LMFDB - Embedded newform 7616.2.a.bm.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7616,2,Mod(1,7616)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7616.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4,0,4,0,-2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.8140661794$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3808)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90211 q^{3} +2.90211 q^{5} +1.00000 q^{7} +0.618034 q^{9} -2.79360 q^{11} +1.95199 q^{13} -5.52015 q^{15} +1.00000 q^{17} +4.42882 q^{19} -1.90211 q^{21} -1.72654 q^{23} +3.42226 q^{25} +4.53077 q^{27} -8.15067 q^{29} -9.64771 q^{31} +5.31375 q^{33} +2.90211 q^{35} +2.61147 q^{37} -3.71290 q^{39} -1.43841 q^{41} -8.99885 q^{43} +1.79360 q^{45} -2.65948 q^{47} +1.00000 q^{49} -1.90211 q^{51} +4.50296 q^{53} -8.10736 q^{55} -8.42412 q^{57} +12.2147 q^{59} +13.8701 q^{61} +0.618034 q^{63} +5.66489 q^{65} -11.8167 q^{67} +3.28408 q^{69} -8.64584 q^{71} +2.37895 q^{73} -6.50953 q^{75} -2.79360 q^{77} -8.59783 q^{79} -10.4721 q^{81} -10.8096 q^{83} +2.90211 q^{85} +15.5035 q^{87} -14.0640 q^{89} +1.95199 q^{91} +18.3510 q^{93} +12.8529 q^{95} +14.0018 q^{97} -1.72654 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7} - 2 q^{9} - 2 q^{11} + 2 q^{13} - 10 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{23} - 6 q^{25} - 6 q^{29} - 16 q^{31} + 4 q^{35} + 8 q^{37} + 10 q^{39} - 10 q^{41} - 4 q^{43} - 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 - 2 * q^9 - 2 * q^11 + 2 * q^13 - 10 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^23 - 6 * q^25 - 6 * q^29 - 16 * q^31 + 4 * q^35 + 8 * q^37 + 10 * q^39 - 10 * q^41 - 4 * q^43 - 2 * q^45 - 14 * q^47 + 4 * q^49 + 14 * q^53 - 2 * q^55 - 10 * q^57 - 12 * q^59 + 2 * q^61 - 2 * q^63 - 8 * q^65 - 22 * q^67 + 10 * q^69 - 16 * q^71 - 14 * q^73 - 20 * q^75 - 2 * q^77 - 10 * q^79 - 24 * q^81 + 6 * q^83 + 4 * q^85 - 22 * q^89 + 2 * q^91 + 6 * q^95 - 12 * q^97 - 4 * 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
$3$	−1.90211	−1.09819	−0.549093	−	0.835761i	$-0.685027\pi$
$3$	−1.90211	−1.09819	−0.549093	+	0.835761i	$0.685027\pi$
$4$	0	0
$5$	2.90211	1.29786	0.648932	−	0.760846i	$-0.275216\pi$
$5$	2.90211	1.29786	0.648932	+	0.760846i	$0.275216\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.618034	0.206011
$10$	0	0
$11$	−2.79360	−0.842303	−0.421152	−	0.906990i	$-0.638374\pi$
$11$	−2.79360	−0.842303	−0.421152	+	0.906990i	$0.638374\pi$
$12$	0	0
$13$	1.95199	0.541384	0.270692	−	0.962666i	$-0.412747\pi$
$13$	1.95199	0.541384	0.270692	+	0.962666i	$0.412747\pi$
$14$	0	0
$15$	−5.52015	−1.42530
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.42882	1.01604	0.508021	−	0.861345i	$-0.330377\pi$
$19$	4.42882	1.01604	0.508021	+	0.861345i	$0.330377\pi$
$20$	0	0
$21$	−1.90211	−0.415075
$22$	0	0
$23$	−1.72654	−0.360009	−0.180005	−	0.983666i	$-0.557611\pi$
$23$	−1.72654	−0.360009	−0.180005	+	0.983666i	$0.557611\pi$
$24$	0	0
$25$	3.42226	0.684452
$26$	0	0
$27$	4.53077	0.871947
$28$	0	0
$29$	−8.15067	−1.51354	−0.756770	−	0.653681i	$-0.773224\pi$
$29$	−8.15067	−1.51354	−0.756770	+	0.653681i	$0.773224\pi$
$30$	0	0
$31$	−9.64771	−1.73278	−0.866390	−	0.499369i	$-0.833565\pi$
$31$	−9.64771	−1.73278	−0.866390	+	0.499369i	$0.833565\pi$
$32$	0	0
$33$	5.31375	0.925005
$34$	0	0
$35$	2.90211	0.490547
$36$	0	0
$37$	2.61147	0.429323	0.214661	−	0.976689i	$-0.431135\pi$
$37$	2.61147	0.429323	0.214661	+	0.976689i	$0.431135\pi$
$38$	0	0
$39$	−3.71290	−0.594540
$40$	0	0
$41$	−1.43841	−0.224641	−0.112321	−	0.993672i	$-0.535828\pi$
$41$	−1.43841	−0.224641	−0.112321	+	0.993672i	$0.535828\pi$
$42$	0	0
$43$	−8.99885	−1.37231	−0.686156	−	0.727455i	$-0.740703\pi$
$43$	−8.99885	−1.37231	−0.686156	+	0.727455i	$0.740703\pi$
$44$	0	0
$45$	1.79360	0.267375
$46$	0	0
$47$	−2.65948	−0.387925	−0.193963	−	0.981009i	$-0.562134\pi$
$47$	−2.65948	−0.387925	−0.193963	+	0.981009i	$0.562134\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.90211	−0.266349
$52$	0	0
$53$	4.50296	0.618529	0.309265	−	0.950976i	$-0.399917\pi$
$53$	4.50296	0.618529	0.309265	+	0.950976i	$0.399917\pi$
$54$	0	0
$55$	−8.10736	−1.09320
$56$	0	0
$57$	−8.42412	−1.11580
$58$	0	0
$59$	12.2147	1.59022	0.795110	−	0.606465i	$-0.207413\pi$
$59$	12.2147	1.59022	0.795110	+	0.606465i	$0.207413\pi$
$60$	0	0
$61$	13.8701	1.77589	0.887944	−	0.459951i	$-0.152133\pi$
$61$	13.8701	1.77589	0.887944	+	0.459951i	$0.152133\pi$
$62$	0	0
$63$	0.618034	0.0778650
$64$	0	0
$65$	5.66489	0.702643
$66$	0	0
$67$	−11.8167	−1.44364	−0.721821	−	0.692080i	$-0.756694\pi$
$67$	−11.8167	−1.44364	−0.721821	+	0.692080i	$0.756694\pi$
$68$	0	0
$69$	3.28408	0.395357
$70$	0	0
$71$	−8.64584	−1.02607	−0.513036	−	0.858367i	$-0.671479\pi$
$71$	−8.64584	−1.02607	−0.513036	+	0.858367i	$0.671479\pi$
$72$	0	0
$73$	2.37895	0.278435	0.139217	−	0.990262i	$-0.455541\pi$
$73$	2.37895	0.278435	0.139217	+	0.990262i	$0.455541\pi$
$74$	0	0
$75$	−6.50953	−0.751655
$76$	0	0
$77$	−2.79360	−0.318361
$78$	0	0
$79$	−8.59783	−0.967332	−0.483666	−	0.875253i	$-0.660695\pi$
$79$	−8.59783	−0.967332	−0.483666	+	0.875253i	$0.660695\pi$
$80$	0	0
$81$	−10.4721	−1.16357
$82$	0	0
$83$	−10.8096	−1.18651	−0.593256	−	0.805014i	$-0.702158\pi$
$83$	−10.8096	−1.18651	−0.593256	+	0.805014i	$0.702158\pi$
$84$	0	0
$85$	2.90211	0.314778
$86$	0	0
$87$	15.5035	1.66215
$88$	0	0
$89$	−14.0640	−1.49079	−0.745393	−	0.666625i	$-0.767738\pi$
$89$	−14.0640	−1.49079	−0.745393	+	0.666625i	$0.767738\pi$
$90$	0	0
$91$	1.95199	0.204624
$92$	0	0
$93$	18.3510	1.90291
$94$	0	0
$95$	12.8529	1.31868
$96$	0	0
$97$	14.0018	1.42166	0.710831	−	0.703363i	$-0.248319\pi$
$97$	14.0018	1.42166	0.710831	+	0.703363i	$0.248319\pi$
$98$	0	0
$99$	−1.72654	−0.173524
$100$	0	0
$101$	6.74186	0.670841	0.335420	−	0.942069i	$-0.391122\pi$
$101$	6.74186	0.670841	0.335420	+	0.942069i	$0.391122\pi$
$102$	0	0
$103$	−13.0135	−1.28226	−0.641130	−	0.767432i	$-0.721534\pi$
$103$	−13.0135	−1.28226	−0.641130	+	0.767432i	$0.721534\pi$
$104$	0	0
$105$	−5.52015	−0.538711
$106$	0	0
$107$	−6.73716	−0.651306	−0.325653	−	0.945489i	$-0.605584\pi$
$107$	−6.73716	−0.651306	−0.325653	+	0.945489i	$0.605584\pi$
$108$	0	0
$109$	−12.7829	−1.22438	−0.612188	−	0.790712i	$-0.709711\pi$
$109$	−12.7829	−1.22438	−0.612188	+	0.790712i	$0.709711\pi$
$110$	0	0
$111$	−4.96731	−0.471476
$112$	0	0
$113$	−0.308340	−0.0290061	−0.0145031	−	0.999895i	$-0.504617\pi$
$113$	−0.308340	−0.0290061	−0.0145031	+	0.999895i	$0.504617\pi$
$114$	0	0
$115$	−5.01062	−0.467243
$116$	0	0
$117$	1.20640	0.111531
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−3.19577	−0.290525
$122$	0	0
$123$	2.73601	0.246698
$124$	0	0
$125$	−4.57878	−0.409539
$126$	0	0
$127$	−17.5343	−1.55592	−0.777959	−	0.628314i	$-0.783745\pi$
$127$	−17.5343	−1.55592	−0.777959	+	0.628314i	$0.783745\pi$
$128$	0	0
$129$	17.1168	1.50705
$130$	0	0
$131$	8.02497	0.701145	0.350573	−	0.936536i	$-0.385987\pi$
$131$	8.02497	0.701145	0.350573	+	0.936536i	$0.385987\pi$
$132$	0	0
$133$	4.42882	0.384028
$134$	0	0
$135$	13.1488	1.13167
$136$	0	0
$137$	−9.63336	−0.823033	−0.411517	−	0.911402i	$-0.635001\pi$
$137$	−9.63336	−0.823033	−0.411517	+	0.911402i	$0.635001\pi$
$138$	0	0
$139$	16.5054	1.39997	0.699983	−	0.714160i	$-0.253191\pi$
$139$	16.5054	1.39997	0.699983	+	0.714160i	$0.253191\pi$
$140$	0	0
$141$	5.05863	0.426014
$142$	0	0
$143$	−5.45309	−0.456010
$144$	0	0
$145$	−23.6542	−1.96437
$146$	0	0
$147$	−1.90211	−0.156884
$148$	0	0
$149$	21.7139	1.77888	0.889438	−	0.457056i	$-0.151096\pi$
$149$	21.7139	1.77888	0.889438	+	0.457056i	$0.151096\pi$
$150$	0	0
$151$	14.0354	1.14219	0.571093	−	0.820885i	$-0.306520\pi$
$151$	14.0354	1.14219	0.571093	+	0.820885i	$0.306520\pi$
$152$	0	0
$153$	0.618034	0.0499651
$154$	0	0
$155$	−27.9987	−2.24891
$156$	0	0
$157$	−16.4958	−1.31651	−0.658253	−	0.752797i	$-0.728704\pi$
$157$	−16.4958	−1.31651	−0.658253	+	0.752797i	$0.728704\pi$
$158$	0	0
$159$	−8.56514	−0.679260
$160$	0	0
$161$	−1.72654	−0.136071
$162$	0	0
$163$	−0.946067	−0.0741017	−0.0370508	−	0.999313i	$-0.511796\pi$
$163$	−0.946067	−0.0741017	−0.0370508	+	0.999313i	$0.511796\pi$
$164$	0	0
$165$	15.4211	1.20053
$166$	0	0
$167$	4.29947	0.332703	0.166351	−	0.986067i	$-0.446801\pi$
$167$	4.29947	0.332703	0.166351	+	0.986067i	$0.446801\pi$
$168$	0	0
$169$	−9.18974	−0.706903
$170$	0	0
$171$	2.73716	0.209316
$172$	0	0
$173$	21.1891	1.61098	0.805489	−	0.592611i	$-0.201903\pi$
$173$	21.1891	1.61098	0.805489	+	0.592611i	$0.201903\pi$
$174$	0	0
$175$	3.42226	0.258699
$176$	0	0
$177$	−23.2338	−1.74636
$178$	0	0
$179$	1.43661	0.107377	0.0536887	−	0.998558i	$-0.482902\pi$
$179$	1.43661	0.107377	0.0536887	+	0.998558i	$0.482902\pi$
$180$	0	0
$181$	−19.2169	−1.42838	−0.714191	−	0.699951i	$-0.753205\pi$
$181$	−19.2169	−1.42838	−0.714191	+	0.699951i	$0.753205\pi$
$182$	0	0
$183$	−26.3826	−1.95026
$184$	0	0
$185$	7.57878	0.557203
$186$	0	0
$187$	−2.79360	−0.204289
$188$	0	0
$189$	4.53077	0.329565
$190$	0	0
$191$	20.9376	1.51499	0.757496	−	0.652840i	$-0.226423\pi$
$191$	20.9376	1.51499	0.757496	+	0.652840i	$0.226423\pi$
$192$	0	0
$193$	−6.42412	−0.462419	−0.231209	−	0.972904i	$-0.574268\pi$
$193$	−6.42412	−0.462419	−0.231209	+	0.972904i	$0.574268\pi$
$194$	0	0
$195$	−10.7753	−0.771633
$196$	0	0
$197$	11.2710	0.803022	0.401511	−	0.915854i	$-0.368485\pi$
$197$	11.2710	0.803022	0.401511	+	0.915854i	$0.368485\pi$
$198$	0	0
$199$	10.6506	0.755002	0.377501	−	0.926009i	$-0.376784\pi$
$199$	10.6506	0.755002	0.377501	+	0.926009i	$0.376784\pi$
$200$	0	0
$201$	22.4767	1.58539
$202$	0	0
$203$	−8.15067	−0.572065
$204$	0	0
$205$	−4.17442	−0.291554
$206$	0	0
$207$	−1.06706	−0.0741659
$208$	0	0
$209$	−12.3724	−0.855816
$210$	0	0
$211$	6.18974	0.426119	0.213060	−	0.977039i	$-0.431657\pi$
$211$	6.18974	0.426119	0.213060	+	0.977039i	$0.431657\pi$
$212$	0	0
$213$	16.4454	1.12682
$214$	0	0
$215$	−26.1157	−1.78107
$216$	0	0
$217$	−9.64771	−0.654929
$218$	0	0
$219$	−4.52503	−0.305773
$220$	0	0
$221$	1.95199	0.131305
$222$	0	0
$223$	6.60543	0.442333	0.221166	−	0.975236i	$-0.429014\pi$
$223$	6.60543	0.442333	0.221166	+	0.975236i	$0.429014\pi$
$224$	0	0
$225$	2.11507	0.141005
$226$	0	0
$227$	4.86253	0.322738	0.161369	−	0.986894i	$-0.448409\pi$
$227$	4.86253	0.322738	0.161369	+	0.986894i	$0.448409\pi$
$228$	0	0
$229$	−7.02748	−0.464389	−0.232194	−	0.972669i	$-0.574591\pi$
$229$	−7.02748	−0.464389	−0.232194	+	0.972669i	$0.574591\pi$
$230$	0	0
$231$	5.31375	0.349619
$232$	0	0
$233$	−14.9010	−0.976194	−0.488097	−	0.872789i	$-0.662309\pi$
$233$	−14.9010	−0.976194	−0.488097	+	0.872789i	$0.662309\pi$
$234$	0	0
$235$	−7.71811	−0.503474
$236$	0	0
$237$	16.3540	1.06231
$238$	0	0
$239$	−15.2775	−0.988220	−0.494110	−	0.869399i	$-0.664506\pi$
$239$	−15.2775	−0.988220	−0.494110	+	0.869399i	$0.664506\pi$
$240$	0	0
$241$	−9.20473	−0.592929	−0.296464	−	0.955044i	$-0.595808\pi$
$241$	−9.20473	−0.592929	−0.296464	+	0.955044i	$0.595808\pi$
$242$	0	0
$243$	6.32688	0.405870
$244$	0	0
$245$	2.90211	0.185409
$246$	0	0
$247$	8.64502	0.550069
$248$	0	0
$249$	20.5612	1.30301
$250$	0	0
$251$	−3.33499	−0.210503	−0.105251	−	0.994446i	$-0.533565\pi$
$251$	−3.33499	−0.210503	−0.105251	+	0.994446i	$0.533565\pi$
$252$	0	0
$253$	4.82328	0.303237
$254$	0	0
$255$	−5.52015	−0.345685
$256$	0	0
$257$	−25.2390	−1.57436	−0.787182	−	0.616721i	$-0.788461\pi$
$257$	−25.2390	−1.57436	−0.787182	+	0.616721i	$0.788461\pi$
$258$	0	0
$259$	2.61147	0.162269
$260$	0	0
$261$	−5.03739	−0.311807
$262$	0	0
$263$	−3.67383	−0.226538	−0.113269	−	0.993564i	$-0.536132\pi$
$263$	−3.67383	−0.226538	−0.113269	+	0.993564i	$0.536132\pi$
$264$	0	0
$265$	13.0681	0.802767
$266$	0	0
$267$	26.7514	1.63716
$268$	0	0
$269$	29.0167	1.76918	0.884591	−	0.466367i	$-0.154438\pi$
$269$	29.0167	1.76918	0.884591	+	0.466367i	$0.154438\pi$
$270$	0	0
$271$	−10.6364	−0.646118	−0.323059	−	0.946379i	$-0.604711\pi$
$271$	−10.6364	−0.646118	−0.323059	+	0.946379i	$0.604711\pi$
$272$	0	0
$273$	−3.71290	−0.224715
$274$	0	0
$275$	−9.56044	−0.576516
$276$	0	0
$277$	7.38623	0.443795	0.221898	−	0.975070i	$-0.428775\pi$
$277$	7.38623	0.443795	0.221898	+	0.975070i	$0.428775\pi$
$278$	0	0
$279$	−5.96261	−0.356972
$280$	0	0
$281$	15.8802	0.947336	0.473668	−	0.880703i	$-0.342930\pi$
$281$	15.8802	0.947336	0.473668	+	0.880703i	$0.342930\pi$
$282$	0	0
$283$	−7.20054	−0.428028	−0.214014	−	0.976831i	$-0.568654\pi$
$283$	−7.20054	−0.428028	−0.214014	+	0.976831i	$0.568654\pi$
$284$	0	0
$285$	−24.4478	−1.44816
$286$	0	0
$287$	−1.43841	−0.0849064
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−26.6329	−1.56125
$292$	0	0
$293$	3.54993	0.207389	0.103695	−	0.994609i	$-0.466934\pi$
$293$	3.54993	0.207389	0.103695	+	0.994609i	$0.466934\pi$
$294$	0	0
$295$	35.4485	2.06389
$296$	0	0
$297$	−12.6572	−0.734444
$298$	0	0
$299$	−3.37019	−0.194903
$300$	0	0
$301$	−8.99885	−0.518685
$302$	0	0
$303$	−12.8238	−0.736707
$304$	0	0
$305$	40.2527	2.30486
$306$	0	0
$307$	−8.41809	−0.480446	−0.240223	−	0.970718i	$-0.577221\pi$
$307$	−8.41809	−0.480446	−0.240223	+	0.970718i	$0.577221\pi$
$308$	0	0
$309$	24.7532	1.40816
$310$	0	0
$311$	8.25835	0.468288	0.234144	−	0.972202i	$-0.424771\pi$
$311$	8.25835	0.468288	0.234144	+	0.972202i	$0.424771\pi$
$312$	0	0
$313$	26.8938	1.52013	0.760063	−	0.649849i	$-0.225168\pi$
$313$	26.8938	1.52013	0.760063	+	0.649849i	$0.225168\pi$
$314$	0	0
$315$	1.79360	0.101058
$316$	0	0
$317$	−18.6789	−1.04911	−0.524557	−	0.851376i	$-0.675769\pi$
$317$	−18.6789	−1.04911	−0.524557	+	0.851376i	$0.675769\pi$
$318$	0	0
$319$	22.7697	1.27486
$320$	0	0
$321$	12.8148	0.715255
$322$	0	0
$323$	4.42882	0.246426
$324$	0	0
$325$	6.68021	0.370552
$326$	0	0
$327$	24.3145	1.34459
$328$	0	0
$329$	−2.65948	−0.146622
$330$	0	0
$331$	−25.9640	−1.42711	−0.713554	−	0.700600i	$-0.752916\pi$
$331$	−25.9640	−1.42711	−0.713554	+	0.700600i	$0.752916\pi$
$332$	0	0
$333$	1.61398	0.0884454
$334$	0	0
$335$	−34.2934	−1.87365
$336$	0	0
$337$	−19.6405	−1.06989	−0.534944	−	0.844888i	$-0.679667\pi$
$337$	−19.6405	−1.06989	−0.534944	+	0.844888i	$0.679667\pi$
$338$	0	0
$339$	0.586497	0.0318541
$340$	0	0
$341$	26.9519	1.45953
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	9.53077	0.513119
$346$	0	0
$347$	−6.39155	−0.343116	−0.171558	−	0.985174i	$-0.554880\pi$
$347$	−6.39155	−0.343116	−0.171558	+	0.985174i	$0.554880\pi$
$348$	0	0
$349$	−26.4474	−1.41570	−0.707850	−	0.706363i	$-0.750335\pi$
$349$	−26.4474	−1.41570	−0.707850	+	0.706363i	$0.750335\pi$
$350$	0	0
$351$	8.84401	0.472058
$352$	0	0
$353$	17.8801	0.951662	0.475831	−	0.879537i	$-0.342147\pi$
$353$	17.8801	0.951662	0.475831	+	0.879537i	$0.342147\pi$
$354$	0	0
$355$	−25.0912	−1.33170
$356$	0	0
$357$	−1.90211	−0.100670
$358$	0	0
$359$	11.7038	0.617705	0.308852	−	0.951110i	$-0.400055\pi$
$359$	11.7038	0.617705	0.308852	+	0.951110i	$0.400055\pi$
$360$	0	0
$361$	0.614487	0.0323414
$362$	0	0
$363$	6.07872	0.319050
$364$	0	0
$365$	6.90398	0.361371
$366$	0	0
$367$	−16.8345	−0.878756	−0.439378	−	0.898302i	$-0.644801\pi$
$367$	−16.8345	−0.878756	−0.439378	+	0.898302i	$0.644801\pi$
$368$	0	0
$369$	−0.888984	−0.0462787
$370$	0	0
$371$	4.50296	0.233782
$372$	0	0
$373$	−14.6561	−0.758866	−0.379433	−	0.925219i	$-0.623881\pi$
$373$	−14.6561	−0.758866	−0.379433	+	0.925219i	$0.623881\pi$
$374$	0	0
$375$	8.70936	0.449749
$376$	0	0
$377$	−15.9100	−0.819407
$378$	0	0
$379$	−27.8491	−1.43051	−0.715256	−	0.698862i	$-0.753690\pi$
$379$	−27.8491	−1.43051	−0.715256	+	0.698862i	$0.753690\pi$
$380$	0	0
$381$	33.3523	1.70869
$382$	0	0
$383$	33.5267	1.71313	0.856566	−	0.516037i	$-0.172593\pi$
$383$	33.5267	1.71313	0.856566	+	0.516037i	$0.172593\pi$
$384$	0	0
$385$	−8.10736	−0.413189
$386$	0	0
$387$	−5.56159	−0.282712
$388$	0	0
$389$	−33.6009	−1.70364	−0.851818	−	0.523838i	$-0.824500\pi$
$389$	−33.6009	−1.70364	−0.851818	+	0.523838i	$0.824500\pi$
$390$	0	0
$391$	−1.72654	−0.0873150
$392$	0	0
$393$	−15.2644	−0.769987
$394$	0	0
$395$	−24.9519	−1.25547
$396$	0	0
$397$	−31.0575	−1.55873	−0.779365	−	0.626570i	$-0.784458\pi$
$397$	−31.0575	−1.55873	−0.779365	+	0.626570i	$0.784458\pi$
$398$	0	0
$399$	−8.42412	−0.421734
$400$	0	0
$401$	−18.5519	−0.926438	−0.463219	−	0.886244i	$-0.653306\pi$
$401$	−18.5519	−0.926438	−0.463219	+	0.886244i	$0.653306\pi$
$402$	0	0
$403$	−18.8322	−0.938100
$404$	0	0
$405$	−30.3913	−1.51016
$406$	0	0
$407$	−7.29541	−0.361620
$408$	0	0
$409$	9.75571	0.482389	0.241194	−	0.970477i	$-0.422461\pi$
$409$	9.75571	0.482389	0.241194	+	0.970477i	$0.422461\pi$
$410$	0	0
$411$	18.3237	0.903843
$412$	0	0
$413$	12.2147	0.601047
$414$	0	0
$415$	−31.3708	−1.53993
$416$	0	0
$417$	−31.3951	−1.53742
$418$	0	0
$419$	18.9350	0.925036	0.462518	−	0.886610i	$-0.346946\pi$
$419$	18.9350	0.925036	0.462518	+	0.886610i	$0.346946\pi$
$420$	0	0
$421$	−6.43539	−0.313642	−0.156821	−	0.987627i	$-0.550125\pi$
$421$	−6.43539	−0.313642	−0.156821	+	0.987627i	$0.550125\pi$
$422$	0	0
$423$	−1.64365	−0.0799170
$424$	0	0
$425$	3.42226	0.166004
$426$	0	0
$427$	13.8701	0.671223
$428$	0	0
$429$	10.3724	0.500783
$430$	0	0
$431$	−10.8690	−0.523540	−0.261770	−	0.965130i	$-0.584306\pi$
$431$	−10.8690	−0.523540	−0.261770	+	0.965130i	$0.584306\pi$
$432$	0	0
$433$	−7.15286	−0.343744	−0.171872	−	0.985119i	$-0.554982\pi$
$433$	−7.15286	−0.343744	−0.171872	+	0.985119i	$0.554982\pi$
$434$	0	0
$435$	44.9929	2.15724
$436$	0	0
$437$	−7.64655	−0.365784
$438$	0	0
$439$	18.3032	0.873564	0.436782	−	0.899567i	$-0.356118\pi$
$439$	18.3032	0.873564	0.436782	+	0.899567i	$0.356118\pi$
$440$	0	0
$441$	0.618034	0.0294302
$442$	0	0
$443$	17.5064	0.831754	0.415877	−	0.909421i	$-0.363475\pi$
$443$	17.5064	0.831754	0.415877	+	0.909421i	$0.363475\pi$
$444$	0	0
$445$	−40.8154	−1.93484
$446$	0	0
$447$	−41.3024	−1.95354
$448$	0	0
$449$	−27.0075	−1.27456	−0.637281	−	0.770631i	$-0.719941\pi$
$449$	−27.0075	−1.27456	−0.637281	+	0.770631i	$0.719941\pi$
$450$	0	0
$451$	4.01834	0.189216
$452$	0	0
$453$	−26.6969	−1.25433
$454$	0	0
$455$	5.66489	0.265574
$456$	0	0
$457$	24.9609	1.16762	0.583812	−	0.811889i	$-0.301561\pi$
$457$	24.9609	1.16762	0.583812	+	0.811889i	$0.301561\pi$
$458$	0	0
$459$	4.53077	0.211478
$460$	0	0
$461$	−28.5207	−1.32834	−0.664172	−	0.747580i	$-0.731216\pi$
$461$	−28.5207	−1.32834	−0.664172	+	0.747580i	$0.731216\pi$
$462$	0	0
$463$	−11.6455	−0.541213	−0.270606	−	0.962690i	$-0.587224\pi$
$463$	−11.6455	−0.541213	−0.270606	+	0.962690i	$0.587224\pi$
$464$	0	0
$465$	53.2568	2.46972
$466$	0	0
$467$	−19.2776	−0.892060	−0.446030	−	0.895018i	$-0.647163\pi$
$467$	−19.2776	−0.892060	−0.446030	+	0.895018i	$0.647163\pi$
$468$	0	0
$469$	−11.8167	−0.545645
$470$	0	0
$471$	31.3768	1.44577
$472$	0	0
$473$	25.1392	1.15590
$474$	0	0
$475$	15.1566	0.695432
$476$	0	0
$477$	2.78298	0.127424
$478$	0	0
$479$	−2.94471	−0.134547	−0.0672737	−	0.997735i	$-0.521430\pi$
$479$	−2.94471	−0.134547	−0.0672737	+	0.997735i	$0.521430\pi$
$480$	0	0
$481$	5.09756	0.232429
$482$	0	0
$483$	3.28408	0.149431
$484$	0	0
$485$	40.6347	1.84513
$486$	0	0
$487$	35.1535	1.59296	0.796478	−	0.604668i	$-0.206694\pi$
$487$	35.1535	1.59296	0.796478	+	0.604668i	$0.206694\pi$
$488$	0	0
$489$	1.79953	0.0813774
$490$	0	0
$491$	4.50715	0.203405	0.101702	−	0.994815i	$-0.467571\pi$
$491$	4.50715	0.203405	0.101702	+	0.994815i	$0.467571\pi$
$492$	0	0
$493$	−8.15067	−0.367088
$494$	0	0
$495$	−5.01062	−0.225211
$496$	0	0
$497$	−8.64584	−0.387819
$498$	0	0
$499$	4.35373	0.194900	0.0974499	−	0.995240i	$-0.468931\pi$
$499$	4.35373	0.194900	0.0974499	+	0.995240i	$0.468931\pi$
$500$	0	0
$501$	−8.17808	−0.365370
$502$	0	0
$503$	−2.24687	−0.100183	−0.0500915	−	0.998745i	$-0.515951\pi$
$503$	−2.24687	−0.100183	−0.0500915	+	0.998745i	$0.515951\pi$
$504$	0	0
$505$	19.5657	0.870660
$506$	0	0
$507$	17.4799	0.776311
$508$	0	0
$509$	−13.9732	−0.619353	−0.309676	−	0.950842i	$-0.600221\pi$
$509$	−13.9732	−0.619353	−0.309676	+	0.950842i	$0.600221\pi$
$510$	0	0
$511$	2.37895	0.105238
$512$	0	0
$513$	20.0660	0.885935
$514$	0	0
$515$	−37.7667	−1.66420
$516$	0	0
$517$	7.42954	0.326751
$518$	0	0
$519$	−40.3041	−1.76915
$520$	0	0
$521$	−17.7337	−0.776928	−0.388464	−	0.921464i	$-0.626994\pi$
$521$	−17.7337	−0.776928	−0.388464	+	0.921464i	$0.626994\pi$
$522$	0	0
$523$	−3.32546	−0.145412	−0.0727060	−	0.997353i	$-0.523163\pi$
$523$	−3.32546	−0.145412	−0.0727060	+	0.997353i	$0.523163\pi$
$524$	0	0
$525$	−6.50953	−0.284099
$526$	0	0
$527$	−9.64771	−0.420261
$528$	0	0
$529$	−20.0191	−0.870394
$530$	0	0
$531$	7.54911	0.327603
$532$	0	0
$533$	−2.80775	−0.121617
$534$	0	0
$535$	−19.5520	−0.845307
$536$	0	0
$537$	−2.73260	−0.117920
$538$	0	0
$539$	−2.79360	−0.120329
$540$	0	0
$541$	1.65988	0.0713637	0.0356819	−	0.999363i	$-0.488640\pi$
$541$	1.65988	0.0713637	0.0356819	+	0.999363i	$0.488640\pi$
$542$	0	0
$543$	36.5527	1.56863
$544$	0	0
$545$	−37.0973	−1.58908
$546$	0	0
$547$	−2.19817	−0.0939869	−0.0469934	−	0.998895i	$-0.514964\pi$
$547$	−2.19817	−0.0939869	−0.0469934	+	0.998895i	$0.514964\pi$
$548$	0	0
$549$	8.57222	0.365853
$550$	0	0
$551$	−36.0979	−1.53782
$552$	0	0
$553$	−8.59783	−0.365617
$554$	0	0
$555$	−14.4157	−0.611912
$556$	0	0
$557$	17.5410	0.743236	0.371618	−	0.928386i	$-0.378803\pi$
$557$	17.5410	0.743236	0.371618	+	0.928386i	$0.378803\pi$
$558$	0	0
$559$	−17.5657	−0.742948
$560$	0	0
$561$	5.31375	0.224347
$562$	0	0
$563$	2.69385	0.113532	0.0567662	−	0.998388i	$-0.481921\pi$
$563$	2.69385	0.113532	0.0567662	+	0.998388i	$0.481921\pi$
$564$	0	0
$565$	−0.894836	−0.0376460
$566$	0	0
$567$	−10.4721	−0.439788
$568$	0	0
$569$	−31.1304	−1.30505	−0.652526	−	0.757766i	$-0.726291\pi$
$569$	−31.1304	−1.30505	−0.652526	+	0.757766i	$0.726291\pi$
$570$	0	0
$571$	18.6992	0.782535	0.391268	−	0.920277i	$-0.372037\pi$
$571$	18.6992	0.782535	0.391268	+	0.920277i	$0.372037\pi$
$572$	0	0
$573$	−39.8257	−1.66374
$574$	0	0
$575$	−5.90868	−0.246409
$576$	0	0
$577$	−13.5891	−0.565723	−0.282862	−	0.959161i	$-0.591284\pi$
$577$	−13.5891	−0.565723	−0.282862	+	0.959161i	$0.591284\pi$
$578$	0	0
$579$	12.2194	0.507821
$580$	0	0
$581$	−10.8096	−0.448459
$582$	0	0
$583$	−12.5795	−0.520989
$584$	0	0
$585$	3.50110	0.144753
$586$	0	0
$587$	−32.6212	−1.34642	−0.673210	−	0.739452i	$-0.735085\pi$
$587$	−32.6212	−1.34642	−0.673210	+	0.739452i	$0.735085\pi$
$588$	0	0
$589$	−42.7280	−1.76058
$590$	0	0
$591$	−21.4386	−0.881867
$592$	0	0
$593$	−31.4668	−1.29219	−0.646094	−	0.763258i	$-0.723598\pi$
$593$	−31.4668	−1.29219	−0.646094	+	0.763258i	$0.723598\pi$
$594$	0	0
$595$	2.90211	0.118975
$596$	0	0
$597$	−20.2587	−0.829132
$598$	0	0
$599$	−6.11264	−0.249756	−0.124878	−	0.992172i	$-0.539854\pi$
$599$	−6.11264	−0.249756	−0.124878	+	0.992172i	$0.539854\pi$
$600$	0	0
$601$	22.5146	0.918391	0.459195	−	0.888335i	$-0.348138\pi$
$601$	22.5146	0.918391	0.459195	+	0.888335i	$0.348138\pi$
$602$	0	0
$603$	−7.30313	−0.297406
$604$	0	0
$605$	−9.27450	−0.377062
$606$	0	0
$607$	−39.8932	−1.61922	−0.809608	−	0.586972i	$-0.800320\pi$
$607$	−39.8932	−1.61922	−0.809608	+	0.586972i	$0.800320\pi$
$608$	0	0
$609$	15.5035	0.628233
$610$	0	0
$611$	−5.19128	−0.210017
$612$	0	0
$613$	3.86002	0.155905	0.0779524	−	0.996957i	$-0.475162\pi$
$613$	3.86002	0.155905	0.0779524	+	0.996957i	$0.475162\pi$
$614$	0	0
$615$	7.94021	0.320180
$616$	0	0
$617$	−12.1932	−0.490878	−0.245439	−	0.969412i	$-0.578932\pi$
$617$	−12.1932	−0.490878	−0.245439	+	0.969412i	$0.578932\pi$
$618$	0	0
$619$	25.8738	1.03995	0.519977	−	0.854180i	$-0.325940\pi$
$619$	25.8738	1.03995	0.519977	+	0.854180i	$0.325940\pi$
$620$	0	0
$621$	−7.82256	−0.313909
$622$	0	0
$623$	−14.0640	−0.563464
$624$	0	0
$625$	−30.3994	−1.21598
$626$	0	0
$627$	23.5337	0.939844
$628$	0	0
$629$	2.61147	0.104126
$630$	0	0
$631$	3.12254	0.124307	0.0621533	−	0.998067i	$-0.480203\pi$
$631$	3.12254	0.124307	0.0621533	+	0.998067i	$0.480203\pi$
$632$	0	0
$633$	−11.7736	−0.467958
$634$	0	0
$635$	−50.8866	−2.01937
$636$	0	0
$637$	1.95199	0.0773406
$638$	0	0
$639$	−5.34342	−0.211383
$640$	0	0
$641$	−34.7538	−1.37269	−0.686346	−	0.727275i	$-0.740787\pi$
$641$	−34.7538	−1.37269	−0.686346	+	0.727275i	$0.740787\pi$
$642$	0	0
$643$	−15.6588	−0.617524	−0.308762	−	0.951139i	$-0.599915\pi$
$643$	−15.6588	−0.617524	−0.308762	+	0.951139i	$0.599915\pi$
$644$	0	0
$645$	49.6750	1.95595
$646$	0	0
$647$	−0.281062	−0.0110497	−0.00552484	−	0.999985i	$-0.501759\pi$
$647$	−0.281062	−0.0110497	−0.00552484	+	0.999985i	$0.501759\pi$
$648$	0	0
$649$	−34.1231	−1.33945
$650$	0	0
$651$	18.3510	0.719234
$652$	0	0
$653$	−25.4043	−0.994149	−0.497074	−	0.867708i	$-0.665592\pi$
$653$	−25.4043	−0.994149	−0.497074	+	0.867708i	$0.665592\pi$
$654$	0	0
$655$	23.2894	0.909991
$656$	0	0
$657$	1.47027	0.0573607
$658$	0	0
$659$	−23.5745	−0.918332	−0.459166	−	0.888350i	$-0.651852\pi$
$659$	−23.5745	−0.918332	−0.459166	+	0.888350i	$0.651852\pi$
$660$	0	0
$661$	−19.6922	−0.765937	−0.382968	−	0.923761i	$-0.625098\pi$
$661$	−19.6922	−0.765937	−0.382968	+	0.923761i	$0.625098\pi$
$662$	0	0
$663$	−3.71290	−0.144197
$664$	0	0
$665$	12.8529	0.498416
$666$	0	0
$667$	14.0725	0.544888
$668$	0	0
$669$	−12.5643	−0.485763
$670$	0	0
$671$	−38.7477	−1.49584
$672$	0	0
$673$	19.9387	0.768579	0.384290	−	0.923213i	$-0.374446\pi$
$673$	19.9387	0.768579	0.384290	+	0.923213i	$0.374446\pi$
$674$	0	0
$675$	15.5055	0.596806
$676$	0	0
$677$	37.5264	1.44225	0.721127	−	0.692802i	$-0.243624\pi$
$677$	37.5264	1.44225	0.721127	+	0.692802i	$0.243624\pi$
$678$	0	0
$679$	14.0018	0.537338
$680$	0	0
$681$	−9.24908	−0.354426
$682$	0	0
$683$	−10.2394	−0.391800	−0.195900	−	0.980624i	$-0.562763\pi$
$683$	−10.2394	−0.391800	−0.195900	+	0.980624i	$0.562763\pi$
$684$	0	0
$685$	−27.9571	−1.06819
$686$	0	0
$687$	13.3671	0.509985
$688$	0	0
$689$	8.78973	0.334862
$690$	0	0
$691$	−45.0051	−1.71208	−0.856038	−	0.516913i	$-0.827081\pi$
$691$	−45.0051	−1.71208	−0.856038	+	0.516913i	$0.827081\pi$
$692$	0	0
$693$	−1.72654	−0.0655859
$694$	0	0
$695$	47.9004	1.81697
$696$	0	0
$697$	−1.43841	−0.0544835
$698$	0	0
$699$	28.3433	1.07204
$700$	0	0
$701$	−10.2087	−0.385576	−0.192788	−	0.981240i	$-0.561753\pi$
$701$	−10.2087	−0.385576	−0.192788	+	0.981240i	$0.561753\pi$
$702$	0	0
$703$	11.5657	0.436210
$704$	0	0
$705$	14.6807	0.552908
$706$	0	0
$707$	6.74186	0.253554
$708$	0	0
$709$	14.1455	0.531244	0.265622	−	0.964077i	$-0.414423\pi$
$709$	14.1455	0.531244	0.265622	+	0.964077i	$0.414423\pi$
$710$	0	0
$711$	−5.31375	−0.199281
$712$	0	0
$713$	16.6572	0.623816
$714$	0	0
$715$	−15.8255	−0.591839
$716$	0	0
$717$	29.0596	1.08525
$718$	0	0
$719$	−23.6932	−0.883608	−0.441804	−	0.897112i	$-0.645661\pi$
$719$	−23.6932	−0.883608	−0.441804	+	0.897112i	$0.645661\pi$
$720$	0	0
$721$	−13.0135	−0.484649
$722$	0	0
$723$	17.5084	0.651146
$724$	0	0
$725$	−27.8937	−1.03595
$726$	0	0
$727$	37.2401	1.38116	0.690579	−	0.723257i	$-0.257356\pi$
$727$	37.2401	1.38116	0.690579	+	0.723257i	$0.257356\pi$
$728$	0	0
$729$	19.3820	0.717851
$730$	0	0
$731$	−8.99885	−0.332834
$732$	0	0
$733$	32.7721	1.21047	0.605233	−	0.796048i	$-0.293080\pi$
$733$	32.7721	1.21047	0.605233	+	0.796048i	$0.293080\pi$
$734$	0	0
$735$	−5.52015	−0.203614
$736$	0	0
$737$	33.0112	1.21598
$738$	0	0
$739$	36.2193	1.33235	0.666173	−	0.745797i	$-0.267931\pi$
$739$	36.2193	1.33235	0.666173	+	0.745797i	$0.267931\pi$
$740$	0	0
$741$	−16.4438	−0.604078
$742$	0	0
$743$	−15.8313	−0.580793	−0.290397	−	0.956906i	$-0.593787\pi$
$743$	−15.8313	−0.580793	−0.290397	+	0.956906i	$0.593787\pi$
$744$	0	0
$745$	63.0163	2.30874
$746$	0	0
$747$	−6.68072	−0.244435
$748$	0	0
$749$	−6.73716	−0.246171
$750$	0	0
$751$	16.6958	0.609238	0.304619	−	0.952474i	$-0.401471\pi$
$751$	16.6958	0.609238	0.304619	+	0.952474i	$0.401471\pi$
$752$	0	0
$753$	6.34354	0.231171
$754$	0	0
$755$	40.7324	1.48240
$756$	0	0
$757$	1.51840	0.0551870	0.0275935	−	0.999619i	$-0.491216\pi$
$757$	1.51840	0.0551870	0.0275935	+	0.999619i	$0.491216\pi$
$758$	0	0
$759$	−9.17442	−0.333010
$760$	0	0
$761$	30.7816	1.11583	0.557916	−	0.829898i	$-0.311601\pi$
$761$	30.7816	1.11583	0.557916	+	0.829898i	$0.311601\pi$
$762$	0	0
$763$	−12.7829	−0.462771
$764$	0	0
$765$	1.79360	0.0648479
$766$	0	0
$767$	23.8430	0.860920
$768$	0	0
$769$	−29.7281	−1.07202	−0.536012	−	0.844211i	$-0.680070\pi$
$769$	−29.7281	−1.07202	−0.536012	+	0.844211i	$0.680070\pi$
$770$	0	0
$771$	48.0074	1.72894
$772$	0	0
$773$	21.8814	0.787019	0.393510	−	0.919320i	$-0.371261\pi$
$773$	21.8814	0.787019	0.393510	+	0.919320i	$0.371261\pi$
$774$	0	0
$775$	−33.0170	−1.18600
$776$	0	0
$777$	−4.96731	−0.178201
$778$	0	0
$779$	−6.37045	−0.228245
$780$	0	0
$781$	24.1531	0.864265
$782$	0	0
$783$	−36.9288	−1.31973
$784$	0	0
$785$	−47.8726	−1.70865
$786$	0	0
$787$	−42.6904	−1.52175	−0.760873	−	0.648900i	$-0.775229\pi$
$787$	−42.6904	−1.52175	−0.760873	+	0.648900i	$0.775229\pi$
$788$	0	0
$789$	6.98804	0.248781
$790$	0	0
$791$	−0.308340	−0.0109633
$792$	0	0
$793$	27.0744	0.961438
$794$	0	0
$795$	−24.8570	−0.881587
$796$	0	0
$797$	−26.3412	−0.933054	−0.466527	−	0.884507i	$-0.654495\pi$
$797$	−26.3412	−0.933054	−0.466527	+	0.884507i	$0.654495\pi$
$798$	0	0
$799$	−2.65948	−0.0940857
$800$	0	0
$801$	−8.69206	−0.307119
$802$	0	0
$803$	−6.64584	−0.234527
$804$	0	0
$805$	−5.01062	−0.176601
$806$	0	0
$807$	−55.1931	−1.94289
$808$	0	0
$809$	8.65929	0.304444	0.152222	−	0.988346i	$-0.451357\pi$
$809$	8.65929	0.304444	0.152222	+	0.988346i	$0.451357\pi$
$810$	0	0
$811$	−39.4372	−1.38483	−0.692414	−	0.721501i	$-0.743453\pi$
$811$	−39.4372	−1.38483	−0.692414	+	0.721501i	$0.743453\pi$
$812$	0	0
$813$	20.2317	0.709557
$814$	0	0
$815$	−2.74559	−0.0961739
$816$	0	0
$817$	−39.8543	−1.39433
$818$	0	0
$819$	1.20640	0.0421549
$820$	0	0
$821$	−4.95450	−0.172913	−0.0864566	−	0.996256i	$-0.527554\pi$
$821$	−4.95450	−0.172913	−0.0864566	+	0.996256i	$0.527554\pi$
$822$	0	0
$823$	10.1166	0.352643	0.176321	−	0.984333i	$-0.443580\pi$
$823$	10.1166	0.352643	0.176321	+	0.984333i	$0.443580\pi$
$824$	0	0
$825$	18.1850	0.633122
$826$	0	0
$827$	2.43956	0.0848318	0.0424159	−	0.999100i	$-0.486495\pi$
$827$	2.43956	0.0848318	0.0424159	+	0.999100i	$0.486495\pi$
$828$	0	0
$829$	33.3853	1.15952	0.579759	−	0.814788i	$-0.303147\pi$
$829$	33.3853	1.15952	0.579759	+	0.814788i	$0.303147\pi$
$830$	0	0
$831$	−14.0494	−0.487370
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	12.4775	0.431803
$836$	0	0
$837$	−43.7115	−1.51089
$838$	0	0
$839$	−30.9726	−1.06929	−0.534647	−	0.845076i	$-0.679555\pi$
$839$	−30.9726	−1.06929	−0.534647	+	0.845076i	$0.679555\pi$
$840$	0	0
$841$	37.4334	1.29081
$842$	0	0
$843$	−30.2060	−1.04035
$844$	0	0
$845$	−26.6697	−0.917464
$846$	0	0
$847$	−3.19577	−0.109808
$848$	0	0
$849$	13.6962	0.470054
$850$	0	0
$851$	−4.50881	−0.154560
$852$	0	0
$853$	3.38771	0.115993	0.0579964	−	0.998317i	$-0.481529\pi$
$853$	3.38771	0.115993	0.0579964	+	0.998317i	$0.481529\pi$
$854$	0	0
$855$	7.94356	0.271664
$856$	0	0
$857$	35.6814	1.21885	0.609427	−	0.792843i	$-0.291400\pi$
$857$	35.6814	1.21885	0.609427	+	0.792843i	$0.291400\pi$
$858$	0	0
$859$	−55.8516	−1.90563	−0.952815	−	0.303551i	$-0.901828\pi$
$859$	−55.8516	−1.90563	−0.952815	+	0.303551i	$0.901828\pi$
$860$	0	0
$861$	2.73601	0.0932430
$862$	0	0
$863$	28.3101	0.963687	0.481843	−	0.876257i	$-0.339967\pi$
$863$	28.3101	0.963687	0.481843	+	0.876257i	$0.339967\pi$
$864$	0	0
$865$	61.4932	2.09083
$866$	0	0
$867$	−1.90211	−0.0645991
$868$	0	0
$869$	24.0189	0.814787
$870$	0	0
$871$	−23.0661	−0.781565
$872$	0	0
$873$	8.65356	0.292879
$874$	0	0
$875$	−4.57878	−0.154791
$876$	0	0
$877$	18.8417	0.636239	0.318120	−	0.948051i	$-0.396949\pi$
$877$	18.8417	0.636239	0.318120	+	0.948051i	$0.396949\pi$
$878$	0	0
$879$	−6.75237	−0.227752
$880$	0	0
$881$	9.85205	0.331924	0.165962	−	0.986132i	$-0.446927\pi$
$881$	9.85205	0.331924	0.165962	+	0.986132i	$0.446927\pi$
$882$	0	0
$883$	48.1062	1.61890	0.809451	−	0.587187i	$-0.199765\pi$
$883$	48.1062	1.61890	0.809451	+	0.587187i	$0.199765\pi$
$884$	0	0
$885$	−67.4270	−2.26653
$886$	0	0
$887$	15.5193	0.521085	0.260543	−	0.965462i	$-0.416099\pi$
$887$	15.5193	0.521085	0.260543	+	0.965462i	$0.416099\pi$
$888$	0	0
$889$	−17.5343	−0.588082
$890$	0	0
$891$	29.2550	0.980080
$892$	0	0
$893$	−11.7784	−0.394148
$894$	0	0
$895$	4.16921	0.139361
$896$	0	0
$897$	6.41049	0.214040
$898$	0	0
$899$	78.6352	2.62263
$900$	0	0
$901$	4.50296	0.150015
$902$	0	0
$903$	17.1168	0.569612
$904$	0	0
$905$	−55.7696	−1.85385
$906$	0	0
$907$	6.54332	0.217268	0.108634	−	0.994082i	$-0.465352\pi$
$907$	6.54332	0.217268	0.108634	+	0.994082i	$0.465352\pi$
$908$	0	0
$909$	4.16670	0.138201
$910$	0	0
$911$	32.1539	1.06531	0.532653	−	0.846334i	$-0.321195\pi$
$911$	32.1539	1.06531	0.532653	+	0.846334i	$0.321195\pi$
$912$	0	0
$913$	30.1979	0.999403
$914$	0	0
$915$	−76.5652	−2.53117
$916$	0	0
$917$	8.02497	0.265008
$918$	0	0
$919$	−10.0708	−0.332205	−0.166103	−	0.986108i	$-0.553118\pi$
$919$	−10.0708	−0.332205	−0.166103	+	0.986108i	$0.553118\pi$
$920$	0	0
$921$	16.0122	0.527619
$922$	0	0
$923$	−16.8766	−0.555500
$924$	0	0
$925$	8.93713	0.293851
$926$	0	0
$927$	−8.04280	−0.264160
$928$	0	0
$929$	−30.5780	−1.00323	−0.501616	−	0.865090i	$-0.667261\pi$
$929$	−30.5780	−1.00323	−0.501616	+	0.865090i	$0.667261\pi$
$930$	0	0
$931$	4.42882	0.145149
$932$	0	0
$933$	−15.7083	−0.514267
$934$	0	0
$935$	−8.10736	−0.265139
$936$	0	0
$937$	−18.1504	−0.592946	−0.296473	−	0.955041i	$-0.595811\pi$
$937$	−18.1504	−0.592946	−0.296473	+	0.955041i	$0.595811\pi$
$938$	0	0
$939$	−51.1550	−1.66938
$940$	0	0
$941$	−11.2139	−0.365564	−0.182782	−	0.983153i	$-0.558510\pi$
$941$	−11.2139	−0.365564	−0.182782	+	0.983153i	$0.558510\pi$
$942$	0	0
$943$	2.48347	0.0808729
$944$	0	0
$945$	13.1488	0.427731
$946$	0	0
$947$	22.8990	0.744118	0.372059	−	0.928209i	$-0.378652\pi$
$947$	22.8990	0.744118	0.372059	+	0.928209i	$0.378652\pi$
$948$	0	0
$949$	4.64368	0.150740
$950$	0	0
$951$	35.5294	1.15212
$952$	0	0
$953$	−41.4874	−1.34391	−0.671955	−	0.740592i	$-0.734545\pi$
$953$	−41.4874	−1.34391	−0.671955	+	0.740592i	$0.734545\pi$
$954$	0	0
$955$	60.7633	1.96625
$956$	0	0
$957$	−43.3106	−1.40003
$958$	0	0
$959$	−9.63336	−0.311077
$960$	0	0
$961$	62.0782	2.00252
$962$	0	0
$963$	−4.16380	−0.134176
$964$	0	0
$965$	−18.6435	−0.600157
$966$	0	0
$967$	−9.33686	−0.300253	−0.150127	−	0.988667i	$-0.547968\pi$
$967$	−9.33686	−0.300253	−0.150127	+	0.988667i	$0.547968\pi$
$968$	0	0
$969$	−8.42412	−0.270622
$970$	0	0
$971$	−44.8227	−1.43843	−0.719215	−	0.694788i	$-0.755498\pi$
$971$	−44.8227	−1.43843	−0.719215	+	0.694788i	$0.755498\pi$
$972$	0	0
$973$	16.5054	0.529137
$974$	0	0
$975$	−12.7065	−0.406934
$976$	0	0
$977$	0.741221	0.0237138	0.0118569	−	0.999930i	$-0.496226\pi$
$977$	0.741221	0.0237138	0.0118569	+	0.999930i	$0.496226\pi$
$978$	0	0
$979$	39.2894	1.25569
$980$	0	0
$981$	−7.90025	−0.252236
$982$	0	0
$983$	34.0128	1.08484	0.542419	−	0.840108i	$-0.317508\pi$
$983$	34.0128	1.08484	0.542419	+	0.840108i	$0.317508\pi$
$984$	0	0
$985$	32.7096	1.04221
$986$	0	0
$987$	5.05863	0.161018
$988$	0	0
$989$	15.5369	0.494044
$990$	0	0
$991$	24.1068	0.765779	0.382890	−	0.923794i	$-0.374929\pi$
$991$	24.1068	0.765779	0.382890	+	0.923794i	$0.374929\pi$
$992$	0	0
$993$	49.3864	1.56723
$994$	0	0
$995$	30.9093	0.979890
$996$	0	0
$997$	−33.1039	−1.04841	−0.524205	−	0.851592i	$-0.675638\pi$
$997$	−33.1039	−1.04841	−0.524205	+	0.851592i	$0.675638\pi$
$998$	0	0
$999$	11.8320	0.374347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bm.1.1		4
4.3	odd	2		7616.2.a.bl.1.4		4
8.3	odd	2		3808.2.a.c.1.1	✓	4
8.5	even	2		3808.2.a.d.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.c.1.1	✓	4	8.3	odd	2
3808.2.a.d.1.4	yes	4	8.5	even	2
7616.2.a.bl.1.4		4	4.3	odd	2
7616.2.a.bm.1.1		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q-1.90211 q^{3} +2.90211 q^{5} +1.00000 q^{7} +0.618034 q^{9} -2.79360 q^{11} +1.95199 q^{13} -5.52015 q^{15} +1.00000 q^{17} +4.42882 q^{19} -1.90211 q^{21} -1.72654 q^{23} +3.42226 q^{25} +4.53077 q^{27} -8.15067 q^{29} -9.64771 q^{31} +5.31375 q^{33} +2.90211 q^{35} +2.61147 q^{37} -3.71290 q^{39} -1.43841 q^{41} -8.99885 q^{43} +1.79360 q^{45} -2.65948 q^{47} +1.00000 q^{49} -1.90211 q^{51} +4.50296 q^{53} -8.10736 q^{55} -8.42412 q^{57} +12.2147 q^{59} +13.8701 q^{61} +0.618034 q^{63} +5.66489 q^{65} -11.8167 q^{67} +3.28408 q^{69} -8.64584 q^{71} +2.37895 q^{73} -6.50953 q^{75} -2.79360 q^{77} -8.59783 q^{79} -10.4721 q^{81} -10.8096 q^{83} +2.90211 q^{85} +15.5035 q^{87} -14.0640 q^{89} +1.95199 q^{91} +18.3510 q^{93} +12.8529 q^{95} +14.0018 q^{97} -1.72654 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7} - 2 q^{9} - 2 q^{11} + 2 q^{13} - 10 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{23} - 6 q^{25} - 6 q^{29} - 16 q^{31} + 4 q^{35} + 8 q^{37} + 10 q^{39} - 10 q^{41} - 4 q^{43} - 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 - 2 * q^9 - 2 * q^11 + 2 * q^13 - 10 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^23 - 6 * q^25 - 6 * q^29 - 16 * q^31 + 4 * q^35 + 8 * q^37 + 10 * q^39 - 10 * q^41 - 4 * q^43 - 2 * q^45 - 14 * q^47 + 4 * q^49 + 14 * q^53 - 2 * q^55 - 10 * q^57 - 12 * q^59 + 2 * q^61 - 2 * q^63 - 8 * q^65 - 22 * q^67 + 10 * q^69 - 16 * q^71 - 14 * q^73 - 20 * q^75 - 2 * q^77 - 10 * q^79 - 24 * q^81 + 6 * q^83 + 4 * q^85 - 22 * q^89 + 2 * q^91 + 6 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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