LMFDB - Embedded newform 8016.2.a.y.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:	$0$
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} - 23x^{6} - 3x^{5} + 163x^{4} + 13x^{3} - 418x^{2} + 4x + 269 $ x^8 - 23x^6 - 3x^5 + 163x^4 + 13x^3 - 418x^2 + 4x + 269
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.72373$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.72373 q^{5} +1.07650 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.72373 q^{5} +1.07650 q^{7} +1.00000 q^{9} +3.31859 q^{11} -6.84649 q^{13} +3.72373 q^{15} +4.33263 q^{17} +2.23962 q^{19} +1.07650 q^{21} -2.07650 q^{23} +8.86618 q^{25} +1.00000 q^{27} +10.1256 q^{29} +7.37662 q^{31} +3.31859 q^{33} +4.00859 q^{35} +0.893756 q^{37} -6.84649 q^{39} -5.72354 q^{41} +7.02378 q^{43} +3.72373 q^{45} -1.79152 q^{47} -5.84115 q^{49} +4.33263 q^{51} -7.33087 q^{53} +12.3575 q^{55} +2.23962 q^{57} -12.1545 q^{59} -8.00262 q^{61} +1.07650 q^{63} -25.4945 q^{65} +1.37654 q^{67} -2.07650 q^{69} +14.4875 q^{71} -5.17841 q^{73} +8.86618 q^{75} +3.57245 q^{77} +4.03811 q^{79} +1.00000 q^{81} -7.67767 q^{83} +16.1336 q^{85} +10.1256 q^{87} -6.68508 q^{89} -7.37023 q^{91} +7.37662 q^{93} +8.33975 q^{95} -4.78308 q^{97} +3.31859 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + q^7 + 8 * q^9	$ 8 q + 8 q^{3} + q^{7} + 8 q^{9} + 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} - 9 q^{23} + 6 q^{25} + 8 q^{27} + 17 q^{29} + 23 q^{31} + 3 q^{33} + 15 q^{35} + 8 q^{37} - 8 q^{39} - 8 q^{41} + 2 q^{43} + 34 q^{47} + 5 q^{49} - 7 q^{51} + 12 q^{53} + 7 q^{55} + 16 q^{59} - 2 q^{61} + q^{63} - 14 q^{65} - 21 q^{67} - 9 q^{69} + 29 q^{71} - 38 q^{73} + 6 q^{75} + 20 q^{77} + 12 q^{79} + 8 q^{81} + 32 q^{83} + 23 q^{85} + 17 q^{87} + 11 q^{89} + 5 q^{91} + 23 q^{93} + 67 q^{95} + 8 q^{97} + 3 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + q^7 + 8 * q^9 + 3 * q^11 - 8 * q^13 - 7 * q^17 + q^21 - 9 * q^23 + 6 * q^25 + 8 * q^27 + 17 * q^29 + 23 * q^31 + 3 * q^33 + 15 * q^35 + 8 * q^37 - 8 * q^39 - 8 * q^41 + 2 * q^43 + 34 * q^47 + 5 * q^49 - 7 * q^51 + 12 * q^53 + 7 * q^55 + 16 * q^59 - 2 * q^61 + q^63 - 14 * q^65 - 21 * q^67 - 9 * q^69 + 29 * q^71 - 38 * q^73 + 6 * q^75 + 20 * q^77 + 12 * q^79 + 8 * q^81 + 32 * q^83 + 23 * q^85 + 17 * q^87 + 11 * q^89 + 5 * q^91 + 23 * q^93 + 67 * q^95 + 8 * q^97 + 3 * 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
$5$	3.72373	1.66530	0.832652	−	0.553797i	$-0.186822\pi$
$5$	3.72373	1.66530	0.832652	+	0.553797i	$0.186822\pi$
$6$	0	0
$7$	1.07650	0.406878	0.203439	−	0.979088i	$-0.434788\pi$
$7$	1.07650	0.406878	0.203439	+	0.979088i	$0.434788\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.31859	1.00059	0.500296	−	0.865854i	$-0.333224\pi$
$11$	3.31859	1.00059	0.500296	+	0.865854i	$0.333224\pi$
$12$	0	0
$13$	−6.84649	−1.89888	−0.949438	−	0.313956i	$-0.898346\pi$
$13$	−6.84649	−1.89888	−0.949438	+	0.313956i	$0.898346\pi$
$14$	0	0
$15$	3.72373	0.961464
$16$	0	0
$17$	4.33263	1.05082	0.525409	−	0.850850i	$-0.323912\pi$
$17$	4.33263	1.05082	0.525409	+	0.850850i	$0.323912\pi$
$18$	0	0
$19$	2.23962	0.513804	0.256902	−	0.966437i	$-0.417298\pi$
$19$	2.23962	0.513804	0.256902	+	0.966437i	$0.417298\pi$
$20$	0	0
$21$	1.07650	0.234911
$22$	0	0
$23$	−2.07650	−0.432980	−0.216490	−	0.976285i	$-0.569461\pi$
$23$	−2.07650	−0.432980	−0.216490	+	0.976285i	$0.569461\pi$
$24$	0	0
$25$	8.86618	1.77324
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	10.1256	1.88028	0.940140	−	0.340788i	$-0.110694\pi$
$29$	10.1256	1.88028	0.940140	+	0.340788i	$0.110694\pi$
$30$	0	0
$31$	7.37662	1.32488	0.662440	−	0.749115i	$-0.269521\pi$
$31$	7.37662	1.32488	0.662440	+	0.749115i	$0.269521\pi$
$32$	0	0
$33$	3.31859	0.577692
$34$	0	0
$35$	4.00859	0.677575
$36$	0	0
$37$	0.893756	0.146933	0.0734663	−	0.997298i	$-0.476594\pi$
$37$	0.893756	0.146933	0.0734663	+	0.997298i	$0.476594\pi$
$38$	0	0
$39$	−6.84649	−1.09632
$40$	0	0
$41$	−5.72354	−0.893867	−0.446933	−	0.894567i	$-0.647484\pi$
$41$	−5.72354	−0.893867	−0.446933	+	0.894567i	$0.647484\pi$
$42$	0	0
$43$	7.02378	1.07112	0.535558	−	0.844499i	$-0.320101\pi$
$43$	7.02378	1.07112	0.535558	+	0.844499i	$0.320101\pi$
$44$	0	0
$45$	3.72373	0.555101
$46$	0	0
$47$	−1.79152	−0.261321	−0.130660	−	0.991427i	$-0.541710\pi$
$47$	−1.79152	−0.261321	−0.130660	+	0.991427i	$0.541710\pi$
$48$	0	0
$49$	−5.84115	−0.834451
$50$	0	0
$51$	4.33263	0.606690
$52$	0	0
$53$	−7.33087	−1.00697	−0.503486	−	0.864003i	$-0.667950\pi$
$53$	−7.33087	−1.00697	−0.503486	+	0.864003i	$0.667950\pi$
$54$	0	0
$55$	12.3575	1.66629
$56$	0	0
$57$	2.23962	0.296645
$58$	0	0
$59$	−12.1545	−1.58238	−0.791190	−	0.611571i	$-0.790538\pi$
$59$	−12.1545	−1.58238	−0.791190	+	0.611571i	$0.790538\pi$
$60$	0	0
$61$	−8.00262	−1.02463	−0.512316	−	0.858797i	$-0.671212\pi$
$61$	−8.00262	−1.02463	−0.512316	+	0.858797i	$0.671212\pi$
$62$	0	0
$63$	1.07650	0.135626
$64$	0	0
$65$	−25.4945	−3.16220
$66$	0	0
$67$	1.37654	0.168171	0.0840855	−	0.996459i	$-0.473203\pi$
$67$	1.37654	0.168171	0.0840855	+	0.996459i	$0.473203\pi$
$68$	0	0
$69$	−2.07650	−0.249981
$70$	0	0
$71$	14.4875	1.71935	0.859674	−	0.510842i	$-0.170666\pi$
$71$	14.4875	1.71935	0.859674	+	0.510842i	$0.170666\pi$
$72$	0	0
$73$	−5.17841	−0.606087	−0.303044	−	0.952977i	$-0.598003\pi$
$73$	−5.17841	−0.606087	−0.303044	+	0.952977i	$0.598003\pi$
$74$	0	0
$75$	8.86618	1.02378
$76$	0	0
$77$	3.57245	0.407119
$78$	0	0
$79$	4.03811	0.454323	0.227162	−	0.973857i	$-0.427055\pi$
$79$	4.03811	0.454323	0.227162	+	0.973857i	$0.427055\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.67767	−0.842734	−0.421367	−	0.906890i	$-0.638450\pi$
$83$	−7.67767	−0.842734	−0.421367	+	0.906890i	$0.638450\pi$
$84$	0	0
$85$	16.1336	1.74993
$86$	0	0
$87$	10.1256	1.08558
$88$	0	0
$89$	−6.68508	−0.708617	−0.354309	−	0.935129i	$-0.615284\pi$
$89$	−6.68508	−0.708617	−0.354309	+	0.935129i	$0.615284\pi$
$90$	0	0
$91$	−7.37023	−0.772610
$92$	0	0
$93$	7.37662	0.764920
$94$	0	0
$95$	8.33975	0.855640
$96$	0	0
$97$	−4.78308	−0.485648	−0.242824	−	0.970070i	$-0.578074\pi$
$97$	−4.78308	−0.485648	−0.242824	+	0.970070i	$0.578074\pi$
$98$	0	0
$99$	3.31859	0.333531
$100$	0	0
$101$	−4.22209	−0.420113	−0.210057	−	0.977689i	$-0.567365\pi$
$101$	−4.22209	−0.420113	−0.210057	+	0.977689i	$0.567365\pi$
$102$	0	0
$103$	17.2482	1.69951	0.849757	−	0.527174i	$-0.176749\pi$
$103$	17.2482	1.69951	0.849757	+	0.527174i	$0.176749\pi$
$104$	0	0
$105$	4.00859	0.391198
$106$	0	0
$107$	−14.6131	−1.41270	−0.706351	−	0.707862i	$-0.749660\pi$
$107$	−14.6131	−1.41270	−0.706351	+	0.707862i	$0.749660\pi$
$108$	0	0
$109$	9.61554	0.921001	0.460501	−	0.887659i	$-0.347670\pi$
$109$	9.61554	0.921001	0.460501	+	0.887659i	$0.347670\pi$
$110$	0	0
$111$	0.893756	0.0848316
$112$	0	0
$113$	10.5985	0.997023	0.498512	−	0.866883i	$-0.333880\pi$
$113$	10.5985	0.997023	0.498512	+	0.866883i	$0.333880\pi$
$114$	0	0
$115$	−7.73232	−0.721042
$116$	0	0
$117$	−6.84649	−0.632958
$118$	0	0
$119$	4.66407	0.427554
$120$	0	0
$121$	0.0130376	0.00118524
$122$	0	0
$123$	−5.72354	−0.516074
$124$	0	0
$125$	14.3966	1.28767
$126$	0	0
$127$	17.6534	1.56649	0.783243	−	0.621716i	$-0.213564\pi$
$127$	17.6534	1.56649	0.783243	+	0.621716i	$0.213564\pi$
$128$	0	0
$129$	7.02378	0.618409
$130$	0	0
$131$	12.2497	1.07026	0.535131	−	0.844769i	$-0.320262\pi$
$131$	12.2497	1.07026	0.535131	+	0.844769i	$0.320262\pi$
$132$	0	0
$133$	2.41095	0.209055
$134$	0	0
$135$	3.72373	0.320488
$136$	0	0
$137$	−11.3635	−0.970852	−0.485426	−	0.874278i	$-0.661336\pi$
$137$	−11.3635	−0.970852	−0.485426	+	0.874278i	$0.661336\pi$
$138$	0	0
$139$	19.1918	1.62783	0.813914	−	0.580986i	$-0.197333\pi$
$139$	19.1918	1.62783	0.813914	+	0.580986i	$0.197333\pi$
$140$	0	0
$141$	−1.79152	−0.150874
$142$	0	0
$143$	−22.7207	−1.90000
$144$	0	0
$145$	37.7051	3.13124
$146$	0	0
$147$	−5.84115	−0.481770
$148$	0	0
$149$	13.7242	1.12433	0.562164	−	0.827026i	$-0.309969\pi$
$149$	13.7242	1.12433	0.562164	+	0.827026i	$0.309969\pi$
$150$	0	0
$151$	−11.7082	−0.952796	−0.476398	−	0.879230i	$-0.658058\pi$
$151$	−11.7082	−0.952796	−0.476398	+	0.879230i	$0.658058\pi$
$152$	0	0
$153$	4.33263	0.350273
$154$	0	0
$155$	27.4686	2.20633
$156$	0	0
$157$	−1.98584	−0.158487	−0.0792437	−	0.996855i	$-0.525251\pi$
$157$	−1.98584	−0.158487	−0.0792437	+	0.996855i	$0.525251\pi$
$158$	0	0
$159$	−7.33087	−0.581376
$160$	0	0
$161$	−2.23534	−0.176170
$162$	0	0
$163$	2.14810	0.168252	0.0841260	−	0.996455i	$-0.473190\pi$
$163$	2.14810	0.168252	0.0841260	+	0.996455i	$0.473190\pi$
$164$	0	0
$165$	12.3575	0.962033
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	33.8745	2.60573
$170$	0	0
$171$	2.23962	0.171268
$172$	0	0
$173$	15.8343	1.20386	0.601928	−	0.798550i	$-0.294399\pi$
$173$	15.8343	1.20386	0.601928	+	0.798550i	$0.294399\pi$
$174$	0	0
$175$	9.54442	0.721490
$176$	0	0
$177$	−12.1545	−0.913587
$178$	0	0
$179$	−8.27008	−0.618134	−0.309067	−	0.951040i	$-0.600017\pi$
$179$	−8.27008	−0.618134	−0.309067	+	0.951040i	$0.600017\pi$
$180$	0	0
$181$	−25.4376	−1.89076	−0.945381	−	0.325968i	$-0.894310\pi$
$181$	−25.4376	−1.89076	−0.945381	+	0.325968i	$0.894310\pi$
$182$	0	0
$183$	−8.00262	−0.591571
$184$	0	0
$185$	3.32811	0.244687
$186$	0	0
$187$	14.3782	1.05144
$188$	0	0
$189$	1.07650	0.0783036
$190$	0	0
$191$	−4.07812	−0.295082	−0.147541	−	0.989056i	$-0.547136\pi$
$191$	−4.07812	−0.295082	−0.147541	+	0.989056i	$0.547136\pi$
$192$	0	0
$193$	−5.24661	−0.377659	−0.188830	−	0.982010i	$-0.560469\pi$
$193$	−5.24661	−0.377659	−0.188830	+	0.982010i	$0.560469\pi$
$194$	0	0
$195$	−25.4945	−1.82570
$196$	0	0
$197$	−24.2812	−1.72997	−0.864983	−	0.501801i	$-0.832671\pi$
$197$	−24.2812	−1.72997	−0.864983	+	0.501801i	$0.832671\pi$
$198$	0	0
$199$	−20.8287	−1.47651	−0.738254	−	0.674523i	$-0.764349\pi$
$199$	−20.8287	−1.47651	−0.738254	+	0.674523i	$0.764349\pi$
$200$	0	0
$201$	1.37654	0.0970936
$202$	0	0
$203$	10.9002	0.765044
$204$	0	0
$205$	−21.3129	−1.48856
$206$	0	0
$207$	−2.07650	−0.144327
$208$	0	0
$209$	7.43238	0.514109
$210$	0	0
$211$	−17.7819	−1.22415	−0.612077	−	0.790798i	$-0.709666\pi$
$211$	−17.7819	−1.22415	−0.612077	+	0.790798i	$0.709666\pi$
$212$	0	0
$213$	14.4875	0.992666
$214$	0	0
$215$	26.1547	1.78373
$216$	0	0
$217$	7.94091	0.539064
$218$	0	0
$219$	−5.17841	−0.349925
$220$	0	0
$221$	−29.6634	−1.99537
$222$	0	0
$223$	15.3349	1.02690	0.513451	−	0.858119i	$-0.328367\pi$
$223$	15.3349	1.02690	0.513451	+	0.858119i	$0.328367\pi$
$224$	0	0
$225$	8.86618	0.591079
$226$	0	0
$227$	17.1663	1.13937	0.569685	−	0.821863i	$-0.307065\pi$
$227$	17.1663	1.13937	0.569685	+	0.821863i	$0.307065\pi$
$228$	0	0
$229$	−8.93152	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$229$	−8.93152	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$230$	0	0
$231$	3.57245	0.235050
$232$	0	0
$233$	−10.8436	−0.710390	−0.355195	−	0.934792i	$-0.615586\pi$
$233$	−10.8436	−0.710390	−0.355195	+	0.934792i	$0.615586\pi$
$234$	0	0
$235$	−6.67116	−0.435178
$236$	0	0
$237$	4.03811	0.262304
$238$	0	0
$239$	1.91518	0.123883	0.0619414	−	0.998080i	$-0.480271\pi$
$239$	1.91518	0.123883	0.0619414	+	0.998080i	$0.480271\pi$
$240$	0	0
$241$	3.61168	0.232649	0.116325	−	0.993211i	$-0.462889\pi$
$241$	3.61168	0.232649	0.116325	+	0.993211i	$0.462889\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−21.7509	−1.38961
$246$	0	0
$247$	−15.3335	−0.975650
$248$	0	0
$249$	−7.67767	−0.486553
$250$	0	0
$251$	14.6131	0.922368	0.461184	−	0.887304i	$-0.347425\pi$
$251$	14.6131	0.922368	0.461184	+	0.887304i	$0.347425\pi$
$252$	0	0
$253$	−6.89104	−0.433236
$254$	0	0
$255$	16.1336	1.01032
$256$	0	0
$257$	−8.65220	−0.539709	−0.269855	−	0.962901i	$-0.586976\pi$
$257$	−8.65220	−0.539709	−0.269855	+	0.962901i	$0.586976\pi$
$258$	0	0
$259$	0.962126	0.0597836
$260$	0	0
$261$	10.1256	0.626760
$262$	0	0
$263$	−5.59793	−0.345183	−0.172592	−	0.984993i	$-0.555214\pi$
$263$	−5.59793	−0.345183	−0.172592	+	0.984993i	$0.555214\pi$
$264$	0	0
$265$	−27.2982	−1.67692
$266$	0	0
$267$	−6.68508	−0.409120
$268$	0	0
$269$	20.3299	1.23953	0.619767	−	0.784786i	$-0.287227\pi$
$269$	20.3299	1.23953	0.619767	+	0.784786i	$0.287227\pi$
$270$	0	0
$271$	18.7500	1.13898	0.569490	−	0.821998i	$-0.307141\pi$
$271$	18.7500	1.13898	0.569490	+	0.821998i	$0.307141\pi$
$272$	0	0
$273$	−7.37023	−0.446067
$274$	0	0
$275$	29.4232	1.77429
$276$	0	0
$277$	−3.65272	−0.219471	−0.109735	−	0.993961i	$-0.535000\pi$
$277$	−3.65272	−0.219471	−0.109735	+	0.993961i	$0.535000\pi$
$278$	0	0
$279$	7.37662	0.441627
$280$	0	0
$281$	−13.4647	−0.803236	−0.401618	−	0.915807i	$-0.631552\pi$
$281$	−13.4647	−0.803236	−0.401618	+	0.915807i	$0.631552\pi$
$282$	0	0
$283$	26.8397	1.59546	0.797728	−	0.603018i	$-0.206035\pi$
$283$	26.8397	1.59546	0.797728	+	0.603018i	$0.206035\pi$
$284$	0	0
$285$	8.33975	0.494004
$286$	0	0
$287$	−6.16137	−0.363694
$288$	0	0
$289$	1.77172	0.104219
$290$	0	0
$291$	−4.78308	−0.280389
$292$	0	0
$293$	3.19272	0.186521	0.0932605	−	0.995642i	$-0.470271\pi$
$293$	3.19272	0.186521	0.0932605	+	0.995642i	$0.470271\pi$
$294$	0	0
$295$	−45.2601	−2.63514
$296$	0	0
$297$	3.31859	0.192564
$298$	0	0
$299$	14.2167	0.822174
$300$	0	0
$301$	7.56107	0.435813
$302$	0	0
$303$	−4.22209	−0.242553
$304$	0	0
$305$	−29.7996	−1.70632
$306$	0	0
$307$	−21.9132	−1.25065	−0.625327	−	0.780363i	$-0.715034\pi$
$307$	−21.9132	−1.25065	−0.625327	+	0.780363i	$0.715034\pi$
$308$	0	0
$309$	17.2482	0.981215
$310$	0	0
$311$	−30.7314	−1.74261	−0.871307	−	0.490737i	$-0.836727\pi$
$311$	−30.7314	−1.74261	−0.871307	+	0.490737i	$0.836727\pi$
$312$	0	0
$313$	−24.8911	−1.40693	−0.703465	−	0.710730i	$-0.748365\pi$
$313$	−24.8911	−1.40693	−0.703465	+	0.710730i	$0.748365\pi$
$314$	0	0
$315$	4.00859	0.225858
$316$	0	0
$317$	24.2167	1.36014	0.680072	−	0.733145i	$-0.261949\pi$
$317$	24.2167	1.36014	0.680072	+	0.733145i	$0.261949\pi$
$318$	0	0
$319$	33.6028	1.88139
$320$	0	0
$321$	−14.6131	−0.815624
$322$	0	0
$323$	9.70346	0.539915
$324$	0	0
$325$	−60.7023	−3.36716
$326$	0	0
$327$	9.61554	0.531740
$328$	0	0
$329$	−1.92857	−0.106326
$330$	0	0
$331$	−10.7698	−0.591963	−0.295982	−	0.955194i	$-0.595647\pi$
$331$	−10.7698	−0.591963	−0.295982	+	0.955194i	$0.595647\pi$
$332$	0	0
$333$	0.893756	0.0489775
$334$	0	0
$335$	5.12587	0.280056
$336$	0	0
$337$	29.4935	1.60661	0.803307	−	0.595565i	$-0.203072\pi$
$337$	29.4935	1.60661	0.803307	+	0.595565i	$0.203072\pi$
$338$	0	0
$339$	10.5985	0.575632
$340$	0	0
$341$	24.4800	1.32567
$342$	0	0
$343$	−13.8235	−0.746397
$344$	0	0
$345$	−7.73232	−0.416294
$346$	0	0
$347$	1.56950	0.0842551	0.0421276	−	0.999112i	$-0.486586\pi$
$347$	1.56950	0.0842551	0.0421276	+	0.999112i	$0.486586\pi$
$348$	0	0
$349$	20.6944	1.10775	0.553873	−	0.832601i	$-0.313149\pi$
$349$	20.6944	1.10775	0.553873	+	0.832601i	$0.313149\pi$
$350$	0	0
$351$	−6.84649	−0.365439
$352$	0	0
$353$	−22.4233	−1.19347	−0.596735	−	0.802438i	$-0.703536\pi$
$353$	−22.4233	−1.19347	−0.596735	+	0.802438i	$0.703536\pi$
$354$	0	0
$355$	53.9475	2.86324
$356$	0	0
$357$	4.66407	0.246849
$358$	0	0
$359$	2.85741	0.150808	0.0754041	−	0.997153i	$-0.475975\pi$
$359$	2.85741	0.150808	0.0754041	+	0.997153i	$0.475975\pi$
$360$	0	0
$361$	−13.9841	−0.736005
$362$	0	0
$363$	0.0130376	0.000684298 0
$364$	0	0
$365$	−19.2830	−1.00932
$366$	0	0
$367$	2.46660	0.128756	0.0643778	−	0.997926i	$-0.479494\pi$
$367$	2.46660	0.128756	0.0643778	+	0.997926i	$0.479494\pi$
$368$	0	0
$369$	−5.72354	−0.297956
$370$	0	0
$371$	−7.89166	−0.409715
$372$	0	0
$373$	−26.4922	−1.37171	−0.685856	−	0.727737i	$-0.740572\pi$
$373$	−26.4922	−1.37171	−0.685856	+	0.727737i	$0.740572\pi$
$374$	0	0
$375$	14.3966	0.743439
$376$	0	0
$377$	−69.3250	−3.57042
$378$	0	0
$379$	−26.1738	−1.34446	−0.672228	−	0.740344i	$-0.734663\pi$
$379$	−26.1738	−1.34446	−0.672228	+	0.740344i	$0.734663\pi$
$380$	0	0
$381$	17.6534	0.904411
$382$	0	0
$383$	9.37212	0.478893	0.239446	−	0.970910i	$-0.423034\pi$
$383$	9.37212	0.478893	0.239446	+	0.970910i	$0.423034\pi$
$384$	0	0
$385$	13.3029	0.677976
$386$	0	0
$387$	7.02378	0.357039
$388$	0	0
$389$	17.1111	0.867566	0.433783	−	0.901017i	$-0.357179\pi$
$389$	17.1111	0.867566	0.433783	+	0.901017i	$0.357179\pi$
$390$	0	0
$391$	−8.99670	−0.454983
$392$	0	0
$393$	12.2497	0.617916
$394$	0	0
$395$	15.0369	0.756586
$396$	0	0
$397$	1.12053	0.0562379	0.0281190	−	0.999605i	$-0.491048\pi$
$397$	1.12053	0.0562379	0.0281190	+	0.999605i	$0.491048\pi$
$398$	0	0
$399$	2.41095	0.120698
$400$	0	0
$401$	31.4021	1.56815	0.784073	−	0.620668i	$-0.213139\pi$
$401$	31.4021	1.56815	0.784073	+	0.620668i	$0.213139\pi$
$402$	0	0
$403$	−50.5040	−2.51578
$404$	0	0
$405$	3.72373	0.185034
$406$	0	0
$407$	2.96601	0.147020
$408$	0	0
$409$	5.16731	0.255507	0.127754	−	0.991806i	$-0.459223\pi$
$409$	5.16731	0.255507	0.127754	+	0.991806i	$0.459223\pi$
$410$	0	0
$411$	−11.3635	−0.560522
$412$	0	0
$413$	−13.0843	−0.643835
$414$	0	0
$415$	−28.5896	−1.40341
$416$	0	0
$417$	19.1918	0.939826
$418$	0	0
$419$	25.5702	1.24919	0.624593	−	0.780951i	$-0.285265\pi$
$419$	25.5702	1.24919	0.624593	+	0.780951i	$0.285265\pi$
$420$	0	0
$421$	27.3730	1.33408	0.667039	−	0.745023i	$-0.267561\pi$
$421$	27.3730	1.33408	0.667039	+	0.745023i	$0.267561\pi$
$422$	0	0
$423$	−1.79152	−0.0871069
$424$	0	0
$425$	38.4139	1.86335
$426$	0	0
$427$	−8.61480	−0.416899
$428$	0	0
$429$	−22.7207	−1.09697
$430$	0	0
$431$	2.71846	0.130944	0.0654718	−	0.997854i	$-0.479145\pi$
$431$	2.71846	0.130944	0.0654718	+	0.997854i	$0.479145\pi$
$432$	0	0
$433$	36.1078	1.73523	0.867615	−	0.497236i	$-0.165652\pi$
$433$	36.1078	1.73523	0.867615	+	0.497236i	$0.165652\pi$
$434$	0	0
$435$	37.7051	1.80782
$436$	0	0
$437$	−4.65057	−0.222467
$438$	0	0
$439$	7.10414	0.339062	0.169531	−	0.985525i	$-0.445775\pi$
$439$	7.10414	0.339062	0.169531	+	0.985525i	$0.445775\pi$
$440$	0	0
$441$	−5.84115	−0.278150
$442$	0	0
$443$	10.0671	0.478302	0.239151	−	0.970982i	$-0.423131\pi$
$443$	10.0671	0.478302	0.239151	+	0.970982i	$0.423131\pi$
$444$	0	0
$445$	−24.8935	−1.18006
$446$	0	0
$447$	13.7242	0.649131
$448$	0	0
$449$	−25.9214	−1.22330	−0.611652	−	0.791127i	$-0.709495\pi$
$449$	−25.9214	−1.22330	−0.611652	+	0.791127i	$0.709495\pi$
$450$	0	0
$451$	−18.9941	−0.894396
$452$	0	0
$453$	−11.7082	−0.550097
$454$	0	0
$455$	−27.4448	−1.28663
$456$	0	0
$457$	−23.2002	−1.08526	−0.542629	−	0.839972i	$-0.682571\pi$
$457$	−23.2002	−1.08526	−0.542629	+	0.839972i	$0.682571\pi$
$458$	0	0
$459$	4.33263	0.202230
$460$	0	0
$461$	0.498036	0.0231959	0.0115979	−	0.999933i	$-0.496308\pi$
$461$	0.498036	0.0231959	0.0115979	+	0.999933i	$0.496308\pi$
$462$	0	0
$463$	−10.3697	−0.481920	−0.240960	−	0.970535i	$-0.577462\pi$
$463$	−10.3697	−0.481920	−0.240960	+	0.970535i	$0.577462\pi$
$464$	0	0
$465$	27.4686	1.27382
$466$	0	0
$467$	−8.20405	−0.379638	−0.189819	−	0.981819i	$-0.560790\pi$
$467$	−8.20405	−0.379638	−0.189819	+	0.981819i	$0.560790\pi$
$468$	0	0
$469$	1.48184	0.0684250
$470$	0	0
$471$	−1.98584	−0.0915028
$472$	0	0
$473$	23.3090	1.07175
$474$	0	0
$475$	19.8569	0.911097
$476$	0	0
$477$	−7.33087	−0.335658
$478$	0	0
$479$	0.232454	0.0106211	0.00531054	−	0.999986i	$-0.498310\pi$
$479$	0.232454	0.0106211	0.00531054	+	0.999986i	$0.498310\pi$
$480$	0	0
$481$	−6.11910	−0.279007
$482$	0	0
$483$	−2.23534	−0.101712
$484$	0	0
$485$	−17.8109	−0.808752
$486$	0	0
$487$	15.4002	0.697852	0.348926	−	0.937150i	$-0.386546\pi$
$487$	15.4002	0.697852	0.348926	+	0.937150i	$0.386546\pi$
$488$	0	0
$489$	2.14810	0.0971403
$490$	0	0
$491$	10.4386	0.471088	0.235544	−	0.971864i	$-0.424313\pi$
$491$	10.4386	0.471088	0.235544	+	0.971864i	$0.424313\pi$
$492$	0	0
$493$	43.8706	1.97583
$494$	0	0
$495$	12.3575	0.555430
$496$	0	0
$497$	15.5957	0.699564
$498$	0	0
$499$	12.3123	0.551175	0.275587	−	0.961276i	$-0.411128\pi$
$499$	12.3123	0.551175	0.275587	+	0.961276i	$0.411128\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	14.2914	0.637223	0.318611	−	0.947885i	$-0.396784\pi$
$503$	14.2914	0.637223	0.318611	+	0.947885i	$0.396784\pi$
$504$	0	0
$505$	−15.7219	−0.699616
$506$	0	0
$507$	33.8745	1.50442
$508$	0	0
$509$	34.2601	1.51855	0.759276	−	0.650769i	$-0.225553\pi$
$509$	34.2601	1.51855	0.759276	+	0.650769i	$0.225553\pi$
$510$	0	0
$511$	−5.57455	−0.246603
$512$	0	0
$513$	2.23962	0.0988817
$514$	0	0
$515$	64.2276	2.83021
$516$	0	0
$517$	−5.94533	−0.261475
$518$	0	0
$519$	15.8343	0.695047
$520$	0	0
$521$	−27.5599	−1.20742	−0.603711	−	0.797204i	$-0.706312\pi$
$521$	−27.5599	−1.20742	−0.603711	+	0.797204i	$0.706312\pi$
$522$	0	0
$523$	−2.72837	−0.119303	−0.0596517	−	0.998219i	$-0.518999\pi$
$523$	−2.72837	−0.119303	−0.0596517	+	0.998219i	$0.518999\pi$
$524$	0	0
$525$	9.54442	0.416553
$526$	0	0
$527$	31.9602	1.39221
$528$	0	0
$529$	−18.6882	−0.812529
$530$	0	0
$531$	−12.1545	−0.527460
$532$	0	0
$533$	39.1862	1.69734
$534$	0	0
$535$	−54.4153	−2.35258
$536$	0	0
$537$	−8.27008	−0.356880
$538$	0	0
$539$	−19.3844	−0.834945
$540$	0	0
$541$	−3.48413	−0.149794	−0.0748971	−	0.997191i	$-0.523863\pi$
$541$	−3.48413	−0.149794	−0.0748971	+	0.997191i	$0.523863\pi$
$542$	0	0
$543$	−25.4376	−1.09163
$544$	0	0
$545$	35.8057	1.53375
$546$	0	0
$547$	−34.2192	−1.46311	−0.731554	−	0.681784i	$-0.761204\pi$
$547$	−34.2192	−1.46311	−0.731554	+	0.681784i	$0.761204\pi$
$548$	0	0
$549$	−8.00262	−0.341544
$550$	0	0
$551$	22.6775	0.966096
$552$	0	0
$553$	4.34702	0.184854
$554$	0	0
$555$	3.32811	0.141270
$556$	0	0
$557$	−14.1158	−0.598104	−0.299052	−	0.954237i	$-0.596670\pi$
$557$	−14.1158	−0.598104	−0.299052	+	0.954237i	$0.596670\pi$
$558$	0	0
$559$	−48.0882	−2.03392
$560$	0	0
$561$	14.3782	0.607050
$562$	0	0
$563$	−36.1826	−1.52492	−0.762458	−	0.647038i	$-0.776008\pi$
$563$	−36.1826	−1.52492	−0.762458	+	0.647038i	$0.776008\pi$
$564$	0	0
$565$	39.4660	1.66035
$566$	0	0
$567$	1.07650	0.0452086
$568$	0	0
$569$	−25.7387	−1.07902	−0.539510	−	0.841979i	$-0.681391\pi$
$569$	−25.7387	−1.07902	−0.539510	+	0.841979i	$0.681391\pi$
$570$	0	0
$571$	−24.9185	−1.04281	−0.521403	−	0.853311i	$-0.674591\pi$
$571$	−24.9185	−1.04281	−0.521403	+	0.853311i	$0.674591\pi$
$572$	0	0
$573$	−4.07812	−0.170366
$574$	0	0
$575$	−18.4106	−0.767775
$576$	0	0
$577$	26.1844	1.09007	0.545036	−	0.838412i	$-0.316516\pi$
$577$	26.1844	1.09007	0.545036	+	0.838412i	$0.316516\pi$
$578$	0	0
$579$	−5.24661	−0.218042
$580$	0	0
$581$	−8.26499	−0.342890
$582$	0	0
$583$	−24.3282	−1.00757
$584$	0	0
$585$	−25.4945	−1.05407
$586$	0	0
$587$	−38.7805	−1.60064	−0.800320	−	0.599573i	$-0.795337\pi$
$587$	−38.7805	−1.60064	−0.800320	+	0.599573i	$0.795337\pi$
$588$	0	0
$589$	16.5208	0.680729
$590$	0	0
$591$	−24.2812	−0.998797
$592$	0	0
$593$	−30.1868	−1.23962	−0.619811	−	0.784751i	$-0.712791\pi$
$593$	−30.1868	−1.23962	−0.619811	+	0.784751i	$0.712791\pi$
$594$	0	0
$595$	17.3677	0.712008
$596$	0	0
$597$	−20.8287	−0.852462
$598$	0	0
$599$	−3.46214	−0.141459	−0.0707296	−	0.997496i	$-0.522533\pi$
$599$	−3.46214	−0.141459	−0.0707296	+	0.997496i	$0.522533\pi$
$600$	0	0
$601$	11.2199	0.457669	0.228835	−	0.973465i	$-0.426508\pi$
$601$	11.2199	0.457669	0.228835	+	0.973465i	$0.426508\pi$
$602$	0	0
$603$	1.37654	0.0560570
$604$	0	0
$605$	0.0485487	0.00197378
$606$	0	0
$607$	−8.59675	−0.348931	−0.174466	−	0.984663i	$-0.555820\pi$
$607$	−8.59675	−0.348931	−0.174466	+	0.984663i	$0.555820\pi$
$608$	0	0
$609$	10.9002	0.441698
$610$	0	0
$611$	12.2657	0.496215
$612$	0	0
$613$	−24.6391	−0.995165	−0.497582	−	0.867417i	$-0.665779\pi$
$613$	−24.6391	−0.995165	−0.497582	+	0.867417i	$0.665779\pi$
$614$	0	0
$615$	−21.3129	−0.859420
$616$	0	0
$617$	12.9004	0.519352	0.259676	−	0.965696i	$-0.416384\pi$
$617$	12.9004	0.519352	0.259676	+	0.965696i	$0.416384\pi$
$618$	0	0
$619$	34.6735	1.39364	0.696822	−	0.717244i	$-0.254597\pi$
$619$	34.6735	1.39364	0.696822	+	0.717244i	$0.254597\pi$
$620$	0	0
$621$	−2.07650	−0.0833270
$622$	0	0
$623$	−7.19647	−0.288321
$624$	0	0
$625$	9.27829	0.371132
$626$	0	0
$627$	7.43238	0.296821
$628$	0	0
$629$	3.87232	0.154399
$630$	0	0
$631$	6.08175	0.242110	0.121055	−	0.992646i	$-0.461372\pi$
$631$	6.08175	0.242110	0.121055	+	0.992646i	$0.461372\pi$
$632$	0	0
$633$	−17.7819	−0.706766
$634$	0	0
$635$	65.7365	2.60868
$636$	0	0
$637$	39.9914	1.58452
$638$	0	0
$639$	14.4875	0.573116
$640$	0	0
$641$	46.7591	1.84687	0.923437	−	0.383749i	$-0.125367\pi$
$641$	46.7591	1.84687	0.923437	+	0.383749i	$0.125367\pi$
$642$	0	0
$643$	−24.1924	−0.954055	−0.477027	−	0.878888i	$-0.658286\pi$
$643$	−24.1924	−0.954055	−0.477027	+	0.878888i	$0.658286\pi$
$644$	0	0
$645$	26.1547	1.02984
$646$	0	0
$647$	25.5727	1.00537	0.502683	−	0.864471i	$-0.332346\pi$
$647$	25.5727	1.00537	0.502683	+	0.864471i	$0.332346\pi$
$648$	0	0
$649$	−40.3358	−1.58332
$650$	0	0
$651$	7.94091	0.311229
$652$	0	0
$653$	10.3856	0.406420	0.203210	−	0.979135i	$-0.434863\pi$
$653$	10.3856	0.406420	0.203210	+	0.979135i	$0.434863\pi$
$654$	0	0
$655$	45.6146	1.78231
$656$	0	0
$657$	−5.17841	−0.202029
$658$	0	0
$659$	−23.3971	−0.911420	−0.455710	−	0.890128i	$-0.650615\pi$
$659$	−23.3971	−0.911420	−0.455710	+	0.890128i	$0.650615\pi$
$660$	0	0
$661$	45.3314	1.76319	0.881593	−	0.472010i	$-0.156471\pi$
$661$	45.3314	1.76319	0.881593	+	0.472010i	$0.156471\pi$
$662$	0	0
$663$	−29.6634	−1.15203
$664$	0	0
$665$	8.97772	0.348141
$666$	0	0
$667$	−21.0258	−0.814123
$668$	0	0
$669$	15.3349	0.592882
$670$	0	0
$671$	−26.5574	−1.02524
$672$	0	0
$673$	−22.6651	−0.873676	−0.436838	−	0.899540i	$-0.643902\pi$
$673$	−22.6651	−0.873676	−0.436838	+	0.899540i	$0.643902\pi$
$674$	0	0
$675$	8.86618	0.341260
$676$	0	0
$677$	−32.4358	−1.24661	−0.623305	−	0.781979i	$-0.714210\pi$
$677$	−32.4358	−1.24661	−0.623305	+	0.781979i	$0.714210\pi$
$678$	0	0
$679$	−5.14897	−0.197599
$680$	0	0
$681$	17.1663	0.657816
$682$	0	0
$683$	−8.10805	−0.310246	−0.155123	−	0.987895i	$-0.549577\pi$
$683$	−8.10805	−0.310246	−0.155123	+	0.987895i	$0.549577\pi$
$684$	0	0
$685$	−42.3148	−1.61676
$686$	0	0
$687$	−8.93152	−0.340759
$688$	0	0
$689$	50.1908	1.91212
$690$	0	0
$691$	0.826869	0.0314556	0.0157278	−	0.999876i	$-0.494993\pi$
$691$	0.826869	0.0314556	0.0157278	+	0.999876i	$0.494993\pi$
$692$	0	0
$693$	3.57245	0.135706
$694$	0	0
$695$	71.4651	2.71083
$696$	0	0
$697$	−24.7980	−0.939291
$698$	0	0
$699$	−10.8436	−0.410144
$700$	0	0
$701$	−30.8174	−1.16396	−0.581979	−	0.813204i	$-0.697721\pi$
$701$	−30.8174	−1.16396	−0.581979	+	0.813204i	$0.697721\pi$
$702$	0	0
$703$	2.00168	0.0754946
$704$	0	0
$705$	−6.67116	−0.251250
$706$	0	0
$707$	−4.54506	−0.170935
$708$	0	0
$709$	10.2855	0.386281	0.193140	−	0.981171i	$-0.438133\pi$
$709$	10.2855	0.386281	0.193140	+	0.981171i	$0.438133\pi$
$710$	0	0
$711$	4.03811	0.151441
$712$	0	0
$713$	−15.3175	−0.573646
$714$	0	0
$715$	−84.6058	−3.16408
$716$	0	0
$717$	1.91518	0.0715238
$718$	0	0
$719$	17.3383	0.646610	0.323305	−	0.946295i	$-0.395206\pi$
$719$	17.3383	0.646610	0.323305	+	0.946295i	$0.395206\pi$
$720$	0	0
$721$	18.5676	0.691494
$722$	0	0
$723$	3.61168	0.134320
$724$	0	0
$725$	89.7756	3.33418
$726$	0	0
$727$	5.69224	0.211114	0.105557	−	0.994413i	$-0.466338\pi$
$727$	5.69224	0.211114	0.105557	+	0.994413i	$0.466338\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.4315	1.12555
$732$	0	0
$733$	−17.4431	−0.644274	−0.322137	−	0.946693i	$-0.604401\pi$
$733$	−17.4431	−0.644274	−0.322137	+	0.946693i	$0.604401\pi$
$734$	0	0
$735$	−21.7509	−0.802294
$736$	0	0
$737$	4.56817	0.168271
$738$	0	0
$739$	20.8704	0.767731	0.383865	−	0.923389i	$-0.374593\pi$
$739$	20.8704	0.767731	0.383865	+	0.923389i	$0.374593\pi$
$740$	0	0
$741$	−15.3335	−0.563292
$742$	0	0
$743$	29.5948	1.08573	0.542863	−	0.839821i	$-0.317340\pi$
$743$	29.5948	1.08573	0.542863	+	0.839821i	$0.317340\pi$
$744$	0	0
$745$	51.1051	1.87235
$746$	0	0
$747$	−7.67767	−0.280911
$748$	0	0
$749$	−15.7310	−0.574797
$750$	0	0
$751$	−33.6241	−1.22696	−0.613480	−	0.789710i	$-0.710231\pi$
$751$	−33.6241	−1.22696	−0.613480	+	0.789710i	$0.710231\pi$
$752$	0	0
$753$	14.6131	0.532530
$754$	0	0
$755$	−43.5980	−1.58669
$756$	0	0
$757$	−33.8076	−1.22876	−0.614379	−	0.789011i	$-0.710593\pi$
$757$	−33.8076	−1.22876	−0.614379	+	0.789011i	$0.710593\pi$
$758$	0	0
$759$	−6.89104	−0.250129
$760$	0	0
$761$	−0.970756	−0.0351899	−0.0175949	−	0.999845i	$-0.505601\pi$
$761$	−0.970756	−0.0351899	−0.0175949	+	0.999845i	$0.505601\pi$
$762$	0	0
$763$	10.3511	0.374735
$764$	0	0
$765$	16.1336	0.583311
$766$	0	0
$767$	83.2156	3.00474
$768$	0	0
$769$	41.1942	1.48550	0.742750	−	0.669569i	$-0.233521\pi$
$769$	41.1942	1.48550	0.742750	+	0.669569i	$0.233521\pi$
$770$	0	0
$771$	−8.65220	−0.311601
$772$	0	0
$773$	−32.7343	−1.17737	−0.588686	−	0.808362i	$-0.700355\pi$
$773$	−32.7343	−1.17737	−0.588686	+	0.808362i	$0.700355\pi$
$774$	0	0
$775$	65.4025	2.34933
$776$	0	0
$777$	0.962126	0.0345161
$778$	0	0
$779$	−12.8186	−0.459273
$780$	0	0
$781$	48.0780	1.72037
$782$	0	0
$783$	10.1256	0.361860
$784$	0	0
$785$	−7.39474	−0.263930
$786$	0	0
$787$	−10.4249	−0.371607	−0.185803	−	0.982587i	$-0.559489\pi$
$787$	−10.4249	−0.371607	−0.185803	+	0.982587i	$0.559489\pi$
$788$	0	0
$789$	−5.59793	−0.199292
$790$	0	0
$791$	11.4093	0.405666
$792$	0	0
$793$	54.7899	1.94565
$794$	0	0
$795$	−27.2982	−0.968167
$796$	0	0
$797$	43.2016	1.53028	0.765139	−	0.643865i	$-0.222670\pi$
$797$	43.2016	1.53028	0.765139	+	0.643865i	$0.222670\pi$
$798$	0	0
$799$	−7.76202	−0.274601
$800$	0	0
$801$	−6.68508	−0.236206
$802$	0	0
$803$	−17.1850	−0.606446
$804$	0	0
$805$	−8.32382	−0.293376
$806$	0	0
$807$	20.3299	0.715645
$808$	0	0
$809$	9.53425	0.335206	0.167603	−	0.985855i	$-0.446397\pi$
$809$	9.53425	0.335206	0.167603	+	0.985855i	$0.446397\pi$
$810$	0	0
$811$	22.0876	0.775601	0.387800	−	0.921743i	$-0.373235\pi$
$811$	22.0876	0.775601	0.387800	+	0.921743i	$0.373235\pi$
$812$	0	0
$813$	18.7500	0.657591
$814$	0	0
$815$	7.99894	0.280191
$816$	0	0
$817$	15.7306	0.550344
$818$	0	0
$819$	−7.37023	−0.257537
$820$	0	0
$821$	−35.7235	−1.24676	−0.623379	−	0.781920i	$-0.714241\pi$
$821$	−35.7235	−1.24676	−0.623379	+	0.781920i	$0.714241\pi$
$822$	0	0
$823$	−28.9617	−1.00954	−0.504770	−	0.863254i	$-0.668423\pi$
$823$	−28.9617	−1.00954	−0.504770	+	0.863254i	$0.668423\pi$
$824$	0	0
$825$	29.4232	1.02439
$826$	0	0
$827$	−42.1646	−1.46621	−0.733103	−	0.680117i	$-0.761929\pi$
$827$	−42.1646	−1.46621	−0.733103	+	0.680117i	$0.761929\pi$
$828$	0	0
$829$	25.4477	0.883835	0.441918	−	0.897056i	$-0.354298\pi$
$829$	25.4477	0.883835	0.441918	+	0.897056i	$0.354298\pi$
$830$	0	0
$831$	−3.65272	−0.126711
$832$	0	0
$833$	−25.3076	−0.876856
$834$	0	0
$835$	−3.72373	−0.128865
$836$	0	0
$837$	7.37662	0.254973
$838$	0	0
$839$	5.50905	0.190194	0.0950968	−	0.995468i	$-0.469684\pi$
$839$	5.50905	0.190194	0.0950968	+	0.995468i	$0.469684\pi$
$840$	0	0
$841$	73.5281	2.53545
$842$	0	0
$843$	−13.4647	−0.463749
$844$	0	0
$845$	126.139	4.33933
$846$	0	0
$847$	0.0140350	0.000482247 0
$848$	0	0
$849$	26.8397	0.921137
$850$	0	0
$851$	−1.85588	−0.0636188
$852$	0	0
$853$	−53.7777	−1.84131	−0.920657	−	0.390373i	$-0.872346\pi$
$853$	−53.7777	−1.84131	−0.920657	+	0.390373i	$0.872346\pi$
$854$	0	0
$855$	8.33975	0.285213
$856$	0	0
$857$	−53.2240	−1.81810	−0.909048	−	0.416691i	$-0.863190\pi$
$857$	−53.2240	−1.81810	−0.909048	+	0.416691i	$0.863190\pi$
$858$	0	0
$859$	−35.3910	−1.20753	−0.603763	−	0.797164i	$-0.706333\pi$
$859$	−35.3910	−1.20753	−0.603763	+	0.797164i	$0.706333\pi$
$860$	0	0
$861$	−6.16137	−0.209979
$862$	0	0
$863$	−29.8390	−1.01573	−0.507866	−	0.861436i	$-0.669565\pi$
$863$	−29.8390	−1.01573	−0.507866	+	0.861436i	$0.669565\pi$
$864$	0	0
$865$	58.9625	2.00479
$866$	0	0
$867$	1.77172	0.0601708
$868$	0	0
$869$	13.4008	0.454592
$870$	0	0
$871$	−9.42447	−0.319336
$872$	0	0
$873$	−4.78308	−0.161883
$874$	0	0
$875$	15.4979	0.523926
$876$	0	0
$877$	−38.7015	−1.30686	−0.653428	−	0.756989i	$-0.726670\pi$
$877$	−38.7015	−1.30686	−0.653428	+	0.756989i	$0.726670\pi$
$878$	0	0
$879$	3.19272	0.107688
$880$	0	0
$881$	19.1289	0.644470	0.322235	−	0.946660i	$-0.395566\pi$
$881$	19.1289	0.644470	0.322235	+	0.946660i	$0.395566\pi$
$882$	0	0
$883$	14.4516	0.486334	0.243167	−	0.969984i	$-0.421814\pi$
$883$	14.4516	0.486334	0.243167	+	0.969984i	$0.421814\pi$
$884$	0	0
$885$	−45.2601	−1.52140
$886$	0	0
$887$	−12.9764	−0.435705	−0.217852	−	0.975982i	$-0.569905\pi$
$887$	−12.9764	−0.435705	−0.217852	+	0.975982i	$0.569905\pi$
$888$	0	0
$889$	19.0038	0.637368
$890$	0	0
$891$	3.31859	0.111177
$892$	0	0
$893$	−4.01234	−0.134268
$894$	0	0
$895$	−30.7955	−1.02938
$896$	0	0
$897$	14.2167	0.474682
$898$	0	0
$899$	74.6929	2.49115
$900$	0	0
$901$	−31.7620	−1.05815
$902$	0	0
$903$	7.56107	0.251617
$904$	0	0
$905$	−94.7228	−3.14869
$906$	0	0
$907$	−32.1886	−1.06881	−0.534403	−	0.845230i	$-0.679463\pi$
$907$	−32.1886	−1.06881	−0.534403	+	0.845230i	$0.679463\pi$
$908$	0	0
$909$	−4.22209	−0.140038
$910$	0	0
$911$	−46.7360	−1.54843	−0.774216	−	0.632921i	$-0.781856\pi$
$911$	−46.7360	−1.54843	−0.774216	+	0.632921i	$0.781856\pi$
$912$	0	0
$913$	−25.4790	−0.843233
$914$	0	0
$915$	−29.7996	−0.985145
$916$	0	0
$917$	13.1868	0.435466
$918$	0	0
$919$	−30.4584	−1.00473	−0.502366	−	0.864655i	$-0.667537\pi$
$919$	−30.4584	−1.00473	−0.502366	+	0.864655i	$0.667537\pi$
$920$	0	0
$921$	−21.9132	−0.722065
$922$	0	0
$923$	−99.1885	−3.26483
$924$	0	0
$925$	7.92421	0.260546
$926$	0	0
$927$	17.2482	0.566505
$928$	0	0
$929$	37.9586	1.24538	0.622691	−	0.782468i	$-0.286040\pi$
$929$	37.9586	1.24538	0.622691	+	0.782468i	$0.286040\pi$
$930$	0	0
$931$	−13.0820	−0.428744
$932$	0	0
$933$	−30.7314	−1.00610
$934$	0	0
$935$	53.5407	1.75097
$936$	0	0
$937$	5.36515	0.175272	0.0876359	−	0.996153i	$-0.472069\pi$
$937$	5.36515	0.175272	0.0876359	+	0.996153i	$0.472069\pi$
$938$	0	0
$939$	−24.8911	−0.812291
$940$	0	0
$941$	31.3068	1.02057	0.510287	−	0.860004i	$-0.329539\pi$
$941$	31.3068	1.02057	0.510287	+	0.860004i	$0.329539\pi$
$942$	0	0
$943$	11.8849	0.387026
$944$	0	0
$945$	4.00859	0.130399
$946$	0	0
$947$	−40.4932	−1.31585	−0.657926	−	0.753082i	$-0.728566\pi$
$947$	−40.4932	−1.31585	−0.657926	+	0.753082i	$0.728566\pi$
$948$	0	0
$949$	35.4540	1.15088
$950$	0	0
$951$	24.2167	0.785279
$952$	0	0
$953$	−19.3095	−0.625497	−0.312749	−	0.949836i	$-0.601250\pi$
$953$	−19.3095	−0.625497	−0.312749	+	0.949836i	$0.601250\pi$
$954$	0	0
$955$	−15.1858	−0.491402
$956$	0	0
$957$	33.6028	1.08622
$958$	0	0
$959$	−12.2328	−0.395018
$960$	0	0
$961$	23.4146	0.755309
$962$	0	0
$963$	−14.6131	−0.470901
$964$	0	0
$965$	−19.5370	−0.628917
$966$	0	0
$967$	−3.40023	−0.109344	−0.0546720	−	0.998504i	$-0.517411\pi$
$967$	−3.40023	−0.109344	−0.0546720	+	0.998504i	$0.517411\pi$
$968$	0	0
$969$	9.70346	0.311720
$970$	0	0
$971$	−10.2669	−0.329482	−0.164741	−	0.986337i	$-0.552679\pi$
$971$	−10.2669	−0.329482	−0.164741	+	0.986337i	$0.552679\pi$
$972$	0	0
$973$	20.6599	0.662326
$974$	0	0
$975$	−60.7023	−1.94403
$976$	0	0
$977$	36.0482	1.15328	0.576642	−	0.816997i	$-0.304363\pi$
$977$	36.0482	1.15328	0.576642	+	0.816997i	$0.304363\pi$
$978$	0	0
$979$	−22.1850	−0.709037
$980$	0	0
$981$	9.61554	0.307000
$982$	0	0
$983$	5.83948	0.186250	0.0931252	−	0.995654i	$-0.470314\pi$
$983$	5.83948	0.186250	0.0931252	+	0.995654i	$0.470314\pi$
$984$	0	0
$985$	−90.4168	−2.88092
$986$	0	0
$987$	−1.92857	−0.0613871
$988$	0	0
$989$	−14.5848	−0.463771
$990$	0	0
$991$	56.3983	1.79155	0.895775	−	0.444508i	$-0.146621\pi$
$991$	56.3983	1.79155	0.895775	+	0.444508i	$0.146621\pi$
$992$	0	0
$993$	−10.7698	−0.341770
$994$	0	0
$995$	−77.5606	−2.45883
$996$	0	0
$997$	−32.8505	−1.04039	−0.520194	−	0.854048i	$-0.674140\pi$
$997$	−32.8505	−1.04039	−0.520194	+	0.854048i	$0.674140\pi$
$998$	0	0
$999$	0.893756	0.0282772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.y.1.8		8
4.3	odd	2		4008.2.a.h.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.h.1.8	✓	8	4.3	odd	2
8016.2.a.y.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.72373 q^{5} +1.07650 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.72373 q^{5} +1.07650 q^{7} +1.00000 q^{9} +3.31859 q^{11} -6.84649 q^{13} +3.72373 q^{15} +4.33263 q^{17} +2.23962 q^{19} +1.07650 q^{21} -2.07650 q^{23} +8.86618 q^{25} +1.00000 q^{27} +10.1256 q^{29} +7.37662 q^{31} +3.31859 q^{33} +4.00859 q^{35} +0.893756 q^{37} -6.84649 q^{39} -5.72354 q^{41} +7.02378 q^{43} +3.72373 q^{45} -1.79152 q^{47} -5.84115 q^{49} +4.33263 q^{51} -7.33087 q^{53} +12.3575 q^{55} +2.23962 q^{57} -12.1545 q^{59} -8.00262 q^{61} +1.07650 q^{63} -25.4945 q^{65} +1.37654 q^{67} -2.07650 q^{69} +14.4875 q^{71} -5.17841 q^{73} +8.86618 q^{75} +3.57245 q^{77} +4.03811 q^{79} +1.00000 q^{81} -7.67767 q^{83} +16.1336 q^{85} +10.1256 q^{87} -6.68508 q^{89} -7.37023 q^{91} +7.37662 q^{93} +8.33975 q^{95} -4.78308 q^{97} +3.31859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + q^7 + 8 * q^9	\( 8 q + 8 q^{3} + q^{7} + 8 q^{9} + 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} - 9 q^{23} + 6 q^{25} + 8 q^{27} + 17 q^{29} + 23 q^{31} + 3 q^{33} + 15 q^{35} + 8 q^{37} - 8 q^{39} - 8 q^{41} + 2 q^{43} + 34 q^{47} + 5 q^{49} - 7 q^{51} + 12 q^{53} + 7 q^{55} + 16 q^{59} - 2 q^{61} + q^{63} - 14 q^{65} - 21 q^{67} - 9 q^{69} + 29 q^{71} - 38 q^{73} + 6 q^{75} + 20 q^{77} + 12 q^{79} + 8 q^{81} + 32 q^{83} + 23 q^{85} + 17 q^{87} + 11 q^{89} + 5 q^{91} + 23 q^{93} + 67 q^{95} + 8 q^{97} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + q^7 + 8 * q^9 + 3 * q^11 - 8 * q^13 - 7 * q^17 + q^21 - 9 * q^23 + 6 * q^25 + 8 * q^27 + 17 * q^29 + 23 * q^31 + 3 * q^33 + 15 * q^35 + 8 * q^37 - 8 * q^39 - 8 * q^41 + 2 * q^43 + 34 * q^47 + 5 * q^49 - 7 * q^51 + 12 * q^53 + 7 * q^55 + 16 * q^59 - 2 * q^61 + q^63 - 14 * q^65 - 21 * q^67 - 9 * q^69 + 29 * q^71 - 38 * q^73 + 6 * q^75 + 20 * q^77 + 12 * q^79 + 8 * q^81 + 32 * q^83 + 23 * q^85 + 17 * q^87 + 11 * q^89 + 5 * q^91 + 23 * q^93 + 67 * q^95 + 8 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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