LMFDB - Embedded newform 3380.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.756.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 2 $ x^3 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.60168$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.60168 q^{3} +1.00000 q^{5} +2.76873 q^{7} +3.76873 q^{9} +O(q^{10})$	$q+2.60168 q^{3} +1.00000 q^{5} +2.76873 q^{7} +3.76873 q^{9} +2.16706 q^{11} +2.60168 q^{15} +5.03630 q^{19} +7.20336 q^{21} +4.93579 q^{23} +1.00000 q^{25} +2.00000 q^{27} -4.43462 q^{29} -3.37041 q^{31} +5.63798 q^{33} +2.76873 q^{35} -3.97209 q^{37} -1.66589 q^{41} -9.80504 q^{43} +3.76873 q^{45} +11.6380 q^{47} +0.665890 q^{49} -12.7408 q^{53} +2.16706 q^{55} +13.1028 q^{57} -8.16706 q^{59} -10.3062 q^{61} +10.4346 q^{63} -7.10284 q^{67} +12.8413 q^{69} +16.1112 q^{71} +15.9721 q^{73} +2.60168 q^{75} +6.00000 q^{77} +11.9442 q^{79} -6.10284 q^{81} -3.97209 q^{83} -11.5375 q^{87} -7.94419 q^{89} -8.76873 q^{93} +5.03630 q^{95} +0.462531 q^{97} +8.16706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^5 + 3 * q^9	$ 3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55} + 30 q^{57} - 24 q^{59} - 6 q^{61} + 24 q^{63} - 12 q^{67} + 24 q^{73} + 18 q^{77} - 12 q^{79} - 9 q^{81} + 12 q^{83} - 18 q^{87} + 24 q^{89} - 18 q^{93} + 18 q^{97} + 24 q^{99}+O(q^{100}) $ 3 * q + 3 * q^5 + 3 * q^9 + 6 * q^11 + 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 - 6 * q^29 + 6 * q^31 - 6 * q^33 + 12 * q^37 - 6 * q^41 - 6 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 - 6 * q^53 + 6 * q^55 + 30 * q^57 - 24 * q^59 - 6 * q^61 + 24 * q^63 - 12 * q^67 + 24 * q^73 + 18 * q^77 - 12 * q^79 - 9 * q^81 + 12 * q^83 - 18 * q^87 + 24 * q^89 - 18 * q^93 + 18 * q^97 + 24 * 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
$3$	2.60168	1.50208	0.751040	−	0.660257i	$-0.229552\pi$
$3$	2.60168	1.50208	0.751040	+	0.660257i	$0.229552\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.76873	1.04648	0.523242	−	0.852184i	$-0.324723\pi$
$7$	2.76873	1.04648	0.523242	+	0.852184i	$0.324723\pi$
$8$	0	0
$9$	3.76873	1.25624
$10$	0	0
$11$	2.16706	0.653392	0.326696	−	0.945130i	$-0.394065\pi$
$11$	2.16706	0.653392	0.326696	+	0.945130i	$0.394065\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.60168	0.671751
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	5.03630	1.15541	0.577704	−	0.816247i	$-0.303949\pi$
$19$	5.03630	1.15541	0.577704	+	0.816247i	$0.303949\pi$
$20$	0	0
$21$	7.20336	1.57190
$22$	0	0
$23$	4.93579	1.02918	0.514592	−	0.857435i	$-0.327944\pi$
$23$	4.93579	1.02918	0.514592	+	0.857435i	$0.327944\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.00000	0.384900
$28$	0	0
$29$	−4.43462	−0.823489	−0.411744	−	0.911299i	$-0.635080\pi$
$29$	−4.43462	−0.823489	−0.411744	+	0.911299i	$0.635080\pi$
$30$	0	0
$31$	−3.37041	−0.605344	−0.302672	−	0.953095i	$-0.597879\pi$
$31$	−3.37041	−0.605344	−0.302672	+	0.953095i	$0.597879\pi$
$32$	0	0
$33$	5.63798	0.981447
$34$	0	0
$35$	2.76873	0.468002
$36$	0	0
$37$	−3.97209	−0.653008	−0.326504	−	0.945196i	$-0.605871\pi$
$37$	−3.97209	−0.653008	−0.326504	+	0.945196i	$0.605871\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.66589	−0.260168	−0.130084	−	0.991503i	$-0.541525\pi$
$41$	−1.66589	−0.260168	−0.130084	+	0.991503i	$0.541525\pi$
$42$	0	0
$43$	−9.80504	−1.49525	−0.747627	−	0.664119i	$-0.768807\pi$
$43$	−9.80504	−1.49525	−0.747627	+	0.664119i	$0.768807\pi$
$44$	0	0
$45$	3.76873	0.561810
$46$	0	0
$47$	11.6380	1.69757	0.848787	−	0.528735i	$-0.177333\pi$
$47$	11.6380	1.69757	0.848787	+	0.528735i	$0.177333\pi$
$48$	0	0
$49$	0.665890	0.0951271
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.7408	−1.75009	−0.875044	−	0.484044i	$-0.839167\pi$
$53$	−12.7408	−1.75009	−0.875044	+	0.484044i	$0.839167\pi$
$54$	0	0
$55$	2.16706	0.292206
$56$	0	0
$57$	13.1028	1.73551
$58$	0	0
$59$	−8.16706	−1.06326	−0.531630	−	0.846977i	$-0.678420\pi$
$59$	−8.16706	−1.06326	−0.531630	+	0.846977i	$0.678420\pi$
$60$	0	0
$61$	−10.3062	−1.31957	−0.659787	−	0.751453i	$-0.729354\pi$
$61$	−10.3062	−1.31957	−0.659787	+	0.751453i	$0.729354\pi$
$62$	0	0
$63$	10.4346	1.31464
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.10284	−0.867751	−0.433875	−	0.900973i	$-0.642854\pi$
$67$	−7.10284	−0.867751	−0.433875	+	0.900973i	$0.642854\pi$
$68$	0	0
$69$	12.8413	1.54592
$70$	0	0
$71$	16.1112	1.91205	0.956026	−	0.293281i	$-0.0947472\pi$
$71$	16.1112	1.91205	0.956026	+	0.293281i	$0.0947472\pi$
$72$	0	0
$73$	15.9721	1.86939	0.934696	−	0.355448i	$-0.115672\pi$
$73$	15.9721	1.86939	0.934696	+	0.355448i	$0.115672\pi$
$74$	0	0
$75$	2.60168	0.300416
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	11.9442	1.34383	0.671913	−	0.740630i	$-0.265473\pi$
$79$	11.9442	1.34383	0.671913	+	0.740630i	$0.265473\pi$
$80$	0	0
$81$	−6.10284	−0.678094
$82$	0	0
$83$	−3.97209	−0.435994	−0.217997	−	0.975949i	$-0.569952\pi$
$83$	−3.97209	−0.435994	−0.217997	+	0.975949i	$0.569952\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−11.5375	−1.23695
$88$	0	0
$89$	−7.94419	−0.842082	−0.421041	−	0.907042i	$-0.638335\pi$
$89$	−7.94419	−0.842082	−0.421041	+	0.907042i	$0.638335\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.76873	−0.909275
$94$	0	0
$95$	5.03630	0.516714
$96$	0	0
$97$	0.462531	0.0469629	0.0234815	−	0.999724i	$-0.492525\pi$
$97$	0.462531	0.0469629	0.0234815	+	0.999724i	$0.492525\pi$
$98$	0	0
$99$	8.16706	0.820820
$100$	0	0
$101$	−12.7408	−1.26776	−0.633880	−	0.773432i	$-0.718539\pi$
$101$	−12.7408	−1.26776	−0.633880	+	0.773432i	$0.718539\pi$
$102$	0	0
$103$	4.13915	0.407842	0.203921	−	0.978987i	$-0.434631\pi$
$103$	4.13915	0.407842	0.203921	+	0.978987i	$0.434631\pi$
$104$	0	0
$105$	7.20336	0.702976
$106$	0	0
$107$	7.06421	0.682923	0.341462	−	0.939896i	$-0.389078\pi$
$107$	7.06421	0.682923	0.341462	+	0.939896i	$0.389078\pi$
$108$	0	0
$109$	14.8692	1.42422	0.712108	−	0.702070i	$-0.247741\pi$
$109$	14.8692	1.42422	0.712108	+	0.702070i	$0.247741\pi$
$110$	0	0
$111$	−10.3341	−0.980870
$112$	0	0
$113$	3.13075	0.294516	0.147258	−	0.989098i	$-0.452955\pi$
$113$	3.13075	0.294516	0.147258	+	0.989098i	$0.452955\pi$
$114$	0	0
$115$	4.93579	0.460265
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.30387	−0.573079
$122$	0	0
$123$	−4.33411	−0.390794
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.1391	−1.43212	−0.716059	−	0.698040i	$-0.754056\pi$
$127$	−16.1391	−1.43212	−0.716059	+	0.698040i	$0.754056\pi$
$128$	0	0
$129$	−25.5096	−2.24599
$130$	0	0
$131$	−8.86925	−0.774910	−0.387455	−	0.921889i	$-0.626646\pi$
$131$	−8.86925	−0.774910	−0.387455	+	0.921889i	$0.626646\pi$
$132$	0	0
$133$	13.9442	1.20911
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	−1.66589	−0.142327	−0.0711633	−	0.997465i	$-0.522671\pi$
$137$	−1.66589	−0.142327	−0.0711633	+	0.997465i	$0.522671\pi$
$138$	0	0
$139$	19.2760	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$139$	19.2760	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$140$	0	0
$141$	30.2783	2.54989
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.43462	−0.368275
$146$	0	0
$147$	1.73243	0.142889
$148$	0	0
$149$	−13.9442	−1.14235	−0.571176	−	0.820828i	$-0.693513\pi$
$149$	−13.9442	−1.14235	−0.571176	+	0.820828i	$0.693513\pi$
$150$	0	0
$151$	6.96370	0.566698	0.283349	−	0.959017i	$-0.408555\pi$
$151$	6.96370	0.566698	0.283349	+	0.959017i	$0.408555\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.37041	−0.270718
$156$	0	0
$157$	12.3341	0.984369	0.492185	−	0.870491i	$-0.336199\pi$
$157$	12.3341	0.984369	0.492185	+	0.870491i	$0.336199\pi$
$158$	0	0
$159$	−33.1475	−2.62877
$160$	0	0
$161$	13.6659	1.07702
$162$	0	0
$163$	−20.5072	−1.60625	−0.803125	−	0.595810i	$-0.796831\pi$
$163$	−20.5072	−1.60625	−0.803125	+	0.595810i	$0.796831\pi$
$164$	0	0
$165$	5.63798	0.438916
$166$	0	0
$167$	19.3039	1.49378	0.746889	−	0.664948i	$-0.231547\pi$
$167$	19.3039	1.49378	0.746889	+	0.664948i	$0.231547\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	18.9805	1.45147
$172$	0	0
$173$	−9.61007	−0.730640	−0.365320	−	0.930882i	$-0.619041\pi$
$173$	−9.61007	−0.730640	−0.365320	+	0.930882i	$0.619041\pi$
$174$	0	0
$175$	2.76873	0.209297
$176$	0	0
$177$	−21.2481	−1.59710
$178$	0	0
$179$	9.87158	0.737836	0.368918	−	0.929462i	$-0.379728\pi$
$179$	9.87158	0.737836	0.368918	+	0.929462i	$0.379728\pi$
$180$	0	0
$181$	1.97209	0.146584	0.0732922	−	0.997311i	$-0.476649\pi$
$181$	1.97209	0.146584	0.0732922	+	0.997311i	$0.476649\pi$
$182$	0	0
$183$	−26.8134	−1.98211
$184$	0	0
$185$	−3.97209	−0.292034
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.53747	0.402792
$190$	0	0
$191$	−5.73850	−0.415223	−0.207611	−	0.978211i	$-0.566569\pi$
$191$	−5.73850	−0.415223	−0.207611	+	0.978211i	$0.566569\pi$
$192$	0	0
$193$	−4.07261	−0.293153	−0.146576	−	0.989199i	$-0.546825\pi$
$193$	−4.07261	−0.293153	−0.146576	+	0.989199i	$0.546825\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.6101	−1.53965	−0.769827	−	0.638253i	$-0.779658\pi$
$197$	−21.6101	−1.53965	−0.769827	+	0.638253i	$0.779658\pi$
$198$	0	0
$199$	11.6101	0.823016	0.411508	−	0.911406i	$-0.365002\pi$
$199$	11.6101	0.823016	0.411508	+	0.911406i	$0.365002\pi$
$200$	0	0
$201$	−18.4793	−1.30343
$202$	0	0
$203$	−12.2783	−0.861767
$204$	0	0
$205$	−1.66589	−0.116351
$206$	0	0
$207$	18.6017	1.29291
$208$	0	0
$209$	10.9139	0.754933
$210$	0	0
$211$	16.2783	1.12064	0.560322	−	0.828275i	$-0.310677\pi$
$211$	16.2783	1.12064	0.560322	+	0.828275i	$0.310677\pi$
$212$	0	0
$213$	41.9163	2.87206
$214$	0	0
$215$	−9.80504	−0.668698
$216$	0	0
$217$	−9.33178	−0.633482
$218$	0	0
$219$	41.5543	2.80798
$220$	0	0
$221$	0	0
$222$	0	0
$223$	11.4370	0.765875	0.382938	−	0.923774i	$-0.374912\pi$
$223$	11.4370	0.765875	0.382938	+	0.923774i	$0.374912\pi$
$224$	0	0
$225$	3.76873	0.251249
$226$	0	0
$227$	−15.9721	−1.06011	−0.530053	−	0.847965i	$-0.677828\pi$
$227$	−15.9721	−1.06011	−0.530053	+	0.847965i	$0.677828\pi$
$228$	0	0
$229$	2.12842	0.140650	0.0703250	−	0.997524i	$-0.477596\pi$
$229$	2.12842	0.140650	0.0703250	+	0.997524i	$0.477596\pi$
$230$	0	0
$231$	15.6101	1.02707
$232$	0	0
$233$	2.86925	0.187971	0.0939853	−	0.995574i	$-0.470039\pi$
$233$	2.86925	0.187971	0.0939853	+	0.995574i	$0.470039\pi$
$234$	0	0
$235$	11.6380	0.759178
$236$	0	0
$237$	31.0749	2.01853
$238$	0	0
$239$	3.16939	0.205011	0.102505	−	0.994732i	$-0.467314\pi$
$239$	3.16939	0.205011	0.102505	+	0.994732i	$0.467314\pi$
$240$	0	0
$241$	4.79664	0.308979	0.154489	−	0.987994i	$-0.450627\pi$
$241$	4.79664	0.308979	0.154489	+	0.987994i	$0.450627\pi$
$242$	0	0
$243$	−21.8776	−1.40345
$244$	0	0
$245$	0.665890	0.0425421
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.3341	−0.654898
$250$	0	0
$251$	−1.00233	−0.0632666	−0.0316333	−	0.999500i	$-0.510071\pi$
$251$	−1.00233	−0.0632666	−0.0316333	+	0.999500i	$0.510071\pi$
$252$	0	0
$253$	10.6961	0.672460
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.4817	1.21523	0.607616	−	0.794231i	$-0.292126\pi$
$257$	19.4817	1.21523	0.607616	+	0.794231i	$0.292126\pi$
$258$	0	0
$259$	−10.9977	−0.683362
$260$	0	0
$261$	−16.7129	−1.03450
$262$	0	0
$263$	−4.19496	−0.258672	−0.129336	−	0.991601i	$-0.541285\pi$
$263$	−4.19496	−0.258672	−0.129336	+	0.991601i	$0.541285\pi$
$264$	0	0
$265$	−12.7408	−0.782663
$266$	0	0
$267$	−20.6682	−1.26487
$268$	0	0
$269$	30.4793	1.85836	0.929179	−	0.369631i	$-0.120516\pi$
$269$	30.4793	1.85836	0.929179	+	0.369631i	$0.120516\pi$
$270$	0	0
$271$	−4.83528	−0.293722	−0.146861	−	0.989157i	$-0.546917\pi$
$271$	−4.83528	−0.293722	−0.146861	+	0.989157i	$0.546917\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.16706	0.130678
$276$	0	0
$277$	−3.00233	−0.180393	−0.0901963	−	0.995924i	$-0.528749\pi$
$277$	−3.00233	−0.180393	−0.0901963	+	0.995924i	$0.528749\pi$
$278$	0	0
$279$	−12.7022	−0.760460
$280$	0	0
$281$	−21.6101	−1.28915	−0.644574	−	0.764542i	$-0.722965\pi$
$281$	−21.6101	−1.28915	−0.644574	+	0.764542i	$0.722965\pi$
$282$	0	0
$283$	−2.47326	−0.147020	−0.0735100	−	0.997294i	$-0.523420\pi$
$283$	−2.47326	−0.147020	−0.0735100	+	0.997294i	$0.523420\pi$
$284$	0	0
$285$	13.1028	0.776146
$286$	0	0
$287$	−4.61241	−0.272262
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	1.20336	0.0705421
$292$	0	0
$293$	−23.9163	−1.39720	−0.698602	−	0.715511i	$-0.746194\pi$
$293$	−23.9163	−1.39720	−0.698602	+	0.715511i	$0.746194\pi$
$294$	0	0
$295$	−8.16706	−0.475504
$296$	0	0
$297$	4.33411	0.251491
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−27.1475	−1.56476
$302$	0	0
$303$	−33.1475	−1.90428
$304$	0	0
$305$	−10.3062	−0.590131
$306$	0	0
$307$	13.5654	0.774217	0.387108	−	0.922034i	$-0.373474\pi$
$307$	13.5654	0.774217	0.387108	+	0.922034i	$0.373474\pi$
$308$	0	0
$309$	10.7687	0.612612
$310$	0	0
$311$	2.12842	0.120692	0.0603458	−	0.998178i	$-0.480780\pi$
$311$	2.12842	0.120692	0.0603458	+	0.998178i	$0.480780\pi$
$312$	0	0
$313$	−22.6077	−1.27787	−0.638933	−	0.769263i	$-0.720624\pi$
$313$	−22.6077	−1.27787	−0.638933	+	0.769263i	$0.720624\pi$
$314$	0	0
$315$	10.4346	0.587924
$316$	0	0
$317$	10.6357	0.597358	0.298679	−	0.954354i	$-0.403454\pi$
$317$	10.6357	0.597358	0.298679	+	0.954354i	$0.403454\pi$
$318$	0	0
$319$	−9.61007	−0.538061
$320$	0	0
$321$	18.3788	1.02581
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	38.6850	2.13929
$328$	0	0
$329$	32.2225	1.77648
$330$	0	0
$331$	1.16472	0.0640190	0.0320095	−	0.999488i	$-0.489809\pi$
$331$	1.16472	0.0640190	0.0320095	+	0.999488i	$0.489809\pi$
$332$	0	0
$333$	−14.9698	−0.820338
$334$	0	0
$335$	−7.10284	−0.388070
$336$	0	0
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	0	0
$339$	8.14521	0.442387
$340$	0	0
$341$	−7.30387	−0.395527
$342$	0	0
$343$	−17.5375	−0.946934
$344$	0	0
$345$	12.8413	0.691355
$346$	0	0
$347$	4.67429	0.250929	0.125464	−	0.992098i	$-0.459958\pi$
$347$	4.67429	0.250929	0.125464	+	0.992098i	$0.459958\pi$
$348$	0	0
$349$	−4.33411	−0.232000	−0.116000	−	0.993249i	$-0.537007\pi$
$349$	−4.33411	−0.232000	−0.116000	+	0.993249i	$0.537007\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.3085	−1.13414	−0.567069	−	0.823670i	$-0.691923\pi$
$353$	−21.3085	−1.13414	−0.567069	+	0.823670i	$0.691923\pi$
$354$	0	0
$355$	16.1112	0.855096
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.7795	0.674474	0.337237	−	0.941420i	$-0.390508\pi$
$359$	12.7795	0.674474	0.337237	+	0.941420i	$0.390508\pi$
$360$	0	0
$361$	6.36435	0.334966
$362$	0	0
$363$	−16.4007	−0.860811
$364$	0	0
$365$	15.9721	0.836018
$366$	0	0
$367$	12.1950	0.636572	0.318286	−	0.947995i	$-0.396893\pi$
$367$	12.1950	0.636572	0.318286	+	0.947995i	$0.396893\pi$
$368$	0	0
$369$	−6.27830	−0.326835
$370$	0	0
$371$	−35.2760	−1.83144
$372$	0	0
$373$	−28.2225	−1.46130	−0.730652	−	0.682750i	$-0.760784\pi$
$373$	−28.2225	−1.46130	−0.730652	+	0.682750i	$0.760784\pi$
$374$	0	0
$375$	2.60168	0.134350
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−23.3146	−1.19759	−0.598795	−	0.800902i	$-0.704354\pi$
$379$	−23.3146	−1.19759	−0.598795	+	0.800902i	$0.704354\pi$
$380$	0	0
$381$	−41.9889	−2.15116
$382$	0	0
$383$	−8.02791	−0.410207	−0.205103	−	0.978740i	$-0.565753\pi$
$383$	−8.02791	−0.410207	−0.205103	+	0.978740i	$0.565753\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	−36.9526	−1.87841
$388$	0	0
$389$	−23.7385	−1.20359	−0.601795	−	0.798651i	$-0.705547\pi$
$389$	−23.7385	−1.20359	−0.601795	+	0.798651i	$0.705547\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−23.0749	−1.16398
$394$	0	0
$395$	11.9442	0.600977
$396$	0	0
$397$	−3.04703	−0.152926	−0.0764630	−	0.997072i	$-0.524363\pi$
$397$	−3.04703	−0.152926	−0.0764630	+	0.997072i	$0.524363\pi$
$398$	0	0
$399$	36.2783	1.81619
$400$	0	0
$401$	−31.2201	−1.55906	−0.779530	−	0.626365i	$-0.784542\pi$
$401$	−31.2201	−1.55906	−0.779530	+	0.626365i	$0.784542\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−6.10284	−0.303253
$406$	0	0
$407$	−8.60774	−0.426670
$408$	0	0
$409$	33.4090	1.65197	0.825986	−	0.563691i	$-0.190619\pi$
$409$	33.4090	1.65197	0.825986	+	0.563691i	$0.190619\pi$
$410$	0	0
$411$	−4.33411	−0.213786
$412$	0	0
$413$	−22.6124	−1.11268
$414$	0	0
$415$	−3.97209	−0.194982
$416$	0	0
$417$	50.1499	2.45585
$418$	0	0
$419$	−30.7408	−1.50179	−0.750894	−	0.660423i	$-0.770377\pi$
$419$	−30.7408	−1.50179	−0.750894	+	0.660423i	$0.770377\pi$
$420$	0	0
$421$	19.2806	0.939680	0.469840	−	0.882752i	$-0.344312\pi$
$421$	19.2806	0.939680	0.469840	+	0.882752i	$0.344312\pi$
$422$	0	0
$423$	43.8605	2.13257
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−28.5351	−1.38091
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.44535	−0.117788	−0.0588942	−	0.998264i	$-0.518757\pi$
$431$	−2.44535	−0.117788	−0.0588942	+	0.998264i	$0.518757\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−11.5375	−0.553179
$436$	0	0
$437$	24.8581	1.18913
$438$	0	0
$439$	11.6659	0.556783	0.278391	−	0.960468i	$-0.410199\pi$
$439$	11.6659	0.556783	0.278391	+	0.960468i	$0.410199\pi$
$440$	0	0
$441$	2.50956	0.119503
$442$	0	0
$443$	20.8074	0.988588	0.494294	−	0.869295i	$-0.335427\pi$
$443$	20.8074	0.988588	0.494294	+	0.869295i	$0.335427\pi$
$444$	0	0
$445$	−7.94419	−0.376590
$446$	0	0
$447$	−36.2783	−1.71590
$448$	0	0
$449$	30.2783	1.42892	0.714461	−	0.699676i	$-0.246672\pi$
$449$	30.2783	1.42892	0.714461	+	0.699676i	$0.246672\pi$
$450$	0	0
$451$	−3.61007	−0.169992
$452$	0	0
$453$	18.1173	0.851225
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.6124	1.05776	0.528882	−	0.848695i	$-0.322611\pi$
$457$	22.6124	1.05776	0.528882	+	0.848695i	$0.322611\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.9419	0.882210	0.441105	−	0.897455i	$-0.354587\pi$
$461$	18.9419	0.882210	0.441105	+	0.897455i	$0.354587\pi$
$462$	0	0
$463$	11.6380	0.540863	0.270431	−	0.962739i	$-0.412834\pi$
$463$	11.6380	0.540863	0.270431	+	0.962739i	$0.412834\pi$
$464$	0	0
$465$	−8.76873	−0.406640
$466$	0	0
$467$	25.0642	1.15983	0.579917	−	0.814676i	$-0.303085\pi$
$467$	25.0642	1.15983	0.579917	+	0.814676i	$0.303085\pi$
$468$	0	0
$469$	−19.6659	−0.908086
$470$	0	0
$471$	32.0894	1.47860
$472$	0	0
$473$	−21.2481	−0.976987
$474$	0	0
$475$	5.03630	0.231081
$476$	0	0
$477$	−48.0168	−2.19854
$478$	0	0
$479$	−18.4407	−0.842577	−0.421288	−	0.906927i	$-0.638422\pi$
$479$	−18.4407	−0.842577	−0.421288	+	0.906927i	$0.638422\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	35.5543	1.61777
$484$	0	0
$485$	0.462531	0.0210025
$486$	0	0
$487$	25.1196	1.13828	0.569140	−	0.822241i	$-0.307276\pi$
$487$	25.1196	1.13828	0.569140	+	0.822241i	$0.307276\pi$
$488$	0	0
$489$	−53.3532	−2.41272
$490$	0	0
$491$	5.73850	0.258975	0.129487	−	0.991581i	$-0.458667\pi$
$491$	5.73850	0.258975	0.129487	+	0.991581i	$0.458667\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	8.16706	0.367082
$496$	0	0
$497$	44.6077	2.00093
$498$	0	0
$499$	−33.3704	−1.49386	−0.746932	−	0.664900i	$-0.768474\pi$
$499$	−33.3704	−1.49386	−0.746932	+	0.664900i	$0.768474\pi$
$500$	0	0
$501$	50.2225	2.24377
$502$	0	0
$503$	−12.6789	−0.565326	−0.282663	−	0.959219i	$-0.591218\pi$
$503$	−12.6789	−0.565326	−0.282663	+	0.959219i	$0.591218\pi$
$504$	0	0
$505$	−12.7408	−0.566959
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.6147	0.780759	0.390380	−	0.920654i	$-0.372344\pi$
$509$	17.6147	0.780759	0.390380	+	0.920654i	$0.372344\pi$
$510$	0	0
$511$	44.2225	1.95629
$512$	0	0
$513$	10.0726	0.444716
$514$	0	0
$515$	4.13915	0.182393
$516$	0	0
$517$	25.2201	1.10918
$518$	0	0
$519$	−25.0023	−1.09748
$520$	0	0
$521$	−16.4346	−0.720014	−0.360007	−	0.932950i	$-0.617226\pi$
$521$	−16.4346	−0.720014	−0.360007	+	0.932950i	$0.617226\pi$
$522$	0	0
$523$	−20.4733	−0.895233	−0.447617	−	0.894226i	$-0.647727\pi$
$523$	−20.4733	−0.895233	−0.447617	+	0.894226i	$0.647727\pi$
$524$	0	0
$525$	7.20336	0.314380
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.36202	0.0592182
$530$	0	0
$531$	−30.7795	−1.33571
$532$	0	0
$533$	0	0
$534$	0	0
$535$	7.06421	0.305412
$536$	0	0
$537$	25.6827	1.10829
$538$	0	0
$539$	1.44302	0.0621553
$540$	0	0
$541$	41.2928	1.77531	0.887657	−	0.460505i	$-0.152332\pi$
$541$	41.2928	1.77531	0.887657	+	0.460505i	$0.152332\pi$
$542$	0	0
$543$	5.13075	0.220182
$544$	0	0
$545$	14.8692	0.636929
$546$	0	0
$547$	−10.4174	−0.445418	−0.222709	−	0.974885i	$-0.571490\pi$
$547$	−10.4174	−0.445418	−0.222709	+	0.974885i	$0.571490\pi$
$548$	0	0
$549$	−38.8413	−1.65771
$550$	0	0
$551$	−22.3341	−0.951465
$552$	0	0
$553$	33.0703	1.40629
$554$	0	0
$555$	−10.3341	−0.438659
$556$	0	0
$557$	20.5845	0.872193	0.436097	−	0.899900i	$-0.356361\pi$
$557$	20.5845	0.872193	0.436097	+	0.899900i	$0.356361\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.6766	0.997850	0.498925	−	0.866645i	$-0.333728\pi$
$563$	23.6766	0.997850	0.498925	+	0.866645i	$0.333728\pi$
$564$	0	0
$565$	3.13075	0.131712
$566$	0	0
$567$	−16.8972	−0.709614
$568$	0	0
$569$	−5.43695	−0.227929	−0.113965	−	0.993485i	$-0.536355\pi$
$569$	−5.43695	−0.227929	−0.113965	+	0.993485i	$0.536355\pi$
$570$	0	0
$571$	−15.2760	−0.639279	−0.319640	−	0.947539i	$-0.603562\pi$
$571$	−15.2760	−0.639279	−0.319640	+	0.947539i	$0.603562\pi$
$572$	0	0
$573$	−14.9297	−0.623698
$574$	0	0
$575$	4.93579	0.205837
$576$	0	0
$577$	−1.76640	−0.0735363	−0.0367682	−	0.999324i	$-0.511706\pi$
$577$	−1.76640	−0.0735363	−0.0367682	+	0.999324i	$0.511706\pi$
$578$	0	0
$579$	−10.5956	−0.440339
$580$	0	0
$581$	−10.9977	−0.456260
$582$	0	0
$583$	−27.6101	−1.14349
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.9139	−0.945760	−0.472880	−	0.881127i	$-0.656786\pi$
$587$	−22.9139	−0.945760	−0.472880	+	0.881127i	$0.656786\pi$
$588$	0	0
$589$	−16.9744	−0.699419
$590$	0	0
$591$	−56.2225	−2.31268
$592$	0	0
$593$	−6.72404	−0.276123	−0.138062	−	0.990424i	$-0.544087\pi$
$593$	−6.72404	−0.276123	−0.138062	+	0.990424i	$0.544087\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	30.2057	1.23624
$598$	0	0
$599$	−19.8669	−0.811740	−0.405870	−	0.913931i	$-0.633031\pi$
$599$	−19.8669	−0.811740	−0.405870	+	0.913931i	$0.633031\pi$
$600$	0	0
$601$	−33.2201	−1.35508	−0.677539	−	0.735487i	$-0.736954\pi$
$601$	−33.2201	−1.35508	−0.677539	+	0.735487i	$0.736954\pi$
$602$	0	0
$603$	−26.7687	−1.09011
$604$	0	0
$605$	−6.30387	−0.256289
$606$	0	0
$607$	14.4174	0.585186	0.292593	−	0.956237i	$-0.405482\pi$
$607$	14.4174	0.585186	0.292593	+	0.956237i	$0.405482\pi$
$608$	0	0
$609$	−31.9442	−1.29444
$610$	0	0
$611$	0	0
$612$	0	0
$613$	11.3364	0.457875	0.228937	−	0.973441i	$-0.426475\pi$
$613$	11.3364	0.457875	0.228937	+	0.973441i	$0.426475\pi$
$614$	0	0
$615$	−4.33411	−0.174768
$616$	0	0
$617$	34.8907	1.40465	0.702323	−	0.711858i	$-0.252146\pi$
$617$	34.8907	1.40465	0.702323	+	0.711858i	$0.252146\pi$
$618$	0	0
$619$	32.6464	1.31217	0.656084	−	0.754688i	$-0.272212\pi$
$619$	32.6464	1.31217	0.656084	+	0.754688i	$0.272212\pi$
$620$	0	0
$621$	9.87158	0.396133
$622$	0	0
$623$	−21.9953	−0.881225
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	28.3946	1.13397
$628$	0	0
$629$	0	0
$630$	0	0
$631$	47.3146	1.88356	0.941782	−	0.336224i	$-0.109150\pi$
$631$	47.3146	1.88356	0.941782	+	0.336224i	$0.109150\pi$
$632$	0	0
$633$	42.3509	1.68330
$634$	0	0
$635$	−16.1391	−0.640463
$636$	0	0
$637$	0	0
$638$	0	0
$639$	60.7190	2.40201
$640$	0	0
$641$	−43.4817	−1.71742	−0.858711	−	0.512460i	$-0.828734\pi$
$641$	−43.4817	−1.71742	−0.858711	+	0.512460i	$0.828734\pi$
$642$	0	0
$643$	20.3062	0.800798	0.400399	−	0.916341i	$-0.368871\pi$
$643$	20.3062	0.800798	0.400399	+	0.916341i	$0.368871\pi$
$644$	0	0
$645$	−25.5096	−1.00444
$646$	0	0
$647$	−30.8027	−1.21098	−0.605490	−	0.795853i	$-0.707023\pi$
$647$	−30.8027	−1.21098	−0.605490	+	0.795853i	$0.707023\pi$
$648$	0	0
$649$	−17.6985	−0.694725
$650$	0	0
$651$	−24.2783	−0.951541
$652$	0	0
$653$	3.61007	0.141273	0.0706366	−	0.997502i	$-0.477497\pi$
$653$	3.61007	0.141273	0.0706366	+	0.997502i	$0.477497\pi$
$654$	0	0
$655$	−8.86925	−0.346550
$656$	0	0
$657$	60.1946	2.34841
$658$	0	0
$659$	−6.74083	−0.262585	−0.131293	−	0.991344i	$-0.541913\pi$
$659$	−6.74083	−0.262585	−0.131293	+	0.991344i	$0.541913\pi$
$660$	0	0
$661$	−26.6682	−1.03727	−0.518637	−	0.854995i	$-0.673560\pi$
$661$	−26.6682	−1.03727	−0.518637	+	0.854995i	$0.673560\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.9442	0.540732
$666$	0	0
$667$	−21.8884	−0.847521
$668$	0	0
$669$	29.7553	1.15041
$670$	0	0
$671$	−22.3341	−0.862199
$672$	0	0
$673$	8.61241	0.331984	0.165992	−	0.986127i	$-0.446917\pi$
$673$	8.61241	0.331984	0.165992	+	0.986127i	$0.446917\pi$
$674$	0	0
$675$	2.00000	0.0769800
$676$	0	0
$677$	8.12842	0.312401	0.156200	−	0.987725i	$-0.450075\pi$
$677$	8.12842	0.312401	0.156200	+	0.987725i	$0.450075\pi$
$678$	0	0
$679$	1.28063	0.0491459
$680$	0	0
$681$	−41.5543	−1.59236
$682$	0	0
$683$	−8.02791	−0.307179	−0.153590	−	0.988135i	$-0.549083\pi$
$683$	−8.02791	−0.307179	−0.153590	+	0.988135i	$0.549083\pi$
$684$	0	0
$685$	−1.66589	−0.0636504
$686$	0	0
$687$	5.53747	0.211268
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−22.1112	−0.841151	−0.420576	−	0.907257i	$-0.638172\pi$
$691$	−22.1112	−0.841151	−0.420576	+	0.907257i	$0.638172\pi$
$692$	0	0
$693$	22.6124	0.858974
$694$	0	0
$695$	19.2760	0.731179
$696$	0	0
$697$	0	0
$698$	0	0
$699$	7.46486	0.282347
$700$	0	0
$701$	6.47932	0.244721	0.122360	−	0.992486i	$-0.460954\pi$
$701$	6.47932	0.244721	0.122360	+	0.992486i	$0.460954\pi$
$702$	0	0
$703$	−20.0047	−0.754490
$704$	0	0
$705$	30.2783	1.14035
$706$	0	0
$707$	−35.2760	−1.32669
$708$	0	0
$709$	2.85246	0.107126	0.0535631	−	0.998564i	$-0.482942\pi$
$709$	2.85246	0.107126	0.0535631	+	0.998564i	$0.482942\pi$
$710$	0	0
$711$	45.0145	1.68817
$712$	0	0
$713$	−16.6357	−0.623010
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.24573	0.307942
$718$	0	0
$719$	−42.0941	−1.56984	−0.784922	−	0.619595i	$-0.787297\pi$
$719$	−42.0941	−1.56984	−0.784922	+	0.619595i	$0.787297\pi$
$720$	0	0
$721$	11.4602	0.426800
$722$	0	0
$723$	12.4793	0.464111
$724$	0	0
$725$	−4.43462	−0.164698
$726$	0	0
$727$	32.8027	1.21659	0.608293	−	0.793713i	$-0.291855\pi$
$727$	32.8027	1.21659	0.608293	+	0.793713i	$0.291855\pi$
$728$	0	0
$729$	−38.6101	−1.43000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−2.38993	−0.0882739	−0.0441370	−	0.999025i	$-0.514054\pi$
$733$	−2.38993	−0.0882739	−0.0441370	+	0.999025i	$0.514054\pi$
$734$	0	0
$735$	1.73243	0.0639017
$736$	0	0
$737$	−15.3923	−0.566981
$738$	0	0
$739$	−41.4988	−1.52656	−0.763280	−	0.646068i	$-0.776412\pi$
$739$	−41.4988	−1.52656	−0.763280	+	0.646068i	$0.776412\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.8046	−1.46029	−0.730145	−	0.683292i	$-0.760548\pi$
$743$	−39.8046	−1.46029	−0.730145	+	0.683292i	$0.760548\pi$
$744$	0	0
$745$	−13.9442	−0.510875
$746$	0	0
$747$	−14.9698	−0.547715
$748$	0	0
$749$	19.5589	0.714667
$750$	0	0
$751$	−15.6659	−0.571656	−0.285828	−	0.958281i	$-0.592269\pi$
$751$	−15.6659	−0.571656	−0.285828	+	0.958281i	$0.592269\pi$
$752$	0	0
$753$	−2.60774	−0.0950315
$754$	0	0
$755$	6.96370	0.253435
$756$	0	0
$757$	32.2736	1.17301	0.586503	−	0.809947i	$-0.300504\pi$
$757$	32.2736	1.17301	0.586503	+	0.809947i	$0.300504\pi$
$758$	0	0
$759$	27.8279	1.01009
$760$	0	0
$761$	−13.2806	−0.481422	−0.240711	−	0.970597i	$-0.577381\pi$
$761$	−13.2806	−0.481422	−0.240711	+	0.970597i	$0.577381\pi$
$762$	0	0
$763$	41.1690	1.49042
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	12.2010	0.439980	0.219990	−	0.975502i	$-0.429397\pi$
$769$	12.2010	0.439980	0.219990	+	0.975502i	$0.429397\pi$
$770$	0	0
$771$	50.6850	1.82538
$772$	0	0
$773$	−27.9721	−1.00609	−0.503043	−	0.864261i	$-0.667786\pi$
$773$	−27.9721	−1.00609	−0.503043	+	0.864261i	$0.667786\pi$
$774$	0	0
$775$	−3.37041	−0.121069
$776$	0	0
$777$	−28.6124	−1.02646
$778$	0	0
$779$	−8.38993	−0.300600
$780$	0	0
$781$	34.9139	1.24932
$782$	0	0
$783$	−8.86925	−0.316961
$784$	0	0
$785$	12.3341	0.440223
$786$	0	0
$787$	10.2336	0.364788	0.182394	−	0.983225i	$-0.441615\pi$
$787$	10.2336	0.364788	0.182394	+	0.983225i	$0.441615\pi$
$788$	0	0
$789$	−10.9139	−0.388547
$790$	0	0
$791$	8.66822	0.308206
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−33.1475	−1.17562
$796$	0	0
$797$	23.3532	0.827214	0.413607	−	0.910456i	$-0.364269\pi$
$797$	23.3532	0.827214	0.413607	+	0.910456i	$0.364269\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−29.9395	−1.05786
$802$	0	0
$803$	34.6124	1.22145
$804$	0	0
$805$	13.6659	0.481659
$806$	0	0
$807$	79.2974	2.79140
$808$	0	0
$809$	37.1364	1.30565	0.652824	−	0.757510i	$-0.273584\pi$
$809$	37.1364	1.30565	0.652824	+	0.757510i	$0.273584\pi$
$810$	0	0
$811$	−20.6296	−0.724403	−0.362201	−	0.932100i	$-0.617975\pi$
$811$	−20.6296	−0.724403	−0.362201	+	0.932100i	$0.617975\pi$
$812$	0	0
$813$	−12.5798	−0.441194
$814$	0	0
$815$	−20.5072	−0.718337
$816$	0	0
$817$	−49.3811	−1.72763
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.6077	0.719215	0.359608	−	0.933104i	$-0.382911\pi$
$821$	20.6077	0.719215	0.359608	+	0.933104i	$0.382911\pi$
$822$	0	0
$823$	−39.8050	−1.38752	−0.693758	−	0.720208i	$-0.744046\pi$
$823$	−39.8050	−1.38752	−0.693758	+	0.720208i	$0.744046\pi$
$824$	0	0
$825$	5.63798	0.196289
$826$	0	0
$827$	−55.8605	−1.94246	−0.971229	−	0.238146i	$-0.923460\pi$
$827$	−55.8605	−1.94246	−0.971229	+	0.238146i	$0.923460\pi$
$828$	0	0
$829$	33.5822	1.16636	0.583178	−	0.812344i	$-0.301809\pi$
$829$	33.5822	1.16636	0.583178	+	0.812344i	$0.301809\pi$
$830$	0	0
$831$	−7.81110	−0.270964
$832$	0	0
$833$	0	0
$834$	0	0
$835$	19.3039	0.668038
$836$	0	0
$837$	−6.74083	−0.232997
$838$	0	0
$839$	−26.4454	−0.912995	−0.456497	−	0.889725i	$-0.650896\pi$
$839$	−26.4454	−0.912995	−0.456497	+	0.889725i	$0.650896\pi$
$840$	0	0
$841$	−9.33411	−0.321866
$842$	0	0
$843$	−56.2225	−1.93641
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.4537	−0.599718
$848$	0	0
$849$	−6.43462	−0.220836
$850$	0	0
$851$	−19.6054	−0.672065
$852$	0	0
$853$	35.7153	1.22287	0.611433	−	0.791296i	$-0.290593\pi$
$853$	35.7153	1.22287	0.611433	+	0.791296i	$0.290593\pi$
$854$	0	0
$855$	18.9805	0.649119
$856$	0	0
$857$	−22.0894	−0.754559	−0.377280	−	0.926099i	$-0.623140\pi$
$857$	−22.0894	−0.754559	−0.377280	+	0.926099i	$0.623140\pi$
$858$	0	0
$859$	−39.2201	−1.33817	−0.669087	−	0.743184i	$-0.733315\pi$
$859$	−39.2201	−1.33817	−0.669087	+	0.743184i	$0.733315\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	−28.9744	−0.986301	−0.493150	−	0.869944i	$-0.664155\pi$
$863$	−28.9744	−0.986301	−0.493150	+	0.869944i	$0.664155\pi$
$864$	0	0
$865$	−9.61007	−0.326752
$866$	0	0
$867$	−44.2285	−1.50208
$868$	0	0
$869$	25.8837	0.878045
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.74316	0.0589970
$874$	0	0
$875$	2.76873	0.0936003
$876$	0	0
$877$	30.9251	1.04427	0.522133	−	0.852864i	$-0.325137\pi$
$877$	30.9251	1.04427	0.522133	+	0.852864i	$0.325137\pi$
$878$	0	0
$879$	−62.2225	−2.09871
$880$	0	0
$881$	−41.2695	−1.39041	−0.695203	−	0.718814i	$-0.744685\pi$
$881$	−41.2695	−1.39041	−0.695203	+	0.718814i	$0.744685\pi$
$882$	0	0
$883$	−15.1973	−0.511430	−0.255715	−	0.966752i	$-0.582311\pi$
$883$	−15.1973	−0.511430	−0.255715	+	0.966752i	$0.582311\pi$
$884$	0	0
$885$	−21.2481	−0.714246
$886$	0	0
$887$	−12.4174	−0.416937	−0.208468	−	0.978029i	$-0.566848\pi$
$887$	−12.4174	−0.416937	−0.208468	+	0.978029i	$0.566848\pi$
$888$	0	0
$889$	−44.6850	−1.49869
$890$	0	0
$891$	−13.2252	−0.443061
$892$	0	0
$893$	58.6124	1.96139
$894$	0	0
$895$	9.87158	0.329970
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.9465	0.498494
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−70.6292	−2.35039
$904$	0	0
$905$	1.97209	0.0655546
$906$	0	0
$907$	11.4709	0.380886	0.190443	−	0.981698i	$-0.439008\pi$
$907$	11.4709	0.380886	0.190443	+	0.981698i	$0.439008\pi$
$908$	0	0
$909$	−48.0168	−1.59262
$910$	0	0
$911$	54.5734	1.80810	0.904048	−	0.427430i	$-0.140581\pi$
$911$	54.5734	1.80810	0.904048	+	0.427430i	$0.140581\pi$
$912$	0	0
$913$	−8.60774	−0.284875
$914$	0	0
$915$	−26.8134	−0.886425
$916$	0	0
$917$	−24.5566	−0.810930
$918$	0	0
$919$	9.05815	0.298801	0.149400	−	0.988777i	$-0.452266\pi$
$919$	9.05815	0.298801	0.149400	+	0.988777i	$0.452266\pi$
$920$	0	0
$921$	35.2928	1.16294
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.97209	−0.130602
$926$	0	0
$927$	15.5993	0.512350
$928$	0	0
$929$	−50.2225	−1.64775	−0.823873	−	0.566774i	$-0.808191\pi$
$929$	−50.2225	−1.64775	−0.823873	+	0.566774i	$0.808191\pi$
$930$	0	0
$931$	3.35362	0.109911
$932$	0	0
$933$	5.53747	0.181289
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.38759	0.176005	0.0880025	−	0.996120i	$-0.471952\pi$
$937$	5.38759	0.176005	0.0880025	+	0.996120i	$0.471952\pi$
$938$	0	0
$939$	−58.8181	−1.91946
$940$	0	0
$941$	58.8302	1.91781	0.958905	−	0.283726i	$-0.0915708\pi$
$941$	58.8302	1.91781	0.958905	+	0.283726i	$0.0915708\pi$
$942$	0	0
$943$	−8.22248	−0.267761
$944$	0	0
$945$	5.53747	0.180134
$946$	0	0
$947$	−12.6403	−0.410755	−0.205377	−	0.978683i	$-0.565842\pi$
$947$	−12.6403	−0.410755	−0.205377	+	0.978683i	$0.565842\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	27.6706	0.897279
$952$	0	0
$953$	32.6077	1.05627	0.528134	−	0.849161i	$-0.322892\pi$
$953$	32.6077	1.05627	0.528134	+	0.849161i	$0.322892\pi$
$954$	0	0
$955$	−5.73850	−0.185693
$956$	0	0
$957$	−25.0023	−0.808211
$958$	0	0
$959$	−4.61241	−0.148942
$960$	0	0
$961$	−19.6403	−0.633558
$962$	0	0
$963$	26.6231	0.857918
$964$	0	0
$965$	−4.07261	−0.131102
$966$	0	0
$967$	20.5845	0.661953	0.330976	−	0.943639i	$-0.392622\pi$
$967$	20.5845	0.661953	0.330976	+	0.943639i	$0.392622\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−60.8349	−1.95228	−0.976142	−	0.217132i	$-0.930330\pi$
$971$	−60.8349	−1.95228	−0.976142	+	0.217132i	$0.930330\pi$
$972$	0	0
$973$	53.3700	1.71096
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.30387	0.233672	0.116836	−	0.993151i	$-0.462725\pi$
$977$	7.30387	0.233672	0.116836	+	0.993151i	$0.462725\pi$
$978$	0	0
$979$	−17.2155	−0.550209
$980$	0	0
$981$	56.0382	1.78916
$982$	0	0
$983$	−40.2504	−1.28379	−0.641894	−	0.766793i	$-0.721851\pi$
$983$	−40.2504	−1.28379	−0.641894	+	0.766793i	$0.721851\pi$
$984$	0	0
$985$	−21.6101	−0.688554
$986$	0	0
$987$	83.8326	2.66842
$988$	0	0
$989$	−48.3956	−1.53889
$990$	0	0
$991$	−50.2271	−1.59552	−0.797759	−	0.602977i	$-0.793981\pi$
$991$	−50.2271	−1.59552	−0.797759	+	0.602977i	$0.793981\pi$
$992$	0	0
$993$	3.03024	0.0961617
$994$	0	0
$995$	11.6101	0.368064
$996$	0	0
$997$	0.273633	0.00866606	0.00433303	−	0.999991i	$-0.498621\pi$
$997$	0.273633	0.00866606	0.00433303	+	0.999991i	$0.498621\pi$
$998$	0	0
$999$	−7.94419	−0.251343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.n.1.3		3
13.5	odd	4		260.2.f.a.181.5	✓	6
13.8	odd	4		260.2.f.a.181.6	yes	6
13.12	even	2		3380.2.a.m.1.3		3
39.5	even	4		2340.2.c.d.181.6		6
39.8	even	4		2340.2.c.d.181.1		6
52.31	even	4		1040.2.k.c.961.1		6
52.47	even	4		1040.2.k.c.961.2		6
65.8	even	4		1300.2.d.c.649.6		6
65.18	even	4		1300.2.d.d.649.6		6
65.34	odd	4		1300.2.f.e.701.2		6
65.44	odd	4		1300.2.f.e.701.1		6
65.47	even	4		1300.2.d.d.649.1		6
65.57	even	4		1300.2.d.c.649.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.5	✓	6	13.5	odd	4
260.2.f.a.181.6	yes	6	13.8	odd	4
1040.2.k.c.961.1		6	52.31	even	4
1040.2.k.c.961.2		6	52.47	even	4
1300.2.d.c.649.1		6	65.57	even	4
1300.2.d.c.649.6		6	65.8	even	4
1300.2.d.d.649.1		6	65.47	even	4
1300.2.d.d.649.6		6	65.18	even	4
1300.2.f.e.701.1		6	65.44	odd	4
1300.2.f.e.701.2		6	65.34	odd	4
2340.2.c.d.181.1		6	39.8	even	4
2340.2.c.d.181.6		6	39.5	even	4
3380.2.a.m.1.3		3	13.12	even	2
3380.2.a.n.1.3		3	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q+2.60168 q^{3} +1.00000 q^{5} +2.76873 q^{7} +3.76873 q^{9} +O(q^{10})\)	\(q+2.60168 q^{3} +1.00000 q^{5} +2.76873 q^{7} +3.76873 q^{9} +2.16706 q^{11} +2.60168 q^{15} +5.03630 q^{19} +7.20336 q^{21} +4.93579 q^{23} +1.00000 q^{25} +2.00000 q^{27} -4.43462 q^{29} -3.37041 q^{31} +5.63798 q^{33} +2.76873 q^{35} -3.97209 q^{37} -1.66589 q^{41} -9.80504 q^{43} +3.76873 q^{45} +11.6380 q^{47} +0.665890 q^{49} -12.7408 q^{53} +2.16706 q^{55} +13.1028 q^{57} -8.16706 q^{59} -10.3062 q^{61} +10.4346 q^{63} -7.10284 q^{67} +12.8413 q^{69} +16.1112 q^{71} +15.9721 q^{73} +2.60168 q^{75} +6.00000 q^{77} +11.9442 q^{79} -6.10284 q^{81} -3.97209 q^{83} -11.5375 q^{87} -7.94419 q^{89} -8.76873 q^{93} +5.03630 q^{95} +0.462531 q^{97} +8.16706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 + 3 * q^9	\( 3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55} + 30 q^{57} - 24 q^{59} - 6 q^{61} + 24 q^{63} - 12 q^{67} + 24 q^{73} + 18 q^{77} - 12 q^{79} - 9 q^{81} + 12 q^{83} - 18 q^{87} + 24 q^{89} - 18 q^{93} + 18 q^{97} + 24 q^{99}+O(q^{100}) \) 3 * q + 3 * q^5 + 3 * q^9 + 6 * q^11 + 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 - 6 * q^29 + 6 * q^31 - 6 * q^33 + 12 * q^37 - 6 * q^41 - 6 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 - 6 * q^53 + 6 * q^55 + 30 * q^57 - 24 * q^59 - 6 * q^61 + 24 * q^63 - 12 * q^67 + 24 * q^73 + 18 * q^77 - 12 * q^79 - 9 * q^81 + 12 * q^83 - 18 * q^87 + 24 * q^89 - 18 * q^93 + 18 * q^97 + 24 * q^99

Introduction

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