LMFDB - Embedded newform 8016.2.a.p.1.2

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$-1.55629$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.33576 q^{5} +0.969164 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.33576 q^{5} +0.969164 q^{7} +1.00000 q^{9} -2.44833 q^{11} +5.19697 q^{13} +3.33576 q^{15} -5.91155 q^{17} +3.40252 q^{19} -0.969164 q^{21} +3.01424 q^{23} +6.12730 q^{25} -1.00000 q^{27} -4.08222 q^{29} +4.76257 q^{31} +2.44833 q^{33} -3.23290 q^{35} -2.59627 q^{37} -5.19697 q^{39} -11.6108 q^{41} +0.104212 q^{43} -3.33576 q^{45} +4.12869 q^{47} -6.06072 q^{49} +5.91155 q^{51} +13.1601 q^{53} +8.16705 q^{55} -3.40252 q^{57} +4.68035 q^{59} -7.22848 q^{61} +0.969164 q^{63} -17.3359 q^{65} +15.7523 q^{67} -3.01424 q^{69} -16.0762 q^{71} +0.416457 q^{73} -6.12730 q^{75} -2.37284 q^{77} -4.60694 q^{79} +1.00000 q^{81} +4.27748 q^{83} +19.7195 q^{85} +4.08222 q^{87} -14.0307 q^{89} +5.03672 q^{91} -4.76257 q^{93} -11.3500 q^{95} -2.73252 q^{97} -2.44833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 9 * q^5 + 4 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9} + 15 q^{11} + 9 q^{15} - 11 q^{17} + 16 q^{19} - 4 q^{21} + 9 q^{23} + 6 q^{25} - 5 q^{27} - q^{29} + 18 q^{31} - 15 q^{33} + 4 q^{35} + 7 q^{37} - 10 q^{41} - 6 q^{43} - 9 q^{45} + 7 q^{47} + 11 q^{49} + 11 q^{51} - 9 q^{53} - 17 q^{55} - 16 q^{57} + 37 q^{59} - 2 q^{61} + 4 q^{63} - 16 q^{65} - 9 q^{69} - 13 q^{71} - 6 q^{73} - 6 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 5 q^{83} + 29 q^{85} + q^{87} - 30 q^{89} + 33 q^{91} - 18 q^{93} - 43 q^{95} - 9 q^{97} + 15 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 9 * q^5 + 4 * q^7 + 5 * q^9 + 15 * q^11 + 9 * q^15 - 11 * q^17 + 16 * q^19 - 4 * q^21 + 9 * q^23 + 6 * q^25 - 5 * q^27 - q^29 + 18 * q^31 - 15 * q^33 + 4 * q^35 + 7 * q^37 - 10 * q^41 - 6 * q^43 - 9 * q^45 + 7 * q^47 + 11 * q^49 + 11 * q^51 - 9 * q^53 - 17 * q^55 - 16 * q^57 + 37 * q^59 - 2 * q^61 + 4 * q^63 - 16 * q^65 - 9 * q^69 - 13 * q^71 - 6 * q^73 - 6 * q^75 - 2 * q^77 - 14 * q^79 + 5 * q^81 + 5 * q^83 + 29 * q^85 + q^87 - 30 * q^89 + 33 * q^91 - 18 * q^93 - 43 * q^95 - 9 * q^97 + 15 * 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
$5$	−3.33576	−1.49180	−0.745899	−	0.666059i	$-0.767980\pi$
$5$	−3.33576	−1.49180	−0.745899	+	0.666059i	$0.767980\pi$
$6$	0	0
$7$	0.969164	0.366309	0.183155	−	0.983084i	$-0.441369\pi$
$7$	0.969164	0.366309	0.183155	+	0.983084i	$0.441369\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.44833	−0.738200	−0.369100	−	0.929390i	$-0.620334\pi$
$11$	−2.44833	−0.738200	−0.369100	+	0.929390i	$0.620334\pi$
$12$	0	0
$13$	5.19697	1.44138	0.720690	−	0.693257i	$-0.243825\pi$
$13$	5.19697	1.44138	0.720690	+	0.693257i	$0.243825\pi$
$14$	0	0
$15$	3.33576	0.861290
$16$	0	0
$17$	−5.91155	−1.43376	−0.716881	−	0.697196i	$-0.754431\pi$
$17$	−5.91155	−1.43376	−0.716881	+	0.697196i	$0.754431\pi$
$18$	0	0
$19$	3.40252	0.780592	0.390296	−	0.920689i	$-0.372373\pi$
$19$	3.40252	0.780592	0.390296	+	0.920689i	$0.372373\pi$
$20$	0	0
$21$	−0.969164	−0.211489
$22$	0	0
$23$	3.01424	0.628513	0.314256	−	0.949338i	$-0.398245\pi$
$23$	3.01424	0.628513	0.314256	+	0.949338i	$0.398245\pi$
$24$	0	0
$25$	6.12730	1.22546
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.08222	−0.758049	−0.379024	−	0.925387i	$-0.623740\pi$
$29$	−4.08222	−0.758049	−0.379024	+	0.925387i	$0.623740\pi$
$30$	0	0
$31$	4.76257	0.855383	0.427691	−	0.903925i	$-0.359327\pi$
$31$	4.76257	0.855383	0.427691	+	0.903925i	$0.359327\pi$
$32$	0	0
$33$	2.44833	0.426200
$34$	0	0
$35$	−3.23290	−0.546460
$36$	0	0
$37$	−2.59627	−0.426825	−0.213412	−	0.976962i	$-0.568458\pi$
$37$	−2.59627	−0.426825	−0.213412	+	0.976962i	$0.568458\pi$
$38$	0	0
$39$	−5.19697	−0.832181
$40$	0	0
$41$	−11.6108	−1.81330	−0.906649	−	0.421886i	$-0.861368\pi$
$41$	−11.6108	−1.81330	−0.906649	+	0.421886i	$0.861368\pi$
$42$	0	0
$43$	0.104212	0.0158921	0.00794606	−	0.999968i	$-0.497471\pi$
$43$	0.104212	0.0158921	0.00794606	+	0.999968i	$0.497471\pi$
$44$	0	0
$45$	−3.33576	−0.497266
$46$	0	0
$47$	4.12869	0.602231	0.301115	−	0.953588i	$-0.402641\pi$
$47$	4.12869	0.602231	0.301115	+	0.953588i	$0.402641\pi$
$48$	0	0
$49$	−6.06072	−0.865817
$50$	0	0
$51$	5.91155	0.827783
$52$	0	0
$53$	13.1601	1.80768	0.903839	−	0.427873i	$-0.140737\pi$
$53$	13.1601	1.80768	0.903839	+	0.427873i	$0.140737\pi$
$54$	0	0
$55$	8.16705	1.10125
$56$	0	0
$57$	−3.40252	−0.450675
$58$	0	0
$59$	4.68035	0.609330	0.304665	−	0.952460i	$-0.401455\pi$
$59$	4.68035	0.609330	0.304665	+	0.952460i	$0.401455\pi$
$60$	0	0
$61$	−7.22848	−0.925512	−0.462756	−	0.886486i	$-0.653139\pi$
$61$	−7.22848	−0.925512	−0.462756	+	0.886486i	$0.653139\pi$
$62$	0	0
$63$	0.969164	0.122103
$64$	0	0
$65$	−17.3359	−2.15025
$66$	0	0
$67$	15.7523	1.92445	0.962225	−	0.272254i	$-0.0877692\pi$
$67$	15.7523	1.92445	0.962225	+	0.272254i	$0.0877692\pi$
$68$	0	0
$69$	−3.01424	−0.362872
$70$	0	0
$71$	−16.0762	−1.90789	−0.953946	−	0.299980i	$-0.903020\pi$
$71$	−16.0762	−1.90789	−0.953946	+	0.299980i	$0.903020\pi$
$72$	0	0
$73$	0.416457	0.0487426	0.0243713	−	0.999703i	$-0.492242\pi$
$73$	0.416457	0.0487426	0.0243713	+	0.999703i	$0.492242\pi$
$74$	0	0
$75$	−6.12730	−0.707519
$76$	0	0
$77$	−2.37284	−0.270410
$78$	0	0
$79$	−4.60694	−0.518321	−0.259160	−	0.965834i	$-0.583446\pi$
$79$	−4.60694	−0.518321	−0.259160	+	0.965834i	$0.583446\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.27748	0.469514	0.234757	−	0.972054i	$-0.424570\pi$
$83$	4.27748	0.469514	0.234757	+	0.972054i	$0.424570\pi$
$84$	0	0
$85$	19.7195	2.13888
$86$	0	0
$87$	4.08222	0.437660
$88$	0	0
$89$	−14.0307	−1.48725	−0.743624	−	0.668598i	$-0.766895\pi$
$89$	−14.0307	−1.48725	−0.743624	+	0.668598i	$0.766895\pi$
$90$	0	0
$91$	5.03672	0.527991
$92$	0	0
$93$	−4.76257	−0.493856
$94$	0	0
$95$	−11.3500	−1.16449
$96$	0	0
$97$	−2.73252	−0.277446	−0.138723	−	0.990331i	$-0.544300\pi$
$97$	−2.73252	−0.277446	−0.138723	+	0.990331i	$0.544300\pi$
$98$	0	0
$99$	−2.44833	−0.246067
$100$	0	0
$101$	4.29904	0.427771	0.213885	−	0.976859i	$-0.431388\pi$
$101$	4.29904	0.427771	0.213885	+	0.976859i	$0.431388\pi$
$102$	0	0
$103$	−3.01050	−0.296633	−0.148317	−	0.988940i	$-0.547386\pi$
$103$	−3.01050	−0.296633	−0.148317	+	0.988940i	$0.547386\pi$
$104$	0	0
$105$	3.23290	0.315499
$106$	0	0
$107$	−14.4071	−1.39279	−0.696396	−	0.717658i	$-0.745214\pi$
$107$	−14.4071	−1.39279	−0.696396	+	0.717658i	$0.745214\pi$
$108$	0	0
$109$	−4.59762	−0.440373	−0.220186	−	0.975458i	$-0.570667\pi$
$109$	−4.59762	−0.440373	−0.220186	+	0.975458i	$0.570667\pi$
$110$	0	0
$111$	2.59627	0.246427
$112$	0	0
$113$	1.31329	0.123544	0.0617718	−	0.998090i	$-0.480325\pi$
$113$	1.31329	0.123544	0.0617718	+	0.998090i	$0.480325\pi$
$114$	0	0
$115$	−10.0548	−0.937614
$116$	0	0
$117$	5.19697	0.480460
$118$	0	0
$119$	−5.72926	−0.525201
$120$	0	0
$121$	−5.00566	−0.455060
$122$	0	0
$123$	11.6108	1.04691
$124$	0	0
$125$	−3.76039	−0.336340
$126$	0	0
$127$	−4.73361	−0.420040	−0.210020	−	0.977697i	$-0.567353\pi$
$127$	−4.73361	−0.420040	−0.210020	+	0.977697i	$0.567353\pi$
$128$	0	0
$129$	−0.104212	−0.00917531
$130$	0	0
$131$	13.8298	1.20832	0.604159	−	0.796863i	$-0.293509\pi$
$131$	13.8298	1.20832	0.604159	+	0.796863i	$0.293509\pi$
$132$	0	0
$133$	3.29760	0.285938
$134$	0	0
$135$	3.33576	0.287097
$136$	0	0
$137$	−2.17605	−0.185912	−0.0929562	−	0.995670i	$-0.529632\pi$
$137$	−2.17605	−0.185912	−0.0929562	+	0.995670i	$0.529632\pi$
$138$	0	0
$139$	−2.38423	−0.202228	−0.101114	−	0.994875i	$-0.532241\pi$
$139$	−2.38423	−0.202228	−0.101114	+	0.994875i	$0.532241\pi$
$140$	0	0
$141$	−4.12869	−0.347698
$142$	0	0
$143$	−12.7239	−1.06403
$144$	0	0
$145$	13.6173	1.13086
$146$	0	0
$147$	6.06072	0.499880
$148$	0	0
$149$	3.20144	0.262273	0.131136	−	0.991364i	$-0.458137\pi$
$149$	3.20144	0.262273	0.131136	+	0.991364i	$0.458137\pi$
$150$	0	0
$151$	−4.11554	−0.334918	−0.167459	−	0.985879i	$-0.553556\pi$
$151$	−4.11554	−0.334918	−0.167459	+	0.985879i	$0.553556\pi$
$152$	0	0
$153$	−5.91155	−0.477921
$154$	0	0
$155$	−15.8868	−1.27606
$156$	0	0
$157$	9.55872	0.762869	0.381434	−	0.924396i	$-0.375430\pi$
$157$	9.55872	0.762869	0.381434	+	0.924396i	$0.375430\pi$
$158$	0	0
$159$	−13.1601	−1.04366
$160$	0	0
$161$	2.92129	0.230230
$162$	0	0
$163$	7.26409	0.568968	0.284484	−	0.958681i	$-0.408178\pi$
$163$	7.26409	0.568968	0.284484	+	0.958681i	$0.408178\pi$
$164$	0	0
$165$	−8.16705	−0.635804
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	14.0085	1.07758
$170$	0	0
$171$	3.40252	0.260197
$172$	0	0
$173$	24.1126	1.83325	0.916624	−	0.399751i	$-0.130903\pi$
$173$	24.1126	1.83325	0.916624	+	0.399751i	$0.130903\pi$
$174$	0	0
$175$	5.93835	0.448897
$176$	0	0
$177$	−4.68035	−0.351797
$178$	0	0
$179$	−21.1778	−1.58290	−0.791450	−	0.611234i	$-0.790673\pi$
$179$	−21.1778	−1.58290	−0.791450	+	0.611234i	$0.790673\pi$
$180$	0	0
$181$	4.43611	0.329733	0.164867	−	0.986316i	$-0.447281\pi$
$181$	4.43611	0.329733	0.164867	+	0.986316i	$0.447281\pi$
$182$	0	0
$183$	7.22848	0.534344
$184$	0	0
$185$	8.66055	0.636736
$186$	0	0
$187$	14.4735	1.05840
$188$	0	0
$189$	−0.969164	−0.0704963
$190$	0	0
$191$	24.2251	1.75287	0.876434	−	0.481522i	$-0.159916\pi$
$191$	24.2251	1.75287	0.876434	+	0.481522i	$0.159916\pi$
$192$	0	0
$193$	2.41344	0.173723	0.0868615	−	0.996220i	$-0.472316\pi$
$193$	2.41344	0.173723	0.0868615	+	0.996220i	$0.472316\pi$
$194$	0	0
$195$	17.3359	1.24145
$196$	0	0
$197$	−17.7201	−1.26250	−0.631252	−	0.775578i	$-0.717459\pi$
$197$	−17.7201	−1.26250	−0.631252	+	0.775578i	$0.717459\pi$
$198$	0	0
$199$	−14.3471	−1.01704	−0.508518	−	0.861051i	$-0.669806\pi$
$199$	−14.3471	−1.01704	−0.508518	+	0.861051i	$0.669806\pi$
$200$	0	0
$201$	−15.7523	−1.11108
$202$	0	0
$203$	−3.95634	−0.277680
$204$	0	0
$205$	38.7307	2.70507
$206$	0	0
$207$	3.01424	0.209504
$208$	0	0
$209$	−8.33051	−0.576234
$210$	0	0
$211$	17.5096	1.20541	0.602704	−	0.797965i	$-0.294090\pi$
$211$	17.5096	1.20541	0.602704	+	0.797965i	$0.294090\pi$
$212$	0	0
$213$	16.0762	1.10152
$214$	0	0
$215$	−0.347625	−0.0237078
$216$	0	0
$217$	4.61571	0.313335
$218$	0	0
$219$	−0.416457	−0.0281416
$220$	0	0
$221$	−30.7222	−2.06660
$222$	0	0
$223$	4.16352	0.278810	0.139405	−	0.990235i	$-0.455481\pi$
$223$	4.16352	0.278810	0.139405	+	0.990235i	$0.455481\pi$
$224$	0	0
$225$	6.12730	0.408486
$226$	0	0
$227$	2.87488	0.190813	0.0954063	−	0.995438i	$-0.469585\pi$
$227$	2.87488	0.190813	0.0954063	+	0.995438i	$0.469585\pi$
$228$	0	0
$229$	−27.4453	−1.81364	−0.906818	−	0.421522i	$-0.861496\pi$
$229$	−27.4453	−1.81364	−0.906818	+	0.421522i	$0.861496\pi$
$230$	0	0
$231$	2.37284	0.156121
$232$	0	0
$233$	20.0204	1.31158	0.655790	−	0.754943i	$-0.272336\pi$
$233$	20.0204	1.31158	0.655790	+	0.754943i	$0.272336\pi$
$234$	0	0
$235$	−13.7723	−0.898406
$236$	0	0
$237$	4.60694	0.299253
$238$	0	0
$239$	19.5149	1.26231	0.631157	−	0.775655i	$-0.282581\pi$
$239$	19.5149	1.26231	0.631157	+	0.775655i	$0.282581\pi$
$240$	0	0
$241$	23.0425	1.48430	0.742148	−	0.670237i	$-0.233807\pi$
$241$	23.0425	1.48430	0.742148	+	0.670237i	$0.233807\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	20.2171	1.29162
$246$	0	0
$247$	17.6828	1.12513
$248$	0	0
$249$	−4.27748	−0.271074
$250$	0	0
$251$	27.0836	1.70950	0.854752	−	0.519036i	$-0.173709\pi$
$251$	27.0836	1.70950	0.854752	+	0.519036i	$0.173709\pi$
$252$	0	0
$253$	−7.37987	−0.463969
$254$	0	0
$255$	−19.7195	−1.23488
$256$	0	0
$257$	25.1437	1.56842	0.784210	−	0.620495i	$-0.213068\pi$
$257$	25.1437	1.56842	0.784210	+	0.620495i	$0.213068\pi$
$258$	0	0
$259$	−2.51622	−0.156350
$260$	0	0
$261$	−4.08222	−0.252683
$262$	0	0
$263$	9.01981	0.556186	0.278093	−	0.960554i	$-0.410298\pi$
$263$	9.01981	0.556186	0.278093	+	0.960554i	$0.410298\pi$
$264$	0	0
$265$	−43.8989	−2.69669
$266$	0	0
$267$	14.0307	0.858663
$268$	0	0
$269$	17.3775	1.05952	0.529762	−	0.848146i	$-0.322281\pi$
$269$	17.3775	1.05952	0.529762	+	0.848146i	$0.322281\pi$
$270$	0	0
$271$	−7.82525	−0.475350	−0.237675	−	0.971345i	$-0.576385\pi$
$271$	−7.82525	−0.475350	−0.237675	+	0.971345i	$0.576385\pi$
$272$	0	0
$273$	−5.03672	−0.304836
$274$	0	0
$275$	−15.0017	−0.904635
$276$	0	0
$277$	−9.82635	−0.590408	−0.295204	−	0.955434i	$-0.595388\pi$
$277$	−9.82635	−0.590408	−0.295204	+	0.955434i	$0.595388\pi$
$278$	0	0
$279$	4.76257	0.285128
$280$	0	0
$281$	8.18040	0.488002	0.244001	−	0.969775i	$-0.421540\pi$
$281$	8.18040	0.488002	0.244001	+	0.969775i	$0.421540\pi$
$282$	0	0
$283$	−19.0666	−1.13339	−0.566695	−	0.823928i	$-0.691778\pi$
$283$	−19.0666	−1.13339	−0.566695	+	0.823928i	$0.691778\pi$
$284$	0	0
$285$	11.3500	0.672316
$286$	0	0
$287$	−11.2527	−0.664228
$288$	0	0
$289$	17.9464	1.05567
$290$	0	0
$291$	2.73252	0.160183
$292$	0	0
$293$	−6.89297	−0.402691	−0.201346	−	0.979520i	$-0.564531\pi$
$293$	−6.89297	−0.402691	−0.201346	+	0.979520i	$0.564531\pi$
$294$	0	0
$295$	−15.6125	−0.908997
$296$	0	0
$297$	2.44833	0.142067
$298$	0	0
$299$	15.6649	0.905926
$300$	0	0
$301$	0.100998	0.00582143
$302$	0	0
$303$	−4.29904	−0.246974
$304$	0	0
$305$	24.1125	1.38068
$306$	0	0
$307$	0.611159	0.0348807	0.0174403	−	0.999848i	$-0.494448\pi$
$307$	0.611159	0.0348807	0.0174403	+	0.999848i	$0.494448\pi$
$308$	0	0
$309$	3.01050	0.171261
$310$	0	0
$311$	10.7954	0.612151	0.306075	−	0.952007i	$-0.400984\pi$
$311$	10.7954	0.612151	0.306075	+	0.952007i	$0.400984\pi$
$312$	0	0
$313$	−23.7741	−1.34379	−0.671895	−	0.740646i	$-0.734520\pi$
$313$	−23.7741	−1.34379	−0.671895	+	0.740646i	$0.734520\pi$
$314$	0	0
$315$	−3.23290	−0.182153
$316$	0	0
$317$	19.3315	1.08577	0.542884	−	0.839808i	$-0.317332\pi$
$317$	19.3315	1.08577	0.542884	+	0.839808i	$0.317332\pi$
$318$	0	0
$319$	9.99463	0.559592
$320$	0	0
$321$	14.4071	0.804128
$322$	0	0
$323$	−20.1142	−1.11918
$324$	0	0
$325$	31.8434	1.76635
$326$	0	0
$327$	4.59762	0.254249
$328$	0	0
$329$	4.00137	0.220603
$330$	0	0
$331$	13.1764	0.724242	0.362121	−	0.932131i	$-0.382053\pi$
$331$	13.1764	0.724242	0.362121	+	0.932131i	$0.382053\pi$
$332$	0	0
$333$	−2.59627	−0.142275
$334$	0	0
$335$	−52.5459	−2.87089
$336$	0	0
$337$	6.38913	0.348038	0.174019	−	0.984742i	$-0.444325\pi$
$337$	6.38913	0.348038	0.174019	+	0.984742i	$0.444325\pi$
$338$	0	0
$339$	−1.31329	−0.0713279
$340$	0	0
$341$	−11.6604	−0.631444
$342$	0	0
$343$	−12.6580	−0.683467
$344$	0	0
$345$	10.0548	0.541332
$346$	0	0
$347$	−24.4772	−1.31401	−0.657003	−	0.753888i	$-0.728176\pi$
$347$	−24.4772	−1.31401	−0.657003	+	0.753888i	$0.728176\pi$
$348$	0	0
$349$	1.21199	0.0648761	0.0324381	−	0.999474i	$-0.489673\pi$
$349$	1.21199	0.0648761	0.0324381	+	0.999474i	$0.489673\pi$
$350$	0	0
$351$	−5.19697	−0.277394
$352$	0	0
$353$	5.84238	0.310959	0.155479	−	0.987839i	$-0.450308\pi$
$353$	5.84238	0.310959	0.155479	+	0.987839i	$0.450308\pi$
$354$	0	0
$355$	53.6263	2.84619
$356$	0	0
$357$	5.72926	0.303225
$358$	0	0
$359$	0.541678	0.0285887	0.0142943	−	0.999898i	$-0.495450\pi$
$359$	0.541678	0.0285887	0.0142943	+	0.999898i	$0.495450\pi$
$360$	0	0
$361$	−7.42283	−0.390675
$362$	0	0
$363$	5.00566	0.262729
$364$	0	0
$365$	−1.38920	−0.0727141
$366$	0	0
$367$	8.92072	0.465658	0.232829	−	0.972518i	$-0.425202\pi$
$367$	8.92072	0.465658	0.232829	+	0.972518i	$0.425202\pi$
$368$	0	0
$369$	−11.6108	−0.604432
$370$	0	0
$371$	12.7543	0.662170
$372$	0	0
$373$	−11.7179	−0.606728	−0.303364	−	0.952875i	$-0.598110\pi$
$373$	−11.7179	−0.606728	−0.303364	+	0.952875i	$0.598110\pi$
$374$	0	0
$375$	3.76039	0.194186
$376$	0	0
$377$	−21.2152	−1.09264
$378$	0	0
$379$	22.6760	1.16479	0.582394	−	0.812907i	$-0.302116\pi$
$379$	22.6760	1.16479	0.582394	+	0.812907i	$0.302116\pi$
$380$	0	0
$381$	4.73361	0.242510
$382$	0	0
$383$	22.5353	1.15150	0.575750	−	0.817626i	$-0.304710\pi$
$383$	22.5353	1.15150	0.575750	+	0.817626i	$0.304710\pi$
$384$	0	0
$385$	7.91521	0.403397
$386$	0	0
$387$	0.104212	0.00529737
$388$	0	0
$389$	−29.2087	−1.48094	−0.740469	−	0.672091i	$-0.765396\pi$
$389$	−29.2087	−1.48094	−0.740469	+	0.672091i	$0.765396\pi$
$390$	0	0
$391$	−17.8189	−0.901138
$392$	0	0
$393$	−13.8298	−0.697623
$394$	0	0
$395$	15.3676	0.773230
$396$	0	0
$397$	27.7288	1.39167	0.695834	−	0.718203i	$-0.255035\pi$
$397$	27.7288	1.39167	0.695834	+	0.718203i	$0.255035\pi$
$398$	0	0
$399$	−3.29760	−0.165087
$400$	0	0
$401$	28.0223	1.39936	0.699682	−	0.714454i	$-0.253325\pi$
$401$	28.0223	1.39936	0.699682	+	0.714454i	$0.253325\pi$
$402$	0	0
$403$	24.7509	1.23293
$404$	0	0
$405$	−3.33576	−0.165755
$406$	0	0
$407$	6.35655	0.315082
$408$	0	0
$409$	−6.52801	−0.322789	−0.161395	−	0.986890i	$-0.551599\pi$
$409$	−6.52801	−0.322789	−0.161395	+	0.986890i	$0.551599\pi$
$410$	0	0
$411$	2.17605	0.107337
$412$	0	0
$413$	4.53603	0.223203
$414$	0	0
$415$	−14.2686	−0.700420
$416$	0	0
$417$	2.38423	0.116756
$418$	0	0
$419$	32.8366	1.60417	0.802087	−	0.597208i	$-0.203723\pi$
$419$	32.8366	1.60417	0.802087	+	0.597208i	$0.203723\pi$
$420$	0	0
$421$	16.1037	0.784846	0.392423	−	0.919785i	$-0.371637\pi$
$421$	16.1037	0.784846	0.392423	+	0.919785i	$0.371637\pi$
$422$	0	0
$423$	4.12869	0.200744
$424$	0	0
$425$	−36.2218	−1.75702
$426$	0	0
$427$	−7.00558	−0.339024
$428$	0	0
$429$	12.7239	0.614317
$430$	0	0
$431$	26.1110	1.25772	0.628861	−	0.777518i	$-0.283522\pi$
$431$	26.1110	1.25772	0.628861	+	0.777518i	$0.283522\pi$
$432$	0	0
$433$	37.1094	1.78336	0.891682	−	0.452662i	$-0.149526\pi$
$433$	37.1094	1.78336	0.891682	+	0.452662i	$0.149526\pi$
$434$	0	0
$435$	−13.6173	−0.652900
$436$	0	0
$437$	10.2560	0.490612
$438$	0	0
$439$	−6.09766	−0.291026	−0.145513	−	0.989356i	$-0.546483\pi$
$439$	−6.09766	−0.291026	−0.145513	+	0.989356i	$0.546483\pi$
$440$	0	0
$441$	−6.06072	−0.288606
$442$	0	0
$443$	35.5045	1.68687	0.843435	−	0.537231i	$-0.180530\pi$
$443$	35.5045	1.68687	0.843435	+	0.537231i	$0.180530\pi$
$444$	0	0
$445$	46.8030	2.21867
$446$	0	0
$447$	−3.20144	−0.151423
$448$	0	0
$449$	19.3795	0.914576	0.457288	−	0.889319i	$-0.348821\pi$
$449$	19.3795	0.914576	0.457288	+	0.889319i	$0.348821\pi$
$450$	0	0
$451$	28.4270	1.33858
$452$	0	0
$453$	4.11554	0.193365
$454$	0	0
$455$	−16.8013	−0.787656
$456$	0	0
$457$	34.2585	1.60254	0.801272	−	0.598300i	$-0.204157\pi$
$457$	34.2585	1.60254	0.801272	+	0.598300i	$0.204157\pi$
$458$	0	0
$459$	5.91155	0.275928
$460$	0	0
$461$	16.0234	0.746282	0.373141	−	0.927775i	$-0.378281\pi$
$461$	16.0234	0.746282	0.373141	+	0.927775i	$0.378281\pi$
$462$	0	0
$463$	17.0377	0.791807	0.395903	−	0.918292i	$-0.370431\pi$
$463$	17.0377	0.791807	0.395903	+	0.918292i	$0.370431\pi$
$464$	0	0
$465$	15.8868	0.736732
$466$	0	0
$467$	37.6433	1.74192	0.870962	−	0.491351i	$-0.163497\pi$
$467$	37.6433	1.74192	0.870962	+	0.491351i	$0.163497\pi$
$468$	0	0
$469$	15.2666	0.704945
$470$	0	0
$471$	−9.55872	−0.440442
$472$	0	0
$473$	−0.255145	−0.0117316
$474$	0	0
$475$	20.8483	0.956584
$476$	0	0
$477$	13.1601	0.602559
$478$	0	0
$479$	−16.3203	−0.745693	−0.372847	−	0.927893i	$-0.621618\pi$
$479$	−16.3203	−0.745693	−0.372847	+	0.927893i	$0.621618\pi$
$480$	0	0
$481$	−13.4928	−0.615217
$482$	0	0
$483$	−2.92129	−0.132924
$484$	0	0
$485$	9.11505	0.413893
$486$	0	0
$487$	0.00252894	0.000114597 0	5.72987e−5	−	1.00000i	$-0.499982\pi$
$487$	0.00252894	0.000114597 0	5.72987e−5		1.00000i	$0.499982\pi$
$488$	0	0
$489$	−7.26409	−0.328494
$490$	0	0
$491$	21.8748	0.987194	0.493597	−	0.869691i	$-0.335682\pi$
$491$	21.8748	0.987194	0.493597	+	0.869691i	$0.335682\pi$
$492$	0	0
$493$	24.1322	1.08686
$494$	0	0
$495$	8.16705	0.367082
$496$	0	0
$497$	−15.5804	−0.698879
$498$	0	0
$499$	−25.1081	−1.12399	−0.561996	−	0.827140i	$-0.689966\pi$
$499$	−25.1081	−1.12399	−0.561996	+	0.827140i	$0.689966\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−4.12839	−0.184076	−0.0920379	−	0.995756i	$-0.529338\pi$
$503$	−4.12839	−0.184076	−0.0920379	+	0.995756i	$0.529338\pi$
$504$	0	0
$505$	−14.3406	−0.638147
$506$	0	0
$507$	−14.0085	−0.622140
$508$	0	0
$509$	−33.5438	−1.48680	−0.743400	−	0.668847i	$-0.766788\pi$
$509$	−33.5438	−1.48680	−0.743400	+	0.668847i	$0.766788\pi$
$510$	0	0
$511$	0.403615	0.0178549
$512$	0	0
$513$	−3.40252	−0.150225
$514$	0	0
$515$	10.0423	0.442517
$516$	0	0
$517$	−10.1084	−0.444567
$518$	0	0
$519$	−24.1126	−1.05843
$520$	0	0
$521$	−14.0186	−0.614165	−0.307083	−	0.951683i	$-0.599353\pi$
$521$	−14.0186	−0.614165	−0.307083	+	0.951683i	$0.599353\pi$
$522$	0	0
$523$	26.8292	1.17316	0.586579	−	0.809892i	$-0.300474\pi$
$523$	26.8292	1.17316	0.586579	+	0.809892i	$0.300474\pi$
$524$	0	0
$525$	−5.93835	−0.259171
$526$	0	0
$527$	−28.1542	−1.22642
$528$	0	0
$529$	−13.9143	−0.604971
$530$	0	0
$531$	4.68035	0.203110
$532$	0	0
$533$	−60.3408	−2.61365
$534$	0	0
$535$	48.0588	2.07776
$536$	0	0
$537$	21.1778	0.913888
$538$	0	0
$539$	14.8387	0.639147
$540$	0	0
$541$	36.9842	1.59008	0.795038	−	0.606559i	$-0.207451\pi$
$541$	36.9842	1.59008	0.795038	+	0.606559i	$0.207451\pi$
$542$	0	0
$543$	−4.43611	−0.190372
$544$	0	0
$545$	15.3366	0.656947
$546$	0	0
$547$	−44.0823	−1.88482	−0.942411	−	0.334458i	$-0.891447\pi$
$547$	−44.0823	−1.88482	−0.942411	+	0.334458i	$0.891447\pi$
$548$	0	0
$549$	−7.22848	−0.308504
$550$	0	0
$551$	−13.8898	−0.591727
$552$	0	0
$553$	−4.46488	−0.189866
$554$	0	0
$555$	−8.66055	−0.367620
$556$	0	0
$557$	−35.7082	−1.51301	−0.756503	−	0.653990i	$-0.773094\pi$
$557$	−35.7082	−1.51301	−0.756503	+	0.653990i	$0.773094\pi$
$558$	0	0
$559$	0.541584	0.0229066
$560$	0	0
$561$	−14.4735	−0.611070
$562$	0	0
$563$	1.44378	0.0608481	0.0304241	−	0.999537i	$-0.490314\pi$
$563$	1.44378	0.0608481	0.0304241	+	0.999537i	$0.490314\pi$
$564$	0	0
$565$	−4.38081	−0.184302
$566$	0	0
$567$	0.969164	0.0407011
$568$	0	0
$569$	12.9059	0.541042	0.270521	−	0.962714i	$-0.412804\pi$
$569$	12.9059	0.541042	0.270521	+	0.962714i	$0.412804\pi$
$570$	0	0
$571$	−24.5471	−1.02726	−0.513631	−	0.858011i	$-0.671700\pi$
$571$	−24.5471	−1.02726	−0.513631	+	0.858011i	$0.671700\pi$
$572$	0	0
$573$	−24.2251	−1.01202
$574$	0	0
$575$	18.4692	0.770217
$576$	0	0
$577$	−37.1787	−1.54777	−0.773885	−	0.633326i	$-0.781689\pi$
$577$	−37.1787	−1.54777	−0.773885	+	0.633326i	$0.781689\pi$
$578$	0	0
$579$	−2.41344	−0.100299
$580$	0	0
$581$	4.14558	0.171988
$582$	0	0
$583$	−32.2203	−1.33443
$584$	0	0
$585$	−17.3359	−0.716749
$586$	0	0
$587$	18.9261	0.781162	0.390581	−	0.920569i	$-0.372274\pi$
$587$	18.9261	0.781162	0.390581	+	0.920569i	$0.372274\pi$
$588$	0	0
$589$	16.2048	0.667705
$590$	0	0
$591$	17.7201	0.728907
$592$	0	0
$593$	17.9146	0.735663	0.367832	−	0.929892i	$-0.380100\pi$
$593$	17.9146	0.735663	0.367832	+	0.929892i	$0.380100\pi$
$594$	0	0
$595$	19.1114	0.783493
$596$	0	0
$597$	14.3471	0.587186
$598$	0	0
$599$	−31.2386	−1.27637	−0.638186	−	0.769882i	$-0.720315\pi$
$599$	−31.2386	−1.27637	−0.638186	+	0.769882i	$0.720315\pi$
$600$	0	0
$601$	40.2423	1.64152	0.820759	−	0.571275i	$-0.193551\pi$
$601$	40.2423	1.64152	0.820759	+	0.571275i	$0.193551\pi$
$602$	0	0
$603$	15.7523	0.641484
$604$	0	0
$605$	16.6977	0.678857
$606$	0	0
$607$	−37.8950	−1.53811	−0.769056	−	0.639182i	$-0.779273\pi$
$607$	−37.8950	−1.53811	−0.769056	+	0.639182i	$0.779273\pi$
$608$	0	0
$609$	3.95634	0.160319
$610$	0	0
$611$	21.4567	0.868044
$612$	0	0
$613$	−0.728292	−0.0294154	−0.0147077	−	0.999892i	$-0.504682\pi$
$613$	−0.728292	−0.0294154	−0.0147077	+	0.999892i	$0.504682\pi$
$614$	0	0
$615$	−38.7307	−1.56177
$616$	0	0
$617$	−7.82010	−0.314826	−0.157413	−	0.987533i	$-0.550315\pi$
$617$	−7.82010	−0.314826	−0.157413	+	0.987533i	$0.550315\pi$
$618$	0	0
$619$	20.0838	0.807237	0.403619	−	0.914927i	$-0.367752\pi$
$619$	20.0838	0.807237	0.403619	+	0.914927i	$0.367752\pi$
$620$	0	0
$621$	−3.01424	−0.120957
$622$	0	0
$623$	−13.5980	−0.544793
$624$	0	0
$625$	−18.0927	−0.723709
$626$	0	0
$627$	8.33051	0.332689
$628$	0	0
$629$	15.3480	0.611965
$630$	0	0
$631$	−41.7515	−1.66210	−0.831050	−	0.556198i	$-0.812260\pi$
$631$	−41.7515	−1.66210	−0.831050	+	0.556198i	$0.812260\pi$
$632$	0	0
$633$	−17.5096	−0.695942
$634$	0	0
$635$	15.7902	0.626614
$636$	0	0
$637$	−31.4974	−1.24797
$638$	0	0
$639$	−16.0762	−0.635964
$640$	0	0
$641$	−6.98395	−0.275850	−0.137925	−	0.990443i	$-0.544043\pi$
$641$	−6.98395	−0.275850	−0.137925	+	0.990443i	$0.544043\pi$
$642$	0	0
$643$	18.8710	0.744200	0.372100	−	0.928193i	$-0.378638\pi$
$643$	18.8710	0.744200	0.372100	+	0.928193i	$0.378638\pi$
$644$	0	0
$645$	0.347625	0.0136877
$646$	0	0
$647$	−18.1800	−0.714731	−0.357365	−	0.933965i	$-0.616325\pi$
$647$	−18.1800	−0.714731	−0.357365	+	0.933965i	$0.616325\pi$
$648$	0	0
$649$	−11.4591	−0.449808
$650$	0	0
$651$	−4.61571	−0.180904
$652$	0	0
$653$	−0.763094	−0.0298622	−0.0149311	−	0.999889i	$-0.504753\pi$
$653$	−0.763094	−0.0298622	−0.0149311	+	0.999889i	$0.504753\pi$
$654$	0	0
$655$	−46.1330	−1.80257
$656$	0	0
$657$	0.416457	0.0162475
$658$	0	0
$659$	−37.9349	−1.47773	−0.738867	−	0.673852i	$-0.764639\pi$
$659$	−37.9349	−1.47773	−0.738867	+	0.673852i	$0.764639\pi$
$660$	0	0
$661$	−9.10415	−0.354111	−0.177055	−	0.984201i	$-0.556657\pi$
$661$	−9.10415	−0.354111	−0.177055	+	0.984201i	$0.556657\pi$
$662$	0	0
$663$	30.7222	1.19315
$664$	0	0
$665$	−11.0000	−0.426562
$666$	0	0
$667$	−12.3048	−0.476444
$668$	0	0
$669$	−4.16352	−0.160971
$670$	0	0
$671$	17.6977	0.683213
$672$	0	0
$673$	46.7357	1.80153	0.900764	−	0.434308i	$-0.143007\pi$
$673$	46.7357	1.80153	0.900764	+	0.434308i	$0.143007\pi$
$674$	0	0
$675$	−6.12730	−0.235840
$676$	0	0
$677$	−1.61292	−0.0619896	−0.0309948	−	0.999520i	$-0.509868\pi$
$677$	−1.61292	−0.0619896	−0.0309948	+	0.999520i	$0.509868\pi$
$678$	0	0
$679$	−2.64826	−0.101631
$680$	0	0
$681$	−2.87488	−0.110166
$682$	0	0
$683$	49.6234	1.89879	0.949394	−	0.314089i	$-0.101699\pi$
$683$	49.6234	1.89879	0.949394	+	0.314089i	$0.101699\pi$
$684$	0	0
$685$	7.25878	0.277344
$686$	0	0
$687$	27.4453	1.04710
$688$	0	0
$689$	68.3926	2.60555
$690$	0	0
$691$	−13.8395	−0.526481	−0.263240	−	0.964730i	$-0.584791\pi$
$691$	−13.8395	−0.526481	−0.263240	+	0.964730i	$0.584791\pi$
$692$	0	0
$693$	−2.37284	−0.0901366
$694$	0	0
$695$	7.95322	0.301683
$696$	0	0
$697$	68.6377	2.59984
$698$	0	0
$699$	−20.0204	−0.757241
$700$	0	0
$701$	35.6587	1.34681	0.673405	−	0.739273i	$-0.264831\pi$
$701$	35.6587	1.34681	0.673405	+	0.739273i	$0.264831\pi$
$702$	0	0
$703$	−8.83388	−0.333176
$704$	0	0
$705$	13.7723	0.518695
$706$	0	0
$707$	4.16648	0.156697
$708$	0	0
$709$	−20.3193	−0.763108	−0.381554	−	0.924347i	$-0.624611\pi$
$709$	−20.3193	−0.763108	−0.381554	+	0.924347i	$0.624611\pi$
$710$	0	0
$711$	−4.60694	−0.172774
$712$	0	0
$713$	14.3555	0.537619
$714$	0	0
$715$	42.4439	1.58731
$716$	0	0
$717$	−19.5149	−0.728797
$718$	0	0
$719$	18.5955	0.693496	0.346748	−	0.937958i	$-0.387286\pi$
$719$	18.5955	0.693496	0.346748	+	0.937958i	$0.387286\pi$
$720$	0	0
$721$	−2.91767	−0.108660
$722$	0	0
$723$	−23.0425	−0.856958
$724$	0	0
$725$	−25.0130	−0.928958
$726$	0	0
$727$	−15.3935	−0.570914	−0.285457	−	0.958392i	$-0.592145\pi$
$727$	−15.3935	−0.570914	−0.285457	+	0.958392i	$0.592145\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.616052	−0.0227855
$732$	0	0
$733$	−34.6865	−1.28117	−0.640587	−	0.767886i	$-0.721309\pi$
$733$	−34.6865	−1.28117	−0.640587	+	0.767886i	$0.721309\pi$
$734$	0	0
$735$	−20.2171	−0.745719
$736$	0	0
$737$	−38.5669	−1.42063
$738$	0	0
$739$	34.3612	1.26400	0.631999	−	0.774969i	$-0.282235\pi$
$739$	34.3612	1.26400	0.631999	+	0.774969i	$0.282235\pi$
$740$	0	0
$741$	−17.6828	−0.649594
$742$	0	0
$743$	−7.57759	−0.277995	−0.138997	−	0.990293i	$-0.544388\pi$
$743$	−7.57759	−0.277995	−0.138997	+	0.990293i	$0.544388\pi$
$744$	0	0
$745$	−10.6793	−0.391257
$746$	0	0
$747$	4.27748	0.156505
$748$	0	0
$749$	−13.9629	−0.510193
$750$	0	0
$751$	8.14033	0.297045	0.148522	−	0.988909i	$-0.452548\pi$
$751$	8.14033	0.297045	0.148522	+	0.988909i	$0.452548\pi$
$752$	0	0
$753$	−27.0836	−0.986983
$754$	0	0
$755$	13.7285	0.499630
$756$	0	0
$757$	38.2174	1.38903	0.694517	−	0.719476i	$-0.255618\pi$
$757$	38.2174	1.38903	0.694517	+	0.719476i	$0.255618\pi$
$758$	0	0
$759$	7.37987	0.267872
$760$	0	0
$761$	39.3206	1.42537	0.712685	−	0.701484i	$-0.247479\pi$
$761$	39.3206	1.42537	0.712685	+	0.701484i	$0.247479\pi$
$762$	0	0
$763$	−4.45585	−0.161313
$764$	0	0
$765$	19.7195	0.712961
$766$	0	0
$767$	24.3237	0.878276
$768$	0	0
$769$	31.7739	1.14580	0.572898	−	0.819627i	$-0.305819\pi$
$769$	31.7739	1.14580	0.572898	+	0.819627i	$0.305819\pi$
$770$	0	0
$771$	−25.1437	−0.905528
$772$	0	0
$773$	20.6413	0.742417	0.371208	−	0.928550i	$-0.378944\pi$
$773$	20.6413	0.742417	0.371208	+	0.928550i	$0.378944\pi$
$774$	0	0
$775$	29.1817	1.04824
$776$	0	0
$777$	2.51622	0.0902687
$778$	0	0
$779$	−39.5059	−1.41545
$780$	0	0
$781$	39.3598	1.40841
$782$	0	0
$783$	4.08222	0.145887
$784$	0	0
$785$	−31.8856	−1.13805
$786$	0	0
$787$	−1.33841	−0.0477092	−0.0238546	−	0.999715i	$-0.507594\pi$
$787$	−1.33841	−0.0477092	−0.0238546	+	0.999715i	$0.507594\pi$
$788$	0	0
$789$	−9.01981	−0.321114
$790$	0	0
$791$	1.27279	0.0452552
$792$	0	0
$793$	−37.5662	−1.33401
$794$	0	0
$795$	43.8989	1.55693
$796$	0	0
$797$	4.64222	0.164436	0.0822180	−	0.996614i	$-0.473800\pi$
$797$	4.64222	0.164436	0.0822180	+	0.996614i	$0.473800\pi$
$798$	0	0
$799$	−24.4069	−0.863456
$800$	0	0
$801$	−14.0307	−0.495750
$802$	0	0
$803$	−1.01963	−0.0359818
$804$	0	0
$805$	−9.74474	−0.343457
$806$	0	0
$807$	−17.3775	−0.611716
$808$	0	0
$809$	−32.7890	−1.15280	−0.576401	−	0.817167i	$-0.695543\pi$
$809$	−32.7890	−1.15280	−0.576401	+	0.817167i	$0.695543\pi$
$810$	0	0
$811$	14.8120	0.520120	0.260060	−	0.965592i	$-0.416258\pi$
$811$	14.8120	0.520120	0.260060	+	0.965592i	$0.416258\pi$
$812$	0	0
$813$	7.82525	0.274443
$814$	0	0
$815$	−24.2313	−0.848785
$816$	0	0
$817$	0.354582	0.0124053
$818$	0	0
$819$	5.03672	0.175997
$820$	0	0
$821$	−14.2710	−0.498063	−0.249031	−	0.968495i	$-0.580112\pi$
$821$	−14.2710	−0.498063	−0.249031	+	0.968495i	$0.580112\pi$
$822$	0	0
$823$	−1.37573	−0.0479548	−0.0239774	−	0.999713i	$-0.507633\pi$
$823$	−1.37573	−0.0479548	−0.0239774	+	0.999713i	$0.507633\pi$
$824$	0	0
$825$	15.0017	0.522291
$826$	0	0
$827$	−35.8388	−1.24624	−0.623118	−	0.782128i	$-0.714134\pi$
$827$	−35.8388	−1.24624	−0.623118	+	0.782128i	$0.714134\pi$
$828$	0	0
$829$	−4.26187	−0.148021	−0.0740105	−	0.997257i	$-0.523580\pi$
$829$	−4.26187	−0.148021	−0.0740105	+	0.997257i	$0.523580\pi$
$830$	0	0
$831$	9.82635	0.340872
$832$	0	0
$833$	35.8283	1.24138
$834$	0	0
$835$	3.33576	0.115439
$836$	0	0
$837$	−4.76257	−0.164619
$838$	0	0
$839$	33.1063	1.14296	0.571479	−	0.820617i	$-0.306370\pi$
$839$	33.1063	1.14296	0.571479	+	0.820617i	$0.306370\pi$
$840$	0	0
$841$	−12.3355	−0.425362
$842$	0	0
$843$	−8.18040	−0.281748
$844$	0	0
$845$	−46.7290	−1.60753
$846$	0	0
$847$	−4.85131	−0.166693
$848$	0	0
$849$	19.0666	0.654363
$850$	0	0
$851$	−7.82580	−0.268265
$852$	0	0
$853$	1.30592	0.0447138	0.0223569	−	0.999750i	$-0.492883\pi$
$853$	1.30592	0.0447138	0.0223569	+	0.999750i	$0.492883\pi$
$854$	0	0
$855$	−11.3500	−0.388162
$856$	0	0
$857$	−46.5410	−1.58981	−0.794905	−	0.606733i	$-0.792480\pi$
$857$	−46.5410	−1.58981	−0.794905	+	0.606733i	$0.792480\pi$
$858$	0	0
$859$	5.58007	0.190390	0.0951948	−	0.995459i	$-0.469653\pi$
$859$	5.58007	0.190390	0.0951948	+	0.995459i	$0.469653\pi$
$860$	0	0
$861$	11.2527	0.383492
$862$	0	0
$863$	−39.1787	−1.33366	−0.666829	−	0.745210i	$-0.732349\pi$
$863$	−39.1787	−1.33366	−0.666829	+	0.745210i	$0.732349\pi$
$864$	0	0
$865$	−80.4339	−2.73483
$866$	0	0
$867$	−17.9464	−0.609493
$868$	0	0
$869$	11.2793	0.382625
$870$	0	0
$871$	81.8643	2.77387
$872$	0	0
$873$	−2.73252	−0.0924819
$874$	0	0
$875$	−3.64444	−0.123204
$876$	0	0
$877$	29.3106	0.989750	0.494875	−	0.868964i	$-0.335214\pi$
$877$	29.3106	0.989750	0.494875	+	0.868964i	$0.335214\pi$
$878$	0	0
$879$	6.89297	0.232494
$880$	0	0
$881$	50.1839	1.69074	0.845370	−	0.534181i	$-0.179380\pi$
$881$	50.1839	1.69074	0.845370	+	0.534181i	$0.179380\pi$
$882$	0	0
$883$	23.6618	0.796282	0.398141	−	0.917324i	$-0.369656\pi$
$883$	23.6618	0.796282	0.398141	+	0.917324i	$0.369656\pi$
$884$	0	0
$885$	15.6125	0.524810
$886$	0	0
$887$	−22.7561	−0.764075	−0.382037	−	0.924147i	$-0.624777\pi$
$887$	−22.7561	−0.764075	−0.382037	+	0.924147i	$0.624777\pi$
$888$	0	0
$889$	−4.58764	−0.153864
$890$	0	0
$891$	−2.44833	−0.0820223
$892$	0	0
$893$	14.0480	0.470097
$894$	0	0
$895$	70.6439	2.36137
$896$	0	0
$897$	−15.6649	−0.523037
$898$	0	0
$899$	−19.4419	−0.648422
$900$	0	0
$901$	−77.7966	−2.59178
$902$	0	0
$903$	−0.100998	−0.00336100
$904$	0	0
$905$	−14.7978	−0.491895
$906$	0	0
$907$	1.12607	0.0373905	0.0186953	−	0.999825i	$-0.494049\pi$
$907$	1.12607	0.0373905	0.0186953	+	0.999825i	$0.494049\pi$
$908$	0	0
$909$	4.29904	0.142590
$910$	0	0
$911$	55.4752	1.83797	0.918987	−	0.394287i	$-0.129008\pi$
$911$	55.4752	1.83797	0.918987	+	0.394287i	$0.129008\pi$
$912$	0	0
$913$	−10.4727	−0.346596
$914$	0	0
$915$	−24.1125	−0.797133
$916$	0	0
$917$	13.4034	0.442619
$918$	0	0
$919$	17.0695	0.563069	0.281535	−	0.959551i	$-0.409157\pi$
$919$	17.0695	0.563069	0.281535	+	0.959551i	$0.409157\pi$
$920$	0	0
$921$	−0.611159	−0.0201384
$922$	0	0
$923$	−83.5474	−2.75000
$924$	0	0
$925$	−15.9081	−0.523057
$926$	0	0
$927$	−3.01050	−0.0988778
$928$	0	0
$929$	30.4339	0.998505	0.499253	−	0.866457i	$-0.333608\pi$
$929$	30.4339	0.998505	0.499253	+	0.866457i	$0.333608\pi$
$930$	0	0
$931$	−20.6217	−0.675850
$932$	0	0
$933$	−10.7954	−0.353425
$934$	0	0
$935$	−48.2800	−1.57892
$936$	0	0
$937$	−3.17063	−0.103580	−0.0517901	−	0.998658i	$-0.516493\pi$
$937$	−3.17063	−0.103580	−0.0517901	+	0.998658i	$0.516493\pi$
$938$	0	0
$939$	23.7741	0.775838
$940$	0	0
$941$	−26.5433	−0.865286	−0.432643	−	0.901565i	$-0.642419\pi$
$941$	−26.5433	−0.865286	−0.432643	+	0.901565i	$0.642419\pi$
$942$	0	0
$943$	−34.9977	−1.13968
$944$	0	0
$945$	3.23290	0.105166
$946$	0	0
$947$	42.5162	1.38159	0.690796	−	0.723050i	$-0.257260\pi$
$947$	42.5162	1.38159	0.690796	+	0.723050i	$0.257260\pi$
$948$	0	0
$949$	2.16432	0.0702566
$950$	0	0
$951$	−19.3315	−0.626868
$952$	0	0
$953$	−54.0577	−1.75110	−0.875550	−	0.483127i	$-0.839501\pi$
$953$	−54.0577	−1.75110	−0.875550	+	0.483127i	$0.839501\pi$
$954$	0	0
$955$	−80.8092	−2.61492
$956$	0	0
$957$	−9.99463	−0.323081
$958$	0	0
$959$	−2.10895	−0.0681015
$960$	0	0
$961$	−8.31792	−0.268320
$962$	0	0
$963$	−14.4071	−0.464264
$964$	0	0
$965$	−8.05065	−0.259160
$966$	0	0
$967$	32.8415	1.05611	0.528056	−	0.849210i	$-0.322921\pi$
$967$	32.8415	1.05611	0.528056	+	0.849210i	$0.322921\pi$
$968$	0	0
$969$	20.1142	0.646161
$970$	0	0
$971$	−0.455903	−0.0146306	−0.00731531	−	0.999973i	$-0.502329\pi$
$971$	−0.455903	−0.0146306	−0.00731531	+	0.999973i	$0.502329\pi$
$972$	0	0
$973$	−2.31071	−0.0740780
$974$	0	0
$975$	−31.8434	−1.01980
$976$	0	0
$977$	−41.1248	−1.31570	−0.657849	−	0.753150i	$-0.728534\pi$
$977$	−41.1248	−1.31570	−0.657849	+	0.753150i	$0.728534\pi$
$978$	0	0
$979$	34.3518	1.09789
$980$	0	0
$981$	−4.59762	−0.146791
$982$	0	0
$983$	−21.9857	−0.701235	−0.350618	−	0.936519i	$-0.614028\pi$
$983$	−21.9857	−0.701235	−0.350618	+	0.936519i	$0.614028\pi$
$984$	0	0
$985$	59.1099	1.88340
$986$	0	0
$987$	−4.00137	−0.127365
$988$	0	0
$989$	0.314119	0.00998840
$990$	0	0
$991$	40.9088	1.29951	0.649756	−	0.760143i	$-0.274871\pi$
$991$	40.9088	1.29951	0.649756	+	0.760143i	$0.274871\pi$
$992$	0	0
$993$	−13.1764	−0.418141
$994$	0	0
$995$	47.8584	1.51721
$996$	0	0
$997$	−16.0296	−0.507663	−0.253832	−	0.967248i	$-0.581691\pi$
$997$	−16.0296	−0.507663	−0.253832	+	0.967248i	$0.581691\pi$
$998$	0	0
$999$	2.59627	0.0821425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.p.1.2		5
4.3	odd	2		501.2.a.b.1.2	✓	5
12.11	even	2		1503.2.a.d.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.b.1.2	✓	5	4.3	odd	2
1503.2.a.d.1.4		5	12.11	even	2
8016.2.a.p.1.2		5	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.33576 q^{5} +0.969164 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.33576 q^{5} +0.969164 q^{7} +1.00000 q^{9} -2.44833 q^{11} +5.19697 q^{13} +3.33576 q^{15} -5.91155 q^{17} +3.40252 q^{19} -0.969164 q^{21} +3.01424 q^{23} +6.12730 q^{25} -1.00000 q^{27} -4.08222 q^{29} +4.76257 q^{31} +2.44833 q^{33} -3.23290 q^{35} -2.59627 q^{37} -5.19697 q^{39} -11.6108 q^{41} +0.104212 q^{43} -3.33576 q^{45} +4.12869 q^{47} -6.06072 q^{49} +5.91155 q^{51} +13.1601 q^{53} +8.16705 q^{55} -3.40252 q^{57} +4.68035 q^{59} -7.22848 q^{61} +0.969164 q^{63} -17.3359 q^{65} +15.7523 q^{67} -3.01424 q^{69} -16.0762 q^{71} +0.416457 q^{73} -6.12730 q^{75} -2.37284 q^{77} -4.60694 q^{79} +1.00000 q^{81} +4.27748 q^{83} +19.7195 q^{85} +4.08222 q^{87} -14.0307 q^{89} +5.03672 q^{91} -4.76257 q^{93} -11.3500 q^{95} -2.73252 q^{97} -2.44833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 9 * q^5 + 4 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9} + 15 q^{11} + 9 q^{15} - 11 q^{17} + 16 q^{19} - 4 q^{21} + 9 q^{23} + 6 q^{25} - 5 q^{27} - q^{29} + 18 q^{31} - 15 q^{33} + 4 q^{35} + 7 q^{37} - 10 q^{41} - 6 q^{43} - 9 q^{45} + 7 q^{47} + 11 q^{49} + 11 q^{51} - 9 q^{53} - 17 q^{55} - 16 q^{57} + 37 q^{59} - 2 q^{61} + 4 q^{63} - 16 q^{65} - 9 q^{69} - 13 q^{71} - 6 q^{73} - 6 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 5 q^{83} + 29 q^{85} + q^{87} - 30 q^{89} + 33 q^{91} - 18 q^{93} - 43 q^{95} - 9 q^{97} + 15 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 9 * q^5 + 4 * q^7 + 5 * q^9 + 15 * q^11 + 9 * q^15 - 11 * q^17 + 16 * q^19 - 4 * q^21 + 9 * q^23 + 6 * q^25 - 5 * q^27 - q^29 + 18 * q^31 - 15 * q^33 + 4 * q^35 + 7 * q^37 - 10 * q^41 - 6 * q^43 - 9 * q^45 + 7 * q^47 + 11 * q^49 + 11 * q^51 - 9 * q^53 - 17 * q^55 - 16 * q^57 + 37 * q^59 - 2 * q^61 + 4 * q^63 - 16 * q^65 - 9 * q^69 - 13 * q^71 - 6 * q^73 - 6 * q^75 - 2 * q^77 - 14 * q^79 + 5 * q^81 + 5 * q^83 + 29 * q^85 + q^87 - 30 * q^89 + 33 * q^91 - 18 * q^93 - 43 * q^95 - 9 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

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