LMFDB - Embedded newform 1638.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$1$
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			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{7} -1.00000 q^{8} -0.561553 q^{10} -2.56155 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.68466 q^{17} +7.68466 q^{19} +0.561553 q^{20} +2.56155 q^{22} +1.43845 q^{23} -4.68466 q^{25} -1.00000 q^{26} -1.00000 q^{28} +5.68466 q^{29} -10.2462 q^{31} -1.00000 q^{32} +5.68466 q^{34} -0.561553 q^{35} -3.43845 q^{37} -7.68466 q^{38} -0.561553 q^{40} -7.12311 q^{41} -10.5616 q^{43} -2.56155 q^{44} -1.43845 q^{46} +1.00000 q^{49} +4.68466 q^{50} +1.00000 q^{52} +4.24621 q^{53} -1.43845 q^{55} +1.00000 q^{56} -5.68466 q^{58} +14.2462 q^{59} -5.68466 q^{61} +10.2462 q^{62} +1.00000 q^{64} +0.561553 q^{65} +1.12311 q^{67} -5.68466 q^{68} +0.561553 q^{70} -8.00000 q^{71} +0.561553 q^{73} +3.43845 q^{74} +7.68466 q^{76} +2.56155 q^{77} -2.87689 q^{79} +0.561553 q^{80} +7.12311 q^{82} -17.1231 q^{83} -3.19224 q^{85} +10.5616 q^{86} +2.56155 q^{88} -10.0000 q^{89} -1.00000 q^{91} +1.43845 q^{92} +4.31534 q^{95} -18.4924 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8} + 3 q^{10} - q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + q^{17} + 3 q^{19} - 3 q^{20} + q^{22} + 7 q^{23} + 3 q^{25} - 2 q^{26} - 2 q^{28} - q^{29} - 4 q^{31} - 2 q^{32} - q^{34} + 3 q^{35} - 11 q^{37} - 3 q^{38} + 3 q^{40} - 6 q^{41} - 17 q^{43} - q^{44} - 7 q^{46} + 2 q^{49} - 3 q^{50} + 2 q^{52} - 8 q^{53} - 7 q^{55} + 2 q^{56} + q^{58} + 12 q^{59} + q^{61} + 4 q^{62} + 2 q^{64} - 3 q^{65} - 6 q^{67} + q^{68} - 3 q^{70} - 16 q^{71} - 3 q^{73} + 11 q^{74} + 3 q^{76} + q^{77} - 14 q^{79} - 3 q^{80} + 6 q^{82} - 26 q^{83} - 27 q^{85} + 17 q^{86} + q^{88} - 20 q^{89} - 2 q^{91} + 7 q^{92} + 21 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8 + 3 * q^10 - q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 + q^17 + 3 * q^19 - 3 * q^20 + q^22 + 7 * q^23 + 3 * q^25 - 2 * q^26 - 2 * q^28 - q^29 - 4 * q^31 - 2 * q^32 - q^34 + 3 * q^35 - 11 * q^37 - 3 * q^38 + 3 * q^40 - 6 * q^41 - 17 * q^43 - q^44 - 7 * q^46 + 2 * q^49 - 3 * q^50 + 2 * q^52 - 8 * q^53 - 7 * q^55 + 2 * q^56 + q^58 + 12 * q^59 + q^61 + 4 * q^62 + 2 * q^64 - 3 * q^65 - 6 * q^67 + q^68 - 3 * q^70 - 16 * q^71 - 3 * q^73 + 11 * q^74 + 3 * q^76 + q^77 - 14 * q^79 - 3 * q^80 + 6 * q^82 - 26 * q^83 - 27 * q^85 + 17 * q^86 + q^88 - 20 * q^89 - 2 * q^91 + 7 * q^92 + 21 * q^95 - 4 * q^97 - 2 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.561553	−0.177579
$11$	−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
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.68466	−1.37873	−0.689366	−	0.724413i	$-0.742111\pi$
$17$	−5.68466	−1.37873	−0.689366	+	0.724413i	$0.742111\pi$
$18$	0	0
$19$	7.68466	1.76298	0.881491	−	0.472201i	$-0.156540\pi$
$19$	7.68466	1.76298	0.881491	+	0.472201i	$0.156540\pi$
$20$	0.561553	0.125567
$21$	0	0
$22$	2.56155	0.546125
$23$	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
$26$	−1.00000	−0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	−10.2462	−1.84027	−0.920137	−	0.391597i	$-0.871923\pi$
$31$	−10.2462	−1.84027	−0.920137	+	0.391597i	$0.871923\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.68466	0.974911
$35$	−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	$0.591210\pi$
$38$	−7.68466	−1.24662
$39$	0	0
$40$	−0.561553	−0.0887893
$41$	−7.12311	−1.11244	−0.556221	−	0.831034i	$-0.687749\pi$
$41$	−7.12311	−1.11244	−0.556221	+	0.831034i	$0.687749\pi$
$42$	0	0
$43$	−10.5616	−1.61062	−0.805311	−	0.592853i	$-0.798002\pi$
$43$	−10.5616	−1.61062	−0.805311	+	0.592853i	$0.798002\pi$
$44$	−2.56155	−0.386169
$45$	0	0
$46$	−1.43845	−0.212087
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.68466	0.662511
$51$	0	0
$52$	1.00000	0.138675
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	0	0
$55$	−1.43845	−0.193960
$56$	1.00000	0.133631
$57$	0	0
$58$	−5.68466	−0.746432
$59$	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	$0.618563\pi$
$62$	10.2462	1.30127
$63$	0	0
$64$	1.00000	0.125000
$65$	0.561553	0.0696521
$66$	0	0
$67$	1.12311	0.137209	0.0686046	−	0.997644i	$-0.478145\pi$
$67$	1.12311	0.137209	0.0686046	+	0.997644i	$0.478145\pi$
$68$	−5.68466	−0.689366
$69$	0	0
$70$	0.561553	0.0671184
$71$	−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	$0.489538\pi$
$74$	3.43845	0.399711
$75$	0	0
$76$	7.68466	0.881491
$77$	2.56155	0.291916
$78$	0	0
$79$	−2.87689	−0.323676	−0.161838	−	0.986817i	$-0.551742\pi$
$79$	−2.87689	−0.323676	−0.161838	+	0.986817i	$0.551742\pi$
$80$	0.561553	0.0627835
$81$	0	0
$82$	7.12311	0.786615
$83$	−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
$86$	10.5616	1.13888
$87$	0	0
$88$	2.56155	0.273062
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	1.43845	0.149968
$93$	0	0
$94$	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	$0.888071\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−4.68466	−0.468466
$101$	−3.12311	−0.310761	−0.155380	−	0.987855i	$-0.549660\pi$
$101$	−3.12311	−0.310761	−0.155380	+	0.987855i	$0.549660\pi$
$102$	0	0
$103$	11.6847	1.15132	0.575662	−	0.817688i	$-0.304744\pi$
$103$	11.6847	1.15132	0.575662	+	0.817688i	$0.304744\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−4.24621	−0.412428
$107$	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	$0.306406\pi$
$110$	1.43845	0.137151
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−12.2462	−1.15203	−0.576013	−	0.817440i	$-0.695392\pi$
$113$	−12.2462	−1.15203	−0.576013	+	0.817440i	$0.695392\pi$
$114$	0	0
$115$	0.807764	0.0753244
$116$	5.68466	0.527807
$117$	0	0
$118$	−14.2462	−1.31147
$119$	5.68466	0.521112
$120$	0	0
$121$	−4.43845	−0.403495
$122$	5.68466	0.514665
$123$	0	0
$124$	−10.2462	−0.920137
$125$	−5.43845	−0.486430
$126$	0	0
$127$	−13.1231	−1.16449	−0.582244	−	0.813014i	$-0.697825\pi$
$127$	−13.1231	−1.16449	−0.582244	+	0.813014i	$0.697825\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.561553	−0.0492514
$131$	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
$134$	−1.12311	−0.0970215
$135$	0	0
$136$	5.68466	0.487455
$137$	−0.561553	−0.0479767	−0.0239883	−	0.999712i	$-0.507636\pi$
$137$	−0.561553	−0.0479767	−0.0239883	+	0.999712i	$0.507636\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	−0.561553	−0.0474599
$141$	0	0
$142$	8.00000	0.671345
$143$	−2.56155	−0.214208
$144$	0	0
$145$	3.19224	0.265101
$146$	−0.561553	−0.0464744
$147$	0	0
$148$	−3.43845	−0.282639
$149$	−23.6155	−1.93466	−0.967330	−	0.253522i	$-0.918411\pi$
$149$	−23.6155	−1.93466	−0.967330	+	0.253522i	$0.918411\pi$
$150$	0	0
$151$	−8.80776	−0.716766	−0.358383	−	0.933575i	$-0.616672\pi$
$151$	−8.80776	−0.716766	−0.358383	+	0.933575i	$0.616672\pi$
$152$	−7.68466	−0.623308
$153$	0	0
$154$	−2.56155	−0.206416
$155$	−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	$0.650877\pi$
$158$	2.87689	0.228873
$159$	0	0
$160$	−0.561553	−0.0443946
$161$	−1.43845	−0.113366
$162$	0	0
$163$	−25.1231	−1.96779	−0.983897	−	0.178738i	$-0.942799\pi$
$163$	−25.1231	−1.96779	−0.983897	+	0.178738i	$0.942799\pi$
$164$	−7.12311	−0.556221
$165$	0	0
$166$	17.1231	1.32901
$167$	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
$170$	3.19224	0.244833
$171$	0	0
$172$	−10.5616	−0.805311
$173$	−3.75379	−0.285395	−0.142698	−	0.989766i	$-0.545578\pi$
$173$	−3.75379	−0.285395	−0.142698	+	0.989766i	$0.545578\pi$
$174$	0	0
$175$	4.68466	0.354127
$176$	−2.56155	−0.193084
$177$	0	0
$178$	10.0000	0.749532
$179$	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	$0.293658\pi$
$182$	1.00000	0.0741249
$183$	0	0
$184$	−1.43845	−0.106044
$185$	−1.93087	−0.141960
$186$	0	0
$187$	14.5616	1.06485
$188$	0	0
$189$	0	0
$190$	−4.31534	−0.313068
$191$	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	$0.561902\pi$
$194$	18.4924	1.32768
$195$	0	0
$196$	1.00000	0.0714286
$197$	−19.1231	−1.36246	−0.681232	−	0.732067i	$-0.738556\pi$
$197$	−19.1231	−1.36246	−0.681232	+	0.732067i	$0.738556\pi$
$198$	0	0
$199$	−21.9309	−1.55464	−0.777319	−	0.629107i	$-0.783421\pi$
$199$	−21.9309	−1.55464	−0.777319	+	0.629107i	$0.783421\pi$
$200$	4.68466	0.331255
$201$	0	0
$202$	3.12311	0.219741
$203$	−5.68466	−0.398985
$204$	0	0
$205$	−4.00000	−0.279372
$206$	−11.6847	−0.814109
$207$	0	0
$208$	1.00000	0.0693375
$209$	−19.6847	−1.36162
$210$	0	0
$211$	4.80776	0.330980	0.165490	−	0.986211i	$-0.447079\pi$
$211$	4.80776	0.330980	0.165490	+	0.986211i	$0.447079\pi$
$212$	4.24621	0.291631
$213$	0	0
$214$	−17.1231	−1.17051
$215$	−5.93087	−0.404482
$216$	0	0
$217$	10.2462	0.695558
$218$	−11.9309	−0.808060
$219$	0	0
$220$	−1.43845	−0.0969801
$221$	−5.68466	−0.382392
$222$	0	0
$223$	17.6155	1.17962	0.589812	−	0.807541i	$-0.299202\pi$
$223$	17.6155	1.17962	0.589812	+	0.807541i	$0.299202\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	12.2462	0.814606
$227$	−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	$0.373043\pi$
$230$	−0.807764	−0.0532624
$231$	0	0
$232$	−5.68466	−0.373216
$233$	−2.63068	−0.172342	−0.0861709	−	0.996280i	$-0.527463\pi$
$233$	−2.63068	−0.172342	−0.0861709	+	0.996280i	$0.527463\pi$
$234$	0	0
$235$	0	0
$236$	14.2462	0.927349
$237$	0	0
$238$	−5.68466	−0.368482
$239$	−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	$0.648900\pi$
$242$	4.43845	0.285314
$243$	0	0
$244$	−5.68466	−0.363923
$245$	0.561553	0.0358763
$246$	0	0
$247$	7.68466	0.488963
$248$	10.2462	0.650635
$249$	0	0
$250$	5.43845	0.343958
$251$	−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
$254$	13.1231	0.823417
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.2462	−0.763898	−0.381949	−	0.924183i	$-0.624747\pi$
$257$	−12.2462	−0.763898	−0.381949	+	0.924183i	$0.624747\pi$
$258$	0	0
$259$	3.43845	0.213655
$260$	0.561553	0.0348260
$261$	0	0
$262$	−5.43845	−0.335988
$263$	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
$266$	7.68466	0.471177
$267$	0	0
$268$	1.12311	0.0686046
$269$	−3.75379	−0.228873	−0.114436	−	0.993431i	$-0.536506\pi$
$269$	−3.75379	−0.228873	−0.114436	+	0.993431i	$0.536506\pi$
$270$	0	0
$271$	13.1231	0.797172	0.398586	−	0.917131i	$-0.369501\pi$
$271$	13.1231	0.797172	0.398586	+	0.917131i	$0.369501\pi$
$272$	−5.68466	−0.344683
$273$	0	0
$274$	0.561553	0.0339246
$275$	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	$0.397924\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	0.561553	0.0335592
$281$	0.246211	0.0146877	0.00734387	−	0.999973i	$-0.497662\pi$
$281$	0.246211	0.0146877	0.00734387	+	0.999973i	$0.497662\pi$
$282$	0	0
$283$	1.75379	0.104252	0.0521260	−	0.998641i	$-0.483400\pi$
$283$	1.75379	0.104252	0.0521260	+	0.998641i	$0.483400\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	2.56155	0.151468
$287$	7.12311	0.420464
$288$	0	0
$289$	15.3153	0.900902
$290$	−3.19224	−0.187455
$291$	0	0
$292$	0.561553	0.0328624
$293$	24.7386	1.44525	0.722623	−	0.691242i	$-0.242936\pi$
$293$	24.7386	1.44525	0.722623	+	0.691242i	$0.242936\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	3.43845	0.199856
$297$	0	0
$298$	23.6155	1.36801
$299$	1.43845	0.0831875
$300$	0	0
$301$	10.5616	0.608758
$302$	8.80776	0.506830
$303$	0	0
$304$	7.68466	0.440745
$305$	−3.19224	−0.182787
$306$	0	0
$307$	9.75379	0.556678	0.278339	−	0.960483i	$-0.410216\pi$
$307$	9.75379	0.556678	0.278339	+	0.960483i	$0.410216\pi$
$308$	2.56155	0.145958
$309$	0	0
$310$	5.75379	0.326793
$311$	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	$0.214984\pi$
$314$	11.4384	0.645509
$315$	0	0
$316$	−2.87689	−0.161838
$317$	−11.1231	−0.624736	−0.312368	−	0.949961i	$-0.601122\pi$
$317$	−11.1231	−0.624736	−0.312368	+	0.949961i	$0.601122\pi$
$318$	0	0
$319$	−14.5616	−0.815290
$320$	0.561553	0.0313918
$321$	0	0
$322$	1.43845	0.0801615
$323$	−43.6847	−2.43068
$324$	0	0
$325$	−4.68466	−0.259858
$326$	25.1231	1.39144
$327$	0	0
$328$	7.12311	0.393308
$329$	0	0
$330$	0	0
$331$	−11.3693	−0.624914	−0.312457	−	0.949932i	$-0.601152\pi$
$331$	−11.3693	−0.624914	−0.312457	+	0.949932i	$0.601152\pi$
$332$	−17.1231	−0.939753
$333$	0	0
$334$	−5.93087	−0.324523
$335$	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	$0.425084\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−3.19224	−0.173123
$341$	26.2462	1.42131
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	10.5616	0.569441
$345$	0	0
$346$	3.75379	0.201805
$347$	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	$0.275215\pi$
$350$	−4.68466	−0.250406
$351$	0	0
$352$	2.56155	0.136531
$353$	2.49242	0.132658	0.0663291	−	0.997798i	$-0.478871\pi$
$353$	2.49242	0.132658	0.0663291	+	0.997798i	$0.478871\pi$
$354$	0	0
$355$	−4.49242	−0.238433
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−16.4924	−0.871652
$359$	−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
$362$	−16.2462	−0.853882
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	0.315342	0.0165057
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	1.43845	0.0749842
$369$	0	0
$370$	1.93087	0.100381
$371$	−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	$0.290614\pi$
$374$	−14.5616	−0.752960
$375$	0	0
$376$	0	0
$377$	5.68466	0.292775
$378$	0	0
$379$	17.7538	0.911951	0.455975	−	0.889992i	$-0.349290\pi$
$379$	17.7538	0.911951	0.455975	+	0.889992i	$0.349290\pi$
$380$	4.31534	0.221372
$381$	0	0
$382$	−13.9309	−0.712765
$383$	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
$386$	5.36932	0.273291
$387$	0	0
$388$	−18.4924	−0.938811
$389$	−20.7386	−1.05149	−0.525745	−	0.850642i	$-0.676213\pi$
$389$	−20.7386	−1.05149	−0.525745	+	0.850642i	$0.676213\pi$
$390$	0	0
$391$	−8.17708	−0.413533
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	19.1231	0.963408
$395$	−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	$0.649181\pi$
$398$	21.9309	1.09930
$399$	0	0
$400$	−4.68466	−0.234233
$401$	8.24621	0.411796	0.205898	−	0.978573i	$-0.433988\pi$
$401$	8.24621	0.411796	0.205898	+	0.978573i	$0.433988\pi$
$402$	0	0
$403$	−10.2462	−0.510400
$404$	−3.12311	−0.155380
$405$	0	0
$406$	5.68466	0.282125
$407$	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	$0.298476\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	11.6847	0.575662
$413$	−14.2462	−0.701010
$414$	0	0
$415$	−9.61553	−0.472008
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	19.6847	0.962808
$419$	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	$0.664235\pi$
$422$	−4.80776	−0.234038
$423$	0	0
$424$	−4.24621	−0.206214
$425$	26.6307	1.29178
$426$	0	0
$427$	5.68466	0.275100
$428$	17.1231	0.827677
$429$	0	0
$430$	5.93087	0.286012
$431$	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	$0.550932\pi$
$434$	−10.2462	−0.491834
$435$	0	0
$436$	11.9309	0.571385
$437$	11.0540	0.528783
$438$	0	0
$439$	21.9309	1.04670	0.523352	−	0.852117i	$-0.324681\pi$
$439$	21.9309	1.04670	0.523352	+	0.852117i	$0.324681\pi$
$440$	1.43845	0.0685753
$441$	0	0
$442$	5.68466	0.270392
$443$	−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
$446$	−17.6155	−0.834119
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	1.68466	0.0795039	0.0397520	−	0.999210i	$-0.487343\pi$
$449$	1.68466	0.0795039	0.0397520	+	0.999210i	$0.487343\pi$
$450$	0	0
$451$	18.2462	0.859181
$452$	−12.2462	−0.576013
$453$	0	0
$454$	1.12311	0.0527100
$455$	−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	$0.480402\pi$
$458$	−11.7538	−0.549218
$459$	0	0
$460$	0.807764	0.0376622
$461$	−9.05398	−0.421686	−0.210843	−	0.977520i	$-0.567621\pi$
$461$	−9.05398	−0.421686	−0.210843	+	0.977520i	$0.567621\pi$
$462$	0	0
$463$	3.68466	0.171241	0.0856203	−	0.996328i	$-0.472713\pi$
$463$	3.68466	0.171241	0.0856203	+	0.996328i	$0.472713\pi$
$464$	5.68466	0.263904
$465$	0	0
$466$	2.63068	0.121864
$467$	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
$471$	0	0
$472$	−14.2462	−0.655735
$473$	27.0540	1.24394
$474$	0	0
$475$	−36.0000	−1.65179
$476$	5.68466	0.260556
$477$	0	0
$478$	16.0000	0.731823
$479$	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
$482$	14.0000	0.637683
$483$	0	0
$484$	−4.43845	−0.201748
$485$	−10.3845	−0.471535
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	5.68466	0.257332
$489$	0	0
$490$	−0.561553	−0.0253684
$491$	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
$494$	−7.68466	−0.345749
$495$	0	0
$496$	−10.2462	−0.460068
$497$	8.00000	0.358849
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	−5.43845	−0.243215
$501$	0	0
$502$	12.8078	0.571638
$503$	−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
$506$	3.68466	0.163803
$507$	0	0
$508$	−13.1231	−0.582244
$509$	−25.0540	−1.11050	−0.555249	−	0.831684i	$-0.687377\pi$
$509$	−25.0540	−1.11050	−0.555249	+	0.831684i	$0.687377\pi$
$510$	0	0
$511$	−0.561553	−0.0248416
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.2462	0.540157
$515$	6.56155	0.289137
$516$	0	0
$517$	0	0
$518$	−3.43845	−0.151077
$519$	0	0
$520$	−0.561553	−0.0246257
$521$	9.68466	0.424293	0.212146	−	0.977238i	$-0.431955\pi$
$521$	9.68466	0.424293	0.212146	+	0.977238i	$0.431955\pi$
$522$	0	0
$523$	38.2462	1.67239	0.836195	−	0.548432i	$-0.184775\pi$
$523$	38.2462	1.67239	0.836195	+	0.548432i	$0.184775\pi$
$524$	5.43845	0.237580
$525$	0	0
$526$	−13.7538	−0.599694
$527$	58.2462	2.53724
$528$	0	0
$529$	−20.9309	−0.910038
$530$	−2.38447	−0.103575
$531$	0	0
$532$	−7.68466	−0.333172
$533$	−7.12311	−0.308536
$534$	0	0
$535$	9.61553	0.415716
$536$	−1.12311	−0.0485108
$537$	0	0
$538$	3.75379	0.161837
$539$	−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	$0.527879\pi$
$542$	−13.1231	−0.563686
$543$	0	0
$544$	5.68466	0.243728
$545$	6.69981	0.286988
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−0.561553	−0.0239883
$549$	0	0
$550$	−12.0000	−0.511682
$551$	43.6847	1.86103
$552$	0	0
$553$	2.87689	0.122338
$554$	−10.4924	−0.445780
$555$	0	0
$556$	12.0000	0.508913
$557$	−21.3693	−0.905447	−0.452724	−	0.891651i	$-0.649548\pi$
$557$	−21.3693	−0.905447	−0.452724	+	0.891651i	$0.649548\pi$
$558$	0	0
$559$	−10.5616	−0.446706
$560$	−0.561553	−0.0237299
$561$	0	0
$562$	−0.246211	−0.0103858
$563$	−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
$566$	−1.75379	−0.0737172
$567$	0	0
$568$	8.00000	0.335673
$569$	41.2311	1.72850	0.864248	−	0.503066i	$-0.167795\pi$
$569$	41.2311	1.72850	0.864248	+	0.503066i	$0.167795\pi$
$570$	0	0
$571$	1.75379	0.0733938	0.0366969	−	0.999326i	$-0.488316\pi$
$571$	1.75379	0.0733938	0.0366969	+	0.999326i	$0.488316\pi$
$572$	−2.56155	−0.107104
$573$	0	0
$574$	−7.12311	−0.297313
$575$	−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	$0.714680\pi$
$578$	−15.3153	−0.637034
$579$	0	0
$580$	3.19224	0.132550
$581$	17.1231	0.710386
$582$	0	0
$583$	−10.8769	−0.450475
$584$	−0.561553	−0.0232372
$585$	0	0
$586$	−24.7386	−1.02194
$587$	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
$590$	−8.00000	−0.329355
$591$	0	0
$592$	−3.43845	−0.141319
$593$	3.75379	0.154150	0.0770748	−	0.997025i	$-0.475442\pi$
$593$	3.75379	0.154150	0.0770748	+	0.997025i	$0.475442\pi$
$594$	0	0
$595$	3.19224	0.130869
$596$	−23.6155	−0.967330
$597$	0	0
$598$	−1.43845	−0.0588225
$599$	−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	$0.627533\pi$
$602$	−10.5616	−0.430457
$603$	0	0
$604$	−8.80776	−0.358383
$605$	−2.49242	−0.101331
$606$	0	0
$607$	−22.5616	−0.915745	−0.457873	−	0.889018i	$-0.651388\pi$
$607$	−22.5616	−0.915745	−0.457873	+	0.889018i	$0.651388\pi$
$608$	−7.68466	−0.311654
$609$	0	0
$610$	3.19224	0.129250
$611$	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	$0.672544\pi$
$614$	−9.75379	−0.393631
$615$	0	0
$616$	−2.56155	−0.103208
$617$	−28.4233	−1.14428	−0.572139	−	0.820156i	$-0.693886\pi$
$617$	−28.4233	−1.14428	−0.572139	+	0.820156i	$0.693886\pi$
$618$	0	0
$619$	31.6847	1.27351	0.636757	−	0.771065i	$-0.280275\pi$
$619$	31.6847	1.27351	0.636757	+	0.771065i	$0.280275\pi$
$620$	−5.75379	−0.231078
$621$	0	0
$622$	−21.1231	−0.846959
$623$	10.0000	0.400642
$624$	0	0
$625$	20.3693	0.814773
$626$	−27.6155	−1.10374
$627$	0	0
$628$	−11.4384	−0.456444
$629$	19.5464	0.779366
$630$	0	0
$631$	5.93087	0.236104	0.118052	−	0.993007i	$-0.462335\pi$
$631$	5.93087	0.236104	0.118052	+	0.993007i	$0.462335\pi$
$632$	2.87689	0.114437
$633$	0	0
$634$	11.1231	0.441755
$635$	−7.36932	−0.292442
$636$	0	0
$637$	1.00000	0.0396214
$638$	14.5616	0.576497
$639$	0	0
$640$	−0.561553	−0.0221973
$641$	3.75379	0.148266	0.0741329	−	0.997248i	$-0.476381\pi$
$641$	3.75379	0.148266	0.0741329	+	0.997248i	$0.476381\pi$
$642$	0	0
$643$	36.8078	1.45156	0.725778	−	0.687929i	$-0.241480\pi$
$643$	36.8078	1.45156	0.725778	+	0.687929i	$0.241480\pi$
$644$	−1.43845	−0.0566828
$645$	0	0
$646$	43.6847	1.71875
$647$	−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
$650$	4.68466	0.183747
$651$	0	0
$652$	−25.1231	−0.983897
$653$	−11.9309	−0.466891	−0.233446	−	0.972370i	$-0.575000\pi$
$653$	−11.9309	−0.466891	−0.233446	+	0.972370i	$0.575000\pi$
$654$	0	0
$655$	3.05398	0.119329
$656$	−7.12311	−0.278111
$657$	0	0
$658$	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	$0.397676\pi$
$662$	11.3693	0.441881
$663$	0	0
$664$	17.1231	0.664505
$665$	−4.31534	−0.167342
$666$	0	0
$667$	8.17708	0.316618
$668$	5.93087	0.229472
$669$	0	0
$670$	−0.630683	−0.0243654
$671$	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	$0.581832\pi$
$674$	−8.56155	−0.329779
$675$	0	0
$676$	1.00000	0.0384615
$677$	30.4924	1.17192	0.585959	−	0.810340i	$-0.300718\pi$
$677$	30.4924	1.17192	0.585959	+	0.810340i	$0.300718\pi$
$678$	0	0
$679$	18.4924	0.709674
$680$	3.19224	0.122417
$681$	0	0
$682$	−26.2462	−1.00502
$683$	−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
$686$	1.00000	0.0381802
$687$	0	0
$688$	−10.5616	−0.402655
$689$	4.24621	0.161768
$690$	0	0
$691$	−16.4924	−0.627401	−0.313701	−	0.949522i	$-0.601569\pi$
$691$	−16.4924	−0.627401	−0.313701	+	0.949522i	$0.601569\pi$
$692$	−3.75379	−0.142698
$693$	0	0
$694$	−20.0000	−0.759190
$695$	6.73863	0.255611
$696$	0	0
$697$	40.4924	1.53376
$698$	−24.2462	−0.917733
$699$	0	0
$700$	4.68466	0.177063
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−26.4233	−0.996573
$704$	−2.56155	−0.0965422
$705$	0	0
$706$	−2.49242	−0.0938036
$707$	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	$0.498528\pi$
$710$	4.49242	0.168598
$711$	0	0
$712$	10.0000	0.374766
$713$	−14.7386	−0.551966
$714$	0	0
$715$	−1.43845	−0.0537949
$716$	16.4924	0.616351
$717$	0	0
$718$	22.7386	0.848598
$719$	−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
$722$	−40.0540	−1.49065
$723$	0	0
$724$	16.2462	0.603786
$725$	−26.6307	−0.989039
$726$	0	0
$727$	21.9309	0.813371	0.406685	−	0.913568i	$-0.366684\pi$
$727$	21.9309	0.813371	0.406685	+	0.913568i	$0.366684\pi$
$728$	1.00000	0.0370625
$729$	0	0
$730$	−0.315342	−0.0116713
$731$	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	$0.502260\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−1.43845	−0.0530219
$737$	−2.87689	−0.105972
$738$	0	0
$739$	42.1080	1.54897	0.774483	−	0.632595i	$-0.218010\pi$
$739$	42.1080	1.54897	0.774483	+	0.632595i	$0.218010\pi$
$740$	−1.93087	−0.0709802
$741$	0	0
$742$	4.24621	0.155883
$743$	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
$746$	−23.6155	−0.864626
$747$	0	0
$748$	14.5616	0.532423
$749$	−17.1231	−0.625665
$750$	0	0
$751$	10.2462	0.373890	0.186945	−	0.982370i	$-0.440141\pi$
$751$	10.2462	0.373890	0.186945	+	0.982370i	$0.440141\pi$
$752$	0	0
$753$	0	0
$754$	−5.68466	−0.207023
$755$	−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	$0.491278\pi$
$758$	−17.7538	−0.644847
$759$	0	0
$760$	−4.31534	−0.156534
$761$	3.12311	0.113212	0.0566062	−	0.998397i	$-0.481972\pi$
$761$	3.12311	0.113212	0.0566062	+	0.998397i	$0.481972\pi$
$762$	0	0
$763$	−11.9309	−0.431926
$764$	13.9309	0.504001
$765$	0	0
$766$	−34.4233	−1.24376
$767$	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	$0.620894\pi$
$770$	−1.43845	−0.0518380
$771$	0	0
$772$	−5.36932	−0.193246
$773$	−24.0691	−0.865706	−0.432853	−	0.901464i	$-0.642493\pi$
$773$	−24.0691	−0.865706	−0.432853	+	0.901464i	$0.642493\pi$
$774$	0	0
$775$	48.0000	1.72421
$776$	18.4924	0.663839
$777$	0	0
$778$	20.7386	0.743516
$779$	−54.7386	−1.96122
$780$	0	0
$781$	20.4924	0.733277
$782$	8.17708	0.292412
$783$	0	0
$784$	1.00000	0.0357143
$785$	−6.42329	−0.229257
$786$	0	0
$787$	−33.3002	−1.18702	−0.593512	−	0.804825i	$-0.702259\pi$
$787$	−33.3002	−1.18702	−0.593512	+	0.804825i	$0.702259\pi$
$788$	−19.1231	−0.681232
$789$	0	0
$790$	1.61553	0.0574779
$791$	12.2462	0.435425
$792$	0	0
$793$	−5.68466	−0.201868
$794$	18.0000	0.638796
$795$	0	0
$796$	−21.9309	−0.777319
$797$	−6.63068	−0.234871	−0.117435	−	0.993081i	$-0.537467\pi$
$797$	−6.63068	−0.234871	−0.117435	+	0.993081i	$0.537467\pi$
$798$	0	0
$799$	0	0
$800$	4.68466	0.165628
$801$	0	0
$802$	−8.24621	−0.291184
$803$	−1.43845	−0.0507617
$804$	0	0
$805$	−0.807764	−0.0284699
$806$	10.2462	0.360907
$807$	0	0
$808$	3.12311	0.109870
$809$	−30.4924	−1.07206	−0.536028	−	0.844200i	$-0.680076\pi$
$809$	−30.4924	−1.07206	−0.536028	+	0.844200i	$0.680076\pi$
$810$	0	0
$811$	37.4384	1.31464	0.657321	−	0.753611i	$-0.271690\pi$
$811$	37.4384	1.31464	0.657321	+	0.753611i	$0.271690\pi$
$812$	−5.68466	−0.199492
$813$	0	0
$814$	−8.80776	−0.308712
$815$	−14.1080	−0.494180
$816$	0	0
$817$	−81.1619	−2.83950
$818$	−23.9309	−0.836723
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−55.6155	−1.94100	−0.970498	−	0.241111i	$-0.922488\pi$
$821$	−55.6155	−1.94100	−0.970498	+	0.241111i	$0.922488\pi$
$822$	0	0
$823$	−28.4924	−0.993183	−0.496592	−	0.867984i	$-0.665415\pi$
$823$	−28.4924	−0.993183	−0.496592	+	0.867984i	$0.665415\pi$
$824$	−11.6847	−0.407054
$825$	0	0
$826$	14.2462	0.495689
$827$	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	$0.348929\pi$
$830$	9.61553	0.333760
$831$	0	0
$832$	1.00000	0.0346688
$833$	−5.68466	−0.196962
$834$	0	0
$835$	3.33050	0.115257
$836$	−19.6847	−0.680808
$837$	0	0
$838$	−24.3153	−0.839960
$839$	−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
$842$	20.2462	0.697731
$843$	0	0
$844$	4.80776	0.165490
$845$	0.561553	0.0193180
$846$	0	0
$847$	4.43845	0.152507
$848$	4.24621	0.145815
$849$	0	0
$850$	−26.6307	−0.913425
$851$	−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	$0.514340\pi$
$854$	−5.68466	−0.194525
$855$	0	0
$856$	−17.1231	−0.585256
$857$	1.50758	0.0514979	0.0257489	−	0.999668i	$-0.491803\pi$
$857$	1.50758	0.0514979	0.0257489	+	0.999668i	$0.491803\pi$
$858$	0	0
$859$	22.2462	0.759031	0.379515	−	0.925185i	$-0.376091\pi$
$859$	22.2462	0.759031	0.379515	+	0.925185i	$0.376091\pi$
$860$	−5.93087	−0.202241
$861$	0	0
$862$	−8.63068	−0.293962
$863$	−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
$866$	6.63068	0.225320
$867$	0	0
$868$	10.2462	0.347779
$869$	7.36932	0.249987
$870$	0	0
$871$	1.12311	0.0380550
$872$	−11.9309	−0.404030
$873$	0	0
$874$	−11.0540	−0.373906
$875$	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	$0.631356\pi$
$878$	−21.9309	−0.740131
$879$	0	0
$880$	−1.43845	−0.0484900
$881$	−26.1771	−0.881928	−0.440964	−	0.897525i	$-0.645363\pi$
$881$	−26.1771	−0.881928	−0.440964	+	0.897525i	$0.645363\pi$
$882$	0	0
$883$	−2.56155	−0.0862031	−0.0431016	−	0.999071i	$-0.513724\pi$
$883$	−2.56155	−0.0862031	−0.0431016	+	0.999071i	$0.513724\pi$
$884$	−5.68466	−0.191196
$885$	0	0
$886$	19.3693	0.650725
$887$	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
$890$	5.61553	0.188233
$891$	0	0
$892$	17.6155	0.589812
$893$	0	0
$894$	0	0
$895$	9.26137	0.309573
$896$	1.00000	0.0334077
$897$	0	0
$898$	−1.68466	−0.0562178
$899$	−58.2462	−1.94262
$900$	0	0
$901$	−24.1383	−0.804162
$902$	−18.2462	−0.607532
$903$	0	0
$904$	12.2462	0.407303
$905$	9.12311	0.303262
$906$	0	0
$907$	−40.4924	−1.34453	−0.672264	−	0.740311i	$-0.734678\pi$
$907$	−40.4924	−1.34453	−0.672264	+	0.740311i	$0.734678\pi$
$908$	−1.12311	−0.0372716
$909$	0	0
$910$	0.561553	0.0186153
$911$	−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
$914$	−2.63068	−0.0870153
$915$	0	0
$916$	11.7538	0.388356
$917$	−5.43845	−0.179593
$918$	0	0
$919$	10.8769	0.358796	0.179398	−	0.983777i	$-0.442585\pi$
$919$	10.8769	0.358796	0.179398	+	0.983777i	$0.442585\pi$
$920$	−0.807764	−0.0266312
$921$	0	0
$922$	9.05398	0.298177
$923$	−8.00000	−0.263323
$924$	0	0
$925$	16.1080	0.529626
$926$	−3.68466	−0.121085
$927$	0	0
$928$	−5.68466	−0.186608
$929$	15.6155	0.512329	0.256164	−	0.966633i	$-0.417541\pi$
$929$	15.6155	0.512329	0.256164	+	0.966633i	$0.417541\pi$
$930$	0	0
$931$	7.68466	0.251855
$932$	−2.63068	−0.0861709
$933$	0	0
$934$	−15.6847	−0.513218
$935$	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	$0.399636\pi$
$938$	1.12311	0.0366707
$939$	0	0
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	0	0
$943$	−10.2462	−0.333663
$944$	14.2462	0.463675
$945$	0	0
$946$	−27.0540	−0.879601
$947$	−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
$950$	36.0000	1.16799
$951$	0	0
$952$	−5.68466	−0.184241
$953$	13.3693	0.433075	0.216537	−	0.976274i	$-0.430524\pi$
$953$	13.3693	0.433075	0.216537	+	0.976274i	$0.430524\pi$
$954$	0	0
$955$	7.82292	0.253144
$956$	−16.0000	−0.517477
$957$	0	0
$958$	−21.3002	−0.688178
$959$	0.561553	0.0181335
$960$	0	0
$961$	73.9848	2.38661
$962$	3.43845	0.110860
$963$	0	0
$964$	−14.0000	−0.450910
$965$	−3.01515	−0.0970613
$966$	0	0
$967$	−17.4384	−0.560783	−0.280391	−	0.959886i	$-0.590464\pi$
$967$	−17.4384	−0.560783	−0.280391	+	0.959886i	$0.590464\pi$
$968$	4.43845	0.142657
$969$	0	0
$970$	10.3845	0.333425
$971$	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
$974$	8.00000	0.256337
$975$	0	0
$976$	−5.68466	−0.181961
$977$	10.3153	0.330017	0.165009	−	0.986292i	$-0.447235\pi$
$977$	10.3153	0.330017	0.165009	+	0.986292i	$0.447235\pi$
$978$	0	0
$979$	25.6155	0.818676
$980$	0.561553	0.0179381
$981$	0	0
$982$	−17.1231	−0.546420
$983$	−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
$986$	32.3153	1.02913
$987$	0	0
$988$	7.68466	0.244482
$989$	−15.1922	−0.483085
$990$	0	0
$991$	33.6155	1.06783	0.533916	−	0.845537i	$-0.320720\pi$
$991$	33.6155	1.06783	0.533916	+	0.845537i	$0.320720\pi$
$992$	10.2462	0.325318
$993$	0	0
$994$	−8.00000	−0.253745
$995$	−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	$0.544189\pi$
$998$	−36.0000	−1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.u.1.2		2
3.2	odd	2		546.2.a.j.1.1	✓	2
12.11	even	2		4368.2.a.be.1.1		2
21.20	even	2		3822.2.a.bo.1.2		2
39.38	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	3.2	odd	2
1638.2.a.u.1.2		2	1.1	even	1	trivial
3822.2.a.bo.1.2		2	21.20	even	2
4368.2.a.be.1.1		2	12.11	even	2
7098.2.a.bl.1.2		2	39.38	odd	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{7} -1.00000 q^{8} -0.561553 q^{10} -2.56155 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.68466 q^{17} +7.68466 q^{19} +0.561553 q^{20} +2.56155 q^{22} +1.43845 q^{23} -4.68466 q^{25} -1.00000 q^{26} -1.00000 q^{28} +5.68466 q^{29} -10.2462 q^{31} -1.00000 q^{32} +5.68466 q^{34} -0.561553 q^{35} -3.43845 q^{37} -7.68466 q^{38} -0.561553 q^{40} -7.12311 q^{41} -10.5616 q^{43} -2.56155 q^{44} -1.43845 q^{46} +1.00000 q^{49} +4.68466 q^{50} +1.00000 q^{52} +4.24621 q^{53} -1.43845 q^{55} +1.00000 q^{56} -5.68466 q^{58} +14.2462 q^{59} -5.68466 q^{61} +10.2462 q^{62} +1.00000 q^{64} +0.561553 q^{65} +1.12311 q^{67} -5.68466 q^{68} +0.561553 q^{70} -8.00000 q^{71} +0.561553 q^{73} +3.43845 q^{74} +7.68466 q^{76} +2.56155 q^{77} -2.87689 q^{79} +0.561553 q^{80} +7.12311 q^{82} -17.1231 q^{83} -3.19224 q^{85} +10.5616 q^{86} +2.56155 q^{88} -10.0000 q^{89} -1.00000 q^{91} +1.43845 q^{92} +4.31534 q^{95} -18.4924 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8} + 3 q^{10} - q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + q^{17} + 3 q^{19} - 3 q^{20} + q^{22} + 7 q^{23} + 3 q^{25} - 2 q^{26} - 2 q^{28} - q^{29} - 4 q^{31} - 2 q^{32} - q^{34} + 3 q^{35} - 11 q^{37} - 3 q^{38} + 3 q^{40} - 6 q^{41} - 17 q^{43} - q^{44} - 7 q^{46} + 2 q^{49} - 3 q^{50} + 2 q^{52} - 8 q^{53} - 7 q^{55} + 2 q^{56} + q^{58} + 12 q^{59} + q^{61} + 4 q^{62} + 2 q^{64} - 3 q^{65} - 6 q^{67} + q^{68} - 3 q^{70} - 16 q^{71} - 3 q^{73} + 11 q^{74} + 3 q^{76} + q^{77} - 14 q^{79} - 3 q^{80} + 6 q^{82} - 26 q^{83} - 27 q^{85} + 17 q^{86} + q^{88} - 20 q^{89} - 2 q^{91} + 7 q^{92} + 21 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8 + 3 * q^10 - q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 + q^17 + 3 * q^19 - 3 * q^20 + q^22 + 7 * q^23 + 3 * q^25 - 2 * q^26 - 2 * q^28 - q^29 - 4 * q^31 - 2 * q^32 - q^34 + 3 * q^35 - 11 * q^37 - 3 * q^38 + 3 * q^40 - 6 * q^41 - 17 * q^43 - q^44 - 7 * q^46 + 2 * q^49 - 3 * q^50 + 2 * q^52 - 8 * q^53 - 7 * q^55 + 2 * q^56 + q^58 + 12 * q^59 + q^61 + 4 * q^62 + 2 * q^64 - 3 * q^65 - 6 * q^67 + q^68 - 3 * q^70 - 16 * q^71 - 3 * q^73 + 11 * q^74 + 3 * q^76 + q^77 - 14 * q^79 - 3 * q^80 + 6 * q^82 - 26 * q^83 - 27 * q^85 + 17 * q^86 + q^88 - 20 * q^89 - 2 * q^91 + 7 * q^92 + 21 * q^95 - 4 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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