LMFDB - Embedded newform 8512.2.a.ca.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8512,2,Mod(1,8512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,0,-3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.41027408.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 12x^{2} - 17x - 7 $ x^6 - 3x^5 - 6x^4 + 15x^3 + 12x^2 - 17*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4256)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.38372$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.38372 q^{3} -1.87017 q^{5} +1.00000 q^{7} -1.08531 q^{9} +1.44821 q^{11} +6.27385 q^{13} -2.58779 q^{15} -4.31837 q^{17} +1.00000 q^{19} +1.38372 q^{21} -4.96568 q^{23} -1.50248 q^{25} -5.65294 q^{27} +0.621241 q^{29} +6.31837 q^{31} +2.00392 q^{33} -1.87017 q^{35} -7.84918 q^{37} +8.68127 q^{39} -1.73328 q^{41} -9.05412 q^{43} +2.02971 q^{45} -3.90815 q^{47} +1.00000 q^{49} -5.97543 q^{51} +8.51353 q^{53} -2.70838 q^{55} +1.38372 q^{57} -7.85342 q^{59} -12.5985 q^{61} -1.08531 q^{63} -11.7331 q^{65} -8.56434 q^{67} -6.87113 q^{69} +12.0257 q^{71} +16.2776 q^{73} -2.07902 q^{75} +1.44821 q^{77} +10.7582 q^{79} -4.56618 q^{81} -9.58475 q^{83} +8.07607 q^{85} +0.859626 q^{87} +6.66661 q^{89} +6.27385 q^{91} +8.74288 q^{93} -1.87017 q^{95} -7.10464 q^{97} -1.57175 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 7 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + 6 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} - 18 q^{27} - 5 q^{29} + 14 q^{31} + 3 q^{33} - 3 q^{35} + q^{37} + 22 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 7 * q^11 - 4 * q^13 + 4 * q^15 - 2 * q^17 + 6 * q^19 - 3 * q^21 - 2 * q^23 + q^25 - 18 * q^27 - 5 * q^29 + 14 * q^31 + 3 * q^33 - 3 * q^35 + q^37 + 22 * q^39 - 5 * q^41 - 15 * q^43 - 22 * q^45 - 7 * q^47 + 6 * q^49 + 4 * q^51 - 15 * q^53 + 20 * q^55 - 3 * q^57 - 23 * q^59 - 9 * q^61 + 3 * q^63 - 20 * q^65 - 2 * q^67 + 2 * q^69 + 23 * q^71 + 2 * q^73 - 4 * q^75 - 7 * q^77 + 17 * q^79 + 6 * q^81 - 22 * q^83 + 14 * q^85 - 10 * q^87 + 3 * q^89 - 4 * q^91 - 10 * q^93 - 3 * q^95 - 21 * q^97 - 3 * q^99

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
$3$	1.38372	0.798893	0.399447	−	0.916756i	$-0.369202\pi$
$3$	1.38372	0.798893	0.399447	+	0.916756i	$0.369202\pi$
$4$	0	0
$5$	−1.87017	−0.836363	−0.418182	−	0.908363i	$-0.637332\pi$
$5$	−1.87017	−0.836363	−0.418182	+	0.908363i	$0.637332\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.08531	−0.361770
$10$	0	0
$11$	1.44821	0.436650	0.218325	−	0.975876i	$-0.429941\pi$
$11$	1.44821	0.436650	0.218325	+	0.975876i	$0.429941\pi$
$12$	0	0
$13$	6.27385	1.74005	0.870026	−	0.493006i	$-0.164102\pi$
$13$	6.27385	1.74005	0.870026	+	0.493006i	$0.164102\pi$
$14$	0	0
$15$	−2.58779	−0.668165
$16$	0	0
$17$	−4.31837	−1.04736	−0.523679	−	0.851915i	$-0.675441\pi$
$17$	−4.31837	−1.04736	−0.523679	+	0.851915i	$0.675441\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.38372	0.301953
$22$	0	0
$23$	−4.96568	−1.03542	−0.517708	−	0.855557i	$-0.673215\pi$
$23$	−4.96568	−1.03542	−0.517708	+	0.855557i	$0.673215\pi$
$24$	0	0
$25$	−1.50248	−0.300496
$26$	0	0
$27$	−5.65294	−1.08791
$28$	0	0
$29$	0.621241	0.115362	0.0576808	−	0.998335i	$-0.481629\pi$
$29$	0.621241	0.115362	0.0576808	+	0.998335i	$0.481629\pi$
$30$	0	0
$31$	6.31837	1.13481	0.567406	−	0.823438i	$-0.307947\pi$
$31$	6.31837	1.13481	0.567406	+	0.823438i	$0.307947\pi$
$32$	0	0
$33$	2.00392	0.348837
$34$	0	0
$35$	−1.87017	−0.316116
$36$	0	0
$37$	−7.84918	−1.29040	−0.645199	−	0.764015i	$-0.723225\pi$
$37$	−7.84918	−1.29040	−0.645199	+	0.764015i	$0.723225\pi$
$38$	0	0
$39$	8.68127	1.39012
$40$	0	0
$41$	−1.73328	−0.270693	−0.135347	−	0.990798i	$-0.543215\pi$
$41$	−1.73328	−0.270693	−0.135347	+	0.990798i	$0.543215\pi$
$42$	0	0
$43$	−9.05412	−1.38074	−0.690370	−	0.723456i	$-0.742552\pi$
$43$	−9.05412	−1.38074	−0.690370	+	0.723456i	$0.742552\pi$
$44$	0	0
$45$	2.02971	0.302571
$46$	0	0
$47$	−3.90815	−0.570063	−0.285031	−	0.958518i	$-0.592004\pi$
$47$	−3.90815	−0.570063	−0.285031	+	0.958518i	$0.592004\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.97543	−0.836728
$52$	0	0
$53$	8.51353	1.16942	0.584711	−	0.811241i	$-0.301208\pi$
$53$	8.51353	1.16942	0.584711	+	0.811241i	$0.301208\pi$
$54$	0	0
$55$	−2.70838	−0.365198
$56$	0	0
$57$	1.38372	0.183279
$58$	0	0
$59$	−7.85342	−1.02243	−0.511214	−	0.859453i	$-0.670804\pi$
$59$	−7.85342	−1.02243	−0.511214	+	0.859453i	$0.670804\pi$
$60$	0	0
$61$	−12.5985	−1.61307	−0.806537	−	0.591184i	$-0.798661\pi$
$61$	−12.5985	−1.61307	−0.806537	+	0.591184i	$0.798661\pi$
$62$	0	0
$63$	−1.08531	−0.136736
$64$	0	0
$65$	−11.7331	−1.45532
$66$	0	0
$67$	−8.56434	−1.04630	−0.523150	−	0.852240i	$-0.675243\pi$
$67$	−8.56434	−1.04630	−0.523150	+	0.852240i	$0.675243\pi$
$68$	0	0
$69$	−6.87113	−0.827187
$70$	0	0
$71$	12.0257	1.42719	0.713594	−	0.700559i	$-0.247066\pi$
$71$	12.0257	1.42719	0.713594	+	0.700559i	$0.247066\pi$
$72$	0	0
$73$	16.2776	1.90515	0.952575	−	0.304305i	$-0.0984241\pi$
$73$	16.2776	1.90515	0.952575	+	0.304305i	$0.0984241\pi$
$74$	0	0
$75$	−2.07902	−0.240065
$76$	0	0
$77$	1.44821	0.165038
$78$	0	0
$79$	10.7582	1.21039	0.605196	−	0.796076i	$-0.293095\pi$
$79$	10.7582	1.21039	0.605196	+	0.796076i	$0.293095\pi$
$80$	0	0
$81$	−4.56618	−0.507353
$82$	0	0
$83$	−9.58475	−1.05206	−0.526032	−	0.850465i	$-0.676321\pi$
$83$	−9.58475	−1.05206	−0.526032	+	0.850465i	$0.676321\pi$
$84$	0	0
$85$	8.07607	0.875972
$86$	0	0
$87$	0.859626	0.0921615
$88$	0	0
$89$	6.66661	0.706659	0.353329	−	0.935499i	$-0.385049\pi$
$89$	6.66661	0.706659	0.353329	+	0.935499i	$0.385049\pi$
$90$	0	0
$91$	6.27385	0.657678
$92$	0	0
$93$	8.74288	0.906594
$94$	0	0
$95$	−1.87017	−0.191875
$96$	0	0
$97$	−7.10464	−0.721367	−0.360684	−	0.932688i	$-0.617457\pi$
$97$	−7.10464	−0.721367	−0.360684	+	0.932688i	$0.617457\pi$
$98$	0	0
$99$	−1.57175	−0.157967
$100$	0	0
$101$	0.416508	0.0414441	0.0207220	−	0.999785i	$-0.493403\pi$
$101$	0.416508	0.0414441	0.0207220	+	0.999785i	$0.493403\pi$
$102$	0	0
$103$	1.97213	0.194319	0.0971597	−	0.995269i	$-0.469024\pi$
$103$	1.97213	0.194319	0.0971597	+	0.995269i	$0.469024\pi$
$104$	0	0
$105$	−2.58779	−0.252543
$106$	0	0
$107$	−13.3939	−1.29484	−0.647421	−	0.762133i	$-0.724152\pi$
$107$	−13.3939	−1.29484	−0.647421	+	0.762133i	$0.724152\pi$
$108$	0	0
$109$	11.2020	1.07296	0.536478	−	0.843914i	$-0.319754\pi$
$109$	11.2020	1.07296	0.536478	+	0.843914i	$0.319754\pi$
$110$	0	0
$111$	−10.8611	−1.03089
$112$	0	0
$113$	−2.66351	−0.250562	−0.125281	−	0.992121i	$-0.539983\pi$
$113$	−2.66351	−0.250562	−0.125281	+	0.992121i	$0.539983\pi$
$114$	0	0
$115$	9.28664	0.865984
$116$	0	0
$117$	−6.80906	−0.629498
$118$	0	0
$119$	−4.31837	−0.395864
$120$	0	0
$121$	−8.90270	−0.809336
$122$	0	0
$123$	−2.39838	−0.216255
$124$	0	0
$125$	12.1607	1.08769
$126$	0	0
$127$	−21.7347	−1.92864	−0.964319	−	0.264742i	$-0.914713\pi$
$127$	−21.7347	−1.92864	−0.964319	+	0.264742i	$0.914713\pi$
$128$	0	0
$129$	−12.5284	−1.10306
$130$	0	0
$131$	1.58834	0.138774	0.0693869	−	0.997590i	$-0.477896\pi$
$131$	1.58834	0.138774	0.0693869	+	0.997590i	$0.477896\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	10.5719	0.909887
$136$	0	0
$137$	−7.99076	−0.682696	−0.341348	−	0.939937i	$-0.610884\pi$
$137$	−7.99076	−0.682696	−0.341348	+	0.939937i	$0.610884\pi$
$138$	0	0
$139$	2.93131	0.248631	0.124315	−	0.992243i	$-0.460327\pi$
$139$	2.93131	0.248631	0.124315	+	0.992243i	$0.460327\pi$
$140$	0	0
$141$	−5.40780	−0.455419
$142$	0	0
$143$	9.08582	0.759794
$144$	0	0
$145$	−1.16182	−0.0964842
$146$	0	0
$147$	1.38372	0.114128
$148$	0	0
$149$	−17.0438	−1.39628	−0.698141	−	0.715960i	$-0.745989\pi$
$149$	−17.0438	−1.39628	−0.698141	+	0.715960i	$0.745989\pi$
$150$	0	0
$151$	−14.1187	−1.14897	−0.574483	−	0.818516i	$-0.694797\pi$
$151$	−14.1187	−1.14897	−0.574483	+	0.818516i	$0.694797\pi$
$152$	0	0
$153$	4.68677	0.378903
$154$	0	0
$155$	−11.8164	−0.949116
$156$	0	0
$157$	−2.74578	−0.219137	−0.109569	−	0.993979i	$-0.534947\pi$
$157$	−2.74578	−0.219137	−0.109569	+	0.993979i	$0.534947\pi$
$158$	0	0
$159$	11.7804	0.934244
$160$	0	0
$161$	−4.96568	−0.391350
$162$	0	0
$163$	3.97102	0.311034	0.155517	−	0.987833i	$-0.450296\pi$
$163$	3.97102	0.311034	0.155517	+	0.987833i	$0.450296\pi$
$164$	0	0
$165$	−3.74765	−0.291754
$166$	0	0
$167$	−10.5482	−0.816247	−0.408123	−	0.912927i	$-0.633817\pi$
$167$	−10.5482	−0.816247	−0.408123	+	0.912927i	$0.633817\pi$
$168$	0	0
$169$	26.3611	2.02778
$170$	0	0
$171$	−1.08531	−0.0829957
$172$	0	0
$173$	4.71688	0.358618	0.179309	−	0.983793i	$-0.442614\pi$
$173$	4.71688	0.358618	0.179309	+	0.983793i	$0.442614\pi$
$174$	0	0
$175$	−1.50248	−0.113577
$176$	0	0
$177$	−10.8670	−0.816811
$178$	0	0
$179$	−5.40152	−0.403729	−0.201864	−	0.979413i	$-0.564700\pi$
$179$	−5.40152	−0.403729	−0.201864	+	0.979413i	$0.564700\pi$
$180$	0	0
$181$	18.8970	1.40460	0.702300	−	0.711881i	$-0.252156\pi$
$181$	18.8970	1.40460	0.702300	+	0.711881i	$0.252156\pi$
$182$	0	0
$183$	−17.4328	−1.28867
$184$	0	0
$185$	14.6793	1.07924
$186$	0	0
$187$	−6.25389	−0.457330
$188$	0	0
$189$	−5.65294	−0.411191
$190$	0	0
$191$	21.9811	1.59049	0.795247	−	0.606285i	$-0.207341\pi$
$191$	21.9811	1.59049	0.795247	+	0.606285i	$0.207341\pi$
$192$	0	0
$193$	2.44679	0.176124	0.0880618	−	0.996115i	$-0.471933\pi$
$193$	2.44679	0.176124	0.0880618	+	0.996115i	$0.471933\pi$
$194$	0	0
$195$	−16.2354	−1.16264
$196$	0	0
$197$	−26.5788	−1.89366	−0.946830	−	0.321735i	$-0.895734\pi$
$197$	−26.5788	−1.89366	−0.946830	+	0.321735i	$0.895734\pi$
$198$	0	0
$199$	16.6781	1.18228	0.591141	−	0.806568i	$-0.298678\pi$
$199$	16.6781	1.18228	0.591141	+	0.806568i	$0.298678\pi$
$200$	0	0
$201$	−11.8507	−0.835883
$202$	0	0
$203$	0.621241	0.0436026
$204$	0	0
$205$	3.24153	0.226398
$206$	0	0
$207$	5.38930	0.374582
$208$	0	0
$209$	1.44821	0.100174
$210$	0	0
$211$	11.7321	0.807671	0.403836	−	0.914832i	$-0.367677\pi$
$211$	11.7321	0.807671	0.403836	+	0.914832i	$0.367677\pi$
$212$	0	0
$213$	16.6403	1.14017
$214$	0	0
$215$	16.9327	1.15480
$216$	0	0
$217$	6.31837	0.428919
$218$	0	0
$219$	22.5237	1.52201
$220$	0	0
$221$	−27.0928	−1.82246
$222$	0	0
$223$	−7.62076	−0.510324	−0.255162	−	0.966898i	$-0.582129\pi$
$223$	−7.62076	−0.510324	−0.255162	+	0.966898i	$0.582129\pi$
$224$	0	0
$225$	1.63066	0.108711
$226$	0	0
$227$	10.1208	0.671739	0.335869	−	0.941909i	$-0.390970\pi$
$227$	10.1208	0.671739	0.335869	+	0.941909i	$0.390970\pi$
$228$	0	0
$229$	−21.4073	−1.41463	−0.707316	−	0.706897i	$-0.750094\pi$
$229$	−21.4073	−1.41463	−0.707316	+	0.706897i	$0.750094\pi$
$230$	0	0
$231$	2.00392	0.131848
$232$	0	0
$233$	−19.7902	−1.29650	−0.648250	−	0.761428i	$-0.724499\pi$
$233$	−19.7902	−1.29650	−0.648250	+	0.761428i	$0.724499\pi$
$234$	0	0
$235$	7.30889	0.476779
$236$	0	0
$237$	14.8864	0.966974
$238$	0	0
$239$	9.80003	0.633911	0.316955	−	0.948440i	$-0.397339\pi$
$239$	9.80003	0.633911	0.316955	+	0.948440i	$0.397339\pi$
$240$	0	0
$241$	4.65591	0.299914	0.149957	−	0.988693i	$-0.452087\pi$
$241$	4.65591	0.299914	0.149957	+	0.988693i	$0.452087\pi$
$242$	0	0
$243$	10.6405	0.682588
$244$	0	0
$245$	−1.87017	−0.119480
$246$	0	0
$247$	6.27385	0.399195
$248$	0	0
$249$	−13.2626	−0.840486
$250$	0	0
$251$	−3.80051	−0.239886	−0.119943	−	0.992781i	$-0.538271\pi$
$251$	−3.80051	−0.239886	−0.119943	+	0.992781i	$0.538271\pi$
$252$	0	0
$253$	−7.19133	−0.452115
$254$	0	0
$255$	11.1750	0.699808
$256$	0	0
$257$	−8.51907	−0.531405	−0.265703	−	0.964055i	$-0.585604\pi$
$257$	−8.51907	−0.531405	−0.265703	+	0.964055i	$0.585604\pi$
$258$	0	0
$259$	−7.84918	−0.487724
$260$	0	0
$261$	−0.674239	−0.0417343
$262$	0	0
$263$	−15.0842	−0.930129	−0.465065	−	0.885277i	$-0.653969\pi$
$263$	−15.0842	−0.930129	−0.465065	+	0.885277i	$0.653969\pi$
$264$	0	0
$265$	−15.9217	−0.978062
$266$	0	0
$267$	9.22474	0.564545
$268$	0	0
$269$	−25.1196	−1.53157	−0.765786	−	0.643096i	$-0.777650\pi$
$269$	−25.1196	−1.53157	−0.765786	+	0.643096i	$0.777650\pi$
$270$	0	0
$271$	12.2124	0.741850	0.370925	−	0.928663i	$-0.379041\pi$
$271$	12.2124	0.741850	0.370925	+	0.928663i	$0.379041\pi$
$272$	0	0
$273$	8.68127	0.525414
$274$	0	0
$275$	−2.17590	−0.131212
$276$	0	0
$277$	−11.6764	−0.701569	−0.350785	−	0.936456i	$-0.614085\pi$
$277$	−11.6764	−0.701569	−0.350785	+	0.936456i	$0.614085\pi$
$278$	0	0
$279$	−6.85739	−0.410541
$280$	0	0
$281$	−19.9192	−1.18828	−0.594139	−	0.804362i	$-0.702507\pi$
$281$	−19.9192	−1.18828	−0.594139	+	0.804362i	$0.702507\pi$
$282$	0	0
$283$	−27.0210	−1.60623	−0.803117	−	0.595822i	$-0.796826\pi$
$283$	−27.0210	−1.60623	−0.803117	+	0.595822i	$0.796826\pi$
$284$	0	0
$285$	−2.58779	−0.153288
$286$	0	0
$287$	−1.73328	−0.102312
$288$	0	0
$289$	1.64833	0.0969604
$290$	0	0
$291$	−9.83086	−0.576295
$292$	0	0
$293$	−25.6906	−1.50086	−0.750431	−	0.660949i	$-0.770154\pi$
$293$	−25.6906	−1.50086	−0.750431	+	0.660949i	$0.770154\pi$
$294$	0	0
$295$	14.6872	0.855122
$296$	0	0
$297$	−8.18662	−0.475036
$298$	0	0
$299$	−31.1539	−1.80168
$300$	0	0
$301$	−9.05412	−0.521871
$302$	0	0
$303$	0.576332	0.0331094
$304$	0	0
$305$	23.5613	1.34912
$306$	0	0
$307$	−11.1372	−0.635634	−0.317817	−	0.948152i	$-0.602950\pi$
$307$	−11.1372	−0.635634	−0.317817	+	0.948152i	$0.602950\pi$
$308$	0	0
$309$	2.72888	0.155240
$310$	0	0
$311$	−11.6914	−0.662960	−0.331480	−	0.943462i	$-0.607548\pi$
$311$	−11.6914	−0.662960	−0.331480	+	0.943462i	$0.607548\pi$
$312$	0	0
$313$	26.8564	1.51802	0.759008	−	0.651082i	$-0.225684\pi$
$313$	26.8564	1.51802	0.759008	+	0.651082i	$0.225684\pi$
$314$	0	0
$315$	2.02971	0.114361
$316$	0	0
$317$	−17.9275	−1.00691	−0.503455	−	0.864022i	$-0.667938\pi$
$317$	−17.9275	−1.00691	−0.503455	+	0.864022i	$0.667938\pi$
$318$	0	0
$319$	0.899685	0.0503727
$320$	0	0
$321$	−18.5335	−1.03444
$322$	0	0
$323$	−4.31837	−0.240281
$324$	0	0
$325$	−9.42634	−0.522879
$326$	0	0
$327$	15.5005	0.857178
$328$	0	0
$329$	−3.90815	−0.215463
$330$	0	0
$331$	9.14251	0.502518	0.251259	−	0.967920i	$-0.419155\pi$
$331$	9.14251	0.502518	0.251259	+	0.967920i	$0.419155\pi$
$332$	0	0
$333$	8.51879	0.466827
$334$	0	0
$335$	16.0167	0.875088
$336$	0	0
$337$	−23.2982	−1.26914	−0.634568	−	0.772867i	$-0.718822\pi$
$337$	−23.2982	−1.26914	−0.634568	+	0.772867i	$0.718822\pi$
$338$	0	0
$339$	−3.68556	−0.200172
$340$	0	0
$341$	9.15030	0.495517
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	12.8501	0.691829
$346$	0	0
$347$	10.7362	0.576347	0.288174	−	0.957578i	$-0.406952\pi$
$347$	10.7362	0.576347	0.288174	+	0.957578i	$0.406952\pi$
$348$	0	0
$349$	10.5423	0.564315	0.282157	−	0.959368i	$-0.408950\pi$
$349$	10.5423	0.564315	0.282157	+	0.959368i	$0.408950\pi$
$350$	0	0
$351$	−35.4657	−1.89302
$352$	0	0
$353$	−37.3810	−1.98959	−0.994795	−	0.101895i	$-0.967510\pi$
$353$	−37.3810	−1.98959	−0.994795	+	0.101895i	$0.967510\pi$
$354$	0	0
$355$	−22.4901	−1.19365
$356$	0	0
$357$	−5.97543	−0.316253
$358$	0	0
$359$	0.960472	0.0506918	0.0253459	−	0.999679i	$-0.491931\pi$
$359$	0.960472	0.0506918	0.0253459	+	0.999679i	$0.491931\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−12.3189	−0.646573
$364$	0	0
$365$	−30.4418	−1.59340
$366$	0	0
$367$	19.6199	1.02415	0.512076	−	0.858940i	$-0.328877\pi$
$367$	19.6199	1.02415	0.512076	+	0.858940i	$0.328877\pi$
$368$	0	0
$369$	1.88115	0.0979287
$370$	0	0
$371$	8.51353	0.442000
$372$	0	0
$373$	6.79997	0.352089	0.176045	−	0.984382i	$-0.443670\pi$
$373$	6.79997	0.352089	0.176045	+	0.984382i	$0.443670\pi$
$374$	0	0
$375$	16.8271	0.868946
$376$	0	0
$377$	3.89757	0.200735
$378$	0	0
$379$	−28.8380	−1.48131	−0.740655	−	0.671886i	$-0.765485\pi$
$379$	−28.8380	−1.48131	−0.740655	+	0.671886i	$0.765485\pi$
$380$	0	0
$381$	−30.0747	−1.54078
$382$	0	0
$383$	−35.6391	−1.82107	−0.910537	−	0.413428i	$-0.864331\pi$
$383$	−35.6391	−1.82107	−0.910537	+	0.413428i	$0.864331\pi$
$384$	0	0
$385$	−2.70838	−0.138032
$386$	0	0
$387$	9.82652	0.499510
$388$	0	0
$389$	2.46878	0.125172	0.0625860	−	0.998040i	$-0.480065\pi$
$389$	2.46878	0.125172	0.0625860	+	0.998040i	$0.480065\pi$
$390$	0	0
$391$	21.4436	1.08445
$392$	0	0
$393$	2.19782	0.110865
$394$	0	0
$395$	−20.1196	−1.01233
$396$	0	0
$397$	−23.3320	−1.17100	−0.585500	−	0.810673i	$-0.699102\pi$
$397$	−23.3320	−1.17100	−0.585500	+	0.810673i	$0.699102\pi$
$398$	0	0
$399$	1.38372	0.0692728
$400$	0	0
$401$	−7.72516	−0.385776	−0.192888	−	0.981221i	$-0.561785\pi$
$401$	−7.72516	−0.385776	−0.192888	+	0.981221i	$0.561785\pi$
$402$	0	0
$403$	39.6405	1.97463
$404$	0	0
$405$	8.53950	0.424331
$406$	0	0
$407$	−11.3672	−0.563453
$408$	0	0
$409$	35.1734	1.73922	0.869608	−	0.493744i	$-0.164372\pi$
$409$	35.1734	1.73922	0.869608	+	0.493744i	$0.164372\pi$
$410$	0	0
$411$	−11.0570	−0.545402
$412$	0	0
$413$	−7.85342	−0.386442
$414$	0	0
$415$	17.9251	0.879907
$416$	0	0
$417$	4.05612	0.198629
$418$	0	0
$419$	33.4426	1.63378	0.816888	−	0.576797i	$-0.195698\pi$
$419$	33.4426	1.63378	0.816888	+	0.576797i	$0.195698\pi$
$420$	0	0
$421$	32.7582	1.59654	0.798268	−	0.602302i	$-0.205750\pi$
$421$	32.7582	1.59654	0.798268	+	0.602302i	$0.205750\pi$
$422$	0	0
$423$	4.24155	0.206231
$424$	0	0
$425$	6.48828	0.314728
$426$	0	0
$427$	−12.5985	−0.609684
$428$	0	0
$429$	12.5723	0.606994
$430$	0	0
$431$	19.0142	0.915884	0.457942	−	0.888982i	$-0.348587\pi$
$431$	19.0142	0.915884	0.457942	+	0.888982i	$0.348587\pi$
$432$	0	0
$433$	−15.0864	−0.725005	−0.362502	−	0.931983i	$-0.618078\pi$
$433$	−15.0864	−0.725005	−0.362502	+	0.931983i	$0.618078\pi$
$434$	0	0
$435$	−1.60764	−0.0770805
$436$	0	0
$437$	−4.96568	−0.237541
$438$	0	0
$439$	−13.4772	−0.643231	−0.321615	−	0.946870i	$-0.604226\pi$
$439$	−13.4772	−0.643231	−0.321615	+	0.946870i	$0.604226\pi$
$440$	0	0
$441$	−1.08531	−0.0516814
$442$	0	0
$443$	−21.2677	−1.01046	−0.505229	−	0.862985i	$-0.668592\pi$
$443$	−21.2677	−1.01046	−0.505229	+	0.862985i	$0.668592\pi$
$444$	0	0
$445$	−12.4677	−0.591023
$446$	0	0
$447$	−23.5839	−1.11548
$448$	0	0
$449$	−29.0154	−1.36932	−0.684660	−	0.728863i	$-0.740049\pi$
$449$	−29.0154	−1.36932	−0.684660	+	0.728863i	$0.740049\pi$
$450$	0	0
$451$	−2.51015	−0.118198
$452$	0	0
$453$	−19.5364	−0.917902
$454$	0	0
$455$	−11.7331	−0.550057
$456$	0	0
$457$	17.1215	0.800910	0.400455	−	0.916317i	$-0.368852\pi$
$457$	17.1215	0.800910	0.400455	+	0.916317i	$0.368852\pi$
$458$	0	0
$459$	24.4115	1.13943
$460$	0	0
$461$	−30.9822	−1.44298	−0.721492	−	0.692422i	$-0.756544\pi$
$461$	−30.9822	−1.44298	−0.721492	+	0.692422i	$0.756544\pi$
$462$	0	0
$463$	−10.8766	−0.505480	−0.252740	−	0.967534i	$-0.581332\pi$
$463$	−10.8766	−0.505480	−0.252740	+	0.967534i	$0.581332\pi$
$464$	0	0
$465$	−16.3506	−0.758242
$466$	0	0
$467$	27.1576	1.25671	0.628353	−	0.777929i	$-0.283729\pi$
$467$	27.1576	1.25671	0.628353	+	0.777929i	$0.283729\pi$
$468$	0	0
$469$	−8.56434	−0.395465
$470$	0	0
$471$	−3.79940	−0.175067
$472$	0	0
$473$	−13.1122	−0.602901
$474$	0	0
$475$	−1.50248	−0.0689386
$476$	0	0
$477$	−9.23981	−0.423062
$478$	0	0
$479$	30.2557	1.38242	0.691208	−	0.722655i	$-0.257079\pi$
$479$	30.2557	1.38242	0.691208	+	0.722655i	$0.257079\pi$
$480$	0	0
$481$	−49.2446	−2.24536
$482$	0	0
$483$	−6.87113	−0.312647
$484$	0	0
$485$	13.2869	0.603325
$486$	0	0
$487$	1.06592	0.0483017	0.0241508	−	0.999708i	$-0.492312\pi$
$487$	1.06592	0.0483017	0.0241508	+	0.999708i	$0.492312\pi$
$488$	0	0
$489$	5.49479	0.248483
$490$	0	0
$491$	11.5877	0.522946	0.261473	−	0.965211i	$-0.415792\pi$
$491$	11.5877	0.522946	0.261473	+	0.965211i	$0.415792\pi$
$492$	0	0
$493$	−2.68275	−0.120825
$494$	0	0
$495$	2.93943	0.132118
$496$	0	0
$497$	12.0257	0.539427
$498$	0	0
$499$	−0.581361	−0.0260253	−0.0130126	−	0.999915i	$-0.504142\pi$
$499$	−0.581361	−0.0260253	−0.0130126	+	0.999915i	$0.504142\pi$
$500$	0	0
$501$	−14.5958	−0.652094
$502$	0	0
$503$	18.1781	0.810520	0.405260	−	0.914202i	$-0.367181\pi$
$503$	18.1781	0.810520	0.405260	+	0.914202i	$0.367181\pi$
$504$	0	0
$505$	−0.778939	−0.0346623
$506$	0	0
$507$	36.4765	1.61998
$508$	0	0
$509$	16.2446	0.720030	0.360015	−	0.932946i	$-0.382772\pi$
$509$	16.2446	0.720030	0.360015	+	0.932946i	$0.382772\pi$
$510$	0	0
$511$	16.2776	0.720079
$512$	0	0
$513$	−5.65294	−0.249583
$514$	0	0
$515$	−3.68820	−0.162522
$516$	0	0
$517$	−5.65981	−0.248918
$518$	0	0
$519$	6.52686	0.286497
$520$	0	0
$521$	−35.8388	−1.57012	−0.785062	−	0.619417i	$-0.787369\pi$
$521$	−35.8388	−1.57012	−0.785062	+	0.619417i	$0.787369\pi$
$522$	0	0
$523$	−6.02738	−0.263559	−0.131779	−	0.991279i	$-0.542069\pi$
$523$	−6.02738	−0.263559	−0.131779	+	0.991279i	$0.542069\pi$
$524$	0	0
$525$	−2.07902	−0.0907359
$526$	0	0
$527$	−27.2851	−1.18856
$528$	0	0
$529$	1.65798	0.0720860
$530$	0	0
$531$	8.52339	0.369884
$532$	0	0
$533$	−10.8744	−0.471020
$534$	0	0
$535$	25.0489	1.08296
$536$	0	0
$537$	−7.47421	−0.322536
$538$	0	0
$539$	1.44821	0.0623786
$540$	0	0
$541$	15.2349	0.654999	0.327500	−	0.944851i	$-0.393794\pi$
$541$	15.2349	0.654999	0.327500	+	0.944851i	$0.393794\pi$
$542$	0	0
$543$	26.1482	1.12213
$544$	0	0
$545$	−20.9496	−0.897382
$546$	0	0
$547$	−2.57224	−0.109981	−0.0549906	−	0.998487i	$-0.517513\pi$
$547$	−2.57224	−0.109981	−0.0549906	+	0.998487i	$0.517513\pi$
$548$	0	0
$549$	13.6733	0.583561
$550$	0	0
$551$	0.621241	0.0264658
$552$	0	0
$553$	10.7582	0.457485
$554$	0	0
$555$	20.3120	0.862198
$556$	0	0
$557$	−39.4445	−1.67132	−0.835660	−	0.549248i	$-0.814914\pi$
$557$	−39.4445	−1.67132	−0.835660	+	0.549248i	$0.814914\pi$
$558$	0	0
$559$	−56.8041	−2.40256
$560$	0	0
$561$	−8.65365	−0.365358
$562$	0	0
$563$	−24.3145	−1.02473	−0.512367	−	0.858767i	$-0.671231\pi$
$563$	−24.3145	−1.02473	−0.512367	+	0.858767i	$0.671231\pi$
$564$	0	0
$565$	4.98120	0.209561
$566$	0	0
$567$	−4.56618	−0.191761
$568$	0	0
$569$	−14.8402	−0.622134	−0.311067	−	0.950388i	$-0.600686\pi$
$569$	−14.8402	−0.622134	−0.311067	+	0.950388i	$0.600686\pi$
$570$	0	0
$571$	14.8847	0.622904	0.311452	−	0.950262i	$-0.399185\pi$
$571$	14.8847	0.622904	0.311452	+	0.950262i	$0.399185\pi$
$572$	0	0
$573$	30.4157	1.27063
$574$	0	0
$575$	7.46085	0.311139
$576$	0	0
$577$	34.7016	1.44465	0.722324	−	0.691555i	$-0.243074\pi$
$577$	34.7016	1.44465	0.722324	+	0.691555i	$0.243074\pi$
$578$	0	0
$579$	3.38568	0.140704
$580$	0	0
$581$	−9.58475	−0.397643
$582$	0	0
$583$	12.3293	0.510629
$584$	0	0
$585$	12.7341	0.526489
$586$	0	0
$587$	−10.3809	−0.428466	−0.214233	−	0.976783i	$-0.568725\pi$
$587$	−10.3809	−0.428466	−0.214233	+	0.976783i	$0.568725\pi$
$588$	0	0
$589$	6.31837	0.260344
$590$	0	0
$591$	−36.7777	−1.51283
$592$	0	0
$593$	14.3376	0.588776	0.294388	−	0.955686i	$-0.404884\pi$
$593$	14.3376	0.588776	0.294388	+	0.955686i	$0.404884\pi$
$594$	0	0
$595$	8.07607	0.331086
$596$	0	0
$597$	23.0779	0.944517
$598$	0	0
$599$	45.0009	1.83869	0.919343	−	0.393457i	$-0.128721\pi$
$599$	45.0009	1.83869	0.919343	+	0.393457i	$0.128721\pi$
$600$	0	0
$601$	26.4220	1.07778	0.538889	−	0.842377i	$-0.318844\pi$
$601$	26.4220	1.07778	0.538889	+	0.842377i	$0.318844\pi$
$602$	0	0
$603$	9.29496	0.378520
$604$	0	0
$605$	16.6495	0.676899
$606$	0	0
$607$	−21.1556	−0.858681	−0.429341	−	0.903143i	$-0.641254\pi$
$607$	−21.1556	−0.858681	−0.429341	+	0.903143i	$0.641254\pi$
$608$	0	0
$609$	0.859626	0.0348338
$610$	0	0
$611$	−24.5191	−0.991938
$612$	0	0
$613$	32.8818	1.32808	0.664041	−	0.747696i	$-0.268840\pi$
$613$	32.8818	1.32808	0.664041	+	0.747696i	$0.268840\pi$
$614$	0	0
$615$	4.48538	0.180868
$616$	0	0
$617$	−33.7894	−1.36031	−0.680155	−	0.733068i	$-0.738088\pi$
$617$	−33.7894	−1.36031	−0.680155	+	0.733068i	$0.738088\pi$
$618$	0	0
$619$	−29.2085	−1.17399	−0.586993	−	0.809592i	$-0.699689\pi$
$619$	−29.2085	−1.17399	−0.586993	+	0.809592i	$0.699689\pi$
$620$	0	0
$621$	28.0707	1.12644
$622$	0	0
$623$	6.66661	0.267092
$624$	0	0
$625$	−15.2301	−0.609205
$626$	0	0
$627$	2.00392	0.0800287
$628$	0	0
$629$	33.8957	1.35151
$630$	0	0
$631$	25.0031	0.995358	0.497679	−	0.867361i	$-0.334186\pi$
$631$	25.0031	0.995358	0.497679	+	0.867361i	$0.334186\pi$
$632$	0	0
$633$	16.2340	0.645243
$634$	0	0
$635$	40.6474	1.61304
$636$	0	0
$637$	6.27385	0.248579
$638$	0	0
$639$	−13.0516	−0.516314
$640$	0	0
$641$	7.82142	0.308927	0.154464	−	0.987998i	$-0.450635\pi$
$641$	7.82142	0.308927	0.154464	+	0.987998i	$0.450635\pi$
$642$	0	0
$643$	13.1758	0.519602	0.259801	−	0.965662i	$-0.416343\pi$
$643$	13.1758	0.519602	0.259801	+	0.965662i	$0.416343\pi$
$644$	0	0
$645$	23.4302	0.922562
$646$	0	0
$647$	16.8290	0.661618	0.330809	−	0.943698i	$-0.392678\pi$
$647$	16.8290	0.661618	0.330809	+	0.943698i	$0.392678\pi$
$648$	0	0
$649$	−11.3734	−0.446444
$650$	0	0
$651$	8.74288	0.342660
$652$	0	0
$653$	−18.8743	−0.738607	−0.369304	−	0.929309i	$-0.620404\pi$
$653$	−18.8743	−0.738607	−0.369304	+	0.929309i	$0.620404\pi$
$654$	0	0
$655$	−2.97046	−0.116065
$656$	0	0
$657$	−17.6662	−0.689226
$658$	0	0
$659$	4.12682	0.160758	0.0803790	−	0.996764i	$-0.474387\pi$
$659$	4.12682	0.160758	0.0803790	+	0.996764i	$0.474387\pi$
$660$	0	0
$661$	−6.03616	−0.234779	−0.117390	−	0.993086i	$-0.537453\pi$
$661$	−6.03616	−0.234779	−0.117390	+	0.993086i	$0.537453\pi$
$662$	0	0
$663$	−37.4889	−1.45595
$664$	0	0
$665$	−1.87017	−0.0725219
$666$	0	0
$667$	−3.08488	−0.119447
$668$	0	0
$669$	−10.5450	−0.407694
$670$	0	0
$671$	−18.2452	−0.704349
$672$	0	0
$673$	14.4452	0.556822	0.278411	−	0.960462i	$-0.410192\pi$
$673$	14.4452	0.556822	0.278411	+	0.960462i	$0.410192\pi$
$674$	0	0
$675$	8.49344	0.326913
$676$	0	0
$677$	12.5170	0.481067	0.240534	−	0.970641i	$-0.422678\pi$
$677$	12.5170	0.481067	0.240534	+	0.970641i	$0.422678\pi$
$678$	0	0
$679$	−7.10464	−0.272651
$680$	0	0
$681$	14.0043	0.536648
$682$	0	0
$683$	47.4786	1.81672	0.908360	−	0.418190i	$-0.137335\pi$
$683$	47.4786	1.81672	0.908360	+	0.418190i	$0.137335\pi$
$684$	0	0
$685$	14.9440	0.570982
$686$	0	0
$687$	−29.6218	−1.13014
$688$	0	0
$689$	53.4126	2.03486
$690$	0	0
$691$	−31.1999	−1.18690	−0.593449	−	0.804871i	$-0.702234\pi$
$691$	−31.1999	−1.18690	−0.593449	+	0.804871i	$0.702234\pi$
$692$	0	0
$693$	−1.57175	−0.0597059
$694$	0	0
$695$	−5.48204	−0.207945
$696$	0	0
$697$	7.48496	0.283513
$698$	0	0
$699$	−27.3842	−1.03576
$700$	0	0
$701$	7.26090	0.274241	0.137120	−	0.990554i	$-0.456215\pi$
$701$	7.26090	0.274241	0.137120	+	0.990554i	$0.456215\pi$
$702$	0	0
$703$	−7.84918	−0.296037
$704$	0	0
$705$	10.1135	0.380896
$706$	0	0
$707$	0.416508	0.0156644
$708$	0	0
$709$	7.41867	0.278614	0.139307	−	0.990249i	$-0.455513\pi$
$709$	7.41867	0.278614	0.139307	+	0.990249i	$0.455513\pi$
$710$	0	0
$711$	−11.6760	−0.437883
$712$	0	0
$713$	−31.3750	−1.17500
$714$	0	0
$715$	−16.9920	−0.635464
$716$	0	0
$717$	13.5605	0.506427
$718$	0	0
$719$	14.2630	0.531921	0.265961	−	0.963984i	$-0.414311\pi$
$719$	14.2630	0.531921	0.265961	+	0.963984i	$0.414311\pi$
$720$	0	0
$721$	1.97213	0.0734458
$722$	0	0
$723$	6.44250	0.239599
$724$	0	0
$725$	−0.933403	−0.0346657
$726$	0	0
$727$	−2.61185	−0.0968681	−0.0484341	−	0.998826i	$-0.515423\pi$
$727$	−2.61185	−0.0968681	−0.0484341	+	0.998826i	$0.515423\pi$
$728$	0	0
$729$	28.4220	1.05267
$730$	0	0
$731$	39.0990	1.44613
$732$	0	0
$733$	−40.8697	−1.50956	−0.754778	−	0.655980i	$-0.772256\pi$
$733$	−40.8697	−1.50956	−0.754778	+	0.655980i	$0.772256\pi$
$734$	0	0
$735$	−2.58779	−0.0954521
$736$	0	0
$737$	−12.4029	−0.456868
$738$	0	0
$739$	29.5011	1.08521	0.542607	−	0.839987i	$-0.317437\pi$
$739$	29.5011	1.08521	0.542607	+	0.839987i	$0.317437\pi$
$740$	0	0
$741$	8.68127	0.318914
$742$	0	0
$743$	−19.0787	−0.699930	−0.349965	−	0.936763i	$-0.613807\pi$
$743$	−19.0787	−0.699930	−0.349965	+	0.936763i	$0.613807\pi$
$744$	0	0
$745$	31.8747	1.16780
$746$	0	0
$747$	10.4024	0.380605
$748$	0	0
$749$	−13.3939	−0.489404
$750$	0	0
$751$	41.0691	1.49863	0.749316	−	0.662213i	$-0.230383\pi$
$751$	41.0691	1.49863	0.749316	+	0.662213i	$0.230383\pi$
$752$	0	0
$753$	−5.25886	−0.191643
$754$	0	0
$755$	26.4044	0.960953
$756$	0	0
$757$	−14.8994	−0.541529	−0.270764	−	0.962646i	$-0.587276\pi$
$757$	−14.8994	−0.541529	−0.270764	+	0.962646i	$0.587276\pi$
$758$	0	0
$759$	−9.95081	−0.361191
$760$	0	0
$761$	−20.2605	−0.734442	−0.367221	−	0.930134i	$-0.619691\pi$
$761$	−20.2605	−0.734442	−0.367221	+	0.930134i	$0.619691\pi$
$762$	0	0
$763$	11.2020	0.405540
$764$	0	0
$765$	−8.76503	−0.316900
$766$	0	0
$767$	−49.2711	−1.77908
$768$	0	0
$769$	25.3531	0.914255	0.457127	−	0.889401i	$-0.348878\pi$
$769$	25.3531	0.914255	0.457127	+	0.889401i	$0.348878\pi$
$770$	0	0
$771$	−11.7880	−0.424536
$772$	0	0
$773$	−2.13170	−0.0766720	−0.0383360	−	0.999265i	$-0.512206\pi$
$773$	−2.13170	−0.0766720	−0.0383360	+	0.999265i	$0.512206\pi$
$774$	0	0
$775$	−9.49324	−0.341007
$776$	0	0
$777$	−10.8611	−0.389640
$778$	0	0
$779$	−1.73328	−0.0621013
$780$	0	0
$781$	17.4157	0.623183
$782$	0	0
$783$	−3.51184	−0.125503
$784$	0	0
$785$	5.13506	0.183278
$786$	0	0
$787$	−1.27696	−0.0455188	−0.0227594	−	0.999741i	$-0.507245\pi$
$787$	−1.27696	−0.0455188	−0.0227594	+	0.999741i	$0.507245\pi$
$788$	0	0
$789$	−20.8723	−0.743074
$790$	0	0
$791$	−2.66351	−0.0947034
$792$	0	0
$793$	−79.0411	−2.80683
$794$	0	0
$795$	−22.0312	−0.781367
$796$	0	0
$797$	15.1112	0.535267	0.267634	−	0.963521i	$-0.413758\pi$
$797$	15.1112	0.535267	0.267634	+	0.963521i	$0.413758\pi$
$798$	0	0
$799$	16.8768	0.597060
$800$	0	0
$801$	−7.23533	−0.255648
$802$	0	0
$803$	23.5733	0.831884
$804$	0	0
$805$	9.28664	0.327311
$806$	0	0
$807$	−34.7586	−1.22356
$808$	0	0
$809$	50.4836	1.77491	0.887455	−	0.460894i	$-0.152471\pi$
$809$	50.4836	1.77491	0.887455	+	0.460894i	$0.152471\pi$
$810$	0	0
$811$	−17.2809	−0.606816	−0.303408	−	0.952861i	$-0.598125\pi$
$811$	−17.2809	−0.606816	−0.303408	+	0.952861i	$0.598125\pi$
$812$	0	0
$813$	16.8986	0.592659
$814$	0	0
$815$	−7.42646	−0.260138
$816$	0	0
$817$	−9.05412	−0.316764
$818$	0	0
$819$	−6.80906	−0.237928
$820$	0	0
$821$	30.5488	1.06616	0.533080	−	0.846065i	$-0.321034\pi$
$821$	30.5488	1.06616	0.533080	+	0.846065i	$0.321034\pi$
$822$	0	0
$823$	56.2052	1.95919	0.979595	−	0.200984i	$-0.0644139\pi$
$823$	56.2052	1.95919	0.979595	+	0.200984i	$0.0644139\pi$
$824$	0	0
$825$	−3.01085	−0.104824
$826$	0	0
$827$	51.1110	1.77730	0.888652	−	0.458581i	$-0.151642\pi$
$827$	51.1110	1.77730	0.888652	+	0.458581i	$0.151642\pi$
$828$	0	0
$829$	−43.9581	−1.52673	−0.763364	−	0.645969i	$-0.776453\pi$
$829$	−43.9581	−1.52673	−0.763364	+	0.645969i	$0.776453\pi$
$830$	0	0
$831$	−16.1570	−0.560479
$832$	0	0
$833$	−4.31837	−0.149623
$834$	0	0
$835$	19.7269	0.682679
$836$	0	0
$837$	−35.7174	−1.23457
$838$	0	0
$839$	22.3591	0.771922	0.385961	−	0.922515i	$-0.373870\pi$
$839$	22.3591	0.771922	0.385961	+	0.922515i	$0.373870\pi$
$840$	0	0
$841$	−28.6141	−0.986692
$842$	0	0
$843$	−27.5626	−0.949307
$844$	0	0
$845$	−49.2997	−1.69596
$846$	0	0
$847$	−8.90270	−0.305900
$848$	0	0
$849$	−37.3896	−1.28321
$850$	0	0
$851$	38.9765	1.33610
$852$	0	0
$853$	55.9657	1.91623	0.958115	−	0.286383i	$-0.0924530\pi$
$853$	55.9657	1.91623	0.958115	+	0.286383i	$0.0924530\pi$
$854$	0	0
$855$	2.02971	0.0694145
$856$	0	0
$857$	28.1719	0.962335	0.481168	−	0.876629i	$-0.340213\pi$
$857$	28.1719	0.962335	0.481168	+	0.876629i	$0.340213\pi$
$858$	0	0
$859$	−38.4594	−1.31222	−0.656110	−	0.754665i	$-0.727799\pi$
$859$	−38.4594	−1.31222	−0.656110	+	0.754665i	$0.727799\pi$
$860$	0	0
$861$	−2.39838	−0.0817367
$862$	0	0
$863$	−8.69372	−0.295938	−0.147969	−	0.988992i	$-0.547274\pi$
$863$	−8.69372	−0.295938	−0.147969	+	0.988992i	$0.547274\pi$
$864$	0	0
$865$	−8.82135	−0.299935
$866$	0	0
$867$	2.28083	0.0774610
$868$	0	0
$869$	15.5801	0.528518
$870$	0	0
$871$	−53.7314	−1.82062
$872$	0	0
$873$	7.71074	0.260969
$874$	0	0
$875$	12.1607	0.411107
$876$	0	0
$877$	−19.0369	−0.642829	−0.321414	−	0.946939i	$-0.604158\pi$
$877$	−19.0369	−0.642829	−0.321414	+	0.946939i	$0.604158\pi$
$878$	0	0
$879$	−35.5487	−1.19903
$880$	0	0
$881$	40.2677	1.35665	0.678327	−	0.734760i	$-0.262705\pi$
$881$	40.2677	1.35665	0.678327	+	0.734760i	$0.262705\pi$
$882$	0	0
$883$	19.4715	0.655269	0.327634	−	0.944805i	$-0.393749\pi$
$883$	19.4715	0.655269	0.327634	+	0.944805i	$0.393749\pi$
$884$	0	0
$885$	20.3230	0.683151
$886$	0	0
$887$	29.0249	0.974560	0.487280	−	0.873246i	$-0.337989\pi$
$887$	29.0249	0.974560	0.487280	+	0.873246i	$0.337989\pi$
$888$	0	0
$889$	−21.7347	−0.728957
$890$	0	0
$891$	−6.61276	−0.221536
$892$	0	0
$893$	−3.90815	−0.130781
$894$	0	0
$895$	10.1017	0.337664
$896$	0	0
$897$	−43.1084	−1.43935
$898$	0	0
$899$	3.92523	0.130914
$900$	0	0
$901$	−36.7646	−1.22481
$902$	0	0
$903$	−12.5284	−0.416919
$904$	0	0
$905$	−35.3405	−1.17476
$906$	0	0
$907$	10.2334	0.339795	0.169898	−	0.985462i	$-0.445656\pi$
$907$	10.2334	0.339795	0.169898	+	0.985462i	$0.445656\pi$
$908$	0	0
$909$	−0.452040	−0.0149932
$910$	0	0
$911$	−37.6447	−1.24722	−0.623612	−	0.781734i	$-0.714336\pi$
$911$	−37.6447	−1.24722	−0.623612	+	0.781734i	$0.714336\pi$
$912$	0	0
$913$	−13.8807	−0.459384
$914$	0	0
$915$	32.6023	1.07780
$916$	0	0
$917$	1.58834	0.0524516
$918$	0	0
$919$	2.87619	0.0948769	0.0474384	−	0.998874i	$-0.484894\pi$
$919$	2.87619	0.0948769	0.0474384	+	0.998874i	$0.484894\pi$
$920$	0	0
$921$	−15.4108	−0.507804
$922$	0	0
$923$	75.4474	2.48338
$924$	0	0
$925$	11.7933	0.387760
$926$	0	0
$927$	−2.14037	−0.0702989
$928$	0	0
$929$	11.1209	0.364865	0.182433	−	0.983218i	$-0.441603\pi$
$929$	11.1209	0.364865	0.182433	+	0.983218i	$0.441603\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−16.1777	−0.529634
$934$	0	0
$935$	11.6958	0.382494
$936$	0	0
$937$	−8.70578	−0.284406	−0.142203	−	0.989838i	$-0.545419\pi$
$937$	−8.70578	−0.284406	−0.142203	+	0.989838i	$0.545419\pi$
$938$	0	0
$939$	37.1619	1.21273
$940$	0	0
$941$	−21.0075	−0.684824	−0.342412	−	0.939550i	$-0.611244\pi$
$941$	−21.0075	−0.684824	−0.342412	+	0.939550i	$0.611244\pi$
$942$	0	0
$943$	8.60693	0.280280
$944$	0	0
$945$	10.5719	0.343905
$946$	0	0
$947$	9.82330	0.319214	0.159607	−	0.987181i	$-0.448977\pi$
$947$	9.82330	0.319214	0.159607	+	0.987181i	$0.448977\pi$
$948$	0	0
$949$	102.123	3.31506
$950$	0	0
$951$	−24.8067	−0.804413
$952$	0	0
$953$	5.85646	0.189709	0.0948547	−	0.995491i	$-0.469761\pi$
$953$	5.85646	0.189709	0.0948547	+	0.995491i	$0.469761\pi$
$954$	0	0
$955$	−41.1082	−1.33023
$956$	0	0
$957$	1.24491	0.0402424
$958$	0	0
$959$	−7.99076	−0.258035
$960$	0	0
$961$	8.92181	0.287800
$962$	0	0
$963$	14.5366	0.468434
$964$	0	0
$965$	−4.57590	−0.147303
$966$	0	0
$967$	32.5497	1.04673	0.523364	−	0.852109i	$-0.324677\pi$
$967$	32.5497	1.04673	0.523364	+	0.852109i	$0.324677\pi$
$968$	0	0
$969$	−5.97543	−0.191959
$970$	0	0
$971$	−37.4823	−1.20286	−0.601432	−	0.798924i	$-0.705403\pi$
$971$	−37.4823	−1.20286	−0.601432	+	0.798924i	$0.705403\pi$
$972$	0	0
$973$	2.93131	0.0939735
$974$	0	0
$975$	−13.0434	−0.417725
$976$	0	0
$977$	21.1317	0.676063	0.338031	−	0.941135i	$-0.390239\pi$
$977$	21.1317	0.676063	0.338031	+	0.941135i	$0.390239\pi$
$978$	0	0
$979$	9.65462	0.308563
$980$	0	0
$981$	−12.1576	−0.388163
$982$	0	0
$983$	−8.78709	−0.280265	−0.140132	−	0.990133i	$-0.544753\pi$
$983$	−8.78709	−0.280265	−0.140132	+	0.990133i	$0.544753\pi$
$984$	0	0
$985$	49.7067	1.58379
$986$	0	0
$987$	−5.40780	−0.172132
$988$	0	0
$989$	44.9599	1.42964
$990$	0	0
$991$	44.7007	1.41996	0.709982	−	0.704219i	$-0.248703\pi$
$991$	44.7007	1.41996	0.709982	+	0.704219i	$0.248703\pi$
$992$	0	0
$993$	12.6507	0.401458
$994$	0	0
$995$	−31.1909	−0.988818
$996$	0	0
$997$	−25.2799	−0.800624	−0.400312	−	0.916379i	$-0.631098\pi$
$997$	−25.2799	−0.800624	−0.400312	+	0.916379i	$0.631098\pi$
$998$	0	0
$999$	44.3709	1.40383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.ca.1.5		6
4.3	odd	2		8512.2.a.cf.1.2		6
8.3	odd	2		4256.2.a.i.1.5	✓	6
8.5	even	2		4256.2.a.n.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.i.1.5	✓	6	8.3	odd	2
4256.2.a.n.1.2	yes	6	8.5	even	2
8512.2.a.ca.1.5		6	1.1	even	1	trivial
8512.2.a.cf.1.2		6	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+1.38372 q^{3} -1.87017 q^{5} +1.00000 q^{7} -1.08531 q^{9} +1.44821 q^{11} +6.27385 q^{13} -2.58779 q^{15} -4.31837 q^{17} +1.00000 q^{19} +1.38372 q^{21} -4.96568 q^{23} -1.50248 q^{25} -5.65294 q^{27} +0.621241 q^{29} +6.31837 q^{31} +2.00392 q^{33} -1.87017 q^{35} -7.84918 q^{37} +8.68127 q^{39} -1.73328 q^{41} -9.05412 q^{43} +2.02971 q^{45} -3.90815 q^{47} +1.00000 q^{49} -5.97543 q^{51} +8.51353 q^{53} -2.70838 q^{55} +1.38372 q^{57} -7.85342 q^{59} -12.5985 q^{61} -1.08531 q^{63} -11.7331 q^{65} -8.56434 q^{67} -6.87113 q^{69} +12.0257 q^{71} +16.2776 q^{73} -2.07902 q^{75} +1.44821 q^{77} +10.7582 q^{79} -4.56618 q^{81} -9.58475 q^{83} +8.07607 q^{85} +0.859626 q^{87} +6.66661 q^{89} +6.27385 q^{91} +8.74288 q^{93} -1.87017 q^{95} -7.10464 q^{97} -1.57175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 7 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + 6 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} - 18 q^{27} - 5 q^{29} + 14 q^{31} + 3 q^{33} - 3 q^{35} + q^{37} + 22 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 7 * q^11 - 4 * q^13 + 4 * q^15 - 2 * q^17 + 6 * q^19 - 3 * q^21 - 2 * q^23 + q^25 - 18 * q^27 - 5 * q^29 + 14 * q^31 + 3 * q^33 - 3 * q^35 + q^37 + 22 * q^39 - 5 * q^41 - 15 * q^43 - 22 * q^45 - 7 * q^47 + 6 * q^49 + 4 * q^51 - 15 * q^53 + 20 * q^55 - 3 * q^57 - 23 * q^59 - 9 * q^61 + 3 * q^63 - 20 * q^65 - 2 * q^67 + 2 * q^69 + 23 * q^71 + 2 * q^73 - 4 * q^75 - 7 * q^77 + 17 * q^79 + 6 * q^81 - 22 * q^83 + 14 * q^85 - 10 * q^87 + 3 * q^89 - 4 * q^91 - 10 * q^93 - 3 * q^95 - 21 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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