LMFDB - Embedded newform 3344.2.a.t.1.3

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.71457$ 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)

$f(q)$	$=$	$q-0.121872 q^{3} -1.06025 q^{5} +3.36889 q^{7} -2.98515 q^{9} -1.00000 q^{11} -2.15993 q^{13} +0.129215 q^{15} +3.67288 q^{17} +1.00000 q^{19} -0.410573 q^{21} -3.15468 q^{23} -3.87587 q^{25} +0.729422 q^{27} +7.17849 q^{29} -4.65295 q^{31} +0.121872 q^{33} -3.57187 q^{35} +2.27446 q^{37} +0.263235 q^{39} -11.3852 q^{41} -9.38838 q^{43} +3.16501 q^{45} +5.77094 q^{47} +4.34940 q^{49} -0.447622 q^{51} -5.65820 q^{53} +1.06025 q^{55} -0.121872 q^{57} +13.7944 q^{59} +6.98152 q^{61} -10.0566 q^{63} +2.29007 q^{65} +4.81332 q^{67} +0.384467 q^{69} -15.2629 q^{71} -8.08806 q^{73} +0.472360 q^{75} -3.36889 q^{77} -13.4291 q^{79} +8.86655 q^{81} -9.96666 q^{83} -3.89418 q^{85} -0.874858 q^{87} -4.61626 q^{89} -7.27655 q^{91} +0.567064 q^{93} -1.06025 q^{95} -4.09907 q^{97} +2.98515 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * 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
$3$	−0.121872	−0.0703629	−0.0351814	−	0.999381i	$-0.511201\pi$
$3$	−0.121872	−0.0703629	−0.0351814	+	0.999381i	$0.511201\pi$
$4$	0	0
$5$	−1.06025	−0.474159	−0.237080	−	0.971490i	$-0.576190\pi$
$5$	−1.06025	−0.474159	−0.237080	+	0.971490i	$0.576190\pi$
$6$	0	0
$7$	3.36889	1.27332	0.636660	−	0.771145i	$-0.280316\pi$
$7$	3.36889	1.27332	0.636660	+	0.771145i	$0.280316\pi$
$8$	0	0
$9$	−2.98515	−0.995049
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.15993	−0.599056	−0.299528	−	0.954087i	$-0.596829\pi$
$13$	−2.15993	−0.599056	−0.299528	+	0.954087i	$0.596829\pi$
$14$	0	0
$15$	0.129215	0.0333632
$16$	0	0
$17$	3.67288	0.890805	0.445402	−	0.895330i	$-0.353061\pi$
$17$	3.67288	0.890805	0.445402	+	0.895330i	$0.353061\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.410573	−0.0895944
$22$	0	0
$23$	−3.15468	−0.657797	−0.328898	−	0.944365i	$-0.606677\pi$
$23$	−3.15468	−0.657797	−0.328898	+	0.944365i	$0.606677\pi$
$24$	0	0
$25$	−3.87587	−0.775173
$26$	0	0
$27$	0.729422	0.140377
$28$	0	0
$29$	7.17849	1.33301	0.666506	−	0.745499i	$-0.267789\pi$
$29$	7.17849	1.33301	0.666506	+	0.745499i	$0.267789\pi$
$30$	0	0
$31$	−4.65295	−0.835694	−0.417847	−	0.908517i	$-0.637215\pi$
$31$	−4.65295	−0.835694	−0.417847	+	0.908517i	$0.637215\pi$
$32$	0	0
$33$	0.121872	0.0212152
$34$	0	0
$35$	−3.57187	−0.603756
$36$	0	0
$37$	2.27446	0.373918	0.186959	−	0.982368i	$-0.440137\pi$
$37$	2.27446	0.373918	0.186959	+	0.982368i	$0.440137\pi$
$38$	0	0
$39$	0.263235	0.0421513
$40$	0	0
$41$	−11.3852	−1.77807	−0.889034	−	0.457841i	$-0.848623\pi$
$41$	−11.3852	−1.77807	−0.889034	+	0.457841i	$0.848623\pi$
$42$	0	0
$43$	−9.38838	−1.43171	−0.715857	−	0.698247i	$-0.753964\pi$
$43$	−9.38838	−1.43171	−0.715857	+	0.698247i	$0.753964\pi$
$44$	0	0
$45$	3.16501	0.471812
$46$	0	0
$47$	5.77094	0.841778	0.420889	−	0.907112i	$-0.361718\pi$
$47$	5.77094	0.841778	0.420889	+	0.907112i	$0.361718\pi$
$48$	0	0
$49$	4.34940	0.621342
$50$	0	0
$51$	−0.447622	−0.0626796
$52$	0	0
$53$	−5.65820	−0.777213	−0.388607	−	0.921404i	$-0.627043\pi$
$53$	−5.65820	−0.777213	−0.388607	+	0.921404i	$0.627043\pi$
$54$	0	0
$55$	1.06025	0.142964
$56$	0	0
$57$	−0.121872	−0.0161423
$58$	0	0
$59$	13.7944	1.79588	0.897939	−	0.440121i	$-0.145064\pi$
$59$	13.7944	1.79588	0.897939	+	0.440121i	$0.145064\pi$
$60$	0	0
$61$	6.98152	0.893892	0.446946	−	0.894561i	$-0.352512\pi$
$61$	6.98152	0.893892	0.446946	+	0.894561i	$0.352512\pi$
$62$	0	0
$63$	−10.0566	−1.26702
$64$	0	0
$65$	2.29007	0.284048
$66$	0	0
$67$	4.81332	0.588041	0.294020	−	0.955799i	$-0.405007\pi$
$67$	4.81332	0.588041	0.294020	+	0.955799i	$0.405007\pi$
$68$	0	0
$69$	0.384467	0.0462844
$70$	0	0
$71$	−15.2629	−1.81137	−0.905685	−	0.423951i	$-0.860643\pi$
$71$	−15.2629	−1.81137	−0.905685	+	0.423951i	$0.860643\pi$
$72$	0	0
$73$	−8.08806	−0.946636	−0.473318	−	0.880892i	$-0.656944\pi$
$73$	−8.08806	−0.946636	−0.473318	+	0.880892i	$0.656944\pi$
$74$	0	0
$75$	0.472360	0.0545434
$76$	0	0
$77$	−3.36889	−0.383920
$78$	0	0
$79$	−13.4291	−1.51090	−0.755448	−	0.655209i	$-0.772581\pi$
$79$	−13.4291	−1.51090	−0.755448	+	0.655209i	$0.772581\pi$
$80$	0	0
$81$	8.86655	0.985172
$82$	0	0
$83$	−9.96666	−1.09398	−0.546992	−	0.837138i	$-0.684227\pi$
$83$	−9.96666	−1.09398	−0.546992	+	0.837138i	$0.684227\pi$
$84$	0	0
$85$	−3.89418	−0.422383
$86$	0	0
$87$	−0.874858	−0.0937946
$88$	0	0
$89$	−4.61626	−0.489323	−0.244661	−	0.969609i	$-0.578677\pi$
$89$	−4.61626	−0.489323	−0.244661	+	0.969609i	$0.578677\pi$
$90$	0	0
$91$	−7.27655	−0.762790
$92$	0	0
$93$	0.567064	0.0588018
$94$	0	0
$95$	−1.06025	−0.108780
$96$	0	0
$97$	−4.09907	−0.416197	−0.208099	−	0.978108i	$-0.566727\pi$
$97$	−4.09907	−0.416197	−0.208099	+	0.978108i	$0.566727\pi$
$98$	0	0
$99$	2.98515	0.300019
$100$	0	0
$101$	−10.8730	−1.08190	−0.540950	−	0.841055i	$-0.681935\pi$
$101$	−10.8730	−1.08190	−0.540950	+	0.841055i	$0.681935\pi$
$102$	0	0
$103$	−8.85864	−0.872867	−0.436434	−	0.899736i	$-0.643759\pi$
$103$	−8.85864	−0.872867	−0.436434	+	0.899736i	$0.643759\pi$
$104$	0	0
$105$	0.435311	0.0424820
$106$	0	0
$107$	18.1669	1.75626	0.878131	−	0.478421i	$-0.158791\pi$
$107$	18.1669	1.75626	0.878131	+	0.478421i	$0.158791\pi$
$108$	0	0
$109$	−0.211303	−0.0202392	−0.0101196	−	0.999949i	$-0.503221\pi$
$109$	−0.211303	−0.0202392	−0.0101196	+	0.999949i	$0.503221\pi$
$110$	0	0
$111$	−0.277193	−0.0263100
$112$	0	0
$113$	−0.198345	−0.0186587	−0.00932935	−	0.999956i	$-0.502970\pi$
$113$	−0.198345	−0.0186587	−0.00932935	+	0.999956i	$0.502970\pi$
$114$	0	0
$115$	3.34476	0.311900
$116$	0	0
$117$	6.44770	0.596090
$118$	0	0
$119$	12.3735	1.13428
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.38754	0.125110
$124$	0	0
$125$	9.41066	0.841715
$126$	0	0
$127$	3.60331	0.319742	0.159871	−	0.987138i	$-0.448892\pi$
$127$	3.60331	0.319742	0.159871	+	0.987138i	$0.448892\pi$
$128$	0	0
$129$	1.14418	0.100739
$130$	0	0
$131$	−16.8741	−1.47430	−0.737150	−	0.675729i	$-0.763829\pi$
$131$	−16.8741	−1.47430	−0.737150	+	0.675729i	$0.763829\pi$
$132$	0	0
$133$	3.36889	0.292119
$134$	0	0
$135$	−0.773371	−0.0665612
$136$	0	0
$137$	−6.10839	−0.521875	−0.260937	−	0.965356i	$-0.584032\pi$
$137$	−6.10839	−0.521875	−0.260937	+	0.965356i	$0.584032\pi$
$138$	0	0
$139$	−11.3347	−0.961398	−0.480699	−	0.876886i	$-0.659617\pi$
$139$	−11.3347	−0.961398	−0.480699	+	0.876886i	$0.659617\pi$
$140$	0	0
$141$	−0.703316	−0.0592299
$142$	0	0
$143$	2.15993	0.180622
$144$	0	0
$145$	−7.61101	−0.632060
$146$	0	0
$147$	−0.530070	−0.0437194
$148$	0	0
$149$	6.75698	0.553554	0.276777	−	0.960934i	$-0.410734\pi$
$149$	6.75698	0.553554	0.276777	+	0.960934i	$0.410734\pi$
$150$	0	0
$151$	−6.46231	−0.525895	−0.262948	−	0.964810i	$-0.584695\pi$
$151$	−6.46231	−0.525895	−0.262948	+	0.964810i	$0.584695\pi$
$152$	0	0
$153$	−10.9641	−0.886395
$154$	0	0
$155$	4.93330	0.396252
$156$	0	0
$157$	0.248382	0.0198230	0.00991152	−	0.999951i	$-0.496845\pi$
$157$	0.248382	0.0198230	0.00991152	+	0.999951i	$0.496845\pi$
$158$	0	0
$159$	0.689576	0.0546869
$160$	0	0
$161$	−10.6278	−0.837585
$162$	0	0
$163$	16.9710	1.32927	0.664637	−	0.747167i	$-0.268586\pi$
$163$	16.9710	1.32927	0.664637	+	0.747167i	$0.268586\pi$
$164$	0	0
$165$	−0.129215	−0.0100594
$166$	0	0
$167$	0.865538	0.0669773	0.0334887	−	0.999439i	$-0.489338\pi$
$167$	0.865538	0.0669773	0.0334887	+	0.999439i	$0.489338\pi$
$168$	0	0
$169$	−8.33471	−0.641132
$170$	0	0
$171$	−2.98515	−0.228280
$172$	0	0
$173$	−4.14483	−0.315125	−0.157563	−	0.987509i	$-0.550364\pi$
$173$	−4.14483	−0.315125	−0.157563	+	0.987509i	$0.550364\pi$
$174$	0	0
$175$	−13.0574	−0.987043
$176$	0	0
$177$	−1.68115	−0.126363
$178$	0	0
$179$	−2.04584	−0.152913	−0.0764567	−	0.997073i	$-0.524361\pi$
$179$	−2.04584	−0.152913	−0.0764567	+	0.997073i	$0.524361\pi$
$180$	0	0
$181$	5.27519	0.392101	0.196051	−	0.980594i	$-0.437188\pi$
$181$	5.27519	0.392101	0.196051	+	0.980594i	$0.437188\pi$
$182$	0	0
$183$	−0.850852	−0.0628968
$184$	0	0
$185$	−2.41150	−0.177297
$186$	0	0
$187$	−3.67288	−0.268588
$188$	0	0
$189$	2.45734	0.178745
$190$	0	0
$191$	16.1899	1.17146	0.585729	−	0.810507i	$-0.300808\pi$
$191$	16.1899	1.17146	0.585729	+	0.810507i	$0.300808\pi$
$192$	0	0
$193$	−20.0620	−1.44409	−0.722045	−	0.691846i	$-0.756798\pi$
$193$	−20.0620	−1.44409	−0.722045	+	0.691846i	$0.756798\pi$
$194$	0	0
$195$	−0.279095	−0.0199864
$196$	0	0
$197$	−23.8398	−1.69851	−0.849257	−	0.527979i	$-0.822950\pi$
$197$	−23.8398	−1.69851	−0.849257	+	0.527979i	$0.822950\pi$
$198$	0	0
$199$	1.90194	0.134825	0.0674125	−	0.997725i	$-0.478526\pi$
$199$	1.90194	0.134825	0.0674125	+	0.997725i	$0.478526\pi$
$200$	0	0
$201$	−0.586609	−0.0413762
$202$	0	0
$203$	24.1835	1.69735
$204$	0	0
$205$	12.0712	0.843087
$206$	0	0
$207$	9.41719	0.654540
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−12.3048	−0.847100	−0.423550	−	0.905873i	$-0.639216\pi$
$211$	−12.3048	−0.847100	−0.423550	+	0.905873i	$0.639216\pi$
$212$	0	0
$213$	1.86012	0.127453
$214$	0	0
$215$	9.95405	0.678860
$216$	0	0
$217$	−15.6753	−1.06411
$218$	0	0
$219$	0.985709	0.0666080
$220$	0	0
$221$	−7.93316	−0.533642
$222$	0	0
$223$	−22.8473	−1.52997	−0.764985	−	0.644048i	$-0.777254\pi$
$223$	−22.8473	−1.52997	−0.764985	+	0.644048i	$0.777254\pi$
$224$	0	0
$225$	11.5700	0.771335
$226$	0	0
$227$	−4.00654	−0.265923	−0.132962	−	0.991121i	$-0.542449\pi$
$227$	−4.00654	−0.265923	−0.132962	+	0.991121i	$0.542449\pi$
$228$	0	0
$229$	−24.6024	−1.62577	−0.812887	−	0.582422i	$-0.802105\pi$
$229$	−24.6024	−1.62577	−0.812887	+	0.582422i	$0.802105\pi$
$230$	0	0
$231$	0.410573	0.0270137
$232$	0	0
$233$	16.2994	1.06781	0.533903	−	0.845545i	$-0.320725\pi$
$233$	16.2994	1.06781	0.533903	+	0.845545i	$0.320725\pi$
$234$	0	0
$235$	−6.11865	−0.399137
$236$	0	0
$237$	1.63664	0.106311
$238$	0	0
$239$	15.4182	0.997324	0.498662	−	0.866797i	$-0.333825\pi$
$239$	15.4182	0.997324	0.498662	+	0.866797i	$0.333825\pi$
$240$	0	0
$241$	24.1441	1.55526	0.777629	−	0.628723i	$-0.216422\pi$
$241$	24.1441	1.55526	0.777629	+	0.628723i	$0.216422\pi$
$242$	0	0
$243$	−3.26885	−0.209697
$244$	0	0
$245$	−4.61146	−0.294615
$246$	0	0
$247$	−2.15993	−0.137433
$248$	0	0
$249$	1.21466	0.0769758
$250$	0	0
$251$	19.9099	1.25670	0.628350	−	0.777931i	$-0.283731\pi$
$251$	19.9099	1.25670	0.628350	+	0.777931i	$0.283731\pi$
$252$	0	0
$253$	3.15468	0.198333
$254$	0	0
$255$	0.474592	0.0297201
$256$	0	0
$257$	−22.3126	−1.39182	−0.695911	−	0.718128i	$-0.744999\pi$
$257$	−22.3126	−1.39182	−0.695911	+	0.718128i	$0.744999\pi$
$258$	0	0
$259$	7.66239	0.476118
$260$	0	0
$261$	−21.4289	−1.32641
$262$	0	0
$263$	−13.7963	−0.850716	−0.425358	−	0.905025i	$-0.639852\pi$
$263$	−13.7963	−0.850716	−0.425358	+	0.905025i	$0.639852\pi$
$264$	0	0
$265$	5.99912	0.368523
$266$	0	0
$267$	0.562593	0.0344301
$268$	0	0
$269$	−11.2028	−0.683048	−0.341524	−	0.939873i	$-0.610943\pi$
$269$	−11.2028	−0.683048	−0.341524	+	0.939873i	$0.610943\pi$
$270$	0	0
$271$	−17.1234	−1.04017	−0.520085	−	0.854115i	$-0.674100\pi$
$271$	−17.1234	−1.04017	−0.520085	+	0.854115i	$0.674100\pi$
$272$	0	0
$273$	0.886808	0.0536721
$274$	0	0
$275$	3.87587	0.233723
$276$	0	0
$277$	−13.7127	−0.823919	−0.411960	−	0.911202i	$-0.635156\pi$
$277$	−13.7127	−0.823919	−0.411960	+	0.911202i	$0.635156\pi$
$278$	0	0
$279$	13.8897	0.831557
$280$	0	0
$281$	25.8599	1.54267	0.771337	−	0.636427i	$-0.219588\pi$
$281$	25.8599	1.54267	0.771337	+	0.636427i	$0.219588\pi$
$282$	0	0
$283$	−28.8711	−1.71621	−0.858103	−	0.513477i	$-0.828357\pi$
$283$	−28.8711	−1.71621	−0.858103	+	0.513477i	$0.828357\pi$
$284$	0	0
$285$	0.129215	0.00765404
$286$	0	0
$287$	−38.3554	−2.26405
$288$	0	0
$289$	−3.50993	−0.206467
$290$	0	0
$291$	0.499562	0.0292848
$292$	0	0
$293$	6.37971	0.372707	0.186353	−	0.982483i	$-0.440333\pi$
$293$	6.37971	0.372707	0.186353	+	0.982483i	$0.440333\pi$
$294$	0	0
$295$	−14.6255	−0.851532
$296$	0	0
$297$	−0.729422	−0.0423254
$298$	0	0
$299$	6.81389	0.394057
$300$	0	0
$301$	−31.6284	−1.82303
$302$	0	0
$303$	1.32511	0.0761256
$304$	0	0
$305$	−7.40217	−0.423847
$306$	0	0
$307$	−10.4299	−0.595264	−0.297632	−	0.954681i	$-0.596197\pi$
$307$	−10.4299	−0.595264	−0.297632	+	0.954681i	$0.596197\pi$
$308$	0	0
$309$	1.07962	0.0614174
$310$	0	0
$311$	8.26224	0.468508	0.234254	−	0.972175i	$-0.424735\pi$
$311$	8.26224	0.468508	0.234254	+	0.972175i	$0.424735\pi$
$312$	0	0
$313$	−24.1167	−1.36315	−0.681577	−	0.731746i	$-0.738706\pi$
$313$	−24.1167	−1.36315	−0.681577	+	0.731746i	$0.738706\pi$
$314$	0	0
$315$	10.6626	0.600767
$316$	0	0
$317$	−11.7330	−0.658989	−0.329495	−	0.944157i	$-0.606878\pi$
$317$	−11.7330	−0.658989	−0.329495	+	0.944157i	$0.606878\pi$
$318$	0	0
$319$	−7.17849	−0.401918
$320$	0	0
$321$	−2.21404	−0.123576
$322$	0	0
$323$	3.67288	0.204365
$324$	0	0
$325$	8.37159	0.464372
$326$	0	0
$327$	0.0257519	0.00142409
$328$	0	0
$329$	19.4416	1.07185
$330$	0	0
$331$	6.95223	0.382129	0.191065	−	0.981577i	$-0.438806\pi$
$331$	6.95223	0.382129	0.191065	+	0.981577i	$0.438806\pi$
$332$	0	0
$333$	−6.78959	−0.372067
$334$	0	0
$335$	−5.10333	−0.278825
$336$	0	0
$337$	29.4416	1.60379	0.801893	−	0.597468i	$-0.203826\pi$
$337$	29.4416	1.60379	0.801893	+	0.597468i	$0.203826\pi$
$338$	0	0
$339$	0.0241727	0.00131288
$340$	0	0
$341$	4.65295	0.251971
$342$	0	0
$343$	−8.92959	−0.482152
$344$	0	0
$345$	−0.407632	−0.0219462
$346$	0	0
$347$	17.2180	0.924311	0.462155	−	0.886799i	$-0.347076\pi$
$347$	17.2180	0.924311	0.462155	+	0.886799i	$0.347076\pi$
$348$	0	0
$349$	4.32405	0.231461	0.115731	−	0.993281i	$-0.463079\pi$
$349$	4.32405	0.231461	0.115731	+	0.993281i	$0.463079\pi$
$350$	0	0
$351$	−1.57550	−0.0840939
$352$	0	0
$353$	23.4421	1.24770	0.623848	−	0.781545i	$-0.285568\pi$
$353$	23.4421	1.24770	0.623848	+	0.781545i	$0.285568\pi$
$354$	0	0
$355$	16.1825	0.858878
$356$	0	0
$357$	−1.50799	−0.0798111
$358$	0	0
$359$	−8.20544	−0.433066	−0.216533	−	0.976275i	$-0.569475\pi$
$359$	−8.20544	−0.433066	−0.216533	+	0.976275i	$0.569475\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.121872	−0.00639662
$364$	0	0
$365$	8.57539	0.448856
$366$	0	0
$367$	−33.1494	−1.73039	−0.865193	−	0.501439i	$-0.832804\pi$
$367$	−33.1494	−1.73039	−0.865193	+	0.501439i	$0.832804\pi$
$368$	0	0
$369$	33.9865	1.76926
$370$	0	0
$371$	−19.0618	−0.989640
$372$	0	0
$373$	−16.5840	−0.858685	−0.429343	−	0.903142i	$-0.641255\pi$
$373$	−16.5840	−0.858685	−0.429343	+	0.903142i	$0.641255\pi$
$374$	0	0
$375$	−1.14690	−0.0592254
$376$	0	0
$377$	−15.5050	−0.798550
$378$	0	0
$379$	−5.79467	−0.297652	−0.148826	−	0.988863i	$-0.547549\pi$
$379$	−5.79467	−0.297652	−0.148826	+	0.988863i	$0.547549\pi$
$380$	0	0
$381$	−0.439143	−0.0224980
$382$	0	0
$383$	−5.89016	−0.300973	−0.150487	−	0.988612i	$-0.548084\pi$
$383$	−5.89016	−0.300973	−0.150487	+	0.988612i	$0.548084\pi$
$384$	0	0
$385$	3.57187	0.182039
$386$	0	0
$387$	28.0257	1.42463
$388$	0	0
$389$	13.8055	0.699964	0.349982	−	0.936756i	$-0.386188\pi$
$389$	13.8055	0.699964	0.349982	+	0.936756i	$0.386188\pi$
$390$	0	0
$391$	−11.5868	−0.585968
$392$	0	0
$393$	2.05649	0.103736
$394$	0	0
$395$	14.2383	0.716405
$396$	0	0
$397$	6.48471	0.325458	0.162729	−	0.986671i	$-0.447970\pi$
$397$	6.48471	0.325458	0.162729	+	0.986671i	$0.447970\pi$
$398$	0	0
$399$	−0.410573	−0.0205544
$400$	0	0
$401$	3.70806	0.185172	0.0925858	−	0.995705i	$-0.470487\pi$
$401$	3.70806	0.185172	0.0925858	+	0.995705i	$0.470487\pi$
$402$	0	0
$403$	10.0500	0.500628
$404$	0	0
$405$	−9.40077	−0.467128
$406$	0	0
$407$	−2.27446	−0.112741
$408$	0	0
$409$	31.0569	1.53567	0.767833	−	0.640650i	$-0.221335\pi$
$409$	31.0569	1.53567	0.767833	+	0.640650i	$0.221335\pi$
$410$	0	0
$411$	0.744442	0.0367206
$412$	0	0
$413$	46.4717	2.28673
$414$	0	0
$415$	10.5672	0.518722
$416$	0	0
$417$	1.38138	0.0676467
$418$	0	0
$419$	11.2871	0.551410	0.275705	−	0.961242i	$-0.411089\pi$
$419$	11.2871	0.551410	0.275705	+	0.961242i	$0.411089\pi$
$420$	0	0
$421$	34.5194	1.68237	0.841186	−	0.540746i	$-0.181858\pi$
$421$	34.5194	1.68237	0.841186	+	0.540746i	$0.181858\pi$
$422$	0	0
$423$	−17.2271	−0.837611
$424$	0	0
$425$	−14.2356	−0.690528
$426$	0	0
$427$	23.5199	1.13821
$428$	0	0
$429$	−0.263235	−0.0127091
$430$	0	0
$431$	−18.1654	−0.874996	−0.437498	−	0.899219i	$-0.644135\pi$
$431$	−18.1654	−0.874996	−0.437498	+	0.899219i	$0.644135\pi$
$432$	0	0
$433$	10.7804	0.518075	0.259037	−	0.965867i	$-0.416595\pi$
$433$	10.7804	0.518075	0.259037	+	0.965867i	$0.416595\pi$
$434$	0	0
$435$	0.927570	0.0444736
$436$	0	0
$437$	−3.15468	−0.150909
$438$	0	0
$439$	1.38742	0.0662180	0.0331090	−	0.999452i	$-0.489459\pi$
$439$	1.38742	0.0662180	0.0331090	+	0.999452i	$0.489459\pi$
$440$	0	0
$441$	−12.9836	−0.618266
$442$	0	0
$443$	7.26792	0.345310	0.172655	−	0.984982i	$-0.444766\pi$
$443$	7.26792	0.345310	0.172655	+	0.984982i	$0.444766\pi$
$444$	0	0
$445$	4.89440	0.232017
$446$	0	0
$447$	−0.823487	−0.0389496
$448$	0	0
$449$	21.4298	1.01134	0.505668	−	0.862728i	$-0.331246\pi$
$449$	21.4298	1.01134	0.505668	+	0.862728i	$0.331246\pi$
$450$	0	0
$451$	11.3852	0.536108
$452$	0	0
$453$	0.787575	0.0370035
$454$	0	0
$455$	7.71498	0.361684
$456$	0	0
$457$	−17.6312	−0.824754	−0.412377	−	0.911013i	$-0.635301\pi$
$457$	−17.6312	−0.824754	−0.412377	+	0.911013i	$0.635301\pi$
$458$	0	0
$459$	2.67908	0.125049
$460$	0	0
$461$	34.2073	1.59320	0.796598	−	0.604510i	$-0.206631\pi$
$461$	34.2073	1.59320	0.796598	+	0.604510i	$0.206631\pi$
$462$	0	0
$463$	11.0009	0.511253	0.255626	−	0.966776i	$-0.417718\pi$
$463$	11.0009	0.511253	0.255626	+	0.966776i	$0.417718\pi$
$464$	0	0
$465$	−0.601231	−0.0278814
$466$	0	0
$467$	−4.38453	−0.202892	−0.101446	−	0.994841i	$-0.532347\pi$
$467$	−4.38453	−0.202892	−0.101446	+	0.994841i	$0.532347\pi$
$468$	0	0
$469$	16.2155	0.748764
$470$	0	0
$471$	−0.0302708	−0.00139481
$472$	0	0
$473$	9.38838	0.431678
$474$	0	0
$475$	−3.87587	−0.177837
$476$	0	0
$477$	16.8905	0.773365
$478$	0	0
$479$	14.3683	0.656505	0.328253	−	0.944590i	$-0.393540\pi$
$479$	14.3683	0.656505	0.328253	+	0.944590i	$0.393540\pi$
$480$	0	0
$481$	−4.91266	−0.223998
$482$	0	0
$483$	1.29523	0.0589349
$484$	0	0
$485$	4.34604	0.197344
$486$	0	0
$487$	−1.01712	−0.0460899	−0.0230449	−	0.999734i	$-0.507336\pi$
$487$	−1.01712	−0.0460899	−0.0230449	+	0.999734i	$0.507336\pi$
$488$	0	0
$489$	−2.06829	−0.0935314
$490$	0	0
$491$	2.41045	0.108782	0.0543910	−	0.998520i	$-0.482678\pi$
$491$	2.41045	0.108782	0.0543910	+	0.998520i	$0.482678\pi$
$492$	0	0
$493$	26.3658	1.18745
$494$	0	0
$495$	−3.16501	−0.142257
$496$	0	0
$497$	−51.4189	−2.30645
$498$	0	0
$499$	−4.07426	−0.182389	−0.0911945	−	0.995833i	$-0.529069\pi$
$499$	−4.07426	−0.182389	−0.0911945	+	0.995833i	$0.529069\pi$
$500$	0	0
$501$	−0.105485	−0.00471272
$502$	0	0
$503$	19.9553	0.889762	0.444881	−	0.895590i	$-0.353246\pi$
$503$	19.9553	0.889762	0.444881	+	0.895590i	$0.353246\pi$
$504$	0	0
$505$	11.5281	0.512993
$506$	0	0
$507$	1.01577	0.0451118
$508$	0	0
$509$	−8.79239	−0.389716	−0.194858	−	0.980831i	$-0.562425\pi$
$509$	−8.79239	−0.389716	−0.194858	+	0.980831i	$0.562425\pi$
$510$	0	0
$511$	−27.2478	−1.20537
$512$	0	0
$513$	0.729422	0.0322048
$514$	0	0
$515$	9.39239	0.413878
$516$	0	0
$517$	−5.77094	−0.253806
$518$	0	0
$519$	0.505138	0.0221731
$520$	0	0
$521$	27.3483	1.19815	0.599075	−	0.800693i	$-0.295535\pi$
$521$	27.3483	1.19815	0.599075	+	0.800693i	$0.295535\pi$
$522$	0	0
$523$	41.7043	1.82360	0.911800	−	0.410634i	$-0.134693\pi$
$523$	41.7043	1.82360	0.911800	+	0.410634i	$0.134693\pi$
$524$	0	0
$525$	1.59133	0.0694512
$526$	0	0
$527$	−17.0897	−0.744441
$528$	0	0
$529$	−13.0480	−0.567304
$530$	0	0
$531$	−41.1783	−1.78699
$532$	0	0
$533$	24.5912	1.06516
$534$	0	0
$535$	−19.2615	−0.832748
$536$	0	0
$537$	0.249331	0.0107594
$538$	0	0
$539$	−4.34940	−0.187342
$540$	0	0
$541$	−20.7584	−0.892472	−0.446236	−	0.894915i	$-0.647236\pi$
$541$	−20.7584	−0.892472	−0.446236	+	0.894915i	$0.647236\pi$
$542$	0	0
$543$	−0.642898	−0.0275894
$544$	0	0
$545$	0.224035	0.00959659
$546$	0	0
$547$	−5.39396	−0.230629	−0.115315	−	0.993329i	$-0.536788\pi$
$547$	−5.39396	−0.230629	−0.115315	+	0.993329i	$0.536788\pi$
$548$	0	0
$549$	−20.8409	−0.889466
$550$	0	0
$551$	7.17849	0.305814
$552$	0	0
$553$	−45.2412	−1.92385
$554$	0	0
$555$	0.293894	0.0124751
$556$	0	0
$557$	−38.0076	−1.61043	−0.805217	−	0.592981i	$-0.797951\pi$
$557$	−38.0076	−1.61043	−0.805217	+	0.592981i	$0.797951\pi$
$558$	0	0
$559$	20.2782	0.857677
$560$	0	0
$561$	0.447622	0.0188986
$562$	0	0
$563$	26.4313	1.11395	0.556973	−	0.830530i	$-0.311963\pi$
$563$	26.4313	1.11395	0.556973	+	0.830530i	$0.311963\pi$
$564$	0	0
$565$	0.210295	0.00884719
$566$	0	0
$567$	29.8704	1.25444
$568$	0	0
$569$	40.6105	1.70248	0.851239	−	0.524778i	$-0.175852\pi$
$569$	40.6105	1.70248	0.851239	+	0.524778i	$0.175852\pi$
$570$	0	0
$571$	−8.44299	−0.353328	−0.176664	−	0.984271i	$-0.556531\pi$
$571$	−8.44299	−0.353328	−0.176664	+	0.984271i	$0.556531\pi$
$572$	0	0
$573$	−1.97309	−0.0824271
$574$	0	0
$575$	12.2271	0.509906
$576$	0	0
$577$	20.9771	0.873288	0.436644	−	0.899634i	$-0.356167\pi$
$577$	20.9771	0.873288	0.436644	+	0.899634i	$0.356167\pi$
$578$	0	0
$579$	2.44499	0.101610
$580$	0	0
$581$	−33.5766	−1.39299
$582$	0	0
$583$	5.65820	0.234339
$584$	0	0
$585$	−6.83619	−0.282642
$586$	0	0
$587$	21.1452	0.872756	0.436378	−	0.899764i	$-0.356261\pi$
$587$	21.1452	0.872756	0.436378	+	0.899764i	$0.356261\pi$
$588$	0	0
$589$	−4.65295	−0.191721
$590$	0	0
$591$	2.90540	0.119512
$592$	0	0
$593$	27.7731	1.14051	0.570253	−	0.821469i	$-0.306845\pi$
$593$	27.7731	1.14051	0.570253	+	0.821469i	$0.306845\pi$
$594$	0	0
$595$	−13.1191	−0.537829
$596$	0	0
$597$	−0.231793	−0.00948667
$598$	0	0
$599$	−23.0129	−0.940281	−0.470140	−	0.882592i	$-0.655797\pi$
$599$	−23.0129	−0.940281	−0.470140	+	0.882592i	$0.655797\pi$
$600$	0	0
$601$	23.3322	0.951738	0.475869	−	0.879516i	$-0.342134\pi$
$601$	23.3322	0.951738	0.475869	+	0.879516i	$0.342134\pi$
$602$	0	0
$603$	−14.3685	−0.585129
$604$	0	0
$605$	−1.06025	−0.0431054
$606$	0	0
$607$	3.22857	0.131044	0.0655218	−	0.997851i	$-0.479129\pi$
$607$	3.22857	0.131044	0.0655218	+	0.997851i	$0.479129\pi$
$608$	0	0
$609$	−2.94730	−0.119430
$610$	0	0
$611$	−12.4648	−0.504273
$612$	0	0
$613$	−17.2202	−0.695519	−0.347759	−	0.937584i	$-0.613057\pi$
$613$	−17.2202	−0.695519	−0.347759	+	0.937584i	$0.613057\pi$
$614$	0	0
$615$	−1.47114	−0.0593220
$616$	0	0
$617$	−27.4776	−1.10621	−0.553103	−	0.833113i	$-0.686557\pi$
$617$	−27.4776	−1.10621	−0.553103	+	0.833113i	$0.686557\pi$
$618$	0	0
$619$	9.53870	0.383393	0.191696	−	0.981454i	$-0.438601\pi$
$619$	9.53870	0.383393	0.191696	+	0.981454i	$0.438601\pi$
$620$	0	0
$621$	−2.30109	−0.0923397
$622$	0	0
$623$	−15.5517	−0.623064
$624$	0	0
$625$	9.40166	0.376066
$626$	0	0
$627$	0.121872	0.00486710
$628$	0	0
$629$	8.35381	0.333088
$630$	0	0
$631$	−1.16647	−0.0464363	−0.0232182	−	0.999730i	$-0.507391\pi$
$631$	−1.16647	−0.0464363	−0.0232182	+	0.999730i	$0.507391\pi$
$632$	0	0
$633$	1.49962	0.0596044
$634$	0	0
$635$	−3.82042	−0.151609
$636$	0	0
$637$	−9.39438	−0.372219
$638$	0	0
$639$	45.5619	1.80240
$640$	0	0
$641$	10.0249	0.395960	0.197980	−	0.980206i	$-0.436562\pi$
$641$	10.0249	0.395960	0.197980	+	0.980206i	$0.436562\pi$
$642$	0	0
$643$	40.7197	1.60583	0.802915	−	0.596094i	$-0.203281\pi$
$643$	40.7197	1.60583	0.802915	+	0.596094i	$0.203281\pi$
$644$	0	0
$645$	−1.21312	−0.0477666
$646$	0	0
$647$	16.6460	0.654423	0.327211	−	0.944951i	$-0.393891\pi$
$647$	16.6460	0.654423	0.327211	+	0.944951i	$0.393891\pi$
$648$	0	0
$649$	−13.7944	−0.541477
$650$	0	0
$651$	1.91038	0.0748735
$652$	0	0
$653$	−34.7416	−1.35954	−0.679771	−	0.733424i	$-0.737921\pi$
$653$	−34.7416	−1.35954	−0.679771	+	0.733424i	$0.737921\pi$
$654$	0	0
$655$	17.8908	0.699053
$656$	0	0
$657$	24.1441	0.941949
$658$	0	0
$659$	22.9537	0.894148	0.447074	−	0.894497i	$-0.352466\pi$
$659$	22.9537	0.894148	0.447074	+	0.894497i	$0.352466\pi$
$660$	0	0
$661$	−22.7141	−0.883476	−0.441738	−	0.897144i	$-0.645638\pi$
$661$	−22.7141	−0.883476	−0.441738	+	0.897144i	$0.645638\pi$
$662$	0	0
$663$	0.966831	0.0375486
$664$	0	0
$665$	−3.57187	−0.138511
$666$	0	0
$667$	−22.6459	−0.876851
$668$	0	0
$669$	2.78445	0.107653
$670$	0	0
$671$	−6.98152	−0.269518
$672$	0	0
$673$	20.9448	0.807364	0.403682	−	0.914899i	$-0.367730\pi$
$673$	20.9448	0.807364	0.403682	+	0.914899i	$0.367730\pi$
$674$	0	0
$675$	−2.82714	−0.108817
$676$	0	0
$677$	11.4153	0.438726	0.219363	−	0.975643i	$-0.429602\pi$
$677$	11.4153	0.438726	0.219363	+	0.975643i	$0.429602\pi$
$678$	0	0
$679$	−13.8093	−0.529952
$680$	0	0
$681$	0.488285	0.0187111
$682$	0	0
$683$	34.9550	1.33752	0.668759	−	0.743479i	$-0.266826\pi$
$683$	34.9550	1.33752	0.668759	+	0.743479i	$0.266826\pi$
$684$	0	0
$685$	6.47643	0.247452
$686$	0	0
$687$	2.99835	0.114394
$688$	0	0
$689$	12.2213	0.465594
$690$	0	0
$691$	15.8717	0.603789	0.301894	−	0.953341i	$-0.402381\pi$
$691$	15.8717	0.603789	0.301894	+	0.953341i	$0.402381\pi$
$692$	0	0
$693$	10.0566	0.382019
$694$	0	0
$695$	12.0177	0.455855
$696$	0	0
$697$	−41.8165	−1.58391
$698$	0	0
$699$	−1.98644	−0.0751339
$700$	0	0
$701$	−18.5830	−0.701869	−0.350935	−	0.936400i	$-0.614136\pi$
$701$	−18.5830	−0.701869	−0.350935	+	0.936400i	$0.614136\pi$
$702$	0	0
$703$	2.27446	0.0857828
$704$	0	0
$705$	0.745693	0.0280844
$706$	0	0
$707$	−36.6298	−1.37760
$708$	0	0
$709$	23.8063	0.894064	0.447032	−	0.894518i	$-0.352481\pi$
$709$	23.8063	0.894064	0.447032	+	0.894518i	$0.352481\pi$
$710$	0	0
$711$	40.0880	1.50342
$712$	0	0
$713$	14.6786	0.549717
$714$	0	0
$715$	−2.29007	−0.0856437
$716$	0	0
$717$	−1.87905	−0.0701745
$718$	0	0
$719$	−37.0144	−1.38040	−0.690202	−	0.723616i	$-0.742478\pi$
$719$	−37.0144	−1.38040	−0.690202	+	0.723616i	$0.742478\pi$
$720$	0	0
$721$	−29.8437	−1.11144
$722$	0	0
$723$	−2.94249	−0.109432
$724$	0	0
$725$	−27.8229	−1.03332
$726$	0	0
$727$	22.0744	0.818694	0.409347	−	0.912379i	$-0.365757\pi$
$727$	22.0744	0.818694	0.409347	+	0.912379i	$0.365757\pi$
$728$	0	0
$729$	−26.2013	−0.970417
$730$	0	0
$731$	−34.4824	−1.27538
$732$	0	0
$733$	−18.7090	−0.691031	−0.345515	−	0.938413i	$-0.612296\pi$
$733$	−18.7090	−0.691031	−0.345515	+	0.938413i	$0.612296\pi$
$734$	0	0
$735$	0.562008	0.0207300
$736$	0	0
$737$	−4.81332	−0.177301
$738$	0	0
$739$	5.91630	0.217635	0.108817	−	0.994062i	$-0.465294\pi$
$739$	5.91630	0.217635	0.108817	+	0.994062i	$0.465294\pi$
$740$	0	0
$741$	0.263235	0.00967017
$742$	0	0
$743$	−14.0422	−0.515159	−0.257579	−	0.966257i	$-0.582925\pi$
$743$	−14.0422	−0.515159	−0.257579	+	0.966257i	$0.582925\pi$
$744$	0	0
$745$	−7.16411	−0.262473
$746$	0	0
$747$	29.7520	1.08857
$748$	0	0
$749$	61.2023	2.23628
$750$	0	0
$751$	−4.91166	−0.179229	−0.0896145	−	0.995977i	$-0.528563\pi$
$751$	−4.91166	−0.179229	−0.0896145	+	0.995977i	$0.528563\pi$
$752$	0	0
$753$	−2.42646	−0.0884250
$754$	0	0
$755$	6.85168	0.249358
$756$	0	0
$757$	49.2829	1.79122	0.895609	−	0.444843i	$-0.146741\pi$
$757$	49.2829	1.79122	0.895609	+	0.444843i	$0.146741\pi$
$758$	0	0
$759$	−0.384467	−0.0139553
$760$	0	0
$761$	−7.25922	−0.263146	−0.131573	−	0.991306i	$-0.542003\pi$
$761$	−7.25922	−0.263146	−0.131573	+	0.991306i	$0.542003\pi$
$762$	0	0
$763$	−0.711856	−0.0257709
$764$	0	0
$765$	11.6247	0.420292
$766$	0	0
$767$	−29.7949	−1.07583
$768$	0	0
$769$	−21.5123	−0.775755	−0.387877	−	0.921711i	$-0.626792\pi$
$769$	−21.5123	−0.775755	−0.387877	+	0.921711i	$0.626792\pi$
$770$	0	0
$771$	2.71928	0.0979325
$772$	0	0
$773$	4.54038	0.163306	0.0816530	−	0.996661i	$-0.473980\pi$
$773$	4.54038	0.163306	0.0816530	+	0.996661i	$0.473980\pi$
$774$	0	0
$775$	18.0342	0.647808
$776$	0	0
$777$	−0.933831	−0.0335010
$778$	0	0
$779$	−11.3852	−0.407917
$780$	0	0
$781$	15.2629	0.546149
$782$	0	0
$783$	5.23615	0.187125
$784$	0	0
$785$	−0.263348	−0.00939928
$786$	0	0
$787$	7.25028	0.258444	0.129222	−	0.991616i	$-0.458752\pi$
$787$	7.25028	0.258444	0.129222	+	0.991616i	$0.458752\pi$
$788$	0	0
$789$	1.68138	0.0598588
$790$	0	0
$791$	−0.668201	−0.0237585
$792$	0	0
$793$	−15.0796	−0.535491
$794$	0	0
$795$	−0.731124	−0.0259303
$796$	0	0
$797$	18.8674	0.668319	0.334159	−	0.942517i	$-0.391548\pi$
$797$	18.8674	0.668319	0.334159	+	0.942517i	$0.391548\pi$
$798$	0	0
$799$	21.1960	0.749860
$800$	0	0
$801$	13.7802	0.486900
$802$	0	0
$803$	8.08806	0.285422
$804$	0	0
$805$	11.2681	0.397149
$806$	0	0
$807$	1.36531	0.0480612
$808$	0	0
$809$	−19.4948	−0.685399	−0.342700	−	0.939445i	$-0.611341\pi$
$809$	−19.4948	−0.685399	−0.342700	+	0.939445i	$0.611341\pi$
$810$	0	0
$811$	−4.28635	−0.150514	−0.0752570	−	0.997164i	$-0.523978\pi$
$811$	−4.28635	−0.150514	−0.0752570	+	0.997164i	$0.523978\pi$
$812$	0	0
$813$	2.08686	0.0731893
$814$	0	0
$815$	−17.9936	−0.630287
$816$	0	0
$817$	−9.38838	−0.328458
$818$	0	0
$819$	21.7216	0.759014
$820$	0	0
$821$	5.37419	0.187561	0.0937803	−	0.995593i	$-0.470105\pi$
$821$	5.37419	0.187561	0.0937803	+	0.995593i	$0.470105\pi$
$822$	0	0
$823$	−26.3196	−0.917445	−0.458723	−	0.888580i	$-0.651693\pi$
$823$	−26.3196	−0.917445	−0.458723	+	0.888580i	$0.651693\pi$
$824$	0	0
$825$	−0.472360	−0.0164455
$826$	0	0
$827$	8.11186	0.282077	0.141039	−	0.990004i	$-0.454956\pi$
$827$	8.11186	0.282077	0.141039	+	0.990004i	$0.454956\pi$
$828$	0	0
$829$	−21.5948	−0.750017	−0.375009	−	0.927021i	$-0.622360\pi$
$829$	−21.5948	−0.750017	−0.375009	+	0.927021i	$0.622360\pi$
$830$	0	0
$831$	1.67120	0.0579733
$832$	0	0
$833$	15.9748	0.553495
$834$	0	0
$835$	−0.917688	−0.0317579
$836$	0	0
$837$	−3.39396	−0.117313
$838$	0	0
$839$	4.35109	0.150216	0.0751082	−	0.997175i	$-0.476070\pi$
$839$	4.35109	0.150216	0.0751082	+	0.997175i	$0.476070\pi$
$840$	0	0
$841$	22.5308	0.776923
$842$	0	0
$843$	−3.15160	−0.108547
$844$	0	0
$845$	8.83689	0.303998
$846$	0	0
$847$	3.36889	0.115756
$848$	0	0
$849$	3.51858	0.120757
$850$	0	0
$851$	−7.17519	−0.245962
$852$	0	0
$853$	19.5022	0.667744	0.333872	−	0.942618i	$-0.391645\pi$
$853$	19.5022	0.667744	0.333872	+	0.942618i	$0.391645\pi$
$854$	0	0
$855$	3.16501	0.108241
$856$	0	0
$857$	13.3462	0.455898	0.227949	−	0.973673i	$-0.426798\pi$
$857$	13.3462	0.455898	0.227949	+	0.973673i	$0.426798\pi$
$858$	0	0
$859$	−21.6016	−0.737036	−0.368518	−	0.929621i	$-0.620135\pi$
$859$	−21.6016	−0.737036	−0.368518	+	0.929621i	$0.620135\pi$
$860$	0	0
$861$	4.67445	0.159305
$862$	0	0
$863$	2.72425	0.0927345	0.0463673	−	0.998924i	$-0.485236\pi$
$863$	2.72425	0.0927345	0.0463673	+	0.998924i	$0.485236\pi$
$864$	0	0
$865$	4.39456	0.149420
$866$	0	0
$867$	0.427763	0.0145276
$868$	0	0
$869$	13.4291	0.455552
$870$	0	0
$871$	−10.3964	−0.352269
$872$	0	0
$873$	12.2363	0.414137
$874$	0	0
$875$	31.7034	1.07177
$876$	0	0
$877$	1.29130	0.0436041	0.0218021	−	0.999762i	$-0.493060\pi$
$877$	1.29130	0.0436041	0.0218021	+	0.999762i	$0.493060\pi$
$878$	0	0
$879$	−0.777508	−0.0262247
$880$	0	0
$881$	24.9873	0.841844	0.420922	−	0.907097i	$-0.361707\pi$
$881$	24.9873	0.841844	0.420922	+	0.907097i	$0.361707\pi$
$882$	0	0
$883$	−15.5046	−0.521772	−0.260886	−	0.965370i	$-0.584015\pi$
$883$	−15.5046	−0.521772	−0.260886	+	0.965370i	$0.584015\pi$
$884$	0	0
$885$	1.78244	0.0599162
$886$	0	0
$887$	9.98029	0.335105	0.167553	−	0.985863i	$-0.446414\pi$
$887$	9.98029	0.335105	0.167553	+	0.985863i	$0.446414\pi$
$888$	0	0
$889$	12.1391	0.407134
$890$	0	0
$891$	−8.86655	−0.297040
$892$	0	0
$893$	5.77094	0.193117
$894$	0	0
$895$	2.16911	0.0725053
$896$	0	0
$897$	−0.830422	−0.0277270
$898$	0	0
$899$	−33.4012	−1.11399
$900$	0	0
$901$	−20.7819	−0.692345
$902$	0	0
$903$	3.85461	0.128274
$904$	0	0
$905$	−5.59303	−0.185919
$906$	0	0
$907$	−11.8887	−0.394758	−0.197379	−	0.980327i	$-0.563243\pi$
$907$	−11.8887	−0.394758	−0.197379	+	0.980327i	$0.563243\pi$
$908$	0	0
$909$	32.4574	1.07654
$910$	0	0
$911$	23.7166	0.785767	0.392883	−	0.919588i	$-0.371478\pi$
$911$	23.7166	0.785767	0.392883	+	0.919588i	$0.371478\pi$
$912$	0	0
$913$	9.96666	0.329848
$914$	0	0
$915$	0.902117	0.0298231
$916$	0	0
$917$	−56.8471	−1.87726
$918$	0	0
$919$	−18.1426	−0.598470	−0.299235	−	0.954179i	$-0.596731\pi$
$919$	−18.1426	−0.598470	−0.299235	+	0.954179i	$0.596731\pi$
$920$	0	0
$921$	1.27111	0.0418845
$922$	0	0
$923$	32.9667	1.08511
$924$	0	0
$925$	−8.81549	−0.289852
$926$	0	0
$927$	26.4443	0.868546
$928$	0	0
$929$	−50.5019	−1.65691	−0.828457	−	0.560053i	$-0.810781\pi$
$929$	−50.5019	−1.65691	−0.828457	+	0.560053i	$0.810781\pi$
$930$	0	0
$931$	4.34940	0.142546
$932$	0	0
$933$	−1.00694	−0.0329656
$934$	0	0
$935$	3.89418	0.127353
$936$	0	0
$937$	−39.6587	−1.29559	−0.647797	−	0.761813i	$-0.724309\pi$
$937$	−39.6587	−1.29559	−0.647797	+	0.761813i	$0.724309\pi$
$938$	0	0
$939$	2.93915	0.0959154
$940$	0	0
$941$	−34.8809	−1.13709	−0.568543	−	0.822653i	$-0.692493\pi$
$941$	−34.8809	−1.13709	−0.568543	+	0.822653i	$0.692493\pi$
$942$	0	0
$943$	35.9166	1.16961
$944$	0	0
$945$	−2.60540	−0.0847537
$946$	0	0
$947$	30.3067	0.984835	0.492417	−	0.870359i	$-0.336113\pi$
$947$	30.3067	0.984835	0.492417	+	0.870359i	$0.336113\pi$
$948$	0	0
$949$	17.4696	0.567088
$950$	0	0
$951$	1.42992	0.0463684
$952$	0	0
$953$	−6.65804	−0.215675	−0.107838	−	0.994169i	$-0.534393\pi$
$953$	−6.65804	−0.215675	−0.107838	+	0.994169i	$0.534393\pi$
$954$	0	0
$955$	−17.1653	−0.555457
$956$	0	0
$957$	0.874858	0.0282801
$958$	0	0
$959$	−20.5785	−0.664513
$960$	0	0
$961$	−9.35006	−0.301615
$962$	0	0
$963$	−54.2309	−1.74757
$964$	0	0
$965$	21.2707	0.684729
$966$	0	0
$967$	20.0020	0.643221	0.321611	−	0.946872i	$-0.395776\pi$
$967$	20.0020	0.643221	0.321611	+	0.946872i	$0.395776\pi$
$968$	0	0
$969$	−0.447622	−0.0143797
$970$	0	0
$971$	0.491941	0.0157871	0.00789357	−	0.999969i	$-0.497487\pi$
$971$	0.491941	0.0157871	0.00789357	+	0.999969i	$0.497487\pi$
$972$	0	0
$973$	−38.1853	−1.22417
$974$	0	0
$975$	−1.02026	−0.0326746
$976$	0	0
$977$	23.6515	0.756679	0.378339	−	0.925667i	$-0.376495\pi$
$977$	23.6515	0.756679	0.378339	+	0.925667i	$0.376495\pi$
$978$	0	0
$979$	4.61626	0.147536
$980$	0	0
$981$	0.630771	0.0201390
$982$	0	0
$983$	−39.5609	−1.26180	−0.630899	−	0.775865i	$-0.717314\pi$
$983$	−39.5609	−1.26180	−0.630899	+	0.775865i	$0.717314\pi$
$984$	0	0
$985$	25.2762	0.805366
$986$	0	0
$987$	−2.36939	−0.0754186
$988$	0	0
$989$	29.6173	0.941777
$990$	0	0
$991$	13.6096	0.432322	0.216161	−	0.976358i	$-0.430646\pi$
$991$	13.6096	0.432322	0.216161	+	0.976358i	$0.430646\pi$
$992$	0	0
$993$	−0.847283	−0.0268877
$994$	0	0
$995$	−2.01654	−0.0639285
$996$	0	0
$997$	41.1821	1.30425	0.652125	−	0.758111i	$-0.273878\pi$
$997$	41.1821	1.30425	0.652125	+	0.758111i	$0.273878\pi$
$998$	0	0
$999$	1.65904	0.0524897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.t.1.3		5
4.3	odd	2		209.2.a.c.1.5	✓	5
12.11	even	2		1881.2.a.k.1.1		5
20.19	odd	2		5225.2.a.h.1.1		5
44.43	even	2		2299.2.a.n.1.1		5
76.75	even	2		3971.2.a.h.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.5	✓	5	4.3	odd	2
1881.2.a.k.1.1		5	12.11	even	2
2299.2.a.n.1.1		5	44.43	even	2
3344.2.a.t.1.3		5	1.1	even	1	trivial
3971.2.a.h.1.1		5	76.75	even	2
5225.2.a.h.1.1		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.121872 q^{3} -1.06025 q^{5} +3.36889 q^{7} -2.98515 q^{9} -1.00000 q^{11} -2.15993 q^{13} +0.129215 q^{15} +3.67288 q^{17} +1.00000 q^{19} -0.410573 q^{21} -3.15468 q^{23} -3.87587 q^{25} +0.729422 q^{27} +7.17849 q^{29} -4.65295 q^{31} +0.121872 q^{33} -3.57187 q^{35} +2.27446 q^{37} +0.263235 q^{39} -11.3852 q^{41} -9.38838 q^{43} +3.16501 q^{45} +5.77094 q^{47} +4.34940 q^{49} -0.447622 q^{51} -5.65820 q^{53} +1.06025 q^{55} -0.121872 q^{57} +13.7944 q^{59} +6.98152 q^{61} -10.0566 q^{63} +2.29007 q^{65} +4.81332 q^{67} +0.384467 q^{69} -15.2629 q^{71} -8.08806 q^{73} +0.472360 q^{75} -3.36889 q^{77} -13.4291 q^{79} +8.86655 q^{81} -9.96666 q^{83} -3.89418 q^{85} -0.874858 q^{87} -4.61626 q^{89} -7.27655 q^{91} +0.567064 q^{93} -1.06025 q^{95} -4.09907 q^{97} +2.98515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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