LMFDB - Embedded newform 8016.2.a.o.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:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.679643$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.416566 q^{5} -4.65960 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.416566 q^{5} -4.65960 q^{7} +1.00000 q^{9} +0.283116 q^{13} +0.416566 q^{15} +0.640714 q^{17} +5.88544 q^{19} -4.65960 q^{21} -5.88544 q^{23} -4.82647 q^{25} +1.00000 q^{27} +5.16687 q^{29} -1.22584 q^{31} -1.94103 q^{35} -6.82816 q^{37} +0.283116 q^{39} +12.4116 q^{41} +3.07617 q^{43} +0.416566 q^{45} -7.82647 q^{47} +14.7119 q^{49} +0.640714 q^{51} -5.46888 q^{53} +5.88544 q^{57} -3.20962 q^{59} -1.16687 q^{61} -4.65960 q^{63} +0.117936 q^{65} +2.29863 q^{67} -5.88544 q^{69} +8.60401 q^{71} +2.26690 q^{73} -4.82647 q^{75} +7.69302 q^{79} +1.00000 q^{81} -9.27646 q^{83} +0.266899 q^{85} +5.16687 q^{87} +3.94103 q^{89} -1.31921 q^{91} -1.22584 q^{93} +2.45167 q^{95} +8.99334 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9 + 8 * q^13 + 5 * q^15 + 10 * q^17 + 2 * q^19 - q^21 - 2 * q^23 + 5 * q^25 + 4 * q^27 + 14 * q^29 - q^31 - 5 * q^35 + 5 * q^37 + 8 * q^39 + 14 * q^41 - 2 * q^43 + 5 * q^45 - 7 * q^47 + 13 * q^49 + 10 * q^51 + 3 * q^53 + 2 * q^57 + 5 * q^59 + 2 * q^61 - q^63 + 6 * q^65 + 7 * q^67 - 2 * q^69 - 2 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + 4 * q^81 - 13 * q^83 - 6 * q^85 + 14 * q^87 + 13 * q^89 + 30 * q^91 - q^93 + 2 * q^95 + 5 * q^97

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
$5$	0.416566	0.186294	0.0931469	−	0.995652i	$-0.470307\pi$
$5$	0.416566	0.186294	0.0931469	+	0.995652i	$0.470307\pi$
$6$	0	0
$7$	−4.65960	−1.76116	−0.880582	−	0.473893i	$-0.842848\pi$
$7$	−4.65960	−1.76116	−0.880582	+	0.473893i	$0.842848\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0.283116	0.0785223	0.0392611	−	0.999229i	$-0.487500\pi$
$13$	0.283116	0.0785223	0.0392611	+	0.999229i	$0.487500\pi$
$14$	0	0
$15$	0.416566	0.107557
$16$	0	0
$17$	0.640714	0.155396	0.0776979	−	0.996977i	$-0.475243\pi$
$17$	0.640714	0.155396	0.0776979	+	0.996977i	$0.475243\pi$
$18$	0	0
$19$	5.88544	1.35021	0.675106	−	0.737720i	$-0.264098\pi$
$19$	5.88544	1.35021	0.675106	+	0.737720i	$0.264098\pi$
$20$	0	0
$21$	−4.65960	−1.01681
$22$	0	0
$23$	−5.88544	−1.22720	−0.613600	−	0.789617i	$-0.710279\pi$
$23$	−5.88544	−1.22720	−0.613600	+	0.789617i	$0.710279\pi$
$24$	0	0
$25$	−4.82647	−0.965295
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.16687	0.959463	0.479732	−	0.877415i	$-0.340734\pi$
$29$	5.16687	0.959463	0.479732	+	0.877415i	$0.340734\pi$
$30$	0	0
$31$	−1.22584	−0.220167	−0.110083	−	0.993922i	$-0.535112\pi$
$31$	−1.22584	−0.220167	−0.110083	+	0.993922i	$0.535112\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.94103	−0.328094
$36$	0	0
$37$	−6.82816	−1.12254	−0.561271	−	0.827632i	$-0.689688\pi$
$37$	−6.82816	−1.12254	−0.561271	+	0.827632i	$0.689688\pi$
$38$	0	0
$39$	0.283116	0.0453349
$40$	0	0
$41$	12.4116	1.93837	0.969183	−	0.246343i	$-0.0792288\pi$
$41$	12.4116	1.93837	0.969183	+	0.246343i	$0.0792288\pi$
$42$	0	0
$43$	3.07617	0.469112	0.234556	−	0.972103i	$-0.424636\pi$
$43$	3.07617	0.469112	0.234556	+	0.972103i	$0.424636\pi$
$44$	0	0
$45$	0.416566	0.0620980
$46$	0	0
$47$	−7.82647	−1.14161	−0.570804	−	0.821086i	$-0.693368\pi$
$47$	−7.82647	−1.14161	−0.570804	+	0.821086i	$0.693368\pi$
$48$	0	0
$49$	14.7119	2.10170
$50$	0	0
$51$	0.640714	0.0897179
$52$	0	0
$53$	−5.46888	−0.751208	−0.375604	−	0.926780i	$-0.622565\pi$
$53$	−5.46888	−0.751208	−0.375604	+	0.926780i	$0.622565\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.88544	0.779546
$58$	0	0
$59$	−3.20962	−0.417857	−0.208928	−	0.977931i	$-0.566998\pi$
$59$	−3.20962	−0.417857	−0.208928	+	0.977931i	$0.566998\pi$
$60$	0	0
$61$	−1.16687	−0.149402	−0.0747011	−	0.997206i	$-0.523800\pi$
$61$	−1.16687	−0.149402	−0.0747011	+	0.997206i	$0.523800\pi$
$62$	0	0
$63$	−4.65960	−0.587055
$64$	0	0
$65$	0.117936	0.0146282
$66$	0	0
$67$	2.29863	0.280822	0.140411	−	0.990093i	$-0.455158\pi$
$67$	2.29863	0.280822	0.140411	+	0.990093i	$0.455158\pi$
$68$	0	0
$69$	−5.88544	−0.708524
$70$	0	0
$71$	8.60401	1.02111	0.510554	−	0.859846i	$-0.329440\pi$
$71$	8.60401	1.02111	0.510554	+	0.859846i	$0.329440\pi$
$72$	0	0
$73$	2.26690	0.265321	0.132660	−	0.991162i	$-0.457648\pi$
$73$	2.26690	0.265321	0.132660	+	0.991162i	$0.457648\pi$
$74$	0	0
$75$	−4.82647	−0.557313
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.69302	0.865533	0.432766	−	0.901506i	$-0.357538\pi$
$79$	7.69302	0.865533	0.432766	+	0.901506i	$0.357538\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.27646	−1.01822	−0.509112	−	0.860700i	$-0.670026\pi$
$83$	−9.27646	−1.01822	−0.509112	+	0.860700i	$0.670026\pi$
$84$	0	0
$85$	0.266899	0.0289493
$86$	0	0
$87$	5.16687	0.553946
$88$	0	0
$89$	3.94103	0.417749	0.208874	−	0.977943i	$-0.433020\pi$
$89$	3.94103	0.417749	0.208874	+	0.977943i	$0.433020\pi$
$90$	0	0
$91$	−1.31921	−0.138291
$92$	0	0
$93$	−1.22584	−0.127113
$94$	0	0
$95$	2.45167	0.251536
$96$	0	0
$97$	8.99334	0.913135	0.456568	−	0.889689i	$-0.349079\pi$
$97$	8.99334	0.913135	0.456568	+	0.889689i	$0.349079\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.3364	−1.22752	−0.613759	−	0.789493i	$-0.710344\pi$
$101$	−12.3364	−1.22752	−0.613759	+	0.789493i	$0.710344\pi$
$102$	0	0
$103$	12.3448	1.21637	0.608183	−	0.793797i	$-0.291899\pi$
$103$	12.3448	1.21637	0.608183	+	0.793797i	$0.291899\pi$
$104$	0	0
$105$	−1.94103	−0.189425
$106$	0	0
$107$	19.0867	1.84518	0.922591	−	0.385779i	$-0.126067\pi$
$107$	19.0867	1.84518	0.922591	+	0.385779i	$0.126067\pi$
$108$	0	0
$109$	17.2553	1.65276	0.826378	−	0.563116i	$-0.190398\pi$
$109$	17.2553	1.65276	0.826378	+	0.563116i	$0.190398\pi$
$110$	0	0
$111$	−6.82816	−0.648100
$112$	0	0
$113$	2.91099	0.273843	0.136921	−	0.990582i	$-0.456279\pi$
$113$	2.91099	0.273843	0.136921	+	0.990582i	$0.456279\pi$
$114$	0	0
$115$	−2.45167	−0.228620
$116$	0	0
$117$	0.283116	0.0261741
$118$	0	0
$119$	−2.98547	−0.273678
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	12.4116	1.11912
$124$	0	0
$125$	−4.09337	−0.366122
$126$	0	0
$127$	21.2636	1.88684	0.943421	−	0.331599i	$-0.107588\pi$
$127$	21.2636	1.88684	0.943421	+	0.331599i	$0.107588\pi$
$128$	0	0
$129$	3.07617	0.270842
$130$	0	0
$131$	14.2030	1.24092	0.620459	−	0.784239i	$-0.286946\pi$
$131$	14.2030	1.24092	0.620459	+	0.784239i	$0.286946\pi$
$132$	0	0
$133$	−27.4238	−2.37795
$134$	0	0
$135$	0.416566	0.0358523
$136$	0	0
$137$	−10.5484	−0.901213	−0.450606	−	0.892723i	$-0.648792\pi$
$137$	−10.5484	−0.901213	−0.450606	+	0.892723i	$0.648792\pi$
$138$	0	0
$139$	−11.9550	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$139$	−11.9550	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$140$	0	0
$141$	−7.82647	−0.659108
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.15234	0.178742
$146$	0	0
$147$	14.7119	1.21342
$148$	0	0
$149$	3.58343	0.293566	0.146783	−	0.989169i	$-0.453108\pi$
$149$	3.58343	0.293566	0.146783	+	0.989169i	$0.453108\pi$
$150$	0	0
$151$	3.74074	0.304417	0.152209	−	0.988348i	$-0.451361\pi$
$151$	3.74074	0.304417	0.152209	+	0.988348i	$0.451361\pi$
$152$	0	0
$153$	0.640714	0.0517986
$154$	0	0
$155$	−0.510642	−0.0410157
$156$	0	0
$157$	−13.7709	−1.09904	−0.549518	−	0.835482i	$-0.685189\pi$
$157$	−13.7709	−1.09904	−0.549518	+	0.835482i	$0.685189\pi$
$158$	0	0
$159$	−5.46888	−0.433710
$160$	0	0
$161$	27.4238	2.16130
$162$	0	0
$163$	−11.2398	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$163$	−11.2398	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.9198	−0.993834
$170$	0	0
$171$	5.88544	0.450071
$172$	0	0
$173$	1.16687	0.0887154	0.0443577	−	0.999016i	$-0.485876\pi$
$173$	1.16687	0.0887154	0.0443577	+	0.999016i	$0.485876\pi$
$174$	0	0
$175$	22.4895	1.70004
$176$	0	0
$177$	−3.20962	−0.241250
$178$	0	0
$179$	−7.77088	−0.580823	−0.290412	−	0.956902i	$-0.593792\pi$
$179$	−7.77088	−0.580823	−0.290412	+	0.956902i	$0.593792\pi$
$180$	0	0
$181$	10.2570	0.762394	0.381197	−	0.924494i	$-0.375512\pi$
$181$	10.2570	0.762394	0.381197	+	0.924494i	$0.375512\pi$
$182$	0	0
$183$	−1.16687	−0.0862574
$184$	0	0
$185$	−2.84438	−0.209123
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.65960	−0.338936
$190$	0	0
$191$	13.5630	0.981381	0.490690	−	0.871334i	$-0.336745\pi$
$191$	13.5630	0.981381	0.490690	+	0.871334i	$0.336745\pi$
$192$	0	0
$193$	−2.56623	−0.184721	−0.0923607	−	0.995726i	$-0.529441\pi$
$193$	−2.56623	−0.184721	−0.0923607	+	0.995726i	$0.529441\pi$
$194$	0	0
$195$	0.117936	0.00844561
$196$	0	0
$197$	−2.92383	−0.208314	−0.104157	−	0.994561i	$-0.533214\pi$
$197$	−2.92383	−0.208314	−0.104157	+	0.994561i	$0.533214\pi$
$198$	0	0
$199$	3.43377	0.243413	0.121707	−	0.992566i	$-0.461163\pi$
$199$	3.43377	0.243413	0.121707	+	0.992566i	$0.461163\pi$
$200$	0	0
$201$	2.29863	0.162133
$202$	0	0
$203$	−24.0756	−1.68977
$204$	0	0
$205$	5.17025	0.361106
$206$	0	0
$207$	−5.88544	−0.409066
$208$	0	0
$209$	0	0
$210$	0	0
$211$	21.8854	1.50666	0.753328	−	0.657645i	$-0.228447\pi$
$211$	21.8854	1.50666	0.753328	+	0.657645i	$0.228447\pi$
$212$	0	0
$213$	8.60401	0.589537
$214$	0	0
$215$	1.28143	0.0873926
$216$	0	0
$217$	5.71191	0.387750
$218$	0	0
$219$	2.26690	0.153183
$220$	0	0
$221$	0.181396	0.0122020
$222$	0	0
$223$	4.65960	0.312030	0.156015	−	0.987755i	$-0.450135\pi$
$223$	4.65960	0.312030	0.156015	+	0.987755i	$0.450135\pi$
$224$	0	0
$225$	−4.82647	−0.321765
$226$	0	0
$227$	22.4467	1.48984	0.744920	−	0.667154i	$-0.232488\pi$
$227$	22.4467	1.48984	0.744920	+	0.667154i	$0.232488\pi$
$228$	0	0
$229$	12.3715	0.817533	0.408766	−	0.912639i	$-0.365959\pi$
$229$	12.3715	0.817533	0.408766	+	0.912639i	$0.365959\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.5940	1.28364	0.641822	−	0.766854i	$-0.278179\pi$
$233$	19.5940	1.28364	0.641822	+	0.766854i	$0.278179\pi$
$234$	0	0
$235$	−3.26024	−0.212675
$236$	0	0
$237$	7.69302	0.499716
$238$	0	0
$239$	−3.43377	−0.222112	−0.111056	−	0.993814i	$-0.535423\pi$
$239$	−3.43377	−0.222112	−0.111056	+	0.993814i	$0.535423\pi$
$240$	0	0
$241$	14.4861	0.933130	0.466565	−	0.884487i	$-0.345491\pi$
$241$	14.4861	0.933130	0.466565	+	0.884487i	$0.345491\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.12848	0.391534
$246$	0	0
$247$	1.66626	0.106022
$248$	0	0
$249$	−9.27646	−0.587872
$250$	0	0
$251$	26.8576	1.69524	0.847618	−	0.530607i	$-0.178036\pi$
$251$	26.8576	1.69524	0.847618	+	0.530607i	$0.178036\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.266899	0.0167139
$256$	0	0
$257$	−3.57846	−0.223218	−0.111609	−	0.993752i	$-0.535600\pi$
$257$	−3.57846	−0.223218	−0.111609	+	0.993752i	$0.535600\pi$
$258$	0	0
$259$	31.8165	1.97698
$260$	0	0
$261$	5.16687	0.319821
$262$	0	0
$263$	−5.52517	−0.340697	−0.170348	−	0.985384i	$-0.554489\pi$
$263$	−5.52517	−0.340697	−0.170348	+	0.985384i	$0.554489\pi$
$264$	0	0
$265$	−2.27815	−0.139945
$266$	0	0
$267$	3.94103	0.241187
$268$	0	0
$269$	29.0039	1.76840	0.884199	−	0.467110i	$-0.154705\pi$
$269$	29.0039	1.76840	0.884199	+	0.467110i	$0.154705\pi$
$270$	0	0
$271$	24.0301	1.45973	0.729863	−	0.683593i	$-0.239584\pi$
$271$	24.0301	1.45973	0.729863	+	0.683593i	$0.239584\pi$
$272$	0	0
$273$	−1.31921	−0.0798422
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.99382	0.179881	0.0899406	−	0.995947i	$-0.471332\pi$
$277$	2.99382	0.179881	0.0899406	+	0.995947i	$0.471332\pi$
$278$	0	0
$279$	−1.22584	−0.0733889
$280$	0	0
$281$	1.72523	0.102919	0.0514593	−	0.998675i	$-0.483613\pi$
$281$	1.72523	0.102919	0.0514593	+	0.998675i	$0.483613\pi$
$282$	0	0
$283$	−18.5372	−1.10192	−0.550960	−	0.834531i	$-0.685738\pi$
$283$	−18.5372	−1.10192	−0.550960	+	0.834531i	$0.685738\pi$
$284$	0	0
$285$	2.45167	0.145225
$286$	0	0
$287$	−57.8331	−3.41378
$288$	0	0
$289$	−16.5895	−0.975852
$290$	0	0
$291$	8.99334	0.527199
$292$	0	0
$293$	14.8331	0.866561	0.433280	−	0.901259i	$-0.357356\pi$
$293$	14.8331	0.866561	0.433280	+	0.901259i	$0.357356\pi$
$294$	0	0
$295$	−1.33702	−0.0778442
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.66626	−0.0963625
$300$	0	0
$301$	−14.3337	−0.826183
$302$	0	0
$303$	−12.3364	−0.708708
$304$	0	0
$305$	−0.486077	−0.0278327
$306$	0	0
$307$	31.9927	1.82592	0.912961	−	0.408047i	$-0.133790\pi$
$307$	31.9927	1.82592	0.912961	+	0.408047i	$0.133790\pi$
$308$	0	0
$309$	12.3448	0.702269
$310$	0	0
$311$	7.07884	0.401404	0.200702	−	0.979652i	$-0.435678\pi$
$311$	7.07884	0.401404	0.200702	+	0.979652i	$0.435678\pi$
$312$	0	0
$313$	23.6496	1.33675	0.668376	−	0.743823i	$-0.266990\pi$
$313$	23.6496	1.33675	0.668376	+	0.743823i	$0.266990\pi$
$314$	0	0
$315$	−1.94103	−0.109365
$316$	0	0
$317$	−13.7007	−0.769506	−0.384753	−	0.923020i	$-0.625713\pi$
$317$	−13.7007	−0.769506	−0.384753	+	0.923020i	$0.625713\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	19.0867	1.06532
$322$	0	0
$323$	3.77088	0.209818
$324$	0	0
$325$	−1.36645	−0.0757971
$326$	0	0
$327$	17.2553	0.954219
$328$	0	0
$329$	36.4683	2.01056
$330$	0	0
$331$	9.40991	0.517215	0.258608	−	0.965982i	$-0.416736\pi$
$331$	9.40991	0.517215	0.258608	+	0.965982i	$0.416736\pi$
$332$	0	0
$333$	−6.82816	−0.374181
$334$	0	0
$335$	0.957530	0.0523155
$336$	0	0
$337$	21.9345	1.19485	0.597423	−	0.801926i	$-0.296191\pi$
$337$	21.9345	1.19485	0.597423	+	0.801926i	$0.296191\pi$
$338$	0	0
$339$	2.91099	0.158103
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−35.9345	−1.94028
$344$	0	0
$345$	−2.45167	−0.131994
$346$	0	0
$347$	−16.6468	−0.893645	−0.446823	−	0.894623i	$-0.647444\pi$
$347$	−16.6468	−0.893645	−0.446823	+	0.894623i	$0.647444\pi$
$348$	0	0
$349$	4.49442	0.240581	0.120291	−	0.992739i	$-0.461617\pi$
$349$	4.49442	0.240581	0.120291	+	0.992739i	$0.461617\pi$
$350$	0	0
$351$	0.283116	0.0151116
$352$	0	0
$353$	9.72523	0.517622	0.258811	−	0.965928i	$-0.416669\pi$
$353$	9.72523	0.517622	0.258811	+	0.965928i	$0.416669\pi$
$354$	0	0
$355$	3.58414	0.190226
$356$	0	0
$357$	−2.98547	−0.158008
$358$	0	0
$359$	9.61854	0.507647	0.253824	−	0.967251i	$-0.418312\pi$
$359$	9.61854	0.507647	0.253824	+	0.967251i	$0.418312\pi$
$360$	0	0
$361$	15.6384	0.823075
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	0.944313	0.0494276
$366$	0	0
$367$	−14.1523	−0.738746	−0.369373	−	0.929281i	$-0.620428\pi$
$367$	−14.1523	−0.738746	−0.369373	+	0.929281i	$0.620428\pi$
$368$	0	0
$369$	12.4116	0.646122
$370$	0	0
$371$	25.4828	1.32300
$372$	0	0
$373$	−28.1149	−1.45574	−0.727868	−	0.685717i	$-0.759489\pi$
$373$	−28.1149	−1.45574	−0.727868	+	0.685717i	$0.759489\pi$
$374$	0	0
$375$	−4.09337	−0.211381
$376$	0	0
$377$	1.46282	0.0753392
$378$	0	0
$379$	−16.5212	−0.848636	−0.424318	−	0.905513i	$-0.639486\pi$
$379$	−16.5212	−0.848636	−0.424318	+	0.905513i	$0.639486\pi$
$380$	0	0
$381$	21.2636	1.08937
$382$	0	0
$383$	15.6907	0.801759	0.400879	−	0.916131i	$-0.368705\pi$
$383$	15.6907	0.801759	0.400879	+	0.916131i	$0.368705\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.07617	0.156371
$388$	0	0
$389$	−21.1842	−1.07408	−0.537040	−	0.843557i	$-0.680458\pi$
$389$	−21.1842	−1.07408	−0.537040	+	0.843557i	$0.680458\pi$
$390$	0	0
$391$	−3.77088	−0.190702
$392$	0	0
$393$	14.2030	0.716445
$394$	0	0
$395$	3.20465	0.161243
$396$	0	0
$397$	−2.03778	−0.102273	−0.0511367	−	0.998692i	$-0.516284\pi$
$397$	−2.03778	−0.102273	−0.0511367	+	0.998692i	$0.516284\pi$
$398$	0	0
$399$	−27.4238	−1.37291
$400$	0	0
$401$	−11.3971	−0.569142	−0.284571	−	0.958655i	$-0.591851\pi$
$401$	−11.3971	−0.569142	−0.284571	+	0.958655i	$0.591851\pi$
$402$	0	0
$403$	−0.347054	−0.0172880
$404$	0	0
$405$	0.416566	0.0206993
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−18.1012	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$409$	−18.1012	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$410$	0	0
$411$	−10.5484	−0.520315
$412$	0	0
$413$	14.9556	0.735915
$414$	0	0
$415$	−3.86425	−0.189689
$416$	0	0
$417$	−11.9550	−0.585437
$418$	0	0
$419$	−15.2047	−0.742796	−0.371398	−	0.928474i	$-0.621121\pi$
$419$	−15.2047	−0.742796	−0.371398	+	0.928474i	$0.621121\pi$
$420$	0	0
$421$	16.2914	0.793992	0.396996	−	0.917820i	$-0.370053\pi$
$421$	16.2914	0.793992	0.396996	+	0.917820i	$0.370053\pi$
$422$	0	0
$423$	−7.82647	−0.380536
$424$	0	0
$425$	−3.09239	−0.150003
$426$	0	0
$427$	5.43715	0.263122
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9.53052	−0.459069	−0.229534	−	0.973301i	$-0.573720\pi$
$431$	−9.53052	−0.459069	−0.229534	+	0.973301i	$0.573720\pi$
$432$	0	0
$433$	−28.5027	−1.36975	−0.684876	−	0.728660i	$-0.740143\pi$
$433$	−28.5027	−1.36975	−0.684876	+	0.728660i	$0.740143\pi$
$434$	0	0
$435$	2.15234	0.103197
$436$	0	0
$437$	−34.6384	−1.65698
$438$	0	0
$439$	38.0222	1.81470	0.907350	−	0.420377i	$-0.138102\pi$
$439$	38.0222	1.81470	0.907350	+	0.420377i	$0.138102\pi$
$440$	0	0
$441$	14.7119	0.700567
$442$	0	0
$443$	−6.29633	−0.299148	−0.149574	−	0.988751i	$-0.547790\pi$
$443$	−6.29633	−0.299148	−0.149574	+	0.988751i	$0.547790\pi$
$444$	0	0
$445$	1.64170	0.0778240
$446$	0	0
$447$	3.58343	0.169491
$448$	0	0
$449$	20.0967	0.948424	0.474212	−	0.880411i	$-0.342733\pi$
$449$	20.0967	0.948424	0.474212	+	0.880411i	$0.342733\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.74074	0.175756
$454$	0	0
$455$	−0.549537	−0.0257627
$456$	0	0
$457$	30.0278	1.40464	0.702322	−	0.711860i	$-0.252147\pi$
$457$	30.0278	1.40464	0.702322	+	0.711860i	$0.252147\pi$
$458$	0	0
$459$	0.640714	0.0299060
$460$	0	0
$461$	−1.21459	−0.0565691	−0.0282845	−	0.999600i	$-0.509004\pi$
$461$	−1.21459	−0.0565691	−0.0282845	+	0.999600i	$0.509004\pi$
$462$	0	0
$463$	−6.80637	−0.316319	−0.158159	−	0.987414i	$-0.550556\pi$
$463$	−6.80637	−0.316319	−0.158159	+	0.987414i	$0.550556\pi$
$464$	0	0
$465$	−0.510642	−0.0236804
$466$	0	0
$467$	−16.5563	−0.766134	−0.383067	−	0.923721i	$-0.625132\pi$
$467$	−16.5563	−0.766134	−0.383067	+	0.923721i	$0.625132\pi$
$468$	0	0
$469$	−10.7107	−0.494574
$470$	0	0
$471$	−13.7709	−0.634529
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−28.4059	−1.30335
$476$	0	0
$477$	−5.46888	−0.250403
$478$	0	0
$479$	−36.6629	−1.67517	−0.837585	−	0.546307i	$-0.816033\pi$
$479$	−36.6629	−1.67517	−0.837585	+	0.546307i	$0.816033\pi$
$480$	0	0
$481$	−1.93316	−0.0881446
$482$	0	0
$483$	27.4238	1.24783
$484$	0	0
$485$	3.74632	0.170112
$486$	0	0
$487$	−9.01223	−0.408383	−0.204192	−	0.978931i	$-0.565457\pi$
$487$	−9.01223	−0.408383	−0.204192	+	0.978931i	$0.565457\pi$
$488$	0	0
$489$	−11.2398	−0.508279
$490$	0	0
$491$	34.5430	1.55890	0.779451	−	0.626463i	$-0.215498\pi$
$491$	34.5430	1.55890	0.779451	+	0.626463i	$0.215498\pi$
$492$	0	0
$493$	3.31048	0.149097
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−40.0913	−1.79834
$498$	0	0
$499$	35.3331	1.58173	0.790864	−	0.611992i	$-0.209631\pi$
$499$	35.3331	1.58173	0.790864	+	0.611992i	$0.209631\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	14.0245	0.625320	0.312660	−	0.949865i	$-0.398780\pi$
$503$	14.0245	0.625320	0.312660	+	0.949865i	$0.398780\pi$
$504$	0	0
$505$	−5.13893	−0.228679
$506$	0	0
$507$	−12.9198	−0.573790
$508$	0	0
$509$	−7.01453	−0.310913	−0.155457	−	0.987843i	$-0.549685\pi$
$509$	−7.01453	−0.310913	−0.155457	+	0.987843i	$0.549685\pi$
$510$	0	0
$511$	−10.5629	−0.467273
$512$	0	0
$513$	5.88544	0.259849
$514$	0	0
$515$	5.14240	0.226601
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1.16687	0.0512198
$520$	0	0
$521$	3.70634	0.162378	0.0811889	−	0.996699i	$-0.474128\pi$
$521$	3.70634	0.162378	0.0811889	+	0.996699i	$0.474128\pi$
$522$	0	0
$523$	−34.4417	−1.50603	−0.753016	−	0.658002i	$-0.771402\pi$
$523$	−34.4417	−1.50603	−0.753016	+	0.658002i	$0.771402\pi$
$524$	0	0
$525$	22.4895	0.981520
$526$	0	0
$527$	−0.785410	−0.0342130
$528$	0	0
$529$	11.6384	0.506018
$530$	0	0
$531$	−3.20962	−0.139286
$532$	0	0
$533$	3.51392	0.152205
$534$	0	0
$535$	7.95087	0.343746
$536$	0	0
$537$	−7.77088	−0.335338
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.07378	0.0891587	0.0445793	−	0.999006i	$-0.485805\pi$
$541$	2.07378	0.0891587	0.0445793	+	0.999006i	$0.485805\pi$
$542$	0	0
$543$	10.2570	0.440168
$544$	0	0
$545$	7.18796	0.307898
$546$	0	0
$547$	−37.0928	−1.58597	−0.792986	−	0.609240i	$-0.791475\pi$
$547$	−37.0928	−1.58597	−0.792986	+	0.609240i	$0.791475\pi$
$548$	0	0
$549$	−1.16687	−0.0498007
$550$	0	0
$551$	30.4093	1.29548
$552$	0	0
$553$	−35.8464	−1.52435
$554$	0	0
$555$	−2.84438	−0.120737
$556$	0	0
$557$	8.75635	0.371019	0.185509	−	0.982643i	$-0.440607\pi$
$557$	8.75635	0.371019	0.185509	+	0.982643i	$0.440607\pi$
$558$	0	0
$559$	0.870913	0.0368357
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.41924	0.101959	0.0509794	−	0.998700i	$-0.483766\pi$
$563$	2.41924	0.101959	0.0509794	+	0.998700i	$0.483766\pi$
$564$	0	0
$565$	1.21262	0.0510153
$566$	0	0
$567$	−4.65960	−0.195685
$568$	0	0
$569$	−1.68627	−0.0706920	−0.0353460	−	0.999375i	$-0.511253\pi$
$569$	−1.68627	−0.0706920	−0.0353460	+	0.999375i	$0.511253\pi$
$570$	0	0
$571$	1.66359	0.0696190	0.0348095	−	0.999394i	$-0.488918\pi$
$571$	1.66359	0.0696190	0.0348095	+	0.999394i	$0.488918\pi$
$572$	0	0
$573$	13.5630	0.566600
$574$	0	0
$575$	28.4059	1.18461
$576$	0	0
$577$	−31.7020	−1.31977	−0.659885	−	0.751366i	$-0.729395\pi$
$577$	−31.7020	−1.31977	−0.659885	+	0.751366i	$0.729395\pi$
$578$	0	0
$579$	−2.56623	−0.106649
$580$	0	0
$581$	43.2246	1.79326
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0.117936	0.00487607
$586$	0	0
$587$	−23.0817	−0.952686	−0.476343	−	0.879260i	$-0.658038\pi$
$587$	−23.0817	−0.952686	−0.476343	+	0.879260i	$0.658038\pi$
$588$	0	0
$589$	−7.21459	−0.297272
$590$	0	0
$591$	−2.92383	−0.120270
$592$	0	0
$593$	21.0156	0.863008	0.431504	−	0.902111i	$-0.357983\pi$
$593$	21.0156	0.863008	0.431504	+	0.902111i	$0.357983\pi$
$594$	0	0
$595$	−1.24365	−0.0509845
$596$	0	0
$597$	3.43377	0.140535
$598$	0	0
$599$	0.274769	0.0112267	0.00561337	−	0.999984i	$-0.498213\pi$
$599$	0.274769	0.0112267	0.00561337	+	0.999984i	$0.498213\pi$
$600$	0	0
$601$	−42.1590	−1.71970	−0.859851	−	0.510546i	$-0.829443\pi$
$601$	−42.1590	−1.71970	−0.859851	+	0.510546i	$0.829443\pi$
$602$	0	0
$603$	2.29863	0.0936074
$604$	0	0
$605$	−4.58222	−0.186294
$606$	0	0
$607$	−36.3658	−1.47604	−0.738022	−	0.674777i	$-0.764240\pi$
$607$	−36.3658	−1.47604	−0.738022	+	0.674777i	$0.764240\pi$
$608$	0	0
$609$	−24.0756	−0.975591
$610$	0	0
$611$	−2.21580	−0.0896417
$612$	0	0
$613$	−28.3291	−1.14420	−0.572102	−	0.820183i	$-0.693872\pi$
$613$	−28.3291	−1.14420	−0.572102	+	0.820183i	$0.693872\pi$
$614$	0	0
$615$	5.17025	0.208484
$616$	0	0
$617$	44.7941	1.80334	0.901672	−	0.432421i	$-0.142340\pi$
$617$	44.7941	1.80334	0.901672	+	0.432421i	$0.142340\pi$
$618$	0	0
$619$	−43.9682	−1.76723	−0.883615	−	0.468214i	$-0.844898\pi$
$619$	−43.9682	−1.76723	−0.883615	+	0.468214i	$0.844898\pi$
$620$	0	0
$621$	−5.88544	−0.236175
$622$	0	0
$623$	−18.3636	−0.735724
$624$	0	0
$625$	22.4272	0.897088
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.37490	−0.174439
$630$	0	0
$631$	−15.8265	−0.630042	−0.315021	−	0.949085i	$-0.602012\pi$
$631$	−15.8265	−0.630042	−0.315021	+	0.949085i	$0.602012\pi$
$632$	0	0
$633$	21.8854	0.869868
$634$	0	0
$635$	8.85770	0.351507
$636$	0	0
$637$	4.16518	0.165030
$638$	0	0
$639$	8.60401	0.340370
$640$	0	0
$641$	−25.4792	−1.00637	−0.503184	−	0.864179i	$-0.667838\pi$
$641$	−25.4792	−1.00637	−0.503184	+	0.864179i	$0.667838\pi$
$642$	0	0
$643$	−20.6623	−0.814841	−0.407420	−	0.913241i	$-0.633572\pi$
$643$	−20.6623	−0.814841	−0.407420	+	0.913241i	$0.633572\pi$
$644$	0	0
$645$	1.28143	0.0504561
$646$	0	0
$647$	−1.24899	−0.0491030	−0.0245515	−	0.999699i	$-0.507816\pi$
$647$	−1.24899	−0.0491030	−0.0245515	+	0.999699i	$0.507816\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	5.71191	0.223868
$652$	0	0
$653$	3.27149	0.128023	0.0640116	−	0.997949i	$-0.479611\pi$
$653$	3.27149	0.128023	0.0640116	+	0.997949i	$0.479611\pi$
$654$	0	0
$655$	5.91647	0.231176
$656$	0	0
$657$	2.26690	0.0884402
$658$	0	0
$659$	20.0805	0.782227	0.391113	−	0.920343i	$-0.372090\pi$
$659$	20.0805	0.782227	0.391113	+	0.920343i	$0.372090\pi$
$660$	0	0
$661$	−24.6738	−0.959698	−0.479849	−	0.877351i	$-0.659309\pi$
$661$	−24.6738	−0.959698	−0.479849	+	0.877351i	$0.659309\pi$
$662$	0	0
$663$	0.181396	0.00704485
$664$	0	0
$665$	−11.4238	−0.442997
$666$	0	0
$667$	−30.4093	−1.17745
$668$	0	0
$669$	4.65960	0.180151
$670$	0	0
$671$	0	0
$672$	0	0
$673$	40.9688	1.57923	0.789615	−	0.613602i	$-0.210280\pi$
$673$	40.9688	1.57923	0.789615	+	0.613602i	$0.210280\pi$
$674$	0	0
$675$	−4.82647	−0.185771
$676$	0	0
$677$	−0.194714	−0.00748345	−0.00374172	−	0.999993i	$-0.501191\pi$
$677$	−0.194714	−0.00748345	−0.00374172	+	0.999993i	$0.501191\pi$
$678$	0	0
$679$	−41.9054	−1.60818
$680$	0	0
$681$	22.4467	0.860160
$682$	0	0
$683$	41.5156	1.58855	0.794275	−	0.607558i	$-0.207851\pi$
$683$	41.5156	1.58855	0.794275	+	0.607558i	$0.207851\pi$
$684$	0	0
$685$	−4.39411	−0.167890
$686$	0	0
$687$	12.3715	0.472003
$688$	0	0
$689$	−1.54833	−0.0589865
$690$	0	0
$691$	−30.0517	−1.14322	−0.571610	−	0.820525i	$-0.693681\pi$
$691$	−30.0517	−1.14322	−0.571610	+	0.820525i	$0.693681\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.98002	−0.188903
$696$	0	0
$697$	7.95228	0.301214
$698$	0	0
$699$	19.5940	0.741112
$700$	0	0
$701$	47.9934	1.81269	0.906344	−	0.422542i	$-0.138862\pi$
$701$	47.9934	1.81269	0.906344	+	0.422542i	$0.138862\pi$
$702$	0	0
$703$	−40.1867	−1.51567
$704$	0	0
$705$	−3.26024	−0.122788
$706$	0	0
$707$	57.4828	2.16186
$708$	0	0
$709$	26.5976	0.998895	0.499448	−	0.866344i	$-0.333536\pi$
$709$	26.5976	0.998895	0.499448	+	0.866344i	$0.333536\pi$
$710$	0	0
$711$	7.69302	0.288511
$712$	0	0
$713$	7.21459	0.270189
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.43377	−0.128236
$718$	0	0
$719$	−0.951068	−0.0354689	−0.0177344	−	0.999843i	$-0.505645\pi$
$719$	−0.951068	−0.0354689	−0.0177344	+	0.999843i	$0.505645\pi$
$720$	0	0
$721$	−57.5217	−2.14222
$722$	0	0
$723$	14.4861	0.538743
$724$	0	0
$725$	−24.9378	−0.926165
$726$	0	0
$727$	5.91558	0.219397	0.109698	−	0.993965i	$-0.465012\pi$
$727$	5.91558	0.219397	0.109698	+	0.993965i	$0.465012\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.97094	0.0728980
$732$	0	0
$733$	−5.27487	−0.194832	−0.0974158	−	0.995244i	$-0.531058\pi$
$733$	−5.27487	−0.194832	−0.0974158	+	0.995244i	$0.531058\pi$
$734$	0	0
$735$	6.12848	0.226052
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−7.61925	−0.280278	−0.140139	−	0.990132i	$-0.544755\pi$
$739$	−7.61925	−0.280278	−0.140139	+	0.990132i	$0.544755\pi$
$740$	0	0
$741$	1.66626	0.0612117
$742$	0	0
$743$	−35.4470	−1.30042	−0.650212	−	0.759753i	$-0.725320\pi$
$743$	−35.4470	−1.30042	−0.650212	+	0.759753i	$0.725320\pi$
$744$	0	0
$745$	1.49274	0.0546896
$746$	0	0
$747$	−9.27646	−0.339408
$748$	0	0
$749$	−88.9365	−3.24967
$750$	0	0
$751$	22.2680	0.812570	0.406285	−	0.913746i	$-0.366824\pi$
$751$	22.2680	0.812570	0.406285	+	0.913746i	$0.366824\pi$
$752$	0	0
$753$	26.8576	0.978745
$754$	0	0
$755$	1.55827	0.0567111
$756$	0	0
$757$	25.1259	0.913216	0.456608	−	0.889668i	$-0.349064\pi$
$757$	25.1259	0.913216	0.456608	+	0.889668i	$0.349064\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.0755	0.691485	0.345743	−	0.938329i	$-0.387627\pi$
$761$	19.0755	0.691485	0.345743	+	0.938329i	$0.387627\pi$
$762$	0	0
$763$	−80.4027	−2.91077
$764$	0	0
$765$	0.266899	0.00964977
$766$	0	0
$767$	−0.908695	−0.0328111
$768$	0	0
$769$	7.11577	0.256601	0.128301	−	0.991735i	$-0.459048\pi$
$769$	7.11577	0.256601	0.128301	+	0.991735i	$0.459048\pi$
$770$	0	0
$771$	−3.57846	−0.128875
$772$	0	0
$773$	38.4478	1.38287	0.691435	−	0.722438i	$-0.256979\pi$
$773$	38.4478	1.38287	0.691435	+	0.722438i	$0.256979\pi$
$774$	0	0
$775$	5.91647	0.212526
$776$	0	0
$777$	31.8165	1.14141
$778$	0	0
$779$	73.0477	2.61721
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.16687	0.184649
$784$	0	0
$785$	−5.73648	−0.204744
$786$	0	0
$787$	−37.5332	−1.33791	−0.668957	−	0.743301i	$-0.733259\pi$
$787$	−37.5332	−1.33791	−0.668957	+	0.743301i	$0.733259\pi$
$788$	0	0
$789$	−5.52517	−0.196701
$790$	0	0
$791$	−13.5641	−0.482283
$792$	0	0
$793$	−0.330359	−0.0117314
$794$	0	0
$795$	−2.27815	−0.0807975
$796$	0	0
$797$	36.2808	1.28513	0.642566	−	0.766230i	$-0.277870\pi$
$797$	36.2808	1.28513	0.642566	+	0.766230i	$0.277870\pi$
$798$	0	0
$799$	−5.01453	−0.177401
$800$	0	0
$801$	3.94103	0.139250
$802$	0	0
$803$	0	0
$804$	0	0
$805$	11.4238	0.402637
$806$	0	0
$807$	29.0039	1.02099
$808$	0	0
$809$	10.4305	0.366716	0.183358	−	0.983046i	$-0.441303\pi$
$809$	10.4305	0.366716	0.183358	+	0.983046i	$0.441303\pi$
$810$	0	0
$811$	−1.68468	−0.0591570	−0.0295785	−	0.999562i	$-0.509416\pi$
$811$	−1.68468	−0.0591570	−0.0295785	+	0.999562i	$0.509416\pi$
$812$	0	0
$813$	24.0301	0.842774
$814$	0	0
$815$	−4.68210	−0.164007
$816$	0	0
$817$	18.1046	0.633400
$818$	0	0
$819$	−1.31921	−0.0460969
$820$	0	0
$821$	1.38085	0.0481920	0.0240960	−	0.999710i	$-0.492329\pi$
$821$	1.38085	0.0481920	0.0240960	+	0.999710i	$0.492329\pi$
$822$	0	0
$823$	12.9488	0.451366	0.225683	−	0.974201i	$-0.427539\pi$
$823$	12.9488	0.451366	0.225683	+	0.974201i	$0.427539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.7924	0.896891	0.448446	−	0.893810i	$-0.351978\pi$
$827$	25.7924	0.896891	0.448446	+	0.893810i	$0.351978\pi$
$828$	0	0
$829$	−23.6347	−0.820866	−0.410433	−	0.911891i	$-0.634622\pi$
$829$	−23.6347	−0.820866	−0.410433	+	0.911891i	$0.634622\pi$
$830$	0	0
$831$	2.99382	0.103854
$832$	0	0
$833$	9.42612	0.326596
$834$	0	0
$835$	0.416566	0.0144159
$836$	0	0
$837$	−1.22584	−0.0423711
$838$	0	0
$839$	37.1047	1.28100	0.640499	−	0.767959i	$-0.278728\pi$
$839$	37.1047	1.28100	0.640499	+	0.767959i	$0.278728\pi$
$840$	0	0
$841$	−2.30347	−0.0794300
$842$	0	0
$843$	1.72523	0.0594201
$844$	0	0
$845$	−5.38197	−0.185145
$846$	0	0
$847$	51.2556	1.76116
$848$	0	0
$849$	−18.5372	−0.636194
$850$	0	0
$851$	40.1867	1.37758
$852$	0	0
$853$	3.93316	0.134669	0.0673345	−	0.997730i	$-0.478551\pi$
$853$	3.93316	0.134669	0.0673345	+	0.997730i	$0.478551\pi$
$854$	0	0
$855$	2.45167	0.0838455
$856$	0	0
$857$	−57.5073	−1.96441	−0.982205	−	0.187810i	$-0.939861\pi$
$857$	−57.5073	−1.96441	−0.982205	+	0.187810i	$0.939861\pi$
$858$	0	0
$859$	8.51851	0.290648	0.145324	−	0.989384i	$-0.453578\pi$
$859$	8.51851	0.290648	0.145324	+	0.989384i	$0.453578\pi$
$860$	0	0
$861$	−57.8331	−1.97095
$862$	0	0
$863$	27.1894	0.925537	0.462768	−	0.886479i	$-0.346856\pi$
$863$	27.1894	0.925537	0.462768	+	0.886479i	$0.346856\pi$
$864$	0	0
$865$	0.486077	0.0165271
$866$	0	0
$867$	−16.5895	−0.563408
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0.650779	0.0220508
$872$	0	0
$873$	8.99334	0.304378
$874$	0	0
$875$	19.0735	0.644802
$876$	0	0
$877$	−40.8576	−1.37966	−0.689831	−	0.723970i	$-0.742315\pi$
$877$	−40.8576	−1.37966	−0.689831	+	0.723970i	$0.742315\pi$
$878$	0	0
$879$	14.8331	0.500309
$880$	0	0
$881$	−10.9645	−0.369404	−0.184702	−	0.982795i	$-0.559132\pi$
$881$	−10.9645	−0.369404	−0.184702	+	0.982795i	$0.559132\pi$
$882$	0	0
$883$	38.3081	1.28917	0.644584	−	0.764533i	$-0.277030\pi$
$883$	38.3081	1.28917	0.644584	+	0.764533i	$0.277030\pi$
$884$	0	0
$885$	−1.33702	−0.0449434
$886$	0	0
$887$	54.4403	1.82793	0.913964	−	0.405796i	$-0.133006\pi$
$887$	54.4403	1.82793	0.913964	+	0.405796i	$0.133006\pi$
$888$	0	0
$889$	−99.0801	−3.32304
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−46.0622	−1.54141
$894$	0	0
$895$	−3.23708	−0.108204
$896$	0	0
$897$	−1.66626	−0.0556349
$898$	0	0
$899$	−6.33374	−0.211242
$900$	0	0
$901$	−3.50398	−0.116735
$902$	0	0
$903$	−14.3337	−0.476997
$904$	0	0
$905$	4.27270	0.142029
$906$	0	0
$907$	1.05231	0.0349414	0.0174707	−	0.999847i	$-0.494439\pi$
$907$	1.05231	0.0349414	0.0174707	+	0.999847i	$0.494439\pi$
$908$	0	0
$909$	−12.3364	−0.409173
$910$	0	0
$911$	−59.5240	−1.97212	−0.986058	−	0.166400i	$-0.946786\pi$
$911$	−59.5240	−1.97212	−0.986058	+	0.166400i	$0.946786\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−0.486077	−0.0160692
$916$	0	0
$917$	−66.1802	−2.18546
$918$	0	0
$919$	−13.2091	−0.435729	−0.217865	−	0.975979i	$-0.569909\pi$
$919$	−13.2091	−0.435729	−0.217865	+	0.975979i	$0.569909\pi$
$920$	0	0
$921$	31.9927	1.05420
$922$	0	0
$923$	2.43593	0.0801798
$924$	0	0
$925$	32.9559	1.08358
$926$	0	0
$927$	12.3448	0.405455
$928$	0	0
$929$	34.0834	1.11824	0.559121	−	0.829086i	$-0.311139\pi$
$929$	34.0834	1.11824	0.559121	+	0.829086i	$0.311139\pi$
$930$	0	0
$931$	86.5861	2.83774
$932$	0	0
$933$	7.07884	0.231751
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.5272	0.670596	0.335298	−	0.942112i	$-0.391163\pi$
$937$	20.5272	0.670596	0.335298	+	0.942112i	$0.391163\pi$
$938$	0	0
$939$	23.6496	0.771774
$940$	0	0
$941$	−32.9415	−1.07386	−0.536932	−	0.843626i	$-0.680417\pi$
$941$	−32.9415	−1.07386	−0.536932	+	0.843626i	$0.680417\pi$
$942$	0	0
$943$	−73.0477	−2.37876
$944$	0	0
$945$	−1.94103	−0.0631418
$946$	0	0
$947$	53.9867	1.75433	0.877166	−	0.480188i	$-0.159431\pi$
$947$	53.9867	1.75433	0.877166	+	0.480188i	$0.159431\pi$
$948$	0	0
$949$	0.641796	0.0208336
$950$	0	0
$951$	−13.7007	−0.444275
$952$	0	0
$953$	−21.7843	−0.705664	−0.352832	−	0.935687i	$-0.614781\pi$
$953$	−21.7843	−0.705664	−0.352832	+	0.935687i	$0.614781\pi$
$954$	0	0
$955$	5.64986	0.182825
$956$	0	0
$957$	0	0
$958$	0	0
$959$	49.1515	1.58718
$960$	0	0
$961$	−29.4973	−0.951527
$962$	0	0
$963$	19.0867	0.615061
$964$	0	0
$965$	−1.06900	−0.0344125
$966$	0	0
$967$	−25.9385	−0.834126	−0.417063	−	0.908878i	$-0.636941\pi$
$967$	−25.9385	−0.834126	−0.417063	+	0.908878i	$0.636941\pi$
$968$	0	0
$969$	3.77088	0.121138
$970$	0	0
$971$	−27.4566	−0.881126	−0.440563	−	0.897722i	$-0.645221\pi$
$971$	−27.4566	−0.881126	−0.440563	+	0.897722i	$0.645221\pi$
$972$	0	0
$973$	55.7054	1.78583
$974$	0	0
$975$	−1.36645	−0.0437615
$976$	0	0
$977$	45.3036	1.44939	0.724695	−	0.689070i	$-0.241981\pi$
$977$	45.3036	1.44939	0.724695	+	0.689070i	$0.241981\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	17.2553	0.550918
$982$	0	0
$983$	−34.6987	−1.10672	−0.553358	−	0.832943i	$-0.686654\pi$
$983$	−34.6987	−1.10672	−0.553358	+	0.832943i	$0.686654\pi$
$984$	0	0
$985$	−1.21797	−0.0388077
$986$	0	0
$987$	36.4683	1.16080
$988$	0	0
$989$	−18.1046	−0.575693
$990$	0	0
$991$	−33.6320	−1.06836	−0.534178	−	0.845372i	$-0.679379\pi$
$991$	−33.6320	−1.06836	−0.534178	+	0.845372i	$0.679379\pi$
$992$	0	0
$993$	9.40991	0.298614
$994$	0	0
$995$	1.43039	0.0453464
$996$	0	0
$997$	36.8954	1.16849	0.584244	−	0.811578i	$-0.301391\pi$
$997$	36.8954	1.16849	0.584244	+	0.811578i	$0.301391\pi$
$998$	0	0
$999$	−6.82816	−0.216033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.o.1.2		4
4.3	odd	2		1002.2.a.i.1.2	✓	4
12.11	even	2		3006.2.a.s.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.i.1.2	✓	4	4.3	odd	2
3006.2.a.s.1.3		4	12.11	even	2
8016.2.a.o.1.2		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.416566 q^{5} -4.65960 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.416566 q^{5} -4.65960 q^{7} +1.00000 q^{9} +0.283116 q^{13} +0.416566 q^{15} +0.640714 q^{17} +5.88544 q^{19} -4.65960 q^{21} -5.88544 q^{23} -4.82647 q^{25} +1.00000 q^{27} +5.16687 q^{29} -1.22584 q^{31} -1.94103 q^{35} -6.82816 q^{37} +0.283116 q^{39} +12.4116 q^{41} +3.07617 q^{43} +0.416566 q^{45} -7.82647 q^{47} +14.7119 q^{49} +0.640714 q^{51} -5.46888 q^{53} +5.88544 q^{57} -3.20962 q^{59} -1.16687 q^{61} -4.65960 q^{63} +0.117936 q^{65} +2.29863 q^{67} -5.88544 q^{69} +8.60401 q^{71} +2.26690 q^{73} -4.82647 q^{75} +7.69302 q^{79} +1.00000 q^{81} -9.27646 q^{83} +0.266899 q^{85} +5.16687 q^{87} +3.94103 q^{89} -1.31921 q^{91} -1.22584 q^{93} +2.45167 q^{95} +8.99334 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9 + 8 * q^13 + 5 * q^15 + 10 * q^17 + 2 * q^19 - q^21 - 2 * q^23 + 5 * q^25 + 4 * q^27 + 14 * q^29 - q^31 - 5 * q^35 + 5 * q^37 + 8 * q^39 + 14 * q^41 - 2 * q^43 + 5 * q^45 - 7 * q^47 + 13 * q^49 + 10 * q^51 + 3 * q^53 + 2 * q^57 + 5 * q^59 + 2 * q^61 - q^63 + 6 * q^65 + 7 * q^67 - 2 * q^69 - 2 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + 4 * q^81 - 13 * q^83 - 6 * q^85 + 14 * q^87 + 13 * q^89 + 30 * q^91 - q^93 + 2 * q^95 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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