LMFDB - Embedded newform 3380.2.a.r.1.2

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.52382$ 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)

$f(q)$	$=$	$q-2.52382 q^{3} -1.00000 q^{5} +4.54727 q^{7} +3.36968 q^{9} +O(q^{10})$	$q-2.52382 q^{3} -1.00000 q^{5} +4.54727 q^{7} +3.36968 q^{9} +2.52395 q^{11} +2.52382 q^{15} +5.14630 q^{17} +7.52781 q^{19} -11.4765 q^{21} +2.63662 q^{23} +1.00000 q^{25} -0.933014 q^{27} -8.61995 q^{29} +8.44094 q^{31} -6.37000 q^{33} -4.54727 q^{35} +6.98381 q^{37} +2.66091 q^{41} -9.74401 q^{43} -3.36968 q^{45} -8.15221 q^{47} +13.6776 q^{49} -12.9884 q^{51} -1.43808 q^{53} -2.52395 q^{55} -18.9989 q^{57} +0.0833281 q^{59} -0.818691 q^{61} +15.3229 q^{63} +5.08761 q^{67} -6.65437 q^{69} -7.78118 q^{71} +0.148454 q^{73} -2.52382 q^{75} +11.4771 q^{77} +7.18836 q^{79} -7.75429 q^{81} -16.9198 q^{83} -5.14630 q^{85} +21.7552 q^{87} +3.05664 q^{89} -21.3034 q^{93} -7.52781 q^{95} +8.27717 q^{97} +8.50490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * 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
$3$	−2.52382	−1.45713	−0.728565	−	0.684977i	$-0.759812\pi$
$3$	−2.52382	−1.45713	−0.728565	+	0.684977i	$0.759812\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.54727	1.71871	0.859353	−	0.511383i	$-0.170867\pi$
$7$	4.54727	1.71871	0.859353	+	0.511383i	$0.170867\pi$
$8$	0	0
$9$	3.36968	1.12323
$10$	0	0
$11$	2.52395	0.760999	0.380499	−	0.924781i	$-0.375752\pi$
$11$	2.52395	0.760999	0.380499	+	0.924781i	$0.375752\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.52382	0.651648
$16$	0	0
$17$	5.14630	1.24816	0.624081	−	0.781360i	$-0.285474\pi$
$17$	5.14630	1.24816	0.624081	+	0.781360i	$0.285474\pi$
$18$	0	0
$19$	7.52781	1.72700	0.863499	−	0.504350i	$-0.168268\pi$
$19$	7.52781	1.72700	0.863499	+	0.504350i	$0.168268\pi$
$20$	0	0
$21$	−11.4765	−2.50438
$22$	0	0
$23$	2.63662	0.549774	0.274887	−	0.961477i	$-0.411360\pi$
$23$	2.63662	0.549774	0.274887	+	0.961477i	$0.411360\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.933014	−0.179559
$28$	0	0
$29$	−8.61995	−1.60068	−0.800342	−	0.599544i	$-0.795349\pi$
$29$	−8.61995	−1.60068	−0.800342	+	0.599544i	$0.795349\pi$
$30$	0	0
$31$	8.44094	1.51604	0.758018	−	0.652233i	$-0.226168\pi$
$31$	8.44094	1.51604	0.758018	+	0.652233i	$0.226168\pi$
$32$	0	0
$33$	−6.37000	−1.10887
$34$	0	0
$35$	−4.54727	−0.768629
$36$	0	0
$37$	6.98381	1.14813	0.574065	−	0.818810i	$-0.305366\pi$
$37$	6.98381	1.14813	0.574065	+	0.818810i	$0.305366\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.66091	0.415564	0.207782	−	0.978175i	$-0.433376\pi$
$41$	2.66091	0.415564	0.207782	+	0.978175i	$0.433376\pi$
$42$	0	0
$43$	−9.74401	−1.48595	−0.742974	−	0.669320i	$-0.766585\pi$
$43$	−9.74401	−1.48595	−0.742974	+	0.669320i	$0.766585\pi$
$44$	0	0
$45$	−3.36968	−0.502323
$46$	0	0
$47$	−8.15221	−1.18912	−0.594561	−	0.804051i	$-0.702674\pi$
$47$	−8.15221	−1.18912	−0.594561	+	0.804051i	$0.702674\pi$
$48$	0	0
$49$	13.6776	1.95395
$50$	0	0
$51$	−12.9884	−1.81873
$52$	0	0
$53$	−1.43808	−0.197535	−0.0987677	−	0.995111i	$-0.531490\pi$
$53$	−1.43808	−0.197535	−0.0987677	+	0.995111i	$0.531490\pi$
$54$	0	0
$55$	−2.52395	−0.340329
$56$	0	0
$57$	−18.9989	−2.51646
$58$	0	0
$59$	0.0833281	0.0108484	0.00542420	−	0.999985i	$-0.498273\pi$
$59$	0.0833281	0.0108484	0.00542420	+	0.999985i	$0.498273\pi$
$60$	0	0
$61$	−0.818691	−0.104823	−0.0524113	−	0.998626i	$-0.516691\pi$
$61$	−0.818691	−0.104823	−0.0524113	+	0.998626i	$0.516691\pi$
$62$	0	0
$63$	15.3229	1.93050
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.08761	0.621550	0.310775	−	0.950483i	$-0.399411\pi$
$67$	5.08761	0.621550	0.310775	+	0.950483i	$0.399411\pi$
$68$	0	0
$69$	−6.65437	−0.801092
$70$	0	0
$71$	−7.78118	−0.923456	−0.461728	−	0.887021i	$-0.652770\pi$
$71$	−7.78118	−0.923456	−0.461728	+	0.887021i	$0.652770\pi$
$72$	0	0
$73$	0.148454	0.0173753	0.00868764	−	0.999962i	$-0.497235\pi$
$73$	0.148454	0.0173753	0.00868764	+	0.999962i	$0.497235\pi$
$74$	0	0
$75$	−2.52382	−0.291426
$76$	0	0
$77$	11.4771	1.30793
$78$	0	0
$79$	7.18836	0.808754	0.404377	−	0.914592i	$-0.367488\pi$
$79$	7.18836	0.808754	0.404377	+	0.914592i	$0.367488\pi$
$80$	0	0
$81$	−7.75429	−0.861587
$82$	0	0
$83$	−16.9198	−1.85719	−0.928597	−	0.371089i	$-0.878984\pi$
$83$	−16.9198	−1.85719	−0.928597	+	0.371089i	$0.878984\pi$
$84$	0	0
$85$	−5.14630	−0.558195
$86$	0	0
$87$	21.7552	2.33240
$88$	0	0
$89$	3.05664	0.324003	0.162002	−	0.986790i	$-0.448205\pi$
$89$	3.05664	0.324003	0.162002	+	0.986790i	$0.448205\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−21.3034	−2.20906
$94$	0	0
$95$	−7.52781	−0.772337
$96$	0	0
$97$	8.27717	0.840419	0.420209	−	0.907427i	$-0.361957\pi$
$97$	8.27717	0.840419	0.420209	+	0.907427i	$0.361957\pi$
$98$	0	0
$99$	8.50490	0.854775
$100$	0	0
$101$	2.46123	0.244902	0.122451	−	0.992475i	$-0.460925\pi$
$101$	2.46123	0.244902	0.122451	+	0.992475i	$0.460925\pi$
$102$	0	0
$103$	−10.6923	−1.05355	−0.526773	−	0.850006i	$-0.676598\pi$
$103$	−10.6923	−1.05355	−0.526773	+	0.850006i	$0.676598\pi$
$104$	0	0
$105$	11.4765	1.11999
$106$	0	0
$107$	−0.880036	−0.0850763	−0.0425381	−	0.999095i	$-0.513544\pi$
$107$	−0.880036	−0.0850763	−0.0425381	+	0.999095i	$0.513544\pi$
$108$	0	0
$109$	−1.54141	−0.147640	−0.0738201	−	0.997272i	$-0.523519\pi$
$109$	−1.54141	−0.147640	−0.0738201	+	0.997272i	$0.523519\pi$
$110$	0	0
$111$	−17.6259	−1.67298
$112$	0	0
$113$	10.7763	1.01375	0.506874	−	0.862020i	$-0.330801\pi$
$113$	10.7763	1.01375	0.506874	+	0.862020i	$0.330801\pi$
$114$	0	0
$115$	−2.63662	−0.245866
$116$	0	0
$117$	0	0
$118$	0	0
$119$	23.4016	2.14522
$120$	0	0
$121$	−4.62969	−0.420881
$122$	0	0
$123$	−6.71566	−0.605531
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.09528	0.540868	0.270434	−	0.962738i	$-0.412833\pi$
$127$	6.09528	0.540868	0.270434	+	0.962738i	$0.412833\pi$
$128$	0	0
$129$	24.5922	2.16522
$130$	0	0
$131$	10.4044	0.909039	0.454520	−	0.890737i	$-0.349811\pi$
$131$	10.4044	0.909039	0.454520	+	0.890737i	$0.349811\pi$
$132$	0	0
$133$	34.2310	2.96820
$134$	0	0
$135$	0.933014	0.0803011
$136$	0	0
$137$	4.89180	0.417934	0.208967	−	0.977923i	$-0.432990\pi$
$137$	4.89180	0.417934	0.208967	+	0.977923i	$0.432990\pi$
$138$	0	0
$139$	−22.4823	−1.90692	−0.953460	−	0.301519i	$-0.902506\pi$
$139$	−22.4823	−1.90692	−0.953460	+	0.301519i	$0.902506\pi$
$140$	0	0
$141$	20.5747	1.73270
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.61995	0.715848
$146$	0	0
$147$	−34.5200	−2.84716
$148$	0	0
$149$	20.4563	1.67584	0.837922	−	0.545790i	$-0.183770\pi$
$149$	20.4563	1.67584	0.837922	+	0.545790i	$0.183770\pi$
$150$	0	0
$151$	−0.0637311	−0.00518637	−0.00259318	−	0.999997i	$-0.500825\pi$
$151$	−0.0637311	−0.00518637	−0.00259318	+	0.999997i	$0.500825\pi$
$152$	0	0
$153$	17.3414	1.40197
$154$	0	0
$155$	−8.44094	−0.677992
$156$	0	0
$157$	−0.0591648	−0.00472187	−0.00236093	−	0.999997i	$-0.500752\pi$
$157$	−0.0591648	−0.00472187	−0.00236093	+	0.999997i	$0.500752\pi$
$158$	0	0
$159$	3.62946	0.287835
$160$	0	0
$161$	11.9894	0.944900
$162$	0	0
$163$	−15.9234	−1.24721	−0.623607	−	0.781738i	$-0.714334\pi$
$163$	−15.9234	−1.24721	−0.623607	+	0.781738i	$0.714334\pi$
$164$	0	0
$165$	6.37000	0.495904
$166$	0	0
$167$	−22.1436	−1.71352	−0.856762	−	0.515711i	$-0.827528\pi$
$167$	−22.1436	−1.71352	−0.856762	+	0.515711i	$0.827528\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	25.3663	1.93981
$172$	0	0
$173$	18.7289	1.42393	0.711966	−	0.702214i	$-0.247805\pi$
$173$	18.7289	1.42393	0.711966	+	0.702214i	$0.247805\pi$
$174$	0	0
$175$	4.54727	0.343741
$176$	0	0
$177$	−0.210305	−0.0158075
$178$	0	0
$179$	8.89212	0.664628	0.332314	−	0.943169i	$-0.392171\pi$
$179$	8.89212	0.664628	0.332314	+	0.943169i	$0.392171\pi$
$180$	0	0
$181$	−3.90674	−0.290386	−0.145193	−	0.989403i	$-0.546380\pi$
$181$	−3.90674	−0.290386	−0.145193	+	0.989403i	$0.546380\pi$
$182$	0	0
$183$	2.06623	0.152740
$184$	0	0
$185$	−6.98381	−0.513460
$186$	0	0
$187$	12.9890	0.949850
$188$	0	0
$189$	−4.24267	−0.308609
$190$	0	0
$191$	22.0068	1.59236	0.796179	−	0.605061i	$-0.206851\pi$
$191$	22.0068	1.59236	0.796179	+	0.605061i	$0.206851\pi$
$192$	0	0
$193$	−6.46062	−0.465046	−0.232523	−	0.972591i	$-0.574698\pi$
$193$	−6.46062	−0.465046	−0.232523	+	0.972591i	$0.574698\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.2295	−1.37004	−0.685021	−	0.728523i	$-0.740207\pi$
$197$	−19.2295	−1.37004	−0.685021	+	0.728523i	$0.740207\pi$
$198$	0	0
$199$	16.3145	1.15651	0.578253	−	0.815858i	$-0.303735\pi$
$199$	16.3145	1.15651	0.578253	+	0.815858i	$0.303735\pi$
$200$	0	0
$201$	−12.8402	−0.905679
$202$	0	0
$203$	−39.1972	−2.75111
$204$	0	0
$205$	−2.66091	−0.185846
$206$	0	0
$207$	8.88458	0.617521
$208$	0	0
$209$	18.9998	1.31424
$210$	0	0
$211$	21.2686	1.46419	0.732094	−	0.681204i	$-0.238543\pi$
$211$	21.2686	1.46419	0.732094	+	0.681204i	$0.238543\pi$
$212$	0	0
$213$	19.6383	1.34560
$214$	0	0
$215$	9.74401	0.664536
$216$	0	0
$217$	38.3832	2.60562
$218$	0	0
$219$	−0.374673	−0.0253180
$220$	0	0
$221$	0	0
$222$	0	0
$223$	9.75386	0.653167	0.326583	−	0.945168i	$-0.394103\pi$
$223$	9.75386	0.653167	0.326583	+	0.945168i	$0.394103\pi$
$224$	0	0
$225$	3.36968	0.224646
$226$	0	0
$227$	−26.0411	−1.72841	−0.864205	−	0.503140i	$-0.832178\pi$
$227$	−26.0411	−1.72841	−0.864205	+	0.503140i	$0.832178\pi$
$228$	0	0
$229$	15.9818	1.05610	0.528052	−	0.849212i	$-0.322923\pi$
$229$	15.9818	1.05610	0.528052	+	0.849212i	$0.322923\pi$
$230$	0	0
$231$	−28.9661	−1.90583
$232$	0	0
$233$	−7.67734	−0.502960	−0.251480	−	0.967863i	$-0.580917\pi$
$233$	−7.67734	−0.502960	−0.251480	+	0.967863i	$0.580917\pi$
$234$	0	0
$235$	8.15221	0.531791
$236$	0	0
$237$	−18.1422	−1.17846
$238$	0	0
$239$	18.1981	1.17714	0.588570	−	0.808446i	$-0.299691\pi$
$239$	18.1981	1.17714	0.588570	+	0.808446i	$0.299691\pi$
$240$	0	0
$241$	−13.7693	−0.886958	−0.443479	−	0.896285i	$-0.646256\pi$
$241$	−13.7693	−0.886958	−0.443479	+	0.896285i	$0.646256\pi$
$242$	0	0
$243$	22.3695	1.43500
$244$	0	0
$245$	−13.6776	−0.873833
$246$	0	0
$247$	0	0
$248$	0	0
$249$	42.7027	2.70617
$250$	0	0
$251$	9.54318	0.602360	0.301180	−	0.953567i	$-0.402619\pi$
$251$	9.54318	0.602360	0.301180	+	0.953567i	$0.402619\pi$
$252$	0	0
$253$	6.65470	0.418377
$254$	0	0
$255$	12.9884	0.813363
$256$	0	0
$257$	−9.56768	−0.596816	−0.298408	−	0.954438i	$-0.596456\pi$
$257$	−9.56768	−0.596816	−0.298408	+	0.954438i	$0.596456\pi$
$258$	0	0
$259$	31.7572	1.97330
$260$	0	0
$261$	−29.0465	−1.79793
$262$	0	0
$263$	21.2418	1.30982	0.654912	−	0.755705i	$-0.272705\pi$
$263$	21.2418	1.30982	0.654912	+	0.755705i	$0.272705\pi$
$264$	0	0
$265$	1.43808	0.0883405
$266$	0	0
$267$	−7.71442	−0.472115
$268$	0	0
$269$	24.1110	1.47008	0.735038	−	0.678026i	$-0.237164\pi$
$269$	24.1110	1.47008	0.735038	+	0.678026i	$0.237164\pi$
$270$	0	0
$271$	2.12699	0.129205	0.0646026	−	0.997911i	$-0.479422\pi$
$271$	2.12699	0.129205	0.0646026	+	0.997911i	$0.479422\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.52395	0.152200
$276$	0	0
$277$	24.5197	1.47325	0.736623	−	0.676303i	$-0.236419\pi$
$277$	24.5197	1.47325	0.736623	+	0.676303i	$0.236419\pi$
$278$	0	0
$279$	28.4433	1.70285
$280$	0	0
$281$	9.63729	0.574913	0.287456	−	0.957794i	$-0.407190\pi$
$281$	9.63729	0.574913	0.287456	+	0.957794i	$0.407190\pi$
$282$	0	0
$283$	−23.5987	−1.40280	−0.701400	−	0.712768i	$-0.747441\pi$
$283$	−23.5987	−1.40280	−0.701400	+	0.712768i	$0.747441\pi$
$284$	0	0
$285$	18.9989	1.12540
$286$	0	0
$287$	12.0999	0.714232
$288$	0	0
$289$	9.48444	0.557908
$290$	0	0
$291$	−20.8901	−1.22460
$292$	0	0
$293$	5.38348	0.314506	0.157253	−	0.987558i	$-0.449736\pi$
$293$	5.38348	0.314506	0.157253	+	0.987558i	$0.449736\pi$
$294$	0	0
$295$	−0.0833281	−0.00485155
$296$	0	0
$297$	−2.35488	−0.136644
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−44.3086	−2.55391
$302$	0	0
$303$	−6.21171	−0.356854
$304$	0	0
$305$	0.818691	0.0468781
$306$	0	0
$307$	−12.1743	−0.694825	−0.347413	−	0.937712i	$-0.612940\pi$
$307$	−12.1743	−0.694825	−0.347413	+	0.937712i	$0.612940\pi$
$308$	0	0
$309$	26.9855	1.53515
$310$	0	0
$311$	−13.2103	−0.749085	−0.374543	−	0.927210i	$-0.622200\pi$
$311$	−13.2103	−0.749085	−0.374543	+	0.927210i	$0.622200\pi$
$312$	0	0
$313$	−24.1485	−1.36495	−0.682476	−	0.730908i	$-0.739097\pi$
$313$	−24.1485	−1.36495	−0.682476	+	0.730908i	$0.739097\pi$
$314$	0	0
$315$	−15.3229	−0.863345
$316$	0	0
$317$	−6.42443	−0.360832	−0.180416	−	0.983590i	$-0.557744\pi$
$317$	−6.42443	−0.360832	−0.180416	+	0.983590i	$0.557744\pi$
$318$	0	0
$319$	−21.7563	−1.21812
$320$	0	0
$321$	2.22105	0.123967
$322$	0	0
$323$	38.7404	2.15557
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.89024	0.215131
$328$	0	0
$329$	−37.0703	−2.04375
$330$	0	0
$331$	21.8105	1.19881	0.599407	−	0.800444i	$-0.295403\pi$
$331$	21.8105	1.19881	0.599407	+	0.800444i	$0.295403\pi$
$332$	0	0
$333$	23.5332	1.28961
$334$	0	0
$335$	−5.08761	−0.277966
$336$	0	0
$337$	5.94670	0.323937	0.161969	−	0.986796i	$-0.448216\pi$
$337$	5.94670	0.323937	0.161969	+	0.986796i	$0.448216\pi$
$338$	0	0
$339$	−27.1975	−1.47716
$340$	0	0
$341$	21.3045	1.15370
$342$	0	0
$343$	30.3651	1.63956
$344$	0	0
$345$	6.65437	0.358259
$346$	0	0
$347$	−20.7097	−1.11175	−0.555877	−	0.831264i	$-0.687618\pi$
$347$	−20.7097	−1.11175	−0.555877	+	0.831264i	$0.687618\pi$
$348$	0	0
$349$	−8.05214	−0.431021	−0.215511	−	0.976502i	$-0.569142\pi$
$349$	−8.05214	−0.431021	−0.215511	+	0.976502i	$0.569142\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.11032	−0.378444	−0.189222	−	0.981934i	$-0.560597\pi$
$353$	−7.11032	−0.378444	−0.189222	+	0.981934i	$0.560597\pi$
$354$	0	0
$355$	7.78118	0.412982
$356$	0	0
$357$	−59.0616	−3.12587
$358$	0	0
$359$	−6.26310	−0.330554	−0.165277	−	0.986247i	$-0.552852\pi$
$359$	−6.26310	−0.330554	−0.165277	+	0.986247i	$0.552852\pi$
$360$	0	0
$361$	37.6679	1.98252
$362$	0	0
$363$	11.6845	0.613278
$364$	0	0
$365$	−0.148454	−0.00777046
$366$	0	0
$367$	15.9522	0.832697	0.416349	−	0.909205i	$-0.363310\pi$
$367$	15.9522	0.832697	0.416349	+	0.909205i	$0.363310\pi$
$368$	0	0
$369$	8.96642	0.466773
$370$	0	0
$371$	−6.53934	−0.339505
$372$	0	0
$373$	−1.42611	−0.0738410	−0.0369205	−	0.999318i	$-0.511755\pi$
$373$	−1.42611	−0.0738410	−0.0369205	+	0.999318i	$0.511755\pi$
$374$	0	0
$375$	2.52382	0.130330
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3.57360	0.183564	0.0917818	−	0.995779i	$-0.470744\pi$
$379$	3.57360	0.183564	0.0917818	+	0.995779i	$0.470744\pi$
$380$	0	0
$381$	−15.3834	−0.788115
$382$	0	0
$383$	4.84620	0.247629	0.123815	−	0.992305i	$-0.460487\pi$
$383$	4.84620	0.247629	0.123815	+	0.992305i	$0.460487\pi$
$384$	0	0
$385$	−11.4771	−0.584925
$386$	0	0
$387$	−32.8342	−1.66906
$388$	0	0
$389$	−33.7772	−1.71257	−0.856285	−	0.516503i	$-0.827233\pi$
$389$	−33.7772	−1.71257	−0.856285	+	0.516503i	$0.827233\pi$
$390$	0	0
$391$	13.5689	0.686207
$392$	0	0
$393$	−26.2589	−1.32459
$394$	0	0
$395$	−7.18836	−0.361686
$396$	0	0
$397$	0.483569	0.0242696	0.0121348	−	0.999926i	$-0.496137\pi$
$397$	0.483569	0.0242696	0.0121348	+	0.999926i	$0.496137\pi$
$398$	0	0
$399$	−86.3929	−4.32506
$400$	0	0
$401$	−18.0902	−0.903380	−0.451690	−	0.892175i	$-0.649179\pi$
$401$	−18.0902	−0.903380	−0.451690	+	0.892175i	$0.649179\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.75429	0.385314
$406$	0	0
$407$	17.6268	0.873726
$408$	0	0
$409$	34.5327	1.70753	0.853766	−	0.520657i	$-0.174313\pi$
$409$	34.5327	1.70753	0.853766	+	0.520657i	$0.174313\pi$
$410$	0	0
$411$	−12.3460	−0.608985
$412$	0	0
$413$	0.378915	0.0186452
$414$	0	0
$415$	16.9198	0.830563
$416$	0	0
$417$	56.7412	2.77863
$418$	0	0
$419$	−23.0215	−1.12467	−0.562337	−	0.826908i	$-0.690098\pi$
$419$	−23.0215	−1.12467	−0.562337	+	0.826908i	$0.690098\pi$
$420$	0	0
$421$	−7.63811	−0.372259	−0.186129	−	0.982525i	$-0.559594\pi$
$421$	−7.63811	−0.372259	−0.186129	+	0.982525i	$0.559594\pi$
$422$	0	0
$423$	−27.4704	−1.33565
$424$	0	0
$425$	5.14630	0.249632
$426$	0	0
$427$	−3.72281	−0.180159
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.3076	−1.07452	−0.537260	−	0.843417i	$-0.680541\pi$
$431$	−22.3076	−1.07452	−0.537260	+	0.843417i	$0.680541\pi$
$432$	0	0
$433$	1.01305	0.0486838	0.0243419	−	0.999704i	$-0.492251\pi$
$433$	1.01305	0.0486838	0.0243419	+	0.999704i	$0.492251\pi$
$434$	0	0
$435$	−21.7552	−1.04308
$436$	0	0
$437$	19.8480	0.949459
$438$	0	0
$439$	−40.1466	−1.91609	−0.958046	−	0.286613i	$-0.907471\pi$
$439$	−40.1466	−1.91609	−0.958046	+	0.286613i	$0.907471\pi$
$440$	0	0
$441$	46.0893	2.19473
$442$	0	0
$443$	−37.0607	−1.76080	−0.880402	−	0.474228i	$-0.842727\pi$
$443$	−37.0607	−1.76080	−0.880402	+	0.474228i	$0.842727\pi$
$444$	0	0
$445$	−3.05664	−0.144899
$446$	0	0
$447$	−51.6281	−2.44192
$448$	0	0
$449$	−11.4726	−0.541424	−0.270712	−	0.962660i	$-0.587259\pi$
$449$	−11.4726	−0.541424	−0.270712	+	0.962660i	$0.587259\pi$
$450$	0	0
$451$	6.71599	0.316244
$452$	0	0
$453$	0.160846	0.00755721
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.21789	0.431195	0.215597	−	0.976482i	$-0.430830\pi$
$457$	9.21789	0.431195	0.215597	+	0.976482i	$0.430830\pi$
$458$	0	0
$459$	−4.80157	−0.224118
$460$	0	0
$461$	3.78265	0.176176	0.0880878	−	0.996113i	$-0.471924\pi$
$461$	3.78265	0.176176	0.0880878	+	0.996113i	$0.471924\pi$
$462$	0	0
$463$	−2.59710	−0.120698	−0.0603488	−	0.998177i	$-0.519221\pi$
$463$	−2.59710	−0.120698	−0.0603488	+	0.998177i	$0.519221\pi$
$464$	0	0
$465$	21.3034	0.987923
$466$	0	0
$467$	−39.2664	−1.81703	−0.908516	−	0.417849i	$-0.862784\pi$
$467$	−39.2664	−1.81703	−0.908516	+	0.417849i	$0.862784\pi$
$468$	0	0
$469$	23.1347	1.06826
$470$	0	0
$471$	0.149321	0.00688037
$472$	0	0
$473$	−24.5934	−1.13080
$474$	0	0
$475$	7.52781	0.345400
$476$	0	0
$477$	−4.84587	−0.221877
$478$	0	0
$479$	−35.8940	−1.64004	−0.820019	−	0.572337i	$-0.806037\pi$
$479$	−35.8940	−1.64004	−0.820019	+	0.572337i	$0.806037\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−30.2592	−1.37684
$484$	0	0
$485$	−8.27717	−0.375847
$486$	0	0
$487$	−30.6683	−1.38971	−0.694856	−	0.719149i	$-0.744532\pi$
$487$	−30.6683	−1.38971	−0.694856	+	0.719149i	$0.744532\pi$
$488$	0	0
$489$	40.1878	1.81735
$490$	0	0
$491$	30.2621	1.36571	0.682856	−	0.730553i	$-0.260738\pi$
$491$	30.2621	1.36571	0.682856	+	0.730553i	$0.260738\pi$
$492$	0	0
$493$	−44.3609	−1.99791
$494$	0	0
$495$	−8.50490	−0.382267
$496$	0	0
$497$	−35.3831	−1.58715
$498$	0	0
$499$	11.5036	0.514973	0.257486	−	0.966282i	$-0.417106\pi$
$499$	11.5036	0.514973	0.257486	+	0.966282i	$0.417106\pi$
$500$	0	0
$501$	55.8866	2.49683
$502$	0	0
$503$	−14.0889	−0.628195	−0.314097	−	0.949391i	$-0.601702\pi$
$503$	−14.0889	−0.628195	−0.314097	+	0.949391i	$0.601702\pi$
$504$	0	0
$505$	−2.46123	−0.109523
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.07854	−0.358075	−0.179037	−	0.983842i	$-0.557298\pi$
$509$	−8.07854	−0.358075	−0.179037	+	0.983842i	$0.557298\pi$
$510$	0	0
$511$	0.675062	0.0298630
$512$	0	0
$513$	−7.02355	−0.310097
$514$	0	0
$515$	10.6923	0.471160
$516$	0	0
$517$	−20.5757	−0.904920
$518$	0	0
$519$	−47.2684	−2.07485
$520$	0	0
$521$	31.6830	1.38806	0.694028	−	0.719948i	$-0.255834\pi$
$521$	31.6830	1.38806	0.694028	+	0.719948i	$0.255834\pi$
$522$	0	0
$523$	5.56717	0.243435	0.121718	−	0.992565i	$-0.461160\pi$
$523$	5.56717	0.243435	0.121718	+	0.992565i	$0.461160\pi$
$524$	0	0
$525$	−11.4765	−0.500876
$526$	0	0
$527$	43.4396	1.89226
$528$	0	0
$529$	−16.0482	−0.697749
$530$	0	0
$531$	0.280789	0.0121852
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0.880036	0.0380473
$536$	0	0
$537$	−22.4421	−0.968449
$538$	0	0
$539$	34.5217	1.48695
$540$	0	0
$541$	3.01207	0.129499	0.0647496	−	0.997902i	$-0.479375\pi$
$541$	3.01207	0.129499	0.0647496	+	0.997902i	$0.479375\pi$
$542$	0	0
$543$	9.85992	0.423130
$544$	0	0
$545$	1.54141	0.0660267
$546$	0	0
$547$	23.1320	0.989053	0.494527	−	0.869163i	$-0.335341\pi$
$547$	23.1320	0.989053	0.494527	+	0.869163i	$0.335341\pi$
$548$	0	0
$549$	−2.75873	−0.117740
$550$	0	0
$551$	−64.8893	−2.76438
$552$	0	0
$553$	32.6874	1.39001
$554$	0	0
$555$	17.6259	0.748177
$556$	0	0
$557$	7.18919	0.304616	0.152308	−	0.988333i	$-0.451329\pi$
$557$	7.18919	0.304616	0.152308	+	0.988333i	$0.451329\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−32.7819	−1.38405
$562$	0	0
$563$	5.32860	0.224574	0.112287	−	0.993676i	$-0.464182\pi$
$563$	5.32860	0.224574	0.112287	+	0.993676i	$0.464182\pi$
$564$	0	0
$565$	−10.7763	−0.453362
$566$	0	0
$567$	−35.2608	−1.48082
$568$	0	0
$569$	39.8437	1.67034	0.835168	−	0.549995i	$-0.185370\pi$
$569$	39.8437	1.67034	0.835168	+	0.549995i	$0.185370\pi$
$570$	0	0
$571$	3.28735	0.137571	0.0687857	−	0.997631i	$-0.478088\pi$
$571$	3.28735	0.137571	0.0687857	+	0.997631i	$0.478088\pi$
$572$	0	0
$573$	−55.5414	−2.32027
$574$	0	0
$575$	2.63662	0.109955
$576$	0	0
$577$	16.3187	0.679358	0.339679	−	0.940541i	$-0.389682\pi$
$577$	16.3187	0.679358	0.339679	+	0.940541i	$0.389682\pi$
$578$	0	0
$579$	16.3055	0.677632
$580$	0	0
$581$	−76.9391	−3.19197
$582$	0	0
$583$	−3.62964	−0.150324
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.3533	0.881344	0.440672	−	0.897668i	$-0.354740\pi$
$587$	21.3533	0.881344	0.440672	+	0.897668i	$0.354740\pi$
$588$	0	0
$589$	63.5418	2.61819
$590$	0	0
$591$	48.5317	1.99633
$592$	0	0
$593$	−26.2461	−1.07780	−0.538898	−	0.842371i	$-0.681159\pi$
$593$	−26.2461	−1.07780	−0.538898	+	0.842371i	$0.681159\pi$
$594$	0	0
$595$	−23.4016	−0.959373
$596$	0	0
$597$	−41.1750	−1.68518
$598$	0	0
$599$	−25.6381	−1.04754	−0.523772	−	0.851858i	$-0.675476\pi$
$599$	−25.6381	−1.04754	−0.523772	+	0.851858i	$0.675476\pi$
$600$	0	0
$601$	14.6633	0.598128	0.299064	−	0.954233i	$-0.403326\pi$
$601$	14.6633	0.598128	0.299064	+	0.954233i	$0.403326\pi$
$602$	0	0
$603$	17.1436	0.698142
$604$	0	0
$605$	4.62969	0.188224
$606$	0	0
$607$	14.8341	0.602100	0.301050	−	0.953608i	$-0.402663\pi$
$607$	14.8341	0.602100	0.301050	+	0.953608i	$0.402663\pi$
$608$	0	0
$609$	98.9268	4.00872
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0.253642	0.0102445	0.00512225	−	0.999987i	$-0.498370\pi$
$613$	0.253642	0.0102445	0.00512225	+	0.999987i	$0.498370\pi$
$614$	0	0
$615$	6.71566	0.270802
$616$	0	0
$617$	9.99816	0.402511	0.201255	−	0.979539i	$-0.435498\pi$
$617$	9.99816	0.402511	0.201255	+	0.979539i	$0.435498\pi$
$618$	0	0
$619$	−6.07953	−0.244357	−0.122178	−	0.992508i	$-0.538988\pi$
$619$	−6.07953	−0.244357	−0.122178	+	0.992508i	$0.538988\pi$
$620$	0	0
$621$	−2.46001	−0.0987167
$622$	0	0
$623$	13.8994	0.556866
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−47.9521	−1.91502
$628$	0	0
$629$	35.9408	1.43305
$630$	0	0
$631$	40.4143	1.60887	0.804434	−	0.594041i	$-0.202468\pi$
$631$	40.4143	1.60887	0.804434	+	0.594041i	$0.202468\pi$
$632$	0	0
$633$	−53.6781	−2.13351
$634$	0	0
$635$	−6.09528	−0.241884
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−26.2201	−1.03725
$640$	0	0
$641$	10.7514	0.424653	0.212327	−	0.977199i	$-0.431896\pi$
$641$	10.7514	0.424653	0.212327	+	0.977199i	$0.431896\pi$
$642$	0	0
$643$	15.7425	0.620823	0.310412	−	0.950602i	$-0.399533\pi$
$643$	15.7425	0.620823	0.310412	+	0.950602i	$0.399533\pi$
$644$	0	0
$645$	−24.5922	−0.968315
$646$	0	0
$647$	−13.4404	−0.528399	−0.264199	−	0.964468i	$-0.585108\pi$
$647$	−13.4404	−0.528399	−0.264199	+	0.964468i	$0.585108\pi$
$648$	0	0
$649$	0.210316	0.00825562
$650$	0	0
$651$	−96.8724	−3.79673
$652$	0	0
$653$	1.21367	0.0474948	0.0237474	−	0.999718i	$-0.492440\pi$
$653$	1.21367	0.0474948	0.0237474	+	0.999718i	$0.492440\pi$
$654$	0	0
$655$	−10.4044	−0.406535
$656$	0	0
$657$	0.500245	0.0195164
$658$	0	0
$659$	−11.8299	−0.460827	−0.230413	−	0.973093i	$-0.574008\pi$
$659$	−11.8299	−0.460827	−0.230413	+	0.973093i	$0.574008\pi$
$660$	0	0
$661$	−44.4477	−1.72882	−0.864409	−	0.502790i	$-0.832307\pi$
$661$	−44.4477	−1.72882	−0.864409	+	0.502790i	$0.832307\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−34.2310	−1.32742
$666$	0	0
$667$	−22.7276	−0.880015
$668$	0	0
$669$	−24.6170	−0.951749
$670$	0	0
$671$	−2.06633	−0.0797699
$672$	0	0
$673$	12.2950	0.473936	0.236968	−	0.971517i	$-0.423846\pi$
$673$	12.2950	0.473936	0.236968	+	0.971517i	$0.423846\pi$
$674$	0	0
$675$	−0.933014	−0.0359117
$676$	0	0
$677$	−7.57026	−0.290949	−0.145474	−	0.989362i	$-0.546471\pi$
$677$	−7.57026	−0.290949	−0.145474	+	0.989362i	$0.546471\pi$
$678$	0	0
$679$	37.6385	1.44443
$680$	0	0
$681$	65.7232	2.51852
$682$	0	0
$683$	16.7077	0.639303	0.319651	−	0.947535i	$-0.396434\pi$
$683$	16.7077	0.639303	0.319651	+	0.947535i	$0.396434\pi$
$684$	0	0
$685$	−4.89180	−0.186906
$686$	0	0
$687$	−40.3351	−1.53888
$688$	0	0
$689$	0	0
$690$	0	0
$691$	18.5620	0.706131	0.353065	−	0.935599i	$-0.385139\pi$
$691$	18.5620	0.706131	0.353065	+	0.935599i	$0.385139\pi$
$692$	0	0
$693$	38.6741	1.46911
$694$	0	0
$695$	22.4823	0.852801
$696$	0	0
$697$	13.6938	0.518691
$698$	0	0
$699$	19.3763	0.732877
$700$	0	0
$701$	25.7585	0.972886	0.486443	−	0.873712i	$-0.338294\pi$
$701$	25.7585	0.972886	0.486443	+	0.873712i	$0.338294\pi$
$702$	0	0
$703$	52.5728	1.98282
$704$	0	0
$705$	−20.5747	−0.774889
$706$	0	0
$707$	11.1919	0.420914
$708$	0	0
$709$	5.33378	0.200314	0.100157	−	0.994972i	$-0.468065\pi$
$709$	5.33378	0.200314	0.100157	+	0.994972i	$0.468065\pi$
$710$	0	0
$711$	24.2225	0.908415
$712$	0	0
$713$	22.2556	0.833478
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−45.9289	−1.71525
$718$	0	0
$719$	−18.5517	−0.691861	−0.345930	−	0.938260i	$-0.612437\pi$
$719$	−18.5517	−0.691861	−0.345930	+	0.938260i	$0.612437\pi$
$720$	0	0
$721$	−48.6208	−1.81073
$722$	0	0
$723$	34.7513	1.29241
$724$	0	0
$725$	−8.61995	−0.320137
$726$	0	0
$727$	14.5977	0.541400	0.270700	−	0.962664i	$-0.412745\pi$
$727$	14.5977	0.541400	0.270700	+	0.962664i	$0.412745\pi$
$728$	0	0
$729$	−33.1938	−1.22940
$730$	0	0
$731$	−50.1456	−1.85470
$732$	0	0
$733$	28.2807	1.04457	0.522286	−	0.852770i	$-0.325079\pi$
$733$	28.2807	1.04457	0.522286	+	0.852770i	$0.325079\pi$
$734$	0	0
$735$	34.5200	1.27329
$736$	0	0
$737$	12.8409	0.472999
$738$	0	0
$739$	2.37192	0.0872523	0.0436262	−	0.999048i	$-0.486109\pi$
$739$	2.37192	0.0872523	0.0436262	+	0.999048i	$0.486109\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.4361	0.969845	0.484923	−	0.874557i	$-0.338848\pi$
$743$	26.4361	0.969845	0.484923	+	0.874557i	$0.338848\pi$
$744$	0	0
$745$	−20.4563	−0.749460
$746$	0	0
$747$	−57.0145	−2.08605
$748$	0	0
$749$	−4.00176	−0.146221
$750$	0	0
$751$	−13.0414	−0.475889	−0.237944	−	0.971279i	$-0.576474\pi$
$751$	−13.0414	−0.475889	−0.237944	+	0.971279i	$0.576474\pi$
$752$	0	0
$753$	−24.0853	−0.877717
$754$	0	0
$755$	0.0637311	0.00231941
$756$	0	0
$757$	−43.6023	−1.58475	−0.792377	−	0.610032i	$-0.791157\pi$
$757$	−43.6023	−1.58475	−0.792377	+	0.610032i	$0.791157\pi$
$758$	0	0
$759$	−16.7953	−0.609630
$760$	0	0
$761$	−17.2976	−0.627038	−0.313519	−	0.949582i	$-0.601508\pi$
$761$	−17.2976	−0.627038	−0.313519	+	0.949582i	$0.601508\pi$
$762$	0	0
$763$	−7.00920	−0.253750
$764$	0	0
$765$	−17.3414	−0.626980
$766$	0	0
$767$	0	0
$768$	0	0
$769$	27.3193	0.985158	0.492579	−	0.870268i	$-0.336054\pi$
$769$	27.3193	0.985158	0.492579	+	0.870268i	$0.336054\pi$
$770$	0	0
$771$	24.1471	0.869638
$772$	0	0
$773$	−21.4675	−0.772131	−0.386066	−	0.922471i	$-0.626166\pi$
$773$	−21.4675	−0.772131	−0.386066	+	0.922471i	$0.626166\pi$
$774$	0	0
$775$	8.44094	0.303207
$776$	0	0
$777$	−80.1497	−2.87535
$778$	0	0
$779$	20.0308	0.717678
$780$	0	0
$781$	−19.6393	−0.702749
$782$	0	0
$783$	8.04253	0.287417
$784$	0	0
$785$	0.0591648	0.00211168
$786$	0	0
$787$	21.4097	0.763174	0.381587	−	0.924333i	$-0.375378\pi$
$787$	21.4097	0.763174	0.381587	+	0.924333i	$0.375378\pi$
$788$	0	0
$789$	−53.6105	−1.90859
$790$	0	0
$791$	49.0027	1.74234
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−3.62946	−0.128724
$796$	0	0
$797$	15.0647	0.533617	0.266809	−	0.963750i	$-0.414031\pi$
$797$	15.0647	0.533617	0.266809	+	0.963750i	$0.414031\pi$
$798$	0	0
$799$	−41.9537	−1.48422
$800$	0	0
$801$	10.2999	0.363930
$802$	0	0
$803$	0.374691	0.0132226
$804$	0	0
$805$	−11.9894	−0.422572
$806$	0	0
$807$	−60.8520	−2.14209
$808$	0	0
$809$	26.7046	0.938885	0.469442	−	0.882963i	$-0.344455\pi$
$809$	26.7046	0.938885	0.469442	+	0.882963i	$0.344455\pi$
$810$	0	0
$811$	−21.8145	−0.766011	−0.383005	−	0.923746i	$-0.625111\pi$
$811$	−21.8145	−0.766011	−0.383005	+	0.923746i	$0.625111\pi$
$812$	0	0
$813$	−5.36814	−0.188269
$814$	0	0
$815$	15.9234	0.557771
$816$	0	0
$817$	−73.3511	−2.56623
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.0913	1.01529	0.507646	−	0.861565i	$-0.330516\pi$
$821$	29.0913	1.01529	0.507646	+	0.861565i	$0.330516\pi$
$822$	0	0
$823$	−20.4597	−0.713182	−0.356591	−	0.934261i	$-0.616061\pi$
$823$	−20.4597	−0.713182	−0.356591	+	0.934261i	$0.616061\pi$
$824$	0	0
$825$	−6.37000	−0.221775
$826$	0	0
$827$	−24.2164	−0.842086	−0.421043	−	0.907041i	$-0.638336\pi$
$827$	−24.2164	−0.842086	−0.421043	+	0.907041i	$0.638336\pi$
$828$	0	0
$829$	−3.77286	−0.131037	−0.0655185	−	0.997851i	$-0.520870\pi$
$829$	−3.77286	−0.131037	−0.0655185	+	0.997851i	$0.520870\pi$
$830$	0	0
$831$	−61.8834	−2.14671
$832$	0	0
$833$	70.3893	2.43885
$834$	0	0
$835$	22.1436	0.766312
$836$	0	0
$837$	−7.87551	−0.272218
$838$	0	0
$839$	12.0985	0.417685	0.208843	−	0.977949i	$-0.433030\pi$
$839$	12.0985	0.417685	0.208843	+	0.977949i	$0.433030\pi$
$840$	0	0
$841$	45.3035	1.56219
$842$	0	0
$843$	−24.3228	−0.837723
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21.0524	−0.723370
$848$	0	0
$849$	59.5591	2.04406
$850$	0	0
$851$	18.4137	0.631212
$852$	0	0
$853$	45.2655	1.54986	0.774930	−	0.632047i	$-0.217785\pi$
$853$	45.2655	1.54986	0.774930	+	0.632047i	$0.217785\pi$
$854$	0	0
$855$	−25.3663	−0.867510
$856$	0	0
$857$	−11.4321	−0.390514	−0.195257	−	0.980752i	$-0.562554\pi$
$857$	−11.4321	−0.390514	−0.195257	+	0.980752i	$0.562554\pi$
$858$	0	0
$859$	−48.9331	−1.66957	−0.834787	−	0.550573i	$-0.814409\pi$
$859$	−48.9331	−1.66957	−0.834787	+	0.550573i	$0.814409\pi$
$860$	0	0
$861$	−30.5379	−1.04073
$862$	0	0
$863$	−7.25636	−0.247009	−0.123505	−	0.992344i	$-0.539413\pi$
$863$	−7.25636	−0.247009	−0.123505	+	0.992344i	$0.539413\pi$
$864$	0	0
$865$	−18.7289	−0.636802
$866$	0	0
$867$	−23.9371	−0.812945
$868$	0	0
$869$	18.1430	0.615461
$870$	0	0
$871$	0	0
$872$	0	0
$873$	27.8914	0.943982
$874$	0	0
$875$	−4.54727	−0.153726
$876$	0	0
$877$	−12.2767	−0.414555	−0.207277	−	0.978282i	$-0.566460\pi$
$877$	−12.2767	−0.414555	−0.207277	+	0.978282i	$0.566460\pi$
$878$	0	0
$879$	−13.5869	−0.458276
$880$	0	0
$881$	−22.2767	−0.750520	−0.375260	−	0.926920i	$-0.622447\pi$
$881$	−22.2767	−0.750520	−0.375260	+	0.926920i	$0.622447\pi$
$882$	0	0
$883$	42.4148	1.42737	0.713686	−	0.700465i	$-0.247024\pi$
$883$	42.4148	1.42737	0.713686	+	0.700465i	$0.247024\pi$
$884$	0	0
$885$	0.210305	0.00706934
$886$	0	0
$887$	−3.44309	−0.115608	−0.0578038	−	0.998328i	$-0.518410\pi$
$887$	−3.44309	−0.115608	−0.0578038	+	0.998328i	$0.518410\pi$
$888$	0	0
$889$	27.7169	0.929594
$890$	0	0
$891$	−19.5714	−0.655667
$892$	0	0
$893$	−61.3683	−2.05361
$894$	0	0
$895$	−8.89212	−0.297231
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−72.7604	−2.42670
$900$	0	0
$901$	−7.40080	−0.246556
$902$	0	0
$903$	111.827	3.72138
$904$	0	0
$905$	3.90674	0.129864
$906$	0	0
$907$	−24.8493	−0.825108	−0.412554	−	0.910933i	$-0.635363\pi$
$907$	−24.8493	−0.825108	−0.412554	+	0.910933i	$0.635363\pi$
$908$	0	0
$909$	8.29357	0.275080
$910$	0	0
$911$	12.5928	0.417218	0.208609	−	0.977999i	$-0.433106\pi$
$911$	12.5928	0.417218	0.208609	+	0.977999i	$0.433106\pi$
$912$	0	0
$913$	−42.7048	−1.41332
$914$	0	0
$915$	−2.06623	−0.0683075
$916$	0	0
$917$	47.3117	1.56237
$918$	0	0
$919$	39.6595	1.30825	0.654124	−	0.756388i	$-0.273038\pi$
$919$	39.6595	1.30825	0.654124	+	0.756388i	$0.273038\pi$
$920$	0	0
$921$	30.7258	1.01245
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.98381	0.229626
$926$	0	0
$927$	−36.0297	−1.18337
$928$	0	0
$929$	41.5553	1.36338	0.681692	−	0.731639i	$-0.261244\pi$
$929$	41.5553	1.36338	0.681692	+	0.731639i	$0.261244\pi$
$930$	0	0
$931$	102.963	3.37447
$932$	0	0
$933$	33.3404	1.09151
$934$	0	0
$935$	−12.9890	−0.424786
$936$	0	0
$937$	−5.57744	−0.182207	−0.0911035	−	0.995841i	$-0.529039\pi$
$937$	−5.57744	−0.182207	−0.0911035	+	0.995841i	$0.529039\pi$
$938$	0	0
$939$	60.9465	1.98891
$940$	0	0
$941$	19.7230	0.642951	0.321475	−	0.946918i	$-0.395821\pi$
$941$	19.7230	0.642951	0.321475	+	0.946918i	$0.395821\pi$
$942$	0	0
$943$	7.01581	0.228466
$944$	0	0
$945$	4.24267	0.138014
$946$	0	0
$947$	−24.4631	−0.794943	−0.397471	−	0.917615i	$-0.630112\pi$
$947$	−24.4631	−0.794943	−0.397471	+	0.917615i	$0.630112\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	16.2141	0.525779
$952$	0	0
$953$	55.5402	1.79912	0.899562	−	0.436794i	$-0.143886\pi$
$953$	55.5402	1.79912	0.899562	+	0.436794i	$0.143886\pi$
$954$	0	0
$955$	−22.0068	−0.712125
$956$	0	0
$957$	54.9090	1.77496
$958$	0	0
$959$	22.2443	0.718306
$960$	0	0
$961$	40.2494	1.29837
$962$	0	0
$963$	−2.96544	−0.0955600
$964$	0	0
$965$	6.46062	0.207975
$966$	0	0
$967$	−18.1787	−0.584589	−0.292294	−	0.956328i	$-0.594419\pi$
$967$	−18.1787	−0.584589	−0.292294	+	0.956328i	$0.594419\pi$
$968$	0	0
$969$	−97.7739	−3.14095
$970$	0	0
$971$	−42.7938	−1.37332	−0.686660	−	0.726979i	$-0.740924\pi$
$971$	−42.7938	−1.37332	−0.686660	+	0.726979i	$0.740924\pi$
$972$	0	0
$973$	−102.233	−3.27743
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.7715	0.888488	0.444244	−	0.895906i	$-0.353472\pi$
$977$	27.7715	0.888488	0.444244	+	0.895906i	$0.353472\pi$
$978$	0	0
$979$	7.71480	0.246566
$980$	0	0
$981$	−5.19406	−0.165834
$982$	0	0
$983$	26.1593	0.834351	0.417176	−	0.908826i	$-0.363020\pi$
$983$	26.1593	0.834351	0.417176	+	0.908826i	$0.363020\pi$
$984$	0	0
$985$	19.2295	0.612701
$986$	0	0
$987$	93.5588	2.97801
$988$	0	0
$989$	−25.6913	−0.816936
$990$	0	0
$991$	50.2217	1.59535	0.797673	−	0.603090i	$-0.206064\pi$
$991$	50.2217	1.59535	0.797673	+	0.603090i	$0.206064\pi$
$992$	0	0
$993$	−55.0459	−1.74683
$994$	0	0
$995$	−16.3145	−0.517205
$996$	0	0
$997$	44.5985	1.41245	0.706224	−	0.707989i	$-0.250397\pi$
$997$	44.5985	1.41245	0.706224	+	0.707989i	$0.250397\pi$
$998$	0	0
$999$	−6.51599	−0.206157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.r.1.2	✓	9
13.5	odd	4		3380.2.f.j.3041.4		18
13.8	odd	4		3380.2.f.j.3041.3		18
13.12	even	2		3380.2.a.s.1.2	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.2	✓	9	1.1	even	1	trivial
3380.2.a.s.1.2	yes	9	13.12	even	2
3380.2.f.j.3041.3		18	13.8	odd	4
3380.2.f.j.3041.4		18	13.5	odd	4

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-2.52382 q^{3} -1.00000 q^{5} +4.54727 q^{7} +3.36968 q^{9} +O(q^{10})\)	\(q-2.52382 q^{3} -1.00000 q^{5} +4.54727 q^{7} +3.36968 q^{9} +2.52395 q^{11} +2.52382 q^{15} +5.14630 q^{17} +7.52781 q^{19} -11.4765 q^{21} +2.63662 q^{23} +1.00000 q^{25} -0.933014 q^{27} -8.61995 q^{29} +8.44094 q^{31} -6.37000 q^{33} -4.54727 q^{35} +6.98381 q^{37} +2.66091 q^{41} -9.74401 q^{43} -3.36968 q^{45} -8.15221 q^{47} +13.6776 q^{49} -12.9884 q^{51} -1.43808 q^{53} -2.52395 q^{55} -18.9989 q^{57} +0.0833281 q^{59} -0.818691 q^{61} +15.3229 q^{63} +5.08761 q^{67} -6.65437 q^{69} -7.78118 q^{71} +0.148454 q^{73} -2.52382 q^{75} +11.4771 q^{77} +7.18836 q^{79} -7.75429 q^{81} -16.9198 q^{83} -5.14630 q^{85} +21.7552 q^{87} +3.05664 q^{89} -21.3034 q^{93} -7.52781 q^{95} +8.27717 q^{97} +8.50490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

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