LMFDB - Embedded newform 1368.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(1,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.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:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.59774$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59774 q^{5} +4.94370 q^{7} +O(q^{10})$	$q+1.59774 q^{5} +4.94370 q^{7} +5.59774 q^{11} -3.49646 q^{13} +7.09419 q^{17} -1.00000 q^{19} -3.19547 q^{23} -2.44724 q^{25} -4.69193 q^{29} -6.00000 q^{31} +7.89872 q^{35} +4.00000 q^{37} +2.50354 q^{41} -8.94370 q^{43} -6.28966 q^{47} +17.4402 q^{49} +5.69901 q^{53} +8.94370 q^{55} -6.99291 q^{59} -3.44724 q^{61} -5.58641 q^{65} -13.3839 q^{67} +13.3839 q^{71} -2.44015 q^{73} +27.6735 q^{77} -9.49646 q^{79} -0.691927 q^{83} +11.3346 q^{85} +8.69193 q^{89} -17.2854 q^{91} -1.59774 q^{95} +16.3909 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 3 * q - 2 * q^5 - 2 * q^7	$ 3 q - 2 q^{5} - 2 q^{7} + 10 q^{11} - 4 q^{13} + 8 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{29} - 18 q^{31} + 24 q^{35} + 12 q^{37} + 14 q^{41} - 10 q^{43} + 8 q^{47} + 29 q^{49} + 10 q^{53} + 10 q^{55} - 8 q^{59} + 24 q^{65} + 16 q^{73} + 16 q^{77} - 22 q^{79} + 18 q^{83} - 10 q^{85} + 6 q^{89} - 4 q^{91} + 2 q^{95} + 22 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 - 2 * q^7 + 10 * q^11 - 4 * q^13 + 8 * q^17 - 3 * q^19 + 4 * q^23 + 3 * q^25 + 6 * q^29 - 18 * q^31 + 24 * q^35 + 12 * q^37 + 14 * q^41 - 10 * q^43 + 8 * q^47 + 29 * q^49 + 10 * q^53 + 10 * q^55 - 8 * q^59 + 24 * q^65 + 16 * q^73 + 16 * q^77 - 22 * q^79 + 18 * q^83 - 10 * q^85 + 6 * q^89 - 4 * q^91 + 2 * q^95 + 22 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	1.59774	0.714529	0.357264	−	0.934003i	$-0.383709\pi$
$5$	1.59774	0.714529	0.357264	+	0.934003i	$0.383709\pi$
$6$	0	0
$7$	4.94370	1.86854	0.934271	−	0.356563i	$-0.116052\pi$
$7$	4.94370	1.86854	0.934271	+	0.356563i	$0.116052\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.59774	1.68778	0.843890	−	0.536516i	$-0.180260\pi$
$11$	5.59774	1.68778	0.843890	+	0.536516i	$0.180260\pi$
$12$	0	0
$13$	−3.49646	−0.969742	−0.484871	−	0.874586i	$-0.661134\pi$
$13$	−3.49646	−0.969742	−0.484871	+	0.874586i	$0.661134\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.09419	1.72059	0.860297	−	0.509793i	$-0.170278\pi$
$17$	7.09419	1.72059	0.860297	+	0.509793i	$0.170278\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.19547	−0.666302	−0.333151	−	0.942874i	$-0.608112\pi$
$23$	−3.19547	−0.666302	−0.333151	+	0.942874i	$0.608112\pi$
$24$	0	0
$25$	−2.44724	−0.489448
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.69193	−0.871269	−0.435634	−	0.900124i	$-0.643476\pi$
$29$	−4.69193	−0.871269	−0.435634	+	0.900124i	$0.643476\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.89872	1.33513
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.50354	0.390988	0.195494	−	0.980705i	$-0.437369\pi$
$41$	2.50354	0.390988	0.195494	+	0.980705i	$0.437369\pi$
$42$	0	0
$43$	−8.94370	−1.36390	−0.681951	−	0.731398i	$-0.738868\pi$
$43$	−8.94370	−1.36390	−0.681951	+	0.731398i	$0.738868\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.28966	−0.917441	−0.458721	−	0.888580i	$-0.651692\pi$
$47$	−6.28966	−0.917441	−0.458721	+	0.888580i	$0.651692\pi$
$48$	0	0
$49$	17.4402	2.49145
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.69901	0.782820	0.391410	−	0.920216i	$-0.371987\pi$
$53$	5.69901	0.782820	0.391410	+	0.920216i	$0.371987\pi$
$54$	0	0
$55$	8.94370	1.20597
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.99291	−0.910400	−0.455200	−	0.890389i	$-0.650432\pi$
$59$	−6.99291	−0.910400	−0.455200	+	0.890389i	$0.650432\pi$
$60$	0	0
$61$	−3.44724	−0.441374	−0.220687	−	0.975345i	$-0.570830\pi$
$61$	−3.44724	−0.441374	−0.220687	+	0.975345i	$0.570830\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.58641	−0.692909
$66$	0	0
$67$	−13.3839	−1.63510	−0.817549	−	0.575859i	$-0.804668\pi$
$67$	−13.3839	−1.63510	−0.817549	+	0.575859i	$0.804668\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.3839	1.58837	0.794186	−	0.607675i	$-0.207898\pi$
$71$	13.3839	1.58837	0.794186	+	0.607675i	$0.207898\pi$
$72$	0	0
$73$	−2.44015	−0.285599	−0.142799	−	0.989752i	$-0.545610\pi$
$73$	−2.44015	−0.285599	−0.142799	+	0.989752i	$0.545610\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	27.6735	3.15369
$78$	0	0
$79$	−9.49646	−1.06843	−0.534217	−	0.845347i	$-0.679394\pi$
$79$	−9.49646	−1.06843	−0.534217	+	0.845347i	$0.679394\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.691927	−0.0759488	−0.0379744	−	0.999279i	$-0.512091\pi$
$83$	−0.691927	−0.0759488	−0.0379744	+	0.999279i	$0.512091\pi$
$84$	0	0
$85$	11.3346	1.22941
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.69193	0.921342	0.460671	−	0.887571i	$-0.347609\pi$
$89$	8.69193	0.921342	0.460671	+	0.887571i	$0.347609\pi$
$90$	0	0
$91$	−17.2854	−1.81200
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.59774	−0.163924
$96$	0	0
$97$	16.3909	1.66425	0.832124	−	0.554590i	$-0.187125\pi$
$97$	16.3909	1.66425	0.832124	+	0.554590i	$0.187125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.49646	0.546918	0.273459	−	0.961884i	$-0.411832\pi$
$101$	5.49646	0.546918	0.273459	+	0.961884i	$0.411832\pi$
$102$	0	0
$103$	0.992912	0.0978346	0.0489173	−	0.998803i	$-0.484423\pi$
$103$	0.992912	0.0978346	0.0489173	+	0.998803i	$0.484423\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.5793	1.98948	0.994739	−	0.102440i	$-0.0326651\pi$
$107$	20.5793	1.98948	0.994739	+	0.102440i	$0.0326651\pi$
$108$	0	0
$109$	−10.3909	−0.995272	−0.497636	−	0.867386i	$-0.665798\pi$
$109$	−10.3909	−0.995272	−0.497636	+	0.867386i	$0.665798\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.0829	−1.79517	−0.897583	−	0.440846i	$-0.854678\pi$
$113$	−19.0829	−1.79517	−0.897583	+	0.440846i	$0.854678\pi$
$114$	0	0
$115$	−5.10552	−0.476092
$116$	0	0
$117$	0	0
$118$	0	0
$119$	35.0715	3.21500
$120$	0	0
$121$	20.3346	1.84860
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.8987	−1.06425
$126$	0	0
$127$	15.3839	1.36510	0.682548	−	0.730841i	$-0.260872\pi$
$127$	15.3839	1.36510	0.682548	+	0.730841i	$0.260872\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.98159	0.609984	0.304992	−	0.952355i	$-0.401346\pi$
$131$	6.98159	0.609984	0.304992	+	0.952355i	$0.401346\pi$
$132$	0	0
$133$	−4.94370	−0.428673
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.48513	−0.468626	−0.234313	−	0.972161i	$-0.575284\pi$
$137$	−5.48513	−0.468626	−0.234313	+	0.972161i	$0.575284\pi$
$138$	0	0
$139$	0.943698	0.0800435	0.0400217	−	0.999199i	$-0.487257\pi$
$139$	0.943698	0.0800435	0.0400217	+	0.999199i	$0.487257\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−19.5722	−1.63671
$144$	0	0
$145$	−7.49646	−0.622547
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.79321	−0.392675	−0.196337	−	0.980536i	$-0.562905\pi$
$149$	−4.79321	−0.392675	−0.196337	+	0.980536i	$0.562905\pi$
$150$	0	0
$151$	−9.49646	−0.772811	−0.386405	−	0.922329i	$-0.626283\pi$
$151$	−9.49646	−0.772811	−0.386405	+	0.922329i	$0.626283\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.58641	−0.769999
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.7974	−1.24501
$162$	0	0
$163$	−2.99291	−0.234423	−0.117211	−	0.993107i	$-0.537396\pi$
$163$	−2.99291	−0.234423	−0.117211	+	0.993107i	$0.537396\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.1955	−1.17586	−0.587930	−	0.808912i	$-0.700057\pi$
$167$	−15.1955	−1.17586	−0.587930	+	0.808912i	$0.700057\pi$
$168$	0	0
$169$	−0.774794	−0.0595995
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.0829	−1.14673	−0.573365	−	0.819300i	$-0.694362\pi$
$173$	−15.0829	−1.14673	−0.573365	+	0.819300i	$0.694362\pi$
$174$	0	0
$175$	−12.0984	−0.914555
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.57932	−0.641249	−0.320624	−	0.947206i	$-0.603893\pi$
$179$	−8.57932	−0.641249	−0.320624	+	0.947206i	$0.603893\pi$
$180$	0	0
$181$	12.8803	0.957386	0.478693	−	0.877982i	$-0.341111\pi$
$181$	12.8803	0.957386	0.478693	+	0.877982i	$0.341111\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.39094	0.469871
$186$	0	0
$187$	39.7114	2.90399
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.28966	0.455104	0.227552	−	0.973766i	$-0.426928\pi$
$191$	6.28966	0.455104	0.227552	+	0.973766i	$0.426928\pi$
$192$	0	0
$193$	9.49646	0.683570	0.341785	−	0.939778i	$-0.388969\pi$
$193$	9.49646	0.683570	0.341785	+	0.939778i	$0.388969\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.4965	1.53156	0.765780	−	0.643103i	$-0.222353\pi$
$197$	21.4965	1.53156	0.765780	+	0.643103i	$0.222353\pi$
$198$	0	0
$199$	−8.94370	−0.634002	−0.317001	−	0.948425i	$-0.602676\pi$
$199$	−8.94370	−0.634002	−0.317001	+	0.948425i	$0.602676\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−23.1955	−1.62800
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.59774	−0.387203
$210$	0	0
$211$	−2.39094	−0.164599	−0.0822996	−	0.996608i	$-0.526226\pi$
$211$	−2.39094	−0.164599	−0.0822996	+	0.996608i	$0.526226\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−14.2897	−0.974547
$216$	0	0
$217$	−29.6622	−2.01360
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.8045	−1.66853
$222$	0	0
$223$	−4.89448	−0.327759	−0.163879	−	0.986480i	$-0.552401\pi$
$223$	−4.89448	−0.327759	−0.163879	+	0.986480i	$0.552401\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.1955	−1.00856	−0.504279	−	0.863541i	$-0.668242\pi$
$227$	−15.1955	−1.00856	−0.504279	+	0.863541i	$0.668242\pi$
$228$	0	0
$229$	6.44015	0.425577	0.212789	−	0.977098i	$-0.431745\pi$
$229$	6.44015	0.425577	0.212789	+	0.977098i	$0.431745\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.89163	0.451486	0.225743	−	0.974187i	$-0.427519\pi$
$233$	6.89163	0.451486	0.225743	+	0.974187i	$0.427519\pi$
$234$	0	0
$235$	−10.0492	−0.655538
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.89163	0.187044	0.0935221	−	0.995617i	$-0.470187\pi$
$239$	2.89163	0.187044	0.0935221	+	0.995617i	$0.470187\pi$
$240$	0	0
$241$	4.89448	0.315281	0.157641	−	0.987497i	$-0.449611\pi$
$241$	4.89448	0.315281	0.157641	+	0.987497i	$0.449611\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	27.8647	1.78021
$246$	0	0
$247$	3.49646	0.222474
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.5906	−1.55215	−0.776074	−	0.630642i	$-0.782792\pi$
$251$	−24.5906	−1.55215	−0.776074	+	0.630642i	$0.782792\pi$
$252$	0	0
$253$	−17.8874	−1.12457
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.50354	−0.156167	−0.0780834	−	0.996947i	$-0.524880\pi$
$257$	−2.50354	−0.156167	−0.0780834	+	0.996947i	$0.524880\pi$
$258$	0	0
$259$	19.7748	1.22875
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.9058	0.795806	0.397903	−	0.917427i	$-0.369738\pi$
$263$	12.9058	0.795806	0.397903	+	0.917427i	$0.369738\pi$
$264$	0	0
$265$	9.10552	0.559347
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.691927	−0.0421875	−0.0210938	−	0.999778i	$-0.506715\pi$
$269$	−0.691927	−0.0421875	−0.0210938	+	0.999778i	$0.506715\pi$
$270$	0	0
$271$	−16.7819	−1.01943	−0.509713	−	0.860344i	$-0.670249\pi$
$271$	−16.7819	−1.01943	−0.509713	+	0.860344i	$0.670249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.6990	−0.826082
$276$	0	0
$277$	−29.3346	−1.76255	−0.881274	−	0.472606i	$-0.843313\pi$
$277$	−29.3346	−1.76255	−0.881274	+	0.472606i	$0.843313\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	32.4667	1.93680	0.968401	−	0.249398i	$-0.0802327\pi$
$281$	32.4667	1.93680	0.968401	+	0.249398i	$0.0802327\pi$
$282$	0	0
$283$	−17.9508	−1.06706	−0.533532	−	0.845780i	$-0.679136\pi$
$283$	−17.9508	−1.06706	−0.533532	+	0.845780i	$0.679136\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.3768	0.730577
$288$	0	0
$289$	33.3276	1.96044
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−22.8803	−1.33668	−0.668341	−	0.743855i	$-0.732995\pi$
$293$	−22.8803	−1.33668	−0.668341	+	0.743855i	$0.732995\pi$
$294$	0	0
$295$	−11.1728	−0.650507
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.1728	0.646141
$300$	0	0
$301$	−44.2149	−2.54851
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.50778	−0.315375
$306$	0	0
$307$	−2.39094	−0.136458	−0.0682291	−	0.997670i	$-0.521735\pi$
$307$	−2.39094	−0.136458	−0.0682291	+	0.997670i	$0.521735\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.89872	−0.447895	−0.223948	−	0.974601i	$-0.571894\pi$
$311$	−7.89872	−0.447895	−0.223948	+	0.974601i	$0.571894\pi$
$312$	0	0
$313$	22.7819	1.28771	0.643854	−	0.765148i	$-0.277334\pi$
$313$	22.7819	1.28771	0.643854	+	0.765148i	$0.277334\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.49646	−0.533374	−0.266687	−	0.963783i	$-0.585929\pi$
$317$	−9.49646	−0.533374	−0.266687	+	0.963783i	$0.585929\pi$
$318$	0	0
$319$	−26.2642	−1.47051
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.09419	−0.394731
$324$	0	0
$325$	8.55668	0.474639
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−31.0942	−1.71428
$330$	0	0
$331$	3.39803	0.186773	0.0933863	−	0.995630i	$-0.470231\pi$
$331$	3.39803	0.186773	0.0933863	+	0.995630i	$0.470231\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−21.3839	−1.16832
$336$	0	0
$337$	−8.39094	−0.457084	−0.228542	−	0.973534i	$-0.573396\pi$
$337$	−8.39094	−0.457084	−0.228542	+	0.973534i	$0.573396\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−33.5864	−1.81881
$342$	0	0
$343$	51.6130	2.78684
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.9887	1.07305	0.536524	−	0.843885i	$-0.319737\pi$
$347$	19.9887	1.07305	0.536524	+	0.843885i	$0.319737\pi$
$348$	0	0
$349$	−21.4331	−1.14729	−0.573643	−	0.819106i	$-0.694470\pi$
$349$	−21.4331	−1.14729	−0.573643	+	0.819106i	$0.694470\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	23.7748	1.26540	0.632702	−	0.774395i	$-0.281946\pi$
$353$	23.7748	1.26540	0.632702	+	0.774395i	$0.281946\pi$
$354$	0	0
$355$	21.3839	1.13494
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−34.8690	−1.84031	−0.920157	−	0.391549i	$-0.871939\pi$
$359$	−34.8690	−1.84031	−0.920157	+	0.391549i	$0.871939\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.89872	−0.204068
$366$	0	0
$367$	5.00709	0.261368	0.130684	−	0.991424i	$-0.458283\pi$
$367$	5.00709	0.261368	0.130684	+	0.991424i	$0.458283\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	28.1742	1.46273
$372$	0	0
$373$	24.7819	1.28316	0.641579	−	0.767057i	$-0.278280\pi$
$373$	24.7819	1.28316	0.641579	+	0.767057i	$0.278280\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.4051	0.844906
$378$	0	0
$379$	13.8874	0.713348	0.356674	−	0.934229i	$-0.383911\pi$
$379$	13.8874	0.713348	0.356674	+	0.934229i	$0.383911\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.7974	0.807212	0.403606	−	0.914933i	$-0.367757\pi$
$383$	15.7974	0.807212	0.403606	+	0.914933i	$0.367757\pi$
$384$	0	0
$385$	44.2149	2.25340
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.9958	−0.658911	−0.329456	−	0.944171i	$-0.606865\pi$
$389$	−12.9958	−0.658911	−0.329456	+	0.944171i	$0.606865\pi$
$390$	0	0
$391$	−22.6693	−1.14643
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.1728	−0.763428
$396$	0	0
$397$	−15.4472	−0.775275	−0.387637	−	0.921812i	$-0.626709\pi$
$397$	−15.4472	−0.775275	−0.387637	+	0.921812i	$0.626709\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−28.8945	−1.44292	−0.721461	−	0.692455i	$-0.756529\pi$
$401$	−28.8945	−1.44292	−0.721461	+	0.692455i	$0.756529\pi$
$402$	0	0
$403$	20.9787	1.04503
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.3909	1.10988
$408$	0	0
$409$	28.7677	1.42247	0.711236	−	0.702954i	$-0.248136\pi$
$409$	28.7677	1.42247	0.711236	+	0.702954i	$0.248136\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−34.5709	−1.70112
$414$	0	0
$415$	−1.10552	−0.0542676
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.08287	−0.346021	−0.173010	−	0.984920i	$-0.555349\pi$
$419$	−7.08287	−0.346021	−0.173010	+	0.984920i	$0.555349\pi$
$420$	0	0
$421$	21.6622	1.05575	0.527875	−	0.849322i	$-0.322989\pi$
$421$	21.6622	1.05575	0.527875	+	0.849322i	$0.322989\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−17.3612	−0.842142
$426$	0	0
$427$	−17.0421	−0.824726
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.8045	−0.809446	−0.404723	−	0.914439i	$-0.632632\pi$
$431$	−16.8045	−0.809446	−0.404723	+	0.914439i	$0.632632\pi$
$432$	0	0
$433$	11.3839	0.547073	0.273537	−	0.961862i	$-0.411807\pi$
$433$	11.3839	0.547073	0.273537	+	0.961862i	$0.411807\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.19547	0.152860
$438$	0	0
$439$	−35.1586	−1.67803	−0.839015	−	0.544108i	$-0.816868\pi$
$439$	−35.1586	−1.67803	−0.839015	+	0.544108i	$0.816868\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.3796	0.873242	0.436621	−	0.899646i	$-0.356175\pi$
$443$	18.3796	0.873242	0.436621	+	0.899646i	$0.356175\pi$
$444$	0	0
$445$	13.8874	0.658326
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.8874	0.938544	0.469272	−	0.883054i	$-0.344516\pi$
$449$	19.8874	0.938544	0.469272	+	0.883054i	$0.344516\pi$
$450$	0	0
$451$	14.0142	0.659902
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−27.6175	−1.29473
$456$	0	0
$457$	−37.1094	−1.73591	−0.867953	−	0.496646i	$-0.834565\pi$
$457$	−37.1094	−1.73591	−0.867953	+	0.496646i	$0.834565\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.2068	0.521952	0.260976	−	0.965345i	$-0.415956\pi$
$461$	11.2068	0.521952	0.260976	+	0.965345i	$0.415956\pi$
$462$	0	0
$463$	−0.0633891	−0.00294594	−0.00147297	−	0.999999i	$-0.500469\pi$
$463$	−0.0633891	−0.00294594	−0.00147297	+	0.999999i	$0.500469\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.7493	−1.93193	−0.965963	−	0.258678i	$-0.916713\pi$
$467$	−41.7493	−1.93193	−0.965963	+	0.258678i	$0.916713\pi$
$468$	0	0
$469$	−66.1657	−3.05525
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−50.0645	−2.30197
$474$	0	0
$475$	2.44724	0.112287
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.79744	0.356274	0.178137	−	0.984006i	$-0.442993\pi$
$479$	7.79744	0.356274	0.178137	+	0.984006i	$0.442993\pi$
$480$	0	0
$481$	−13.9858	−0.637699
$482$	0	0
$483$	0	0
$484$	0	0
$485$	26.1884	1.18915
$486$	0	0
$487$	10.5035	0.475961	0.237981	−	0.971270i	$-0.423515\pi$
$487$	10.5035	0.475961	0.237981	+	0.971270i	$0.423515\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.31516	0.194740	0.0973702	−	0.995248i	$-0.468957\pi$
$491$	4.31516	0.194740	0.0973702	+	0.995248i	$0.468957\pi$
$492$	0	0
$493$	−33.2854	−1.49910
$494$	0	0
$495$	0	0
$496$	0	0
$497$	66.1657	2.96794
$498$	0	0
$499$	11.8382	0.529950	0.264975	−	0.964255i	$-0.414636\pi$
$499$	11.8382	0.529950	0.264975	+	0.964255i	$0.414636\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.5722	0.694332	0.347166	−	0.937804i	$-0.387144\pi$
$503$	15.5722	0.694332	0.347166	+	0.937804i	$0.387144\pi$
$504$	0	0
$505$	8.78188	0.390789
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.6848	0.695218	0.347609	−	0.937640i	$-0.386994\pi$
$509$	15.6848	0.695218	0.347609	+	0.937640i	$0.386994\pi$
$510$	0	0
$511$	−12.0634	−0.533653
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.58641	0.0699056
$516$	0	0
$517$	−35.2079	−1.54844
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.2783	1.32652	0.663259	−	0.748390i	$-0.269173\pi$
$521$	30.2783	1.32652	0.663259	+	0.748390i	$0.269173\pi$
$522$	0	0
$523$	−0.880309	−0.0384932	−0.0192466	−	0.999815i	$-0.506127\pi$
$523$	−0.880309	−0.0384932	−0.0192466	+	0.999815i	$0.506127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.5651	−1.85417
$528$	0	0
$529$	−12.7890	−0.556042
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.75353	−0.379158
$534$	0	0
$535$	32.8803	1.42154
$536$	0	0
$537$	0	0
$538$	0	0
$539$	97.6254	4.20502
$540$	0	0
$541$	9.43307	0.405559	0.202780	−	0.979224i	$-0.435002\pi$
$541$	9.43307	0.405559	0.202780	+	0.979224i	$0.435002\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.6020	−0.711150
$546$	0	0
$547$	22.4894	0.961576	0.480788	−	0.876837i	$-0.340351\pi$
$547$	22.4894	0.961576	0.480788	+	0.876837i	$0.340351\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.69193	0.199883
$552$	0	0
$553$	−46.9476	−1.99642
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.98159	−0.126334	−0.0631670	−	0.998003i	$-0.520120\pi$
$557$	−2.98159	−0.126334	−0.0631670	+	0.998003i	$0.520120\pi$
$558$	0	0
$559$	31.2713	1.32263
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.3980	−0.817529	−0.408765	−	0.912640i	$-0.634040\pi$
$563$	−19.3980	−0.817529	−0.408765	+	0.912640i	$0.634040\pi$
$564$	0	0
$565$	−30.4894	−1.28270
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.3081	−0.474059	−0.237030	−	0.971502i	$-0.576174\pi$
$569$	−11.3081	−0.474059	−0.237030	+	0.971502i	$0.576174\pi$
$570$	0	0
$571$	10.9929	0.460039	0.230020	−	0.973186i	$-0.426121\pi$
$571$	10.9929	0.460039	0.230020	+	0.973186i	$0.426121\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.82009	0.326120
$576$	0	0
$577$	44.1023	1.83600	0.918002	−	0.396575i	$-0.129801\pi$
$577$	44.1023	1.83600	0.918002	+	0.396575i	$0.129801\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.42068	−0.141914
$582$	0	0
$583$	31.9016	1.32123
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.40226	0.0991521	0.0495760	−	0.998770i	$-0.484213\pi$
$587$	2.40226	0.0991521	0.0495760	+	0.998770i	$0.484213\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.7677	1.09922	0.549609	−	0.835422i	$-0.314777\pi$
$593$	26.7677	1.09922	0.549609	+	0.835422i	$0.314777\pi$
$594$	0	0
$595$	56.0350	2.29721
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.60197	0.351467	0.175734	−	0.984438i	$-0.443770\pi$
$599$	8.60197	0.351467	0.175734	+	0.984438i	$0.443770\pi$
$600$	0	0
$601$	29.4965	1.20319	0.601593	−	0.798803i	$-0.294533\pi$
$601$	29.4965	1.20319	0.601593	+	0.798803i	$0.294533\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	32.4894	1.32088
$606$	0	0
$607$	−12.6161	−0.512074	−0.256037	−	0.966667i	$-0.582417\pi$
$607$	−12.6161	−0.512074	−0.256037	+	0.966667i	$0.582417\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21.9915	0.889682
$612$	0	0
$613$	−17.3346	−0.700139	−0.350070	−	0.936724i	$-0.613842\pi$
$613$	−17.3346	−0.700139	−0.350070	+	0.936724i	$0.613842\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.6664	0.912516	0.456258	−	0.889848i	$-0.349189\pi$
$617$	22.6664	0.912516	0.456258	+	0.889848i	$0.349189\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	42.9703	1.72157
$624$	0	0
$625$	−6.77479	−0.270992
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28.3768	1.13146
$630$	0	0
$631$	2.95787	0.117751	0.0588755	−	0.998265i	$-0.481248\pi$
$631$	2.95787	0.117751	0.0588755	+	0.998265i	$0.481248\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.5793	0.975401
$636$	0	0
$637$	−60.9787	−2.41607
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.3081	0.446642	0.223321	−	0.974745i	$-0.428310\pi$
$641$	11.3081	0.446642	0.223321	+	0.974745i	$0.428310\pi$
$642$	0	0
$643$	−21.7256	−0.856773	−0.428387	−	0.903596i	$-0.640918\pi$
$643$	−21.7256	−0.856773	−0.428387	+	0.903596i	$0.640918\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.0942	1.06518	0.532591	−	0.846373i	$-0.321218\pi$
$647$	27.0942	1.06518	0.532591	+	0.846373i	$0.321218\pi$
$648$	0	0
$649$	−39.1445	−1.53655
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.1912	1.25974	0.629870	−	0.776700i	$-0.283108\pi$
$653$	32.1912	1.25974	0.629870	+	0.776700i	$0.283108\pi$
$654$	0	0
$655$	11.1547	0.435851
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.9703	−0.427341	−0.213670	−	0.976906i	$-0.568542\pi$
$659$	−10.9703	−0.427341	−0.213670	+	0.976906i	$0.568542\pi$
$660$	0	0
$661$	10.9929	0.427575	0.213787	−	0.976880i	$-0.431420\pi$
$661$	10.9929	0.427575	0.213787	+	0.976880i	$0.431420\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.89872	−0.306299
$666$	0	0
$667$	14.9929	0.580528
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19.2967	−0.744943
$672$	0	0
$673$	−10.2783	−0.396201	−0.198100	−	0.980182i	$-0.563477\pi$
$673$	−10.2783	−0.396201	−0.198100	+	0.980182i	$0.563477\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−36.4667	−1.40153	−0.700765	−	0.713392i	$-0.747158\pi$
$677$	−36.4667	−1.40153	−0.700765	+	0.713392i	$0.747158\pi$
$678$	0	0
$679$	81.0319	3.10972
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.5651	1.16954	0.584771	−	0.811198i	$-0.301184\pi$
$683$	30.5651	1.16954	0.584771	+	0.811198i	$0.301184\pi$
$684$	0	0
$685$	−8.76379	−0.334847
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−19.9264	−0.759134
$690$	0	0
$691$	−9.82401	−0.373723	−0.186861	−	0.982386i	$-0.559832\pi$
$691$	−9.82401	−0.373723	−0.186861	+	0.982386i	$0.559832\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.50778	0.0571934
$696$	0	0
$697$	17.7606	0.672731
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−34.0531	−1.28617	−0.643085	−	0.765795i	$-0.722346\pi$
$701$	−34.0531	−1.28617	−0.643085	+	0.765795i	$0.722346\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.1728	1.02194
$708$	0	0
$709$	8.76771	0.329278	0.164639	−	0.986354i	$-0.447354\pi$
$709$	8.76771	0.329278	0.164639	+	0.986354i	$0.447354\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.1728	0.718028
$714$	0	0
$715$	−31.2713	−1.16948
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.1013	0.898826	0.449413	−	0.893324i	$-0.351633\pi$
$719$	24.1013	0.898826	0.449413	+	0.893324i	$0.351633\pi$
$720$	0	0
$721$	4.90866	0.182808
$722$	0	0
$723$	0	0
$724$	0	0
$725$	11.4823	0.426441
$726$	0	0
$727$	−37.8240	−1.40281	−0.701407	−	0.712761i	$-0.747445\pi$
$727$	−37.8240	−1.40281	−0.701407	+	0.712761i	$0.747445\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−63.4483	−2.34672
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−74.9193	−2.75969
$738$	0	0
$739$	−13.0421	−0.479762	−0.239881	−	0.970802i	$-0.577108\pi$
$739$	−13.0421	−0.479762	−0.239881	+	0.970802i	$0.577108\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.7890	0.505868	0.252934	−	0.967484i	$-0.418604\pi$
$743$	13.7890	0.505868	0.252934	+	0.967484i	$0.418604\pi$
$744$	0	0
$745$	−7.65827	−0.280577
$746$	0	0
$747$	0	0
$748$	0	0
$749$	101.738	3.71742
$750$	0	0
$751$	19.8874	0.725701	0.362851	−	0.931847i	$-0.381803\pi$
$751$	19.8874	0.725701	0.362851	+	0.931847i	$0.381803\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.1728	−0.552196
$756$	0	0
$757$	−24.4543	−0.888808	−0.444404	−	0.895827i	$-0.646584\pi$
$757$	−24.4543	−0.888808	−0.444404	+	0.895827i	$0.646584\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−42.4696	−1.53952	−0.769760	−	0.638333i	$-0.779624\pi$
$761$	−42.4696	−1.53952	−0.769760	+	0.638333i	$0.779624\pi$
$762$	0	0
$763$	−51.3697	−1.85971
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.4504	0.882853
$768$	0	0
$769$	7.54567	0.272104	0.136052	−	0.990702i	$-0.456559\pi$
$769$	7.54567	0.272104	0.136052	+	0.990702i	$0.456559\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22.5035	−0.809396	−0.404698	−	0.914450i	$-0.632623\pi$
$773$	−22.5035	−0.809396	−0.404698	+	0.914450i	$0.632623\pi$
$774$	0	0
$775$	14.6835	0.527445
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.50354	−0.0896988
$780$	0	0
$781$	74.9193	2.68082
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.19547	0.114051
$786$	0	0
$787$	24.7819	0.883379	0.441689	−	0.897168i	$-0.354379\pi$
$787$	24.7819	0.883379	0.441689	+	0.897168i	$0.354379\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−94.3399	−3.35434
$792$	0	0
$793$	12.0531	0.428019
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.11260	−0.287363	−0.143682	−	0.989624i	$-0.545894\pi$
$797$	−8.11260	−0.287363	−0.143682	+	0.989624i	$0.545894\pi$
$798$	0	0
$799$	−44.6201	−1.57854
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−13.6593	−0.482028
$804$	0	0
$805$	−25.2401	−0.889598
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.6948	0.657273	0.328637	−	0.944456i	$-0.393411\pi$
$809$	18.6948	0.657273	0.328637	+	0.944456i	$0.393411\pi$
$810$	0	0
$811$	−31.0460	−1.09017	−0.545087	−	0.838379i	$-0.683503\pi$
$811$	−31.0460	−1.09017	−0.545087	+	0.838379i	$0.683503\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.78188	−0.167502
$816$	0	0
$817$	8.94370	0.312900
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.5977	−0.614165	−0.307083	−	0.951683i	$-0.599353\pi$
$821$	−17.5977	−0.614165	−0.307083	+	0.951683i	$0.599353\pi$
$822$	0	0
$823$	14.7327	0.513549	0.256774	−	0.966471i	$-0.417340\pi$
$823$	14.7327	0.513549	0.256774	+	0.966471i	$0.417340\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.5638	1.58441	0.792204	−	0.610257i	$-0.208934\pi$
$827$	45.5638	1.58441	0.792204	+	0.610257i	$0.208934\pi$
$828$	0	0
$829$	−25.7890	−0.895688	−0.447844	−	0.894112i	$-0.647808\pi$
$829$	−25.7890	−0.895688	−0.447844	+	0.894112i	$0.647808\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	123.724	4.28678
$834$	0	0
$835$	−24.2783	−0.840187
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.97026	−0.102545	−0.0512725	−	0.998685i	$-0.516328\pi$
$839$	−2.97026	−0.102545	−0.0512725	+	0.998685i	$0.516328\pi$
$840$	0	0
$841$	−6.98582	−0.240891
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.23792	−0.0425856
$846$	0	0
$847$	100.528	3.45419
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.7819	−0.438157
$852$	0	0
$853$	54.5567	1.86798	0.933992	−	0.357293i	$-0.116300\pi$
$853$	54.5567	1.86798	0.933992	+	0.357293i	$0.116300\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.27125	−0.316700	−0.158350	−	0.987383i	$-0.550617\pi$
$857$	−9.27125	−0.316700	−0.158350	+	0.987383i	$0.550617\pi$
$858$	0	0
$859$	−5.85236	−0.199680	−0.0998399	−	0.995004i	$-0.531833\pi$
$859$	−5.85236	−0.199680	−0.0998399	+	0.995004i	$0.531833\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.5496	1.61861	0.809303	−	0.587391i	$-0.199845\pi$
$863$	47.5496	1.61861	0.809303	+	0.587391i	$0.199845\pi$
$864$	0	0
$865$	−24.0984	−0.819371
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−53.1586	−1.80328
$870$	0	0
$871$	46.7961	1.58562
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−58.8237	−1.98860
$876$	0	0
$877$	33.6622	1.13669	0.568346	−	0.822790i	$-0.307584\pi$
$877$	33.6622	1.13669	0.568346	+	0.822790i	$0.307584\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.6664	1.03318	0.516589	−	0.856233i	$-0.327201\pi$
$881$	30.6664	1.03318	0.516589	+	0.856233i	$0.327201\pi$
$882$	0	0
$883$	−30.6059	−1.02997	−0.514985	−	0.857199i	$-0.672203\pi$
$883$	−30.6059	−1.02997	−0.514985	+	0.857199i	$0.672203\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.7606	1.13357	0.566785	−	0.823866i	$-0.308187\pi$
$887$	33.7606	1.13357	0.566785	+	0.823866i	$0.308187\pi$
$888$	0	0
$889$	76.0531	2.55074
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.28966	0.210476
$894$	0	0
$895$	−13.7075	−0.458191
$896$	0	0
$897$	0	0
$898$	0	0
$899$	28.1516	0.938907
$900$	0	0
$901$	40.4299	1.34692
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.5793	0.684080
$906$	0	0
$907$	−7.87322	−0.261426	−0.130713	−	0.991420i	$-0.541727\pi$
$907$	−7.87322	−0.261426	−0.130713	+	0.991420i	$0.541727\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.02974	−0.166643	−0.0833213	−	0.996523i	$-0.526553\pi$
$911$	−5.02974	−0.166643	−0.0833213	+	0.996523i	$0.526553\pi$
$912$	0	0
$913$	−3.87322	−0.128185
$914$	0	0
$915$	0	0
$916$	0	0
$917$	34.5149	1.13978
$918$	0	0
$919$	−33.7890	−1.11460	−0.557298	−	0.830313i	$-0.688162\pi$
$919$	−33.7890	−1.11460	−0.557298	+	0.830313i	$0.688162\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−46.7961	−1.54031
$924$	0	0
$925$	−9.78897	−0.321859
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.38385	0.0454028	0.0227014	−	0.999742i	$-0.492773\pi$
$929$	1.38385	0.0454028	0.0227014	+	0.999742i	$0.492773\pi$
$930$	0	0
$931$	−17.4402	−0.571578
$932$	0	0
$933$	0	0
$934$	0	0
$935$	63.4483	2.07498
$936$	0	0
$937$	−33.2362	−1.08578	−0.542890	−	0.839804i	$-0.682670\pi$
$937$	−33.2362	−1.08578	−0.542890	+	0.839804i	$0.682670\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31.4880	1.02648	0.513239	−	0.858245i	$-0.328445\pi$
$941$	31.4880	1.02648	0.513239	+	0.858245i	$0.328445\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.6778	1.51682	0.758412	−	0.651776i	$-0.225976\pi$
$947$	46.6778	1.51682	0.758412	+	0.651776i	$0.225976\pi$
$948$	0	0
$949$	8.53189	0.276957
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.4894	1.44115	0.720576	−	0.693376i	$-0.243877\pi$
$953$	44.4894	1.44115	0.720576	+	0.693376i	$0.243877\pi$
$954$	0	0
$955$	10.0492	0.325185
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−27.1168	−0.875648
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.1728	0.488430
$966$	0	0
$967$	−21.9858	−0.707016	−0.353508	−	0.935431i	$-0.615011\pi$
$967$	−21.9858	−0.707016	−0.353508	+	0.935431i	$0.615011\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	0	0
$973$	4.66536	0.149565
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.69901	−0.182328	−0.0911638	−	0.995836i	$-0.529059\pi$
$977$	−5.69901	−0.182328	−0.0911638	+	0.995836i	$0.529059\pi$
$978$	0	0
$979$	48.6551	1.55502
$980$	0	0
$981$	0	0
$982$	0	0
$983$	49.1586	1.56792	0.783959	−	0.620813i	$-0.213197\pi$
$983$	49.1586	1.56792	0.783959	+	0.620813i	$0.213197\pi$
$984$	0	0
$985$	34.3456	1.09434
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.5793	0.908770
$990$	0	0
$991$	−2.90866	−0.0923966	−0.0461983	−	0.998932i	$-0.514711\pi$
$991$	−2.90866	−0.0923966	−0.0461983	+	0.998932i	$0.514711\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.2897	−0.453013
$996$	0	0
$997$	−58.0882	−1.83967	−0.919835	−	0.392305i	$-0.871678\pi$
$997$	−58.0882	−1.83967	−0.919835	+	0.392305i	$0.871678\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.a.m.1.3	✓	3
3.2	odd	2		1368.2.a.o.1.1	yes	3
4.3	odd	2		2736.2.a.bc.1.3		3
12.11	even	2		2736.2.a.be.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.a.m.1.3	✓	3	1.1	even	1	trivial
1368.2.a.o.1.1	yes	3	3.2	odd	2
2736.2.a.bc.1.3		3	4.3	odd	2
2736.2.a.be.1.1		3	12.11	even	2

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59774 q^{5} +4.94370 q^{7} +O(q^{10})\)	\(q+1.59774 q^{5} +4.94370 q^{7} +5.59774 q^{11} -3.49646 q^{13} +7.09419 q^{17} -1.00000 q^{19} -3.19547 q^{23} -2.44724 q^{25} -4.69193 q^{29} -6.00000 q^{31} +7.89872 q^{35} +4.00000 q^{37} +2.50354 q^{41} -8.94370 q^{43} -6.28966 q^{47} +17.4402 q^{49} +5.69901 q^{53} +8.94370 q^{55} -6.99291 q^{59} -3.44724 q^{61} -5.58641 q^{65} -13.3839 q^{67} +13.3839 q^{71} -2.44015 q^{73} +27.6735 q^{77} -9.49646 q^{79} -0.691927 q^{83} +11.3346 q^{85} +8.69193 q^{89} -17.2854 q^{91} -1.59774 q^{95} +16.3909 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 3 * q - 2 * q^5 - 2 * q^7	\( 3 q - 2 q^{5} - 2 q^{7} + 10 q^{11} - 4 q^{13} + 8 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{29} - 18 q^{31} + 24 q^{35} + 12 q^{37} + 14 q^{41} - 10 q^{43} + 8 q^{47} + 29 q^{49} + 10 q^{53} + 10 q^{55} - 8 q^{59} + 24 q^{65} + 16 q^{73} + 16 q^{77} - 22 q^{79} + 18 q^{83} - 10 q^{85} + 6 q^{89} - 4 q^{91} + 2 q^{95} + 22 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 - 2 * q^7 + 10 * q^11 - 4 * q^13 + 8 * q^17 - 3 * q^19 + 4 * q^23 + 3 * q^25 + 6 * q^29 - 18 * q^31 + 24 * q^35 + 12 * q^37 + 14 * q^41 - 10 * q^43 + 8 * q^47 + 29 * q^49 + 10 * q^53 + 10 * q^55 - 8 * q^59 + 24 * q^65 + 16 * q^73 + 16 * q^77 - 22 * q^79 + 18 * q^83 - 10 * q^85 + 6 * q^89 - 4 * q^91 + 2 * q^95 + 22 * q^97

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