LMFDB - Embedded newform 3344.2.a.ba.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 $ x^7 - x^6 - 14x^5 + 10x^4 + 59x^3 - 27x^2 - 66*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.5
Root			$0.456669$ 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)

$f(q)$	$=$	$q+0.835165 q^{3} +0.221953 q^{5} -4.69915 q^{7} -2.30250 q^{9} +O(q^{10})$	$q+0.835165 q^{3} +0.221953 q^{5} -4.69915 q^{7} -2.30250 q^{9} +1.00000 q^{11} +5.89016 q^{13} +0.185367 q^{15} +7.06513 q^{17} -1.00000 q^{19} -3.92457 q^{21} -1.06348 q^{23} -4.95074 q^{25} -4.42846 q^{27} -7.62662 q^{29} -0.901295 q^{31} +0.835165 q^{33} -1.04299 q^{35} -2.71758 q^{37} +4.91925 q^{39} +0.788714 q^{41} -0.714571 q^{43} -0.511046 q^{45} -3.96368 q^{47} +15.0820 q^{49} +5.90055 q^{51} -9.69714 q^{53} +0.221953 q^{55} -0.835165 q^{57} +7.33476 q^{59} +8.15179 q^{61} +10.8198 q^{63} +1.30734 q^{65} -7.86697 q^{67} -0.888178 q^{69} +3.13400 q^{71} -6.49076 q^{73} -4.13468 q^{75} -4.69915 q^{77} -12.0148 q^{79} +3.20900 q^{81} -16.3902 q^{83} +1.56812 q^{85} -6.36948 q^{87} -12.9487 q^{89} -27.6788 q^{91} -0.752730 q^{93} -0.221953 q^{95} +8.14414 q^{97} -2.30250 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * 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
$3$	0.835165	0.482183	0.241091	−	0.970502i	$-0.422495\pi$
$3$	0.835165	0.482183	0.241091	+	0.970502i	$0.422495\pi$
$4$	0	0
$5$	0.221953	0.0992603	0.0496301	−	0.998768i	$-0.484196\pi$
$5$	0.221953	0.0992603	0.0496301	+	0.998768i	$0.484196\pi$
$6$	0	0
$7$	−4.69915	−1.77611	−0.888057	−	0.459734i	$-0.847945\pi$
$7$	−4.69915	−1.77611	−0.888057	+	0.459734i	$0.847945\pi$
$8$	0	0
$9$	−2.30250	−0.767500
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.89016	1.63364	0.816818	−	0.576895i	$-0.195736\pi$
$13$	5.89016	1.63364	0.816818	+	0.576895i	$0.195736\pi$
$14$	0	0
$15$	0.185367	0.0478616
$16$	0	0
$17$	7.06513	1.71355	0.856773	−	0.515694i	$-0.172466\pi$
$17$	7.06513	1.71355	0.856773	+	0.515694i	$0.172466\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.92457	−0.856411
$22$	0	0
$23$	−1.06348	−0.221750	−0.110875	−	0.993834i	$-0.535365\pi$
$23$	−1.06348	−0.221750	−0.110875	+	0.993834i	$0.535365\pi$
$24$	0	0
$25$	−4.95074	−0.990147
$26$	0	0
$27$	−4.42846	−0.852258
$28$	0	0
$29$	−7.62662	−1.41623	−0.708114	−	0.706098i	$-0.750454\pi$
$29$	−7.62662	−1.41623	−0.708114	+	0.706098i	$0.750454\pi$
$30$	0	0
$31$	−0.901295	−0.161877	−0.0809387	−	0.996719i	$-0.525792\pi$
$31$	−0.901295	−0.161877	−0.0809387	+	0.996719i	$0.525792\pi$
$32$	0	0
$33$	0.835165	0.145384
$34$	0	0
$35$	−1.04299	−0.176297
$36$	0	0
$37$	−2.71758	−0.446768	−0.223384	−	0.974731i	$-0.571710\pi$
$37$	−2.71758	−0.446768	−0.223384	+	0.974731i	$0.571710\pi$
$38$	0	0
$39$	4.91925	0.787711
$40$	0	0
$41$	0.788714	0.123176	0.0615882	−	0.998102i	$-0.480383\pi$
$41$	0.788714	0.123176	0.0615882	+	0.998102i	$0.480383\pi$
$42$	0	0
$43$	−0.714571	−0.108971	−0.0544855	−	0.998515i	$-0.517352\pi$
$43$	−0.714571	−0.108971	−0.0544855	+	0.998515i	$0.517352\pi$
$44$	0	0
$45$	−0.511046	−0.0761823
$46$	0	0
$47$	−3.96368	−0.578163	−0.289081	−	0.957305i	$-0.593350\pi$
$47$	−3.96368	−0.578163	−0.289081	+	0.957305i	$0.593350\pi$
$48$	0	0
$49$	15.0820	2.15458
$50$	0	0
$51$	5.90055	0.826242
$52$	0	0
$53$	−9.69714	−1.33200	−0.666002	−	0.745950i	$-0.731996\pi$
$53$	−9.69714	−1.33200	−0.666002	+	0.745950i	$0.731996\pi$
$54$	0	0
$55$	0.221953	0.0299281
$56$	0	0
$57$	−0.835165	−0.110620
$58$	0	0
$59$	7.33476	0.954904	0.477452	−	0.878658i	$-0.341560\pi$
$59$	7.33476	0.954904	0.477452	+	0.878658i	$0.341560\pi$
$60$	0	0
$61$	8.15179	1.04373	0.521865	−	0.853028i	$-0.325236\pi$
$61$	8.15179	1.04373	0.521865	+	0.853028i	$0.325236\pi$
$62$	0	0
$63$	10.8198	1.36317
$64$	0	0
$65$	1.30734	0.162155
$66$	0	0
$67$	−7.86697	−0.961104	−0.480552	−	0.876966i	$-0.659564\pi$
$67$	−7.86697	−0.961104	−0.480552	+	0.876966i	$0.659564\pi$
$68$	0	0
$69$	−0.888178	−0.106924
$70$	0	0
$71$	3.13400	0.371937	0.185969	−	0.982556i	$-0.440458\pi$
$71$	3.13400	0.371937	0.185969	+	0.982556i	$0.440458\pi$
$72$	0	0
$73$	−6.49076	−0.759687	−0.379843	−	0.925051i	$-0.624022\pi$
$73$	−6.49076	−0.759687	−0.379843	+	0.925051i	$0.624022\pi$
$74$	0	0
$75$	−4.13468	−0.477432
$76$	0	0
$77$	−4.69915	−0.535518
$78$	0	0
$79$	−12.0148	−1.35177	−0.675884	−	0.737008i	$-0.736238\pi$
$79$	−12.0148	−1.35177	−0.675884	+	0.737008i	$0.736238\pi$
$80$	0	0
$81$	3.20900	0.356556
$82$	0	0
$83$	−16.3902	−1.79906	−0.899528	−	0.436863i	$-0.856089\pi$
$83$	−16.3902	−1.79906	−0.899528	+	0.436863i	$0.856089\pi$
$84$	0	0
$85$	1.56812	0.170087
$86$	0	0
$87$	−6.36948	−0.682880
$88$	0	0
$89$	−12.9487	−1.37256	−0.686281	−	0.727336i	$-0.740758\pi$
$89$	−12.9487	−1.37256	−0.686281	+	0.727336i	$0.740758\pi$
$90$	0	0
$91$	−27.6788	−2.90152
$92$	0	0
$93$	−0.752730	−0.0780545
$94$	0	0
$95$	−0.221953	−0.0227719
$96$	0	0
$97$	8.14414	0.826912	0.413456	−	0.910524i	$-0.364321\pi$
$97$	8.14414	0.826912	0.413456	+	0.910524i	$0.364321\pi$
$98$	0	0
$99$	−2.30250	−0.231410
$100$	0	0
$101$	−15.3713	−1.52950	−0.764750	−	0.644327i	$-0.777138\pi$
$101$	−15.3713	−1.52950	−0.764750	+	0.644327i	$0.777138\pi$
$102$	0	0
$103$	8.83682	0.870718	0.435359	−	0.900257i	$-0.356622\pi$
$103$	8.83682	0.870718	0.435359	+	0.900257i	$0.356622\pi$
$104$	0	0
$105$	−0.871069	−0.0850076
$106$	0	0
$107$	9.04360	0.874277	0.437139	−	0.899394i	$-0.355992\pi$
$107$	9.04360	0.874277	0.437139	+	0.899394i	$0.355992\pi$
$108$	0	0
$109$	−18.5950	−1.78107	−0.890537	−	0.454911i	$-0.849671\pi$
$109$	−18.5950	−1.78107	−0.890537	+	0.454911i	$0.849671\pi$
$110$	0	0
$111$	−2.26963	−0.215424
$112$	0	0
$113$	−4.42423	−0.416196	−0.208098	−	0.978108i	$-0.566727\pi$
$113$	−4.42423	−0.416196	−0.208098	+	0.978108i	$0.566727\pi$
$114$	0	0
$115$	−0.236042	−0.0220110
$116$	0	0
$117$	−13.5621	−1.25382
$118$	0	0
$119$	−33.2001	−3.04345
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.658706	0.0593935
$124$	0	0
$125$	−2.20859	−0.197543
$126$	0	0
$127$	0.338657	0.0300510	0.0150255	−	0.999887i	$-0.495217\pi$
$127$	0.338657	0.0300510	0.0150255	+	0.999887i	$0.495217\pi$
$128$	0	0
$129$	−0.596784	−0.0525439
$130$	0	0
$131$	4.55211	0.397720	0.198860	−	0.980028i	$-0.436276\pi$
$131$	4.55211	0.397720	0.198860	+	0.980028i	$0.436276\pi$
$132$	0	0
$133$	4.69915	0.407468
$134$	0	0
$135$	−0.982909	−0.0845953
$136$	0	0
$137$	−0.654263	−0.0558974	−0.0279487	−	0.999609i	$-0.508898\pi$
$137$	−0.654263	−0.0558974	−0.0279487	+	0.999609i	$0.508898\pi$
$138$	0	0
$139$	−11.0479	−0.937072	−0.468536	−	0.883444i	$-0.655218\pi$
$139$	−11.0479	−0.937072	−0.468536	+	0.883444i	$0.655218\pi$
$140$	0	0
$141$	−3.31033	−0.278780
$142$	0	0
$143$	5.89016	0.492560
$144$	0	0
$145$	−1.69275	−0.140575
$146$	0	0
$147$	12.5960	1.03890
$148$	0	0
$149$	−7.57743	−0.620767	−0.310384	−	0.950611i	$-0.600457\pi$
$149$	−7.57743	−0.620767	−0.310384	+	0.950611i	$0.600457\pi$
$150$	0	0
$151$	6.95296	0.565824	0.282912	−	0.959146i	$-0.408700\pi$
$151$	6.95296	0.565824	0.282912	+	0.959146i	$0.408700\pi$
$152$	0	0
$153$	−16.2675	−1.31515
$154$	0	0
$155$	−0.200045	−0.0160680
$156$	0	0
$157$	−14.7427	−1.17659	−0.588297	−	0.808645i	$-0.700201\pi$
$157$	−14.7427	−1.17659	−0.588297	+	0.808645i	$0.700201\pi$
$158$	0	0
$159$	−8.09871	−0.642270
$160$	0	0
$161$	4.99744	0.393853
$162$	0	0
$163$	4.94015	0.386942	0.193471	−	0.981106i	$-0.438025\pi$
$163$	4.94015	0.386942	0.193471	+	0.981106i	$0.438025\pi$
$164$	0	0
$165$	0.185367	0.0144308
$166$	0	0
$167$	−22.6032	−1.74909	−0.874544	−	0.484946i	$-0.838839\pi$
$167$	−22.6032	−1.74909	−0.874544	+	0.484946i	$0.838839\pi$
$168$	0	0
$169$	21.6940	1.66877
$170$	0	0
$171$	2.30250	0.176077
$172$	0	0
$173$	−8.18620	−0.622385	−0.311193	−	0.950347i	$-0.600728\pi$
$173$	−8.18620	−0.622385	−0.311193	+	0.950347i	$0.600728\pi$
$174$	0	0
$175$	23.2643	1.75861
$176$	0	0
$177$	6.12573	0.460438
$178$	0	0
$179$	14.0830	1.05261	0.526305	−	0.850296i	$-0.323577\pi$
$179$	14.0830	1.05261	0.526305	+	0.850296i	$0.323577\pi$
$180$	0	0
$181$	−14.3260	−1.06484	−0.532422	−	0.846479i	$-0.678718\pi$
$181$	−14.3260	−1.06484	−0.532422	+	0.846479i	$0.678718\pi$
$182$	0	0
$183$	6.80809	0.503268
$184$	0	0
$185$	−0.603175	−0.0443463
$186$	0	0
$187$	7.06513	0.516653
$188$	0	0
$189$	20.8100	1.51371
$190$	0	0
$191$	−0.394967	−0.0285788	−0.0142894	−	0.999898i	$-0.504549\pi$
$191$	−0.394967	−0.0285788	−0.0142894	+	0.999898i	$0.504549\pi$
$192$	0	0
$193$	9.03423	0.650298	0.325149	−	0.945663i	$-0.394586\pi$
$193$	9.03423	0.650298	0.325149	+	0.945663i	$0.394586\pi$
$194$	0	0
$195$	1.09184	0.0781884
$196$	0	0
$197$	7.85313	0.559512	0.279756	−	0.960071i	$-0.409746\pi$
$197$	7.85313	0.559512	0.279756	+	0.960071i	$0.409746\pi$
$198$	0	0
$199$	−7.81540	−0.554019	−0.277009	−	0.960867i	$-0.589343\pi$
$199$	−7.81540	−0.554019	−0.277009	+	0.960867i	$0.589343\pi$
$200$	0	0
$201$	−6.57022	−0.463427
$202$	0	0
$203$	35.8386	2.51538
$204$	0	0
$205$	0.175057	0.0122265
$206$	0	0
$207$	2.44865	0.170193
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	21.5955	1.48670	0.743348	−	0.668905i	$-0.233237\pi$
$211$	21.5955	1.48670	0.743348	+	0.668905i	$0.233237\pi$
$212$	0	0
$213$	2.61741	0.179342
$214$	0	0
$215$	−0.158601	−0.0108165
$216$	0	0
$217$	4.23533	0.287513
$218$	0	0
$219$	−5.42086	−0.366308
$220$	0	0
$221$	41.6147	2.79931
$222$	0	0
$223$	6.95854	0.465979	0.232989	−	0.972479i	$-0.425149\pi$
$223$	6.95854	0.465979	0.232989	+	0.972479i	$0.425149\pi$
$224$	0	0
$225$	11.3991	0.759938
$226$	0	0
$227$	−7.67819	−0.509619	−0.254810	−	0.966991i	$-0.582013\pi$
$227$	−7.67819	−0.509619	−0.254810	+	0.966991i	$0.582013\pi$
$228$	0	0
$229$	5.83925	0.385869	0.192934	−	0.981212i	$-0.438200\pi$
$229$	5.83925	0.385869	0.192934	+	0.981212i	$0.438200\pi$
$230$	0	0
$231$	−3.92457	−0.258218
$232$	0	0
$233$	−21.8431	−1.43099	−0.715494	−	0.698619i	$-0.753798\pi$
$233$	−21.8431	−1.43099	−0.715494	+	0.698619i	$0.753798\pi$
$234$	0	0
$235$	−0.879751	−0.0573886
$236$	0	0
$237$	−10.0343	−0.651799
$238$	0	0
$239$	−4.21004	−0.272325	−0.136162	−	0.990687i	$-0.543477\pi$
$239$	−4.21004	−0.272325	−0.136162	+	0.990687i	$0.543477\pi$
$240$	0	0
$241$	−17.4276	−1.12261	−0.561304	−	0.827610i	$-0.689700\pi$
$241$	−17.4276	−1.12261	−0.561304	+	0.827610i	$0.689700\pi$
$242$	0	0
$243$	15.9654	1.02418
$244$	0	0
$245$	3.34750	0.213864
$246$	0	0
$247$	−5.89016	−0.374782
$248$	0	0
$249$	−13.6885	−0.867474
$250$	0	0
$251$	−2.14594	−0.135451	−0.0677254	−	0.997704i	$-0.521574\pi$
$251$	−2.14594	−0.135451	−0.0677254	+	0.997704i	$0.521574\pi$
$252$	0	0
$253$	−1.06348	−0.0668602
$254$	0	0
$255$	1.30964	0.0820130
$256$	0	0
$257$	−5.23899	−0.326799	−0.163399	−	0.986560i	$-0.552246\pi$
$257$	−5.23899	−0.326799	−0.163399	+	0.986560i	$0.552246\pi$
$258$	0	0
$259$	12.7703	0.793510
$260$	0	0
$261$	17.5603	1.08695
$262$	0	0
$263$	−12.2597	−0.755967	−0.377983	−	0.925812i	$-0.623382\pi$
$263$	−12.2597	−0.755967	−0.377983	+	0.925812i	$0.623382\pi$
$264$	0	0
$265$	−2.15231	−0.132215
$266$	0	0
$267$	−10.8143	−0.661826
$268$	0	0
$269$	−9.37273	−0.571465	−0.285733	−	0.958309i	$-0.592237\pi$
$269$	−9.37273	−0.571465	−0.285733	+	0.958309i	$0.592237\pi$
$270$	0	0
$271$	13.3131	0.808716	0.404358	−	0.914601i	$-0.367495\pi$
$271$	13.3131	0.808716	0.404358	+	0.914601i	$0.367495\pi$
$272$	0	0
$273$	−23.1163	−1.39906
$274$	0	0
$275$	−4.95074	−0.298541
$276$	0	0
$277$	−14.5641	−0.875073	−0.437537	−	0.899201i	$-0.644149\pi$
$277$	−14.5641	−0.875073	−0.437537	+	0.899201i	$0.644149\pi$
$278$	0	0
$279$	2.07523	0.124241
$280$	0	0
$281$	20.2411	1.20748	0.603741	−	0.797180i	$-0.293676\pi$
$281$	20.2411	1.20748	0.603741	+	0.797180i	$0.293676\pi$
$282$	0	0
$283$	−14.6230	−0.869248	−0.434624	−	0.900612i	$-0.643119\pi$
$283$	−14.6230	−0.869248	−0.434624	+	0.900612i	$0.643119\pi$
$284$	0	0
$285$	−0.185367	−0.0109802
$286$	0	0
$287$	−3.70629	−0.218775
$288$	0	0
$289$	32.9161	1.93624
$290$	0	0
$291$	6.80170	0.398723
$292$	0	0
$293$	27.6524	1.61547	0.807737	−	0.589543i	$-0.200692\pi$
$293$	27.6524	1.61547	0.807737	+	0.589543i	$0.200692\pi$
$294$	0	0
$295$	1.62797	0.0947841
$296$	0	0
$297$	−4.42846	−0.256965
$298$	0	0
$299$	−6.26404	−0.362259
$300$	0	0
$301$	3.35788	0.193545
$302$	0	0
$303$	−12.8376	−0.737499
$304$	0	0
$305$	1.80931	0.103601
$306$	0	0
$307$	0.756065	0.0431509	0.0215755	−	0.999767i	$-0.493132\pi$
$307$	0.756065	0.0431509	0.0215755	+	0.999767i	$0.493132\pi$
$308$	0	0
$309$	7.38020	0.419845
$310$	0	0
$311$	−1.13352	−0.0642761	−0.0321381	−	0.999483i	$-0.510232\pi$
$311$	−1.13352	−0.0642761	−0.0321381	+	0.999483i	$0.510232\pi$
$312$	0	0
$313$	−10.9012	−0.616172	−0.308086	−	0.951359i	$-0.599688\pi$
$313$	−10.9012	−0.616172	−0.308086	+	0.951359i	$0.599688\pi$
$314$	0	0
$315$	2.40148	0.135308
$316$	0	0
$317$	8.32326	0.467481	0.233741	−	0.972299i	$-0.424903\pi$
$317$	8.32326	0.467481	0.233741	+	0.972299i	$0.424903\pi$
$318$	0	0
$319$	−7.62662	−0.427009
$320$	0	0
$321$	7.55289	0.421561
$322$	0	0
$323$	−7.06513	−0.393114
$324$	0	0
$325$	−29.1606	−1.61754
$326$	0	0
$327$	−15.5299	−0.858803
$328$	0	0
$329$	18.6260	1.02688
$330$	0	0
$331$	2.47466	0.136020	0.0680098	−	0.997685i	$-0.478335\pi$
$331$	2.47466	0.136020	0.0680098	+	0.997685i	$0.478335\pi$
$332$	0	0
$333$	6.25723	0.342894
$334$	0	0
$335$	−1.74610	−0.0953994
$336$	0	0
$337$	−30.3242	−1.65187	−0.825933	−	0.563768i	$-0.809351\pi$
$337$	−30.3242	−1.65187	−0.825933	+	0.563768i	$0.809351\pi$
$338$	0	0
$339$	−3.69496	−0.200683
$340$	0	0
$341$	−0.901295	−0.0488079
$342$	0	0
$343$	−37.9788	−2.05066
$344$	0	0
$345$	−0.197134	−0.0106133
$346$	0	0
$347$	10.4332	0.560081	0.280041	−	0.959988i	$-0.409652\pi$
$347$	10.4332	0.560081	0.280041	+	0.959988i	$0.409652\pi$
$348$	0	0
$349$	4.89049	0.261782	0.130891	−	0.991397i	$-0.458216\pi$
$349$	4.89049	0.261782	0.130891	+	0.991397i	$0.458216\pi$
$350$	0	0
$351$	−26.0843	−1.39228
$352$	0	0
$353$	34.1254	1.81631	0.908155	−	0.418634i	$-0.137491\pi$
$353$	34.1254	1.81631	0.908155	+	0.418634i	$0.137491\pi$
$354$	0	0
$355$	0.695600	0.0369186
$356$	0	0
$357$	−27.7276	−1.46750
$358$	0	0
$359$	−10.5366	−0.556103	−0.278051	−	0.960566i	$-0.589689\pi$
$359$	−10.5366	−0.556103	−0.278051	+	0.960566i	$0.589689\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.835165	0.0438348
$364$	0	0
$365$	−1.44064	−0.0754067
$366$	0	0
$367$	−31.1277	−1.62485	−0.812427	−	0.583062i	$-0.801854\pi$
$367$	−31.1277	−1.62485	−0.812427	+	0.583062i	$0.801854\pi$
$368$	0	0
$369$	−1.81601	−0.0945378
$370$	0	0
$371$	45.5684	2.36579
$372$	0	0
$373$	28.9881	1.50095	0.750473	−	0.660901i	$-0.229826\pi$
$373$	28.9881	1.50095	0.750473	+	0.660901i	$0.229826\pi$
$374$	0	0
$375$	−1.84454	−0.0952516
$376$	0	0
$377$	−44.9220	−2.31360
$378$	0	0
$379$	26.4204	1.35712	0.678561	−	0.734544i	$-0.262604\pi$
$379$	26.4204	1.35712	0.678561	+	0.734544i	$0.262604\pi$
$380$	0	0
$381$	0.282835	0.0144901
$382$	0	0
$383$	13.6280	0.696360	0.348180	−	0.937428i	$-0.386800\pi$
$383$	13.6280	0.696360	0.348180	+	0.937428i	$0.386800\pi$
$384$	0	0
$385$	−1.04299	−0.0531557
$386$	0	0
$387$	1.64530	0.0836353
$388$	0	0
$389$	−15.7277	−0.797428	−0.398714	−	0.917075i	$-0.630543\pi$
$389$	−15.7277	−0.797428	−0.398714	+	0.917075i	$0.630543\pi$
$390$	0	0
$391$	−7.51360	−0.379979
$392$	0	0
$393$	3.80176	0.191774
$394$	0	0
$395$	−2.66671	−0.134177
$396$	0	0
$397$	16.8523	0.845790	0.422895	−	0.906179i	$-0.361014\pi$
$397$	16.8523	0.845790	0.422895	+	0.906179i	$0.361014\pi$
$398$	0	0
$399$	3.92457	0.196474
$400$	0	0
$401$	30.3744	1.51682	0.758412	−	0.651775i	$-0.225975\pi$
$401$	30.3744	1.51682	0.758412	+	0.651775i	$0.225975\pi$
$402$	0	0
$403$	−5.30877	−0.264449
$404$	0	0
$405$	0.712247	0.0353919
$406$	0	0
$407$	−2.71758	−0.134706
$408$	0	0
$409$	−1.31844	−0.0651925	−0.0325962	−	0.999469i	$-0.510378\pi$
$409$	−1.31844	−0.0651925	−0.0325962	+	0.999469i	$0.510378\pi$
$410$	0	0
$411$	−0.546417	−0.0269528
$412$	0	0
$413$	−34.4672	−1.69602
$414$	0	0
$415$	−3.63785	−0.178575
$416$	0	0
$417$	−9.22683	−0.451840
$418$	0	0
$419$	30.3257	1.48151	0.740753	−	0.671778i	$-0.234469\pi$
$419$	30.3257	1.48151	0.740753	+	0.671778i	$0.234469\pi$
$420$	0	0
$421$	8.27204	0.403154	0.201577	−	0.979473i	$-0.435393\pi$
$421$	8.27204	0.403154	0.201577	+	0.979473i	$0.435393\pi$
$422$	0	0
$423$	9.12638	0.443740
$424$	0	0
$425$	−34.9776	−1.69666
$426$	0	0
$427$	−38.3065	−1.85378
$428$	0	0
$429$	4.91925	0.237504
$430$	0	0
$431$	−22.9574	−1.10582	−0.552909	−	0.833241i	$-0.686482\pi$
$431$	−22.9574	−1.10582	−0.552909	+	0.833241i	$0.686482\pi$
$432$	0	0
$433$	11.4207	0.548846	0.274423	−	0.961609i	$-0.411513\pi$
$433$	11.4207	0.548846	0.274423	+	0.961609i	$0.411513\pi$
$434$	0	0
$435$	−1.41372	−0.0677829
$436$	0	0
$437$	1.06348	0.0508730
$438$	0	0
$439$	−5.43642	−0.259466	−0.129733	−	0.991549i	$-0.541412\pi$
$439$	−5.43642	−0.259466	−0.129733	+	0.991549i	$0.541412\pi$
$440$	0	0
$441$	−34.7264	−1.65364
$442$	0	0
$443$	34.7494	1.65100	0.825498	−	0.564405i	$-0.190894\pi$
$443$	34.7494	1.65100	0.825498	+	0.564405i	$0.190894\pi$
$444$	0	0
$445$	−2.87401	−0.136241
$446$	0	0
$447$	−6.32840	−0.299323
$448$	0	0
$449$	−11.3505	−0.535662	−0.267831	−	0.963466i	$-0.586307\pi$
$449$	−11.3505	−0.535662	−0.267831	+	0.963466i	$0.586307\pi$
$450$	0	0
$451$	0.788714	0.0371391
$452$	0	0
$453$	5.80687	0.272830
$454$	0	0
$455$	−6.14338	−0.288006
$456$	0	0
$457$	6.89453	0.322513	0.161256	−	0.986913i	$-0.448445\pi$
$457$	6.89453	0.322513	0.161256	+	0.986913i	$0.448445\pi$
$458$	0	0
$459$	−31.2877	−1.46038
$460$	0	0
$461$	13.5620	0.631643	0.315822	−	0.948819i	$-0.397720\pi$
$461$	13.5620	0.631643	0.315822	+	0.948819i	$0.397720\pi$
$462$	0	0
$463$	18.6923	0.868704	0.434352	−	0.900743i	$-0.356977\pi$
$463$	18.6923	0.868704	0.434352	+	0.900743i	$0.356977\pi$
$464$	0	0
$465$	−0.167071	−0.00774771
$466$	0	0
$467$	14.7156	0.680957	0.340479	−	0.940252i	$-0.389411\pi$
$467$	14.7156	0.680957	0.340479	+	0.940252i	$0.389411\pi$
$468$	0	0
$469$	36.9681	1.70703
$470$	0	0
$471$	−12.3126	−0.567333
$472$	0	0
$473$	−0.714571	−0.0328560
$474$	0	0
$475$	4.95074	0.227155
$476$	0	0
$477$	22.3277	1.02231
$478$	0	0
$479$	16.3467	0.746901	0.373450	−	0.927650i	$-0.378175\pi$
$479$	16.3467	0.746901	0.373450	+	0.927650i	$0.378175\pi$
$480$	0	0
$481$	−16.0070	−0.729856
$482$	0	0
$483$	4.17368	0.189909
$484$	0	0
$485$	1.80762	0.0820796
$486$	0	0
$487$	24.8625	1.12663	0.563314	−	0.826243i	$-0.309526\pi$
$487$	24.8625	1.12663	0.563314	+	0.826243i	$0.309526\pi$
$488$	0	0
$489$	4.12584	0.186577
$490$	0	0
$491$	−17.5025	−0.789878	−0.394939	−	0.918707i	$-0.629234\pi$
$491$	−17.5025	−0.789878	−0.394939	+	0.918707i	$0.629234\pi$
$492$	0	0
$493$	−53.8830	−2.42677
$494$	0	0
$495$	−0.511046	−0.0229698
$496$	0	0
$497$	−14.7271	−0.660603
$498$	0	0
$499$	3.13006	0.140121	0.0700603	−	0.997543i	$-0.477681\pi$
$499$	3.13006	0.140121	0.0700603	+	0.997543i	$0.477681\pi$
$500$	0	0
$501$	−18.8774	−0.843380
$502$	0	0
$503$	8.98965	0.400829	0.200414	−	0.979711i	$-0.435771\pi$
$503$	8.98965	0.400829	0.200414	+	0.979711i	$0.435771\pi$
$504$	0	0
$505$	−3.41170	−0.151819
$506$	0	0
$507$	18.1180	0.804650
$508$	0	0
$509$	23.6416	1.04790	0.523949	−	0.851750i	$-0.324458\pi$
$509$	23.6416	1.04790	0.523949	+	0.851750i	$0.324458\pi$
$510$	0	0
$511$	30.5011	1.34929
$512$	0	0
$513$	4.42846	0.195521
$514$	0	0
$515$	1.96136	0.0864277
$516$	0	0
$517$	−3.96368	−0.174323
$518$	0	0
$519$	−6.83683	−0.300103
$520$	0	0
$521$	36.0036	1.57735	0.788673	−	0.614813i	$-0.210768\pi$
$521$	36.0036	1.57735	0.788673	+	0.614813i	$0.210768\pi$
$522$	0	0
$523$	−7.08081	−0.309622	−0.154811	−	0.987944i	$-0.549477\pi$
$523$	−7.08081	−0.309622	−0.154811	+	0.987944i	$0.549477\pi$
$524$	0	0
$525$	19.4295	0.847973
$526$	0	0
$527$	−6.36777	−0.277384
$528$	0	0
$529$	−21.8690	−0.950827
$530$	0	0
$531$	−16.8883	−0.732889
$532$	0	0
$533$	4.64565	0.201225
$534$	0	0
$535$	2.00725	0.0867810
$536$	0	0
$537$	11.7616	0.507550
$538$	0	0
$539$	15.0820	0.649630
$540$	0	0
$541$	−4.23609	−0.182124	−0.0910619	−	0.995845i	$-0.529026\pi$
$541$	−4.23609	−0.182124	−0.0910619	+	0.995845i	$0.529026\pi$
$542$	0	0
$543$	−11.9646	−0.513449
$544$	0	0
$545$	−4.12720	−0.176790
$546$	0	0
$547$	−9.53317	−0.407609	−0.203805	−	0.979012i	$-0.565331\pi$
$547$	−9.53317	−0.407609	−0.203805	+	0.979012i	$0.565331\pi$
$548$	0	0
$549$	−18.7695	−0.801063
$550$	0	0
$551$	7.62662	0.324905
$552$	0	0
$553$	56.4593	2.40089
$554$	0	0
$555$	−0.503750	−0.0213830
$556$	0	0
$557$	2.85948	0.121160	0.0605800	−	0.998163i	$-0.480705\pi$
$557$	2.85948	0.121160	0.0605800	+	0.998163i	$0.480705\pi$
$558$	0	0
$559$	−4.20894	−0.178019
$560$	0	0
$561$	5.90055	0.249121
$562$	0	0
$563$	−12.8091	−0.539841	−0.269920	−	0.962883i	$-0.586997\pi$
$563$	−12.8091	−0.539841	−0.269920	+	0.962883i	$0.586997\pi$
$564$	0	0
$565$	−0.981969	−0.0413118
$566$	0	0
$567$	−15.0796	−0.633284
$568$	0	0
$569$	−10.5273	−0.441327	−0.220664	−	0.975350i	$-0.570822\pi$
$569$	−10.5273	−0.441327	−0.220664	+	0.975350i	$0.570822\pi$
$570$	0	0
$571$	24.9055	1.04226	0.521132	−	0.853476i	$-0.325510\pi$
$571$	24.9055	1.04226	0.521132	+	0.853476i	$0.325510\pi$
$572$	0	0
$573$	−0.329863	−0.0137802
$574$	0	0
$575$	5.26499	0.219565
$576$	0	0
$577$	−26.7799	−1.11486	−0.557430	−	0.830224i	$-0.688213\pi$
$577$	−26.7799	−1.11486	−0.557430	+	0.830224i	$0.688213\pi$
$578$	0	0
$579$	7.54507	0.313562
$580$	0	0
$581$	77.0200	3.19533
$582$	0	0
$583$	−9.69714	−0.401615
$584$	0	0
$585$	−3.01014	−0.124454
$586$	0	0
$587$	−18.7991	−0.775920	−0.387960	−	0.921676i	$-0.626820\pi$
$587$	−18.7991	−0.775920	−0.387960	+	0.921676i	$0.626820\pi$
$588$	0	0
$589$	0.901295	0.0371372
$590$	0	0
$591$	6.55866	0.269787
$592$	0	0
$593$	−3.68216	−0.151208	−0.0756041	−	0.997138i	$-0.524089\pi$
$593$	−3.68216	−0.151208	−0.0756041	+	0.997138i	$0.524089\pi$
$594$	0	0
$595$	−7.36886	−0.302094
$596$	0	0
$597$	−6.52715	−0.267138
$598$	0	0
$599$	−43.2659	−1.76780	−0.883899	−	0.467677i	$-0.845091\pi$
$599$	−43.2659	−1.76780	−0.883899	+	0.467677i	$0.845091\pi$
$600$	0	0
$601$	−14.4072	−0.587683	−0.293842	−	0.955854i	$-0.594934\pi$
$601$	−14.4072	−0.587683	−0.293842	+	0.955854i	$0.594934\pi$
$602$	0	0
$603$	18.1137	0.737647
$604$	0	0
$605$	0.221953	0.00902366
$606$	0	0
$607$	6.06496	0.246169	0.123084	−	0.992396i	$-0.460721\pi$
$607$	6.06496	0.246169	0.123084	+	0.992396i	$0.460721\pi$
$608$	0	0
$609$	29.9312	1.21287
$610$	0	0
$611$	−23.3467	−0.944508
$612$	0	0
$613$	−3.51855	−0.142113	−0.0710565	−	0.997472i	$-0.522637\pi$
$613$	−3.51855	−0.142113	−0.0710565	+	0.997472i	$0.522637\pi$
$614$	0	0
$615$	0.146202	0.00589542
$616$	0	0
$617$	7.17642	0.288912	0.144456	−	0.989511i	$-0.453857\pi$
$617$	7.17642	0.288912	0.144456	+	0.989511i	$0.453857\pi$
$618$	0	0
$619$	8.27421	0.332568	0.166284	−	0.986078i	$-0.446823\pi$
$619$	8.27421	0.332568	0.166284	+	0.986078i	$0.446823\pi$
$620$	0	0
$621$	4.70956	0.188988
$622$	0	0
$623$	60.8480	2.43783
$624$	0	0
$625$	24.2635	0.970539
$626$	0	0
$627$	−0.835165	−0.0333533
$628$	0	0
$629$	−19.2001	−0.765557
$630$	0	0
$631$	9.09560	0.362090	0.181045	−	0.983475i	$-0.442052\pi$
$631$	9.09560	0.362090	0.181045	+	0.983475i	$0.442052\pi$
$632$	0	0
$633$	18.0358	0.716859
$634$	0	0
$635$	0.0751660	0.00298287
$636$	0	0
$637$	88.8356	3.51980
$638$	0	0
$639$	−7.21604	−0.285462
$640$	0	0
$641$	16.4386	0.649285	0.324642	−	0.945837i	$-0.394756\pi$
$641$	16.4386	0.649285	0.324642	+	0.945837i	$0.394756\pi$
$642$	0	0
$643$	−7.04594	−0.277865	−0.138932	−	0.990302i	$-0.544367\pi$
$643$	−7.04594	−0.277865	−0.138932	+	0.990302i	$0.544367\pi$
$644$	0	0
$645$	−0.132458	−0.00521553
$646$	0	0
$647$	−18.1024	−0.711679	−0.355840	−	0.934547i	$-0.615805\pi$
$647$	−18.1024	−0.711679	−0.355840	+	0.934547i	$0.615805\pi$
$648$	0	0
$649$	7.33476	0.287914
$650$	0	0
$651$	3.53719	0.138634
$652$	0	0
$653$	37.2985	1.45960	0.729802	−	0.683659i	$-0.239612\pi$
$653$	37.2985	1.45960	0.729802	+	0.683659i	$0.239612\pi$
$654$	0	0
$655$	1.01035	0.0394778
$656$	0	0
$657$	14.9450	0.583059
$658$	0	0
$659$	−13.2223	−0.515068	−0.257534	−	0.966269i	$-0.582910\pi$
$659$	−13.2223	−0.515068	−0.257534	+	0.966269i	$0.582910\pi$
$660$	0	0
$661$	−17.6212	−0.685385	−0.342692	−	0.939448i	$-0.611339\pi$
$661$	−17.6212	−0.685385	−0.342692	+	0.939448i	$0.611339\pi$
$662$	0	0
$663$	34.7552	1.34978
$664$	0	0
$665$	1.04299	0.0404454
$666$	0	0
$667$	8.11073	0.314049
$668$	0	0
$669$	5.81153	0.224687
$670$	0	0
$671$	8.15179	0.314696
$672$	0	0
$673$	32.2335	1.24251	0.621255	−	0.783609i	$-0.286623\pi$
$673$	32.2335	1.24251	0.621255	+	0.783609i	$0.286623\pi$
$674$	0	0
$675$	21.9241	0.843861
$676$	0	0
$677$	−10.2788	−0.395045	−0.197523	−	0.980298i	$-0.563290\pi$
$677$	−10.2788	−0.395045	−0.197523	+	0.980298i	$0.563290\pi$
$678$	0	0
$679$	−38.2706	−1.46869
$680$	0	0
$681$	−6.41255	−0.245730
$682$	0	0
$683$	−25.4018	−0.971973	−0.485986	−	0.873966i	$-0.661540\pi$
$683$	−25.4018	−0.971973	−0.485986	+	0.873966i	$0.661540\pi$
$684$	0	0
$685$	−0.145215	−0.00554840
$686$	0	0
$687$	4.87674	0.186059
$688$	0	0
$689$	−57.1177	−2.17601
$690$	0	0
$691$	2.58092	0.0981828	0.0490914	−	0.998794i	$-0.484367\pi$
$691$	2.58092	0.0981828	0.0490914	+	0.998794i	$0.484367\pi$
$692$	0	0
$693$	10.8198	0.411010
$694$	0	0
$695$	−2.45212	−0.0930141
$696$	0	0
$697$	5.57236	0.211068
$698$	0	0
$699$	−18.2426	−0.689998
$700$	0	0
$701$	−30.0612	−1.13540	−0.567698	−	0.823237i	$-0.692166\pi$
$701$	−30.0612	−1.13540	−0.567698	+	0.823237i	$0.692166\pi$
$702$	0	0
$703$	2.71758	0.102496
$704$	0	0
$705$	−0.734737	−0.0276718
$706$	0	0
$707$	72.2321	2.71657
$708$	0	0
$709$	−6.90683	−0.259391	−0.129696	−	0.991554i	$-0.541400\pi$
$709$	−6.90683	−0.259391	−0.129696	+	0.991554i	$0.541400\pi$
$710$	0	0
$711$	27.6640	1.03748
$712$	0	0
$713$	0.958506	0.0358963
$714$	0	0
$715$	1.30734	0.0488916
$716$	0	0
$717$	−3.51608	−0.131310
$718$	0	0
$719$	−13.1525	−0.490506	−0.245253	−	0.969459i	$-0.578871\pi$
$719$	−13.1525	−0.490506	−0.245253	+	0.969459i	$0.578871\pi$
$720$	0	0
$721$	−41.5256	−1.54649
$722$	0	0
$723$	−14.5549	−0.541302
$724$	0	0
$725$	37.7574	1.40227
$726$	0	0
$727$	46.8434	1.73733	0.868663	−	0.495404i	$-0.164980\pi$
$727$	46.8434	1.73733	0.868663	+	0.495404i	$0.164980\pi$
$728$	0	0
$729$	3.70675	0.137287
$730$	0	0
$731$	−5.04854	−0.186727
$732$	0	0
$733$	−31.9162	−1.17885	−0.589426	−	0.807822i	$-0.700646\pi$
$733$	−31.9162	−1.17885	−0.589426	+	0.807822i	$0.700646\pi$
$734$	0	0
$735$	2.79571	0.103121
$736$	0	0
$737$	−7.86697	−0.289784
$738$	0	0
$739$	−30.0302	−1.10468	−0.552339	−	0.833620i	$-0.686264\pi$
$739$	−30.0302	−1.10468	−0.552339	+	0.833620i	$0.686264\pi$
$740$	0	0
$741$	−4.91925	−0.180713
$742$	0	0
$743$	42.6871	1.56604	0.783019	−	0.621998i	$-0.213679\pi$
$743$	42.6871	1.56604	0.783019	+	0.621998i	$0.213679\pi$
$744$	0	0
$745$	−1.68183	−0.0616175
$746$	0	0
$747$	37.7384	1.38078
$748$	0	0
$749$	−42.4972	−1.55282
$750$	0	0
$751$	−30.7211	−1.12103	−0.560514	−	0.828145i	$-0.689396\pi$
$751$	−30.7211	−1.12103	−0.560514	+	0.828145i	$0.689396\pi$
$752$	0	0
$753$	−1.79222	−0.0653120
$754$	0	0
$755$	1.54323	0.0561638
$756$	0	0
$757$	−49.0714	−1.78353	−0.891765	−	0.452498i	$-0.850533\pi$
$757$	−49.0714	−1.78353	−0.891765	+	0.452498i	$0.850533\pi$
$758$	0	0
$759$	−0.888178	−0.0322388
$760$	0	0
$761$	−24.8178	−0.899645	−0.449822	−	0.893118i	$-0.648513\pi$
$761$	−24.8178	−0.899645	−0.449822	+	0.893118i	$0.648513\pi$
$762$	0	0
$763$	87.3806	3.16339
$764$	0	0
$765$	−3.61061	−0.130542
$766$	0	0
$767$	43.2029	1.55997
$768$	0	0
$769$	−38.9862	−1.40588	−0.702940	−	0.711250i	$-0.748130\pi$
$769$	−38.9862	−1.40588	−0.702940	+	0.711250i	$0.748130\pi$
$770$	0	0
$771$	−4.37542	−0.157577
$772$	0	0
$773$	0.998099	0.0358991	0.0179496	−	0.999839i	$-0.494286\pi$
$773$	0.998099	0.0358991	0.0179496	+	0.999839i	$0.494286\pi$
$774$	0	0
$775$	4.46208	0.160283
$776$	0	0
$777$	10.6653	0.382617
$778$	0	0
$779$	−0.788714	−0.0282586
$780$	0	0
$781$	3.13400	0.112143
$782$	0	0
$783$	33.7742	1.20699
$784$	0	0
$785$	−3.27218	−0.116789
$786$	0	0
$787$	−18.7420	−0.668082	−0.334041	−	0.942559i	$-0.608412\pi$
$787$	−18.7420	−0.668082	−0.334041	+	0.942559i	$0.608412\pi$
$788$	0	0
$789$	−10.2389	−0.364514
$790$	0	0
$791$	20.7901	0.739211
$792$	0	0
$793$	48.0153	1.70507
$794$	0	0
$795$	−1.79753	−0.0637519
$796$	0	0
$797$	5.05128	0.178925	0.0894627	−	0.995990i	$-0.471485\pi$
$797$	5.05128	0.178925	0.0894627	+	0.995990i	$0.471485\pi$
$798$	0	0
$799$	−28.0039	−0.990708
$800$	0	0
$801$	29.8144	1.05344
$802$	0	0
$803$	−6.49076	−0.229054
$804$	0	0
$805$	1.10920	0.0390940
$806$	0	0
$807$	−7.82777	−0.275551
$808$	0	0
$809$	−4.98214	−0.175163	−0.0875813	−	0.996157i	$-0.527914\pi$
$809$	−4.98214	−0.175163	−0.0875813	+	0.996157i	$0.527914\pi$
$810$	0	0
$811$	−25.3145	−0.888913	−0.444456	−	0.895800i	$-0.646603\pi$
$811$	−25.3145	−0.888913	−0.444456	+	0.895800i	$0.646603\pi$
$812$	0	0
$813$	11.1187	0.389949
$814$	0	0
$815$	1.09648	0.0384080
$816$	0	0
$817$	0.714571	0.0249997
$818$	0	0
$819$	63.7303	2.22692
$820$	0	0
$821$	5.81543	0.202960	0.101480	−	0.994838i	$-0.467642\pi$
$821$	5.81543	0.202960	0.101480	+	0.994838i	$0.467642\pi$
$822$	0	0
$823$	−47.7881	−1.66579	−0.832895	−	0.553431i	$-0.813318\pi$
$823$	−47.7881	−1.66579	−0.832895	+	0.553431i	$0.813318\pi$
$824$	0	0
$825$	−4.13468	−0.143951
$826$	0	0
$827$	39.0692	1.35857	0.679284	−	0.733875i	$-0.262290\pi$
$827$	39.0692	1.35857	0.679284	+	0.733875i	$0.262290\pi$
$828$	0	0
$829$	−7.57440	−0.263070	−0.131535	−	0.991312i	$-0.541991\pi$
$829$	−7.57440	−0.263070	−0.131535	+	0.991312i	$0.541991\pi$
$830$	0	0
$831$	−12.1634	−0.421945
$832$	0	0
$833$	106.557	3.69197
$834$	0	0
$835$	−5.01684	−0.173615
$836$	0	0
$837$	3.99135	0.137961
$838$	0	0
$839$	28.0343	0.967850	0.483925	−	0.875110i	$-0.339211\pi$
$839$	28.0343	0.967850	0.483925	+	0.875110i	$0.339211\pi$
$840$	0	0
$841$	29.1653	1.00570
$842$	0	0
$843$	16.9046	0.582227
$844$	0	0
$845$	4.81504	0.165642
$846$	0	0
$847$	−4.69915	−0.161465
$848$	0	0
$849$	−12.2126	−0.419136
$850$	0	0
$851$	2.89008	0.0990708
$852$	0	0
$853$	−27.9605	−0.957349	−0.478674	−	0.877992i	$-0.658883\pi$
$853$	−27.9605	−0.957349	−0.478674	+	0.877992i	$0.658883\pi$
$854$	0	0
$855$	0.511046	0.0174774
$856$	0	0
$857$	17.8805	0.610787	0.305393	−	0.952226i	$-0.401212\pi$
$857$	17.8805	0.610787	0.305393	+	0.952226i	$0.401212\pi$
$858$	0	0
$859$	−7.76017	−0.264774	−0.132387	−	0.991198i	$-0.542264\pi$
$859$	−7.76017	−0.264774	−0.132387	+	0.991198i	$0.542264\pi$
$860$	0	0
$861$	−3.09536	−0.105490
$862$	0	0
$863$	−31.9699	−1.08827	−0.544135	−	0.838998i	$-0.683142\pi$
$863$	−31.9699	−1.08827	−0.544135	+	0.838998i	$0.683142\pi$
$864$	0	0
$865$	−1.81695	−0.0617782
$866$	0	0
$867$	27.4903	0.933621
$868$	0	0
$869$	−12.0148	−0.407574
$870$	0	0
$871$	−46.3377	−1.57009
$872$	0	0
$873$	−18.7519	−0.634655
$874$	0	0
$875$	10.3785	0.350858
$876$	0	0
$877$	−2.12224	−0.0716630	−0.0358315	−	0.999358i	$-0.511408\pi$
$877$	−2.12224	−0.0716630	−0.0358315	+	0.999358i	$0.511408\pi$
$878$	0	0
$879$	23.0944	0.778953
$880$	0	0
$881$	−15.1859	−0.511627	−0.255814	−	0.966726i	$-0.582343\pi$
$881$	−15.1859	−0.511627	−0.255814	+	0.966726i	$0.582343\pi$
$882$	0	0
$883$	−29.5964	−0.995999	−0.498000	−	0.867177i	$-0.665932\pi$
$883$	−29.5964	−0.995999	−0.498000	+	0.867177i	$0.665932\pi$
$884$	0	0
$885$	1.35962	0.0457032
$886$	0	0
$887$	−23.2340	−0.780121	−0.390061	−	0.920789i	$-0.627546\pi$
$887$	−23.2340	−0.780121	−0.390061	+	0.920789i	$0.627546\pi$
$888$	0	0
$889$	−1.59140	−0.0533740
$890$	0	0
$891$	3.20900	0.107506
$892$	0	0
$893$	3.96368	0.132640
$894$	0	0
$895$	3.12575	0.104482
$896$	0	0
$897$	−5.23151	−0.174675
$898$	0	0
$899$	6.87384	0.229255
$900$	0	0
$901$	−68.5116	−2.28245
$902$	0	0
$903$	2.80438	0.0933240
$904$	0	0
$905$	−3.17970	−0.105697
$906$	0	0
$907$	35.4985	1.17871	0.589354	−	0.807875i	$-0.299382\pi$
$907$	35.4985	1.17871	0.589354	+	0.807875i	$0.299382\pi$
$908$	0	0
$909$	35.3924	1.17389
$910$	0	0
$911$	38.5075	1.27581	0.637905	−	0.770115i	$-0.279801\pi$
$911$	38.5075	1.27581	0.637905	+	0.770115i	$0.279801\pi$
$912$	0	0
$913$	−16.3902	−0.542436
$914$	0	0
$915$	1.51107	0.0499546
$916$	0	0
$917$	−21.3911	−0.706395
$918$	0	0
$919$	−20.9347	−0.690572	−0.345286	−	0.938497i	$-0.612218\pi$
$919$	−20.9347	−0.690572	−0.345286	+	0.938497i	$0.612218\pi$
$920$	0	0
$921$	0.631439	0.0208066
$922$	0	0
$923$	18.4598	0.607610
$924$	0	0
$925$	13.4540	0.442366
$926$	0	0
$927$	−20.3468	−0.668276
$928$	0	0
$929$	−0.536958	−0.0176170	−0.00880851	−	0.999961i	$-0.502804\pi$
$929$	−0.536958	−0.0176170	−0.00880851	+	0.999961i	$0.502804\pi$
$930$	0	0
$931$	−15.0820	−0.494294
$932$	0	0
$933$	−0.946677	−0.0309928
$934$	0	0
$935$	1.56812	0.0512832
$936$	0	0
$937$	34.4984	1.12701	0.563507	−	0.826112i	$-0.309452\pi$
$937$	34.4984	1.12701	0.563507	+	0.826112i	$0.309452\pi$
$938$	0	0
$939$	−9.10430	−0.297107
$940$	0	0
$941$	17.8959	0.583391	0.291695	−	0.956511i	$-0.405781\pi$
$941$	17.8959	0.583391	0.291695	+	0.956511i	$0.405781\pi$
$942$	0	0
$943$	−0.838778	−0.0273144
$944$	0	0
$945$	4.61884	0.150251
$946$	0	0
$947$	−16.0979	−0.523111	−0.261556	−	0.965188i	$-0.584235\pi$
$947$	−16.0979	−0.523111	−0.261556	+	0.965188i	$0.584235\pi$
$948$	0	0
$949$	−38.2316	−1.24105
$950$	0	0
$951$	6.95130	0.225411
$952$	0	0
$953$	29.2828	0.948562	0.474281	−	0.880374i	$-0.342708\pi$
$953$	29.2828	0.948562	0.474281	+	0.880374i	$0.342708\pi$
$954$	0	0
$955$	−0.0876640	−0.00283674
$956$	0	0
$957$	−6.36948	−0.205896
$958$	0	0
$959$	3.07448	0.0992802
$960$	0	0
$961$	−30.1877	−0.973796
$962$	0	0
$963$	−20.8229	−0.671008
$964$	0	0
$965$	2.00517	0.0645488
$966$	0	0
$967$	39.4644	1.26909	0.634544	−	0.772887i	$-0.281188\pi$
$967$	39.4644	1.26909	0.634544	+	0.772887i	$0.281188\pi$
$968$	0	0
$969$	−5.90055	−0.189553
$970$	0	0
$971$	−27.0245	−0.867257	−0.433628	−	0.901092i	$-0.642767\pi$
$971$	−27.0245	−0.867257	−0.433628	+	0.901092i	$0.642767\pi$
$972$	0	0
$973$	51.9159	1.66435
$974$	0	0
$975$	−24.3539	−0.779950
$976$	0	0
$977$	−55.2922	−1.76895	−0.884477	−	0.466584i	$-0.845484\pi$
$977$	−55.2922	−1.76895	−0.884477	+	0.466584i	$0.845484\pi$
$978$	0	0
$979$	−12.9487	−0.413843
$980$	0	0
$981$	42.8149	1.36697
$982$	0	0
$983$	−36.0221	−1.14893	−0.574463	−	0.818531i	$-0.694789\pi$
$983$	−36.0221	−1.14893	−0.574463	+	0.818531i	$0.694789\pi$
$984$	0	0
$985$	1.74302	0.0555373
$986$	0	0
$987$	15.5557	0.495145
$988$	0	0
$989$	0.759929	0.0241643
$990$	0	0
$991$	−50.7944	−1.61354	−0.806768	−	0.590868i	$-0.798785\pi$
$991$	−50.7944	−1.61354	−0.806768	+	0.590868i	$0.798785\pi$
$992$	0	0
$993$	2.06675	0.0655863
$994$	0	0
$995$	−1.73465	−0.0549921
$996$	0	0
$997$	−41.8525	−1.32548	−0.662741	−	0.748849i	$-0.730607\pi$
$997$	−41.8525	−1.32548	−0.662741	+	0.748849i	$0.730607\pi$
$998$	0	0
$999$	12.0347	0.380761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.ba.1.5		7
4.3	odd	2		209.2.a.d.1.4	✓	7
12.11	even	2		1881.2.a.p.1.4		7
20.19	odd	2		5225.2.a.n.1.4		7
44.43	even	2		2299.2.a.q.1.4		7
76.75	even	2		3971.2.a.i.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.4	✓	7	4.3	odd	2
1881.2.a.p.1.4		7	12.11	even	2
2299.2.a.q.1.4		7	44.43	even	2
3344.2.a.ba.1.5		7	1.1	even	1	trivial
3971.2.a.i.1.4		7	76.75	even	2
5225.2.a.n.1.4		7	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)

\(f(q)\)	\(=\)	\(q+0.835165 q^{3} +0.221953 q^{5} -4.69915 q^{7} -2.30250 q^{9} +O(q^{10})\)	\(q+0.835165 q^{3} +0.221953 q^{5} -4.69915 q^{7} -2.30250 q^{9} +1.00000 q^{11} +5.89016 q^{13} +0.185367 q^{15} +7.06513 q^{17} -1.00000 q^{19} -3.92457 q^{21} -1.06348 q^{23} -4.95074 q^{25} -4.42846 q^{27} -7.62662 q^{29} -0.901295 q^{31} +0.835165 q^{33} -1.04299 q^{35} -2.71758 q^{37} +4.91925 q^{39} +0.788714 q^{41} -0.714571 q^{43} -0.511046 q^{45} -3.96368 q^{47} +15.0820 q^{49} +5.90055 q^{51} -9.69714 q^{53} +0.221953 q^{55} -0.835165 q^{57} +7.33476 q^{59} +8.15179 q^{61} +10.8198 q^{63} +1.30734 q^{65} -7.86697 q^{67} -0.888178 q^{69} +3.13400 q^{71} -6.49076 q^{73} -4.13468 q^{75} -4.69915 q^{77} -12.0148 q^{79} +3.20900 q^{81} -16.3902 q^{83} +1.56812 q^{85} -6.36948 q^{87} -12.9487 q^{89} -27.6788 q^{91} -0.752730 q^{93} -0.221953 q^{95} +8.14414 q^{97} -2.30250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * q^99

Introduction

L-functions

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