
$424$	0	0
$425$	−90.7817	−4.40356
$426$	0	0
$427$	2.09848	0.101553
$428$	0	0
$429$	14.0859	0.680073
$430$	0	0
$431$	−10.5358	−0.507493	−0.253747	−	0.967271i	$-0.581663\pi$
$431$	−10.5358	−0.507493	−0.253747	+	0.967271i	$0.581663\pi$
$432$	0	0
$433$	25.4815	1.22456	0.612281	−	0.790641i	$-0.290252\pi$
$433$	25.4815	1.22456	0.612281	+	0.790641i	$0.290252\pi$
$434$	0	0
$435$	18.1454	0.870006
$436$	0	0
$437$	27.7465	1.32730
$438$	0	0
$439$	−10.6257	−0.507139	−0.253570	−	0.967317i	$-0.581605\pi$
$439$	−10.6257	−0.507139	−0.253570	+	0.967317i	$0.581605\pi$
$440$	0	0
$441$	−5.92516	−0.282150
$442$	0	0
$443$	18.2108	0.865221	0.432611	−	0.901581i	$-0.357593\pi$
$443$	18.2108	0.865221	0.432611	+	0.901581i	$0.357593\pi$
$444$	0	0
$445$	54.2898	2.57358
$446$	0	0
$447$	−15.6452	−0.739991
$448$	0	0
$449$	−9.78490	−0.461778	−0.230889	−	0.972980i	$-0.574163\pi$
$449$	−9.78490	−0.461778	−0.230889	+	0.972980i	$0.574163\pi$
$450$	0	0
$451$	7.23402	0.340637
$452$	0	0
$453$	−1.41219	−0.0663506
$454$	0	0
$455$	−19.1890	−0.899595
$456$	0	0
$457$	11.5068	0.538266	0.269133	−	0.963103i	$-0.413263\pi$
$457$	11.5068	0.538266	0.269133	+	0.963103i	$0.413263\pi$
$458$	0	0
$459$	−7.46321	−0.348353
$460$	0	0
$461$	8.77664	0.408769	0.204385	−	0.978891i	$-0.434481\pi$
$461$	8.77664	0.408769	0.204385	+	0.978891i	$0.434481\pi$
$462$	0	0
$463$	36.0679	1.67622	0.838109	−	0.545503i	$-0.183661\pi$
$463$	36.0679	1.67622	0.838109	+	0.545503i	$0.183661\pi$
$464$	0	0
$465$	−45.9459	−2.13069
$466$	0	0
$467$	−37.0965	−1.71662	−0.858311	−	0.513129i	$-0.828486\pi$
$467$	−37.0965	−1.71662	−0.858311	+	0.513129i	$0.828486\pi$
$468$	0	0
$469$	2.80091	0.129334
$470$	0	0
$471$	−22.9926	−1.05944
$472$	0	0
$473$	0.0132244	0.000608057 0
$474$	0	0
$475$	−58.9195	−2.70341
$476$	0	0
$477$	5.05303	0.231362
$478$	0	0
$479$	−22.8211	−1.04272	−0.521361	−	0.853336i	$-0.674575\pi$
$479$	−22.8211	−1.04272	−0.521361	+	0.853336i	$0.674575\pi$
$480$	0	0
$481$	−5.47317	−0.249555
$482$	0	0
$483$	−5.93873	−0.270222
$484$	0	0
$485$	10.7202	0.486780
$486$	0	0
$487$	30.6526	1.38900	0.694502	−	0.719491i	$-0.255625\pi$
$487$	30.6526	1.38900	0.694502	+	0.719491i	$0.255625\pi$
$488$	0	0
$489$	0.534913	0.0241896
$490$	0	0
$491$	−39.1263	−1.76575	−0.882873	−	0.469612i	$-0.844394\pi$
$491$	−39.1263	−1.76575	−0.882873	+	0.469612i	$0.844394\pi$
$492$	0	0
$493$	−32.6877	−1.47218
$494$	0	0
$495$	−13.0623	−0.587106
$496$	0	0
$497$	3.19337	0.143242
$498$	0	0
$499$	−31.3754	−1.40456	−0.702278	−	0.711903i	$-0.747834\pi$
$499$	−31.3754	−1.40456	−0.702278	+	0.711903i	$0.747834\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−4.52554	−0.201784	−0.100892	−	0.994897i	$-0.532170\pi$
$503$	−4.52554	−0.201784	−0.100892	+	0.994897i	$0.532170\pi$
$504$	0	0
$505$	−79.2139	−3.52497
$506$	0	0
$507$	6.95933	0.309075
$508$	0	0
$509$	−15.5695	−0.690105	−0.345052	−	0.938583i	$-0.612139\pi$
$509$	−15.5695	−0.690105	−0.345052	+	0.938583i	$0.612139\pi$
$510$	0	0
$511$	11.0848	0.490362
$512$	0	0
$513$	−4.84381	−0.213859
$514$	0	0
$515$	−28.0567	−1.23633
$516$	0	0
$517$	16.1069	0.708380
$518$	0	0
$519$	14.8264	0.650808
$520$	0	0
$521$	2.97556	0.130361	0.0651807	−	0.997873i	$-0.479238\pi$
$521$	2.97556	0.130361	0.0651807	+	0.997873i	$0.479238\pi$
$522$	0	0
$523$	−17.0981	−0.747649	−0.373825	−	0.927499i	$-0.621954\pi$
$523$	−17.0981	−0.747649	−0.373825	+	0.927499i	$0.621954\pi$
$524$	0	0
$525$	12.6109	0.550383
$526$	0	0
$527$	82.7685	3.60545
$528$	0	0
$529$	9.81280	0.426643
$530$	0	0
$531$	−8.22992	−0.357148
$532$	0	0
$533$	10.2504	0.443995
$534$	0	0
$535$	47.0792	2.03541
$536$	0	0
$537$	−5.50306	−0.237474
$538$	0	0
$539$	18.6815	0.804668
$540$	0	0
$541$	25.9408	1.11528	0.557641	−	0.830082i	$-0.311706\pi$
$541$	25.9408	1.11528	0.557641	+	0.830082i	$0.311706\pi$
$542$	0	0
$543$	2.71026	0.116308
$544$	0	0
$545$	8.85195	0.379176
$546$	0	0
$547$	−17.2904	−0.739285	−0.369642	−	0.929174i	$-0.620520\pi$
$547$	−17.2904	−0.739285	−0.369642	+	0.929174i	$0.620520\pi$
$548$	0	0
$549$	2.02411	0.0863868
$550$	0	0
$551$	−21.2151	−0.903795
$552$	0	0
$553$	14.7880	0.628849
$554$	0	0
$555$	5.07544	0.215441
$556$	0	0
$557$	9.42843	0.399495	0.199748	−	0.979847i	$-0.435988\pi$
$557$	9.42843	0.399495	0.199748	+	0.979847i	$0.435988\pi$
$558$	0	0
$559$	0.0187386	0.000792557 0
$560$	0	0
$561$	23.5308	0.993472
$562$	0	0
$563$	−16.6116	−0.700096	−0.350048	−	0.936732i	$-0.613835\pi$
$563$	−16.6116	−0.700096	−0.350048	+	0.936732i	$0.613835\pi$
$564$	0	0
$565$	−58.5850	−2.46469
$566$	0	0
$567$	1.03675	0.0435392
$568$	0	0
$569$	−24.9448	−1.04574	−0.522871	−	0.852412i	$-0.675139\pi$
$569$	−24.9448	−1.04574	−0.522871	+	0.852412i	$0.675139\pi$
$570$	0	0
$571$	−11.2408	−0.470413	−0.235206	−	0.971945i	$-0.575577\pi$
$571$	−11.2408	−0.470413	−0.235206	+	0.971945i	$0.575577\pi$
$572$	0	0
$573$	1.73921	0.0726564
$574$	0	0
$575$	−69.6777	−2.90576
$576$	0	0
$577$	−32.5782	−1.35625	−0.678124	−	0.734947i	$-0.737207\pi$
$577$	−32.5782	−1.35625	−0.678124	+	0.734947i	$0.737207\pi$
$578$	0	0
$579$	−2.46438	−0.102416
$580$	0	0
$581$	3.82888	0.158849
$582$	0	0
$583$	−15.9317	−0.659825
$584$	0	0
$585$	−18.5089	−0.765249
$586$	0	0
$587$	−25.2451	−1.04198	−0.520988	−	0.853564i	$-0.674436\pi$
$587$	−25.2451	−1.04198	−0.520988	+	0.853564i	$0.674436\pi$
$588$	0	0
$589$	53.7188	2.21344
$590$	0	0
$591$	−14.6400	−0.602211
$592$	0	0
$593$	−17.1283	−0.703375	−0.351688	−	0.936117i	$-0.614392\pi$
$593$	−17.1283	−0.703375	−0.351688	+	0.936117i	$0.614392\pi$
$594$	0	0
$595$	−32.0557	−1.31416
$596$	0	0
$597$	−2.76829	−0.113299
$598$	0	0
$599$	22.8295	0.932787	0.466394	−	0.884577i	$-0.345553\pi$
$599$	22.8295	0.932787	0.466394	+	0.884577i	$0.345553\pi$
$600$	0	0
$601$	10.7578	0.438821	0.219410	−	0.975633i	$-0.429587\pi$
$601$	10.7578	0.438821	0.219410	+	0.975633i	$0.429587\pi$
$602$	0	0
$603$	2.70164	0.110019
$604$	0	0
$605$	−4.38810	−0.178402
$606$	0	0
$607$	20.3748	0.826987	0.413494	−	0.910507i	$-0.364308\pi$
$607$	20.3748	0.826987	0.413494	+	0.910507i	$0.364308\pi$
$608$	0	0
$609$	4.54079	0.184002
$610$	0	0
$611$	22.8230	0.923322
$612$	0	0
$613$	29.9549	1.20987	0.604933	−	0.796277i	$-0.293200\pi$
$613$	29.9549	1.20987	0.604933	+	0.796277i	$0.293200\pi$
$614$	0	0
$615$	−9.50553	−0.383300
$616$	0	0
$617$	−8.18017	−0.329321	−0.164661	−	0.986350i	$-0.552653\pi$
$617$	−8.18017	−0.329321	−0.164661	+	0.986350i	$0.552653\pi$
$618$	0	0
$619$	−5.66485	−0.227690	−0.113845	−	0.993499i	$-0.536317\pi$
$619$	−5.66485	−0.227690	−0.113845	+	0.993499i	$0.536317\pi$
$620$	0	0
$621$	−5.72825	−0.229867
$622$	0	0
$623$	13.5857	0.544300
$624$	0	0
$625$	62.1407	2.48563
$626$	0	0
$627$	15.2721	0.609908
$628$	0	0
$629$	−9.14307	−0.364558
$630$	0	0
$631$	15.9628	0.635469	0.317734	−	0.948180i	$-0.397078\pi$
$631$	15.9628	0.635469	0.317734	+	0.948180i	$0.397078\pi$
$632$	0	0
$633$	11.5671	0.459750
$634$	0	0
$635$	−82.2535	−3.26413
$636$	0	0
$637$	26.4712	1.04883
$638$	0	0
$639$	3.08019	0.121850
$640$	0	0
$641$	30.3348	1.19815	0.599076	−	0.800692i	$-0.295535\pi$
$641$	30.3348	1.19815	0.599076	+	0.800692i	$0.295535\pi$
$642$	0	0
$643$	−26.3000	−1.03717	−0.518586	−	0.855026i	$-0.673541\pi$
$643$	−26.3000	−1.03717	−0.518586	+	0.855026i	$0.673541\pi$
$644$	0	0
$645$	−0.0173769	−0.000684213 0
$646$	0	0
$647$	11.6090	0.456395	0.228197	−	0.973615i	$-0.426717\pi$
$647$	11.6090	0.456395	0.228197	+	0.973615i	$0.426717\pi$
$648$	0	0
$649$	25.9482	1.01855
$650$	0	0
$651$	−11.4977	−0.450631
$652$	0	0
$653$	−38.3792	−1.50189	−0.750947	−	0.660362i	$-0.770403\pi$
$653$	−38.3792	−1.50189	−0.750947	+	0.660362i	$0.770403\pi$
$654$	0	0
$655$	21.1844	0.827743
$656$	0	0
$657$	10.6919	0.417131
$658$	0	0
$659$	10.0104	0.389950	0.194975	−	0.980808i	$-0.437537\pi$
$659$	10.0104	0.389950	0.194975	+	0.980808i	$0.437537\pi$
$660$	0	0
$661$	−0.712609	−0.0277173	−0.0138586	−	0.999904i	$-0.504411\pi$
$661$	−0.712609	−0.0277173	−0.0138586	+	0.999904i	$0.504411\pi$
$662$	0	0
$663$	33.3425	1.29492
$664$	0	0
$665$	−20.8049	−0.806781
$666$	0	0
$667$	−25.0888	−0.971444
$668$	0	0
$669$	0.906902	0.0350629
$670$	0	0
$671$	−6.38182	−0.246368
$672$	0	0
$673$	10.7589	0.414727	0.207363	−	0.978264i	$-0.433512\pi$
$673$	10.7589	0.414727	0.207363	+	0.978264i	$0.433512\pi$
$674$	0	0
$675$	12.1639	0.468188
$676$	0	0
$677$	−35.3284	−1.35778	−0.678891	−	0.734239i	$-0.737539\pi$
$677$	−35.3284	−1.35778	−0.678891	+	0.734239i	$0.737539\pi$
$678$	0	0
$679$	2.68268	0.102952
$680$	0	0
$681$	25.9521	0.994487
$682$	0	0
$683$	−1.52908	−0.0585087	−0.0292544	−	0.999572i	$-0.509313\pi$
$683$	−1.52908	−0.0585087	−0.0292544	+	0.999572i	$0.509313\pi$
$684$	0	0
$685$	−65.5625	−2.50501
$686$	0	0
$687$	−7.53366	−0.287427
$688$	0	0
$689$	−22.5748	−0.860033
$690$	0	0
$691$	48.2878	1.83696	0.918478	−	0.395472i	$-0.129419\pi$
$691$	48.2878	1.83696	0.918478	+	0.395472i	$0.129419\pi$
$692$	0	0
$693$	−3.26876	−0.124170
$694$	0	0
$695$	−22.5481	−0.855296
$696$	0	0
$697$	17.1236	0.648601
$698$	0	0
$699$	12.0894	0.457263
$700$	0	0
$701$	16.5715	0.625896	0.312948	−	0.949770i	$-0.398683\pi$
$701$	16.5715	0.625896	0.312948	+	0.949770i	$0.398683\pi$
$702$	0	0
$703$	−5.93407	−0.223808
$704$	0	0
$705$	−21.1645	−0.797102
$706$	0	0
$707$	−19.8228	−0.745514
$708$	0	0
$709$	43.3937	1.62969	0.814843	−	0.579682i	$-0.196823\pi$
$709$	43.3937	1.62969	0.814843	+	0.579682i	$0.196823\pi$
$710$	0	0
$711$	14.2639	0.534937
$712$	0	0
$713$	63.5274	2.37912
$714$	0	0
$715$	58.3569	2.18242
$716$	0	0
$717$	24.7634	0.924806
$718$	0	0
$719$	−18.1916	−0.678433	−0.339217	−	0.940708i	$-0.610162\pi$
$719$	−18.1916	−0.678433	−0.339217	+	0.940708i	$0.610162\pi$
$720$	0	0
$721$	−7.02104	−0.261477
$722$	0	0
$723$	19.3715	0.720435
$724$	0	0
$725$	53.2760	1.97862
$726$	0	0
$727$	34.2077	1.26869	0.634347	−	0.773048i	$-0.281269\pi$
$727$	34.2077	1.26869	0.634347	+	0.773048i	$0.281269\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.0313032	0.00115779
$732$	0	0
$733$	40.8885	1.51025	0.755125	−	0.655581i	$-0.227576\pi$
$733$	40.8885	1.51025	0.755125	+	0.655581i	$0.227576\pi$
$734$	0	0
$735$	−24.5475	−0.905449
$736$	0	0
$737$	−8.51802	−0.313765
$738$	0	0
$739$	7.17167	0.263814	0.131907	−	0.991262i	$-0.457890\pi$
$739$	7.17167	0.263814	0.131907	+	0.991262i	$0.457890\pi$
$740$	0	0
$741$	21.6401	0.794970
$742$	0	0
$743$	−41.3616	−1.51741	−0.758705	−	0.651435i	$-0.774167\pi$
$743$	−41.3616	−1.51741	−0.758705	+	0.651435i	$0.774167\pi$
$744$	0	0
$745$	−64.8169	−2.37471
$746$	0	0
$747$	3.69318	0.135126
$748$	0	0
$749$	11.7813	0.430479
$750$	0	0
$751$	−49.3180	−1.79964	−0.899820	−	0.436262i	$-0.856302\pi$
$751$	−49.3180	−1.79964	−0.899820	+	0.436262i	$0.856302\pi$
$752$	0	0
$753$	10.2306	0.372825
$754$	0	0
$755$	−5.85062	−0.212926
$756$	0	0
$757$	−2.53486	−0.0921312	−0.0460656	−	0.998938i	$-0.514668\pi$
$757$	−2.53486	−0.0921312	−0.0460656	+	0.998938i	$0.514668\pi$
$758$	0	0
$759$	18.0606	0.655559
$760$	0	0
$761$	−40.7976	−1.47891	−0.739457	−	0.673204i	$-0.764918\pi$
$761$	−40.7976	−1.47891	−0.739457	+	0.673204i	$0.764918\pi$
$762$	0	0
$763$	2.21515	0.0801938
$764$	0	0
$765$	−30.9196	−1.11790
$766$	0	0
$767$	36.7679	1.32761
$768$	0	0
$769$	−3.40923	−0.122940	−0.0614701	−	0.998109i	$-0.519579\pi$
$769$	−3.40923	−0.122940	−0.0614701	+	0.998109i	$0.519579\pi$
$770$	0	0
$771$	−10.9078	−0.392835
$772$	0	0
$773$	24.9168	0.896194	0.448097	−	0.893985i	$-0.352102\pi$
$773$	24.9168	0.896194	0.448097	+	0.893985i	$0.352102\pi$
$774$	0	0
$775$	−134.900	−4.84575
$776$	0	0
$777$	1.27010	0.0455646
$778$	0	0
$779$	11.1136	0.398186
$780$	0	0
$781$	−9.71156	−0.347507
$782$	0	0
$783$	4.37985	0.156523
$784$	0	0
$785$	−95.2566	−3.39986
$786$	0	0
$787$	−19.5963	−0.698534	−0.349267	−	0.937023i	$-0.613569\pi$
$787$	−19.5963	−0.698534	−0.349267	+	0.937023i	$0.613569\pi$
$788$	0	0
$789$	6.25855	0.222810
$790$	0	0
$791$	−14.6606	−0.521269
$792$	0	0
$793$	−9.04287	−0.321122
$794$	0	0
$795$	20.9344	0.742465
$796$	0	0
$797$	−23.0064	−0.814927	−0.407464	−	0.913221i	$-0.633587\pi$
$797$	−23.0064	−0.814927	−0.407464	+	0.913221i	$0.633587\pi$
$798$	0	0
$799$	38.1265	1.34882
$800$	0	0
$801$	13.1042	0.463014
$802$	0	0
$803$	−33.7106	−1.18962
$804$	0	0
$805$	−24.6038	−0.867169
$806$	0	0
$807$	19.5130	0.686889
$808$	0	0
$809$	−34.6584	−1.21852	−0.609262	−	0.792969i	$-0.708534\pi$
$809$	−34.6584	−1.21852	−0.609262	+	0.792969i	$0.708534\pi$
$810$	0	0
$811$	16.0577	0.563863	0.281931	−	0.959435i	$-0.409025\pi$
$811$	16.0577	0.563863	0.281931	+	0.959435i	$0.409025\pi$
$812$	0	0
$813$	−2.09461	−0.0734613
$814$	0	0
$815$	2.21611	0.0776269
$816$	0	0
$817$	0.0203166	0.000710786 0
$818$	0	0
$819$	−4.63175	−0.161846
$820$	0	0
$821$	−39.9424	−1.39400	−0.696999	−	0.717072i	$-0.745482\pi$
$821$	−39.9424	−1.39400	−0.696999	+	0.717072i	$0.745482\pi$
$822$	0	0
$823$	−40.6350	−1.41645	−0.708223	−	0.705989i	$-0.750503\pi$
$823$	−40.6350	−1.41645	−0.708223	+	0.705989i	$0.750503\pi$
$824$	0	0
$825$	−38.3516	−1.33523
$826$	0	0
$827$	−35.2313	−1.22511	−0.612556	−	0.790428i	$-0.709858\pi$
$827$	−35.2313	−1.22511	−0.612556	+	0.790428i	$0.709858\pi$
$828$	0	0
$829$	−7.26479	−0.252317	−0.126158	−	0.992010i	$-0.540265\pi$
$829$	−7.26479	−0.252317	−0.126158	+	0.992010i	$0.540265\pi$
$830$	0	0
$831$	15.7238	0.545451
$832$	0	0
$833$	44.2207	1.53216
$834$	0	0
$835$	−4.14293	−0.143372
$836$	0	0
$837$	−11.0902	−0.383333
$838$	0	0
$839$	18.5154	0.639223	0.319611	−	0.947549i	$-0.396448\pi$
$839$	18.5154	0.639223	0.319611	+	0.947549i	$0.396448\pi$
$840$	0	0
$841$	−9.81693	−0.338515
$842$	0	0
$843$	4.80654	0.165546
$844$	0	0
$845$	28.8320	0.991852
$846$	0	0
$847$	−1.09810	−0.0377311
$848$	0	0
$849$	−8.76619	−0.300855
$850$	0	0
$851$	−7.01759	−0.240560
$852$	0	0
$853$	−19.6985	−0.674465	−0.337232	−	0.941421i	$-0.609491\pi$
$853$	−19.6985	−0.674465	−0.337232	+	0.941421i	$0.609491\pi$
$854$	0	0
$855$	−20.0676	−0.686296
$856$	0	0
$857$	41.7550	1.42632	0.713161	−	0.701000i	$-0.247263\pi$
$857$	41.7550	1.42632	0.713161	+	0.701000i	$0.247263\pi$
$858$	0	0
$859$	−1.33973	−0.0457112	−0.0228556	−	0.999739i	$-0.507276\pi$
$859$	−1.33973	−0.0457112	−0.0228556	+	0.999739i	$0.507276\pi$
$860$	0	0
$861$	−2.37870	−0.0810660
$862$	0	0
$863$	5.12540	0.174470	0.0872352	−	0.996188i	$-0.472197\pi$
$863$	5.12540	0.174470	0.0872352	+	0.996188i	$0.472197\pi$
$864$	0	0
$865$	61.4249	2.08851
$866$	0	0
$867$	38.6995	1.31431
$868$	0	0
$869$	−44.9726	−1.52559
$870$	0	0
$871$	−12.0698	−0.408970
$872$	0	0
$873$	2.58759	0.0875768
$874$	0	0
$875$	30.7701	1.04022
$876$	0	0
$877$	24.9158	0.841348	0.420674	−	0.907212i	$-0.361794\pi$
$877$	24.9158	0.841348	0.420674	+	0.907212i	$0.361794\pi$
$878$	0	0
$879$	11.2792	0.380440
$880$	0	0
$881$	4.58480	0.154466	0.0772330	−	0.997013i	$-0.475391\pi$
$881$	4.58480	0.154466	0.0772330	+	0.997013i	$0.475391\pi$
$882$	0	0
$883$	−0.991563	−0.0333688	−0.0166844	−	0.999861i	$-0.505311\pi$
$883$	−0.991563	−0.0333688	−0.0166844	+	0.999861i	$0.505311\pi$
$884$	0	0
$885$	−34.0960	−1.14612
$886$	0	0
$887$	6.69971	0.224954	0.112477	−	0.993654i	$-0.464122\pi$
$887$	6.69971	0.224954	0.112477	+	0.993654i	$0.464122\pi$
$888$	0	0
$889$	−20.5835	−0.690347
$890$	0	0
$891$	−3.15291	−0.105626
$892$	0	0
$893$	24.7450	0.828060
$894$	0	0
$895$	−22.7988	−0.762079
$896$	0	0
$897$	25.5914	0.854473
$898$	0	0
$899$	−48.5734	−1.62001
$900$	0	0
$901$	−37.7118	−1.25636
$902$	0	0
$903$	−0.00434846	−0.000144708 0
$904$	0	0
$905$	11.2284	0.373245
$906$	0	0
$907$	36.0972	1.19859	0.599293	−	0.800530i	$-0.295448\pi$
$907$	36.0972	1.19859	0.599293	+	0.800530i	$0.295448\pi$
$908$	0	0
$909$	−19.1202	−0.634178
$910$	0	0
$911$	−25.8299	−0.855783	−0.427891	−	0.903830i	$-0.640743\pi$
$911$	−25.8299	−0.855783	−0.427891	+	0.903830i	$0.640743\pi$
$912$	0	0
$913$	−11.6442	−0.385368
$914$	0	0
$915$	8.38574	0.277224
$916$	0	0
$917$	5.30128	0.175064
$918$	0	0
$919$	50.1478	1.65422	0.827111	−	0.562038i	$-0.189983\pi$
$919$	50.1478	1.65422	0.827111	+	0.562038i	$0.189983\pi$
$920$	0	0
$921$	−0.138289	−0.00455677
$922$	0	0
$923$	−13.7610	−0.452950
$924$	0	0
$925$	14.9018	0.489968
$926$	0	0
$927$	−6.77219	−0.222428
$928$	0	0
$929$	−26.6182	−0.873316	−0.436658	−	0.899628i	$-0.643838\pi$
$929$	−26.6182	−0.873316	−0.436658	+	0.899628i	$0.643838\pi$
$930$	0	0
$931$	28.7003	0.940615
$932$	0	0
$933$	−9.20714	−0.301428
$934$	0	0
$935$	97.4866	3.18815
$936$	0	0
$937$	30.7296	1.00389	0.501947	−	0.864899i	$-0.332617\pi$
$937$	30.7296	1.00389	0.501947	+	0.864899i	$0.332617\pi$
$938$	0	0
$939$	−27.8661	−0.909374
$940$	0	0
$941$	−27.7925	−0.906009	−0.453004	−	0.891508i	$-0.649648\pi$
$941$	−27.7925	−0.906009	−0.453004	+	0.891508i	$0.649648\pi$
$942$	0	0
$943$	13.1429	0.427991
$944$	0	0
$945$	4.29517	0.139722
$946$	0	0
$947$	−4.51582	−0.146744	−0.0733722	−	0.997305i	$-0.523376\pi$
$947$	−4.51582	−0.146744	−0.0733722	+	0.997305i	$0.523376\pi$
$948$	0	0
$949$	−47.7670	−1.55058
$950$	0	0
$951$	3.46368	0.112317
$952$	0	0
$953$	25.9218	0.839690	0.419845	−	0.907596i	$-0.362084\pi$
$953$	25.9218	0.839690	0.419845	+	0.907596i	$0.362084\pi$
$954$	0	0
$955$	7.20542	0.233162
$956$	0	0
$957$	−13.8093	−0.446390
$958$	0	0
$959$	−16.4066	−0.529798
$960$	0	0
$961$	91.9924	2.96750
$962$	0	0
$963$	11.3637	0.366191
$964$	0	0
$965$	−10.2098	−0.328664
$966$	0	0
$967$	−4.56606	−0.146835	−0.0734173	−	0.997301i	$-0.523390\pi$
$967$	−4.56606	−0.146835	−0.0734173	+	0.997301i	$0.523390\pi$
$968$	0	0
$969$	36.1504	1.16132
$970$	0	0
$971$	−53.9165	−1.73026	−0.865131	−	0.501546i	$-0.832765\pi$
$971$	−53.9165	−1.73026	−0.865131	+	0.501546i	$0.832765\pi$
$972$	0	0
$973$	−5.64252	−0.180891
$974$	0	0
$975$	−54.3432	−1.74038
$976$	0	0
$977$	−52.1129	−1.66724	−0.833619	−	0.552340i	$-0.813735\pi$
$977$	−52.1129	−1.66724	−0.833619	+	0.552340i	$0.813735\pi$
$978$	0	0
$979$	−41.3163	−1.32048
$980$	0	0
$981$	2.13664	0.0682176
$982$	0	0
$983$	31.4212	1.00218	0.501090	−	0.865395i	$-0.332933\pi$
$983$	31.4212	1.00218	0.501090	+	0.865395i	$0.332933\pi$
$984$	0	0
$985$	−60.6527	−1.93256
$986$	0	0
$987$	−5.29630	−0.168583
$988$	0	0
$989$	0.0240262	0.000763989 0
$990$	0	0
$991$	−46.8107	−1.48699	−0.743495	−	0.668741i	$-0.766834\pi$
$991$	−46.8107	−1.48699	−0.743495	+	0.668741i	$0.766834\pi$
$992$	0	0
$993$	10.3902	0.329724
$994$	0	0
$995$	−11.4688	−0.363587
$996$	0	0
$997$	54.3626	1.72168	0.860841	−	0.508874i	$-0.169938\pi$
$997$	54.3626	1.72168	0.860841	+	0.508874i	$0.169938\pi$
$998$	0	0
$999$	1.22509	0.0387600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.z.1.8		8
4.3	odd	2		501.2.a.d.1.4	✓	8
12.11	even	2		1503.2.a.f.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.d.1.4	✓	8	4.3	odd	2
1503.2.a.f.1.5		8	12.11	even	2
8016.2.a.z.1.8		8	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.14293 q^{5} +1.03675 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.14293 q^{5} +1.03675 q^{7} +1.00000 q^{9} -3.15291 q^{11} -4.46759 q^{13} +4.14293 q^{15} -7.46321 q^{17} -4.84381 q^{19} +1.03675 q^{21} -5.72825 q^{23} +12.1639 q^{25} +1.00000 q^{27} +4.37985 q^{29} -11.0902 q^{31} -3.15291 q^{33} +4.29517 q^{35} +1.22509 q^{37} -4.46759 q^{39} -2.29440 q^{41} -0.00419434 q^{43} +4.14293 q^{45} -5.10859 q^{47} -5.92516 q^{49} -7.46321 q^{51} +5.05303 q^{53} -13.0623 q^{55} -4.84381 q^{57} -8.22992 q^{59} +2.02411 q^{61} +1.03675 q^{63} -18.5089 q^{65} +2.70164 q^{67} -5.72825 q^{69} +3.08019 q^{71} +10.6919 q^{73} +12.1639 q^{75} -3.26876 q^{77} +14.2639 q^{79} +1.00000 q^{81} +3.69318 q^{83} -30.9196 q^{85} +4.37985 q^{87} +13.1042 q^{89} -4.63175 q^{91} -11.0902 q^{93} -20.0676 q^{95} +2.58759 q^{97} -3.15291 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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