LMFDB - Embedded newform 4368.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.561553 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.561553 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.56155 q^{11} +1.00000 q^{13} +0.561553 q^{15} +5.68466 q^{17} -7.68466 q^{19} -1.00000 q^{21} +1.43845 q^{23} -4.68466 q^{25} -1.00000 q^{27} -5.68466 q^{29} +10.2462 q^{31} +2.56155 q^{33} -0.561553 q^{35} -3.43845 q^{37} -1.00000 q^{39} +7.12311 q^{41} +10.5616 q^{43} -0.561553 q^{45} +1.00000 q^{49} -5.68466 q^{51} -4.24621 q^{53} +1.43845 q^{55} +7.68466 q^{57} +14.2462 q^{59} -5.68466 q^{61} +1.00000 q^{63} -0.561553 q^{65} -1.12311 q^{67} -1.43845 q^{69} -8.00000 q^{71} +0.561553 q^{73} +4.68466 q^{75} -2.56155 q^{77} +2.87689 q^{79} +1.00000 q^{81} -17.1231 q^{83} -3.19224 q^{85} +5.68466 q^{87} +10.0000 q^{89} +1.00000 q^{91} -10.2462 q^{93} +4.31534 q^{95} -18.4924 q^{97} -2.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{15} - q^{17} - 3 q^{19} - 2 q^{21} + 7 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} + 4 q^{31} + q^{33} + 3 q^{35} - 11 q^{37} - 2 q^{39} + 6 q^{41} + 17 q^{43} + 3 q^{45} + 2 q^{49} + q^{51} + 8 q^{53} + 7 q^{55} + 3 q^{57} + 12 q^{59} + q^{61} + 2 q^{63} + 3 q^{65} + 6 q^{67} - 7 q^{69} - 16 q^{71} - 3 q^{73} - 3 q^{75} - q^{77} + 14 q^{79} + 2 q^{81} - 26 q^{83} - 27 q^{85} - q^{87} + 20 q^{89} + 2 q^{91} - 4 q^{93} + 21 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^15 - q^17 - 3 * q^19 - 2 * q^21 + 7 * q^23 + 3 * q^25 - 2 * q^27 + q^29 + 4 * q^31 + q^33 + 3 * q^35 - 11 * q^37 - 2 * q^39 + 6 * q^41 + 17 * q^43 + 3 * q^45 + 2 * q^49 + q^51 + 8 * q^53 + 7 * q^55 + 3 * q^57 + 12 * q^59 + q^61 + 2 * q^63 + 3 * q^65 + 6 * q^67 - 7 * q^69 - 16 * q^71 - 3 * q^73 - 3 * q^75 - q^77 + 14 * q^79 + 2 * q^81 - 26 * q^83 - 27 * q^85 - q^87 + 20 * q^89 + 2 * q^91 - 4 * q^93 + 21 * q^95 - 4 * q^97 - 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.561553	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	−0.561553	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.561553	0.144992
$16$	0	0
$17$	5.68466	1.37873	0.689366	−	0.724413i	$-0.257889\pi$
$17$	5.68466	1.37873	0.689366	+	0.724413i	$0.257889\pi$
$18$	0	0
$19$	−7.68466	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$19$	−7.68466	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.43845	0.299937	0.149968	−	0.988691i	$-0.452083\pi$
$23$	1.43845	0.299937	0.149968	+	0.988691i	$0.452083\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	0	0
$31$	10.2462	1.84027	0.920137	−	0.391597i	$-0.128077\pi$
$31$	10.2462	1.84027	0.920137	+	0.391597i	$0.128077\pi$
$32$	0	0
$33$	2.56155	0.445909
$34$	0	0
$35$	−0.561553	−0.0949197
$36$	0	0
$37$	−3.43845	−0.565277	−0.282639	−	0.959226i	$-0.591210\pi$
$37$	−3.43845	−0.565277	−0.282639	+	0.959226i	$0.591210\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	0	0
$43$	10.5616	1.61062	0.805311	−	0.592853i	$-0.201998\pi$
$43$	10.5616	1.61062	0.805311	+	0.592853i	$0.201998\pi$
$44$	0	0
$45$	−0.561553	−0.0837114
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.68466	−0.796011
$52$	0	0
$53$	−4.24621	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$54$	0	0
$55$	1.43845	0.193960
$56$	0	0
$57$	7.68466	1.01786
$58$	0	0
$59$	14.2462	1.85470	0.927349	−	0.374197i	$-0.122082\pi$
$59$	14.2462	1.85470	0.927349	+	0.374197i	$0.122082\pi$
$60$	0	0
$61$	−5.68466	−0.727846	−0.363923	−	0.931429i	$-0.618563\pi$
$61$	−5.68466	−0.727846	−0.363923	+	0.931429i	$0.618563\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	−1.12311	−0.137209	−0.0686046	−	0.997644i	$-0.521855\pi$
$67$	−1.12311	−0.137209	−0.0686046	+	0.997644i	$0.521855\pi$
$68$	0	0
$69$	−1.43845	−0.173169
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	0.561553	0.0657248	0.0328624	−	0.999460i	$-0.489538\pi$
$73$	0.561553	0.0657248	0.0328624	+	0.999460i	$0.489538\pi$
$74$	0	0
$75$	4.68466	0.540938
$76$	0	0
$77$	−2.56155	−0.291916
$78$	0	0
$79$	2.87689	0.323676	0.161838	−	0.986817i	$-0.448258\pi$
$79$	2.87689	0.323676	0.161838	+	0.986817i	$0.448258\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.1231	−1.87951	−0.939753	−	0.341856i	$-0.888945\pi$
$83$	−17.1231	−1.87951	−0.939753	+	0.341856i	$0.888945\pi$
$84$	0	0
$85$	−3.19224	−0.346247
$86$	0	0
$87$	5.68466	0.609459
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−10.2462	−1.06248
$94$	0	0
$95$	4.31534	0.442745
$96$	0	0
$97$	−18.4924	−1.87762	−0.938811	−	0.344434i	$-0.888071\pi$
$97$	−18.4924	−1.87762	−0.938811	+	0.344434i	$0.888071\pi$
$98$	0	0
$99$	−2.56155	−0.257446
$100$	0	0
$101$	3.12311	0.310761	0.155380	−	0.987855i	$-0.450340\pi$
$101$	3.12311	0.310761	0.155380	+	0.987855i	$0.450340\pi$
$102$	0	0
$103$	−11.6847	−1.15132	−0.575662	−	0.817688i	$-0.695256\pi$
$103$	−11.6847	−1.15132	−0.575662	+	0.817688i	$0.695256\pi$
$104$	0	0
$105$	0.561553	0.0548019
$106$	0	0
$107$	17.1231	1.65535	0.827677	−	0.561205i	$-0.189662\pi$
$107$	17.1231	1.65535	0.827677	+	0.561205i	$0.189662\pi$
$108$	0	0
$109$	11.9309	1.14277	0.571385	−	0.820682i	$-0.306406\pi$
$109$	11.9309	1.14277	0.571385	+	0.820682i	$0.306406\pi$
$110$	0	0
$111$	3.43845	0.326363
$112$	0	0
$113$	12.2462	1.15203	0.576013	−	0.817440i	$-0.304608\pi$
$113$	12.2462	1.15203	0.576013	+	0.817440i	$0.304608\pi$
$114$	0	0
$115$	−0.807764	−0.0753244
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	5.68466	0.521112
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	−7.12311	−0.642269
$124$	0	0
$125$	5.43845	0.486430
$126$	0	0
$127$	13.1231	1.16449	0.582244	−	0.813014i	$-0.302175\pi$
$127$	13.1231	1.16449	0.582244	+	0.813014i	$0.302175\pi$
$128$	0	0
$129$	−10.5616	−0.929893
$130$	0	0
$131$	5.43845	0.475159	0.237580	−	0.971368i	$-0.423646\pi$
$131$	5.43845	0.475159	0.237580	+	0.971368i	$0.423646\pi$
$132$	0	0
$133$	−7.68466	−0.666344
$134$	0	0
$135$	0.561553	0.0483308
$136$	0	0
$137$	0.561553	0.0479767	0.0239883	−	0.999712i	$-0.492364\pi$
$137$	0.561553	0.0479767	0.0239883	+	0.999712i	$0.492364\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.56155	−0.214208
$144$	0	0
$145$	3.19224	0.265101
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	23.6155	1.93466	0.967330	−	0.253522i	$-0.0815889\pi$
$149$	23.6155	1.93466	0.967330	+	0.253522i	$0.0815889\pi$
$150$	0	0
$151$	8.80776	0.716766	0.358383	−	0.933575i	$-0.383328\pi$
$151$	8.80776	0.716766	0.358383	+	0.933575i	$0.383328\pi$
$152$	0	0
$153$	5.68466	0.459577
$154$	0	0
$155$	−5.75379	−0.462155
$156$	0	0
$157$	−11.4384	−0.912887	−0.456444	−	0.889752i	$-0.650877\pi$
$157$	−11.4384	−0.912887	−0.456444	+	0.889752i	$0.650877\pi$
$158$	0	0
$159$	4.24621	0.336746
$160$	0	0
$161$	1.43845	0.113366
$162$	0	0
$163$	25.1231	1.96779	0.983897	−	0.178738i	$-0.0572014\pi$
$163$	25.1231	1.96779	0.983897	+	0.178738i	$0.0572014\pi$
$164$	0	0
$165$	−1.43845	−0.111983
$166$	0	0
$167$	5.93087	0.458944	0.229472	−	0.973315i	$-0.426300\pi$
$167$	5.93087	0.458944	0.229472	+	0.973315i	$0.426300\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.68466	−0.587661
$172$	0	0
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	0	0
$175$	−4.68466	−0.354127
$176$	0	0
$177$	−14.2462	−1.07081
$178$	0	0
$179$	16.4924	1.23270	0.616351	−	0.787472i	$-0.288610\pi$
$179$	16.4924	1.23270	0.616351	+	0.787472i	$0.288610\pi$
$180$	0	0
$181$	16.2462	1.20757	0.603786	−	0.797147i	$-0.293658\pi$
$181$	16.2462	1.20757	0.603786	+	0.797147i	$0.293658\pi$
$182$	0	0
$183$	5.68466	0.420222
$184$	0	0
$185$	1.93087	0.141960
$186$	0	0
$187$	−14.5616	−1.06485
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	13.9309	1.00800	0.504001	−	0.863703i	$-0.331861\pi$
$191$	13.9309	1.00800	0.504001	+	0.863703i	$0.331861\pi$
$192$	0	0
$193$	−5.36932	−0.386492	−0.193246	−	0.981150i	$-0.561902\pi$
$193$	−5.36932	−0.386492	−0.193246	+	0.981150i	$0.561902\pi$
$194$	0	0
$195$	0.561553	0.0402136
$196$	0	0
$197$	19.1231	1.36246	0.681232	−	0.732067i	$-0.261444\pi$
$197$	19.1231	1.36246	0.681232	+	0.732067i	$0.261444\pi$
$198$	0	0
$199$	21.9309	1.55464	0.777319	−	0.629107i	$-0.216579\pi$
$199$	21.9309	1.55464	0.777319	+	0.629107i	$0.216579\pi$
$200$	0	0
$201$	1.12311	0.0792178
$202$	0	0
$203$	−5.68466	−0.398985
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	1.43845	0.0999790
$208$	0	0
$209$	19.6847	1.36162
$210$	0	0
$211$	−4.80776	−0.330980	−0.165490	−	0.986211i	$-0.552921\pi$
$211$	−4.80776	−0.330980	−0.165490	+	0.986211i	$0.552921\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	−5.93087	−0.404482
$216$	0	0
$217$	10.2462	0.695558
$218$	0	0
$219$	−0.561553	−0.0379462
$220$	0	0
$221$	5.68466	0.382392
$222$	0	0
$223$	−17.6155	−1.17962	−0.589812	−	0.807541i	$-0.700798\pi$
$223$	−17.6155	−1.17962	−0.589812	+	0.807541i	$0.700798\pi$
$224$	0	0
$225$	−4.68466	−0.312311
$226$	0	0
$227$	−1.12311	−0.0745431	−0.0372716	−	0.999305i	$-0.511867\pi$
$227$	−1.12311	−0.0745431	−0.0372716	+	0.999305i	$0.511867\pi$
$228$	0	0
$229$	11.7538	0.776712	0.388356	−	0.921509i	$-0.373043\pi$
$229$	11.7538	0.776712	0.388356	+	0.921509i	$0.373043\pi$
$230$	0	0
$231$	2.56155	0.168538
$232$	0	0
$233$	2.63068	0.172342	0.0861709	−	0.996280i	$-0.472537\pi$
$233$	2.63068	0.172342	0.0861709	+	0.996280i	$0.472537\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.87689	−0.186874
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.561553	−0.0358763
$246$	0	0
$247$	−7.68466	−0.488963
$248$	0	0
$249$	17.1231	1.08513
$250$	0	0
$251$	−12.8078	−0.808419	−0.404209	−	0.914666i	$-0.632453\pi$
$251$	−12.8078	−0.808419	−0.404209	+	0.914666i	$0.632453\pi$
$252$	0	0
$253$	−3.68466	−0.231652
$254$	0	0
$255$	3.19224	0.199906
$256$	0	0
$257$	12.2462	0.763898	0.381949	−	0.924183i	$-0.375253\pi$
$257$	12.2462	0.763898	0.381949	+	0.924183i	$0.375253\pi$
$258$	0	0
$259$	−3.43845	−0.213655
$260$	0	0
$261$	−5.68466	−0.351872
$262$	0	0
$263$	13.7538	0.848095	0.424047	−	0.905640i	$-0.360609\pi$
$263$	13.7538	0.848095	0.424047	+	0.905640i	$0.360609\pi$
$264$	0	0
$265$	2.38447	0.146477
$266$	0	0
$267$	−10.0000	−0.611990
$268$	0	0
$269$	3.75379	0.228873	0.114436	−	0.993431i	$-0.463494\pi$
$269$	3.75379	0.228873	0.114436	+	0.993431i	$0.463494\pi$
$270$	0	0
$271$	−13.1231	−0.797172	−0.398586	−	0.917131i	$-0.630499\pi$
$271$	−13.1231	−0.797172	−0.398586	+	0.917131i	$0.630499\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	10.4924	0.630429	0.315214	−	0.949021i	$-0.397924\pi$
$277$	10.4924	0.630429	0.315214	+	0.949021i	$0.397924\pi$
$278$	0	0
$279$	10.2462	0.613425
$280$	0	0
$281$	−0.246211	−0.0146877	−0.00734387	−	0.999973i	$-0.502338\pi$
$281$	−0.246211	−0.0146877	−0.00734387	+	0.999973i	$0.502338\pi$
$282$	0	0
$283$	−1.75379	−0.104252	−0.0521260	−	0.998641i	$-0.516600\pi$
$283$	−1.75379	−0.104252	−0.0521260	+	0.998641i	$0.516600\pi$
$284$	0	0
$285$	−4.31534	−0.255619
$286$	0	0
$287$	7.12311	0.420464
$288$	0	0
$289$	15.3153	0.900902
$290$	0	0
$291$	18.4924	1.08405
$292$	0	0
$293$	−24.7386	−1.44525	−0.722623	−	0.691242i	$-0.757064\pi$
$293$	−24.7386	−1.44525	−0.722623	+	0.691242i	$0.757064\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	2.56155	0.148636
$298$	0	0
$299$	1.43845	0.0831875
$300$	0	0
$301$	10.5616	0.608758
$302$	0	0
$303$	−3.12311	−0.179418
$304$	0	0
$305$	3.19224	0.182787
$306$	0	0
$307$	−9.75379	−0.556678	−0.278339	−	0.960483i	$-0.589784\pi$
$307$	−9.75379	−0.556678	−0.278339	+	0.960483i	$0.589784\pi$
$308$	0	0
$309$	11.6847	0.664717
$310$	0	0
$311$	21.1231	1.19778	0.598891	−	0.800831i	$-0.295608\pi$
$311$	21.1231	1.19778	0.598891	+	0.800831i	$0.295608\pi$
$312$	0	0
$313$	27.6155	1.56092	0.780461	−	0.625205i	$-0.214984\pi$
$313$	27.6155	1.56092	0.780461	+	0.625205i	$0.214984\pi$
$314$	0	0
$315$	−0.561553	−0.0316399
$316$	0	0
$317$	11.1231	0.624736	0.312368	−	0.949961i	$-0.398878\pi$
$317$	11.1231	0.624736	0.312368	+	0.949961i	$0.398878\pi$
$318$	0	0
$319$	14.5616	0.815290
$320$	0	0
$321$	−17.1231	−0.955719
$322$	0	0
$323$	−43.6847	−2.43068
$324$	0	0
$325$	−4.68466	−0.259858
$326$	0	0
$327$	−11.9309	−0.659779
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.3693	0.624914	0.312457	−	0.949932i	$-0.398848\pi$
$331$	11.3693	0.624914	0.312457	+	0.949932i	$0.398848\pi$
$332$	0	0
$333$	−3.43845	−0.188426
$334$	0	0
$335$	0.630683	0.0344579
$336$	0	0
$337$	8.56155	0.466377	0.233189	−	0.972431i	$-0.425084\pi$
$337$	8.56155	0.466377	0.233189	+	0.972431i	$0.425084\pi$
$338$	0	0
$339$	−12.2462	−0.665123
$340$	0	0
$341$	−26.2462	−1.42131
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.807764	0.0434886
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	24.2462	1.29787	0.648935	−	0.760844i	$-0.275215\pi$
$349$	24.2462	1.29787	0.648935	+	0.760844i	$0.275215\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−2.49242	−0.132658	−0.0663291	−	0.997798i	$-0.521129\pi$
$353$	−2.49242	−0.132658	−0.0663291	+	0.997798i	$0.521129\pi$
$354$	0	0
$355$	4.49242	0.238433
$356$	0	0
$357$	−5.68466	−0.300864
$358$	0	0
$359$	−22.7386	−1.20010	−0.600050	−	0.799963i	$-0.704852\pi$
$359$	−22.7386	−1.20010	−0.600050	+	0.799963i	$0.704852\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	0	0
$363$	4.43845	0.232958
$364$	0	0
$365$	−0.315342	−0.0165057
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	7.12311	0.370814
$370$	0	0
$371$	−4.24621	−0.220452
$372$	0	0
$373$	23.6155	1.22277	0.611383	−	0.791335i	$-0.290614\pi$
$373$	23.6155	1.22277	0.611383	+	0.791335i	$0.290614\pi$
$374$	0	0
$375$	−5.43845	−0.280840
$376$	0	0
$377$	−5.68466	−0.292775
$378$	0	0
$379$	−17.7538	−0.911951	−0.455975	−	0.889992i	$-0.650710\pi$
$379$	−17.7538	−0.911951	−0.455975	+	0.889992i	$0.650710\pi$
$380$	0	0
$381$	−13.1231	−0.672317
$382$	0	0
$383$	34.4233	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$383$	34.4233	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$384$	0	0
$385$	1.43845	0.0733101
$386$	0	0
$387$	10.5616	0.536874
$388$	0	0
$389$	20.7386	1.05149	0.525745	−	0.850642i	$-0.323787\pi$
$389$	20.7386	1.05149	0.525745	+	0.850642i	$0.323787\pi$
$390$	0	0
$391$	8.17708	0.413533
$392$	0	0
$393$	−5.43845	−0.274333
$394$	0	0
$395$	−1.61553	−0.0812860
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	7.68466	0.384714
$400$	0	0
$401$	−8.24621	−0.411796	−0.205898	−	0.978573i	$-0.566012\pi$
$401$	−8.24621	−0.411796	−0.205898	+	0.978573i	$0.566012\pi$
$402$	0	0
$403$	10.2462	0.510400
$404$	0	0
$405$	−0.561553	−0.0279038
$406$	0	0
$407$	8.80776	0.436585
$408$	0	0
$409$	23.9309	1.18331	0.591653	−	0.806193i	$-0.298476\pi$
$409$	23.9309	1.18331	0.591653	+	0.806193i	$0.298476\pi$
$410$	0	0
$411$	−0.561553	−0.0276994
$412$	0	0
$413$	14.2462	0.701010
$414$	0	0
$415$	9.61553	0.472008
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	24.3153	1.18788	0.593941	−	0.804509i	$-0.297571\pi$
$419$	24.3153	1.18788	0.593941	+	0.804509i	$0.297571\pi$
$420$	0	0
$421$	−20.2462	−0.986740	−0.493370	−	0.869820i	$-0.664235\pi$
$421$	−20.2462	−0.986740	−0.493370	+	0.869820i	$0.664235\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−26.6307	−1.29178
$426$	0	0
$427$	−5.68466	−0.275100
$428$	0	0
$429$	2.56155	0.123673
$430$	0	0
$431$	8.63068	0.415725	0.207863	−	0.978158i	$-0.433349\pi$
$431$	8.63068	0.415725	0.207863	+	0.978158i	$0.433349\pi$
$432$	0	0
$433$	−6.63068	−0.318650	−0.159325	−	0.987226i	$-0.550932\pi$
$433$	−6.63068	−0.318650	−0.159325	+	0.987226i	$0.550932\pi$
$434$	0	0
$435$	−3.19224	−0.153056
$436$	0	0
$437$	−11.0540	−0.528783
$438$	0	0
$439$	−21.9309	−1.04670	−0.523352	−	0.852117i	$-0.675319\pi$
$439$	−21.9309	−1.04670	−0.523352	+	0.852117i	$0.675319\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−19.3693	−0.920264	−0.460132	−	0.887851i	$-0.652198\pi$
$443$	−19.3693	−0.920264	−0.460132	+	0.887851i	$0.652198\pi$
$444$	0	0
$445$	−5.61553	−0.266202
$446$	0	0
$447$	−23.6155	−1.11698
$448$	0	0
$449$	−1.68466	−0.0795039	−0.0397520	−	0.999210i	$-0.512657\pi$
$449$	−1.68466	−0.0795039	−0.0397520	+	0.999210i	$0.512657\pi$
$450$	0	0
$451$	−18.2462	−0.859181
$452$	0	0
$453$	−8.80776	−0.413825
$454$	0	0
$455$	−0.561553	−0.0263260
$456$	0	0
$457$	2.63068	0.123058	0.0615291	−	0.998105i	$-0.480402\pi$
$457$	2.63068	0.123058	0.0615291	+	0.998105i	$0.480402\pi$
$458$	0	0
$459$	−5.68466	−0.265337
$460$	0	0
$461$	9.05398	0.421686	0.210843	−	0.977520i	$-0.432379\pi$
$461$	9.05398	0.421686	0.210843	+	0.977520i	$0.432379\pi$
$462$	0	0
$463$	−3.68466	−0.171241	−0.0856203	−	0.996328i	$-0.527287\pi$
$463$	−3.68466	−0.171241	−0.0856203	+	0.996328i	$0.527287\pi$
$464$	0	0
$465$	5.75379	0.266826
$466$	0	0
$467$	15.6847	0.725799	0.362900	−	0.931828i	$-0.381787\pi$
$467$	15.6847	0.725799	0.362900	+	0.931828i	$0.381787\pi$
$468$	0	0
$469$	−1.12311	−0.0518602
$470$	0	0
$471$	11.4384	0.527056
$472$	0	0
$473$	−27.0540	−1.24394
$474$	0	0
$475$	36.0000	1.65179
$476$	0	0
$477$	−4.24621	−0.194421
$478$	0	0
$479$	21.3002	0.973230	0.486615	−	0.873616i	$-0.338231\pi$
$479$	21.3002	0.973230	0.486615	+	0.873616i	$0.338231\pi$
$480$	0	0
$481$	−3.43845	−0.156780
$482$	0	0
$483$	−1.43845	−0.0654516
$484$	0	0
$485$	10.3845	0.471535
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−25.1231	−1.13611
$490$	0	0
$491$	17.1231	0.772755	0.386377	−	0.922341i	$-0.373726\pi$
$491$	17.1231	0.772755	0.386377	+	0.922341i	$0.373726\pi$
$492$	0	0
$493$	−32.3153	−1.45541
$494$	0	0
$495$	1.43845	0.0646534
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	−5.93087	−0.264972
$502$	0	0
$503$	−5.12311	−0.228428	−0.114214	−	0.993456i	$-0.536435\pi$
$503$	−5.12311	−0.228428	−0.114214	+	0.993456i	$0.536435\pi$
$504$	0	0
$505$	−1.75379	−0.0780426
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	25.0540	1.11050	0.555249	−	0.831684i	$-0.312623\pi$
$509$	25.0540	1.11050	0.555249	+	0.831684i	$0.312623\pi$
$510$	0	0
$511$	0.561553	0.0248416
$512$	0	0
$513$	7.68466	0.339286
$514$	0	0
$515$	6.56155	0.289137
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3.75379	−0.164773
$520$	0	0
$521$	−9.68466	−0.424293	−0.212146	−	0.977238i	$-0.568045\pi$
$521$	−9.68466	−0.424293	−0.212146	+	0.977238i	$0.568045\pi$
$522$	0	0
$523$	−38.2462	−1.67239	−0.836195	−	0.548432i	$-0.815225\pi$
$523$	−38.2462	−1.67239	−0.836195	+	0.548432i	$0.815225\pi$
$524$	0	0
$525$	4.68466	0.204455
$526$	0	0
$527$	58.2462	2.53724
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	14.2462	0.618233
$532$	0	0
$533$	7.12311	0.308536
$534$	0	0
$535$	−9.61553	−0.415716
$536$	0	0
$537$	−16.4924	−0.711701
$538$	0	0
$539$	−2.56155	−0.110334
$540$	0	0
$541$	−4.06913	−0.174946	−0.0874728	−	0.996167i	$-0.527879\pi$
$541$	−4.06913	−0.174946	−0.0874728	+	0.996167i	$0.527879\pi$
$542$	0	0
$543$	−16.2462	−0.697192
$544$	0	0
$545$	−6.69981	−0.286988
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	−5.68466	−0.242615
$550$	0	0
$551$	43.6847	1.86103
$552$	0	0
$553$	2.87689	0.122338
$554$	0	0
$555$	−1.93087	−0.0819609
$556$	0	0
$557$	21.3693	0.905447	0.452724	−	0.891651i	$-0.350452\pi$
$557$	21.3693	0.905447	0.452724	+	0.891651i	$0.350452\pi$
$558$	0	0
$559$	10.5616	0.446706
$560$	0	0
$561$	14.5616	0.614789
$562$	0	0
$563$	−31.0540	−1.30877	−0.654385	−	0.756162i	$-0.727072\pi$
$563$	−31.0540	−1.30877	−0.654385	+	0.756162i	$0.727072\pi$
$564$	0	0
$565$	−6.87689	−0.289313
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−41.2311	−1.72850	−0.864248	−	0.503066i	$-0.832205\pi$
$569$	−41.2311	−1.72850	−0.864248	+	0.503066i	$0.832205\pi$
$570$	0	0
$571$	−1.75379	−0.0733938	−0.0366969	−	0.999326i	$-0.511684\pi$
$571$	−1.75379	−0.0733938	−0.0366969	+	0.999326i	$0.511684\pi$
$572$	0	0
$573$	−13.9309	−0.581970
$574$	0	0
$575$	−6.73863	−0.281020
$576$	0	0
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	0	0
$579$	5.36932	0.223141
$580$	0	0
$581$	−17.1231	−0.710386
$582$	0	0
$583$	10.8769	0.450475
$584$	0	0
$585$	−0.561553	−0.0232174
$586$	0	0
$587$	11.3693	0.469262	0.234631	−	0.972085i	$-0.424612\pi$
$587$	11.3693	0.469262	0.234631	+	0.972085i	$0.424612\pi$
$588$	0	0
$589$	−78.7386	−3.24437
$590$	0	0
$591$	−19.1231	−0.786619
$592$	0	0
$593$	−3.75379	−0.154150	−0.0770748	−	0.997025i	$-0.524558\pi$
$593$	−3.75379	−0.154150	−0.0770748	+	0.997025i	$0.524558\pi$
$594$	0	0
$595$	−3.19224	−0.130869
$596$	0	0
$597$	−21.9309	−0.897571
$598$	0	0
$599$	−17.4384	−0.712516	−0.356258	−	0.934388i	$-0.615948\pi$
$599$	−17.4384	−0.712516	−0.356258	+	0.934388i	$0.615948\pi$
$600$	0	0
$601$	−19.1231	−0.780048	−0.390024	−	0.920805i	$-0.627533\pi$
$601$	−19.1231	−0.780048	−0.390024	+	0.920805i	$0.627533\pi$
$602$	0	0
$603$	−1.12311	−0.0457364
$604$	0	0
$605$	2.49242	0.101331
$606$	0	0
$607$	22.5616	0.915745	0.457873	−	0.889018i	$-0.348612\pi$
$607$	22.5616	0.915745	0.457873	+	0.889018i	$0.348612\pi$
$608$	0	0
$609$	5.68466	0.230354
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−25.5464	−1.03181	−0.515905	−	0.856646i	$-0.672544\pi$
$613$	−25.5464	−1.03181	−0.515905	+	0.856646i	$0.672544\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	28.4233	1.14428	0.572139	−	0.820156i	$-0.306114\pi$
$617$	28.4233	1.14428	0.572139	+	0.820156i	$0.306114\pi$
$618$	0	0
$619$	−31.6847	−1.27351	−0.636757	−	0.771065i	$-0.719725\pi$
$619$	−31.6847	−1.27351	−0.636757	+	0.771065i	$0.719725\pi$
$620$	0	0
$621$	−1.43845	−0.0577229
$622$	0	0
$623$	10.0000	0.400642
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	−19.6847	−0.786130
$628$	0	0
$629$	−19.5464	−0.779366
$630$	0	0
$631$	−5.93087	−0.236104	−0.118052	−	0.993007i	$-0.537665\pi$
$631$	−5.93087	−0.236104	−0.118052	+	0.993007i	$0.537665\pi$
$632$	0	0
$633$	4.80776	0.191091
$634$	0	0
$635$	−7.36932	−0.292442
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−3.75379	−0.148266	−0.0741329	−	0.997248i	$-0.523619\pi$
$641$	−3.75379	−0.148266	−0.0741329	+	0.997248i	$0.523619\pi$
$642$	0	0
$643$	−36.8078	−1.45156	−0.725778	−	0.687929i	$-0.758520\pi$
$643$	−36.8078	−1.45156	−0.725778	+	0.687929i	$0.758520\pi$
$644$	0	0
$645$	5.93087	0.233528
$646$	0	0
$647$	−2.24621	−0.0883077	−0.0441538	−	0.999025i	$-0.514059\pi$
$647$	−2.24621	−0.0883077	−0.0441538	+	0.999025i	$0.514059\pi$
$648$	0	0
$649$	−36.4924	−1.43245
$650$	0	0
$651$	−10.2462	−0.401581
$652$	0	0
$653$	11.9309	0.466891	0.233446	−	0.972370i	$-0.425000\pi$
$653$	11.9309	0.466891	0.233446	+	0.972370i	$0.425000\pi$
$654$	0	0
$655$	−3.05398	−0.119329
$656$	0	0
$657$	0.561553	0.0219083
$658$	0	0
$659$	3.36932	0.131250	0.0656250	−	0.997844i	$-0.479096\pi$
$659$	3.36932	0.131250	0.0656250	+	0.997844i	$0.479096\pi$
$660$	0	0
$661$	16.2462	0.631904	0.315952	−	0.948775i	$-0.397676\pi$
$661$	16.2462	0.631904	0.315952	+	0.948775i	$0.397676\pi$
$662$	0	0
$663$	−5.68466	−0.220774
$664$	0	0
$665$	4.31534	0.167342
$666$	0	0
$667$	−8.17708	−0.316618
$668$	0	0
$669$	17.6155	0.681056
$670$	0	0
$671$	14.5616	0.562143
$672$	0	0
$673$	−13.1922	−0.508523	−0.254262	−	0.967135i	$-0.581832\pi$
$673$	−13.1922	−0.508523	−0.254262	+	0.967135i	$0.581832\pi$
$674$	0	0
$675$	4.68466	0.180313
$676$	0	0
$677$	−30.4924	−1.17192	−0.585959	−	0.810340i	$-0.699282\pi$
$677$	−30.4924	−1.17192	−0.585959	+	0.810340i	$0.699282\pi$
$678$	0	0
$679$	−18.4924	−0.709674
$680$	0	0
$681$	1.12311	0.0430375
$682$	0	0
$683$	−47.6847	−1.82460	−0.912301	−	0.409519i	$-0.865696\pi$
$683$	−47.6847	−1.82460	−0.912301	+	0.409519i	$0.865696\pi$
$684$	0	0
$685$	−0.315342	−0.0120486
$686$	0	0
$687$	−11.7538	−0.448435
$688$	0	0
$689$	−4.24621	−0.161768
$690$	0	0
$691$	16.4924	0.627401	0.313701	−	0.949522i	$-0.398431\pi$
$691$	16.4924	0.627401	0.313701	+	0.949522i	$0.398431\pi$
$692$	0	0
$693$	−2.56155	−0.0973053
$694$	0	0
$695$	6.73863	0.255611
$696$	0	0
$697$	40.4924	1.53376
$698$	0	0
$699$	−2.63068	−0.0995016
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	26.4233	0.996573
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.12311	0.117456
$708$	0	0
$709$	0.246211	0.00924666	0.00462333	−	0.999989i	$-0.498528\pi$
$709$	0.246211	0.00924666	0.00462333	+	0.999989i	$0.498528\pi$
$710$	0	0
$711$	2.87689	0.107892
$712$	0	0
$713$	14.7386	0.551966
$714$	0	0
$715$	1.43845	0.0537949
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	−34.8769	−1.30069	−0.650344	−	0.759640i	$-0.725375\pi$
$719$	−34.8769	−1.30069	−0.650344	+	0.759640i	$0.725375\pi$
$720$	0	0
$721$	−11.6847	−0.435159
$722$	0	0
$723$	14.0000	0.520666
$724$	0	0
$725$	26.6307	0.989039
$726$	0	0
$727$	−21.9309	−0.813371	−0.406685	−	0.913568i	$-0.633316\pi$
$727$	−21.9309	−0.813371	−0.406685	+	0.913568i	$0.633316\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	60.0388	2.22062
$732$	0	0
$733$	−0.384472	−0.0142008	−0.00710040	−	0.999975i	$-0.502260\pi$
$733$	−0.384472	−0.0142008	−0.00710040	+	0.999975i	$0.502260\pi$
$734$	0	0
$735$	0.561553	0.0207132
$736$	0	0
$737$	2.87689	0.105972
$738$	0	0
$739$	−42.1080	−1.54897	−0.774483	−	0.632595i	$-0.781990\pi$
$739$	−42.1080	−1.54897	−0.774483	+	0.632595i	$0.781990\pi$
$740$	0	0
$741$	7.68466	0.282303
$742$	0	0
$743$	46.1080	1.69154	0.845768	−	0.533550i	$-0.179143\pi$
$743$	46.1080	1.69154	0.845768	+	0.533550i	$0.179143\pi$
$744$	0	0
$745$	−13.2614	−0.485859
$746$	0	0
$747$	−17.1231	−0.626502
$748$	0	0
$749$	17.1231	0.625665
$750$	0	0
$751$	−10.2462	−0.373890	−0.186945	−	0.982370i	$-0.559859\pi$
$751$	−10.2462	−0.373890	−0.186945	+	0.982370i	$0.559859\pi$
$752$	0	0
$753$	12.8078	0.466741
$754$	0	0
$755$	−4.94602	−0.180004
$756$	0	0
$757$	1.50758	0.0547938	0.0273969	−	0.999625i	$-0.491278\pi$
$757$	1.50758	0.0547938	0.0273969	+	0.999625i	$0.491278\pi$
$758$	0	0
$759$	3.68466	0.133745
$760$	0	0
$761$	−3.12311	−0.113212	−0.0566062	−	0.998397i	$-0.518028\pi$
$761$	−3.12311	−0.113212	−0.0566062	+	0.998397i	$0.518028\pi$
$762$	0	0
$763$	11.9309	0.431926
$764$	0	0
$765$	−3.19224	−0.115416
$766$	0	0
$767$	14.2462	0.514401
$768$	0	0
$769$	−20.5616	−0.741469	−0.370734	−	0.928739i	$-0.620894\pi$
$769$	−20.5616	−0.741469	−0.370734	+	0.928739i	$0.620894\pi$
$770$	0	0
$771$	−12.2462	−0.441037
$772$	0	0
$773$	24.0691	0.865706	0.432853	−	0.901464i	$-0.357507\pi$
$773$	24.0691	0.865706	0.432853	+	0.901464i	$0.357507\pi$
$774$	0	0
$775$	−48.0000	−1.72421
$776$	0	0
$777$	3.43845	0.123354
$778$	0	0
$779$	−54.7386	−1.96122
$780$	0	0
$781$	20.4924	0.733277
$782$	0	0
$783$	5.68466	0.203153
$784$	0	0
$785$	6.42329	0.229257
$786$	0	0
$787$	33.3002	1.18702	0.593512	−	0.804825i	$-0.297741\pi$
$787$	33.3002	1.18702	0.593512	+	0.804825i	$0.297741\pi$
$788$	0	0
$789$	−13.7538	−0.489648
$790$	0	0
$791$	12.2462	0.435425
$792$	0	0
$793$	−5.68466	−0.201868
$794$	0	0
$795$	−2.38447	−0.0845685
$796$	0	0
$797$	6.63068	0.234871	0.117435	−	0.993081i	$-0.462533\pi$
$797$	6.63068	0.234871	0.117435	+	0.993081i	$0.462533\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	−1.43845	−0.0507617
$804$	0	0
$805$	−0.807764	−0.0284699
$806$	0	0
$807$	−3.75379	−0.132140
$808$	0	0
$809$	30.4924	1.07206	0.536028	−	0.844200i	$-0.319924\pi$
$809$	30.4924	1.07206	0.536028	+	0.844200i	$0.319924\pi$
$810$	0	0
$811$	−37.4384	−1.31464	−0.657321	−	0.753611i	$-0.728310\pi$
$811$	−37.4384	−1.31464	−0.657321	+	0.753611i	$0.728310\pi$
$812$	0	0
$813$	13.1231	0.460247
$814$	0	0
$815$	−14.1080	−0.494180
$816$	0	0
$817$	−81.1619	−2.83950
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	55.6155	1.94100	0.970498	−	0.241111i	$-0.0775116\pi$
$821$	55.6155	1.94100	0.970498	+	0.241111i	$0.0775116\pi$
$822$	0	0
$823$	28.4924	0.993183	0.496592	−	0.867984i	$-0.334585\pi$
$823$	28.4924	0.993183	0.496592	+	0.867984i	$0.334585\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	1.93087	0.0671429	0.0335715	−	0.999436i	$-0.489312\pi$
$827$	1.93087	0.0671429	0.0335715	+	0.999436i	$0.489312\pi$
$828$	0	0
$829$	26.3153	0.913970	0.456985	−	0.889475i	$-0.348929\pi$
$829$	26.3153	0.913970	0.456985	+	0.889475i	$0.348929\pi$
$830$	0	0
$831$	−10.4924	−0.363978
$832$	0	0
$833$	5.68466	0.196962
$834$	0	0
$835$	−3.33050	−0.115257
$836$	0	0
$837$	−10.2462	−0.354161
$838$	0	0
$839$	−38.7386	−1.33741	−0.668703	−	0.743530i	$-0.733150\pi$
$839$	−38.7386	−1.33741	−0.668703	+	0.743530i	$0.733150\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	0.246211	0.00847997
$844$	0	0
$845$	−0.561553	−0.0193180
$846$	0	0
$847$	−4.43845	−0.152507
$848$	0	0
$849$	1.75379	0.0601899
$850$	0	0
$851$	−4.94602	−0.169548
$852$	0	0
$853$	−2.63068	−0.0900729	−0.0450364	−	0.998985i	$-0.514340\pi$
$853$	−2.63068	−0.0900729	−0.0450364	+	0.998985i	$0.514340\pi$
$854$	0	0
$855$	4.31534	0.147582
$856$	0	0
$857$	−1.50758	−0.0514979	−0.0257489	−	0.999668i	$-0.508197\pi$
$857$	−1.50758	−0.0514979	−0.0257489	+	0.999668i	$0.508197\pi$
$858$	0	0
$859$	−22.2462	−0.759031	−0.379515	−	0.925185i	$-0.623909\pi$
$859$	−22.2462	−0.759031	−0.379515	+	0.925185i	$0.623909\pi$
$860$	0	0
$861$	−7.12311	−0.242755
$862$	0	0
$863$	−7.36932	−0.250854	−0.125427	−	0.992103i	$-0.540030\pi$
$863$	−7.36932	−0.250854	−0.125427	+	0.992103i	$0.540030\pi$
$864$	0	0
$865$	−2.10795	−0.0716725
$866$	0	0
$867$	−15.3153	−0.520136
$868$	0	0
$869$	−7.36932	−0.249987
$870$	0	0
$871$	−1.12311	−0.0380550
$872$	0	0
$873$	−18.4924	−0.625874
$874$	0	0
$875$	5.43845	0.183853
$876$	0	0
$877$	−23.7538	−0.802108	−0.401054	−	0.916054i	$-0.631356\pi$
$877$	−23.7538	−0.802108	−0.401054	+	0.916054i	$0.631356\pi$
$878$	0	0
$879$	24.7386	0.834413
$880$	0	0
$881$	26.1771	0.881928	0.440964	−	0.897525i	$-0.354637\pi$
$881$	26.1771	0.881928	0.440964	+	0.897525i	$0.354637\pi$
$882$	0	0
$883$	2.56155	0.0862031	0.0431016	−	0.999071i	$-0.486276\pi$
$883$	2.56155	0.0862031	0.0431016	+	0.999071i	$0.486276\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	19.5076	0.655000	0.327500	−	0.944851i	$-0.393794\pi$
$887$	19.5076	0.655000	0.327500	+	0.944851i	$0.393794\pi$
$888$	0	0
$889$	13.1231	0.440135
$890$	0	0
$891$	−2.56155	−0.0858152
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−9.26137	−0.309573
$896$	0	0
$897$	−1.43845	−0.0480284
$898$	0	0
$899$	−58.2462	−1.94262
$900$	0	0
$901$	−24.1383	−0.804162
$902$	0	0
$903$	−10.5616	−0.351466
$904$	0	0
$905$	−9.12311	−0.303262
$906$	0	0
$907$	40.4924	1.34453	0.672264	−	0.740311i	$-0.265322\pi$
$907$	40.4924	1.34453	0.672264	+	0.740311i	$0.265322\pi$
$908$	0	0
$909$	3.12311	0.103587
$910$	0	0
$911$	−50.4233	−1.67060	−0.835299	−	0.549796i	$-0.814706\pi$
$911$	−50.4233	−1.67060	−0.835299	+	0.549796i	$0.814706\pi$
$912$	0	0
$913$	43.8617	1.45161
$914$	0	0
$915$	−3.19224	−0.105532
$916$	0	0
$917$	5.43845	0.179593
$918$	0	0
$919$	−10.8769	−0.358796	−0.179398	−	0.983777i	$-0.557415\pi$
$919$	−10.8769	−0.358796	−0.179398	+	0.983777i	$0.557415\pi$
$920$	0	0
$921$	9.75379	0.321398
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	16.1080	0.529626
$926$	0	0
$927$	−11.6847	−0.383775
$928$	0	0
$929$	−15.6155	−0.512329	−0.256164	−	0.966633i	$-0.582459\pi$
$929$	−15.6155	−0.512329	−0.256164	+	0.966633i	$0.582459\pi$
$930$	0	0
$931$	−7.68466	−0.251855
$932$	0	0
$933$	−21.1231	−0.691539
$934$	0	0
$935$	8.17708	0.267419
$936$	0	0
$937$	18.9848	0.620208	0.310104	−	0.950703i	$-0.399636\pi$
$937$	18.9848	0.620208	0.310104	+	0.950703i	$0.399636\pi$
$938$	0	0
$939$	−27.6155	−0.901199
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	0	0
$943$	10.2462	0.333663
$944$	0	0
$945$	0.561553	0.0182673
$946$	0	0
$947$	−45.7926	−1.48806	−0.744030	−	0.668146i	$-0.767088\pi$
$947$	−45.7926	−1.48806	−0.744030	+	0.668146i	$0.767088\pi$
$948$	0	0
$949$	0.561553	0.0182288
$950$	0	0
$951$	−11.1231	−0.360691
$952$	0	0
$953$	−13.3693	−0.433075	−0.216537	−	0.976274i	$-0.569476\pi$
$953$	−13.3693	−0.433075	−0.216537	+	0.976274i	$0.569476\pi$
$954$	0	0
$955$	−7.82292	−0.253144
$956$	0	0
$957$	−14.5616	−0.470708
$958$	0	0
$959$	0.561553	0.0181335
$960$	0	0
$961$	73.9848	2.38661
$962$	0	0
$963$	17.1231	0.551784
$964$	0	0
$965$	3.01515	0.0970613
$966$	0	0
$967$	17.4384	0.560783	0.280391	−	0.959886i	$-0.409536\pi$
$967$	17.4384	0.560783	0.280391	+	0.959886i	$0.409536\pi$
$968$	0	0
$969$	43.6847	1.40335
$970$	0	0
$971$	14.2462	0.457183	0.228591	−	0.973522i	$-0.426588\pi$
$971$	14.2462	0.457183	0.228591	+	0.973522i	$0.426588\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	4.68466	0.150029
$976$	0	0
$977$	−10.3153	−0.330017	−0.165009	−	0.986292i	$-0.552765\pi$
$977$	−10.3153	−0.330017	−0.165009	+	0.986292i	$0.552765\pi$
$978$	0	0
$979$	−25.6155	−0.818676
$980$	0	0
$981$	11.9309	0.380923
$982$	0	0
$983$	−32.1771	−1.02629	−0.513145	−	0.858302i	$-0.671520\pi$
$983$	−32.1771	−1.02629	−0.513145	+	0.858302i	$0.671520\pi$
$984$	0	0
$985$	−10.7386	−0.342161
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.1922	0.483085
$990$	0	0
$991$	−33.6155	−1.06783	−0.533916	−	0.845537i	$-0.679280\pi$
$991$	−33.6155	−1.06783	−0.533916	+	0.845537i	$0.679280\pi$
$992$	0	0
$993$	−11.3693	−0.360794
$994$	0	0
$995$	−12.3153	−0.390423
$996$	0	0
$997$	−8.73863	−0.276755	−0.138378	−	0.990380i	$-0.544189\pi$
$997$	−8.73863	−0.276755	−0.138378	+	0.990380i	$0.544189\pi$
$998$	0	0
$999$	3.43845	0.108788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.be.1.1		2
4.3	odd	2		546.2.a.j.1.1	✓	2
12.11	even	2		1638.2.a.u.1.2		2
28.27	even	2		3822.2.a.bo.1.2		2
52.51	odd	2		7098.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.j.1.1	✓	2	4.3	odd	2
1638.2.a.u.1.2		2	12.11	even	2
3822.2.a.bo.1.2		2	28.27	even	2
4368.2.a.be.1.1		2	1.1	even	1	trivial
7098.2.a.bl.1.2		2	52.51	odd	2

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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.561553 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.561553 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.56155 q^{11} +1.00000 q^{13} +0.561553 q^{15} +5.68466 q^{17} -7.68466 q^{19} -1.00000 q^{21} +1.43845 q^{23} -4.68466 q^{25} -1.00000 q^{27} -5.68466 q^{29} +10.2462 q^{31} +2.56155 q^{33} -0.561553 q^{35} -3.43845 q^{37} -1.00000 q^{39} +7.12311 q^{41} +10.5616 q^{43} -0.561553 q^{45} +1.00000 q^{49} -5.68466 q^{51} -4.24621 q^{53} +1.43845 q^{55} +7.68466 q^{57} +14.2462 q^{59} -5.68466 q^{61} +1.00000 q^{63} -0.561553 q^{65} -1.12311 q^{67} -1.43845 q^{69} -8.00000 q^{71} +0.561553 q^{73} +4.68466 q^{75} -2.56155 q^{77} +2.87689 q^{79} +1.00000 q^{81} -17.1231 q^{83} -3.19224 q^{85} +5.68466 q^{87} +10.0000 q^{89} +1.00000 q^{91} -10.2462 q^{93} +4.31534 q^{95} -18.4924 q^{97} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{15} - q^{17} - 3 q^{19} - 2 q^{21} + 7 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} + 4 q^{31} + q^{33} + 3 q^{35} - 11 q^{37} - 2 q^{39} + 6 q^{41} + 17 q^{43} + 3 q^{45} + 2 q^{49} + q^{51} + 8 q^{53} + 7 q^{55} + 3 q^{57} + 12 q^{59} + q^{61} + 2 q^{63} + 3 q^{65} + 6 q^{67} - 7 q^{69} - 16 q^{71} - 3 q^{73} - 3 q^{75} - q^{77} + 14 q^{79} + 2 q^{81} - 26 q^{83} - 27 q^{85} - q^{87} + 20 q^{89} + 2 q^{91} - 4 q^{93} + 21 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^15 - q^17 - 3 * q^19 - 2 * q^21 + 7 * q^23 + 3 * q^25 - 2 * q^27 + q^29 + 4 * q^31 + q^33 + 3 * q^35 - 11 * q^37 - 2 * q^39 + 6 * q^41 + 17 * q^43 + 3 * q^45 + 2 * q^49 + q^51 + 8 * q^53 + 7 * q^55 + 3 * q^57 + 12 * q^59 + q^61 + 2 * q^63 + 3 * q^65 + 6 * q^67 - 7 * q^69 - 16 * q^71 - 3 * q^73 - 3 * q^75 - q^77 + 14 * q^79 + 2 * q^81 - 26 * q^83 - 27 * q^85 - q^87 + 20 * q^89 + 2 * q^91 - 4 * q^93 + 21 * q^95 - 4 * q^97 - q^99

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