LMFDB - Embedded newform 3344.2.a.x.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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 $ x^6 - 2x^5 - 12x^4 + 28x^3 + 16x^2 - 60*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.18868$ 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)

$f(q)$	$=$	$q+2.18868 q^{3} -3.62873 q^{5} +4.31127 q^{7} +1.79031 q^{9} +O(q^{10})$	$q+2.18868 q^{3} -3.62873 q^{5} +4.31127 q^{7} +1.79031 q^{9} -1.00000 q^{11} -6.10370 q^{13} -7.94212 q^{15} -0.694815 q^{17} -1.00000 q^{19} +9.43598 q^{21} +7.79640 q^{23} +8.16767 q^{25} -2.64762 q^{27} +6.02709 q^{29} +6.26529 q^{31} -2.18868 q^{33} -15.6444 q^{35} +10.1139 q^{37} -13.3590 q^{39} -2.04380 q^{41} +2.16767 q^{43} -6.49655 q^{45} +10.2074 q^{47} +11.5870 q^{49} -1.52073 q^{51} -1.95798 q^{53} +3.62873 q^{55} -2.18868 q^{57} -5.89831 q^{59} +6.84803 q^{61} +7.71852 q^{63} +22.1487 q^{65} +8.35725 q^{67} +17.0638 q^{69} -0.566034 q^{71} +7.29948 q^{73} +17.8764 q^{75} -4.31127 q^{77} +13.9103 q^{79} -11.1657 q^{81} +7.05772 q^{83} +2.52129 q^{85} +13.1914 q^{87} -14.3590 q^{89} -26.3147 q^{91} +13.7127 q^{93} +3.62873 q^{95} -2.89842 q^{97} -1.79031 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * 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
$3$	2.18868	1.26363	0.631817	−	0.775118i	$-0.282309\pi$
$3$	2.18868	1.26363	0.631817	+	0.775118i	$0.282309\pi$
$4$	0	0
$5$	−3.62873	−1.62282	−0.811408	−	0.584480i	$-0.801299\pi$
$5$	−3.62873	−1.62282	−0.811408	+	0.584480i	$0.801299\pi$
$6$	0	0
$7$	4.31127	1.62951	0.814753	−	0.579808i	$-0.196872\pi$
$7$	4.31127	1.62951	0.814753	+	0.579808i	$0.196872\pi$
$8$	0	0
$9$	1.79031	0.596771
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.10370	−1.69286	−0.846431	−	0.532498i	$-0.821253\pi$
$13$	−6.10370	−1.69286	−0.846431	+	0.532498i	$0.821253\pi$
$14$	0	0
$15$	−7.94212	−2.05065
$16$	0	0
$17$	−0.694815	−0.168517	−0.0842587	−	0.996444i	$-0.526852\pi$
$17$	−0.694815	−0.168517	−0.0842587	+	0.996444i	$0.526852\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	9.43598	2.05910
$22$	0	0
$23$	7.79640	1.62566	0.812830	−	0.582500i	$-0.197926\pi$
$23$	7.79640	1.62566	0.812830	+	0.582500i	$0.197926\pi$
$24$	0	0
$25$	8.16767	1.63353
$26$	0	0
$27$	−2.64762	−0.509534
$28$	0	0
$29$	6.02709	1.11920	0.559602	−	0.828762i	$-0.310954\pi$
$29$	6.02709	1.11920	0.559602	+	0.828762i	$0.310954\pi$
$30$	0	0
$31$	6.26529	1.12528	0.562639	−	0.826703i	$-0.309786\pi$
$31$	6.26529	1.12528	0.562639	+	0.826703i	$0.309786\pi$
$32$	0	0
$33$	−2.18868	−0.381000
$34$	0	0
$35$	−15.6444	−2.64439
$36$	0	0
$37$	10.1139	1.66271	0.831354	−	0.555744i	$-0.187566\pi$
$37$	10.1139	1.66271	0.831354	+	0.555744i	$0.187566\pi$
$38$	0	0
$39$	−13.3590	−2.13916
$40$	0	0
$41$	−2.04380	−0.319189	−0.159594	−	0.987183i	$-0.551019\pi$
$41$	−2.04380	−0.319189	−0.159594	+	0.987183i	$0.551019\pi$
$42$	0	0
$43$	2.16767	0.330566	0.165283	−	0.986246i	$-0.447146\pi$
$43$	2.16767	0.330566	0.165283	+	0.986246i	$0.447146\pi$
$44$	0	0
$45$	−6.49655	−0.968449
$46$	0	0
$47$	10.2074	1.48890	0.744451	−	0.667677i	$-0.232711\pi$
$47$	10.2074	1.48890	0.744451	+	0.667677i	$0.232711\pi$
$48$	0	0
$49$	11.5870	1.65529
$50$	0	0
$51$	−1.52073	−0.212944
$52$	0	0
$53$	−1.95798	−0.268949	−0.134475	−	0.990917i	$-0.542935\pi$
$53$	−1.95798	−0.268949	−0.134475	+	0.990917i	$0.542935\pi$
$54$	0	0
$55$	3.62873	0.489298
$56$	0	0
$57$	−2.18868	−0.289897
$58$	0	0
$59$	−5.89831	−0.767895	−0.383948	−	0.923355i	$-0.625436\pi$
$59$	−5.89831	−0.767895	−0.383948	+	0.923355i	$0.625436\pi$
$60$	0	0
$61$	6.84803	0.876800	0.438400	−	0.898780i	$-0.355545\pi$
$61$	6.84803	0.876800	0.438400	+	0.898780i	$0.355545\pi$
$62$	0	0
$63$	7.71852	0.972442
$64$	0	0
$65$	22.1487	2.74720
$66$	0	0
$67$	8.35725	1.02100	0.510501	−	0.859877i	$-0.329460\pi$
$67$	8.35725	1.02100	0.510501	+	0.859877i	$0.329460\pi$
$68$	0	0
$69$	17.0638	2.05424
$70$	0	0
$71$	−0.566034	−0.0671759	−0.0335880	−	0.999436i	$-0.510693\pi$
$71$	−0.566034	−0.0671759	−0.0335880	+	0.999436i	$0.510693\pi$
$72$	0	0
$73$	7.29948	0.854339	0.427170	−	0.904172i	$-0.359511\pi$
$73$	7.29948	0.854339	0.427170	+	0.904172i	$0.359511\pi$
$74$	0	0
$75$	17.8764	2.06419
$76$	0	0
$77$	−4.31127	−0.491315
$78$	0	0
$79$	13.9103	1.56503	0.782513	−	0.622635i	$-0.213938\pi$
$79$	13.9103	1.56503	0.782513	+	0.622635i	$0.213938\pi$
$80$	0	0
$81$	−11.1657	−1.24064
$82$	0	0
$83$	7.05772	0.774685	0.387343	−	0.921936i	$-0.373393\pi$
$83$	7.05772	0.774685	0.387343	+	0.921936i	$0.373393\pi$
$84$	0	0
$85$	2.52129	0.273473
$86$	0	0
$87$	13.1914	1.41426
$88$	0	0
$89$	−14.3590	−1.52205	−0.761027	−	0.648720i	$-0.775305\pi$
$89$	−14.3590	−1.52205	−0.761027	+	0.648720i	$0.775305\pi$
$90$	0	0
$91$	−26.3147	−2.75853
$92$	0	0
$93$	13.7127	1.42194
$94$	0	0
$95$	3.62873	0.372300
$96$	0	0
$97$	−2.89842	−0.294290	−0.147145	−	0.989115i	$-0.547008\pi$
$97$	−2.89842	−0.294290	−0.147145	+	0.989115i	$0.547008\pi$
$98$	0	0
$99$	−1.79031	−0.179933
$100$	0	0
$101$	−7.57491	−0.753732	−0.376866	−	0.926268i	$-0.622998\pi$
$101$	−7.57491	−0.753732	−0.376866	+	0.926268i	$0.622998\pi$
$102$	0	0
$103$	−3.29026	−0.324199	−0.162099	−	0.986774i	$-0.551827\pi$
$103$	−3.29026	−0.324199	−0.162099	+	0.986774i	$0.551827\pi$
$104$	0	0
$105$	−34.2406	−3.34154
$106$	0	0
$107$	−10.9079	−1.05451	−0.527255	−	0.849707i	$-0.676779\pi$
$107$	−10.9079	−1.05451	−0.527255	+	0.849707i	$0.676779\pi$
$108$	0	0
$109$	5.42741	0.519852	0.259926	−	0.965629i	$-0.416302\pi$
$109$	5.42741	0.519852	0.259926	+	0.965629i	$0.416302\pi$
$110$	0	0
$111$	22.1360	2.10105
$112$	0	0
$113$	11.9439	1.12359	0.561794	−	0.827277i	$-0.310111\pi$
$113$	11.9439	1.12359	0.561794	+	0.827277i	$0.310111\pi$
$114$	0	0
$115$	−28.2910	−2.63815
$116$	0	0
$117$	−10.9275	−1.01025
$118$	0	0
$119$	−2.99553	−0.274600
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.47323	−0.403337
$124$	0	0
$125$	−11.4946	−1.02811
$126$	0	0
$127$	−7.72813	−0.685761	−0.342880	−	0.939379i	$-0.611403\pi$
$127$	−7.72813	−0.685761	−0.342880	+	0.939379i	$0.611403\pi$
$128$	0	0
$129$	4.74433	0.417715
$130$	0	0
$131$	10.5909	0.925335	0.462668	−	0.886532i	$-0.346892\pi$
$131$	10.5909	0.925335	0.462668	+	0.886532i	$0.346892\pi$
$132$	0	0
$133$	−4.31127	−0.373834
$134$	0	0
$135$	9.60749	0.826881
$136$	0	0
$137$	−8.74829	−0.747417	−0.373708	−	0.927546i	$-0.621914\pi$
$137$	−8.74829	−0.747417	−0.373708	+	0.927546i	$0.621914\pi$
$138$	0	0
$139$	−10.2696	−0.871055	−0.435527	−	0.900175i	$-0.643438\pi$
$139$	−10.2696	−0.871055	−0.435527	+	0.900175i	$0.643438\pi$
$140$	0	0
$141$	22.3407	1.88143
$142$	0	0
$143$	6.10370	0.510417
$144$	0	0
$145$	−21.8707	−1.81626
$146$	0	0
$147$	25.3603	2.09168
$148$	0	0
$149$	−21.5725	−1.76729	−0.883643	−	0.468161i	$-0.844917\pi$
$149$	−21.5725	−1.76729	−0.883643	+	0.468161i	$0.844917\pi$
$150$	0	0
$151$	0.461397	0.0375479	0.0187740	−	0.999824i	$-0.494024\pi$
$151$	0.461397	0.0375479	0.0187740	+	0.999824i	$0.494024\pi$
$152$	0	0
$153$	−1.24394	−0.100566
$154$	0	0
$155$	−22.7350	−1.82612
$156$	0	0
$157$	−4.52449	−0.361093	−0.180547	−	0.983566i	$-0.557787\pi$
$157$	−4.52449	−0.361093	−0.180547	+	0.983566i	$0.557787\pi$
$158$	0	0
$159$	−4.28539	−0.339853
$160$	0	0
$161$	33.6124	2.64903
$162$	0	0
$163$	−11.9622	−0.936953	−0.468477	−	0.883476i	$-0.655197\pi$
$163$	−11.9622	−0.936953	−0.468477	+	0.883476i	$0.655197\pi$
$164$	0	0
$165$	7.94212	0.618293
$166$	0	0
$167$	9.83576	0.761114	0.380557	−	0.924758i	$-0.375732\pi$
$167$	9.83576	0.761114	0.380557	+	0.924758i	$0.375732\pi$
$168$	0	0
$169$	24.2552	1.86578
$170$	0	0
$171$	−1.79031	−0.136909
$172$	0	0
$173$	−12.7818	−0.971783	−0.485891	−	0.874019i	$-0.661505\pi$
$173$	−12.7818	−0.971783	−0.485891	+	0.874019i	$0.661505\pi$
$174$	0	0
$175$	35.2130	2.66185
$176$	0	0
$177$	−12.9095	−0.970338
$178$	0	0
$179$	8.13311	0.607897	0.303949	−	0.952688i	$-0.401695\pi$
$179$	8.13311	0.607897	0.303949	+	0.952688i	$0.401695\pi$
$180$	0	0
$181$	18.5543	1.37913	0.689564	−	0.724225i	$-0.257802\pi$
$181$	18.5543	1.37913	0.689564	+	0.724225i	$0.257802\pi$
$182$	0	0
$183$	14.9881	1.10795
$184$	0	0
$185$	−36.7004	−2.69827
$186$	0	0
$187$	0.694815	0.0508099
$188$	0	0
$189$	−11.4146	−0.830290
$190$	0	0
$191$	10.6265	0.768910	0.384455	−	0.923144i	$-0.374389\pi$
$191$	10.6265	0.768910	0.384455	+	0.923144i	$0.374389\pi$
$192$	0	0
$193$	−0.759689	−0.0546836	−0.0273418	−	0.999626i	$-0.508704\pi$
$193$	−0.759689	−0.0546836	−0.0273418	+	0.999626i	$0.508704\pi$
$194$	0	0
$195$	48.4763	3.47146
$196$	0	0
$197$	1.68678	0.120178	0.0600891	−	0.998193i	$-0.480862\pi$
$197$	1.68678	0.120178	0.0600891	+	0.998193i	$0.480862\pi$
$198$	0	0
$199$	−24.2153	−1.71658	−0.858290	−	0.513166i	$-0.828473\pi$
$199$	−24.2153	−1.71658	−0.858290	+	0.513166i	$0.828473\pi$
$200$	0	0
$201$	18.2913	1.29017
$202$	0	0
$203$	25.9844	1.82375
$204$	0	0
$205$	7.41641	0.517984
$206$	0	0
$207$	13.9580	0.970147
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	28.4049	1.95547	0.977735	−	0.209841i	$-0.0672947\pi$
$211$	28.4049	1.95547	0.977735	+	0.209841i	$0.0672947\pi$
$212$	0	0
$213$	−1.23887	−0.0848858
$214$	0	0
$215$	−7.86588	−0.536448
$216$	0	0
$217$	27.0113	1.83365
$218$	0	0
$219$	15.9762	1.07957
$220$	0	0
$221$	4.24094	0.285277
$222$	0	0
$223$	8.82043	0.590660	0.295330	−	0.955395i	$-0.404570\pi$
$223$	8.82043	0.590660	0.295330	+	0.955395i	$0.404570\pi$
$224$	0	0
$225$	14.6227	0.974845
$226$	0	0
$227$	17.2796	1.14689	0.573443	−	0.819246i	$-0.305607\pi$
$227$	17.2796	1.14689	0.573443	+	0.819246i	$0.305607\pi$
$228$	0	0
$229$	2.15539	0.142432	0.0712162	−	0.997461i	$-0.477312\pi$
$229$	2.15539	0.142432	0.0712162	+	0.997461i	$0.477312\pi$
$230$	0	0
$231$	−9.43598	−0.620842
$232$	0	0
$233$	11.9733	0.784398	0.392199	−	0.919880i	$-0.371715\pi$
$233$	11.9733	0.784398	0.392199	+	0.919880i	$0.371715\pi$
$234$	0	0
$235$	−37.0399	−2.41622
$236$	0	0
$237$	30.4451	1.97762
$238$	0	0
$239$	−18.7692	−1.21408	−0.607038	−	0.794673i	$-0.707643\pi$
$239$	−18.7692	−1.21408	−0.607038	+	0.794673i	$0.707643\pi$
$240$	0	0
$241$	14.9468	0.962807	0.481404	−	0.876499i	$-0.340127\pi$
$241$	14.9468	0.962807	0.481404	+	0.876499i	$0.340127\pi$
$242$	0	0
$243$	−16.4953	−1.05817
$244$	0	0
$245$	−42.0462	−2.68624
$246$	0	0
$247$	6.10370	0.388369
$248$	0	0
$249$	15.4471	0.978918
$250$	0	0
$251$	12.1955	0.769771	0.384885	−	0.922964i	$-0.374241\pi$
$251$	12.1955	0.769771	0.384885	+	0.922964i	$0.374241\pi$
$252$	0	0
$253$	−7.79640	−0.490155
$254$	0	0
$255$	5.51830	0.345569
$256$	0	0
$257$	−6.96669	−0.434570	−0.217285	−	0.976108i	$-0.569720\pi$
$257$	−6.96669	−0.434570	−0.217285	+	0.976108i	$0.569720\pi$
$258$	0	0
$259$	43.6036	2.70939
$260$	0	0
$261$	10.7904	0.667908
$262$	0	0
$263$	−25.0396	−1.54401	−0.772005	−	0.635617i	$-0.780746\pi$
$263$	−25.0396	−1.54401	−0.772005	+	0.635617i	$0.780746\pi$
$264$	0	0
$265$	7.10498	0.436455
$266$	0	0
$267$	−31.4273	−1.92332
$268$	0	0
$269$	1.58130	0.0964134	0.0482067	−	0.998837i	$-0.484649\pi$
$269$	1.58130	0.0964134	0.0482067	+	0.998837i	$0.484649\pi$
$270$	0	0
$271$	−5.57867	−0.338880	−0.169440	−	0.985540i	$-0.554196\pi$
$271$	−5.57867	−0.338880	−0.169440	+	0.985540i	$0.554196\pi$
$272$	0	0
$273$	−57.5944	−3.48577
$274$	0	0
$275$	−8.16767	−0.492529
$276$	0	0
$277$	30.5385	1.83488	0.917440	−	0.397874i	$-0.130252\pi$
$277$	30.5385	1.83488	0.917440	+	0.397874i	$0.130252\pi$
$278$	0	0
$279$	11.2168	0.671533
$280$	0	0
$281$	−15.0369	−0.897029	−0.448514	−	0.893776i	$-0.648047\pi$
$281$	−15.0369	−0.897029	−0.448514	+	0.893776i	$0.648047\pi$
$282$	0	0
$283$	−29.1158	−1.73076	−0.865378	−	0.501120i	$-0.832921\pi$
$283$	−29.1158	−1.73076	−0.865378	+	0.501120i	$0.832921\pi$
$284$	0	0
$285$	7.94212	0.470450
$286$	0	0
$287$	−8.81139	−0.520120
$288$	0	0
$289$	−16.5172	−0.971602
$290$	0	0
$291$	−6.34371	−0.371875
$292$	0	0
$293$	5.88816	0.343990	0.171995	−	0.985098i	$-0.444979\pi$
$293$	5.88816	0.343990	0.171995	+	0.985098i	$0.444979\pi$
$294$	0	0
$295$	21.4034	1.24615
$296$	0	0
$297$	2.64762	0.153630
$298$	0	0
$299$	−47.5869	−2.75202
$300$	0	0
$301$	9.34540	0.538660
$302$	0	0
$303$	−16.5791	−0.952442
$304$	0	0
$305$	−24.8496	−1.42289
$306$	0	0
$307$	−19.2140	−1.09660	−0.548300	−	0.836282i	$-0.684725\pi$
$307$	−19.2140	−1.09660	−0.548300	+	0.836282i	$0.684725\pi$
$308$	0	0
$309$	−7.20132	−0.409669
$310$	0	0
$311$	−26.7459	−1.51662	−0.758310	−	0.651894i	$-0.773975\pi$
$311$	−26.7459	−1.51662	−0.758310	+	0.651894i	$0.773975\pi$
$312$	0	0
$313$	−16.0435	−0.906834	−0.453417	−	0.891298i	$-0.649795\pi$
$313$	−16.0435	−0.906834	−0.453417	+	0.891298i	$0.649795\pi$
$314$	0	0
$315$	−28.0084	−1.57809
$316$	0	0
$317$	18.5664	1.04280	0.521398	−	0.853314i	$-0.325411\pi$
$317$	18.5664	1.04280	0.521398	+	0.853314i	$0.325411\pi$
$318$	0	0
$319$	−6.02709	−0.337453
$320$	0	0
$321$	−23.8739	−1.33251
$322$	0	0
$323$	0.694815	0.0386605
$324$	0	0
$325$	−49.8530	−2.76535
$326$	0	0
$327$	11.8789	0.656902
$328$	0	0
$329$	44.0069	2.42618
$330$	0	0
$331$	34.6196	1.90286	0.951431	−	0.307861i	$-0.0996132\pi$
$331$	34.6196	1.90286	0.951431	+	0.307861i	$0.0996132\pi$
$332$	0	0
$333$	18.1070	0.992255
$334$	0	0
$335$	−30.3262	−1.65690
$336$	0	0
$337$	2.84625	0.155045	0.0775224	−	0.996991i	$-0.475299\pi$
$337$	2.84625	0.155045	0.0775224	+	0.996991i	$0.475299\pi$
$338$	0	0
$339$	26.1414	1.41980
$340$	0	0
$341$	−6.26529	−0.339284
$342$	0	0
$343$	19.7760	1.06780
$344$	0	0
$345$	−61.9199	−3.33366
$346$	0	0
$347$	−10.0153	−0.537651	−0.268826	−	0.963189i	$-0.586636\pi$
$347$	−10.0153	−0.537651	−0.268826	+	0.963189i	$0.586636\pi$
$348$	0	0
$349$	−0.712288	−0.0381279	−0.0190640	−	0.999818i	$-0.506069\pi$
$349$	−0.712288	−0.0381279	−0.0190640	+	0.999818i	$0.506069\pi$
$350$	0	0
$351$	16.1603	0.862571
$352$	0	0
$353$	32.4528	1.72729	0.863645	−	0.504101i	$-0.168176\pi$
$353$	32.4528	1.72729	0.863645	+	0.504101i	$0.168176\pi$
$354$	0	0
$355$	2.05398	0.109014
$356$	0	0
$357$	−6.55626	−0.346994
$358$	0	0
$359$	−19.6749	−1.03840	−0.519200	−	0.854653i	$-0.673770\pi$
$359$	−19.6749	−1.03840	−0.519200	+	0.854653i	$0.673770\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.18868	0.114876
$364$	0	0
$365$	−26.4878	−1.38644
$366$	0	0
$367$	31.5887	1.64891	0.824457	−	0.565924i	$-0.191480\pi$
$367$	31.5887	1.64891	0.824457	+	0.565924i	$0.191480\pi$
$368$	0	0
$369$	−3.65905	−0.190482
$370$	0	0
$371$	−8.44138	−0.438255
$372$	0	0
$373$	20.4557	1.05916	0.529579	−	0.848260i	$-0.322350\pi$
$373$	20.4557	1.05916	0.529579	+	0.848260i	$0.322350\pi$
$374$	0	0
$375$	−25.1580	−1.29915
$376$	0	0
$377$	−36.7876	−1.89466
$378$	0	0
$379$	−6.22508	−0.319761	−0.159880	−	0.987136i	$-0.551111\pi$
$379$	−6.22508	−0.319761	−0.159880	+	0.987136i	$0.551111\pi$
$380$	0	0
$381$	−16.9144	−0.866550
$382$	0	0
$383$	0.592813	0.0302913	0.0151457	−	0.999885i	$-0.495179\pi$
$383$	0.592813	0.0302913	0.0151457	+	0.999885i	$0.495179\pi$
$384$	0	0
$385$	15.6444	0.797314
$386$	0	0
$387$	3.88080	0.197272
$388$	0	0
$389$	32.2424	1.63475	0.817377	−	0.576103i	$-0.195427\pi$
$389$	32.2424	1.63475	0.817377	+	0.576103i	$0.195427\pi$
$390$	0	0
$391$	−5.41705	−0.273952
$392$	0	0
$393$	23.1802	1.16929
$394$	0	0
$395$	−50.4765	−2.53975
$396$	0	0
$397$	−25.1889	−1.26420	−0.632098	−	0.774888i	$-0.717806\pi$
$397$	−25.1889	−1.26420	−0.632098	+	0.774888i	$0.717806\pi$
$398$	0	0
$399$	−9.43598	−0.472390
$400$	0	0
$401$	−22.3272	−1.11497	−0.557484	−	0.830188i	$-0.688233\pi$
$401$	−22.3272	−1.11497	−0.557484	+	0.830188i	$0.688233\pi$
$402$	0	0
$403$	−38.2414	−1.90494
$404$	0	0
$405$	40.5174	2.01332
$406$	0	0
$407$	−10.1139	−0.501325
$408$	0	0
$409$	24.1005	1.19169	0.595847	−	0.803098i	$-0.296817\pi$
$409$	24.1005	1.19169	0.595847	+	0.803098i	$0.296817\pi$
$410$	0	0
$411$	−19.1472	−0.944461
$412$	0	0
$413$	−25.4292	−1.25129
$414$	0	0
$415$	−25.6105	−1.25717
$416$	0	0
$417$	−22.4768	−1.10069
$418$	0	0
$419$	17.0480	0.832851	0.416425	−	0.909170i	$-0.363283\pi$
$419$	17.0480	0.832851	0.416425	+	0.909170i	$0.363283\pi$
$420$	0	0
$421$	4.12963	0.201266	0.100633	−	0.994924i	$-0.467913\pi$
$421$	4.12963	0.201266	0.100633	+	0.994924i	$0.467913\pi$
$422$	0	0
$423$	18.2744	0.888533
$424$	0	0
$425$	−5.67502	−0.275279
$426$	0	0
$427$	29.5237	1.42875
$428$	0	0
$429$	13.3590	0.644980
$430$	0	0
$431$	5.86783	0.282643	0.141322	−	0.989964i	$-0.454865\pi$
$431$	5.86783	0.282643	0.141322	+	0.989964i	$0.454865\pi$
$432$	0	0
$433$	15.5840	0.748920	0.374460	−	0.927243i	$-0.377828\pi$
$433$	15.5840	0.748920	0.374460	+	0.927243i	$0.377828\pi$
$434$	0	0
$435$	−47.8679	−2.29509
$436$	0	0
$437$	−7.79640	−0.372952
$438$	0	0
$439$	−13.1937	−0.629699	−0.314850	−	0.949142i	$-0.601954\pi$
$439$	−13.1937	−0.629699	−0.314850	+	0.949142i	$0.601954\pi$
$440$	0	0
$441$	20.7444	0.987830
$442$	0	0
$443$	4.14411	0.196893	0.0984463	−	0.995142i	$-0.468613\pi$
$443$	4.14411	0.196893	0.0984463	+	0.995142i	$0.468613\pi$
$444$	0	0
$445$	52.1050	2.47002
$446$	0	0
$447$	−47.2152	−2.23320
$448$	0	0
$449$	−35.9971	−1.69881	−0.849405	−	0.527742i	$-0.823039\pi$
$449$	−35.9971	−1.69881	−0.849405	+	0.527742i	$0.823039\pi$
$450$	0	0
$451$	2.04380	0.0962390
$452$	0	0
$453$	1.00985	0.0474468
$454$	0	0
$455$	95.4889	4.47659
$456$	0	0
$457$	7.94870	0.371825	0.185912	−	0.982566i	$-0.440476\pi$
$457$	7.94870	0.371825	0.185912	+	0.982566i	$0.440476\pi$
$458$	0	0
$459$	1.83960	0.0858654
$460$	0	0
$461$	−9.33968	−0.434992	−0.217496	−	0.976061i	$-0.569789\pi$
$461$	−9.33968	−0.434992	−0.217496	+	0.976061i	$0.569789\pi$
$462$	0	0
$463$	−13.0328	−0.605686	−0.302843	−	0.953040i	$-0.597936\pi$
$463$	−13.0328	−0.605686	−0.302843	+	0.953040i	$0.597936\pi$
$464$	0	0
$465$	−49.7596	−2.30755
$466$	0	0
$467$	−17.1169	−0.792076	−0.396038	−	0.918234i	$-0.629615\pi$
$467$	−17.1169	−0.792076	−0.396038	+	0.918234i	$0.629615\pi$
$468$	0	0
$469$	36.0304	1.66373
$470$	0	0
$471$	−9.90264	−0.456290
$472$	0	0
$473$	−2.16767	−0.0996695
$474$	0	0
$475$	−8.16767	−0.374758
$476$	0	0
$477$	−3.50539	−0.160501
$478$	0	0
$479$	−39.8917	−1.82270	−0.911349	−	0.411635i	$-0.864958\pi$
$479$	−39.8917	−1.82270	−0.911349	+	0.411635i	$0.864958\pi$
$480$	0	0
$481$	−61.7319	−2.81473
$482$	0	0
$483$	73.5666	3.34740
$484$	0	0
$485$	10.5176	0.477578
$486$	0	0
$487$	21.3504	0.967478	0.483739	−	0.875212i	$-0.339278\pi$
$487$	21.3504	0.967478	0.483739	+	0.875212i	$0.339278\pi$
$488$	0	0
$489$	−26.1814	−1.18397
$490$	0	0
$491$	17.0293	0.768524	0.384262	−	0.923224i	$-0.374456\pi$
$491$	17.0293	0.768524	0.384262	+	0.923224i	$0.374456\pi$
$492$	0	0
$493$	−4.18771	−0.188605
$494$	0	0
$495$	6.49655	0.291998
$496$	0	0
$497$	−2.44033	−0.109464
$498$	0	0
$499$	4.94667	0.221443	0.110722	−	0.993851i	$-0.464684\pi$
$499$	4.94667	0.221443	0.110722	+	0.993851i	$0.464684\pi$
$500$	0	0
$501$	21.5273	0.961769
$502$	0	0
$503$	3.22196	0.143660	0.0718301	−	0.997417i	$-0.477116\pi$
$503$	3.22196	0.143660	0.0718301	+	0.997417i	$0.477116\pi$
$504$	0	0
$505$	27.4873	1.22317
$506$	0	0
$507$	53.0868	2.35767
$508$	0	0
$509$	10.1751	0.451003	0.225502	−	0.974243i	$-0.427598\pi$
$509$	10.1751	0.451003	0.225502	+	0.974243i	$0.427598\pi$
$510$	0	0
$511$	31.4700	1.39215
$512$	0	0
$513$	2.64762	0.116895
$514$	0	0
$515$	11.9395	0.526115
$516$	0	0
$517$	−10.2074	−0.448921
$518$	0	0
$519$	−27.9753	−1.22798
$520$	0	0
$521$	−33.3792	−1.46237	−0.731186	−	0.682179i	$-0.761033\pi$
$521$	−33.3792	−1.46237	−0.731186	+	0.682179i	$0.761033\pi$
$522$	0	0
$523$	2.17534	0.0951208	0.0475604	−	0.998868i	$-0.484855\pi$
$523$	2.17534	0.0951208	0.0475604	+	0.998868i	$0.484855\pi$
$524$	0	0
$525$	77.0700	3.36361
$526$	0	0
$527$	−4.35321	−0.189629
$528$	0	0
$529$	37.7838	1.64277
$530$	0	0
$531$	−10.5598	−0.458257
$532$	0	0
$533$	12.4748	0.540342
$534$	0	0
$535$	39.5819	1.71127
$536$	0	0
$537$	17.8008	0.768160
$538$	0	0
$539$	−11.5870	−0.499089
$540$	0	0
$541$	24.6958	1.06176	0.530878	−	0.847448i	$-0.321862\pi$
$541$	24.6958	1.06176	0.530878	+	0.847448i	$0.321862\pi$
$542$	0	0
$543$	40.6093	1.74271
$544$	0	0
$545$	−19.6946	−0.843624
$546$	0	0
$547$	9.80330	0.419159	0.209579	−	0.977792i	$-0.432791\pi$
$547$	9.80330	0.419159	0.209579	+	0.977792i	$0.432791\pi$
$548$	0	0
$549$	12.2601	0.523249
$550$	0	0
$551$	−6.02709	−0.256763
$552$	0	0
$553$	59.9708	2.55022
$554$	0	0
$555$	−80.3254	−3.40962
$556$	0	0
$557$	−12.1796	−0.516065	−0.258033	−	0.966136i	$-0.583074\pi$
$557$	−12.1796	−0.516065	−0.258033	+	0.966136i	$0.583074\pi$
$558$	0	0
$559$	−13.2308	−0.559603
$560$	0	0
$561$	1.52073	0.0642051
$562$	0	0
$563$	−10.6703	−0.449698	−0.224849	−	0.974394i	$-0.572189\pi$
$563$	−10.6703	−0.449698	−0.224849	+	0.974394i	$0.572189\pi$
$564$	0	0
$565$	−43.3412	−1.82338
$566$	0	0
$567$	−48.1384	−2.02162
$568$	0	0
$569$	−20.8260	−0.873072	−0.436536	−	0.899687i	$-0.643795\pi$
$569$	−20.8260	−0.873072	−0.436536	+	0.899687i	$0.643795\pi$
$570$	0	0
$571$	−21.5755	−0.902907	−0.451454	−	0.892295i	$-0.649094\pi$
$571$	−21.5755	−0.902907	−0.451454	+	0.892295i	$0.649094\pi$
$572$	0	0
$573$	23.2581	0.971621
$574$	0	0
$575$	63.6784	2.65557
$576$	0	0
$577$	15.5212	0.646157	0.323079	−	0.946372i	$-0.395282\pi$
$577$	15.5212	0.646157	0.323079	+	0.946372i	$0.395282\pi$
$578$	0	0
$579$	−1.66272	−0.0691001
$580$	0	0
$581$	30.4277	1.26235
$582$	0	0
$583$	1.95798	0.0810912
$584$	0	0
$585$	39.6530	1.63945
$586$	0	0
$587$	−4.08383	−0.168558	−0.0842789	−	0.996442i	$-0.526859\pi$
$587$	−4.08383	−0.168558	−0.0842789	+	0.996442i	$0.526859\pi$
$588$	0	0
$589$	−6.26529	−0.258157
$590$	0	0
$591$	3.69182	0.151861
$592$	0	0
$593$	−45.1039	−1.85220	−0.926098	−	0.377283i	$-0.876858\pi$
$593$	−45.1039	−1.85220	−0.926098	+	0.377283i	$0.876858\pi$
$594$	0	0
$595$	10.8700	0.445626
$596$	0	0
$597$	−52.9996	−2.16913
$598$	0	0
$599$	1.29026	0.0527186	0.0263593	−	0.999653i	$-0.491609\pi$
$599$	1.29026	0.0527186	0.0263593	+	0.999653i	$0.491609\pi$
$600$	0	0
$601$	−29.0011	−1.18298	−0.591489	−	0.806313i	$-0.701460\pi$
$601$	−29.0011	−1.18298	−0.591489	+	0.806313i	$0.701460\pi$
$602$	0	0
$603$	14.9621	0.609303
$604$	0	0
$605$	−3.62873	−0.147529
$606$	0	0
$607$	11.0119	0.446958	0.223479	−	0.974709i	$-0.428259\pi$
$607$	11.0119	0.446958	0.223479	+	0.974709i	$0.428259\pi$
$608$	0	0
$609$	56.8715	2.30455
$610$	0	0
$611$	−62.3029	−2.52051
$612$	0	0
$613$	1.94010	0.0783600	0.0391800	−	0.999232i	$-0.487525\pi$
$613$	1.94010	0.0783600	0.0391800	+	0.999232i	$0.487525\pi$
$614$	0	0
$615$	16.2321	0.654543
$616$	0	0
$617$	15.1760	0.610964	0.305482	−	0.952198i	$-0.401182\pi$
$617$	15.1760	0.610964	0.305482	+	0.952198i	$0.401182\pi$
$618$	0	0
$619$	−35.8548	−1.44113	−0.720564	−	0.693389i	$-0.756117\pi$
$619$	−35.8548	−1.44113	−0.720564	+	0.693389i	$0.756117\pi$
$620$	0	0
$621$	−20.6419	−0.828330
$622$	0	0
$623$	−61.9057	−2.48020
$624$	0	0
$625$	0.872460	0.0348984
$626$	0	0
$627$	2.18868	0.0874074
$628$	0	0
$629$	−7.02726	−0.280195
$630$	0	0
$631$	1.43092	0.0569640	0.0284820	−	0.999594i	$-0.490933\pi$
$631$	1.43092	0.0569640	0.0284820	+	0.999594i	$0.490933\pi$
$632$	0	0
$633$	62.1691	2.47100
$634$	0	0
$635$	28.0433	1.11286
$636$	0	0
$637$	−70.7239	−2.80218
$638$	0	0
$639$	−1.01338	−0.0400886
$640$	0	0
$641$	−27.8685	−1.10074	−0.550369	−	0.834922i	$-0.685513\pi$
$641$	−27.8685	−1.10074	−0.550369	+	0.834922i	$0.685513\pi$
$642$	0	0
$643$	1.90753	0.0752256	0.0376128	−	0.999292i	$-0.488025\pi$
$643$	1.90753	0.0752256	0.0376128	+	0.999292i	$0.488025\pi$
$644$	0	0
$645$	−17.2159	−0.677874
$646$	0	0
$647$	44.9164	1.76584	0.882922	−	0.469519i	$-0.155573\pi$
$647$	44.9164	1.76584	0.882922	+	0.469519i	$0.155573\pi$
$648$	0	0
$649$	5.89831	0.231529
$650$	0	0
$651$	59.1191	2.31706
$652$	0	0
$653$	23.6282	0.924642	0.462321	−	0.886713i	$-0.347017\pi$
$653$	23.6282	0.924642	0.462321	+	0.886713i	$0.347017\pi$
$654$	0	0
$655$	−38.4317	−1.50165
$656$	0	0
$657$	13.0683	0.509844
$658$	0	0
$659$	10.7391	0.418334	0.209167	−	0.977880i	$-0.432925\pi$
$659$	10.7391	0.418334	0.209167	+	0.977880i	$0.432925\pi$
$660$	0	0
$661$	−46.0837	−1.79245	−0.896224	−	0.443603i	$-0.853700\pi$
$661$	−46.0837	−1.79245	−0.896224	+	0.443603i	$0.853700\pi$
$662$	0	0
$663$	9.28206	0.360485
$664$	0	0
$665$	15.6444	0.606665
$666$	0	0
$667$	46.9896	1.81945
$668$	0	0
$669$	19.3051	0.746378
$670$	0	0
$671$	−6.84803	−0.264365
$672$	0	0
$673$	−22.6538	−0.873242	−0.436621	−	0.899646i	$-0.643825\pi$
$673$	−22.6538	−0.873242	−0.436621	+	0.899646i	$0.643825\pi$
$674$	0	0
$675$	−21.6249	−0.832341
$676$	0	0
$677$	24.3654	0.936438	0.468219	−	0.883613i	$-0.344896\pi$
$677$	24.3654	0.936438	0.468219	+	0.883613i	$0.344896\pi$
$678$	0	0
$679$	−12.4959	−0.479547
$680$	0	0
$681$	37.8194	1.44924
$682$	0	0
$683$	26.6471	1.01962	0.509811	−	0.860286i	$-0.329715\pi$
$683$	26.6471	1.01962	0.509811	+	0.860286i	$0.329715\pi$
$684$	0	0
$685$	31.7452	1.21292
$686$	0	0
$687$	4.71746	0.179982
$688$	0	0
$689$	11.9509	0.455294
$690$	0	0
$691$	−39.6176	−1.50713	−0.753563	−	0.657376i	$-0.771666\pi$
$691$	−39.6176	−1.50713	−0.753563	+	0.657376i	$0.771666\pi$
$692$	0	0
$693$	−7.71852	−0.293202
$694$	0	0
$695$	37.2655	1.41356
$696$	0	0
$697$	1.42007	0.0537888
$698$	0	0
$699$	26.2057	0.991192
$700$	0	0
$701$	1.92410	0.0726722	0.0363361	−	0.999340i	$-0.488431\pi$
$701$	1.92410	0.0726722	0.0363361	+	0.999340i	$0.488431\pi$
$702$	0	0
$703$	−10.1139	−0.381451
$704$	0	0
$705$	−81.0684	−3.05321
$706$	0	0
$707$	−32.6575	−1.22821
$708$	0	0
$709$	−6.91616	−0.259742	−0.129871	−	0.991531i	$-0.541456\pi$
$709$	−6.91616	−0.259742	−0.129871	+	0.991531i	$0.541456\pi$
$710$	0	0
$711$	24.9037	0.933961
$712$	0	0
$713$	48.8466	1.82932
$714$	0	0
$715$	−22.1487	−0.828313
$716$	0	0
$717$	−41.0797	−1.53415
$718$	0	0
$719$	12.3035	0.458843	0.229422	−	0.973327i	$-0.426317\pi$
$719$	12.3035	0.458843	0.229422	+	0.973327i	$0.426317\pi$
$720$	0	0
$721$	−14.1852	−0.528284
$722$	0	0
$723$	32.7137	1.21664
$724$	0	0
$725$	49.2273	1.82826
$726$	0	0
$727$	27.6098	1.02399	0.511995	−	0.858989i	$-0.328907\pi$
$727$	27.6098	1.02399	0.511995	+	0.858989i	$0.328907\pi$
$728$	0	0
$729$	−2.60577	−0.0965099
$730$	0	0
$731$	−1.50613	−0.0557062
$732$	0	0
$733$	−45.0293	−1.66319	−0.831597	−	0.555380i	$-0.812573\pi$
$733$	−45.0293	−1.66319	−0.831597	+	0.555380i	$0.812573\pi$
$734$	0	0
$735$	−92.0257	−3.39442
$736$	0	0
$737$	−8.35725	−0.307843
$738$	0	0
$739$	−37.8963	−1.39404	−0.697018	−	0.717053i	$-0.745490\pi$
$739$	−37.8963	−1.39404	−0.697018	+	0.717053i	$0.745490\pi$
$740$	0	0
$741$	13.3590	0.490757
$742$	0	0
$743$	−23.3103	−0.855172	−0.427586	−	0.903975i	$-0.640636\pi$
$743$	−23.3103	−0.855172	−0.427586	+	0.903975i	$0.640636\pi$
$744$	0	0
$745$	78.2807	2.86798
$746$	0	0
$747$	12.6355	0.462309
$748$	0	0
$749$	−47.0270	−1.71833
$750$	0	0
$751$	14.9643	0.546056	0.273028	−	0.962006i	$-0.411975\pi$
$751$	14.9643	0.546056	0.273028	+	0.962006i	$0.411975\pi$
$752$	0	0
$753$	26.6919	0.972709
$754$	0	0
$755$	−1.67428	−0.0609334
$756$	0	0
$757$	−13.3436	−0.484980	−0.242490	−	0.970154i	$-0.577964\pi$
$757$	−13.3436	−0.484980	−0.242490	+	0.970154i	$0.577964\pi$
$758$	0	0
$759$	−17.0638	−0.619377
$760$	0	0
$761$	−11.6962	−0.423986	−0.211993	−	0.977271i	$-0.567995\pi$
$761$	−11.6962	−0.423986	−0.211993	+	0.977271i	$0.567995\pi$
$762$	0	0
$763$	23.3990	0.847102
$764$	0	0
$765$	4.51390	0.163200
$766$	0	0
$767$	36.0015	1.29994
$768$	0	0
$769$	41.2234	1.48655	0.743276	−	0.668985i	$-0.233271\pi$
$769$	41.2234	1.48655	0.743276	+	0.668985i	$0.233271\pi$
$770$	0	0
$771$	−15.2478	−0.549137
$772$	0	0
$773$	−12.9999	−0.467574	−0.233787	−	0.972288i	$-0.575112\pi$
$773$	−12.9999	−0.467574	−0.233787	+	0.972288i	$0.575112\pi$
$774$	0	0
$775$	51.1728	1.83818
$776$	0	0
$777$	95.4341	3.42368
$778$	0	0
$779$	2.04380	0.0732269
$780$	0	0
$781$	0.566034	0.0202543
$782$	0	0
$783$	−15.9574	−0.570273
$784$	0	0
$785$	16.4181	0.585988
$786$	0	0
$787$	−4.63820	−0.165334	−0.0826669	−	0.996577i	$-0.526344\pi$
$787$	−4.63820	−0.165334	−0.0826669	+	0.996577i	$0.526344\pi$
$788$	0	0
$789$	−54.8037	−1.95106
$790$	0	0
$791$	51.4934	1.83089
$792$	0	0
$793$	−41.7983	−1.48430
$794$	0	0
$795$	15.5505	0.551520
$796$	0	0
$797$	−11.0271	−0.390600	−0.195300	−	0.980744i	$-0.562568\pi$
$797$	−11.0271	−0.390600	−0.195300	+	0.980744i	$0.562568\pi$
$798$	0	0
$799$	−7.09226	−0.250906
$800$	0	0
$801$	−25.7072	−0.908318
$802$	0	0
$803$	−7.29948	−0.257593
$804$	0	0
$805$	−121.970	−4.29888
$806$	0	0
$807$	3.46095	0.121831
$808$	0	0
$809$	−18.8659	−0.663290	−0.331645	−	0.943404i	$-0.607604\pi$
$809$	−18.8659	−0.663290	−0.331645	+	0.943404i	$0.607604\pi$
$810$	0	0
$811$	13.9642	0.490348	0.245174	−	0.969479i	$-0.421155\pi$
$811$	13.9642	0.490348	0.245174	+	0.969479i	$0.421155\pi$
$812$	0	0
$813$	−12.2099	−0.428221
$814$	0	0
$815$	43.4076	1.52050
$816$	0	0
$817$	−2.16767	−0.0758371
$818$	0	0
$819$	−47.1115	−1.64621
$820$	0	0
$821$	−30.8485	−1.07662	−0.538310	−	0.842747i	$-0.680937\pi$
$821$	−30.8485	−1.07662	−0.538310	+	0.842747i	$0.680937\pi$
$822$	0	0
$823$	9.48137	0.330500	0.165250	−	0.986252i	$-0.447157\pi$
$823$	9.48137	0.330500	0.165250	+	0.986252i	$0.447157\pi$
$824$	0	0
$825$	−17.8764	−0.622376
$826$	0	0
$827$	−16.7454	−0.582296	−0.291148	−	0.956678i	$-0.594037\pi$
$827$	−16.7454	−0.582296	−0.291148	+	0.956678i	$0.594037\pi$
$828$	0	0
$829$	1.59130	0.0552682	0.0276341	−	0.999618i	$-0.491203\pi$
$829$	1.59130	0.0552682	0.0276341	+	0.999618i	$0.491203\pi$
$830$	0	0
$831$	66.8389	2.31862
$832$	0	0
$833$	−8.05085	−0.278945
$834$	0	0
$835$	−35.6913	−1.23515
$836$	0	0
$837$	−16.5881	−0.573368
$838$	0	0
$839$	5.12222	0.176839	0.0884193	−	0.996083i	$-0.471818\pi$
$839$	5.12222	0.176839	0.0884193	+	0.996083i	$0.471818\pi$
$840$	0	0
$841$	7.32587	0.252616
$842$	0	0
$843$	−32.9110	−1.13352
$844$	0	0
$845$	−88.0154	−3.02782
$846$	0	0
$847$	4.31127	0.148137
$848$	0	0
$849$	−63.7251	−2.18704
$850$	0	0
$851$	78.8516	2.70300
$852$	0	0
$853$	−51.2020	−1.75312	−0.876561	−	0.481290i	$-0.840168\pi$
$853$	−51.2020	−1.75312	−0.876561	+	0.481290i	$0.840168\pi$
$854$	0	0
$855$	6.49655	0.222177
$856$	0	0
$857$	−7.92412	−0.270683	−0.135341	−	0.990799i	$-0.543213\pi$
$857$	−7.92412	−0.270683	−0.135341	+	0.990799i	$0.543213\pi$
$858$	0	0
$859$	39.4668	1.34659	0.673296	−	0.739373i	$-0.264878\pi$
$859$	39.4668	1.34659	0.673296	+	0.739373i	$0.264878\pi$
$860$	0	0
$861$	−19.2853	−0.657241
$862$	0	0
$863$	52.1231	1.77429	0.887145	−	0.461491i	$-0.152685\pi$
$863$	52.1231	1.77429	0.887145	+	0.461491i	$0.152685\pi$
$864$	0	0
$865$	46.3817	1.57703
$866$	0	0
$867$	−36.1509	−1.22775
$868$	0	0
$869$	−13.9103	−0.471873
$870$	0	0
$871$	−51.0102	−1.72841
$872$	0	0
$873$	−5.18907	−0.175624
$874$	0	0
$875$	−49.5563	−1.67531
$876$	0	0
$877$	20.6344	0.696773	0.348386	−	0.937351i	$-0.386730\pi$
$877$	20.6344	0.696773	0.348386	+	0.937351i	$0.386730\pi$
$878$	0	0
$879$	12.8873	0.434677
$880$	0	0
$881$	42.3008	1.42515	0.712574	−	0.701597i	$-0.247529\pi$
$881$	42.3008	1.42515	0.712574	+	0.701597i	$0.247529\pi$
$882$	0	0
$883$	47.1397	1.58638	0.793188	−	0.608977i	$-0.208420\pi$
$883$	47.1397	1.58638	0.793188	+	0.608977i	$0.208420\pi$
$884$	0	0
$885$	46.8451	1.57468
$886$	0	0
$887$	−5.41860	−0.181939	−0.0909694	−	0.995854i	$-0.528997\pi$
$887$	−5.41860	−0.181939	−0.0909694	+	0.995854i	$0.528997\pi$
$888$	0	0
$889$	−33.3180	−1.11745
$890$	0	0
$891$	11.1657	0.374066
$892$	0	0
$893$	−10.2074	−0.341578
$894$	0	0
$895$	−29.5129	−0.986506
$896$	0	0
$897$	−104.152	−3.47755
$898$	0	0
$899$	37.7615	1.25942
$900$	0	0
$901$	1.36043	0.0453226
$902$	0	0
$903$	20.4541	0.680669
$904$	0	0
$905$	−67.3284	−2.23807
$906$	0	0
$907$	−22.1148	−0.734309	−0.367154	−	0.930160i	$-0.619668\pi$
$907$	−22.1148	−0.734309	−0.367154	+	0.930160i	$0.619668\pi$
$908$	0	0
$909$	−13.5615	−0.449805
$910$	0	0
$911$	−18.0321	−0.597429	−0.298715	−	0.954343i	$-0.596558\pi$
$911$	−18.0321	−0.597429	−0.298715	+	0.954343i	$0.596558\pi$
$912$	0	0
$913$	−7.05772	−0.233576
$914$	0	0
$915$	−54.3879	−1.79801
$916$	0	0
$917$	45.6604	1.50784
$918$	0	0
$919$	36.9395	1.21852	0.609261	−	0.792970i	$-0.291466\pi$
$919$	36.9395	1.21852	0.609261	+	0.792970i	$0.291466\pi$
$920$	0	0
$921$	−42.0532	−1.38570
$922$	0	0
$923$	3.45490	0.113720
$924$	0	0
$925$	82.6066	2.71609
$926$	0	0
$927$	−5.89059	−0.193472
$928$	0	0
$929$	10.2792	0.337250	0.168625	−	0.985680i	$-0.446067\pi$
$929$	10.2792	0.337250	0.168625	+	0.985680i	$0.446067\pi$
$930$	0	0
$931$	−11.5870	−0.379750
$932$	0	0
$933$	−58.5382	−1.91645
$934$	0	0
$935$	−2.52129	−0.0824551
$936$	0	0
$937$	1.42527	0.0465615	0.0232807	−	0.999729i	$-0.492589\pi$
$937$	1.42527	0.0465615	0.0232807	+	0.999729i	$0.492589\pi$
$938$	0	0
$939$	−35.1141	−1.14591
$940$	0	0
$941$	32.0967	1.04632	0.523161	−	0.852234i	$-0.324753\pi$
$941$	32.0967	1.04632	0.523161	+	0.852234i	$0.324753\pi$
$942$	0	0
$943$	−15.9343	−0.518892
$944$	0	0
$945$	41.4205	1.34741
$946$	0	0
$947$	8.61405	0.279919	0.139959	−	0.990157i	$-0.455303\pi$
$947$	8.61405	0.279919	0.139959	+	0.990157i	$0.455303\pi$
$948$	0	0
$949$	−44.5538	−1.44628
$950$	0	0
$951$	40.6360	1.31771
$952$	0	0
$953$	6.48103	0.209941	0.104971	−	0.994475i	$-0.466525\pi$
$953$	6.48103	0.209941	0.104971	+	0.994475i	$0.466525\pi$
$954$	0	0
$955$	−38.5609	−1.24780
$956$	0	0
$957$	−13.1914	−0.426416
$958$	0	0
$959$	−37.7162	−1.21792
$960$	0	0
$961$	8.25380	0.266252
$962$	0	0
$963$	−19.5286	−0.629300
$964$	0	0
$965$	2.75671	0.0887415
$966$	0	0
$967$	−55.2454	−1.77657	−0.888286	−	0.459292i	$-0.848103\pi$
$967$	−55.2454	−1.77657	−0.888286	+	0.459292i	$0.848103\pi$
$968$	0	0
$969$	1.52073	0.0488528
$970$	0	0
$971$	−30.1518	−0.967618	−0.483809	−	0.875174i	$-0.660747\pi$
$971$	−30.1518	−0.967618	−0.483809	+	0.875174i	$0.660747\pi$
$972$	0	0
$973$	−44.2750	−1.41939
$974$	0	0
$975$	−109.112	−3.49439
$976$	0	0
$977$	22.6608	0.724984	0.362492	−	0.931987i	$-0.381926\pi$
$977$	22.6608	0.724984	0.362492	+	0.931987i	$0.381926\pi$
$978$	0	0
$979$	14.3590	0.458917
$980$	0	0
$981$	9.71676	0.310232
$982$	0	0
$983$	−17.3274	−0.552658	−0.276329	−	0.961063i	$-0.589118\pi$
$983$	−17.3274	−0.552658	−0.276329	+	0.961063i	$0.589118\pi$
$984$	0	0
$985$	−6.12087	−0.195027
$986$	0	0
$987$	96.3169	3.06580
$988$	0	0
$989$	16.9000	0.537389
$990$	0	0
$991$	−27.1931	−0.863818	−0.431909	−	0.901917i	$-0.642160\pi$
$991$	−27.1931	−0.863818	−0.431909	+	0.901917i	$0.642160\pi$
$992$	0	0
$993$	75.7711	2.40452
$994$	0	0
$995$	87.8709	2.78569
$996$	0	0
$997$	21.0570	0.666881	0.333441	−	0.942771i	$-0.391790\pi$
$997$	21.0570	0.666881	0.333441	+	0.942771i	$0.391790\pi$
$998$	0	0
$999$	−26.7776	−0.847207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.x.1.5		6
4.3	odd	2		836.2.a.d.1.2	✓	6
12.11	even	2		7524.2.a.r.1.6		6
44.43	even	2		9196.2.a.k.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.d.1.2	✓	6	4.3	odd	2
3344.2.a.x.1.5		6	1.1	even	1	trivial
7524.2.a.r.1.6		6	12.11	even	2
9196.2.a.k.1.2		6	44.43	even	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)

\(f(q)\)	\(=\)	\(q+2.18868 q^{3} -3.62873 q^{5} +4.31127 q^{7} +1.79031 q^{9} +O(q^{10})\)	\(q+2.18868 q^{3} -3.62873 q^{5} +4.31127 q^{7} +1.79031 q^{9} -1.00000 q^{11} -6.10370 q^{13} -7.94212 q^{15} -0.694815 q^{17} -1.00000 q^{19} +9.43598 q^{21} +7.79640 q^{23} +8.16767 q^{25} -2.64762 q^{27} +6.02709 q^{29} +6.26529 q^{31} -2.18868 q^{33} -15.6444 q^{35} +10.1139 q^{37} -13.3590 q^{39} -2.04380 q^{41} +2.16767 q^{43} -6.49655 q^{45} +10.2074 q^{47} +11.5870 q^{49} -1.52073 q^{51} -1.95798 q^{53} +3.62873 q^{55} -2.18868 q^{57} -5.89831 q^{59} +6.84803 q^{61} +7.71852 q^{63} +22.1487 q^{65} +8.35725 q^{67} +17.0638 q^{69} -0.566034 q^{71} +7.29948 q^{73} +17.8764 q^{75} -4.31127 q^{77} +13.9103 q^{79} -11.1657 q^{81} +7.05772 q^{83} +2.52129 q^{85} +13.1914 q^{87} -14.3590 q^{89} -26.3147 q^{91} +13.7127 q^{93} +3.62873 q^{95} -2.89842 q^{97} -1.79031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * q^99

Introduction

L-functions

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Varieties

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Database

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