LMFDB - Embedded newform 6848.2.a.bv.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6848,2,Mod(1,6848)] 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("6848.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6848, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6848 = 2^{6} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6848.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,3,0,-5,0,-4,0,6,0,-2,0,-18,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.6815553042$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 7x^{4} + 29x^{3} - 12x^{2} - 20x + 8 $ x^7 - x^6 - 10x^5 + 7x^4 + 29x^3 - 12x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 107)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.400681$ of defining polynomial
Character	$\chi$	$=$	6848.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05242 q^{3} -0.858034 q^{5} -3.08104 q^{7} +1.21244 q^{9} -1.56312 q^{11} +4.04020 q^{13} -1.76105 q^{15} -0.343798 q^{17} +1.63409 q^{19} -6.32361 q^{21} +7.84845 q^{23} -4.26378 q^{25} -3.66883 q^{27} +3.98461 q^{29} -6.10060 q^{31} -3.20819 q^{33} +2.64364 q^{35} -7.64770 q^{37} +8.29220 q^{39} +0.592274 q^{41} +3.65195 q^{43} -1.04031 q^{45} +3.80419 q^{47} +2.49283 q^{49} -0.705619 q^{51} -12.6409 q^{53} +1.34121 q^{55} +3.35384 q^{57} -8.55816 q^{59} -8.95223 q^{61} -3.73558 q^{63} -3.46663 q^{65} +13.1646 q^{67} +16.1083 q^{69} -7.08789 q^{71} +3.52777 q^{73} -8.75108 q^{75} +4.81605 q^{77} -3.98353 q^{79} -11.1673 q^{81} -17.2874 q^{83} +0.294990 q^{85} +8.17811 q^{87} +2.34142 q^{89} -12.4480 q^{91} -12.5210 q^{93} -1.40210 q^{95} +2.44185 q^{97} -1.89519 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 6 q^{9} - 2 q^{11} - 18 q^{13} + 9 q^{15} - q^{17} - 4 q^{19} + 11 q^{21} + 8 q^{25} + 3 q^{27} + 3 q^{29} - 4 q^{31} - 6 q^{33} - 19 q^{35} - 25 q^{37} - 5 q^{39} + 11 q^{43}+ \cdots - 52 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 6 * q^9 - 2 * q^11 - 18 * q^13 + 9 * q^15 - q^17 - 4 * q^19 + 11 * q^21 + 8 * q^25 + 3 * q^27 + 3 * q^29 - 4 * q^31 - 6 * q^33 - 19 * q^35 - 25 * q^37 - 5 * q^39 + 11 * q^43 - 25 * q^45 + 9 * q^47 + 13 * q^49 - 8 * q^51 - 8 * q^53 + 13 * q^55 - 3 * q^57 - 19 * q^59 - 25 * q^61 + 11 * q^63 + 8 * q^65 - 24 * q^67 + 19 * q^71 + 30 * q^73 - 41 * q^75 - 6 * q^77 + 21 * q^79 - 17 * q^81 + 12 * q^83 - 2 * q^85 + 15 * q^87 - 22 * q^89 - 2 * q^91 - 9 * q^93 + 44 * q^95 - 4 * q^97 - 52 * 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
$3$	2.05242	1.18497	0.592483	−	0.805583i	$-0.298148\pi$
$3$	2.05242	1.18497	0.592483	+	0.805583i	$0.298148\pi$
$4$	0	0
$5$	−0.858034	−0.383724	−0.191862	−	0.981422i	$-0.561453\pi$
$5$	−0.858034	−0.383724	−0.191862	+	0.981422i	$0.561453\pi$
$6$	0	0
$7$	−3.08104	−1.16453	−0.582263	−	0.813001i	$-0.697832\pi$
$7$	−3.08104	−1.16453	−0.582263	+	0.813001i	$0.697832\pi$
$8$	0	0
$9$	1.21244	0.404147
$10$	0	0
$11$	−1.56312	−0.471300	−0.235650	−	0.971838i	$-0.575722\pi$
$11$	−1.56312	−0.471300	−0.235650	+	0.971838i	$0.575722\pi$
$12$	0	0
$13$	4.04020	1.12055	0.560275	−	0.828307i	$-0.310695\pi$
$13$	4.04020	1.12055	0.560275	+	0.828307i	$0.310695\pi$
$14$	0	0
$15$	−1.76105	−0.454701
$16$	0	0
$17$	−0.343798	−0.0833832	−0.0416916	−	0.999131i	$-0.513275\pi$
$17$	−0.343798	−0.0833832	−0.0416916	+	0.999131i	$0.513275\pi$
$18$	0	0
$19$	1.63409	0.374886	0.187443	−	0.982276i	$-0.439980\pi$
$19$	1.63409	0.374886	0.187443	+	0.982276i	$0.439980\pi$
$20$	0	0
$21$	−6.32361	−1.37992
$22$	0	0
$23$	7.84845	1.63651	0.818257	−	0.574852i	$-0.194940\pi$
$23$	7.84845	1.63651	0.818257	+	0.574852i	$0.194940\pi$
$24$	0	0
$25$	−4.26378	−0.852756
$26$	0	0
$27$	−3.66883	−0.706067
$28$	0	0
$29$	3.98461	0.739924	0.369962	−	0.929047i	$-0.379371\pi$
$29$	3.98461	0.739924	0.369962	+	0.929047i	$0.379371\pi$
$30$	0	0
$31$	−6.10060	−1.09570	−0.547850	−	0.836577i	$-0.684553\pi$
$31$	−6.10060	−1.09570	−0.547850	+	0.836577i	$0.684553\pi$
$32$	0	0
$33$	−3.20819	−0.558474
$34$	0	0
$35$	2.64364	0.446857
$36$	0	0
$37$	−7.64770	−1.25727	−0.628637	−	0.777699i	$-0.716387\pi$
$37$	−7.64770	−1.25727	−0.628637	+	0.777699i	$0.716387\pi$
$38$	0	0
$39$	8.29220	1.32782
$40$	0	0
$41$	0.592274	0.0924976	0.0462488	−	0.998930i	$-0.485273\pi$
$41$	0.592274	0.0924976	0.0462488	+	0.998930i	$0.485273\pi$
$42$	0	0
$43$	3.65195	0.556917	0.278459	−	0.960448i	$-0.410176\pi$
$43$	3.65195	0.556917	0.278459	+	0.960448i	$0.410176\pi$
$44$	0	0
$45$	−1.04031	−0.155081
$46$	0	0
$47$	3.80419	0.554897	0.277449	−	0.960740i	$-0.410511\pi$
$47$	3.80419	0.554897	0.277449	+	0.960740i	$0.410511\pi$
$48$	0	0
$49$	2.49283	0.356119
$50$	0	0
$51$	−0.705619	−0.0988064
$52$	0	0
$53$	−12.6409	−1.73636	−0.868180	−	0.496249i	$-0.834710\pi$
$53$	−12.6409	−1.73636	−0.868180	+	0.496249i	$0.834710\pi$
$54$	0	0
$55$	1.34121	0.180849
$56$	0	0
$57$	3.35384	0.444227
$58$	0	0
$59$	−8.55816	−1.11418	−0.557089	−	0.830453i	$-0.688082\pi$
$59$	−8.55816	−1.11418	−0.557089	+	0.830453i	$0.688082\pi$
$60$	0	0
$61$	−8.95223	−1.14622	−0.573108	−	0.819480i	$-0.694262\pi$
$61$	−8.95223	−1.14622	−0.573108	+	0.819480i	$0.694262\pi$
$62$	0	0
$63$	−3.73558	−0.470639
$64$	0	0
$65$	−3.46663	−0.429982
$66$	0	0
$67$	13.1646	1.60832	0.804158	−	0.594415i	$-0.202616\pi$
$67$	13.1646	1.60832	0.804158	+	0.594415i	$0.202616\pi$
$68$	0	0
$69$	16.1083	1.93922
$70$	0	0
$71$	−7.08789	−0.841177	−0.420589	−	0.907251i	$-0.638176\pi$
$71$	−7.08789	−0.841177	−0.420589	+	0.907251i	$0.638176\pi$
$72$	0	0
$73$	3.52777	0.412895	0.206447	−	0.978458i	$-0.433810\pi$
$73$	3.52777	0.412895	0.206447	+	0.978458i	$0.433810\pi$
$74$	0	0
$75$	−8.75108	−1.01049
$76$	0	0
$77$	4.81605	0.548840
$78$	0	0
$79$	−3.98353	−0.448182	−0.224091	−	0.974568i	$-0.571941\pi$
$79$	−3.98353	−0.448182	−0.224091	+	0.974568i	$0.571941\pi$
$80$	0	0
$81$	−11.1673	−1.24081
$82$	0	0
$83$	−17.2874	−1.89754	−0.948770	−	0.315966i	$-0.897671\pi$
$83$	−17.2874	−1.89754	−0.948770	+	0.315966i	$0.897671\pi$
$84$	0	0
$85$	0.294990	0.0319962
$86$	0	0
$87$	8.17811	0.876786
$88$	0	0
$89$	2.34142	0.248190	0.124095	−	0.992270i	$-0.460397\pi$
$89$	2.34142	0.248190	0.124095	+	0.992270i	$0.460397\pi$
$90$	0	0
$91$	−12.4480	−1.30491
$92$	0	0
$93$	−12.5210	−1.29837
$94$	0	0
$95$	−1.40210	−0.143853
$96$	0	0
$97$	2.44185	0.247932	0.123966	−	0.992286i	$-0.460439\pi$
$97$	2.44185	0.247932	0.123966	+	0.992286i	$0.460439\pi$
$98$	0	0
$99$	−1.89519	−0.190474
$100$	0	0
$101$	−14.8606	−1.47868	−0.739340	−	0.673332i	$-0.764862\pi$
$101$	−14.8606	−1.47868	−0.739340	+	0.673332i	$0.764862\pi$
$102$	0	0
$103$	14.0057	1.38002	0.690012	−	0.723798i	$-0.257605\pi$
$103$	14.0057	1.38002	0.690012	+	0.723798i	$0.257605\pi$
$104$	0	0
$105$	5.42587	0.529510
$106$	0	0
$107$	1.00000	0.0966736
$107$	1.00000	0.0966736
$108$	0	0
$109$	7.27844	0.697148	0.348574	−	0.937281i	$-0.386666\pi$
$109$	7.27844	0.697148	0.348574	+	0.937281i	$0.386666\pi$
$110$	0	0
$111$	−15.6963	−1.48983
$112$	0	0
$113$	9.98665	0.939465	0.469732	−	0.882809i	$-0.344350\pi$
$113$	9.98665	0.939465	0.469732	+	0.882809i	$0.344350\pi$
$114$	0	0
$115$	−6.73423	−0.627971
$116$	0	0
$117$	4.89850	0.452867
$118$	0	0
$119$	1.05926	0.0971019
$120$	0	0
$121$	−8.55664	−0.777877
$122$	0	0
$123$	1.21560	0.109607
$124$	0	0
$125$	7.94863	0.710947
$126$	0	0
$127$	20.1182	1.78520	0.892601	−	0.450848i	$-0.148878\pi$
$127$	20.1182	1.78520	0.892601	+	0.450848i	$0.148878\pi$
$128$	0	0
$129$	7.49535	0.659929
$130$	0	0
$131$	−19.0407	−1.66360	−0.831798	−	0.555078i	$-0.812688\pi$
$131$	−19.0407	−1.66360	−0.831798	+	0.555078i	$0.812688\pi$
$132$	0	0
$133$	−5.03470	−0.436564
$134$	0	0
$135$	3.14798	0.270935
$136$	0	0
$137$	10.4337	0.891409	0.445705	−	0.895180i	$-0.352953\pi$
$137$	10.4337	0.891409	0.445705	+	0.895180i	$0.352953\pi$
$138$	0	0
$139$	−8.04313	−0.682209	−0.341105	−	0.940025i	$-0.610801\pi$
$139$	−8.04313	−0.682209	−0.341105	+	0.940025i	$0.610801\pi$
$140$	0	0
$141$	7.80780	0.657535
$142$	0	0
$143$	−6.31534	−0.528115
$144$	0	0
$145$	−3.41893	−0.283927
$146$	0	0
$147$	5.11635	0.421989
$148$	0	0
$149$	−5.43794	−0.445493	−0.222747	−	0.974876i	$-0.571502\pi$
$149$	−5.43794	−0.445493	−0.222747	+	0.974876i	$0.571502\pi$
$150$	0	0
$151$	−2.26716	−0.184499	−0.0922494	−	0.995736i	$-0.529406\pi$
$151$	−2.26716	−0.184499	−0.0922494	+	0.995736i	$0.529406\pi$
$152$	0	0
$153$	−0.416834	−0.0336990
$154$	0	0
$155$	5.23452	0.420446
$156$	0	0
$157$	−23.6776	−1.88968	−0.944840	−	0.327531i	$-0.893783\pi$
$157$	−23.6776	−1.88968	−0.944840	+	0.327531i	$0.893783\pi$
$158$	0	0
$159$	−25.9445	−2.05753
$160$	0	0
$161$	−24.1814	−1.90576
$162$	0	0
$163$	7.44006	0.582751	0.291375	−	0.956609i	$-0.405887\pi$
$163$	7.44006	0.582751	0.291375	+	0.956609i	$0.405887\pi$
$164$	0	0
$165$	2.75274	0.214300
$166$	0	0
$167$	−16.5490	−1.28060	−0.640299	−	0.768125i	$-0.721190\pi$
$167$	−16.5490	−1.28060	−0.640299	+	0.768125i	$0.721190\pi$
$168$	0	0
$169$	3.32323	0.255633
$170$	0	0
$171$	1.98123	0.151509
$172$	0	0
$173$	−19.9068	−1.51348	−0.756742	−	0.653714i	$-0.773210\pi$
$173$	−19.9068	−1.51348	−0.756742	+	0.653714i	$0.773210\pi$
$174$	0	0
$175$	13.1369	0.993056
$176$	0	0
$177$	−17.5650	−1.32026
$178$	0	0
$179$	2.86289	0.213982	0.106991	−	0.994260i	$-0.465878\pi$
$179$	2.86289	0.213982	0.106991	+	0.994260i	$0.465878\pi$
$180$	0	0
$181$	3.68801	0.274128	0.137064	−	0.990562i	$-0.456233\pi$
$181$	3.68801	0.274128	0.137064	+	0.990562i	$0.456233\pi$
$182$	0	0
$183$	−18.3738	−1.35823
$184$	0	0
$185$	6.56199	0.482447
$186$	0	0
$187$	0.537399	0.0392985
$188$	0	0
$189$	11.3038	0.822232
$190$	0	0
$191$	−2.54895	−0.184436	−0.0922179	−	0.995739i	$-0.529396\pi$
$191$	−2.54895	−0.184436	−0.0922179	+	0.995739i	$0.529396\pi$
$192$	0	0
$193$	−2.29500	−0.165198	−0.0825988	−	0.996583i	$-0.526322\pi$
$193$	−2.29500	−0.165198	−0.0825988	+	0.996583i	$0.526322\pi$
$194$	0	0
$195$	−7.11499	−0.509515
$196$	0	0
$197$	23.8635	1.70021	0.850103	−	0.526617i	$-0.176540\pi$
$197$	23.8635	1.70021	0.850103	+	0.526617i	$0.176540\pi$
$198$	0	0
$199$	−19.9864	−1.41680	−0.708400	−	0.705811i	$-0.750583\pi$
$199$	−19.9864	−1.41680	−0.708400	+	0.705811i	$0.750583\pi$
$200$	0	0
$201$	27.0194	1.90580
$202$	0	0
$203$	−12.2768	−0.861661
$204$	0	0
$205$	−0.508191	−0.0354936
$206$	0	0
$207$	9.51577	0.661392
$208$	0	0
$209$	−2.55428	−0.176683
$210$	0	0
$211$	−13.9182	−0.958171	−0.479086	−	0.877768i	$-0.659032\pi$
$211$	−13.9182	−0.958171	−0.479086	+	0.877768i	$0.659032\pi$
$212$	0	0
$213$	−14.5473	−0.996767
$214$	0	0
$215$	−3.13350	−0.213703
$216$	0	0
$217$	18.7962	1.27597
$218$	0	0
$219$	7.24048	0.489267
$220$	0	0
$221$	−1.38901	−0.0934351
$222$	0	0
$223$	0.937849	0.0628030	0.0314015	−	0.999507i	$-0.490003\pi$
$223$	0.937849	0.0628030	0.0314015	+	0.999507i	$0.490003\pi$
$224$	0	0
$225$	−5.16957	−0.344638
$226$	0	0
$227$	−13.5014	−0.896122	−0.448061	−	0.894003i	$-0.647885\pi$
$227$	−13.5014	−0.896122	−0.448061	+	0.894003i	$0.647885\pi$
$228$	0	0
$229$	−22.1489	−1.46364	−0.731820	−	0.681498i	$-0.761329\pi$
$229$	−22.1489	−1.46364	−0.731820	+	0.681498i	$0.761329\pi$
$230$	0	0
$231$	9.88458	0.650358
$232$	0	0
$233$	4.30702	0.282162	0.141081	−	0.989998i	$-0.454942\pi$
$233$	4.30702	0.282162	0.141081	+	0.989998i	$0.454942\pi$
$234$	0	0
$235$	−3.26412	−0.212928
$236$	0	0
$237$	−8.17589	−0.531081
$238$	0	0
$239$	12.2987	0.795538	0.397769	−	0.917486i	$-0.369785\pi$
$239$	12.2987	0.795538	0.397769	+	0.917486i	$0.369785\pi$
$240$	0	0
$241$	−21.9315	−1.41274	−0.706368	−	0.707845i	$-0.749667\pi$
$241$	−21.9315	−1.41274	−0.706368	+	0.707845i	$0.749667\pi$
$242$	0	0
$243$	−11.9136	−0.764255
$244$	0	0
$245$	−2.13894	−0.136652
$246$	0	0
$247$	6.60205	0.420078
$248$	0	0
$249$	−35.4811	−2.24852
$250$	0	0
$251$	15.5593	0.982092	0.491046	−	0.871134i	$-0.336615\pi$
$251$	15.5593	0.982092	0.491046	+	0.871134i	$0.336615\pi$
$252$	0	0
$253$	−12.2681	−0.771289
$254$	0	0
$255$	0.605445	0.0379144
$256$	0	0
$257$	−21.0087	−1.31049	−0.655243	−	0.755418i	$-0.727434\pi$
$257$	−21.0087	−1.31049	−0.655243	+	0.755418i	$0.727434\pi$
$258$	0	0
$259$	23.5629	1.46413
$260$	0	0
$261$	4.83111	0.299038
$262$	0	0
$263$	−30.3403	−1.87086	−0.935432	−	0.353508i	$-0.884989\pi$
$263$	−30.3403	−1.87086	−0.935432	+	0.353508i	$0.884989\pi$
$264$	0	0
$265$	10.8463	0.666284
$266$	0	0
$267$	4.80558	0.294097
$268$	0	0
$269$	−4.44775	−0.271184	−0.135592	−	0.990765i	$-0.543294\pi$
$269$	−4.44775	−0.271184	−0.135592	+	0.990765i	$0.543294\pi$
$270$	0	0
$271$	3.02256	0.183608	0.0918038	−	0.995777i	$-0.470737\pi$
$271$	3.02256	0.183608	0.0918038	+	0.995777i	$0.470737\pi$
$272$	0	0
$273$	−25.5486	−1.54627
$274$	0	0
$275$	6.66481	0.401903
$276$	0	0
$277$	−15.5342	−0.933360	−0.466680	−	0.884426i	$-0.654550\pi$
$277$	−15.5342	−0.933360	−0.466680	+	0.884426i	$0.654550\pi$
$278$	0	0
$279$	−7.39660	−0.442823
$280$	0	0
$281$	−23.9934	−1.43132	−0.715662	−	0.698447i	$-0.753875\pi$
$281$	−23.9934	−1.43132	−0.715662	+	0.698447i	$0.753875\pi$
$282$	0	0
$283$	−0.375834	−0.0223410	−0.0111705	−	0.999938i	$-0.503556\pi$
$283$	−0.375834	−0.0223410	−0.0111705	+	0.999938i	$0.503556\pi$
$284$	0	0
$285$	−2.87771	−0.170461
$286$	0	0
$287$	−1.82482	−0.107716
$288$	0	0
$289$	−16.8818	−0.993047
$290$	0	0
$291$	5.01170	0.293791
$292$	0	0
$293$	1.19887	0.0700389	0.0350194	−	0.999387i	$-0.488851\pi$
$293$	1.19887	0.0700389	0.0350194	+	0.999387i	$0.488851\pi$
$294$	0	0
$295$	7.34319	0.427537
$296$	0	0
$297$	5.73483	0.332769
$298$	0	0
$299$	31.7093	1.83380
$300$	0	0
$301$	−11.2518	−0.648544
$302$	0	0
$303$	−30.5001	−1.75219
$304$	0	0
$305$	7.68132	0.439831
$306$	0	0
$307$	−3.53770	−0.201907	−0.100954	−	0.994891i	$-0.532189\pi$
$307$	−3.53770	−0.201907	−0.100954	+	0.994891i	$0.532189\pi$
$308$	0	0
$309$	28.7456	1.63528
$310$	0	0
$311$	27.8999	1.58206	0.791030	−	0.611777i	$-0.209545\pi$
$311$	27.8999	1.58206	0.791030	+	0.611777i	$0.209545\pi$
$312$	0	0
$313$	−4.32943	−0.244714	−0.122357	−	0.992486i	$-0.539045\pi$
$313$	−4.32943	−0.244714	−0.122357	+	0.992486i	$0.539045\pi$
$314$	0	0
$315$	3.20525	0.180596
$316$	0	0
$317$	−1.93923	−0.108918	−0.0544589	−	0.998516i	$-0.517343\pi$
$317$	−1.93923	−0.108918	−0.0544589	+	0.998516i	$0.517343\pi$
$318$	0	0
$319$	−6.22845	−0.348726
$320$	0	0
$321$	2.05242	0.114555
$322$	0	0
$323$	−0.561796	−0.0312592
$324$	0	0
$325$	−17.2265	−0.955556
$326$	0	0
$327$	14.9384	0.826097
$328$	0	0
$329$	−11.7209	−0.646192
$330$	0	0
$331$	24.7851	1.36231	0.681157	−	0.732138i	$-0.261477\pi$
$331$	24.7851	1.36231	0.681157	+	0.732138i	$0.261477\pi$
$332$	0	0
$333$	−9.27238	−0.508123
$334$	0	0
$335$	−11.2957	−0.617150
$336$	0	0
$337$	−26.2295	−1.42881	−0.714406	−	0.699731i	$-0.753303\pi$
$337$	−26.2295	−1.42881	−0.714406	+	0.699731i	$0.753303\pi$
$338$	0	0
$339$	20.4968	1.11323
$340$	0	0
$341$	9.53599	0.516403
$342$	0	0
$343$	13.8868	0.749816
$344$	0	0
$345$	−13.8215	−0.744124
$346$	0	0
$347$	16.2012	0.869726	0.434863	−	0.900497i	$-0.356797\pi$
$347$	16.2012	0.869726	0.434863	+	0.900497i	$0.356797\pi$
$348$	0	0
$349$	9.68150	0.518239	0.259119	−	0.965845i	$-0.416568\pi$
$349$	9.68150	0.518239	0.259119	+	0.965845i	$0.416568\pi$
$350$	0	0
$351$	−14.8228	−0.791183
$352$	0	0
$353$	0.951831	0.0506609	0.0253304	−	0.999679i	$-0.491936\pi$
$353$	0.951831	0.0506609	0.0253304	+	0.999679i	$0.491936\pi$
$354$	0	0
$355$	6.08164	0.322780
$356$	0	0
$357$	2.17404	0.115063
$358$	0	0
$359$	15.4999	0.818052	0.409026	−	0.912523i	$-0.365869\pi$
$359$	15.4999	0.818052	0.409026	+	0.912523i	$0.365869\pi$
$360$	0	0
$361$	−16.3298	−0.859461
$362$	0	0
$363$	−17.5619	−0.921758
$364$	0	0
$365$	−3.02695	−0.158438
$366$	0	0
$367$	2.63718	0.137660	0.0688299	−	0.997628i	$-0.478073\pi$
$367$	2.63718	0.137660	0.0688299	+	0.997628i	$0.478073\pi$
$368$	0	0
$369$	0.718096	0.0373826
$370$	0	0
$371$	38.9471	2.02204
$372$	0	0
$373$	−8.98443	−0.465196	−0.232598	−	0.972573i	$-0.574723\pi$
$373$	−8.98443	−0.465196	−0.232598	+	0.972573i	$0.574723\pi$
$374$	0	0
$375$	16.3140	0.842449
$376$	0	0
$377$	16.0986	0.829123
$378$	0	0
$379$	1.01229	0.0519977	0.0259988	−	0.999662i	$-0.491723\pi$
$379$	1.01229	0.0519977	0.0259988	+	0.999662i	$0.491723\pi$
$380$	0	0
$381$	41.2910	2.11540
$382$	0	0
$383$	−4.16498	−0.212821	−0.106410	−	0.994322i	$-0.533936\pi$
$383$	−4.16498	−0.212821	−0.106410	+	0.994322i	$0.533936\pi$
$384$	0	0
$385$	−4.13234	−0.210603
$386$	0	0
$387$	4.42777	0.225076
$388$	0	0
$389$	−2.14928	−0.108973	−0.0544865	−	0.998515i	$-0.517352\pi$
$389$	−2.14928	−0.108973	−0.0544865	+	0.998515i	$0.517352\pi$
$390$	0	0
$391$	−2.69828	−0.136458
$392$	0	0
$393$	−39.0796	−1.97131
$394$	0	0
$395$	3.41800	0.171978
$396$	0	0
$397$	−10.2162	−0.512738	−0.256369	−	0.966579i	$-0.582526\pi$
$397$	−10.2162	−0.512738	−0.256369	+	0.966579i	$0.582526\pi$
$398$	0	0
$399$	−10.3333	−0.517314
$400$	0	0
$401$	8.64190	0.431556	0.215778	−	0.976442i	$-0.430771\pi$
$401$	8.64190	0.431556	0.215778	+	0.976442i	$0.430771\pi$
$402$	0	0
$403$	−24.6476	−1.22779
$404$	0	0
$405$	9.58193	0.476130
$406$	0	0
$407$	11.9543	0.592553
$408$	0	0
$409$	−21.2166	−1.04909	−0.524546	−	0.851382i	$-0.675765\pi$
$409$	−21.2166	−1.04909	−0.524546	+	0.851382i	$0.675765\pi$
$410$	0	0
$411$	21.4143	1.05629
$412$	0	0
$413$	26.3681	1.29749
$414$	0	0
$415$	14.8332	0.728133
$416$	0	0
$417$	−16.5079	−0.808396
$418$	0	0
$419$	−16.5527	−0.808650	−0.404325	−	0.914615i	$-0.632494\pi$
$419$	−16.5527	−0.808650	−0.404325	+	0.914615i	$0.632494\pi$
$420$	0	0
$421$	−8.84459	−0.431059	−0.215529	−	0.976497i	$-0.569148\pi$
$421$	−8.84459	−0.431059	−0.215529	+	0.976497i	$0.569148\pi$
$422$	0	0
$423$	4.61235	0.224260
$424$	0	0
$425$	1.46588	0.0711055
$426$	0	0
$427$	27.5822	1.33480
$428$	0	0
$429$	−12.9617	−0.625799
$430$	0	0
$431$	28.3430	1.36523	0.682616	−	0.730777i	$-0.260842\pi$
$431$	28.3430	1.36523	0.682616	+	0.730777i	$0.260842\pi$
$432$	0	0
$433$	29.9692	1.44023	0.720114	−	0.693855i	$-0.244089\pi$
$433$	29.9692	1.44023	0.720114	+	0.693855i	$0.244089\pi$
$434$	0	0
$435$	−7.01710	−0.336444
$436$	0	0
$437$	12.8251	0.613506
$438$	0	0
$439$	−13.6019	−0.649183	−0.324592	−	0.945854i	$-0.605227\pi$
$439$	−13.6019	−0.649183	−0.324592	+	0.945854i	$0.605227\pi$
$440$	0	0
$441$	3.02241	0.143924
$442$	0	0
$443$	−30.1900	−1.43437	−0.717185	−	0.696883i	$-0.754570\pi$
$443$	−30.1900	−1.43437	−0.717185	+	0.696883i	$0.754570\pi$
$444$	0	0
$445$	−2.00902	−0.0952365
$446$	0	0
$447$	−11.1609	−0.527895
$448$	0	0
$449$	−15.9602	−0.753207	−0.376603	−	0.926375i	$-0.622908\pi$
$449$	−15.9602	−0.753207	−0.376603	+	0.926375i	$0.622908\pi$
$450$	0	0
$451$	−0.925797	−0.0435941
$452$	0	0
$453$	−4.65317	−0.218625
$454$	0	0
$455$	10.6808	0.500725
$456$	0	0
$457$	10.6928	0.500187	0.250093	−	0.968222i	$-0.419539\pi$
$457$	10.6928	0.500187	0.250093	+	0.968222i	$0.419539\pi$
$458$	0	0
$459$	1.26134	0.0588741
$460$	0	0
$461$	30.1188	1.40277	0.701387	−	0.712781i	$-0.252565\pi$
$461$	30.1188	1.40277	0.701387	+	0.712781i	$0.252565\pi$
$462$	0	0
$463$	16.5292	0.768176	0.384088	−	0.923296i	$-0.374516\pi$
$463$	16.5292	0.768176	0.384088	+	0.923296i	$0.374516\pi$
$464$	0	0
$465$	10.7434	0.498215
$466$	0	0
$467$	39.1468	1.81150	0.905750	−	0.423813i	$-0.139308\pi$
$467$	39.1468	1.81150	0.905750	+	0.423813i	$0.139308\pi$
$468$	0	0
$469$	−40.5608	−1.87293
$470$	0	0
$471$	−48.5965	−2.23921
$472$	0	0
$473$	−5.70845	−0.262475
$474$	0	0
$475$	−6.96739	−0.319686
$476$	0	0
$477$	−15.3263	−0.701744
$478$	0	0
$479$	−2.08979	−0.0954849	−0.0477425	−	0.998860i	$-0.515203\pi$
$479$	−2.08979	−0.0954849	−0.0477425	+	0.998860i	$0.515203\pi$
$480$	0	0
$481$	−30.8983	−1.40884
$482$	0	0
$483$	−49.6305	−2.25827
$484$	0	0
$485$	−2.09519	−0.0951375
$486$	0	0
$487$	33.4056	1.51375	0.756877	−	0.653557i	$-0.226724\pi$
$487$	33.4056	1.51375	0.756877	+	0.653557i	$0.226724\pi$
$488$	0	0
$489$	15.2702	0.690540
$490$	0	0
$491$	16.1338	0.728110	0.364055	−	0.931377i	$-0.381392\pi$
$491$	16.1338	0.728110	0.364055	+	0.931377i	$0.381392\pi$
$492$	0	0
$493$	−1.36990	−0.0616973
$494$	0	0
$495$	1.62614	0.0730896
$496$	0	0
$497$	21.8381	0.979572
$498$	0	0
$499$	−21.5343	−0.964007	−0.482003	−	0.876169i	$-0.660091\pi$
$499$	−21.5343	−0.964007	−0.482003	+	0.876169i	$0.660091\pi$
$500$	0	0
$501$	−33.9655	−1.51747
$502$	0	0
$503$	19.6399	0.875700	0.437850	−	0.899048i	$-0.355740\pi$
$503$	19.6399	0.875700	0.437850	+	0.899048i	$0.355740\pi$
$504$	0	0
$505$	12.7509	0.567406
$506$	0	0
$507$	6.82067	0.302917
$508$	0	0
$509$	−33.7799	−1.49727	−0.748633	−	0.662984i	$-0.769290\pi$
$509$	−33.7799	−1.49727	−0.748633	+	0.662984i	$0.769290\pi$
$510$	0	0
$511$	−10.8692	−0.480826
$512$	0	0
$513$	−5.99519	−0.264694
$514$	0	0
$515$	−12.0174	−0.529549
$516$	0	0
$517$	−5.94641	−0.261523
$518$	0	0
$519$	−40.8571	−1.79343
$520$	0	0
$521$	27.3215	1.19698	0.598489	−	0.801131i	$-0.295768\pi$
$521$	27.3215	1.19698	0.598489	+	0.801131i	$0.295768\pi$
$522$	0	0
$523$	−4.31540	−0.188699	−0.0943496	−	0.995539i	$-0.530077\pi$
$523$	−4.31540	−0.188699	−0.0943496	+	0.995539i	$0.530077\pi$
$524$	0	0
$525$	26.9625	1.17674
$526$	0	0
$527$	2.09737	0.0913629
$528$	0	0
$529$	38.5982	1.67818
$530$	0	0
$531$	−10.3763	−0.450291
$532$	0	0
$533$	2.39291	0.103648
$534$	0	0
$535$	−0.858034	−0.0370960
$536$	0	0
$537$	5.87585	0.253562
$538$	0	0
$539$	−3.89661	−0.167839
$540$	0	0
$541$	38.0043	1.63393	0.816967	−	0.576685i	$-0.195654\pi$
$541$	38.0043	1.63393	0.816967	+	0.576685i	$0.195654\pi$
$542$	0	0
$543$	7.56936	0.324832
$544$	0	0
$545$	−6.24515	−0.267513
$546$	0	0
$547$	2.79784	0.119627	0.0598135	−	0.998210i	$-0.480949\pi$
$547$	2.79784	0.119627	0.0598135	+	0.998210i	$0.480949\pi$
$548$	0	0
$549$	−10.8540	−0.463239
$550$	0	0
$551$	6.51121	0.277387
$552$	0	0
$553$	12.2734	0.521919
$554$	0	0
$555$	13.4680	0.571684
$556$	0	0
$557$	41.9379	1.77697	0.888483	−	0.458910i	$-0.151760\pi$
$557$	41.9379	1.77697	0.888483	+	0.458910i	$0.151760\pi$
$558$	0	0
$559$	14.7546	0.624054
$560$	0	0
$561$	1.10297	0.0465674
$562$	0	0
$563$	12.3159	0.519054	0.259527	−	0.965736i	$-0.416433\pi$
$563$	12.3159	0.519054	0.259527	+	0.965736i	$0.416433\pi$
$564$	0	0
$565$	−8.56888	−0.360496
$566$	0	0
$567$	34.4070	1.44496
$568$	0	0
$569$	5.52854	0.231769	0.115884	−	0.993263i	$-0.463030\pi$
$569$	5.52854	0.231769	0.115884	+	0.993263i	$0.463030\pi$
$570$	0	0
$571$	−28.7910	−1.20487	−0.602433	−	0.798169i	$-0.705802\pi$
$571$	−28.7910	−1.20487	−0.602433	+	0.798169i	$0.705802\pi$
$572$	0	0
$573$	−5.23153	−0.218550
$574$	0	0
$575$	−33.4641	−1.39555
$576$	0	0
$577$	2.16381	0.0900807	0.0450404	−	0.998985i	$-0.485658\pi$
$577$	2.16381	0.0900807	0.0450404	+	0.998985i	$0.485658\pi$
$578$	0	0
$579$	−4.71031	−0.195754
$580$	0	0
$581$	53.2633	2.20973
$582$	0	0
$583$	19.7593	0.818346
$584$	0	0
$585$	−4.20308	−0.173776
$586$	0	0
$587$	34.4416	1.42156	0.710778	−	0.703416i	$-0.248343\pi$
$587$	34.4416	1.42156	0.710778	+	0.703416i	$0.248343\pi$
$588$	0	0
$589$	−9.96891	−0.410762
$590$	0	0
$591$	48.9780	2.01469
$592$	0	0
$593$	23.5676	0.967806	0.483903	−	0.875122i	$-0.339219\pi$
$593$	23.5676	0.967806	0.483903	+	0.875122i	$0.339219\pi$
$594$	0	0
$595$	−0.908878	−0.0372604
$596$	0	0
$597$	−41.0206	−1.67886
$598$	0	0
$599$	34.3462	1.40335	0.701675	−	0.712497i	$-0.252436\pi$
$599$	34.3462	1.40335	0.701675	+	0.712497i	$0.252436\pi$
$600$	0	0
$601$	5.58348	0.227755	0.113878	−	0.993495i	$-0.463673\pi$
$601$	5.58348	0.227755	0.113878	+	0.993495i	$0.463673\pi$
$602$	0	0
$603$	15.9613	0.649996
$604$	0	0
$605$	7.34189	0.298490
$606$	0	0
$607$	−6.51339	−0.264370	−0.132185	−	0.991225i	$-0.542199\pi$
$607$	−6.51339	−0.264370	−0.132185	+	0.991225i	$0.542199\pi$
$608$	0	0
$609$	−25.1971	−1.02104
$610$	0	0
$611$	15.3697	0.621790
$612$	0	0
$613$	−31.5232	−1.27321	−0.636606	−	0.771189i	$-0.719662\pi$
$613$	−31.5232	−1.27321	−0.636606	+	0.771189i	$0.719662\pi$
$614$	0	0
$615$	−1.04302	−0.0420587
$616$	0	0
$617$	23.6858	0.953553	0.476776	−	0.879025i	$-0.341805\pi$
$617$	23.6858	0.953553	0.476776	+	0.879025i	$0.341805\pi$
$618$	0	0
$619$	25.6089	1.02931	0.514653	−	0.857398i	$-0.327921\pi$
$619$	25.6089	1.02931	0.514653	+	0.857398i	$0.327921\pi$
$620$	0	0
$621$	−28.7946	−1.15549
$622$	0	0
$623$	−7.21401	−0.289023
$624$	0	0
$625$	14.4987	0.579948
$626$	0	0
$627$	−5.24247	−0.209364
$628$	0	0
$629$	2.62926	0.104836
$630$	0	0
$631$	18.7159	0.745067	0.372534	−	0.928019i	$-0.378489\pi$
$631$	18.7159	0.745067	0.372534	+	0.928019i	$0.378489\pi$
$632$	0	0
$633$	−28.5661	−1.13540
$634$	0	0
$635$	−17.2621	−0.685025
$636$	0	0
$637$	10.0716	0.399049
$638$	0	0
$639$	−8.59363	−0.339959
$640$	0	0
$641$	11.5357	0.455632	0.227816	−	0.973704i	$-0.426841\pi$
$641$	11.5357	0.455632	0.227816	+	0.973704i	$0.426841\pi$
$642$	0	0
$643$	12.5190	0.493701	0.246851	−	0.969054i	$-0.420604\pi$
$643$	12.5190	0.493701	0.246851	+	0.969054i	$0.420604\pi$
$644$	0	0
$645$	−6.43126	−0.253231
$646$	0	0
$647$	22.1925	0.872478	0.436239	−	0.899831i	$-0.356310\pi$
$647$	22.1925	0.872478	0.436239	+	0.899831i	$0.356310\pi$
$648$	0	0
$649$	13.3775	0.525111
$650$	0	0
$651$	38.5778	1.51198
$652$	0	0
$653$	0.545398	0.0213431	0.0106715	−	0.999943i	$-0.496603\pi$
$653$	0.545398	0.0213431	0.0106715	+	0.999943i	$0.496603\pi$
$654$	0	0
$655$	16.3376	0.638363
$656$	0	0
$657$	4.27721	0.166870
$658$	0	0
$659$	−0.416896	−0.0162400	−0.00811999	−	0.999967i	$-0.502585\pi$
$659$	−0.416896	−0.0162400	−0.00811999	+	0.999967i	$0.502585\pi$
$660$	0	0
$661$	−28.6717	−1.11520	−0.557600	−	0.830110i	$-0.688278\pi$
$661$	−28.6717	−1.11520	−0.557600	+	0.830110i	$0.688278\pi$
$662$	0	0
$663$	−2.85084	−0.110718
$664$	0	0
$665$	4.31994	0.167520
$666$	0	0
$667$	31.2730	1.21090
$668$	0	0
$669$	1.92486	0.0744195
$670$	0	0
$671$	13.9935	0.540211
$672$	0	0
$673$	6.25365	0.241060	0.120530	−	0.992710i	$-0.461541\pi$
$673$	6.25365	0.241060	0.120530	+	0.992710i	$0.461541\pi$
$674$	0	0
$675$	15.6431	0.602102
$676$	0	0
$677$	9.78422	0.376038	0.188019	−	0.982165i	$-0.439793\pi$
$677$	9.78422	0.376038	0.188019	+	0.982165i	$0.439793\pi$
$678$	0	0
$679$	−7.52343	−0.288723
$680$	0	0
$681$	−27.7107	−1.06188
$682$	0	0
$683$	24.7545	0.947205	0.473602	−	0.880739i	$-0.342953\pi$
$683$	24.7545	0.947205	0.473602	+	0.880739i	$0.342953\pi$
$684$	0	0
$685$	−8.95245	−0.342055
$686$	0	0
$687$	−45.4589	−1.73436
$688$	0	0
$689$	−51.0718	−1.94568
$690$	0	0
$691$	−41.1103	−1.56391	−0.781955	−	0.623334i	$-0.785778\pi$
$691$	−41.1103	−1.56391	−0.781955	+	0.623334i	$0.785778\pi$
$692$	0	0
$693$	5.83918	0.221812
$694$	0	0
$695$	6.90128	0.261780
$696$	0	0
$697$	−0.203622	−0.00771275
$698$	0	0
$699$	8.83983	0.334353
$700$	0	0
$701$	−20.9689	−0.791986	−0.395993	−	0.918254i	$-0.629600\pi$
$701$	−20.9689	−0.791986	−0.395993	+	0.918254i	$0.629600\pi$
$702$	0	0
$703$	−12.4970	−0.471334
$704$	0	0
$705$	−6.69935	−0.252312
$706$	0	0
$707$	45.7860	1.72196
$708$	0	0
$709$	33.2811	1.24990	0.624948	−	0.780666i	$-0.285120\pi$
$709$	33.2811	1.24990	0.624948	+	0.780666i	$0.285120\pi$
$710$	0	0
$711$	−4.82979	−0.181131
$712$	0	0
$713$	−47.8802	−1.79313
$714$	0	0
$715$	5.41877	0.202651
$716$	0	0
$717$	25.2422	0.942686
$718$	0	0
$719$	−49.7380	−1.85492	−0.927458	−	0.373928i	$-0.878011\pi$
$719$	−49.7380	−1.85492	−0.927458	+	0.373928i	$0.878011\pi$
$720$	0	0
$721$	−43.1522	−1.60707
$722$	0	0
$723$	−45.0128	−1.67404
$724$	0	0
$725$	−16.9895	−0.630975
$726$	0	0
$727$	20.6762	0.766838	0.383419	−	0.923575i	$-0.374747\pi$
$727$	20.6762	0.766838	0.383419	+	0.923575i	$0.374747\pi$
$728$	0	0
$729$	9.05028	0.335195
$730$	0	0
$731$	−1.25553	−0.0464376
$732$	0	0
$733$	−2.70726	−0.0999949	−0.0499975	−	0.998749i	$-0.515921\pi$
$733$	−2.70726	−0.0999949	−0.0499975	+	0.998749i	$0.515921\pi$
$734$	0	0
$735$	−4.39000	−0.161928
$736$	0	0
$737$	−20.5780	−0.757999
$738$	0	0
$739$	−4.12593	−0.151775	−0.0758873	−	0.997116i	$-0.524179\pi$
$739$	−4.12593	−0.151775	−0.0758873	+	0.997116i	$0.524179\pi$
$740$	0	0
$741$	13.5502	0.497779
$742$	0	0
$743$	31.7860	1.16611	0.583057	−	0.812431i	$-0.301857\pi$
$743$	31.7860	1.16611	0.583057	+	0.812431i	$0.301857\pi$
$744$	0	0
$745$	4.66593	0.170947
$746$	0	0
$747$	−20.9600	−0.766885
$748$	0	0
$749$	−3.08104	−0.112579
$750$	0	0
$751$	46.7601	1.70630	0.853150	−	0.521666i	$-0.174689\pi$
$751$	46.7601	1.70630	0.853150	+	0.521666i	$0.174689\pi$
$752$	0	0
$753$	31.9342	1.16375
$754$	0	0
$755$	1.94530	0.0707967
$756$	0	0
$757$	−30.2678	−1.10010	−0.550051	−	0.835131i	$-0.685392\pi$
$757$	−30.2678	−1.10010	−0.550051	+	0.835131i	$0.685392\pi$
$758$	0	0
$759$	−25.1793	−0.913952
$760$	0	0
$761$	9.68827	0.351200	0.175600	−	0.984462i	$-0.443814\pi$
$761$	9.68827	0.351200	0.175600	+	0.984462i	$0.443814\pi$
$762$	0	0
$763$	−22.4252	−0.811847
$764$	0	0
$765$	0.357658	0.0129311
$766$	0	0
$767$	−34.5767	−1.24849
$768$	0	0
$769$	23.5689	0.849915	0.424958	−	0.905213i	$-0.360289\pi$
$769$	23.5689	0.849915	0.424958	+	0.905213i	$0.360289\pi$
$770$	0	0
$771$	−43.1187	−1.55288
$772$	0	0
$773$	−5.10704	−0.183687	−0.0918436	−	0.995773i	$-0.529276\pi$
$773$	−5.10704	−0.183687	−0.0918436	+	0.995773i	$0.529276\pi$
$774$	0	0
$775$	26.0116	0.934364
$776$	0	0
$777$	48.3611	1.73494
$778$	0	0
$779$	0.967827	0.0346760
$780$	0	0
$781$	11.0792	0.396446
$782$	0	0
$783$	−14.6189	−0.522436
$784$	0	0
$785$	20.3162	0.725116
$786$	0	0
$787$	40.3189	1.43721	0.718606	−	0.695417i	$-0.244781\pi$
$787$	40.3189	1.43721	0.718606	+	0.695417i	$0.244781\pi$
$788$	0	0
$789$	−62.2711	−2.21691
$790$	0	0
$791$	−30.7693	−1.09403
$792$	0	0
$793$	−36.1688	−1.28439
$794$	0	0
$795$	22.2612	0.789524
$796$	0	0
$797$	−32.5063	−1.15143	−0.575715	−	0.817650i	$-0.695276\pi$
$797$	−32.5063	−1.15143	−0.575715	+	0.817650i	$0.695276\pi$
$798$	0	0
$799$	−1.30787	−0.0462691
$800$	0	0
$801$	2.83883	0.100305
$802$	0	0
$803$	−5.51435	−0.194597
$804$	0	0
$805$	20.7485	0.731288
$806$	0	0
$807$	−9.12867	−0.321345
$808$	0	0
$809$	40.5004	1.42392	0.711959	−	0.702221i	$-0.247808\pi$
$809$	40.5004	1.42392	0.711959	+	0.702221i	$0.247808\pi$
$810$	0	0
$811$	5.17273	0.181639	0.0908196	−	0.995867i	$-0.471051\pi$
$811$	5.17273	0.181639	0.0908196	+	0.995867i	$0.471051\pi$
$812$	0	0
$813$	6.20358	0.217569
$814$	0	0
$815$	−6.38382	−0.223616
$816$	0	0
$817$	5.96761	0.208780
$818$	0	0
$819$	−15.0925	−0.527375
$820$	0	0
$821$	−6.50706	−0.227098	−0.113549	−	0.993532i	$-0.536222\pi$
$821$	−6.50706	−0.227098	−0.113549	+	0.993532i	$0.536222\pi$
$822$	0	0
$823$	−42.8219	−1.49268	−0.746339	−	0.665566i	$-0.768190\pi$
$823$	−42.8219	−1.49268	−0.746339	+	0.665566i	$0.768190\pi$
$824$	0	0
$825$	13.6790	0.476242
$826$	0	0
$827$	27.2737	0.948401	0.474200	−	0.880417i	$-0.342737\pi$
$827$	27.2737	0.948401	0.474200	+	0.880417i	$0.342737\pi$
$828$	0	0
$829$	45.7752	1.58984	0.794919	−	0.606715i	$-0.207513\pi$
$829$	45.7752	1.58984	0.794919	+	0.606715i	$0.207513\pi$
$830$	0	0
$831$	−31.8828	−1.10600
$832$	0	0
$833$	−0.857031	−0.0296944
$834$	0	0
$835$	14.1996	0.491397
$836$	0	0
$837$	22.3820	0.773637
$838$	0	0
$839$	50.1044	1.72980	0.864898	−	0.501948i	$-0.167383\pi$
$839$	50.1044	1.72980	0.864898	+	0.501948i	$0.167383\pi$
$840$	0	0
$841$	−13.1228	−0.452512
$842$	0	0
$843$	−49.2445	−1.69607
$844$	0	0
$845$	−2.85144	−0.0980926
$846$	0	0
$847$	26.3634	0.905857
$848$	0	0
$849$	−0.771370	−0.0264734
$850$	0	0
$851$	−60.0226	−2.05755
$852$	0	0
$853$	−31.6634	−1.08413	−0.542067	−	0.840335i	$-0.682358\pi$
$853$	−31.6634	−1.08413	−0.542067	+	0.840335i	$0.682358\pi$
$854$	0	0
$855$	−1.69997	−0.0581376
$856$	0	0
$857$	−33.3618	−1.13962	−0.569809	−	0.821777i	$-0.692983\pi$
$857$	−33.3618	−1.13962	−0.569809	+	0.821777i	$0.692983\pi$
$858$	0	0
$859$	−25.5566	−0.871981	−0.435990	−	0.899951i	$-0.643602\pi$
$859$	−25.5566	−0.871981	−0.435990	+	0.899951i	$0.643602\pi$
$860$	0	0
$861$	−3.74531	−0.127640
$862$	0	0
$863$	−12.6757	−0.431486	−0.215743	−	0.976450i	$-0.569217\pi$
$863$	−12.6757	−0.431486	−0.215743	+	0.976450i	$0.569217\pi$
$864$	0	0
$865$	17.0807	0.580761
$866$	0	0
$867$	−34.6486	−1.17673
$868$	0	0
$869$	6.22675	0.211228
$870$	0	0
$871$	53.1878	1.80220
$872$	0	0
$873$	2.96059	0.100201
$874$	0	0
$875$	−24.4901	−0.827916
$876$	0	0
$877$	51.9996	1.75590	0.877950	−	0.478752i	$-0.158911\pi$
$877$	51.9996	1.75590	0.877950	+	0.478752i	$0.158911\pi$
$878$	0	0
$879$	2.46059	0.0829938
$880$	0	0
$881$	24.5788	0.828081	0.414040	−	0.910258i	$-0.364117\pi$
$881$	24.5788	0.828081	0.414040	+	0.910258i	$0.364117\pi$
$882$	0	0
$883$	6.62659	0.223002	0.111501	−	0.993764i	$-0.464434\pi$
$883$	6.62659	0.223002	0.111501	+	0.993764i	$0.464434\pi$
$884$	0	0
$885$	15.0713	0.506617
$886$	0	0
$887$	−14.7796	−0.496250	−0.248125	−	0.968728i	$-0.579814\pi$
$887$	−14.7796	−0.496250	−0.248125	+	0.968728i	$0.579814\pi$
$888$	0	0
$889$	−61.9851	−2.07891
$890$	0	0
$891$	17.4559	0.584794
$892$	0	0
$893$	6.21637	0.208023
$894$	0	0
$895$	−2.45645	−0.0821101
$896$	0	0
$897$	65.0809	2.17299
$898$	0	0
$899$	−24.3085	−0.810735
$900$	0	0
$901$	4.34591	0.144783
$902$	0	0
$903$	−23.0935	−0.768504
$904$	0	0
$905$	−3.16444	−0.105189
$906$	0	0
$907$	−21.5048	−0.714054	−0.357027	−	0.934094i	$-0.616210\pi$
$907$	−21.5048	−0.714054	−0.357027	+	0.934094i	$0.616210\pi$
$908$	0	0
$909$	−18.0175	−0.597604
$910$	0	0
$911$	−25.4159	−0.842065	−0.421033	−	0.907045i	$-0.638332\pi$
$911$	−25.4159	−0.842065	−0.421033	+	0.907045i	$0.638332\pi$
$912$	0	0
$913$	27.0224	0.894310
$914$	0	0
$915$	15.7653	0.521185
$916$	0	0
$917$	58.6653	1.93730
$918$	0	0
$919$	20.7751	0.685306	0.342653	−	0.939462i	$-0.388675\pi$
$919$	20.7751	0.685306	0.342653	+	0.939462i	$0.388675\pi$
$920$	0	0
$921$	−7.26086	−0.239253
$922$	0	0
$923$	−28.6365	−0.942581
$924$	0	0
$925$	32.6081	1.07215
$926$	0	0
$927$	16.9811	0.557732
$928$	0	0
$929$	0.734942	0.0241127	0.0120563	−	0.999927i	$-0.496162\pi$
$929$	0.734942	0.0241127	0.0120563	+	0.999927i	$0.496162\pi$
$930$	0	0
$931$	4.07351	0.133504
$932$	0	0
$933$	57.2625	1.87469
$934$	0	0
$935$	−0.461106	−0.0150798
$936$	0	0
$937$	−18.0314	−0.589061	−0.294531	−	0.955642i	$-0.595163\pi$
$937$	−18.0314	−0.589061	−0.294531	+	0.955642i	$0.595163\pi$
$938$	0	0
$939$	−8.88583	−0.289978
$940$	0	0
$941$	−31.5482	−1.02844	−0.514221	−	0.857658i	$-0.671919\pi$
$941$	−31.5482	−1.02844	−0.514221	+	0.857658i	$0.671919\pi$
$942$	0	0
$943$	4.64843	0.151374
$944$	0	0
$945$	−9.69906	−0.315511
$946$	0	0
$947$	−31.4358	−1.02153	−0.510764	−	0.859721i	$-0.670637\pi$
$947$	−31.4358	−1.02153	−0.510764	+	0.859721i	$0.670637\pi$
$948$	0	0
$949$	14.2529	0.462669
$950$	0	0
$951$	−3.98011	−0.129064
$952$	0	0
$953$	42.2834	1.36969	0.684847	−	0.728687i	$-0.259869\pi$
$953$	42.2834	1.36969	0.684847	+	0.728687i	$0.259869\pi$
$954$	0	0
$955$	2.18709	0.0707725
$956$	0	0
$957$	−12.7834	−0.413229
$958$	0	0
$959$	−32.1466	−1.03807
$960$	0	0
$961$	6.21726	0.200557
$962$	0	0
$963$	1.21244	0.0390703
$964$	0	0
$965$	1.96919	0.0633903
$966$	0	0
$967$	−30.8891	−0.993328	−0.496664	−	0.867943i	$-0.665442\pi$
$967$	−30.8891	−0.993328	−0.496664	+	0.867943i	$0.665442\pi$
$968$	0	0
$969$	−1.15304	−0.0370411
$970$	0	0
$971$	−24.5489	−0.787811	−0.393906	−	0.919151i	$-0.628876\pi$
$971$	−24.5489	−0.787811	−0.393906	+	0.919151i	$0.628876\pi$
$972$	0	0
$973$	24.7812	0.794450
$974$	0	0
$975$	−35.3561	−1.13230
$976$	0	0
$977$	−23.9232	−0.765373	−0.382686	−	0.923878i	$-0.625001\pi$
$977$	−23.9232	−0.765373	−0.382686	+	0.923878i	$0.625001\pi$
$978$	0	0
$979$	−3.65993	−0.116972
$980$	0	0
$981$	8.82467	0.281750
$982$	0	0
$983$	3.26031	0.103988	0.0519939	−	0.998647i	$-0.483442\pi$
$983$	3.26031	0.103988	0.0519939	+	0.998647i	$0.483442\pi$
$984$	0	0
$985$	−20.4757	−0.652410
$986$	0	0
$987$	−24.0562	−0.765716
$988$	0	0
$989$	28.6622	0.911404
$990$	0	0
$991$	7.31531	0.232379	0.116189	−	0.993227i	$-0.462932\pi$
$991$	7.31531	0.232379	0.116189	+	0.993227i	$0.462932\pi$
$992$	0	0
$993$	50.8695	1.61430
$994$	0	0
$995$	17.1490	0.543661
$996$	0	0
$997$	−28.0435	−0.888146	−0.444073	−	0.895991i	$-0.646467\pi$
$997$	−28.0435	−0.888146	−0.444073	+	0.895991i	$0.646467\pi$
$998$	0	0
$999$	28.0581	0.887720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6848.2.a.bv.1.6		7
4.3	odd	2		6848.2.a.bu.1.2		7
8.3	odd	2		107.2.a.b.1.4	✓	7
8.5	even	2		1712.2.a.t.1.2		7
24.11	even	2		963.2.a.f.1.4		7
40.19	odd	2		2675.2.a.g.1.4		7
56.27	even	2		5243.2.a.g.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
107.2.a.b.1.4	✓	7	8.3	odd	2
963.2.a.f.1.4		7	24.11	even	2
1712.2.a.t.1.2		7	8.5	even	2
2675.2.a.g.1.4		7	40.19	odd	2
5243.2.a.g.1.4		7	56.27	even	2
6848.2.a.bu.1.2		7	4.3	odd	2
6848.2.a.bv.1.6		7	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:	\( N \)	\(=\)	\( 6848 = 2^{6} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6848.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05242 q^{3} -0.858034 q^{5} -3.08104 q^{7} +1.21244 q^{9} -1.56312 q^{11} +4.04020 q^{13} -1.76105 q^{15} -0.343798 q^{17} +1.63409 q^{19} -6.32361 q^{21} +7.84845 q^{23} -4.26378 q^{25} -3.66883 q^{27} +3.98461 q^{29} -6.10060 q^{31} -3.20819 q^{33} +2.64364 q^{35} -7.64770 q^{37} +8.29220 q^{39} +0.592274 q^{41} +3.65195 q^{43} -1.04031 q^{45} +3.80419 q^{47} +2.49283 q^{49} -0.705619 q^{51} -12.6409 q^{53} +1.34121 q^{55} +3.35384 q^{57} -8.55816 q^{59} -8.95223 q^{61} -3.73558 q^{63} -3.46663 q^{65} +13.1646 q^{67} +16.1083 q^{69} -7.08789 q^{71} +3.52777 q^{73} -8.75108 q^{75} +4.81605 q^{77} -3.98353 q^{79} -11.1673 q^{81} -17.2874 q^{83} +0.294990 q^{85} +8.17811 q^{87} +2.34142 q^{89} -12.4480 q^{91} -12.5210 q^{93} -1.40210 q^{95} +2.44185 q^{97} -1.89519 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 6 q^{9} - 2 q^{11} - 18 q^{13} + 9 q^{15} - q^{17} - 4 q^{19} + 11 q^{21} + 8 q^{25} + 3 q^{27} + 3 q^{29} - 4 q^{31} - 6 q^{33} - 19 q^{35} - 25 q^{37} - 5 q^{39} + 11 q^{43}+ \cdots - 52 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 6 * q^9 - 2 * q^11 - 18 * q^13 + 9 * q^15 - q^17 - 4 * q^19 + 11 * q^21 + 8 * q^25 + 3 * q^27 + 3 * q^29 - 4 * q^31 - 6 * q^33 - 19 * q^35 - 25 * q^37 - 5 * q^39 + 11 * q^43 - 25 * q^45 + 9 * q^47 + 13 * q^49 - 8 * q^51 - 8 * q^53 + 13 * q^55 - 3 * q^57 - 19 * q^59 - 25 * q^61 + 11 * q^63 + 8 * q^65 - 24 * q^67 + 19 * q^71 + 30 * q^73 - 41 * q^75 - 6 * q^77 + 21 * q^79 - 17 * q^81 + 12 * q^83 - 2 * q^85 + 15 * q^87 - 22 * q^89 - 2 * q^91 - 9 * q^93 + 44 * q^95 - 4 * q^97 - 52 * q^99

Introduction

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