LMFDB - Embedded newform 3344.2.a.z.1.2

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:	6.6.106392688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 $ x^6 - 2x^5 - 9x^4 + 12x^3 + 25x^2 - 10*x - 16
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			1.2
Root			$2.50058$ 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)

$f(q)$	$=$	$q-1.50058 q^{3} -2.88388 q^{5} -1.88388 q^{7} -0.748274 q^{9} +O(q^{10})$	$q-1.50058 q^{3} -2.88388 q^{5} -1.88388 q^{7} -0.748274 q^{9} +1.00000 q^{11} -1.08333 q^{13} +4.32748 q^{15} -4.80304 q^{17} -1.00000 q^{19} +2.82691 q^{21} -8.51652 q^{23} +3.31677 q^{25} +5.62457 q^{27} -7.97528 q^{29} +4.32748 q^{31} -1.50058 q^{33} +5.43289 q^{35} -7.21003 q^{37} +1.62562 q^{39} -8.45503 q^{41} -3.83075 q^{43} +2.15793 q^{45} -4.50575 q^{47} -3.45099 q^{49} +7.20732 q^{51} -6.88436 q^{53} -2.88388 q^{55} +1.50058 q^{57} -1.50998 q^{59} +3.50460 q^{61} +1.40966 q^{63} +3.12419 q^{65} +7.29221 q^{67} +12.7797 q^{69} +7.56562 q^{71} +9.30612 q^{73} -4.97707 q^{75} -1.88388 q^{77} -0.234538 q^{79} -6.19527 q^{81} +4.64910 q^{83} +13.8514 q^{85} +11.9675 q^{87} -7.24132 q^{89} +2.04086 q^{91} -6.49371 q^{93} +2.88388 q^{95} -8.05525 q^{97} -0.748274 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9	$ 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + q^15 - q^17 - 6 * q^19 + 5 * q^21 + 4 * q^23 + 5 * q^25 + 16 * q^27 - 7 * q^29 + q^31 + 4 * q^33 + 32 * q^35 + 5 * q^37 + 13 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 7 * q^49 + 35 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 18 * q^59 + 16 * q^61 + 6 * q^63 + 10 * q^65 + 18 * q^67 - 2 * q^69 + 7 * q^71 - 21 * q^73 + 23 * q^75 + 3 * q^77 + 22 * q^79 - 10 * q^81 + 16 * q^83 + 3 * q^87 - 13 * q^89 + 7 * q^91 - 9 * q^93 + 3 * q^95 - 15 * q^97 + 6 * 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
$3$	−1.50058	−0.866358	−0.433179	−	0.901308i	$-0.642608\pi$
$3$	−1.50058	−0.866358	−0.433179	+	0.901308i	$0.642608\pi$
$4$	0	0
$5$	−2.88388	−1.28971	−0.644856	−	0.764304i	$-0.723082\pi$
$5$	−2.88388	−1.28971	−0.644856	+	0.764304i	$0.723082\pi$
$6$	0	0
$7$	−1.88388	−0.712040	−0.356020	−	0.934478i	$-0.615867\pi$
$7$	−1.88388	−0.712040	−0.356020	+	0.934478i	$0.615867\pi$
$8$	0	0
$9$	−0.748274	−0.249425
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.08333	−0.300461	−0.150231	−	0.988651i	$-0.548002\pi$
$13$	−1.08333	−0.300461	−0.150231	+	0.988651i	$0.548002\pi$
$14$	0	0
$15$	4.32748	1.11735
$16$	0	0
$17$	−4.80304	−1.16491	−0.582454	−	0.812864i	$-0.697907\pi$
$17$	−4.80304	−1.16491	−0.582454	+	0.812864i	$0.697907\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.82691	0.616882
$22$	0	0
$23$	−8.51652	−1.77582	−0.887909	−	0.460020i	$-0.847842\pi$
$23$	−8.51652	−1.77582	−0.887909	+	0.460020i	$0.847842\pi$
$24$	0	0
$25$	3.31677	0.663355
$26$	0	0
$27$	5.62457	1.08245
$28$	0	0
$29$	−7.97528	−1.48097	−0.740486	−	0.672071i	$-0.765405\pi$
$29$	−7.97528	−1.48097	−0.740486	+	0.672071i	$0.765405\pi$
$30$	0	0
$31$	4.32748	0.777239	0.388619	−	0.921398i	$-0.372952\pi$
$31$	4.32748	0.777239	0.388619	+	0.921398i	$0.372952\pi$
$32$	0	0
$33$	−1.50058	−0.261217
$34$	0	0
$35$	5.43289	0.918326
$36$	0	0
$37$	−7.21003	−1.18532	−0.592661	−	0.805452i	$-0.701923\pi$
$37$	−7.21003	−1.18532	−0.592661	+	0.805452i	$0.701923\pi$
$38$	0	0
$39$	1.62562	0.260307
$40$	0	0
$41$	−8.45503	−1.32045	−0.660227	−	0.751066i	$-0.729540\pi$
$41$	−8.45503	−1.32045	−0.660227	+	0.751066i	$0.729540\pi$
$42$	0	0
$43$	−3.83075	−0.584184	−0.292092	−	0.956390i	$-0.594351\pi$
$43$	−3.83075	−0.584184	−0.292092	+	0.956390i	$0.594351\pi$
$44$	0	0
$45$	2.15793	0.321686
$46$	0	0
$47$	−4.50575	−0.657232	−0.328616	−	0.944464i	$-0.606582\pi$
$47$	−4.50575	−0.657232	−0.328616	+	0.944464i	$0.606582\pi$
$48$	0	0
$49$	−3.45099	−0.492999
$50$	0	0
$51$	7.20732	1.00923
$52$	0	0
$53$	−6.88436	−0.945640	−0.472820	−	0.881159i	$-0.656764\pi$
$53$	−6.88436	−0.945640	−0.472820	+	0.881159i	$0.656764\pi$
$54$	0	0
$55$	−2.88388	−0.388863
$56$	0	0
$57$	1.50058	0.198756
$58$	0	0
$59$	−1.50998	−0.196583	−0.0982913	−	0.995158i	$-0.531338\pi$
$59$	−1.50998	−0.196583	−0.0982913	+	0.995158i	$0.531338\pi$
$60$	0	0
$61$	3.50460	0.448718	0.224359	−	0.974507i	$-0.427971\pi$
$61$	3.50460	0.448718	0.224359	+	0.974507i	$0.427971\pi$
$62$	0	0
$63$	1.40966	0.177600
$64$	0	0
$65$	3.12419	0.387508
$66$	0	0
$67$	7.29221	0.890885	0.445442	−	0.895311i	$-0.353046\pi$
$67$	7.29221	0.890885	0.445442	+	0.895311i	$0.353046\pi$
$68$	0	0
$69$	12.7797	1.53849
$70$	0	0
$71$	7.56562	0.897874	0.448937	−	0.893563i	$-0.351803\pi$
$71$	7.56562	0.897874	0.448937	+	0.893563i	$0.351803\pi$
$72$	0	0
$73$	9.30612	1.08920	0.544600	−	0.838696i	$-0.316682\pi$
$73$	9.30612	1.08920	0.544600	+	0.838696i	$0.316682\pi$
$74$	0	0
$75$	−4.97707	−0.574702
$76$	0	0
$77$	−1.88388	−0.214688
$78$	0	0
$79$	−0.234538	−0.0263876	−0.0131938	−	0.999913i	$-0.504200\pi$
$79$	−0.234538	−0.0263876	−0.0131938	+	0.999913i	$0.504200\pi$
$80$	0	0
$81$	−6.19527	−0.688363
$82$	0	0
$83$	4.64910	0.510305	0.255153	−	0.966901i	$-0.417874\pi$
$83$	4.64910	0.510305	0.255153	+	0.966901i	$0.417874\pi$
$84$	0	0
$85$	13.8514	1.50239
$86$	0	0
$87$	11.9675	1.28305
$88$	0	0
$89$	−7.24132	−0.767578	−0.383789	−	0.923421i	$-0.625381\pi$
$89$	−7.24132	−0.767578	−0.383789	+	0.923421i	$0.625381\pi$
$90$	0	0
$91$	2.04086	0.213940
$92$	0	0
$93$	−6.49371	−0.673367
$94$	0	0
$95$	2.88388	0.295880
$96$	0	0
$97$	−8.05525	−0.817886	−0.408943	−	0.912560i	$-0.634103\pi$
$97$	−8.05525	−0.817886	−0.408943	+	0.912560i	$0.634103\pi$
$98$	0	0
$99$	−0.748274	−0.0752043
$100$	0	0
$101$	−6.06788	−0.603777	−0.301888	−	0.953343i	$-0.597617\pi$
$101$	−6.06788	−0.603777	−0.301888	+	0.953343i	$0.597617\pi$
$102$	0	0
$103$	−13.7616	−1.35597	−0.677987	−	0.735074i	$-0.737147\pi$
$103$	−13.7616	−1.35597	−0.677987	+	0.735074i	$0.737147\pi$
$104$	0	0
$105$	−8.15246	−0.795599
$106$	0	0
$107$	3.53873	0.342102	0.171051	−	0.985262i	$-0.445284\pi$
$107$	3.53873	0.342102	0.171051	+	0.985262i	$0.445284\pi$
$108$	0	0
$109$	5.68927	0.544933	0.272467	−	0.962165i	$-0.412161\pi$
$109$	5.68927	0.544933	0.272467	+	0.962165i	$0.412161\pi$
$110$	0	0
$111$	10.8192	1.02691
$112$	0	0
$113$	−5.78942	−0.544623	−0.272311	−	0.962209i	$-0.587788\pi$
$113$	−5.78942	−0.544623	−0.272311	+	0.962209i	$0.587788\pi$
$114$	0	0
$115$	24.5606	2.29029
$116$	0	0
$117$	0.810626	0.0749424
$118$	0	0
$119$	9.04835	0.829461
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.6874	1.14399
$124$	0	0
$125$	4.85423	0.434175
$126$	0	0
$127$	−9.09017	−0.806622	−0.403311	−	0.915063i	$-0.632141\pi$
$127$	−9.09017	−0.806622	−0.403311	+	0.915063i	$0.632141\pi$
$128$	0	0
$129$	5.74833	0.506112
$130$	0	0
$131$	19.8530	1.73457	0.867283	−	0.497815i	$-0.165864\pi$
$131$	19.8530	1.73457	0.867283	+	0.497815i	$0.165864\pi$
$132$	0	0
$133$	1.88388	0.163353
$134$	0	0
$135$	−16.2206	−1.39605
$136$	0	0
$137$	15.0136	1.28269	0.641347	−	0.767251i	$-0.278376\pi$
$137$	15.0136	1.28269	0.641347	+	0.767251i	$0.278376\pi$
$138$	0	0
$139$	−8.10427	−0.687395	−0.343698	−	0.939080i	$-0.611679\pi$
$139$	−8.10427	−0.687395	−0.343698	+	0.939080i	$0.611679\pi$
$140$	0	0
$141$	6.76122	0.569398
$142$	0	0
$143$	−1.08333	−0.0905924
$144$	0	0
$145$	22.9998	1.91003
$146$	0	0
$147$	5.17847	0.427113
$148$	0	0
$149$	13.1339	1.07597	0.537986	−	0.842954i	$-0.319185\pi$
$149$	13.1339	1.07597	0.537986	+	0.842954i	$0.319185\pi$
$150$	0	0
$151$	1.79627	0.146178	0.0730892	−	0.997325i	$-0.476714\pi$
$151$	1.79627	0.146178	0.0730892	+	0.997325i	$0.476714\pi$
$152$	0	0
$153$	3.59399	0.290557
$154$	0	0
$155$	−12.4799	−1.00241
$156$	0	0
$157$	16.9881	1.35580	0.677898	−	0.735156i	$-0.262891\pi$
$157$	16.9881	1.35580	0.677898	+	0.735156i	$0.262891\pi$
$158$	0	0
$159$	10.3305	0.819263
$160$	0	0
$161$	16.0441	1.26445
$162$	0	0
$163$	11.1884	0.876341	0.438170	−	0.898892i	$-0.355627\pi$
$163$	11.1884	0.876341	0.438170	+	0.898892i	$0.355627\pi$
$164$	0	0
$165$	4.32748	0.336894
$166$	0	0
$167$	−15.2962	−1.18366	−0.591828	−	0.806064i	$-0.701593\pi$
$167$	−15.2962	−1.18366	−0.591828	+	0.806064i	$0.701593\pi$
$168$	0	0
$169$	−11.8264	−0.909723
$170$	0	0
$171$	0.748274	0.0572219
$172$	0	0
$173$	−10.0067	−0.760798	−0.380399	−	0.924822i	$-0.624213\pi$
$173$	−10.0067	−0.760798	−0.380399	+	0.924822i	$0.624213\pi$
$174$	0	0
$175$	−6.24841	−0.472335
$176$	0	0
$177$	2.26584	0.170311
$178$	0	0
$179$	6.93780	0.518555	0.259278	−	0.965803i	$-0.416516\pi$
$179$	6.93780	0.518555	0.259278	+	0.965803i	$0.416516\pi$
$180$	0	0
$181$	13.8801	1.03170	0.515851	−	0.856678i	$-0.327476\pi$
$181$	13.8801	1.03170	0.515851	+	0.856678i	$0.327476\pi$
$182$	0	0
$183$	−5.25892	−0.388751
$184$	0	0
$185$	20.7929	1.52872
$186$	0	0
$187$	−4.80304	−0.351233
$188$	0	0
$189$	−10.5960	−0.770747
$190$	0	0
$191$	2.46370	0.178267	0.0891335	−	0.996020i	$-0.471590\pi$
$191$	2.46370	0.178267	0.0891335	+	0.996020i	$0.471590\pi$
$192$	0	0
$193$	11.6753	0.840409	0.420204	−	0.907430i	$-0.361958\pi$
$193$	11.6753	0.840409	0.420204	+	0.907430i	$0.361958\pi$
$194$	0	0
$195$	−4.68808	−0.335721
$196$	0	0
$197$	−19.0357	−1.35624	−0.678119	−	0.734952i	$-0.737205\pi$
$197$	−19.0357	−1.35624	−0.678119	+	0.734952i	$0.737205\pi$
$198$	0	0
$199$	−6.26600	−0.444185	−0.222092	−	0.975026i	$-0.571289\pi$
$199$	−6.26600	−0.444185	−0.222092	+	0.975026i	$0.571289\pi$
$200$	0	0
$201$	−10.9425	−0.771825
$202$	0	0
$203$	15.0245	1.05451
$204$	0	0
$205$	24.3833	1.70300
$206$	0	0
$207$	6.37269	0.442932
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	20.8366	1.43445	0.717226	−	0.696840i	$-0.245411\pi$
$211$	20.8366	1.43445	0.717226	+	0.696840i	$0.245411\pi$
$212$	0	0
$213$	−11.3528	−0.777880
$214$	0	0
$215$	11.0474	0.753429
$216$	0	0
$217$	−8.15246	−0.553425
$218$	0	0
$219$	−13.9645	−0.943636
$220$	0	0
$221$	5.20327	0.350010
$222$	0	0
$223$	−27.9736	−1.87325	−0.936624	−	0.350337i	$-0.886067\pi$
$223$	−27.9736	−1.87325	−0.936624	+	0.350337i	$0.886067\pi$
$224$	0	0
$225$	−2.48185	−0.165457
$226$	0	0
$227$	25.1395	1.66857	0.834283	−	0.551337i	$-0.185882\pi$
$227$	25.1395	1.66857	0.834283	+	0.551337i	$0.185882\pi$
$228$	0	0
$229$	21.9050	1.44752	0.723762	−	0.690050i	$-0.242411\pi$
$229$	21.9050	1.44752	0.723762	+	0.690050i	$0.242411\pi$
$230$	0	0
$231$	2.82691	0.185997
$232$	0	0
$233$	−16.1431	−1.05757	−0.528783	−	0.848757i	$-0.677352\pi$
$233$	−16.1431	−1.05757	−0.528783	+	0.848757i	$0.677352\pi$
$234$	0	0
$235$	12.9941	0.847639
$236$	0	0
$237$	0.351942	0.0228611
$238$	0	0
$239$	−0.996425	−0.0644534	−0.0322267	−	0.999481i	$-0.510260\pi$
$239$	−0.996425	−0.0644534	−0.0322267	+	0.999481i	$0.510260\pi$
$240$	0	0
$241$	−22.6038	−1.45604	−0.728020	−	0.685556i	$-0.759559\pi$
$241$	−22.6038	−1.45604	−0.728020	+	0.685556i	$0.759559\pi$
$242$	0	0
$243$	−7.57724	−0.486080
$244$	0	0
$245$	9.95225	0.635826
$246$	0	0
$247$	1.08333	0.0689305
$248$	0	0
$249$	−6.97633	−0.442107
$250$	0	0
$251$	−23.7686	−1.50026	−0.750131	−	0.661290i	$-0.770009\pi$
$251$	−23.7686	−1.50026	−0.750131	+	0.661290i	$0.770009\pi$
$252$	0	0
$253$	−8.51652	−0.535429
$254$	0	0
$255$	−20.7851	−1.30161
$256$	0	0
$257$	−1.27407	−0.0794741	−0.0397371	−	0.999210i	$-0.512652\pi$
$257$	−1.27407	−0.0794741	−0.0397371	+	0.999210i	$0.512652\pi$
$258$	0	0
$259$	13.5828	0.843997
$260$	0	0
$261$	5.96769	0.369391
$262$	0	0
$263$	−12.7381	−0.785466	−0.392733	−	0.919652i	$-0.628470\pi$
$263$	−12.7381	−0.785466	−0.392733	+	0.919652i	$0.628470\pi$
$264$	0	0
$265$	19.8537	1.21960
$266$	0	0
$267$	10.8661	0.664997
$268$	0	0
$269$	−7.92406	−0.483139	−0.241569	−	0.970384i	$-0.577662\pi$
$269$	−7.92406	−0.483139	−0.241569	+	0.970384i	$0.577662\pi$
$270$	0	0
$271$	−2.18100	−0.132486	−0.0662431	−	0.997804i	$-0.521101\pi$
$271$	−2.18100	−0.132486	−0.0662431	+	0.997804i	$0.521101\pi$
$272$	0	0
$273$	−3.06247	−0.185349
$274$	0	0
$275$	3.31677	0.200009
$276$	0	0
$277$	−13.3873	−0.804364	−0.402182	−	0.915560i	$-0.631748\pi$
$277$	−13.3873	−0.804364	−0.402182	+	0.915560i	$0.631748\pi$
$278$	0	0
$279$	−3.23814	−0.193862
$280$	0	0
$281$	22.5787	1.34693	0.673465	−	0.739219i	$-0.264805\pi$
$281$	22.5787	1.34693	0.673465	+	0.739219i	$0.264805\pi$
$282$	0	0
$283$	−14.1317	−0.840044	−0.420022	−	0.907514i	$-0.637978\pi$
$283$	−14.1317	−0.840044	−0.420022	+	0.907514i	$0.637978\pi$
$284$	0	0
$285$	−4.32748	−0.256338
$286$	0	0
$287$	15.9283	0.940217
$288$	0	0
$289$	6.06917	0.357010
$290$	0	0
$291$	12.0875	0.708582
$292$	0	0
$293$	−12.0224	−0.702355	−0.351178	−	0.936309i	$-0.614219\pi$
$293$	−12.0224	−0.702355	−0.351178	+	0.936309i	$0.614219\pi$
$294$	0	0
$295$	4.35460	0.253535
$296$	0	0
$297$	5.62457	0.326370
$298$	0	0
$299$	9.22619	0.533564
$300$	0	0
$301$	7.21668	0.415963
$302$	0	0
$303$	9.10531	0.523087
$304$	0	0
$305$	−10.1069	−0.578717
$306$	0	0
$307$	−25.2989	−1.44388	−0.721941	−	0.691954i	$-0.756750\pi$
$307$	−25.2989	−1.44388	−0.721941	+	0.691954i	$0.756750\pi$
$308$	0	0
$309$	20.6504	1.17476
$310$	0	0
$311$	−16.6945	−0.946657	−0.473328	−	0.880886i	$-0.656948\pi$
$311$	−16.6945	−0.946657	−0.473328	+	0.880886i	$0.656948\pi$
$312$	0	0
$313$	−8.39640	−0.474593	−0.237296	−	0.971437i	$-0.576261\pi$
$313$	−8.39640	−0.474593	−0.237296	+	0.971437i	$0.576261\pi$
$314$	0	0
$315$	−4.06529	−0.229053
$316$	0	0
$317$	10.9712	0.616206	0.308103	−	0.951353i	$-0.400306\pi$
$317$	10.9712	0.616206	0.308103	+	0.951353i	$0.400306\pi$
$318$	0	0
$319$	−7.97528	−0.446530
$320$	0	0
$321$	−5.31013	−0.296382
$322$	0	0
$323$	4.80304	0.267248
$324$	0	0
$325$	−3.59315	−0.199312
$326$	0	0
$327$	−8.53718	−0.472107
$328$	0	0
$329$	8.48831	0.467976
$330$	0	0
$331$	18.8667	1.03701	0.518504	−	0.855075i	$-0.326489\pi$
$331$	18.8667	1.03701	0.518504	+	0.855075i	$0.326489\pi$
$332$	0	0
$333$	5.39508	0.295648
$334$	0	0
$335$	−21.0299	−1.14898
$336$	0	0
$337$	18.4177	1.00327	0.501637	−	0.865078i	$-0.332731\pi$
$337$	18.4177	1.00327	0.501637	+	0.865078i	$0.332731\pi$
$338$	0	0
$339$	8.68746	0.471838
$340$	0	0
$341$	4.32748	0.234346
$342$	0	0
$343$	19.6884	1.06308
$344$	0	0
$345$	−36.8551	−1.98421
$346$	0	0
$347$	5.74801	0.308570	0.154285	−	0.988026i	$-0.450693\pi$
$347$	5.74801	0.308570	0.154285	+	0.988026i	$0.450693\pi$
$348$	0	0
$349$	−34.4243	−1.84269	−0.921346	−	0.388744i	$-0.872909\pi$
$349$	−34.4243	−1.84269	−0.921346	+	0.388744i	$0.872909\pi$
$350$	0	0
$351$	−6.09325	−0.325234
$352$	0	0
$353$	8.11721	0.432035	0.216018	−	0.976389i	$-0.430693\pi$
$353$	8.11721	0.432035	0.216018	+	0.976389i	$0.430693\pi$
$354$	0	0
$355$	−21.8184	−1.15800
$356$	0	0
$357$	−13.5777	−0.718610
$358$	0	0
$359$	23.0264	1.21529	0.607645	−	0.794209i	$-0.292114\pi$
$359$	23.0264	1.21529	0.607645	+	0.794209i	$0.292114\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.50058	−0.0787598
$364$	0	0
$365$	−26.8378	−1.40475
$366$	0	0
$367$	−21.1810	−1.10564	−0.552821	−	0.833300i	$-0.686449\pi$
$367$	−21.1810	−1.10564	−0.552821	+	0.833300i	$0.686449\pi$
$368$	0	0
$369$	6.32668	0.329354
$370$	0	0
$371$	12.9693	0.673334
$372$	0	0
$373$	7.78939	0.403320	0.201660	−	0.979456i	$-0.435366\pi$
$373$	7.78939	0.403320	0.201660	+	0.979456i	$0.435366\pi$
$374$	0	0
$375$	−7.28413	−0.376151
$376$	0	0
$377$	8.63985	0.444975
$378$	0	0
$379$	16.7545	0.860619	0.430309	−	0.902681i	$-0.358404\pi$
$379$	16.7545	0.860619	0.430309	+	0.902681i	$0.358404\pi$
$380$	0	0
$381$	13.6405	0.698823
$382$	0	0
$383$	23.7500	1.21357	0.606785	−	0.794866i	$-0.292459\pi$
$383$	23.7500	1.21357	0.606785	+	0.794866i	$0.292459\pi$
$384$	0	0
$385$	5.43289	0.276886
$386$	0	0
$387$	2.86645	0.145710
$388$	0	0
$389$	0.00246725	0.000125095 0	6.25473e−5	−	1.00000i	$-0.499980\pi$
$389$	0.00246725	0.000125095 0	6.25473e−5		1.00000i	$0.499980\pi$
$390$	0	0
$391$	40.9052	2.06866
$392$	0	0
$393$	−29.7910	−1.50276
$394$	0	0
$395$	0.676380	0.0340324
$396$	0	0
$397$	35.4514	1.77926	0.889628	−	0.456687i	$-0.150964\pi$
$397$	35.4514	1.77926	0.889628	+	0.456687i	$0.150964\pi$
$398$	0	0
$399$	−2.82691	−0.141522
$400$	0	0
$401$	−14.6214	−0.730157	−0.365078	−	0.930977i	$-0.618958\pi$
$401$	−14.6214	−0.730157	−0.365078	+	0.930977i	$0.618958\pi$
$402$	0	0
$403$	−4.68808	−0.233530
$404$	0	0
$405$	17.8664	0.887789
$406$	0	0
$407$	−7.21003	−0.357388
$408$	0	0
$409$	−6.49695	−0.321254	−0.160627	−	0.987015i	$-0.551352\pi$
$409$	−6.49695	−0.321254	−0.160627	+	0.987015i	$0.551352\pi$
$410$	0	0
$411$	−22.5290	−1.11127
$412$	0	0
$413$	2.84462	0.139975
$414$	0	0
$415$	−13.4075	−0.658146
$416$	0	0
$417$	12.1611	0.595530
$418$	0	0
$419$	−26.1834	−1.27914	−0.639571	−	0.768732i	$-0.720888\pi$
$419$	−26.1834	−1.27914	−0.639571	+	0.768732i	$0.720888\pi$
$420$	0	0
$421$	−5.45746	−0.265980	−0.132990	−	0.991117i	$-0.542458\pi$
$421$	−5.45746	−0.265980	−0.132990	+	0.991117i	$0.542458\pi$
$422$	0	0
$423$	3.37154	0.163930
$424$	0	0
$425$	−15.9306	−0.772747
$426$	0	0
$427$	−6.60226	−0.319506
$428$	0	0
$429$	1.62562	0.0784854
$430$	0	0
$431$	−18.2241	−0.877823	−0.438912	−	0.898530i	$-0.644636\pi$
$431$	−18.2241	−0.877823	−0.438912	+	0.898530i	$0.644636\pi$
$432$	0	0
$433$	−13.9249	−0.669188	−0.334594	−	0.942362i	$-0.608599\pi$
$433$	−13.9249	−0.669188	−0.334594	+	0.942362i	$0.608599\pi$
$434$	0	0
$435$	−34.5129	−1.65477
$436$	0	0
$437$	8.51652	0.407400
$438$	0	0
$439$	4.64620	0.221751	0.110876	−	0.993834i	$-0.464634\pi$
$439$	4.64620	0.221751	0.110876	+	0.993834i	$0.464634\pi$
$440$	0	0
$441$	2.58229	0.122966
$442$	0	0
$443$	−26.6366	−1.26554	−0.632771	−	0.774339i	$-0.718083\pi$
$443$	−26.6366	−1.26554	−0.632771	+	0.774339i	$0.718083\pi$
$444$	0	0
$445$	20.8831	0.989954
$446$	0	0
$447$	−19.7084	−0.932176
$448$	0	0
$449$	30.5263	1.44063	0.720314	−	0.693649i	$-0.243998\pi$
$449$	30.5263	1.44063	0.720314	+	0.693649i	$0.243998\pi$
$450$	0	0
$451$	−8.45503	−0.398132
$452$	0	0
$453$	−2.69544	−0.126643
$454$	0	0
$455$	−5.88560	−0.275921
$456$	0	0
$457$	−24.1980	−1.13194	−0.565968	−	0.824427i	$-0.691497\pi$
$457$	−24.1980	−1.13194	−0.565968	+	0.824427i	$0.691497\pi$
$458$	0	0
$459$	−27.0150	−1.26095
$460$	0	0
$461$	16.1065	0.750153	0.375076	−	0.926994i	$-0.377616\pi$
$461$	16.1065	0.750153	0.375076	+	0.926994i	$0.377616\pi$
$462$	0	0
$463$	29.7887	1.38440	0.692200	−	0.721705i	$-0.256641\pi$
$463$	29.7887	1.38440	0.692200	+	0.721705i	$0.256641\pi$
$464$	0	0
$465$	18.7271	0.868448
$466$	0	0
$467$	−33.9385	−1.57049	−0.785244	−	0.619187i	$-0.787462\pi$
$467$	−33.9385	−1.57049	−0.785244	+	0.619187i	$0.787462\pi$
$468$	0	0
$469$	−13.7377	−0.634346
$470$	0	0
$471$	−25.4919	−1.17460
$472$	0	0
$473$	−3.83075	−0.176138
$474$	0	0
$475$	−3.31677	−0.152184
$476$	0	0
$477$	5.15139	0.235866
$478$	0	0
$479$	3.50720	0.160248	0.0801241	−	0.996785i	$-0.474468\pi$
$479$	3.50720	0.160248	0.0801241	+	0.996785i	$0.474468\pi$
$480$	0	0
$481$	7.81083	0.356143
$482$	0	0
$483$	−24.0754	−1.09547
$484$	0	0
$485$	23.2304	1.05484
$486$	0	0
$487$	−32.6977	−1.48168	−0.740838	−	0.671684i	$-0.765571\pi$
$487$	−32.6977	−1.48168	−0.740838	+	0.671684i	$0.765571\pi$
$488$	0	0
$489$	−16.7890	−0.759225
$490$	0	0
$491$	29.9128	1.34995	0.674973	−	0.737842i	$-0.264155\pi$
$491$	29.9128	1.34995	0.674973	+	0.737842i	$0.264155\pi$
$492$	0	0
$493$	38.3056	1.72520
$494$	0	0
$495$	2.15793	0.0969919
$496$	0	0
$497$	−14.2527	−0.639323
$498$	0	0
$499$	23.2622	1.04136	0.520680	−	0.853752i	$-0.325679\pi$
$499$	23.2622	1.04136	0.520680	+	0.853752i	$0.325679\pi$
$500$	0	0
$501$	22.9531	1.02547
$502$	0	0
$503$	−7.52711	−0.335617	−0.167809	−	0.985820i	$-0.553669\pi$
$503$	−7.52711	−0.335617	−0.167809	+	0.985820i	$0.553669\pi$
$504$	0	0
$505$	17.4991	0.778698
$506$	0	0
$507$	17.7464	0.788145
$508$	0	0
$509$	−24.4275	−1.08273	−0.541365	−	0.840788i	$-0.682092\pi$
$509$	−24.4275	−1.08273	−0.541365	+	0.840788i	$0.682092\pi$
$510$	0	0
$511$	−17.5316	−0.775554
$512$	0	0
$513$	−5.62457	−0.248331
$514$	0	0
$515$	39.6869	1.74881
$516$	0	0
$517$	−4.50575	−0.198163
$518$	0	0
$519$	15.0159	0.659123
$520$	0	0
$521$	−39.7937	−1.74339	−0.871697	−	0.490045i	$-0.836980\pi$
$521$	−39.7937	−1.74339	−0.871697	+	0.490045i	$0.836980\pi$
$522$	0	0
$523$	13.4389	0.587640	0.293820	−	0.955861i	$-0.405073\pi$
$523$	13.4389	0.587640	0.293820	+	0.955861i	$0.405073\pi$
$524$	0	0
$525$	9.37621	0.409211
$526$	0	0
$527$	−20.7851	−0.905411
$528$	0	0
$529$	49.5311	2.15353
$530$	0	0
$531$	1.12988	0.0490325
$532$	0	0
$533$	9.15958	0.396745
$534$	0	0
$535$	−10.2053	−0.441212
$536$	0	0
$537$	−10.4107	−0.449254
$538$	0	0
$539$	−3.45099	−0.148645
$540$	0	0
$541$	1.49194	0.0641437	0.0320719	−	0.999486i	$-0.489789\pi$
$541$	1.49194	0.0641437	0.0320719	+	0.999486i	$0.489789\pi$
$542$	0	0
$543$	−20.8282	−0.893823
$544$	0	0
$545$	−16.4072	−0.702807
$546$	0	0
$547$	10.2877	0.439869	0.219934	−	0.975515i	$-0.429416\pi$
$547$	10.2877	0.439869	0.219934	+	0.975515i	$0.429416\pi$
$548$	0	0
$549$	−2.62240	−0.111921
$550$	0	0
$551$	7.97528	0.339758
$552$	0	0
$553$	0.441842	0.0187890
$554$	0	0
$555$	−31.2013	−1.32442
$556$	0	0
$557$	−39.2207	−1.66184	−0.830918	−	0.556395i	$-0.812184\pi$
$557$	−39.2207	−1.66184	−0.830918	+	0.556395i	$0.812184\pi$
$558$	0	0
$559$	4.14996	0.175525
$560$	0	0
$561$	7.20732	0.304293
$562$	0	0
$563$	−40.1331	−1.69141	−0.845705	−	0.533650i	$-0.820820\pi$
$563$	−40.1331	−1.69141	−0.845705	+	0.533650i	$0.820820\pi$
$564$	0	0
$565$	16.6960	0.702406
$566$	0	0
$567$	11.6711	0.490142
$568$	0	0
$569$	−33.6336	−1.40999	−0.704997	−	0.709210i	$-0.749052\pi$
$569$	−33.6336	−1.40999	−0.704997	+	0.709210i	$0.749052\pi$
$570$	0	0
$571$	1.25726	0.0526145	0.0263073	−	0.999654i	$-0.491625\pi$
$571$	1.25726	0.0526145	0.0263073	+	0.999654i	$0.491625\pi$
$572$	0	0
$573$	−3.69697	−0.154443
$574$	0	0
$575$	−28.2474	−1.17800
$576$	0	0
$577$	47.2793	1.96826	0.984132	−	0.177437i	$-0.0567806\pi$
$577$	47.2793	1.96826	0.984132	+	0.177437i	$0.0567806\pi$
$578$	0	0
$579$	−17.5197	−0.728094
$580$	0	0
$581$	−8.75836	−0.363358
$582$	0	0
$583$	−6.88436	−0.285121
$584$	0	0
$585$	−2.33775	−0.0966540
$586$	0	0
$587$	−29.6527	−1.22390	−0.611950	−	0.790897i	$-0.709615\pi$
$587$	−29.6527	−1.22390	−0.611950	+	0.790897i	$0.709615\pi$
$588$	0	0
$589$	−4.32748	−0.178311
$590$	0	0
$591$	28.5645	1.17499
$592$	0	0
$593$	24.3518	1.00001	0.500003	−	0.866023i	$-0.333332\pi$
$593$	24.3518	1.00001	0.500003	+	0.866023i	$0.333332\pi$
$594$	0	0
$595$	−26.0944	−1.06977
$596$	0	0
$597$	9.40260	0.384823
$598$	0	0
$599$	−30.3821	−1.24138	−0.620690	−	0.784056i	$-0.713148\pi$
$599$	−30.3821	−1.24138	−0.620690	+	0.784056i	$0.713148\pi$
$600$	0	0
$601$	−9.07411	−0.370140	−0.185070	−	0.982725i	$-0.559251\pi$
$601$	−9.07411	−0.370140	−0.185070	+	0.982725i	$0.559251\pi$
$602$	0	0
$603$	−5.45657	−0.222209
$604$	0	0
$605$	−2.88388	−0.117246
$606$	0	0
$607$	35.6353	1.44639	0.723196	−	0.690643i	$-0.242672\pi$
$607$	35.6353	1.44639	0.723196	+	0.690643i	$0.242672\pi$
$608$	0	0
$609$	−22.5454	−0.913585
$610$	0	0
$611$	4.88121	0.197473
$612$	0	0
$613$	−17.3009	−0.698777	−0.349388	−	0.936978i	$-0.613611\pi$
$613$	−17.3009	−0.698777	−0.349388	+	0.936978i	$0.613611\pi$
$614$	0	0
$615$	−36.5890	−1.47541
$616$	0	0
$617$	−15.7903	−0.635695	−0.317847	−	0.948142i	$-0.602960\pi$
$617$	−15.7903	−0.635695	−0.317847	+	0.948142i	$0.602960\pi$
$618$	0	0
$619$	20.2011	0.811950	0.405975	−	0.913884i	$-0.366932\pi$
$619$	20.2011	0.811950	0.405975	+	0.913884i	$0.366932\pi$
$620$	0	0
$621$	−47.9017	−1.92223
$622$	0	0
$623$	13.6418	0.546547
$624$	0	0
$625$	−30.5829	−1.22332
$626$	0	0
$627$	1.50058	0.0599272
$628$	0	0
$629$	34.6300	1.38079
$630$	0	0
$631$	−1.49437	−0.0594900	−0.0297450	−	0.999558i	$-0.509470\pi$
$631$	−1.49437	−0.0594900	−0.0297450	+	0.999558i	$0.509470\pi$
$632$	0	0
$633$	−31.2669	−1.24275
$634$	0	0
$635$	26.2150	1.04031
$636$	0	0
$637$	3.73855	0.148127
$638$	0	0
$639$	−5.66116	−0.223952
$640$	0	0
$641$	18.8967	0.746377	0.373188	−	0.927756i	$-0.378265\pi$
$641$	18.8967	0.746377	0.373188	+	0.927756i	$0.378265\pi$
$642$	0	0
$643$	28.4030	1.12011	0.560053	−	0.828457i	$-0.310781\pi$
$643$	28.4030	1.12011	0.560053	+	0.828457i	$0.310781\pi$
$644$	0	0
$645$	−16.5775	−0.652739
$646$	0	0
$647$	21.5350	0.846630	0.423315	−	0.905983i	$-0.360866\pi$
$647$	21.5350	0.846630	0.423315	+	0.905983i	$0.360866\pi$
$648$	0	0
$649$	−1.50998	−0.0592719
$650$	0	0
$651$	12.2334	0.479464
$652$	0	0
$653$	16.2729	0.636808	0.318404	−	0.947955i	$-0.396853\pi$
$653$	16.2729	0.636808	0.318404	+	0.947955i	$0.396853\pi$
$654$	0	0
$655$	−57.2538	−2.23709
$656$	0	0
$657$	−6.96353	−0.271673
$658$	0	0
$659$	−4.66279	−0.181637	−0.0908183	−	0.995867i	$-0.528948\pi$
$659$	−4.66279	−0.181637	−0.0908183	+	0.995867i	$0.528948\pi$
$660$	0	0
$661$	−23.1724	−0.901304	−0.450652	−	0.892700i	$-0.648808\pi$
$661$	−23.1724	−0.901304	−0.450652	+	0.892700i	$0.648808\pi$
$662$	0	0
$663$	−7.80789	−0.303233
$664$	0	0
$665$	−5.43289	−0.210678
$666$	0	0
$667$	67.9216	2.62994
$668$	0	0
$669$	41.9764	1.62290
$670$	0	0
$671$	3.50460	0.135294
$672$	0	0
$673$	46.2401	1.78242	0.891212	−	0.453588i	$-0.149856\pi$
$673$	46.2401	1.78242	0.891212	+	0.453588i	$0.149856\pi$
$674$	0	0
$675$	18.6554	0.718047
$676$	0	0
$677$	−33.6361	−1.29274	−0.646370	−	0.763024i	$-0.723714\pi$
$677$	−33.6361	−1.29274	−0.646370	+	0.763024i	$0.723714\pi$
$678$	0	0
$679$	15.1751	0.582368
$680$	0	0
$681$	−37.7237	−1.44557
$682$	0	0
$683$	4.79055	0.183305	0.0916526	−	0.995791i	$-0.470785\pi$
$683$	4.79055	0.183305	0.0916526	+	0.995791i	$0.470785\pi$
$684$	0	0
$685$	−43.2973	−1.65430
$686$	0	0
$687$	−32.8701	−1.25407
$688$	0	0
$689$	7.45803	0.284128
$690$	0	0
$691$	16.4497	0.625777	0.312888	−	0.949790i	$-0.398703\pi$
$691$	16.4497	0.625777	0.312888	+	0.949790i	$0.398703\pi$
$692$	0	0
$693$	1.40966	0.0535485
$694$	0	0
$695$	23.3718	0.886541
$696$	0	0
$697$	40.6098	1.53821
$698$	0	0
$699$	24.2239	0.916231
$700$	0	0
$701$	−32.6582	−1.23348	−0.616742	−	0.787165i	$-0.711548\pi$
$701$	−32.6582	−1.23348	−0.616742	+	0.787165i	$0.711548\pi$
$702$	0	0
$703$	7.21003	0.271931
$704$	0	0
$705$	−19.4986	−0.734359
$706$	0	0
$707$	11.4312	0.429913
$708$	0	0
$709$	−14.2212	−0.534090	−0.267045	−	0.963684i	$-0.586047\pi$
$709$	−14.2212	−0.534090	−0.267045	+	0.963684i	$0.586047\pi$
$710$	0	0
$711$	0.175499	0.00658172
$712$	0	0
$713$	−36.8551	−1.38023
$714$	0	0
$715$	3.12419	0.116838
$716$	0	0
$717$	1.49521	0.0558397
$718$	0	0
$719$	−26.9647	−1.00561	−0.502807	−	0.864399i	$-0.667699\pi$
$719$	−26.9647	−1.00561	−0.502807	+	0.864399i	$0.667699\pi$
$720$	0	0
$721$	25.9253	0.965508
$722$	0	0
$723$	33.9187	1.26145
$724$	0	0
$725$	−26.4522	−0.982410
$726$	0	0
$727$	21.4897	0.797008	0.398504	−	0.917167i	$-0.369529\pi$
$727$	21.4897	0.797008	0.398504	+	0.917167i	$0.369529\pi$
$728$	0	0
$729$	29.9560	1.10948
$730$	0	0
$731$	18.3992	0.680521
$732$	0	0
$733$	−49.5639	−1.83069	−0.915343	−	0.402676i	$-0.868080\pi$
$733$	−49.5639	−1.83069	−0.915343	+	0.402676i	$0.868080\pi$
$734$	0	0
$735$	−14.9341	−0.550852
$736$	0	0
$737$	7.29221	0.268612
$738$	0	0
$739$	−32.5018	−1.19560	−0.597800	−	0.801646i	$-0.703958\pi$
$739$	−32.5018	−1.19560	−0.597800	+	0.801646i	$0.703958\pi$
$740$	0	0
$741$	−1.62562	−0.0597185
$742$	0	0
$743$	−23.7010	−0.869506	−0.434753	−	0.900550i	$-0.643164\pi$
$743$	−23.7010	−0.869506	−0.434753	+	0.900550i	$0.643164\pi$
$744$	0	0
$745$	−37.8766	−1.38769
$746$	0	0
$747$	−3.47880	−0.127283
$748$	0	0
$749$	−6.66654	−0.243590
$750$	0	0
$751$	40.5798	1.48078	0.740388	−	0.672179i	$-0.234642\pi$
$751$	40.5798	1.48078	0.740388	+	0.672179i	$0.234642\pi$
$752$	0	0
$753$	35.6666	1.29976
$754$	0	0
$755$	−5.18023	−0.188528
$756$	0	0
$757$	−29.9224	−1.08755	−0.543773	−	0.839232i	$-0.683005\pi$
$757$	−29.9224	−1.08755	−0.543773	+	0.839232i	$0.683005\pi$
$758$	0	0
$759$	12.7797	0.463873
$760$	0	0
$761$	−23.8651	−0.865108	−0.432554	−	0.901608i	$-0.642388\pi$
$761$	−23.8651	−0.865108	−0.432554	+	0.901608i	$0.642388\pi$
$762$	0	0
$763$	−10.7179	−0.388015
$764$	0	0
$765$	−10.3646	−0.374734
$766$	0	0
$767$	1.63580	0.0590655
$768$	0	0
$769$	35.3464	1.27463	0.637313	−	0.770605i	$-0.280046\pi$
$769$	35.3464	1.27463	0.637313	+	0.770605i	$0.280046\pi$
$770$	0	0
$771$	1.91183	0.0688530
$772$	0	0
$773$	−48.9820	−1.76176	−0.880880	−	0.473340i	$-0.843048\pi$
$773$	−48.9820	−1.76176	−0.880880	+	0.473340i	$0.843048\pi$
$774$	0	0
$775$	14.3533	0.515585
$776$	0	0
$777$	−20.3821	−0.731203
$778$	0	0
$779$	8.45503	0.302933
$780$	0	0
$781$	7.56562	0.270719
$782$	0	0
$783$	−44.8575	−1.60308
$784$	0	0
$785$	−48.9916	−1.74859
$786$	0	0
$787$	8.01895	0.285845	0.142922	−	0.989734i	$-0.454350\pi$
$787$	8.01895	0.285845	0.142922	+	0.989734i	$0.454350\pi$
$788$	0	0
$789$	19.1145	0.680495
$790$	0	0
$791$	10.9066	0.387793
$792$	0	0
$793$	−3.79664	−0.134822
$794$	0	0
$795$	−29.7920	−1.05661
$796$	0	0
$797$	−22.9829	−0.814096	−0.407048	−	0.913407i	$-0.633442\pi$
$797$	−22.9829	−0.814096	−0.407048	+	0.913407i	$0.633442\pi$
$798$	0	0
$799$	21.6413	0.765614
$800$	0	0
$801$	5.41849	0.191453
$802$	0	0
$803$	9.30612	0.328406
$804$	0	0
$805$	−46.2693	−1.63078
$806$	0	0
$807$	11.8907	0.418571
$808$	0	0
$809$	23.7249	0.834122	0.417061	−	0.908878i	$-0.363060\pi$
$809$	23.7249	0.834122	0.417061	+	0.908878i	$0.363060\pi$
$810$	0	0
$811$	−15.1527	−0.532083	−0.266042	−	0.963962i	$-0.585716\pi$
$811$	−15.1527	−0.532083	−0.266042	+	0.963962i	$0.585716\pi$
$812$	0	0
$813$	3.27275	0.114780
$814$	0	0
$815$	−32.2659	−1.13023
$816$	0	0
$817$	3.83075	0.134021
$818$	0	0
$819$	−1.52712	−0.0533620
$820$	0	0
$821$	−39.3391	−1.37295	−0.686473	−	0.727155i	$-0.740842\pi$
$821$	−39.3391	−1.37295	−0.686473	+	0.727155i	$0.740842\pi$
$822$	0	0
$823$	8.43831	0.294141	0.147071	−	0.989126i	$-0.453016\pi$
$823$	8.43831	0.294141	0.147071	+	0.989126i	$0.453016\pi$
$824$	0	0
$825$	−4.97707	−0.173279
$826$	0	0
$827$	9.58576	0.333330	0.166665	−	0.986014i	$-0.446700\pi$
$827$	9.58576	0.333330	0.166665	+	0.986014i	$0.446700\pi$
$828$	0	0
$829$	−12.3387	−0.428540	−0.214270	−	0.976775i	$-0.568737\pi$
$829$	−12.3387	−0.428540	−0.214270	+	0.976775i	$0.568737\pi$
$830$	0	0
$831$	20.0886	0.696867
$832$	0	0
$833$	16.5752	0.574298
$834$	0	0
$835$	44.1124	1.52657
$836$	0	0
$837$	24.3402	0.841321
$838$	0	0
$839$	−19.5182	−0.673844	−0.336922	−	0.941533i	$-0.609386\pi$
$839$	−19.5182	−0.673844	−0.336922	+	0.941533i	$0.609386\pi$
$840$	0	0
$841$	34.6051	1.19328
$842$	0	0
$843$	−33.8810	−1.16692
$844$	0	0
$845$	34.1059	1.17328
$846$	0	0
$847$	−1.88388	−0.0647309
$848$	0	0
$849$	21.2057	0.727778
$850$	0	0
$851$	61.4044	2.10491
$852$	0	0
$853$	−32.0583	−1.09766	−0.548828	−	0.835935i	$-0.684926\pi$
$853$	−32.0583	−1.09766	−0.548828	+	0.835935i	$0.684926\pi$
$854$	0	0
$855$	−2.15793	−0.0737997
$856$	0	0
$857$	3.94409	0.134728	0.0673639	−	0.997728i	$-0.478541\pi$
$857$	3.94409	0.134728	0.0673639	+	0.997728i	$0.478541\pi$
$858$	0	0
$859$	−19.0054	−0.648455	−0.324227	−	0.945979i	$-0.605104\pi$
$859$	−19.0054	−0.648455	−0.324227	+	0.945979i	$0.605104\pi$
$860$	0	0
$861$	−23.9016	−0.814564
$862$	0	0
$863$	14.9638	0.509373	0.254687	−	0.967024i	$-0.418028\pi$
$863$	14.9638	0.509373	0.254687	+	0.967024i	$0.418028\pi$
$864$	0	0
$865$	28.8582	0.981210
$866$	0	0
$867$	−9.10725	−0.309298
$868$	0	0
$869$	−0.234538	−0.00795616
$870$	0	0
$871$	−7.89985	−0.267676
$872$	0	0
$873$	6.02753	0.204001
$874$	0	0
$875$	−9.14479	−0.309150
$876$	0	0
$877$	−49.9928	−1.68814	−0.844068	−	0.536237i	$-0.819846\pi$
$877$	−49.9928	−1.68814	−0.844068	+	0.536237i	$0.819846\pi$
$878$	0	0
$879$	18.0405	0.608491
$880$	0	0
$881$	−45.5061	−1.53314	−0.766570	−	0.642160i	$-0.778038\pi$
$881$	−45.5061	−1.53314	−0.766570	+	0.642160i	$0.778038\pi$
$882$	0	0
$883$	−16.3052	−0.548715	−0.274357	−	0.961628i	$-0.588465\pi$
$883$	−16.3052	−0.548715	−0.274357	+	0.961628i	$0.588465\pi$
$884$	0	0
$885$	−6.53441	−0.219652
$886$	0	0
$887$	26.9400	0.904556	0.452278	−	0.891877i	$-0.350611\pi$
$887$	26.9400	0.904556	0.452278	+	0.891877i	$0.350611\pi$
$888$	0	0
$889$	17.1248	0.574347
$890$	0	0
$891$	−6.19527	−0.207549
$892$	0	0
$893$	4.50575	0.150779
$894$	0	0
$895$	−20.0078	−0.668786
$896$	0	0
$897$	−13.8446	−0.462257
$898$	0	0
$899$	−34.5129	−1.15107
$900$	0	0
$901$	33.0659	1.10158
$902$	0	0
$903$	−10.8292	−0.360372
$904$	0	0
$905$	−40.0287	−1.33060
$906$	0	0
$907$	−34.1279	−1.13320	−0.566600	−	0.823993i	$-0.691742\pi$
$907$	−34.1279	−1.13320	−0.566600	+	0.823993i	$0.691742\pi$
$908$	0	0
$909$	4.54044	0.150597
$910$	0	0
$911$	−47.0120	−1.55758	−0.778789	−	0.627286i	$-0.784166\pi$
$911$	−47.0120	−1.55758	−0.778789	+	0.627286i	$0.784166\pi$
$912$	0	0
$913$	4.64910	0.153863
$914$	0	0
$915$	15.1661	0.501376
$916$	0	0
$917$	−37.4007	−1.23508
$918$	0	0
$919$	30.5621	1.00815	0.504075	−	0.863660i	$-0.331833\pi$
$919$	30.5621	1.00815	0.504075	+	0.863660i	$0.331833\pi$
$920$	0	0
$921$	37.9629	1.25092
$922$	0	0
$923$	−8.19605	−0.269776
$924$	0	0
$925$	−23.9140	−0.786289
$926$	0	0
$927$	10.2975	0.338213
$928$	0	0
$929$	−28.7378	−0.942855	−0.471428	−	0.881905i	$-0.656261\pi$
$929$	−28.7378	−0.942855	−0.471428	+	0.881905i	$0.656261\pi$
$930$	0	0
$931$	3.45099	0.113102
$932$	0	0
$933$	25.0513	0.820143
$934$	0	0
$935$	13.8514	0.452989
$936$	0	0
$937$	−39.3706	−1.28618	−0.643091	−	0.765789i	$-0.722348\pi$
$937$	−39.3706	−1.28618	−0.643091	+	0.765789i	$0.722348\pi$
$938$	0	0
$939$	12.5994	0.411167
$940$	0	0
$941$	26.0169	0.848127	0.424064	−	0.905632i	$-0.360603\pi$
$941$	26.0169	0.848127	0.424064	+	0.905632i	$0.360603\pi$
$942$	0	0
$943$	72.0075	2.34489
$944$	0	0
$945$	30.5577	0.994041
$946$	0	0
$947$	48.2165	1.56683	0.783413	−	0.621502i	$-0.213477\pi$
$947$	48.2165	1.56683	0.783413	+	0.621502i	$0.213477\pi$
$948$	0	0
$949$	−10.0816	−0.327262
$950$	0	0
$951$	−16.4632	−0.533855
$952$	0	0
$953$	20.4822	0.663483	0.331741	−	0.943370i	$-0.392364\pi$
$953$	20.4822	0.663483	0.331741	+	0.943370i	$0.392364\pi$
$954$	0	0
$955$	−7.10502	−0.229913
$956$	0	0
$957$	11.9675	0.386855
$958$	0	0
$959$	−28.2838	−0.913330
$960$	0	0
$961$	−12.2729	−0.395900
$962$	0	0
$963$	−2.64794	−0.0853286
$964$	0	0
$965$	−33.6703	−1.08388
$966$	0	0
$967$	−21.7691	−0.700046	−0.350023	−	0.936741i	$-0.613826\pi$
$967$	−21.7691	−0.700046	−0.350023	+	0.936741i	$0.613826\pi$
$968$	0	0
$969$	−7.20732	−0.231532
$970$	0	0
$971$	−18.5628	−0.595708	−0.297854	−	0.954611i	$-0.596271\pi$
$971$	−18.5628	−0.595708	−0.297854	+	0.954611i	$0.596271\pi$
$972$	0	0
$973$	15.2675	0.489453
$974$	0	0
$975$	5.39180	0.172676
$976$	0	0
$977$	10.8339	0.346607	0.173303	−	0.984868i	$-0.444556\pi$
$977$	10.8339	0.346607	0.173303	+	0.984868i	$0.444556\pi$
$978$	0	0
$979$	−7.24132	−0.231434
$980$	0	0
$981$	−4.25713	−0.135920
$982$	0	0
$983$	11.6111	0.370336	0.185168	−	0.982707i	$-0.440717\pi$
$983$	11.6111	0.370336	0.185168	+	0.982707i	$0.440717\pi$
$984$	0	0
$985$	54.8967	1.74916
$986$	0	0
$987$	−12.7373	−0.405434
$988$	0	0
$989$	32.6247	1.03740
$990$	0	0
$991$	34.1635	1.08524	0.542620	−	0.839978i	$-0.317432\pi$
$991$	34.1635	1.08524	0.542620	+	0.839978i	$0.317432\pi$
$992$	0	0
$993$	−28.3109	−0.898421
$994$	0	0
$995$	18.0704	0.572870
$996$	0	0
$997$	−13.4776	−0.426838	−0.213419	−	0.976961i	$-0.568460\pi$
$997$	−13.4776	−0.426838	−0.213419	+	0.976961i	$0.568460\pi$
$998$	0	0
$999$	−40.5533	−1.28305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.z.1.2		6
4.3	odd	2		1672.2.a.i.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.i.1.5	✓	6	4.3	odd	2
3344.2.a.z.1.2		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)

\(f(q)\)	\(=\)	\(q-1.50058 q^{3} -2.88388 q^{5} -1.88388 q^{7} -0.748274 q^{9} +O(q^{10})\)	\(q-1.50058 q^{3} -2.88388 q^{5} -1.88388 q^{7} -0.748274 q^{9} +1.00000 q^{11} -1.08333 q^{13} +4.32748 q^{15} -4.80304 q^{17} -1.00000 q^{19} +2.82691 q^{21} -8.51652 q^{23} +3.31677 q^{25} +5.62457 q^{27} -7.97528 q^{29} +4.32748 q^{31} -1.50058 q^{33} +5.43289 q^{35} -7.21003 q^{37} +1.62562 q^{39} -8.45503 q^{41} -3.83075 q^{43} +2.15793 q^{45} -4.50575 q^{47} -3.45099 q^{49} +7.20732 q^{51} -6.88436 q^{53} -2.88388 q^{55} +1.50058 q^{57} -1.50998 q^{59} +3.50460 q^{61} +1.40966 q^{63} +3.12419 q^{65} +7.29221 q^{67} +12.7797 q^{69} +7.56562 q^{71} +9.30612 q^{73} -4.97707 q^{75} -1.88388 q^{77} -0.234538 q^{79} -6.19527 q^{81} +4.64910 q^{83} +13.8514 q^{85} +11.9675 q^{87} -7.24132 q^{89} +2.04086 q^{91} -6.49371 q^{93} +2.88388 q^{95} -8.05525 q^{97} -0.748274 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9	\( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + q^15 - q^17 - 6 * q^19 + 5 * q^21 + 4 * q^23 + 5 * q^25 + 16 * q^27 - 7 * q^29 + q^31 + 4 * q^33 + 32 * q^35 + 5 * q^37 + 13 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 7 * q^49 + 35 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 18 * q^59 + 16 * q^61 + 6 * q^63 + 10 * q^65 + 18 * q^67 - 2 * q^69 + 7 * q^71 - 21 * q^73 + 23 * q^75 + 3 * q^77 + 22 * q^79 - 10 * q^81 + 16 * q^83 + 3 * q^87 - 13 * q^89 + 7 * q^91 - 9 * q^93 + 3 * q^95 - 15 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

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