LMFDB - Embedded newform 3344.2.a.v.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:	$6$
Coefficient field:	6.6.576096652.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 11x^{3} + 16x^{2} - 6x - 4 $ x^6 - 2x^5 - 8x^4 + 11x^3 + 16x^2 - 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.393583$ 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)

$f(q)$	$=$	$q-1.39358 q^{3} -3.68793 q^{5} +0.935893 q^{7} -1.05793 q^{9} -1.00000 q^{11} -0.935893 q^{13} +5.13944 q^{15} +5.08152 q^{17} -1.00000 q^{19} -1.30424 q^{21} -0.699469 q^{23} +8.60085 q^{25} +5.65506 q^{27} +10.3529 q^{29} -1.63536 q^{31} +1.39358 q^{33} -3.45151 q^{35} +3.78098 q^{37} +1.30424 q^{39} +3.35846 q^{41} -5.45297 q^{43} +3.90156 q^{45} +0.658952 q^{47} -6.12410 q^{49} -7.08152 q^{51} -3.54292 q^{53} +3.68793 q^{55} +1.39358 q^{57} -8.23478 q^{59} +2.14708 q^{61} -0.990105 q^{63} +3.45151 q^{65} -11.9457 q^{67} +0.974768 q^{69} +6.67265 q^{71} +5.27906 q^{73} -11.9860 q^{75} -0.935893 q^{77} +2.98471 q^{79} -4.70702 q^{81} -7.41103 q^{83} -18.7403 q^{85} -14.4276 q^{87} +11.8282 q^{89} -0.875895 q^{91} +2.27901 q^{93} +3.68793 q^{95} +2.22644 q^{97} +1.05793 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + q^5 + 4 * q^9 - 6 * q^11 - 7 * q^15 + 3 * q^17 - 6 * q^19 + 7 * q^21 - 7 * q^23 + 3 * q^25 - 13 * q^27 - 4 * q^29 - 7 * q^31 + 4 * q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 7 * q^41 - 21 * q^43 + 11 * q^45 - 16 * q^47 + 2 * q^49 - 15 * q^51 + 17 * q^53 - q^55 + 4 * q^57 - 19 * q^59 - 6 * q^61 - 2 * q^63 + 6 * q^65 - 14 * q^67 + q^69 - q^71 - 5 * q^73 - 18 * q^75 + 2 * q^81 - 11 * q^83 - 26 * q^85 - 14 * q^87 + 12 * q^89 - 44 * q^91 - 6 * q^93 - q^95 + 14 * q^97 - 4 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.39358	−0.804586	−0.402293	−	0.915511i	$-0.631787\pi$
$3$	−1.39358	−0.804586	−0.402293	+	0.915511i	$0.631787\pi$
$4$	0	0
$5$	−3.68793	−1.64929	−0.824647	−	0.565648i	$-0.808626\pi$
$5$	−3.68793	−1.64929	−0.824647	+	0.565648i	$0.808626\pi$
$6$	0	0
$7$	0.935893	0.353734	0.176867	−	0.984235i	$-0.443404\pi$
$7$	0.935893	0.353734	0.176867	+	0.984235i	$0.443404\pi$
$8$	0	0
$9$	−1.05793	−0.352642
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.935893	−0.259570	−0.129785	−	0.991542i	$-0.541429\pi$
$13$	−0.935893	−0.259570	−0.129785	+	0.991542i	$0.541429\pi$
$14$	0	0
$15$	5.13944	1.32700
$16$	0	0
$17$	5.08152	1.23245	0.616224	−	0.787571i	$-0.288661\pi$
$17$	5.08152	1.23245	0.616224	+	0.787571i	$0.288661\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.30424	−0.284609
$22$	0	0
$23$	−0.699469	−0.145849	−0.0729247	−	0.997337i	$-0.523233\pi$
$23$	−0.699469	−0.145849	−0.0729247	+	0.997337i	$0.523233\pi$
$24$	0	0
$25$	8.60085	1.72017
$26$	0	0
$27$	5.65506	1.08832
$28$	0	0
$29$	10.3529	1.92248	0.961241	−	0.275709i	$-0.0889125\pi$
$29$	10.3529	1.92248	0.961241	+	0.275709i	$0.0889125\pi$
$30$	0	0
$31$	−1.63536	−0.293720	−0.146860	−	0.989157i	$-0.546917\pi$
$31$	−1.63536	−0.293720	−0.146860	+	0.989157i	$0.546917\pi$
$32$	0	0
$33$	1.39358	0.242592
$34$	0	0
$35$	−3.45151	−0.583411
$36$	0	0
$37$	3.78098	0.621590	0.310795	−	0.950477i	$-0.399405\pi$
$37$	3.78098	0.621590	0.310795	+	0.950477i	$0.399405\pi$
$38$	0	0
$39$	1.30424	0.208846
$40$	0	0
$41$	3.35846	0.524503	0.262251	−	0.965000i	$-0.415535\pi$
$41$	3.35846	0.524503	0.262251	+	0.965000i	$0.415535\pi$
$42$	0	0
$43$	−5.45297	−0.831570	−0.415785	−	0.909463i	$-0.636493\pi$
$43$	−5.45297	−0.831570	−0.415785	+	0.909463i	$0.636493\pi$
$44$	0	0
$45$	3.90156	0.581610
$46$	0	0
$47$	0.658952	0.0961180	0.0480590	−	0.998844i	$-0.484696\pi$
$47$	0.658952	0.0961180	0.0480590	+	0.998844i	$0.484696\pi$
$48$	0	0
$49$	−6.12410	−0.874872
$50$	0	0
$51$	−7.08152	−0.991611
$52$	0	0
$53$	−3.54292	−0.486657	−0.243329	−	0.969944i	$-0.578239\pi$
$53$	−3.54292	−0.486657	−0.243329	+	0.969944i	$0.578239\pi$
$54$	0	0
$55$	3.68793	0.497281
$56$	0	0
$57$	1.39358	0.184585
$58$	0	0
$59$	−8.23478	−1.07208	−0.536039	−	0.844193i	$-0.680080\pi$
$59$	−8.23478	−1.07208	−0.536039	+	0.844193i	$0.680080\pi$
$60$	0	0
$61$	2.14708	0.274906	0.137453	−	0.990508i	$-0.456108\pi$
$61$	2.14708	0.274906	0.137453	+	0.990508i	$0.456108\pi$
$62$	0	0
$63$	−0.990105	−0.124741
$64$	0	0
$65$	3.45151	0.428107
$66$	0	0
$67$	−11.9457	−1.45939	−0.729697	−	0.683770i	$-0.760339\pi$
$67$	−11.9457	−1.45939	−0.729697	+	0.683770i	$0.760339\pi$
$68$	0	0
$69$	0.974768	0.117348
$70$	0	0
$71$	6.67265	0.791897	0.395949	−	0.918273i	$-0.370416\pi$
$71$	6.67265	0.791897	0.395949	+	0.918273i	$0.370416\pi$
$72$	0	0
$73$	5.27906	0.617868	0.308934	−	0.951084i	$-0.400028\pi$
$73$	5.27906	0.617868	0.308934	+	0.951084i	$0.400028\pi$
$74$	0	0
$75$	−11.9860	−1.38402
$76$	0	0
$77$	−0.935893	−0.106655
$78$	0	0
$79$	2.98471	0.335807	0.167903	−	0.985803i	$-0.446300\pi$
$79$	2.98471	0.335807	0.167903	+	0.985803i	$0.446300\pi$
$80$	0	0
$81$	−4.70702	−0.523002
$82$	0	0
$83$	−7.41103	−0.813466	−0.406733	−	0.913547i	$-0.633332\pi$
$83$	−7.41103	−0.813466	−0.406733	+	0.913547i	$0.633332\pi$
$84$	0	0
$85$	−18.7403	−2.03267
$86$	0	0
$87$	−14.4276	−1.54680
$88$	0	0
$89$	11.8282	1.25378	0.626892	−	0.779107i	$-0.284327\pi$
$89$	11.8282	1.25378	0.626892	+	0.779107i	$0.284327\pi$
$90$	0	0
$91$	−0.875895	−0.0918187
$92$	0	0
$93$	2.27901	0.236323
$94$	0	0
$95$	3.68793	0.378374
$96$	0	0
$97$	2.22644	0.226061	0.113030	−	0.993592i	$-0.463944\pi$
$97$	2.22644	0.226061	0.113030	+	0.993592i	$0.463944\pi$
$98$	0	0
$99$	1.05793	0.106326
$100$	0	0
$101$	−4.93425	−0.490976	−0.245488	−	0.969400i	$-0.578948\pi$
$101$	−4.93425	−0.490976	−0.245488	+	0.969400i	$0.578948\pi$
$102$	0	0
$103$	4.48420	0.441842	0.220921	−	0.975292i	$-0.429094\pi$
$103$	4.48420	0.441842	0.220921	+	0.975292i	$0.429094\pi$
$104$	0	0
$105$	4.80997	0.469405
$106$	0	0
$107$	−3.83294	−0.370545	−0.185272	−	0.982687i	$-0.559317\pi$
$107$	−3.83294	−0.370545	−0.185272	+	0.982687i	$0.559317\pi$
$108$	0	0
$109$	−12.0662	−1.15574	−0.577868	−	0.816131i	$-0.696115\pi$
$109$	−12.0662	−1.15574	−0.577868	+	0.816131i	$0.696115\pi$
$110$	0	0
$111$	−5.26912	−0.500122
$112$	0	0
$113$	0.199224	0.0187415	0.00937073	−	0.999956i	$-0.497017\pi$
$113$	0.199224	0.0187415	0.00937073	+	0.999956i	$0.497017\pi$
$114$	0	0
$115$	2.57959	0.240548
$116$	0	0
$117$	0.990105	0.0915352
$118$	0	0
$119$	4.75575	0.435959
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.68029	−0.422007
$124$	0	0
$125$	−13.2797	−1.18777
$126$	0	0
$127$	5.22878	0.463979	0.231990	−	0.972718i	$-0.425476\pi$
$127$	5.22878	0.463979	0.231990	+	0.972718i	$0.425476\pi$
$128$	0	0
$129$	7.59917	0.669069
$130$	0	0
$131$	−10.7848	−0.942271	−0.471135	−	0.882061i	$-0.656156\pi$
$131$	−10.7848	−0.942271	−0.471135	+	0.882061i	$0.656156\pi$
$132$	0	0
$133$	−0.935893	−0.0811522
$134$	0	0
$135$	−20.8555	−1.79495
$136$	0	0
$137$	1.92971	0.164866	0.0824332	−	0.996597i	$-0.473731\pi$
$137$	1.92971	0.164866	0.0824332	+	0.996597i	$0.473731\pi$
$138$	0	0
$139$	12.3719	1.04937	0.524687	−	0.851295i	$-0.324182\pi$
$139$	12.3719	1.04937	0.524687	+	0.851295i	$0.324182\pi$
$140$	0	0
$141$	−0.918305	−0.0773352
$142$	0	0
$143$	0.935893	0.0782633
$144$	0	0
$145$	−38.1807	−3.17074
$146$	0	0
$147$	8.53445	0.703910
$148$	0	0
$149$	0.770299	0.0631054	0.0315527	−	0.999502i	$-0.489955\pi$
$149$	0.770299	0.0631054	0.0315527	+	0.999502i	$0.489955\pi$
$150$	0	0
$151$	−13.4413	−1.09383	−0.546917	−	0.837187i	$-0.684199\pi$
$151$	−13.4413	−1.09383	−0.546917	+	0.837187i	$0.684199\pi$
$152$	0	0
$153$	−5.37586	−0.434613
$154$	0	0
$155$	6.03110	0.484430
$156$	0	0
$157$	−14.4123	−1.15022	−0.575112	−	0.818074i	$-0.695042\pi$
$157$	−14.4123	−1.15022	−0.575112	+	0.818074i	$0.695042\pi$
$158$	0	0
$159$	4.93735	0.391558
$160$	0	0
$161$	−0.654628	−0.0515919
$162$	0	0
$163$	−19.3130	−1.51271	−0.756357	−	0.654159i	$-0.773023\pi$
$163$	−19.3130	−1.51271	−0.756357	+	0.654159i	$0.773023\pi$
$164$	0	0
$165$	−5.13944	−0.400105
$166$	0	0
$167$	9.63137	0.745298	0.372649	−	0.927972i	$-0.378450\pi$
$167$	9.63137	0.745298	0.372649	+	0.927972i	$0.378450\pi$
$168$	0	0
$169$	−12.1241	−0.932623
$170$	0	0
$171$	1.05793	0.0809016
$172$	0	0
$173$	0.677247	0.0514901	0.0257451	−	0.999669i	$-0.491804\pi$
$173$	0.677247	0.0514901	0.0257451	+	0.999669i	$0.491804\pi$
$174$	0	0
$175$	8.04947	0.608483
$176$	0	0
$177$	11.4759	0.862578
$178$	0	0
$179$	−5.72548	−0.427942	−0.213971	−	0.976840i	$-0.568640\pi$
$179$	−5.72548	−0.427942	−0.213971	+	0.976840i	$0.568640\pi$
$180$	0	0
$181$	−9.35081	−0.695041	−0.347520	−	0.937672i	$-0.612976\pi$
$181$	−9.35081	−0.695041	−0.347520	+	0.937672i	$0.612976\pi$
$182$	0	0
$183$	−2.99214	−0.221185
$184$	0	0
$185$	−13.9440	−1.02518
$186$	0	0
$187$	−5.08152	−0.371597
$188$	0	0
$189$	5.29253	0.384975
$190$	0	0
$191$	−17.4681	−1.26394	−0.631972	−	0.774991i	$-0.717754\pi$
$191$	−17.4681	−1.26394	−0.631972	+	0.774991i	$0.717754\pi$
$192$	0	0
$193$	−16.7647	−1.20675	−0.603374	−	0.797459i	$-0.706177\pi$
$193$	−16.7647	−1.20675	−0.603374	+	0.797459i	$0.706177\pi$
$194$	0	0
$195$	−4.80997	−0.344449
$196$	0	0
$197$	17.0575	1.21530	0.607649	−	0.794205i	$-0.292113\pi$
$197$	17.0575	1.21530	0.607649	+	0.794205i	$0.292113\pi$
$198$	0	0
$199$	20.4336	1.44850	0.724249	−	0.689539i	$-0.242187\pi$
$199$	20.4336	1.44850	0.724249	+	0.689539i	$0.242187\pi$
$200$	0	0
$201$	16.6473	1.17421
$202$	0	0
$203$	9.68919	0.680048
$204$	0	0
$205$	−12.3858	−0.865059
$206$	0	0
$207$	0.739986	0.0514326
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−7.80329	−0.537201	−0.268600	−	0.963252i	$-0.586561\pi$
$211$	−7.80329	−0.537201	−0.268600	+	0.963252i	$0.586561\pi$
$212$	0	0
$213$	−9.29889	−0.637149
$214$	0	0
$215$	20.1102	1.37150
$216$	0	0
$217$	−1.53052	−0.103899
$218$	0	0
$219$	−7.35682	−0.497127
$220$	0	0
$221$	−4.75575	−0.319907
$222$	0	0
$223$	−0.928430	−0.0621723	−0.0310861	−	0.999517i	$-0.509897\pi$
$223$	−0.928430	−0.0621723	−0.0310861	+	0.999517i	$0.509897\pi$
$224$	0	0
$225$	−9.09905	−0.606604
$226$	0	0
$227$	−9.79093	−0.649847	−0.324923	−	0.945740i	$-0.605339\pi$
$227$	−9.79093	−0.649847	−0.324923	+	0.945740i	$0.605339\pi$
$228$	0	0
$229$	14.6197	0.966098	0.483049	−	0.875593i	$-0.339529\pi$
$229$	14.6197	0.966098	0.483049	+	0.875593i	$0.339529\pi$
$230$	0	0
$231$	1.30424	0.0858130
$232$	0	0
$233$	−8.36058	−0.547720	−0.273860	−	0.961770i	$-0.588300\pi$
$233$	−8.36058	−0.547720	−0.273860	+	0.961770i	$0.588300\pi$
$234$	0	0
$235$	−2.43017	−0.158527
$236$	0	0
$237$	−4.15945	−0.270185
$238$	0	0
$239$	−22.7011	−1.46841	−0.734206	−	0.678927i	$-0.762445\pi$
$239$	−22.7011	−1.46841	−0.734206	+	0.678927i	$0.762445\pi$
$240$	0	0
$241$	−16.4595	−1.06025	−0.530124	−	0.847920i	$-0.677855\pi$
$241$	−16.4595	−1.06025	−0.530124	+	0.847920i	$0.677855\pi$
$242$	0	0
$243$	−10.4056	−0.667516
$244$	0	0
$245$	22.5853	1.44292
$246$	0	0
$247$	0.935893	0.0595494
$248$	0	0
$249$	10.3279	0.654503
$250$	0	0
$251$	14.9560	0.944016	0.472008	−	0.881594i	$-0.343529\pi$
$251$	14.9560	0.944016	0.472008	+	0.881594i	$0.343529\pi$
$252$	0	0
$253$	0.699469	0.0439752
$254$	0	0
$255$	26.1162	1.63546
$256$	0	0
$257$	−17.8714	−1.11479	−0.557394	−	0.830248i	$-0.688199\pi$
$257$	−17.8714	−1.11479	−0.557394	+	0.830248i	$0.688199\pi$
$258$	0	0
$259$	3.53860	0.219878
$260$	0	0
$261$	−10.9526	−0.677948
$262$	0	0
$263$	−9.61150	−0.592670	−0.296335	−	0.955084i	$-0.595765\pi$
$263$	−9.61150	−0.592670	−0.296335	+	0.955084i	$0.595765\pi$
$264$	0	0
$265$	13.0660	0.802641
$266$	0	0
$267$	−16.4835	−1.00878
$268$	0	0
$269$	24.1855	1.47462	0.737308	−	0.675557i	$-0.236097\pi$
$269$	24.1855	1.47462	0.737308	+	0.675557i	$0.236097\pi$
$270$	0	0
$271$	−16.6190	−1.00953	−0.504765	−	0.863257i	$-0.668421\pi$
$271$	−16.6190	−1.00953	−0.504765	+	0.863257i	$0.668421\pi$
$272$	0	0
$273$	1.22063	0.0738761
$274$	0	0
$275$	−8.60085	−0.518650
$276$	0	0
$277$	−8.59648	−0.516513	−0.258256	−	0.966076i	$-0.583148\pi$
$277$	−8.59648	−0.516513	−0.258256	+	0.966076i	$0.583148\pi$
$278$	0	0
$279$	1.73009	0.103578
$280$	0	0
$281$	20.8058	1.24117	0.620584	−	0.784140i	$-0.286896\pi$
$281$	20.8058	1.24117	0.620584	+	0.784140i	$0.286896\pi$
$282$	0	0
$283$	−24.8703	−1.47838	−0.739192	−	0.673495i	$-0.764792\pi$
$283$	−24.8703	−1.47838	−0.739192	+	0.673495i	$0.764792\pi$
$284$	0	0
$285$	−5.13944	−0.304434
$286$	0	0
$287$	3.14315	0.185535
$288$	0	0
$289$	8.82180	0.518930
$290$	0	0
$291$	−3.10273	−0.181885
$292$	0	0
$293$	−1.88487	−0.110115	−0.0550576	−	0.998483i	$-0.517534\pi$
$293$	−1.88487	−0.110115	−0.0550576	+	0.998483i	$0.517534\pi$
$294$	0	0
$295$	30.3693	1.76817
$296$	0	0
$297$	−5.65506	−0.328140
$298$	0	0
$299$	0.654628	0.0378581
$300$	0	0
$301$	−5.10339	−0.294155
$302$	0	0
$303$	6.87629	0.395033
$304$	0	0
$305$	−7.91830	−0.453401
$306$	0	0
$307$	24.7291	1.41137	0.705683	−	0.708527i	$-0.250640\pi$
$307$	24.7291	1.41137	0.705683	+	0.708527i	$0.250640\pi$
$308$	0	0
$309$	−6.24911	−0.355499
$310$	0	0
$311$	−26.9935	−1.53066	−0.765332	−	0.643636i	$-0.777425\pi$
$311$	−26.9935	−1.53066	−0.765332	+	0.643636i	$0.777425\pi$
$312$	0	0
$313$	1.18039	0.0667195	0.0333598	−	0.999443i	$-0.489379\pi$
$313$	1.18039	0.0667195	0.0333598	+	0.999443i	$0.489379\pi$
$314$	0	0
$315$	3.65144	0.205735
$316$	0	0
$317$	33.7737	1.89692	0.948459	−	0.316899i	$-0.102642\pi$
$317$	33.7737	1.89692	0.948459	+	0.316899i	$0.102642\pi$
$318$	0	0
$319$	−10.3529	−0.579650
$320$	0	0
$321$	5.34153	0.298135
$322$	0	0
$323$	−5.08152	−0.282743
$324$	0	0
$325$	−8.04947	−0.446504
$326$	0	0
$327$	16.8153	0.929888
$328$	0	0
$329$	0.616708	0.0340002
$330$	0	0
$331$	−14.8244	−0.814824	−0.407412	−	0.913245i	$-0.633569\pi$
$331$	−14.8244	−0.814824	−0.407412	+	0.913245i	$0.633569\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	44.0548	2.40697
$336$	0	0
$337$	−23.7922	−1.29604	−0.648022	−	0.761622i	$-0.724404\pi$
$337$	−23.7922	−1.29604	−0.648022	+	0.761622i	$0.724404\pi$
$338$	0	0
$339$	−0.277636	−0.0150791
$340$	0	0
$341$	1.63536	0.0885598
$342$	0	0
$343$	−12.2828	−0.663206
$344$	0	0
$345$	−3.59488	−0.193542
$346$	0	0
$347$	−19.4448	−1.04385	−0.521927	−	0.852990i	$-0.674787\pi$
$347$	−19.4448	−1.04385	−0.521927	+	0.852990i	$0.674787\pi$
$348$	0	0
$349$	−36.6940	−1.96418	−0.982092	−	0.188404i	$-0.939668\pi$
$349$	−36.6940	−1.96418	−0.982092	+	0.188404i	$0.939668\pi$
$350$	0	0
$351$	−5.29253	−0.282494
$352$	0	0
$353$	23.4755	1.24947	0.624737	−	0.780835i	$-0.285206\pi$
$353$	23.4755	1.24947	0.624737	+	0.780835i	$0.285206\pi$
$354$	0	0
$355$	−24.6083	−1.30607
$356$	0	0
$357$	−6.62754	−0.350767
$358$	0	0
$359$	18.9503	1.00016	0.500078	−	0.865980i	$-0.333305\pi$
$359$	18.9503	1.00016	0.500078	+	0.865980i	$0.333305\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.39358	−0.0731442
$364$	0	0
$365$	−19.4688	−1.01905
$366$	0	0
$367$	−12.5792	−0.656631	−0.328315	−	0.944568i	$-0.606481\pi$
$367$	−12.5792	−0.656631	−0.328315	+	0.944568i	$0.606481\pi$
$368$	0	0
$369$	−3.55300	−0.184962
$370$	0	0
$371$	−3.31579	−0.172147
$372$	0	0
$373$	−27.4973	−1.42376	−0.711879	−	0.702302i	$-0.752156\pi$
$373$	−27.4973	−1.42376	−0.711879	+	0.702302i	$0.752156\pi$
$374$	0	0
$375$	18.5063	0.955663
$376$	0	0
$377$	−9.68919	−0.499019
$378$	0	0
$379$	28.7184	1.47517	0.737583	−	0.675257i	$-0.235967\pi$
$379$	28.7184	1.47517	0.737583	+	0.675257i	$0.235967\pi$
$380$	0	0
$381$	−7.28674	−0.373311
$382$	0	0
$383$	26.3536	1.34661	0.673303	−	0.739367i	$-0.264875\pi$
$383$	26.3536	1.34661	0.673303	+	0.739367i	$0.264875\pi$
$384$	0	0
$385$	3.45151	0.175905
$386$	0	0
$387$	5.76884	0.293246
$388$	0	0
$389$	−2.13833	−0.108417	−0.0542087	−	0.998530i	$-0.517264\pi$
$389$	−2.13833	−0.108417	−0.0542087	+	0.998530i	$0.517264\pi$
$390$	0	0
$391$	−3.55436	−0.179752
$392$	0	0
$393$	15.0295	0.758138
$394$	0	0
$395$	−11.0074	−0.553844
$396$	0	0
$397$	32.9260	1.65251	0.826253	−	0.563299i	$-0.190468\pi$
$397$	32.9260	1.65251	0.826253	+	0.563299i	$0.190468\pi$
$398$	0	0
$399$	1.30424	0.0652939
$400$	0	0
$401$	21.9766	1.09746	0.548729	−	0.836001i	$-0.315112\pi$
$401$	21.9766	1.09746	0.548729	+	0.836001i	$0.315112\pi$
$402$	0	0
$403$	1.53052	0.0762408
$404$	0	0
$405$	17.3592	0.862584
$406$	0	0
$407$	−3.78098	−0.187416
$408$	0	0
$409$	−19.4032	−0.959425	−0.479712	−	0.877426i	$-0.659259\pi$
$409$	−19.4032	−0.959425	−0.479712	+	0.877426i	$0.659259\pi$
$410$	0	0
$411$	−2.68921	−0.132649
$412$	0	0
$413$	−7.70687	−0.379230
$414$	0	0
$415$	27.3314	1.34164
$416$	0	0
$417$	−17.2413	−0.844311
$418$	0	0
$419$	9.54182	0.466148	0.233074	−	0.972459i	$-0.425121\pi$
$419$	9.54182	0.466148	0.233074	+	0.972459i	$0.425121\pi$
$420$	0	0
$421$	34.9016	1.70100	0.850499	−	0.525977i	$-0.176300\pi$
$421$	34.9016	1.70100	0.850499	+	0.525977i	$0.176300\pi$
$422$	0	0
$423$	−0.697122	−0.0338952
$424$	0	0
$425$	43.7053	2.12002
$426$	0	0
$427$	2.00944	0.0972437
$428$	0	0
$429$	−1.30424	−0.0629695
$430$	0	0
$431$	14.5320	0.699983	0.349992	−	0.936753i	$-0.386184\pi$
$431$	14.5320	0.699983	0.349992	+	0.936753i	$0.386184\pi$
$432$	0	0
$433$	−24.3163	−1.16857	−0.584283	−	0.811550i	$-0.698624\pi$
$433$	−24.3163	−1.16857	−0.584283	+	0.811550i	$0.698624\pi$
$434$	0	0
$435$	53.2080	2.55113
$436$	0	0
$437$	0.699469	0.0334601
$438$	0	0
$439$	−12.2402	−0.584194	−0.292097	−	0.956389i	$-0.594353\pi$
$439$	−12.2402	−0.584194	−0.292097	+	0.956389i	$0.594353\pi$
$440$	0	0
$441$	6.47885	0.308516
$442$	0	0
$443$	3.50983	0.166757	0.0833785	−	0.996518i	$-0.473429\pi$
$443$	3.50983	0.166757	0.0833785	+	0.996518i	$0.473429\pi$
$444$	0	0
$445$	−43.6215	−2.06786
$446$	0	0
$447$	−1.07348	−0.0507737
$448$	0	0
$449$	−19.1263	−0.902625	−0.451312	−	0.892366i	$-0.649044\pi$
$449$	−19.1263	−0.902625	−0.451312	+	0.892366i	$0.649044\pi$
$450$	0	0
$451$	−3.35846	−0.158144
$452$	0	0
$453$	18.7315	0.880083
$454$	0	0
$455$	3.23024	0.151436
$456$	0	0
$457$	35.4333	1.65750	0.828750	−	0.559619i	$-0.189052\pi$
$457$	35.4333	1.65750	0.828750	+	0.559619i	$0.189052\pi$
$458$	0	0
$459$	28.7363	1.34129
$460$	0	0
$461$	18.8922	0.879899	0.439950	−	0.898022i	$-0.354996\pi$
$461$	18.8922	0.879899	0.439950	+	0.898022i	$0.354996\pi$
$462$	0	0
$463$	22.6999	1.05496	0.527478	−	0.849569i	$-0.323138\pi$
$463$	22.6999	1.05496	0.527478	+	0.849569i	$0.323138\pi$
$464$	0	0
$465$	−8.40485	−0.389765
$466$	0	0
$467$	0.971666	0.0449633	0.0224817	−	0.999747i	$-0.492843\pi$
$467$	0.971666	0.0449633	0.0224817	+	0.999747i	$0.492843\pi$
$468$	0	0
$469$	−11.1799	−0.516238
$470$	0	0
$471$	20.0847	0.925454
$472$	0	0
$473$	5.45297	0.250728
$474$	0	0
$475$	−8.60085	−0.394634
$476$	0	0
$477$	3.74815	0.171616
$478$	0	0
$479$	23.9664	1.09505	0.547526	−	0.836789i	$-0.315570\pi$
$479$	23.9664	1.09505	0.547526	+	0.836789i	$0.315570\pi$
$480$	0	0
$481$	−3.53860	−0.161346
$482$	0	0
$483$	0.912279	0.0415101
$484$	0	0
$485$	−8.21097	−0.372841
$486$	0	0
$487$	−16.5519	−0.750036	−0.375018	−	0.927017i	$-0.622364\pi$
$487$	−16.5519	−0.750036	−0.375018	+	0.927017i	$0.622364\pi$
$488$	0	0
$489$	26.9143	1.21711
$490$	0	0
$491$	8.24711	0.372187	0.186093	−	0.982532i	$-0.440417\pi$
$491$	8.24711	0.372187	0.186093	+	0.982532i	$0.440417\pi$
$492$	0	0
$493$	52.6083	2.36936
$494$	0	0
$495$	−3.90156	−0.175362
$496$	0	0
$497$	6.24488	0.280121
$498$	0	0
$499$	−1.67368	−0.0749242	−0.0374621	−	0.999298i	$-0.511927\pi$
$499$	−1.67368	−0.0749242	−0.0374621	+	0.999298i	$0.511927\pi$
$500$	0	0
$501$	−13.4221	−0.599656
$502$	0	0
$503$	38.4859	1.71600	0.858001	−	0.513648i	$-0.171706\pi$
$503$	38.4859	1.71600	0.858001	+	0.513648i	$0.171706\pi$
$504$	0	0
$505$	18.1972	0.809764
$506$	0	0
$507$	16.8960	0.750376
$508$	0	0
$509$	−26.1695	−1.15994	−0.579972	−	0.814637i	$-0.696936\pi$
$509$	−26.1695	−1.15994	−0.579972	+	0.814637i	$0.696936\pi$
$510$	0	0
$511$	4.94064	0.218561
$512$	0	0
$513$	−5.65506	−0.249677
$514$	0	0
$515$	−16.5374	−0.728727
$516$	0	0
$517$	−0.658952	−0.0289807
$518$	0	0
$519$	−0.943800	−0.0414282
$520$	0	0
$521$	−17.1490	−0.751311	−0.375656	−	0.926759i	$-0.622582\pi$
$521$	−17.1490	−0.751311	−0.375656	+	0.926759i	$0.622582\pi$
$522$	0	0
$523$	7.71709	0.337445	0.168722	−	0.985664i	$-0.446036\pi$
$523$	7.71709	0.337445	0.168722	+	0.985664i	$0.446036\pi$
$524$	0	0
$525$	−11.2176	−0.489576
$526$	0	0
$527$	−8.31012	−0.361994
$528$	0	0
$529$	−22.5107	−0.978728
$530$	0	0
$531$	8.71179	0.378059
$532$	0	0
$533$	−3.14315	−0.136145
$534$	0	0
$535$	14.1356	0.611137
$536$	0	0
$537$	7.97893	0.344316
$538$	0	0
$539$	6.12410	0.263784
$540$	0	0
$541$	−6.84605	−0.294335	−0.147167	−	0.989112i	$-0.547016\pi$
$541$	−6.84605	−0.294335	−0.147167	+	0.989112i	$0.547016\pi$
$542$	0	0
$543$	13.0311	0.559220
$544$	0	0
$545$	44.4994	1.90615
$546$	0	0
$547$	23.9144	1.02251	0.511253	−	0.859430i	$-0.329182\pi$
$547$	23.9144	1.02251	0.511253	+	0.859430i	$0.329182\pi$
$548$	0	0
$549$	−2.27146	−0.0969434
$550$	0	0
$551$	−10.3529	−0.441048
$552$	0	0
$553$	2.79337	0.118786
$554$	0	0
$555$	19.4321	0.824849
$556$	0	0
$557$	−37.5532	−1.59118	−0.795590	−	0.605835i	$-0.792839\pi$
$557$	−37.5532	−1.59118	−0.795590	+	0.605835i	$0.792839\pi$
$558$	0	0
$559$	5.10339	0.215851
$560$	0	0
$561$	7.08152	0.298982
$562$	0	0
$563$	17.5457	0.739461	0.369731	−	0.929139i	$-0.379450\pi$
$563$	17.5457	0.739461	0.369731	+	0.929139i	$0.379450\pi$
$564$	0	0
$565$	−0.734726	−0.0309102
$566$	0	0
$567$	−4.40526	−0.185004
$568$	0	0
$569$	−43.3795	−1.81856	−0.909282	−	0.416181i	$-0.863368\pi$
$569$	−43.3795	−1.81856	−0.909282	+	0.416181i	$0.863368\pi$
$570$	0	0
$571$	−24.9640	−1.04471	−0.522356	−	0.852728i	$-0.674947\pi$
$571$	−24.9640	−1.04471	−0.522356	+	0.852728i	$0.674947\pi$
$572$	0	0
$573$	24.3432	1.01695
$574$	0	0
$575$	−6.01603	−0.250886
$576$	0	0
$577$	−3.88246	−0.161629	−0.0808144	−	0.996729i	$-0.525752\pi$
$577$	−3.88246	−0.161629	−0.0808144	+	0.996729i	$0.525752\pi$
$578$	0	0
$579$	23.3630	0.970932
$580$	0	0
$581$	−6.93593	−0.287751
$582$	0	0
$583$	3.54292	0.146733
$584$	0	0
$585$	−3.65144	−0.150968
$586$	0	0
$587$	3.84313	0.158623	0.0793114	−	0.996850i	$-0.474728\pi$
$587$	3.84313	0.158623	0.0793114	+	0.996850i	$0.474728\pi$
$588$	0	0
$589$	1.63536	0.0673839
$590$	0	0
$591$	−23.7711	−0.977812
$592$	0	0
$593$	18.1126	0.743794	0.371897	−	0.928274i	$-0.378707\pi$
$593$	18.1126	0.743794	0.371897	+	0.928274i	$0.378707\pi$
$594$	0	0
$595$	−17.5389	−0.719025
$596$	0	0
$597$	−28.4759	−1.16544
$598$	0	0
$599$	−28.6039	−1.16873	−0.584363	−	0.811493i	$-0.698655\pi$
$599$	−28.6039	−1.16873	−0.584363	+	0.811493i	$0.698655\pi$
$600$	0	0
$601$	−34.6660	−1.41405	−0.707027	−	0.707187i	$-0.749964\pi$
$601$	−34.6660	−1.41405	−0.707027	+	0.707187i	$0.749964\pi$
$602$	0	0
$603$	12.6376	0.514643
$604$	0	0
$605$	−3.68793	−0.149936
$606$	0	0
$607$	−36.4105	−1.47786	−0.738928	−	0.673784i	$-0.764668\pi$
$607$	−36.4105	−1.47786	−0.738928	+	0.673784i	$0.764668\pi$
$608$	0	0
$609$	−13.5027	−0.547157
$610$	0	0
$611$	−0.616708	−0.0249493
$612$	0	0
$613$	−38.6118	−1.55952	−0.779758	−	0.626081i	$-0.784658\pi$
$613$	−38.6118	−1.55952	−0.779758	+	0.626081i	$0.784658\pi$
$614$	0	0
$615$	17.2606	0.696014
$616$	0	0
$617$	−12.5159	−0.503871	−0.251935	−	0.967744i	$-0.581067\pi$
$617$	−12.5159	−0.503871	−0.251935	+	0.967744i	$0.581067\pi$
$618$	0	0
$619$	5.87425	0.236106	0.118053	−	0.993007i	$-0.462335\pi$
$619$	5.87425	0.236106	0.118053	+	0.993007i	$0.462335\pi$
$620$	0	0
$621$	−3.95554	−0.158730
$622$	0	0
$623$	11.0699	0.443506
$624$	0	0
$625$	5.97031	0.238813
$626$	0	0
$627$	−1.39358	−0.0556544
$628$	0	0
$629$	19.2131	0.766078
$630$	0	0
$631$	−38.9417	−1.55024	−0.775122	−	0.631811i	$-0.782312\pi$
$631$	−38.9417	−1.55024	−0.775122	+	0.631811i	$0.782312\pi$
$632$	0	0
$633$	10.8745	0.432224
$634$	0	0
$635$	−19.2834	−0.765238
$636$	0	0
$637$	5.73150	0.227090
$638$	0	0
$639$	−7.05916	−0.279256
$640$	0	0
$641$	−26.8864	−1.06195	−0.530975	−	0.847388i	$-0.678174\pi$
$641$	−26.8864	−1.06195	−0.530975	+	0.847388i	$0.678174\pi$
$642$	0	0
$643$	−47.5320	−1.87448	−0.937239	−	0.348687i	$-0.886628\pi$
$643$	−47.5320	−1.87448	−0.937239	+	0.348687i	$0.886628\pi$
$644$	0	0
$645$	−28.0252	−1.10349
$646$	0	0
$647$	−27.7865	−1.09240	−0.546201	−	0.837654i	$-0.683926\pi$
$647$	−27.7865	−1.09240	−0.546201	+	0.837654i	$0.683926\pi$
$648$	0	0
$649$	8.23478	0.323243
$650$	0	0
$651$	2.13291	0.0835954
$652$	0	0
$653$	31.1045	1.21721	0.608606	−	0.793472i	$-0.291729\pi$
$653$	31.1045	1.21721	0.608606	+	0.793472i	$0.291729\pi$
$654$	0	0
$655$	39.7736	1.55408
$656$	0	0
$657$	−5.58486	−0.217886
$658$	0	0
$659$	−43.2523	−1.68487	−0.842436	−	0.538796i	$-0.818879\pi$
$659$	−43.2523	−1.68487	−0.842436	+	0.538796i	$0.818879\pi$
$660$	0	0
$661$	35.5341	1.38212	0.691058	−	0.722799i	$-0.257145\pi$
$661$	35.5341	1.38212	0.691058	+	0.722799i	$0.257145\pi$
$662$	0	0
$663$	6.62754	0.257392
$664$	0	0
$665$	3.45151	0.133844
$666$	0	0
$667$	−7.24152	−0.280393
$668$	0	0
$669$	1.29385	0.0500229
$670$	0	0
$671$	−2.14708	−0.0828873
$672$	0	0
$673$	−9.65878	−0.372319	−0.186159	−	0.982520i	$-0.559604\pi$
$673$	−9.65878	−0.372319	−0.186159	+	0.982520i	$0.559604\pi$
$674$	0	0
$675$	48.6383	1.87209
$676$	0	0
$677$	38.7057	1.48758	0.743790	−	0.668413i	$-0.233026\pi$
$677$	38.7057	1.48758	0.743790	+	0.668413i	$0.233026\pi$
$678$	0	0
$679$	2.08371	0.0799655
$680$	0	0
$681$	13.6445	0.522858
$682$	0	0
$683$	−15.9434	−0.610056	−0.305028	−	0.952343i	$-0.598666\pi$
$683$	−15.9434	−0.610056	−0.305028	+	0.952343i	$0.598666\pi$
$684$	0	0
$685$	−7.11664	−0.271913
$686$	0	0
$687$	−20.3738	−0.777309
$688$	0	0
$689$	3.31579	0.126322
$690$	0	0
$691$	14.5517	0.553571	0.276786	−	0.960932i	$-0.410731\pi$
$691$	14.5517	0.553571	0.276786	+	0.960932i	$0.410731\pi$
$692$	0	0
$693$	0.990105	0.0376110
$694$	0	0
$695$	−45.6269	−1.73073
$696$	0	0
$697$	17.0660	0.646423
$698$	0	0
$699$	11.6512	0.440688
$700$	0	0
$701$	23.6087	0.891689	0.445844	−	0.895110i	$-0.352903\pi$
$701$	23.6087	0.891689	0.445844	+	0.895110i	$0.352903\pi$
$702$	0	0
$703$	−3.78098	−0.142603
$704$	0	0
$705$	3.38664	0.127548
$706$	0	0
$707$	−4.61793	−0.173675
$708$	0	0
$709$	46.3389	1.74029	0.870146	−	0.492793i	$-0.164024\pi$
$709$	46.3389	1.74029	0.870146	+	0.492793i	$0.164024\pi$
$710$	0	0
$711$	−3.15761	−0.118419
$712$	0	0
$713$	1.14388	0.0428388
$714$	0	0
$715$	−3.45151	−0.129079
$716$	0	0
$717$	31.6359	1.18146
$718$	0	0
$719$	46.6570	1.74001	0.870007	−	0.493040i	$-0.164114\pi$
$719$	46.6570	1.74001	0.870007	+	0.493040i	$0.164114\pi$
$720$	0	0
$721$	4.19673	0.156294
$722$	0	0
$723$	22.9377	0.853060
$724$	0	0
$725$	89.0436	3.30699
$726$	0	0
$727$	−26.9900	−1.00100	−0.500502	−	0.865736i	$-0.666851\pi$
$727$	−26.9900	−1.00100	−0.500502	+	0.865736i	$0.666851\pi$
$728$	0	0
$729$	28.6221	1.06008
$730$	0	0
$731$	−27.7094	−1.02487
$732$	0	0
$733$	22.1162	0.816879	0.408439	−	0.912785i	$-0.366073\pi$
$733$	22.1162	0.816879	0.408439	+	0.912785i	$0.366073\pi$
$734$	0	0
$735$	−31.4745	−1.16095
$736$	0	0
$737$	11.9457	0.440024
$738$	0	0
$739$	−28.9847	−1.06622	−0.533109	−	0.846046i	$-0.678976\pi$
$739$	−28.9847	−1.06622	−0.533109	+	0.846046i	$0.678976\pi$
$740$	0	0
$741$	−1.30424	−0.0479126
$742$	0	0
$743$	−4.37615	−0.160546	−0.0802728	−	0.996773i	$-0.525579\pi$
$743$	−4.37615	−0.160546	−0.0802728	+	0.996773i	$0.525579\pi$
$744$	0	0
$745$	−2.84081	−0.104079
$746$	0	0
$747$	7.84031	0.286862
$748$	0	0
$749$	−3.58722	−0.131074
$750$	0	0
$751$	−52.4195	−1.91282	−0.956408	−	0.292034i	$-0.905668\pi$
$751$	−52.4195	−1.91282	−0.956408	+	0.292034i	$0.905668\pi$
$752$	0	0
$753$	−20.8425	−0.759541
$754$	0	0
$755$	49.5704	1.80405
$756$	0	0
$757$	5.87571	0.213556	0.106778	−	0.994283i	$-0.465947\pi$
$757$	5.87571	0.213556	0.106778	+	0.994283i	$0.465947\pi$
$758$	0	0
$759$	−0.974768	−0.0353819
$760$	0	0
$761$	−11.6463	−0.422179	−0.211090	−	0.977467i	$-0.567701\pi$
$761$	−11.6463	−0.422179	−0.211090	+	0.977467i	$0.567701\pi$
$762$	0	0
$763$	−11.2927	−0.408823
$764$	0	0
$765$	19.8258	0.716804
$766$	0	0
$767$	7.70687	0.278279
$768$	0	0
$769$	−23.6432	−0.852595	−0.426298	−	0.904583i	$-0.640182\pi$
$769$	−23.6432	−0.852595	−0.426298	+	0.904583i	$0.640182\pi$
$770$	0	0
$771$	24.9053	0.896943
$772$	0	0
$773$	14.9649	0.538251	0.269126	−	0.963105i	$-0.413265\pi$
$773$	14.9649	0.538251	0.269126	+	0.963105i	$0.413265\pi$
$774$	0	0
$775$	−14.0655	−0.505247
$776$	0	0
$777$	−4.93133	−0.176910
$778$	0	0
$779$	−3.35846	−0.120329
$780$	0	0
$781$	−6.67265	−0.238766
$782$	0	0
$783$	58.5462	2.09227
$784$	0	0
$785$	53.1515	1.89706
$786$	0	0
$787$	−52.1218	−1.85794	−0.928971	−	0.370153i	$-0.879305\pi$
$787$	−52.1218	−1.85794	−0.928971	+	0.370153i	$0.879305\pi$
$788$	0	0
$789$	13.3944	0.476854
$790$	0	0
$791$	0.186453	0.00662950
$792$	0	0
$793$	−2.00944	−0.0713573
$794$	0	0
$795$	−18.2086	−0.645794
$796$	0	0
$797$	−0.228796	−0.00810438	−0.00405219	−	0.999992i	$-0.501290\pi$
$797$	−0.228796	−0.00810438	−0.00405219	+	0.999992i	$0.501290\pi$
$798$	0	0
$799$	3.34847	0.118461
$800$	0	0
$801$	−12.5133	−0.442136
$802$	0	0
$803$	−5.27906	−0.186294
$804$	0	0
$805$	2.41422	0.0850902
$806$	0	0
$807$	−33.7045	−1.18645
$808$	0	0
$809$	−6.19090	−0.217661	−0.108830	−	0.994060i	$-0.534710\pi$
$809$	−6.19090	−0.217661	−0.108830	+	0.994060i	$0.534710\pi$
$810$	0	0
$811$	32.7083	1.14854	0.574271	−	0.818665i	$-0.305286\pi$
$811$	32.7083	1.14854	0.574271	+	0.818665i	$0.305286\pi$
$812$	0	0
$813$	23.1599	0.812253
$814$	0	0
$815$	71.2252	2.49491
$816$	0	0
$817$	5.45297	0.190775
$818$	0	0
$819$	0.926632	0.0323791
$820$	0	0
$821$	−45.1037	−1.57413	−0.787065	−	0.616869i	$-0.788401\pi$
$821$	−45.1037	−1.57413	−0.787065	+	0.616869i	$0.788401\pi$
$822$	0	0
$823$	36.4985	1.27226	0.636129	−	0.771583i	$-0.280535\pi$
$823$	36.4985	1.27226	0.636129	+	0.771583i	$0.280535\pi$
$824$	0	0
$825$	11.9860	0.417299
$826$	0	0
$827$	−8.61147	−0.299450	−0.149725	−	0.988728i	$-0.547839\pi$
$827$	−8.61147	−0.299450	−0.149725	+	0.988728i	$0.547839\pi$
$828$	0	0
$829$	22.0910	0.767253	0.383626	−	0.923488i	$-0.374675\pi$
$829$	22.0910	0.767253	0.383626	+	0.923488i	$0.374675\pi$
$830$	0	0
$831$	11.9799	0.415579
$832$	0	0
$833$	−31.1197	−1.07823
$834$	0	0
$835$	−35.5199	−1.22922
$836$	0	0
$837$	−9.24806	−0.319660
$838$	0	0
$839$	−4.14847	−0.143221	−0.0716106	−	0.997433i	$-0.522814\pi$
$839$	−4.14847	−0.143221	−0.0716106	+	0.997433i	$0.522814\pi$
$840$	0	0
$841$	78.1822	2.69594
$842$	0	0
$843$	−28.9946	−0.998626
$844$	0	0
$845$	44.7129	1.53817
$846$	0	0
$847$	0.935893	0.0321577
$848$	0	0
$849$	34.6588	1.18949
$850$	0	0
$851$	−2.64468	−0.0906585
$852$	0	0
$853$	−32.1956	−1.10236	−0.551178	−	0.834388i	$-0.685821\pi$
$853$	−32.1956	−1.10236	−0.551178	+	0.834388i	$0.685821\pi$
$854$	0	0
$855$	−3.90156	−0.133430
$856$	0	0
$857$	8.02947	0.274282	0.137141	−	0.990552i	$-0.456209\pi$
$857$	8.02947	0.274282	0.137141	+	0.990552i	$0.456209\pi$
$858$	0	0
$859$	10.7062	0.365289	0.182645	−	0.983179i	$-0.441534\pi$
$859$	10.7062	0.365289	0.182645	+	0.983179i	$0.441534\pi$
$860$	0	0
$861$	−4.38025	−0.149278
$862$	0	0
$863$	−13.5645	−0.461742	−0.230871	−	0.972984i	$-0.574158\pi$
$863$	−13.5645	−0.461742	−0.230871	+	0.972984i	$0.574158\pi$
$864$	0	0
$865$	−2.49764	−0.0849223
$866$	0	0
$867$	−12.2939	−0.417523
$868$	0	0
$869$	−2.98471	−0.101250
$870$	0	0
$871$	11.1799	0.378815
$872$	0	0
$873$	−2.35541	−0.0797185
$874$	0	0
$875$	−12.4283	−0.420155
$876$	0	0
$877$	27.3923	0.924972	0.462486	−	0.886627i	$-0.346958\pi$
$877$	27.3923	0.924972	0.462486	+	0.886627i	$0.346958\pi$
$878$	0	0
$879$	2.62672	0.0885972
$880$	0	0
$881$	−46.5748	−1.56914	−0.784572	−	0.620037i	$-0.787117\pi$
$881$	−46.5748	−1.56914	−0.784572	+	0.620037i	$0.787117\pi$
$882$	0	0
$883$	35.9121	1.20854	0.604269	−	0.796781i	$-0.293465\pi$
$883$	35.9121	1.20854	0.604269	+	0.796781i	$0.293465\pi$
$884$	0	0
$885$	−42.3222	−1.42264
$886$	0	0
$887$	40.9815	1.37603	0.688013	−	0.725699i	$-0.258483\pi$
$887$	40.9815	1.37603	0.688013	+	0.725699i	$0.258483\pi$
$888$	0	0
$889$	4.89358	0.164125
$890$	0	0
$891$	4.70702	0.157691
$892$	0	0
$893$	−0.658952	−0.0220510
$894$	0	0
$895$	21.1152	0.705802
$896$	0	0
$897$	−0.912279	−0.0304601
$898$	0	0
$899$	−16.9307	−0.564671
$900$	0	0
$901$	−18.0034	−0.599780
$902$	0	0
$903$	7.11201	0.236673
$904$	0	0
$905$	34.4852	1.14633
$906$	0	0
$907$	34.2438	1.13705	0.568523	−	0.822668i	$-0.307515\pi$
$907$	34.2438	1.13705	0.568523	+	0.822668i	$0.307515\pi$
$908$	0	0
$909$	5.22007	0.173139
$910$	0	0
$911$	−55.1081	−1.82581	−0.912906	−	0.408169i	$-0.866167\pi$
$911$	−55.1081	−1.82581	−0.912906	+	0.408169i	$0.866167\pi$
$912$	0	0
$913$	7.41103	0.245269
$914$	0	0
$915$	11.0348	0.364800
$916$	0	0
$917$	−10.0934	−0.333313
$918$	0	0
$919$	1.06736	0.0352088	0.0176044	−	0.999845i	$-0.494396\pi$
$919$	1.06736	0.0352088	0.0176044	+	0.999845i	$0.494396\pi$
$920$	0	0
$921$	−34.4621	−1.13557
$922$	0	0
$923$	−6.24488	−0.205553
$924$	0	0
$925$	32.5197	1.06924
$926$	0	0
$927$	−4.74395	−0.155812
$928$	0	0
$929$	−12.0458	−0.395210	−0.197605	−	0.980282i	$-0.563316\pi$
$929$	−12.0458	−0.395210	−0.197605	+	0.980282i	$0.563316\pi$
$930$	0	0
$931$	6.12410	0.200709
$932$	0	0
$933$	37.6178	1.23155
$934$	0	0
$935$	18.7403	0.612873
$936$	0	0
$937$	38.1579	1.24656	0.623282	−	0.781997i	$-0.285799\pi$
$937$	38.1579	1.24656	0.623282	+	0.781997i	$0.285799\pi$
$938$	0	0
$939$	−1.64497	−0.0536816
$940$	0	0
$941$	−17.7232	−0.577761	−0.288881	−	0.957365i	$-0.593283\pi$
$941$	−17.7232	−0.577761	−0.288881	+	0.957365i	$0.593283\pi$
$942$	0	0
$943$	−2.34914	−0.0764984
$944$	0	0
$945$	−19.5185	−0.634936
$946$	0	0
$947$	−37.0110	−1.20270	−0.601348	−	0.798987i	$-0.705369\pi$
$947$	−37.0110	−1.20270	−0.601348	+	0.798987i	$0.705369\pi$
$948$	0	0
$949$	−4.94064	−0.160380
$950$	0	0
$951$	−47.0664	−1.52623
$952$	0	0
$953$	6.61111	0.214155	0.107077	−	0.994251i	$-0.465851\pi$
$953$	6.61111	0.214155	0.107077	+	0.994251i	$0.465851\pi$
$954$	0	0
$955$	64.4210	2.08462
$956$	0	0
$957$	14.4276	0.466378
$958$	0	0
$959$	1.80600	0.0583188
$960$	0	0
$961$	−28.3256	−0.913729
$962$	0	0
$963$	4.05497	0.130670
$964$	0	0
$965$	61.8270	1.99028
$966$	0	0
$967$	17.8631	0.574438	0.287219	−	0.957865i	$-0.407269\pi$
$967$	17.8631	0.574438	0.287219	+	0.957865i	$0.407269\pi$
$968$	0	0
$969$	7.08152	0.227491
$970$	0	0
$971$	−37.1698	−1.19284	−0.596418	−	0.802674i	$-0.703410\pi$
$971$	−37.1698	−1.19284	−0.596418	+	0.802674i	$0.703410\pi$
$972$	0	0
$973$	11.5788	0.371199
$974$	0	0
$975$	11.2176	0.359251
$976$	0	0
$977$	−1.03235	−0.0330278	−0.0165139	−	0.999864i	$-0.505257\pi$
$977$	−1.03235	−0.0330278	−0.0165139	+	0.999864i	$0.505257\pi$
$978$	0	0
$979$	−11.8282	−0.378030
$980$	0	0
$981$	12.7652	0.407561
$982$	0	0
$983$	−30.4875	−0.972402	−0.486201	−	0.873847i	$-0.661618\pi$
$983$	−30.4875	−0.972402	−0.486201	+	0.873847i	$0.661618\pi$
$984$	0	0
$985$	−62.9070	−2.00438
$986$	0	0
$987$	−0.859434	−0.0273561
$988$	0	0
$989$	3.81418	0.121284
$990$	0	0
$991$	2.54227	0.0807579	0.0403790	−	0.999184i	$-0.487143\pi$
$991$	2.54227	0.0807579	0.0403790	+	0.999184i	$0.487143\pi$
$992$	0	0
$993$	20.6591	0.655595
$994$	0	0
$995$	−75.3576	−2.38900
$996$	0	0
$997$	−20.4065	−0.646282	−0.323141	−	0.946351i	$-0.604739\pi$
$997$	−20.4065	−0.646282	−0.323141	+	0.946351i	$0.604739\pi$
$998$	0	0
$999$	21.3817	0.676487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.v.1.3		6
4.3	odd	2		1672.2.a.j.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.j.1.4	✓	6	4.3	odd	2
3344.2.a.v.1.3		6	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)

\(f(q)\)	\(=\)	\(q-1.39358 q^{3} -3.68793 q^{5} +0.935893 q^{7} -1.05793 q^{9} -1.00000 q^{11} -0.935893 q^{13} +5.13944 q^{15} +5.08152 q^{17} -1.00000 q^{19} -1.30424 q^{21} -0.699469 q^{23} +8.60085 q^{25} +5.65506 q^{27} +10.3529 q^{29} -1.63536 q^{31} +1.39358 q^{33} -3.45151 q^{35} +3.78098 q^{37} +1.30424 q^{39} +3.35846 q^{41} -5.45297 q^{43} +3.90156 q^{45} +0.658952 q^{47} -6.12410 q^{49} -7.08152 q^{51} -3.54292 q^{53} +3.68793 q^{55} +1.39358 q^{57} -8.23478 q^{59} +2.14708 q^{61} -0.990105 q^{63} +3.45151 q^{65} -11.9457 q^{67} +0.974768 q^{69} +6.67265 q^{71} +5.27906 q^{73} -11.9860 q^{75} -0.935893 q^{77} +2.98471 q^{79} -4.70702 q^{81} -7.41103 q^{83} -18.7403 q^{85} -14.4276 q^{87} +11.8282 q^{89} -0.875895 q^{91} +2.27901 q^{93} +3.68793 q^{95} +2.22644 q^{97} +1.05793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + q^5 + 4 * q^9 - 6 * q^11 - 7 * q^15 + 3 * q^17 - 6 * q^19 + 7 * q^21 - 7 * q^23 + 3 * q^25 - 13 * q^27 - 4 * q^29 - 7 * q^31 + 4 * q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 7 * q^41 - 21 * q^43 + 11 * q^45 - 16 * q^47 + 2 * q^49 - 15 * q^51 + 17 * q^53 - q^55 + 4 * q^57 - 19 * q^59 - 6 * q^61 - 2 * q^63 + 6 * q^65 - 14 * q^67 + q^69 - q^71 - 5 * q^73 - 18 * q^75 + 2 * q^81 - 11 * q^83 - 26 * q^85 - 14 * q^87 + 12 * q^89 - 44 * q^91 - 6 * q^93 - q^95 + 14 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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