LMFDB - Embedded newform 3380.2.a.l.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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q-0.445042 q^{3} +1.00000 q^{5} +0.445042 q^{7} -2.80194 q^{9} +O(q^{10})$	$q-0.445042 q^{3} +1.00000 q^{5} +0.445042 q^{7} -2.80194 q^{9} -3.24698 q^{11} -0.445042 q^{15} +2.93900 q^{17} +3.04892 q^{19} -0.198062 q^{21} -2.80194 q^{23} +1.00000 q^{25} +2.58211 q^{27} +2.15883 q^{29} -0.939001 q^{31} +1.44504 q^{33} +0.445042 q^{35} -3.00000 q^{37} -4.58211 q^{41} +4.63102 q^{43} -2.80194 q^{45} +7.39373 q^{47} -6.80194 q^{49} -1.30798 q^{51} -13.0586 q^{53} -3.24698 q^{55} -1.35690 q^{57} +3.17629 q^{59} -11.2470 q^{61} -1.24698 q^{63} -11.9879 q^{67} +1.24698 q^{69} -3.40581 q^{71} +0.317667 q^{73} -0.445042 q^{75} -1.44504 q^{77} -8.18060 q^{79} +7.25667 q^{81} +9.62565 q^{83} +2.93900 q^{85} -0.960771 q^{87} -5.04354 q^{89} +0.417895 q^{93} +3.04892 q^{95} -13.8605 q^{97} +9.09783 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9}+O(q^{10}) $ 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9	$ 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9} - 5 q^{11} - q^{15} - q^{17} - 5 q^{21} - 4 q^{23} + 3 q^{25} + 2 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} + q^{35} - 9 q^{37} - 8 q^{41} - q^{43} - 4 q^{45} - 10 q^{47} - 16 q^{49} - 9 q^{51} - 8 q^{53} - 5 q^{55} + 17 q^{59} - 29 q^{61} + q^{63} - 17 q^{67} - q^{69} + 3 q^{71} - 16 q^{73} - q^{75} - 4 q^{77} - 13 q^{79} - 5 q^{81} + 17 q^{83} - q^{85} + 10 q^{87} - 9 q^{89} + 7 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) $ 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9 - 5 * q^11 - q^15 - q^17 - 5 * q^21 - 4 * q^23 + 3 * q^25 + 2 * q^27 - 2 * q^29 + 7 * q^31 + 4 * q^33 + q^35 - 9 * q^37 - 8 * q^41 - q^43 - 4 * q^45 - 10 * q^47 - 16 * q^49 - 9 * q^51 - 8 * q^53 - 5 * q^55 + 17 * q^59 - 29 * q^61 + q^63 - 17 * q^67 - q^69 + 3 * q^71 - 16 * q^73 - q^75 - 4 * q^77 - 13 * q^79 - 5 * q^81 + 17 * q^83 - q^85 + 10 * q^87 - 9 * q^89 + 7 * q^93 - 6 * q^97 + 9 * 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
$3$	−0.445042	−0.256945	−0.128473	−	0.991713i	$-0.541007\pi$
$3$	−0.445042	−0.256945	−0.128473	+	0.991713i	$0.541007\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.445042	0.168210	0.0841050	−	0.996457i	$-0.473197\pi$
$7$	0.445042	0.168210	0.0841050	+	0.996457i	$0.473197\pi$
$8$	0	0
$9$	−2.80194	−0.933979
$10$	0	0
$11$	−3.24698	−0.979001	−0.489501	−	0.872003i	$-0.662821\pi$
$11$	−3.24698	−0.979001	−0.489501	+	0.872003i	$0.662821\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.445042	−0.114909
$16$	0	0
$17$	2.93900	0.712812	0.356406	−	0.934331i	$-0.384002\pi$
$17$	2.93900	0.712812	0.356406	+	0.934331i	$0.384002\pi$
$18$	0	0
$19$	3.04892	0.699470	0.349735	−	0.936849i	$-0.386272\pi$
$19$	3.04892	0.699470	0.349735	+	0.936849i	$0.386272\pi$
$20$	0	0
$21$	−0.198062	−0.0432207
$22$	0	0
$23$	−2.80194	−0.584244	−0.292122	−	0.956381i	$-0.594361\pi$
$23$	−2.80194	−0.584244	−0.292122	+	0.956381i	$0.594361\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.58211	0.496926
$28$	0	0
$29$	2.15883	0.400885	0.200443	−	0.979705i	$-0.435762\pi$
$29$	2.15883	0.400885	0.200443	+	0.979705i	$0.435762\pi$
$30$	0	0
$31$	−0.939001	−0.168650	−0.0843248	−	0.996438i	$-0.526873\pi$
$31$	−0.939001	−0.168650	−0.0843248	+	0.996438i	$0.526873\pi$
$32$	0	0
$33$	1.44504	0.251550
$34$	0	0
$35$	0.445042	0.0752258
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.58211	−0.715605	−0.357802	−	0.933797i	$-0.616474\pi$
$41$	−4.58211	−0.715605	−0.357802	+	0.933797i	$0.616474\pi$
$42$	0	0
$43$	4.63102	0.706224	0.353112	−	0.935581i	$-0.385123\pi$
$43$	4.63102	0.706224	0.353112	+	0.935581i	$0.385123\pi$
$44$	0	0
$45$	−2.80194	−0.417688
$46$	0	0
$47$	7.39373	1.07849	0.539243	−	0.842150i	$-0.318710\pi$
$47$	7.39373	1.07849	0.539243	+	0.842150i	$0.318710\pi$
$48$	0	0
$49$	−6.80194	−0.971705
$50$	0	0
$51$	−1.30798	−0.183154
$52$	0	0
$53$	−13.0586	−1.79374	−0.896869	−	0.442297i	$-0.854164\pi$
$53$	−13.0586	−1.79374	−0.896869	+	0.442297i	$0.854164\pi$
$54$	0	0
$55$	−3.24698	−0.437823
$56$	0	0
$57$	−1.35690	−0.179725
$58$	0	0
$59$	3.17629	0.413518	0.206759	−	0.978392i	$-0.433708\pi$
$59$	3.17629	0.413518	0.206759	+	0.978392i	$0.433708\pi$
$60$	0	0
$61$	−11.2470	−1.44003	−0.720014	−	0.693959i	$-0.755865\pi$
$61$	−11.2470	−1.44003	−0.720014	+	0.693959i	$0.755865\pi$
$62$	0	0
$63$	−1.24698	−0.157105
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.9879	−1.46456	−0.732279	−	0.681005i	$-0.761543\pi$
$67$	−11.9879	−1.46456	−0.732279	+	0.681005i	$0.761543\pi$
$68$	0	0
$69$	1.24698	0.150119
$70$	0	0
$71$	−3.40581	−0.404196	−0.202098	−	0.979365i	$-0.564776\pi$
$71$	−3.40581	−0.404196	−0.202098	+	0.979365i	$0.564776\pi$
$72$	0	0
$73$	0.317667	0.0371801	0.0185901	−	0.999827i	$-0.494082\pi$
$73$	0.317667	0.0371801	0.0185901	+	0.999827i	$0.494082\pi$
$74$	0	0
$75$	−0.445042	−0.0513890
$76$	0	0
$77$	−1.44504	−0.164678
$78$	0	0
$79$	−8.18060	−0.920390	−0.460195	−	0.887818i	$-0.652220\pi$
$79$	−8.18060	−0.920390	−0.460195	+	0.887818i	$0.652220\pi$
$80$	0	0
$81$	7.25667	0.806296
$82$	0	0
$83$	9.62565	1.05655	0.528276	−	0.849073i	$-0.322839\pi$
$83$	9.62565	1.05655	0.528276	+	0.849073i	$0.322839\pi$
$84$	0	0
$85$	2.93900	0.318779
$86$	0	0
$87$	−0.960771	−0.103005
$88$	0	0
$89$	−5.04354	−0.534614	−0.267307	−	0.963611i	$-0.586134\pi$
$89$	−5.04354	−0.534614	−0.267307	+	0.963611i	$0.586134\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.417895	0.0433337
$94$	0	0
$95$	3.04892	0.312812
$96$	0	0
$97$	−13.8605	−1.40732	−0.703662	−	0.710534i	$-0.748453\pi$
$97$	−13.8605	−1.40732	−0.703662	+	0.710534i	$0.748453\pi$
$98$	0	0
$99$	9.09783	0.914367
$100$	0	0
$101$	−6.57002	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$101$	−6.57002	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$102$	0	0
$103$	−1.45712	−0.143575	−0.0717873	−	0.997420i	$-0.522870\pi$
$103$	−1.45712	−0.143575	−0.0717873	+	0.997420i	$0.522870\pi$
$104$	0	0
$105$	−0.198062	−0.0193289
$106$	0	0
$107$	−10.5646	−1.02132	−0.510661	−	0.859782i	$-0.670599\pi$
$107$	−10.5646	−1.02132	−0.510661	+	0.859782i	$0.670599\pi$
$108$	0	0
$109$	0.518122	0.0496271	0.0248136	−	0.999692i	$-0.492101\pi$
$109$	0.518122	0.0496271	0.0248136	+	0.999692i	$0.492101\pi$
$110$	0	0
$111$	1.33513	0.126725
$112$	0	0
$113$	−8.11529	−0.763423	−0.381711	−	0.924282i	$-0.624665\pi$
$113$	−8.11529	−0.763423	−0.381711	+	0.924282i	$0.624665\pi$
$114$	0	0
$115$	−2.80194	−0.261282
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.30798	0.119902
$120$	0	0
$121$	−0.457123	−0.0415567
$122$	0	0
$123$	2.03923	0.183871
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.20344	−0.550466	−0.275233	−	0.961378i	$-0.588755\pi$
$127$	−6.20344	−0.550466	−0.275233	+	0.961378i	$0.588755\pi$
$128$	0	0
$129$	−2.06100	−0.181461
$130$	0	0
$131$	−6.62565	−0.578885	−0.289443	−	0.957195i	$-0.593470\pi$
$131$	−6.62565	−0.578885	−0.289443	+	0.957195i	$0.593470\pi$
$132$	0	0
$133$	1.35690	0.117658
$134$	0	0
$135$	2.58211	0.222232
$136$	0	0
$137$	19.3327	1.65171	0.825853	−	0.563885i	$-0.190694\pi$
$137$	19.3327	1.65171	0.825853	+	0.563885i	$0.190694\pi$
$138$	0	0
$139$	−22.9191	−1.94398	−0.971988	−	0.235029i	$-0.924482\pi$
$139$	−22.9191	−1.94398	−0.971988	+	0.235029i	$0.924482\pi$
$140$	0	0
$141$	−3.29052	−0.277112
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.15883	0.179281
$146$	0	0
$147$	3.02715	0.249675
$148$	0	0
$149$	12.9638	1.06203	0.531016	−	0.847362i	$-0.321810\pi$
$149$	12.9638	1.06203	0.531016	+	0.847362i	$0.321810\pi$
$150$	0	0
$151$	15.2446	1.24059	0.620293	−	0.784370i	$-0.287014\pi$
$151$	15.2446	1.24059	0.620293	+	0.784370i	$0.287014\pi$
$152$	0	0
$153$	−8.23490	−0.665752
$154$	0	0
$155$	−0.939001	−0.0754224
$156$	0	0
$157$	1.19567	0.0954248	0.0477124	−	0.998861i	$-0.484807\pi$
$157$	1.19567	0.0954248	0.0477124	+	0.998861i	$0.484807\pi$
$158$	0	0
$159$	5.81163	0.460892
$160$	0	0
$161$	−1.24698	−0.0982758
$162$	0	0
$163$	19.8823	1.55730	0.778652	−	0.627457i	$-0.215904\pi$
$163$	19.8823	1.55730	0.778652	+	0.627457i	$0.215904\pi$
$164$	0	0
$165$	1.44504	0.112496
$166$	0	0
$167$	−19.0030	−1.47050	−0.735248	−	0.677799i	$-0.762934\pi$
$167$	−19.0030	−1.47050	−0.735248	+	0.677799i	$0.762934\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−8.54288	−0.653290
$172$	0	0
$173$	8.24459	0.626824	0.313412	−	0.949617i	$-0.398528\pi$
$173$	8.24459	0.626824	0.313412	+	0.949617i	$0.398528\pi$
$174$	0	0
$175$	0.445042	0.0336420
$176$	0	0
$177$	−1.41358	−0.106251
$178$	0	0
$179$	17.5623	1.31266	0.656332	−	0.754472i	$-0.272107\pi$
$179$	17.5623	1.31266	0.656332	+	0.754472i	$0.272107\pi$
$180$	0	0
$181$	−12.5821	−0.935221	−0.467610	−	0.883935i	$-0.654885\pi$
$181$	−12.5821	−0.935221	−0.467610	+	0.883935i	$0.654885\pi$
$182$	0	0
$183$	5.00538	0.370008
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−9.54288	−0.697844
$188$	0	0
$189$	1.14914	0.0835880
$190$	0	0
$191$	−6.31096	−0.456645	−0.228323	−	0.973586i	$-0.573324\pi$
$191$	−6.31096	−0.456645	−0.228323	+	0.973586i	$0.573324\pi$
$192$	0	0
$193$	−0.356896	−0.0256899	−0.0128450	−	0.999918i	$-0.504089\pi$
$193$	−0.356896	−0.0256899	−0.0128450	+	0.999918i	$0.504089\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.219833	0.0156624	0.00783121	−	0.999969i	$-0.497507\pi$
$197$	0.219833	0.0156624	0.00783121	+	0.999969i	$0.497507\pi$
$198$	0	0
$199$	9.36121	0.663598	0.331799	−	0.943350i	$-0.392344\pi$
$199$	9.36121	0.663598	0.331799	+	0.943350i	$0.392344\pi$
$200$	0	0
$201$	5.33513	0.376311
$202$	0	0
$203$	0.960771	0.0674329
$204$	0	0
$205$	−4.58211	−0.320028
$206$	0	0
$207$	7.85086	0.545672
$208$	0	0
$209$	−9.89977	−0.684782
$210$	0	0
$211$	−10.8659	−0.748041	−0.374020	−	0.927420i	$-0.622021\pi$
$211$	−10.8659	−0.748041	−0.374020	+	0.927420i	$0.622021\pi$
$212$	0	0
$213$	1.51573	0.103856
$214$	0	0
$215$	4.63102	0.315833
$216$	0	0
$217$	−0.417895	−0.0283685
$218$	0	0
$219$	−0.141375	−0.00955325
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−20.1444	−1.34897	−0.674483	−	0.738290i	$-0.735633\pi$
$223$	−20.1444	−1.34897	−0.674483	+	0.738290i	$0.735633\pi$
$224$	0	0
$225$	−2.80194	−0.186796
$226$	0	0
$227$	−19.9215	−1.32224	−0.661120	−	0.750281i	$-0.729918\pi$
$227$	−19.9215	−1.32224	−0.661120	+	0.750281i	$0.729918\pi$
$228$	0	0
$229$	0.200455	0.0132465	0.00662323	−	0.999978i	$-0.497892\pi$
$229$	0.200455	0.0132465	0.00662323	+	0.999978i	$0.497892\pi$
$230$	0	0
$231$	0.643104	0.0423131
$232$	0	0
$233$	−11.7922	−0.772536	−0.386268	−	0.922387i	$-0.626236\pi$
$233$	−11.7922	−0.772536	−0.386268	+	0.922387i	$0.626236\pi$
$234$	0	0
$235$	7.39373	0.482314
$236$	0	0
$237$	3.64071	0.236490
$238$	0	0
$239$	18.4765	1.19515	0.597573	−	0.801815i	$-0.296132\pi$
$239$	18.4765	1.19515	0.597573	+	0.801815i	$0.296132\pi$
$240$	0	0
$241$	9.30021	0.599079	0.299540	−	0.954084i	$-0.403167\pi$
$241$	9.30021	0.599079	0.299540	+	0.954084i	$0.403167\pi$
$242$	0	0
$243$	−10.9758	−0.704100
$244$	0	0
$245$	−6.80194	−0.434560
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−4.28382	−0.271476
$250$	0	0
$251$	13.6015	0.858518	0.429259	−	0.903181i	$-0.358775\pi$
$251$	13.6015	0.858518	0.429259	+	0.903181i	$0.358775\pi$
$252$	0	0
$253$	9.09783	0.571976
$254$	0	0
$255$	−1.30798	−0.0819088
$256$	0	0
$257$	−30.2258	−1.88543	−0.942717	−	0.333594i	$-0.891739\pi$
$257$	−30.2258	−1.88543	−0.942717	+	0.333594i	$0.891739\pi$
$258$	0	0
$259$	−1.33513	−0.0829607
$260$	0	0
$261$	−6.04892	−0.374419
$262$	0	0
$263$	14.2959	0.881523	0.440761	−	0.897624i	$-0.354708\pi$
$263$	14.2959	0.881523	0.440761	+	0.897624i	$0.354708\pi$
$264$	0	0
$265$	−13.0586	−0.802184
$266$	0	0
$267$	2.24459	0.137366
$268$	0	0
$269$	−16.5526	−1.00923	−0.504614	−	0.863345i	$-0.668365\pi$
$269$	−16.5526	−1.00923	−0.504614	+	0.863345i	$0.668365\pi$
$270$	0	0
$271$	11.2174	0.681411	0.340705	−	0.940170i	$-0.389334\pi$
$271$	11.2174	0.681411	0.340705	+	0.940170i	$0.389334\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.24698	−0.195800
$276$	0	0
$277$	−17.8455	−1.07223	−0.536115	−	0.844145i	$-0.680109\pi$
$277$	−17.8455	−1.07223	−0.536115	+	0.844145i	$0.680109\pi$
$278$	0	0
$279$	2.63102	0.157515
$280$	0	0
$281$	−4.60925	−0.274965	−0.137482	−	0.990504i	$-0.543901\pi$
$281$	−4.60925	−0.274965	−0.137482	+	0.990504i	$0.543901\pi$
$282$	0	0
$283$	22.6015	1.34352	0.671759	−	0.740769i	$-0.265539\pi$
$283$	22.6015	1.34352	0.671759	+	0.740769i	$0.265539\pi$
$284$	0	0
$285$	−1.35690	−0.0803756
$286$	0	0
$287$	−2.03923	−0.120372
$288$	0	0
$289$	−8.36227	−0.491898
$290$	0	0
$291$	6.16852	0.361605
$292$	0	0
$293$	−27.8756	−1.62851	−0.814255	−	0.580507i	$-0.802854\pi$
$293$	−27.8756	−1.62851	−0.814255	+	0.580507i	$0.802854\pi$
$294$	0	0
$295$	3.17629	0.184931
$296$	0	0
$297$	−8.38404	−0.486492
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.06100	0.118794
$302$	0	0
$303$	2.92394	0.167976
$304$	0	0
$305$	−11.2470	−0.644000
$306$	0	0
$307$	−22.1317	−1.26312	−0.631561	−	0.775326i	$-0.717585\pi$
$307$	−22.1317	−1.26312	−0.631561	+	0.775326i	$0.717585\pi$
$308$	0	0
$309$	0.648481	0.0368908
$310$	0	0
$311$	7.86592	0.446035	0.223018	−	0.974814i	$-0.428409\pi$
$311$	7.86592	0.446035	0.223018	+	0.974814i	$0.428409\pi$
$312$	0	0
$313$	−20.7928	−1.17528	−0.587640	−	0.809122i	$-0.699943\pi$
$313$	−20.7928	−1.17528	−0.587640	+	0.809122i	$0.699943\pi$
$314$	0	0
$315$	−1.24698	−0.0702593
$316$	0	0
$317$	−16.7791	−0.942408	−0.471204	−	0.882024i	$-0.656180\pi$
$317$	−16.7791	−0.942408	−0.471204	+	0.882024i	$0.656180\pi$
$318$	0	0
$319$	−7.00969	−0.392467
$320$	0	0
$321$	4.70171	0.262424
$322$	0	0
$323$	8.96077	0.498591
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.230586	−0.0127514
$328$	0	0
$329$	3.29052	0.181412
$330$	0	0
$331$	−9.61356	−0.528409	−0.264205	−	0.964467i	$-0.585109\pi$
$331$	−9.61356	−0.528409	−0.264205	+	0.964467i	$0.585109\pi$
$332$	0	0
$333$	8.40581	0.460636
$334$	0	0
$335$	−11.9879	−0.654970
$336$	0	0
$337$	−6.81269	−0.371111	−0.185555	−	0.982634i	$-0.559408\pi$
$337$	−6.81269	−0.371111	−0.185555	+	0.982634i	$0.559408\pi$
$338$	0	0
$339$	3.61165	0.196158
$340$	0	0
$341$	3.04892	0.165108
$342$	0	0
$343$	−6.14244	−0.331661
$344$	0	0
$345$	1.24698	0.0671351
$346$	0	0
$347$	19.1468	1.02785	0.513926	−	0.857835i	$-0.328191\pi$
$347$	19.1468	1.02785	0.513926	+	0.857835i	$0.328191\pi$
$348$	0	0
$349$	20.7222	1.10923	0.554616	−	0.832107i	$-0.312865\pi$
$349$	20.7222	1.10923	0.554616	+	0.832107i	$0.312865\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.23191	−0.172017	−0.0860087	−	0.996294i	$-0.527411\pi$
$353$	−3.23191	−0.172017	−0.0860087	+	0.996294i	$0.527411\pi$
$354$	0	0
$355$	−3.40581	−0.180762
$356$	0	0
$357$	−0.582105	−0.0308083
$358$	0	0
$359$	27.6582	1.45974	0.729871	−	0.683585i	$-0.239580\pi$
$359$	27.6582	1.45974	0.729871	+	0.683585i	$0.239580\pi$
$360$	0	0
$361$	−9.70410	−0.510742
$362$	0	0
$363$	0.203439	0.0106778
$364$	0	0
$365$	0.317667	0.0166275
$366$	0	0
$367$	−36.5327	−1.90699	−0.953496	−	0.301405i	$-0.902544\pi$
$367$	−36.5327	−1.90699	−0.953496	+	0.301405i	$0.902544\pi$
$368$	0	0
$369$	12.8388	0.668360
$370$	0	0
$371$	−5.81163	−0.301725
$372$	0	0
$373$	−15.1371	−0.783767	−0.391884	−	0.920015i	$-0.628176\pi$
$373$	−15.1371	−0.783767	−0.391884	+	0.920015i	$0.628176\pi$
$374$	0	0
$375$	−0.445042	−0.0229819
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7.13169	−0.366330	−0.183165	−	0.983082i	$-0.558634\pi$
$379$	−7.13169	−0.366330	−0.183165	+	0.983082i	$0.558634\pi$
$380$	0	0
$381$	2.76079	0.141440
$382$	0	0
$383$	−7.05861	−0.360678	−0.180339	−	0.983605i	$-0.557719\pi$
$383$	−7.05861	−0.360678	−0.180339	+	0.983605i	$0.557719\pi$
$384$	0	0
$385$	−1.44504	−0.0736462
$386$	0	0
$387$	−12.9758	−0.659599
$388$	0	0
$389$	5.85623	0.296923	0.148461	−	0.988918i	$-0.452568\pi$
$389$	5.85623	0.296923	0.148461	+	0.988918i	$0.452568\pi$
$390$	0	0
$391$	−8.23490	−0.416457
$392$	0	0
$393$	2.94869	0.148742
$394$	0	0
$395$	−8.18060	−0.411611
$396$	0	0
$397$	26.5773	1.33388	0.666939	−	0.745113i	$-0.267604\pi$
$397$	26.5773	1.33388	0.666939	+	0.745113i	$0.267604\pi$
$398$	0	0
$399$	−0.603875	−0.0302316
$400$	0	0
$401$	−6.64609	−0.331890	−0.165945	−	0.986135i	$-0.553067\pi$
$401$	−6.64609	−0.331890	−0.165945	+	0.986135i	$0.553067\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.25667	0.360587
$406$	0	0
$407$	9.74094	0.482840
$408$	0	0
$409$	39.0170	1.92927	0.964633	−	0.263597i	$-0.0849090\pi$
$409$	39.0170	1.92927	0.964633	+	0.263597i	$0.0849090\pi$
$410$	0	0
$411$	−8.60388	−0.424398
$412$	0	0
$413$	1.41358	0.0695579
$414$	0	0
$415$	9.62565	0.472504
$416$	0	0
$417$	10.2000	0.499495
$418$	0	0
$419$	15.8944	0.776492	0.388246	−	0.921556i	$-0.373081\pi$
$419$	15.8944	0.776492	0.388246	+	0.921556i	$0.373081\pi$
$420$	0	0
$421$	11.8726	0.578636	0.289318	−	0.957233i	$-0.406571\pi$
$421$	11.8726	0.578636	0.289318	+	0.957233i	$0.406571\pi$
$422$	0	0
$423$	−20.7168	−1.00728
$424$	0	0
$425$	2.93900	0.142562
$426$	0	0
$427$	−5.00538	−0.242227
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.6093	0.559198	0.279599	−	0.960117i	$-0.409798\pi$
$431$	11.6093	0.559198	0.279599	+	0.960117i	$0.409798\pi$
$432$	0	0
$433$	−18.6528	−0.896396	−0.448198	−	0.893934i	$-0.647934\pi$
$433$	−18.6528	−0.896396	−0.448198	+	0.893934i	$0.647934\pi$
$434$	0	0
$435$	−0.960771	−0.0460655
$436$	0	0
$437$	−8.54288	−0.408661
$438$	0	0
$439$	37.3454	1.78240	0.891199	−	0.453612i	$-0.149865\pi$
$439$	37.3454	1.78240	0.891199	+	0.453612i	$0.149865\pi$
$440$	0	0
$441$	19.0586	0.907553
$442$	0	0
$443$	22.9323	1.08955	0.544773	−	0.838583i	$-0.316616\pi$
$443$	22.9323	1.08955	0.544773	+	0.838583i	$0.316616\pi$
$444$	0	0
$445$	−5.04354	−0.239087
$446$	0	0
$447$	−5.76941	−0.272884
$448$	0	0
$449$	−31.3497	−1.47948	−0.739742	−	0.672890i	$-0.765053\pi$
$449$	−31.3497	−1.47948	−0.739742	+	0.672890i	$0.765053\pi$
$450$	0	0
$451$	14.8780	0.700578
$452$	0	0
$453$	−6.78448	−0.318763
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.7633	1.20516	0.602578	−	0.798060i	$-0.294140\pi$
$457$	25.7633	1.20516	0.602578	+	0.798060i	$0.294140\pi$
$458$	0	0
$459$	7.58881	0.354215
$460$	0	0
$461$	23.1487	1.07814	0.539071	−	0.842261i	$-0.318776\pi$
$461$	23.1487	1.07814	0.539071	+	0.842261i	$0.318776\pi$
$462$	0	0
$463$	14.8616	0.690678	0.345339	−	0.938478i	$-0.387764\pi$
$463$	14.8616	0.690678	0.345339	+	0.938478i	$0.387764\pi$
$464$	0	0
$465$	0.417895	0.0193794
$466$	0	0
$467$	10.8925	0.504044	0.252022	−	0.967722i	$-0.418905\pi$
$467$	10.8925	0.504044	0.252022	+	0.967722i	$0.418905\pi$
$468$	0	0
$469$	−5.33513	−0.246353
$470$	0	0
$471$	−0.532123	−0.0245189
$472$	0	0
$473$	−15.0368	−0.691394
$474$	0	0
$475$	3.04892	0.139894
$476$	0	0
$477$	36.5894	1.67531
$478$	0	0
$479$	23.3817	1.06833	0.534167	−	0.845379i	$-0.320625\pi$
$479$	23.3817	1.06833	0.534167	+	0.845379i	$0.320625\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0.554958	0.0252515
$484$	0	0
$485$	−13.8605	−0.629375
$486$	0	0
$487$	−39.2573	−1.77892	−0.889458	−	0.457017i	$-0.848918\pi$
$487$	−39.2573	−1.77892	−0.889458	+	0.457017i	$0.848918\pi$
$488$	0	0
$489$	−8.84846	−0.400141
$490$	0	0
$491$	36.8810	1.66442	0.832208	−	0.554464i	$-0.187077\pi$
$491$	36.8810	1.66442	0.832208	+	0.554464i	$0.187077\pi$
$492$	0	0
$493$	6.34481	0.285756
$494$	0	0
$495$	9.09783	0.408917
$496$	0	0
$497$	−1.51573	−0.0679898
$498$	0	0
$499$	23.1360	1.03571	0.517855	−	0.855469i	$-0.326731\pi$
$499$	23.1360	1.03571	0.517855	+	0.855469i	$0.326731\pi$
$500$	0	0
$501$	8.45712	0.377836
$502$	0	0
$503$	12.1172	0.540280	0.270140	−	0.962821i	$-0.412930\pi$
$503$	12.1172	0.540280	0.270140	+	0.962821i	$0.412930\pi$
$504$	0	0
$505$	−6.57002	−0.292362
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.7560	1.27459	0.637294	−	0.770621i	$-0.280054\pi$
$509$	28.7560	1.27459	0.637294	+	0.770621i	$0.280054\pi$
$510$	0	0
$511$	0.141375	0.00625407
$512$	0	0
$513$	7.87263	0.347585
$514$	0	0
$515$	−1.45712	−0.0642085
$516$	0	0
$517$	−24.0073	−1.05584
$518$	0	0
$519$	−3.66919	−0.161059
$520$	0	0
$521$	7.24937	0.317601	0.158800	−	0.987311i	$-0.449237\pi$
$521$	7.24937	0.317601	0.158800	+	0.987311i	$0.449237\pi$
$522$	0	0
$523$	2.75004	0.120251	0.0601253	−	0.998191i	$-0.480850\pi$
$523$	2.75004	0.120251	0.0601253	+	0.998191i	$0.480850\pi$
$524$	0	0
$525$	−0.198062	−0.00864415
$526$	0	0
$527$	−2.75973	−0.120216
$528$	0	0
$529$	−15.1491	−0.658658
$530$	0	0
$531$	−8.89977	−0.386217
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−10.5646	−0.456750
$536$	0	0
$537$	−7.81594	−0.337283
$538$	0	0
$539$	22.0858	0.951301
$540$	0	0
$541$	−43.8286	−1.88434	−0.942170	−	0.335137i	$-0.891218\pi$
$541$	−43.8286	−1.88434	−0.942170	+	0.335137i	$0.891218\pi$
$542$	0	0
$543$	5.59956	0.240300
$544$	0	0
$545$	0.518122	0.0221939
$546$	0	0
$547$	41.8015	1.78730	0.893651	−	0.448763i	$-0.148135\pi$
$547$	41.8015	1.78730	0.893651	+	0.448763i	$0.148135\pi$
$548$	0	0
$549$	31.5133	1.34496
$550$	0	0
$551$	6.58211	0.280407
$552$	0	0
$553$	−3.64071	−0.154819
$554$	0	0
$555$	1.33513	0.0566729
$556$	0	0
$557$	−19.0446	−0.806946	−0.403473	−	0.914992i	$-0.632197\pi$
$557$	−19.0446	−0.806946	−0.403473	+	0.914992i	$0.632197\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	4.24698	0.179308
$562$	0	0
$563$	−5.92453	−0.249689	−0.124845	−	0.992176i	$-0.539843\pi$
$563$	−5.92453	−0.249689	−0.124845	+	0.992176i	$0.539843\pi$
$564$	0	0
$565$	−8.11529	−0.341413
$566$	0	0
$567$	3.22952	0.135627
$568$	0	0
$569$	9.23059	0.386966	0.193483	−	0.981104i	$-0.438022\pi$
$569$	9.23059	0.386966	0.193483	+	0.981104i	$0.438022\pi$
$570$	0	0
$571$	−34.1081	−1.42738	−0.713690	−	0.700462i	$-0.752977\pi$
$571$	−34.1081	−1.42738	−0.713690	+	0.700462i	$0.752977\pi$
$572$	0	0
$573$	2.80864	0.117333
$574$	0	0
$575$	−2.80194	−0.116849
$576$	0	0
$577$	−25.8595	−1.07654	−0.538272	−	0.842771i	$-0.680923\pi$
$577$	−25.8595	−1.07654	−0.538272	+	0.842771i	$0.680923\pi$
$578$	0	0
$579$	0.158834	0.00660090
$580$	0	0
$581$	4.28382	0.177723
$582$	0	0
$583$	42.4010	1.75607
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.54766	−0.352800	−0.176400	−	0.984319i	$-0.556445\pi$
$587$	−8.54766	−0.352800	−0.176400	+	0.984319i	$0.556445\pi$
$588$	0	0
$589$	−2.86294	−0.117965
$590$	0	0
$591$	−0.0978347	−0.00402438
$592$	0	0
$593$	−12.7885	−0.525162	−0.262581	−	0.964910i	$-0.584574\pi$
$593$	−12.7885	−0.525162	−0.262581	+	0.964910i	$0.584574\pi$
$594$	0	0
$595$	1.30798	0.0536219
$596$	0	0
$597$	−4.16613	−0.170508
$598$	0	0
$599$	−9.02236	−0.368644	−0.184322	−	0.982866i	$-0.559009\pi$
$599$	−9.02236	−0.368644	−0.184322	+	0.982866i	$0.559009\pi$
$600$	0	0
$601$	15.0737	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$601$	15.0737	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$602$	0	0
$603$	33.5894	1.36787
$604$	0	0
$605$	−0.457123	−0.0185847
$606$	0	0
$607$	44.0863	1.78941	0.894705	−	0.446658i	$-0.147386\pi$
$607$	44.0863	1.78941	0.894705	+	0.446658i	$0.147386\pi$
$608$	0	0
$609$	−0.427583	−0.0173266
$610$	0	0
$611$	0	0
$612$	0	0
$613$	33.9135	1.36975	0.684877	−	0.728659i	$-0.259856\pi$
$613$	33.9135	1.36975	0.684877	+	0.728659i	$0.259856\pi$
$614$	0	0
$615$	2.03923	0.0822296
$616$	0	0
$617$	−32.5827	−1.31173	−0.655865	−	0.754878i	$-0.727696\pi$
$617$	−32.5827	−1.31173	−0.655865	+	0.754878i	$0.727696\pi$
$618$	0	0
$619$	−20.0713	−0.806733	−0.403366	−	0.915039i	$-0.632160\pi$
$619$	−20.0713	−0.806733	−0.403366	+	0.915039i	$0.632160\pi$
$620$	0	0
$621$	−7.23490	−0.290326
$622$	0	0
$623$	−2.24459	−0.0899275
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.40581	0.175951
$628$	0	0
$629$	−8.81700	−0.351557
$630$	0	0
$631$	−27.4494	−1.09274	−0.546371	−	0.837543i	$-0.683991\pi$
$631$	−27.4494	−1.09274	−0.546371	+	0.837543i	$0.683991\pi$
$632$	0	0
$633$	4.83579	0.192205
$634$	0	0
$635$	−6.20344	−0.246176
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.54288	0.377510
$640$	0	0
$641$	−29.1758	−1.15238	−0.576188	−	0.817317i	$-0.695460\pi$
$641$	−29.1758	−1.15238	−0.576188	+	0.817317i	$0.695460\pi$
$642$	0	0
$643$	41.1511	1.62284	0.811420	−	0.584464i	$-0.198695\pi$
$643$	41.1511	1.62284	0.811420	+	0.584464i	$0.198695\pi$
$644$	0	0
$645$	−2.06100	−0.0811518
$646$	0	0
$647$	5.59850	0.220100	0.110050	−	0.993926i	$-0.464899\pi$
$647$	5.59850	0.220100	0.110050	+	0.993926i	$0.464899\pi$
$648$	0	0
$649$	−10.3134	−0.404835
$650$	0	0
$651$	0.185981	0.00728916
$652$	0	0
$653$	31.4359	1.23018	0.615092	−	0.788456i	$-0.289119\pi$
$653$	31.4359	1.23018	0.615092	+	0.788456i	$0.289119\pi$
$654$	0	0
$655$	−6.62565	−0.258885
$656$	0	0
$657$	−0.890084	−0.0347255
$658$	0	0
$659$	18.7006	0.728474	0.364237	−	0.931306i	$-0.381330\pi$
$659$	18.7006	0.728474	0.364237	+	0.931306i	$0.381330\pi$
$660$	0	0
$661$	46.5623	1.81106	0.905531	−	0.424280i	$-0.139473\pi$
$661$	46.5623	1.81106	0.905531	+	0.424280i	$0.139473\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.35690	0.0526182
$666$	0	0
$667$	−6.04892	−0.234215
$668$	0	0
$669$	8.96508	0.346610
$670$	0	0
$671$	36.5187	1.40979
$672$	0	0
$673$	27.8176	1.07229	0.536145	−	0.844126i	$-0.319880\pi$
$673$	27.8176	1.07229	0.536145	+	0.844126i	$0.319880\pi$
$674$	0	0
$675$	2.58211	0.0993853
$676$	0	0
$677$	33.1366	1.27354	0.636771	−	0.771053i	$-0.280270\pi$
$677$	33.1366	1.27354	0.636771	+	0.771053i	$0.280270\pi$
$678$	0	0
$679$	−6.16852	−0.236726
$680$	0	0
$681$	8.86592	0.339743
$682$	0	0
$683$	−14.8780	−0.569291	−0.284645	−	0.958633i	$-0.591876\pi$
$683$	−14.8780	−0.569291	−0.284645	+	0.958633i	$0.591876\pi$
$684$	0	0
$685$	19.3327	0.738666
$686$	0	0
$687$	−0.0892109	−0.00340361
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−4.34614	−0.165335	−0.0826675	−	0.996577i	$-0.526344\pi$
$691$	−4.34614	−0.165335	−0.0826675	+	0.996577i	$0.526344\pi$
$692$	0	0
$693$	4.04892	0.153806
$694$	0	0
$695$	−22.9191	−0.869373
$696$	0	0
$697$	−13.4668	−0.510092
$698$	0	0
$699$	5.24804	0.198499
$700$	0	0
$701$	43.8049	1.65449	0.827245	−	0.561842i	$-0.189907\pi$
$701$	43.8049	1.65449	0.827245	+	0.561842i	$0.189907\pi$
$702$	0	0
$703$	−9.14675	−0.344976
$704$	0	0
$705$	−3.29052	−0.123928
$706$	0	0
$707$	−2.92394	−0.109966
$708$	0	0
$709$	−25.9963	−0.976311	−0.488155	−	0.872757i	$-0.662330\pi$
$709$	−25.9963	−0.976311	−0.488155	+	0.872757i	$0.662330\pi$
$710$	0	0
$711$	22.9215	0.859625
$712$	0	0
$713$	2.63102	0.0985326
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.22282	−0.307087
$718$	0	0
$719$	−25.0941	−0.935853	−0.467926	−	0.883768i	$-0.654999\pi$
$719$	−25.0941	−0.935853	−0.467926	+	0.883768i	$0.654999\pi$
$720$	0	0
$721$	−0.648481	−0.0241507
$722$	0	0
$723$	−4.13898	−0.153930
$724$	0	0
$725$	2.15883	0.0801771
$726$	0	0
$727$	3.08516	0.114422	0.0572112	−	0.998362i	$-0.481779\pi$
$727$	3.08516	0.114422	0.0572112	+	0.998362i	$0.481779\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	0	0
$731$	13.6106	0.503405
$732$	0	0
$733$	−25.6558	−0.947618	−0.473809	−	0.880628i	$-0.657121\pi$
$733$	−25.6558	−0.947618	−0.473809	+	0.880628i	$0.657121\pi$
$734$	0	0
$735$	3.02715	0.111658
$736$	0	0
$737$	38.9245	1.43380
$738$	0	0
$739$	27.7681	1.02147	0.510733	−	0.859740i	$-0.329374\pi$
$739$	27.7681	1.02147	0.510733	+	0.859740i	$0.329374\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.64609	−0.280508	−0.140254	−	0.990116i	$-0.544792\pi$
$743$	−7.64609	−0.280508	−0.140254	+	0.990116i	$0.544792\pi$
$744$	0	0
$745$	12.9638	0.474955
$746$	0	0
$747$	−26.9705	−0.986798
$748$	0	0
$749$	−4.70171	−0.171797
$750$	0	0
$751$	25.2825	0.922571	0.461286	−	0.887252i	$-0.347388\pi$
$751$	25.2825	0.922571	0.461286	+	0.887252i	$0.347388\pi$
$752$	0	0
$753$	−6.05323	−0.220592
$754$	0	0
$755$	15.2446	0.554807
$756$	0	0
$757$	−52.6034	−1.91190	−0.955952	−	0.293524i	$-0.905172\pi$
$757$	−52.6034	−1.91190	−0.955952	+	0.293524i	$0.905172\pi$
$758$	0	0
$759$	−4.04892	−0.146966
$760$	0	0
$761$	−9.36526	−0.339490	−0.169745	−	0.985488i	$-0.554294\pi$
$761$	−9.36526	−0.339490	−0.169745	+	0.985488i	$0.554294\pi$
$762$	0	0
$763$	0.230586	0.00834778
$764$	0	0
$765$	−8.23490	−0.297733
$766$	0	0
$767$	0	0
$768$	0	0
$769$	21.3392	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$769$	21.3392	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$770$	0	0
$771$	13.4517	0.484453
$772$	0	0
$773$	7.80684	0.280793	0.140396	−	0.990095i	$-0.455162\pi$
$773$	7.80684	0.280793	0.140396	+	0.990095i	$0.455162\pi$
$774$	0	0
$775$	−0.939001	−0.0337299
$776$	0	0
$777$	0.594187	0.0213163
$778$	0	0
$779$	−13.9705	−0.500544
$780$	0	0
$781$	11.0586	0.395708
$782$	0	0
$783$	5.57434	0.199210
$784$	0	0
$785$	1.19567	0.0426753
$786$	0	0
$787$	3.83399	0.136667	0.0683335	−	0.997663i	$-0.478232\pi$
$787$	3.83399	0.136667	0.0683335	+	0.997663i	$0.478232\pi$
$788$	0	0
$789$	−6.36227	−0.226503
$790$	0	0
$791$	−3.61165	−0.128415
$792$	0	0
$793$	0	0
$794$	0	0
$795$	5.81163	0.206117
$796$	0	0
$797$	−38.1704	−1.35207	−0.676033	−	0.736871i	$-0.736302\pi$
$797$	−38.1704	−1.35207	−0.676033	+	0.736871i	$0.736302\pi$
$798$	0	0
$799$	21.7302	0.768759
$800$	0	0
$801$	14.1317	0.499319
$802$	0	0
$803$	−1.03146	−0.0363994
$804$	0	0
$805$	−1.24698	−0.0439503
$806$	0	0
$807$	7.36658	0.259316
$808$	0	0
$809$	49.9439	1.75593	0.877967	−	0.478721i	$-0.158899\pi$
$809$	49.9439	1.75593	0.877967	+	0.478721i	$0.158899\pi$
$810$	0	0
$811$	−38.3381	−1.34623	−0.673117	−	0.739536i	$-0.735045\pi$
$811$	−38.3381	−1.34623	−0.673117	+	0.739536i	$0.735045\pi$
$812$	0	0
$813$	−4.99223	−0.175085
$814$	0	0
$815$	19.8823	0.696447
$816$	0	0
$817$	14.1196	0.493982
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.5308	0.995732	0.497866	−	0.867254i	$-0.334117\pi$
$821$	28.5308	0.995732	0.497866	+	0.867254i	$0.334117\pi$
$822$	0	0
$823$	−18.8006	−0.655348	−0.327674	−	0.944791i	$-0.606265\pi$
$823$	−18.8006	−0.655348	−0.327674	+	0.944791i	$0.606265\pi$
$824$	0	0
$825$	1.44504	0.0503099
$826$	0	0
$827$	−1.96748	−0.0684158	−0.0342079	−	0.999415i	$-0.510891\pi$
$827$	−1.96748	−0.0684158	−0.0342079	+	0.999415i	$0.510891\pi$
$828$	0	0
$829$	20.5536	0.713857	0.356929	−	0.934132i	$-0.383824\pi$
$829$	20.5536	0.713857	0.356929	+	0.934132i	$0.383824\pi$
$830$	0	0
$831$	7.94198	0.275504
$832$	0	0
$833$	−19.9909	−0.692644
$834$	0	0
$835$	−19.0030	−0.657625
$836$	0	0
$837$	−2.42460	−0.0838064
$838$	0	0
$839$	−46.5585	−1.60738	−0.803690	−	0.595049i	$-0.797133\pi$
$839$	−46.5585	−1.60738	−0.803690	+	0.595049i	$0.797133\pi$
$840$	0	0
$841$	−24.3394	−0.839291
$842$	0	0
$843$	2.05131	0.0706509
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.203439	−0.00699025
$848$	0	0
$849$	−10.0586	−0.345210
$850$	0	0
$851$	8.40581	0.288148
$852$	0	0
$853$	−37.7622	−1.29295	−0.646477	−	0.762933i	$-0.723758\pi$
$853$	−37.7622	−1.29295	−0.646477	+	0.762933i	$0.723758\pi$
$854$	0	0
$855$	−8.54288	−0.292160
$856$	0	0
$857$	43.1487	1.47393	0.736965	−	0.675931i	$-0.236258\pi$
$857$	43.1487	1.47393	0.736965	+	0.675931i	$0.236258\pi$
$858$	0	0
$859$	−14.8025	−0.505056	−0.252528	−	0.967590i	$-0.581262\pi$
$859$	−14.8025	−0.505056	−0.252528	+	0.967590i	$0.581262\pi$
$860$	0	0
$861$	0.907542	0.0309290
$862$	0	0
$863$	49.0723	1.67044	0.835221	−	0.549914i	$-0.185340\pi$
$863$	49.0723	1.67044	0.835221	+	0.549914i	$0.185340\pi$
$864$	0	0
$865$	8.24459	0.280324
$866$	0	0
$867$	3.72156	0.126391
$868$	0	0
$869$	26.5623	0.901063
$870$	0	0
$871$	0	0
$872$	0	0
$873$	38.8364	1.31441
$874$	0	0
$875$	0.445042	0.0150452
$876$	0	0
$877$	−24.4470	−0.825515	−0.412758	−	0.910841i	$-0.635434\pi$
$877$	−24.4470	−0.825515	−0.412758	+	0.910841i	$0.635434\pi$
$878$	0	0
$879$	12.4058	0.418438
$880$	0	0
$881$	−21.6974	−0.731004	−0.365502	−	0.930811i	$-0.619103\pi$
$881$	−21.6974	−0.731004	−0.365502	+	0.930811i	$0.619103\pi$
$882$	0	0
$883$	26.4711	0.890824	0.445412	−	0.895326i	$-0.353057\pi$
$883$	26.4711	0.890824	0.445412	+	0.895326i	$0.353057\pi$
$884$	0	0
$885$	−1.41358	−0.0475171
$886$	0	0
$887$	−38.3631	−1.28811	−0.644054	−	0.764980i	$-0.722749\pi$
$887$	−38.3631	−1.28811	−0.644054	+	0.764980i	$0.722749\pi$
$888$	0	0
$889$	−2.76079	−0.0925939
$890$	0	0
$891$	−23.5623	−0.789365
$892$	0	0
$893$	22.5429	0.754369
$894$	0	0
$895$	17.5623	0.587041
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.02715	−0.0676091
$900$	0	0
$901$	−38.3793	−1.27860
$902$	0	0
$903$	−0.917231	−0.0305235
$904$	0	0
$905$	−12.5821	−0.418243
$906$	0	0
$907$	32.3655	1.07468	0.537340	−	0.843366i	$-0.319429\pi$
$907$	32.3655	1.07468	0.537340	+	0.843366i	$0.319429\pi$
$908$	0	0
$909$	18.4088	0.610581
$910$	0	0
$911$	25.3927	0.841297	0.420648	−	0.907224i	$-0.361803\pi$
$911$	25.3927	0.841297	0.420648	+	0.907224i	$0.361803\pi$
$912$	0	0
$913$	−31.2543	−1.03437
$914$	0	0
$915$	5.00538	0.165473
$916$	0	0
$917$	−2.94869	−0.0973743
$918$	0	0
$919$	−36.3720	−1.19980	−0.599900	−	0.800075i	$-0.704793\pi$
$919$	−36.3720	−1.19980	−0.599900	+	0.800075i	$0.704793\pi$
$920$	0	0
$921$	9.84953	0.324553
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	0	0
$927$	4.08277	0.134096
$928$	0	0
$929$	59.1202	1.93967	0.969835	−	0.243761i	$-0.0783812\pi$
$929$	59.1202	1.93967	0.969835	+	0.243761i	$0.0783812\pi$
$930$	0	0
$931$	−20.7385	−0.679678
$932$	0	0
$933$	−3.50066	−0.114607
$934$	0	0
$935$	−9.54288	−0.312085
$936$	0	0
$937$	19.1758	0.626447	0.313223	−	0.949679i	$-0.398591\pi$
$937$	19.1758	0.626447	0.313223	+	0.949679i	$0.398591\pi$
$938$	0	0
$939$	9.25368	0.301983
$940$	0	0
$941$	−26.6276	−0.868034	−0.434017	−	0.900905i	$-0.642904\pi$
$941$	−26.6276	−0.868034	−0.434017	+	0.900905i	$0.642904\pi$
$942$	0	0
$943$	12.8388	0.418088
$944$	0	0
$945$	1.14914	0.0373817
$946$	0	0
$947$	−34.9154	−1.13460	−0.567299	−	0.823512i	$-0.692012\pi$
$947$	−34.9154	−1.13460	−0.567299	+	0.823512i	$0.692012\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	7.46740	0.242147
$952$	0	0
$953$	32.9989	1.06894	0.534470	−	0.845187i	$-0.320511\pi$
$953$	32.9989	1.06894	0.534470	+	0.845187i	$0.320511\pi$
$954$	0	0
$955$	−6.31096	−0.204218
$956$	0	0
$957$	3.11960	0.100843
$958$	0	0
$959$	8.60388	0.277834
$960$	0	0
$961$	−30.1183	−0.971557
$962$	0	0
$963$	29.6015	0.953894
$964$	0	0
$965$	−0.356896	−0.0114889
$966$	0	0
$967$	−24.5996	−0.791069	−0.395534	−	0.918451i	$-0.629441\pi$
$967$	−24.5996	−0.791069	−0.395534	+	0.918451i	$0.629441\pi$
$968$	0	0
$969$	−3.98792	−0.128110
$970$	0	0
$971$	13.9046	0.446219	0.223109	−	0.974793i	$-0.428379\pi$
$971$	13.9046	0.446219	0.223109	+	0.974793i	$0.428379\pi$
$972$	0	0
$973$	−10.2000	−0.326996
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.00537681	0.000172019 0	8.60097e−5	−	1.00000i	$-0.499973\pi$
$977$	0.00537681	0.000172019 0	8.60097e−5		1.00000i	$0.499973\pi$
$978$	0	0
$979$	16.3763	0.523388
$980$	0	0
$981$	−1.45175	−0.0463507
$982$	0	0
$983$	−6.69692	−0.213599	−0.106799	−	0.994281i	$-0.534060\pi$
$983$	−6.69692	−0.213599	−0.106799	+	0.994281i	$0.534060\pi$
$984$	0	0
$985$	0.219833	0.00700445
$986$	0	0
$987$	−1.46442	−0.0466130
$988$	0	0
$989$	−12.9758	−0.412608
$990$	0	0
$991$	−44.3846	−1.40992	−0.704962	−	0.709245i	$-0.749036\pi$
$991$	−44.3846	−1.40992	−0.704962	+	0.709245i	$0.749036\pi$
$992$	0	0
$993$	4.27844	0.135772
$994$	0	0
$995$	9.36121	0.296770
$996$	0	0
$997$	7.14782	0.226374	0.113187	−	0.993574i	$-0.463894\pi$
$997$	7.14782	0.226374	0.113187	+	0.993574i	$0.463894\pi$
$998$	0	0
$999$	−7.74632	−0.245083

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.l.1.2	yes	3
13.5	odd	4		3380.2.f.g.3041.3		6
13.8	odd	4		3380.2.f.g.3041.4		6
13.12	even	2		3380.2.a.k.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.k.1.2	✓	3	13.12	even	2
3380.2.a.l.1.2	yes	3	1.1	even	1	trivial
3380.2.f.g.3041.3		6	13.5	odd	4
3380.2.f.g.3041.4		6	13.8	odd	4

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)

\(f(q)\)	\(=\)	\(q-0.445042 q^{3} +1.00000 q^{5} +0.445042 q^{7} -2.80194 q^{9} +O(q^{10})\)	\(q-0.445042 q^{3} +1.00000 q^{5} +0.445042 q^{7} -2.80194 q^{9} -3.24698 q^{11} -0.445042 q^{15} +2.93900 q^{17} +3.04892 q^{19} -0.198062 q^{21} -2.80194 q^{23} +1.00000 q^{25} +2.58211 q^{27} +2.15883 q^{29} -0.939001 q^{31} +1.44504 q^{33} +0.445042 q^{35} -3.00000 q^{37} -4.58211 q^{41} +4.63102 q^{43} -2.80194 q^{45} +7.39373 q^{47} -6.80194 q^{49} -1.30798 q^{51} -13.0586 q^{53} -3.24698 q^{55} -1.35690 q^{57} +3.17629 q^{59} -11.2470 q^{61} -1.24698 q^{63} -11.9879 q^{67} +1.24698 q^{69} -3.40581 q^{71} +0.317667 q^{73} -0.445042 q^{75} -1.44504 q^{77} -8.18060 q^{79} +7.25667 q^{81} +9.62565 q^{83} +2.93900 q^{85} -0.960771 q^{87} -5.04354 q^{89} +0.417895 q^{93} +3.04892 q^{95} -13.8605 q^{97} +9.09783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9}+O(q^{10}) \) 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9	\( 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9} - 5 q^{11} - q^{15} - q^{17} - 5 q^{21} - 4 q^{23} + 3 q^{25} + 2 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} + q^{35} - 9 q^{37} - 8 q^{41} - q^{43} - 4 q^{45} - 10 q^{47} - 16 q^{49} - 9 q^{51} - 8 q^{53} - 5 q^{55} + 17 q^{59} - 29 q^{61} + q^{63} - 17 q^{67} - q^{69} + 3 q^{71} - 16 q^{73} - q^{75} - 4 q^{77} - 13 q^{79} - 5 q^{81} + 17 q^{83} - q^{85} + 10 q^{87} - 9 q^{89} + 7 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9 - 5 * q^11 - q^15 - q^17 - 5 * q^21 - 4 * q^23 + 3 * q^25 + 2 * q^27 - 2 * q^29 + 7 * q^31 + 4 * q^33 + q^35 - 9 * q^37 - 8 * q^41 - q^43 - 4 * q^45 - 10 * q^47 - 16 * q^49 - 9 * q^51 - 8 * q^53 - 5 * q^55 + 17 * q^59 - 29 * q^61 + q^63 - 17 * q^67 - q^69 + 3 * q^71 - 16 * q^73 - q^75 - 4 * q^77 - 13 * q^79 - 5 * q^81 + 17 * q^83 - q^85 + 10 * q^87 - 9 * q^89 + 7 * q^93 - 6 * q^97 + 9 * q^99

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