LMFDB - Embedded newform 3344.2.a.bb.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 $ x^9 - x^8 - 22x^7 + 22x^6 + 152x^5 - 136x^4 - 341x^3 + 169x^2 + 196*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.438092$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.438092 q^{3} +3.19099 q^{5} -3.98768 q^{7} -2.80808 q^{9} +O(q^{10})$	$q+0.438092 q^{3} +3.19099 q^{5} -3.98768 q^{7} -2.80808 q^{9} -1.00000 q^{11} +5.23234 q^{13} +1.39795 q^{15} +2.24978 q^{17} +1.00000 q^{19} -1.74697 q^{21} +2.71057 q^{23} +5.18243 q^{25} -2.54447 q^{27} +6.72953 q^{29} +4.03835 q^{31} -0.438092 q^{33} -12.7246 q^{35} -5.79575 q^{37} +2.29224 q^{39} -6.31351 q^{41} -0.982247 q^{43} -8.96055 q^{45} +2.38198 q^{47} +8.90157 q^{49} +0.985609 q^{51} +0.773120 q^{53} -3.19099 q^{55} +0.438092 q^{57} -2.37823 q^{59} +1.97086 q^{61} +11.1977 q^{63} +16.6963 q^{65} +15.3638 q^{67} +1.18748 q^{69} +8.69237 q^{71} -0.457046 q^{73} +2.27038 q^{75} +3.98768 q^{77} +1.37587 q^{79} +7.30952 q^{81} -2.14599 q^{83} +7.17902 q^{85} +2.94815 q^{87} +14.3559 q^{89} -20.8649 q^{91} +1.76917 q^{93} +3.19099 q^{95} +10.7901 q^{97} +2.80808 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.438092	0.252932	0.126466	−	0.991971i	$-0.459636\pi$
$3$	0.438092	0.252932	0.126466	+	0.991971i	$0.459636\pi$
$4$	0	0
$5$	3.19099	1.42705	0.713527	−	0.700627i	$-0.247096\pi$
$5$	3.19099	1.42705	0.713527	+	0.700627i	$0.247096\pi$
$6$	0	0
$7$	−3.98768	−1.50720	−0.753600	−	0.657333i	$-0.771684\pi$
$7$	−3.98768	−1.50720	−0.753600	+	0.657333i	$0.771684\pi$
$8$	0	0
$9$	−2.80808	−0.936025
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.23234	1.45119	0.725595	−	0.688122i	$-0.241565\pi$
$13$	5.23234	1.45119	0.725595	+	0.688122i	$0.241565\pi$
$14$	0	0
$15$	1.39795	0.360948
$16$	0	0
$17$	2.24978	0.545651	0.272826	−	0.962064i	$-0.412042\pi$
$17$	2.24978	0.545651	0.272826	+	0.962064i	$0.412042\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.74697	−0.381220
$22$	0	0
$23$	2.71057	0.565194	0.282597	−	0.959239i	$-0.408804\pi$
$23$	2.71057	0.565194	0.282597	+	0.959239i	$0.408804\pi$
$24$	0	0
$25$	5.18243	1.03649
$26$	0	0
$27$	−2.54447	−0.489683
$28$	0	0
$29$	6.72953	1.24964	0.624821	−	0.780768i	$-0.285172\pi$
$29$	6.72953	1.24964	0.624821	+	0.780768i	$0.285172\pi$
$30$	0	0
$31$	4.03835	0.725308	0.362654	−	0.931924i	$-0.381871\pi$
$31$	4.03835	0.725308	0.362654	+	0.931924i	$0.381871\pi$
$32$	0	0
$33$	−0.438092	−0.0762620
$34$	0	0
$35$	−12.7246	−2.15086
$36$	0	0
$37$	−5.79575	−0.952816	−0.476408	−	0.879224i	$-0.658061\pi$
$37$	−5.79575	−0.952816	−0.476408	+	0.879224i	$0.658061\pi$
$38$	0	0
$39$	2.29224	0.367053
$40$	0	0
$41$	−6.31351	−0.986005	−0.493003	−	0.870028i	$-0.664101\pi$
$41$	−6.31351	−0.986005	−0.493003	+	0.870028i	$0.664101\pi$
$42$	0	0
$43$	−0.982247	−0.149791	−0.0748957	−	0.997191i	$-0.523862\pi$
$43$	−0.982247	−0.149791	−0.0748957	+	0.997191i	$0.523862\pi$
$44$	0	0
$45$	−8.96055	−1.33576
$46$	0	0
$47$	2.38198	0.347448	0.173724	−	0.984794i	$-0.444420\pi$
$47$	2.38198	0.347448	0.173724	+	0.984794i	$0.444420\pi$
$48$	0	0
$49$	8.90157	1.27165
$50$	0	0
$51$	0.985609	0.138013
$52$	0	0
$53$	0.773120	0.106196	0.0530981	−	0.998589i	$-0.483090\pi$
$53$	0.773120	0.106196	0.0530981	+	0.998589i	$0.483090\pi$
$54$	0	0
$55$	−3.19099	−0.430273
$56$	0	0
$57$	0.438092	0.0580267
$58$	0	0
$59$	−2.37823	−0.309619	−0.154810	−	0.987944i	$-0.549476\pi$
$59$	−2.37823	−0.309619	−0.154810	+	0.987944i	$0.549476\pi$
$60$	0	0
$61$	1.97086	0.252342	0.126171	−	0.992008i	$-0.459731\pi$
$61$	1.97086	0.252342	0.126171	+	0.992008i	$0.459731\pi$
$62$	0	0
$63$	11.1977	1.41078
$64$	0	0
$65$	16.6963	2.07093
$66$	0	0
$67$	15.3638	1.87699	0.938494	−	0.345296i	$-0.112222\pi$
$67$	15.3638	1.87699	0.938494	+	0.345296i	$0.112222\pi$
$68$	0	0
$69$	1.18748	0.142956
$70$	0	0
$71$	8.69237	1.03159	0.515797	−	0.856711i	$-0.327496\pi$
$71$	8.69237	1.03159	0.515797	+	0.856711i	$0.327496\pi$
$72$	0	0
$73$	−0.457046	−0.0534932	−0.0267466	−	0.999642i	$-0.508515\pi$
$73$	−0.457046	−0.0534932	−0.0267466	+	0.999642i	$0.508515\pi$
$74$	0	0
$75$	2.27038	0.262161
$76$	0	0
$77$	3.98768	0.454438
$78$	0	0
$79$	1.37587	0.154797	0.0773987	−	0.997000i	$-0.475339\pi$
$79$	1.37587	0.154797	0.0773987	+	0.997000i	$0.475339\pi$
$80$	0	0
$81$	7.30952	0.812168
$82$	0	0
$83$	−2.14599	−0.235553	−0.117776	−	0.993040i	$-0.537577\pi$
$83$	−2.14599	−0.235553	−0.117776	+	0.993040i	$0.537577\pi$
$84$	0	0
$85$	7.17902	0.778674
$86$	0	0
$87$	2.94815	0.316075
$88$	0	0
$89$	14.3559	1.52172	0.760859	−	0.648917i	$-0.224778\pi$
$89$	14.3559	1.52172	0.760859	+	0.648917i	$0.224778\pi$
$90$	0	0
$91$	−20.8649	−2.18723
$92$	0	0
$93$	1.76917	0.183454
$94$	0	0
$95$	3.19099	0.327389
$96$	0	0
$97$	10.7901	1.09557	0.547787	−	0.836618i	$-0.315470\pi$
$97$	10.7901	1.09557	0.547787	+	0.836618i	$0.315470\pi$
$98$	0	0
$99$	2.80808	0.282222
$100$	0	0
$101$	4.62813	0.460516	0.230258	−	0.973130i	$-0.426043\pi$
$101$	4.62813	0.460516	0.230258	+	0.973130i	$0.426043\pi$
$102$	0	0
$103$	9.43808	0.929961	0.464981	−	0.885321i	$-0.346061\pi$
$103$	9.43808	0.929961	0.464981	+	0.885321i	$0.346061\pi$
$104$	0	0
$105$	−5.57456	−0.544021
$106$	0	0
$107$	8.00763	0.774127	0.387063	−	0.922053i	$-0.373489\pi$
$107$	8.00763	0.774127	0.387063	+	0.922053i	$0.373489\pi$
$108$	0	0
$109$	−10.8144	−1.03583	−0.517915	−	0.855432i	$-0.673292\pi$
$109$	−10.8144	−1.03583	−0.517915	+	0.855432i	$0.673292\pi$
$110$	0	0
$111$	−2.53907	−0.240998
$112$	0	0
$113$	−0.556365	−0.0523385	−0.0261692	−	0.999658i	$-0.508331\pi$
$113$	−0.556365	−0.0523385	−0.0261692	+	0.999658i	$0.508331\pi$
$114$	0	0
$115$	8.64942	0.806563
$116$	0	0
$117$	−14.6928	−1.35835
$118$	0	0
$119$	−8.97138	−0.822405
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.76590	−0.249393
$124$	0	0
$125$	0.582128	0.0520671
$126$	0	0
$127$	8.92981	0.792393	0.396196	−	0.918166i	$-0.370330\pi$
$127$	8.92981	0.792393	0.396196	+	0.918166i	$0.370330\pi$
$128$	0	0
$129$	−0.430314	−0.0378871
$130$	0	0
$131$	−17.7768	−1.55317	−0.776584	−	0.630014i	$-0.783049\pi$
$131$	−17.7768	−1.55317	−0.776584	+	0.630014i	$0.783049\pi$
$132$	0	0
$133$	−3.98768	−0.345775
$134$	0	0
$135$	−8.11938	−0.698805
$136$	0	0
$137$	19.2157	1.64171	0.820855	−	0.571137i	$-0.193497\pi$
$137$	19.2157	1.64171	0.820855	+	0.571137i	$0.193497\pi$
$138$	0	0
$139$	0.418269	0.0354771	0.0177386	−	0.999843i	$-0.494353\pi$
$139$	0.418269	0.0354771	0.0177386	+	0.999843i	$0.494353\pi$
$140$	0	0
$141$	1.04353	0.0878808
$142$	0	0
$143$	−5.23234	−0.437550
$144$	0	0
$145$	21.4739	1.78331
$146$	0	0
$147$	3.89970	0.321642
$148$	0	0
$149$	20.6867	1.69472	0.847359	−	0.531020i	$-0.178191\pi$
$149$	20.6867	1.69472	0.847359	+	0.531020i	$0.178191\pi$
$150$	0	0
$151$	−12.0649	−0.981829	−0.490915	−	0.871208i	$-0.663337\pi$
$151$	−12.0649	−0.981829	−0.490915	+	0.871208i	$0.663337\pi$
$152$	0	0
$153$	−6.31754	−0.510743
$154$	0	0
$155$	12.8863	1.03505
$156$	0	0
$157$	−13.8328	−1.10398	−0.551989	−	0.833851i	$-0.686131\pi$
$157$	−13.8328	−1.10398	−0.551989	+	0.833851i	$0.686131\pi$
$158$	0	0
$159$	0.338698	0.0268605
$160$	0	0
$161$	−10.8089	−0.851860
$162$	0	0
$163$	−19.3745	−1.51753	−0.758764	−	0.651366i	$-0.774196\pi$
$163$	−19.3745	−1.51753	−0.758764	+	0.651366i	$0.774196\pi$
$164$	0	0
$165$	−1.39795	−0.108830
$166$	0	0
$167$	−5.31331	−0.411157	−0.205578	−	0.978641i	$-0.565908\pi$
$167$	−5.31331	−0.411157	−0.205578	+	0.978641i	$0.565908\pi$
$168$	0	0
$169$	14.3774	1.10595
$170$	0	0
$171$	−2.80808	−0.214739
$172$	0	0
$173$	20.7402	1.57685	0.788423	−	0.615133i	$-0.210898\pi$
$173$	20.7402	1.57685	0.788423	+	0.615133i	$0.210898\pi$
$174$	0	0
$175$	−20.6659	−1.56219
$176$	0	0
$177$	−1.04188	−0.0783128
$178$	0	0
$179$	21.8488	1.63306	0.816529	−	0.577304i	$-0.195895\pi$
$179$	21.8488	1.63306	0.816529	+	0.577304i	$0.195895\pi$
$180$	0	0
$181$	18.4549	1.37174	0.685872	−	0.727722i	$-0.259421\pi$
$181$	18.4549	1.37174	0.685872	+	0.727722i	$0.259421\pi$
$182$	0	0
$183$	0.863416	0.0638255
$184$	0	0
$185$	−18.4942	−1.35972
$186$	0	0
$187$	−2.24978	−0.164520
$188$	0	0
$189$	10.1465	0.738051
$190$	0	0
$191$	10.6411	0.769966	0.384983	−	0.922924i	$-0.374207\pi$
$191$	10.6411	0.769966	0.384983	+	0.922924i	$0.374207\pi$
$192$	0	0
$193$	−8.70961	−0.626932	−0.313466	−	0.949599i	$-0.601490\pi$
$193$	−8.70961	−0.626932	−0.313466	+	0.949599i	$0.601490\pi$
$194$	0	0
$195$	7.31453	0.523805
$196$	0	0
$197$	−16.0399	−1.14280	−0.571398	−	0.820673i	$-0.693599\pi$
$197$	−16.0399	−1.14280	−0.571398	+	0.820673i	$0.693599\pi$
$198$	0	0
$199$	5.71521	0.405141	0.202570	−	0.979268i	$-0.435071\pi$
$199$	5.71521	0.405141	0.202570	+	0.979268i	$0.435071\pi$
$200$	0	0
$201$	6.73076	0.474751
$202$	0	0
$203$	−26.8352	−1.88346
$204$	0	0
$205$	−20.1464	−1.40708
$206$	0	0
$207$	−7.61150	−0.529036
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−8.87144	−0.610735	−0.305367	−	0.952235i	$-0.598779\pi$
$211$	−8.87144	−0.610735	−0.305367	+	0.952235i	$0.598779\pi$
$212$	0	0
$213$	3.80805	0.260924
$214$	0	0
$215$	−3.13434	−0.213760
$216$	0	0
$217$	−16.1036	−1.09318
$218$	0	0
$219$	−0.200228	−0.0135302
$220$	0	0
$221$	11.7716	0.791843
$222$	0	0
$223$	−16.2433	−1.08773	−0.543866	−	0.839172i	$-0.683040\pi$
$223$	−16.2433	−1.08773	−0.543866	+	0.839172i	$0.683040\pi$
$224$	0	0
$225$	−14.5527	−0.970177
$226$	0	0
$227$	23.6882	1.57224	0.786121	−	0.618073i	$-0.212086\pi$
$227$	23.6882	1.57224	0.786121	+	0.618073i	$0.212086\pi$
$228$	0	0
$229$	−28.3114	−1.87087	−0.935435	−	0.353499i	$-0.884992\pi$
$229$	−28.3114	−1.87087	−0.935435	+	0.353499i	$0.884992\pi$
$230$	0	0
$231$	1.74697	0.114942
$232$	0	0
$233$	−13.1863	−0.863862	−0.431931	−	0.901907i	$-0.642168\pi$
$233$	−13.1863	−0.863862	−0.431931	+	0.901907i	$0.642168\pi$
$234$	0	0
$235$	7.60089	0.495827
$236$	0	0
$237$	0.602757	0.0391533
$238$	0	0
$239$	27.5665	1.78313	0.891564	−	0.452894i	$-0.149608\pi$
$239$	27.5665	1.78313	0.891564	+	0.452894i	$0.149608\pi$
$240$	0	0
$241$	−2.12177	−0.136675	−0.0683375	−	0.997662i	$-0.521769\pi$
$241$	−2.12177	−0.136675	−0.0683375	+	0.997662i	$0.521769\pi$
$242$	0	0
$243$	10.8356	0.695107
$244$	0	0
$245$	28.4048	1.81472
$246$	0	0
$247$	5.23234	0.332926
$248$	0	0
$249$	−0.940140	−0.0595790
$250$	0	0
$251$	−8.29144	−0.523351	−0.261675	−	0.965156i	$-0.584275\pi$
$251$	−8.29144	−0.523351	−0.261675	+	0.965156i	$0.584275\pi$
$252$	0	0
$253$	−2.71057	−0.170412
$254$	0	0
$255$	3.14507	0.196952
$256$	0	0
$257$	−9.22849	−0.575658	−0.287829	−	0.957682i	$-0.592933\pi$
$257$	−9.22849	−0.575658	−0.287829	+	0.957682i	$0.592933\pi$
$258$	0	0
$259$	23.1116	1.43608
$260$	0	0
$261$	−18.8970	−1.16970
$262$	0	0
$263$	−16.7623	−1.03361	−0.516805	−	0.856103i	$-0.672879\pi$
$263$	−16.7623	−1.03361	−0.516805	+	0.856103i	$0.672879\pi$
$264$	0	0
$265$	2.46702	0.151548
$266$	0	0
$267$	6.28918	0.384892
$268$	0	0
$269$	−10.4883	−0.639485	−0.319742	−	0.947504i	$-0.603596\pi$
$269$	−10.4883	−0.639485	−0.319742	+	0.947504i	$0.603596\pi$
$270$	0	0
$271$	−14.6179	−0.887971	−0.443986	−	0.896034i	$-0.646436\pi$
$271$	−14.6179	−0.887971	−0.443986	+	0.896034i	$0.646436\pi$
$272$	0	0
$273$	−9.14073	−0.553222
$274$	0	0
$275$	−5.18243	−0.312512
$276$	0	0
$277$	−19.6524	−1.18080	−0.590400	−	0.807110i	$-0.701030\pi$
$277$	−19.6524	−1.18080	−0.590400	+	0.807110i	$0.701030\pi$
$278$	0	0
$279$	−11.3400	−0.678907
$280$	0	0
$281$	−21.9831	−1.31140	−0.655700	−	0.755022i	$-0.727626\pi$
$281$	−21.9831	−1.31140	−0.655700	+	0.755022i	$0.727626\pi$
$282$	0	0
$283$	29.1374	1.73204	0.866019	−	0.500011i	$-0.166671\pi$
$283$	29.1374	1.73204	0.866019	+	0.500011i	$0.166671\pi$
$284$	0	0
$285$	1.39795	0.0828072
$286$	0	0
$287$	25.1763	1.48611
$288$	0	0
$289$	−11.9385	−0.702265
$290$	0	0
$291$	4.72707	0.277106
$292$	0	0
$293$	−31.1656	−1.82071	−0.910357	−	0.413824i	$-0.864193\pi$
$293$	−31.1656	−1.82071	−0.910357	+	0.413824i	$0.864193\pi$
$294$	0	0
$295$	−7.58892	−0.441844
$296$	0	0
$297$	2.54447	0.147645
$298$	0	0
$299$	14.1826	0.820204
$300$	0	0
$301$	3.91689	0.225766
$302$	0	0
$303$	2.02755	0.116479
$304$	0	0
$305$	6.28899	0.360106
$306$	0	0
$307$	−3.74105	−0.213513	−0.106757	−	0.994285i	$-0.534047\pi$
$307$	−3.74105	−0.213513	−0.106757	+	0.994285i	$0.534047\pi$
$308$	0	0
$309$	4.13474	0.235217
$310$	0	0
$311$	−15.0222	−0.851828	−0.425914	−	0.904764i	$-0.640047\pi$
$311$	−15.0222	−0.851828	−0.425914	+	0.904764i	$0.640047\pi$
$312$	0	0
$313$	23.9925	1.35614	0.678068	−	0.734999i	$-0.262817\pi$
$313$	23.9925	1.35614	0.678068	+	0.734999i	$0.262817\pi$
$314$	0	0
$315$	35.7318	2.01326
$316$	0	0
$317$	9.18161	0.515691	0.257845	−	0.966186i	$-0.416987\pi$
$317$	9.18161	0.515691	0.257845	+	0.966186i	$0.416987\pi$
$318$	0	0
$319$	−6.72953	−0.376781
$320$	0	0
$321$	3.50808	0.195802
$322$	0	0
$323$	2.24978	0.125181
$324$	0	0
$325$	27.1162	1.50414
$326$	0	0
$327$	−4.73769	−0.261995
$328$	0	0
$329$	−9.49858	−0.523674
$330$	0	0
$331$	16.3647	0.899483	0.449741	−	0.893159i	$-0.351516\pi$
$331$	16.3647	0.899483	0.449741	+	0.893159i	$0.351516\pi$
$332$	0	0
$333$	16.2749	0.891860
$334$	0	0
$335$	49.0258	2.67856
$336$	0	0
$337$	0.673860	0.0367075	0.0183537	−	0.999832i	$-0.494157\pi$
$337$	0.673860	0.0367075	0.0183537	+	0.999832i	$0.494157\pi$
$338$	0	0
$339$	−0.243739	−0.0132381
$340$	0	0
$341$	−4.03835	−0.218689
$342$	0	0
$343$	−7.58283	−0.409435
$344$	0	0
$345$	3.78924	0.204006
$346$	0	0
$347$	28.4650	1.52808	0.764041	−	0.645167i	$-0.223212\pi$
$347$	28.4650	1.52808	0.764041	+	0.645167i	$0.223212\pi$
$348$	0	0
$349$	14.7954	0.791980	0.395990	−	0.918255i	$-0.370402\pi$
$349$	14.7954	0.791980	0.395990	+	0.918255i	$0.370402\pi$
$350$	0	0
$351$	−13.3135	−0.710623
$352$	0	0
$353$	19.1901	1.02139	0.510694	−	0.859763i	$-0.329389\pi$
$353$	19.1901	1.02139	0.510694	+	0.859763i	$0.329389\pi$
$354$	0	0
$355$	27.7373	1.47214
$356$	0	0
$357$	−3.93029	−0.208013
$358$	0	0
$359$	−23.5088	−1.24074	−0.620372	−	0.784307i	$-0.713019\pi$
$359$	−23.5088	−1.24074	−0.620372	+	0.784307i	$0.713019\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.438092	0.0229939
$364$	0	0
$365$	−1.45843	−0.0763378
$366$	0	0
$367$	4.63616	0.242006	0.121003	−	0.992652i	$-0.461389\pi$
$367$	4.63616	0.242006	0.121003	+	0.992652i	$0.461389\pi$
$368$	0	0
$369$	17.7288	0.922926
$370$	0	0
$371$	−3.08295	−0.160059
$372$	0	0
$373$	−16.4555	−0.852031	−0.426016	−	0.904716i	$-0.640083\pi$
$373$	−16.4555	−0.852031	−0.426016	+	0.904716i	$0.640083\pi$
$374$	0	0
$375$	0.255025	0.0131695
$376$	0	0
$377$	35.2112	1.81347
$378$	0	0
$379$	25.2389	1.29643	0.648217	−	0.761456i	$-0.275515\pi$
$379$	25.2389	1.29643	0.648217	+	0.761456i	$0.275515\pi$
$380$	0	0
$381$	3.91208	0.200422
$382$	0	0
$383$	−3.24423	−0.165772	−0.0828862	−	0.996559i	$-0.526414\pi$
$383$	−3.24423	−0.165772	−0.0828862	+	0.996559i	$0.526414\pi$
$384$	0	0
$385$	12.7246	0.648508
$386$	0	0
$387$	2.75823	0.140208
$388$	0	0
$389$	−19.5091	−0.989151	−0.494576	−	0.869135i	$-0.664676\pi$
$389$	−19.5091	−0.989151	−0.494576	+	0.869135i	$0.664676\pi$
$390$	0	0
$391$	6.09819	0.308399
$392$	0	0
$393$	−7.78788	−0.392846
$394$	0	0
$395$	4.39039	0.220904
$396$	0	0
$397$	28.0599	1.40828	0.704142	−	0.710059i	$-0.251332\pi$
$397$	28.0599	1.40828	0.704142	+	0.710059i	$0.251332\pi$
$398$	0	0
$399$	−1.74697	−0.0874578
$400$	0	0
$401$	−3.71630	−0.185583	−0.0927915	−	0.995686i	$-0.529579\pi$
$401$	−3.71630	−0.185583	−0.0927915	+	0.995686i	$0.529579\pi$
$402$	0	0
$403$	21.1300	1.05256
$404$	0	0
$405$	23.3246	1.15901
$406$	0	0
$407$	5.79575	0.287285
$408$	0	0
$409$	−33.7346	−1.66807	−0.834035	−	0.551711i	$-0.813975\pi$
$409$	−33.7346	−1.66807	−0.834035	+	0.551711i	$0.813975\pi$
$410$	0	0
$411$	8.41825	0.415242
$412$	0	0
$413$	9.48362	0.466658
$414$	0	0
$415$	−6.84783	−0.336147
$416$	0	0
$417$	0.183240	0.00897331
$418$	0	0
$419$	−36.0785	−1.76255	−0.881274	−	0.472605i	$-0.843314\pi$
$419$	−36.0785	−1.76255	−0.881274	+	0.472605i	$0.843314\pi$
$420$	0	0
$421$	−35.7259	−1.74117	−0.870587	−	0.492015i	$-0.836260\pi$
$421$	−35.7259	−1.74117	−0.870587	+	0.492015i	$0.836260\pi$
$422$	0	0
$423$	−6.68879	−0.325220
$424$	0	0
$425$	11.6593	0.565560
$426$	0	0
$427$	−7.85914	−0.380330
$428$	0	0
$429$	−2.29224	−0.110671
$430$	0	0
$431$	−2.97024	−0.143071	−0.0715357	−	0.997438i	$-0.522790\pi$
$431$	−2.97024	−0.143071	−0.0715357	+	0.997438i	$0.522790\pi$
$432$	0	0
$433$	−23.2232	−1.11604	−0.558019	−	0.829828i	$-0.688438\pi$
$433$	−23.2232	−1.11604	−0.558019	+	0.829828i	$0.688438\pi$
$434$	0	0
$435$	9.40753	0.451056
$436$	0	0
$437$	2.71057	0.129664
$438$	0	0
$439$	7.18824	0.343076	0.171538	−	0.985178i	$-0.445126\pi$
$439$	7.18824	0.343076	0.171538	+	0.985178i	$0.445126\pi$
$440$	0	0
$441$	−24.9963	−1.19030
$442$	0	0
$443$	4.75543	0.225937	0.112969	−	0.993599i	$-0.463964\pi$
$443$	4.75543	0.225937	0.112969	+	0.993599i	$0.463964\pi$
$444$	0	0
$445$	45.8094	2.17158
$446$	0	0
$447$	9.06266	0.428649
$448$	0	0
$449$	14.7151	0.694447	0.347223	−	0.937782i	$-0.387125\pi$
$449$	14.7151	0.694447	0.347223	+	0.937782i	$0.387125\pi$
$450$	0	0
$451$	6.31351	0.297292
$452$	0	0
$453$	−5.28554	−0.248336
$454$	0	0
$455$	−66.5796	−3.12130
$456$	0	0
$457$	−7.06904	−0.330676	−0.165338	−	0.986237i	$-0.552871\pi$
$457$	−7.06904	−0.330676	−0.165338	+	0.986237i	$0.552871\pi$
$458$	0	0
$459$	−5.72449	−0.267196
$460$	0	0
$461$	11.8409	0.551486	0.275743	−	0.961231i	$-0.411076\pi$
$461$	11.8409	0.551486	0.275743	+	0.961231i	$0.411076\pi$
$462$	0	0
$463$	−24.2656	−1.12772	−0.563858	−	0.825871i	$-0.690684\pi$
$463$	−24.2656	−1.12772	−0.563858	+	0.825871i	$0.690684\pi$
$464$	0	0
$465$	5.64539	0.261799
$466$	0	0
$467$	1.64300	0.0760288	0.0380144	−	0.999277i	$-0.487897\pi$
$467$	1.64300	0.0760288	0.0380144	+	0.999277i	$0.487897\pi$
$468$	0	0
$469$	−61.2659	−2.82900
$470$	0	0
$471$	−6.06004	−0.279232
$472$	0	0
$473$	0.982247	0.0451638
$474$	0	0
$475$	5.18243	0.237786
$476$	0	0
$477$	−2.17098	−0.0994023
$478$	0	0
$479$	−24.8814	−1.13686	−0.568430	−	0.822731i	$-0.692449\pi$
$479$	−24.8814	−1.13686	−0.568430	+	0.822731i	$0.692449\pi$
$480$	0	0
$481$	−30.3253	−1.38272
$482$	0	0
$483$	−4.73529	−0.215463
$484$	0	0
$485$	34.4313	1.56344
$486$	0	0
$487$	10.6295	0.481669	0.240834	−	0.970566i	$-0.422579\pi$
$487$	10.6295	0.481669	0.240834	+	0.970566i	$0.422579\pi$
$488$	0	0
$489$	−8.48781	−0.383832
$490$	0	0
$491$	−15.2782	−0.689495	−0.344748	−	0.938695i	$-0.612036\pi$
$491$	−15.2782	−0.689495	−0.344748	+	0.938695i	$0.612036\pi$
$492$	0	0
$493$	15.1399	0.681869
$494$	0	0
$495$	8.96055	0.402747
$496$	0	0
$497$	−34.6623	−1.55482
$498$	0	0
$499$	35.0522	1.56915	0.784576	−	0.620032i	$-0.212880\pi$
$499$	35.0522	1.56915	0.784576	+	0.620032i	$0.212880\pi$
$500$	0	0
$501$	−2.32772	−0.103995
$502$	0	0
$503$	−27.4686	−1.22476	−0.612382	−	0.790562i	$-0.709788\pi$
$503$	−27.4686	−1.22476	−0.612382	+	0.790562i	$0.709788\pi$
$504$	0	0
$505$	14.7683	0.657182
$506$	0	0
$507$	6.29860	0.279731
$508$	0	0
$509$	39.1973	1.73739	0.868695	−	0.495347i	$-0.164959\pi$
$509$	39.1973	1.73739	0.868695	+	0.495347i	$0.164959\pi$
$510$	0	0
$511$	1.82255	0.0806250
$512$	0	0
$513$	−2.54447	−0.112341
$514$	0	0
$515$	30.1168	1.32711
$516$	0	0
$517$	−2.38198	−0.104760
$518$	0	0
$519$	9.08610	0.398835
$520$	0	0
$521$	12.4402	0.545015	0.272507	−	0.962154i	$-0.412147\pi$
$521$	12.4402	0.545015	0.272507	+	0.962154i	$0.412147\pi$
$522$	0	0
$523$	−38.1962	−1.67020	−0.835101	−	0.550097i	$-0.814591\pi$
$523$	−38.1962	−1.67020	−0.835101	+	0.550097i	$0.814591\pi$
$524$	0	0
$525$	−9.05354	−0.395129
$526$	0	0
$527$	9.08538	0.395765
$528$	0	0
$529$	−15.6528	−0.680556
$530$	0	0
$531$	6.67826	0.289812
$532$	0	0
$533$	−33.0344	−1.43088
$534$	0	0
$535$	25.5523	1.10472
$536$	0	0
$537$	9.57179	0.413053
$538$	0	0
$539$	−8.90157	−0.383418
$540$	0	0
$541$	−24.4612	−1.05167	−0.525835	−	0.850587i	$-0.676247\pi$
$541$	−24.4612	−1.05167	−0.525835	+	0.850587i	$0.676247\pi$
$542$	0	0
$543$	8.08495	0.346959
$544$	0	0
$545$	−34.5086	−1.47819
$546$	0	0
$547$	6.20346	0.265241	0.132620	−	0.991167i	$-0.457661\pi$
$547$	6.20346	0.265241	0.132620	+	0.991167i	$0.457661\pi$
$548$	0	0
$549$	−5.53432	−0.236199
$550$	0	0
$551$	6.72953	0.286688
$552$	0	0
$553$	−5.48653	−0.233311
$554$	0	0
$555$	−8.10215	−0.343917
$556$	0	0
$557$	−15.2030	−0.644170	−0.322085	−	0.946711i	$-0.604384\pi$
$557$	−15.2030	−0.644170	−0.322085	+	0.946711i	$0.604384\pi$
$558$	0	0
$559$	−5.13945	−0.217376
$560$	0	0
$561$	−0.985609	−0.0416124
$562$	0	0
$563$	32.1315	1.35418	0.677090	−	0.735900i	$-0.263241\pi$
$563$	32.1315	1.35418	0.677090	+	0.735900i	$0.263241\pi$
$564$	0	0
$565$	−1.77536	−0.0746899
$566$	0	0
$567$	−29.1480	−1.22410
$568$	0	0
$569$	−32.9127	−1.37977	−0.689885	−	0.723919i	$-0.742339\pi$
$569$	−32.9127	−1.37977	−0.689885	+	0.723919i	$0.742339\pi$
$570$	0	0
$571$	−30.9880	−1.29681	−0.648403	−	0.761297i	$-0.724563\pi$
$571$	−30.9880	−1.29681	−0.648403	+	0.761297i	$0.724563\pi$
$572$	0	0
$573$	4.66180	0.194749
$574$	0	0
$575$	14.0474	0.585815
$576$	0	0
$577$	14.7221	0.612891	0.306446	−	0.951888i	$-0.400860\pi$
$577$	14.7221	0.612891	0.306446	+	0.951888i	$0.400860\pi$
$578$	0	0
$579$	−3.81561	−0.158571
$580$	0	0
$581$	8.55751	0.355025
$582$	0	0
$583$	−0.773120	−0.0320194
$584$	0	0
$585$	−46.8846	−1.93844
$586$	0	0
$587$	−29.1247	−1.20210	−0.601052	−	0.799210i	$-0.705251\pi$
$587$	−29.1247	−1.20210	−0.601052	+	0.799210i	$0.705251\pi$
$588$	0	0
$589$	4.03835	0.166397
$590$	0	0
$591$	−7.02695	−0.289050
$592$	0	0
$593$	21.6083	0.887347	0.443673	−	0.896189i	$-0.353675\pi$
$593$	21.6083	0.887347	0.443673	+	0.896189i	$0.353675\pi$
$594$	0	0
$595$	−28.6276	−1.17362
$596$	0	0
$597$	2.50379	0.102473
$598$	0	0
$599$	16.3301	0.667232	0.333616	−	0.942709i	$-0.391731\pi$
$599$	16.3301	0.667232	0.333616	+	0.942709i	$0.391731\pi$
$600$	0	0
$601$	15.2703	0.622890	0.311445	−	0.950264i	$-0.399187\pi$
$601$	15.2703	0.622890	0.311445	+	0.950264i	$0.399187\pi$
$602$	0	0
$603$	−43.1427	−1.75691
$604$	0	0
$605$	3.19099	0.129732
$606$	0	0
$607$	22.4833	0.912569	0.456285	−	0.889834i	$-0.349180\pi$
$607$	22.4833	0.912569	0.456285	+	0.889834i	$0.349180\pi$
$608$	0	0
$609$	−11.7563	−0.476388
$610$	0	0
$611$	12.4633	0.504213
$612$	0	0
$613$	−36.3850	−1.46958	−0.734788	−	0.678297i	$-0.762718\pi$
$613$	−36.3850	−1.46958	−0.734788	+	0.678297i	$0.762718\pi$
$614$	0	0
$615$	−8.82596	−0.355897
$616$	0	0
$617$	9.74677	0.392390	0.196195	−	0.980565i	$-0.437141\pi$
$617$	9.74677	0.392390	0.196195	+	0.980565i	$0.437141\pi$
$618$	0	0
$619$	11.3859	0.457638	0.228819	−	0.973469i	$-0.426514\pi$
$619$	11.3859	0.457638	0.228819	+	0.973469i	$0.426514\pi$
$620$	0	0
$621$	−6.89698	−0.276766
$622$	0	0
$623$	−57.2465	−2.29353
$624$	0	0
$625$	−24.0546	−0.962183
$626$	0	0
$627$	−0.438092	−0.0174957
$628$	0	0
$629$	−13.0392	−0.519905
$630$	0	0
$631$	16.6073	0.661128	0.330564	−	0.943784i	$-0.392761\pi$
$631$	16.6073	0.661128	0.330564	+	0.943784i	$0.392761\pi$
$632$	0	0
$633$	−3.88650	−0.154475
$634$	0	0
$635$	28.4950	1.13079
$636$	0	0
$637$	46.5760	1.84541
$638$	0	0
$639$	−24.4088	−0.965598
$640$	0	0
$641$	32.2215	1.27267	0.636336	−	0.771412i	$-0.280449\pi$
$641$	32.2215	1.27267	0.636336	+	0.771412i	$0.280449\pi$
$642$	0	0
$643$	−30.0697	−1.18583	−0.592917	−	0.805264i	$-0.702024\pi$
$643$	−30.0697	−1.18583	−0.592917	+	0.805264i	$0.702024\pi$
$644$	0	0
$645$	−1.37313	−0.0540669
$646$	0	0
$647$	−33.1572	−1.30355	−0.651773	−	0.758414i	$-0.725974\pi$
$647$	−33.1572	−1.30355	−0.651773	+	0.758414i	$0.725974\pi$
$648$	0	0
$649$	2.37823	0.0933538
$650$	0	0
$651$	−7.05486	−0.276502
$652$	0	0
$653$	−19.0661	−0.746114	−0.373057	−	0.927808i	$-0.621690\pi$
$653$	−19.0661	−0.746114	−0.373057	+	0.927808i	$0.621690\pi$
$654$	0	0
$655$	−56.7257	−2.21646
$656$	0	0
$657$	1.28342	0.0500710
$658$	0	0
$659$	−17.6031	−0.685719	−0.342859	−	0.939387i	$-0.611395\pi$
$659$	−17.6031	−0.685719	−0.342859	+	0.939387i	$0.611395\pi$
$660$	0	0
$661$	−18.0022	−0.700206	−0.350103	−	0.936711i	$-0.613853\pi$
$661$	−18.0022	−0.700206	−0.350103	+	0.936711i	$0.613853\pi$
$662$	0	0
$663$	5.15704	0.200283
$664$	0	0
$665$	−12.7246	−0.493441
$666$	0	0
$667$	18.2409	0.706290
$668$	0	0
$669$	−7.11606	−0.275123
$670$	0	0
$671$	−1.97086	−0.0760841
$672$	0	0
$673$	−10.3402	−0.398584	−0.199292	−	0.979940i	$-0.563864\pi$
$673$	−10.3402	−0.398584	−0.199292	+	0.979940i	$0.563864\pi$
$674$	0	0
$675$	−13.1865	−0.507550
$676$	0	0
$677$	−20.9421	−0.804872	−0.402436	−	0.915448i	$-0.631836\pi$
$677$	−20.9421	−0.804872	−0.402436	+	0.915448i	$0.631836\pi$
$678$	0	0
$679$	−43.0276	−1.65125
$680$	0	0
$681$	10.3776	0.397671
$682$	0	0
$683$	−21.8938	−0.837742	−0.418871	−	0.908046i	$-0.637574\pi$
$683$	−21.8938	−0.837742	−0.418871	+	0.908046i	$0.637574\pi$
$684$	0	0
$685$	61.3172	2.34281
$686$	0	0
$687$	−12.4030	−0.473204
$688$	0	0
$689$	4.04523	0.154111
$690$	0	0
$691$	37.2751	1.41801	0.709005	−	0.705203i	$-0.249144\pi$
$691$	37.2751	1.41801	0.709005	+	0.705203i	$0.249144\pi$
$692$	0	0
$693$	−11.1977	−0.425365
$694$	0	0
$695$	1.33469	0.0506278
$696$	0	0
$697$	−14.2040	−0.538015
$698$	0	0
$699$	−5.77680	−0.218499
$700$	0	0
$701$	18.5562	0.700858	0.350429	−	0.936589i	$-0.386036\pi$
$701$	18.5562	0.700858	0.350429	+	0.936589i	$0.386036\pi$
$702$	0	0
$703$	−5.79575	−0.218591
$704$	0	0
$705$	3.32989	0.125411
$706$	0	0
$707$	−18.4555	−0.694090
$708$	0	0
$709$	36.0300	1.35313	0.676567	−	0.736381i	$-0.263467\pi$
$709$	36.0300	1.35313	0.676567	+	0.736381i	$0.263467\pi$
$710$	0	0
$711$	−3.86355	−0.144894
$712$	0	0
$713$	10.9462	0.409940
$714$	0	0
$715$	−16.6963	−0.624408
$716$	0	0
$717$	12.0767	0.451011
$718$	0	0
$719$	52.4774	1.95708	0.978538	−	0.206065i	$-0.0660660\pi$
$719$	52.4774	1.95708	0.978538	+	0.206065i	$0.0660660\pi$
$720$	0	0
$721$	−37.6360	−1.40164
$722$	0	0
$723$	−0.929529	−0.0345696
$724$	0	0
$725$	34.8753	1.29524
$726$	0	0
$727$	27.1977	1.00871	0.504353	−	0.863497i	$-0.331731\pi$
$727$	27.1977	1.00871	0.504353	+	0.863497i	$0.331731\pi$
$728$	0	0
$729$	−17.1815	−0.636353
$730$	0	0
$731$	−2.20984	−0.0817338
$732$	0	0
$733$	−5.67116	−0.209469	−0.104735	−	0.994500i	$-0.533399\pi$
$733$	−5.67116	−0.209469	−0.104735	+	0.994500i	$0.533399\pi$
$734$	0	0
$735$	12.4439	0.459001
$736$	0	0
$737$	−15.3638	−0.565933
$738$	0	0
$739$	16.6032	0.610759	0.305379	−	0.952231i	$-0.401217\pi$
$739$	16.6032	0.610759	0.305379	+	0.952231i	$0.401217\pi$
$740$	0	0
$741$	2.29224	0.0842077
$742$	0	0
$743$	−51.5391	−1.89079	−0.945394	−	0.325931i	$-0.894322\pi$
$743$	−51.5391	−1.89079	−0.945394	+	0.325931i	$0.894322\pi$
$744$	0	0
$745$	66.0110	2.41846
$746$	0	0
$747$	6.02610	0.220483
$748$	0	0
$749$	−31.9318	−1.16676
$750$	0	0
$751$	−0.509955	−0.0186085	−0.00930427	−	0.999957i	$-0.502962\pi$
$751$	−0.509955	−0.0186085	−0.00930427	+	0.999957i	$0.502962\pi$
$752$	0	0
$753$	−3.63241	−0.132372
$754$	0	0
$755$	−38.4991	−1.40112
$756$	0	0
$757$	−35.8962	−1.30467	−0.652334	−	0.757932i	$-0.726210\pi$
$757$	−35.8962	−1.30467	−0.652334	+	0.757932i	$0.726210\pi$
$758$	0	0
$759$	−1.18748	−0.0431028
$760$	0	0
$761$	−48.3656	−1.75325	−0.876626	−	0.481172i	$-0.840211\pi$
$761$	−48.3656	−1.75325	−0.876626	+	0.481172i	$0.840211\pi$
$762$	0	0
$763$	43.1243	1.56120
$764$	0	0
$765$	−20.1592	−0.728859
$766$	0	0
$767$	−12.4437	−0.449317
$768$	0	0
$769$	47.3727	1.70830	0.854152	−	0.520023i	$-0.174077\pi$
$769$	47.3727	1.70830	0.854152	+	0.520023i	$0.174077\pi$
$770$	0	0
$771$	−4.04293	−0.145602
$772$	0	0
$773$	35.3323	1.27082	0.635408	−	0.772177i	$-0.280832\pi$
$773$	35.3323	1.27082	0.635408	+	0.772177i	$0.280832\pi$
$774$	0	0
$775$	20.9284	0.751772
$776$	0	0
$777$	10.1250	0.363232
$778$	0	0
$779$	−6.31351	−0.226205
$780$	0	0
$781$	−8.69237	−0.311037
$782$	0	0
$783$	−17.1231	−0.611929
$784$	0	0
$785$	−44.1404	−1.57544
$786$	0	0
$787$	−15.9372	−0.568101	−0.284050	−	0.958809i	$-0.591678\pi$
$787$	−15.9372	−0.568101	−0.284050	+	0.958809i	$0.591678\pi$
$788$	0	0
$789$	−7.34345	−0.261434
$790$	0	0
$791$	2.21861	0.0788845
$792$	0	0
$793$	10.3122	0.366197
$794$	0	0
$795$	1.08078	0.0383313
$796$	0	0
$797$	−23.3849	−0.828335	−0.414167	−	0.910201i	$-0.635927\pi$
$797$	−23.3849	−0.828335	−0.414167	+	0.910201i	$0.635927\pi$
$798$	0	0
$799$	5.35893	0.189585
$800$	0	0
$801$	−40.3123	−1.42437
$802$	0	0
$803$	0.457046	0.0161288
$804$	0	0
$805$	−34.4911	−1.21565
$806$	0	0
$807$	−4.59485	−0.161746
$808$	0	0
$809$	44.3607	1.55964	0.779820	−	0.626004i	$-0.215310\pi$
$809$	44.3607	1.55964	0.779820	+	0.626004i	$0.215310\pi$
$810$	0	0
$811$	13.7634	0.483299	0.241650	−	0.970364i	$-0.422312\pi$
$811$	13.7634	0.483299	0.241650	+	0.970364i	$0.422312\pi$
$812$	0	0
$813$	−6.40396	−0.224597
$814$	0	0
$815$	−61.8239	−2.16560
$816$	0	0
$817$	−0.982247	−0.0343645
$818$	0	0
$819$	58.5901	2.04731
$820$	0	0
$821$	−24.5018	−0.855118	−0.427559	−	0.903987i	$-0.640626\pi$
$821$	−24.5018	−0.855118	−0.427559	+	0.903987i	$0.640626\pi$
$822$	0	0
$823$	−2.33038	−0.0812318	−0.0406159	−	0.999175i	$-0.512932\pi$
$823$	−2.33038	−0.0812318	−0.0406159	+	0.999175i	$0.512932\pi$
$824$	0	0
$825$	−2.27038	−0.0790444
$826$	0	0
$827$	−9.86168	−0.342924	−0.171462	−	0.985191i	$-0.554849\pi$
$827$	−9.86168	−0.342924	−0.171462	+	0.985191i	$0.554849\pi$
$828$	0	0
$829$	−34.4628	−1.19694	−0.598471	−	0.801144i	$-0.704225\pi$
$829$	−34.4628	−1.19694	−0.598471	+	0.801144i	$0.704225\pi$
$830$	0	0
$831$	−8.60957	−0.298663
$832$	0	0
$833$	20.0265	0.693879
$834$	0	0
$835$	−16.9547	−0.586743
$836$	0	0
$837$	−10.2754	−0.355171
$838$	0	0
$839$	−26.7027	−0.921878	−0.460939	−	0.887432i	$-0.652487\pi$
$839$	−26.7027	−0.921878	−0.460939	+	0.887432i	$0.652487\pi$
$840$	0	0
$841$	16.2866	0.561606
$842$	0	0
$843$	−9.63060	−0.331695
$844$	0	0
$845$	45.8781	1.57825
$846$	0	0
$847$	−3.98768	−0.137018
$848$	0	0
$849$	12.7649	0.438089
$850$	0	0
$851$	−15.7098	−0.538526
$852$	0	0
$853$	−9.79178	−0.335264	−0.167632	−	0.985850i	$-0.553612\pi$
$853$	−9.79178	−0.335264	−0.167632	+	0.985850i	$0.553612\pi$
$854$	0	0
$855$	−8.96055	−0.306444
$856$	0	0
$857$	8.33973	0.284880	0.142440	−	0.989803i	$-0.454505\pi$
$857$	8.33973	0.284880	0.142440	+	0.989803i	$0.454505\pi$
$858$	0	0
$859$	39.9083	1.36166	0.680828	−	0.732444i	$-0.261620\pi$
$859$	39.9083	1.36166	0.680828	+	0.732444i	$0.261620\pi$
$860$	0	0
$861$	11.0295	0.375885
$862$	0	0
$863$	−35.7088	−1.21554	−0.607771	−	0.794112i	$-0.707936\pi$
$863$	−35.7088	−1.21554	−0.607771	+	0.794112i	$0.707936\pi$
$864$	0	0
$865$	66.1817	2.25025
$866$	0	0
$867$	−5.23016	−0.177626
$868$	0	0
$869$	−1.37587	−0.0466732
$870$	0	0
$871$	80.3886	2.72386
$872$	0	0
$873$	−30.2996	−1.02548
$874$	0	0
$875$	−2.32134	−0.0784756
$876$	0	0
$877$	−7.58918	−0.256268	−0.128134	−	0.991757i	$-0.540899\pi$
$877$	−7.58918	−0.256268	−0.128134	+	0.991757i	$0.540899\pi$
$878$	0	0
$879$	−13.6534	−0.460517
$880$	0	0
$881$	33.8610	1.14081	0.570403	−	0.821365i	$-0.306787\pi$
$881$	33.8610	1.14081	0.570403	+	0.821365i	$0.306787\pi$
$882$	0	0
$883$	5.59510	0.188290	0.0941451	−	0.995558i	$-0.469988\pi$
$883$	5.59510	0.188290	0.0941451	+	0.995558i	$0.469988\pi$
$884$	0	0
$885$	−3.32464	−0.111757
$886$	0	0
$887$	12.0075	0.403172	0.201586	−	0.979471i	$-0.435390\pi$
$887$	12.0075	0.403172	0.201586	+	0.979471i	$0.435390\pi$
$888$	0	0
$889$	−35.6092	−1.19429
$890$	0	0
$891$	−7.30952	−0.244878
$892$	0	0
$893$	2.38198	0.0797100
$894$	0	0
$895$	69.7195	2.33046
$896$	0	0
$897$	6.21330	0.207456
$898$	0	0
$899$	27.1762	0.906376
$900$	0	0
$901$	1.73935	0.0579461
$902$	0	0
$903$	1.71595	0.0571034
$904$	0	0
$905$	58.8895	1.95755
$906$	0	0
$907$	−7.10915	−0.236055	−0.118028	−	0.993010i	$-0.537657\pi$
$907$	−7.10915	−0.236055	−0.118028	+	0.993010i	$0.537657\pi$
$908$	0	0
$909$	−12.9961	−0.431055
$910$	0	0
$911$	−35.7037	−1.18292	−0.591458	−	0.806335i	$-0.701448\pi$
$911$	−35.7037	−1.18292	−0.591458	+	0.806335i	$0.701448\pi$
$912$	0	0
$913$	2.14599	0.0710219
$914$	0	0
$915$	2.75515	0.0910826
$916$	0	0
$917$	70.8882	2.34093
$918$	0	0
$919$	−9.46092	−0.312087	−0.156043	−	0.987750i	$-0.549874\pi$
$919$	−9.46092	−0.312087	−0.156043	+	0.987750i	$0.549874\pi$
$920$	0	0
$921$	−1.63893	−0.0540044
$922$	0	0
$923$	45.4814	1.49704
$924$	0	0
$925$	−30.0361	−0.987580
$926$	0	0
$927$	−26.5028	−0.870467
$928$	0	0
$929$	−25.0864	−0.823058	−0.411529	−	0.911397i	$-0.635005\pi$
$929$	−25.0864	−0.823058	−0.411529	+	0.911397i	$0.635005\pi$
$930$	0	0
$931$	8.90157	0.291737
$932$	0	0
$933$	−6.58108	−0.215455
$934$	0	0
$935$	−7.17902	−0.234779
$936$	0	0
$937$	46.4200	1.51648	0.758238	−	0.651978i	$-0.226061\pi$
$937$	46.4200	1.51648	0.758238	+	0.651978i	$0.226061\pi$
$938$	0	0
$939$	10.5109	0.343011
$940$	0	0
$941$	−36.5606	−1.19184	−0.595920	−	0.803044i	$-0.703213\pi$
$941$	−36.5606	−1.19184	−0.595920	+	0.803044i	$0.703213\pi$
$942$	0	0
$943$	−17.1133	−0.557284
$944$	0	0
$945$	32.3775	1.05324
$946$	0	0
$947$	21.6683	0.704127	0.352063	−	0.935976i	$-0.385480\pi$
$947$	21.6683	0.704127	0.352063	+	0.935976i	$0.385480\pi$
$948$	0	0
$949$	−2.39142	−0.0776288
$950$	0	0
$951$	4.02239	0.130435
$952$	0	0
$953$	34.0715	1.10369	0.551843	−	0.833948i	$-0.313925\pi$
$953$	34.0715	1.10369	0.551843	+	0.833948i	$0.313925\pi$
$954$	0	0
$955$	33.9558	1.09878
$956$	0	0
$957$	−2.94815	−0.0953002
$958$	0	0
$959$	−76.6261	−2.47439
$960$	0	0
$961$	−14.6918	−0.473928
$962$	0	0
$963$	−22.4860	−0.724602
$964$	0	0
$965$	−27.7923	−0.894666
$966$	0	0
$967$	−22.7675	−0.732153	−0.366077	−	0.930585i	$-0.619299\pi$
$967$	−22.7675	−0.732153	−0.366077	+	0.930585i	$0.619299\pi$
$968$	0	0
$969$	0.985609	0.0316623
$970$	0	0
$971$	−35.9500	−1.15369	−0.576846	−	0.816853i	$-0.695717\pi$
$971$	−35.9500	−1.15369	−0.576846	+	0.816853i	$0.695717\pi$
$972$	0	0
$973$	−1.66792	−0.0534711
$974$	0	0
$975$	11.8794	0.380445
$976$	0	0
$977$	−9.97123	−0.319008	−0.159504	−	0.987197i	$-0.550989\pi$
$977$	−9.97123	−0.319008	−0.159504	+	0.987197i	$0.550989\pi$
$978$	0	0
$979$	−14.3559	−0.458815
$980$	0	0
$981$	30.3676	0.969563
$982$	0	0
$983$	−8.23876	−0.262776	−0.131388	−	0.991331i	$-0.541943\pi$
$983$	−8.23876	−0.262776	−0.131388	+	0.991331i	$0.541943\pi$
$984$	0	0
$985$	−51.1832	−1.63083
$986$	0	0
$987$	−4.16125	−0.132454
$988$	0	0
$989$	−2.66246	−0.0846611
$990$	0	0
$991$	21.2222	0.674145	0.337072	−	0.941479i	$-0.390563\pi$
$991$	21.2222	0.674145	0.337072	+	0.941479i	$0.390563\pi$
$992$	0	0
$993$	7.16922	0.227508
$994$	0	0
$995$	18.2372	0.578158
$996$	0	0
$997$	−47.7857	−1.51339	−0.756693	−	0.653770i	$-0.773186\pi$
$997$	−47.7857	−1.51339	−0.756693	+	0.653770i	$0.773186\pi$
$998$	0	0
$999$	14.7471	0.466578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.bb.1.6		9
4.3	odd	2		1672.2.a.k.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.k.1.4	✓	9	4.3	odd	2
3344.2.a.bb.1.6		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+0.438092 q^{3} +3.19099 q^{5} -3.98768 q^{7} -2.80808 q^{9} +O(q^{10})\)	\(q+0.438092 q^{3} +3.19099 q^{5} -3.98768 q^{7} -2.80808 q^{9} -1.00000 q^{11} +5.23234 q^{13} +1.39795 q^{15} +2.24978 q^{17} +1.00000 q^{19} -1.74697 q^{21} +2.71057 q^{23} +5.18243 q^{25} -2.54447 q^{27} +6.72953 q^{29} +4.03835 q^{31} -0.438092 q^{33} -12.7246 q^{35} -5.79575 q^{37} +2.29224 q^{39} -6.31351 q^{41} -0.982247 q^{43} -8.96055 q^{45} +2.38198 q^{47} +8.90157 q^{49} +0.985609 q^{51} +0.773120 q^{53} -3.19099 q^{55} +0.438092 q^{57} -2.37823 q^{59} +1.97086 q^{61} +11.1977 q^{63} +16.6963 q^{65} +15.3638 q^{67} +1.18748 q^{69} +8.69237 q^{71} -0.457046 q^{73} +2.27038 q^{75} +3.98768 q^{77} +1.37587 q^{79} +7.30952 q^{81} -2.14599 q^{83} +7.17902 q^{85} +2.94815 q^{87} +14.3559 q^{89} -20.8649 q^{91} +1.76917 q^{93} +3.19099 q^{95} +10.7901 q^{97} +2.80808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * q^99

Introduction

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Varieties

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