LMFDB - Embedded newform 8016.2.a.ba.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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 16x^{7} + 45x^{6} + 67x^{5} - 166x^{4} - 83x^{3} + 152x^{2} + 51x - 10 $ x^9 - 3x^8 - 16x^7 + 45x^6 + 67x^5 - 166x^4 - 83x^3 + 152x^2 + 51x - 10
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			1.2
Root			$-1.99382$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.99382 q^{5} +4.65862 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.99382 q^{5} +4.65862 q^{7} +1.00000 q^{9} -0.962769 q^{11} +0.976361 q^{13} +2.99382 q^{15} +2.56202 q^{17} -8.42193 q^{19} -4.65862 q^{21} -6.29084 q^{23} +3.96298 q^{25} -1.00000 q^{27} -3.04318 q^{29} +10.1421 q^{31} +0.962769 q^{33} -13.9471 q^{35} +10.7136 q^{37} -0.976361 q^{39} +0.413133 q^{41} +5.35521 q^{43} -2.99382 q^{45} +8.47126 q^{47} +14.7028 q^{49} -2.56202 q^{51} -13.1750 q^{53} +2.88236 q^{55} +8.42193 q^{57} +0.101385 q^{59} -14.3711 q^{61} +4.65862 q^{63} -2.92305 q^{65} +11.7656 q^{67} +6.29084 q^{69} -12.5261 q^{71} -11.0294 q^{73} -3.96298 q^{75} -4.48518 q^{77} +1.47183 q^{79} +1.00000 q^{81} +4.56878 q^{83} -7.67023 q^{85} +3.04318 q^{87} +8.20803 q^{89} +4.54850 q^{91} -10.1421 q^{93} +25.2138 q^{95} +11.1603 q^{97} -0.962769 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} + 6 q^{15} - 9 q^{17} + 8 q^{19} - 11 q^{21} + 7 q^{23} - q^{25} - 9 q^{27} - 9 q^{29} + 25 q^{31} + q^{33} - 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} + 24 q^{43} - 6 q^{45} + 16 q^{47} + 4 q^{49} + 9 q^{51} - 26 q^{53} + 29 q^{55} - 8 q^{57} + 4 q^{59} - 20 q^{61} + 11 q^{63} - 8 q^{65} + 25 q^{67} - 7 q^{69} + 15 q^{71} - 10 q^{73} + q^{75} - 20 q^{77} + 34 q^{79} + 9 q^{81} + 4 q^{83} - 13 q^{85} + 9 q^{87} + 13 q^{89} + 21 q^{91} - 25 q^{93} + 7 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9 - q^11 - 4 * q^13 + 6 * q^15 - 9 * q^17 + 8 * q^19 - 11 * q^21 + 7 * q^23 - q^25 - 9 * q^27 - 9 * q^29 + 25 * q^31 + q^33 - 5 * q^35 - 6 * q^37 + 4 * q^39 - 4 * q^41 + 24 * q^43 - 6 * q^45 + 16 * q^47 + 4 * q^49 + 9 * q^51 - 26 * q^53 + 29 * q^55 - 8 * q^57 + 4 * q^59 - 20 * q^61 + 11 * q^63 - 8 * q^65 + 25 * q^67 - 7 * q^69 + 15 * q^71 - 10 * q^73 + q^75 - 20 * q^77 + 34 * q^79 + 9 * q^81 + 4 * q^83 - 13 * q^85 + 9 * q^87 + 13 * q^89 + 21 * q^91 - 25 * q^93 + 7 * q^95 - 4 * q^97 - 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
$5$	−2.99382	−1.33888	−0.669439	−	0.742867i	$-0.733466\pi$
$5$	−2.99382	−1.33888	−0.669439	+	0.742867i	$0.733466\pi$
$6$	0	0
$7$	4.65862	1.76079	0.880397	−	0.474238i	$-0.157276\pi$
$7$	4.65862	1.76079	0.880397	+	0.474238i	$0.157276\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.962769	−0.290286	−0.145143	−	0.989411i	$-0.546364\pi$
$11$	−0.962769	−0.290286	−0.145143	+	0.989411i	$0.546364\pi$
$12$	0	0
$13$	0.976361	0.270794	0.135397	−	0.990791i	$-0.456769\pi$
$13$	0.976361	0.270794	0.135397	+	0.990791i	$0.456769\pi$
$14$	0	0
$15$	2.99382	0.773002
$16$	0	0
$17$	2.56202	0.621381	0.310690	−	0.950511i	$-0.399440\pi$
$17$	2.56202	0.621381	0.310690	+	0.950511i	$0.399440\pi$
$18$	0	0
$19$	−8.42193	−1.93212	−0.966062	−	0.258311i	$-0.916834\pi$
$19$	−8.42193	−1.93212	−0.966062	+	0.258311i	$0.916834\pi$
$20$	0	0
$21$	−4.65862	−1.01659
$22$	0	0
$23$	−6.29084	−1.31173	−0.655865	−	0.754878i	$-0.727696\pi$
$23$	−6.29084	−1.31173	−0.655865	+	0.754878i	$0.727696\pi$
$24$	0	0
$25$	3.96298	0.792596
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.04318	−0.565104	−0.282552	−	0.959252i	$-0.591181\pi$
$29$	−3.04318	−0.565104	−0.282552	+	0.959252i	$0.591181\pi$
$30$	0	0
$31$	10.1421	1.82158	0.910788	−	0.412875i	$-0.135475\pi$
$31$	10.1421	1.82158	0.910788	+	0.412875i	$0.135475\pi$
$32$	0	0
$33$	0.962769	0.167597
$34$	0	0
$35$	−13.9471	−2.35749
$36$	0	0
$37$	10.7136	1.76130	0.880651	−	0.473766i	$-0.157106\pi$
$37$	10.7136	1.76130	0.880651	+	0.473766i	$0.157106\pi$
$38$	0	0
$39$	−0.976361	−0.156343
$40$	0	0
$41$	0.413133	0.0645205	0.0322602	−	0.999480i	$-0.489729\pi$
$41$	0.413133	0.0645205	0.0322602	+	0.999480i	$0.489729\pi$
$42$	0	0
$43$	5.35521	0.816662	0.408331	−	0.912834i	$-0.366111\pi$
$43$	5.35521	0.816662	0.408331	+	0.912834i	$0.366111\pi$
$44$	0	0
$45$	−2.99382	−0.446293
$46$	0	0
$47$	8.47126	1.23566	0.617830	−	0.786312i	$-0.288012\pi$
$47$	8.47126	1.23566	0.617830	+	0.786312i	$0.288012\pi$
$48$	0	0
$49$	14.7028	2.10039
$50$	0	0
$51$	−2.56202	−0.358754
$52$	0	0
$53$	−13.1750	−1.80973	−0.904865	−	0.425699i	$-0.860028\pi$
$53$	−13.1750	−1.80973	−0.904865	+	0.425699i	$0.860028\pi$
$54$	0	0
$55$	2.88236	0.388658
$56$	0	0
$57$	8.42193	1.11551
$58$	0	0
$59$	0.101385	0.0131992	0.00659962	−	0.999978i	$-0.497899\pi$
$59$	0.101385	0.0131992	0.00659962	+	0.999978i	$0.497899\pi$
$60$	0	0
$61$	−14.3711	−1.84003	−0.920014	−	0.391886i	$-0.871823\pi$
$61$	−14.3711	−1.84003	−0.920014	+	0.391886i	$0.871823\pi$
$62$	0	0
$63$	4.65862	0.586931
$64$	0	0
$65$	−2.92305	−0.362560
$66$	0	0
$67$	11.7656	1.43740	0.718698	−	0.695323i	$-0.244739\pi$
$67$	11.7656	1.43740	0.718698	+	0.695323i	$0.244739\pi$
$68$	0	0
$69$	6.29084	0.757328
$70$	0	0
$71$	−12.5261	−1.48657	−0.743285	−	0.668975i	$-0.766733\pi$
$71$	−12.5261	−1.48657	−0.743285	+	0.668975i	$0.766733\pi$
$72$	0	0
$73$	−11.0294	−1.29090	−0.645449	−	0.763803i	$-0.723330\pi$
$73$	−11.0294	−1.29090	−0.645449	+	0.763803i	$0.723330\pi$
$74$	0	0
$75$	−3.96298	−0.457606
$76$	0	0
$77$	−4.48518	−0.511134
$78$	0	0
$79$	1.47183	0.165594	0.0827969	−	0.996566i	$-0.473615\pi$
$79$	1.47183	0.165594	0.0827969	+	0.996566i	$0.473615\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.56878	0.501489	0.250744	−	0.968053i	$-0.419325\pi$
$83$	4.56878	0.501489	0.250744	+	0.968053i	$0.419325\pi$
$84$	0	0
$85$	−7.67023	−0.831953
$86$	0	0
$87$	3.04318	0.326263
$88$	0	0
$89$	8.20803	0.870049	0.435025	−	0.900419i	$-0.356740\pi$
$89$	8.20803	0.870049	0.435025	+	0.900419i	$0.356740\pi$
$90$	0	0
$91$	4.54850	0.476812
$92$	0	0
$93$	−10.1421	−1.05169
$94$	0	0
$95$	25.2138	2.58688
$96$	0	0
$97$	11.1603	1.13316	0.566579	−	0.824007i	$-0.308266\pi$
$97$	11.1603	1.13316	0.566579	+	0.824007i	$0.308266\pi$
$98$	0	0
$99$	−0.962769	−0.0967620
$100$	0	0
$101$	−5.03063	−0.500566	−0.250283	−	0.968173i	$-0.580524\pi$
$101$	−5.03063	−0.500566	−0.250283	+	0.968173i	$0.580524\pi$
$102$	0	0
$103$	−7.19615	−0.709057	−0.354529	−	0.935045i	$-0.615359\pi$
$103$	−7.19615	−0.709057	−0.354529	+	0.935045i	$0.615359\pi$
$104$	0	0
$105$	13.9471	1.36110
$106$	0	0
$107$	14.9736	1.44755	0.723777	−	0.690034i	$-0.242405\pi$
$107$	14.9736	1.44755	0.723777	+	0.690034i	$0.242405\pi$
$108$	0	0
$109$	−1.32527	−0.126938	−0.0634690	−	0.997984i	$-0.520216\pi$
$109$	−1.32527	−0.126938	−0.0634690	+	0.997984i	$0.520216\pi$
$110$	0	0
$111$	−10.7136	−1.01689
$112$	0	0
$113$	8.09741	0.761740	0.380870	−	0.924629i	$-0.375625\pi$
$113$	8.09741	0.761740	0.380870	+	0.924629i	$0.375625\pi$
$114$	0	0
$115$	18.8337	1.75625
$116$	0	0
$117$	0.976361	0.0902646
$118$	0	0
$119$	11.9355	1.09412
$120$	0	0
$121$	−10.0731	−0.915734
$122$	0	0
$123$	−0.413133	−0.0372509
$124$	0	0
$125$	3.10465	0.277688
$126$	0	0
$127$	15.8311	1.40479	0.702393	−	0.711789i	$-0.252115\pi$
$127$	15.8311	1.40479	0.702393	+	0.711789i	$0.252115\pi$
$128$	0	0
$129$	−5.35521	−0.471500
$130$	0	0
$131$	5.80759	0.507412	0.253706	−	0.967281i	$-0.418351\pi$
$131$	5.80759	0.507412	0.253706	+	0.967281i	$0.418351\pi$
$132$	0	0
$133$	−39.2346	−3.40207
$134$	0	0
$135$	2.99382	0.257667
$136$	0	0
$137$	−20.6048	−1.76039	−0.880195	−	0.474612i	$-0.842589\pi$
$137$	−20.6048	−1.76039	−0.880195	+	0.474612i	$0.842589\pi$
$138$	0	0
$139$	7.58360	0.643233	0.321616	−	0.946870i	$-0.395774\pi$
$139$	7.58360	0.643233	0.321616	+	0.946870i	$0.395774\pi$
$140$	0	0
$141$	−8.47126	−0.713409
$142$	0	0
$143$	−0.940010	−0.0786076
$144$	0	0
$145$	9.11074	0.756605
$146$	0	0
$147$	−14.7028	−1.21266
$148$	0	0
$149$	12.7087	1.04114	0.520569	−	0.853819i	$-0.325720\pi$
$149$	12.7087	1.04114	0.520569	+	0.853819i	$0.325720\pi$
$150$	0	0
$151$	−14.4023	−1.17204	−0.586021	−	0.810296i	$-0.699306\pi$
$151$	−14.4023	−1.17204	−0.586021	+	0.810296i	$0.699306\pi$
$152$	0	0
$153$	2.56202	0.207127
$154$	0	0
$155$	−30.3637	−2.43887
$156$	0	0
$157$	−7.84981	−0.626483	−0.313241	−	0.949674i	$-0.601415\pi$
$157$	−7.84981	−0.626483	−0.313241	+	0.949674i	$0.601415\pi$
$158$	0	0
$159$	13.1750	1.04485
$160$	0	0
$161$	−29.3067	−2.30969
$162$	0	0
$163$	−0.151488	−0.0118654	−0.00593272	−	0.999982i	$-0.501888\pi$
$163$	−0.151488	−0.0118654	−0.00593272	+	0.999982i	$0.501888\pi$
$164$	0	0
$165$	−2.88236	−0.224392
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.0467	−0.926671
$170$	0	0
$171$	−8.42193	−0.644041
$172$	0	0
$173$	10.9592	0.833212	0.416606	−	0.909087i	$-0.363220\pi$
$173$	10.9592	0.833212	0.416606	+	0.909087i	$0.363220\pi$
$174$	0	0
$175$	18.4620	1.39560
$176$	0	0
$177$	−0.101385	−0.00762059
$178$	0	0
$179$	−14.6334	−1.09375	−0.546874	−	0.837215i	$-0.684182\pi$
$179$	−14.6334	−1.09375	−0.546874	+	0.837215i	$0.684182\pi$
$180$	0	0
$181$	−4.60022	−0.341932	−0.170966	−	0.985277i	$-0.554689\pi$
$181$	−4.60022	−0.341932	−0.170966	+	0.985277i	$0.554689\pi$
$182$	0	0
$183$	14.3711	1.06234
$184$	0	0
$185$	−32.0746	−2.35817
$186$	0	0
$187$	−2.46663	−0.180378
$188$	0	0
$189$	−4.65862	−0.338865
$190$	0	0
$191$	17.2571	1.24868	0.624339	−	0.781153i	$-0.285368\pi$
$191$	17.2571	1.24868	0.624339	+	0.781153i	$0.285368\pi$
$192$	0	0
$193$	−11.6499	−0.838580	−0.419290	−	0.907852i	$-0.637721\pi$
$193$	−11.6499	−0.838580	−0.419290	+	0.907852i	$0.637721\pi$
$194$	0	0
$195$	2.92305	0.209324
$196$	0	0
$197$	10.8741	0.774747	0.387373	−	0.921923i	$-0.373382\pi$
$197$	10.8741	0.774747	0.387373	+	0.921923i	$0.373382\pi$
$198$	0	0
$199$	−9.11256	−0.645972	−0.322986	−	0.946404i	$-0.604687\pi$
$199$	−9.11256	−0.645972	−0.322986	+	0.946404i	$0.604687\pi$
$200$	0	0
$201$	−11.7656	−0.829881
$202$	0	0
$203$	−14.1770	−0.995031
$204$	0	0
$205$	−1.23685	−0.0863851
$206$	0	0
$207$	−6.29084	−0.437244
$208$	0	0
$209$	8.10838	0.560868
$210$	0	0
$211$	18.1269	1.24791	0.623953	−	0.781462i	$-0.285526\pi$
$211$	18.1269	1.24791	0.623953	+	0.781462i	$0.285526\pi$
$212$	0	0
$213$	12.5261	0.858272
$214$	0	0
$215$	−16.0326	−1.09341
$216$	0	0
$217$	47.2482	3.20742
$218$	0	0
$219$	11.0294	0.745301
$220$	0	0
$221$	2.50146	0.168266
$222$	0	0
$223$	14.2420	0.953716	0.476858	−	0.878980i	$-0.341776\pi$
$223$	14.2420	0.953716	0.476858	+	0.878980i	$0.341776\pi$
$224$	0	0
$225$	3.96298	0.264199
$226$	0	0
$227$	−10.5377	−0.699410	−0.349705	−	0.936860i	$-0.613718\pi$
$227$	−10.5377	−0.699410	−0.349705	+	0.936860i	$0.613718\pi$
$228$	0	0
$229$	20.2304	1.33686	0.668431	−	0.743774i	$-0.266966\pi$
$229$	20.2304	1.33686	0.668431	+	0.743774i	$0.266966\pi$
$230$	0	0
$231$	4.48518	0.295103
$232$	0	0
$233$	23.4478	1.53612	0.768059	−	0.640379i	$-0.221223\pi$
$233$	23.4478	1.53612	0.768059	+	0.640379i	$0.221223\pi$
$234$	0	0
$235$	−25.3615	−1.65440
$236$	0	0
$237$	−1.47183	−0.0956056
$238$	0	0
$239$	−8.49855	−0.549726	−0.274863	−	0.961483i	$-0.588632\pi$
$239$	−8.49855	−0.549726	−0.274863	+	0.961483i	$0.588632\pi$
$240$	0	0
$241$	−18.5989	−1.19806	−0.599031	−	0.800726i	$-0.704447\pi$
$241$	−18.5989	−1.19806	−0.599031	+	0.800726i	$0.704447\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−44.0175	−2.81217
$246$	0	0
$247$	−8.22285	−0.523207
$248$	0	0
$249$	−4.56878	−0.289535
$250$	0	0
$251$	2.98372	0.188331	0.0941655	−	0.995557i	$-0.469982\pi$
$251$	2.98372	0.188331	0.0941655	+	0.995557i	$0.469982\pi$
$252$	0	0
$253$	6.05663	0.380777
$254$	0	0
$255$	7.67023	0.480329
$256$	0	0
$257$	4.66532	0.291014	0.145507	−	0.989357i	$-0.453519\pi$
$257$	4.66532	0.291014	0.145507	+	0.989357i	$0.453519\pi$
$258$	0	0
$259$	49.9105	3.10129
$260$	0	0
$261$	−3.04318	−0.188368
$262$	0	0
$263$	11.4032	0.703151	0.351576	−	0.936159i	$-0.385646\pi$
$263$	11.4032	0.703151	0.351576	+	0.936159i	$0.385646\pi$
$264$	0	0
$265$	39.4437	2.42301
$266$	0	0
$267$	−8.20803	−0.502323
$268$	0	0
$269$	−17.5202	−1.06823	−0.534114	−	0.845413i	$-0.679355\pi$
$269$	−17.5202	−1.06823	−0.534114	+	0.845413i	$0.679355\pi$
$270$	0	0
$271$	4.07499	0.247538	0.123769	−	0.992311i	$-0.460502\pi$
$271$	4.07499	0.247538	0.123769	+	0.992311i	$0.460502\pi$
$272$	0	0
$273$	−4.54850	−0.275288
$274$	0	0
$275$	−3.81544	−0.230079
$276$	0	0
$277$	22.9380	1.37821	0.689105	−	0.724661i	$-0.258004\pi$
$277$	22.9380	1.37821	0.689105	+	0.724661i	$0.258004\pi$
$278$	0	0
$279$	10.1421	0.607192
$280$	0	0
$281$	−12.4485	−0.742615	−0.371308	−	0.928510i	$-0.621091\pi$
$281$	−12.4485	−0.742615	−0.371308	+	0.928510i	$0.621091\pi$
$282$	0	0
$283$	21.2448	1.26287	0.631436	−	0.775428i	$-0.282466\pi$
$283$	21.2448	1.26287	0.631436	+	0.775428i	$0.282466\pi$
$284$	0	0
$285$	−25.2138	−1.49354
$286$	0	0
$287$	1.92463	0.113607
$288$	0	0
$289$	−10.4361	−0.613886
$290$	0	0
$291$	−11.1603	−0.654229
$292$	0	0
$293$	3.52870	0.206149	0.103074	−	0.994674i	$-0.467132\pi$
$293$	3.52870	0.206149	0.103074	+	0.994674i	$0.467132\pi$
$294$	0	0
$295$	−0.303530	−0.0176722
$296$	0	0
$297$	0.962769	0.0558655
$298$	0	0
$299$	−6.14213	−0.355209
$300$	0	0
$301$	24.9479	1.43797
$302$	0	0
$303$	5.03063	0.289002
$304$	0	0
$305$	43.0245	2.46357
$306$	0	0
$307$	15.5946	0.890031	0.445016	−	0.895523i	$-0.353198\pi$
$307$	15.5946	0.890031	0.445016	+	0.895523i	$0.353198\pi$
$308$	0	0
$309$	7.19615	0.409374
$310$	0	0
$311$	0.111395	0.00631664	0.00315832	−	0.999995i	$-0.498995\pi$
$311$	0.111395	0.00631664	0.00315832	+	0.999995i	$0.498995\pi$
$312$	0	0
$313$	−15.8427	−0.895484	−0.447742	−	0.894163i	$-0.647772\pi$
$313$	−15.8427	−0.895484	−0.447742	+	0.894163i	$0.647772\pi$
$314$	0	0
$315$	−13.9471	−0.785830
$316$	0	0
$317$	18.3265	1.02932	0.514658	−	0.857395i	$-0.327919\pi$
$317$	18.3265	1.02932	0.514658	+	0.857395i	$0.327919\pi$
$318$	0	0
$319$	2.92988	0.164042
$320$	0	0
$321$	−14.9736	−0.835745
$322$	0	0
$323$	−21.5771	−1.20058
$324$	0	0
$325$	3.86930	0.214630
$326$	0	0
$327$	1.32527	0.0732877
$328$	0	0
$329$	39.4644	2.17574
$330$	0	0
$331$	6.26802	0.344521	0.172261	−	0.985051i	$-0.444893\pi$
$331$	6.26802	0.344521	0.172261	+	0.985051i	$0.444893\pi$
$332$	0	0
$333$	10.7136	0.587101
$334$	0	0
$335$	−35.2241	−1.92450
$336$	0	0
$337$	−9.78190	−0.532854	−0.266427	−	0.963855i	$-0.585843\pi$
$337$	−9.78190	−0.532854	−0.266427	+	0.963855i	$0.585843\pi$
$338$	0	0
$339$	−8.09741	−0.439791
$340$	0	0
$341$	−9.76450	−0.528778
$342$	0	0
$343$	35.8843	1.93757
$344$	0	0
$345$	−18.8337	−1.01397
$346$	0	0
$347$	−2.68557	−0.144169	−0.0720845	−	0.997399i	$-0.522965\pi$
$347$	−2.68557	−0.144169	−0.0720845	+	0.997399i	$0.522965\pi$
$348$	0	0
$349$	−16.1287	−0.863351	−0.431675	−	0.902029i	$-0.642077\pi$
$349$	−16.1287	−0.863351	−0.431675	+	0.902029i	$0.642077\pi$
$350$	0	0
$351$	−0.976361	−0.0521143
$352$	0	0
$353$	20.8777	1.11121	0.555604	−	0.831447i	$-0.312487\pi$
$353$	20.8777	1.11121	0.555604	+	0.831447i	$0.312487\pi$
$354$	0	0
$355$	37.5008	1.99034
$356$	0	0
$357$	−11.9355	−0.631692
$358$	0	0
$359$	5.52417	0.291555	0.145777	−	0.989317i	$-0.453432\pi$
$359$	5.52417	0.291555	0.145777	+	0.989317i	$0.453432\pi$
$360$	0	0
$361$	51.9289	2.73310
$362$	0	0
$363$	10.0731	0.528699
$364$	0	0
$365$	33.0202	1.72836
$366$	0	0
$367$	26.8197	1.39998	0.699988	−	0.714155i	$-0.253189\pi$
$367$	26.8197	1.39998	0.699988	+	0.714155i	$0.253189\pi$
$368$	0	0
$369$	0.413133	0.0215068
$370$	0	0
$371$	−61.3775	−3.18656
$372$	0	0
$373$	−2.13709	−0.110655	−0.0553273	−	0.998468i	$-0.517620\pi$
$373$	−2.13709	−0.110655	−0.0553273	+	0.998468i	$0.517620\pi$
$374$	0	0
$375$	−3.10465	−0.160324
$376$	0	0
$377$	−2.97124	−0.153027
$378$	0	0
$379$	15.5192	0.797167	0.398583	−	0.917132i	$-0.369502\pi$
$379$	15.5192	0.797167	0.398583	+	0.917132i	$0.369502\pi$
$380$	0	0
$381$	−15.8311	−0.811054
$382$	0	0
$383$	31.7717	1.62346	0.811730	−	0.584033i	$-0.198526\pi$
$383$	31.7717	1.62346	0.811730	+	0.584033i	$0.198526\pi$
$384$	0	0
$385$	13.4278	0.684346
$386$	0	0
$387$	5.35521	0.272221
$388$	0	0
$389$	27.7661	1.40780	0.703898	−	0.710301i	$-0.251441\pi$
$389$	27.7661	1.40780	0.703898	+	0.710301i	$0.251441\pi$
$390$	0	0
$391$	−16.1172	−0.815084
$392$	0	0
$393$	−5.80759	−0.292954
$394$	0	0
$395$	−4.40640	−0.221710
$396$	0	0
$397$	24.2186	1.21550	0.607748	−	0.794130i	$-0.292073\pi$
$397$	24.2186	1.21550	0.607748	+	0.794130i	$0.292073\pi$
$398$	0	0
$399$	39.2346	1.96419
$400$	0	0
$401$	36.8639	1.84089	0.920447	−	0.390867i	$-0.127825\pi$
$401$	36.8639	1.84089	0.920447	+	0.390867i	$0.127825\pi$
$402$	0	0
$403$	9.90235	0.493271
$404$	0	0
$405$	−2.99382	−0.148764
$406$	0	0
$407$	−10.3147	−0.511281
$408$	0	0
$409$	−7.05203	−0.348700	−0.174350	−	0.984684i	$-0.555782\pi$
$409$	−7.05203	−0.348700	−0.174350	+	0.984684i	$0.555782\pi$
$410$	0	0
$411$	20.6048	1.01636
$412$	0	0
$413$	0.472316	0.0232411
$414$	0	0
$415$	−13.6781	−0.671433
$416$	0	0
$417$	−7.58360	−0.371371
$418$	0	0
$419$	−27.1446	−1.32610	−0.663051	−	0.748574i	$-0.730739\pi$
$419$	−27.1446	−1.32610	−0.663051	+	0.748574i	$0.730739\pi$
$420$	0	0
$421$	22.1971	1.08182	0.540910	−	0.841080i	$-0.318080\pi$
$421$	22.1971	1.08182	0.540910	+	0.841080i	$0.318080\pi$
$422$	0	0
$423$	8.47126	0.411887
$424$	0	0
$425$	10.1532	0.492504
$426$	0	0
$427$	−66.9494	−3.23991
$428$	0	0
$429$	0.940010	0.0453841
$430$	0	0
$431$	−13.0366	−0.627952	−0.313976	−	0.949431i	$-0.601661\pi$
$431$	−13.0366	−0.627952	−0.313976	+	0.949431i	$0.601661\pi$
$432$	0	0
$433$	7.48494	0.359703	0.179852	−	0.983694i	$-0.442438\pi$
$433$	7.48494	0.359703	0.179852	+	0.983694i	$0.442438\pi$
$434$	0	0
$435$	−9.11074	−0.436826
$436$	0	0
$437$	52.9810	2.53443
$438$	0	0
$439$	31.8500	1.52012	0.760059	−	0.649854i	$-0.225170\pi$
$439$	31.8500	1.52012	0.760059	+	0.649854i	$0.225170\pi$
$440$	0	0
$441$	14.7028	0.700131
$442$	0	0
$443$	−8.08246	−0.384009	−0.192005	−	0.981394i	$-0.561499\pi$
$443$	−8.08246	−0.384009	−0.192005	+	0.981394i	$0.561499\pi$
$444$	0	0
$445$	−24.5734	−1.16489
$446$	0	0
$447$	−12.7087	−0.601102
$448$	0	0
$449$	−37.0405	−1.74805	−0.874024	−	0.485884i	$-0.838498\pi$
$449$	−37.0405	−1.74805	−0.874024	+	0.485884i	$0.838498\pi$
$450$	0	0
$451$	−0.397751	−0.0187294
$452$	0	0
$453$	14.4023	0.676679
$454$	0	0
$455$	−13.6174	−0.638394
$456$	0	0
$457$	17.4599	0.816739	0.408370	−	0.912817i	$-0.366097\pi$
$457$	17.4599	0.816739	0.408370	+	0.912817i	$0.366097\pi$
$458$	0	0
$459$	−2.56202	−0.119585
$460$	0	0
$461$	−12.6672	−0.589972	−0.294986	−	0.955502i	$-0.595315\pi$
$461$	−12.6672	−0.589972	−0.294986	+	0.955502i	$0.595315\pi$
$462$	0	0
$463$	15.9913	0.743177	0.371588	−	0.928398i	$-0.378813\pi$
$463$	15.9913	0.743177	0.371588	+	0.928398i	$0.378813\pi$
$464$	0	0
$465$	30.3637	1.40808
$466$	0	0
$467$	30.8847	1.42917	0.714587	−	0.699547i	$-0.246615\pi$
$467$	30.8847	1.42917	0.714587	+	0.699547i	$0.246615\pi$
$468$	0	0
$469$	54.8114	2.53096
$470$	0	0
$471$	7.84981	0.361700
$472$	0	0
$473$	−5.15583	−0.237065
$474$	0	0
$475$	−33.3760	−1.53139
$476$	0	0
$477$	−13.1750	−0.603243
$478$	0	0
$479$	18.4760	0.844188	0.422094	−	0.906552i	$-0.361295\pi$
$479$	18.4760	0.844188	0.422094	+	0.906552i	$0.361295\pi$
$480$	0	0
$481$	10.4603	0.476950
$482$	0	0
$483$	29.3067	1.33350
$484$	0	0
$485$	−33.4120	−1.51716
$486$	0	0
$487$	11.9533	0.541656	0.270828	−	0.962628i	$-0.412703\pi$
$487$	11.9533	0.541656	0.270828	+	0.962628i	$0.412703\pi$
$488$	0	0
$489$	0.151488	0.00685051
$490$	0	0
$491$	−6.63468	−0.299419	−0.149709	−	0.988730i	$-0.547834\pi$
$491$	−6.63468	−0.299419	−0.149709	+	0.988730i	$0.547834\pi$
$492$	0	0
$493$	−7.79668	−0.351145
$494$	0	0
$495$	2.88236	0.129553
$496$	0	0
$497$	−58.3542	−2.61754
$498$	0	0
$499$	42.3783	1.89711	0.948557	−	0.316605i	$-0.102543\pi$
$499$	42.3783	1.89711	0.948557	+	0.316605i	$0.102543\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	24.3222	1.08447	0.542237	−	0.840226i	$-0.317578\pi$
$503$	24.3222	1.08447	0.542237	+	0.840226i	$0.317578\pi$
$504$	0	0
$505$	15.0608	0.670197
$506$	0	0
$507$	12.0467	0.535014
$508$	0	0
$509$	−7.02710	−0.311471	−0.155735	−	0.987799i	$-0.549775\pi$
$509$	−7.02710	−0.311471	−0.155735	+	0.987799i	$0.549775\pi$
$510$	0	0
$511$	−51.3820	−2.27301
$512$	0	0
$513$	8.42193	0.371837
$514$	0	0
$515$	21.5440	0.949342
$516$	0	0
$517$	−8.15587	−0.358695
$518$	0	0
$519$	−10.9592	−0.481055
$520$	0	0
$521$	23.5312	1.03092	0.515461	−	0.856913i	$-0.327621\pi$
$521$	23.5312	1.03092	0.515461	+	0.856913i	$0.327621\pi$
$522$	0	0
$523$	43.3371	1.89500	0.947500	−	0.319755i	$-0.103601\pi$
$523$	43.3371	1.89500	0.947500	+	0.319755i	$0.103601\pi$
$524$	0	0
$525$	−18.4620	−0.805749
$526$	0	0
$527$	25.9843	1.13189
$528$	0	0
$529$	16.5747	0.720638
$530$	0	0
$531$	0.101385	0.00439975
$532$	0	0
$533$	0.403367	0.0174717
$534$	0	0
$535$	−44.8283	−1.93810
$536$	0	0
$537$	14.6334	0.631476
$538$	0	0
$539$	−14.1554	−0.609715
$540$	0	0
$541$	−25.0364	−1.07640	−0.538201	−	0.842817i	$-0.680896\pi$
$541$	−25.0364	−1.07640	−0.538201	+	0.842817i	$0.680896\pi$
$542$	0	0
$543$	4.60022	0.197414
$544$	0	0
$545$	3.96763	0.169955
$546$	0	0
$547$	−4.78599	−0.204634	−0.102317	−	0.994752i	$-0.532626\pi$
$547$	−4.78599	−0.204634	−0.102317	+	0.994752i	$0.532626\pi$
$548$	0	0
$549$	−14.3711	−0.613342
$550$	0	0
$551$	25.6294	1.09185
$552$	0	0
$553$	6.85670	0.291576
$554$	0	0
$555$	32.0746	1.36149
$556$	0	0
$557$	−5.00703	−0.212155	−0.106077	−	0.994358i	$-0.533829\pi$
$557$	−5.00703	−0.212155	−0.106077	+	0.994358i	$0.533829\pi$
$558$	0	0
$559$	5.22862	0.221147
$560$	0	0
$561$	2.46663	0.104141
$562$	0	0
$563$	36.9366	1.55669	0.778346	−	0.627836i	$-0.216059\pi$
$563$	36.9366	1.55669	0.778346	+	0.627836i	$0.216059\pi$
$564$	0	0
$565$	−24.2422	−1.01988
$566$	0	0
$567$	4.65862	0.195644
$568$	0	0
$569$	−28.3118	−1.18689	−0.593447	−	0.804873i	$-0.702233\pi$
$569$	−28.3118	−1.18689	−0.593447	+	0.804873i	$0.702233\pi$
$570$	0	0
$571$	−6.56835	−0.274877	−0.137438	−	0.990510i	$-0.543887\pi$
$571$	−6.56835	−0.274877	−0.137438	+	0.990510i	$0.543887\pi$
$572$	0	0
$573$	−17.2571	−0.720925
$574$	0	0
$575$	−24.9305	−1.03967
$576$	0	0
$577$	−33.2317	−1.38345	−0.691727	−	0.722160i	$-0.743150\pi$
$577$	−33.2317	−1.38345	−0.691727	+	0.722160i	$0.743150\pi$
$578$	0	0
$579$	11.6499	0.484154
$580$	0	0
$581$	21.2842	0.883018
$582$	0	0
$583$	12.6845	0.525339
$584$	0	0
$585$	−2.92305	−0.120853
$586$	0	0
$587$	13.5810	0.560546	0.280273	−	0.959920i	$-0.409575\pi$
$587$	13.5810	0.560546	0.280273	+	0.959920i	$0.409575\pi$
$588$	0	0
$589$	−85.4161	−3.51951
$590$	0	0
$591$	−10.8741	−0.447300
$592$	0	0
$593$	5.20329	0.213674	0.106837	−	0.994277i	$-0.465928\pi$
$593$	5.20329	0.213674	0.106837	+	0.994277i	$0.465928\pi$
$594$	0	0
$595$	−35.7327	−1.46490
$596$	0	0
$597$	9.11256	0.372952
$598$	0	0
$599$	10.2141	0.417337	0.208668	−	0.977986i	$-0.433087\pi$
$599$	10.2141	0.417337	0.208668	+	0.977986i	$0.433087\pi$
$600$	0	0
$601$	17.4052	0.709971	0.354986	−	0.934872i	$-0.384486\pi$
$601$	17.4052	0.709971	0.354986	+	0.934872i	$0.384486\pi$
$602$	0	0
$603$	11.7656	0.479132
$604$	0	0
$605$	30.1570	1.22606
$606$	0	0
$607$	−8.73864	−0.354691	−0.177345	−	0.984149i	$-0.556751\pi$
$607$	−8.73864	−0.354691	−0.177345	+	0.984149i	$0.556751\pi$
$608$	0	0
$609$	14.1770	0.574482
$610$	0	0
$611$	8.27101	0.334609
$612$	0	0
$613$	27.9839	1.13026	0.565130	−	0.825002i	$-0.308826\pi$
$613$	27.9839	1.13026	0.565130	+	0.825002i	$0.308826\pi$
$614$	0	0
$615$	1.23685	0.0498745
$616$	0	0
$617$	−35.8615	−1.44373	−0.721864	−	0.692035i	$-0.756714\pi$
$617$	−35.8615	−1.44373	−0.721864	+	0.692035i	$0.756714\pi$
$618$	0	0
$619$	−25.4287	−1.02207	−0.511033	−	0.859561i	$-0.670737\pi$
$619$	−25.4287	−1.02207	−0.511033	+	0.859561i	$0.670737\pi$
$620$	0	0
$621$	6.29084	0.252443
$622$	0	0
$623$	38.2381	1.53198
$624$	0	0
$625$	−29.1097	−1.16439
$626$	0	0
$627$	−8.10838	−0.323817
$628$	0	0
$629$	27.4484	1.09444
$630$	0	0
$631$	34.3659	1.36808	0.684042	−	0.729442i	$-0.260220\pi$
$631$	34.3659	1.36808	0.684042	+	0.729442i	$0.260220\pi$
$632$	0	0
$633$	−18.1269	−0.720479
$634$	0	0
$635$	−47.3956	−1.88084
$636$	0	0
$637$	14.3552	0.568774
$638$	0	0
$639$	−12.5261	−0.495523
$640$	0	0
$641$	−27.6147	−1.09071	−0.545357	−	0.838204i	$-0.683606\pi$
$641$	−27.6147	−1.09071	−0.545357	+	0.838204i	$0.683606\pi$
$642$	0	0
$643$	1.24094	0.0489379	0.0244689	−	0.999701i	$-0.492211\pi$
$643$	1.24094	0.0489379	0.0244689	+	0.999701i	$0.492211\pi$
$644$	0	0
$645$	16.0326	0.631281
$646$	0	0
$647$	25.0085	0.983186	0.491593	−	0.870825i	$-0.336415\pi$
$647$	25.0085	0.983186	0.491593	+	0.870825i	$0.336415\pi$
$648$	0	0
$649$	−0.0976106	−0.00383155
$650$	0	0
$651$	−47.2482	−1.85180
$652$	0	0
$653$	27.7223	1.08486	0.542429	−	0.840101i	$-0.317505\pi$
$653$	27.7223	1.08486	0.542429	+	0.840101i	$0.317505\pi$
$654$	0	0
$655$	−17.3869	−0.679363
$656$	0	0
$657$	−11.0294	−0.430300
$658$	0	0
$659$	−10.4052	−0.405330	−0.202665	−	0.979248i	$-0.564960\pi$
$659$	−10.4052	−0.405330	−0.202665	+	0.979248i	$0.564960\pi$
$660$	0	0
$661$	−14.4554	−0.562251	−0.281126	−	0.959671i	$-0.590708\pi$
$661$	−14.4554	−0.562251	−0.281126	+	0.959671i	$0.590708\pi$
$662$	0	0
$663$	−2.50146	−0.0971485
$664$	0	0
$665$	117.461	4.55496
$666$	0	0
$667$	19.1441	0.741264
$668$	0	0
$669$	−14.2420	−0.550628
$670$	0	0
$671$	13.8360	0.534134
$672$	0	0
$673$	9.62407	0.370981	0.185490	−	0.982646i	$-0.440613\pi$
$673$	9.62407	0.370981	0.185490	+	0.982646i	$0.440613\pi$
$674$	0	0
$675$	−3.96298	−0.152535
$676$	0	0
$677$	28.4117	1.09195	0.545974	−	0.837802i	$-0.316160\pi$
$677$	28.4117	1.09195	0.545974	+	0.837802i	$0.316160\pi$
$678$	0	0
$679$	51.9917	1.99526
$680$	0	0
$681$	10.5377	0.403805
$682$	0	0
$683$	3.05896	0.117048	0.0585239	−	0.998286i	$-0.481361\pi$
$683$	3.05896	0.117048	0.0585239	+	0.998286i	$0.481361\pi$
$684$	0	0
$685$	61.6873	2.35695
$686$	0	0
$687$	−20.2304	−0.771838
$688$	0	0
$689$	−12.8636	−0.490064
$690$	0	0
$691$	20.0725	0.763594	0.381797	−	0.924246i	$-0.375305\pi$
$691$	20.0725	0.763594	0.381797	+	0.924246i	$0.375305\pi$
$692$	0	0
$693$	−4.48518	−0.170378
$694$	0	0
$695$	−22.7040	−0.861211
$696$	0	0
$697$	1.05845	0.0400918
$698$	0	0
$699$	−23.4478	−0.886878
$700$	0	0
$701$	−25.9984	−0.981947	−0.490974	−	0.871174i	$-0.663359\pi$
$701$	−25.9984	−0.981947	−0.490974	+	0.871174i	$0.663359\pi$
$702$	0	0
$703$	−90.2291	−3.40305
$704$	0	0
$705$	25.3615	0.955168
$706$	0	0
$707$	−23.4358	−0.881393
$708$	0	0
$709$	9.35327	0.351270	0.175635	−	0.984455i	$-0.443802\pi$
$709$	9.35327	0.351270	0.175635	+	0.984455i	$0.443802\pi$
$710$	0	0
$711$	1.47183	0.0551979
$712$	0	0
$713$	−63.8024	−2.38942
$714$	0	0
$715$	2.81423	0.105246
$716$	0	0
$717$	8.49855	0.317384
$718$	0	0
$719$	36.5227	1.36207	0.681034	−	0.732252i	$-0.261531\pi$
$719$	36.5227	1.36207	0.681034	+	0.732252i	$0.261531\pi$
$720$	0	0
$721$	−33.5241	−1.24850
$722$	0	0
$723$	18.5989	0.691701
$724$	0	0
$725$	−12.0601	−0.447899
$726$	0	0
$727$	26.4767	0.981966	0.490983	−	0.871169i	$-0.336638\pi$
$727$	26.4767	0.981966	0.490983	+	0.871169i	$0.336638\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.7201	0.507458
$732$	0	0
$733$	−27.7117	−1.02355	−0.511777	−	0.859118i	$-0.671013\pi$
$733$	−27.7117	−1.02355	−0.511777	+	0.859118i	$0.671013\pi$
$734$	0	0
$735$	44.0175	1.62361
$736$	0	0
$737$	−11.3275	−0.417255
$738$	0	0
$739$	−35.5867	−1.30908	−0.654538	−	0.756029i	$-0.727137\pi$
$739$	−35.5867	−1.30908	−0.654538	+	0.756029i	$0.727137\pi$
$740$	0	0
$741$	8.22285	0.302074
$742$	0	0
$743$	−41.8212	−1.53427	−0.767135	−	0.641485i	$-0.778319\pi$
$743$	−41.8212	−1.53427	−0.767135	+	0.641485i	$0.778319\pi$
$744$	0	0
$745$	−38.0477	−1.39396
$746$	0	0
$747$	4.56878	0.167163
$748$	0	0
$749$	69.7564	2.54884
$750$	0	0
$751$	−38.9341	−1.42073	−0.710363	−	0.703836i	$-0.751469\pi$
$751$	−38.9341	−1.42073	−0.710363	+	0.703836i	$0.751469\pi$
$752$	0	0
$753$	−2.98372	−0.108733
$754$	0	0
$755$	43.1179	1.56922
$756$	0	0
$757$	−13.3933	−0.486787	−0.243394	−	0.969928i	$-0.578261\pi$
$757$	−13.3933	−0.486787	−0.243394	+	0.969928i	$0.578261\pi$
$758$	0	0
$759$	−6.05663	−0.219842
$760$	0	0
$761$	53.8875	1.95342	0.976710	−	0.214564i	$-0.0688332\pi$
$761$	53.8875	1.95342	0.976710	+	0.214564i	$0.0688332\pi$
$762$	0	0
$763$	−6.17394	−0.223512
$764$	0	0
$765$	−7.67023	−0.277318
$766$	0	0
$767$	0.0989887	0.00357427
$768$	0	0
$769$	−17.2334	−0.621453	−0.310727	−	0.950499i	$-0.600572\pi$
$769$	−17.2334	−0.621453	−0.310727	+	0.950499i	$0.600572\pi$
$770$	0	0
$771$	−4.66532	−0.168017
$772$	0	0
$773$	−50.4756	−1.81548	−0.907741	−	0.419532i	$-0.862194\pi$
$773$	−50.4756	−1.81548	−0.907741	+	0.419532i	$0.862194\pi$
$774$	0	0
$775$	40.1930	1.44377
$776$	0	0
$777$	−49.9105	−1.79053
$778$	0	0
$779$	−3.47937	−0.124662
$780$	0	0
$781$	12.0597	0.431530
$782$	0	0
$783$	3.04318	0.108754
$784$	0	0
$785$	23.5009	0.838784
$786$	0	0
$787$	−29.2874	−1.04398	−0.521991	−	0.852951i	$-0.674811\pi$
$787$	−29.2874	−1.04398	−0.521991	+	0.852951i	$0.674811\pi$
$788$	0	0
$789$	−11.4032	−0.405965
$790$	0	0
$791$	37.7228	1.34127
$792$	0	0
$793$	−14.0314	−0.498268
$794$	0	0
$795$	−39.4437	−1.39892
$796$	0	0
$797$	−1.07432	−0.0380543	−0.0190272	−	0.999819i	$-0.506057\pi$
$797$	−1.07432	−0.0380543	−0.0190272	+	0.999819i	$0.506057\pi$
$798$	0	0
$799$	21.7035	0.767815
$800$	0	0
$801$	8.20803	0.290016
$802$	0	0
$803$	10.6188	0.374730
$804$	0	0
$805$	87.7389	3.09239
$806$	0	0
$807$	17.5202	0.616741
$808$	0	0
$809$	22.8689	0.804028	0.402014	−	0.915634i	$-0.368310\pi$
$809$	22.8689	0.804028	0.402014	+	0.915634i	$0.368310\pi$
$810$	0	0
$811$	−20.4412	−0.717788	−0.358894	−	0.933378i	$-0.616846\pi$
$811$	−20.4412	−0.717788	−0.358894	+	0.933378i	$0.616846\pi$
$812$	0	0
$813$	−4.07499	−0.142916
$814$	0	0
$815$	0.453527	0.0158864
$816$	0	0
$817$	−45.1012	−1.57789
$818$	0	0
$819$	4.54850	0.158937
$820$	0	0
$821$	40.3421	1.40795	0.703974	−	0.710226i	$-0.251407\pi$
$821$	40.3421	1.40795	0.703974	+	0.710226i	$0.251407\pi$
$822$	0	0
$823$	48.6160	1.69465	0.847323	−	0.531078i	$-0.178213\pi$
$823$	48.6160	1.69465	0.847323	+	0.531078i	$0.178213\pi$
$824$	0	0
$825$	3.81544	0.132836
$826$	0	0
$827$	39.6734	1.37958	0.689790	−	0.724009i	$-0.257703\pi$
$827$	39.6734	1.37958	0.689790	+	0.724009i	$0.257703\pi$
$828$	0	0
$829$	47.4157	1.64681	0.823407	−	0.567451i	$-0.192071\pi$
$829$	47.4157	1.64681	0.823407	+	0.567451i	$0.192071\pi$
$830$	0	0
$831$	−22.9380	−0.795710
$832$	0	0
$833$	37.6687	1.30514
$834$	0	0
$835$	−2.99382	−0.103606
$836$	0	0
$837$	−10.1421	−0.350562
$838$	0	0
$839$	8.74261	0.301828	0.150914	−	0.988547i	$-0.451778\pi$
$839$	8.74261	0.301828	0.150914	+	0.988547i	$0.451778\pi$
$840$	0	0
$841$	−19.7391	−0.680658
$842$	0	0
$843$	12.4485	0.428749
$844$	0	0
$845$	36.0658	1.24070
$846$	0	0
$847$	−46.9267	−1.61242
$848$	0	0
$849$	−21.2448	−0.729119
$850$	0	0
$851$	−67.3974	−2.31035
$852$	0	0
$853$	−2.82336	−0.0966701	−0.0483350	−	0.998831i	$-0.515392\pi$
$853$	−2.82336	−0.0966701	−0.0483350	+	0.998831i	$0.515392\pi$
$854$	0	0
$855$	25.2138	0.862293
$856$	0	0
$857$	−39.6895	−1.35577	−0.677883	−	0.735170i	$-0.737103\pi$
$857$	−39.6895	−1.35577	−0.677883	+	0.735170i	$0.737103\pi$
$858$	0	0
$859$	0.796057	0.0271611	0.0135805	−	0.999908i	$-0.495677\pi$
$859$	0.796057	0.0271611	0.0135805	+	0.999908i	$0.495677\pi$
$860$	0	0
$861$	−1.92463	−0.0655912
$862$	0	0
$863$	−7.43568	−0.253113	−0.126557	−	0.991959i	$-0.540393\pi$
$863$	−7.43568	−0.253113	−0.126557	+	0.991959i	$0.540393\pi$
$864$	0	0
$865$	−32.8099	−1.11557
$866$	0	0
$867$	10.4361	0.354427
$868$	0	0
$869$	−1.41703	−0.0480695
$870$	0	0
$871$	11.4875	0.389238
$872$	0	0
$873$	11.1603	0.377720
$874$	0	0
$875$	14.4634	0.488952
$876$	0	0
$877$	−13.0468	−0.440558	−0.220279	−	0.975437i	$-0.570697\pi$
$877$	−13.0468	−0.440558	−0.220279	+	0.975437i	$0.570697\pi$
$878$	0	0
$879$	−3.52870	−0.119020
$880$	0	0
$881$	39.0090	1.31425	0.657123	−	0.753783i	$-0.271773\pi$
$881$	39.0090	1.31425	0.657123	+	0.753783i	$0.271773\pi$
$882$	0	0
$883$	3.17514	0.106852	0.0534261	−	0.998572i	$-0.482986\pi$
$883$	3.17514	0.106852	0.0534261	+	0.998572i	$0.482986\pi$
$884$	0	0
$885$	0.303530	0.0102030
$886$	0	0
$887$	19.6773	0.660698	0.330349	−	0.943859i	$-0.392834\pi$
$887$	19.6773	0.660698	0.330349	+	0.943859i	$0.392834\pi$
$888$	0	0
$889$	73.7513	2.47354
$890$	0	0
$891$	−0.962769	−0.0322540
$892$	0	0
$893$	−71.3444	−2.38745
$894$	0	0
$895$	43.8097	1.46440
$896$	0	0
$897$	6.14213	0.205080
$898$	0	0
$899$	−30.8642	−1.02938
$900$	0	0
$901$	−33.7547	−1.12453
$902$	0	0
$903$	−24.9479	−0.830214
$904$	0	0
$905$	13.7723	0.457805
$906$	0	0
$907$	13.3941	0.444745	0.222372	−	0.974962i	$-0.428620\pi$
$907$	13.3941	0.444745	0.222372	+	0.974962i	$0.428620\pi$
$908$	0	0
$909$	−5.03063	−0.166855
$910$	0	0
$911$	−14.3696	−0.476085	−0.238043	−	0.971255i	$-0.576506\pi$
$911$	−14.3696	−0.476085	−0.238043	+	0.971255i	$0.576506\pi$
$912$	0	0
$913$	−4.39868	−0.145575
$914$	0	0
$915$	−43.0245	−1.42234
$916$	0	0
$917$	27.0554	0.893447
$918$	0	0
$919$	16.6199	0.548241	0.274121	−	0.961695i	$-0.411613\pi$
$919$	16.6199	0.548241	0.274121	+	0.961695i	$0.411613\pi$
$920$	0	0
$921$	−15.5946	−0.513860
$922$	0	0
$923$	−12.2300	−0.402554
$924$	0	0
$925$	42.4577	1.39600
$926$	0	0
$927$	−7.19615	−0.236352
$928$	0	0
$929$	−17.7181	−0.581313	−0.290657	−	0.956827i	$-0.593874\pi$
$929$	−17.7181	−0.581313	−0.290657	+	0.956827i	$0.593874\pi$
$930$	0	0
$931$	−123.826	−4.05822
$932$	0	0
$933$	−0.111395	−0.00364691
$934$	0	0
$935$	7.38466	0.241504
$936$	0	0
$937$	56.4997	1.84576	0.922882	−	0.385083i	$-0.125827\pi$
$937$	56.4997	1.84576	0.922882	+	0.385083i	$0.125827\pi$
$938$	0	0
$939$	15.8427	0.517008
$940$	0	0
$941$	28.4198	0.926460	0.463230	−	0.886238i	$-0.346690\pi$
$941$	28.4198	0.926460	0.463230	+	0.886238i	$0.346690\pi$
$942$	0	0
$943$	−2.59895	−0.0846335
$944$	0	0
$945$	13.9471	0.453699
$946$	0	0
$947$	−27.1922	−0.883628	−0.441814	−	0.897107i	$-0.645665\pi$
$947$	−27.1922	−0.883628	−0.441814	+	0.897107i	$0.645665\pi$
$948$	0	0
$949$	−10.7687	−0.349567
$950$	0	0
$951$	−18.3265	−0.594276
$952$	0	0
$953$	−40.0549	−1.29751	−0.648753	−	0.760999i	$-0.724709\pi$
$953$	−40.0549	−1.29751	−0.648753	+	0.760999i	$0.724709\pi$
$954$	0	0
$955$	−51.6647	−1.67183
$956$	0	0
$957$	−2.92988	−0.0947095
$958$	0	0
$959$	−95.9902	−3.09968
$960$	0	0
$961$	71.8622	2.31814
$962$	0	0
$963$	14.9736	0.482518
$964$	0	0
$965$	34.8778	1.12276
$966$	0	0
$967$	−28.2604	−0.908795	−0.454397	−	0.890799i	$-0.650145\pi$
$967$	−28.2604	−0.908795	−0.454397	+	0.890799i	$0.650145\pi$
$968$	0	0
$969$	21.5771	0.693158
$970$	0	0
$971$	−57.3946	−1.84188	−0.920940	−	0.389704i	$-0.872577\pi$
$971$	−57.3946	−1.84188	−0.920940	+	0.389704i	$0.872577\pi$
$972$	0	0
$973$	35.3291	1.13260
$974$	0	0
$975$	−3.86930	−0.123917
$976$	0	0
$977$	32.6415	1.04429	0.522147	−	0.852855i	$-0.325131\pi$
$977$	32.6415	1.04429	0.522147	+	0.852855i	$0.325131\pi$
$978$	0	0
$979$	−7.90244	−0.252563
$980$	0	0
$981$	−1.32527	−0.0423127
$982$	0	0
$983$	6.11976	0.195190	0.0975950	−	0.995226i	$-0.468885\pi$
$983$	6.11976	0.195190	0.0975950	+	0.995226i	$0.468885\pi$
$984$	0	0
$985$	−32.5551	−1.03729
$986$	0	0
$987$	−39.4644	−1.25617
$988$	0	0
$989$	−33.6888	−1.07124
$990$	0	0
$991$	31.2483	0.992636	0.496318	−	0.868141i	$-0.334685\pi$
$991$	31.2483	0.992636	0.496318	+	0.868141i	$0.334685\pi$
$992$	0	0
$993$	−6.26802	−0.198910
$994$	0	0
$995$	27.2814	0.864878
$996$	0	0
$997$	15.4045	0.487866	0.243933	−	0.969792i	$-0.421562\pi$
$997$	15.4045	0.487866	0.243933	+	0.969792i	$0.421562\pi$
$998$	0	0
$999$	−10.7136	−0.338963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.ba.1.2		9
4.3	odd	2		4008.2.a.i.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.i.1.2	✓	9	4.3	odd	2
8016.2.a.ba.1.2		9	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.99382 q^{5} +4.65862 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.99382 q^{5} +4.65862 q^{7} +1.00000 q^{9} -0.962769 q^{11} +0.976361 q^{13} +2.99382 q^{15} +2.56202 q^{17} -8.42193 q^{19} -4.65862 q^{21} -6.29084 q^{23} +3.96298 q^{25} -1.00000 q^{27} -3.04318 q^{29} +10.1421 q^{31} +0.962769 q^{33} -13.9471 q^{35} +10.7136 q^{37} -0.976361 q^{39} +0.413133 q^{41} +5.35521 q^{43} -2.99382 q^{45} +8.47126 q^{47} +14.7028 q^{49} -2.56202 q^{51} -13.1750 q^{53} +2.88236 q^{55} +8.42193 q^{57} +0.101385 q^{59} -14.3711 q^{61} +4.65862 q^{63} -2.92305 q^{65} +11.7656 q^{67} +6.29084 q^{69} -12.5261 q^{71} -11.0294 q^{73} -3.96298 q^{75} -4.48518 q^{77} +1.47183 q^{79} +1.00000 q^{81} +4.56878 q^{83} -7.67023 q^{85} +3.04318 q^{87} +8.20803 q^{89} +4.54850 q^{91} -10.1421 q^{93} +25.2138 q^{95} +11.1603 q^{97} -0.962769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} + 6 q^{15} - 9 q^{17} + 8 q^{19} - 11 q^{21} + 7 q^{23} - q^{25} - 9 q^{27} - 9 q^{29} + 25 q^{31} + q^{33} - 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} + 24 q^{43} - 6 q^{45} + 16 q^{47} + 4 q^{49} + 9 q^{51} - 26 q^{53} + 29 q^{55} - 8 q^{57} + 4 q^{59} - 20 q^{61} + 11 q^{63} - 8 q^{65} + 25 q^{67} - 7 q^{69} + 15 q^{71} - 10 q^{73} + q^{75} - 20 q^{77} + 34 q^{79} + 9 q^{81} + 4 q^{83} - 13 q^{85} + 9 q^{87} + 13 q^{89} + 21 q^{91} - 25 q^{93} + 7 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9 - q^11 - 4 * q^13 + 6 * q^15 - 9 * q^17 + 8 * q^19 - 11 * q^21 + 7 * q^23 - q^25 - 9 * q^27 - 9 * q^29 + 25 * q^31 + q^33 - 5 * q^35 - 6 * q^37 + 4 * q^39 - 4 * q^41 + 24 * q^43 - 6 * q^45 + 16 * q^47 + 4 * q^49 + 9 * q^51 - 26 * q^53 + 29 * q^55 - 8 * q^57 + 4 * q^59 - 20 * q^61 + 11 * q^63 - 8 * q^65 + 25 * q^67 - 7 * q^69 + 15 * q^71 - 10 * q^73 + q^75 - 20 * q^77 + 34 * q^79 + 9 * q^81 + 4 * q^83 - 13 * q^85 + 9 * q^87 + 13 * q^89 + 21 * q^91 - 25 * q^93 + 7 * q^95 - 4 * q^97 - q^99

Introduction

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