LMFDB - Embedded newform 3344.2.a.u.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:	$5$
Coefficient field:	5.5.3979184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 18x^{2} + 5x - 12 $ x^5 - 2x^4 - 9x^3 + 18x^2 + 5x - 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			$1.85873$ 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)

$f(q)$	$=$	$q-2.35480 q^{3} +2.85873 q^{5} -0.141265 q^{7} +2.54511 q^{9} +O(q^{10})$	$q-2.35480 q^{3} +2.85873 q^{5} -0.141265 q^{7} +2.54511 q^{9} +1.00000 q^{11} +5.42599 q^{13} -6.73176 q^{15} +2.00000 q^{17} -1.00000 q^{19} +0.332652 q^{21} +1.40384 q^{23} +3.17236 q^{25} +1.07119 q^{27} -5.80295 q^{29} +10.7761 q^{31} -2.35480 q^{33} -0.403840 q^{35} +0.313629 q^{37} -12.7772 q^{39} +8.33578 q^{41} -4.16450 q^{43} +7.27578 q^{45} -1.09021 q^{47} -6.98004 q^{49} -4.70961 q^{51} +2.00000 q^{53} +2.85873 q^{55} +2.35480 q^{57} -3.73068 q^{59} +5.05434 q^{61} -0.359535 q^{63} +15.5115 q^{65} +0.969988 q^{67} -3.30577 q^{69} -7.10655 q^{71} -11.4635 q^{73} -7.47030 q^{75} -0.141265 q^{77} +1.71747 q^{79} -10.1578 q^{81} +13.3433 q^{83} +5.71747 q^{85} +13.6648 q^{87} -7.87522 q^{89} -0.766505 q^{91} -25.3755 q^{93} -2.85873 q^{95} +12.8652 q^{97} +2.54511 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9}+O(q^{10}) $ 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9	$ 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9} + 5 q^{11} + 4 q^{13} - q^{15} + 10 q^{17} - 5 q^{19} + 2 q^{21} - 5 q^{23} + 6 q^{25} - 7 q^{27} + 16 q^{29} - q^{31} - q^{33} + 10 q^{35} - q^{37} + 4 q^{39} + 28 q^{41} - 4 q^{43} + 12 q^{45} + 4 q^{47} - q^{49} - 2 q^{51} + 10 q^{53} + 7 q^{55} + q^{57} + q^{59} - 16 q^{61} - 12 q^{63} + 24 q^{65} + 9 q^{67} - 7 q^{69} - 7 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} - 6 q^{79} + 5 q^{81} - 4 q^{83} + 14 q^{85} + 2 q^{87} + 31 q^{89} + 12 q^{91} - 7 q^{93} - 7 q^{95} + 13 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9 + 5 * q^11 + 4 * q^13 - q^15 + 10 * q^17 - 5 * q^19 + 2 * q^21 - 5 * q^23 + 6 * q^25 - 7 * q^27 + 16 * q^29 - q^31 - q^33 + 10 * q^35 - q^37 + 4 * q^39 + 28 * q^41 - 4 * q^43 + 12 * q^45 + 4 * q^47 - q^49 - 2 * q^51 + 10 * q^53 + 7 * q^55 + q^57 + q^59 - 16 * q^61 - 12 * q^63 + 24 * q^65 + 9 * q^67 - 7 * q^69 - 7 * q^71 + 8 * q^73 + 8 * q^75 - 8 * q^77 - 6 * q^79 + 5 * q^81 - 4 * q^83 + 14 * q^85 + 2 * q^87 + 31 * q^89 + 12 * q^91 - 7 * q^93 - 7 * q^95 + 13 * q^97 + 8 * 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
$3$	−2.35480	−1.35955	−0.679774	−	0.733422i	$-0.737922\pi$
$3$	−2.35480	−1.35955	−0.679774	+	0.733422i	$0.737922\pi$
$4$	0	0
$5$	2.85873	1.27847	0.639233	−	0.769014i	$-0.279252\pi$
$5$	2.85873	1.27847	0.639233	+	0.769014i	$0.279252\pi$
$6$	0	0
$7$	−0.141265	−0.0533933	−0.0266966	−	0.999644i	$-0.508499\pi$
$7$	−0.141265	−0.0533933	−0.0266966	+	0.999644i	$0.508499\pi$
$8$	0	0
$9$	2.54511	0.848368
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.42599	1.50490	0.752450	−	0.658650i	$-0.228872\pi$
$13$	5.42599	1.50490	0.752450	+	0.658650i	$0.228872\pi$
$14$	0	0
$15$	−6.73176	−1.73813
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.332652	0.0725907
$22$	0	0
$23$	1.40384	0.292721	0.146360	−	0.989231i	$-0.453244\pi$
$23$	1.40384	0.292721	0.146360	+	0.989231i	$0.453244\pi$
$24$	0	0
$25$	3.17236	0.634473
$26$	0	0
$27$	1.07119	0.206150
$28$	0	0
$29$	−5.80295	−1.07758	−0.538790	−	0.842440i	$-0.681119\pi$
$29$	−5.80295	−1.07758	−0.538790	+	0.842440i	$0.681119\pi$
$30$	0	0
$31$	10.7761	1.93544	0.967719	−	0.252030i	$-0.0810982\pi$
$31$	10.7761	1.93544	0.967719	+	0.252030i	$0.0810982\pi$
$32$	0	0
$33$	−2.35480	−0.409919
$34$	0	0
$35$	−0.403840	−0.0682614
$36$	0	0
$37$	0.313629	0.0515603	0.0257802	−	0.999668i	$-0.491793\pi$
$37$	0.313629	0.0515603	0.0257802	+	0.999668i	$0.491793\pi$
$38$	0	0
$39$	−12.7772	−2.04598
$40$	0	0
$41$	8.33578	1.30183	0.650915	−	0.759150i	$-0.274385\pi$
$41$	8.33578	1.30183	0.650915	+	0.759150i	$0.274385\pi$
$42$	0	0
$43$	−4.16450	−0.635081	−0.317540	−	0.948245i	$-0.602857\pi$
$43$	−4.16450	−0.635081	−0.317540	+	0.948245i	$0.602857\pi$
$44$	0	0
$45$	7.27578	1.08461
$46$	0	0
$47$	−1.09021	−0.159024	−0.0795118	−	0.996834i	$-0.525336\pi$
$47$	−1.09021	−0.159024	−0.0795118	+	0.996834i	$0.525336\pi$
$48$	0	0
$49$	−6.98004	−0.997149
$50$	0	0
$51$	−4.70961	−0.659477
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	2.85873	0.385472
$56$	0	0
$57$	2.35480	0.311902
$58$	0	0
$59$	−3.73068	−0.485693	−0.242846	−	0.970065i	$-0.578081\pi$
$59$	−3.73068	−0.485693	−0.242846	+	0.970065i	$0.578081\pi$
$60$	0	0
$61$	5.05434	0.647142	0.323571	−	0.946204i	$-0.395117\pi$
$61$	5.05434	0.647142	0.323571	+	0.946204i	$0.395117\pi$
$62$	0	0
$63$	−0.359535	−0.0452972
$64$	0	0
$65$	15.5115	1.92396
$66$	0	0
$67$	0.969988	0.118503	0.0592514	−	0.998243i	$-0.481129\pi$
$67$	0.969988	0.118503	0.0592514	+	0.998243i	$0.481129\pi$
$68$	0	0
$69$	−3.30577	−0.397968
$70$	0	0
$71$	−7.10655	−0.843392	−0.421696	−	0.906737i	$-0.638565\pi$
$71$	−7.10655	−0.843392	−0.421696	+	0.906737i	$0.638565\pi$
$72$	0	0
$73$	−11.4635	−1.34170	−0.670852	−	0.741591i	$-0.734071\pi$
$73$	−11.4635	−1.34170	−0.670852	+	0.741591i	$0.734071\pi$
$74$	0	0
$75$	−7.47030	−0.862596
$76$	0	0
$77$	−0.141265	−0.0160987
$78$	0	0
$79$	1.71747	0.193230	0.0966152	−	0.995322i	$-0.469198\pi$
$79$	1.71747	0.193230	0.0966152	+	0.995322i	$0.469198\pi$
$80$	0	0
$81$	−10.1578	−1.12864
$82$	0	0
$83$	13.3433	1.46462	0.732310	−	0.680971i	$-0.238442\pi$
$83$	13.3433	1.46462	0.732310	+	0.680971i	$0.238442\pi$
$84$	0	0
$85$	5.71747	0.620147
$86$	0	0
$87$	13.6648	1.46502
$88$	0	0
$89$	−7.87522	−0.834772	−0.417386	−	0.908729i	$-0.637054\pi$
$89$	−7.87522	−0.834772	−0.417386	+	0.908729i	$0.637054\pi$
$90$	0	0
$91$	−0.766505	−0.0803515
$92$	0	0
$93$	−25.3755	−2.63132
$94$	0	0
$95$	−2.85873	−0.293300
$96$	0	0
$97$	12.8652	1.30626	0.653131	−	0.757245i	$-0.273455\pi$
$97$	12.8652	1.30626	0.653131	+	0.757245i	$0.273455\pi$
$98$	0	0
$99$	2.54511	0.255793
$100$	0	0
$101$	5.69958	0.567129	0.283565	−	0.958953i	$-0.408483\pi$
$101$	5.69958	0.567129	0.283565	+	0.958953i	$0.408483\pi$
$102$	0	0
$103$	12.2300	1.20506	0.602530	−	0.798096i	$-0.294159\pi$
$103$	12.2300	1.20506	0.602530	+	0.798096i	$0.294159\pi$
$104$	0	0
$105$	0.950965	0.0928046
$106$	0	0
$107$	−4.62726	−0.447334	−0.223667	−	0.974666i	$-0.571803\pi$
$107$	−4.62726	−0.447334	−0.223667	+	0.974666i	$0.571803\pi$
$108$	0	0
$109$	8.04430	0.770505	0.385252	−	0.922811i	$-0.374114\pi$
$109$	8.04430	0.770505	0.385252	+	0.922811i	$0.374114\pi$
$110$	0	0
$111$	−0.738536	−0.0700987
$112$	0	0
$113$	−19.2944	−1.81507	−0.907534	−	0.419978i	$-0.862038\pi$
$113$	−19.2944	−1.81507	−0.907534	+	0.419978i	$0.862038\pi$
$114$	0	0
$115$	4.01321	0.374233
$116$	0	0
$117$	13.8097	1.27671
$118$	0	0
$119$	−0.282531	−0.0258995
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−19.6291	−1.76990
$124$	0	0
$125$	−5.22473	−0.467314
$126$	0	0
$127$	−16.5537	−1.46891	−0.734453	−	0.678659i	$-0.762561\pi$
$127$	−16.5537	−1.46891	−0.734453	+	0.678659i	$0.762561\pi$
$128$	0	0
$129$	9.80659	0.863422
$130$	0	0
$131$	−3.22202	−0.281509	−0.140754	−	0.990045i	$-0.544953\pi$
$131$	−3.22202	−0.281509	−0.140754	+	0.990045i	$0.544953\pi$
$132$	0	0
$133$	0.141265	0.0122493
$134$	0	0
$135$	3.06224	0.263556
$136$	0	0
$137$	−2.45489	−0.209736	−0.104868	−	0.994486i	$-0.533442\pi$
$137$	−2.45489	−0.209736	−0.104868	+	0.994486i	$0.533442\pi$
$138$	0	0
$139$	−6.98790	−0.592706	−0.296353	−	0.955078i	$-0.595771\pi$
$139$	−6.98790	−0.592706	−0.296353	+	0.955078i	$0.595771\pi$
$140$	0	0
$141$	2.56723	0.216200
$142$	0	0
$143$	5.42599	0.453744
$144$	0	0
$145$	−16.5891	−1.37765
$146$	0	0
$147$	16.4366	1.35567
$148$	0	0
$149$	11.8441	0.970309	0.485154	−	0.874429i	$-0.338763\pi$
$149$	11.8441	0.970309	0.485154	+	0.874429i	$0.338763\pi$
$150$	0	0
$151$	11.8441	0.963861	0.481931	−	0.876209i	$-0.339936\pi$
$151$	11.8441	0.963861	0.481931	+	0.876209i	$0.339936\pi$
$152$	0	0
$153$	5.09021	0.411519
$154$	0	0
$155$	30.8059	2.47439
$156$	0	0
$157$	11.7550	0.938153	0.469077	−	0.883158i	$-0.344587\pi$
$157$	11.7550	0.938153	0.469077	+	0.883158i	$0.344587\pi$
$158$	0	0
$159$	−4.70961	−0.373496
$160$	0	0
$161$	−0.198314	−0.0156293
$162$	0	0
$163$	4.38277	0.343285	0.171643	−	0.985159i	$-0.445093\pi$
$163$	4.38277	0.343285	0.171643	+	0.985159i	$0.445093\pi$
$164$	0	0
$165$	−6.73176	−0.524067
$166$	0	0
$167$	14.7982	1.14512	0.572560	−	0.819863i	$-0.305950\pi$
$167$	14.7982	1.14512	0.572560	+	0.819863i	$0.305950\pi$
$168$	0	0
$169$	16.4414	1.26472
$170$	0	0
$171$	−2.54511	−0.194629
$172$	0	0
$173$	15.1779	1.15395	0.576976	−	0.816761i	$-0.304233\pi$
$173$	15.1779	1.15395	0.576976	+	0.816761i	$0.304233\pi$
$174$	0	0
$175$	−0.448145	−0.0338766
$176$	0	0
$177$	8.78501	0.660322
$178$	0	0
$179$	4.38703	0.327902	0.163951	−	0.986468i	$-0.447576\pi$
$179$	4.38703	0.327902	0.163951	+	0.986468i	$0.447576\pi$
$180$	0	0
$181$	14.8516	1.10391	0.551957	−	0.833873i	$-0.313881\pi$
$181$	14.8516	1.10391	0.551957	+	0.833873i	$0.313881\pi$
$182$	0	0
$183$	−11.9020	−0.879819
$184$	0	0
$185$	0.896583	0.0659181
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	−0.151322	−0.0110070
$190$	0	0
$191$	−5.35007	−0.387118	−0.193559	−	0.981089i	$-0.562003\pi$
$191$	−5.35007	−0.387118	−0.193559	+	0.981089i	$0.562003\pi$
$192$	0	0
$193$	−19.1684	−1.37977	−0.689886	−	0.723918i	$-0.742339\pi$
$193$	−19.1684	−1.37977	−0.689886	+	0.723918i	$0.742339\pi$
$194$	0	0
$195$	−36.5265	−2.61572
$196$	0	0
$197$	9.09238	0.647805	0.323903	−	0.946090i	$-0.395005\pi$
$197$	9.09238	0.647805	0.323903	+	0.946090i	$0.395005\pi$
$198$	0	0
$199$	15.8284	1.12205	0.561023	−	0.827800i	$-0.310408\pi$
$199$	15.8284	1.12205	0.561023	+	0.827800i	$0.310408\pi$
$200$	0	0
$201$	−2.28413	−0.161110
$202$	0	0
$203$	0.819756	0.0575356
$204$	0	0
$205$	23.8298	1.66434
$206$	0	0
$207$	3.57292	0.248335
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	13.3190	0.916916	0.458458	−	0.888716i	$-0.348402\pi$
$211$	13.3190	0.916916	0.458458	+	0.888716i	$0.348402\pi$
$212$	0	0
$213$	16.7345	1.14663
$214$	0	0
$215$	−11.9052	−0.811929
$216$	0	0
$217$	−1.52228	−0.103339
$218$	0	0
$219$	26.9944	1.82411
$220$	0	0
$221$	10.8520	0.729984
$222$	0	0
$223$	−2.17751	−0.145817	−0.0729085	−	0.997339i	$-0.523228\pi$
$223$	−2.17751	−0.145817	−0.0729085	+	0.997339i	$0.523228\pi$
$224$	0	0
$225$	8.07400	0.538267
$226$	0	0
$227$	14.0465	0.932297	0.466149	−	0.884706i	$-0.345641\pi$
$227$	14.0465	0.932297	0.466149	+	0.884706i	$0.345641\pi$
$228$	0	0
$229$	4.52153	0.298791	0.149395	−	0.988778i	$-0.452267\pi$
$229$	4.52153	0.298791	0.149395	+	0.988778i	$0.452267\pi$
$230$	0	0
$231$	0.332652	0.0218869
$232$	0	0
$233$	−0.911962	−0.0597446	−0.0298723	−	0.999554i	$-0.509510\pi$
$233$	−0.911962	−0.0597446	−0.0298723	+	0.999554i	$0.509510\pi$
$234$	0	0
$235$	−3.11662	−0.203306
$236$	0	0
$237$	−4.04430	−0.262706
$238$	0	0
$239$	−21.7682	−1.40807	−0.704035	−	0.710165i	$-0.748620\pi$
$239$	−21.7682	−1.40807	−0.704035	+	0.710165i	$0.748620\pi$
$240$	0	0
$241$	−9.84508	−0.634177	−0.317089	−	0.948396i	$-0.602705\pi$
$241$	−9.84508	−0.634177	−0.317089	+	0.948396i	$0.602705\pi$
$242$	0	0
$243$	20.7060	1.32829
$244$	0	0
$245$	−19.9541	−1.27482
$246$	0	0
$247$	−5.42599	−0.345248
$248$	0	0
$249$	−31.4209	−1.99122
$250$	0	0
$251$	28.8579	1.82149	0.910747	−	0.412964i	$-0.135506\pi$
$251$	28.8579	1.82149	0.910747	+	0.412964i	$0.135506\pi$
$252$	0	0
$253$	1.40384	0.0882587
$254$	0	0
$255$	−13.4635	−0.843119
$256$	0	0
$257$	2.10024	0.131010	0.0655048	−	0.997852i	$-0.479134\pi$
$257$	2.10024	0.131010	0.0655048	+	0.997852i	$0.479134\pi$
$258$	0	0
$259$	−0.0443049	−0.00275297
$260$	0	0
$261$	−14.7691	−0.914186
$262$	0	0
$263$	29.2474	1.80347	0.901737	−	0.432286i	$-0.142293\pi$
$263$	29.2474	1.80347	0.901737	+	0.432286i	$0.142293\pi$
$264$	0	0
$265$	5.71747	0.351221
$266$	0	0
$267$	18.5446	1.13491
$268$	0	0
$269$	27.2791	1.66323	0.831617	−	0.555350i	$-0.187416\pi$
$269$	27.2791	1.66323	0.831617	+	0.555350i	$0.187416\pi$
$270$	0	0
$271$	−1.20506	−0.0732024	−0.0366012	−	0.999330i	$-0.511653\pi$
$271$	−1.20506	−0.0732024	−0.0366012	+	0.999330i	$0.511653\pi$
$272$	0	0
$273$	1.80497	0.109242
$274$	0	0
$275$	3.17236	0.191301
$276$	0	0
$277$	−25.7061	−1.54453	−0.772266	−	0.635299i	$-0.780877\pi$
$277$	−25.7061	−1.54453	−0.772266	+	0.635299i	$0.780877\pi$
$278$	0	0
$279$	27.4262	1.64197
$280$	0	0
$281$	19.7071	1.17562	0.587812	−	0.808998i	$-0.299989\pi$
$281$	19.7071	1.17562	0.587812	+	0.808998i	$0.299989\pi$
$282$	0	0
$283$	−3.11769	−0.185327	−0.0926636	−	0.995697i	$-0.529538\pi$
$283$	−3.11769	−0.185327	−0.0926636	+	0.995697i	$0.529538\pi$
$284$	0	0
$285$	6.73176	0.398755
$286$	0	0
$287$	−1.17756	−0.0695090
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−30.2950	−1.77593
$292$	0	0
$293$	26.8515	1.56868	0.784340	−	0.620331i	$-0.213002\pi$
$293$	26.8515	1.56868	0.784340	+	0.620331i	$0.213002\pi$
$294$	0	0
$295$	−10.6650	−0.620941
$296$	0	0
$297$	1.07119	0.0621566
$298$	0	0
$299$	7.61723	0.440516
$300$	0	0
$301$	0.588300	0.0339091
$302$	0	0
$303$	−13.4214	−0.771039
$304$	0	0
$305$	14.4490	0.827348
$306$	0	0
$307$	−4.51352	−0.257600	−0.128800	−	0.991671i	$-0.541113\pi$
$307$	−4.51352	−0.257600	−0.128800	+	0.991671i	$0.541113\pi$
$308$	0	0
$309$	−28.7993	−1.63834
$310$	0	0
$311$	−1.37274	−0.0778410	−0.0389205	−	0.999242i	$-0.512392\pi$
$311$	−1.37274	−0.0778410	−0.0389205	+	0.999242i	$0.512392\pi$
$312$	0	0
$313$	−22.6764	−1.28175	−0.640874	−	0.767646i	$-0.721428\pi$
$313$	−22.6764	−1.28175	−0.640874	+	0.767646i	$0.721428\pi$
$314$	0	0
$315$	−1.02782	−0.0579109
$316$	0	0
$317$	15.3966	0.864757	0.432378	−	0.901692i	$-0.357675\pi$
$317$	15.3966	0.864757	0.432378	+	0.901692i	$0.357675\pi$
$318$	0	0
$319$	−5.80295	−0.324903
$320$	0	0
$321$	10.8963	0.608172
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	17.2132	0.954818
$326$	0	0
$327$	−18.9428	−1.04754
$328$	0	0
$329$	0.154009	0.00849079
$330$	0	0
$331$	21.3513	1.17358	0.586788	−	0.809741i	$-0.300392\pi$
$331$	21.3513	1.17358	0.586788	+	0.809741i	$0.300392\pi$
$332$	0	0
$333$	0.798219	0.0437422
$334$	0	0
$335$	2.77294	0.151502
$336$	0	0
$337$	−13.1456	−0.716088	−0.358044	−	0.933705i	$-0.616556\pi$
$337$	−13.1456	−0.716088	−0.358044	+	0.933705i	$0.616556\pi$
$338$	0	0
$339$	45.4346	2.46767
$340$	0	0
$341$	10.7761	0.583557
$342$	0	0
$343$	1.97490	0.106634
$344$	0	0
$345$	−9.45032	−0.508788
$346$	0	0
$347$	−23.0346	−1.23656	−0.618281	−	0.785958i	$-0.712170\pi$
$347$	−23.0346	−1.23656	−0.618281	+	0.785958i	$0.712170\pi$
$348$	0	0
$349$	28.1194	1.50520	0.752598	−	0.658481i	$-0.228801\pi$
$349$	28.1194	1.50520	0.752598	+	0.658481i	$0.228801\pi$
$350$	0	0
$351$	5.81226	0.310235
$352$	0	0
$353$	−28.4868	−1.51620	−0.758099	−	0.652140i	$-0.773871\pi$
$353$	−28.4868	−1.51620	−0.758099	+	0.652140i	$0.773871\pi$
$354$	0	0
$355$	−20.3157	−1.07825
$356$	0	0
$357$	0.665305	0.0352117
$358$	0	0
$359$	13.8587	0.731436	0.365718	−	0.930726i	$-0.380823\pi$
$359$	13.8587	0.731436	0.365718	+	0.930726i	$0.380823\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.35480	−0.123595
$364$	0	0
$365$	−32.7712	−1.71532
$366$	0	0
$367$	−28.4446	−1.48480	−0.742399	−	0.669958i	$-0.766312\pi$
$367$	−28.4446	−1.48480	−0.742399	+	0.669958i	$0.766312\pi$
$368$	0	0
$369$	21.2154	1.10443
$370$	0	0
$371$	−0.282531	−0.0146683
$372$	0	0
$373$	−8.42178	−0.436063	−0.218032	−	0.975942i	$-0.569964\pi$
$373$	−8.42178	−0.436063	−0.218032	+	0.975942i	$0.569964\pi$
$374$	0	0
$375$	12.3032	0.635335
$376$	0	0
$377$	−31.4868	−1.62165
$378$	0	0
$379$	31.0227	1.59353	0.796765	−	0.604289i	$-0.206543\pi$
$379$	31.0227	1.59353	0.796765	+	0.604289i	$0.206543\pi$
$380$	0	0
$381$	38.9808	1.99705
$382$	0	0
$383$	11.6375	0.594649	0.297325	−	0.954776i	$-0.403906\pi$
$383$	11.6375	0.594649	0.297325	+	0.954776i	$0.403906\pi$
$384$	0	0
$385$	−0.403840	−0.0205816
$386$	0	0
$387$	−10.5991	−0.538783
$388$	0	0
$389$	−17.5168	−0.888135	−0.444067	−	0.895993i	$-0.646465\pi$
$389$	−17.5168	−0.888135	−0.444067	+	0.895993i	$0.646465\pi$
$390$	0	0
$391$	2.80768	0.141990
$392$	0	0
$393$	7.58722	0.382724
$394$	0	0
$395$	4.90979	0.247038
$396$	0	0
$397$	−18.2320	−0.915040	−0.457520	−	0.889199i	$-0.651262\pi$
$397$	−18.2320	−0.915040	−0.457520	+	0.889199i	$0.651262\pi$
$398$	0	0
$399$	−0.332652	−0.0166534
$400$	0	0
$401$	22.9830	1.14772	0.573858	−	0.818955i	$-0.305446\pi$
$401$	22.9830	1.14772	0.573858	+	0.818955i	$0.305446\pi$
$402$	0	0
$403$	58.4709	2.91264
$404$	0	0
$405$	−29.0383	−1.44293
$406$	0	0
$407$	0.313629	0.0155460
$408$	0	0
$409$	−5.14711	−0.254508	−0.127254	−	0.991870i	$-0.540616\pi$
$409$	−5.14711	−0.254508	−0.127254	+	0.991870i	$0.540616\pi$
$410$	0	0
$411$	5.78080	0.285146
$412$	0	0
$413$	0.527015	0.0259327
$414$	0	0
$415$	38.1450	1.87247
$416$	0	0
$417$	16.4551	0.805812
$418$	0	0
$419$	−16.7561	−0.818591	−0.409295	−	0.912402i	$-0.634225\pi$
$419$	−16.7561	−0.818591	−0.409295	+	0.912402i	$0.634225\pi$
$420$	0	0
$421$	−17.7102	−0.863141	−0.431571	−	0.902079i	$-0.642040\pi$
$421$	−17.7102	−0.863141	−0.431571	+	0.902079i	$0.642040\pi$
$422$	0	0
$423$	−2.77470	−0.134911
$424$	0	0
$425$	6.34473	0.307765
$426$	0	0
$427$	−0.714003	−0.0345530
$428$	0	0
$429$	−12.7772	−0.616887
$430$	0	0
$431$	−19.9484	−0.960881	−0.480440	−	0.877027i	$-0.659523\pi$
$431$	−19.9484	−0.960881	−0.480440	+	0.877027i	$0.659523\pi$
$432$	0	0
$433$	34.3623	1.65135	0.825674	−	0.564147i	$-0.190795\pi$
$433$	34.3623	1.65135	0.825674	+	0.564147i	$0.190795\pi$
$434$	0	0
$435$	39.0641	1.87298
$436$	0	0
$437$	−1.40384	−0.0671548
$438$	0	0
$439$	17.2190	0.821820	0.410910	−	0.911676i	$-0.365211\pi$
$439$	17.2190	0.821820	0.410910	+	0.911676i	$0.365211\pi$
$440$	0	0
$441$	−17.7649	−0.845950
$442$	0	0
$443$	−38.6040	−1.83413	−0.917065	−	0.398738i	$-0.869448\pi$
$443$	−38.6040	−1.83413	−0.917065	+	0.398738i	$0.869448\pi$
$444$	0	0
$445$	−22.5132	−1.06723
$446$	0	0
$447$	−27.8906	−1.31918
$448$	0	0
$449$	15.2480	0.719596	0.359798	−	0.933030i	$-0.382846\pi$
$449$	15.2480	0.719596	0.359798	+	0.933030i	$0.382846\pi$
$450$	0	0
$451$	8.33578	0.392517
$452$	0	0
$453$	−27.8906	−1.31041
$454$	0	0
$455$	−2.19123	−0.102727
$456$	0	0
$457$	24.4114	1.14192	0.570958	−	0.820980i	$-0.306572\pi$
$457$	24.4114	1.14192	0.570958	+	0.820980i	$0.306572\pi$
$458$	0	0
$459$	2.14238	0.0999975
$460$	0	0
$461$	−12.3071	−0.573198	−0.286599	−	0.958051i	$-0.592525\pi$
$461$	−12.3071	−0.573198	−0.286599	+	0.958051i	$0.592525\pi$
$462$	0	0
$463$	16.6112	0.771990	0.385995	−	0.922501i	$-0.373858\pi$
$463$	16.6112	0.771990	0.385995	+	0.922501i	$0.373858\pi$
$464$	0	0
$465$	−72.5419	−3.36405
$466$	0	0
$467$	−7.05468	−0.326452	−0.163226	−	0.986589i	$-0.552190\pi$
$467$	−7.05468	−0.326452	−0.163226	+	0.986589i	$0.552190\pi$
$468$	0	0
$469$	−0.137026	−0.00632726
$470$	0	0
$471$	−27.6808	−1.27546
$472$	0	0
$473$	−4.16450	−0.191484
$474$	0	0
$475$	−3.17236	−0.145558
$476$	0	0
$477$	5.09021	0.233065
$478$	0	0
$479$	−26.0721	−1.19127	−0.595633	−	0.803257i	$-0.703099\pi$
$479$	−26.0721	−1.19127	−0.595633	+	0.803257i	$0.703099\pi$
$480$	0	0
$481$	1.70175	0.0775931
$482$	0	0
$483$	0.466991	0.0212488
$484$	0	0
$485$	36.7782	1.67001
$486$	0	0
$487$	11.5757	0.524545	0.262273	−	0.964994i	$-0.415528\pi$
$487$	11.5757	0.524545	0.262273	+	0.964994i	$0.415528\pi$
$488$	0	0
$489$	−10.3206	−0.466713
$490$	0	0
$491$	−24.1340	−1.08915	−0.544577	−	0.838711i	$-0.683310\pi$
$491$	−24.1340	−1.08915	−0.544577	+	0.838711i	$0.683310\pi$
$492$	0	0
$493$	−11.6059	−0.522703
$494$	0	0
$495$	7.27578	0.327022
$496$	0	0
$497$	1.00391	0.0450315
$498$	0	0
$499$	−13.6502	−0.611067	−0.305534	−	0.952181i	$-0.598835\pi$
$499$	−13.6502	−0.611067	−0.305534	+	0.952181i	$0.598835\pi$
$500$	0	0
$501$	−34.8469	−1.55685
$502$	0	0
$503$	25.7833	1.14962	0.574810	−	0.818287i	$-0.305076\pi$
$503$	25.7833	1.14962	0.574810	+	0.818287i	$0.305076\pi$
$504$	0	0
$505$	16.2936	0.725055
$506$	0	0
$507$	−38.7163	−1.71945
$508$	0	0
$509$	−40.5713	−1.79829	−0.899146	−	0.437648i	$-0.855812\pi$
$509$	−40.5713	−1.79829	−0.899146	+	0.437648i	$0.855812\pi$
$510$	0	0
$511$	1.61940	0.0716380
$512$	0	0
$513$	−1.07119	−0.0472941
$514$	0	0
$515$	34.9624	1.54063
$516$	0	0
$517$	−1.09021	−0.0479474
$518$	0	0
$519$	−35.7409	−1.56885
$520$	0	0
$521$	8.12291	0.355871	0.177936	−	0.984042i	$-0.443058\pi$
$521$	8.12291	0.355871	0.177936	+	0.984042i	$0.443058\pi$
$522$	0	0
$523$	35.4698	1.55099	0.775493	−	0.631357i	$-0.217502\pi$
$523$	35.4698	1.55099	0.775493	+	0.631357i	$0.217502\pi$
$524$	0	0
$525$	1.05529	0.0460568
$526$	0	0
$527$	21.5521	0.938826
$528$	0	0
$529$	−21.0292	−0.914314
$530$	0	0
$531$	−9.49496	−0.412046
$532$	0	0
$533$	45.2299	1.95912
$534$	0	0
$535$	−13.2281	−0.571901
$536$	0	0
$537$	−10.3306	−0.445799
$538$	0	0
$539$	−6.98004	−0.300652
$540$	0	0
$541$	−1.65962	−0.0713525	−0.0356763	−	0.999363i	$-0.511359\pi$
$541$	−1.65962	−0.0713525	−0.0356763	+	0.999363i	$0.511359\pi$
$542$	0	0
$543$	−34.9727	−1.50082
$544$	0	0
$545$	22.9965	0.985063
$546$	0	0
$547$	16.9321	0.723963	0.361982	−	0.932185i	$-0.382100\pi$
$547$	16.9321	0.723963	0.361982	+	0.932185i	$0.382100\pi$
$548$	0	0
$549$	12.8638	0.549015
$550$	0	0
$551$	5.80295	0.247214
$552$	0	0
$553$	−0.242619	−0.0103172
$554$	0	0
$555$	−2.11128	−0.0896187
$556$	0	0
$557$	−33.8412	−1.43390	−0.716949	−	0.697125i	$-0.754462\pi$
$557$	−33.8412	−1.43390	−0.716949	+	0.697125i	$0.754462\pi$
$558$	0	0
$559$	−22.5966	−0.955733
$560$	0	0
$561$	−4.70961	−0.198840
$562$	0	0
$563$	−37.1967	−1.56765	−0.783827	−	0.620979i	$-0.786735\pi$
$563$	−37.1967	−1.56765	−0.783827	+	0.620979i	$0.786735\pi$
$564$	0	0
$565$	−55.1577	−2.32050
$566$	0	0
$567$	1.43494	0.0602618
$568$	0	0
$569$	−1.67265	−0.0701211	−0.0350606	−	0.999385i	$-0.511162\pi$
$569$	−1.67265	−0.0701211	−0.0350606	+	0.999385i	$0.511162\pi$
$570$	0	0
$571$	3.99672	0.167257	0.0836287	−	0.996497i	$-0.473349\pi$
$571$	3.99672	0.167257	0.0836287	+	0.996497i	$0.473349\pi$
$572$	0	0
$573$	12.5984	0.526305
$574$	0	0
$575$	4.45349	0.185723
$576$	0	0
$577$	12.5478	0.522374	0.261187	−	0.965288i	$-0.415886\pi$
$577$	12.5478	0.522374	0.261187	+	0.965288i	$0.415886\pi$
$578$	0	0
$579$	45.1378	1.87586
$580$	0	0
$581$	−1.88495	−0.0782009
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	39.4783	1.63223
$586$	0	0
$587$	−23.1429	−0.955210	−0.477605	−	0.878575i	$-0.658495\pi$
$587$	−23.1429	−0.955210	−0.477605	+	0.878575i	$0.658495\pi$
$588$	0	0
$589$	−10.7761	−0.444020
$590$	0	0
$591$	−21.4108	−0.880722
$592$	0	0
$593$	12.8322	0.526956	0.263478	−	0.964665i	$-0.415130\pi$
$593$	12.8322	0.526956	0.263478	+	0.964665i	$0.415130\pi$
$594$	0	0
$595$	−0.807680	−0.0331117
$596$	0	0
$597$	−37.2728	−1.52547
$598$	0	0
$599$	−24.5282	−1.00220	−0.501098	−	0.865390i	$-0.667070\pi$
$599$	−24.5282	−1.00220	−0.501098	+	0.865390i	$0.667070\pi$
$600$	0	0
$601$	13.2101	0.538851	0.269425	−	0.963021i	$-0.413166\pi$
$601$	13.2101	0.538851	0.269425	+	0.963021i	$0.413166\pi$
$602$	0	0
$603$	2.46872	0.100534
$604$	0	0
$605$	2.85873	0.116224
$606$	0	0
$607$	−30.7247	−1.24708	−0.623538	−	0.781793i	$-0.714306\pi$
$607$	−30.7247	−1.24708	−0.623538	+	0.781793i	$0.714306\pi$
$608$	0	0
$609$	−1.93036	−0.0782223
$610$	0	0
$611$	−5.91548	−0.239315
$612$	0	0
$613$	−27.2812	−1.10188	−0.550939	−	0.834545i	$-0.685730\pi$
$613$	−27.2812	−1.10188	−0.550939	+	0.834545i	$0.685730\pi$
$614$	0	0
$615$	−56.1145	−2.26275
$616$	0	0
$617$	−33.9097	−1.36515	−0.682576	−	0.730814i	$-0.739141\pi$
$617$	−33.9097	−1.36515	−0.682576	+	0.730814i	$0.739141\pi$
$618$	0	0
$619$	33.5126	1.34698	0.673492	−	0.739195i	$-0.264794\pi$
$619$	33.5126	1.34698	0.673492	+	0.739195i	$0.264794\pi$
$620$	0	0
$621$	1.50378	0.0603445
$622$	0	0
$623$	1.11250	0.0445712
$624$	0	0
$625$	−30.7979	−1.23192
$626$	0	0
$627$	2.35480	0.0940418
$628$	0	0
$629$	0.627258	0.0250104
$630$	0	0
$631$	−11.1858	−0.445298	−0.222649	−	0.974899i	$-0.571470\pi$
$631$	−11.1858	−0.445298	−0.222649	+	0.974899i	$0.571470\pi$
$632$	0	0
$633$	−31.3636	−1.24659
$634$	0	0
$635$	−47.3227	−1.87795
$636$	0	0
$637$	−37.8737	−1.50061
$638$	0	0
$639$	−18.0869	−0.715507
$640$	0	0
$641$	−23.0254	−0.909449	−0.454725	−	0.890632i	$-0.650262\pi$
$641$	−23.0254	−0.909449	−0.454725	+	0.890632i	$0.650262\pi$
$642$	0	0
$643$	−23.3859	−0.922250	−0.461125	−	0.887335i	$-0.652554\pi$
$643$	−23.3859	−0.922250	−0.461125	+	0.887335i	$0.652554\pi$
$644$	0	0
$645$	28.0345	1.10386
$646$	0	0
$647$	−26.1402	−1.02768	−0.513838	−	0.857887i	$-0.671777\pi$
$647$	−26.1402	−1.02768	−0.513838	+	0.857887i	$0.671777\pi$
$648$	0	0
$649$	−3.73068	−0.146442
$650$	0	0
$651$	3.58468	0.140495
$652$	0	0
$653$	−4.13649	−0.161873	−0.0809367	−	0.996719i	$-0.525791\pi$
$653$	−4.13649	−0.161873	−0.0809367	+	0.996719i	$0.525791\pi$
$654$	0	0
$655$	−9.21089	−0.359899
$656$	0	0
$657$	−29.1759	−1.13826
$658$	0	0
$659$	25.6597	0.999559	0.499779	−	0.866153i	$-0.333414\pi$
$659$	25.6597	0.999559	0.499779	+	0.866153i	$0.333414\pi$
$660$	0	0
$661$	29.2228	1.13663	0.568317	−	0.822809i	$-0.307595\pi$
$661$	29.2228	1.13663	0.568317	+	0.822809i	$0.307595\pi$
$662$	0	0
$663$	−25.5543	−0.992447
$664$	0	0
$665$	0.403840	0.0156602
$666$	0	0
$667$	−8.14641	−0.315430
$668$	0	0
$669$	5.12762	0.198245
$670$	0	0
$671$	5.05434	0.195121
$672$	0	0
$673$	−17.3166	−0.667506	−0.333753	−	0.942661i	$-0.608315\pi$
$673$	−17.3166	−0.667506	−0.333753	+	0.942661i	$0.608315\pi$
$674$	0	0
$675$	3.39820	0.130797
$676$	0	0
$677$	−47.6442	−1.83111	−0.915557	−	0.402188i	$-0.868250\pi$
$677$	−47.6442	−1.83111	−0.915557	+	0.402188i	$0.868250\pi$
$678$	0	0
$679$	−1.81741	−0.0697456
$680$	0	0
$681$	−33.0767	−1.26750
$682$	0	0
$683$	−36.3714	−1.39171	−0.695857	−	0.718180i	$-0.744975\pi$
$683$	−36.3714	−1.39171	−0.695857	+	0.718180i	$0.744975\pi$
$684$	0	0
$685$	−7.01789	−0.268140
$686$	0	0
$687$	−10.6473	−0.406220
$688$	0	0
$689$	10.8520	0.413428
$690$	0	0
$691$	26.9855	1.02658	0.513288	−	0.858216i	$-0.328427\pi$
$691$	26.9855	1.02658	0.513288	+	0.858216i	$0.328427\pi$
$692$	0	0
$693$	−0.359535	−0.0136576
$694$	0	0
$695$	−19.9766	−0.757754
$696$	0	0
$697$	16.6716	0.631480
$698$	0	0
$699$	2.14749	0.0812256
$700$	0	0
$701$	10.0352	0.379022	0.189511	−	0.981879i	$-0.439310\pi$
$701$	10.0352	0.379022	0.189511	+	0.981879i	$0.439310\pi$
$702$	0	0
$703$	−0.313629	−0.0118287
$704$	0	0
$705$	7.33904	0.276404
$706$	0	0
$707$	−0.805153	−0.0302809
$708$	0	0
$709$	−3.63248	−0.136421	−0.0682104	−	0.997671i	$-0.521729\pi$
$709$	−3.63248	−0.136421	−0.0682104	+	0.997671i	$0.521729\pi$
$710$	0	0
$711$	4.37114	0.163931
$712$	0	0
$713$	15.1279	0.566543
$714$	0	0
$715$	15.5115	0.580096
$716$	0	0
$717$	51.2599	1.91434
$718$	0	0
$719$	−22.8315	−0.851471	−0.425735	−	0.904848i	$-0.639985\pi$
$719$	−22.8315	−0.851471	−0.425735	+	0.904848i	$0.639985\pi$
$720$	0	0
$721$	−1.72768	−0.0643421
$722$	0	0
$723$	23.1832	0.862194
$724$	0	0
$725$	−18.4091	−0.683696
$726$	0	0
$727$	−5.07700	−0.188296	−0.0941478	−	0.995558i	$-0.530013\pi$
$727$	−5.07700	−0.188296	−0.0941478	+	0.995558i	$0.530013\pi$
$728$	0	0
$729$	−18.2852	−0.677231
$730$	0	0
$731$	−8.32901	−0.308059
$732$	0	0
$733$	32.5179	1.20107	0.600537	−	0.799597i	$-0.294953\pi$
$733$	32.5179	1.20107	0.600537	+	0.799597i	$0.294953\pi$
$734$	0	0
$735$	46.9880	1.73318
$736$	0	0
$737$	0.969988	0.0357300
$738$	0	0
$739$	30.5536	1.12393	0.561966	−	0.827160i	$-0.310045\pi$
$739$	30.5536	1.12393	0.561966	+	0.827160i	$0.310045\pi$
$740$	0	0
$741$	12.7772	0.469380
$742$	0	0
$743$	−27.6773	−1.01538	−0.507690	−	0.861540i	$-0.669500\pi$
$743$	−27.6773	−1.01538	−0.507690	+	0.861540i	$0.669500\pi$
$744$	0	0
$745$	33.8592	1.24051
$746$	0	0
$747$	33.9602	1.24254
$748$	0	0
$749$	0.653671	0.0238846
$750$	0	0
$751$	−42.2574	−1.54199	−0.770997	−	0.636839i	$-0.780241\pi$
$751$	−42.2574	−1.54199	−0.770997	+	0.636839i	$0.780241\pi$
$752$	0	0
$753$	−67.9547	−2.47641
$754$	0	0
$755$	33.8592	1.23226
$756$	0	0
$757$	0.994204	0.0361350	0.0180675	−	0.999837i	$-0.494249\pi$
$757$	0.994204	0.0361350	0.0180675	+	0.999837i	$0.494249\pi$
$758$	0	0
$759$	−3.30577	−0.119992
$760$	0	0
$761$	38.3271	1.38936	0.694679	−	0.719320i	$-0.255546\pi$
$761$	38.3271	1.38936	0.694679	+	0.719320i	$0.255546\pi$
$762$	0	0
$763$	−1.13638	−0.0411398
$764$	0	0
$765$	14.5516	0.526113
$766$	0	0
$767$	−20.2426	−0.730919
$768$	0	0
$769$	−37.5257	−1.35321	−0.676606	−	0.736345i	$-0.736550\pi$
$769$	−37.5257	−1.35321	−0.676606	+	0.736345i	$0.736550\pi$
$770$	0	0
$771$	−4.94566	−0.178114
$772$	0	0
$773$	−8.13798	−0.292703	−0.146351	−	0.989233i	$-0.546753\pi$
$773$	−8.13798	−0.292703	−0.146351	+	0.989233i	$0.546753\pi$
$774$	0	0
$775$	34.1856	1.22798
$776$	0	0
$777$	0.104329	0.00374280
$778$	0	0
$779$	−8.33578	−0.298660
$780$	0	0
$781$	−7.10655	−0.254292
$782$	0	0
$783$	−6.21605	−0.222143
$784$	0	0
$785$	33.6045	1.19940
$786$	0	0
$787$	−15.8233	−0.564041	−0.282021	−	0.959408i	$-0.591005\pi$
$787$	−15.8233	−0.564041	−0.282021	+	0.959408i	$0.591005\pi$
$788$	0	0
$789$	−68.8720	−2.45191
$790$	0	0
$791$	2.72564	0.0969125
$792$	0	0
$793$	27.4248	0.973883
$794$	0	0
$795$	−13.4635	−0.477502
$796$	0	0
$797$	−1.70989	−0.0605674	−0.0302837	−	0.999541i	$-0.509641\pi$
$797$	−1.70989	−0.0605674	−0.0302837	+	0.999541i	$0.509641\pi$
$798$	0	0
$799$	−2.18042	−0.0771378
$800$	0	0
$801$	−20.0433	−0.708194
$802$	0	0
$803$	−11.4635	−0.404539
$804$	0	0
$805$	−0.566927	−0.0199815
$806$	0	0
$807$	−64.2369	−2.26124
$808$	0	0
$809$	21.8083	0.766737	0.383369	−	0.923595i	$-0.374764\pi$
$809$	21.8083	0.766737	0.383369	+	0.923595i	$0.374764\pi$
$810$	0	0
$811$	−19.1410	−0.672133	−0.336066	−	0.941838i	$-0.609097\pi$
$811$	−19.1410	−0.672133	−0.336066	+	0.941838i	$0.609097\pi$
$812$	0	0
$813$	2.83769	0.0995221
$814$	0	0
$815$	12.5292	0.438878
$816$	0	0
$817$	4.16450	0.145698
$818$	0	0
$819$	−1.95084	−0.0681677
$820$	0	0
$821$	9.11875	0.318247	0.159123	−	0.987259i	$-0.449133\pi$
$821$	9.11875	0.318247	0.159123	+	0.987259i	$0.449133\pi$
$822$	0	0
$823$	−4.68797	−0.163412	−0.0817062	−	0.996656i	$-0.526037\pi$
$823$	−4.68797	−0.163412	−0.0817062	+	0.996656i	$0.526037\pi$
$824$	0	0
$825$	−7.47030	−0.260082
$826$	0	0
$827$	−41.8350	−1.45475	−0.727373	−	0.686242i	$-0.759259\pi$
$827$	−41.8350	−1.45475	−0.727373	+	0.686242i	$0.759259\pi$
$828$	0	0
$829$	41.4318	1.43898	0.719492	−	0.694501i	$-0.244375\pi$
$829$	41.4318	1.43898	0.719492	+	0.694501i	$0.244375\pi$
$830$	0	0
$831$	60.5329	2.09986
$832$	0	0
$833$	−13.9601	−0.483688
$834$	0	0
$835$	42.3042	1.46400
$836$	0	0
$837$	11.5432	0.398991
$838$	0	0
$839$	−35.3199	−1.21938	−0.609689	−	0.792641i	$-0.708706\pi$
$839$	−35.3199	−1.21938	−0.609689	+	0.792641i	$0.708706\pi$
$840$	0	0
$841$	4.67423	0.161180
$842$	0	0
$843$	−46.4063	−1.59832
$844$	0	0
$845$	47.0016	1.61690
$846$	0	0
$847$	−0.141265	−0.00485393
$848$	0	0
$849$	7.34154	0.251961
$850$	0	0
$851$	0.440285	0.0150928
$852$	0	0
$853$	−7.73034	−0.264682	−0.132341	−	0.991204i	$-0.542249\pi$
$853$	−7.73034	−0.264682	−0.132341	+	0.991204i	$0.542249\pi$
$854$	0	0
$855$	−7.27578	−0.248826
$856$	0	0
$857$	44.9489	1.53543	0.767713	−	0.640794i	$-0.221395\pi$
$857$	44.9489	1.53543	0.767713	+	0.640794i	$0.221395\pi$
$858$	0	0
$859$	9.51950	0.324801	0.162401	−	0.986725i	$-0.448076\pi$
$859$	9.51950	0.324801	0.162401	+	0.986725i	$0.448076\pi$
$860$	0	0
$861$	2.77292	0.0945008
$862$	0	0
$863$	−0.0224450	−0.000764037 0	−0.000382019	−	1.00000i	$-0.500122\pi$
$863$	−0.0224450	−0.000764037 0	−0.000382019		1.00000i	$0.500122\pi$
$864$	0	0
$865$	43.3895	1.47529
$866$	0	0
$867$	30.6125	1.03965
$868$	0	0
$869$	1.71747	0.0582612
$870$	0	0
$871$	5.26315	0.178335
$872$	0	0
$873$	32.7433	1.10819
$874$	0	0
$875$	0.738073	0.0249514
$876$	0	0
$877$	0.313013	0.0105697	0.00528485	−	0.999986i	$-0.498318\pi$
$877$	0.313013	0.0105697	0.00528485	+	0.999986i	$0.498318\pi$
$878$	0	0
$879$	−63.2300	−2.13269
$880$	0	0
$881$	13.5495	0.456493	0.228246	−	0.973603i	$-0.426701\pi$
$881$	13.5495	0.456493	0.228246	+	0.973603i	$0.426701\pi$
$882$	0	0
$883$	23.9821	0.807062	0.403531	−	0.914966i	$-0.367783\pi$
$883$	23.9821	0.807062	0.403531	+	0.914966i	$0.367783\pi$
$884$	0	0
$885$	25.1140	0.844199
$886$	0	0
$887$	0.0179402	0.000602374 0	0.000301187	−	1.00000i	$-0.499904\pi$
$887$	0.0179402	0.000602374 0	0.000301187		1.00000i	$0.499904\pi$
$888$	0	0
$889$	2.33847	0.0784297
$890$	0	0
$891$	−10.1578	−0.340298
$892$	0	0
$893$	1.09021	0.0364825
$894$	0	0
$895$	12.5414	0.419212
$896$	0	0
$897$	−17.9371	−0.598902
$898$	0	0
$899$	−62.5330	−2.08559
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−1.38533	−0.0461010
$904$	0	0
$905$	42.4569	1.41132
$906$	0	0
$907$	−29.4028	−0.976305	−0.488152	−	0.872758i	$-0.662329\pi$
$907$	−29.4028	−0.976305	−0.488152	+	0.872758i	$0.662329\pi$
$908$	0	0
$909$	14.5060	0.481134
$910$	0	0
$911$	−37.6260	−1.24660	−0.623302	−	0.781981i	$-0.714209\pi$
$911$	−37.6260	−1.24660	−0.623302	+	0.781981i	$0.714209\pi$
$912$	0	0
$913$	13.3433	0.441600
$914$	0	0
$915$	−34.0246	−1.12482
$916$	0	0
$917$	0.455159	0.0150307
$918$	0	0
$919$	21.3192	0.703255	0.351627	−	0.936140i	$-0.385628\pi$
$919$	21.3192	0.703255	0.351627	+	0.936140i	$0.385628\pi$
$920$	0	0
$921$	10.6284	0.350219
$922$	0	0
$923$	−38.5601	−1.26922
$924$	0	0
$925$	0.994946	0.0327136
$926$	0	0
$927$	31.1267	1.02234
$928$	0	0
$929$	30.7342	1.00836	0.504179	−	0.863599i	$-0.331795\pi$
$929$	30.7342	1.00836	0.504179	+	0.863599i	$0.331795\pi$
$930$	0	0
$931$	6.98004	0.228762
$932$	0	0
$933$	3.23254	0.105829
$934$	0	0
$935$	5.71747	0.186981
$936$	0	0
$937$	−10.1091	−0.330249	−0.165124	−	0.986273i	$-0.552803\pi$
$937$	−10.1091	−0.330249	−0.165124	+	0.986273i	$0.552803\pi$
$938$	0	0
$939$	53.3986	1.74260
$940$	0	0
$941$	27.4478	0.894773	0.447386	−	0.894341i	$-0.352355\pi$
$941$	27.4478	0.894773	0.447386	+	0.894341i	$0.352355\pi$
$942$	0	0
$943$	11.7021	0.381073
$944$	0	0
$945$	−0.432589	−0.0140721
$946$	0	0
$947$	36.5490	1.18768	0.593842	−	0.804582i	$-0.297611\pi$
$947$	36.5490	1.18768	0.593842	+	0.804582i	$0.297611\pi$
$948$	0	0
$949$	−62.2010	−2.01913
$950$	0	0
$951$	−36.2559	−1.17568
$952$	0	0
$953$	−23.2561	−0.753339	−0.376670	−	0.926348i	$-0.622931\pi$
$953$	−23.2561	−0.753339	−0.376670	+	0.926348i	$0.622931\pi$
$954$	0	0
$955$	−15.2944	−0.494917
$956$	0	0
$957$	13.6648	0.441721
$958$	0	0
$959$	0.346792	0.0111985
$960$	0	0
$961$	85.1236	2.74592
$962$	0	0
$963$	−11.7769	−0.379504
$964$	0	0
$965$	−54.7974	−1.76399
$966$	0	0
$967$	46.9842	1.51091	0.755455	−	0.655200i	$-0.227416\pi$
$967$	46.9842	1.51091	0.755455	+	0.655200i	$0.227416\pi$
$968$	0	0
$969$	4.70961	0.151294
$970$	0	0
$971$	−12.5236	−0.401901	−0.200951	−	0.979601i	$-0.564403\pi$
$971$	−12.5236	−0.401901	−0.200951	+	0.979601i	$0.564403\pi$
$972$	0	0
$973$	0.987149	0.0316465
$974$	0	0
$975$	−40.5338	−1.29812
$976$	0	0
$977$	31.7614	1.01614	0.508069	−	0.861316i	$-0.330359\pi$
$977$	31.7614	1.01614	0.508069	+	0.861316i	$0.330359\pi$
$978$	0	0
$979$	−7.87522	−0.251693
$980$	0	0
$981$	20.4736	0.653672
$982$	0	0
$983$	−23.7840	−0.758591	−0.379295	−	0.925276i	$-0.623834\pi$
$983$	−23.7840	−0.758591	−0.379295	+	0.925276i	$0.623834\pi$
$984$	0	0
$985$	25.9927	0.828197
$986$	0	0
$987$	−0.362661	−0.0115436
$988$	0	0
$989$	−5.84630	−0.185901
$990$	0	0
$991$	−50.9003	−1.61690	−0.808450	−	0.588565i	$-0.799693\pi$
$991$	−50.9003	−1.61690	−0.808450	+	0.588565i	$0.799693\pi$
$992$	0	0
$993$	−50.2782	−1.59553
$994$	0	0
$995$	45.2492	1.43450
$996$	0	0
$997$	−4.56186	−0.144476	−0.0722378	−	0.997387i	$-0.523014\pi$
$997$	−4.56186	−0.144476	−0.0722378	+	0.997387i	$0.523014\pi$
$998$	0	0
$999$	0.335956	0.0106292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.u.1.2		5
4.3	odd	2		1672.2.a.g.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.g.1.4	✓	5	4.3	odd	2
3344.2.a.u.1.2		5	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)

\(f(q)\)	\(=\)	\(q-2.35480 q^{3} +2.85873 q^{5} -0.141265 q^{7} +2.54511 q^{9} +O(q^{10})\)	\(q-2.35480 q^{3} +2.85873 q^{5} -0.141265 q^{7} +2.54511 q^{9} +1.00000 q^{11} +5.42599 q^{13} -6.73176 q^{15} +2.00000 q^{17} -1.00000 q^{19} +0.332652 q^{21} +1.40384 q^{23} +3.17236 q^{25} +1.07119 q^{27} -5.80295 q^{29} +10.7761 q^{31} -2.35480 q^{33} -0.403840 q^{35} +0.313629 q^{37} -12.7772 q^{39} +8.33578 q^{41} -4.16450 q^{43} +7.27578 q^{45} -1.09021 q^{47} -6.98004 q^{49} -4.70961 q^{51} +2.00000 q^{53} +2.85873 q^{55} +2.35480 q^{57} -3.73068 q^{59} +5.05434 q^{61} -0.359535 q^{63} +15.5115 q^{65} +0.969988 q^{67} -3.30577 q^{69} -7.10655 q^{71} -11.4635 q^{73} -7.47030 q^{75} -0.141265 q^{77} +1.71747 q^{79} -10.1578 q^{81} +13.3433 q^{83} +5.71747 q^{85} +13.6648 q^{87} -7.87522 q^{89} -0.766505 q^{91} -25.3755 q^{93} -2.85873 q^{95} +12.8652 q^{97} +2.54511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9}+O(q^{10}) \) 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9	\( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9} + 5 q^{11} + 4 q^{13} - q^{15} + 10 q^{17} - 5 q^{19} + 2 q^{21} - 5 q^{23} + 6 q^{25} - 7 q^{27} + 16 q^{29} - q^{31} - q^{33} + 10 q^{35} - q^{37} + 4 q^{39} + 28 q^{41} - 4 q^{43} + 12 q^{45} + 4 q^{47} - q^{49} - 2 q^{51} + 10 q^{53} + 7 q^{55} + q^{57} + q^{59} - 16 q^{61} - 12 q^{63} + 24 q^{65} + 9 q^{67} - 7 q^{69} - 7 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} - 6 q^{79} + 5 q^{81} - 4 q^{83} + 14 q^{85} + 2 q^{87} + 31 q^{89} + 12 q^{91} - 7 q^{93} - 7 q^{95} + 13 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9 + 5 * q^11 + 4 * q^13 - q^15 + 10 * q^17 - 5 * q^19 + 2 * q^21 - 5 * q^23 + 6 * q^25 - 7 * q^27 + 16 * q^29 - q^31 - q^33 + 10 * q^35 - q^37 + 4 * q^39 + 28 * q^41 - 4 * q^43 + 12 * q^45 + 4 * q^47 - q^49 - 2 * q^51 + 10 * q^53 + 7 * q^55 + q^57 + q^59 - 16 * q^61 - 12 * q^63 + 24 * q^65 + 9 * q^67 - 7 * q^69 - 7 * q^71 + 8 * q^73 + 8 * q^75 - 8 * q^77 - 6 * q^79 + 5 * q^81 - 4 * q^83 + 14 * q^85 + 2 * q^87 + 31 * q^89 + 12 * q^91 - 7 * q^93 - 7 * q^95 + 13 * q^97 + 8 * q^99

Introduction

L-functions

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